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+{"text":"\n\n\n\\section{Method}\nOur method consists of a multi-scale light field encoded using a progressive neural network to reduce aliasing and enable adaptive streaming and rendering.\n\n\n\n\\subsection{Multi-scale Light Field}\n\nWe propose encoding multiple representations of the light field which produce anti-aliased representations of a light field, similar to mipmaps for conventional images. For this, we generate an image pyramid for each image view with area downsampling and encode each resolution as a separate light field representation. In contrast to bilinear or nearest pixel downsampling, area downsampling creates an anti-aliased view as shown in \\autoref{fig:image_downsampling_methods} where each pixel in the downsampled image is an average over a corresponding area in the full-resolution image.\n\n\\begin{figure}[!htb]\n    \\centering\n    \\captionsetup[subfigure]{labelformat=empty}\n    \\begin{subfigure}{0.52\\linewidth}\n        \\includegraphics[width=\\linewidth]{figures\/aliasing\/aliasing_fullres.pdf}\n        \\vspace{-5.5mm}\n        \\caption{Single-scale Light Field Network}\n    \\end{subfigure}\\\\\n    \\vspace{1mm}\n    \\begin{subfigure}{0.52\\linewidth}\n        \\includegraphics[width=\\linewidth]{figures\/aliasing\/aliasing_mipnet.pdf}\n        \\vspace{-5.5mm}\n        \\caption{Multi-scale Light Field Network}\n    \\end{subfigure}\n    \\caption{Rendering a single full-scale LFN at a lower (1\/8) resolution results in aliasing, unlike the multi-scale LFN at the same (1\/8) resolution. When changing viewpoints (3 shown above), the aliasing results in flickering.}\n    \\label{fig:aliasing_figure}\n\\end{figure}\n\nAs with mipmaps, lower-resolution representations trade-off sharpness for less aliasing. In this case, where a light field is rendered into an image, sharpness may be preferable over some aliasing. However, in the case of videos where the light field is viewed from different poses or the light field is dynamic, sampling from a high-resolution light field leads to flickering. With a multi-scale representation, rendering light fields at smaller resolutions can be done at a lower scale to reduce flickering.\n\n\\subsection{Progressive Model}\n\n\\begin{figure}[!htbp]\n  \\centering\n  \\includegraphics[width=\\linewidth]{figures\/model_illustration_subnet_lods.pdf}\n  \\caption{Using subsets of neurons to encode different levels of detail within a single model.}\n  \\label{fig:model_illustration_subnet}\n\\end{figure}\n\nFor light field streaming, we wish to have a compact representation that can simultaneously encode multiple levels of detail into a single progressive model. Specifically, we seek a model with the following properties:\nGiven specific subsets of network weights, we should be able to render a complete image with reduced details.\nProgressive levels of details should share weights with lower levels to eliminate encoding redundancy.\n\nWith the above properties, our model offers several benefits over a standard full-scale network or encoding multi-scale light fields using multiple networks.\nFirst, our model composes all scales within a single network hence requiring less storage space than storing four separate models.\nSecond, our model is progressive, thus only parameters necessary for the desired level of detail are needed in a streaming scenario.\nThird, our model allows rendering different levels of detail across pixels for dithered transitions and foveation.\nFourth, our model allows for faster inference of lower levels of detail by utilizing fewer parameters.\nModel sizes and training times appear in \\autoref{tab:nonprogressive_vs_progressive}.\n\n\n\\begin{figure}[!htbp]\n    \\centering\n    \\includegraphics[width=0.5\\linewidth]{figures\/subnet_layer_illustration.pdf}\n    \\caption{Illustration of a single layer where a subset of the weight matrix and bias vector are used, evaluating half of the neurons and reducing the operations down to approximately a quarter.}\n    \\label{fig:subnet_layer_illustration}\n\\end{figure}\n\n\\subsubsection{Subset of neurons}\nTo encode multiple levels of detail within a unified representation, we propose using subsets of neurons as shown in \\autoref{fig:model_illustration_subnet}. For each level of detail, we assign a fixed number of neurons to evaluate, with increasing levels of detail using an increasing proportion of the neurons in the neural network. An illustration of the matrix subset operation applied to each layer is shown in \\autoref{fig:subnet_layer_illustration}. For example, a single linear layer with $512$ neurons will have a $512 \\times 512$ weight matrix and a bias vector of size $512$. For a low level of detail, we can assign a neuron subset size of $25\\%$ or $128$ neurons. Then each linear layer would use the top-left $128 \\times 128$ submatrix for the weights and the top $128$ subvector for the bias. To keep the input and output dimensions the same, the first hidden layer uses every column of the weight matrix and the final output layer uses every row of the weight matrix.\nUnlike using a subset of layers, as is done with early exiting~\\cite{bolukbasi2017adaptive,yeo2018neuraladaptive}, using a subset of neurons preserves the number of sequential non-linear activations between all levels of details. Since layer widths are typically larger than model depths, using subsets of neurons also allows more granular control over the quantity and quality of levels. \n\n\\subsubsection{Training} \\label{sec:mipnet_training}\nWith multiple levels of detail at varying resolutions, a modified training scheme is required to efficiently encode all levels of detail together. \nOn one hand, generating all levels of detail at each iteration heavily slows down training as each iteration requires an independent forward pass through the network. \nOn the other hand, selecting a single level of detail at each iteration or applying progressive training compromises the highest level of detail whose full weights would only get updated a fraction of the time.\n\nWe adopt an intermediate approach that focuses on training the highest level of detail. \nDuring training, each batch does a full forward pass through the entire network, producing pixels at the highest level of detail along with a forward pass at a randomly selected lower level of detail. The squared L2 losses for both the full detail and reduced detail are added together for a combined backward pass. By training at the highest level of detail, we ensure that every weight gets updated at each batch and that the highest level of detail is trained to the same number of iterations as a standard model. Specifically, given a batch of rays $\\{\\mathbf{r}_i\\}_{i=1}^{b}$, corresponding ground truth colors $\\{\\mathbf\\{\\mathbf{y}_i^c\\}_{i=1}^{b}\\}_{c=1}^{4}$ for four levels of details, and a random lower level of detail $k \\in \\{1,2,3\\}$, our loss function is:\n$$L = \\frac{1}{b}\\sum_{i=1}^b \\left[ \\left\\Vert f_4(\\mathbf{r}_i) - \\mathbf{y}_i^4\\right\\Vert^2_2 + \\left\\Vert f_k(\\mathbf{r}_i) - \\mathbf{y}_i^k\\right\\Vert^2_2 \\right]$$\nAs rays are sampled from the highest-resolution representation, each ray will not correspond to a unique pixel in the lower-resolution images in the image pyramid. Thus for lower levels of detail, corresponding colors are sampled using bilinear interpolation from the area downsampled training images.\n\n\\subsection{Adaptive Rendering}\nOur proposed model allows dynamic levels of detail on a per-ray\/per-pixel level or on a per-object level. As aliasing appears primarily at smaller scales, selecting the level of detail based on the distance to the viewer reduces aliasing and flickering for distant objects.\n\n\\subsubsection{Distance-based Level of Detail}\nWhen rendering light fields from different distances, the space between rays in object space is proportional to the distance between the object and the camera. Due to this, aliasing and flickering artifacts can occur at large distances when the object appears small. By selecting the level of detail based on the distance of the object from the camera or viewer, aliasing and flickering can be reduced. With fewer pixels to render, rendering smaller objects involves fewer forward passes through the light field network. Hence, the amount of operations is typically proportional to the number of pixels. By rendering smaller representations at lower LODs, we further improve the performance as pixels from lower LODs only perform a forward pass using a subset of weights.\n\n\\begin{figure}[!htbp]\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/lod_transition_zoom.pdf}\n    \\caption{With per-pixel level of detail, dithering can be applied to transition between levels of detail.}\n    \\label{fig:transition_example}\n\\end{figure}\n\n\\subsubsection{Dithered Transitions}\nWith per-pixel LOD, transitioning between levels can be achieved with dithered rendering. By increasing the fraction of pixels rendered at the new level of detail in each frame, dithering creates the appearance of a fading transition between levels as shown in \\autoref{fig:transition_example}.\n\n\\begin{figure}[!htbp]\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{figures\/foveation_example.pdf}\n    \\caption{An example of foveated rendering from our progressive multi-scale LFN. Foveated rendering  generates peripheral pixels at lower levels of detail to further improve performance in eye-tracked environments. In this example, foveation is centered over the left eye of the subject. }\n    \\label{fig:foveation_example}\n\\end{figure}\n\n\\subsubsection{Foveated Rendering}\nIn virtual and augmented reality applications where real-time gaze information is available through an eye-tracker, foveated rendering~\\cite{Meng20203D,meng2018Kernel,li2021logrectilinear} can be applied to better focus rendering compute. As each pixel can be rendered at a separate level of detail, peripheral pixels can be rendered at a lower level of detail to improve the performance. \\autoref{fig:foveation_example} shows an example of foveation from our progressive multi-scale LFN.\n\n\\newcommand{0.24\\linewidth}{0.24\\linewidth}\n\\newcommand{\\addDatasetResults}[2]{\n    \\begin{subfigure}[b]{0.24\\linewidth}\n        \\centering\n        \\includegraphics[width=0.8\\linewidth]{figures\/#2\/comparison2_r8.pdf}\n        \\caption{Dataset #1 at 1\/8 scale}\n        \\label{fig:qualitative_comparison_#1_r8}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[b]{0.24\\linewidth}\n        \\centering\n        \\includegraphics[width=0.8\\linewidth]{figures\/#2\/comparison2_r4.pdf}\n        \\caption{Dataset #1 at 1\/4 scale}\n        \\label{fig:qualitative_comparison_#1_r4}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[b]{0.24\\linewidth}\n        \\centering\n        \\includegraphics[width=0.8\\linewidth]{figures\/#2\/comparison2_r2.pdf}\n        \\caption{Dataset #1 at 1\/2 scale}\n        \\label{fig:qualitative_comparison_#1_r2}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[b]{0.24\\linewidth}\n        \\centering\n        \\includegraphics[width=0.8\\linewidth]{figures\/#2\/comparison2_r1.pdf}\n        \\caption{Dataset #1 at full scale}\n        \\label{fig:qualitative_comparison_#1_r1}\n    \\end{subfigure}\n}\n\\begin{figure*}[!htbp]\n    \\centering\n    \\addDatasetResults{C}{david5}\n    \\caption{Qualitative results of one of our datasets at four scales. Single-scale LFNs (left) trained at the full-resolution exhibit aliasing and flickering at lower scales. Our progressive multi-scale LFNs (right) encode all four scales into a single model and have reduced aliasing and flickering at lower scales.}\n    \\label{fig:qualitative_comparison}\n\\end{figure*}\n\n\\newcommand{\\datasetScaledResults}[2]{\n    \\begin{subfigure}[b]{0.17\\linewidth}\n      \\centering\n      \\includegraphics[width=\\linewidth]{figures\/#2\/mipnet_lod_scaled.pdf}\n      \\caption{Dataset #1}\n      \\label{fig:qualitative_scaled_#2}\n    \\end{subfigure}\n}\n\\begin{figure*}[!htbp]\n    \\centering\n    \\datasetScaledResults{A}{jon}\n    \\hfill\n    \\datasetScaledResults{B}{david4}\n    \\hfill\n    \\datasetScaledResults{C}{david5}\n    \\hfill\n    \\datasetScaledResults{D}{sida1}\n    \\hfill\n    \\datasetScaledResults{E}{maria}\n    \\caption{Scaled qualitative results from our progressive multi-scale LFNs at four levels of detail.}\n    \\label{fig:qualtiative_scaled}\n\\end{figure*}\n\n\n\\section{Conclusion}\nIn this paper, we propose a method to encode multi-scale light field networks targeted for streaming.\nMulti-scale LFNs encode multiple levels of details at different scales to reduce aliasing and flickering at lower resolutions.\nOur multi-scale LFN features a progressive representation that encodes all levels of details into a single network using subsets of network weights. \nWith our method, light field networks can be progressively downloaded and rendered based on the desired level of detail for real-time streaming of 6DOF content.\nWe hope progressive multi-scale LFNs will help improve the real-time streaming of photorealistic characters and objects to real-time 3D desktop, AR, and VR applications~\\cite{li2020meteovis,du2019geollery}.\n\\section{Experiments}\nTo evaluate our progressive multi-scale LFN, we conduct experiments with four levels of detail. We first compare the quality when rendering at different scales from a standard single-scale LFN and our multi-scale LFN. We then perform an ablation comparing multiple LFNs trained at separate scales to our progressive network. Finally, we benchmark our multi-scale LFN to evaluate the speedup gained by utilizing fewer neurons when rendering at lower levels of detail. Additional details are in the supplementary material.\n\n\\subsection{Experimental Setup}\nFor our evaluation, we use datasets of real people consisting of $240$ synchronized images from our capture studio. Images are captured at $4032 \\times 3040$ resolution. Our datasets are processed with COLMAP~\\cite{schoenberger2016sfm,schoenberger2016mvs} to extract camera parameters and background matting~\\cite{lin2020matting} to focus on the foreground subject. In each dataset, images are split into groups of 216 training poses, 12 validation poses, and 12 test poses.\n\nWe quantitatively evaluate the encoding quality by computing the PSNR and SSIM metrics. Both metrics are computed on cropped images that tightly bound the subject to reduce the contribution of empty background pixels. We also measure the render time to determine the relative speedup at different levels of detail. Metrics are averaged across all images in the test split of each dataset. \n\nWhen training LFNs, we use a batch size of $8192$ rays with $67\\%$ of rays sampled from the foreground and $33\\%$ sampled from the background. For each epoch, we iterate through training images in batches of two images. In each image batch, we train on $50\\%$ of all foreground rays sampling rays across the two viewpoints. For our multi-scale progressive model and single-scale model, we train using $100$ epochs. For our ablation LFNs at lower resolutions, we train until the validation PSNR matches that of our progressive multi-scale LFN with a limit of $1000$ epochs. Each LFN is trained using a single NVIDIA RTX 2080 TI GPU.\n\n\n\\begin{table}[!htbp]\n  \\caption{Model Parameters at Different Levels of Detail.}\n  \\label{tab:model_parameters}\n  \\centering\n  \\scalebox{0.87}{\n  \\begin{tabular}{lrrrr}\n    \\toprule\n    Level of Detail & 1 & 2 & 3 & 4\\\\\n    \\midrule\n    Model Layers & 10 & 10 & 10 & 10\\\\\n    Layer Width  & 128 & 256 & 384 & 512\\\\\n    Parameters   & 136,812 & 533,764 & 1,193,860 & 2,116,100\\\\\n    Full Size (MB)    & 0.518 & 2.036 & 4.554 & 8.072\\\\\n    Target Scale    & $1\/8$ & $1\/4$ & $1\/2$ & $1$\\\\\n    Train Width & 504 & 1008 & 2016 & 4032\\\\\n    Train Height & 380 & 760 & 1520 & 3040\\\\\n    \\bottomrule\n  \\end{tabular}\n  }\n\\end{table}\n\n\n\\subsection{Rendering Quality}\nWe first evaluate the rendering quality by comparing a standard single-scale LFN trained at the highest resolution with our multi-scale progressive LFN. \nFor the single-scale LFN, we train a model with $10$ layers and $512$ neurons per hidden layer on each of our datasets and render them at multiple scales: $1\/8$, $1\/4$, $1\/2$, and full scale. \nFor our multi-scale progressive LFN, we train an equivalently sized model with four LODs corresponding to each of the target scales. The lowest LOD targets the $1\/8$ scale and utilizes $128$ neurons from each hidden layer. Each increasing LOD uses an additional $128$ neurons. Model parameters for each LOD are laid out in \\autoref{tab:model_parameters}.\nNote that our qualitative results are cropped to the subject and hence have a smaller resolution.\nQualitative results are shown in \\autoref{fig:qualitative_comparison} with renders from both models placed side-by-side. \nQuantitative results are shown in \\autoref{tab:quantiative_quality}.\n\nAt the full scale, both single-scale and multi-scale models are trained to produce full-resolution images. As a result, the quality of the renders from both models is visually similar as shown in \\autoref{fig:qualitative_comparison}.\nAt $1\/2$ scale, the resolution is still sufficiently high so little to no aliasing can be seen in renders from either model. At $1\/4$ scale, we begin to see significant aliasing around figure outlines and shape borders in the single-scale model.\nThis is especially evident in \\autoref{fig:qualitative_comparison_C_r4} where the outlines in the cartoon graphic have an inconsistent thickness. \nOur multi-scale model has much less aliasing with the trade-off of having more blurriness. At the lowest level of detail, we see severe aliasing along the fine textures as well as along the borders of the object in the single-scale model. With a moving camera, the aliasing seen at the two lowest scales appears as flickering artifacts.\n\n\\begin{table}[!htbp]\n  \\caption{Average Rendering Quality Over All Datasets.}\n  \\newcommand{0.9}{0.9}\n  \\label{tab:quantiative_quality}\n  \\centering\n  \\begin{subfigure}[h]{\\linewidth}\n      \\centering\n      \\scalebox{0.9}{\n      \\begin{tabular}{lcccc}\n          \\toprule\n          Model & LOD 1 & LOD 2 & LOD 3 & LOD 4 \\\\ \n          \\midrule\nSingle-scale LFN & 26.95 & 28.05 & 28.21 & 27.75 \\\\\nMultiple LFNs & 29.13 & 29.88 & 29.27 & 27.75 \\\\\nMulti-scale LFN & 29.37 & 29.88 & 29.01 & 28.12 \\\\\n          \\bottomrule\n      \\end{tabular}\n      }\n      \\caption{PSNR (dB) at 1\/8, 1\/4, 1\/2, and 1\/1 scale.}\n      \\label{fig:quantiative_quality_psnr}\n  \\end{subfigure}\\\\\n  \\vspace{2mm}\n  \\begin{subfigure}[h]{\\linewidth}\n      \\centering\n      \\scalebox{0.9}{\n      \\begin{tabular}{lcccc}\n          \\toprule\n           Model & LOD 1 & LOD 2 & LOD 3 & LOD 4 \\\\ \n          \\midrule\nSingle-scale LFN & 0.8584 & 0.8662 & 0.8527 & 0.8480 \\\\\nMultiple LFNs & 0.8133 & 0.8572 & 0.8532 & 0.8480 \\\\\nMulti-scale LFN & 0.8834 & 0.8819 & 0.8626 & 0.8570 \\\\\n           \\bottomrule\n      \\end{tabular}\n      }\n      \\caption{SSIM at 1\/8, 1\/4, 1\/2, and 1\/1 scale.}\n      \\label{fig:quantiative_quality_ssim}\n  \\end{subfigure}\n\\end{table}\n\n\\subsection{Progressive Model Ablation}\nWe next perform an ablation experiment to determine the benefits and drawbacks of our progressive representation.\nFor a non-progressive comparison, we represent each scale of a multi-scale light field using a separate LFN. \nEach LFN is trained using images downscaled to the appropriate resolution.\nThis ablation helps determine the training overhead our progressive LFNs encounter by compressing four resolutions into a single model and how the rendering quality compares to having separate models.\n\n\\begin{table}[!htbp]\n  \\caption{Multiple LFNs \\textit{vs} Progressive Multi-scale LFN.}\n  \\newcommand{0.83}{0.83}\n  \\label{tab:nonprogressive_vs_progressive}\n  \\centering\n  \\begin{subfigure}[t]{\\linewidth}\n  \\centering\n  \\scalebox{0.83}{\n  \\begin{tabular}{lccccc}\n    \\toprule\n    Model & LOD 1 & LOD 2 & LOD 3 & LOD 4 & Total\\\\\n    \\midrule\n    Multiple LFNs & 0.518 & 2.036 & 4.554 & 8.072 & 15.180\\\\\n    Multi-scale LFN & \\multicolumn{4}{c}{\\leavevmode\\leaders\\hrule height 0.7ex depth \\dimexpr0.4pt-0.7ex\\hfill\\kern0pt\\;8.072\\;\\leavevmode\\leaders\\hrule height 0.7ex depth \\dimexpr0.4pt-0.7ex\\hfill\\kern0pt} & 8.072\\\\\n    \\bottomrule\n  \\end{tabular}\n  }\n  \\caption{Model Size (MB).}\n  \\label{tab:nonprogressive_vs_progressive_model_size}\n  \\end{subfigure}\\\\\n  \\vspace{2mm}\n  \\begin{subfigure}[t]{\\linewidth}\n  \\centering\n  \\scalebox{0.83}{\n  \\begin{tabular}{lccccc}\n    \\toprule\n    Model & LOD 1 & LOD 2 & LOD 3 & LOD 4 & Total\\\\\n    \\midrule\nMultiple LFNs & 3.51 & 6.36 & 4.09 & 11.35 & 25.31\\\\\nMulti-scale LFN & \\multicolumn{4}{c}{\\leavevmode\\leaders\\hrule height 0.7ex depth \\dimexpr0.4pt-0.7ex\\hfill\\kern0pt\\;17.78\\;\\leavevmode\\leaders\\hrule height 0.7ex depth \\dimexpr0.4pt-0.7ex\\hfill\\kern0pt} & 17.78\\\\\n    \\bottomrule\n  \\end{tabular}\n  }\n  \\caption{Average Training Time Over All Datasets (hours).}\n  \\label{tab:nonprogressive_vs_progressive_training_time}\n  \\end{subfigure}\n\\end{table}\n\nProgressive and non-progressive model sizes and training times are shown in \\autoref{tab:nonprogressive_vs_progressive}. In terms of model size, using multiple LFNs adds up to $15$ MB compared to $8$ MB for our progressive model. This $47\\%$ savings could allow storing or streaming of more light fields where model size is a concern. \nFor the training time, we observe that training a multi-scale network takes on average $17.78$ hours. Additional offline training time for our multi-scale network compared to training a single standard LFN is expected since each training iteration involves two forward passes. Encoding a multi-scale light field using multiple independent LFNs requires training each network separately, totaling $25.31$ hours. Despite the higher compression achieved by our progressive multi-scale LFN, we observe little to no training overhead. On average, training our multi-scale progressive LFN saves $30\\%$ of training time compared to naively training multiple light field networks at different resolutions.\n\n\nQuantitative PSNR and SSIM quality metrics are shown in \\autoref{tab:quantiative_quality}. We observe that utilizing multiple LFNs to encode each scale of each light field yields better PSNR results compared to rendering from a single-scale LFN. However, utilizing multiple LFNs increases the total model size.\nOur multi-scale LFN achieves the superior PSNR and SSIM results between these two setups without increasing the total model size.\nIn an on-demand streaming scenario, using only a single-scale model would incur visual artifacts and unnecessary computation at lower scales. Streaming separate models resolves these issues but requires more bandwidth. Neither option is ideal for battery life in mobile devices. Our multi-scale model alleviates the aliasing and flickering artifacts without incurring the drawbacks of storing and transmitting multiple models.\n\n\n\n\\begin{table}[!htbp]\n  \\caption{Training Ablation Average Rendering Quality Results.}\n  \\newcommand{0.9}{0.9}\n  \\label{tab:ablation_quality}\n  \\centering\n  \\begin{subfigure}[h]{\\linewidth}\n      \\centering\n      \\scalebox{0.9}{\n      \\begin{tabular}{lcccc}\n          \\toprule\n          Ablation & LOD 1 & LOD 2 & LOD 3 & LOD 4 \\\\ \n          \\midrule\nProgressive Training & 26.30 & 30.33 & 29.08 & 28.10 \\\\\n108 Training Views   & 29.04 & 29.63 & 28.93 & 27.98 \\\\\nOurs                 & 29.37 & 29.88 & 29.01 & 28.12 \\\\\n          \\bottomrule\n      \\end{tabular}\n      }\n      \\caption{PSNR (dB) at 1\/8, 1\/4, 1\/2, and 1\/1 scale.}\n      \\label{fig:ablation_quality_psnr}\n  \\end{subfigure}\\\\\n  \\vspace{2mm}\n  \\begin{subfigure}[h]{\\linewidth}\n      \\centering\n      \\scalebox{0.9}{\n      \\begin{tabular}{lcccc}\n          \\toprule\n           Model & LOD 1 & LOD 2 & LOD 3 & LOD 4 \\\\ \n          \\midrule\nProgressive Training & 0.7947 & 0.8719 & 0.8447 & 0.8390 \\\\\n108 Training Views   & 0.8765 & 0.8771 & 0.8604 & 0.8566 \\\\\nOurs                 & 0.8834 & 0.8819 & 0.8626 & 0.8570 \\\\\n           \\bottomrule\n      \\end{tabular}\n      }\n      \\caption{SSIM at 1\/8, 1\/4, 1\/2, and 1\/1 scale.}\n      \\label{fig:ablation_quality_ssim}\n  \\end{subfigure}\n\\end{table}\n\n\\subsection{Training Ablation}\nTo evaluate our training strategy, we perform two ablation experiments.\nFirst, we compare our strategy to a coarse-to-fine progressive training strategy where lower levels of detail are trained and then frozen as higher levels are trained.\nProgressive training takes on average $24.20$ hours to train.\nSecond, we use half the number of training views to evaluate how the quality degrades with fewer training views.\nBoth experiments use our progressive multi-scale model architecture.\nOur results are shown in~\\autoref{tab:ablation_quality} with additional details in the supplementary material.\n\n\n\\begin{figure}[!htbp]\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/rendering_pipeline.pdf}\n    \\caption{Overview of our rendering pipeline which utilizes an auxiliary network to skip evaluation of empty rays.}\n    \\label{fig:rendering_pipeline}\n\\end{figure}\n\n\\begin{table}[!htbp]\n  \\caption{Average Model Rendering Performance for our Multi-scale LFN (milliseconds per frame).}\n  \\label{tab:performance_table_v2}\n  \\begin{subfigure}[t]{\\linewidth}\n      \\centering\n      \\begin{tabular}{cccc}\n        \\hline\n         LOD 1 & LOD 2 & LOD 3 & LOD 4\\\\\n        \\hline \n3.7 & 3.9 & 4.7 & 5.9\\\\\n        \\hline\n      \\end{tabular}\n      \\caption{Rendering each LOD at 1\/8 scale.}\n      \\label{tab:performance_table_v2_8thscale}\n  \\end{subfigure}\\\\%\n  \\begin{subfigure}[!ht]{\\linewidth}\n      \\centering\n      \\begin{tabular}{cccc}\n        \\hline\n        LOD 1 & LOD 2 & LOD 3 & LOD 4\\\\\n        \\hline\n4 & 11 & 58 & 305\\\\\n        \\hline\n      \\end{tabular}\n      \\caption{Rendering at 1\/8, 1\/4, 1\/2, and 1\/1 scale.}\n      \\label{tab:performance_table_v2_multiscale}\n  \\end{subfigure}\n\\end{table}\n\n\\subsection{Level of Detail Rendering Speedup}\nWe evaluate the rendering speed of our progressive model at different levels of detail to determine the observable speedup from rendering with the appropriate level of detail. For optimal rendering performance, we render using half-precision floating-point values and skip empty rays using an auxiliary neural network encoding only the ray occupancy as shown in \\autoref{fig:rendering_pipeline}. Our auxiliary network is made up of three layers with a width of 16 neurons per hidden layer. Evaluation is performed by rendering rays from all training poses with intrinsics scaled to the corresponding resolution as shown in \\autoref{tab:model_parameters}. \nAverage rendering time results are shown in \\autoref{tab:performance_table_v2}. \nRendering times for lower LODs (1\/8 scale) from our multi-scale network are reduced from $\\sim5.9$ to $\\sim3.7$ msec.\n\\section{Background and Related Works}\nOur work builds upon existing work in neural 3D representations, adaptive neural network inference, and levels of detail. In this section, we provide some background on neural representations and focus on prior work most related to ours. For a more comprehensive overview of neural rendering, we refer interested readers to a recent survey~\\cite{tewari2021advances}.\n\n\n\\begin{table}[!htbp]\n  \\centering\n  \\caption{Neural 3D Representations Comparison}\n  \\label{tab:comparison_table}\n  \\scalebox{0.80}{\n      \\begin{tabular}{lcccc}\n        \\toprule\n        Model & Real-time & Model Size & Multi-scale & Progressive\\\\\n        \\midrule\n        NeRF & \\resizebox{\\widthof{\\scrossmark}*\\ratio{\\widthof{x}}{\\widthof{\\normalsize x}}}{!}{\\scrossmark} & $\\leq 10$ MB & \\resizebox{\\widthof{\\scrossmark}*\\ratio{\\widthof{x}}{\\widthof{\\normalsize x}}}{!}{\\scrossmark}  & \\resizebox{\\widthof{\\scrossmark}*\\ratio{\\widthof{x}}{\\widthof{\\normalsize x}}}{!}{\\scrossmark} \\\\\n        mip-NeRF & \\resizebox{\\widthof{\\scrossmark}*\\ratio{\\widthof{x}}{\\widthof{\\normalsize x}}}{!}{\\scrossmark}  & $\\leq 10$ MB & \\resizebox{\\widthof{\\scheckmark}*\\ratio{\\widthof{x}}{\\widthof{\\normalsize x}}}{!}{\\scheckmark} & \\resizebox{\\widthof{\\scrossmark}*\\ratio{\\widthof{x}}{\\widthof{\\normalsize x}}}{!}{\\scrossmark} \\\\\n        Plenoctree & \\resizebox{\\widthof{\\scheckmark}*\\ratio{\\widthof{x}}{\\widthof{\\normalsize x}}}{!}{\\scheckmark} & $\\geq 30$ MB& \\resizebox{\\widthof{\\scrossmark}*\\ratio{\\widthof{x}}{\\widthof{\\normalsize x}}}{!}{\\scrossmark} &\\resizebox{\\widthof{\\scrossmark}*\\ratio{\\widthof{x}}{\\widthof{\\normalsize x}}}{!}{\\scrossmark} \\\\\n        LFN & \\resizebox{\\widthof{\\scheckmark}*\\ratio{\\widthof{x}}{\\widthof{\\normalsize x}}}{!}{\\scheckmark} &  $\\leq 10$ MB & \\resizebox{\\widthof{\\scrossmark}*\\ratio{\\widthof{x}}{\\widthof{\\normalsize x}}}{!}{\\scrossmark} & \\resizebox{\\widthof{\\scrossmark}*\\ratio{\\widthof{x}}{\\widthof{\\normalsize x}}}{!}{\\scrossmark} \\\\\n        Ours & \\resizebox{\\widthof{\\scheckmark}*\\ratio{\\widthof{x}}{\\widthof{\\normalsize x}}}{!}{\\scheckmark} & $\\leq 10$ MB & \\resizebox{\\widthof{\\scheckmark}*\\ratio{\\widthof{x}}{\\widthof{\\normalsize x}}}{!}{\\scheckmark} & \\resizebox{\\widthof{\\scheckmark}*\\ratio{\\widthof{x}}{\\widthof{\\normalsize x}}}{!}{\\scheckmark}\\\\\n      \\bottomrule\n      \\end{tabular}\n  }\n\\end{table}\n\n\n\\subsection{Neural 3D Representations}\nTraditionally, 3D content has been encoded in a wide variety of formats such as meshes, point clouds, voxels, multi-plane images, and even side-by-side video.\nTo represent the full range of visual effects from arbitrary viewpoints, researchers have considered using radiance fields and light fields which can be encoded using neural networks. With neural networks, 3D data such as signed distance fields, radiance fields, and light fields can be efficiently encoded as coordinate functions without explicit limitations on the resolution.\n\n\\subsubsection{Neural Radiance Field (NeRF)}\nEarly neural representations focused on signed distance functions and radiance fields. Neural Radiance Fields (NeRF)~\\cite{mildenhall2020nerf} use differentiable volume rendering to encode multiple images of a scene into a radiance field encoded as a neural network. With a sufficiently dense set of images, NeRF is able to synthesize new views by using the neural network to interpolate radiance values across different positions and viewing directions. Since the introduction of NeRF, followup works have added support for deformable scenes \\cite{pumarola2020d,park2021nerfies,park2021hypernerf}, real-time inference \\cite{garbin2021fastnerf,reiser2021kilonerf,neff2021donerf,yu2021plenoxels},\nimproved quality \\cite{shen2021snerf}, generalization across scenes \\cite{yu2020pixelnerf,rebain2021lolnerf},\ngenerative modelling \\cite{schwarz2020graf,niemeyer2021campari,xie2021fignerf}, videos \\cite{li2021neural,xian2021space,wang2022fourier}, sparse views \\cite{Chibane_2021_CVPR,niemeyer2021regnerf,Jain_2021_ICCV}, fast training \\cite{mueller2022instant,yu2021plenoxels}, and even large-scale scenes \\cite{rematas2021urban,xiangli2021citynerf,turki2021meganerf}. \n\nAmong NeRF research, Mip-NeRF~\\cite{barron2021mipnerf,barron2022mipnerf360} and BACON~\\cite{lindell2021bacon} share a similar goal to our work in offering a multi-scale representation to reduce aliasing. \nMip-NeRF approximates the radiance across conical regions along the ray, cone-tracing, using integrated positional encoding (IPE). With cone-tracing, Mip-NeRF reduces aliasing while also slightly reducing training time.\nBACON~\\cite{lindell2021bacon} provide a multi-scale scene representation through band-limited networks using Multiplicative Filter Networks~\\cite{fathony2021multiplicative} and multiple exit points. Each scale in BACON has band-limited outputs in Fourier space.\n\n\\subsubsection{Light Field Network (LFN)}\nOur work builds upon Light Field Networks (LFNs)~\\cite{sitzmann2021lfns,feng2021signet,li2022neulf,chandramouli2021light}. Light field networks encode light fields~\\cite{levoy1996lightfieldrendering} using a coordinate-wise representation, where for each ray $\\mathbf{r}$, a neural network $f$ is used to directly encode the color $c=f(\\mathbf{r})$. To represent rays, Feng and Varshney \\cite{feng2021signet} pair the traditional two-plane parameterization with Gegenbauer polynomials while Sitzmann \\etal~\\cite{sitzmann2021lfns} use Pl\\\"{u}cker coordinates which combine the ray direction $\\mathbf{r}_d$ and ray origin $\\mathbf{r}_o$ into a 6-dimensional vector $(\\mathbf{r}_d, \\mathbf{r}_o \\times \\mathbf{r}_d)$. Light fields can also be combined with explicit representations~\\cite{ost2021point}. In our work, we adopt the Pl\\\"{u}cker coordinate parameterization which can represent rays in all directions.\n\nCompared to NeRF, LFNs are faster to render as they only require a single forward pass per ray, enabling real-time rendering. However, LFNs are worse at view synthesis as they lack the self-supervision that volume rendering provides with explicit 3D coordinates. As a result, LFNs are best trained from very dense camera arrays or from a NeRF teacher model such as in~\\cite{attal2022learning}. Similar to NeRFs, LFNs are encoded as multi-layer perceptrons, hence they remain compact compared to voxel and octree NeRF representations~\\cite{yu2021plenoctrees,hedman2021snerg,garbin2021fastnerf,yu2021plenoxels} which use upwards of $50$ MB. Therefore LFNs strike a balance between fast rendering times and storage compactness which could make them suitable for streaming 3D scenes over the internet.\n\n\n\\subsection{Levels of Detail}\nLevel of detail methods are commonly used in computer graphics to improve performance and quality. For 2D textures, one of the most common techniques is mipmapping~\\cite{williams1983mipmap} which both reduces aliasing and improves performance with cache coherency by encoding textures at multiple resolutions. For neural representations, techniques for multiple levels of detail have also been proposed for signed distance functions as well as neural radiance fields.\n\nFor signed distance fields, Takikawa \\etal \\cite{takikawa2021neural} propose storing learned features within nodes of an octree to represent a signed distance function with multiple levels of detail. \nBuilding upon octree features, Chen \\etal \\cite{chen2021multiresolution} develop Multiresolution Deep Implicit Functions (MDIF) to represent 3D shapes which combines a global SDF representation with local residuals. \nFor radiance fields, Yang \\etal~\\cite{yang2021recursivenerf} develop Recursive-NeRF which grows a neural representation with multi-stage training which is able to reduce NeRF rendering time. \nBACON~\\cite{lindell2021bacon} offers levels of detail for various scene representations including 2D images and radiance fields with different scales encoding different bandwidths in Fourier space.\nPINs~\\cite{Landgraf2022PINs} uses a progressive fourier encoding to improve reconstruction for scenes with a wide frequency spectrum.\nFor more explicit representations, Variable Bitrate Neural Fields~\\cite{takikawa2022variable} use a vector-quantized auto-decoder to compress feature grids into lookup indices and a learned codebook. Levels of detail are obtained by learning compressed feature grids at multiple resolutions in a sparse octree.\nIn parallel, Streamable Neural Fields~\\cite{cho2022streamable} propose using progressive neural networks~\\cite{rusu2016progressive} to represent spectrally, spatially, or temporally growing neural representations.\n\n\\subsection{Adaptive Inference}\nCurrent methods use adaptive neural network inference based on available computational resources or the difficulty of each input example. Bolukbasi \\etal~\\cite{bolukbasi2017adaptive} propose two schemes for adaptive inference. Early exiting allows intermediate layers to route outputs directly to an exit head while network selection dynamically routes features to different networks. Both methods allow easy examples to be classified by a subset of neural network layers. Yeo \\etal \\cite{yeo2018neuraladaptive} apply early exiting to upscale streamed video. By routing intermediate CNN features to exit layers, a single super-resolution network can be partially downloaded and applied on a client device based on available compute. Similar ideas have also been used to adapt NLP to input complexity~\\cite{zhou2020bert,weijie2020fastbert}.\nYang \\etal~\\cite{yang2020resolutionadaptive} develop Resolution Adaptive Network (RANet) which extracts features and performs classification of an image progressively from low-resolution to high-resolution. Doing so sequentially allows efficient early exiting as many images can be classified with only coarse features. Slimmable neural networks~\\cite{yu2018slimmable} train a model at different widths with a switchable batch normalization and observe better performance than individual models at detection and segmentation tasks.\n\n\\section{Introduction}\n\nVolumetric images and video allow users to view 3D scenes from novel viewpoints with a full 6 degrees of freedom (6DOF). Compared to conventional 2D and 360 images and videos, volumetric content enables additional immersion with motion parallax and binocular stereo while also allowing users to freely explore the full scene.\n\n\nTraditionally, scenes generated with 3D modeling have been commonly represented as meshes and materials. These classical representations are suitable for real-time rendering as well as streaming for 3D games and AR effects. However, real captured scenes are challenging to represent using the mesh and material representation, requiring complex reconstruction and texture fusion techniques \\cite{duo2016fusion4d,du2018montage4d}. To produce better photo-realistic renders, existing view-synthesis techniques have attempted to adapt these representations with multi-plane images (MPI)~\\cite{zhou2018stereo}, layered-depth images (LDI)~\\cite{shade1998ldi}, and multi-sphere images (MSI)~\\cite{attal2020matryodshka}. These representations allow captured scenes to be accurately represented with 6DOF but only within a limited area.\n\n\n\\begin{figure}[!tbp]\n    \\includegraphics[width=\\linewidth]{figures\/maria\/teaser_v3_vertical.pdf}\n    \\caption{Our progressive multi-scale light field network is suited for streaming, reduces aliasing and flicking, and renders more efficiently at lower resolutions.}\n    \\label{fig:teaser}\n    \\vspace{-5mm}\n\\end{figure}\n\nRadiance fields and light fields realistically represent captured scenes from an arbitrarily large set of viewpoints.\nRecently, neural representations such as neural radiance fields (NeRFs)~\\cite{mildenhall2020nerf} and light field networks (LFNs)~\\cite{sitzmann2021lfns} have been found to compactly represent 3D scenes and perform view-synthesis allowing re-rendering from arbitrary viewpoints. Given a dozen or more photographs capturing a scene from different viewpoints from a conventional camera, a NeRF or LFN can be optimized to accurately reproduce the original images and stored in less than 10 MB. Once trained, NeRFs will also produce images from different viewpoints with photo-realistic quality.\n\nWhile NeRFs are able to encode and reproduce real-life 3D scenes from arbitrary viewpoints with photorealistic accuracy, they suffer from several drawbacks which limit their use in a full on-demand video streaming pipeline. Some notable drawbacks include slow rendering, aliasing at smaller scales, and a lack of progressive decoding. While some of these drawbacks have been independently addressed in follow-up work~\\cite{barron2021mipnerf,yu2021plenoctrees}, approaches to resolving them cannot always be easily combined and may also introduce additional limitations.\n\nIn this paper, we extend light field networks with multiple levels of detail and anti-aliasing at lower scales making them more suitable for on-demand streaming. This is achieved by encoding multiple reduced-resolution representations into a single network. With multiple levels of details, light field networks can be progressively streamed and rendered based on the desired scale or resource limitations. For free-viewpoint rendering and dynamic content, anti-aliasing is essential to reducing flickering when rendering at smaller scales. In summary, the contributions of our progressive multi-scale light field network are:\n\\begin{enumerate}\n\\item Composing multiple levels of detail within a single network takes up less space than storing all the levels independently.\n\\item The progressive nature of our model requires us to only download the parameters necessary up to the desired level of detail. \n\\item Our model allows rendering among multiple levels of detail simultaneously. This makes for a smooth transition between multiple levels of detail as well as spatially adaptive rendering (including foveated rendering) without requiring independent models at multiple levels of detail on the GPU.\n\\item Our model allows for faster inference of lower levels of detail by using fewer parameters.\n\\item We are able to encode light fields at multiple scales to reduce aliasing when rendering at lower resolutions.\n\\end{enumerate}\n\n\\section{Discussion}\nThe primary contribution of our paper is the identification of a representation more suitable for on-demand streaming. \nThis is achieved by enhancing light field networks, which support real-time rendering, to progressively support multiple levels of detail at different resolutions.\nOur representation inherits some of the limitations of LFNs such as long training times and worse view-synthesis quality compared to NeRF-based models.\nSince our method does not support continuous LODs, we require the user to decide the LODs ahead of time. For intermediate resolutions, rendering to the nearest LOD is sufficient to avoid aliasing and flickering.\nWhile our light field datasets could also be rendered with classical techniques~\\cite{levoy1996lightfieldrendering,xiao2014cvpr,peter2001wavelet}, doing so over the internet would not be as bandwidth-efficient.\nFuture work may combine our multi-scale light field networks with PRIF~\\cite{Feng2022PRIF} to encode color and geometry simultaneously or with VIINTER~\\cite{Feng2022Viinter} for dynamic light fields.\n\n\n\n\\section{Method}\nOur method consists of a multi-scale light field encoded using a progressive neural network to reduce aliasing and enable adaptive streaming and rendering.\n\n\n\n\\subsection{Multi-scale Light Field}\n\nWe propose encoding multiple representations of the light field which produce anti-aliased representations of a light field, similar to mipmaps for conventional images. For this, we generate an image pyramid for each image view with area downsampling and encode each resolution as a separate light field representation. In contrast to bilinear or nearest pixel downsampling, area downsampling creates an anti-aliased view as shown in \\autoref{fig:image_downsampling_methods} where each pixel in the downsampled image is an average over a corresponding area in the full-resolution image.\n\n\\begin{figure}[!htb]\n    \\centering\n    \\captionsetup[subfigure]{labelformat=empty}\n    \\begin{subfigure}{0.52\\linewidth}\n        \\includegraphics[width=\\linewidth]{figures\/aliasing\/aliasing_fullres.pdf}\n        \\vspace{-5.5mm}\n        \\caption{Single-scale Light Field Network}\n    \\end{subfigure}\\\\\n    \\vspace{1mm}\n    \\begin{subfigure}{0.52\\linewidth}\n        \\includegraphics[width=\\linewidth]{figures\/aliasing\/aliasing_mipnet.pdf}\n        \\vspace{-5.5mm}\n        \\caption{Multi-scale Light Field Network}\n    \\end{subfigure}\n    \\caption{Rendering a single full-scale LFN at a lower (1\/8) resolution results in aliasing, unlike the multi-scale LFN at the same (1\/8) resolution. When changing viewpoints (3 shown above), the aliasing results in flickering.}\n    \\label{fig:aliasing_figure}\n\\end{figure}\n\nAs with mipmaps, lower-resolution representations trade-off sharpness for less aliasing. In this case, where a light field is rendered into an image, sharpness may be preferable over some aliasing. However, in the case of videos where the light field is viewed from different poses or the light field is dynamic, sampling from a high-resolution light field leads to flickering. With a multi-scale representation, rendering light fields at smaller resolutions can be done at a lower scale to reduce flickering.\n\n\\subsection{Progressive Model}\n\n\\begin{figure}[!htbp]\n  \\centering\n  \\includegraphics[width=\\linewidth]{figures\/model_illustration_subnet_lods.pdf}\n  \\caption{Using subsets of neurons to encode different levels of detail within a single model.}\n  \\label{fig:model_illustration_subnet}\n\\end{figure}\n\nFor light field streaming, we wish to have a compact representation that can simultaneously encode multiple levels of detail into a single progressive model. Specifically, we seek a model with the following properties:\nGiven specific subsets of network weights, we should be able to render a complete image with reduced details.\nProgressive levels of details should share weights with lower levels to eliminate encoding redundancy.\n\nWith the above properties, our model offers several benefits over a standard full-scale network or encoding multi-scale light fields using multiple networks.\nFirst, our model composes all scales within a single network hence requiring less storage space than storing four separate models.\nSecond, our model is progressive, thus only parameters necessary for the desired level of detail are needed in a streaming scenario.\nThird, our model allows rendering different levels of detail across pixels for dithered transitions and foveation.\nFourth, our model allows for faster inference of lower levels of detail by utilizing fewer parameters.\nModel sizes and training times appear in \\autoref{tab:nonprogressive_vs_progressive}.\n\n\n\\begin{figure}[!htbp]\n    \\centering\n    \\includegraphics[width=0.5\\linewidth]{figures\/subnet_layer_illustration.pdf}\n    \\caption{Illustration of a single layer where a subset of the weight matrix and bias vector are used, evaluating half of the neurons and reducing the operations down to approximately a quarter.}\n    \\label{fig:subnet_layer_illustration}\n\\end{figure}\n\n\\subsubsection{Subset of neurons}\nTo encode multiple levels of detail within a unified representation, we propose using subsets of neurons as shown in \\autoref{fig:model_illustration_subnet}. For each level of detail, we assign a fixed number of neurons to evaluate, with increasing levels of detail using an increasing proportion of the neurons in the neural network. An illustration of the matrix subset operation applied to each layer is shown in \\autoref{fig:subnet_layer_illustration}. For example, a single linear layer with $512$ neurons will have a $512 \\times 512$ weight matrix and a bias vector of size $512$. For a low level of detail, we can assign a neuron subset size of $25\\%$ or $128$ neurons. Then each linear layer would use the top-left $128 \\times 128$ submatrix for the weights and the top $128$ subvector for the bias. To keep the input and output dimensions the same, the first hidden layer uses every column of the weight matrix and the final output layer uses every row of the weight matrix.\nUnlike using a subset of layers, as is done with early exiting~\\cite{bolukbasi2017adaptive,yeo2018neuraladaptive}, using a subset of neurons preserves the number of sequential non-linear activations between all levels of details. Since layer widths are typically larger than model depths, using subsets of neurons also allows more granular control over the quantity and quality of levels. \n\n\\subsubsection{Training} \\label{sec:mipnet_training}\nWith multiple levels of detail at varying resolutions, a modified training scheme is required to efficiently encode all levels of detail together. \nOn one hand, generating all levels of detail at each iteration heavily slows down training as each iteration requires an independent forward pass through the network. \nOn the other hand, selecting a single level of detail at each iteration or applying progressive training compromises the highest level of detail whose full weights would only get updated a fraction of the time.\n\nWe adopt an intermediate approach that focuses on training the highest level of detail. \nDuring training, each batch does a full forward pass through the entire network, producing pixels at the highest level of detail along with a forward pass at a randomly selected lower level of detail. The squared L2 losses for both the full detail and reduced detail are added together for a combined backward pass. By training at the highest level of detail, we ensure that every weight gets updated at each batch and that the highest level of detail is trained to the same number of iterations as a standard model. Specifically, given a batch of rays $\\{\\mathbf{r}_i\\}_{i=1}^{b}$, corresponding ground truth colors $\\{\\mathbf\\{\\mathbf{y}_i^c\\}_{i=1}^{b}\\}_{c=1}^{4}$ for four levels of details, and a random lower level of detail $k \\in \\{1,2,3\\}$, our loss function is:\n$$L = \\frac{1}{b}\\sum_{i=1}^b \\left[ \\left\\Vert f_4(\\mathbf{r}_i) - \\mathbf{y}_i^4\\right\\Vert^2_2 + \\left\\Vert f_k(\\mathbf{r}_i) - \\mathbf{y}_i^k\\right\\Vert^2_2 \\right]$$\nAs rays are sampled from the highest-resolution representation, each ray will not correspond to a unique pixel in the lower-resolution images in the image pyramid. Thus for lower levels of detail, corresponding colors are sampled using bilinear interpolation from the area downsampled training images.\n\n\\subsection{Adaptive Rendering}\nOur proposed model allows dynamic levels of detail on a per-ray\/per-pixel level or on a per-object level. As aliasing appears primarily at smaller scales, selecting the level of detail based on the distance to the viewer reduces aliasing and flickering for distant objects.\n\n\\subsubsection{Distance-based Level of Detail}\nWhen rendering light fields from different distances, the space between rays in object space is proportional to the distance between the object and the camera. Due to this, aliasing and flickering artifacts can occur at large distances when the object appears small. By selecting the level of detail based on the distance of the object from the camera or viewer, aliasing and flickering can be reduced. With fewer pixels to render, rendering smaller objects involves fewer forward passes through the light field network. Hence, the amount of operations is typically proportional to the number of pixels. By rendering smaller representations at lower LODs, we further improve the performance as pixels from lower LODs only perform a forward pass using a subset of weights.\n\n\\begin{figure}[!htbp]\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/lod_transition_zoom.pdf}\n    \\caption{With per-pixel level of detail, dithering can be applied to transition between levels of detail.}\n    \\label{fig:transition_example}\n\\end{figure}\n\n\\subsubsection{Dithered Transitions}\nWith per-pixel LOD, transitioning between levels can be achieved with dithered rendering. By increasing the fraction of pixels rendered at the new level of detail in each frame, dithering creates the appearance of a fading transition between levels as shown in \\autoref{fig:transition_example}.\n\n\\begin{figure}[!htbp]\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{figures\/foveation_example.pdf}\n    \\caption{An example of foveated rendering from our progressive multi-scale LFN. Foveated rendering  generates peripheral pixels at lower levels of detail to further improve performance in eye-tracked environments. In this example, foveation is centered over the left eye of the subject. }\n    \\label{fig:foveation_example}\n\\end{figure}\n\n\\subsubsection{Foveated Rendering}\nIn virtual and augmented reality applications where real-time gaze information is available through an eye-tracker, foveated rendering~\\cite{Meng20203D,meng2018Kernel,li2021logrectilinear} can be applied to better focus rendering compute. As each pixel can be rendered at a separate level of detail, peripheral pixels can be rendered at a lower level of detail to improve the performance. \\autoref{fig:foveation_example} shows an example of foveation from our progressive multi-scale LFN.\n\n\\newcommand{0.24\\linewidth}{0.24\\linewidth}\n\\newcommand{\\addDatasetResults}[2]{\n    \\begin{subfigure}[b]{0.24\\linewidth}\n        \\centering\n        \\includegraphics[width=0.8\\linewidth]{figures\/#2\/comparison2_r8.pdf}\n        \\caption{Dataset #1 at 1\/8 scale}\n        \\label{fig:qualitative_comparison_#1_r8}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[b]{0.24\\linewidth}\n        \\centering\n        \\includegraphics[width=0.8\\linewidth]{figures\/#2\/comparison2_r4.pdf}\n        \\caption{Dataset #1 at 1\/4 scale}\n        \\label{fig:qualitative_comparison_#1_r4}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[b]{0.24\\linewidth}\n        \\centering\n        \\includegraphics[width=0.8\\linewidth]{figures\/#2\/comparison2_r2.pdf}\n        \\caption{Dataset #1 at 1\/2 scale}\n        \\label{fig:qualitative_comparison_#1_r2}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[b]{0.24\\linewidth}\n        \\centering\n        \\includegraphics[width=0.8\\linewidth]{figures\/#2\/comparison2_r1.pdf}\n        \\caption{Dataset #1 at full scale}\n        \\label{fig:qualitative_comparison_#1_r1}\n    \\end{subfigure}\n}\n\\begin{figure*}[!htbp]\n    \\centering\n    \\addDatasetResults{C}{david5}\n    \\caption{Qualitative results of one of our datasets at four scales. Single-scale LFNs (left) trained at the full-resolution exhibit aliasing and flickering at lower scales. Our progressive multi-scale LFNs (right) encode all four scales into a single model and have reduced aliasing and flickering at lower scales.}\n    \\label{fig:qualitative_comparison}\n\\end{figure*}\n\n\\newcommand{\\datasetScaledResults}[2]{\n    \\begin{subfigure}[b]{0.17\\linewidth}\n      \\centering\n      \\includegraphics[width=\\linewidth]{figures\/#2\/mipnet_lod_scaled.pdf}\n      \\caption{Dataset #1}\n      \\label{fig:qualitative_scaled_#2}\n    \\end{subfigure}\n}\n\\begin{figure*}[!htbp]\n    \\centering\n    \\datasetScaledResults{A}{jon}\n    \\hfill\n    \\datasetScaledResults{B}{david4}\n    \\hfill\n    \\datasetScaledResults{C}{david5}\n    \\hfill\n    \\datasetScaledResults{D}{sida1}\n    \\hfill\n    \\datasetScaledResults{E}{maria}\n    \\caption{Scaled qualitative results from our progressive multi-scale LFNs at four levels of detail.}\n    \\label{fig:qualtiative_scaled}\n\\end{figure*}\n\n\n\\section{Conclusion}\nIn this paper, we propose a method to encode multi-scale light field networks targeted for streaming.\nMulti-scale LFNs encode multiple levels of details at different scales to reduce aliasing and flickering at lower resolutions.\nOur multi-scale LFN features a progressive representation that encodes all levels of details into a single network using subsets of network weights. \nWith our method, light field networks can be progressively downloaded and rendered based on the desired level of detail for real-time streaming of 6DOF content.\nWe hope progressive multi-scale LFNs will help improve the real-time streaming of photorealistic characters and objects to real-time 3D desktop, AR, and VR applications~\\cite{li2020meteovis,du2019geollery}.\n\\section{Experiments}\nTo evaluate our progressive multi-scale LFN, we conduct experiments with four levels of detail. We first compare the quality when rendering at different scales from a standard single-scale LFN and our multi-scale LFN. We then perform an ablation comparing multiple LFNs trained at separate scales to our progressive network. Finally, we benchmark our multi-scale LFN to evaluate the speedup gained by utilizing fewer neurons when rendering at lower levels of detail. Additional details are in the supplementary material.\n\n\\subsection{Experimental Setup}\nFor our evaluation, we use datasets of real people consisting of $240$ synchronized images from our capture studio. Images are captured at $4032 \\times 3040$ resolution. Our datasets are processed with COLMAP~\\cite{schoenberger2016sfm,schoenberger2016mvs} to extract camera parameters and background matting~\\cite{lin2020matting} to focus on the foreground subject. In each dataset, images are split into groups of 216 training poses, 12 validation poses, and 12 test poses.\n\nWe quantitatively evaluate the encoding quality by computing the PSNR and SSIM metrics. Both metrics are computed on cropped images that tightly bound the subject to reduce the contribution of empty background pixels. We also measure the render time to determine the relative speedup at different levels of detail. Metrics are averaged across all images in the test split of each dataset. \n\nWhen training LFNs, we use a batch size of $8192$ rays with $67\\%$ of rays sampled from the foreground and $33\\%$ sampled from the background. For each epoch, we iterate through training images in batches of two images. In each image batch, we train on $50\\%$ of all foreground rays sampling rays across the two viewpoints. For our multi-scale progressive model and single-scale model, we train using $100$ epochs. For our ablation LFNs at lower resolutions, we train until the validation PSNR matches that of our progressive multi-scale LFN with a limit of $1000$ epochs. Each LFN is trained using a single NVIDIA RTX 2080 TI GPU.\n\n\n\\begin{table}[!htbp]\n  \\caption{Model Parameters at Different Levels of Detail.}\n  \\label{tab:model_parameters}\n  \\centering\n  \\scalebox{0.87}{\n  \\begin{tabular}{lrrrr}\n    \\toprule\n    Level of Detail & 1 & 2 & 3 & 4\\\\\n    \\midrule\n    Model Layers & 10 & 10 & 10 & 10\\\\\n    Layer Width  & 128 & 256 & 384 & 512\\\\\n    Parameters   & 136,812 & 533,764 & 1,193,860 & 2,116,100\\\\\n    Full Size (MB)    & 0.518 & 2.036 & 4.554 & 8.072\\\\\n    Target Scale    & $1\/8$ & $1\/4$ & $1\/2$ & $1$\\\\\n    Train Width & 504 & 1008 & 2016 & 4032\\\\\n    Train Height & 380 & 760 & 1520 & 3040\\\\\n    \\bottomrule\n  \\end{tabular}\n  }\n\\end{table}\n\n\n\\subsection{Rendering Quality}\nWe first evaluate the rendering quality by comparing a standard single-scale LFN trained at the highest resolution with our multi-scale progressive LFN. \nFor the single-scale LFN, we train a model with $10$ layers and $512$ neurons per hidden layer on each of our datasets and render them at multiple scales: $1\/8$, $1\/4$, $1\/2$, and full scale. \nFor our multi-scale progressive LFN, we train an equivalently sized model with four LODs corresponding to each of the target scales. The lowest LOD targets the $1\/8$ scale and utilizes $128$ neurons from each hidden layer. Each increasing LOD uses an additional $128$ neurons. Model parameters for each LOD are laid out in \\autoref{tab:model_parameters}.\nNote that our qualitative results are cropped to the subject and hence have a smaller resolution.\nQualitative results are shown in \\autoref{fig:qualitative_comparison} with renders from both models placed side-by-side. \nQuantitative results are shown in \\autoref{tab:quantiative_quality}.\n\nAt the full scale, both single-scale and multi-scale models are trained to produce full-resolution images. As a result, the quality of the renders from both models is visually similar as shown in \\autoref{fig:qualitative_comparison}.\nAt $1\/2$ scale, the resolution is still sufficiently high so little to no aliasing can be seen in renders from either model. At $1\/4$ scale, we begin to see significant aliasing around figure outlines and shape borders in the single-scale model.\nThis is especially evident in \\autoref{fig:qualitative_comparison_C_r4} where the outlines in the cartoon graphic have an inconsistent thickness. \nOur multi-scale model has much less aliasing with the trade-off of having more blurriness. At the lowest level of detail, we see severe aliasing along the fine textures as well as along the borders of the object in the single-scale model. With a moving camera, the aliasing seen at the two lowest scales appears as flickering artifacts.\n\n\\begin{table}[!htbp]\n  \\caption{Average Rendering Quality Over All Datasets.}\n  \\newcommand{0.9}{0.9}\n  \\label{tab:quantiative_quality}\n  \\centering\n  \\begin{subfigure}[h]{\\linewidth}\n      \\centering\n      \\scalebox{0.9}{\n      \\begin{tabular}{lcccc}\n          \\toprule\n          Model & LOD 1 & LOD 2 & LOD 3 & LOD 4 \\\\ \n          \\midrule\nSingle-scale LFN & 26.95 & 28.05 & 28.21 & 27.75 \\\\\nMultiple LFNs & 29.13 & 29.88 & 29.27 & 27.75 \\\\\nMulti-scale LFN & 29.37 & 29.88 & 29.01 & 28.12 \\\\\n          \\bottomrule\n      \\end{tabular}\n      }\n      \\caption{PSNR (dB) at 1\/8, 1\/4, 1\/2, and 1\/1 scale.}\n      \\label{fig:quantiative_quality_psnr}\n  \\end{subfigure}\\\\\n  \\vspace{2mm}\n  \\begin{subfigure}[h]{\\linewidth}\n      \\centering\n      \\scalebox{0.9}{\n      \\begin{tabular}{lcccc}\n          \\toprule\n           Model & LOD 1 & LOD 2 & LOD 3 & LOD 4 \\\\ \n          \\midrule\nSingle-scale LFN & 0.8584 & 0.8662 & 0.8527 & 0.8480 \\\\\nMultiple LFNs & 0.8133 & 0.8572 & 0.8532 & 0.8480 \\\\\nMulti-scale LFN & 0.8834 & 0.8819 & 0.8626 & 0.8570 \\\\\n           \\bottomrule\n      \\end{tabular}\n      }\n      \\caption{SSIM at 1\/8, 1\/4, 1\/2, and 1\/1 scale.}\n      \\label{fig:quantiative_quality_ssim}\n  \\end{subfigure}\n\\end{table}\n\n\\subsection{Progressive Model Ablation}\nWe next perform an ablation experiment to determine the benefits and drawbacks of our progressive representation.\nFor a non-progressive comparison, we represent each scale of a multi-scale light field using a separate LFN. \nEach LFN is trained using images downscaled to the appropriate resolution.\nThis ablation helps determine the training overhead our progressive LFNs encounter by compressing four resolutions into a single model and how the rendering quality compares to having separate models.\n\n\\begin{table}[!htbp]\n  \\caption{Multiple LFNs \\textit{vs} Progressive Multi-scale LFN.}\n  \\newcommand{0.83}{0.83}\n  \\label{tab:nonprogressive_vs_progressive}\n  \\centering\n  \\begin{subfigure}[t]{\\linewidth}\n  \\centering\n  \\scalebox{0.83}{\n  \\begin{tabular}{lccccc}\n    \\toprule\n    Model & LOD 1 & LOD 2 & LOD 3 & LOD 4 & Total\\\\\n    \\midrule\n    Multiple LFNs & 0.518 & 2.036 & 4.554 & 8.072 & 15.180\\\\\n    Multi-scale LFN & \\multicolumn{4}{c}{\\leavevmode\\leaders\\hrule height 0.7ex depth \\dimexpr0.4pt-0.7ex\\hfill\\kern0pt\\;8.072\\;\\leavevmode\\leaders\\hrule height 0.7ex depth \\dimexpr0.4pt-0.7ex\\hfill\\kern0pt} & 8.072\\\\\n    \\bottomrule\n  \\end{tabular}\n  }\n  \\caption{Model Size (MB).}\n  \\label{tab:nonprogressive_vs_progressive_model_size}\n  \\end{subfigure}\\\\\n  \\vspace{2mm}\n  \\begin{subfigure}[t]{\\linewidth}\n  \\centering\n  \\scalebox{0.83}{\n  \\begin{tabular}{lccccc}\n    \\toprule\n    Model & LOD 1 & LOD 2 & LOD 3 & LOD 4 & Total\\\\\n    \\midrule\nMultiple LFNs & 3.51 & 6.36 & 4.09 & 11.35 & 25.31\\\\\nMulti-scale LFN & \\multicolumn{4}{c}{\\leavevmode\\leaders\\hrule height 0.7ex depth \\dimexpr0.4pt-0.7ex\\hfill\\kern0pt\\;17.78\\;\\leavevmode\\leaders\\hrule height 0.7ex depth \\dimexpr0.4pt-0.7ex\\hfill\\kern0pt} & 17.78\\\\\n    \\bottomrule\n  \\end{tabular}\n  }\n  \\caption{Average Training Time Over All Datasets (hours).}\n  \\label{tab:nonprogressive_vs_progressive_training_time}\n  \\end{subfigure}\n\\end{table}\n\nProgressive and non-progressive model sizes and training times are shown in \\autoref{tab:nonprogressive_vs_progressive}. In terms of model size, using multiple LFNs adds up to $15$ MB compared to $8$ MB for our progressive model. This $47\\%$ savings could allow storing or streaming of more light fields where model size is a concern. \nFor the training time, we observe that training a multi-scale network takes on average $17.78$ hours. Additional offline training time for our multi-scale network compared to training a single standard LFN is expected since each training iteration involves two forward passes. Encoding a multi-scale light field using multiple independent LFNs requires training each network separately, totaling $25.31$ hours. Despite the higher compression achieved by our progressive multi-scale LFN, we observe little to no training overhead. On average, training our multi-scale progressive LFN saves $30\\%$ of training time compared to naively training multiple light field networks at different resolutions.\n\n\nQuantitative PSNR and SSIM quality metrics are shown in \\autoref{tab:quantiative_quality}. We observe that utilizing multiple LFNs to encode each scale of each light field yields better PSNR results compared to rendering from a single-scale LFN. However, utilizing multiple LFNs increases the total model size.\nOur multi-scale LFN achieves the superior PSNR and SSIM results between these two setups without increasing the total model size.\nIn an on-demand streaming scenario, using only a single-scale model would incur visual artifacts and unnecessary computation at lower scales. Streaming separate models resolves these issues but requires more bandwidth. Neither option is ideal for battery life in mobile devices. Our multi-scale model alleviates the aliasing and flickering artifacts without incurring the drawbacks of storing and transmitting multiple models.\n\n\n\n\\begin{table}[!htbp]\n  \\caption{Training Ablation Average Rendering Quality Results.}\n  \\newcommand{0.9}{0.9}\n  \\label{tab:ablation_quality}\n  \\centering\n  \\begin{subfigure}[h]{\\linewidth}\n      \\centering\n      \\scalebox{0.9}{\n      \\begin{tabular}{lcccc}\n          \\toprule\n          Ablation & LOD 1 & LOD 2 & LOD 3 & LOD 4 \\\\ \n          \\midrule\nProgressive Training & 26.30 & 30.33 & 29.08 & 28.10 \\\\\n108 Training Views   & 29.04 & 29.63 & 28.93 & 27.98 \\\\\nOurs                 & 29.37 & 29.88 & 29.01 & 28.12 \\\\\n          \\bottomrule\n      \\end{tabular}\n      }\n      \\caption{PSNR (dB) at 1\/8, 1\/4, 1\/2, and 1\/1 scale.}\n      \\label{fig:ablation_quality_psnr}\n  \\end{subfigure}\\\\\n  \\vspace{2mm}\n  \\begin{subfigure}[h]{\\linewidth}\n      \\centering\n      \\scalebox{0.9}{\n      \\begin{tabular}{lcccc}\n          \\toprule\n           Model & LOD 1 & LOD 2 & LOD 3 & LOD 4 \\\\ \n          \\midrule\nProgressive Training & 0.7947 & 0.8719 & 0.8447 & 0.8390 \\\\\n108 Training Views   & 0.8765 & 0.8771 & 0.8604 & 0.8566 \\\\\nOurs                 & 0.8834 & 0.8819 & 0.8626 & 0.8570 \\\\\n           \\bottomrule\n      \\end{tabular}\n      }\n      \\caption{SSIM at 1\/8, 1\/4, 1\/2, and 1\/1 scale.}\n      \\label{fig:ablation_quality_ssim}\n  \\end{subfigure}\n\\end{table}\n\n\\subsection{Training Ablation}\nTo evaluate our training strategy, we perform two ablation experiments.\nFirst, we compare our strategy to a coarse-to-fine progressive training strategy where lower levels of detail are trained and then frozen as higher levels are trained.\nProgressive training takes on average $24.20$ hours to train.\nSecond, we use half the number of training views to evaluate how the quality degrades with fewer training views.\nBoth experiments use our progressive multi-scale model architecture.\nOur results are shown in~\\autoref{tab:ablation_quality} with additional details in the supplementary material.\n\n\n\\begin{figure}[!htbp]\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/rendering_pipeline.pdf}\n    \\caption{Overview of our rendering pipeline which utilizes an auxiliary network to skip evaluation of empty rays.}\n    \\label{fig:rendering_pipeline}\n\\end{figure}\n\n\\begin{table}[!htbp]\n  \\caption{Average Model Rendering Performance for our Multi-scale LFN (milliseconds per frame).}\n  \\label{tab:performance_table_v2}\n  \\begin{subfigure}[t]{\\linewidth}\n      \\centering\n      \\begin{tabular}{cccc}\n        \\hline\n         LOD 1 & LOD 2 & LOD 3 & LOD 4\\\\\n        \\hline \n3.7 & 3.9 & 4.7 & 5.9\\\\\n        \\hline\n      \\end{tabular}\n      \\caption{Rendering each LOD at 1\/8 scale.}\n      \\label{tab:performance_table_v2_8thscale}\n  \\end{subfigure}\\\\%\n  \\begin{subfigure}[!ht]{\\linewidth}\n      \\centering\n      \\begin{tabular}{cccc}\n        \\hline\n        LOD 1 & LOD 2 & LOD 3 & LOD 4\\\\\n        \\hline\n4 & 11 & 58 & 305\\\\\n        \\hline\n      \\end{tabular}\n      \\caption{Rendering at 1\/8, 1\/4, 1\/2, and 1\/1 scale.}\n      \\label{tab:performance_table_v2_multiscale}\n  \\end{subfigure}\n\\end{table}\n\n\\subsection{Level of Detail Rendering Speedup}\nWe evaluate the rendering speed of our progressive model at different levels of detail to determine the observable speedup from rendering with the appropriate level of detail. For optimal rendering performance, we render using half-precision floating-point values and skip empty rays using an auxiliary neural network encoding only the ray occupancy as shown in \\autoref{fig:rendering_pipeline}. Our auxiliary network is made up of three layers with a width of 16 neurons per hidden layer. Evaluation is performed by rendering rays from all training poses with intrinsics scaled to the corresponding resolution as shown in \\autoref{tab:model_parameters}. \nAverage rendering time results are shown in \\autoref{tab:performance_table_v2}. \nRendering times for lower LODs (1\/8 scale) from our multi-scale network are reduced from $\\sim5.9$ to $\\sim3.7$ msec.\n\\section{Background and Related Works}\nOur work builds upon existing work in neural 3D representations, adaptive neural network inference, and levels of detail. In this section, we provide some background on neural representations and focus on prior work most related to ours. For a more comprehensive overview of neural rendering, we refer interested readers to a recent survey~\\cite{tewari2021advances}.\n\n\n\\begin{table}[!htbp]\n  \\centering\n  \\caption{Neural 3D Representations Comparison}\n  \\label{tab:comparison_table}\n  \\scalebox{0.80}{\n      \\begin{tabular}{lcccc}\n        \\toprule\n        Model & Real-time & Model Size & Multi-scale & Progressive\\\\\n        \\midrule\n        NeRF & \\resizebox{\\widthof{\\scrossmark}*\\ratio{\\widthof{x}}{\\widthof{\\normalsize x}}}{!}{\\scrossmark} & $\\leq 10$ MB & \\resizebox{\\widthof{\\scrossmark}*\\ratio{\\widthof{x}}{\\widthof{\\normalsize x}}}{!}{\\scrossmark}  & \\resizebox{\\widthof{\\scrossmark}*\\ratio{\\widthof{x}}{\\widthof{\\normalsize x}}}{!}{\\scrossmark} \\\\\n        mip-NeRF & \\resizebox{\\widthof{\\scrossmark}*\\ratio{\\widthof{x}}{\\widthof{\\normalsize x}}}{!}{\\scrossmark}  & $\\leq 10$ MB & \\resizebox{\\widthof{\\scheckmark}*\\ratio{\\widthof{x}}{\\widthof{\\normalsize x}}}{!}{\\scheckmark} & \\resizebox{\\widthof{\\scrossmark}*\\ratio{\\widthof{x}}{\\widthof{\\normalsize x}}}{!}{\\scrossmark} \\\\\n        Plenoctree & \\resizebox{\\widthof{\\scheckmark}*\\ratio{\\widthof{x}}{\\widthof{\\normalsize x}}}{!}{\\scheckmark} & $\\geq 30$ MB& \\resizebox{\\widthof{\\scrossmark}*\\ratio{\\widthof{x}}{\\widthof{\\normalsize x}}}{!}{\\scrossmark} &\\resizebox{\\widthof{\\scrossmark}*\\ratio{\\widthof{x}}{\\widthof{\\normalsize x}}}{!}{\\scrossmark} \\\\\n        LFN & \\resizebox{\\widthof{\\scheckmark}*\\ratio{\\widthof{x}}{\\widthof{\\normalsize x}}}{!}{\\scheckmark} &  $\\leq 10$ MB & \\resizebox{\\widthof{\\scrossmark}*\\ratio{\\widthof{x}}{\\widthof{\\normalsize x}}}{!}{\\scrossmark} & \\resizebox{\\widthof{\\scrossmark}*\\ratio{\\widthof{x}}{\\widthof{\\normalsize x}}}{!}{\\scrossmark} \\\\\n        Ours & \\resizebox{\\widthof{\\scheckmark}*\\ratio{\\widthof{x}}{\\widthof{\\normalsize x}}}{!}{\\scheckmark} & $\\leq 10$ MB & \\resizebox{\\widthof{\\scheckmark}*\\ratio{\\widthof{x}}{\\widthof{\\normalsize x}}}{!}{\\scheckmark} & \\resizebox{\\widthof{\\scheckmark}*\\ratio{\\widthof{x}}{\\widthof{\\normalsize x}}}{!}{\\scheckmark}\\\\\n      \\bottomrule\n      \\end{tabular}\n  }\n\\end{table}\n\n\n\\subsection{Neural 3D Representations}\nTraditionally, 3D content has been encoded in a wide variety of formats such as meshes, point clouds, voxels, multi-plane images, and even side-by-side video.\nTo represent the full range of visual effects from arbitrary viewpoints, researchers have considered using radiance fields and light fields which can be encoded using neural networks. With neural networks, 3D data such as signed distance fields, radiance fields, and light fields can be efficiently encoded as coordinate functions without explicit limitations on the resolution.\n\n\\subsubsection{Neural Radiance Field (NeRF)}\nEarly neural representations focused on signed distance functions and radiance fields. Neural Radiance Fields (NeRF)~\\cite{mildenhall2020nerf} use differentiable volume rendering to encode multiple images of a scene into a radiance field encoded as a neural network. With a sufficiently dense set of images, NeRF is able to synthesize new views by using the neural network to interpolate radiance values across different positions and viewing directions. Since the introduction of NeRF, followup works have added support for deformable scenes \\cite{pumarola2020d,park2021nerfies,park2021hypernerf}, real-time inference \\cite{garbin2021fastnerf,reiser2021kilonerf,neff2021donerf,yu2021plenoxels},\nimproved quality \\cite{shen2021snerf}, generalization across scenes \\cite{yu2020pixelnerf,rebain2021lolnerf},\ngenerative modelling \\cite{schwarz2020graf,niemeyer2021campari,xie2021fignerf}, videos \\cite{li2021neural,xian2021space,wang2022fourier}, sparse views \\cite{Chibane_2021_CVPR,niemeyer2021regnerf,Jain_2021_ICCV}, fast training \\cite{mueller2022instant,yu2021plenoxels}, and even large-scale scenes \\cite{rematas2021urban,xiangli2021citynerf,turki2021meganerf}. \n\nAmong NeRF research, Mip-NeRF~\\cite{barron2021mipnerf,barron2022mipnerf360} and BACON~\\cite{lindell2021bacon} share a similar goal to our work in offering a multi-scale representation to reduce aliasing. \nMip-NeRF approximates the radiance across conical regions along the ray, cone-tracing, using integrated positional encoding (IPE). With cone-tracing, Mip-NeRF reduces aliasing while also slightly reducing training time.\nBACON~\\cite{lindell2021bacon} provide a multi-scale scene representation through band-limited networks using Multiplicative Filter Networks~\\cite{fathony2021multiplicative} and multiple exit points. Each scale in BACON has band-limited outputs in Fourier space.\n\n\\subsubsection{Light Field Network (LFN)}\nOur work builds upon Light Field Networks (LFNs)~\\cite{sitzmann2021lfns,feng2021signet,li2022neulf,chandramouli2021light}. Light field networks encode light fields~\\cite{levoy1996lightfieldrendering} using a coordinate-wise representation, where for each ray $\\mathbf{r}$, a neural network $f$ is used to directly encode the color $c=f(\\mathbf{r})$. To represent rays, Feng and Varshney \\cite{feng2021signet} pair the traditional two-plane parameterization with Gegenbauer polynomials while Sitzmann \\etal~\\cite{sitzmann2021lfns} use Pl\\\"{u}cker coordinates which combine the ray direction $\\mathbf{r}_d$ and ray origin $\\mathbf{r}_o$ into a 6-dimensional vector $(\\mathbf{r}_d, \\mathbf{r}_o \\times \\mathbf{r}_d)$. Light fields can also be combined with explicit representations~\\cite{ost2021point}. In our work, we adopt the Pl\\\"{u}cker coordinate parameterization which can represent rays in all directions.\n\nCompared to NeRF, LFNs are faster to render as they only require a single forward pass per ray, enabling real-time rendering. However, LFNs are worse at view synthesis as they lack the self-supervision that volume rendering provides with explicit 3D coordinates. As a result, LFNs are best trained from very dense camera arrays or from a NeRF teacher model such as in~\\cite{attal2022learning}. Similar to NeRFs, LFNs are encoded as multi-layer perceptrons, hence they remain compact compared to voxel and octree NeRF representations~\\cite{yu2021plenoctrees,hedman2021snerg,garbin2021fastnerf,yu2021plenoxels} which use upwards of $50$ MB. Therefore LFNs strike a balance between fast rendering times and storage compactness which could make them suitable for streaming 3D scenes over the internet.\n\n\n\\subsection{Levels of Detail}\nLevel of detail methods are commonly used in computer graphics to improve performance and quality. For 2D textures, one of the most common techniques is mipmapping~\\cite{williams1983mipmap} which both reduces aliasing and improves performance with cache coherency by encoding textures at multiple resolutions. For neural representations, techniques for multiple levels of detail have also been proposed for signed distance functions as well as neural radiance fields.\n\nFor signed distance fields, Takikawa \\etal \\cite{takikawa2021neural} propose storing learned features within nodes of an octree to represent a signed distance function with multiple levels of detail. \nBuilding upon octree features, Chen \\etal \\cite{chen2021multiresolution} develop Multiresolution Deep Implicit Functions (MDIF) to represent 3D shapes which combines a global SDF representation with local residuals. \nFor radiance fields, Yang \\etal~\\cite{yang2021recursivenerf} develop Recursive-NeRF which grows a neural representation with multi-stage training which is able to reduce NeRF rendering time. \nBACON~\\cite{lindell2021bacon} offers levels of detail for various scene representations including 2D images and radiance fields with different scales encoding different bandwidths in Fourier space.\nPINs~\\cite{Landgraf2022PINs} uses a progressive fourier encoding to improve reconstruction for scenes with a wide frequency spectrum.\nFor more explicit representations, Variable Bitrate Neural Fields~\\cite{takikawa2022variable} use a vector-quantized auto-decoder to compress feature grids into lookup indices and a learned codebook. Levels of detail are obtained by learning compressed feature grids at multiple resolutions in a sparse octree.\nIn parallel, Streamable Neural Fields~\\cite{cho2022streamable} propose using progressive neural networks~\\cite{rusu2016progressive} to represent spectrally, spatially, or temporally growing neural representations.\n\n\\subsection{Adaptive Inference}\nCurrent methods use adaptive neural network inference based on available computational resources or the difficulty of each input example. Bolukbasi \\etal~\\cite{bolukbasi2017adaptive} propose two schemes for adaptive inference. Early exiting allows intermediate layers to route outputs directly to an exit head while network selection dynamically routes features to different networks. Both methods allow easy examples to be classified by a subset of neural network layers. Yeo \\etal \\cite{yeo2018neuraladaptive} apply early exiting to upscale streamed video. By routing intermediate CNN features to exit layers, a single super-resolution network can be partially downloaded and applied on a client device based on available compute. Similar ideas have also been used to adapt NLP to input complexity~\\cite{zhou2020bert,weijie2020fastbert}.\nYang \\etal~\\cite{yang2020resolutionadaptive} develop Resolution Adaptive Network (RANet) which extracts features and performs classification of an image progressively from low-resolution to high-resolution. Doing so sequentially allows efficient early exiting as many images can be classified with only coarse features. Slimmable neural networks~\\cite{yu2018slimmable} train a model at different widths with a switchable batch normalization and observe better performance than individual models at detection and segmentation tasks.\n\n\\section{Introduction}\n\nVolumetric images and video allow users to view 3D scenes from novel viewpoints with a full 6 degrees of freedom (6DOF). Compared to conventional 2D and 360 images and videos, volumetric content enables additional immersion with motion parallax and binocular stereo while also allowing users to freely explore the full scene.\n\n\nTraditionally, scenes generated with 3D modeling have been commonly represented as meshes and materials. These classical representations are suitable for real-time rendering as well as streaming for 3D games and AR effects. However, real captured scenes are challenging to represent using the mesh and material representation, requiring complex reconstruction and texture fusion techniques \\cite{duo2016fusion4d,du2018montage4d}. To produce better photo-realistic renders, existing view-synthesis techniques have attempted to adapt these representations with multi-plane images (MPI)~\\cite{zhou2018stereo}, layered-depth images (LDI)~\\cite{shade1998ldi}, and multi-sphere images (MSI)~\\cite{attal2020matryodshka}. These representations allow captured scenes to be accurately represented with 6DOF but only within a limited area.\n\n\n\\begin{figure}[!tbp]\n    \\includegraphics[width=\\linewidth]{figures\/maria\/teaser_v3_vertical.pdf}\n    \\caption{Our progressive multi-scale light field network is suited for streaming, reduces aliasing and flicking, and renders more efficiently at lower resolutions.}\n    \\label{fig:teaser}\n    \\vspace{-5mm}\n\\end{figure}\n\nRadiance fields and light fields realistically represent captured scenes from an arbitrarily large set of viewpoints.\nRecently, neural representations such as neural radiance fields (NeRFs)~\\cite{mildenhall2020nerf} and light field networks (LFNs)~\\cite{sitzmann2021lfns} have been found to compactly represent 3D scenes and perform view-synthesis allowing re-rendering from arbitrary viewpoints. Given a dozen or more photographs capturing a scene from different viewpoints from a conventional camera, a NeRF or LFN can be optimized to accurately reproduce the original images and stored in less than 10 MB. Once trained, NeRFs will also produce images from different viewpoints with photo-realistic quality.\n\nWhile NeRFs are able to encode and reproduce real-life 3D scenes from arbitrary viewpoints with photorealistic accuracy, they suffer from several drawbacks which limit their use in a full on-demand video streaming pipeline. Some notable drawbacks include slow rendering, aliasing at smaller scales, and a lack of progressive decoding. While some of these drawbacks have been independently addressed in follow-up work~\\cite{barron2021mipnerf,yu2021plenoctrees}, approaches to resolving them cannot always be easily combined and may also introduce additional limitations.\n\nIn this paper, we extend light field networks with multiple levels of detail and anti-aliasing at lower scales making them more suitable for on-demand streaming. This is achieved by encoding multiple reduced-resolution representations into a single network. With multiple levels of details, light field networks can be progressively streamed and rendered based on the desired scale or resource limitations. For free-viewpoint rendering and dynamic content, anti-aliasing is essential to reducing flickering when rendering at smaller scales. In summary, the contributions of our progressive multi-scale light field network are:\n\\begin{enumerate}\n\\item Composing multiple levels of detail within a single network takes up less space than storing all the levels independently.\n\\item The progressive nature of our model requires us to only download the parameters necessary up to the desired level of detail. \n\\item Our model allows rendering among multiple levels of detail simultaneously. This makes for a smooth transition between multiple levels of detail as well as spatially adaptive rendering (including foveated rendering) without requiring independent models at multiple levels of detail on the GPU.\n\\item Our model allows for faster inference of lower levels of detail by using fewer parameters.\n\\item We are able to encode light fields at multiple scales to reduce aliasing when rendering at lower resolutions.\n\\end{enumerate}\n\n\\section{Discussion}\nThe primary contribution of our paper is the identification of a representation more suitable for on-demand streaming. \nThis is achieved by enhancing light field networks, which support real-time rendering, to progressively support multiple levels of detail at different resolutions.\nOur representation inherits some of the limitations of LFNs such as long training times and worse view-synthesis quality compared to NeRF-based models.\nSince our method does not support continuous LODs, we require the user to decide the LODs ahead of time. For intermediate resolutions, rendering to the nearest LOD is sufficient to avoid aliasing and flickering.\nWhile our light field datasets could also be rendered with classical techniques~\\cite{levoy1996lightfieldrendering,xiao2014cvpr,peter2001wavelet}, doing so over the internet would not be as bandwidth-efficient.\nFuture work may combine our multi-scale light field networks with PRIF~\\cite{Feng2022PRIF} to encode color and geometry simultaneously or with VIINTER~\\cite{Feng2022Viinter} for dynamic light fields.","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Supplemental Material\\\\for ``Noncommuting Momenta of Topological Solitons\"}\n\n\nHere we extend our analysis for homogeneous spaces $G\/H$ to general K\\\"ahler manifold.\n\nSuppose ${\\cal M}$ is a K\\\"ahler manifold, whose metric $g_{a\\bar{b}} = \\partial_a \\bar{\\partial}_{\\bar{b}} K$ is given by the K\\\"ahler potential $K(z^a, \\bar{z}^{\\bar{b}})$.  Then the K\\\"ahler form $\\omega = i g_{a\\bar{b}}\\mathrm{d}z^a \\wedge\\mathrm{d}\\bar{z}^{\\bar{b}}$ is a nontrivial element of $H^2({\\cal M})$.  The general form of the Lagrangian in $2+1$ dimension is \n\\begin{equation}\n\\mathcal{L} = \\frac{i}{2}\\big(\\bar{\\partial}_{\\bar{b}} K \\dot{\\bar{z}}^{\\bar{b}}-\\partial_a K \\dot{z}^a\\big)\n+ g^T_{a\\bar{b}} \\dot{z}^a \\dot{\\bar{z}}^{\\bar{b}}\n- g^S_{a\\bar{b}} \\nabla_i{z}^a \\nabla_i{\\bar{z}}^{\\bar{b}}\\ .\n\\end{equation}\nHere we made it clear that the K\\\"ahler metrics $g^T_{a\\bar{b}}=\\partial_a \\bar{\\partial}_{\\bar{b}} K^T$, $g^S_{a\\bar{b}}=\\partial_a \\bar{\\partial}_{\\bar{b}} K^S$ may in general be different from $g_{a\\bar{b}}$ obtained from $K$. It is convenient to introduce a complex notation for the spatial coordinates $w = x+iy$, $\\nabla = \\frac{1}{2} (\\nabla_x - i \\nabla_y)$.  The energy $E$ of a static field configuration is given by\n\\begin{eqnarray}\nE&=& \\int d^2 x\\,g^S_{a\\bar{b}} \\nabla_i{z}^a \\nabla_i{\\bar{z}}^{\\bar{b}}\\nonumber\\\\\n&=& \\int d^2 x\\,2g^S_{a\\bar{b}}\\left[\\left(\\nabla z^a \\bar{\\nabla}{\\bar{z}}^{\\bar{b}} - \\bar{\\nabla}{z}^a \\nabla\\bar{z}^{\\bar{b}}\\right)+2 \\bar{\\nabla}{z}^a \\nabla\\bar{z}^{\\bar{b}} \\right] \\nonumber \\\\\n&\\geq& \\int d^2 x\\,ig^S_{a\\bar{b}}\\epsilon^{ij}\\nabla_iz^a \\nabla_j\\bar{z}^{\\bar{b}} \n= 2\\pi\\hbar n_0 N^S\\ .\n\\end{eqnarray}\nHere we used the fact that $g^S_{a\\bar{b}}$ is positive definite to derive the inequality.  The last expression is nothing but the pull-back of the K\\\"ahler form $\\omega^S = i g_{a\\bar{b}}^S\\mathrm{d}z^a \\wedge\\mathrm{d}\\bar{z}^{\\bar{b}}$.  To minimize the energy for a fixed $N^S>0$, we need $\\bar{\\nabla}{z}^a = 0$, namely $z^a(w)$ is a holomorphic map ${\\mathbb C}\\rightarrow {\\cal M}$ (anti-holomorphic for $N^S<0$).  Expanding the solution for its translational moduli, $z^a = z^a(w-w_0(t))$ with $w_0=x_0+iy_0$, the first term in the Lagrangian becomes\n\\begin{eqnarray}\n\\lefteqn{\n\\int d^2 x\\,\\frac{i}{2}\\big(\\bar{\\partial}_{\\bar{b}} K \\dot{\\bar{z}}^{\\bar{b}}-\\partial_a K \\dot{z}^a\\big)}\\nonumber \\\\\n&=&\\left[\\int d^2 x\\,2g_{a\\bar{b}}(\\nabla z^a\\bar{\\nabla}\\bar{z}^{\\bar{b}}-\\bar{\\nabla}z^a\\nabla\\bar{z}^{\\bar{b}})\\right]\\frac{1}{4i}(\\bar{w}_0\\dot{w}_0 - \\dot{\\bar{w}}_0w_0) \\nonumber \\\\\n&=&\\left[\\int d^2 x\\,ig_{a\\bar{b}}\\epsilon^{ij}\\nabla_iz^a\\nabla_j\\bar{z}^{\\bar{b}}\\right]\\frac{1}{2}\\epsilon_{ij} x_0^i \\dot{x}_0^j \\nonumber \\\\\n&=&(2\\pi \\hbar n_0 N) \\frac{1}{2} \\epsilon_{ij} x_0^i \\dot{x}_0^j,\n\\end{eqnarray}\nfor a nontrivial K\\\"ahler class.  The skyrmion for a ferromagnet is a special case of this general consideration.\n\nWe believe the analysis would extend to even wider class of target manifolds as long as they permit presymplectic structure and have a nontrivial $H^2$.\n\n\n\\end{document}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIt was shown by Schwinger\\cite{Schwinger} that electrons get an\nanomalous magnetic moment $\\mu^\\prime=\\alpha\/2\\pi\\mu_B$ (being\n$\\mu_B$ the Bohr magneton) due to QED radiative corrections. By\nconsidering the  propagation of electromagnetic radiation in vacuum\nin presence of an external magnetic field, Shabad\n\\cite{shabad1,shabad2} showed the drastic departure of the photon\ndispersion equation from the light cone curve near the energy\nthresholds for free pair creation. Three different modes of photon\npropagation were found according to the eigenvalues of the\npolarization operator in presence of a magnetic field. Some years\nlater, the same property was obtained \\cite{usov1,usov2} in the\nvicinity of the threshold for positronium creation. As a result, the\nproblem of the propagation of light in empty space, in presence of\nan external magnetic field is similar to the problem of the\ndispersion of light in an anisotropic medium\\cite{proceeding}, where\nthe role of the medium is played by the polarized vacuum in the\nexternal magnetic field. An anisotropy is created by the preferred\ndirection in space along $\\textbf{B}$. In this context we can\nmention other characteristics of magnetized quantum vacuum, as the\nbirefringence\\cite{adler,uso3}, (which plays an important role in\nthe photon splitting and capture effect), and the vacuum\nmagnetization \\cite{Elizabeth}.\n\nFrom the previous paragraph we may conjecture that, similar to the\ncase of the electron\\cite{Schwinger},  a photon anomalous magnetic\nmoment might also exist, as a consequence of its radiative\ncorrections in presence of the magnetic field. We want to show in\nwhat follows that this conjecture is true: the photon having a non\nvanishing momentum component orthogonal to a constant magnetic\nfield, exhibits such anomalous magnetic moment due to  its\ninteraction  with the virtual electron-positron pairs in the\nmagnetized vacuum where it propagates.\n\nIn order to calculate this photon intrinsic property we recall that\nSchwinger's result \\cite{Schwinger} was obtained by a weak field\napproximation of the electron Green's function in the self energy\noperator. In place of using such approximation in the calculation of\nthe photon anomalous magnetic moment, we prefer to use the exact\nform of the polarization operator eigenvalue calculated by Batalin\nand Shabad \\cite{batalin, shabad1} which provides information about\nthe three photon propagation modes in the external classical\nmagnetic field. From it we get the photon equation of motion, in\nwhich we may take the weak as well as the strong field limits. In\nany case, we get that the photon acquires a magnetic moment along\nthe external field, which is proportional to the electron anomalous\nmagnetic moment.\n\nFor the zero field case the photon magnetic moment strictly\nvanishes, but for small fields  $\\vert\\textbf{B}\\vert \\ll B_c$\n(where $B_c=m^2\/e\\simeq 4.4 \\times 10^{13}$G is the Schwinger\ncritical field) it is significant for a wide range of frequencies.\nThis might be interesting in an astrophysical context, for instance,\nin the propagation and light deflection near a dense magnetized\nobject.\n\nMoreover, the photon anomalous magnetic moment in the case of strong\nmagnetic fields, may play an important role in the propagation of\nelectromagnetic radiation through the magnetospheres of neutron\nstars \\cite{shabad3} and in other stellar objects where large\nmagnetic fields arise $\\textbf{B}\\gtrsim \\textbf{B}_c $. The free\nand bound pair creation of electrons and positrons at the\nthresholds\\cite{Hugo1,AO,Leinson}  is related to a production and\npropagation of $\\gamma$ radiation\\cite{denisov,harding}.\n\nThe paper has the following structure: in the  Sec. II from the\ngeneral solution of the dispersion equation of the polarization\noperator  we analyze the interaction term and defined a magnetic\nmoment of the photon. In section  III we obtain from \\cite{shabad1}\nthe three eigenvalues of the polarization operator calculated in one\nloop approximation in the weak field limit and the expression of the\nmagnetic moment under this field regime.\n\nIn Sec. IV and V the magnetic moment is  obtained in the strong\nmagnetic field limit for the photon and the photon-positronium mixed\nstate  when both particles are created in the Landau ground state\n$n^\\prime=n=0$. We will refer also to the case when one of the\nparticles appear in the first excited state $n=1$, and the other in\nthe Landau ground state. We discuss mainly the so-called second mode\nof propagation near the first pair creation threshold\n$n^\\prime=n=0$, since as it was studied in \\cite{shabad3}, in the\nrealistic conditions of production and propagation of $\\gamma-$\nquanta  the third mode decays into the second mode  via the photon\nsplitting $\\gamma\\rightarrow\\gamma\\gamma$ process \\cite{adler,uso3}.\nMoreover, higher thresholds are damped by the quasi-stationarity of\nthe electron and positron states on excited Landau levels $n\\geq1$\nor $n^\\prime\\geq1$, which may fall down to the ground state with a\nphoton emitted.\n\nFinally in the Sec. VI the results are analyzed and some remarks and\nconclusions are given in Sec. VI on the basis of comparing the\nphoton behavior with that of a neutral massive particle with\nnon-vanishing magnetic moment which interacts with the external\nfield $\\vert\\bf{B}\\vert \\simeq B_c$.\n\n\n\n\\section{The Interaction Energy and The Magnetic Moment of the Radiation}\n\nIn paper\\cite{batalin} it was shown that the presence of the\nconstant magnetic field, creates, in addition to the photon momentum\nfour-vector $k_\\mu$, three other four-vectors, $F_{\\mu \\rho}k^\\rho$,\n$F^2_{\\mu \\rho}k^{\\rho}$, $F^{*}_{\\mu \\rho}k^{\\rho}$, where $F_{\\mu\n\\nu}=\\partial_\\mu A_\\nu-\\partial_\\nu A_\\mu$ is the electromagnetic\nfield tensor and $F^*_{\\mu \\nu}=\\frac{i}{2}\\epsilon_{\\mu \\nu \\rho\n\\kappa}F^{\\rho \\kappa}$ its dual pseudotensor. One get from these\nfour-vectors three basic independent scalars $k^2$, $kF^2k$,\n$kF^{*2}k$, which in addition to the field invariant ${\\cal\nF}=\\frac{1}{4}F_{\\mu \\rho}F^{\\rho \\mu}=\\frac{1}{2}B^2$, are a set of\nfour basic scalars of our problem.\n\nIn correspondence to each eigenvalue $\\pi^{(i)}_{n,n^\\prime}$,\n$i=1,2,3$, the polarization tensor has  an eigenvector\n$a^{(i)}_\\mu$(x). The basic vectors \\cite{shabad1} are obtained by\nnormalizing the set of four vectors $C^{1}_\\mu= k^2 F^2_{\\mu\n\\lambda}k^\\lambda-k_\\mu (kF^2 k)$, $C^{2}_\\mu=F^{*}_{\\mu\n\\lambda}k^\\lambda$, $C^{3}_\\mu=F_{\\mu \\lambda}k^\\lambda$,\n$C^{4}_\\mu=k_\\mu$. The first three satisfy in general the\nfour-dimensional transversality condition $C^{1,2,3}_\\mu k^{\\mu}=0$,\nwhereas it is $C^{4}_\\mu C^{{4}\\mu}=0$ only in the light cone. By\nconsidering $a^{(i)}_\\mu (x)$ as the electromagnetic four vector\ndescribing the eigenmodes, it is easy to obtain the corresponding\nelectric and magnetic fields of each mode ${\\bf e}^{(i)}=\n\\frac{\\partial }{\\partial x_0}\\vec{a}^{(i)}-\\frac{\\partial\n}{\\partial {\\bf x}}a^{(i)}_0$, ${\\bf\nh}^{(i)}=\\nabla\\times\\vec{a}^{(i)}$. Up to a factor of\nproportionality, we rewrite them from \\cite{shabad1}(see also\n\\cite{Hugo1}),\n\\begin{eqnarray}\n{\\bf e}^{(1)}=-\\textbf{k}_{\\perp} k^2 \\omega, \\ \\ {\\bf h}^{(1)}=\n[{\\bf k}\\times {\\bf k}_{\\parallel} ]k^2\\label{wavectors1}, \\\\\n\\textbf{e}^{(2)}_{\\perp}= {\\bf k}_{\\perp} k_{\\parallel}^2, \\ \\\n\\textbf{e}^{(2)}_{\\parallel}=\\textbf{k}_{\\parallel}(k_{\\parallel}^2-\\omega^2),\\\n\\ \\textbf{h}^{(2)}=[{\\bf k}_{\\perp}\\times {\\bf k}_{\\parallel}]\n\\omega\\label{wavectors2},\\\\ \\textbf{e}^{(3)}=[{\\bf k}_{\\perp}\\times {\\bf\nk}_{\\parallel} ] \\omega, \\ \\  \\textbf{h}^{(3)}_{\\perp}=-{\\bf\nk}_{\\perp}k_{\\parallel}^2,\\ \\ \\textbf{h}^{(3)}_{\\parallel}=-{\\bf\nk}_{\\parallel} k_{\\perp}^2. \\label{wavectors3}\n\\end{eqnarray}\nThe previous formulae refer to the reference frame which is at\nrest or moving parallel to $B$. The vectors ${\\bf k}_{\\perp}$ and\n${\\bf k}_{\\parallel}$ are the components of $\\textbf{k}$ across\nand along $\\bf B_z$ (Here  the photon four-momentum squared,\n$k^2=k_\\perp^2+k_\\parallel^2-\\omega^2$). It is easy to see that\nthe mode $i=3$ is a transverse plane polarized wave with its\nelectric field orthogonal to the plane determined by the vectors\n$(\\textbf{B}, \\textbf{k})$. For propagation orthogonal to $B$, the\nmode $a^{(1)}_\\mu$ is a pure longitudinal and non physical\nelectric wave, whereas $a^{(2)}_\\mu$ is transverse. For\npropagation parallel to $B$, the mode $a^{(2)}_\\mu$ becomes purely\nelectric longitudinal (and non physical), whereas $a^{(1)}_\\mu$ is\ntransverse \\cite{shabad1,shabad2}.\nIn  paper \\cite{shabad1} it was  shown that  the dispersion law of\na photon propagating in  vacuum in a strong magnetic field is\ngiven for each mode by the solution of the equations\n\\begin{equation}\nk^2=\\pi_{n,n^{\\prime}}^{(i)}\\left(k^2+\\frac{kF^2k}{2\\mathcal{F}},\\frac{kF^2k}{2\\mathcal{F}},B\\right),\\\n\\\\ \\ i=1,2,3 \\label{egg}\n\\end{equation}\nThe $\\pi^\\prime$s are the  eigenvalues of the polarization\noperator $\\Pi_{\\mu\\nu}(k)$ with the electron and the positron in\nthe Landau levels $n$ and $n^{\\prime}$ or viceversa.\n\nBy solving (\\ref{egg}) for $z_1=k^2+\\frac{kF^2k}{2\\mathcal{F}}$ in\nterms of $kF^2k\/2\\mathcal{F}$ it results\n\\begin{equation}\n\\omega^2=\\vert\\textbf{k}\\vert^2+f_i\\left(\\frac{kF^2k}{2\\mathcal{F}},B\\right)\n\\label{eg2}\n\\end{equation}\nThe term $ f_i\\left(kF^2k\/2\\mathcal{F},B\\right)$ is due to the\ninteraction of the photon with  the virtual $e^{\\pm}$ pairs in the\nexternal field, leading to the magnetization of\nvacuum\\cite{Elizabeth}. Moreover it causes a drastic  departure of\nthe photon dispersion equation from the light cone curve near the\nenergy thresholds for free pair creation.\n\nThis characteristic  stems from the arising of bound states in the\nexternal field, leading to a singular behavior of the polarization\noperator $\\Pi_{\\mu\\nu}(k)$ near the pair creation thresholds  for\nelectrons and positrons. These particles, coming from the photon\ndecay in the external field, appear in Landau levels $n$ and\n$n^{\\prime}$ (cyclotron resonance), or either still stronger\nsingular behavior of $\\Pi_{\\mu\\nu}$ near the thresholds of an\n$e^+e^-$ bound state (due to positronium formation).\n\nTo understand the different behavior of $\\Pi_{\\mu\\nu}(k)$ in\npresence of an external magnetic field, as compared to the zero\nfield case, we recall that in the latter problem the polarization\noperator is rotationally invariant with regard to the only\nsignificant four-vector, $k_\\mu$, whereas in the magnetic field case\nthis symmetry is reduced to axial. Thus, it is invariant under\nrotations in the plane perpendicular to the external field\n$\\textbf{B}$.\n\nThe presence of the interaction energy of the photon with the\nelectron-positron field opens the possibility of defining a magnetic\nmoment for the photon, for this we expand the dispersion equation in\nlinear terms of $\\Delta B=B-B_r$ around some field value $B_r$,\n\\begin{equation}\n\\label{exp}\n\\omega(B)=\\omega(B_r)+\\left.\\left(\\frac{\\partial\\omega}{\\partial\nkF^2k}\\frac{\\partial kF^2k}{\\partial\nB}+\\frac{\\partial\\omega}{\\partial B}\\right)\\right\\vert_{B=B_r}\\Delta\nB\n\\end{equation}\n\nThis means that, in the rest frame, where no electric field exists\nthe photon exhibits  a longitudinal magnetic moment given by\n\\begin{equation}\n\\mu_\\gamma=-\\left.\\left(-2k_\\perp^2\\textbf{B}_z\\frac{\\partial\\omega}{\\partial\nkF^2k}+\\frac{\\partial\\omega}{\\partial\n\\textbf{B}_z}\\right)\\right\\vert_{B=B_r}\\cdot\\vec{\\kappa}_\\parallel\\label{treee}\n\\end{equation}\nwhere $\\vec{\\kappa}_\\parallel$  is an unit vector in the direction\nalong the magnetic field $B_r=B_z$.\n\nThe modulus of the magnetic moment along $\\textbf{B}$  can be\nexpressed as $\\mu_\\gamma=\\mu^\\prime g_\\gamma$ where the factor\n$g_\\gamma$ is a sort of gyromagnetic ratio. As in the case of the\nelectron\\cite{lipman} $\\mu_{\\gamma}$ is not a constant of motion,\nbut is a quantum average.\n\nAs different from the classical theory of  propagation of\nelectromagnetic radiation  in presence of a constant external\nmagnetic field, it is expected that $\\mu_\\gamma$  be different from\nzero due to radiative corrections, which are dependent on\n$\\vert\\textbf{B}\\vert=B_z=B$. The magnetic moment is induced by the\nexternal field on the photon through its interaction with the\npolarized electron-positron virtual quanta of vacuum and is oriented\nalong $z$.\n\n\nThe gauge invariance property $\\pi^{(i)}(0,0)=0$ implies that the\nfunction  $f_i(kF^2k\/2\\mathcal{F},B)$ vanishes when\n$kF^2k\/2\\mathcal{F}=0$, this means that the anomalous magnetic\nmoment of the photon is a magnitude subject to the gauge invariance\nproperty of the theory and therefore when the propagation is\nparallel to $\\bf{B}$, $k_\\perp=0$, is cancelled. In every mode,\nincluding positronium formation\n\\begin{equation}\n\\mu_\\gamma=0\\ \\ \\textrm{if} \\ \\ k_\\perp=0\\label{gauge}\n\\end{equation}\ntherefore the magnetic moment of the radiation is determined\nessentially by the perpendicular photon momentum component and\nthis determines the optical properties of the quantum vacuum.\n\nParticularly  interesting is the case when $B_r=B_z^r\\to0$. If\n$\\bf{B}$ is assumed small $(\\vert\\textbf{B}\\vert\\ll B_c)$, the\ndispersion law can be written as\n\\begin{equation}\n\\omega=\\vert\\bf{k}\\vert-\\mu_\\gamma\\cdot\\bf{B} \\label{de0}\n\\end{equation}\nThe first term of (\\ref{de0}) corresponds to the light cone\nequation, whereas the second contains the dipole moment\ncontribution of the virtual pairs $e^\\pm$.\n\nBy substitution of (\\ref{de0}) in (\\ref{wavectors2}) and\n(\\ref{wavectors3}) we obtain that the electric and magnetic fields\nof the radiation corresponding to the second and third modes are\nincreased by the factors\n\\begin{equation}\n\\Delta\\textbf{e}^{(2)}_{\\parallel}=2\\mu_\\gamma^{(2)}B_z\\vert\\textbf{k}\\vert\\textbf{k}_{\\parallel},\n\\end{equation}\n\\begin{equation}\n\\Delta\\textbf{h}^{(2)}=\\Delta\\textbf{e}^{(3)}=-\\mu_\\gamma^{(2,3)}B_z[{\\bf\nk}_{\\perp}\\times {\\bf k}_{\\parallel}].\n\\end{equation}\nTherefore,  the magnetic moment of the photon leads to linear\neffects in quantum electrodynamics and in consequence the refraction\nindex $n^{(i)}=\\vert\\textbf{k}\\vert\/\\omega_i$ in mode $i$  is given\nby\n\\begin{equation}\nn^{(i)}=1+\\frac{\\mu_\\gamma^{(i)}}{\\vert\\textbf{k}\\vert}\nB_z\\label{in}\n\\end{equation}\nthe gauge invariant property (\\ref{gauge}) implies that the\nrefraction index for parallel propagation, $k_\\perp=0$, be exactly\nunity: for any mode $n_i=1$. This can be reinterpreted by saying\nthat for parallel propagation to $B_z$ the refraction index is\nequal to unity  due to the vanishing of the photon magnetic\nmoment.\n\nThe components of the group velocity,\n($\\textbf{v}^{(i)}=\\nabla_\\textbf{k} \\omega_i$),\n$\\textrm{v}_{\\perp\\parallel}$ can be written as\n \\begin{equation}\n\\textrm{v}_\\perp^{(i)}=\\frac{\\partial \\omega}{\\partial\nk_\\perp}=\\frac{k_\\perp}{\\vert\\textbf{k}\\vert}\\left(1-\\frac{\\vert\\textbf{k}\\vert}{k_\\perp}\\frac{\\partial\\mu_\\gamma^{(i)}}{\\partial\nk_\\perp}B_z\\right)\\label{vper}\n\\end{equation}\nand\n\\begin{equation}\n\\textrm{v}_\\parallel^{(i)}=\\frac{\\partial \\omega_i}{\\partial\nk_\\parallel}=\\frac{\nk_\\parallel}{\\omega_i}\\simeq\\frac{k_\\parallel}{\\vert\\textbf{k}\\vert}\\left(1+\\frac{\\mu_\\gamma^{(i)}}{\\vert\\textbf{k}\\vert}B_z\\right).\n\\label{vpara}\n\\end{equation}\n\nIt follows from (\\ref{vper}) and (\\ref{vpara}) that  the angle\n$\\theta^{(i)}$ between the direction of the group velocity and the\nexternal magnetic field  satisfies the relation\n\\[\n\\tan\\theta^{(i)}=\\frac{\\textrm{v}_\\perp^{(i)}}{\\textrm{v}_\\parallel^{(i)}}=\\left(1-\\frac{\\vert\\textbf{k}\\vert}{k_\\perp}\\frac{\\partial\\mu_\\gamma^{(i)}}{\\partial\nk_\\perp}B_z\\right)\\left(1+\\frac{\\mu_\\gamma^{(i)}}{\\vert\\textbf{k}\\vert}B_z\\right)^{-1}\\tan\\vartheta,\n\\] being $\\vartheta$ the angle between the photon momentum and\n$\\textbf{B}$, with $\\tan \\vartheta=k_\\perp\/k_\\parallel$.\n\n\\section{The Polarization Eigenvalues in Weak Field Limit}\nIn this paper we shall only deal with the transparency region (we do\nnot consider the absorption of the photon to create observable\n$e^{\\pm}$ pairs), $\\omega^2-k_\\parallel^2\\leq k_{\\perp}^{\\prime 2}$\n)\\textit{i.e.}, we will keep our discussion within the kinematic\ndomain, where $\\pi_{1,2,3}$ are real, where\n\\begin{equation}\nk_{\\perp}^{\\prime\n2}=m_0^2\\left[(1+2\\frac{B}{B_c}n)^{1\/2}+(1+2\\frac{B}{B_c}n^{\\prime})^{1\/2}\\right]^2\n\\end{equation}\nis the pair creation squared threshold energy, with the electron and\npositron in Landau levels $n,n^{\\prime}\\neq 0$). We will be\ninterested in a photon whose energy is near the pair creation\nthreshold energy.\n\nIn the limit $B\\ll B_c$ and in one loop approximation the first\nand third modes with energies range less than  the first cyclotron\nresonance, whose energy is given by $2m_0$, does not show any\nsingular behavior and in this sense they behave similarly to the\neigenvalues does not contribute. It follows that the dispersion\nlaw for the first and third modes are given by\n$\\omega_1=\\omega_3=\\vert\\bf{k}\\vert$.\n\nNevertheless, from the calculations made in appendix A it is seen\nthat the second eigenvalue can be expressed as in the low frequency\nlimit $4m_0^2\\gg k^2+\\frac{kF^2k}{\\mathcal{F}}$ as\n\\begin{equation}\n\\pi_2=-\\frac{2\\mu^{\\prime}B}{m_0}\\frac{kF^2k}{2\\mathcal{F}}\\exp\\left(\\frac{kF^2k}{4m_0^2\\mathcal{F}}\\frac{B_c}{B}\\right),\\label{pi22}\\\\\n\\end{equation}\nHere $\\mu^\\prime=(\\alpha\/2\\pi)\\mu_B$ is the anomalous magnetic\nmoment of the electron.\n\nThis means that the dispersion equation for the second mode has the\nsolution\n\\begin{equation}\n\\omega^2\\simeq\nk_\\parallel^2-\\frac{kF^2k}{2\\mathcal{F}}\\left(1-\\frac{2\\mu^{\\prime}B}{m_0}\\exp\\left(\\frac{kF^2k}{4m_0^2\\mathcal{F}}\\frac{B_c}{B}\\right)\\right)\n\\label{de1}\n\\end{equation}\n\n\\begin{figure}[!htbp]\n\\includegraphics[width=3.5in]{edc.eps}\n\\caption{\\label{fig:edc} Dispersion curves  for second mode  in the\nfirst cyclotron resonance  for different values of weak field. The\ndotted line represent the light cone curve, the dashed line\ncorrespond to the threshold energy for pair formation $k_\\perp^2 =\n4m_0^2$.}\n\\end{figure}\n\nFor values of  energies and magnetic fields for which the\nexponential factor in (\\ref{pi22}) is of order unity we get\n\\[\n\\omega^2=k_\\parallel^2-\\frac{kF^2k}{2\\mathcal{F}}\\left(1-\\frac{2\\mu^{\\prime}B}{m_0}\\right)\n\\]\nwhich we can be approximated as\n\\[\n\\omega=\\vert\\textbf{k}\\vert+\\frac{\\mu^{\\prime}B}{m_0\\vert\\textbf{k}\\vert}\\frac{kF^2k}{2\\mathcal{F}}\n\\]\nIn such case the  magnetic moment is determined by\n\\begin{equation}\n\\mu_\\gamma=-\\frac{\\mu^{\\prime}}{m_0\\vert\\textbf{k}\\vert}\\frac{kF^2k}{2\\mathcal{F}}\\label{lemm}\n\\end{equation}\nin the case of perpendicular propagation\n\\begin{equation}\n\\mu_\\gamma^{max}=\\frac{\\mu^{\\prime}k_\\perp}{m_0}\n\\end{equation}\n\nFor perpendicular propagation to $B_z$, $(k_\\parallel=0)$, and\nalthough the present approximation is not strictly valid for\n$k_\\perp\\to 2m_0$ the photon anomalous magnetic moment is of order\n\\begin{equation}\n\\mu_\\gamma \\sim 2\\mu^{\\prime}\n\\end{equation}\n\nBelow we will get a larger value near the first pair creation\nthreshold. In the table below we show some $\\mu_\\gamma^{max}$ values\nsuch that $\n\\exp\\left[-\\frac{k_\\perp^2}{2m_0}\\frac{B_c}{B}\\right]\\sim 1 $\ncorresponding to  ranges of $X$- rays energies and magnetic field\nvalues.\n\n\\begin{widetext}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\n\\cline{1-9} \\multicolumn{1}{|c|}{}&\n\\multicolumn{1}{|c|}{$\\omega=m_010^{-8}$} &\n\\multicolumn{1}{|c|}{$B=1$ G} &\n\\multicolumn{1}{|c|}{$\\omega=m_010^{-6}$} &\n\\multicolumn{1}{|c|}{$B=10^4$ G} &\n\\multicolumn{1}{|c|}{$\\omega=m_010^{-4}$} &\n\\multicolumn{1}{|c|}{$B=10^8$ G}&\n\\multicolumn{1}{|c|}{$\\omega=m_010^{-2}$}&\n\\multicolumn{1}{|c|}{$B=10^{12}$ G}\\\\\n\\hline \\multicolumn{1}{|c|}{$\\mu_{\\gamma}^{max}$} &\n\\multicolumn{2}{|c|}{$10^{-8}\\mu^{\\prime}$} &\n\\multicolumn{2}{|c|}{$10^{-6}\\mu^{\\prime}$} &\n\\multicolumn{2}{|c|}{$10^{-4}\\mu^{\\prime}$} &\n\\multicolumn{2}{|c|}{$10^{-2}\\mu^{\\prime}$} \\\\\n\\hline\n\\end{tabular}\n\\\\\n\\\\\n\\end{widetext}\n\nAccording to (\\ref{in}) the refraction index in the weak field\napproximation and low frequency limit is given by\n\\[\nn^{(2)}=1+\\frac{\\mu^\\prime k_\\perp^2}{m_0\\vert\\textbf{k}\\vert^2}B.\n\\]\n\nIt must be noticed that when the propagation is perpendicular to\n$B_z$   the refraction index is maximum\n\\[\nn_\\perp^{(2)}=1+\\frac{\\mu^\\prime}{m_0}B\n\\]\n\nFrom (\\ref{vper}) and (\\ref{vpara}) we obtain that the absolute\nvalue  of the group velocity is given by\n\\begin{equation}\n\\textrm{v}^{(2)}\\simeq\n1-\\frac{\\mu_\\gamma^{(2)}}{\\vert\\textbf{k}\\vert}B.\\label{vel}\n\\end{equation}\nin the last expression we neglected the  term $B_z$ squared. In the\nparticular case $k_\\parallel=0$\n\\begin{equation}\n\\textrm{v}_\\perp^{(2)}=1-\\frac{\\mu^\\prime}{m_0}B.\n\\end{equation}\n\nIn the asymptotic region of supercritical magnetic fields $B\\gg B_c$\nand  restricted energy of longitudinal motion\\cite{proceeding}\n$\\omega^2-k_\\parallel^2\\ll(B\/B_c)m_0^2$, in the low frequency limit\nthe behavior of the photon propagating in the second mode is equal\nto the case of weak magnetic field (\\ref{pi22}). When the magnetic\nfield  $B\\sim B_c$, the photon magnetic moment can be approximated\nby (\\ref{lemm}). For a typical $X$ ray photon of wavelength $\\sim$\n\\AA{}, and propagation perpendicular  to $B_z$, its magnetic moment\nhas values of order $\\mu_\\gamma\\sim 10^{-2}\\mu^\\prime$ in this case.\nThis behavior is also present in the photon-positronium mixed state.\n\n\\section{The Cyclotron Resonance}\n\nSimilarly to the case of strong magnetic field $(B\\gtrsim B_c)$, the\nsingularity corresponding to cyclotron resonance appears in the\nsecond mode when $B\\ll B_c$. The corresponding eigenvalue near of\nthe first pair creation threshold  is given by\n\\begin{equation}\n\\pi_2=\\frac{2\\alpha m_0^3\nB}{B_c}\\exp\\left(\\frac{kF^2k}{4m_0^2\\mathcal{F}}\\frac{B_c}{B}\\right)\\left[4m_0^2+\\left(k^2+\\frac{kF^2k}{2\\mathcal{F}}\\right)\\right]^{1\/2},\\label{pi2}\\\\\n\\end{equation}\n\nThe term $\\exp(kF^2k B_c\/4m_0^2\\mathcal{F}B)$ plays an important\nrole in the cyclotron resonance, both in the weak and the large\nfield regime. The solution of the equation (\\ref{egg}) for the\nsecond mode (first obtained by Shabad \\cite{shabad2}) is shown\nschematically in Fig \\ref{fig:edc}. In this picture, it is noted the\ndeparture of the dispersion law from the light cone curve. For the\nfirst threshold the deflection increases with increasing external\nmagnetic field. In the vicinity of the first threshold the solutions\nof (\\ref{egg}) and (\\ref{pi2})  are similar to the case of strong\nmagnetic field regime, therefore, in order to show  the results in a\nmore compact form let us use the form used in \\cite{Hugo2}. In it\nthe eigenvalues of $\\Pi_{\\mu\\nu}(k\\vert A_\\mu^{ext})$  near the\nthresholds can be written approximately  as\n\\begin{equation}\n\\pi_{n,n^{\\prime}}^{(i)}\\approx-\\frac{2\\pi\\phi_{n,n^{\\prime}}^{(i)}}{\\vert\\Lambda\\vert}\n\\label{eg5}\n\\end{equation}\nwith $\\vert\\Lambda\\vert=((k_\\perp^{\\prime 2 }-k_\\perp^{\\prime \\prime\n2})(k_\\perp^{\\prime 2}+k^2+\\frac{kF^2k}{\\mathcal{F}}))^{1\/2}$\n and\n\\begin{eqnarray}\nk_\\perp^{\\prime\\prime\n2}=m_0^2\\left[(1+2\\frac{B}{B_c}n)^{1\/2}-(1+2\\frac{B}{B_c}n^{\\prime})^{1\/2}\\right]^2,\n\\end{eqnarray}\n\\noindent where $k_{\\perp}^{\\prime 2}$ is the squared threshold\nenergy for $e^{\\pm}$ pair production, and $k_{\\perp}^{\\prime\\prime\n2}$ is the squared threshold energy for excitation between Landau\nlevels $n,n^{\\prime}$ of an electron or positron. The functions\n$\\phi_{n,n^{\\prime}}^{(i)}$ are rewritten from \\cite{Hugo1} in the\nAppendix B.\n\nIn the vicinity of the first resonance $n=n^{\\prime}=0$ and\nconsidering $k_\\perp\\neq0$ and $k_\\parallel\\neq0$, according to\n\\cite{shabad1,shabad2} the physical eigenwaves are described by the\nsecond and third modes, but only the second mode has a singular\nbehavior near the threshold and the function $\\phi^{(2)}_{n n'}$ has\nthe structure\n\\begin{equation}\n\\phi_{0,0}^{(2)}\\simeq-\\frac{2\\alpha B m_0^4}{\\pi\nB_c}\\textrm{exp}\\left(\\frac{kF^2k}{4m_0^2\\mathcal{F}}\\frac{B_c}{B}\\right)\n\\end{equation}\nIn this case\n $k_{\\perp}^{\\prime\\prime 2}=0$ and $k_{\\perp}^{\\prime\n2}=4m_0^2$ is the threshold energy.\n\nWhen $\\textbf{B}=\\textbf{B}_z$, the scalar\n$kF^2k\/2\\mathcal{F}=-k_\\perp^2$ and the approximation of the modes\n(\\ref{eg5}) turns the dispersion equation (\\ref{eg1}) into a cubic\nequation in the variable $z_1=\\omega^2-k_\\parallel^2$ that can be\nsolved by applying the Cardano formula. We will refer in the\nfollowing to (\\ref{eg2}) as the real solution of this equation.\n\nWe would define the function\n\\begin{equation}\n\\Lambda^{*}=(k_\\perp^{\\prime\\prime 2}- k_\\perp^{\\prime 2})\n(k_\\perp^2-k_\\perp^{\\prime 2})\n\\end{equation}\nto simplify the form of the  solutions (\\ref{eg2}) of the equation\n({\\ref{eg1}}). The functions  $f_{i}$ are dependent on\n$k_{\\perp}^{2},k_{\\perp}^{\\prime 2},k_{\\perp}^{\\prime\\prime 2}, B$,\nand are\n\\begin{equation}\nf_i^{(1)}=\\frac{1}{3}\\left[2k_\\perp^{2}+k_\\perp^{\\prime\n2}+\\frac{\\Lambda^{* 2}}{(k_\\perp^{\\prime\\prime 2}-k_\\perp^{\\prime\n2})\\mathcal{G}^{1\/3}}+\n\\frac{\\mathcal{G}^{1\/3}}{k_\\perp^{\\prime\\prime 2}-k_{\\perp}^{\\prime\n2}}\\right] \\label{fi}\n\\end{equation}\nwhere\n\\[\n\\mathcal{G} =6 \\pi\\sqrt{3}D-\\Lambda^{*3}+54 \\pi^2\n\\phi_{n,n^{\\prime}}^{(i) 2}(k_\\perp^{\\prime\\prime 2}-k_\\perp^{\\prime\n2})^2\n\\]\nwith\n\\[\nD=\\sqrt{-(k_\\perp^{\\prime 2}-k_\\perp^{\\prime\\prime\n2})^2\\Lambda^{*3}\\phi_{n,n^{\\prime}}^{(i) 2}\\left[1-\\frac{27 \\pi^2\n\\phi_{n,n^{\\prime}}^{(i)2}(k_\\perp^{\\prime\\prime2}-k_\\perp^{\\prime\n2})^2}{\\Lambda^{*3}}\\right]}\n\\]\nThe  solution $z_1=f_i^{(1)}$, where $z_1=\\omega^2-k^2_{\\parallel}$,\nconcern the values of $k_\\perp^2$ exceeding the root $k_\\perp^{\\ast\n2}$ of the equation $D=0$, approximately equal to\n\\[\nk_\\perp^{\\star 2}\\simeq k_\\perp^{\\prime\n2}-3\\left(\\frac{\\pi^2\\phi_{n,n^{\\prime}}^{ (i)2}(k_\\perp^{\\prime\n2})}{k_\\perp^{\\prime\\prime 2}-k_\\perp^{\\prime 2}}\\right)^{1\/3}\n\\]\n(for $k_\\perp^{2}<k_\\perp^{*2}$, $D$ becomes complex). Besides\n(\\ref{fi}), there are two other solutions of the above- mentioned\ncubic equation resulting from substituting (\\ref{eg5}) in\n(\\ref{eg1}). These are complex solutions and are located in the\nsecond sheet of the complex plane of the variable\n$z_1=\\omega^2-k_\\parallel^2$. At $k_\\perp^2>k_\\perp^{\\ast 2}$ these\ntwo complex solutions are given by\n\\[\nf_i^{(2)}=\\frac{1}{6}\\left[2(2k_\\perp^{2}+k_\\perp^{\\prime\n2})-\\frac{(1+i\\sqrt {3})\\Lambda^{* 2}}{(k_\\perp^{\\prime\\prime\n2}-k_\\perp^{\\prime 2})\\mathcal{G}^{1\/3}}-\n\\frac{(1-i\\sqrt{3})\\mathcal{G}^{1\/3}}{k_\\perp^{\\prime\\prime\n2}-k_\\perp^{\\prime 2}}\\right]\n\\]\nand\n\\[\nf_i^{(3)}=\\frac{1}{6}\\left[2(2k_\\perp^{2}+k_\\perp^{\\prime\n2})-\\frac{(1-i\\sqrt {3})\\Lambda^{* 2}}{(k_\\perp^{\\prime\\prime\n2}-k_{\\perp}^{\\prime 2})\\mathcal{G}^{1\/3}}-\n\\frac{(1+i\\sqrt{3})\\mathcal{G}^{1\/3}}{k_\\perp^{\\prime\\prime\n2}-k_{\\perp}^{\\prime 2}}\\right]\n\\]\nbut they are not interesting to us in the present context.\n\nWe should define the functions\n\\[\nm_n=\\frac{k_\\perp^{\\prime}+k_\\perp^{\\prime\\prime }}{2}\\ \\\n\\textrm{and}\\ \\ m_{n^{\\prime}}=\\frac{k_\\perp^{\\prime\n}-k_\\perp^{\\prime\\prime }}{2},\n\\]\nwhich are positive for all possible values of $n$ and $n^\\prime$.\n\nNow the   magnetic moment of the photon can be derived by taking the\nimplicit derivative $\\partial\\omega\/\\partial B_z$ and\n$\\partial\\omega\/\\partial kF^2k$ in the dispersion equation, from\n(\\ref{egg}) and (\\ref{eg5})  it is obtained that\n\\begin{widetext}\n\\begin{equation}\n\\mu_\\gamma^{(i)}=\\frac{\\pi}{2\\omega(\\vert\n\\Lambda\\vert^3-4\\pi\\phi_{n,n^\\prime}^{(i)}m_n\nm_{n^\\prime})}\\left[\\phi_{n,n^{\\prime}}^{(i)}\\left(A\\frac{\\partial\nm_n}{\\partial B}+Q\\frac{\\partial m_{n^\\prime}}{\\partial\nB}\\right)-2\\Lambda^2\\frac{\\partial\n\\phi_{n,n^{\\prime}}^{(i)}}{\\partial B}\\right] \\label{mm2}\n\\end{equation}\nbeing\n\\[\nA=-4m_{n^\\prime}[z_1-(m_n+m_{n^\\prime})(3m_n+m_{n^\\prime})]>0\n\\]\nand\n\\[\nQ=-4m_{n}[z_1-(m_n+m_{n^\\prime})(m_n+3m_{n^\\prime})]>0.\n\\]\n\\end{widetext}\n\nThe expression  (\\ref{mm2}) contains terms with paramagnetic and\ndiamagnetic behavior, in which, as is typical in the relativistic\ncase, the dependence of $\\mu$ on $B$ is non-linear. It contains also\ndiamagnetic terms, depending on the sign of $\\phi_{n,\nn^{\\prime}}^{(i)}$ and its derivatives with regard to $B$: if\n$\\phi_{n, n^{\\prime}}^{(i)}>0$ the first term in the bracket,\ncontributes paramagnetically ($\\frac{\\partial m_n}{\\partial B}\\geq\n0$ and $\\frac{\\partial m_{n^\\prime}}{\\partial B}\\geq 0$), if\n$\\phi_{n, n^{\\prime}}^{(i)}<0$ the sign of this term is opposite and\nits contribution is diamagnetic. The second term of (\\ref{mm2}) will\nbe paramagnetic or diamagnetic depending on the sign of the\nderivative of the $\\phi_{n, n^{\\prime}}^{(i)}$ with regard to $B$.\n\nIn particular if $n=n^\\prime$\n\\begin{equation}\n\\mu_\\gamma^{(i)}=\\frac{\\pi}{\\omega(\\vert\n\\Lambda\\vert^3-4\\pi\\phi_{n}^{(i)}m_n^2)}\\left[\\phi_{n}^{(i)}A\\frac{\\partial\nm_n}{\\partial B}-\\Lambda^2\\frac{\\partial \\phi_{n}^{(i)}}{\\partial\nB}\\right] \\label{mm3}\n\\end{equation}\nwith $A=-4m_{n}[z_1-8m_n^2]>0$\n\nIn the vicinity of the first threshold  $k_\\perp^{\\prime\\prime\n2}=0$, $k_\\perp^{\\prime 2}=4m_0^2$ and $\\partial m_n\/\\partial B=0$\nwhen $n=0$, therefore for the second mode the absolute value of the\nmagnetic moment is given by\n\\begin{equation}\n\\mu_{\\gamma}^{(2)}=-\\frac{\\pi\\vert\\Lambda\\vert^2}{\\omega\\left(\\vert\\Lambda\\vert^3-4m_0^2\\pi\\phi_{00}^{(2)}\\right)}\\frac{\\partial\n\\phi_{00}^{(2)}}{\\partial B}\\label{FRR}\n\\end{equation}\nOne can write the (\\ref{FRR}) in explicit form as\n\\begin{widetext}\n\\begin{equation}\n\\mu_\\gamma^{(2)}=\\frac{\\alpha\nm_0^3\\left(4m_0^2+k^2+\\frac{kF^2k}{2\\mathcal{F}}\\right)\\exp\\left(\\frac{kF^2k}{4m_0^2\n\\mathcal{F}}\\frac{B_c}{B}\\right)}{\\omega\nB_c\\left[(4m_0^2+k^2+\\frac{kF^2k}{2\\mathcal{F}})^{3\/2}+\\alpha m_0^3\n\\frac{B}{B_c} \\exp\\left(\\frac{kF^2k}{4m_0^2\n\\mathcal{F}}\\frac{B_c}{B}\\right)\\right]}\\left(1-\\frac{kF^2k}{4\nm_0^2\\mathcal{F}}\\frac{B_c}{B}\\right).\\label{FRR1}\n\\end{equation}\n\\end{widetext}\nif we consider for simplicity propagation perpendicular to the field\n$B$, and $\\omega$ near the threshold $2m_0$, the function\n$\\mu_{\\gamma}^{(2)}=f(X)$, where $X=\\sqrt{4m_0^2-\\omega^2}$ has a\nmaximum for $X= {\\pi\\phi_{00}^{(2)}\/m_0}^{1\/3}$, which is very near\nthe threshold.\n\nThus, for perpendicular propagation the expression (\\ref{FRR1}) has\na maximum value when $k_\\perp^2 \\simeq k_\\perp^{\\prime 2}$.\nTherefore in a vicinity of the first pair creation threshold the\nmagnetic moment of the photon has a paramagnetic behavior and a\nresonance peak whose value is given by\n\\begin{equation}\n\\mu_{\\gamma}^{(2)}=\\frac{m_0^2(B+2B_c)}{3m_\\gamma\nB^2}\\left[2\\alpha\\frac{B}{B_c}\\exp\\left(-\\frac{2B_c}{B}\\right)\\right]^{2\/3}\n\\label{mumax}\n\\end{equation}\nWe would like to define the \"dynamical mass\" $m_{\\gamma}$ of the\nphoton in presence of a strong magnetic field by the equation\n\\begin{equation}\nm_{\\gamma}^{(2)}=\\omega (k_\\perp^{\\prime\n2})=\\sqrt{4m_0^2-m_0^2\\left[2\\alpha\n\\frac{B}{B_c}\\exp\\left(-\\frac{2B_c}{B}\\right)\\right]^{2\/3}}\n\\label{dm}\n\\end{equation}\n\nFrom (\\ref{mumax}) it is seen that  near the pair creation\nthreshold, the magnetic moment induced by the external field in the\nphoton as a result of its interaction with the polarized\nelectron-positron quanta of vacuum, has a peak (Fig. 3).  This is a\nresonance peak, and we understand it as due to the interaction of\nthe photon with the polarized $e^{\\pm}$ pairs.\n\nThe expression (\\ref{mumax}) presents a maximum when $B\\simeq B_c$,\nin such case the magnetic moment is given by\n\\begin{equation}\n\\mu_\\gamma^{(2)}\\approx 3\\mu^\\prime\n\\left(\\frac{1}{2\\alpha}\\right)^{1\/3}\\approx 12.85\\mu^\\prime\n\\end{equation}\n\nThe arising of a photon dynamical mass is a consequence of the\nradiative corrections, which become significant for photon energies\nnear the pair creation threshold and magnetic fields large enough to\nmake significant the exponential term $\\exp(kF^2k\nB_c\/4m_0^2\\mathcal{F}B) =e^{-k_\\perp^2\/eB}$. This means that the\nmassless photon coexists with the massive pair, leading to a\nbehavior very similar to that of a neutral vector particle\n\\cite{Osipov} bearing a magnetic moment.One should notice from\n(\\ref{dm}) that the dynamical mass decrease with the increasing\nfield intensity, which suggests that the interaction energy of the\nphoton with its environment increase for magnetic fields far greater\nthan $B_c$ near the first threshold, when the photon coexist with\nthe virtual pair.\n\nThe  problem of  neutral vector particles is  studied elsewhere\n\\cite{Herman}. The energy eigenvalues being\n\\begin{equation}\nE(p, B,\\eta) =\\sqrt{p_3^2+p_{\\perp}^2+M^2\n+\\eta(\\sqrt{p_{\\perp}^2+M^2}) qB} \\label{spec},\n\\end{equation}\n\\noindent where the second square root expresses the dependence of\nthe eigenvalues on the \"transverse energy\", proportional to the\nscalar $pF^2p\/{\\cal F}$. We have $\\eta =0,\\pm 1$, and $q$ being a\nquantity having the dimensions of magnetic moment. In what follows\nwe exclude the value $\\eta=0$, since it corresponds to the case of\nno interaction with the external field $B$. We have that for\n$\\eta=-1$ the magnetic moment is $\\mu= \\pm \\frac{q}{\\sqrt{M^2 -\nMqB}}$, which is divergent at the threshold $M =qB$. The behavior of\nthe photon near the critical field $B_c$ closely resembles this\nbehavior, since it has a maximum value at the threshold.\n\n\nFor $B_z\\gg B_c$ although the vacuum is strongly polarized, the\nphoton shows a weaker polarization, \\emph{i.e} the contribution from\nthe singular behavior near the thresholds decreases. Actually, the\npropagation is being decreased due to an increasing in the imaginary\npart of the photon energy $\\Gamma$ (we have  the total frequency\n$\\omega=\\omega_r + i\\Gamma$, where $\\omega_r$ is the real part of\nit). As the modes are bent to propagate parallel to $B$, they\npropagate in an increasingly absorbent medium.  But for $B>>B_c$ we\nare in a region beyond QED and new phenomena related to the standard\nmodel may appear, as for instance, the creation of $\\mu^{\\pm}$\npairs, and their subsequent decay according to the allowed channels.\n\n\nAs opposite to the second mode case, the polarization eigenvalue\nfrom the third mode in supercritical magnetic field does not\nmanifest a singular behavior in the first resonance\n\\cite{proceeding}. In this case the eigenvalues are given by\n\\begin{widetext}\n\\begin{equation}\n\\pi_3=\\frac{\\alpha\nk^2}{3\\pi}\\left(\\ln\\frac{B}{B_c}-C\\right)+\\frac{\\alpha}{3\\pi}\\left(0.21\n\\frac{kF^2k}{2\\mathcal{F}}-1.21\\left(k^2+\\frac{kF^2k}{2\\mathcal{F}}\\right)\n\\right)\n\\end{equation}\nIn this case the dispersion equation has the solution\n\\begin{equation}\n\\omega^2=\\vert\\textbf{k}\\vert^2+\\frac{kF^2k}{2\\mathcal{F}}\\left[1-\\frac{\\alpha}{3\\pi}\\left(\\ln\\frac{B}{B_c}-C-1.21\\right)\\right]^{-1}\n\\label{3m}\n\\end{equation}\n\\end{widetext}\nbeing $C$ is Euler constant. For fields $B\\sim B_c$ so that the\nlogarithmic terms are small, the corresponding magnetic moment is\ngiven by\n\\begin{equation}\n\\mu_\\gamma^{(3)}=\\frac{2\\mu^{\\prime}}{3m_0}\\frac{k_\\perp^2}{\\omega}\n\\label{3mm}\n\\end{equation}\nfor perpendicular propagation to $B_z$ and for photons with energies\nnear of $m_0$ the magnetic moment of the photon propagating in the\nthird mode has a value of $\\mu_\\gamma^{(3)}\\sim\\mu^{\\prime}$.\n\n\\section{Magnetic Moment For The Photon-Positronium  Mixed States}\n\nIn the case of positronium formation, by following\n\\cite{shabad3,Leinson}, neglecting the retardation effect and in the\nlowest adiabatic approximation,  the Bethe-Salpeter  equation is\nreduced  to a Schr\\\"odinger equation in the variable $(z^e-z^p)$\nwhich  governs  the relative motion along $\\textbf{B}$ of the\nelectron and positron.\n\nUnder such approximations, the conservation law induced by the\ntraslational invariance takes the form $p_x=p_x^p+p_x^e=k_\\perp$.\nTherefore the binding energy depends on the distance between the\n$y$-coordinates of the centers of the electron and positron orbits\n$\\vert y_0^p-y_0^e\\vert=p_x\/\\sqrt{eB}$.\n\nThe  Schr\\\"odinger equation  mentioned  includes the attractive\ncoulomb force whose potential in our case has the form\n\\[\nV_{nn^\\prime}(z^e-z^p)=-\\frac{e^2}{\\sqrt{(z^e-z^p)^2+L^4p_x^2}}\n\\]\nwhere $L=(eB)^{-1\/2}$ is the radius of the electron orbit. The\neigenvalue of this equation\n$\\Delta\\varepsilon_{n,n^{\\prime}}(n_c,k_\\perp^2)$ is the binding\nenergy of the particles which is numbered by a discrete number\n$n_c$ that identifies the Coulomb-bound state for\n$\\Delta\\varepsilon_{n,n^{\\prime}}(n_c,k_\\perp^2)>0$   and by a\ncontinuous one in the opposite case.\n\nThe energy of the pair which does not move along the external\nmagnetic field is given by\n\\begin{equation}\n\\varepsilon_{n,n^\\prime}(n_c,k_\\perp ^2)=k_\\perp^{\\prime}+\\Delta \\varepsilon_{n,n^{\\prime}}(n_c,k_\\perp^2)\n\\end{equation}\n\nIn this paper we will consider the case in which the Coulomb state\nis $n_c=0$, here the binding energy is given by  the expression of\nthe eigenvalues of this equation,\n\\begin{equation}\n\\Delta \\varepsilon_{n,n^{\\prime}}(0,k_\\perp^2)=-\\frac{\\alpha^2\nM_r}{2}\\left(2\\ln\\left[\\frac{a_{nn^\\prime}^B}{2\n\\sqrt{L^2+L^4P_x^2}}\\right]\\right)^{-2},\n\\end{equation}\nwhere  $a_B^{nn^\\prime}=1\/e^2M_r$ is Bohr radius and\n$M_r=m_nm_{n^\\prime}\/(m_n+m_{n^\\prime})$ the reduced mass of the\nbound pair.\n\nThe dispersion equation of the positronium is given by\n\\begin{equation}\nk_\\perp^2+k_\\parallel^2-\\omega^2=-\\frac{2\\pi \\Phi_{nn^\\prime\nn_c}^{(i)}}{\\varepsilon_{nn^\\prime}^2-\\omega^2+k_\\parallel^2}\n\\label{p1}\n\\end{equation}\n\nFor each set of discrete quantum number $n$, $n^\\prime$, $n_c$,\nequation (\\ref{p1}) is quadratic with regard to the variable\n$z_1=\\omega^2-k_\\parallel^2$ and its solutions are\n\\begin{eqnarray}\nf_i=\\frac{1}{2}\\left(\\varepsilon_{nn^\\prime\nn_c}^2(k_\\perp^2)+[k_\\perp^2\\pm\\right.\n\\\\\\nonumber\\left.(\\varepsilon_{nn^\\prime\nn_c}^2(k_\\perp^2)-k_\\perp^2)^2-8\\pi\\Phi_{nn^\\prime\nn_c}^{(i)}(k_\\perp^2)]^{1\/2}\\right)\n\\end{eqnarray}\n\nAt the first cyclotron resonance $n=n^{\\prime}=0$, the function\n$\\Phi_{000}$ that define the second mode has the structure\n\\begin{equation}\n\\Phi_{000}^{(2)}=\\phi_{00}^{(2)}\\varepsilon_{000}(k_\\perp^2)\\vert \\psi_{000}(0)\\vert^2\n\\end{equation}\nwith\n\\[\n\\Delta\\varepsilon_{00}(0,k_\\perp^2)=-\\alpha^2 m_0\\left(\\ln\\left[\\frac{1}{\\alpha}\\sqrt{\\frac{ B_c(1+\\frac{k_\\perp^2B_c}{m_0^2B})}{B}}\\right]\\right)^2\n\\]\nwhere\n\\[\n\\vert\\psi_{000}(0)\\vert^2=\\alpha\\left\\vert\\ln\\left[\\frac{1}{\\alpha}\\sqrt{\\frac{B}{B_c\\left(1+\\frac{k_\\perp^2B_c}{m_0^2B}\\right)}}\\right]\\right\\vert\n\\]\n is the wave function squared of the longitudinal motion\\cite{Loudon} at $ z^e=z^p$.\n\nIn the case of photon-positronium mixed states we obtain from\n(\\ref{treee}) and (\\ref{p1}) that\n\\begin{equation}\n\\mu_{\\gamma}^{P}=\\frac{\\pi\\Upsilon}{\\omega\n(k_\\parallel^2-\\omega^2-\\varepsilon^2)\\left\n(1+\\frac{2\\pi\\Phi_{nn^\\prime\nn_c}^{(i)}}{k_\\parallel^2-\\omega^2-\\varepsilon^2} \\right)}\n\\label{mps}\n\\end{equation}\nbeing\n\\[\n\\Upsilon=\\varepsilon_{nn^\\prime}(n_c) \\Phi_{n n^{\\prime}\nn_c}^{(i)}\\frac{\\partial\\varepsilon}{\\partial\nB}-\\frac{\\partial\\Phi_{n n^{\\prime} n_c}^{(i)}}{\\partial B}\n\\]\n\nFollowing the  reasoning of (\\ref{mumax}), for  magnetic fields\n$B\\gg B_c$ one can define the dynamical mass of the\nphoton-positronium mixed state in the first threshold for\npositronium energy, $\\varepsilon^2 \\sim 3.996m_0^2$ and\nperpendicular propagation as\n\\begin{equation}\nm_{\\gamma}^{P}=\\sqrt{\\varepsilon_{00}^2-2m_0^2\\alpha\\left[\\frac{B}{B_c}\\ln\\left(\\frac{1}{2\\alpha}\\frac{B}{B_c}\\right)\\right]^{1\/2}}\n\\end{equation}\nin this regime\n\\begin{equation}\n\\mu_{\\gamma}^{P}=\\frac{m_0^2\\alpha\\left(1+\\ln\\left[\\frac{B}{2\\alpha\nB_c}\\right]\\right)}{2B_cm_\\gamma^P\\sqrt{\\frac{B}{B_c}\\ln\\left[\\frac{B}{2\\alpha\nB_c}\\right]}}\n\\end{equation}\nwhen $k_\\parallel=0$.\n\nIn a similar way to the free pair creation, the behavior of the\nmagnetic moment of the magnetic moment of the mixed state for $n,n'\n\\neq 0$ can be paramagnetic for some values of Landau numbers and\nintervals in momentum space,  and diamagnetic in other ones.\n\n\n\\section{Results and discussion}\n\nFor perpendicular propagation the dependence of the magnetic moment\nwith regard $k_\\perp^2$ in the first resonance $k_\\perp^{\\prime 2}\n=4m_0^2$ is displayed in FIG.  \\ref{fig:Phvk} for free pair creation\nand photon-positronium mixed state. Both\n curves  show the same qualitative behavior. As it\nwas expected, near the threshold energy appears the peak\ncharacteristic of the resonance. The result shows that near the pair\ncreation threshold the magnetic moment of a photon may have values\ngreater than the anomalous  magnetic moment of the electron.\n(Numerical calculations range from 0 to more than $12\\mu^\\prime$).\nThis result  is related to the probability of the pair creation,\nwhich is maximum in the first resonant form when $k_\\perp=2m_0$, and\nincrease with $B$, because the medium is becoming absorbent.\n\\begin{figure}[!htbp]\n\\includegraphics[width=3in]{GMk.eps}\n\\caption{\\label{fig:Phvk} Magnetic moment curves of the photon\n(dark) and photon-positronium mixed states (dashed) with regard to\nperpendicular momentum squared,  for the second mode\n$k_\\perp^{\\prime 2}=4m_0^2$ with $n=n^{\\prime}=0$.}\n\\end{figure}\n\nWe observe that near the thresholds the behavior of the curves is\nthe same for all  pairs $\\omega$, $k_\\parallel$ satisfying the\ncondition $\\omega^{2}-k_\\parallel^2=k_\\perp^{\\prime 2}$.\n\nIn all curves the magnetic moment decrease for momentum values\n$k_\\perp^2>4m_0^2$. Therefore the vacuum polarization decreases,\nthus, the magnetic moment tends to vanish as it is shown in FIG.\n\\ref{fig:Phvk}. We interpret these results in the sense that for\nphotons with squared transversal component of the momenta greater\nthan $4m_0^2$ the probability of free pair creation in the Landau\nground state (and positronium creation) decrease very fast, since\nthat region of momenta is to be considered inside the transparency\nregion corresponding to the next thresholds, i.e., $n=0, n'=1$ or\nvice-versa.\n\n\\begin{figure}[!htbp]\n\\includegraphics[width=3in]{GMB.eps}\n\\caption{\\label{fig:mb1} Magnetic moment curves for the photon and\nphoton-positronium mixed states with regard to external magnetic\nfield strength. The propagation is perpendicular to $\\textbf B$ and\nthe values of the perpendicular momentum are equal to the absolute\nvalues of the free and bound threshold energies. The first\n(continuous) curve refers to the photon whereas the second (dashed)\ncorresponds to the photon-positronium mixed state.}\n\\end{figure}\n\nThe behavior of the magnetic moment with regard to the field is\nshown in the FIG. \\ref{fig:mb1} for the case of photon and\nphoton-positronium mixed state. The picture was obtained in the\nfield interval $0 \\leq B \\leq 10B_c$ by considering perpendicular\npropagation and taking  the values of the momentum squared as equal\nto the absolute values of the threshold energies of free and bound\npair creation. Here, as opposite to the case of low frequency, the\nmagnetic moment of the photon tend to vanish when $B\\rightarrow0$.\nWe note  that, again, the magnetic moment of the photon is greater\nthan $\\mu^\\prime$. In correspondence with figs. \\ref{fig:Phvk} the\nmagnetic moment of the photon not considering Coulomb interaction,\nis greater than the corresponding to photon-positronium mixed state.\nThis result is to be expected  due to the existence of the binding\nenergy, in the latter case it entails a decrease of the threshold\nenergy. Each curve has a maximum value, this maximum for the free\npair creation  is approximately $B\\approx 1.5 B_c$, whereas for\nphoton-positron bound state  the value is $B\\approx B_c$.\n\n\\begin{figure}[!htbp]\n\\includegraphics[width=3in]{masa1.eps}\n\\caption{\\label{fig:mass} Photon dynamical mass dependence with\nregard the external  magnetic field of the photon-positronium\n(dashed) and photon-free pair (dark) mixed states. Both curves were\nobtained near the corresponding first thresholds for each process.}\n\\end{figure}\n\n\nIn fig. \\ref{fig:mass} we display the photon dynamical mass\ndependence on the magnetic field, by considering perpendicular\npropagation near the thresholds, for free and bound pair creation\nand for the second mode. It is shown that for that mode, the\ndynamical mass decreases with increasing magnetic field.\n\nFor magnetic field values  $B>1.5B_c$ the dynamical mass of the\nphoton-positronium mixed state is greater than the corresponding\nto the case of free pair creation which suggests that the latter\nis more probable that the bound state case.\n\n\\begin{figure}[!htbp]\n\\includegraphics[width=3in]{EFk.eps}\n\\caption{\\label{su} Curves for the modulus of the photon magnetic\nmoment with regard to the perpendicular momentum squared, near the\npair creation threshold for the second mode $k_\\perp^{\\prime 2}=7.46\nm_0^2$ with $n=0$, $n^{\\prime}=1$ and for perpendicular propagation.\nThe value of the external magnetic field used for calculation are\n$B=B_c$. The dashed line correspond to  photon-positronium mixed\nstate, whereas the dark line to the free pair creation.}\n\\end{figure}\n\nWe have found that when the particles are created in excited states\nthe behavior of the magnetic moment reaches higher\n values with regard the case analyzed previously when\n$k_\\perp^2=k_\\perp^{\\prime 2}$ and $B\\backsim B_c$. Calculation\npoints out that these values may be of order $10^{2}\\mu^\\prime$. The\nnew values obtained come fundamentally due to the fact of the\nthreshold energies, which depend on the  magnetic field $\\textbf{B}$\n(see Fig.2). The new behavior of the photon and photon-positronium\nmixed state is shown in the Fig.\\ref{su} and Fig. \\ref{suB}.\n\n\n\\begin{figure}[!htbp]\n\\includegraphics[width=3in]{EFB.eps}\n\\caption{\\label{suB}Curves for the modulus of the photon magnetic\nmoment (dark) and photon-positronium mixed states(dashed) plotted\nwith regard to the external magnetic field strength when the\npropagation is perpendicular to $\\textbf B$ and the values of the\nperpendicular momentum squared is equal to the  values of the free\nand bound threshold energies for $n=0$ and $n^\\prime=1$.}\n\\end{figure}\n\nIn this case the dynamical mass Fig.\\ref{m1}, as different from the\nprevious case, increases with increasing magnetic field. This means\nthat the magnetic field confine the virtual particles near the\nthreshold  when these tend to be created in excited states. The\ndynamical mass of the positronium in such conditions is always\nsmaller  than the corresponding to free pair creation, which\nsuggests that the bound state pair creation in this configuration is\nmore probable.\n\n\\begin{figure}[!htbp]\n\\includegraphics[width=3in]{masa.eps}\n\\caption{\\label{m1}Photon dynamical mass dependence with\nregard the external  magnetic field of the photon-positronium\n(dashed) and photon-free pair (dark)  being $n=0$ and $n^\\prime=1$. Both curves\nwere obtained in the corresponding energy thresholds.}\n\\end{figure}\n\n\\section{Conclusion}\n\nWe conclude, first, that a photon propagating in vacuum in presence\nof an external magnetic field, exhibits a nonzero magnetic moment\nand a sort of dynamical mass due to the magnetic field. This\nphenomenon occurs whenever the photon has a nonzero perpendicular\nmomentum component to the external magnetic field. The values of\nthis anomalous magnetic moment depend on the propagation mode and\nmagnetic field regime. The maximum value taken by the photon\nmagnetic moment is greater than the anomalous magnetic moment of the\nelectron in a strong magnetic field.\n\nSecond, in the small field and low frequency approximations, the\nmagnetic moment of the photon also exists and is slowly dependent of\nthe magnetic field intensity in some range of frequencies, whereas\nthe high frequency limit it depends on $\\vert\\bf{B}\\vert$. In both\ncases it vanishes when $B\\to0.$\n\nUnder these conditions, the behavior of the photon is similar to a\nvector neutral massive particle.\n\n\\section{Acknowledgement}\nBoth authors are greatly indebted to Professor A.E. Shabad, from\nP. N.Lebedev Physical Institute in Moscow for several comments and\nilluminating discussions.\n\n\\section{Appendix A}\nThe three eigenvalues $\\pi_i=1,2,3$ of the polarization operator in\none loop approximation, calculated using the exact propagator of\nelectron in an external magnetic field, can be expressed as linear\ncombination of three functions $\\Sigma_i$. In what follows we will\ncall $x=B\/B_c$\n\\begin{eqnarray}\n&\\pi_1&=-\\frac{1}{2}k^2\\Sigma_1,\\\\\n&\\pi_2&=-\\frac{1}{2}\\left(\\left(\\frac{kF^2k}{2\\mathcal{F}}+k^2\\right)\\Sigma_2-\\frac{kF^2k}{2\\mathcal{F}}\\Sigma_1\\right),\\\\\n&\\pi_3&=-\\frac{1}{2}\\left(\\left(\\frac{kF^2k}{2\\mathcal{F}}+k^2\\right)\\Sigma_1-\\frac{kF^2k}{2\\mathcal{F}}\\Sigma_3\\right).\\label{eg1}\n\\end{eqnarray}\nwhere $\\mathcal{F}=\\frac{B^2}{2}$ and\n\nWe express\n\\begin{equation}\n\\Sigma_i=\\Sigma_i^{(1)}+\\Sigma_i^{(2)},\\label{aS}\n\\end{equation}\nbeing\n\\begin{equation}\n\\Sigma_i^{(1)}(x)=\\frac{2\\alpha}{\\pi}\\int_0^\\infty\ndte^{-t\/x}\\int_{-1}^1d\\eta \\left[\\frac{\\sigma_i(t,\\eta)}{\\sinh\nt}-\\lim_{t\\rightarrow0}\\frac{\\sigma_i(t,\\eta)}{\\sinh t}\\right]\n\\end{equation}\nand\n\\begin{widetext}\n\\begin{eqnarray}\n\\Sigma_i^{(2)}(x,kF^2k, k^2,\n\\mathcal{F})=\\frac{2\\alpha}{\\pi}\\int_0^\\infty\ndte^{-t\/x}\\int_{-1}^1d\\eta\\frac{\\sigma_i(t,\\eta)}{\\sinh t}\n\\left[\\exp\\left(\\frac{kF^2k}{2\\mathcal{F}}\\frac{M(t,\\eta)}{m_0^2x}-\\left(\\frac{kF^2k}{2\\mathcal{F}}+k^2\\right)\\frac{1-\\eta^2}{4m_0^2x}t\\right)-1\\right],\n\\label{S2}\n\\end{eqnarray}\n\\end{widetext}\nwhere\n\\begin{equation}\nM(t,\\eta)=\\frac{\\cosh t -\\cosh\\eta t}{2\\sinh t},\n\\end{equation}\n\\begin{equation}\n\\sigma_1(t,\\eta)=\\frac{1-\\eta }{2}\\frac{\\sinh(1+\\eta)t}{\\sinh t},\n\\label{ss1}\n\\end{equation}\n\\begin{equation}\n\\sigma_2(t,\\eta)=\\frac{1-\\eta^2}{2}\\cosh t \\label{ss2},\n\\end{equation}\n\\begin{equation}\n\\sigma_3(t,\\eta)=\\frac{\\cosh t -\\cosh\\eta t}{2\\sinh^2 t}.\n\\label{ss3}\n\\end{equation}\n\nExpress $\\Sigma_i^{(1)}$ as\n\\begin{equation}\n\\Sigma_i^{(1)}(x)=\\frac{2\\alpha}{\\pi}\\int_0^\\infty\ndte^{-t\/x}\\left[\\frac{g_i(t)}{\\sinh t}-\\frac{1)}{3t}\\right].\n\\end{equation}\nHere\n\\[\ng_i(t)=\\int_{-1}^1d\\eta\\sigma_i(t,\\eta)d\\eta\n\\]\nand in explicit form\n\\begin{eqnarray}\n&g_1(t)&=\\frac{1}{4t\\sinh\nt}\\left(\\frac{\\sinh2t}{t}-2\\right),\\label{g1}\\\\\n&g_2(t)&=\\frac{\\cosh t}{3},\\label{g2}\\\\\n&g_3(t)&=\\frac{1}{\\sinh^2 t}\\left(\\cosh t-\\frac{\\sinh\nt}{t}\\right).\\label{g3}\n\\end{eqnarray}\n\nLet\n\\begin{equation}\nu_i(t)=\\frac{g_i(t)}{\\sinh t}-\\frac{1}{3t}\\label{e69}\n\\end{equation}\nThe asymptotic expansion of (\\ref{e69}) in powers of $\\exp(-t)$ have\nthe form\n\\begin{equation}\nu_1(t)=0, \\ \\ u_2(t)=1\/3,\\ \\ u_3(t)=0\n\\end{equation}\n\nOur next purpose is to analyze the  behavior of $\\pi_{i}$ when\n$x\\rightarrow0$. In this case the behavior of $\\Sigma_i^{(1)}(x)$ is\ndetermined by the the factor $\\exp(-t\/x)$ at the integrand which\ntends to zero when $x\\rightarrow0$. Taking in account the expansion\n\\[\n\\exp(-t\/x)\\simeq \\exp(-t\/\\epsilon)+\\frac{\\exp(-t\/\\epsilon\n)t}{\\epsilon^2}(x-\\epsilon),\n\\]integrating by $t$  and taking\nthe limit when $\\epsilon\\rightarrow0$ we obtain that\n\n\\begin{eqnarray}\n&\\Sigma_1^{(1)}&(x)=0,\\ \\ \\Sigma_2^{(1)}(x)\\simeq\\frac{2\\alpha\nB}{3\\pi B_c},\\ \\ \\Sigma_3^{(1)}(x)=0.\\label{ddd}\n\\end{eqnarray}\n\n\nThe functions $\\Sigma_i^{(2)}$  depend of three arguments, as\nindicated in (\\ref{S2}). The asymptotic expansion of\n(\\ref{ss1}),(\\ref{ss2}), (\\ref{ss3}) in powers of $\\exp(-t)$ and\n$\\exp(t\\eta)$ produces an expansion of (\\ref{S2}) into a sum of\ncontributions coming from the thresholds, the singular behavior in\nthe  threshold points originating from the divergencies of the\n$t-$integration in ($\\ref{S2}$) near $t=\\infty$ as it was made in\n\\cite{shabad1}. The leading term in the expansion of (\\ref{ss1}),\n(\\ref{ss2}), (\\ref{ss3}) at $t\\rightarrow\\infty$ are\n\\begin{eqnarray}\n&\\left.\\left(\\frac{\\sigma_1(t,\\eta)}{\\sinh\nt}\\right)\\right\\vert_{t\\rightarrow\n\\infty}&=\\frac{1-\\eta}{2}\\exp\\left(-t(1-\\eta)\\right),\\\\\n&\\left.\\left(\\frac{\\sigma_2(t,\\eta)}{\\sinh\nt}\\right)\\right\\vert_{t\\rightarrow \\infty}&=\\frac{1-\\eta^2}{4},\\\\\n&\\left.\\left(\\frac{\\sigma_3(t,\\eta)}{\\sinh\nt}\\right)\\right\\vert_{t\\rightarrow \\infty}&=2\\exp\\left(-2t)\\right).\n\\end{eqnarray}\nThe function $M(\\infty,\\eta)=1\/2$, one obtains near the lowest\nsingular threshold ($n=0$, $n^\\prime=1$ or viceversa for $i=1$,\n$n=n^\\prime=1$ for $i=1$, and $n=n^\\prime=0$ for $i=2$). Taking into\naccount this expansion we can write the expression (\\ref{S2}) for\nthe first and their modes as\n\\begin{widetext}\n\\begin{equation}\n\\Sigma_1^{(2)}=\\frac{4\\alpha e B}{\\pi}\\int_{-1}^1d\\eta(1-\\eta)\n\\left[\\frac{\\exp\\left(\\frac{kF^2k}{4m_0^2\\mathcal{F}}\\frac{B_c}{B}\\right)}{4m_0^2+4(1-\\eta)eB+\\left(k^2+\\frac{kF^2k}{2\\mathcal{F}}\\right)\\left(1-\\eta^2\\right)}-\\frac{1}{4\nm_0^2+4(1-\\eta)eB}\\right], \\label{SS21}\n\\end{equation}\n\\begin{equation}\n\\Sigma_2^{(2)}=\\frac{2\\alpha e B}{\\pi}\\int_{-1}^1d\\eta(1-\\eta^2)\n\\exp\\left(\\frac{kF^2k}{4m_0^2\\mathcal{F}}\\frac{B_c}{B}\\right)\\left(\\frac{1}{4m_0^2+\\left(k^2+\\frac{kF^2k}{2\\mathcal{F}}\\right)\\left(1-\\eta^2\\right)}\\right)-\\frac{2\\alpha\neB}{3\\pi m_0^2}, \\label{SS22}\n\\end{equation}\n\\begin{equation}\n\\Sigma_3^{(2)}=\\frac{16\\alpha e B}{\\pi}\\int_{-1}^1d\\eta\n\\exp\\left(\\frac{kF^2k}{4m_0^2\\mathcal{F}}\\frac{B_c}{B}\\right)\\left[\\frac{1}{4m_0^2+8eB+\\left(k^2+\\frac{kF^2k}{2\\mathcal{F}}\\right)\\left(1-\\eta^2\\right)}-\\frac{1}{4m_0^2+8eB}\\right].\n\\label{SS23}\n\\end{equation}\n\nIn limit $x\\to 0$ we get only one expression with singularity in the\nfirst threshold\n\\begin{equation}\n\\Sigma_2^{(2)}=\\frac{2\\alpha B}{3 \\pi\nB_c}\\left(3m_0^2\\int_{0}^1d\\eta(1-\\eta^2)\\frac{\\exp\\left(\\frac{kF^2k}{4m_0^2\\mathcal{F}}\n\\frac{B_c}{B}\\right)}{4m_0^2+\\left(k^2+\\frac{kF^2k}{2\\mathcal{F}}\\right)\\left(1-\\eta^2\\right)}-1\\right).\\label{As}\n\\end{equation}\n\nBy carrying out the integration on $\\eta$ we obtain\n\\begin{equation}\n\\Sigma_2^{(2)}=\\frac{2\\alpha B}{3 \\pi\nB_c}\\left(3m_0^2\\exp\\left(\\frac{kF^2k}{4m_0^2\\mathcal{F}}\\frac{B_c}{B}\\right)\n\\left[\\frac{2}{k^2+\\frac{kF^2k}{2\\mathcal{F}}}-\\frac{8m_0^2\\arctan\\left(\\frac{k^2+\\frac{kF^2k}{2\\mathcal{F}}}{4m_0^2+k^2+\\frac{kF^2k}{2\\mathcal{F}}}\n\\right)^{1\/2}}{\\left(4m_0^2+k^2+\\frac{kF^2k}{2\\mathcal{F}}\\right)^{1\/2}\\left(k^2+\\frac{kF^2k}{2\\mathcal{F}}\\right)^{3\/2}}\\right]-1\\right).\\label{As0}\n\\end{equation}\n\nIts behavior in the low frequency limit is given by\n\\begin{equation}\n\\Sigma_2^{(2)}=\\frac{2\\alpha B}{3 \\pi\nB_c}\\left(3m_0^2\\exp\\left(\\frac{kF^2k}{4m_0^2\\mathcal{F}}\\frac{B_c}{B}\\right)\\left[\\frac{2}{k^2+\\frac{kF^2k}{2\\mathcal{F}}}-\\frac{8m_0^2}{\\left(4m_0^2+k^2+\\frac{kF^2k}{2\\mathcal{F}}\\right)\\left(k^2+\\frac{kF^2k}{2\\mathcal{F}}\\right)}\\right]-1\\right),\\label{As}\n\\end{equation}\nwhich we can express as\n\\begin{equation}\n\\Sigma_2^{(2)}=\\frac{2\\alpha B}{3 \\pi\nB_c}\\left(\\frac{3m_0^2}{k^2+\\frac{kF^2k}{2\\mathcal{F}}}\\exp\\left(\\frac{kF^2k}{4m_0^2\\mathcal{F}}\n\\frac{B_c}{B}\\right)\\left[2-\\frac{8m_0^2}{\\left(4m_0^2+k^2+\\frac{kF^2k}{2\\mathcal{F}}\\right)}\\right]-1\\right)\\label{As}\n\\end{equation}\n\\end{widetext}\ntherefore\n\\begin{equation}\n\\Sigma_2^{(2)}=\\frac{2\\alpha B}{3 \\pi\nB_c}\\left(\\frac{3}{2}\\exp\\left(\\frac{kF^2k}{4m_0^2\\mathcal{F}}\\frac{B_c}{B}\\right)-1\\right).\\label{As}\n\\end{equation}\n\nUnder such condition, by substituting the last one and the second\nexpression of (\\ref{ddd}) in (\\ref{eg1}), we have that the second\neigenvalue of the polarization operator can be written as\n\\begin{equation}\n\\pi_2=-\\frac{2\\mu^{\\prime}B}{m_0}\\left(\\frac{kF^2k}{2\\mathcal{F}}+k^2\\right)\\exp\\left(\\frac{kF^2k}{4m_0^2\\mathcal{F}}\\frac{B_c}{B}\\right).\n\\end{equation}\nIn the same approximation\n$\\frac{kF^2k}{2\\mathcal{F}}+k^2\\approx\\frac{kF^2k}{2\\mathcal{F}}$\nand\n\\begin{equation}\n\\pi_2=-\\frac{2\\mu^{\\prime}B}{m_0}\\frac{kF^2k}{2\\mathcal{F}}\\exp\\left(\\frac{kF^2k}{4m_0^2\\mathcal{F}}\\frac{B_c}{B}\\right).\n\\end{equation}\n\nNow, the behavior  $\\pi_2$ near of the first threshold can be\ndetermined from (\\ref{As0}) by taking into account  the point\n$\\omega^2-k_\\parallel=4m_0^2-\\epsilon$ with $\\epsilon>0$ and\n$\\epsilon\\rightarrow0$\n\\begin{eqnarray}\n\\pi_2=\\frac{2\\alpha m_0^3\nB}{B_c}\\exp\\left(\\frac{kF^2k}{4m_0^2\\mathcal{F}}\\frac{B_c}{B}\\right)\\left[4m_0^2+\\left(k^2+\\frac{kF^2k}{2\\mathcal{F}}\\right)\\right]^{1\/2}.\n\\end{eqnarray}\n\n\n\n\\section{Appendix B}\n\\begin{widetext}\nThe function $\\phi_{n,n^\\prime}^{(i)}$ is given by\n\\begin{equation}\n\\phi_{n,n^\\prime}^{(1)}=-\\frac{e^2}{4 \\pi^2}\\frac{e B\nk^2}{z_1}\\left[(2eB(n+n^\\prime)+z_1)F_{n,n^\\prime}^{(2)}-4k_\\perp^2N_{n,n^\\prime}^{(1)}\\right],\n\\end{equation}\n\\begin{equation}\n\\phi_{n,n^\\prime}^{(2)}=-\\frac{e^2}{4 \\pi^2}e B\n\\left[\\left(\\frac{2e^2B^2(n-n^\\prime)^2}{z_1}+2m^2+eB(n+n^\\prime)\\right)F_{n,n^\\prime}^{(1)}+2eB\n(n n^\\prime)^{1\/2} G_{n,n^\\prime}^{(1)}\\right],\n\\end{equation}\n\\begin{equation}\n\\phi_{n,n^\\prime}^{(3)}=-\\frac{e^2}{4 \\pi^2}\\frac{e B\nk^2}{z_1}\\left[(2eB(n+n^\\prime)+z_1)F_{n,n^\\prime}^{(2)}+4k_\\perp^2N_{n,n^\\prime}^{(1)}\\right],\n\\end{equation}\nwhere, calling $y=\\frac{kF^2k}{4m_0^2\\mathcal{F}}\\frac{B_c}{B}$ and\n$z_1=k^2+\\frac{kF^2k}{2\\mathcal{F}}$\n\\[\nF_{n,n^\\prime}^{(1)}=\\left\\{[L_{n^\\prime-1}^{n-n^\\prime}(y)]^2+\\frac{n^\\prime}{n}[L_{n^\\prime}^{n-n^\\prime}(y)]^2\\right\\}\\frac{(n^\\prime-1)!}{(n-1)!}y^{n-n^\\prime}\\exp[-y],\n\\]\n\\[\nF_{n,n^\\prime}^{(2,3)}=\\left\\{\\frac{y}{n}[L_{n^\\prime-1}^{n-n^\\prime+1}(y)]^2\\pm\\frac{n^\\prime}{x}[L_{n^\\prime}^{n-n^\\prime-1}(y)]^2\\right\\}\\frac{(n^\\prime-1)!}{(n-1)!}y^{n-n^\\prime}\\exp[-y],\n\\]\n\\[\nG_{n,n^\\prime}^{(1)}=2\\left(\\frac{n^\\prime}{n}\\right)^{1\/2}\\frac{(n^\\prime-1)!}{(n-1)!}y^{n-n^\\prime}L_{n^\\prime-1}^{n-n^\\prime}(y)L_{n^\\prime}^{n-n^\\prime}(y)\\exp[-y],\n\\]\n\\[\nN_{n,n^\\prime}^{(1)}=\\frac{n^\\prime!}{(n-1)!}y^{n-n^\\prime-1}L_{n^\\prime-1}^{n-n^\\prime+1}(y)L_{n}^{n-n^\\prime-1}(y)\\exp[-y].\n\\]\n\nHere $L_n^m(y)$ are generalized Laguerre polynomials. Laguerre\npolynomials with $-1$ for lower index must be taken as zero.\n\\end{widetext}\n\n\\bibliographystyle{apsrev}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:intro}\n\nConsider the problem of recovering a vector $\\mathbf{x}^0 \\in {\\mathbb{R}}^N$\nfrom noisy linear measurements of the form\n\\begin{equation} \\label{eq:yAx}\n    \\mathbf{y} = \\mathbf{A}\\mathbf{x}^0 + \\mathbf{w} \\in {\\mathbb{R}}^M ,\n   \n\\end{equation}\nwhere $\\mathbf{A}$ is a known matrix and $\\mathbf{w}$ is an unknown, unstructured noise vector.\nIn the statistics literature, this problem is known as \\emph{standard linear regression}, and in the signal processing literature this is known as \\emph{solving a linear inverse problem}, or as \\emph{compressive sensing} when $M\\ll N$ and $\\mathbf{x}^0$ is sparse.\n\n\n\\subsection{Problem Formulations} \\label{sec:formulations}\n\nOne approach to recovering $\\mathbf{x}^0$ is \\emph{regularized quadratic loss minimization},\nwhere an estimate $\\hat{\\mathbf{x}}$ of $\\mathbf{x}^0$ is computed by solving an optimization problem of the form\n\\begin{equation}\n\\hat{\\mathbf{x}}\n= \\mathop{\\mathrm{arg\\,min}}_{\\mathbf{x}\\in{\\mathbb{R}}^N} \\frac{1}{2}\\|\\mathbf{y}-\\mathbf{A}\\mathbf{x}\\|_2^2 + f(\\mathbf{x}) .\n\\label{eq:RQLM}\n\\end{equation}\nHere, the penalty function or ``regularization'' $f(\\mathbf{x})$ is chosen to promote a desired structure in $\\hat{\\mathbf{x}}$.\nFor example, the choice $f(\\mathbf{x})=\\lambda\\|\\mathbf{x}\\|_1$ with $\\lambda>0$ promotes sparsity in $\\hat{\\mathbf{x}}$.\n\nAnother approach is through the Bayesian methodology.\nHere, one presumes a prior density $p(\\mathbf{x})$ and likelihood function $p(\\mathbf{y}|\\mathbf{x})$ and then aims to compute the posterior density\n\\begin{equation} \\label{eq:Bayes}\np(\\mathbf{x}|\\mathbf{y}) = \\frac{p(\\mathbf{y}|\\mathbf{x})p(\\mathbf{x})}{\\int p(\\mathbf{y}|\\mathbf{x})p(\\mathbf{x})\\dif\\mathbf{x}}\n\\end{equation}\nor, in practice, a summary of it \\cite{pereyra2016stoch}.\nExample summaries include the maximum \\emph{a posteriori} (MAP) estimate\n\\begin{equation} \\label{eq:MAP}\n\\hat{\\mathbf{x}}_{\\text{\\sf MAP}} = \\mathop{\\mathrm{arg\\,max}}_\\mathbf{x} p(\\mathbf{x}|\\mathbf{y}) ,\n\\end{equation}\nthe minimum mean-squared error (MMSE) estimate\n\\begin{equation} \\label{eq:MMSE}\n\\hat{\\mathbf{x}}_{\\text{\\sf MMSE}} = \\mathop{\\mathrm{arg\\,min}}_{\\tilde{\\mathbf{x}}} \\int \\|\\mathbf{x}-\\tilde{\\mathbf{x}}\\|^2 p(\\mathbf{x}|\\mathbf{y}) \\dif\\mathbf{x}\n= \\mathbb{E}[\\mathbf{x}|\\mathbf{y}] ,\n\\end{equation}\nor the posterior marginal densities $\\{p(x_n|\\mathbf{y})\\}_{n=1}^N$.\n\nNote that, if the noise $\\mathbf{w}$ is modeled as\n$\\mathbf{w} \\sim {\\mathcal N}(\\mathbf{0},\\gamma_w^{-1}\\mathbf{I})$,\ni.e., additive white Gaussian noise (AWGN) with some precision $\\gamma_w>0$,\nthen\nthe regularized quadratic loss minimization problem \\eqref{eq:RQLM}\nis equivalent to\nMAP estimation under the prior\n$p(\\mathbf{x}) \\propto \\exp[ -\\gamma_w f(\\mathbf{x}) ]$,\nwhere $\\propto$ denotes equality up to a scaling that is independent of $\\mathbf{x}$.\nThus we focus on MAP, MMSE, and marginal posterior inference in the sequel.\n\n\n\\subsection{Approximate Message Passing} \\label{sec:amp}\n\nRecently, the so-called \\emph{approximate message passing} (AMP) algorithm \\cite{DonohoMM:09,DonohoMM:10-ITW1} was proposed as an iterative method to recover $\\mathbf{x}^0$ from measurements of the form \\eqref{eq:yAx}.\nThe AMP iterations are specified in Algorithm~\\ref{algo:amp}.\nThere,\\footnote{The subscript ``1'' in $\\mathbf{g}_1$ is used promote notational consistency with Vector AMP algorithm presented in the sequel.}\n$\\mathbf{g}_1(\\cdot,\\gamma_k):{\\mathbb{R}}^N\\rightarrow{\\mathbb{R}}^N$ is a \\emph{denoising} function parameterized by $\\gamma_k$,\nand $\\bkt{\\mathbf{g}_1'(\\mathbf{r}_k,\\gamma_k)}$ is its \\emph{divergence} at $\\mathbf{r}_k$.\nIn particular, $\\mathbf{g}_1'(\\mathbf{r}_k,\\gamma_k)\\in{\\mathbb{R}}^N$ is the diagonal of the Jacobian,\n\\begin{align}\n\\mathbf{g}_1'(\\mathbf{r}_k,\\gamma_k)\n= \\mathop{\\mathrm{diag}}\\left[\\frac{\\partial \\mathbf{g}_1(\\mathbf{r}_k,\\gamma_k)}{\\partial \\mathbf{r}_k}\\right]\n\\label{eq:jacobian} ,\n\\end{align}\nand $\\bkt{\\cdot}$ is the empirical averaging operation\n\\begin{align}\n\\bkt{\\mathbf{u}} := \\frac{1}{N} \\sum_{n=1}^N u_n\n\\label{eq:bkt} .\n\\end{align}\n\n\\begin{algorithm}[t]\n\\caption{AMP}\n\\begin{algorithmic}[1]  \\label{algo:amp}\n\\REQUIRE{Matrix $\\mathbf{A}\\!\\in\\!{\\mathbb{R}}^{M\\times N}$, measurement vector $\\mathbf{y}$,\ndenoiser $\\mathbf{g}_1(\\cdot,\\gamma_k)$, and number of iterations $K_{\\rm it}$.}\n\\STATE{Set $\\mathbf{v}_{-1}=\\mathbf{0}$ and select initial $\\mathbf{r}_0,\\gamma_0$.}\n\\FOR{$k=0,1,\\dots,K_{\\rm it}$}\n    \\STATE{$\\hat{\\mathbf{x}}_{k} = \\mathbf{g}_1(\\mathbf{r}_k,\\gamma_k)$}\n        \\label{line:x}\n    \\STATE{$\\alpha_k = \\bkt{\\mathbf{g}_1'(\\mathbf{r}_k,\\gamma_k)}$}\n        \\label{line:a}\n    \\STATE{$\\mathbf{v}_k = \\mathbf{y} - \\mathbf{A}\\hat{\\mathbf{x}}_k\n        + \\frac{N}{M}\\alpha_{k-1}\\mathbf{v}_{k-1}$}\n        \\label{line:v}\n    \\STATE{$\\mathbf{r}_{k\\! + \\! 1} = \\hat{\\mathbf{x}}_k + \\mathbf{A}^{\\text{\\sf T}}\\mathbf{v}_k$}\n        \\label{line:r}\n    \\STATE{Select $\\gamma_{k\\! + \\! 1}$}\n        \\label{line:gamma}\n\\ENDFOR\n\\STATE{Return $\\hat{\\mathbf{x}}_{K_{\\rm it}}$.}\n\\end{algorithmic}\n\\end{algorithm}\n\nWhen $\\mathbf{A}$ is a large i.i.d.\\ sub-Gaussian matrix,\n$\\mathbf{w}\\sim{\\mathcal N}(\\mathbf{0},\\gamma_{w0}^{-1}\\mathbf{I})$,\nand $\\mathbf{g}_1(\\cdot,\\gamma_k)$ is \\emph{separable}, i.e.,\n\\begin{align}\n[\\mathbf{g}_1(\\mathbf{r}_k,\\gamma_k)]_n\n= g_1(r_{kn},\\gamma_k)\n~ \\forall n\n\\label{eq:g1sep} ,\n\\end{align}\nwith identical Lipschitz components $g_1(\\cdot,\\gamma_k):{\\mathbb{R}}\\rightarrow {\\mathbb{R}}$,\nAMP displays a remarkable behavior, which is that $\\mathbf{r}_k$ behaves like a white-Gaussian-noise corrupted version of the true signal $\\mathbf{x}^0$ \\cite{DonohoMM:09}.\nThat is,\n\\begin{align}\n\\mathbf{r}_k\n&= \\mathbf{x}^0 + {\\mathcal N}(\\mathbf{0},\\tau_k\\mathbf{I}) ,\n\\label{eq:unbiased}\n\\end{align}\nfor some variance $\\tau_k>0$.\nMoreover, the variance $\\tau_k$ can be predicted through the following \\emph{state evolution} (SE):\n\\begin{subequations} \\label{eq:ampSE}\n\\begin{align}\n{\\mathcal E}(\\gamma_k,\\tau_k)\n&= \\frac{1}{N}\\mathbb{E}\\left[\\big\\|\\mathbf{g}_1\\big(\\mathbf{x}^0+{\\mathcal N}(\\mathbf{0},\\tau_k\\mathbf{I}),\\gamma_k\\big)-\\mathbf{x}^0\\big\\|^2\\right] \\\\\n\\tau_{k\\! + \\! 1}\n&= \\gamma_{w0}^{-1} + \\frac{N}{M}{\\mathcal E}(\\gamma_k,\\tau_k),\n\\end{align}\n\\end{subequations}\nwhere ${\\mathcal E}(\\gamma_k,\\tau_k)$ is the \\textb{MSE of the} AMP estimate $\\hat{\\mathbf{x}}_k$.\n\nThe AMP SE \\eqref{eq:ampSE} was rigorously established\nfor i.i.d.\\ Gaussian $\\mathbf{A}$ in \\cite{BayatiM:11} and for i.i.d.\\ sub-Gaussian $\\mathbf{A}$ in \\cite{BayLelMon:15}\nin the \\emph{large-system limit} (i.e., $N,M\\rightarrow\\infty$ and $N\/M\\rightarrow\\delta\\in (0,1)$) under some mild regularity conditions.\nBecause the SE \\eqref{eq:ampSE} holds for generic $g_1(\\cdot,\\gamma_k)$ and generic $\\gamma_k$-update rules, it can be used to characterize the application of AMP to many problems, as further discussed in Section~\\ref{sec:bayes}.\n\n\n\\subsection{Limitations, Modifications, and Alternatives to AMP} \\label{sec:alternatives}\n\nAn important limitation of AMP's SE is that it holds only under large i.i.d.\\ sub-Gaussian $\\mathbf{A}$.\nAlthough recent analysis \\cite{rush2016finite} has rigorously analyzed AMP's performance under finite-sized i.i.d.\\ Gaussian $\\mathbf{A}$, there remains the important question of how AMP behaves with general $\\mathbf{A}$.\n\nUnfortunately, it turns out that the AMP Algorithm~\\ref{algo:amp} is somewhat fragile with regard to the construction of $\\mathbf{A}$.\nFor example, AMP diverges with even mildly ill-conditioned or non-zero-mean $\\mathbf{A}$ \\cite{RanSchFle:14-ISIT,Caltagirone:14-ISIT,Vila:ICASSP:15}.\nAlthough damping \\cite{RanSchFle:14-ISIT,Vila:ICASSP:15}, mean-removal \\cite{Vila:ICASSP:15}, sequential updating \\cite{manoel2015swamp}, and direct free-energy minimization \\cite{rangan2015admm} all help to prevent AMP from diverging, such strategies are limited in effectiveness.\n\nMany other algorithms for standard linear regression \\eqref{eq:yAx} have been designed using approximations of belief propagation (BP) and\/or free-energy minimization.\nAmong these are the Adaptive Thouless-Anderson-Palmer (ADATAP) \\cite{opper2001adaptive}, Expectation Propagation (EP) \\cite{Minka:01,seeger2005expectation}, Expectation Consistent Approximation (EC) \\cite{OppWin:05,kabashima2014signal,fletcher2016expectation}, (S-transform AMP) S-AMP \\cite{cakmak2014samp,cakmak2015samp},\nand (Orthogonal AMP) OAMP \\cite{ma2016orthogonal} approaches.\nAlthough numerical experiments suggest that some of these algorithms are more robust than AMP Algorithm~\\ref{algo:amp} to the choice of $\\mathbf{A}$, their convergence has not been rigorously analyzed.\nIn particular, there remains the question of whether there exists an AMP-like algorithm with a rigorous SE analysis that holds for a larger class of matrices than i.i.d.\\ sub-Gaussian.\nIn the sequel, we describe one such algorithm.\n\n\n\\subsection{Contributions}\n\nIn this paper, we propose a computationally efficient iterative algorithm for the estimation of the vector $\\mathbf{x}^0$ from noisy linear measurements $\\mathbf{y}$ of the form in \\eqref{eq:yAx}.\n(See Algorithm~\\ref{algo:vampSVD}.)\nWe call the algorithm ``\\emph{vector AMP}'' (VAMP) because\ni) its behavior can be rigorously characterized by a scalar SE under large random $\\mathbf{A}$,\nand\nii) it can be derived using an approximation of BP on a factor graph with vector-valued variable nodes.\nWe outline VAMP's derivation in Section~\\ref{sec:vamp} with the aid of some background material that is reviewed in Section~\\ref{sec:back}.\n\nIn Section~\\ref{sec:SE}, we establish the VAMP SE in the case of\nlarge \\emph{right-orthogonally invariant} random $\\mathbf{A}$\nand separable Lipschitz denoisers $\\mathbf{g}_1(\\cdot,\\gamma_k)$,\nusing techniques similar to those used by Bayati and Montanari in \\cite{BayatiM:11}.\nImportantly, these right-orthogonally invariant $\\mathbf{A}$ allow arbitrary singular values and arbitrary left singular vectors, making VAMP much more robust than AMP in regards to the construction of $\\mathbf{A}$.\nIn Section~\\ref{sec:replica}, we establish that the asymptotic MSE predicted by VAMP's SE agrees with the MMSE predicted by the replica method \\cite{tulino2013support} when VAMP's priors are matched to the true data.\nFinally, in Section~\\ref{sec:num}, we present numerical experiments demonstrating that VAMP's empirical behavior matches its SE at moderate dimensions, even when $\\mathbf{A}$ is highly ill-conditioned or non-zero-mean.\n\n\n\\subsection{Relation to Existing Work}\n\nThe idea to construct algorithms from graphical models with vector-valued nodes is not new, and in fact underlies the EC- and EP-based algorithms described in~\\cite{Minka:01,seeger2005expectation,OppWin:05,kabashima2014signal,fletcher2016expectation}.\nThe use of vector-valued nodes is also central to the derivation of S-AMP~\\cite{cakmak2014samp,cakmak2015samp}.\nIn the sequel, we present a simple derivation of VAMP \\textb{that uses the EP methodology from \\cite{Minka:01,seeger2005expectation}, which passes approximate messages between the nodes of a factor graph.\nBut we note that VAMP can also be derived using the EC methodology, which formulates a variational optimization problem using a constrained version of the Kullback-Leibler distance and then relaxes the density constraints to moment constraints.\nFor more details on the latter approach, we refer the interested reader to the discussion of ``diagonal restricted EC'' in \\cite[App.~D]{OppWin:05} and ``uniform diagonalized EC'' in \\cite{fletcher2016expectation}.\n}\n\nIt was recently shown \\cite{kabashima2014signal} that, for large right-orthogonally invariant $\\mathbf{A}$, the fixed points of diagonal-restricted EC are ``good'' in the sense that they are consistent with a certain replica prediction of the MMSE that is derived in \\cite{kabashima2014signal}.\nSince the fixed points of ADATAP and S-AMP are known \\cite{cakmak2014samp} to coincide with those of diagonal-restricted EC (and thus VAMP), all of these algorithms can be understood to have good fixed points.\nThe trouble is that these algorithms do not necessarily converge to their fixed points.\nFor example, S-AMP diverges with even mildly ill-conditioned or non-zero-mean $\\mathbf{A}$, as demonstrated in Section~\\ref{sec:num}.\n\\emph{Our main contribution is establishing that VAMP's behavior can be exactly predicted by an SE analysis analogous to that for AMP.\nThis SE analysis then provides precise convergence guarantees for large right-orthogonally invariant $\\mathbf{A}$.}\nThe numerical results presented in Section~\\ref{sec:num} confirm that, in practice, VAMP's convergence is remarkably robust, even with very ill-conditioned or mean-perturbed matrices $\\mathbf{A}$ of finite dimension.\n\nThe main insight that leads to both the VAMP algorithm and its SE analysis\ncomes from a consideration of the singular value decomposition (SVD) of $\\mathbf{A}$.\nSpecifically, take the ``economy\" SVD,\n\\begin{align}\n\\mathbf{A}\n&=\\overline{\\mathbf{U}}\\mathrm{Diag}(\\overline{\\mathbf{s}})\\overline{\\mathbf{V}}^{\\text{\\sf T}}\n\\label{eq:econSVD} ,\n\\end{align}\nwhere $\\overline{\\mathbf{s}}\\in{\\mathbb{R}}^R$ for $R:=\\mathrm{rank}(\\mathbf{A})\\leq\\min(M,N)$.\nThe VAMP iterations can be performed by matrix-vector multiplications with $\\overline{\\mathbf{V}}\\in{\\mathbb{R}}^{N\\times R}$ and $\\overline{\\mathbf{V}}^{\\text{\\sf T}}$, yielding a structure\nvery similar to that of AMP.\nComputationally, the SVD form of VAMP\n(i.e., Algorithm~\\ref{algo:vampSVD})\nhas the benefit that, once the SVD has been computed,\nVAMP's per-iteration cost will be dominated by $O(RN)$ floating-point operations (flops), as opposed to $O(N^3)$ for the EC methods from\n\\cite[App.~D]{OppWin:05} or \\cite{fletcher2016expectation}.\nFurthermore, if these matrix-vector multiplications have fast implementations (e.g., $O(N)$ when $\\overline{\\mathbf{V}}$ is a discrete wavelet transform), then the complexity of VAMP reduces accordingly.\nWe emphasize that VAMP uses a single SVD, not a per-iteration SVD.\nIn many applications, this SVD can be computed off-line.\nIn the case that SVD complexity may be an issue, we note that it costs $O(MNR)$ flops\nby classical methods or $O(MN\\log R)$ by modern approaches \\cite{halko2011matrix}.\n\nThe SVD offers more than just a fast algorithmic implementation.\nMore importantly, it connects VAMP to AMP in such a way that the Bayati and Montanari's SE analysis of AMP \\cite{BayatiM:11} can be extended to obtain a rigorous SE for VAMP.\nIn this way, the SVD can be viewed as a proof technique.\nSince it will be useful for derivation\/interpretation in the sequel, we note that the VAMP iterations can also be written without an explicit SVD\n(see Algorithm~\\ref{algo:vamp}),\nin which case they coincide with the uniform-diagonalization variant of the generalized EC method from \\cite{fletcher2016expectation}.\nIn this latter implementation, the linear MMSE (LMMSE) estimate \\eqref{eq:g2slr} must be computed at each iteration, as well as the trace of its covariance matrix \\eqref{eq:a2slr}, which both involve the inverse of an $N\\times N$ matrix.\n\nThe OAMP-LMMSE algorithm from \\cite{ma2016orthogonal}\nis similar to VAMP and diagonal-restricted EC,\nbut different in that it approximates certain variance terms.\nThis difference can be seen by comparing\nequations (30)-(31) in \\cite{ma2016orthogonal} to\nlines~\\ref{line:gamtilsvd} and \\ref{line:gamsvd} in Algorithm~\\ref{algo:vampSVD}\n(or lines~\\ref{line:gam1} and \\ref{line:gam2} in Algorithm~\\ref{algo:vamp}).\nFurthermore, OAMP-LMMSE differs from VAMP in its reliance on matrix inversion (see, e.g., the comments in the Conclusion of \\cite{ma2016orthogonal}).\n\n\n\\textb{Shortly after the initial publication of this work, \\cite{takeuchi2017rigorous} proved a very similar result for the complex case using a fully probabilistic analysis.}\n\n\\subsection{Notation}\n\nWe use\ncapital boldface letters like $\\mathbf{A}$ for matrices,\nsmall boldface letters like $\\mathbf{a}$ for vectors,\n$(\\cdot)^{\\text{\\sf T}}$ for transposition,\nand $a_n=[\\mathbf{a}]_n$ to denote the $n$th element of $\\mathbf{a}$.\nAlso, we use\n$\\|\\mathbf{a}\\|_p=(\\sum_n |a_n|^p)^{1\/p}$ for the $\\ell_p$ norm of $\\mathbf{a}$,\n$\\|\\mathbf{A}\\|_2$ for the spectral norm of $\\mathbf{A}$,\n$\\mathrm{Diag}(\\mathbf{a})$ for the diagonal matrix created from vector $\\mathbf{a}$, and\n$\\mathop{\\mathrm{diag}}(\\mathbf{A})$ for the vector extracted from the diagonal of matrix $\\mathbf{A}$.\nLikewise, we use\n$\\mathbf{I}_N$ for the $N\\times N$ identity matrix,\n$\\mathbf{0}$ for the matrix of all zeros, and\n$\\mathbf{1}$ for the matrix of all ones.\nFor a random vector $\\mathbf{x}$, we denote\nits probability density function (pdf) by $p(\\mathbf{x})$,\nits expectation by $\\mathbb{E}[\\mathbf{x}]$,\nand\nits covariance matrix by $\\mathrm{Cov}[\\mathbf{x}]$.\nSimilarly, we use\n$p(\\mathbf{x}|\\mathbf{y})$, $\\mathbb{E}[\\mathbf{x}|\\mathbf{y}]$, and $\\mathrm{Cov}[\\mathbf{x}|\\mathbf{y}]$ for the\n\\emph{conditional} pdf, expectation, and covariance, respectively.\nAlso, we use\n$\\mathbb{E}[\\mathbf{x}|b]$ and $\\mathrm{Cov}[\\mathbf{x}|b]$ to denote the\nexpectation and covariance of $\\mathbf{x}\\sim b(\\mathbf{x})$,\ni.e., $\\mathbf{x}$ distributed according to the pdf $b(\\mathbf{x})$.\nWe refer to\nthe Dirac delta pdf using $\\delta(\\mathbf{x})$\nand to\nthe pdf of a Gaussian random vector $\\mathbf{x}\\in{\\mathbb{R}}^N$ with mean $\\mathbf{a}$ and covariance $\\mathbf{C}$ using ${\\mathcal N}(\\mathbf{x};\\mathbf{a},\\mathbf{C})=\\exp( -(\\mathbf{x}-\\mathbf{a})^{\\text{\\sf T}}\\mathbf{C}^{-1}(\\mathbf{x}-\\mathbf{a})\/2 )\/\\sqrt{(2\\pi)^N|\\mathbf{C}|}$.\nFinally,\n$p(\\mathbf{x})\\propto f(\\mathbf{x})$ says that functions $p(\\cdot)$ and $f(\\cdot)$ are equal up to a scaling that is invariant to $\\mathbf{x}$.\n\n\n\\section{Background on the AMP Algorithm} \\label{sec:back}\n\nIn this section, we provide background on the AMP algorithm that will be useful in the sequel.\n\n\n\\subsection{Applications to Bayesian Inference} \\label{sec:bayes}\n\nWe first detail the application of the AMP Algorithm~\\ref{algo:amp} to the Bayesian inference problems from Section~\\ref{sec:formulations}.\nSuppose that the prior on $\\mathbf{x}$ is i.i.d., so that it takes the form\n\\begin{equation}\np(\\mathbf{x})=\\prod_{n=1}^N p(x_n)\n\\label{eq:pxiid} .\n\\end{equation}\nThen AMP can be applied to MAP problem \\eqref{eq:MAP} by choosing the scalar denoiser as\n\\begin{equation}\ng_1(r_{kn},\\gamma_k)\n= \\mathop{\\mathrm{arg\\,min}}_{x_n\\in{\\mathbb{R}}} \\left[ \\frac{\\gamma_k}{2}|x_n-r_{kn}|^2 - \\ln p(x_n) \\right]\n\\label{eq:gmapsca} .\n\\end{equation}\nLikewise, AMP can be applied to the MMSE problem \\eqref{eq:MMSE} by choosing\n\\begin{equation}\ng_1(r_{kn},\\gamma_k)\n= \\mathbb{E}[x_n|r_{kn},\\gamma_k]\n\\label{eq:g1mmsesca} ,\n\\end{equation}\nwhere the expectation in \\eqref{eq:g1mmsesca} is with respect to the conditional density\n\\begin{align}\np(x_n|r_{kn},\\gamma_k)\n\\propto \\exp\\left[ -\\frac{\\gamma_k}{2}|r_{kn}-x_n|^2 +\\ln p(x_n) \\right]\n\\label{eq:pxr1sca} .\n\\end{align}\nIn addition, $p(x_n|r_{kn},\\gamma_k)$ in \\eqref{eq:pxr1sca} acts as AMP's iteration-$k$ approximation of the marginal posterior $p(x_n|\\mathbf{y})$.\nFor later use, we note that the derivative of the MMSE scalar denoiser \\eqref{eq:g1mmsesca} w.r.t.\\ its first argument can be expressed as\n\\begin{equation}\n    g_1'(r_{kn},\\gamma_k) = \\gamma_k\\mathrm{var}\\left[ x_n | r_{kn},\\gamma_k \\right]\n\\label{eq:g1dervar} ,\n\\end{equation}\nwhere the variance is computed with respect to the density~\\eqref{eq:pxr1sca}\n(see, e.g., \\cite{Rangan:11-ISIT}).\n\nIn \\eqref{eq:gmapsca}-\\eqref{eq:pxr1sca}, $\\gamma_k$ can be interpreted as an estimate of $\\tau_k^{-1}$, the iteration-$k$ precision of $\\mathbf{r}_k$ from \\eqref{eq:unbiased}.\nIn the case that $\\tau_k$ is known, the ``matched'' assignment\n\\begin{align}\n\\gamma_k=\\tau_k^{-1}\n\\label{eq:gamma_matched}\n\\end{align}\nleads to the interpretation of \\eqref{eq:gmapsca} and \\eqref{eq:g1mmsesca} as the scalar MAP and MMSE denoisers of $r_{kn}$, respectively.\nSince, in practice, $\\tau_k$ is usually not known, it has been suggested to use\n\\begin{align}\n\\gamma_{k\\! + \\! 1} &= \\frac{M}{\\|\\mathbf{v}_k\\|^2}\n\\label{eq:gamma_amp_practical} ,\n\\end{align}\nalthough other choices are possible \\cite{Montanari:12-bookChap}.\n\n\n\\subsection{Relation of AMP to IST} \\label{sec:ista}\n\nThe AMP Algorithm~\\ref{algo:amp} is closely related to the well-known \\emph{iterative soft thresholding} (IST) algorithm \\cite{ChamDLL:98,DaubechiesDM:04} that can be used\\footnote{The IST algorithm is guaranteed to converge \\cite{DaubechiesDM:04} when $\\|\\mathbf{A}\\|_2<1$.} to solve \\eqref{eq:RQLM} with convex $f(\\cdot)$.\nIn particular, if the term\n\\begin{align}\n\\frac{N}{M}\\alpha_{k-1}\\mathbf{v}_{k-1}\n\\label{eq:onsager}\n\\end{align}\nis removed from line~\\ref{line:v} of Algorithm~\\ref{algo:amp}, then what remains is the IST algorithm.\n\nThe term \\eqref{eq:onsager} is known as the \\emph{Onsager} term in the statistical physics literature \\cite{ThoulessAP:77}.\nUnder large i.i.d.\\ sub-Gaussian $\\mathbf{A}$, the Onsager correction ensures the behavior in \\eqref{eq:unbiased}.\nWhen \\eqref{eq:unbiased} holds, the denoiser $g_{1}(\\cdot,\\gamma_k)$ can be optimized accordingly, in which case each iteration of AMP becomes very productive.\nAs a result, AMP converges much faster than ISTA for i.i.d.\\ Gaussian $\\mathbf{A}$ (see, e.g., \\cite{Montanari:12-bookChap} for a comparison).\n\n\n\\subsection{Derivations of AMP} \\label{sec:amp_deriv}\n\nThe AMP algorithm can be derived in several ways.\nOne way is through approximations of loopy belief propagation (BP) \\cite{Pearl:88,YedidiaFW:03} on a bipartite factor graph constructed from the factorization\n\\begin{align}\np(\\mathbf{y},\\mathbf{x})\n&= \\left[ \\prod_{m=1}^M {\\mathcal N}(y_m;\\mathbf{a}_m^{\\text{\\sf T}}\\mathbf{x},\\gamma_w^{-1}) \\right] \\left[ \\prod_{n=1}^N p(x_n) \\right]\n\\label{eq:amp_factors} ,\n\\end{align}\nwhere $\\mathbf{a}_m^{\\text{\\sf T}}$ denotes the $m$th row of $\\mathbf{A}$.\nWe refer the reader to \\cite{DonohoMM:10-ITW1,Rangan:11-ISIT} for details on the message-passing derivation of AMP, noting connections to the general framework of \\emph{expectation propagation} (EP) \\cite{Minka:01,seeger2005expectation}.\nAMP can also be derived through a ``free-energy'' approach, where one\ni) proposes a cost function, ii) derives conditions on its stationary points, and iii) constructs an algorithm whose fixed points coincide with those stationary points.\nWe refer the reader to \\cite{RanSRFC:13-ISIT,Krzakala:14-ISITbethe,cakmak2014samp} for details, and note connections to the general framework of \\emph{expectation consistent approximation} (EC) \\cite{OppWin:05,fletcher2016expectation}.\n\n\n\n\n\\section{The Vector AMP Algorithm} \\label{sec:vamp}\n\nThe \\emph{Vector AMP} (VAMP) algorithm is stated in Algorithm~\\ref{algo:vampSVD}.\nIn line~\\ref{line:dsvd}, ``$\\overline{\\mathbf{s}}^2$'' refers to the componentwise square of vector $\\overline{\\mathbf{s}}$.\nAlso, $\\mathrm{Diag}(\\mathbf{a})$ denotes the diagonal matrix whose diagonal components are given by the vector $\\mathbf{a}$.\n\n\\begin{algorithm}[t]\n\\caption{Vector AMP (SVD Form)}\n\\begin{algorithmic}[1]  \\label{algo:vampSVD}\n\\REQUIRE{\nMatrix $\\mathbf{A} \\in {\\mathbb{R}}^{M \\times N}$; measurements $\\mathbf{y} \\in {\\mathbb{R}}^M$;\ndenoiser $\\mathbf{g}_1(\\cdot,\\gamma_k)$;\nassumed noise precision $\\gamma_w \\geq 0$; and\nnumber of iterations $K_{\\rm it}$. }\n\\STATE{Compute economy SVD $\\overline{\\mathbf{U}}\\mathrm{Diag}(\\overline{\\mathbf{s}})\\overline{\\mathbf{V}}^{\\text{\\sf T}}=\\mathbf{A}$\nwith $\\overline{\\mathbf{U}}^{\\text{\\sf T}}\\overline{\\mathbf{U}}=\\mathbf{I}_R$,\n$\\overline{\\mathbf{V}}^{\\text{\\sf T}}\\overline{\\mathbf{V}}=\\mathbf{I}_R$,\n$\\overline{\\mathbf{s}}\\in{\\mathbb{R}}_+^{R}$},\n$R=\\mathrm{rank}(\\mathbf{A})$.\n\\STATE{Compute preconditioned $\\tilde{\\mathbf{y}}:=\\mathrm{Diag}(\\overline{\\mathbf{s}})^{-1}\\overline{\\mathbf{U}}^{\\text{\\sf T}}\\mathbf{y}$}\n\\STATE{Select initial $\\mathbf{r}_{0}$ and $\\gamma_{0}\\geq 0$.}\n\\FOR{$k=0,1,\\dots,K_{\\rm it}$}\n    \\STATE{$\\hat{\\mathbf{x}}_{k} = \\mathbf{g}_1(\\mathbf{r}_{k},\\gamma_{k})$}\n        \\label{line:xsvd}\n    \\STATE{$\\alpha_{k} = \\bkt{ \\mathbf{g}_1'(\\mathbf{r}_{k},\\gamma_{k}) }$}\n        \\label{line:asvd}\n    \\STATE{$\\tilde{\\mathbf{r}}_k = (\\hat{\\mathbf{x}}_{k} - \\alpha_{k}\\mathbf{r}_{k})\/(1-\\alpha_{k})$}\n        \\label{line:xtilsvd}\n    \\STATE{$\\tilde{\\gamma}_{k} = \\gamma_{k}(1-\\alpha_{k})\/\\alpha_{k}$}\n        \\label{line:gamtilsvd}\n    \\STATE{$\\mathbf{d}_k=\\gamma_w\\mathrm{Diag}\\big(\\gamma_w\\overline{\\mathbf{s}}^2+\\tilde{\\gamma}_{k}\\mathbf{1}\\big)^{-1}\\overline{\\mathbf{s}}^2$}\n        \\label{line:dsvd}\n    \\STATE{$\\gamma_{k\\! + \\! 1} = \\tilde{\\gamma}_{k}\\bkt{\\mathbf{d}_k}\/(\\frac{N}{R}-\\bkt{\\mathbf{d}_k})$}\n        \\label{line:gamsvd}\n    \\STATE{$\\mathbf{r}_{k\\! + \\! 1} = \\tilde{\\mathbf{r}}_k + \\frac{N}{R}\n        \\overline{\\mathbf{V}}\\mathrm{Diag}\\big(\\mathbf{d}_k\/\\bkt{\\mathbf{d}_k}\\big)\\big(\\tilde{\\mathbf{y}}-\\overline{\\mathbf{V}}^{\\text{\\sf T}}\\tilde{\\mathbf{r}}_k\\big)$}\n        \\label{line:rsvd}\n\\ENDFOR\n\\STATE{Return $\\hat{\\mathbf{x}}_{K_{\\rm it}}$.}\n\\end{algorithmic}\n\\end{algorithm}\n\n\\subsection{Relation of VAMP to AMP} \\label{sec:vamp2amp}\n\nA visual examination of VAMP Algorithm~\\ref{algo:vampSVD} shows many similarities with AMP Algorithm~\\ref{algo:amp}.\nIn particular, the denoising and divergence steps in lines~\\ref{line:xsvd}-\\ref{line:asvd} of Algorithm~\\ref{algo:vampSVD} are identical to those in lines~\\ref{line:x}-\\ref{line:a} of Algorithm~\\ref{algo:amp}.\nLikewise, an Onsager term $\\alpha_k\\mathbf{r}_k$ is visible in line~\\ref{line:xtilsvd} of Algorithm~\\ref{algo:vampSVD}, analogous to the one in line~\\ref{line:v} of Algorithm~\\ref{algo:amp}.\nFinally, the per-iteration computational complexity of each algorithm is dominated by two matrix-vector multiplications: those involving $\\mathbf{A}$ and $\\mathbf{A}^{\\text{\\sf T}}$ in Algorithm~\\ref{algo:amp} and those involving $\\overline{\\mathbf{V}}$ and $\\overline{\\mathbf{V}}^{\\text{\\sf T}}$ in Algorithm~\\ref{algo:vampSVD}.\n\nThe most important similarity between the AMP and VAMP algorithms is not obvious from visual inspection and will be established rigorously in the sequel.\nIt is the following: for certain large random $\\mathbf{A}$,\nthe VAMP quantity $\\mathbf{r}_k$ behaves\nlike a white-Gaussian-noise corrupted version of the true signal $\\mathbf{x}^0$, i.e.,\n\\begin{align}\n\\mathbf{r}_k\n&= \\mathbf{x}^0 + {\\mathcal N}(\\mathbf{0},\\tau_k\\mathbf{I}) ,\n\\label{eq:unbiased2}\n\\end{align}\nfor some variance $\\tau_k>0$.\nMoreover, the noise variance $\\tau_k$ can be tracked through a scalar SE formalism whose details will be provided in the sequel.\nFurthermore, the VAMP quantity $\\gamma_k$ can be interpreted as an estimate of $\\tau_k^{-1}$ in \\eqref{eq:unbiased2}, analogous to the AMP quantity $\\gamma_k$ discussed around \\eqref{eq:gamma_matched}.\n\nIt should be emphasized that the class of matrices $\\mathbf{A}$ under which the VAMP SE holds is much bigger than the class under which the AMP SE holds.\nIn particular, VAMP's SE holds for large random matrices $\\mathbf{A}$ whose right singular%\n  \\footnote{We use several forms of SVD in this paper.\n  Algorithm~\\ref{algo:vampSVD} uses the ``economy'' SVD\n  $\\mathbf{A}=\\overline{\\mathbf{U}}\\mathrm{Diag}(\\overline{\\mathbf{s}})\\overline{\\mathbf{V}}^{\\text{\\sf T}}\\in{\\mathbb{R}}^{M\\times N}$,\n  where $\\overline{\\mathbf{s}}\\in{\\mathbb{R}}_+^R$ with $R=\\mathrm{rank}(\\mathbf{A})$, so that\n  $\\overline{\\mathbf{U}}$ and\/or $\\overline{\\mathbf{V}}$ may be tall.\n  The discussion in Section~\\ref{sec:vamp2amp} uses the ``standard'' SVD\n  $\\mathbf{A}=\\mathbf{U}\\mathbf{S}\\mathbf{V}^{\\text{\\sf T}}$, where $\\mathbf{S}\\in{\\mathbb{R}}^{M\\times N}$ and both\n  $\\mathbf{U}$ and $\\mathbf{V}$ are orthogonal.\n  Finally, the state-evolution proof in Section~\\ref{sec:SE} uses the\n  standard SVD on square $\\mathbf{A}\\in{\\mathbb{R}}^{N\\times N}$.}\nvector matrix $\\mathbf{V}\\in{\\mathbb{R}}^{N\\times N}$ is uniformly distributed on the group of orthogonal matrices.\nNotably, VAMP's SE holds for \\emph{arbitrary} (i.e., deterministic) left singular vector matrices $\\mathbf{U}$ and singular values, apart from some mild regularity conditions that will be detailed in the sequel.\nIn contrast, AMP's SE is known to hold \\cite{BayatiM:11,BayLelMon:15} only for large i.i.d.\\ sub-Gaussian matrices $\\mathbf{A}$, which implies i) random orthogonal $\\mathbf{U}$ and $\\mathbf{V}$ and ii) a particular distribution on the singular values of $\\mathbf{A}$.\n\n\\subsection{EP Derivation of VAMP}\n\n\\textb{As with AMP (i.e., Algorithm~\\ref{algo:amp}), VAMP (i.e., Algorithm~\\ref{algo:vampSVD}) can be derived in many ways.}\nHere we present a very simple derivation based on an EP-like approximation of the sum-product (SP) belief-propagation algorithm.\nUnlike the AMP algorithm, whose message-passing derivation uses a loopy factor graph with \\emph{scalar}-valued nodes, the VAMP algorithm uses a non-loopy graph with \\emph{vector}-valued nodes, hence the name ``vector AMP.''\nWe note that VAMP can also be derived using the ``diagonal restricted'' or ``uniform diagonalization'' EC approach \\cite{OppWin:05,fletcher2016expectation},\nbut that derivation is much more complicated.\n\nTo derive VAMP, we start with the factorization\n\\begin{align}\np(\\mathbf{y},\\mathbf{x})\n&= p(\\mathbf{x}) {\\mathcal N}(\\mathbf{y};\\mathbf{A}\\mathbf{x},\\gamma_w^{-1}\\mathbf{I}) ,\n\\end{align}\nand split $\\mathbf{x}$ into two identical variables $\\mathbf{x}_1=\\mathbf{x}_2$, giving an equivalent factorization\n\\begin{align}\np(\\mathbf{y},\\mathbf{x}_1,\\mathbf{x}_2)\n&= p(\\mathbf{x}_1) \\delta(\\mathbf{x}_1-\\mathbf{x}_2) {\\mathcal N}(\\mathbf{y};\\mathbf{A}\\mathbf{x}_2,\\gamma_w^{-1}\\mathbf{I})\n\\label{eq:vamp_factors} ,\n\\end{align}\nwhere $\\delta(\\cdot)$ is the Dirac delta distribution.\nThe factor graph corresponding to \\eqref{eq:vamp_factors} is shown in Figure~\\ref{fig:fg_split}.\n\\begin{figure}[t]\n  \\centering\n  \\newcommand{0.8}{0.8}\n  \\psfrag{px}[b][Bl][0.8]{$p(\\mathbf{x}_1)$}\n  \\psfrag{x1}[t][Bl][0.8]{$\\mathbf{x}_1$}\n  \\psfrag{del}[b][Bl][0.8]{$\\delta(\\mathbf{x}_1-\\mathbf{x}_2)$}\n  \\psfrag{x2}[t][Bl][0.8]{$\\mathbf{x}_2$}\n  \\psfrag{py|x}[b][Bl][0.8]{${\\mathcal N}(\\mathbf{y};\\mathbf{A}\\mathbf{x}_2,\\gamma_w^{-1}\\mathbf{I})$}\n  \\includegraphics[width=2.0in]{figures\/fg_split.eps}\n  \\caption{The factor graph used for the derivation of VAMP.\n           The circles represent variable nodes and\n           the squares represent factor nodes from \\eqref{eq:vamp_factors}.}\n  \\label{fig:fg_split}\n\\end{figure}\nWe then pass messages on this factor graph according to the following rules.\n\\begin{enumerate}\n\\item \\label{rule:b}\n\\emph{\\underline{Approximate beliefs}:}\nThe approximate belief $b_{\\textsf{app}}(\\mathbf{x})$ on variable node $\\mathbf{x}$\nis ${\\mathcal N}(\\mathbf{x};\\hat{\\mathbf{x}},\\eta^{-1}\\mathbf{I})$, where\n$\\hat{\\mathbf{x}} = \\mathbb{E}[\\mathbf{x}|b_{\\textsf{sp}}]$ and\n$\\eta^{-1} = \\bkt{\\mathop{\\mathrm{diag}}(\\mathrm{Cov}[\\mathbf{x}|b_{\\textsf{sp}}])}$\nare the mean and average variance of the corresponding SP belief\n$b_{\\textsf{sp}}(\\mathbf{x}) \\propto \\prod_i \\msg{f_i}{\\mathbf{x}}(\\mathbf{x})$,\ni.e., the normalized product of all messages impinging on the node.\nSee Figure~\\ref{fig:ep_rules}(a) for an illustration.\n\n\\item \\label{rule:v2f}\n\\emph{\\underline{Variable-to-factor messages}:}\nThe message from\na variable node $\\mathbf{x}$ to a connected factor node $f_i$ is\n$\\msg{\\mathbf{x}}{f_i}(\\mathbf{x}) \\propto b_{\\textsf{app}}(\\mathbf{x})\/\\msg{f_i}{\\mathbf{x}}(\\mathbf{x})$,\ni.e., the ratio of the most recent approximate belief $b_{\\textsf{app}}(\\mathbf{x})$ to\nthe most recent message from $f_i$ to $\\mathbf{x}$.\nSee Figure~\\ref{fig:ep_rules}(b) for an illustration.\n\n\\item \\label{rule:f2v}\n\\emph{\\underline{Factor-to-variable messages}:}\nThe message from a factor node $f$ to a connected variable node\n$\\mathbf{x}_i$ is\n$\\msg{f}{\\mathbf{x}_i}(\\mathbf{x}_i)\\propto\n \\int f(\\mathbf{x}_i,\\{\\mathbf{x}_j\\}_{j\\neq i}\\}) \\prod_{j\\neq i} \\msg{\\mathbf{x}_j}{f}(\\mathbf{x}_j) \\dif\\mathbf{x}_j$.\nSee Figure~\\ref{fig:ep_rules}(c) for an illustration.\n\\end{enumerate}\n\\begin{figure}[t]\n  \\centering\n  \\newcommand{0.8}{0.8}\n  \\newcommand{0.6}{0.6}\n  \\psfrag{x}[b][Bl][0.8]{$\\mathbf{x}$}\n  \\psfrag{x1}[b][Bl][0.8]{$\\mathbf{x}_2$}\n  \\psfrag{x2}[b][Bl][0.8]{$\\mathbf{x}_3$}\n  \\psfrag{y}[b][Bl][0.8]{$\\mathbf{x}_1$}\n  \\psfrag{f}[bl][Bl][0.7]{$f(\\mathbf{x}_1,\\mathbf{x}_2,\\mathbf{x}_3)$}\n  \\psfrag{f1}[b][Bl][0.8]{$f_1(\\mathbf{x})$}\n  \\psfrag{f2}[b][Bl][0.8]{$f_2(\\mathbf{x})$}\n  \\psfrag{f3}[b][Bl][0.8]{$f_3(\\mathbf{x})$}\n  \\psfrag{m1}[t][Bl][0.6]{$\\msg{f_1}{\\mathbf{x}}(\\mathbf{x})$}\n  \\psfrag{m2}[b][Bl][0.6]{$\\msg{f_2}{\\mathbf{x}}(\\mathbf{x})$}\n  \\psfrag{m3}[t][Bl][0.6]{$\\msg{f_3}{\\mathbf{x}}(\\mathbf{x})$}\n  \\psfrag{m6}[t][Bl][0.6]{$\\msg{\\mathbf{x}}{f_1}(\\mathbf{x})$}\n  \\psfrag{m7}[b][Bl][0.6]{$\\msg{\\mathbf{x}_2}{f}(\\mathbf{x}_2)$}\n  \\psfrag{m8}[t][Bl][0.6]{$\\msg{\\mathbf{x}_3}{f}(\\mathbf{x}_3)$}\n  \\psfrag{m9}[t][Bl][0.6]{$\\msg{f}{\\mathbf{x}_1}(\\mathbf{x}_1)$}\n  \\psfrag{a}[t][Bl][0.8]{(a)}\n  \\psfrag{b}[t][Bl][0.8]{(b)}\n  \\psfrag{c}[t][Bl][0.8]{(c)}\n  \\includegraphics[width=3.2in]{figures\/ep_rules}\n  \\caption{Factor graphs to illustrate\n           (a) messaging through a factor node and\n           (b) messaging through a variable node.}\n  \\label{fig:ep_rules}\n\\end{figure}\n\nBy applying the above message-passing rules to the factor graph in Figure~\\ref{fig:fg_split}, one obtains Algorithm~\\ref{algo:vamp}.\n(See Appendix~\\ref{sec:EP} for a detailed derivation.)\nLines~\\ref{line:x2}--\\ref{line:a2} of Algorithm~\\ref{algo:vamp} use\n\\begin{align}\n\\mathbf{g}_2(\\mathbf{r}_{2k},\\gamma_{2k})\n&:= \\left( \\gamma_w \\mathbf{A}^{\\text{\\sf T}}\\mathbf{A} + \\gamma_{2k}\\mathbf{I}\\right)^{-1}\n        \\left( \\gamma_w\\mathbf{A}^{\\text{\\sf T}}\\mathbf{y} + \\gamma_{2k}\\mathbf{r}_{2k} \\right)\n\\label{eq:g2slr} ,\n\\end{align}\nwhich can be recognized as the\nMMSE estimate of a random vector $\\mathbf{x}_2$ under\nlikelihood ${\\mathcal N}(\\mathbf{y};\\mathbf{A}\\mathbf{x}_2,\\gamma_w^{-1}\\mathbf{I})$\nand prior $\\mathbf{x}_2\\sim {\\mathcal N}(\\mathbf{r}_{2k},\\gamma_{2k}^{-1}\\mathbf{I})$.\nSince this estimate is linear in $\\mathbf{r}_{2k}$,\nwe will refer to it as the ``LMMSE'' estimator.\nFrom \\eqref{eq:jacobian}-\\eqref{eq:bkt} and \\eqref{eq:g2slr}, it follows that\nline~\\ref{line:a2} of Algorithm~\\ref{algo:vamp} uses\n\\begin{align}\n\\bkt{\\mathbf{g}'_2(\\mathbf{r}_{2k},\\gamma_{2k})}\n&= \\frac{\\gamma_{2k}}{N} \\mathrm{Tr}\\left[\n        \\left( \\gamma_w \\mathbf{A}^{\\text{\\sf T}}\\mathbf{A} + \\gamma_{2k}\\mathbf{I}\\right)^{-1} \\right]\n\\label{eq:a2slr} .\n\\end{align}\n\n\\textb{%\nAlgorithm~\\ref{algo:vamp} is merely a restatement of VAMP Algorithm~\\ref{algo:vampSVD}.\nTheir equivalence can then be seen by substituting the ``economy'' SVD $\\mathbf{A}=\\overline{\\mathbf{U}}\\mathrm{Diag}(\\overline{\\mathbf{s}})\\overline{\\mathbf{V}}^{\\text{\\sf T}}$ into Algorithm~\\ref{algo:vamp}, simplifying, and equating\n$\\hat{\\mathbf{x}}_k \\equiv \\hat{\\mathbf{x}}_{1k}$,\n$\\mathbf{r}_k \\equiv \\mathbf{r}_{1k}$,\n$\\gamma_k \\equiv \\gamma_{1k}$,\n$\\tilde{\\gamma}_k \\equiv \\gamma_{2k}$,\nand\n$\\alpha_k \\equiv \\alpha_{1k}$.\n}\n\nAs presented in Algorithm~\\ref{algo:vamp}, the steps of VAMP exhibit an elegant symmetry.\nThe first half of the steps perform denoising on $\\mathbf{r}_{1k}$ and then Onsager correction in $\\mathbf{r}_{2k}$, while the second half of the steps perform LMMSE estimation $\\mathbf{r}_{2k}$ and Onsager correction in $\\mathbf{r}_{1,k\\! + \\! 1}$.\n\n\n\\begin{algorithm}[t]\n\\caption{Vector AMP (LMMSE form)}\n\\begin{algorithmic}[1]  \\label{algo:vamp}\n\\REQUIRE{\nLMMSE estimator\n$\\mathbf{g}_2(\\mathbf{r}_{2k},\\gamma_{2k})$ from \\eqref{eq:g2slr},\ndenoiser $\\mathbf{g}_1(\\cdot,\\gamma_{1k})$,\nand\nnumber of iterations $K_{\\rm it}$.  }\n\\STATE{ Select initial $\\mathbf{r}_{10}$ and $\\gamma_{10}\\geq 0$.}\n\\FOR{$k=0,1,\\dots,K_{\\rm it}$}\n    \\STATE{\/\/ Denoising }\n    \\STATE{$\\hat{\\mathbf{x}}_{1k} = \\mathbf{g}_1(\\mathbf{r}_{1k},\\gamma_{1k})$}\n        \\label{line:x1}\n    \\STATE{$\\alpha_{1k} = \\bkt{ \\mathbf{g}_1'(\\mathbf{r}_{1k},\\gamma_{1k}) }$}\n        \\label{line:a1}\n    \\STATE{$\\eta_{1k} = \\gamma_{1k}\/\\alpha_{1k}$}\n        \\label{line:eta1}\n    \\STATE{$\\gamma_{2k} = \\eta_{1k} - \\gamma_{1k}$}\n        \\label{line:gam2}\n    \\STATE{$\\mathbf{r}_{2k} = (\\eta_{1k}\\hat{\\mathbf{x}}_{1k} - \\gamma_{1k}\\mathbf{r}_{1k})\/\\gamma_{2k}$}\n        \\label{line:r2}\n    \\STATE{ }\n    \\STATE{\/\/ LMMSE estimation }\n    \\STATE{$\\hat{\\mathbf{x}}_{2k} = \\mathbf{g}_2(\\mathbf{r}_{2k},\\gamma_{2k})$}\n        \\label{line:x2}\n    \\STATE{$\\alpha_{2k} = \\bkt{ \\mathbf{g}_2'(\\mathbf{r}_{2k},\\gamma_{2k}) } $}\n        \\label{line:a2}\n    \\STATE{$\\eta_{2k} = \\gamma_{2k}\/\\alpha_{2k}$}\n        \\label{line:eta2}\n    \\STATE{$\\gamma_{1,k\\! + \\! 1} = \\eta_{2k} - \\gamma_{2k}$}\n        \\label{line:gam1}\n    \\STATE{$\\mathbf{r}_{1,k\\! + \\! 1} = (\\eta_{2k}\\hat{\\mathbf{x}}_{2k} - \\gamma_{2k}\\mathbf{r}_{2k})\/\\gamma_{1,k\\! + \\! 1}$}\n        \\label{line:r1}\n\\ENDFOR\n\\STATE{Return $\\hat{\\mathbf{x}}_{1K_{\\rm it}}$.}\n\\end{algorithmic}\n\\end{algorithm}\n\n\\subsection{Implementation Details} \\label{sec:implementation}\n\nFor practical implementation with finite-dimensional $\\mathbf{A}$, we find that it helps to make some small enhancements to VAMP.\nIn the discussion below we will refer to Algorithm~\\ref{algo:vampSVD}, but the same approaches apply to Algorithm~\\ref{algo:vamp}.\n\nFirst, we suggest to clip the precisions $\\gamma_k$ and $\\tilde{\\gamma}_k$ to a positive interval $[\\gamma_{\\min},\\gamma_{\\max}]$.\nIt is possible, though uncommon, for line~\\ref{line:asvd} of Algorithm~\\ref{algo:vampSVD} to return a negative $\\alpha_k$, which will lead to negative $\\gamma_k$ and $\\tilde{\\gamma}_k$ if not accounted for.\nFor the numerical results in Section~\\ref{sec:num}, we used $\\gamma_{\\min}=1\\times 10^{-11}$ and $\\gamma_{\\max}=1\\times 10^{11}$.\n\nSecond, we find that a small amount of damping can be helpful when $\\mathbf{A}$ is highly ill-conditioned.\nIn particular, we suggest to replace lines~\\ref{line:xsvd} and \\ref{line:gamsvd} of Algorithm~\\ref{algo:vampSVD} with the damped versions\n\\begin{align}\n\\hat{\\mathbf{x}}_k\n&= \\rho \\mathbf{g}_1(\\mathbf{r}_k,\\gamma_k) + (1-\\rho) \\hat{\\mathbf{x}}_{k\\! - \\! 1}\n\\label{eq:xdamp}\\\\\n\\gamma_{k\\! + \\! 1}\n&= \\rho \\tilde{\\gamma}_{k}\\bkt{\\mathbf{d}_k}R\/(N-\\bkt{\\mathbf{d}_k}R) + (1-\\rho) \\gamma_k\n\\label{eq:gamdamp}\n\\end{align}\nfor all iterations $k>1$, where $\\rho\\in(0,1]$ is a suitably chosen damping parameter.\nNote that, when $\\rho=1$, the damping has no effect.\nFor the numerical results in Section~\\ref{sec:num}, we used $\\rho=0.97$.\n\nThird, rather than requiring VAMP to complete $K_{\\rm it}$ iterations, we suggest that the iterations are stopped when the normalized difference $\\|\\mathbf{r}_{1k}-\\mathbf{r}_{1,k\\! - \\! 1}\\|\/\\|\\mathbf{r}_{1k}\\|$ falls below a tolerance $\\tau$.\nFor the numerical results in Section~\\ref{sec:num}, we used $\\tau=1\\times 10^{-4}$.\n\n\\textb{We note that the three minor modifications described above are standard features of many AMP implementations, such as the one in the GAMPmatlab toolbox \\cite{GAMP-code}.  However, as discussed in Section~\\ref{sec:alternatives}, they are not enough to stabilize AMP for in the case of ill-conditioned or non-zero-mean $\\mathbf{A}$.}\n\nFinally, we note that the VAMP algorithm requires the user to choose\nthe measurement-noise precision $\\gamma_w$ and\nthe denoiser $\\mathbf{g}_1(\\cdot,\\gamma_k)$.\nIdeally, the true noise precision $\\gamma_{w0}$ is known and the signal $\\mathbf{x}^0$ is i.i.d.\\ with known prior $p(x_j)$, in which case the MMSE denoiser can be straightforwardly designed.\nIn practice, however, $\\gamma_{w0}$ and $p(x_j)$ are usually unknown.\nFortunately, there is a simple expectation-maximization (EM)-based method to estimate both quantities on-line, whose details are given in \\cite{fletcher2016emvamp}.\nThe numerical results in \\cite{fletcher2016emvamp} show that the convergence and asymptotic performance of EM-VAMP is nearly identical to that of VAMP with known $\\gamma_{w0}$ and $p(x_j)$.\nFor the numerical results in Section~\\ref{sec:num}, however, we assume that $\\gamma_{w0}$ and $p(x_j)$ are known.\n\nMatlab implementations of VAMP and EM-VAMP can be found in the public-domain GAMPmatlab toolbox \\cite{GAMP-code}.\n\n\n\n\n\\section{State Evolution} \\label{sec:SE}\n\n\\subsection{Large-System Analysis} \\label{sec:large}\nOur primary goal is to understand the behavior of the VAMP algorithm for a certain class of matrices in the high-dimensional regime.\nWe begin with an overview of our analysis framework and follow with more details in later sections.\n\n\\subsubsection{Linear measurement model}\nOur analysis considers a sequence of problems indexed by the signal\ndimension $N$.  For each $N$, we assume that there is a ``true\" vector $\\mathbf{x}^0\\in{\\mathbb{R}}^N$\nwhich is observed through measurements of the form,\n\\begin{equation} \\label{eq:yAxslr}\n    \\mathbf{y} = \\mathbf{A}\\mathbf{x}^0 + \\mathbf{w} \\in {\\mathbb{R}}^N, \\quad \\mathbf{w} \\sim {\\mathcal N}(\\mathbf{0}, \\gamma_{w0}^{-1}\\mathbf{I}_N),\n\\end{equation}\nwhere $\\mathbf{A}\\in{\\mathbb{R}}^{N\\times N}$ is a known transform and $\\mathbf{w}$ is Gaussian noise with precision $\\gamma_{w0}$.\nNote that we use $\\gamma_{w0}$ to denote the ``true\" noise precision\nto distinguish it from $\\gamma_w$, which is the noise precision postulated by the estimator.\n\nFor the transform $\\mathbf{A}$,\nour key assumption is that it can be modeled as a large, \\emph{right-orthogonally invariant} random matrix.\nSpecifically, we assume that it has an SVD of the form\n\\begin{equation} \\label{eq:ASVD}\n    \\mathbf{A}=\\mathbf{U}\\mathbf{S}\\mathbf{V}^{\\text{\\sf T}}, \\quad \\mathbf{S} = \\mathrm{Diag}(\\mathbf{s}),\n\\end{equation}\nwhere $\\mathbf{U}$ and $\\mathbf{V}$ are $N\\times N$ orthogonal matrices\nsuch that $\\mathbf{U}$ is deterministic and\n$\\mathbf{V}$ is Haar distributed (i.e.\\ uniformly distributed on the set of orthogonal matrices).\nWe refer to $\\mathbf{A}$ as ``right-orthogonally invariant'' because the distribution of $\\mathbf{A}$ is identical to that of $\\mathbf{A}\\mathbf{V}_0$ for any fixed orthogonal matrix $\\mathbf{V}_0$.\nWe will discuss the distribution of the singular values $\\mathbf{s}\\in{\\mathbb{R}}^N$ below.\n\nAlthough we have assumed that $\\mathbf{A}$ is square to streamline the analysis,\nwe make this assumption without loss of generality.\nFor example, by setting\n$$\n\\mathbf{U}=\\begin{bmatrix}\\mathbf{U}_0&\\mathbf{0}\\\\\\mathbf{0}&\\mathbf{I}\\end{bmatrix},\n\\quad\n\\mathbf{s}=\\begin{bmatrix}\\mathbf{s}_0\\\\\\mathbf{0}\\end{bmatrix},\n$$\nour formulation can model a wide rectangular matrix whose SVD is $\\mathbf{U}_0\\mathbf{S}_0\\mathbf{V}^{\\text{\\sf T}}$ with $\\mathop{\\mathrm{diag}}(\\mathbf{S}_0)=\\mathbf{s}_0$.\nA similar manipulation allows us to model a tall rectangular matrix.\n\n\\subsubsection{Denoiser}\nOur analysis applies to a fairly general class of denoising\nfunctions $\\mathbf{g}_1(\\cdot,\\gamma_{1k})$ indexed by the parameter $\\gamma_{1k}\\geq 0$.  Our main assumption is that the denoiser is separable,\nmeaning that it is of the form \\eqref{eq:g1sep} for some scalar denoiser $g_1(\\cdot,\\gamma_{1k})$.\nAs discussed above, this separability assumption will occur for the MAP and MMSE denoisers\nunder the assumption of an i.i.d.\\ prior.  However, we do not require the denoiser to be\nMAP or MMSE for any particular prior.\nWe will impose certain Lipschitz continuity conditions on $g_1(\\cdot,\\textb{\\gamma_{1k}})$ in the sequel.\n\n\\subsubsection{Asymptotic distributions}\nIt remains to describe the distributions of the\ntrue vector $\\mathbf{x}^0$ and the singular-value vector $\\mathbf{s}$.\nA simple model would be to assume that they\nare random i.i.d.\\ sequences that grow with $N$.  However, following the\nBayati-Montanari analysis \\cite{BayatiM:11},\nwe will consider a more general framework where each of these vectors is modeled as deterministic sequence\nfor which the empirical distribution of the components converges in distribution.\nWhen the vectors $\\mathbf{x}^0$ and $\\mathbf{s}$ are i.i.d.\\ random sequences, they will satisfy this condition almost surely.\nDetails of this analysis framework are reviewed in Appendix~\\ref{sec:empConv}.\n\nUsing the definitions in Appendix~\\ref{sec:empConv},\nwe assume that the components of the singular-value vector $\\mathbf{s}\\in{\\mathbb{R}}^N$ in \\eqref{eq:ASVD}\nconverge empirically with second-order moments as\n\\begin{equation} \\label{eq:Slim}\n    \\lim_{N \\rightarrow \\infty} \\{ s_n \\}_{n=1}^N \\stackrel{PL(2)}{=} S,\n\\end{equation}\nfor some positive random variable $S$.  We assume that $\\mathbb{E}[S] > 0$ and $S \\in [0,S_{max}]$\nfor some finite maximum value $S_{max}$.\nAdditionally, we assume that\nthe components of the true vector, $\\mathbf{x}^0$, and the initial input to the denoiser, $\\mathbf{r}_{10}$,\nconverge empirically as\n\\begin{equation} \\label{eq:RX0lim}\n    \\lim_{N \\rightarrow \\infty} \\{ (r_{10,n}, x^0_n) \\}_{n=1}^N \\stackrel{PL(2)}{=} (R_{10},X^0),\n\\end{equation}\nfor some random variables $(R_{10},X^0)$.\nNote that the convergence with second-order moments\nrequires that $\\mathbb{E} [(X^0)^2] < \\infty$ and $\\mathbb{E} [R^2_{10}] < \\infty$, so they have bounded\nsecond moments.\nWe also assume that the initial second-order term, \\textb{if dependent on $N$,} converges as\n\\begin{equation} \\label{eq:gam10lim}\n    \\lim_{N \\rightarrow \\infty}\\gamma_{10}\\textb{(N)} = \\overline{\\gamma}_{10},\n\\end{equation}\nfor some $\\overline{\\gamma}_{10} > 0$.\n\nAs stated above, most of our analysis will apply to general separable denoisers\n$g_1(\\cdot,\\gamma_{1k})$.  However, some results will apply specifically to MMSE denoisers.\nUnder the assumption that the components of the true vector $\\mathbf{x}^0$ are asymptotically\ndistributed like the random variable $X^0$\\textb{, as in \\eqref{eq:RX0lim}}, the MMSE denoiser~\\eqref{eq:g1mmsesca}\nand its derivative \\eqref{eq:g1dervar} reduce to\n\\begin{align} \\label{eq:g1mmsex0}\n\\begin{split}\n    g_1(r_1,\\gamma_1) &= \\mathbb{E}\\left[ X^0 | R_1=r_1 \\right], \\\\\n    g_1'(r_1,\\gamma_1) &= \\gamma_1\\mathrm{var}\\left[ X^0 | R_1=r_1 \\right],\n\\end{split}\n\\end{align}\nwhere $R_1$ is the random variable representing $X^0$ corrupted by AWGN noise,\ni.e.,\n\\[\n    R_1 = X^0 + P, \\quad P \\sim {\\mathcal N}(0,\\gamma_1^{-1}),\n\\]\nwith $P$ being independent of $X^0$.  Thus, the MMSE denoiser and its derivative\ncan be computed from the\nposterior mean and variance of $X^0$ under an AWGN measurement.\n\n\\subsection{Error Functions}\n\nBefore describing the state evolution (SE) equations and the\nanalysis in the LSL, we need to introduce two\nkey functions:  \\emph{error functions} and \\emph{sensitivity functions}.\nWe begin by describing the error functions.\n\nThe error functions, in essence,\ndescribe the mean squared error (MSE)\nof the denoiser and LMMSE estimators under AWGN measurements.\n\\textb{Recall from Section~\\ref{sec:large}, that we have assumed that\nthe denoiser $\\mathbf{g}_1(\\cdot,\\gamma_1)$ is separable with some componentwise function\n$g_1(\\cdot,\\gamma_1)$.}\nFor this function $g_1(\\cdot,\\gamma_1)$, define the error function as\n\\begin{align}\n    \\MoveEqLeft {\\mathcal E}_1(\\gamma_1,\\tau_1)\n    := \\mathbb{E}\\left[ (g_1(R_1,\\gamma_1)-X^0)^2 \\right],  \\nonumber \\\\\n    & R_1 = X^0 + P, \\quad P \\sim {\\mathcal N}(0,\\tau_1). \\label{eq:eps1}\n\\end{align}\nThe function ${\\mathcal E}_1(\\gamma_1,\\tau_1)$ thus represents the MSE of the\nestimate $\\hat{X} = g_1(R_1,\\gamma_1)$ from a measurement $R_1$\ncorrupted by Gaussian noise of variance $\\tau_1$.\nFor the LMMSE estimator, we define the error function as\n\\begin{align}\n    \\MoveEqLeft {\\mathcal E}_2(\\gamma_2,\\tau_2)\n    := \\lim_{N \\rightarrow \\infty}\n        \\frac{1}{N} \\mathbb{E} \\left[ \\| \\mathbf{g}_2(\\mathbf{r}_2,\\gamma_2) -\\mathbf{x}^0 \\|^2 \\right], \\nonumber \\\\\n    & \\mathbf{r}_2 = \\mathbf{x}^0 + \\mathbf{q}, \\quad \\mathbf{q} \\sim {\\mathcal N}(0,\\tau_2 \\mathbf{I}), \\nonumber \\\\\n    & \\mathbf{y} = \\mathbf{A}\\mathbf{x}^0 + \\mathbf{w}, \\quad \\mathbf{w} \\sim {\\mathcal N}(0,\\gamma_{w0}^{-1} \\mathbf{I}),\n    \\label{eq:eps2}\n\\end{align}\nwhich is the average per component error of the vector estimate under Gaussian noise.\nNote that ${\\mathcal E}_2(\\gamma_2,\\tau_2)$ is implicitly a function of the noise precision levels $\\gamma_{w0}$ and $\\gamma_w$ \\textb{(through $\\mathbf{g}_2$ from \\eqref{eq:g2slr})}, but this dependence is omitted to simplify the notation.\n\nWe will say that both estimators are ``matched\" when\n\\[\n    \\tau_1 = \\gamma_1^{-1}, \\quad \\tau_2 = \\gamma_2^{-1}, \\quad \\gamma_w = \\gamma_{w0},\n\\]\nso that the noise levels used by the estimators both match the true noise levels.\nUnder the matched condition, we will use the simplified notation\n\\[\n    {\\mathcal E}_1(\\gamma_1) := {\\mathcal E}_1(\\gamma_1,\\gamma_1^{-1}), \\quad\n    {\\mathcal E}_2(\\gamma_2) := {\\mathcal E}_2(\\gamma_2,\\gamma_2^{-1}).\n\\]\nThe following lemma establishes some basic properties of the error functions.\n\n\\begin{lemma} \\label{lem:errfn} Recall the error functions ${\\mathcal E}_1,{\\mathcal E}_2$ defined above.\n\\begin{enumerate}[(a)]\n\\item  For the MMSE denoiser \\eqref{eq:g1mmsex0} under the matched condition\n$\\tau_1=\\gamma_1^{-1}$, the error function  is the conditional variance\n\\begin{equation} \\label{eq:E1match}\n    {\\mathcal E}_1(\\gamma_1) = \\mathrm{var}\\left[ X^0 | R_1 = X^0 \\textb{+ P} \\right],~\\textb{P\\sim{\\mathcal N}(0,\\gamma_1^{-1})} .\n\\end{equation}\n\n\\item The LMMSE error function is given by\n\\begin{equation} \\label{eq:eps2Q}\n    {\\mathcal E}_2(\\gamma_2,\\tau_2) = \\lim_{N \\rightarrow \\infty} \\frac{1}{N} \\mathrm{Tr}\\left[ \\mathbf{Q}^{-2}\n        \\tilde{\\mathbf{Q}} \\right] ,\n\\end{equation}\nwhere $\\mathbf{Q}$ and $\\tilde{\\mathbf{Q}}$ are the matrices\n\\begin{equation} \\label{eq:QQtdef}\n    \\mathbf{Q} : = \\gamma_w \\mathbf{A}^{\\text{\\sf T}}\\mathbf{A} + \\gamma_2\\mathbf{I}, \\quad\n    \\tilde{\\mathbf{Q}} :=\n        \\frac{\\gamma_w^2}{\\gamma_{w0}}\\mathbf{A}^{\\text{\\sf T}}\\mathbf{A} + \\tau_2\\gamma_2^2\\mathbf{I}.\n\\end{equation}\nUnder the matched condition $\\tau_2 = \\gamma_2^{-1}$ and $\\gamma_w = \\gamma_{w0}$,\n\\begin{equation} \\label{eq:eps2Qmatch}\n    {\\mathcal E}_2(\\gamma_2) = \\lim_{N \\rightarrow \\infty} \\frac{1}{N} \\mathrm{Tr}\\left[ \\mathbf{Q}^{-1} \\right].\n\\end{equation}\n\n\\item The LMMSE error function is also given by\n\\begin{equation} \\label{eq:eps2S}\n    {\\mathcal E}_2(\\gamma_2,\\tau_2) = \\mathbb{E}\\left[ \\frac{\n    \\gamma_w^2 S^2\/\\gamma_{w0} + \\tau_2\\gamma_2^2}{(\\gamma_w S^2 + \\gamma_2)^2}  \\right],\n\\end{equation}\nwhere $S$ is the random variable \\eqref{eq:Slim} representing the distribution of the\nsingular values of $\\mathbf{A}$.  For the matched condition\n$\\tau_2 = \\gamma_2^{-1}$ and $\\gamma_w = \\gamma_{w0}$,\n\\begin{equation} \\label{eq:eps2Smatch}\n    {\\mathcal E}_2(\\gamma_2) = \\mathbb{E}\\left[\n        \\frac{1}{\\gamma_wS^2+ \\gamma_2} \\right].\n\\end{equation}\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof} See Appendix~\\ref{sec:errsenspf}.\n\\end{proof}\n\n\\subsection{Sensitivity Functions}\nThe sensitivity functions describe the expected divergence of the estimator.\nFor the denoiser, the sensitivity function is defined as\n\\begin{align}\n    \\MoveEqLeft A_1(\\gamma_1,\\tau_1)\n    := \\mathbb{E}\\left[ g_1'(R_1,\\gamma_1) \\right],  \\nonumber \\\\\n    & R_1 = X^0 + P, \\quad P \\sim {\\mathcal N}(0,\\tau_1), \\label{eq:sens1}\n\\end{align}\nwhich is the average derivative under a Gaussian noise input.  For the\nLMMSE estimator, the sensitivity is defined as\n\\begin{align}\n    \\MoveEqLeft A_2(\\gamma_2)\n    := \\lim_{N \\rightarrow \\infty}\n        \\frac{1}{N} \\mathrm{Tr}\\left[ \\frac{\\partial \\mathbf{g}_2(\\mathbf{r}_2,\\gamma_2)}{\\partial \\mathbf{r}_2}\n            \\right].\n\\end{align}\n\n\\begin{lemma} \\label{lem:sens}\nFor the sensitivity functions above:\n\\begin{enumerate}[(a)]\n\\item  For the MMSE denoiser \\eqref{eq:g1mmsex0} under the matched condition\n$\\tau_1=\\gamma_1^{-1}$, the sensitivity function  is given by\n\\begin{equation} \\label{eq:A1match}\n    A_1(\\gamma_1,\\gamma_1^{-1}) = \\gamma_1\\mathrm{var}\\left[ X^0 | R_1 = X^0 + {\\mathcal N}(0,\\gamma_1^{-1}) \\right],\n\\end{equation}\nwhich is the ratio of the conditional variance to the measurement variance $\\gamma_1^{-1}$.\n\n\\item The LMMSE estimator's sensitivity function is given by\n\\[\n    A_2(\\gamma_2) = \\lim_{N \\rightarrow \\infty} \\frac{1}{N}\n        \\gamma_2\\mathrm{Tr}\\left[ (\\gamma_w \\mathbf{A}^{\\text{\\sf T}}\\mathbf{A} + \\gamma_2\\mathbf{I})^{-1} \\right].\n\\]\n\n\\item The LMMSE estimator's sensitivity function can also be written as\n\\[\n    A_2(\\gamma_2) = \\mathbb{E}\\left[ \\frac{\\gamma_2}{\\gamma_wS^2+ \\gamma_2} \\right].\n\\]\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof} See Appendix~\\ref{sec:errsenspf}.\n\\end{proof}\n\n\n\\subsection{State Evolution Equations}\n\nWe can now describe our main result, which is the SE equations\nfor VAMP.\nFor a given iteration $k \\geq 1$, consider the set of components,\n\\[\n     \\{ (\\hat{x}_{1k,n},r_{1k,n},x^0_n), ~ n=1,\\ldots,N \\}.\n\\]\nThis set represents the components of the true vector $\\mathbf{x}^0$,\nits corresponding estimate $\\hat{\\mathbf{x}}_{1k}$ and the denoiser input\n$\\mathbf{r}_{1k}$.  \\textb{Theorem~\\ref{thm:se}} below will \nshow that, under certain assumptions,\nthese components converge empirically as\n\\begin{equation} \\label{eq:limrx1}\n    \\lim_{N \\rightarrow \\infty} \\{ (\\hat{x}_{1k,n},r_{1k,n},x^0_n) \\}\n    \\stackrel{PL(2)}{=} (\\hat{X}_{1k},R_{1k},X^0),\n\\end{equation}\nwhere the random variables $(\\hat{X}_{1k},R_{1k},X^0)$ are given by\n\\begin{subequations} \\label{eq:RX0var}\n\\begin{align}\n    R_{1k} &= X^0 + P_k, \\quad P_k \\sim {\\mathcal N}(0,\\tau_{1k}), \\\\\n    \\hat{X}_{1k} &= g_1(R_{1k},\\overline{\\gamma}_{1k}),\n\\end{align}\n\\end{subequations}\nfor constants $\\overline{\\gamma}_{1k}$ and $\\tau_{1k}$ that will be defined below.\nThus, each component $r_{1k,n}$ appears as the true component $x^0_n$ plus\nGaussian noise.  The corresponding estimate $\\hat{x}_{1k,n}$ then appears as the\ndenoiser output with $r_{1k,n}$ as the input.  Hence, the asymptotic behavior\nof any component $x^0_n$ and its corresponding $\\hat{x}_{1k,n}$ is identical to\na simple scalar system.  We will refer to \\eqref{eq:limrx1}-\\eqref{eq:RX0var} as the denoiser's \\emph{scalar equivalent model}.\n\nFor the LMMSE estimation function,\nwe define the transformed error and transformed noise,\n\\begin{equation} \\label{eq:qerrdef}\n    \\mathbf{q}_k := \\mathbf{V}^{\\text{\\sf T}}(\\mathbf{r}_{2k}-\\mathbf{x}^0), \\quad {\\boldsymbol \\xi} := \\mathbf{U}^{\\text{\\sf T}}\\mathbf{w},\n\\end{equation}\nwhere $\\mathbf{U}$ and $\\mathbf{V}$ are the matrices in the SVD decomposition \\eqref{eq:ASVD}.\n\\textb{Theorem~\\ref{thm:se} will also show that}  \nthese transformed errors and singular values $s_n$ converge as,\n\\begin{equation} \\label{eq:limqxi}\n    \\lim_{N \\rightarrow \\infty} \\{ (q_{k,n},\\xi_n,s_n) \\}\n    \\stackrel{PL(2)}{=} (Q_k,\\Xi,S),\n\\end{equation}\nto a set of random variables $(Q_k,\\Xi,S)$.\nThese random variables are independent, with\n$S$ defined in the limit \\eqref{eq:Slim} and\n\\begin{equation} \\label{eq:QXivar}\n    Q_k \\sim {\\mathcal N}(0,\\tau_{2k}), \\quad \\Xi \\sim {\\mathcal N}(0,\\gamma_{w0}^{-1}),\n\\end{equation}\nwhere  $\\tau_{2k}$ is a variance that will be defined below and $\\gamma_{w0}$\nis the noise precision in the measurement model \\eqref{eq:yAxslr}.\nThus \\eqref{eq:limqxi}-\\eqref{eq:QXivar} is a scalar equivalent model for the LMMSE estimator.\n\nThe variance terms are defined recursively through what are called \\emph{state evolution}\nequations,\n\\begin{subequations} \\label{eq:se}\n\\begin{align}\n    \\overline{\\alpha}_{1k} &= A_1(\\overline{\\gamma}_{1k},\\tau_{1k}) \\label{eq:a1se} \\\\\n    \\overline{\\eta}_{1k} &= \\frac{\\overline{\\gamma}_{1k}}{\\overline{\\alpha}_{1k}}, \\quad\n    \\overline{\\gamma}_{2k} = \\overline{\\eta}_{1k} - \\overline{\\gamma}_{1k} \\label{eq:eta1se} \\\\\\\n    \\tau_{2k} &= \\frac{1}{(1-\\overline{\\alpha}_{1k})^2}\\left[\n        {\\mathcal E}_1(\\overline{\\gamma}_{1k},\\tau_{1k}) - \\overline{\\alpha}_{1k}^2\\tau_{1k} \\right],\n            \\label{eq:tau2se} \\\\\n    \\overline{\\alpha}_{2k} &= A_2(\\overline{\\gamma}_{2k},\\tau_{2k}) \\label{eq:a2se} \\\\\n    \\overline{\\eta}_{2k} &= \\frac{\\overline{\\gamma}_{2k}}{\\overline{\\alpha}_{2k}}, \\quad\n    \\overline{\\gamma}_{1,k\\! + \\! 1} = \\overline{\\eta}_{2k} - \\overline{\\gamma}_{2k} \\label{eq:eta2se} \\\\\n    \\tau_{1,k\\! + \\! 1} &= \\frac{1}{(1-\\overline{\\alpha}_{2k})^2}\\left[\n        {\\mathcal E}_2(\\overline{\\gamma}_{2k},\\tau_{2k}) - \\overline{\\alpha}_{2k}^2\\tau_{2k} \\right],\n            \\label{eq:tau1se}\n\\end{align}\n\\end{subequations}\nwhich are initialized with\n\\begin{equation}\n    \\tau_{10} = \\mathbb{E}[(R_{10}-X^0)^2],\n\\end{equation}\nand $\\overline{\\gamma}_{10}$ defined from the limit \\eqref{eq:gam10lim}.\n\n\n\\begin{theorem} \\label{thm:se}\nUnder the above assumptions and definitions, assume additionally that for all iterations $k$:\n\\begin{enumerate}[(i)]\n\\item The solution $\\overline{\\alpha}_{1k}$ from the SE equations \\eqref{eq:se} satisfies\n\\begin{equation} \\label{eq:asecon}\n    \\overline{\\alpha}_{1k} \\in (0,1).\n\\end{equation}\n\\item The functions $A_i(\\gamma_i,\\tau_i)$ and ${\\mathcal E}_i(\\gamma_i,\\tau_i)$\nare continuous at $(\\gamma_i,\\tau_i)=(\\overline{\\gamma}_{ik},\\tau_{ik})$.\n\\item The denoiser function $g_1(r_1,\\gamma_1)$ and its derivative\n $g_1'(r_1,\\gamma_1)$\nare uniformly Lipschitz in $r_1$ at $\\gamma_1=\\overline{\\gamma}_{1k}$.\n(See  Appendix~\\ref{sec:empConv} for a precise definition of uniform Lipschitz continuity.)\n\\end{enumerate}\nThen, for any fixed iteration $k \\geq 0$,\n\\begin{equation} \\label{eq:aglim}\n    \\lim_{N \\rightarrow \\infty} (\\alpha_{ik},\\eta_{ik},\\gamma_{ik}) =\n    (\\overline{\\alpha}_{ik},\\overline{\\eta}_{ik}, \\overline{\\gamma}_{ik})\n\\end{equation}\nalmost surely.\nIn addition, the empirical limit \\eqref{eq:limrx1} holds almost surely for all $k > 0$,\nand \\eqref{eq:limqxi} holds almost surely for all $k \\geq 0$.\n\\end{theorem}\n\n\\subsection{Mean Squared Error}\nOne important\nuse of the scalar equivalent model is to predict the asymptotic performance\nof the VAMP algorithm in the LSL.  For example, define the asymptotic\nmean squared error (MSE) of the iteration-$k$ estimate $\\hat{\\mathbf{x}}_{ik}$ as\n\\begin{equation} \\label{eq:mseLim}\n    \\mbox{\\small \\sffamily MSE}_{ik} := \\lim_{N \\rightarrow \\infty} \\frac{1}{N}\\|\\hat{\\mathbf{x}}_{ik}-\\mathbf{x}^0\\|^2.\n\\end{equation}\nFor this MSE, we claim that\n\\begin{equation} \\label{eq:mseEcal}\n    \\mbox{\\small \\sffamily MSE}_{ik} = {\\mathcal E}_i(\\overline{\\gamma}_{ik},\\tau_{ik}).\n\\end{equation}\nTo prove \\eqref{eq:mseEcal} for $i=1$, we write\n\\begin{align*}\n    \\mbox{\\small \\sffamily MSE}_{1k}\n    &=\\lim_{N \\rightarrow \\infty}\n        \\frac{1}{N} \\sum_{n=1}^N (\\hat{x}_{1k,n}-x^0_n)^2     \\\\\n    &\\stackrel{(a)}{=} \\mathbb{E}[(\\hat{X}_{1k}-X^0)^2] \\\\\n    &\\stackrel{(b)}{=} \\mathbb{E}[(g_1(R_1,\\overline{\\gamma}_{1k})-X^0)^2]\n     \\stackrel{(c)}{=} {\\mathcal E}_1(\\overline{\\gamma}_{1k},\\tau_{1k})\n\\end{align*}\nwhere (a) and (b) follow from the convergence in \\eqref{eq:limrx1} and the scalar equivalent model \\eqref{eq:limrx1},\nand where (c) follows from \\eqref{eq:eps1}.  Using the scalar equivalent model~\\eqref{eq:limqxi},\nthe definition of ${\\mathcal E}_2(\\cdot)$ in \\eqref{eq:eps2}, and calculations similar to the proof\nof Lemma~\\ref{lem:errfn}, one can also show that \\eqref{eq:mseEcal} holds for $i=2$.\n\nInterestingly, this type of calculation can be used to compute any other componentwise distortion metric.\nSpecifically, given any distortion function $d(x,\\hat{x})$ that is pseudo-Lipschitz of order two,\nits average value is given by\n\\[\n    \\lim_{N \\rightarrow \\infty}\n        \\frac{1}{N} \\sum_{n=1}^N d(x^0_n,\\hat{x}_{1k,n}) = \\mathbb{E}\\left[ d(X^0,\\hat{X}_{1k}) \\right],\n\\]\nwhere the expectation is from  the scalar equivalent model \\eqref{eq:limrx1}.\n\n\\subsection{Contractiveness of the Denoiser}\nAn essential requirement of Theorem~\\ref{thm:se} is the condition~\\eqref{eq:asecon}\nthat $\\overline{\\alpha}_{1k} \\in (0,1)$.  This assumption requires that, in a certain average,\nthe denoiser function $g_1(\\cdot,\\gamma_1)$ is increasing (i.e., $g_1'(r_{1n},\\gamma_1) > 0$)\nand is a contraction (i.e., $g_1'(r_{1n},\\gamma_1) < 1$).\nIf these conditions are not met,\nthen $\\overline{\\alpha}_{1k} \\leq 0$ or $\\overline{\\alpha}_{1k} \\geq 1$,\nand either the estimated precision $\\overline{\\eta}_{1k}$ or $\\overline{\\gamma}_{2k}$\nin \\eqref{eq:eta1se} may be negative, causing subsequent updates to be invalid.\nThus, $\\overline{\\alpha}_{1k}$ must be in the range $(0,1)$.\nThere are two important conditions under which\nthis increasing contraction property are provably guaranteed:\n\n\\noindent\n\\paragraph*{Strongly convex penalties}  Suppose that  $g_1(r_{1n},\\gamma_1)$ is the\neither the MAP denoiser \\eqref{eq:gmapsca} or the MMSE denoiser\n\\eqref{eq:g1mmsesca} for a density $p(x_n)$ that is strongly log-concave.\nThat is, there exists constants $c_1,c_2 > 0$ such that\n\\[\n    c_1 \\leq -\\frac{\\partial^2}{\\partial x_n^2} \\ln p(x_n) \\leq c_2.\n\\]\nThen, using results from log-concave functions \\cite{brascamp2002extensions},\nit is shown in \\cite{rangan2015admm} that\n\\[\n    g_1'(r_{1n},\\gamma_1) \\in\n        \\left[ \\frac{\\gamma_1}{c_2 + \\gamma_1}, \\frac{\\gamma_1}{c_1 + \\gamma_1}\\right]\n        \\subset (0,1),\n\\]\nfor all $r_{1n}$ and $\\gamma_1 > 0$.  Hence, from the definition of the\nsensitivity function \\eqref{eq:sens1},\nthe sensitivity $\\overline{\\alpha}_{1k}$ in \\eqref{eq:a1se} will be in the range $(0,1)$.\n\n\\noindent\n\\paragraph*{Matched MMSE denoising}  Suppose that  $g_1(r_{1n},\\gamma_1)$ is the\nMMSE denoiser in the matched condition where $\\overline{\\gamma}_{1k}=\\tau_{1k}^{-1}$\nfor some iteration $k$.\nFrom \\eqref{eq:A1match},\n\\[\n    A_1(\\gamma_1,\\gamma_1^{-1}) = \\gamma_1\\mathrm{var}\\left[ X^0 | R_1 = X^0 + {\\mathcal N}(0,\\gamma_1^{-1})\n        \\right].\n\\]\nSince the conditional variance is positive, $A_1(\\gamma_1,\\gamma_1^{-1}) > 0$.\nAlso, since the variance is bounded above by the MSE of a linear estimator,\n\\begin{align*}\n    \\MoveEqLeft \\gamma_1\\mathrm{var}\\left[ X^0 | R_1 = X^0 + {\\mathcal N}(0,\\gamma_1^{-1})\\right]\\\\\n    &\\leq \\gamma_1\\frac{\\gamma_1^{-1}\\tau_{x_0}}{\\tau_{x_0}+\\gamma_1^{-1}}\n    = \\frac{\\gamma_1\\tau_{x_0}}{1+ \\gamma_1\\tau_{x_0}} < 1,\n\\end{align*}\nwhere $\\tau_{x0} = \\mathrm{var}(X^0)$.\nThus, we have $A_1(\\gamma_1,\\gamma_1^{-1}) \\in (0,1)$ and\n$\\overline{\\alpha}_{1k} \\in (0,1)$.\n\n\\medskip\nIn the case when the prior is not log-concave and the estimator uses an denoiser\nthat is not perfectly matched, $\\overline{\\alpha}_{1k}$ may not be in the valid range $(0,1)$.\nIn these cases, VAMP may obtain invalid (i.e.\\ negative) variance estimates.\n\n\n\\section{MMSE Denoising, Optimality, and Connections to the Replica Method} \\label{sec:replica}\n\nAn important special case of the VAMP algorithm\nis when we apply the MMSE optimal denoiser under matched $\\gamma_w$.  In this case,\nthe SE equations simplify considerably.\n\n\\begin{theorem} \\label{thm:seMmse}  Consider the SE equations \\eqref{eq:se} with\nthe MMSE optimal denoiser\n\\eqref{eq:g1mmsex0}, matched $\\gamma_w=\\gamma_{w0}$, and matched initial condition $\\overline{\\gamma}_{10} = \\tau_{10}^{-1}$.\nThen, for all iterations $k \\geq 0$,\n\\begin{subequations} \\label{eq:sematch}\n\\begin{align}\n    \\overline{\\eta}_{1k} &= \\frac{1}{{\\mathcal E}_1(\\overline{\\gamma}_{1k})}, \\quad\n    \\overline{\\gamma}_{2k} = \\tau_{2k}^{-1} = \\overline{\\eta}_{1k} - \\overline{\\gamma}_{1k},\n        \\label{eq:eta1sematch} \\\\\n    \\overline{\\eta}_{2k} &= \\frac{1}{{\\mathcal E}_2(\\overline{\\gamma}_{2k})}, \\quad\n    \\overline{\\gamma}_{1,k\\! + \\! 1} = \\tau_{1,k\\! + \\! 1}^{-1} = \\overline{\\eta}_{2k} - \\overline{\\gamma}_{2k}.\n        \\label{eq:eta2sematch}\n\\end{align}\nIn addition, for estimators $i=1,2$,  $\\overline{\\eta}_{ik}$ is the inverse MSE:\n\\begin{equation} \\label{eq:etammse}\n    \\overline{\\eta}_{ik}^{-1} = \\lim_{N \\rightarrow \\infty} \\frac{1}{N} \\|\\hat{\\mathbf{x}}_{ik}-\\mathbf{x}^0\\|^2.\n\\end{equation}\n\\end{subequations}\n\\end{theorem}\n\\begin{proof}  See Appendix~\\ref{sec:seMmsePf}.\n\\end{proof}\n\n\nIt is useful to compare this result with\nthe work \\cite{tulino2013support}, which uses the \\emph{replica method} from statistical physics\nto predict the asymptotic MMSE error in the LSL.  To state the result,\ngiven a positive semidefinite matrix $\\mathbf{C}$, we define its Stieltjes transform as\n\\begin{equation} \\label{eq:stieltjes}\n    S_{\\mathbf{C}}(\\omega) = \\frac{1}{N} \\mathrm{Tr}\\left[ (\\mathbf{C} - \\omega \\mathbf{I}_N)^{-1} \\right]\n     = \\frac{1}{N} \\sum_{n=1}^N \\frac{1}{\\lambda_n - \\omega},\n\\end{equation}\nwhere $\\lambda_n$ are the eigenvalues of $\\mathbf{C}$. Also, let $R_{\\mathbf{C}}(\\omega)$\ndenote the so-called $R$-transform of $\\mathbf{C}$, given by\n\\begin{equation} \\label{eq:rtrans}\n    R_{\\mathbf{C}}(\\omega) =  S_{\\mathbf{C}}^{-1}(-\\omega) - \\frac{1}{\\omega},\n\\end{equation}\nwhere the inverse $S_{\\mathbf{C}}^{-1}(\\cdot)$ is in terms of composition of functions.\nThe Stieltjes and $R$-transforms are discussed in detail in \\cite{TulinoV:04}.\nThe Stieltjes and $R$-transforms can be\nextended to random matrix sequences by taking limits as $N \\rightarrow\\infty$\n(for matrix sequences where these limits converge almost surely).\n\nNow suppose that $\\hat{\\mathbf{x}} = \\mathbb{E}[\\mathbf{x}^0|\\mathbf{y}]$ is the MMSE estimate of $\\mathbf{x}^0$ given $\\mathbf{y}$.\nLet $\\overline{\\eta}^{-1}$ be the asymptotic inverse MSE\n\\[\n    \\overline{\\eta}^{-1} := \\lim_{N \\rightarrow \\infty} \\frac{1}{N} \\|\\hat{\\mathbf{x}}-\\mathbf{x}^0\\|^2.\n\\]\nUsing a so-called replica symmetric analysis, it is argued in\n\\cite{tulino2013support} that this MSE should satisfy the fixed point\nequations\n\\begin{equation} \\label{eq:fixrep}\n    \\overline{\\gamma}_1 = R_{\\mathbf{C}}(-\\overline{\\eta}^{-1}), \\quad\n    \\overline{\\eta}^{-1} =  {\\mathcal E}_1( \\overline{\\gamma}_1 ),\n\\end{equation}\nwhere $\\mathbf{C}=\\gamma_{w0}\\mathbf{A}^{\\text{\\sf T}}\\mathbf{A}$.\nA similar result is given in \\cite{kabashima2014signal}.\n\n\\begin{theorem} \\label{thm:replica}  Let $\\overline{\\gamma}_i,\\overline{\\eta}_i$ be any fixed point\nsolutions to the SE equations \\eqref{eq:sematch} of VAMP under MMSE denoising and matched $\\gamma_w=\\gamma_{w0}$.\nThen $\\overline{\\eta}_1=\\overline{\\eta}_2$.  If we define $\\overline{\\eta}:=\\overline{\\eta}_i$ as the common value,\nthen $\\overline{\\gamma}_1$ and $\\overline{\\eta}$ satisfy the replica fixed point equation \\eqref{eq:fixrep}.\n\\end{theorem}\n\\begin{proof}  Note that we have dropped the iteration index $k$ since we are discussing\na fixed point.  First, \\eqref{eq:sematch} shows that, at any fixed point,\n\\[\n    \\overline{\\gamma}_1+\\overline{\\gamma}_2 = \\overline{\\eta}_1 = \\overline{\\eta}_2,\n\\]\nso that $\\overline{\\eta}_1=\\overline{\\eta}_2$.  Also, in the matched case, \\eqref{eq:eps2Smatch} shows that\n\\[\n    {\\mathcal E}_2(\\overline{\\gamma}_2) = S_{\\mathbf{C}}(-\\overline{\\gamma}_2) .\n\\]\nSince $\\overline{\\eta}^{-1} = {\\mathcal E}_2(\\overline{\\gamma}_2)$, we have that\n\\[\n    \\overline{\\gamma}_1 = \\overline{\\eta}-\\overline{\\gamma}_2 = \\overline{\\eta}+S_\\mathbf{C}^{-1}(\\overline{\\eta}^{-1}) = R_\\mathbf{C}(-\\overline{\\eta}^{-1}).\n\\]\nAlso,  $\\overline{\\eta}^{-1}=\\overline{\\eta}_1^{-1} = {\\mathcal E}(\\overline{\\gamma}_1)$.\n\\end{proof}\n\nThe consequence of Theorem~\\ref{thm:replica} is that, if the replica equations \\eqref{eq:fixrep}\nhave a unique fixed point, then the MSE achieved by the VAMP algorithm\nexactly matches the Bayes optimal MSE as predicted by the replica method.  Hence, if this\nreplica prediction is correct, then the VAMP method provides a computationally efficient\nmethod for finding MSE optimal estimates under very general priors---including priors\nfor which the associated penalty functions are not convex.\n\nThe replica method, however, is generally heuristic.\nBut in the case of i.i.d.\\ Gaussian matrices,\nit has recently been proven that the replica prediction is correct\n\\cite{reeves2016replica,barbier2016mutual}.\n\n\n\n\n\\section{Numerical Experiments} \\label{sec:num}\n\nIn this section, we present numerical experiments that compare\nthe VAMP\\footnote{A Matlab implementation of VAMP can be found in the public-domain GAMPmatlab toolbox \\cite{GAMP-code}.}\nAlgorithm~\\ref{algo:vampSVD} to\nthe VAMP state evolution from Section~\\ref{sec:SE},\nthe replica prediction from \\cite{tulino2013support},\nthe AMP Algorithm~\\ref{algo:amp} from \\cite{DonohoMM:10-ITW1},\nthe S-AMP algorithm from \\cite[Sec.~IV]{cakmak2014samp},\nthe adaptively damped (AD) GAMP algorithm from \\cite{Vila:ICASSP:15},\nand the support-oracle MMSE estimator, whose MSE lower bounds that achievable by any practical method.\nIn all cases, we consider the recovery of vectors $\\mathbf{x}^0\\in{\\mathbb{R}}^N$\nfrom AWGN-corrupted measurements $\\mathbf{y}\\in{\\mathbb{R}}^M$ constructed from \\eqref{eq:yAx}, where\n$\\mathbf{x}^0$ was drawn i.i.d.\\ zero-mean Bernoulli-Gaussian with $\\Pr\\{x^0_j\\neq 0\\}=0.1$,\nwhere $\\mathbf{w}\\sim{\\mathcal N}(\\mathbf{0},\\mathbf{I}\/\\gamma_{w0})$,\nand where $M=512$ and $N=1024$.\nAll methods under test were matched to the true signal and noise statistics.\nWhen computing the support-oracle MMSE estimate, the support of $\\mathbf{x}^0$ is assumed to be known, in which case the problem reduces to estimating the non-zero coefficients of $\\mathbf{x}^0$.\nSince these non-zero coefficients are Gaussian, their MMSE estimate can be computed in closed form.\nFor VAMP we used the implementation enhancements described in Section~\\ref{sec:implementation}.\nFor line~\\ref{line:gamma} of AMP Algorithm~\\ref{algo:amp}, we used $1\/\\gamma_{k\\! + \\! 1}=1\/\\gamma_{w0}+\\frac{N}{M}\\alpha_k\/\\gamma_k$, as specified in \\cite[Eq.\\ (25)]{DonohoMM:10-ITW1}.\nFor the AMP, S-AMP, and AD-GAMP algorithms, we allowed a maximum of $1000$ iterations, and for the VAMP algorithm we allowed a maximum of $100$ iterations.\n\n\\subsection{Ill-conditioned $\\mathbf{A}$} \\label{sec:ill}\n\nFirst we investigate algorithm robustness to the condition number of $\\mathbf{A}$.\nFor this study, realizations of $\\mathbf{A}$ were constructed from the SVD $\\mathbf{A}=\\overline{\\mathbf{U}}\\mathrm{Diag}(\\overline{\\mathbf{s}})\\overline{\\mathbf{V}}^{\\text{\\sf T}}\\in{\\mathbb{R}}^{M\\times N}$ with geometric singular values $\\overline{\\mathbf{s}}\\in{\\mathbb{R}}^M$.\nThat is, $\\bar{s}_i\/\\bar{s}_{i-1}=\\rho~\\forall i$, with $\\rho$ chosen to achieve a desired condition number $\\kappa(\\mathbf{A}):=\\bar{s}_1\/\\bar{s}_M$ and with $\\bar{s}_1$ chosen so that $\\|\\mathbf{A}\\|_F^2=N$.\nThe singular vector matrices $\\overline{\\mathbf{U}},\\overline{\\mathbf{V}}$ were drawn uniformly at random from the group of orthogonal matrices, i.e., from the Haar distribution.\nFinally, the signal and noise variances were set to achieve a signal-to-noise ratio (SNR) $\\mathbb{E}[\\|\\mathbf{A}\\mathbf{x}\\|^2]\/\\mathbb{E}[\\|\\mathbf{w}\\|^2]$ of $40$~dB.\n\nFigure~\\ref{fig:nmse_vs_cond} plots the median normalized MSE (NMSE) achieved by each algorithm over $500$ independent realizations of $\\{\\mathbf{A},\\mathbf{x},\\mathbf{w}\\}$, where $\\text{NMSE}(\\hat{\\mathbf{x}}):=\\|\\hat{\\mathbf{x}}-\\mathbf{x}^0\\|^2\/\\|\\mathbf{x}^0\\|^2$.\nTo enhance visual clarity, NMSEs were clipped to a maximum value of $1$.\nAlso, error bars are shown that (separately) quantify the positive and negative standard deviations of VAMP's NMSE from the median value.\nThe NMSE was evaluated for condition numbers $\\kappa(\\mathbf{A})$ ranging from $1$ (i.e., row-orthogonal $\\mathbf{A}$) to $1\\times10^6$ (i.e., highly ill-conditioned $\\mathbf{A}$).\n\n\\begin{figure}[t]\n\\centering\n\\psfrag{SNR=40dB, N=1024, M=512, rho=0.2, U=Haar, V=Haar, isCmplx=0, median of 500}{}\n\\psfrag{condition number}[t][t][0.7]{\\sf condition number $\\kappa(\\mathbf{A})$}\n\\psfrag{median NMSE [dB]}[b][b][0.7]{\\sf median NMSE [dB]}\n\\psfrag{damped GAMP}[lB][lB][0.42]{\\sf AD-GAMP}\n\\includegraphics[width=3.15in]{figures\/nmse_vs_cond.eps}\n\\caption{NMSE versus condition number $\\kappa(\\mathbf{A})$ at final algorithm iteration.  The reported NMSE is the median over $500$ realizations, with error bars shown on the VAMP trace.\n\\label{fig:nmse_vs_cond}}\n\\end{figure}\n\nIn Figure~\\ref{fig:nmse_vs_cond}, we see that AMP and S-AMP diverged for even mildly ill-conditioned $\\mathbf{A}$.\nWe also see that, while adaptive damping helped to extend the operating range of AMP, it had a limited effect.\nIn contrast, Figure~\\ref{fig:nmse_vs_cond} shows that VAMP's NMSE stayed relatively close to the replica prediction for all condition numbers $\\kappa(\\mathbf{A})$.\nThe small gap between VAMP and the replica prediction is due to finite-dimensional effects; the SE analysis from Section~\\ref{sec:SE} establishes that this gap closes in the large-system limit.\nFinally, Figure~\\ref{fig:nmse_vs_cond} shows that the oracle bound is close to the replica prediction at small $\\kappa(\\mathbf{A})$ but not at large $\\kappa(\\mathbf{A})$.\n\nFigure~\\ref{fig:nmse_vs_iter_cond}(a) plots NMSE versus algorithm iteration for condition number $\\kappa(\\mathbf{A})=1$ and Figure~\\ref{fig:nmse_vs_iter_cond}(b) plots the same for $\\kappa(\\mathbf{A})=1000$, again with error bars on the VAMP traces.\nBoth figures show that the VAMP trajectory stayed very close to the VAMP-SE trajectory at every iteration.\nThe figures also show that VAMP converges a bit quicker than AMP, S-AMP, and AD-GAMP when $\\kappa(\\mathbf{A})=1$, and that VAMP's convergence rate is relatively insensitive to the condition number $\\kappa(\\mathbf{A})$.\n\n\\begin{figure}[t]\n\\centering\n\\psfrag{condition number=1}[b][b][0.7]{(a)}\n\\psfrag{condition number=1000}[b][b][0.7]{(b)}\n\\psfrag{iterations}[t][t][0.7]{\\sf iteration}\n\\psfrag{median NMSE [dB]}[b][b][0.7]{\\sf median NMSE [dB]}\n\\psfrag{damped GAMP}[lB][lB][0.42]{\\sf AD-GAMP}\n\\includegraphics[width=3.15in]{figures\/nmse_vs_iter_cond.eps}\n\\caption{NMSE versus algorithm iteration for condition number $\\kappa(\\mathbf{A})=1$ in (a) and $\\kappa(\\mathbf{A})=1000$ in (b).  The reported NMSE is the median over $500$ realizations, with error bars shown on the VAMP traces.\n\\label{fig:nmse_vs_iter_cond}}\n\\end{figure}\n\n\n\\subsection{Non-zero-mean $\\mathbf{A}$} \\label{sec:nzmean}\n\nIn this section, we investigate algorithm robustness to the componentwise mean of $\\mathbf{A}$.\nFor this study, realizations of $\\mathbf{A}$ were constructed by first drawing an i.i.d.\\ ${\\mathcal N}(\\mu,1\/M)$ matrix and then scaling it so that $\\|\\mathbf{A}\\|_F^2=N$ (noting that essentially no scaling is needed when $\\mu\\approx 0$).\nAs before, the signal and noise variances were set to achieve an SNR of $40$~dB.\nFor AD-GAMP, we used the mean-removal trick proposed in \\cite{Vila:ICASSP:15}.\n\nFigure~\\ref{fig:nmse_vs_mean} plots the NMSE achieved by each algorithm over $200$ independent realizations of $\\{\\mathbf{A},\\mathbf{x},\\mathbf{w}\\}$.\nThe NMSE was evaluated for mean parameters $\\mu$ between $0.001$ and $10$.\nNote that, when $\\mu>0.044$, the mean is larger than the standard deviation.\nThus, the values of $\\mu$ that we consider are quite extreme relative to past studies like \\cite{Caltagirone:14-ISIT}.\n\n\\begin{figure}[t]\n\\centering\n\\psfrag{SNR=40dB, N=1024, M=512, rho=0.2, isCmplx=0, median of 200}{}\n\\psfrag{nzmean}[t][t][0.7]{\\sf mean $\\mu$ of $\\mathbf{A}$}\n\\psfrag{median NMSE [dB]}[b][b][0.7]{\\sf median NMSE [dB]}\n\\psfrag{damped GAMP}[lB][lB][0.42]{\\sf MAD-GAMP}\n\\includegraphics[width=3.15in]{figures\/nmse_vs_mean.eps}\n\\caption{NMSE versus mean $\\mu$ at final algorithm iteration.  The reported NMSE is the median over $200$ realizations, with error bars shown on the VAMP trace.\n\\label{fig:nmse_vs_mean}}\n\\end{figure}\n\nFigure~\\ref{fig:nmse_vs_mean} shows that AMP and S-AMP diverged for even mildly mean-perturbed $\\mathbf{A}$.\nIn contrast, the figure shows that VAMP and mean-removed AD-GAMP (MAD-GAMP) closely matched the replica prediction for all mean parameters $\\mu$.\nIt also shows a relatively small gap between the replica prediction and the oracle bound, especially for small $\\mu$.\n\n\\begin{figure}[t]\n\\centering\n\\psfrag{nzmean=0.001}[b][b][0.7]{(a)}\n\\psfrag{nzmean=1}[b][b][0.7]{(b)}\n\\psfrag{iterations}[t][t][0.7]{\\sf iteration}\n\\psfrag{median NMSE [dB]}[b][b][0.7]{\\sf median NMSE [dB]}\n\\psfrag{damped GAMP}[lB][lB][0.42]{\\sf MAD-GAMP}\n\\includegraphics[width=3.15in]{figures\/nmse_vs_iter_mean.eps}\n\\caption{NMSE versus algorithm iteration when $\\mathbf{A}$ has mean $\\mu=0.001$ in (a) and $\\mu=1$ in (b).  The reported NMSE is the median over $200$ realizations, with error bars shown on the VAMP traces.\n\\label{fig:nmse_vs_iter_mean}}\n\\end{figure}\n\nFigure~\\ref{fig:nmse_vs_iter_mean}(a) plots NMSE versus algorithm iteration for matrix mean $\\mu=0.001$ and Figure~\\ref{fig:nmse_vs_iter_mean}(b) plots the same for $\\mu=1$.\nWhen $\\mu=0.001$, VAMP closely matched its SE at all iterations and converged noticeably quicker than AMP, S-AMP, and MAD-VAMP.\nWhen $\\mu=1$, there was a small but noticeable gap between VAMP and its SE for the first few iterations, although the gap closed after about $10$ iterations.\nThis gap may be due to the fact that the random matrix $\\mathbf{A}$ used for this experiment was not right-\\textb{orthogonally} invariant, since the dominant singular vectors are close to (scaled versions of) the $\\mathbf{1}$s vector for sufficiently large $\\mu$.\n\n\n\\subsection{Row-orthogonal $\\mathbf{A}$} \\label{sec:SNR}\n\nIn this section we investigate algorithm NMSE versus SNR for row-orthogonal $\\mathbf{A}$, i.e., $\\mathbf{A}$ constructed as in Section~\\ref{sec:ill} but with $\\kappa(\\mathbf{A})=1$.\nPrevious studies \\cite{cakmak2015samp,kabashima2014signal} have demonstrated that, when $\\mathbf{A}$ is \\textb{orthogonally} invariant but not i.i.d.\\ Gaussian (e.g., row-orthogonal),\nthe fixed points of S-AMP and diagonal-restricted EC are better than those of AMP because the former approaches exploit the singular-value spectrum of $\\mathbf{A}$, whereas AMP does not.\n\nTable~\\ref{tab:nmse_vs_snr} reports the NMSE achieved by VAMP, S-AMP, and AMP at three levels of SNR: $10$~dB, $20$~dB, and $30$~dB.\nThe NMSEs reported in the table were computed from an average of $1000$ independent realizations of $\\{\\mathbf{A},\\mathbf{x},\\mathbf{w}\\}$.\nSince the NMSE differences between the algorithms are quite small, the table also reports the standard error on each NMSE estimate to confirm its accuracy.\n\nTable~\\ref{tab:nmse_vs_snr} shows that VAMP and S-AMP gave nearly identical NMSE at all tested SNRs, which is expected because these two algorithms share the same fixed points.\nThe table also shows that VAMP's NMSE was strictly better than AMP's NMSE at low SNR (as expected), but that the NMSE difference narrows as the SNR increases.\nFinally, the table reports the replica prediction of the NMSE, which is about $3\\%$ lower (i.e., $-0.15$~dB) than VAMP's empirical NMSE at each SNR.\nWe attribute this difference to finite-dimensional effects.\n\n\\begin{table}[t]\n\\centering\n\\begin{tabular}{@{}c@{~} | @{~}c @{~~} r@{}l @{~~} r@{}l @{~~} r@{}l@{}}\nSNR &replica  &VAMP&(stderr)      &S-AMP&(stderr)      &AMP&(stderr) \\\\\\hline\n10 dB&5.09e-02&5.27e-02&(4.3e-04)&5.27e-02&(4.3e-04)&5.42e-02&(4.2e-04)\\\\\n20 dB&3.50e-03&3.57e-03&(2.7e-05)&3.58e-03&(2.7e-05)&3.62e-03&(2.6e-05)\\\\\n30 dB&2.75e-04&2.84e-04&(2.2e-06)&2.85e-04&(2.2e-06)&2.85e-04&(2.1e-06)\\\\\n\\end{tabular}\n\\medskip\n\n\\caption{Average NMSE versus SNR for row-orthogonal $\\mathbf{A}$, where the average was computed from $1000$ realizations. Standard error deviations are also reported.\n\\label{tab:nmse_vs_snr}}\n\\end{table}\n\n\\subsection{Discussion}\n\nOur numerical results confirm what is already known about the \\emph{fixed points} of diagonally restricted EC (via VAMP) and S-AMP.\nThat is,\nwhen $\\mathbf{A}$ is large and right-\\textb{orthogonally} invariant,\nthey agree with each other and with the replica prediction;\nand when $\\mathbf{A}$ is large i.i.d.\\ Gaussian (which is a special case of right-\\textb{orthogonally} invariant \\cite{TulinoV:04}),\nthey furthermore agree with the fixed points of AMP\n\\cite{cakmak2015samp,kabashima2014signal}.\n\nBut our numerical results also clarify that it is not enough for an algorithm to have good fixed points, because it may not converge to its fixed points.\nFor example, although the fixed points of S-AMP are good (i.e., replica matching) for \\emph{any} large right-\\textb{orthogonally} invariant $\\mathbf{A}$, our numerical results indicate that S-AMP converges only for a small subset of large right-\\textb{orthogonally} invariant $\\mathbf{A}$: those with singular-value spectra similar (or flatter than) i.i.d.\\ Gaussian $\\mathbf{A}$.\n\nThe SE analysis from Section~\\ref{sec:SE} establishes that, in the large-system limit and under matched priors, VAMP is guaranteed to converge to a fixed point that is also a fixed point of the replica equation \\eqref{eq:fixrep}.\nOur numerical results suggest that, even with large but finite-dimensional right \\textb{orthogonally} invariant $\\mathbf{A}$ (i.e., $512\\times 1024$ in our simulations),\nVAMP attains NMSEs that are very close to the replica prediction.\n\n\n\n\\section{Conclusions} \\label{sec:conc}\n\nIn this paper, we considered the standard linear regression (SLR) problem \\eqref{eq:yAx}, where the goal is to recover the vector $\\mathbf{x}^0$ from noisy linear measurements $\\mathbf{y}=\\mathbf{A}\\mathbf{x}^0+\\mathbf{w}$.\nOur work is inspired by Donoho, Maleki, and Montanari's AMP algorithm \\cite{DonohoMM:09}, which offers a computationally efficient approach to SLR.\nAMP has the desirable property that its behavior is rigorously characterized under large i.i.d.\\ sub-Gaussian $\\mathbf{A}$ by a scalar state evolution whose fixed points, when unique, are Bayes optimal \\cite{BayatiM:11}.\nA major shortcoming of AMP, however, is its fragility with respect to the i.i.d.\\ sub-Gaussian model on $\\mathbf{A}$: even small perturbations from this model can cause AMP to diverge.\n\nIn response, we proposed a vector AMP (VAMP) algorithm that (after performing an initial SVD) has similar complexity to AMP but is much more robust with respect to the matrix $\\mathbf{A}$.\nOur main contribution is establishing that VAMP's behavior can be rigorously characterized by a scalar state-evolution that holds for large, right-\\textb{orthogonally} invariant $\\mathbf{A}$.\nThe fixed points of VAMP's state evolution are, in fact,\nconsistent with the replica prediction of the minimum mean-squared\nerror recently derived in \\cite{tulino2013support}.\nWe also showed how VAMP can be derived as an approximation of belief propagation on a factor graph with vector-valued nodes, hence the name ``vector AMP.''\nFinally, we presented numerical experiments to demonstrate VAMP's robust convergence for ill-conditioned and mean-perturbed matrices $\\mathbf{A}$ that cause earlier AMP algorithms to diverge.\n\nAs future work, it would be interesting to extend VAMP to the generalized linear model, where the outputs $\\mathbf{A}\\mathbf{x}^0$ are non-linearly mapped to $\\mathbf{y}$.\nAlso, it would be interesting to design and analyze extensions of VAMP that are robust to \\textb{more general models for $\\mathbf{A}$, such as the case where $\\mathbf{A}$ is statistically coupled to $\\mathbf{x}^0$.}\n\n\\appendices\n\n\\section{Message-Passing Derivation of VAMP} \\label{sec:EP}\n\nIn this appendix, we detail the message-passing derivation of Algorithm~\\ref{algo:vamp}.\nBelow, we will use $k$ to denote the VAMP iteration and $n$ to index the elements of $N$-dimensional vectors like $\\mathbf{x}_{1},\\mathbf{r}_{1k}$ and $\\hat{\\mathbf{x}}_{1k}$.\nWe start by initializing the message-passing with\n$\\msg{\\delta}{\\mathbf{x}_1}(\\mathbf{x}_1)={\\mathcal N}(\\mathbf{x}_1;\\mathbf{r}_{10},\\gamma_{10}^{-1}\\mathbf{I})$.\nThe following steps are then repeated for $k=0,1,2,\\dots$.\n\nFrom Rule~\\ref{rule:b}, we first set\nthe approximate belief on $\\mathbf{x}_1$ as\n${\\mathcal N}(\\mathbf{x}_1;\\hat{\\mathbf{x}}_{1k},\\eta_{1k}^{-1}\\mathbf{I})$,\nwhere\n$\\hat{\\mathbf{x}}_{1k} = \\mathbb{E}[\\mathbf{x}_1|b_{\\textsf{sp}}(\\mathbf{x}_1)]$ and\n$\\eta_{1k}^{-1} = \\bkt{\\mathop{\\mathrm{diag}}(\\mathrm{Cov}[\\mathbf{x}_1|b_{\\textsf{sp}}(\\mathbf{x}_1)])}$\nfor the SP belief\n$b_{\\textsf{sp}}(\\mathbf{x}_1)\\propto p(\\mathbf{x}_1){\\mathcal N}(\\mathbf{x}_1;\\mathbf{r}_{1k},\\gamma_{1k}^{-1}\\mathbf{I})$.\nWith an i.i.d.\\ prior $p(\\mathbf{x}_1)$ as in \\eqref{eq:pxiid},\nwe have that $[\\hat{\\mathbf{x}}_{1k}]_n= g_1(r_{1k,n},\\gamma_{1k})$\nfor the conditional-mean estimator $g_1(\\cdot,\\gamma_{1k})$\ngiven in \\eqref{eq:g1mmsesca},\nyielding line~\\ref{line:x1} of Algorithm~\\ref{algo:vamp}.\nFurthermore, from \\eqref{eq:g1dervar} we see that\nthe corresponding conditional covariance is\n$\\gamma_{1k}^{-1}g_1'(r_{1k,n},\\gamma_{1k})$,\nyielding lines~\\ref{line:a1}-\\ref{line:eta1} of Algorithm~\\ref{algo:vamp}.\n\nNext, Rule~\\ref{rule:v2f} says to set the message $\\msg{\\mathbf{x}_1}{\\delta}(\\mathbf{x}_1)$\nproportional to\n${\\mathcal N}(\\mathbf{x}_1;\\hat{\\mathbf{x}}_{1k},\\eta_{1k}^{-1}\\mathbf{I})\n \/{\\mathcal N}(\\mathbf{x}_1;\\mathbf{r}_{1k},\\gamma_{1k}^{-1}\\mathbf{I})$.\nSince\n\\begin{align}\n\\lefteqn{\n{\\mathcal N}(\\mathbf{x};\\hat{\\mathbf{x}},\\eta^{-1}\\mathbf{I})\/{\\mathcal N}(\\mathbf{x};\\mathbf{r},\\gamma^{-1}\\mathbf{I})\n}\\nonumber\\\\\n&\\propto {\\mathcal N}\\big(\\mathbf{x};(\\hat{\\mathbf{x}}\\eta-\\mathbf{r}\\gamma)\/(\\eta-\\gamma),(\\eta-\\gamma)^{-1}\\mathbf{I}\\big)\n\\label{eq:gauss_div},\n\\end{align}\nwe have\n$\\msg{\\mathbf{x}_1}{\\delta}(\\mathbf{x}_1)={\\mathcal N}(\\mathbf{x}_1;\\mathbf{r}_{2k},\\gamma_{2k}^{-1}\\mathbf{I})$\nfor\n$\\mathbf{r}_{2k}=(\\hat{\\mathbf{x}}_{1k}\\eta_{1k}-\\mathbf{r}_{1k}\\gamma_{1k})\/(\\eta_{1k}-\\gamma_{1k})$\nand\n$\\gamma_{2k}=\\eta_{1k}-\\gamma_{1k}$,\nyielding lines~\\ref{line:gam2}-\\ref{line:r2} of Algorithm~\\ref{algo:vamp}.\nRule~\\ref{rule:f2v} then implies that the message $\\msg{\\mathbf{x}_1}{\\delta}(\\mathbf{x}_1)$\nwill flow rightward through the $\\delta$ node unchanged, manifesting as\n$\\msg{\\delta}{\\mathbf{x}_2}(\\mathbf{x}_2)={\\mathcal N}(\\mathbf{x}_2;\\mathbf{r}_{2k},\\gamma_{2k}^{-1}\\mathbf{I})$\non the other side.\n\nRule~\\ref{rule:b} then says to set the approximate belief on $\\mathbf{x}_2$ at\n${\\mathcal N}(\\mathbf{x}_2;\\hat{\\mathbf{x}}_{2k},\\eta_{2k}^{-1}\\mathbf{I})$,\nwhere\n$\\hat{\\mathbf{x}}_{2k} = \\mathbb{E}[\\mathbf{x}_2|b_{\\textsf{sp}}(\\mathbf{x}_2)]$ and\n$\\eta_{2k}^{-1} = \\bkt{\\mathop{\\mathrm{diag}}(\\mathrm{Cov}[\\mathbf{x}_2|b_{\\textsf{sp}}(\\mathbf{x}_2)])}$\nfor the SP belief\n$b_{\\textsf{sp}}(\\mathbf{x}_2)\\propto {\\mathcal N}(\\mathbf{x}_2;\\mathbf{r}_{2k},\\gamma_{2k}^{-1}\\mathbf{I})\n {\\mathcal N}(\\mathbf{y};\\mathbf{A}\\mathbf{x}_2,\\gamma_w^{-1}\\mathbf{I})$.\nUsing standard manipulations, it can be shown that this belief is Gaussian with mean\n\\begin{align}\n\\hat{\\mathbf{x}}_{2k}\n&= \\left( \\gamma_w \\mathbf{A}^{\\text{\\sf T}}\\mathbf{A} + \\gamma_{2k}\\mathbf{I}\\right)^{-1}\n        \\left( \\gamma_w\\mathbf{A}^{\\text{\\sf T}}\\mathbf{y} + \\gamma_{2k}\\mathbf{r}_{2k} \\right)\n\\label{eq:x2}\n\\end{align}\nand covariance $(\\gamma_w\\mathbf{A}^{\\text{\\sf T}}\\mathbf{A}+\\gamma_{2k}\\mathbf{I})^{-1}$.\nThe equivalence between \\eqref{eq:x2} and \\eqref{eq:g2slr}\nexplains line~\\ref{line:x2} of Algorithm~\\ref{algo:vamp}.\nFurthermore, it can be seen by inspection that the average of the diagonal of this covariance matrix coincides with\n$\\gamma_{2k}^{-1} \\bkt{\\mathbf{g}_2'(\\mathbf{r}_{2k},\\gamma_{2k})}$\nfor $\\bkt{\\mathbf{g}_2'(\\mathbf{r}_{2k},\\gamma_{2k})}$ from \\eqref{eq:a2slr},\nthus explaining lines~\\ref{line:a2}-\\ref{line:eta2} of Algorithm~\\ref{algo:vamp}.\n\nRule~\\ref{rule:v2f} then says to set the message $\\msg{\\mathbf{x}_2}{\\delta}(\\mathbf{x}_2)$ at\n${\\mathcal N}(\\mathbf{x}_2;\\hat{\\mathbf{x}}_{2k},\\eta_{2k}^{-1}\\mathbf{I})\n \/{\\mathcal N}(\\mathbf{x}_2;\\mathbf{r}_{2k},\\gamma_{2k}^{-1}\\mathbf{I})$,\nwhich \\eqref{eq:gauss_div} simplifies to\n${\\mathcal N}(\\mathbf{x}_2;\\mathbf{r}_{1,k\\! + \\! 1},\\gamma_{1,k\\! + \\! 1}^{-1}\\mathbf{I})$\nfor\n$\\mathbf{r}_{1,k\\! + \\! 1}=(\\hat{\\mathbf{x}}_{2k}\\eta_{2k}-\\mathbf{r}_{2k}\\gamma_{2k})\/(\\eta_{2k}-\\gamma_{2k})$\nand\n$\\gamma_{1,k\\! + \\! 1}=\\eta_{2k}-\\gamma_{2k}$,\nyielding lines~\\ref{line:gam1}-\\ref{line:r1} of Algorithm~\\ref{algo:vamp}.\nFinally, Rule~\\ref{rule:f2v} implies that the message $\\msg{\\mathbf{x}_2}{\\delta}(\\mathbf{x}_2)$\nflows left through the $\\delta$ node unchanged, manifesting as\n$\\msg{\\delta}{\\mathbf{x}_1}(\\mathbf{x}_1)={\\mathcal N}(\\mathbf{x}_1;\\mathbf{r}_{1k\\! + \\! 1},\\gamma_{1,k\\! + \\! 1}^{-1}\\mathbf{I})$\non the other side.\nThe above messaging sequence is then repeated with $k\\leftarrow k+1$.\n\n\n\\section{Convergence of Vector Sequences} \\label{sec:empConv}\nWe review some definitions from the Bayati-Montanari paper \\cite{BayatiM:11}, since we will use the same\nanalysis framework in this paper.\nFix a dimension $r > 0$, and suppose that, for each $N$,\n$\\mathbf{x}(N)$ is a vector of the form\n\\[\n    \\mathbf{x}(N) = (\\mathbf{x}_1(N),\\ldots,\\mathbf{x}_N(N)),\n\\]\n\\textb{with vector sub-components} $\\mathbf{x}_n(N) \\in {\\mathbb{R}}^r$.  Thus, the total dimension\nof $\\mathbf{x}(N)$ is $rN$.  In this case, we will say that\n$\\mathbf{x}(N)$ is a \\emph{block vector sequence that scales with $N$\nunder blocks $\\mathbf{x}_n(N) \\in {\\mathbb{R}}^r$.}\nWhen $r=1$, so that the blocks are scalar, we will simply say that\n$\\mathbf{x}(N)$ is a \\emph{vector sequence that scales with $N$}.\nSuch vector sequences can be deterministic or random.\nIn most cases, we will omit the notational dependence on $N$ and simply write $\\mathbf{x}$.\n\nNow, given $p \\geq 1$,\na function $\\mathbf{f}:{\\mathbb{R}}^s \\rightarrow {\\mathbb{R}}^r$ is called \\emph{pseudo-Lipschitz of order $p$},\nif there exists a constant $C > 0$ such that for all $\\mathbf{x}_1,\\mathbf{x}_2 \\in{\\mathbb{R}}^s$,\n\\[\n    \\|\\mathbf{f}(\\mathbf{x}_1)-\\mathbf{f}(\\mathbf{x}_2)\\| \\leq C\\|\\mathbf{x}_1-\\mathbf{x}_2\\|\\left[ 1 + \\|\\mathbf{x}_1\\|^{p-1}\n    + \\|\\mathbf{x}_2\\|^{p-1} \\right].\n\\]\nObserve that in the case $p=1$, pseudo-Lipschitz continuity reduces to\nthe standard Lipschitz continuity.\n\n\\textb{Now suppose that $\\mathbf{x}=\\mathbf{x}(N)$ is a block vector sequence,\nwhich may be deterministic or random.}\nGiven $p \\geq 1$, we will say that $\\mathbf{x}=\\mathbf{x}(N)$ converges\n\\emph{empirically with $p$-th order moments} if there exists a random variable\n$X \\in {\\mathbb{R}}^r$ such that\n\\begin{enumerate}[(i)]\n\\item $\\mathbb{E}|X|^p < \\infty$; and\n\\item for any scalar-valued pseudo-Lipschitz continuous function $f(\\cdot)$ of order $p$,\n\\begin{equation} \\label{eq:plplim}\n    \\lim_{N \\rightarrow \\infty} \\frac{1}{N} \\sum_{n=1}^N f(x_n(N)) = \\mathbb{E}\\left[ f(X) \\right] \\mbox{ a.s.}.\n\\end{equation}\n\\end{enumerate}\nThus, the empirical mean of the components $f(x_n(N))$ converges to\nthe expectation $\\mathbb{E}[ f(X) ]$.\n\\textb{When $\\mathbf{x}$ converges empirically with $p$-th order moments},\nwe will write, with some abuse of notation,\n\\begin{equation} \\label{eq:plLim}\n    \\lim_{N \\rightarrow \\infty} \\left\\{ x_n \\right\\}_{n=1}^N \\stackrel{PL(p)}{=} X,\n\\end{equation}\nwhere, as usual, we have omitted the dependence $x_n=x_n(N)$.\n\\textb{Note that the almost sure convergence in condition (ii) applies to the case\nwhere $\\mathbf{x}(N)$ is a random vector sequence.  Importantly,\nthis condition holds pointwise over each function $f(\\cdot)$.\nIt is shown in \\cite[Lemma 4]{BayatiM:11} that, if condition (i) is true and\ncondition (ii) is true for any bounded continuous functions $f(x)$\nas well as $f(x)=x^p$, then condition (ii) holds for all pseudo-Lipschitz functions\nof order $p$. }\n\nWe conclude with one final definition.\nLet ${\\bm{\\phi}}(\\mathbf{r},\\gamma)$ be a function on $\\mathbf{r} \\in {\\mathbb{R}}^s$ and $\\gamma \\in {\\mathbb{R}}$.\nWe say that ${\\bm{\\phi}}(\\mathbf{r},\\gamma)$ is \\emph{uniformly Lipschitz continuous} in $\\mathbf{r}$\nat $\\gamma=\\overline{\\gamma}$ if there exists constants\n$L_1$ and $L_2 \\geq 0$ and an open neighborhood $U$ of $\\overline{\\gamma}$, such that\n\\begin{equation} \\label{eq:unifLip1}\n    \\|{\\bm{\\phi}}(\\mathbf{r}_1,\\gamma)-{\\bm{\\phi}}(\\mathbf{r}_2,\\gamma)\\| \\leq L_1\\|\\mathbf{r}_1-\\mathbf{r}_2\\|,\n\\end{equation}\nfor all $\\mathbf{r}_1,\\mathbf{r}_2 \\in {\\mathbb{R}}^s$ and $\\gamma \\in U$; and\n\\begin{equation} \\label{eq:unifLip2}\n    \\|{\\bm{\\phi}}(\\mathbf{r},\\gamma_1)-{\\bm{\\phi}}(\\mathbf{r},\\gamma_2)\\| \\leq L_2\\left(1+\\|\\mathbf{r}\\|\\right)|\\gamma_1-\\gamma_2|,\n\\end{equation}\nfor all $\\mathbf{r} \\in {\\mathbb{R}}^s$ and $\\gamma_1,\\gamma_2 \\in U$.\n\n\\section{Proof of Lemmas~\\ref{lem:errfn} and \\ref{lem:sens}} \\label{sec:errsenspf}\n\nFor Lemma~\\ref{lem:errfn}, part (a) follows immediately from\n\\eqref{eq:g1mmsex0} and \\eqref{eq:eps1}.\nTo prove part (b), suppose\n\\[\n    \\mathbf{y} = \\mathbf{A}\\mathbf{x}^0 + \\mathbf{w}, \\quad \\mathbf{r}_2 = \\mathbf{x}^0+\\mathbf{q}.\n\\]\nThen, the error is given by\n\\begin{align}\n    \\MoveEqLeft \\mathbf{g}_2(\\mathbf{r}_{2},\\gamma_2) - \\mathbf{x}^0\n    \\stackrel{(a)}{=} \\left( \\gamma_w \\mathbf{A}^{\\text{\\sf T}}\\mathbf{A} + \\gamma_2\\mathbf{I}\\right)^{-1}\n        \\nonumber \\\\\n    & \\quad \\times \\left( \\gamma_w\\mathbf{A}^{\\text{\\sf T}}\\mathbf{A}\\mathbf{x}^0 + \\gamma_w\\mathbf{A}^{\\text{\\sf T}}\\mathbf{w}\n    + \\gamma_2\\mathbf{r}_{2} \\right) -\\mathbf{x}^0 \\nonumber \\\\\n   &\\stackrel{(b)}{=}\n   \\left( \\gamma_w \\mathbf{A}^{\\text{\\sf T}}\\mathbf{A} + \\gamma_2\\mathbf{I}\\right)^{-1}\n    \\left( \\gamma_2\\mathbf{q}+ \\gamma_w\\mathbf{A}^{\\text{\\sf T}}\\mathbf{w}\\right),  \\nonumber \\\\\n   &\\stackrel{(c)}{=} \\mathbf{Q}^{-1}\n    \\left( \\gamma_2\\mathbf{q}+ \\gamma_w\\mathbf{A}^{\\text{\\sf T}}\\mathbf{w}\\right), \\nonumber\n\\end{align}\nwhere (a) follows by substituting $\\mathbf{y} = \\mathbf{A}\\mathbf{x}^0 + \\mathbf{w}$ into \\eqref{eq:g2slr};\npart (b) follows from the substitution $\\textb{\\mathbf{r}_2} = \\mathbf{x}^0+\\mathbf{q}$ and collecting the terms\nwith $\\mathbf{x}^0$; and (c) follows from the definition of $\\mathbf{Q}$ in \\eqref{eq:QQtdef}.\nHence, the error covariance matrix is given\n\\begin{align*}\n    \\MoveEqLeft \\mathbb{E}\\left[ (\\mathbf{g}_2(\\mathbf{r}_{2},\\gamma_2) - \\mathbf{x}^0)(\\mathbf{g}_2(\\mathbf{r}_{2},\\gamma_2) - \\mathbf{x}^0)^{\\text{\\sf T}}\n        \\right] \\\\\n    &= \\mathbf{Q}^{-1}\\left[ \\gamma_2^2\\mathbb{E}[\\mathbf{q}\\qbf^{\\text{\\sf T}}] + \\gamma_w^2\\mathbf{A}\\mathbb{E}[\\mathbf{w}\\wbf^{\\text{\\sf T}}]\\mathbf{A}^{\\text{\\sf T}}\n    \\right]\\mathbf{Q}^{-1} \\\\\n    &= \\mathbf{Q}^{-1}\\tilde{\\mathbf{Q}}\\mathbf{Q}^{-1},\n\\end{align*}\nwhere we have used the\nthe fact that $\\mathbf{q}$ and $\\mathbf{w}$ are independent Gaussians\nwith variances $\\tau_2$ and $\\gamma_{w0}^{-1}$.    This proves \\eqref{eq:eps2Q}.\n\\textb{Then,} under the matched condition, \\textb{we have that} $\\mathbf{Q}=\\tilde{\\mathbf{Q}}$, which proves \\eqref{eq:eps2Qmatch}.\nPart (c) of Lemma~\\ref{lem:errfn} follows from part (b) by using the SVD \\eqref{eq:ASVD}.\n\nFor Lemma~\\ref{lem:sens}, part (a) follows from averaging \\eqref{eq:g1mmsex0} over $r_1$.\nPart (b) follows by taking the derivative in \\eqref{eq:g2slr} and part (c)\nfollows from using the SVD~\\eqref{eq:ASVD}.\n\n\n\\section{Orthogonal Matrices Under Linear Constraints}\n\n\nIn preparation for proving Theorem~\\ref{thm:se},\nwe derive various results on orthogonal matrices subject to linear constraints.\nTo this end, suppose $\\mathbf{V} \\in {\\mathbb{R}}^{N \\times N}$ is an orthogonal matrix\nsatisfying linear constraints\n\\begin{equation} \\label{eq:AVB}\n    \\mathbf{A} = \\mathbf{V}\\mathbf{B},\n\\end{equation}\nfor some matrices $\\mathbf{A}, \\mathbf{B} \\in {\\mathbb{R}}^{N \\times s}$ for some $s$.  Assume $\\mathbf{A}$ and $\\mathbf{B}$\nare full column rank (hence $s \\leq N$).  Let\n\\begin{equation} \\label{eq:UABdef}\n    \\mathbf{U}_\\mathbf{A} = \\mathbf{A}(\\mathbf{A}^{\\text{\\sf T}}\\mathbf{A})^{-1\/2}, \\quad\n    \\mathbf{U}_\\mathbf{B} = \\mathbf{B}(\\mathbf{B}^{\\text{\\sf T}}\\mathbf{B})^{-1\/2}.\n\\end{equation}\nAlso, let $\\mathbf{U}_{\\mathbf{A}^\\perp}$ and $\\mathbf{U}_{\\mathbf{B}^\\perp}$ be any $N \\times (N-s)$\nmatrices whose columns are\nan orthonormal bases for $\\mathrm{Range}(\\mathbf{A})^\\perp$ and\n$\\mathrm{Range}(\\mathbf{B})^\\perp$, respectively.\nDefine\n\\begin{equation} \\label{eq:Vtdef}\n    \\tilde{\\mathbf{V}} := \\mathbf{U}_{\\mathbf{A}^\\perp}^{\\text{\\sf T}}\\mathbf{V}\\mathbf{U}_{\\mathbf{B}^\\perp},\n\\end{equation}\nwhich has dimension $(N-s) \\times (N-s)$.\n\n\\begin{lemma}  \\label{lem:orthogRep}\nUnder the above definitions $\\tilde{\\mathbf{V}}$ satisfies\n\\begin{equation} \\label{eq:VVtrep}\n    \\mathbf{V} = \\mathbf{A}(\\mathbf{A}^{\\text{\\sf T}}\\mathbf{A})^{-1}\\mathbf{B}^{\\text{\\sf T}} + \\mathbf{U}_{\\mathbf{A}^\\perp}\\tilde{\\mathbf{V}}\\mathbf{U}_{\\mathbf{B}^\\perp}^{\\text{\\sf T}}.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\n\\textb{Let $\\mathbf{P}_\\mathbf{A}:=\\mathbf{U}_\\mathbf{A}\\mathbf{U}_\\mathbf{A}^{\\text{\\sf T}}$ and $\\mathbf{P}_\\mathbf{A}^\\perp:=\\mathbf{U}_{\\mathbf{A}^\\perp}\\mathbf{U}_{\\mathbf{A}^\\perp}^{\\text{\\sf T}}$ are\nthe orthogonal projections onto $\\mathrm{Range}(\\mathbf{A})$ and $\\mathrm{Range}(\\mathbf{A})^\\perp$ respectively.\nDefine $\\mathbf{P}_\\mathbf{B}$ and $\\mathbf{P}_\\mathbf{B}^\\perp$ similarly.  Since, $\\mathbf{A}=\\mathbf{V}\\mathbf{B}$, we have $\\mathbf{V}^{\\text{\\sf T}}\\mathbf{A} = \\mathbf{B}$ and therefore,\n\\begin{equation} \\label{eq:PAVPB}\n    \\mathbf{P}_\\mathbf{A}^\\perp\\mathbf{V}\\mathbf{P}_{\\mathbf{B}} = \\mathbf{0}, \\quad\n    \\mathbf{P}_\\mathbf{A}\\mathbf{V}\\mathbf{P}_{\\mathbf{B}}^\\perp = \\mathbf{0}.\n\\end{equation}\nTherefore,\n\\begin{align}\n    \\mathbf{V} &= (\\mathbf{P}_\\mathbf{A}+\\mathbf{P}_\\mathbf{A}^\\perp)\\mathbf{V}(\\mathbf{P}_\\mathbf{B}+\\mathbf{P}_\\mathbf{B}^\\perp) \\nonumber \\\\\n    &= (\\mathbf{P}_\\mathbf{A}\\mathbf{V}\\mathbf{P}_\\mathbf{B} + \\mathbf{P}_\\mathbf{A}^\\perp\\mathbf{V}\\mathbf{P}_\\mathbf{B}^\\perp). \\label{eq:PVP1}\n\\end{align}\nNow,\n\\begin{align}\n    \\MoveEqLeft \\mathbf{P}_\\mathbf{A}\\mathbf{V}\\mathbf{P}_\\mathbf{B} = \\mathbf{P}_\\mathbf{A}\\mathbf{V}\\mathbf{B}(\\mathbf{B}\\Bbf^{\\text{\\sf T}})^{-1}\\mathbf{B}^{\\text{\\sf T}} \\nonumber \\\\\n    &= \\mathbf{P}_{\\mathbf{A}}\\mathbf{A}(\\mathbf{B}\\Bbf^{\\text{\\sf T}})^{-1}\\mathbf{B}^{\\text{\\sf T}} \\nonumber \\\\\n    &= \\mathbf{A}(\\mathbf{B}\\Bbf^{\\text{\\sf T}})^{-1}\\mathbf{B}^{\\text{\\sf T}} = \\mathbf{A}(\\mathbf{A}^{\\text{\\sf T}}\\mathbf{A})^{-1}\\mathbf{B}^{\\text{\\sf T}}, \\label{eq:PVP2}\n\\end{align}\nwhere, in the last step we used the fact that\n\\[\n    \\mathbf{A}^{\\text{\\sf T}}\\mathbf{A}=\\mathbf{B}^{\\text{\\sf T}}\\mathbf{V}^{\\text{\\sf T}}\\mathbf{V}\\mathbf{B} = \\mathbf{B}^{\\text{\\sf T}}\\mathbf{B}.\n\\]\nAlso,  using the definition of $\\tilde{\\mathbf{V}}$ in \\eqref{eq:Vtdef},\n\\begin{equation} \\label{eq:PVP3}\n    \\mathbf{P}_{\\mathbf{A}^\\perp}\\mathbf{V}\\mathbf{P}_{\\mathbf{B}^\\perp}^{\\text{\\sf T}}  =\\mathbf{U}_{\\mathbf{A}^\\perp}\\tilde{\\mathbf{V}}\\mathbf{U}_{\\mathbf{B}^\\perp}^{\\text{\\sf T}}.\n\\end{equation}\nSubstituting \\eqref{eq:PVP2} and \\eqref{eq:PVP3} into \\eqref{eq:PVP1} obtains \\eqref{eq:VVtrep}.\nTo prove that $\\tilde{\\mathbf{V}}$ is orthogonal,\n\\begin{align*}\n    \\MoveEqLeft \\tilde{\\mathbf{V}}^{\\text{\\sf T}}\\tilde{\\mathbf{V}} \\stackrel{(a)}{=}\n    \\mathbf{U}_{\\mathbf{B}^\\perp}^{\\text{\\sf T}}\\mathbf{V}\\mathbf{P}_\\mathbf{A}\\mathbf{V}\\mathbf{U}_{\\mathbf{B}^\\perp}\n    \\\\\n    &\\stackrel{(b)}{=}\n    \\mathbf{U}_{\\mathbf{B}^\\perp}^{\\text{\\sf T}}\\mathbf{V}^{\\text{\\sf T}}\\mathbf{V}\\mathbf{U}_{\\mathbf{B}^\\perp} \\stackrel{(c)}{=} \\mathbf{I},\n\\end{align*}\nwhere (a) uses \\eqref{eq:Vtdef}; (b) follows from \\eqref{eq:PAVPB} and\n(c) follows from the fact that $\\mathbf{V}$ and $\\mathbf{U}_{\\mathbf{B}^\\perp}$ have orthonormal\ncolumns.\n}\n\\end{proof}\n\n\\begin{lemma} \\label{lem:orthogLin}  Let $\\mathbf{V} \\in {\\mathbb{R}}^{N \\times N}$ be a random matrix\nthat is Haar distributed.  Suppose that $\\mathbf{A}$ and $\\mathbf{B}$ are deterministic and $G$ is\nthe event that $\\mathbf{V}$ satisfies linear constraints \\eqref{eq:AVB}.\nThen, the conditional distribution given $G$,\n$\\tilde{\\mathbf{V}}$ is Haar distributed matrix independent of $G$.  Thus,\n\\[\n    \\left. \\mathbf{V} \\right|_{G} \\stackrel{d}{=}\n    \\mathbf{A}(\\mathbf{A}^{\\text{\\sf T}}\\mathbf{A})^{-1}\\mathbf{B}^{\\text{\\sf T}} + \\mathbf{U}_{\\mathbf{A}^\\perp}\\tilde{\\mathbf{V}}\\mathbf{U}_{\\mathbf{B}^\\perp}^{\\text{\\sf T}},\n\\]\nwhere $\\tilde{\\mathbf{V}}$ is Haar distributed and independent of $G$.\n\\end{lemma}\n\\begin{proof}   Let $O_N$  be the set\nof $N \\times N$ orthogonal matrices and let ${\\mathcal L}$ be the set of matrices $\\mathbf{V} \\in O_N$\nthat satisfy the linear constraints \\eqref{eq:AVB}.  If $p_{\\mathbf{V}}(\\mathbf{V})$ is the uniform density\non $O_N$ (i.e.\\ the Haar measure), the conditional density on $\\mathbf{V}$ given the event $G$,\n\\[\n    p_{\\mathbf{V}|G}(\\mathbf{V}|G) = \\frac{1}{Z}p_{\\mathbf{V}}(\\mathbf{V})\\indic{\\mathbf{V} \\in {\\mathcal L}},\n\\]\nwhere $Z$ is the normalization constant.  Now let $\\phi:\\tilde{\\mathbf{V}} \\mapsto \\mathbf{V}$\nbe the mapping described by \\eqref{eq:VVtrep} which maps $O_{N-s}$ to ${\\mathcal L}$.\nThis mapping is invertible.  Since $\\phi$ is affine, the conditional density on\n$\\tilde{\\mathbf{V}}$ is given by\n\\begin{align}\n    \\MoveEqLeft p_{\\tilde{\\mathbf{V}}|G}(\\tilde{\\mathbf{V}}|G) \\propto\n    p_{\\mathbf{V}|G}(\\phi(\\tilde{\\mathbf{V}})|G) \\nonumber \\\\\n    &\\propto p_{\\mathbf{V}}(\\phi(\\tilde{\\mathbf{V}}))\\indic{\\phi(\\tilde{\\mathbf{V}} \\in {\\mathcal L})}\n    = p_{\\mathbf{V}}(\\phi(\\tilde{\\mathbf{V}})), \\label{eq:ptvg}\n\\end{align}\nwhere in the last step we used the fact that, for any matrix $\\tilde{\\mathbf{V}}$,\n$\\phi(\\tilde{\\mathbf{V}}) \\in {\\mathcal L}$ (i.e.\\ satisfies the linear constraints \\eqref{eq:AVB}).\nNow to show that $\\tilde{\\mathbf{V}}$ is conditionally Haar distributed, we need to show that for any\northogonal matrix $\\mathbf{W}_0 \\in O_{N-s}$,\n\\begin{equation} \\label{eq:pvtorthog}\n    p_{\\tilde{\\mathbf{V}}|G}(\\mathbf{W}_0\\tilde{\\mathbf{V}}|G)=p_{\\tilde{\\mathbf{V}}|G}(\\tilde{\\mathbf{V}}|G).\n\\end{equation}\nTo prove this, given $\\mathbf{W}_0 \\in O_{N-s}$, define the matrix,\n\\[\n    \\mathbf{W} = \\mathbf{U}_{\\mathbf{A}}\\mathbf{U}_{\\mathbf{A}}^{\\text{\\sf T}} + \\mathbf{U}_{\\mathbf{A}^\\perp}\\mathbf{W}_0\\mathbf{U}_{\\mathbf{A}^\\perp}^{\\text{\\sf T}}.\n\\]\nOne can verify that $\\mathbf{W} \\in O_N$ (i.e.\\ it is orthogonal) and\n\\begin{equation} \\label{eq:pwvt}\n    \\phi(\\mathbf{W}_0\\tilde{\\mathbf{V}}) =\\mathbf{W}\\phi(\\tilde{\\mathbf{V}}).\n\\end{equation}\nHence,\n\\begin{align}\n    \\MoveEqLeft p_{\\tilde{\\mathbf{V}}|G}(\\mathbf{W}_0\\tilde{\\mathbf{V}}|G)\n    \\stackrel{(a)}{\\propto} p_{\\mathbf{V}}(\\phi(\\mathbf{W}_0\\tilde{\\mathbf{V}})) \\nonumber \\\\\n    &\\stackrel{(b)}{\\propto} p_{\\mathbf{V}}(\\mathbf{W}\\phi(\\tilde{\\mathbf{V}}))\n    \\stackrel{(c)}{\\propto} p_{\\mathbf{V}}(\\phi(\\tilde{\\mathbf{V}})), \\nonumber\n\\end{align}\nwhere (a) follows from \\eqref{eq:ptvg}; (b) follows from \\eqref{eq:pwvt}; and\n(c) follows from the orthogonal invariance of $\\mathbf{V}$.  Hence, the conditional density\nof $\\tilde{\\mathbf{V}}$ is invariant under orthogonal transforms and is thus Haar distributed.\n\\end{proof}\n\nWe will use Lemma~\\ref{lem:orthogLin} in conjunction with the following\nsimple result.\n\n\\begin{lemma} \\label{lem:orthogGaussLim} Fix a dimension $s \\geq 0$,\nand suppose that $\\mathbf{x}(N)$ and $\\mathbf{U}(N)$ are sequences such that\nfor each $N$,\n\\begin{enumerate}[(i)]\n\\item $\\mathbf{U}=\\mathbf{U}(N) \\in {\\mathbb{R}}^{N\\times (N-s)}$ is a deterministic\nmatrix with $\\mathbf{U}^{\\text{\\sf T}}\\mathbf{U} = \\mathbf{I}$;\n\\item $\\mathbf{x}=\\mathbf{x}(N) \\in {\\mathbb{R}}^{N-s}$ a random vector that\nis isotropically distributed  in that $\\mathbf{V}\\mathbf{x}\\stackrel{d}{=}\\mathbf{x}$ for any orthogonal $(N-s)\\times(N-s)$ matrix\n$\\mathbf{V}$.\n\\item The \\textb{normalized squared Euclidean norm} converges almost surely as\n\\[\n    \\lim_{N \\rightarrow \\infty} \\frac{1}{N} \\|\\mathbf{x}\\|^2 = \\tau,\n\\]\nfor some $\\tau > 0$.\n\\end{enumerate}\nThen, if we define $\\mathbf{y} = \\mathbf{U}\\mathbf{x}$, we have that the components of $\\mathbf{y}$\nconverge empirically to a Gaussian random variable\n\\begin{equation} \\label{eq:ygausslim}\n    \\lim_{N \\rightarrow \\infty} \\{ y_n \\} \\stackrel{PL(2)}{=} Y \\sim {\\mathcal N}(0,\\tau).\n\\end{equation}\n\\end{lemma}\n\\begin{proof}  Since $\\mathbf{x}$ is isotropically distributed, it can be generated as a normalized\nGaussian, i.e.\\\n\\[\n    \\mathbf{x} \\stackrel{d}{=} \\frac{\\|\\mathbf{x}\\|}{\\|\\mathbf{w}_0\\|}\\mathbf{w}_0, \\quad \\mathbf{w}_0 \\sim {\\mathcal N}(\\mathbf{0},\\mathbf{I}_{N-s}).\n\\]\nFor each $N$, let $\\mathbf{U}_\\perp$ be an $N \\times s$ matrix such that $\\mathbf{S} := [\\mathbf{U} ~ \\mathbf{U}_{\\perp}]$ is\northogonal.  That is, the $s$ columns of $\\mathbf{U}_\\perp$ are an orthonormal basis\nof the orthogonal complement of the $\\mathrm{Range}(\\mathbf{U})$.  If we let $\\mathbf{w}_1 \\sim {\\mathcal N}(0,\\mathbf{I}_s)$,\nthen if we define\n\\[\n    \\mathbf{w} = \\left[ \\begin{array}{c} \\mathbf{w}_0 \\\\ \\mathbf{w}_1 \\end{array} \\right],\n\\]\nso that $\\mathbf{w} \\sim {\\mathcal N}(0,\\mathbf{I}_N)$.\nWith this definition, we can write $\\mathbf{y}$ as\n\\begin{equation} \\label{eq:yux}\n  \\mathbf{y} = \\mathbf{U}\\mathbf{x} \\stackrel{d}{=} \\frac{\\|\\mathbf{x}\\|}{\\|\\mathbf{w}_0\\|}\\left[ \\mathbf{S}\\mathbf{w} - \\mathbf{U}_{\\perp}\\mathbf{w}_1 \\right].\n\\end{equation}\nNow,\n\\[\n    \\lim_{N \\rightarrow\\infty} \\frac{\\|\\mathbf{x}\\|}{\\|\\mathbf{w}_0\\|} = \\sqrt{\\tau},\n\\]\nalmost surely.  Also, since $\\mathbf{w} \\sim {\\mathcal N}(0,\\mathbf{I})$ and $\\mathbf{S}$ is orthogonal,\n$\\mathbf{S}\\mathbf{w} \\sim {\\mathcal N}(0,\\mathbf{I})$.  Finally, since $\\mathbf{w}_1$ is $s$-dimensional,\n\\[\n    \\lim_{N \\rightarrow \\infty} \\frac{1}{N} \\|\\mathbf{U}_{\\perp}\\mathbf{w}_1\\|^2 =\n    \\lim_{N \\rightarrow \\infty} \\frac{1}{N} \\|\\mathbf{w}_1\\|^2 = 0,\n\\]\nalmost surely.  Substituting these properties into \\eqref{eq:yux},\nwe obtain \\eqref{eq:ygausslim}.\n\\end{proof}\n\n\\section{A General Convergence Result}\n\n\nTo analyze the VAMP method, we a consider the\nfollowing more general recursion.  For each dimension $N$, we are given\nan orthogonal matrix $\\mathbf{V} \\in {\\mathbb{R}}^{N \\times N}$,\nand an initial vector $\\mathbf{u}_0 \\in {\\mathbb{R}}^N$.  \n\\textb{Also, we are given disturbance vectors \n\\[\n    \\mathbf{w}^p=(w_1^p,\\ldots,w_n^p), \\quad\n    \\mathbf{w}^q=(w_1^q,\\ldots,w_n^q),\n\\]\nwhere the components $w_n^p \\in {\\mathbb{R}}^{n_p}$ and $w_n^q \\in {\\mathbb{R}}^{n_q}$ \nfor some finite dimensions $n_p$ and $n_q$ that do not grow with $N$.\n}\nThen, we generate a sequence of iterates by the following recursion:%\n\\begin{subequations}\n\\label{eq:algoGen}\n\\begin{align}\n    \\mathbf{p}_k &= \\mathbf{V}\\mathbf{u}_k \\label{eq:pupgen} \\\\\n    \\alpha_{1k} &= \\bkt{ \\mathbf{f}_p'(\\mathbf{p}_k,\\mathbf{w}^p,\\gamma_{1k})},\n    \\quad \\gamma_{2k} = \\Gamma_1(\\gamma_{1k},\\alpha_{1k})\n        \\label{eq:alpha1gen}  \\\\\n    \\mathbf{v}_k &= C_1(\\alpha_{1k})\\left[\n        \\mathbf{f}_p(\\mathbf{p}_k,\\mathbf{w}^p,\\gamma_{1k})- \\alpha_{1k} \\mathbf{p}_{k} \\right] \\label{eq:vupgen} \\\\\n    \\mathbf{q}_k &= \\mathbf{V}^{\\text{\\sf T}}\\mathbf{v}_k \\label{eq:qupgen} \\\\\n    \\alpha_{2k} &= \\bkt{ \\mathbf{f}_q'(\\mathbf{q}_k,\\mathbf{w}^q,\\gamma_{2k})},\n    \\quad \\gamma_{1,k\\! + \\! 1} =\\Gamma_2(\\gamma_{2k},\\alpha_{2k})\n        \\label{eq:alpha2gen} \\\\\n    \\mathbf{u}_{k\\! + \\! 1} &= C_2(\\alpha_{2k})\\left[\n        \\mathbf{f}_q(\\mathbf{q}_k,\\mathbf{w}^q,\\gamma_{2k}) - \\alpha_{2k}\\mathbf{q}_{k} \\right], \\label{eq:uupgen}\n\\end{align}\n\\end{subequations}\nwhich is initialized with some vector $\\mathbf{u}_0$ and scalar $\\gamma_{10}$.\nHere, $\\mathbf{f}_p(\\cdot)$ and $\\mathbf{f}_q(\\cdot)$ are separable functions, meaning\n\\begin{align}\\label{eq:fpqcomp}\n\\begin{split}\n    \\left[ \\mathbf{f}_p(\\mathbf{p},\\mathbf{w}^p,\\gamma_1)\\right]_n = f_p(p_n,w^p_n,\\gamma_1)~\\forall n, \\\\\n    \\left[ \\mathbf{f}_q(\\mathbf{q},\\mathbf{w}^q,\\gamma_2)\\right]_n = f_q(q_n,w^q_n,\\gamma_2)~\\forall n,\n\\end{split}\n\\end{align}\nfor scalar-valued functions $f_p(\\cdot)$ and $f_q(\\cdot)$.\nThe functions $\\Gamma_i(\\cdot)$ and $C_i(\\cdot)$ are also scalar-valued.\n\\textb{In the recursion \\eqref{eq:algoGen}, the variables $\\gamma_{1k}$\nand $\\gamma_{2k}$ \nrepresent some parameter of the update functions $\\mathbf{f}_p(\\cdot)$ and $\\mathbf{f}_q(\\cdot)$,\nand the functions $\\Gamma_i(\\cdot)$ represent how these parameters are updated.}\n\nSimilar to our analysis of the VAMP, we consider the following\nlarge-system limit (LSL) analysis.\nWe consider a sequence of runs of the recursions indexed by $N$.\nWe model the initial condition $\\mathbf{u}_0$ and disturbance vectors $\\mathbf{w}^p$ and $\\mathbf{w}^q$\nas deterministic sequences that scale with $N$ and assume that their components\nconverge empirically as\n\\begin{equation} \\label{eq:U0lim}\n    \\lim_{N \\rightarrow \\infty} \\{ u_{0n} \\} \\stackrel{PL(2)}{=} U_0,\n\\end{equation}\nand\n\\begin{equation} \\label{eq:WpqLim}\n    \\lim_{N \\rightarrow \\infty} \\{ w^p_n \\} \\stackrel{PL(2)}{=} W^p, \\quad\n    \\lim_{N \\rightarrow \\infty} \\{ w^q_n \\} \\stackrel{PL(2)}{=} W^q,\n\\end{equation}\nto random variables $U_0$, $W^p$ and $W^q$.  \\textb{The\nvectors $W_p$ and $W_q$ are random vectors in ${\\mathbb{R}}^{n_p}$ and ${\\mathbb{R}}^{n_q}$,\nrespectively.} We assume that the\ninitial constant converges as\n\\begin{equation} \\label{eq:gam10limgen}\n    \\lim_{N \\rightarrow \\infty} \\gamma_{10} = \\overline{\\gamma}_{10},\n\\end{equation}\nfor some $\\overline{\\gamma}_{10}$.\n  The matrix $\\mathbf{V} \\in {\\mathbb{R}}^{N \\times N}$\nis assumed to be uniformly distributed on the set of orthogonal matrices\nindependent of $\\mathbf{r}_0$, $\\mathbf{w}^p$ and $\\mathbf{w}^q$.\nSince $\\mathbf{r}_0$, $\\mathbf{w}^p$ and\n$\\mathbf{w}^q$ are deterministic, the only randomness is in the matrix $\\mathbf{V}$.\n\nUnder the above assumptions, define the SE equations\n\\begin{subequations} \\label{eq:segen}\n\\begin{align}\n    \\overline{\\alpha}_{1k} &= \\mathbb{E}\\left[ f_p'(P_k,W^p,\\overline{\\gamma}_{1k})\\right],\n        \\label{eq:a1segen} \\\\\n    \\tau_{2k} &= C_1^2(\\overline{\\alpha}_{1k}) \\left\\{\n        \\mathbb{E}\\left[ f_p^2(P_k,W^p,\\overline{\\gamma}_{1k})\\right]  - \\overline{\\alpha}_{1k}^2\\tau_{1k} \\right\\}\n        \\label{eq:tau2segen} \\\\\n    \\overline{\\gamma}_{2k} &= \\Gamma_1(\\overline{\\gamma}_{1k},\\overline{\\alpha}_{1k}) \\label{eq:gam2segen} \\\\\n    \\overline{\\alpha}_{2k} &= \\mathbb{E}\\left[ f_q'(Q_k,W^q,\\overline{\\gamma}_{2k} \\right],\n        \\label{eq:a2segen} \\\\\n    \\tau_{1,k\\! + \\! 1} &= C_2^2(\\overline{\\alpha}_{2k})\\left\\{\n        \\mathbb{E}\\left[ f_q^2(Q_k,W^q,\\overline{\\gamma}_{2k})\\right] - \\overline{\\alpha}_{2k}^2\\tau_{2k}\\right\\}\n        \\label{eq:tau1segen} \\\\\n    \\gamma_{1,k\\! + \\! 1} &= \\Gamma_2(\\overline{\\gamma}_{2k},\\overline{\\alpha}_{2k}), \\label{eq:gam1segen}\n\\end{align}\n\\end{subequations}\nwhich are initialized with $\\overline{\\gamma}_{10}$ in \\eqref{eq:gam10limgen} and\n\\begin{equation} \\label{eq:tau10gen}\n    \\tau_{10} = \\mathbb{E}[U_0^2],\n\\end{equation}\nwhere $U_0$ is the random variable in \\eqref{eq:U0lim}.\nIn the SE equations~\\eqref{eq:segen},\nthe expectations are taken with respect to random variables\n\\[\n    P_k \\sim {\\mathcal N}(0,\\tau_{1k}), \\quad Q_k \\sim {\\mathcal N}(0,\\tau_{2k}),\n\\]\nwhere $P_k$ is independent of $W^p$ and $Q_k$ is independent of $W^q$.\n\n\\begin{theorem} \\label{thm:genConv}  Consider the recursions \\eqref{eq:algoGen}\nand SE equations \\eqref{eq:segen} under the above assumptions.  Assume additionally that,\nfor all $k$:\n\\begin{enumerate}[(i)]\n\\item For $i=1,2$, the functions\n\\[\n    C_i(\\alpha_i), \\quad \\Gamma_i(\\gamma_i,\\alpha_i),\n\\]\nare continuous at the points $(\\gamma_i,\\alpha_i)=(\\overline{\\gamma}_{ik},\\overline{\\alpha}_{ik})$\nfrom the SE equations; and\n\\item The function $f_p(p,w^p,\\gamma_1)$ and its derivative $f_p'(p,w^p,\\gamma_1)$\nare uniformly Lipschitz continuous in $(p,w^p)$ at $\\gamma_1=\\overline{\\gamma}_{1k}$.\n\\item The function $f_q(q,w^q,\\gamma_2)$ and its derivative $f_q'(q,w^q,\\gamma_2)$\nare uniformly Lipschitz continuous in $(q,w^q)$ at $\\gamma_2=\\overline{\\gamma}_{2k}$.\n\\end{enumerate}\nThen,\n\\begin{enumerate}[(a)]\n\\item For any fixed $k$, almost surely the components of $(\\mathbf{w}^p,\\mathbf{p}_0,\\ldots,\\mathbf{p}_k)$\nempirically converge as\n\\begin{equation} \\label{eq:Pconk}\n    \\lim_{N \\rightarrow \\infty} \\left\\{ (w^p_n,p_{0n},\\ldots,p_{kn}) \\right\\}\n    \\stackrel{PL(2)}{=} (W^p,P_0,\\ldots,P_k),\n\\end{equation}\nwhere $W^p$ is the random variable in the limit \\eqref{eq:WpqLim} and\n$(P_0,\\ldots,P_k)$ is a zero mean Gaussian random vector independent of $W^p$,\nwith $\\mathbb{E}[P_k^2] = \\tau_{1k}$.  In addition, we have that\n\\begin{equation} \\label{eq:ag1limgen}\n    \\lim_{N \\rightarrow \\infty} (\\alpha_{1k},\\gamma_{1k}) = (\\overline{\\alpha}_{1k},\\overline{\\gamma}_{1k}),\n\\end{equation}\nalmost surely.\n\n\\item For any fixed $k$, almost surely the components of $(\\mathbf{w}^q,\\mathbf{q}_0,\\ldots,\\mathbf{q}_k)$\nempirically converge as\n\\begin{equation} \\label{eq:Qconk}\n    \\lim_{N \\rightarrow \\infty} \\left\\{ (w^q_n,q_{0n},\\ldots,q_{kn}) \\right\\}\n    \\stackrel{PL(2)}{=} (W^q,Q_0,\\ldots,Q_k),\n\\end{equation}\nwhere $W^q$ is the random variable in the limit \\eqref{eq:WpqLim} and\n$(Q_0,\\ldots,Q_k)$ is a zero mean Gaussian random vector independent of $W^q$,\nwith $\\mathbb{E}[P_k^2] = \\tau_{2k}$.  In addition, we have that\n\\begin{equation} \\label{eq:ag2limgen}\n    \\lim_{N \\rightarrow \\infty} (\\alpha_{2k},\\gamma_{2k}) = (\\overline{\\alpha}_{2k},\\overline{\\gamma}_{2k}),\n\\end{equation}\nalmost surely.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}  We will prove this in the next Appendix,\nAppendix~\\ref{sec:genConvPf}.\n\\end{proof}\n\n\\section{Proof of Theorem~\\ref{thm:genConv}} \\label{sec:genConvPf}\n\n\\subsection{Induction Argument}\nWe use an induction argument.   Given iterations $k, \\ell \\geq 0$,\ndefine the hypothesis, $H_{k,\\ell}$ as the statement:\n\\begin{itemize}\n\\item Part (a) of Theorem~\\ref{thm:genConv} is true up to $k$; and\n\\item Part (b) of Theorem~\\ref{thm:genConv} is true up to $\\ell$.\n\\end{itemize}\nThe induction argument will then follow by showing the following three facts:\n\\begin{itemize}\n\\item $H_{0,-1}$ is true;\n\\item If $H_{k,k\\! - \\! 1}$ is true, then so is $H_{k,k}$;\n\\item If $H_{k,k}$ is true, then so is $H_{k\\! + \\! 1,k}$.\n\\end{itemize}\n\n\\subsection{Induction Initialization}\nWe first show that the hypothesis $H_{0,-1}$ is true.  That is,\nwe must show \\eqref{eq:Pconk} and \\eqref{eq:ag1limgen} for $k=0$.\nThis is a special case of Lemma~\\ref{lem:orthogGaussLim}.\nSpecifically, for each $N$, let $\\mathbf{U}=\\mathbf{I}_N$, the $N \\times N$ identity\nmatrix, which trivially satisfies property (i) of Lemma~~\\ref{lem:orthogGaussLim}\nwith $s=0$.  Let $\\mathbf{x} = \\mathbf{p}_0$.  Since $\\mathbf{p}_0 = \\mathbf{V}\\mathbf{u}_0$ and $\\mathbf{V}$ is\nHaar distributed independent of $\\mathbf{u}_0$, we have that $\\mathbf{p}_0$ is orthogonally\ninvariant and satisfies property (ii) of Lemma~\\ref{lem:orthogGaussLim}.\nAlso,\n\\[\n    \\lim_{N \\rightarrow \\infty} \\|\\mathbf{p}_0\\|^2 \\stackrel{(a)}{=}\n     \\lim_{N \\rightarrow \\infty} \\|\\mathbf{u}_0\\|^2 \\stackrel{(b)}{=} \\mathbb{E} [U_0^2]\n     \\stackrel{(c)}{=} \\tau_{10},\n\\]\nwhere (a) follows from the fact that  $\\mathbf{p}_0=\\mathbf{V}\\mathbf{u}_0$ and $\\mathbf{V}$ is orthogonal;\n(b) follows from the assumption~\\eqref{eq:U0lim} and (c) follows from the definition\n\\eqref{eq:tau10gen}.  This proves property (iii) of Lemma~\\ref{lem:orthogGaussLim}.\nHence, $\\mathbf{p}_0 = \\mathbf{U}\\mathbf{p}_0$,  we have that the components of $\\mathbf{p}_0$ converge\nempirically as\n\\[\n    \\lim_{N \\rightarrow \\infty} \\{ p_{0n} \\} \\stackrel{PL(2)}{=} P_0 \\sim {\\mathcal N}(0,\\tau_{10}),\n\\]\nfor a Gaussian random variable $P_0$.  Moreover, since $\\mathbf{V}$ is independent of $\\mathbf{w}^p$,\nand the components of $\\mathbf{w}^p$ converge empirically as \\eqref{eq:WpqLim},\nwe have that the components of $\\mathbf{p}_n,\\mathbf{w}^p$ almost surely converge empirically as\n\\[\n    \\lim_{N \\rightarrow \\infty} \\{ w^p_n, p_{0n} \\} \\stackrel{PL(2)}{=} (W^p,P_0),\n\\]\nwhere $W^p$ is independent of $P_0$.  This  proves \\eqref{eq:Pconk} for $k=0$.\n\nNow, we have assumed in \\eqref{eq:gam10limgen} that $\\gamma_{10} \\rightarrow \\overline{\\gamma}_{10}$\nas $N \\rightarrow \\infty$.  Also, since\n $f_p'(p,w^p,\\gamma_1)$ is uniformly Lipschitz continuous in $(p,w^p)$\nat $\\gamma_1 = \\overline{\\gamma}_{10}$, we have that\n$\\alpha_{10} = \\bkt{ \\mathbf{f}_p'(\\mathbf{p}_0,\\mathbf{w}^p,\\gamma_{10}) }$\nconverges to $\\overline{\\alpha}_{10}$ in \\eqref{eq:a1segen} almost surely.\nThis proves \\eqref{eq:ag1limgen}.\n\n\n\n\\subsection{The Induction Recursion}\nWe next show the implication $H_{k,k\\! - \\! 1} \\Rightarrow H_{k,k}$. The implication\n$H_{k,k} \\Rightarrow H_{k\\! + \\! 1,k}$ is proven similarly.  Hence, fix $k$ and assume\nthat $H_{k,k\\! - \\! 1}$ holds.\nSince $\\Gamma_1(\\gamma_i,\\alpha_i)$ is continuous at $(\\overline{\\gamma}_{1k},\\overline{\\alpha}_{1k})$,\nthe limits \\eqref{eq:ag1limgen} combined with \\eqref{eq:gam2segen} show that\n\\[\n    \\lim_{N \\rightarrow\\infty} \\gamma_{2k} =\\lim_{N \\rightarrow\\infty} \\Gamma_1(\\gamma_{1k},\\alpha_{1k}) =\n        \\overline{\\gamma}_{2k}.\n\\]\nIn addition, the induction hypothesis shows that for $\\ell=0,\\ldots,k$,\nthe components of $(\\mathbf{w}^p,\\mathbf{p}_\\ell)$ almost surely converge empirically as\n\\[\n    \\lim_{N \\rightarrow\\infty} \\{ (w^p_n,p_{\\ell n})\\}  \\stackrel{PL(2)}{=} (W^p,P_\\ell),\n\\]\nwhere $P_\\ell \\sim {\\mathcal N}(0,\\tau_{1\\ell})$ for $\\tau_{1\\ell}$ given by the SE equations.\nSince $f_p(\\cdot)$ is Lipschitz continuous \\textb{and} $C_1(\\alpha_{1\\ell})$ is continuous\nat $\\alpha_{1\\ell}=\\overline{\\alpha}_{1\\ell}$, \\textb{one may observe that} the definition of $\\mathbf{v}_\\ell$ in \\eqref{eq:vupgen}\nand the limits \\eqref{eq:ag1limgen} show that\n\\[\n    \\lim_{N \\rightarrow\\infty} \\{ (w^p_n,p_{\\ell n},v_{\\ell n})\\}  \\stackrel{PL(2)}{=} (W^p,P_\\ell ,V_\\ell ),\n\\]\nwhere $V_\\ell $ is the random variable\n\\begin{equation} \\label{eq:Velldef}\n    V_\\ell  = g_p(P_\\ell ,W^p,\\overline{\\gamma}_{1\\ell },\\overline{\\alpha}_{1\\ell}),\n\\end{equation}\nand $g_p(\\cdot)$ is the function\n\\begin{equation} \\label{eq:gpdef}\n    g_p(p,w^p,\\gamma_1,\\alpha_1) := C_1(\\alpha_1)\\left[ f_p(p,w^p,\\gamma_1)-\\alpha_1 p \\right].\n\\end{equation}\n\\textb{\nSimilarly, we have the limit\n\\[\n    \\lim_{N \\rightarrow\\infty} \\{ (w^q_n,q_{\\ell n},u_{\\ell n})\\}  \n    \\stackrel{PL(2)}{=} (W^q,Q_\\ell ,U_\\ell ),\n\\]\nwhere $U_\\ell $ is the random variable,\n\\begin{equation} \\label{eq:Uelldef}\n    U_\\ell  = g_q(Q_\\ell ,W^q,\\overline{\\gamma}_{2\\ell },\\overline{\\alpha}_{2\\ell})\n\\end{equation}\nand $g_q(\\cdot)$ is the function\n\\begin{equation} \\label{eq:gqdef}\n    g_q(q,w^q,\\gamma_2,\\alpha_2) := C_2(\\alpha_1)\\left[ f_q(q,w^q,\\gamma_2)-\\alpha_2 q \\right].\n\\end{equation}\n}\n\n\nWe next introduce the notation\n\\[\n    \\mathbf{U}_k := \\left[ \\mathbf{u}_0 \\cdots \\mathbf{u}_k \\right] \\in {\\mathbb{R}}^{N \\times (k\\! + \\! 1)},\n\\]\nto represent the first $k\\! + \\! 1$ values of the vector $\\mathbf{u}_\\ell$.\nWe define the matrices $\\mathbf{V}_k$, $\\mathbf{Q}_k$ and $\\mathbf{P}_k$ similarly.\nUsing this notation, let $G_k$ be the \\textb{tuple of random matrices},\n\\begin{equation} \\label{eq:Gdef}\n    G_k := \\left\\{ \\mathbf{U}_k, \\mathbf{P}_k, \\mathbf{V}_k, \\mathbf{Q}_{k\\! - \\! 1} \\right\\}.\n\\end{equation}\nWith some abuse of notation, we will also use $G_k$\nto denote the sigma-algebra generated by these variables.\nThe set \\eqref{eq:Gdef} contains all the outputs of the algorithm\n\\eqref{eq:algoGen} immediately \\emph{before} \\eqref{eq:qupgen} in iteration $k$.\n\nNow, the actions of the matrix $\\mathbf{V}$ in the recursions \\eqref{eq:algoGen}\nare through the matrix-vector multiplications \\eqref{eq:pupgen} and \\eqref{eq:qupgen}.\nHence, if we define the matrices\n\\begin{equation} \\label{eq:ABdef}\n    \\mathbf{A}_k := \\left[ \\mathbf{P}_k ~ \\mathbf{V}_{k\\! - \\! 1} \\right], \\quad\n    \\mathbf{B}_k := \\left[ \\mathbf{U}_k ~ \\mathbf{Q}_{k\\! - \\! 1} \\right],\n\\end{equation}\nthe output of the recursions in the set $G_k$ will be unchanged for all\nmatrices $\\mathbf{V}$ satisfying the linear constraints\n\\begin{equation} \\label{eq:ABVconk}\n    \\mathbf{A}_k = \\mathbf{V}\\mathbf{B}_k.\n\\end{equation}\nHence, the conditional distribution of $\\mathbf{V}$ given $G_k$ is precisely\nthe uniform distribution on the set of orthogonal matrices satisfying\n\\eqref{eq:ABVconk}.  The matrices $\\mathbf{A}_k$ and $\\mathbf{B}_k$ are of dimensions\n$N \\times s$ where $s=2k+1$.\nFrom Lemma~\\ref{lem:orthogLin}, this conditional distribution is given by\n\\begin{equation} \\label{eq:Vconk}\n    \\left. \\mathbf{V} \\right|_{G_k} \\stackrel{d}{=}\n    \\mathbf{A}_k(\\mathbf{A}^{\\text{\\sf T}}_k\\mathbf{A}_k)^{-1}\\mathbf{B}_k^{\\text{\\sf T}} + \\mathbf{U}_{\\mathbf{A}_k^\\perp}\\tilde{\\mathbf{V}}\\mathbf{U}_{\\mathbf{B}_k^\\perp}^{\\text{\\sf T}},\n\\end{equation}\nwhere $\\mathbf{U}_{\\mathbf{A}_k^\\perp}$ and $\\mathbf{U}_{\\mathbf{B}_k^\\perp}$ are $N \\times (N-s)$ matrices\nwhose columns are an orthonormal basis for $\\mathrm{Range}(\\mathbf{A}_k)^\\perp$ and $\\mathrm{Range}(\\mathbf{B}_k)^\\perp$.\nThe matrix $\\tilde{\\mathbf{V}}$ is  Haar distributed on the set of $(N-s)\\times(N-s)$\northogonal matrices and independent of $G_k$.\n\nUsing \\eqref{eq:Vconk} we can write $\\mathbf{q}_k$ in \\eqref{eq:qupgen} as a sum of two terms\n\\begin{equation} \\label{eq:qpart}\n    \\mathbf{q}_k = \\mathbf{V}^{\\text{\\sf T}}\\mathbf{v}_k = \\mathbf{q}_k^{\\rm det} + \\mathbf{q}_k^{\\rm ran},\n\\end{equation}\nwhere $\\mathbf{q}_k^{\\rm det}$ is what we will call the \\emph{deterministic} part:\n\\begin{equation} \\label{eq:qkdet}\n    \\mathbf{q}_k^{\\rm det} = \\mathbf{B}_k(\\mathbf{A}^{\\text{\\sf T}}_k\\mathbf{A}_k)^{-1}\\mathbf{A}_k^{\\text{\\sf T}}\\mathbf{v}_k,\n\\end{equation}\nand $\\mathbf{q}_k^{\\rm ran}$ is what we will call the \\emph{random} part:\n\\begin{equation} \\label{eq:qkran}\n    \\mathbf{q}_k^{\\rm ran} = \\mathbf{U}_{\\mathbf{B}_k^\\perp}\\tilde{\\mathbf{V}}^{\\text{\\sf T}} \\mathbf{U}_{\\mathbf{A}_k^\\perp}^{\\text{\\sf T}} \\mathbf{v}_k.\n\\end{equation}\nThe next few lemmas will evaluate the asymptotic distributions of the two\nterms in \\eqref{eq:qpart}.\n\n\\begin{lemma} \\label{lem:qconvdet}\nUnder the induction hypothesis $H_{k,k\\! - \\! 1}$, there \\textb{exist} constants\n$\\beta_{k,0},\\ldots,\\beta_{k,k\\! - \\! 1}$ such that the components of $\\mathbf{q}_k^{\\rm det}$\nalong with $(\\mathbf{q}_0,\\ldots,\\mathbf{q}_{k\\! - \\! 1})$ converge empirically as\n\\begin{align}\n    \\MoveEqLeft \\lim_{N \\rightarrow \\infty} \\left\\{ w^q_n,q_{0n},\\ldots,q_{k\\! - \\! 1,n},q_{kn}^{\\rm det}) \\right\\}\n    \\nonumber \\\\\n    &\\stackrel{PL(2)}{=} (W^q,Q_0,\\ldots,Q_{k\\! - \\! 1},Q_k^{\\rm det}),\n        \\label{eq:qconvdet}\n\\end{align}\nwhere $Q_\\ell$, $\\ell=0,\\ldots,k\\! - \\! 1$ are the Gaussian random variables in induction hypothesis\n\\eqref{eq:Qconk} and $Q_k^{\\rm det}$ is a linear combination,\n\\begin{equation} \\label{eq:Qkdetlim}\n    Q_k^{\\rm det} = \\beta_{k0}Q_0 + \\cdots + \\beta_{k,k\\! - \\! 1}Q_{k\\! - \\! 1}.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nWe evaluate the asymptotic values of various terms in \\eqref{eq:qkdet}.\nUsing the definition of $\\mathbf{A}_k$ in \\eqref{eq:ABdef},\n\\[\n    \\mathbf{A}^{\\text{\\sf T}}_k\\mathbf{A}_k = \\left[ \\begin{array}{cc}\n        \\mathbf{P}_k^{\\text{\\sf T}}\\mathbf{P}_k & \\mathbf{P}_k^{\\text{\\sf T}}\\mathbf{V}_{k\\! - \\! 1} \\\\\n        \\mathbf{V}_{k\\! - \\! 1}^{\\text{\\sf T}}\\mathbf{P}_k & \\mathbf{V}_{k\\! - \\! 1}^{\\text{\\sf T}}\\mathbf{V}_{k\\! - \\! 1}\n        \\end{array} \\right]\n\\]\nWe can then easily evaluate the asymptotic value of these\nterms as follows.  For example, the asymptotic value of the\n$(i,j)$ component of the matrix $\\mathbf{P}_k^{\\text{\\sf T}}\\mathbf{P}_k$ is given by\n\\begin{align*}\n    \\MoveEqLeft \\lim_{N \\rightarrow \\infty} \\frac{1}{N} \\left[ \\mathbf{P}_k^{\\text{\\sf T}}\\mathbf{P}_k \\right]_{ij}\n        \\stackrel{(a)}{=} \\frac{1}{N} \\mathbf{p}_i^{\\text{\\sf T}}\\mathbf{p}_j \\\\\n        &= \\frac{1}{N} \\sum_{n=1}^N p_{in}p_{jn}\n        \\stackrel{(b)}{=} E(P_iP_j) \\stackrel{(c)}{=} \\left[ \\mathbf{Q}^p_k\\right]_{ij},\n\\end{align*}\nwhere (a) follows since the $i$-th column of $\\mathbf{P}_k$ is precisely the vector\n$\\mathbf{p}_i$; (b) follows due to convergence assumption in \\eqref{eq:Pconk};\nand in (c), we use $\\mathbf{Q}^p_k$ to denote the covariance matrix of $(P_0,\\ldots,P_k)$.\nSimilarly\n\\[\n    \\lim_{N \\rightarrow \\infty}  \\frac{1}{N}  \\mathbf{V}_{k\\! - \\! 1}^{\\text{\\sf T}}\\mathbf{V}_{k\\! - \\! 1} = \\mathbf{Q}^v_{k\\! - \\! 1},\n\\]\nwhere $\\mathbf{Q}^v_{k\\! - \\! 1}$ has the components,\n\\[\n    \\left[ \\mathbf{Q}^v_{k\\! - \\! 1} \\right]_{ij} = \\mathbb{E} \\left[ V_iV_j \\right],\n\\]\nwhere $V_i$ is the random variable in \\eqref{eq:Velldef}.\nFinally, the expectation for the cross-terms are given by\n\\begin{align*}\n    \\mathbb{E}[V_iX_j]\n    &\\stackrel{(a)}{=}\n    \\mathbb{E}[g_p(P_i,W^p,\\overline{\\gamma}_{1i},\\overline{\\alpha}_{1i})X_j] \\nonumber \\\\\n    &\\stackrel{(b)}{=} \\mathbb{E}\\left[ g_p'(P_i,W^p,\\overline{\\gamma}_{1i},\\overline{\\alpha}_{1i})\\right]\n        \\mathbb{E}[X_iX_j] \\nonumber \\\\\n     &\\stackrel{(c)}{=}\n     \\mathbb{E}[X_iX_j]C_1(\\overline{\\alpha}_{1i})\\left( \\mathbb{E}\n       \n        \\left[ f_p'(P_i,W^p,\\overline{\\gamma}_{1i}) \\right]-\\overline{\\alpha}_{1i}\n        \\right) \\\\\n     &\\stackrel{(d)}{=} 0,\n\\end{align*}\nwhere (a) follows from \\eqref{eq:Velldef};\n(b) follows from Stein's Lemma; and  (c) follows from the definition of $g_p(\\cdot)$\nin \\eqref{eq:gpdef};  and (d) follows from \\eqref{eq:a1segen}.\nThe above calculations show that\n\\begin{equation} \\label{eq:AAlim}\n    \\lim_{N \\rightarrow \\infty} \\frac{1}{N}\\mathbf{A}^{\\text{\\sf T}}_k\\mathbf{A}_k \n        \\stackrel{a.s.}{=} \\left[ \\begin{array}{cc}\n        \\mathbf{Q}_k^p & \\mathbf{0} \\\\\n        \\mathbf{0} & \\mathbf{Q}^v_{k\\! - \\! 1}\n        \\end{array} \\right],\n\\end{equation}\nA similar calculation shows that\n\\begin{equation} \\label{eq:Aslim}\n    \\lim_{N \\rightarrow \\infty} \\frac{1}{N} \\mathbf{A}^{\\text{\\sf T}}_k\\mathbf{s}_k = \\left[\n    \\begin{array}{c} \\mathbf{0} \\\\ \\mathbf{b}^s_k \\end{array} \\right],\n\\end{equation}\nwhere \\textb{$\\mathbf{b}^v_k$} is the vector of correlations\n\\begin{equation}\n    \\mathbf{b}^v_k = \\mat{ \\mathbb{E}[V_0V_k] & \\mathbb{E}[V_1V_k] & \\cdots & \\mathbb{E}[V_{k\\! - \\! 1}V_k] }^{\\text{\\sf T}}.\n\\end{equation}\nCombining \\eqref{eq:AAlim} and \\eqref{eq:Aslim} shows that\n\\begin{equation} \\label{eq:Vsmult1}\n    \\lim_{N \\rightarrow \\infty} (\\mathbf{A}^{\\text{\\sf T}}_k\\mathbf{A}_k)^{-1}\\mathbf{A}_k^{\\text{\\sf T}} \\mathbf{v}_k \n    \\stackrel{a.s.}{=}\n    \\left[ \\begin{array}{c} \\mathbf{0} \\\\ \\mathbf{\\beta}_k \\end{array} \\right],\n\\end{equation}\nwhere \n\\[\n    \\mathbf{\\beta}_k := \\left[ \\mathbf{Q}^v_{k-1}\\right]^{-1}\\mathbf{b}^v_k.\n\\]\nTherefore,\n\\begin{align}\n    \\MoveEqLeft \\mathbf{q}_k^{\\rm det} = \\mathbf{B}_k(\\mathbf{A}^{\\text{\\sf T}}_k\\mathbf{A}_k)^{-1}\\mathbf{A}_k^{\\text{\\sf T}}\\mathbf{v}_k \\nonumber \\\\\n    &= \\left[ \\mathbf{U}_k ~ \\mathbf{Q}_{k\\! - \\! 1} \\right]\n    \\left[ \\begin{array}{c} \\mathbf{0} \\\\ \\mathbf{\\beta}_k \\end{array} \\right]\n    +{\\boldsymbol \\xi} \\nonumber \\\\\n    &= \\sum_{\\ell=0}^{k\\! - \\! 1} \\beta_{k\\ell} \\mathbf{q}_\\ell + {\\boldsymbol \\xi}, \\label{eq:qbetasum}\n\\end{align}\nwhere \\textb{${\\boldsymbol \\xi} \\in {\\mathbb{R}}^N$ is the error,\n\\begin{equation} \\label{eq:xisdef}\n    {\\boldsymbol \\xi} = \\mathbf{B}_k \\mathbf{s}, \\quad\n    \\mathbf{s} := (\\mathbf{A}^{\\text{\\sf T}}_k\\mathbf{A}_k)^{-1}\\mathbf{A}_k^{\\text{\\sf T}} \\mathbf{v}_k -\n    \\left[ \\begin{array}{c} \\mathbf{0} \\\\ \\mathbf{\\beta}_k \\end{array} \\right].\n\\end{equation}\nWe next need to bound the norm of the error term ${\\boldsymbol \\xi}$.\nSince ${\\boldsymbol \\xi} = \\mathbf{B}_k\\mathbf{s}$, the definition of $\\mathbf{B}_k$ in \n\\eqref{eq:ABdef} shows that\n\\begin{equation} \\label{eq:xisum}\n    {\\boldsymbol \\xi} = \\sum_{i=0}^k s_i \\mathbf{u}_i + \\sum_{j=0}^{k\\! - \\! 1} s_{k+j+1}\\mathbf{q}_j,\n\\end{equation}\nwhere we have indexed the components of $\\mathbf{s}$ in \\eqref{eq:xisdef}\nas $\\mathbf{s}=(s_0,\\ldots,s_{2k})$.\nFrom \\eqref{eq:Vsmult1}, the components $s_j \\rightarrow 0$ almost surely,\nand therefore\n\\[\n    \\lim_{N \\rightarrow \\infty} \\max_{j=0,\\ldots,2k} |s_j| \\stackrel{a.s.}{=} 0.\n\\]\nAlso, by the induction hypothesis,\n\\[\n    \\lim_{N \\rightarrow \\infty} \\frac{1}{N} \\|\\mathbf{u}_i\\|^2 \\stackrel{a.s.}{=} E(U_i^2),\n    \\quad\n    \\lim_{N \\rightarrow \\infty} \\frac{1}{N} \\|\\mathbf{q}_j\\|^2 \\stackrel{a.s.}{=} E(Q_j^2).\n\\]\nTherefore, from \\eqref{eq:xisum},\n\\begin{align}\n    \\MoveEqLeft \\lim_{N \\rightarrow \\infty} \\frac{1}{N}\\|{\\boldsymbol \\xi}\\|^2\n       \\leq\n        \\lim_{N \\rightarrow \\infty} \n        \\left[ \\max_{j=0,\\ldots,2k} |s_j|^2 \\right] \\nonumber \\\\\n        & \\times \n        \\frac{1}{N} \\left[ \\sum_i \\|\\mathbf{u}_i\\|^2 + \\sum_j \\|\\mathbf{q}_j\\|^2 \\right] \\stackrel{a.s.}{=} 0.\n    \\label{eq:xilimbnd}\n\\end{align}\nTherefore, if $f(q_1,\\cdots,q_k)$ is\npseudo-Lipschitz continuous of order 2,\n\\begin{align*}\n    \\MoveEqLeft \\lim_{N \\rightarrow \\infty} \\frac{1}{N} \\sum_{n=1}^N\n        f(q_{0n},\\cdots,q_{k\\! - \\! 1,n},q^{\\rm det}_k) \\\\\n        &\\stackrel{(a)}{=} \\lim_{N \\rightarrow \\infty} \\frac{1}{N} \\sum_{n=1}^N\n        f\\left( q_{0n},\\cdots,q_{k\\! - \\! 1,n},\n            \\sum_{\\ell=0}^{k\\! - \\! 1} \\beta_{k\\ell} q_{\\ell n} \\right)\\\\\n        &\\stackrel{(b)}{=} \\mathbb{E}\\left[\n        f\\left(Q_0,\\cdots,Q_{k\\! - \\! 1}, \\sum_{\\ell=0}^{k\\! - \\! 1} \\beta_{k\\ell} Q_{\\ell n}\n            \\right) \\right],\n\\end{align*}\nwhere (a) follows from the \\eqref{eq:qbetasum}, the bound\n\\eqref{eq:xilimbnd}, and the pseudo-Lipschitz continuity of $f(\\cdot)$;\nand (b) follows from the fact that $f(\\cdot)$ is pseudo-Lipschitz continuous\nand the induction hypothesis that\n\\[\n    \\lim_{N \\rightarrow \\infty} \\{q_{0n},\\cdots,q_{k\\! - \\! 1,n}\\} \\stackrel{PL(2)}{=}\n    (Q_0,\\ldots,Q_{k\\! - \\! 1}).\n\\]\nThis proves \\eqref{eq:qconvdet}.\n}\n\n\\end{proof}\n\n\n\\begin{lemma} \\label{lem:rhoconv}\nUnder the induction hypothesis $H_{k,k\\! - \\! 1}$, the following limit\nholds almost surely\n\\begin{equation}\n    \\lim_{N \\rightarrow \\infty} \\frac{1}{N} \\| \\mathbf{U}_{\\mathbf{A}_k^\\perp}^{\\text{\\sf T}}\\mathbf{s}_k\\|^2 =\n    \\rho_k,\n\\end{equation}\nfor some constant $\\rho_k \\geq 0$.\n\\end{lemma}\n\\begin{proof}  From \\eqref{eq:ABdef}, the matrix  $\\mathbf{A}_k$ has $s=2k+1$ columns.\nFrom Lemma~\\ref{lem:orthogLin},\n$\\mathbf{U}_{\\mathbf{A}_k^\\perp}$ is an orthonormal basis of $N-s$ in the $\\mathrm{Range}(\\mathbf{A}_k)^\\perp$.\nHence, the energy $\\| \\mathbf{U}_{\\mathbf{A}_k^\\perp}\\mathbf{s}_k\\|^2$ is precisely\n\\[\n    \\| \\mathbf{U}_{\\mathbf{A}_k^\\perp}\\mathbf{s}_k\\|^2 = \\mathbf{s}_k^{\\text{\\sf T}}\\mathbf{s}_k - \\mathbf{s}_k^{\\text{\\sf T}}\\mathbf{A}_k\n        (\\mathbf{A}_k^{\\text{\\sf T}}\\mathbf{A}_k)^{-1}\\mathbf{A}_k^{\\text{\\sf T}}\\mathbf{s}_k.\n\\]\nUsing similar calculations as the previous lemma, we have\n\\[\n    \\lim_{N \\rightarrow \\infty} \\frac{1}{N} \\| \\mathbf{U}_{\\mathbf{A}_k}\\mathbf{s}_k\\|^2\n    = \\mathbb{E}[S_k^2] - (\\mathbf{b}^s_k)^{\\text{\\sf T}}\\left[ \\mathbf{Q}^s_k \\right]^{-1}\\mathbf{b}^s_k.\n\\]\nHence, the lemma is proven if we define $\\rho_k$ as the right hand side of this\nequation.\n\\end{proof}\n\n\\begin{lemma} \\label{lem:qconvran}\nUnder the induction hypothesis $H_{k,k\\! - \\! 1}$, the components of\nthe ``random\" part $\\mathbf{q}_k^{\\rm ran}$ along with the components\nof $(\\mathbf{w}^q,\\mathbf{q}_0,\\ldots,\\mathbf{q}_{k\\! - \\! 1})$\nalmost surely converge empirically as\n\\begin{align}\n    \\MoveEqLeft  \\lim_{N \\rightarrow \\infty}\n        \\left\\{ (w^q_n,q_{0n},\\ldots,q_{k\\! - \\! 1,n},q_{kn}^{\\rm ran}) \\right\\}\n        \\nonumber \\\\\n        &\\stackrel{PL(2)}{=}  (W^q,Q_0,\\ldots,Q_{k\\! - \\! 1},U_k), \\label{eq:qranlim}\n\\end{align}\nwhere $U_k \\sim {\\mathcal N}(0,\\rho_k)$ is a Gaussian random variable\nindependent of $(W^q,Q_0,\\ldots,Q_{k\\! - \\! 1})$ and $\\rho_k$ is the constant\nin Lemma~\\ref{lem:rhoconv}.\n\\end{lemma}\n\\begin{proof}\nThis is a direct application of Lemma~\\ref{lem:orthogGaussLim}.\nLet $\\mathbf{x} = \\tilde{\\mathbf{V}}^{\\text{\\sf T}}\\mathbf{U}_{\\mathbf{A}_k^\\perp}^{\\text{\\sf T}}\\mathbf{s}_k$ so that\n\\[\n    \\mathbf{q}_k^{\\rm det} = \\mathbf{U}_{\\mathbf{B}_k^\\perp}\\mathbf{x}_k.\n\\]\nFor each $N$, $\\mathbf{U}_{\\mathbf{B}_k^\\perp} \\in {\\mathbb{R}}^{N \\times (N-s)}$ is a matrix\nwith orthonormal columns spanning $\\mathrm{Range}(\\mathbf{B}_k)^\\perp$.\nAlso, since $\\tilde{\\mathbf{V}}$ is uniformly distributed on the set of\n$(N-s)\\times (N-s)$ orthogonal matrices, and independent of $G_k$,\nthe conditional distribution $\\mathbf{x}_k$ given $G_k$ is orthogonally invariant in that\n\\[\n    \\left. \\mathbf{U} \\mathbf{x}_k \\right|_{G_k} \\stackrel{d}{=} \\left. \\mathbf{x}_k \\right|_{G_k},\n\\]\nfor any orthogonal matrix $\\mathbf{U}$.  Lemma~\\ref{lem:rhoconv} also shows that\n\\[\n    \\lim_{N \\rightarrow \\infty} \\frac{1}{N} \\|\\mathbf{x}_k\\|^2 = \\rho_k,\n\\]\nalmost surely.\nThe limit \\eqref{eq:qranlim} now follows from\nLemma~\\ref{lem:orthogGaussLim}.\n\\end{proof}\n\nUsing the partition \\eqref{eq:qpart} and Lemmas~\\ref{lem:qconvdet} and \\ref{lem:qconvran},\nthe components of $(\\mathbf{w}^q,\\mathbf{q}_0,\\ldots,\\mathbf{q}_k)$\nalmost surely converge empirically as\n\\begin{align*}\n    \\lefteqn{ \\lim_{N \\rightarrow \\infty} \\{ (w^q_n,q_{0n},\\ldots,q_{kn}) \\} }\\\\\n    &\\stackrel{PL(2)}{=} \\lim_{N \\rightarrow \\infty} \\{ (w^q_n,q_{0n},\\ldots,q^{\\rm det}_{kn} + q^{\\rm ran}_{kn}) \\}\n    \\\\\n    &\\stackrel{PL(2)}{=}  (W^q,Q_0,\\ldots,Q_k),\n\\end{align*}\nwhere $Q_k$ is the random variable\n\\[\n    Q_k = \\beta_{k0}Q_0 + \\cdots + \\beta_{k,k\\! - \\! 1}Q_{k\\! - \\! 1} + U_k.\n\\]\nSince $(Q_0,\\ldots,Q_{k\\! - \\! 1})$ is jointly Gaussian and $U_k$ is Gaussian independent of\n$(Q_0,\\ldots,Q_{k\\! - \\! 1})$ we have that $(Q_0,\\ldots,Q_k)$ is Gaussian.  This proves\n\\eqref{eq:Qconk}.\n\nNow the function $\\Gamma_1(\\gamma_1,\\alpha_1)$ is assumed to be\ncontinuous at $(\\overline{\\gamma}_{1k},\\overline{\\alpha}_{1k})$.  Also, the induction hypothesis assumes that\n$\\alpha_{1k} \\rightarrow \\overline{\\alpha}_{1k}$ and $\\gamma_{1k} \\rightarrow \\overline{\\gamma}_{1k}$ almost surely.\nHence,\n\\begin{equation} \\label{eq:gam2limpf}\n    \\lim_{N \\rightarrow \\infty} \\gamma_{2k} = \\lim_{N \\rightarrow \\infty} \\Gamma_1(\\gamma_{1k},\\alpha_{1k})\n    = \\overline{\\gamma}_{2k}.\n\\end{equation}\nIn addition, since we have assumed that $\\mathbf{f}_q'(\\mathbf{q},\\mathbf{w}^q,\\gamma_1)$ is Lipschitz\ncontinuous in $(\\mathbf{q},\\mathbf{w}^q)$ and continuous in $\\gamma_1$,\n\\begin{align}\n    \\lim_{N \\rightarrow \\infty} \\alpha_{2k}\n    &= \\lim_{N \\rightarrow \\infty} \\bkt{\\mathbf{f}_q'(\\mathbf{q}_k,\\mathbf{w}^q,\\gamma_{1k})} \\nonumber \\\\\n    &= \\mathbb{E}\\left[ f_q'(Q_k,W^q,\\overline{\\gamma}_{1k}) \\right] = \\overline{\\alpha}_{1k}. \\label{eq:a2limpf}\n\\end{align}\nThe limits \\eqref{eq:gam2limpf} and \\eqref{eq:a2limpf} prove \\eqref{eq:ag2limgen}.\n\nFinally, we need to show that $\\mathbb{E}[Q_k^2] = \\tau_{2k}$ is the variance from the SE\nequations.\n\\begin{align}\n \\mathbb{E}[Q_k^2] &\\stackrel{(a)}{=} \\lim_{N \\rightarrow \\infty} \\frac{1}{N}\n        \\|\\mathbf{q}_k\\|^2 \\nonumber \\\\\n         & \\stackrel{(b)}{=} \\lim_{N \\rightarrow \\infty} \\frac{1}{N}\n        \\|\\mathbf{v}_k\\|^2 \\nonumber \\\\\n         & \\stackrel{(c)}{=} \\mathbb{E}\\left[ g_p(P_k,W^p,\\overline{\\gamma}_{1k},\\overline{\\alpha}_{1k}) \\right] \\nonumber \\\\\n         & \\stackrel{(d)}{=} C_1^2(\\overline{\\alpha}_{1k})\n         \\mathbb{E}\\left[ \\left(f_p(P_k,W^p,\\overline{\\gamma}_{1k}) - \\overline{\\alpha}_{1k}P_k\\right)^2 \\right] \\nonumber \\\\\n         & = C_1^2(\\overline{\\alpha}_{1k})\\Bigl\\{\n         \\mathbb{E}\\left[ f_p^2(P_k,W^p,\\overline{\\gamma}_{1k})\\right]  \\nonumber \\\\\n         & \\quad - 2\\overline{\\alpha}_{1k}\n            \\mathbb{E}\\left[ P_k f_p(P_k,W^p,\\overline{\\gamma}_{1k})\\right] + \\overline{\\alpha}^2_{1k}\\mathbb{E}\\left[ P_k^2 \\right]\n            \\Bigr\\} \\nonumber \\\\\n         & \\stackrel{(e)}{=} C_1^2(\\overline{\\alpha}_{1k})\\Bigl\\{\n         \\mathbb{E}\\left[ f_p^2(P_k,W^p,\\overline{\\gamma}_{1k})\\right]  \\nonumber \\\\\n         & \\quad - 2\\overline{\\alpha}_{1k}\\tau_{1k}\n            \\mathbb{E}\\left[ f_p'(P_k,W^p,\\overline{\\gamma}_{1k})\\right] + \\overline{\\alpha}^2_{1k}\\tau_{1k}   \\Bigr\\}\n            \\nonumber \\\\\n         & \\stackrel{(f)}{=} C_1^2(\\overline{\\alpha}_{1k})\\left\\{\n         \\mathbb{E}\\left[ f_p^2(P_k,W^p,\\overline{\\gamma}_{1k})\\right]- \\overline{\\alpha}_{1k}^2\\tau_{1k} \\right\\} \\nonumber \\\\\n         & \\stackrel{(g)}{=} \\tau_{2k},\n\\end{align}\nwhere (a) follows from the fact that the components of $\\mathbf{q}_k$ converge empirically\nto $Q_k$;\n(b) follows from \\eqref{eq:qupgen} and the fact that $\\mathbf{V}$ is orthogonal;\n(c) follows from the limit \\eqref{eq:Velldef}; and\n(d) follows from \\eqref{eq:gpdef};\n(e) follows from Stein's Lemma and the fact that $\\mathbb{E} [P_k^2] = \\tau_{1k}$;\n(f) follows from the definition of $\\overline{\\alpha}_{1k}$ in \\eqref{eq:a1segen};\nand (g) follows from \\eqref{eq:tau2segen}.  Thus, $\\mathbb{E} [Q_k^2] = \\tau_{2k}$,\nand we have proven the implication $H_{k,k\\! - \\! 1} \\Rightarrow H_{k,k}$.\n\n\\section{Proof of Theorem~\\ref{thm:se} }\n\nTheorem~\\ref{thm:se} is essentially a special case of Theorem~\\ref{thm:genConv}.\nWe need to simply rewrite the recursions in Algorithm~\\ref{algo:vamp} in the form\n\\eqref{eq:algoGen}.\nTo this end, define the error terms\n\\begin{equation} \\label{eq:pvslr}\n    \\mathbf{p}_k := \\mathbf{r}_{1k}-\\mathbf{x}^0, \\quad\n    \\mathbf{v}_k := \\mathbf{r}_{2k}-\\mathbf{x}^0,\n\\end{equation}\nand their transforms,\n\\begin{equation} \\label{eq:uqslr}\n    \\mathbf{u}_k := \\mathbf{V}^{\\text{\\sf T}}\\mathbf{p}_k, \\quad\n    \\mathbf{q}_k := \\mathbf{V}^{\\text{\\sf T}}\\mathbf{v}_k.\n\\end{equation}\nAlso, define the disturbance terms\n\\begin{equation} \\label{eq:wpqslr}\n    \\mathbf{w}^q := ({\\boldsymbol \\xi},\\mathbf{s}), \\quad\n    \\mathbf{w}^p := \\mathbf{x}^0, \\quad {\\boldsymbol \\xi} := \\mathbf{U}^{\\text{\\sf T}}\\mathbf{w},\n\\end{equation}\nand the componentwise update functions\n\\begin{subequations} \\label{eq:fqpslr}\n\\begin{align}\n    f_q(q,(\\xi,s),\\gamma_2) &:= \\frac{\\gamma_w s\\xi + \\gamma_2 q}{\n        \\gamma_w s^2 + \\gamma_2}, \\label{eq:fqslr} \\\\\n    f_p(p,x^0,\\gamma_1) &= g_1(p+x^0,\\gamma_1) - x^0.\n        \\label{eq:fpslr}\n\\end{align}\n\\end{subequations}\nWith these definitions, we claim that the outputs satisfy the recursions:\n\\begin{subequations} \\label{eq:gecslr}\n\\begin{align}\n    \\mathbf{p}_k &= \\mathbf{V}\\mathbf{u}_k \\label{eq:pupslr} \\\\\n    \\alpha_{1k} &= \\bkt{ \\mathbf{f}_p'(\\mathbf{p}_k,\\mathbf{x}^0,\\gamma_{1k})},\n    \\quad \\gamma_{2k} = \\frac{(1-\\alpha_{1k})\\gamma_{1k}}{\\alpha_{1k}}\n        \\label{eq:alpha1slr}  \\\\\n    \\mathbf{v}_k &= \\frac{1}{1-\\alpha_{1k}}\\left[\n        \\mathbf{f}_p(\\mathbf{p}_k,\\mathbf{x}^0,\\gamma_{1k})- \\alpha_{1k} \\mathbf{p}_{k} \\right] \\label{eq:vupslr} \\\\\n    \\mathbf{q}_k &= \\mathbf{V}^{\\text{\\sf T}}\\mathbf{v}_k \\label{eq:qupslr} \\\\\n    \\alpha_{2k} &= \\bkt{ \\mathbf{f}_q'(\\mathbf{q}_k,\\mathbf{w}^q,\\gamma_{2k})},\n    \\quad \\gamma_{1,k\\! + \\! 1} = \\frac{(1-\\alpha_{2k})\\gamma_{2k}}{\\alpha_{2k}}\n        \\label{eq:alpha2slr} \\\\\n    \\mathbf{u}_{k\\! + \\! 1} &= \\frac{1}{1-\\alpha_{2k}}\\left[\n        \\mathbf{f}_q(\\mathbf{q}_k,\\mathbf{w}^q,\\gamma_{2k}) - \\alpha_{2k}\\mathbf{q}_{k} \\right] \\label{eq:uupslr}\n\\end{align}\n\\end{subequations}\nBefore we prove \\eqref{eq:gecslr}, we can see that \\eqref{eq:gecslr} is a special\ncase of the general recursions in \\eqref{eq:algoGen} if we define\n\\[\n    C_i(\\alpha_i) = \\frac{1}{1-\\alpha_i}, \\quad \\Gamma_i(\\gamma_i,\\alpha_i) =\n        \\gamma_i\\left[\\frac{1}{\\alpha_i}-1 \\right].\n\\]\nIt is also straightforward to verify the continuity assumptions in Theorem~\\ref{thm:genConv}.\nThe assumption of Theorem~\\ref{thm:se} states that $\\overline{\\alpha}_{ik} \\in (0,1)$.  Since\n$\\overline{\\gamma}_{10} > 0$, $\\overline{\\gamma}_{ik} > 0$ for all $k$ and $i$.  Therefore,\n$C_i(\\alpha_i)$ and $\\Gamma_i(\\gamma_i,\\alpha_i)$ are continuous at all points\n$(\\gamma_i,\\alpha_i) = (\\overline{\\gamma}_{ik},\\overline{\\alpha}_{ik})$.\nAlso, since $s \\in [0,S_{max}]$ and $\\gamma_{2k} > 0$ for all $k$, the function\n$f_q(q,(\\xi,s),\\gamma_2)$ in \\eqref{eq:fqpslr} is uniformly Lipschitz continuous\nin $(q,\\xi,s)$ at all $\\gamma_2 = \\overline{\\gamma}_{2k}$.\nSimilarly, since the denoiser function $g_1(r_1,\\gamma_1)$ is assumed be to uniformly\nLipschitz continuous in $r_1$ at all $\\gamma_1 = \\overline{\\gamma}_{1k}$, so is the function\n$f_p(r_1,x^0,\\gamma_1)$ in \\eqref{eq:fpslr}.  Hence all the conditions of Theorem~\\ref{thm:genConv}\nare satisfied.  The SE equations \\eqref{eq:se} immediately\nfrom the general SE equations \\eqref{eq:segen}.  In addition, the limits\n\\eqref{eq:limrx1} and and \\eqref{eq:limqxi} are special cases of the limits\n\\eqref{eq:Pconk} and \\eqref{eq:Qconk}.  This proves Theorem~\\ref{thm:se}.\n\nSo, it remains only to show that the updates in\n\\eqref{eq:gecslr} indeed hold.\nEquations \\eqref{eq:pupslr} and \\eqref{eq:qupslr} follow immediately\nfrom the definitions \\eqref{eq:pvslr} and \\eqref{eq:uqslr}.\nNext, observe that we can rewrite the\nLMMSE estimation function \\eqref{eq:g2slr} as\n\\begin{align}\n    \\lefteqn{ \\mathbf{g}_2(\\mathbf{r}_{2k},\\gamma_{2k}) }\\nonumber\\\\\n    &\\stackrel{(a)}{=} \\left( \\gamma_w \\mathbf{A}^{\\text{\\sf T}}\\mathbf{A} + \\gamma_{2k}\\mathbf{I}\\right)^{-1}\n    \\left( \\gamma_w\\mathbf{A}^{\\text{\\sf T}}\\mathbf{A}\\mathbf{x}^0 + \\gamma_w\\mathbf{A}^{\\text{\\sf T}}\\mathbf{w}\n    + \\gamma_{2k}\\mathbf{r}_{2k} \\right) \\nonumber \\\\\n   &\\stackrel{(b)}{=} \\mathbf{x}^0 +\n   \\left( \\gamma_w \\mathbf{A}^{\\text{\\sf T}}\\mathbf{A} + \\gamma_{2k}\\mathbf{I}\\right)^{-1}\n    \\left( \\gamma_{2k}(\\mathbf{r}_{2k}-\\mathbf{x}^0)+ \\gamma_w\\mathbf{A}^{\\text{\\sf T}}\\mathbf{w}\\right) \\nonumber \\\\\n    &\\stackrel{(d)}{=} \\mathbf{x}^0 +\n    \\mathbf{V}\\left( \\gamma_w \\mathbf{S}^2 + \\gamma_{2k}\\mathbf{I}\\right)^{-1}\n    \\left( \\gamma_{2k}\\mathbf{q}_k + \\mathbf{S}{\\boldsymbol \\xi} \\right), \\nonumber  \\\\\n    &\\stackrel{(d)}{=} \\mathbf{x}^0 + \\mathbf{V} \\mathbf{f}_q(\\mathbf{q}_k,\\mathbf{w}^q,\\gamma_{2k}),   \\label{eq:g2slrV}\n\\end{align}\nwhere (a) follows by substituting \\eqref{eq:yAxslr} into \\eqref{eq:g2slr};\n(b) is a simple algebraic manipulation;\n(c) follows from the SVD definition \\eqref{eq:ASVD} and the definitions\n${\\boldsymbol \\xi}$ in \\eqref{eq:wpqslr} and $\\mathbf{q}_k$ in \\eqref{eq:uqslr}; and\n(d) follows from the definition of componentwise function\n$f_q(\\cdot)$ in \\eqref{eq:fqslr}. Therefore, the divergence $\\alpha_{2k}$ satisfies\n\\begin{align}\n    \\alpha_{2k}\n    &\\stackrel{(a)}{=}\n    \\frac{1}{N}\n        \\mathrm{Tr}\\left[ \\frac{\\partial \\mathbf{g}_2(\\mathbf{r}_{2k},\\gamma_{2k})}{\\partial \\mathbf{r}_{2k}} \\right]\n        \\nonumber \\\\\n    & \\stackrel{(b)}{=}    \\frac{1}{N}\n        \\mathrm{Tr}\\left[ \\mathbf{V} \\mathrm{Diag}(\\mathbf{f}_q'(\\mathbf{q}_{k},\\mathbf{w}^q,\\gamma_{2k}))\n        \\frac{\\partial \\mathbf{q}_k}{\\partial \\mathbf{r}_{2k}}  \\right]\n        \\nonumber \\\\\n    & \\stackrel{(c)}{=}    \\frac{1}{N}\n        \\mathrm{Tr}\\left[ \\mathbf{V} \\mathrm{Diag}(\\mathbf{f}_q'(\\mathbf{q}_{k},\\mathbf{w}^q,\\gamma_{2k})) \\mathbf{V}^{\\text{\\sf T}}  \\right] \\nonumber \\\\\n    & \\stackrel{(d)}{=}  \\bkt{ \\mathbf{f}_q'(\\mathbf{q}_{k},\\mathbf{w}^q,\\gamma_{2k}) },\n    \\label{eq:a2pf}\n\\end{align}\nwhere\n(a) follows from line~\\ref{line:a2} of Algorithm~\\ref{algo:vamp} and \\eqref{eq:jacobian}--\\eqref{eq:bkt};\n(b) follows from \\eqref{eq:g2slrV}; (c) follows from \\eqref{eq:uqslr}; and\n(d) follows from $\\mathbf{V}^{\\text{\\sf T}}\\mathbf{V}=\\mathbf{I}$ and \\eqref{eq:jacobian}--\\eqref{eq:bkt}.\nAlso, from lines~\\ref{line:eta2}-\\ref{line:gam1} of Algorithm~\\ref{algo:vamp},\n\\begin{equation} \\label{eq:g2upa}\n    \\gamma_{1,k\\! + \\! 1} = \\eta_{2k}-\\gamma_{2k} = \\gamma_{2k}\\left[ \\frac{1}{\\alpha_{2k}}-1 \\right].\n\\end{equation}\nEquations \\eqref{eq:a2pf} and \\eqref{eq:g2upa} prove \\eqref{eq:alpha2slr}.\nIn addition,\n\\begin{align}\n    \\lefteqn{ \\mathbf{p}_{k\\! + \\! 1} \\stackrel{(a)}{=} \\mathbf{r}_{1,k\\! + \\! 1} - \\mathbf{x}^0 }\\nonumber\\\\\n    &\\stackrel{(b)}{=}  \\frac{1}{1-\\alpha_{2k}}\\left[\n        \\mathbf{g}_2(\\mathbf{r}_{2k},\\gamma_{2k}) - \\alpha_{2k}\\mathbf{r}_{2k} \\right]\n        -\\mathbf{x}^0 \\nonumber \\\\\n    &\\stackrel{(c)}{=}  \\frac{1}{1-\\alpha_{2k}}\\left[\n        \\mathbf{x}^0 + \\mathbf{V} \\mathbf{f}_q(\\mathbf{q}_k,\\mathbf{w}^q,\\gamma_{2k})\n        - \\alpha_{2k}(\\mathbf{x}^0 + \\mathbf{v}_{k}) \\right]\n        -\\mathbf{x}^0 \\nonumber \\\\\n    &\\stackrel{(d)}{=}  \\frac{1}{1-\\alpha_{2k}}\\left[\n        \\mathbf{V} \\mathbf{f}_q(\\mathbf{q}_k,\\mathbf{w}^q,\\gamma_{2k})\n        - \\alpha_{2k}\\mathbf{v}_{k} \\right] \\nonumber \\\\\n    &\\stackrel{(e)}{=}  \\mathbf{V} \\left[ \\frac{1}{1-\\alpha_{2k}}\\left[\n        \\mathbf{f}_q(\\mathbf{q}_k,\\mathbf{w}^q,\\gamma_{2k}) -  \\alpha_{2k}\\mathbf{q}_{k} \\right] \\right]\n        \\label{eq:pupslr2} ,\n\\end{align}\nwhere (a) follows from \\eqref{eq:pvslr};\n(b) follows from\nlines~\\ref{line:x2}-\\ref{line:r1} of Algorithm~\\ref{algo:vamp};\n(c) follows from \\eqref{eq:g2slrV} and the definition of $\\mathbf{v}_k$ in \\eqref{eq:pvslr};\n(d) follows from collecting the terms with $\\mathbf{x}^0$;\nand (e) follows from the definition\n$\\mathbf{q}_k=\\mathbf{V}^{\\text{\\sf T}}\\mathbf{v}_k$ in \\eqref{eq:uqslr}.\nCombining \\eqref{eq:pupslr2} with\n$\\mathbf{u}_{k\\! + \\! 1} = \\mathbf{V}^{\\text{\\sf T}}\\mathbf{p}_{k\\! + \\! 1}$ proves \\eqref{eq:uupslr}.\n\nThe derivation for the updates for $\\mathbf{v}_k$ are similar.  First,\n\\begin{align}\n    \\MoveEqLeft \\alpha_{1k} \\stackrel{(a)}{=}\n    \\bkt{ \\mathbf{g}_1'(\\mathbf{r}_{1k},\\gamma_{1k}) }\n     \\stackrel{(b)}{=}  \\bkt{ \\mathbf{f}_p'(\\mathbf{p}_{k},\\mathbf{x}^0) },\n    \\label{eq:a1pf}\n\\end{align}\nwhere (a) follows from line~\\ref{line:a1} of Algorithm~\\ref{algo:vamp}\nand (b) follows from the vectorization\nof $\\mathbf{f}_p(\\cdot)$ in \\eqref{eq:fpslr} and the fact that $\\mathbf{p}_k=\\mathbf{r}_{1k}+\\mathbf{x}^0$.\nAlso, from lines~\\ref{line:eta1}-\\ref{line:gam2} of Algorithm~\\ref{algo:vamp},\n\\begin{equation} \\label{eq:g1upa}\n    \\gamma_{2k} = \\eta_{1k}-\\gamma_{1k} = \\gamma_{1k}\\left[ \\frac{1}{\\alpha_{1k}}-1 \\right].\n\\end{equation}\nEquations \\eqref{eq:a1pf} and \\eqref{eq:g1upa} prove \\eqref{eq:alpha1slr}.\nAlso,\n\\begin{align}\n    \\lefteqn{ \\mathbf{v}_{k} \\stackrel{(a)}{=} \\mathbf{r}_{2k} - \\mathbf{x}^0 } \\nonumber \\\\\n    &\\stackrel{(b)}{=}  \\frac{1}{1-\\alpha_{1k}}\\left[\n        \\mathbf{g}_1(\\mathbf{r}_{1k},\\gamma_{1k}) - \\alpha_{1k}\\mathbf{r}_{1k} \\right]   -\\mathbf{x}^0 \\nonumber \\\\\n    &\\stackrel{(c)}{=}  \\frac{1}{1-\\alpha_{1k}}\\left[\n        \\mathbf{f}_p(\\mathbf{p}_{k},\\mathbf{x}^0,\\gamma_{1k}) +\\mathbf{x}^0 - \\alpha_{1k}(\\mathbf{p}_{k}+\\mathbf{x}^0) \\right]\n     -\\mathbf{x}^0 \\nonumber \\\\\n    &\\stackrel{(d)}{=}  \\frac{1}{1-\\alpha_{1k}}\\left[\n        \\mathbf{f}_p(\\mathbf{p}_{k},\\mathbf{x}^0,\\gamma_{1k}) - \\alpha_{1k}\\mathbf{p}_{k}\\right]\n\\end{align}\nwhere (a) is the definition of $\\mathbf{v}_k$ in \\eqref{eq:pvslr};\n(b) follows from lines~\\ref{line:x1}-\\ref{line:r2} of Algorithm~\\ref{algo:vamp};\n(c) follows from the vectorization of $f_p(\\cdot)$ in \\eqref{eq:fpslr} and the definition of $\\mathbf{p}_k$ in \\eqref{eq:pvslr};\nand (d) follows from collecting the terms with $\\mathbf{x}^0$.\nThis proves \\eqref{eq:vupslr}.  All together, we have proven \\eqref{eq:gecslr} and\nthe proof is complete.\n\n\\section{Proof of Theorem~\\ref{thm:seMmse}} \\label{sec:seMmsePf}\nWe use induction.  Suppose that, for some $k$,\n$\\overline{\\gamma}_{1k} = \\tau_{1k}^{-1}$.  From \\eqref{eq:a1se}, \\eqref{eq:A1match}\nand \\eqref{eq:E1match},\n\\begin{equation} \\label{eq:a1matchpf}\n    \\overline{\\alpha}_{1k} = \\overline{\\gamma}_{1k}{\\mathcal E}_1(\\overline{\\gamma}_{1k}).\n\\end{equation}\nHence, from \\eqref{eq:eta1se}, $\\overline{\\eta}_{1k}^{-1} = {\\mathcal E}_1(\\overline{\\gamma}_{1k})$ and\n$\\overline{\\gamma}_{2k} = \\overline{\\eta}_{1k} - \\overline{\\gamma}_{1k}$.  Also,\n\\begin{align*}\n      \\tau_{2k}\n      &\\stackrel{(a)}{=} \\frac{1}{(1-\\overline{\\alpha}_{1k})^2}\\left[\n        {\\mathcal E}_1(\\overline{\\gamma}_{1k},\\tau_{1k}) - \\overline{\\alpha}_{1k}^2\\tau_{1k} \\right] \\\\\n      &\\stackrel{(b)}{=} \\frac{1}{(1-\\overline{\\gamma}_{1k}{\\mathcal E}_1(\\overline{\\gamma}_{1k}))^2}\\left[\n        {\\mathcal E}_1(\\overline{\\gamma}_{1k},\\tau_{1k}) - \\overline{\\gamma}_{1k}{\\mathcal E}_1^2(\\overline{\\gamma}_{1k}) \\right] \\\\\n      &\\stackrel{(c)}{=} \\frac{{\\mathcal E}_1(\\overline{\\gamma}_{1k},\\tau_{1k})}\n      {1-\\overline{\\gamma}_{1k}{\\mathcal E}_1(\\overline{\\gamma}_{1k})} \\\\\n      &\\stackrel{(d)}{=} \\frac{1}{\\overline{\\eta}_{1k} - \\overline{\\gamma}_{1k}},\n\\end{align*}\nwhere (a) follows from \\eqref{eq:tau2se};\n(b) follows from \\eqref{eq:a1matchpf} and the matched condition $\\overline{\\gamma}_{1k} = \\tau_{1k}^{-1}$;\n(c) follows from canceling terms in the fraction and (d) follows from the fact that\n$\\overline{\\eta}_{1k}^{-1} = {\\mathcal E}_1(\\overline{\\gamma}_{1k})$ and $\\overline{\\gamma}_{1k} = \\overline{\\eta}_{1k}\/\\overline{\\alpha}_{1k}$.\nThis proves \\eqref{eq:eta1sematch}.  A similar argument shows that \\eqref{eq:eta2sematch} holds if\n$\\overline{\\gamma}_{2k} = \\tau_{2k}^{-1}$.    Finally, \\eqref{eq:etammse} follows from\n\\eqref{eq:sematch} and \\eqref{eq:mseEcal}.\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nA broad class of both Bayesian \n\\citep{neal, williams1997, hazan2015steps, lee2018deep, matthews2018, matthews2018b_arxiv, Borovykh2018, garriga2018deep, novak2018bayesian, yang2017mean, yang2018a, pretorius2019expected, yang2019scaling, yang2019wide, neuraltangents2020, hron2020, Hu2020InfinitelyWG} and gradient descent trained \\citep{Jacot2018ntk, li2018learning, allen2018convergence, du2018gradient, du2018gradienta, zou2018stochastic, lee2019wide, chizat2019lazy, arora2019on, sohl2020infinite, Huang2020OnTN, du2019graph, yang2019scaling, yang2019wide, neuraltangents2020, hron2020} neural networks converge to Gaussian Processes (GPs) or closely-related kernel methods as their intermediate layers are made infinitely wide. \nThe predictions of these infinite width networks are described by the Neural Network Gaussian Process (NNGP) \\citep{lee2018deep,matthews2018} kernel for Bayesian networks, and by the Neural Tangent Kernel (NTK) \\citep{Jacot2018ntk} and weight space linearization \\citep{lee2019wide,chizat2019lazy} for gradient descent trained networks. \n\nThis correspondence has been key to recent breakthroughs in our understanding of neural networks \\citep{xiao18a, valle-perez2018deep, wei2019regularization, xiao2019disentangling, NIPS2019_9449, ben2019role, yang2019fine, ober2020global, Hu2020Provable, lewkowycz2020large, lewkowycz2020training}. \nIt has also enabled practical advances in kernel methods \\citep{garriga2018deep, novak2018bayesian, arora2019on, li2019enhanced, Arora2020Harnessing, Shankar2020NeuralKW, neuraltangents2020, hron2020}, Bayesian deep learning \\citep{wang2018function, Cheng_2019_CVPR, Carvalho2020ScalableUF}, active learning \\citep{ijcai2019-499}, and semi-supervised learning \\citep{Hu2020InfinitelyWG}.\nThe NNGP, NTK, and related large width limits \\citep{cho2009kernel, daniely2016toward, poole2016exponential, chen2018rnn, li2018on, daniely2017sgd,  pretorius2018critical, hayou2018selection, karakida2018universal, blumenfeld2019mean, hayou2019meanfield, schoenholz2016deep, pennington2017resurrecting, xiao18a, yang2017mean, geiger2019disentangling, geiger2020scaling, antognini2019finite,  Dyer2020Asymptotics, huang2019dynamics, yaida2019non} are unique in giving an exact theoretical description of large scale neural networks.\nBecause of this, we believe they will continue to play a transformative role in deep learning theory.\n\nInfinite networks are a newly active field, and \nfoundational\nempirical questions remain unanswered. \nIn this work, we perform an extensive and in-depth empirical study of finite and infinite width neural networks. \nIn so doing, we provide quantitative answers to questions about the factors of variation that drive performance in finite networks and kernel methods, uncover surprising new behaviors, and develop best practices that improve the performance of both finite and infinite width networks.\nWe believe our results will both ground and motivate future work in wide networks.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Experiment design}\\label{sec:experimental_design}\n\n\n\\definecolor{ntk_param}{RGB}{255, 140, 0}\n\\definecolor{standard_param}{RGB}{65, 105, 225}\n\n\n\\vspace{-0.15cm}\\begin{figure}[t]\n\\hfill\\hfill Finite gradient descent (GD) \\hfill \\quad\\quad Infinite GD \\quad Infinite Bayesian\n\\vspace{-0.25cm}\n\\centering\n\\includegraphics[width=1.02\\columnwidth]{figures\/main_plot_nonlinear_split.pdf}\n\\caption{\n\n\\textbf{CIFAR-10 test accuracy for finite and infinite networks and their variations}. Starting from the \nfinite width %\n\\texttt{base} network of given architecture class described in \\sref{sec:experimental_design}, performance changes from \\textbf{centering} (\\texttt{+C}), \\textbf{large learning rate} (\\texttt{+LR}), allowing \\textbf{underfitting} by early stopping (\\texttt{+U}), input preprocessing with \\textbf{ZCA regularization} (\\texttt{+ZCA}), multiple initialization \\textbf{ensembling} (\\texttt{+Ens}), and some combinations are shown, for {\\color{standard_param}\\textbf{Standard}} and {\\color{ntk_param}\\textbf{NTK}} parameterizations. \nThe performance of the \\textbf{linearized} (\\texttt{lin}) base network is also shown. \nSee \\Tabref{tab:main-table} for precise values for each of these experiments, as well as for additional experimental conditions not shown here.}\n\\label{fig:tricks_vs_accuracy}\n\\end{figure}\n\n\nTo systematically develop a phenomenology of infinite and finite neural networks, we first establish base cases for each architecture where infinite-width kernel methods, linearized weight-space networks, and nonlinear gradient descent based training can be directly compared. In the finite-width settings, the base case uses \nmini-batch gradient descent at a constant small learning rate~\\cite{lee2019wide} with MSE loss (implementation details in~\\sref{app:batch-size}). In the kernel-learning setting we compute the NNGP and NTK for the entire dataset and do exact inference as described in~\\cite[page 16]{rasmussen2006gaussian}. Once this one-to-one comparison has been established, we augment the base setting with a wide range of interventions. We discuss each of these interventions in detail below. Some interventions will approximately preserve the correspondence (for example, data augmentation), while others explicitly break the correspondence in a way that has been hypothesized in the literature to affect performance (for example, large learning rates~\\cite{lewkowycz2020large}). \nWe additionally explore linearizing the base model around its initialization, in which case its training dynamics become exactly described by a constant kernel. This differs from the kernel setting described above due to finite width effects.\n\nWe use MSE loss to allow for easier comparison to kernel methods, whose predictions can be evaluated in closed form for MSE. \nSee \\Tabref{tab:xent-vs-mse} and \\Figref{fig:xent-vs-mse} for a comparison of MSE to softmax-cross-entropy loss. \nSoftmax-cross-entropy provides a consistent small benefit over MSE, and will be interesting to consider in future work.\n\nArchitectures we work with are built from either Fully-Connected (\\texttt{FCN})\nor Convolutional (\\texttt{CNN}) layers. In all cases we use ReLU nonlinearities with critical initialization with small bias variance ($\\sigma_w^2=2.0, \\sigma_b^2=0.01$). Except if otherwise stated, we consider \\texttt{FCN}s with 3-layers of width 2048 and \\texttt{CNN}s with 8-layers of 512 channels per layer. For convolutional networks we must collapse the spatial dimensions of image-shaped data before the final readout layer. To do this we either: flatten the image into a one-dimensional vector (\\texttt{VEC}) or apply global average pooling to the spatial dimensions (\\texttt{GAP}). \nFinally, we compare two ways of parameterizing the weights and biases of the network: the standard parameterization (STD), which is used in work on finite-width networks, and the NTK parameterization (NTK) which has been \nused in \nmost infinite-width studies to date (see~\\cite{sohl2020infinite} for the standard parameterization at infinite width).\n\nExcept where noted, for all kernel experiments we optimize over diagonal kernel regularization\nindependently for each experiment.\nFor finite width networks, except where noted we use a small learning rate corresponding to the base case. See \\sref{app hyperparameters} for details.\n\nThe experiments described in this paper are often very compute intensive. For example, to compute the NTK or NNGP for the entirety of CIFAR-10 for \\texttt{CNN-GAP} architectures one must explicitly evaluate the entries in a $6\\times10^7$-by-$6\\times10^7$ \nkernel matrix. Typically this takes around 1200 GPU hours with double precision, and so we implement our experiments via massively distributed compute infrastructure based on beam~\\cite{beam}. All experiments use the Neural Tangents library~\\cite{neuraltangents2020}, built on top of JAX~\\cite{jax2018github}. \n\nTo be as systematic as possible while also tractable given this large computational requirement, we evaluated every intervention for every architecture and focused on a single dataset, CIFAR-10~\\cite{krizhevsky2009learning}. However, to ensure robustness of our results across dataset, we evaluate several key claims on CIFAR-100 and Fashion-MNIST~\\cite{xiao2017\/online}. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Observed empirical phenomena}\n\n\n\n\\subsection{NNGP\/NTK can outperform finite networks}\n\\label{sec:infinite-vs-finite}\nA common \nassumption\nin the study of infinite networks is that they underperform the corresponding finite network in the large data regime.\nWe carefully examine this assumption, by comparing kernel methods against the base case of a finite width architecture trained with small learning rate and no regularization (\\sref{sec:experimental_design}), and then individually examining the effects of common training practices which break (large LR, L2 regularization) or improve (ensembling) the infinite width correspondence to kernel methods.\nThe results of these experiments are \nsummarized in \\Figref{fig:tricks_vs_accuracy} and \\Tabref{tab:main-table}.\n\nFirst focusing on base finite networks, we observe that infinite \\texttt{FCN} and \\texttt{CNN-VEC} outperform their respective finite networks. On the other hand, infinite \\texttt{CNN-GAP} networks perform worse than their finite-width counterparts in the base case, consistent with observations in~\\citet{arora2019on}. We emphasize that architecture plays a key role in relative performance, in line with an observation made in \\citet{geiger2019disentangling} in the study of lazy training. For example, infinite-\\texttt{FCN}s outperform finite-width networks even when combined with various tricks such as high learning rate, L2, and underfitting. Here the performance becomes similar only after ensembling (\\sref{sec:ensemble_of_networks}). \n\nOne interesting observation is that ZCA regularization preprocessing (\\sref{sec:zca}) can provide significant improvements to the \\texttt{CNN-GAP} kernel, closing the gap to within 1-2\\%. \n\n\\subsection{NNGP typically outperforms NTK}\n\\label{sec:nngp-vs-ntk}\nRecent %\nevaluations of infinite width networks have put significant emphasis on the NTK, without explicit comparison against the respective NNGP models \\citep{arora2019on, li2019enhanced, du2019graph, Arora2020Harnessing}. Combined with the view of NNGPs as ``weakly-trained'' \\citep{lee2019wide, arora2019on} (i.e. having only the last layer learned), one might expect NTK to be a more effective model class than NNGP.\nOn the contrary, we usually\nobserve that NNGP inference achieves better performance. This can be seen in \\Tabref{tab:main-table} where SOTA performance among fixed kernels is attained with the NNGP across all architectures. \nIn \\Figref{fig:nngp-vs-ntk} we show that this trend persists across CIFAR-10, CIFAR-100, and Fashion-MNIST (see~\\Figref{fig:nngp-vs-ntk-uci} for similar trends on UCI regression tasks). \nIn addition to producing stronger models, NNGP kernels require about half the memory and compute as the corresponding NTK, \nand some of the most performant kernels do not have an associated NTK at all~\\cite{Shankar2020NeuralKW}. Together these results suggest that when approaching a new problem where the goal is to maximize performance, \npractitioners should start with the NNGP.\n\nWe emphasize that both tuning of the diagonal regularizer (\\Figref{fig reg compare}) and sufficient numerical precision (\\sref{sec:diag-reg}, \\Figref{app kernel spectra}) were crucial to achieving an accurate comparison of these kernels.\n\n\n\n\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{figures\/nngp_vs_ntk.pdf}\n\\caption{\n\n\\textbf{NNGP often outperforms NTK in image classification tasks when diagonal regularization is carefully tuned.} \nThe performance of the NNGP and NT kernels are plotted against each other \nfor a variety of data pre-processing configurations (\\sref{sec:zca}),\nwhile regularization (\\Figref{fig reg compare}) is independently tuned for each.\n}\n\\label{fig:nngp-vs-ntk}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{figures\/ensemble_valid.pdf}\n\\caption{\n\n\\textbf{Centering can accelerate training and improve performance}. Validation accuracy throughout training for several finite width architectures. See \\Figref{fig:training_curves} for training accuracy. \n}\n\\label{fig:validation_curves}\n\\end{figure}\n\n\n\\subsection{Centering and ensembling finite networks both lead to kernel-like performance}\n\\label{sec:ensemble_of_networks}\nFor overparameterized neural networks, some randomness from the initial parameters persists throughout training and the resulting learned functions are themselves random. This excess variance in the network's predictions generically increases the total test error through the variance term of the bias-variance decomposition. For infinite-width kernel systems this variance is eliminated by using the mean predictor. For finite-width models, the variance can be large, and test performance can be significantly improved by \\emph{ensembling} a collection of models~\\cite{geiger2019disentangling, geiger2020scaling}. %\nIn \\Figref{fig:ensemble}, we examine the effect of ensembling. For \\texttt{FCN}, ensembling closes the gap with kernel methods, suggesting that finite width \\texttt{FCN}s underperform \\texttt{FCN} kernels primarily due to variance.\nFor \\texttt{CNN} models, ensembling also improves test performance, and ensembled \\texttt{CNN-GAP} models significantly outperform the best kernel methods. \nThe observation that ensembles of finite width \\texttt{CNN}s can outperform infinite width networks while ensembles of finite \\texttt{FCN}s cannot (see \\Figref{fig:ensemble}) is consistent with earlier findings in~\\cite{geiger2020scaling}.\n\nPrediction variance can also be reduced by \\emph{centering} the model, i.e. subtracting the model's initial predictions: $f_\\text{centered}(t) = f(\\theta(t)) - f(\\theta(0))$. A similar variance reduction technique has been studied in~\\cite{chizat2019lazy, zhang2019type, hu2020Simple, bai2020Beyond}. \nIn \\Figref{fig:validation_curves}, we observe that centering significantly speeds up training and improves generalization for \\texttt{FCN} and \\texttt{CNN-VEC} models, but has little-to-no effect on \\texttt{CNN-GAP} architectures.\nWe observe that the scale posterior variance of \\texttt{CNN-GAP}, in the infinite-width kernel, is small relative to the prior variance given more data, consistent with centering and ensembles having small effect.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{figures\/ensemble_network_performance.pdf}\n\\caption{\n\n\\textbf{Ensembling base networks enables them to match the performance of kernel methods, and exceed kernel performance for nonlinear \\texttt{CNN}s.} See \\Figref{fig app ensemble} for test MSE.\n}\n\\label{fig:ensemble}\n\\end{figure}\n\n\n\\subsection{Large LRs and L2 regularization drive differences between finite networks and kernels}\\label{sec:l2_lr}\nIn practice, L2 regularization (a.k.a. weight decay) or larger learning rates can break the correspondence between kernel methods and finite width neural network training even at large widths. \n\n\\citet{lee2019wide} \nderives \na critical learning rate $\\eta_{\\text{critical}}$ such that wide network training dynamics are equivalent to linearized training for $\\eta< \\eta_{\\text{critical}}$.\n\\citet{lewkowycz2020large} argues that even at large width a learning rate $\\eta \\in (\\eta_{\\text{critical}}, c\\cdot\\eta_{\\text{critical}})$ for a constant $c>1$ forces the network to move away from its initial high curvature minimum and converge to a lower curvature minimum, while \\citet{li2019towards} argues that large initial learning rates enable networks to learn `hard-to-generalize' patterns.\n\nIn \\Figref{fig:tricks_vs_accuracy} (and \\Tabref{tab:main-table}), we observe that the effectiveness of a large learning rate (LR) is highly sensitive to both architecture and paramerization: LR improves performance of \\texttt{FCN} and \\texttt{CNN-GAP} by about $ 1\\%$ for STD parameterization and about $2\\%$ for NTK parameterization. In stark contrast, it has little effect on \\texttt{CNN-VEC} with NTK parameterization and surprisingly, a huge performance boost on \\texttt{CNN-VEC} with STD parameterization ($+5\\%$).      \n\nL2 regularization (\\eqref{eq:l2-reg}) regularizes the squared distance between the parameters and the origin and encourages the network to converge to minima with smaller Euclidean norms. Such minima are different from those obtained by NT kernel-ridge regression (i.e. adding a diagonal regularization term to the NT kernel) \\citep{wei2019regularization},\nwhich essentially penalizes the deviation of the network's parameters from initialization \\cite{hu2019understanding}.  See~\\Figref{fig:reg-compare-sm} for a comparison.\n\nL2 regularization consistently improves (+$1$-$2\\%$) performance for all architectures and parameterizations. \nEven with a well-tuned L2 regularization, finite width \\texttt{CNN-VEC} and \\texttt{FCN} still underperform NNGP\/NTK. \nCombining L2 with early stopping produces a dramatic additional $10\\% - 15\\%$ performance boost for finite width \\texttt{CNN-VEC}, outperforming NNGP\/NTK.\nFinally, we note that L2+LR together provide a superlinear performance gain for all cases except \\texttt{FCN} and \\texttt{CNN-GAP} with NTK-parameterization. \nUnderstanding the nonlinear interactions between L2, LR, and early stopping on finite width networks is an important research question (e.g. see~\\cite{lewkowycz2020large,lewkowycz2020training} for LR\/L2 effect on the training dynamics). \n\n\\subsection{Improving L2 regularization for networks using the standard parameterization}\n\\label{sec improved standard}\n\nWe find that L2 regularization provides dramatically more benefit (by up to $6\\%$) to finite width networks with the NTK parameterization than to those that use the standard parameterization (see \\Tabref{tab:main-table}). There is a bijective mapping between weights in networks with the two parameterizations, which preserves the function computed by both networks: $W^l_\\text{STD} =  \\nicefrac{W^l_\\text{NTK}\\,}{\\sqrt{n^l}}$, where $W^l$ is the $l$th layer weight matrix, and $n^l$ is the width of the preceding activation vector. \nMotivated by the improved performance of the L2 regularizer in the NTK parameterization, we use this mapping to construct a regularizer for standard parameterization networks that produces the same penalty as vanilla L2 regularization would produce on the equivalent NTK-parameterized network. This modified regularizer is\n    ${R}^{\\text{STD}}_{\\text{Layerwise}} = \\frac{\\lambda}{2} \\sum_l n^l  \\norm{W^l_\\text{STD}}^2$.\nThis can be thought of as a layer-wise regularization constant $\\lambda^l = \\lambda n^l$. \nThe improved performance of this regularizer is illustrated in \\Figref{fig reg compare}.\n\n\\begin{figure}\n\\centering\n\\begin{overpic}[width=\\columnwidth]{figures\/network_l2_ntk.pdf}\n\\end{overpic}\n\\ \\ \n\\vspace{-0.45cm}\n\\caption{\n\n\\textbf{Layerwise scaling motivated by NTK makes L2 regularization more helpful in standard parameterization networks.}\nSee \\sref{sec improved standard} for introduction of the improved regularizer, \\Figref{fig:l2-init} for further analysis on L2 regularization to initial weights, and \\Figref{fig:reg-compare-sm} for effects on varying widths.\n\\label{fig reg compare}\n}\n\\end{figure}\n\n\n\\subsection{Performance can be non-monotonic in width beyond double descent}\n\\label{sec:perf_vs_width}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{figures\/network_vs_width_ntk.pdf}\n\\caption{\n\n\\textbf{Finite width networks generally perform better with increasing width, but \\texttt{CNN-VEC} shows surprising non-monotonic behavior.}\n {\\bf L2}: non-zero weight decay allowed during training {\\bf LR}: large learning rate allowed. Dashed lines are allowing underfitting (\\textbf{U}). See \\Figref{fig:width-combined} for plots for the standard parameterization, and \\sref{sec:equivariance} for discussion of \\texttt{CNN-VEC} results.\n }\n\\label{fig:width}\n\\end{figure}\n\nDeep learning practitioners have repeatedly found that increasing the number of parameters in their models leads to improved performance~\\citep{lawrence1998size, bartlett1998sample, Neyshabur2014InSO, canziani2016analysis, novak2018sensitivity, parkoptimal, novak2018bayesian}. \nWhile this behavior is consistent with a Bayesian perspective on generalization \\citep{mackay1995probable,smith2017bayesian, wilson2020bayesian},\nit seems at odds with classic generalization theory which primarily considers worst-case overfitting \\citep{haussler1992decision, NIPS1988_154, Vapnik1998StatisticalLT, bartlett2002rademacher, bousquet2002stability, mukherjee2004statistical, poggio2004general}. This has led to a great deal of work on the interplay of overparameterization and generalization \\citep{zhang2016understanding, advani2017high, neyshabur2018towards, neyshabur2018the, NIPS2018_8038, allen2019learning, ghorbani2019limitations, ghorbani2019linearized, arora2019fine, brutzkus19b}.\nOf particular interest has been the phenomenon of double descent, in which performance increases overall with parameter account, but drops dramatically when the neural network is roughly critically parameterized~\\citep{opper1990ability, belkin2019reconciling, nakkiran2019deep}.\n\nEmpirically, we find that in most cases (\\texttt{FCN} and \\texttt{CNN-GAP} in both parameterizations, \\texttt{CNN-VEC} with standard parameterization) increasing width leads to monotonic improvements in performance. \nHowever, we also find a more complex dependence on width in specific relatively simple settings. \nFor example, in \\Figref{fig:width} for \\texttt{CNN-VEC}\nwith NTK parameterization the performance depends non-monotonically on the width, and the optimal width has an intermediate value.\\footnote{Similar behavior was observed in~\\cite{andreassen2020} for \\texttt{CNN-VEC} and in~\\cite{aitchison2019bigger} for finite width Bayesian networks.} This nonmonotonicity is distinct from double-descent-like behavior, as all widths correspond to overparameterized models.\n\n\n\\subsection{Diagonal regularization of kernels behaves like early stopping}\\label{sec:diag-reg}\n\\vspace{-0.1cm}\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.245\\columnwidth]{figures\/fc_diag_reg.pdf}\n\\includegraphics[width=0.245\\columnwidth]{figures\/cv_diag_reg.pdf}\n\\includegraphics[width=0.495\\columnwidth]{figures\/cg_diag_reg.pdf}\n\\caption{\n\n\\textbf{Diagonal kernel regularization acts similarly to early stopping.}\nSolid lines corresponds to NTK inference with varying diagonal regularization $\\varepsilon$. Dashed lines correspond to predictions after gradient descent evolution to time $\\tau = \\eta t$ (with $\\eta=\\nicefrac{m}{\\textrm{tr}({\\cal K})}$). \nLine color indicates varying training set size $m$. \nPerforming early stopping at time $t$ corresponds closely to regularizing with coefficient $\\varepsilon = \\nicefrac{K m}{\\eta t}$, where $K=10$ denotes number of output classes.\n}\n\\label{fig:diag-reg}\n\\end{figure}\n\nWhen performing kernel inference, it is common to add a diagonal regularizer to the training kernel matrix, ${\\cal K}_{\\textrm{reg}} = {\\cal K} + \\varepsilon \\tfrac{\\textrm{tr}({\\cal K})}{m} I$. For linear regression, \\citet{ali2019continuous} proved that the inverse of a kernel regularizer is related to early stopping time under gradient flow. With kernels, gradient flow dynamics correspond directly to training of a wide neural network \\citep{Jacot2018ntk, lee2019wide}. \n\nWe experimentally explore the relationship between early stopping, kernel regularization, and generalization in \\Figref{fig:diag-reg}. \nWe observe a close relationship between regularization and early stopping, and find that in most cases the best validation performance occurs with early stopping and non-zero $\\varepsilon$. \nWhile \\citet{ali2019continuous} do not consider a $\\tfrac{\\textrm{tr}({\\cal K})}{m}$ scaling on the kernel regularizer, we found it useful since experiments become invariant under scale of ${\\cal K}$.\n\n\n\\subsection{\nFloating point precision determines critical dataset size for failure of kernel methods\n}\n\\label{sec:kernel eigs}\n\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=\\linewidth]{figures\/kernel_spectrum_powerlaw.png}\n\\caption{\n\\textbf{Tail eigenvalues of infinite network kernels show power-law decay.} \nThe red dashed line shows the predicted scale of noise in the eigenvalues due to floating point precision, for kernel matrices of increasing width. \nEigenvalues for CNN-GAP architectures decay fast, and may be overwhelmed by \\texttt{float32} quantization noise \nfor dataset sizes of $O(10^4)$. For \\texttt{float64}, quantization noise is not predicted to become significant until a dataset size of $O(10^{10})$ (\\Figref{app kernel spectra}).\n}\n\\label{fig:kernel_spectra}\n\\end{figure}\n\n\nWe observe empirically that kernels become sensitive to \\texttt{float32} vs. \\texttt{float64} numerical precision at a critical dataset size. For instance, GAP models suffer \\texttt{float32} numerical precision errors at a dataset size of $\\sim{10}^4$.\n This phenomena can be understood with a simple random noise model (see \\sref{app:noise-model} for details). The key insight is that kernels with fast eigenvalue decay suffer from floating point noise. Empirically, the tail eigenvalue of the NNGP\/NTK follows a power law (see \\Figref{fig:kernel_spectra})\n and measuring their decay trend provides good indication of critical dataset size\n\\begin{equation}\n    m^* \\gtrsim\n    \\left(\\nicefrac{C}{\\pp{\\sqrt{2} \\sigma_n}}\\right)^{\\tfrac{2}{2\\alpha - 1}} \\quad \\textrm{if } \\alpha > \\tfrac{1}{2}\\ \\qquad \\left(\\infty  \\quad \\textrm{otherwise}\\right)\\,,\n\\label{eq:critical-m}\n\\end{equation}\nwhere $\\sigma_n$ is the typical noise scale, e.g. \\texttt{float32} epsilon, and the kernel eigenvalue decay  is modeled as $\\lambda_i \\sim C \\, i^{-\\alpha}$ as $i$ increases. \nBeyond this critical dataset size, the smallest eigenvalues in the kernel become dominated by floating point noise.\n\n\\subsection{Linearized \\texttt{CNN-GAP} models perform poorly due to poor conditioning}\n\\label{sec:cnn-gap-conditioning}\n\nWe observe that the linearized \\texttt{CNN-GAP} \nconverges {\\em extremely} slowly on the training set (\\Figref{fig:training_curves}), \nleading to poor validation performance (\\Figref{fig:validation_curves}). \nEven after training for more than 10M steps with varying L2 regularization strengths and LRs, the best training accuracy was below 90\\%, and test accuracy $\\sim$70\\% -- worse\nthan both the corresponding infinite and nonlinear finite width networks.\n\nThis is caused by \npoor conditioning of pooling networks. \\citet{xiao2019disentangling} (Table 1) show that the conditioning at initialization of a \\texttt{CNN-GAP} network is worse than that of \\texttt{FCN} or \\texttt{CNN-VEC} networks by a factor of the number of pixels (1024 for CIFAR-10). This poor conditioning of the kernel eigenspectrum can be seen in \\Figref{fig:kernel_spectra}. For linearized networks, in addition to slowing training by a factor of 1024, this leads to numerical instability when using \\texttt{float32}.\n   \n\n\n\\subsection{\nRegularized ZCA whitening improves accuracy\n}\\label{sec:zca}\n\n\nZCA whitening~\\cite{bell1997independent} (see \\Figref{fig cifar zca} for an illustration) is a data preprocessing technique that was once common~\\cite{goodfellow2013maxout,zagoruyko2016wide}, but has fallen out of favor. However it was recently shown to dramatically improve accuracy in some kernel methods by~\\citet{Shankar2020NeuralKW}, in combination with a small regularization parameter in the denominator (see \\sref{app ZCA}). \nWe investigate the utility of ZCA whitening as a preprocessing step for both finite and infinite width neural networks. \nWe observe that while pure ZCA whitening is detrimental for both kernels and finite networks (consistent with predictions in \\citep{wadia2020whitening}), with tuning of the regularization parameter it provides performance benefits for both kernel methods and finite network training (\\Figref{fig:zca}).  \n \n\n\\begin{figure}\n\\centering\n\\begin{overpic}[width=\\linewidth]{figures\/kernel_zca.pdf}\n \\put (0,0) {\\textbf{\\small(a)}}\n\\end{overpic}   \n\n\\vspace{0.2cm}\n\\begin{overpic}[width=\\linewidth]{figures\/network_zca.pdf}\n \\put (0,0) {\\textbf{\\small(b)}}\n\\end{overpic}   \n\n\\caption{\n\n\\textbf{Regularized ZCA whitening improves image classification performance for both finite and infinite width networks.} \nAll plots show performance as a function of ZCA regularizaiton strength.\n(\\textbf{a}) ZCA whitening of inputs to kernel methods on CIFAR-10, Fashion-MNIST, and CIFAR-100. (\\textbf{b}) ZCA whitening of inputs to finite width networks (training curves in \\Figref{fig:app-zca-training}).\n}\n\\label{fig:zca}\n\\end{figure}\n\n\\subsection{Equivariance \nis only beneficial for narrow networks far from the kernel regime\n}\\label{sec:equivariance}\n    \n    Due to weight sharing between spatial locations, outputs of a convolutional layer are translation-{\\em equivariant} (up to edge effects), i.e. if an input image is translated, the activations are translated in the same spatial direction. However, the vast majority of contemporary \\texttt{CNN}s utilize weight sharing in conjunction with pooling layers, making the network outputs approximately translation-\\textit{invariant} (\\texttt{CNN-GAP}). \n    The impact of equivariance alone (\\texttt{CNN-VEC})\n    on generalization is not well understood -- \n    it is a property of internal representations only, and does not translate into meaningful statements about the classifier outputs. \n    Moreover, in the infinite-width limit it is guaranteed to have no impact on the outputs \\citep{novak2018bayesian, yang2019scaling}. In the finite regime it has been reported both to provide substantial benefits by \\citet{lecun1989generalization, novak2018bayesian} and no significant benefits by \\citet{bartunov2018assessing}.\n    \n    We conjecture that equivariance can only be leveraged far from the kernel regime. Indeed, as observed in \\Figref{fig:tricks_vs_accuracy} and discussed in \\sref{sec:l2_lr}, multiple kernel correspondence-breaking tricks are required for a meaningful boost in performance over NNGP or NTK (which are mathematically guaranteed to not benefit from equivariance), and the boost is largest at a moderate\n    width (\\Figref{fig:width}). \n    Otherwise, even large ensembles of equivariant models (see \\texttt{CNN-VEC LIN} in \\Figref{fig:ensemble}) perform comparably to their infinite width, equivariance-agnostic counterparts. Accordingly, prior work that managed to extract benefits from equivariant models \\citep{lecun1989generalization, novak2018bayesian} tuned networks far outside the kernel regime (extremely small size and \\texttt{+LR+L2+U} respectively). We further confirm this phenomenon in a controlled setting in \\Figref{fig:crop_translate}.\n    \n\n    \\definecolor{darkred_f}{RGB}{247, 129, 191}\n    \\definecolor{darkblue_f}{RGB}{55, 126, 184}\n    \\definecolor{darkorange_f}{RGB}{255, 127, 0}\n    \\definecolor{darkgreen_f}{RGB}{77, 175, 74}\n\n    \\begin{figure}\n        \\centering\n        \\includegraphics[width=0.4\\textwidth]{figures\/crop_4.pdf}\n        \\includegraphics[width=0.59\\textwidth]{figures\/translate_4.pdf}\n        \\caption{\n        \n        \\textbf{Equivariance is only leveraged in a \\texttt{CNN} model outside of the kernel regime.} \n        If a \\texttt{CNN} model is able to utilize equivariance effectively, we expect it to be more robust to crops and translations than an {\\color{darkred_f}\\texttt{FCN}}. Surprisingly, performance of a {\\color{darkgreen_f}wide \\texttt{CNN-VEC}} degrades with the  magnitude of the input perturbation as fast as that of an  {\\color{darkred_f}\\texttt{FCN}}, indicating that equivariance is not exploited.\n        In contrast, performance of a {\\color{darkorange_f}narrow model with weight decay (\\texttt{CNN-VEC+L2+narrow})} falls off much slower.\n        {\\color{darkblue_f}Translation-invariant \\texttt{CNN-GAP}} remains, as expected, the most robust. Details in \\sref{sec:equivariance}, \\sref{app hyperparameters}. \n        }\n        \\label{fig:crop_translate}\n    \\end{figure}\n\n\\subsection{Ensembling kernel predictors enables practical data augmentation with NNGP\/NTK}\\label{sec:data-augmentation}\nFinite width neural network often are trained with data augmentation (DA) to improve performance. We observe that the \\texttt{FCN} and \\texttt{CNN-VEC} architectures (both finite and infinite networks) benefit from DA, and that DA can cause \\texttt{CNN-VEC} to become competitive with \\texttt{CNN-GAP} (\\Tabref{tab:main-table}). While \\texttt{CNN-VEC} possess translation equivariance but not invariance (\\sref{sec:equivariance}), we believe it can effectively leverage equivariance to learn invariance from data.\n\nFor kernels, expanding a dataset with augmentation is computationally challenging, since kernel computation is quadratic in dataset size, and inference is cubic. \n\\citet{li2019enhanced, Shankar2020NeuralKW} incorporated flip augmentation by doubling the training set size. \nExtending this strategy to more augmentations such as crop or mixup~\\cite{zhang2018mixup}, or to broader augmentations strategies like AutoAugment~\\cite{cubuk2019autoaugment} and RandAugment~\\cite{cubuk2019randaugment}, becomes rapidly infeasible.\n\nHere we introduce a straightforward method for ensembling kernel predictors to enable more extensive data augmentation. \nMore sophisticated approximation approaches such as the Nystr\u00f6m method~\\citep{williams2001using} might yield even better performance. \nThe strategy involves constructing a set of augmented batches, performing kernel inference for each of them, and then performing ensembling \nof the resulting predictions. This is equivalent to replacing the kernel with a block diagonal approximation, where each block corresponds to one of the batches, and the union of all augmented batches is the full augmented dataset. See \\sref{app kernel ensembling} for more details.\nThis method achieves SOTA for a kernel method corresponding to the infinite width limit of each architecture class we studied (\\Figref{fig:kerne-da-ens} and \\Tabref{tab:sota-kernel-table}).\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{figures\/da_ensemble.pdf}\n\\caption{\n\n\\textbf{Ensembling kernel predictors makes predictions from large augmented datasets computationally tractable.} \nWe used standard crop by 4 and flip data augmentation (DA) common for training neural networks for CIFAR-10. We observed that DA ensembling improves accuracy and is much more effective for NNGP compared to NTK. In the last panel, we applied data augmentation by ensemble to the Myrtle architecture studied in \\citet{Shankar2020NeuralKW}. We observe improvements over our base setting, but do not reach the reported best performance. We believe techniques such as leave-one-out tilt and ZCA augmentation also used in~\\cite{Shankar2020NeuralKW} contribute to this difference.}\n\\label{fig:kerne-da-ens}\n\\end{figure}\n\n\n\n\n\n\\begin{table}\n\\centering\n\\caption{\n\n\\textbf{CIFAR-10 test accuracy for kernels of the  corresponding architecture type}}\n\\label{tab:sota-kernel-table}\n\\vspace{0.1cm}\n\\resizebox{\\textwidth}{!}{\n\\begin{tabular}{@{}llll@{}}\n\\toprule\nArchitecture  & Method & \n\\begin{tabular}[c]{@{}l@{}}NTK\n\\end{tabular} & \n\\begin{tabular}[c]{@{}l@{}}NNGP\n\\end{tabular} \\\\ \n\\midrule \\midrule\n{}{}{\\textbf{FC}}             \n                                         & \\citet{novak2018bayesian}                     & -                 & 59.9          \\\\\n                                         &  ZCA Reg (this work)                      & 59.7             & 59.7          \\\\\n                                         & DA Ensemble (this work)               & \\textbf{61.5}     & \\textbf{62.4} \\\\\n\\midrule\n{\\textbf{CNN-VEC}}        & \\citet{novak2018bayesian}                     & \\textbf{-}        & 67.1          \\\\\n                                         & \\citet{li2019enhanced}                    & 66.6             & 66.8         \\\\\n                                         & ZCA Reg (this work)                      & 69.8             & 69.4         \\\\\n                                         &  Flip Augmentation, \\citet{li2019enhanced}            & 69.9             & 70.5          \\\\\n                                         & DA Ensemble (this work)               & \\textbf{70.5}     & \\textbf{73.2} \\\\\n\\midrule\n{\\textbf{CNN-GAP}}        & \\citet{arora2019on, li2019enhanced}             & 77.6             & 78.5          \\\\\n                                         &  ZCA Reg (this work)                      & 83.2    &  83.5 \\\\\n                                         & Flip Augmentation, \\citet{li2019enhanced}           & 79.7             & 80.0          \\\\\n                                         & DA Ensemble (this work)               & \\textbf{83.7 (32 ens)}            & \\textbf{84.8 (32 ens)}         \\\\\n \\midrule\n{\\textbf{Myrtle}\n\\tablefootnote{The normalized Gaussian Myrtle kernel used in~\\citet{Shankar2020NeuralKW} does not have a corresponding finite-width neural network, and was additionally tuned on the test set for the case of CIFAR-10.} \n} \n                                         & Myrtle ZCA and Flip Augmentation, \\citet{Shankar2020NeuralKW} & -    & \\textbf{89.8}               \\\\\n\\bottomrule\n\\end{tabular}\n}\n\\end{table}\n\\section{Discussion}\n\n\nWe performed an in-depth investigation of the phenomenology of finite and infinite width neural networks \nthrough a series of controlled interventions. \nWe \nquantified\nphenomena having to do with \ngeneralization, architecture dependendence, deviations between infinite and finite networks, numerical stability, data augmentation, data preprocessing, ensembling, network topology, and failure modes of linearization.\nWe further developed best practices that improve performance for both finite and infinite networks.\nWe believe our experiments provide firm empirical ground for future studies.\n\nThe careful study of other architectural components such as self-attention, normalization, and residual connections would be an interesting extension to this work, especially in light of results such as \\citet{Goldblum2020Truth} which empirically observes that the large width behavior of Residual Networks does not conform to the infinite-width limit.\nAnother interesting future direction would be incorporating systematic finite-width corrections, such as those in~\\citet{yaida2019non, Dyer2020Asymptotics, antognini2019finite, huang2019dynamics}.\n\n\n\\section*{Broader Impact}\n\nDeveloping theoretical understanding of neural networks is crucial both for understanding their biases, and predicting when and how they will fail. \nUnderstanding biases in models is of critical importance if we hope to prevent them from perpetuating and exaggerating existing racial, gender, and other social biases \\citep{hardt2016equality, barocas2016big, doshi2017towards, barocas-hardt-narayanan}. \nUnderstanding model failure has a direct impact on human safety, as neural networks increasingly do things like drive cars and control the electrical grid~\\citep{bojarski2016end, rudin2011machine, ozay2015machine}. \n\nWe believe that wide neural networks are currently the most promising direction for the development of neural network theory. \nWe further believe that the experiments we present in this paper will provide empirical underpinnings that allow better theory to be developed. \nWe thus believe that this paper will in a small way aid the engineering of safer and more just machine learning models.\n\n\n\\begin{ack}\nWe thank Yasaman Bahri and Ethan Dyer for discussions and feedback on the project.\nWe are also grateful to Atish Agarwala and Gamaleldin Elsayed for providing valuable feedbacks on a\ndraft. \n\nWe acknowledge the Python community~\\cite{van1995python} for developing the core set of tools that enabled this work, including NumPy~\\cite{numpy}, SciPy~\\cite{scipy}, Matplotlib~\\cite{matplotlib}, Pandas~\\cite{pandas}, Jupyter~\\cite{jupyter}, JAX~\\cite{jaxrepro}, Neural Tangents~\\cite{neuraltangents2020}, Apache Beam~\\cite{beam}, Tensorflow datasets~\\cite{TFDS} and Google Colaboratory~\\cite{colab}.\n\\end{ack}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\small\n\n\n\\section{Glossary}\nWe use the following abbreviations in this work:\n\n\\begin{itemize}\n    \\item{\\bf L2}: L2 reguarization a.k.a. weight decay;\n    \\item {\\bf LR}: using large learning rate;\n    \\item {\\bf  U}: allowing underfitting; \n    \\item {\\bf DA}: using data augmentation;\n    \\item {\\bf  C}: centering the network so that the logits are always zero at initialization;\n    \\item {\\bf  Ens}: neural network ensembling logits over multiple initialization;\n    \\item {\\bf  ZCA}: zero-phase component analysis regularization preprocessing;\n    \\item {\\bf FCN}: fully-connected neural network.;\n    \\item {\\bf  CNN-VEC}: convolutional neural network with a vectorized readout layer;\n    \\item {\\bf  CNN-GAP}: convolutional neural network with a global average pooling readout layer;\n    \\item {\\bf  NNGP}: neural network Gaussian process;\n    \\item {\\bf  NTK}: neural tangent kernel.\n\\end{itemize}\n\n\\section{Main table}\n\n\n   \n\\begin{table}[h]\n\\centering\n\\caption{\\textbf{CIFAR-10 classification accuracy for nonlinear and linearized finite neural networks, as well as for NTK and NNGP kernel methods}.\nStarting from \\texttt{Base} network of given architecture class described in \\sref{sec:experimental_design}, performance change of \\textbf{centering} (\\texttt{+C}), \\textbf{large learning rate} (\\texttt{+LR}), allowing \\textbf{underfitting} by early stopping (\\texttt{+U}), input preprocessing with \\textbf{ZCA regularization} (\\texttt{+ZCA}), multiple initialization \\textbf{ensembling} (\\texttt{+Ens}), and some combinations are shown, for {\\color{standard_param}\\textbf{Standard}} and {\\color{ntk_param}\\textbf{NTK}} parameterization. See also~\\Figref{fig:tricks_vs_accuracy}.\n}\n\\vspace{0.1cm}\n\\label{tab:main-table}\n\\resizebox{\\columnwidth}{!}{%\n\\begin{tabular}{@{}lc|ccccccccc|cc|cc@{}}\n\\toprule\n{} &\n  Param &\n  Base &\n  +C &\n  +LR &\n  +L2 &\n  \\begin{tabular}[c]{@{}l@{}}+L2 \\\\ +U\\end{tabular}\n &\n  \\begin{tabular}[c]{@{}c@{}}+L2 \\\\ +LR\\end{tabular} &\n  \\begin{tabular}[c]{@{}l@{}}+L2 \\\\+LR\\\\ +U \\end{tabular} &\n  +ZCA &\n  \\begin{tabular}[c]{@{}c@{}} Best \\\\ w\/o DA\\end{tabular} &\n  +Ens &\n  \\begin{tabular}[c]{@{}l@{}}+Ens \\\\+C \\end{tabular}\n&\n \\begin{tabular}[c]{@{}l@{}} +DA\\\\ +U \\end{tabular}&\n  \\begin{tabular}[c]{@{}l@{}}+DA \\\\+L2\\\\ +LR\\\\ +U\\end{tabular} \\\\\n\\midrule\\midrule\nFCN &\n  \\begin{tabular}[c]{@{}c@{}}STD\\\\ NTK\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}47.82\\\\ 46.16\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}53.22\\\\ 51.74\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}49.07\\\\ 48.14\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}49.82\\\\ 54.27\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}49.82\\\\ 54.27\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}55.32\\\\ 55.11\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}55.32\\\\ 55.44\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}44.29\\\\ 44.86\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}55.90\\\\ 55.44\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}58.11\\\\ 58.14\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}58.25\\\\ 58.31\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}} 65.29\\\\ 61.87\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}} 67.43\\\\ 69.35\\end{tabular} \\\\\n \\midrule\nCNN-VEC &\n  \\begin{tabular}[c]{@{}c@{}}STD\\\\ NTK\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}56.68\\\\ 60.73\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}60.82\\\\ 58.09\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}62.16\\\\ 60.73\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}57.15\\\\ 61.30\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}67.07\\\\ 75.85\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}62.16\\\\ 76.93\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}68.99\\\\ 77.47\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}57.39\\\\ 61.35\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}68.99\\\\ 77.47\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}67.30\\\\ 71.32\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}65.65\\\\ 67.23\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}76.73\\\\ 83.92\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}83.01\\\\ 85.63\\end{tabular} \\\\\n  \\midrule\nCNN-GAP &\n  \\begin{tabular}[c]{@{}c@{}}STD\\\\ NTK\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}80.26\\\\ 80.61\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}81.25\\\\ 81.73\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}80.93\\\\ 82.44\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}81.67\\\\ 81.17\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}81.10\\\\ 81.17\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}83.69\\\\ 82.44\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}83.01\\\\ 82.43\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}84.90\\\\ 83.75\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}84.22\\\\ 83.92\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}84.15\\\\ 85.22\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}84.62\\\\ 85.75\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}84.36\\\\ 84.07\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}86.45\\\\ 86.68\\end{tabular}\n\\end{tabular}%\n}\n\\\\\n\\vspace{0.2cm}\n\\resizebox{\\columnwidth}{!}{%\n\\begin{tabular}{@{}lc|cccccc||ccc|ccc@{}}\n\\toprule\n &\n  Param &\n  Lin Base &\n  +C &\n  +L2 &\n  \\begin{tabular}[c]{@{}l@{}}+L2 \\\\+U \\end{tabular} &\n  +Ens &\n  \\begin{tabular}[c]{@{}l@{}}+Ens \\\\+C \\end{tabular}&\n  NTK & +ZCA & \\begin{tabular}[c]{@{}c@{}}+DA \\\\ +ZCA\\end{tabular}&\n  NNGP &+ZCA &\n\\begin{tabular}[c]{@{}c@{}}+DA \\\\ +ZCA\\end{tabular}\\\\\n  \\midrule\\midrule\nFCN &\n  \\begin{tabular}[c]{@{}c@{}}STD\\\\ NTK\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}43.09\\\\ 48.61\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}51.48\\\\ 52.12\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}44.16\\\\ 51.77\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}50.77\\\\ 51.77\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}57.85\\\\ 58.04\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}57.99\\\\ 58.16\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}58.05\\\\ 58.28\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}59.65\\\\ 59.68\\end{tabular}  &\n  \\begin{tabular}[c]{@{}c@{}}-\\\\ 61.54\\end{tabular} &\n  58.61 &\n  59.70 &\n  62.40 \\\\\n  \\midrule\nCNN-VEC &\n  \\begin{tabular}[c]{@{}c@{}}STD\\\\ NTK\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}52.43\\\\ 55.88\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}60.61\\\\ 58.94\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}58.41\\\\ 58.52\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}58.41\\\\ 58.50\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}64.58\\\\ 65.45\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}64.67\\\\ 65.54\\end{tabular} &\n  {\\begin{tabular}[c]{@{}c@{}}66.64\\\\ 66.78\\end{tabular}} &\n  \\begin{tabular}[c]{@{}c@{}}69.65\\\\ 69.79\\end{tabular}&\n  \\begin{tabular}[c]{@{}c@{}}-\\\\ 70.52\\end{tabular} &\n  66.69 &\n  69.44 &\n  73.23 \\\\\n  \\midrule\nCNN-GAP &\n  \\begin{tabular}[c]{@{}c@{}}STD\\\\ NTK\\end{tabular} &\n  \\multicolumn{6}{c||}{\\begin{tabular}[c]{@{}c@{}} \\textgreater 70.00* (Train accuracy 86.22 after 14M steps)\\\\ \\textgreater 68.59* (Train accuracy 79.90 after 14M steps)\\end{tabular}} &\n  \\begin{tabular}[c]{@{}c@{}}76.97\\\\ 77.00\\end{tabular} &\n  \\begin{tabular}[c]{@{}c@{}}83.24\\\\ 83.24\\end{tabular} &\n   \\begin{tabular}[c]{@{}c@{}}-\\\\ 83.74\\end{tabular}\n   &\n  78.0 &\n  83.45 & 84.82\n\\end{tabular}%\n}\n\\end{table}\n\n\n\n\n\n\n\n\\section{Experimental details}\n\nFor all experiments, we use Neural Tangents (NT) library~\\cite{neuraltangents2020} built on top of JAX~\\cite{jaxrepro}. First we describe experimental settings that is mostly common and then describe specific details and hyperparameters for each experiments.\n\n\\textbf{Finite width neural networks}\nWe train finite width networks with Mean Squared Error (MSE) loss \n$${\\cal L } = \\frac{1}{2 |{\\cal D}| K} \\sum_{(x_i, y_i)\\in {\\cal D}} \\|f(x_i) - y_i\\|^2\\,,$$\nwhere $K$ is the number of classes and $\\|\\cdot \\|$ is the $L^2$ norm in $\\mathbb R^{K}$. For the experiments with \\texttt{+L2}, we add L2 regularization to the loss \n\\begin{equation}\\label{eq:l2-reg}\n    {R}_{\\text{L2}} = \\frac{\\lambda}{2} \\sum_l  \\norm{W^l}^2\\,,\n\\end{equation}\nand tune $\\lambda$ using grid-search optimizing for the validation accuracy.\n\nWe optimize the loss using mini-batch SGD with constant learning rate. We use batch-size of $100$ for \\texttt{FCN} and $40$ for both \\texttt{CNN-VEC} and \\texttt{CNN-GAP} (see \\sref{app:batch-size} for further details on this choice). \nLearning rate is parameterized with learning rate factor $c$ with respect to the critical learning rate\n\\begin{equation}\n    \\eta = c\\,  \\eta_\\text{critical}\\,.\n\\end{equation}\nIn practice, we compute empirical NTK $\\hat \\Theta (x, x') = \\sum_j \\partial_j f(x) \\partial_j f(x')$ on 16 random points in the training set to estimate $\\eta_\\text{critical}$~\\cite{lee2019wide} by maximum eigenvalue of $\\hat \\Theta (x, x)$. This is readily available in NT library~\\cite{neuraltangents2020} using \\texttt{nt.monte\\_carlo\\_kernel\\_fn} and \\texttt{nt.predict.max\\_learning\\_rate}. \nBase case considered without large learning rate indicates $c \\leq 1$, and large learning rate (\\texttt{+LR}) runs are allowing $c > 1$. Note that for linearized networks $\\eta_\\text{critical}$ is strict upper-bound for the learning rates and no $c >1$ is allowed~\\cite{lee2019wide, yang2019fine, lewkowycz2020large}.\n\n\nTraining steps are chosen to be large enough, such that learning rate factor $c \\leq 1$ can reach above $99\\%$ accuracy on $5k$ random subset of training data for 5 logarithmic spaced measurements. For different learning rates, physical time $t=\\eta \\times \\text{(\\# of steps)}$ roughly determines learning dynamics and small learning rate trials need larger number of steps. \nAchieving termination criteria was possible for all of the trials except for linearized \\texttt{CNN-GAP} and data augmented training of \\texttt{FCN}, \\texttt{CNN-VEC}. In these cases, we report best achieved performance without fitting the training set. \n\n\n\\textbf{NNGP \/ NTK} For inference, except for data augmentation ensembles for which default zero regularization was chosen, we grid search over diagonal regularization in the range \\texttt{numpy.logspace(-7, 2, 14)} and $0$. Diagonal regularization is parameterized as\n$${\\cal K}_{\\textrm{reg}} = {\\cal K} + \\varepsilon \\tfrac{\\textrm{tr}({\\cal K})}{m} I$$\nwhere ${\\cal K}$ is either NNGP or NTK for the training set. We work with this parameterization since $\\varepsilon$ is invariant to scale of ${\\cal K}$.\n\n\n\\textbf{Dataset}\nFor all our experiments (unless specified) we use train\/valid\/test split of 45k\/5k\/10k  for CIFAR-10\/100 and 50k\/10k\/10k for Fashion-MNIST. For all our experiments, inputs are standardized with per channel mean and standard deviation. ZCA regularized whitening is applied as described in \\sref{app ZCA}. \nOutput is encoded as mean subtracted one-hot-encoding for the MSE loss, e.g. for a label in class $c$, $-0.1 \\cdot \\bf{1} + e_c$. For the softmax-cross-entropy loss in~\\sref{app:xent-vs-mse}, we use standard one-hot-encoded output.\n\nFor data augmentation, we use widely-used augmentation for CIFAR-10; horizontal flips with 50\\% probability and random crops by 4-pixels with zero-padding. \n\n\n\\textbf{Details of architecture choice:}\nWe only consider ReLU activation (with the exception of Myrtle-kernel which use scaled Gaussian activation~\\cite{Shankar2020NeuralKW}) and choose critical initialization weight variance of $\\sigma_w^2=2$ with small bias variance $\\sigma_b^2=0.01$. \nFor convolution layers, we exclusively consider $3 \\times 3$ filters with stride $1$ and \\texttt{SAME} (zero) padding so that image size does not change under convolution operation. \n\n\n\\subsection{Hyperparameter configurations for all experiments}\n\\label{app hyperparameters}\n\n\nWe used grid-search for tuning hyperparameters and use accuracy on validation set for deciding on  hyperparameter configuration or measurement steps (for underfitting \/ early stopping). All reported numbers unless specified is test set performance.\n\n\\textbf{\\Figref{fig:tricks_vs_accuracy}, Table~\\ref{tab:main-table}}: We grid-search over L2 regularization strength $\\lambda \\in \\{0\\} \\cup \\{10^{-k} | k \\text{ from -9 to -3}\\}$ and learning rate factor $c \\in \\{2^k | k\\text{ from -2  to  5}\\}$. For linearized networks same search space is used except that $c>1$ configuration is infeasible and training diverges. For non-linear, centered runs $c \\in \\{2^k | k\\text{ from 0  to  4}\\}$ is used. Network ensembles uses base configuration with $\\lambda=0$, $c=1$ with 64 different initialization seed. Kernel ensemble is over 50 predictors for \\texttt{FCN} and \\texttt{CNN-VEC} and 32 predictors for \\texttt{CNN-GAP}. Finite networks trained with data-augmentation has different learning rate factor range of $c \\in \\{1, 4, 8\\}$. \n\n\n\\textbf{\\Figref{fig:nngp-vs-ntk}}: Each datapoint corresponds to either standard preprocessed or ZCA regularization preprocessed (as described in~\\sref{sec:zca}) with regularization strength was varied in $\\{10^{-k}| k \\in [-6, -5, ..., 4, 5]\\}$ for \\texttt{FCN} and \\texttt{CNN-VEC}, $\\{10^{-k}| k \\in [-3, -2, ..., 2, 3]\\}$ for \\texttt{CNN-GAP}. \n\n\n\\textbf{\\Figref{fig:validation_curves}, \\Figref{fig:ensemble}, \\Figref{fig:training_curves}, \\Figref{fig app ensemble}}: Learning rate factors are $c=1$ for non-linear networks and $c=0.5$ for linearized networks. While we show NTK parameterized runs, we also observe similar trends for STD parameterized networks. Shaded regions show range of minimum and maximum performance across 64 different seeds. Solid line indicates the mean performance. \n\n\\textbf{\\Figref{fig reg compare}}\nWhile \\texttt{FCN} is the base configuration, \\texttt{CNN-VEC} is a narrow network with 64 channels per layer since moderate width benefits from L2 more for the NTK parameterization~\\Figref{fig:width-combined}. For \\texttt{CNN-GAP} 128 channel networks is used. All networks with different L2 strategy are trained with \\texttt{+LR} ($c>1$).\n\n\\textbf{\\Figref{fig:width}, \\Figref{fig:reg-compare-sm},  \\Figref{fig:width-combined}}: \n$\\lambda \\in \\{0, 10^{-9}, 10^{-7}, 10^{-5}, 10^{-3}\\}$ and $c \\in \\{2^k | k \\text{ from} -2 \\text{ to } 5\\}$. \n\n\n\\textbf{\\Figref{fig:diag-reg}}: We use 640 subset of validation set for evaluation. \\texttt{CNN-GAP} is a variation of the base model with 3 convolution layers with $\\sigma_b^2 = 0.1$ while \\texttt{FCN} and \\texttt{CNN-VEC} is the base model.\nTraining evolution is computed using analytic time-evolution described in~\\citet{lee2019wide} and implemented in NT library via \\texttt{nt.predict.gradient\\_descent\\_mse} with 0 diagonal regularization. \n\n\n\\textbf{\\Figref{fig:zca}}: Kernel experiments details are same as in \\Figref{fig:nngp-vs-ntk}. Finite networks are base configuration with $c=1$ and $\\lambda=0$. \n\n\\textbf{\\Figref{fig:crop_translate}}: Evaluated networks uses NTK parameterization with $c=1$. {\\color{darkorange_f}\\texttt{CNN-VEC+L2+narrow}} uses 128 channels instead of 512 of the base {\\color{darkgreen_f}\\texttt{CNN-VEC}} and {\\color{darkblue_f}\\texttt{CNN-GAP}} networks, and trained with L2 regularization strength $\\lambda=10^{-7}$. \\emph{Crop} transformation uses zero-padding while \\emph{Translate} transformation uses circular boundary condition after shifting images. Each transformation is applied to the test set inputs where shift direction is chosen randomly. Each points correspond to average accuracy over 20 random seeds. {\\color{darkred_f}\\texttt{FCN}} had 2048 hidden units.\n\n\n\\textbf{\\Figref{fig:kerne-da-ens}, Table~\\ref{tab:sota-kernel-table}}: For all data augmentation ensembles, first instance is taken to be from non-augmented training set. Further details on kernel ensemble is described in~\\sref{app kernel ensembling}. For all kernels, inputs are preprocessed with optimal ZCA regularization observed in~\\Figref{fig:zca} (10 for \\texttt{FCN}, 1 for \\texttt{CNN-VEC}, \\texttt{CNN-GAP} and \\texttt{Myrtle}.). We ensemble over 50 different augmented draws for \\texttt{FCN} and \\texttt{CNN-VEC}, whereas for \\texttt{CNN-GAP}, we ensemble over 32 draws of augmented training set.\n\n\n\n\\textbf{\\Figref{fig:xent-vs-mse}, Table~\\ref{tab:xent-vs-mse}}:\nDetails for MSE trials are same as ~\\Figref{fig:tricks_vs_accuracy} and Table~\\ref{tab:main-table}. Trials with softmax-cross-entropy loss was tuned with same hyperparameter range as MSE except that learning rate factor range was $c\\in \\{1, 4, 8\\}$.\n\n\n\\textbf{\\Figref{fig:bs}}: We present result with NTK parameterized networks with $\\lambda=0$. \\texttt{FCN} network is width 1024 with $\\eta=10.0$ for MSE loss and $\\eta=2.0$ for softmax-cross-entropy loss. \\texttt{CNN-GAP} uses 256 channels with $\\eta=5.0$ for MSE loss and $\\eta=0.2$ for softmax-cross-entropy loss. Random seed was fixed to be the same across all runs for comparison.\n\n\\textbf{\\Figref{fig:l2-init}}: NTK pamareterization with $c=4$ was used for both L2 to zero or initialization. Random seed was fixed to be the same across all runs for comparison.\n\n\\section{Noise model}\n\\label{app:noise-model}\nIn this section, we provide details on noise model discussed in~\\sref{sec:kernel eigs}. Consider a random $m \\times m$ Hermitian matrix $N$ with entries order of $\\sigma_n$ which is considered as noise perturbation to the kernel matrix\n\\begin{equation}\n    \\tilde K = K + N\\,.\n\\end{equation}\nEigenvalues of this random matrix $N$ follow Wigner's semi-circle law, and the smallest eigenvalue is given by $\\lambda_{\\min}(N) \\approx - \\sqrt{2m} \\sigma_n$. When the smallest eigenvalue of $K$ is smaller (in order) than $|\\lambda_{\\min}(N)|$, one needs to add diagonal regularizer larger than the order of $|\\lambda_{\\min}(N)|$ to ensure positive definiteness. For estimates, let us use machine precision\\footnote{\\texttt{np.finfo(np.float32).eps}, \\texttt{np.finfo(np.float64).eps}} $\\epsilon_{32} \\approx 10^{-7}$  and $\\epsilon_{64} \\approx 2 \\times 10^{-16}$ which we use as proxy values for $\\sigma_n$. \nNote that noise scale is relative to elements in $K$ which is assume to be $O(1)$. Naively scaling $K$ by multiplicative constant will also scale $\\sigma_n$.\n\nEmpirically one can model tail $i^{\\textrm{th}}$ eigenvalues of infinite width kernel matrix of size $m \\times m$ as\n\\begin{equation}\n    \\lambda_i \\approx C \\frac{ m} {i^{\\alpha}} \\,.\n\\end{equation}\nNote that we are considering $O(1)$ entries for $K$ and typical eigenvalues scale linearly with dataset size $m$. For a given dataset size, the power law observed is $\\alpha$ and $C$ is dataset-size independent constant. Thus the smallest eigenvalue is order $\\lambda_\\text{min}(K) \\sim C m^{1- \\alpha}$. \n\nIn the noise model, we can apply Weyl's inequality which says\n\\begin{equation}\n    \\lambda_\\text{min}(K) - \\sqrt{2 m} \\sigma_n \\leq \\lambda_\\text{min} (\\tilde K ) \\leq \\lambda_\\text{min}(K)  + \\sqrt{2 m} \\sigma_n \\,.\n\\end{equation}\n\nConsider the worst-case where negative eigenvalue noise affecting the kernel's smallest eigenvalue. In that case perturbed matrices minimum eigenvalue could become negative, breaking positive semi-definiteness(PSD) of the kernel.  \n\nThis model allows to predict critical dataset size ($m^*$) over which PSD can be broken under specified noise scale and kernel eigenvalue decay. With condition that perturbed smallest eigenvalue becomes negative \n\\begin{equation}\n    C m^{1-\\alpha} \\lesssim \\sqrt{2 m}\\sigma_n\\,,\n\\end{equation}\nwe obtain\n\\begin{equation}\n    m^* \\gtrsim\n    \\begin{cases}\n    \\left(\\frac{C}{\\sqrt{2} \\sigma_n}\\right)^{\\tfrac{2}{2\\alpha - 1}} & \\textrm{if } \\alpha > \\tfrac{1}{2}\\\\\n    \\infty & \\textrm{else}\n    \\end{cases}\n\\end{equation}\n\nWhen PSD is broken, one way to preserve PSD is to add diagonal regularizer (\\sref{sec:diag-reg}). \nFor CIFAR-10 with $m=50k$, typical negative eigenvalue from \\texttt{float32} noise is around $4 \\times 10^{-5}$ and $7 \\times 10^{-14}$  with \\texttt{float64} noise scale, considering $\\sqrt{2 m} \\sigma_n$.  Note that \\citet{arora2019on} regularized kernel with regularization strength $5 \\times 10^{-5}$ which is on par with typical negative eigenvalue introduced due to \\texttt{float32} noise. Of course, this only applies if kernel eigenvalue decay is sufficiently fast that full dataset size is above $m^*$. \n\n\nWe observe that \\texttt{FCN} and \\texttt{CNN-VEC} kernels with small $\\alpha$ would not suffer from increasing dataset-size under \\texttt{float32} precision. On the other-hand, worse conditioning of \\texttt{CNN-GAP} not only affects the training time (\\sref{sec:cnn-gap-conditioning}) but also required precision. One could add sufficiently large diagonal regularization to mitigate effect from the noise at the expense of losing information and generalization strength included in eigen-directions with small eigenvalues.  \n\n\n\\begin{figure}[t!]\n\\centering\n\n\\begin{overpic}[width=0.86\\linewidth]{figures\/kernel_spectrum.png}\n \\put (0,0) {\\textbf{\\small(a)}}\n \\end{overpic}\\\\\n\\vspace{0.3cm}\n   \\begin{overpic}[width=0.43\\linewidth]{figures\/critical_dataset_size_by_exponent}\n \\put (0,0) {\\textbf{\\small(b)}}\n \\end{overpic}\n \\begin{overpic}[width=0.43\\linewidth]{figures\/critical_dataset_size_by_noise}\n \\put (0,0) {\\textbf{\\small(c)}}\n \\end{overpic}\n \n\\caption{\\textbf{The CNN-GAP architecture has poor\nkernel conditioning} \n\\textbf{(a)} Eigenvalue spectrum of infinite network kernels on 10k datapoints. Dashed lines are noise eigenvalue scale from \\texttt{float32} precision. Eigenvalue for CNN-GAP's NNGP decays fast and negative eigenvalue may occur when dataset size is $O(10^4)$ in \\texttt{float32} but is well-behaved with higher precision. \n\\textbf{(b-c)} Critical dataset size as function of eigenvalue decay exponent $\\alpha$ or noise strength $\\sigma_n$ given by \\eqref{eq:critical-m}. \n}\n\\label{app kernel spectra}\n\\end{figure}\n\n\n\\section{Data augmentation via kernel ensembling}\n\\label{app kernel ensembling}\n\n\nWe start considering general ensemble averaging of predictors. \nConsider a sequence of training sets $\\{{\\cal D}_i\\}$ each consisting of $m$ input-output pairs $\\{(x_1, y_1), \\dots, (x_m, y_m)\\}$ from a data-generating distribution. For a learning algorithm, which we use NNGP\/NTK inference for this study, will give prediction $\\mu (x^*, {\\cal D}_i)$ of unseen test point $x^*$. It is possible to obtain better predictor by averaging output of different predictors\n\\begin{equation}\n    \\hat \\mu(x^*) = \\frac{1}{E}\\sum_i^E \\mu (x^*, {\\cal D}_i)\\, , \n\\end{equation}\nwhere $E$ denotes the cardinality of $\\{{\\cal D}_i\\}$.  \nThis ensemble averaging is simple type of committee machine which has long history~\\cite{clemen1989combining,dietterich2000ensemble}. While more sophisticated ensembling method exists (e.g.~\\cite{freund1995desicion, breiman1996bagging,breiman2001random, opitz1996generating,opitz1999popular, rokach2010ensemble}), \nwe strive for simplicity and considered naive averaging. One alternative we considered is generalizing average by\n\\begin{equation}\n    \\hat \\mu_w(x^*) = \\frac{1}{E}\\sum_i^E w_i  \\,\\mu (x^*, {\\cal D}_i)\\,,\n\\end{equation}\nwere $w_i$ in general is set of weights satisfying $\\sum_i w_i = 1$. We can utilize posterior variance $\\sigma_i^2$ from NNGP or NTK with MSE loss via Inverse-variance weighting (IVW) where weights are given  as\n\\begin{equation}\n    w_i = \\frac{\\sigma_i^{-2}}{\\sum_j \\sigma_j^{-2}} \\,.\n\\end{equation}\nIn simple bagging setting~\\cite{breiman1996bagging}, we observe small improvements with IVW over naive averaging. This indicates posterior variance for different draw of  $\\{{\\cal D}_i\\}$ was quite similar. \n\nApplication to data augmentation (DA) is simple as we consider process of generating $\\{{\\cal D}_i\\}$ from a (stochastic) data augmentation transformation ${\\cal T}$. We consider action of ${\\cal T}(x, y) = T(x, y)$ be stochastic (e.g. $T$ is a random crop operator) with probability $p$ augmentation transformation (which itself could be stochastic) and probability $(1 - p)$ of $T = \\text{Id}$. Considering ${\\cal D}_0$ as clean un-augmented training set, we can imagine dataset generating process ${\\cal D}_i \\sim {\\cal T} ({\\cal D}_0)$, where we overloaded definition of ${\\cal T}$ on training-set to be data generating distribution. \n\nFor experiments in~\\sref{sec:data-augmentation}, we took $T$ to be standard augmentation strategy of horizontal flip and random crop by 4-pixels with augmentation fraction $p = 0.5$ (see~\\Figref{fig:kerne-da-ens-frac} for effect of augmentation fraction on kernel ensemble). In this framework, it is trivial to generalize the DA transformation to be quite general (e.g. learned augmentation strategy studied by~\\citet{cubuk2019autoaugment, cubuk2019randaugment}).\n\n\n\n\n\n\n\n\n\n\\section{ZCA whitening}\n\\label{app ZCA}\nConsider $m$ (flattened) $d$-dimensional training set inputs $X$ (a $d\\times m$ matrix) with  data covariance\n\\begin{equation}\n    \\Sigma_X = \\frac{1}{d} X X^T \\,.\n\\end{equation}\nThe goal of whitening is to find a whitening transformation $W$, a $d\\times d$ matrix, such that the features of transformed input \n\\begin{equation}\n    Y = W X\n\\end{equation}\nare uncorrelated, e.g. $\\Sigma_Y \\equiv \\frac 1 d YY^T= I$. Note that $\\Sigma_X$ is constructed only from training set while $W$ is applied to both training set and test set inputs.\nWhitening transformation can be efficiently computed by eigen-decomposition\\footnote{For PSD matricies, it is numerically more reliable to obtain via SVD.}\n\\begin{equation}\n    \\Sigma_X = U D U^T\\,\n\\end{equation}\nwhere $D$ is diagonal matrix with eigenvalues, and $U$ contains eigenvector of $\\Sigma_X$ as its columns. \n\nWith this ZCA whitening transformation is obtained by following whitening matrix\n\\begin{align}\n    W_\\text{ZCA} &= U \\sqrt{\\left(D + \\epsilon  \\tfrac{tr(D)}{d} I_d\\right)^{-1} } \\, U^T \\,.\n\\end{align}\n\nHere, we introduced trivial reparameterization of conventional regularizer such that regularization strength $\\epsilon$ is input scale invariant. It is easy to check $\\epsilon \\rightarrow 0$ corresponds to whitening with $\\Sigma_Y = I$. In \\sref{sec:zca}, we study the benefit of taking non-zero regularization strength for both kernels and finite networks. We denote transformation with non-zero regularizer, ZCA regularization preprocessing. ZCA transformation preserves spatial and chromatic structure of original image as illustrated in~\\Figref{app ZCA}. Therefore image inputs are reshaped to have the same shape as original image. \n\nIn practice, we standardize both training and test set per (RGB channel) features\nof the training set before and after the ZCA whitening. This ensures transformed inputs are mean zero and variance of order 1. \n\n\n\n\n\\begin{figure}\n\\centering\n  \\begin{overpic}[width=\\linewidth]{figures\/zca_fig.pdf}\n \\put (0,1) {\\textbf{\\small(a)}}\n \\put (48.5,1) {\\textbf{\\small(b)}}\n\\end{overpic}    \n    \\caption{\n    \\textbf{Illustration of ZCA whitening.} Whitening is a linear transformation of a dataset\n    that removes correlations between feature dimensions, setting all non-zero eigenvalues of the covariance matrix to 1. \n    ZCA whitening is a specific choice of the linear transformation that rescales the data in the directions given by the eigenvectors of the covariance matrix, but without additional rotations or flips. \n    {\\em (a)} A toy 2d dataset before and after ZCA whitening. Red arrows indicate the eigenvectors of the covariance matrix of the unwhitened data.\n    {\\em (b)} ZCA whitening of CIFAR-10 images preserves spatial and chromatic structure, while equalizing the variance across all feature directions. \n    Figure reproduced with permission from \\citet{wadia2020whitening}. See also \\sref{sec:zca}.\n    }\n\\label{fig cifar zca}\n\\end{figure}\n\n\n\n\n\n\n\n\\section{MSE vs Softmax-cross-entropy loss training of neural networks}\n\\label{app:xent-vs-mse}\nOur focus was mainly on fininte networks trained with MSE loss for simple comparison with kernel methods that gives closed form solution. Here we present comparison of MSE vs softmax-cross-entropy trained networks. See Table~\\ref{tab:xent-vs-mse} and \\Figref{fig:xent-vs-mse}. \n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=.4\\columnwidth]{figures\/xent_vs_mse.pdf}\n\\caption{\\textbf{MSE trained networks are competitive while there is a clear benefit to using Cross-entropy loss}}\n\\label{fig:xent-vs-mse}\n\\end{figure}\n\n\\begin{table}\n\\centering\n\\caption{Effects of MSE vs softmax-cross-entropy loss on base networks with various interventions}\\label{tab:xent-vs-mse}\n\\begin{tabular}{llllllll}\n\\toprule\nArchitecture &     Type & Param &   Base & +LR+U & +L2+U &  +L2+LR+U &   Best \\\\\n\\midrule \\midrule\nFCN &          MSE &   STD &  47.82  & 49.07 &  49.82 &  55.32 &  55.90 \\\\\n  &            &   NTK &  46.16 & 49.17 & 54.27 &  55.44 &  55.44 \\\\\n  &       XENT &   STD &  55.01 &  57.28 & 53.98 &  57.64 &  57.64 \\\\\n  &            &   NTK &  53.39 &  56.59 & 56.31 &  58.99 &  58.99 \\\\\n  &      MSE+DA &   STD &  65.29 &  66.11 & 65.28 &  67.43 &  67.43 \\\\\n  &            &   NTK &  61.87 &  62.12 & 67.58 &  69.35 &  69.35 \\\\\n  &    XENT+DA &   STD &  64.15 &  64.15 & 67.93 &  67.93 &  67.93 \\\\\n  &            &   NTK &  62.88 &  62.88 &  67.90 &  67.90 &  67.90 \\\\\n \\midrule\nCNN-VEC &          MSE &   STD &  56.68 &  63.51 & 67.07 &  68.99 &  68.99 \\\\\n  &           &   NTK &  60.73 &  61.58 & 75.85 &  77.47 &  77.47 \\\\\n &     XENT &   STD &  64.31 &  65.30 & 64.57 &  66.95 &  66.95 \\\\\n &      &   NTK &  67.13 &  73.23 & 72.93 &  74.05 &  74.05 \\\\\n &       MSE+DA &   STD &  76.73 & 81.84 &  76.66 &  83.01 &  83.01 \\\\\n &        &   NTK &  83.92 &  84.76 & 84.87 &  85.63 &  85.63 \\\\\n &   XENT+DA &   STD &  81.84 & 83.86 & 81.78 &  84.37 &  84.37 \\\\\n &           &   NTK &  86.83 & 88.59 & 87.49 &  88.83 &  88.83 \\\\\n \\midrule\nCNN-GAP &          MSE &   STD &  80.26 & 80.93 & 81.10 &  83.01 &  84.22 \\\\\n &           &   NTK &  80.61 & 82.44 & 81.17 &  82.43 &  83.92 \\\\\n &     XENT &   STD &  83.66 & 83.80 & 84.59 &  83.87 &  83.87 \\\\\n &        &   NTK &  83.87 &  84.40 & 84.51 &  84.51 &  84.51 \\\\\n &       MSE+DA &   STD &  84.36 &  83.88 & 84.89 &  86.45 &  86.45 \\\\\n &        &   NTK &  84.07 &  85.54 & 85.39 &  86.68 &  86.68 \\\\\n &  XENT+DA &   STD &  86.04 &  86.01 & 86.42 &  87.26 &  87.26 \\\\\n &   &   NTK &  86.87 &  87.31 & 86.39 &  88.26 &  88.26 \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\n\\section{Comment on batch size}\n\\label{app:batch-size}\nCorrespondence between NTK and gradient descent training is direct in the full batch gradient descent (GD) setup (see~\\cite{Dyer2020Asymptotics} for extensions to mini-batch SGD setting). Therefore base comparison between finite networks and kernels is the full batch setting. While it is possible to train our base models with GD, for full CIFAR-10 large emprical study becomes impractical. In practice, we use mini-batch SGD with batch-size $100$ for FCN and $40$ for CNNs. \n\nWe studied batch size effect of training dynamics in \\Figref{fig:bs} and found that for these batch-size choices does not affecting training dynamics compared to much larger batch size. \n\\citet{shallue2018measuring, mccandlish2018empirical} observed that universally for wide variety of deep learning models  there are batch size beyond which one could gain training speed benefit in number of steps. We observe that maximal useful batch-size in workloads we study is quite small.\n\n\\begin{figure}\n\\centering\n\\begin{overpic}[width=0.8\\linewidth]{figures\/bs_fc_mse.pdf}\n \\put (0,0) {\\textbf{\\small(a)}}\n\\end{overpic}\\\\\n\\vspace{0.2cm}\n\\begin{overpic}[width=0.8\\linewidth]{figures\/bs_fc_xent.pdf}\n \\put (0,0) {\\textbf{\\small(b)}}\n\\end{overpic}\n\\\\\n\\vspace{0.2cm}\n\\begin{overpic}[width=0.8\\linewidth]{figures\/bs_cg_mse.pdf}\n \\put (0,0) {\\textbf{\\small(c)}}\n\\end{overpic}\\\\ \n\\vspace{0.2cm}\n\\begin{overpic}[width=0.8\\linewidth]{figures\/bs_cg_xent.pdf}\n \\put (0,0) {\\textbf{\\small(d)}}\n\\end{overpic}\n\n    \\caption{\\textbf{Batch size does not affect training dynamics for moderately large batch size.}}\n    \\label{fig:bs}\n\\end{figure}\n\n\n\\section{Addtional tables and plots}\n\n\\begin{table}[h]\n\\centering\n\\caption{\\textbf{CIFAR-10 classification mean squared error(MSE) for nonlinear and linearized finite neural networks, as well as for NTK and NNGP kernel methods}.\nStarting from \\texttt{Base} network of given architecture class described in \\sref{sec:experimental_design}, performance change of \\textbf{centering} (\\texttt{+C}), \\textbf{large learning rate} (\\texttt{+LR}), allowing \\textbf{underfitting} by early stopping (\\texttt{+U}), input preprocessing with \\textbf{ZCA regularization} (\\texttt{+ZCA}), multiple initialization \\textbf{ensembling} (\\texttt{+Ens}), and some combinations are shown, for {\\color{standard_param}\\textbf{Standard}} and {\\color{ntk_param}\\textbf{NTK}} parameterization. See also Table~\\ref{tab:main-table} and \\Figref{fig:tricks_vs_accuracy} for accuracy comparison.}\n\\vspace{0.3cm}\n\\label{tab:main-table-mse}\n\\resizebox{\\columnwidth}{!}{%\n\n\\begin{tabular}{@{}lc|ccccccccc|cc|cc@{}}\n\\toprule\n{} &\n  Param &\n  Base &\n  +C &\n  +LR &\n  +L2 &\n  \\begin{tabular}[c]{@{}l@{}}+L2 \\\\ +U\\end{tabular}\n &\n  \\begin{tabular}[c]{@{}c@{}}+L2 \\\\ +LR\\end{tabular} &\n  \\begin{tabular}[c]{@{}l@{}}+L2 \\\\+LR\\\\ +U \\end{tabular} &\n  +ZCA &\n  \\begin{tabular}[c]{@{}c@{}} Best \\\\ w\/o DA\\end{tabular} &\n  +Ens &\n  \\begin{tabular}[c]{@{}l@{}}+Ens \\\\+C \\end{tabular}\n&\n \\begin{tabular}[c]{@{}l@{}} +DA\\\\ +U \\end{tabular}&\n  \\begin{tabular}[c]{@{}l@{}}+DA \\\\+L2\\\\ +LR\\\\ +U\\end{tabular} \\\\\n  \n  \n\\midrule \\midrule\n      FCN &   STD &  0.0443 &  0.0363 &  0.0406 &  0.0411 &  0.0355 &  0.0337 &  0.0329 &  0.0483 &  0.0319 &   0.0301   &  0.0304     &0.0267 &  0.0242 \\\\\n  &   NTK &  0.0465 &  0.0371 &  0.0423 &  0.0338 &  0.0336 &  0.0308 &  0.0308 &  0.0484 &  0.0308 &    0.0300  &  0.0302     &  0.0281 &  0.0225 \\\\\n  \\midrule\n     CNN-VEC &   STD &  0.0381 &  0.0330 &  0.0340 &  0.0377 &  0.0279 &  0.0340 &  0.0265 &  0.0383 &  0.0265 &   0.0278   &  0.0287     &  0.0228 &  0.0183 \\\\\n &   NTK &  0.0355 &  0.0353 &  0.0355 &  0.0355 &  0.0231 &  0.0246 &  0.0227 &  0.0361 &  0.0227 &   0.0254    &    0.0278   & 0.0164 &  0.0143 \\\\\n \\midrule\n    CNN-GAP &   STD &  0.0209 &  0.0201 &  0.0207 &  0.0201 &  0.0201 &  0.0179 &  0.0177 &  0.0190 &  0.0159 & 0.0172     &    0.0165   &  0.0185 &  0.0149 \\\\\n &   NTK &  0.0209 &  0.0201 &  0.0195 &  0.0205 &  0.0181 &  0.0175 &  0.0170 &  0.0194 &  0.0161 & 0.0163     &   0.0157    &  0.0186 &  0.0145 \\\\\n\\bottomrule\n\\end{tabular}%\n}\n\\\\\n\\vspace{0.2cm}\n\n\\resizebox{\\columnwidth}{!}{%\n\\begin{tabular}{@{}lc|cccccc||ccc|ccc@{}}\n\\toprule\n &\n  Param &\n  Lin Base &\n  +C &\n  +L2 &\n  \\begin{tabular}[c]{@{}l@{}}+L2 \\\\+U \\end{tabular} &\n  +Ens &\n  \\begin{tabular}[c]{@{}l@{}}+Ens \\\\+C \\end{tabular}&\n  NTK & +ZCA & \\begin{tabular}[c]{@{}c@{}}+DA \\\\ +ZCA\\end{tabular}&\n  NNGP &+ZCA &\n\\begin{tabular}[c]{@{}c@{}}+DA \\\\ +ZCA\\end{tabular}\\\\\n  \\midrule\\midrule\n  FCN &   STD &  0.0524 &  0.0371 &  0.0508 &  0.0350 & 0.0309 & 0.0305 & 0.0306 & 0.0302 & - & \\multirow{2}*{0.0309} & \\multirow{2}*{0.0308} & \\multirow{2}*{0.0297}  \\\\\n &   NTK &  0.0399 &  0.0366 &  0.0370 &  0.0368 &  0.0305 & 0.0304 &0.0305 & 0.0302 & 0.0298 \\\\\n \\midrule\n  CNN-VEC &   STD &  0.0436 &  0.0322 &  0.0351 &  0.0351 & 0.0293 & 0.0291 & 0.0287 & 0.0277 & - & \\multirow{2}*{0.0286} & \\multirow{2}*{0.0281} & \\multirow{2}*{0.0256}\\\\\n &   NTK &  0.0362 &  0.0337 &  0.0342 &  0.0339 &  0.0286 &0.0286 & 0.0283 & 0.0274 & 0.0273 \\\\\n \\midrule\n  CNN-GAP &   STD &  \\multicolumn{6}{c||}{< 0.0272* (Train accuracy 86.22 after 14M steps)} &  0.0233 & 0.0200& - & \\multirow{2}*{0.0231} & \\multirow{2}*{0.0204} & \\multirow{2}*{0.0191} \\\\\n &   NTK &  \\multicolumn{6}{c||}{< 0.0276* (Train accuracy 79.90 after 14M steps)}  & 0.0232 & 0.0200& 0.0195\\\\\n \\bottomrule\n\\end{tabular}%\n}\n\\end{table}\n\n    \n\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=\\columnwidth]{figures\/nngp_vs_ntk_uci.pdf}\n\\caption{\\textbf{On UCI dataset NNGP often outperforms NTK on RMSE.}\nWe evaluate predictive performance of FC NNGP and NTK on UCI regression dataset in the standard 20-fold splits first utilized in~\\cite{hernandez2015probabilistic, gal2016dropout}. We plot average RMSE across the splits. Different scatter points are varying hyperparameter settings of (depth, weight variance, bias variance). In the tabular data setting, dominance of NNGP is not as prominent across varying dataset as in image classification domain. \n}\n\\label{fig:nngp-vs-ntk-uci}\n\\end{figure}\n\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=\\columnwidth]{figures\/ensemble_training_curve.pdf}\n\\caption{\\textbf{Centering can accelerate training}. Validation (top) and training (bottom) accuracy throughout training for several finite width architectures. See also \\sref{sec:ensemble_of_networks} and \\Figref{fig:validation_curves}.\n}\n\\label{fig:training_curves}\n\\end{figure}\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=\\columnwidth]{figures\/ensemble_network_performance_combined.pdf}\n\\caption{\\textbf{Ensembling base networks causes them to match kernel performance, or exceed it for nonlinear CNNs.} See also \\sref{sec:ensemble_of_networks} and \\Figref{fig:ensemble}.\n}\n\\label{fig app ensemble}\n\\end{figure}\n\n\n\n\n\\begin{figure}[h]\n\\centering\n\\begin{overpic}[width=\\linewidth]{figures\/network_l2.pdf}\n\\end{overpic}    \n\\\\\n\\caption{\n\\textbf {Performance of nonlinear and linearized networks as a function of L2 regularization for a variety of widths.} Dashed lines are NTK parameterized networks while solid lines are networks with standard parameterization. We omit linearized \\texttt{CNN-GAP} plots as they did not converge even with extensive compute budget.\nL2 regularization is more helpful in networks with an NTK parameterization than a standard parameterization \n\\label{fig:reg-compare-sm}\n}\n\\end{figure}\n\n\n\n\n\\begin{figure}[h]\n\\centering\n\\begin{overpic}[width=.6\\columnwidth]{figures\/l2_to_init_training_curve.pdf}\n \\put (0,0) {\\textbf{\\small(a)}}\n\\end{overpic} \n\\begin{overpic}[width=.32\\columnwidth]{figures\/l2_to_init.pdf}\n \\put (0,0) {\\textbf{\\small(b)}}\n\\end{overpic}    \n\\caption{\\textbf{L2 regularization to initial weights does not provide performance benefit.} {\\bf (a)} Comparing training curves of L2 regularization to either 0 or initial weights. {\\bf (b)} Peak performance of after L2 regularization to either 0 or initial weights. Increasing L2 regularization to initial weights do not provide performance benefits, instead performance remains flat until model's capacity deteriorates. \n}\n\\label{fig:l2-init}\n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{figures\/network_vs_width.pdf}\n\\caption{\n\\textbf{Finite width networks generally perform better with increasing width, but \\texttt{CNN-VEC} shows surprising non-monotonic behavior.} See also~\\sref{sec:perf_vs_width} and \\Figref{fig:width}\n {\\bf L2}: non-zero weight decay allowed during training {\\bf LR}: large learning rate allowed. Dashed lines are allowing underfitting (\\textbf{U}).}\n\\label{fig:width-combined}\n\\end{figure}\n\n\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=\\columnwidth]{figures\/zca_train_curves_std.pdf}\n\\includegraphics[width=\\columnwidth]{figures\/zca_train_curves_ntk.pdf}\n\\caption{\\textbf{ZCA regularization helps finite network training.}\n(\\textbf{upper}) Standard parameterization, (\\textbf{lower}) NTK parameterization. See also \\sref{sec:zca} and \\Figref{fig:zca}.\n}\n\\label{fig:app-zca-training}\n\\end{figure}\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=\\columnwidth]{figures\/da_ensemble_fraction.pdf}\n\\caption{\\textbf{Data augmentation ensemble for infinite network kernels with varying augmentation fraction.} See also \\sref{sec:data-augmentation}.}\n\\label{fig:kerne-da-ens-frac}\n\\end{figure}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nOver the last  decades numerical simulations have became a powerful tool to study the validity of cosmological models.\n$\\Lambda$-CDM scenarios have been found to be able to reproduce successfully the global properties of the observed structure  at large scales.\nHowever, at galactic scale, the so-called concordance ($\\Lambda$-CDM) paradigm has been challenged by several  observational results. In this respect, the cuspy inner profile obtained in many of the numerical simulations has been claimed  not to be consistent with the rotation curves observed for low surface brightness (Flores \\& Primack 1994; Moore 1994; Dutton, van den Bosch \\& Courteau 2008; Salucci, Yegorova \\& Drory 2008) and dwarf galaxies (e.g. Gnedin \\& Zhao 2002). There have been many attempts to explain the possible erasement of the inner cusp via different mechanisms: interactions through dynamical friction with the substructure (e.g. El-Zant et al. 2001, Tonini, Lapi \\& Salucci 2006), SN feedback  (Mashchenko et al. 2006) etc., but the problem is not yet solved.\n\u00b7 \n\nAnother important  pending issue is the overabundance of small dark matter (DM) subhaloes (e.g. Moore et al. 1999; Stadel et al. 2009). The total number of subhaloes found in the simulations is much larger than the number of known satellite galaxies surrounding the Milky Way.\nThe inner shape of the DM  density profiles and the abundance of subhaloes are of particular interest because they can be  used to perform several observational diagnostics such as gravitational lensing (Zackrisson \\& Riahm 2009). Also, the prospect of detecting DM particles annihilation makes it essential a deep understanding of the small scale distribution of the DM in the central regions of our Galaxy (Diemand et al. 2008, Springel et al. 2008). \n\nAlthough the ``universality'' of the spherically-averaged density profiles (Navarro, Frenk and White 1996, hereafter NFW) of $\\Lambda$-CDM  haloes has gained a broad consensus over the past couple of decades, new evidences based on high resolution simulations have been found against it (Navarro et al. 2004, Merrit et al. 2006, Gao et al. 2008). Merrit et al. (2006) tested different fitting functions and found that the profiles were better described by a de Vaucouleurs' law instead of the NFW two parameter formula. They also found a systematic variation in the profile shape with halo mass. Navarro  et al. (2008, hereafter N08) analysed different galaxy-sized haloes simulated with unprecedent high resolution and found small but significant deviations from the so-called NFW universal profile. \n\nThe contraction of the DM haloes due to the infall and condensation of baryons in the central regions is a well accepted process\n (e.g. Barnes \\& White 1984). Many attempts to predict their effects have been made through  models based on the adiabatic contraction (AC) hypothesis  such as the one developed by Blumenthal et al. (1986, hereafter B86). However, it has been shown that this kind of models overestimates the level of contraction. More recently,  Gnedin et al. (2004) and Sellwood \\& McGaugh (2005) developed AC models based on the work of Young (1980) which considered the possibility of radial motions. However, all  AC-based models  missed the hierarchical characteristic of the assembly of the galaxy as suggested by the current large scale observations (e.g. S\\'anchez et al. 2006), which might have non negligible consequences on the final distribution of\nthe DM (e.g. Debatistta et al. 2008).\n\nIn a fully cosmological context, simulations  have already shown how the DM haloes concentrate  when  baryons are included, providing hints of a possible dependence on the assembly history  (e.g. Tissera \\& Dom\\'{\\i}nguez-Tenreiro et al. 1998; Gnedin et al. 2004; O\\~norbe et al. 2007). Recently Romano-D\\'{\\i}az et al. (2008) analysed the evolution of the central DM profile   in cosmologically grown galactic haloes, claiming that  when baryons are present the cusp is gradually levelled off. They suggest that this effect could be associated with the action of the subhaloes that heat up the cusp region of the DM halo through dynamical friction, and force it to expand (Ma \\& Boylan-Kolchin 2004; Debatistta et al. 2008). \n\nIn order to help shed light on these issues, we study a set of intermediate resolution cosmological simulations where different baryonic structures have been able to  form from identical initial conditions via the modification of the physics of baryons. \nThe set of simulations studied in our work are those analysed by Scannapieco et al. (2008, hereafter S08) where  the star formation activity and the SN feedback were modified in order to study the role played by each of these processes. As a consequence, a variety of galaxies were obtained, each one exhibiting different morphological and dynamical properties. These experiments allow us to analyse how the dark matter evolves when baryons are assembled in a different fashion but governed by the same underlying merger tree. \nThe first results of this analysis were reported by  Pedrosa, Tissera \\& Scannapieco (2009) where it is clearly shown that the final structure of the dark matter halo  depends on the way baryons are put together and not solely on the amount of baryons gathered in the centre. \nWe also found hints for a  re-distribution of angular momentum related to the accretion of satellites.  \nIn this paper, we extend this work and analyse in detail the DM haloes considering the evolution of baryons.\n\n\n\n This paper is organized as follows. In Section 2, we describe the numerical experiments and summarize the main features of the simulated galaxies. In Section 3, we analyse the DM density profiles. In Section 4, we study the interaction with satellites. In Section 5, rotation curves and the AC hypothesis are analysed. In Section 6 we summarize our main results.\n\n\\section{The Numerical Experiments}\n\n\nWe analysed a set of six realizations of a  $\\approx 10^{12} h^{-1}$  M$_{\\odot}$ mass  halo, run with an\nextended version of the code GADGET-2  according to Scannapieco et al. (2005, 2006). This extended GADGET-2 code was designed to improve the representation of the ISM and SN feedback by including a new multiphase model for the gas component, metal-dependent\n cooling, chemical enrichment and energy feedback by SN events.\n\n\n\nThe initial  condition corresponds to an $\\approx 10^{12} h^{-1}\\ $M$_{\\odot}$ halo extracted from a cosmological simulation and re-simulated with higher resolution. This halo was required to have no major mergers since $z=1$. The simulations have been run from $z=38$ to $z=0$ and are consistent with \n a $\\Lambda$CDM universe with \n$\\Omega_{\\Lambda}=0.7$, \n$\\Omega_{m}=0.3$, $\\Omega_{b}=0.04$, \n$\\sigma_{8}=0.9$\n and $H_{0}= 100 h \\ {\\rm km} \\ {\\rm s}^{-1}\\ {\\rm Mpc}^{-1}$, with $h=0.7$. The dark matter particle mass \nis $1.6\\times 10^{7} h^{-1}\\ $M$_{\\odot}$ while initially the gas mass particle is  $2.4\\times 10^{6} h^{-1}\\  $M$_{\\odot}$.\n The maximum gravitational softening used is $\\epsilon_{g}=0.8 h^{-1}$ kpc. \n\nThe analysed simulations  have the same initial condition but have been run using  different input\nparameters for the Star Formation (SF) and SN feedback models as it can be seen in Table ~\\ref{tab1}. The version of {\\small GADGET-2} used to run these\nsimulations\nincludes the multiphase model for the interstellar medium, the SF algorithm and SN feedback\npresented by Scannapieco et al. (2005, 2006). This set of simulations was performed by \nS08 who varied the star formation efficiency ($c$), the fraction of SN energy ($\\epsilon_{\\rm c}$) injected into the cold phase (and correspondingly the fraction of SN energy that is pumped into the hot phase) and the total energy released during a SN explosion ($E_{\\rm SN}$). \nAs a result, DM haloes host baryonic structures with different morphologies since the transformation of gas into stars has been \nregulated differently in each run.\n\nIn order to be able to assess the effects of galaxy formation  on the dark matter haloes, for this work, we  performed a pure gravitational run (DM-only) of the same initial condition used by S08, with a dark matter  particle of $1.84\\times 10^{7} h^{-1}\\ $M$_{\\odot}$.\n\n\\subsection{SF and SN feedback parameters and resulting morphology}\n\nThe hydrodynamical simulations used in this work have been analysed in detail by S08, particularly the different properties of the\nsimulated galaxy at $z=0$. In this section, we only summarize their main characteristics to facilitate the interpretation \nof our results.\n\nThe first point to note is that when  SN feedback is not included, the gas collapses and concentrates at the centre of the potential well very efficiently. In this case, the star formation  follows the gas collapse and there is no mechanism to  regulate the SF activity. As a result, a stellar spheroidal component is formed very early,  consuming most of the gas reservoir and preventing the formation of a disc structure at later times, as it  is the case in the NF halo.\n On the contrary, when the SN feedback is included, the SF activity gets self-regulated as a consequence of the heating and pressurizing of the interstellar gas. In this case, disc-like components can be formed, populated mainly by young stars (S08). \nDepending on the combination of SF and SN parameters, the baryons settle down determining structures with different morphologies\nand disc components of different sizes. In some cases, the systems are dominated by a spheroidal component.\n\nS08 adopted the standard value of $E_{\\rm SN}=10^{51}$ erg per SN event and, then, varied it from 0.3 to 3  $\\times10^{51}$ erg as shown in Table ~\\ref{tab1}. As larger values for the SN energy are assumed, the SF is more strongly quenched and more violent\nwinds are able to develop,  resulting in lower final stellar and gas masses. In E-3, where  the unrealistic value of $3\\times10^{51}$ erg per SN is assumed,  most of the gas is blown out, producing  the most DM dominated system  with the less concentrated DM profile in our set of hydrodynamical simulations.\nAlternatively, S08 varied the fraction  $\\epsilon_{\\rm c}$ of energy pumped into the cold phase producing a stronger inhibition of the \nstar formation activity due to increased strength of the induced galactic winds (F-0.9). This particular combination of parameters\nresults in a very extended disc structure.\nA decrease in the star formation efficiency (C-0.01) produces a slower rate of transformation of gas into stars with weaker starbursts so that the energy injected into the ISM is not able to generate strong galactic winds. The regulation of the star formation\nactivity is not enough to prevent an important early consumption of gas into stars.\n\nIn summary, the final simulated galaxies have the following characteristics.\nAt $z=0$, the galaxy formed in  NF run is dominated by an extended spheroid, with most of its stars\nformed at $ z > 2$. The E-0.7 and F-0.9 runs have been able to produce galaxies with important disc components\nas a result of the regulation of SF by SN feedback. These systems have a half mass radius ($r_{d}$) of 5.72 kpc $h^{-1}$ and 9.74 kpc $h^{-1}$, respectively. And, they also show the largest disc to spheroid mass ratios ($D\/S$): 0.82 for E-0.7 and 0.98 for F-0.9. The E-0.3 run was able to develop a small ($r_{\\rm d}=4.75$) and thick disc ($D\/S = 0.6$). In the case of C-0.01 run the disc component is almost negligible. In the E-3 run, the large amount of energy assumed per SN  triggers  violent outflows which expel a significant amount of the gas content of the main galaxy. \nThe stellar masses of the final simulated galaxies vary from 15.9 to 1.3 $\\times 10^{10} h^{-1} $M$_{\\odot}$, as summarized in Table \\ref{tab1}.\n\n\n\n\\begin{table*}\n  \\begin{center}\n  \\caption{Main characteristics of  DM haloes and their main central galaxies. For each simulation we provide the values of SF and SN parameters: $\\epsilon_{\\rm c}$, $c$ and $E_{SN} $ (see Section 2 for details).\n We show the virial radius $r_{\\rm 200}$, the virial mass $M_{\\rm 200}$, total stellar mass $M_{\\rm s}$ of the central galaxy,\nthe  total-mass-to-stellar\nmass ratio $M_{\\rm t}\/M_{\\rm s}$, the $n$ and $r_{-2}$ Einasto parameters, the\n inner logarithm slope \n$\\gamma_{\\rm inner}$, the logarithmic slope of the baryonic circular velocity $LS$ and the ratio \n$V_{\\rm max}\/V_{\\rm 200}$. $M_{\\rm t}$ and $M_{\\rm s}$ are evaluated within twice \nthe  optical radius defined as that  enclosing $83\\%$ of the baryons in the central region. Bootstraps errors for $n$ \nand $r_{-2}$ are shown within parenthesis.}\n  \\label{tab1}\n {\\scriptsize\n  \\begin{tabular}{|l|c|c|c|c|c|c|c|c|c|c|c|c|}\\hline\n{\\bf Run}  & {\\bf $\\epsilon_{\\rm c}$}  & {\\bf $E_{\\rm SN}$}  & {\\bf $c$}  & {\\bf $r_{\\rm 200}$} & {\\bf $M_{\\rm 200}$} & {\\bf $M_{\\rm s}$} &  {\\bf $M_{\\rm t}\/M_{\\rm s}$} &  {\\bf $n$} & {\\bf $r_{-2}$} & {\\bf $\\gamma_{\\rm inner}$} & {$ LS $} & {$V_{\\rm max}\/V_{\\rm 200} $}\\\\\n   & & & &kpc $h^{-1}$& $10^{10} h^{-1}\\ $M$_{\\odot}$& $10^{10} h^{-1}\\ $M$_{\\odot}$ & & &kpc $h^{-1}$ &&\\\\ \\hline\nNF & - & - & 0.1 & 217.5 & 224.4 & 15.9 &  4.1 &  5.973 (1)& 16.89 (1) & 1.24  & -0.22 & 1.46 \\\\ \nF-0.9 & 0.9 & 1 & 0.1 & 212.0 & 212.3 & 6.7 &  7.7 &  6.476 (1)& 17.83 (1) & 1.24  & -0.08 & 1.25 \\\\ \nC-0.01 & 0.5 & 1 & 0.01 & 207.7 & 202.2 & 10.0 &  3.7 &  6.499 (1)& 16.56 (1) & 1.24 & -0.20 & 1.43 \\\\ \nE-0.3 & 0.5 & 0.3 & 0.1 & 214.7 & 220.8 & 13.2 &  3.8 &  6.764 (1)& 14.76 (1) & 1.24 & -0.17 & 1.47 \\\\ \nE-0.7 & 0.5 & 0.7 & 0.1 & 211.1 & 209.0& 7.5 &  5.1 &  6.887 (4)& 15.35 (1)& 1.28 & -0.08 & 1.33 \\\\ \nE-3 & 0.5 & 3 & 0.1 & 205.8 & 194.0 & 1.3 & 37.2 & 5.585 (1)& 21.90 (1)& 1.15 & -0.19 & 1.17 \\\\ \nDM-only & - & - & - & 217.0 & 227.4 & - & - & 5.239 (3) & 24.06 (1) & 1.08 & - & - \\\\ \n\n  \\end{tabular}\n  }\n \\end{center}\n\\vspace{1mm}\n\\end{table*}\n\n\\section{Dark matter density profiles}\n\n\nAll analysed DM haloes have been identified at their virial radius ($r_{200}$) defined as  the one that enclosed an sphere of mean density $200 \\times \\rho_{crit}$ where $\\rho_{crit}$ is the closure density of the Universe.\nWe have checked that, at $z=0$, all virialised haloes are  relaxed as indicated by a relax parameter of $\\approx  0.002$ (Neto et al. 2007).\nOn average,  DM  haloes have more than a total of 120000 particles within their virial radius.\n\nIn order to construct the DM profiles, we first eliminate the substructures within $r_{200}$, which affects the profiles mainly in the outer regions near $r_{200}$. \nThe determination of the centre of mass  of the haloes is of great importance for the analysis of the dark matter distribution since a displacement could produce an artificial erasement of the inner cusp. We used the shrinking sphere method proposed by  Power et al. (2003) in order to find it. We calculate spherically-averaged density profiles between three times $\\epsilon_{g}$ and the virial radius. We use three $\\epsilon_{g}$ as the inner radius to increase the numerical robustness of our fittings. \n\nWe calculate spherically-averaged DM profiles using logarithmic binning of the DM distribution cleaned of substructures. \nThen, we fit the NFW, Jaffe (Jaffe, W. 1983)  and Einasto (Einasto, J. 1965) expressions to the DM profiles finding that the Einasto's model provides the best fit in all cases. The Einasto's formula can be written as:\n\n\\begin{equation}\nln\\left( \\frac{\\rho\\left( r\\right) }{\\rho_{-2}} \\right) = \\left(\\ -2n\\right) \\left[ \\left( \\frac{r}{r_{-2}}\\right) ^{1\/n}-1\\right]  \n\\label{eq1}\n\\end{equation}\n\n\nwhere  $n$, $r_{-2}$  and $\\rho_{-2}$ indicate the sharpness of the profiles and the radius and density where their logarithmic slope takes the isothermal value. The number of free parameters is reduced to two by imposing the extra constrain of \nobtaining the total mass at the virial radius. \n The fitting values obtained for the different haloes are shown in Table \\ref{tab1}. \nWe estimated bootstrap errors for $n$ and $r_{-2}$ by fitting the Einasto's formula to 100 \nrandomly-generated realizations\nof the DM  profiles and by estimating the standard dispersion over the generated set of  parameters. \n\n\\begin{figure}\n\\resizebox{8cm}{!}{\\includegraphics{Rho_dm_err_col.ps}}\n\\hspace*{-0.2cm}\n\\caption{Spherically-averaged DM profiles for the NF (black line), E-0.7 (magenta line), F-0.9 (red line), E-0.3 (green line), C-0.01 (blue line), E-3 (cyan line)  and  DM-only (black thick dashed line)  experiments. The  inner most bin corresponds to three times the  gravitational softenings.\nThe errorbars have been estimated by the boostrap resampling technique. } \n\\label{fig1}\n\\end{figure}\n\n\\begin{figure*}\n\\hspace*{-0.2cm}\\resizebox{7cm}{!}{\\includegraphics{ft_cont_nf_61.ps}}\n\\hspace*{-0.2cm}\\resizebox{7cm}{!}{\\includegraphics{ft_cont_E03_61.ps}}\n\\hspace*{-0.2cm}\\resizebox{7cm}{!}{\\includegraphics{ft_cont_c001_61.ps}}\n\\hspace*{-0.2cm}\\resizebox{7cm}{!}{\\includegraphics{ft_cont_run13_61.ps}}\n\\hspace*{-0.2cm}\\resizebox{7cm}{!}{\\includegraphics{ft_cont_F09_61.ps}}\n\\hspace*{-0.2cm}\\resizebox{7cm}{!}{\\includegraphics{ft_cont_E3_61.ps}}\n\\hspace*{-0.2cm}\\\\\n\\caption{Age-radial distance maps of the stars in the NF (upper left panel), E-0.3 (upper right panel), C-0.01 (middle left panel), E-0.7 (middle right panel), F-0.9 (lower left panel) and E-3 (lower right panel) experiments. This figure shows the inside-out formation of the disc in E-0.7 and F-0.9, and the outside-in formation of the spheroid in NF and C-0.01.} \n\\label{histories}\n\\end{figure*}\n\nWhen baryons are present, the shape of the DM profiles in the central regions changes significantly in comparison to the DM-only case  as it can be appreciated from   Fig.~\\ref{fig1}. In the case of the DM-only run, its profile is sharper (i.e. smaller $n$ value)\nthan those of haloes with baryons, indicating the increase in the DM concentration in the latter cases. Interestingly, all  haloes, except for the E-3 (the most DM dominated) and the DM-only cases, present a nearly isothermal behaviour in the region dominated by baryons, in agreement with the results found by Tissera et al. (2009) who analysed several dark matter haloes with higher numerical resolution as part of the Aquarius project. \n\nHowever, as it can be seen from Table \\ref{tab1}, each DM halo has different fitting parameters. In order to understand the origin of these differences, we  correlate their properties with those of the galaxy they  host. In fact, the comparison between the  E-0.7 and the NF profiles shows that the DM distribution in E-0.7  is more concentrated than in NF, although it hosts a galaxy a factor of two less massive\nthan the later.  This finding suggests that the total amount of baryons collected within the central region of a halo is  not the only relevant factor  affecting the response of the DM to the presence of baryons.\n\n\nIn Fig. ~\\ref{histories}, we display the density-contour maps of  the age of the stars associated to each simulated galaxy as a function of tridimensional\ndistance to the centre of mass. These maps provide a picture of both the star formation history and the final stellar mass distribution in each galaxy.\nThose systems with an important disc structure (F-0.9 and E-0.7) have most of the stars  younger than 8 Gyr located\noutside the central region, while systems dominated by an old stellar spheroidal component (NF, E-0.3, C-0.01) have most of their stars in the inner region.\n\nThe NF and  E-0.7 runs have produced very different galaxies as shown in  Fig.\\ref{histories}.\n  The galaxy in the  NF run is dominated by\nold stars, determining a spatially extended spheroid with  $78\\%$ of the final stellar mass older than 10 Gyr.\nThe system in  E-0.7 has a compact old spheroid and an important disc component populated by younger stars. \nThis disc is able to survive the interaction with satellites at lower redshifts. \nTheir different star formation histories and morphologies are the result of the action of the  SN feedback in E-0.7 which was successful at regulating the transformation of gas into stars, preventing the formation of an early, extended spheroid and assuring the existence of gas to form a disc later on. The SN feedback has also affected the formation of stars in the satellite systems that merged with the main object as we will discussed in detail in Section 4. \n\nA similar  trend is  found when comparing the E-0.3 and C-0.01 runs. In these two cases the feedback was not that efficient at regulating the SF activity. As a result the final stellar masses are only slightly lower than the NF case.\n But interestingly, the E-0.3  has a more concentrated DM profile than  C-0.01. We note that while C-0.01 formed an old extended spheroid,  E-0.3 has an old  spheroid but it was able to develop  a smaller and thicker disc. \n\nFor the F-0.9 run, we found that it has also a more concentrated profile than the NF case. And, again, this can be linked to a similar pattern in  its formation history: F-0.9 and E-0.7 both have compact old spheroids and  extended, inside-out-formed discs.\nIn these last two runs, the total amount of stars within the central region is approximately a factor of two lower than\nin the case of the NF run. \n\nIn E-3, where an extreme  value for the energy per SN was assumed, we obtained the less concentrated  DM profile among the cases with baryons. This profile is weakly more concentrated than the DM-only one. \nAs expected, the galaxy in E-3 is also the most DM dominated one in the central region, as shown by the total\nto stellar mass ratio $M_{\\rm t}\/M_{\\rm s}=37$ (Table~\\ref{tab1}). \nIn this simulation, most of the gas has been blown away and  only a\n small fraction of stars has been formed in a bursty fashion as shown in Fig. ~\\ref{histories}.\n\n\\subsection{Velocity Dispersion}\n\nIn Fig.~\\ref{Sigma}, we plotted the velocity dispersion $\\sigma$ as a function of the radius from three times the gravitational softening. We found that when baryons are included the $\\sigma^2$ profiles increase  in the central regions compared to the DM-only cases so that\nthe 'temperature inversion' typical of the NFW profiles is lost (e.g. Tissera et al. 1998; Romano-D\\'{\\i}az et al. 2008; see also Tissera et al. 2009 for high resolution simulations).\n  N08 found a similarity between the $\\sigma ^2$ profiles and $\\rho r^2$ and they proposed that this may be due to a scaling relation between densities and velocity dispersions in haloes. We found that this similarity only holds for the E-3 run, where most of the baryons have been blown away due to the strong galactic winds produced as a consequence of the extreme high energy assumed per SN.\n\n\n\n\\begin{figure}\n\\hspace*{-0.2cm}\\resizebox{7cm}{!}{\\includegraphics{sigma_61.ps}}\n\\hspace*{-0.2cm}\\\\\n\\caption{Velocity dispersion  as a function of radius from three times the gravitational softening: NF (black line), E-0.7 (magenta line), F-0.9 (red line), E-0.3 (green line), C-0.01 (blue line), E-3 (cyan line)  and  DM-only (black thick dashed line).} \n\\label{Sigma}\n\\end{figure}\n\n\nFrom Fig.~\\ref{Sigma} and Table ~\\ref{tab1}, we can see that there is a correlation between the inner slope of the $\\sigma^2$ profile and the stellar mass in the simulated galaxies, so that the higher the mass, the steeper the inner profile. From the analysis of the profiles of the progenitor system as a function of redshift, we found that, in the DM-only case, the temperature inversion is present from at least $z \\approx 2$ to $z=0$, but in the runs with baryons, the inversion profile is never at place (Romano-D\\'{\\i}az et al. 2008).\n\n\\subsubsection{Velocity Anisotropy}\n\nTo provide a measure of the velocity structure of the haloes, we calculate the anisotropy parameter $\\beta = 1 - \\frac{\\sigma_{t}^{2}}{2 \\sigma_{r}^{2}}$, where $\\sigma_{t}$ and $\\sigma_{r}$ are the dispersions of the tangential and radial motions. For an isotropic distribution, $\\beta$ should be zero while for a system dominated by radial motions, it should have positive values. Fig.~\\ref{Beta} shows the anisotropy parameter as a function of radius for the haloes at $z=0$. When baryons are included the central anisotropy tends to increase slightly in most cases, although because of the high level of noise, this result should be confirmed by higher numerical resolution. Interestingly, from $r \\approx 10$ kpc $h^{-1}$, we found those haloes  hosting spheroid-dominated galaxies to have weaker level of radial anisotropy compared with the DM-only case (upper panel of Fig.~\\ref{Beta}). Conversely, those haloes hosting disc-dominated systems have higher velocity anisotropies (lower panel of Fig.~\\ref{Beta}).\n\n\n\\begin{figure}\n\\hspace*{-0.2cm}\\resizebox{7cm}{!}{\\includegraphics{Beta_sph.ps}}\n\\hspace*{-0.2cm}\\resizebox{7cm}{!}{\\includegraphics{Beta_dsk.ps}}\n\\hspace*{-0.2cm}\\\\\n\\caption{ Velocity anisotropy parameter $\\beta$ as a function of radius  for the NF (black line), E-0.3 (green line), C-0.01 (blue line) runs (upper panel), and, for E-0.7 (magenta line), F-0.9 (red line) and E-3 (cyan line) runs (lower panel), at $z=0$. In both panels, the dashed line corresponds to the DM-only run.} \n\\label{Beta}\n\\end{figure}\n\n\n\\section{Interaction with satellites}\n\n\\begin{figure}\n\\resizebox{8cm}{!}{\\includegraphics{deltav2_full.ps}}\n\\hspace*{-0.2cm}\n\\caption{Central halo mass concentration $\\Delta_{v\/2}$ as function of redshift for NF (black line), E-0.7 (magenta line), F-0.9 (red line), E-0.3 (green line), C-0.01 (blue line), E-3 (cyan line)  and  DM-only (black dashed line) haloes. \nThe dotted line is the expected growth for a \nconstant density due to the expansion of the Universe.} \n\\label{delta_v2}\n\\end{figure}\n\nThe analysis of the DM profiles suggests a connection between the DM evolution and the history of formation of the baryonic structures. We can follow the formation of a halo and its galaxy with time but it is not possible to reliably estimate the\nDM profiles when the system get smaller because of the high numerical noise present in our simulations.\nThen, in order to assess the evolution of the different runs,  we calculated the concentration parameter proposed by Alam, Bullock \\& Weinberg (2002), $\\Delta_{v\/2}$, as a function of redshift. This parameter measures the mean DM density normalized to the cosmic closure density within the radius at which the circular rotational speed due to the DM alone rises to half its maximum value. This parameter has the advantage of being independent of a specific density profile and, as it is an integrated quantity, it can be estimated more robustly at any time. In  Fig.~\\ref{delta_v2}, we show $\\Delta_{v\/2}$  as a function of redshift for the progenitor haloes for our set of simulations.  The dotted line shows the expected relation for a constant density perturbation due to the expansion of the Universe alone (i.e. hereafter critical relation). All haloes increase their concentration as they grow with time.\nThe DM-only run has the lowest  concentration, as expected, at all times. We note that the relation flattens between $z \\approx 1$ and $z \\approx 1.8$, coinciding with the close approaching of satellites as we will discuss in more detail later on.\nFrom Fig.~\\ref{delta_v2},  we can see that haloes  hosting baryons follow different paths between them. The NF run shows a DM halo\nwhich is always more concentrated than the DM-only one but it has a stronger flattening of the relation during the same period of time. The other haloes do not show such a strong change in the slope, except for C-0.01 run. Note that a flat slope\nis indicating an expansion of the mass distribution in the central regions.\nAnother interesting case is that of the F-0.9 halo which host the most extended and important disc. This system does not show a \nchange in the slope and follows the critical  relation  even closer than the DM-only case.\nA similar behaviour can be observed for  E-3 which shows\na slightly higher concentration driven by the presence of the baryons that have been able to settle in. \nIn these runs, the entrances of satellites presumably cause weaker effects on the dark matter distribution compared to the  DM-only run, since they are less massive due to the strong action of SN feedback and are easily disrupted as they fall in.\n\nRecall that the merger trees of these haloes are the same with the only difference being the  fraction of \nbaryons and the gas reservoir in each substructure, which depend on the SF and the SN feedback parameters adopted in each run. So, in order to understand the origin of the different evolution of  halo concentration, we \nanalyse the satellites within the virial radius of the progenitor objects as a function of redshift\n\nIt has been shown in previous works (e.g. Barnes \\& Hernquist 1992) that when two systems collide, orbital angular momentum can be transfered from the baryonic clumps to the dark matter haloes. Hence, it might be possible that if the properties\nof the satellite distribution were affected by the choice of SF and SN parameters, \n they might also transfer different amount of angular momentum to the DM. \n\nTo illustrate the differences  in the satellite distribution in each run, in Fig.~\\ref{coseno} we show the distribution  of the cosine of the angle between the total angular momentum ($J$)  of the  main stellar system and the angular momentum of each stellar particle ($J_i$) at $z \\approx 1.6$, when the $\\Delta_{v\/2}$\nshows a change in the slope (Fig.~\\ref{delta_v2}). As we can see,  at this redshift there is no disc structure within any of the systems (i.e. there are no particles ordered at cosine $\\approx 1$). And secondly, the distribution of stellar clumps  surrounding the main systems is very different.  It can be seen that, in the NF case, the satellites are clearly more massive and have been able to survive further in the halo since stars are more gravitationally bounded.\nThose systems that later on are able to develop a disc component (E-0.7 and F-0.9) show smaller stellar satellites.\n\nThis trend can be quantified from Fig.~\\ref{massint} where we show the cumulative mass of the satellites for each component (dark matter, gas and stars) as a function of the distance to the centre of mass at $z \\approx 1.6$. For the stellar component, the satellites of the NF run are the most massive one, while those in  E-0.7 and F-0.9 are less massive due to the SN feedback action.\nThe most diffuse and smallest satellites are found in the E-3 halo, as expected.\nThe gas component behaves similarly to the stellar one, so that  those systems with weaker or no SN feedback have also the larger fraction of gas   per satellite (i.e. systems run with strong feedback blow away  important fractions of gas).\n\n\n\\begin{figure*}\n\\hspace*{-0.2cm}\\resizebox{5cm}{!}{\\includegraphics{Cos2_44_nf.ps}}\n\\hspace*{-0.2cm}\\resizebox{5cm}{!}{\\includegraphics{Cos2_44_E03.ps}}\n\\hspace*{-0.2cm}\\resizebox{5cm}{!}{\\includegraphics{Cos2_44_c001.ps}}\n\\hspace*{-0.2cm}\\resizebox{5cm}{!}{\\includegraphics{Cos2_44_run13.ps}}\n\\hspace*{-0.2cm}\\resizebox{5cm}{!}{\\includegraphics{Cos2_44_F09.ps}}\n\\hspace*{-0.2cm}\\resizebox{5cm}{!}{\\includegraphics{Cos2_44_E3.ps}}\n\\hspace*{-0.2cm}\\\\\n\\caption{Cosine of the angle between the total  angular momentum of the simulated galaxies and the angular momentum of each stellar particles within the virial radius at $z \\approx 1.6$ for   NF , E03 and C-0.01  (upper panels) and  E-0.7 , F-0.9 and  E-3 (lower panel).} \n\\label{coseno}\n\\end{figure*}\n\n\\begin{figure*}\n\\hspace*{-0.2cm}\\resizebox{5cm}{!}{\\includegraphics{Mass_sat_DM_44.ps}}\n\\hspace*{-0.2cm}\\resizebox{5cm}{!}{\\includegraphics{Mass_sat_stars_44.ps}}\n\\hspace*{-0.2cm}\\resizebox{5cm}{!}{\\includegraphics{Mass_sat_gas_44.ps}}\n\\hspace*{-0.2cm}\\\\\n\\caption{Integrated  DM, stellar and gas mass of  the satellites within the virial radius at $z \\approx 1.6$ (see Fig.8 for color code).} \n\\label{massint}\n\\end{figure*}\n\n\nFig.~\\ref{sat_mas} shows the total baryonic (left) and dark matter (right) mass of the satellites within the virial radius as a function of the redshift. It can be seen that both components, baryonic and DM, present noticeable mergers or disintegration episodes from $z \\approx 2$ indicated by a decrease in the total mass in the identified substructure within the virial radius at a given redshift. An increase of the total mass in satellites implies that new substructure has entered the virial radius. From this figure we can also see that the DM associated to the satellites varies slightly from halo to halo as expected since they all share the same merger tree. However the baryonic mass shows more important differences. In the case of the NF halo, the total baryonic mass in satellites within the virial radius changes weakly with time. However, the rest of the haloes not only have a lower baryonic content, but also experience larger changes as a function of redshift.\nAfter $z  \\approx 2$, the main period of satellites accretion is around $z \\approx 0.8$. Before and after that time, there are mainly  merger events with the main galaxy  or satellite tidal disruptions which feed the background halo.  We can not differentiate between these processes with the help of this plot since it provides the total mass associated to the subhaloes which can be individualized at a given redshift. However, we carried out a thorough analysis of these subhaloes following their progenitors in time in each run to be sure that we\nwere actually quantifying these effects, without being contaminated by fly-bye intruders.\n\nIn order to improve our understanding of  the angular momentum content of the central galaxy and the DM,  we analysed\n the  specific angular momentum of each mass component of the systems defined at $z \\approx 1.6$  as a function of time. \nWe chose the systems at this redshift as a reference one because, from this time,\n we detect the larger differences in the evolution of the central mass concentration of our haloes\n(Fig.~\\ref{delta_v2}) and also the most  important interaction events with satellites (Fig.~\\ref{sat_mas}).\nWe selected   the stars and gas  components within 1.5 times the optical radius ($\\approx$ 12  $h^{-1}$ kpc)  and of the DM within the virial radius, without including subhaloes, at $z \\approx 1.6$. We estimated the cumulative  mass in  bins containing a growing fraction of the total selected mass. We analysed the specific angular momentum content of each mass component  with redshift. \n\n\nIn Fig.~\\ref{Jtotal}, we show the distributions for the NF and E-0.7 cases.\nThe stellar mass that was within 1.5 the optical radius  at $z \\approx 1.6$ shows an increase of the specific angular momentum in each\nmass bin, even in the lowest one.\nBoth systems show the same trend although the acquisition of angular momentum is much larger for the stars in the NF halo for all mass bins, producing a stronger stellar migration\\footnote{Note that the detection of stellar migration does not affect the fact that the discs are mainly formed inside-out as shown in Fig.~\\ref{histories} and S08.} (Roskar et al. 2008).\nThe associated gas mass in the NF case at the same redshift shows also an increase in its specific angular momentum  which is, then, partially lost at low redshifts.  In E-0.7, the gas component in outer regions gains  larger fraction of angular momentum. This could be explained  by the action of the SN feedback which triggers important gas outflows (S08).\nHence, we found that baryons determining the galaxy at $z \\approx 1.6$ acquired angular momentum which produced its expansion.\nBecause baryons dominate the central regions, the global potential well  also changes, probably acting against further DM contraction.\n\n\nFinally, we also measured the specific angular momentum content of the DM halo which host the galaxy at $z \\approx 1.6$.\nAs it can be seen from Fig.~\\ref{Jtotal}, these DM particles increase their angular momentum content as a function of redshift and in all mass bins, even in the most  central one. \nThe DM particles in the NF case acquire a larger fraction of angular momentum than those in the E-0.7 case.\n Note that these DM particles represent the halo at $z \\approx 1.6$ without substructure so the increase of angular momentum is expected because of the angular momentum transfer from the infalling \nsatellites via dynamical friction (the total angular momentum of the  DM halo is expected to be conserved  as \nshown in S08). \n\n\n\n\\begin{figure*}\n\\hspace*{-0.2cm}\\resizebox{7cm}{!}{\\includegraphics{mbar_rvir_z.ps}}\n\\hspace*{-0.2cm}\\resizebox{7cm}{!}{\\includegraphics{mdm_rvir_z.ps}}\n\\hspace*{-0.2cm}\\\\\n\\caption{Baryonic (left panel) and DM (right panel) total mass in satellites  within the virial radius as a function of the redshift for the  NF (black line), E-0.7 (magenta line), F-0.9 (red line), E-0.3 (green line), C-0.01 (blue line), E-3 (cyan line)  and  DM-only (black thick dashed line) runs.} \n\\label{sat_mas}\n\\end{figure*}\n\n\\begin{figure*}\n\n\\hspace*{-0.2cm}\\resizebox{6cm}{!}{\\includegraphics{Jtotal2_m_star_bg_nf.ps}}\n\\hspace*{-0.2cm}\\resizebox{6cm}{!}{\\includegraphics{Jtotal2_m_gas_bg_nf.ps}}\n\\hspace*{-0.2cm}\\resizebox{6cm}{!}{\\includegraphics{Jtotal2_m_dm_bg_nf.ps}}\\\\\n\\hspace*{-0.2cm}\\resizebox{6cm}{!}{\\includegraphics{Jtotal2_m_star_bg_run13.ps}}\n\\hspace*{-0.2cm}\\resizebox{6cm}{!}{\\includegraphics{Jtotal2_m_gas_bg_run13.ps}}\n\\hspace*{-0.2cm}\\resizebox{6cm}{!}{\\includegraphics{Jtotal2_m_dm_bg_run13.ps}}\n\\caption{Time evolution of the specific angular momentum  of  stellar (left panels) and gas (middle panels) components within 1.5 the optical radius ($ \\approx 12 h^{-1}$ kpc ) and\nof the DM halo (right panels) within the virial radius measured at  $z\\approx 1.6$, in  mass bins containing a growing fraction of the total mass a function of redshift for the NF (upper panels) and E-0.7 (lower panels) runs. The lines represent a growing fraction of mass, from $10\\%$ for the lower one up to $100\\%$ for the upper one.} \n\\label{Jtotal}\n\\end{figure*}\n\n\n\n\\section{Rotation curves}\n\nA well-known problem of numerical simulations is the inability of CDM scenarios to produce systems with flat rotation curves comparable to that of the Milky Way because of the catastrophic concentration of baryons at the central region. Dutton et al. (2007, 2008), among others, found that, in the absence of baryons, there appears to be a reasonable agreement between theory and observation because the former predicts $V_{max} \\approx   V_{200}$, where  $V_{max}$ is the maximum of the total circular velocity and $V_{200}$\nis the circular velocity at the virial radius. However, when the effects of baryons are taken into account through the AC hypothesis, results do not match observations since $V_{max}$ is significantly increased by a factor of two (Navarro \\& Steinmetz 2000; Dutton et al 2007). These values are too high for matching the Tully-Fisher relation.\nDutton et al (2008) proposed that a net halo expansion that reverse the contraction would be required in order to lower the $\\frac{V_{max}}{V_{200}} $ ratio. A mechanism suggested in this work  is SN feedback since a net halo expansion could result from the rapid removal of the disc mass. However, while Gnedin \\& Zao (2002) found this effect to be too weak to reconcile observations and theory, Read \\& Gilmore (2005) claimed that if this process is repeated several times, a reduction in the halo density could be accounted for. \n\nFrom our simulations, we can estimate the circular velocities for each component and analyse how the  $\\frac{V_{max}}{V_{200}} $ratio\nvaries for different combinations of the SF and the  SN feedback parameters. The circular velocity of baryons shown in Fig.~\\ref{Vcir} (left panel) reflects the presence of a concentrated dominating spheroid  (NF, C-0.01 and E-0.3) or a dominating disc component (E-0.7 and F-0.9). The dark matter\ndistributions (middle panel) vary between haloes in agreement with Fig. 1,  but in  the central region the shape of the total circular velocity (right panel) is\ndetermined mainly by the baryonic component.  As it can be seen from this plot, only when an important disc component is at place, the total velocity  distribution gets flat in the baryonic dominated region.\n\n\nIn Fig.~\\ref{Vcir_E07_red} (upper panels), we show  the evolution of the circular velocity for the baryonic component of the NF and E-0.7 cases. In the E-0.7 case, the curves become flatter in an inside-out process (see Fig.~\\ref{histories} and S08). We can also see how the baryons  in the very central region moves outwards, contributing to produce a flatter curve.\nIn the case of  the NF run, the baryonic circular velocity is very sharp from high redshift and it also shows an outward\ndisplacement of baryons located in the central region  which is explained by the increase\nin the angular momentum content of the the stellar component as shown in the previous section.\n\n We have quantified the flattening of the baryonic circular velocity curve by measuring its logarithmic slope ($LS$) between the radius at the maximum velocity curve and at the optical radius for each simulated halo. From Table \\ref{tab1}, we can see that  the lower absolute values correspond to the E-0.7 and F-0.9 runs when the disc component is the dominating one.\n In Fig.~\\ref{Vmax200} (upper panel), we show \n$LS$ as a function of the shape parameter $n$ of the dark matter profiles. As we can see, galaxies with lower absolute values of $LS$ tend to have  haloes with the largest $n$ parameters. The less concentrated cases are found\nwhen either spheroid-dominating systems were able to form (NF and C-0.01) or in the E-3 run because of its low baryonic content due to the action of very violent galactic winds.\n\n\n\nWe also estimated the $\\frac{V_{max}}{V_{200}} $ ratio as a function of shape parameter $n$ of the dark matter profiles as displayed in  Fig.~\\ref{Vmax200} (lower panel). As it can be seen,  we found values for this ratio between $\\approx 1.15$ and $1.5$. Again, those systems where a disc-like galaxy was able to form show \nthe lowest ratios, indicating the existence of flat circular velocity curves in agreement with observational results.  The small disc in E-3 has the lowest ratio which is produced mainly for the fact that\nthe halo has weakly contracted during its assembly due to the very small fraction of baryons retained in the central region.\n\n\\begin{figure*}\n\\hspace*{-0.2cm}\\resizebox{6cm}{!}{\\includegraphics{Vcir_bar.ps}}\n\\hspace*{-0.2cm}\\resizebox{6cm}{!}{\\includegraphics{Vcir_dark.ps}}\n\\hspace*{-0.2cm}\\resizebox{6cm}{!}{\\includegraphics{Vcir_total.ps}}\n\\caption{Baryonic (left panel), dark matter (middle panel) and total  (right panel) circular  velocities for the NF (black), E-0.7 (magenta), F-0.9 (red), C-0.01 (blue), E-0.3 (green) and E-3 (cyan) runs.} \n\\label{Vcir}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\hspace*{-0.2cm}\\resizebox{6cm}{!}{\\includegraphics{Vrot_bar_nf_red.ps}}\n\\hspace*{-0.2cm}\\resizebox{6cm}{!}{\\includegraphics{Vrot_bar_E07_red.ps}}\\\\\n\\hspace*{-0.2cm}\\resizebox{6cm}{!}{\\includegraphics{Vrot_dark_nf_red.ps}}\n\\hspace*{-0.2cm}\\resizebox{6cm}{!}{\\includegraphics{Vrot_dark_E07_red.ps}}\n\\caption{Baryonic (upper panels) and dark matter (lower panels) circular velocities as a function of redshift for the NF (left panels) and E-0.7 (right panels) simulations. The redshift decreases  from red to green colors starting at $z \\approx 2 $ (red) and ending at $z=0$ (light green). } \n\\label{Vcir_E07_red}\n\\end{figure*}\n\n\n\\begin{figure}\n\\hspace*{-0.2cm}\\resizebox{7cm}{!}{\\includegraphics{LS_n_fil.ps}}\n\\hspace*{-0.2cm}\\resizebox{7cm}{!}{\\includegraphics{Vmax200_n.ps}}\n\\hspace*{-0.2cm}\\\\\n\\caption{Baryonic logarithmic slope  (upper panel) and the ratio between the maximum total circular velocity and the virial velocity as a function  the Einasto's shape parameter ($n$) for  NF (plus), E-0.7 (diamond), F-0.9 (asterisk), C-0.01 (triangle), E-0.3 (square), E-3 (cross) runs.} \n\\label{Vmax200}\n\\end{figure}\n\n\\subsection{Adiabatic contraction prescription}\n\nThe prediction of the effects of baryons on the DM haloes  based on AC  hypothesis (e.g. B86) is widely used.  However the main assumptions of this approximation, namely that halo particles move on circular orbits,  is not realistic. Several studies (Gnedin et al. 2004; Sellwood \\& McGaugh 2005) have reported the AC hypothesis to overestimate the level of compression. Possible alternative models have been  developed to calculate the contraction of the DM halo originated by the accumulation of baryons in the central region.\nHowever, as we claimed in Pedrosa et al. (2009), the response of the DM halo to the presence of baryons\ndepends strongly on the way baryons are assembled (see also Romano-D\\'{\\i}az et al. 2008). Actually, this has been the main discussion of the current paper.\nIn this section, we aim at comparing  different proposed prescriptions found in the literature to predict\nthe effects of baryons, and the statistical motivated prescription of Abadi et al. (2009, hereafter A09).\n\nIn Fig.~\\ref{AC2}, we show the DM circular velocity obtained for each of our runs (red solid lines), the corresponding one from the DM-only run (black lines) and the velocities estimated by applying different AC models to the DM-only run taking into account the  baryonic distributions of the corresponding hydrodynamical runs. We found that, as expected, the B86 model (violet lines) largely overpredicts the level of concentration and also changes the shape of the DM distribution when compared to the DM profiles obtained from the cosmological runs.  The recipes of Gnedin et al. (blue lines) and A09 (green lines)  also overpredict the level of contraction, although the disagreement is not so large. Tissera et al. (2009) found a similar behaviour for their simulated haloes, which are roughly one order of magnitude higher in numerical resolution.\n\n\nFrom the mass distribution of our simulations  we can estimate the best fitting for the relation between $\\frac{r_{f}}{r_{i}}$ and $\\frac{M_{i}}{M_{f}}$ (A09), where $r_i$ and $M_i$ correspond to the radii that contains a given number of particles and the total mass within that radii in the DM-only run, and $M_f$ is the final total (baryonic and DM) mass at $r_f$ estimated in the same way from each of our simulations including baryons. Following A09, we assume a function of the form:\n\n\\begin{equation}\n\\frac{r_{f}}{r_{i}} =  1 + a \\times ((\\frac{M_{i}}{M_{f}})^x - b)\n\\label{eq2}\n\\end{equation}\n\nWe found that $x=4 $ is the exponent that best represent our mass relations. Keeping $x=4$ fixed, we fit the  $a$ and $b$ parameters in order to reproduce the level and shape of the contraction that we obtained in our simulations (Fig. \\ref{AC1}). \nFor the spheroid-dominated systems (NF, C-0.01 and E-3), we found that they are better reproduced with values of $a=0.14$ and $b=1.25$, while for haloes hosting important disc structures (E-0.7, F-0.9 and E-0.3), we get  $a=0.15$ and $b=1.40$. We have included the E-0.3 case among the ones with disc structure as it was able to develop a small and thick disc with a $D\/S =0.6$ and, it presents a similar behaviour for the relation between radius and masses ratios than the other two runs with disc galaxies. The largest contraction predicted for haloes hosting a disc structure is in agreement with our previous results and discussion.\n\nConsistently with previous findings,  Fig. ~\\ref{AC2} shows that the response of the DM halo to the presence of baryons does not depend solely on the amount collected at the central region but is the result of the joint evolution of baryons and DM as they are assembled. Although it can be seen from the residuals in Fig. ~\\ref{AC2} that our prescription provides a good prediction of the level of contraction, a larger sample of  galaxies is needed in order to claim its general validity.\n\n\\begin{figure*}\n\\hspace*{-0.2cm}\\resizebox{6cm}{!}{\\includegraphics{AC_vel_f12_nf.ps}}\n\\hspace*{-0.2cm}\\resizebox{6cm}{!}{\\includegraphics{AC_vel_f12_E03.ps}}\n\\hspace*{-0.2cm}\\resizebox{6cm}{!}{\\includegraphics{AC_vel_f12_c001.ps}}\n\\hspace*{-0.2cm}\\resizebox{6cm}{!}{\\includegraphics{AC_vel_f12_run13.ps}}\n\\hspace*{-0.2cm}\\resizebox{6cm}{!}{\\includegraphics{AC_vel_f12_F09.ps}}\n\\hspace*{-0.2cm}\\resizebox{6cm}{!}{\\includegraphics{AC_vel_f12_E3.ps}}\n\\hspace*{-0.2cm}\n\\caption{ Circular velocity obtained from our simulations with baryons (red lines), the DM-only run (black line), the B86  (dotted- dashed violet lines), the prescriptions of Gnedin et el. (2004) (blue dashed lines) and the A09 (green dashed lines), and our approximation (equation 2, red dotted lines). In the small, lower plots we show the residuals for each  velocity curve respect to that of the DM-only run.}\n\\label{AC2}\n\\end{figure*}\n\n\\begin{figure}\n\\hspace*{-0.2cm}\\resizebox{7cm}{!}{\\includegraphics{f_our_ab_n4.ps}}\n\\hspace*{-0.2cm}\n\\caption{$\\frac{r_{f}}{r_{i}}$ ratio vs $\\frac{M_{i}}{M_{f}}$, NF (black lines), E-0.7 (magenta line), F-0.9 (red line), E-0.3 (green line), C-0.01 (blue line), E-3 (cyan line). The dashed lines correspond to the fits obtained according to Eq.2}\n\\label{AC1}\n\\end{figure}\n\n\n\n\\section{Conclusions}\n\nWe have studied the DM distribution in a set of runs of a  $\\approx 10^{12} M_{\\sun}$ halo extracted from a cosmological simulation, where the physics that regulates the  SF activity and SN feedback was varied allowing the formation of galaxies with different morphologies at $z=0$.\nSince the underlying DM merger tree is the same in all our runs, the differences in the properties of the DM and baryons can be directly ascribed to the variations in the baryonic physics.\nFor the same reasons, all simulations have been run with the same numerical resolution so they are affected by resolution in a similar way. Because we are studying the DM distribution principally in the central regions, we have estimated all quantities outside three gravitational softenings.\nAnd as a further test of the robustness of our results against numerical artifacts, we have studied the DM profiles in two haloes selected from a fully cosmological simulation with higher numerical resolution. These haloes host galaxies with different morphologies and reproduce remarkably well  our findings (appendix A).\n  \n Our main results can be summarized as follows:\n\n\\begin{enumerate}\n\n\\item[1. ]We found that the Einasto's model provides the best fit for the spherically-averaged density profiles of our DM haloes. When baryons are present, the haloes become more concentrated in the central regions. However, the amount of baryons collected within the inner regions does not by itself  determine the response of the DM halo to the assembly process of the galaxy.\n\n\\item[2. ]When baryons are included, the velocity dispersion increases in the central region compared to the dissipationless case and no ``temperature inversion'' is observed, except for the  E-3 run where the fraction of baryons remaining in the halo is very small due to its strong SN feedback. The slope of the inner velocity dispersion profiles increases with increasing baryonic mass collected at the centre. We found that haloes hosting spheroidal galaxies tend to have weaker levels of velocity anisotropy than the DM-only run. While those haloes with an important disc galaxy show the highest levels of velocity anisotropy.\n\n\\item[3. ]The formation history plays an important role in the final distribution of its DM halo. We observed that those systems that are able to develop inside-out-formed discs, although they host in the central regions a lower amount of baryons than  galaxies that formed  old extended spheroids, have  more concentrated DM  profiles.\nSince all our runs shared the same merger tree, the differences between them can be directly ascribed to their different baryonic evolutions  which are determined mainly by the SN feedback.\n\n\\item[4. ]We followed the evolution of the DM distribution in the central regions with redshift via the concentration parameter $\\Delta_{v\/2}$. We found that all haloes increase their concentration as they grow in time. As expected, the dissipationless run has the lowest concentration at all times. Also, we observed that  haloes present a flattening in this relation with respect to the critical relation indicating an expansion of the mass distribution in the central regions. This flattening differs among the different haloes being larger for those systems hosting an spheroidal galaxy. This trend can be linked with close approaches of satellites and their properties. \n\n\\item[5. ] The analysis of  the satellites within the virial radius of the progenitor objects as a function of redshift yields that in the NF case, the satellites are clearly more massive and have been able to survive further in the halo, since stars are more gravitationally bounded. The systems that, at lower redshifts, developed a disc component (e.g E-0.7 and F-0.9) show less massive satellites at all redshifts as  a consequence of the action of SN feedback. As expected, the most diffuse satellites correspond to the E-3 case. By analysing the mass of the satellites within the virial radius, we found noticeable mergers or disintegration episodes which can be correlated with features in the specific\nangular momentum content of the mass components. \n\nWe studied the specific angular momentum content of main baryonic component and its halo from  $z \\approx 1.6$, when the $\\Delta_{v\/2}$ starts to\nclearly indicate an expansion of the central DM concentration. We selected the  NF and E-0.7 haloes as case studies.\nFor these cases, we found that even the inner $10\\%$ of the stellar mass gains angular momentum as a function of redshift, although\nthe increase is more important for the NF case. The gas component shows a similar behaviour but, in this case, it is in the E-0.7 run where the\ngas acquired larger amount of angular momentum  induced by the SN feedback which we know is successful at\ndriving galactic outflows (S08).\nThe DM halo (without substructure) increases its angular momentum content at all mass bins. Again, the larger profits are measured for\nthe NF case which has the most massive orbiting satellites. Hence, the mass components identified at $ z \\approx 1.6$  in NF case  gained more angular momentum up to $z=0$ than their counterparts in E-0.7, except for the gas component which is anyway less massive than the other ones. The fact that stars migrate also contributes to change the inner potential well since they are the dominating mass component in the central region. This, on its turn,  probably acts\nagainst further DM contraction.  Both effects could explain  the evolution of $\\Delta_{v\/2}$ and the fact that, when the halo hosts a disc-dominated galaxy, it is more concentrated than when it hosts a spheroid-dominated one. \n\n\n\\item[6. ]The baryonic rotation curves of our simulated galaxies reflect the presence of a concentrated dominating spheroid or a dominating disc component. The total circular velocity in the central regions is mainly determined by the baryonic component. When an important disc component is present the total velocity distribution gets flat in the central regions. The evolution of the baryonic circular velocity with time of the E-0.7 run shows that as the disc forms, the curve became flatter out to larger radii. We  quantified the flattening of the curve through the LS and correlated it with the shape parameter. Galaxies with lower values of LS tend to have haloes with larger $n$ parameter. \nWe found values for $\\frac{V_{max}}{V_{200}} $ between $\\approx 1.15$ and $1.5$. The systems where a disc galaxy was able to form present the lowest ratios, in agreement with observational results.\n\n\\item[7. ]We have compared our simulated haloes with different AC prescriptions. We found, as expected, that the Blumenthal et al (1986) model overpredicts the level of concentration and also changes the shape of the DM distribution. The recipes of Gnedin et al. (2004) and A09 are an  improvement over the B86 approach. However, they overpredict the level of contraction when the haloes host an spheroid-dominated galaxy.\n From our analysis, we have obtained a prescription that provides a better representation for the contraction of our haloes,\ndepending on the morphology of the galaxy. However, the 'universality' of this prescription should be\ntested with a larger statistical sample.\n\n\n\nAll our findings indicate that the response of the DM halo to the presence of baryons is the result of the joint evolution of baryons and DM during the assembly of the galaxy and in this sense,  the cosmological context for galaxy formation cannot be ignored.\n\n\\end{enumerate}\n\n\n\\section*{acknowledgement}\nSP and PBT  acknowledge productive discussions with M. Abadi,  O. Valenzuela, M.E. De Rossi, T. Tecce and C. Artale.\nThis work was partially supported by PICT 32342(2005), PICT  Max Planck 245(2006) of Foncyt and DAAD-Mincyt collaboration(2007).\nWe thank the anonymous referee for her\/his useful comments.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nAt sufficiently high angular resolution ($\\sim$1 mas), the inner\njets of some quasars have been found to exhibit Faraday Rotation\nMeasures in excess of 1000 radians m$^{-2}$\\ (Udomprasert et~al.\\ 1997; Cotton\net~al.\\ 1997; Taylor 1998).  This RM is most likely produced by\norganized magnetic fields and ionized gas close to the center of\nactivity.  Beyond a projected distance from the core of 20 pc, the RMs in the\njet fall below 100 radians m$^{-2}$ and are thus consistent with a purely\ngalactic origin.\n\nIn \\S2 a sample of 40 strong, compact galaxies, quasars and BL\nLacertae objects is defined.  Four of these sources were reported on\nin Taylor 1998 (Paper I).  In \\S3 I describe observations of four more\nquasars from this sample, and the results are presented in \\S4.  In\n\\S5 the RM observations are discussed in the context of unified schemes.\n\nI assume H$_0 = 50$ km s$^{-1}$ Mpc$^{-1}$ and q$_0$=0.5 throughout.\n\n\\section{Sample Selection}\n\nThe sample of target sources was drawn from the 15 GHz VLBA survey of\nKellermann et~al. (1998).  This is a sample of 132 strong, compact AGN\ndrawn from the complete 5 GHz 1 Jansky catalogue of K\\\"uhr et~al.\\\n(1981) as supplemented by Stickel, Meisenheimer \\& K\\\"uhr (1994).  To\nreduce the sample size to a tractable level and facilitate scheduling,\nan initial flux limit of $S_{15}$ $>$ 2 Jy, where $S_{15}$ is the\ntotal 15 GHz flux measured by Kellermann et~al., and declination limit\nof $\\delta > -$10{$^\\circ$}\\ was imposed.  In addition, the somewhat weaker\nquasars 3C\\,380 and 3C\\,395  were also\nobserved.  The final sample consists of 40 objects and is presented in\nTable 1.  In the sample there are 25 quasars, 5 galaxies, 8 \nBL Lacertae objects, and 2 as yet unidentified objects.  Redshifts\nare available for all but 3 sources.  \n\n\\section{Observations}\n\n\\subsection{Data Reduction Procedures}\n\nThe observations, performed on 1998 Aug.\\ 3 (1998.58), were carried\nout at multiple frequencies (or IFs) between 4.6 and 15.2 GHz (see Table 2)\nusing the 10 element VLBA.\\footnote{The National Radio Astronomy\nObservatory is operated by Associated Universities, Inc., under\ncooperative agreement with the National Science Foundation} Both right\nand left circular polarizations were recorded using 1 bit sampling\nacross a bandwidth of 8 MHz.  The VLBA correlator produced 16\nfrequency channels across each IF during every 2 second integration.\n\nAmplitude calibration for each antenna was derived from measurements\nof the antenna gain and system temperatures during each run.  Global\nfringe fitting was performed using the AIPS task FRING, an\nimplementation of the Schwab \\& Cotton (1983) algorithm.  The fringe\nfitting was performed on each IF and polarization independently using\na solution interval of 2 minutes, and a point source model was\nassumed. Next, a short segment of the cross hand data from the\nstrongly polarized calibrator 3C\\,279 (1253{\\tt -}055) was fringe\nfitted in order to determine the right-left delay difference, and the\ncorrection obtained was applied to the rest of the data.  Once delay\nand rate solutions were applied the data were averaged in frequency.\nThe data from all sources were edited and averaged over 20 second\nintervals using {\\caps Difmap} (Shepherd, Pearson \\& Taylor 1994;\nShepherd 1997) and then were subsequently self-calibrated using AIPS\nand {\\caps Difmap} in combination.\n\nNext, the strong, compact calibrator 1638+398 was used to determine\nthe feed polarizations of the antennas using the AIPS task LPCAL.  I\nassumed that the VLBA antennas had good quality feeds with relatively\npure polarizations, which allowed me to use a linearized model to fit\nthe feed polarizations.  Once these were determined, the solutions\nwere applied to observations of 3C\\,279, and to the compact source\n0954+658.  The polarization angle calibration of the VLBA was set\nusing component C4 of 3C\\,279 which has been shown to be stable over\nmany years (Taylor 1998). A small RM of $-$40 radians m$^{-2}$\\ was assumed for\nC4, consistent with the value of $-$64 $\\pm$ 50 radians m$^{-2}$\\ found by Taylor\n(1998).  The absolute electric vector position angle (EVPA)\ncalibration is estimated to be good to about 3{$^\\circ$}\\ at all frequencies\nobserved.  This translates into an absolute uncertainty in the RM \n(determined by observations between 8 and 15 GHz) of\n50 radians m$^{-2}$.  The polarization calibration of the VLBA was verified by\nobserving the compact source 0954+658 with the VLA on 1998 August 4.\nThe flux density scale and polarization angle calibration at the VLA\nwere set using 3C\\,286.  Reasonable agreement was found between the\npolarized flux densities and polarization angles measured with the VLA\nand VLBA (see Table 3).  The discrepancies are consistent with the\nmoderately low signal-to-noise ratio of the observations of 0954+658.\n\n\\section{Results}\n\nIn order to compare images of the total intensity, polarized\nintensity, and polarization angle obtained at different frequencies\nwith a fixed array of antennas, it is necessary to taper the\nobservations made at higher frequencies (and hence higher intrinsic\nresolution) in order to match the resolution of the lowest frequency\nobserved.  Various combinations of images were produced covering the\nranges from 5-8 GHz, 8-15 GHz, and 12-15 GHz. The heavy taper\nrequired to match the 15 GHz to the 5 GHz resolution made this\ncombination of limited utility.  And as the cores were \ngenerally found to depolarize at 5 GHz, all of the results\npresented here have been derived from the 7 frequencies in \nthe range 8--15 GHz (see Table 2).  Faraday Rotation Measures were obtained\nby performing a least-squares fit of a $\\lambda^2$-relation\nto measurements of the polarization angle at each pixel\nin matched-resolution images.  Representative fits are shown\nfor each source in Figures 1--5, and the RM distributions for \neach target sources is shown in Figure 6.\n\nIn the process of fringe-fitting the VLBA data, the brightest\ncomponent of each source (in almost all cases the flat-spectrum core\ncomponent) is shifted to the map center.  If this core component is\nthe location in the jet where the synchrotron self absorption optical\ndepth is unity, then its position will vary with frequency (Blandford\n\\& K\\\"onigl 1979), such that higher frequencies look ``deeper'' down\nthe jet towards the true center of activity.  This frequency dependent\nshift of the core component can be measured by using optically thin\nfeatures in the jet to register simultaneous, multi-frequency\nobservations (see e.g. Lobanov 1996).  The amount of this shift is\nsmall, $< 0.2$ mas between 5 and 15 GHz for 3C\\,345 (Lobanov 1996).\nFor all the observations presented here, the registration of the\nmulti-frequency observations were found to be within 10\\% of the beam\nsize, and no attempt to correct for a core shift was made.\n\n\n\\subsection{4C\\,73.18 (1928+738)}\n\nThis relatively bright, nearby quasar ($V$ = 15.5, $z$=0.3021, 1 mas = 5.52\npc -- Lawrence et~al.\\ 1996) is included in the PR sample (Pearson \\&\nReadhead 1988), and as such was imaged with VLBI in the early 1980s.\nThese first VLBI observations were reported on by Eckart et~al.\\ (1985)\nwho found a southward jet extending 17 mas with superluminal\ncomponents moving at 0.6 mas y$^{-1}$ (14 c) relative to an\nassumed stationary core component.  Hummel et~al.\\ (1992) monitored\n1928+738 in six epochs at 22 GHz between 1985.10 and 1989.71 and found\nrelative superluminal motions in the range 0.3 -- 0.4 mas y$^{-1}$.\nRos et~al.\\ (1999) were able to align three epochs taken at 5 or 8.4\nGHz between 1985.77 and 1991.89 relative to the nearby BL Lac object\n2007+777.  Their astrometry suggests that in some epochs the core\ncomponent is weak or undetected at 8.4 GHz.\n\n\\placefigure{fig1}\n\nCawthorne et~al.\\ (1993) measured the linear polarization of 1928+738\nat 5 GHz (epoch 1984.23) and found a fractional polarization ranging\nfrom a level of 1\\% at the northernmost component to 10\\% in the knots\n$\\sim$10 mas to the south.  At both 8 and 15 GHz (epoch 1998.59) I\nfind 1\\% or less fractional polarization from the northernmost\ncomponent (A) (see Fig.~1 and Table 4).  The fractional polarization\nincreases along the jet to 17\\% at component C, $\\sim$10 mas to the\nsouth.  The RM of component A is high at 2200 radians m$^{-2}$\\ in the rest\nframe of the source.  However the depolarization of 0.3, indicating\nstronger polarization at 8.4 GHz, suggests that frequency dependent\nsubstructure is present which could influence the observed RM.  In the\njet the RMs fall to background levels ($-$32 $\\pm$ 50) and are roughly\nin agreement with the RM of 33 $\\pm$ 4 measured on kiloparsec scales\nwith the VLA by Wrobel (1993).\n\n\\subsection{3C\\,395 (1901+319)}\n\nThis source was identified with a magnitude 17 quasar at $z$=0.635 (1\nmas = 7.75 pc) by Gelderman \\& Whittle (1994).  The parsec to\nkiloparsec scale radio structure has been reported on by Lara et~al.\\ (1997).\nThey find a bright, unresolved core (component A) and a straight jet\nextending 15 mas to the east.  While the inner 15 mas of the jet is straight, \nafter that it appears to bend by 180{$^\\circ$}\\ and form a knotty\njet traveling more than 500 mas in the opposite apparent\ndirection.  Given the likely orientation of the radio source\nclose to the line-of-sight, a large intrinsic bend in\nthe source is not required.  On the arscecond scale, a halo\nis seen which Lara et~al.\\ interpret to to be the radio lobes\nseen in projection against the jet.  \n\n\\placefigure{fig2}\n\nNo VLBI polarimetric observations of 3C\\,395 have been previously\nreported.  I find the core and inner jet component (marked with an A\nin Fig.~2) to be polarized at the $\\sim$1.5\\% level on average, and\nto have an average RM of 300 radians m$^{-2}$ (see Table 4).  There is a strong\ngradient in RM (see Figures 2 and 6), however, such that the western\nend reaches RMs of 1200 radians m$^{-2}$, while the eastern end is consistent\nwith an RM of 0 radians m$^{-2}$. The western end, presumably closest to the\ncenter of activity, has just 1\\% polarization at 8.3 GHz.  The \npoor fits of the RM at the western end are probably related to \nthe low fractional polarization, and blending with the more \nstrongly polarized eastern end.\nThe \npolarization increases toward the eastern end to a maximum at the edge\nof 10\\%.  The $\\sim$16 mas distant component (B), is $\\sim$11\\%\npolarized at 8.3 GHz and has a small RM of 68 $\\pm$ 40 radians m$^{-2}$.  \nThe integrated RM measured by Simard-Normandin et~al.\\ (1981) \nwas 169 $\\pm$ 1 radians m$^{-2}$\\ for this moderately low galactic latitude\n($b = 12${$^\\circ$}) source.\n\n\\subsection{2134+004}\n\nThis magnitude 17 quasar lies at a redshift of 1.94 (1 mas = 8.24 pc)\nand has been observed to vary in the optical by more than 3 magnitudes\n(Gottlieb \\& Liller 1978).  Although variations in total intensity are\nonly $\\sim$15\\% on timescales of years (Aller, Aller \\& Hughes 1994),\nlarge variations were reported for the parsec-scale structure at 10.7\nGHz between 1987 and 1989 requiring apparent velocities of more than\n60 c (Pauliny-Toth et~al.\\ 1990).  These structures are quite\ndifferent, however, from the 15 GHz images reported by Kellermann\net~al.\\ (1998).  Observations presented here support the suggestion by\nKellermann et~al.\\ that the Pauliny-Toth et~al.\\ structures were the\nresult of inadequate sampling of the ($u$,$v$) plane and the close\nproximity of this source to the celestial equator.\n\n\\placefigure{fig3}\n\nNo VLBI polarimetric observations of 2134+004 have been \npreviously reported.  I find two unresolved components\nseparated by 1.7 mas (see Fig.~3).  Component A is the brighter of the\ntwo at 15 GHz and has a flatter spectral index of \n0.0 $\\pm$ 0.1, compared to $-$0.6 $\\pm$ 0.1 for component B.  \nOn the basis of its flatter spectrum I identify\ncomponent A as the core component.  Component A is polarized \nat the $\\sim$5\\% level and exhibits an RM of 1100 radians m$^{-2}$ (see Table 4).\nComponent B is polarized at the $\\sim$3\\% level and has a \nsignificant RM of 340 $\\pm$ 20 radians m$^{-2}$.  After correcting \nfor the effects of the different RM, both components have a\nprojected magnetic field direction near 90{$^\\circ$}\\ (see Fig. 7),\nvery nearly parallel to the jet direction.  \n\nGiven the strong polarized flux density of component B (96 mJy at 15\nGHz), there is some hope that this component may be stable enough to\nuse for absolute electric vector position angle calibration of VLBI\nobservations at frequencies of 8 GHz and higher.  Such usage would be\nsimilar to how these and many other VLBI polarimetric observations\nhave made use of component C4 in 3C\\,279 (described in \\S3.1).  The\nposition on the sky of 2134+004 makes it a possible alternate for\nobservations scheduled in such a way as to make observations of\n3C\\,279 (1253{\\tt -}055) impractical.  One disadvantage of 2134+004 is that\ncomponent B is only 1.7 mas (14 pc from the core component (A)), and\nstill has a considerable observed RM of 340 radians m$^{-2}$ (3000 radians m$^{-2}$\\ in\nthe rest frame of the source).  As an example, an RM of 340\nradians m$^{-2}$\\ causes a rotation of 17{$^\\circ$}\\ between 8.4 and 15 GHz.  Further\nobservations are also required to check on the stability of the\npolarization properties of 2134+004.\n\n\\subsection{CTA\\,102 (2230+114)}\n\nThe highly polarized quasar (HPQ) CTA\\,102 lies at a\nredshift of 1.037 (1 mas = 8.54 pc).  CTA\\,102 has the distinction of\nbeing the first source demonstrated to be variable in the radio\n(Scholomitski 1965).  The short timescale ($\\sim$100 days) of the\nvariations implied a small angular size and a high brightness\ntemperature.  CTA\\,102 was also one of the targets of early VLBI\nobservations by Kellermann et~al.\\ (1968) who confirmed that the source\nflux density was dominated by a compact ($<$4 mas) component.  A\nmulti-wavelength, global VLBI campaign on CTA\\,102 is reported on by\nRantakyr\\\"o et~al.\\ (1996), who find a strong compact component,\npresumed to be the core, and a jet extending 15 mas to the southeast.\nUsing their own observations from 1983 to 1992 and those available in\nthe literature Rantakyr\\\"o et~al.\\ find a range of apparent jet\nvelocities from 0 to 30 $\\pm$ 12 c.  These velocities are at\nthe extreme end of those measured in compact jets (see Vermeulen \\&\nCohen 1994), and given the rather coarse sampling of the VLBI\nobservations, may be the result of misidentifying components.\n\n\\placefigure{fig4}\n\nNo VLBI polarimetric observations of 2230+114 have been previously\nreported.  We find a similar source structure to that seen by\nRantakyr\\\"o et~al.\\ (1996) at 22 GHz, and by Kellermann et~al.\\ (1998)\nat 15 GHz, consisting of a core (component A), and a wiggling\none-sided jet extending approximately 15 mas to the southeast (Fig.~4).  The\nfractional polarization of the core is 1.1\\% at 15 GHz, but only 0.5\\%\nat 8.4 GHz, indicating significant depolarization.  The fractional\npolarization of the inner jet component B is much higher, $\\sim$13\\%\nat both 8.4 and 15 GHz.  The jet fades in total intensity with\nincreasing distance from the core and is polarized at the 5--10\\%\nlevel.  The RMs reach $-$1000 radians m$^{-2}$\\ in the western part of component\nA (Fig.~6).  In component B the polarization angles are well fit by a RM of\n$-90\\,\\pm\\,20$ radians m$^{-2}$ (Fig.~4).  The proximity of the more strongly polarized\ncomponent B,may have partly obscured the\nRM of component A.  The true RM of component A may be considerably\nlarger. In the jet (e.g., component C) the RMs range\nbetween $\\pm$400 radians m$^{-2}$, but are not particularly well fit owing to a\nlow SNR between 12 and 15 GHz.  This is also reflected in the\nanomalously low ratio of $D_{15\/8}$ for component C.  On the large\nscale, Simard-Normandin et~al.\\ (1981) measured an RM of\n$-53\\,\\pm\\,0.4$ radians m$^{-2}$.\n\n\\subsection{Variable rotation measures in 3C\\,279}\n\nThe strong quasar 3C\\,279 was observed briefly in order to use the\nstable jet component C4 to set the absolute \nEVPA.  As mentioned in \\S3.1, the derived corrections were checked\nusing contemporaneous VLBA and VLA observations of the compact source\n0954+658 and found to be within 13{$^\\circ$}\\ at all frequencies.\nComparing these epoch 1998.59 observations to similar\nobservations made in 1997.07 (Paper I), the properties of C4\nappear stable to within $\\sim$10\\% in total intensity and polarized intensity\nand to within 13{$^\\circ$}\\ in position angle.  In contrast, at 15 GHz the core\nof 3C\\,279 increased in total intensity by 14\\%, in polarized intensity\nby 90\\% and decreased in RM by 75\\% (see Fig.~5).  To my knowledge,\nthis is the first observation of variability in the RM of an \nextragalactic radio source.  \n\n\\placefigure{fig5}\n\nTaylor (1998) suggested that 3C\\,279 could be useful for absolute EVPA\ncalibration even at frequencies as low as 2 GHz under the assumption\nthat component C4 dominated the polarized flux density.  Given the\nhighly variable nature of the core polarization properties, one can\nimagine that this assumption may sometimes be invalid, so it is\nprobably unwise to attempt to tie EVPA calibration to C4 in\nobservations where C4 cannot be clearly resolved from the core (e.g.\nbelow 5 GHz with the VLBA).  Also the uncertainty of the determination\nof the RM based on high frequency observations extrapolates to large\nuncertainties in the EVPA of C4 below 8 GHz.\n\n\\section{Discussion}\n\nIt is possible that the RMs measured for the core and inner jet\ncomponents are not produced by a Faraday screen, but have some other\norigin (Taylor 1998 and references therein).  One possibility is a\nfrequency dependent substructure to the inner components.  Certainly\nin the cores of many of these sources higher frequency observations\nhave shown them to be composed of multiple components.  Blending of\nnearby components will produce similar effects at the interface where\nthey merge. It seems unlikely, however, that the changes in EVPA due\nto this substructure, or blending of nearby components, would\nreproduce a $\\lambda^2$-law, especially within the 8 GHz band where\nthe change in frequency is quite modest (see the clear changes in EVPA\nat 8 GHz in 1928+738 shown in Fig.~1).  It is more likely that these\nobservations sample the RM of the component that dominates the\npolarized flux density within the telescope beam.  An analogous, but more extreme, situation is\nfound in single dish observations of extended sources having high RMs\nsuch as Hydra~A, for which Simard-Normandin et~al.\\ (1981) find an RM\nof $-$871 $\\pm$ 8 radians m$^{-2}$\\ whereas detailed imaging of the RM structure\nby Taylor \\& Perley (1993) reveal RMs between $-$12000 and $+$3000\nradians m$^{-2}$.  Higher resolution observations are in progress in order to\nbetter resolve the RM structure in a few of the quasars in this\nsample.  For the remainder of this discussion we proceed with the\nhypothesis that the RMs are the result of a magnetized plasma\nsomewhere along the line-of-sight.\n\n\\placefigure{fig6}\n\\placefigure{fig7}\n\nEvidence is presented for high Faraday Rotation Measures ($|$RM$| >$\n1000 radians m$^{-2}$) near the center of activity in 7 of 8 bright quasars \n(with 3C\\,345 being the only exception).  The\njets of 7 of these quasars have $|$RM$|$s less than 100 radians m$^{-2}$\nbeyond a projected distance from the nucleus of 20 pc --\nsmall enough to be produced by the passage of the radiation through\nthe ISM of our Galaxy.  For the smallest source, 2134+004, which only\nextends over $\\sim$14 pc, the RMs are 1120 and 340 radians m$^{-2}$\\ \ntowards the core and jet component respectively.\n\n\\subsection{A cartoon model and implications for the Unified Scheme}\n\nIn Fig.~8 a simple cartoon schematic for the AGN environment is shown.\nThe details of the equatorial region are based on discussions in Peck,\nTaylor \\& Conway (1999).  According to unified schemes (see for\nexample the review by Antonucci 1993), core-dominated quasars are\nviewed such that the jet axis makes only a small angle to the\nline-of-sight, and they therefore exhibit one-sided jets, apparent\nsuperluminal motions, and broad optical emission lines.  While jet\ncomponents are within $\\sim$100 pc of the center of activity\nthey are viewed through ionized gas which acts as a Faraday screen.\nOnce the jet components move farther from the nuclear environment the\nRM rapidly drops.\n\n\\placefigure{fig8}\n\nAn observer looking at an AGN nearly edge-on through the denser,\nmulti-phase disk would likely see a galaxy with narrow optical\nemission lines and symmetric parsec-scale radio structures.\nThe much smaller ($\\sim$1 pc) broad line region is hidden from\nview by the molecular disk.  While the\ncores of lobe-dominated FR II radio galaxies (generally unified with\nquasars) have not yet been observed with VLBI polarimetry, there is\nanother class of bright sources known as Compact Symmetric Objects\nthat are likely to be youthful versions of FR II radio galaxies\n(Readhead et~al.\\ 1996).  For the CSOs, which often have intrinsic\nsizes less than $\\sim$100 pc, all the components will be viewed\nthrough a dense multi-phase medium and extremely high RMs ($>>$ 50000\nradians m$^{-2}$) are therefore to be expected if the model shown in Fig.~8 is\ncorrect.  Unfortunately such extreme RMs will depolarize the radio\nsource and hence cannot be measured directly.  Recent VLBA polarimetry\nof 20 CSOs by Peck \\& Taylor (1999) place typical limits on the\nfractional polarization in CSOs of $<$1\\%.  No polarized flux on the\nparsec-scale has ever been reported for a CSO.\n\nAn observer looking at an AGN even closer to the jet axis than the\noptimal angle for apparent superluminal motions (1\/$\\gamma$, where\n$\\gamma$ is the Lorentz factor of the jet) would see slower motions,\nand greater Doppler beaming.  BL Lac objects are even more highly core\ndominated than quasars.  There is some evidence that BL Lac objects\nhave smaller apparent motions than quasars (Gabuzda et~al.\\ 1994,\nBritzen et~al.\\ 1999).  BL Lac objects have also been observed to have\nmore strongly polarized cores (Gabuzda et~al.\\ 1992) compared to quasar\ncores.  This might be achieved if the relativistic jet evacuates a\ncone through the ionized gas in the nuclear region such that cores of\nBL Lacs are not viewed through a Faraday screen (see Fig.~8).  In fact\nfrom lower resolution studies we already know that the BL Lac cores\n(which dominate over the jet in integrated polarization) must have\nmoderate to low RMs (Rusk 1988; Gabuzda et~al.\\ 1992).  Future studies\nof the RM distribution of BL Lac objects on the parsec scale will be\nof interest to directly confirm these low RMs.\n\n\\placefigure{fig9}\n\nIf the RM observed in the core and inner jets is dependent on the\norientation of the source as suggested above, then it might be\nexpected to correlate with another orientation dependent parameter,\nsuch as $R$ -- the ratio of the core flux density to the lobe flux\ndensity at an emitted frequency of 5 GHz.  Here the term ``core flux\ndensity'' refers to the arcsecond-scale core which includes all the\nVLBI components.  Figure 9 shows the RM plotted against $R$ for the 8\nquasars studied in this paper and in paper I.  Although 3C\\,345 has\nboth the lowest RM and the highest measured $R$ value, the results for\nthe sample as a whole are inconclusive.  One oddity is the high RM,\nbut lack of any kpc-scale extended emission around 2134+004, even\nthough the core component is not dominant on the parsec-scale like it\nis for the rest of the sources shown.  Indeed computing $R$ on the\nbasis of the parsec-scale emission ($-$0.5) would shift this point to\nthe top-left corner of the plot.  I suggest that the lack of extended\nemission for 2134+004 is due to an intrinsically small size.  If\n2134+004 is viewed at a larger angle to the line-of-sight then one\nalso expects much slower apparent motion in the jet.  Slow motion has\nbeen confirmed in 2134+004 ($<$4 c -- K. Kellermann, private\ncommunication), in sharp contrast to the average speeds of 12 -- 14 c\nreported for 3C\\,345, 3C\\,380 and 3C\\,273 (Vermeulen \\& Cohen 1994 and\nreferences therein).\n\n\n\n\\section{Conclusions}\n\nA sample of 40 strong, compact AGN is identified for a multi-frequency\npolarimetric study.  These observations allow imaging of the\nFaraday Rotation Measure distribution on the parsec scale.  This is a\nrelatively new way of probing the AGN environment.  Rest frame Faraday\nRotation Measures in excess of 1000 radians m$^{-2}$\\ are found in the nuclear\nregions of 7 out of the 8 quasars studied so far.  In all cases the\nhigh RMs are confined to the nuclear region within a projected\ndistance of 20 pc of the center of activity.  The high RMs\nseen in these sources probably originate in the same region that\nproduces the narrow optical emission lines.  There is some indication\nthat higher nuclear RMs are seen when an AGN is viewed at larger\nangles to the line of sight, and that correspondingly the lowest\nnuclear RMs are seen when the line-of-sight is closely aligned to the\njet axis.  These findings are consistent with the predictions of\nunified schemes, but the details of the correlation of the nuclear\nRM with other orientation-dependent observables are as yet unclear.\nFurther multi-frequency polarimetric observations are needed for the\nremainder of the sample.\n\n\\acknowledgments\n\nI am grateful to Alison Peck and Ken Kellermann for many useful\ndiscussions and to Alison Peck again for her help making Fig.~8. \nI thank the referee, John Wardle, for a thorough review including\nnumerous constructive suggestions.  This\nresearch has made use of data from the University of Michigan Radio\nAstronomy Observatory which is supported by the National Science\nFoundation and by funds from the University of Michigan.  This\nresearch has made use of the NASA\/IPAC Extragalactic Database (NED)\nwhich is operated by the Jet Propulsion Laboratory, Caltech, under\ncontract with NASA.\n \n\\clearpage\n\n\\renewcommand{\\baselinestretch}{1.2}\n\\normalsize\n\\begin{center}\nTABLE 1 \\\\\nVLBA R{\\sc otation} M{\\sc easure} I{\\sc maging}\n\\end{center}\n\\begin{center}\n\\begin{tabular}{l l c r r r r r r}\n\\hline\n\\hline\nSource & Name & ID & Mag & z & $S_{15}$ \\\\\n(1) & (2) & (3) & (4) & (5) & (6) \\\\\n\\hline\n\\noalign{\\vskip2pt}\n0133{\\tt +}476 &   DA55 & Q  & 18.0 & 0.86 & 2.22 \\\\\n0202{\\tt +}149 &        & G? & 22.1 & -    & 2.29 \\\\\n0212{\\tt +}735 &        & BL & 19.0 & 2.37 & 2.69 \\\\\n0336{\\tt -}019 &  CTA26 & Q  & 18.4 & 0.85 & 2.23 \\\\\n0355{\\tt +}508 &        & EF & -    &  -   & 3.23 \\\\\n0415{\\tt +}379 &  3C111 & G  & 18.0 & 0.05 & 5.98 \\\\\n0420{\\tt -}014 &        & Q  & 17.8 & 0.92 & 4.20 \\\\\n0430{\\tt +}052 &  3C120 & G  & 14.2 & 0.03 & 3.01 \\\\\n0458{\\tt -}020 &        & Q  & 18.4 & 2.29 & 2.33 \\\\\n0528{\\tt +}134 &        & Q  & 20.0 & 2.06 & 7.95 \\\\\n0552{\\tt +}398 &  DA193 & Q  & 18.0 & 2.37 & 5.02 \\\\\n0605{\\tt -}085 &        & Q  & 18.5 & 0.87 & 2.80 \\\\\n0736{\\tt +}017 &        & Q  & 16.5 & 0.19 & 2.58 \\\\\n0748{\\tt +}126 &        & Q  & 17.8 & 0.89 & 3.25 \\\\\n0923{\\tt +}392 & 4C39.25 & Q & 17.9 & 0.70 & 10.84 \\\\\n1055{\\tt +}018 &        & BL & 18.3 & 0.89 & 2.15 \\\\\n1226{\\tt +}023 & 3C273  & Q & 12.9 & 0.16 & 25.72 \\\\\n1228{\\tt +}126 &  M87 & G & 9.6 & 0.00 & 2.40 \\\\\n1253{\\tt -}055 & 3C279  & Q & 17.8 & 0.54 & 21.56 \\\\\n1308{\\tt +}326 &        & BL & 19.0 & 1.00 & 3.31 \\\\\n\\end{tabular}\n\\end{center}\n\\clearpage\n\\begin{center}\n{\\sc Table 1 Continued}\\\\\n\\smallskip\n\\begin{tabular}{l l c r r r r r r}\n\\hline\n\\hline\nSource & Name & ID & Mag & z & $S_{15}$ \\\\\n(1) & (2) & (3) & (4) & (5) & (6) \\\\\n\\hline\n\\noalign{\\vskip2pt}\n1546{\\tt +}027 &        & Q & 18.0 & 0.41 & 2.83 \\\\\n1548{\\tt +}056 &        & Q & 17.7 & 1.42 & 2.83 \\\\\n1611{\\tt +}343 &  DA406 & Q & 17.5 & 1.40 & 4.05 \\\\\n1641{\\tt +}399 & 3C345 & Q & 16.0 & 0.59 & 8.48 \\\\\n1741{\\tt -}038 &       & Q & 18.6 & 1.05 & 4.06 \\\\\n1749{\\tt +}096 &       & BL & 16.8 & 0.32 & 5.58 \\\\\n1803{\\tt +}784 &       & BL & 17.0 & 0.68 & 2.05 \\\\\n1823{\\tt +}568 &       & BL & 18.4 & 0.66 & 2.31 \\\\\n1828{\\tt +}487$^1$ & 3C380 & Q & 16.8 & 0.69 & 1.82 \\\\\n1901{\\tt +}319 & 3C395 & Q & 17.5 & 0.64 & 1.09 \\\\\n1928{\\tt +}738 &       & Q & 16.5 & 0.30 & 3.04 \\\\\n2005{\\tt +}403 &       & Q & 19.5 & 1.74 & 2.51 \\\\\n2021{\\tt +}317 &       & EF & - & - &  2.02 \\\\\n2021{\\tt +}614 &       & G & 19.5 & 0.23 & 2.21 \\\\\n2134{\\tt +}004 &       & Q & 16.8 & 1.93 & 5.51 \\\\\n2200{\\tt +}420 & BL Lac & BL & 14.5 & 0.07 & 3.23 \\\\\n2201{\\tt +}315 &       & Q & 15.5 & 0.30 & 3.10 \\\\\n2223{\\tt -}052 & 3C446 & BL & 17.2 & 1.40 & 3.92 \\\\\n2230{\\tt +}114 & CTA102 & Q & 17.3 & 1.04 & 2.33 \\\\\n2251{\\tt +}158 & 3C454.3 & Q & 16.1 & 0.86 & 8.86 \\\\\n\\hline\n\\end{tabular}\\\\\n\\medskip\n{\\sc Notes to Table 1}\\\\\n\\end{center}\n\\vspace{-0.1cm}\n$^1$ Not in the Kellermann et al.\\ (1998) survey.\nCol.(1).---B1950 source name.\nCol.(2).---Alternate common name.\nCol.(3).---Optical magnitude.\nCol.(4).---Optical identification from the literature (NED). Key \nto identifications: Q--quasar; G--galaxy; BL--BL Lac object; EF--empty \nfield.\nCol.(5).---Redshift.\nCol.(6).---Total flux density at 15 GHz measured by Kellermann et~al.\\\n(1998), or in the case of 3C\\,380 by Taylor (1998).\n\\smallskip\n\\clearpage\n\n\\begin{center}\nTABLE 2 \\\\\n\\smallskip\nVLBA O{\\sc bservational} P{\\sc arameters}\n\\smallskip\n \n\\begin{tabular}{l r r r r r r r r}\n\\hline\n\\hline\nSource & Frequency & BW & Scans & Time \\\\\n(1) & (2) & (3) & (4) & (5) \\\\\n\\hline\n\\noalign{\\vskip2pt}\n0954+658 & 4.616, 4.654, 4.854, 5.096 & 8 & 3 & 10 \\\\ \n         & 8.114, 8.209, 8.369, 8.594 & 8 & 3 & 10 \\\\\n         & 12.115, 12.591 & 16 & 2 & 10 \\\\\n         & 15.165 & 32 & 2 & 10 \\\\\n3C\\,279 & 4.616, 4.654, 4.854, 5.096 & 8 & 2 & 8 \\\\ \n       & 8.114, 8.209, 8.369, 8.594 & 8 & 2 & 7 \\\\\n       & 12.115, 12.591 & 16 & 2 & 7 \\\\\n       & 15.165 & 32 & 2 & 7 \\\\\n1638+398 & 4.616, 4.654, 4.854, 5.096 & 8 & 7 & 25 \\\\ \n         & 8.114, 8.209, 8.369, 8.594 & 8 & 7 & 25 \\\\\n         & 12.115, 12.591 & 16 & 7 & 25 \\\\\n         & 15.165 & 32 & 7 & 25 \\\\\n1928+738 & 4.616, 4.654, 4.854, 5.096 & 8 & 8 & 28 \\\\ \n       & 8.114, 8.209, 8.369, 8.594 & 8 & 8 & 28 \\\\\n       & 12.115, 12.591 & 16 & 8 & 28 \\\\\n       & 15.165 & 32 & 8 & 28 \\\\\n3C\\,395 & 4.616, 4.654, 4.854, 5.096 & 8 & 7 & 25 \\\\ \n       & 8.114, 8.209, 8.369, 8.594 & 8 & 7 & 25 \\\\\n       & 12.115, 12.591 & 16 & 7 & 25 \\\\\n       & 15.165 & 32 & 7 & 25 \\\\\n2134+004 & 4.616, 4.654, 4.854, 5.096 & 8 & 7 & 25 \\\\ \n       & 8.114, 8.209, 8.369, 8.594 & 8 & 7 & 25 \\\\\n       & 12.115, 12.591 & 16 & 7 & 25 \\\\\n       & 15.165 & 32 & 7 & 25 \\\\\n\\end{tabular}\n\\end{center}\n\\smallskip\n\\clearpage\n\n\\begin{center}\nTABLE 2 Continued\\\\\n\\smallskip\n\\begin{tabular}{l r r r r r r r r}\n\\hline\n\\hline\nSource & Frequency & BW & Scans & Time \\\\\n(1) & (2) & (3) & (4) & (5) \\\\\n\\hline\n\\noalign{\\vskip2pt}\n2230+114 & 4.616, 4.654, 4.854, 5.096 & 8 & 7 & 25 \\\\ \n       & 8.114, 8.209, 8.369, 8.594 & 8 & 7 & 25 \\\\\n       & 12.115, 12.591 & 16 & 7 & 25 \\\\\n       & 15.165 & 32 & 7 & 25 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\begin{center}\n{\\sc Notes to Table 2}\n\\end{center}\nCol.(1).---Source name.\nCol.(2).---Observing frequency in GHz.\nCol.(3).---Total spanned bandwidth in MHz.\nCol.(4).---Number of scans (each 2 -- 4 minutes duration).\nCol.(5).---Total integration time on source in minutes.\n\\bigskip\n \n\\begin{center}\n\\clearpage \nTABLE 3 \\\\\n\\smallskip\nP{\\sc olarization} A{\\sc ngle} C{\\sc alibration}\n\\smallskip\n \n\\begin{tabular}{l r r r r r r r r r}\n\\hline\n\\hline\nSource & $\\nu$ & $S_{\\rm VLA}$ & $S_{\\rm VLBA}$ & $P_{\\rm VLA}$ &\n$P_{\\rm VLBA}$ & $\\chi_{\\rm VLA}$ & $\\chi_{\\rm VLBA}$ \\\\\n(1) & (2) & (3) & (4) & (5) & (6) & (7) & (8) \\\\\n\\hline\n\\noalign{\\vskip2pt}\n0954+658 & 4.7 & 0.33 & 0.29 & 13 & 8.4 & $-$12 & $-$25 \\\\\n         & 8.4 & 0.34 & 0.34 & 10 & 5.5 & $-$22 & $-$29 \\\\\n         & 15.2 & 0.32 & 0.32 & 13 & 17 & $-$15 & $-$28 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\smallskip\n\\begin{center}\n{\\sc Notes to Table 3}\n\\end{center}\nCol.(1).---Source name.\nCol.(2).---Observing band in GHz.\nCol.(3).---Integrated VLA flux density in Jy.\nCol.(4).---Integrated VLBA flux density in Jy.\nCol.(5).---Integrated VLBA polarized flux density in mJy.\nCol.(6).---Integrated VLA polarized flux density in mJy.\nCol.(7).---VLA polarization angle (E-vector) in degrees.\nCol.(8).---VLBA peak polarization angle (E-vector) in degrees.\n\\bigskip\n \n\\begin{center}\n\\clearpage \nTABLE 4 \\\\\n\\smallskip\nC{\\sc omponent} P{\\sc ositions,} F{\\sc lux} D{\\sc ensities, and},\nR{\\sc otation} M{\\sc easures}\n\\smallskip\n \n\\begin{tabular}{l r r r r r r r r r r r r r }\n\\hline\n\\hline\nComp & $r$ & $I_{\\rm 15}$ & $P_{\\rm 15}$ & $m_{\\rm 15}$ & $\\chi_{\\rm 15}$ &\n       $I_{\\rm 8}$ & $P_{\\rm 8}$ & $m_{\\rm 8}$ & $\\chi_{\\rm 8}$ &\n        RM & BPA & $D_{\\rm 15\/8}$ \\\\\n(1) & (2) & (3) & (4) & (5) &\n      (6) & (7) & (8) & (9) & \n      (10) & (11) & (12) & (13) \\\\\n\\hline\n\\noalign{\\vskip2pt}\n1928+738 \\\\\nA & 0.0 & 2310 & 6.0 & 0.3 & $-$29 & 1970 & 19 & 1.0 & $-$86 & $-$1300 & 78 & 0.3 \\\\\nB & 2.6 & 420 & 36.3 & 8.6 & 51 & 480 & 44 & 9.2 & 48 & $-$140 & 32 & 0.9 \\\\\nC & 11.1 & 17 & 3.0 & 17 & 79 & 24 & 4.0 & 17 & 85 & $-$32 & 6 & 1.0 \\\\\n3C\\,395 \\\\\nA & 0.0 & 890 & 13.0 & 1.5 & 19 & 834 & 13.2 & 1.6 & 30 & 300 & $-$81 & 0.9 \\\\\nB & 15.9 & 51 & 2.6 & 5.1 & 4.9 & 84 & 9.6 & 11.4 & 7.0 & 68 & $-$89 & 0.5 \\\\\n2134+004 \\\\\nA & 0.0 & 3170 & 183 & 5.8 & 14.5 & 3240 & 140 & 4.3 & 77 & 1120 & 84 & 1.3 \\\\\nB & 1.7 & 2610 &  96 & 3.7 &  3.5 & 3810 & 103 & 2.7 & 24 & 340 & 89 & 1.4 \\\\\n2230+114 \\\\\nA & 0.0 & 4490 & 48.1 & 1.1 & 49    & 2740 & 13.1 & 0.5 & 15 & $-$610  & $-$26 & 2.2 \\\\\nB & 2.3 & 580 & 75.6 & 13.0 & 85    & 710 & 97.6 & 13.7 & 82 & $-$90 & 0 & 0.9 \\\\\nC & 7.4 & 200 & 7.4 &  3.7 & 61     & 310 & 20.6 & 6.6 & 75 & $-$185 & $-$1 & 0.6 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\smallskip\n\\begin{center}\n{\\sc Notes to Table 4}\n\\end{center}\nCol.(1).---Component name.\nCol.(2).---Distance from the core in mas.\nCol.(3).---15.165 GHz total intensity in mJy\/beam.\nCol.(4).---15.165 GHz polarized intensity in mJy\/beam.\nCol.(5).---15.165 GHz fractional polarization in \\%.\nCol.(6).---15.165 GHz electric vector polarization angle in degrees.\nCol.(7).---8.369 GHz total intensity in mJy\/beam.\nCol.(8).---8.369 GHz polarized intensity in mJy\/beam.\nCol.(9).---8.369 GHz fractional polarization in \\%.\nCol.(10).---8.369 GHz electric vector polarization angle in degrees.\nCol.(11).---Observed Faraday Rotation Measure in rad m$^{-2}$ derived\nfrom a linear least-squares fit to the polarization angle measurements\nbetween 8.114 and 15.165 GHz as discussed in the text.  If generated\nin the rest frame of the source these values are larger by a factor of\n$(1+z)^2$.\nCol.(12).---Magnetic Field polarization angle corrected for RM.\nCol.(13).---Depolarization ($m_{15}\/m_{8}$).\n\\bigskip\n\n\\renewcommand{\\baselinestretch}{1.5}\n\\normalsize\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\IEEEPARstart{H}{ardware} implementation of biologically inspired neuromorphic systems has gained a lot of interest in the past few years \\cite{IBM_PCM,ICM_Truenorth}. Neuromorphic engineering aims to achieve intelligent computing by taking inspiration from complex neural\/synaptic circuits and their biophysical mechanisms, using nano-electronic\/magnetic devices \\cite{b1,b2}. The nanodevices when made to follow certain conductance (weight) modulation rules within a connected network lead to equivalent trained neural networks that may be used for different training and inference tasks \\cite{b4,b8}. Various emerging non-volatile nanodevices are currently being investigated for synaptic applications; Conductive-Bridge Memory (CBRAM) \\cite{b4}, Phase Change Memory (PCM) \\cite{b5}, Magnetic-Tunnel Junction (MTJ) \\cite{b6}, domain walls \\cite{sengupta2016hybrid} and oxide based resistive switching memory (OxRAM) \\cite{b7,b8}. Among emerging spintronic nanodevices, magnetic skyrmions have gained a significant amount of interest owing to their small size, topological stability and ultralow current densities \\cite{b9,b25}. Skyrmions are topologically stable spin textures which have been experimentally observed in Heavy metal (HM)-Ferromagnetic Metal (FM) heterostructure and bulk ferromagnets \\cite{b24,b23}. They have been proposed as possible post-Moore candidates for logic \\cite{b18}, \\cite{b17} and storage \\cite{b19} applications. They have also been proposed for emulating the properties of leaky-integrate and fire neurons in \\cite{li2017magnetic,chen2018compact}. However for a device to behave as analog (multilevel) synapse, it should have multiple programmable non-volatile conductance states \\cite{b7}. Controlled conductance variation using skyrmions on nanotracks was first reported in \\cite{b26}. However in \\cite{b26} the conductance states reported were a strong function of the inter programming-pulse delay, thus making the intermediate synaptic weights transient or not truly non-volatile. Also to the best of our knowledge no work has yet shown a pathway for implementing online unsupervised learning with multilevel skyrmion on nanotrack synapses.\n\nIn this work we present a modified nanotrack structure which makes the overall synapse a 4-terminal device and completely decouples the spike transmission and programming paths. This simplifies the synaptic circuit and results in a 1T1N like configuration. We then study in detail the impact of two different programming parameters: pulse width and pulse amplitude, on the number of distinct conductance levels and energy per transition of the proposed Skyrmion on Nanotrack (Sk-N) synapse. Using this optimized nanotrack and programming parameters we propose a '1 Transistor\/1 nanotrack' (1T1N) Skyrmion on Nanotrack (Sk-N) based synaptic bit-cell that changes the conductance of the synapse according to a modified version of the biological unsupervised STDP learning rule \\cite{ambrogio2016neuromorphic}. We also design peripheral subthreshold CMOS neuron circuit comprising of following blocks: current scaling circuit, Differential Pair Integrator adopted from \\cite{qiao2015reconfigurable,livi2009current}, comparator and a custom spike generator unit. The current scaling circuit used for this work is based on the Gilbert normalizer circuit \\cite{liu2002analog} and was recently shown to work with two-terminal differential memristive synapses \\cite{nair2017differential}. In \\cite{nair2017differential}, the circuit was included inside each synapse bit-cell. In this work, we show how the same circuit can be used effectively for 4-Terminal non-differential Sk-N based synapses by coupling it with each post-neuron rather than the synaptic bit-cells, thus saving silicon footprint. The spike generator unit is able to generate custom pre and post-neuronal spikes that lead to the modified STDP learning rule. On combining all the circuits we demonstrate spike transmission, neuronal integration and simplified STDP based conductance modulation in a single synaptic bit-cell through circuit simulations. The circuit simulations were done in CADENCE Spectre, using TSMC 65 nm technology node PDK. The characteristics of our 1T1N synaptic bit-cell and programming circuit were used in system-level simulations to demonstrate online unsupervised learning in a Spiking Neural Network (SNN) in the MATLAB based neuromorphic hardware simulator, MASTISK \\cite{mastisk}. The task involved pattern recognition in noisy video streams and multi-class classification of handwritten digits (MNIST).\n\n\n\\section{Skyrmion on Nanotrack}\n\n\\subsection{Skyrmion Physics}\n\nThe creation of a skyrmion can be attributed to the competition between ferromagnetic exchange coupling and Dzyaloshinskii\u2013Moriya interaction (DMI) in magnetic systems. These systems have breaking or lacking inversion symmetry in bulk lattices \\cite{b23} or at the interface of thin films \\cite{b24} respectively. The DMI interaction between two neighbouring atomic spins in a lattice having large Spin - Orbit Coupling (SOC) is give by:\n\\begin{align}\n\\label{eq:DMI}\n\\boldsymbol{H} = &-\\boldsymbol{D} . (\\boldsymbol{S_{i}}\\times\\boldsymbol{S_{j}}),\n\\end{align}\n\nwhere D denotes the DMI vector, and S$_{i}$ and S$_{j}$ are the\nspins on site i and j , respectively. In this work we are mainly concerned with skyrmions created in interface of ultrathin films (also known as nanotracks) comprising of a Ferromagnetic-Metal (FM) layer deposited on a Heavy-Metal (HM) layer. In such systems spin polarised current is used to disturb the magnetization equilibrium of the metal layer and induce topological transition of the magnetic textures. This when facilitated by DMI leads to creation of skyrmions. Skyrmions on nanotracks can be moved by either Current-In-Plane (CIP) or Current-Perpendicular-to-Plane (CPP). In this work the latter case is implemented due to its energy efficiency \\cite{sampaio2013nucleation}. According to this method a charge current flowing through the HM layer induces a vertical spin current due to Spin-Hall Effect (SHE) which is responsible for moving the skyrmions in the FM layer by exerting spin torque. Due to the non-linear spin texture of skyrmions, there will be definite change in the site-dependent spin mixing of magnetic states in the ferromagnetic environment. This can be detected using Tunneling Magneto Resistance (TMR) measurements \\cite{gould2004tunneling}.\n\n\n\\subsection{Skyrmion on Nanotrack Synapse}\n\nThe concept of skyrmion on nanotrack synapse used for this work is shown in Fig. \\ref{skyrmion1}. It has ultrathin metallic films comprising of a Ferromagnetic Metal (FM) layer deposited on top of a Heavy Metal (HM) layer. The FM layer has Perpendicular Magnetic Anisotropy (PMA) in the direction specified by $\\hat{y}$ (Fig. \\ref{skyrmion1}). A metallic capping layer with a stronger PMA than the FM layer is used as an energy barrier for the skyrmions, thus seperating the nanotrack into two parts: pre-synapse and post-synapse. Please note that the terminologies pre-synapse and post-synapse have been adopted from \\cite{b26}. They just refer to the two sides of the synapse separated by the PMA barrier and have no relation with pre-neuron and post-neuron. MTJ like structures are used at either ends of the nanotrack for nucleating (write-MTJ) and reading (read-MTJ) skyrmions respectively. The write-MTJ (pinned layer is the portion of FM layer below it) is used to introduce spin-polarised current into the pre-synapse region in order to create skyrmions. The read-MTJ has its free layer separated from the FM layer with an insulating magnetic material that allows magnetic coupling but prevents current flowing across it. The pinned and free layers of the Read-MTJ are separated by non-magnetic spacer. This decouples the read and write\/program paths of the nanotrack completely.\nThe benefit obtained by doing so is discussed in Section \\ref{2t1n}. The read current (I$_{Syn}$) flowing through the write-MTJ (shown in Fig. \\ref{skyrmion1}) will be proportional to the conductance that is determined by the spin configurations of the post-synaptic FM layer and the free and pinned layers of the read-MTJ. Since the spin direction of the FM layer is opposite to that of the pinned layer of read-MTJ, it is in Anti-parallel (AP) state in the absence of skyrmions. As the net spin of a skyrmion in the $\\hat{y}$ direction is zero, its presence in the post-synaptic region is equivalent to removal of a portion of the FM layer along with its spin. This leads to a less AP nature of the read-MTJ and therefore an increase in conductance. Therefore higher the number of skyrmions in the post-synapse, higher is the conductance of the post-synapse and the corresponding read current flowing through it. Programming current I$_{P}$ (charge current) when passed through the HM layer, the spin current caused by it as a consequence of Spin Hall Effect (SHE) is responsible for moving the skyrmions from one end of the nanotrack to the other. \n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=\\columnwidth]{skyr1.png}\n\\caption{Schematic of proposed skyrmion on nanotrack based synaptic device with completely decoupled read and program paths.}\n\\label{skyrmion1}\n\\end{figure}\n\n\\subsection{Micro-Magnetic Simulation Framework}\n\nThe skyrmion-based artificial synaptic device was numerically simulated by 3D magnetization dynamics in the Object Oriented MicroMagnetic Framework (OOMMF) software \\cite{b22} which used the extended Landau-Lifshitz-Gilbert equation including spin transfer torques as follows \\cite{wiesendanger2016nanoscale}:\n\n\\begin{align}\n\\label{eq:LLGS-CPP}\n\\frac{d\\boldsymbol{m}}{dt} = &-\\left|\\gamma_{0}\\right|\\boldsymbol{m}\\times\\boldsymbol{h}_{\\text{eff}}+\\alpha(\\boldsymbol{m}\\times\\frac{d\\boldsymbol{m}}{dt}) \\\\ \\notag\n&+\\frac{u}{t}\\boldsymbol{m}\\times (\\boldsymbol{p}\\times \\boldsymbol{m}),\n\\end{align}\n\nwhere $\\gamma$ is the gyromagnetic ratio, $\\boldsymbol{m}$ =$\\frac{\\boldsymbol{M}}{\\boldsymbol{M_{S}}}$ is the reduced magnetization, $\\boldsymbol{h}_{eff}$ = $\\frac{\\boldsymbol{H}_{eff}}{\\boldsymbol{M_{S}}}$ is the reduced effective field, $\\alpha$ is the Gilbert Damping Factor, $u=|\\frac{\\gamma_{0}\\hbar}{\\mu_{0}e}|\\frac{jP}{2M_{\\text{S}}}$, $\\hbar$ is the reduced Planck constant, $j$ is the applied current density, $P=0.6$ is the spin polarization, $\\mu_{0}$ is the permeability of free space, $e$ is the electron charge, $\\boldsymbol{p}$ is the unit spin polarization direction and $\\boldsymbol{p}=-\\hat{y}$ was set for driving the skyrmions. The parameters were extracted from experiments \\cite{wiesendanger2016nanoscale} for a 0.4 nm thick Co layer on a Pt substrate.\n\nIn order to calculate the conductance of the post-synaptic region Tunneling Magneto Resistance (TMR) has been employed. This is done by dividing the read - MTJ into multiple 2 nm $\\times$ 2 nm cells in the x-y plane and calculating the conductance of each cell by Julliere's model \\cite{julliere1975tunneling}: \n\n\\begin{align}\n\\label{eq:julliere1}\nG = G_{0}(\\frac{1+p^{2}\\cos\\theta}{1+p^{2}}) \\\\ \\notag\n   putting \\ \\theta \\Rightarrow \\pi - \\theta \\\\ \n\\label{eq:julliere2}  = G_{0}(\\frac{1-p^{2}\\cos\\theta}{1+p^{2}}),\n\\end{align}\n\nwhere G$_{0}$ is the conductance when the magnetization is perfectly\nparallel to the reference layer, p is the spin polarization and $\\theta$ is the magnetization of each cell with respect to the reference layer. However since in this work the spins of the reference layer of MTJ and FM layer are oppositely oriented, $\\theta$ is replaced by $\\pi - \\theta$ leading to the form shown in Eq. \\ref{eq:julliere2}. As a result the introduction of skyrmions in the post-synapse leads to $\\theta$ being less than $\\pi$ which gives a higher value of G according to Eq. \\ref{eq:julliere2}. \n\n\n\\begin{table}[htbp]\n  \\centering\n  \\caption{Device parameters for the HM-FM heterostructure}\n    \\begin{tabular}{|c|c|p{8.215em}|}\n    \\hline\n    Parameter & Description & \\multicolumn{1}{c|}{Value} \\\\\n    \\hline\n    M$_{s}$    & Saturation Magnetization & \\multicolumn{1}{c|}{580 kA\/m} \\\\\n    \\hline\n    A     & Exchange Constant & \\multicolumn{1}{c|}{15 pJ\/m} \\\\\n    \\hline\n    D     & DMI Factor & \\multicolumn{1}{c|}{3 mJ\/m2} \\\\\n    \\hline\n    $\\alpha$ & Gilbert Damping Factor & \\multicolumn{1}{c|}{0.3} \\\\\n    \\hline\n    K     & Magnetic anisotropy & \\multicolumn{1}{c|}{0.8 MJ\/m$^{3}$} \\\\\n    \\hline\n    P     & Spin Polarization & \\multicolumn{1}{c|}{0.6} \\\\\n    \\hline\n    t$_{f}$    & Thin Film Thickness & \\multicolumn{1}{c|}{1 nm} \\\\\n    \\hline\n    l $\\times$ w & Nanotrack Length and Width & 820 nm $\\times$ 280 nm \\\\\n    \\hline\n    $\\rho$  & Resistivity of HM layer & 100 $\\mu$$\\Omega$cm \\cite{b27} \\\\\n    \\hline\n    \\end{tabular}%\n  \\label{tab:device_params}%\n\\end{table}%\n\n\\section{Nanotrack Programming parameters}\n\\label{opt}\nIn the nanotrack simulations it was found that when skyrmions try to flow across the PMA barrier, there exists a competition between; driving force due to current, repulsive force due to nanotrack edge, PMA barrier and inter-skyrmion (Sk-Sk) interactions. If repulsive and driving forces are not optimized it leads to leaky movement of skyrmions across the barrier after the programming pulses are removed. This makes the conductance levels dependent on the inter-pulse delay as was the case in \\cite{b26} and thus cannot be used for stable temporal long-term (LTP\/LTD) non-volatile synaptic emulation. Each skyrmion crossing over to the post-synapse region increases the conductance of the read path. Thus, maximum number of conductance levels that can be achieved is the total number of skyrmions nucleated in pre-synapse. If multiple skyrmions enter the post-synapse under the influence of a single pulse then the effective number of distinct conductance states is reduced. The maximum number of stable skyrmions that are created during nucleation depends on the width of nanotrack \\cite{b26}. Therefore the number of conductance levels is proportional to the width of the nanotrack. Increasing the length, allows more room for sequential movement of skyrmions in response to driving currents. However a large width and length would also incur synaptic area overheads. Therefore dimension of the nanotrack was fixed at: 820 nm $\\times$ 280 nm $\\times$ 1 nm. It allowed nucleation of 17 skyrmions in the pre-synapse during initiation. The spin polarisation value was set to 0.6 so as to improve the control of driving current on the movement of skyrmions across the PMA barrier. The micromagnetic simulation parameters are listed in Table \\ref{tab:device_params}. \n\n\\begin{figure}[!t]\n\\centering\n\\begin{subfigure}{0.85\\linewidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{opt_curr_den_1.png}\n  \\caption{}\n  \\label{fig:sub1}\n\\end{subfigure}\n\\begin{subfigure}{0.85\\linewidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{opt_duty_cycle_1.png}\n  \\caption{}\n  \\label{fig:sub2}\n\\end{subfigure}\n\\caption{Variation of number of conductance states and energy per conductance state transition with (a) current density and (b) pulse width of programming pulse. The red circle represents the programming parameter value chosen for other analysis in this work.}\n\\label{fig:master_opt}\n\\end{figure}\n\n\n\\begin{figure}[!b]\n\\centering\n\\includegraphics[scale = 0.40]{complete_contrast.png}\n\\caption{(a) Micromagnetic simulations showing non-volatile conductance modulation of our skyrmionic synapse. First 7 pulses (5 MA\/cm$^{2}$, 2 ns width) move the skyrmions from pre to post-synapse region (LTP) and the next 7 vice-versa (LTD). (b) Evolution of post-synapse conductance.}\n\\label{conductance}\n\\end{figure}\n\nThe variation of conductance levels and programming energy per pulse with current density is shown in Fig. \\ref{fig:master_opt} (a). As the current density increases, the number of conductance states decrease. This is because with higher values of current, the events when multiple skyrmions cross the barrier increases. However if the current density is lower than a certain threshold (J$_{th}$) then it fails to move the skyrmions across the barrier. The current pulse width was kept at a constant value of 2 ns. The energy values are calculated using Equation \\ref{eq:energy}:\n\\begin{align}\n\\label{eq:energy}\nE_{Spike} = \\rho\\times t\\times l\\times w\\times J^{2}\\times T_{W},\n\\end{align}\nwhere $\\rho$ is the resistivity of Pt thin film \\cite{b27}, t and w are the thickness and width of HM layer and J and T$_{W}$ are the current density amplitude and width of the pulses used for programming. The variation of conductance levels and energy for different programming pulse widths is shown in Fig. \\ref{fig:master_opt} (b). Current density of the pulses used was 5 MA\/cm$^{2}$. As the pulse width was decreased, the time window allowed for multiple skyrmions to cross the barrier decreased and therefore the number of distinct conductance levels increased. It was also found that just after a pulse ended, there was some transient component in the skyrmions' velocity. The maximum inter pulse delay required for the skyrmion movement across the barrier to stabilise was found to be 5 ns. This puts a maximum limit on the operating frequency of neuromorphic systems built with this nanotrack. In all the simulations only those conductance states are reported which were non-volatile and did not involve any skyrmion crossing barrier after removal of programming pulse.\nThe programming parameters used in this work are: current density = 5 MA\/cm$^{2}$ and pulse width = 2 ns and has been circled in red in Fig. \\ref{fig:master_opt}. Using these parameters, consecutive potentiating (depressing) pulses which moved skyrmions from pre to post-synaptic (post to pre-synaptic) region were applied. The position of skyrmions in the nanotrack after different pulses are shown in Fig. \\ref{conductance} (a) whereas the conductance and post-synaptic skyrmion population variation with pulses is shown in Fig. \\ref{conductance} (b). \nThe conductance modulation curve is nearly linear and symmetrical in both LTP and LTD phases, which is a desirable property in electronic synapses \\cite{sung2018effect}.\n\n\n\n\\begin{figure*}\n\\centering\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n  \\includegraphics[scale=0.1]{skyr2.png}\n  \\caption{}\n  \\label{fig:sub1}\n\\end{subfigure}%\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n  \\hbox{\\hspace{2cm} \\includegraphics[height=4.4cm,keepaspectratio]{skyr3.png}}\n  \\caption{}\n  \\label{fig:sub2}\n\\end{subfigure}\n\\caption{(a) Proposed 1T1N skyrmionic synapse and different components of post-neuron, (b) 1T1N synaptic bit-cells in crossbar arrangement.}\n\\label{fig:master_synapse}\n\\end{figure*}\n\n\\section{1T1N Synapse and Simplified STDP}\n\\label{2t1n}\nThe proposed 1T1N synaptic circuit along with the pre and post-neurons are shown in Fig. \\ref{fig:master_synapse} (a). The proposed pre\/post-neuronal spikes (programming methodology) are shown in Fig. \\ref{spikes}. In our proposed synaptic design there are four possible modes of operation; (1) idle, (2) spike transmission, (3) potentiation and (4) depression. The different components of a neuron used in this work are: Current scaling circuit, integrator, comparator and a spike generator. The circuit and functioning of each of these components are discussed in detail in Section \\ref{ref1}. Spike transmission occurs when there is a pre-spike i.e. V$^{Pre}_{S1}$ is HIGH. The pre-spike voltage applied across the terminals B and C result in a current proportional to the conductance of the synapse (I$_{Syn}$), flowing out of C and entering the post neuron through the current scaling circuit. Consequently a current I$_{Spike}$ enters the integrator and keeps getting integrated in the form of voltage (V$_{mem}$) across a capacitor. When V$_{mem}$ crosses a threshold, two different spikes are generated: V$_{P}$ - V$_{N}$ which is the post-neuronal spike for the synapse under consideration and V$^{Post}_{S1}$ which is the pre-neuronal spike for the next layer of synapses. Although during spike transmission the gate of T1 receives a HIGH voltage, if there is no post-neuronal spike it will be in cutoff and no significant current will flow between A and D. In the case when only post spike occurs, the synaptic circuit remains in idle mode. This is because there is no current flow through the synapse as the transistor T1 is off. Plasticity (LTP or LTD) occurs when both the pre and post-spikes have temporal coincidence. When the positive part of post-spike coincides with the pre-spike as shown in Fig. \\ref{spikes} (a), LTP occurs (conductance of read-MTJ increases). At the onset of the pre-spike, T1 is in cutoff and spike transmission takes place from B to C. As soon as the post-spike arrives, T1 starts conducting and a current (whose magnitude depends on HM layer resistance and V$_{P}$ - V$_{N}$) flows from A to D, moving the skyrmions to the right. Its worth noting that spike transmission takes place between B and C even when the post-neuronal spike is driving skyrmions across the barrier in the HM layer. This is the unique advantage provided by our 4 Terminal decoupled read-program nanotrack. In case of a 3-terminal nanotrack \\cite{sengupta2016hybrid,b26} separate transistors would be required for switching off spike transmission during programming mode, since both operations take place through a common node. LTD occurs when the negative part of the post-spike coincides with the pre-spike (Fig. \\ref{spikes} (b)). This results in current flowing from D to A, thus moving the skyrmions from  post to pre-synaptic region and thus reducing the conductance of read-MTJ. For certain range of temporal spacing between pre and post-neuronal spikes, portions of both the positive and negative parts of the post-neuronal spike coincides with the pre-neuronal spike (shown in Fig. \\ref{spikes} (c)). In such a case both LTP and LTD would take place in degrees proportional to their extent of their overlap. Based on the micromagnetic simulations and nanotrack parameters described in the previous section, the characteristics of pre\/post neuron spikes used are; t$_{S2}$ = 2 ns, t$_{S1}$ = 22 ns and t$_{w}$ = 17 ns. Therefore the temporal conditions for which LTP, LTD and both occur are: 3 $\\leqslant$ $\\Delta$t \\textless 22, -21 \\textless $\\Delta$t $\\leqslant$ -2 and -2 \\textless $\\Delta$t \\textless 3 respectively, where $\\Delta$t = t$_{post}$-t$_{pre}$. The resultant STDP characteristic curve showing the percentage change in conductance for different values of $\\Delta$t is shown in Fig. \\ref{stdp}. The chosen spike scheme leads to synaptic programming energy consumption of $\\sim$ 1.2 fJ\/ per spike, (Equation \\ref{eq:energy}). The arrangement of our proposed 1T1N synaptic bit-cell in crossbar arrangement is shown in Fig. \\ref{fig:master_synapse} (b). All bit-cells in a column share the same post-neuron whereas all those in a row share the same pre-neuron. Therefore currents from all the synaptic bit-cells in a single column that receive pre-neuronal spike contribute to I$_{Syn}$ that enters the post-neuron below. All the bit-cells in a column receive post-neuronal spike across V$_{P}$ and V$_{N}$, but only those are programmed whose transistor is ON due to simultaneous presence of pre-spike. Also note that due to the decoupled write and read paths in our proposed synaptic structure, conductance modulation and spike transmission can both occur simultaneously, thus leading to faster unsupervised learning. Whereas in the structure proposed in \\cite{b26} due to a shared node between the two paths only one function (spike transmission or conductance modulation) could occur at a time. Considering that the same learning rule is being implemented, an extra transistor would be required to switch off the spike transmission path when the programming pulses are being applied in that case. \n\n\n\n\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{spikes.png}\n\\caption{Proposed pre and post-spike pulse shapes and their different orientation in time leading to (a) LTP, (b) LTD and (c) combination of LTP and LTD.}\n\\label{spikes}\n\\end{figure}\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale = 0.37\t]{stdp.png}\n\\caption{Simplified STDP characteristics used in our proposed design. The percentage change in conductance is shown with respect to an initial state where 5 skyrmions are present in post-synaptic region. }\n\\label{stdp}\n\\end{figure}\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=\\columnwidth]{dpi_skyr.png}\n\\caption{Subthreshold current scaling and normalizing block used in our neuron circuit. The sum of current from synapses in previous layer (I$_{1}$, I$_{2}$ .. etc) enter the circuit block as I$_{Syn}$ and the resultant output current I$_{Spike}$ is passed to the integrator stage of the neuron.}\n\\label{dpi}\n\\end{figure}\n\n\\section{Subthreshold CMOS neuron circuit}\n\\label{ref1}\nIn this section we present the schematics, functioning and outputs of different building blocks of the CMOS based neuron circuit. All simulations were done in CADENCE Spectre, using TSMC 65 nm technology node PDK. The various circuit parameters used for simulation has been detailed in Table \\ref{tab:circuit_params}. \n\n\\subsection{Current Scaling Circuit}\n\\label{ref11}\nThe CMOS circuit involved in scaling and normalizing the current entering the post-neuron during spike transmission is shown in Fig. \\ref{dpi}. The total current from synapses I$_{Syn}$, in the previous layer enter the post-neuron via this current scaling circuit block as I$_{Spike}$. The transistors N2-N6 operate in their subthreshold regime. This is ensured by making the bias current (I$_{B}$) controlled by V$_{B}$ and is very small, of the order of a few hundreds of nano-amperes. In that case the currents I$_{Spike1}$, I$_{ref}$ and I$_{B}$ are given by equation \\ref{eq:dpi1}. \nThe currents I$_{Syn}$ and I$_{Spike}$ are determined by equation \\ref{eq:dpi1}:\n\\begin{align}\n\\label{eq:dpi1}\nI_{Spike1} = I_{0}e^{\\frac{kV_{x}-V_{C}}{V_{T}}}, \\hspace{2mm} I_{ref} = I_{0}e^{\\frac{kV_{ref}-V_{C}}{V_{T}}}, \\hspace{2mm} I_{b} = I_{0}e^{\\frac{kV_{B}}{V_{T}}}\\\\ \\notag\n\\end{align}\nOn applying Kirchoff's law at node C, one can eliminate V$_{C}$ to get the relation between I$_{Spike1}$ and node voltage V$_{x}$, as shown in equation \\ref{eq:dpi2}.\n\\begin{align}\n\\label{eq:dpi2}\nI_{Spike1} = I_{b}\\frac{e^{\\frac{kV_{x}}{V_{T}}}}{e^{\\frac{kV_{x}}{V_{T}}}+e^{\\frac{kV_{ref}}{V_{T}}}},\\\\ \\notag\n\\end{align}\nVoltage V$_{x}$ is a function of the current I$_{Syn}$ and the exact relation between them depends on whether the transistor N1 operates in subthreshold or superthreshold regime. The PMOS transistors N5 and N6 are used as to make a current mirror that converts the sinking current flowing through N2 into a sourcing current flowing through N6 and therefore $I_{Spike1} \\approx I_{Spike}$. As I$_{Spike}$ gets integrated in the neuron's integrator block, it's input impedance keeps increasing. Therefore in order to prevent the transistor N6 from going into cutoff region while trying to source the same amount of current into a larger impedance load, it's supply voltage (Vdd2) was kept at a slightly higher level (650 mV) as compared to Vdd1 (600 mV).\n\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[scale = 0.3]{R_var_DPI1.png}\n\\caption{Current response of the scaling and normalizing block for different values of cumulative resistance of the synapses connecting it to neurons in previous layer. }\n\\label{dpi_res}\n\\end{figure}\n\nVariation of I$_{Syn}$ and I$_{Spike}$ with resistance of synapse is shown in Fig. \\ref{dpi_res}. As more number of synapses (attached to different pre-neurons) are connected in parallel, the effective resistance of the path terminating in N1 decreases. Thus Fig. \\ref{dpi_res} shows how values of I$_{Syn}$ and I$_{Spike}$ are affected when the number of synapses connecting a post-neuron to the previous neuron layer increases. The maximum resistance for which a current value has been plotted is 22 k$\\Omega$, since that is the maximum resistance of a single nanotrack synapse when there are zero skyrmions in post-synaptic region.  \n\n\\begin{figure*}\n\\centering\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n  \\includegraphics[width=1.2\\linewidth]{neuron1.png}\n  \\caption{}\n  \\label{fig:sub1}\n\\end{subfigure}%\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n  \\includegraphics[width=0.8\\linewidth]{timing1.png}\n  \\caption{}\n  \\label{fig:sub2}\n\\end{subfigure}\n\\caption{(a) Subthreshold CMOS circuit of neuron used for this work, (b) Timing diagram showing the global signals (CLK, POS, NEG) and output spikes of the neuron (V$_{S1}$, V$_{P}$-V$_{N}$) for different temporal placements of spiking event.}\n\\label{fig:master_neuron}\n\\end{figure*}\n\n\n\\subsection{Integrator, Comparator and Spike Generator}\n\\label{ref12}\nThe CMOS circuit that constitutes the remaining of our post-neuron is shown in Fig. \\ref{fig:master_neuron} (a). It starts with a Differential Pair Integrator filter for integrating incoming current signal (M1-M3). It forms the integrator block of our post-neuron. The voltage accumulated across the capacitance (C$_{mem}$) acts as the neuron's membrane potential. The inverters constituted by M5-M6 and M7-M8 play the role of comparator and spike event generating block. The circuit beyond it comprising of the Flip-Flop, AND gates, M9-M10 and R1-R2 is responsible for generating the spikes as specified in Fig. \\ref{spikes}, whenever a spike event is encountered.\n\nThe expression governing the dynamics of a generic non-linear integrate and fire system is given by equation \\ref{eq:generic_LIF} \\cite{abbott1993asynchronous}. \n\\begin{align}\n\\label{eq:generic_LIF}\n\\tau\\frac{du}{dt} = F(u) + G(u)I\\\\ \\notag\n\\end{align}\nAssuming all transistors to be in subthreshold saturation and applying translinear principle, one gets the expression for I$_{out}$ as shown in equation \\ref{eq:neuron_LIF1}, where the currents I$_{out}$, I$_{t}$, I$_{\\tau}$ are given by equation \\ref{eq:neuron_current} and $\\tau = C_{mem}V_{T}\/kI_{\\tau}$.\n\\begin{align}\n\\label{eq:neuron_current}\nI_{out} = I_{0}e^{\\frac{kV_{mem}}{V_{T}}}, \\hspace{2mm} I_{t} = I_{0}e^{\\frac{kV_{thr}}{V_{T}}}, \\hspace{2mm} I_{\\tau} = I_{0}e^{\\frac{kV_{tau}}{V_{T}}}\\\\ \\notag\n\\end{align}\n\\vspace{-5mm}\n\\begin{align}\n\\label{eq:neuron_LIF1}\n\\tau\\frac{dI_{out}}{dt} = \\frac{I_{t}I_{out}I_{Spike}}{I_{\\tau}(I_{out}+I_{t})} - I_{out}\\\\ \\notag\n\\end{align}\nOn replacing $I_{out}$ with $I_{0}e^{\\frac{kV_{mem}}{V_{T}}}$ (see equation \\ref{eq:neuron_current}), one obtains the temporal dynamics of V$_{mem}$ shown in equation \\ref{eq:neuron_LIF2}.\n\\begin{align}\n\\label{eq:neuron_LIF2}\n\\tau\\frac{dV_{mem}}{dt} = \\frac{I_{t}I_{Spike}}{I_{\\tau}(I_{out}+I_{t})} - \\frac{V_{T}}{k}\\\\ \\notag\n\\end{align}\nIt is worth noting here that I$_{Spike}$ is the input current to the DPI integrator block and I$_{out}$ is a function of V$_{mem}$. Therefore both equation \\ref{eq:neuron_LIF1} and \\ref{eq:neuron_LIF2} represent non-linear integrate and fire dynamics (see equation \\ref{eq:generic_LIF}) in variables I$_{out}$ and V$_{mem}$ respectively.\n\nThe threshold voltage of the neuron is determined by the switching voltage of the inverter (M5-M6). This can be controlled by changing the dimensions (W\/L ratio) of M5 and M6. Therefore as V$_{mem}$ approaches the threshold, the output of the first inverter changes drastically to 0. In order to invert this output and provide spike events such that a high voltage corresponds to V$_{mem}$ reaching its threshold, a second inverter M7-M8 has been used. The second inverter also sharpens the response of the first inverter, thus resulting in positive spike events only when V$_{mem}$ crosses the threshold. \n\nIn order to generate spikes of fixed pulse widths and desired shapes we make use of synchronous circuits, following the inverters. Three different clock signals have been used for this purpose: CLK, POS and NEG with widths as indicated in Fig. \\ref{fig:master_neuron} (b). The Flip-Flop (FF) is used to generate the spike (V$_{S1}$) which acts as the pre-neuronal spike for the next layer of synapses. It's pulse width is same as that of CLK i.e. 22 ns. This spike is also connected to gate of M4 so as to discharge C$_{mem}$, thus reseting the accumulated membrane potential. C$_{ref}$ determines the time taken by C$_{mem}$ to discharge. Following this AND gates are used to make the outputs: V$^{AND}_{P}$ and V$^{AND}_{N}$ HIGH only when HIGH of POS and NEG overlap with HIGH of V$_{S1}$ respectively. The outputs from the AND gates are then connected to source follower circuits, so as to be able to drive larger loads. The final outputs of the circuit V$_{P}$ and V$_{N}$ are connected to the pre and post-synaptic regions of our nanotrack synapse. Therefore the differential voltage V$_{P}$ - V$_{N}$ represent the post-neuronal spike, used to drive skyrmions in the synapses in the previous layer. As we saw in Section \\ref{opt}, different current amplitudes and pulse widths may lead to different conductance levels and energy dissipation in the synapse, one can control the amplitude of V$_{P}$ - V$_{N}$ by varying the values of R1, R2 and W\/L ratios of M9 and M10, whereas the temporal parameters of the spikes (t$_{S1}$, t$_{w}$, t$_{S2}$ discussed in Section \\ref{2t1n}) can be controlled by the pulse widths and duty ratios of CLK, POS, NEG. The time evolution of global clock signals, spike events and output spikes (V$_{S1}$ and V$_{P}$ - V$_{N}$) are shown in Fig. \\ref{fig:master_neuron} (b). The temporal parameters of CLK, POS and NEG (shown in Fig. \\ref{fig:master_neuron} (b)) were chosen so as to generate spikes with parameters as discussed in Section \\ref{2t1n}. \n\nIn order to demonstrate the working of the proposed 1T-1N STDP powered skyrmionic synapse with neuron circuit, we considered a single synaptic bit-cell between a pre and post-neuron. The pre-neuron was modeled as a pulse source, generating (V$^{PRE}_{S1}$) with pulse width 23 ns and a frequency $\\sim$ 27 MHz. The post-neuron was simulated with the circuit discussed in this section. The HM layer was modeled by a resistance of 3 k$\\Omega$. The time evolution of V$^{POST}_{S1}$, V$_{mem}$ (membrane potential), programming current flowing through HM layer and resistance of read MTJ is shown in Fig. \\ref{neuron_res}. It is worth noting here that the sign of the programming current shown in Fig. \\ref{neuron_res} depends on the time ($\\Delta$t) with which the pre and post-neuronal spikes are separated (exact dependence has been discussed in Section \\ref{2t1n}). Accordingly current which is positive, negative or both might flow through the synapse, resulting in LTP, LTD or a combination of LTP and LTD respectively. The energy consumed in the neuron circuit (including the current scaling block, integrator, comparator and spike generation blocks) was found to be 0.25 pJ\/spike for a supply voltage (V$_{dd}$) of 600 mV and neuron firing rate of $\\sim$ 2.3 MHz.\n\n\\begin{table}[htbp]\n  \\centering\n  \\caption{Circuit parameters for CMOS Neuron Circuit}\n    \\begin{tabular}{|P{14.43em}|P{7.23em}|}\n    \\hline\n    \\textbf{Parameter Description} & \\textbf{Value} \\\\\n    \\hline\n     V$_{B}$   & 450 mV  \\\\\n    \\hline\n     V$_{ref}$   & 450 mV \\\\\n    \\hline\n     V$_{dd}$   & 600 mV \\\\\n    \\hline\n     V$_{dd2}$   & 650 mV \\\\\n    \\hline\n     V$_{thr}$   & 350 mV \\\\\n    \\hline\n     V$_{tau}$   & 270 mV \\\\\n    \\hline\n     C$_{mem}$   & 25 fF \\\\\n    \\hline\n     C$_{ref}$   & 5 fF \\\\\n    \\hline\n    W\/L ratio of N1 & 10 $\\mu$m \/ 65 nm \\\\\n    \\hline\n    W\/L ratio of N2-N6, M1-M3, M8 & 200 nm \/ 65 nm \\\\\n    \\hline\n    W\/L ratio of M5 & 240 nm \/ 65 nm \\\\\n    \\hline\n    W\/L ratio of M6 & 400 nm \/ 65 nm \\\\\n    \\hline\n    W\/L ratio of M7 & 400 nm \/ 65 nm \\\\\n    \\hline\n     W\/L ratio of M9-M10 & 16 $\\mu$m \/ 65 nm \\\\\n    \\hline\n     W\/L ratio of T1 & 2.7 $\\mu$m \/ 65 nm \\\\\n    \\hline\n    \\end{tabular}%\n  \\label{tab:circuit_params}%\n\\end{table}%\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=\\columnwidth]{neuron_res.png}\n\\caption{Circuit simulation result showing pre-neuronal spikes, output spike and membrane potential of post-neuron, programming current flowing through synapse and its resistance when a single 1T1N synaptic bit-cell is connected between a pre-neuron and post-neuron.}\n\\label{neuron_res}\n\\end{figure}\n\n\\section{Unsupervised learning with skyrmion synapse}\nIn order to validate the working of our proposed skyrmion on nanotrack synaptic bit-cell for unsupervised learning we simulated two applications. In the first, we simulated a fully connected feed forward two-layer Spiking Neural Network (SNN) (see Fig. \\ref{network}) powered by Spike-Timing Dependent Plasticity (STDP). All simulations were done on the MATLAB based neuromorphic simulator named MASTISK \\cite{mastisk}. The network has 90,000 spiking pixels in the first layer and 2 LIF neurons in the output layer. The two layers are connected by excitatory synapses in an all-to-all fashion. The output neurons are connected to each other through inhibitory synapses. This is done to implement Winner-Take-All mechanism \\cite{gupta2009hebbian}. Video stream comprising of binary 300 $\\times$ 300 frames were used for training the SNN as shown in Fig. \\ref{network} (b)-(d). The video stream comprises of mostly gaussian noise frames (see Fig. \\ref{network} (b)) with two complex patterns (IIT-D logo and BUAA logo) embedded at different timestamps (Fig. \\ref{network} (c)-(d)). The input layer encodes the frames into spikes using poisson spike encoding \\cite{b14} with the mean firing rate proportional to the intensities of the pixels. Fig. \\ref{fullresult} (a) shows the rastor plot of input neurons during the entire training period. Spiking activity of the neurons are denoted by dots in Fig. \\ref{fullresult} (a) and the color denotes the kind of frame that was presented. Black dots represent noisy frame whereas the red and blue dots represent the frames containing the IIT-D logo and BUAA logo respectively. The input video frames at specific timestamps are illustrated in Fig. \\ref{fullresult} (b). Fig. \\ref{fullresult} (c) and (d) show the evolution of conductance of the synapses connecting the output neurons 1 and 2 to the input layer respectively, thus depicting the representations learned by each of the output neurons. Neuron 1 became selective to BUAA logo, while neuron 2 got selective to IIT-D logo. Initially the output neurons fire randomly, however from 1000 ns onwards as the occurrence of patterns increases the spiking becomes more specific to occurrence of a particular input pattern (Fig. \\ref{fullresult} (e)). In order to study the degree of selectiveness of the neurons to different patterns, we separately noted the conductance of the synapses connecting the output neurons to different types of input pixels. The input pixels could either be pattern pixels and carry information for the two different patterns or be background pixel that do not carry information regarding any pattern. The averaged conductance of the synapses between the input pixels of the three types and two output neurons are shown in Fig. \\ref{cond2}. The logo with maximum averaged conductance is the one for which a particular neuron gets selective. The difference between the conductances of the two patterns for a particular neuron depicts how well it can differentiate between the two patterns. The low conductance of the background synapses (noise) shows that the network is able to get selective to patterns and not background noise. The system achieves a low false positive spike rate of 6.5$\\times$10$^{-4}$ (inset of Fig. \\ref{cond2}). Since output neuron firing activity starts around 350 ns there is a transient in the false positive spike rate of either neuron. Ultra-low on-line unsupervised learning synaptic programming power consumption of $\\sim$ 1 nW \/ synapse and neuron firing power consumption of $\\sim$ 1.64 $\\mu$W \/ neuron was achieved (Total programming events: $\\sim$ 4.6$\\times$10$^5$; Total post-neuron firing events: 40; synaptic programming energy: 1.2 fJ; CMOS neuron energy per firing event: 0.25 pJ; duration: 3050 ns; total synapses: 1.8$\\times$10$^5$). The power takes in account both: energy spent in moving skyrmions in the nanotrack, and integration of I$_{Spike}$ in DPI integrator, comparator and the spike generation circuit over the entire duration of learning. \n\n\\begin{table}[htbp]\n  \\centering\n  \\caption{MNIST Classification Accuracy for different synapses}\n    \\begin{tabular}{|P{14.43em}|P{7.23em}|}\n    \\hline\n    \\textbf{Synapse Type} & \\textbf{Accuracy} \\\\\n    \\hline\n   Redundancy = 2    & 73.8 \\% \\\\\n    \\hline\n   Redundancy = 4    & 82.17 \\% \\\\\n    \\hline\n    Redundancy = 6    & 84.97 \\% \\\\\n    \\hline\n    Ideal Synapse   & 85.5 \\% \\\\\n    \\hline\n    \\end{tabular}\n  \\label{tab:mnist}\n\\end{table}\n\n\nFor the second application, we simulated a 3-layer SNN comprising of input, output (excitatory) and inhibitory neurons respectively inspired from \\cite{diehl2015unsupervised}. The network was trained on 60,000 training images of the MNIST database. In the inference mode, the skyrmion on nanotrack characterized by its properties depicted in Fig. \\ref{conductance}, was used as synapse. We varied the number of nanotracks connected in parallel to constitute a single synapse in order to see how performance varied with the redundancy. The different levels of redundancy and their corresponding classification accuracies have been shown in Table \\ref{tab:mnist}. The synapse with redundancy = 6 almost reaches the performance level of ideal synapse (infinite dynamic range and conductance levels).\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale = 0.2]{network.jpg}\n\\caption{(a) SNN topology simulated for this work (90 K neurons, 180 K synapses).(b-d) Frames of video dataset used for unsupervized learning application.}\n\\label{network}\n\\end{figure}\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale = 0.3]{whole_result.jpg}\n\\caption{(a) Input Rastor plot showing spiking activity of layer-1 neurons. (blue='IITD' logo, red = 'BUAA' logo occurence. (b) frames of input video data. (c)-(d) synaptic weight maps for Neuron 1 and 2 respectively, at timestamps: 75ns, 1125ns, 2175ns and 2925ns. (e) rastor plot for output neurons.} \n\\label{fullresult}\n\\end{figure}\n\n\\begin{table*}[htbp]\n  \\centering\n  \\caption{Comparison with other synapses}\n    \\begin{tabular}{|P{6.5em}|P{6.855em}|P{6.43em}|P{6.645em}|P{6.355em}|P{6.355em}|p{6.07em}|p{6.57em}|}\n    \\hline\n    \\textbf{Device} & \\textbf{Dimension} & \\textbf{Programming Energy} & \\textbf{Programming Time} & \\textbf{Synapse Configuration} & \\multicolumn{1}{p{6.355em}|}{\\textbf{Terminals}} & \\textbf{Type} & \\textbf{Programming Neuron Spikes} \\\\\n    \\hline\n    Ag-Si memristor \\cite{jo2010nanoscale} & 100 nm X 100 nm & 25.2 pJ-403.2 pJ & 300 us & 1R    & 2     & Experimental & Identical \\\\\n    \\hline\n    PCM \\cite{kuzum2011nanoelectronic} & length: 500 nm, BE diameter: 75 nm & LTD (avg) = 13.5 pJ           LTP (avg) =  50 pJ & LTD transition: 75 ns LTP transition: 5 us  & 1R    & 2     & Experimental & Non-Identical \\\\\n    \\hline\n    AlOx\/HfO2 Bilayer RRAM \\cite{woo2016improved} & 21 nm thick & LTP (avg) = 3.24 nJ   LTD (avg) = 4 nJ & 100 us & 1T1R  & 3     & Experimental & Identical \\\\\n    \\hline\n    Domain Wall synapse \\cite{sengupta2016hybrid} & 320 nm X 20 nm & 48 fJ\/event & 1 ns   & 3T1R  & 5     & Simulation & Pres-Spikes = Complex exponential Post-Spikes =  identical pulses \\\\\n    \\hline\n   1T1N Skyrmion Synapse [This Work] & 820 nm X 280 nm & 1.2 fJ\/event & 2 ns   & 1T1R  & 4     & Simulation & Identical  \\\\\n    \\hline\n    \\end{tabular}%\n  \\label{tab:lit_comp}%\n\\end{table*}%\n\n\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[scale = 0.35]{allNeuron_Cond.png}\n\\caption{Averaged conductance of the synapses corresponding to the input video for: (a) Neuron 1, and (b) Neuron 2. Inset shows false positive spike rate evolution with time.}\n\\label{cond2}\n\\end{figure}\n\n\\section{Discussion}\nTable \\ref{tab:lit_comp} compares our proposed skyrmion synapse with other nanodevice based synapses. It clearly highlights the ultra-low energy (1.2 fJ\/event) and sub-nanosecond time (2 ns) consumed in changing conductance states in our synapse. Since synapses based on nanotrack type structure (domain wall or skyrmion based) usually have 3 or more terminals, it becomes difficult to use nanotrack based synapses in 1R configuration (synaptic bit-cell containing only nanotrack and no transistor). For instance domain wall on nanotrack based synapse proposed in \\cite{sengupta2016hybrid} had 3 transistors apart from the nanotrack in their synaptic bit-cell. However we show that using our proposed skyrmion on nanotrack configuration and neuron circuit one can implement a modified version of STDP learning rule with only 1 transistor along with the nanotrack in a bit-cell.\nA limitation of our synapse is the large dimensions of the nanotrack (1.5 \\% larger than \\cite{sengupta2016hybrid}). The length of the nanotracks determines the degree with which the skyrmions can freely move in the nanotrack without interacting with each other. It also has an impact on number of distinct conductance levels that can be achieved. In order to achieve same number of conductance states with shorter length it would require scaling down skyrmion sizes. In our simulations we used skyrmions with diameter $\\sim$ 15 nm. Recently studies have shown that size of skyrmions can even be reduced to 1-3 nm \\cite{wiesendanger2016nanoscale,romming2015field}. This opens possibilities of reducing the nanotrack length further without losing number of conductance levels. \n\nThis work is based on simulations done holistically at the device, circuit and architectural level with parameters calibrated to experimental results \\cite{wiesendanger2016nanoscale}. It clearly highlights the benefits of neuromorphic systems built with hybrid CMOS-skyrmion circuits.\n\n\n\\section{Conclusion}\nWe illustrate an approach for practical realization of multi-level synapse using hybrid skyrmion-CMOS based spiking neuromorphic systems based on simulations done at device, circuit and system level. Firstly through micromagnetic simulations we study in detail the impact of different programming parameters of a HM\/FM nanotrack on various synaptic performance parameters and demonstrate true non-volatile multi-level conductance states, independent of inter-spike delay or frequency. We use a read-MTJ on top of the post-synaptic region of the nanotrack separated by insulating magnetic material from the FM layer beneath it. This makes our synapse 4-terminal with completely decoupled read and program paths. Leveraging the conductance modulation, we propose a 1T1N synaptic architecture and programming methodology to implement a modified version of the biological STDP rule, while consuming $\\sim$ 1.2 fJ energy per programing event. We design a custom subthreshold synchronous spike generator circuit which when coupled with a current scaling circuit, Differential Pair Integrator and inverter based thresholding circuit, can perform the desired functionalities of a Leaky-Integrate and Fire neuron at extremely low energy (0.25 pJ\/output spike). Unsupervised learning is demonstrated by simulating feed-forward SNN for pattern extraction and multi-class classification application. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\ifCLASSOPTIONcaptionsoff\n  \\newpage\n\\fi\n\n\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\subsection*{Keywords}\n\nHeat conduction; Transient; Mixed boundary conditions; Wiener-Hopf; Cagniard-de Hoop\n\n\\section{Introduction} \\label{intro}\n\nTraditionally, great interest has been shown in determining the disturbances that are generated when loads are applied on the surface of a half-space. Lamb \\cite{Lam-04} obtained the exact solution when an impulsive, concentrated load is applied along a line of the free surface of an isotropic linear elastic medium. de Hoop reappraised this problem \\cite{DeH-60}, modifying the method originally devised by Cagniard \\cite{Cag-39}, \\cite{Cag-62}, leading to the now well-known \\textit{Cagniard-de Hoop} (CdH) technique. This method has been used widely since, allowing exact solutions to be obtained for a wide range of transient elasticity problems. The method can also be useful in order to render solutions into integral forms that are rapidly convergent when calculated numerically.\n\nTransient \\textit{thermoelastic} half-space problems were considered by Danilovskaya \\cite{Dan-50}, Boley and Tolins \\cite{Bol-62} and Achenbach \\cite{Ach-63} but in these problems the forcing was such that the CdH technique was not required. The extension of these problems to inhomogeneous media was considered by Baczynski \\cite{Bac-03} and Parnell \\cite{Par-06}. A purely thermal, transient problem that employed the CdH method was solved in \\cite{She-01}. The thermoelastic Lamb problem was studied by Nayfeh and Nemat-Nasser \\cite{Nay-72} who used generalized thermoelasticity in order to retain a finite thermal wave speed, employing the CdH technique to determine the solution.\n\nAll of the above problems are of fundamental importance in an array of applications where a number of alternative boundary conditions on the surface can arise. What appears to be rather lacking in the literature however are studies of transient problems with \\textit{mixed} boundary conditions, where in the context of the thermal problems the condition takes the form, e.g.\n\\begin{align}\n\\cT(0,y,t) &= f(y,t) & \\textnormal{for $y>0$}, &&\n\\deriv{\\cT}{x}(0,y,t) &= g(y,t) &&  \\textnormal{for $y<0$}, \\label{mixed1}\n\\end{align}\nwhere $\\cT(x,y,t)$ is the temperature field, $f(y,t)$ and $g(y,t)$ are two specified functions, $t$ is time and with reference to Fig.\\ \\ref{fig:setdomaindescription} $x$ and $y$ are Cartesian coordinates. The half-space resides in $x\\geq 0$ and $y$ runs parallel to the surface, which is defined by $x=0$.\n\nGenerally such problems lead to the propagation of a thermal disturbance into the half-space. Indeed, such thermal front problems are of importance in a number of applications including defect sizing \\cite{Alm-94}, transient thermography \\cite{Sha-13}, solar cell manufacturing \\cite{Pil-02} and thermal insulation \\cite{Ree-00}. Caflisch and Keller \\cite{Caf-81},  Levine \\cite{Lev-82} and Satapathy and Sahoo \\cite{Sat-02} studied front propagation in the thermal context with mixed boundary conditions but in the context of steady problems with applications in quenching. Kozlov et al.\\ \\cite{Koz-01} considered a transient half-space problem with mixed boundary conditions and made progress by using cylindrical coordinates due to the special form of the boundary condition chosen.\n\nMixed boundary conditions are generally difficult to handle even in steady problems and analytical or semi-analytical solutions are frequently only possible by the application of the Wiener-Hopf method \\cite{Nob-88}. This method exploits the analyticity properties of functions in order to yield an explicit or approximate solution in the Fourier transform domain. Contour integration then yields the solution in the physical domain.\n\n\nHere we shall consider a rather general mixed boundary value problem in the context of thermal front propagation and determine solutions using the Wiener-Hopf method and Cagniard-de Hoop technique. This problem of mixed boundary conditions of the form \\eqref{mixed1} is of particular interest in analyzing the field close to the location of the change in boundary condition type, i.e.\\ $x=y=0$ in \\eqref{mixed1}.\n\nWe obtain a solution in single integral form by using a deformation of the Laplace contour in a similar manner to the Cagniard-de Hoop method. Although it appears that we cannot obtain an explicit solution, the solution determined can be evaluated rapidly on a desktop PC and therefore it is of great utility due to its general form and its ability to circumvent a direct numerical simulation of the problem. Although similar problems, involving a discontinuous temperature boundary condition have been considered in the building insulation literature, see e.g.\\ Claesson and Hegentoft \\cite{Cla-91} and Hegentoft and Claesson \\cite{Heg-91} to the authors' knowledge it does not appear that the solution we provide has been written down anywhere in the literature before now.\n\nIn this paper we shall first set out the problem description in section \\ref{problemdescription} before determining the solution in the transform domain in section \\ref{WH}. In section \\ref{CDH} we describe how we deform the Laplace contour onto a steepest descent path, in the manner of the Cagniard-de Hoop technique in order to obtain a solution in terms of a single integral along the deformed contour path, with an integrand that decays exponentially. In section \\ref{sec:particularexamples} we illustrate the efficacy of the scheme by determining the solution for a number of different boundary conditions, with validation provided by finite element solutions.\n\n\n\n\\section{Problem description} \\label{problemdescription}\n\nAssume that the problem under consideration is two-dimensional, being independent of $z$ and define the two dimensional half-space domain $\\mathcal{D}=\\{(x,y):0\\leq x <\\infty, -\\infty < y < \\infty\\}$. We seek solutions to the anisotropic heat equation:\n\\begin{align}\n\\frac{k}{\\rho c_V}\\left(\\derivtwo{\\cT}{x}+\\ell\\derivtwo{\\cT}{y}\\right) &= \\deriv{\\cT}{t}\n\\end{align}\nwhere $k$ and $k\\ell$ are the thermal conductivities ($\\ell>0$) in the $x$ and $y$ directions respectively, $c_V$ is the specific heat at constant volume, $\\rho$ is the mass density, $t$ is time and $\\cT=\\cT(x,y,t)$ is the temperature field. We can combine the constants as $\\ka=k\/(\\rho c_V)$, the thermal diffusivity.\n\nIt is convenient to non-dimensionalise the governing equation, using coordinates with a ``hat'' and scale the $y$ coordinate to remove the anisotropy coefficient. Write $(\\hat{x},\\hat{y},\\hat{T},\\hat{t})=(x\/x^*,y\/y^*,(\\cT-\\cT^*)\/\\cT^*,t\/t^*)$, where\n\\begin{align}\nx^*=1[\\text{m}], \\quad y^*=\\frac{1}{\\sqrt{\\ell}} [\\text{m}], \\quad t^*=\\frac{(x^*)^2}{\\ka} [\\text{s}],\n\\end{align}\nand $\\cT^*$ is the reference temperature in Kelvin. Upon doing so and ``dropping hats'' we find\n\\begin{align}\n\\nabla^2 \\cT &= \\deriv{\\cT}{t}    \\label{heat}\n\\end{align}\nwhere $\\nabla^2 = \\pa^2\/\\pa x^2 + \\pa^2\/\\pa y^2$. We wish to solve \\eqref{heat} on the (scaled) domain $\\mathcal{D}$ with boundary $\\partial \\mathcal{D}=\\partial \\mathcal{D}^-\\cup\\partial \\mathcal{D}^+$ as illustrated in Fig.\\ \\ref{fig:setdomaindescription}. We consider homogeneous initial conditions of the form\n\\begin{align}\n\\cT(x\\geq 0,y,t=0) &= 0 \\label{2.4}\n\\end{align}\nand boundary conditions of the form \\eqref{mixed1} but simplify by removing the $y$ dependence, i.e.\\\n\\begin{align}\n\\cT\\Big|_{\\partial \\mathcal{D}^+}=\\cT(x=0,y>0,t>0)= T_0 f_0(t), \\\\\n\\deriv{\\cT}{x}\\Big|_{\\partial \\mathcal{D}^-}= \\deriv{\\cT}{x}(x=0,y<0,t>0) = T_0'g_0(t), \\label{2.6}\n\\end{align}\nwhere $T_0$ and $T_0'$ are real constants and $f_0(t)$ and $g_0(t)$ are piecewise continuous functions of time.\n\nWe therefore have a mixed boundary value problem, which in general are not straightforward to solve even in the steady context so that the time dependence adds an additional element of complexity. Furthermore we allow for the fact that we could have a step change at $t=0$ on $x=0$, leading to a propagating discontinuity front in the half-space.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.6]{domainD.eps}\n\\end{center}\n\\caption{Domain $\\mathcal{D}$ of the problem and its boundaries $\\partial \\mathcal{D}^+$ and $\\partial \\mathcal{D}^-$} on which Dirichlet and Neumann boundary conditions are imposed respectively.\n\\label{fig:setdomaindescription}\n\\end{figure}\n\nIn order to determine $\\cT$ it is convenient to introduce an alternative problem (for convergence issues as will be shown), giving rise to a different temperature distribution $T$, depending on a small parameter $\\epsilon$ and where $T$ converges to $\\cT$ as $\\epsilon\\rightarrow 0$. This problem is described as follows\n\\begin{align}\n\\nabla^2 T &= \\deriv{T}{t}\\label{eq:newheateq}\\\\\nT(x\\geq0,y,t=0)&=0\\\\\nT\\Big|_{\\partial \\mathcal{D}^+}=T(x=0, y>0,t>0)&=T_0f_0(t)e^{-\\epsilon y}\\\\\n \\deriv{T}{x}\\Big|_{\\partial \\mathcal{D}^-}=\\deriv{T}{x}(x=0,y<0,t>0)&=T_0'g_0(t)e^{\\epsilon y}. \\label{eq:newderbc}\n\\end{align}\nAs is easily seen, we recover the solution to our original problem by taking the limit as $\\epsilon$ tends to zero:\n\\begin{align}\n\\cT(x,y,t)=\\lim_{\\epsilon \\rightarrow 0} T(x,y,t).\n\\end{align}\n\n\n\\section{Solution in the transform domains via the Wiener-Hopf technique} \\label{WH}\n\nDefine the Laplace transform in time for any function $\\phi(x,y,t)$ by\n\\begin{align}\n\\mathcal{L}(\\phi(x,y,t)) = \\tilde{\\phi}(x,y,s) &= \\int_0^{\\infty}\\phi(x,y,t)e^{-st}{dt}\n\\end{align}\n\nand hence applying this to the governing scaled equations (\\ref{eq:newheateq})-(\\ref{eq:newderbc}) we have\n\\begin{align}\n\\nabla^2 \\tilde{T} &= s\\tilde{T} \\label{heatLT}\n\\end{align}\nand boundary conditions become\n\\begin{align}\n\\tilde{T}(x=0,y>0,s) &= \\tilde{f}_0(s)T_0e^{-\\epsilon y} \\\\\n\\deriv{\\tilde{T}}{x}(x=0,y<0,s) &= \\tilde{g}_0(s)T_0'e^{\\epsilon y}.\n\\end{align}\nAlthough $s\\in\\mathbb{C}$ the set of complex numbers, for the sake of the analysis to follow we can assume it to be real and positive. This allows us to scale the $(x,y)$ variables to simplify the governing equations. The derivation goes through retaining explicit dependence on $s$ in the governing equation, but the algebra becomes rather heavy and tedious and does not render any greater understanding of the problem; both approaches lead to the same result. As such we will rescale $x$ and $y$ in order to eliminate $s$ from the governing equation. Thus define\n\\begin{align}\nx_0 &= x\\sqrt{s}, & y_0 &= y\\sqrt{s} \\label{x0y0}\n\\end{align}\nand therefore we obtain\n\\begin{align}\n\\nabla^2_0 \\tilde{T} &= \\tilde{T} \\label{heatLT2}\n\\end{align}\nwhere $\\nabla_0^2 = \\pa^2\/\\pa x_0^2 + \\pa^2\/\\pa y_0^2$. The boundary conditions become\n\\begin{align}\n\\tilde{T}(x_0=0,y_0>0,s) &= \\tilde{f}_0(s)T_0e^{-\\epsilon y} \\\\\n\\deriv{\\tilde{T}}{x_0}(x_0=0,y_0<0,s) &= \\frac{\\tilde{g}_0(s)}{\\sqrt{s}}T_0'e^{\\epsilon y}.\n\\end{align}\n\nNext define the Fourier transform in $y_0$ as\n\\begin{align}\n\\mathcal{F}(\\tilde{T}(x_0,y_0,s)) = \\Theta(x_0,\\al,s) &= \\int_{-\\infty}^{\\infty}\\tilde{T}(x_0,y_0,s)e^{i\\al y_0}{dy_0}\n\\end{align}\nand define $\\Theta^+$ and $\\Theta^-$ as\n\\begin{align}\n\\Theta^-(x_0,\\al,s)= \\int_{-\\infty}^{0}\\tilde{T}(x_0,y_0,s)e^{i\\al y_0}{dy_0} \\quad \\text{and} \\quad \\Theta^+(x_0,\\al,s)=\\int_{0}^{\\infty}\\tilde{T}(x_0,y_0,s)e^{i\\al y_0}{dy_0},\n\\end{align}\nso that $\\Theta=\\Theta^-+\\Theta^+$. Applying the Fourier transform to the governing (Laplace transformed) equation \\eqref{heatLT2} we find that\n\\begin{align}\n\\Theta'' &= (\\al^2+1)\\Theta \\label{heattrans}\n\\end{align}\nand to the boundary conditions, we find that\n\\begin{align}\n\\Theta^+(x_0=0,\\al,s) &= \\frac{i T_0}{(\\al+i\\eps)}\\tilde{f}_0(s) \\label{BC1}\\\\\n\\deriv{\\Theta^-}{x_0}(x_0=0,\\al,s) &= -\\frac{i T_0'}{(\\al-i\\eps)}\\frac{\\tilde{g}_0(s)}{\\sqrt{s}} \\label{BC2}.\n\\end{align}\nWith reference to Fig.\\ \\ref{fig:setdescription}, we note that $\\Theta^+$ is analytic on $\\Omega^+=\\{\\al \\in \\mathbb{C}, \\Im(\\al)>-\\epsilon\\}$, $\\Theta^-$ is analytic on $\\Omega^-=\\{\\al \\in \\mathbb{C}, \\Im(\\al)<\\epsilon\\}$ and $\\Theta^+$, $\\Theta^-$ and $\\Theta$ are analytic on the strip $\\mathcal{S}=\\Omega^- \\cap \\Omega^+$. The superscript $+$ and $-$ notation thus indicates analyticity in the domains $\\Omega^+$ and $\\Omega^-$ respectively.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.6]{domainOmega.eps}\n\\end{center}\n\\caption{Illustrating the domains $\\Omega^+$, $\\Omega^-$ and $\\mathcal{S}$ on which the functions $\\Theta^+, \\Theta^-$ and $\\Theta$ are analytic, respectively.}\n\\label{fig:setdescription}\n\\end{figure}\n\nThe solution of \\eqref{heattrans} is\n\\begin{align}\n\\Theta(x_0,\\al,s) &= A_1(\\al,s) \\exp(-(\\al^2+1)^{1\/2}x_0) \\label{Thetasol}\n\\end{align}\nwhere the branch of the square root function in the exponent is chosen so that its real part satisfies $\\Re(\\al^2+1)^{1\/2}>0$, ensuring that the solution decays as $x_0\\rightarrow \\infty$.  For conciseness let us introduce $\\Phi^+$ and $\\Psi^-$, analytic on $\\Omega^+$ and $\\Omega^-$ respectively, as\n\\begin{align}\n\\Phi^+(\\al,s)=\\deriv{\\Theta^+}{x}\\Big|_{x=0} \\quad \\text{and} \\quad \\Psi^-(\\al,s)=\\Theta^{-}\\Big|_{x=0}. \\label{concise}\n\\end{align}\nImposing the boundary conditions \\eqref{BC1}-\\eqref{BC2}, employing \\eqref{concise} and eliminating $A_1$ between the two resulting equations, we arrive at\n\\begin{align}\n-\\Phi^+ &= K(\\al)\\Psi^- + \\frac{K(\\al)iT_0}{(\\al+i\\eps)}\\tilde{f}_0(s)-\\frac{iT_0'}{(\\al-i\\eps)}\\frac{\\tilde{g}_0(s)}{\\sqrt{s}}. \\label{elimA1}\n\\end{align}\nThe \\textit{kernel} here $K(\\al)=(\\al^2+1)^{1\/2}$ is easily factorized as $K(\\al)=K^-\/K^+$ where\n\\begin{align}\nK^+(\\al) = (\\al+i)^{-1\/2} \\quad \\text{and} \\quad  K^-(\\al) = (\\al-i)^{1\/2}. \\label{Kminus}\n\\end{align}\n\nMultiplying \\eqref{elimA1} by $K^+$  we obtain\n\\begin{align}\n-K^+(\\al)\\Phi^+ &= K^-(\\al)\\Psi^- + S(\\al,s) \\label{elimA12}\n\\end{align}\nwhere\n\\begin{align}\nS(\\al,s) &= \\frac{K^-(\\al)iT_0}{(\\al+i\\eps)}\\tilde{f}_0(s)-\\frac{K^+(\\al)iT_0'}{(\\al-i\\eps)}\\frac{\\tilde{g}_0(s)}{\\sqrt{s}}  \\label{S} \\\\\n &= S^-(\\al,s) + S^+(\\al,s),\n\\end{align}\nwhere we have indicated that we wish to determine a sum factorization of the function $S$.  One can employ the pole removal method \\cite{Vei-07} in order to show quite straightforwardly that\n\\begin{align}\nS^+(\\al,s)&=iT_0f_0(s)L_1^++iT_0'\\frac{g_0(s)}{\\sqrt{s}}L_2^+, \\\\\nS^-(\\al,s)&=iT_0f_0(s)L_1^-+iT_0'\\frac{g_0(s)}{\\sqrt{s}}L_2^-,\\label{Sminus}\n\\end{align}\nwhere\n\\begin{align}\nL_1^- &= \\frac{(\\al-i)^{1\/2}-c_-}{\\al+i\\eps}, &  \\quad L_1^+=\\frac{c_-}{\\al+i\\eps}, \\label{eq:L1}, \\\\\nL_2^+ &= -\\frac{(\\al+i)^{-1\/2}-c_+}{\\al-i\\eps}, & L_2^-=-\\frac{c_+}{\\al-i\\eps} \\label{eq:L2}\n\\end{align}\nand\n\\begin{align}\nc_- &= [-i(\\eps+1)]^{1\/2}, & c_+ &= [i(\\eps+1)]^{-1\/2}.\n\\end{align}\n\n\nReferring to \\eqref{elimA1} we can therefore define a function $E(\\al)$ such that\n\\begin{align}\nE(\\al)=\\left\\{ \\begin{array}{ccc}\n-(K^+\\Phi^++S^+)&\\text{on}&\\Omega^+\\backslash \\mathcal{S} \\\\\n-(K^+\\Phi^++S^+)=(K^-\\Psi^-+S^-)&\\text{on}&\\mathcal{S}\\\\\n(K^-\\Psi^-+S^-)&\\text{on}&\\Omega^-\\backslash \\mathcal{S},\n\\end{array} \\right. \\label{EE}\n\\end{align}\nand therefore is analytic on the whole $\\alpha$-complex plane. It can be shown that as $|\\al|\\rightarrow\\infty$, $E(\\alpha)=O(\\al^{-1\/2})$ when $\\al\\in\\Omega^+$ and $E(\\al)=O(\\al^{1\/2})$ when $\\al\\in\\Omega^-$. As such $E(\\al)=o(\\al)$ as $|\\al|\\rightarrow\\infty$. This together with the analyticity of $E(\\al)$ and the extended Liouville theorem (see for example \\cite{Nob-88}), implies that $E(\\al)$ is constant. However, we also know that $E(\\al)\\rightarrow 0$ as $|\\al|\\rightarrow\\infty$ and $\\al\\in\\Omega^+$. We can then conclude that $E(\\al)=0$ everywhere. Given this, we therefore have from \\eqref{EE}\n\\begin{align}\n\\Phi^+ = -\\frac{S^+}{K^+}\\quad \\text{and} \\quad \\Psi^- &= -\\frac{S^-}{K^-}.\n\\end{align}\nFrom the original expressions for $A_1$ determined from the boundary conditions we can show that\n\\begin{align}\nA_1(\\al,s) &= \\frac{iT_0}{(\\al+i\\eps)}\\tilde{f}_0(s) +\\Psi^- \\\\\n&= \\frac{iT_0 c_- \\tilde{f}_0(s)}{(\\al+i\\eps)(\\al-i)^{1\/2}} + \\frac{iT_0' c_+ \\tilde{g}_0(s)}{(\\al-i\\eps)(\\al-i)^{1\/2}\\sqrt{s}}\n\\end{align}\nwhere  \\eqref{Kminus}, \\eqref{eq:L1}, \\eqref{eq:L2} and \\eqref{Sminus} have all been used. Referring to \\eqref{Thetasol}, the solution in transform space is therefore\n\\begin{align}\n\\Theta(x_0,\\al,s)\n &= \\left(\\frac{iT_0 f \\tilde{f}_0(s)}{(\\al+i\\eps)(\\al-i)^{1\/2}} + \\frac{iT_0' g \\tilde{g}_0(s)}{(\\al-i\\eps)(\\al-i)^{1\/2}\\sqrt{s}}\\right)\\exp(-(\\al^2+1)^{1\/2}x_0).\n\\end{align}\n\n\n\nFormally inverting the Fourier transform and using \\eqref{x0y0} gives the Laplace transformed solution as\n\\begin{align}\n\\tilde{T}(x,y,\\al) &= \\frac{iT_0\\tilde{f}_0(s)}{2\\pi}I_1(x,y,s)+\\frac{iT_0'\\tilde{g}_0(s)}{2\\pi\\sqrt{s}}I_2(x,y,s), \\label{eq:defI1I2raph}\n\\end{align}\nwhere\n\\begin{align}\nI_1=c_-\\int_{-\\infty}^{\\infty}\\frac{e^{-[(\\al^2+1)^{1\/2}x+i\\al y]\\sqrt{s}}}{(\\al+i\\eps)(\\al-i)^{1\/2}} \\, \\mathrm{d}\\al \\quad \\text{and} \\quad I_2=c_+\\int_{-\\infty}^{\\infty}\\frac{e^{-[(\\al^2+1)^{1\/2}x+i\\al y]\\sqrt{s}}}{(\\al-i\\eps)(\\al-i)^{1\/2}} \\, \\mathrm{d}\\al \\label{I1I2}\n\\end{align}\n\n\\section{Semi-analytical inversion via a Cagniard-de Hoop approach} \\label{CDH}\n\nMotivated by the Cagniard-de Hoop technique, let us introduce polar coordinates $r$ and $\\theta$ related to $x$ and $y$ in the usual manner, i.e.\\ $x=r\\cos\\theta, y=r\\sin\\theta$ where $\\theta\\in[-\\pi\/2,\\pi\/2]$ and introduce the parameter $\\beta$ via the expression\n\\begin{align}\n\\beta r &= (\\al^2+1)^{1\/2} x + i\\al y\n\\end{align}\nso that\n\\begin{align}\n\\beta &= (\\al^2+1)^{1\/2} \\cos\\theta + i\\al \\sin\\theta.\n\\end{align}\nInverting for $\\alpha$ we therefore determine the two paths $B_+$ and $B_-$ in the right and left halves of the $\\alpha$-plane\n\\begin{align}\n\\al_{\\pm} &= -i\\be\\sin\\theta \\pm \\sqrt{\\be^2-1}\\cos\\theta.\n\\end{align}\nIn the $\\alpha$-plane, with $\\be\\in[1,\\infty)$ these paths start at $\\al=-i\\sin\\theta$ and move off to infinity either in the upper half-plane ($\\theta\\in(-\\pi\/2,0)$, see Fig.\\ \\ref{fig:alphaplane1}) or the lower half-plane ($\\theta\\in(0,\\pi\/2)$, see Fig.\\ \\ref{fig:Blowerraph}).\n\nSince $B^{\\pm}$ are steepest descent paths for the integrals, the idea is to deform the integrals \\eqref{I1I2} from the real line onto these to aid convergence. In classical Cagniard de Hoop problems this frequently permits one to write the $\\al$ integral in the form of a Laplace transform of a function that is independent of $s$ and thus we can determine the inverse transform immediately, thus rendering explicit solutions. Here we are not so fortunate, the function will not be independent of $s$ but nevertheless we are able to make significant progress due to the fact that the inverse Laplace transform integral can be determined analytically in many important cases as we shall see shortly. This leaves the solution in a single integral form that is rapidly convergent.\n\nAt this point note that\n\\begin{align}\n\\lim_{\\eps\\rightarrow 0} c_- = (e^{-i\\pi\/2})^{1\/2} = e^{-i\\pi\/4} = c \\quad \\text{and} \\quad\n\\lim_{\\eps\\rightarrow 0} c_+ = (e^{i\\pi\/2})^{-1\/2} = e^{-i\\pi\/4} = c\n\\end{align}\nand let us consider the case of negative and positive $\\theta$ separately.\n\n\n\\subsection{The case of $\\theta\\in[-\\pi\/2,0)$} \\label{sec:negtheta}\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.6]{pathpostheta.eps}\n\\end{center}\n\\caption{Illustrating the location of the paths $B^\\pm$ when $\\theta\\in[-\\pi\/2,0)$.  The poles at $\\al=\\pm i\\eps$ and branch point at $\\al=i$ are also depicted where the branch cut passes to infinity vertically along the imaginary axis.}\n\\label{fig:alphaplane1}\n\\end{figure}\n\n\\subsubsection{Evaluation of $I_1$}\n\nReferring to \\eqref{I1I2} and Fig.\\ \\ref{fig:alphaplane1}, we see that the integrand of $I_1$ has a pole at $\\al=-i\\eps$ and a branch point at $\\al=i$. Deforming the contour from the real axis to the $B^\\pm$ contour in the $\\al$-plane we do not cross any singularities and hence we find that\n\\begin{align}\nI_1\n&=c_-\\int_{B^+}\\frac{e^{-[(\\al^2+1)^{1\/2}x+i\\al y]\\sqrt{s}}}{(\\al+i\\eps)(\\al-i)^{1\/2}}\\, \\mathrm{d}\\al-c_-\\int_{B^-}\\frac{e^{-[(\\al^2+1)^{1\/2}x+i\\al y]\\sqrt{s}}}{(\\al+i\\eps)(\\al-i)^{1\/2}}\\, \\mathrm{d}\\al \\nonumber\\\\\n&=I_1^+-I_1^- \\label{I1s}\n\\end{align}\nOn $B^\\pm$, we have $\\al=\\al_\\pm=-i\\be\\sin\\theta \\pm \\sqrt{\\be^2-1}\\cos\\theta$ and so we have\n\\begin{align}\n\\mathrm{d}\\al=(-i\\sin\\theta\\pm\\be\\cos\\theta(\\be^2-1)^{-1\/2})\\mathrm{d}\\be.\n\\end{align}\nNoting that $\\be\\in[1,\\infty)$ is a parametrization of the paths $B^{\\pm}$ we can rewrite $I_1^\\pm$ as follows, taking the limit as $\\eps\\rightarrow 0$,\n\\begin{align}\n\\lim_{\\eps \\rightarrow 0} I_1^\\pm=c\\int_1^\\infty  \\frac{(-i\\sin\\theta\\pm\\be\\cos\\theta(\\be^2-1)^{-1\/2})}{\\al_\\pm(\\al_\\pm-i)^{1\/2}} e^{-\\be r \\sqrt{s}} \\,\\mathrm{d}\\be\n\\end{align}\nTherefore from \\eqref{I1s} we determine the form\n\\begin{align}\n\\lim_{\\eps \\rightarrow 0} I_1\n&= c\\int_1^\\infty \\mathcal{F}(\\be,\\theta) e^{-\\be r \\sqrt{s}} \\,\\mathrm{d}\\be, \\label{eq:limL1raph}\n\\end{align}\nwhere\n\\begin{align}\n\\mathcal{F}(\\be,\\theta)=&-i\\sin\\theta\\left(\\frac{1}{\\al_+(\\al_+-i)^{1\/2}}-\\frac{1}{\\al_-(\\al_--i)^{1\/2}} \\right)\\nonumber \\\\\n&+\\frac{\\be}{(\\be^2-1)^{1\/2}}\\cos\\theta\\left( \\frac{1}{\\al_+(\\al_+-i)^{1\/2}}+\\frac{1}{\\al_-(\\al_--i)^{1\/2}}\\right). \\label{eq:calFraph}\n\\end{align}\n\nAs an aside, we note by deforming into the lower half-plane that\n\\begin{align}\n\\int_1^{\\infty}\\mathcal{F}(\\be,\\theta)\\,\\mathrm{d}\\be &=\\lim_{\\epsilon \\rightarrow 0} \\int_{-\\infty}^{\\infty} \\frac{1}{(\\al+i \\epsilon)(\\al-i)^{1\/2}}\\,\\mathrm{d}\\al\\\\\n &= - \\frac{2i\\pi}{c}. \\label{eq:easynegraph}\n\\end{align}\n\n\\subsubsection{Evaluation of $I_2$}\n\nReferring to \\eqref{I1I2} and Fig.\\ \\ref{fig:alphaplane1}, we see that the integrand of $I_2$ has a simple pole at $\\al=i\\eps$ and a branch point at $\\al=i$. Deforming the contour from the real axis to the $B^\\pm$ contour in the $\\al$-plane we cross the simple pole and pick up its residue. Accounting for this contribution we find that\n\\begin{align}\nI_2\n&= c_+\\int_{B^+} \\frac{e^{-[(\\al^2+1)^{1\/2}x+i\\al y]\\sqrt{s}}}{(\\al-i\\eps)(\\al-i)^{1\/2}} \\, \\mathrm{d}\\al-c_+\\int_{B^-} \\frac{e^{-[(\\al^2+1)^{1\/2}x+i\\al y]\\sqrt{s}}}{(\\al-i\\eps)(\\al-i)^{1\/2}} \\, \\mathrm{d}\\al+R_2\\nonumber\\\\\n&=I_2^+-I_2^-+R_2 \\label{I2s}\n\\end{align}\nwhere\n\\begin{align}\nR_2\n&=2\\pi i c_+\\frac{e^{-(((i\\eps)^2+1)^{1\/2} \\cos\\theta + i(i\\eps) \\sin\\theta)r\\sqrt{s}}}{(i\\eps-i)^{1\/2}}\n\\end{align}\nand in particular,\n\\begin{align}\n\\lim_{\\eps \\rightarrow 0} R_2\n&=2i\\pi e^{-x\\sqrt{s}}. \\label{R2s}\n\\end{align}\nNoting that $\\be\\in[1,\\infty)$ is a parametrization of the paths $B^{\\pm}$ we can rewrite $I_2^\\pm$ as follows, taking the limit as $\\eps\\rightarrow 0$,\n\\begin{align}\n\\lim_{\\eps \\rightarrow 0} I_2^\\pm= c\\int_1^\\infty  \\frac{(-i\\sin\\theta\\pm\\be\\cos\\theta(\\be^2-1)^{-1\/2})}{\\al_\\pm(\\al_\\pm-i)^{1\/2}} e^{-\\be r \\sqrt{s}} \\,\\mathrm{d}\\be.\n\\end{align}\nTherefore from \\eqref{I2s} and \\eqref{R2s} we determine the form\n\\begin{align}\n\\lim_{\\eps \\rightarrow 0} I_2\n&= c\\int_1^\\infty \\mathcal{F}(\\be,\\theta) e^{-\\be r \\sqrt{s}} \\,\\mathrm{d}\\be+2\\pi i e^{-x\\sqrt{s}}. \\label{eq:limL2raph}\n\\end{align}\n\n\n\\subsubsection{An expression for $\\mathcal{T}(r,\\theta,t)$}\n\nWe show in Appendix \\ref{appendix} that\n\\begin{align}\nc\\mathcal{F}(\\be,\\theta)=\\frac{\\sqrt{2}}{i}\\mathcal{G}(\\be,\\theta), \\label{eq:calGraph}\n\\end{align}\nwhere $\\mathcal{G}$ is a real-valued function. As such, using this together with \\eqref{eq:defI1I2raph}, \\eqref{eq:limL1raph}, \\eqref{eq:limL2raph} and \\eqref{eq:calGraph} we find the following expression for $\\tilde{\\mathcal{T}}=\\lim_{\\eps \\rightarrow 0}\\tilde{T}$\n\\begin{align}\n\\tilde{\\mathcal{T}}(r,\\theta,s)&=\\frac{iT_0\\tilde{f}_0(s)}{2\\pi}\\lim_{\\eps \\rightarrow 0}I_1(x,y,s)+\\frac{iT_0'\\tilde{g}_0(s)}{2\\pi\\sqrt{s}}\\lim_{\\eps \\rightarrow 0}I_2(x,y,s)\\nonumber\\\\\n&= \\frac{1}{\\sqrt{2}\\pi}\\left(T_0\\tilde{f}_0(s) + T_0'\\frac{\\tilde{g}(s)}{\\sqrt{s}}\\right) \\int_1^\\infty \\mathcal{G}(\\be,\\theta) e^{-\\be r \\sqrt{s}} \\,\\mathrm{d}\\be -\\frac{T_0'\\tilde{g}_0(s)}{\\sqrt{s}} e^{-x\\sqrt{s}}.\n\\end{align}\nFinally we recover $\\mathcal{T}$ by taking the inverse Laplace Transform,\n\\begin{align}\n\\mathcal{T}(r,\\theta,t)\n&=\\frac{T_0}{\\sqrt{2}\\pi}\\int_1^\\infty \\mathcal{G}(\\be,\\theta) \\mathcal{T}_1(r,\\be,t)\\, \\mathrm{d}\\be+\\frac{T_0'}{\\sqrt{2}\\pi}\\int_1^\\infty\\mathcal{G}(\\be,\\theta)\\mathcal{T}_2(r,\\be,t)\\, \\mathrm{d}\\be \\nonumber \\\\\n& \\hspace{4cm}-T_0'\\mathcal{T}_2(r,\\cos\\theta,t),\n\\end{align}\nwhere\n\\begin{align}\n\\mathcal{T}_1(r,\\be,t) &= \\frac{1}{2i\\pi}\\int_{\\si-i\\infty}^{\\si+i\\infty}\\tilde{f}_0(s)e^{-\\be r \\sqrt{s}}e^{st}\\, \\mathrm{d}s, \\label{eq:calT1raph}\\\\\n\\mathcal{T}_2(r,\\be,t) &=\\frac{1}{2i\\pi}\\int_{\\si-i\\infty}^{\\si+i\\infty}\\frac{\\tilde{g}_0(s)}{\\sqrt{s}}e^{-\\be r \\sqrt{s}}e^{st} \\label{eq:calT2raph}\n\\end{align}\nand where as usual $\\si\\in\\mathbb{R}$ is chosen here such that all singularities of the integrands are to the left of the line $s=\\si$.\n\n\\subsection{The case of $\\theta\\in(0,\\pi\/2]$} \\label{sub:postheta}\n\nThis case follows entirely analogously to the negative $\\theta$ scenario, the difference here being that the paths $B_{\\pm}$ reside in the lower half of the complex $\\al$-plane and as such the pole that leads to a contribution to the integral is at $\\al=-i\\eps$. We find that\n\\begin{align}\n\\cT(r,\\theta,t) &= \\frac{T_0}{\\sqrt{2}\\pi} \\int_1^{\\infty} \\mathcal{G}(\\be,\\theta) \\cT_1(r,\\be,t) \\, \\mathrm{d}\\be\n+\\frac{T_0'}{\\sqrt{2}\\pi}\\int_1^{\\infty} \\mathcal{G}(\\be,\\theta) \\cT_2(r,\\be,t) \\, \\mathrm{d}\\be \\nonumber \\\\\n& \\hspace{4cm} + T_0 \\cT_1(r,\\cos\\theta,t).\n\\end{align}\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[scale=0.6]{pathnegtheta.eps}\n\\end{center}\n\\caption{Illustrating the location of the paths $B^\\pm$ when $\\theta\\in(0,\\pi\/2]$. The poles at $\\al=\\pm i\\eps$ and branch point at $\\al=i$ are also depicted where the branch cut passes to infinity vertically along the imaginary axis.}\n\\label{fig:Blowerraph}\n\\end{figure}\n\nAs an aside, for $\\theta\\in(0,\\pi\/2]$, analogously to the derivation of \\eqref{eq:easynegraph} we have\n\\begin{align}\n\\int_1^{\\infty}\\mathcal{F}(\\be,\\theta)\\, \\mathrm{d}\\be &= 0  \\label{eq:intFzero}\n\\end{align}\nTherefore using, \\eqref{eq:easynegraph}, \\eqref{eq:calGraph} and \\eqref{eq:intFzero} we have\n\\begin{align}\n\\frac{1}{\\pi\\sqrt{2}}\\int_1^{\\infty}\\mathcal{G}(\\be,\\theta)\\, \\mathrm{d}\\be&=1- H(\\theta)\n\\label{eq:magicidentity}\n\\end{align}\nwhere $H(\\theta)$ is the Heaviside step function.\n\n\n\n\\subsection{A summary of the solution}\n\nCombining the results from sections \\ref{sec:negtheta} and \\ref{sub:postheta}, we can write down the solution for all values of $\\theta$,\n\\begin{align}\n\\cT(r,\\theta,t)\n&= \\frac{T_0}{\\pi \\sqrt{2}} \\int_1^{\\infty} \\mathcal{G} (\\beta, \\theta)\n  \\mathcal{T}_1 (r, \\beta, t) \\mathrm{d} \\beta + \\frac{T_0'}{\\pi \\sqrt{2}}\n  \\int_1^{\\infty} \\mathcal{G} (\\beta, \\theta) \\mathcal{T}_2 (r, \\beta, t)\n  \\mathrm{d} \\beta \\nonumber\\\\\n  & + H(\\theta) T_0 \\mathcal{T}_1 (r, \\cos \\theta, t) - (1\n  - H(\\theta)) T_0' \\mathcal{T}_2 (r, \\cos \\theta, t).\n\\label{eq:generalsolution1}\n\\end{align}\nBoth integrals in \\eqref{eq:generalsolution1} and the additional term have a discontinuity at $\\theta=0$ and as such this form is not particularly ``clean''. We are able to improve upon this form, using \\eqref{eq:magicidentity} to generate\n\\begin{multline}\n\\cT(r,\\theta,t) = \\frac{T_0}{\\pi \\sqrt{2}} \\int_1^{\\infty} \\mathcal{G} (\\beta, \\theta)\n  \\{ \\mathcal{T}_1 (r, \\beta, t) -\\mathcal{T}_1 (r, \\cos \\theta, t) \\}\n  \\mathrm{d} \\beta \\\\\n  + \\frac{T_0'}{\\pi \\sqrt{2}} \\int_1^{\\infty} \\mathcal{G} (\\beta, \\theta)\n  \\{ \\mathcal{T}_2 (r, \\beta, t) -\\mathcal{T}_2 (r, \\cos \\theta, t) \\}\n  \\mathrm{d} \\beta\n  + T_0 \\mathcal{T}_1 (r, \\cos \\theta, t).  \\label{eq:niceexpre}\n\\end{multline}\nEach term in this expression is now continuous across $\\theta=0$. We shall discuss this aspect further in the context of specific examples in the next section.\n\n\\section{Some specific boundary conditions} \\label{sec:particularexamples}\n\n\\subsection{Perfect insulator on $y<0$}\n\nLet us now assume that $T_0'=0$ so that we have a perfect insulator on $y<0$. We shall consider a variety of temperature profiles for $y>0$. The solution is therefore obtained by setting $T_0'=0$ in \\eqref{eq:generalsolution1} or equivalently \\eqref{eq:niceexpre}. As such only $\\cT_1$ enters the analysis.\n\n\n\n\\subsubsection{Step temperature change}\\label{sub:stepchange}\n\nTake the simplest form, $f_0(t)=1$ so that $\\tilde{f}_0(s)=1\/s$ and we need to determine $\\cT_1$ defined in \\eqref{eq:calT1raph}. It transpires that it is convenient to differentiate the expression for $\\cT_1$ with respect to $t$, which enables the inverse Laplace integral to be evaluated analytically in this case\n\\begin{align}\n\\frac{\\mathrm{d}\\cT_1}{\\mathrm{d}t}(r,\\be,t) &= \\frac{1}{2\\pi i}\\int_{\\si-i\\infty}^{\\si+i\\infty}e^{-\\be r\\sqrt{s}}e^{st}\\lit ds = \\frac{r\\be}{2\\sqrt{\\pi}t^{3\/2}}e^{-r^2\\be^2\/(4t)}.\n\\end{align}\nWe then integrate (definitely) with respect to $t$ with a lower limit of $t=0$, finding that\n\\begin{align}\n\\cT_1(r,\\be,t) &= \\tn{Erfc}\\left(\\frac{r\\be}{2\\sqrt{t}}\\right).\n\\label{eq:Tcal1_simple}\n\\end{align}\nAppealing to \\eqref{eq:generalsolution1} we have as our solution\n\\begin{align}\n\\cT(r,\\theta,t) &= \\frac{T_0}{\\sqrt{2}\\pi}\\int_1^{\\infty} \\mathcal{G}(\\be,\\theta) \\tn{Erfc}\\left(\\frac{r\\be}{2\\sqrt{t}}\\right) \\, \\mathrm{d}\\be\n+H(\\theta)T_0 \\tn{Erfc}\\left(\\frac{r\\cos\\theta}{2\\sqrt{t}}\\right)\n\\label{eq:discontinuous_step_T}\n\\end{align}\nnoting that $G(\\be,\\theta)$ is the real function defined in \\eqref{calG}. Each term of \\eqref{eq:discontinuous_step_T} is easily computed numerically, and, in Fig.\\ \\ref{localcoords} we plot the resulting temperature profile on the horizontal axis against $y$ running vertically, at time $t=0.02$ for two values of $x$, $x=0.05$ and $x=0.2$. The circles are results taken from a finite element solution of the same problem in COMSOL and provide validation of the present semi-analytical scheme.\n\\begin{figure}[h!]\n\\psfrag{y}{$y$}\n\\psfrag{T}{$\\cT$}\n\\begin{center}\n\\includegraphics[scale=0.9]{heat1.eps}\n\\end{center}\n\\caption{Thermal field $\\cT(x,y,t)$ at $x=0.05$ (red) and $x=0.2$ (blue) when $t=0.02$ for the case of a perfect insulator on $y<0$ and a step change temperature increase at $t=0$ on $y>0$. Circles are predictions from solutions to the same problem using finite elements methods in COMSOL, which provides validation of the present semi-analytical method.}\n\\label{localcoords}\n\\end{figure}\n\nNote that the temperature profile is continuous across the $x$-axis, i.e.\\ $\\theta=0$. Rather interestingly, both terms on the right hand side of \\eqref{eq:discontinuous_step_T} are discontinuous across $\\theta=0$, as is shown in Fig.\\ \\ref{fig:discontinuous_terms} but the two discontinuities compensate exactly to yield a continuous temperature profile.\n\\begin{figure}[h!]\n\\centerline{\n\\includegraphics[width=0.5\\textwidth]{Integral1_dis.eps}\n\\includegraphics[width=0.5\\textwidth]{Integral2_dis.eps}}\n\\caption{Plot of the first (left) and\n  second (right) terms on the right hand side of \\eqref{eq:discontinuous_step_T} for $r=0.05$, $t=0.02$ and $\\theta \\in [- \\pi \/ 2,\n  \\pi\/2]$.}\n\\label{fig:discontinuous_terms}\n\\end{figure}\nIf instead of the form \\eqref{eq:generalsolution1}, we use \\eqref{eq:niceexpre}, the solution is written as\n\\begin{multline}\n\\cT(r,\\theta,t) =\\frac{T_0}{\\pi \\sqrt{2}} \\int_1^{\\infty} \\mathcal{G} (\\beta, \\theta)\n  \\left\\{ \\tn{Erfc}\\left(\\frac{r\\be}{2\\sqrt{t}}\\right) -\\tn{Erfc}\\left(\\frac{r\\cos(\\theta)}{2\\sqrt{t}}\\right)\\right\\}\n  \\mathrm{d} \\beta\\\\\n  +T_0 \\tn{Erfc}\\left(\\frac{r\\cos(\\theta)}{2\\sqrt{t}}\\right).\n\\label{eq:continuous_step_T}\n\\end{multline}\nThis expression is also straightforward to evaluate numerically and (obviously) gives the same results as those presented in Fig.\\ \\ref{localcoords}, but this time, as shown in Fig.\\ \\ref{fig:continuous_terms}, both terms on the right hand side of \\eqref{eq:continuous_step_T} are continuous across $\\theta=0$.\n\n\\begin{figure}[h!]\n \\centerline{\\includegraphics[width=0.5\\textwidth]{Continuous_term_1.eps}\\includegraphics[width=0.5\\textwidth]{Continuous_term_2.eps}}\n  \\caption{Plot of the first (left) and\n  second (right) terms on the right hand side of \\eqref{eq:continuous_step_T} for $r=0.05$, $t=0.02$ and $\\theta \\in [- \\pi \/ 2,\n  \\pi\/2]$.}\n\\label{fig:continuous_terms}\n\\end{figure}\n\nFinally, in Fig.\\ \\ref{fig:temp1}, we plot the two dimensional temperature contour profile on the $(x,y)$ plane at $t=0.02$ illustrating how the distribution spreads out from the upper half-plane.\n\n\\begin{figure*}[h!]\\centering\n\\includegraphics[scale=1.0]{perfect_2D.eps}\n\\caption{Contour plot of the temperature profile when $t=0.02, T_0=1, T_0'=0$, i.e.\\ a perfect insulator on $y<0$ and step change in temperature on $y>0$. We see how the thermal field propagates into $y<0$.}\n\\label{fig:temp1}\n\\end{figure*}\n\n\\subsection{Imperfect insulator on $y<0$}\n\nLet us now consider the case when $T_0'\\neq 0$ and let us take $f(t)=g(t)=1$ so that $\\tilde{f}_0(s)=\\tilde{g}_0(s)=1\/s$ and now both $\\cT_1$ and $\\cT_2$ play a role. Once again it is convenient to differentiate with respect to $t$ in order to evaluate the inverse Laplace transforms\nand subsequently integrating these expressions definitely with respect to $t$ with a lower limit of $t=0$ yields\n\\begin{align}\n\\cT_1(r,\\be,t) &= \\tn{Erfc}\\left(\\frac{r\\be}{2\\sqrt{t}}\\right), \\\\\n\\cT_2(r,\\be,t) &= 2\\sqrt{\\frac{t}{\\pi}} e^{-\\be^2r^2\/(4t)}- r \\be \\tn{Erfc}\\left(\\frac{r\\be}{2\\sqrt{t}}\\right).\n\\end{align}\nEither of the expressions \\eqref{eq:generalsolution1} or \\eqref{eq:niceexpre} then recover the temperature profile. Both formulations are easily computed numerically and give rise to a continuous temperature profile, as seen in Fig.\\ \\ref{fig:tempprof5.2} where the profile is plotted at $t=0.02$ for $x=0.05$ and $x=0.2$. The contour plot of the thermal field at $t=0.02$ is given in Fig.\\ \\ref{fig:temp2}.\n\n\\begin{figure}[h!]\n\\psfrag{y}{$y$}\n\\psfrag{T}{$\\cT$}\n\\begin{center}\n\\includegraphics[scale=0.9]{Imperfect_line.eps}\n\\end{center}\n\\caption{Thermal field $\\cT(x,y,t)$ at $x=0.05$ (red) and $x=0.2$ (blue) when $t=0.02$ for the case of an imperfect insulator on $y<0$ and a step change temperature increase at $t=0$ on $y>0$. Circles are predictions from solutions to the same problem using finite elements methods in COMSOL, which provides validation of the present semi-analytical method.}\n\\label{fig:tempprof5.2}\n\\end{figure}\n\n\n\\begin{figure*}[hbt!]\\centering\n\\includegraphics{imperfect_2D.eps}\n\\caption{Contour plot of the temperature profile when $t=0.02, T_0=1, T_0'=1$, i.e.\\ an imperfect insulator on $y<0$ and step change in temperature on $y>0$. We see how the thermal field propagates into $y<0$.}\n\\label{fig:temp2}\n\\end{figure*}\n\n\\subsection{Continuous ramp up and down superposed on a step change in $y>0$}\n\nThus far we have considered only cases where $f_0$ and $g_0$ are constant, accommodating for the step change at $t=0$ of course. Let us now consider the case when these functions can be unsteady and in particular when $f_0(t)$ is a step and superposed general ramp up and down profile given by\n\\begin{align}\n  f_0 (t) & =  H(t) + (t - a) H(t - a) + 2 (b - t)\n  H(t - b) + (t - (2 b - a)) H(t - (2 b - a)) \\label{f0ramp}\n\\end{align}\nand illustrated in Fig.\\ \\ref{fig:rampupdown}.\n\n\n\n\\begin{figure}[h]\n\\psfrag{f}{$f_0(t)$}\n\\psfrag{t}{$t$}\n\\psfrag{0}{$0$}\n\\psfrag{a}{$a$}\n\\psfrag{b}{$b$}\n\\psfrag{d}{$2b-a$}\n\\psfrag{1}{$1$}\n\\psfrag{c}{$1+b-a$}\n\\psfrag{e}{$0$}\n  \\centering\\includegraphics[width=0.8\\textwidth]{ramp_profile.eps}\n  \\caption{Plot of $f_0 (t)$ as defined by \\eqref{f0ramp}. A ramp up and down is superposed on a step change unit temperature profile.}\n\t\\label{fig:rampupdown}\n\\end{figure}\n\n\nThe Laplace transform of $f_0 (t)$ is\n\\begin{align}\n  \\tilde{f}_0 (s) & =  \\frac{1}{s} + \\frac{e^{- as}}{s^2} - 2 \\frac{e^{-\n  bs}}{s^2} + \\frac{e^{- (2 b - a) s}}{s^2} \\nonumber \\\\\n  & =  \\tilde{f}_0^{(1)} (s) + \\tilde{f}_0^{(2)} (s) + \\tilde{f}_0^{(3)} (s)\n  + \\tilde{f}_0^{(4)} (s)\n\\end{align}\nand the resulting expression for $\\mathcal{T}_1$ is\n\\begin{align}\n  \\mathcal{T}_1 (r, \\beta, t) & = \\frac{1}{2 i \\pi} \\int_{\\si - i \\infty}^{\\si +\n  i \\infty} (\\tilde{f}_0^{(1)} (s) + \\tilde{f}_0^{(2)} (s) + \\tilde{f}_0^{(3)}\n  (s) + \\tilde{f}_0^{(4)} (s)) e^{- \\beta r \\sqrt{s}} e^{st} \\mathd s \\nonumber \\\\\n  & =  \\mathcal{T}_1^{(1)} (r, \\beta, t) +\\mathcal{T}_1^{(2)} (r, \\beta, t)\n  +\\mathcal{T}_1^{(3)} (r, \\beta, t) +\\mathcal{T}_1^{(4)} (r, \\beta, t).\n\\end{align}\nThe case of $\\mathcal{T}_1^{(1)}$ has already been dealt with in Section \\ref{sub:stepchange}\nand is thus given by \\eqref{eq:Tcal1_simple}. Moreover, $\\mathcal{T}_1^{(2)} (r,\n\\beta, t), \\hspace{1em} \\mathcal{T}_1^{(3)} (r, \\beta, t)$ and\n$\\mathcal{T}_1^{(4)} (r, \\beta, t)$ have a very similar structure, so we only\nneed to consider only the case of $\\mathcal{T}_1^{(2)} (r, \\beta, t)$ in detail. In fact, we have\n\\begin{align}\n  \\mathcal{T}_1^{(2)} (r, \\beta, t) & =  \\frac{1}{2 i \\pi}\\int_{\\si - i \\infty}^{\\si + i \\infty}\n  \\tilde{f}_0^{(2)} (s) e^{- \\beta r \\sqrt{s}} e^{st} \\mathd s \\nonumber \\\\\n  & =   \\frac{1}{2 i \\pi}\\int_{\\si - i \\infty}^{\\si + i \\infty} \\frac{e^{- as}}{s^2} e^{- \\beta r\n  \\sqrt{s}} e^{st} \\mathd s\n\\end{align}\nDifferentiating $\\mathcal{T}_1^{(2)}$ twice with respect to time, we obtain\n\\begin{align}\n  \\frac{\\partial^2 \\mathcal{T}_1^{(2)}}{\\partial t^2} (r, \\beta, t) & =\n  H(t - a) \\frac{r \\beta}{2 \\sqrt{\\pi} (t - a)^{3 \/ 2}} e^{-\n  \\frac{r^2 \\beta^2}{4 (t - a)}},\n\\end{align}\nwhich, following the same reasoning as in Section \\ref{sub:stepchange} implies that\n\\begin{align}\n  \\frac{\\partial \\mathcal{T}_1^{(2)}}{\\partial t^{}} (r, \\beta, t) & =\n  H(t - a) \\tmop{Erfc} \\left( \\frac{r \\beta}{2 \\sqrt{t - a}}\n  \\right)\n\\end{align}\nand by definite integration with respect to time, we obtain\n\\begin{align}\n  \\mathcal{T}_1^{(2)} (r, \\beta, t) & =  H(t - a) \\left\\{ \\left( t\n  - a + \\frac{r^2 \\beta^2}{2} \\right) \\tmop{Erfc} \\left( \\frac{r \\beta}{2\n  \\sqrt{t - a}} \\right) - \\frac{r \\beta \\sqrt{t - a}}{\\sqrt{\\pi}} e^{-\n  \\frac{r^2 \\beta^2}{4 t}} \\right\\}.\n\\end{align}\nThe functions $\\mathcal{T}_1^{(3)} (r, \\beta, t)$ and $\\mathcal{T}_1^{(4)} (r,\n\\beta, t)$ are obtained in the same manner. Using these expressions in the general formulation \\eqref{eq:generalsolution1} or \\eqref{eq:niceexpre} enables the solution to be computed rather rapidly. In Fig.\\ \\ref{rampfigs} we plot the resulting thermal field at the locations $(x,y)=(0.1,0.03)$ and $(0.2,0.03)$ in the cases when $T_0=1$ with $T_0'=0$ (left) and $T_0=1$ with $T_0'=1$ (right). We also plot the associated solutions determined previously where no ramp up and down is present in the boundary condition.\n\n\\begin{figure}[h!]\n  \\centerline{\\includegraphics[width=0.5\\textwidth]{Evolution_ramp.eps}\\quad\\includegraphics[width=0.5\\textwidth]{Evolution_ramp_with.eps}}\n  \\caption{Illustrating the effect of the ramp up and down on the evolution of the temperature profile at\n  two different locations $(x, y) = (0.1, 0.03)$ (red) and $(x, y) = (0.2,\n  0.03)$ (blue) for $T_0 = 1$ and $T_0'=0$ (left) or $T_0 = 1$ and $T_0' = 1$ (right). The\n  dashed lines correspond to the profile without the unsteady ramp, and the\n  plain lines correspond to the profile with the unsteady ramp.}\n  \\label{rampfigs}\n\\end{figure}\n\n\n\\section{Concluding remarks} \\label{sec:conc}\n\nEmploying the Wiener-Hopf technique and a Cagniard-de Hoop-type integral, a rapidly convergent integral expression has been determined for a class of transient thermal mixed boundary value problems. The integral is easily computable on a standard desktop PC for a wide range of transient boundary forcings of interest. Here we illustrated the computation for a number of cases, including step-changes in temperature and ramp up and down boundary profiles. Such quasi-analytical expressions are of great utility in order to speed up computations and enable asymptotic analysis close to locations of interest. Future work could include extensions to full elastodynamics and coupled thermoelasticity. In these cases matrix Wiener-Hopf problems will result in general.\n\n\n\\subsection*{Acknowledgements}\n\nParnell is thankful to Universit\\'{e} Paris-Est Cr\\'eteil (UPEC) for funding his visiting position in April 2013 when this work was initiated. He also gratefully acknowledges the Engineering and Physical Sciences Research Council for funding his research fellowship (EP\/L018039\/1). Abrahams thanks the Royal Society for a Wolfson Research Merit award (2013-2018).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{Introduction} Introduction}\nSince it was believed, based on the Mermin-Wagner theorem, that\ntwo-dimensional (2D) crystals are not stable at finite temperature\n\\cite{Mermin68}, it came as a surprise that monolayer graphene could\nbe stabilized on a Si\/SiO$_2$ substrate\n\\cite{Novoselov04,Zhang05,Geim07}. Due to its peculiar properties\nlike, e.g., a linear dispersion leading to Klein tunneling\n\\cite{Katsnelson06}, high room-temperature mobility allowing quantum\nHall steps at 300 K  \\cite{Novoselov07}, a low spin-orbit\ninteraction beneficial for spintronic devices \\cite{Tombros07} or tunable spin-polarized edge states \\cite{Son06},\ngraphene studies have become a major issue in solid-state physics.\nAlready the first transport results not exhibiting weak localization\nlead to the speculation of a curved surface \\cite{Morozov06}\nacting as a phase-breaking field \\cite{McCann06}. Such a curvature was\nindeed observed by microscopic electron diffraction of suspended\nmonolayer graphene sheets \\cite{Meyer07}. The lateral wavelength of\nthe isotropic curvature is estimated to be\n$\\lambda =10-25$~nm with an amplitude of $A\\simeq 1$~nm. The rippling has\nbeen reproduced theoretically by Monte-Carlo simulations with a\npreferential wavelength of 8 nm barely depending on temperature\n\\cite{Fasolino07}. It is argued that the anharmonic coupling between\nbending and stretching modes in graphene causes the rippling and is responsible for\nthe stability of the 2D crystal. Moreover, it has been shown experimentally\nthat rippling  can ultimately limit the mobility of graphene at 300 K, if defects\nare avoided \\cite{Morozov08}\n\nHowever, the corrugation properties of graphene deposited on a substrate as\ntypically used in transport experiments are not clarified.\nIn particular, previous scanning probe studies\n\\cite{Stolyarova07,Ishigami07,Tikhonenko08,Zhang08} revealed only\ncorrugations, which are attributed to the roughness of the underlying\nsubstrate, except for a torn flake manipulated by AFM \\cite{Tikhonenko08}. Thus, it is unclear, if the intrinsic tendency for rippling\npersists on the substrate, thereby ultimately limiting mobility and influencing\nweak localization, which has meanwhile been found, at least, for some samples\n\\cite{Tikhonenko08}. Here, we demonstrate that a regular short wave-length\ncorrugation with $\\lambda\\simeq 15$~nm and $A\\simeq 1$~nm, which is\nnot induced by the substrate, can even prevail the substrate corrugation.\nSince $\\lambda$ and $A$ are close to the values\nfound on suspended graphene\n\\cite{Meyer07}, we attribute this additional corrugation to the intrinsic rippling\nof graphene \\cite{Fasolino07}.\nMoreover, we find that the long-range corrugation implied by the substrate\nhas lower amplitude on graphene than on SiO$_2$ suggesting that our high-mobility\ngraphene is partly suspended between hills of the substrate, which\nmight favor the development of the intrinsic short-scale rippling.\n\n\nThe graphene sample is fabricated by the mechanical exfoliation\ntechnique as described in \\cite{Novoselov04,Lemme07}. Using an\noptical microscope, a graphene flake containing a monolayer region\nis identified. Raman spectroscopy is used to confirm the number of\nlayers with the help of the 2D line \\cite{Ferrari06}.\nFig.~\\ref{fig1}(a) shows three spectra measured in different areas\nof the flake visible in Fig.~\\ref{fig1}(b). In addition to a large\nmonolayer area, we find smaller bi- and multilayer areas as marked\nby 1L, 2L and MuL, respectively. Next, gold contacts with a 10~nm Cr\nseed layer were deposited and structured with a lift-off process.\nExcept for one side, the graphene sample is completely surrounded by\nthe gold electrode as shown in Fig.~\\ref{fig1}(b). The mobility of\ngraphene samples prepared identically but with several contacts\nis $\\mu=1-1.5$ m$^2$\/(Vs) at 300 K \\cite{Lemme08}, i.e. comparable with high-mobility\nvalues obtained by other groups \\cite{Morozov08,Tan07}. In order to\nremove residual resist and adsorbates, the sample was rinsed in\nisopropanol and acetone, baked out to 150$^\\circ$ C in air for four\nhours and, additionally, baked out to the same temperature in\nultrahigh-vacuum (UHV) for three hours.\nThe gold film served as the electrical contact for scanning tunneling microscopy (STM) and was used to prepare the STM tip (etched W) by voltage pulses.\nIn order to find the graphene within the UHV-STM, we used an optical\nmicroscope with a focal length of 30~cm and 5~$\\mu$m\nlateral resolution. Fig.~\\ref{fig1}(c) shows a microscopic image of\nthe tip approaching the graphene flake. The home-built STM\nfeatures an $xy$ stage for lateral positioning and operates in UHV\n($2\\cdot 10^{-7}$ Pa) as described elsewhere\n\\cite{Wiebe04,Geringer08}. The $xy$ stage has been checked to move\naccurately within 10 \\%. Topographic images are recorded applying\na bias $V$ to the tip. Spectroscopic $dI\/dV$ curves are\nmeasured by lock-in technique using a modulation voltage $V_{\\rm\nmod}$ after stabilizing the tip at voltage $V_{\\rm stab}$ and\ncurrent $I_{\\rm stab}$. STM images\nobtained at the intentionally irregular edge of the gold contact are\ndisplayed in the insets of Fig.~\\ref{fig1}(b). They are used for\norientation by STM and aid in finding monolayer, bilayer and\nmultilayer areas on the flake.\n\n\\begin{figure}\n\\includegraphics[width=\\linewidth]{Figure1}\n\\caption{\\label{fig1} (Color online) (a) Raman spectra using laser light of wavelength 532.1~nm.\nThe spectra are recorded on different areas of the graphene flake and the attribution to monolayer, bilayer and multilayer is marked.\n(b) Optical image of the graphene sample with gold contacts. The different areas of monolayer (1L), bilayer (2L) and multilayer (MuL) as identified by Raman spectroscopy are indicated. Insets show two STM images acquired at the edge of the gold contact, which are used for orientation. (c) Microscopic image of the STM tip in tunneling contact with the graphene in UHV recorded with a long-range optical microscope.}\n\\end{figure}\n\nAtomic force microscopy images are taken under ambient conditions in\ntapping mode using either a commercial cantilever with a silicon tip or an\nultrasharp tungsten tip with a tip radius of 1~nm, which is attached at the\nbottom of a Si cantilever \\cite{cantilever}. Imaging has been carried out in the attractive regime \\cite{holscher07} with an oscillation frequency slightly above resonance.\n\nFig.~\\ref{fig2}(b) and (c) show two atomically resolved STM images\nrecorded on the monolayer (1L) and the multilayer (MuL) of graphene,\nrespectively. One observes a hexagonal pattern on the monolayer and\na triangular pattern on the multilayer in accordance with previous\nresults \\cite{Stolyarova07, Zhang08}. The atomic resolution is still\nperceptible at lower resolution in Fig.~\\ref{fig2}(a) and (d), but\nan additional irregular corrugation appears. Line sections shown in\nFig.~\\ref {fig2}(e) reveal that the corrugation height on this\nmonolayer region is about 0.6~nm, a factor of three larger than on\nthe multilayer region. Notice that corrugation heights pretended by\ncharged defects are typically smaller by more than an order of magnitude \\cite{Wittneven98}.\nA representative $dI\/dV\/(I\/V)$ curve of the\nmonolayer, which is known to represent the local density of states\n(LDOS) \\cite{Stroscio86}, is shown in Fig.~\\ref{fig2}(f). As\nexpected for graphene, it shows a V-like shape with the Dirac point\nat about $V=- 20$ mV.\nIt does not change significantly across the area depicted in\nFig.~\\ref{fig2}(d). Note that graphene on SiC(0001)\nexhibits a more complicated $dI\/dV$ spectrum\n\\cite{brar07,rutter07,lauffer08}. Note also, that we\ndid not observe the phonon gap found at 4 K recently \\cite{Zhang08},\nalthough we used several different micro- and macrotips. This is tentatively attributed\nto the different temperature of the two experiments.\nThe good representation of the graphene LDOS by $dI\/dV\/(I\/V)$ and the continuous atomic resolution across Fig.~\\ref{fig2}(d) demonstrates the absence of resist on this part of the surface. Indeed, we found several large areas without resist, but still also areas where a remaining resist is apparent within the STM images.\nThe fact that the Dirac point is observed\nclose to 0 V shows a negligible influence of charging adsorbates,\nwhich are probably removed in UHV.\n\n\\begin{figure*}\n\\includegraphics[width=\\linewidth]{Figure2}\n\\caption{\\label{fig2} (Color online) (a) Constant current STM image\nof multilayer graphene (0.4\\,V, 258\\,pA) (raw data). (b) Higher\nresolution STM image of the same area as in (a) (0.4\\,V, 198\\,pA).\n(c) High resolution STM image of graphene monolayer area (1\\,V,\n394\\,pA). (d) Lower resolution STM image of monolayer graphene\n(0.5\\,V, 292\\,pA) (raw data). (e) Line section along the lines\ndepicted in (a)(MuL) and (d)(1L). (f) Normalized dI\/dV spectrum of\nmonolayer graphene: $V_{\\rm stab}$=0.7\\,V, $I_{\\rm stab}$=300\\,pA,\n$V_{\\rm mod}$=20\\,mV.}\n\\end{figure*}\n\nFig.~\\ref{fig3}(a) shows a large area STM image of the graphene\nmonolayer. One observes a rather regular corrugation with amplitudes\nof more than 1~nm and a preferential distance between hills of about\n15~nm. The histogram \\cite{EPAPS} is Gaussian with a full width at\nhalf maximum (FWHM) of 0.78~nm. The rms roughness is determined to\nbe 0.36~nm. The latter two values fluctuate across the monolayer\narea by about 25 \\% (average rms roughness 0.32~nm), but the\ncorrugation length scale remains constant. For comparison,\nFig.~\\ref{fig3}(c) shows an AFM image of the bare SiO$_2$ surface\nrecorded in tapping-mode. It is obvious that the corrugation on the\nsubstrate is lower and exhibits a larger length scale than the\ncorrugation on graphene. The FWHM of the histogram of\nFig.~\\ref{fig3}(c) is 0.48~nm and the rms roughness is 0.22~nm\n(average of all data $(0.25 \\pm 0.05)$~nm). We used four different\ntips, including two ultrasharp tips with a tip radius of 1~nm\n\\cite{cantilever}, but always observed the same corrugation height\nand length scale, which, in addition, is in good agreement with\nprevious results \\cite{Ishigami07}.\n\n\\begin{figure*}\n\\includegraphics[width=\\linewidth]{Figure3}\n\\caption{\\label{fig3} (Color online) (a), (b) 3D and 2D constant current STM image of\nmonolayer graphene (1\\,V, 207\\,pA). (c) 3D tapping-mode AFM image of the ${\\rm SiO_2}$ substrate (resonance frequency 326.4~kHz, force constant 47~N\/m, excitation frequency 326.5~kHz, oscillation amplitude 18~nm, constant amplitude feedback, setpoint 90\\%).}\n\\end{figure*}\n\nCareful inspection of the graphene monolayer in Fig.~\\ref{fig3}(a) and (b)\nshows a modulation on top of the short-range corrugation exhibiting\na similar length scale as the corrugation on the SiO$_2$ substrate.\nIn order to disentangle the two contributions, we used\ntwo-dimensional autocorrelation functions of the images (a) and (c)\n\\cite{horcas07}, which are shown in Fig.~\\ref{fig4}(a) and (b).\nNote that both figures use the same color scale, i.\\ e.\\ the\nintensities are directly comparable. A long-range structure\nrepresented by a central spot surrounded by four bright areas is\nvisible in both images although with weaker intensity for graphene\n(a). Fig.~\\ref{fig4}(c) shows the autocorrelation function of the\ngraphene after removing the long-range part by high-pass filtering\nusing a smooth (first order Butterworth) cutoff at wavelength\n$\\lambda=20$~nm. A preferential short-range distance is clearly\nvisible by the strong spots around the center, but there is\nadditional multiple correlation albeit not with high symmetry.\nNotice that creep and drift effects are carefully removed from the\nSTM-image using the atomic resolution. We checked the preferential\ndirections visible in both correlation patterns. They are rotating arbitrarily across the SiO$_2$, but we find indication for a preferential orientation on graphene \\cite{EPAPS}.\nFig.~\\ref{fig4}(d) shows\nradial line sections averaged over all angles of the correlation\nfunctions. The SiO$_2$ (black curve) exhibits a correlation length\nof about 25~nm comparable to \\cite{Ishigami07} and, in addition, a\nvery weak maximum at about 50~nm indicating a slight preferential\ndistance between hills. The gray curve shows the radial line section\nof the high-pass filtered image of graphene visible in\nFig.~\\ref{fig4}(c). It exhibits a damped oscillation with a wave\nlength of about 14~nm looking very similar to correlation functions\nof liquids. For comparison, the radial line section of the\nautocorrelation function of the AFM image after applying the same\nhigh-pass filter is shown as a dotted line and does not exhibit any\nstructure. We observe the damped oscillation also after high-pass\nfiltering the original STM image in Fig.~\\ref{fig3}(a) and\ndetermining the autocorrelation function afterwards. In addition, we\nchecked that the obtained wave length is quite robust with respect\nto slight parameter changes of the filtering process. Depending on\nthe filtering procedure and the selection of the minima or maxima\nused for determining the wave length, $\\lambda$ is varying by at\nmaximum $\\pm 2$ \\,nm with a mean value of $\\lambda = 15$~nm.\n\n\\begin{figure}\n\\includegraphics[width=\\linewidth]{Figure4}\n\\caption{\\label{fig4} (Color online) (a) 2D autocorrelation function of the\ndrift corrected STM image of monolayer graphene as shown in\nFig.~\\ref{fig3}(a). (b) 2D autocorrelation function of the AFM image\nof the SiO$_2$ surface as shown in Fig.~\\ref{fig3}(c), same color scale as in (a). (c) FFT high\npass filtered image (1.\\ order Butterworth, 20\\,nm) of the 2D autocorrelation\nfunction shown in (a). (d) Radial line sections of the unfiltered AFM autocorrelation image (b) (black), of the high pass filtered STM autocorrelation image (c) (gray), and of the high pass filtered AFM autocorrelation image (dotted). The inset shows a larger view of the area marked by the dashed line. The black and gray graphs are offset vertically for clarity.}\n\\end{figure}\n\nFrom these results, we conclude that monolayer graphene can exhibit\na rather regular, liquid-like short-scale corrugation not induced by the\nsubstrate. It is similar in height and wavelength to the one\nobserved on suspended graphene \\cite{Meyer07} and calculated for\nfreely suspended graphene by atomic Monte-Carlo methods\n\\cite{Fasolino07}. This observation suggests that the graphene might be\npartly suspended even on the SiO$_2$.\nIn order to verify this, we compared the Fourier components of the\nimages of graphene and SiO$_2$ within the large wavelengths region\n$\\lambda =40-60$~nm \\cite{EPAPS}. We indeed find that graphene\nexhibits an  up to 40~\\% lower amplitude at all wavelengths within\nthis wavelength range. This implies that the graphene does not\nfollow the substrate corrugations exactly, but instead includes\nareas not in contact with the substrate. We believe that this partly\nfree-standing configuration is the prerequisite for the intrinsic\nrippling of graphene. Indeed, we have also found other graphene\nflakes prepared nominally identically by the same person which did\nnot show the intrinsic rippling. This highlights that tiny details\nof the preparation process can have large impacts on the physical\nproperties, maybe explaining the contradicting weak localization\nproperties in different studies \\cite{Tikhonenko08,Morozov06}. A\ncomparison of Raman data of several samples indicates that samples\nwith intrinsic rippling are more frequent than samples without\n\\cite{EPAPS}. Future systematic studies using our method to\ndetermine the intrinsic rippling are required.\n\nIn summary, we have investigated high-mobility graphene flakes on SiO$_2$ by\nUHV-STM and scanning tunneling spectroscopy. A mesoscopic corrugation decreasing in\namplitude with increasing thickness\nnot induced by the substrate is identified. It is rather regular exhibiting liquid-like\ncorrelation properties with a preferential wavelength\nof about 15~nm and a rms roughness of 0.32~nm. Since the long-range\ncorrugation of the substrate is additionally visible on graphene,\nbut with smaller amplitude than on the substrate, we conclude that\nthe rippled graphene is partly freely suspended.\n\nWe gratefully acknowledge useful\ndiscussions with G. G\\\"untherodt, U. D. Schwarz and H. Kurz and financial support by FOR\n912, project TP 6 of the Deutsche Forschungsgemeinschaft and by the\nGerman Federal Ministry of Education and Research (BMBF) under\ncontract number NKNF 03X5508 (\"ALEGRA\").\n\n\\section*{Supplementary information}\n\n\\subsection*{Histogram of Corrugation}\nFigure \\ref{figs1} shows representative histograms of the height values on the SiO$_2$ substrate\nand the graphene sample as obtained\nby atomic force microscopy (AFM) and scanning tunneling microscopy (STM), respectively. The histograms are based on\nimages of 200 $\\times$ 200 nm$^2$ including 512 $\\times$ 512 measurement points each. These images are shown in\nFig. 3 of the main text. The histogram curves are normalized in\norder to enclose the same integral area. Gaussian fits are added as guides to the eye. Obviously, the\nheight histogram of Graphene is broader exhibiting a full width at half maximum (FWHM) of 0.78 nm, while\nSiO$_2$ exhibits a FWHM of only 0.48 nm. While the latter value is quite robust across the surface, the former\nvalue is changing by about 25 \\% across the surface, but is on average larger than the FWHM on SiO$_2$.\nThis demonstrates that the rippling of graphene is not limited to substrate corrugations.\n\n\\begin{figure}\n\\includegraphics[width=\\linewidth]{Figure_supp1}\n\\caption{\\label{figs1} (Color online) Height histograms of SiO$_2$ (blue) and graphene (red) corresponding to the images in Fig.\\ 3 of the main text. On average, the graphene histogram exhibits a larger FWHM value compared to SiO$_2$.}\n\\end{figure}\n\n\\subsection*{Tip Dependence of Substrate Corrugation}\nFigure \\ref{figs2} shows two AFM images of the SiO$_2$ substrate obtained in tapping mode by two different tips.\nRepresentative scanning electron microscopy (SEM) images of the tips provided by the\nsupplier are added as insets. They are cross-checked by SEM images after the measurements. These images are in agreement\nwith the supplier images,\nbut due to a moderate background pressure, the ultrasharp W-tip fastly burnt away during SEM imaging.\nThe left AFM image is measured by a cantilever with a Si tip having an apex radius of about 10 nm, while the right AFM image is measured\nwith a W tip of nominal tip apex radius of 1 nm mounted at the bottom of the Si tip. Both images exhibit a very similar lateral scale of the corrugation.\nThe rms values are 220 pm and 270 pm, respectively, showing that the tip properties do not significantly change the\nobserved corrugation of the substrate. We cross-checked this result using more than ten images obtained with both tips, which lead to fluctuations in rms values of less than $\\pm$ 10\\%.\n\n\\begin{figure*}\n\\includegraphics[width=\\linewidth]{Figure_supp2}\n\\caption{\\label{figs2} (Color online) AFM images of SiO$_2$ with tips of different radii: Si tip NSC15 with 10 nm (a) and W\/Si tip DP15\/HiRes-W\/AlBS with 1 nm (b), respectively. Both images show the same color scale. Insets: SEM images of similar tips, with kind permission of MikroMasch (www.spmtips.com).}\n\\end{figure*}\n\n\\subsection*{Comparison of Correlation Functions}\nFigure \\ref{figs3} shows a comparison of the correlation functions of the topographic images of graphene (top row) and SiO$_2$ (bottom row).\nThe left column shows\nthe correlation functions directly obtained from the topography. The middle and right column show the high-frequency\nand low-frequency part of the correlation function separated by a smooth first-order butterworth filter with cutoff\nwavelength $\\lambda=20$ nm. While the large wavelengths visible in the right columns are quite similar in both images, the rather regular short wavelength pattern (middle column) is only observed on graphene. Note that correlation functions within the same columns are scaled to the same contrast, i.\\ e., their brightness is directly comparable.\n\n\\begin{figure*}\n\\includegraphics[width=\\linewidth]{Figure_supp3}\n\\caption{\\label{figs3} (Color online) 2D autocorrelation images of graphene (top row) and SiO$_2$ (bottom row). Left: autocorrelation calculated from original images. Middle and right: high-frequency and low-frequency part, obtained by high pass or low pass filtering, respectively, using a 20 nm first-order Butterworth filter. Note that the color scale is identical within each column. Drift effects have been removed from the graphene images by using the atomic resolution.}\n\\end{figure*}\n\n\\subsection*{Reproducibility of Correlation Functions}\nFigure \\ref{figs4} shows three images (top row) obtained in different areas of the monolayer graphene, separated by several $\\mu$m.\nThey exhibit different rms values of 270 pm, 220 pm and 360 pm (left to right), respectively. The middle row shows the\ncorresponding correlation functions. The bottom row shows the same correlation functions after high-pass\nfiltering using the identical procedure as in Figure 3. Notice that the scales of the images are different.\nWhile the long-range corrugation (middle row) is oriented in different directions, the short range corrugations\n(bottom row) exhibit the same\nwave length of about 15 nm in all three areas and show a similar orientation as marked by the white lines. The latter indicates that the the short-scale corrugation is related to the atomic directions in graphene. But since the directions of the rippling are fluctuating slightly across the sample similar to the orientations of molecules in liquid crystals, more data are required in order to substantiate the relation between atomic directions and preferred directions of rippling. \n\n\\begin{figure*}\n\\includegraphics[width=\\linewidth]{Figure_supp4}\n\\caption{\\label{figs4} (Color online) Different graphene\nareas (top row), separated by several $\\mu$m, with autocorrelation images (middle) and high-pass filtered autocorrelation images (bottom row). The preferential orientation of long-range corrugation varies arbitrarily, while the short-range corrugation shows similar orientation. Inset: Autocorrelation of the atomic resolution image.}\n\\end{figure*}\n\nFigure \\ref{figs5} shows topographic images of different areas of the SiO$_2$ substrate (top row) and the corresponding correlation functions\n(bottom row). They exhibit similar length scales of corrugation but clearly different orientations, which might be due to the polishing procedure of the sample.\n\n\\begin{figure*}\n\\includegraphics[width=\\linewidth]{Figure_supp5}\n\\caption{\\label{figs5} (Color online) AFM images of different SiO$_2$ areas (top) with corresponding 2D autocorrelation (bottom). The preferential orientation varies arbitrarily.}\n\\end{figure*}\n\n\\subsection*{Wave Length Dependence of Corrugations}\nFigure \\ref{figs6} shows the radially averaged Fourrier transforms (FT) of the topographic images of graphene (black curve) and SiO$_2$ (grey curve). The FT curves are based on images\nof the same lateral size and resolution. The curves are averaged using 10 images of 200 $\\times$ 200 nm$^2$ from different areas of the surfaces. Obviously,\nthe corrugation of SiO$_2$ is larger at large wave length (small wave number) down to 30 nm by up to 66 \\%, while the corrugation on the\ngraphene is larger at smaller wavelengths (larger wave numbers) by up to a factor of 2.6. The individual FTs of different SiO$_2$ areas are nearly indistinguishable, i.e they are not deviating from the width of the grey curve in Fig. 5. However, the FTs of different graphene areas are shifted vertically due to the fluctuating rms value of corrugation, which leads to crossing points of\nthe two curves varying between wavelengths of 20 nm and 40 nm. Importantly, all of the graphene curves show one\ncrossing point with the averaged SiO$_2$ curve.\n\n\\begin{figure}\n\\includegraphics[width=\\linewidth]{Figure_supp6}\n\\caption{\\label{figs6} Radial averages of the Fourier transformations of graphene (black) and SiO$_2$ (grey) images, averaged over 10 images. The crossing point of individual curves varies between 20 and 40 nm.}\n\\end{figure}\n\n\\subsection*{Raman investigations on monolayer graphene}\n\nIn order to observe the influence of the graphene film morphology\non Raman scattering, we compared Raman spectra (laser wave length\n532.1 nm), taken in identical experimental conditions, of a\nmonolayer flake which is partly suspended and one that follows the\nsubstrate (Fig.\\ \\ref{figs7}). The surface structure of these samples was\ndetermined using STM measurements (Insets Fig.\\ \\ref{figs7}a). The Raman\nspectra were recorded before and after the STM measurements giving\nthe same result. A difference between both spectra regarding the\npositions of the 2D and G peaks is clearly visible. The wave number\nof the 2D peak is 2686.7 cm$^{-1}$ for the partly suspended graphene\nflake and 2670.5 cm$^{-1}$ for the flake following the substrate.\nHence, it results that the change in morphology causes a shift of\nthe 2D peak by 16.2 cm$^{-1}$ (Fig.\\ \\ref{figs7}a). A similar effect was\nobserved at the G peak which is shifted by 5.5 cm$^{-1}$ to lower\nvalues  for the flake following the substrate (Fig.\\ \\ref{figs7}b). The D peak\n(about 1350 cm$^{-1}$) is barely visible for both samples indicating\na good quality of our graphene. Raman spectra of several of our\ngraphene samples (about 15) prepared in nominally identical\nconditions show more frequently a non shifted 2D line with an\naverage value of 2692.4$\\,\\pm\\,$5.3 cm$^{-1}$. Hence, we conclude\nthat our samples mostly exhibit intrinsic rippling. If we compare\nwith wave numbers of the 2D line within the literature using only\nthe data taken with a similar laser wave length (514 - 532 nm)\n\\cite{Ferrari06,Gupta06,Graf07}, we find 2680 - 2700\ncm$^{-1}$. Hence, we believe that the majority of samples\ninvestigated so far exhibit intrinsic rippling as displayed in the\nright inset of Figure 7a.\n\n\\begin{figure*}\n\\includegraphics[width=\\linewidth]{Figure_supp7}\n\\caption{\\label{figs7} (Color online) (a)  Comparison of Raman\nspectra at 532.1 nm laser wave length for partly suspended and\nfollowing the substrate graphene  monolayer flake.  The 2D peak\nposition is 2686.7 cm$^{-1}$ for partly suspended graphene and\n2670.5 cm$^{-1}$ for the flake following the substrate. The\nexperimental conditions for both samples are identical. Left inset:\nSTM measurement of the graphene flake following the substrate (0.4\nV, 0.2 nA). Right inset: STM measurement of the partly suspended\ngraphene flake (1 V, 0.2 nA). (b)  Raman spectra of the D and G band\n for the same samples. The D line intensity (1350 cm$^{-1}$, marked by a dashed line) is very weak for both\nsamples. The G line from the flake following the substrate is\nshifted to lower values by 5.5 cm$^{-1}$ with respect to the G peak\nof the partly suspended graphene.}\n\\end{figure*}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nLet $\\overline{\\cal M}_{g,n}$ be the moduli space\nof genus $g$ stable curves---curves with only nodal singularities and finite automorphism group---with $n$ labeled points disjoint from nodes.   Define $\\psi_i=c_1(L_i)\\in H^{2}(\\overline{\\mathcal{M}}_{g,n},\\mathbb{Q}) $ the first Chern class of the line bundle $L_i\\to\\overline{\\mathcal{M}}_{g,n}$ with fibre above $[(C,p_1,\\ldots,p_n)]$ given by $T_{p_i}^*C$.  Consider the natural maps given by the forgetful map \n\\begin{equation}\\label{forgetful}\n\\overline{\\cal M}_{g,n+1}\\stackrel{\\pi}{\\longrightarrow}\\overline{\\cal M}_{g,n}\n\\end{equation}\nand the gluing maps \n\\begin{equation}\\label{gluing}\n\\overline{\\cal M}_{g-1,n+2}\\stackrel{\\phi_{\\text{irr}}}{\\longrightarrow}\\overline{\\cal M}_{g,n},\\qquad\\overline{\\cal M}_{h,|I|+1}\\times\\overline{\\cal M}_{g-h,|J|+1}\\stackrel{\\phi_{h,I}}{\\longrightarrow}\\overline{\\cal M}_{g,n},\\qquad I\\sqcup J=\\{1,...,n\\}.\n\\end{equation}\n\nIn this paper we construct cohomology classes $\\Theta_{g,n}\\in H^*(\\overline{\\cal M}_{g,n})$ for $g\\geq 0$, $n\\geq 0$ and $2g-2+n>0$ satisfying the following four properties:\n\\begin{enumerate}[(i)]\n\\setlength{\\itemindent}{20pt}\n\\item $\\Theta_{g,n}\\in H^*(\\overline{\\cal M}_{g,n})$ is of pure degree, \\label{pure}\n\\item $\\phi_{\\text{irr}}^*\\Theta_{g,n}=\\Theta_{g-1,n+2}$,\\quad $\\phi_{h,I}^*\\Theta_{g,n}=\\pi_1^*\\Theta_{h,|I|+1}\\cdot\\pi_2^* \\Theta_{g-h,|J|+1}$,  \\label{glue}\n\\item $\\Theta_{g,n+1}=\\psi_{n+1}\\cdot\\pi^*\\Theta_{g,n}$,  \\label{forget}\n\\item  $\\Theta_{1,1}=3\\psi_1$.  \\label{base}\n\\end{enumerate}\nThe properties \\eqref{pure}-\\eqref{base} uniquely define intersection numbers of the classes $\\Theta_{g,n}$ with the classes $\\psi_i$.  It is not clear if they uniquely define the classes $\\Theta_{g,n}$ themselves.\n\\begin{remark}  \\label{gluestab}\nOne can replace \\eqref{glue} by the equivalent property\n$$\\phi_{\\Gamma}^*\\Theta_{g,n}=\\Theta_{\\Gamma}.$$\nfor any stable graph $\\Gamma$ of genus $g$ and with $n$ external edges.  Here\n$$\\phi_{\\Gamma}:\\overline{\\cal M}_{\\Gamma}=\\prod_{v\\in V(\\Gamma)}\\overline{\\cal M}_{g(v),n(v)}\\to\\overline{\\cal M}_{g,n},\\quad \\Theta_{\\Gamma}=\\prod_{v\\in V(\\Gamma)}\\pi_v^*\\Theta_{g(v),n(v)}\\in H^*(\\overline{\\cal M}_{\\Gamma})$$\nwhere $\\pi_v$ is projection onto the factor $\\overline{\\cal M}_{g(v),n(v)}$.\nThis generalises \\eqref{glue} from 1-edge stable graphs where $\\phi_{\\Gamma_{\\text{irr}}}=\\phi_{\\text{irr}}$ and $\\phi_{\\Gamma_{h,I}}=\\phi_{h,I}$.   See Section~\\ref{pixrel} for more on stable graphs.\n\\end{remark}\n\\begin{remark}\nThe sequence of classes $\\Theta_{g,n}$ satisfies many properties of a cohomological field theory (CohFT).  It is essentially a 1-dimensional CohFT with vanishing genus zero classes, not to be confused with the Hodge class which is trivial in genus zero but does not vanish there.  The trivial cohomology class $1\\in H^0(\\overline{\\cal M}_{g,n})$, which is a trivial example of a CohFT, satisfies conditions \\eqref{pure} and \\eqref{glue}.  In this case, the forgetful map property \\eqref{forget} is replaced by $\\Theta_{g,n+1}=\\pi^*\\Theta_{g,n}$ and the initial value property \\eqref{base} is replaced by $\\Theta_{1,1}=1$.  \n\\end{remark}\n\\begin{remark}\nThe existence proof of $\\Theta_{g,n}$---see Section~\\ref{existence}---requires the initial value property \\eqref{base} given by $\\Theta_{1,1}=3\\psi_1$.\nThe existence of $\\Theta_{g,n}$ with \\eqref{base} replaced by $\\Theta_{1,1}=\\lambda\\psi_1$ for general $\\lambda\\in\\mathbb{C}$ is unknown.  One can of course replace $\\Theta_{g,n}$ by $\\lambda^{2g-2+n}\\Theta_{g,n}$ but this would change property \\eqref{forget}.\n\\end{remark}\n\n\\begin{theorem}  \\label{main}\nThere exists a class $\\Theta_{g,n}$ satisfying \\eqref{pure} - \\eqref{base} and furthermore any such class satisfies the following properties.\n\\begin{enumerate}[(I)]\n\\item \n$\\Theta_{g,n}\\in H^{4g-4+2n}(\\overline{\\cal M}_{g,n})$. \\label{degree}\n\\item $\\Theta_{0,n}=0$ for all $n$ and $\\phi_{\\Gamma}^*\\Theta_{g,n}=0$ for any $\\Gamma$ with a genus 0 vertex. \\label{genus0}\n\\item $\\Theta_{g,n}\\in H^*(\\overline{\\cal M}_{g,n})^{S_n}$, i.e. it is symmetric under the $S_n$ action.  \\label{symmetric}\n\\item The intersection numbers\n$\\displaystyle \\int_{\\overline{\\cal M}_{g,n}}\\Theta_{g,n}\\prod_{i=1}^n\\psi_i^{m_i}\\prod_{j=1}^N\\kappa_{\\ell_j}$\nare uniquely determined.  \\label{unique}\n\\item \n$\\displaystyle Z^{\\Theta}(\\hbar,t_0,t_1,...)=\\exp\\sum_{g,n,\\vec{k}}\\frac{\\hbar^{g-1}}{n!}\\int_{\\overline{\\cal M}_{g,n}}\\Theta_{g,n}\\cdot\\prod_{j=1}^n\\psi_j^{k_j}\\prod t_{k_j}$\nis a tau function of the KdV hierarchy. \\label{thetatau}\n\\end{enumerate}\n\\end{theorem}\n\nThe main content of Theorem~\\ref{main} is the existence of $\\Theta_{g,n}$ which is constructed via the push-forward of a class over the moduli space of spin curves in Section~\\ref{existence}, and the KdV property \\eqref{thetatau} proven in Section~\\ref{sec:kdv}.  The non-constructive uniqueness result \\eqref{unique}---which relies on the existence of non-explicit tautological relations---follows from the more general property that the intersection numbers $\\int_{\\overline{\\cal M}_{g,n}}\\Theta_{g,n}\\prod_{i=1}^n\\psi_i^{m_i}$ are uniquely determined by {\\em any} initial value $\\int_{\\overline{\\cal M}_{1,1}}\\Theta_{1,1}\\in\\mathbb{C}$.  For the initial value $\\int_{\\overline{\\cal M}_{1,1}}\\Theta_{1,1}=\\frac{1}{8}$, \\eqref{unique} is strengthened by \\eqref{thetatau} which allows one to recursively calculate all intersection numbers $\\int_{\\overline{\\cal M}_{g,n}}\\Theta_{g,n}\\prod_{i=1}^n\\psi_i^{m_i}$ via recursive relations coming out of the KdV hierarchy.  The proofs of properties~\\eqref{degree} - \\eqref{unique} are straightforward and presented in Section~\\ref{sec:unique}.  \n\nThe proof of \\eqref{thetatau} does not directly use the KdV hierarchy.  Instead it identifies the proposed KdV tau function $Z^{\\Theta}$ with a known KdV tau function---the Brezin-Gross-Witten KdV tau function $Z^{\\text{BGW}}$ defined in \\cite{BGrExt,GWiPos}.  This identification of  $Z^{\\Theta}(\\hbar,t_0,t_1,...)$ with $Z^{\\text{BGW}}(\\hbar,t_0,t_1,...)$ is stated as Theorem~\\ref{tauf} in Section~\\ref{intkdv}.  The proof of Theorem~\\ref{tauf} uses a set of tautological relations, known as Pixton's relations, obtained from the moduli space of 3-spin curves and proven in \\cite{PPZRel}.  Just as tautological relations give topological recursion relations for Gromov-Witten invariants, the intersections of $\\Theta_{g,n}$ with Pixton's relations produce topological recursion relations satisfied by $\\Theta_{g,n}$ that are enough to uniquely determine all intersection numbers $\\int_{\\overline{\\cal M}_{g,n}}\\Theta_{g,n}\\prod_{i=1}^n\\psi_i^{m_i}\\prod_{j=1}^N\\kappa_{\\ell_j}$.  The strategy of the proof of \\eqref{thetatau} is to show that the coefficients of $Z^{\\text{BGW}}(\\hbar,t_0,t_1,...)$ satisfy the same relations as the corresponding coefficients of $Z^{\\Theta}(\\hbar,t_0,t_1,...)$, given by $\\int_{\\overline{\\cal M}_{g,n}}\\Theta_{g,n}\\prod_{i=1}^n\\psi_i^{m_i}$.  \\\\\n\n\n\n\n\\noindent {\\em Acknowledgements.}   I would like to thank Dimitri Zvonkine for his ongoing interest in this work which benefited immensely from many conversations together.  I would also like to thank Alessandro Chiodo, Oliver Leigh, Ran Tessler and Ravi Vakil for useful conversations, and the Institut Henri Poincar\\'e where part of this work was carried out.\n\n\n\n\\section{Existence}   \\label{existence}\nThe existence of a cohomology class $\\Theta_{g,n}\\in H^*(\\overline{\\cal M}_{g,n})$ satisfying \\eqref{pure} - \\eqref{base} can be proven using the moduli space of stable twisted spin curves $\\overline{\\cal M}_{g,n}^{\\rm spin}$ which consist of pairs $(\\Sigma,\\theta)$ given by a twisted stable curve $\\Sigma$ equipped with an orbifold line bundle $\\theta$ together with an isomorphism $\\theta^{\\otimes 2}\\cong \\omega_{\\Sigma}^{\\text{log}}$.  See precise definitions below.  We first construct a cohomology class on $\\overline{\\cal M}_{g,n}^{\\rm spin}$ and then push it forward to a cohomology class on $\\overline{\\cal M}_{g,n}$.\n\n\n\nA stable twisted curve, with group $\\mathbb{Z}_2$, is a 1-dimensional orbifold, or stack, $\\mathcal{C}$ such that generic points of $\\mathcal{C}$ have trivial isotropy group and non-trivial orbifold points have isotropy group $\\mathbb{Z}_2$.  A stable twisted curve is equipped with a map which forgets the orbifold structure $\\rho:\\mathcal{C}\\to C$ where $C$ is a stable curve known as the the coarse curve of $\\mathcal{C}$.  We say that $\\mathcal{C}$ is smooth if its coarse curve $C$ is smooth.  Each nodal point of $\\mathcal{C}$ (corresponding to a nodal point of $C$) has non-trivial isotropy group and all other points of $\\mathcal{C}$ with non-trivial isotropy group are labeled points of $\\mathcal{C}$.  \n\nA line bundle $L$ over $\\mathcal{C}$ is a locally equivariant bundle over the local charts, such that at each nodal point there is an equivariant isomorphism of fibres.  Hence each orbifold point $p$ associates a representation of $\\mathbb{Z}_2$ on $L|_p$ acting by multiplication by $\\exp(2\\pi i\\lambda_p)$ for $\\lambda_p=0$ or $\\frac12$.  One says $L$ is {\\em banded} by $\\lambda_p$.  The equivariant isomorphism at nodes guarantees that the representations agree on each local irreducible component at the node.  \n\nThe sheaf of local sections $\\mathcal{O}_\\mathcal{C}(L)$ pushes forward to a sheaf $|L|:=\\rho_*\\mathcal{O}_\\mathcal{C}(L)$ on $C$ which can be identified with the local sections of $L$ invariant under the $\\mathbb{Z}_2$ action.  Away from nodal points $|L|$ is locally free, hence a line bundle.  The pull-back bundle $\\rho^*(|L|)=L\\otimes\\bigotimes_{i\\in I}\\mathcal{O}(-p_i)$ where $L$ is banded by the non-trivial representation precisely at $p_i$ for $i\\in I$.   Hence $\\deg|L|=\\deg L-\\frac12 |I|$.  At nodal points, the push-forward $|L|$ is locally free when $L$ is banded by the trivial representation, and $|L|$ is a torsion-free sheaf that is not locally free when $L$ is banded by the non-trivial representation.  See \\cite{FJRQua} for a nice description of these ideas.\n\nThe canonical bundle $\\omega_\\mathcal{C}$ of $\\mathcal{C}$ is generated by $dz$ for any local coordinate $z$.  At an orbifold point $x=z^2$ the canonical bundle $\\omega_\\mathcal{C}$ is generated by $dz$ hence it is banded by $\\frac12$ i.e. $dz\\mapsto-dz$ under $z\\mapsto -z$.  Over the coarse curve $\\omega_C$ is generated by $dx=2zdz$.  In other words $\\rho^*\\omega_C\\not\\cong\\omega_\\mathcal{C}$ however $\\omega_C\\cong\\rho_*\\omega_\\mathcal{C}$.  Moreover, $\\deg\\omega_C=-\\chi=2g-2$ and \n$$\\deg\\omega_\\mathcal{C}=-\\chi^{\\text{orb}}=2g-2+\\frac12n.$$\nFor $\\omega_\\mathcal{C}^{\\text{log}}=\\omega_\\mathcal{C}(p_1,...,p_n)$, locally $\\frac{dx}{x}=2\\frac{dz}{z}$ so $\\rho^*\\omega_C^{\\text{log}}\\cong\\omega_\\mathcal{C}^{\\text{log}}$ and $\\deg\\omega_C^{\\text{log}}=2g-2+n=\\deg\\omega_\\mathcal{C}^{\\text{log}}$.\n\n\n\nFollowing \\cite{AJaMod}, define\n$$\\overline{\\cal M}_{g,n}^{\\rm spin}=\\{(\\mathcal{C},\\theta,p_1,...,p_n,\\phi)\\mid \\phi:\\theta^2\\stackrel{\\cong}{\\longrightarrow}\\omega_{\\mathcal{C}}^{\\text{log}}\\}.\n$$\nHere $\\omega_{\\mathcal{C}}^{\\text{log}}$ and $\\theta$ are line bundles over the stable twisted curve $\\mathcal{C}$ with labeled orbifold points $p_j$ and $\\deg\\theta=g-1+\\frac12n$.  The relation $\\theta^2\\stackrel{\\cong}{\\longrightarrow}\\omega_{\\mathcal{C}}^{\\text{log}}$ is possible because the representation associated to $\\omega_{\\mathcal{C}}^{\\text{log}}$ at $p_i$ is trivial---$dz\/z\\stackrel{z\\mapsto-z}{\\longrightarrow}dz\/z$.  We require the representations associated to $\\theta$ at each $p_i$ to be non-trivial, i.e $\\lambda_{p_i}=\\frac12$.  At nodal points $p$, both types $\\lambda_p=0$ or $\\frac12$ can occur.  The equivariant isomorphism of fibres over nodal points forces the balanced condition $\\lambda_{p_+}=\\lambda_{p_-}$ for $p_\\pm$ corresponding to $p$ on each irreducible component.  Among the $2^{2g}$ different spin structures on a twisted curve $\\mathcal{C}$, some will have $\\lambda_p=0$ and some will have $\\lambda_p=\\frac12$.   \n\nThe forgetful map\n$$f:\\overline{\\cal M}_{g,n+1}^{\\rm spin}\\to\\overline{\\cal M}_{g,n}^{\\rm spin}\n$$\nis defined via  $f(\\mathcal{C},\\theta,p_1,...,p_{n+1},\\phi)=(\\rho(\\mathcal{C}),\\rho_*\\theta,p_1,...,p_n,\\rho_*\\phi)$ where $\\rho$ forgets the label and orbifold structure at $p_{n+1}$.  As described above, the push-forward $\\rho_*\\theta$ consists of local sections invariant under the $\\mathbb{Z}_2$ action.  Since the representation at $p_{n+1}$ is given by multiplication by $-1$, any invariant local section must vanish at $p_{n+1}$.  In other words $\\rho_*\\theta=\\rho_*\\{\\theta(-p_{n+1})\\}$, $\\rho^*\\rho_*\\theta=\\{\\theta(-p_{n+1})\\}$ and $\\deg\\rho_*\\theta=\\deg\\theta-\\frac12$.\n\nTautological line bundles $L_{p_i}\\to\\overline{\\cal M}_{g,n}^{\\rm spin}$, $i=1,...,n$ are defined analogously to those defined over $\\overline{\\cal M}_{g,n}$.  For a family $\\pi:\\mathcal{C}\\to S$ with sections $p_i:S\\to\\mathcal{C}$, $i=1,...,n$, they are defined by $L_{p_i}:=p_i^*(\\omega_{\\mathcal{C}\/S})$. \n\nWe can now define a vector bundle over $\\overline{\\cal M}_{g,n}^{\\rm spin}$ using the dual bundle $\\theta^{\\vee}$ on each stable twisted curve.  Denote by $\\mathcal{E}$ the universal spin structure over $\\overline{\\cal M}_{g,n}^{\\rm spin}$.  Given a map $S\\to\\overline{\\cal M}_{g,n}^{\\rm spin}$, $\\mathcal{E}$ pulls back to $\\theta$ giving a family $(\\mathcal{C},\\theta,p_1,...,p_n,\\phi)$ where $\\pi:\\mathcal{C}\\to S$ has stable twisted curve fibres, $p_i:S\\to\\mathcal{C}$ are sections with orbifold isotropy $\\mathbb{Z}_2$ and $\\phi:\\theta^2\\stackrel{\\cong}{\\longrightarrow}\\omega_{\\mathcal{C}\/S}^{\\text{log}}=\\omega_{\\mathcal{C}\/S}(p_1,..,p_n)$.  Consider the push-forward sheaf $\\pi_*\\mathcal{E}^{\\vee}$ over $\\overline{\\cal M}_{g,n}^{\\rm spin}$.  We have\n$$\\deg\\theta^{\\vee}=1-g-\\frac12n<0.\n$$\nFurthermore, for any irreducible component $\\mathcal{C}'\\stackrel{i}{\\to}\\mathcal{C}$, the pole structure on sections of the log canonical bundle at nodes yields $i^*\\omega_{\\mathcal{C}\/S}^{\\text{log}}=\\omega_{\\mathcal{C}'\/S}^{\\text{log}}$.  Hence $\\phi':(\\theta|_{\\mathcal{C}'})^2\\stackrel{\\cong}{\\longrightarrow}\\omega_{\\mathcal{C}'\/S}^{\\text{log}}$, where $\\phi'=i^*\\circ\\phi|_{\\mathcal{C}'}$.  Since the irreducible component $\\mathcal{C}'$ is stable its log canonical bundle has negative degree and\n$$\\deg\\theta^{\\vee}|_{\\mathcal{C}'}<0.\n$$\nNegative degree of $\\theta^{\\vee}$ restricted to any irreducible component implies that $R^0\\pi_*\\mathcal{E}^{\\vee}=0$ and the following definition makes sense.\n\\begin{definition}  \\label{obsbun}\nDefine a bundle $E_{g,n}=R\\pi_*\\mathcal{E}^\\vee$ over $\\overline{\\cal M}_{g,n}^{\\rm spin}$ with fibre $H^1(\\theta^{\\vee})^{\\vee}$.   \n\\end{definition} \nIt is a bundle of rank $2g-2+n$ by the following Riemann-Roch calculation.   Orbifold Riemann-Roch takes into account the representation information in terms of its band $\\lambda_{p_i}$.\n$$h^0(\\theta^{\\vee})-h^1(\\theta^{\\vee})=1-g+\\deg \\theta^{\\vee}-\\sum_{i=1}^n\\lambda_{p_i}=1-g+1-g-\\frac12n-\\frac12n=2-2g-n.\n$$\nHere we used the requirement that $\\lambda_{p_i}=\\frac12$ for $i=1,...,n$.  Since $\\deg\\theta^{\\vee}=1-g-\\frac12n<0$, and the restriction of $\\theta^{\\vee}$ to any irreducible component also has negative degree, we have $h^0(\\theta^{\\vee})=0$ and hence $h^1(\\theta^{\\vee})=2g-2+n$.  Thus $H^1(\\theta^{\\vee})^{\\vee}$ gives fibres of a rank $2g-2+n$ vector bundle.\n\nThe analogue of the boundary maps $\\phi_{\\text{irr}}$ and $\\phi_{h,I}$ defined in \\eqref{gluing} are multivalued maps (binary relations) defined as follows.  Consider a node $p\\in\\mathcal{C}$ for $(\\mathcal{C},\\theta,p_1,...,p_n,\\phi)\\in\\overline{\\cal M}_{g,n}^{\\rm spin}$.  Denote the normalisation by $\\nu:\\tilde{\\mathcal{C}}\\to\\mathcal{C}$ with points $p_\\pm\\in\\tilde{\\mathcal{C}}$ that map to the node $p=\\nu(p_\\pm)$.  When $\\tilde{\\mathcal{C}}$ is not connected, then since $\\theta$ must be banded by $\\lambda_p=0$ at an even number of orbifold points, $\\lambda_{p_i}=\\frac12$ forces $\\lambda_{p_\\pm}=\\frac12$.  Hence it decomposes into two spin structures $\\theta_1$ and $\\theta_2$.  Any two spin structures $\\theta_1$ and $\\theta_2$ can glue, but not uniquely, to give a spin structure on $\\mathcal{C}$.  This gives rise to a multivalued map, as described in \\cite{FJRWit}, which uses the fibre product:\n$$\\begin{array}{ccc}\\left(\\overline{\\cal M}_{h,|I|+1}\\times\\overline{\\cal M}_{g-h,|J|+1}\\right)\\times_{\\overline{\\cal M}_{g,n}}\\overline{\\cal M}^{\\rm spin}_{g,n}&\\to&\\overline{\\cal M}^{\\rm spin}_{g,n}\\\\\n\\downarrow&&\\downarrow\\\\\n\\overline{\\cal M}_{h,|I|+1}\\times\\overline{\\cal M}_{g-h,|J|+1}&\\to&\\overline{\\cal M}_{g,n}\\end{array}\n$$\nand is given by\n$$\n\\begin{array}{c}\n\\left(\\overline{\\cal M}_{h,|I|+1}\\times\\overline{\\cal M}_{g-h,|J|+1}\\right)\n\\times_{\\overline{\\cal M}_{g,n}}\\overline{\\cal M}^{\\rm spin}_{g,n}\\\\\\hspace{1.5cm}\\swarrow\\hat{\\nu}\\hspace{3cm}\\phi_{h,I}\\searrow\\\\\n\\qquad\\overline{\\cal M}^{\\rm spin}_{h,|I|+1}\\times\\overline{\\cal M}^{\\rm spin}_{g-h,|J|+1}\\hspace{1cm}\\dashrightarrow\\hspace{1cm}\\overline{\\cal M}^{\\rm spin}_{g,n}\\end{array}\n$$\nwhere $I\\sqcup J=\\{1,...,n\\}$.  The map $\\hat{\\nu}$ is given by the pull back of the spin structure obtained from $\\overline{\\cal M}^{\\rm spin}_{g,n}$ to the normalisation defined by the points of $\\overline{\\cal M}_{h,|I|+1}$ and $\\overline{\\cal M}_{g-h,|J|+1}$. The broken arrow $\\dashrightarrow$ represents the multiply-defined map $\\phi_{h,I}\\circ\\hat{\\nu}^{-1}$.  \n\nWhen $\\tilde{\\mathcal{C}}$ is connected, a spin structure $\\theta$ on $\\mathcal{C}$ pulls back to a spin structure $\\tilde{\\theta}=\\nu^*\\theta$ on $\\tilde{\\mathcal{C}}$.  As above, any spin structure $\\tilde{\\theta}$ glues non-uniquely, to give a spin structure on $\\mathcal{C}$, and defines a multiply-defined map which uses the fibre product:\n$$\\begin{array}{ccc}\\overline{\\cal M}_{g-1,n+2}\\times_{\\overline{\\cal M}_{g,n}}\\overline{\\cal M}^{\\rm spin}_{g,n}&\\to&\\overline{\\cal M}^{\\rm spin}_{g,n}\\\\\n\\downarrow&&\\downarrow\\\\\n\\overline{\\cal M}_{g-1,n+2}&\\to&\\overline{\\cal M}_{g,n}\\end{array}.\n$$\nThere are two cases corresponding to the following decomposition of the fibre product:\n$$\\overline{\\cal M}_{g-1,n+2}\\times_{\\overline{\\cal M}_{g,n}}\\overline{\\cal M}^{\\rm spin}_{g,n}=\\bigsqcup_{i\\in\\{0,1\\}}\\left(\\overline{\\cal M}_{g-1,n+2}\\times_{\\overline{\\cal M}_{g,n}}\\overline{\\cal M}^{\\rm spin}_{g,n}\\right)^{(i)}$$\nwhich depends on the behaviour of $\\theta$ at the nodal point $p_\\pm$.   The superscript $\\cdot^{(0)}$, respectively $\\cdot^{(1)}$, denotes those $\\theta$ banded by $\\lambda_{p_\\pm}=\\frac12$, respectively $\\lambda_{p_\\pm}=0$.\n\nIn the first case, when $\\theta$ is banded by $\\lambda_{p_\\pm}=\\frac12$, we have:  \n$$\n\\begin{array}{c}\\left(\\overline{\\cal M}_{g-1,n+2}\\times_{\\overline{\\cal M}_{g,n}}\\overline{\\cal M}^{\\rm spin}_{g,n}\\right)^{(0)}\\\\\\swarrow\\hat{\\nu}\\hspace{2cm}\\phi_{\\text{irr}}\\searrow\\\\\\overline{\\cal M}^{\\rm spin}_{g-1,n+2}\\hspace{1cm}\\dashrightarrow\\hspace{1cm}\\overline{\\cal M}^{\\rm spin}_{g,n}\\end{array}\n$$\nwhere $\\hat{\\nu}$ is given by pull back of the spin structure obtained from $\\overline{\\cal M}^{\\rm spin}_{g,n}$ to the normalisation defined by the point of $\\overline{\\cal M}_{g-1,n+2}$. \n\nIn the second case when $\\theta$ is banded by $\\lambda_{p_\\pm}=0$, we have:\n$$\n\\begin{array}{c}\\left(\\overline{\\cal M}_{g-1,n+2}\\times_{\\overline{\\cal M}_{g,n}}\\overline{\\cal M}^{\\rm spin}_{g,n}\\right)^{(1)}\\\\\\swarrow\\hat{\\nu}\\hspace{2cm}\\phi_{\\text{irr},2}\\searrow\\\\\\overline{\\cal M}^{\\rm spin}_{g-1,n,2}\\hspace{1cm}\\dashrightarrow\\hspace{1cm}\\overline{\\cal M}^{\\rm spin}_{g,n}\\end{array}\n$$\nwhere $\\overline{\\cal M}^{\\rm spin}_{g-1,n,2}$ consists of spin structures banded, as usual, by $\\lambda_{p_i}=\\frac12$ for $i=1,...,n$ and banded by $\\lambda_{p_i}=0$ for $i=n+1,n+2$.  \n\nThe bundle $E_{g-1,n,2}\\to\\overline{\\cal M}^{\\rm spin}_{g-1,n,2}$ is still defined, with fibre $H^1(\\tilde{\\theta}^{\\vee})$, because $H^0(\\tilde{\\theta}^{\\vee})=0$ since the band $\\lambda_{p_{n+1}}=0=\\lambda_{p_{n+2}}$ does not affect $\\deg\\tilde{\\theta}^{\\vee}=1-(g-1)-\\frac12(n+2)<0$ and also negative degree on all irreducible components.  It does affect the rank of the bundle.   By Riemann-Roch $h^0(\\tilde{\\theta}^{\\vee})-h^1(\\tilde{\\theta}^{\\vee})=1-(g-1)+\\deg\\tilde{\\theta}^{\\vee}-\\frac12n=2-2g-n+1$ hence $\\dim H^1(\\tilde{\\theta}^{\\vee})=\\dim H^1(\\theta^{\\vee})-1$ when $\\tilde{\\theta}=\\nu^*\\theta$.\n\nThe bundle $E_{g,n}$ behaves naturally with respect to the boundary divisors.\n\\begin{lemma}   \\label{pullback}\n$$\n\\phi_{\\text{irr}}^*E_{g,n}\\equiv  \\hat\\nu^*E_{g-1,n+2},\\quad\\phi_{h,I}^*E_{g,n}\\cong \\hat\\nu^*\\left(\\pi_1^*E_{h,|I|+1}\\oplus\\pi_2^*E_{g-h,|J|+1}\\right)\n$$\nwhere $\\pi_i$ is projection from $\\overline{\\cal M}^{\\rm spin}_{h,|I|+1}\\times\\overline{\\cal M}^{\\rm spin}_{g-h,|J|+1}$ onto the $i$th factor, $i=1,2$. \n\\end{lemma}\n\\begin{proof}\nA spin structure $\\tilde{\\theta}$ on a connected normalisation $\\tilde{\\mathcal{C}}$ has $\\deg\\tilde{\\theta}^{\\vee}=1-(g-1)-\\frac12(n+2)<0$, and also negative degree on all irreducible components, hence $H^0(\\tilde{\\theta}^{\\vee})=0$.\nBy Riemann-Roch \n$$h^0(\\tilde{\\theta}^{\\vee})-h^1(\\tilde{\\theta}^{\\vee})=1-(g-1)+\\deg\\tilde{\\theta}^{\\vee}-\\frac12(n+2)=2-2g-n.$$ \nHence $\\dim H^1(\\tilde{\\theta}^{\\vee})=\\dim H^1(\\theta^{\\vee})$ and the natural map\n$$ 0\\to H^1(\\mathcal{C},\\theta^{\\vee})\\to H^1(\\tilde{\\mathcal{C}},\\tilde{\\theta}^{\\vee})\n$$\nis an isomorphism.  In other words $\\phi_{\\text{irr}}^*E_{g,n}\\equiv  \\nu^*E_{g-1,n+2}$.\n\nThe argument is analogous when $\\tilde{\\mathcal{C}}$ is not connected and $\\lambda_{p_\\pm}=\\frac12$.  Again $\\deg\\theta_i^{\\vee}<0$, and it has negative degree on all irreducible components, hence $H^0(\\theta_i^{\\vee})=0$ for $i=1,2$.  By Riemann-Roch $\\dim H^1(\\theta_1^{\\vee})+\\dim H^1(\\theta_2^{\\vee})=\\dim H^1(\\theta^{\\vee})$ so the natural map\n$$ 0\\to H^1(\\mathcal{C},\\theta^{\\vee})\\to H^1(\\tilde{\\mathcal{C}}_1,\\theta_1^{\\vee})\\oplus H^1(\\tilde{\\mathcal{C}}_2,\\theta_2^{\\vee})\n$$\nis an isomorphism.  In other words $\\phi_{h,I}^*E_{g,n}\\cong \\hat\\nu^*\\left(\\pi_1^*E_{h,|I|+1}\\oplus\\pi_2^*E_{g-h,|J|+1}\\right)$.\n\\end{proof}\nThe following lemma describes the pull-back of $E_{g,n}$ to the fibre product $\\overline{\\cal M}_{g-1,n+2}\\times_{\\overline{\\cal M}_{g,n}}\\overline{\\cal M}^{\\rm spin}_{g,n}$ in terms of the bundle $E_{g-1,n,2}\\to\\overline{\\cal M}^{\\rm spin}_{g-1,n,2}$ defined above.\n\\begin{lemma}  \\label{pullback1}\n$$0\\to \\hat\\nu^*E_{g-1,n,2}\\to \\phi_{\\text{irr},2}^*E_{g,n}\\to \\mathcal{O}_{\\overline{\\cal M}_{g-1,n+2}\\times_{\\overline{\\cal M}_{g,n}}\\overline{\\cal M}^{\\rm spin}_{g,n}} \\to 0.\n$$\n\\end{lemma}\n\\begin{proof}\nWhen the bundle $\\theta$ is banded by $\\lambda_{p_\\pm}=0$, the map between sheaves of local holomorphic sections\n$$\\mathcal{O}_C({\\theta},U)\\to\\mathcal{O}_{\\tilde{C}}({\\nu^*\\theta},\\nu^{-1}U)$$ \nis not surjective whenever $U\\ni p$.  The image consists of local sections that agree, under an identification of fibres, at $p_+$ and $p_-$.  Hence the dual bundle $\\theta^\\vee$ on $\\mathcal{C}$ is a quotient sheaf\n\\begin{equation}  \\label{normcomp}\n0\\to I\\to\\nu^*\\theta^\\vee\\to\\theta^\\vee\\to 0\n\\end{equation} \nwhere $\\mathcal{O}_{\\tilde{C}}(I,U)$ is generated by the element of the dual that sends a local section $s\\in \\mathcal{O}_{\\tilde{C}}({\\nu^*\\theta},\\nu^{-1}U)$ to $s(p_+)-s(p_-)$.  Note that evaluation $s(p_\\pm)$ only makes sense after a choice of trivialisation of $\\nu^*\\theta$ at $p_+$ and $p_-$, but the ideal $I$ is independent of this choice.  The complex \\eqref{normcomp} splits as follows.  We can choose a representative $\\phi$ upstairs of any element from the quotient space so that $\\phi(p_+)=0$, i.e. $\\mathcal{O}_C(\\theta^\\vee,U)$ corresponds to elements of $\\mathcal{O}_{\\tilde{C}}({\\nu^*\\theta^\\vee},\\nu^{-1}U)$ that vanish at $p_+$.  This is achieved by adding the appropriate multiple of $s(p_+)-s(p_-)$ to a given $\\phi\\in\\mathcal{O}_{\\tilde{C}}({\\nu^*\\theta^\\vee},\\nu^{-1}U)$.  (Note that $\\phi(p_-)$ is arbitrary.  One could instead arrange $\\phi(p_-)=0$ with $\\phi(p_+)$ arbitrary.)  In other words we can identify $\\theta^\\vee$ with $\\nu^*\\theta^\\vee(-p_+)$ in the complex:\n$$0\\to \\nu^*\\theta^\\vee(-p_+)\\to \\nu^*\\theta^\\vee\\to \\nu^*\\theta^\\vee|_{p_+}\\to 0.\n$$\nIn a family $\\pi:C\\to S$, $R^0\\pi_*(\\nu^*\\theta^\\vee)=0=R^0\\pi_*(\\nu^*\\theta^\\vee(-p_+))$ since $\\deg\\nu^*\\theta^\\vee<0$, and it has negative degree on all irreducible components.  Also $R^1\\pi_*(\\nu^*\\theta^\\vee|_{p_+})=0$ since $p_+$ has relative dimension 0.  Thus\n$$0\\to R^0\\pi_*(\\nu^*\\theta^\\vee|_{p_+})\\to R^1\\pi_*(\\nu^*\\theta^\\vee(-p_+))\\to R^1\\pi_*(\\nu^*\\theta^\\vee) \\to 0.\n$$\nFurthermore, we have $\\nu^*L|_{p_+}\\cong\\mathbb{C}$ canonically via evaluation, hence $R^0\\pi_*(\\nu^*L|_{p_+})\\cong\\mathcal{O}_S$.  Since $\\hat\\nu^*E_{g-1,n,2}=R^1\\pi_*(\\nu^*\\theta^\\vee)$ and $\\phi_{\\text{irr},2}^*E_{g,n}=R^1\\pi_*(\\nu^*\\theta^\\vee(-p_+))$, take the dual of the sequence, and  the result follows.\n\\end{proof}\n\\begin{remark}\nWhen $\\lambda_{p_\\pm}=\\frac12$ we have $\\lambda_{p_+}+\\lambda_{p_-}=1$.  We see from above that $\\lambda_{p_\\pm}=0$ really wants one of $\\lambda_{p_\\pm}$ to be 1 to preserve $\\lambda_{p_+}+\\lambda_{p_-}=1$.\n\\end{remark}\n\n\n\n\n\n\n\\begin{definition}  \\label{fundclass}\nFor $2g-2+n>0$ define the Euler class\n$$\\Omega_{g,n}:=c_{2g-2+n}(E_{g,n})\\in H^{4g-4+2n}(\\overline{\\cal M}_{g,n}^{\\rm spin},\\mathbb{Q}).\n$$\n\\end{definition} \nNote that $\\Omega_{0,n}=0$ for $n=3,4,...$ because rank$(E_{0,n})=n-2$ is greater than $\\dim\\overline{\\cal M}_{0,n}^{\\rm spin}=n-3$ so its top Chern class vanishes.  Nevertheless, it would be interesting to know if the bundles $E_{0,n}$ carry non-trivial information.\n\nThe cohomology classes $\\Omega_{g,n}$ behave well with respect to inclusion of strata.\n\\begin{lemma} \\label{omeganat}\n$$\\phi_{\\text{irr}}^*\\Omega_{g,n}=\\hat\\nu^*\\Omega_{g-1,n+2},\\quad \\phi_{h,I}^*\\Omega_{g,n}=\\hat\\nu^*\\left(\\pi_1^*\\Omega_{h,|I|+1}\\cdot\\pi_2^*\\Omega_{g-h,|J|+1}\\right),\\quad\\phi_{\\text{irr},2}^*\\Omega_{g,n}=0.$$ \n\\end{lemma}\n \n\\begin{proof} \nThis is an immediate application of Lemma~\\ref{pullback}:\n$$\n\\phi_{\\text{irr}}^*E_{g,n}\\equiv  \\hat\\nu^*E_{g-1,n+2},\\quad\\phi_{h,I}^*E_{g,n}\\cong \\hat\\nu^*\\left(\\pi_1^*E_{h,|I|+1}\\oplus\\pi_2^*E_{g-h,|J|+1}\\right)\n$$\nand the naturality of $c_{2g-2+n}=c_{\\rm top}$.  We have \n$$\\phi_{\\text{irr}}^*c_{\\rm top}(E_{g,n})=\\hat\\nu^*c_{\\rm top}(E_{g-1,n+2}),\\quad\\phi_{h,I}^*c_{\\rm top}(E_{g,n})=\\hat\\nu^*\\left(\\pi_1^*c_{\\rm top}(E_{h,|I|+1})\\cdot\\pi_2^*c_{\\rm top}(E_{g-h,|J|+1})\\right).\n$$\n\nAlso, $\\phi_{\\text{irr},2}^*\\Omega_{g,n}=0$ follows immediately from the exact sequence of Lemma~\\ref{pullback1}\n$$0\\to E_{g-1,n,2}\\to \\phi_{\\text{irr},2}^*E_{g,n}\\to \\mathcal{O}_{\\overline{\\cal M}^{\\rm spin}_{g-1,n,2}} \\to 0\n$$\nwhich implies $\\phi_{\\text{irr},2}^*c_{2g-2+n}(E_{g,n})=c_{2g-3+n}(E_{g-1,n,2})\\cdot c_1(\\mathcal{O}_{\\overline{\\cal M}^{\\rm spin}_{g-1,n,2}})=0$.\n\\end{proof}\nA consequence of Lemma~\\ref{omeganat} is that $\\phi_{\\Gamma}^*\\Omega_{g,n}=0$ for any stable graph $\\Gamma$ (defined as for $\\overline{\\cal M}_{g,n}$ with extra data on edges) that contains a genus 0 vertex.\n\nConsider the map $\\pi:\\overline{\\cal M}_{g,n+1}^{\\rm spin}\\to\\overline{\\cal M}_{g,n}^{\\rm spin}$ that forgets the point $p_{n+1}$.\n\\begin{lemma}  \\label{omegaforget}\n$$\\Omega_{g,n+1}=\\psi_{p_{n+1}}\\cdot \\pi^*\\Omega_{g,n}.$$ \n\\end{lemma}\n \n\\begin{proof} \nTensor the exact sequence of sheaves \n$$0\\to \\mathcal{O}(-p_{n+1})\\to\\mathcal{O}\\to\\mathcal{O} |_{p_{n+1}}\\to 0$$\nwith $\\omega_{\\mathcal{C}\/S}\\otimes\\theta$ over a family $\\pi:\\mathcal{C}\\to S$ where $S\\to\\overline{\\cal M}_{g,n+1}^{\\rm spin}$ to get:\n$$0\\to \\omega_{\\mathcal{C}\/S}\\otimes\\theta(-p_{n+1})\\to\\omega_{\\mathcal{C}\/S}\\otimes\\theta\\to\\omega_{\\mathcal{C}\/S}\\otimes\\theta |_{p_{n+1}}\\to 0$$\nhence\n$$0\\to R^0\\pi_*\\left(\\omega_{\\mathcal{C}\/S}\\otimes\\theta(-p_{n+1})\\right)\\to R^0\\pi_*\\left(\\omega_{\\mathcal{C}\/S}\\otimes\\theta\\right)\\to R^0\\pi_*\\left(\\omega_{\\mathcal{C}\/S}\\otimes\\theta |_{p_{n+1}}\\right)\\to R^1\\pi_*\\left(\\omega_{\\mathcal{C}\/S}\\otimes\\theta(-p_{n+1})\\right).$$\nWe have $R^1\\pi_*\\left(\\omega_{\\mathcal{C}\/S}\\otimes\\theta(-p_{n+1})\\right)=0$ (check) which leaves a short exact sequence.  By Serre duality, the first two terms are\n$R^1\\pi_*\\left(\\theta^\\vee(p_{n+1})\\right)^\\vee\\to R^1\\pi_*\\left(\\theta^\\vee\\right)^\\vee$.  \n\nRecall that the forgetful map $(\\mathcal{C},\\theta,p_1,...,p_{n+1},\\phi)\\mapsto(\\pi(\\mathcal{C}),\\pi_*\\theta,p_1,...,p_n,\\pi_*\\phi)$ pushes forward $\\theta$ via $\\pi$ which forgets the orbifold structure at $p_{n+1}$.  As described earlier, $\\pi^*\\pi_*\\theta=\\{\\theta(-p_{n+1})\\}$ but $\\pi_*\\theta_{g,n+1}=\\theta_{g,n}$, where the ${g,n}$ subscript which is usually omitted is temporarily included for clarity.  Hence $\\theta^\\vee(p_{n+1})=\\pi^*\\theta^\\vee$ and $R^1\\pi_*\\left(\\theta^\\vee(p_{n+1})\\right)=R^1\\pi_*\\left(\\pi^*\\theta^\\vee\\right)=\\pi^*R^1\\pi_*\\left(\\theta^\\vee\\right)$.  Thus the first two terms of the short exact sequence become $\\pi^*E_{g,n}\\to E_{g,n+1}$.\n\nFor the third term of the short exact sequence, the residue map produces a canonical isomorphism $\\omega_{\\mathcal{C}\/S}^{\\text{log}}\\cong\\mathbb{C}$, as proven in \\cite{FJRWit}.  Thus $\\omega_{\\mathcal{C}\/S}^{\\text{log}}|_{p_{n+1}}=\\mathcal{O}_{p_{n+1}}$, and hence also $\\theta |_{p_{n+1}}=\\mathcal{O}_{p_{n+1}}$.  The triviality of $\\theta |_{p_{n+1}}$ implies $R^0\\pi_*\\left(\\omega_{\\mathcal{C}\/S}\\otimes\\theta |_{p_{n+1}}\\right)=R^0\\pi_*\\left(\\omega_{\\mathcal{C}\/S}|_{p_{n+1}}\\right)=L_{p_{n+1}}$.  Hence the short exact sequence becomes:\n$$0\\to\\pi^*E_{g,n}\\to E_{g,n+1}\\to L_{p_{n+1}}\\to 0.\n$$\nSince $c_1(L_{p_{n+1}})=\\psi_{p_{n+1}}$, we have $c_{2g-2+n+1}(E_{g,n+1})=\\psi_{p_{n+1}}\\cdot \\pi^*c_{2g-2+n}(E_{g,n})$ as required.  \n\\end{proof} \n\\begin{definition}\nFor $p:\\overline{\\cal M}_{g,n}^{\\rm spin}\\to\\overline{\\cal M}_{g,n}$ define\n$$\\Theta_{g,n}=2^np_*\\Omega_{g,n}\\in H^{4g-4+2n}(\\overline{\\cal M}_{g,n}).\n$$\n\\end{definition}\nLemma~\\ref{omegaforget} and the relation\n$$\\psi_{n+1}=\\frac12p^*\\psi_{n+1}$$\nproven in \\cite[Prop. 2.4.1]{FJRWit}, together with the factor of $2^n$ in the definition of $\\Omega_{g,n}$, immediately gives property \\eqref{forget} of $\\Theta_{g,n}$\n$$\\Theta_{g,n+1}=\\psi_{n+1}\\cdot\\pi^*\\Theta_{g,n}.$$  \nProperty \\eqref{base} of $\\Theta_{g,n}$ is given by the following calculation.\n\\begin{proposition}  \\label{theta11}\n$\\Theta_{1,1}=3\\psi_1\\in H^2(\\overline{\\cal M}_{1,1})$.\n\\end{proposition}\n\\begin{proof}\nA one-pointed twisted elliptic curve $(\\mathcal{E},p)$ is a one-pointed elliptic curve $(E,p)$ such that $p$ has isotropy $\\mathbb{Z}_2$.  The degree of the divisor $p$ in $\\mathcal{E}$ is $\\frac12$ and the degree of every other point in $\\mathcal{E}$ is 1.  If $dz$ is a holomorphic differential on $E$ (where $E=\\mathbb{C}\/\\Lambda$ and $z$ is the identity function on the universal cover $\\mathbb{C}$) then locally near $p$ we have $z=t^2$ so $dz=2tdt$ vanishes at $p$.  In particular, the canonical divisor $(\\omega_{\\mathcal{E}})=p$ has degree $\\frac12$ and $(\\omega^{\\text{log}}_{\\mathcal{E}})=(\\omega_{\\mathcal{E}}(p))=2p$ has degree 1.  \n\nA spin structure on $\\mathcal{E}$ is a degree $\\frac12$ line bundle $\\mathcal{L}$ satisfying $\\mathcal{L}^2=\\omega^{\\text{log}}_{\\mathcal{E}}$.  Line bundles on $\\mathcal{E}$ correspond to divisors on $\\mathcal{E}$ up to linear equivalence.  Note that meromorphic functions on $\\mathcal{E}$ are exactly the meromorphic functions on $E$.  The four spin structures on $\\mathcal{E}$ are given by the divisors $\\theta_0=p$ and $\\theta_i=q_i-p$, $i=1,2,3$, where $q_i$ is a non-trivial order 2 element in the group $E$ with identity $p$.  Clearly $\\theta_0^2=2p=\\omega^{\\text{log}}_{\\mathcal{E}}$.  For $i=1,2,3$, $\\theta_i^2=2q_i-2p\\sim 2p$ since there is a meromorphic function $\\wp(z)-\\wp(q_i)$ on $E$ with a double pole at $p$ and a double zero at $q_i$.  Its divisor on $\\mathcal{E}$ is $2q_i-4p$, since $p$ has isotropy $\\mathbb{Z}_2$, hence $2q_i-2p\\sim 2p$.\n\nSince $H^2(\\overline{\\cal M}_{1,1})$ is generated by $\\psi_1$ it is enough to calculate $\\int_{\\overline{\\cal M}_{1,1}}\\Theta_{1,1}$.  The Chern character of the push-forward bundle $E_{1,1}$ is calculated via the Grothendieck-Riemann-Roch theorem\n$$ \\text{Ch}(R\\pi_*\\mathcal{E}^\\vee)=\\pi_*(\\text{Ch}(\\mathcal{E}^\\vee)\\text{Td}(\\omega_\\pi^\\vee)).\n$$\nIn fact we need to use the orbifold Grothendieck-Riemann-Roch theorem \\cite{ToeThe}.  The calculation we need is a variant of the calculation in \\cite[Theorem 6.3.3]{FJRWit} which applies to $\\mathcal{E}$ such that $\\mathcal{E}^2=\\omega_{\\mathcal{C}}^{\\text{log}}$ instead of $\\mathcal{E}^\\vee$.  Importantly, this means that the Todd class has been worked out, and it remains to adjust the $\\text{Ch}(\\mathcal{E}^\\vee)$ term.  We get\n$$\\int_{\\overline{\\cal M}_{1,1}}p_*c_1(E_{1,1})=2\\int_{\\overline{\\cal M}_{1,1}}\\left[\\frac{11}{24}\\kappa_1+\\frac{1}{24}\\psi_1+\\frac12 \\left(-\\frac{1}{24}+\\frac{1}{12}\\right)(i_{\\Gamma})_*(1)\\right]\n=2\\left(\\frac{11}{24^2}+\\frac{1}{24^2}+\\frac12\\cdot \\frac{1}{24}\\cdot\\frac12\\right)=\\frac{1}{16}$$\nwhich agrees with\n$$\\int_{\\overline{\\cal M}_{1,1}}\\frac32\\psi_1=\\frac32\\cdot\\frac{1}{24}=\\frac{1}{16}.\n$$\nHence $p_*c_1(E)=\\frac32\\psi_1$ and $\\Theta_{1,1}=2p_*c_1(E)=3\\psi_1$.\n\n\n\\end{proof}\n\\begin{proposition}\nThe pushforward $p_*2^n\\Omega_{g,n}=\\Theta_{g,n}\\in H^{4g-4+2n}(\\overline{\\cal M}_{g,n})$ satisfies property \\eqref{glue}.\n\\end{proposition}\n\\begin{proof}\nThe two properties \\eqref{glue} of $\\Theta_{g,n}$, follow from the analogous properties for $\\Omega_{g,n}$.  This uses the relationship between compositions of pull-backs and push-forwards in the following diagrams:\n\\begin{center}\n\\begin{tikzpicture}[scale=0.5]\n\\draw (0,0) node {$\\overline{\\cal M}_{g-1,n+2}^{\\rm spin}$};\n\\draw [->, line width=1pt,dashed] (2,0)--(4,0);\n\\draw (3,.5) node {$\\phi_{\\text{irr}}\\circ\\hat{\\nu}^{-1}$};\n\\draw (3,-3.5) node {$\\phi_{\\text{irr}}$};\n\\draw (6,0) node {$\\overline{\\cal M}_{g,n}^{\\rm spin}$};\n\\draw [->, line width=1pt] (0,-1)--(0,-3);\n\\draw (-.5,-2) node {$p$};\n\\draw (0,-4) node {$\\overline{\\cal M}_{g-1,n+2}$};\n\\draw [->, line width=1pt] (2,-4)--(4,-4);\n\\draw (6,-4) node {$\\overline{\\cal M}_{g,n}$};\n\\draw [->, line width=1pt] (6,-1)--(6,-3);\n\\draw (5.5,-2) node {$p$};\n\\end{tikzpicture}\n\\hspace{3cm}\n\\begin{tikzpicture}[scale=0.5]\n\\draw (0,0) node {$\\overline{\\cal M}_{h,|I|+1}^{\\rm spin}\\times \\overline{\\cal M}_{g-h,|J|+1}^{\\rm spin}$};\n\\draw [->, line width=1pt,dashed] (4,0)--(6,0);\n\\draw (5,.5) node {$\\phi_{h,I}\\circ\\hat{\\nu}^{-1}$};\n\\draw (5,-3.5) node {$\\phi_{h,I}$};\n\\draw (8,0) node {$\\overline{\\cal M}_{g,n}^{\\rm spin}$};\n\\draw [->, line width=1pt] (0,-1)--(0,-3);\n\\draw (-.5,-2) node {$p$};\n\\draw (0,-4) node {$\\overline{\\cal M}_{h,|I|+1}\\times\\overline{\\cal M}_{g-h,|J|+1}$};\n\\draw [->, line width=1pt] (4,-4)--(6,-4);\n\\draw (8,-4) node {$\\overline{\\cal M}_{g,n}$};\n\\draw [->, line width=1pt] (8,-1)--(8,-3);\n\\draw (7.5,-2) node {$p$};\n\\end{tikzpicture}\n\\end{center}\nwhere the broken arrows signify multiply-defined maps which are defined above using fibre products.\n\nOn cohomology, we have $\\phi_{\\text{irr}}^*p_*=4p_*\\hat\\nu_*\\phi_{\\text{irr}}^*$ and $\\phi_{h,I}^*p_*=4p_*\\hat\\nu_*\\phi_{h,I}^*$ where the factor of 4 is due to the degree of $\\hat\\nu$ ramification of $p$---see (39) in \\cite{JKVMod}.  Hence \n$$\\phi_{\\text{irr}}^*\\Theta_{g,n}=\\phi_{\\text{irr}}^*p_*2^n\\Omega_{g,n}=4p_*\\hat\\nu_*\\phi_{\\text{irr}}^*2^n\\Omega_{g,n}=p_*2^{n+2}\\Omega_{g-1,n+2}=\\Theta_{g-1,n+2}$$\nand similarly $\\phi_{h,I}^*\\Theta_{g,n}=\\pi_1^*\\Theta_{h,|I|+1}\\cdot\\pi_2^* \\Theta_{g-h,|J|+1}$ uses $4\\cdot2^n=2^{n+2}=2^{|I|+1+|J|+1}$.\n\\end{proof}\n\n\\begin{remark}\nThe construction of $\\Omega_{g,n}$ should also follow from the cosection construction in \\cite{CLLWit} using the moduli space of spin curves with fields \n$$\\overline{\\cal M}_{g,n}(\\mathbb{Z}_2)^p=\\{(C,\\theta,\\rho)\\mid (C,\\theta)\\in\\overline{\\cal M}_{g,n}^{\\rm spin},\\ \\rho\\in H^0(\\theta)\\}.$$\nA cosection of the pull-back of $E_{g,n}$ to $\\overline{\\cal M}_{g,n}(\\mathbb{Z}_2)^p$ is given by $\\rho^{-3}$ since it pairs well with $H^1(\\theta)$---we have $\\rho^{-3}\\in H^0((\\theta^\\vee)^3)$ while $H^1(\\theta)\\cong H^0(\\omega\\otimes\\theta^\\vee)^\\vee=H^0((\\theta^\\vee)^3)^\\vee$.  Using the cosection $\\rho^{-3}$ a virtual fundamental class is constructed in \\cite{CLLWit} that likely gives rise to $\\Omega_{g,n}\\in H^{4g-4+2n}(\\overline{\\cal M}_{g,n}^{\\rm spin})$.  The virtual fundamental class is constructed away from the zero set of $\\rho$.\n\\end{remark}\n\n\n\\section{Uniqueness}   \\label{sec:unique}\nThe degree property \\eqref{degree} of Theorem~\\ref{main}, $\\Theta_{g,n}\\in H^{4g-4+2n}(\\overline{\\cal M}_{g,n})$, proven below, enables a reduction argument which leads to uniqueness of intersection numbers $\\displaystyle \\int_{\\overline{\\cal M}_{g,n}}\\Theta_{g,n}\\prod_{i=1}^n\\psi_i^{m_i}\\prod_{j=1}^N\\kappa_{\\ell_j}$ which is property \\eqref{unique} of Theorem~\\ref{main}.  In this section we prove these two results together with properties \\eqref{genus0} and \\eqref{symmetric} which are immediate consequences.\nWe leave the proof of the final property \\eqref{thetatau} of Theorem~\\ref{main} until Section~\\ref{intkdv}.\n\nWe first prove the following lemma which will be needed later.\n\\begin{lemma}  \\label{nonzero}\nProperties \\eqref{pure}-\\eqref{base} imply that $\\Theta_{g,n}\\neq 0$ for $g>0$.\n\\end{lemma}\n\\begin{proof}\nWe have $\\Theta_{1,1}=3\\psi_1\\neq 0$ by \\eqref{base} hence $\\Theta_{1,n}=3\\psi_1\\psi_2...\\psi_n$ by \\eqref{forget} together with \nthe equality $\\psi_n\\psi_i=\\psi_n\\pi^*\\psi_i$ for $i<n$.  Hence $\\int_{\\overline{\\cal M}_{g,n}}\\Theta_{1,n}=(n-1)!\/8$ and in particular $\\Theta_{1,n}\\neq 0$.  \n\nNow we proceed by induction on $g$.  Assume $\\Theta_{h,n}\\neq 0$ for $0<h<g$.  For $g>1$, let $\\Gamma$ be the stable graph consisting of a genus $g-1$ vertex attached by a single edge to a genus 1 vertex with $n$ labeled leaves (denoted {\\em ordinary} leaves in Section~\\ref{twloop}).  Then by \\eqref{glue}\n$$\\phi_{\\Gamma}^*\\Theta_{g,n}=\\Theta_{g-1,1}\\otimes\\Theta_{1,n+1}\n$$\nwhich is non-zero since $\\Theta_{g-1,1}\\neq 0$ by the inductive hypothesis and $\\Theta_{1,n+1}\\neq 0$ by the calculation above.\n\\end{proof}\n\\begin{remark}\nWe can replace \\eqref{base} by any initial condition $\\Theta_{1,1}=\\lambda\\psi_1$ for $\\lambda\\neq 0$ and the proof of Lemma~\\ref{nonzero} still applies, if $\\Theta_{g,n}$ exists.\n\\end{remark}\n\n\\begin{proof}[Proof of \\eqref{degree}]\n\n\nWrite $d(g,n)=\\text{degree}(\\Theta_{g,n})$ which exists by \\eqref{pure}.  The degree here is half the cohomological degree so $\\Theta_{g,n}\\in H^{2d(g,n)}(\\overline{\\cal M}_{g,n})$.  Using \\eqref{glue}, $\\phi_{\\text{irr}}^*\\Theta_{g,n}=\\Theta_{g-1,n+2}$ implies that $d(g,n)=d(g-1,n+2)$ since $\\Theta_{g-1,n+2}\\neq 0$ by Lemma~\\ref{nonzero}.  Hence \n$d(g,n)=f(2g-2+n)$ \nis a function of $2g-2+n$.  Similarly, using \\eqref{glue}, $\\phi_{h,I}^*\\Theta_{g,n}=\\Theta_{h,|I|+1}\\otimes \\Theta_{g-h,|J|+1}$ implies that \n$f(a+b)=f(a)+f(b)=(a+b)f(1)$ since $\\Theta_{h,|I|+1}\\neq 0$ and $\\Theta_{g-h,|J|+1}\\neq 0$ again by Lemma~\\ref{nonzero}.\nHence $d(g,n)=(2g-2+n)k$ for an integer $k$.  But  $d(g,n)\\leq 3g-3+n$ implies $k\\leq 1$.  When $k=0$, this gives $\\Theta_{g,n}\\in H^0(\\overline{\\cal M}_{g,n})$, and $\\Theta_{g,n}$ is a topological field theory.  But $\\deg\\Theta_{g,n}=0$ contradicts  \\eqref{base} hence $k=1$ and $\\deg\\Theta_{g,n}=2g-2+n$.\n\nThe proof above used only properties \\eqref{pure}, \\eqref{glue} and \\eqref{base}.  Alternatively, one can use \\eqref{forget} in place of the second part of \\eqref{glue} given by $\\phi_{h,I}^*\\Theta_{g,n}=\\Theta_{h,|I|+1}\\otimes \\Theta_{g-h,|J|+1}$.\n\\end{proof}\n\\begin{proof}[Proof of \\eqref{genus0}]\nThis is an immediate consequence of \\eqref{degree} since $\\deg\\Theta_{0,n}=n-2>n-3=\\dim\\overline{\\cal M}_{0,n}$ hence $\\Theta_{0,n}=0$.  For any stable graph $\\Gamma$ with a genus 0 vertex, Remark~\\ref{gluestab} gives $\\phi_{\\Gamma}^*\\Theta_{g,n}=\\Theta_{\\Gamma}$.  Furthermore $0=\\Theta_{\\Gamma}=\\prod_{v\\in V(\\Gamma)}\\pi_v^*\\Theta_{g(v),n(v)}$ since the genus 0 vertex contributes a factor of 0 to the product.\n\\end{proof}\n\\begin{proof}[Proof of \\eqref{symmetric}]\nProperty \\eqref{forget} implies that $\\Theta_{g,n}=\\pi^*\\Theta_g\\cdot\\prod_{i=1}^n\\psi_i$ where $\\pi:\\overline{\\cal M}_{g,n}\\to\\overline{\\cal M}_{g}$ is the forgetful map.  Since $\\pi^*\\omega\\in H^*(\\overline{\\cal M}_{g,n})^{S_n}$ for any $\\omega\\in H^*(\\overline{\\cal M}_{g})$ and clearly $\\prod_{i=1}^n\\psi_i\\in H^*(\\overline{\\cal M}_{g,n})^{S_n}$ hence we have $\\Theta_{g,n}\\in H^*(\\overline{\\cal M}_{g,n})^{S_n}$ as required.\n\\end{proof}\n\\begin{proof}[Proof of \\eqref{unique}]\nThe uniqueness result follows from the more general result given in the following proposition.\n\\end{proof}\n\\begin{proposition}   \\label{th:unique}\nFor any $\\Theta_{g,n}$ satisfying \\eqref{pure} - \\eqref{forget} above, the intersection numbers\n\\begin{equation}  \\label{corr}\n\\int_{\\overline{\\cal M}_{g,n}}\\Theta_{g,n}\\prod_{i=1}^n\\psi_i^{m_i}\\prod_{j=1}^N\\kappa_{\\ell_j}\n\\end{equation}\nare uniquely determined from the initial condition $\\Theta_{1,1}=\\lambda\\psi_1$ for $\\lambda\\in\\mathbb{C}$.\n\\end{proposition}\n\\begin{proof}\nFor $n>0$, we will push forward the integral \\eqref{corr} via the forgetful map $\\pi:\\overline{\\cal M}_{g,n}\\to\\overline{\\cal M}_{g,n-1}$ as follows.  Consider first the case when there are no $\\kappa$ classes.  The presence of $\\psi_n$ in $\\Theta_{g,n}=\\psi_n\\cdot\\pi^*\\Theta_{g,n-1}$ gives\n$$\\Theta_{g,n}\\psi_k=\\Theta_{g,n}\\pi^*\\psi_k,\\quad k<n\n$$\nsince $\\psi_n\\psi_k=\\psi_n\\pi^*\\psi_k$ for $k<n$.  Hence\n\\begin{align*}\n\\int_{\\overline{\\cal M}_{g,n}}\\Theta_{g,n}\\prod_{i=1}^n\\psi_i^{m_i}&=\n\\int_{\\overline{\\cal M}_{g,n}}\\pi^*\\Big(\\Theta_{g,n-1}\\prod_{i=1}^{n-1}\\psi_i^{m_i}\\Big)\\psi_n^{m_n+1}\n=\\int_{\\overline{\\cal M}_{g,n-1}}\\pi_*\\left\\{\\pi^*\\Big(\\Theta_{g,n-1}\\prod_{i=1}^{n-1}\\psi_i^{m_i}\\Big)\\psi_n^{m_n+1}\\right\\}\\\\\n&=\\int_{\\overline{\\cal M}_{g,n-1}}\\Theta_{g,n-1}\\prod_{i=1}^{n-1}\\psi_i^{m_i}\\kappa_{m_n}\n\\end{align*}\nso we have reduced an intersection number over $\\overline{\\cal M}_{g,n}$ to an intersection number over $\\overline{\\cal M}_{g,n-1}$.\nIn the presence of $\\kappa$ classes, replace $\\kappa_{\\ell_j}$ by $\\kappa_{\\ell_j}=\\pi^*\\kappa_{\\ell_j}+\\psi_n^{\\ell_j}$ and repeat the push-forward as above on all summands.  By induction, we see that\n$$\\int_{\\overline{\\cal M}_{g,n}}\\Theta_{g,n}\\prod_{i=1}^n\\psi_i^{m_i}\\prod_{j=1}^N\\kappa_{\\ell_j}=\\int_{\\overline{\\cal M}_{g}}\\Theta_{g}\\cdot p(\\kappa_1,\\kappa_2,...,\\kappa_{3g-3})$$\ni.e. the intersection number \\eqref{corr} reduces to an intersection number over $\\overline{\\cal M}_{g}$ of $\\Theta_g$ times a polynomial in the $\\kappa$ classes.  \n\nBy \\eqref{degree}, $\\deg\\Theta_g=2g-2$, so we may assume the polynomial $p$ consists only of terms of homogeneous degree $g-1$ (where $\\deg\\kappa_r=r$).  But by Looijenga's theorem \\cite{LooTau}, and the stronger result given by Pixton's relations \\cite{PPZRel}, a homogeneous degree $g-1$ monomial in the $\\kappa$ classes is equal in cohomology to the sum of boundary terms.  Now property \\eqref{glue} of $\\Theta_g$, shows that the pull-back of $\\Theta_g$ to these boundary terms is $\\Theta_{g',n'}$ for $g'<g$ so we have expressed \\eqref{corr} as a sum of integrals of $\\theta_{g',n'}$ against $\\psi$ and $\\kappa$ classes.  By induction, one can reduce to the integral $\\int_{\\overline{\\cal M}_{1,1}}\\Theta_{1,1}=\\frac{\\lambda}{24}$ and the proposition is proven.\n\\end{proof}\n\\begin{remark}   \\label{removekappa}\nThe intersection numbers $\\int_{\\overline{\\cal M}_{g,n}}\\Theta_{g,n}\\prod_{i=1}^n\\psi_i^{m_i}\\prod_{j=1}^N\\kappa_{\\ell_j}$ are directly related to the intersection numbers $\\int_{\\overline{\\cal M}_{g,n}}\\Theta_{g,n}\\prod_{i=1}^n\\psi_i^{m_i}$ with no $\\kappa$ classes.  This essentially reverses the reduction shown in the proof of Proposition~\\ref{th:unique}.  Explicitly,\nfor $\\pi:\\overline{\\cal M}_{g,n+N}\\to\\overline{\\cal M}_{g,n}$ and ${\\bf m}=(m_1,...,m_N)$ define a polynomial in $\\kappa$ classes by \n$$R_{\\bf m}(\\kappa_1,\\kappa_2,...)=\\pi_*\\Big(\\psi_{n+1}^{m_1+1}...\\psi_{n+N}^{m_N+1}\\Big)$$\nso, for example, $R_{(m_1,m_2)}=\\kappa_{m_1}\\kappa_{m_1}+\\kappa_{m_1+m_2}$.\nThen\n\\begin{equation}  \\label{thetakappa}\n\\Theta_{g,n}\\cdot R_{\\bf m}=\\Theta_{g,n}\\cdot \\pi_*\\Big(\\psi_{n+1}^{m_1+1}...\\psi_{n+N}^{m_N+1}\\Big)\n=\\pi_*\\Big(\\pi^*\\Theta_{g,n}\\cdot\\psi_{n+1}^{m_1+1}...\\psi_{n+N}^{m_N+1}\\Big)\n=\\pi_*\\Big(\\Theta_{g,n+N}\\cdot\\psi_{n+1}^{m_1}...\\psi_{n+N}^{m_N}\\Big).\n\\end{equation}\nThe polynomials $R_{\\bf m}(\\kappa_1,\\kappa_2,...)$ generate all polynomials in the $\\kappa_i$ so \\eqref{thetakappa} can be used to remove any $\\kappa$ class.\n\\end{remark}\n\\begin{remark} \nExistence of classes $\\Theta_{g,n}$ satisfying \\eqref{pure} - \\eqref{forget} and the initial condition $\\Theta_{1,1}=\\lambda\\psi_1$ for a general $\\lambda\\in\\mathbb{C}$ is an interesting problem.  One approach is to look inside the strata algebra---see Section~\\ref{pixrel}---which consists of push-forwards of classes involving $\\kappa$ and $\\psi$ classes from lower strata.\n\\end{remark}\nThe intersection numbers $\\int_{\\overline{\\cal M}_{g,n}}\\Theta_{g,n}\\cdot\\prod_{j=1}^n\\psi_j^{k_j}$ satisfy a dilaton equation which is realised as a homogeneity condition on the partition function  \n\\begin{equation}  \\label{eq:tauf}\nZ^{\\Theta}(\\hbar,t_0,t_1,...)=\\exp\\sum_{g,n,\\vec{k}}\\frac{\\hbar^{g-1}}{n!}\\int_{\\overline{\\cal M}_{g,n}}\\Theta_{g,n}\\cdot\\prod_{j=1}^n\\psi_j^{k_j}\\prod t_{k_j}.\n\\end{equation}\n\\begin{proposition}  \\label{th:dilaton}\nThe function $Z^{\\Theta}(\\hbar,t_0,t_1,...)$ is homogeneous of degree 0 with respect to $\\{q=1-t_0,t_1,t_2,...\\}$ with $\\deg q=1$ and $\\deg t_i=2i+1$ for $i>0$.  Equivalently it satisfies the dilaton equation:\n\\begin{equation}  \\label{dilaton}\n\\frac{\\partial}{\\partial t_0}Z(\\hbar,t_0,t_1,...)=\\sum_{i=0}^{\\infty}(2i+1)t_i \\frac{\\partial}{\\partial t_i}Z(\\hbar,t_0,t_1,...)\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nWe have\n\\begin{align*} \n\\int_{\\overline{\\cal M}_{g,n+1}}\\Theta_{g,n+1}\\cdot\\prod_{j=1}^n\\psi_j^{k_j}&=\\int_{\\overline{\\cal M}_{g,n+1}}\\pi^*\\Theta_{g,n}\\cdot\\psi_{n+1}\\cdot\\prod_{j=1}^n\\psi_j^{k_j}=\\int_{\\overline{\\cal M}_{g,n+1}}\\pi^*\\Theta_{g,n}\\cdot\\psi_{n+1}\\cdot\\prod_{j=1}^n\\pi^*\\psi_j^{k_j}\\\\\n&=\\int_{\\overline{\\cal M}_{g,n}}\\Theta_{g,n}\\cdot\\prod_{j=1}^n\\psi_j^{k_j}\\cdot\\pi_*\\psi_{n+1}=\n(2g-2+n)\\int_{\\overline{\\cal M}_{g,n}}\\Theta_{g,n}\\cdot\\prod_{j=1}^n\\psi_j^{k_j}.\\nonumber\n\\end{align*}\nwhere we have used $\\psi_{n+1}\\cdot\\psi_j=\\psi_{n+1}\\cdot\\pi^*\\psi_j$ for $j=1,...,n$ and $\\pi_*(\\pi^*\\omega\\cdot\\psi_{n+1})=\\omega\\cdot\\pi_*\\psi_{n+1}$.  But this exactly agrees with the dilaton equation \\eqref{dilaton} via the correspondence \\eqref{eq:tauf}.\n\\end{proof}\nProposition~\\ref{th:dilaton} together with the initial condition $\\Theta_{1,1}=\\lambda\\psi_1$, or $\\int_{\\overline{\\cal M}_{1,1}}\\Theta_{1,1}=\\frac{\\lambda}{24}$, gives \n\\begin{equation} \\label{genus1}\n\\log Z^{\\Theta}=-\\frac{\\lambda}{24}\\log(1-t_0)+O(\\hbar).\n\\end{equation}\nThe following example demonstrates Proposition~\\ref{th:unique} with an explicit genus 2 relation.\n\\begin{example}  \\label{gen2rel}\nA genus two Pixton relation first proven by Mumford \\cite[equation (8.5)]{MumTow}, relating $\\kappa_1$ and the divisors $\\cal M_{\\Gamma_1}\\cong{\\overline{\\cal M}_{1,1}}\\times{\\overline{\\cal M}_{1,1}}$ and $\\cal M_{\\Gamma_2}\\cong{\\overline{\\cal M}_{1,2}}$ in $\\overline{\\cal M}_{2}$, labeled by stable graphs $\\Gamma_i$ is given by\n$$\\kappa_1-\\frac{7}{5}[\\cal M_{\\Gamma_1}]-\\frac{1}{5}[\\cal M_{\\Gamma_2}]=0$$\nwhich induces the relation \n$$\\Theta_{2}\\cdot\\kappa_1-\\frac{7}{5}\\Theta_{2}\\cdot[\\cal M_{\\Gamma_1}]-\\frac{1}{5}\\Theta_{2}\\cdot[\\cal M_{\\Gamma_2}]=0.$$ \nProperty \\eqref{glue} of $\\Theta_{g,n}$ yields\n$$\\int_{\\overline{\\cal M}_{2}}\\Theta_{2}\\cdot[\\cal M_{\\Gamma_1}]=\\int_{\\cal M_{\\Gamma_1}}\\Theta_{2}=\\int_{\\overline{\\cal M}_{1,1}}\\Theta_{1,1}\\cdot\\int_{\\overline{\\cal M}_{1,1}}\\Theta_{1,1}\\cdot\\frac{1}{|\\text{Aut}(\\Gamma_1)|},\\quad\\int_{\\cal M_{\\Gamma_2}}\\Theta_{2}=\\int_{\\overline{\\cal M}_{1,2}}\\Theta_{1,2}\\cdot\\frac{1}{|\\text{Aut}(\\Gamma_2)|}\n$$\nhence the relation on the level of intersection numbers is given by\n\\begin{equation*}  \n\\int_{\\overline{\\cal M}_{2}}\\Theta_{2}\\cdot\\kappa_1-\\frac{7}{5}\\cdot\\int_{\\overline{\\cal M}_{1,1}}\\Theta_{1,1}\\cdot\\int_{\\overline{\\cal M}_{1,1}}\\Theta_{1,1}\\cdot\\frac{1}{|\\text{Aut}(\\Gamma_1)|}-\\frac{1}{5}\\cdot\\int_{\\overline{\\cal M}_{1,2}}\\Theta_{1,2}\\cdot\\frac{1}{|\\text{Aut}(\\Gamma_2)|}=0.\n\\end{equation*}\nWe have $\\int_{\\overline{\\cal M}_{1,1}}\\Theta_{1,1}=\\frac{\\lambda}{24}=\\int_{\\overline{\\cal M}_{1,2}}\\Theta_{1,1}$ from \\eqref{genus1}, and $|\\text{Aut}(\\Gamma_1)|=2=|\\text{Aut}(\\Gamma_2)|$.\nHence\n\\begin{align*}  \n\\int_{\\overline{\\cal M}_{2}}\\Theta_{2}\\cdot\\kappa_1 &=\\frac{7}{5}\\cdot\\int_{\\overline{\\cal M}_{1,1}}\\Theta_{1,1}\\cdot\\int_{\\overline{\\cal M}_{1,1}}\\Theta_{1,1}\\cdot\\frac{1}{|\\text{Aut}(\\Gamma_1)|}+\\frac{1}{5}\\cdot\\int_{\\overline{\\cal M}_{1,2}}\\Theta_{1,2}\\cdot\\frac{1}{|\\text{Aut}(\\Gamma_2)|}\\\\&=\\frac{7}{5}\\cdot\\left(\\frac{\\lambda}{24}\\right)^2\\cdot\\frac{1}{2}+\\frac{1}{5}\\cdot\\frac{\\lambda}{24}\\cdot\\frac{1}{2}=\\frac{7\\lambda^2+24\\lambda}{5760}. \\nonumber\n\\end{align*} \n\\end{example}\nFrom now on we will specialise to the case $\\lambda=3$ for which we have a proof of existence of $\\Theta_{g,n}$.  From example~\\ref{gen2rel}, we have\n$\\int_{\\overline{\\cal M}_{2,1}}\\Theta_{2,1}\\cdot\\psi_1=\\int_{\\overline{\\cal M}_{2,1}}\\pi^*\\Theta_2\\cdot\\psi_1^2=  \\int_{\\overline{\\cal M}_{2}}\\Theta_{2}\\cdot\\kappa_1=\\frac{3}{128}$.\nAll genus 2 terms can be obtained from $\\int_{\\overline{\\cal M}_{2,1}}\\Theta_{2,1}\\cdot\\psi_1$ and \\eqref{dilaton}.  Combining this with  \\eqref{genus1} we have\n\\begin{equation}  \\label{lowgtheta} \n\\log Z^{\\Theta}=-\\frac{1}{8}\\log(1-t_0)+\\hbar\\frac{3}{128}\\frac{t_1}{(1-t_0)^3}+O(\\hbar^2).\n\\end{equation} \n\n\\section{KdV tau functions}   \\label{sec:kdv}\nIn this section we prove property~\\ref{thetatau} of Theorem~\\ref{main} which states that a generating function for the intersection numbers $\\int_{\\overline{\\cal M}_{g,n}}\\Theta_{g,n}\\prod_{i=1}^n\\psi_i^{m_i}$ is a tau function of the KdV hierarchy, analogous to the theorem conjectured by Witten and proven by Kontsevich for the intersection numbers $\\int_{\\overline{\\cal M}_{g,n}}\\prod_{i=1}^n\\psi_i^{m_i}$.\n\nA tau function $Z(t_0,t_1,...)$ of the KdV hierarchy (equivalently the KP hierarchy in odd times $p_{2k+1}=t_k\/(2k+1)!!$) gives rise to a solution $U=\\hbar\\frac{\\partial^2}{\\partial t_0^2}\\log Z$ of the KdV hierarchy \n\\begin{equation}\\label{kdv}\nU_{t_1}=UU_{t_0}+\\frac{\\hbar}{12}U_{t_0t_0t_0},\\quad U(t_0,0,0,...)=f(t_0).\n\\end{equation}\nThe first equation in the hierarchy is the KdV equation \\eqref{kdv}, and later equations $U_{t_k}=P_k(U,U_{t_0},U_{t_0t_0},...)$ for $k>1$ determine $U$ uniquely from $U(t_0,0,0,...)$.  See \\cite{MJDSol} for the full definition.\n\nThe Brezin-Gross-Witten solution $U^{\\text{BGW}}=\\hbar\\frac{\\partial^2}{\\partial t_0^2}\\log Z^{\\text{BGW}}$ of the KdV hierarchy arises out of a unitary matrix model studied in \\cite{BGrExt,GWiPos}.  It is defined by the initial condition\n$$\nU^{\\text{BGW}}(t_0,0,0,...)=\\frac{\\hbar}{8(1-t_0)^2}.\n$$\nThe low genus $g$ terms (= coefficient of $\\hbar^{g-1}$) of $\\log Z^{\\text{BGW}}$ are\n\\begin{align}  \\label{lowg}\n\\log Z^{\\text{BGW}}&=-\\frac{1}{8}\\log(1-t_0)+\\hbar\\frac{3}{128}\\frac{t_1}{(1-t_0)^3}+\\hbar^2\\frac{15}{1024}\\frac{t_2}{(1-t_0)^5}+\\hbar^2\\frac{63}{1024}\\frac{t_1^2}{(1-t_0)^6}+O(\\hbar^3)\\\\\n&=(\\frac{1}{8}t_0+\\frac{1}{16}t_0^2+\\frac{1}{24}t_0^3+...)+\\hbar(\\frac{3}{128}t_1+\\frac{9}{128}t_0t_1+...)+\\hbar^2(\\frac{15}{1024}t_2+\\frac{63}{1024}t_1^2+...)+...\\nonumber\n\\end{align}\n\nThe tau function $Z^{\\text{BGW}}(\\hbar,t_0,t_1,...)$ shares many properties of the famous Kontsevich-Witten tau function $Z^{\\text{KW}}(\\hbar,t_0,t_1,...)$ introduced in \\cite{WitTwo}.  The Kontsevich-Witten tau function $Z^{\\text{KW}}$ is defined by the initial condition $U^{\\text{KW}}(t_0,0,0,...)=t_0$ for $U^{\\text{KW}}=\\hbar\\frac{\\partial^2}{\\partial t_0^2}\\log Z^{\\text{KW}}$.  The low genus terms of $\\log Z^{\\text{KW}}$ are \n$$\\log Z^{\\text{KW}}(\\hbar,t_0,t_1,...)=\\hbar^{-1}(\\frac{t_0^3}{3!}+\\frac{t_0^3t_1}{3!}+\\frac{t_0^4t_2}{4!}+...)+\\frac{t_1}{24}+...\n$$\n\\begin{theorem}[Witten-Kontsevich 1992 \\cite{KonInt,WitTwo}]\n$$Z^{\\text{KW}}(\\hbar,t_0,t_1,...)=\\exp\\sum_{g,n}\\hbar^{g-1}\\frac{1}{n!}\\sum_{\\vec{k}\\in\\mathbb{N}^n}\\int_{\\overline{\\cal M}_{g,n}}\\prod_{i=1}^n\\psi_i^{k_i}t_{k_i}\n$$\nis a tau function of the KdV hierarchy\n\\end{theorem}\nThe main aim of this section is to prove that $Z^{\\Theta}(\\hbar,t_0,t_1,...)=Z^{\\text{BGW}}(\\hbar,t_0,t_1,...)$ which implies property~\\ref{thetatau} of Theorem~\\ref{main}.  Agreement of $Z^{\\Theta}(\\hbar,t_0,t_1,...)$ and $Z^{\\text{BGW}}(\\hbar,t_0,t_1,...)$ up to genus 2 is clear from \\eqref{lowgtheta} and \\eqref{lowg} and can be extended to genus 3 using Appendix~\\ref{sec:calc}.  Furthermore, $Z^{\\text{BGW}}(\\hbar,t_0,t_1,...)$ also satisfies the homogeneity property \\eqref{dilaton}---see \\cite{DNoTop}---which is apparent in the low genus terms given in \\eqref{lowg}.  The homogeneity property satisfied by both $Z^{\\Theta}$ and $Z^{\\text{BGW}}$ reduces the proof that they are equal to checking only finitely many coefficients for each genus.  \n\\begin{theorem}  \\label{tauf}\n$$Z^{\\Theta}(\\hbar,t_0,t_1,...)=Z^{\\text{BGW}}(\\hbar,t_0,t_1,...).$$ \n\\end{theorem}\n{\\em Outline of the proof of Theorem~\\ref{tauf}.} We summarise here Sections~\\ref{twloop}-\\ref{sec:TR} which contain the proof of Theorem~\\ref{tauf}.\n\nThe equality of $Z^{\\Theta}$ and $Z^{\\text{BGW}}$ up to genus two used the relation between coefficients of $Z^{\\Theta}(\\hbar,t_0,t_1,...)$\n\\begin{equation}  \\label{relg2}\n\\int_{\\overline{\\cal M}_{2,1}}\\Theta_{2,1}\\cdot\\psi_1-\\frac{7}{10}\\cdot\\int_{\\overline{\\cal M}_{1,1}}\\Theta_{1,1}\\cdot\\int_{\\overline{\\cal M}_{1,1}}\\Theta_{1,1}-\\frac{1}{10}\\cdot\\int_{\\overline{\\cal M}_{1,2}}\\Theta_{1,2}=0\n\\end{equation}\narising from the genus two Pixton relation.  The main idea of the proof is to show that the coefficients of $Z^{\\text{BGW}}(\\hbar,t_0,t_1,...)$ also satisfy \\eqref{relg2}, and more generally a set of relations arising from Pixton relations.  \n\nAssociated to the $A_2$ Frobenius manifold is a partition function $Z_{A_2}(\\hbar,\\{t^{\\alpha}_k\\})$, defined in Section~\\ref{contw}.  Pixton's relations on the level of intersection numbers arise due to unexpected vanishing of some of the coefficients in $Z_{A_2}(\\hbar,\\{t^{\\alpha}_k\\})$.  Briefly, the partition function $Z_{A_2}(\\hbar,\\{t^{\\alpha}_k\\})$ is constructed in two ways---out of intersections of cohomology classes in $H^*(\\overline{\\cal M}_{g,n})$ which form a  CohFT and out of a graphical construction that associates coefficients of $Z^{\\text{KW}}(\\hbar,t_0,t_1,...)$ to vertices of the graphs, described in Sections~\\ref{twloop} and \\ref{sec:A2part}.  The vanishing of coefficients in $Z_{A_2}(\\hbar,\\{t^{\\alpha}_k\\})$ is clear only from one of these constructions hence leads to relations among intersection numbers.  There is a third construction of $Z_{A_2}(\\hbar,\\{t^{\\alpha}_k\\})$ via topological recursion, defined in Section~\\ref{sec:TR}, applied to a spectral curve naturally associated to the $A_2$ Frobenius manifold.  This construction gives a second method to prove vanishing of certain coefficients of $Z_{A_2}(\\hbar,\\{t^{\\alpha}_k\\})$.\n\nThe partition function $Z^{\\Theta}_{A_2}(\\hbar,\\{t^{\\alpha}_k\\})$ is built out of the $A_2$ CohFT cohomology classes in $H^*(\\overline{\\cal M}_{g,n})$ times $\\Theta_{g,n}$.  It stores relations between intersection numbers involving $\\Theta_{g,n}$ such as \\eqref{relg2}.  So for example, the coefficient of $\\hbar t_1^1$\nin $\\log Z^{\\Theta}_{A_2}(\\hbar,\\{t^{\\alpha}_k\\})$ vanishes and is also the relation \\eqref{relg2}.  We can construct $Z^{\\Theta}_{A_2}(\\hbar,\\{t^{\\alpha}_k\\})$ via the graphical construction that produces $Z_{A_2}(\\hbar,\\{t^{\\alpha}_k\\})$ although with vertex contributions coming from coefficients of $Z^{\\Theta}(\\hbar,t_0,t_1,...)$ in place of coefficients of $Z^{\\text{KW}}(\\hbar,t_0,t_1,...)$.\n\nThe idea is to produce a partition function $Z^{\\text{BGW}}_{A_2}(\\hbar,\\{t^{\\alpha}_k\\})$ using the graphical construction of $Z_{A_2}(\\hbar,\\{t^{\\alpha}_k\\})$ with $Z^{\\text{KW}}$ replaced by $Z^{\\text{BGW}}$.  Vanishing of coefficients of $Z^{\\text{BGW}}_{A_2}(\\hbar,\\{t^{\\alpha}_k\\})$ are relations between coefficients of $Z^{\\text{BGW}}$ analogous to each of the relations satisfied by the coefficients of $Z^{\\Theta}_{A_2}(\\hbar,\\{t^{\\alpha}_k\\})$ such as \\eqref{relg2}.  To prove vanishing of primary coefficients of $Z^{\\text{BGW}}_{A_2}(\\hbar,\\{t^{\\alpha}_k\\})$ for $n\\leq g-1$, which is enough to prove that the coefficients of $Z^{\\text{BGW}}$ satisfy the same relations as the coefficients of $Z^{\\Theta}$, we use the third construction of $Z_{A_2}(\\hbar,\\{t^{\\alpha}_k\\})$ via topological recursion.  We construct a spectral curve to get $Z^{\\text{BGW}}_{A_2}(\\hbar,\\{t^{\\alpha}_k\\})$ and this construction allows us to prove vanishing of certain coefficients.  To conclude, we have vanishing of certain coefficients of $Z^{\\Theta}_{A_2}(\\hbar,\\{t^{\\alpha}_k\\})$ due to a cohomological viewpoint, and we have vanishing of corresponding coefficients of $Z^{\\text{BGW}}_{A_2}(\\hbar,\\{t^{\\alpha}_k\\})$ due to the topological recursion viewpoint, and this shows that coefficients of $Z^{\\text{BGW}}$ and $Z^{\\Theta}$ satisfy the same relations.\n\n\n\n\n\n\\subsection{Twisted loop group action} \\label{twloop}\n\nGivental \\cite{GiventalMain} showed how to build partition functions of cohomological field theories which are sequences of cohomology classes in $H^*(\\overline{\\cal M}_{g,n})$---see Section~\\ref{sec:cohft}---out of the basic building block $Z^{\\text{KW}}(\\hbar,t_0,t_1,...)$.  This construction can be immediately adapted to allow one to use in place of $Z^{\\text{KW}}(\\hbar,t_0,t_1,...)$ the building blocks $Z^{\\text{BGW}}(\\hbar,t_0,t_1,...)$ and $Z^{\\Theta}(\\hbar,t_0,t_1,...)$ (where the latter two will eventually be shown to coincide).  Givental defined an action by elements of the twisted loop group and translations on the basic building block partition functions above.  This action was interpreted as an action on sequences of cohomology classes in $H^*(\\overline{\\cal M}_{g,n})$ independently, by Katzarkov-Kontsevich-Pantev, Kazarian and Teleman---see \\cite{PPZRel,ShaBCOV}.\n\nConsider an element of the loop group $LGL(N,\\mathbb{C})$ given by a formal series\n$$R(z)  =  \\sum_{k=0}^\\infty R_k z^k$$\nwhere $R_k$ are $N\\times N$ matrices and $R_0=I$.  We further require $R(z)$ to lie in the twisted loop group $L^{(2)}GL(N,\\mathbb{C})\\subset LGL(N,\\mathbb{C})$, the subgroup which is defined to consist of elements satisfying\n$$ R(z)R(-z)^T=I.\n$$\nDefine \n$$\\mathcal{E}(w,z)=\\frac{I-R^{-1}(z)R^{-1}(w)^T}{w+z}=\\sum_{i,j\\geq 0}\\mathcal{E}_{ij}w^iz^j$$\nwhich has the power series expansion on the right since the numerator $I-R^{-1}(z)R^{-1}(w)^T$ vanishes at $w=-z$ since $R^{-1}(z)$ is also an element of the twisted loop group.\n\nGivental's action is defined via weighted sums over graphs.  Consider the following set of decorated graphs with vertices labeled by the set $\\{1,...,N\\}$.\n\\begin{definition}\nFor a graph $\\gamma$ denote by\n$$V(\\gamma),\\quad E(\\gamma),\\quad H(\\gamma),\\quad L(\\gamma)=L^*(\\gamma)\\sqcup L^\\bullet(\\gamma)$$\nits set of vertices, edges, half-edges and leaves.  The disjoint splitting of $L(\\gamma)$ into ordinary leaves $L^*$ and dilaton leaves $L^\\bullet$ is part of the structure on $\\gamma$.  The set of half-edges consists of leaves and oriented edges so there is an injective map $L(\\gamma)\\to H(\\gamma)$ and a multiply-defined map $E(\\gamma)\\to H(\\gamma)$ denoted by $E(\\gamma)\\ni e\\mapsto \\{e^+,e^-\\}\\subset H(\\gamma)$.\nThe map sending a half-edge to its vertex is given by $v:H(\\gamma)\\to V(\\gamma)$.  Decorate $\\gamma$ by  functions:\n\\begin{align*}\ng&:V(\\gamma)\\to\\mathbb{N}\\\\\n\\alpha&:V(\\gamma)\\to\\{1,...,N\\}\\\\\np&:L^*(\\gamma)\\stackrel{\\cong}{\\to}\\{1,2,...,n\\}\\\\\nk&:H(\\gamma)\\to\\mathbb{N}\n\\end{align*}\nsuch that  $k|_{L^\\bullet(\\gamma)}>1$ and $n=|L^*(\\gamma)|$.  We write $g_v=g(v)$, $\\alpha_v=\\alpha(v)$, $\\alpha_\\ell=\\alpha(v(\\ell))$, $p_\\ell=p(\\ell)$, $k_\\ell=k(\\ell)$.\nThe {\\em genus} of $\\gamma$ is $g(\\gamma)=\\displaystyle b_1(\\gamma)+\\hspace{-2mm}\\sum_{v\\in V(\\gamma)}\\hspace{-2mm}g(v)$.  We say $\\gamma$ is {\\em stable} if any vertex labeled by $g=0$ is of valency $\\geq 3$ and there are no isolated vertices labeled by $g=1$.   We write $n_v$ for the valency of the vertex $v$.\nDefine $\\Gamma_{g,n}$ to be the set of all stable, connected, genus $g$, decorated graphs with $n$ ordinary leaves.\n\\end{definition}\nGiven a sequence of classes $\\Omega=\\{\\Omega_{g,n}\\in H^*(\\overline{\\cal M}_{g,n})\\otimes (V^*)^{\\otimes n}\\mid g,n\\in\\mathbb{N},2g-2+n>0\\}$, following \\cite{PPZRel,ShaBCOV} define a new sequence of classes $R\\Omega=\\{(R\\Omega)_{g,n}\\}$ by a weighted sum over stable graphs, with weights defined as follows.\n\\begin{enumerate}[(i)]\n\\item {\\em Vertex weight:} $w(v)=\\Omega_{g(v),n_v}$ at each vertex $v$ \n\\item {\\em Leaf weight:} $w(\\ell)=R^{-1}(\\psi_{p(\\ell)})$ at each leaf $\\ell$\n\\item {\\em Edge weight:} $w(e)=\\mathcal{E}(\\psi_e',\\psi_e'')$ at each edge $e$\n\\end{enumerate}\nThen\n$$(R\\Omega)_{g,n}=\\sum_{\\Gamma\\in G_{g,n}}\\frac{1}{|{\\rm Aut}(\\Gamma)|}(p_{\\Gamma})_*\\hspace{-6mm}\\prod_{\\begin{array}{c}v\\in V(\\Gamma)\\\\\\ell\\in L^*(\\Gamma)\\\\ e\\in E(\\Gamma)\\end{array}} \\hspace{-4mm}w(v) w(\\ell)w(e)\n$$\nThis defines an action of the twisted loop group on sequences of cohomology classes.  It is applied in \\cite{PPZRel,ShaBCOV} to cohomological field theories $\\Omega_{g,n}$ whereas a generalised notion of cohomological field theory is used here.\n \nDefine a translation action of $T(z)\\in zV[[z]]$ on the sequence $\\Omega_{g,n}$ as follows.\n\\begin{enumerate}[(iv)]\n\\item {\\em Dilaton leaf weight:}  $w(\\ell)=T(\\psi_{p(\\ell)})$ at each dilaton leaf $\\ell\\in L^\\bullet$.\n\\end{enumerate}\nSo the translation action is given by\n\\begin {equation}  \\label{transl}\n(T\\Omega)_{g,n}(v_1\\otimes...\\otimes v_n)=\\sum_{m\\geq 0}\\frac{1}{m!}p_*\\Omega_{g,n+m}(v_1\\otimes...\\otimes v_n\\otimes T(\\psi_{n+1})\\otimes...\\otimes  T(\\psi_{n+m}))\n\\end{equation}\nwhere $p:\\overline{\\cal M}_{g,n+m}\\to\\overline{\\cal M}_{g,n}$ is the forgetful map.\nTo ensure the sum \\eqref{transl} is finite, one usually requires $T(z)\\in z^2V[[z]]$, so that $\\dim\\overline{\\cal M}_{g,n+m}=3g-3+n+m$ grows more slowly in $m$ than the degree $2m$ coming from $T$.  Instead, we control growth of the degree of $\\Omega_{g,n}$ in $n$ to ensure $T(z)\\in zV[[z]]$ produces a finite sum.\n\nThe partition function of a sequence $\\Omega=\\{\\Omega_{g,n}\\in H^*(\\overline{\\cal M}_{g,n})\\otimes (V^*)^{\\otimes n}\\mid g,n\\in\\mathbb{N},2g-2+n>0\\}$ is defined by:\n$$\nZ_{\\Omega}(\\hbar,\\{t^{\\alpha}_k\\})=\\exp\\sum_{g,n,\\vec{k}}\\frac{\\hbar^{g-1}}{n!}\\int_{\\overline{\\cal M}_{g,n}}\\Omega_{g,n}(e_{\\alpha_1}\\otimes...\\otimes e_{\\alpha_n})\\cdot\\prod_{j=1}^n\\psi_j^{k_j}\\prod t^{\\alpha_j}_{k_j}\n$$\nwhere $\\{e_1,...,e_N\\}$ is a basis of $V$, $\\alpha\\in\\{1,...,N\\}$ and $k\\in\\mathbb{N}$.  For $\\dim V=1$ and $\\Omega_{g,n}=1\\in H^*(\\overline{\\cal M}_{g,n})$, $Z_{\\Omega}(\\hbar,\\{t_k\\})=Z^{\\text{KW}}(\\hbar,\\{t_k\\})$.  Similarly, for $\\Omega_{g,n}=\\Theta_{g,n}\\in H^*(\\overline{\\cal M}_{g,n})$, $Z_{\\Omega}(\\hbar,\\{t_k\\})=Z^{\\Theta}(\\hbar,\\{t_k\\})$.\n\nThe action on sequences of cohomology classes above immediately gives rise to an action on partition functions, which store correlators of the sequence of cohomology classes, known as the Givental action \\cite{DSSGiv,GivGro,ShaBCOV}.  It gives a graphical construction of the partition functions $Z_{R\\Omega}$ and $Z_{T\\Omega}$ obtained from $Z_{\\Omega}$.  \n\nThe graphical expansions can be conveniently expressed via the action of differential operators $\\hat{R}$ and $\\hat{T}$: $Z_{R\\Omega}=\\hat{R}Z_{\\Omega}$, $Z_{T\\Omega}=\\hat{T}Z_{\\Omega}$ as follows.  Givental used $R(z)$ to produce a differential operator $\\hat{R}$, a so-called quantisation of $R(z)$, which acts on a product of tau-functions to produce a generating series for the correlators of the cohomological field theory.  Put $R(z)=\\displaystyle\\exp(\\sum_{\\ell>0} r_\\ell z^\\ell)$.\n$$\n\\hat{R}=\\exp\\left\\{\\sum_{\\ell=1}^{\\infty}\\sum_{\\alpha,\\beta}\n\\left(\\sum_{k=0}^{\\infty}u^{k,\\beta}(r_k)_\\beta^\\alpha\\frac{\\partial}{\\partial u^{k+\\ell,\\alpha}}+\\frac{\\hbar}{2}\\sum_{m=0}^{\\ell-1}(-1)^{m+1}(r_\\ell)^{\\alpha}_{\\beta}\\frac{\\partial^2}{\\partial u^{m,\\alpha}\\partial u^{\\ell-m-1,\\beta}}\\right)\n\\right\\}.\n$$\nThe action of the differential operator $\\hat{R}$ on $Z_{\\Omega}(\\hbar,\\{t^{\\alpha}_k\\})$ is equivalent to the weighted sum over graphs.  The first term $u^{k,\\beta}(r_k)_\\beta^\\alpha\\frac{\\partial}{\\partial u^{k+\\ell,\\alpha}}$ gives ordinary leaf contributions, and the second term $\\frac{\\hbar}{2}(-1)^{m+1}(r_\\ell)^{\\alpha}_{\\beta}\\frac{\\partial^2}{\\partial u^{m,\\alpha}\\partial u^{\\ell-m-1,\\beta}}$ gives edge contributions.  \n\nAssociated to $T(z)=\\sum_{\\ell>0}v_\\ell z^\\ell=\\exp(t_\\ell z^\\ell)-I\\in zV[[z]]$\nis the differential operator\n$$\\hat{T}=\\exp\\left\\{\\sum_{\\ell=1}^{\\infty}\\sum_{\\alpha,\\beta}\n\\sum_{k=0}^{\\infty}(t_k)_\\beta^\\alpha\\frac{\\partial}{\\partial u^{k+\\ell,\\alpha}}\\right\\}\n$$\nwhich is a translation operator.  It gives dilaton leaf contributions to the weighted sum of graphs.  The differential operators $\\hat{R}$ and $\\hat{T}$ act on a product of functions $Z^{\\text{BGW}}(\\hbar,t_0,t_1,...)$, $Z^{\\text{KW}}(\\hbar,t_0,t_1,...)$ and $Z^{\\Theta}(\\hbar,t_0,t_1,...)$ where the TFT is encoded via rescaling of the variables.  \n\n\\subsubsection{Pixton's relations}  \\label{pixrel}\nDual to any point $(C,p_1,...,p_n)\\in\\overline{\\cal M}_{g,n}$ is its stable graph $\\Gamma$ with vertices $V(\\Gamma)$ representing irreducible components of $C$, internal edges representing nodal singularities and a (labeled) external edge for each $p_i$.  Each vertex is labeled by a genus $g(v)$ and has valency $n(v)$.  The genus of a stable graph is $g(\\Gamma)=\\displaystyle h_1(\\Gamma)+\\hspace{-3mm}\\sum_{v\\in V(\\Gamma)}g(v)$.   For a given stable graph $\\Gamma$ of genus $g$ and with $n$ external edges we have\n$$\\phi_{\\Gamma}:\\overline{\\cal M}_{\\Gamma}=\\prod_{v\\in V(\\Gamma)}\\overline{\\cal M}_{g(v),n(v)}\\to\\overline{\\cal M}_{g,n}.$$\n\nThe {\\em strata algebra} $S_{g,n}$ is a finite-dimensional vector space over $\\mathbb{Q}$ with basis given by isomorphism classes of pairs $(\\Gamma, \\omega)$, for $\\Gamma$ a stable graph of genus $g$ with $n$ external edges and $\\omega\\in H^*(\\overline{\\cal M}_{\\Gamma})$ a product of $\\kappa$ and $\\psi$ classes in each $\\overline{\\cal M}_{g(v),n(v)}$ for each vertex $v\\in V(\\Gamma)$.  There is a natural map $q:S_{g,n}\\to H^*(\\overline{\\cal M}_{g,n})$ defined by the push-forward $q(\\Gamma, \\omega)=\\phi_{\\Gamma*}(\\omega)\\in H^*(\\overline{\\cal M}_{g,n})$.  The map $q$ allows one to define a multiplication on $S_{g,n}$, essentially coming from intersection theory in $\\overline{\\cal M}_{g,n}$, which can be described purely graphically.  The image $q(S_{g,n})\\subset H^*(\\overline{\\cal M}_{g,n})$ is the tautological ring $RH^*(\\overline{\\cal M}_{g,n})$ and an element of the kernel of $q$ is a tautological relation.  See \\cite{PPZRel}, Section~0.3 for a good description of $S_{g,n}$.  \n\nThe main result of \\cite{PPZRel} is the construction of elements $R^d_{g,A}\\in S_{g,n}$ for $A=(a_1,...,a_n)$, $a_i\\in\\{0,1\\}$ satisfying $q(R^d_{g,A})=0$ which push forward to tautological relations in $H^{2d}(\\overline{\\cal M}_{g,n})$.   The elements $R^d_{g,A}$ are constructed out of the elements $R\\Omega)_{g,n}$ defined above.  The element $R^1_2\\in H^{2}(\\overline{\\cal M}_{2})$ is given in Example~\\ref{gen2rel}.\n\n\n\n\n\n\\subsection{Construction of elements of the twisted loop group}  \\label{contw}\nConsider the linear system\n\\begin{equation}   \\label{compat} \n\\left(\\frac{d}{dz}-U-\\frac{V}{z}\\right)Y=0.\n\\end{equation}\nwhere $Y(z)\\in\\mathbb{C}^N$, $U=\\mathrm{diag}(u_1,...,u_N)$ for $u_i$ distinct and $V$ is skew symmetric.  An asymptotic solution of \\eqref{compat} as $z\\to\\infty$\n$$Y=R(z^{-1})e^{zU},\\quad R(z)=I+R_1z+R_2z^2+...$$\ndefines a power series $R(z)$ with coefficients given by $N\\times N$ matrices which is easily shown to satisfy $R(z)R^T(-z)=I$ hence $R(z)\\in L^{(2)}GL(N,\\mathbb{C})$.  Substitute $Y=R(z^{-1})e^{zU}$ into \\eqref{compat} and send $z\\mapsto z^{-1}$ to get\n$$\n0=\\left(\\frac{d}{dz}+\\frac{U}{z^2}+\\frac{V}{z}\\right)R(z)e^{U\/z}\n=\\left(\\frac{d}{dz}R(z)+\\frac{1}{z^2}[U,R(z)]+\\frac{1}{z}VR(z)\\right)e^{U\/z}\n$$\nor equivalently\n\\begin{equation}   \\label{teleman}\n[R_{k+1},U]=(k+V)R_k,\\quad k=0,1,...\n\\end{equation}\nWe will describe two natural solutions of \\eqref{compat} using the data of a Frobenius manifold, and using the data of a Riemann surface equipped with a meromorphic function.  These will give two constructions of elements $R(z)$ of the twisted loop group.  The construction of an element $R(z)$ in two ways using both of these viewpoints will be crucial to proving that $Z^{\\Theta}$ and $Z^{\\rm BGW}$ coincide.\n\n\n\nDubrovin \\cite{DubGeo} proved that there is a Frobenius manifold structure on the space of pairs $(U,V)$ where $V$ varies in the $u_i$ so that the monodromy of $Y$ around 0 remains fixed.  Conversely, he associated a family of linear systems \\eqref{compat} depending on the canonical coordinates $(u_1,...,u_N)$ of any semisimple Frobenius manifold $M$ as follows.  \n\nRecall that a Frobenius manifold is a complex manifold $M$ equipped with an associative product on its tangent bundle compatible with a flat metric---a nondegenerate symmetric bilinear form---on the manifold \\cite{DubGeo}.  It is encoded by a single function $F(t_1,...,t_{N})$, known as the {\\em prepotential}, that satisfies a nonlinear partial differential equation known as the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equation:\n$$F_{ijm}\\eta^{mn}F_{k\\ell n}=F_{i\\ell m}\\eta^{mn}F_{jkn},\\quad \\eta_{ij}=F_{0ij}\n$$\nwhere $\\eta^{ik}\\eta_{kj}=\\delta_{ij}$, $F_i=\\frac{\\partial}{\\partial t_i}F$ and $\\{t_1,...,t_{N}\\}$ are (flat) local coordinates of $M$ which correspond to the $k=0$ coordinates above via $t_{\\alpha}=t^{\\alpha}_0$, $\\alpha=1,...,N$.  It comes equipped with the two vector fields $1\\!\\!1$, the unit for the product, and the Euler vector field $E$ which describes symmetries of the Frobenius manifold, neatly encoded by $E\\cdot F(t_1,...,t_{N})=c\\cdot F(t_1,...,t_{N})$ for some $c\\in\\mathbb{C}$.  Multiplication by the Euler vector field $E$ produces an endomorphism $U$ with eigenvalues $\\{u_1,...,u_N\\}$ known as {\\em canonical coordinates} on $M$.  They give rise to vector fields  $\\partial\/\\partial u_i$ with respect to which the metric $\\eta$, product $\\raisebox{-0.5ex}{\\scalebox{1.8}{$\\cdot$}}$ and Euler vector field $E$ are diagonal\n$$\\frac{\\partial}{\\partial u_i}\\raisebox{-0.5ex}{\\scalebox{1.8}{$\\cdot$}}\\frac{\\partial}{\\partial u_j}=\\delta_{ij}\\frac{\\partial}{\\partial u_i},\\quad \\eta\\left(\\frac{\\partial}{\\partial u_i},\\frac{\\partial}{\\partial u_j}\\right)=\\delta_{ij}\\Delta_i,\\quad E=\\sum u_i\\frac{\\partial}{\\partial u_i}.$$\n\nThe differential equation \\eqref{compat} in $z$ is defined at any point of the Frobenius manifold using $U$, the endomorphism defined by multiplication by the Euler vector field $E$, and the endomorphism $V=[\\Gamma,U]$ where $\\Gamma_{ij}=\\frac{\\partial_{u_i}\\Delta_j}{2\\sqrt{\\Delta_i\\Delta_j}}$ for $i\\neq j$ are the so-called rotation coefficients of the metric $\\eta$ in the normalised canonical basis.  Hence associated to each point of the Frobenius manifold is an element $R(z)=\\sum R_kz^k$ of the twisted loop group.  From $(H=T_pM,\\raisebox{-0.5ex}{\\scalebox{1.8}{$\\cdot$}},E,V)$ the endomorphisms $R_k$ of $H$ are defined recursively from $R_1$ by $R_0=I$ and \\eqref{teleman}. \n\n\\begin{example} \nThe 2-dimensional versal deformation space of the $A_2$ singularity has a Frobenius manifold structure \\cite{DubGeo,SaiPer} defined by the prepotential \n$$F(t_1,t_2)=\\frac12 t_1^2t_2+\\frac{1}{72}t_2^4.\n$$\nThe prepotential determines the flat metric with respect to the basis $\\{\\frac{\\partial}{\\partial t_1},\\frac{\\partial}{\\partial t_2}\\}$ by $\\eta_{ij}=F_{1ij}$ hence\n$$\\eta=\\left(\\begin{array}{cc}0&1\\\\1&0\\end{array}\\right)\n$$\nand the (commutative) product by $\\eta(\\partial\/\\partial t_i\\raisebox{-0.5ex}{\\scalebox{1.8}{$\\cdot$}}\\partial\/\\partial t_j,\\partial\/\\partial t_k)=F_{ijk}$ hence\n$$\\frac{\\partial}{\\partial t_1}\\raisebox{-0.5ex}{\\scalebox{1.8}{$\\cdot$}}\\frac{\\partial}{\\partial t_1}=\\frac{\\partial}{\\partial t_1},\\quad \\frac{\\partial}{\\partial t_1}\\raisebox{-0.5ex}{\\scalebox{1.8}{$\\cdot$}}\\frac{\\partial}{\\partial t_2}=\\frac{\\partial}{\\partial t_2},\\quad \\frac{\\partial}{\\partial t_2}\\raisebox{-0.5ex}{\\scalebox{1.8}{$\\cdot$}}\\frac{\\partial}{\\partial t_2}=\\frac13 t_2\\frac{\\partial}{\\partial t_1}.\n$$\nThe unit and Euler vector fields are\n$$1\\!\\!1=\\frac{\\partial}{\\partial t_1},\\quad E=t_1\\frac{\\partial}{\\partial t_1}+\\frac23t_2\\frac{\\partial}{\\partial t_2}.$$ \nand $E\\cdot F(t_1,t_2)=\\frac83 F(t_1,t_2)$.\nThe canonical coordinates are  \n$$ u_1=t_1+\\frac{2}{3\\sqrt{3}}t_2^{3\/2},\\quad  u_2=t_1-\\frac{2}{3\\sqrt{3}}t_2^{3\/2}.\n$$\nWith respect to the normalised canonical basis, the rotation coefficients $\\Gamma_{12}=\\dfrac{-i\\sqrt{3}}{8}t_2^{-3\/2}=\\Gamma_{21}$ give rise to\n$V=[\\Gamma,U]=\\frac{i\\sqrt{3}}{2}t_2^{-3\/2}\\left(\\begin{array}{cc}0&-1\\\\1&0\\end{array}\\right)\n$.  In canonical coordinates we have\n\\begin{equation}   \\label{A2V}  U=\\left(\\begin{array}{cc}u_1&0\\\\0&u_2\\end{array}\\right),\\quad V=\\frac{2i}{3(u_1-u_2)}\\left(\\begin{array}{cc}0&1\\\\-1&0\\end{array}\\right).\n\\end{equation}\nThe metric $\\eta$ applied to the vector fields  $\\partial\/\\partial u_i=\\frac12\\left(\\partial\/\\partial t_1-(-1)^i(3\/t_2)^{1\/2}\\partial\/\\partial t_2\\right)$ is $\\eta\\left(\\frac{\\partial}{\\partial u_i},\\frac{\\partial}{\\partial u_j}\\right)=\\delta_{ij}\\Delta_i$ where $\\Delta_1=\\frac{\\sqrt{3}}{2}t_2^{-1\/2}=-\\Delta_2$.  \n\\end{example} \n\nWe consider three natural bases of the tangent space $H=T_pM$ at any point $p$ of a Frobenius manifold.  The flat basis $\\{\\partial\/\\partial t_i\\}$ which gives a constant metric $\\eta$, the canonical basis $\\{\\partial\/\\partial u_i\\}$ which gives a trivial product $\\raisebox{-0.5ex}{\\scalebox{1.8}{$\\cdot$}}$, and the normalised canonical basis $\\{v_i\\}$, for $v_i=\\Delta_i^{-1\/2}\\partial\/\\partial u_i$, which gives a trivial metric $\\eta$.    (A different choice of square root of $\\Delta_i$ would simply give a different choice of normalised canonical basis.)  The transition matrix $\\Psi$ from flat coordinates to normalised canonical coordinates sends the metric $\\eta$ to the dot product, i.e. $\\Psi^T\\Psi=\\eta$.  The TFT structure on $H$ induced from $\\eta$ and $\\raisebox{-0.5ex}{\\scalebox{1.8}{$\\cdot$}}$ is diagonal in the normalised canonical basis.  It is given by\n$\n\\Omega_{g,n}(v_i^{\\otimes n})=\\Delta_i^{1-g-\\frac12 n}\n$$\nand vanishes on mixed products of $v_i$ and $v_j$, $i\\neq j$.  We find the normalised canonical basis most useful for comparisons with topological recursion---see Section~\\ref{sec:TR}\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n\\iffalse\nThe coefficients $R_k$ are defined using $\\Psi$, the transition matrix from flat coordinates to normalised canonical coordinates, via the initial condition $R_0=I$ and the inductive equation\n\\begin{equation}\\label{eqdefR}\nd\\left( R(z) \\Psi \\right) = {\\left[R(z),dU\\right] \\over z} \\Psi\n\\end{equation}\nwhich uniquely determines $R(z)$ up to left multiplication by a diagonal matrix $D(z)$ independent of $u$ with $D(0)=I$.\nUsing $d\\Psi = \\left[\\Gamma,dU\\right] \\Psi$ one can write\n$$\nd\\left(R(z) \\Psi \\right) = d\\left[R(z)\\right] \\Psi + R(z) d \\Psi = \nd\\left[R(z)\\right] \\Psi + R(z)  \\left[\\Gamma,dU\\right] \\Psi .\n$$\nTogether with equation \\eqref{eqdefR} and the invertibility of $\\Psi$, this gives\n\\begin{equation}  \\label{eqdefR2}\ndR(z) = {\\left[R(z),dU\\right] \\over z} - R(z)  \\left[\\Gamma,dU\\right].\n\\end{equation}\nThis re-expresses the equation for $R(z)$ in terms of the rotation coefficients, which uses less information than the full metric, encoded in $\\Psi$.   Since $dU(1\\!\\!1) =I$, an immediate consequence of \\eqref{eqdefR2} is\n\\begin{equation} \\label{unR}\n1\\!\\!1\\cdot R(z)=0.\n\\end{equation}\n\n\n\n\n\nIf the theory is homogenous, then invariance under the action of the Euler field \n\\begin{equation} \\label{homR}\n(z\\partial_z+E)\\cdot R(z)=0\n\\end{equation}\nfixes the diagonal ambiguity in $R(z)$.\n\n\nThe first non-trivial term $R_1$ of $R(z)$ is given by the rotation coefficients\n\\begin{equation} \\label{R1=Gam}\nR_1=\\Gamma.\n\\end{equation}\nThis follows from comparing the constant (in $z$) term in \\eqref{eqdefR} which is\n$$d\\Psi=[R_1,dU]\\Psi\n$$\nto equation \\eqref{psipde} given by $d\\Psi = \\left[\\Gamma,dU\\right] \\Psi$.  Since $dU$ is diagonal with distinct diagonal terms, we see that \\eqref{R1=Gam} holds for off-diagonal terms, and the ambiguity in the diagonal term for both is unimportant---it can be fixed in $R_1$ by \\eqref{eqdefR} together with $E\\cdot R_1=-R_1$.\n\\fi\n\n\n\\begin{example}  \\label{A2RT}\nOn the $A_2$ Frobenius manifold restrict to the point $(u_1,u_2)=(2,-2)$, or equivalently $(t_1,t_2)=(0,3)$.   Then $\\Delta_1=1\/2=-\\Delta_2$ determines the TFT.  (In the flat basis the TFT structure at $(t_1,t_2)=(0,3)$ on $H$ is given by $\\Omega_{g,n}((\\partial\/\\partial t_1)^{\\otimes n_0}\\otimes (\\partial\/\\partial t_2)^{\\otimes n_1})=2^g\\cdot\\delta^{\\rm odd}_{g+n_1}$. )\nAt the point $(u_1,u_2)=(2,-2)$ the element $R(z)\\in  L^{(2)}GL(2,\\mathbb{C})$ satisfying \\eqref{teleman} with $U$ and $V$ given by\n$$U=\\left(\\begin{array}{cc}2&0\\\\0&-2\\end{array}\\right),\\quad V=\\frac{1}{6}\\left(\\begin{array}{cc}0&i\\\\-i&0\\end{array}\\right)\n$$\nis\n$$ R(z)=\\sum_m\\frac{(6m)!}{(6m-1)(3m)!(2m)!}\\left(\\begin{array}{cc}-1&(-1)^m6mi\\\\-6mi&(-1)^{m-1}\\end{array}\\right)\\left(\\frac{z}{1728}\\right)^m.$$\nWe also have\n$$T_0(z) = 1\\!\\!1-R^{-1}(z)(1\\!\\!1),\\quad 1\\!\\!1=\\frac{1}{\\sqrt{2}}\\left(\\begin{array}{c}1\\\\i\\end{array}\\right)\n$$\n$$T(z) = zT_0(z).\n$$\n\\end{example}\n\n\\begin{remark}\nEquation~\\eqref{teleman} defines $R(z)$ as an endomorphism valued power series over an $N$-dimensional vector space $H$ equipped with a symmetric, bilinear, nondegenerate form $\\eta$.  The matrix $R(z)$ here---using a basis for $H$ so that $\\eta$ is dot product---is related to the matrix $R(z)$ in \\cite{PPZRel} by conjugation by the transition matrix $\\Psi$ from flat coordinates to normalised canonical coordinates\n$$ R(z)=\\Psi\\sum_m\\frac{(6m)!}{(3m)!(2m)!}\\left(\\begin{array}{cc}\\frac{1+6m}{1-6m}&0\\\\0&1\\end{array}\\right)\\left(\\begin{array}{cc}0&1\\\\1&0\\end{array}\\right)^m\n\\left(\\frac{z}{1728}\\right)^m\\Psi^{-1},\\quad \\Psi=\\frac{1}{\\sqrt{2}}\\left(\\begin{array}{cc}1&1\\\\i&-i\\end{array}\\right).\n$$  \n\\end{remark}\n\n\\subsubsection{Spectral curves and the twisted loop group.}   \\label{specLG2}\nAn element of the twisted loop group $R(z)\\in L^{(2)}GL(N,\\mathbb{C})$ can be naturally defined from a Riemann surface $\\Sigma$ equipped with a bidifferential $B(p_1,p_2)$ on $\\Sigma\\times\\Sigma$ and a meromorphic function $x:\\Sigma\\to\\mathbb{C}$, for $N=$ the number of zeros of $dx$.  A basic example is the function $x=z^2$ on $\\Sigma=\\mathbb{C}$ which gives rise to the constant element $R(z)=1\\in GL(1,\\mathbb{C})$.  More generally, any function $x$ that looks like this example locally---$x=s^2+c$ for $s$ a local coordinate around a zero of $dx$ and $c\\in\\mathbb{C}$---gives $R(z)=I+R_1z+...\\in L^{(2)}GL(N,\\mathbb{C})$ which is in some sense a deformation of $I\\in GL(N,\\mathbb{C})$, or $N$ copies of the basic example.\n\n\\begin{definition} \\label{bidiff} On any compact Riemann surface with a choice of $\\mathcal A$-cycles $(\\Sigma,\\{ {\\mathcal A}_i\\}_{i=1,...,g})$, define a {\\em fundamental normalised bidifferential of the second kind}  $B(p,p')$ to be a symmetric tensor product of differentials on $\\Sigma\\times\\Sigma$, uniquely defined by the properties that it has a double pole on the diagonal of zero residue, double residue equal to $1$, no further singularities and normalised by $\\int_{p\\in{\\mathcal A}_i}B(p,p')=0$, $i=1,...,g$, \\cite{FayThe}.  On a rational curve, which is sufficient for this paper, $B$ is the Cauchy kernel\n$$\nB(z_1,z_2)=\\frac{dz_1dz_2}{(z_1-z_2)^2}.$$  \n\\end{definition}\nThe bidifferential $B(p,p')$ acts as a kernel for producing meromorphic differentials on the Riemann surface $\\Sigma$ via $\\omega(p)=\\int_{\\Lambda}\\lambda(p')B(p,p')$ where $\\lambda$ is a function defined along the contour $\\Lambda\\subset\\Sigma$.  Depending on the choice of $(\\Lambda,\\lambda)$, $\\omega$ can be a differential of the 1st kind (holomorphic), 2nd kind (zero residues) or 3rd kind (simple poles).  A fundamental example is $B({\\cal P},p)$ which is a normalised (trivial $\\mathcal A$-periods) differential of the second kind holomorphic on $\\Sigma\\backslash{\\cal P}$ with a double pole at a simple zero ${\\cal P}$ of $dx$.  The expression $B({\\cal P},p)$\nis an integral over a closed contour around ${\\cal P}$ defined as follows.\n\\begin{definition}\\label{evaluationform}\nFor a Riemann surface equipped with a meromorphic function $(\\Sigma,x)$ we define evaluation of any meromorphic differential $\\omega$ at a simple zero ${\\cal P}$ of $dx$ by\n$$\n\\omega({\\cal P}):=\\mathop{\\,\\rm Res\\,}_{p={\\cal P}}\\frac{\\omega(p)}{\\sqrt{2(x(p)-x({\\cal P}))}\n$$\nwhere we choose a branch of $\\sqrt{x(p)-x({\\cal P})}$ once and for all at ${\\cal P}$ to remove the $\\pm1$ ambiguity.\n\\end{definition}\n\n\n\nShramchenko \\cite{ShrRie} constructed a solution of \\eqref{compat} with $V=[\\mathcal{B},U]$ for $\\mathcal{B}_{ij}=B({\\cal P}_i,{\\cal P}_j)$ (defined for $i\\neq j$) given by\n$$Y(z)^i_j= -\\frac{\\sqrt{z}}{\\sqrt{2\\pi}}\\int_{\\Gamma_j} B({\\cal P}_i,p)\\cdot e^{-x(p)\\over z}.$$\nThe proof in \\cite{ShrRie} is indirect, showing that $Y(z)^i_j$ satisfies an associated set of PDEs in $u_i$, and using the Rauch variational formula to calculate $\\partial_{u_k}B({\\cal P}_i,p)$.  Rather than giving this proof, we will work directly with the associated element $R(z)$ of the twisted loop group.\n\\begin{definition}  \\label{BtoR}\nDefine the asymptotic series for $z$ near 0 by\n\\begin{equation}  \\label{eq:BtoR}\nR^{-1}(z)^i_j = -\\frac{\\sqrt{z}}{\\sqrt{2\\pi}}\\int_{\\Gamma_j} B({\\cal P}_i,p)\\cdot e^{u_j-x(p)\\over z}\n\\end{equation}\nwhere $\\Gamma_j$ is a path of steepest descent for $-x(p)\/z$ containing $u_j=x({\\cal P}_j)$.  \n\\end{definition}\nNote that the asymptotic expansion of the contour integral \\eqref{eq:BtoR} for $z$ near $0$ depends only the intersection of $\\Gamma_j$ with a neighbourhood of $p={\\cal P}_j$.  When $i=j$, the integrand has zero residue at $p={\\cal P}_j$ so we deform $\\Gamma_j$ to go around ${\\cal P}_j$ to get a well-defined integral.  Locally, this is the same as defining $\\int_\\mathbb{R} s^{-2}\\exp(-s^2)ds=-2\\sqrt{\\pi}$ by  integrating the analytic function $z^{-2}\\exp(-z^2)$ along the real line in $\\mathbb{C}$ deformed to avoid 0.\n\\begin{lemma} [\\cite{ShrRie}]  \\label{eynlem}\nThe asymptotic series $R(z)$ defined in \\eqref{eq:BtoR} satisfies the twisted loop group condition\n\\begin{equation}\\label{twloopgp}\nR(z) R^T(-z) = Id.\n\\end{equation}\n \\end{lemma}\n \\begin{proof}\nThe proof here is taken from \\cite{DNOPSPri}.  We have\n \\begin{align}  \\label{unberg}\n\\sum_{i=1}^N\\mathop{\\,\\rm Res\\,}_{q={\\cal P}_i}\\frac{B(p,q)B(p',q)}{dx(q)}&=-\\mathop{\\,\\rm Res\\,}_{q=p}\\frac{B(p,q)B(p',q)}{dx(q)}-\\mathop{\\,\\rm Res\\,}_{q=p'}\\frac{B(p,q)B(p',q)}{dx(q)}\\\\ \\nonumber\n &=-d_p\\left(\\frac{B(p,p')}{dx(p)}\\right)-d_{p'}\\left(\\frac{B(p,p')}{dx(p')}\\right)\n \\end{align}\nwhere the first equality uses the fact that the only poles of the integrand are $\\{p,p',{\\cal P}_i,i=1,...,N\\}$, and the second equality uses the Cauchy formula satisfied by the Bergman kernel.  Define the Laplace transform of the Bergman kernel by\n$$\n\\check{B}^{i,j}(z_1,z_2)  =\n\\frac{ e^{ \\frac{u_i }{ z_1}+ \\frac{u_j}{ z_2}}}{2 \\pi \\sqrt{z_1z_2}} \\int_{\\Gamma_i}\n\\int_{\\Gamma_j} B(p,p') e^{- \\frac{x(p) }{ z_1}- \\frac{x(p')}{ z_2}} .\n$$\nThe Laplace transform of the LHS of \\eqref{unberg} is\n \\begin{align*} \n \\frac{ e^{ \\frac{u_i }{ z_1}+ \\frac{u_j}{ z_2}}}{2 \\pi \\sqrt{z_1z_2}} \\int_{\\Gamma_i}\n\\int_{\\Gamma_j} e^{- \\frac{x(p) }{ z_1}- \\frac{x(p')}{ z_2}}\\sum_{k=1}^N\\mathop{\\,\\rm Res\\,}_{q={\\cal P}_k}\\frac{B(p,q)B(p',q)}{dx(q)}&\n=\\sum_{k=1}^N\\frac{ e^{ \\frac{u_i }{ z_1}+ \\frac{u_j}{ z_2}}}{2 \\pi \\sqrt{z_1z_2}} \\int_{\\Gamma_i}\ne^{- \\frac{x(p) }{ z_1}}B(p,{\\cal P}_k) \\int_{\\Gamma_j}e^{- \\frac{x(p')}{ z_2}}B(p',{\\cal P}_k)\\\\\n&=\\sum_{k=1}^N\\frac{\\left[R^{-1}(z_1)\\right]^k_i\\left[R^{-1}(z_2)\\right]_{j}^k}{ z_1z_2}.\n \\end{align*}\nSince the Laplace transform satisfies\n$\\displaystyle\\int_{\\Gamma_i} d\\left(\\frac{\\omega(p)}{dx(p)}\\right)e^{- \\frac{x(p) }{ z}}=\\frac{1}{z}\\int_{\\Gamma_i} \\omega(p)e^{- \\frac{x(p) }{ z}}\n$\nfor any differential $\\omega(p)$, by integration by parts,\nthen the Laplace transform of the RHS of \\eqref{unberg} is\n$$ -\\frac{ e^{ \\frac{u_i }{ z_1}+ \\frac{u_j}{ z_2}}}{2 \\pi \\sqrt{z_1z_2}} \\int_{\\Gamma_i}\n\\int_{\\Gamma_j} e^{- \\frac{x(p) }{ z_1}- \\frac{x(p')}{ z_2}}\n\\left\\{d_p\\left(\\frac{B(p,p')}{dx(p)}\\right)+d_{p'}\\left(\\frac{B(p,p')}{dx(p')}\\right)\\right\\}=-\\left(\\frac{1}{z_1}+\\frac{1}{z_2}\\right)\\check{B}^{i,j}(z_1,z_2).\n $$\nPutting the two sides together yields the following result due to Eynard \\cite{EynInv}\n\\begin{equation}\\label{shape2}\n\\check{B}^{i,j}(z_1,z_2) =    - \\frac{\\sum_{k=1}^{N} \\left[R^{-1}(z_1)\\right]^k_i \\left[R^{-1}(z_2)\\right]^k_j }{ z_1+z_2}.\n\\end{equation}\nEquation \\eqref{twloopgp} is an immediate consequence of \\eqref{shape2} and the finiteness of $\\check{B}^{i,j}(z_1,z_2)$ at $z_2=-z_1$.\n\\end{proof}\n\\begin{example}  \\label{specA2B}\nLet $\\Sigma\\cong\\mathbb{C}$ be a rational curve equipped with the meromorphic function $x=z^3-3z$ and bidifferential $B(z_1,z_2)=dz_1dz_2\/(z_1-z_2)^2$.\n\nChoose a local coordinate $t$ around $z=-1={\\cal P}_1$ so that $x(t)=\\frac12 t^2+2$.  Then\n\\begin{align*}\nB({\\cal P}_1,t)&=\\frac{-i}{\\sqrt{6}}\\frac{dz}{(z+1)^2}=dt\\left(t^{-2}-\\frac{1}{144}+\\frac{35}{41472}t^2+...+{\\rm odd\\ terms}\\right)\\\\\nB({\\cal P}_2,t)&=\\frac{1}{\\sqrt{6}}\\frac{dz}{(z-1)^2}=dt\\left(-\\frac{i}{24}+\\frac{35i}{3456}t^2+...+{\\rm odd\\ terms}\\right)\n\\end{align*}\nChoose a local coordinate $s$ around $z=1={\\cal P}_2$ so that $x(s)=\\frac12 s^2-2$.  Then\n\\begin{align*}\nB({\\cal P}_1,s)&=\\frac{-i}{\\sqrt{6}}\\frac{dz}{(z+1)^2}=ds\\left(-\\frac{i}{24}-\\frac{35i}{3456}s^2+...+{\\rm odd\\ terms}\\right)\\\\\nB({\\cal P}_2,s)&=\\frac{1}{\\sqrt{6}}\\frac{dz}{(z-1)^2}=ds\\left(s^{-2}+\\frac{1}{144}+\\frac{35}{41472}s^2+...+{\\rm odd\\ terms}\\right)\n\\end{align*}\n\nThe odd terms are annihilated by the Laplace transform, and we get\n\\begin{align*}R^{-1}(z)^1_1 &= -\\frac{\\sqrt{z}}{\\sqrt{2\\pi}}\\int_{\\Gamma_1} B({\\cal P}_1,t)\\cdot e^{-\\frac12t^2\\over z}=1+\\frac{1}{144}z-\\frac{35}{41472}z^2+...\\\\\nR^{-1}(z)^1_2 &= -\\frac{\\sqrt{z}}{\\sqrt{2\\pi}}\\int_{\\Gamma_1} B({\\cal P}_2,t)\\cdot e^{-\\frac12t^2\\over z}=\\frac{i}{24}z-\\frac{35i}{3456}z^2+...\\\\\nR^{-1}(z)^2_1 &= -\\frac{\\sqrt{z}}{\\sqrt{2\\pi}}\\int_{\\Gamma_2} B({\\cal P}_1,s)\\cdot e^{-\\frac12s^2\\over z}=\\frac{i}{24}z+\\frac{35i}{3456}z^2+...\\\\\nR^{-1}(z)^2_2 &= -\\frac{\\sqrt{z}}{\\sqrt{2\\pi}}\\int_{\\Gamma_2} B({\\cal P}_2,s)\\cdot e^{-\\frac12s^2\\over z}=1-\\frac{1}{144}z-\\frac{35}{41472}z^2+...\n\\end{align*}\nHence $R^{-1}(z)=I-R_1z+(R_1^2-R_2)z^2+...=I-R_1^Tz+R_2^Tz^2+...$ gives \n$$ R_1=\\frac{1}{144}\\left(\\begin{array}{cc}-1&-6i\\\\-6i&1\\end{array}\\right),\\quad\nR_2=\\frac{35}{41472}\\left(\\begin{array}{cc}-1&12i\\\\-12i&-1\\end{array}\\right)\n$$\nwhich agrees with Example~\\ref{A2RT} for the $A_2$ Frobenius manifold.\n\\end{example}\n\\subsection{Partition Functions from the $A_2$ singularity}   \\label{sec:A2part}\nUsing the twisted loop group action and the translation action defined in the previous section, we construct in \\eqref{ZA2}, \\eqref{ZtA2} and \\eqref{ZBGWA2} below, three partition functions out of $Z^{\\text{BGW}}(\\hbar,t_0,t_1,...)$, $Z^{\\text{KW}}(\\hbar,t_0,t_1,...)$ and $Z^{\\Theta}(\\hbar,t_0,t_1,...)$, denoted by\n$$Z_{A_2}(\\hbar,\\{t^{\\alpha}_k\\}),\\quad Z^{\\Theta}_{A_2}(\\hbar,\\{t^{\\alpha}_k\\}),\\quad Z^{\\text{BGW}}_{A_2}(\\hbar,\\{t^{\\alpha}_k\\})$$\nwhere $\\alpha\\in\\{1,2\\}$, $k\\in\\mathbb{N}$, i.e. $\\{t^{\\alpha}_k\\}=\\{t^1_0,t^2_0,t^1_1,t^2_1,t^1_2,t^2_2,...\\}$.  \n\nThe three partition functions above use the element $R(z)$ of the twisted loop group arising out of the $A_2$ Frobenius manifold.  \nThe partition function $Z_{A_2}(\\hbar,\\{t^{\\alpha}_k\\})$ is defined via the graphical action defined in Section~\\ref{twloop} (and represented here via the equivalent action by differential operators) using $R(z)$ and $T(z)$ defined from the $A_2$ Frobenius manifold in Example~\\ref{A2RT}\n\\begin{align}   \\label{ZA2}\nZ_{A_2}(\\hbar,\\{t^{\\alpha}_k\\})&:=\\hat{\\Psi}\\cdot\\hat{R}\\cdot \\hat{T}\\cdot Z^{\\text{KW}}(\\tfrac{1}{2}\\hbar,\\{\\frac{1}{\\sqrt{2}}v^{1}_k\\})Z^{\\text{KW}}(-\\tfrac{1}{2}\\hbar,\\{\\frac{i}{\\sqrt{2}}v^{2}_k\\})\\\\\n&=\\exp\\sum_{g,n,\\vec{k}}\\frac{\\hbar^{g-1}}{n!}\\int_{\\overline{\\cal M}_{g,n}}\\Omega^{A_2}_{g,n}(e_{\\alpha_1}\\otimes...\\otimes e_{\\alpha_n})\\cdot\\prod_{j=1}^n\\psi_j^{k_j}\\prod t^{\\alpha_j}_{k_j}\n\\nonumber\n\\end{align}\nwhere $\\hat{R}$ and $\\hat{T}$ are differential operators acting on copies of $Z^{\\text{KW}}$, which can be expressed as a sum over stable graphs, and $\\hat{\\Psi}$ is the linear change of coordinates $v^i_k=\\Psi^i_{\\alpha}t^{\\alpha}_k$ (also expressible as a differential operator).\nSimilarly, define\n\\begin{align}   \\label{ZtA2}\nZ^{\\Theta}_{A_2}(\\hbar,\\{t^{\\alpha}_k\\})&:=\\hat{\\Psi}\\cdot\\hat{R}\\cdot \\hat{T}_0\\cdot Z^{\\Theta}(\\tfrac{1}{2}\\hbar,\\{\\frac{1}{\\sqrt{2}}v^{1}_k\\})Z^{\\Theta}(-\\tfrac{1}{2}\\hbar,\\{\\frac{i}{\\sqrt{2}}v^{2}_k\\})\\\\\n&=\\exp\\sum_{g,n,\\vec{k},\\vec{\\alpha}}\\frac{\\hbar^{g-1}}{n!}\\int_{\\overline{\\cal M}_{g,n}}\\Theta_{g,n}\\cdot\\Omega^{A_2}_{g,n}(e_{\\alpha_1},...,e_{\\alpha_n})\\cdot\\prod_{j=1}^n\\psi_j^{k_j}\\prod t_{k_j}^{\\alpha_j}.\n\\nonumber\n\\end{align}\nThe operators $\\hat{\\Psi}$ and $\\hat{R}$ in \\eqref{ZtA2} coincide with the operators in \\eqref{ZA2} associated to the $A_2$ singularity.  The translation operator $\\hat{T}_0=z^{-1}\\hat{T}$ is related to the translation operator in \\eqref{ZA2} essentially by the shift $m_i+1\\to m_i$ we saw in \\eqref{thetakappa}.  The graphical expression \\eqref{A2coh} for $\\Omega^{A_2}_{g,n}(e_{\\alpha_1},...,e_{\\alpha_n})$ immediately gives rise to a graphical expression for $\\Theta_{g,n}\\cdot\\Omega^{A_2}_{g,n}(e_{\\alpha_1},...,e_{\\alpha_n})$.  This produces a graphical expression for $Z^{\\Theta}_{A_2}$ which is expressed in \\eqref{ZtA2} as a differential operator acting on two copies of $Z^{\\Theta}$ in place of $Z^{\\text{KW}}$.  This is a sum over graphs $G'_{g,n}$ via the structure shown in \\eqref{thetarel} (which also applies when the expression is non-zero).  \n\nFinally, define\n\\begin{equation}   \\label{ZBGWA2}\nZ^{\\text{BGW}}_{A_2}\n:=\\hat{\\Psi}\\cdot\\hat{R}\\cdot \\hat{T}_0\\cdot Z^{\\text{BGW}}(\\tfrac{1}{2}\\hbar,\\{\\frac{1}{\\sqrt{2}}v^{1}_k\\})Z^{\\text{BGW}}(-\\tfrac{1}{2}\\hbar,\\{\\frac{i}{\\sqrt{2}}v^{2}_k\\}).\n\\end{equation}\nEquivalently, we have replaced each $\\int_{\\overline{\\cal M}_{g,n+N}}\\Theta_{g,n+N}\\prod_{i=1}^{n+N}\\psi_i^{k_i}$ in $Z^{\\Theta}_{A_2}$ with the corresponding coefficients from $F_g^{\\text{BGW}}$.\nTo prove Theorem~\\ref{tauf}, we will show that $Z^{\\text{BGW}}_{A_2}$ has many vanishing terms from which it will follow that $Z^{\\text{BGW}}_{A_2}=Z^{\\Theta}_{A_2}$ and $Z^{\\text{BGW}}=Z^{\\Theta}$.\n\n\n\n\nThe $A_2$ Frobenius manifold and its associated cohomological field theory $\\Omega^{A_2}_{g,n}\\in H^*(\\overline{\\cal M}_{g,n})\\otimes (V^*)^{\\otimes n}$ for $H=\\mathbb{C}^2$ is used to produce Pixton's relations among tautological cohomology classes over $\\overline{\\cal M}_{g,n}$.  \nThe class $\\Omega^{A_2}_{g,n}\\in H^*(\\overline{\\cal M}_{g,n})\\otimes (V^*)^{\\otimes n}$ is defined by the Givental-Teleman theorem---see \\cite{DSSGiv, GivGro, TelStr}---via a sum over stable graphs:\n\\begin{equation}  \\label{A2coh}\n\\Omega^{A_2}_{g,n}=\\sum_{\\Gamma\\in G_{g,n}}(\\phi_{\\Gamma})_*\\omega^R_{\\Gamma}\\in H^*(\\overline{\\cal M}_{g,n})\n\\end{equation}\nwhere $\\omega^R_{\\Gamma}$ is built out of contributions to edges and vertices of $\\Gamma$ using $R$, $\\psi$ and $\\kappa$ classes and a topological field theory at vertices that takes in vectors of $H$.\n\nThe key idea behind Pixton's relations among tautological classes  \\cite{PPZRel} is a degree bound on the cohomological classes $\\deg\\Omega^{A_2}_{g,n}\\leq\\frac13(g-1+n )<3g-3+n$.  The construction of $\\Omega^{A_2}_{g,n}$ using $R$ in \\eqref{A2coh} does not know about this degree bound and produces classes in the degrees where $\\Omega^{A_2}_{g,n}$ vanishes.  This leads to sums of tautological classes representing the zero class, i.e. relations, expressed as sums over stable graphs $G_{g,n}$:\n\\begin{equation}  \\label{pixtonrel} \n0=\\sum_{\\Gamma\\in G_{g,n}}(\\phi_{\\Gamma})_*\\omega^R_{\\Gamma}\\in H^*(\\overline{\\cal M}_{g,n})=\\sum_{\\Gamma\\in G'_{g,n}}(\\phi_{\\Gamma})_*\\tilde{\\omega}^R_{\\Gamma}\n\\end{equation}\nfor classes $\\omega^R_{\\Gamma},\\tilde{\\omega}^R_{\\Gamma}\\in H^*(\\overline{\\cal M}_{\\Gamma})$.  Here we denote by $G'_{g,n}$ the set of all stable graphs of genus $g$ with $n$ labeled points and any number of extra leaves, known as {\\em dilaton leaves} at each vertex.  So the set $G_{g,n}$ is finite whereas $G'_{g,n}$ is infinite.  Nevertheless, all sums in \\eqref{pixtonrel} are finite.  The classes $\\omega_{\\Gamma}$ appearing in \\eqref{pixtonrel} consist of products of $\\psi$ and $\\kappa$ classes associated to each vertex of $\\Gamma$.  The classes $\\tilde{\\omega}^R_{\\Gamma}$ consist of products of only $\\psi$ classes, again associated to each vertex of $\\Gamma$.\n\nPixton's relations \\eqref{pixtonrel} induce relations between intersection numbers of $\\psi$ classes with $\\Theta_{g,n}$:\n\\begin{equation}  \\label{thetarel}\n 0=\\Theta_{g,n}\\cdot\\sum_{\\Gamma\\in G_{g,n}}(\\phi_{\\Gamma})_*\\omega^R_{\\Gamma}\n=\\sum_{\\Gamma\\in G_{g,n}}(\\phi_{\\Gamma})_*\\Big(\\Theta_{\\Gamma}\\cdot\\omega^R_{\\Gamma}\\Big)\n=\\sum_{\\Gamma\\in G'_{g,n}}(\\phi_{\\Gamma})_*\\Big(\\Theta_{\\Gamma}\\cdot\\tilde{\\omega}^R_{\\Gamma}\\Big).\n\\end{equation}\nThe second equality uses $\\Theta_{g,n}\\cdot(\\phi_{\\Gamma})_*=(\\phi_{\\Gamma})_*\\Theta_{\\Gamma}$.  The final equality uses Remark~\\ref{removekappa} to replace $\\kappa$ classes by $\\psi$ classes, such as in Example~\\ref{gen2rel} below where the $\\kappa_1$ term is replaced by $\\int_{\\overline{\\cal M}_{2,1}}\\Theta_{2,1}\\cdot\\psi_1=\\int_{\\overline{\\cal M}_{2}}\\Theta_{2}\\cdot\\kappa_1$.  The classes $\\omega_{\\Gamma}$ in \\eqref{thetarel} are linear combinations of monomials $\\prod\\psi_i^{k_i}\\cdot R_{\\bf m}$, which form a basis for products of $\\psi$ and $\\kappa$ classes.  The final equality in \\eqref{thetarel} is obtained by substituting (linear combinations of) the following expression into the sum over $G_{g,n}$\n$$\\Theta_{g,n}(\\phi_{\\Gamma})_*\\Big(\\prod\\psi_i^{k_i}\\cdot R_{\\bf m}\\Big)\n=(\\phi_{\\Gamma})_*\\Big(\\Theta_{\\Gamma}\\prod\\psi_i^{k_i}\\cdot R_{\\bf m}\\Big)\n=(\\phi_{\\Gamma})_*\\Big(\\prod\\psi_i^{k_i}\\cdot \\pi_*(\\Theta_{\\Gamma_{(N)}}\\psi_{n+1}^{m_1}...\\psi_{n+N}^{m_N})\\Big)\n$$\nwhere $\\Gamma_{(N)}$ is obtained from $\\Gamma$ by adding $N$ (dilaton) leaves to the vertex of $\\Gamma$ on which $\\prod\\psi_i^{k_i}\\cdot R_{\\bf m}$ is defined.   The final term in \\eqref{thetarel} is a sum over graphs of intersection numbers of $\\Theta_{g,n}$ with $\\psi$ classes only.  \n\n{\\em Primary invariants} of a partition function are those coefficients of $\\prod_{i=1}^nt^{\\alpha_i}_{k_i}$ with all $k_i=0$.  They correspond to intersections in $\\overline{\\cal M}_{g,n}$ with no $\\psi$ classes.  The primary invariants of $Z^{\\Theta}_{A_2}(\\hbar,\\{t^{\\alpha}_k\\})$ vanish for $n<2g-2$.  This uses $\\deg\\Omega^{A_2}_{g,n}\\leq\\frac13(g-1+n)$ so $\\deg\\Omega^{A_2}_{g,n}\\cdot\\Theta_{g,n}\\leq\\frac13(g-1+n)+2g-2+n<3g-3+n$ when $n<2g-2$.  These vanishing coefficients correspond to top intersections of $\\psi$ classes with the relations \\eqref{thetarel}.  For $n\\leq g-1$ these relations are sufficient to uniquely determine the intersection numbers $\\int_{\\overline{\\cal M}_{g,n}}\\Theta_{g,n}\\prod_{i=1}^n\\psi_i^{m_i}$ via the recursive method in Section~\\ref{sec:unique}.  For $n\\geq g$ the dilaton equation \\eqref{dilaton} determines the intersection numbers from the lower ones.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Topological recursion}   \\label{sec:TR}\n\nTopological recursion is a procedure which takes as input a spectral curve, defined below, and produces a collection of symmetric tensor products of meromorphic 1-forms $\\omega_{g,n}$ on $C^n$.   The correlators store enumerative information in different ways.  Periods of the correlators store top intersection numbers of tautological classes in the moduli space of stable curves $\\overline{\\cal M}_{g,n}$ and local expansions of the correlators can serve as generating functions for enumerative problems.  \n\nA spectral curve $S=(C,x,y,B)$ is a Riemann surface $C$ equipped with two meromorphic functions $x, y: C\\to \\mathbb{C}$ and a bidifferential $B(p_1,p_2)$ defined in \\eqref{bidiff}, which is the Cauchy kernel in this paper.   Topological recursion, as developed by Chekhov, Eynard, Orantin \\cite{CEyHer,EOrInv}, is a procedure that produces from a spectral curve $S=(C,x,y,B)$ a symmetric tensor product of meromorphic 1-forms $\\omega_{g,n}$ on $C^n$ for integers $g\\geq 0$ and $n\\geq 1$, which we refer to as {\\em correlation differentials} or {\\em correlators}.  The correlation differentials $\\omega_{g,n}$ are defined by\n\\[\n\\omega_{0,1}(p_1) = -y(p_1) \\, d x(p_1) \\qquad \\text{and} \\qquad \\omega_{0,2}(p_1, p_2) = B(p_1,p_2)\n\\]\nand for $2g-2+n>0$ they are defined recursively via the following equation.\n\\[\n\\omega_{g,n}(p_1, p_L) = \\sum_{d x(\\alpha) = 0} \\mathop{\\text{Res}}_{p=\\alpha} K(p_1, p) \\Bigg[ \\omega_{g-1,n+1}(p, \\hat{p}, p_L) + \\mathop{\\sum_{g_1+g_2=g}}_{I \\sqcup J = L}^\\circ \\omega_{g_1,|I|+1}(p, p_I) \\, \\omega_{g_2,|J|+1}(\\hat{p}, p_J) \\Bigg]\n\\]\nHere, we use the notation $L = \\{2, 3, \\ldots, n\\}$ and $p_I = \\{p_{i_1}, p_{i_2}, \\ldots, p_{i_k}\\}$ for $I = \\{i_1, i_2, \\ldots, i_k\\}$. The outer summation is over the zeroes $\\alpha$ of $dx$ and $p \\mapsto \\hat{p}$ is the involution defined locally near $\\alpha$ satisfying $x(\\hat{p}) = x(p)$ and $\\hat{p} \\neq p$. The symbol $\\circ$ over the inner summation means that we exclude any term that involves $\\omega_{0,1}$. Finally, the recursion kernel is given by\n\\[\nK(p_1,p) = \\frac{1}{2}\\frac{\\int_{\\hat{p}}^p \\omega_{0,2}(p_1, \\,\\cdot\\,)}{[y(p)-y(\\hat{p})] \\, d x(p)}.\n\\]\nwhich is well-defined in the vicinity of each zero of $dx$.   It acts on differentials in $p$ and produces differentials in $p_1$ since the quotient of a differential in $p$ by the differential $dx(p)$ is a meromorphic function.  For $2g-2+n>0$, each $\\omega_{g,n}$ is a symmetric tensor product of meromorphic 1-forms on $C^n$ with residueless poles at the zeros of $dx$ and holomorphic elsewhere.  A zero $\\alpha$ of $dx$ is {\\em regular}, respectively irregular, if $y$ is regular, respectively has a simple pole, at $\\alpha$.  The order of the pole in each variable of $\\omega_{g,n}$ at a regular, respectively irregular, zero of $dx$ is $6g-4+2n$, respectively $2g$.  Define $\\Phi(p)$ up to an additive constant by $d\\Phi(p)=y(p)dx(p)$.  For $2g-2+n>0$, the invariants satisfy the dilaton equation~\\cite{EOrInv}\n\\[\n\\sum_{\\alpha}\\mathop{\\,\\rm Res\\,}_{p=\\alpha}\\Phi(p)\\, \\omega_{g,n+1}(p,p_1, \\ldots ,p_n)=(2g-2+n) \\,\\omega_{g,n}(p_1, \\ldots, p_n),\n\\] \nwhere the sum is over the zeros $\\alpha$ of $dx$.  This enables the definition of the so-called {\\em symplectic invariants}\n\\[ F_g=\\sum_{\\alpha}\\mathop{\\,\\rm Res\\,}_{p=\\alpha}\\Phi(p)\\omega_{g,1}(p).\\]\nThe correlators $\\omega_{g,n}$ are normalised differentials of the second kind in each variable---they have zero $\\mathcal{A}$-periods, and poles only at the zeros ${\\cal P}_i$ of $dx$ of zero residue.  Their principle parts are skew-invariant under the local involution $p\\mapsto\\hat{p}$.  A basis of such normalised differentials of the second kind is constructed from $x$ and $B$ in the following definition. \n\\begin{definition}\\label{auxdif}\nFor a Riemann surface $C$ equipped with a meromorphic function $x:C\\to\\mathbb{C}$ and bidifferential $B(p_1,p_2)$ define the auxiliary differentials on $C$ as follows.  For each zero ${\\cal P}_i$ of $dx$,  define\n\\begin{equation}  \\label{Vdiff}\nV^i_0(p)=B({\\cal P}_i,p),\\quad V^i_{k+1}(p)=d\\left(\\frac{V^i_k(p)}{dx(p)}\\right),\\ i=1,...,N,\\quad k=0,1,2,...\n\\end{equation}\nwhere evaluation $B({\\cal P}_i,p)$ at  ${\\cal P}_i$ is given in Definition~\\ref{evaluationform}.\n\\end{definition}\nThe correlators $\\omega_{g,n}$ are polynomials in the auxiliary differentials $V^i_k(p)$.  To any spectral curve $S$, one can define a partition function $Z^S$ by assembling the polynomials built out of the correlators $\\omega_{g,n}$ \\cite{DOSSIde,EynInv,NScPol}.\n\\begin{definition}  \\label{TRpart}\n$$Z^S(\\hbar,\\{u^{\\alpha}_k\\}):=\\left.\\exp\\sum_{g,n}\\frac{\\hbar^{g-1}}{n!}\\omega^S_{g,n}\\right|_{V^{\\alpha}_k(p_i)=u^{\\alpha}_k}.\n$$\n\\end{definition}\nAs usual define $F_g$ to be the contribution from $\\omega_{g,n}$:\n$$\\log Z^S(\\hbar,\\{u^{\\alpha}_k\\})=\\sum_{g\\geq 0}\\hbar^{g-1}F_g^S(\\{u^{\\alpha}_k\\}).$$\n\n\\subsubsection{From topological recursion to Givental reconstruction}   \\label{sec:TR2Giv}\nThe relation between Givental's reconstruction of CohFTs defined in Section~\\ref{twloop} and topological recursion was proven in \\cite{DOSSIde}.  The $A_2$ case is treated in \\cite{DNOPSDub}.  Recall that the input data for Givental's reconstruction is an element $R(z)\\in L^{(2)}GL(N,\\mathbb{C})$ and $T(z)=z\\left(1\\!\\!1-R^{-1}(z)1\\!\\!1\\right)\\in z\\ \\mathbb{C}^N[[z]]$ defined by a vector $1\\!\\!1\\in\\mathbb{C}^N$.  Its output is a CohFT or its partition function $Z(\\hbar,\\{t^{\\alpha}_k\\})$.  The input data for topological recursion is a spectral curve $S=(C,x,y,B)$.  Its output is the correlators $\\omega_{g,n}$ which can be assembled into a partition function $Z^S(\\hbar,\\{t^{\\alpha}_k\\})$.  This is summarised in the following diagram.\n\\begin{center}\n\\begin{tikzpicture}[scale=0.5]\n\\draw (1,4) node {$R(z)\\in L^{(2)}GL(N,\\mathbb{C})$};\n\\draw (1,3) node {$T(z)\\in z\\cdot \\mathbb{C}^N[[z]]$};\n\\draw [->, line width=1pt] (5,4)--(13,4);\n\\draw (9,5) node {Givental reconstruction};\n\\draw (15.5,4) node {$Z(\\hbar,\\{t^{\\alpha}_k\\})$};\n\\draw [<->, line width=1pt] (1,2.5)--(1,-1.5);\n\\draw [<->, line width=1pt] (15.5,2.5)--(15.5,-1.5);\n\\draw (1,-2.5) node {$S=(C,x,y,B)$};\n\\draw (15.5,-2.5) node {$Z^S(\\hbar,\\{t^{\\alpha}_k\\})$};\n\\draw [->, line width=1pt] (5,-2.5)--(13,-2.5);\n\\draw (9,-2) node {topological recursion};\n\\end{tikzpicture}\n\\end{center}\nThe left arrow in the diagram, i.e. a correspondence between the input data, which produces the same output $Z(\\hbar,\\{t^{\\alpha}_k\\})=Z^S(\\hbar,\\{t^{\\alpha}_k\\})$ is the main result of \\cite{DOSSIde}.   Given $R(z)$ and $T(z)$ arising from a CohFT, it was proven in \\cite{DOSSIde} that there exists a local spectral curve $S$, which is a collection of disks neighbourhoods of zeros of $dx$ on which $B$ and $y$ are define locally, giving the partition function $Z$ of the CohFT.  We will use the converse of this result proven in \\cite{DNOPSPri}, beginning instead from $S$, which builds on the construction of \\cite{DOSSIde}.\n\nRecall from Section~\\ref{specLG2} the construction\n$$(C,x,B)\\mapsto R(z)\\in L^{(2)}GL(N,\\mathbb{C})$$\nof an element $R(z)$ from part of the data of a spectral curve $S$.  We will now associate a (locally defined) function $y$ on $C$ to the unit vector $1\\!\\!1=\\{\\Delta_i^{1\/2}\\}\\in\\mathbb{C}^N$ in normalised canonical coordinates to get the full data of a spectral curve $S=(C,x,y,B)$.\nDefine $dy$ by\n\\begin{equation} \\label{ytoT}\n\\sum_{k=1}^N R^{-1}(z)^i_k \\cdot\\Delta_k^{1\/2} = \\frac{1}{\\sqrt{2\\pi z}}\\int_{\\Gamma_i}dy(p)\\cdot e^{(u_i-x(p)) \\over z}.\n\\end{equation}\nThis defines $y$ locally around each ${\\cal P}_i$.  In fact, it only defines the part of $y$ skew-invariant under the local involution $p\\mapsto\\hat{p}$ defined by $x$ since the Laplace transform annihilates invariant parts, but topological recursion depends only on this skew-invariant part of $y$.  The $z\\to0$ limit of \\eqref{ytoT} gives\n\\begin{equation} \\label{yunit}\ndy({\\cal P}_i)=\\Delta_i^{1\/2}\n\\end{equation}\nhence conversely $y$ determines the unit $1\\!\\!1=\\{\\Delta_i^{1\/2}\\}\\in\\mathbb{C}^N$ and $R^{-1}(z)1\\!\\!1$ which produces the translation.  Equation \\eqref{yunit} is a strong condition on $y$ since it shows that $y(p)$ is determined by $dy({\\cal P}_i)$, $i=1,...,N$ and $B(p,p')$.  If a globally defined differential $dy$ on a compact curve $C$ satisfies the condition\n\\begin{equation} \\label{DOSStest}\nd\\left(\\frac{dy}{dx}(p)\\right)=-\\sum_{i=1}^n \\mathop{\\,\\rm Res\\,}_{p'={\\cal P}_i}\\frac{dy}{dx}(p')B(p',p)\n\\end{equation}\nthen $dy$ satisfies \\eqref{ytoT} for $\\Delta_i^{1\/2}$ defined by \\eqref{yunit} and $R(z)$ defined by \\eqref{eq:BtoR}.  This is immediate by taking the Laplace transform of \\eqref{DOSStest} and using the fact that $\\displaystyle\\mathop{\\,\\rm Res\\,}_{p'={\\cal P}_i}\\frac{dy}{dx}(p')B(p',p)=dy({\\cal P}_i)B({\\cal P}_i,p)$.  Note that if the poles of $dy$ are dominated by the poles of $dx$, equivalently $dy\/dx$ has poles only at the zeros of $dx$, then \\eqref{DOSStest} is satisfied as a consequence of the Cauchy formula.    This allows us to produce spectral curves $S=(C,x,y,B)$ that give rise to $R(z)$ and $1\\!\\!1$---see Example~\\ref{specA2rel} below.\n\nThe result of \\cite{DOSSIde} was generalised in \\cite{CNoTop} to show that the differential operators $\\hat{\\Psi}$, $\\hat{R}$ and $\\hat{T}_0$ acting on copies of $Z^{\\text{BGW}}$\narises by applying topological recursion to an irregular spectral curve.  Equivalently, periods of the correlators of an irregular spectral curve store linear combinations of coefficients of $\\log Z^{\\text{BGW}}$.  The appearance of $Z^{\\text{BGW}}$ is due to its relationship with topological recursion applied to the curve $x=\\frac{1}{2}z^2$, $y=\\frac{1}{z}$ \\cite{DNoTop}.\n\n\\subsubsection{Examples}\nWe demonstrate topological recursion with four key examples of rational spectral curves equipped with the bidifferential $B(p_1,p_2)$ given by the Cauchy kernel. The spectral curves in the Examples~\\ref{specBes} and \\ref{specAiry}, denoted $S_{\\text{Airy}}$ and $S_{\\text{Bes}}$, have partition functions $Z^{\\text{KW}}$ and $Z^{\\text{BGW}}$ respectively.  \nAny spectral curve at regular, respectively irregular, zeros of $dx$ is locally isomorphic to $S_{\\text{Airy}}$, respectively $S_{\\text{Bes}}$. \nA consequence is that the tau functions $Z^{\\text{KW}}$ and $Z^{\\text{BGW}}$ are fundamental to the correlators produced from topological recursion.  Moreover, they are built out of $Z^{\\text{KW}}$ and $Z^{\\text{BGW}}$ via Givental reconstruction described in Section~\\ref{twloop} where $R$ and $T$ are obtained from the spectral curve  as described in Section~\\ref{sec:TR2Giv}.\n\nExample~\\ref{specA2} with spectral curve $S_{A_2}$ has partition function $Z_{A_2}$ corresponding to the $A_2$ Frobenius manifold---defined in \\eqref{ZA2}.  Example~\\ref{specA2rel} with spectral curve $S^{\\text{BGW}}_{A_2}$ has partition function $Z^{\\text{BGW}}_{A_2}$ which will be shown to have vanishing primary terms for $n\\leq g-1$ which correspond to relations among coefficients of $Z^{\\text{BGW}}$.   In each example we use a global rational parameter $z$ for the curve $C\\cong\\mathbb{C}$. \n\n\\begin{example}   \\label{specBes}\nTopological recursion applied to the Bessel curve\n$$\nS_{\\text{Bes}}=\\left(\\mathbb{C},x=\\frac{1}{2}z^2,\\ y=\\frac{1}{z},\\ B=\\frac{dzdz'}{(z-z')^2}\\right)\n$$\nproduces correlators\n$$\n\\omega_{g,n}^{\\text{Bes}}=\\sum_{\\vec{k}\\in\\mathbb{Z}_+^n}C_g(k_1,...,k_n)\\prod_{i=1}^n(2k_i+1)!!\\frac{dz_i}{z_i^{2k_i+2}}\n$$\nwhere $C_g(k_1,...,k_n)\\neq 0$ only for $\\sum_{i=1}^n k_i=g-1$.   Define $\\xi_k(z)=(2k+1)!!z^{-(2k+2)}dz$.  It is proven in \\cite{DNoTop} that\n$$Z^{\\text{BGW}}(\\hbar,t_0,t_1,...)=\\left.\\exp\\sum_{g,n}\\frac{\\hbar^{g-1}}{n!}\\omega_{g,n}^{\\text{Bes}}\\right|_{\\xi_k(z_i)=t_k}\\hspace{-8mm}=\\hspace{3mm}\\exp\\sum_{g,n,\\vec{k}}\\frac{\\hbar^{g-1}}{n!}C_g(k_1,...,k_n)\\prod t_{k_j}.$$\n\\end{example}\n\\begin{example}   \\label{specAiry}\nTopological recursion applied to the Airy curve \n$$\nS_{\\text{Airy}}=\\left(\\mathbb{C},x=\\frac{1}{2}z^2,\\ y=z,\\ B=\\frac{dzdz'}{(z-z')^2}\\right)\n$$\nproduces correlators which are proven in \\cite{EOrTop} to store intersection numbers\n$$\\omega_{g,n}^{\\text{Airy}}=\\sum_{\\vec{k}\\in\\mathbb{Z}_+^n}\\int_{\\overline{\\cal M}_{g,n}}\\prod_{i=1}^n\\psi_i^{k_i}(2k_i+1)!!\\frac{dz_i}{z_i^{2k_i+2}}\n$$\nand the coefficient is non-zero only for $\\sum_{i=1}^n k_i=3g-3+n$.  Hence\n$$Z^{\\text{KW}}(\\hbar,t_0,t_1,...)=\\left.\\exp\\sum_{g,n}\\frac{\\hbar^{g-1}}{n!}\\omega_{g,n}^{\\text{Airy}}\\right|_{\\xi_k(z_i)=t_k}\\hspace{-8mm}=\\hspace{3mm}\\exp\\sum_{g,n,\\vec{k}}\\frac{\\hbar^{g-1}}{n!}\\int_{\\overline{\\cal M}_{g,n}}\\prod_{i=1}^n\\psi_i^{k_i}\\prod t_{k_j}.$$\n\\end{example}\nFor the next two examples, define a collection of differentials $\\xi^{\\alpha}_k(z)$ on $\\mathbb{C}$ for $\\alpha\\in\\{1,2\\}$, $k\\in\\{0,1,2,...\\}$ by\n\\begin{equation}   \\label{flatdiff}\n\\xi^{\\alpha}_0=\\frac{dz}{(1-z)^2}-(-1)^{\\alpha}\\frac{dz}{(1+z)^2},\\quad \\xi^\\alpha_{k+1}(p)=d\\left(\\frac{\\xi^\\alpha_k(p)}{dx(p)}\\right),\\ \\alpha=1,2,\\quad k=0,1,2,...\n\\end{equation}\nThese are linear combinations of the $V^i_k(p)$ defined in \\eqref{Vdiff} with $x=z^3-3z$. The $V^i_k(p)$ correspond to normalised canonical coordinates while the $\\xi^{\\alpha}_k(p)$ correspond to flat coordinates.\n\\begin{example}  \\label{specA2}\nConsider the spectral curve  \n$$\nS_{A_2}=\\left(\\mathbb{C},x=z^3-3z,\\ y=z\\sqrt{-3},\\ B=\\frac{dzdz'}{(z-z')^2}\\right).\n$$\nExample~\\ref{specA2B} shows that $(\\mathbb{C},x=z^3-3z,B)$ produces the $R(z)$ associated to the $A_2$ Frobenius manifold at the point $(u_1,u_2)=(2,-2)$ calculated in Example~\\ref{A2RT}.  \n$$R^{-1}(z)=I- \\frac{1}{144}\\left(\\begin{array}{cc}-1&-6i\\\\-6i&1\\end{array}\\right)z+\\frac{35}{41472}\\left(\\begin{array}{cc}-1&-12i\\\\12i&-1\\end{array}\\right)z^2+...\n$$\nIt remains to show that $y$ gives rise to $1\\!\\!1$ and $R^{-1}(z)1\\!\\!1$.  The local expansions of $dy=\\sqrt{-3} dz$ around $z=-1={\\cal P}_1$ and $z=1={\\cal P}_2$ in the respective local coordinates $t$ and $s$, such that $x(t)=\\frac12 t^2+2$ and $x(s)=\\frac12 s^2-2$ are:\n$$ dy=\\sqrt{-3} dz=\\left(\\frac{1}{\\sqrt{2} }-\\frac{5}{144\\sqrt{2}}t^2+\\frac{385}{124416\\sqrt{2}}t^4+...+{\\rm odd\\ terms}\\right)dt\n$$\n$$ dy=\\left(\\frac{i}{\\sqrt{2}}+\\frac{5i}{144\\sqrt{2}}s^2+\\frac{385i}{124416\\sqrt{2}}s^4+...+{\\rm odd\\ terms}\\right)ds\n$$\nhence the Laplace transforms are:\n$$\\frac{1}{\\sqrt{2\\pi z}}\\int_{\\Gamma_1}dy(p)\\cdot e^{(u_1-x(p)) \\over z}=\\frac{1}{\\sqrt{2}}-\\frac{5}{144\\sqrt{2}}z+\\frac{385}{41472\\sqrt{2}}z^2+...\n$$\n$$\\frac{1}{\\sqrt{2\\pi z}}\\int_{\\Gamma_2}dy(p)\\cdot e^{(u_2-x(p)) \\over z}=\\frac{i}{\\sqrt{2}}+\\frac{5i}{144\\sqrt{2}}z+\\frac{385i}{41472\\sqrt{2}}z^2+...\n$$\nwhich indeed gives \n$$\\left\\{\\frac{1}{\\sqrt{2\\pi z}}\\int_{\\Gamma_k}dy(p)\\cdot e^{(u_k-x(p)) \\over z}\\right\\}=R^{-1}(z)1\\!\\!1=\\frac{1}{\\sqrt{2}}\\left(\\begin{array}{c}1\\\\i\\end{array}\\right)+\\frac{5}{144\\sqrt{2}}\\left(\\begin{array}{c}-1\\\\i\\end{array}\\right)z+\\frac{385}{41472\\sqrt{2}}\\left(\\begin{array}{c}1\\\\i\\end{array}\\right)z^2+...\n$$\nNote that $dy({\\cal P}_1)=\\frac{1}{\\sqrt{2}}=\\Delta_1^{1\/2}$ and $dy({\\cal P}_2)=\\frac{i}{\\sqrt{2}}=\\Delta_2^{1\/2}$---the first terms in the expansions in $t$ and $s$ above---gives the unit $1\\!\\!1$ (and the TFT).  These first terms are enough to produce the entire local expansions of $dy$, and hence prove that the series for $dy$ gives $R^{-1}(z)1\\!\\!1$.  The poles of $dy$ are dominated by the poles of $dx$, i.e. $dy\/dx$ has poles only at the zeros ${\\cal P}_1$ and ${\\cal P}_2$ of $dx$, hence $dy$ satisfies \\eqref{DOSStest} so its Laplace transform satisfies \\eqref{ytoT} as required.\n\nApply topological recursion to $S_{A_2}$ to produce correlators $\\omega^{A_2}_{g,n}$.  Then the partition function associated to $S_{A_2}$ coincides with the partition function of the $A_2$ Frobenius manifold:\n$$Z^{A_2}(\\hbar,\\{t^{\\alpha}_k\\})=\\left.\\exp\\sum_{g,n}\\frac{\\hbar^{g-1}}{n!}\\omega^{A_2}_{g,n}\\right|_{\\xi^{\\alpha}_k(z_i)=t^{\\alpha}_k}.\n$$  \nThis was first proven in \\cite{DNOPSDub} via the superpotential construction of Dubrovin \\cite{DubGeo}.\n\n\n\nThe coefficients of $\\log Z^{A_2}(\\hbar,\\{t^{\\alpha}_k\\})$ are obtained from $\\omega_{g,n}$ by\n\\begin{equation}  \\label{insert}\n\\left.\\frac{\\partial^n}{\\partial t^{\\alpha_1}_{k_1}...\\partial t^{\\alpha_n}_{k_n}}F_g^{A_2}(\\{t^{\\alpha}_k\\})\\right|_{t^{\\alpha}_k=0}=\\mathop{\\,\\rm Res\\,}_{z_1=\\infty}...\\mathop{\\,\\rm Res\\,}_{z_n=\\infty}\\prod_{i=1}^n p_{\\alpha_i,k_i}(z_i)\\omega_{g,n}(z_1,...,z_n)\n\\end{equation}\nfor the polynomials in $z$ given by $p_{\\alpha,k}(z)=\\sqrt{-3}\\frac{(-1)^{\\alpha}}{\\alpha}z^{3k+\\alpha}+\\text{lower order terms}$.  The lower order terms (and the top coefficient) will not be important here because we will only consider vanishing of \\eqref{insert} arising from high enough order vanishing of $\\omega_{g,n}(z_1,...,z_n)$ at $z_i=\\infty$ so that the integrand in \\eqref{insert} is holomorphic at $z_i=\\infty$.  Equation~\\eqref{insert} is a special case of the more general phenomena, proven in \\cite{DNOPSPri}, that periods of $\\omega_{g,n}$ are dual to insertions of vectors in a CohFT.  In this case it follows from the easily verified fact that the residues are dual to the differentials $\\xi^{\\alpha}_k$ defined in \\eqref{flatdiff}.  \n\n\n\nRecall that vanishing of coefficients of $\\log Z^{A_2}(\\hbar,\\{t^{\\alpha}_k\\})$ correspond to relations among top intersection classes of tautological relations.   The correlators $\\omega^{A_2}_{g,n}$ are holomorphic at $z=\\infty$ (their only poles occur at $z=\\pm1$) and in fact have high order vanishing there.  The high order vanishing at $z=\\infty$ together with \\eqref{insert} proves vanishing of some coefficients of $\\log Z^{A_2}(\\hbar,\\{t^{\\alpha}_k\\})$.  For example:\n$$\\omega^{A_2}_{2,1}(z) = \\frac{35}{243}\\frac{z(11z^4+14z^2+2)}{(z^2-1)^{10}}dz\\quad\\Rightarrow\\quad\\mathop{\\,\\rm Res\\,}_{z=\\infty}z^m\\omega^{A_2}_{2,1}(z)=0,\\quad m\\in\\{0,1,2,...,13\\}\n$$\nHence \\eqref{insert} vanishes for $k_1=0,1,2,3$ which gives relations between intersection numbers\n$$\\int_{\\overline{\\cal M}_{2,1}}R^{(m)}\\psi_1^{4-m}=0,\\quad m=1,2,3,4$$\nwhere $R^{(m)}$ is a relation between cohomology classes in $H^{2m}(\\overline{\\cal M}_{2,1})$ proven in \\cite{PPZRel}, such as $R^{(2)}=\\psi_1^2+$ boundary terms $=0$.\n\nNote that if one rescales $y\\mapsto\\lambda y$ then the correlators rescale by $\\omega^{A_2}_{g,n}\\mapsto\\lambda^{2-2g-n}\\omega^{A_2}_{g,n}$ so the change does not affect the vanishing terms.  The coefficient $\\sqrt{-3}$ is chosen for convenience to get precise agreement with the associated topological field theory and Frobenius manifold.\n\\end{example}\nThe next example produces $Z^{\\text{BGW}}_{A_2}$, defined in \\eqref{ZBGWA2}, which we recall replaces factors of $Z^{\\text{KW}}$ with $Z^{\\text{BGW}}$ in the partition function $Z_{A_2}$ of the $A_2$ Frobenius manifold. \n\\begin{example}   \\label{specA2rel}\nThe following spectral curve $S^{\\text{BGW}}_{A_2}$ shares $(C,x,B)$ with the spectral curve $S_{A_2}$, since this produces the correct operator $R(z)$ used in the construction of both $Z_{A_2}$ and $Z^{\\text{BGW}}_{A_2}$.  We replace the function $y$ in $S_{A_2}$ by $dy\/dx$ which produces the required shift on the Laplace transform. \n$$\nS^{\\text{BGW}}_{A_2}=\\left(\\mathbb{C},x=z^3-3z,\\ y=\\frac{\\sqrt{-3}}{3z^2-3},\\ B=\\frac{dzdz'}{(z-z')^2}\\right).\n$$\nAs in the previous example, we have:\n$$R^{-1}(z)=I- \\frac{1}{144}\\left(\\begin{array}{cc}-1&-6i\\\\-6i&1\\end{array}\\right)z+\\frac{35}{41472}\\left(\\begin{array}{cc}-1&-12i\\\\12i&-1\\end{array}\\right)z^2+...\n$$\nThe local expansions of $dy=\\frac{2}{\\sqrt{-3}}\\frac{zdz}{(z^2-1)^2}$ around $z=-1={\\cal P}_1$ and $z=1={\\cal P}_2$ in the respective local coordinates $t$ and $s$, such that $x(t)=\\frac12 t^2+2$ and $x(s)=\\frac12 s^2-2$ are:\n$$ dy=\\frac{2}{\\sqrt{-3}}\\frac{zdz}{(z^2-1)^2}=\\left(-\\frac{1}{\\sqrt{2}}t^{-2}-\\frac{5}{144\\sqrt{2}}+\\frac{385}{41472\\sqrt{2}}t^2+...+{\\rm odd\\ terms}\\right)dt\n$$\n$$ dy=\\left(-\\frac{i}{\\sqrt{2}}s^{-2}+\\frac{5i}{144\\sqrt{2}}+\\frac{385i}{41472\\sqrt{2}}s^2+...+{\\rm odd\\ terms}\\right)ds\n$$\nand the Laplace transforms are a shift of the Laplace transforms in the previous example:\n$$\\frac{1}{\\sqrt{2\\pi z}}\\int_{\\Gamma_1}dy(p)\\cdot e^{(u_1-x(p)) \\over z}=\\frac{1}{\\sqrt{2}}z^{-1}-\\frac{5}{144\\sqrt{2}}+\\frac{385}{41472\\sqrt{2}}z+...\n$$\n$$\\frac{1}{\\sqrt{2\\pi z}}\\int_{\\Gamma_2}dy(p)\\cdot e^{(u_2-x(p)) \\over z}=\\frac{i}{\\sqrt{2}}z^{-1}+\\frac{5i}{144\\sqrt{2}}+\\frac{385i}{41472\\sqrt{2}}z+...\n$$\nApply topological recursion to $S^{\\text{BGW}}_{A_2}$ to produce correlators $\\omega_{g,n}$ and partition function $Z^{S^{\\text{BGW}}_{A_2}}$.  It is proven in \\cite{CNoTop} that topological recursion applied to an irregular spectral curve, i.e. $y$ has simple poles at the zeros of $dx$, produces a partition function $Z^S$ with translation term encoded in the Laplace transform of $dy$ and in this case $T_0(z)=1\\!\\!1-R^{-1}(z)1\\!\\!1$.  This is exactly the construction of $Z^{\\text{BGW}}_{A_2}$ defined in Section~\\ref{twloop} hence we have:\n$$Z^{\\text{BGW}}_{A_2}(\\hbar,\\{t^{\\alpha}_k\\})=Z^{S^{\\text{BGW}}_{A_2}}(\\hbar,\\{t^{\\alpha}_k\\}).$$ \nLemma~\\ref{vanA2BGW} below proves that the sums of the orders of vanishing of $\\omega^{{\\text{BGW}},A_2}_{g,n}(z_1,...,z_n)$ at $z_i=\\infty$ is bounded below by $2g-2$.  \nExplicitly in low genus, \n$$\\omega^{{\\text{BGW}},A_2}_{1,1}(z)=\\frac{z^2+1}{4\\sqrt{-3}(z^2-1)^2}dz,\\quad\\omega^{{\\text{BGW}},A_2}_{2,1}(z)=\\frac{-5z^2-1}{16\\sqrt{-3}(z-1)^4(z+1)^4}dz.\n$$\nWe find that $-\\displaystyle\\mathop{\\,\\rm Res\\,}_{z=\\infty}\\sqrt{-3}\\cdot z\\cdot\\omega_{1,1}(z)=\\tfrac{1}{4}$ agrees with the graphical expansion\n\\begin{center}\n\\begin{tikzpicture}\n\\node(0) at (1,0)   {$e_0$}    ;\n\\node(1) at (2,0)[shape=circle,draw]        {1};\n\\path [-] (0)      edge   node   {}     (1);\n\\end{tikzpicture}\n\\end{center}\nwhich contributes $2^g\\cdot\\int_{\\overline{\\cal M}_{1,1}}\\Theta_{1,1}=2\\cdot\\frac{1}{8}=\\frac{1}{4}$.\n\n\nFrom the vanishing of $\\omega_{2,1}(z)$ at $z=\\infty$ it immediately follows that $\\displaystyle\\mathop{\\,\\rm Res\\,}_{z=\\infty}\\frac{\\sqrt{-3}}{2}z\\cdot\\omega_{2,1}(z)=0$ which signifies a relation between coefficients of $Z^{\\text{BGW}}(\\hbar,t_0,t_1,...)$.  We will write the relations using $\\Theta_{g,n}$ however the relations are between coefficients of $Z^{\\text{BGW}}(\\hbar,t_0,t_1,...)$ and what we are showing here is that these coefficients satisfy the same relations as intersection numbers involving $\\Theta_{g,n}$, or equivalently coefficients of $Z^{\\Theta}(\\hbar,t_0,t_1,...)$.  The graphical expansion of this is given by:\n\\begin{center}\n\\begin{tikzpicture}\n  \n\\node(0) at (-5,0)   {}    ;\n\\node(1) at (-4,0)[shape=circle,draw]        {2};\n\\path [-] (0)      edge   node   {}     (1);\n\\node(5) at (-2.5,0)   {}    ;\n\\node(6) at (-1.5,0)[shape=circle,draw]        {2};\n\\node(7) at (-.5,.5)   {};\n\\tikzset{decoration={snake,amplitude=.4mm,segment length=2mm,\n                       post length=0mm,pre length=0mm}}\n  \\draw[decorate] (6) -- (7);\n\\node(2) at (6,0)(text) {};\n\\path [-] (5)      edge   node   {}     (6);\n\n\\node(3) at (1,0)[shape=circle,draw]        {1};\n\\node(4) at (3,0)[shape=circle,draw]        {1};\n\\path [-] (3)      edge   node [above]  {}     (4);\n\\node(10) at (0,0)        {};\n\\path [-] (10)      edge   node   {}         (3);\n\\circledarrow{}{text}{.8cm};\n\\node(8) at (5.25,0)[shape=circle,draw]        {1};\n\\node(11) at (4.25,0)        {};\n\\path [-] (11)      edge   node   {}         (8);\n\n\\end{tikzpicture}\n\\end{center}\n(plus graphs containing genus 0 vertices on which $\\Theta_{2,1}$ vanishes)\t  \nwhich contributes \n$$2^2\\cdot\\frac{60}{1728}\\cdot\\int_{\\overline{\\cal M}_{2,1}}\\Theta_{2,1}\\cdot\\psi_1+2^2\\cdot\\frac{-60}{1728}\\cdot\\int_{\\overline{\\cal M}_{2,1}}\\Theta_{2,1}\\cdot\\kappa_1+2^2\\cdot\\frac{84}{1728}\\cdot\\int_{\\overline{\\cal M}_{1,2}}\\Theta_{1,2}\\cdot\\int_{\\overline{\\cal M}_{1,1}}\\Theta_{1,1}+\\frac{2}{2}\\cdot\\frac{84-60}{1728}\\cdot\\int_{\\overline{\\cal M}_{1,3}}\\Theta_{1,3}\n$$\nwhich agrees with the expansion in weighted graphs of $\\displaystyle\\mathop{\\,\\rm Res\\,}_{z=\\infty}\\frac{\\sqrt{-3}}{2}z\\cdot\\omega_{2,1}(z)=0$ given by\n$$\\frac{5}{1536}-\\frac{15}{1536}+\\frac{7}{2304}+\\frac{1}{288}=0. \n$$\n\n\n\n\n\n\n\n\n\n\n\n   \n\n\n\n\n\n\\end{example}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsubsection{Intersection numbers and the Brezin-Gross-Witten tau function.}  \\label{intkdv}\n\nWe are finally in a position to prove that $Z^{\\text{BGW}}(\\hbar,t_0,t_1,...)$ is a generating function for the intersection numbers $\\int_{\\overline{\\cal M}_{g,n}}\\Theta_{g,n}\\prod_{i=1}^n\\psi_i^{k_i}$, or\n$$Z^{\\text{BGW}}(\\hbar,t_0,t_1,...)=Z^{\\Theta}(\\hbar,t_0,t_1,...)$$ \nstated as Theorem~\\ref{tauf} below.   {\\em A priori} the coefficients of $Z^{\\text{BGW}}_{A_2}$ have nothing to do with integration of cohomology classes over $\\overline{\\cal M}_{g,n}$, nevertheless we will see that $Z^{\\text{BGW}}_{A_2}$ and $Z^{\\Theta}_{A_2}$ share the same relations which will be used to prove their coincidence.  This relies on the spectral curve $x=z^3-3z$, $y=\\frac{1}{\\sqrt{-3}(z^2-1)}$ analysed in Example~\\ref{specA2rel}. \n\n\n\n\n\n\\begin{proof}[Proof of Theorem~\\ref{tauf}]  The proof is a rather beautiful application of Pixton's relations proven by Pandharipande, Pixton and Zvonkine, \\cite{PPZRel}.  We use Pixton's relations to induce relations among the intersection numbers $\\int_{\\overline{\\cal M}_{g,n}}\\Theta_{g,n}\\prod_{i=1}^n\\psi_i^{m_i}$.  \nThe key idea of the proof of the theorem is to show that the coefficients of $Z^{\\text{BGW}}$ satisfy the same relations as those, such as \\eqref{relg2}, induced by Pixton's relations on the intersection numbers $\\int_{\\overline{\\cal M}_{g,n}}\\Theta_{g,n}\\prod_{i=1}^n\\psi_i^{m_i}$.  The proof here brings together all of the results of Section~\\ref{sec:kdv}.\\\\\n\nRecall from Section~\\ref{sec:A2part} that the partition functions $Z^{\\Theta}_{A_2}(\\hbar,\\{t^{\\alpha}_k\\})$ and $Z^{\\text{BGW}}_{A_2}(\\hbar,\\{t^{\\alpha}_k\\})$ have the same structure since they are built out of the same element $R(z)$, arising from the $A_2$ Frobenius manifold, and the same translation $T_0(z)=1\\!\\!1-R^{-1}(z)1\\!\\!1$, but with {\\em a  priori} different vertex contributions.  Vanishing coefficients in both partition functions produce the same relations between coefficients of $Z^{\\Theta}(\\hbar,t_0,t_1,...)$, respectively $Z^{\\text{BGW}}(\\hbar,t_0,t_1,...)$.  Enough of these relations will prove that $Z^{\\Theta}(\\hbar,t_0,t_1,...)=Z^{\\text{BGW}}(\\hbar,t_0,t_1,...)$.  Vanishing of certain coefficients of $Z^{\\Theta}_{A_2}(\\hbar,\\{t^{\\alpha}_k\\})$ is a consequence of the cohomological viewpoint---it comes from $\\Theta_{g,n}R^{(g-1)}=0$ for Pixton relations $0=R^{(g-1)}\\in H^{2g-2}(\\overline{\\cal M}_{g,n})$.  The vanishing of corresponding coefficients of $Z^{\\text{BGW}}_{A_2}(\\hbar,\\{t^{\\alpha}_k\\})$ uses topological recursion applied to the spectral curve $S^{\\text{BGW}}_{A_2}$, defined in Example~\\ref{specA2rel}, which produces the partition function $Z^{\\text{BGW}}_{A_2}(\\hbar,\\{t^{\\alpha}_k\\})$ out of correlators $\\omega^{{\\text{BGW}},A_2}_{g,n}(z_1,...,z_n)$.   We know that $\\omega^{{\\text{BGW}},A_2}_{g,n}(z_1,...,z_n)$ is holomorphic at $z_i=\\infty$ for each $i=1,...,n$.  The next lemma shows its order of vanishing there.\n\\begin{lemma}  \\label{vanA2BGW}\n$$\\sum_{i=1}^n\\text{ord}_{z_i=\\infty\\ }\\omega^{{\\text{BGW}},A_2}_{g,n}(z_1,...,z_n)\\geq 2g-2\n$$\nwhere $\\text{ord}_{z=\\infty\\ }\\eta(z)$ is the order of vanishing the differential at $z=\\infty$.\n\\end{lemma}\n\\begin{proof}\nWe can make the rational differential\n$$\\omega^{A_2}_{g,n}(z_1,...,z_n)=\\frac{p_{g,n}(z_1,....,z_n)}{\\prod_{i=1}^n(z_i^2-1)^{2g}}dz_1...dz_n\n$$\nhomogeneous by applying topological recursion to $x(z)=z^3-3Q^2z$ and $y=\\sqrt{-3}\/x'(z)$ which are homogeneous in $z$ and $Q$.  Then $\\omega^{A_2}_{g,n}(Q,z_1,...,z_n)$ is homogeneous in $z$ and $Q$ of degree $2-2g-n$:\n$$\\omega^{A_2}_{g,n}(Q,z_1,...,z_n)=\\lambda^{2-2g-n}\\omega^{A_2}_{g,n}(\\lambda Q,\\lambda z_1,...,\\lambda z_n).\n$$\nThe degree of homogeneity uses the fact that $(z,Q)\\mapsto(\\lambda z,\\lambda Q)$ $\\Rightarrow$ $ydx\\mapsto\\lambda ydx$ $\\Rightarrow$ $\\omega_{g,n}\\mapsto\\lambda^{2-2g-n}\\omega_{g,n}$\nbecause $ydx$ appears in the kernel $K(p_1,p)$ with homogeneous degree $-1$ which easily leads to degree $2-2g-n$ for $\\omega_{g,n}$.  The degree $2-2g-n$ homogeneity of\n$$\\omega^{A_2}_{g,n}(Q,z_1,...,z_n)=\\frac{p_{g,n}(Q,z_1,....,z_n)}{\\prod_{i=1}^n(z_i^2-Q^2)^{2g}}dz_1...dz_n\n$$\nimplies that $\\deg p_{g,n}(Q,z_1,....,z_n)=4gn-n+2-2g-n$.  But we also know that $\\omega^{A_2}_{g,n}(Q,z_1,...,z_n)$ is well-defined as $Q\\to 0$---the limit becomes $\\omega_{g,n}$ of the spectral curve $x(z)=z^3$ and $y=\\sqrt{-3}\/x'(z)$ using the topological recursion defined by Bouchard and Eynard \\cite{BEyThi}---so $\\deg p_{g,n}(z_1,....,z_n)\\leq 4gn-n+2-2g-n$.  Note that $dz_i$ is homogeneous of degree 1 but has a pole of order 2 at $z_i=\\infty$, hence\n$$\\sum_{i=1}^n\\text{ord}_{z_i=\\infty\\ }\\omega^{{\\text{BGW}},A_2}_{g,n}(z_1,...,z_n)=4gn-\\deg p_{g,n}(z_1,....,z_n)-2n\\geq 2g-2.\n$$\n\\end{proof}\nThe primary coefficients of $Z^{\\text{BGW}}_{A_2}(\\hbar,\\{t^{\\alpha}_k\\})$, those where $k=0$, correspond to $$ \\mathop{\\,\\rm Res\\,}_{z_1=\\infty}...\\mathop{\\,\\rm Res\\,}_{z_n=\\infty}\\prod_{i=1}^nz_i^{\\epsilon_i}\\omega^{A_2}_{g,n}(z_1,...,z_n)$$ for $\\epsilon_i=1$ or 2.  Different choices of $\\epsilon_i$ give different relations (except half which vanish for parity reasons).  We have:\n\\begin{corollary}\n$$n< 2g-2\\Rightarrow \\mathop{\\,\\rm Res\\,}_{z_1=\\infty}...\\mathop{\\,\\rm Res\\,}_{z_n=\\infty}\\prod_{i=1}^nz_i^{\\epsilon_i}\\omega^{A_2}_{g,n}(z_1,...,z_n)=0$$\n\\end{corollary}\n\\begin{proof}\nSince $n< 2g-2$ and $\\sum_{i=1}^n\\text{ord}_{z_i=\\infty\\ }\\omega^{{\\text{BGW}},A_2}_{g,n}(z_1,...,z_n)\\geq 2g-2$ by Lemma~\\ref{vanA2BGW}, there exists an $i$ such that $\\text{ord}_{z_i=\\infty\\ }\\omega^{{\\text{BGW}},A_2}_{g,n}(z_1,...,z_n)\\geq 2$.  Hence $z_i^{\\epsilon_i}\\omega^{A_2}_{g,n}(z_1,...,z_n)$ is holomorphic at $z_i=\\infty$, so \n$$\\displaystyle\\mathop{\\,\\rm Res\\,}_{z_i=\\infty}z_i^{\\epsilon_i}\\omega^{A_2}_{g,n}(z_1,...,z_n)=0$$ \nand the multiple residue vanishes as required.\n\\end{proof}\nHence the primary coefficients of $Z^{\\text{BGW}}_{A_2}(\\hbar,\\{t^{\\alpha}_k\\})$ vanish for $n< 2g-2$ proving that the coefficients of $Z^{\\Theta}(\\hbar,t_0,t_1,...)$ and $Z^{\\text{BGW}}(\\hbar,t_0,t_1,...)$ for $n<2g-2$ satisfy the same relations.  Pixton's relations give enough relations among coefficients of $Z^{\\Theta}(\\hbar,t_0,t_1,...)$ to calculate them, and hence the same relations also uniquely determine the coefficients of $Z^{\\text{BGW}}(\\hbar,t_0,t_1,...)$ and we have $Z^{\\Theta}(\\hbar,t_0,t_1,...)=Z^{\\text{BGW}}(\\hbar,t_0,t_1,...)$ up to coefficients with $n< 2g-2$.  The dilaton equation \\eqref{dilaton} then proves equality of coefficients for all $n$ giving $Z^{\\Theta}(\\hbar,t_0,t_1,...)=Z^{\\text{BGW}}(\\hbar,t_0,t_1,...)$.\n\n\n\n\n\\end{proof}\n\n\n\n\n\n\n \n\n\n\\section{Cohomological field theories}  \\label{sec:cohft}\n\nThe class $\\Theta_{g,n}$ combines with known enumerative invariants, such as Gromov-Witten invariants, to give rise to new invariants.  More generally, $\\Theta_{g,n}$ pairs with any cohomological field theory, which is fundamentally related to the moduli space of curves $\\overline{\\cal M}_{g,n}$, retaining many of the properties of the cohomological field theory, and is in particular often calculable.\n\nA {\\em cohomological field theory} is a pair $(V,\\eta)$ composed of a finite-dimensional complex vector space $V$ equipped with a metric $\\eta$ and a sequence of $S_n$-equivariant maps. \n\\[ \\Omega_{g,n}:V^{\\otimes n}\\to H^*(\\overline{\\cal M}_{g,n})\\]\nthat satisfy compatibility conditions from inclusion of strata:\n$$\\phi_{\\text{irr}}:\\overline{\\cal M}_{g-1,n+2}\\to\\overline{\\cal M}_{g,n},\\quad \\phi_{h,I}:\\overline{\\cal M}_{h,|I|+1}\\times\\overline{\\cal M}_{g-h,|J|+1}\\to\\overline{\\cal M}_{g,n},\\quad I\\sqcup J=\\{1,...,n\\}$$\ngiven by\n\\begin{align}\n\\phi_{\\text{irr}}^*\\Omega_{g,n}(v_1\\otimes...\\otimes v_n)&=\\Omega_{g-1,n+2}(v_1\\otimes...\\otimes v_n\\otimes\\Delta) \n  \\label{glue1}\n\\\\\n\\phi_{h,I}^*\\Omega_{g,n}(v_1\\otimes...\\otimes v_n)&=\\Omega_{h,|I|+1}\\otimes \\Omega_{g-h,|J|+1}\\big(\\bigotimes_{i\\in I}v_i\\otimes\\Delta\\otimes\\bigotimes_{j\\in J}v_j\\big) \\label{glue2}\n\\end{align}\nwhere $\\Delta\\in V\\otimes V$ is dual to the metric $\\eta\\in V^*\\otimes V^*$.  When $n=0$, $\\Omega_g:=\\Omega_{g,0}\\in H^*(\\overline{\\cal M}_{g})$.\nThere exists a vector $1\\!\\!1\\in V$ compatible with the forgetful map $\\pi:\\overline{\\cal M}_{g,n+1}\\to\\overline{\\cal M}_{g,n}$ by\n\\begin{equation}   \\label{cohforget}\n\\Omega_{g,n+1}(1\\!\\!1\\otimes v_1\\otimes...\\otimes v_n)=\\pi^*\\Omega_{g,n}(v_1\\otimes...\\otimes v_n)\n\\end{equation}\nfor $2g-2+n>0$,  and\n$$\\Omega_{0,3}(1\\!\\!1\\otimes v_1\\otimes v_2)=\\eta(v_1,v_2).\n$$\n\nFor a one-dimensional CohFT, i.e. $\\dim V=1$, identify $\\Omega_{g,n}$ with the image $\\Omega_{g,n}(1\\!\\!1^{\\otimes n})$, so we write $\\Omega_{g,n}\\in H^*(\\overline{\\cal M}_{g,n})$.  A trivial example of a CohFT is $\\Omega_{g,n}=1\\in H^0(\\overline{\\cal M}_{g,n})$ which is a topological field theory as we now describe.\n\nA two-dimensional topological field theory (TFT) is a vector space $V$ and a sequence of symmetric linear maps\n\\[ \\Omega^0_{g,n}:V^{\\otimes n}\\to \\mathbb{C}\\]\nfor integers $g\\geq 0$ and $n>0$ satisfying the following conditions.  The map $\\Omega^0_{0,2}=\\eta$ defines a metric $\\eta$, and together with $\\Omega^0_{0,3}$ it defines a\nproduct $\\raisebox{-0.5ex}{\\scalebox{1.8}{$\\cdot$}}$ on $V$ via\n\\begin{equation}   \\label{prod}\n\\eta(v_1\\raisebox{-0.5ex}{\\scalebox{1.8}{$\\cdot$}} v_2,v_3)=\\Omega^0_{0,3}(v_1,v_2,v_3)\n\\end{equation}\nwith identity $1\\!\\!1$ given by the dual of $\\Omega^0_{0,1}=1\\!\\!1^*=\\eta(1\\!\\!1,\\cdot)$.  It satisfies \n$$\\Omega^0_{g,n+1}(1\\!\\!1\\otimes v_1\\otimes...\\otimes v_n)=\\Omega^0_{g,n}(v_1\\otimes...\\otimes v_n)$$ \nand the gluing conditions \n$$\n\\Omega^0_{g,n}(v_1\\otimes...\\otimes v_n)=\\Omega^0_{g-1,n+2}(v_1\\otimes...\\otimes v_n\\otimes\\Delta)=\\Omega^0_{g_1,|I|+1}\\otimes \\Omega^0_{g_2,|J|+1}\\big(\\bigotimes_{i\\in I}v_i\\otimes\\Delta\\otimes\\bigotimes_{j\\in J}v_j\\big)$$\nfor $g=g_1+g_2$ and $I\\sqcup J=\\{1,...,n\\}$.\n\nConsider the natural isomorphism $H^0({\\overline{\\cal M}_{g,n})}\\cong\\mathbb{C}$.  The degree zero part of a CohFT $\\Omega_{g,n}$ is a TFT: \n$$\\Omega^0_{g,n}:V^{\\otimes n}\\stackrel{\\Omega_{g,n}}{\\to} H^*({\\overline{\\cal M}_{g,n}})\\to H^0({\\overline{\\cal M}_{g,n}}).\n$$\nWe often write $\\Omega_{0,3}=\\Omega^0_{0,3}$ interchangeably.  Associated to $\\Omega_{g,n}$ is the product \\eqref{prod} built from $\\eta$ and $\\Omega_{0,3}$.\n\n\n\nGiven a CohFT $\\Omega=\\{\\Omega_{g,n}\\}$ and a basis $\\{e_1,...,e_N\\}$ of $V$, the partition function of $\\Omega$ is defined as in \\eqref{ZA2}.\n\\begin{equation}   \\label{partfun}\nZ_{\\Omega}(\\hbar,\\{t^{\\alpha}_k\\})=\\exp\\sum_{g,n,\\vec{k}}\\frac{\\hbar^{g-1}}{n!}\\int_{\\overline{\\cal M}_{g,n}}\\Omega_{g,n}(e_{\\alpha_1}\\otimes...\\otimes e_{\\alpha_n})\\cdot\\prod_{j=1}^n\\psi_j^{k_j}\\prod t^{\\alpha_j}_{k_j}\n\\end{equation}\nfor $\\alpha_i\\in\\{1,...,N\\}$ and $k_j\\in\\mathbb{N}$.\n\n \n\\begin{remark}  \\label{theco}\nThe class $\\Theta_{g,n}$ defined in the introduction satisfies properties~\\eqref{glue1} and \\eqref{glue2} of a one-dimensional CohFT.  In place of property~\\eqref{cohforget}, it satisfies $\\Theta_{g,n+1}(1\\!\\!1\\otimes v_1\\otimes...\\otimes v_n)=\\psi_{n+1}\\cdot\\pi^*\\Theta_{g,n}(v_1\\otimes...\\otimes v_n)$ and $\\Theta_{0,3}=0$.\n\\end{remark}\n\\begin{definition}  \\label{cohftheta}\nFor any CohFT $\\Omega$ defined on $(V,\\eta)$ define $\\Omega^\\Theta=\\{\\Omega^\\Theta_{g,n}\\}$ to be the sequence of $S_n$-equivariant maps $\\Omega^\\Theta_{g,n}:V^{\\otimes n}\\to H^*(\\overline{\\cal M}_{g,n})$ given by $\\Omega^\\Theta_{g,n}(v_1\\otimes...\\otimes v_n)=\\Theta_{g,n}\\cdot\\Omega_{g,n}(v_1\\otimes...\\otimes v_n)$.   \n\\end{definition}\nThis is essentially to the tensor products of CohFTs, albeit involving $\\Theta_{g,n}$.  The tensor products of CohFTs is obtained as above by cup product on $H^*(\\overline{\\cal M}_{g,n})$, generalising Gromov-Witten invariants of target products and the K\\\"unneth formula $H^*(X_1\\times X_2)\\cong H^*X_1\\otimes H^*X_2$.  \n\nGeneralising Remark~\\ref{theco}, $\\Omega^\\Theta_{g,n}$ satisfies properties~\\eqref{glue1} and \\eqref{glue2} of a CohFT on $(V,\\eta)$.  In place of property~\\eqref{cohforget}, it satisfies \n$$\\Omega^\\Theta_{g,n+1}(1\\!\\!1\\otimes v_1\\otimes...\\otimes v_n)=\\psi_{n+1}\\cdot\\pi^*\\Omega^\\Theta_{g,n}(v_1\\otimes...\\otimes v_n)$$ and $\\Omega^\\Theta_{0,3}=0$.\n\n The product defined in \\eqref{prod} is {\\em semisimple} if it is diagonal $V\\cong\\mathbb{C}\\oplus\\mathbb{C}\\oplus...\\oplus\\mathbb{C}$, i.e. there is a canonical basis $\\{ u_1,...,u_N\\}\\subset V$ such that $u_i\\cdot u_j=\\delta_{ij}u_i$.  The metric is then necessarily diagonal with respect to the same basis, $\\eta(u_i,u_j)=\\delta_{ij}\\eta_i$ for some $\\eta_i\\in\\mathbb{C} \\setminus \\{0\\}$, $i=1,...,N$.  The Givental-Teleman theorem \\cite{GiventalMain,TelStr} states that the twisted loop group action defined in Section~\\ref{twloop} acts transitively on semisimple CohFTs.  In particular, a semisimple homogeneous CohFT is uniquely determined by its underlying TFT.  The tau function $Z^{\\text{BGW}}$ appears in a generalisation of Givental's decomposition of CohFTs \\cite{CNoTop} which, combined with the result $Z^{\\text{BGW}}=Z^{\\Theta}$, generalises Givental's action on CohFTs to allow one to replace the TFT term by the classes $\\Theta_{g,n}$.  For example, in \\cite{DNoTopI} the enumeration of bipartite dessins d'enfant is shown to satisfy topological recursion for the curve $xy^2+xy+1=0$ which has both a regular and irregular zero of $dx$.  Underlying this example is a generalised CohFT built out of the classes $\\Theta_{g,n}\\in H^*(\\overline{\\cal M}_{g,n})$ combined with the class $1\\in H^*(\\overline{\\cal M}_{g,n})$ via Givental's action.\n\n\n\n\\subsection{Gromov-Witten invariants}\nLet $X$ be a projective algebraic variety and consider $(C,x_1,\\dots,x_n)$ a connected smooth curve of genus $g$ with $n$ distinct marked points.  For $\\beta \\in H_2(X,\\mathbb{Z})$ the moduli space of stable maps $\\cal M^g_n(X,\\beta)$ is defined by:\n$$\\cal M_{g,n}(X,\\beta)=\\{(C,x_1,\\dots,x_n)\\stackrel{\\pi}{\\rightarrow} X\\mid \\pi_\\ast [C]=\\beta\\}\/\\sim$$\nwhere $\\pi$ is a morphism from a connected nodal curve $C$ containing distinct points $\\{x_1,\\dots,x_n\\}$ that avoid the nodes.  Any genus zero irreducible component of $C$ with fewer than three distinguished points (nodal or marked), or genus one irreducible component of $C$ with no distinguished point, must not be collapsed to a point.  We quotient by isomorphisms of the domain $C$ that fix each $x_i$.    The moduli space of stable maps has irreducible components of different dimensions but it has a virtual class of dimension\n$$ \\dim[\\cal M_{g,n}(X,\\beta)]^{\\text{virt}}=(\\dim X-3)(1-g)+\\langle c_1(X),\\beta\\rangle +n.\n$$\nFor $i=1,\\dots,n$ there exist evaluation maps:\n\\begin{equation}\nev_i:\\cal M_{g,n}(X,\\beta)\\longrightarrow X, \\quad ev_i(\\pi)=\\pi(x_i)\n\\end{equation}\nand classes $\\gamma\\in H^*(X,\\mathbb{Z})$ pull back to classes in $H^*(\\cal M_{g,n}(X,\\beta),\\mathbb{Q})$\n\\begin{equation}\nev_i^\\ast:H^*(X,\\mathbb{Z})\\longrightarrow H^*(\\cal M_{g,n}(X,\\beta),\\mathbb{Q}).\n\\end{equation}\nThe forgetful map $p:\\cal M_{g,n}(X,\\beta)\\to \\overline{\\cal M}_{g,n}$ maps a stable map to its domain curve followed by contraction of unstable components.  The push-forward map $p_*$ on cohomology defines a CohFT $\\Omega_X$ on the even part of the cohomology $V=H^{\\text{even}}(X;\\mathbb{C})$ (and a generalisation of a CohFT on $H^*(X;\\mathbb{C})$) equipped with the metric  \n$$\\eta(\\alpha,\\beta)=\\int_X\\alpha\\wedge\\beta.$$\nWe have $(\\Omega_X)_{g,n}:H^{\\text{even}}(X)^{\\otimes n}\\to H^*(\\overline{\\cal M}_{g,n})$ defined by \n$$(\\Omega_X)_{g,n}(\\alpha_1,...\\alpha_n)=\\sum_{\\beta}p_*\\left(\\prod_{i=1}^nev_i^\\ast(\\alpha_i)\\cap[\\cal M_{g,n}(X,\\beta)]^{\\text{virt}}\\right)\\in H^*(\\overline{\\cal M}_{g,n}).$$\nNote that it is the dependence of $p=p(g,n,\\beta)$ on $\\beta$ (which is suppressed) that allows $(\\Omega_X)_{g,n}(\\alpha_1,...\\alpha_n)$ to be composed of different degree terms.  The partition function of the CohFT $\\Omega_X$ with respect to a chosen basis $e_{\\alpha}$ of $H^{\\text{even}}(X;\\mathbb{C})$ is\n$$Z_{\\Omega_X}(\\hbar,\\{t^{\\alpha}_k\\})=\\exp\\sum_{\\Small{\\begin{array}{c}g,n,\\vec{k}\\\\ \\vec{\\alpha},\\beta\\end{array}}}\\frac{\\hbar^{g-1}}{n!}\\int_{\\overline{\\cal M}_{g,n}}p_*\\left(\\prod_{i=1}^nev_i^\\ast(e_{\\alpha_i})\\cap[\\cal M_{g,n}(X,\\beta)]^{\\text{virt}}\\right)\\cdot\\prod_{j=1}^n\\psi_j^{k_j}\\prod t^{\\alpha_j}_{k_j}.\n$$\nIt stores {\\em ancestor} invariants.  These are different to {\\em descendant} invariants which use in place of $\\psi_j=c_1(L_j)$, $\\Psi_j=c_1(\\mathcal{L}_j)$ for line bundles $\\mathcal{L}_j\\to \\cal M_{g,n}(X,\\beta)$ defined as the cotangent bundle over the $i$th marked point.\n\nFollowing Definition~\\ref{cohftheta}, we define $\\Omega_X^{\\Theta}$ by\n$$(\\Omega_X^{\\Theta})_{g,n}(\\alpha_1,...\\alpha_n)=\\Theta_{g,n}\\cdot\\sum_{\\beta}p_*\\left(\\prod_{i=1}^nev_i^\\ast(\\alpha_i)\\right)\\in H^*(\\overline{\\cal M}_{g,n}).$$\nand\n$$Z^{\\Theta}_{\\Omega_X}(\\hbar,\\{t^{\\alpha}_k\\})=\\exp\\sum_{\\Small{\\begin{array}{c}g,n,\\vec{k}\\\\ \\vec{\\alpha},\\beta\\end{array}}}\\frac{\\hbar^{g-1}}{n!}\\int_{\\overline{\\cal M}_{g,n}}\\Theta_{g,n}\\cdot p_*\\left(\\prod_{i=1}^nev_i^\\ast(e_{\\alpha_i})\\right)\\cdot\\prod_{j=1}^n\\psi_j^{k_j}\\prod t^{\\alpha_j}_{k_j}.\n$$\nThe invariants $\\Omega_X^{\\Theta}$ have not been defined directly from a moduli space $\\cal M_{g,n}^{\\Theta}(X,\\beta)$, say obtained by restricting the domain of a stable map to a $g-1$-dimensional subvariety $X_{g,n}\\subset\\overline{\\cal M}_{g,n}$ arising as the push-forward of the zero set of a section of the bundle $E_{g,n}$ defined in Definition~\\ref{obsbun}.  Nevertheless, it is instructive for comparison purposes to write the virtual dimension of this yet to be defined $\\cal M_{g,n}^{\\Theta}(X,\\beta)$.  We have\n$$ \\dim[\\cal M_{g,n}^{\\Theta}(X,\\beta)]^{\\text{virt}}=(\\dim X-1)(1-g)+\\langle c_1(X),\\beta\\rangle.\n$$\nAgain we emphasise that the invariants of $X$ stored in $Z^{\\Theta}_{\\Omega_X}(\\hbar,\\{t^{\\alpha}_k\\})$ are rigorously defined, and the purpose of the dimension formula is for a comparison with usual Gromov-Witten invariants.  We see that the virtual dimension is independent of $n$.   Elliptic curves now take the place of Calabi-Yau 3-folds to give virtual dimension zero moduli spaces, independent of genus and degree.  The invariants of a target curve $X$ are trivial when the genus of $X$ is greater than 1 and computable when $X=\\mathbb{P}^1$ \\cite{NorGro} producing results anologous to the usual Gromov-Witten invariants in \\cite{NScGro}.  For $c_1(X)=0$ and $\\dim X>1$, the invariants vanish for $g>1$, while for $g=1$ it seems to predict an invariant associated to maps of elliptic curves to $X$.\n\n \n\n\\subsubsection{Weil-Petersson volumes} \nA fundamental example of a 1-dimensional CohFT is given by \n$$\\Omega_{g,n}=\\exp(2\\pi^2\\kappa_1)\\in H^*(\\overline{\\cal M}_{g,n}).$$  \nIts partition function stores Weil-Petersson volumes \n$$V_{g,n}=\\frac{(2\\pi^2)^{3g-3+n}}{(3g-3+n)!}\\int_{\\overline{\\cal M}_{g,n}}\\kappa_1^{3g-3+n}$$\nand deformed Weil-Petersson volumes studied by Mirzakhani \\cite{MirSim}.\nWeil-Petersson volumes of the subvariety of $\\overline{\\cal M}_{g,n}$ dual to $\\Theta_{g,n}$ make sense even before we find such a subvariety.  They are given by \n$$V^{\\Theta}_{g,n}=\\frac{(2\\pi^2)^{g-1}}{(g-1)!}\\int_{\\overline{\\cal M}_{g,n}}\\Theta_{g,n}\\cdot\\kappa_1^{g-1}$$ \nwhich are calculable since they are given by a translation of $Z^{\\text{BGW}}$.  If we include $\\psi$ classes, we get polynomials $V^{\\Theta}_{g,n}(L_1,...,L_n)$ which give the deformed volumes analogous to Mirzakhani's volumes.  \n\n\n\n\\subsubsection{ELSV formula}\nAnother example of a 1-dimensional CohFT is given by \n$$\\Omega_{g,n}=c(E^\\vee)=1-\\lambda_1+...+(-1)^{g}\\lambda_{g}\\in H^*(\\overline{\\cal M}_{g,n})$$ \nwhere $\\lambda_i=c_i(E)$ is the $i$th Chern class of the Hodge bundle $E\\to\\overline{\\cal M}_{g,n}$ defined to have fibres $H^0(\\omega_C)$ over a nodal curve $C$.\n\nHurwitz \\cite{HurRie} studied the problem of connected curves $\\Sigma$ of genus $g$ covering $\\mathbb{P}^1$, branched over $r+1$ fixed points $\\{p_1,p_2,...,p_r,p_{r+1}\\}$ with arbitrary profile $\\mu=(\\mu_1,...,\\mu_n)$ over $p_{r+1}$.  Over the other $r$ branch points one specifies simple ramification, i.e. the partition $(2,1,1,....)$.   The Riemann-Hurwitz formula determines the number $r$ of simple branch points via $2-2g-n=|\\mu|-r$.  \n\\begin{definition} \nDefine the simple Hurwitz number $H_{g,\\mu}$ to be the weighted count of genus $g$ connected covers of $\\mathbb{P}^1$ with ramification $\\mu=(\\mu_1,...,\\mu_n)$ over $\\infty$ and simple ramification elsewhere.\nEach cover $\\pi$ is counted with weight $1\/|{\\rm Aut}(\\pi)|$.\n\\end{definition}\nCoefficients of the partition function of the CohFT $\\Omega_{g,n}=c(E^\\vee)$ appear naturally in the ELSV formula \\cite{ELSVHur} which relates the Hurwitz numbers $H_{g,\\mu}$ to the Hodge classes.  The ELSV formula is:\n\\[ H_{g,\\mu}=\\frac{r(g,\\mu)!}{|{\\rm Aut\\ }\\mu|}\\prod_{i=1}^n\\frac{\\mu_i^{\\mu_i}}{\\mu_i!}\\int_{\\overline{\\cal M}_{g,n}}\\frac{1-\\lambda_1+...+(-1)^g\\lambda_g}{(1-\\mu_1\\psi_1)...(1-\\mu_n\\psi_n)}\\]\nwhere $\\mu=(\\mu_1,...,\\mu_n)$ and $r(g,\\mu)=2g-2+n+|\\mu|$.\n\nUsing $\\Omega^{\\Theta}_{g,n}=\\Theta\\cdot c(E^\\vee)$ we can define an analogue of the ELSV formula:\n$$H^{\\Theta}_{g,\\mu}=\\frac{(2g-2+n+|\\mu|)!}{|{\\rm Aut\\ }\\mu|}\\prod_{i=1}^n\\frac{\\mu_i^{\\mu_i}}{\\mu_i!}\\int_{\\overline{\\cal M}_{g,n}}\\Theta_{g,n}\\cdot\\frac{1-\\lambda_1+...+(-1)^{g-1}\\lambda_{g-1}}{(1-\\mu_1\\psi_1)...(1-\\mu_n\\psi_n)}.\n$$\nIt may be that $H^{\\Theta}_{g,\\mu}$ has an interpretation of enumerating simple Hurwitz covers. \nNote that it makes sense to set all $\\mu_i=0$, and in particular there are non-trivial primary invariants over $\\overline{\\cal M}_g$, unlike for simple Hurwitz numbers. \nAn example calculation:\n$$\\int_{\\overline{\\cal M}_2}\\Theta_2\\lambda_1=\\frac{1}{5}\\cdot\\frac{1}{8}\\cdot\\frac{1}{8}\\cdot\\frac{1}{2}+\\frac{1}{10}\\cdot\\frac{1}{8}\\cdot\\frac{1}{2}=\\frac{1}{128}\\quad\\quad\\Leftarrow\\quad\n\\lambda_1=\\frac{1}{10}(2\\delta_{1,1}+\\delta_{\\text{irr}}).\n$$\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nOur world is bursting with knowledge. Nearly every discipline has been subdivided into numerous sub-disciplines. By July 27 2017, there are 42.5 million entries on Wikipedia\\footnote{\\url{https:\/\/en.wikipedia.org\/wiki\/Wikipedia:Size_of_Wikipedia}}, which, undoubtedly, stands for only a small portion of human knowledge. People learn whenever and wherever possible. People's learning of knowledge is not confined to childhood or the classroom but takes place throughout life and in a range of situations; it can take the form of formal learning or informal learning \\cite{Paradise2009}, such as daily interactions with others and with the world around us. Lifelong learning is the ``ongoing, voluntary, and self-motivated\" pursuit of knowledge for either personal or professional reasons \\cite{Cliath2000}. According to Tough's study, almost 70\\% of learning projects are self-planned \\cite{Tough1979}. \n\nAs people learn eternally, one significant issue is to evaluate how much knowledge an individual is possessing at a particular time. E.g., suppose we have a large database that records all the entries of Wikipedia and a person's understanding degree of each entry (on a scale of 1 to 10). With this information, a lot of new applications are becoming practical. The following are some examples:\n\n\n\\subsubsection*{1. Determine a person's knowledge state}\nIf we set a threshold to the understanding grades, and assume the subject has understood a knowledge entry if its grade is larger than the threshold, then we can determine a person's knowledge state and knowledge composition at a particular time. As Figure 1 illustrates.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.8\\columnwidth]{knowledge-composition}\n\t\\caption{A person's knowledge composition}\n\\end{figure}\n\n\\subsubsection*{2. Discover a person's expertises and deficiencies}\t\nExpertise finding is critical for an organization or project. Since the participation of experts plays an important role for the success of an organization or project. With the understanding evaluation database, it is convenient to discover a person's domain-level expertise and topic-level expertise. E.g., if a person has a question of what is Poincare Conjecture, he do not need to ask a mathematician (domain-level expert), who is not always available. Instead, he can ask one of his friends who is not a mathematician but has topic-level expertise in it. Besides expertise, we can also discover a person's deficiencies in a field, so he can remedy the deficiencies.\n\n\\subsubsection*{3. Make personalized recommendations}\nIf we know a person's understanding degree to each piece of knowledge, it is natural to make personalized recommendations to the subject based on the information. E.g., suppose we know a person has good understanding in the topics of Deep Learning, then we can recommend latest papers about Deep Learning to the subject, or recommend the person as a reviewer of a paper related to Deep Learning. In addition, we can recommend learning materials to the subject to help him practice meaningful learning, which will be discussed in details in Section 4.\n\n\n\\subsection{Procedural and Conceptual Knowledge}\nStudies of knowledge indicate that knowledge may be classified into two major categories: procedural and  conceptual knowledge \\cite{hiebert2013conceptual,mccormick1997conceptual}. Procedural knowledge is the knowledge exercised in the accomplishment of a task, and thus includes knowledge which cannot be easily articulated by the individual, since it is typically nonconscious (or tacit). It is commonly referred to as ``know-how\". Such as knowing how to cook delicious food, how to drive an airplane, or how to play basketball etc. Conceptual knowledge is quite different from procedural knowledge. It involves understanding of the principles that govern a domain and of the interrelations between pieces of knowledge in a domain; it concerns understanding and interpreting concepts and the relations between concepts. It is commonly referred to as ``know-why\", such as knowing why something happens in a particular way. In this article, we only deal with evaluating a person's understanding of conceptual knowledge.\n\n\\subsection{The framework}\nAt present, assessment of one's understanding of conceptual knowledge is primarily through tests \\cite{pisa2000measuring,hunt2003concept,qian2004evaluation} or interviews, which have some limitations such as low-efficiency and not-comprehensive. E.g., it needs other people's cooperation to accomplish the assessment, which is time-consuming; moreover, only a small portion of topics of a domain is evaluated during an assessment, which cannot comprehensively reflect a person's knowledge state of the domain.\n\n\nWe propose a new method named Individual Conceptual Knowledge Evaluation Model (ICKEM) to evaluate one's understanding of a piece of conceptual knowledge quantitatively. It has the advantage of evaluating a person's understanding of conceptual knowledge independently, comprehensively, and automatically. It keeps track of one's daily learning activities (reading, listening, discussing, writing etc.), dividing them into a sequence of learning sessions, then analyzes the text content of each learning session to discover the involved knowledge topics and their shares in the session. Then a learning experience is inserted to the involved knowledge topics' learning histories (It maintains a leaning history for \\emph{each} knowledge topic). Therefore, after a period of time (e.g., several years or decades of years), a knowledge topic's leaning history that records the subject's each leaning experience about the topic is generated. Based on the learning history, the subject's familiarity degree to a knowledge topic is evaluated. Finally, it estimates one's understanding degree to a topic, by comprehensively evaluating one's familiarity degrees to the topic itself and other topics that are essential to understand the topic. Figure 2 is the framework of ICKEM. Each hexagon of the diagram indicates a processing step; the following rectangle indicates the results of the process.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.9\\columnwidth]{2flowchart}\n\t\\caption{The framework of ICKEM}\n\\end{figure}\n\nThe remainder of this paper is organized as follows. Section 2 discusses how to calculate a person's familiarity degree to a knowledge topic. Section 3 introduces a method to estimate a person's understanding degree to a topic, by checking the familiarity degrees of the topic's understanding tree. It also presents an algorithm to generate a knowledge topic's understanding tree. In Section 4, we introduce an utilization of ICKEM, computer-aided incremental meaningful learning (CAIML), which helps an individual practice meaningful learning. Section 5 discusses related issues about evaluating one's understanding of conceptual knowledge. We cover related work in Section 6, before concluding in Section 7. \n\n\n\\section{Evaluate familiarity degree}\nThis section introduces the procedures that are devised to evaluate a person's familiarity degree to a knowledge topic. It starts by presenting a formal definition of knowledge and learning, then discusses how to divide a person's daily learning activities into a series of learning sessions, and analyze the text learning content to obtain a topic's share in a session. After these procedures, a knowledge topic's learning history can be generated. Finally, based on the learning history, the subject's familiarity degree to a topic is calculated.\n\n\\subsection{Definition of knowledge and learning}\nKnowledge is conventionally defined as beliefs that are true and justified. To be `true' means that it is in accord with the way in which objects, people, processes and events exist and behave in the real world. However, exactly what evidence is necessary and sufficient to allow a true belief to be `justified' has been a topic of discussion (largely among philosophers) for more than 2,000 years \\cite{hunt2003concept}. In ICKEM, a Knowledge Point is defined as a piece of conceptual knowledge that is explicitly defined and has been widely accepted, such as Bayes' theorem, Euler's formula, mass-energy equivalence, Maxwell's equations, gravitational wave, and the expectation-maximization algorithm etc.\n\nLearning is the process of acquiring, modifying, or reinforcing knowledge, behaviors, skills, values, or preferences in memory \\cite{terry2015learning, hunt2003concept, bransford1999people}. An individual's possessing of knowledge is the product of all the experiences from the beginning of his\/her life to the moment at hand \\cite{hunt2003concept,benassi2014applying}. Learning produces changes in the organism and the changes produced are relatively permanent \\cite{schacter2011psychology}.\n\n\\subsection{Discriminate learning sessions}\nIn ICKEM, a person's daily activities are classified into two categories: learning activities and non-learning activities. An activity is recognized as a learning activity if its content involves at least one Knowledge Point of a predefined set. In addition, the learning activities are divided into a sequence of learning sessions, since it is essential to know how many times and how long for each time the individual has learned a Knowledge Point. An individual can employ many methods to learn conceptual knowledge, such as reading, listening, discussing, and writing. Different strategies should be used to discriminate learning sessions for different learning methods. E.g., for leaning by reading documents on a computer, Algorithm 1 is devised to discriminate learning sessions. For other learning methods, it is more complicated to divide learning sessions. However, there are already some attempts for detecting human daily activities \\cite{Ni2013,Piyathilaka2013,Sung2012}. Sung et al. devised an algorithm for recognizing human daily activities from RGB-D Images \\cite{Sung2012}. They tested their algorithm on detecting and recognizing twelve different activities (brushing teeth, cooking, working on computer, talking on phone, drinking water, talking on a chair etc.) performed by four people and achieved good performance.\n\nAlgorithm 1 works by periodically checking the occurrence of the following events: \n\\begin{enumerate}\n\t\\item A document is opened;\n\t\\item The foreground window has switched to an opened document from another application (APP), such as a game APP;\n\t\\item After the computer being idled for a period of time, there are some mouse or keyboard inputs detected, which indicates the individual has come back from other things. Meanwhile, the foreground window is an opened document;\n\t\\item A document is closed;\n\t\\item The foreground window has switched to another APP from a document;\n\t\\item The foreground window is a document, but the computer has idled for a certain period of time without any mouse or keyboard inputs detected, the individual is assumed to have left to do other things.\n\\end{enumerate}\n\nOccurrence of Event 1, 2, or 3 indicates a learning session has started; if Event 4, 5, or 6 occurred, a learning session is assumed being terminated. The duration of a learning session equals the interval between its start and stop time. \n\n\\begin{algorithm}\n\t\\caption{An algorithm to discriminate one's learning sessions when reading.}\n\t\\label{alg1}\n\t\\begin{algorithmic}[1]\t\t\t\n\t\t\\WHILE{The Reader APP is running}\t\t\n\t\t\\IF{Event 4 \\textbf{OR} Event 5 \\textbf{OR} Event 6 occurred}\t\t\t\t\t\n\t\t\\STATE Record that a document's learning session has stopped;\n\t\t\\ELSE\n\t\t\\IF{Event 1 \\textbf{OR} Event 2 \\textbf{OR} Event 3 occurred}\t\t\t\n\t\t\\STATE Record that a learning session about the current document has started;\n\t\t\\ENDIF\n\t\t\\ELSE\n\t\t\\IF{There is no APP and document switch}\t\t\t\n\t\t\\STATE Check and record if there is a Page switch;\n\t\t\\ENDIF\t\t\t\n\t\t\\ENDIF\n\t\t\\STATE Keep silent for $ T $ seconds;\n\t\t\\ENDWHILE\t\t\n\t\\end{algorithmic}\t\n\\end{algorithm}\n\nFigure 3 shows some examples of discriminated learning sessions. Attribute ``\\textit{did}'' means document ID, which indexes a document uniquely. Attribute ``\\textit{actiontype}'' indicates the type of an action. ``\\textit{Doc Act}'' means a document has been activated. ``\\textit{Page Act}'' is defined  similarly. ``\\textit{Doc DeAct}'' means a document has been deactivated. That is to say, a learning session has stopped. Attribute ``\\textit{page}'' indicates a page number. Attribute ``\\textit{duration}'' records how long a page has been activated in seconds. If two learning sessions' interval is less than a certain threshold, such as 30 minutes, and their learning material is the same (e.g., the same document), they are merged into one session. Therefore, ``Session 2'' and ``Session 3'' are merged into one session.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.8\\columnwidth]{4learningsession}\n\t\\caption{Some examples of discriminated learning sessions.}\n\\end{figure}\n\n\\subsection{Capture the text learning content}\n\nMost learning processes are associated with a piece of learning material. E.g., reading a book, the book is the learning material; attending a course or discussion, the course and discussion contents can be regarded as the learning material. Some learning materials are text or can be converted to text. E.g., discussing a piece of knowledge with others. The discussion contents can be converted to text by exploiting Speech Recognition. Similarly, if one is reading a printed book, the contents of the book can be captured by wearable computers like Google Glass and then converted to text through Optical Character Recognition (OCR). If the book is electronic, no conversion is needed; the text can be extracted directly. Since Algorithm 1 records the accurate set of pages a person has read during a learning session, only the text content of related pages is extracted.\n\n\\subsection{Calculate a Knowledge Point's share}\nA learning session may involve many topics, it is necessary to know how much a learning session involves a topic. To calculate a Knowledge Point's share, maybe the simplest way is to deem the text learning content as a bag of words, and calculate a Knowledge Point's share based on its Term Frequency (TF) or normalized TF. Just like Equation 1 and 2. $ N_{i} $ is term $ i $'s normalized TF. It is calculated with Equation 1, where $ T_{i} $ is term $ i $'s TF, $ Max(TF) $ is the maximum TF of the captured text content, $ \\alpha $ is a constant regulatory factor.\n\n\\begin{equation}\n\tN_{i} = \\alpha + (1 - \\alpha) * T_{i} \/ Max(TF)\n\\end{equation}\n\\begin{equation}\n\t\\xi_{i} = \\frac{N_{i}}{\\sum_{j=1}^mN_{j}}\n\\end{equation}\n\nAnother method of discovering a Knowledge Point's share is to analyze the text learning content with topic model. A topic model is a type of statistical model for discovering the abstract \"topics\" that occur in a collection of documents. Topic model is a frequently used text-mining tool for discovery of hidden semantic structures in a text body \\cite{blei2012probabilistic,steyvers2007}. An early topic model was described in \\cite{Papadimitriou1998}. Another one, called probabilistic latent semantic analysis (PLSA), was created by Hofmann in 1999 \\cite{hofmann1999probabilistic}. Latent Dirichlet allocation (LDA) is the most common topic model currently in use \\cite{blei2003latent}. It is a generalization of PLSA. It introduces sparse Dirichlet prior distributions over document-topic and topic-word distributions. Other topic models are generally extensions on LDA.\n\nThe inputs of a probabilistic topic model are a collection of $ N $ documents, a vocabulary set $ V $, and the number of topics $ k $. The outputs of a probabilistic topic model are the following:\n\n\\begin{itemize}\n\t\\item k topics, each is word distribution : $ \\{\\theta_{1},...,\\theta_{k}\\} $;\n\t\\item Coverage of topics in each document $ d_{i} $: $ \\{\\pi_{i1},...,\\pi_{ik}\\} $;\\\\\n\t$ \\pi_{ij} $ is the probability of document $ d_{i}$ covering topic $ \\theta_{j} $.\t\n\\end{itemize}\n\nThe subject's $ N $ learning sessions'  text-contents can be deemed as a collection of $ N $ documents, then the document collection is analyzed with a topic model like LDA.\nBased on the outputs of topic model, Equation 3 is used to calculate a Knowledge Point's share in a learning session. Only the top $ m $ terms of each topic are considered. For document $ d $, $ \\varphi_{ij} $ is the share of term $ i $ which comes from topic $ j $, $ \\pi_{j} $ is topic $j $'s share of document $ d $, $ p(t_{i}|\\theta_{j}) $ is term $ i $'s share of topic $ j $. A Knowledge Point's share equals its term share. Therefore, with topic model, Knowledge Points involved in each learning session can be discovered; their shares can also be calculated.\n\n\\begin{equation}\n\\varphi_{ij} = \\frac{\\pi_{j}p(t_{i}|\\theta_{j})}{\\sum_{j=1}^k\\sum_{i=1}^m\\pi_{j}p(t_{i}|\\theta_{j})}\n\\end{equation}\n\n\n\\subsection{The subject's state during a session}\nThe subject's physical and psychological status may influence the effect of a learning session. E.g., if the subject is tired, or severely injured, the learning effect is usually worse than being in a normal state. Similarly, if the subject is in a state of severe depression, anxiety, or scatterbrained, the learning effect cannot be good either. One issue is how to collect values for these attributes. Fortunately, there are already systems for monitoring a person's physical and psychological status \\cite{Yang2005,Cheng2012,DeRemer2015,Kurniawan2013,Lin2014}. Another issue is studies need to be conducted to decide how these attributes would affect the learning effect. Preferable to have some equations to calculate a ``physical and psychological status factor\", describing how the learning effect is affected by the subject's physical and psychological state during a session.\n\nOn the other hand, if we just want a rough estimation of the subject's familiarity degree to a Knowledge Point, these physical and psychological status attributes may be ignored. Since we are observing the subject in a long time rang, typically several years or decades of years, we can assume the subject is in ``normal'' state most of the time, and ignore its fluctuation.\n\n\\subsection{Effectiveness of a learning method}\nDifferent learning methods may have different levels of effectiveness. E.g., learning by reading a book and learning by discussing with others, do these two methods have equivalent levels of memory retention after the learning experience? (supposing other conditions are equivalent, such as learning content, session duration, physical and psychological status etc.) These is no consensus about the effectiveness of different learning methods. The National Training Laboratories (NTL) Institute proposes a \\emph{learning pyramid} model, which maps a range of learning methods onto a triangular image in proportion to their effectiveness in promoting student retention of the material taught \\cite{sousa2001brain,magennis2005teaching}. However, the credibility and research base of the model have been questioned by other researchers \\cite{lalley2007learning,letrud2012rebuttal}. Further research is necessary to make a systematic comparison between the effectiveness of different learning methods. Preferable to have a ``learning method factor\" depicting the effectiveness of a method. Similarly, if we just want a rough estimation of the subject's familiarity degrees, we can omit the difference among different learning methods. As \\cite{lalley2007learning} concluded, there is no learning method being consistently superior to the others and all being effective in certain contexts. \\cite{lalley2007learning} also stressed the importance of reading: reading is not only an effective learning method, it is also the main foundation for becoming a ``life-long learner\".\n\n\\subsection{A Knowledge Point's learning history}\nWith the discriminated learning sessions, a Knowledge Point's share in each session, the subject's physical and psychological status during a session, and the ``learning method factor\", after a period of time, the subject's learning histories about each Knowledge Point can be generated. Figure 4 shows an exemplary learning history. It records a person's each learning experience about a Knowledge Point. ``LCT\" stands for ``learning cessation time\". It is used to calculate the interval between the learning time and the evaluation time, which is then used to estimate how much information has been lost due to memory decay. ``Duration\" is the length of a learning session. ``Proportion\" is the Knowledge Point's share during a learning session. ``PPS factor\" stands for the ``physical and psychological status factor\". It is a number between 0 and 1 that is calculated based on the subject's average physical and psychological status during a session. ``LM factor\" stands for the ``learning method factor\". It is also a number between 0 and 1 that is allocated to a learning method according to its effectiveness level.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=1.0\\columnwidth]{LHIS-vldb}\n\t\\caption{An exemplary learning history of a Knowledge Point}\n\\end{figure}\n\n\\subsection{Memory retention of a learning experience}\nPeople learn all the time; meanwhile, people forget all the time. Human memory declines over time. Interestingly, most researchers report there is no age difference for memory decay \\cite{rubin1996one}. To calculate the familiarity degree, we need to address how the effect of a learning experience decays over time. However, there is no consensus of how human memory decays. Psychologists have suggested many functions to describe the monotonic loss of information with time \\cite{rubin1996one}. But there is no unanimous agreement of which one is the best. The search for a general description of forgetting is one of the oldest unresolved problems in experimental psychology \\cite{averell2011form}. \n\nWe propose to use Ebbinghaus' forgetting curve equation \\cite{ebbinghaus1913memory} to describe the memory retention of a learning experience over time. Since it is the most well-known forgetting curve equation and its soundness has been proved by many studies \\cite{murre2015replication,finkenbinder1913curve,heller1991replikation}.\nEbbinghaus found Equation 4 can be used to describe the proportion of memory retention after a period of time\\footnote{It can be found at \\url{http:\/\/psychclassics.yorku.ca\/Ebbinghaus\/memory7.htm}}, where $ t $ is the time in minutes counting from one minute before the end of the learning, $ k $ and $ c $ are two constants that equal 1.84 and 1.25, respectively. Figure 5 shows the percentage of memory retention over time calculated by Equation 4. The Y axis is the percentage of memory retention; the X axis is the time-since-learning in minutes. It can be seen that memory retention declines drastically during the first 24 hours (1,440 minutes), then the speed tends to be steady.\n\n\\begin{equation}\nb(t) = k\/((\\log t)^c + k)\n\\end{equation}\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.8\\columnwidth]{3forgettingcurve}\n\t\\caption{The percentage of memory retention over time calculated by Equation 4}\n\\end{figure}\n\n\\subsection{Calculate the familiarity degree}\n\nBased on the learning history, Equation 5 is utilized to calculate the subject's Familiarity Measure to a Knowledge Point  $ k_{i} $ at time $ t $. A Familiarity Measure is defined as a score that depicts a person's familiarity degree to a Knowledge Point. Its unit is defined as \\emph{gl}. The input is $ k_{i} $'s learning history--a sequence of $ m $ learning sessions (like Figure 4). $ d_{j} $ is session $ j $'s duration; $ \\xi_{ij} $ is Knowledge Point $ k_{i} $'s share in session $ j $; $ t_{j} $ is session $ j $'s ``learning cessation time\", $ b(t-t_{j}) $ calculates the percent of memory retention of session $ j $ at time $ t $ with Equation 4; $ F^{pps}_{j} $ is the ``physical and psychological status factor\" of session $ j $; $ F^{lm}_{j} $ is the ``learning method factor\" of session $ j $.\n\n\\begin{equation}\n\tF(k_{i},t) = \\sum_{j=1}^md_{j}*\\xi_{ij}*b(t-t_{j})*F^{pps}_{j}*F^{lm}_{j}\n\\end{equation}\n\nThe computation hypothesizes each learning experience about a Knowledge Point contributes some effect to the subject's current understanding of it, and the learning effect declines over time according to Ebbinghaus' forgetting curve. Other attributes (the subject's physical and psychological status, learning method) that may affect the learning effect are counted in as numeric factors. The Familiarity Measure of a Knowledge Point is calculated as the additive effects of all the learning experiences about it. If we omit the influence of subject physical and psychological status and the difference among learning methods, Equation 5 can be simplified to Equation 6.\n\n\\begin{equation}\n\tF(k_{i},t) = \\sum_{j=1}^md_{j}*\\xi_{ij}*b(t-t_{j})\n\\end{equation}\n\nTable 1 compares Familiarity Measures calculated by Equation 5 and Equation 6 at different times based on the learning history of Figure 4. The evaluation times are an hour, a day, a month, a year, and 10 years after last learning of the knowledge point.\n\n\\begin{table*}\n\t\\centering\n\t\\begin{tabular}{ c  c  c  c c c }\n\t\t\\hline\n\t\t& 5\/17\/2016 16:25 & 5\/18\/2016 15:25 & 6\/17\/2016 15:25 &5\/17\/2017 15:25 & 5\/17\/2026 15:25\\\\ \\hline\n\t\tEquation 5 & 25\t&23.5&\t21.9&\t19.4&\t16.6\\\\ \\hline\t\t\n\t\tEquation 6 & 42.6&\t40.2&\t37.6\t&33.3\t&28.5\\\\  \\hline\n\t\\end{tabular}\n\t\\caption{Familiarity Measures calculated at different times}\n\\end{table*}\n\n\n\\section{Estimate understanding degree}\nUnderstanding is quite subtle to measure. We hypothesize that if a person is familiar with a Knowledge Point itself and all the background knowledge that is essential to understand it, he should have understood the Knowledge Point. Because Familiarity Measure depicts the cumulated effects of one's learning experiences about a topic, high levels of Familiarity Measures imply intensive learning activities about the suite of knowledge topics. Intensive learning activities usually result in a good understanding.\n\nThe background Knowledge Points that are essential to understand a Knowledge Point can be extracted by analyzing its definition. Table 2 lists eight reduced documents, each of them is a definition of a Knowledge Point in Probability Theory or Stochastic Process, the texts are quoted from Wikipedia and other websites. The third column of Table 2 lists the involved Knowledge Points in the documents, which are deemed as the background knowledge to understand the host Knowledge Point.\n\n\\begin{table*}\n\t\\centering\n\t\\begin{tabular}{|C{1cm}|L{12cm}|C{2.5cm}|}\n\t\t\\hline\n\t\tDoc & \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad Content & Knowledge Points  \\\\ \\hline\n\t\tD1 & A Strictly Stationary Process (SSP) is a Stochastic Process (SP) whose Joint Probability Distribution (JPD) does not change when shifted in time. &  SSP, JPD,\\qquad\\qquad Time, SP \\\\ \\hline\n\t\t\\qquad \\qquad \\qquad \\qquad\\qquad \\qquad D2 &  A Stochastic Process (SP) is a Probability Model (PM) used to describe phenomena that evolve over time or space.  In probability theory, a stochastic process is a Time Sequence (TS) representing the evolution of some system represented by a variable whose change is subject to a Random Variation (RaV).  &  SP, PM, TS, Time, Space, System,\\qquad \\qquad Variable, RaV \\\\ \\hline\n\t\t\\qquad \\qquad \\qquad\\qquad \\qquad \\qquad D3 &  In the study of probability, given at least two Random Variables (RV) X, Y, ... that are defined on a Probability Space (PS), the Joint Probability Distribution (JPD) for X, Y, ... is a Probability Distribution (PD) that gives the probability that each of X, Y, ... falls in any particular range or discrete set of values specified for that variable.  &  JPD, RV,\\qquad\\qquad PS, PD,\\qquad \\qquad Variable, Probability \\\\ \\hline\n\t\t\\qquad \\qquad D4 &  A Probability model (PM) is a mathematical representation of a random phenomenon. It is defined by its Sample Space (SS), events within the SS, and probabilities associated with each event.  &  PM, SS,\\qquad\\qquad Event, Probability \\\\ \\hline\n\t\t\n\t\tD5 &  In probability and statistics, a Random variable (RV) is a variable quantity whose possible values depend, in some clearly-defined way, on a set of random events.  &  RV, Variable, Event \\\\ \\hline\n\t\t\\qquad \\qquad D6 &  A Probability Space (PS) is a Mathematical Construct (MC) that models a real-world process consisting of states that occur randomly. It consists of three parts: a Sample Space (SS), a set of events, and the assignment of probabilities to the events.  &  PS, MC, SS, Probability, Event \\\\ \\hline\n\t\tD7 & A Probability Distribution (PD) is a table or an equation that links each outcome of a statistical experiment with its probability of occurrence. &  PD,\\qquad\\qquad Probability \\\\ \\hline\n\t\tD8 & The Sample Space (SS) is the set of all possible outcomes of the samples. &  SS, Sample \\\\  \t \n\t\t\\hline \n\t\\end{tabular}\n\t\\caption{A list of documents and their involved Knowledge Points}\n\\end{table*}\n\nAn Understanding Tree is a treelike data structure which compiles the background Knowledge Points that are essential to understand the root Knowledge Point. Figure 6 illustrates four Understanding Trees based on the definitions of Table 2. The nodes of the tree can be further interpreted by other Knowledge Points until they are Basic Knowledge Points (BKP). A BKP is a Knowledge Point that is simple enough so that it is not interpreted by other Knowledge Points. Figure 7 shows a fully extended Understanding Tree based on the definitions of Table 2. Figure 8 shows an exemplary Understanding Tree with all the redundant nodes eliminated. Each node is tagged with a Familiarity Measure calculated with Equation 5 or 6. The leaf nodes of Figure 7 and Figure 8 are BKPs. The height and number of nodes of an Understanding Tree characterize the complexity degree of it. Understanding Tree can be used to evaluate a person's topic-level expertise, such as evaluating a person's understanding to a specific algorithm like the Quicksort algorithm.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.99\\columnwidth]{12example-trees}\n\t\\caption{Some examples of Understanding Trees}\n\\end{figure}\n\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=1.3\\columnwidth]{9full}\n\t\\caption{A fully extended Understanding Tree}\n\\end{figure*}\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=1.3\\columnwidth]{10full-concise-scores}\n\t\\caption{A standard Understanding Tree tagged with Familiarity Measures}\n\\end{figure*}\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=1.3\\columnwidth]{10full-concise-percent}\n\t\\caption{Familiarity Measures transformed into percentages}\n\\end{figure*}\n\n\\subsection{Calculation of understanding degree}\nIf all the Familiarity Measures of an Understanding Tree exceed a threshold (such as 100), it is assumed that the person has understood the root Knowledge Point. Then the Knowledge Point is classified as ``Understood''; otherwise, it is classified as ``Not Understood''. Due to the differences of people's intelligence and talent, different people may have different thresholds. \n\nIf a Familiarity Measure is less than the threshold, a percentage is calculated by dividing it by the threshold, indicating the subject's percent of familiarity of the node; if the Familiarity Measure is greater than the threshold, the percentage is set to 1, implying the quantity of familiarity of this node is large enough for understanding the root, extra familiarity is good but not necessary. Equation 7 is used to calculate the percentage of familiarity, $ F(k_{i},t) $ is the subject's Familiarity Measure to Knowledge Point $ k_{i} $ at time $ t $, $ f_{T} $ is the threshold.\n\nIf a Knowledge Point is classified as ``Not Understood'', a percent of understanding is calculated with Equation 8. $ PU(k_{r},t) $ is the subject's percent of understanding of the root Knowledge Point at time $ t $, $ PF(k_{r},t) $ is the percent of familiarity of the root, $ \\frac{1}{m}\\sum_{j=1}^mPF(k_{j},t) $ calculates the average percent of familiarities of its descendants (not including the root). E.g., Figure 9 is the Understanding Tree of Figure 8 with Familiarity Measures transformed into percentages (assuming $ f_{T} $ equals 100). The $ PF(k_{r},t) $ of it equals 85\\%, and $ \\frac{1}{m}\\sum_{j=1}^mPF(k_{j},t) $ equals 89\\%, so the $ PU(k_{r},t) $ equals 76\\%, indicating the subject's understanding of the root Knowledge Point is 76\\%. If $ PU(k_{r},t) $ is less than 100\\%, the Knowledge Point is classified as ``Not Understood\". Thus it can be seen that we are using a conservative strategy for estimating the subject's understanding. For a Knowledge Point to be classified as ``Understood\", the subject must be familiar with \\emph{every} node of its Understanding Tree. This strategy should guarantee a good precision of the Knowledge Points that are classified as ``Understood\", and thus guarantees the quality of the estimated topic-level expertise of the subject. \n\n\n\\begin{equation}  \n\tPF(k_{i},t)=\\left\\{\n\t\\begin{array}{cl}\n\t\t 1                  &    F(k_{i},t) \\geq f_{T} \\\\\n\t\t F(k_{i},t)\/f_{T}   &    F(k_{i},t) < f_{T}\n\t\\end{array}\n\t\\right.  \n\\end{equation}\n\n\\begin{equation}\n\tPU(k_{r},t) = PF(k_{r},t)*\\frac{1}{m}\\sum_{j=1}^mPF(k_{j},t)\n\\end{equation}\n\n\nIf $ PU(k_{r},t) $ equals 100\\%, the subject is assumed having understood the root Knowledge Point, then the average Familiarity Measure of the Understanding Tree divided by the threshold features the magnitude of understanding. Since people usually are very familiar with the BKPs, it may be preferable to exclude them or normalize their effects when computing the average Familiarity Measure. \n\n\\subsection{Construction of Understanding Tree}\nAn Understanding Tree can be constructed manually by a group of experts, or generated automatically by machines. Algorithm 2 shows the steps to generate an Understanding Tree automatically. Table 3 illustrates three definitions of the Central Limit Theorem (CLT)\\footnote{They can be found at \\url{http:\/\/www.math.uah.edu\/stat\/sample\/CLT.html}, \\url{http:\/\/sphweb.bumc.bu.edu\/otlt\/MPH-Modules\/BS\/BS704_Probability\/BS704_Probability12.html}, \\url{http:\/\/stattrek.com\/statistics\/dictionary.aspx}}, the third column lists the involved Knowledge Points in each definition. According to the rule mentioned in Algorithm 2, the child nodes of CLT are selected as \\emph{sample}, \\emph{distribution}, \\emph{mean}, \\emph{independent}, and \\emph{normal} (Knowledge Points that are not BKPs can be further extended). To be safe, the generated Understanding Tree can be inspected by human experts. Understanding Tree is a static structure, once constructed, it is not likely to change them.\n\n\\begin{algorithm} \n\t\\caption{ An algorithm to construct Understanding Tree}  \n\t\\begin{algorithmic}[1]\n\t\t\\REQUIRE ~~\\\\\n\t\t\n\t\tThe set of all Knowledge Points, $ \\Omega $;\\\\\n\t\tThe set of all BKPs, $ B $;\\\\\n\t\tThe root Knowledge Point, $ R_k $;\n\t\t\n\t\t\\ENSURE ~~\\\\\n\t\t$ R_k $'s Understanding Tree;\n\t\t\\STATE Search definitions of $ R_k $ from a library of authoritative documents;\n\t\t\\STATE Discover involved Knowledge Points for each definition;\n\t\t\\STATE Select Knowledge Points according to some rules, e.g., more than half of the definitions have referenced the Knowledge Point; \n\t\t\\STATE Recursively extend non-BKP Knowledge Points that are selected;\n\t\t\\RETURN All the selected Knowledge Points;\t\t\n\t\\end{algorithmic}\n\\end{algorithm} \n\n\\begin{table*}\n\t\\centering\n\t\\begin{tabular}{|C{1cm}|L{12cm}|C{3.5cm}|}\n\t\t\\hline\n\t\t& \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad Content & Knowledge Points  \\\\ \\hline\n\t\t\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad 1 & The Central Limit Theorem (CLT) states that the sampling distribution of the mean of any independent, random variable will be normal or nearly normal, if the sample size is large enough. &  CLT,  sample, distribution, mean, independent, random variable, normal, size \\\\ \\hline\n\t\t\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad 2 & The Central Limit Theorem (CLT) states that the distribution of the sum (or average) of a large number of independent, identically distributed variables will be approximately normal, regardless of the underlying distribution.  &  CLT,  distribution, sum,  average, independent, variable, normal \\\\ \\hline\n\t\t\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad 3 &  The Central Limit Theorem (CLT) states that if you have a population with mean $ \\mu $ and standard deviation $ \\sigma $ and take sufficiently large random samples from the population with replacement, then the distribution of the sample means will be approximately normally distributed.  &  CLT,  population, standard deviation, random, replacement, distribution, sample, mean, normal \\\\  \n\t\t\\hline \n\t\\end{tabular}\n\t\\caption{Three definitions of the Central Limit Theorem (CLT)}\n\\end{table*}\n\n\\section{Facilitate meaningful learning}\nKnowing one's understanding degrees of knowledge helps practice meaningful learning. Meaningful learning is the concept that learned knowledge is fully understood to the extent that it relates to other knowledge, implies there is a comprehensive knowledge of the context of the facts learned \\cite{Mayer2002}. ICKEM helps people practice Computer-aided Incremental Meaningful Learning (CAIML). CAIML is defined as a strategy of letting the computer estimate a person's current knowledge states, then recommend the optimum learning material for the subject to accomplish a meaningful learning. The optimum material introduces some new knowledge blended with old knowledge the subject has known, meanwhile, interpreting the new knowledge.\n\nFor example, a college student, who has comprehended the basic knowledge of Advanced Mathematics and Computer Science, wants to be an expert of Artificial Intelligence (AI) by teaching himself. A professor recommends him 1,000 documents related to AI, and asserts if he can fully understand the contents of the documents, he will be an expert of AI. The question is: what is the best sequence for the student to learn the documents? Some documents are easy to understand, and should be read in the beginning; some documents are intricate, learning them in the first place is painful and frustrating, and should be put at the end of the learning process. If a computer knows a person's knowledge states at any time, it is not difficult to make the learning plan, and recommend the optimum document for current learning.\n\\subsection{An algorithm to facilitate CAIML}\nAlgorithm 3 is devised to facilitate CAIML. It recommends the optimum document for the subject to learn, by analyzing his current knowledge states. It searches for the document which has the least Knowledge Points that are not understood by the subject, which implies it is the easiest document to understand at present. In the document, the ``Understood\" Knowledge Points server as interpretations to the ``Not Understood\" ones when learning the document. Therefore, the algorithm facilitates a meaningful learning.\n\n\\begin{algorithm}\n\t\\caption{ An algorithm to recommend the optimum document(s) for CAIML}  \n\t\\begin{algorithmic}[1]\n\t\t\\REQUIRE ~~\\\\\n\t\t\n\t\tThe set of documents to be learned;\\\\\n\t\tThe set of all Knowledge Points, tagged with the subject's Familiarity Measures to them;\\\\\n\t\tThe set of all non-BKP Knowledge Points' Understanding Trees;\n\t\t\n\t\t\\ENSURE ~~\\\\\n\t\tThe optimum document(s) for current learning to practice a meaningful learning;\n\t\t\\STATE Extract involved Knowledge Points in each document, classify them as ``Understood\" and ``Not Understood\", according to the rule mentioned in Section 3.1;\n\t\t\\STATE Count the number of ``Not Understood\" Knowledge Points for each document;\n\t\t\\RETURN The document(s) with the least ``Not Understood\" Knowledge Points;\t\t\n\t\\end{algorithmic}\n\\end{algorithm}\n\nAn alternative method to recommend the optimum document is to estimate the subject's anticipated percentage of understanding of each document, then return the one that is most approaching 100\\%. A person's understanding of a document is calculated with Equation 9. Suppose the document involves $ n $ Knowledge Points, $ PU(d,t) $ is the percent of understanding of the document at time $ t $; $ \\xi_{i} $ is Knowledge Point $ k_{i} $'s share of the document, its calculation has been discussed in Section 2.4; $ PU(k_{i},t) $  is Knowledge Point $ k_{i} $'s percent of understanding at time $ t $. If a person has understood all the Knowledge Points the document contains, its $ PU(d,t) $ is 100\\%. It is possible that a person learns a ``100\\% understood\" document to strengthen his knowledge. As Equation 9 can estimate a person's understanding degree to document, it can also be used to recommend appropriate reviewers for a paper.\n\n\\begin{equation}\n\tPU(d,t)  = \\frac{\\sum_{i=1}^n\\xi_{i}*PU(k_{i},t)}{\\sum_{i=1}^n\\xi_{i}}\n\\end{equation}\n\n\nPeople's memory fades away over time, the Familiarity Measures decrease accordingly. It is also possible that the computer recommends a document that had been tagged ``100\\% understood\", because the $ PU(d,t) $ has abated.\n\\subsection{An example of CAIML}\nHere is an example to illustrate the logic of CAIML. Suppose a person wants to fully understand the suite of documents listed in Table 2 by learning them. Figure 7 shows all the Knowledge Points involved in the documents, the third column of Table 2 lists ones referenced by each of them. The subject is assumed to have understood the BKPs before the beginning of the learning, which are the leaf nodes of Figure 7 (if not, he can learn them first). Table 4 shows the number of ``Not Understood\" Knowledge Points before the beginning of the learning and after each learning, for each document. E.g., there are 8 ``Not Understood\" Knowledge Points before the starting of the learning for D1, they are SSP, SP, JPD, PM, SS, RV, PS, and PD. According to Algorithm 3, the subject should first learn either of D5, D7 or D8. After learning of D5, the subject is assumed to have understood RV. Therefore, the number of ``Not Understood\" Knowledge Points for D1 becomes 7. The first column of Table 4 suggests one of the optimum learning sequences of the documents, which is D5, D8, D4, D2, D7, D6, D3, and D1.\n\n\n\\begin{table}\n\t\\centering\n\t\\begin{tabular}{ c  c  c  c  c  c  c  c  c }\n\t\t\\hline\n\t\t& D1 & D2 & D3 & D4 & D5 & D6 & D7 & D8\\\\ \\hline\n\t\tBefore starting & 8 & 3 & 5 & 2 & 1 & 2 & 1 & 1\\\\ \\hline\n\t\tD5 & 7 & 3 & 4 & 2 & 0 & 2 & 1 & 1\\\\ \\hline\n\t\tD8 & 6 & 2 & 3 & 1 & 0 & 1 & 1 & 0\\\\ \\hline\n\t\tD4 & 5 & 1 & 3 & 0 & 0 & 1 & 1 & 0\\\\ \\hline\n\t\tD2 & 4 & 0 & 3 & 0 & 0 & 1 & 1 & 0\\\\ \\hline\n\t\tD7 & 3 & 0 & 2 & 0 & 0 & 1 & 0 & 0\\\\ \\hline\n\t\tD6 & 2 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\\\ \\hline\n\t\tD3 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\ \\hline\n\t\tD1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\   \t \n\t\t\\hline \n\t\\end{tabular}\n\t\\caption{An example of CAIML}\n\\end{table}\n\n\n\\section{Discussion}\nAssessing a person's understanding of conceptual knowledge is not easy; as an experimental model aimed to accomplish this, a great deal of thought and research are required to realize its potential.\n\n\\subsection{Dealing with similar Knowledge Points}\nThere is a problem of how to deal with Knowledge Points that are different but have little distinction, such as ``random variation\" and ``random variable\". One solution is to homogenize them to the same Knowledge Point; another is to compensate one Knowledge Point's Familiarity Measure with others' Familiarity Measures, because learning others helps to understand it. The compensation can be calculated with Equation 10. $ F_{k_{i}} $ is Knowledge Point $ k_{i} $'s Familiarity Measure, each of its sibling contributes $ 1\/c_{j} $ of its Familiarity Measure to $ k_{i} $ ($ c_{j} $ is coefficient to be determined). \n\n\\begin{equation}\n\tF_{k_{i}\\_new} = F_{k_{i}\\_old} + \\sum_{j=1}^m{\\frac{1}{c_{j}}}F_{k_{j}}\n\\end{equation}\n\n\\subsection{Trade-offs of evaluating one's understanding of conceptual knowledge}\nQuantitatively assessing one's understanding of conceptual knowledge seems to be a good thing, but there are risks that it introduces some harmful effects. For example, if the Familiarity Measures and understanding degrees calculated are inaccurate, it may lead to wrong decisions. In addition, it cannot detect a person's talent and potential in a field. On the other hand, traditional exams or interviews have their limitations. For example, it needs other people's cooperation to accomplish the evaluation; it only assesses one's knowledge in a particular field at a time, and the evaluation is not comprehensive. Since it only assesses questions being asked, not all of the topics in a field are evaluated. ICKEM assesses one's knowledge independently, comprehensively, and automatically. Therefore, the methods of evaluating one's understanding of knowledge should be used cooperatively, complementing one another.\n\n\\subsection{Privacy issues}\nRecording one's learning history will inevitably violates his privacy. To protect privacy, the learning histories can be password protected or encrypted and stored in personal storage; they should not be revealed to other people. The only information that can be viewed by the outside world is the individual's knowledge measures of some Knowledge Points. The Knowledge Points that may involve privacy are separated from others; every output of them should be authorized by the owner. A person may choose to keep all of his knowledge measures private.\n\n\\subsection{The success criteria for ICKEM}\nFor ICKEM to be a successful model of evaluating a person's understanding of conceptual knowledge, it should have a good performance of discovering a person's domain-level expertise and topic-level expertise. To evaluate its performance, firstly, a group of subjects' learning histories over a long time (several years or decades of years) should be recorded.\n\nFor evaluating its ability of discovering domain-level expertise, each subject's domain-level expertise, such as Mathematics, Chemistry, Computer Science, Biology etc., is determined through traditional methods like tests and interviews. For each domain, a set of knowledge topics that are distinctive for the domain is selected. Their Familiarity Measures are calculated based on the subjects' learning histories. Then randomly select a number of knowledge topics from each domain and calculate the average Familiarity Measure of them. The domain which has the highest average Familiarity Measure is a subject's domain-level expertise. Finally, compare the domain-level expertise discovered by ICKEM with the expertise discovered by traditional methods. If they are equivalent, it proves ICKEM's capability of discovering domain-level expertise.\n\nFor evaluating ICKEM's ability of discovering topic-level expertise, randomly select a set of knowledge topics from each domain, then evaluate a subject's understanding degrees about them, classifying them as ``Understood'' and ``Not Understood'' according to the method mentioned in Section 3.1, then use traditional methods to evaluate a subject's understanding degrees about them, classifying them as ``Understood'' and ``Not Understood'' too. By comparing the results of both classification methods, we can obtain the precision and recall of ICKEM's capability of discovering topic-level expertise.\n\n\\section{Related work}\nMany research fields focus on the collection of personal information, such as lifelogging, expertise finding, and personal informatics. Bush envisioned the `memex' system, in which individuals could compress and store personally experienced information, such as books, records, and communications \\cite{bush1979we}. Inspired by `memex', Gemmell et al. developed a project called MyLifeBits to store all of a person's digital media, including documents, images, audio, and video \\cite{gemmell2002mylifebits}. In \\cite{Liu2017}, a person's reading history about an electronic document is used as attributes for re-finding the document. ICKEM is similar to `memex' and MyLifeBits in that it records an individual's digital history, although for a different purpose. `Memex' and MyLifeBits are mainly for re-finding or reviewing personal data; ICKEM is for quantitatively evaluating a person's knowledge.\n\nPersonal informatics is a class of tools that help people collect personally relevant information for the purpose of self-reflection and gaining self-knowledge \\cite{li2010stage,wolf2009know,yau2009self}. Various tools have been developed to help people collect and analyze different kinds of personal information, such as location \\cite{lindqvist2011m}, finances \\cite{kaye2014money}, food \\cite{cordeiro2015barriers}, weight \\cite{kay2013there,maitland2011designing}, and physical activity \\cite{fritz2014persuasive}. ICKEM facilitates a new type of personal informatics tool that helps people discover their expertise and deficiencies in a more accurate way, by quantitatively assessing an individual's understanding of knowledge.\n\nExpertise is one's expert skill or knowledge in a particular field. Expertise finding is the use of tools for finding and assessing individual expertise \\cite{mcdonald1998just,mattox1999enterprise,vivacqua1999agents}. As an important link of knowledge sharing, expertise finding has been heavily studied in many research communities \\cite{ackerman2013sharing, Cheng2014,maybury2002awareness,tang2008arnetminer,Aslay2013,guy2013mining}. Many sources of data have been exploited to assess an individual's expertise, such as one's publications, documents, emails, web search behavior, other people's recommendations, social media etc. ICKEM provides a new source of data to analyze one's expertise--one's learning history about a topic, which is more comprehensive and straightforward than other data sources. Because one's expertise is mainly obtained through learning (Including ``Informal Learning\", which occurs through the experience of day-to-day situations, such as a casual conversation, play, exploring, etc.)\n\n\\section{Conclusion}\nPeople's pursuing of knowledge is never stopping. Most conceptual knowledge is transmitted through language; it is hard to imagine how a person can obtain conceptual knowledge without using language. A piece of written or spoken language can be converted into text.  We propose a new method to estimate a person's understanding of a piece of conceptual knowledge, by analyzing the text content of one's all learning experiences about a knowledge topic. The computation of familiarity degree takes into account the total time the subject has devoted to a knowledge topic, a topic's share in a learning session, the subject's physical and psychological status during a session, the memory decay of each learning experience over time, and the difference among learning methods. To estimate a person's understanding degree to a knowledge topic, it comprehensively evaluates one's familiarity degrees to the topic itself and other topics that are essential to understand the topic.\nQuantitatively evaluating a person's understanding of knowledge facilitates many applications, such as personalized recommendation, meaningful learning, expertise and deficiency finding etc. With the prevailing of wearable computers like Google Glass and Apple Watch, and maturing of technologies like Speech Recognition and Optical Character Recognition (OCR), it is practicable to analysis people's daily learning activities like talking, listening, and reading. Therefore, ICKEM is technically feasible and beneficial.\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nOn November 12, 2014, the ESA\/Rosetta descent module Philae landed at the Abydos site on the surface of comet 67P\/Churyumov-Gerasimenko (hereafter 67P\/C-G). As part of the scientific payload aboard Philae, the Ptolemy mass spectrometer (Wright~et~al. 2007) performed the analysis of several samples from the surface at Agilkia (Wright~et~al. 2015) and atmosphere at Abydos (Morse~et~al. 2015). The main molecules detected by Ptolemy on the Abydos site were H$_2$O, CO and CO$_2$, with a measured CO\/CO$_2$ molar ratio of 0.07~$\\pm$~0.04. Meanwhile, the CO\/CO$_2$ ratio has also been sampled in 67P\/C-G's coma between August and September 2014 by the ROSINA Double Focusing Mass Spectrometer (DFMS; Balsiger~et~al. 2007; H{\\\"a}ssig~et~al. 2013) aboard the Rosetta spacecraft. Strong variations of the CO and CO$_2$ production rates, dominated by the diurnal changes on the comet, have been measured by ROSINA, giving CO\/CO$_2$ ratios ranging between 0.50~$\\pm$~0.18 and 1.62~$\\pm$~1.34 over the August-September 2014 time period (H{\\\"a}ssig~et~al. 2015). Large fluctuations correlated with the sampled latitudes have also been observed and explained either by seasonal variations  or by a compositional heterogeneity in the nucleus (H{\\\"a}ssig~et~al. 2015). Further investigation of the coma heterogeneity performed by Luspay-Kuti~et~al. (2015) in the southern hemisphere of 67P\/C-G at a later time period led to conclusions in favor of compositional heterogeneity. This latter hypothesis is also reinforced by the Ptolemy measurement of the CO\/CO$_2$ ratio at the Abydos site, which is found outside the range covered by the ROSINA measurements (Morse~et~al. 2015).\n\n\nHere, we aim at investigating the physico-chemical properties of the Abydos subsurface which can reproduce the CO\/CO$_2$ ratio observed by Ptolemy, assuming that the composition of the solid phase located beneath the landing site initially corresponds to the value in the coma. To investigate the possibility of a heterogeneous nucleus for 67P\/C-G, we have employed a comet nucleus model with i) an updated set of thermodynamic parameters relevant for this comet and ii) an appropriate parameterization of the illumination at the Abydos site. This allows us to mimic the thermal evolution of the subsurface of this location. By searching for the matching conditions between the properties of the Abydos subsurface and the Ptolemy data, we provide several constraints on the structural properties and composition of Philae's landing site in different cases.\n\n\n\\section{The comet nucleus model}\n\nThe one-dimensional comet nucleus model used in this work is described in Marboeuf~et~al. (2012). This model considers an initially homogeneous sphere composed of a predefined porous mixture of ices and dust in specified proportions. It describes heat transmission, gas diffusion, sublimation\/recondensation of volatiles within the nucleus, water ice phase transition, dust release, and dust mantle formation. The model takes into account different phase changes of water ice, within: amorphous ice, crystalline ice and clathrates. The use of a 1D model is a good approximation for the study of a specific point at the surface of the comet nucleus, here the Abydos landing site. However, since 67P\/C-G's shape is far from being a sphere (Sierks~et~al. 2015), we have parameterized the model in a way that correctly reproduces the illumination conditions at Abydos. This has been made possible via the use of the 3D shape model developed by Jorda~et~al. (2014) which gives the coordinates of Abydos on the surface of 67P\/C-G's nucleus, as well as the radius corresponding to the Abydos landing site and the normal to the surface at this specific location. The Abydos landing site is located just outside the Hatmethit depression, at the very edge of the area illuminated during the mapping and close observation phases of the Rosetta mission, roughly until December 2014. It is also at the edge of a relatively flat region of the small lobe illuminated throughout the perihelion passage of the comet. Geomorphologically, Abydos is interpreted as being a rough deposit region composed of meter-size boulders (Lucchetti~et~al. 2016). Other geometric parameters specific to 67P\/C-G, such as the obliquity and the argument of the subsolar meridian at perihelion, are calculated from the orientation of the spin axis computed during the shape reconstruction process (Jorda~et~al. 2014). Table~1 summarizes the main parameters used in this work. The porosity and dust\/ice ratio of the cometary material are set in the range of measurement of 80~$\\pm$~5~\\% (Kofman~et~al. 2015) and 4~$\\pm$~2 (Rotundi~et~al. 2014), respectively. These two parameters are linked through the density of the cometary material, and are set to be compatible with the preliminary value determined by Jorda~et~al. (2014) (510~$\\pm$~20~kg\/m$^3$). 67P\/C-G's thermal inertia is estimated to be in the 10--150~W~K$^{-1}$~m$^{-2}$~s$^{1\/2}$ range based on the measurement obtained by the Rosetta\/VIRTIS instrument (Leyrat~et~al. 2015). According to the same study, regions surrounding Abydos are characterized by a thermal inertia in the lower half of this range. We have therefore chosen a low thermal inertia close to 50~W~K$^{-1}$~m$^{-2}$~s$^{1\/2}$.\n\nIn addition to water ice and dust, the solid phase of our model includes CO and CO$_2$ volatiles. Although coma abundances do not necessarily reflect those in the nucleus, they constitute the most relevant constraint available on its composition. We thus considered the CO\/CO$_2$ ratio (1.62~$\\pm$~1.34) measured by ROSINA on August 7, 2014 at 18 hours, as representative of the bulk ice composition in the nucleus, and more specifically in the Abydos subsurface. This ratio is derived from the CO\/H$_2$O and CO$_2$\/H$_2$O ROSINA measurements performed at this date, which are equal to 0.13~$\\pm$~0.07 and 0.08~$\\pm$~0.05, respectively (H{\\\"a}ssig~et~al. 2015). We selected the date of August 7 because i) the corresponding ROSINA measurements were performed at high northern latitudes where the Abydos site is located, and ii) the large CO\/CO$_2$ range obtained at this moment covers all values measured by ROSINA at other dates (including the late value obtained by Le~Roy~et~al. (2015) on October 20 for the northern hemisphere, namely CO\/CO$_2$~=~1.08).\n\nThe three main phases of ices identified in the literature, namely crystalline ice, amorphous ice and clathrate phase, are considered in this work. Outgassing of volatiles in 67P\/C-G could then result from the sublimation of ices, amorphous-to-crystalline ice phase transition, or destabilization of clathrates in the crystalline ice, amorphous ice and clathrates cases, respectively. Because the properties of volatiles trapping in the nucleus matrix strongly depend on the considered icy phase, the following models have been considered:\n\n\\paragraph{Crystalline ice model} Water ice is fully crystalline, meaning that no volatile species are trapped in the water ice structure. Here, CO and CO$_2$ are condensed in the pores of the matrix made of water ice and dust;\n\n\\paragraph{Amorphous ice model} The matrix itself is made of amorphous water ice with a volatile trapping efficiency not exceeding $\\sim$10\\%. In this case, the cumulated mole fraction of volatiles is higher than this value, implying that an extra amount of volatiles is crystallized in the pores. With this in mind, we consider different distributions of CO and CO$_2$ in the both phases of this model;\n\n\\paragraph{Clathrate model} Water ice is exclusively used to form clathrates. Similarly to amorphous ice, clathrates have a maximum trapping capacity ($\\sim$17\\%). The extra amount of volatiles, if any, also crystallizes in the pores. In our case, however, CO is fully trapped in clathrates and escapes only when water ice sublimates. In contrast, we assume that solid CO$_2$ exists in the form of crystalline CO$_2$ in the pores of the nucleus because this molecule is expected to condense in this form in the protosolar nebula (Mousis~et~al. 2008).\n\n\\begin{deluxetable}{lcl}\n\\tablecaption{Modeling parameters for the nucleus}\n\\tablehead{\n\\colhead{Parameter} & \\colhead{Value} & \\colhead{Reference}\n}\n\\startdata\nRotation period\t(hr)\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t& 12.4\t\t\t\t& Mottola~et~al. (2014)\t\t\\\\\nObliquity ($\\degree$)\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t& 52.25\t\t\t\t&\t\t\t\t\t\t\t\\\\\nSubsolar meridian $\\Phi$ ($\\degree$) \\footnote{Argument of subsolar meridian at perihelion.}\t\t\t& -111\t\t\t\t&\t\t\t\t\t\t\t\\\\\nCo-latitude ($\\degree$) \\footnote{Angle between the normal to the surface and the equatorial plane.}\t& -21\t\t\t\t&\t\t\t\t\t\t\t\\\\\nInitial radius (km)\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t& 2.43\t\t\t\t&\t\t\t\t\t\t\t\\\\\nBolometric albedo (\\%)\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t& 1.5\t\t\t\t& Fornasier~et~al. (2015)\t\\\\\nDust\/ice mass ratio\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t& 4~$\\pm$~2\t\t\t& Rotundi~et~al. (2014)\t\t\\\\\nPorosity (\\%)\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t& 80~$\\pm$~5\t\t& Kofman~et~al. (2015)\t\t\\\\\nDensity (kg\/m$^3$)\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t& 510~$\\pm$~20\t\t& Jorda~et~al. (2014)\t\t\\\\\nI (W~K$^{-1}$~m$^{-2}$~s$^{1\/2}$) \\footnote{Thermal inertia.}\t\t\t\t\t\t\t\t\t\t\t& 50\t\t\t\t& Leyrat~et~al. (2015)\t\t\\\\\nCO\/CO$_2$ initial ratio\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t& 1.62~$\\pm$~1.34\t& H{\\\"a}ssig~et~al. (2015)\t\\\\\n\\enddata\n\\end{deluxetable}\n\n\n\\section{Thermal evolution of the subsurface at Abydos}\n\nOur results show that the illumination at the surface of the Abydos site is a critical parameter for the evolution of the nucleus, regardless the considered ice structure. Consequently, all three models described in Section~2 present the same behavior up to a given point. We first describe the characteristics displayed in common by the three models by presenting the thermal evolution of the crystalline ice model, before discussing the variations resulting from the different assumptions on the nature of ices. Figure~1 shows the time evolution of the nucleus stratigraphy, which corresponds to the structural differentiation occurring in the subsurface of the Abydos site. This differentiation results from the sublimation of the different ices. After each perihelion passage, the sublimation interfaces of CO and CO$_2$ reach deeper layers beneath the nucleus surface, with a progression of $\\sim$20 m per orbit. The CO sublimation interface always propagates deeper than its CO$_2$ counterpart because of the higher volatility of the former molecule. On the other hand, because surface ablation is significant, the progression of these interfaces is stopped by the propagation of the water sublimation front after perihelion. This allows the Abydos region to present a ``fresh'' surface after each perihelion.\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=9cm]{figure1.pdf}\n\\caption{Stratigraphy of the Abydos subsurface represented during 35 years of evolution ($\\sim$5 orbits of 67P\/C-G), showing the interfaces of sublimation of considered volatile species. The surface ablation occurs at each perihelion (represented by the vertical lines) and reaches all interfaces.}\n\\end{center}\n\\end{figure}\n\nAt the surface of the Abydos site, the outgassing rates of CO and CO$_2$ vary with the illumination conditions, reaching maxima at perihelion and minima at aphelion. Because the sublimation interface of CO$_2$ is closer to the surface, its production rate is more sensitive to the illumination conditions than that of CO. As a result, the outgassing rate of  CO$_2$ presents important variations with illumination while that of CO is less affected. This difference strongly impacts the evolution of the CO\/CO$_2$ outgassing ratio at the surface of Abydos (see Figure~2). Close to perihelion, this ratio crosses the range of values measured by Ptolemy (0.07~$\\pm$~0.04) and reaches a minimum. Note that the CO\/CO$_2$ outgassing ratio presents spikes during a certain period after perihelion. These spikes appear when the CO and CO$_2$ interfaces of sublimation are dragged out to the surface by ablation and result from temperature variations induced by diurnal variations of the insolation.\n\nWe define $\\Delta t$ as the time difference existing between the Ptolemy CO\/CO$_2$ observations (here 0.07 measured on November 12, 2014) and the epoch at which our model reproduces these data (see Figure~2). In each case investigated, we vary the input parameters of the model to minimize the value of $\\Delta t$ (see Table~2 for details). We have also defined the quantity $f_t = \\Delta t \/ P_{orb}$, namely the fraction of 67P\/C-G's orbital period ($P_{orb} = 6.44$ yr) corresponding to $\\Delta t$. The results of our simulations are indicated below.\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=9cm]{figure2.pdf}\n\\caption{Evolution of the CO\/CO$_2$ outgassing ratio at Abydos in the case of the crystalline ice model, during one orbit. The green line and green area represent the Ptolemy central value and its range of uncertainty, respectively. The blue dots correspond to the measurement epoch (November 12, 2014). Vertical lines correspond to passages at perihelion. See text for a  description of $\\Delta t$.}\n\\end{center}\n\\end{figure}\n\n\\begin{deluxetable}{lccc}\n\\tablecaption{Optimized set of parameters for the three models}\n\\tablehead{\n\\colhead{} & \\colhead{Crystalline} & \\colhead{Amorphous} & \\colhead{Clathrate}\\\\ \n\\colhead{} & \\colhead{ice model} & \\colhead{ice model} & \\colhead{model}\n}\n\\startdata\nParameters\t\t\t\t\t\t\t\t\t\t&\t\t\t\t\t&\t\t\t\t\t&\t\t\t\t\\\\\n\\hline\nDust\/ice mass ratio\t\t\t\t\t\t\t\t& 6\t\t\t\t\t& 4\t\t\t\t\t& 4\t\t\t\t\\\\\nPorosity (\\%)\t\t\t\t\t\t\t\t\t& 78\t\t\t\t& 76\t\t\t\t& 76\t\t\t\\\\\nDensity (kg\/m$^3$)\t\t\t\t\t\t\t\t& 516\t\t\t\t& 510\t\t\t\t& 519\t\t\t\\\\\nI (W K$^{-1}$ m$^{-2}$ s$^{1\/2}$) \\footnote{Thermal inertia resulting from the different water ice conductivities.}\t\t& $\\sim$60\t& 40--60\t& $\\sim$50\t\\\\\nCO\/CO$_2$ initial ratio\t\t\t\t\t\t\t& 0.46\t\t\t\t& 1.62\t\t\t\t& 0.46\t\t\t\\\\\nCO\/H$_2$O initial abundance\t\t\t\t\t\t& 6\\%\t\t\t\t& 13\\% \\footnote{2.8\\% trapped in amorphous ice, 10.2\\% condensed in the pores (Marboeuf~et~al. 2012).}\t& 6\\%\t\\\\\nCO$_2$\/H$_2$O initial abundance\t\t\t\t\t& 13\\%\t\t\t\t& 8\\% \\footnote{4.2\\% trapped in amorphous ice, 3.2\\% condensed in the pores (Marboeuf~et~al. 2012).}\t& 13\\%\t\\\\\n\\hline\nResults\t\t\t\t\t\t\t\t\t\t\t&\t\t\t\t\t&\t\t\t\t\t&\t\t\t\t\\\\\n\\hline\n$\\Delta t$ (day)\t\t\t\t\t\t\t\t& 52\t\t\t\t& 4\t\t\t\t\t& 34\t\t\t\\\\\n$f_t$ (\\%)\t\t\t\t\t\t\t\t\t\t& 2.2\t\t\t\t& 0.17\t\t\t\t& 1.4\t\t\t\\\\\n\\enddata\n\\end{deluxetable}\n\n\n\\paragraph{Crystalline ice model} In this case, we find that the initial CO\/CO$_2$ ratio and the dust\/ice ratio adopted in the nucleus have a strong influence on $\\Delta t$. This quantity is minimized i) when the adopted dust\/ice ratio becomes higher than those found by Rotundi~et~al. (2014) and ii) if the selected initial CO\/CO$_2$ ratio is lower than the nominal value found by ROSINA (H{\\\"a}ssig~et~al. 2015). Figure~3 shows the evolution of $\\Delta t$ as a function of the dust\/ice ratio for two different values of the initial CO\/CO$_2$ ratio, namely 1.62 (the central value) and 0.46 (close to the lower limit). These results confirm the aforementioned trend: with CO\/CO$_2$ = 0.46 and a dust\/ice ratio of 6 or higher, the Ptolemy's measurement epoch is matched with $\\Delta t$~=~52~$\\pm$~27 days or lower, which corresponds to $f_t$ lower than $\\sim$2\\%. These results can be explained by the thermal conductivity of crystalline water ice, which is in the 3--20~W~m$^{-1}$~K$^{-1}$ range (Klinger 1980), considering the temperatures in the comet. Because dust has a conductivity of 4~W~m$^{-1}$~K$^{-1}$ (Ellsworth~\\&~Schubert 1983), the global conductivity decreases with the increase of the dust\/ice ratio in the nucleus. Since heating in the nucleus is mostly provided by surface illumination, a low conductivity increases the temperature gradient between the upper and deeper layers (where the CO$_2$ and the CO sublimation interfaces are located, respectively). This gradient enhances more the sublimation rate of CO$_2$ ice than that of CO ice, leading to smaller CO\/CO$_2$ outgassing ratios at the surface and to a smaller $\\Delta t$ (see Figure~2).\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=9cm]{figure3.pdf}\n\\caption{Evolution of $\\Delta t$ and $f_t$ as a function of the dust\/ice mass ratio for two different values of the initial CO\/CO$_2$ ratio in the case of the crystalline ice model (see text).}\n\\end{center}\n\\end{figure}\n\n\\paragraph{Amorphous ice model} Here, values of $\\Delta t$ are significantly lower than those obtained with the crystalline ice model; a $\\Delta t$ of 4~$\\pm$~9 days is obtained using the central values listed in Table~2 for the initial CO\/CO$_2$ and dust\/ice ratios, leading to $f_t \\sim 0.2\\%$. Higher values of the CO\/CO$_2$ ratio never allow our model matching the Ptolemy data: the CO\/CO$_2$ outgassing ratio is increased sufficiently high so that even its minimum becomes higher than the range of measurements performed by Ptolemy. On the other hand, lower CO\/CO$_2$ ratios increase $\\Delta t$. Interestingly, the results of this model are poorly affected by the considered dust\/ice ratio. Within the 0.12--1.35~W~m$^{-1}$~K$^{-1}$ range (Klinger 1980), the conductivity of amorphous water ice lays under those of dust (4~W~m$^{-1}$~K$^{-1}$) and crystalline ices (3--20~W~m$^{-1}$~K$^{-1}$). Since amorphous ice dominates the volatile phase, the mean conductivity is never too far from to that of dust, irrespective of the dust\/ice ratio considered within the observed range.\n\n\\paragraph{Clathrate model} In this case, low CO\/CO$_2$ ratios (still within the range given in Table~1) are required to get values of $\\Delta t$ below 50 days (i.e., $f_t$ below 2\\%). Similarly to the case of the amorphous ice model, $\\Delta t$ is poorly sensitive to the variation of the dust\/ice ratio because the conductivity of clathrates (0.5~W~m$^{-1}$~K$^{-1}$; Krivchikov~et~al. 2005a, 2005b) is small compared to that of dust.\n\n\n\\section{Discussion}\n\nOur goal was to investigate the possibility of recovering the value of the CO\/CO$_2$ outgassing ratio measured by the Ptolemy instrument at the surface of Abydos. Interestingly, all considered models match the Ptolemy value with $f_t$ lower than $\\sim$2\\%, provided that an optimized set of parameters is adopted for the Abydos region. Despite the fact it is poorly sensitive to the adopted dust\/ice ratio, a nucleus model dominated by amorphous ice (and possibly including a smaller fraction of crystalline ices) gives the best results ($\\Delta t$~$\\leq$~4~days, i.e. $f_t$~$\\leq$~0.2\\%) for a primordial CO\/CO$_2$ ratio equal to the central value measured by the ROSINA instrument. On the other hand, the crystalline ice and clathrate models require a primordial CO\/CO$_2$ ratio close to the lower limit sampled by ROSINA to obtain values of $\\Delta t$ under 50 days (i.e. $f_t$ under 2\\%). We stress the fact that the CO\/CO$_2$ range of validity is used under the assumption that the ROSINA measurements correspond to the bulk nucleus abundances, which is not necessarily true. A second requirement to minimize the value of $\\Delta t$ in the crystalline ice model is the necessity to adopt a dust\/ice ratio at least equal to or higher than the upper limit determined by Rotundi~et~al. (2014) for 67P\/C-G. This is supported by the pictures taken at Abydos by the CIVA instrument (Bibring~et~al. 2007) aboard the Philae module. The very low reflectance of 3--5\\% of the Abydos region (Bibring~et~al. 2015) is in agreement with the OSIRIS and VIRTIS reflectance measurements in the visible (Sierks~et~al. 2015) and near-IR (Capaccioni~et~al. 2015), which are consistent with a low ice content in the upper surface layer (Capaccioni~et~al. 2015).\n\nSurface illumination can also greatly influence the CO\/CO$_2$ outgassing ratio on 67P\/C-G. To quantify this effect, we have simulated a point of 67P\/C-G's nucleus which is more illuminated in comparison to Abydos. We have performed a set of simulations at a co-latitude of -52.25$\\degree$, a point that receives permanent sunlight around perihelion. At the date when the CO\/CO$_2$ outgassing ratio at Abydos is equal to 0.07 (the central value of Ptolemy's range of measurement), we obtain a different value for the CO\/CO$_2$ outgassing ratio at this new location, irrespective of the adopted model. For the crystalline ice model, the outgassing ratio at the illuminated site reaches a value of 0.11, which is still within Ptolemy's range of measurement. This implies that a different illumination cannot explain a strong variation of the CO\/CO$_2$ outgassing ratio if the nucleus presents a homogeneous composition. In this case, the difference between the Ptolemy and ROSINA measurements is clearly due to a heterogeneity in the nucleus composition. On the other hand, the CO\/CO$_2$ outgassing ratio at the illuminated site is equal to 0.74 and 0.76 in the cases of the amorphous ice and clathrate models, respectively. These values are within ROSINA's range of measurements, implying that the difference of illumination is sufficient to explain the difference with the CO\/CO$_2$ ratio sampled at Abydos, assuming a homogeneous nucleus.\n\nIn summary, all possible water ice structures are able to reproduce the observations made by Ptolemy, assuming that the primordial CO\/CO$_2$ ratio is the one inferred by ROSINA. Each case requires a unique set of input parameters taken from the range of values inferred by Rosetta and which describes the structure and composition of the material. According to our simulations, a heterogeneity in the composition of 67P\/C-G's nucleus is possible only if the nucleus is composed of crystalline ices. However, if we consider different ice phases like amorphous ice or clathrates, the difference between the Ptolemy and ROSINA measurements could simply originate from the variation of illumination between different regions of the nucleus.\n\nIn the upcoming months, the Philae module could awaken and allow the Ptolemy mass spectrometer to perform additional measurements of the CO\/CO$_2$ ratio. By comparing these new values with the different CO\/CO$_2$ outgassing ratios predicted by our three models at the same date, we would be able to see which model is the most reliable, and thus to determine which water ice structure is dominant at the surface of 67P\/C-G's nucleus.\n\n\n\\acknowledgements\nO.M. acknowledges support from CNES. This work has been partly carried out thanks to the support of the A*MIDEX project (n\\textsuperscript{o} ANR-11-IDEX-0001-02) funded by the ``Investissements d'Avenir'' French Government program, managed by the French National Research Agency (ANR). Funding and operation of the Ptolemy instrument was provided by the Science and Technology Facilities Council (Consolidated Grant ST\/L000776\/1) and UK Space Agency (Post-launch support ST\/K001973\/1). A.L.-K. acknowledges support from the NASA Jet Propulsion Laboratory (subcontract No. 1496541).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{introduction}\n\nIn 1962, John Olson isolated a water-soluble bacteriochlorophyll\n({\\it BChl a}) protein from green sulfur bacteria \\cite{olson63}. In\n1975, Roger Fenna and Brian Matthews resolved the X-ray structure of\nthis protein (Fenna-Matthews-Olson  (FMO)) from {\\it\nProsthecochloris aestuarii} and found that the protein consists of\nthree identical subunits related by $C_3$ symmetry, each containing\nseven {\\it BChl a} pigments \\cite{fenna75}.  In photosynthetic\nmembranes of green sulfur bacteria, this protein channels the\nexcitations from the chlorosomes to the reaction center. Since it\nwas the first photosynthetic antenna complex of which the X-ray\nstructure became available, it triggered a wide variety of studies\nof spectroscopic and theoretical nature, and it therefore has become\none of the most widely studied and well-characterized\npigment-protein complexes.\n\nThe excitation-energy transfer can be elucidated with 2D\necho-spectroscopy using three laser pulses which hit the sample\nwithin several femtoseconds time spacings. The experimentally\nmeasured 2D echo-spectra of the complex show wave-like energy\ntransfer with oscillation periods roughly consistent with the\neigenenergy spacings  of the excited system\\cite{engel07}. This has\nbrought a long-standing question again into the scientific focus\nthat whether nontrivial quantum coherence effects exist in natural\nbiological systems under physiological conditions\n\\cite{lee07,adolphs06}.\n\nMany studies have attempted to unravel the precise role of quantum\ncoherence in the EET of light-harvesting complexes\n\\cite{mohseni08,plenio08,caruso09,chin10,olaya08,ishizaki09,yang10,hoyer09,sarovar11,shi11}.\nThe interplay of coherent dynamics, which leads to a delocalization\nof an initial excitation arriving at the FMO complex from the\nantenna, and the coupling to a vibronic environment with slow and\nfast fluctuations, has led to studies of environmentally assisted\ntransport in the FMO. An interesting finding is that the\nenvironmental decoherence and noise  play  a crucial role in the\nexcitation energy transfer in the FMO\n\\cite{mohseni08,plenio08,caruso09,chin10,caruso101,shabani11,ghosh11,yi01}.\n\nIn these studies, the FMO complex is treated using the  so-called\nFrenkel exciton Hamiltonian,\n\\begin{equation}\nH=\\sum_{j=1}^7 E_j|j\\rangle\\langle\nj|+\\sum_{i>j=1}^7J_{ij}(|i\\rangle\\langle j|+h.c.),\n\\end{equation}\nwhere $|j\\rangle$ represents the state with only the $j$-th site\nbeing excited and all other sites in their electronic ground state.\n$E_j$ is the on-site energy of site $j$, and $J_{ij}$ denotes the\ninter-site  coupling between sites $i$ and $j$. The inter-site\ncouplings $J_{ij}$ given by $J_{ij}=\\int\nd\\vec{r}\\Phi_i(\\vec{r})\\rho_j(\\vec{r})$ represent  the Coulomb\ncouplings between the transition densities of the BChls\n\\cite{adolphs08}, where $\\Phi_i(\\vec{r})$ is the electrostatic\npotential  of the transition density of {\\it BChl i} and\n$\\rho_j(\\vec{r})$ is the transition density of {\\it BChl j}.  It is\neasy to find that this calculation can only give real inter-site\ncouplings $J_{ij}.$ In fact, the couplings, which has been used in\nprevious studies of 2D echo-spectra and facilitate comparisons\nbetween different approaches, are all real.\n\nWe should emphasize that the inter-site couplings originate from\ndipole-dipole couplings between different chromophores. They are in\ngeneral complex from the aspect of quantum mechanics, regardless  it\nis difficult to calculate.   In this paper, we will shed light on\nthe effects of  complex inter-site couplings by addressing  the\nfollowing questions. First, we present a theoretical framework which\nelucidates how complex couplings  could assist the excitation\ntransfer. We optimize the relative phases in the couplings  for\nmaximal transfer efficiency and study the robustness of excitation\ntransfer against the fluctuation of the on-site energy and\ninter-site couplings. Second, we present a study on the coherence in\nthe initially excited states by showing the dependence of the\ntransfer efficiency on the initial states.\n\nThis paper is organized as follows. In Sec. {\\rm II}, we introduce\nthe model  to describe the FMO complex for the dynamics of the\nexciton transport. In Sec. {\\rm III}, we optimize the phases\nphenomenologically introduced  in the couplings  for maximal\nexcitation energy transfer efficiency and explore how the efficiency\ndepends on the initial state of the FMO. The fluctuations in the\nlocal on-site energy and in the inter-site couplings affect the\ntransfer efficiency, this effect is studied in Sec. {\\rm IV}.\nFinally, we conclude  in Sec. {\\rm V}.\n\n\\section{model}\nSince the FMO complex is arranged as a trimer with three different\nsubunits interacting weakly with each other, we  restrict our study\nto a single subunit which contains eight bacteriochlorophyll\nmolecules (BChls)\\cite{tronrud09, busch11}. The presence of the 8th\nBChl chromophore has just been suggested by the recent\ncrystallographic data\\cite{tronrud09}, and the recent experimental\ndata and theoretical studies indicated that the 8th BChl is the\nclosest site to the baseplate and should be the point at which\nenergy flows into the FMO complex\\cite{busch11,ritschel11}. However,\nthe eighth BChl is loosely bound, it usually detaches from the\nothers when the system is isolated from its environment to perform\nexperiments. Thus we do not take the eighth into account and the EET\nwill be explored by  using the seven-site model\\cite{tronrud09}.\n\nWe describe the FMO complex by a generic one-dimensional Frenkel\nexciton model, consisting of a regular chain of 7 optically active\nsites, which are modeled as two-level systems with parallel\ntransition dipoles. The corresponding Hamiltonian reads,\n\\begin{equation}\nH=\\sum_{j=1}^7 E_j|j\\rangle\\langle\nj|+\\sum_{i>j=1}^7(J_{ij}|i\\rangle\\langle j|+J_{ij}^*|j\\rangle\\langle\ni|),\n\\end{equation}\nwhere $|j\\rangle$ represents the state where only the $j$-th site is\nexcited and all other sites are in their electronic ground state.\n$E_j$ is the on-site energy of site $j$, and $J_{ij}$ denotes the\nexcitonic coupling between sites $i$ and $j$, $J_{ij}=J_{ji}^*$. The\non-site  and the inter-site energies from Ref.\\cite{adolphs06} (in\nunits of $\\mbox{cm}^{-1}$) will be adapted to study the quantum\ndynamics of the FMO. As afromentioned, a phase $\\phi_{ij}$ is added\nto the inter-site coupling $J_{ij}$ stronger than 15 $cm^{-1}$,\n\\begin{widetext}\n\\begin{equation}\nH \\!=\\!\\! \\left(\\!\\!\\begin{array}{ccccccc}\n\\mathbf{215}   & \\!\\mathbf{-104.1e^{i\\phi_{12}}} & 5.1  & -4.3  &   4.7 & \\mathbf{-15.1e^{i\\phi_{16}}} &  -7.8 \\\\\n\\!\\mathbf{-104.1e^{-i\\phi_{12}}} &  \\mathbf{220.0} &\\mathbf{ 32.6e^{i\\phi_{23}}} & 7.1   &   5.4 &   8.3 &   0.8 \\\\\n5.1 &  \\mathbf{ 32.6e^{-i\\phi_{23}} }&  0.0 & \\mathbf{-46.8e^{i\\phi_{34}}} &   1.0 &  -8.1 &   5.1 \\\\\n-4.3 &    7.1 &\\!\\mathbf{-46.8e^{-i\\phi_{34}}} & \\mathbf{125.0} &\\!\n\\mathbf{-70.7e^{i\\phi_{45}}} &\\! -14.7 &\n\\mathbf{ -61.5e^{i\\phi_{47}}}\\\\\n4.7 &    5.4 &  1.0 & \\!\\mathbf{-70.7e^{-i\\phi_{45}}} & \\mathbf{450.0} & \\mathbf{ 89.7e^{i\\phi_{56}}} &  -2.5 \\\\\n\\mathbf{-15.1e^{-i\\phi_{16}}} &    8.3 & -8.1 & -14.7 &  \\mathbf{89.7e^{-i\\phi_{56}}} & \\mathbf{330.0} & \\mathbf{ 32.7e^{i\\phi_{67}}} \\\\\n-7.8 &    0.8 &  5.1 & \\mathbf{-61.5e^{-i\\phi_{47}}} &  -2.5 &\n\\mathbf{32.7e^{-i\\phi_{67}}} & \\mathbf{280.0}\n\\end{array}\\!\\!\n\\right). \\label{ha}\n\\end{equation}\n\\end{widetext}\nHere the zero energy has been shifted by 12230 $\\mbox{cm}^{-1}$ for\nall sites, corresponding to a wavelength of $\\sim 800 \\mbox{nm}$. We\nnote that in units of  $\\hbar=1$, we have 1 ps$^{-1}$=5.3 cm$^{-1}$.\nThen by dividing $|J_{ij}|$ and $E_{j}$ by 5.3, all elements of the\nHamiltonian  are rescaled by units of ps$^{-1}$. We can find from\nthe Hamiltonian $H$ that in the Fenna-Matthew-Olson complex (FMO),\nthere are two dominating exciton energy transfer (EET) pathways:\n$1\\rightarrow 2\\rightarrow 3$ and $6\\rightarrow (5,7)\\rightarrow 4\n\\rightarrow 3$. Although the nearest neighbor terms dominate the\nsite to site coupling, significant hopping matrix elements exist\nbetween more distant sites. This indicates that coherent transport\nitself may not explain why the excitation energy transfer is so\nefficient.\n\nWe phenomenologically introduce a decoherence scheme to describe the\nexcitation population transfer from site 3 to the reaction center\n(site 8). The  master equation to describe the dynamics of the FMO\ncomplex  is then given by,\n\\begin{eqnarray}\n\\frac{d\\rho}{dt} = -i[H,\\rho]+ {\\cal L}_{38}(\\rho)\\;,\n\\label{masterE}\n\\end{eqnarray}\nwhere $\\mathcal{L}_{38}(\\rho)=\\Gamma \\left[P_{83}\\rho(t)\nP_{38}-\\frac{1}{2}P_3\\rho(t)-\\frac{1}{2}\\rho(t)P_3\\right]$ with\n$P_{38}=|3\\rangle\\langle 8|=P_{83}^{\\dagger}$ characterizes the\nexcitation trapping at site 8 via site 3.\n\nWe shall use the population $P_8$ at time $T$ in the reaction center\ngiven by $P_8(T)=\\Tr(|8\\rangle\\langle 8|\\rho(T))$ to quantify the\nexcitation transfer efficiency. Clearly, the Liouvillian\n$L_{38}(\\rho)$ plays an essential role in the excitation transfer.\nWith this term, we will show in the next section that the complex\ninter-site coupling can enhance the excitation energy transfer.\n\n\n\\section{simulation results}\nIt has been shown that the Hamiltonian with real inter-site\ncouplings can not give high excitation energy transfer efficiency.\nIndeed, the previous  numerical simulations shown that $\\Gamma$ can\nbe optimized to 87.14 with all $\\phi_{ij}=0$ to obtain the highest\nexcitation transfer efficiency $p_8=0.6781$ in this case\n\\cite{yi12}.\n\nWith complex inter-site couplings, the EET efficiency can reach\nalmost $90\\%$ by optimizing the phases added to $J_{ij}$. In units\nof $\\pi$, these phases are  $\\phi_{12}=1.2566, \\phi_{16}=3.3510,\n\\phi_{23}=1.8850, \\phi_{34}=1.8850, \\phi_{45}=1.8850,\n\\phi_{47}=0.8378, \\phi_{56}=3.1416, \\phi_{67}=0.3142.$ With these\nphases, the population on each site as a function of time is plotted\nin Fig. \\ref{fig1}. Two observations can be made from Fig.\n\\ref{fig1}: (1) The population shows oscillatory behavior in the\nwhole energy transfer process, this is different from the situation\nwhere decoherence dominates the mechanism of EET \\cite{yi12},  (2)\nthe excitation on site 1 and 2 dominates the population, while the\npopulation at site 3 is almost zero. The first observation can be\nunderstood as the quantum coherence lasts the whole EET, and the\nsecond observation suggests that the site 1 and site 2 play\nimportant role in the excitation energy transfer. The excitation on\nsite 3 is trapped and transferred to the reaction center almost\nimmediately as to obtain a high transfer efficiency, as Fig.\n\\ref{fig1} shows.\n\\begin{figure}\n\\includegraphics*[width=0.8\\columnwidth,\nheight=0.5\\columnwidth]{fmo_cf1.eps}\n\\includegraphics*[width=0.8\\columnwidth,\nheight=0.5\\columnwidth]{fmo_cf6.eps} \\caption{Population on each\nsite as a function of time. Site 8 represents the reaction center.\nThe phases are optimized for the EET efficiency. The exciton is\nassumed initially on the site 1. The upper panel shows the\npopulation without decoherence, while the bottom panel shows that\nwith $\\gamma_2=2.4$ and $\\gamma_{j\\neq 2}=0,$ see\nEq.(\\ref{masterE1}). $\\Gamma=44$ is chosen throughout this paper.}\n\\label{fig1}\n\\end{figure}\n\nSpatial and temporal relaxation of exciton  shows that the site 1\nand 6 were populated initially with large contribution\n\\cite{adolphs06}. Then it is interesting to study how the initial\nstates affect the excitation transfer efficiency. In the following,\nwe shall shed light on this issue. Two cases are considered. First,\nwe calculate the excitation transfer efficiency with exciton\ninitially in a superposition of site 1 and 6 and analyze  the effect\nof transfer efficiency on the initial states. Second, we extend this\nstudy to initial states where sites 1 and 2 are initially excited.\nThese analyses are performed by numerically calculate the  transfer\nefficiency at time $T= 5 $ps, selected results are shown in Fig.\n\\ref{fig2} and Fig. \\ref{fig3}.\n\n\\begin{figure}\n\\includegraphics*[width=0.8\\columnwidth,\nheight=0.5\\columnwidth]{fmo_cf2.eps}\n\\includegraphics*[width=0.8\\columnwidth,\nheight=0.5\\columnwidth]{fmo_cf3.eps}\\caption{The dependence of the\ntransfer efficiency on the initial states. Top panel:  the initial\nstate site  is a superposition of $|1\\rangle$  and $|6\\rangle$.\nNamely, $|\\psi(t=0)\\rangle=\\cos\\theta|1\\rangle+\\sin\\theta\\exp(i\n\\phi)|6\\rangle.$ Lower panel: the initial state is\n$|\\psi(t=0)\\rangle=\\cos\\theta|1\\rangle+\\sin\\theta\\exp(i\n\\phi)|2\\rangle.$ The phases in $J_{ij}$  are optimized for maximal\ntransfer efficiency, i.e., they take the same values  as in\nFig.\\ref{fig1}. } \\label{fig2}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics*[width=0.8\\columnwidth,\nheight=0.5\\columnwidth]{fmo_cf4.eps} \\caption{The transfer\nefficiency versus initial states: the case of classical\nsuperposition. The phases (couplings)  are the same as in Fig.\n\\ref{fig1}. The initial state is  $p|1\\rangle\\langle\n1|+(1-p)|6\\rangle\\langle 6|$ for solid line, and it is $p\n|1\\rangle\\langle 1|+(1-p)|2\\rangle\\langle 2|$ for dotted line.}\n\\label{fig3}\n\\end{figure}\n\nIn Fig. \\ref{fig2} we present the transfer efficiency as a function\nof $\\theta$ and $\\phi$, which characterize the pure initial states\nof the FMO complex through\n$|\\psi(t=0)\\rangle=\\cos\\theta|1\\rangle+\\sin\\theta\\exp(i\n\\phi)|6\\rangle$ (upper panel) and\n$|\\psi(t=0)\\rangle=\\cos\\theta|1\\rangle+\\sin\\theta\\exp(i\n\\phi)|2\\rangle$ (lower panel). We find the FMO complex efficient to\ntransfer excitation initially at site 1 is not good at EET with site\n6 occupied. Namely, coherent superposition of sites 1 and 6\ndecreases the exciton transfer efficiency. For exciton initially in\na superposition of 1 and 6, the transfer efficiency is more\nsensitive to the population ratio (characterized by $\\theta$) but\nnot to the relative phase $\\phi$.   For exciton initially excited on\nsite 1 and 2, both the population ratio and the relative phase\naffect the energy transfer, the transfer efficiency reaches its\nmaximum with $\\theta=\\frac 3 4\\pi$ and $\\phi=\\frac 1 2\\pi$ and it\narrives at its minimum with $\\theta=\\frac {\\pi}{4}$ and $\\phi=\\frac\n1 2\\pi$. This observation suggests that a properly superposition of\nsite 1 and site 2 can enhance  the EET, although the increase of\nefficiency is small.\n\nFig. \\ref{fig3} shows the dependence of the transfer efficiency on\nthe  mixing rate $p$, where the initial state is a classical mixing\nof site 1 and 6 (or 2). Obviously, the population mixing of site 1\nand site 6 does not favors the transfer efficiency. While  the\nclassical mixing of site 1 and site  2  increase the transfer\nefficiency a little bit (not clear from the figure).\n\nIt is recently suggested that decoherence is  an essential feature\nof EET in the FMO complex, where decoherence arises from\ninteractions with the protein cage, the reaction center and the\nsurrounding environment. Further examination  shown that dephasing\ncan increase the efficiency of transport in the FMO, whereas  other\ntypes of decoherence (for example dissipation) block the EET. These\neffects of decoherence  can be understood as  the\nfluctuation-induced-broadening of energy levels, which bridges the\non-site energy gap and  changes equivalently  the coupling between\nthe sites.\n\nWe now examine if the decoherence can further increase the EET\nefficiency with the complex inter-site couplings.  The decoherence\ncan be described by an added Liouvillian term\n\\begin{equation}\n\\mathcal{L}(\\rho)=\\sum_{j=1}^7 \\gamma_j \\left[P_j \\rho(t)\nP_j-\\frac{1}{2}P_j\\rho(t)-\\frac{1}{2}\\rho(t)P_j\\right],\n\\label{masterE1}\n\\end{equation}\nto Eq. (\\ref{masterE}) with $P_j=|j\\rangle\\langle j|$,\n$j=1,2,3,...,7$. The decoherence may come from site-environment\ncouplings, where the environment models the thermal phonon bath and\nradiation fields.\n\\begin{figure}\n\\includegraphics*[width=0.8\\columnwidth,\nheight=0.5\\columnwidth]{fmo_cf5.eps} \\caption{EET efficiency as a\nfunction of $\\gamma_2$. The other decoherence rates $\\gamma_j$  are\nzero, the phases in the inter-site couplings are optimized for EET\nefficiency with $\\gamma_2=0.$} \\label{fig4}\n\\end{figure}\nOptimizing  the decoherence rates  $\\gamma_j, (j=1,2,...,7)$ with\n$\\Gamma=44$ for the EET  efficiency, we find   $\\gamma_2=2.4$ and\n$\\gamma_j=0, (j\\neq 2)$ yields a transfer efficiency $P_8=0.9344,$\nsee Fig. \\ref{fig4}.  Non-zero $\\gamma_j, (j=1,3,4,5,6,7)$ can\n increase the EET efficiency, but the amplitude of the\nincrease is not evident, for example, $\\gamma_1=2.4, \\gamma_3=2.0,\n\\gamma_6=1.8,$ the other $\\gamma_j=0$ and $\\Gamma=44$ yields\n$P_8=0.9374$.\n\n\\section{effect of fluctuations}\nThe site energies are among the most debated properties of the FMO\ncomplex. These values are needed for exciton calculations of the\nlinear spectra and simulations of dynamics. They depend on local\ninteractions between the {\\it BChl a} molecule and the protein\nenvelope and include electrostatic interactions and ligation. Since\nthe interactions are difficult to identify and even harder to\nquantify, the site energies are usually treated as independent\nparameters  obtained from a simultaneous fit to several optical\nspectra. There are many approaches to obtain the site energies by\nfitting  the spectra.  One of the main differences between those\napproaches is whether they restrict the interactions to {\\it BChl a}\nmolecules within a subunits or whether they include interactions in\nthe whole trimer. Since the site energies and inter-site couplings\nmay be different and fluctuate due to the couplings to its\nsurroundings, it is interesting to ask how the fluctuations in the\nsite energies and couplings affect the transfer efficiency. In the\nfollowing, we will study this issue and show that the exciton\ntransfer in the FMO complex remains essentially unaffected in the\npresence of random variations in site\n energies and inter-site couplings.  This strongly suggests that the\nexperimental results recorded for samples at low temperature would\nalso be observable at higher temperatures.\n\nTo be specific, we consider two types of fluctuations in the site\nenergies and inter-site couplings. In the first one, it has zero\nmean, while the another type of fluctuations has nonzero positive\nmean. For the fluctuations with zero mean, the Hamiltonian in Eq.\n(\\ref{ha}) takes the following changes, $H_{jj}\\rightarrow\nH_{jj}(1+r_1\\cdot(\\mbox {rand}(1)-0.5))$ and $H_{i\\neq j}\\rightarrow\nH_{i\\neq j}(e^{i\\phi_{ij}}+r_2\\cdot(\\mbox {rand}(1)-0.5)).$ For the\nfluctuations with nonzero positive mean, the Hamiltonian takes the\nfollowing changes, $H_{jj}\\rightarrow H_{jj}(1+r_1\\cdot \\mbox\n{rand}(1))$  and $H_{i\\neq j}\\rightarrow H_{i\\neq\nj}(e^{i\\phi_{ij}}+r_2\\cdot\\mbox {rand}(1)),$ where $\\mbox{rand(1)}$\ndenotes a random number between 0 and 1. So a 100\\% static disorder\nmay appear in the on-site energies and inter-site couplings.\n\n\\begin{figure}\n\\includegraphics*[width=0.8\\columnwidth,\nheight=0.5\\columnwidth]{fmo_cf7.eps} \\caption{Effect of fluctuations\nin the Hamiltonian on the transfer efficiency. The plot is for  the\nfluctuations with zero mean. The resulting $P_8$ is an averaged\nresult over 20 independent runs. The fluctuations with non-zero mean\nhas a similar effect on the EET, which is not shown here.}\n\\label{fig5}\n\\end{figure}\n\nWith these arrangements, we numerically calculated the transfer\nefficiency and present the results in Fig. \\ref{fig5}. Each value of\ntransfer efficiency is a result averaged over 20 fluctuations. From\nthe numerical simulations, we notice that the results for\nfluctuations with non-zero mean are almost the same as those for\nzero-mean fluctuations, we hence present here only the results for\nzero-mean fluctuations.  Two observations are obvious. (1) With\nzero-mean fluctuations in the site energies and couplings increase,\nthe transfer efficiency fluctuates greatly. (2) The efficiency\ndecreases with both $r_2$ and $r_1$. Moreover, the EET efficiency is\nsensitive to the fluctuations in the site energy more than that in\nthe inter-site couplings.\n\n\nThese feature  can be interpreted as follows.  Since  the network\nwith Hamiltonian Eq.(\\ref{ha}) is optimal for EET efficiency, any\nchanges in the site-energy and inter-site couplings would diminish\nthe excitation energy transfer, leading to the decrease in the EET\nefficiency. This feature is different from the case  that the\ndecoherence dominates the mechanism of EET. As Ref. \\cite{yi12}\nshown, the efficiency increases with $r_1$ but decreases with $r_2$.\nThe physics behind this difference is as follows.  For the case that\nthe decoherence dominates the mechanism of EET, the energy gap\nbetween neighboring sites blocks the energy transfer, whereas the\ninter-site couplings (representing the overlap of different sites)\ntogether with the decoherence  favor the transport. Whereas for the\npresent case  the phases in the inter-site couplings need to {\\it\nmatch} the energy spacing between sites. As a result, any mismatches\nwould results in decrease in the EET efficiency.\n\n\\begin{figure}\n\\includegraphics*[width=0.8\\columnwidth,\nheight=0.5\\columnwidth]{fmo_cf10.eps}\n\\includegraphics*[width=0.8\\columnwidth,\nheight=0.5\\columnwidth]{fmo_cf9.eps} \\caption{Effect of fluctuations\nin the Hamiltonian on the transfer efficiency.  The fluctuations are\nGaussian with mean $\\mu$  and the standard deviation $\\sigma$.\nUpper: The fluctuations happen only in the on-site energies; Bottom:\nThe fluctuations in both the on-site energies and couplings. The\nresulting fidelity is an averaged result over 15 independent runs.}\n\\label{fig6}\n\\end{figure}\n\nIt was suggested\\cite{adolphs06}  that the fluctuations in the\non-site energies are Gaussian. To examine the effect of Gaussian\nfluctuations, we introduce a Gaussian function,\n$y(x|\\sigma,\\mu)=\\frac{1}{\\sigma\\sqrt{2\\pi}}e^{-\\frac{(x-\\mu)^2}{2\\sigma^2}},$\nwhere $\\mu$ is the mean, while $\\sigma$ denotes the standard\ndeviation. We assume that the fluctuations enter only into  the\non-site energies, namely,  $H_{jj}$ is replaced by\n$H_{jj}(1+y(x|\\sigma,\\mu))$, $j=1,2,...,7$. With these notations, we\nplot the effect of fluctuations on the transfer efficiency in Fig.\n\\ref{fig6}(upper panel). We find that the dependence of the\nefficiency on  the variation is subtle: When the mean is zero, the\nefficiency reach its maximum at $\\sigma=0.5,$ while when the mean is\nlarge, small variation favors the EET efficiency. Moreover, we find\nthat the efficiency  decreases as the mean increases. This again can\nbe understood as  mismatch  between the site energy and the\ninter-site couplings caused by the fluctuations. When these Gaussian\nfluctuations occur in both the on-site energies and the couplings,\nwe find from Fig.\\ref{fig6}(bottom panel) that the transfer\nefficiency decreases as the mean  and variation increases.\n\nAbsorption spectroscopy and 2D echo-spectroscopy are used to study\ntime-resolved processes such as energy transfer or vibrational decay\nas well as to measure intermolecular couplings strengths. They\nprovide a tool that gives direct insight into the energy states of a\nquantum system. In turn,  the eigenenergies are essential  to\ncalculate the absorption spectroscopy and 2D echo-spectroscopy. It\nis easy to find that the eigenenergies and energy gaps for the FMO\nwith complex inter-site couplings, given by (96.2851, 62.2958,\n61.4307, 48.2046, 22.6379, 18.9438,  -4.1374),  are almost the same\nas in the system with real inter-site couplings given by (96.8523,\n62.6424, 57.9484, 50.6357, 22.8218, 19.2387,  -4.4789). The maximal\ndifference is at the 7th eigenenergy, it is about 8.25\\% to its\neigenenergy. This suggests that the absorption spectroscopy and 2D\necho-spectroscopy may not change much due to the introduced phases\nin the inter-site couplings.\n\n\\section{Conclusion}\nExcitation  energy transfer (EET) has been an interesting subject\nfor decades not only for its phenomenal efficiency but also for its\nfundamental role in Nature.  Recent studies show that the quantum\ncoherence itself cannot explain the very high excitation energy\ntransfer efficiency in the Fenna-Matthews-Olson  Complex. In this\npaper, we show that this is not the case when the inter-site\ncouplings are complex. Based on the Frenkel exciton Hamiltonian and\nthe phenomenologically introduced decoherence, we have optimized the\nphases in the inter-site couplings  for maximal energy transfer\nefficiency, which can reach about 89.72\\% at time 5ps without\ndecoherence and 93.44\\% with only dephasing at site 2. By\nconsidering different mixing of exciton on site 1 and site 6 (site\n2) as the initial states, we have examined the effect of initial\nstates on the energy transfer efficiency. The results suggest that\nany mixing of site 1 and site 6 decreases the energy transfer\nefficiency, whereas a coherent superposition (or classically mixing)\nof site 1 and site 2 does not change much the EET efficiency. The\nfluctuations in the site energies and inter-site couplings diminish\nthe EET due to the mismatch caused by the fluctuations.\n\nFinishing this work, we have noticed a closely related\npreprint\\cite{zhu12}. While they focus on bi-pathway EET in the FMO\n(i.e., neglecting the inter-site couplings weaker than 15 $cm^{-1}$)\nand a general representation for complex network, we are mainly\nconcerned about the phases added to the inter-site couplings,\ninitial excitations and fluctuations in the site energies and\ninter-site couplings. In this respect, both works are complementary\nto each other.\n\n\\ \\ \\\\\nStimulating discussions with Jiangbin Gong are acknowledged. This\nwork is supported by the NSF of China under Grants Nos 61078011 and\n11175032 as well as the National Research Foundation and Ministry of\nEducation, Singapore under academic research grant No. WBS:\nR-710-000-008-271.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nDuring the past few decades, deep learning has achieved remarkable success in a wide range of applications, including computer vision \\citep{krizhevsky2017imagenet,he2016deep,chen2020simple,he2020momentum,chen2020big}, neural language processing \\citep{kim2014convolutional,vaswani2017attention,lample2019cross}, autonomous vehicles \\citep{grigorescu2020survey,al2017deep,fujiyoshi2019deep}, and protein structure prediction \\citep{senior2020improved,jumper2021highly}. An important reason for this success is due to powerful computing resources, which allow deep neural networks to directly handle giant datasets and bypass complicated manual feature extraction, which causes potential loss of data information \\citep{szegedy2016rethinking,szegedy2017inception}. For example, the powerful large language model GPT-3 contains $175$ billion parameters and is trained on $45$ terabytes of text data with thousands of graphics processing units (GPUs) \\citep{brown2020language}. However, massive data are generated every day \\citep{sagiroglu2013big}, which pose a significant threat to training efficiency and data storage, and deep learning might reach a bottleneck due to the mismatch between the volume of data and computing resources \\citep{strubell2019energy}. Recently, many methods have been proposed to improve training efficiency in deep learning from several perspectives as follows:\n\\begin{itemize}\n    \\item Quantization: This approach sacrifices data at the byte level during the training process for acceleration \\citep{gupta2015deep,micikevicius2018mixed,faraone2018syq,gong2019mixed}.\n    \\item Model pruning: This approach removes trainable parameters that have little influence on the final performance  \\citep{sun2017meprop,lym2019prunetrain,anwar2017structured,luo2017thinet}.\n    \\item Optimization: This approach designs the training algorithms for fast convergence \\citep{zhang2016efficient} or less memory cost \\citep{bernstein2018signsgd,karimireddy2019error}.\n    \\item Dataset reduction: this approach generates few representative data to constitute a new training set. Depending on whether the generated data are natural or synthetic, dataset reduction can be classified into coreset selection \\citep{mirzasoleiman2020coresets,Coleman2020Selection} and dataset distillation \\citep{wang2018dataset}.\n\\end{itemize}\n\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[width=0.9\\columnwidth]{figures\/dd.pdf}\n    \\caption{An illustration of dataset distillation. The models trained on the large original dataset and small synthetic dataset demonstrate comparable performance on the test set.}\n    \\label{fig:dd}\n\\end{figure*}\n\nThe above narrative shows that efficient deep learning is an extensive task and is beyond the scope of this paper; readers can refer to \\citep{sze2017efficient,menghani2021efficient} for more details. In this survey, we focus on dataset distillation to improve training efficiency by synthesizing training data. To better appreciate dataset distillation, we briefly introduce other cognate methods in dataset reduction. For a support vector machine (SVM), its hyperplane is solely determined by ``support vectors'', and removing all other points in the training dataset does not influence on the convergence result \\citep{cortes1995support}. Therefore, the selection of ``support vectors'' is a favorable method to reduce the training burden for SVMs \\citep{nalepa2019selecting}. When the scenario expands to deep learning, the problem of selecting ``support vectors'' generalizes to the well-known {\\it coreset selection}; this is the algorithm \nselects a few prototype examples from the original training dataset as the coreset, and then the model is solely trained on the small coreset to save training costs while avoiding large performance drops.\nHowever, elements in the coreset are unmodified and constrained by the original data, which considerably restricts the coreset's expressiveness, especially when the coreset budget is limited. Recently, a novel approach, {\\it dataset distillation} (DD) \\footnote{Dataset distillation is also referred to as dataset condensation in some studies.}, has attracted growing attention from the deep learning community and has been leveraged in many practical applications  \\citep{masarczyk2020reducing,liu2022meta,chen2022bidirectional, chen2023bidirectional}. Different from coreset selection, dataset distillation removes the restriction of uneditable elements and carefully modifies a small number of examples to preserve more information, as shown in Figure \\ref{fig:dd_sample} for synthetic examples. By distilling the knowledge of the large original dataset into a small synthetic set, models trained on the distilled dataset can acquire a comparable generalization performance. A general illustration for dataset distillation is presented in Figure \\ref{fig:dd}.\n\n\n\nDue to the property of extremely high dimensions in the deep learning regime, the data information is hardly disentangled to specific concepts, and thus distilling numerous high-dimensional data into a few points is not a trivial task. Based on the objectives applied to mimic target data, dataset distillation methods can be grouped into meta-learning frameworks and data matching frameworks, and these techniques in each framework can be further classified in a much more detailed manner.  In the meta-learning framework, the distilled data are considered hyperparameters and optimized in a nested loop fashion according to the distilled-data-trained model's risk {\\it w.r.t.} the target data \\citep{maclaurin2015gradient,wang2018dataset}. The data matching framework updates distilled data by imitating the influence of target data on model training from parameter or feature space \\citep{zhao2021dataset,cazenavette2022distillation,zhao2023distribution}. Figure \\ref{fig:tree diagram} presents these different categories of DD algorithms in a tree diagram.\n\n\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.9\\columnwidth]{figures\/dd_survey_framework.pdf}\n    \\caption{The schematic structure of dataset distillation and the relationship between the adjacent sections. The body of this survey mainly contains the fundamentals of dataset distillation, taxonomy of distillation schemes, types of factorized DD, distillation algorithms, performance comparison, applications, challenges, and future directions. Note that `Section' is abbreviated as `Sec.' in this figure.}\n    \\label{fig:structure}\n\\end{figure}\n\n\\begin{figure}[t]\n    \\centering\n    \\scalebox{0.7}{\n    \\begin{forest}\n  [\\textbf{Dataset Distillation}\n    [\\textbf{Meta-Learning Framework}\n     [\\textbf{Backpropagation Through Time}\n       [DD \\citep{wang2018dataset} \\\\ LD \\citep{bohdal2020flexible} \\\\ SLDD \\citep{sucholutsky2021soft} \\\\ \\blue{GTN} \\citep{such2020generative} \\\\ \\blue{RTP} \\citep{deng2022remember}]\n     ]\n     [\\textbf{Kernel Ridge Regression}\n       [KIP \\citep{nguyen2020dataset,nguyen2021dataset} \\\\ FRePo \\citep{zhou2022dataset} \\\\ RFAD \\citep{loo2022efficient}]\n     ]\n    ]\n    [\\textbf{Data Matching Framework}\n      [\\textbf{Gradient Match}\n        [DC \\citep{zhao2021dataset} \\\\ DCC \\citep{lee2022dataset} \\\\ \\blue{IDC} \\citep{kim2022dataset}]\n      ]\n      [\\textbf{Trajectory Match}\n        [MTT \\citep{cazenavette2022distillation} \\\\ FTD \\citep{du2022minimizing} \\\\ TESLA \\citep{cui2022scaling} \\\\ \\blue{Haba} \\citep{liu2022dataset}]\n      ]\n      [\\textbf{Distribution Match}\n        [DM \\citep{zhao2023distribution} \\\\ CAFE \\citep{wang2022cafe} \\\\ \\blue{IT-GAN} \\citep{zhao2022synthesizing} \\\\ \\blue{KFS} \\citep{lee2022kfs}]\n      ]\n    ]\n  ]\n\\end{forest}}\n    \\caption{Tree diagram for different categories of dataset distillation algorithms. The factorized DD methods are marked in blue.}\n    \\label{fig:tree diagram}\n\\end{figure}\n\nApart from directly considering the synthetic examples as optimization objectives, in some studies, a proxy model is designed, which consists of latent codes and decoders to generate highly informative examples and to resort to learning the latent codes and decoders. For example, \\citet{such2020generative} employed a network to generate highly informative data from noise and optimized the network with the meta-learning framework. \\citet{zhao2022synthesizing} optimized the vectors and put them into the generator of a well-trained generative adversarial network (GAN) to produce the synthetic examples. Moreover, \\citet{kim2022dataset,deng2022remember,liu2022dataset,lee2022kfs} learned a couple of latent codes and decoders, and then synthetic data were generated according to the different combinations of latent codes and decodes. With this factorization of synthetic data, the compression ratio of DD can be further decreased, and the performance can also be improved due to the intraclass information extracted by latent codes.\n\n\n\nIn this paper, we present a comprehensive survey on dataset distillation, and the main objectives of this survey are as follows: (1) present a clear and systematic overview of dataset distillation; (2) review the recent progress and state-of-the-art algorithms and discuss various applications in dataset distillation; (3) give a comprehensive performance comparison {\\it w.r.t.} different dataset distillation algorithms; and (4) provide a detailed discussion on the limitation and promising directions of dataset distillation to benefit future studies. Recently, \\citet{larasati2022review} published a short review on dataset distillation. However, these authors only provided a general overview of DD and focus on the aspect of applications. Different from \\citet{larasati2022review}, our paper gives a systematic, comprehensive survey on dataset distillation from a wide aspect of distillation frameworks, algorithms, applications, limitations, and promising directions.\n\nThe rest of this paper is organized as follows. We first provide soem background with respect to DD in Section 2. DD methods under the meta-learning framework are described and comprehensively analyzed in Section 3, and a description of the data matching framework follows in Section 4. Section 5 discusses different types of factorized dataset distillation. Next, we report the performance of various distillation algorithms in Section 6. Applications with dataset distillation are shown in Section 7, and finally we discuss challenges and future directions in Section 8.\n\n\n\n\n\n\n\\section{Background}\nBefore introducing dataset distillation, we first define some notations used in this paper.\nFor a dataset $\\mathcal{T}=\\{(\\boldsymbol{x}_i, y_i)\\}_{i=1}^{m}$, $\\boldsymbol{x}_i\\in \\mathbb{R}^d$, $d$ is the dimension of the input data, and $y_i$ is the label. We assume that $(\\boldsymbol{x}_i,y_i)$ for $1\\leq i \\leq m$ are independent and identically distributed (i.i.d.) random variables drawn from the data generating distribution $\\mathcal{D}$. We employ $f_{\\boldsymbol{\\theta}}$ to denote the neural network parameterized by $\\boldsymbol{\\theta}$, and $f_{\\boldsymbol{\\theta}}(\\boldsymbol{x})$ is the prediction or output of $f_{\\boldsymbol{\\theta}}$ at $\\boldsymbol{x}$. Moreover, we also define the loss between the prediction and ground truth as $\\ell(f_{\\boldsymbol{\\theta}}(\\boldsymbol{x}),y)$, and the expected risk in terms of $\\boldsymbol{\\theta}$ is defined as\n\\begin{equation}\n    \\mathcal{R}_{\\mathcal{D}}(\\boldsymbol{\\theta}) = \\mathbb{E}_{(\\boldsymbol{x},y)\\sim \\mathcal{D}}\\left[\\ell\\left(f_{\\boldsymbol{\\theta}}\\left(\\boldsymbol{x}\\right),y\\right) \\right].\n\\end{equation}\nSince the data generating distribution $\\mathcal{D}$ is unknown, evaluating the expected risk $\\mathcal{R}_{\\mathcal{D}}$ is impractical. Therefore, a practical way to estimate the expected risk is by the empirical risk $\\mathcal{R}_\\mathcal{T}$, which is defined as\n\\begin{equation}\n    \\mathcal{R}_{\\mathcal{T}}(\\boldsymbol{\\theta}) = \\mathbb{E}_{(\\boldsymbol{x},y)\\sim \\mathcal{T}}\\left[\\ell\\left(f_{\\boldsymbol{\\theta}}\\left(\\boldsymbol{x}\\right),y\\right) \\right] = \\frac{1}{m}\\sum_{i=1}^{m} \\ell \\left(f_{\\boldsymbol{\\theta}}\\left(\\boldsymbol{x}_i\\right),y_i\\right).\n\\end{equation}\n\nFor the training algorithm $\\texttt{alg}$, $\\texttt{alg}(\\mathcal{T}, \\boldsymbol{\\theta}^{(0)})$ denote the learned parameters returned by empirical risk minimization (ERM) on the dataset $\\mathcal{T}$ with the initialized parameter $\\boldsymbol{\\theta}^{(0)}$:\n\\begin{equation}\n    \\texttt{alg}(\\mathcal{T}, \\boldsymbol{\\theta}^{(0)}) = \\arg\\min_{\\boldsymbol{\\theta}} \\mathcal{R}_{\\mathcal{T}}(\\boldsymbol{\\theta}).\n\\end{equation}\nIn the deep learning paradigm, gradient descent is the dominant training algorithm to train a neural network by minimizing the empirical risk step by step. Specifically, the network's parameters are initialized with $\\boldsymbol{\\theta}^{(0)}$, and then the parameter is iteratively updated according to the gradient of empirical risk:\n\\begin{equation}\n\\label{eq:gradient descent}\n    \\boldsymbol{\\theta}^{(k+1)} = \\boldsymbol{\\theta}^{(k)} - \\eta \\boldsymbol{g}_{\\mathcal{T}}^{(k)},\n\\end{equation}\nwhere $\\eta$ is the learning rate and $ \\boldsymbol{g}_{\\mathcal{T}}^{(k)}=\\nabla_{\\boldsymbol{\\theta}^{(k)}} \\mathcal{R}_{\\mathcal{T}}(\\boldsymbol{\\theta}^{(k)})$ is the gradient.\nWe omit $\\boldsymbol{\\theta}^{(0)}$ and use $\\texttt{alg}(\\mathcal{T})$ if there is no ambiguity for the sake of simplicity. Then, the model $f$ trained on the dataset $\\mathcal{T}$ can be denoted as $f_{\\texttt{alg}(\\mathcal{T})}$.\n\nBecause deep learning models are commonly extremely overparameterized, {\\it i.e.,} the number of model parameters overwhelms the number of training examples, the empirical risk readily reaches zero. In this case, the generalization error, which measures the difference between the expected risk and the empirical risk, can be solely equal to the expected risk, which is reflected by test loss or test error in practical pipelines.\n\n\n\n\n\\subsection{Formalizing Dataset Distillation}\n\nGiven a target dataset (source training dataset) $\\mathcal{T}=\\{(\\boldsymbol{x}_i,y_i)\\}_{i=1}^m$, the objective of dataset distillation is to extract the knowledge of $\\mathcal{T}$ into a small synthetic dataset $\\mathcal{S}=\\{(\\boldsymbol{s}_j,y_j)\\}_{j=1}^n$, where $n\\ll m$, and the model trained on the small distilled dataset $\\mathcal{S}$ can achieve a comparable generalization performance to the large original dataset $\\mathcal{T}$:\n\\begin{equation}\n    \\mathbb{E}_{(\\boldsymbol{x},y)\\sim \\mathcal{D} \\atop \\boldsymbol{\\theta}^{(0)}\\sim \\mathbf{P}}\\left[\\ell\\left(f_{\\texttt{alg}(\\mathcal{T})}\\left(\\boldsymbol{x}\\right),y\\right) \\right] \\simeq \\mathbb{E}_{(\\boldsymbol{x},y)\\sim \\mathcal{D} \\atop \\boldsymbol{\\theta}^{(0)}\\sim \\mathbf{P} }\\left[\\ell\\left(f_{\\texttt{alg}(\\mathcal{S})}\\left(\\boldsymbol{x}\\right),y\\right) \\right].\n\\end{equation}\n\n\\iffalse\nIntuitively, if the parameters learned on the distilled dataset $\\mathcal{S}$ can also have a minimal loss on the target dataset $\\mathcal{T}$, the models trained on $\\mathcal{S}$ and $\\mathcal{T}$ are expected to possess a similar generalization performance. Therefore, \\citet{wang2018dataset} convert the objective of DD to minimize the term $\\mathcal{R}_{\\mathcal{T}}\\left(\\texttt{alg}\\left(\\mathcal{S}\\right)\\right)$, and the dataset distillation can be formulated to a bilevel optimization problem as below:\n\\begin{equation}\n\\mathcal{S}^\\ast = \\arg\\min_{\\mathcal{S}}\\mathcal{R}_{\\mathcal{T}}\\left({\\texttt{alg}(\\mathcal{S})}\\right) \\quad  \\text{(outer loop)}\n\\end{equation}\nsubject to\n\\begin{equation}\n    \\texttt{alg}(\\mathcal{S})=\\arg\\min _{\\boldsymbol{\\theta}} \\mathcal{R}_\\mathcal{S}\\left({\\boldsymbol{\\theta}}\\right) \\quad \\text{(inner loop)}\n\\end{equation}\nThe inner loop optimizes the model parameters based on the synthetic dataset and is often realized by gradient descent for neural networks or regression for kernel method. During the outer loop iteration, the synthetic set is updated by minimizing the model's risk in terms of the target dataset. With the nested loop, the synthetic dataset gradually converges to one of the optima.\n\\fi\n\n\n\nBecause the training algorithm $\\texttt{alg}\\left(\\mathcal{S},\\boldsymbol{\\theta}^{(0)}\\right)$ is determined by the training set $\\mathcal{S}$ and the initialized network parameter $\\boldsymbol{\\theta}^{(0)}$, many dataset distillation algorithms will take expectation on $\\mathcal{S}$ {\\it w.r.t.} $\\boldsymbol{\\theta}^{(0)}$ to improve the robustness of the distilled dataset $\\mathcal{S}$ to different parameter initialization, and the objective function has the form of $\\mathbb{E}_{ \\boldsymbol{\\theta}^{(0)}\\sim \\mathbf{P}}\\left[\\mathcal{L} \\left(\\mathcal{S} \\right ) \\right]$. In the following narrative, we omit this expectation {\\it w.r.t.} initialization $\\boldsymbol{\\theta}^{(0)}$ for the sake of simplicity.\n\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[width=0.9\\columnwidth]{figures\/synthetic_image_samples.pdf}\n    \\caption{Example synthetic images distilled from CIFAR-10\/100 and Tiny ImageNet by matching the training trajectory \\citep{cazenavette2022distillation}.}\n    \\label{fig:dd_sample}\n\\end{figure*}\n\n\\section{Meta-Learning Framework}\nBy assuming $\\mathcal{R}_{\\mathcal{D}}\\left({\\texttt{alg}(\\mathcal{S})}\\right) \\simeq \\mathcal{R}_{\\mathcal{T}}\\left({\\texttt{alg}(\\mathcal{S})}\\right)$, the target dataset $\\mathcal{T}$ is employed as the validation set in terms of the model ${\\texttt{alg}(\\mathcal{S})}$. In consequent, the objective of DD can be converted to minimize $\\mathcal{R}_{\\mathcal{T}}\\left({\\texttt{alg}(\\mathcal{S})}\\right)$ to improve the generalization performance of ${\\texttt{alg}(\\mathcal{S})}$ in terms of $\\mathcal{D}$. {To this end, dataset distillation falls into the meta-learning area: the hyperparameter $\\mathcal{S}$ is updated by the meta (or outer) algorithm, and the base (or inner) algorithm solves the conventional supervised learning problem {\\it w.r.t.} the synthetic dataset $\\mathcal{S}$} \\citep{hospedales2021meta}; and the dataset distillation task can be formulated to a bilevel optimization problem as follows:\n\\begin{equation}\n\\mathcal{S}^\\ast = \\arg\\min_{\\mathcal{S}}\\mathcal{R}_{\\mathcal{T}}\\left({\\texttt{alg}(\\mathcal{S})}\\right) \\quad  \\text{(outer loop)}\n\\end{equation}\nsubject to\n\\begin{equation}\n\\label{eq: inner loop}\n    \\texttt{alg}(\\mathcal{S})=\\arg\\min _{\\boldsymbol{\\theta}} \\mathcal{R}_\\mathcal{S}\\left({\\boldsymbol{\\theta}}\\right) \\quad \\text{(inner loop)}\n\\end{equation}\nThe inner loop optimizes the model parameters based on the synthetic dataset and is often realized by gradient descent for neural networks or regression for kernel method. During the outer loop iteration, the synthetic set is updated by minimizing the model's risk in terms of the target dataset. With the nested loop, the synthetic dataset gradually converges to one of the optima. \n\nAccording to the model and optimization methods used in the inner lop, the meta-learning framework of DD can be further classified into two sub-categories of backpropagation through time time (BPTT) approach and kernel ridge regression (KRR) approach.\n\n\\subsection{Backpropagation Through Time Approach}\n\nAs shown in the above formulation of bilevel optimization, the objective function of DD can be directly defined as the meta-loss of\n\\begin{equation}\n    \\mathcal{L}(\\mathcal{S}) = \\mathcal{R}_{\\mathcal{T}}\\left(\\texttt{alg}\\left(\\mathcal{S}\\right)\\right).\n\\end{equation}\nThen the distilled dataset is updated by $\\mathcal{S} = \\mathcal{S} - \\alpha \\nabla_\\mathcal{S} \\mathcal{L}(\\mathcal{S})$ with the step size $\\alpha$. However, in the inner loop of Eq. \\ref{eq: inner loop}, the neural network is trained by iterative gradient descent (Eq. \\ref{eq:gradient descent}), which yields a series of intermediate parameter states of $\\{\\boldsymbol{\\mathcal{\\theta}}^{(0)}, \\boldsymbol{\\mathcal{\\theta}}^{(1)}, \\cdots, \\boldsymbol{\\mathcal{\\theta}}^{(T)} \\}$, backpropagation through time \\citep{werbos1990backpropagation} is required to recursively compute the meta-gradient $\\nabla_\\mathcal{S}\\mathcal{L}(\\mathcal{S})$: \n\\begin{equation}\n    \\nabla_\\mathcal{S}\\mathcal{L}(\\mathcal{S}) = \\frac{\\partial \\mathcal{L}}{\\partial \\mathcal{S}} = \\frac{\\partial \\mathcal{L}}{\\partial \\boldsymbol{\\theta}^{(T)}}\\left( \\sum_{k=0}^{k=T} \\frac{\\partial \\boldsymbol{\\theta}^{(T)}}{\\partial \\boldsymbol{\\theta}^{(k)}} \\cdot \\frac{\\partial \\boldsymbol{\\theta}^{(k)}}{\\partial \\mathcal{S}} \\right), \\qquad \\frac{\\partial \\boldsymbol{\\theta}^{(T)}}{\\partial \\boldsymbol{\\theta}^{(k)}} = \\prod_{i=k+1}^{T} \\frac{\\partial \\boldsymbol{\\theta}^{(i)}}{\\partial \\boldsymbol{\\theta}^{(i-1)}}.\n\\end{equation}\n\nWe present the meta-gradient backpropagation of BPTT in Figure \\ref{figure:metaloss}. Due to the requirement of unrolling the recursive computation graph, BPTT is both computationally expensive and memory demanding, which also severely affects the final distillation performance. \n\n\nTo alleviate the inefficiency in unrolling the long parameter path of $\\{\\boldsymbol{\\mathcal{\\theta}}^{(0)}, \\cdots, \\boldsymbol{\\mathcal{\\theta}}^{(T)} \\}$, \\citet{wang2018dataset} adopted a single-step optimization {\\it w.r.t.} the model parameter from $\\boldsymbol{\\theta}^{(0)}$ to $\\boldsymbol{\\theta}^{(1)}$, and the meta-loss was computed based on $\\boldsymbol{\\theta}^{(1)}$ and the target dataset $\\mathcal{T}$: {\n\\begin{equation}\n    \\boldsymbol{\\theta}^{(1)} = \\boldsymbol{\\theta}^{(0)} - \\eta \\nabla_{ \\boldsymbol{\\theta}^{(0)}}\\mathcal{R}_{\\mathcal{S}}(\\boldsymbol{\\theta}^{(0)}) \\quad \\text{and} \\quad \\mathcal{L} = \\mathcal{R}_{\\mathcal{T}}(\\boldsymbol{\\theta}^{(1)}).\n\\end{equation}\nTherefore, the distilled data $\\mathcal{S}$ and learning rate $\\eta$ can be efficiently updated via the short-range BPTT as follows:\n\\begin{equation}\n    \\mathcal{S} =  \\mathcal{S} - \\alpha_{\\boldsymbol{s}_i } \\nabla \\mathcal{L} \\quad \\text{and} \\quad \\eta = \\eta - \\alpha_\\eta \\nabla \\mathcal{L}.\n\\end{equation}}\nUnlike freezing distilled labels, \\citet{sucholutsky2021soft} extended the work of \\citet{wang2018dataset} by learning a soft-label in the synthetic dataset $\\mathcal{S}$; {\\it i.e.}, the label $y$ in the synthetic dataset is also trainable for better information compression, {\\it i.e.}, $y_i = y_i - \\alpha \\nabla_{y_i} \\mathcal{L}$ for $(\\boldsymbol{s}_i, y_i)\\in \\mathcal{S}$. Similarly, \\citet{bohdal2020flexible} also extended the standard example distillation to label distillation by solely optimizing the labels of synthetic datasets. Moreover, these researchers provided improvements on the efficiency of long inner loop optimization via (1) iteratively updating the model parameters $\\boldsymbol{\\theta}$ and the distilled labels $y$, {{\\it i.e.}, one outer step followed by only one inner step for faster convergence}; and (2) fixing the feature extractor of neural networks and solely updating the last linear layer with ridge regression to avoid second-order meta-gradient computation. \nAlthough the BPTT framework has been shown to underperform other algorithms, \\citet{deng2022remember} empirically demonstrated that adding momentum term and longer unrolled trajectory ($200$ steps) in the inner loop optimization can considerably enhance the distillation performance; and the inner loop of model training becomes\n\\begin{equation}\n    \\boldsymbol{\\theta}^{(k+1)} = \\boldsymbol{\\theta}^{(k)} - \\eta \\boldsymbol{m}^{(k+1)},\n\\end{equation}\nwhere\n\\begin{equation}\n    \\boldsymbol{m}^{(k+1)} = \\beta \\boldsymbol{m}^{(k)} +  \\nabla_{\\boldsymbol{\\theta}^{(k)}} \\mathcal{R}_{\\mathcal{T}}(\\boldsymbol{\\theta}^{(k)}) \\quad \\text{s.t.} \\quad \\boldsymbol{m}^{(0)} = \\boldsymbol{0}.\n\\end{equation}\n\n\n\n\\begin{figure}[t]\n\\centering\n\n\\subfigure[]{\n\\begin{minipage}[b]{0.35\\textwidth}\n\\centering\n    \t\t\\includegraphics[width=0.9\\columnwidth]{figures\/metaloss.pdf}\n    \t\t\\end{minipage}\n\t\t\\label{figure:metaloss}   \n    \t}\n\\subfigure[]{\n\\begin{minipage}[b]{0.3\\textwidth}\n\\centering\n    \t\t\\includegraphics[width=0.9\\columnwidth]{figures\/kernelloss.pdf}\n    \t\t\\end{minipage}\n\t\t\\label{figure:kernelloss}   \n    \t}\n\\subfigure[]{\n\\begin{minipage}[b]{0.26\\textwidth}\n\\centering\n    \t\t\\includegraphics[width=0.9\\columnwidth]{figures\/matchloss.pdf}\n    \t\t\\end{minipage}\n\t\t\\label{figure:matchloss}   \n    \t}\n\\caption{ (a) Meta-gradient backpropagation in BPTT \\citep{zhou2022dataset}; (b) Meta-gradient backpropagation in kernel ridge regression; and (c) Meta-gradient backpropagation in gradient matching.}\n\\label{figure:high dimension}\n\\end{figure}\n\n\\subsection{Kernel Ridge Regression Approach}\nAlthough multistep gradient descent can gradually approach the optimal network parameters in terms of the synthetic dataset during the inner loop, this iterative algorithm makes the meta-gradient backpropagation highly inefficient, as shown in BPTT. \nConsidering the existence of closed-form solutions in the kernel regression regime, \\citet{nguyen2020dataset} replaced the neural network in the inner loop with a kernel model, which bypasses the recursive backpropagation of the meta-gradient. For the regression model $f(\\boldsymbol{x})=\\boldsymbol{w}^\\top\\psi(\\boldsymbol{x})$, where $\\psi(\\cdot)$ is a nonlinear mapping and the corresponding kernel is $K(\\boldsymbol{x},\\boldsymbol{x}^\\prime)=\\langle \\psi(\\boldsymbol{x}), \\psi(\\boldsymbol{x}^\\prime)\\rangle$, there exists a closed-form solution for $\\boldsymbol{w}$ when the regression model is trained on $\\mathcal{S}$ with kernel ridge regression (KRR):\n\\begin{equation}\n   \\boldsymbol{w} =  \\psi(X_{s})^\\top\\left(\\mathbf{K}_{X_s X_s}+\\lambda I\\right)^{-1}y_{s},\n\\end{equation}\nwhere $\\mathbf{K}_{X_s X_s} = [K(\\boldsymbol{s}_i,\\boldsymbol{s}_j)]_{ij}\\in \\mathbb{R}^{n\\times n}$ is called the {\\it kernel matrix} or {\\it Gram matrix} associated with $K$ and the dataset $\\mathcal{S}$, and $\\lambda > 0$ is a fixed regularization parameter \\citep{petersen2008matrix}. Therefore, the mean square error (MSE) of predicting $\\mathcal{T}$ with the model trained on $\\mathcal{S}$ is\n\\begin{equation}\n\\label{eq:mse}\n    \\mathcal{L}(\\mathcal{S}) = \\frac{1}{2}\\left\\|y_t-\\mathbf{K}_{X_t X_s}\\left(\\mathbf{K}_{X_s X_s}+\\lambda I\\right)^{-1} y_s\\right\\|^2,\n\\end{equation}\nwhere $\\mathbf{K}_{X_t X_s}=[K(\\boldsymbol{x}_i,\\boldsymbol{s}_j)]_{ij}\\in \\mathbb{R}^{m\\times n}$. Then the distilled dataset is updated via the meta-gradient of the above loss. Due to the closed-form solution in KRR, $\\boldsymbol{\\theta}$ does not require an iterative update and the backward pass of the gradient thus bypasses the recursive computation graph, as shown in Figure \\ref{figure:kernelloss}. \n\n\nIn the KRR regime, the synthetic dataset $\\mathcal{S}$ can be directly updated by backpropagating the meta-gradient through the kernel function. Although this formulation is solid, this algorithm is designed in the KRR scenario and only employs simple kernels, which causes performance drops when the distilled dataset is transferred to train neural networks. \\citet{jacot2018neural} proposed the neural tangent kernel (NTK) theory that proves the equivalence between training infinite-width neural networks and kernel regression. With this equivalence, \\citet{nguyen2021dataset} employed infinite-width networks as the kernel for dataset distillation, which narrows the gap between the scenarios of KRR and deep learning. \n\nHowever, every entry in the kernel matrix must be calculated separately via the kernel function, and thus computing the kernel matrix $\\mathbf{K}_{X_s X_t}$ has the time complexity of $\\mathcal{O}(|\\mathcal{T}||\\mathcal{S}|)$, which is severely inefficient for large-scale datasets with huge $|\\mathcal{T}|$. To tackle this problem, \\citet{loo2022efficient} replaced the NTK kernel with neural network Gaussian process (NNGP) kernel that only considers the training dynamic of the last-layer classifier for speed up. With this replacement, the random features $\\psi(\\boldsymbol{x})$ and $\\psi(\\boldsymbol{s})$ can be explicitly computed via multiple sampling from the Gaussian process, and thus the kernel matrix computation can be decomposed into random feature calculation and random feature matrix multiplication. Because matrix multiplication requires negligible amounts of time for small distilled datasets, the time complexity of kernel matrix computation degrades to $\\mathcal{O}(|\\mathcal{T}|+|\\mathcal{S}|)$. In addition, these authors demonstrated two issues of MSE loss (Eq. \\ref{eq:mse}) used in \\citep{nguyen2020dataset} and \\citep{nguyen2021dataset}: (1) over-influence on corrected data: the correctly classified examples in $\\mathcal{T}$ can induce larger loss than the incorrectly classified examples; and (2) unclear probabilistic interpretation for classification tasks. To overcome these issues, they propose to apply a cross-entropy (CE) loss with Platt scaling \\citep{platt1999probabilistic} to replace the MSE loss:\n\\begin{equation}\n    \\mathcal{L}_{\\tau} = \\text{CE}(y_t, \\hat{y}_t \/ \\tau),\n\\end{equation}\nwhere $\\tau$ is a positive learned temperature scaling parameter, and $\\hat{y}_t = \\mathbf{K}_{X_t X_s}\\left(\\mathbf{K}_{X_s X_s}+\\lambda I\\right)^{-1} y_s$ is still calculated using the KRR formula.\n\nA similar efficient method was also proposed by \\citet{zhou2022dataset}, which also focused on solving the last-layer in neural networks with KRR. Specifically, the neural network $f_{\\boldsymbol{\\theta}} = g_{\\boldsymbol{\\theta}_2} \\circ h_{\\boldsymbol{\\theta}_1}$ can be decomposed with the feature extractor $h$ and the linear classifier $g$. Then these coworkers fixed the feature extractor $h$ and the linear classifier $g$ possesses a closed-form solution with KRR; and the the distilled data are accordingly optimized with the MSE loss:\n\\begin{equation}\n    \\mathcal{L}(\\mathcal{S}) = \\frac{1}{2}\\left\\|y_t-\\mathbf{K}_{X_t X_s}^{\\boldsymbol{\\theta}_1^{(k)}}\\left(\\mathbf{K}_{X_s X_s}^{\\boldsymbol{\\theta}_1^{(k)}}+\\lambda I\\right)^{-1} y_s\\right\\|^2\n\\end{equation}\nand\n\\begin{equation}\n    \\boldsymbol{\\theta}_1^{(k)} = \\boldsymbol{\\theta}_1^{(k-1)} - \\eta \\nabla_{ \\boldsymbol{\\theta}_1^{(k-1)}}\\mathcal{R}_{\\mathcal{S}}(\\boldsymbol{\\theta}_1^{(k-1)}),\n\\end{equation}\nwhere the kernel matrices $\\mathbf{K}_{X_t X_s}^{\\boldsymbol{\\theta}_1}$ and $\\mathbf{K}_{X_s X_s}^{\\boldsymbol{\\theta}_1}$ are induced by $h_{\\boldsymbol{\\theta}_1}(\\cdot)$. Notably, although the feature extractor $h_{\\boldsymbol{\\theta}_1}$ is continuously updated, the classifier $g_{\\boldsymbol{\\theta}_2}$ is directly solved by KRR, and thus the meta-gradient backpropagation side-steps the recursive computation graph.\n\n\n\\subsection{Discussion}\n\n\n\nFrom the loss surface perspective \\citep{li2018visualizing}, the meta-learning framework of minimizing $\\mathcal{R}_{\\mathcal{T}}\\left(\\texttt{alg}\\left(\\mathcal{S}\\right)\\right)$ can be considered to mimic the local minima of target data with the distilled data. However, the loss landscape {\\it w.r.t.} parameters is closely related to the network architecture, while only one type of small network is used in the BPTT approach. Consequently, there is a moderate performance drop when the distilled dataset is employed to train other complicated networks. Moreover, a long unrolled trajectory and second-order gradient computation are also two key challenges for BPTT approach, which hinder its efficiency. The KRR approach compensates for these shortcomings by replacing networks with the nonparametric kernel model which admits closed-form solution. Although KRR is nonparametric and does not involve neural networks during the distillation process, previous research has shown that the training dynamic of neural networks is equal to the kernel method when the width of networks becomes infinite \\citep{jacot2018neural,golikov2022neural,bietti2019inductive,bietti2019inductive}, which partially guarantees the feasibility of the kernel regression approach and explains its decent performance when transferred to wide neural networks.\n\n\n\n\n\n\n\n\n\n\\section{Data Matching Framework}\n\n\n\nAlthough it is not feasible to explicitly extract information from target data and then inject them into synthetic data, information distillation can be achieved by implicitly aligning the byproducts of target data and synthetic data from different aspects; and this byproduct matching allows synthetic data to imitate the influence of target data on model training. The objective function of data matching can be summarized as follows.\n\\begin{equation}\n    \\mathcal{L}(\\mathcal{S}) = \\sum_{k=0}^{T} D\\left( \\phi(\\mathcal{S},\\boldsymbol{\\theta}^{(k)}), \\phi(\\mathcal{T},\\boldsymbol{\\theta}^{(k)}) \\right)\n\\end{equation}\nsubject to\n\\begin{equation}\n\\label{eq:dm_update}\n    \\boldsymbol{\\theta}^{(k)} = \\boldsymbol{\\theta}^{(k-1)} - \\eta \\nabla_{\\boldsymbol{\\theta}^{(k-1)}} \\mathcal{R}_{\\mathcal{S}}\\left(\\boldsymbol{\\theta}^{(k-1)}\\right),\n\\end{equation}\nwhere $D(\\cdot,\\cdot)$ is a distance function, and $\\phi(\\cdot)$ maps the dataset $\\mathcal{S}$ or $\\mathcal{T}$ to other informative spaces, such as gradient, parameter, and feature spaces. In practical implementation, the full datasets of $\\mathcal{S}$ and $\\mathcal{T}$ are often replaced with random sampled batches of $\\mathcal{B}_\\mathcal{S}$ and $\\mathcal{B}_\\mathcal{T}$ for memory saving and faster convergence. \n\nCompared to the aforementioned meta-learning framework, the data matching loss not only focuses on the final parameter $\\texttt{alg}(\\mathcal{S})$ but also supervises the intermediate states, as shown in the sum operation $\\sum_{k=0}^{T}$. By this, the distilled data can better imitate the influence of target data on training networks at different training stages.\n\n\n\\iffalse\nTo circumvent the computation and memory inefficiency brought by the recursive computation graph, recent methods attempt to explore a substitute for the meta-loss $\\mathcal{R}_{\\mathcal{T}}\\left( \\texttt{alg}\\left(\\mathcal{S}\\right) \\right)$.\nWith the common sense that similar model parameters induce similar performance, the objective of DD is converted to match the model parameters trained on $\\mathcal{S}$ and $\\mathcal{T}$. As the final well-trained parameters can be regarded as an accumulation of parameter gradients, the matching objective can be further converted from parameters to their gradients induced by $\\mathcal{S}$ and $\\mathcal{T}$. With the parameter or gradient matching, $\\mathcal{S}$ can mimic the influence of $\\mathcal{T}$ on network parameters, and consequently helps learn a satisfying model for prediction. \n\nExcept for shortening the distance between parameters or gradients, other works also study DD from feature perspective \\citep{zhao2021distribution}: to represent $\\mathcal{T}$ with $\\mathcal{S}$, the synthetic points in $\\mathcal{S}$ should be as close to the examples in $\\mathcal{T}$ as possible in terms of feature space, and consequently distribution or feature matching are developed for this distillation. Compared to the above parameter or gradient matching, distribution matching can more fully grasp the characteristic of original data distribution with the synthetic dataset $\\mathcal{S}$.\n\n\n\nOverall, these methods achieve dataset distillation from $\\mathcal{T}$ to $\\mathcal{S}$ by matching their byproducts from different perspectives of parameter, gradient, and feature space, and the objective function can be summarized as follows. \n\n\n\n\n\n\n\n\n\n\n\\begin{equation}\n    \\mathcal{L}(\\mathcal{S}) = \\sum_{k=0}^{T-1} D\\left( \\phi(\\mathcal{S},\\boldsymbol{\\theta}^{(k)}), \\phi(\\mathcal{T},\\boldsymbol{\\theta}^{(k)}) \\right),\n\\end{equation}\nwhere $D(\\cdot,\\cdot)$ is a distance function, and $\\phi(\\cdot)$ maps the dataset $\\mathcal{S}$ or $\\mathcal{T}$ to gradient, feature, or parameter space. For example, $\\phi(\\mathcal{S},\\boldsymbol{\\theta}^{(k)}) = \\boldsymbol{g}_{\\mathcal{S}}^{(k)}$ in the gradient match. For distribution matching, the feature extractor, usually a set of pre-trained networks, is employed to achieve this mapping to feature space.\n\nCompared to the meta-loss approach, the loss $\\mathcal{L}^{(k)}(\\mathcal{S}) = D\\left( \\phi(\\mathcal{S},\\boldsymbol{\\theta}^{(k)}), \\phi(\\mathcal{T},\\boldsymbol{\\theta}^{(k)}) \\right) $ is only related to the current parameter $\\boldsymbol{\\theta}^{(k)}$ and does not acquire previous states, which decomposes the recursive computation graph and the backward propagation of the meta-gradient $\\nabla_{\\mathcal{S}}\\mathcal{L}$ consequently becomes more efficient, as shown in Figure \\ref{figure:matchloss}.\n\\fi\n\n\\subsection{Gradient Matching Approach}\nTo achieve comparable generalization performance, a natural approach is to imitate the model parameters, {\\it i.e.}, matching the training trajectories introduced by $\\mathcal{S}$ and $\\mathcal{T}$. With a fixed parameter initialization, the training trajectory of $\\{\\boldsymbol{\\mathcal{\\theta}}^{(0)}, \\boldsymbol{\\mathcal{\\theta}}^{(1)}, \\cdots, \\boldsymbol{\\mathcal{\\theta}}^{(T)} \\}$ is equal to a series of gradients $\\{\\boldsymbol{g}^{(0)}, \\cdots, \\boldsymbol{g}^{(T)}\\}$. Therefore, matching the gradients induced by $\\mathcal{S}$ and $\\mathcal{T}$ is a convincing proxy to mimic the influence on model parameters \\citep{zhao2021dataset}, and the objective function is formulated as\n\\begin{equation}\n    \\mathcal{L}(\\mathcal{S}) =  \\sum_{k=0}^{T-1} D\\left( \\nabla_{\\boldsymbol{\\theta}^{(k)}} \\mathcal{R}_{\\mathcal{S}}\\left(\\boldsymbol{\\theta}^{(k)}\\right), \\nabla_{\\boldsymbol{\\theta}^{(k)}} \\mathcal{R}_{\\mathcal{T}}\\left(\\boldsymbol{\\theta}^{(k)}\\right) \\right).\n\\end{equation}\nThe difference $D$ between gradients is measured in the layer-wise aspect:\n\\begin{equation}\n    D\\left( \\boldsymbol{g}_{\\mathcal{S}}, \\boldsymbol{g}_{\\mathcal{T}} \\right) = \\sum_{l=1}^L \\texttt{dis}\\left( \\boldsymbol{g}_{\\mathcal{S}}^{l}, \\boldsymbol{g}_{\\mathcal{T}}^{l} \\right),\n\\end{equation}\nwhere $\\boldsymbol{g}^{l}$ denotes the gradient of $i$-th layer, and $\\texttt{dis}$ is the sum of cosine distance as follows:\n\\begin{equation}\n\\label{eq:dis}\n    \\texttt{dis}\\left(\\boldsymbol{A}, \\boldsymbol{B}\\right) = \\sum_{i=1}^{\\texttt{out}} \\left(1 - \\frac{\\boldsymbol{A}_i \\boldsymbol{B}_i}{ \\| \\boldsymbol{A}_i \\| \\| \\boldsymbol{B}_i \\|} \\right),\n\\end{equation}\nwhere $\\texttt{out}$ denotes the number of output channels for specific layer; and $\\boldsymbol{A}_i$ and $\\boldsymbol{B}_i$ are the flatten gradients in the $i$-th channel.\n\nTo improve the convergence speed in practical implementation, \\citet{zhao2021dataset} proposed to match gradient in class-wise as follows:\n\\begin{equation}\n\\label{eq:interclass_gm}\n    \\mathcal{L}^{(k)}=\\sum_{c=1}^C D\\left(\\nabla_{\\boldsymbol{\\theta}^{(k)}} \\mathcal{R}\\left(\\mathcal{B}_\\mathcal{S}^c, \\boldsymbol{\\theta}^{(k)}\\right), \\nabla_{\\boldsymbol{\\theta}^{(k)}} \\mathcal{R}\\left(\\mathcal{B}_\\mathcal{T}^c,  \\boldsymbol{\\theta}^{(k)}\\right) \\right),\n\\end{equation}\nwhere $c$ is the class index and $\\mathcal{B}_\\mathcal{S}^c$ and $\\mathcal{B}_\\mathcal{T}^c$ denotes the batch examples belong to the $c$-th class. However, according to \\citet{lee2022dataset}, the class-wise gradient matching pays much attention to the class-common features and overlooks the class-discriminative features in the target dataset, and the distilled synthetic dataset $\\mathcal{S}$ does not possess enough class-discriminative information, especially when the target dataset is fine-grained; {\\it i.e.}, class-common features are the dominant. Based on this finding, these coworkers proposed an improved objective function of\n\\begin{equation}\n\\label{eq:intraclass_gm}\n    \\mathcal{L}^{(k)}=D\\left(\\sum_{c=1}^C\\nabla_{\\boldsymbol{\\theta}^{(k)}} \\mathcal{R}\\left(\\mathcal{B}_\\mathcal{S}^c, \\boldsymbol{\\theta}^{(k)}\\right), \\sum_{c=1}^C \\nabla_{\\boldsymbol{\\theta}^{(k)}} \\mathcal{R}\\left(\\mathcal{B}_\\mathcal{T}^c, \\boldsymbol{\\theta}^{(k)}\\right) \\right)\n\\end{equation}\nto better capture constractive signals between different classes through the sum of loss gradients between classes. A similar approach was proposed by \\citet{jiang2022delving}, which employed both intraclass and interclass gradient matching by combining Eq. \\ref{eq:interclass_gm} and Eq. \\ref{eq:intraclass_gm}. Moreover, these researchers measure the difference of gradients by considering the magnitude instead of only considering the angle, {\\it i.e.}, the cosine distance, and the distance function of Eq. \\ref{eq:dis} is improved to\n\\begin{equation}\n    \\texttt{dis}\\left(\\boldsymbol{A}, \\boldsymbol{B}\\right) = \\sum_{i=1}^{\\texttt{out}} \\left(1 - \\frac{\\boldsymbol{A}_i \\boldsymbol{B}_i}{ \\| \\boldsymbol{A}_i \\| \\| \\boldsymbol{B}_i \\|} + \\| \\boldsymbol{A}_i - \\boldsymbol{B}_i \\|\\right).\n\\end{equation}\nTo alleviate easy overfitting on the small dataset $\\mathcal{S}$, \\citet{kim2022dataset} proposed to perform inner loop optimization on the target dataset $\\mathcal{T}$ instead of $\\mathcal{S}$, {\\it i.e.}, replaced the parameter update of Eq. \\ref{eq:dm_update} with\n\\begin{equation}\n    \\boldsymbol{\\theta}^{(k+1)} = \\boldsymbol{\\theta}^{(k)} - \\eta \\nabla_{\\boldsymbol{\\theta}^{(k)}} \\mathcal{R}_\\mathcal{T}\\left(\\boldsymbol{\\theta}^{(k)}\\right).\n\\end{equation}\n\nAlthough data augmentation facilitates a large performance increase in conventional network training, conducting augmentation on distilled datasets demonstrates no improvement on the final test accuracy, because the characteristics of synthetic images are different from those of natural images and are not optimized under the supervision of various transformations. To leverage data augmentation on synthetic datasets, \\citet{zhao2021dsa} designed data Siamese augmentation (DSA) that homologously augments the distilled data and the target data during the distillation process as follows:\n\\begin{equation}\n    \\mathcal{L} = D\\left(\\nabla_{\\boldsymbol{\\theta}} \\mathcal{R}\\left(\\mathcal{A}_{\\omega}\\left(\\mathcal{B}_\\mathcal{S} \\right), \\boldsymbol{\\theta}\\right), \\nabla_{\\boldsymbol{\\theta}} \\mathcal{R}\\left(\\mathcal{A}_{\\omega}\\left(\\mathcal{B}_\\mathcal{T} \\right), \\boldsymbol{\\theta}\\right) \\right),\n\\end{equation}\nwhere $\\mathcal{A}$ is a family of image transformations such as cropping and flipping that are parameterized with $\\omega^{\\mathcal{S}}$ and $\\omega^{\\mathcal{T}}$ for synthetic and target data, respectively. In DSA, the augmented form of distilled data has a consistent correspondence {\\it w.r.t.} the augmented form of the target data, {\\it i.e.}, $\\omega^{\\mathcal{S}} = \\omega^{\\mathcal{T}} = \\omega$; and $\\omega$ is randomly picked from $\\mathcal{A}$ at different iterations. Notably, the transformation $\\mathcal{A}$ requires to be differentiable {\\it w.r.t.} the synthetic dataset $\\mathcal{S}$ for backpropagation:\n\\begin{equation}\n\\frac{\\partial D(\\cdot)}{\\partial \\mathcal{S}}=\\frac{\\partial D(\\cdot)}{\\partial \\nabla_\\theta \\mathcal{L}(\\cdot)} \\frac{\\partial \\nabla_\\theta \\mathcal{L}(\\cdot)}{\\partial \\mathcal{A}(\\cdot)} \\frac{\\partial \\mathcal{A}(\\cdot)}{\\partial \\mathcal{S}}.\n\\end{equation}\n\nThrough setting $\\omega^{\\mathcal{S}} = \\omega^{\\mathcal{T}}$, DSA permits the knowledge transfer from the transformation of target images to the corresponding transformation of synthetic images. Consequently, the augmented synthetic images also possess meaningful characteristics of the natural images. Due to its superior compatibility, DSA has been widely used in many data matching methods. \n\n\\begin{figure}[t]\n\\centering\n\n\\subfigure[Trajectory matching]{\n\\begin{minipage}[b]{0.62\\textwidth}\n\\centering\n    \t\t\\includegraphics[width=0.9\\columnwidth]{figures\/mtt_illustration.pdf}\n    \t\t\\end{minipage}\n\t\t\\label{figure:mtt_illustration}   \n    \t}\n\\subfigure[Distribution matching]{\n\\begin{minipage}[b]{0.3\\textwidth}\n\\centering\n    \t\t\\includegraphics[width=0.9\\columnwidth]{figures\/dm_illustration.pdf}\n    \t\t\\end{minipage}\n\t\t\\label{figure:dm_illustration}   \n    \t}\n\n\\caption{(a) An illustration of trajectory matching \\citep{cazenavette2022distillation}; (b) Compared to gradient matching approach, distribution matching approach can more comprehensively cover the data distribution in feature space.}\n\\label{figure:match illustration}\n\\end{figure}\n\n\\subsection{Trajectory Matching Approach}\n\nUnlike circuitously matching the gradients, \\citet{cazenavette2022distillation} directly matched the long-range training trajectory between the target dataset and the synthetic dataset. Specifically, they train models on the target dataset $\\mathcal{T}$ and collect the expert training trajectory into a buffer in advance. Then the ingredients in the buffer are randomly selected to initialize the networks for training $\\mathcal{S}$. After collecting the trajectories of $\\mathcal{S}$, the synthetic dataset is updated by matching their parameters, as shown in Figure \\ref{figure:mtt_illustration}. The objective loss of trajectory matching is defined as \n\\begin{equation}\n\\label{eq:mtt_loss}\n    \\mathcal{L}= \\frac{\\| \\boldsymbol{\\theta}^{(k+N)} - \\boldsymbol{\\theta}_{\\mathcal{T}}^{(k+M)} \\|_2 ^2}{ \\|\\boldsymbol{\\theta}_{\\mathcal{T}}^{(k)} - \\boldsymbol{\\theta}_{\\mathcal{T}}^{(k+M)} \\|_2 ^2},\n\\end{equation}\nwhere $\\boldsymbol{\\theta}_{\\mathcal{T}}$ denotes the target parameter by training the model on $\\mathcal{T}$, which is stored in the buffer, and $\\boldsymbol{\\theta}^{(k+N)}$ are the parameter obtained by training the model on $\\mathcal{S}$ for $N$ epochs with the initialization of $\\boldsymbol{\\theta}_{\\mathcal{T}}^{(k)}$. The denominator in the loss function is for normalization.\n\n\nAlthough trajectory matching demonstrated empirical success, \\citet{du2022minimizing} argued that $\\boldsymbol{\\theta}^{(k)}$ is generally induced by training on $\\mathcal{S}$ for $k$ epochs during natural learning process, while it was directly initialized with $\\boldsymbol{\\theta}_{\\mathcal{T}}^{(k)}$ in trajectory matching, which causes the {\\it accumulated trajectory error} that measures the difference between the parameters from real trajectories of $\\mathcal{S}$ and $\\mathcal{T}$. Because the accumulated trajectory error is induced by the mismatch between $\\boldsymbol{\\theta}_{\\mathcal{T}}^{(k)}$ and real $\\boldsymbol{\\theta}^{(k)}$ from the natural training, these researchers alleviate this error by adding random noise to when initialize $\\boldsymbol{\\theta}^{(k)}$ to improve robustness {\\it w.r.t.} $\\boldsymbol{\\theta}_{\\mathcal{T}}^{(k)}$.\n\n\n\n\nCompared to matching gradients, while trajectory matching side-steps second-order gradient computation, unfortunately, unrolling $N$ SGD updates are required during meta-gradient backpropagation due to the existence of $\\boldsymbol{\\theta}^{(k+N)}$ in Eq. \\ref{eq:mtt_loss}. The unrolled gradient computation significantly increases the memory burden and impedes scalability. By disentangling the meta-gradient {\\it w.r.t.} synthetic examples into two passes, \\citet{cui2022scaling} greatly reduced the memory required by trajectory matching and successfully scaled trajectory matching approach to the large ImageNet-1K dataset \\citep{russakovsky2015imagenet}. Motivated by knowledge distillation \\citep{gou2021knowledge}, these scholars also proposed assigning soft-labels to synthetic examples with pretrained models in the buffer; and the soft-labels help learn intraclass information and consequently improve distillation performance.\n\n\n\n\n\n\n\n\\subsection{Distribution Matching Approach}\n\n\\begin{wrapfigure}{r}{0.5\\textwidth}\n    \\centering\n    \\includegraphics[width=0.45\\columnwidth]{figures\/dm.pdf}\n    \\caption{An illustration of distribution matching. $\\psi$ is the feature extractor, and red dashed lines denote the backpropagation of gradients.}\n    \\label{fig:dm}\n\\end{wrapfigure}\nAlthough the above parameter-wise matching shows satisfying performance, \\citet{zhao2023distribution} visualized the distilled data in two dimension and revealed that there is a large distribution discrepancy between the distilled data and the target data. In other words, the distilled dataset cannot comprehensively cover the data distribution in feature space, as shown in Figure \\ref{figure:dm_illustration}. Based on this discovery, these researchers proposed to match the synthetic and target data from the distribution perspective for dataset distillation. Specifically, they employed the pretrained feature extractor $\\psi_{\\boldsymbol{v}}$ with the parameter $\\boldsymbol{v}$ to achieve mapping from the input space to the feature space. The synthetic data are optimized according to the objective function \n\\begin{equation}\n\\label{eq:dm}\n    \\mathcal{L}(\\mathcal{S})=\\sum_{c=1}^C \\|\\psi_{\\boldsymbol{v}}(\\mathcal{B}_\\mathcal{S}^c) - \\psi_{\\boldsymbol{v}}(\\mathcal{B}_\\mathcal{T}^c)  \\|^2,\n\\end{equation}\nwhere $c$ denotes the class index. An illustration of distribution matching is also presented Figure \\ref{fig:dm}. As shown in Eq. \\ref{eq:dm}, distribution matching does not rely on the model parameters and drops bilevel optimization for less memory requirement, whereas \nit empirically underperforms the above gradient and trajectory matching approaches. \n\n\\citet{wang2022cafe} improved the distribution matching from several aspects: (1) using multiple-layer features other than only the last-layer features for matching, and the feature matching loss is formulated as follows:\n\\begin{equation}\n    \\mathcal{L}_{\\text{f}} = \\sum_{c=1}^C \\sum_{l=1}^L |f_{\\boldsymbol{\\theta}}^l (\\mathcal{B}_\\mathcal{S}^c) - f_{\\boldsymbol{\\theta}}^l (\\mathcal{T}_\\mathcal{S}^c)|^2,\n\\end{equation}\nwhere $f_{\\boldsymbol{\\theta}}^l$ denotes the $l$-th layer features and $L$ is the total number of network layers; (2) recalling the bilevel optimization that updates $\\mathcal{S}$ with different model parameters by inserting the inner loop of $\\boldsymbol{\\theta}^{(k+1)} = \\boldsymbol{\\theta}^{(k)} -  \\eta\\nabla_{\\boldsymbol{\\theta}^{(k)}} \\mathcal{R}_{\\mathcal{S}}(\\boldsymbol{\\theta}^{(k)})$;\nand (3) proposing the discrimination loss in the last-layer feature space to enlarge the class distinction of synthetic data. Specifically, the synthetic feature center $\\bar{f}^\\mathcal{S}_{c,L}$ of each category $c$ is obtained by averaging the batch $\\mathcal{B}_\\mathcal{S}^c$. The objective of discrimination loss is to improve the classification ability of the feature center $\\bar{\\boldsymbol{F}}_{L}^{\\mathcal{S}} = [\\bar{f}^\\mathcal{S}_{1,L}, \\bar{f}^\\mathcal{S}_{2,L}, \\cdots, \\bar{f}^\\mathcal{S}_{C,L}]$ on predicting the real data feature $\\boldsymbol{F}_L^{\\mathcal{T}}=[{f}^\\mathcal{T}_{1,L}, {f}^\\mathcal{T}_{1,L}, \\cdots, {f}^\\mathcal{T}_{C,L}]$. The logits are first calculated by\n\\begin{equation}\n    \\mathbf{O} = \\left\\langle \\mathbf{\\boldsymbol{F}}_{L}^{\\mathcal{T}}, \\left(\\bar{\\boldsymbol{F}}_{L}^\\mathcal{S}\\right)^\\top \\right \\rangle,\n\\end{equation}\nwhere $\\mathbf{O}\\in \\mathbb{R}^{N\\times C}$ contains the logits of $N=C\\times |\\mathcal{B}_\\mathcal{T}^c|$ target data points. Then the probability $p_i$ in terms of the ground-truth label is derived via $p_i = \\text{softmax}(\\mathbf{O}_i)$; and the classification loss is\n\\begin{equation}\n    \\mathcal{L}_{\\text{d}} = \\frac{1}{N} \\sum_{i=1}^N \\log p_i.\n\\end{equation}\nTherefore, the total loss in \\citep{wang2022cafe} for dataset distillation is $\\mathcal{L}_{\\text{total}} = \\mathcal{L}_{\\text{f}} + \\beta \\mathcal{L}_{\\text{d}}$, where $\\beta$ is a positive factor for balancing the feature matching and discrimination loss.\n\n\n\n\n\n\\subsection{Discussion}\n\nThe gradient matching approach can be considered as to be short-range parameter matching, while its backpropagation requires second-order gradient computation. Although the trajectory matching approach make ups for this imperfection, the long-range trajectory introduces a recursive computation graph during meta-gradient backpropagation. Different from matching in parameter space, distribution matching approach employs the feature space as the match proxy and also bypasses second-order gradient computation. Although distribution matching has advantages in scalability {\\it w.r.t.} high-dimensional data, it empirically underperforms trajectory matching, which might be attributed to the mismatch between the comprehensive distribution and decent distillation performance. In detail, distribution matching achieves DD by mimicking features of the target data, so that the distilled data are evenly distributed in the feature space. However, not all features are equally important and distribution matching might waste the budget on imitating less informative features, thereby undermining the distillation performance. \n\n\n\n\n\n\n\\iffalse\n\\section{Distillation schemes}\n\\label{sec:distillation schemes}\n\nIn this section, we taxonomise the dataset distillation schemes according to their optimized objectives. Because dataset distillation can be formulated as a complicated bilevel problem and the closed-form solution is intractable, the distilled dataset is optimized by the nested outer loop and inner loop gradient descent: the inner loop, {\\it i.e.}, model training, optimizes the parameter $\\boldsymbol{\\theta}$ according to the distilled dataset, and the outer loop optimization is responsible for the synthetic dataset updating and is the core of dataset distillation algorithms. The outer loop optimization can be concisely summarized as three steps: (1) compute the objective function $\\mathcal{L}(\\mathcal{S})$ that is induced by the distilled dataset $\\mathcal{S}$ and can reflect the distillation performance; (2) get the gradient $\\nabla_{\\mathcal{S}}\\mathcal{L}$ {\\it w.r.t.} the distilled data; and (3) update the distilled dataset with the gradient. Therefore, the key to dataset distillation is the design of objective function $\\mathcal{L}(\\mathcal{S})$, which closely relates to the computation of gradients and determines the distillation performance and also efficiency. In this survey, we divide the objective function of dataset distillation into three categories: (1) meta-loss approach; (2) data matching approach; and (3) kernel regression approach.\n\\fi\n\n\n\n\n\n\n\\iffalse\n\\subsection{Meta-loss Approach}\n\nMotivated by meta-learning, which strives to obtain a set of satisfied hyperparameters for fast model training \\citep{maclaurin2015gradient,finn2017model}, most pioneered dataset distillation methods regard the distilled dataset as hyperparameters and directly define the objective function as the meta-loss of \n\\begin{equation}\n    \\mathcal{L}(\\mathcal{S}) = \\mathcal{R}_{\\mathcal{T}}\\left(\\texttt{alg}\\left(\\mathcal{S}\\right)\\right).\n\\end{equation}\nThen the distilled dataset is updated by $\\mathcal{S} = \\mathcal{S} - \\alpha \\nabla_\\mathcal{S} \\mathcal{L}(\\mathcal{S})$ with the step size $\\alpha$. However, as the neural network is adopted in the distillation process, the training algorithm $\\texttt{alg}$ on the model parameter $\\boldsymbol{\\theta}$ is iterative (Eq. \\ref{eq:gradient descent}) in the inner loop. Hence, the computation of meta-gradient $\\nabla_\\mathcal{S}\\mathcal{L}(\\mathcal{S})$ requires unrolling the recursive computation graph, which retraces to the previous states of $\\{\\boldsymbol{\\mathcal{\\theta}}^{(0)}, \\boldsymbol{\\mathcal{\\theta}}^{(1)}, \\cdots, \\boldsymbol{\\mathcal{\\theta}}^{(T)} \\}$, as shown in Figure \\ref{figure:metaloss}. For this reason, albeit the meta-loss approach is brief and clear, this unroll is both computing and memory expensive, which also severely affects the final distillation performance.\n\n\n\n\\subsection{Data Matching Approach}\n\n\nTo circumvent the computation and memory inefficiency brought by the recursive computation graph, recent methods attempt to explore a substitute for the meta-loss $\\mathcal{R}_{\\mathcal{T}}\\left( \\texttt{alg}\\left(\\mathcal{S}\\right) \\right)$.\nWith the common sense that similar model parameters induce similar performance, the objective of DD is converted to match the model parameters trained on $\\mathcal{S}$ and $\\mathcal{T}$. As the final well-trained parameters can be regarded as an accumulation of parameter gradients, the matching objective can be further converted from parameters to their gradients induced by $\\mathcal{S}$ and $\\mathcal{T}$. With the parameter or gradient matching, $\\mathcal{S}$ can mimic the influence of $\\mathcal{T}$ on network parameters, and consequently helps learn a satisfying model for prediction. \n\nExcept for shortening the distance between parameters or gradients, other works also study DD from feature perspective \\citep{zhao2021distribution}: to represent $\\mathcal{T}$ with $\\mathcal{S}$, the synthetic points in $\\mathcal{S}$ should be as close to the examples in $\\mathcal{T}$ as possible in terms of feature space, and consequently distribution or feature matching are developed for this distillation. Compared to the above parameter or gradient matching, distribution matching can more fully grasp the characteristic of original data distribution with the synthetic dataset $\\mathcal{S}$.\n\n\n\nOverall, these methods achieve dataset distillation from $\\mathcal{T}$ to $\\mathcal{S}$ by matching their byproducts from different perspectives of parameter, gradient, and feature space, and the objective function can be summarized as follows. \n\n\n\n\n\n\n\n\n\\begin{equation}\n    \\mathcal{L}(\\mathcal{S}) = \\sum_{k=0}^{T-1} D\\left( \\phi(\\mathcal{S},\\boldsymbol{\\theta}^{(k)}), \\phi(\\mathcal{T},\\boldsymbol{\\theta}^{(k)}) \\right),\n\\end{equation}\nwhere $D(\\cdot,\\cdot)$ is a distance function, and $\\phi(\\cdot)$ maps the dataset $\\mathcal{S}$ or $\\mathcal{T}$ to gradient, feature, or parameter space. For example, $\\phi(\\mathcal{S},\\boldsymbol{\\theta}^{(k)}) = \\boldsymbol{g}_{\\mathcal{S}}^{(k)}$ in the gradient matching. For distribution matching, the feature extractor, usually a set of pre-trained networks, is employed to achieve this mapping to feature space.\n\nCompared to the meta-loss approach, the loss $\\mathcal{L}^{(k)}(\\mathcal{S}) = D\\left( \\phi(\\mathcal{S},\\boldsymbol{\\theta}^{(k)}), \\phi(\\mathcal{T},\\boldsymbol{\\theta}^{(k)}) \\right) $ is only related to the current parameter $\\boldsymbol{\\theta}^{(k)}$ and does not acquire previous states, which decomposes the recursive computation graph and the backward propagation of the meta-gradient $\\nabla_{\\mathcal{S}}\\mathcal{L}$ consequently becomes more efficient, as shown in Figure \\ref{figure:matchloss}.\n\n\n\\subsection{Kernel Regression Approach}\n\nUnlike training neural networks where the optimum is intractable, some methods consider dataset distillation in the kernel regression regime that admits a closed-form solution. For the regression model $f(\\boldsymbol{x})=\\boldsymbol{w}^\\top\\psi(\\boldsymbol{x})$, where $\\psi(\\cdot)$ is a non-linear mapping and the corresponding kernel is $K(\\boldsymbol{x},\\boldsymbol{x}^\\prime)=\\langle \\psi(\\boldsymbol{x}), \\psi(\\boldsymbol{x}^\\prime)\\rangle$, there exists a closed-form solution for $\\boldsymbol{w}$ when the regression model is trained on $\\mathcal{S}$ with kernel ridge regression (KRR):\n\\begin{equation}\n   \\boldsymbol{w} =  \\psi(X_{s})^\\top\\left(\\mathbf{K}_{X_s X_s}+\\lambda I\\right)^{-1}y_{s},\n\\end{equation}\nwhere $\\mathbf{K}_{X_s X_s} = [K(\\boldsymbol{s}_i,\\boldsymbol{s}_j)]_{ij}\\in \\mathbb{R}^{n\\times n}$ is called the {\\it kernel matrix} or {\\it Gram matrix} associated to $K$ and the dataset $\\mathcal{S}$, and $\\lambda > 0$ is a fixed regularization parameter. Therefore, the mean square error (MSE) of predicting $\\mathcal{T}$ with the model trained on $\\mathcal{S}$ is\n\\begin{equation}\n    \\mathcal{L}(\\mathcal{S}) = \\frac{1}{2}\\left\\|y_t-\\mathbf{K}_{X_t X_s}\\left(\\mathbf{K}_{X_s X_s}+\\lambda I\\right)^{-1} y_s\\right\\|^2,\n\\end{equation}\nwhere $\\mathbf{K}_{X_t X_s}=[K(\\boldsymbol{x}_i,\\boldsymbol{s}_j)]_{ij}\\in \\mathbb{R}^{m\\times n}$. Then the distilled dataset is updated via the gradient of the above loss. Due to the closed-form solution in KRR, $\\boldsymbol{\\theta}$ does not require an iterative update and the backward pass of gradient thus bypasses the recursive computation graph, as shown in Figure \\ref{figure:kernelloss}. \n\nIt is worth noting that the KRR is non-parametric and does not involve neural networks during the distillation process. However, previous research has shown that the training dynamic of neural networks is equal to the kernel method when the width of networks becomes infinite \\citep{jacot2018neural}, which partially guarantees the feasibility of the kernel regression approach and explains its decent performance when transferred to the neural networks.\n\n\n\\fi\n\n\\iffalse\n\\begin{table}[t]\n    \n    \\centering\n    \\caption{A summary of dataset distillation algorithms}\n    \\begin{tabular}{c|c}\n    \\toprule\n       Methods  & Distillation schemes   \\\\\n    \\hline\n       DD (\\citet{wang2018dataset})  & BPTT  \\\\ \n       LD (\\citet{bohdal2020flexible}) & BPTT   \\\\\n       SLDD (\\citet{sucholutsky2021soft}) & BPTT   \\\\\n       DC (\\citet{zhao2021dataset})& Gradient match   \\\\\n      \n       DCC (\\citet{lee2022dataset}) & Gradient match  \\\\\n       MTT (\\citet{cazenavette2022distillation}) & Trajectory match  \\\\\n       FTD (\\citet{du2022minimizing})& Trajectory match   \\\\\n       TESLA (\\citet{cui2022scaling})& Trajectory match   \\\\\n       DM (\\citet{zhao2021distribution}) & Distribution match  \\\\\n       CAFE (\\citet{wang2022cafe})& Distribution match   \\\\\n       KIP (\\citet{nguyen2020dataset,nguyen2021dataset}) & Kernel ridge regression   \\\\\n       FRePo (\\citet{zhou2022dataset})& Kernel ridge regression   \\\\\n       RFAD (\\citet{loo2022efficient})& Kernel ridge regression   \\\\\n       GTN (\\citet{such2020generative}) & BPTT  \\\\\n       GAN (\\citet{zhao2022synthesizing}) & Distribution match \\\\\n       IDC (\\citet{kim2022dataset}) & Gradient match  \\\\\n       \\citet{deng2022remember} & BPTT  \\\\\n       Haba (\\citet{liu2022dataset}) & Trajectory match  \\\\\n       KFS (\\citet{lee2022kfs}) & Distribution match   \\\\\n    \\bottomrule\n    \\end{tabular}\n    \\label{tab:my_label}\n\\end{table}\n\\fi\n\n\\iffalse\n\\begin{table}[t]\n    \n    \\centering\n    \\caption{A summary of dataset distillation algorithms}\n    \\begin{tabular}{c|c|c}\n    \\toprule\n       Methods  & Distillation schemes & Objective function  \\\\\n    \\hline\n       DD (\\citet{wang2018dataset})  & BPTT & $\\mathcal{R}_{\\mathcal{T}}\\left(\\texttt{alg}\\left(\\mathcal{S}\\right)\\right)$  \\\\ \n       LD (\\citet{bohdal2020flexible}) & BPTT & $\\mathcal{R}_{\\mathcal{T}}\\left(\\texttt{alg}\\left(\\mathcal{S}\\right)\\right)$  \\\\\n       SLDD (\\citet{sucholutsky2021soft}) & BPTT & $\\mathcal{R}_{\\mathcal{T}}\\left(\\texttt{alg}\\left(\\mathcal{S}\\right)\\right)$   \\\\\n       DC (\\citet{zhao2021dataset})& Gradient match & $\\sum_{c=1}^C D\\left(\\nabla_{\\boldsymbol{\\theta}_t}\\mathcal{R}\\left(B_c^{\\mathcal{S}}, \\boldsymbol{\\theta}_t\\right), \\nabla_{\\boldsymbol{\\theta}_t}\\mathcal{R}\\left(B_c^{\\mathcal{T}}, \\boldsymbol{\\theta}_t\\right)\\right)$  \\\\\n      \n       DCC (\\citet{lee2022dataset}) & Gradient match &  $ D\\left( \\sum_{c=1}^C \\nabla_{\\boldsymbol{\\theta}_t}\\mathcal{R}\\left(B_c^{\\mathcal{S}}, \\boldsymbol{\\theta}_t\\right), \\sum_{c=1}^C \\nabla_{\\boldsymbol{\\theta}_t}\\mathcal{R}\\left(B_c^{\\mathcal{T}}, \\boldsymbol{\\theta}_t\\right)\\right)$ \\\\\n       MTT (\\citet{cazenavette2022distillation}) & Trajectory match & $\\| \\boldsymbol{\\theta}^{(k+N)} - \\boldsymbol{\\theta}_{\\mathcal{T}}^{(k+M)} \\|_2 ^2 \/ \\|\\boldsymbol{\\theta}_{\\mathcal{T}}^{(k)} - \\boldsymbol{\\theta}_{\\mathcal{T}}^{(k+M)} \\|_2 ^2$ \\\\\n      \n      \n       DM (\\citet{zhao2021distribution}) & Distribution match & $\\sum_{c=1}^{C}\\left\\|\\frac{1}{\\left|B_c^{\\mathcal{T}}\\right|}  \\psi_{\\boldsymbol{\\vartheta}}(B_c^{\\mathcal{S}})-\\frac{1}{\\left|B_c^{\\mathcal{S}}\\right|} \\psi_{\\boldsymbol{\\vartheta}}(B_c^{\\mathcal{T}})\\right\\|^2$ \\\\\n      \n       KIP (\\citet{nguyen2020dataset,nguyen2021dataset}) & Kernel regression & $\\frac{1}{2}\\left\\|y_t-\\mathbf{K}_{X_t X_s}\\left(\\mathbf{K}_{X_s X_s}+\\lambda I\\right)^{-1} y_s\\right\\|^2$  \\\\\n      \n      \n      \n      \n      \n      \n      \n      \n    \\bottomrule\n    \\end{tabular}\n    \\label{tab:my_label}\n\\end{table}\n\\fi\n\n\\iffalse\n\\section{Distillation Algorithms}\n\nAlbeit the objective of DD is to match the performance induced by $\\mathcal{S}$ and $\\mathcal{T}$ and is easy to describe, the realization of DD is not trivial due to many ingredients like high-dimensional data, intractable solution of neural networks, {\\it etc}. \nIn the following parts, we will review recently proposed dataset distillation algorithms in detail according to the taxonomy in Section \\ref{sec:distillation schemes}. Because the training algorithm $\\texttt{alg}\\left(\\mathcal{S},\\boldsymbol{\\theta}^{(0)}\\right)$ is both determined by the training set $\\mathcal{S}$ and the initialized parameter $\\boldsymbol{\\theta}^{(0)}$, many dataset distillation algorithms will take expectation {\\it w.r.t.} $\\boldsymbol{\\theta}^{(0)}$ in order to improve the robustness of the distilled dataset $\\mathcal{S}$ to different parameter initialization. In the following narrative, we will omit this expectation {\\it w.r.t.} initialization for the sake of simplicity.\n\n    \n\\subsection{Meta-loss Approach}\n\nFrom the above illustration in Section \\ref{sec:distillation schemes}, the most direct objective is to match the loss $\\mathcal{R}_{\\mathcal{T}}\\left(\\texttt{alg}\\left(\\mathcal{S}\\right)\\right)$ and $\\mathcal{R}_{\\mathcal{T}}\\left(\\texttt{alg}\\left(\\mathcal{T}\\right)\\right)$ {\\it w.r.t} the large target dataset $\\mathcal{T}$. As the empirical risk $\\mathcal{R}_{\\mathcal{T}}\\left(\\texttt{alg}\\left(\\mathcal{T}\\right)\\right)$ easily achieves zero due to the overparameterization, the objective is the same as minimizing the meta-loss $\\mathcal{R}_{\\mathcal{T}}\\left(\\texttt{alg}\\left(\\mathcal{S}\\right)\\right)$. \\citet{wang2018dataset} first realize this optimization via the hyperparameter optimization\\citep{maclaurin2015gradient}. To alleviate the inefficiency in unrolling the long parameter path of $\\{\\boldsymbol{\\mathcal{\\theta}}^{(0)}, \\boldsymbol{\\mathcal{\\theta}}^{(1)}, \\cdots, \\boldsymbol{\\mathcal{\\theta}}^{(T)} \\}$, they employ a single step optimization{\\it w.r.t.} the model parameter from $\\boldsymbol{\\theta}^{(0)}$ to $\\boldsymbol{\\theta}^{(1)}$, and the loss is computed based on $\\boldsymbol{\\theta}^{(1)}$ and the target dataset $\\mathcal{T}$. Besides, both the data $\\boldsymbol{s}\\in \\mathcal{S}$ and learning rate $\\eta$ are learnable in \\citet{wang2018dataset}. Unlike freezing labels, \\citet{sucholutsky2021soft} extend \\citet{wang2018dataset} by learning a soft-label in the synthetic dataset $\\mathcal{S}$, {\\it i.e.}, the label $y$ in the synthetic dataset is also trainable. Similarly, \\citep{bohdal2020flexible} also extend the standard example distillation to label distillation by solely optimizing the labels of synthetic datasets. Moreover, they provide improvements on the efficiency of long inner loop optimization via (1) introducing shorter but more frequent inner loop and (2) employing ridge regression to estimate the solution of classifiers to avoid second-order gradient computation. Albeit the BPTT framework has been shown to underperform other algorithms, \\citet{deng2022remember} empirically demonstrate that adding momentum term and longer trajectory in the inner loop optimization can considerably increase the distillation performance.\n\n\n\\iffalse\n\\begin{figure}[t]\n\\centering\n\n\\subfigure[]{\n\\begin{minipage}[b]{0.35\\textwidth}\n\\centering\n    \t\t\\includegraphics[width=\\columnwidth]{figures\/standard_dd.png}\n    \t\t\\end{minipage}\n\t\t\\label{figure:standard_dd}   \n    \t}\n\\subfigure[]{\n\\begin{minipage}[b]{0.26\\textwidth}\n\\centering\n    \t\t\\includegraphics[width=\\columnwidth]{figures\/standard_dd.png}\n    \t\t\\end{minipage}\n\t\t\\label{figure:xx1}   \n    \t}\n\\subfigure[]{\n\\begin{minipage}[b]{0.3\\textwidth}\n\\centering\n    \t\t\\includegraphics[width=\\columnwidth]{figures\/standard_dd.png}\n    \t\t\\end{minipage}\n\t\t\\label{figure:xx2}   \n    \t}\n\\caption{xxx.}\n\\label{figure:xx4}\n\\end{figure}\n\\fi\n\n   \n\n\\subsection{Gradient Matching Approach}\nAs optimizing the meta-loss is not easy, some substitutes of objective have been proposed to distil the dataset. To achieve comparable generalization performance, the intuition is to employ the distilled dataset to imitate the effect on model parameters, {\\it i.e.}, matching the training trajectories introduced by $\\mathcal{S}$ and $\\mathcal{T}$. With a fixed parameter initialization, the training trajectory of $\\{\\boldsymbol{\\mathcal{\\theta}}^{(0)}, \\boldsymbol{\\mathcal{\\theta}}^{(1)}, \\cdots, \\boldsymbol{\\mathcal{\\theta}}^{(T)} \\}$ is equal to a series of gradients $\\{\\boldsymbol{g}^{(0)}, \\cdots, \\boldsymbol{g}^{(T)}\\}$. Therefore, matching the gradients induced by $\\mathcal{S}$ and $\\mathcal{T}$ is a convincing proxy to mimic the influence on model parameters \\citep{zhao2021dataset}, and the objective function can be formulated as\n\\begin{equation}\n    \\mathcal{L}(\\mathcal{S}) =  \\sum_{k=0}^{T-1} D\\left( \\boldsymbol{g}_{\\mathcal{S}}^{(k)}, \\boldsymbol{g}_{\\mathcal{T}}^{(k)} \\right),\n\\end{equation}\nwhere $\\boldsymbol{g}_{\\mathcal{S}}^{(k)}$ and $\\boldsymbol{g}_{\\mathcal{T}}^{(k)}$ denote the gradient {\\it w.r.t.} model parameters generated by $\\mathcal{S}$ and $\\mathcal{T}$ in the $k$-th training epoch, respectively. It is worth noting that these two gradients are induced on the same parameter $\\boldsymbol{\\theta}^{(k)}$ to be cohesive with the bilevel optimization. Concretely, the gradient matching is class-wise: $\\mathcal{L}^{(k)}=\\sum_{c=1}^C D\\left(\\nabla_{\\boldsymbol{\\theta}^{(k)}} \\mathcal{R}\\left(\\mathcal{S}_c, \\boldsymbol{\\theta}^{(k)}\\right), \\nabla_{\\boldsymbol{\\theta}^{(k)}} \\mathcal{R}\\left(\\mathcal{T}_c, \\boldsymbol{\\theta}^{(k)}\\right) \\right)$, where $c$ is the class index and $\\mathcal{T}_c$ denotes the examples belong to the $c$-th class. According to \\citet{lee2022dataset}, the class-wise gradient matching pays much attention to the class-common features and overlooks the class-discriminative features in the target dataset, and the distilled synthetic dataset $\\mathcal{S}$ does not possess enough class-discriminative information, especially when the target dataset is fine-grained, {\\it i.e.}, class-common features are the dominant. Based on this finding, they propose a improved objective function of $\\mathcal{L}^{(k)}=D\\left(\\sum_{c=1}^C\\nabla_{\\boldsymbol{\\theta}^{(k)}} \\mathcal{R}\\left(\\mathcal{S}_c, \\boldsymbol{\\theta}^{(k)}\\right), \\sum_{c=1}^C \\nabla_{\\boldsymbol{\\theta}^{(k)}} \\mathcal{R}\\left(\\mathcal{T}_c, \\boldsymbol{\\theta}^{(k)}\\right) \\right)$ to better capture constractive signals between different classes. A similar approach is proposed by \\citet{jiang2022delving}, which considers both intraclass and inter-class gradient matching. To alleviate easy overfitting on the small dataset $\\mathcal{S}$, \\citet{kim2022dataset} propose to do inner loop optimization on the target dataset $\\mathcal{T}$. In spite of data augmentation brings a large performance increase, conducting augmentation on distilled datasets has no improvement on the final test accuracy, because the synthetic images have different characteristics compared to natural images and also are not optimized under the supervision of various transformations. To leverage data augmentation on synthetic datasets. \\citet{zhao2021dsa} design data siamese augmentation (DSA) that homologously augments the distilled data and the target data during the distillation process. In DSA, the augmented form of distilled data has a consistent correspondence {\\it w.r.t.} the augmented form of the target data, which permits the knowledge transfer from the transformation of target images to the corresponding transformation of synthetic images. Consequently, the augmented synthetic images also possess meaningful characteristics of the natural images. Due to its superior compatibility, DSA has been widely equipped in many data matching methods. \n\n\\subsection{Trajectory matching approach}\n\nUnlike circuitously matching the gradients, \\citet{cazenavette2022distillation} directly matches the long-range training trajectory between the target dataset and the synthetic dataset. Concretely, they collect the typical training trajectory {\\it w.r.t.} the target dataset into the buffer in advance, and then ingredients in the buffer are randomly selected to initialize the networks for training $\\mathcal{S}$. After collecting the trajectory of $\\mathcal{S}$, the synthetic dataset is updated by matching their trajectory. The objective loss of matching trajectory is defined as $\\mathcal{L}=\\| \\boldsymbol{\\theta}^{(k+N)} - \\boldsymbol{\\theta}_{\\mathcal{T}}^{(k+M)} \\|_2 ^2 \/ \\|\\boldsymbol{\\theta}_{\\mathcal{T}}^{(k)} - \\boldsymbol{\\theta}_{\\mathcal{T}}^{(k+M)} \\|_2 ^2 $, where $\\boldsymbol{\\theta}_{\\mathcal{T}}$ denote the target parameter by training the model on $\\mathcal{T}$ and is stored in the buffer, and $\\boldsymbol{\\theta}^{k+N}$ are the parameter by training the model on $\\mathcal{S}$ for $N$ epochs with the initialization of $\\boldsymbol{\\theta}_{\\mathcal{T}}^{(k)}$. The denominator in the loss function is for normalization.\n\n\nAlbeit trajectory matching received empirical success, \\citet{du2022minimizing} propose that there exists an accumulated trajectory error when matching the trajectory due to the segmented alignment from $\\boldsymbol{\\theta}_{\\mathcal{T}}^{(k)}$ to $\\boldsymbol{\\theta}_{\\mathcal{T}}^{(k+M)}$, and they alleviate this by adding random noise when initialize the distilled network to improve robustness {\\it w.r.t.} the accumulated trajectory error.\n\nCompared to matching gradients, while trajectory matching side-steps second-order gradient computation, it, unfortunately, requires unrolling $N$ SGD updates during the meta-gradient backpropagation as the existence of $\\boldsymbol{\\theta}^{(k+N)}$. The unrolled gradient computation significantly increases the memory burden and impedes scalability. By disentangling the meta-gradient {\\it w.r.t.} synthetic examples into two passes, \\citet{cui2022scaling} greatly reduce the memory required by trajectory matching. Motivated by knowledge distillation \\citep{gou2021knowledge}, they propose to assign soft-label to synthetic examples with pre-trained models in the buffer, and the soft-label helps learn intraclass information and consequently improves distillation performance.\n\n\n\n\n\n\n\n\\subsection{Distribution Matching Approach}\nAlbeit the parameter-wise match shows a satisfying performance, \\citet{zhao2021distribution} visualise the distilled data in 2-dimension and reveal that there is a large distribution discrepancy between the distilled data and the target data. In other words, the distilled dataset can not comprehensively cover the data distribution. Based on this discovery, they propose to match the synthetic and target data from the distribution perspective for dataset distillation. Concretely, they employ the pre-trained feature extractor $\\psi_{\\boldsymbol{v}}$ with the parameter $\\boldsymbol{v}$ to achieve the mapping from input space to feature space. The synthetic data is optimized according to the objective function $\\mathcal{L}(\\mathcal{S})=\\sum_{c=1}^C \\|\\psi_{\\boldsymbol{v}}(\\mathcal{S}_c) - \\psi_{\\boldsymbol{v}}(\\mathcal{T}_c)  \\|^2$, where $c$ denotes the class index. Though this distribution matching drops bilevel optimization for boosting, it empirically underperforms the above gradient and trajectory matching  approaches. \\citet{wang2022cafe} improve the distribution alignment from several aspects: (1) using multiple-layer features other than only the last-layer outputs for matching; (2) proposing the discrimination loss to enlarge the class distinction of synthetic data; and (3) recalling the bilevel optimization that updates $\\mathcal{S}$ with different model parameters for better generalization. Besides, \\citet{lee2022kfs} analyze that the subsampling in distribution matching is biased, and they propose to use full batch training to mitigate this problem.\n\n\n\n\n\n\n\\subsection{Kernel Regression Approach}\nBecause the solutions of deep non-linear neural networks are commonly intractable, multi-step gradient descent is used to get the optimal parameters in terms of the synthetic dataset. Nevertheless, the iterative algorithm makes the loss backpropagation in DD very inefficient. Different from distilling datasets with deep networks, \\citet{nguyen2020dataset} considers the dataset distillation in the simple kernel regression regime that admits a closed-form solution. The regression model is defined as $f(\\boldsymbol{x}) = \\boldsymbol{w}^\\top \\psi(\\boldsymbol{x})$, where $\\psi(\\cdot)$ is a non-linear mapping, and the weight $\\boldsymbol{w}$ is learnable. Given the dataset $\\mathcal{S}$, the weight can be directly solved with $\\boldsymbol{w} =  \\psi(X_{s})^\\top\\left(\\mathbf{K}_{X_s X_s}+\\lambda I\\right)^{-1}y_{s}$ \\citep{petersen2008matrix}, where  $\\mathbf{K}_{X_s X_s} = [K(\\boldsymbol{s}_i,\\boldsymbol{s}_j)]_{ij}\\in \\mathbb{R}^{n\\times n}$ is the kernel matrix associated to the kernel $K(\\boldsymbol{x},\\boldsymbol{x}^\\prime)=\\langle \\psi(\\boldsymbol{x}), \\psi(\\boldsymbol{x}^\\prime)\\rangle$ and the dataset $\\mathcal{S}$. Therefore, the MSE of predicting the target dataset $\\mathcal{T}$ can be derived as $\\mathcal{L}(\\mathcal{S}) = \\frac{1}{2}\\left\\|y_t-\\mathbf{K}_{X_t X_s}\\left(\\mathbf{K}_{X_s X_s}+\\lambda I\\right)^{-1} y_s\\right\\|^2$. In the kernel ridge regression regime, the synthetic dataset $\\mathcal{S}$ can be directly updated by backpropagating the meta-gradient through the kernel function. Albeit this formulation is solid, this algorithm is designed in the KRR scenario and only employs simple kernels, which causes performance drops when the distilled dataset is transferred to train neural networks. \\citet{jacot2018neural} propose the neural tangent kernel (NTK) theory that proves the equivalence between training infinite-width neural networks and kernel regression. With this equivalence, \\citet{nguyen2021dataset} employ the infinite-width networks as the kernel for dataset distillation, which narrows the gap between the scenarios of KRR and deep learning. However, computing the kernel matrix $\\mathbf{K}_{X_s X_t}$ and its corresponding inverse is expensive and almost intractable for large target datasets due to the complexity of $\\mathcal{O}(|\\mathcal{T}||\\mathcal{S}|)$. \\citet{loo2022efficient} replace NTK kernel with neural network Gaussian process (NNGP) kernel that only considers the training of last-layer classifier for speed up. With random features in NNGP, the computation of kernel matrix can be decomposed, and the complexity of $\\mathbf{K}_{X_s X_t}$ becomes $\\mathcal{O}(|\\mathcal{T}|+|\\mathcal{S}|)$. A similar method is also proposed by \\citet{zhou2022dataset}, which fixes the feature extractor in the neural networks to improve the efficiency of meta-gradient backpropagation.\n\\fi\n\n\n\\section{Factorized Dataset Distillation}\n\nIn representation learning, although the image data are in the extremely high-dimensional space, they may lie on a low-dimensional manifold and rely on a few features, and one can recover the source image from the low-dimensional features with specific decoders \\citep{bengio2013representation,zhang2018network}. For this reason, it is plausible to implicitly learn synthetic datasets by optimizing their factorized features and corresponding decoders, which is termed as {\\it factorized dataset distillation}, as shown in Figure \\ref{figure:disentange}. To be in harmony with decoders, we recall the notion of a feature as code in terms of the dataset. By factorizing synthetic datasets with a combination of codes and decoders, the compression ratio can be further decreased, and information redundancy in distilled images can also be reduced. According to the learnability of the code and decoder, we classify factorized dataset distillation into three categories of code-based DD, decoder-based DD, and code-decoder DD.\n\n\\begin{figure*}[t]\n\\centering\n\n\\subfigure[Nonfactorized dataset distillation]{\n\\begin{minipage}[b]{0.4\\textwidth}\n\\centering\n    \t\t\\includegraphics[width=0.95\\columnwidth]{figures\/vanilla_dd.pdf}\n    \t\t\\end{minipage}\n\t\t\\label{figure:vanilla_dd}   \n    \t}\n\\subfigure[Factorized dataset distillation]{\n\\begin{minipage}[b]{0.56\\textwidth}\n\\centering\n    \t\t\\includegraphics[width=0.95\\columnwidth]{figures\/disentangled_dd.pdf}\n    \t\t\\end{minipage}\n\t\t\\label{figure:disentangled_dd}   \n    \t}\n\\caption{Schematic diagrams of nonfactorized dataset distillation and factorized dataset distillation.}\n\\label{figure:disentange}\n\\end{figure*}\n\nCode-based DD aims to learn a series of low-dimensional codes for generating highly informative images through a specific generator. In \\citep{zhao2022synthesizing}, the vectors that are put into the GAN generator are learned to producing informative images. Concretely, these investigators inverse real examples with a GAN generator and collect corresponding latent codes. Then, the latent vectors are further optimized with the distribution matching algorithm. By this, the optimized latent vectors can induce more informative synthetic examples with the pretrained GAN generator. In addition to use the GAN, \\citet{kim2022dataset} employed a deterministic multi-formation function $\\texttt{multi-form}(\\cdot)$ as the decoder to create synthetic data from fewer condensed data, {\\it i.e.}, the synthetic data are generated by $\\mathcal{S} = \\texttt{multi-form} (\\mathcal{C})$. Then the condensed data $\\mathcal{C}$ is optimized in an end-to-end fashion by the gradient matching approach.\n\nDifferent from code-based DD, decoder-based DD solely learns decoders that are used to produce highly informative data. In \\citep{such2020generative}, a generative teaching network (GTN) that employs a trainable network was proposed to generate synthetic images from random noise based on given labels, and the meta-gradients are backpropagated via BPTT to update the GTN rather than the synthetic data.\n\nNaturally, code-decoder DD combines code-based and decoder-based DD and allows the training on both codes and decoders. \\citet{deng2022remember} generated synthetic images via the matrix multiplication between codes and decodes, which they call the {\\it memory} and {\\it addressing function}. Specifically, they use the memory $\\mathcal{M} = \\{\\boldsymbol{b}_1, \\cdots, \\boldsymbol{b}_K\\}$ to store the bases $\\boldsymbol{b}_i \\in \\mathbb{R}^{d}$ that have the same dimension as target data; and $r$ addressing functions of $\\mathcal{A} = \\{\\boldsymbol{A}_1,\\cdots,\\boldsymbol{A}_r\\}$ are used to recover the synthetic images according to the given one-hot label $\\boldsymbol{y}$ as follows:\n\\begin{equation}\n\\label{eq:memory}\n    \\boldsymbol{s}_i^\\top = \\boldsymbol{y}^T \\boldsymbol{A}_i [\\boldsymbol{b}_1, \\cdots, \\boldsymbol{b}_K] ^\\top,\n\\end{equation}\nwhere the one-hot label $\\boldsymbol{y} \\in \\mathbb{R}^{C\\times 1}$, and $\\boldsymbol{A}_i \\in \\mathbb{R}^{C\\times K}$, and $C$ is the number of classes. By plugging different addressing function $\\boldsymbol{A}_i$ into Eq. \\ref{eq:memory}, a total number of $r$ synthetic images can be generated from the bases for each category. During the distillation process, these two elements of $\\mathcal{M}$ and $\\mathcal{A}$ were optimized with the meta-learning framework. Different from matrix multiplication in \\citep{deng2022remember}, \\citet{liu2022dataset} employ {\\it hallucinator} networks as decoders to generate synthetic data from {\\it basis}. Specifically, a hallucinator network is consist of three parts of a encoder $\\texttt{enc}$, an affine transformation with trainable scale $\\sigma$ and shift $\\mu$, and a decoder $\\texttt{dec}$, then a basis $\\boldsymbol{b}$ is put into the hallucinator network to generate corresponding synthetic image $\\boldsymbol{s}$ as follows:\n\\begin{equation}\n    \\mathbf{f}_1 = \\texttt{enc}(\\boldsymbol{b}), \\quad \\mathbf{f}_2 = \\sigma \\times \\mathbf{f}_1 + \\mu, \\quad \\boldsymbol{s} = \\texttt{dec}(\\mathbf{f}_2),\n\\end{equation}\nwhere the multiplication is element-wise operation. In addition, to enlarge the knowledge divergence encoded by different hallucinator networks for efficient compression, these researchers proposed an adversarial contrastive constraint that maximizes the divergence of features of different generated synthetic data. A similar code-decoder factorization method is also presented in \\citet{lee2022kfs}, where they adopted an improved distribution matching to optimize the latent codes and decoders.\n\nThrough implicitly learning the latent codes and decoders, factorized DD possesses advantages such as more compact representation and shared representation across classes that consequently improve the dataset distillation performance. Notably, this code-decoder factorization is compatible with the aforementioned distillation approaches. Therefore, the exploration of synthetic data generation and distillation frameworks can promote dataset distillation in parallel.\n\n\n\n\n\n\n   \n    \n   \n\n   \n\n   \n\n   \n\n   \n    \n\n\n\\section{Performance Comparison}\n\\label{sec:performance comparison}\n\n\\begin{table}[t]\n    \n    \\centering\n    \\caption{Performance comparison of different dataset distillation methods on different datasets. Abbreviations of GM, TM, DM are for gradient matching, trajectory matching, and distribution matching, respectively. Whole denotes the test accuracy in terms of the whole target dataset.}\n    \\resizebox{\\linewidth}{!}{\n    \\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c}\n    \\toprule\n\n    \\multirow{2}{*}{Methods} & \\multirow{2}{*}{Schemes} & \\multicolumn{3}{c|}{MNIST} & \\multicolumn{3}{c|}{FashionMNIST} & \\multicolumn{3}{c|}{SVHN} & \\multicolumn{3}{c|}{CIFAR-10} & \\multicolumn{3}{c|}{CIFAR-100} & \\multicolumn{3}{c}{Tiny ImageNet} \\\\\n    \\cline{3-20}\n     & & $1$ & $10$ & $50$ & $1$ & $10$ & $50$ & $1$ & $10$ & $50$ & $1$ & $10$ & $50$ & $1$ & $10$ & $50$ & $1$ & $10$ & $50$ \\\\\n\n    \\hline\n    Random & - & \\format{64.9}{3.5} & \\format{95.1}{0.9} & \\format{97.9}{0.2}  & \\format{51.4}{3.8} & \\format{73.8}{0.7} & \\format{82.5}{0.7} & \\format{14.6}{1.6} & \\format{35.1}{4.1} & \\format{70.9}{0.9} & \\format{14.4}{2.0} & \\format{26.0}{1.2} & \\format{43.4}{1.0}  & \\format{4.2}{0.3} & \\format{14.6}{0.5} & \\format{30.0}{0.4} & \\format{1.4}{0.1} & \\format{5.0}{0.2} & \\format{15.0}{0.4} \\\\\n\n    Herding & - & \\format{89.2}{1.6} & \\format{93.7}{0.3} & \\format{94.8}{0.2}  & \\format{67.0}{1.9} & \\format{71.1}{0.7} & \\format{71.9}{0.8} & \\format{20.9}{1.3} & \\format{50.5}{3.3} & \\format{72.6}{0.8} & \\format{21.5}{1.2} & \\format{31.6}{0.7} & \\format{40.4}{0.6}  & \\format{8.4}{0.3} & \\format{17.3}{0.3} & \\format{33.7}{0.5} & \\format{2.8}{0.2} & \\format{6.3}{0.2} & \\format{16.7}{0.3} \\\\\n\n    \n    DD \\citep{wang2018dataset} & BPTT & - & \\format{79.5}{8.1} & -  & - & - & - &- & - & - & \n    - & \\format{36.8}{1.2} & -  & - & - & - & \n    - & - & - \\\\\n\n\n    LD \\citep{bohdal2020flexible} & BPTT & \\format{60.9}{3.2} & \\format{87.3}{0.7} & \\format{93.3}{0.3}  & \n   - & - & - & - & - & - & \n   \\format{25.7}{0.7} & \\format{38.3}{0.4} & \\format{42.5}{0.4}  & \\format{11.5}{0.4} & - & - & \n   - & - & - \\\\\n\n    DC \\citep{zhao2021dataset} & GM & \\format{91.7}{0.5} & \\format{94.7}{0.2} & \\format{98.8}{0.2}  & \n    \\format{70.5}{0.6} & \\format{82.3}{0.4} & \\format{83.6}{0.4} & \\format{31.2}{1.4} & \\format{76.1}{0.6} & \\format{82.3}{0.3} & \\format{28.3}{0.5} & \\format{44.9}{0.5} & \\format{53.9}{0.5} & \\format{13.9}{0.4} & \\format{32.3}{0.3} & \\format{42.8}{0.4} & - & - & - \\\\\n\n    DSA \\citep{zhao2021dsa} & GM & \\format{88.7}{0.6} & \\format{97.8}{0.1} & \\orformat{99.2}{0.1}  & \n    \\format{70.6}{0.6} & \\format{86.6}{0.3} & \\format{88.7}{0.2} & \\format{27.5}{1.4} & \\orformat{79.2}{0.5} & \\format{84.4}{0.4} & \\format{28.8}{0.7} & \\format{52.1}{0.5} & \\format{60.6}{0.5}  & \\format{13.9}{0.4} & \\format{32.3}{0.3} & \\format{42.8}{0.4} & - & - & -\\\\\n\n    DCC \\citep{lee2022dataset} & GM & - & - & -  & \n    - & - & - & \n    \\format{47.5}{2.6} & \\orformat{80.5}{0.6} & \\orformat{87.2}{0.3} & \\format{34.0}{0.7} & \\format{54.5}{0.5} & \\format{64.2}{0.4}  & \\format{14.6}{0.3} & \\format{33.5}{0.3} & \\format{39.3}{0.4} & - & - & - \\\\\n\n    MTT \\citep{cazenavette2022distillation} & TM & \\format{91.4}{0.9} & \\format{97.3}{0.1} & \\format{98.5}{0.1}  & \n    \\format{75.1}{0.9} & \\orformat{87.2}{0.3} & \\format{88.3}{0.1} & - & - & - & \n    \\format{46.3}{0.8} & \\format{65.3}{0.7} & \\format{71.6}{0.2}  & \\format{24.3}{0.3} & \\format{40.1}{0.4} & \\format{47.7}{0.2} & \\format{8.8}{0.3} & \\format{23.2}{0.2} & \\orformat{28.0}{0.3} \\\\\n\n    FTD \\citep{du2022minimizing} & TM & - & - & -  & - & - & - & - & - & - & \n    \\format{46.8}{0.3} & \\orformat{66.6}{0.3} & \\orformat{73.8}{0.2}  & \\format{25.2}{0.2} & \\orformat{43.4}{0.3} & \\format{50.7}{0.3} & \\orformat{10.4}{0.3} & \\orformat{24.5}{0.2} & - \\\\\n\n    TESLA \\citep{cui2022scaling} & TM & - & - & -  & - & - & - & - & - & - & \n    \\format{48.5}{0.8} & \\orformat{66.4}{0.8} & \\orformat{72.6}{0.7}  & \\format{24.8}{0.4} & \\format{41.7}{0.3} & \\format{47.9}{0.3} & \\format{7.7}{0.2} & \\format{18.8}{1.3} & \\orformat{27.9}{1.2} \\\\\n\n    DM \\citep{zhao2023distribution} & DM & \\format{89.2}{1.6} & \\format{97.3}{0.3} & \\format{94.8}{0.2}  & \n   - & - & - & - & - & - & \n   \\format{26.0}{0.8} & \\format{48.9}{0.6} & \\format{63.0}{0.4}  & \\format{11.4}{0.3} & \\format{29.7}{0.3} & \\format{43.6}{0.4} & \\format{3.9}{0.2} & \\format{12.9}{0.4} & \\format{24.1}{0.3} \\\\\n\n    CAFE \\citep{wang2022cafe} & DM & \\format{90.8}{0.5} & \\format{97.5}{0.1} & \\format{98.9}{0.2}  & \n    \\format{73.7}{0.7} & \\format{83.0}{0.3} & \\format{88.2}{0.3} & \\format{42.9}{3.0} & \\format{77.9}{0.6} & \\format{82.3}{0.4} & \\format{31.6}{0.8} & \\format{50.9}{0.5} & \\format{62.3}{0.4}  & \\format{14.0}{0.3} & \\format{31.5}{0.2} & \\format{42.9}{0.2} & - & - & - \\\\\n\n    KIP \\citep{nguyen2020dataset,nguyen2021dataset} & KRR & \\format{90.1}{0.1} & \\format{97.5}{0.0} & \\format{98.3}{0.1}  & \n    \\format{73.5}{0.5} & \\format{86.8}{0.1} & \\format{88.0}{0.1} & \\orformat{57.3}{0.1} & \\format{75.0}{0.1} & \\orformat{85.0}{0.1} & \\orformat{49.9}{0.2} & \\format{62.7}{0.3} & \\format{68.6}{0.2}  & \\format{15.7}{0.2} & \\format{28.3}{0.1} & - & \n    - & - & - \\\\\n\n    FRePo \\citep{zhou2022dataset} & KRR & \\orformat{93.0}{0.4} & \\orformat{98.6}{0.1} & \\orformat{99.2}{0.0}  & \n    \\orformat{75.6}{0.3} & \\format{86.2}{0.2} & \\orformat{89.6}{0.1} & - & - & - & \n    \\format{46.8}{0.7} & \\format{65.5}{0.4} & \\format{71.7}{0.2}  & \\orformat{28.7}{0.1} & \\orformat{42.5}{0.2} & \\format{44.3}{0.2} & \\orformat{15.4}{0.3} & \\orformat{25.4}{0.2} & - \\\\\n\n    RFAD \\citep{loo2022efficient} & KRR & \\orformat{94.4}{1.5} & \\orformat{98.5}{0.1} & \\format{98.8}{0.1}  & \n    \\orformat{78.6}{1.3} & \\orformat{87.0}{0.5} & \\orformat{88.8}{0.4} & \\orformat{52.2}{2.2} & \\format{74.9}{0.4} & \\format{80.9}{0.3} & \\orformat{53.6}{1.2} & \\format{66.3}{0.5} & \\format{71.1}{0.4}  & \\orformat{26.3}{1.1} & \\format{33.0}{0.3} & - &\n    - & - & - \\\\\n\n    IDC$^\\ast$ \\citep{kim2022dataset} & GM & - & - & -  & - & - & - & \\format{68.1}{0.1} & \\format{87.3}{0.2} & \\format{90.2}{0.1} & \\format{50.0}{0.4} & \\format{67.5}{0.5} & \\format{74.5}{0.1}  & - & \\format{44.8}{0.2} & - &\n    - & - & -\\\\\n\n    RTP$^\\ast$ \\citep{deng2022remember} & BPTT & \\blformat{98.7}{0.7} & \\blformat{99.3}{0.5} & \\blformat{99.4}{0.4}  & \n    \\blformat{88.5}{0.1} & \\blformat{90.0}{0.7} & \\blformat{91.2}{0.3} & \\blformat{87.3}{0.1} & \\format{89.1}{0.2} & \\format{89.5}{0.2} & \\blformat{66.4}{0.4} & \\format{71.2}{0.4} & \\format{73.6}{0.4}  & \\format{34.0}{0.4} & \\format{42.9}{0.7} & - & \n    \\format{16.0}{0.7} & - & - \\\\\n\n    HaBa$^\\ast$ \\citep{liu2022dataset} & TM & - & - & -  & - & - & - & \\format{69.8}{1.3} & \\format{83.2}{0.4} & \\format{88.3}{0.1} & \\format{48.3}{0.8} & \\format{69.9}{0.4} & \\format{74.0}{0.2}  & \\format{33.4}{0.4} & \\format{40.2}{0.2} & \\blformat{47.0}{0.2} & - & - & - \\\\\n\n    KFS$^\\ast$ \\citep{lee2022kfs} & DM & - & - & -  & - & - & - & \n    \\format{82.9}{0.4} & \\blformat{91.4}{0.2} & \\blformat{92.2}{0.1} & \\format{59.8}{0.5} & \\blformat{72.0}{0.3} & \\blformat{75.0}{0.2}  & \\blformat{40.0}{0.5} & \\blformat{50.6}{0.2} & - & \n    \\blformat{22.7}{0.3} & \\blformat{27.8}{0.2} & - \\\\\n   \\cline{3-20}\n   Whole & - & \\multicolumn{3}{c|}{\\format{99.6}{0.0}} & \\multicolumn{3}{c|}{\\format{93.5}{0.1}} & \\multicolumn{3}{c|}{\\format{95.4}{0.1}} & \\multicolumn{3}{c|}{\\format{84.8}{0.1}} & \\multicolumn{3}{c|}{\\format{56.2}{0.3}} & \\multicolumn{3}{c}{\\format{37.6}{0.4}}\\\\\n    \\bottomrule\n    \\end{tabular}\n    }\n    \\label{tab:dd performance}\n\\end{table}\n\n\\begin{table}[t]\n    \\centering\n    \\caption{Performance comparison of different dataset distillation methods on ImageNet-1K.}\n    \\begin{tabular}{c|c|c|c | c }\n    \\toprule\n    Methods & IPC=$1$ & IPC = $2$ & IPC = $10$ & IPC = $50$  \\\\\n    \\hline\n    Random & \\format{0.5}{0.1}  & \\format{0.9}{0.1} & \\format{3.6}{0.1} & \\format{15.3}{2.3} \\\\\n    DM \\citep{zhao2023distribution} & \\format{1.5}{0.1} & \\format{1.7}{0.1} & - & - \\\\\n\n    FRePo \\citep{zhou2022dataset} & \\format{7.5}{0.3} & \\format{9.7}{0.2} & - & - \\\\\n    TESLA \\citep{cui2022scaling} & \\bformat{7.7}{0.2} & \\bformat{10.5}{0.3} & \\bformat{17.8}{1.3} & \\bformat{27.9}{1.2} \\\\\n    \\cline{2-5}\n    Whole & \\multicolumn{4}{c}{\\format{33.8}{0.3}}  \\\\\n    \\bottomrule\n    \\end{tabular}\n    \\label{tab:dd performance imagenet}\n\\end{table}\n\nTo demonstrate the effectiveness of dataset distillation, we collect and summarize the classification performance of some characteristic dataset distillation approaches on the following image datasets: MNIST \\citep{lecun1998gradient}, FashionMNIST \\citep{xiao2017\/online} , SVHN \\citep{netzer2011reading}, CIFAR-10\/100 \\citep{krizhevsky2009learning}, Tiny ImageNet \\citep{le2015tiny}, and ImageNet-1K \\citep{russakovsky2015imagenet}. The details of these datasets are presented as follows. Recently, \\citet{cui2022dc} publish a benchmark in terms of dataset distillation, however, it only contains five DD methods, and we provide a comprehensive comparison of over $15$ existing dataset distillation methods in this survey.\n\nMNIST is a black-and-white dataset that consists of $60,000$ training images and $10,000$ test images from $10$ different classes; the size of  each example image is $28\\times 28$ pixels. SVHN is a colorful dataset and consists of $73,257$ digits and $26,032$ digits for training and testing, respectively, and the example images in SVHN are $32\\times 32$ RGB images. CIFAR-10\/100 are composed of $50,000$ training images and $10,000$ test images from $10$ and $100$ different classes, respectively. The RGB images in CIFAR-10\/100 are comprised of $32\\times 32$ pixels. Tiny ImageNet consists of $100,000$ and $10,000$ $64\\times 64$ RGB images from $200$ different classes for training and testing, respectively. ImageNet-1K is a large image dataset that consists of over $1,000,000$ high-resolution RGB images from $1,000$ different classes.\n\nAn important factor affecting the test accuracy {\\it w.r.t.} the distilled dataset is the {\\it distillation budget}, which constrains the size of the distilled dataset by the notion of images allocated per class (IPC). Usually, the distilled dataset is set to have the same number of classes as the target dataset. Therefore, for a target dataset with $100$ classes, setting the distillation budget to $\\text{IPC}=10$ suggests that there are a total of $10\\times 100 = 1,000$ images in the distilled dataset. \n\n\\begin{table*}[t]\n    \\centering\n    \\caption{Cross-architecture performance {\\it w.r.t.} ResNet-18 (RN-18) and ResNet-152 (RN-152) on CIFAR-10. The results of DC, DSA, MTT, DM, and KIP are derived from \\citet{cui2022dc}. The results of FTD and CAFE are collected from \\citet{du2022minimizing}; and the result of TESLA is derived from \\citet{cui2022scaling}.}\n    \\resizebox{\\linewidth}{!}{\n    \\begin{tabular}{c|ccc|ccc|ccc}\n    \\toprule\n    \\multirow{2}{*}{Methods} & \\multicolumn{3}{c|}{IPC=$1$} & \\multicolumn{3}{c|}{IPC=$10$} & \\multicolumn{3}{c}{IPC=$50$}   \\\\ \n    \\cline{2-10}\n     & \\multicolumn{1}{c}{ConvNet} & \\multicolumn{1}{c}{RN-18} & \\multicolumn{1}{c|}{RN-152} & \\multicolumn{1}{c}{ConvNet} & \\multicolumn{1}{c}{RN-18} & \\multicolumn{1}{c|}{RN-152} & \\multicolumn{1}{c}{ConvNet} & \\multicolumn{1}{c}{RN-18} & \\multicolumn{1}{c}{RN-152}  \\\\\n    \\hline\n    \n    DC \\citep{zhao2021dataset} & \\format{28.3}{0.5} & \\format{25.6}{0.6} & \\orformat{15.3}{0.4} & \\format{44.9}{0.5} & \\format{42.1}{0.6} & \\format{16.1}{1.0}& \\format{53.9}{0.5}& \\format{45.9}{1.4}& \\format{19.7}{1.2}\\\\\n    \n    DSA \\citep{zhao2021dsa} & \\format{28.8}{0.5} &  \\format{25.6}{0.6} & \\format{15.1}{0.7} & \\format{52.1}{0.5} & \\format{42.1}{0.6}& \\format{16.1}{1.0} & \\format{60.6}{0.5} & \\format{49.5}{0.7} & \\format{20.0}{1.2}\\\\\n\n    MTT \\citep{cazenavette2022distillation} & \\format{46.3}{0.8}& \\orformat{34.2}{1.4} & \\format{13.4}{0.9} & \\format{65.3}{0.7} & \\format{38.8}{0.7} & \\format{15.9}{0.2}& \\format{71.6}{0.2}& \\format{60.0}{0.7}& \\format{20.9}{1.6}\\\\\n\n    FTD \\citep{du2022minimizing} & -& -&- &-&-&-& \\orformat{73.8}{0.2}& \\orformat{65.7}{0.3}&-\\\\\n\n    TESLA \\citep{cui2022scaling} & -& -& -& \\orformat{66.4}{0.8}& \\orformat{48.9}{2.2}&-&-&-&-\\\\\n\n    DM \\citep{zhao2023distribution}& \\format{26.0}{0.8}& \\format{20.6}{0.5} & \\format{14.1}{0.6} & \\format{48.9}{0.6}& \\format{38.2}{1.1}& \\format{15.6}{1.5}& \\format{63.0}{0.4}& \\format{52.8}{0.4}& \\orformat{21.7}{1.3}\\\\\n\n    CAFE \\citep{wang2022cafe}&- & -& -&-&-&-& \\format{55.5}{0.4}& \\format{25.3}{0.9}&-\\\\\n    \n    KIP \\citep{nguyen2021dataset} & \\orformat{49.9}{0.2} & \\format{27.6}{1.1} & \\format{14.2}{0.8} & \\format{62.7}{0.3}& \\format{45.2}{1.4} & \\orformat{16.6}{1.4}& \\format{68.6}{0.2}& \\format{60.0}{0.7}& \\format{20.9}{1.6}\\\\\n\n    \\bottomrule\n    \\end{tabular}}\n    \\label{tab:cross architecture 1}\n\\end{table*}\n\n\\begin{table*}[t]\n    \\centering\n    \\caption{Cross-architecture performance {\\it w.r.t.} ResNet-10 (RN-10) and DenseNet-121 (DN-121) on CIFAR-10. The results are derived from \\citet{lee2022kfs}}\n    \\resizebox{\\linewidth}{!}{\n    \\begin{tabular}{c|ccc|ccc|ccc}\n    \\toprule\n    \\multirow{2}{*}{Methods} & \\multicolumn{3}{c|}{IPC=$1$} & \\multicolumn{3}{c|}{IPC=$10$} & \\multicolumn{3}{c}{IPC=$50$}   \\\\ \n    \\cline{2-10}\n     & \\multicolumn{1}{c}{ConvNet} & \\multicolumn{1}{c}{RN-10} & \\multicolumn{1}{c|}{DN-121} & \\multicolumn{1}{c}{ConvNet} & \\multicolumn{1}{c}{RN-10} & \\multicolumn{1}{c|}{RN-121} & \\multicolumn{1}{c}{ConvNet} & \\multicolumn{1}{c}{RN-10} & \\multicolumn{1}{c}{RN-121}  \\\\\n    \\hline\n    \n    \n    DSA \\citep{zhao2021dsa} & \\format{28.8}{0.7} &  \\format{25.1}{0.8} & \\format{25.9}{1.8} & \\format{52.1}{0.5} & \\format{31.4}{0.9}& \\format{32.9}{1.0} & \\format{60.6}{0.5} & \\format{49.0}{0.7} & \\format{53.4}{0.8}\\\\\n\n\n    DM \\citep{zhao2023distribution}& \\format{26.0}{0.8}& \\format{13.7}{1.6} & \\format{12.9}{1.8} & \\format{48.9}{0.6}& \\format{31.7}{1.1}& \\format{32.2}{0.8}& \\format{63.0}{0.4}& \\format{49.1}{0.7}& \\format{53.7}{0.7}\\\\\n\n    IDC \\citep{kim2022dataset}& \\format{50.0}{0.4} & \\format{41.9}{0.6}&  \\format{39.8}{1.2} & \\format{67.5}{0.5}& \\format{63.5}{0.1}& \\format{61.6}{0.6}& \\format{74.5}{0.1}& \\format{72.4}{0.5}& \\format{71.8}{0.6}\\\\\n    \n    KFS \\citep{lee2022kfs} & \\orformat{59.8}{0.5} & \\orformat{47.0}{0.8} & \\orformat{49.5}{1.3} & \\orformat{72.0}{0.3}& \\orformat{70.3}{0.3} & \\orformat{71.4}{0.4}& \\orformat{75.0}{0.2}& \\orformat{75.1}{0.3}& \\orformat{76.3}{0.4}\\\\\n\n    \\bottomrule\n    \\end{tabular}}\n    \\label{tab:cross architecture 2}\n\\end{table*}\n\n\\subsection{Standard Benchmark}\n\\label{sec:standard benchmark}\n\nFor a comprehensive comparison, we also present the test accuracy {\\it w.r.t.} a random select search, coreset approach, and the original target dataset. For the coreset approach, the Herding algorithm \\citep{rebuffi2017icarl} is employed in this survey due to its superior performance. Notably, for most of parametric DD methods, ConvNet \\citep{gidaris2018dynamic} is the default architecture to distill the synthetic data; and all the test accuracy of distilled data are obtained by training the same architecture of ConvNet for fair comparison. \n\nThe empirical comparison results associated with MNIST, FashionMNIST, SVHN, CIFAR-10\/100, and Tiny ImageNet are presented in Table \\ref{tab:dd performance}. In spite of severe scale issues and few implementations on ImageNet-1K, we present the DD results of ImageNet-1K in Table \\ref{tab:dd performance imagenet} for completion. As shown in these tables, many methods are not tested on some datasets due to the lack of meaning for easy datasets or scalability problems for large datasets. We preserve these blanks in tables for a more fair and comprehensive comparison. To make the comparison clear, we use the orange number for the best of two nonfactorized DD methods and blue number for the best factorized DD methods in each category in Table \\ref{tab:dd performance}. In addition, we note the factorized DD methods with $^\\ast$. Based on the performance comparison in the tables, we observe  the followings:\n\\begin{itemize}\n    \\item Dataset distillation can be realized on many datasets with various sizes.\n    \\item Dataset distillation methods outperforms random pick and coreset selection by large margins.\n    \\item KRR and trajectory matching approaches possess advanced DD performance {\\it w.r.t.} three larger datasets of CIFAR-10\/100 and Tiny ImageNet among nonfactorized DD methods.\n    \\item Factorized dataset distillation by optimizing the latent codes and decoders can largely improve the test accuracy of distilled data.\n    \\item KFS \\citep{lee2022kfs} has the best overall performance among different datasets of CIFAR-10\/100 and Tiny ImageNet.\n   \n\\end{itemize}\n\n\n\\subsection{Cross-Architecture Transferability}\n\\label{sec:cross architecture transferability}\n\nThe standard benchmark in Table \\ref{tab:dd performance} only shows the performance of DD methods in terms of the specific ConvNet, while the distilled dataset is usually used to train other unseen network architectures. Therefore, measuring the DD performance on different architecture is also important for more comprehensive evaluation. However, there is not a standard set of architectures to evaluate cross-architecture transferability of DD algorithms, and the choice of architectures are highly different in the DD literature. \n\nIn this survey, we collect the DD results on four popular architectures of ResNet-10\/18\/152 \\citep{he2016deep} and DenseNet-121 \\citep{huang2017densely} to evaluate the cross-architecture transferability, as shown in Tables \\ref{tab:cross architecture 1} and \\ref{tab:cross architecture 2}. By investigating these tables, we can obtain several observations as follows:\n\n\\begin{itemize}\n    \\item There is a significant performance drop for nonfactorized DD methods when the distilled data are used to train unseen architectures (Table \\ref{tab:cross architecture 1}).\n    \\item The distilled data are not always possess the best performance across different architectures, which underlines the importance of evaluation on multiple architectures (Table \\ref{tab:cross architecture 1}).\n    \\item The factorized DD methods (IDC and KFS) possess better cross-architecture transferasbility {\\it i.e.}, suffer a smaller accuracy drop when encounter unseen architectures (Table \\ref{tab:cross architecture 2}).\n\\end{itemize}\n\n\n\n\\iffalse\n\\begin{table}\n    \n    \\centering\n    \\caption{Performance comparison of different dataset distillation methods on MNIST.}\n    \\begin{tabular}{c|c|c|c|c}\n    \\toprule\n\n    \\multirow{2}{*}{Methods} & \\multirow{2}{*}{Distillation schemes} & \\multicolumn{3}{c}{Accuracy} \\\\\n    \\cline{3-5}\n     & & IPC=$1$ & IPC=$10$ & IPC=$50$ \\\\\n\n    \\hline\n    Random & - & $64.9\\pm 3.5$ & $95.1\\pm 0.9$ & $97.9\\pm 0.2$ \\\\\n    Herding & - & $89.2\\pm 1.6$ & $93.7\\pm 0.3$ & $94.8\\pm 0.2$ \\\\\n   DD (\\citet{wang2018dataset})  & BPTT & - & $79.5 \\pm 8.1$ & - \\\\\n   LD (\\citet{bohdal2020flexible}) & BPTT & $60.9\\pm 3.2$ & $87.3\\pm 0.7$ & $93.3\\pm 0.3$ \\\\\n   DC (\\citet{zhao2021dataset})& Gradient match & $91.7\\pm 0.5$ & $97.4\\pm 0.2$ & $98.8\\pm 0.2$ \\\\\n   DSA (\\citet{zhao2021dsa})& Gradient match & $88.7\\pm 0.6$ & $97.8\\pm 0.1$ & \\bm{$99.2\\pm 0.1$} \\\\\n   DCC (\\citet{lee2022dataset}) & Gradient match  & - & - & - \\\\\n   MTT (\\citet{cazenavette2022distillation}) & Trajectory match & $91.4\\pm 0.9$ & $97.3 \\pm 0.1$ & $98.5\\pm 0.1$ \\\\\n   FTD (\\citet{du2022minimizing})& Trajectory match & - & - & - \\\\\n   DM (\\citet{zhao2021distribution}) & Distribution match & $89.2\\pm 1.6$ & $93.7\\pm 0.3$ & $94.8\\pm 0.2$ \\\\\n   CAFE (\\citet{wang2022cafe})& Distribution match & $90.8\\pm 0.5$ & $97.5\\pm 0.1$ & $98.9\\pm 0.2$ \\\\\n   KIP (\\citet{nguyen2020dataset,nguyen2021dataset}) & Kernel regression & $90.1\\pm 0.1$ & $97.5\\pm 0.0$ & $98.3\\pm 0.1$ \\\\\n   FRePo (\\citet{zhou2022dataset})& Kernel regression & \\bm{$93.0\\pm 0.4$} & \\bm{$98.6\\pm 0.1$} & \\bm{$99.2\\pm 0.0$} \\\\\n   RFAD (\\citet{loo2022efficient})& Kernel regression & \\bm{$94.4\\pm 1.5$} & \\bm{$98.5\\pm 0.1$} & $98.8\\pm 0.1$ \\\\\n   IDC (\\citet{kim2022dataset})$^\\ast$ & Gradient match & - & - & - \\\\\n   \\citet{deng2022remember}$^\\ast$ & BPTT & $89.2\\pm 1.6$ & $93.7\\pm 0.3$ & $94.8\\pm 0.2$ \\\\\n   Haba (\\citet{liu2022dataset})$^\\ast$ & Trajectory match & - & - & - \\\\\n   KFS (\\citet{lee2022kfs})$^\\ast$ & Distribution match & - & - & - \\\\\n   \\cline{3-5}\n   Whole dataset & - & \\multicolumn{3}{c}{$99.6 \\pm 0.0$} \\\\\n    \\bottomrule\n    \\end{tabular}\n    \\label{tab:dd on mnist}\n\\end{table}\n\n\n\\begin{table}\n    \n    \\centering\n    \\caption{Performance comparison of different dataset distillation methods on FashionMNIST.}\n    \\begin{tabular}{c|c|c|c|c}\n    \\toprule\n\n    \\multirow{2}{*}{Methods} & \\multirow{2}{*}{Distillation schemes} & \\multicolumn{3}{c}{Accuracy} \\\\\n    \\cline{3-5}\n     & & IPC=$1$ & IPC=$10$ & IPC=$50$ \\\\\n\n    \\hline\n    Random & - & $51.4\\pm 3.8$ & $73.8\\pm 0.7$ & $82.5\\pm 0.7$ \\\\\n    Herding & - & $67.0\\pm 1.9$ & $71.1\\pm 0.7$ & $71.9\\pm 0.8$ \\\\\n   DD (\\citet{wang2018dataset})  & BPTT & - & - & - \\\\\n   LD (\\citet{bohdal2020flexible}) & BPTT & - & - & - \\\\\n   DC (\\citet{zhao2021dataset})& Gradient match & $70.6\\pm 0.6$ & $82.3\\pm 0.4$ & $83.6\\pm 0.4$ \\\\\n   DSA (\\citet{zhao2021dsa})& Gradient match & $70.6\\pm 0.6$ & $86.6\\pm 0.3$ & $88.7\\pm 0.2$ \\\\\n   DCC (\\citet{lee2022dataset}) & Gradient match  & - & - & - \\\\\n   MTT (\\citet{cazenavette2022distillation}) & Trajectory match & $75.1\\pm 0.9$ & \\bm{$87.2 \\pm 0.3$} & $88.3\\pm 0.1$ \\\\\n   FTD (\\citet{du2022minimizing})& Trajectory match & - & - & - \\\\\n   DM (\\citet{zhao2021distribution}) & Distribution match  & - & - & - \\\\\n   CAFE (\\citet{wang2022cafe})& Distribution match & $73.7\\pm 0.7$ & $83.0\\pm 0.3$ & $88.2\\pm 0.3$ \\\\\n   KIP (\\citet{nguyen2020dataset,nguyen2021dataset}) & Kernel regression & $73.5\\pm 0.5$ & $86.8\\pm 0.1$ & $88.0\\pm 0.1$ \\\\\n   FRePo (\\citet{zhou2022dataset})& Kernel regression & $75.6\\pm 0.3$ & $86.2\\pm 0.2$ & \\bm{$89.6\\pm 0.1$} \\\\\n   RFAD (\\citet{loo2022efficient})& Kernel regression & \\bm{$78.6\\pm 1.3$} & $87.0\\pm 0.5$ & $88.8\\pm 0.4$ \\\\\n   IDC (\\citet{kim2022dataset})$^\\ast$ & Gradient match & - & - & - \\\\\n   \\citet{deng2022remember}$^\\ast$ & BPTT & \\bm{ $88.5\\pm 0.1$} & \\bm{$90.0\\pm 0.7$} & \\bm{$91.2\\pm 0.3$} \\\\\n   Haba (\\citet{liu2022dataset})$^\\ast$ & Trajectory match & - & - & - \\\\\n   KFS (\\citet{lee2022kfs})$^\\ast$ & Distribution match & - & - & - \\\\\n   \\cline{3-5}\n   Whole dataset & - & \\multicolumn{3}{c}{$93.5 \\pm 0.1$} \\\\\n    \\bottomrule\n    \\end{tabular}\n    \\label{tab:dd on fmnist}\n\\end{table}\n\n\n\\begin{table}\n    \n    \\centering\n    \\caption{Performance comparison of different dataset distillation methods on SVHN.}\n    \\begin{tabular}{c|c|c|c|c}\n    \\toprule\n\n    \\multirow{2}{*}{Methods} & \\multirow{2}{*}{Distillation schemes} & \\multicolumn{3}{c}{Accuracy} \\\\\n    \\cline{3-5}\n     & & IPC=$1$ & IPC=$10$ & IPC=$50$ \\\\\n\n    \\hline\n    Random & - & $14.6\\pm 1.6$ & $35.1\\pm 4.1$ & $70.9\\pm 0.9$ \\\\\n    Herding & - & $20.9\\pm 1.3$ & $50.5\\pm 3.3$ & $72.6\\pm 0.8$ \\\\\n   DD (\\citet{wang2018dataset})  & BPTT & - & - & - \\\\\n   LD (\\citet{bohdal2020flexible}) & BPTT & - & - & - \\\\\n   DC (\\citet{zhao2021dataset})& Gradient match & $31.2\\pm 1.4$ & $76.1\\pm 0.6$ & $82.3\\pm 0.3$ \\\\\n   DSA (\\citet{zhao2021dsa})& Gradient match & $27.5\\pm 1.4$ & $79.2\\pm 0.5$ & $84.4\\pm 0.4$ \\\\\n   DCC (\\citet{lee2022dataset}) & Gradient match  & $47.5\\pm 2.6$ & $80.5\\pm 0.6$ & $87.2\\pm 0.3$ \\\\\n   MTT (\\citet{cazenavette2022distillation}) & Trajectory match & - & - & - \\\\\n   FTD (\\citet{du2022minimizing})& Trajectory match & - & - & - \\\\\n   DM (\\citet{zhao2021distribution}) & Distribution match  & - & - & - \\\\\n   CAFE (\\citet{wang2022cafe})& Distribution match & $42.9\\pm 3.0$ & $77.9\\pm 0.6$ & $82.3\\pm 0.4$ \\\\\n   KIP (\\citet{nguyen2020dataset,nguyen2021dataset}) & Kernel regression & $57.3\\pm 0.1$ & $75.0\\pm 0.1$ & $80.5\\pm 0.1$ \\\\\n   FRePo (\\citet{zhou2022dataset})& Kernel regression & - & - & - \\\\\n   RFAD (\\citet{loo2022efficient})& Kernel regression & $52.2\\pm 2.2$ & $74.9\\pm 0.4$ & $80.9\\pm 0.3$ \\\\\n   IDC (\\citet{kim2022dataset})$^\\ast$ & Gradient match & $68.1\\pm 0.1$ & $87.3\\pm 0.2$ & \\bm{$90.2\\pm 0.1$} \\\\\n   \\citet{deng2022remember}$^\\ast$ & BPTT & \\bm{$87.3\\pm 0.1$} & \\bm{$89.1\\pm 0.2$} & {$89.5\\pm 0.2$} \\\\\n   Haba (\\citet{liu2022dataset})$^\\ast$ & Trajectory match & $69.8\\pm 1.3$ & $83.2\\pm 0.4$ & $88.3\\pm 0.1$ \\\\\n   KFS (\\citet{lee2022kfs})$^\\ast$ & Distribution match & \\bm{$82.9\\pm 0.4$} & \\bm{$91.4\\pm 0.2$} & \\bm{$92.2\\pm 0.1$} \\\\\n   \\cline{3-5}\n   Whole dataset & - & \\multicolumn{3}{c}{$95.4 \\pm 0.1$} \\\\\n    \\bottomrule\n    \\end{tabular}\n    \\label{tab:dd on svhn}\n\\end{table}\n\n\\begin{table}\n    \n    \\centering\n    \\caption{Performance comparison of different dataset distillation methods on CIFAR-10.}\n    \\begin{tabular}{c|c|c|c|c}\n    \\toprule\n\n    \\multirow{2}{*}{Methods} & \\multirow{2}{*}{Distillation schemes} & \\multicolumn{3}{c}{Accuracy} \\\\\n    \\cline{3-5}\n     & & IPC=$1$ & IPC=$10$ & IPC=$50$ \\\\\n\n    \\hline\n    Random & - & $14.4\\pm 2.0$ & $26.0\\pm 1.2$ & $43.4\\pm 1.0$ \\\\\n    Herding & - & $21.5\\pm 1.2$ & $31.6\\pm 0.7$ & $40.4\\pm 0.6$ \\\\\n   DD (\\citet{wang2018dataset})  & BPTT & - & $36.8\\pm 1.2$ & - \\\\\n   LD (\\citet{bohdal2020flexible}) & BPTT & $25.7\\pm 0.7$ & $38.3\\pm 0.4$ & $42.5\\pm 0.4$ \\\\\n   DC (\\citet{zhao2021dataset})& Gradient match & $28.3\\pm 0.5$ & $44.9\\pm 0.5$ & $53.9\\pm 0.5$ \\\\\n   DSA (\\citet{zhao2021dsa})& Gradient match & $28.8\\pm 0.7$ & $52.1\\pm 0.5$ & $60.6\\pm 0.5$ \\\\\n   DCC (\\citet{lee2022dataset}) & Gradient match  & $47.5\\pm 2.6$ & $80.5\\pm 0.6$ & $87.2\\pm 0.3$ \\\\\n   MTT (\\citet{cazenavette2022distillation}) & Trajectory match & $46.3\\pm 0.8$ & $65.3\\pm 0.7$ & $71.6\\pm 0.2$ \\\\\n   FTD (\\citet{du2022minimizing})& Trajectory match & $46.8\\pm 0.3$ & $66.6\\pm 0.3$ & $73.8\\pm 0.2$ \\\\\n   TESLA (\\citet{cui2022scaling})& Trajectory match & $48.5\\pm 0.8$ & $66.4\\pm 0.8$ & $72.6\\pm 0.7$ \\\\\n   DM (\\citet{zhao2021distribution}) & Distribution match  & $26.0\\pm 0.8$ & $48.9\\pm 0.6$ & $63.0\\pm 0.4$ \\\\\n   CAFE (\\citet{wang2022cafe})& Distribution match & $31.6\\pm 0.8$ & $50.9\\pm 0.5$ & $62.3\\pm 0.4$ \\\\\n   KIP (\\citet{nguyen2020dataset,nguyen2021dataset}) & Kernel regression & $49.9\\pm 0.2$ & $62.7\\pm 0.3$ & $68.6\\pm 0.2$ \\\\\n   FRePo (\\citet{zhou2022dataset})& Kernel regression & $46.8\\pm 0.7$ & $65.5\\pm 0.4$ & $71.7\\pm 0.2$ \\\\\n   RFAD (\\citet{loo2022efficient})& Kernel regression & $53.6\\pm 1.2$ & $66.3\\pm 0.5$ & $71.1\\pm 0.4$ \\\\\n   IDC (\\citet{kim2022dataset})$^\\ast$ & Gradient match & $50.0\\pm 0.4$ & $67.5\\pm 0.5$ & \\bm{$74.5\\pm 0.1$} \\\\\n   \\citet{deng2022remember}$^\\ast$ & BPTT & \\bm{$66.4\\pm 0.4$} & \\bm{$71.2\\pm 0.4$} & {$73.6\\pm 0.4$} \\\\\n   Haba (\\citet{liu2022dataset})$^\\ast$ & Trajectory match & $48.3\\pm 0.8$ & $69.9\\pm 0.4$ & $74.0\\pm 0.2$ \\\\\n   KFS (\\citet{lee2022kfs})$^\\ast$ & Distribution match & \\bm{$59.8\\pm 0.5$} & \\bm{$72.0\\pm 0.3$} & \\bm{$75.0\\pm 0.2$} \\\\\n   \\cline{3-5}\n   Whole dataset & - & \\multicolumn{3}{c}{$84.8 \\pm 0.1$} \\\\\n    \\bottomrule\n    \\end{tabular}\n    \\label{tab:dd on cifar-10}\n\\end{table}\n\n\\begin{table}\n    \n    \\centering\n    \\caption{Performance comparison of different dataset distillation methods on CIFAR-100.}\n    \\begin{tabular}{c|c|c|c|c}\n    \\toprule\n\n    \\multirow{2}{*}{Methods} & \\multirow{2}{*}{Distillation schemes} & \\multicolumn{3}{c}{Accuracy} \\\\\n    \\cline{3-5}\n     & & IPC=$1$ & IPC=$10$ & IPC=$50$ \\\\\n\n    \\hline\n    Random & - & $4.2 \\pm 0.3$ & $14.6\\pm 0.5$ & $30.0\\pm 0.4$ \\\\\n    Herding & - & $8.4 \\pm 0.3$ & $17.3\\pm 0.3$ & $33.7\\pm 0.5$ \\\\\n   DD (\\citet{wang2018dataset})  & BPTT & - & - & - \\\\\n   LD (\\citet{bohdal2020flexible}) & BPTT & $11.5\\pm 0.4$ & - & - \\\\\n   DC (\\citet{zhao2021dataset})& Gradient match & $12.8\\pm 0.3$ & $26.6\\pm 0.3$ & $32.1\\pm 0.3$ \\\\\n   DSA (\\citet{zhao2021dsa})& Gradient match & $13.9\\pm 0.4$ & $32.3\\pm 0.3$ & $42.8\\pm 0.4$ \\\\\n   DCC (\\citet{lee2022dataset}) & Gradient match  & $14.6\\pm 0.3$ & $33.5\\pm 0.3$ & $39.3\\pm 0.4$ \\\\\n   MTT (\\citet{cazenavette2022distillation}) & Trajectory match & $24.3\\pm 0.3$ & $40.1\\pm 0.4$ & \\bm{$47.7\\pm 0.2$} \\\\\n   FTD (\\citet{du2022minimizing})& Trajectory match & $25.2\\pm 0.2$ & $43.4\\pm 0.3$ & \\bm{$50.7\\pm 0.3$} \\\\\n   TESLA (\\citet{cui2022scaling})& Trajectory match & $24.8\\pm 0.4$ & $41.7\\pm 0.3$ & $47.9\\pm 0.3$ \\\\\n   DM (\\citet{zhao2021distribution}) & Distribution match  & $11.4\\pm 0.3$ & $29.7\\pm 0.3$ & $43.6\\pm 0.4$ \\\\\n   CAFE (\\citet{wang2022cafe})& Distribution match & $14.0\\pm 0.3$ & $31.5\\pm 0.2$ & $42.9\\pm 0.2$ \\\\\n   KIP (\\citet{nguyen2020dataset,nguyen2021dataset}) & Kernel regression & $15.7\\pm 0.2$ & $28.3\\pm 0.1$ & - \\\\\n   FRePo (\\citet{zhou2022dataset})& Kernel regression & $28.7\\pm 0.1$ & $42.5\\pm 0.2$ & $44.3\\pm 0.2$ \\\\\n   RFAD (\\citet{loo2022efficient})& Kernel regression & $26.3\\pm 1.1$ & $33.0\\pm 0.3$ & - \\\\\n   IDC (\\citet{kim2022dataset})$^\\ast$ & Gradient match & - & \\bm{$44.8\\pm 0.2$} & - \\\\\n   \\citet{deng2022remember}$^\\ast$ & BPTT & \\bm{$34.0\\pm 0.4$} & {$42.9\\pm 0.7$} & - \\\\\n   Haba (\\citet{liu2022dataset})$^\\ast$ & Trajectory match & $33.4\\pm 0.4$ & $40.2\\pm 0.2$ & $47.0\\pm 0.2$ \\\\\n   KFS (\\citet{lee2022kfs})$^\\ast$ & Distribution match & \\bm{$40.0\\pm 0.5$} & \\bm{$50.6\\pm 0.2$} & - \\\\\n   \\cline{3-5}\n   Whole dataset & - & \\multicolumn{3}{c}{$56.2 \\pm 0.3$} \\\\\n    \\bottomrule\n    \\end{tabular}\n    \\label{tab:dd on cifar-100}\n\\end{table}\n\n\\begin{table}\n    \n    \\centering\n    \\caption{Performance comparison of different dataset distillation methods on Tiny ImageNet.}\n    \\begin{tabular}{c|c|c|c|c}\n    \\toprule\n\n    \\multirow{2}{*}{Methods} & \\multirow{2}{*}{Distillation schemes} & \\multicolumn{3}{c}{Accuracy} \\\\\n    \\cline{3-5}\n     & & IPC=$1$ & IPC=$10$ & IPC=$50$ \\\\\n\n    \\hline\n    Random & - & $1.4 \\pm 0.1$ & $5.0 \\pm 0.2$ & $15.0\\pm 0.4$ \\\\\n    Herding & - & $2.8 \\pm 0.2$ & $6.3 \\pm 0.2$ & $16.7\\pm 0.3$ \\\\\n   DD (\\citet{wang2018dataset})  & BPTT & - & - & - \\\\\n   LD (\\citet{bohdal2020flexible}) & BPTT & - & - & - \\\\\n   DC (\\citet{zhao2021dataset})& Gradient match & - & - & - \\\\\n   DSA (\\citet{zhao2021dsa})& Gradient match & - & - & - \\\\\n   DCC (\\citet{lee2022dataset}) & Gradient match  & - & - & - \\\\\n   MTT (\\citet{cazenavette2022distillation}) & Trajectory match & $8.8 \\pm 0.3$ & $23.2\\pm 0.2$ & \\bm{$28.0\\pm 0.3$} \\\\\n   FTD (\\citet{du2022minimizing})& Trajectory match & $10.4\\pm 0.3$ & $24.5\\pm 0.2$ & - \\\\\n   TESLA (\\citet{cui2022scaling})& Trajectory match & $7.7\\pm 0.2$ & $18.8\\pm 1.3$ & \\bm{$27.9\\pm 1.2$} \\\\\n   DM (\\citet{zhao2021distribution}) & Distribution match  & $3.9 \\pm 0.2$ & $12.9\\pm 0.4$ & $24.1\\pm 0.3$ \\\\\n   CAFE (\\citet{wang2022cafe})& Distribution match & - & - & - \\\\\n   KIP (\\citet{nguyen2020dataset,nguyen2021dataset}) & Kernel regression & - & - & - \\\\\n   FRePo (\\citet{zhou2022dataset})& Kernel regression & $15.4\\pm 0.3$ & \\bm{$25.4\\pm 0.2$} & - \\\\\n   RFAD (\\citet{loo2022efficient})& Kernel regression & - & - & - \\\\\n   IDC (\\citet{kim2022dataset})$^\\ast$ & Gradient match & - & - & - \\\\\n   \\citet{deng2022remember}$^\\ast$ & BPTT & \\bm{$16.0\\pm 0.7$} & - & - \\\\\n   Haba (\\citet{liu2022dataset})$^\\ast$ & Trajectory match & - & - & - \\\\\n   KFS (\\citet{lee2022kfs})$^\\ast$ & Distribution match & \\bm{$22.7\\pm 0.3$} & \\bm{$27.8\\pm 0.2$} & - \\\\\n   \\cline{3-5}\n   Whole dataset & - & \\multicolumn{3}{c}{$37.6 \\pm 0.4$} \\\\\n    \\bottomrule\n    \\end{tabular}\n    \\label{tab:dd on ti}\n\\end{table}\n\\fi\n\n\n\n\n\\section{Application}\nDue to this superior performance in compressing massive datasets, dataset distillation has been widely employed in many application domains that limit training efficiency and storage, including continual learning and neural architecture search. Furthermore, due to the correspondence between examples and gradients, dataset distillation can also benefit privacy preservation, federated learning, and adversarial robustness. In this section, we briefly review these applications {\\it w.r.t} dataset distillation.\n\n\\subsection{Continual Learning}\n\nDuring to the training process, when there is a shift in the training data distribution, the model will suddenly lose its ability to predict the previous data distribution. This phenomenon is referred to as {\\it catastrophic forgetting} and is common in deep learning. To overcome this problem, continual learning has been developed to incrementally learn new tasks while preserving performance on old tasks \\citep{rebuffi2017icarl,castro2018end}. A common method used in continual learning is the replay-based strategy, which allows a limited memory to store a few training examples for rehearsal in the following training. Therefore, the key to the replay-based strategy is to select highly informative training examples to store. Benefiting from extracting the essence of datasets, the dataset distillation technique has been employed to compress data for memory with limited storage \\citep{zhao2021dataset,carta2022distilled}. Because the incoming data have a changing distribution, the frequency is high for updating the elements in memory, which leads to strict requirements for the efficiency of dataset distillation algorithms. To conveniently embed dataset distillation into the replay-based strategy, \\citet{wiewel2021condensed} and \\citet{sangermano2022sample} decomposed the process of synthetic data generation by linear or nonlinear combination, and thus fewer parameters were optimized during dataset distillation. In addition to enhancing replay-based methods, dataset distillation can also learn a sequence of stable datasets, and the network trained on these stable datasets will not suffer from catastrophic forgetting \\citep{masarczyk2020reducing}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Neural Architecture Search}\n\nFor a given dataset, the technique of neural architecture search (NAS) aims to find an optimal architecture from thousands of network candidates for better generalization. The NAS process usually includes training the network candidates on a small proxy of the original dataset to save training time, and the generalization ranking can be estimated according to these trained network candidates. Therefore, it is important to design the proxy dataset such that models trained on it can reflect the model's true performance in terms of the original data, but the size of the proxy dataset requires control for the sake of efficiency. To construct proxy datasets, conventional methods have been developed, including random selection or greedy search, without altering the original data \\citep{c2022speeding,li2020sgas}. \\citet{such2020generative} first proposed optimizing a highly informative dataset as the proxy for network candidate selection. More subsequent works have considered NAS as an auxiliary task for testing the proposed dataset distillation algorithms \\citep{zhao2021dataset,zhao2021dsa,zhao2023distribution,du2022minimizing,cui2022dc}. Through simulation on the synthetic dataset, a fairly accurate generalization ranking can be collected for selecting the optimal architecture while considerably reducing the training time.\n\n\n\n\n\\subsection{Privacy Protection}\nAs overparameterized neural networks easily memorize all the training data, there exists the risk for privacy leakage by inferring the well-trained networks \\citep{shokri2017membership,long2018understanding,salem2018ml,hu2022membership}. With a synthetic dataset, the network avoids explicitly training on the original dataset and consequently helps protect data privacy \\citep{zhou2022dataset}. In addition, \\citet{chen2022privacy} employed gradient matching to distil private datasets by adding differential privacy noise to the gradients of synthetic data. Theoretically, \\citet{dong2022privacy} and \\citet{zheng2023differentially} built the connection between dataset distillation and differential privacy, and proved the superiority of dataset distillation in privacy preservation over conventional private data generation methods. However, \\citet{carlini2022no} argued that there exists flaws in both experimental and theoretical evidences in \\citet{dong2022privacy}: (1) membership inference baseline should include all training examples, while only $1\\%$ of training points are used to match the synthetic sample size, which thus contributes to the high attack accuracy of baseline; and (2) the privacy analysis is based on a assumption stating that the output of learning algorithms follows a exponential distribution in terms of loss; and the assumption has already implied differential privacy.\n\n\n\\subsection{Federated Learning}\n\nFederated learning has received increasing attention during training neural networks in the past few years due to its advantages in distributed training and private data protection \\citep{yang2019federated}. The federated learning framework consists of multiple clients and one central server, and each client possesses exclusive data for training the corresponding local model \\citep{mcmahan2017communication}. In one round of federated learning, clients transmit the induced gradients or model parameters to the server after training with their exclusive data. Then the central server aggregates the received gradients or parameters to update the model parameters and broadcast new parameters to clients for the next round. In the federated learning scenario, the data distributed in different clients are often non-i.i.d data, which causes a biased minimum {\\it w.r.t.} the local model and significantly hinders the convergence speed. Consequently, the data heterogeneity remarkably imposes a communication burden between the server and clients. \\citet{goetz2020federated} proposed distilling a small set of synthetic data from original exclusive data by matching gradients, and then the synthetic data instead of a large number of gradients were transmitted to the server for model updating. Other distillation approaches such as BPTT  \\citep{hu2022fedsynth,zhou2020distilled}, KRR \\citep{song2022federated}, and distribution matching \\citep{xiong2022feddm}, were also employed to compress the exclusive data to alleviate the communication cost at each transition round. However, \\citet{liu2022meta} discovered that synthetic data generated by dataset distillation are still heterogeneous. To address the problem, these researchers proposed two strategies of dynamic weight assignment and meta knowledge sharing during the distillation process, which significantly accelerate the convergence speed of federated learning. Apart from compressing the local data, \\citet{pi2022dynafed} also distilled data via trajectory matching in the server, which allows the synthetic data to possess global information. Then the distilled data can be used to fine-tune the server model for convergence speed up. \n\n\n\n\n\\subsection{Adversarial Robustness}\nDeep learning networks with standard training are fragile with respect to adversarial attacks, and adding imperceptible perturbations to the input can completely fool the network \\citep{goodfellow2014explaining}. Adversarial training has been widely used to address this problem by continuously feeding adversarial examples during the training process \\citep{madry2017towards}. However, the generation of adversarial examples requires multistep gradient ascent, which considerably undermines training efficiency. Recently, \\citet{tsilivis2022can} and \\citet{wu2022towards} employed dataset distillation to extract the information of adversarial examples and generate robust datasets. Then standard network training on the distilled robust dataset is sufficient to achieve satisfactory robustness to adversarial perturbations, which substantially saves computing resources compared to expensive adversarial training.\n\n\n\n\n\n\\subsection{Other Applications}\n\nApart from the aforementioned applications, we also summarize other applications in terms of dataset distillation as follows.\n\nBecause of their small size, synthetic datasets have been applied to explainability algorithms \\citep{loo2022efficient}. In particular, due to the small size of synthetic data, it is easy to measure how the synthetic examples influence the prediction of test examples. If the test and training images both rely on the same synthetic images, then the training image will greatly influence the prediction of the test image. In other words, the synthetic set becomes a bridge to connect the training and testing examples. \n\nUsing the characteristic of capturing the essence of datasets, \\citep{cazenavette2022textures,chen2022fashion} utilized dataset distillation for visual design. \\citet{cazenavette2022textures} generated the representative textures by randomly cropping the synthetic images during the distillation process. In addition to extracting texture,  \\citet{chen2022fashion} imposed the synthetic images to model the outfit compatibility through dataset distillation.\n\n\nIn addition to the continuous image domain, dataset distillation has also been extended to discrete data such as graphs \\citep{jin2022graph,jin2022condensing,liu2022graph} and recommender systems \\citep{sachdeva2022data}. \\citet{jin2022graph} first formulated this dataset distillation {\\it w.r.t.} graph data, and successfully distilled a small, condensed graph via gradient matching. Due to the computational inefficiency in distilling pairwise relations for graph nodes, \\citet{jin2022condensing} turned to learning a probabilistic graph model, which allows a differentiable optimization {\\it w.r.t.} the discrete graph. These authors also employed one-step strategy for further distillation speedup. Moreover, \\citet{liu2022dataset} accelerated the graph distillation through distribution matching from the perspective of receptive fields. For the recommender system,  \\citet{sachdeva2022data} distilled a massive dataset to a continuous prior for tackling the discrete data problem.\n\nDue to their small size and abstract visual information, distilled data can also be applied in medicine, especially in medical image sharing \\citep{li2023sharing}. Empirical studies on gastric X-ray images have shown the advantages of DD in medical image transition and the anonymization of patient information \\citep{li2020soft,li2022compressed}. Through dataset distillation, hospitals can share their valuable medical data at lower cost and risk to collaboratively build powerful computer-aided diagnosis systems.\n\n\nFor backdoor attack, \\citet{liu2023backdoor} considered to inject backdoor triggers into the small distilled dataset. Because the synthetic data has a small size, directly trigger insertion is less effective and also perceptible. Therefore, these coworkers turned to insert triggers into the target dataset during the dataset distillation procedure. In addition, they propose to iteratively optimize the triggers during the distillation process to preserve the triggers' information for better backdoor attack performance.\n\n\n\n\n\n\n\n\n\n\n\\section{Challenges and Directions}\n\nDue to its superiority in compressing datasets and training speedup, dataset distillation has promising prospects in a wide range of areas. In the following, we discuss existing challenges in terms of dataset distillation. Furthermore, we investigate plausible directions and provide insights into dataset distillation to promote future studies.\n\n\n\n\\subsection{Challenges}\n\nFor dataset distillation, there are two main ingredients: (1) measure the difference between the knowledge of synthetic data and real (original) data; and (2) update the synthetic data to narrow the knowledge difference. Based on these ingredients, we discuss the challenges of dataset distillation from the perspectives of (1) the definition of the difference in the knowledge between two types of datasets, (2) the form of synthetic datasets, (3) the evaluation of synthetic datasets, and (4) the theory behind dataset distillation.\n\n\nDataset distillation methods aim to transfer the knowledge of a target dataset into a small synthetic dataset to achieve a comparable generalization with respect to the original data distribution. Although the knowledge of datasets is an abstract notion, many DD approaches indirectly measure the difference in the knowledge base via various proxy losses. However, the existing definition {\\it w.r.t.} the knowledge difference is inconsistent with the original comparable generalization. For example, the meta-loss $\\mathcal{R}_{\\mathcal{T}}(\\texttt{alg}(\\mathcal{S}))$ in the meta-learning framework measure the knowledge difference in the error in terms of a specific target dataset. Through this, one can only obtain a similar training error other than the test error {\\it w.r.t.} the target dataset. As for the data matching framework, it intuitively makes knowledge concrete with a series of parameters or features, which are matched for knowledge transmission. Therefore, a more consistent definition of knowledge difference should be investigated to improve the dataset distillation performance.\n\nAlthough most dataset distillation methods directly update the synthetic images, optimizing factorized codes and decoders that cooperatively generate synthetic data outperform the direct optimization by a large margin, as discussed in Section \\ref{sec:standard benchmark}. From this, the predefined form of the code and decoder has a remarkable influence on the distillation performance. Many Factorization methods on synthetic data are based on intuition and lack rigorous analysis, and only some vague reasons are proposed, such as decreasing redundant information across different classes and learning more compact representations. Hence, it is necessary to explore the influence of the predefined factorization on the dataset distillation for learning a more informative distilled dataset.\n\nIn practical dataset distillation, the optimization of synthetic data is closely related to the specific network architecture (or kernels in KRR approach), and there is naturally a performance drop when the learned synthetic dataset is applied to other unseen architectures; please see Section \\ref{sec:cross architecture transferability}. Therefore, it is one-sided to compare different DD methods only on one or a few architectures. To compare different DD methods in a more comprehensive scenario, it is essential to set a reasonable baseline consisting of multiple handpicked network architectures with different complexity. \n\nAlthough various DD approaches have been emerging, there have been few investigations that discuss the theory behind dataset distillation. Nevertheless, developing the theory is extremely necessary and will considerably promote the dataset distillation by directly improving the performance and also suggesting the correct direction of development. For example, the theory can help derive a better definition of the dataset knowledge and consequently increase the performance. In addition, the theoretical correlation between the distillation budget (the size of synthetic datasets) and performance can provide an upper bound on the test error, which can offer a holistic understanding and prevent researchers from blindly improving the DD performance. Hence, a solid theory is indispensable to take the development of DD to the next stage.\n\n\n\n\n\n\n\n\n\n\n\\subsection{Future Directions}\n\nAlthough the existing DD methods have shown impressive results in compressing massive datasets, there are still many areas that are worth exploring to promote dataset distillation. In this section, we will discuss some promising directions that shed light on future studies.\n\n\\textbf{Fine-tune with synthetic data.} With the rapid development of foundational models \\citep{bommasani2021opportunities}, the mainstream deep learning pipeline gradually converts from scratch learning to pretrained learning or transfer learning. The foundational models of BERT \\citep{devlin2018bert}, GPT \\citep{radford2018improving,radford2019language,brown2020language}, {\\it etc.}, are trained with billions of data points to learn an appropriate latent representation. With only fine-tuning on specific datasets, the foundational models can achieve extraordinary performance on various downstream tasks and outperform models learned from scratch by a large margin. To this end, conducting dataset distillation in the fine-tuning regime is a promising direction to cooperate with foundational models. With synthetic data, the fine-tuning process will be faster while the downstream performance is preserved. Despite possessing a similar goal, distilling datasets based on pretrained models might be more challenging than the initialized models because the pretrained models are harder to collect for repeated distillation to enhance synthetic datasets' robustness {\\it w.r.t.} model initialization. Apart from condensing datasets in the fine-tuning stage, it is also an interesting topic to distil datasets in the vanilla training stage so that models trained on synthetic datasets can be easier or faster to fine-tune by specific datasets.\n\n\n\n\n\n\n\\textbf{Initialization augmentation.} Benefiting from excellent efficiency and compatibility, data augmentation has become an essential strategy for enhancing the model's generalization {\\it w.r.t.} data distribution \\citep{shorten2019survey, chawla2002smote,perez2017effectiveness}. From basic cropping and flipping \\citep{zagoruyko2016wide} to the mixup method \\citep{zhang2018mixup}, data augmentation efficiently constructs variants of training images with prior rules to improve the model's robustness to patterns. In dataset distillation, synthetic datasets are also sensitive to the parameter initialization and network architecture adopted in the distillation process \\citep{wang2018dataset,zhao2021dataset}. Existing works alleviate this problem by trivial repeated distillation {\\it w.r.t.} different initializations and then taking an expectation \\citep{zhou2022dataset}. However, the random selection of initialization is not an elaborate strategy and has limited improvement. \\citet{zhang2022accelerating} showed that employing early-stage models trained with a few epochs as the initialization can achieve better distillation performance, and they further proposed weight perturbation methods to efficiently generate early-stage models for repeated distillation. Therefore, it is important for many dataset distillation algorithms to investigate how to design the initialization augmentation so that synthetic datasets achieve better generalization to different initializations and architectures.\n\n\n\n\\textbf{Preprocessing of target datasets.} To better extract knowledge from training data, preprocessing, such as normalization, is a common approach that reshapes the structure of data by well-designed mappings \\citep{garcia2016big}. In existing dataset distillation methods, only a few works discuss the influence of preprocessing on final distillation performance \\citep{nguyen2020dataset}. \\citet{nguyen2020dataset,nguyen2021dataset} and \\citet{cazenavette2022distillation} preprocessed target datasets with zero-phase compenent analysis (ZCA) whitening \\citep{kessy2018optimal} before distillation and achieved satisfying empirical results. Through elaborate preprocessing, more explicit information can emerge, and the target datasets might be easier to distill. In addition, this preprocessing is compatible with existing dataset distillation algorithms and thus can be deployed without extra effort. For this reason, it is useful to investigate the preprocessing of target datasets for distillation acceleration and also performance improvement.\n\n\n\\section*{Acknowledge}\n\nWe sincerely thank \\href{https:\/\/github.com\/Guang000\/Awesome-Dataset-Distillation}{Awesome-Dataset-Distillation} for its comprehensive and timely DD publication summary. We also thank Can Chen and Guang Li for their kind feedback and suggestions.\n\n\n\\bibliographystyle{plainnat}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nIn a combinatorial optimisation problem, such as MAX-CUT or the TRAVELLING-SALESPERSON \\cite{Pap81}, the aim is to evolve from some initial state to a final state that encodes the solution to the optimisation problem. One approach might be to evolve adiabatically, encoding each of the initial and final states as the ground state of some Hamiltonian and interpolating sufficiently slowly between them. In practice this approach is limited by the minimum spectral gap of the interpolating Hamiltonian \\cite{Ami09, Rei04}. This approach is known as adiabatic quantum optimisation (AQO) \\cite{Apo89,Far00, Kad98,Fin94,Alb18}.\n\nIn the absence of mature hardware, AQO has relied on the adiabatic principle as a guiding design tenet. In turn AQO has lead to Quantum Annealing (QA). Similar to AQO, QA attempts to interpolate continuously between the initial and final Hamiltonians. QA denotes a broader approach than AQO to find (or approximate) the solution of the optimisation problem. This could include: thermal effects \\cite{Dic13}; intentionally operating diabatically \\cite{Cro21,Fry21,Som12}; or adding new terms to the Hamiltonian \\cite{Far11, Zen16, Far02, Fein22, Cro14, Cho21}. QA has gone on to help inspire a number of other approaches, such as: the gate-based Quantum Approximate Optimisation Algorithm (QAOA) \\cite{Far14};  continuous-time quantum walks (QW) for optimisation \\cite{Cal19,Ken20}; and combinations of different approaches \\cite{Cal21, Mor19}.  In short, the adiabatic principle has been a successful design-principle for developing new heuristic quantum algorithms for optimisation. \n\nAn alternative approach for evolving from an initial state to the final state might be via Hamiltonians for optimal state transfer \\cite{Bro06}. These are Hamiltonians that transfer the system from the initial state to the final state in the shortest possible time. In this paper we use these Hamiltonians as our underlying design principle. Such Hamiltonians typically require knowledge of the final state (i.e., the solution to the optimisation problem). In the absence of this information we therefore focus on how the behaviour of these Hamiltonians might be approximated to find approximate solutions to optimisation problems. The result is a rapid continuous-time approach, with a single variational parameter. \n\nThe framework of this paper is as follows: Sections \\ref{sec:QAframe}, \\ref{sec:hamdes}, \\ref{sec:met}, and \\ref{sec:prob} introduce the requisite background material and introduce the Hamiltonians considered in the rest of the paper (Sec.\\ \\ref{sec:hamdes}). Sections \\ref{sec:H1}, \\ref{sec:QZ}, and \\ref{sec:lpa} provide analytical and numerical evidence for the performance of these Hamiltonians. Throughout the paper we adopt the convention $\\hbar=1$. The corresponding Pauli matrices are denoted by $X,Y$ and $Z$. The identity is denoted by $I$. The commutator is denoted by $[\\cdot,\\cdot]$. The simulations make use of the Python packages QuTiP \\cite{Joh12,Joh13} and Qiskit \\cite{Qiskit}.\n\n\\section{The QA-framework}\n\\label{sec:QAframe}\nBefore elaborating on our new approach, in this section we provide a very brief overview of the adiabatic influenced algorithms mentioned in the introduction.  \n\nIn QA, typically the optimisation problem is encoded as finding the ground-state, $\\ket{\\psi_f}$, of an Ising Hamiltonian, $H_f$. The system is initialised in the ground-state, $\\ket{\\psi_i}$, of an easy to prepare Hamiltonian $H_i$. Usually, $H_i$ is taken to be the transverse-field Hamiltonian. The optimisation problem is solved by evolving the system to the ground-state of $H_f$. We refer to this set-up as the QA-framework. We utilise this encoding of the optimisation problem for our new approach. \n\nFor the adiabatic influenced approaches (AQO, QA, QAOA and QW) the system is subjected to the following Hamiltonian: \n\\begin{equation}\n\\label{eq:QAham}\n    H(t)=A(t) H_i + B(t) H_f,\n\\end{equation}\nwhere $t\\in[0,T]$ denotes time. The design philosophy behind how the schedules $A(t)$ and $B(t)$ are chosen, is what distinguishes these near-term intermediate-scale quantum (NISQ) \\cite{Pre18} approaches. For AQO, the schedules are chosen to interpolate adiabatically between the two ground states \\cite{Apo89,Far00, Kad98,Fin94}. In QA, this restriction is loosened to typically continuous, monotonically decreasing for $A(t)$ and increasing for $B(t)$, schedules \\cite{Cro21,Hau20}. In QW $A(t)$ and $B(t)$ are taken to be constants over the whole evolution \\cite{Cal19,Ken20}. \n\nQAOA \\cite{Far14} is a gate-based design-philosophy for determining the schedules. In QAOA either $A(t)=1$ and $B(t)=0$, or $A(t)=0$ and $B(t)=1$. The switching parameter, $p$, controls the number of times the schedules alternate between the two parameter settings. The duration between switching is either determined prior to the evolution by a classical computer or by using a classical variational outer-loop attempting to minimise $\\langle H_f \\rangle$ by varying $2p$ free-parameters. QAOA again relies on the adiabatic principle to provide a guarantee of finding the ground-state in the limit of infinite $p$. However, far from this limit, the variational method allows QAOA to exploit non-adiabatic evolution \\cite{Zho20}. QAOA has been further generalised, retaining its switching framework to the `Quantum Alternating Operator Ansatz' \\cite{Had19}. It has also become a popular choice for benchmarking the performance of quantum hardware \\cite{Har21, Gra22, Ott17}. \n\n\nRecently Brady et al.\\ applied optimal control-theory to help design and investigate schedules \\cite{Bra21_1,Bra21,fei22}. They demonstrated for $0\\leq A(t) \\leq1$ and $B(t)=1-A(t)$, that the optimal schedule starts and finishes with a `bang'  (i.e., one of the controls takes its maximum value)\\cite{Bra21_1}. This claim was further investigated in \\cite{Venuti_2021} and applied to open quantum systems.\n\nClearly there are many design-philosophies for tacking problems within the QA-framework. Although, some have been inspired by the adiabatic theorem, practical implementation might have little resemblance to this original idea. \n\nThe next section provides a brief introduction to Hamiltonians for optimal state-transfer as well as the motivation for the choice of Hamiltonians used in this paper. The bulk of this paper will examine the performance of this approach on a variety of optimisation problems.\n\n\n\\section{Hamiltonian design}\n\\label{sec:hamdes}\n\nThe aim of optimal state-transfer is to find a Hamiltonian that transfers the system from an initial state (i.e., $\\ket{\\psi_i}$) to a known final state (i.e., $\\ket{\\psi_f}$) in the shortest possible time. We shall refer to this Hamiltonian as the optimal Hamiltonian (although it is by no means unique).\n\nOne notable approach to finding the optimal Hamiltonian comes from Nielsen et al. \\cite{Nie05,Nie06,Dow07} who investigated the use of differential geometry, at the level of unitaries, to find geodesics connecting the identity to the desired unitary. The length of the geodesic was linked to the computational complexity of the problem \\cite{Nie06}. A second approach comes from Carnali et al.. Inspired by the brachistochrone problem, they developed a variational approach at both the level of state-vectors \\cite{Car06} and unitaries \\cite{Car07}. The quantum brachistochrone problem has generated a considerable amount of literature and interest \\cite{Rez09,Wan15,Wak20,Wan21, San21, Yan22}. \n\nBoth approaches allow for constraints to be imposed on the Hamiltonian. Both concluded that if the only constraint is on the total energy of the Hamiltonian, the optimal Hamiltonian is constant in time. In the rest of this section we outline the geometric argument put forward by Brody et al.\\ \\cite{Bro06} to find the optimal Hamiltonian in this case. The reader is invited to refer to the original work for the explicit details. \n\nThe optimal Hamiltonian will generate evolution in a straight line in the (complex-projective) space in which the states live.  Intuitively, a line in this space between $\\ket{\\psi_i}$ and $\\ket{\\psi_f}$ consists of superpositions of the two states. Hence, the line in the complex-projective space can be represented on the Bloch sphere.\n\nIf, without loss of generality, $\\ket{\\psi_i}$ and $\\ket{\\psi_f}$ are placed in the traditional $z-x$ plane of the Bloch sphere, it is clear that the optimal Hamiltonian generates rotations in this plane. The optimal Hamiltonian is then (reminiscent of the cross-product):\n\\begin{equation}\n    H_{opt}=-i \\left(\\ket{\\psi_i}\\bra{\\psi_f}-\\ket{\\psi_f}\\bra{\\psi_i}\\right).\n    \\label{eq:optHam}\n\\end{equation}\nThis can be scaled to meet the condition on the energy of the Hamiltonian. It then remains to calculate the time required to transfer between the two states. This will depend on how far apart the states are and how fast the evolution is. This is encapsulated in the Anandan-Aharonov relationship \\cite{Ana90}:\n\\begin{equation}\n    \\frac{ds}{dt}=2\\delta E(t).\n\\end{equation}\n\nThe left-hand-side denotes the speed of the state, $\\ket{\\psi(t)}$, where $ds$ is the infinitesimal distance between $\\ket{\\psi\\left(t+dt\\right)}$ and $\\ket{\\psi\\left(t\\right)}$ \\footnote{The distance is measured by the Fubini-study metric, $ds^2=4\\left(1-\\abs{\\bra{\\psi\\left(t\\right)}\\ket{\\psi\\left(t+dt\\right)}}^2\\right)$, on the complex-projective space.}.  Under evolution by the Schr\\\"odinger equation, with Hamiltonian $H$, the instantaneous speed of the evolution is given by the uncertainty in the energy, $\\delta E(t)^2=\\bra{\\psi(t)}H(t)^2\\ket{\\psi(t)}-\\bra{\\psi(t)}H(t)\\ket{\\psi(t)}^2$.\n\nSince the optimal Hamiltonian is constant in time, $\\delta E$ can be evaluated using the initial state.  Therefore, the time of evolution is:\n\\begin{equation}\nT=\\frac{\\arccos{\\abs{\\bra{\\psi_f}\\ket{\\psi_i}}}}{ \\sqrt{1-\\abs{\\bra{\\psi_f}\\ket{\\psi_i}}^2}}.\n\\end{equation}\n\n In summary, $e^{-i H_{opt} T}\\ket{\\psi_i}$ generates the state $\\ket{\\psi_f}$. The goal of the rest of this paper is to harness some of the physics behind this expression for computation.  To this end we primarily focus on Hamiltonians which are constant in time. Appendix \\ref{sec:opHamQa} contains more details on what optimal Hamiltonians might look like within the QA-framework. \n \n In the style of a variational quantum eigensolver (VQE) \\cite{Per14,Fed21}, we allow $T$ to be a variational parameter that needs to be optimised in our new approach. In this paper we select the $T$ that minimises the final value of $\\langle H_f \\rangle$. There is scope to extend this to different metrics \\cite{Li20,Bar20}. As $T$ is a variational parameter the Hamiltonian is only important up to some constant factor. Rewriting Eq.\\ \\ref{eq:optHam} up to some constant gives:\n\\begin{equation}\n    \\label{eq:optHamCom}\n    H_{opt}\\propto\\frac{1}{2i}\\left[\\ket{\\psi_i}\\bra{\\psi_i},\\ket{\\psi_f}\\bra{\\psi_f}\\right],\n\\end{equation}\nassuming $\\ket{\\psi_i}$ and $\\ket{\\psi_f}$ have a non-zero overlap (this is a given in the standard QA-framework). This equation (Eq.\\ \\ref{eq:optHamCom}) provides the starting point for all the Hamiltonians considered in this paper. \n\nThe optimal Hamiltonian (i.e., Eq.\\ \\ref{eq:optHamCom}) requires knowledge of the final state. In practice, when attempting to solve an optimisation problem, one doesn't have direct access to $\\ket{\\psi_f}$.  Instead one has easy access to $H_i$ and $H_f$. Therefore, we make the pragmatic substitutions $\\ket{\\psi_i}\\bra{\\psi_i} \\rightarrow H_i$ and $\\ket{\\psi_f}\\bra{\\psi_f} \\rightarrow H_f$ into Eq.\\ \\ref{eq:optHamCom}\n\n\\begin{align}\nH_{opt}\\propto\\frac{1}{2i}&\\left[\\ket{\\psi_i}\\bra{\\psi_i},\\ket{\\psi_f}\\bra{\\psi_f}\\right] \\nonumber \\\\\n    & \\hspace{0.6cm}\\downarrow\\hspace{1.3cm}\\downarrow \\nonumber\\\\\n    H_1=\\frac{1}{2i}&\\left[\\hspace{0.4cm}H_i\\hspace{0.4cm},\\hspace{0.6cm}H_f\\hspace{0.4cm}\\right].\n\\end{align}\nThis Hamiltonian is the most amenable to NISQ implementation of all the Hamiltonians considered in this paper, therefore the bulk of the paper is devoted to demonstrating its performance. The results can be seen in Sec.\\ \\ref{sec:H1}.\n\nThe substitutions $\\ket{\\psi_i}\\bra{\\psi_i}\\rightarrow H_i$ and $\\ket{\\psi_f}\\bra{\\psi_f}\\rightarrow H_f$ introduce errors, such that the evolution under $H_1$ no longer closely follows the evolution under $H_{opt}$. In Sec.\\ \\ref{sec:QZ} we try to correct for this error by adding a new term to the Hamiltonian\n\\begin{equation}H_{1,improved}=H_1+H_{QZ}.\n\\end{equation}\nThe proposed form of $H_{QZ}$ is motivated by the quantum Zermello problem \\cite{Bro15,Brody_2015,Rus14,Rus15}.\n\nFinally, in Sec.\\ \\ref{sec:lpa}  we exploit our knowledge of the initial state and propose the substitution $\\ket{\\psi_f}\\bra{\\psi_f}\\rightarrow f(H_f)$, where $f(\\cdot)$ is some real function:\n\n\\begin{align}\n    H_{opt}\\propto\\frac{1}{2i}&\\left[\\ket{\\psi_i}\\bra{\\psi_i},\\ket{\\psi_f}\\bra{\\psi_f}\\right] \\nonumber\\\\\n    & \\hspace{2.1cm}\\downarrow \\nonumber\\\\\n    H_{\\psi_i}=\\frac{1}{2i}&\\left[\\ket{\\psi_i}\\bra{\\psi_i},\\hspace{0.2cm}f\\left(H_f\\right)\\hspace{0.1cm}\\right].\n\\end{align}\n\n\n\\section{Metrics used to assess the performance}\n\\label{sec:met}\nOnce a problem and algorithm have been determined it remains to decide on the metric (or metrics) by which to assess the performance. A common choice is the ground-state probability, $P_{gs}$. This is a reasonable measure for an exact solver. It fails to capture the performance of an approximate solver (a solver that finds \\textit{good enough} solutions). A common measure for approximate solvers, such as QAOA, is the approximation ratio. The approximation ratio is a measure on the final distribution produced by the approach. Here we define the approximation ratio to be $\\langle H_f \\rangle\/E_{min}$ where $E_{min}$ is the energy of the ground-state solution and the expectation is with respect to the final state. If the approach finds the ground-state exactly, then $\\langle H_f \\rangle\/E_{min}=1$. Random guessing (for all the problem Hamiltonians considered in Sec.\\ \\ref{sec:prob} ) has an approximation ratio of $0$. If an algorithm for a specific problem is cited to have a approximation ratio with no explicit reference to time, this refers to an optimised approximation ratio. \n\nIn Sec.\\ \\ref{sec:H1Num}, we consider the width of the final energy distribution. Wider distributions may mean the algorithm is harder to optimise in practice. We use $\\sigma=\\sqrt{\\langle H_f^2\\rangle-\\langle H_f \\rangle}\/E_{min}$ to measure the width of the distribution.\n\nOf crucial interest for NISQ implementations is the duration of each run. For continuous-time approaches this is simply the time of each run. We would like to compare this approach to QAOA, which has the following ansatz:\n\\begin{equation}\n    \\label{eq:QAOA}\n    \\ket{\\psi_{QAOA}}=\\prod_{k=1}^p \\left(e^{-i \\beta_k H_i}e^{-i \\gamma_k H_f}\\right) \\ket{+}\n\\end{equation}\nwhere the $\\beta_k$s and $\\gamma_k$s are the variational parameters. We take the time of a QAOA run to be the sum of the variational parameters. In Sec.\\ \\ref{sec:QAOA} we compare the optimal time of QAOA $p=1$ and $H_1$, the optimal time being the time that maximises the approximation ratio. To make a fair comparison between the two approaches with different problem sizes, we fix the energy of the Hamiltonians in Sec.\\ \\ref{sec:QAOA} to be: \n\\begin{equation}\n     \\label{eq:norm}\n    \\frac{1}{2^n}\\Tr{H_*^2}=n,\n\\end{equation}\nfor $*=i,f,1$, where $n$ is the number of qubits.\n\nThese are by no means all possible metrics to assess the performance of an algorithm. For example time to solution is another popular metric choice \\cite{Alb18,Kadowaki_2022}.\n\n\n\n\\section{Problems considered}\n\\label{sec:prob}\nThe Hamiltonians proposed in Sec.\\ \\ref{sec:hamdes} need to be applied to optimisation problems to assess their performance. Algorithms are unlikely to work uniformly well on all problems; for instance QA is believed to work better on problems with tall, thin barriers in the energy landscape \\cite{Kat15}. Consequently the apparent performance of a heuristic algorithm will, in general, be dependent on the optimisation problem considered as well as the algorithm. With this caveat in mind this paper makes use of two canonical optimisation problems: MAX-CUT and a Sherrington-Kirkpatrick-inspired problem. This section briefly outlines these problems. Readers familiar with these problems, may skip this section.\n\n\\subsection{MAX-CUT}\n\nMAX-CUT seeks to find the maximum cut of a graph, $G=(V,E)$. A cut separates the nodes of the graph into two disjoint sets. The value of the cut is equal to the number of edges between the two disjoint sets. In general, MAX-CUT is an NP-hard problem \\cite{Gar76}. Indeed, finding very good approximations for MAX-CUT is a computationally hard problem \\cite{Pap88}.\n\n\nThe corresponding Ising formulation of this problem is:\n\\begin{equation}\n    H_f=\\sum_{(i,j) \\in E} Z_i Z_j, \n\\end{equation}\nup to a constant offset (i.e. a term proportional to the identity) and multiplicative factor. In this paper we explore MAX-CUT on a range of graphs.\n\n\\subsubsection{Two-regular graphs}\n\nMAX-CUT on two-regular graphs (also known as the Ring of Disagrees or the anti-ferromagnetic ring) is a well studied problem in the context of QA and QAOA \\cite{Far00,Wan18,mbe19,Far14}. The problem Hamiltonian\n\\begin{equation}\n    H_f=\\sum_{i=1}^n Z_i Z_{i+1}\n\\end{equation}\nwith $n+1=1$, consists of nearest-neighbour terms only. The performance of QA and QAOA on this problem has been understood by applying the Jordan-Wigner transformation to map the problem onto free fermions \\cite{Far00,Wan18}. In Sec.\\ \\ref{sec:RoD} we follow the approach laid out by \\cite{Wan18} to apply this technique to $H_1$. A further useful tutorial for tackling the Ising chain with the Jordan-Wigner transformations can be found in \\cite{Mbe20}.\n\nAlternatively, provided $p$ is sufficiently small, the performance of QAOA can be understood in terms of locality \\cite{Far14}. Due to the structure of the ansatz in QAOA (i.e.\\ Eq.\\ \\ref{eq:QAOA}), to find the expectation of a term in $H_f$, such as $Z_iZ_{i+1}$, it is only necessary to consider a subgraph. This subgraph consists of all nodes connected by no more than $p$ edges to a node in the support of the expectation value being calculated. For two regular graphs this is a chain consisting of $2p+2$ nodes. Provided this subgraph is smaller than the problem graph (i.e. the two-regular graph has more than $2p+2$ nodes), then QAOA is operating locally.\n\nFor a given $p$, all the subgraphs are identical for a two-regular graph. Hence, the performance of QAOA for this problem depends only on its performance on this subgraph. As a direct consequence of the locality, the approximation ratio of QAOA will not change as the size of the two-regular graph is scaled. By optimising over this subgraph with a classical resource, it is therefore possible to find the optimal time and approximation ratio of QAOA for this problem.\n\n\n\\subsubsection{Three-regular graphs}\n\\label{sec:prob3reg}\nMAX-CUT on three-regular graphs was considered in the original QAOA paper by Farhi et al.\\ \\cite{Far14}. The local-nature of QAOA allowed them to calculate explicit bounds on the performance of their algorithm. To this end, the graph was broken down into subgraphs to measure local expectation values (i.e., $\\langle Z_iZ_{i+1}\\rangle$). For QAOA $p=1$, there are three distinct subgraphs. By simulating QAOA for the subgraphs and bounding the proportion of subgraphs in the problem, they calculated a lower bound on the performance.  The small number of relevant subgraphs for three-regular graphs, makes this approach particularly amenable. They found that QAOA $p=1$  will produce a distribution whose average will correspond to at least $0.6924$ times the best-cut. This lower-bound is saturated by triangle-free graphs. These graphs consist of a single subgraph. Therefore, the performance of QAOA $p=1$ is largely dependent of the proportion of edges in this problem that belong to this subgraph. We refer to the performance of a local approach being limited by its performance on one subgraph (or possibly a small handful of subgraphs) as being dominated by this subgraph.\n\nThis method was later extended by Braida et al.\\  \\cite{Bra22} who applied it to QA by using an approach inspired by Lieb-Robinson bounds. By operating QA on short times it can be treated as a local algorithm. By then simulating local sub-graphs (as in the QAOA case) and calculating the bounds outlined in the paper, the worst-case performance for each sub-graph can be calculated.  This approach does not necessarily find a tight bound, but it is nonetheless impressive, with bounds in continuous-time quantum computation a rarity. They show that QA finds at least $0.5933$ times the best cut. The authors conjecture that $0.6963$ times the best cut might be a tighter bound on the performance but are unable to rigorously show that this is the case.\n\nIn Sec.\\ \\ref{sec:3reg} we make use of the approach formulated by Braida et al.\\ to prove a lower bound for $H_1$ for this problem.\n\n\\subsubsection{Random graph instances}\n\nTo generalise the MAX-CUT instances discussed above, we also consider MAX-CUT on randomly generated graphs. These graphs do not have fixed degree. The graphs are generated by selecting an edge between any two nodes with probability $p$. MAX-CUT undergoes a computational phase change for random graphs at $p=1\/2$ (harder problems appear for $p>1\/2$) \\cite{Gam18,Cop03,Pol22}. We set $p=2\/3$ for this paper (note that this is no guarantee of hardness for the problem instances considered). \n\n\\subsection{Sherrington-Kirkpatrick inspired model}\n\nIn a MAX-CUT problem all the couplers in the graph are set to the same value. Here we introduce a second problem, the Sherrington-Kirkpatrick model (SKM), where this is not the case. The problem is to find the ground-state of \n\\begin{equation}\n    H_f=\\sum_{i,j} J_{i,j} Z_i Z_j \n\\end{equation}\nwhere the $J_{i,j}$'s are randomly selected from a normal distribution with mean 0 and variance 1 \\cite{She75}. Each qubit is coupled to every other qubit. \n\nTo further distinguish the SKM from the MAX-CUT problems we introduce bias terms to the Hamiltonian,\n\\begin{equation}\n    H_f=\\sum_{i,j} J_{i,j} Z_i Z_j+\\sum_i^n h_i Z_i,\n\\end{equation}\nwhere the $h_{i}$'s are also randomly selected from a normal distribution with mean 0 and variance 1. Here we use the SKM to give an indicative idea of the performance of the proposed Hamiltonians on a wider range of problem instances, rather than just MAX-CUT.\n\n\nThe rest of the paper will examine the performance of each of the Hamiltonians introduced in Sec.\\ \\ref{sec:hamdes}. Each Hamiltonian will be applied to one or more of the problems outlined in Sec.\\ \\ref{sec:prob}. The performance will be optimised to give the best approximation ratio for each problem instance. The discussion of the performance will utilise the metrics outlined Sec.\\ \\ref{sec:met}. \n\n\n\\section{Taking the commutator between the initial and final Hamiltonian}\n\\label{sec:H1}\nIn Sec.\\ \\ref{sec:hamdes}. we motivated the Hamiltonian\n\\begin{equation}\n    \\label{eq:sqham}\n    H_1=\\frac{1}{2i}\\left[H_i,H_f\\right]\n\\end{equation}\nby substituting out the projectors in Eq.\\ \\ref{eq:optHamCom} for easily accessible Hamiltonians. In this section we explore the effectiveness of these substitutions. We begin by demonstrating that Eq.\\ref{eq:sqham} generates the optimal rotation for a single qubit (Sec.\\ \\ref{sec:singleQ}). In Sec.\\ \\ref{sec:SERand} we show that $H_1$ has the potential to outperform random guessing within the QA-framework. The rest of the section analyses the performance of $H_1$ on the problems outlined in Sec.\\ \\ref{sec:prob}.\n\n\n\\subsection{The optimal approach for a single qubit}\n\\label{sec:singleQ}\n\nHere we outline a simple geometric argument which shows that $H_1$ generates the optimal rotation for a single qubit (hence the name $H_1$). The eigenstates of $H_i$ and $H_f$ can be represented as points on the surface of the Bloch sphere, see Fig.\\ \\ref{fig:bloch}. Since these points lie in a plane, the aim is to write down a Hamiltonian that generates rotation in this plane. By writing the Hamiltonians in the Pauli basis, it is clear the Hamiltonian that generates the correct rotation is:\n\\begin{equation*}\n    H_1=\\frac{1}{2i}\\left[H_i,H_f\\right].\n\\end{equation*}\nFor the full details see Appendix \\ref{app:sq}. Again, by using the Anandan-Aharonov relationship:\n\\begin{equation}\n    \\frac{d\\theta}{dt}=2\\delta E,\n\\end{equation}\nwhere $\\delta E$ is the uncertainty in the energy and $\\theta$ is the distance between the desired states (Fig.\\ \\ref{fig:bloch}), we can calculate the time required to transfer between the two ground-states. \n\n\\begin{figure}\n\\begin{tikzpicture}[line cap=round, line join=round, >=Triangle]\n  \\clip(-2.19,-2.49) rectangle (2.66,2.58);\n\n  \\draw [shift={(0,0)}, lightgray, fill, fill opacity=0.1] (0,0) -- (-135.7:0.4) arc (-135.7:-41:0.4) -- cycle;\n  \\draw(0,0) circle (2cm);\n  \\draw [rotate around={0.:(0.,0.)},dash pattern=on 3pt off 3pt] (0,0) ellipse (2cm and 0.9cm);\n\n  \\draw [->] (0,0) -- (0,2);\n  \\draw [->] (0,0) -- (-0.81,-0.79);\n  \\draw [->] (0,0) -- (0.90,-0.8);\n  \\draw [->, style=dashed] (0,0) -- (0.81,0.79);\n  \\draw [->, style=dashed] (0,0) -- (-0.9,0.8);\n\n  \\draw[ ->] (0.5,1.3);\n  \\draw (0,1.2) ellipse (0.5cm and 0.225cm);\n  \\draw (-0.08,-0.3) node[anchor=north west] {$\\theta$};\n \n  \\draw (-1.05,-0.75) node[anchor=north west] {$\\mathbf {\\hat{m}}$};\n  \\draw (0.8,-0.7) node[anchor=north west] {$\\mathbf {\\hat{n}}$};\n  \\draw (-0.5,2.6) node[anchor=north west] {$\\mathbf {\\hat{k}}=\\mathbf {\\hat{m}}\\cross\\mathbf {\\hat{n}}$};\n\\end{tikzpicture}\n\\caption{The geometric intuition behind finding the Hamiltonian for optimally transferring between the ground-states of $H_i$ and $H_f$ on the Bloch sphere. The vectors $\\pm \\hat{m}$ ($\\pm \\hat{n}$) are the eigenvectors of $H_i$ ($H_f$). The aim is to generate a rotation of $\\theta$ around $\\hat{k}$ to map $\\pm \\hat{m}$ to $\\pm \\hat{n}$. The handedness of the cross-product takes into account the direction.}\n\\label{fig:bloch}\n\\end{figure}\n\nHaving established $H_1$ as the optimal Hamiltonian for a single qubit, the next section investigates its performance on larger problems. \n\n\\subsection{Application to larger problems}\n\\subsubsection{Outperforming random guessing for short times}\n\\label{sec:SERand}\n\nIn this section we demonstrate that $H_1$ can always do better than random guessing within the QA-framework. Starting with the time-dependent Schr\\\"odinger equation:\n\\begin{equation*}\n    \\ket{\\dot{\\psi}(t)}=-i H_1 \\ket{\\psi(t)},\n\\end{equation*}\nwe expand $\\ket{\\psi(t)}$ in terms of the eigenbasis of $H_f$, so $\\ket{\\psi(t)}=\\sum_k c_{k}(t) \\ket{k}$ where $\\ket{k}$ are the eigenvectors of $H_f$ with associated eigenvalue $E_k$. The eigenvalues are ordered such that $E_0\\leq E_1 \\leq E_2 \\dots$. Substituting this into the Schr\\\"odinger equation gives:\n\\begin{align*}\n    \\sum_k \\dot{c}_k(t) \\ket{k}&=-\\frac{i}{2i}\\sum_k c_k(t) \\left(H_i H_f -H_f H_i \\right)\\ket{k}\\\\\n    &=-\\frac{1}{2}\\sum_k c_k(t) E_k \\left(H_i-H_f H_i\\right)\\ket{k}\n\\end{align*}\n\nActing with $\\bra{j}$ on each side, to find $\\dot{c}_j(t)$, gives:\n\\begin{equation}\n    \\dot{c}_j(t)=-\\frac{1}{2}\\sum_k c_k(t) \\underbrace{\\left(E_k-E_j\\right)}_{\\text{\"Velocity\"}}\\overbrace{\\bra{j}H_i\\ket{k}}^{\\substack{\\text{How the basis}\\\\ \\text{states of }H_f \\\\ \\text{are connected}}}.\n    \\label{eq:TDSEcoef}\n\\end{equation}\n\nIn the standard QA-framework $H_i=-\\sum_k^n X_k$, $c_k(0)=1\/\\sqrt{2^n}$, for all $k$, and the basis states (e.g. $\\ket{k}$), correspond to computational basis states. Accordingly, $H_1$ connects computational basis states which are a Hamming-distance of one away.\n\nThe difference in energy of the computational basis states intuitively provides something akin to a velocity, with greater rates of change between states which are further apart in energy.\n\nFocusing on the derivative of the ground-state amplitude at $t=0$ we have:\n\\begin{equation}\n    \\dot{c}_0(0)=-\\frac{1}{2}\\sum_k \\underbrace{c_k(0)}_{>0}\\underbrace{\\left(E_k-E_0\\right)}_{\\geq 0}\\underbrace{\\bra{0}-\\sum_j X_j\\ket{k}}_{=0 \\text{ or }-1},\n\\end{equation}\nso $\\dot{c}_0(0)\\geq0$, with equality if all states in a Hamming-distance of one have the same energy as $\\ket{0}$. In this case the above logic can be repeated for these states. Hence, at $t=0$, the ground state amplitude is increasing - meaning $H_1$ can do better than random guessing by measuring on short times. This is evidence that $H_1$ is capturing something of the of the optimal Hamiltonian for short times. Indeed, for short times all the amplitudes flow from higher-energy states to lower-energy states. \n\nThe above logic can be extended to the case where $H_i$ is any stoquastic Hamiltonian in the computational basis \\cite{Bra10,Alb18} and $\\ket{\\psi_i}$ the corresponding ground-state.  That is to say, we require $H_i$ to have non-positive off-diagonal elements in the computational basis (i.e. stoquastic) and as a consequence we can can write the ground-state of $H_i$  with real non-negative amplitudes \\cite{Alb18}. Consequently, for any stoquastic choice of $H_i$ the ground-state amplitude is increasing at $t=0$ and can do better than the initial value of $c_0$ at short times. \n\nWe could also take a generalised version of $H_1$:\n\\begin{equation}\n    H_{1,gen}=\\frac{1}{2i}\\left[f\\left(H_i\\right),g\\left(H_f\\right)\\right],\n\\end{equation}\nwhere $f$ and $g$ are real functions. Eq.\\ \\ref{eq:TDSEcoef} becomes:\n\\begin{equation}\n    \\dot{c}_j(t)=-\\frac{1}{2}\\sum_k c_k(t) \\left(g\\left(E_k\\right)-g\\left(E_j\\right)\\right)\\bra{j}f\\left(H_i\\right)\\ket{k}.\n\\end{equation}\n\nThe function acting on $H_i$ (i.e., $f(\\cdot)$) controls how the computational-basis states are connected, while the function acting on $H_f$ (i.e., $g(\\cdot)$) controls the velocity between computational basis states. If $f(\\cdot)$ is the identity and $H_i$ stoquastic, then any monotonic function for $g(\\cdot)$ (e.g., $H_f^3$, $H_f^5$, $\\exp{H_f}$,...) will do better than $c_0(0)$ for short $t$. Taking $H_i$ to be the transverse-field Hamiltonian, that is better than random-guessing. \n\nThe above analysis demonstrates that $H_1$ has potential for tackling generic problems within the QA-framework. The next sections apply $H_1$ to specific examples in an attempt to quantify the success of this approach. For the rest of this paper we take $H_i$ to be the transverse-field Hamiltonian.\n\n\\subsubsection{MAX-CUT on two-regular graphs}\n\\label{sec:RoD}\n\nHere we study the performance of $H_1$ on MAX-CUT with two-regular graphs. The explicit form of $H_1$ is then:\n\\begin{equation}\n    H_1=\\sum_{j=1}^n \\left(Y_jZ_{j+1}+Z_jY_{j+1}\\right).\n\\end{equation}\nThis can be solved analytically by mapping the problem onto free fermions via the Jordan-Wigner transformation. Details can be found in Appendix \\ \\ref{app:ferm}.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.48\\textwidth]{images\/RoDtd.png}\n    \\caption{A time-domain plot of the ground-state probability (in pink) and approximation ratio (in blue) for $H_1$ applied to MAX-CUT on a 2-regular graph with 400 qubits. Random guessing corresponds to a ground-state probability of $2^{-399}\\approx 10^{-120}$. The dashed purple line shows the location of the optimal time, corresponding to the maximum in approximation ratio.}\n    \\label{fig:RoDtd}\n\\end{figure}\n\nA time domain plot for the approximation ratio and ground-state probability is shown in Fig.\\ \\ref{fig:RoDtd} for 400 qubits. The peak in approximation ratio corresponding to the optimal time. As expected from the previous section (Sec.\\ \\ref{sec:SERand}) the approximation ratio is increasing at $t=0$.  There is a clear peak in ground-state probability at a time of $t\\approx 0.275$. This peak remains present for larger problem sizes too. The peak also occurs at a later time than the peak in approximation ratio. Further insight into this phenomena may be found in Sec.\\ \\ref{sec:lpa}.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.48\\textwidth]{images\/RoDeC.png}\n    \\caption{The approximation ratio for $H_1$ applied to MAX-CUT on two-regular graphs with an even number of qubits (solid line). The dashed lines show the performance of QAOA for this problem when $p < n\/2$ \\cite{Far14,mbe19}. The corresponding optimal times can be found in Fig. \\ref{fig:RoDts}. The approximation ratio for $H_1$ freezes out at $0.5819$, with a corresponding time of $0.2301$.}. \n    \\label{fig:RodeC}\n\\end{figure}\n\nThe key result of this section is shown in Fig.\\ \\ref{fig:RodeC}, showing the optimal approximation ratio versus problem size for even numbers of qubits only. The corresponding plot for odd numbers of qubits can be found in Appendix \\ref{app:ferm}. Notably the approximation ratio saturates for large problem sizes, achieving an approximation ratio of $0.5819$, compared to $0.5$ for QAOA $p=1$. This is despite QAOA $p=1$ having two variational parameters, compared to the single variational parameter for $H_1$. This behaviour is suggestive of $H_1$ optimising locally, since its approximation ratio is largely independent of problem size.\n\nDespite MAX-CUT on two-regular graphs being an easy problem, the ground-state probability scales exponentially with problem size (Fig.\\ \\ref{fig:RoDegsp}). This is not necessarily a problem, as we present $H_1$ as an approximate approach only. Optimising the performance to give the best ground-state probability provides a small gain in performance but does not change the overall exponential scaling. The optimal times for both approximation ratio and ground-state probability can be found in Fig.\\ \\ref{fig:RoDts}. Both times freeze out at constant values for problem sizes greater than 10 qubits. \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.48\\textwidth]{images\/RoDegsp.png}\n    \\caption{The ground-state probability for different problem sizes under the evolution of $H_1$. The blue line shows the ground-state probability for times that maximise the approximation ratio. The optimised ground-state probability is shown in pink. The dashed purple line shows the probability of randomly guessing the ground-state. Only even qubit numbers are plotted here.}\n    \\label{fig:RoDegsp}\n\\end{figure}\n\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.48\\textwidth]{images\/Rodts.png}\n    \\caption{The optimal times for MAX-CUT on two-regular graphs. The blue (pink) line shows the time that optimises the approximation ratio (ground state probability). Only even numbers of qubits are plotted.}\n    \\label{fig:RoDts}\n\\end{figure}\n\n\nWe have demonstrated with MAX-CUT on two-regular graphs, at large problem sizes, that $H_1$ can provide a better approximation ratio than QAOA $p=1$. We explore this comparison with QAOA $p=1$ further in Sec.\\ \\ref{sec:QAOA}. \n\n\n\n\n\n\n\\subsubsection{Performance on MAX-CUT problems with three-regular graphs}\n\\label{sec:3reg}\n\nIn Sec.\\ \\ref{sec:prob3reg} we outlined the approach taken by Braida et al.\\ to prove bounds on QA on MAX-CUT with three-regular graphs. Here we apply this approach to $H_1$. For details of the method the reader is referred to \\cite{Bra22} and for explicit details of this computation to Appendix \\ref{app:LRB}. We find that that $H_1$ finds at least $0.6003$ times the best cut. Hence $H_1$ has a marginally better worst-case than QA (which is 0.5933 times the best cut \\cite{Bra22}), when this method is applied. Both bounds are not necessarily (and unlikely to be) tight. Resorting to numerical simulation gives Fig.\\ \\ref{fig:MC3Reg}. Here we can see, for the random instances considered, that $H_1$ never does worse than the QAOA $p=1$ worst bound. Direct comparisons to QAOA $p=1$ can be found in Sec.\\ \\ref{sec:QAOA}. Fig.\\ \\ref{fig:MC3Reg} also shows that the approximation ratio of $H_1$ on three-regular graphs has little dependence on the problem size, again suggesting that it is optimising locally.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.48\\textwidth]{images\/3RegPer.png}\n    \\caption{The performance of $H_1$ on randomly generated  instances of three-regular graphs. For each problem size, 100 instances were generated. After accounting for graph isomorphisms the number of samples in order of ascending problem size were $[1,2,6,15,46,87,97]$. Disconnected graphs were allowed. The y-axis shows the average cut-value from sampling $H_1$. The final time has been numerically optimised to give the best approximation ratio.\n    }\n    \\label{fig:MC3Reg}\n\\end{figure}\n\n\n\\subsubsection{Numerical simulations on random instances of MAX-CUT and Sherrington-Kirkpatrick problems}\n\\label{sec:H1Num}\n\n\\begin{figure*}\n    \\centering\n    \\begin{subfigure}[t]{0.3\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{images\/MCRandc.png}\n        \\caption{Approximation ratio for randomly generated MAX-CUT.}\n        \\label{fig:MCRandc}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[t]{0.3\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{images\/MCRandgsp_log.png}\n        \\caption{Ground-state probability for randomly generated MAX-CUT.}\n        \\label{fig:MCRandgsp}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[t]{0.3\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{images\/MCRAndVar.png}\n        \\caption{Width of the final distribution, $\\sigma$, for randomly generated instances of MAX-CUT.}\n        \\label{fig:MCRandVar}\n    \\end{subfigure}\n    \\hfill\n    \\\\\n    \\begin{subfigure}[t]{0.3\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{images\/SKc.png}\n        \\caption{Approximation ratio for the SKM.}\n        \\label{fig:SKc}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[t]{0.3\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{images\/SKgsp_log.png}\n        \\caption{Ground-state probability for the SKM.}\n        \\label{fig:SKgsp}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[t]{0.3\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{images\/SKVar.png}\n        \\caption{Width of the final distribution, $\\sigma$, for the SKM.}\n        \\label{fig:SKVar}\n    \\end{subfigure}\n    \\caption{Performance of $H_1$ on 100 randomly-generated instances of MAX-CUT and SKM.}\n    \\label{fig:NumPer}\n\\end{figure*}\n\nIn the previous two sections we established the performance of $H_1$ on problems with a high degree of structure, allowing for more  analytical investigation. In this section we explore the performance of $H_1$ numerically on MAX-CUT with random graphs and on the SKM. Consequently, we are restricted to exploring problem sizes that can be simulated classically.\n\nFor each problem we consider 100 randomly-generated instances for each problem size. The results can be seen in Fig.\\ \\ref{fig:NumPer}. For each instance the time has been numerically optimised to maximise the approximation ratio within the interval $[0,2\\pi)$. From Fig.\\ \\ref{fig:NumPer} we draw some conclusions from the simulations, with the caveat that either much larger sizes need to be simulated and\/or analytic work is required to fully substantiate the claims. On all the problems considered $H_1$ performed better than random guessing (which results in an approximation ratio of 0). \n\nFocusing first on MAX-CUT: there appears to be some evidence that the distribution of approximation ratios is becoming smaller as the problem size is increased (Fig.\\ \\ref{fig:MCRandc}). In addition the approximation ratio tends to a constant value, independent of the problem size. From analysing the regular graphs, it is reasonable to assume that $H_1$ is optimising locally. Therefore, we would expect the performance of $H_1$ to depend on the subgraphs in the problem. If the performance is limited by one, or a certain combination of subgraphs, then that would explain the constant approximation ratio. As the problem size is increased the chance of having an atypical combination of subgraphs is likely to decrease, resulting in a smaller distribution.\n \nThe ground-state probability shows a clear exponential dependence on problem size (Fig.\\ \\ref{fig:MCRandgsp}).\n %\nThe width of the final energy distribution is tending to a constant value around $0.3$ (Fig.\\ \\ref{fig:MCRandVar}), meaning that $H_1$ finds neither a computational basis state nor a superposition of degenerate computational basis states. \n\n\nMoving on to the SKM, the approximation ratio also appears to have little dependence on the problem size for more than 7 qubits, an indicator that the dynamics under $H_1$ is approximately local for these times. The ground-state probability also appears to decline exponentially (Fig.\\ \\ref{fig:SKgsp}). Finally, $\\sigma$ is non-zero (Fig.\\ \\ref{fig:SKVar}), suggesting the final state is not a computational basis state.  \n\nWe have explained some characteristics of the data in Fig.\\ \\ref{fig:NumPer} by treating $H_1$ as a local algorithm. This insight helps to motivate the next section on choosing good initial starting guesses for the time $T$.\n\n\n\\subsubsection{Estimating the optimal time}\nPresenting the application of $H_1$ as a variational approach begs the question of how to find good initial guesses for the time, $T$, at which to measure the system. As previously noted the optimal time corresponds to the maximum approximation-ratio.\n\nFrom Sec.\\ \\ref{sec:SERand} we expect the first turning point in approximation ratio after $t=0$ to be a local maximum. Consequently, we might expect $T$ to be small. Further to this, throughout the work so far we have seen evidence of $H_1$ behaving locally. This local behaviour will allow us to  motivate good initial guesses for $H_1$.  As $H_1$ is optimising locally, its performance does not depend on the graph as a whole, but only on subgraphs.\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.48\\textwidth]{images\/3RegOt.png}\n    \\caption{The optimal time for the three-regular MAX-CUT instances considered in Fig.\\ \\ref{fig:MC3Reg}. The optimal time was found by dividing the interval $[0,2 \\pi] $ into 1000 time steps.}\n    \\label{fig:3RegOt}\n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n\\begin{tikzpicture}[scale=1.8, auto,swap]\n\n\\foreach \\pos\/\\name in { {(-1,0)\/0}, {(0,0)\/1}, {(1,0)\/2}, {(2,0)\/3}}\n        \\node[vertex] (\\name) at \\pos {$\\name$};\n  \n\n\\foreach \\source\/ \\dest  in {0\/1, 1\/2, 2\/3}\n        \\path[edge] (\\source) -- node {}(\\dest);\n\n\\end{tikzpicture}\n\\caption{A subgraph of two-regular graphs}\n\\label{fig:subgraph_RD}\n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n\\begin{tikzpicture}[scale=1.8, auto,swap]\n\n\\foreach \\pos\/\\name in { {(-1,1)\/0}, {(-1,-1)\/1}, {(0,0)\/2}, {(2,0)\/3},{(3,1)\/4},{(3,-1)\/5}}\n        \\node[vertex] (\\name) at \\pos {$\\name$};\n  \n\n\\foreach \\source\/ \\dest  in {0\/2, 1\/2, 2\/3,3\/4,3\/5}\n        \\path[edge] (\\source) -- node {}(\\dest);\n\n\\end{tikzpicture}\n\\caption{A small subgraph contained in three-regular graphs which is prevalent in graphs that $H_1$ performs worse on.}\n\\label{fig:subgraph_3Reg}\n\\end{figure}\n\n\n\n\\begin{figure}\n    \\centering\n    \\begin{subfigure}[t]{0.48\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{images\/MCRandOt.png}\n        \\caption{Optimal times for MAX-CUT on randomly generated graphs. The optimal time was found by dividing the interval $[0,2 \\pi] $ into 1000 time steps.}\n        \\label{fig:MCRandOt}\n    \\end{subfigure}\n    \\\\\n    \\begin{subfigure}[t]{0.48\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{images\/SKOt.png}\n        \\caption{Optimal times for the SKM. The optimal time was found by dividing the interval $[0,2 \\pi] $ into 1000 time steps.}\n        \\label{fig:SKOt}\n    \\end{subfigure}\n\\caption{Optimal time for $H_1$ on 20 randomly generated instances of MAX-CUT and SK.}\n\\label{fig:NumTime}\n\\end{figure}\n\n\n\\begin{figure*}\n    \\centering\n    \\begin{subfigure}[t]{0.3\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{images\/QAOA_Comp_3_reg.png}\n        \\caption{Approximation ratio comparison for MAX-CUT on three-regular graphs.}\n        \\label{fig:comp3regc}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[t]{0.3\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{images\/QAOA_Comp_rand.png}\n        \\caption{Approximation ratio comparison for MAX-CUT on randomly generated graphs.}\n        \\label{fig:comprandc}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[t]{0.3\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{images\/QAOA_Comp_sk.png}\n        \\caption{Approximation ratio comparison for the SKM.}\n        \\label{fig:compskc}\n    \\end{subfigure}\n    \\hfill\n    \\\\\n    \\begin{subfigure}[t]{0.3\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{images\/QAOA_comp_time_3_reg.png}\n        \\caption{Optimal time comparison for MAX-CUT on three-regular graphs.}\n        \\label{fig:comp3regt}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[t]{0.3\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{images\/QAOA_comp_time_rand.png}\n        \\caption{Optimal time comparison for MAX-CUT on randomly generated graphs.}\n        \\label{fig:comprandt}\n    \\end{subfigure}\n    \\hfill\n    \\begin{subfigure}[t]{0.3\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{images\/QAOA_comp_time_sk.png}\n        \\caption{Optimal time comparison for the SKM.}\n        \\label{fig:compskt}\n    \\end{subfigure}\n    \\caption{Comparison of $H_1$ (y-axis on the above plots) with QAOA $p=1$ (x-axis on the above plots)  for the problem instances considered in Sec.\\ \\ref{sec:3reg} and Sec.\\ \\ref{sec:H1Num}. The top line of figures compares approximation ratios. The bottom line of figures compares the optimal times (i.e.\\ the time that maximises the approximation ratios) of the two approaches. }\n    \\label{fig:comph1QAOA}\n\\end{figure*}\n\nThe optimal time for MAX-CUT on two-regular graphs was $T=0.23$. Under the assumption that $H_1$ is behaving locally we can estimate the optimal times by considering a smaller subgraph. The subgraph in question is shown in Fig.\\ \\ref{fig:subgraph_RD}. By numerically optimising $\\langle Z_1Z_2 \\rangle$ for this subgraph with $H_f=Z_0Z_1+Z_1Z_2+Z_2Z_3$, we can find good estimates for the optimal $T$. Optimising this subgraph, within the interval $T\\in[0,1)$, gives $T=0.22$. This estimate matches the optimal time well. Choosing a larger subgraph will give a better estimate on the time.\n\nFig.\\ \\ref{fig:3RegOt} shows the optimal time for the larger instances of three-regular graphs considered in Fig.\\ \\ref{fig:MC3Reg}. The range of optimal times varied very little between problem instances and problem sizes, centered around $T=0.176$. As with the two-regular case we can examine subgraphs. In this case we consider the subgraph shown in Fig.\\ \\ref{fig:subgraph_3Reg}. This is the subgraph that saturates the Lieb-Robinson bound.  Numerically optimising $\\langle Z_2Z_3 \\rangle$ for this subgraph with $H_f=Z_0Z_2+Z_1Z_2+Z_2Z_3+Z_3Z_4+Z_3Z_5$ gives a time of $T=0.19$. This again is a good estimate of the optimal time.\n\nFor problems with well understood local structure, such as regular graphs, we have shown that we can exploit this knowledge to provide reasonable estimates of the optimal times. These subgraphs are also of the same size used in finding the optimal time in QAOA $p=1$ \\cite{Far14}.\n\nFor the MAX-CUT and SKM problems in Sec.\\ \\ref{sec:H1Num}, the optimal times can be found in Fig.\\ \\ref{fig:NumTime}. It appears that the optimal time tends to a constant value (or a small range of values), with $T<1$. The optimal times are clustered together, suggesting good optimal times might be transferable between problem instances. This approach is common within the QAOA literature \\cite{Gal21}.\n\nSo far we have explored the performance of $H_1$. We have demonstrated that $H_1$ can provide a better approximation ratio for MAX-CUT on two-regular graphs. The intuition gained by studying QAOA $p=1$ has been transferable to the understanding of $H_1$. In the final part of this section we make some direct comparisons between QAOA $p=1$ and $H_1$.\n\n\n\n\\subsection{Direct numerical comparisons to QAOA p=1}\n\\label{sec:QAOA}\n\nWe have established in the previous sections that $H_1$ operates in a local fashion. Calculating the optimal time for sub-graphs the same size as those involved in QAOA $p=1$ gave good estimates for the optimal time for larger problem sizes. Therefore it is reasonable to assume that $H_1$ sees a similar proportion of the graph as QAOA $p=1$. Both approaches are variational with short run-times too. Since both approaches are using comparable resources, in this section we attempt to compare the two. Again we focus on the problems outlined in Sec.\\ \\ref{sec:prob}. \n\nFor two-regular graphs, the optimal time for QAOA $p=1$ is 2.4 times longer than the optimal time for $H_1$ for large problem sizes, despite providing a poorer approximation ratio.\n\nThe results comparing  $H_1$ and QAOA $p=1$, for all the problem instances considered in Sec.\\ \\ref{sec:3reg} and Sec.\\ \\ref{sec:H1Num}, are shown in Fig.\\ \\ref{fig:comph1QAOA}. \n\nThe approximation ratios for each approach are largely correlated, suggesting in general that harder problems for QAOA $p=1$ corresponded to harder problems for $H_1$.For all the MAX-CUT instances considered, $H_1$ gave a greater than or equal to approximation ratio compared to QAOA $p=1$ (Fig.\\ \\ref{fig:comp3regc} and Fig.\\ \\ref{fig:comprandc}). For a handful of problems with the SKM, QAOA $p=1$ outperformed $H_1$, but for the vast majority of problems instances $H_1$ performed better (Fig.\\ \\ref{fig:compskc}).\n\nTurning now to the optimal time, $H_1$ had in the majority of cases the shorter optimal time (Figs.\\ \\ref{fig:comp3regt}, \\ref{fig:comprandt} and \\ref{fig:compskt}), though there are more exceptions to this with the SKM (Fig.\\ \\ref{fig:compskt}). In Appendix \\ref{app:QAOA_better} we elaborate further on the exceptions, that is the MAX-CUT problems that have longer run-times than QAOA p=1 and the SK problems that do better for QAOA p=1. \n\nThis section has numerically demonstrated that $H_1$ provides a better approximation ratio than QAOA $p=1$ in a significantly shorter time for the majority of instances considered, justifing our description of this this approach as rapid, which is crucial for NISQ implementation \\cite{Pre18}. Given that, $H_1$ tends to provide a better approximation ratio, in a shorter time, with fewer variational parameters, it raises the question - does QAOA $p=1$, the foundation of any QAOA circuit, make effective use of its afforded resources?\n\n\n\\section{An improvement inspired by the Quantum Zermello problem}\n\\label{sec:QZ}\n\\subsection{The approach}\n\\label{sec:QZapp}\nWith QAOA it is clear how to get better approximation ratios, that is by increasing $p$. It is less clear how to do this with $H_1$. One suggestion might be to append this Hamiltonian to a QAOA circuit. However, the aim of this paper is to explore how Hamiltonians for optimal state-transfer can provide a guiding design principle.  Therefore, in this section we explore adding another term, motivated by this new design principle, to $H_1$ to improve the approximation ratio.\n\nIn Sec.\\ \\ref{sec:hamdes} we motivated $H_1$ from the optimal Hamiltonian by making the pragmatic substitutions $\\ket{\\psi_i}\\bra{\\psi_i} \\rightarrow H_i$ and $\\ket{\\psi_f}\\bra{\\psi_f} \\rightarrow H_f$. Subsequently, we demonstrated that $H_1$ provides a reasonable performance. However, $H_1$ no longer closely followed the evolution under the optimal Hamiltonian. To partially correct for this error we add another term to the Hamiltonian:\n\\begin{equation}\n    \\label{eq:corQZ}\n    H_{1,improved}=H_1+H_{QZ}.\n\\end{equation}\n\n Again, we make use of Hamiltonians for optimal state-transfer to motivate the form of $H_{QZ}$. Finding the optimal Hamiltonian in the presence of an uncontrollable term in the Hamiltonian is known as the quantum Zermello (QZ) problem \\cite{Bro15,Brody_2015,Rus14,Rus15}. \n\nIn the rest of this section we expand on the details of the QZ problem. From the exact form of the optimal correction, $H_{cor}$, we then apply a series of approximations so that $H_{cor}$ is time-independent and does not rely on knowledge of $\\ket{\\psi_f}$. This final Hamiltonian will be $H_{QZ}$ in Eq.\\ \\ref{eq:corQZ}.\n\n\n\\begin{figure}\n    \\centering\n   \n\\begin{tikzpicture}\n\n\\fill[black] (-1,5) circle (0.15cm) node[anchor=south east] {$\\ket{\\psi_i}$};\n\n\\fill[black] (5,-1) circle (0.15cm) node[anchor=north west] {$\\ket{\\psi_f}$};\n\n\\draw[myblue,ultra thick,dashed] (5,-1) .. controls (4,5) and (4,0) .. (0,2);\n\n\\draw[myblue, ultra thick] (-0.1,1.9) -- (0.1,2.1);\n\n\\draw[myblue, ultra thick] (-0.1,2.1) -- (0.1,1.9)node[anchor=south west] {$e^{i H_{QW}T} \\ket{\\psi_f}$};\n\n\\draw[mypink, ultra thick](-1,5)--(-2\/3,4) node[anchor=south west] {$e^{-i H t} \\ket{\\psi_i}$};\n\n\\draw[mypink, ultra thick, dashed](-2\/3,4)--(-1\/3,3);\n\\end{tikzpicture}\n    \\caption{A cartoon of the evolution of states in the QZ problem for constant $H_{QW}$. In the interaction picture, with background Hamiltonian $H_{QW}$, it appears the final state is moving under the influence of this Hamiltonian. In this frame Eq. \\ref{eq:optHam} can then be applied. It then remains to move out of the interaction picture to get Eq.\\ \\ref{eq:opHQZ}. }\n    \\label{fig:QZcartoon}\n\\end{figure}\n\nThe QZ problem, like the quantum brachistochrone problem, asks what is the Hamiltonian that transfers the system from $\\ket{\\psi_i}$ to the final state $\\ket{\\psi_f}$ in the shortest possible time. Unlike the quantum brachistochrone problem, part of the Hamiltonian is uncontrollable. In the case of a constant uncontrollable term, the total Hamiltonian can be written as:\n\\begin{equation}\n    H_{opt | QW}=H_{QW}+H_{cor}(t),\n\\end{equation}\nwhere $H_{QW}$ is the constant `quantum wind' that cannot be changed and $H(t)$ is the Hamiltonian we are free to vary. Typically, $H_{QW}$ is understood as a noise term  \\cite{Lap10,Muk13}. Instead, here we will take $H_{QW}$ to be $H_1$ to provide a favourable quantum wind that $H_{cor}(t)$ can provide an improvement on. \n\nThe optimal form of $H_{cor}(t)$ is (up to some factor) \\cite{Brody_2015}:\n\n\\begin{multline}\n    H_{cor}(t)=-i e^{-iH_{1}t}\\left(\\ket{\\psi_i}\\bra{\\psi_f}e^{-iH_{1}T} \\right. \\\\\n    \\left. -e^{i H_{1}T}\\ket{\\psi_f}\\bra{\\psi_i} \\right)e^{iH_{1}t},\n    \\label{eq:opHQZ}\n\\end{multline}\nwhere $t$ is the time and $T$ is the final time. The motivation for this equation can be found in Fig.\\ \\ref{fig:QZcartoon}. This Hamiltonian requires knowledge of the final state, so we introduce a series of approximations to make $H_{cor}(t)$ more amenable for implementation. \n\n Since we know that the optimum evolution under $H_1$ tends to be short, we make the assumption that the total time $T$ is small. Therefore, we approximate the optimal correction with $H_{cor}(t)$ with\n\\begin{equation}\n    H_{cor}(0)=-i \\left(\\ket{\\psi_i}\\bra{\\psi_f}e^{-iH_{1}T} \\right. \\\\\n    \\left. -e^{i H_{1}T}\\ket{\\psi_f}\\bra{\\psi_i} \\right).\n\\end{equation}\n\nNote that this Hamiltonian is time-independent. We can estimate the error in this step by:\n\\begin{align*}\n    \\delta&=\\left\\|H_{cor}(t)-H_{cor}(0)\\right\\|\\\\\n    &=\\left\\|\\int_0^t ds\\, e^{-iH_{1}s}\\left[H_{1},H_{cor}(0)\\right]e^{iH_{1}s}\\right\\|\\\\\n    &\\leq \\int_0^t ds\\, \\left\\|e^{-iH_{1}s}\\left[H_{1},H_{cor}(0)\\right]e^{iH_{1}s}\\right\\|\\\\\n    &=\\left\\|\\left[H_{1},H_{cor}(0)\\right]\\right\\|t\n\\end{align*}\n\nHence, $\\delta \\leq \\left\\|\\left[H_{1},H_{cor}(0)\\right]\\right\\|t$.\n\n\nIntroducing the commutator structure (Sec.\\ \\ref{sec:hamdes}) with the same pragmatic substitutions as before for $H_1$ gives:\n\\begin{equation}\n   \\label{eq:infT}\n    H_{QZ}=-i \\left[H_i,e^{i H_{1}T}H_fe^{-iH_{1}T} \\right],\n\\end{equation}\nwhere we have introduced the subscript $QZ$ to distinguish this Hamiltonian from $H_{cor}(0)$ prior to the substitutions. Quantifying the error in this step remains an open challenge. Expanding this expression in $T$ gives:\n\n\\begin{multline}\n    \\label{eq:QZexp}\n    H_{QZ}=-i \\left[H_i, H_f+iT \\left[H_{1},H_f\\right]\\right.\\\\\n    \\left.-T^2\\left[H_{1},\\left[H_{1},H_f\\right]\\right]\/2+\\mathcal{O}\\left(T^3\\right)\\right].\n\\end{multline}\n\nFrom the QZ problem we have motivated the form of the correction $H_{QZ}$ in Eq.\\ \\ref{eq:corQZ}.  In spite of this we have no guarantee on its performance - to this end we carry out numerical simulations. \n\nIn both its philosophy and structure $H_{QZ}$ is reminiscent of shortcuts to adiabaticity (STA) \\cite{Gue19}. In STA the aim is to modify the Hamiltonian in Eq. \\ref{eq:QAham}  to reach the adiabatic result in a shorter time, typically by appending to the standard QA Hamiltonian. The approach here is distinctly different  for three key reasons, besides the initial starting point of Hamiltonian. \n\\begin{enumerate}\n    \\item The aim here is to do something distinctly different from $H_1$, not to recover its behaviour in a shorter time.\n    \\item In the QZ-inspired approach we make use of the excited states, with the aim of finding approximate solutions, as opposed to exact solutions. \n    \\item Here we only consider time-independent Hamiltonians.\n\\end{enumerate}\n\nA clear downside to $H_{QZ}$ is the increased complexity, compared to say QAOA. However, if $H_{QZ}$ is decomposed into a QAOA-style circuit, the single free parameter in $H_{QZ}$ might translate to fewer free parameters in the QAOA circuit, allowing for easier optimisation. \n\n\n\\subsection{Numerical simulations}\n\\subsubsection{MAX-CUT on two-regular graphs}\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.48\\textwidth]{images\/QZperRoD_std.png}\n    \\caption{Numerically optimised performance of Eq.\\ \\ref{eq:QZexp}. Each point has been optimised in the time interval [0,0.3] by considering 3000 divisions. The legend shows the order of $T$, with `exp' referring to Eq.\\  \\ref{eq:infT}. The dashed lines show the asymptotic performance of QAOA. }\n    \\label{fig:SerPer}\n\\end{figure}\n\nHere we focus on applying Eq.\\ \\ref{eq:QZexp} up to various orders in $T$ to MAX-CUT on two-regular graphs with an even number of qubits.  We focus on this problem as it is trivial to scale and the performance of QAOA and $H_1$ on this problem is well understood. \n\nThe results for MAX-CUT with two-regular graphs can be seen in Fig.\\ \\ref{fig:SerPer}. Increasing the expansion in $T$ appears to improve the approximation ratio. But the improvement is capped, shown by the data labelled `exp'. Notably, this approach with a single variational parameter at order $T^2$ is performing better than QAOA $p=3$ (with 6 variational parameters) for 10 qubits. \n\nThe optimal $T$ for the QZ-inspired Hamiltonians can be seen in Fig.\\ \\ref{fig:Sertime}. Again, the optimal time for each order in $T$ appears to be tending to a constant value, suggesting this approach is still acting in a local fashion. This is consistent with the approximation ratio plateauing. As we can see the QZ-inspired approach is still operating in a rapid fashion. \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.48\\textwidth]{images\/QZperRoD_t_std.png}\n    \\caption{The corresponding optimal times for Fig.\\ \\ref{fig:SerPer}. The norm of each Hamiltonian, for each problem size has been fixed so Eq.\\ \\ref{eq:norm} is true, to make comparisons fair.}\n    \\label{fig:Sertime}\n\\end{figure}\n\n\\subsubsection{Random instances on MAX-CUT and Sherrington-Kirkpatrick problems}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.48\\textwidth]{images\/BoxMacCutIter.png}\n    \\caption{The approximation ratio for the QZ-inspired approach on 100 random MAX-CUT instances. The x-axis label refers to the order of $T$ in the expansion of Eq.\\ \\ref{eq:QZexp}, with $0$ being $H_1$ and $e$ referring to the full exponential (i.e.\\ Eq.\\ \\ref{eq:infT}).}\n    \\label{fig:qzmc}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.48\\textwidth]{images\/BoxSKIter.png}\n    \\caption{The approximation ratio for the QZ-inspired approach on 100 instances of the SKM. The x-axis label refers to the order of $T$ in the expansion of Eq.\\ \\ref{eq:QZexp}, with $0$ being $H_1$ and $e$ referring to the full exponential (i.e.\\ Eq.\\ \\ref{eq:infT}).}\n    \\label{fig:qzsk}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.48\\textwidth]{images\/BoxMacCutIter_time.png}\n    \\caption{The optimal time for the QZ-inspired approach on 100 random Max-CUT instances. The x-axis label refers to the order of $T$ in the expansion of Eq.\\ \\ref{eq:QZexp}, with $0$ being $H_1$ and $e$ referring to the full exponential (i.e.\\ Eq.\\ \\ref{eq:infT}). The norm of each Hamiltonian, for each problem size has been fixed, according to Eq.\\ \\ref{eq:norm}, ensuring a fair comparison.}\n    \\label{fig:QZmct}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.48\\textwidth]{images\/BoxSKIterpng_time.png}\n    \\caption{The optimal time for the QZ-inspired approach on 100 random Max-CUT instances. The x-axis label refers to the order of $T$ in the expansion of Eq.\\ \\ref{eq:QZexp}, with $0$ being $H_1$ and $e$ referring to the full exponential (i.e.\\ Eq.\\ \\ref{eq:infT}). The norm of each Hamiltonian, for each problem size has been fixed, according to Eq.\\ \\ref{eq:norm}, ensuring a fair comparison.}\n    \\label{fig:QZSKt}\n\\end{figure}\n\n\nTo complete this section we examine the performance of the QZ-inspired approach (Eq.\\ \\ref{eq:QZexp}) to the randomly generated instances of MAX-CUT and SKM, detailed in Sec.\\ \\ref{sec:prob}. \n\nThe results for different orders in $T$ for the approximation ratio can be seen in Fig.\\ \\ref{fig:qzmc} for MAX-CUT and Fig.\\ \\ref{fig:qzsk} for the SKM. All the QZ-inspired approaches provide an improvement on the original $H_1$ Hamiltonian, indexed by 0 in the figures. However, the performance is not monotonically increasing with the order of the expansion. This is not unusual for a Taylor series of an oscillatory function. Consequently, achieving better approximation ratios is not as simple as increasing the order of $T$. At the same time, this means that it is not necessary to go to high orders in $T$, with very non-local terms, to achieve a significant gain in performance. For example, in both cases going to first order achieves a substantial improvement. \n\nThe optimal times for the QZ-inspired approach can be found in Fig.\\ \\ref{fig:QZmct} for MAX-CUT and Fig.\\ \\ref{fig:QZSKt} for the SKM. For clarity we only show the optimal times for the larger problem instances. As with $H_1$ the optimal times are clustered for a given order. The lack of dependence on problem size for optimal times and approximation ratios suggests that the QZ-inspired approach is still optimising locally. Compared to the $H_1$ case, the operators have a larger support. Despite optimising locally, they are optimising less locally than $H_1$, hence the increased performance.\n\nHere we have numerically demonstrated that the QZ-inspired approach can provide an improvement over $H_1$, suggesting how this new design philosophy might be extended. The numerics also suggest that going to first order may provide the best possible advantage. \n\n\n\\section{Using knowledge of the initial state}\n\\label{sec:lpa}\n\nAs introduced in Sec.\\ \\ref{sec:hamdes}, in this section we exploit our knowledge of the initial state and evaluate the performance of \n\\begin{equation}\n    H_{\\psi_i}=-i\\left[\\ket{\\psi_i}\\bra{\\psi_i},f(H_f)\\right] \n\\end{equation}\nwithin the QA-framework. We take $f(\\cdot)$ to be a real function such that:\n\\begin{equation}\n    f(H_f)=\\sum_k f(E_k)\\ket{E_k}\\bra{E_k},\n\\end{equation}\nwhere $\\ket{E_k}$ and $E_k$ are the eigenvectors and associated eigenvalues of $H_f$.\n\nEvolution under $H_{\\psi_i}$ can be calculated analytically - the details can be found in Appendix \\ref{app:expHam}. By evolving $\\ket{\\psi_i}$ the state:\n\\begin{equation}\n    \\ket{\\omega}=\\frac{1}{\\sqrt{\\Tr f^2(H_f)}}\\sum_k f(E_k)\\ket{E_k},\n\\end{equation}\ncan be reached. Indeed $H_{\\psi_i}$ will generate linear superpositions of $\\ket{\\omega}$ and $\\ket{\\psi_i}$ only.\n\nAssuming that the state $\\ket{\\omega}$ is prepared, then the probability of finding the ground-state is \n\\begin{equation}\n    P_{gs}=\\frac{g f^2(E_{gs})}{\\Tr f^2(H_f)},\n\\end{equation}\nwhere $g$ is the ground-state degeneracy and $E_{gs}$ the associated energy. If $f(\\cdot)$ is the identity, then $\\Tr H_f$ scales approximately as $2^n$ and $E_{gs}$ might scale with $n$. Hence the ground state probability will scale as  $\\sim n^2\/2^n$. Indeed if $f(H_f)=H_f^m$, where $m$ is some  positive integer, then the ground state probability might scale as $\\sim n^{2m}\/2^n$. This is an improvement over random guessing, but still with exponential scaling. This may be of some practical benefit, depending on the computational cost of calculating $f(H_f)$. \nIf $f(\\cdot)$ is the projector onto the ground-state then $P_{gs}=1$ (as expected). \n\nCalculating the expectation of $H_f$ for $\\ket{\\omega}$ gives:\n\\begin{equation}\n    \\langle H_f \\rangle =\\frac{1}{\\Tr f^2(H_f)}\\sum_k E_k f^2(E_k).\n\\end{equation}\nHere we can see that $\\langle H_f \\rangle$ will be dominated by states for which $f^2(E_k)$ is large. If $f(\\cdot)$ is the identify this most likely means low energy states and high energy states. Hence, we do not expect a good approximation ratio. This provides some insight into Sec.\\ \\ref{sec:RoD} where the observed peak in ground-state probability did not coincide with the optimal approximation ratio.\n\nThis approach has the potential to provide a modest practical speed-up with a polynomial prefactor on the hardest problems. However, the success of this approach depends on the (unlikely) feasibility of implementing $H_{\\psi_i}$ and $f(H_f)$. It does however provide further evidence of the power of commutators for designing algorithms to tackle optimisation problems.\n\n\n\n\\section{Discussion and Conclusion}\n\nDesigning quantum algorithms to tackle combinatorial optimisation problems, especially within the NISQ framework, remains a challenge. Many algorithms have used AQO in their inspiration, such as in the choice of Hamiltonians. In this paper we have explored using optimal Hamiltonians as a guiding design principle. \n\nWith $H_1$, the commutator between $H_i$ and $H_f$, we demonstrated that we can outperform QAOA p=1, with fewer resources. The short run-times which do not appear to scale with problem size suggest that this approach is acting locally. An effective Lieb-Robinson bound prevents the information about the problem propagating instantaneously \\cite{Lieb1972,WangZ20}. This helps provide some insights into the performance of $H_1$:\n\\begin{itemize}\n    \\item In the local regime, the effective local Hilbert space is smaller than the global Hilbert space, consequently $H_1$ will be a better approximation of the optimal Hamiltonian. This accounts for why we might expect $H_1$ to work better on short run-times.\n    \\item A local algorithm is unlikely to be able to solve an optimisation problem, as it cannot see the whole graph. It follows that such an approach would have poor scaling of the ground-state probability. \n\\end{itemize}\n\nDue to the local nature of $H_1$ we were able to utilise some analytical tricks to assist the numerical assessment of its performance, allowing for some guarantee of the performance of the approach on large problem sizes. The techniques used had already been developed or deployed by the continuous-time quantum computing community in the context of QA\/QAOA, indicative of the wide applicability of the tools being developed to assess these algorithms.\n\nLocal approaches have clear advantages in NISQ-era computations. The short run-times put fewer demands on the coherence times of the device. The local nature can also help mitigate some errors. If, for example, there is a control misspecification in one part of the Hamiltonian this is unlikely to propagate through the whole system and affect the entire computation. \n\nBuoyed by the relative success of utilizing Hamiltonians for optimal state-transfer we turned to the quantum Zermello problem to help find improvements. These Hamiltonians compromised a single variational parameter and short run-times, for increased complexity in the Hamiltonian. Again, the saturation of the optimal time suggest these approaches are still operating locally.\n\nThe success of this approach, within the NISQ era, will depend on the feasibility of implementing these Hamiltonians. This might be achieved through decomposition into a product formula \\cite{Childs_2013} for gate based approaches, resulting in a QAOA like circuit. Alternatively, one could attempt to explicitly engineer the interactions involved. Indeed, for exponentially scaling problems, implementing these Hamiltonians for short times could be less challenging then maintaining coherence for exponentially increasing times.\n\nAlthough the results of this paper are not fully conclusive, it has shown that by considering Hamiltonians for optimal state-transfer we can develop promising new algorithms. We hope the results presented in this paper will encourage further work into the success of these Hamiltonians. There is scope for taking this work further. This could include changing the choice of $H_i$, exploiting our observation that any stoquastic Hamiltonian can lead to an increase in ground state probability. For $H_f$ we have only explored problems with trivial Ising encodings. There is scope to explore new encodings such as LHZ \\cite{Lec15}  or Domain-wall \\cite{Cha19} encodings. Such encodings will result in different $H_1$ and presumably distinct dynamics.\n\n\\section*{Acknowledgments}\nWe gratefully acknowledge Filip Wudarski, Glen Mbeng, Henry Chew, Natasha Feinstein, and Sougato Bose for inspiring discussion and helpful comments. This work was supported by the Engineering and Physical Sciences Research Council through the Centre for Doctoral Training in Delivering Quantum Tech-\nnologies [grant number EP\/S021582\/1] and the ESPRC Hub in Quantum Computing and Simulation [grant number EP\/T001062\/1].\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:intro}\n\nAlgorithms with exponential speedups over classical counterparts \\cite{shorsalgorithm,groversalgorithm}, \nsimulation of quantum systems \\cite{sethlloyd, du_hydrogen,lidar_thermalrate} and testing basic \nprinciples of quantum mechanics \\cite{jharana_nohiding,mahesh_legget} makes \nquantum Information processing (QIP) and quantum computation  intensively investigated fields of physics over last decade. \nThe idea of simulating quantum systems in a quantum computer was proposed \nby Feynman \\cite{feynman_qs} in 1982 and is one of the most important practical application of the \nquantum computer. Quantum simulation has the potential to revolutionize the physics and chemistry and draws attention \nrecently by solving problems like -- molecular Hydrogen simulation \\cite{du_hydrogen}, \ncalculations of thermal rate constants of chemical reactions \\cite{lidar_thermalrate} and quantum chemical dynamics \n\\cite{kassal_chemicaldynamics}. \n\n\nDzyaloshinsky-Moriya (DM) interaction is an anisotropic antisymmetric exchange interaction arising from spin-orbit \ncoupling \\cite{dzyaloshinsky, moriya}.\nIt was proposed by Dzyaloshinski to explain the weak ferromagnetism of antiferromagnetic crystals \n($\\alpha$-$Fe_2O_3$, $MnCO_3$)\\cite{dzyaloshinsky}.\nDM interaction is crucial in the description of many antiferromagnetic systems \\cite{dender,kohgi,greven} and is \nimportant in the entanglement properties of the system. Here we present \na generic unitary operator decomposition which will help to simulate the Hamiltonian -- DM interaction in the presence of Heisenberg \nXY interaction -- in a two qubit system with almost any basic interaction between them.\n\n\\begin{comment}\nEntanglement behavior in Ising model with DM interaction \\cite{kargarian_ent}, role of DM interaction in \nentanglement transfer \\cite{maruyama_et}, quantum phase interference of spins\\cite{wernsdorfer} and significance of \nDM interaction in quantum dots \\cite{kavokin_quantumdots} are the recent investigations in DM interactions. \n\\end{comment}\n\n\n\n \nLong-lasting coherence and high fidelity controls in nuclear magnetic resonance (NMR) are ideal for quantum \ninformation processing. Experimental implementation quantum algorithms  \n(Deutsch-Jozsa algorithm, Grover's search algorithm and Shor's algorithm of factorization), testing basic \nprinciples of quantum mechanics (nohiding theorem \\cite{jharana_nohiding} and Leggett-Garg inequality \n\\cite{mahesh_legget}) and quantum simulation \n(hydrogen molecule \\cite{du_hydrogen} and system with competing two and three Body interactions \n\\cite{suter_2and3bodyinteractions}) were performed in Liquid state NMR (LSNMR).\n\nGenetic algorithms (GA) are stochastic global search method based on the mechanics of natural biological \nevolution \\cite{whitely_ga}. It was first proposed by John Holland in 1975 \\cite{holland_ga}. GA operates on \na population of solutions of a specific problem by encoding the solutions to a simple chromosome-like \ndata structure, and applies recombination operators. GAs are attractive in engineering design and applications because \nthey are easy to use and are likely to find the globally best design or solution, which is superior to \nother design or solution \\cite{rasheed_ga}. Here we used Genetic algorithm optimization for solving \nUOD for generic DM Hamiltonian with Heisenberg-XY interaction. \n\nSection \\ref{sec:theory} deals with theoretical discussion of DM Hamiltonian simulation followed by experimental \nimplementation in Section \\ref{sec:expt}.\n\n\n\n\\section{Theory} \\label{sec:theory}\n\n\nDM interaction in the presence of Heisenberg XY interaction $H(J,D)$ is, \n\\begin{equation} \\label{eqn:dmh}\nH(J,D)=J(\\sigma_{1x}\\sigma_{2x}+\\sigma_{1y}\\sigma_{2y})+D(\\sigma_{1x}\\sigma_{2y}-\\sigma_{1y}\\sigma_{2x}),\n\\end{equation}\nwhere J and D respectively represents the strength of Heisenberg and DM interactions.\n\nExperimental simulation of $H(J,D)$ (Eqn. \\ref{eqn:dmh}) in a quantum system (with Hamiltonian $H_{sys}$)  requires \nUOD of evolution operator $U(J,D,t)$,\n\\begin{equation} \\label{eqn:dmu}\nU(D,J,t)=exp(-i H(J,D) \\times t),\n\\end{equation}\nin terms  of Single Qubit Rotations \n$R^n(\\theta,\\phi)$ ($\\theta$ angle rotation along $\\phi$ axis on $n^{th}$ spin),\n\\begin{equation} \\label{eqn:usqr}\nR^n(\\theta,\\phi) = exp(-i \\theta\/2 \\times [Cos\\phi ~\\sigma_{nx} +Sin\\phi ~\\sigma_{ny}]),\n\\end{equation}\nand evolution under system Hamiltonian $U_{sys}$  (Eqn. \\ref{eqn:sysu}),\n\\begin{equation} \\label{eqn:sysu}\nU_{sys}(t)=exp(-i H_{sys} \\times t).\n\\end{equation}\n\nWithout losing generality, Eqn. \\ref{eqn:dmu} can be written  as,\n\\begin{equation} \\label{eqn:dmu1}\n\\begin{split}\nU(\\gamma,\\tau)=exp(-i [(\\sigma_{1x}\\sigma_{2x}+\\sigma_{1y}\\sigma_{2y})+ \\\\ \n\\gamma (\\sigma_{1x}\\sigma_{2y}-\\sigma_{1y}\\sigma_{2x})]~ \\tau),\n\\end{split}\n\\end{equation}\nwhere $\\gamma$ represents the relative ratio of interaction strengths ($\\gamma = D\/J$) and $\\tau=J\\times t$.\n\nEqn. \\ref{eqn:dmu1} forms the complete unitary operator for the Hamiltonian (Eqn. \\ref{eqn:dmh}) with  \n$\\gamma$ and $\\tau$ varies from 0 to $\\infty$. We performed UOD for Eqn. \\ref{eqn:dmu1} using \nGenetic algorithm optimization \\cite{vsmanu_ga}.\nIn an operator optimization (as shown in \\cite{vsmanu_ga}), optimization is performed for a constant unitary \nmatrix-- corresponds to a single fidelity point. \nHere optimization has to be performed for a two dimensional fidelity profile generated by $\\gamma$ and $\\tau$. We name it as Fidelity Profile \nOptimization (FPO). FPO for the present case is explained in following steps.\n\nIn the first step, we performed Fidelity Profile Optimization with following assumptions -- (a).the range of \n$\\tau$ is from 0 to 15, (b). the range of $\\gamma$ is from 0 to 1 and (c). the system Hamiltonian ($H_{sys}$) is \ngiven by Eqn. \\ref{eqn:zzh}. \n\\begin{equation} \\label{eqn:zzh}\nH_{sys}=H_{zz}=J_{zz} (\\sigma_{1z}\\sigma_{2z}).\n\\end{equation}\nwhere $J_{zz}$ is the strength of zz-interaction.\n\nThe optimization procedure using Genetic algorithm is explained in the Supporting information. \n\nThe optimized UOD (Eqn. \\ref{eqn:uh1}) has seven SQRs and two system Hamiltonian evolutions. \n\\begin{equation} \\label{eqn:uh1}\n\\begin{split}\nU(\\gamma,\\tau) = R^1(\\tfrac{\\pi}{2},-\\tfrac{\\pi}{2})  R^1(\\tfrac{\\pi}{2},\\theta_2) R^2(\\pi,\\pi)  \t\nU_{zz}(\\tfrac{\\pi}{4})\\\\  R^1(\\theta_1,\\theta_2+\\tfrac{\\pi}{2})R^2(\\pi-\\theta_1,0)\t\t\t\nU_{zz}(\\tfrac{\\pi}{4})\\\\  R^1(\\tfrac{\\pi}{2},\\theta_2+\\pi) R^2(\\tfrac{\\pi}{2},\\tfrac{\\pi}{2}), \n\\end{split}\n\\end{equation}\nwhere  $\\theta_1$ and $\\theta_2$ (Eqn. \\ref{eqn:th1}) impart  $\\gamma$ and $\\tau$ dependence to UOD \nand $U_{zz}$ is given by Eqn. \\ref{eqn:sysu} with $J_{zz} \\times t =\\pi\/4$.\n\\begin{equation} \\label{eqn:th1}\n\\begin{split}\n\\theta_1=[0.8423-0.3455~Cos(1.117~\\gamma)+ \\\\0.01806~Sin(1.117 \\gamma)]~\\tau \\\\\n\\theta_2=1.345~exp(-0.8731 \\gamma)+1.796.\n\\end{split}\n\\end{equation}\nThe fidelity \\cite{vsmanu_ga} profile of UOD is \nshown in Fig. \\ref{fig:fp1}. The minimum point in fidelity profile is greater than 99.99 \\%. \nIt should be noted that, the total length of UOD (Eqn. \\ref{eqn:uh1}) \nis invariant under $\\gamma$ and $\\tau$ (with the assumption -- all the SQRs are instantaneous).\n\n\n\\begin{figure}\n\\includegraphics[width=0.35\\textwidth, height=0.27\\textwidth]{h1.jpeg}\n\\caption{Fidelity profile of UOD given in Eqn. \\ref{eqn:uh1}.}  \\label{fig:fp1}\n\\end{figure}\nFor generalizing the assumption on $\\tau$, we solved Eqn. \\ref{eqn:pf} numerically and find the \nperiod P($\\gamma$) of U($\\gamma$,$\\tau$) (Eqn. \\ref{eqn:p}). \n\\begin{equation} \\label{eqn:pf}\n H(\\gamma,\\tau+n \\times P(\\gamma))= H(\\gamma,\\tau).\n\\end{equation}\n\\begin{equation} \\label{eqn:p}\n P(\\gamma)=3.008~\\gamma^3-6.627~\\gamma^2-0.1498~\\gamma+12.59.\n\\end{equation}\nEqn. \\ref{eqn:p} has a maximum value of 12.59 at $\\gamma$=0. Since the maximum value of period is \nless than 15 (FOP performed till $\\tau$=15), UOD (Eqn. \\ref{eqn:uh1}) can be used for any value of $\\tau$.\nSame argument can be used for extending the range of $\\tau$ to $-\\infty$.\n\nIn order to incorporate the range of $\\gamma$ from 0 to $\\infty$, we performed  FPO for the \noperator Eqn. \\ref{eqn:dmu2}),\n\\begin{equation}\\label{eqn:dmu2}\n\\begin{split} \nU'(\\gamma',\\tau')=exp(-i [\\gamma' (\\sigma_{1x}\\sigma_{2x}+\\sigma_{1y}\\sigma_{2y})+ \\\\ \n(\\sigma_{1x}\\sigma_{2y}-\\sigma_{1y}\\sigma_{2x})]~ \\tau'),\n\\end{split}\n\\end{equation}\nand the optimized unitary decomposition are,\n\\begin{equation} \\label{eqn:uh2}\n\\begin{split}\nU'(\\gamma',\\tau') = R^1(\\tfrac{\\pi}{2},\\tfrac{\\pi}{2}) R^2(\\tfrac{\\pi}{2},\\theta_3)  \t\nU_{zz}(\\tfrac{\\pi}{4}) R^1(\\theta_2+\\theta_3,0)\\\\ R^2(\\theta_1,\\theta_4)\t\t\t\nU_{zz}(\\tfrac{\\pi}{4}) R^1(\\tfrac{\\pi}{2},\\tfrac{\\pi}{2}) R^2(\\tfrac{\\pi}{2},\\theta_3),\n\\end{split}\n\\end{equation}\nwhere  $\\theta_1 \\cdots \\theta_4$ (Eqn. \\ref{eqn:th2}) impart  $\\gamma$ and $\\tau$ dependence to UOD.\n\n\\begin{small}\\begin{align} \\label{eqn:th2}\n\\begin{split}\n&\\theta=[ 0.09812~ exp(-2.42 \\gamma)+0.4023 ~exp(0.5524 \\gamma)] \\tau,\\\\\n&\\theta_1=-\\theta+3.142, \\\\\n&\\theta_2=\\theta-[1.242 ~exp(-0.9617 \\gamma)+ 0.3546~exp(-0.1145 \\gamma)], \\\\\n&\\theta_3=1.259~exp(-0.957 \\gamma)+3.479~exp(-0.0087 \\gamma), \\\\\n&\\theta_4=1.256~exp(-0.959 \\gamma)+ 1.912~exp(-0.0166 \\gamma),\n\\end{split}\n\\end{align}\n\\end{small}\nwhere $\\gamma'$ varies from 0 to 1 and $\\tau$ from 0 to 15.\n\nEqn. \\ref{eqn:dmu2} satisfy the same periodicity relation as shown in Eqn. \\ref{eqn:p} and hence \ncan use the same reasoning for extending $\\tau$ range from 0 to $+\\infty$.\n\nFor $\\gamma > 1$, Eqn. \\ref{eqn:dmu1} can be written as,\n\\begin{equation} \\label{eqn:dmut}\n\\begin{split}\nU(\\gamma'',\\tau'')=exp(-i [\\gamma''(\\sigma_{1x}\\sigma_{2x}+\\sigma_{1y}\\sigma_{2y})+ \\\\ \n (\\sigma_{1x}\\sigma_{2y}-\\sigma_{1y}\\sigma_{2x})]~  \\tau''),\n\\end{split}\n\\end{equation}\nwhere $\\gamma''=1\/\\gamma$ and $\\tau''=\\gamma \\times \\tau$.\n\n\nEqn. \\ref{eqn:dmu2} and Eqn. \\ref{eqn:dmut} are equivalent and hence UOD for Eqn. \\ref{eqn:dmu1} can be shown as,\n\\begin{equation}\\label{eqn:uod}\nU(\\gamma, \\tau) =   \\left\\{ \n  \\begin{array}{l l}\n    \\text{Eqn. \\ref{eqn:dmu1}} & \\quad \\text{if $\\gamma \\leqslant $  1}\\\\\n    \\text{Eqn. \\ref{eqn:dmu2}} & \\quad \\text{if $\\gamma >$  1}\\\\\n  \\end{array} \\right.\n\\end{equation}\n\nThe UOD optimization given  Eqn. \\ref{eqn:uh1} is based on $H_{sys}=H_{zz}$ (Eqn. \\ref{eqn:zzh}). \nIt can be generalized to almost any interaction by \\textit{term isolation} procedure by Bremner et al. \\cite{bremner,hill}. \n\nAs an example consider the case,\n\\begin{equation} \\label{eqn:hsystst}\nH_{sys}= J(\\sigma_x\\sigma_x+\\sigma_y\\sigma_y+\\sigma_z\\sigma_z).\n\\end{equation}\n\nThe $H_{zz}$ terms can be isolated from Eqn. \\ref{eqn:hsystst} and is shown in Eqn. \\ref{eqn:hsyststiso}.\n\n\\begin{equation} \\label{eqn:hsyststiso}\n\\begin{split}\nexp(-iJ_{zz}\\sigma_z\\sigma_z t)=R^1(\\pi,z)exp(-iH_{sys}t)R^1(\\pi,z)\\\\exp(-iH_{sys}t),\n\\end{split}\n\\end{equation}\nwhere $R^1(\\pi,z) $ represents a $\\pi$-SQR on spin 1 along Z axis.\n\nCombinig all the steps above forms most generic UOD of the Hamiltonian -- DM interaction in the presence of Heisenberg XY interaction.\n\n\\section{Experimental Quantum Simulation} \\label{sec:expt}\nWe performed Quantum simulation experiments in a two qubit NMR system $^{13}CHCl_3$ (dissolved in Acetone-D6) \n(Fig. \\ref{fig:chcl3}) with $^{13}C$ and $^1H$ spins act as two qubit system with scalar coupling (zz interaction-- Eqn. \\ref{eqn:zzh}) \nbetween them. The system Hamiltonian is zz interaction (Eqn. \\ref{eqn:zzh}). We performed all the experiments in Bruker AV-500 \nspectrometer.\n\n\\begin{figure}\n\\includegraphics[width=0.14\\textwidth, height=0.12\\textwidth]{chcl3.png}\n\\caption{$^{13}C$ labeled Chloroform used for quantum simulation. $^{13}C$ and $^1H$ act as qubits with zz \ninteraction ($J_{CH}$=215.1 Hz) between them.}  \\label{fig:chcl3}\n\\end{figure}\n\nQuantum computation experiments in NMR starts with (i). preparation of pseudo pure states \n\\cite{cory_pps, gershenfeld_pps, Mahesh_sallt}, (ii). processing the state by evolving under different average \nHamiltonians \\cite{ernstbook} and (iii). read-out by quantum state tomography \\cite{avik_adiabatic}. \nHere we studied the entanglement dynamics (quantified by concurrence \\cite{coffman_concurrence}) of a Bell state \n(Eqn. \\ref{eqn:bell2}) in the Hamiltonian given in  Eqn. \\ref{eqn:dmh}.\n\n\\begin{equation} \\label{eqn:bell2}\n |\\phi\\rangle_-=\\frac{1}{\\sqrt{2}}(|01\\rangle-|10\\rangle)\n\\end{equation}\n\nUsing the unitary operator decompositions shown in Eqn. \\ref{eqn:uod}, we have simulated the Hamiltonian $H(\\gamma, \\tau)$ \nfor $\\gamma$=\\{0.33, 0.66, 0.99\\} and studied the entanglement dynamics of the singlet Bell state (Eqn. \\ref{eqn:bell2}) under these Hamiltonians.\n\nIn experimental implementation $R^n(\\theta,\\phi)$ is implemented by hard pulse \\cite{spindynamics} with suitable length \n(determined by $\\theta$) along the axis $\\phi$ on $^1H$ or $^{13}C$ spin and $U_{zz}(\\theta')$ is implemented by system \nHamiltonian evolution for a time determined by $\\theta'$ \\cite{ernstbook}. The experimental simulation results \n(Fig. \\ref{fig:ed})  shows a good agreement with the theoretical simulation. Average experimental deviation ($AED$) \nin concurrence -- calculated \nusing the formula Eqn. \\ref{eqn:cd} -- is 3.83\\%, which is mainly due to decoherence effects, static and rf inhomogeneities. \n\n\n\\begin{equation} \\label{eqn:cd}\n AED= \\displaystyle \\sum_{i=1}^{n} \\frac{|C_{es}(i)-C_{ts}(i)|}{C_{ts}(i)}\n\\end{equation}\n\n\n\nwhere $C_{es}(i)$ and $C_{ts}(i)$ are the concurrence in experimental and theoretical simulations and $n$ is the \nnumber of experimental points (here we performed simulation for $n=16$).\n\\subsubsection{Entanglement Preservation}\n\nHou et al. \\cite{hou2012preservation} demonstrated a mechanism for entanglement preservation of a quantum state in a Hamiltonian of the \ntype given in Eqn. \\ref{eqn:dmu}. \n\nPreservation of initial entanglement of a  quantum state is performed by free evolution interrupted with a certain operator $O$, which makes the \nstate to go back to its initial state. The operator sequence for preservation is given in Eqn. \\ref{eqn:epp}.\n\n\\begin{equation} \\label{eqn:epp}\n OU{(\\gamma, \\tau)}OU{(\\gamma, \\tau)}\\equiv I,\n\\end{equation}\nwhere $O=I_1\\otimes \\sigma_{2z}$.\n\nWe performed entanglement preservation experiment for singlet state (Eqn. \\ref{eqn:bell2}) in $H(\\gamma,\\tau)$ with $\\gamma$=\\{0.33, 0.66, 0.99\\}.\nThe experimental results (Fig. \\ref{fig:ep}) shows excellent entanglement preservation and good agreement with the theoretical simulation. \nThe experimental deviation of concurrence (Eqn. \\ref{eqn:cd}) is less than 2\\%.\n\n\\begin{figure}[t]\n \\subfigure[~]{\\includegraphics[width=0.35\\textwidth, height=0.25\\textwidth]{ed.jpg}  \\label{fig:ed}} \\\\\n \\subfigure[~]{\\includegraphics[width=0.35\\textwidth, height=0.25\\textwidth]{ep.jpeg} \\label{fig:ep}}\n\\caption{(a).Entanglement (concurrence) dynamics of $^{13}C-H$ system under the Hamiltonian Eqn. \\ref{eqn:dmh}, \n(b). Entanglement preservation experiment using  Eqn. \\ref{eqn:epp}. Starting from singlet state, the concurrence \nsustains at 1 with the preservation procedure.}\n\\end{figure}\n\\section*{Conclusion}\nWe have performed Fidelity Profile Optimization for the Hamiltonian -- DM interaction in the presence \nof Heisenberg XY interaction. The optimized UOD can be used for all  relative strengths ($\\gamma$) of the  \ninteractions and length is invariant under $\\gamma$ or evolution time. Using these decompositions, \nwe have experimentally verified the entanglement preservation mechanism \nsuggested by Hou et al.\n\n\\section*{Acknowledgments}\n\nWe thank Prof. Apoorva Patel for discussions and suggestions,\nand the NMR Research Centre for use of the AV-500 NMR spectrometer.\nV.S.M. thanks UGC-India for support through a fellowship.\n\n\n\\bibliographystyle{unsrtnat}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{introsec}\\eqnoset\n\n\n\nIn this paper, for any bounded domain $\\Og$ with smooth boundary in ${\\rm I\\kern -1.6pt{\\rm R}}^n$, $n\\ge2$, we consider the following elliptic system of $m$ equations ($m\\ge2$) \n\\beqno{e1}\\left\\{\\begin{array}{l} -\\Div(A(x,u)Du)=\\hat{f}(x,u,Du),\\quad x\\in \\Og,\\\\\\mbox{$u=0$ or $\\frac{\\partial u}{\\partial \\nu}=0$ on $\\partial \\Og$}. \\end{array}\\right.\\end{equation}\nwhere $A(x,u)$ is a $m\\times m$ matrix in $x\\in\\Og$ and $u\\in{\\rm I\\kern -1.6pt{\\rm R}}^m$, $\\hat{f}:\\Og\\times{\\rm I\\kern -1.6pt{\\rm R}}^m\\times {\\rm I\\kern -1.6pt{\\rm R}}^{mn}\\to{\\rm I\\kern -1.6pt{\\rm R}}^m$ is a vector valued function. \n\nWe say that $u$ is a strong solution if $u$ is continuous on $\\bar{\\Og}$ with $Du\\in L^\\infty_{loc}(\\Og)$ and $D^2u\\in L^2_{loc}(\\Og)$. Hence, $u$ solves \\mref{e1} almost everywhere in $\\Og$.\n\n\nThe strongly coupled system \\mref{e1} appears in many applications, ranging from differential geometry to physical models. For instance, it describes the steady states of Maxwell-Stephan systems describing the diffusive transport of multicomponent mixtures, models in reaction and diffusion in electrolysis and diffusion of polymers, or population dynamics, among others. \n\nIt is always assumed that the matrix $A(x,u)$ is elliptic in the sense that there exist two scalar positive continuous functions $\\llg_1(x,u), \\llg_2(x,u)$ such that \\beqno{genelliptic} \\llg_1(x,u)|\\zeta|^2 \\le \\myprod{A(x,u)\\zeta,\\zeta}\\le \\llg_2(x,u)|\\zeta|^2 \\quad \\mbox{for all } x\\in\\Og,\\,u\\in{\\rm I\\kern -1.6pt{\\rm R}}^m,\\,\\zeta \\in{\\rm I\\kern -1.6pt{\\rm R}}^{nm}.\\end{equation} If there exist positive constants $c_1,c_2$ such that $c_1\\le \\llg_1(x,u)$ and  $\\llg_2(x,u)\\le c_2$ then we say that $A(x,u)$ is {\\em regular elliptic}. If $c_1\\le \\llg_1(x,u)$ and  $\\llg_2(x,u)\/\\llg_1(x,u)\\le c_2$, we say that $A(x,u)$ is {\\em uniform elliptic}. On the other hand, if we allow $c_1=0$ and $\\llg_1(x,u)$ tend to zero (respectively, $\\infty$) when $|u|\\to\\infty$ then we say that $A(x,u)$ is {\\em singular} (respectively, {\\em degenerate}). \n\nThe first fundamental problem in the study of \\mref{e1} is the existence and regularity of its solutions. One can decide to work with either weak or strong solutions. In the first case, the existence of a weak solution can be achieved via Galerkin or variational methods \\cite{Gius} but its regularity (e.g., boundedness, H\\\"older continuity of the solution and its higher derivatives) is an open issue and difficult to address. Several works (see \\cite{Gius} and the reference therein) have been done along this line to establish only {\\em partial regularity} of {\\em bounded} weak solutions, wherever they are VMO. The assumption on the boundedness of weak solutions is a very severe and hard to check one, as maximum principles do not generally exist for systems (i.e. $m>1$) like \\mref{e1}. One usually needs to use ad hoc techniques on the case by case basis to show that $u$ is bounded.  Even for bounded weak solutions, we only know that they are partially regular, i.e. H\\\"older continuous almost everywhere. Techniques in this approach rely heavily on the fact that $A(x,u)$ is {\\em regular elliptic}, and hence the boundedness of weak solutions.\n\nIn our recent work \\cite{dleJFA}, we choose a different approach making use of fixed point theory and discussing the existence of {\\em strong} solutions of \\mref{e1} under the weakest assumption that they are a-priori VMO, not necessarily bounded, and general structural conditions on the data of \\mref{e1} which are independent of $x$, we assumed only that $A(u)$ is {\\em uniformly elliptic}.  Applications were presented in \\cite{dleJFA} when $\\llg_1(u)$ has a positive polynomial growth in $|u|$ and, without the boundedness assumption on the solutions, so \\mref{e1} can be degenerate as $|u|\\to\\infty$.\n\nIn this paper, we will establish much stronger results than those in \\cite{dleJFA} under much more general assumptions on the structure of \\mref{e1} below. Beside the minor fact that the data can depend on $x$, we allow further that:\n\\begin{itemize}\n\t\\item $A(x,u)$ can be either degenerate or singular as $|u|$ tend to infinity;\n\t\\item  $\\hat{f}(x,u,Du)$ can have a quadratic growth in $Du$.\n\t\n\\end{itemize} \n\n\nMost remarkably, the key assumption in \\cite{dleJFA} that $u$ is VMO will be replaced by a more versatile one in this paper: $K(u)$ is VMO for some suitable map $K:{\\rm I\\kern -1.6pt{\\rm R}}^m\\to{\\rm I\\kern -1.6pt{\\rm R}}^m$. This allows us to consider the singular case where one may not be able to estimate the BMO norm of $u$  but that of $K(u)$ in small balls. Examples of this case in applications will be provided in \\refsec{res}.\n\nOne of the key ingredients in the proof in \\cite{dleJFA} is the {\\em local} weighted Gagliardo-Nirenberg inequality involving BMO norm \\cite[Lemma 2.4]{dleANS}. This allows us to consider VMO solutions in \\cite{dleJFA}. In this paper, we make use of a new version of this inequality reported in our work \\cite{dleGNnew} replacing the BMO norm of $u$ by that of $K(u)$ for some suitable map $K:{\\rm I\\kern -1.6pt{\\rm R}}^m\\to{\\rm I\\kern -1.6pt{\\rm R}}^m$.\n\n\n\nWe consider the following structural conditions on the data of \\mref{e1}.\n\n\n\n\n\\begin{description}\n\n\\item[A)] $A(x,u)$ is $C^1$ in $x\\in\\Og$, $u\\in{\\rm I\\kern -1.6pt{\\rm R}}^m$ and there exist a constant $C_*>0$  and  scalar $C^1$ positive functions $\\llg(u),\\og(x)$  such that  for all $u\\in{\\rm I\\kern -1.6pt{\\rm R}}^m$,  $\\zeta\\in{\\rm I\\kern -1.6pt{\\rm R}}^{mn}$ and $x\\in\\Og$ \n\\beqno{A1} \\llg(u)\\og(x)|\\zeta|^2 \\le \\myprod{A(x,u)\\zeta,\\zeta} \\mbox{ and } |A(x,u)|\\le C_*\\llg(u)\\og(x).\\end{equation} \n\nIn addition, there is a constant $C$ such that $|\\llg_u(u)||u|\\le C\\llg(u)$ and  \\beqno{Axcond} |A_u(x,u)|\\le C|\\llg_u(u)|\\og(x),\\; |A_x(x,u)|\\le C|\\llg(u)||D\\og|.\\end{equation}\n\n\n\\end{description}\n\nHere and throughout this paper, if $B$ is a $C^1$ (vector valued) function in $u\\in {\\rm I\\kern -1.6pt{\\rm R}}^m$ then we abbreviate its  derivative $\\frac{\\partial B}{\\partial u}$ by $B_u$. Also, with a slight abuse of notations, $A(x,u)\\zeta$, $\\myprod{A(x,u)\\zeta,\\zeta}$ in \\mref{genelliptic}, \\mref{A1} should be understood in the following way: For $A(x,u)=[a_{ij}(x,u)]$, $\\zeta\\in{\\rm I\\kern -1.6pt{\\rm R}}^{mn}$ we write $\\zeta=[\\zeta_i]_{i=1}^m$ with $\\zeta_i=(\\zeta_{i,1},\\ldots\\zeta_{i,n})$ and $$A(x,u)\\zeta=[\\Sigma_{j=1}^m a_{ij}\\zeta_j]_{i=1}^m,\\; \\myprod{A(x,u)\\zeta,\\zeta}=\\Sigma_{i,j=1}^m a_{ij}\\myprod{\\zeta_i,\\zeta_j}.$$\n\n\n\nWe also assume that $A(x,u)$ is regular elliptic for {\\em bounded} $u$.\n\n\\begin{description}\\item[AR)] $\\og\\in C^1(\\Og)$ and there are positive numbers $\\mu_*,\\mu_{**}$ such that \n\\beqno{Aregcond2} \\mu_*\\le\\og(x)\\le \\mu_{**}, \\; |D\\og(x)|\\le \\mu_{**} \\quad \\forall x\\in\\Og.\\end{equation} For any bounded set $K\\subset{\\rm I\\kern -1.6pt{\\rm R}}^m$ there is a constant $\\llg_*(K)>0$ such that\n\\beqno{Aregcond1} \\llg_*(K)\\le\\llg(u) \\quad \\forall u\\in K.\\end{equation} \\end{description}\n\nConcerning the reaction term $\\hat{f}(x,u,Du)$, which may have linear or {\\em quadratic} growth in $Du$, we assume the following condition.\n\\begin{description} \\item[F)] There exist a constant $C$ and a nonegative differentiable function $f:{\\rm I\\kern -1.6pt{\\rm R}}^m\\to{\\rm I\\kern -1.6pt{\\rm R}}$ such that $\\hat{f}$ satisfies: For  any diffrentiable vector valued functions $u:{\\rm I\\kern -1.6pt{\\rm R}}^n\\to{\\rm I\\kern -1.6pt{\\rm R}}^m$ and $p:{\\rm I\\kern -1.6pt{\\rm R}}^n\\to{\\rm I\\kern -1.6pt{\\rm R}}^{mn}$ we assume either that\n\\begin{description}\\item[f.1)] $\\hat{f}$ has a linear growth in $p$ \n\\beqno{FUDU11}|\\hat{f}(x,u,p)| \\le C\\llg(u)|p|\\og(x) + f(u)\\og(x),\\end{equation} $$|D\\hat{f}(x,u,p)| \\le  C(\\llg(u)|Dp|+ |\\llg_u(u)||p|^2)\\og+ C\\llg(u)|p||D\\og|+C|D(f(u)\\og(x))|;$$\\end{description} or \\begin{description} \n\\item[f.2)] $\\llg_{uu}(u)$ exists and $\\hat{f}$ has a quadratic growth in $p$ \\beqno{FUDU112}|\\hat{f}(x,u,p)| \\le C|\\llg_u(u)||p|^2\\og(x) + f(u)\\og(x),\\end{equation}  $$\\begin{array}{lll}|D\\hat{f}(x,u,p)| &\\le&  C(|\\llg_u(u)||p||Dp|+ |\\llg_{uu}(u)||p|^3)\\og+ C|\\llg_u(u)||p|^2|D\\og|\\\\&&+C|D(f(u)\\og(x))|.\\end{array}$$ Furthermore, we assume that \\beqno{llgquadcond} |\\llg_{uu}(u)|\\llg(u)\\le C|\\llg_u(u)|^2.\\end{equation}\n\\end{description}\n\\end{description}\n\nBy a formal differentiation of \\mref{FUDU11} and \\mref{FUDU112}, one can see that  the growth conditions for $\\hat{f}$ naturally implies those of  $D\\hat{f}$ in the above assumption. The condition \\mref{llgquadcond} is verified easily if $\\llg(u)$ has a polynomial growth in $|u|$.\n\n\n\n\nWe organize our paper as follows. In \\refsec{res} we state our main results and their applications which are actually consequences of the most general but technical \\reftheo{gentheo1} in \\refsec{w12est} where we deal with general map $K$. The proof of the results in \\refsec{res}, which provide examples of the map $K$, will thus be provided in \\refsec{proofmainsec}. In \\refsec{GNsec} we state the new version of the local weighted Gagliardo-Nirenberg inequality in \\cite{dleGNnew} to prepare for the proof the main technical theorem in \\refsec{w12est}. We collect some elementary but useful facts in our proof in \\refsec{appsec}.\n\n\n\n\\section{Preliminaries and Main Results}\\eqnoset\\label{res}\n\n\n\n\n\n\n\n\nWe state the main results of this paper in this section. In fact, these results are consequences of our main technical results in \\refsec{w12est} assuming general conditions A) and F) and, roughly speaking, some a~priori knowledge on the smallness of the BMO norm of $K(u)$ for a general map $K:{\\rm I\\kern -1.6pt{\\rm R}}^m\\to{\\rm I\\kern -1.6pt{\\rm R}}^m$ and any strong solution $u$ to \\mref{e1}.\n\n\n\n\nWe define the measure $d\\mu =\\og(x)dx$ and recall that a vector valued function $f\\in L^1(\\Og,\\mu)$ is said to be in $BMO(\\Og,\\mu)$ if \\beqno{bmodef} [f]_{*,\\mu}:=\\sup_{B_R\\subset\\Og}\\mitmu{B_R}{|f-f_{B_R}|}<\\infty,\\quad f_{B_R}:=\\frac{1}{\\mu(B_R)}\\iidmu{B_R}{f}.\\end{equation} We then define $$\\|f\\|_{BMO(\\Og,\\mu)}:=[f]_{*,\\mu}+\\|f\\|_{L^1(\\Og,\\mu)}.$$\n\n\n\n\nThroughout this paper, in  our statements and proofs, we use $C,C_1,\\ldots$ to denote various constants which can change from line to line but depend only on the parameters of the hypotheses in an obvious way. We will write $C(a,b,\\ldots)$ when the dependence of  a constant $C$ on its parameters is needed to emphasize that $C$ is bounded in terms of its parameters. We also write $a\\lesssim b$ if there is a universal constant $C$ such that $a\\le Cb$. In the same way, $a\\sim b$ means $a\\lesssim b$ and $b\\lesssim a$.\n\nTo begin, as in \\cite{dleANS} with $A$ is independent of $x$, we assume that the eigenvalues of the matrix $A(x,u)$ are not too far apart. Namely, for $C_*$ defined in \\mref{A1} of A) we assume\n\\begin{description}\\item[SG)] $(n-2)\/n <C_*^{-1}$. \\end{description} Here $C_*$ is, in certain sense, the ratio of the largest and smallest eigenvalues of $A(x,u)$. This condition seems to be necessary as we deal with systems, cf.  \\cite{dleN}. \n\n\nIn this section we assume further the following growth conditions on $\\llg(u)$ and $f(u)$. \\begin{description}\\item[AF)] There is a constant $k$ such that for all $u\\in{\\rm I\\kern -1.6pt{\\rm R}}^m$\n\\beqno{llucond} \\llg(u)\\sim (\\llg_0+|u|)^k \\mbox{ and }|\\llg_u(u)|\\sim (\\llg_0+|u|)^{k-1}. \\end{equation}\n\nIf $\\hat{f}$ depends on $x$, then we also assume that there is a constant $C$ such that\n\\beqno{ffucond} |f(u)|\\le C|f_u(u)|(1+|u|).\\end{equation}\n\\end{description}\n\nWe imbed \\mref{e1} in the following family of systems \n\\beqno{e1famzzz}\\left\\{\\begin{array}{l} -\\Div(A(x,\\sg u)Du)=\\hat{f}(x,\\sg u,\\sg Du),\\quad x\\in \\Og, \\sg\\in[0,1],\\\\\\mbox{$u=0$ or $\\frac{\\partial u}{\\partial \\nu}=0$ on $\\partial \\Og$}. \\end{array}\\right.\\end{equation}\n\n\nOur first main result is the following.\n\n\n\\btheo{dleNL-mainthm} \nAssume that A), AR), F), AF) and SG) hold. Suppose also that the following integrability conditions hold for {\\em any strong} solution $u$ of \\mref{e1famzzz}: There exist $r_0>n\/2$, $\\bg_0\\in(0,1)$    such that the following quantities\n\\beqno{llgmainhyp009} \\|\\llg^{-1}(u)\\|_{L^{\\frac n2}(\\Og,\\mu)},\\;\\||f_u(u)|\\llg^{-1}(u)\\|_{L^{r_0}(\\Og,\\mu)},\\; \\|(\\llg(u)|u|^2)^{\\bg_0}\\|_{L^{1}(\\Og,\\mu)},\\end{equation} \n\\beqno{llgmainhyp0099}\\iidmu{\\Og}{(|f_u(u)|+\\llg(u))|Du|^{2}}\\end{equation} are bounded by some constant $C_0$.\n\nDefine $K_0:{\\rm I\\kern -1.6pt{\\rm R}}^m\\to{\\rm I\\kern -1.6pt{\\rm R}}^m$ by $K_0(u)=|\\log(|U|)||U|^{-1}U$, $U=[\\llg_0+|u_i|]_{i=1}^m$. We assume that the BMO norms of $K_0(u)$ and  $\\log(\\llg_0+|u|)$ are small in small balls in the following sense: For any $\\eg>0$ there is $R_\\eg>0$ such that  \\beqno{mainloghyp} \\|\\log(\\llg_0+|u|)\\|_{BMO(B_R,\\mu)},\\;\\|K_0(u)\\|_{BMO(B_R,\\mu)}<\\eg \\quad \\mbox{for all $B_R\\subset\\Og$ with $R\\le R_\\eg$}.\\end{equation} \n\nThen \\mref{e1} has a strong solution $u$.\n\n \\end{theorem}\n \n\n\nThe condition \\mref{mainloghyp} on the smallness of the BMO norm of $K_0(u)$ in small balls is the most crucial one in applications. The next result is more applicable in the checking of this condition. \n\n\\bcoro{dlec1coro} The conclusion of \\reftheo{dleNL-mainthm} holds if \\mref{mainloghyp} is replaced by: There exist $\\ag\\in(0,1)$ and a constant $C_0$ such that \n\\beqno{pis20az5} \\iidmu{\\Og}{(\\llg_0+|u|)^{-n\\ag}|Du|^{n}}\\le C_0.\\end{equation}\n\n\\end{corol}\n\nIn \\cite{dleJFA}, we consider the case $\\llg(u)\\sim(\\llg_0+|u|)^k$ with $k>0$ and assume that $u$ has small BMO norm in small balls, which can be verified by establishing that $\\|Du\\|_{L^n(\\Og)}$ is bounded. The assumption \\mref{pis20az5} is of course much weaker, especially when $|u|$ is large, and can apply to the case $k<0$.\n\n\n\nWe present an application of \\refcoro{dlec1coro}. This example concerns cross diffusion systems with polynomial growth data on planar domains. This type of systems occurs in many applications in mathematical biology and ecology. We will see that the assumptions of the corollary can be verified by a very simple integrability assumption on the solutions. \n\n\n\\bcoro{2dthm1} Let $n=2$. Suppose A), F) and $f(u)\\lesssim (\\llg_0+|u|)^l$ and $\\llg(u)\\sim(\\llg_0+|u|)^{k}$ for some $k,l$ satisfying \\beqno{klcond} k>\\frac{-2C_*}{C_*-1} \\mbox{ and } l-k<\\frac{C_*+1}{C_*-1}.\\end{equation} \nIf $\\hat{f}$ has a quadratic growth in $Du$ as in \\mref{FUDU112} of f.2) then we assume further that \\beqno{FUDU112z} |\\hat{f}(x,u,p)| \\le \\eg_0|\\llg_u(u)||p|^2 + f(u),\\end{equation} with $\\eg_0$ being sufficiently small.\n\n\nIf there is a constant $C_0$ such that for {\\em any strong} solution $u$ of \\mref{e1famzzz} \\beqno{keyn2lnorm0} \\|u\\|_{L^{l_0}(\\Og,\\mu)}\\le C_0\\quad \\mbox{for some $l_0>\\max\\{l,l-k-1\\}$},\\end{equation} then \\mref{e1} has a strong solution $u$.  \\end{corol}\n\nThe assumption \\mref{keyn2lnorm0} is a very weak one. For example, if $k\\ge-1$ then we see from the growth condition $|f(u)|\\lesssim (\\llg_0+|u|)^l$ that  \\mref{keyn2lnorm0} simply requires that $l_0>l$, or equivalently, $f(u)\\in L^r(\\Og,\\mu)$ for some $r>1$. \n\nThis result greatly generalizes \\cite[Corollary 3.9]{dleJFA} in many aspects. Beside the fact that we allow quadratic growth in $Du$ for $\\hat{f}(x,u,Du)$ and $k<0$, we also consider a much general relation between the the growths of $f(u)$ and $\\llg(u)$ in \\mref{klcond}, while we assume in \\cite{dleJFA} that $f(u)\\lesssim \\llg(u)|u|$ (i.e. $l-k=1$).\n\n\n\nIn the second main result, we consider the following generalized SKT system (see \\cite{dleJFA,SKT,yag}) with Dirichlet or Neumann boundary conditions on a bounded domain $\\Og\\subset{\\rm I\\kern -1.6pt{\\rm R}}^n$ with $n\\le4$.\n\\beqno{genSKT} -\\Delta(P_i(u))=B_i(u,Du)+ f_i(u), \\quad i=1,\\ldots,m.\\end{equation} Here, $P_i:{\\rm I\\kern -1.6pt{\\rm R}}^m\\to{\\rm I\\kern -1.6pt{\\rm R}}$ are $C^2$ functions. The functions $B_i, f_i$ are $C^1$ functions on ${\\rm I\\kern -1.6pt{\\rm R}}^m\\times{\\rm I\\kern -1.6pt{\\rm R}}^{mn}$ and ${\\rm I\\kern -1.6pt{\\rm R}}^m$ respectively. We will assume that $B_i(u,Du)$ has linear growth in $Du$.\n\nBy a different choice of the map $K$ in the main technical theorem, we have the following.\n\n\\btheo{n3SKT} Assume that the matrix $A(u)=\\mbox{diag}[(P_i)_u(u)]$ satisfies the condition A) with $\\llg(u)\\sim (\\llg_0+|u|)^k$ for some $k\\ge-1$. Moreover, $\\hat{f}(u,Du)=\\mbox{diag}[B_i(u,Du)+f_i(u)]$ satisfies the following special version f.1) of F)  $$|B_i(u,Du)|\\le C\\llg(u)|Du|, \\; |f_i(u)|\\le f(u).$$\n\n\nThus, \\mref{genSKT} can be written as \\mref{e1}. Assume that there exist $r_0>n\/2$ and a constant $C_0$  such that for {\\em any strong} solution $u$ of \\mref{e1famzzz}\n\\beqno{fuhyp0}\\|f_u(u)\\llg^{-1}(u)\\|_{L^{r_0}(\\Og)}\\le C_0,\\end{equation} and the following conditions. \\begin{description}\\item[i)] If $k\\ge0$ then  $\\|u\\|_{L^1(\\Og)}\\le C_0$. \\item[ii)] If $k\\in[-1,0)$ then $\\|u\\|_{L^{-kn\/2}(\\Og)}\\le C_0$. Furthermore,\n\\beqno{fuhyp0k}\\|\\llg^{-2}(u)f_u(u)\\|_{L^\\frac{n}{2}(\\Og)}\\le C_0.\\end{equation}\n\\end{description}\n\nThen \n\\mref{genSKT} has a strong solution for $n=2,3,4$.\n\\end{theorem}\n\nThe above result generalizes \\cite[Corollary 3.10]{dleJFA} where we assumed that $\\hat{f}$ is independent of $Du$, $k>0$ and $f(u)\\lesssim \\llg(u)|u|$. In this case, it is natural to assume that $|f_u(u)|\\lesssim\\llg(u)$ so that \\mref{fuhyp0} obviously holds. We also have $|\\llg^{-2}(u)f_u(u)|\\lesssim \\llg^{-1}(u)$ so that \\mref{fuhyp0k} is in fact a consequence of the assumption $\\|u\\|_{L^{-kn\/2}(\\Og)}\\le C_0$ in ii).\n\nWe should remark that all the assumptions on strong solutions of the family \\mref{e1famzzz} can be checked by considering the case $\\sg=1$ (i.e. \\mref{e1} or \\mref{genSKT}) because these systems satisfy the same structural conditions uniformly with respect to the parameter $\\sg\\in[0,1]$.\n\n\n\\section{A general  local weighted Gagliardo-Nirenberg inequality} \\eqnoset\\label{GNsec}\nIn this section, we present a  local weighted Gagliardo-Nirenberg inequality in our recent work \\cite{dleGNnew}, which will be one of the main ingredients of the proof of our main technical theorem in \\refsec{w12est}. \nThis inequality generalizes \\cite[Lemma 2.4]{dleANS} by replacing the Lebesgue measure with general one and the BMO norm of $u$ with that of $K(u)$ where $K$ is a suitable map on ${\\rm I\\kern -1.6pt{\\rm R}}^m$, and so the applications of our main technical theorem in the next section will be much more versatile than those in \\cite{dleJFA,dleANS}.\n\nLet us begin by describing the assumptions in \\cite{dleGNnew} for this general inequality.\nWe need to recall some well known notions from Harmonic Analysis.\n\nLet $\\og\\in L^1(\\Og)$ be a nonnegative function and  define the measure $d\\mu=\\og(x)dx$. For any $\\mu$-measurable subset $A$ of $\\Og$  and any  locally $\\mu$-integrable function $U:\\Og\\to{\\rm I\\kern -1.6pt{\\rm R}}^m$ we denote by  $\\mu(A)$ the measure of $A$ and $U_A$ the average of $U$ over $A$. That is, $$U_A=\\mitmu{A}{U(x)} =\\frac{1}{\\mu(A)}\\iidmu{A}{U(x)}.$$\n\n\n\n\n\nWe say that $\\Og$ and $\\mu$ support a $q_*$-Poincar\\'e inequality if the following holds. \\begin{description} \\item[P)]  There exist $q_*\\in(0,2]$, $\\tau_*\\ge 1$ and some constant $C_P$ such that \\beqno{Pineq2} \n\\mitmu{B}{|h-h_{B}|}\\le C_Pl(B)\n\\left(\\mitmu{\\tau_*B}{|Dh|^{q_*}}\\right)^\\frac{1}{q_*}\\end{equation}\nfor  any cube $B\\subset\\Og$ with side length $l(B)$ and any function $u\\in C^1(B)$. \\end{description}\n\nHere and throughout this section,  we denote by $l(B)$ the side length of $B$ and by $\\tau B$ the cube which is concentric with $B$ and has side length $\\tau l(B)$. We also write $B_R(x)$ for a cube centered at $x$ with side length $R$ and sides parallel to to standard axes of ${\\rm I\\kern -1.6pt{\\rm R}}^n$. We will omit $x$ in the notation $B_R(x)$ if no ambiguity can arise.\n\nWe consider the following conditions on the density $\\og(x)$.\n\n\\begin{description} \\item[LM.1)]  For some $N\\in(0,n]$ and any ball $B_r$ we have $\\mu(B_r)\\le C_\\mu r^N$. \nAssume also that $\\mu$ supports the 2-Poincar\\'e inequality \\mref{Pineq2} in P). Furthermore, $\\mu$ is doubling and satisfies  the following inequality for some $s_*>0$ \\beqno{fracmu} \\left(\\frac{r}{r_0}\\right)^{s_*}\\le C_\\mu\\frac{\\mu(B_r(x))}{\\mu(B_{r_0}(x_0))},\\end{equation} where $B_r(x), B_{r_0}(x_0)$ are any cubes with $x\\in B_{r_0}(x_0)$.\n\n\\item[LM.2)]  $\\og=\\og_0^2$ for some $\\og_0\\in C^1(\\Og)$ and $d\\mu=\\og_0^2 dx$  also supports a Hardy type inequality: There is a constant $C_H$ such that for any function $u\\in C^1_0(B)$\\beqno{lehr1m} \\iidx{\\Og}{|u|^2|D\\og_0|^2}\\le C_H\\iidx{\\Og}{|Du|^2\\og_0^2}.\\end{equation} \n\\end{description}\n\n\n\nFor $\\cg\\in(1,\\infty)$ we say that a nonnegative locally integrable function $w$ belongs to the class $A_\\cg$ or $w$ is an $A_\\cg$ weight on $\\Og$ if the quantity\n\\beqno{aweight} [w]_{\\cg,\\Og} := \\sup_{B\\subset\\Og} \\left(\\mitmu{B}{w}\\right) \\left(\\mitmu{B}{w^{1-\\cg'}}\\right)^{\\cg-1} \\quad\\mbox{is finite}.\\end{equation}\nHere, $\\cg'=\\cg\/(\\cg-1)$. For more details on these classes we refer the reader to \\cite{OP,st}. If the domain $\\Og$ is specified we simply denote $ [w]_{\\cg,\\Og}$ by $ [w]_{\\cg}$.\n\n\n\n\n\nWe assume the following hypotheses.\n\n\\begin{description}\n\\item[A.1)] Let $K:\\mbox{dom}(K)\\to{\\rm I\\kern -1.6pt{\\rm R}}^m$ be a $C^1$ map on a domain $\\mbox{dom}(K)\\subset{\\rm I\\kern -1.6pt{\\rm R}}^m$ such that $\\DKTmU=(K_U(U)^{-1})^T$ exists and $\\mccK_U\\in L^\\infty(\\mbox{dom}(K))$.\n\nFurthermore, let $\\Fg,\\LLg:\\mbox{dom}(K)\\to{\\rm I\\kern -1.6pt{\\rm R}}^+$ be $C^1$ positive functions. We assume that for all $U\\in \\mbox{dom}(K)$\n\\beqno{kappamain} |\\DKTmU|\\lesssim \\LLg(U)\\Fg^{-1}(U),\\end{equation}\n\\beqno{logcondszmain}|\\Fg_U(U)||\\mccK(U)|\\lesssim \\Fg(U).\\end{equation}\n\\end{description}\n\n\nLet $\\Og_*$ be a proper subset of $\\Og$ and $\\og_*$ be a function in $C^1(\\Og)$ satisfying  \n\\beqno{subogm} \\og_*\\equiv 1 \\mbox{ in $\\Og_*$ and } \\og_*\\le 1 \\mbox{ in $\\Og$}.\\end{equation}\n\nFor any $U\\in C^2(\\Og,\\mbox{dom}(K))$  we denote\n\\beqno{Idefm} I_1:=\\iidmu{\\Og}{\\Fg^2(U)|DU|^{2p+2}},\\;\nI_2:=\\iidmu{\\Og}{\\LLg^2(U)|DU|^{2p-2}|D^2U|^2},\\end{equation}\n\\beqno{Idef1zm} \\myIbar:=\\iidmu{\\Og}{|\\LLg_U(U)|^2|DU|^{2p+2}},\\;I_{1,*}:=\\iidmu{\\Og_*}{\\Fg^2(U)|DU|^{2p+2}},\\end{equation} \\beqno{I0*m} \\breve{I}_{0,*}:=\\sup_\\Og|D\\og_*|^2\\iidmu{\\Og}{\\LLg^2(U)|DU|^{2p}}.\\end{equation} \n\nBy \\refrem{PSrem} below, the assumption PS) in \\cite{dleGNnew} that $\\mu$ supports a Poincar\\'e-Sobolev inequality  is then satisfied. We established the following local weighted Gagliardo-Nirenberg inequality in \\cite{dleGNnew}.\n\n\\btheo{GNlocalog1m} Suppose LM.1)-LM.2), A.1). Let $U\\in C^2(\\Og,\\mbox{dom}(K))$ and satisfy\n\\beqno{boundaryzm}\\myprod{\\og_*\\og_0^2 \\Fg^2(U)\\DKTmU DU,\\vec{\\nu}}=0\\end{equation} on $\\partial\\Og$ where $\\vec{\\nu}$ is the outward normal vector of $\\partial\\Og$.  Let $\\myPi(x):=\\LLg^{p+1}(U(x))\\Fg^{-p}(U(x))$ and assume that $[\\myPi^{\\ag}]_{\\bg+1}$ is finite for some $\\ag>2\/(p+2)$ and $\\bg<p\/(p+2)$.\n\n\n\nThen, for any $\\eg>0$ there are constants $C, C([\\myPi^{\\ag}]_{\\bg+1})$ such that\n\\beqno{GNlocog11m}I_{1,*}\\le  \\eg I_1+\\eg^{-1}C\\|K(U)\\|_{BMO(\\mu)}^2[I_2+\\myIbar+C([\\myPi^{\\ag}]_{\\bg+1}) [I_2+\\myIbar+\\breve{I}_{0,*}]].\\end{equation}\nHere, $C$ also depends on $C_{P},C_\\mu$ and $C_H$.\n\n\\end{theorem}\n\n\n\nFor our purpose in this paper we need only a special case of \\reftheo{GNlocalog1m} where  $\\Og,\\Og_*$ are concentric balls $B_s,B_t$, $0<s<t$.\nWe let $\\og_*$ be a cutoff function for $B_s,B_{t}$:  $\\og_*$ is a $C^1$ function satisfying $\\og_*\\equiv1$ in $B_s$ and $\\og_*\\equiv0$ outside $B_t$ and $|D\\og_*|\\le 1\/(t-s)$. The condition \\mref{boundaryzm} of the above theorem is clearly satisfied on the boundary of $\\Og=B_t$. We also consider only the case $\\Fg(U)\\sim|\\LLg_U(U)|$.\n\n\n\n\nWe then have the following corollary.\n\n\\bcoro{GNlocalog1mcoro} Suppose that LM.1)-LM.2), A.1) hold for   $\\Fg(U)=|\\LLg_U(U)|$. Accordingly, define  $\\myPi_p(x):=\\LLg^{p+1}(U(x))|\\LLg_U(U(x))|^{-p}$ and let $B_t(x_0)$ be any ball in $\\Og$ and assume that \\begin{description}\\item[A.2)]  $[\\myPi_p^{\\ag}]_{\\bg+1,B_t(x_0)}$ is finite for some $\\ag>2\/(p+2)$ and $\\bg<p\/(p+2)$.\\end{description}\n\n\n We denote (compare with \\mref{Idefm}-\\mref{I0*m}) \\beqno{Idefmt} I_{0}(t,x_0):=\\iidmu{B_t(x_0)}{\\LLg^2(U)|DU|^{2p}},\\; I_1(t,x_0):=\\iidmu{B_t(x_0)}{|\\LLg_U(U)|^2|DU|^{2p+2}},\\;\\end{equation}\n\\beqno{I0*mt} \nI_2(t,x_0):=\\iidmu{B_t(x_0)}{\\LLg^2(U)|DU|^{2p-2}|D^2U|^2}.\\end{equation} \n\n\nThen, for any $\\eg>0$ and any ball $B_s(x_0)$, $0<s<t$, there are constants $C, C([\\myPi_p^{\\ag}]_{\\bg+1,B_t(x_0)})$ with $C$ also depending on $C_{PS},C_\\mu$ and $C_H$ such that for $$C_{\\eg,U,\\myPi}=\\eg+\\eg^{-1}C\\|K(U)\\|_{BMO(B_t(x_0),\\mu)}^2[1+C([\\myPi_p^{\\ag}]_{\\bg+1,B_t(x_0)})]$$ we have\n\\beqno{GNlocog11mcoro}I_{1}(s,x_0)\\le  C_{\\eg,U,\\myPi}[I_1(t,x_0)+I_2(t,x_0)+(t-s)^{-2}I_{0}(t,x_0)].\\end{equation} \n\n\\end{corol}\n\nWe end this section by some remarks on  the measure $\\mu$.\n\\brem{PSrem} As we assume in LM.1) that\n$\\mu$ is doubling and supports a $2$-Poincar\\'e inequality \\mref{Pineq2}, from \\cite[Section 3]{Haj} we see that  an improved $q_*$-Poincar\\'e inequality also holds for any $q_*\\in(p_1,2]$ for some $p_1<2$. That is, see \\cite[eqn. (5)]{Haj}, there are some constants $C_P$ and $\\tau_*\\ge1$ such that the following $q_*$-Poincar\\'e inequality holds true\n\\beqno{Pineq} \n\\mitmu{B}{|h-h_{B}|}\\le C_Pl(B)\n\\left(\\mitmu{\\tau_*B}{|Dh|^{q_*}}\\right)^\\frac{1}{q_*}.\\end{equation} \nThis and the assumption \\mref{fracmu} and \\cite[1) of Theorem 5.1]{Haj}  show that the following Poincar\\'e-Sobolev inequality holds for {\\em some} $\\pi_*>2$ and $q_*<2$ and some constant $C_{PS}$\n\\beqno{PSineq} \\frac{1}{l(B)}\n\\left(\\mitmu{B}{|u-u_{B}|^{\\pi_*}}\\right)^\\frac1{\\pi_*}  \\le C_{PS}\n\\left(\\mitmu{\\tau_*B}{|Du|^{2}}\\right)^\\frac{1}{2},\\quad \\pi_*>2.\\end{equation}\\erem  In fact, if $q_*<s_*$, the exponent in \\mref{fracmu}, then \\cite[1) of Theorem 5.1]{Haj} establishes \\mref{PSineq}\nfor  $\\pi_*=s_*q_*\/(s_*-q_*)$. Thus, $\\pi_*>2$ if $s_*<2q_*\/(2-q_*)$. This is the case if we choose $q_*<2$ and closed to 2. Hence, the assumption PS) in \\cite{dleGNnew} that $\\mu$ supports a Poincar\\'e-Sobolev inequality \\mref{PSineq} is then satisfied for some $q_*<2$ and $\\pi_*>2$  (the dimensional parameters $d,n$ in that paper are now denoted by $n,N$ respectively).\n\n\\brem{PSremH}  If $q_*=s_*$, \\cite[2) of Theorem 5.1]{Haj} shows that \\mref{PSineq} holds true\nfor any $\\pi_*>1$. In addition, if $q_*>s_*$ then  the H\\\"older norm of $u$ is bounded in terms of $\\|Du\\|_{L^{q_*}(\\Og,\\mu)}$. \n\n\\erem\n\n\n\n\\section{The main technical theorem}\\eqnoset\\label{w12est}\n\nIn this section, we establish the main result of this paper. We consider the following system\n\\beqno{gensys}\\left\\{\\begin{array}{l} -\\Div(A(x,u)Du)=\\hat{f}(x,u,Du),\\quad x\\in \\Og,\\\\\\mbox{$u=0$ or $\\frac{\\partial u}{\\partial \\nu}=0$ on $\\partial \\Og$}. \\end{array}\\right.\\end{equation}\n\nWe imbed this system in the following family of systems \n\\beqno{gensysfam}\\left\\{\\begin{array}{l} -\\Div(A(x,\\sg u)Du)=\\hat{f}(x,\\sg u,\\sg Du),\\quad x\\in \\Og, \\sg\\in[0,1],\\\\\\mbox{$u=0$ or $\\frac{\\partial u}{\\partial \\nu}=0$ on $\\partial \\Og$}. \\end{array}\\right.\\end{equation}\n\n\nFirst of all, we will assume that the system \\mref{gensys} satisfies the structural conditions A) and F). Additional assumptions serving the purpose of this paper then follow  for the validity of the local weighted Gagliardo-Nirenberg inequality of \\refcoro{GNlocalog1mcoro} with  $\\LLg(U)=\\llg^\\frac12(U)$.\n\n\n\\begin{description} \n\\item[H)] There is a $C^1$ map $K:{\\rm I\\kern -1.6pt{\\rm R}}^m\\to{\\rm I\\kern -1.6pt{\\rm R}}^m$ such that $\\mathbb{K}(u)=(K_u(u)^{-1})^T$ exists and $\\mathbb{K}_U\\in L^\\infty({\\rm I\\kern -1.6pt{\\rm R}}^m)$.\nFurthermore,  for all $u\\in {\\rm I\\kern -1.6pt{\\rm R}}^m$\n\\beqno{kappamainz} |\\mathbb{K}(u)|\\lesssim \\llg(u)|\\llg_u(u)|^{-1}.\\end{equation}\n\\end{description}\n\n\\brem{polyrem} We can see that the condition H) implies the condition A.1) in \\reftheo{GNlocalog1m}, and then \\refcoro{GNlocalog1mcoro}\nwith $\\LLg(u)=\\llg^\\frac12(u)$ and $\\Fg(u)=|\\LLg_u(u)|$, \\mref{GNlocog11mcoro} is then applicable. Indeed, the assumption \\mref{kappamain} in this case is \\mref{kappamainz}.  It is not difficult to see that the assumption in f.2) that  $|\\llg_{uu}(u)|\\llg(u)\\lesssim |\\llg_u(u)|^2$ and \\mref{kappamainz} imply $|\\Fg_u(u)||\\mathbb{K}(u)|\\lesssim \\Fg(u)$, which gives \\mref{logcondszmain} of A.1). Hence, A.1) holds by H). In particular, if $\\llg$ has a polynomial growth in $u$, i.e. $\\llg(u)\\sim (\\llg_0+|u|)^k$ for some $k\\ne0$ and $\\llg_0\\ge0$, then H) reduced to the simple condition $|\\mathbb{K}(u)|\\lesssim|u|$.\n\\erem\n\n\nFor any strong solution $u$ of \\mref{gensysfam} we will consider  the following assumptions. The exponents $s_*,\\pi_*$ are defined in \\mref{fracmu} and  in the Poincar\\'e-Sobolev inequality \\mref{PSineq}.\n\n\\begin{description} \n\n\\item[M.0)] There exist a constant $C_0$  and some $r_0>r_*:={\\pi_*}\/({\\pi_*}-2)$ such that\n\\beqno{llgmainhyp0} \\|\\llg^{-1}(u)\\|_{L^{r_*}(\\Og,\\mu)},\\;\\|\\llg^{-1}(u)f_u(u)\\|_{L^{r_0}(\\Og,\\mu)}\\le C_0,\\end{equation}\n\\beqno{llgB3}\\iidmu{\\Og}{(|f_u(u)|+\\llg(u))|Du|^{2}}\\le C_0.\\end{equation}\n\n\\item[M.1)] For any given  \n$\\mu_0>0$ \nthere is positive $R_{\\mu_0}$ sufficiently small in terms of the constants in A) and F) such that \\beqno{Keymu0} \\sup_{x_0\\in\\bar{\\Og}}\\|K(u)\\|_{BMO(B_{R}(x_0)\\cap\\Og,\\mu)}^2 \\le \\mu_0.\\end{equation}  \n\nFurthermore, for  $\\myPi_p(x):= \\llg^{p+\\frac12}(u)|\\llg_u(u)|^{-p}$ and any $p\\in[1,s_*\/2]$ there exist some $\\ag>2\/(p+2)$, $\\bg<p\/(p+2)$  such that $[\\myPi_p^{\\ag}]_{\\bg+1,B_{R_{\\mu_0}}(x_0)\\cap\\Og}\\le C_0$. \n \\end{description}\n\n\nThe main theorem of this section is the following.\n\n\\btheo{gentheo1} Assume A), F), AR) and H).  Moreover, if $\\hat{f}$ has a quadratic growth in $Du$ as in f.2) then we assume also that $n\\le3$. Suppose also that any strong solution $u$ to \\mref{gensysfam} satisfies M.0), M.1) for some constant $C_0$ and \\beqno{hypoiterpis1z00}\\iidmu{\\Og}{|f_u(u)|(1+f(u)^{s_*}|f_u(u)|^{-s_*})}\\le C_0,\\end{equation}\nuniformly in $\\sg$.  Then the system \\mref{gensys} has a strong solution. \\end{theorem}\n\nThe condition \\mref{hypoiterpis1z00} is not needed if $\\hat{f}$ is independent of $x$ (see \\refrem{hypoiterpis1zrem}). On the other hand,  we can assume a mild condition on the growth of $f(u)$ and show that \\mref{hypoiterpis1z00} can be greatly relaxed in the following result.\n\n\\bcoro{gentheo1coro} Assume as in \\reftheo{gentheo1} and in addition that \n\\beqno{specfcond}f(u)\\lesssim|f_u(u)|(1+|u|).\\end{equation} Then the conclusion of the theorem still holds if we replace \\mref{hypoiterpis1z00} by\n\\beqno{hypoiterpis1z11} \\|f_u(u)\\|_{L^1(\\Og,\\mu)}\\le C_0,\\end{equation} and assume that there exists $\\bg_0\\in(0,1)$ such that\n\\beqno{hypoiterpis1z11z}\\iidmu{\\Og}{(\\llg(u)|u|^2)^{\\bg_0}}\\le C_0.\\end{equation}\n\\end{corol}\n\nWe only discuss the existence of strong solutions in this paper so that the condition on the regularity of $\\og$ in AR) seems to be necessary. Under this condition, for $\\og_0=\\sqrt{\\og}$, the density $\\og$ clearly satisfies LM.1)-LM.2) and supports the Hardy and Poincar\\'e-Sobolev inequalities \\mref{lehr1m} and \\mref{PSineq},  with $s_*=n$ and $\\pi_*=2n\/(n-2)$. However, in the proof of this section, we prefer to keep the notations $s_*,\\pi_*$ as in \\refsec{GNsec} because many results in this section will be applicable in our future works where we will consider systems which are degenerate and singular in $u,x$ altogether, see \\refrem{murem1}.\n\nThe proof of \\reftheo{gentheo1} relies on the Leray Schauder fixed point index theorem. We obtain the existence of a strong solution $u$ of \\mref{gensys} as a fixed point of a nonlinear map defined on an appropriate Banach space $\\mX$. The proof will be based on several lemmas and we will sketch the main steps below.\n\n\nWe will show in \\reflemm{dleANSpropt0} that there exist  $p>s_*\/2$ and a constant $M_*$ depending only on the constants in A) and F) such that any strong solution $u$ of \\mref{gensysfam} will satisfy \\beqno{M*def} \\|Du\\|_{L^{2p}(\\Og,\\mu)}\\le M_*.\\end{equation}  This and \\refrem{PSremH} imply that there are positive constants $\\ag,M_0$ such that \\beqno{M0def} \\|u\\|_{C^{\\ag}(\\Og)}\\le M_0.\\end{equation}\n\nFor $\\sg\\in[0,1]$ and any $u\\in {\\rm I\\kern -1.6pt{\\rm R}}^m$ and $\\zeta\\in{\\rm I\\kern -1.6pt{\\rm R}}^{mn}$ we  define  the vector valued functions $F^{(\\sg)}$ and $f^{(\\sg)}$ by \\beqno{Bfdef}F^{(\\sg)}(x,u,\\zeta):=\\int_{0}^{1}\\partial_\\zeta F(\\sg,u,t\\zeta)\\,dt,\\quad f^{(\\sg)}(x,u):=\\int_{0}^{1}\\partial_u F(\\sg,x,tu,0)\\,dt.\\end{equation}\n\n\n\n  \n For   any given $u,w\\in W^{1,2}(\\Og)$ we write \\beqno{Bfdefalin}\\mathbf{\\hat{f}}(\\sg,x, u,w)=F^{(\\sg)}(x,u,Du)Dw+f^{(\\sg)}(x,u)w+\\hat{f}(x,0,0).\\end{equation}\n\n\nFor any given $u\\in W^{1,2}(\\Og)$ satisfying \\mref{M0def} we consider the following linear systems, noting that $\\mathbf{\\hat{f}}(\\sg,x,u,w)$ is linear in $w,Dw$\n\\beqno{Tmapdef}\\left\\{\\begin{array}{l} -\\Div(A(x,\\sg u)Dw)+Lw=\\mathbf{\\hat{f}}(\\sg,x,u,w)+Lu\\quad x\\in \\Og, \\\\\\mbox{$w=0$ or $\\frac{\\partial w}{\\partial \\nu}=0$ on $\\partial \\Og$}. \\end{array}\\right.\\end{equation}\nHere, $L$ is a suitable positive definite matrix depending on the constant $M_0$ such that the above system has a unique weak solution $w$ if $u$ satisfies \\mref{M0def}. We then define $T_\\sg(u)=w$ and apply the Leray-Schauder fixed point theorem to establish the existence of a fixed point of $T_1$.\nIt is clear from \\mref{Bfdefalin} that $\\hat{f}(x,\\sg u,\\sg Du)=\\mathbf{\\hat{f}}(\\sg,x,u,u)$.  \nTherefore,  from  the definition of $T_\\sg$ we see that a fixed point of $T_\\sg$ is a weak solution of \\mref{gensysfam}. By an appropriate choice of $\\mX$, we will show that these fixed points are strong solutions of \\mref{gensysfam}, and so a fixed point of $T_1$ is a strong solution of \\mref{gensys}.\n\n\nFrom the proof of Leray-Schauder fixed point theorem in \\cite[Theorem 11.3]{GT}, we need to  find some ball $B_M$ of radius $M$ and centered at $0$ of $\\mX$ such that $T_\\sg: \\bar{B}_M\\to \\mX$ is compact and  that $T_\\sg$ has no fixed point on the boundary of $B_M$.  The topological degree $\\mbox{ind}(T_\\sg, B_M)$ is then well defined and invariant by homotopy so that $\\mbox{ind}(T_1, B_M)=\\mbox{ind}(T_0, B_M)$. It is easy to see that the latter is nonzero because the linear system $$\\left\\{\\begin{array}{l} -\\Div(A(x,0)Du)=\\mathbf{\\hat{f}}(x,0,0)\\quad x\\in \\Og, \\\\\\mbox{$u=0$ or $\\frac{\\partial u}{\\partial \\nu}=0$ on $\\partial \\Og$}, \\end{array}\\right.$$ has a unique solution in $B_M$. Hence, $T_1$ has a fixed point in $B_M$.\n\nTherefore, the theorem is proved as we will establish the following claims.\n\n\\begin{description} \\item[Claim 1] There exist a Banach space $\\mX$ and $M>0$ such that the map $T_\\sg:\\bar{B}_M\\to\\mX$ is well defined and compact. \n\\item[Claim 2] $T_\\sg$ has no fixed point on the boundary of $\\bar{B}_M$. That is,  $\\|u\\|_\\mX< M$ for any fixed points of $u=T_\\sg(u)$.\n\\end{description}\n\n\nThe following lemma establishes Claim 1.\n\n\\blemm{Tmaplem} Suppose that there exist $p>s_*\/2$ and a constant $M_*$  such that any strong solution $u$ of \\mref{gensysfam} satisfies  \\beqno{M*defz} \\|Du\\|_{W^{1,2p}(\\Og,\\mu)}\\le M_*.\\end{equation} Then, there exist  $M,\\bg>0$  such that for $\\mX=C^{\\bg}(\\Og)\\cap W^{1,2}(\\Og)$ the map $T_\\sg:\\bar{B}_M\\to\\mX$ is well defined and compact for all $\\sg\\in[0,1]$. Moreover, $T_\\sg$ has no fixed points on $\\partial B_M$. \\end{lemma}\n\n\n{\\bf Proof:~~} \nFor some constant $M_0>0$ we consider $u:\\Og_{R}\\to{\\rm I\\kern -1.6pt{\\rm R}}^m$ satisfying \n\\beqno{Xdefstartzz} \\|u\\|_{C(\\Og)}\\le M_0,\\; \\|Du\\|_{L^2(\\Og)}\\le M_0,\\end{equation}\nand write the system \\mref{Tmapdef} as a linear elliptic system for $w$\n\\beqno{LSUsys} -\\Div(\\mathbf{a}(x) Dw) +\\mathbf{b}(x)Dw+\\mathbf{g}(x)w+Lw =\\mathbf{f}(x),\\end{equation} where $ \\mathbf{a}(x)=A(x,\\sg u)$, $\\mathbf{b}(x)=F^{(\\sg)}(x,u,Du)$, $\\mathbf{g}(x)=f^{(\\sg)}(x,u)$, and $\\mathbf{f}(x)=\\hat{f}(x,0,0)+Lu$.\n\nThe matrix $\\mathbf{a}(x)$ is then regular elliptic with uniform ellipticity constants by A), AR) because $u$ is bounded. From the theory of {\\em linear} elliptic systems it is well known that if the operator $\\mathcal{L}(w)=-\\Div(\\mathbf{a}(x) Dw) +\\mathbf{g}(x)w +Lw$ is monotone and there exist positive constants $m$ and $q$ such that  \\beqno{LSUcond}\\|\\mathbf{b}\\|_{L^q(\\Og)},\\; \\|\\mathbf{g}\\|_{L^q(\\Og)},\\; \\|\\mathbf{f}\\|_{L^q(\\Og)}\\le m,\\; \\mbox{$q> n\/2$},\\end{equation} then the system \\mref{LSUsys}  has a unique weak solution $w$. \n\nIt is easy to find  a matrix $L$ such that $\\mathcal{L}(w)$ is monotone. Because the matrix $\\mathbf{a}$ is regular elliptic and $\\mathbf{g}$ is bounded (see below). We just need to choose a positive definite matrix $L$ satisfying $\\myprod{Lw,w}\\ge l_0|w|^2$ for some $l_0>0$ and sufficiently large in terms of $M_0$.\n\nNext, we will show that \\mref{LSUcond} holds  by F) and \\mref{Xdefstartzz}. We consider the two cases  f.1) and  f.2). If f.1) holds then from the definition \\mref{Bfdef} there is a constant $C(|u|)$ such that  $$|\\mathbf{b}(x)|=|F^{(\\sg)}(x,u,\\zeta)|\\le C(|u|),\\; |\\mathbf{g}(x)|=|f^{(\\sg)}(x,u)|\\le C(|u|).$$\n \n From  \\mref{Xdefstartzz}, we see that $\\|u\\|_\\infty\\le M_0$ and so there is  a constant $m$ depending on $M_0$ such that \\mref{LSUcond} holds for any $q$ and $n$.\n \n \nIf f.2) holds then\n\\beqno{Bfdeff2}|F^{(\\sg)}(x,u,\\zeta)|\\le C(|u|)|\\zeta|,\\; |f^{(\\sg)}(x,u)|\\le C(|u|).\\end{equation} Therefore, $\\|\\mathbf{b}\\|_{L^2(\\Og)}$ is bounded by $C\\|Du\\|_{L^2(\\Og)}$. Again, if  $n\\le3$ then  \\mref{Xdefstartzz} implies the condition \\mref{LSUcond} for $q=2$.\n\n \n\nIn both cases,  \\mref{LSUsys} (or \\mref{Tmapdef}) has a unique weak solution $w$. \nWe then define $T_\\sg(u)=w$. Moreover, from the regularity theory of linear systems, $w\\in C^{\\ag_0}(\\Og)$ for some $\\ag_0>0$ depending on $M_0$.\n\nThe bound in the assumption \\mref{M*defz}  and \\refrem{PSremH} imply that $u$ is H\\\"older continuous and provide positive constants $\\ag, C(M_*)$ such that $\\|u\\|_{C^{\\ag}(\\Og)}\\le C(M_*)$. Also, the assumption \\mref{llgB3} and AR), that $\\llg(u),\\og$ are bounded from below, yield that $\\|Du\\|_{L^2(\\Og)}\\le C(C_0)$. Thus, there is a constant $M_1$, depending on $M_*,C_0$ such that any strong solution $u$ of \\mref{gensysfam} satisfies \n\\beqno{Xdefstart} \\|u\\|_{C^{\\ag}(\\Og)}\\le M_1,\\; \\|Du\\|_{L^2(\\Og)}\\le M_1.\\end{equation}\n\nIt is well known that there is a constant $c_0>1$, depending on $\\ag$ and the diameter of $\\Og$, such that $\\|\\cdot\\|_{C^{\\bg}(\\Og)}\\le c_0\\|\\cdot\\|_{C^{\\ag}(\\Og)}$ for all $\\bg\\in(0,\\ag)$. We now let $M_0$, the constant in \\mref{Xdefstartzz}, be $M=(c_0+1)M_1$ and define $\\ag_0$ in the previous argument accordingly. \n\nDefine $\\mX=C^{\\bg}(\\Og)\\cap W^{1,2}(\\Og)$ for some positive $\\bg<\\min\\{\\ag,\\ag_0\\}$. The space $\\mX$ is equipped with the norm $$\\|u\\|_\\mX = \\max\\{\\|u\\|_{C^{\\bg}(\\Og)},\\|Du\\|_{L^2(\\Og)}\\}.$$ \n \n\nWe now see that $T_\\sg$ is well defined and maps the ball $\\bar{B}_M$ of $\\mX$ into $\\mX$. Moreover, from the definition $M=(c_0+1)M_1$, it is clear that $T_\\sg$ has no fixed point on the boundary of $B_M$ because such a fixed points $u$ satisfies \\mref{Xdefstart} which implies $\\|u\\|_{\\mX}\\le c_0M_1<M$. \n \nFinally, we need only show that $T_\\sg$ is compact.  If $u$ belongs to a bounded set $K$ of $\\bar{B}_M$ then $\\|u\\|_\\mX\\le C(K)$ for some constant $C(K)$ and there is a constant $C_1(K)$  such that $\\|w\\|_{C^{\\ag_0}(\\Og)}\\le C_1(K)$. Thus $T_\\sg(K)$ is compact in $C^{\\bg}(\\Og)$ because $\\bg<\\ag_0$.  So, we need only show that $T(K)$ is precompact in $W^{1,2}(\\Og)$.  We will discuss only the quadratic growth case where \\mref{Bfdeff2} holds because the case $\\hat{f}$ has linear growth is similar and easier.\n \n \nFirst of all, for $u\\in K$ we easily see that $\\|Dw\\|_{L^2(\\Og)}$ is uniformly bounded by a constant depending on $K$. The argument is standard by testing the linear system \\mref{Tmapdef} by $w$ and using the boundedness of $\\|w\\|_{L^\\infty}$ and $\\|u\\|_{L^\\infty}$, \\mref{Bfdeff2}, AR) and Young's inequality.\n\n\nLet $\\{u_n\\}$ be a sequence in $K$ and $w_n=T_\\sg(u_n)$.  We have, writing $W=w_n-w_m$  $$-\\Div(A(x,\\sg u_n)DW)=\\Div((A(x,\\sg u_n)-A(x,\\sg u_m)Dw_m)+\\Psi_{m,n},$$ where $\\Psi_{m,n}$ is defined by $$\\begin{array}{lll}\\Psi_{m,n}&=&F^{(\\sg)}(x,u_n,Du_m)Dw_n-F^{(\\sg)}(x,u_m,Du_m)Dw_m+\\\\&&f^{(\\sg)}(x,u_n)u_n-f^{(\\sg)}(x,u_m)u_m +L(u_n-u_m+w_m-w_n).\\end{array}$$\n\nTesting the above system with $W$ and using AR), we have\n$$\\llg_*(K)\\mu_*\\iidx{\\Og}{|DW|^2} \\le \\iidx{\\Og}{[|A(x,\\sg u_n)-A(x,\\sg u_m)||Dw_m||DW|+|\\Psi_{m,n}||W|]}.$$\nBy Young's inequality, we find a constant $C$ depending on $K$ and $\\mu_*$ such that\n$$\\iidx{\\Og}{|DW|^2}\\le C\\iidx{\\Og}{[|A(x,\\sg u_n)-A(x,\\sg u_m)||Dw_m|^2} +\\sup_\\Og|W|\\|\\Psi_{m,n}\\|_{L^1(\\Og)}.$$\nBy \\mref{Bfdeff2}, it is clear that   $|\\Psi_{n,m}|\\le C(K)[(|Du_n|+|Du_m|)(|Dw_n|+|Dw_m|)+1]$.\nUsing the fact that $\\|Dw_n\\|_{L^2(\\Og)}$ and $\\|Du_n\\|_{L^2(\\Og)}$ are uniformly bounded, we see that $\\|\\Psi_{m,n}\\|_{L^1(\\Og)}$ is bounded. Hence, $$\\iidx{\\Og}{|Dw_n-Dw_m|^2}\\le C(K)\\max\\{\\sup_\\Og|A(x,\\sg u_n)-A(x,\\sg u_m)|,\\sup_\\Og|w_n-w_m|\\}.$$\n\nSince $u_n,w_n$ are bounded in $C^\\bg(\\Og)$, passing to subsequences we can assume that $u_n,w_n$ converge in $C^0(\\Og)$. Thus, $\\|A(x,\\sg u_n)-A(x,\\sg u_m)\\|_\\infty,\\|w_n-w_m\\|_\\infty\\to0$. We then see from the above estimate that $Dw_n$ converges in $L^{2}(\\Og)$. Thus, $T_\\sg(K)$ is precompact in $W^{1,2}(\\Og)$.\n\n\nHence, \n$T_\\sg:\\mX\\to\\mX$ is a compact map. The proof is complete. \\begin{flushright} Q.E.D. \\end{flushright}\n\nWe now turn to Claim 2, the hardest part of the proof, and provide a uniform estimate for the fixed points of $T_\\sg$ and justify the key assumption \\mref{M*defz} of \\reflemm{Tmaplem}. To this end, we first have the following lemma. \n\n\\blemm{Dulocbound} A fixed point of $T_\\sg$ is also a strong solution of \\mref{gensysfam}. \n \\end{lemma}\n \nThe  lemma can easily follow from the results in \\cite{Gius}. However, there are some remarks need be made here as we are also considering quadratic growth $\\hat{f}$ in this paper and our system is not necessarily variational.\n\n\\brem{Dulocboundrem} If $u$ is a fixed point of $T_\\sg$ in $\\mX$ then it solves \\mref{gensysfam} weakly and is continuous. Thus, $u$ is bounded and belongs to $VMO(\\Og)$. By AR), the system \\mref{gensysfam} is regular elliptic. We can adapt the proof in \\cite{GiaS}, which deals with parabolic systems, to our elliptic case.\nIf $\\hat{f}$ satisfies a quadratic growth in $Du$ then,  because $u$ is bounded, the condition \\cite[(0.4)]{GiaS} that $|\\hat{f}|\\le a|Du|^2+b$ is satisfied here. The proof of \\cite[Theorems 2.1 and 3.2]{GiaS} assumed the 'smallness condition' (see \\cite[(0.6)]{GiaS}) $2aM<\\llg_0$, where $M=\\sup |u|$. This 'smallness condition' was needed because only weak bounded solutions, which are not necessarily continuous, were considered in \\cite{GiaS}. In our case,  $u$ is continuous so that we do not require this 'smallness condition'.  Indeed, a careful checking of the arguments of the proof in \\cite[Lemma 2.1 and page 445]{GiaS} shows that if $R$ is small and one knows that the solution $u$ is continuous then these argument still hold as long we can absorb the integrals involving $|Du|^2, |Dw|^2$ (see the estimate after \\cite[(3.7)]{GiaS}) on the right hand sides to the left right hand sides of the estimates. Thus, \\cite[Theorems 2.1 and 3.2]{GiaS} apply to our case and yield that $u\\in C^{a}(\\Og)$ for all $a\\in(0,1)$ and that, since $A(x,u)$ is differentiable,  $Du$ is locally H\\\"older continuous in $\\Og$. Therefore, $u$ is also a strong solution (see \\cite[Chapter 10]{Gius}).\\erem\n\n\nThanks to \\reflemm{Dulocbound}, we need only consider a strong solution $u$ of \\mref{gensysfam} and establish \\mref{M*defz} for some $p>s_*\/2$. Because the data of \\mref{Tmapdef} satisfy the structural conditions A), F) with the same set of constants and the assumptions of the theorem are assumed to be uniform for all $\\sg\\in[0,1]$, we will only present the proof for the case $\\sg=1$ in the sequel.\n\n\n\nLet $u$ be a strong solution  of \\mref{e1} on $\\Og$.  We begin with an energy estimate for $Du$. For $p\\ge1$ and any ball $B_s$ with center $x_0\\in\\bar{\\Og}$ we denote $\\Og_s=B_s\\cap\\Og$ and \n\\beqno{Hdef}\\ccH_{p}(s):=\n\\iidmu{\\Og_s}{\\llg(u)|Du|^{2p-2}|D^2u|^2},\\end{equation}\n\\beqno{Bdef}\\ccB_{p}(s):= \\iidmu{\\Og_s}{\\frac{|\\llg_u(u)|^2}{\\llg(u)}|Du|^{2p+2}},\\end{equation} \\beqno{Cdef}\\ccC_{p}(s):=\\iidmu{\\Og_s}{(|f_u(u)|+\\llg(u))|Du|^{2p}},\\end{equation} and \\beqno{cIdef}\\mccIee_{\\og,p}(s):=\\iidx{B_s}{(\\llg(u)|Du|^{2p}|D\\og_0|^2+|f(u)||Du|^{2p-1}|D\\og_0|\\og_0)}.\\end{equation}\n\n\n\n\\blemm{dleANSenergy}  Assume A), F). \nLet $u$ be any strong solution  of \\mref{gensys} on $\\Og$ and  $p$ be any number in $[1,\\max\\{1,s_*\/2\\}]$.\n\nThere is a constant $C$, which depends only on the parameters in A) and F), such that for any two concentric balls $B_s,B_t$ with center $x_0\\in\\bar{\\Og}$ and $s<t$ \\beqno{keydupANSenergy}\\ccH_{p}(s)\\le C\\ccB_{p}(t)+C(1+(t-s)^{-2})[\\ccC_{p}(t)+\\mccIee_{\\og,p}(t)].\\end{equation}\n\n\n\\end{lemma}\n\n{\\bf Proof:~~} The proof is similar to the energy estimate of $Du$ for the parabolic case in \\cite[Lemma 3.2]{dleANS}. As we consider $A, \\hat{f}$ depending on $x$, there are some extra terms in the estimate which need some extra attention.\nWe follow the proof of  \\cite[Lemma 3.2]{dleANS} with $W=U=u$ and $\\bg(u)=1$. Roughly speaking, we differentiated the system in $x$ to obtain\n\\beqno{ga2zzz} -\\Div(A(x,u)D^2u+A_u(x,u)DuDu+A_x(x,u)Du)=D\\hat{f}(x,u,Du).\\end{equation} \n\n\nFor any two concentric balls $B_s,B_t$, with $s<t$, let $\\psi$ be a cutoff function for $B_s,B_{t}$. That is, $\\psi$ is a $C^1$ function satisfying $\\psi\\equiv1$ in $B_s$ and $\\psi\\equiv0$ outside $B_t$ and $|D\\psi|\\le 1\/(t-s)$. We then\ntest \\mref{ga2zzz} with $|Du|^{2p-2}Du\\psi^2(x)$ and obtain, using integration by parts and Young's inequality\n\\beqno{keyenergystart}\\begin{array}{ll}\\lefteqn{\n\\iidx{\\Og_t}{\\llg(u)|Du|^{2p-2}|D^2u|^2\\psi^2\\og}\\le  C\\iidx{\\Og_t}{[\\frac{|\\llg_u(u)|^2}{\\llg(u)}|Du|^{2p+2}+|D\\psi|^{2}\\llg(u)|Du|^{2p}]\\og}}\\hspace{1cm}&\\\\& +\\iidx{\\Og_t}{[|A_x(x,u)||Du|^{2p-1}|D^2u|\\psi^2+C(p)|D\\hat{f}(x,u,Du)||Du|^{2p-1}\\psi^2]}.\\end{array}\\end{equation}\n\n\n\nHere, integrals in the first line of \\mref{keyenergystart} result from the same argument in the proof of \\cite[Lemma 3.2]{dleANS} using the spectral gap condition SG) we are assuming here (see also \\reflemm{SGrem} in the Appendix).  Meanwhile, the integral in the last line comes from the presence of $x$ in $A,\\hat{f}$ and we will discuss it below. \n\nWe consider the integrand $|A_x(x,u)||Du|^{2p-1}|D^2u|\\psi^2$ on the right hand side of \\mref{keyenergystart}. As $|A_x(x,u)|\\le C_*\\llg(u)|D\\og|$ and $\\og=\\og_0^2$, we have to deal with the integral $$C_*\\iidx{\\Og}{\\llg(u)|Du|^{2p-1}|D^2u|\\psi^2|D\\og_0|\\og_0}.$$\nBy Young's inequality this integral can be estimated by   \\beqno{Axest2} \\frac12\\iidx{\\Og}{ \\llg(u)|Du|^{2p-2}|D^2u|^2\\psi^2\\og_0^2}+\\frac12C_*^2 \\iidx{\\Og}{\\llg(u)|Du|^{2p}\\psi^2|D\\og_0|^2}.\\end{equation} \n\n\n\n\n\nNext, we consider the integral of $|D\\hat{f}(x,u,Du)||Du|^{2p-1}\\psi^2$. First, if $\\hat{f}$ has a linear growth in $Du$ then by f.1) in F) with $p=Du$, $$|D\\hat{f}(x,u,Du)| \\lesssim  (\\llg(u)|D^2u|+ |\\llg_u(u)||Du|^2)\\og+\\llg(u)|Du||D\\og|+ |f_u(u)||Du|\\og +|f(u)||D\\og|.$$ Therefore, using   Young's inequality,  we get\n\\beqno{f1duest}\\begin{array}{ll}\\lefteqn{|D\\hat{f}(x,u,Du)||Du|^{2p-1}\\lesssim\\llg(u)|Du|^{2p}|D\\og_0|^2+|f(u)||Du|^{2p-1}|D\\og_0|\\og_0+}\\hspace{.2cm}&\\\\& \\left\\{\\eg\\llg(u)|Du|^{2p-2}|D^2u|^2+ C(\\eg)\\llg(u)|Du|^{2p}+\\frac{|\\llg_u(u)|^2}{\\llg(u)}|Du|^{2p+2}+|f_u(u)||Du|^{2p}\\right\\}\\og_0^2.\\end{array}\\end{equation}\n\n\nSimilarly, if $\\hat{f}$ has a quadratic growth in $Du$ then by f.2) in F) $$\\begin{array}{lll}|D\\hat{f}(x,u,Du)| &\\lesssim&  (|\\llg_u(u)||Du||D^2u|+ |\\llg_{uu}(u)||Du|^3)\\og+ |\\llg_u||Du|^2|D\\og|+\\\\&&\n|f_u(u)||Du|\\og +|f(u)||D\\og|.\\end{array}$$\n\nWe then have to deal with three extra terms which can be handled by  Young's inequality and the assumption \\mref{llgquadcond} that $|\\llg_{uu}(u)|\\llg(u)\\lesssim |\\llg_u(u)|^2$ as follows.\n$$|\\llg_u(u)||Du||D^2u||Du|^{2p-1}\\le \\eg\\llg(u)|Du|^{2p-2}|D^2u|^2 + C(\\eg)\\frac{|\\llg_u(u)|^2}{\\llg(u)}|Du|^{2p+2},$$\n$$|\\llg_u||Du|^2|D\\og||Du|^{2p-1}\\lesssim \\frac{|\\llg_u(u)|^2}{\\llg(u)}|Du|^{2p+2}\\og+\\llg(u)|Du|^{2p}|D\\og|^2\\og^{-1},$$\n$$|\\llg_{uu}(u)||Du|^3|Du|^{2p-1}\\lesssim \\frac{|\\llg_u(u)|^2}{\\llg(u)}|Du|^{2p+2}.$$ We then get the same terms as in \\mref{f1duest} for the linear growth case. \n\nFinally, we use the definitions \\mref{Hdef}-\\mref{cIdef} to see that the integral in the last line of \\mref{keyenergystart} can be estimated by\n\\beqno{energyDu}(\\eg+\\frac12)\\iidmu{\\Og_t}{\\llg(u)|Du|^{2p-2}|D^2u|^2\\psi^2}+C(\\eg)[\\ccB_p(t)+(1+(t-s)^{-2})(\\ccC_p(t)+\\mccIee_{\\og,p}(t))].\\end{equation}\nWe then choose $\\eg$  sufficiently small  so that the first integral can be absorbed to the left hand side of \\mref{keyenergystart}. We then obtain \\mref{keydupANSenergy} and complete the proof. \\begin{flushright} Q.E.D. \\end{flushright}\n\n\n\n\n\nNext, if the condition \\mref{Keymu0} of M.1) holds then we combine the energy estimate and the local Gagliardo-Nirenberg inequality \\mref{GNlocog11mcoro} to have the following stronger estimate.\n\n\\blemm{dleANSprop}  In addition to the assumptions of \\reflemm{dleANSenergy}, we suppose that M.1) holds for some $p$. That is, for any given  \n$\\mu_0>0$ there exist a constant $C_0$ and a positive $R_{\\mu_0}$ sufficiently small in terms of the constants in A) and F) such that\n\\beqno{Keymu01} \\sup_{x_0\\in\\bar{\\Og}}[\\myPi_p^{\\ag}]_{\\bg+1,\\Og_R(x_0)}\\le C_0,\\;\\|K(u)\\|_{BMO(\\Og_{R}(x_0),\\mu)}^2 \\le \\mu_0.\\end{equation}\n\nThen for sufficiently small $\\mu_0$ there is a constant $C$ depending only on the parameters of A) and F) such that for $2R<R_{\\mu_0}$ we have \\beqno{keydupANSppb}\\ccB_{p}(R)+\\ccH_{p}(R)\\le C(1+R^{-2})[\\ccC_{p}(2R)+\\mccIee_{\\og,p}(2R)].\\end{equation}\n\n \\end{lemma}\n\n \n{\\bf Proof:~~}  Recall the energy estimate \\mref{keydupANSenergy} in \\reflemm{dleANSenergy} \n\\beqno{keyiteration1} \\ccH_{p}(s)\\le C\\ccB_{p}(t)+C(1+(t-s)^{-2})[\\ccC_{p}(t)+\\mccIee_{\\og,p}(t)], \\; 0<s<t.\\end{equation} \n\nWe apply \\refcoro{GNlocalog1mcoro} to estimate $\\ccB_{p}(t)$, the integral  on the right hand side of \\mref{keyiteration1}. We let $\\LLg(u)=\\llg^\\frac{1}{2}(u)$ in \\refcoro{GNlocalog1mcoro} and note that $\\myPi_p$ defined there is now comparable to the $\\myPi_p=\\llg^{p+\\frac12}(u)|\\llg_u(u)|^{-p}$ in M.1). We compare the definitions \\mref{Idefmt} and \\mref{I0*mt} with those in \\mref{Hdef}-\\mref{Cdef} to see that for $U(x)=u(x)$ $$\\ccB_{p}(t)=I_1(t,x_0),\\;\\ccC_{p}(t)=I_0(t,x_0),\\;\\ccH_{p}(t)=I_2(t,x_0).$$\nHence, for any $\\eg>0$ we can use \\mref{GNlocog11mcoro} obtain a constant $C$ such that (using the bound $[\\myPi_p^{\\ag}]_{\\bg+1,B_{R_{\\mu_0}}(x_0)\\cap\\Og}\\le C_0$ and the definitions of $\\mu_0$ in \\mref{Keymu01}  and $C(\\eg,U,\\myPi)$  in \\refcoro{GNlocalog1mcoro})\n$$ \\ccB_{p}(s) \\le \\eg\\ccB_{p}(t)+C\\eg^{-1}\\mu_0\\ccH_{p}(t)+C\\eg^{-1}\\mu_0(t-s)^{-2}\\ccC_{p}(t)\\quad 0<s<t\\le R_{\\mu_0}.$$\n\nDefine $F(t):=\\ccB_{p}(t)$, $G(t):=\\ccH_{p}(t)$, $g(t):=\\ccC_{p}(t)$ and $\\eg_0=\\eg+C\\eg^{-1}\\mu_0$. The above yields \\beqno{keyiteration2}F(s)\\le \\eg_0[F(t)+G(t)] +C(t-s)^{-2}g(t).\\end{equation} \n\nNow, for $h(t):=\\mccIee_{\\og,p}(t)$ the energy estimate \\mref{keyiteration1}  implies\n \\beqno{Giter}G(s)\\le C[F(t)+(1+(t-s)^{-2})(g(t)+h(t))].\\end{equation}\n\nAs $\\eg_0=\\eg+C\\eg^{-1}\\mu_0$, it is clear that we can choose and fix some $\\eg$ sufficiently small and then $\\mu_0$ small in terms of $C,\\eg$ to have $2C\\eg_0<1$. Thus, if $\\mu_0$ is sufficiently small in terms of the  constants in A),F), then we can apply a simple iteration argument \\cite[Lemma 3.11]{dleANS} to the two inequalities \\mref{keyiteration2} and \\mref{Giter} and obtain for $0<s<t\\le R_{\\mu_0}$ $$ F(s)+G(s) \\le C(1+(t-s)^{-2})[g(t)+h(t)].$$ \n\nFor any $R<R_{{\\mu_0}}\/2$ we take $t=2R$ and $s=\\frac32R$ in the above to obtain $$\\ccB_{p}(\\frac32R)+\\ccH_{p}(\\frac32R)\\le C(1+R^{-2})[\\ccC_{p}(2R)+\\mccIee_{\\og,p}(2R)].$$ Combining this and \\mref{keyiteration1} with $s=R$ and $t=\\frac32R$, we see that\n$$\\ccB_{p}(R)+\\ccH_{p}(R)\\le C(1+R^{-2})[\\ccC_{p}(2R)+\\mccIee_{\\og,p}(2R)].$$\nThis is \\mref{keydupANSppb} and the proof is complete. \\begin{flushright} Q.E.D. \\end{flushright}\n\n\n\n\n\n\n\nFinally, we have the following lemma giving a uniform bound for strong solutions.\n\n\\blemm{dleANSpropt0}  Assume as in \\reflemm{dleANSprop} and AR). \nWe assume further that there exists a constant $C_0$ such that for $r_*={\\pi_*}\/({\\pi_*}-2)$ and some $r_0>r_*$ \\beqno{llgmainhyp} \\|\\llg^{-1}(u)\\|_{L^{r_*}(\\Og,\\mu)},\\;\\||f_u(u)|\\llg^{-1}(u)\\|_{L^{r_0}(\\Og,\\mu)}\\le C_0 ,\\end{equation}\n\\beqno{hypoiterpis1}\\iidmu{\\Og}{(|f_u(u)|+\\llg(u))|Du|^{2}}\\le C_0,\\end{equation} and\n\\beqno{hypoiterpis1z}\\iidmu{\\Og}{|f_u(u)|(1+f(u)^{s_*}|f_u(u)|^{-s_*})}\\le C_0.\\end{equation}\n\nThen there exist $p>s_*\/2$ and a constant $M_*$ depending only on the parameters of A) and F), $\\mu_0$, $R_{\\mu_0}$, $C_0$ and the geometry of $\\Og$  such that \n\\beqno{keydupANSt0}\\iidmu{\\Og}{|Du|^{2p}}\\le   M_*.\\end{equation}\n \\end{lemma}\n\n\n{\\bf Proof:~~} First of all, by the condition AR), there is a constant $C_\\og$ such that $|D\\og_0|\\le C_\\og\\og_0$ and therefore we have from the the definition \\mref{cIdef} that \\beqno{cIdef1}\\mccIee_{\\og,p}(s)\\le C_\\og\\iidx{B_s}{(\\llg(u)|Du|^{2p}+f(u)|Du|^{2p-1})\\og_0^2}.\\end{equation} By Young's inequality, $f(u)|Du|^{2p-1}\\lesssim |f_u(u)||Du|^{2p}+(f(u)|f_u(u)|^{-1})^{2p}|f_u(u)|$. It follows easily from the assumption  \\mref{hypoiterpis1z} that the integral of $(f(u)|f_u(u)|^{-1})^{2p}|f_u(u)|$ is bounded by $C_0$ for any $p\\in[1,s_*\/2]$. We then have from \\mref{keydupANSppb} and \\mref{cIdef1} that\n\\beqno{keydupANSppbbb}\\ccB_{p}(R)+\\ccH_{p}(R)\\le C(1+R^{-2})[\\ccC_{p}(2R)+C_0].\\end{equation}\n\n\nThe main idea of the proof is to show that the above estimate is self-improving in the sense that if it is true for some exponent $p\\ge1$ then it is also true for $\\cg_*p$ with some fixed $\\cg_*>1$ and $R$ being replaced by $R\/2$. To this end, assume that for some $p\\ge1$ we can find a constant $C(C_0,R,p)$ such that \\beqno{piter1} \\ccC_{p}(2R)\\le C(C_0,R,p).\\end{equation} \n\nIt then clearly follows from \\mref{keydupANSppbbb}, and the definition of $\\ccB_{p}(R),\\ccH_{p}(R)$ that \\beqno{keydupANSpp1} \\iidmu{\\Og_R}{[V^2+\n|DV|^2]}\\le C(C_0,R,p), \\quad \\mbox{where $V=\\llg^\\frac12(u)|Du|^{p}$}.\\end{equation}\n\n\nLet $\\pi_*$ be the exponent in the Poincar\\'e-Sobolev inequality \\mref{PSineq}. We have \n\\beqno{genPSpi}\\iidmu{\\Og_R}{|V|^{\\pi_*}}\\lesssim  \\iidmu{\\Og_R}{|DV|^{2}}+\\mypar{\\iidmu{\\Og_R}{|V|^2}}^{{\\pi_*}\/2}.\\end{equation} We see that \\mref{keydupANSpp1} and the above inequality imply\n\\beqno{piter1az} \\iidmu{\\Og_R}{\\llg^{{\\pi_*}\/2}(u)|Du|^{p {\\pi_*}}} \\le C(C_0,R,p).\\end{equation}\n\nLet $\\cg_*\\in(1,{\\pi_*}\/2)$.  We  denote $\\bg_*=\\pi_*\/(\\pi_*-2\\cg_*)$ and $g(u)=\\max\\{|f_u(u)|,\\llg(u)\\}$. We write $g(u)|Du|^{2\\cg_* p}= \\llg(u)^{\\cg_*}|Du|^{2\\cg_* p}g(u)\\llg(u)^{-\\cg_*}$ and  and use H\\\"older's inequality and \\mref{piter1az} to get\n\\beqno{piter1z1}\\iidmu{\\Og_R}{g(u)|Du|^{2\\cg_* p}}\\le C(C_0,R,p)^\\frac{2\\cg_*}{\\pi_*}\\mypar{\\iidmu{\\Og_R}{(g(u)\\llg(u)^{-\\cg_*})^{\\bg_*}}}^{1-\\frac{2\\cg_*}{{\\pi_*}}}.\\end{equation} Again, as $(g(u)\\llg(u)^{-\\cg_*})^{\\bg_*}=(g(u)\\llg(u)^{-1})^{\\bg_*}\\llg(u)^{-(\\cg_*-1)\\bg_*}$, the last integral can be bounded via H\\\"older's inequality by $$\\mypar{\\iidmu{\\Og_R}{(g(u)\\llg(u)^{-1})^{\\bg_*\\ag}}}^\\frac1\\ag\\mypar{\\iidmu{\\Og_R}{(\\llg(u)^{-(\\cg_*-1)\\bg_*\\ag'}}}^\\frac{1}{\\ag'}$$\nBy the assumption \\mref{llgmainhyp},  $|f_u(u)|\\llg^{-1}(u)$ is in $L^r_0(\\Og,\\mu)$ for some $r_0>r_*=\\pi_*\/(\\pi_*-2)$ and $\\llg^{-1}(u)\\in L^{r_*}(\\Og,\\mu)$.  We can find $\\ag,\\cg_*>1$ such that $\\ag\\bg_*<r_0$ and $(\\cg_*-1)\\bg_*\\ag'<r_*$ and see that, from the definition $g(u)=\\max\\{|f_u(u)|,\\llg(u)\\}$, the above integrals are bounded by some constant $C(C_0)$. Hence, this fact and \\mref{piter1z1} imply\\beqno{piter1z}\\ccC_{\\cg_*p}(R)=\\iidmu{\\Og_R}{(|f_u(u)|+\\llg(u))|Du|^{2\\cg_* p}}\\le C(C_0,R,p).\\end{equation}\n\n \n\nWe just show that if \\mref{piter1} holds true for some $p\\ge 1$ and $R>0$ then \\mref{piter1z}  provides some fixed  $\\cg_{*}>1$ such that \\mref{piter1} remains true for the new exponent $\\cg_{*} p$  and $R\/2$. By the assumption \\mref{hypoiterpis1}, \\mref{piter1} holds for $p=1$. It is now clear that, as long as the energy estimate \\mref{keydupANSenergy} is valid by \\reflemm{dleANSenergy}), we can repeat the argument $k_0$ times to find  a number $p>s_*\/2$ such that \\mref{piter1} holds. It follows  that \nthere is a constant $C$ depending only on the parameters of A) and F), $\\mu_0$, $R_{\\mu_0}$ and $k_0$ \nsuch that for some $p>s_*\/2$\n\\beqno{keydupANSt0z}\\iidmu{\\Og_{R_0}}{\\llg^{{\\pi_*}\/2}(u)|Du|^{{\\pi_*}p}}\\le   C\\mbox{ for  $R_0=2^{-k_0}R_{\\mu_0}$}.\\end{equation}\n\nWe now write $|Du|^{2p}=\\llg(u)|Du|^{2p}\\llg^{-1}(u)$ and have\n$$\\iidmu{\\Og_{R_0}}{|Du|^{2p}}\\le \\mypar{\\iidmu{\\Og_{R_0}}{\\llg^{{\\pi_*}\/2}(u)|Du|^{{\\pi_*}p}}}^\\frac{2}{{\\pi_*}}\\mypar{\\iidmu{\\Og_{R_0}}{\\llg(u)^{-{{\\pi_*}\/({\\pi_*}-2)}}}}^{1-\\frac{2}{{\\pi_*}}}.$$ By \\mref{llgmainhyp},  the last integral in the above estimate is bounded.\nUsing \\mref{keydupANSt0z} and summing the above inequalities over a finite covering of balls $B_{R_0}$ for $\\Og$, we find a constant $C$, depending also on the geometry of $\\Og$, and obtain the desired estimate \\mref{keydupANSt0}. The lemma is proved. \\begin{flushright} Q.E.D. \\end{flushright}\n\n\\brem{hypoiterpis1zrem} The condition \\mref{hypoiterpis1z} is void if $\\hat{f}$ is independent of $x$. Indeed, it was used only to estimate $\\mccIee_{\\og,p}$, which results from \\mref{Giter} in the proof of \\reflemm{dleANSprop}, and obtain \\mref{keydupANSppbbb}. If $\\hat{f}$ is independent of $x$ then $\\mccIee_{\\og,p}=0$ from the proof of the energy estimate in \\reflemm{dleANSenergy} (see \\mref{f1duest}) so that \\mref{hypoiterpis1z} is not needed. \\erem\n\n\n\n\n\\brem{murem1} It is also important to note that the estimate of \\reflemm{dleANSpropt0}, based on those in  \\reflemm{dleANSenergy}, \\reflemm{dleANSprop}, is  {\\em independent} of lower\/upper bounds of the function $\\llg_*$ in AR) but the integrals in M.0). The assumption AR) was used only in \\reflemm{Tmaplem} to define the map $T_\\sg$ and \\reflemm{Dulocbound} to show that fixed points of $T_\\sg$ are strong solutions. \\erem\n\n\n\n\nWe are ready to provide the proof of the main theorem of this section. \n\n{\\bf Proof of \\reftheo{gentheo1}:} \nIt is now clear that the assumptions M.0) and M.1) of our theorem allow us to\napply \\reflemm{dleANSpropt0} and obtain a priori uniform bound  for any continuous strong solution $u$ of \\mref{gensysfam}. The uniform estimate \\mref{keydupANSt0} shows that the assumption \\mref{M*defz} of \\reflemm{Tmaplem} holds true so that the map $T_\\sg$ is well defined and compact on a ball $\\bar{B}_M$ of $\\mX$ for some $M$ depending on the bound $M_*$ provided by \\reflemm{dleANSpropt0}. Combining with \\reflemm{Dulocbound}, the fixed points of $T_\\sg$ are strong solutions of the system \\mref{gensysfam} so that $T_\\sg$ does not have a fixed point on the boundary of $\\bar{B}_M$. Thus, by the Leray-Schauder fixed point theorem, $T_1$ has a fixed point in $B_M$ which is a strong solution to \\mref{gensys}. The proof is complete.\n\\begin{flushright} Q.E.D. \\end{flushright}\n\n{\\bf Proof of \\refcoro{gentheo1coro}:} We just need to show that \\reflemm{dleANSpropt0} remains true with the condition \\mref{hypoiterpis1z}, which is \\mref{hypoiterpis1z00}, being replaced by \\mref{hypoiterpis1z11} and \\mref{hypoiterpis1z11z}. We revisit the proof of \\reflemm{dleANSpropt0}. First of all,  we use Young's inequality in the estimate \\mref{cIdef1} for $\\mccIee_{\\og,p}(s)$ to get $f(u)|Du|^{2p-1}\\lesssim |f_u(u)||Du|^{2p}+(f(u)|f_u(u)|^{-1})^{2p}|f_u(u)|$. Using \\mref{specfcond} and \\mref{hypoiterpis1z11},  \\mref{keydupANSppbbb} now yields $$\\ccB_{p}(R)+\\ccH_{p}(R)\\le C(1+R^{-2})[\\ccC_{p}(2R)+C_0+I_p(2R)], \\quad I_p(s)=\\iidmu{\\Og_s}{|u|^{2p}|f_u(u)|}.$$ \n\nAs in \\mref{piter1}, using the same idea, we will first assume for some $p\\ge1$ that \\beqno{piter111} \\ccC_{p}(2R)+I_p(2R)\\le C(C_0,R,p)\\end{equation} and show that this assumption is self-improving, i.e., if it holds for some $p\\ge1$ then it remains true for $\\cg_*p$ for some fixed $\\cg_*>1$ and $2R$ being replaced by $R$. We need only consider $I_p(R)$ and assume first that \\beqno{improvep1} \\iidmu{\\Og_R}{(|f_u(u)|+\\llg(u))|u|^{2}}\\le C(C_0,R).\\end{equation}\n\nDenote $g(u)=|f_u(u)|$ and $\\cg_*=2-\\frac{1}{r_0}-\\frac{2}{\\pi_*}$. Since $r_0>\\frac{\\pi_*}{\\pi_*-2}$ it is clear that $\\cg_*>1$. We then define $r_1=\\frac{\\pi_*}{(\\cg_*-1)(\\pi_*-2)}$, $r_2=\\frac{\\pi_*}{2\\cg_*}$ and note that\n$$\\frac{1}{r_0}+\\frac{1}{r_1}+\\frac{1}{r_2}=\\frac{1}{r_0}+\\frac{(\\cg_*-1)(\\pi_*-2)}{\\pi_*}+\\frac{2\\cg_*}{\\pi_*}=\\frac{1}{r_0}+\\cg_*-1+\\frac{2}{\\pi_*}=1.$$ Therefore, writing\n$g(u)|u|^{2p\\cg_*}=g(u)\\llg^{-1}(u)\\llg^{1-\\cg_*}(u)\\llg^{\\cg_*}|u|^{2p\\cg_*}$, we can use  H\\\"older's inequality to see that $I_{p\\cg_*}(R)$ can be bounded by \\beqno{fullgup}\\mypar{\\iidmu{\\Og_R}{(g(u)\\llg(u)^{-1})^{r_0}}}^\\frac{1}{r_0}\\mypar{\\iidmu{\\Og_R}{\\llg(u)^{-\\frac{\\pi_*}{\\pi_*-2}}}}^\\frac{1}{r_1}\\mypar{\\iidmu{\\Og_R}{\\llg(u)^{\\frac{\\pi_*}{2}}|u|^{p\\pi_*}}}^\\frac{1}{r_2}.\n\\end{equation}\nThanks to \\mref{llgmainhyp}, the first two integrals are bounded by $C(C_0)$. We estimate the third integral. Let $U=\\llg(u)^{\\frac{\\pi_*}{4}}|u|^\\frac{p\\pi_*}{2}$. Because $|\\llg_u(u)||u|\\lesssim\\llg(u)$, $|DU|^2\\lesssim \\llg(u)^{\\frac{\\pi_*}{2}}|u|^{p\\pi_*-1}|Du|$. By Poincar\\'e and Young's inequalities, we have for any $\\bg\\in(0,1]$\n$$\\iidmu{\\Og_R}{U^2}\\le R^2\\iidmu{\\Og_R}{\\llg(u)^{\\frac{\\pi_*}{2}}(\\eg|u|^{p\\pi_*}+C(\\eg)|Du|^{p\\pi_*})}+R^{n-n\/\\bg}\\mypar{\\iidmu{\\Og_R}{U^{2\\bg}}}^\\frac{1}{\\bg}.$$\n\nChoosing $\\eg$ small and $\\bg=2\/\\pi_*$, we see that the right hand side is bounded by $$ CR^2\\iidmu{\\Og_R}{\\llg(u)^{\\frac{\\pi_*}{2}}|Du|^{p\\pi_*}}+L_p(R)^\\frac{\\pi_*}{2},\\; \\mbox{where } L_p(s)=\\iidmu{\\Og_s}{\\llg(u)|u|^{2p}}.$$\n\nPutting these estimates together and using \\mref{piter1az}, which holds because of \\mref{piter111}, we see that the third integral in \\mref{fullgup} is bounded by a constant $C(C_0,R,p)$ if $L_p(R)\\le C(C_0,R,p)$. We then have $I_{\\cg_* p}(R)\\le C(C_0,R,p)$.\n\nOn the other hand, we can show that the estimate $L_p(R)\\le C(C_0,R,p)$ is also self improving. We repeat the argument in \\mref{fullgup} with $g(u)=\\llg(u)$ and, of course, $r_0=\\infty$ to see that if $L_p(R)\\le C(C_0,R,p)$ then $L_{\\cg_* p}(R)\\le C(C_0,R,p)$ for $\\cg_*=2-\\frac{2}{\\pi_*}>1$ because $\\pi_*>2$.\n\nHence, the estimate \\mref{piter111} remains true for $p,2R$ being replaced by $\\cg_*p,R$ respectively for some fixed $\\cg_*>1$. The proof of \\reflemm{dleANSpropt0} can continue with \\mref{improvep1} replacing \\mref{hypoiterpis1z}.\n\nFinally, we show that \\mref{improvep1} comes from the assumptions \\mref{hypoiterpis1} and \\mref{hypoiterpis1z11}. From \\mref{fullgup} with $p=1$, we see that $I_1(R)$ can be estimated in terms of the integral of $\\llg(u)^{\\frac{\\pi_*}{2}}|u|^{\\pi_*}$ over $\\Og_R$. The latter can be estimated by $L_1(R)$ via the Poincar\\'e-Sobolev inequality \\mref{genPSpi}, using $V=\\llg(u)^{\\frac{1}{2}}|u|$ and the assumption on the integrability of $|DV|^2\\sim\\llg(u)|Du|^2$ in \\mref{hypoiterpis1}. Hence, we need only consider $L_1(R)$. By the interpolation inequality, we have $$\\iidmu{\\Og_R}{\\llg(u)|u|^{2}}\\le R^2\\iidmu{\\Og_R}{\\llg(u)|Du|^2}+R^{n-n\/\\bg}\\mypar{\\iidmu{\\Og_R}{(\\llg(u)|u|^{2})^\\bg}}^\\frac{1}{\\bg}.$$ We then use \\mref{hypoiterpis1} and \\mref{hypoiterpis1z11} to see that $L_1(R)\\le C(C_0,R)$ and conclude the proof. \\begin{flushright} Q.E.D. \\end{flushright}\n\n\n\\section{Proof of the main results} \\label{proofmainsec} \\eqnoset\n\nWe now present the proof of the results in \\refsec{res} which are in fact just applications of the main technical theorem with different choices of the map $K$ verifying the conditions H) and M.1). Again, since the systems in \\mref{e1famzzz} satisfy the same set of conditions uniformly with respect to $\\sg\\in[0,1]$ we will only present the proof for $\\sg=1$ in the sequel.\n\n\n\n{\\bf Proof of \\reftheo{dleNL-mainthm}:} This theorem is a consequence of \\refcoro{gentheo1coro} whose integrability conditions in M.0) are already assumed in \\mref{llgmainhyp009} and \\mref{llgmainhyp0099}. We need only verify the condition M.1) of \\refcoro{gentheo1coro} (or \\reftheo{gentheo1}) with the map $K$ being defined by\n\\beqno{Kmapdef} K_{\\eg_0}(u)=(|\\log(|U|)|+\\eg_0)|U|^{-1}U, \\quad U=[\\llg_0+|u_i|]_{i=1}^m.\\end{equation} This map satisfies for any $\\eg_0>0$ the condition H) of \\reftheo{gentheo1} (see \\reflemm{H2lemm} and \\refrem{H2lemmrem} in the Appendix). We need only check the condition M.1). Because $$\\|K_{\\eg_0}(u)\\|_{BMO(B_R,\\mu)}\\le \\|K_0(u)\\|_{BMO(B_R,\\mu)}+\\eg_0\\||U|^{-1}U\\|_{BMO(B_R,\\mu)},$$ and $\\||U|^{-1}U\\|_{BMO(B_R,\\mu)}=1$ and so $\\|K_{\\eg_0}(u)\\|_{BMO(B_R,\\mu)}<\\mu_0$ for any given $\\mu_0>0$ if $\\eg_0<\\mu_0\/2$ and $R$ is small, thanks to the assumption \\mref{mainloghyp}. Therefore, the smallness condition \\mref{Keymu0} of M.1) holds.\n\nNext, we consider $\\myPi_p=\\llg^{p+\\frac12}(u)|\\llg_u(u)|^{-p}\\sim (\\llg_0+|u|)^{k_p}$ for some number $k_p$ because $\\llg(u), |\\llg_u(u)|$ have polynomial growths.  As we assume in \\mref{mainloghyp} that $\\log(\\llg_0+|u|)$ has small BMO norm in small balls,  $[\\myPi_p^\\ag]_{\\bg+1,B_R}\\sim [(\\llg_0+|u|)^{k_p\\ag}]_{\\bg+1,B_R}$ is bounded (see \\reflemm{Westlemm} following this proof) for any given $\\ag,\\bg>0$ if $R$ is sufficiently small.  Thus, M.1) is verified and \\refcoro{gentheo1coro} applies here to complete the proof. \\begin{flushright} Q.E.D. \\end{flushright}\n\n\\newcommand{\\mmc}{\\mathbf{c}}\n\n\nWe have the following lemma which was used in the above proof to establish that $[\\myPi_p^\\ag]_{\\bg+1,B_R}$ is bounded. This lemma will be frequently referred to in the rest of this section. \n\n\\blemm{Westlemm} Let $\\mu$ be a doubling measure and $U$ be a nonnegative function on a ball $B\\subset \\Og$. There is a constant $\\mmc_2$ depending only on the doubling constant of $\\mu$ such that for any given $l$ and $\\bg>0$ if $[\\log U]_{*,\\mu}$ is sufficiently small then $[U^l]_{\\bg+1}\\le \\mmc_2^{1+\\bg}$.\n\\end{lemma}\n\n{\\bf Proof:~~} We first recall the John-Nirenberg inequality (\\cite[Chapter 9]{Graf}): For any BMO($\\mu$) function $v$ \nthere are constants $\\mmc_1,\\mmc_2$, which depend only on the doubling constant of $\\mu$, such that  \n\\beqno{JNineq}\\mitmu{B}{e^{\\frac{\\mmc_1}{[v]_{*,\\mu}}|v-v_B|}} \\le \\mmc_2.\\end{equation}\n\n\nFor any $\\bg>0$ we know that $e^v$ is an $A_{\\bg+1}$ weight with $[e^v]_{\\bg+1}\\le \\mmc_2^{1+\\bg}$ (e.g. see \\cite[Chapter 9]{Graf}) if  \\beqno{Acgcond}\\sup_B \\mitmu{B}{e^{(v-v_B)}} \\le \\mmc_2,\\; \\sup_B \\mitmu{B}{e^{-\\frac{1}{\\bg}(v-v_B)}} \\le \\mmc_2.\\end{equation}\n\nIt is clear that \\mref{Acgcond} follows from \\mref{JNineq} if $\\mmc_1[v]_{*,\\mu}^{-1}\\ge \\max\\{1,\\bg^{-1}\\}$. Therefore, for $v= l\\log U$ we see that if $[\\log U]_{*,\\mu}\\le \\mmc_1\\bg|l|^{-1}$ then $[U^l]_{\\bg+1}\\le \\mmc_2^{1+\\bg}$. Hence, for any given $l$ and $\\bg>0$ if $[\\log U]_{*,\\mu}$ is sufficiently small then $[U^l]_{\\bg+1}\\le \\mmc_2^{1+\\bg}$.\n\\begin{flushright} Q.E.D. \\end{flushright}\n\n\n{\\bf Proof of \\refcoro{dlec1coro}:}  We just need to show that the assumption \\mref{pis20az5} implies \\mref{mainloghyp} of \\reftheo{dleNL-mainthm}. For $U=[\\llg_0+|u_i|]_{i=1}^m$ we have from the calculation in \\refrem{H2lemmrem} that\n$$(K_0)_u(u) = \\frac{|\\log(|U|)|}{|U|}\\left[I + \\left(\\frac{\\mbox{sign}(\\log(|U|))}{|\\log(|U|)|}-1\\right)\\zeta\\zeta^T\\right]\\mbox{diag}[\\mbox{sign} (u_i)],$$ where $\\zeta=|U|^{-1}U$. It is clear that $|(K_0)_u(u)|\\le C(1+|\\log(|U|))|U|^{-1}$. Since $|U|$ is bounded from below by $\\llg_0$, for any $\\ag\\in(0,1)$ there is a constant $C(\\ag)$ such that $|(K_0)_u(u)|\\le C(\\ag)|U|^{-\\ag}$. Therefore, $|D(K_0(u))|$ and $|D(\\log(\\llg_0+|u|))|$ can be bounded by $C(\\llg_0+|u|)^{-\\ag}|Du|$. \nIt follows from the assumption \\mref{pis20az5} that \\beqno{pis20az5zz} \\iidmu{\\Og}{|DK_0(u)|^{n}},\\; \\iidmu{\\Og}{|D(\\log(\\llg_0+|u|))|^{n}}\\le C(C_0).\\end{equation}\n\nFrom the Poincar\\'e-Sobolev inequality, using the assumption that $\\og$ is bounded from below and above by AR) \n$$\\left(\\mitmu{B_r}{|K_0(u)-K_0(u)_{B_r}|^2}\\right)^\\frac12 \\le C(n)\n\\left(\\iidmu{B_r}{|D(K_0(u))|^{n}}\\right)^\\frac{1}{n}.$$ The  continuity of the integral of $|D(K_0(u))|^{n}$ and the uniform bound \\mref{pis20az5zz} show that the last integral is small if $r$ is. The same argument applies to the function $\\log(\\llg_0+|u|)$. We then see that the BMO norms of $K_0(u)$ and  $\\log(\\llg_0+|u|)$ are small in small balls, and so \\mref{mainloghyp} of \\reftheo{dleNL-mainthm} holds. The proof is complete.\n\\begin{flushright} Q.E.D. \\end{flushright}\n\nFor the proof of \\refcoro{2dthm1} we first need the following lemma.\n\n\\blemm{2dlemma} Suppose A), F) and \\mref{FUDU112z} if $\\hat{f}$ has a quadratic growth in $Du$ with $\\eg_0$ being sufficiently small. For any $s$ satisfying \\beqno{sconds}s>-1 \\mbox{ and } C_*^{-1}>s\/(s+2)\\end{equation} and any ${\\ag_0}\\in(0,1)$ we have for $U:=\\llg_0+|u|$ that \\beqno{nequal2est}\\iidmu{\\Og}{U^{k+s}|Du|^2} \\le C\\mypar{\\iidmu{\\Og}{U^{{\\ag_0}(k+s+2)}}}^\\frac1{\\ag_0}+C\\iidmu{\\Og}{U^sf(u)|u|}.\\end{equation} \\end{lemma}\n\n{\\bf Proof:~~}  \nLet $X=[\\llg_0+|u_i|]_{i=1}^m$ and test the system with $|X|^su$ to get \\beqno{llgDu2}\\iidmu{\\Og}{\\myprod{A(u)Du,D(|X|^su)}} \\le \\iidmu{\\Og}{\\myprod{\\hat{f}(u,Du),|X|^su}}.\\end{equation}  We note that $\\myprod{A(u)Du,D(|X|^su)}=\\myprod{A(u) DX,D(|X|^sX)}$ so that, by  the assumption \\mref{sconds} on $s$, there is $c_0>0$ such that $\\myprod{A(u)Du,D(|X|^su)}\\ge c_0\\llg(u)|X|^s|DX|^2$ (see \\mref{SGrem0} in the Appendix). Because $|DX|=|Du|$ and $|X|\\sim U$, the above yields \\beqno{fuduu}\\iidmu{\\Og}{\\llg(u)U^s|Du|^2}\\le C\\iidmu{\\Og}{U^s\\myprod{\\hat{f}(u,Du),u}}.\\end{equation}\n\nIf $\\hat{f}$ satifies f.1) then a simple use of Young's inequality gives $$|\\myprod{\\hat{f}(u,Du),u}| \\le \\eg\\llg(u)|Du|^2 + C(\\eg)\\llg(u)|u|^2+f(u)|u|.$$ \n\nIf f.2) holds with \\mref{FUDU112z} then $|\\myprod{\\hat{f}(u,Du),u}| \\le C|\\llg_u(u)||u||Du|^2 + f(u)|u|$. Because $|\\llg_u(u)||u|\\lesssim \\llg(u)$, we obtain the above inequality again with $\\eg=C\\eg_0$. Therefore, if $\\eg_0$ is sufficiently small then \\mref{fuduu} and the fact that $\\llg(u)\\sim U^k$  imply $$\\iidmu{\\Og}{U^{k+s}|Du|^2} \\le C\\iidmu{\\Og}{U^{k+s+2}}+C\\iidmu{\\Og}{U^sf(u)|u|}.$$\nWe apply the interpolation inequality $\\|w\\|_{L^2(\\Og,\\mu)}^2 \\le \\eg\\|Dw\\|_{L^2(\\Og,\\mu)}^2+C(\\eg)\\|w^{\\ag_0}\\|_{L^2(\\Og,\\mu)}^{1\/{\\ag_0}}$ with ${\\ag_0}\\in(0,1)$ and $w=U^{(k+s+2)\/2}$ to the first integral on the right hand side, noting also that $|Dw|^2\\sim U^{k+s}|Du|^2$. For $\\eg$ sufficiently small, we derive  \\mref{nequal2est} from the above estimate  and complete the proof. \\begin{flushright} Q.E.D. \\end{flushright}\n\n{\\bf Proof of \\refcoro{2dthm1}:} We apply \\refcoro{dlec1coro} here. We will verify first the condition \\mref{pis20az5} and then the integrability assumptions \\mref{llgmainhyp009} and \\mref{llgmainhyp0099}.\n\nIn the sequel we denote $U=\\llg_0+|u|$. From \\mref{nequal2est} of \\reflemm{2dlemma} we see that  if  there exist positive numbers $\\ag_0\\in(0,1)$, $C_0>0$ and $s$ satisfies \\mref{sconds} such that if\n\\beqno{nequal2conda}\\|U^{p}\\|_{L^1(\\Og)}\\le C_0,\\;p=\\ag_0(s+k+2)\\mbox{ and }p=s+l+1,\\end{equation} then \\mref{nequal2est}, with the assumption that $f(u)\\lesssim U^l$, implies  \\beqno{nequal2condb}\\iidx{\\Og}{U^{k+s}|Du|^2}\\le C(C_0).\\end{equation}\n\nWe will first show that  the assumption \\mref{keyn2lnorm0}, that $\\|u\\|_{L^{l_0}(\\Og,\\mu)}\\le C_0$ for some $l_0>\\max\\{l,l-k-1\\}$, implies \\mref{nequal2condb} for some $s=s_0>\\max\\{-1,-k-2\\}$. Indeed, for any such $l_0$ we have $s_0+l+1\\le l_0$ if $s_0$ is close to $\\max\\{-1,-k-2\\}$. Moreover, $s_0$ satisfies \\mref{sconds}. This clearly holds if $s_0\\le0$, i.e. $k\\ge-2$. Otherwise, the assumption $k>-2C_*\/(C_*-1)$ yields that $C_*^{-1}>s_0\/(s_0+2)$ if $s_0$ is close to $-k-2>0$.  Thus, \\mref{nequal2est} holds for such $s_0$. We also choose $\\ag_0\\in(0,1)$ sufficiently small such that $\\ag_0(s_0+k+2)\\le l_0$. With these choices of $\\ag_0, s_0$ and the assumption \\mref{keyn2lnorm0}, we see that \\mref{nequal2conda} and then \\mref{nequal2condb} hold for $s=s_0$.\n\nAs $k+s_0>-2$, we can find $\\ag\\in(0,1)$ such that $-2\\ag\\le k+s_0$. Therefore, \\mref{nequal2condb} with $s=s_0$ yields \\mref{pis20az5} of \\refcoro{dlec1coro} for $n=2$ because $$\\iidx{\\Og}{(\\llg_0+|u|)^{-2\\ag}|Du|^2} \\le \\iidx{\\Og}{(\\llg_0+|u|)^{k+s_0}|Du|^2}\\le C(C_0).$$\n\nWe now check the integrability conditions \\mref{llgmainhyp009} and \\mref{llgmainhyp0099} of \\reftheo{dleNL-mainthm} which read\n\\beqno{llgmainhyp00} \\|\\llg^{-1}(u)\\|_{L^{\\frac n2}(\\Og,\\mu)},\\;\\||f_u(u)|\\llg^{-1}(u)\\|_{L^{r_0}(\\Og,\\mu)}\\le C_0,\\end{equation}\n\\beqno{llgB30}\\iidmu{\\Og}{(|f_u(u)|+\\llg(u))|Du|^{2}}\\le C_0,\\end{equation} \n\\beqno{llgB40}\\iidmu{\\Og}{(\\llg(u)|u|^2)^{\\bg_0}}\\le C_0.\\end{equation}\n\n\n\n\nBecause $n=2$, we have the inequality $\\|w\\|_{L^q(\\Og)} \\le C\\|Dw\\|_{L^2(\\Og)}+C\\|w^\\bg\\|_{L^1(\\Og)}^\\frac{1}{\\bg}$ which holds for all $q\\ge1$ and $\\bg\\in(0,1)$. Applying this to $w=|U|^{(k+s_0)\/2+1}$ and using \\mref{nequal2condb} and the assumption \\mref{keyn2lnorm0}, we see that $\\|U^q\\|_{L^1(\\Og)}\\le C(C_0)$ for all $q\\ge1$. By H\\\"older's inequality this is also true for $q\\ge0$. It is also true for $q<0$ because $U\\ge\\llg_0>0$. We then have\\beqno{n2allq} \\|U^q\\|_{L^1(\\Og)}\\le C(C_0,\\llg_0,q) \\quad \\mbox{for all $q$}.\\end{equation}\n\nThe above then immediately implies the integrability conditions \\mref{llgmainhyp00} and \\mref{llgB40} because $\\llg(u)$ and $|f_u(u)|$ are powers of $U$.\n\n\nSimilarly, \\mref{n2allq} implies that \\mref{nequal2conda} holds for any $p$ so that \\mref{nequal2condb} is valid if $s\\ge0$ and $C_*^{-1}>s\/(s+2)$.  To verify  \\mref{llgB30} we need to find a constant $C(C_0)$ such that \\beqno{llgB30zz}\\iidx{\\Og}{(\\llg_0+|u|)^{l-1}|Du|^2}+\\iidx{\\Og}{(\\llg_0+|u|)^k|Du|^{2}}\\le C(C_0).\\end{equation} Letting $s=0$ in \\mref{nequal2condb}, we see that the second integral on the left hand side is bounded by a constant $C(C_0)$. If $l\\le 1+k$ then the first integral in \\mref{llgB30zz} is bounded by the second one and we obtain the desired bound. If $l>k+1$ we let $s=l-k-1$ in \\mref{nequal2condb}. The condition on $s$ in \\mref{sconds} holds because $$\\frac{s}{s+2}<C_*^{-1} \\Leftrightarrow \\frac{l-k-1}{l-k+1}<C_*^{-1} \\Leftrightarrow l-k<\\frac{C_*+1}{C_*-1},$$ which is assumed in \\mref{klcond}. Hence, \\mref{llgB30zz} holds. We have verified all assumptions of \\reftheo{dleNL-mainthm} and the  proof is complete. \\begin{flushright} Q.E.D. \\end{flushright}\n\nWe now give the proof of \\reftheo{n3SKT}. The case $n=2$ is similar and easier so that we will consider only $n=3,4$. The proof is again based on \\reftheo{gentheo1} using the new map  \n\\beqno{Kn34def}K(u)=[K_i(u)]_{i=1}^m, \\mbox{ where } K_i(u)=\\log(\\llg_0^{k+1}+|P_i(u)|).\\end{equation}\n\nWe first check the conditions H) of \\reftheo{gentheo1} in the following lemma.\n\\blemm{newKlem} For any $k\\ge-1$ there exists a constant $C(k)$ such that for $\\mathbb{K}(u)=K_u^{-1}(u)^T$ $$|\\mathbb{K}(u)|\\le C(k) \\llg(u)|\\llg_u(u)|^{-1},\\; |\\mathbb{K}_u(u)|\\le C(k).$$\n\\end{lemma}\n{\\bf Proof:~~} As $A(u)=P_u(u)$, we have from the definition \\mref{Kn34def} that  $$K_u(u)=[\\frac{\\partial K_i(u)}{\\partial u_j}(K_i(u))^{-1}]=\\mbox{diag}[\\mbox{sign}P_i(u)(\\llg_0^{k+1}+|P_i(u)|)^{-1}]A(u),$$ and so \n$K_u^{-1}(u)=A(u)^{-1}\\mbox{diag}[\\llg_0^{k+1}\\mbox{sign}P_i(u)+P_i(u)]$. We  show that there is a cosntant $C(k)$ such that $|K_u^{-1}(u)|$ is bounded by $C(k)\\llg(u)|\\llg_u(u)|^{-1}$. Since  $\\llg(u), C_*\\llg(u)$ are the smallest and largest eigenvalues of $A(u)$, $|A^{-1}(u)|\\sim \\llg^{-1}(u)$. Using the facts that $\\llg(u)\\sim (\\llg_0+|u|)^k$, $|\\llg_u(u)|\\sim (\\llg_0+|u|)^{k-1}$ and $(\\llg_0+|u|)^{k+1}\\ge \\llg_0^{k+1}$ (because $k\\ge-1$),  it follows easily that $\\llg_0^{k+1}|\\llg_u(u)|\\lesssim \\llg(u)^{2}$. We collect these facts in the following. \\beqno{Allgk1}|P_i(u)|\\lesssim |A(u)||u|,\\;|A^{-1}(u)|\\sim \\llg^{-1}(u),\\; \\llg_0^{k+1}|\\llg_u(u)|\\lesssim \\llg(u)^{2}.\\end{equation}\nWe then have  $|A(u)^{-1}\\mbox{diag}[P_i(u)]|\\lesssim |u|\\lesssim \\llg(u)|\\llg_u(u)|^{-1}$ and $\\llg_0^{k+1}|A(u)^{-1}|\\sim \\llg_0^{k+1}\\llg(u)^{-1}\\lesssim \\llg(u)|\\llg_u(u)|^{-1}$. Therefore  $|\\mathbb{K}(u)|=|K_u^{-1}(u)^T|\\lesssim \\llg(u)|\\llg_u(u)|^{-1}$. \n\nIn addition, as $|\\mathbb{K}_u(u)|= |(K_u^{-1}(u))_u|$ and $A(u)=\\mbox{diag}[(P_i(u))_u]$ $$|\\mathbb{K}_u(u)|\\lesssim |A^{-1}(u)|^2|A_u(u)||\\mbox{diag}[\\llg_0^{k+1}+|P_i(u)|]|+1.$$\nUsing \\mref{Allgk1} and the fact that $|A_u(u)|\\lesssim |\\llg_u(u)|$,  we easily see that $|\\mathbb{K}_u(u)|$ is bounded by some constant $C(k)$. The lemma is proved. \\begin{flushright} Q.E.D. \\end{flushright}\n\nNext, we need to show that $K(u)$ has small BMO norm in small balls by establishing that $DK(u)\\in L^n(\\Og)$ and using the Poincar\\'e-Sobolev inequality as in the proof of \\refcoro{dlec1coro}.  To this end, we need\nthe following lemma.\n\\blemm{DU6lem} Assume that there exist $r_0>n\/2$ and $C_0$ such that\n\\beqno{llgn3}  \\|f_u(u)\\llg^{-1}(u)\\|_{L^{r_0}(\\Og)}\\le C_0.\\end{equation}\nThen for any $\\bg_0\\in(0,1]$  there exists a constant $C(C_0,\\bg_0)$,  such that \\beqno{DUUest}\\|DP(u)\\|_{L^\\frac{2n}{n-2}(\\Og)}\\le C(C_0,\\bg_0)\\||P(u)|^{\\bg_0}\\|_{L^1(\\Og)}.\\end{equation} \\end{lemma}\n{\\bf Proof:~~} In the sequel, we write $U=[U_i]_{i=1}^m$, $U_i=P_i(u)$. Multiplying the $i$-th equation in \\mref{genSKT} by $-\\Delta U_i$, integrating over $\\Og$ and summing the results, we get\n$$\\iidx{\\Og}{|\\Delta U|^2}=-\\sum_i\\iidx{\\Og}{\\myprod{B_i(u,Du),\\Delta U_i}}-\\sum_i\\iidx{\\Og}{\\myprod{f_i(u),\\Delta U_i}}.$$ Applying integration by parts to the last integral, we have $$\\iidx{\\Og}{|\\Delta U|^2}=-\\sum_i\\iidx{\\Og}{\\myprod{B_i(u,Du),\\Delta U_i}}+\\sum_{i,j}\\iidx{\\Og}{\\myprod{(f_i)_{u_j}(u)Du_j,D U_i}}.$$\n\nThe condition A) implies $\\llg(u)|Du|^2\\le \\myprod{A(u)Du,Du}=\\myprod{DU,Du}$ and so Young's inequality yields $\\llg(u)|Du|^2\\le \\frac12\\llg^{-1}(u)|DU|^2+\\frac12\\llg(u)|Du|^2$. We then have $\\llg(u)|Du|\\le |DU|$.\nUsing this fact, the assumption that $|B_i(u,Du)|\\le C\\llg(u)|Du|$ and applying Young's inequality to the first integral on the right hand side of the above, we get $$\\|\\Delta U\\|_{L^2(\\Og)}^2\\lesssim\\iidx{\\Og}{|DU|^2}+\\iidx{\\Og}{|f_u(u)\\llg^{-1}(u)||DU|^2}.$$\nBy H\\\"older's inequality and \\mref{llgn3}, the last integral is estimated by\n$$ \\mypar{\\iidx{\\Og}{|f_u(u)\\llg^{-1}(u)|^{r_0}}}^\\frac{1}{r_0}\\|DU\\|_{L^{2r'_0}(\\Og)}^2 \\le C(C_0)\\|DU\\|_{L^{2r'_0}(\\Og)}^2.$$\n\nUsing Schauder's inequality $\\|D^2 U\\|_{L^2(\\Og)}\\le C\\|\\Delta U\\|_{L^2(\\Og)}$, we obtain from the above two inequalities that\n\\beqno{D2UPP}\\|D^2U\\|_{L^2(\\Og)}^2\\lesssim \\|DU\\|_{L^2(\\Og)}^2+C(C_0)\\|DU\\|_{L^{2r'_0}(\\Og)}^2.\\end{equation}\n\nWe recall the following interpolating Sobolev inequality: for any  $\\eg>0$  \\beqno{intineq}\\|W\\|_{L^p(\\Og)}\\le \\eg\\|DW\\|_{L^2(\\Og)} + C(\\eg)\\|W^\\bg\\|_{L^1(\\Og)}^\\frac{1}{\\bg} \\mbox{ for any $p\\in[1,\\frac{2n}{n-2})$ and $\\bg\\in(0,1]$}.\\end{equation} \n\nBecause $r_0>n\/2$, $2r'_0<2n\/(n-2)$ so that we can apply \\mref{intineq} to $W=DU$ with $p=2$, $p=2r'_0$, $\\bg=1$ and $\\eg$ is sufficiently small in \\mref{D2UPP} to see that  $\\|D^2U\\|_{L^2(\\Og)}^2\\le C(C_0)\\|DU\\|_{L^1(\\Og)}^2$. As $\\|D U\\|_{L^1(\\Og)}\\le \\eg\\|D^2U\\|_{L^2(\\Og)} + C(\\eg)\\|U\\|_{L^1(\\Og)}$,\nwe obtain for small $\\eg$ that $\\|D^2U\\|_{L^2(\\Og)}\\le C(C_0)\\|U\\|_{L^1(\\Og)}$.\nSobolev's embedding theorem then yields $$\\|D U\\|_{L^\\frac{2n}{n-2}(\\Og)}\\le C(C_0)\\|U\\|_{L^1(\\Og)}.$$\nApplying \\mref{intineq} again, with $W=|U|$, $p=1$ and $\\bg=\\bg_0$, to estimate the norm $\\|U\\|_{L^1(\\Og)}$ and noting that $\\|DU\\|_{L^2(\\Og)}\\lesssim\\|DU\\|_{L^\\frac{2n}{n-2}(\\Og)}$, we obtain \\mref{DUUest}. \n\\begin{flushright} Q.E.D. \\end{flushright}\n\n\n{\\bf Proof of \\reftheo{n3SKT}:} The proof is again based on \\reftheo{gentheo1}. The assumptions i) and ii) state \\beqno{uintcondk}\\left\\{\\begin{array}{ll}\\|u\\|_{L^1(\\Og)}\\le C_0 & \\mbox{if $k\\ge0$,}\\\\\\|u\\|_{L^{-kn\/2}(\\Og)}\\le C_0 &\\mbox{if $k\\in[-1,0)$,}\\end{array} \\right.\\end{equation} and clearly imply\n \\beqno{fuhyp0z}\\|\\llg^{-1}(u)\\|_{L^{\\frac{n}{2}}(\\Og)}\\le C(C_0,\\llg_0).\\end{equation} This and the assumption \\mref{fuhyp0} provide the integrability condition \\mref{llgmainhyp0} of M.0). Concerning the integrability condition \\mref{llgB3} in M.0), we use H\\\"older's inequality, writing $\\llg(u)|Du|^2= \\llg^{-1}(u)\\llg^2(u)|Du|^2$, and \\mref{fuhyp0z} and \\mref{DUUest} to have\n $$\\iidx{\\Og}{\\llg(u)|Du|^2}\\le\\|\\llg^{-1}(u)\\|_{L^\\frac{n}{2}(\\Og)}\\|\\llg(u)Du\\|_{L^\\frac{2n}{n-2}(\\Og)}^2\\le C_0\\|DP(u)\\|_{L^\\frac{2n}{n-2}(\\Og)}^2\\le C(C_0).$$\n \n Similarly, the integral of $|f_u(u)||Du|^2$ can be estimated by\n $$\\|\\llg^{-2}(u)f_u(u)\\|_{L^\\frac{n}{2}(\\Og)}\\|\\llg(u)Du\\|_{L^\\frac{2n}{n-2}(\\Og)}^2\\le  C(C_0)\\|\\llg^{-2}(u)f_u(u)\\|_{L^\\frac{n}{2}(\\Og)}.$$\n \n If $k\\ge0$ then $\\llg^{-2}(u)|f_u(u)|\\le C(\\llg_0)\\llg^{-1}(u)|f_u(u)|$ so that the last norm in the above is bounded, thanks to \\mref{fuhyp0}. If $k<0$ then this norm is bounded by the assumption \\mref{fuhyp0k}. We conclude that the condition \\mref{llgB3} in M.0) holds.\n \n \nWe discuss the condition M.1). First of all, \nbecause $|P(u)|\\le \\llg(u)|u|\\le (\\llg_0+|u|)^{k+1}$, \\mref{uintcondk} also shows that for any positive and sufficiently small $\\bg_0$ \\beqno{Puhypz} \\|(\\llg_0^{k+1}+|P(u)|)^{\\bg_0}\\|_{L^1(\\Og)}\\le C(\\llg_0,C_0).\\end{equation}\n\nNext, using the Poincar\\'e-Sobolev inequality as in the proof of \\refcoro{dlec1coro}, we show that $K(u)$ has small BMO norm in small balls by estimating\n\\beqno{3dlog}\\|DK_i(u)\\|_{L^n(\\Og)} \\le \\|(\\llg_0^{k+1}+|P_i(u)|)^{-1}DP_i(u)\\|_{L^n(\\Og)}\\le C(\\llg_0)\\|DP_i(u)\\|_{L^n(\\Og)}.\\end{equation}\nBecause $n\\le4$, $n\\le 2n\/(n-2)$. By the assumption \\mref{fuhyp0}, $\\|f_u(u)\\llg^{-1}(u)\\|_{L^{r_0}(\\Og)}\\le C_0$, \\reflemm{DU6lem} shows that $\\|DP(u)\\|_{L^n(\\Og)}\\le C(C_0,\\bg_0)\\||P(u)|^{\\bg_0}\\|_{L^1(\\Og)}$. This and \\mref{Puhypz} and \\mref{3dlog} provide a constant $C(\\llg_0,C_0)$ such that  $\\|DK(u)\\|_{L^n(\\Og)}\\le C(\\llg_0,C_0)$. We then see that $K(u)$ has small BMO norm in small balls.\n\n\n\nConcerning the weight $\\myPi_p$, because $|Du|\\lesssim\\llg^{-1}(u)|DP(u)|$, $k+1\\ge0$, we have\n$$|D(\\log(\\llg_0+|u|))|=\\frac{|Du|}{\\llg_0+|u|}\\lesssim \\frac{|DP(u)|}{(\\llg_0+|u|)\\llg(u)}\\sim \\frac{|DP(u)|}{(\\llg_0+|u|)^{k+1}}\\le \\llg_0^{-k-1}|DP(u)|.$$ \\reflemm{DU6lem} then shows that $D\\log(\\llg_0+|u|)\\in L^n(\\Og)$ so that $\\log(\\llg_0+|u|)$ is has small BMO norm in small balls. \\reflemm{Westlemm} applies to yield that $[\\myPi_p^\\ag]_{\\bg+1,B_R}\\sim [(\\llg_0+|u|)^{k_p\\ag}]_{\\cg,B_R}$ is bounded for any given $\\ag,\\bg>0$ if $R$ is sufficiently small. The assumption M.1) of \\reftheo{gentheo1} is verified.\n\n\nWe thus establish the conditions of \\reftheo{gentheo1} and complete the proof. \\begin{flushright} Q.E.D. \\end{flushright}\n\n\\section{Appendix}\\label{appsec}\\eqnoset\n\nLet $m,l$ be any integers. For $X=[X_i]_{i=1}^m$, $X_i\\in {\\rm I\\kern -1.6pt{\\rm R}}^l$ and for any $C^1$ function $k:{\\rm I\\kern -1.6pt{\\rm R}}^+\\to{\\rm I\\kern -1.6pt{\\rm R}}$ we consider the maps\n\\beqno{kdef} K(X)=k(|X|)|X|^{-1}X,\\;\\zeta=|X|^{-1}X=[\\zeta_i]_{i=1}^m.\\end{equation}\n\nWe see that $D_X(|X|)=\\zeta$ and $D_X\\zeta=|X|^{-1}(I-\\zeta\\zeta^T)$, where $\\zeta\\zeta^T=[\\myprod{\\zeta_i,\\zeta_j}]$. Hence, \n$$D_XK(X)= k(|X|)D_X\\zeta+D_Xk(|X|)\\zeta^T=k(|X|)|X|^{-1}(I-\\zeta\\zeta^T)+k'(|X|)\\zeta\\zeta^T.$$ \nWe then introduce the notations \\beqno{kudef}b=k'(|X|)|X|\/k(|X|),\\; \\ccK(\\zeta)=I+(b-1)\\zeta\\zeta^T.\\end{equation} Therefore, the calculation for $D_XK(X)$ yields \\beqno{kxdef} D_XK(X)=k(|X|)|X|^{-1}(I+(b-1)\\zeta\\zeta^T)=k(|X|)|X|^{-1}\\ccK(\\zeta).\\end{equation}\n\n\nIf $k(|X|)\\ne 0$ and $k'(|X|)\\ne 0$ then $\\ccK(\\zeta)$ is invertible. \nWe can use the Serman-Morrison formula $(I+wv^T)^{-1}=I-(1+v^Tw)^{-1}wv^T$, setting $w=(b-1)\\zeta$ and $v=\\zeta$, to see that  \\beqno{DXKinv} (D_XK(X))^{-1}=\\frac{|X|}{k(|X|)}(I+(b^{-1}-1)\\zeta\\zeta^T).\\end{equation} \n\nOtherwise, if $k(|X|)=0$ (resp. $k'(|X|)=0$) then $D_XK(X)= k'(|X|)\\zeta\\zeta^T$ (resp. $D_XK(X)= k(|X|)|X|^{-1}(I-\\zeta\\zeta^T)$) and $D_XK(X)$ is not invertible.\n\nThe following lemma was used in the checking of the condition H) for the map $K_{\\eg_0}(u)$ in the proof of \\reftheo{dleNL-mainthm}.\n\\blemm{H2lemm} For any $\\eg_0,\\llg_0>0$ let $k(t)=|\\log(t)|+\\eg_0$ and $X(u)=[\\llg_0+|u_i|]_{i=1}^m$ in \\mref{kdef}. There exists a constant $C(\\eg_0)$ such that the map $\\mathbb{K}(u)=(K_u(X(u))^{-1})^T$ satisfies \\beqno{H2check}|\\mathbb{K}(u)|\\le C(\\eg_0)|X|,\\; \\|\\mathbb{K}_u(u)\\|_{L^\\infty({\\rm I\\kern -1.6pt{\\rm R}}^m)}\\le C(\\eg_0).\\end{equation}\n\n\\end{lemma}\n\n{\\bf Proof:~~} As $k'(t)=\\mbox{sign}(\\log t) t^{-1}$, we have $$b^{-1}=k(|X|)(k'(|X|)|X|)^{-1}=\\mbox{sign}(\\log(|X|)) (|\\log(|X|)|+\\eg_0).$$ \n\nDefine $X_u=\\mbox{diag}[\\mbox{sign} u_i]$.  We have from \\mref{DXKinv} and the definition $X=[\\llg_0+|u_i|]_{i=1}^m$ that\n$$\\mathbb{K}(u)=(X_u)^{-1}\\frac{|X|}{|\\log(|X|)|+\\eg_0}(I+(\\mbox{sign}(\\log(|X|)) (|\\log(|X|)|+\\eg_0)-1)\\zeta\\zeta^T).$$\n\nAs $\\eg_0>0$, we easily see that $|\\mathbb{K}(u)|\\le C(\\eg_0)|X|$ for some constant $C(\\eg_0)$. A straightforward calculation also shows that $\\|\\mathbb{K}_u(u)\\|_{L^\\infty({\\rm I\\kern -1.6pt{\\rm R}}^m)}\\le C(\\eg_0)$.  \\begin{flushright} Q.E.D. \\end{flushright}\n\n\\brem{H2lemmrem}\nIf $\\llg(u)\\sim (\\llg_0+|u|)^k\\sim |X(u)|^k$, with $k\\ne0$, and $\\LLg(u)= \\llg^\\frac12(u)$ and $\\Fg(u)=|\\LLg_u(u)|$. We then have $\\LLg(u)\\Fg^{-1}(u)\\sim |X(u)|$ and $\\Fg(u)|\\Fg_u(u)|^{-1}\\sim |X(u)|$. We obtain from \\mref{H2check} that $|\\mathbb{K}(u)|\\lesssim \\LLg(u)\\Fg^{-1}(u)$ and  $|\\mathbb{K}(u)||\\Fg_u(u)|\\lesssim\\Fg(u)$.\nThus, the assumptions on the map $K$ for the local Gagliardo-Nirenberg inequality are verified here. \\erem\n\n\nWe then need that $K(U)$ is BMO and $\\myPi^\\ag$ is a weight.  By \\mref{kudef} $$K_U(U) = \\frac{|\\log(|X|)|+\\eg_0}{|X|}\\left(I + (\\frac{X_s}{|\\log(|X|)|+\\eg_0}-1)\\zeta\\zeta^T\\right)X_U, \\quad X=[\\llg_0+|U_i|]_{i=1}^m.$$\n\nRecall that $$\\myPi=\\LLg^{p+1}(U)\\Fg^{-p}(U)\\sim |U|^p\\llg^\\frac12(U),\\; \\ag>2\/(p+2),\\; \\bg<p\/(p+2).$$\n\n\nWe note some properties of $K$.\n\n\\blemm{Klemm} Let $X_i:\\Og\\to {\\rm I\\kern -1.6pt{\\rm R}}^l$ be $C^1$ maps on a domain $\\Og\\subset {\\rm I\\kern -1.6pt{\\rm R}}^n$ for $i=1,\\ldots,m$. If $b=k'(|X|)|X|\/k(|X|)>0$ then \\beqno{DXDK}\\myprod{DX,D(K(X))}\\ge0.\\end{equation} Moreover, for $\\ag=1-(\\frac{b-1}{b+1})^2$\n\\beqno{DXDK1}\\myprod{DX,D(K(X))}\\ge \\ag^\\frac12 |DX||D(K(X))|. \\end{equation}\n\n\\end{lemma}\n\n{\\bf Proof:~~} By \\mref{kxdef} $D(K(X))=\\mmk(|X|)\\ccK(\\zeta)DX$, where $\\mmk(t):=k(t)t^{-1}$, and so\n$$\\myprod{DX,D(K(X))}=\\mmk(|X|)(|DX|^2+(b-1)\\myprod{DX,\\zeta\\zeta^T DX}).$$ Note that $|\\zeta\\zeta^T|\\le1$ so that $\\myprod{DX,D(K(X))}\\ge0$ if $s=b-1>-1$. This gives \\mref{DXDK}.\n\nSince $\\zeta\\zeta^T$ is a projection, i.e. $(\\zeta\\zeta^T)^2=\\zeta\\zeta^T$, we have, setting $J=\\myprod{\\zeta\\zeta^T DX,DX}$\n$$\\begin{array}{lll}|D(K(X))|^2&=&|K_X(X)DX|^2=\\mmk^2(|X|)\\myprod{\\ccK(\\zeta)DX,\\ccK(\\zeta)DX}\\\\&=&\n\\mmk^2(|X|)(|DX|^2+(2s+s^2)J).\\end{array}$$\nHence, we can write $\\myprod{DX,D(K(X))}^2-\\ag |DX|^2|D(K(X))|^2$ as\n$$\\mmk^2(|X|)[(1-\\ag)|DX|^4+(2s-\\ag(2s+s^2))|DX|^2 J+s^2J^2].$$\nIf we choose $\\ag=1-(\\frac{s}{s+2})^2$ then the above is $\\mmk^2(|X|)\\left(\\frac{s}{s+2}|DX|^2-sJ\\right)^2\\ge0$.\nTherefore $\\myprod{DX,D(K(X))}^2\\ge \\ag |DX|^2|D(K(X))|^2$. This and \\mref{DXDK} yield \\mref{DXDK1}. \\begin{flushright} Q.E.D. \\end{flushright}\n\nWe now consider a matrix $A$ satisfying for some positive $\\llg,\\LLg$ and any vector $\\chi$\\beqno{Aellcond} \\myprod{A\\chi,\\chi}\\ge \\llg|\\chi|^2,\\quad |A\\chi|\\le \\LLg|\\chi|.\\end{equation} \n\n\\blemm{AuK} Assume \\mref{Aellcond} and that $b>0$ (see \\mref{kudef}). For $\\kappa=\\llg\/\\LLg^2$ and $\\nu=\\llg\/\\LLg$ we have $$\\myprod{\\kappa A DX,D(K(X))}\\ge (\\ag^\\frac12-(1-\\nu^2)^\\frac12) |DX||D(K(X))|.$$\n\\end{lemma}\n\n{\\bf Proof:~~} From \\mref{Aellcond} with $\\chi=DX$, we note that $$\\begin{array}{lll}|\\kappa ADX-DX|^2 &=& \\kappa^2|ADX|^2-2\\kappa\\myprod{ADX,DX}+|DX|^2\\\\&\\le& (\\kappa^2\\LLg^2-2\\kappa\\llg+1)|DX|^2=(1-\\nu^2)|DX|^2.\\end{array}$$\n\nTherefore, using \\mref{DXDK1}$$\\begin{array}{lll}\\myprod{\\kappa ADX,D(K(X))}&=&\\myprod{\\kappa ADX-DX,D(K(X))}+\\myprod{DX,D(K(X))}\\\\&\\ge& -|\\kappa ADX-DX||DK(X)|+\\ag^\\frac12 |DX||D(K(X))|.\\end{array}$$ As $|\\kappa ADX-DX|^2\\le (1-\\nu^2)|DX|^2$, we obtain the lemma. \\begin{flushright} Q.E.D. \\end{flushright}\n\nLet $k(t)=|t|^{s+1}$ then $K(X)=|X|^sX$ and $b=s+1$. The above lemma then gives the following result which was used in the energy estimate.\n\\blemm{SGrem} Assume \\mref{Aellcond}. If $s>-1$ and $\\nu=\\frac{\\llg}{\\LLg}>\\frac{s}{s+2}$, then  \\beqno{SGrem0}\\myprod{A DX,D(|X|^sX)}\\ge  c_0\\frac{\\LLg^2}{\\llg}|X|^s|DX|^{2},\\end{equation} where $c_0=(1-(\\frac{s}{s+2})^2)^\\frac12-(1-\\nu^2)^\\frac12>0$.\n\\end{lemma}\n\n\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Appendix}\n\\irlabel{cut|cut}\n\nThis appendix briefly discusses generalized uses and forms of the differential ghost axioms and how it generalizes the differential auxiliaries proof rule \\cite{DBLP:journals\/lmcs\/Platzer12}.\n\n\\paragraph{Differential Lipschitz Ghosts}\nThe differential ghost axiom \\irref{DG} generalizes to arbitrary Lipschitz-continuous differential equations \\m{\\pevolve{\\D{y}=g(x,y)}}:\n\\[\n      \\cinferenceRule[DLG|DG$_\\ell$]{differential Lipschitz ghost variables} %\n      {\\linferenceRule[lpmil]\n        {\\big({\\dbox{\\pevolvein{\\D{x}=f(x)}{q(x)}}{p(x)}} \\lbisubjunct \n        {\\lexists{y}{\\dbox{\\pevolvein{\\D{x}=f(x)\\syssep\\D{y}=g(x,y)}{q(x)}}{p(x)}}}\\big)}\n        {\\lexists{\\ell}{\\lforall{x,y,z}{\\abs{g(x,y)-g(x,z)} \\leq \\ell\\abs{y-z}}}}\n      }\n      {}%\n\\]\nThe soundness argument for \\irref{DLG} is an extension of the soundness proof for \\irref{DG}.\nThe direction ``$\\lylpmi$'' of \\irref{DG} is sound for all differential equations.\nThe proof for the direction ``$\\limply$'' extends the proof for \\irref{DG} with an adaptation of the function $F$ from \\rref{eq:diffghost-extra-ODE} to the differential equation \\m{\\pevolve{\\D{y}=g(x,y)}}:\n\\begin{equation}\n\\begin{aligned}\n  y(0) &=d\\\\\n  \\D{y}(t) &= F(t,y(t)) \\mdefeq \\ivaluation{\\imodif[state]{\\Iff[t]}{y}{y(t)}}{g(x,y)}\n\\end{aligned}\n\\label{eq:Lipschitz-diffghost-extra-ODE}\n\\end{equation}\nThis function $F(t,\\delta)$ is still continuous on $[0,r]\\times\\reals$ since it is a composition of the continuous evaluation (of the, by assumption, continuous term $g(x,y)$) with the (continuous) composition of the continuous function $\\iget[state]{\\Iff[t]}$ of $t$ with the continuous modification of the value of variable $y$ to $\\delta$.\nBy assumption $F(t,y)$ is Lipschitz in $y$, since there is an $\\ell\\in\\reals$ such that for all $t,a,b\\in\\reals$:\n\\begin{multline*}\n\\abs{F(t,a)-F(t,b)} = \\abs{ \\ivaluation{\\imodif[state]{\\Iff[t]}{y}{a}}{g(x,y)} - \\ivaluation{\\imodif[state]{\\Iff[t]}{y}{b}}{g(x,y)} }\n= \\abs{\\ivaluation{\\imodif[state]{\\imodif[state]{\\Iff[t]}{y}{a}}{z}{b}}{g(x,y) - g(x,z)}}\n\\\\\n= \\ivaluation{\\imodif[state]{\\imodif[state]{\\Iff[t]}{y}{a}}{z}{b}}{\\underbrace{\\abs{g(x,y) - g(x,z)}}_{\\leq\\ell\\ivaluation{\\imodif[state]{\\imodif[state]{\\Iff[t]}{y}{a}}{z}{b}}{\\abs{y-z}}}}\n\\leq \\ell \\abs{a-b}\n\\end{multline*}\nThis establishes the only two properties of $F$ that the soundness proof of \\irref{DG} was based on.\nThe existence of a solution \\(y:[0,r]\\to\\reals\\) of \\rref{eq:Lipschitz-diffghost-extra-ODE} is, thus, established again by Picard-Lindel\\\"of as needed for the soundness proof.\n\n\n\\paragraph{Differential Auxiliaries Rule}\nThe differential auxiliaries proof rule \\cite{DBLP:journals\/lmcs\/Platzer12} is derivable from \\irref{DG} and monotonicity \\irref{M}.\n\n\\begin{calculuscollections}{10cm}\n\\begin{calculus}\n\\cinferenceRule[diffaux|$DA$]{differential auxiliary variables}\n{\\linferenceRule[sequent]\n  {\\lsequent[s]{}{p(x)\\lbisubjunct\\lexists{y}{r(x,y)}}\n  &\\lsequent{r(x,y)} {\\dbox{\\hevolvein{\\D{x}=f(x)\\syssep\\D{y}=g(x,y)}{q(x)}}{r(x,y)}}}\n  {\\lsequent{p(x)} {\\dbox{\\hevolvein{\\D{x}=f(x)}{q(x)}}{p(x)}}}\n}{}%\n\\end{calculus}\n\\end{calculuscollections}\n\n\\noindent\nwhere $y$ is new and \\m{\\hevolve{\\D{y}=g(x,y),y(0)=y_0}} has a solution $y:[0,\\infty)\\to\\reals^n$ for each $y_0$.\n\nThe derivation proceeds as follows (the middle premise uses \\irref{vacuousexists} with $y\\not\\in p(x)$):\n\n{%\n\\renewcommand{\\linferPremissSeparation}{~}%\n\\begin{sequentdeduction}[array]\n\\linfer[cut]\n  {\\lsequent{} {p(x)\\lbisubjunct\\lexists{y}{r(x,y)}}\n  !\\linfer[existsinst]%\n   {\\linfer[DG]\n    {\\linfer[existsinst]%\n     {\\linfer[M]\n        {\\linfer[vacuousexists]%\n         {\\lsequent{\\lexists{y}{r(x,y)}} {p(x)}}\n         {\\lsequent{r(x,y)} {p(x)}}\n        !\\lsequent{r(x,y)} {\\dbox{\\pevolvein{\\D{x}=f(x)\\syssep\\D{y}=g(x,y)}{q(x)}}{r(x,y)}}\n        }\n        {\\lsequent{r(x,y)} {\\dbox{\\pevolvein{\\D{x}=f(x)\\syssep\\D{y}=g(x,y)}{q(x)}}{p(x)}}}\n      }\n      {\\lsequent{r(x,y)} {\\lexists{y}{\\dbox{\\pevolvein{\\D{x}=f(x)\\syssep\\D{y}=g(x,y)}{q(x)}}{p(x)}}}}\n      }\n    {\\lsequent{r(x,y)} {\\dbox{\\pevolvein{\\D{x}=f(x)}{q(x)}}{p(x)}}}\n  }\n  {\\lsequent{\\lexists{y}{r(x,y)}} {\\dbox{\\pevolvein{\\D{x}=f(x)}{q(x)}}{p(x)}}}\n  }\n  {\\lsequent{p(x)} {\\dbox{\\pevolvein{\\D{x}=f(x)}{q(x)}}{p(x)}}}\n\\end{sequentdeduction}\n}%\nUsing the following duals of \\irref{allinst} and  \\irref{vacuousall} as well as monotonicity rule \\irref{M} \\cite{DBLP:journals\/tocl\/Platzer15} that derives from \\irref{G+K}:\n\n\\begin{calculuscollections}{\\columnwidth}\n  \\begin{calculus}\n      \\cinferenceRule[existsinst|$\\exists$i]{existential instantiation}\n       {p(f) \\limply (\\lexists{x}{p(x)})}\n       {}\n       \\cinferenceRule[vacuousexists|V$_\\exists$]{vacuous existential quantifier}\n       {\\lexists{x}{p} \\limply p}\n       {}%\n      \\cinferenceRule[M|M]{$\\dbox{}{}$ monotonic \/ $\\dbox{}{}$-generalization} %\n      {\\linferenceRule[formula]\n        {\\phi\\limply\\psi}\n        {\\dbox{\\alpha}{\\phi}\\limply\\dbox{\\alpha}{\\psi}}\n      }{}\n  \\end{calculus}\n\\end{calculuscollections}\n\n\\section{Introduction}\n\n\\emph{Differential dynamic logic} (\\dL) \\cite{DBLP:journals\/jar\/Platzer08,DBLP:conf\/lics\/Platzer12b} is a logic for proving correctness properties of hybrid systems.\nIt has a sound and complete proof calculus relative to differential equations \\cite{DBLP:journals\/jar\/Platzer08,DBLP:conf\/lics\/Platzer12b} and a sound and complete proof calculus relative to discrete systems \\cite{DBLP:conf\/lics\/Platzer12b}.\nBoth sequent calculi \\cite{DBLP:journals\/jar\/Platzer08} and Hilbert-type axiomatizations \\cite{DBLP:conf\/lics\/Platzer12b} have been presented for \\dL but only the former has been implemented.\nThe implementation of \\dL's sequent calculus in \\KeYmaera\\iflongversion \\cite{DBLP:conf\/cade\/PlatzerQ08}\\fi makes it straightforward for users to prove properties of hybrid systems, because it provides rules performing natural decompositions for each operator.\nThe downside is that the implementation of the rule schemata and their side conditions on occurrence constraints and relations of reading and writing of variables as well as rule applications in context is nontrivial and inflexible in \\KeYmaera.\n\nThe goal of this paper is to identify how to make it straightforward to implement the axioms and proof rules of differential dynamic logic by writing down a finite list of \\emph{axioms} (concrete formulas, not axiom schemata that represent an infinite list of axioms subject to sophisticated soundness-critical schema variable matching implementations).\nThey require multiple axioms to be combined with one another to obtain the effect that a user would want for proving a hybrid system conjecture.\nThis paper argues that this is still a net win for hybrid systems, because a substantially simpler prover core is easier to implement correctly, and the need to combine multiple axioms to obtain user-level proof steps can be achieved equally well by appropriate tactics, which are not soundness-critical.\n\nTo achieve this goal, this paper follows observations for differential game logic \\cite{DBLP:journals\/corr\/Platzer14:dGL} that highlight the significance and elegance of \\emph{uniform substitutions}, a classical proof rule for first-order logic \\cite[\\S35,40]{Church_1956}.\nUniform substitutions uniformly instantiate predicate and function symbols with formulas and terms, respectively, as functions of their  arguments.\nIn the presence of the nontrivial binding structure that nondeterminism and differential equations of hybrid programs induce for the dynamic modalities of differential dynamic logic, flexible but sound uniform substitutions become more complex for \\dL, but can still be read off elegantly from its static semantics.\nIn fact, \\dL's static semantics is solely captured\\footnote{\nThis approach is dual to other successful ways of solving the intricacies and subtleties of substitutions \\cite{DBLP:journals\/jsl\/Church40,DBLP:journals\/jsl\/Henkin53} by imposing occurrence side conditions on axiom schemata and proof rules, which is what uniform substitutions can get rid of.\n}\nin the implementation of uniform substitution (and bound variable renaming), thereby leading to a completely modular proof calculus.\n\nThis paper introduces a static and dynamic semantics for \\emph{differential-form} \\dL, proves coincidence lemmas and uniform substitution lemmas, culminating in a soundness proof for uniform substitutions (\\rref{sec:usubst}).\nIt exploits the new \\emph{differential forms} that this paper adds to \\dL for internalizing differential invariants \\cite{DBLP:journals\/logcom\/Platzer10}, differential cuts \\cite{DBLP:journals\/logcom\/Platzer10,DBLP:journals\/lmcs\/Platzer12},  differential ghosts \\cite{DBLP:journals\/lmcs\/Platzer12}, differential substitutions, total differentials and Lie-derivations \\cite{DBLP:journals\/logcom\/Platzer10,DBLP:journals\/lmcs\/Platzer12} as first-class citizens in \\dL, culminating in entirely modular axioms for differential equations and a superbly modular soundness proof (\\rref{sec:dL-axioms}).\nThis approach is to be contrasted with earlier approaches for differential invariants that were based on complex built-in rules \\cite{DBLP:journals\/logcom\/Platzer10,DBLP:journals\/lmcs\/Platzer12}.\nThe relationship to related work from previous presentations of differential dynamic logic \\cite{DBLP:journals\/jar\/Platzer08,DBLP:conf\/lics\/Platzer12b} continues to apply except that \\dL now internalizes differential equation reasoning axiomatically via differential forms.\n\n\\newsavebox{\\Rval}%\n\\sbox{\\Rval}{$\\scriptstyle\\mathbb{R}$}\n\\irlabel{qear|\\usebox{\\Rval}}\n\n\\newsavebox{\\USarg}%\n\\sbox{\\USarg}{$\\boldsymbol{\\cdot}$}\n\n\\newsavebox{\\UScarg}%\n\\sbox{\\UScarg}{$\\boldsymbol{\\_}$}\n\n\\newsavebox{\\Lightningval}%\n\\sbox{\\Lightningval}{$\\scriptstyle\\textcolor{red}{\\lightning}$}\n\\irlabel{clash|clash\\usebox{\\Lightningval}} %\n\\irlabel{unsound|\\usebox{\\Lightningval}}  %\n\n\n\\section{Differential-Form Differential Dynamic Logic}\n\\subsection{Syntax}\nFormulas and hybrid programs (\\HPs) of \\dL are defined by simultaneous induction based on the following definition of terms.\nSimilar simultaneous inductions are used throughout the proofs for \\dL.\nThe set of all \\emph{variables} is $\\mathcal{V}$.\nFor any $V\\subseteq\\mathcal{V}$ is \\(\\D{V}\\mdefeq\\{\\D{x} : x\\in V\\}\\) the set of \\emph{differential symbols} $\\D{x}$ for the variables in $V$.\nFunction symbols are written $f,g,h$, predicate symbols $p,q,r$, and variables $x,y,z\\in\\mathcal{V}$ with differential symbols $\\D{x},\\D{y},\\D{z}\\in\\D{\\mathcal{V}}$.\nProgram constants are $a,b,c$.\n\n\\begin{definition}[Terms]\n\\emph{Terms} are defined by this grammar\n(with $\\theta,\\eta,\\theta_1,\\dots,\\theta_k$ as terms, $x\\in\\mathcal{V}$ as variable, $\\D{x}\\in\\D{\\mathcal{V}}$ differential symbol, and $f$ function symbol):\n\\[\n  \\theta,\\eta ~\\mathrel{::=}~\n  x\n  ~|~ \\D{x}\n  ~|~\n  f(\\theta_1,\\dots,\\theta_k)\n  ~|~\n  \\theta+\\eta\n  ~|~\n  \\theta\\cdot\\eta\n  ~|~ \\der{\\theta}\n\\]\n\\end{definition}\nNumber literals such as 0,1 are allowed as function symbols without arguments that are always interpreted as the numbers they denote.\nBeyond differential symbols $\\D{x}$, \\emph{differential-form \\dL} allows \\emph{differentials} \\(\\der{\\theta}\\) of terms $\\theta$ as terms for the purpose of axiomatically internalizing reasoning about differential equations.\n\\begin{definition}[Hybrid program]\n\\emph{Hybrid programs} (\\HPs) are defined by the following grammar (with $\\alpha,\\beta$ as \\HPs, program constant $a$, variable $x$, term $\\theta$ possibly containing $x$, and formula $\\psi$ of first-order logic of real arithmetic):\n\\[\n  \\alpha,\\beta ~\\mathrel{::=}~\n  a~|~\n  \\pupdate{\\pumod{x}{\\theta}}\n  ~|~ \\Dupdate{\\Dumod{\\D{x}}{\\theta}}\n  ~|~\n  \\ptest{\\psi}\n  ~|~\n  \\pevolvein{\\D{x}=\\genDE{x}}{\\psi}\n  ~|~\n  \\alpha\\cup\\beta\n  ~|~\n  \\alpha;\\beta\n  ~|~\n  \\prepeat{\\alpha}\n\\]\n\\end{definition}\n\\emph{Assignments} \\m{\\pupdate{\\pumod{x}{\\theta}}} of $\\theta$ to variable $x$, \\emph{tests} \\m{\\ptest{\\psi}} of the formula $\\psi$ in the current state, \\emph{differential equations} \\(\\pevolvein{\\D{x}=\\genDE{x}}{\\psi}\\) restricted to the evolution domain constraint $\\psi$, \\emph{nondeterministic choices} \\(\\pchoice{\\alpha}{\\beta}\\), \\emph{sequential compositions} \\(\\alpha;\\beta\\), and \\emph{nondeterministic repetition} \\(\\prepeat{\\alpha}\\) are as usual in \\dL \\cite{DBLP:journals\/jar\/Platzer08,DBLP:conf\/lics\/Platzer12b}.\nThe effect of the \\emph{differential assignment} \\m{\\Dupdate{\\Dumod{\\D{x}}{\\theta}}} to differential symbol $\\D{x}$ is similar to the effect of the assignment \\m{\\pupdate{\\pumod{x}{\\theta}}} to variable $x$, except that it changes the value of the differential symbol $\\D{x}$ around instead of the value of $x$.\nIt is not to be confused with the differential equation \\(\\pevolve{\\D{x}=\\genDE{x}}\\), which will follow said differential equation continuously for an arbitrary amount of time.\nThe differential assignment \\m{\\Dupdate{\\Dumod{\\D{x}}{\\theta}}}, instead, only assigns the value of $\\theta$ to the differential symbol $\\D{x}$ discretely once at an instant of time.\nProgram constants $a$ are uninterpreted, i.e.\\ their behavior depends on the interpretation in the same way that the values of function symbols  $f$ and predicate symbols $p$ depends on their interpretation.\n\n\\begin{definition}[\\dL formula]\nThe \\emph{formulas of (differential-form) differential dynamic logic} ({\\dL}) are defined by the grammar\n(with \\dL formulas $\\phi,\\psi$, terms $\\theta,\\eta,\\theta_1,\\dots,\\theta_k$, predicate symbol $p$, \\predicational symbol $C$, variable $x$, \\HP $\\alpha$):\n  \\[\n  \\phi,\\psi ~\\mathrel{::=}~\n  \\theta\\geq\\eta ~|~\n  p(\\theta_1,\\dots,\\theta_k) ~|~\n  \\contextapp{C}{\\phi} ~|~\n  \\lnot \\phi ~|~\n  \\phi \\land \\psi ~|~\n  \\lforall{x}{\\phi} ~|~ \n  \\lexists{x}{\\phi} ~|~\n  \\dbox{\\alpha}{\\phi}\n  ~|~ \\ddiamond{\\alpha}{\\phi}\n  \\]\n\\end{definition}\nOperators $>,\\leq,<,\\lor,\\limply,\\lbisubjunct$ are definable, e.g., \\(\\phi\\limply\\psi\\) as \\(\\lnot(\\phi\\land\\lnot\\psi)\\).\nLikewise \\(\\dbox{\\alpha}{\\phi}\\) is equivalent to \\(\\lnot\\ddiamond{\\alpha}{\\lnot\\phi}\\) and \\(\\lforall{x}{\\phi}\\) equivalent to \\(\\lnot\\lexists{x}{\\lnot\\phi}\\).\nThe modal formula \\(\\dbox{\\alpha}{\\phi}\\) expresses that $\\phi$ holds after all runs of $\\alpha$, while the dual \\(\\ddiamond{\\alpha}{\\phi}\\) expresses that there is a run of $\\alpha$ after which $\\phi$ holds.\n\\emph{\\Predicational symbols} $C$ (with formula $\\phi$ as argument), i.e.\\ higher-order predicate symbols that bind all variables of $\\phi$, are unnecessary but internalize contextual congruence reasoning efficiently.\n\n\\subsection{Dynamic Semantics}\n\nA state is a mapping from variables $\\mathcal{V}$ \nand differential symbols $\\D{\\mathcal{V}}$ to $\\reals$.\nThe set of states is denoted \\(\\linterpretations{\\Sigma}{V}\\).\nLet\n\\m{\\iget[state]{\\imodif[state]{\\vdLint[const=I,state=\\nu]}{x}{r}}} denote the state that agrees with state~$\\iget[state]{\\vdLint[const=I,state=\\nu]}$ except for the value of variable~\\m{x}, which is changed to~\\m{r \\in \\reals}, and accordingly for \\m{\\iget[state]{\\imodif[state]{\\vdLint[const=I,state=\\nu]}{x'}{r}}}. %\nThe interpretation of a function symbol $f$ with arity $n$ (i.e.\\ with $n$ arguments) is a smooth function \\(\\iget[const]{\\vdLint[const=I,state=\\nu]}(f):\\reals^n\\to\\reals\\) of $n$ arguments.\n\n\\begin{definition}[Semantics of terms] \\label{def:dL-valuationTerm}\nFor each interpretation $\\iget[const]{\\vdLint[const=I,state=\\nu]}$, the \\emph{semantics of a term} $\\theta$ in a state $\\iget[state]{\\vdLint[const=I,state=\\nu]}\\in\\linterpretations{\\Sigma}{V}$ is its value in $\\reals$.\nIt is defined inductively as follows\n\\begin{compactenum}\n\\item \\m{\\ivaluation{\\vdLint[const=I,state=\\nu]}{x} = \\iget[state]{\\vdLint[const=I,state=\\nu]}(x)} for variable $x\\in\\mathcal{V}$\n\\item \\m{\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\D{x}} = \\iget[state]{\\vdLint[const=I,state=\\nu]}(\\D{x})} for differential symbol $\\D{x}\\in\\D{\\mathcal{V}}$\n\\item \\(\\ivaluation{\\vdLint[const=I,state=\\nu]}{f(\\theta_1,\\dots,\\theta_k)} = \\iget[const]{\\vdLint[const=I,state=\\nu]}(f)\\big(\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\theta_1},\\dots,\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\theta_k}\\big)\\) for function symbol $f$\n\\item \\m{\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\theta+\\eta} = \\ivaluation{\\vdLint[const=I,state=\\nu]}{\\theta} + \\ivaluation{\\vdLint[const=I,state=\\nu]}{\\eta}}\n\\item \\m{\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\theta\\cdot\\eta} = \\ivaluation{\\vdLint[const=I,state=\\nu]}{\\theta} \\cdot \\ivaluation{\\vdLint[const=I,state=\\nu]}{\\eta}}\n\\item\n\\m{\n\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\der{\\theta}}\n\\displaystyle\n=\n\\newcommand{\\vdLint[const=I,state=]}{\\vdLint[const=I,state=]}\n\\sum_x \\iget[state]{\\vdLint[const=I,state=\\nu]}(\\D{x}) \\Dp[x]{\\ivaluation{\\vdLint[const=I,state=]}{\\theta}}(\\iget[state]{\\vdLint[const=I,state=\\nu]})\n=\n\\def\\Im{\\imodif[state]{\\vdLint[const=I,state=\\nu]}{x}{X}}%\n\\sum_x \\iget[state]{\\vdLint[const=I,state=\\nu]}(\\D{x})\n \\itimes \\Dp[X]{\\ivaluation{\\Im}{\\theta}}\n}\n\\end{compactenum}\n\\end{definition}\n\n\\noindent\nTime-derivatives are undefined in an isolated state $\\iget[state]{\\vdLint[const=I,state=\\nu]}$.\nThe clou is that differentials can still be given a local semantics:\n\\m{\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\der{\\theta}}} is the sum of all (analytic) spatial partial derivatives of the value of $\\theta$ by all variables $x$ (or rather their values $X$) multiplied by the corresponding tangent described by the value $\\iget[state]{\\vdLint[const=I,state=\\nu]}(\\D{x})$ of differential symbol $\\D{x}$.\nThat sum over all variables $x\\in\\mathcal{V}$ has finite support, because $\\theta$ only mentions finitely many variables $x$ and the partial derivative by variables $x$ that do not occur in $\\theta$ is 0.\nThe spatial derivatives exist since $\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\theta}$ is a composition of smooth functions, so smooth.\nThus, the semantics of \\m{\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\der{\\theta}}} is the \\emph{differential}\\footnote{%\nA slight abuse of notation rewrites the differential as \\(\\ivalues{\\vdLint[const=I,state=\\nu]}{\\der{\\theta}} = d\\ivalues{\\vdLint[const=I,state=\\nu]}{\\theta} = \\sum_{i=1}^n \\Dp[x^i]{\\ivalues{\\vdLint[const=I,state=\\nu]}{\\theta}} dx^i\\)\nwhen $x^1,\\dots,x^n$ are the variables in $\\theta$ and their differentials \\(dx^i\\) form the basis of the cotangent space, which, when evaluated at a point $\\iget[state]{\\vdLint[const=I,state=\\nu]}$ whose values \\(\\iget[state]{\\vdLint[const=I,state=\\nu]}(\\D{x})\\) determine the tangent vector alias vector field, coincides with \\rref{def:dL-valuationTerm}.\n}\nof (the value of) $\\theta$, hence a differential one-form giving a real value for each tangent vector (i.e.\\ vector field) described by the values \\(\\iget[state]{\\vdLint[const=I,state=\\nu]}(\\D{x})\\).\nThe values \\m{\\iget[state]{\\vdLint[const=I,state=\\nu]}(\\D{x})} of the differential symbols $\\D{x}$ describe an arbitrary tangent vector or vector field.\nAlong the flow of (the vector field of a) differential equation, though, the value of the differential \\m{\\der{\\theta}} coincides with the analytic time-derivative of $\\theta$ (\\rref{lem:differentialLemma}).\nThe interpretation of predicate symbol $p$ with arity $n$ is an $n$-ary relation \\(\\iget[const]{\\vdLint[const=I,state=\\nu]}(p)\\subseteq\\reals^n\\).\nThe interpretation of \\predicational symbol $C$ is a functional \\(\\iget[const]{\\vdLint[const=I,state=\\nu]}(C)\\) mapping subsets \\(M\\subseteq\\linterpretations{\\Sigma}{V}\\) to subsets \\(\\iget[const]{\\vdLint[const=I,state=\\nu]}(C)(M)\\subseteq\\linterpretations{\\Sigma}{V}\\).\n\n\\begin{definition}[\\dL semantics] \\label{def:dL-valuation}\nThe \\emph{semantics of a \\dL formula} $\\phi$, for each interpretation $\\iget[const]{\\vdLint[const=I,state=\\nu]}$ with a corresponding set of states $\\linterpretations{\\Sigma}{V}$, is the subset \\m{\\imodel{\\vdLint[const=I,state=\\nu]}{\\phi}\\subseteq\\linterpretations{\\Sigma}{V}} of states in which $\\phi$ is true.\nIt is defined inductively as follows\n\\begin{compactenum}\n\\item \\(\\imodel{\\vdLint[const=I,state=\\nu]}{\\theta\\geq\\eta} = \\{\\iget[state]{\\vdLint[const=I,state=\\nu]} \\in \\linterpretations{\\Sigma}{V} \\with \\ivaluation{\\vdLint[const=I,state=\\nu]}{\\theta}\\geq\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\eta}\\}\\)\n\\item \\(\\imodel{\\vdLint[const=I,state=\\nu]}{p(\\theta_1,\\dots,\\theta_k)} = \\{\\iget[state]{\\vdLint[const=I,state=\\nu]} \\in \\linterpretations{\\Sigma}{V} \\with (\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\theta_1},\\dots,\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\theta_k})\\in\\iget[const]{\\vdLint[const=I,state=\\nu]}(p)\\}\\)\n\\item \\(\\imodel{\\vdLint[const=I,state=\\nu]}{\\contextapp{C}{\\phi}} = \\iget[const]{\\vdLint[const=I,state=\\nu]}(C)\\big(\\imodel{\\vdLint[const=I,state=\\nu]}{\\phi}\\big)\\) for \\predicational symbol $C$\n\\item \\(\\imodel{\\vdLint[const=I,state=\\nu]}{\\lnot\\phi} = \\scomplement{(\\imodel{\\vdLint[const=I,state=\\nu]}{\\phi})}\n= \\linterpretations{\\Sigma}{V}\\setminus\\imodel{\\vdLint[const=I,state=\\nu]}{\\phi}\\) \n\\item \\(\\imodel{\\vdLint[const=I,state=\\nu]}{\\phi\\land\\psi} = \\imodel{\\vdLint[const=I,state=\\nu]}{\\phi} \\cap \\imodel{\\vdLint[const=I,state=\\nu]}{\\psi}\\)\n\\item\n\\(\\def\\Im{\\imodif[state]{\\vdLint[const=I,state=\\nu]}{x}{r}}%\n\\imodel{\\vdLint[const=I,state=\\nu]}{\\lexists{x}{\\phi}} =  \\{\\iget[state]{\\vdLint[const=I,state=\\nu]} \\in \\linterpretations{\\Sigma}{V} \\with \\iget[state]{\\Im} \\in \\imodel{\\vdLint[const=I,state=\\nu]}{\\phi} ~\\text{for some}~r\\in\\reals\\}\\)\n\n\\item \\(\\imodel{\\vdLint[const=I,state=\\nu]}{\\ddiamond{\\alpha}{\\phi}} = \\iaccess[\\alpha]{\\vdLint[const=I,state=\\nu]}\\compose\\imodel{\\vdLint[const=I,state=\\nu]}{\\phi}\\)\n\\(=\\{\\iget[state]{\\vdLint[const=I,state=\\nu]} \\with \\imodels{\\vdLint[const=I,state=\\omega]}{\\phi} ~\\text{for some}~\\iget[state]{\\vdLint[const=I,state=\\omega]}~\\text{such that}~\\iaccessible[\\alpha]{\\vdLint[const=I,state=\\nu]}{\\vdLint[const=I,state=\\omega]}\\}\\)\n\n\\item \\(\\imodel{\\vdLint[const=I,state=\\nu]}{\\dbox{\\alpha}{\\phi}} = \\imodel{\\vdLint[const=I,state=\\nu]}{\\lnot\\ddiamond{\\alpha}{\\lnot\\phi}}\\)\n\\(=\\{\\iget[state]{\\vdLint[const=I,state=\\nu]} \\with \\imodels{\\vdLint[const=I,state=\\omega]}{\\phi} ~\\text{for all}~\\iget[state]{\\vdLint[const=I,state=\\omega]}~\\text{such that}~\\iaccessible[\\alpha]{\\vdLint[const=I,state=\\nu]}{\\vdLint[const=I,state=\\omega]}\\}\\)\n\n\\end{compactenum}\nA \\dL formula $\\phi$ is \\emph{valid in $\\iget[const]{\\vdLint[const=I,state=\\nu]}$}, written \\m{\\iget[const]{\\vdLint[const=I,state=\\nu]}\\models{\\phi}}, iff \\m{\\imodel{\\vdLint[const=I,state=\\nu]}{\\phi}=\\linterpretations{\\Sigma}{V}}, i.e.\\ \\m{\\imodels{\\vdLint[const=I,state=\\nu]}{\\phi}} for all $\\iget[state]{\\vdLint[const=I,state=\\nu]}$.\nFormula $\\phi$ is \\emph{valid}, written \\m{\\entails\\phi}, iff \\m{\\iget[const]{\\vdLint[const=I,state=\\nu]}\\models{\\phi}} for all interpretations $\\iget[const]{\\vdLint[const=I,state=\\nu]}$.\n\\end{definition}\n\n\\noindent\nThe interpretation of a program constant $a$ is a state-transition relation \\(\\iget[const]{\\vdLint[const=I,state=\\nu]}(a)\\subseteq\\linterpretations{\\Sigma}{V}\\times\\linterpretations{\\Sigma}{V}\\),\nwhere \\(\\related{\\iget[const]{\\vdLint[const=I,state=\\nu]}(a)}{\\iget[state]{\\vdLint[const=I,state=\\nu]}}{\\iget[state]{\\vdLint[const=I,state=\\omega]}}\\) iff $a$ can run from initial state $\\iget[state]{\\vdLint[const=I,state=\\nu]}$ to final state $\\iget[state]{\\vdLint[const=I,state=\\omega]}$.\n\n\\begin{definition}[Transition semantics of \\HPs] \\label{def:HP-transition}\nFor each interpretation $\\iget[const]{\\vdLint[const=I,state=\\nu]}$, each \\HP $\\alpha$ is interpreted semantically as a binary transition relation \\m{\\iaccess[\\alpha]{\\vdLint[const=I,state=\\nu]}\\subseteq\\linterpretations{\\Sigma}{V}\\times\\linterpretations{\\Sigma}{V}} on states, defined inductively by\n\\begin{compactenum}\n\\item \\m{\\iaccess[a]{\\vdLint[const=I,state=\\nu]} = \\iget[const]{\\vdLint[const=I,state=\\nu]}(a)} for program constants $a$\n\\item\n\\(\\def\\Im{\\imodif[state]{\\vdLint[const=I,state=\\nu]}{x}{r}}%\n\\iaccess[\\pupdate{\\pumod{x}{\\theta}}]{\\vdLint[const=I,state=\\nu]} = \\{(\\iget[state]{\\vdLint[const=I,state=\\nu]},\\iget[state]{\\Im}) \\with r=\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\theta}\\}\n= \\{(\\iget[state]{\\vdLint[const=I,state=\\nu]},\\iget[state]{\\vdLint[const=I,state=\\omega]}) \\with \n \\iget[state]{\\vdLint[const=I,state=\\omega]}=\\iget[state]{\\vdLint[const=I,state=\\nu]}~\\text{except}~\\ivaluation{\\vdLint[const=I,state=\\omega]}{x}=\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\theta}\\}\n\\)\n\n\\item\n\\(\\def\\Im{\\imodif[state]{\\vdLint[const=I,state=\\nu]}{x'}{r}}%\n\\iaccess[\\Dupdate{\\Dumod{\\D{x}}{\\theta}}]{\\vdLint[const=I,state=\\nu]} = \\{(\\iget[state]{\\vdLint[const=I,state=\\nu]},\\iget[state]{\\Im}) \\with r=\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\theta}\\}\n= \\{(\\iget[state]{\\vdLint[const=I,state=\\nu]},\\iget[state]{\\vdLint[const=I,state=\\omega]}) \\with \n \\iget[state]{\\vdLint[const=I,state=\\omega]}=\\iget[state]{\\vdLint[const=I,state=\\nu]}~\\text{except}~\\ivaluation{\\vdLint[const=I,state=\\omega]}{\\D{x}}=\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\theta}\\}\n\\)\n\n\n\\item \\m{\\iaccess[\\ptest{\\psi}]{\\vdLint[const=I,state=\\nu]} = \\{(\\iget[state]{\\vdLint[const=I,state=\\nu]},\\iget[state]{\\vdLint[const=I,state=\\nu]}) \\with \\imodels{\\vdLint[const=I,state=\\nu]}{\\psi}\\}}\n\\item\n  \\m{\\iaccess[\\pevolvein{\\D{x}=\\genDE{x}}{\\psi}]{\\vdLint[const=I,state=\\nu]} = \\{(\\iget[state]{\\vdLint[const=I,state=\\nu]},\\iget[state]{\\vdLint[const=I,state=\\omega]}) \\with\n  \\imodels{\\DALint[const=I,flow=\\varphi]}{\\D{x}=\\genDE{x}\\land\\psi}},\n  i.e.\n  \\(\\imodels{\\Iff[\\zeta]}{\\D{x}=\\genDE{x}\\land\\psi}\\)\n  for all \\(0\\leq \\zeta\\leq r\\),\n  for some function \\m{\\iget[flow]{\\DALint[const=I,flow=\\varphi]}:[0,r]\\to\\linterpretations{\\Sigma}{V}} of some duration $r$ for which all\n  \\(\\iget[state]{\\Iff[\\zeta]}(\\D{x}) = \\D[t]{\\iget[state]{\\Iff[t]}(x)}(\\zeta)\\) exist\n  and \\(\\iget[state]{\\vdLint[const=I,state=\\nu]}=\\iget[state]{\\Iff[0]}\\) on $\\scomplement{\\{\\D{x}\\}}$ and \\(\\iget[state]{\\vdLint[const=I,state=\\omega]}=\\iget[state]{\\Iff[r]}\\)$\\}$;\n   i.e., $\\iget[flow]{\\DALint[const=I,flow=\\varphi]}$ solves the differential equation\n   and satisfies $\\psi$ at all times.\n   In case $r=0$, the only condition is that \\(\\iget[state]{\\vdLint[const=I,state=\\nu]}=\\iget[state]{\\vdLint[const=I,state=\\omega]}\\) on $\\scomplement{\\{\\D{x}\\}}$ and \\(\\iget[state]{\\vdLint[const=I,state=\\omega]}(\\D{x})=\\ivaluation{\\vdLint[const=I,state=\\omega]}{\\genDE{x}}\\) and \\(\\imodels{\\vdLint[const=I,state=\\omega]}{\\psi}\\).\n\n\\item \\m{\\iaccess[\\pchoice{\\alpha}{\\beta}]{\\vdLint[const=I,state=\\nu]} = \\iaccess[\\alpha]{\\vdLint[const=I,state=\\nu]} \\cup \\iaccess[\\beta]{\\vdLint[const=I,state=\\nu]}}\n\n\\item\n\\newcommand{\\iconcat[state=\\mu]{\\I}}{\\iconcat[state=\\mu]{\\vdLint[const=I,state=\\nu]}}\n\\m{\\iaccess[\\alpha;\\beta]{\\vdLint[const=I,state=\\nu]} = \\iaccess[\\alpha]{\\vdLint[const=I,state=\\nu]} \\compose\\iaccess[\\beta]{\\vdLint[const=I,state=\\nu]}}\n\\(= \\{(\\iget[state]{\\vdLint[const=I,state=\\nu]},\\iget[state]{\\vdLint[const=I,state=\\omega]}) : (\\iget[state]{\\vdLint[const=I,state=\\nu]},\\iget[state]{\\iconcat[state=\\mu]{\\I}}) \\in \\iaccess[\\alpha]{\\vdLint[const=I,state=\\nu]},  (\\iget[state]{\\iconcat[state=\\mu]{\\I}},\\iget[state]{\\vdLint[const=I,state=\\omega]}) \\in \\iaccess[\\beta]{\\vdLint[const=I,state=\\nu]}\\}\\)\n\n\\item \\m{\\iaccess[\\prepeat{\\alpha}]{\\vdLint[const=I,state=\\nu]} = \\displaystyle\n\\closureTransitive{\\big(\\iaccess[\\alpha]{\\vdLint[const=I,state=\\nu]}\\big)}\n=\n\\cupfold_{n\\in\\naturals}\\iaccess[{\\prepeat[n]{\\alpha}}]{\\vdLint[const=I,state=\\nu]}} \nwith \\m{\\prepeat[n+1]{\\alpha} \\mequiv \\prepeat[n]{\\alpha};\\alpha} and \\m{\\prepeat[0]{\\alpha}\\mequiv\\,\\ptest{\\ltrue}}\n\\end{compactenum}\nwhere $\\closureTransitive{\\rho}$ denotes the reflexive transitive closure of relation $\\rho$.\n\\end{definition}\nThe initial values \\(\\iget[state]{\\vdLint[const=I,state=\\nu]}(\\D{x})\\) of differential symbols $\\D{x}$ do \\emph{not} influence the behavior of\\\\ \\(\\iaccessible[\\pevolvein{\\D{x}=\\genDE{x}}{\\psi}]{\\vdLint[const=I,state=\\nu]}{\\vdLint[const=I,state=\\omega]}\\), because they may not be compatible with the time-derivatives for the differential equation, e.g. in \\m{\\Dupdate{\\Dumod{\\D{x}}{1}};\\pevolve{\\D{x}=2}}, with a $\\D{x}$ mismatch.\n\\iflongversion\nThe final values \\(\\iget[state]{\\vdLint[const=I,state=\\omega]}(\\D{x})\\) will coincide with the derivatives, though.\n\\fi\n\nFunctions and predicates are interpreted by $\\iget[const]{\\vdLint[const=I,state=\\nu]}$ and are only influenced indirectly by $\\iget[state]{\\vdLint[const=I,state=\\nu]}$ through the values of their arguments.\nSo \\(p(e)\\limply\\dbox{\\pupdate{\\pumod{x}{x+1}}}{p(e)}\\) is valid if $x$ is not in $e$ since the change in $x$ does not change whether $p(e)$ is true (\\rref{lem:coincidence-term}).\nBy contrast \\(p(x)\\limply\\dbox{\\pupdate{\\pumod{x}{x+1}}}{p(x)}\\) is invalid, since it is false when \\(\\iget[const]{\\vdLint[const=I,state=\\nu]}(p)=\\{d \\with d\\leq5\\}\\) and \\(\\iget[state]{\\vdLint[const=I,state=\\nu]}(x)=4.5\\).\nIf the semantics of $p$ were to depend on the state $\\iget[state]{\\vdLint[const=I,state=\\nu]}$, then there would be no discernible relationship between the truth-values of $p$ in different states, so not even \\(p\\limply\\dbox{\\pupdate{\\pumod{x}{x+1}}}{p}\\) would be valid.\n\n\\subsection{Static Semantics}\n\nThe static semantics of \\dL and \\HPs defines some aspects of their behavior that can be read off directly from their syntactic structure without running their programs or evaluating their dynamical effects.\nThe most important aspects of the static semantics concern free or bound occurrences of variables\\iflongversion (which are closely related to the notions of scope and definitions\/uses in compilers)\\fi.\nBound variables $x$ are those that are bound by $\\lforall{x}{}$or $\\lexists{x}{}$, but also those that are bound by modalities such as \\(\\dbox{\\pupdate{\\pumod{x}{5y}}}{}\\)\nor \\(\\ddiamond{\\pevolve{\\D{x}=1}}{}\\)\nor \\(\\dbox{\\pchoice{\\pumod{x}{1}}{\\pevolve{\\D{x}=1}}}{}\\)\nor \\(\\dbox{\\pchoice{\\pumod{x}{1}}{\\ptest{\\ltrue}}}{}\\).\n\nThe notions of free and bound variables are defined by simultaneous induction in the subsequent definitions: free variables for terms ($\\freevars{\\theta}$), formulas ($\\freevars{\\phi}$), and \\HPs ($\\freevars{\\alpha}$), as well as bound variables for formulas ($\\boundvars{\\phi}$) and for \\HPs ($\\boundvars{\\alpha}$).\nFor \\HPs, there will be a need to distinguish must-bound variables ($\\mustboundvars{\\alpha}$) that are bound\/written to on all executions of $\\alpha$ from (may-)bound variables ($\\boundvars{\\alpha}$) which are bound on some (not necessarily all) execution paths of $\\alpha$, such as in \\(\\dbox{\\pchoice{\\pumod{x}{1}}{(\\pupdate{\\pumod{x}{0}};\\pupdate{\\pumod{y}{x+1}})}}{}\\), which has bound variables \\(\\{x,y\\}\\) but must-bound variables only $\\{x\\}$, because $y$ is not written to in the first choice.\n\n\n\\begin{definition}[Bound variable] \\label{def:boundvars}\n  The set $\\boundvars{\\phi}\\subseteq\\mathcal{V}\\cup\\D{\\mathcal{V}}$ of \\emph{bound variables} of \\dL formula $\\phi$ is defined inductively as\n  \\begin{align*}\n  \\boundvars{\\theta\\geq\\eta} = \\boundvars{p(\\theta_1,\\dots,\\theta_k)} &= \\emptyset\\\\\n  \\boundvars{\\contextapp{C}{\\phi}} &= \\mathcal{V}\\cup\\D{\\mathcal{V}}\\\\ %\n  \\boundvars{\\lnot\\phi} &= \\boundvars{\\phi}\\\\\n  \\boundvars{\\phi\\land\\psi} &= \\boundvars{\\phi}\\cup\\boundvars{\\psi}\\\\\n  \\boundvars{\\lforall{x}{\\phi}} = \\boundvars{\\lexists{x}{\\phi}} &= \\{x\\}\\cup\\boundvars{\\phi}\\\\\n  \\boundvars{\\dbox{\\alpha}{\\phi}} = \\boundvars{\\ddiamond{\\alpha}{\\phi}} &= \\boundvars{\\alpha}\\cup\\boundvars{\\phi}\n  \\end{align*}\n\\end{definition}\n\n\\begin{definition}[Free variable] \\label{def:freevars}\n  The set $\\freevars{\\theta}\\subseteq\\mathcal{V}\\cup\\D{\\mathcal{V}}$ of \\emph{free variables} of term $\\theta$, i.e.\\ those that occur in $\\theta$, is defined inductively as\n  \\begin{align*}\n  \\freevars{x} &= \\{x\\}\\\\\n  \\freevars{\\D{x}} &= \\{\\D{x}\\}\\\\\n  \\freevars{f(\\theta_1,\\dots,\\theta_k)} &= \\freevars{\\theta_1}\\cup\\dots\\cup\\freevars{\\theta_k}\\\\\n  \\freevars{\\theta+\\eta} = \\freevars{\\theta\\cdot\\eta} &= \\freevars{\\theta}\\cup\\freevars{\\eta}\n  \\\\\n  \\freevars{\\der{\\theta}} &= \\freevars{\\theta} \\cup \\D{\\freevars{\\theta}}\n  \\end{align*}\n  The set $\\freevars{\\phi}$ of \\emph{free variables} of \\dL formula $\\phi$, i.e.\\ all those that occur in $\\phi$ outside the scope of quantifiers or modalities binding it, is defined inductively as\n  \\begin{align*}\n  \\freevars{\\theta\\geq\\eta} &= \\freevars{\\theta}\\cup\\freevars{\\eta}\\\\\n  \\freevars{p(\\theta_1,\\dots,\\theta_k)} &= \\freevars{\\theta_1}\\cup\\dots\\cup\\freevars{\\theta_k}\\\\\n  \\freevars{\\contextapp{C}{\\phi}} &= \\mathcal{V}\\cup\\D{\\mathcal{V}}\\\\ %\n  \\freevars{\\lnot\\phi} &= \\freevars{\\phi}\\\\\n  \\freevars{\\phi\\land\\psi} &= \\freevars{\\phi}\\cup\\freevars{\\psi}\\\\\n  \\freevars{\\lforall{x}{\\phi}} = \\freevars{\\lexists{x}{\\phi}} &= \\freevars{\\phi}\\setminus\\{x\\}\\\\\n  \\freevars{\\dbox{\\alpha}{\\phi}} = \\freevars{\\ddiamond{\\alpha}{\\phi}} &= \\freevars{\\alpha}\\cup(\\freevars{\\phi}\\setminus\\mustboundvars{\\alpha})\n  \\end{align*}\n\\end{definition}\nSoundness requires that\n\\(\\freevars{\\dbox{\\alpha}{\\phi}}\\) is not defined as \\(\\freevars{\\alpha}\\cup(\\freevars{\\phi}\\setminus\\boundvars{\\alpha})\\),\notherwise \\(\\dbox{\\pchoice{\\pupdate{\\pumod{x}{1}}}{\\pupdate{\\pumod{y}{2}}}}{x\\geq1}\\) would have no free variables,\nbut its truth-value depends on the initial value of $x$,\ndemanding \\(\\freevars{\\dbox{\\pchoice{\\pupdate{\\pumod{x}{1}}}{\\pupdate{\\pumod{y}{2}}}}{x\\geq1}}=\\{x\\}\\).\n\\iflongversion\n  The simpler definition \n  \\(\\freevars{\\dbox{\\alpha}{\\phi}} = \\freevars{\\alpha}\\cup\\freevars{\\phi}\\)\n  would be correct, but the results would be less precise then.\n  Likewise for $\\ddiamond{\\alpha}{\\phi}$.\n\\fi\n\\iflongversion\nSoundness requires \\(\\freevars{\\der{\\theta}}\\) not to be defined as \\(\\D{\\freevars{\\theta}}\\),\nbecause\nthe value of \\(\\der{x y}\\) depends on $\\{x,\\D{x},y,\\D{y}\\}$, since \\(\\der{x y}\\) equals \\(\\D{x} y + x \\D{y}\\) (\\rref{lem:derivationLemma}).\n\\fi\n\nThe static semantics defines which variables are free so may be read ($\\freevars{\\alpha}$), which are bound ($\\boundvars{\\alpha}$) so may be written to somewhere in $\\alpha$, and which are must-bound ($\\mustboundvars{\\alpha}$) so must be written to on all execution paths of $\\alpha$.\n\n\\begin{definition}[Bound variable] \\label{def:boundvars-HP}\n  The set $\\boundvars{\\alpha}\\subseteq\\mathcal{V}\\cup\\D{\\mathcal{V}}$ of \\emph{bound variables} of \\HP $\\alpha$, i.e.\\ all those that may potentially be written to, is defined inductively:\n  \\begin{align*}\n  \\boundvars{a} &= \\mathcal{V}\\cup\\D{\\mathcal{V}} &&\\text{for program constant $a$}\\\\\n  \\boundvars{\\pupdate{\\pumod{x}{\\theta}}} &= \\{x\\}\n  \\\\\n  \\boundvars{\\Dupdate{\\Dumod{\\D{x}}{\\theta}}} &= \\{\\D{x}\\}\n  \\\\\n  \\boundvars{\\ptest{\\psi}} &= \\emptyset\n  \\\\\n  \\boundvars{\\pevolvein{\\D{x}=\\genDE{x}}{\\psi}} &= \\{x,\\D{x}\\}\n  \\\\\n  \\boundvars{\\alpha\\cup\\beta} = \\boundvars{\\alpha;\\beta} &= \\boundvars{\\alpha}\\cup\\boundvars{\\beta}\n  \\\\\n  \\boundvars{\\prepeat{\\alpha}} &= \\boundvars{\\alpha}\n  \\end{align*}\n\\end{definition}\n\n\\begin{definition}[Must-bound variable] \\label{def:mustboundvar}\n  The set $\\mustboundvars{\\alpha}\\subseteq\\boundvars{\\alpha}\\subseteq\\mathcal{V}\\cup\\D{\\mathcal{V}}$ of \\emph{must-bound variables} of \\HP $\\alpha$, i.e.\\ all those that must be written to on all paths of $\\alpha$, is defined inductively as\n  \\begin{align*}\n  \\mustboundvars{a} &= \\emptyset &&\\text{for program constant $a$}\\\\\n    \\mustboundvars{\\alpha} &= \\boundvars{\\alpha} &&\\text{for other atomic \\HPs $\\alpha$}\n    \\\\\n  \\mustboundvars{\\alpha\\cup\\beta} &= \\mustboundvars{\\alpha}\\cap\\mustboundvars{\\beta}\n  \\\\\n  \\mustboundvars{\\alpha;\\beta} &= \\mustboundvars{\\alpha}\\cup\\mustboundvars{\\beta}\n  \\\\\n  \\mustboundvars{\\prepeat{\\alpha}} &= \\emptyset\n  \\end{align*}\n\\end{definition}\n\\iflongversion\nObviously, \\(\\mustboundvars{\\alpha}\\subseteq\\boundvars{\\alpha}\\).\nIf $\\alpha$ is only built by sequential compositions from atomic programs without program constants, then \\(\\mustboundvars{\\alpha}=\\boundvars{\\alpha}\\).\n\\fi\n\n\\begin{definition}[Free variable] \\label{def:freevars-HP}\n  The set $\\freevars{\\alpha}\\subseteq\\mathcal{V}\\cup\\D{\\mathcal{V}}$ of \\emph{free variables} of \\HP $\\alpha$, i.e.\\ all those that may potentially be read, is defined inductively as\n  \\begin{align*}\n  \\freevars{a} &= \\mathcal{V}\\cup\\D{\\mathcal{V}} &&\\hspace{-10pt}\\text{for program constant $a$}\\\\\n  \\freevars{\\pupdate{\\pumod{x}{\\theta}}}\n  = \\freevars{\\Dupdate{\\Dumod{\\D{x}}{\\theta}}} &= \\freevars{\\theta}\n  \\\\\n  \\freevars{\\ptest{\\psi}} &= \\freevars{\\psi}\n  \\\\\n  \\freevars{\\pevolvein{\\D{x}=\\genDE{x}}{\\psi}} &= \\{x\\}\\cup\\freevars{\\genDE{x}}\\cup\\freevars{\\psi}\n  \\\\\n  \\freevars{\\pchoice{\\alpha}{\\beta}} &= \\freevars{\\alpha}\\cup\\freevars{\\beta}\n  \\\\\n  \\freevars{\\alpha;\\beta} &= \\freevars{\\alpha}\\cup(\\freevars{\\beta}\\setminus\\mustboundvars{\\alpha})\n  \\\\\n  \\freevars{\\prepeat{\\alpha}} &= \\freevars{\\alpha}\n  \\end{align*}\n\\iflongversion\nThe \\emph{variables} of \\HP $\\alpha$, whether free or bound, are \\(\\vars{\\alpha}=\\freevars{\\alpha}\\cup\\boundvars{\\alpha}\\).\n\\fi\n\\end{definition}\n\\iflongversion\n  The simpler definition \n  \\(\\freevars{\\alpha\\cup\\beta} = \\freevars{\\alpha}\\cup\\freevars{\\beta}\\)\n  would be correct, but the results would be less precise then.\n\\fi\nUnlike $x$, the left-hand side $\\D{x}$ of differential equations is not added to the free variables of \\(\\freevars{\\pevolvein{\\D{x}=\\genDE{x}}{\\psi}}\\), because its behavior does not depend on the initial value of differential symbols $\\D{x}$, only the initial value of $x$.\n  Free and bound variables are the set of all variables $\\mathcal{V}$ and differential symbols $\\D{\\mathcal{V}}$ for program constants $a$, because their effect depends on the interpretation $\\iget[const]{\\vdLint[const=I,state=\\nu]}$, so may read and write all \\(\\freevars{a}=\\boundvars{a}=\\mathcal{V}\\cup\\D{\\mathcal{V}}\\) but not on all paths \\(\\mustboundvars{a}=\\emptyset\\).\n  Subsequent results about free and bound variables are, thus, vacuously true when program constants occur.\nCorresponding observations hold for \\predicational symbols.\n\nThe static semantics defines which variables are readable or writable.\nThere may not be any run of $\\alpha$ in which a variable is read or written to.\nIf $x\\not\\in\\freevars{\\alpha}$, though, then $\\alpha$  cannot read the value of $x$.\nIf $x\\not\\in\\boundvars{\\alpha}$, it cannot write to $x$.\n\\iflongversion\n\\rref{def:freevars-HP} is parsimonious.\nFor example, $x$ is not a free variable of the following program\n\\[\n(\\pchoice{\\pupdate{\\pumod{x}{1}}}{\\pupdate{\\pumod{x}{2}}}); \\pupdate{\\pumod{z}{x+y}}\n\\]\nbecause $x$ is never actually read, since $x$ must have been defined on \\emph{every} execution path of the first part before being read by the second part.\nNo execution of the above program, thus, depends on the initial value of $x$, which is why it is not a free variable.\nThis would have been different for the simpler definition\n\\[\n\\freevars{\\alpha;\\beta} = \\freevars{\\alpha}\\cup\\freevars{\\beta}\n\\]\nThere is a limit to the precision with which any static analysis can determine which variables are really read or written \\cite{DBLP:journals\/ams\/Rice53}.\nThe static semantics in \\rref{def:freevars-HP} will, e.g., call $x$ a free variable of the following program even though no execution could read it, because they fail test $\\ptest{\\lfalse}$ when running the branch reading $x$:\n\\[\n\\pupdate{\\pumod{z}{0}}; \\prepeat{(\\ptest{\\lfalse};\\pupdate{\\pumod{z}{z+x}})}\n\\]\n\\fi\n\nThe \\dfn{signature}, i.e.\\ set of function, predicate, \\predicational symbols, and program constants in $\\phi$ is denoted by \\(\\intsigns{\\phi}\\) (accordingly for terms and programs).\nIt is defined like $\\freevars{\\phi}$ except that all occurrences are free.\nVariables in $\\mathcal{V}\\cup\\D{\\mathcal{V}}$ are interpreted by state $\\iget[state]{\\vdLint[const=I,state=\\nu]}$.\nThe symbols in $\\intsigns{\\phi}$ are interpreted by interpretation $\\iget[const]{\\vdLint[const=I,state=\\nu]}$.\n\n\\subsection{Correctness of Static Semantics}\nThe following result reflects that \\HPs have bounded effect: for a variable $x$ to be modified during a run of $\\alpha$, $x$ needs the be a bound variable in \\HP $\\alpha$, i.e.\\ \\(x\\in\\boundvars{\\alpha}\\), so that $\\alpha$ can write to $x$.\nThe converse is not true, because $\\alpha$ may bind a variable $x$, e.g. by having an assignment to $x$, that never actually changes the value of $x$, such as \\(\\pupdate{\\pumod{x}{x}}\\) or because the assignment can never be executed.\n\\iflongversion\nThe following program, for example, binds $x$ but will never change the value of $x$ because there is no way of satisfying the test $\\ptest{\\lfalse}$:\n\\(\n\\pchoice{(\\ptest{\\lfalse}; \\pupdate{\\pumod{x}{42}})}{\\pupdate{\\pumod{z}{x+1}}}\n\\).\n\\fi\n\\begin{lemma}[Bound effect lemma] \\label{lem:bound}\n  If \\(\\iaccessible[\\alpha]{\\vdLint[const=I,state=\\nu]}{\\vdLint[const=I,state=\\omega]}\\), then \\(\\iget[state]{\\vdLint[const=I,state=\\nu]}=\\iget[state]{\\vdLint[const=I,state=\\omega]}\\) on $\\scomplement{\\boundvars{\\alpha}}$.\n\\end{lemma}\n\\begin{proofatend}\nThe proof is by a straightforward structural induction on $\\alpha$.\n\\begin{compactitem}\n\\item Since \\(\\boundvars{a} = \\mathcal{V}\\cup\\D{\\mathcal{V}}\\), the statement is vacuously true for program constant $a$, because \\(\\scomplement{\\boundvars{a}}=\\emptyset\\).\n\n\\item \\m{\\iaccessible[\\pupdate{\\pumod{x}{\\theta}}]{\\vdLint[const=I,state=\\nu]}{\\vdLint[const=I,state=\\omega]} = \\{(\\iget[state]{\\vdLint[const=I,state=\\nu]},\\iget[state]{\\vdLint[const=I,state=\\omega]}) \\with \\iget[state]{\\vdLint[const=I,state=\\omega]}=\\iget[state]{\\vdLint[const=I,state=\\nu]}~\\text{except that}~\\ivaluation{\\vdLint[const=I,state=\\omega]}{x}=\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\theta}\\}} \nimplies that \\(\\iget[state]{\\vdLint[const=I,state=\\nu]}=\\iget[state]{\\vdLint[const=I,state=\\omega]}\\) except for \\(\\{x\\}=\\boundvars{\\pupdate{\\pumod{x}{\\theta}}}\\).\n\n\\item \\m{\\iaccessible[\\Dupdate{\\Dumod{\\D{x}}{\\theta}}]{\\vdLint[const=I,state=\\nu]}{\\vdLint[const=I,state=\\omega]} = \\{(\\iget[state]{\\vdLint[const=I,state=\\nu]},\\iget[state]{\\vdLint[const=I,state=\\omega]}) \\with \\iget[state]{\\vdLint[const=I,state=\\omega]}=\\iget[state]{\\vdLint[const=I,state=\\nu]}~\\text{except that}~\\ivaluation{\\vdLint[const=I,state=\\omega]}{\\D{x}}=\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\theta}\\}} \nimplies that \\(\\iget[state]{\\vdLint[const=I,state=\\nu]}=\\iget[state]{\\vdLint[const=I,state=\\omega]}\\) except for \\(\\{\\D{x}\\}=\\boundvars{\\Dupdate{\\Dumod{\\D{x}}{\\theta}}}\\).\n\n\\item \\m{\\iaccessible[\\ptest{\\psi}]{\\vdLint[const=I,state=\\nu]}{\\vdLint[const=I,state=\\nu]} = \\{(\\iget[state]{\\vdLint[const=I,state=\\nu]},\\iget[state]{\\vdLint[const=I,state=\\nu]}) \\with \\imodels{\\vdLint[const=I,state=\\nu]}{\\psi} ~\\text{i.e.}~ \\iget[state]{\\vdLint[const=I,state=\\nu]}\\in\\imodel{\\vdLint[const=I,state=\\nu]}{\\psi}\\}}\nfits to \\(\\boundvars{\\ptest{\\psi}}=\\emptyset\\)\n\n\\item\n  \\m{\\iaccessible[\\pevolvein{\\D{x}=\\genDE{x}}{\\psi}]{\\vdLint[const=I,state=\\nu]}{\\vdLint[const=I,state=\\omega]}} \n  implies that \\(\\iget[state]{\\vdLint[const=I,state=\\nu]}=\\iget[state]{\\vdLint[const=I,state=\\omega]}\\) except for the variables with differential equations, which are \\(\\{x,\\D{x}\\}=\\boundvars{\\pevolvein{\\D{x}=\\genDE{x}}{\\psi}}\\)\n  observing that \\(\\iget[state]{\\vdLint[const=I,state=\\nu]}(\\D{x})\\) and \\(\\iget[state]{\\vdLint[const=I,state=\\omega]}(\\D{x})\\) may differ by \\rref{def:HP-transition}.\n\n\\item \\m{\\iaccessible[\\pchoice{\\alpha}{\\beta}]{\\vdLint[const=I,state=\\nu]}{\\vdLint[const=I,state=\\omega]} = \\iaccess[\\alpha]{\\vdLint[const=I,state=\\nu]} \\cup \\iaccess[\\beta]{\\vdLint[const=I,state=\\nu]}}\nimplies \\(\\iaccessible[\\alpha]{\\vdLint[const=I,state=\\nu]}{\\vdLint[const=I,state=\\omega]}\\) or \\(\\iaccessible[\\beta]{\\vdLint[const=I,state=\\nu]}{\\vdLint[const=I,state=\\omega]}\\),\nwhich, by induction hypothesis,\nimplies \\(\\iget[state]{\\vdLint[const=I,state=\\nu]}=\\iget[state]{\\vdLint[const=I,state=\\omega]}\\) on $\\scomplement{\\boundvars{\\alpha}}$\nor\n\\(\\iget[state]{\\vdLint[const=I,state=\\nu]}=\\iget[state]{\\vdLint[const=I,state=\\omega]}\\) on $\\scomplement{\\boundvars{\\beta}}$, respectively.\nEither case implies \\(\\iget[state]{\\vdLint[const=I,state=\\nu]}=\\iget[state]{\\vdLint[const=I,state=\\omega]}\\) on $\\scomplement{(\\boundvars{\\alpha}\\cup\\boundvars{\\beta})}=\\scomplement{\\boundvars{\\pchoice{\\alpha}{\\beta}}}$.\n\n\\item\n\\newcommand{\\iconcat[state=\\mu]{\\I}}{\\dLint[state=\\mu]}%\n\\m{\\iaccessible[\\alpha;\\beta]{\\vdLint[const=I,state=\\nu]}{\\vdLint[const=I,state=\\omega]} = \\iaccess[\\alpha]{\\vdLint[const=I,state=\\nu]} \\compose\\iaccess[\\beta]{\\vdLint[const=I,state=\\nu]}},\ni.e.\\ there is a $\\iget[state]{\\iconcat[state=\\mu]{\\I}}$ such that \\(\\iaccessible[\\alpha]{\\vdLint[const=I,state=\\nu]}{\\iconcat[state=\\mu]{\\I}}\\) and \\(\\iaccessible[\\beta]{\\iconcat[state=\\mu]{\\I}}{\\vdLint[const=I,state=\\omega]}\\).\nSo, by induction hypothesis, \\(\\iget[state]{\\vdLint[const=I,state=\\nu]}=\\iget[state]{\\iconcat[state=\\mu]{\\I}}\\) on $\\scomplement{\\boundvars{\\alpha}}$ and \\(\\iget[state]{\\iconcat[state=\\mu]{\\I}}=\\iget[state]{\\vdLint[const=I,state=\\omega]}\\) on $\\scomplement{\\boundvars{\\beta}}$.\nBy transitivity, \\(\\iget[state]{\\vdLint[const=I,state=\\nu]}=\\iget[state]{\\vdLint[const=I,state=\\omega]}\\) on $\\scomplement{(\\boundvars{\\alpha}\\cup\\boundvars{\\beta})}=\\scomplement{\\boundvars{\\alpha;\\beta}}$.\n\n\\item\n\\renewcommand{\\iconcat[state=\\mu]{\\I}}[1][]{\\dLint[state=\\nu_{#1}]}%\n\\m{\\iaccessible[\\prepeat{\\alpha}]{\\vdLint[const=I,state=\\nu]}{\\vdLint[const=I,state=\\omega]} = \\displaystyle\\cupfold_{n\\in\\naturals}\\iaccess[{\\prepeat[n]{\\alpha}}]{\\vdLint[const=I,state=\\nu]}}, then there is an $n\\in\\naturals$ and a sequence \\(\\iget[state]{\\iconcat[state=\\mu]{\\I}[0]}=\\iget[state]{\\vdLint[const=I,state=\\nu]},\\iget[state]{\\iconcat[state=\\mu]{\\I}[1]},\\dots,\\iget[state]{\\iconcat[state=\\mu]{\\I}[n]}=\\iget[state]{\\vdLint[const=I,state=\\omega]}\\) such that \\(\\iaccessible[\\alpha]{\\iconcat[state=\\mu]{\\I}[i]}{\\iconcat[state=\\mu]{\\I}[i+1]}\\) for all $i<n$.\nBy $n$ uses of the induction hypothesis, \\(\\iget[state]{\\iconcat[state=\\mu]{\\I}[i]}=\\iget[state]{\\iconcat[state=\\mu]{\\I}[i+1]}\\) on $\\scomplement{\\boundvars{\\alpha}}$ for all $i<n$.\nThus, \\(\\iget[state]{\\vdLint[const=I,state=\\nu]}=\\iget[state]{\\iconcat[state=\\mu]{\\I}[0]}=\\iget[state]{\\iconcat[state=\\mu]{\\I}[n]}=\\iget[state]{\\vdLint[const=I,state=\\omega]}\\) on $\\scomplement{\\boundvars{\\alpha}}=\\scomplement{\\boundvars{\\prepeat{\\alpha}}}$.\n\n\\end{compactitem}\n\\end{proofatend}\nSimilarly, only $\\boundvars{\\phi}$ change their value during the evaluation of formulas.\n\nThe value of a term only depends on the values of its free variables.\nWhen evaluating a term $\\theta$ in two states $\\iget[state]{\\vdLint[const=I,state=\\nu]}$, $\\iget[state]{\\Ialt}$ that differ widely but agree on the free variables $\\freevars{\\theta}$ of $\\theta$, the values of $\\theta$ in both states coincide.\nAccordingly for different interpretations \\(\\iget[const]{\\vdLint[const=I,state=\\nu]},\\iget[const]{\\Ialt}\\) that agree on the symbols $\\intsigns{\\theta}$ that occur in $\\theta$.\n\n\\begin{lemma}[Coincidence lemma] \\label{lem:coincidence-term}\n  If \\(\\iget[state]{\\vdLint[const=I,state=\\nu]}=\\iget[state]{\\Ialt}\\) on $\\freevars{\\theta}$\n  and \\(\\iget[const]{\\vdLint[const=I,state=\\nu]}=\\iget[const]{\\Ialt}\\) on $\\intsigns{\\theta}$, then\n  \\m{\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\theta}=\\ivaluation{\\Ialt}{\\theta}}.\n\\end{lemma}\n\\begin{proofatend}\nThe proof is by structural induction on $\\theta$.\n\\begin{compactitem}\n\\item \\m{\\ivaluation{\\vdLint[const=I,state=\\nu]}{x} = \\iget[state]{\\vdLint[const=I,state=\\nu]}(x)}\n= \\(\\iget[state]{\\Ialt}(x) = \\ivaluation{\\Ialt}{x}\\) for variable $x$ since \\(\\iget[state]{\\vdLint[const=I,state=\\nu]}=\\iget[state]{\\Ialt}\\) on \\(\\freevars{x}=\\{x\\}\\).\n\n\n\\item \\m{\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\D{x}} = \\iget[state]{\\vdLint[const=I,state=\\nu]}(\\D{x})}\n= \\(\\iget[state]{\\Ialt}(\\D{x}) = \\ivaluation{\\Ialt}{\\D{x}}\\) for differential symbol $\\D{x}$ since \\(\\iget[state]{\\vdLint[const=I,state=\\nu]}=\\iget[state]{\\Ialt}\\) on \\(\\freevars{\\D{x}}=\\{\\D{x}\\}\\).\n\n\\item \\m{\\ivaluation{\\vdLint[const=I,state=\\nu]}{f(\\theta_1,\\dots,\\theta_k)} = \\iget[const]{\\vdLint[const=I,state=\\nu]}(f)(\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\theta_1},\\dots,\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\theta_k})}\n\\(\\stackrel{\\text{IH}}{=} \\iget[const]{\\Ialt}(f)({\\ivaluation{\\Ialt}{\\theta_1},\\dots,\\ivaluation{\\Ialt}{\\theta_k}})\n= \\ivaluation{\\Ialt}{f(\\theta_1,\\dots,\\theta_k)}\\)\nby induction hypothesis, because \\(\\freevars{\\theta_i}\\subseteq\\freevars{f(\\theta_1,\\dots,\\theta_k)}\\)\nand $\\iget[const]{\\vdLint[const=I,state=\\nu]}$ and $\\iget[const]{\\Ialt}$ were assumed to agree on the function symbol $f$ that occurs in the term.\n\n\\item \\m{\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\theta+\\eta} = \\ivaluation{\\vdLint[const=I,state=\\nu]}{\\theta} + \\ivaluation{\\vdLint[const=I,state=\\nu]}{\\eta}}\n\\(\\stackrel{\\text{IH}}{=} \\ivaluation{\\Ialt}{\\theta} + \\ivaluation{\\Ialt}{\\eta} = \\ivaluation{\\Ialt}{\\theta+\\eta}\\)\nby induction hypothesis, because \\(\\freevars{\\theta}\\subseteq\\freevars{\\theta+\\eta}\\) and \\(\\freevars{\\eta}\\subseteq\\freevars{\\theta+\\eta}\\).\n\n\\item \\m{\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\theta\\cdot\\eta} = \\ivaluation{\\vdLint[const=I,state=\\nu]}{\\theta} \\cdot \\ivaluation{\\vdLint[const=I,state=\\nu]}{\\eta}}\n\\(\\stackrel{\\text{IH}}{=} \\ivaluation{\\Ialt}{\\theta} \\cdot \\ivaluation{\\Ialt}{\\eta} = \\ivaluation{\\Ialt}{\\theta\\cdot\\eta}\\)\nby induction hypothesis, because \\(\\freevars{\\theta}\\subseteq\\freevars{\\theta\\cdot\\eta}\\) and \\(\\freevars{\\eta}\\subseteq\\freevars{\\theta\\cdot\\eta}\\).\n\n\\item\n{\\def\\Im{\\imodif[state]{\\vdLint[const=I,state=\\nu]}{x}{X}}%\n\\def\\Ialtm{\\vdLint[const=J,state=\\modif{\\tilde{\\nu}}{x}{X}]}%\n\\m{\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\der{\\theta}}\n\\displaystyle\n= \\sum_x \\iget[state]{\\vdLint[const=I,state=\\nu]}(\\D{x}) \\itimes \\Dp[X]{\\ivaluation{\\Im}{\\theta}}\n= \\sum_x \\iget[state]{\\Ialt}(\\D{x}) \\itimes \\Dp[X]{\\ivaluation{\\Im}{\\theta}}\n\\stackrel{\\text{IH}}{=} \\sum_x \\iget[state]{\\Ialt}(\\D{x}) \\itimes \\Dp[X]{\\ivaluation{\\Ialtm}{\\theta}}\n}\nsince \\(\\iget[state]{\\vdLint[const=I,state=\\nu]}=\\iget[state]{\\Ialt}\\) on $\\freevars{\\der{\\theta}}$,\nwhich includes all differential symbols $\\D{x}$ for all \\(x\\in\\freevars{\\theta}\\) (the others have partial derivative 0 so do not contribute to the sum),\nand by induction hypothesis on the simpler term $\\theta$, because \\(\\freevars{\\theta}\\subseteq\\freevars{\\der{\\theta}}\\).\nNote that partial derivatives are functional, i.e. the partial derivatives by $X$ of \\(\\ivaluation{\\Im}{\\theta}\\) and \\(\\ivaluation{\\Ialtm}{\\theta}\\) agree since,\nby induction hypothesis, \\(\\ivaluation{\\Im}{\\theta}=\\ivaluation{\\Ialtm}{\\theta}\\) for all $X$ since \\(\\iget[state]{\\Im}=\\iget[state]{\\Ialtm}\\) on \\(\\{x\\}\\cup\\freevars{\\theta}\\) since $x$ is interpreted to be $X$ in both states and \\(\\iget[state]{\\vdLint[const=I,state=\\nu]}=\\iget[state]{\\Ialt}\\) on $\\freevars{\\theta}$ already.\n}\n\n\\end{compactitem}\n\\end{proofatend}\n\nBy a more subtle argument, the values of \\dL formulas also only depend on the values of their free variables.\nWhen evaluating \\dL formula $\\phi$ in two states $\\iget[state]{\\vdLint[const=I,state=\\nu]}$, $\\iget[state]{\\Ialt}$ that differ but agree on the free variables $\\freevars{\\phi}$ of $\\phi$, the (truth) values of $\\phi$ in both states coincide.\n\\rref{lem:coincidence} and~\\ref{lem:coincidence-HP} are proved by simultaneous induction.\n\\begin{lemma}[Coincidence lemma] \\label{lem:coincidence}\n  If \\(\\iget[state]{\\vdLint[const=I,state=\\nu]}=\\iget[state]{\\Ialt}\\) on $\\freevars{\\phi}$\n  and \\(\\iget[const]{\\vdLint[const=I,state=\\nu]}=\\iget[const]{\\Ialt}\\) on $\\intsigns{\\phi}$, then\n  \\m{\\imodels{\\vdLint[const=I,state=\\nu]}{\\phi}} iff \\m{\\imodels{\\Ialt}{\\phi}}.\n\\end{lemma}\n\\begin{proofatend}\nThe proof is by structural induction on $\\phi$.\n\\begin{enumerate}\n\\item \\(\\imodels{\\vdLint[const=I,state=\\nu]}{p(\\theta_1,\\dots,\\theta_k)}\\) iff \\((\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\theta_1},\\dots,\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\theta_k})\\in\\iget[const]{\\vdLint[const=I,state=\\nu]}(p)\\)\niff \\((\\ivaluation{\\Ialt}{\\theta_1},\\dots,\\ivaluation{\\Ialt}{\\theta_k})\\in\\iget[const]{\\Ialt}(p)\\) iff \\(\\imodels{\\Ialt}{p(\\theta_1,\\dots,\\theta_k)}\\)\nby \\rref{lem:coincidence-term} since $\\freevars{\\theta_i}\\subseteq\\freevars{p(\\theta_1,\\dots,\\theta_k)}$\nand $\\iget[const]{\\vdLint[const=I,state=\\nu]}$ and $\\iget[const]{\\Ialt}$ were assumed to agree on the function symbol $p$ that occurs in the formula.\n\n\\item \\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\theta\\geq\\eta}\\) iff \\(\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\theta}\\geq\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\eta}\\)\niff \\(\\ivaluation{\\Ialt}{\\theta}\\geq\\ivaluation{\\Ialt}{\\eta}\\) iff \\(\\imodels{\\Ialt}{\\theta\\geq\\eta}\\)\nby \\rref{lem:coincidence-term} since $\\freevars{\\theta}\\cup\\freevars{\\eta}\\subseteq\\freevars{\\theta\\geq\\eta}$\nand the interpretation of $\\geq$ is fixed.\n\n\\item \\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\contextapp{C}{\\phi}} = \\iget[const]{\\vdLint[const=I,state=\\nu]}(C)\\big(\\imodel{\\vdLint[const=I,state=\\nu]}{\\phi}\\big)\\)\niff(IH) \\(\\imodels{\\Ialt}{\\contextapp{C}{\\phi}} = \\iget[const]{\\Ialt}(C)\\big(\\imodel{\\Ialt}{\\phi}\\big)\\)\nsince \\(\\iget[state]{\\vdLint[const=I,state=\\nu]}=\\iget[state]{\\Ialt}\\) on \\(\\freevars{\\contextapp{C}{\\phi}}=\\mathcal{V}\\cup\\D{\\mathcal{V}}\\), so \\(\\iget[state]{\\vdLint[const=I,state=\\nu]}=\\iget[state]{\\Ialt}\\),\nand \\(\\iget[const]{\\vdLint[const=I,state=\\nu]}=\\iget[const]{\\Ialt}\\) on \\(\\intsigns{\\contextapp{C}{\\phi}}=\\{C\\}\\cup\\intsigns{\\phi}\\), so \\(\\iget[const]{\\vdLint[const=I,state=\\nu]}(C)=\\iget[const]{\\Ialt}(C)\\) and, by induction hypothesis, \\(\\imodel{\\vdLint[const=I,state=\\nu]}{\\phi}=\\imodel{\\Ialt}{\\phi}\\).\n\n\\item \\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\lnot\\phi}\\) iff \\(\\inonmodels{\\vdLint[const=I,state=\\nu]}{\\phi}\\)\niff(IH) \\(\\inonmodels{\\Ialt}{\\phi}\\) iff \\(\\imodels{\\Ialt}{\\lnot\\phi}\\)\nby induction hypothesis as $\\freevars{\\lnot\\phi}=\\freevars{\\phi}$.\n\n\\item \\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\phi\\land\\psi}\\) iff \\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\phi}\\cap\\imodel{\\vdLint[const=I,state=\\nu]}{\\psi}\\)\niff(IH) \\(\\imodels{\\Ialt}{\\phi}\\cap\\imodel{\\Ialt}{\\psi}\\) iff \\(\\imodels{\\Ialt}{\\phi\\land\\psi}\\)\nby induction hypothesis using $\\freevars{\\phi\\land\\psi}=\\freevars{\\phi}\\cup\\freevars{\\psi}$.\n\n\\item\n{\\def\\Im{\\imodif[state]{\\vdLint[const=I,state=\\nu]}{x}{r}}%\n\\def\\Imalt{\\dLint[const=I,state=\\tilde{\\nu}_x^r]}%\n\\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\lexists{x}{\\phi}}\\) iff \\(\\iget[state]{\\Im} \\in \\imodel{\\vdLint[const=I,state=\\nu]}{\\phi} ~\\text{for some}~r\\in\\reals\\)\niff \\(\\iget[state]{\\Imalt} \\in \\imodel{\\vdLint[const=I,state=\\nu]}{\\phi} ~\\text{for some}~r\\in\\reals\\) iff(H) \\(\\imodels{\\Ialt}{\\lexists{x}{\\phi}}\\)\nfor the same $r$\nby induction hypothesis using that \\(\\iget[state]{\\Im}=\\iget[state]{\\Imalt}\\) on $\\freevars{\\phi}\\subseteq\\{x\\}\\cup\\freevars{\\lexists{x}{\\phi}}$.\n}\n\n\\item The case \\(\\lforall{x}{\\phi}\\) follows from the equivalence \\(\\lforall{x}{\\phi} \\mequiv \\lnot\\lexists{x}{\\lnot\\phi}\\) using \\(\\freevars{\\lnot\\lexists{x}{\\lnot\\phi}} = \\freevars{\\lforall{x}{\\phi}}\\).\n\n\\item \\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\ddiamond{\\alpha}{\\phi}}\\) iff there is a $\\iget[state]{\\vdLint[const=I,state=\\omega]}$ such that \\(\\iaccessible[\\alpha]{\\vdLint[const=I,state=\\nu]}{\\vdLint[const=I,state=\\omega]}\\) and \\(\\imodels{\\vdLint[const=I,state=\\omega]}{\\phi}\\).\nSince \\(\\iget[state]{\\vdLint[const=I,state=\\nu]}=\\iget[state]{\\Ialt}\\) on $\\freevars{\\ddiamond{\\alpha}{\\phi}}\\supseteq\\freevars{\\alpha}$\nand \\(\\iaccessible[\\alpha]{\\vdLint[const=I,state=\\nu]}{\\vdLint[const=I,state=\\omega]}\\),\n\\rref{lem:coincidence-HP} implies with \\(\\iget[const]{\\vdLint[const=I,state=\\nu]}=\\iget[const]{\\Ialt}\\) on $\\intsigns{\\alpha}$ that there is an $\\iget[state]{\\Italt}$ such that\n  \\(\\iaccessible[\\alpha]{\\Ialt}{\\Italt}\\)\n  and \\(\\iget[state]{\\vdLint[const=I,state=\\omega]}=\\iget[state]{\\Italt}\\) on $\\freevars{\\ddiamond{\\alpha}{\\phi}}\\cup\\mustboundvars{\\alpha}\n  = \\freevars{\\alpha}\\cup(\\freevars{\\phi}\\setminus\\mustboundvars{\\alpha})\\cup\\mustboundvars{\\alpha}\n  = \\freevars{\\alpha}\\cup\\freevars{\\phi}\\cup\\mustboundvars{\\alpha}\n  \\supseteq \\freevars{\\phi}$.\n\n\\begin{center}\n\\includegraphics{images\/fig-usubst-coincidence-diamond}\n\\end{center}\n\nSince, \\(\\iget[state]{\\vdLint[const=I,state=\\omega]}=\\iget[state]{\\Italt}\\) on $\\freevars{\\phi}$ and \\(\\iget[const]{\\vdLint[const=I,state=\\nu]}=\\iget[const]{\\Ialt}\\) on $\\intsigns{\\phi}$,\nthe induction hypothesis implies that\n\\(\\imodels{\\Italt}{\\phi}\\) since \\(\\imodels{\\vdLint[const=I,state=\\omega]}{\\phi}\\).\nSince \\(\\iaccessible[\\alpha]{\\Ialt}{\\Italt}\\), this implies\n\\m{\\imodels{\\Ialt}{\\ddiamond{\\alpha}{\\phi}}}.\n\n\\item \\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\dbox{\\alpha}{\\phi}} = \\imodel{\\vdLint[const=I,state=\\nu]}{\\lnot\\ddiamond{\\alpha}{\\lnot\\phi}}\\)\niff \\(\\inonmodels{\\vdLint[const=I,state=\\nu]}{\\ddiamond{\\alpha}{\\lnot\\phi}}\\)\niff \\(\\inonmodels{\\Ialt}{\\ddiamond{\\alpha}{\\lnot\\phi}}\\) iff \\(\\imodels{\\Ialt}{\\dbox{\\alpha}{\\phi}}\\)\nby induction hypothesis using $\\freevars{\\ddiamond{\\alpha}{\\lnot\\phi}}=\\freevars{\\dbox{\\alpha}{\\phi}}$.\n\n\\end{enumerate}\n\\end{proofatend}\n\nIn a sense, the runs of an \\HP $\\alpha$ also only depend on the values of its free variables, because its behavior cannot depend on the values of variables that it never reads.\nThat is, if \\(\\iget[state]{\\vdLint[const=I,state=\\nu]}=\\iget[state]{\\Ialt}\\) on $\\freevars{\\alpha}$\n  and \\(\\iaccessible[\\alpha]{\\vdLint[const=I,state=\\nu]}{\\vdLint[const=I,state=\\omega]}\\),\n  then there is an $\\iget[state]{\\Italt}$ such that\n  \\(\\iaccessible[\\alpha]{\\Ialt}{\\Italt}\\)\n  and $\\iget[state]{\\vdLint[const=I,state=\\omega]}$ and $\\iget[state]{\\Italt}$ agree in some sense.\nThere is a subtlety, though.\nThe resulting states $\\iget[state]{\\vdLint[const=I,state=\\omega]}$ and $\\iget[state]{\\Italt}$ will only continue to agree on $\\freevars{\\alpha}$ and the variables that are bound on the particular path that $\\alpha$ took for the transition \\(\\iaccessible[\\alpha]{\\vdLint[const=I,state=\\nu]}{\\vdLint[const=I,state=\\omega]}\\).\nOn variables $z$ that are neither free (so the initial states $\\iget[state]{\\vdLint[const=I,state=\\nu]}$ and $\\iget[state]{\\Ialt}$ have not been assumed to coincide) nor bound on the particular path that $\\alpha$ took, $\\iget[state]{\\vdLint[const=I,state=\\omega]}$ and $\\iget[state]{\\Italt}$ may continue to disagree, because $z$ has not been written to.\n\\iflongversion\n\\begin{example}\nLet \\(\\iaccessible[\\alpha]{\\vdLint[const=I,state=\\nu]}{\\vdLint[const=I,state=\\omega]}\\).\nIt is not enough to assume \\(\\iget[state]{\\vdLint[const=I,state=\\nu]}=\\iget[state]{\\Ialt}\\) only on $\\freevars{\\alpha}$ in order to guarantee \\(\\iget[state]{\\vdLint[const=I,state=\\omega]}=\\iget[state]{\\Italt}\\) on $\\vars{\\alpha}$ for some $\\iget[state]{\\Italt}$ such that\n  \\(\\iaccessible[\\alpha]{\\Ialt}{\\Italt}\\), because\n\\[\n\\alpha \\mdefequiv \\pchoice{\\pupdate{\\pumod{x}{1}}}{\\pupdate{\\pumod{y}{2}}}\n\\]\nwill force the final states to agree only on either $x$ or on $y$, whichever one was assigned to during the respective run of $\\alpha$, not on both \\(\\boundvars{\\alpha}=\\{x,y\\}\\),\neven though any initial states \\(\\iget[state]{\\vdLint[const=I,state=\\nu]},\\iget[state]{\\Ialt}\\) agree on $\\freevars{\\alpha}=\\emptyset$.\nNote that this can only happen because \\(\\mustboundvars{\\alpha}=\\emptyset\\neq\\boundvars{\\alpha}=\\{x,y\\}\\).\n\\end{example}\n\\fi\nYet, $\\iget[state]{\\vdLint[const=I,state=\\omega]}$ and $\\iget[state]{\\Italt}$ agree on the variables that are bound on \\emph{all} paths of $\\alpha$, rather than somewhere in $\\alpha$.\nThat is on the must-bound variables of $\\alpha$.\nIf initial states agree on (at least) all free variables $\\freevars{\\alpha}$ that \\HP $\\alpha$ may read, then the final states agree on those as well as on all variables that $\\alpha$ must write, i.e.\\ on $\\mustboundvars{\\alpha}$.\n\n\\begin{lemma}[Coincidence lemma] \\label{lem:coincidence-HP}\n  If \\(\\iget[state]{\\vdLint[const=I,state=\\nu]}=\\iget[state]{\\Ialt}\\) on $V\\supseteq\\freevars{\\alpha}$\n  and \\(\\iget[const]{\\vdLint[const=I,state=\\nu]}=\\iget[const]{\\Ialt}\\) on $\\intsigns{\\alpha}$\n  and \\(\\iaccessible[\\alpha]{\\vdLint[const=I,state=\\nu]}{\\vdLint[const=I,state=\\omega]}\\),\n  then there is an $\\iget[state]{\\Italt}$ such that\n  \\(\\iaccessible[\\alpha]{\\Ialt}{\\Italt}\\)\n  and \\(\\iget[state]{\\vdLint[const=I,state=\\omega]}=\\iget[state]{\\Italt}\\) on $V\\cup\\mustboundvars{\\alpha}$.\n\\iflongversion\n\\begin{center}\n  \\vspace{-\\baselineskip}\n\\begin{tikzpicture}\n  \\matrix (m) [matrix of math nodes,row sep=3em,column sep=4em,minimum width=2em]\n  {\n     \\iget[state]{\\vdLint[const=I,state=\\nu]} & \\iget[state]{\\vdLint[const=I,state=\\omega]} \\\\\n     \\iget[state]{\\Ialt} & \\iget[state]{\\Italt} \\\\};\n  \\path\n    (m-1-1) edge [similar state] node [left,align=right] {on $V\\supseteq\\freevars{\\alpha}$} (m-2-1)\n            edge [transition] node [below] {$\\alpha$} (m-1-2)\n    (m-2-1) edge [transition,transition exists] node [above] {$\\alpha$}\n            node [below] {$\\exists$} (m-2-2)\n    (m-1-2) edge [similar state] node [right,align=right] {on $V\\cup\\mustboundvars{\\alpha}$} (m-2-2);\n  \\path\n    (m-1-1) edge[similar state,bend left=30] node[above] {on $\\scomplement{\\boundvars{\\alpha}}$} (m-1-2)\n    (m-2-1) edge[similar state,bend right=30] node[below] {on $\\scomplement{\\boundvars{\\alpha}}$} (m-2-2);\n\\end{tikzpicture}\n\\end{center}\n\\fi\n\\end{lemma}\n\\begin{proofatend}\nThe proof is by induction on the structural complexity of $\\alpha$, where $\\prepeat{\\alpha}$ is considered to be structurally more complex than \\HPs of any length but with less nested repetitions, which induces a well-founded order on \\HPs.\nFor atomic programs $\\alpha$, for which \\(\\boundvars{\\alpha}=\\mustboundvars{\\alpha}\\), it is enough to conclude agreement on $\\vars{\\alpha}\\mdefeq \\freevars{\\alpha}\\cup\\boundvars{\\alpha}=\\freevars{\\alpha}\\cup\\mustboundvars{\\alpha}$, because any variable in $V\\setminus\\vars{\\alpha}$ is in \\(\\scomplement{\\boundvars{\\alpha}}\\), which remains unchanged by $\\alpha$ according to \\rref{lem:bound}.\n\n\\begin{compactitem}\n\n\\item Since \\(\\freevars{a} = \\mathcal{V}\\cup\\D{\\mathcal{V}}\\) so $\\iget[state]{\\vdLint[const=I,state=\\nu]}=\\iget[state]{\\Ialt}$, the statement is vacuously true for program constant $a$.\n\n\\item \\m{\\iaccessible[\\pupdate{\\pumod{x}{\\theta}}]{\\vdLint[const=I,state=\\nu]}{\\vdLint[const=I,state=\\omega]} = \\{(\\iget[state]{\\vdLint[const=I,state=\\nu]},\\iget[state]{\\vdLint[const=I,state=\\omega]}) \\with \\iget[state]{\\vdLint[const=I,state=\\omega]}=\\iget[state]{\\vdLint[const=I,state=\\nu]}~\\text{except that}~\\ivaluation{\\vdLint[const=I,state=\\omega]}{x}=\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\theta}\\}}\nthen there is a transition \\(\\iaccessible[\\pupdate{\\pumod{x}{\\theta}}]{\\Ialt}{\\Italt}\\)\nand \\(\\iget[state]{\\Italt}(x)=\\ivaluation{\\Italt}{x}=\\ivaluation{\\Ialt}{\\theta}\n=\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\theta}=\\ivaluation{\\vdLint[const=I,state=\\omega]}{x} = \\iget[state]{\\vdLint[const=I,state=\\nu]}(x)\\)\nby \\rref{lem:coincidence}, since \\(\\iget[state]{\\vdLint[const=I,state=\\nu]}=\\iget[state]{\\Ialt}\\) on $\\freevars{\\pupdate{\\pumod{x}{\\theta}}}=\\freevars{\\theta}$ and \\(\\iget[const]{\\vdLint[const=I,state=\\nu]}=\\iget[const]{\\Ialt}\\) on $\\intsigns{\\theta}$.\nSo, \\(\\iget[state]{\\vdLint[const=I,state=\\omega]}=\\iget[state]{\\Italt}\\) on $\\boundvars{\\pupdate{\\pumod{x}{\\theta}}}=\\{x\\}$.\nAlso, \\(\\iget[state]{\\vdLint[const=I,state=\\nu]}=\\iget[state]{\\vdLint[const=I,state=\\omega]}\\) on $\\scomplement{\\boundvars{\\pupdate{\\pumod{x}{\\theta}}}}$\nand \\(\\iget[state]{\\Ialt}=\\iget[state]{\\Italt}\\) on $\\scomplement{\\boundvars{\\pupdate{\\pumod{x}{\\theta}}}}$ by \\rref{lem:bound}.\nSince \\(\\iget[state]{\\vdLint[const=I,state=\\nu]}=\\iget[state]{\\Ialt}\\) on $\\freevars{\\pupdate{\\pumod{x}{\\theta}}}$, these imply\n\\(\\iget[state]{\\vdLint[const=I,state=\\omega]}=\\iget[state]{\\Italt}\\) on $\\freevars{\\pupdate{\\pumod{x}{\\theta}}}\\setminus\\boundvars{\\pupdate{\\pumod{x}{\\theta}}}$.\nSince \\(\\iget[state]{\\vdLint[const=I,state=\\omega]}=\\iget[state]{\\Italt}\\) on $\\boundvars{\\pupdate{\\pumod{x}{\\theta}}}$ had been shown already, this implies\n\\(\\iget[state]{\\vdLint[const=I,state=\\omega]}=\\iget[state]{\\Italt}\\) on $\\vars{\\pupdate{\\pumod{x}{\\theta}}}$.\n\n\\item\n{\\def\\Im{\\imodif[state]{\\vdLint[const=I,state=\\nu]}{x'}{r}}%\n\\def\\Ialtm{\\vdLint[const=J,state=\\modif{\\tilde{\\nu}}{x'}{r}]}%\n\\m{\\iaccessible[\\Dupdate{\\Dumod{\\D{x}}{\\theta}}]{\\vdLint[const=I,state=\\nu]}{\\vdLint[const=I,state=\\omega]} = \\{(\\iget[state]{\\vdLint[const=I,state=\\nu]},\\iget[state]{\\Im}) \\with r=\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\theta}\\}}.\nAs \\(\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\theta}=\\ivaluation{\\Ialt}{\\theta}\\) by \\rref{lem:coincidence-term} since \\(\\freevars{\\theta}\\subseteq\\freevars{\\Dupdate{\\Dumod{\\D{x}}{\\theta}}}\\),\nthis implies\n\\(\\iaccessible[\\Dupdate{\\Dumod{\\D{x}}{\\theta}}]{\\Ialt}{\\Ialtm} = \\{(\\iget[state]{\\Ialt},\\iget[state]{\\Ialtm}) \\with r=\\ivaluation{\\Ialt}{\\theta}\\}\\).\nBy construction \\(\\iget[state]{\\vdLint[const=I,state=\\omega]}=\\iget[state]{\\Ialtm}\\) on $\\boundvars{\\Dupdate{\\Dumod{\\D{x}}{\\theta}}}=\\{\\D{x}\\}$\nand they continue to agree on $\\freevars{\\Dupdate{\\Dumod{\\D{x}}{\\theta}}}\\setminus\\boundvars{{\\Dupdate{\\Dumod{\\D{x}}{\\theta}}}}$.\n}\n\n\\item \\m{\\iaccessible[\\ptest{\\psi}]{\\vdLint[const=I,state=\\nu]}{\\vdLint[const=I,state=\\omega]} = \\{(\\iget[state]{\\vdLint[const=I,state=\\nu]},\\iget[state]{\\vdLint[const=I,state=\\nu]}) \\with \\imodels{\\vdLint[const=I,state=\\nu]}{\\psi} ~\\text{i.e.}~ \\iget[state]{\\vdLint[const=I,state=\\nu]}\\in\\imodel{\\vdLint[const=I,state=\\nu]}{\\psi}\\}}\nthen \\(\\iget[state]{\\vdLint[const=I,state=\\omega]}=\\iget[state]{\\vdLint[const=I,state=\\nu]}\\) by \\rref{def:HP-transition}.\nSince, \\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\psi}\\) and \\(\\iget[state]{\\vdLint[const=I,state=\\nu]}=\\iget[state]{\\Ialt}\\) on $\\freevars{\\ptest{\\psi}}$ and \\(\\iget[const]{\\vdLint[const=I,state=\\nu]}=\\iget[const]{\\Ialt}\\) on $\\intsigns{\\psi}$,\n\\rref{lem:coincidence} implies that \\(\\imodels{\\Ialt}{\\psi}\\), so \\(\\iaccessible[\\ptest{\\psi}]{\\Ialt}{\\Ialt}\\).\nSo \\(\\iget[state]{\\vdLint[const=I,state=\\nu]}=\\iget[state]{\\Ialt}\\) on $\\vars{\\ptest{\\psi}}$ since \\(\\boundvars{\\ptest{\\psi}}=\\emptyset\\).\n\n\\item\n  \\m{\\iaccessible[\\pevolvein{\\D{x}=\\genDE{x}}{\\psi}]{\\vdLint[const=I,state=\\nu]}{\\vdLint[const=I,state=\\omega]}} \n  implies that there is an $\\iget[state]{\\Italt}$ reached from $\\iget[state]{\\Ialt}$ by following the differential equation for the same amount it took to reach $\\iget[state]{\\vdLint[const=I,state=\\omega]}$ from $\\iget[state]{\\vdLint[const=I,state=\\nu]}$.\n  The solution will be the same, because \\(\\iget[const]{\\vdLint[const=I,state=\\nu]}=\\iget[const]{\\Ialt}\\) on $\\intsigns{\\pevolvein{\\D{x}=\\genDE{x}}{\\psi}}$ and \\(\\iget[state]{\\vdLint[const=I,state=\\nu]}=\\iget[state]{\\Ialt}\\) on $\\freevars{\\pevolvein{\\D{x}=\\genDE{x}}{\\psi}}$, which, using \\rref{lem:coincidence}, contains all the variables whose values the differential equation solution depends on.\n  Thus, both solutions agree on all variables that evolve during the continuous evolution, i.e.\\ $\\boundvars{\\pevolvein{\\D{x}=\\genDE{x}}{\\psi}}$.\n  Thus,\n  \\m{\\iaccessible[\\pevolvein{\\D{x}=\\genDE{x}}{\\psi}]{\\Ialt}{\\Italt}}\n  and \\(\\iget[state]{\\vdLint[const=I,state=\\omega]}=\\iget[state]{\\Italt}\\) on $\\vars{\\pevolvein{\\D{x}=\\genDE{x}}{\\psi}}$.\n\n\\item \\m{\\iaccessible[\\pchoice{\\alpha}{\\beta}]{\\vdLint[const=I,state=\\nu]}{\\vdLint[const=I,state=\\omega]} = \\iaccess[\\alpha]{\\vdLint[const=I,state=\\nu]} \\cup \\iaccess[\\beta]{\\vdLint[const=I,state=\\nu]}}\nimplies \\(\\iaccessible[\\alpha]{\\vdLint[const=I,state=\\nu]}{\\vdLint[const=I,state=\\omega]}\\) or \\(\\iaccessible[\\beta]{\\vdLint[const=I,state=\\nu]}{\\vdLint[const=I,state=\\omega]}\\),\nwhich since \\(V\\supseteq\\freevars{\\pchoice{\\alpha}{\\beta}}\\supseteq\\freevars{\\alpha}\\)\nand \\(V\\supseteq\\freevars{\\pchoice{\\alpha}{\\beta}}\\supseteq\\freevars{\\beta}\\) implies, by induction hypothesis,\nthat there is an $\\iget[state]{\\Italt}$ such that\n  \\(\\iaccessible[\\alpha]{\\Ialt}{\\Italt}\\)\n  and \\(\\iget[state]{\\vdLint[const=I,state=\\omega]}=\\iget[state]{\\Italt}\\) on $V\\cup\\mustboundvars{\\alpha}$\nor that there is an $\\iget[state]{\\Italt}$ such that\n  \\(\\iaccessible[\\beta]{\\Ialt}{\\Italt}\\)\n  and \\(\\iget[state]{\\vdLint[const=I,state=\\omega]}=\\iget[state]{\\Italt}\\) on $V\\cup\\mustboundvars{\\beta}$, respectively.\n  In either case, there is a $\\iget[state]{\\Italt}$ such that\n  \\(\\iaccessible[\\pchoice{\\alpha}{\\beta}]{\\Ialt}{\\Italt}\\)\n  and \\(\\iget[state]{\\vdLint[const=I,state=\\omega]}=\\iget[state]{\\Italt}\\) on $V\\cup\\mustboundvars{\\pchoice{\\alpha}{\\beta}}$,\n  because \\(\\iaccess[\\alpha]{\\Ialt}\\subseteq\\iaccess[\\pchoice{\\alpha}{\\beta}]{\\Ialt}\\)\n  and \\(\\iaccess[\\beta]{\\Ialt}\\subseteq\\iaccess[\\pchoice{\\alpha}{\\beta}]{\\Ialt}\\) \n  and \\(\\mustboundvars{\\pchoice{\\alpha}{\\beta}}=\\mustboundvars{\\alpha}\\cap\\mustboundvars{\\beta}\\).\n\n\\begin{center}\n\\includegraphics{images\/fig-usubst-coincidence-choice}\n\\end{center}\n\n\\item\n\\newcommand{\\iconcat[state=\\mu]{\\I}}{\\iconcat[state=\\mu]{\\vdLint[const=I,state=\\nu]}}%\n\\newcommand{\\Izalt}{\\vdLint[const=J,state=\\tilde{\\mu}]}%\n\\m{\\iaccessible[\\alpha;\\beta]{\\vdLint[const=I,state=\\nu]}{\\vdLint[const=I,state=\\omega]} = \\iaccess[\\alpha]{\\vdLint[const=I,state=\\nu]} \\compose \\iaccess[\\beta]{\\vdLint[const=I,state=\\nu]}},\ni.e.\\ there is a $\\iget[state]{\\iconcat[state=\\mu]{\\I}}$ such that \\(\\iaccessible[\\alpha]{\\vdLint[const=I,state=\\nu]}{\\iconcat[state=\\mu]{\\I}}\\) and \\(\\iaccessible[\\beta]{\\iconcat[state=\\mu]{\\I}}{\\vdLint[const=I,state=\\omega]}\\).\nSince \\(V\\supseteq\\freevars{\\alpha;\\beta}\\supseteq\\freevars{\\alpha}\\), by induction hypothesis, there is a $\\iget[state]{\\Izalt}$ such that\n  \\(\\iaccessible[\\alpha]{\\Ialt}{\\Izalt}\\)\n  and \\(\\iget[state]{\\iconcat[state=\\mu]{\\I}}=\\iget[state]{\\Izalt}\\) on $V\\cup\\mustboundvars{\\alpha}$.\n  Since \\(V\\supseteq\\freevars{\\alpha;\\beta}\\),\n  so\n  \\(V\\cup\\mustboundvars{\\alpha} \\supseteq \\freevars{\\alpha;\\beta} \\cup \\mustboundvars{\\alpha}\n  = \\freevars{\\alpha}\\cup(\\freevars{\\beta}\\setminus\\mustboundvars{\\alpha}) \\cup \\mustboundvars{\\alpha}\n  = \\freevars{\\alpha}\\cup\\freevars{\\beta}\\cup\\mustboundvars{\\alpha}\n  \\supseteq \\freevars{\\beta}\\)\n  by \\rref{def:freevars-HP},\n  and since \\(\\iaccessible[\\beta]{\\iconcat[state=\\mu]{\\I}}{\\vdLint[const=I,state=\\omega]}\\), the induction hypothesis implies that\n  there is an $\\iget[state]{\\Italt}$ such that\n  \\(\\iaccessible[\\beta]{\\Izalt}{\\Italt}\\)\n  and \\(\\iget[state]{\\vdLint[const=I,state=\\omega]}=\\iget[state]{\\Italt}\\) on $(V\\cup\\mustboundvars{\\alpha})\\cup\\mustboundvars{\\beta} = V\\cup\\mustboundvars{\\alpha;\\beta}$.\n\n  \\begin{center}\n  \\includegraphics{images\/fig-usubst-coincidence-compose}\n\\end{center}\n\n\\item\n\\renewcommand{\\iconcat[state=\\mu]{\\I}}[1][]{\\dLint[state=\\nu_{#1}]}%\n\\m{\\iaccessible[\\prepeat{\\alpha}]{\\vdLint[const=I,state=\\nu]}{\\vdLint[const=I,state=\\omega]} = \\displaystyle\\cupfold_{n\\in\\naturals}\\iaccess[{\\prepeat[n]{\\alpha}}]{\\vdLint[const=I,state=\\nu]}}\niff there is an $n\\in\\naturals$ such that \\(\\iaccessible[\\alpha^n]{\\vdLint[const=I,state=\\nu]}{\\vdLint[const=I,state=\\omega]}\\).\nThe case $n=0$ follows from the assumption \\(\\iget[state]{\\vdLint[const=I,state=\\nu]}=\\iget[state]{\\Ialt}\\) on $V\\supseteq\\freevars{\\alpha}$, since \\(\\iget[state]{\\vdLint[const=I,state=\\omega]}=\\iget[state]{\\vdLint[const=I,state=\\nu]}\\) holds in that case and $\\mustboundvars{\\prepeat{\\alpha}}=\\emptyset$.\nThe case $n>0$ proceeds as follows.\nSince \\(\\freevars{\\prepeat[n]{\\alpha}}=\\freevars{\\prepeat{\\alpha}}=\\freevars{\\alpha}\\), the induction hypothesis applied to the structurally simpler \\HP $\\prepeat[n]{\\alpha}$ implies\nthat there is an $\\iget[state]{\\Italt}$ such that\n  \\(\\iaccessible[\\alpha^n]{\\Ialt}{\\Italt}\\)\n  and \\(\\iget[state]{\\vdLint[const=I,state=\\omega]}=\\iget[state]{\\Italt}\\) on $V\\cup\\mustboundvars{\\prepeat[n]{\\alpha}} \\supseteq V = V\\cup\\mustboundvars{\\prepeat{\\alpha}}$,\n  since $\\mustboundvars{\\prepeat{\\alpha}}=\\emptyset$.\n  Since \\(\\iaccess[{\\prepeat[n]{\\alpha}}]{\\Ialt}\\subseteq\\iaccess[\\prepeat{\\alpha}]{\\Ialt}\\), this concludes the proof.\n\n\\end{compactitem}\n\\end{proofatend}\n\n\\iflongversion\nWhen assuming $\\iget[state]{\\vdLint[const=I,state=\\nu]}$ and $\\iget[state]{\\Ialt}$ to agree on all $\\vars{\\alpha}$, whether free or bound, $\\iget[state]{\\vdLint[const=I,state=\\omega]}$ and $\\iget[state]{\\Italt}$ will continue to agree on $\\vars{\\alpha}$:\n\\begin{corollary}[Coincidence lemma] \\label{cor:coincidence-HP}\n  If \\(\\iget[state]{\\vdLint[const=I,state=\\nu]}=\\iget[state]{\\Ialt}\\) on $\\vars{\\alpha}$\n  and \\(\\iget[const]{\\vdLint[const=I,state=\\nu]}=\\iget[const]{\\Ialt}\\) on $\\intsigns{\\alpha}$\n  and \\(\\iaccessible[\\alpha]{\\vdLint[const=I,state=\\nu]}{\\vdLint[const=I,state=\\omega]}\\),\n  then there is an $\\iget[state]{\\Italt}$ such that\n  \\(\\iaccessible[\\alpha]{\\Ialt}{\\Italt}\\)\n  and \\(\\iget[state]{\\vdLint[const=I,state=\\omega]}=\\iget[state]{\\Italt}\\) on $\\vars{\\alpha}$.\n  The same continues to hold if \\(\\iget[state]{\\vdLint[const=I,state=\\nu]}=\\iget[state]{\\Ialt}\\) only on $\\vars{\\alpha}\\setminus\\mustboundvars{\\alpha}$.\n\\end{corollary}\n\\begin{proofatend}\nBy \\rref{lem:coincidence-HP} with \\(V=\\vars{\\alpha}\\supseteq\\freevars{\\alpha}\\) or \\(V=\\vars{\\alpha}\\setminus\\mustboundvars{\\alpha}\\), respectively.\n\n\\end{proofatend}\n\n\n\\begin{remark}\n  Using hybrid computation sequences, the agreement in \\rref{lem:coincidence-HP} continues to hold for\n  \\(\\iget[state]{\\vdLint[const=I,state=\\omega]}=\\iget[state]{\\Italt}\\) on $V\\cup W$, where $W$ is the set of must-bound variables on the hybrid computation sequence that $\\alpha$ actually took for the transition \\(\\iaccessible[\\alpha]{\\vdLint[const=I,state=\\nu]}{\\vdLint[const=I,state=\\omega]}\\), which could be larger than $\\mustboundvars{\\alpha}$.\n\\end{remark}\n\\fi\n\n\n\\section{Uniform Substitutions} \\label{sec:usubst}\n\nThe uniform substitution rule \\irref{US0} from first-order logic \\cite[\\S35,40]{Church_1956} substitutes \\emph{all} occurrences of predicate $p(\\usarg)$ by a formula $\\mapply{\\psi}{\\usarg}$, i.e.\\ it replaces all occurrences of $p(\\theta)$, for any (vectorial) term $\\theta$, by the corresponding $\\mapply{\\psi}{\\theta}$ simultaneously:\n\\[\n      \\cinferenceRule[US0|US$_1$]{uniform substitution}\n      {\\linferenceRule[formula]\n        {\\preusubst[\\phi]{p}}\n        {\\usubst[\\phi]{p}{\\psi}}\n      }{}%\n      \\qquad\\qquad\n      \\cinferenceRule[US|US]{uniform substitution}\n      {\\linferenceRule[formula]\n        {\\phi}\n        {\\applyusubst{\\sigma}{\\phi}}\n      }{}%\n\\]\nRule \\irref{US0} \\cite{DBLP:journals\/corr\/Platzer14:dGL} requires all relevant substitutions of $\\mapply{\\psi}{\\theta}$ for $p(\\theta)$ to be admissible and requires that no $p(\\theta)$ occurs in the scope of a quantifier or modality binding a variable of $\\mapply{\\psi}{\\theta}$ other than the occurrences in $\\theta$; see \\cite[\\S35,40]{Church_1956}.\n\nThis section considers a constructive definition of this proof rule that is more general: \\irref{US}.\nThe \\dL calculus uses uniform substitutions that affect terms, formulas, and programs.\nA \\dfn{uniform substitution} $\\sigma$ is a mapping\nfrom expressions of the\nform \\(f(\\usarg)\\) to terms $\\applysubst{\\sigma}{f(\\usarg)}$,\nfrom \\(p(\\usarg)\\) to formulas $\\applysubst{\\sigma}{p(\\usarg)}$,\nfrom \\(\\contextapp{C}{\\uscarg}\\) to formulas $\\applysubst{\\sigma}{\\contextapp{C}{\\uscarg}}$,\nand from program constants \\(a\\) to \\HPs $\\applysubst{\\sigma}{a}$.\nVectorial extensions are accordingly for uniform substitutions of other arities $k\\geq0$.\nHere $\\usarg$ is a reserved function symbol of arity zero and $\\uscarg$ a reserved \\predicational symbol of arity zero.\nFigure~\\ref{fig:usubst} defines the result $\\applyusubst{\\sigma}{\\phi}$ of applying to a \\dL formula~$\\phi$ the \\dfn{uniform substitution} $\\sigma$ that uniformly replaces all occurrences of function~$f$ by a (instantiated) term and all occurrences of predicate~$p$ or \\predicational~$C$ by a (instantiated) formula \nas well as of program constant $a$ by a program.\nThe notation $\\applysubst{\\sigma}{f(\\usarg)}$ denotes the replacement for $f(\\usarg)$ according to $\\sigma$, i.e.\\ the value $\\applysubst{\\sigma}{f(\\usarg)}$ of function $\\sigma$ at $f(\\usarg)$.\nBy contrast, $\\applyusubst{\\sigma}{\\phi}$ denotes the result of applying $\\sigma$ to $\\phi$ according to \\rref{fig:usubst} (likewise for $\\applyusubst{\\sigma}{\\theta}$ and $\\applyusubst{\\sigma}{\\alpha}$).\nThe notation $f\\in\\replacees{\\sigma}$ signifies that $\\sigma$ replaces $f$, i.e.\\ \\(\\applysubst{\\sigma}{f(\\usarg)} \\neq f(\\usarg)\\).\nFinally, $\\sigma$ is a total function when augmented with \\(\\applysubst{\\sigma}{g(\\usarg)}=g(\\usarg)\\) for all $g\\not\\in\\replacees{\\sigma}$.\nAccordingly for predicate symbols, \\predicational{}s, and program constants.\n\n\\begin{definition}[Admissible uniform substitution] \\label{def:usubst-admissible}\n  \\index{admissible|see{substitution, admissible}}\n  The uniform substitution~$\\sigma$ is $U$-\\emph{\\text{admissible}\\xspace} for $\\phi$ (or $\\theta$ or $\\alpha$, respectively) with respect to the set $U\\subseteq\\mathcal{V}\\cup\\D{\\mathcal{V}}$ iff\n  \\(\\freevars{\\restrict{\\sigma}{\\intsigns{\\phi}}}\\cap U=\\emptyset\\),\n  where \\({\\restrict{\\sigma}{\\intsigns{\\phi}}}\\) is the restriction of $\\sigma$ that only replaces symbols that occur in $\\phi$\n  and\n  \\(\\freevars{\\sigma}=\\cupfold_{f\\in\\replacees{\\sigma}} \\freevars{\\applysubst{\\sigma}{f(\\usarg)}} \\cup \\cupfold_{p\\in\\replacees{\\sigma}} \\freevars{\\applysubst{\\sigma}{p(\\usarg)}}\\)\n  are the \\emph{free variables} that $\\sigma$ introduces. \n  The uniform substitution~$\\sigma$ is \\emph{\\text{admissible}\\xspace} for $\\phi$ (or $\\theta$ or $\\alpha$, respectively) iff all admissibility conditions during its application according to \\rref{fig:usubst} hold, \n  which check that the bound variables $U$ of each operator are not free in the substitution on its arguments, i.e.\\ $\\sigma$ is $U$-admissible.\n  %\n  Otherwise the substitution clashes so its result $\\applyusubst{\\sigma}{\\phi}$ ($\\applyusubst{\\sigma}{\\theta}$ or $\\applyusubst{\\sigma}{\\alpha}$) is not defined.\n  %\n\\end{definition}\n\n\\irref{US} is only applicable if $\\sigma$ is admissible for $\\phi$.\nIn all subsequent results, all applications of uniform substitutions are required to be defined (no clash).\n\n\\begin{figure}[tbhp]\n  \\begin{displaymath}\n    \\begin{array}{@{}rcll@{}}\n    \\applyusubst{\\sigma}{x} &=& x & \\text{for variable $x\\in\\mathcal{V}$}\\\\\n    \\applyusubst{\\sigma}{\\D{x}} &=& \\D{x} & \\text{for differential symbol $\\D{x}\\in\\D{\\mathcal{V}}$}\\\\\n    \\applyusubst{\\sigma}{f(\\theta)} &=& (\\applyusubst{\\sigma}{f})(\\applyusubst{\\sigma}{\\theta})\n  \\mdefeq \\applyusubst{\\{\\usarg\\mapsto\\applyusubst{\\sigma}{\\theta}\\}}{\\applysubst{\\sigma}{f(\\usarg)}} &\n  \\text{for function symbol}~f\\in\\replacees{\\sigma}\n    \\\\\n  \\applyusubst{\\sigma}{g(\\theta)} &=& g(\\applyusubst{\\sigma}{\\theta}) &\\text{for function symbol}~g\\not\\in\\replacees{\\sigma}\n  \\\\\n  \\applyusubst{\\sigma}{\\theta+\\eta} &=& \\applyusubst{\\sigma}{\\theta} + \\applyusubst{\\sigma}{\\eta}\n  \\\\\n  \\applyusubst{\\sigma}{\\theta\\cdot\\eta} &=& \\applyusubst{\\sigma}{\\theta} \\cdot \\applyusubst{\\sigma}{\\eta}\n  \\\\\n  \\applyusubst{\\sigma}{\\der{\\theta}} &=& \\der{\\applyusubst{\\sigma}{\\theta}} &\\text{if $\\sigma$ $\\mathcal{V}\\cup\\D{\\mathcal{V}}$-admissible for $\\theta$}\n  \\\\\n  \\hline\n  \\applyusubst{\\sigma}{\\theta\\geq\\eta} &\\mequiv& \\applyusubst{\\sigma}{\\theta} \\geq \\applyusubst{\\sigma}{\\eta}\\\\\n    \\applyusubst{\\sigma}{p(\\theta)} &\\mequiv& (\\applyusubst{\\sigma}{p})(\\applyusubst{\\sigma}{\\theta})\n  \\mdefequiv \\applyusubst{\\{\\usarg\\mapsto\\applyusubst{\\sigma}{\\theta}\\}}{\\applysubst{\\sigma}{p(\\usarg)}} &\n  \\text{for predicate symbol}~p\\in\\replacees{\\sigma}\\\\\n  \\applyusubst{\\sigma}{q(\\theta)} &\\mequiv& q(\\applyusubst{\\sigma}{\\theta}) &\\text{for predicate symbol}~q\\not\\in\\replacees{\\sigma}\\\\\n  \\applyusubst{\\sigma}{\\contextapp{C}{\\phi}} &\\mequiv& \\contextapp{\\applyusubst{\\sigma}{C}}{\\applyusubst{\\sigma}{\\phi}}\n  \\mdefequiv \\applyusubst{\\{\\uscarg\\mapsto\\applyusubst{\\sigma}{\\phi}\\}}{\\applysubst{\\sigma}{\\contextapp{C}{\\uscarg}}} &\n  \\text{if $\\sigma$ $\\mathcal{V}\\cup\\D{\\mathcal{V}}$-admissible for $\\phi$, $C\\in\\replacees{\\sigma}$}\n  %\n  %\n  \\\\\n  \\applyusubst{\\sigma}{\\contextapp{C}{\\phi}} &\\mequiv& \\contextapp{C}{\\applyusubst{\\sigma}{\\phi}} &\n  \\text{if $\\sigma$ $\\mathcal{V}\\cup\\D{\\mathcal{V}}$-admissible for $\\phi$, $C\\not\\in\\replacees{\\sigma}$}\n  %\n  \\\\\n    \\applyusubst{\\sigma}{\\lnot\\phi} &\\mequiv& \\lnot\\applyusubst{\\sigma}{\\phi}\\\\\n    \\applyusubst{\\sigma}{\\phi\\land\\psi} &\\mequiv& \\applyusubst{\\sigma}{\\phi} \\land \\applyusubst{\\sigma}{\\psi}\\\\\n    \\applyusubst{\\sigma}{\\lforall{x}{\\phi}} &=& \\lforall{x}{\\applyusubst{\\sigma}{\\phi}} & \\text{if $\\sigma$ $\\{x\\}$-admissible for $\\phi$}\\\\\n    \\applyusubst{\\sigma}{\\lexists{x}{\\phi}} &=& \\lexists{x}{\\applyusubst{\\sigma}{\\phi}} & \\text{if $\\sigma$ $\\{x\\}$-admissible for $\\phi$}\\\\\n    \\applyusubst{\\sigma}{\\dbox{\\alpha}{\\phi}} &=& \\dbox{\\applyusubst{\\sigma}{\\alpha}}{\\applyusubst{\\sigma}{\\phi}} & \\text{if $\\sigma$ $\\boundvars{\\applyusubst{\\sigma}{\\alpha}}$-admissible for $\\phi$}\\\\\n    \\applyusubst{\\sigma}{\\ddiamond{\\alpha}{\\phi}} &=& \\ddiamond{\\applyusubst{\\sigma}{\\alpha}}{\\applyusubst{\\sigma}{\\phi}} & \\text{if $\\sigma$ $\\boundvars{\\applyusubst{\\sigma}{\\alpha}}$-admissible for $\\phi$}\n    \\\\\n  \\hline\n    \\applyusubst{\\sigma}{a} &\\mequiv& \\applysubst{\\sigma}{a} &\\text{for program constant $a\\in\\replacees{\\sigma}$}\\\\\n    \\applyusubst{\\sigma}{b} &\\mequiv& b &\\text{for program constant $b\\not\\in\\replacees{\\sigma}$}\\\\\n    \\applyusubst{\\sigma}{\\pupdate{\\umod{x}{\\theta}}} &\\mequiv& \\pupdate{\\umod{x}{\\applyusubst{\\sigma}{\\theta}}}\\\\\n    \\applyusubst{\\sigma}{\\Dupdate{\\Dumod{\\D{x}}{\\theta}}} &\\mequiv& \\Dupdate{\\Dumod{\\D{x}}{\\applyusubst{\\sigma}{\\theta}}}\\\\\n    \\applyusubst{\\sigma}{\\pevolvein{\\D{x}=\\genDE{x}}{\\psi}} &\\mequiv&\n    \\hevolvein{\\D{x}=\\applyusubst{\\sigma}{\\genDE{x}}}{\\applyusubst{\\sigma}{\\psi}} & \\text{if $\\sigma$ $\\{x,\\D{x}\\}$-admissible for $\\genDE{x},\\psi$}\\\\\n    \\applyusubst{\\sigma}{\\ptest{\\psi}} &\\mequiv& \\ptest{\\applyusubst{\\sigma}{\\psi}}\\\\\n    \\applyusubst{\\sigma}{\\pchoice{\\alpha}{\\beta}} &\\mequiv& \\pchoice{\\applyusubst{\\sigma}{\\alpha}} {\\applyusubst{\\sigma}{\\beta}}\\\\\n    \\applyusubst{\\sigma}{\\alpha;\\beta} &\\mequiv& \\applyusubst{\\sigma}{\\alpha}; \\applyusubst{\\sigma}{\\beta} &\\text{if $\\sigma$ $\\boundvars{\\applyusubst{\\sigma}{\\alpha}}$-admissible for $\\beta$}\\\\\n    \\applyusubst{\\sigma}{\\prepeat{\\alpha}} &\\mequiv& \\prepeat{(\\applyusubst{\\sigma}{\\alpha})} &\\text{if $\\sigma$ $\\boundvars{\\applyusubst{\\sigma}{\\alpha}}$-admissible for $\\alpha$}\n    \\end{array}%\n  \\end{displaymath}%\n  \\vspace{-\\baselineskip}\n  \\caption{Recursive application of uniform substitution~$\\sigma$}%\n  \\index{substitution!uniform|textbf}%\n  \\label{fig:usubst}\n\\end{figure}\n\n\\subsection{Correctness of Uniform Substitutions}\n\nLet\n\\m{\\iget[const]{\\imodif[const]{\\vdLint[const=I,state=\\nu]}{p}{R}}} denote the interpretation that agrees with interpretation~$\\iget[const]{\\vdLint[const=I,state=\\nu]}$ except for the interpretation of predicate symbol~$p$, which is changed to~\\m{R \\subseteq \\reals}.\nAccordingly for predicate symbols of other arities, for function symbols $f$, and \\predicational{s} $C$.\n\n\\begin{corollary}[Substitution adjoints] \\label{cor:adjointUsubst}\n\\def\\Ialta{\\iadjointSubst{\\sigma}{\\Ialt}}%\nThe \\emph{adjoint interpretation} $\\iget[const]{\\Ia}$ to substitution $\\sigma$ for $\\iportray{\\vdLint[const=I,state=\\nu]}$ is the interpretation that agrees with $\\iget[const]{\\vdLint[const=I,state=\\nu]}$ except that for each function symbol $f\\in\\replacees{\\sigma}$, predicate symbol $p\\in\\replacees{\\sigma}$, \\predicational $C\\in\\replacees{\\sigma}$, and program constant $a\\in\\replacees{\\sigma}$:\n\\begin{align*}\n  \\iget[const]{\\Ia}(f) &: \\reals\\to\\reals;\\, d\\mapsto\\ivaluation{\\imodif[const]{\\vdLint[const=I,state=\\nu]}{\\,\\usarg}{d}}{\\applysubst{\\sigma}{f}(\\usarg)}\\\\\n  \\iget[const]{\\Ia}(p) &= \\{d\\in\\reals \\with \\imodels{\\imodif[const]{\\vdLint[const=I,state=\\nu]}{\\,\\usarg}{d}}{\\applysubst{\\sigma}{p}(\\usarg)}\\}\n  \\\\\n  \\iget[const]{\\Ia}(C) &: \\powerset{\\reals}\\to\\powerset{\\reals};\\, R\\mapsto\\imodel{\\imodif[const]{\\vdLint[const=I,state=\\nu]}{\\,\\uscarg}{R}}{\\applysubst{\\sigma}{\\contextapp{C}{\\uscarg}}}\n  \\\\\n  \\iget[const]{\\Ia}(a) &= \\iaccess[\\applysubst{\\sigma}{a}]{\\vdLint[const=I,state=\\nu]}\n\\end{align*}\nIf \\(\\iget[state]{\\vdLint[const=I,state=\\nu]}=\\iget[state]{\\vdLint[const=I,state=\\omega]}\\) on \\(\\freevars{\\sigma}\\),\nthen \\(\\iget[const]{\\Ia}=\\iget[const]{\\Ita}\\).\nIf $\\sigma$ is $U$-admissible for $\\phi$ (or $\\theta$ or $\\alpha$, respectively) and \\(\\iget[state]{\\vdLint[const=I,state=\\nu]}=\\iget[state]{\\vdLint[const=I,state=\\omega]}\\) on $\\scomplement{U}$, then\n\\begin{align*}\n  \\ivalues{\\Ia}{\\theta} &= \\ivalues{\\Ita}{\\theta}\n  ~\\text{i.e.}~\n  \\ivaluation{\\iconcat[state=\\mu]{\\Ia}}{\\theta} = \\ivaluation{\\iconcat[state=\\mu]{\\Ita}}{\\theta} ~\\text{for all}~\\mu\n  \\\\\n  \\imodel{\\Ia}{\\phi} &= \\imodel{\\Ita}{\\phi}\\\\\n  \\iaccess[\\alpha]{\\Ia} &= \\iaccess[\\alpha]{\\Ita}\n\\end{align*}\n\\end{corollary}\n\\begin{proofatend}\nFor well-definedness of $\\iget[const]{\\Ia}$, note that $\\iget[const]{\\Ia}(f)$ is a smooth function since \\({\\applysubst{\\sigma}{f}(\\usarg)}\\) has smooth values.\nFirst, \\(\\iget[const]{\\Ia}(a) = \\iaccess[\\applysubst{\\sigma}{a}]{\\vdLint[const=I,state=\\nu]} = \\iget[const]{\\Ita}(a)\\) holds because the adjoint to $\\sigma$ for $\\iportray{\\vdLint[const=I,state=\\nu]}$ in the case of programs is independent of $\\iget[state]{\\Ia}$ (the program has access to its respective initial state at runtime).\nLikewise \\(\\iget[const]{\\Ia}(C) = \\iget[const]{\\Ita}(C)\\) for \\predicational{s}.\nBy \\rref{lem:coincidence-term},\n\\(\\ivaluation{\\imodif[const]{\\vdLint[const=I,state=\\nu]}{\\,\\usarg}{d}}{\\applysubst{\\sigma}{f}(\\usarg)}\n= \\ivaluation{\\imodif[const]{\\vdLint[const=I,state=\\omega]}{\\,\\usarg}{d}}{\\applysubst{\\sigma}{f}(\\usarg)}\\) when \\(\\iget[state]{\\vdLint[const=I,state=\\nu]}=\\iget[state]{\\vdLint[const=I,state=\\omega]}\\) on \\(\\freevars{\\applysubst{\\sigma}{f}(\\usarg)}\\).\nAlso \\(\\imodels{\\imodif[const]{\\vdLint[const=I,state=\\nu]}{\\,\\usarg}{d}}{\\applysubst{\\sigma}{p}(\\usarg)}\\)\niff \\(\\imodels{\\imodif[const]{\\vdLint[const=I,state=\\omega]}{\\,\\usarg}{d}}{\\applysubst{\\sigma}{p}(\\usarg)}\\)\nby \\rref{lem:coincidence} when \\(\\iget[state]{\\vdLint[const=I,state=\\nu]}=\\iget[state]{\\vdLint[const=I,state=\\omega]}\\) on \\(\\freevars{\\applysubst{\\sigma}{p}(\\usarg)}\\).\nThus, \\(\\iget[const]{\\Ia}=\\iget[const]{\\Ita}\\) when $\\iget[state]{\\vdLint[const=I,state=\\nu]}=\\iget[state]{\\vdLint[const=I,state=\\omega]}$ on $\\freevars{\\sigma}$.\n\nIf $\\sigma$ is $U$-admissible for $\\phi$ (or $\\theta$ or $\\alpha$), then\n  \\(\\freevars{\\applysubst{\\sigma}{f(\\usarg)}}\\cap U=\\emptyset\\)\n  so\n  \\(\\freevars{\\applysubst{\\sigma}{f(\\usarg)}}\\subseteq\\scomplement{U}\\)\n  for every function symbol $f\\in\\intsigns{\\phi}$ (or $\\theta$ or $\\alpha$) and likewise for predicate symbols $p\\in\\intsigns{\\phi}$.\n  Since \\(\\iget[state]{\\vdLint[const=I,state=\\nu]}=\\iget[state]{\\vdLint[const=I,state=\\omega]}\\) on $\\scomplement{U}$,\n  so \\(\\iget[const]{\\Ita}=\\iget[const]{\\Ia}\\) on the function and predicate symbols in $\\intsigns{\\phi}$ (or $\\theta$ or $\\alpha$).\n  Finally \\(\\iget[const]{\\Ita}=\\iget[const]{\\Ia}\\) implies that\n  \\(\\imodels{\\Ita}{\\phi}\\) iff \\(\\imodels{\\Ia}{\\phi}\\) by \\rref{lem:coincidence}\n  and that \\(\\ivalues{\\Ia}{\\theta} = \\ivalues{\\Ita}{\\theta}\\) by \\rref{lem:coincidence-term}\n  and that \\(\\iaccess[\\alpha]{\\Ita} = \\iaccess[\\alpha]{\\Ia}\\) by \\rref{lem:coincidence-HP}.\n\n\\end{proofatend}\n\nSubstituting equals for equals is sound by the compositional semantics of \\dL.\nThe more general uniform substitutions are still sound, because interpretations of uniform substitutes correspond to interpretations of their adjoints.\nThe semantic modification of adjoint interpretations has the same effect as the syntactic uniform substitution, proved by simultaneous induction.\n\\iflongversion\nRecall that all substitutions in the following are assumed to be defined (not clash), otherwise the lemmas make no claim.\n\\fi\n\n\\begin{lemma}[Uniform substitution lemma] \\label{lem:usubst-term}\nThe uniform substitution $\\sigma$ and its adjoint interpretation $\\iportray{\\Ia}$ to $\\sigma$ for $\\iportray{\\vdLint[const=I,state=\\nu]}$ have the same \\emph{term} semantics:\n\\[\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\sigma}{\\theta}} = \\ivaluation{\\Ia}{\\theta}\\]\n\\end{lemma}\n\\begin{proofatend}\n\\def\\Im{\\vdLint[const=\\usebox{\\ImnI},state=\\nu\\oplus(x\\mapsto\\usebox{\\Imnx})]}%\n\nThe proof is by structural induction on $\\theta$.\n\\begin{compactitem}\n\\item\n  \\(\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\sigma}{x}} \n  = \\ivaluation{\\vdLint[const=I,state=\\nu]}{x} = \\iget[state]{\\vdLint[const=I,state=\\nu]}(x) \\)\n  = \\(\\ivaluation{\\Ia}{x}\\)\n  \\(\\text{since}~ x\\not\\in\\replacees{\\sigma}\\)\n  for variable $x\\in\\mathcal{V}$\n\n\\item\n  \\(\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\sigma}{\\D{x}}} \n  = \\ivaluation{\\vdLint[const=I,state=\\nu]}{\\D{x}} = \\iget[state]{\\vdLint[const=I,state=\\nu]}(\\D{x}) \\)\n  = \\(\\ivaluation{\\Ia}{\\D{x}}\\)\n  \\(\\text{as}~ \\D{x}\\not\\in\\replacees{\\sigma}\\)\n  for differential symbol $\\D{x}\\in\\D{\\mathcal{V}}$\n\n\\item\n  \\(\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\sigma}{f(\\theta)}}\n  = \\ivaluation{\\vdLint[const=I,state=\\nu]}{(\\applyusubst{\\sigma}{f})\\big(\\applyusubst{\\sigma}{\\theta}\\big)}\n  = \\ivaluation{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\{\\usarg\\mapsto\\applyusubst{\\sigma}{\\theta}\\}}{\\applysubst{\\sigma}{f(\\usarg)}}}\\)\n  \\(\\stackrel{\\text{IH}}{=} \\ivaluation{\\Iminner}{\\applysubst{\\sigma}{f(\\usarg)}}\\)\n    \\(= (\\iget[const]{\\Ia}(f))(d)\\)\n  \\(= (\\iget[const]{\\Ia}(f))(\\ivaluation{\\Ia}{\\theta})\n  = \\ivaluation{\\Ia}{f(\\theta)}\\)\n  with\n  \\(d\\mdefeq\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\sigma}{\\theta}} \n  \\stackrel{\\text{IH}}{=} \\ivaluation{\\Ia}{\\theta}\\)\n  by using the induction hypothesis twice,\n  once for \\(\\applyusubst{\\sigma}{\\theta}\\) on the smaller $\\theta$ \n  and once for\n  \\(\\applyusubst{\\{\\usarg\\mapsto\\applyusubst{\\sigma}{\\theta}\\}}{\\applysubst{\\sigma}{f(\\usarg)}}\\)\n  on the possibly bigger term\n  \\({\\applysubst{\\sigma}{f(\\usarg)}}\\)\n  but the structurally simpler uniform substitution\n  \\(\\applyusubst{\\{\\usarg\\mapsto\\applyusubst{\\sigma}{\\theta}\\}}{\\dots}\\)\n  that is a substitution on the symbol $\\usarg$ of arity zero, not a substitution of functions with arguments.\n  For well-foundedness of the induction note that the $\\usarg$ substitution only happens for function symbols $f$ with at least one argument $\\theta$\n  (\\(\\text{for}~f\\in\\replacees{\\sigma}\\)).\n  \n\\item\n  \\(\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\sigma}{g(\\theta)}}\n  = \\ivaluation{\\vdLint[const=I,state=\\nu]}{g(\\applyusubst{\\sigma}{\\theta})}\\)\n  = \\(\\iget[const]{\\vdLint[const=I,state=\\nu]}(g)\\big(\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\sigma}{\\theta}}\\big)\\)\n  \\(\\stackrel{\\text{IH}}{=} \\iget[const]{\\vdLint[const=I,state=\\nu]}(g)\\big(\\ivaluation{\\Ia}{\\theta}\\big)\n  = \\iget[const]{\\Ia}(g)\\big(\\ivaluation{\\Ia}{\\theta}\\big)\n  = \\ivaluation{\\Ia}{g(\\theta)}\\)\n  by induction hypothesis and since \\(\\iget[const]{\\vdLint[const=I,state=\\nu]}(g)=\\iget[const]{\\Ia}(g)\\) as the interpretation of $g$ does not change in $\\iget[const]{\\Ia}$\n  \\(\\text{for}~g\\not\\in\\replacees{\\sigma}\\).\n\n\\item\n  \\(\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\sigma}{\\theta+\\eta}}\n  = \\ivaluation{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\sigma}{\\theta} + \\applyusubst{\\sigma}{\\eta}}\n  = \\ivaluation{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\sigma}{\\theta}} + \\ivaluation{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\sigma}{\\eta}}\\)\n  \\(\\stackrel{\\text{IH}}{=} \\ivaluation{\\Ia}{\\theta} + \\ivaluation{\\Ia}{\\eta}\n  = \\ivaluation{\\Ia}{\\theta+\\eta}\\)\n  by induction hypothesis.\n\n\\item\n  \\(\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\sigma}{\\theta\\cdot\\eta}}\n  = \\ivaluation{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\sigma}{\\theta} \\cdot \\applyusubst{\\sigma}{\\eta}}\n  = \\ivaluation{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\sigma}{\\theta}} \\cdot \\ivaluation{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\sigma}{\\eta}}\\)\n  \\(\\stackrel{\\text{IH}}{=} \\ivaluation{\\Ia}{\\theta} \\cdot \\ivaluation{\\Ia}{\\eta}\n  = \\ivaluation{\\Ia}{\\theta\\cdot\\eta}\\)\n  by induction hypothesis.\n\n\\item\n\\def\\Im{\\imodif[state]{\\vdLint[const=I,state=\\nu]}{x}{X}}%\n\\def\\Ima{\\iadjointSubst{\\sigma}{\\Im}}%\n\\def\\Iam{\\imodif[state]{\\Ia}{x}{X}}%\n\\(\n\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\sigma}{\\der{\\theta}}}\n= \\ivaluation{\\vdLint[const=I,state=\\nu]}{\\der{\\applyusubst{\\sigma}{\\theta}}}\n= \\sum_x \\iget[state]{\\vdLint[const=I,state=\\nu]}(\\D{x}) \\itimes \\Dp[X]{\\ivaluation{\\Im}{\\applyusubst{\\sigma}{\\theta}}}\n\\stackrel{\\text{IH}}{=} \\sum_x \\iget[state]{\\vdLint[const=I,state=\\nu]}(\\D{x}) \\itimes \\Dp[X]{\\ivaluation{\\Ima}{\\theta}}\n= \\sum_x \\iget[state]{\\vdLint[const=I,state=\\nu]}(\\D{x}) \\itimes \\Dp[X]{\\ivaluation{\\Iam}{\\theta}}\n= \\ivaluation{\\Ia}{\\der{\\theta}}\n\\)\nby induction hypothesis,\nprovided $\\sigma$ is $\\mathcal{V}\\cup\\D{\\mathcal{V}}$-admissible for $\\theta$, i.e. does not introduce any variables or differential symbols, \nso that \\rref{cor:adjointUsubst} implies \\(\\iget[const]{\\Ia}=\\iget[const]{\\Ita}\\) for all $\\iget[state]{\\vdLint[const=I,state=\\nu]},\\iget[state]{\\vdLint[const=I,state=\\omega]}$ (that agree on $\\scomplement{(\\mathcal{V}\\cup\\D{\\mathcal{V}})}=\\emptyset$, which imposes no condition on $\\iget[state]{\\vdLint[const=I,state=\\nu]},\\iget[state]{\\vdLint[const=I,state=\\omega]}$).\n\n\\end{compactitem}\n\\end{proofatend}\n\n\n\n\\begin{lemma}[Uniform substitution lemma] \\label{lem:usubst}\nThe uniform substitution $\\sigma$ and its adjoint interpretation $\\iportray{\\Ia}$ to $\\sigma$ for $\\iportray{\\vdLint[const=I,state=\\nu]}$ have the same \\emph{formula} semantics:\n\\[\\imodels{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\sigma}{\\phi}} ~\\text{iff}~ \\imodels{\\Ia}{\\phi}\\]\n\\end{lemma}\n\\begin{proofatend}\nThe proof is by structural induction on $\\phi$.\n\\begin{compactitem}\n\\item\n  \\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\sigma}{\\theta\\geq\\eta}}\\)\n  iff \\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\sigma}{\\theta} \\geq \\applyusubst{\\sigma}{\\eta}}\\)\n  iff \\(\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\sigma}{\\theta}} \\geq \\ivaluation{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\sigma}{\\eta}}\\),\n  by \\rref{lem:usubst-term},\n  iff \\(\\ivaluation{\\Ia}{\\theta} \\geq \\ivaluation{\\Ia}{\\eta}\\)\n  iff \\(\\ivaluation{\\Ia}{\\theta\\geq\\eta}\\)\n\n\\item\n  \\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\sigma}{p(\\theta)}}\\)\n  iff \\(\\imodels{\\vdLint[const=I,state=\\nu]}{(\\applyusubst{\\sigma}{p})\\big(\\applyusubst{\\sigma}{\\theta}\\big)}\\)\n  iff \\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\{\\usarg\\mapsto\\applyusubst{\\sigma}{\\theta}\\}}{\\applysubst{\\sigma}{p(\\usarg)}}}\\)\n  iff, by IH, \\(\\imodels{\\Iminner}{\\applysubst{\\sigma}{p(\\usarg)}}\\)\n  iff \\(d \\in \\iget[const]{\\Ia}(p)\\)\n  iff \\((\\ivaluation{\\Ia}{\\theta}) \\in \\iget[const]{\\Ia}(p)\\)\n  iff \\(\\imodels{\\Ia}{p(\\theta)}\\)\n  with \\(d\\mdefeq\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\sigma}{\\theta}} = \\ivaluation{\\Ia}{\\theta}\\)\n  by using \\rref{lem:usubst-term} for \\(\\applyusubst{\\sigma}{\\theta}\\)\n  and by using the induction hypothesis\n  for \\(\\applyusubst{\\{\\usarg\\mapsto\\applyusubst{\\sigma}{\\theta}\\}}{\\applysubst{\\sigma}{p(\\usarg)}}\\)\n  on the possibly bigger formula \\({\\applysubst{\\sigma}{p(\\usarg)}}\\) but the structurally simpler uniform substitution \\(\\applyusubst{\\{\\usarg\\mapsto\\applyusubst{\\sigma}{\\theta}\\}}{\\dots}\\) that is a mere substitution on symbol $\\usarg$ of arity zero, not a substitution of predicates\n  (\\(\\text{for}~p\\in\\replacees{\\sigma}\\)).\n\n\\item\n  \\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\sigma}{q(\\theta)}}\\)\n  iff \\(\\imodels{\\vdLint[const=I,state=\\nu]}{q(\\applyusubst{\\sigma}{\\theta})}\\)\n  iff \\(\\big(\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\sigma}{\\theta}}\\big)\\in \\iget[const]{\\vdLint[const=I,state=\\nu]}(q)\\)\n  so, by \\rref{lem:usubst-term}, iff \\(\\big(\\ivaluation{\\Ia}{\\theta}\\big) \\in \\iget[const]{\\vdLint[const=I,state=\\nu]}(q)\\)\n  iff \\(\\big(\\ivaluation{\\Ia}{\\theta}\\big) \\in \\iget[const]{\\Ia}(q)\\)\n  iff \\(\\imodels{\\Ia}{q(\\theta)}\\)\n  since \\(\\iget[const]{\\vdLint[const=I,state=\\nu]}(q)=\\iget[const]{\\Ia}(q)\\) as the interpretation of $q$ does not change in $\\iget[const]{\\Ia}$\n  (\\(\\text{for}~q\\not\\in\\replacees{\\sigma}\\))\n\n\\item\n\\def\\ImM{\\imodif[const]{\\vdLint[const=I,state=\\nu]}{\\uscarg}{R}}%\n\\let\\ImN\\ImM%\n\\def\\IaM{\\imodif[const]{\\Ia}{\\uscarg}{R}}%\n  For the case \\({\\applyusubst{\\sigma}{\\contextapp{C}{\\phi}}}\\), first show \n  \\(\\imodel{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\sigma}{\\phi}} = \\imodel{\\Ia}{\\phi}\\).\n  By induction hypothesis for the smaller $\\phi$:\n  \\(\\imodels{\\vdLint[const=I,state=\\omega]}{\\applyusubst{\\sigma}{\\phi}}\\)\n  iff\n  \\(\\imodels{\\Ita}{\\phi}\\),\n  where \n  \\(\\imodel{\\Ita}{\\phi}=\\imodel{\\Ia}{\\phi}\\)\n  by \\rref{cor:adjointUsubst}\n  for all $\\iget[state]{\\Ia},\\iget[state]{\\Ita}$\n  (that agree on $\\scomplement{(\\mathcal{V}\\cup\\D{\\mathcal{V}})}=\\emptyset$, which imposes no condition on $\\iget[state]{\\vdLint[const=I,state=\\nu]},\\iget[state]{\\vdLint[const=I,state=\\omega]}$)\n  since $\\sigma$ is $\\mathcal{V}\\cup\\D{\\mathcal{V}}$-admissible for $\\phi$.\n  The proof then proceeds:\n\n  \\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\sigma}{\\contextapp{C}{\\phi}}}\\)\n  \\(=\\imodel{\\vdLint[const=I,state=\\nu]}{\\contextapp{\\applyusubst{\\sigma}{C}}{\\applyusubst{\\sigma}{\\phi}}}\\)\n  \\(= \\imodel{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\{\\uscarg\\mapsto\\applyusubst{\\sigma}{\\phi}\\}}{\\applysubst{\\sigma}{\\contextapp{C}{\\uscarg}}}}\\),\n  so, by induction hypothesis for the structurally simpler uniform substitution ${\\{\\uscarg\\mapsto\\applyusubst{\\sigma}{\\phi}\\}}$ that is a mere substitution on symbol $\\uscarg$ of arity zero, iff\n  \\(\\imodels{\\ImM}{\\applysubst{\\sigma}{\\contextapp{C}{\\uscarg}}}\\)\n  since the adjoint to \\(\\{\\uscarg\\mapsto\\applyusubst{\\sigma}{\\phi}\\}\\) is $\\iget[const]{\\ImM}$ with \\(R\\mdefeq\\imodel{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\sigma}{\\phi}}\\).\n  \n  Also\n  \\(\\imodels{\\Ia}{\\contextapp{C}{\\phi}}\\)\n  \\(= \\iget[const]{\\Ia}(C)\\big(\\imodel{\\Ia}{\\phi}\\big)\\)\n  \\(= \\imodel{\\ImN}{\\applysubst{\\sigma}{\\contextapp{C}{\\uscarg}}}\\)\n  for \\(R=\\imodel{\\Ia}{\\phi}=\\imodel{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\sigma}{\\phi}}\\) by induction hypothesis.\n  Both sides are, thus, equivalent.\n  \n\\item\n  The case \\({\\applyusubst{\\sigma}{\\contextapp{C}{\\phi}}}\\) for $C\\not\\in\\replacees{\\sigma}$ again first shows\n  \\(\\imodel{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\sigma}{\\phi}} = \\imodel{\\Ia}{\\phi}\\)\n  for all $\\iget[state]{\\vdLint[const=I,state=\\nu]}$ using that $\\sigma$ is $\\mathcal{V}\\cup\\D{\\mathcal{V}}$-admissible for $\\phi$.\n  Then\n  \\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\sigma}{\\contextapp{C}{\\phi}}}\\)\n  \\(= \\imodel{\\vdLint[const=I,state=\\nu]}{\\contextapp{C}{\\applyusubst{\\sigma}{\\phi}}}\\)\n  \\(= \\iget[const]{\\vdLint[const=I,state=\\nu]}(C)\\big(\\imodel{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\sigma}{\\phi}}\\big)\\)\n  \\(= \\iget[const]{\\vdLint[const=I,state=\\nu]}(C)\\big(\\imodel{\\Ia}{\\phi}\\big)\\)\n  \\(= \\iget[const]{\\Ia}(C)\\big(\\imodel{\\Ia}{\\phi}\\big)\\)\n  \\(= \\imodel{\\Ia}{\\contextapp{C}{\\phi}}\\)\n  iff \\(\\imodels{\\Ia}{\\contextapp{C}{\\phi}}\\)\n\n\\item\n  \\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\sigma}{\\lnot\\phi}}\\)\n  iff \\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\lnot\\applyusubst{\\sigma}{\\phi}}\\)\n  iff \\(\\inonmodels{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\sigma}{\\phi}}\\),\n  by induction hypothesis,\n  iff \\(\\inonmodels{\\Ia}{\\phi}\\)\n  iff \\(\\imodels{\\Ia}{\\lnot\\phi}\\)\n\n\\item\n  \\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\sigma}{\\phi\\land\\psi}}\\)\n  iff \\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\sigma}{\\phi} \\land \\applyusubst{\\sigma}{\\psi}}\\)\n  iff \\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\sigma}{\\phi}}\\) and \\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\sigma}{\\psi}}\\),\n  by induction hypothesis,\n  iff \\(\\imodels{\\Ia}{\\phi}\\) and \\(\\imodels{\\Ia}{\\psi}\\)\n  iff \\(\\imodels{\\Ia}{\\phi\\land\\psi}\\)\n\n\\item\n\\def\\Imd{\\imodif[state]{\\vdLint[const=I,state=\\nu]}{x}{d}}%\n\\def\\Iamd{\\imodif[state]{\\Ia}{x}{d}}%\n\\def\\Imda{\\iadjointSubst{\\sigma}{\\Imd}}%\n  \\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\sigma}{\\lexists{x}{\\phi}}}\\)\n  iff \\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\lexists{x}{\\applyusubst{\\sigma}{\\phi}}}\\)\n  (provided $\\sigma$ is $\\{x\\}$-admissible for $\\phi$)\n  iff \\(\\imodels{\\Imd}{\\applyusubst{\\sigma}{\\phi}}\\) for some $d$,\n  so, by induction hypothesis,\n  iff \\(\\imodels{\\Imda}{\\phi}\\) for some $d$,\n  which is equivalent to\n  \\(\\imodels{\\Iamd}{\\phi}\\) by \\rref{cor:adjointUsubst} as $\\sigma$ is $\\{x\\}$-admissible for $\\phi$ and $\\iget[state]{\\vdLint[const=I,state=\\nu]}=\\iget[state]{\\Imd}$ on $\\scomplement{\\{x\\}}$.\n  Thus, this is equivalent to\n  \\(\\imodels{\\Ia}{\\lexists{x}{\\phi}}\\).\n  \n\\item The case\n  \\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\sigma}{\\lforall{x}{\\phi}}}\\)\n  follows by duality \\(\\lforall{x}{\\phi} \\mequiv \\lnot\\lexists{x}{\\lnot\\phi}\\), which is respected in the definition of uniform substitutions.\n\n\\item\n  \\newcommand{\\Iat}{\\iconcat[state=\\omega]{\\Ia}}%\n  \\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\sigma}{\\ddiamond{\\alpha}{\\phi}}}\\)\n  iff \\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\ddiamond{\\applyusubst{\\sigma}{\\alpha}}{\\applyusubst{\\sigma}{\\phi}}}\\)\n  (provided $\\sigma$ is $\\boundvars{\\applyusubst{\\sigma}{\\alpha}}$-admissible for $\\phi$)\n  iff there is a $\\iget[state]{\\vdLint[const=I,state=\\omega]}$ such that\n  \\(\\iaccessible[\\applyusubst{\\sigma}{\\alpha}]{\\vdLint[const=I,state=\\nu]}{\\vdLint[const=I,state=\\omega]}\\) and \\(\\imodels{\\vdLint[const=I,state=\\omega]}{\\applyusubst{\\sigma}{\\phi}}\\),\n  which, by \\rref{lem:usubst-HP} and induction hypothesis, respectively, is equivalent to:\n  there is a $\\iget[state]{\\Ita}$ such that\n  \\(\\iaccessible[\\alpha]{\\Ia}{\\Ita}\\) and \\(\\imodels{\\Ita}{\\phi}\\),\n  which is equivalent to\n  \\(\\imodels{\\Ia}{\\ddiamond{\\alpha}{\\phi}}\\),\n  because \\(\\imodels{\\Ita}{\\phi}\\) is equivalent to \\(\\imodels{\\Iat}{\\phi}\\) by \\rref{cor:adjointUsubst}\n  as $\\sigma$ is $\\boundvars{\\applyusubst{\\sigma}{\\alpha}}$-admissible for $\\phi$ and \\(\\iget[state]{\\Ia}=\\iget[state]{\\Iat}\\) on $\\scomplement{\\boundvars{\\applyusubst{\\sigma}{\\alpha}}}$ by \\rref{lem:bound} since\n  \\(\\iaccessible[\\applyusubst{\\sigma}{\\alpha}]{\\vdLint[const=I,state=\\nu]}{\\vdLint[const=I,state=\\omega]}\\).\n\n\\item The case\n  \\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\sigma}{\\dbox{\\alpha}{\\phi}}}\\)\n  follows by duality \\(\\dbox{\\alpha}{\\phi} \\mequiv \\lnot\\ddiamond{\\alpha}{\\lnot\\phi}\\), which is respected in the definition of uniform substitutions.\n\n\\end{compactitem}\n\\end{proofatend}\n\n\n\\begin{lemma}[Uniform substitution lemma] \\label{lem:usubst-HP}\nThe uniform substitution $\\sigma$ and its adjoint interpretation $\\iportray{\\Ia}$ to $\\sigma$ for $\\iportray{\\vdLint[const=I,state=\\nu]}$ have the same \\emph{program} semantics:\n\\[\n\\iaccessible[{\\applyusubst{\\sigma}{\\alpha}}]{\\vdLint[const=I,state=\\nu]}{\\vdLint[const=I,state=\\omega]}\n~\\text{iff}~\n\\iaccessible[\\alpha]{\\Ia}{\\Ita}\n\\]\n\\end{lemma}\n\\begin{proofatend}\nThe proof is by structural induction on $\\alpha$.\n\\begin{compactitem}\n\\item\n  \\(\\iaccessible[\\applyusubst{\\sigma}{a}]{\\vdLint[const=I,state=\\nu]}{\\vdLint[const=I,state=\\omega]} = \\iaccess[\\applysubst{\\sigma}{a}]{\\vdLint[const=I,state=\\nu]} = \\iget[const]{\\Ia}(a) = \\iaccess[a]{\\Ia}\\)\n  for program constant $a\\in\\replacees{\\sigma}$\n  (the proof is accordingly for $a\\not\\in\\replacees{\\sigma}$).\n\n\\item \n  \\(\\iaccessible[\\applyusubst{\\sigma}{\\pumod{x}{\\theta}}]{\\vdLint[const=I,state=\\nu]}{\\vdLint[const=I,state=\\omega]}\n  = \\iaccess[\\pumod{x}{\\applyusubst{\\sigma}{\\theta}}]{\\vdLint[const=I,state=\\nu]}\\)\n  iff \\(\\iget[state]{\\vdLint[const=I,state=\\omega]} = \\modif{\\iget[state]{\\vdLint[const=I,state=\\nu]}}{x}{\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\sigma}{\\theta}}}\\)\n  = \\(\\modif{\\iget[state]{\\vdLint[const=I,state=\\nu]}}{x}{\\ivaluation{\\Ia}{\\theta}}\\)\n  by \\rref{lem:usubst}, which is, thus, equivalent to\n  \\(\\iaccessible[\\pumod{x}{\\theta}]{\\Ia}{\\Ita}\\).\n\n\\item\n  \\(\\iaccessible[\\applyusubst{\\sigma}{\\Dumod{\\D{x}}{\\theta}}]{\\vdLint[const=I,state=\\nu]}{\\vdLint[const=I,state=\\omega]}\n  = \\iaccess[\\Dumod{\\D{x}}{\\applyusubst{\\sigma}{\\theta}}]{\\vdLint[const=I,state=\\nu]}\\)\n  iff \\(\\iget[state]{\\vdLint[const=I,state=\\omega]} = \\modif{\\iget[state]{\\vdLint[const=I,state=\\nu]}}{\\D{x}}{\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\sigma}{\\theta}}}\\)\n  = \\(\\modif{\\iget[state]{\\vdLint[const=I,state=\\nu]}}{\\D{x}}{\\ivaluation{\\Ia}{\\theta}}\\)\n  by \\rref{lem:usubst}, which is, thus, equivalent to\n  \\(\\iaccessible[\\Dumod{\\D{x}}{\\theta}]{\\Ia}{\\Ita}\\).\n\n\\item\n  \\(\\iaccessible[\\applyusubst{\\sigma}{\\ptest{\\psi}}]{\\vdLint[const=I,state=\\nu]}{\\vdLint[const=I,state=\\omega]}\n  = \\iaccess[\\ptest{\\applyusubst{\\sigma}{\\psi}}]{\\vdLint[const=I,state=\\nu]}\\)\n  iff \\(\\iget[state]{\\vdLint[const=I,state=\\omega]}=\\iget[state]{\\vdLint[const=I,state=\\nu]}\\)\n  and \\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\sigma}{\\psi}}\\),\n  iff, by \\rref{lem:usubst},\n  \\(\\iget[state]{\\Ita}=\\iget[state]{\\Ia}\\)     and\n  \\(\\imodels{\\Ia}{\\psi}\\),\n  which is equivalent to\n  \\(\\iaccessible[\\ptest{\\psi}]{\\Ia}{\\Ita}\\).\n\n\\item\n\\newcommand{\\iconcat[state=\\varphi(t)]{\\I}}{\\iconcat[state=\\varphi(t)]{\\vdLint[const=I,state=\\nu]}}\n\\def\\Izetaa{\\iadjointSubst{\\sigma}{\\iconcat[state=\\varphi(t)]{\\I}}}%\n\\newcommand{\\iconcat[state=\\varphi(t)]{\\Ia}}{\\iconcat[state=\\varphi(t)]{\\Ia}}\n  \\(\\iaccessible[\\applyusubst{\\sigma}{\\pevolvein{\\D{x}=\\genDE{x}}{\\psi}}]{\\vdLint[const=I,state=\\nu]}{\\vdLint[const=I,state=\\omega]}\n  = \\iaccess[\\pevolvein{\\D{x}=\\applyusubst{\\sigma}{\\genDE{x}}}\n  {\\applyusubst{\\sigma}{\\psi}}]{\\vdLint[const=I,state=\\nu]}\\)\n  (provided $\\sigma$ $\\{x,\\D{x}\\}$-admissible for $\\genDE{x},\\psi$)\n  iff \\(\\mexists{\\varphi:[0,T]\\to\\linterpretations{\\Sigma}{V}}\\)\n  with \\(\\varphi(0)=\\iget[state]{\\vdLint[const=I,state=\\nu]}, \\varphi(T)=\\iget[state]{\\vdLint[const=I,state=\\omega]}\\) and for all $t\\geq0$:\n  \\(\\D{\\varphi}(t) = \\ivaluation{\\iconcat[state=\\varphi(t)]{\\I}}{\\applyusubst{\\sigma}{\\genDE{x}}}\n  = \\ivaluation{\\Izetaa}{\\genDE{x}}\\) by \\rref{lem:usubst-term}\n  as well as\n  \\(\\imodels{\\iconcat[state=\\varphi(t)]{\\I}}{\\applyusubst{\\sigma}{\\psi}}\\),\n  which, by \\rref{lem:usubst}, is equivalent to\n  \\(\\imodels{\\Izetaa}{\\psi}\\).\n  \n  Also\n  \\(\\iaccessible[\\pevolvein{\\D{x}=\\genDE{x}}{\\psi}]{\\Ia}{\\Ita}\\)\n  iff \\(\\mexists{\\varphi:[0,T]\\to\\linterpretations{\\Sigma}{V}}\\)\n  with \\(\\varphi(0)=\\iget[state]{\\vdLint[const=I,state=\\nu]}, \\varphi(T)=\\iget[state]{\\vdLint[const=I,state=\\omega]}\\) and for all $t\\geq0$:\n  \\(\\D{\\varphi}(t) = \\ivaluation{\\iconcat[state=\\varphi(t)]{\\Ia}}{\\genDE{x}}\\)\n  and\n  \\(\\imodels{\\iconcat[state=\\varphi(t)]{\\Ia}}{\\psi}\\).\n  Finally,\n  \\(\\ivalues{\\iconcat[state=\\varphi(t)]{\\Ia}}{\\genDE{x}}=\\ivalues{\\Izetaa}{\\genDE{x}}\\) and\n  \\(\\imodel{\\Izetaa}{\\psi}=\\imodel{\\iconcat[state=\\varphi(t)]{\\Ia}}{\\psi}\\)\n  by \\rref{cor:adjointUsubst}\n  since $\\sigma$ is $\\{x,\\D{x}\\}$-admissible for $\\genDE{x},\\psi$ and \\(\\iget[state]{\\vdLint[const=I,state=\\nu]}=\\iget[state]{\\iconcat[state=\\varphi(t)]{\\Ia}}\\) on $\\scomplement{\\boundvars{\\pevolvein{\\D{x}=\\genDE{x}}{\\psi}}}=\\scomplement{\\{x,\\D{x}\\}}$ by \\rref{lem:bound}.\n  \n\\item  \n  \\(\\iaccessible[\\applyusubst{\\sigma}{\\pchoice{\\alpha}{\\beta}}]{\\vdLint[const=I,state=\\nu]}{\\vdLint[const=I,state=\\omega]}\n  = \\iaccess[\\pchoice{\\applyusubst{\\sigma}{\\alpha}}{\\applyusubst{\\sigma}{\\beta}}]{\\vdLint[const=I,state=\\nu]}\\)\n  = \\(\\iaccess[\\applyusubst{\\sigma}{\\alpha}]{\\vdLint[const=I,state=\\nu]} \\cup \\iaccess[\\applyusubst{\\sigma}{\\beta}]{\\vdLint[const=I,state=\\nu]}\\),\n  which, by induction hypothesis, is equivalent to\n  \\(\\iaccessible[\\alpha]{\\Ia}{\\Ita}\\) or \\(\\iaccessible[\\beta]{\\Ia}{\\Ita}\\),\n  which is equivalent to\n  \\(\\iaccessible[\\alpha]{\\Ia}{\\Ita} \\cup \\iaccess[\\beta]{\\Ia} = \\iaccess[\\pchoice{\\alpha}{\\beta}]{\\Ia}\\).\n  \n\n\\item\n{\\newcommand{\\iconcat[state=\\mu]{\\I}}{\\iconcat[state=\\mu]{\\vdLint[const=I,state=\\nu]}}%\n\\newcommand{\\Iza}{\\iadjointSubst{\\sigma}{\\iconcat[state=\\mu]{\\I}}}%\n\\newcommand{\\Iaz}{\\iconcat[state=\\mu]{\\Ia}}%\n  \\(\\iaccessible[\\applyusubst{\\sigma}{\\alpha;\\beta}]{\\vdLint[const=I,state=\\nu]}{\\vdLint[const=I,state=\\omega]}\n  = \\iaccess[\\applyusubst{\\sigma}{\\alpha}; \\applyusubst{\\sigma}{\\beta}]{\\vdLint[const=I,state=\\nu]}\\)\n  = \\(\\iaccess[\\applyusubst{\\sigma}{\\alpha}]{\\vdLint[const=I,state=\\nu]} \\compose \\iaccess[\\applyusubst{\\sigma}{\\beta}]{\\vdLint[const=I,state=\\nu]}\\)\n  (provided $\\sigma$ is $\\boundvars{\\applyusubst{\\sigma}{\\alpha}}$-admissible for $\\beta$)\n  iff there is a $\\iget[state]{\\iconcat[state=\\mu]{\\I}}$ such that\n  \\(\\iaccessible[\\applyusubst{\\sigma}{\\alpha}]{\\vdLint[const=I,state=\\nu]}{\\iconcat[state=\\mu]{\\I}}\\) and \\(\\iaccessible[\\applyusubst{\\sigma}{\\beta}]{\\iconcat[state=\\mu]{\\I}}{\\vdLint[const=I,state=\\omega]}\\),\n  which, by induction hypothesis, is equivalent to\n  \\(\\iaccessible[\\alpha]{\\Ia}{\\Iza}\\) and \\(\\iaccessible[\\beta]{\\Iza}{\\Ita}\\).\n  Yet, \\(\\iaccess[\\beta]{\\Iza}=\\iaccess[\\beta]{\\Ia}\\)\n  by \\rref{cor:adjointUsubst}, because $\\sigma$ is $\\boundvars{\\applyusubst{\\sigma}{\\alpha}}$-admissible for $\\beta$ and \\(\\iget[state]{\\vdLint[const=I,state=\\nu]}=\\iget[state]{\\vdLint[const=I,state=\\omega]}\\) on $\\scomplement{\\boundvars{\\applyusubst{\\sigma}{\\alpha}}}$ by \\rref{lem:bound} since \\(\\iaccessible[\\applyusubst{\\sigma}{\\alpha}]{\\vdLint[const=I,state=\\nu]}{\\iconcat[state=\\mu]{\\I}}\\).\n  Finally,\n  \\(\\iaccessible[\\alpha]{\\Ia}{\\Iaz}\\) and \\(\\iaccessible[\\beta]{\\Iaz}{\\Ita}\\) for some $\\iget[state]{\\Iaz}$\n  is equivalent to \\(\\iaccessible[\\alpha;\\beta]{\\Ia}{\\Ita}\\).\n\n}\n\n\\item\n\\newcommand{\\iconcat[state=\\mu]{\\I}}[1][]{\\iconcat[state=\\nu_{#1}]{\\vdLint[const=I,state=\\nu]}}%\n\\newcommand{\\Iza}[1][]{\\iadjointSubst{\\sigma}{\\iconcat[state=\\mu]{\\I}[#1]}}%\n\\newcommand{\\Iaz}[1][]{\\iconcat[state=\\nu_{#1}]{\\Ia}}%\n  \\(\\iaccessible[\\applyusubst{\\sigma}{\\prepeat{\\alpha}}]{\\vdLint[const=I,state=\\nu]}{\\vdLint[const=I,state=\\omega]}\n  = \\iaccess[\\prepeat{(\\applyusubst{\\sigma}{\\alpha})}]{\\vdLint[const=I,state=\\nu]}\n  = \\closureTransitive{\\big(\\iaccess[\\applyusubst{\\sigma}{\\alpha}]{\\vdLint[const=I,state=\\nu]}\\big)}\n  = \\cupfold_{n\\in\\naturals} (\\iaccess[\\applyusubst{\\sigma}{\\alpha}]{\\vdLint[const=I,state=\\nu]})^n\n  \\)\n  (provided $\\sigma$ is $\\boundvars{\\applyusubst{\\sigma}{\\alpha}}$-admissible for $\\alpha$)\n  iff there are $n\\in\\naturals$ and \\(\\iget[state]{\\iconcat[state=\\mu]{\\I}[0]}=\\iget[state]{\\Ia},\\iget[state]{\\iconcat[state=\\mu]{\\I}[1]},\\dots,\\iget[state]{\\iconcat[state=\\mu]{\\I}[n]}=\\iget[state]{\\Ita}\\) such that\n  \\(\\iaccessible[\\applyusubst{\\sigma}{\\alpha}]{\\iconcat[state=\\mu]{\\I}[i]}{\\iconcat[state=\\mu]{\\I}[i+1]}\\) for all $i<n$.\n  By $n$ uses of the induction hypothesis, this is equivalent to\n  \\(\\iaccessible[\\alpha]{\\Iza[i]}{\\Iza[i+1]}\\) for all $i<n$,\n  which is equivalent to\n  \\(\\iaccessible[\\alpha]{\\Iaz[i]}{\\Iza[i+1]}\\) \n  by \\rref{cor:adjointUsubst}\n  since $\\sigma$ is $\\boundvars{\\applyusubst{\\sigma}{\\alpha}}$-admissible for $\\alpha$\n  and \\(\\iget[state]{\\Iza[i+1]}=\\iget[state]{\\Iza[i]}\\) on $\\scomplement{\\boundvars{\\applyusubst{\\sigma}{\\alpha}}}$ by \\rref{lem:bound} as \\(\\iaccessible[\\applyusubst{\\sigma}{\\alpha}]{\\iconcat[state=\\mu]{\\I}[i]}{\\iconcat[state=\\mu]{\\I}[i+1]}\\) for all $i<n$.\n  Thus, this is equivalent to\n  \\(\\iaccessible[\\prepeat{\\alpha}]{\\Ia}{\\Ita}\n  = \\closureTransitive{\\big(\\iaccess[\\alpha]{\\Ia}\\big)}\\).\n\n\\end{compactitem}\n\\end{proofatend}\n\n\\subsection{Soundness}\n\nThe uniform substitution lemmas are the key insights for the soundness of \\irref{US}.\n\\irref{US} is only applicable if the uniform substitution is defined (does not clash).\n\\begin{theorem}[Soundness of uniform substitution] \\label{thm:usubst-sound}\n  \\irref{US} is sound and so is its special case \\irref{US0}.\n  That is, if their premise is valid, then so is their conclusion.\n\\end{theorem}\n\\begin{proofatend}\n\\def\\Ia{\\iadjointSubst{\\sigma}{\\vdLint[const=I,state=\\nu]}}%\nLet the premise $\\phi$ of \\irref{US} be valid, i.e.\\ \\m{\\imodels{\\vdLint[const=I,state=\\nu]}{\\phi}} for all interpretations and states $\\iportray{\\vdLint[const=I,state=\\nu]}$.\nTo show that the conclusion is valid, consider any interpretation and state $\\iportray{\\vdLint[const=I,state=\\nu]}$ and show \\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\sigma}{\\phi}}\\).\nBy \\rref{lem:usubst}, \\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\sigma}{\\phi}}\\) iff \\(\\imodels{\\Ia}{\\phi}\\).\nThe latter holds, because \\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\phi}\\) for all $\\iportray{\\vdLint[const=I,state=\\nu]}$, including for $\\iportray{\\Ia}$, by premise.\nThe rule \\irref{US0} is the special case of \\irref{US} where $\\sigma$ only substitutes predicate symbol $p$.\n\n\\end{proofatend}\n\n\n\\section{Differential Dynamic Logic Axioms} \\label{sec:dL-axioms}\n\nProof rules and axioms for a Hilbert-type axiomatization of \\dL from prior work \\cite{DBLP:conf\/lics\/Platzer12b} are shown in \\rref{fig:dL}, except that, thanks to rule \\irref{US}, axioms and rules now comprise the finite list of \\dL formulas in \\rref{fig:dL} as opposed to an infinite collection of axioms from a finite list of axiom schemata along with schema variables, side conditions, and implicit instantiation rules.\n\\begin{figure}[tbhp]\n  \\renewcommand*{\\irrulename}[1]{\\text{#1}}%\n    \\iflongversion\\else\\linferenceRulevskipamount=0.4em\\fi\n  \\begin{calculuscollections}{\\columnwidth}\n    \\begin{calculus}\n      \\cinferenceRule[diamond|$\\didia{\\cdot}$]{diamond axiom}\n      {\\linferenceRule[equiv]\n        {\\lnot\\dbox{a}{\\lnot p(\\bar{x})}}\n        {\\ddiamond{a}{p(\\bar{x})}}\n      }\n      {}\n      \\cinferenceRule[assignb|$\\dibox{:=}$]{assignment \/ substitution axiom}\n      {\\linferenceRule[equiv]\n        {p(f)}\n        {\\dbox{\\pupdate{\\umod{x}{f}}}{p(x)}}\n      }\n      {}%\n      \\cinferenceRule[testb|$\\dibox{?}$]{test}\n      {\\linferenceRule[equiv]\n        {(q \\limply p)}\n        {\\dbox{\\ptest{q}}{p}}\n      }{}\n      \\cinferenceRule[choiceb|$\\dibox{\\cup}$]{axiom of nondeterministic choice}\n      {\\linferenceRule[equiv]\n        {\\dbox{a}{p(\\bar{x})} \\land \\dbox{b}{p(\\bar{x})}}\n        {\\dbox{\\pchoice{a}{b}}{p(\\bar{x})}}\n      }{}\n      \\cinferenceRule[composeb|$\\dibox{{;}}$]{composition} %\n      {\\linferenceRule[equiv]\n        {\\dbox{a}{\\dbox{b}{p(\\bar{x})}}}\n        {\\dbox{a;b}{p(\\bar{x})}}\n      }{}\n      \\cinferenceRule[iterateb|$\\dibox{{}^*}$]{iteration\/repeat unwind} %\n      {\\linferenceRule[equiv]\n        {p(\\bar{x}) \\land \\dbox{a}{\\dbox{\\prepeat{a}}{p(\\bar{x})}}}\n        {\\dbox{\\prepeat{a}}{p(\\bar{x})}}\n      }{}\n      %\n      \\cinferenceRule[K|K]{K axiom \/ modal modus ponens} %\n      {\\linferenceRule[impl]\n        {\\dbox{a}{(p(\\bar{x})\\limply q(\\bar{x}))}}\n        {(\\dbox{a}{p(\\bar{x})}\\limply\\dbox{a}{q(\\bar{x})})}\n      }{}\n      \\cinferenceRule[I|I]{loop induction}\n      {\\linferenceRule[impl]\n        {\\dbox{\\prepeat{a}}{(p(\\bar{x})\\limply\\dbox{a}{p(\\bar{x})})}}\n        {(p(\\bar{x})\\limply\\dbox{\\prepeat{a}}{p(\\bar{x})})}\n      }{}\n      \\cinferenceRule[V|V]{vacuous $\\dbox{}{}$}\n      {\\linferenceRule[impl]\n        {p}\n        {\\dbox{a}{p}}\n      }{}%\n    \\end{calculus}\n    \\qquad\n    \\begin{calculus}\n      \\cinferenceRule[G|G]{$\\dbox{}{}$ generalisation} %\n      {\\linferenceRule[formula]\n        {p(\\bar{x})}\n        {\\dbox{a}{p(\\bar{x})}}\n      }{}\n      \\cinferenceRule[gena|$\\forall{}$]{$\\forall{}$ generalisation}\n      {\\linferenceRule[formula]\n        {p(x)}\n        {\\lforall{x}{p(x)}}\n      }{}%\n      %\n      %\n      \\cinferenceRule[MP|MP]{modus ponens}\n      {\\linferenceRule[formula]\n        {p\\limply q \\quad p}\n        {q}\n      }{}%\n      \\cinferenceRule[CT|CT]{congterm congruence on terms}\n      {\\linferenceRule[formula]\n        {f(\\bar{x}) = g(\\bar{x})}\n        {c(f(\\bar{x})) = c(g(\\bar{x}))}\n      }{}%\n      \\cinferenceRule[CQ|CQ]{congequal congruence of equations on formulas (convert term congruence to formula congruence: term congruence on formulas)}\n      {\\linferenceRule[formula]\n        {f(\\bar{x}) = g(\\bar{x})}\n        {p(f(\\bar{x})) \\lbisubjunct p(g(\\bar{x}))}\n      }{}%\n      \\cinferenceRule[CE|CE]{congequiv congruence of equivalences on formulas}\n      {\\linferenceRule[formula]\n        {p(\\bar{x}) \\lbisubjunct q(\\bar{x})}\n        {\\contextapp{C}{p(\\bar{x})} \\lbisubjunct \\contextapp{C}{q(\\bar{x})}}\n      }{}%\n      %\n      \\iflongversion\n      \\cinferenceRule[USagain|US]{uniform substitution}\n      {\\linferenceRule[formula]\n        {\\phi}\n        {\\applyusubst{\\sigma}{\\phi}}\n      }{}%\n      \\fi\n    \\end{calculus}%\n  \\end{calculuscollections}\n  \\caption{Differential dynamic logic axioms and proof rules}\n  \\label{fig:dL}\n\\end{figure}\nSoundness of the axioms in \\rref{fig:dL} follows from the soundness of corresponding axiom schemata \\cite{DBLP:conf\/lics\/Platzer12b}, but would be easier to prove standalone, because it is a finite list of formulas without the need to prove soundness for all their instantiations.\nThe rules in \\rref{fig:dL} are \\emph{axiomatic rules}, i.e.\\ pairs of concrete formulas instantiated by \\irref{US}.\nFurther, $\\bar{x}$ is the vector of all relevant variables, which is finite-dimensional, or, in practice, considered as a built-in vectorial term.\nProofs in the uniform substitution \\dL calculus use \\irref{US} (and bound renaming such as \\(\\lforall{x}{p(x)}\\lbisubjunct\\lforall{y}{p(y)}\\)) to instantiate the axioms from \\rref{fig:dL} to the required form.\n\\irref{CT+CQ+CE} are congruence rules, which are included for efficiency to use axioms in any context even if not needed for completeness.\n\n\\paragraph{Real Quantifiers.}\n\nBesides (decidable) real arithmetic (whose use is denoted \\irref{qear}), complete axioms for first-order logic can be adopted to express\nuniversal instantiation \\irref{allinst}\\iflongversion (if $p$ is true of all $x$ it is also true of constant symbol $f$)\\fi,\ndistributivity \\irref{alldist},\nand vacuous quantification \\irref{vacuousall}\\iflongversion (predicate $p$ of arity zero does not depend on $x$)\\fi.\n\n  \\begin{calculuscollections}{\\columnwidth}\n    \\iflongversion\\else\\linferenceRulevskipamount=0.2em\\fi\n    \\begin{calculus}\n      \\cinferenceRule[allinst|$\\forall$i]{universal instantiation}\n       {(\\lforall{x}{p(x)}) \\limply p(f)}\n       {}\n       \\cinferenceRule[alldist|$\\forall{\\limply}$]{$\\forall$ distributes over $\\limply$}\n       {\\lforall{x}{(p(x)\\limply q(x))} \\limply (\\lforall{x}{p(x)} \\limply \\lforall{x}{q(x)})}\n       {}\n       \\cinferenceRule[vacuousall|V$_\\forall$]{vacuous universal quantifier}\n       {p \\limply \\lforall{x}{p}}\n       {}%\n    \\end{calculus}\n  \\end{calculuscollections}\n\\vspace{-\\baselineskip}\n\n\\paragraph{The Significance of Clashes.}\n\n\\iflongversion\nThis section illustrates how soundness-critical it is for \\irref{US} to produce substitution clashes by showing unsound na\\\"ive proof attempts that \\irref{US} prevents successfully.\n\\fi\n\\irref{US} clashes for substitutions that introduce a free variable into a bound context.\nEven an occurrence of $p(x)$ in a context where $x$ is bound does not allow mentioning $x$ in the replacement except in the $\\usarg$ places:\n\\[\n\\linfer[clash]\n{\\dbox{\\pupdate{\\umod{x}{f}}}{p(x)} \\lbisubjunct p(f)}\n{\\dbox{\\pupdate{\\umod{x}{x+1}}}{x\\neq x} \\lbisubjunct x+1\\neq x}\n\\qquad\n\\sigma=\\usubstlist{\\usubstmod{f}{x+1},\\usubstmod{p(\\usarg)}{(\\usarg\\neq x)}}\n\\]\n\\irref{US} can directly handle even nontrivial binding structures, though, e.g. from \\irref{assignb}\nwith the substitution \\(\\sigma=\\usubstlist{\\usubstmod{f}{x^2},\\usubstmod{p(\\usarg)}{\\dbox{\\prepeat{(z:=\\usarg+z)};z:=\\usarg+yz}{y\\geq\\usarg}}}\\):\n\\[\n\\linfer[US]\n{\\dbox{\\pupdate{\\umod{x}{f}}}{p(x)} \\lbisubjunct p(f)}\n{\\dbox{x:=x^2}{\\dbox{\\prepeat{(z:=x{+}z)};z:=x{+}yz}{y{\\geq}x}} \\lbisubjunct\n\\dbox{\\prepeat{(z:=x^2{+}z)};z:=x^2{+}yz}{y{\\geq}x^2}\n}\n\\]\n\n\\iflongversion\nSimilarly from \\irref{assignb}\nwith \\(\\usubstlist{\\usubstmod{f}{x^2},\\usubstmod{p(\\usarg)}{\\dbox{(\\pchoice{y:=y+1}{\\prepeat{z:=\\usarg+z}});z:=\\usarg+yz}{y>\\usarg}}}\\):\n{\\footnotesize\n\\[\n\\hspace{-1cm}\n\\linfer[US]\n{\\dbox{\\pupdate{\\umod{x}{f}}}{p(x)} \\lbisubjunct p(f)}\n{\\dbox{x:=x^2}{\\dbox{(\\pchoice{y:=y{+}1}{\\prepeat{z:=x{+}z}});z:=x{+}yz}{y{>}x}} \\lbisubjunct\n\\dbox{(\\pchoice{y:=y{+}1}{\\prepeat{z:=x^2{+}z}});z:=x^2{+}yz}{y{>}x^2}\n}\n\\]\n}%\nIt is soundness-critical that \\irref{US} clashes when trying to instantiate $p$ in \\irref{vacuousall} with a formula that mentions the bound variable $x$:\n\\[\n\\linfer[clash]\n{p \\limply \\lforall{x}{p}}\n{x\\geq0 \\limply \\lforall{x}{x\\geq0}}\n\\qquad\n\\usubstlist{\\usubstmod{p}{x\\geq0}}\n\\]\nIt is soundness-critical that \\irref{US} clashes when substituting $p$ in vacuous program axiom \\irref{V} with a formula with a free occurrence of a variable bound by $a$:\n\\[\n\\linfer[clash]\n{p \\limply \\dbox{a}{p}}\n{x\\geq0 \\limply \\dbox{x:=x-1}{x\\geq0}}\n\\qquad\n\\usubstlist{\\usubstmod{a}{x:=x-1},\\usubstmod{p}{x\\geq0}}\n\\]\nG\\\"odel's generalization rule \\irref{G} uses $p(\\bar{x})$ instead of $p$ from \\irref{V}, so allows the proof:\n\\[\n\\linfer[US]\n{(-x)^2\\geq0}\n{\\dbox{x:=x-1}{(-x)^2\\geq0}}\n\\]\n\n\\noindent\nLet \\(\\bar{x}=(x,y)\\), \n\\(\\usubstlist{\\usubstmod{a}{x:=x+1},\\usubstmod{b}{x:=0;y:=0},\\usubstmod{p(\\bar{x})}{x\\geq y}}\\), \\irref{US} derives:%\n\\vspace{-\\baselineskip}\n\\begin{sequentdeduction}\n  \\linfer[US]\n  {\\linfer[choiceb]\n    {\\lclose}\n    {\\dbox{\\pchoice{a}{b}}{p(\\bar{x})} \\lbisubjunct \\dbox{a}{p(\\bar{x})} \\land \\dbox{b}{p(\\bar{x})}}\n  }\n  {\\dbox{\\pchoice{x:=x+1}{(x:=0;y:=0)}}{x\\geq y} \\lbisubjunct \\dbox{x:=x+1}{x\\geq0} \\land \\dbox{x:=0;y:=0}{x\\geq y}}\n\\end{sequentdeduction}\n\n\\noindent\nWith \\(\\bar{x}=(x,y)\\) and\n\\(\\usubstlist{\\usubstmod{a}{\\pchoice{x:=x+1}{y:=0}},\\usubstmod{b}{y:=y+1},\\usubstmod{p(\\bar{x})}{x\\geq y}}\\) \\irref{US} derives:\n\\vspace{-\\baselineskip}\n\\begin{sequentdeduction}\n\\hspace{-0.5cm}\n  \\linfer[US]\n  {\\linfer[composeb]\n    {\\lclose}\n    {\\dbox{a;b}{p(\\bar{x})} \\lbisubjunct \\dbox{a}{\\dbox{b}{p(\\bar{x})}}}\n  }\n  {\\dbox{(\\pchoice{x:=x+1}{y:=0});y:=y+1}{x\\geq y} \\lbisubjunct \\dbox{\\pchoice{x:=x+1}{y:=0}}{\\dbox{y:=y+1}{x\\geq y}}}\n\\end{sequentdeduction}\n\nNot all axioms fit to the uniform substitution framework.\nThe Barcan axiom was used in a completeness proof for the Hilbert-type calculus for differential dynamic logic \\cite{DBLP:conf\/lics\/Platzer12b} (but not in the completeness proof for its sequent calculus \\cite{DBLP:journals\/jar\/Platzer08}):\n\\[\n      \\cinferenceRule[B|B]{Barcan$\\dbox{}{}\\forall{}$} %\n      {\\linferenceRule[impl]\n        {\\lforall{x}{\\dbox{\\alpha}{p(x)}}}\n        {\\dbox{\\alpha}{\\lforall{x}{p(x)}}}\n      }{\\m{x\\not\\in\\alpha}}\n\\]\n\\irref{B} is unsound without the restriction \\(x\\not\\in\\alpha\\), though, so that the following would be an unsound axiom:\n\\begin{equation}\n{\\lforall{x}{\\dbox{a}{p(x)}}\\limply{\\dbox{a}{\\lforall{x}{p(x)}}}}\n\\label{eq:unsound-B-attempt}\n\\end{equation}\nbecause $x\\not\\in a$ cannot be enforced for program constants, since their effect might very well depend on the value of $x$ or since they might write to $x$.\nIn \\rref{eq:unsound-B-attempt}, $x$ cannot be written by $a$ without violating soundness:\n\\[\n\\linfer[unsound]\n  {\\lforall{x}{\\dbox{a}{p(x)}}\\limply{\\dbox{a}{\\lforall{x}{p(x)}}}}\n  {\\lforall{x}{\\dbox{x:=0}{x\\geq0}}\\limply{\\dbox{x:=0}{\\lforall{x}{x\\geq0}}}}\n\\qquad\n\\usubstlist{\\usubstmod{a}{x:=0},\\usubstmod{p(\\usarg)}{\\usarg\\geq0}}\n\\]\nnor can $x$ be read by $a$ in \\rref{eq:unsound-B-attempt} without violating soundness:\n\\[\n\\linfer[unsound]\n  {\\lforall{x}{\\dbox{a}{p(x)}}\\limply{\\dbox{a}{\\lforall{x}{p(x)}}}}\n  {\\lforall{x}{\\dbox{\\ptest{y=x^2}}{y=x^2}}\\limply{\\dbox{\\ptest{y=x^2}}{\\lforall{x}{y=x^2}}}}\n\\qquad\n\\usubstlist{\\usubstmod{a}{\\ptest{y=x^2}},\\usubstmod{p(\\usarg)}{y=\\usarg^2}}\n\\]\n\nThus, the completeness proof for differential dynamic logic from prior work \\cite{DBLP:conf\/lics\/Platzer12b} does not directly carry over.\nA more general completeness result for differential game logic \\cite{DBLP:journals\/corr\/Platzer14:dGL} implies, however, that \\irref{B} is unnecessary for completeness.\n\\fi\n\n\n\\section{Differential Equations and Differential Axioms} \\label{sec:differential}\n\n\\rref{sec:dL-axioms} leverages the first-order features of \\dL and \\irref{US} to obtain a finite list of axioms without side-conditions.\nThey lack axioms for differential equations, though.\nClassical calculi for \\dL have axioms for replacing differential equations with a quantifier for time $t\\geq0$ and an assignment for their solutions $\\solf(t)$ \\cite{DBLP:journals\/jar\/Platzer08,DBLP:conf\/lics\/Platzer12b}.\nBesides being limited to simple differential equations, such axioms have the inherent side-condition ``if $\\solf(t)$ is a solution of the differential equation \\(\\pevolve{\\D{x}=\\genDE{x}}\\) with symbolic initial value $x$''.\nSuch a side-condition is more difficult than occurrence and read\/write conditions, but equally soundness-critical.\nThis section leverages \\irref{US} and the new differential forms in \\dL to obtain a logically internalized version of differential invariants and related proof rules for differential equations \\cite{DBLP:journals\/logcom\/Platzer10,DBLP:journals\/lmcs\/Platzer12} as axioms (without schema variables and free of side-conditions).\nThese axioms can prove properties of more general ``unsolvable'' differential equations. They can also prove all properties of differential equations that can be proved with solutions \\cite{DBLP:journals\/lmcs\/Platzer12} while guaranteeing correctness of the solution as part of the proof.\n\n\n\\subsection{Differentials: Invariants, Cuts, Effects, and Ghosts} \\label{sec:diffind}\n\nFigure~\\ref{fig:dL-ODE} shows differential equation axioms for differential weakening (\\irref{DW}), differential cuts (\\irref{DC}), differential effect (\\irref{DE}), differential invariants (\\irref{DI}) \\cite{DBLP:journals\/logcom\/Platzer10}, differential ghosts (\\irref{DG}) \\cite{DBLP:journals\/lmcs\/Platzer12}, solutions (\\irref{DS}), differential substitutions (\\irref{Dassignb}), and differential axioms (\\irref{Dplus+Dtimes+Dcompose}).\nAxioms identifying \\(\\der{x}=\\D{x}\\) for variables $x\\in\\mathcal{V}$ and \\(\\der{f}=0\\) for functions $f$ and number literals of arity 0 are used implicitly.\nSome axioms use reverse implications \\((\\phi\\lylpmi\\psi)\\mequiv(\\psi\\limply\\phi)\\) for emphasis.%\n\\begin{figure}[tbhp]\n  \\begin{calculuscollections}{\\columnwidth}\n  \\renewcommand*{\\irrulename}[1]{\\text{#1}}%\n    \\iflongversion\\else\\linferenceRulevskipamount=0.4em\\fi\n    \\begin{calculus}\n      \\cinferenceRule[DW|DW]{differential evolution domain} %\n      {\\dbox{\\pevolvein{\\D{x}=f(x)}{q(x)}}{q(x)}}\n      {}\n      \\cinferenceRule[DC|DC]{differential cut} %\n      {\\linferenceRule[lpmi]\n        {\\big(\\dbox{\\pevolvein{\\D{x}=f(x)}{q(x)}}{p(x)} \\lbisubjunct \\dbox{\\pevolvein{\\D{x}=f(x)}{q(x)\\land r(x)}}{p(x)}\\big)}\n        {\\dbox{\\pevolvein{\\D{x}=f(x)}{q(x)}}{r(x)}}\n      }\n      {}%\n      \\cinferenceRule[DE|DE]{differential effect} %\n      {\\linferenceRule[viuqe]\n        {\\dbox{\\pevolvein{\\D{x}=f(x)}{q(x)}}{p(x,\\D{x})}}\n        {\\dbox{\\pevolvein{\\D{x}=f(x)}{q(x)}}{\\dbox{\\Dupdate{\\Dumod{\\D{x}}{f(x)}}}{p(x,\\D{x})}}}\n      }\n      {}%\n      \\cinferenceRule[DI|DI]{differential induction} %\n      {\\linferenceRule[lpmi]\n        {\\dbox{\\pevolvein{\\D{x}=f(x)}{q(x)}}{p(x)}}\n        {\\big(q(x)\\limply p(x)\\land\\dbox{\\pevolvein{\\D{x}=f(x)}{q(x)}}{\\der{p(x)}}}\\big)\n      }\n      {}%\n      \\cinferenceRule[DG|DG]{differential ghost variables} %\n      {\\linferenceRule[viuqe]\n        {\\dbox{\\pevolvein{\\D{x}=f(x)}{q(x)}}{p(x)}}\n        {\\lexists{y}{\\dbox{\\pevolvein{\\D{x}=f(x)\\syssep\\D{y}=a(x)y+b(x)}{q(x)}}{p(x)}}}\n      }\n      {}%\n      \\cinferenceRule[DS|DS]{(constant) differential equation solution} %\n      {\\linferenceRule[viuqe]\n        {\\dbox{\\pevolvein{\\D{x}=f}{q(x)}}{p(x)}}\n        {\\lforall{t{\\geq}0}{\\big((\\lforall{0{\\leq}s{\\leq}t}{q(x+f\\itimes s)}) \\limply \\dbox{\\pupdate{\\pumod{x}{x+f\\itimes t}}}{p(x)}\\big)}}\n        %\n      }\n      {}%\n      \\cinferenceRule[Dassignb|$\\dibox{':=}$]{differential assignment}\n      {\\linferenceRule[equiv]\n        {p(f)}\n        {\\dbox{\\Dupdate{\\Dumod{\\D{x}}{f}}}{p(\\D{x})}}\n      }\n      {}%\n      \\cinferenceRule[Dplus|$+'$]{derive sum}\n      {\\linferenceRule[eq]\n        {\\der{f(\\bar{x})}+\\der{g(\\bar{x})}}\n        {\\der{f(\\bar{x})+g(\\bar{x})}}\n      }\n      {}\n      \\cinferenceRule[Dtimes|$\\cdot'$]{derive product}\n      {\\linferenceRule[eq]\n        {\\der{f(\\bar{x})}\\cdot g(\\bar{x})+f(\\bar{x})\\cdot\\der{g(\\bar{x})}}\n        {\\der{f(\\bar{x})\\cdot g(\\bar{x})}}\n      }\n      {}\n      \\cinferenceRule[Dcompose|$\\compose'$]{derive composition}\n      {\n        \\dbox{\\pupdate{\\pumod{y}{g(x)}}}{\\dbox{\\Dupdate{\\Dumod{\\D{y}}{1}}}\n        {\\big( \\der{f(g(x))} = \\der{f(y)}\\stimes\\der{g(x)}\\big)}}\n      }\n      {}%\n   \\end{calculus}%\n\\end{calculuscollections}%\n  \\caption{Differential equation axioms and differential axioms}\n  \\label{fig:dL-ODE}\n\\end{figure}\n\n\\emph{Differential weakening} axiom \\irref{DW} internalizes that differential equations can never leave their evolution domain $q(x)$.\n\\irref{DW} implies\\footnote{%\n\\(\\dbox{\\pevolvein{\\D{x}=f(x)}{q(x)}}{(q(x)\\limply p(x))} \\limply \\dbox{\\pevolvein{\\D{x}=f(x)}{q(x)}}{p(x)}\\) derives by \\irref{K} from \\irref{DW}.\nThe converse\n\\(\\dbox{\\pevolvein{\\D{x}=f(x)}{q(x)}}{p(x)} \\limply \\dbox{\\pevolvein{\\D{x}=f(x)}{q(x)}}{(q(x)\\limply p(x))}\\)\nderives by \\irref{K} since \\irref{G} derives \\(\\dbox{\\pevolvein{\\D{x}=f(x)}{q(x)}}{\\big(p(x)\\limply(q(x)\\limply p(x))\\big)}\\).\n}\n\\({\\dbox{\\pevolvein{\\D{x}=f(x)}{q(x)}}{p(x)}} \\lbisubjunct\n        {\\dbox{\\pevolvein{\\D{x}=f(x)}{q(x)}}{(q(x)\\limply p(x))}} \\irlabel{diffweaken|DW}\\)\nalso called \\irref{diffweaken},\nwhose (right) assumption is best proved by \\irref{G}\\iflongversion yielding premise \\(q(x)\\limply p(x)\\)\\fi.\nThe \\emph{differential cut} axiom \\irref{DC} is a cut for differential equations.\nIt internalizes that differential equations staying in $r(x)$ stay in $p(x)$ iff $p(x)$ always holds after the differential equation that is restricted to the smaller evolution domain \\(\\pevolvein{}{q(x)\\land r(x)}\\).\n\\irref{DC} is a differential variant of modal modus ponens \\irref{K}.\n\n\\emph{Differential effect} axiom \\irref{DE} internalizes that the effect on differential symbols along a differential equation is a differential assignment assigning the right-hand side $f(x)$ to the left-hand side $\\D{x}$.\nAxiom \\irref{DI} internalizes \\emph{differential invariants},\ni.e.\\ that a differential equation stays in $p(x)$ if it starts in $p(x)$ and if its differential $\\der{p(x)}$ always holds after the differential equation \\(\\pevolvein{\\D{x}=f(x)}{q(x)}\\).\nThe differential equation also vacuously stays in $p(x)$ if it starts outside $q(x)$, since it is stuck then.\nThe (right) assumption of \\irref{DI} is best proved by \\irref{DE} to select the appropriate vector field \\(\\D{x}=f(x)\\) for the differential $\\der{p(x)}$\nand a subsequent \\irref{diffweaken+G} to make the evolution domain constraint $q(x)$ available as an assumption.\nFor simplicity, this paper focuses on atomic postconditions for which \\(\\der{\\theta\\geq\\eta} \\mequiv \\der{\\theta>\\eta} \\mequiv \\der{\\theta}\\geq\\der{\\eta}\\)\nand \\(\\der{\\theta=\\eta} \\mequiv \\der{\\theta\\neq\\eta} \\mequiv \\der{\\theta}=\\der{\\eta}\\), etc.\nAxiom \\irref{DG} internalizes \\emph{differential ghosts},\ni.e. that additional differential equations can be added if their solution exists long enough.\nAxiom \\irref{DS} solves differential equations with the help of \\irref{DG+DC}.\nVectorial generalizations to systems of differential equations are possible for the axioms in \\rref{fig:dL-ODE}.\n\nThe following proof proves a property of a differential equation using differential invariants without having to solve that differential equation.\nOne use of \\irref{US} is shown explicitly, other uses of \\irref{US} are similar for \\irref{DI+DE+Dassignb} instances.\n\\vspace{-\\baselineskip}\n{\\footnotesize\n\\renewcommand{\\linferPremissSeparation}{~}%\n\\let\\orgcdot\\cdot%\n\\def\\cdot{{\\orgcdot}}%\n\\begin{sequentdeduction}[array]\n\\linfer[DI]\n{\\linfer[DE]\n {\\linfer[CE]%\n  {\\linfer[G]\n    {\\linfer[Dassignb]\n      {\\linfer[qear]\n        {\\lclose}\n        {\\lsequent{}{x^3\\cdot x + x\\cdot x^3\\geq0}}\n      }%\n      {\\lsequent{}{\\dbox{\\Dupdate{\\Dumod{\\D{x}}{x^3}}}{\\D{x}\\cdot x+x\\cdot\\D{x}\\geq0}}}\n      %\n    }%\n    {\\lsequent{}{\\dbox{\\pevolve{\\D{x}=x^3}}{\\dbox{\\Dupdate{\\Dumod{\\D{x}}{x^3}}}{\\D{x}\\cdot x+x\\cdot\\D{x}\\geq0}}}}\n    !\n    \\linfer%\n    {\\linfer[CQ] %\n      {\\linfer%\n        {\\linfer[US]\n          {\\linfer[Dtimes]\n            {\\lclose}\n            {\\lseqalign{\\der{f(\\bar{x})\\cdot g(\\bar{x})}} {= \\der{f(\\bar{x})}\\cdot g(\\bar{x}) + f(\\bar{x})\\cdot\\der{g(\\bar{x})}}}\n          }\n        {\\lseqalign{\\der{x\\cdot x}} {= \\der{x}\\cdot x + x\\cdot\\der{x}}}\n      }\n      {\\lseqalign{\\der{x\\cdot x}} {= \\D{x}\\cdot x + x\\cdot\\D{x}}}\n      }\n      {\\lseqalign{\\der{x\\cdot x}\\geq0}{\\lbisubjunct\\D{x}\\cdot x+x\\cdot\\D{x}\\geq0}}\n    }\n    {\\lseqalign{\\der{x\\cdot x\\geq1}}{\\lbisubjunct\\D{x}\\cdot x+x\\cdot\\D{x}\\geq0}}\n  }%\n  {\\lsequent{}{\\dbox{\\pevolve{\\D{x}=x^3}}{\\dbox{\\Dupdate{\\Dumod{\\D{x}}{x^3}}}{\\der{x\\cdot x\\geq1}}}}}\n }%\n  {\\lsequent{}{\\dbox{\\pevolve{\\D{x}=x^3}}{\\der{x\\cdot x\\geq1}}}}\n}%\n{\\lsequent{x\\cdot x\\geq1} {\\dbox{\\pevolve{\\D{x}=x^3}}{x\\cdot x\\geq1}}}\n\\end{sequentdeduction}\n}%\nPrevious calculi \\cite{DBLP:journals\/logcom\/Platzer10,DBLP:journals\/lmcs\/Platzer12} collapse this proof into a single proof step with complicated built-in operator implementations that silently perform the same reasoning in an opaque way.\nThe approach presented here combines separate axioms to achieve the same effect in a modular way, because they have individual responsibilities internalizing separate logical reasoning principles in \\emph{differential-form} \\dL.\n\\iflongversion Tactics combining the axioms as indicated make the axiomatic way equally convenient.\nClever cuts or \\irref{MP} enable proofs in which the main argument remains as fast \\cite{DBLP:journals\/logcom\/Platzer10,DBLP:journals\/lmcs\/Platzer12} while the additional premises subsequently check soundness.\nBoth \\irref{CQ} and also \\irref{CE} simplify the proof substantially but are not necessary:\n\\vspace{-\\baselineskip}\n{\\footnotesize\n\\begin{sequentdeduction}%\n\\hspace*{-0.7cm}\n\\linfer[MP]\n  {\\linfer\n    {\\lclose}\n    {\\sdots\\limply(\\der{x\\cdot x}\\geq0 \\lbisubjunct \\D{x}\\cdot x + x\\cdot\\D{x}\\geq0)}\n  &\\linfer%\n    {\\linfer[US]\n      {\\linfer[Dtimes]\n        {\\lclose}\n        {\\der{f(\\bar{x})\\cdot g(\\bar{x})} = \\der{f(\\bar{x})}\\cdot g(\\bar{x}) + f(\\bar{x})\\cdot\\der{g(\\bar{x})}}\n      }\n      {\\der{x\\cdot x} = \\der{x}\\cdot x + x\\cdot\\der{x}}\n    }\n    {\\der{x\\cdot x} = \\D{x}\\cdot x + x\\cdot\\D{x}}\n  }\n  {\\linfer[G]\n    {\\der{x\\cdot x}\\geq0 \\lbisubjunct \\D{x}\\cdot x + x\\cdot\\D{x}\\geq0}\n    {\\linfer[K]\n      {\\dbox{\\Dupdate{\\Dumod{\\D{x}}{x^3}}}{(\\der{x\\cdot x}\\geq0 \\lbisubjunct \\D{x}\\cdot x + x\\cdot\\D{x}\\geq0)}}\n      {\\dbox{\\Dupdate{\\Dumod{\\D{x}}{x^3}}}{\\der{x\\cdot x}\\geq0} \\lbisubjunct \\dbox{\\Dupdate{\\Dumod{\\D{x}}{x^3}}}{\\D{x}\\cdot x + x\\cdot\\D{x}\\geq0}}\n    }\n}\n\\end{sequentdeduction}\n\\vspace{-\\baselineskip}\n\\begin{sequentdeduction}[array]\n\\linfer[DI]\n{\\linfer[DE]\n {\\linfer[G]\n  {\\linfer[MP]\n    {\\text{use proof above}\n    !\\linfer[Dassignb]\n      {\\linfer[qear]\n        {\\lclose}\n        {\\lsequent{}{x^3\\cdot x + x\\cdot x^3\\geq0}}\n      }%\n      {\\lsequent{}{\\dbox{\\Dupdate{\\Dumod{\\D{x}}{x^3}}}{\\D{x}\\cdot x + x\\cdot\\D{x}\\geq0}}}\n      %\n    }%\n    {\\lsequent{}{\\dbox{\\Dupdate{\\Dumod{\\D{x}}{x^3}}}{\\der{x\\cdot x}\\geq0}}}\n  }%\n  {\\lsequent{}{\\dbox{\\pevolve{\\D{x}=x^3}}{\\dbox{\\Dupdate{\\Dumod{\\D{x}}{x^3}}}{\\der{x\\cdot x}\\geq0}}}}\n }%\n  {\\lsequent{}{\\dbox{\\pevolve{\\D{x}=x^3}}{\\der{x\\cdot x}\\geq0}}}\n}%\n{\\lsequent{x\\cdot x\\geq1} {\\dbox{\\pevolve{\\D{x}=x^3}}{x\\cdot x\\geq1}}}\n\\end{sequentdeduction}\n}%\nThe proof uses (implicit) cuts with equivalences predicting the outcome of the right branch, which is simple but inconvenient.\n\\newcommand{\\mydiffcond}[1][x,\\D{x}]{j(#1)}%\nA constructive direct proof uses a free function symbol $\\mydiffcond$, instead, which is ultimately instantiated by \\irref{US} as in \\rref{thm:dL-sound}.\n\nThe same technique is helpful for invariant search, in which case a free predicate symbol $p(\\bar{x})$ is used and instantiated by \\irref{US} lazily when the proof closes.\n{\\footnotesize\n\\begin{sequentdeduction}[array]\n\\linfer[DI]\n{\\linfer[DE]\n {\\linfer[CE]%\n  {\\linfer[G]\n    {\\linfer[Dassignb]\n      {\\linfer%\n        {\\linfer[qear]\n          {\\lclose}\n          {\\lsequent{}{x^3\\cdot x + x\\cdot x^3\\geq0}}\n        }\n        {\\lsequent{}\\mydiffcond[x,x^3]\\geq0}\n      }%\n      {\\lsequent{}{\\dbox{\\Dupdate{\\Dumod{\\D{x}}{x^3}}}{\\mydiffcond\\geq0}}}\n      %\n    }%\n    {\\lsequent{}{\\dbox{\\pevolve{\\D{x}=x^3}}{\\dbox{\\Dupdate{\\Dumod{\\D{x}}{x^3}}}{\\mydiffcond\\geq0}}}}\n    !\n    \\linfer%\n    {\\linfer[CQ] %\n      {\\linfer%\n        {\\linfer%\n        {\\linfer[US]\n          {\\linfer[Dtimes]\n            {\\lclose}\n            {\\lseqalign{\\der{f(\\bar{x})\\cdot g(\\bar{x})}} {= \\der{f(\\bar{x})}\\cdot g(\\bar{x}) + f(\\bar{x})\\cdot\\der{g(\\bar{x})}}}\n          }\n        {\\lseqalign{\\der{x\\cdot x}} {= \\der{x}\\cdot x + x\\cdot\\der{x}}}\n        }\n      {\\lseqalign{\\der{x\\cdot x}} {= \\D{x} \\cdot x + x\\cdot\\D{x}}}\n      }\n      {\\lseqalign{\\der{x\\cdot x}} {= \\mydiffcond}}\n      }\n      {\\lseqalign{\\der{x\\cdot x}\\geq0}{\\lbisubjunct\\mydiffcond\\geq0}}\n    }\n    {\\lseqalign{\\der{x\\cdot x\\geq1}}{\\lbisubjunct\\mydiffcond\\geq0}}\n  }%\n  {\\lsequent{}{\\dbox{\\pevolve{\\D{x}=x^3}}{\\dbox{\\Dupdate{\\Dumod{\\D{x}}{x^3}}}{\\der{x\\cdot x\\geq1}}}}}\n }%\n  {\\lsequent{}{\\dbox{\\pevolve{\\D{x}=x^3}}{\\der{x\\cdot x\\geq1}}}}\n}%\n{\\lsequent{x\\cdot x\\geq1} {\\dbox{\\pevolve{\\D{x}=x^3}}{x\\cdot x\\geq1}}}\n\\end{sequentdeduction}\n}%\n\\fi\n\n\\iflongversion\nProofs based entirely on equivalences for solving differential equations involve \\irref{DG} for introducing a time variable, \\irref{DC} to cut the solutions in, \\irref{DW} to export the solution to the postcondition, inverse \\irref{DC} to remove the evolution domain constraints again, inverse \\irref{DG} to remove the original differential equations, and finally \\irref{DS} to solve the differential equation for time:\n\\def\\prem{\\phi}%\n{\\footnotesize\n\\begin{sequentdeduction}[array]\n\\linfer[DG]\n {\\linfer%\n   {\\linfer[DC]\n     {\\linfer[DC]\n       {\\linfer[diffweaken]\n         {\\linfer[G+K]%\n           {\\linfer[DC]\n             {\\linfer[DC]\n               {\\linfer[DG]\n                 {\\linfer[DG]\n                   {\\linfer[DS]\n                     {\\linfer[assignb]\n                       {\\linfer[qear]\n                         {\\lclose}\n                         {\\lsequent{\\prem} {\\lforall{s{\\geq}0}{(x_0+\\frac{a}{2}s^2+v_0s\\geq0)}}}\n                       }%\n                       {\\lsequent{\\prem} {\\lforall{s{\\geq}0}{\\dbox{\\pupdate{\\pumod{t}{0+1s}}}{x_0+\\frac{a}{2}t^2+v_0t\\geq0}}}}\n                     }%\n                     {\\lsequent{\\prem} {\\dbox{\\pevolve{\\D{t}=1}}{x_0+\\frac{a}{2}t^2+v_0t\\geq0}}}\n                   }%\n                   {\\lsequent{\\prem} {\\dbox{\\pevolve{\\D{v}=a\\syssep\\D{t}=1}}{x_0+\\frac{a}{2}t^2+v_0t\\geq0}}}\n                 }%\n                 {\\lsequent{\\prem} {\\dbox{\\pevolve{\\D{x}=v\\syssep\\D{v}=a\\syssep\\D{t}=1}}{x_0+\\frac{a}{2}t^2+v_0t\\geq0}}}\n               }%\n               {\\lsequent{\\prem} {\\dbox{\\pevolvein{\\D{x}=v\\syssep\\D{v}=a\\syssep\\D{t}=1}{v=v_0+at}}{x_0+\\frac{a}{2}t^2+v_0t\\geq0}}}\n             }%\n             {\\lsequent{\\prem} {\\dbox{\\pevolvein{\\D{x}=v\\syssep\\D{v}=a\\syssep\\D{t}=1}{v=v_0+at\\land x=x_0+\\frac{a}{2}t^2+v_0t}}{x_0+\\frac{a}{2}t^2+v_0t\\geq0}}} \n           }%\n           {\\lsequent{\\prem} {\\dbox{\\pevolvein{\\D{x}=v\\syssep\\D{v}=a\\syssep\\D{t}=1}{v=v_0+at\\land x=x_0+\\frac{a}{2}t^2+v_0t}}{(x{=}x_0{+}\\frac{a}{2}t^2{+}v_0t\\limply x{\\geq}0)}}}\n         }%\n         {\\lsequent{\\prem} {\\dbox{\\pevolvein{\\D{x}=v\\syssep\\D{v}=a\\syssep\\D{t}=1}{v=v_0+at\\land x=x_0+\\frac{a}{2}t^2+v_0t}}{x\\geq0}}} \n       }%\n       {\\lsequent{\\prem} {\\dbox{\\pevolvein{\\D{x}=v\\syssep\\D{v}=a\\syssep\\D{t}=1}{v=v_0+at}}{x\\geq0}}} \n     }%\n     {\\lsequent{\\prem} {\\dbox{\\pevolve{\\D{x}=v\\syssep\\D{v}=a\\syssep\\D{t}=1}}{x\\geq0}}} \n   }%\n   {\\lsequent{\\prem} {\\lexists{t}{\\dbox{\\pevolve{\\D{x}=v\\syssep\\D{v}=a\\syssep\\D{t}=1}}{x\\geq0}}}}\n }%\n {\\lsequent{\\prem} {\\dbox{\\pevolve{\\D{x}=v\\syssep\\D{v}=a}}{x\\geq0}}} \n\\end{sequentdeduction}\n}%\nwhere $\\prem$ is \\({a\\geq0\\land v=v_0\\geq0 \\land x=x_0\\geq0}\\). %\nThe existential quantifier for $t$ is instantiated by $0$, leading to $\\dbox{\\pupdate{\\pumod{t}{0}}}{}$ (suppressed in the proof for readability reasons).\nThe 4 uses of \\irref{DC} lead to 2 additional premises proving that \\(v=v_0+at\\) and then \\(x=x_0+\\frac{a}{2}t^2+v_0t\\) are differential invariants (using \\irref{DI+DE+diffweaken}).\nShortcuts using \\irref{diffweaken} are possible but the above proof generalize to $\\ddiamond{}{}$ because it is an equivalence proof.\nThe additional premise for \\irref{DC} with \\(v=v_0+at\\) proves as follows:\n{\\footnotesize\n\\begin{sequentdeduction}[array]\n\\linfer[DI]\n{\\linfer[DE]\n {\\linfer[G]\n   {\\linfer[CE]%\n    {\\linfer[Dassignb]\n      {\\linfer[qear]\n        {\\lclose}\n        {\\lsequent{}{a=0+a\\cdot1}}\n      }%\n      {\\lsequent{} {\\dbox{\\Dupdate{\\Dumod{\\D{v}}{a}}}{\\dbox{\\Dupdate{\\Dumod{\\D{t}}{1}}}{\\D{v}=0+a\\D{t}}}}}\n      %\n    !\n    \\linfer%\n    {\\linfer[CQ] %\n      {\\linfer[Dtimes]%\n        {\\linfer[US]\n          {\\linfer[Dplus]\n            {\\lclose}\n            {\\lseqalign{\\der{f(\\bar{x})+ g(\\bar{x})}} {= \\der{f(\\bar{x})} + \\der{g(\\bar{x})}}}\n          }\n        {\\lseqalign{\\der{v_0+at}} {= \\der{v_0}+\\der{a t}}}\n      }\n      {\\lseqalign{\\der{v_0+at}} {= 0+a(\\D{t})}}\n      }\n      {\\lseqalign{\\D{v}=\\der{v_0+at}}{\\lbisubjunct\\D{v}=0+a\\D{t}}}\n    }\n    {\\lseqalign{\\der{v=v_0+at}}{\\lbisubjunct\\D{v}=0+a\\D{t}}}\n    }%\n    {\\lsequent{} {\\dbox{\\Dupdate{\\Dumod{\\D{v}}{a}}}{\\dbox{\\Dupdate{\\Dumod{\\D{t}}{1}}}{\\der{v=v_0+at}}}}}\n  }%\n  {\\lsequent{} {\\dbox{\\pevolve{\\D{x}=v\\syssep\\D{v}=a\\syssep\\D{t}=1}}{\\dbox{\\Dupdate{\\Dumod{\\D{v}}{a}}}{\\dbox{\\Dupdate{\\Dumod{\\D{t}}{1}}}{\\der{v=v_0+at}}}}}}\n }%\n  {\\lsequent{} {\\dbox{\\pevolve{\\D{x}=v\\syssep\\D{v}=a\\syssep\\D{t}=1}}{\\der{v=v_0+at}}}}\n}%\n{\\lsequent{\\prem} {\\dbox{\\pevolve{\\D{x}=v\\syssep\\D{v}=a\\syssep\\D{t}=1}}{v=v_0+at}}}\n\\end{sequentdeduction}\n}\nThe additional premise for \\irref{DC} with \\(x=x_0+\\frac{a}{2}t^2+v_0t\\) proves as follows:\n\n{\\tiny%\n\\begin{sequentdeduction}[array]\n\\linfer[DI]\n{\\linfer[DE]\n {\\linfer[diffweaken]\n {\\linfer[G]\n   {\\linfer[CE]%\n    {\\linfer[Dassignb]\n      {\\linfer[qear]\n        {\\lclose}\n        {\\lsequent{}{{v=v_0+at} \\limply v=at\\cdot1+v_0\\cdot1}}\n      }%\n      {\\lsequent{} {{v=v_0+at} \\limply \\dbox{\\Dupdate{\\Dumod{\\D{x}}{v}}}{\\dbox{\\Dupdate{\\Dumod{\\D{t}}{1}}}{\\D{x}=at\\D{t}+v_0\\D{t}}}}}\n      %\n    !\n    \\linfer%\n    {\\linfer[CQ] %\n      {\\linfer[Dplus+Dtimes]%\n        {\\linfer%\n          {\\lclose}\n          {\\lseqalign{2\\frac{a}{2}t\\D{t}+v_0\\D{t}} {= at\\D{t}+v_o\\D{t}}}\n        }\n      {\\lseqalign{\\der{x_0+\\frac{a}{2}t^2+v_0t}} {= at\\D{t}+v_o\\D{t}}}\n      }\n      {\\lseqalign{\\D{x}=\\der{x_0+\\frac{a}{2}t^2+v_0t}}{\\lbisubjunct\\D{x}=at\\D{t}+v_o\\D{t}}}\n    }\n    {\\lseqalign{\\der{x=x_0+\\frac{a}{2}t^2+v_0t}}{\\lbisubjunct\\D{x}=at\\D{t}+v_o\\D{t}}}\n    }%\n    {\\lsequent{} {{v=v_0+at} \\limply \\dbox{\\Dupdate{\\Dumod{\\D{x}}{v}}}{\\dbox{\\Dupdate{\\Dumod{\\D{t}}{1}}}{\\der{x=x_0+\\frac{a}{2}t^2+v_0t}}}}}\n  }%\n  {\\lsequent{} {\\dbox{\\pevolvein{\\D{x}=v\\syssep\\D{v}=a\\syssep\\D{t}=1}{v=v_0+at}}{({v=v_0+at}\\limply\\dbox{\\Dupdate{\\Dumod{\\D{x}}{v}}}{\\dbox{\\Dupdate{\\Dumod{\\D{t}}{1}}}{\\der{x=x_0+\\frac{a}{2}t^2+v_0t}}})}}}\n  }%\n  {\\lsequent{} {\\dbox{\\pevolvein{\\D{x}=v\\syssep\\D{v}=a\\syssep\\D{t}=1}{v=v_0+at}}{\\dbox{\\Dupdate{\\Dumod{\\D{x}}{v}}}{\\dbox{\\Dupdate{\\Dumod{\\D{t}}{1}}}{\\der{x=x_0+\\frac{a}{2}t^2+v_0t}}}}}}\n }%\n  {\\lsequent{} {\\dbox{\\pevolvein{\\D{x}=v\\syssep\\D{v}=a\\syssep\\D{t}=1}{v=v_0+at}}{\\der{x=x_0+\\frac{a}{2}t^2+v_0t}}}}\n}%\n{\\lsequent{\\prem} {\\dbox{\\pevolvein{\\D{x}=v\\syssep\\D{v}=a\\syssep\\D{t}=1}{v=v_0+at}}{x=x_0+\\frac{a}{2}t^2+v_0t}}}\n\\end{sequentdeduction}\n}%\n\\fi\n\n\n\\subsection{Differential Substitution Lemmas}\n\nThe key insight for the soundness of \\irref{DI} is that the analytic time-derivative of the value of a term $\\eta$ along a differential equation \\(\\pevolvein{\\D{x}=\\genDE{x}}{\\psi}\\) agrees with the values of its differential $\\der{\\eta}$ along the vector field of that differential equation.\n\n\\begin{lemma}[Differential lemma] \\label{lem:differentialLemma}\n  %\n  If \\m{\\imodels{\\DALint[const=I,flow=\\varphi]}{\\D{x}=\\genDE{x}\\land\\psi}}\n  holds for some flow \\m{\\iget[flow]{\\DALint[const=I,flow=\\varphi]}:[0,r]\\to\\linterpretations{\\Sigma}{V}} \nof any duration $r>0$,\n  then for all $0\\leq\\zeta\\leq r$:\n  \\[\n  \\ivaluation{\\Iff[\\zeta]}{\\der{\\eta}}\n  = \\D[t]{\\ivaluation{\\Iff[t]}{\\eta}} (\\zeta)\n  \\]\n\\end{lemma}\n\\begin{proofatend}\n\\def\\Ifzm{\\imodif[state]{\\Iff[\\zeta]}{x}{X}}%\n\\newcommand{\\vdLint[const=I,state=]}{\\vdLint[const=I,state=]}\nBy chain rule \\cite[\\S3.10]{Walter:Ana2}:\n\\[\n\\D[t]{\\ivaluation{\\Iff[t]}{\\eta}} (\\zeta)\n=\n\\D{(\\ivaluation{\\vdLint[const=I,state=]}{\\eta} \\compose \\iget[flow]{\\DALint[const=I,flow=\\varphi]})}(\\zeta) = (\\gradient{\\ivaluation{\\vdLint[const=I,state=]}{\\eta}})\\big(\\iget[flow]{\\DALint[const=I,flow=\\varphi]}(\\zeta)\\big)\n\\stimes \\D{\\iget[flow]{\\DALint[const=I,flow=\\varphi]}}(\\zeta)\n= \\sum_x \\Dp[x]{\\ivaluation{\\vdLint[const=I,state=]}{\\eta}}\\big(\\iget[flow]{\\DALint[const=I,flow=\\varphi]}(\\zeta)\\big) \\D{\\iget[flow]{\\DALint[const=I,flow=\\varphi]}}(\\zeta)(x)\n\\]\nwhere \\((\\gradient{\\ivaluation{\\vdLint[const=I,state=]}{\\eta}})\\big(\\iget[flow]{\\DALint[const=I,flow=\\varphi]}(\\zeta)\\big)\\), the spatial gradient \\(\\gradient{\\ivaluation{\\vdLint[const=I,state=]}{\\eta}}\\)\nat \\(\\iget[flow]{\\DALint[const=I,flow=\\varphi]}(\\zeta)\\),\nis the vector of\n\\(\\Dp[x]{\\ivaluation{\\vdLint[const=I,state=]}{\\eta}}\\big(\\iget[flow]{\\DALint[const=I,flow=\\varphi]}(\\zeta)\\big)\n= \\Dp[X]{\\ivaluation{\\Ifzm}{\\eta}}\\).\nChain rule and \\rref{def:dL-valuationTerm} and\\rref{def:HP-transition}, thus, imply:\n\\[\n\\ivaluation{\\Iff[\\zeta]}{\\der{\\eta}}\n= \\sum_x \\iget[state]{\\Iff[\\zeta]}(\\D{x}) \\itimes \\Dp[X]{\\ivaluation{\\Ifzm}{\\eta}}\n= \\sum_x \\Dp[X]{\\ivaluation{\\Ifzm}{\\eta}} \\itimes \\D[t]{\\iget[state]{\\Iff[t]}(x)}(\\zeta)\n= \\D[t]{\\ivaluation{\\Iff[t]}{\\eta}} (\\zeta)\n\n\\]\n\\end{proofatend}\n\nThe key insight for the soundness of differential effects \\irref{DE} is that differential assignments mimicking the differential equation are vacuous along that differential equation.\nThe differential substitution resulting from a subsequent use of \\irref{Dassignb} is crucial to relay the values of the time-derivatives of the state variables $x$ along a differential equation by way of their corresponding differential symbol $\\D{x}$.\nIn combination, this makes it possible to soundly substitute the right-hand side of a differential equation for its left-hand side in a proof.\n\n\\begin{lemma}[Differential assignment] \\label{lem:differentialAssignLemma}\n  %\n  If \\m{\\imodels{\\DALint[const=I,flow=\\varphi]}{\\D{x}=\\genDE{x}\\land\\psi}}\n  for some flow \\m{\\iget[flow]{\\DALint[const=I,flow=\\varphi]}:[0,r]\\to\\linterpretations{\\Sigma}{V}} \nof any duration $r\\geq0$,\n  then\n  \\[\n  \\imodels{\\DALint[const=I,flow=\\varphi]}{\\phi \\lbisubjunct \\dbox{\\Dupdate{\\Dumod{\\D{x}}{\\genDE{x}}}}{\\phi}}\n  \\]\n\\end{lemma}\n\\begin{proofatend}\n\\m{\\imodels{\\DALint[const=I,flow=\\varphi]}{\\D{x}=\\genDE{x}\\land\\psi}} implies\n\\(\\imodels{\\Iff[\\zeta]}{\\D{x}=\\genDE{x}\\land\\psi}\\),\ni.e. \\(\\iget[state]{\\Iff[\\zeta]}(\\D{x}) = \\ivaluation{\\Iff[\\zeta]}{\\genDE{x}}\\) and \\(\\imodels{\\Iff[\\zeta]}{\\psi}\\)\nfor all $0\\leq \\zeta\\leq r$.\nConsequently \\(\\iaccessible[\\Dupdate{\\Dumod{\\D{x}}{\\genDE{x}}}]{\\Iff[\\zeta]}{\\Iff[\\zeta]}\\)\ndoes not change the state, so that $\\phi$ and \\(\\dbox{\\Dupdate{\\Dumod{\\D{x}}{\\genDE{x}}}}{\\phi}\\) are equivalent along $\\iget[flow]{\\DALint[const=I,flow=\\varphi]}$.\n\n\\end{proofatend}\n\nThe final insights for differential invariant reasoning for differential equations are syntactic ways of computing differentials, which can be internalized as axioms (\\irref{Dplus+Dtimes+Dcompose}), since differentials are syntactically represented in differential-form \\dL. \n\\begin{lemma}[Derivations] \\label{lem:derivationLemma}\n  The following equations of differentials are valid:\n  \\begin{align}%\n    \\der{f} & = 0\n      &&\\text{for arity 0 functions\/numbers}~f\n    \\label{eq:Dconstant}\\\\\n    %\n    \\der{x} & =  \\D{x}\n      &&\\text{for variables}~x\\in\\mathcal{V}\\label{eq:Dpolynomial}\\\\\n    \\der{\\theta+\\eta} & = \\der{\\theta} + \\der{\\eta}\n    \\label{eq:Dadditive}\\\\\n    \\der{\\theta\\cdot \\eta} & = \\der{\\theta}\\cdot \\eta + \\theta\\cdot\\der{\\eta}\n    \\label{eq:DLeibniz}\n    \\\\\n    \\dbox{\\pupdate{\\pumod{y}{\\theta}}}{\\dbox{\\Dupdate{\\Dumod{\\D{y}}{1}}}\n    {}&{\\big( \\der{f(\\theta)} = \\der{f(y)}\\stimes\\der{\\theta}\\big)}}\n    &&\\text{for $y,\\D{y}\\not\\in\\theta$}\n    \\label{eq:Dcompose}\n    %\n    %\n  \\end{align}%\n\\end{lemma}\n\\begin{proofatend}\n\\def\\Im{\\imodif[state]{\\vdLint[const=I,state=\\nu]}{x}{X}}%\n\\edef\\Imyypyval{\\lenvelope\\theta\\renvelope^I \\nu}%\n\\def\\Imyyp{\\vdLint[const=I,state=\\modif{\\nu}{y}{\\Imyypyval}\\modif{}{\\D{y}}{1}]{\\vdLint[const=I,state=\\nu]}}%\n\\newcommand{\\vdLint[const=I,state=]}{\\vdLint[const=I,state=]}%\nThe proof shows each equation separately.\nThe first parts consider any constant function (i.e. arity 0) or number literal $f$ for \\rref{eq:Dconstant} and align the differential \\(\\der{x}\\) of a term that happens to be a variable $x\\in\\mathcal{V}$ with its corresponding differential symbol $\\D{x}\\in\\D{\\mathcal{V}}$ for \\rref{eq:Dpolynomial}.\nThe other cases exploit linearity for \\rref{eq:Dadditive} and Leibniz properties of partial derivatives for \\rref{eq:DLeibniz}.\nCase \\rref{eq:Dcompose} exploits the chain rule and assignments and differential assignments for the fresh $y,\\D{y}$ to mimic partial derivatives.\nEquation \\rref{eq:Dcompose} generalizes to functions $f$ of arity $n>1$, in which case $\\stimes$ is the (definable) Euclidean scalar product.\n\\iflongversion\n\\def\\aptag#1{\\tag{#1}}%\n\\else%\n\\def\\aptag#1{\\notag&\\hspace*{-8pt}(#1)}%\n\\fi\n\\begin{align}\n\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\der{f}}\n&= \\sum_x \\iget[state]{\\vdLint[const=I,state=\\nu]}(\\D{x}) \\itimes \\Dp[X]{\\ivaluation{\\Im}{f}}\n= \\sum_x \\iget[state]{\\vdLint[const=I,state=\\nu]}(\\D{x}) \\itimes \\Dp[X]{\\iget[const]{\\Im}(f)}\n= 0\n\\aptag{\\ref{eq:Dconstant}}\n\\\\\n\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\der{x}}\n&=\n\\def\\Imy{\\imodif[state]{\\vdLint[const=I,state=\\nu]}{y}{X}}%\n\\sum_y \\iget[state]{\\vdLint[const=I,state=\\nu]}(\\D{y}) \\itimes \\Dp[X]{\\ivaluation{\\Imy}{x}}\n= \\iget[state]{\\vdLint[const=I,state=\\nu]}(\\D{x}) \\itimes \\Dp[X]{\\ivaluation{\\Im}{x}}\n= \\iget[state]{\\vdLint[const=I,state=\\nu]}(\\D{x}) \\itimes \\Dp[X]{X}\n= \\iget[state]{\\vdLint[const=I,state=\\nu]}(\\D{x})\n= \\ivaluation{\\vdLint[const=I,state=\\nu]}{\\D{x}}\n\\aptag{\\ref{eq:Dpolynomial}}\n\\\\\n\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\der{\\theta+\\eta}}\n&= \\sum_x \\iget[state]{\\vdLint[const=I,state=\\nu]}(\\D{x}) \\itimes \\Dp[X]{\\ivaluation{\\Im}{\\theta+\\eta}}\n= \\sum_x \\iget[state]{\\vdLint[const=I,state=\\nu]}(\\D{x}) \\itimes \\Dp[X]{(\\ivaluation{\\Im}{\\theta}+\\ivaluation{\\Im}{\\eta})}\n\\notag\n\\\\&= \\sum_x \\iget[state]{\\vdLint[const=I,state=\\nu]}(\\D{x}) \\itimes \\Big(\\Dp[X]{\\ivaluation{\\Im}{\\theta}} + \\Dp[X]{\\ivaluation{\\Im}{\\eta}}\\Big)\n\\notag\n\\\\&= \\sum_x \\iget[state]{\\vdLint[const=I,state=\\nu]}(\\D{x}) \\itimes \\Dp[X]{\\ivaluation{\\Im}{\\theta}} + \\sum_x \\iget[state]{\\vdLint[const=I,state=\\nu]}(\\D{x}) \\itimes \\Dp[X]{\\ivaluation{\\Im}{\\eta}}\n\\notag\n\\\\&= \\ivaluation{\\vdLint[const=I,state=\\nu]}{\\der{\\theta}} + \\ivaluation{\\vdLint[const=I,state=\\nu]}{\\der{\\eta}}\n= \\ivaluation{\\vdLint[const=I,state=\\nu]}{\\der{\\theta} + \\der{\\eta}}\n\\aptag{\\ref{eq:Dadditive}}\n\\\\\n\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\der{\\theta\\cdot\\eta}}\n&= \\sum_x \\iget[state]{\\vdLint[const=I,state=\\nu]}(\\D{x}) \\itimes \\Dp[X]{\\ivaluation{\\Im}{\\theta\\cdot\\eta}}\n= \\sum_x \\iget[state]{\\vdLint[const=I,state=\\nu]}(\\D{x}) \\itimes \\Dp[X]{(\\ivaluation{\\Im}{\\theta}\\cdot\\ivaluation{\\Im}{\\eta})}\n\\notag\n\\\\\n&= \\sum_x \\iget[state]{\\vdLint[const=I,state=\\nu]}(\\D{x}) \\itimes \\Big(\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\eta} \\itimes \\Dp[X]{\\ivaluation{\\Im}{\\theta}}\n+ \\ivaluation{\\vdLint[const=I,state=\\nu]}{\\theta} \\itimes \\Dp[X]{\\ivaluation{\\Im}{\\eta}}\\Big)\n\\notag\n\\\\\n&=\n\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\eta} \\sum_x \\iget[state]{\\vdLint[const=I,state=\\nu]}(\\D{x}) \\itimes \\Dp[X]{\\ivaluation{\\Im}{\\theta}}\n+ \\ivaluation{\\vdLint[const=I,state=\\nu]}{\\theta} \\sum_x \\iget[state]{\\vdLint[const=I,state=\\nu]}(\\D{x}) \\itimes \\Dp[X]{\\ivaluation{\\Im}{\\eta}}\n\\notag\n\\\\\n&= \n\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\der{\\theta}}\\cdot \\ivaluation{\\vdLint[const=I,state=\\nu]}{\\eta} + \\ivaluation{\\vdLint[const=I,state=\\nu]}{\\theta}\\cdot\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\der{\\eta}}\n= \\ivaluation{\\vdLint[const=I,state=\\nu]}{\\der{\\theta}\\cdot \\eta + \\theta\\cdot\\der{\\eta}}\n\\aptag{\\ref{eq:DLeibniz}}\n\\intertext{Proving that\n\\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\dbox{\\pupdate{\\pumod{y}{\\theta}}}{\\dbox{\\Dupdate{\\Dumod{\\D{y}}{1}}}{\\big(\\der{f(\\theta)} = \\der{f(y)}\\stimes\\der{\\theta}\\big)}}}\\)\nrequires showing that\n\\newline\n\\(\\imodels{\\Imyyp}{\\der{f(\\theta)} = \\der{f(y)}\\stimes\\der{\\theta}}\\),\ni.e.\\\n\\(\\ivaluation{\\Imyyp}{\\der{f(\\theta)}} = \\ivaluation{\\Imyyp}{\\der{f(y)}\\stimes\\der{\\theta}}\\).\nThis is equivalent to\n\\(\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\der{f(\\theta)}} = \\ivaluation{\\Imyyp}{\\der{f(y)}}\\stimes\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\der{\\theta}}\\)\nby \\rref{lem:coincidence-term} since\n\\(\\iget[state]{\\vdLint[const=I,state=\\nu]}=\\iget[state]{\\Imyyp}\\) on $\\scomplement{\\{y,\\D{y}\\}}$ and\n\\(y,\\D{y}\\not\\in\\freevars{\\theta}\\) by assumption,\nso \\(y,\\D{y}\\not\\in\\freevars{\\der{f(\\theta)}}\\) and \\(y,\\D{y}\\not\\in\\freevars{\\der{\\theta}}\\).\nThe latter equation proves using the chain rule and a fresh variable $z$ when denoting \\(\\ivaluation{\\vdLint[const=I,state=]}{f} \\mdefeq \\iget[const]{\\vdLint[const=I,state=]}(f)\\):}\n    \\ivaluation{\\vdLint[const=I,state=\\nu]}{\\der{f(\\theta)}} & =\n    \\sum_x \\iget[state]{\\vdLint[const=I,state=\\nu]}(\\D{x}) \\Dp[x]{\\ivaluation{\\vdLint[const=I,state=]}{f(\\theta)}}(\\iget[state]{\\vdLint[const=I,state=\\nu]})\n    = \\sum_x \\iget[state]{\\vdLint[const=I,state=\\nu]}(\\D{x}) \\Dp[x]{(\\ivaluation{\\vdLint[const=I,state=]}{f}\\compose\\ivaluation{\\vdLint[const=I,state=]}{\\theta})}(\\iget[state]{\\vdLint[const=I,state=\\nu]})\n  \\notag\n    \\\\&\n    %\n     \\stackrel{\\text{chain}}{=} \\sum_x \\iget[state]{\\vdLint[const=I,state=\\nu]}(\\D{x}) \\Dp[y]{\\ivaluation{\\vdLint[const=I,state=]}{f}}\\big(\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\theta}\\big) \\stimes \\Dp[x]{\\ivaluation{\\vdLint[const=I,state=]}{\\theta}}(\\iget[state]{\\vdLint[const=I,state=\\nu]})\n\\notag\n     \\\\&\n     = \\Dp[y]{\\ivaluation{\\vdLint[const=I,state=]}{f}}\\big(\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\theta}\\big) \\stimes \\sum_x \\iget[state]{\\vdLint[const=I,state=\\nu]}(\\D{x}) \\Dp[x]{\\ivaluation{\\vdLint[const=I,state=]}{\\theta}}(\\iget[state]{\\vdLint[const=I,state=\\nu]})\n     = \\Dp[y]{\\ivaluation{\\vdLint[const=I,state=]}{f}}\\big(\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\theta}\\big) \\stimes \\ivaluation{\\vdLint[const=I,state=\\nu]}{\\der{\\theta}}\n\\notag\n     \\\\&= \\Dp[y]{\\iget[const]{\\vdLint[const=I,state=]}(f)}\\big(\\ivaluation{\\vdLint[const=I,state=\\nu]}{\\theta}\\big) \\stimes \\ivaluation{\\vdLint[const=I,state=\\nu]}{\\der{\\theta}}\n\\notag\n    \\\\&=\n    \\Dp[z]{\\iget[const]{\\vdLint[const=I,state=]}(f)}(\\Imyypyval) \\itimes 1 \\stimes \\ivaluation{\\vdLint[const=I,state=\\nu]}{\\der{\\theta}}\n\\notag\n    \\\\&=\n    \\Dp[z]{\\iget[const]{\\vdLint[const=I,state=]}(f)}\\big(\\ivaluation{\\Imyyp}{y}\\big) \\itimes \\Dp[y]{\\ivaluation{\\vdLint[const=I,state=]}{y}}(\\iget[state]{\\Imyyp}) \\stimes \\ivaluation{\\vdLint[const=I,state=\\nu]}{\\der{\\theta}}\n\\notag\n    \\\\&\\stackrel{\\text{chain}}{=}\n    \\Dp[y]{(\\iget[const]{\\vdLint[const=I,state=]}(f)\\compose\\ivaluation{\\vdLint[const=I,state=]}{y})}(\\iget[state]{\\Imyyp}) \\stimes \\ivaluation{\\vdLint[const=I,state=\\nu]}{\\der{\\theta}}\n\\notag\n    \\\\&=\n    \\left(\\Dp[y]{\\ivaluation{\\vdLint[const=I,state=]}{f(y)}}(\\iget[state]{\\Imyyp})\\right) \\stimes \\ivaluation{\\vdLint[const=I,state=\\nu]}{\\der{\\theta}}\n\\notag\n    \\\\&=\n    \\left(\\iget[state]{\\Imyyp}(\\D{y}) \\Dp[y]{\\ivaluation{\\vdLint[const=I,state=]}{f(y)}}(\\iget[state]{\\Imyyp})\\right) \\stimes \\ivaluation{\\vdLint[const=I,state=\\nu]}{\\der{\\theta}}\n\\notag\n    \\\\&=\n    \\left(\\sum_{x\\in\\{y\\}} \\iget[state]{\\Imyyp}(\\D{x}) \\Dp[x]{\\ivaluation{\\vdLint[const=I,state=]}{f(y)}}(\\iget[state]{\\Imyyp})\\right) \\stimes \\ivaluation{\\vdLint[const=I,state=\\nu]}{\\der{\\theta}}\n\\notag\n    \\\\&=\n    \\ivaluation{\\Imyyp}{\\der{f(y)}} \\stimes \\ivaluation{\\vdLint[const=I,state=\\nu]}{\\der{\\theta}}\n    %\n    \\aptag{\\ref{eq:Dcompose}}\n    \n\\end{align}\n\n\\end{proofatend}\n\n\\subsection{Soundness}\n\n\\begin{theorem}[Soundness] \\label{thm:dL-sound}\n  The \\dL axioms and proof rules in \\rref{fig:dL}, \\ref{fig:dL-ODE} are sound, i.e.\\ the axioms are valid formulas and the conclusion of a rule is valid if its premises are.\n  All \\irref{US} instances of the proof rules (with $\\freevars{\\sigma}=\\emptyset$) are sound.\n\\end{theorem}\n\\begin{proofatend}\nThe axioms (and most proof rules) in \\rref{fig:dL} are special instances of corresponding axiom schemata and proof rules for differential dynamic logic \\cite{DBLP:conf\/lics\/Platzer12b} and, thus, sound.\nAll proof rules except \\irref{US} are even \\emph{locally sound}, i.e. for all $\\iget[const]{\\vdLint[const=I,state=\\nu]}$: if all their premises $\\phi_j$ are valid in $\\iget[const]{\\vdLint[const=I,state=\\nu]}$ (\\m{\\iget[const]{\\vdLint[const=I,state=\\nu]}\\models{\\phi_j}}) then their conclusion $\\psi$ is, too (\\m{\\iget[const]{\\vdLint[const=I,state=\\nu]}\\models{\\psi}}).\nLocal soundness implies soundness.\nIn addition, local soundness implies that \\irref{US} can be used to soundly instantiate proof rules just like it soundly instantiates axioms (\\rref{thm:usubst-sound}).\nIf\n\\begin{equation}\n\\linfer\n{\\phi_1 \\quad \\dots \\quad \\phi_n}\n{\\psi}\n\\label{eq:proofrule}\n\\end{equation}\nis a locally sound proof rule then its substitution instance is locally sound:\n\\begin{equation}\n\\linfer\n{\\applyusubst{\\sigma}{\\phi_1} \\quad \\dots \\quad \\applyusubst{\\sigma}{\\phi_n}}\n{\\applyusubst{\\sigma}{\\psi}}\n\\label{eq:usubstituted-proofrule}\n\\end{equation}\nwhere $\\sigma$ is any uniform substitution (for which the above results are defined, i.e.\\ no clash) with $\\freevars{\\sigma}=\\emptyset$.\nTo show this, consider any $\\iget[const]{\\vdLint[const=I,state=\\nu]}$ in which all premises of \\rref{eq:usubstituted-proofrule} are valid, i.e.\\\n\\(\\iget[const]{\\vdLint[const=I,state=\\nu]}\\models{\\applyusubst{\\sigma}{\\phi_j}}\\) for all $j$.\nThat is, \\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\sigma}{\\phi_j}}\\) for all $\\iget[state]{\\vdLint[const=I,state=\\nu]}$ and all $j$.\nBy \\rref{lem:usubst}, \\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\sigma}{\\phi_j}}\\) is equivalent to\n\\(\\imodels{\\Ia}{\\phi_j}\\),\nwhich, thus, also holds for all $\\iget[state]{\\vdLint[const=I,state=\\nu]}$ and all $j$.\nBy \\rref{cor:adjointUsubst}, \\(\\imodel{\\Ia}{\\phi_j}=\\imodel{\\Ita}{\\phi_j}\\) for any $\\iget[state]{\\Ita}$, since $\\freevars{\\sigma}=\\emptyset$.\nConsequently, all premises of \\rref{eq:proofrule} are valid in $\\iget[const]{\\Ita}$, i.e. \\(\\iget[const]{\\Ita}\\models{\\phi_j}\\) for all $j$.\nThus, \\(\\iget[const]{\\Ita}\\models{\\psi}\\) by local soundness of \\rref{eq:proofrule}.\nThat is, \\(\\imodels{\\Ia}{\\psi}=\\imodel{\\Ita}{\\psi}\\) by \\rref{cor:adjointUsubst} for all $\\iget[state]{\\Ia}$.\nBy \\rref{lem:usubst}, \\(\\imodels{\\Ia}{\\psi}\\) is equivalent to \\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\applyusubst{\\sigma}{\\psi}}\\),\nwhich continues to hold for all $\\iget[state]{\\vdLint[const=I,state=\\nu]}$.\nThus, \\(\\iget[const]{\\vdLint[const=I,state=\\nu]}\\models{\\applyusubst{\\sigma}{\\psi}}\\), i.e.\\ the conclusion of \\rref{eq:usubstituted-proofrule} is valid in $\\iget[const]{\\vdLint[const=I,state=\\nu]}$, hence \\rref{eq:usubstituted-proofrule} locally sound.\nConsequently, all \\irref{US} instances of the locally sound proof rules of \\dL with $\\freevars{\\sigma}=\\emptyset$ are locally sound.\nNote that \\irref{gena+MP} can be augmented soundly to use $p(\\bar{x})$ instead of $p(x)$ or $p$, respectively, such that the $\\freevars{\\sigma}=\\emptyset$ requirement will be met during \\irref{US} instances of all rules.\n\n\\begin{compactitem}\n\\item[\\irref{DW}]\nSoundness of \\irref{DW} uses that differential equations can never leave their evolution domain by \\rref{def:HP-transition}.\nTo show \\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\dbox{\\pevolvein{\\D{x}=f(x)}{q(x)}}{q(x)}}\\), consider any $\\iget[flow]{\\DALint[const=I,flow=\\varphi]}$ of any duration $r\\geq0$ solving \\(\\imodels{\\DALint[const=I,flow=\\varphi]}{\\D{x}=f(x)\\land q(x)}\\).\nThen \\(\\imodels{\\DALint[const=I,flow=\\varphi]}{q(x)}\\) hence \\(\\imodels{\\Iff[r]}{q(x)}\\).\n\n\n\\item[\\irref{DC}]\nSoundness of \\irref{DC} is a stronger version of  soundness for the differential cut rule \\cite{DBLP:journals\/logcom\/Platzer10}.\n\\irref{DC} is a differential version of the modal modus ponens \\irref{K}.\nThe core is that if $r(x)$ always holds after the differential equation\nand $p(x)$ always holds after the differential equation \\(\\pevolvein{\\D{x}=f(x)}{q(x)\\land r(x)}\\) that is restricted to $r(x)$,\nthen $p(x)$ always holds after the differential equation \\(\\pevolvein{\\D{x}=f(x)}{q(x)}\\) without that additional restriction.\nLet \\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\dbox{\\pevolvein{\\D{x}=f(x)}{q(x)}}{r(x)}}\\).\nSince all restrictions of solutions are solutions, this is equivalent to\n\\(\\imodels{\\DALint[const=I,flow=\\varphi]}{r(x)}\\) for all $\\iget[flow]{\\DALint[const=I,flow=\\varphi]}$ of any duration solving \\(\\imodels{\\DALint[const=I,flow=\\varphi]}{\\D{x}=f(x)\\land q(x)}\\) and starting in \\(\\iget[state]{\\Iff[0]}=\\iget[state]{\\vdLint[const=I,state=\\nu]}\\) on $\\scomplement{\\{\\D{x}\\}}$.\nConsequently, for all $\\iget[flow]{\\DALint[const=I,flow=\\varphi]}$ starting in \\(\\iget[state]{\\Iff[0]}=\\iget[state]{\\vdLint[const=I,state=\\nu]}\\) on $\\scomplement{\\{\\D{x}\\}}$:\n\\(\\imodels{\\DALint[const=I,flow=\\varphi]}{\\D{x}=f(x)\\land q(x)}\\) is equivalent to\n\\(\\imodels{\\DALint[const=I,flow=\\varphi]}{\\D{x}=f(x)\\land q(x) \\land r(x)}\\).\nHence, \\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\dbox{\\pevolvein{\\D{x}=f(x)}{q(x)\\land r(x)}}{p(x)}}\\)\nis equivalent to \\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\dbox{\\pevolvein{\\D{x}=f(x)}{q(x)}}{p(x)}}\\).\n\n\\item[\\irref{DE}]\nSoundness of \\irref{DE} is genuine to differential-form \\dL leveraging \\rref{lem:differentialAssignLemma}.\nConsider any state $\\iget[state]{\\vdLint[const=I,state=\\nu]}$.\nThen\n\\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\dbox{\\pevolvein{\\D{x}=f(x)}{q(x)}}{p(x,\\D{x})}}\\)\niff\n\\(\\imodels{\\Iff[r]}{p(x,\\D{x})}\\)\nfor all solutions $\\iget[flow]{\\DALint[const=I,flow=\\varphi]}:[0,r]\\to\\linterpretations{\\Sigma}{V}$ of\n\\(\\imodels{\\DALint[const=I,flow=\\varphi]}{\\D{x}=f(x)\\land q(x)}\\) of any duration $r$ starting in \\(\\iget[state]{\\Iff[0]}=\\iget[state]{\\vdLint[const=I,state=\\nu]}\\) on $\\scomplement{\\{\\D{x}\\}}$.\nThat is equivalent to: for all $\\iget[flow]{\\DALint[const=I,flow=\\varphi]}$,\nif\n\\(\\imodels{\\DALint[const=I,flow=\\varphi]}{\\D{x}=f(x)\\land q(x)}\\)\nthen\n\\(\\imodels{\\DALint[const=I,flow=\\varphi]}{p(x,\\D{x})}\\).\nBy \\rref{lem:differentialAssignLemma}, \\(\\imodels{\\DALint[const=I,flow=\\varphi]}{p(x,\\D{x})}\\) iff\n\\(\\imodels{\\DALint[const=I,flow=\\varphi]}{\\dbox{\\Dupdate{\\Dumod{\\D{x}}{f(x)}}}{p(x,\\D{x})}}\\),\nso, that is equivalent to\n\\(\\imodels{\\Iff[r]}{\\dbox{\\Dupdate{\\Dumod{\\D{x}}{f(x)}}}{p(x,\\D{x})}}\\)\nfor all solutions $\\iget[flow]{\\DALint[const=I,flow=\\varphi]}:[0,r]\\to\\linterpretations{\\Sigma}{V}$ of\n\\(\\imodels{\\DALint[const=I,flow=\\varphi]}{\\D{x}=f(x)\\land q(x)}\\) of any duration $r$ starting in \\(\\iget[state]{\\Iff[0]}=\\iget[state]{\\vdLint[const=I,state=\\nu]}\\) on $\\scomplement{\\{\\D{x}\\}}$,\nwhich is, consequently, equivalent to\n\\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\dbox{\\pevolvein{\\D{x}=f(x)}{q(x)}}{\\dbox{\\Dupdate{\\Dumod{\\D{x}}{f(x)}}}{p(x,\\D{x})}}}\\).\n\n\n\\item[\\irref{DI}]\nSoundness of \\irref{DI} has some relation to the soundness proof for differential invariants \\cite{DBLP:journals\/logcom\/Platzer10}, yet is generalized to leverage differentials.\nThe proof is only shown for \\m{p(x)\\mdefequiv g(x)\\geq0}, in which case \\(\\der{p(x)} \\mequiv (\\der{g(x)}\\geq0)\\).\nConsider a state $\\iget[state]{\\vdLint[const=I,state=\\nu]}$ in which\n\\\\\n\\(\\imodels{\\vdLint[const=I,state=\\nu]}{q(x) \\limply (p(x) \\land \\dbox{\\pevolvein{\\D{x}=f(x)}{q(x)}}{\\der{p(x)}}})\\).\nIf \\(\\inonmodels{\\vdLint[const=I,state=\\nu]}{q(x)}\\), there is nothing to show, because there is no solution of \\({\\pevolvein{\\D{x}=f(x)}{q(x)}}\\) for any duration, so the consequence holds vacuously.\nOtherwise, \\(\\imodels{\\vdLint[const=I,state=\\nu]}{q(x)}\\) implies \\(\\imodels{\\vdLint[const=I,state=\\nu]}{p(x) \\land \\dbox{\\pevolvein{\\D{x}=f(x)}{q(x)}}{\\der{p(x)}}}\\).\nTo show that\n\\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\dbox{\\pevolvein{\\D{x}=f(x)}{q(x)}}{p(x)}}\\) consider any solution $\\iget[flow]{\\DALint[const=I,flow=\\varphi]}$ of any duration $r\\geq0$.\nThe case $r=0$ follows from \\(\\imodels{\\vdLint[const=I,state=\\nu]}{p(x)}\\) by \\rref{lem:coincidence} since \\(\\freevars{p(x)}=\\{x\\}\\) is disjoint from \\(\\{\\D{x}\\}\\), which is changed by evolutions of any duration.\nThat leaves the case \\(r>0\\).\n\nLet \\(\\imodels{\\DALint[const=I,flow=\\varphi]}{\\pevolvein{\\D{x}=f(x)}{q(x)}}\\),\nwhich, by\n\\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\dbox{\\pevolvein{\\D{x}=f(x)}{q(x)}}{\\der{p(x)}}}\\),\nimplies\n\\(\\imodels{\\DALint[const=I,flow=\\varphi]}{\\der{p(x)}}\\).\nSince \\(r>0\\), \\rref{lem:differentialLemma} implies\n\\(0\\leq\\ivaluation{\\Iff[\\zeta]}{\\der{g(x)}}\n= \\D[t]{\\ivaluation{\\Iff[t]}{g(x)}} (\\zeta)\\)\nfor all $\\zeta$.\nTogether with \\(\\imodels{\\Iff[0]}{p(x)}\\) (by \\rref{lem:coincidence} and \\(\\freevars{p(x)}\\cap\\{\\D{x}\\}=\\emptyset\\)), i.e.\\\n\\(\\imodels{\\Iff[0]}{g(x)\\geq0}\\),\nthis implies \\(\\imodels{\\Iff[\\zeta]}{g(x)\\geq0}\\) for all $\\zeta$, including $r$,\nby the mean-value theorem since \\(\\ivaluation{\\Iff[t]}{g(x)}\\) is continuous in $t$ on $[0,r]$ and differentiable on $(0,r)$.\n\n\\item[\\irref{DG}]\n\\def\\Imd{\\imodif[state]{\\vdLint[const=I,state=\\nu]}{x}{d}}%\n\\newcommand{\\DALint[const=I,flow=\\tilde{\\varphi}]}{\\DALint[const=I,flow=\\tilde{\\varphi}]}\n\\newcommand{\\Iffy}[1][t]{\\vdLint[const=I,state=\\tilde{\\varphi}(#1)]}\nSoundness of \\irref{DG} is a constructive variation of the soundness proof for differential auxiliaries \\cite{DBLP:journals\/lmcs\/Platzer12}.\nLet \\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\lexists{y}{\\dbox{\\pevolvein{\\D{x}=f(x)\\syssep\\D{y}=a(x)y+b(x)}{q(x)}}{p(x)}}}\\),\nthat is,\n\\\\\n\\(\\imodels{\\Imd}{\\dbox{\\pevolvein{\\D{x}=f(x)\\syssep\\D{y}=a(x)y+b(x)}{q(x)}}{p(x)}}\\) for some $d$.\nIn order to show that\n\\\\\n\\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\dbox{\\pevolvein{\\D{x}=f(x)}{q(x)}}{p(x)}}\\),\nconsider any \\(\\iget[flow]{\\DALint[const=I,flow=\\varphi]}:[0,r]\\to\\linterpretations{\\Sigma}{V}\\) such that\n\\(\\imodels{\\DALint[const=I,flow=\\varphi]}{\\D{x}=f(x)\\land q(x)}\\) and \\(\\iget[state]{\\Iff[0]}=\\iget[state]{\\vdLint[const=I,state=\\nu]}\\) on $\\scomplement{\\{\\D{x}\\}}$.\nBy modifying the values of $y,\\D{y}$ along $\\iget[flow]{\\DALint[const=I,flow=\\varphi]}$, this function can be augmented to a solution \n\\(\\iget[flow]{\\DALint[const=I,flow=\\tilde{\\varphi}]}:[0,r]\\to\\linterpretations{\\Sigma}{V}\\) such that\n\\(\\imodels{\\DALint[const=I,flow=\\tilde{\\varphi}]}{\\D{x}=f(x)\\land\\D{y}=a(x)y+b(x)\\land q(x)}\\) and \\(\\iget[state]{\\Iffy[0]}(y)=d\\).\nThe assumption then implies \\(\\imodels{\\Iffy[r]}{p(x)}\\), which, by \\rref{lem:coincidence}, is equivalent to \\(\\imodels{\\Iff[r]}{p(x)}\\) since \\(y,\\D{y}\\not\\in\\freevars{p(x)}\\) and \\(\\iget[state]{\\Iff[r]}=\\iget[state]{\\Iffy[r]}\\) on $\\scomplement{\\{y,\\D{y}\\}}$,\nwhich implies \\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\dbox{\\pevolvein{\\D{x}=f(x)}{q(x)}}{p(x)}}\\), since $\\iget[flow]{\\DALint[const=I,flow=\\varphi]}$ was arbitrary.\nThe construction of the modification $\\iget[flow]{\\DALint[const=I,flow=\\tilde{\\varphi}]}$ of $\\iget[flow]{\\DALint[const=I,flow=\\varphi]}$ on $\\{y,\\D{y}\\}$ proceeds as follows.\nBy Picard-Lindel\\\"of \\cite[\\S10.VII]{Walter:ODE}, there is a solution \\(y:[0,r]\\to\\reals\\) of the initial-value problem\n\\begin{equation}\n\\begin{aligned}\n  y(0) &=d\\\\\n  \\D{y}(t) &= F(t,y(t)) \\mdefeq y(t) \\ivaluation{\\Iff[t]}{a(x)} + \\ivaluation{\\Iff[t]}{b(x)}\n\\end{aligned}\n\\label{eq:diffghost-extra-ODE}\n\\end{equation}\nbecause \\(F(t,y)\\) is continuous on \\([0,r]\\times\\reals\\)\n(since \\(\\ivaluation{\\Iff[t]}{a(x)}\\) and \\(\\ivaluation{\\Iff[t]}{b(x)}\\) are continuous in $t$ as compositions of the, by \\rref{def:dL-valuationTerm} smooth, evaluation function and the continuous solution $\\iget[state]{\\Iff[t]}$ of a differential equation)\nand because \\(F(t,y)\\) satisfies the Lipschitz condition\n\\[\n\\norm{F(t,y)-F(t,z)} = \\norm{(y-z)\\ivaluation{\\Iff[t]}{a(x)}} \\leq \\norm{y-z} \\max_{t\\in[0,r]} \\ivaluation{\\Iff[t]}{a(x)}\n\\]\nwhere the maximum exists, because it is a maximum of a continuous function on the compact set $[0,r]$.\nThe modification $\\iget[flow]{\\DALint[const=I,flow=\\tilde{\\varphi}]}$ agrees with $\\iget[flow]{\\DALint[const=I,flow=\\varphi]}$ on $\\scomplement{\\{y,\\D{y}\\}}$ and is defined as \\(\\iget[state]{\\Iffy[t]}(y) = y(t)\\) and \\(\\iget[state]{\\Iffy[t]}(\\D{y}) = F(t,y(t)) = \\D{y}(t)\\) on $\\{y,\\D{y}\\}$, respectively, for the solution $y(t)$ of \\rref{eq:diffghost-extra-ODE}.\nBy construction, \\(\\iget[state]{\\Iffy[0]}(y)=d\\) and\n\\(\\imodels{\\DALint[const=I,flow=\\tilde{\\varphi}]}{\\D{x}=f(x)\\land\\D{y}=a(x)y+b(x)\\land q(x)}\\),\nbecause \\(\\iget[state]{\\Iff[t]}=\\iget[state]{\\Iffy[t]}\\) on $\\scomplement{\\{y,\\D{y}\\}}$ so that\n\\(\\pevolvein{\\D{x}=f(x)}{q(x)}\\) continues to hold along $\\iget[flow]{\\DALint[const=I,flow=\\tilde{\\varphi}]}$ by \\rref{lem:coincidence-term} because \\(y,\\D{y}\\not\\in\\freevars{\\pevolvein{\\D{x}=f(x)}{q(x)}}\\),\nand because \\(\\D{y}=a(x)y+b(x)\\) holds along $\\iget[flow]{\\DALint[const=I,flow=\\tilde{\\varphi}]}$ by \\rref{eq:diffghost-extra-ODE}.\n\n\\def\\Imyd{\\imodif[state]{\\vdLint[const=I,state=\\nu]}{y}{d}}%\n\\newcommand{\\Ifrest}{\\DALint[const=I,flow=\\restrict{\\varphi}{\\scomplement{\\{y,\\D{y}\\}}}]}%\n\\newcommand*{\\Iffrest}[1][\\zeta]{\\vdLint[const=I,state=\\restrict{\\varphi}{\\scomplement{\\{y,\\D{y}\\}}}(#1)]}%\nConversely, let \\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\dbox{\\pevolvein{\\D{x}=f(x)}{q(x)}}{p(x)}}\\).\nThis direction shows a stronger version of\n\\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\lexists{y}{\\dbox{\\pevolvein{\\D{x}=f(x)\\syssep\\D{y}=a(x)y+b(x)}{q(x)}}{p(x)}}}\\)\nby showing that\n\\\\\n\\(\\imodels{\\Imyd}{\\dbox{\\pevolvein{\\D{x}=f(x)\\syssep\\D{y}=\\eta}{q(x)}}{p(x)}}\\)\nfor all $d\\in\\reals$ and all terms $\\eta$.\nConsider any \\(\\iget[flow]{\\DALint[const=I,flow=\\varphi]}:[0,r]\\to\\linterpretations{\\Sigma}{V}\\) such that\n\\(\\imodels{\\DALint[const=I,flow=\\varphi]}{\\D{x}=f(x)\\land\\D{y}=\\eta\\land q(x)}\\)\nwith \\(\\iget[state]{\\Iff[0]}=\\iget[state]{\\Imyd}\\) on $\\scomplement{\\{\\D{x},\\D{y}\\}}$.\nThen the restriction \\(\\iget[flow]{\\Ifrest}\\) of $\\iget[flow]{\\DALint[const=I,flow=\\varphi]}$ to $\\scomplement{\\{y,\\D{y}\\}}$ with \\(\\iget[state]{\\Iffrest[t]}=\\iget[state]{\\Imyd}\\) on $\\{y,\\D{y}\\}$ for all $t\\in[0,r]$ still solves\n\\(\\imodels{\\Ifrest}{\\D{x}=f(x)\\land q(x)}\\) by \\rref{lem:coincidence-term} since \\(\\iget[flow]{\\Ifrest}=\\iget[flow]{\\DALint[const=I,flow=\\varphi]}\\) on $\\scomplement{\\{y,\\D{y}\\}}$ and \\(y,\\D{y}\\not\\in\\freevars{\\pevolvein{\\D{x}=f(x)}{q(x)}}\\).\nIt also satisfies \\(\\iget[state]{\\Iffrest[0]}=\\iget[state]{\\Imyd}\\) on $\\scomplement{\\{\\D{x}\\}}$,\nbecause \\(\\iget[state]{\\Iff[0]}=\\iget[state]{\\Imyd}\\) on $\\scomplement{\\{\\D{x},\\D{y}\\}}$ yet \\(\\iget[state]{\\Iffrest[t]}(\\D{y})=\\iget[state]{\\Imyd}(\\D{y})\\).\nThus, by assumption, \\(\\imodels{\\Iffrest[r]}{p(x)}\\),\nwhich implies \\(\\imodels{\\Iff[r]}{p(x)}\\)\nby \\rref{lem:coincidence}, because\n\\(\\iget[flow]{\\DALint[const=I,flow=\\varphi]}=\\iget[flow]{\\Ifrest}\\) on $\\scomplement{\\{y,\\D{y}\\}}$ and $y,\\D{y}\\not\\in\\freevars{p(x)}$,\n\n\\item[\\irref{DS}]\n\\def\\iconcat[state=\\varphi(t)]{\\I}{\\imodif[state]{\\vdLint[const=I,state=\\nu]}{t}{\\zeta}}%\n\\def\\constODE{f}%\nSoundness of the solution axiom \\irref{DS} follows from existence and uniqueness of global solutions of constant differential equations.\nConsider any state $\\iget[state]{\\vdLint[const=I,state=\\nu]}$.\nThere is a unique \\cite[\\S10.VII]{Walter:ODE} global solution $\\iget[flow]{\\DALint[const=I,flow=\\varphi]}:[0,\\infty)\\to\\linterpretations{\\Sigma}{V}$ defined as \\(\\iget[state]{\\Iff[\\zeta]}(x) \\mdefeq \\ivaluation{\\iconcat[state=\\varphi(t)]{\\I}}{x+\\constODE t}\\)\nand \\(\\iget[state]{\\Iff[\\zeta]}(\\D{x}) \\mdefeq \\D[t]{\\iget[state]{\\Iff[t]}(x)}(\\zeta) = \\iget[const]{\\iconcat[state=\\varphi(t)]{\\I}}(\\constODE)\\)\nand \\(\\iget[state]{\\Iff[\\zeta]} = \\iget[state]{\\vdLint[const=I,state=\\nu]}\\) on $\\scomplement{\\{x,\\D{x}\\}}$.\nThis solution satisfies\n\\(\\iget[state]{\\Iff[0]}=\\iget[state]{\\vdLint[const=I,state=\\nu]}(x)\\) on $\\scomplement{\\{\\D{x}\\}}$ \nand\n  \\m{\\imodels{\\DALint[const=I,flow=\\varphi]}{\\D{x}=\\constODE}},\n  i.e.\n  \\(\\imodels{\\Iff[\\zeta]}{\\D{x}=\\constODE}\\)\n  for all \\(0\\leq \\zeta\\leq r\\).\nAll solutions of \\(\\pevolve{\\D{x}=\\constODE}\\) from initial state $\\iget[state]{\\vdLint[const=I,state=\\nu]}$ are restrictions of $\\iget[flow]{\\DALint[const=I,flow=\\varphi]}$ to subintervals of $[0,\\infty)$.\nThe (unique) state $\\iget[state]{\\vdLint[const=I,state=\\omega]}$ that satisfies \\(\\iaccessible[\\pupdate{\\pumod{x}{x+\\constODE t}}]{\\iconcat[state=\\varphi(t)]{\\I}}{\\vdLint[const=I,state=\\omega]}\\) agrees with \\(\\iget[state]{\\vdLint[const=I,state=\\omega]}=\\iget[state]{\\Iff[\\zeta]}\\) on $\\scomplement{\\{\\D{x}\\}}$, so that, by $\\D{x}\\not\\in\\freevars{p(x)}$, \\rref{lem:coincidence} implies that \\(\\imodels{\\vdLint[const=I,state=\\omega]}{p(x)}\\) iff \\(\\imodels{\\Iff[\\zeta]}{p(x)}\\).\n\nFirst consider axiom\n\\({\\dbox{\\pevolve{\\D{x}=\\constODE}}{p(x)}} \\lbisubjunct {\\lforall{t{\\geq}0}{\\dbox{\\pupdate{\\pumod{x}{x+\\constODE t}}}{p(x)}}}\\) for $q(x)\\mequiv\\ltrue$.\nIf\n\\\\\n\\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\dbox{\\pevolve{\\D{x}=\\constODE}}{p(x)}}\\),\nthen \\(\\imodels{\\Iff[\\zeta]}{p(x)}\\) for all $\\zeta\\geq0$,\nbecause the restriction of $\\iget[flow]{\\DALint[const=I,flow=\\varphi]}$ to $[0,\\zeta)$ solves \\(\\D{x}=\\constODE\\) from $\\iget[state]{\\vdLint[const=I,state=\\nu]}$,\nthus \\(\\imodels{\\vdLint[const=I,state=\\omega]}{p(x)}\\),\nwhich implies\n\\(\\imodels{\\iconcat[state=\\varphi(t)]{\\I}}{\\dbox{\\pupdate{\\pumod{x}{x+\\constODE t}}}{p(x)}}\\),\nso \\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\lforall{t{\\geq}0}{\\dbox{\\pupdate{\\pumod{x}{x+\\constODE t}}}{p(x)}}}\\) as $\\zeta\\geq0$ was arbitrary.\n\nConversely, \\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\lforall{t{\\geq}0}{\\dbox{\\pupdate{\\pumod{x}{x+\\constODE t}}}{p(x)}}}\\)\nimplies \\(\\imodels{\\iconcat[state=\\varphi(t)]{\\I}}{\\dbox{\\pupdate{\\pumod{x}{x+\\constODE t}}}{p(x)}}\\)\nfor all $\\zeta\\geq0$,\ni.e. $\\imodels{\\vdLint[const=I,state=\\omega]}{p(x)}$ when \\(\\iaccessible[\\pupdate{\\pumod{x}{x+\\constODE t}}]{\\iconcat[state=\\varphi(t)]{\\I}}{\\vdLint[const=I,state=\\omega]}\\).\n\\rref{lem:coincidence} again implies \\(\\imodels{\\Iff[\\zeta]}{p(x)}\\) for all $\\zeta\\geq0$, so \\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\dbox{\\pevolve{\\D{x}=\\constODE}}{p(x)}}\\), since all solutions are restrictions of $\\iget[flow]{\\DALint[const=I,flow=\\varphi]}$.\n\nSoundness of \\irref{DS} now follows using that all solutions $\\iget[flow]{\\DALint[const=I,flow=\\varphi]}:[0,r]\\to\\linterpretations{\\Sigma}{V}$ of \\(\\pevolvein{\\D{x}=f(x)}{q(x)}\\) satisfy \\(\\imodels{\\Iff[\\zeta]}{q(x)}\\) for all $0\\leq\\zeta\\leq r$, which, using \\rref{lem:coincidence} as above, is equivalent to \\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\lforall{0{\\leq}s{\\leq}t}{q(x+\\constODE s)}}\\) when \\(\\iget[state]{\\vdLint[const=I,state=\\nu]}(t)=r\\).\n\n\\item[\\irref{Dassignb}]\nSoundness of \\irref{Dassignb} follows from the semantics of differential assignments (\\rref{def:HP-transition}) and compositionality.\nIn detail: \\m{\\Dupdate{\\Dumod{\\D{x}}{f}}} changes the value of symbol $\\D{x}$ to the value of $f$.\nThe predicate $p$ has the same value for arguments $\\D{x}$ and $f$ that have the same value.\n\n\\item[\\irref{Dplus+Dtimes+Dcompose}] Soundness of the derivation axioms \\irref{Dplus+Dtimes+Dcompose} follows from \\rref{lem:derivationLemma}, since they are special instances of \\rref{eq:Dadditive} and \\rref{eq:DLeibniz} and \\rref{eq:Dcompose}, respectively.\nFor \\irref{Dcompose} observe that $y,\\D{y}\\not\\in g(x)$.\n\n\\item[\\irref{G}]\nLet the premise \\(p(\\bar{x})\\) be valid in some $\\iget[const]{\\vdLint[const=I,state=\\nu]}$, i.e.\\ \\m{\\iget[const]{\\vdLint[const=I,state=\\nu]}\\models{p(\\bar{x})}}, i.e.\\ \\(\\imodels{\\vdLint[const=I,state=\\omega]}{p(\\bar{x})}\\) for all $\\iget[state]{\\vdLint[const=I,state=\\omega]}$.\nThen, the conclusion \\(\\dbox{a}{p(\\bar{x})}\\) is valid in the same $\\iget[const]{\\vdLint[const=I,state=\\nu]}$,\ni.e.\\ \\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\dbox{a}{p(\\bar{x})}}\\) for all $\\iget[state]{\\vdLint[const=I,state=\\nu]}$,\nbecause \\(\\imodels{\\vdLint[const=I,state=\\omega]}{p(\\bar{x})}\\) for all $\\iget[state]{\\vdLint[const=I,state=\\omega]}$, so also for all $\\iget[state]{\\vdLint[const=I,state=\\omega]}$ with \\(\\iaccessible[a]{\\vdLint[const=I,state=\\nu]}{\\vdLint[const=I,state=\\omega]}\\).\nThus, \\irref{G} is locally sound.\n\n\\item[\\irref{gena}]\n\\def\\Imd{\\imodif[state]{\\vdLint[const=I,state=\\nu]}{x}{d}}%\nLet the premise \\(p(x)\\) be valid in some $\\iget[const]{\\vdLint[const=I,state=\\nu]}$, i.e.\\ \\m{\\iget[const]{\\vdLint[const=I,state=\\nu]}\\models{p(x)}}, i.e.\\ \\(\\imodels{\\vdLint[const=I,state=\\omega]}{p(x)}\\) for all $\\iget[state]{\\vdLint[const=I,state=\\omega]}$.\nThen, the conclusion \\(\\lforall{x}{p(x)}\\) is valid in the same $\\iget[const]{\\vdLint[const=I,state=\\nu]}$,\ni.e.\\ \\(\\imodels{\\vdLint[const=I,state=\\nu]}{\\lforall{x}{p(x)}}\\) for all $\\iget[state]{\\vdLint[const=I,state=\\nu]}$, i.e.\\ \\(\\imodels{\\Imd}{p(x)}\\) for all $d\\in\\reals$,\nbecause \\(\\imodels{\\vdLint[const=I,state=\\omega]}{p(x)}\\) for all $\\iget[state]{\\vdLint[const=I,state=\\omega]}$, so in particular for all $\\iget[state]{\\vdLint[const=I,state=\\omega]}=\\iget[state]{\\Imd}$ for any $d\\in\\reals$.\nThus, \\irref{gena} is locally sound.\n\n\\item[\\irref{CQ}]\nLet the premise \\(f(\\bar{x})=g(\\bar{x})\\) be valid in some $\\iget[const]{\\vdLint[const=I,state=\\nu]}$, i.e. \\m{\\iget[const]{\\vdLint[const=I,state=\\nu]}\\models{f(\\bar{x})=g(\\bar{x})}}, i.e.\\ \\(\\imodels{\\vdLint[const=I,state=\\nu]}{f(\\bar{x})=g(\\bar{x})}\\) for all $\\iget[state]{\\vdLint[const=I,state=\\nu]}$,\ni.e.\\ \\(\\ivaluation{\\vdLint[const=I,state=\\nu]}{f(\\bar{x})}=\\ivaluation{\\vdLint[const=I,state=\\nu]}{g(\\bar{x})}\\) for all $\\iget[state]{\\vdLint[const=I,state=\\nu]}$.\nConsequently, \\(\\ivaluation{\\vdLint[const=I,state=\\nu]}{f(\\bar{x})}\\in\\iget[const]{\\vdLint[const=I,state=\\nu]}(p)\\) iff \\(\\ivaluation{\\vdLint[const=I,state=\\nu]}{g(\\bar{x})}\\in\\iget[const]{\\vdLint[const=I,state=\\nu]}(p)\\).\nSo, \\(\\iget[const]{\\vdLint[const=I,state=\\nu]}\\models {p(f(\\bar{x})) \\lbisubjunct p(g(\\bar{x}))}\\).\nThus, \\irref{CQ} is locally sound.\n\n\\item[\\irref{CE}]\nLet the premise \\(p(\\bar{x})\\lbisubjunct q(\\bar{x})\\) be valid in some $\\iget[const]{\\vdLint[const=I,state=\\nu]}$, i.e. \\m{\\iget[const]{\\vdLint[const=I,state=\\nu]}\\models{p(\\bar{x}) \\lbisubjunct q(\\bar{x})}}, i.e.\\ \\(\\imodels{\\vdLint[const=I,state=\\nu]}{p(\\bar{x}) \\lbisubjunct q(\\bar{x})}\\) for all $\\iget[state]{\\vdLint[const=I,state=\\nu]}$.\nConsequently, \\(\\imodel{\\vdLint[const=I,state=\\nu]}{p(\\bar{x})} = \\imodel{\\vdLint[const=I,state=\\nu]}{q(\\bar{x})}\\).\nThus,\n\\(\\imodel{\\vdLint[const=I,state=\\nu]}{\\contextapp{C}{p(\\bar{x})}}\n= \\iget[const]{\\vdLint[const=I,state=\\nu]}(C)\\big(\\imodel{\\vdLint[const=I,state=\\nu]}{p(\\bar{x})}\\big)\n= \\iget[const]{\\vdLint[const=I,state=\\nu]}(C)\\big(\\imodel{\\vdLint[const=I,state=\\nu]}{q(\\bar{x})}\\big)\n= \\imodel{\\vdLint[const=I,state=\\nu]}{\\contextapp{C}{q(\\bar{x})}}\\).\nThis implies\n\\(\\iget[const]{\\vdLint[const=I,state=\\nu]}\\models{\\contextapp{C}{p(\\bar{x})} \\lbisubjunct \\contextapp{C}{q(\\bar{x})}}\\),\nhence the conclusion is valid in $\\iget[const]{\\vdLint[const=I,state=\\nu]}$.\nThus, \\irref{CE} is locally sound.\n\n\\item[\\irref{CT}]\nRule \\irref{CT} is a (locally sound) derived rule and only included for comparison.\n\\irref{CT} is derivable from \\irref{CQ} using \\(p(\\usarg) \\mdefequiv (c(\\usarg)=c(g(\\bar{x})))\\) and reflexivity of $=$.\n\n\\item[\\irref{MP}]\nModus ponens \\irref{MP} is locally sound with respect to the interpretation $\\iget[const]{\\vdLint[const=I,state=\\nu]}$ and the state $\\iget[state]{\\vdLint[const=I,state=\\nu]}$, which implies local soundness and thus soundness.\nIf \\(\\imodels{\\vdLint[const=I,state=\\nu]}{p\\limply q}\\) and \\(\\imodels{\\vdLint[const=I,state=\\nu]}{p}\\) then \\(\\imodels{\\vdLint[const=I,state=\\nu]}{q}\\).\n\n\\item[\\irref{US}]\nUniform substitution is sound by \\rref{thm:usubst-sound}, just not necessarily locally sound.\n\n\\end{compactitem}\n\\end{proofatend}\n\n\n\\section{Conclusions}\n\nWith differential forms for local reasoning about differential equations, uniform substitutions lead to a simple and modular proof calculus for differential dynamic logic that is entirely based on axioms and axiomatic rules, instead of soundness-critical schema variables with side-conditions in axiom schemata.\nThe \\irref{US} calculus is straightforward to implement and enables flexible reasoning with axioms by contextual equivalence.\nEfficiency can be regained by tactics that combine multiple axioms and rebalance the proof to obtain short proof search branches.\nContextual equivalence rewriting for implications is possible when adding monotone \\predicational{s} $C$ whose substitution instances limit $\\uscarg$ to positive polarity.\n\n\\paragraph*{Acknowledgment.}\nI thank the anonymous reviewers of the conference version \\cite{DBLP:conf\/cade\/Platzer15} for their helpful feedback.\n\nThis material is based upon work supported by the National Science Foundation by \nNSF CAREER Award CNS-1054246.\nThe views and conclusions contained in this document are those of the author and should not be interpreted as representing the official policies, either expressed or implied, of any sponsoring institution or government. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the author(s) and do not necessarily reflect the views of any sponsoring institution or government.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nQuantum critical metals (QCM) constitute a rich and ever-evolving field of modern condensed matter physics. The broad interest in their behavior is fueled not only by the increasing number of experimentally probed systems,\n such as cuprates and iron-based materials \\cite{PhysRevB.81.184519, hussey2008phenomenology, doi:10.1146\/annurev-conmatphys-031119-050558,RevModPhys.73.797}, but also by the advent of modern numerical techniques, in particular Quantum Monte-Carlo \\cite{berg2012sign, PhysRevX.6.031028, Lederer4905, doi:10.1146\/annurev-conmatphys-031218-013339}. One exciting aspect of QCMs is their prototypical non-Fermi liquid behavior, and it is a theorist's dream to determine the associated critical  behavior exactly.\n So far, this has only been achieved in the special cases  \n such as the SYK model \\cite{PhysRevLett.70.3339, Kitaev2015}, a particular matrix large $N$ theory~\\cite{PhysRevLett.123.096402}, a scalar large $N$ model with a particular dispersion of a critical boson \\cite{PhysRevB.82.045121}, and models obtained by dimensional regularization \\cite{PhysRevB.88.245106}.\n  Besides, a fully  self-consistent description of a spin density wave quantum critical point (QCP) in (2+1)D at the lowest energies has also been  proposed in Ref.\\ \\cite{PhysRevX.7.021010} and Ref.~\\cite{PhysRevB.90.045121} presented\n the general description of critical theories with a chiral Fermi surface. \n\nA {particular example of a} QCM, which defies complete understanding despite years of study \\cite{POLCHINSKI1994617, PhysRevB.50.14048, PhysRevB.50.17917, PhysRevB.64.195109, PhysRevLett.91.066402, PhysRevB.73.045127, PhysRevB.73.085101, PhysRevB.74.195126,  doi:10.1146\/annurev-conmatphys-070909-103925,  metlitski2010quantum, holder2015anomalous, PhysRevB.82.045121, PhysRevB.88.245106,PhysRevB.96.155125, PhysRevB.103.235129},\nis the one at an Ising nematic QCP in (2+1)D.\nHere, the fermionic self-energy both at one and two loop level scales as $\\omega^{2\/3}$, which is a hallmark of a non-Fermi liquid. However, the loop expansion lacks a control parameter. It was originally   believed \\cite{POLCHINSKI1994617,PhysRevB.50.14048} that one can justify the expansion by extending the model to  $N$ fermion flavors   and taking the limit $N \\to \\infty$. Indeed, at large $N$, the prefactor for $\\omega^{2\/3}$ at the two-loop order is parametrically smaller than the one-loop result  \\cite{PhysRevB.50.14048,PhysRevB.74.195126}. However, it was later recognized by S.-S.\\ Lee \\cite{PhysRevB.80.165102} that this does not hold  beyond the two-loop order. He analyzed a simplified ``one-patch'' model  without backscattering  and  explicitly demonstrated that planar diagrams, which emerge at three-loop and higher orders, are not small in $1\/N$.  Subsequent studies of  the Ising nematic  model within the ``two-patch theory'', which includes backscattering, found more contributions from planar diagrams, not small in $1\/N$, and discovered  additional logarithmic singularities (Refs.~\\cite{metlitski2010quantum, holder2015anomalous}).\n\nThere are several equivalent characterizations of planar diagrams, which can all be invoked to extract their\nspecial behaviour at large $N$. The original name-giving criterion in the high-energy literature \\cite{HOOFT1974461} is that they can be drawn on a plane without holes.  From the condensed matter perspective, the planar diagrams\ndescribe a subset of scattering processes which have a (1+1)-dimensional structure, despite the original problem being (2+1)D -- more specifically, in these diagrams  the curvature term in the fermionic dispersion cancels out \\cite{metlitski2010quantum},\\footnote{We note that the two-dimensionality still shows up via Landau damping, without it the fermionic self-energy would be more singular than $\\omega^{2\/3}$ \\cite{PhysRevB.73.085101, PhysRevLett.97.226403} }.\n\nBecause the leading one-loop self-energy $\\omega^{2\/3}$  also comes from (1+1)D processes\n\\cite{PhysRevLett.95.026402, PhysRevB.73.045128}, the $N \\to \\infty$ limit at a QCP should be described by some effective (1+1)-dimensional theory.  However, the detailed structure of higher-loop planar diagrams is not yet known, and the story is further complicated by apparent UV-divergences at higher-loop orders~\\cite{metlitski2010quantum,holder2015anomalous}.  As a result, the form of fermionic and bosonic propagators at a QCP in the large $N$ limit remains unclear.\n\n\n\nIn this paper we analyze the scattering rate in a (2+1)D Fermi liquid away from a nematic\n  QCP.   Our first goal is to understand the  role of planar diagrams in a Fermi liquid regime. In a generic  2D\n   Fermi liquid, the imaginary part of the one-loop self-energy $\\text{Im}\\left[\\Sigma (\\omega)\\right]$ at $k=k_F$ scales as $\\omega^2 \\ln(\\omega)$  (as $ (\\omega^2 + (\\pi T)^2)\\ln\\left[{\\text{max}} \\left(\\omega, T\\right)\\right]$ at a finite temperature $T$). This non-analytic form comes from forward and backscattering processes~\\cite{PhysRevB.71.205112},\n which can be regarded as effectively one-dimensional. We analyze $\\text{Im}\\left[\\Sigma (\\omega)\\right]$ at $T=0$ beyond one-loop order, and show that it contains a series of higher powers of $\\ln(\\omega)$. The logarithms come from the Cooper part of backscattering, which is manifestly (1+1)D,  and we argue that these terms can be equally viewed as coming from planar processes.  We demonstrate this explicitly by analyzing the system behavior in the  limit of large $N$, when only planar processes survive. We explicitly  sum up series of logarithms and obtain the scattering rate to logarithmic accuracy to all orders in the interaction. We show, however, that in a Fermi liquid not all planar processes are (1+1)D -- the ones that do not contain the highest power of $\\ln (\\omega)$ at any loop order involve scattering with deviations from a single direction by momenta $q \\sim M$, where $M^2$  scales with  the distance from a QCP \\footnote{At a QCP, all planar processes become (1+1)D, and, simultaneously, the argument of the logarithm becomes of order one. More specifically, it becomes\na function of the combination $q^3\/\\omega$, where $q$ and $\\omega$ are relevant internal momenta and frequency in a\nplanar diagram. Both are infinitesimally small at vanishing external frequency, and typical $q^3$ are of order $\\omega$.Then the effective dimension of the full $N = \\infty$ theory becomes (1+1)D, with no separation into the leading and subleading terms.}.\n\n\nFor a toy model with repulsive interaction in the particle-particle channel, the full $\\text{Im} \\left[ \\Sigma(\\omega) \\right]$ to logarithmic accuracy\nremains finite and is reduced, roughly by a half, compared to the one-loop expression. A similar result holds for the $T^2$\nterm in the specific heat of a 2D Fermi gas~\\cite{PhysRevB.74.075102, PhysRevB.76.165111}.\n For the realistic case of an attractive pairing interaction,\n the ground state is  an $s$-wave superconductor~\\cite{PhysRevLett.77.3009,Moon2010,PhysRevLett.114.097001, PhysRevB.90.165144, PhysRevB.90.174510, PhysRevB.91.115111,PhysRevB.94.115138, CHOWDHURY2020168125}.\n We show that, in this case, the prefactor of the $\\omega^2$ term in  $\\text{Im} \\left[ \\Sigma(\\omega) \\right]$ increases with $\\ln(\\omega)$, and the summation of the logarithms breaks down at some energy scale $\\omega  \\geq \\Delta_0$, where $\\Delta_0$ is a pairing gap.\n\n\nOur second goal is then to correctly incorporate the superconducting ground state and obtain $\\text{Im} \\left[ \\Sigma(\\omega) \\right]$\nfor $\\omega \\simeq \\Delta_0$. This is of relevance both for experiments and for\nQuantum Monte-Carlo studies, which found~\\cite{PhysRevX.6.031028,Lederer4905} that superconductivity must be included to understand the data for the self-energy at the lowest frequencies\/temperatures.\n We argue that in a 2D Fermi liquid, the frequency range where logarithmic renormalizations from planar diagrams are relevant, but superconductivity can be neglected, can be made wide even near a QCP,\n  by restricting to a  weak coupling. We show that in this case one can incorporate superconductivity in a\n     controllable way at $\\omega \\simeq \\Delta_0$ \\footnote{Exactly at a QCP, the scale $\\Delta_0$ is of the same order as the upper edge of the non-Fermi-liquid~\\cite{CHUBUKOV2020168142}, and there is no sizable frequency range for the renormalization of the self-energy by planar diagrams without including superconducting fluctuations, unless one makes specific assumptions about the shape of the Fermi surface \\cite{PhysRevB.80.165102}.}.\n\n\nThe fermionic self-energy at $\\omega \\simeq \\Delta_0$ has been analyzed in the past for a  conventional $s$-wave superconductor  with momentum-independent  pairing interaction  ~\\cite{PhysRevB.42.10211,Larkin2008}.    We show that our case is qualitatively different, because the interaction, mediated by soft nematic fluctuations, gives rise to near-equal attraction in a large number of pairing channels, and $s$-wave is only slightly preferable\n over other pairing symmetries ~\\cite{PhysRevLett.114.097001, PhysRevB.98.220501}. In this case, we show that $\\text{Im} \\left[ \\Sigma(\\omega) \\right]$ has the same form as in a pure $s$-wave superconductor between $\\Delta_0$ (below which $\\text{Im} \\left[ \\Sigma(\\omega) \\right] = 0$) and slightly over $3\\Delta_0$, where the system realizes that attraction in the $s$-wave channel is  larger than that in  other channels.\nAt larger $\\omega$,   all pairing channels contribute equally, and  the form of $\\text{Im} \\left[ \\Sigma(\\omega) \\right]$ changes qualitatively.\n\n\n The\n paper is structured as follows: In Sec.\\ \\ref{modelsec}, we present the\nmodel, recapitulate one-loop results for $\\Sigma[\\omega]$  and discuss the relevant energy scales. In Sec.\\ \\ref{planarnonplanarsec}, we analyze the structure of three-loop planar diagrams and demonstrate that the theory is effectively one-dimensional at logarithmic order, but not beyond. In the next Sec.\\ \\ref{flselfsecrep}, we sum up the series of the leading $\\omega^2 (\\ln{\\omega})^n$ contributions to the self-energy from backscattering at all loop orders.\nIn Sec.\\ \\ref{repulsive}, we consider a toy model with repulsive pairing interaction and show that the summation of the leading logarithms converges at any frequency.\nThe realistic case of an attractive interaction is analyzed in Sec. \\ref{attractivenormalstatesec}, where we\nshow that the convergence only holds  up to  $\\omega  \\sim \\Delta_0$. In Sec.\\ \\ref{scscatesec}, we demonstrate how the results get modified once we add superconductivity and obtain the expressions for $\\text{Im} \\left[ \\Sigma(\\omega) \\right]$ at $\\omega \\simeq \\Delta_0$. Conclusions and an outlook are presented in Sec.\\ \\ref{concsec}. Various technical details are relegated to the Appendices.\n\n\n\\section{Model and one-loop results}\n\\label{modelsec}\n\nWe consider a system of interacting $N$-flavor fermions in two dimensions  at $T=0$.\nTwo spin projections are incorporated  into $N$ such that the physical case of spin-1\/2  fermions corresponds to $N = 2$. We assume that the system is {isotropic and} close to a QCP towards a nematic order {($d$-wave Pomeranchuk instability)} and that near a QCP the dominant interaction between fermions is mediated by soft fluctuations of a nematic order parameter.  The corresponding $T = 0$ Euclidean action \\cite{PhysRevLett.91.066402, PhysRevB.73.045127} is given by\n\\begin{align}\n\\label{mainaction}\n\\mathcal{S} &= \\mathcal{S}_0 + \\mathcal{S}_{\\text{int}} \\ , \\\\ \\notag\n\\mathcal{S}_0 &= \\sum_{\\sigma = 1}^N \\int_{k} \\bar\\psi_\\sigma(k) (-i\\omega_m + \\xi_{\\boldsymbol{k}})\\psi_\\sigma(k)\\  ,  \\\\ \\notag\n\\mathcal{S}_\\text{int} &=\n\\frac{g}{V}  \\sum_{\\sigma, \\sigma^\\prime} \\int_{k,p,q}  D_0({\\boldsymbol{q}})\nf_{\\boldsymbol{q}}({\\boldsymbol{k}}) f_{\\boldsymbol{q}}({\\boldsymbol{p}})  \\times  \\\\ & \\notag \\quad \\bar\\psi_\\sigma(k + q\/2) \\bar\\psi_{\\sigma^\\prime} (p - q\/2) \\psi_{\\sigma^\\prime}(p+q\/2) \\psi_\\sigma(k-q\/2) \\ , \\\\  \\notag\n\\int_k &\\equiv \\int \\frac{d\\omega_m}{2\\pi} \\frac{d{\\boldsymbol{k}}}{(2\\pi)^2} \\ .\n\\end{align}\nHere $\\xi_{\\boldsymbol{k}} = v_F(|{\\boldsymbol{k}}| - k_F)$ is the flavor-independent fermionic dispersion, linearized around the Fermi momentum,\n$\\omega_m$ is a Matsubara frequency, $g$ is  the coupling,  which   we set to be small compared to the Fermi energy, $g\/E_F \\ll 1$, and\n\\begin{align}\nD_0({\\boldsymbol{q}}) = - \\frac{1}{q^2 + M^2} \\ , \\quad q \\equiv |{\\boldsymbol{q}}| \\ ,\n\\end{align}\nis the bare bosonic propagator. The parameter $M$ (bosonic mass) measures the deviation from the quantum critical point, and has units of momentum. Near a QCP,\n\\begin{align}\n\\epsilon \\equiv \\frac{M}{k_F}  \\ll 1 \\ .\n\\end{align}\n{Finally, $f_{\\boldsymbol{q}}({\\boldsymbol{k}}) = \\cos(2 \\theta_{{\\boldsymbol{k}}{\\boldsymbol{q}}})$ (with $ \\theta_{{\\boldsymbol{k}}{\\boldsymbol{q}}} = \\measuredangle({\\boldsymbol{k}},{\\boldsymbol{q}}))$ is the $d$-wave nematic form factor (not related to pairing symmetry), which specifies the symmetry of the ordered  state. At a general boson-fermion vertex with incoming and outgoing fermionic momenta ${\\boldsymbol{k}}_{\\text{in}}, {\\boldsymbol{k}}_\\text{out}$, the half-sum $({\\boldsymbol{k}}_{\\text{in}} + {\\boldsymbol{k}}_{\\text{out}})\/2$ corresponds to ${\\boldsymbol{k}}$ in the above defintion of $f$, while ${\\boldsymbol{k}}_\\text{out} - {\\boldsymbol{k}}_\\text{in}$ corresponds to ${\\boldsymbol{q}}$. In our analysis, all momenta will fulfill $|{\\boldsymbol{k}}_{\\text{in}}|, |{\\boldsymbol{k}}_\\text{out}| \\simeq k_F$, resulting in ${\\boldsymbol{k}} \\perp {\\boldsymbol{q}}$ and  $f \\simeq - 1$. (see Fig.\\ \\ref{formfactor}). Since all relevant expressions contain $f^2$, we can set $f = 1$ from the start. The form factor is important at high energies of order $\\mathcal{O}(E_F)$ or in the ordered state only, which are not considered in this paper. For a short discussion of lattice effects, see Sec.\\ \\ref{concsec}. }\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{fig1.pdf}\n\\caption{{Definition of form factor, see main text}}\n\\label{formfactor}\n\\end{figure}\n\n \n \n \n \n \nThe action of Eq.\\ (\\ref{mainaction}) is obtained from a microscopic Hamiltonian with 4-fermion interaction by integrating out fermions with energies above a certain cutoff. To account for screening from fermions with energies smaller than the cutoff, the bare propagator  $D_0({\\boldsymbol{q}})$ should be dressed by a particle-hole bubble $\\Pi_{\\text{ph}}$, see Fig.\\ \\ref{diagtable}(a) and  App.\\ \\ref{bubblesapp}. The dressed propagator $D(\\omega_m,{\\boldsymbol{q}})$ is\n\\begin{align}\n\\label{Landaudampingdef}\n&D(\\omega_m,{\\boldsymbol{q}}) = - \\frac{1}{q^2 + M^2 + g \\Pi_{\\text{ph}} (\\omega_m, {\\boldsymbol{q}}) }\n , &\\\\ \\notag\n &{\\text {where}}  \\quad\n  \\Pi_{\\text{ph}} (\\omega_m , {\\boldsymbol{q}}) = N\\rho \\frac{|\\omega_m|}{\\sqrt{\\omega^2_m + (v_F q)^2}} \\ .  &\n\\end{align}\n$\\rho = m\/(2\\pi)$ the density of states per flavor, and $m = k_F\/v_F$ the fermionic mass.\n   We assume, following earlier works \\cite{metlitski2010quantum, doi:10.1080\/0001873021000057123}, that the\nstatic part of $\\Pi_{\\text{ph}}$ is already incorporated into $D(\\omega_m, {\\boldsymbol{q}})$.\n   At $\\omega_m \\ll v_F q$, which will be relevant to our analysis, $\\Pi_{\\text{ph}} (\\omega_m , {\\boldsymbol{q}}) \\approx N \\rho |\\omega_m|\/(v_F q)$ describes (in real frequencies) the Landau damping of\n a boson due to a decay into the particle-hole continuum.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=.5\\columnwidth]{fig2.pdf}\n\\caption{One-loop diagrams. (a) Particle-hole bubble $\\Pi_{\\text{ph}}$, which renormalizes the bare interaction $D_0$ (dashed wavy lines). (b) One-loop self-energy $\\Sigma^{(1)}$. Full wavy line represents the renormalized interaction $D$. (c) Diagram which effectively determines $\\Sigma^{(1)}(\\omega)$ for $\\omega \\ll {\\omega_{\\text{FL}}}$ (interaction lines are static).}\n\\label{diagtable}\n\\end{figure}\n\nWith these ingredients, we can evaluate the fermionic self-energy for the external momentum ${\\boldsymbol{k}}$ on the Fermi surface,\n$\\Sigma(\\omega_m,{\\boldsymbol{k}}_F) \\equiv  \\Sigma(\\omega_m)$\n$(\\text{with}\\  G^{-1} = G^{-1}_0 - \\Sigma$). The one-loop  self-energy  $\\Sigma^{(1)}(\\omega_m)$, represented by the diagram in Fig.\\ \\ref{diagtable}(b), has been obtained before, see, e.g., Refs.\\ \\cite{PhysRevB.74.195126, metlitski2010quantum},\nand we just present the result:\n\\begin{align}\n\\label{sigma1resultmain}\n& \\Sigma^{(1)}(\\omega_m) \\simeq \\\\& \\notag  \\begin{dcases} - i\n\\lambda  \\omega_m +  i \\text{sign}(\\omega_m) \\frac{2\\lambda}{\\pi}\n\\frac{\\omega^2_m}{{\\omega_{\\text{FL}}}} \\ln\\left(\\frac{{\\omega_{\\text{FL}}}}{|\\omega_m|}\\right), \\! &\\omega_m \\ll {\\omega_{\\text{FL}}} \\\\\n\\omega_{\\text{IN}}^{1\/3}  \\omega^{2\/3}_m, \\! &\\omega_m \\gg {\\omega_{\\text{FL}}}.\n\\end{dcases}\n\\end{align}\nThe first line in Eq.\\ \\eqref{sigma1resultmain} has a familiar Fermi-liquid form\n in 2D. It contains\ntwo  parameters\n\\begin{align}\n\\label{lambdadef}\n\\lambda \\equiv \\frac{g}{4 \\pi v_F M} , \\quad \\quad {\\omega_{\\text{FL}}}  \\equiv \\frac{ v_F M^3}{N g \\rho}  \\ .\n\\end{align}\nHere, $\\lambda$ is an effective coupling, which will be used to control perturbation theory, and ${\\omega_{\\text{FL}}}$\n  serves as UV cutoff for the logarithms and also marks the crossover from a Fermi-liquid to a non-Fermi-liquid regime.\n  The second line  in \\eqref{sigma1resultmain}  is the  one-loop formula for the self-energy in the non-Fermi liquid regime.\n The scale\n \\beq\n\\label{win}\n\\omega_{\\text{IN}} = \\frac{8}{3\\sqrt{3}} \\lambda^3 {\\omega_{\\text{FL}}} \\\n \\eeq\n is the upper boundary of the true non-Fermi-liquid region, where $\\Sigma(\\omega_m) > \\omega_m$.\n  The relevant energy scales are summarized in Fig.\\ \\ref{scales} for both $\\lambda \\ll 1$, $\\lambda \\gg 1$.\n\nIn deriving  Eq.\\ \\eqref{sigma1resultmain} we assumed that typical internal frequencies $\\omega^\\prime_m$ and momenta $q$ satisfy\n$\\omega^\\prime_m \\ll v_F q$. This condition is naturally satisfied at a QCP, where typical $\\omega^\\prime_m \\sim q^{3}$, see e.g.\\ \\cite{PhysRevX.10.031053}, but in a weakly coupled Fermi liquid at $\\lambda \\ll 1$ we must additionally require  $\\lambda > \\epsilon\/N$, such that ${\\omega_{\\text{FL}}}\n \\leq v_F M$.\n\n\nFor $\\omega \\ll {\\omega_{\\text{FL}}}$, the Landau damping is small compared to the mass term $M^2$ in $D(\\omega_m,{\\boldsymbol{q}})$, and the Fermi-liquid form of $\\Sigma^{(1)}(\\omega_m)$  is obtained by expanding in the Landau damping. In effect, this reduces the evaluation of the diagram in Fig.\\ \\ref{diagtable}(b) to the evaluation of the two-loop diagram with static interactions $D_0({\\boldsymbol{q}})$ of Fig.\\ \\ref{diagtable}(c).  Let us now zoom into  the $\\omega^2$-part of the Fermi-liquid self-energy at these frequencies:\n \\begin{align}\n\\label{sigmaomegaqdef}\n\\Sigma^{(1)}_{\\omega^2} (\\omega_m) \\equiv  i \\text{sign}(\\omega_m)\n\\frac{2\\lambda}{\\pi} \\frac{\\omega^2_m}{{\\omega_{\\text{FL}}}} \\ln\\left(\\frac{{\\omega_{\\text{FL}}}}{|\\omega_m|}\\right).\n\\end{align}\n Upon analytical continuation $i\\omega_m  \\rightarrow \\omega + i\\delta$,  it maps to\nthe imaginary part of the retarded self-energy\n\\begin{align}\n\\text{Im}\\left[\\Sigma^R(\\omega)\\right] =  -\\frac{2 \\lambda}{\\pi} \\frac{\\omega^2}{{\\omega_{\\text{FL}}}} \\ln\\left(\\frac{{\\omega_{\\text{FL}}}}{|\\omega|}\\right),\n\\label{a_1}\n\\end{align}\nwhich determines the fermionic scattering rate.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=\\columnwidth]{fig3.pdf}\n\\caption{Energy scales. For $\\lambda \\ll 1$, the self-energy is always smaller than the bare frequency, $\\Sigma(\\omega_m)< \\omega_m$, and there is no true non-Fermi-liquid. The green sliver marks the superconducting region for attractive interactions discussed in Sec.\\ \\ref{scscatesec}, with $\\Delta_0 \\simeq {\\omega_{\\text{FL}}} \\exp(-\\lambda)$. For $\\lambda \\gg 1$, a true non-Fermi-liquid region (red) develops where  $\\Sigma(\\omega_m)> \\omega_m$. Superconductivity in the strong coupling case is outside the scope of this paper. The scale $v_F M > {\\omega_{\\text{FL}}}$ is not explicitly shown. }\n\\label{scales}\n\\end{figure}\n\n\n\\begin{figure}[b]\n\\centering\n\\includegraphics[width=\\columnwidth]{fig4.pdf}\n\\caption{One-dimensional scattering at two-loop level. (a) Most singular momentum points (forward and backscattering) which contribute to the logarithm in $\\Sigma_{\\omega^2}^{(1)}$. The grayed out diagram is also part of backscattering, but its contribution can be neglected since it involves a large momentum transfer $\\sim 2k_F$. (b) Way to extract the logarithm from the forward and backscattering processes (see main text).}\n\\label{oneloop1ddiag}\n\\end{figure}\n\n\nFor our future considerations it is important that the $\\omega_m^2 \\ln{(\\omega_m)}$ dependence in  Eq.~(\\ref{sigmaomegaqdef}) can be traced to effectively one-dimensional scattering, embedded in two dimensions (see Appendix \\ref{oneloopfermiapp} for technical details).  Namely,  the processes that contribute to Eq.\\ (\\ref{sigmaomegaqdef}) are forward scattering and backscattering, with momentum deviations  from ${\\boldsymbol{k}}$ and ${-{\\boldsymbol{k}}}$  on the order of external $\\omega_m$  (see Fig.~\\ref{oneloop1ddiag}(a)). As sketched in Fig.\\ \\ref{oneloop1ddiag}(b), instead of evaluating the closed particle-hole bubble in the diagram one can explicitly obtain one half of $\\omega_m^2 \\ln{(\\omega_m)}$\n from either of these two  scattering processes (the first diagram describes forward scattering and the second one backscattering). In each case,  the integration over the internal momentum ${\\boldsymbol{q}}_1$ (assuming $|{\\boldsymbol{q}}_1| \\ll k_F$) and corresponding frequency in the blue boxes in Fig.\\ \\ref{oneloop1ddiag}(b) produces a Landau damping term $\\sim \\omega_m^\\prime\/q$ as the leading dynamical contribution,  where $q = |{\\boldsymbol{q}}|$ and $\\omega_m^\\prime$  are the momentum and the frequency of the remaining internal fermion in the diagrams in Fig.\\ \\ref{oneloop1ddiag}. The integral over the angle between ${\\boldsymbol{k}},{\\boldsymbol{q}}$ (or, equivalently, the component of ${\\boldsymbol{q}}$ parallel to ${\\boldsymbol{k}}$) restricts internal frequencies to $\\omega_m^\\prime \\in (0,\\omega_m)$, and each diagram yields\n\\begin{align}\n\\label{schematic1}\n\\Sigma_{\\omega^2}^{(1)} (\\omega_m) \\propto \\int_0^{\\omega_m} d\\omega_m^\\prime \\int_{\\omega^\\prime_m\/v_F}^{{\\omega_{\\text{FL}}}\/v_F} dq \\frac{\\omega_m^\\prime}{q} \\propto \\omega_m^2 \\ln(\\omega_m) \\ ,\n\\end{align}\nas in Eq.~\\eqref{sigmaomegaqdef}.\n\nRelevant $q$ in the integral are of order $\\omega'_m\/v_F$ (to logarithmic accuracy), and relevant $\\omega'_m$ are of order of external $\\omega_m$, which we set to be the smallest scale in the problem. Then relevant $q^2$ are much smaller than $M^2$,  and hence  the static part of $D(\\omega_m, q)$ can be approximated by $1\/M^2$, i.e., at this order  static interaction can be treated as momentum-independent. This explains why the prefactor in Eq.~\\eqref{sigmaomegaqdef} is $ \\lambda\/{\\omega_{\\text{FL}}} \\propto 1\/M^4$ .\n\n\n\n\\section{Structure and effective dimensionality of higher order diagrams}\n\\label{planarnonplanarsec}\n\nThere are  multiple  self-energy diagrams at higher\n loop  orders, but we have two tuning knobs to single out the most important ones: small external $\\omega_m$ and large $N$. Let us focus on large $N$ first. At the QCP, the diagrams with the highest power of $N$ are the planar ones, as discussed in Refs.~\\cite{PhysRevB.73.045128, metlitski2010quantum,PhysRevB.82.045121}. We  show   below  that the same holds away from a QCP. We discuss the general structure of planar diagrams in Appendix \\ref{planarstructureapp}.\n\nWe compute the self-energy order-by-order in the loop expansion on the Matsubara axis, and then convert the\n result onto the real axis.  We associate the full dynamical $D(\\omega_m, {\\boldsymbol{q}})$ to each bosonic propagator in the diagrams and absorb the $-i\\lambda \\omega_m$ term in the self-energy into the bare fermionic propagator.\n\nThe first planar diagrams beyond one-loop appear at the three-loop order. There are two such diagrams, we show them in Figs. \\ref{planar3loop}(a),(b).  Each diagram can be computed explicitly (App \\ref{threeloopApp}), and the result is\n (for $\\omega_m>0$)\n \\begin{align}\n\\label{forwardresult3loop}\n\\Sigma^{(3)}_{\\omega^2, \\text{a}}(\\omega_m) &=\ni  \\frac{1}{\\pi^2} \\frac{\\lambda^2}{1+\\lambda} \\frac{\\omega^2_m}{{\\omega_{\\text{FL}}}}\\times(0.56329\\hdots)\\  ,\n\\\\\n\\label{backscatteringhere}\n\\Sigma^{(3)}_{\\omega^2,\\text{b}}(\\omega_m) &= i \\frac{1}{2\\pi} \\frac{\\lambda^2}{1+\\lambda} \\frac{\\omega^2_m}{{\\omega_{\\text{FL}}}} \\ln^2\\left(\\frac{{\\omega_{\\text{FL}}}}{\\omega_m}\\right) \\ .\n\\end{align}\nComparing these two terms with the one-loop result, Eq.~(\\ref{sigmaomegaqdef}), we find that the power of $N$ is the same, and there is an additional factor $\\lambda\/(1+\\lambda)$. Besides,  $\\Sigma^{(3)}_{\\omega^2,\\text{a}}( \\omega_m)$ contains no logarithm, while $\\Sigma^{(3)}_{\\omega^2,\\text{b}}(\\omega_m)$ contains $\\ln^2{(\\omega_m)}$.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=.85\\columnwidth]{fig5.pdf}\n\\caption{Three-loop diagrams. Labels indicate four-momenta (a) Planar diagram in the forward scattering channel (b) Planar diagram in the  backscattering channel (c) Non-planar diagram. }\n\\label{planar3loop}\n\\end{figure}\n\n\nFor comparison, consider one of the non-planar diagrams,  like the one in Fig.\\ \\ref{planar3loop}(c). Evaluating the corresponding integrals, we obtain\n\\begin{align}\n\\label{backresult3loop_1}\n\\Sigma^{(3)}_\\text{c}(\\omega_m) =\n- i \\frac{2}{N} \\frac{\\lambda}{(1+\\lambda)\\pi} \\frac{\\omega^2_m}{{\\omega_{\\text{FL}}}} \\ln\\left(\\frac{{\\omega_{\\text{FL}}}}{\\omega_m}\\right)  \\ .\n\\end{align}\nComparing this result with Eq.~(\\ref{sigmaomegaqdef}), we see that $\\Sigma^{(3)}_\\text{c}(\\omega_m)$ contains an additional factor $1\/N$.\n\nWe now look more closely at the two planar diagrams and explore another knob: small external $\\omega_m$.\nWe remind the reader that  the $\\omega^2_m \\ln{(\\omega_m)}$  term in the one-loop self-energy,\n Eq.~(\\ref{sigmaomegaqdef}),  comes from forward scattering and backscattering.\n The issue we\nwant to address is  whether the self-energies $\\Sigma^{(3)}_{\\omega^2,\\text{a}}(\\omega_m)$ and $\\Sigma^{(3)}_{\\omega^2,\\text{b}}(\\omega_m)$ can be cast into the same form as Eq.~(\\ref{sigmaomegaqdef})\nwith dressed static forward scattering and backscattering amplitudes. Simple experimentation shows that this holds if  two of the three internal momenta and frequencies in Fig.~\\ref{planar3loop} (a),(b), (either $q$ and $q_1$, or $q$ and $q_2$,  where $q = ( \\omega'_m, {\\boldsymbol{q}})$), scale with $\\omega_m$ and hence are infinitesimally small at $\\omega_m \\to 0$. At a QCP,  $\\omega_m$ is the only scale in the problem, and all internal momenta\/frequencies in the planar diagrams are necessary infinitesimally small at $\\omega_m \\to 0$. However, in a Fermi liquid there exists an internal momentum scale $M$ (energy scale $v_F M$), and some internal momenta may be of order $M$. We illustrate this  in Fig.~\\ref{planar3loopvertices}, where a blue box labels a  dressed vertex.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=.85\\columnwidth]{fig6.pdf}\n\\caption{Three-loop diagrams as in Fig.\\ \\ref{planar3loop}, with the dressed vertices marked by blue boxes.}\n\\label{planar3loopvertices}\n\\end{figure}\n\nThe computation of the dressed vertices at vanishing external $q$ and $q_2$ is straightforward and we skip the details. For the planar diagram of Fig.~\\ref{planar3loopvertices}(b),  we find that,\n to logarithmic accuracy, the result can be  expressed via the dressed  static backscattering amplitude, i.e., the sum of the backscattering part of  $\\Sigma^{(1)}_{\\omega^2}(\\omega_m)$  and    $\\Sigma^{(3)}_{\\omega^2,\\text{b}}(\\omega_m)$ can be re-expressed as\n  \\begin{align}\n\\label{sigmaomegaqdef_new}\n&\\Sigma_{\\omega^2, \\text{bs}}(\\omega_m)  \\equiv \\\\ & \\notag  i \\text{sign}(\\omega_m)\n\\frac{2\\lambda}{\\pi {\\omega_{\\text{FL}}}}  \\int_0^{|\\omega_m|} d\\omega_m^\\prime\\omega_m^\\prime \\int_{\\omega_m^\\prime}^{{\\omega_{\\text{FL}}}} \\frac{dz}{z} \\left[\\frac{M^2}{g}\\Gamma_{\\text{bs}} (z)\\right]^2 ,\n\\end{align}\nwhere\n\\beq\n\\Gamma_{\\text{bs}} (z) = - \\frac{g}{M^2} \\left[1 + \\frac{1}{2} \n \\lambda Z  \\ln\\left(\\frac{{\\omega_{\\text{FL}}}}{z}\\right) + ...  \\right]\n\\label{a_2}\n\\eeq\n is the dressed backscattering amplitude and $Z =1\/(1+\\lambda)$  is the quasiparticle spectral weight.\nHowever, if we go beyond logarithmic accuracy, we find that $\\Sigma^{(3)}_{\\omega^2,\\text{b}}(\\omega_m)$ contains contributions that come from $|{\\boldsymbol{q}}| \\sim M$ and cannot be expressed via the  backscattering vertex. Only very close to a QCP, where $M$ is vanishingly small, these non-logarithmic terms can be absorbed into the dressed backscattering amplitude.\n\n For the diagram of Fig.~\\ref{planar3loopvertices}(a), which is a candidate for the forward-scattering contribution,  the dressed vertex (the blue box) vanishes at external $q, q_2=0$, and to reproduce $\\Sigma^{(3)}_{\\omega^2,\\text{a}}(\\omega_m)$ one has to keep $|{\\boldsymbol{q}}_2| \\sim M$. This implies that $\\Sigma^{(3)}_{\\omega^2,\\text{a}}(\\omega_m)$ does not come from forward scattering.  This again holds at a finite $M$.   The vanishing of the dressed vertex  at $q, q_2 =0$ also explains why the three-loop self-energy  $\\Sigma^{(3)}_{\\omega^2,\\text{a}}(\\omega_m)$ does not contain a logarithm.\n\nWe therefore see that in a Fermi liquid, only planar diagrams survive in the limit $N \\to \\infty$, but only a portion of the self-energy  from these diagrams comes from forward scattering and backscattering. The converse is also true: one can easily verify that the self-energy $\\Sigma^{(3)}_\\text{a}(\\omega_m)$ from the non-planar diagram in Fig.~\\ref{planar3loopvertices}(c) does come from backscattering. However, this vertex renormalization is small in $1\/N$.\n\n\n Below we keep  $M$ finite and compute the full self-energy to all orders in the interaction to logarithmic accuracy.\n To this accuracy, we can neglect renormalizations from ``near-forward-scattering\"  and  ``near-back-scattering\"  processes and compute the renormalization of the backscattering amplitude keeping only the largest power of $\\ln{(\\omega_m)}$ at each order in the loop expansion. The fact that the self-energy can be obtained from the renormalized backscattering amplitude (to the logarithmic accuracy) again implies that the problem is effectively one-dimensional: the relevant internal fermions have momenta $\\pm {\\boldsymbol{k}}$ up to small deviations of order $\\omega$, i.e., they only move along a single direction. \n We note that for this calculation we do not need to set $N \\to \\infty$ as the $1\/N$ terms are outside the logarithmic accuracy.\n\n\\section{The full result for the self-energy to all orders in the interaction}\n\\label{flselfsecrep}\n\nLet us now compute the fully renormalized backscattering contribution to the self-energy by extracting the most logarithmically singular contributions to $\\Sigma_{\\omega^2, \\text{bs}}$ at each loop order. One can straightforwardly verify that these contributions arise from the diagrams shown in Fig.\\ \\ref{maindiagram}, which particle-particle (Cooper) ``bubbles\".  We label  them as  ``Cooper diagrams''.\n\\begin{figure*}\n\\centering\n\\includegraphics[width=.8\\textwidth]{fig7.pdf}\n\\caption{Diagram contributing to $\\Sigma(\\omega)$ with 4 Cooper bubbles. Labels indicate four-momenta, interaction lines are RPA dressed. }\n\\label{maindiagram}\n\\end{figure*}\nAn example of a  Cooper diagram is shown in Fig.\\ \\ref{maindiagram}. A generic  diagram with\n$n$ Cooper bubbles contains $\\ln({\\omega_{\\text{FL}}}\/|\\omega_m|)^n$. Our goal is to sum up this series.  We will see that the series\nis not geometrical, because the interaction is momentum-dependent.\n\nWe proceed with the evaluation of the Cooper diagrams. An inspection of these diagrams\nyields the following general strategy for the calculation of the $O(\\omega^2_m)$ term in the self-energy to logarithmic accuracy (see App.\\ \\ref{maincalcdetailssec} for details):\n    we select one cross-section, from  which we take the Landau-damping term (this gives the overall $\\omega_m^2$ in $\\Sigma_{\\omega^2, \\text{bs}}$) and sum up series of Cooper logarithms on both sides of this cross-section,\nwhich produces a factor $\\Gamma^2_{\\text{bs}}$. For the calculation of the Landau damping in the selected cross-section with internal momentum ${\\boldsymbol{p}}_i$ in the Cooper bubble, we set  ${{\\boldsymbol{q}}}$  in Fig.\\ \\ref{maindiagram} nearly transverse to the external ${{\\boldsymbol{k}}}$ and ${\\boldsymbol{p}}_i$ nearly parallel to ${{\\boldsymbol{k}}}$. For the other Cooper bubbles, we only use a static interaction between fermions on the Fermi surface, \n \\bea\n\\label{interactionform}\nD(p_1 - p_2) \\simeq  -g k^2_F \\left[\\epsilon^2 +  4 \\sin^2\\!\\left(\\frac{\\phi_{1} - \\phi_{2}}{2}\\right) \\right] \\ , \n\\eea\nwhere $\\phi_i$ are the angles of ${\\boldsymbol{p}}_i$ measured relative to ${\\boldsymbol{k}}$. We  evaluate each particle-particle bubble $\\Pi_\\text{pp}$  to logarithmic accuracy and for $v_F q \\gg \\omega_m'$, where $\\omega'_m$ is the frequency component of $q$ in Fig.\\ \\ref{maindiagram}. \nUnder these conditions, the integration of the two fermionic propagators  in the particle-particle bubble with $q-p_j$ and $p_j$ over internal frequency $\\omega_{jm}$ and over $|{\\boldsymbol{p}}_j|$  yields (see App.\\ \\ref{bubblesapp})\n\\begin{align}\n\\label{Pippmain}\n \\Pi_{\\text{pp}} (|{\\boldsymbol{q}}|, \\phi_j) = Z \\rho\\ln\\left(\\frac{{\\omega_{\\text{FL}}}}{v_F |{\\boldsymbol{q}}| |\\sin{\\phi_j}| }\\right) \\ . \n \\end{align} \n We assume and verify that typical values of {\\it all} $\\phi_j$ are of order $\\epsilon$.\n To logarithmic accuracy, we can then approximate $\\Pi_{\\text{pp}} (|{\\boldsymbol{q}}|, \\phi_j) \\simeq \\Pi_{\\text{pp}} (z) = Z \\rho\n \\ln({\\omega_{\\text{FL}}}\/z)$, where $z =v_F |{\\boldsymbol{q}}|\\epsilon$. Then the remaining integrations over $\\phi_{j}$ in the Cooper ladder \n involve only the interactions.   This leads to the following expression for the backscattering amplitude, with $L(z) = \\ln({\\omega_{\\text{FL}}}\/z)$: \n \\begin{widetext}\n\\begin{align}\n\\label{Tfirst}\n&\n\\Gamma_{\\text{bs}} (z) \\equiv - \\frac{g}{k_F^2} \\sum_{n = 0}^\\infty \\left(\\frac{g\\rho Z L(z)}{k^2_F}\\right)^n\n  \\cdot \\frac{1}{(2\\pi)^n} \\int_0^{2\\pi} d\\phi_1 d\\phi_2 \\hdots d\\phi_n\\frac{1}{\\epsilon^2 + 4 \\sin^2\\left(\\frac{\\phi_1}{2}\\right)}  \\times \\frac{1}{\\epsilon^2 + 4 \\sin^2\\left(\\frac{\\phi_2 - \\phi_1}{2}\\right)} \\times \\hdots  \\\\ & \\notag  \\times \\frac{1}{\\epsilon^2 + 4\\sin^2\\left(\\frac{\\phi_n - \\phi_{n-1}}{2}\\right)} \\times \\frac{1}{\\epsilon^2 + 4 \\sin^2\\left(\\frac{\\phi_{n}}{2}\\right)} \\  .\n\\end{align}\nTo leading order in $\\epsilon$, the integrand  can be approximated as \n\\begin{align}\n\\label{phiint}\n&\\frac{1}{(2\\pi)^n \\epsilon^{n+1}} \\int_{-\\infty}^\\infty d\\phi_1 \\hdots d\\phi_n \\frac{\\epsilon}{\\epsilon^2+ \\phi_1^2} \\times \\frac{\\epsilon}{\\epsilon^2+(\\phi_1 - \\phi_2)^2} \\times \\frac{\\epsilon}{\\epsilon^2+ (\\phi_2 - \\phi_3)^2} \\times \\hdots \\times \\frac{\\epsilon}{\\epsilon^2 + (\\phi_{n-1} - \\phi_n)^2} \\times  \\frac{\\epsilon}{\\epsilon^2+\\phi_n^2} \\notag  \\\\ &  = \\frac{1}{(2\\pi)^n \\epsilon^{n+2}} \\times \\frac{\\pi^n}{(n+1)} \n= \\frac{1}{n+1} \\frac{1}{(2\\epsilon)^{n}} \\times \\frac{k^2_F}{M^2} \\ .\n\\end{align}\n\\end{widetext}\nThe integrals in Eq.\\ \\eqref{phiint} were solved by going to Fourier space, where the convolutions of Lorentzians become products of exponentials. The expression for the backscattering amplitude then reduces to  \n\\begin{align}\n\\label{Tsumfirst}\n\\Gamma_{\\text{bs}}(z)\n \\simeq - \\frac{g}{M^2} \\sum_{n = 0}^\\infty \\left(\n\\tilde{\\lambda} \n  L(z)\\right)^n  \\frac{1}{n+1} , \\quad  \\tilde\\lambda \\equiv  \\lambda Z = \\frac{\\lambda}{1+ \\lambda}\\  .\n\\end{align}\nWe see that\n $\\tilde{\\lambda}$ becomes the effective coupling controlling the perturbation theory. \n As long as $\\tilde{\\lambda} L(z) < 1$, the summation in Eq.\\  \\eqref{Tsumfirst} converges, and we obtain\n\\begin{align}\n\\label{Tlognormal} \\Gamma_{\\text{bs}}(z) =  \\frac{g}{M^2}\\left(\\frac{\\ln\\left(1- \\tilde{\\lambda} L(z)\\right)}{\\tilde{\\lambda} L(z)} \\right) \\ .\n\\end{align}\nThe full self-energy from backscattering then becomes\n\\begin{align}\n\\label{Sigmap2}\n&\\Sigma_{\\omega^2, \\text{bs}}(\\omega_m) = i  \\text{sign}(\\omega_m) \\frac{\\lambda}{\\pi} \\frac{\\omega^2_m}{{\\omega_{\\text{FL}}}}   \\int_{\\omega_m}^{{\\omega_{\\text{FL}}}}  \\frac{dz}{z}\n \\left[\\frac{M^2}{g} \\Gamma_{\\text{bs}} (z)\\right]^2  \\notag \\\\\n & =   i  \\text{sign}(\\omega_m) \\frac{\\lambda}{\\pi}  \\frac{\\omega^2_m}{{\\omega_{\\text{FL}}}}\n \\int_{\\omega_m}^{{\\omega_{\\text{FL}}}}  \\frac{dz}{z} \\left(\\frac{\\ln\\left(1- \\tilde{\\lambda} L(z)\\right)}{\\tilde{\\lambda} L(z)}\\right)^2  \\ .\n\\end{align}\nAt $\\tilde{\\lambda} \\rightarrow 0$, we get \n\\begin{align}\n\\label{wcresult}\n\\Sigma_{\\omega^2, \\text{bs}}(\\omega_m) = i \\text{sign}(\\omega_m) \\frac{\\lambda}{\\pi} \\frac{\\omega^2_m}{{\\omega_{\\text{FL}}}} \\ln\\left(\\frac{{\\omega_{\\text{FL}}}}{|\\omega_m|}\\right) \\ .\n\\end{align}\nThis is exactly a half of the one-loop result, which is expected, as Eq.\\ \\eqref{Sigmap2} only contains the backscattering contribution. {The weak-coupling result \\eqref{wcresult} is only valid for $\\lambda \\ln({\\omega_{\\text{FL}}}\/|\\omega_m|) \\ll 1$; for small frequencies, $\\lambda \\ln({\\omega_{\\text{FL}}}\/|\\omega_m|)$ becomes $O(1)$, and one has to solve the full integral \\eqref{Sigmap2} instead. }\n\n\\subsection{Repulsive pairing interaction}\n\\label{repulsive}\n\nWe first analyze Eq.~(\\ref{Sigmap2}) for a toy model, in which we assume that ${\\tilde \\lambda}$ is negative, i.e., that the pairing interaction is repulsive.  In this case,  the series in Eq.\\ (\\ref{Tsumfirst}) converges for all $z$, even the smallest ones. The integral over $z$ in Eq.~\\eqref{Sigmap2} can then be evaluated exactly, and the result is\n\\begin{align}\n\\label{Sigmawithxrep}\n\\Sigma_{\\omega^2, \\text{bs}}(\\omega_m) &= i  \\text{sign}(\\omega_m) \n \\frac{\\omega^2_m }{ Z \\pi {\\omega_{\\text{FL}}}} f_\\text{bs;r}(\\ell) \\ ,\n \\end{align}\n  where\n  \\beq\n   \\ell = \n   |\\tilde{\\lambda}| \\ln({\\omega_{\\text{FL}}}\/|\\omega_m|)\n\\eeq\n and\n \\beq\nf_\\text{bs;r}(\\ell) =\n- \\frac{(1+\\ell) \\ln^2(1+\\ell)}{\\ell} - 2\\text{Li}_2(-\\ell) \\ ,\n\\eeq\nwith  $\\text{Li}_2$ the Polylogarithm.\nThe forward scattering contribution, in the same units, is\n\\begin{align}\n\\Sigma_{\\text{fw}}(\\omega_m) = i\\text{sign}(\\omega_m) \n \\frac{\\omega^2_m}{Z \\pi {\\omega_{\\text{FL}}}}  f_\\text{fs}(\\ell), \\quad f_{\\text{fs}}(\\ell) = \\ell \\ .\n\\end{align}\nA plot of $f_\\text{bs;r}, f_\\text{fs}$ is shown in Fig.\\ \\ref{frepulsiveplot}. For small $\\ell$, i.e., fairly large frequencies, $f_\\text{bs;r} = f_{\\text{fs}}$ as expected. For smaller frequencies (increasing $\\ell$), $f_\\text{bs;r}(\\ell)$ does not grow linearly, but is bounded, and can be approximated by the limiting form\n\\begin{align}\nf_\\text{bs;r}(\\ell) = \\frac{\\pi^2}{3} - \\frac{\\ln^2(\\ell)}{\\ell} + \\mathcal{O} \\left( \\frac{\\ln(\\ell)}{\\ell}\n\\right) \\ .\n\\end{align}\nTherefore, for a repulsive pairing interaction, the backscattering rate incures a logarithmic suppression compared to forward scattering; at large $\\ell$ the full rate is reduced  to the forward scattering part, i.e., by a half compared to the one-loop result.\n\n Upon analytical continuation to the real axis, $i\\omega_m \\to \\omega + i\\delta$, Eq.~(\\ref{Sigmawithxrep}) becomes the expression for the scattering rate  at a low frequency \n \\begin{align}\n\\label{Imsigmarep}\n{\\text{Im}} \\left[\\Sigma_{\\text{bs;r}}^R (\\omega)\\right] &= - \n\\frac{\\omega^2}{Z\\pi {\\omega_{\\text{FL}}}} f_\\text{\nbs;r}(\\ell) \\ ,\n\\end{align}\n where now $\\ell = |{\\tilde \\lambda}| \\ln\\left({\\omega_{\\text{FL}}}\/|\\omega|\\right)$.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{fig8.pdf}\\caption{The functions\n$f_\\text{bs;r}(\\ell)$ and $f_\\text{fs}(\\ell)$ \n for the toy model with repulsive pairing interaction.  }\n\\label{frepulsiveplot}\n\\end{figure}\n\n\\subsection{Attractive pairing interaction}\n\\label{attractivenormalstatesec}\nLet us now consider the physical case of attractive pairing interaction, $\\tilde{\\lambda} >0$. Now we have\n\\begin{align}\n\\label{sigmaattleading}\n\\Sigma_{\\omega^2, \\text{bs}} &= i\\text{sign}(\\omega_m) \\frac{\\omega^2_m}{Z\\pi {\\omega_{\\text{FL}}}}\nf_\\text{bs;a}(\\ell) \\ , \\nonumber \\\\\nf_\\text{bs;a}(\\ell) &= \\frac{ (\\ell - 1)\\ln^2(1-\\ell)}{\\ell} + 2\\text{Li}_{2}(\\ell) \\ ,\n\\end{align}\nand  $f_{\\text{bs;a}}(\\ell)$ is well-defined  only for $\\ell \\leq 1$.\nAt larger $\\ell$ the results must be modified by including superconductivity, as we show below. A plot of $f_\\text{bs;a}(\\ell)$ is presented in Fig.\\ \\ref{fattractiveplot}. Again, for small $\\ell$, \n$f_\\text{bs;a} \\simeq \\ell$, but at\n$\\ell \\lesssim 1$ it grows faster.  In the limit $\\ell \\rightarrow 1$,\n\\begin{align}\nf_\\text{bs;a}(\\ell) = \\frac{\\pi^2}{3} - (1 - \\ell) \\ln^2(1- \\ell) + \\mathcal{O}\\left(\\ln(1-\\ell) (1- \\ell)\\right) .\n\\end{align}\n$\nf_\\text{bs;a}(1) = \\pi^2\/3$ is finite, but the slope $df_\\text{bs;a}(\\ell)\/d\\ell$  is logarithmically divergent at\n  $\\ell \\rightarrow 1$. As a curiosity, we note that $f_{\\text{bs;r}}$ and $f_{\\text{bs;a}}$ are related as\n\\beq\n\\label{fssym}\nf_{\\text{bs;r}}(\\ell) =f_\\text{bs;a}\\left(\\frac{\\ell}{1+\\ell} \\right) \\ .\n\\eeq\nThis can be derived by a simple substitution in the integral defining $f_\\text{bs;a}$. \n\n \\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{fig9.pdf}\n\\caption{\n The function  $f_{\\text{bs;a}}$ to leading order in $\\epsilon$,  from Eq. (\\ref{sigmaattleading}). \n  $\\ell = 1$ corresponds to $\\omega = \\Delta_0$. The inset shows the modification of \n  $f_\\text{bs;a}$ due to extra contributions beyond the leading order in $\\epsilon$,  \n   see  Eq.~\\eqref{fbsstarall}.\n}\n\\label{fattractiveplot}\n\\end{figure}\nLet us look at the behavior of\n$f_{\\text{bs;a}} (\\ell)$  as\n $ \\ell \\rightarrow 1$ more carefully. We recall that\n$f_{\\text{bs;a}} (\\ell)$   has been obtained using the expression for the backscattering amplitude $\\Gamma_{\\text{bs}} (z)$, Eq.~\\eqref{Tlognormal},  valid to the leading order in $\\epsilon = M\/k_F$. We now\ncheck whether the dependence of $f_{\\text{bs;a}} (\\ell)$ on $1-\\ell$ gets modified if we compute $\\Gamma_{\\text{bs}} (z)$ beyond  $\\mathcal{O}(\\epsilon)$.  The reasoning for doing this is the following:  to leading order in $\\epsilon$, partial amplitudes in pairing channels with different angular momentum are all equal, and this gives rise to the appearance of \n  the factor $1\/(n+1)$ in the series for $\\Gamma_{\\text{bs}} (z)$ in \\eqref{Tlognormal}.\n Beyond the order $\\mathcal{O}(\\epsilon)$, partial pairing amplitudes do differ, and the largest one is in the $s$-wave channel.\n If we would only include this channel and ignore all other pairing components, we would find that the logarithmical series \n becomes geometrical and $f_{\\text{bs;a}} (\\ell) \\propto 1\/(1-\\ell)$ (see Eq.~(\\ref{sigmaattsubleading_1}) below).  For a small but finite $\\epsilon$, we therefore expect a crossover from \n  $f_{\\text{bs;a}} (\\ell)= \\pi^2\/3- (1-\\ell) \\ln^2(1- \\ell) +...$ at $1-\\ell\\leq 1$ to  $\n   f_{\\text{bs;a}} (\\ell) \\propto 1\/(1-\\ell)$ at the smallest $1- \\ell$. To describe this crossover,  we now evaluate $\\Gamma_{\\text{bs}}(z)$ in next-to-leading order in $\\epsilon$.\n  This can be achieved by angular momentum decomposition. \n  Namely,\n  the effective interaction on the Fermi surface, \n\\begin{align}\nV(\\phi) &= \n\\tilde{\\lambda} \\frac{2\\epsilon}{\\epsilon^2 + 4 \\sin^2(\\phi\/2)}\\ , \n\\end{align}\n can be decomposed into angular momenta as\n\\begin{align}\n\\label{Veffangular}\nV_m = \\int_0^{2\\pi} \\frac{d\\phi}{2\\pi} V(\\phi) \\exp(-i\\phi m) \\ .\n\\end{align}\n To a very good approximation (see  App.\\ \\ref{Tmatrixapp})\n \\begin{align}\n\\label{Veffangular_1}\nV_m  \\approx \\tilde{\\lambda} \\exp(- |m| \\epsilon) \\ .\n\\end{align}\nAt $\\epsilon \\rightarrow 0$, all components  $V_m$ become degenerate. However, for a nonzero $\\epsilon$,\n   this is not quite the case yet, and  the $s$-wave  component \n     $V_0 \\simeq \\tilde{\\lambda}$ is largest.\nA straightforward analysis then shows that the series for\n $\\Gamma_{\\text{bs}}(z)$ can be rewritten as\n\\begin{align}\n\\notag\n \\Gamma_{\\text{bs}}(z) &= - \\frac{1}{\\rho} \\sum_m V_m \\sum_{n = 0}^\\infty \\left[ V_m L(z)\\right]^{n} \\\\ &= - \\frac{1}{\\rho} \\frac{V_0}{1-V_0 L(z)} - \\frac{2}{\\rho} \\sum_{m = 1}^\\infty \\frac{V_m}{1- V_m L(z)},\n\\label{remsum}\n\\end{align}\nwhere in the second line we singled out the $s$-wave part. The remaining sum in Eq.\\ \\eqref{remsum} can be evaluated using the Euler-Maclaurin formula, and  the result is\n\\begin{align}\n\\label{Tnextto}\n&\\Gamma_{\\text{bs}}(z) \\simeq    \\\\ & \\notag -\\frac{g}{M^2}\n\\left[-\\frac{\\ln\\left( 1- \\tilde{\\lambda} L(z)\n +\n  \\epsilon\n \\right)}{\\tilde{\\lambda} L(z)} + \\frac{\\epsilon}{2} \\frac{1}{1- \\tilde{\\lambda} L(z)} \\right] .\n\\end{align}\nEq.\\ \\eqref{Tnextto} shows that the $\\ln\\left( 1- \\tilde{\\lambda} L(z)\\right)$ dependence of the\nbackscattering amplitude is the consequence of the near degeneracy of all\npartial pairing amplitudes. Once this degeneracy is lifted at a finite $\\epsilon$,\n  the backscattering amplitude develops a conventional $s$-wave pole,\n albeit with a small weight $\\epsilon\/2$. Inserting this\n  $\\Gamma_{\\text{bs}} (z)$  into the integral for  $\\Sigma_{\\omega^2, \\text{bs}}$, Eq.~(\\ref{Sigmap2}), we obtain, \n \\begin{align}\n\\label{sigmaattsubleading_1}\n&\\Sigma_{\\omega^2, \\text{bs}} = i  \\text{sign}(\\omega_m) \\frac{\\omega_m^2}{Z \\pi {\\omega_{\\text{FL}}}}\nf^*_\\text{bs;a}(\\ell),  \n\\end{align} \nwhere \\begin{align}\n\\label{fbsstarall}\nf^*_\\text{bs;a}(\\ell) \\simeq f_\\text{bs;a}(\\ell) + \\frac{\\epsilon^2}{4(1-\\ell)} \\ .\n\\end{align} \n At $1-\\ell \\lesssim \\epsilon$, \n \\begin{align}  \\notag \nf^*_{\\text{bs;a}}(\\ell) \\simeq   &\\frac{\\epsilon^2}{4(1- \\ell)}  + \\frac{\\pi^2}{3}  + {\\epsilon} \\ln{(1-\\ell + \\epsilon)}\\ln{(1-\\ell)}\n \\\\ & - (1-\\ell +\\epsilon) \\ln^2 (1-\\ell + \\epsilon) \\ .\n \\label{a_5}\n\\end{align}\n$f_\\text{bs;a}(\\ell)$ and $f^*_\\text{bs;a}(\\ell)$ are plotted in the inset of Fig.\\ \\ref{fattractiveplot}. \nWe see that the self-energy diverges as $1\/(1-\\ell)$  at $\\ell \\to 1$, as is expected for an $s$-wave superconductor\n~\\cite{PhysRevB.42.10211,Larkin2008}, but in our case the divergent term appears with the prefactor $\\epsilon^2$ \\footnote{We note that the prefactor of the $s$-wave term gets renormalized ($N \\rightarrow N-1$ in the expression for ${\\omega_{\\text{FL}}}$) by ladder diagrams without a loop, in which a line, expressing an outgoing fermion, is attached to the upper part of the Cooper ladder \\cite{haussmann1993crossover}. Such a diagram has the same number of logarithms, but involves a large momentum transfer $2k_F$ and therefore renormalizes the $s$-wave part only}.\n\n\n\n\n\nLike before, upon analytical continuation to the real axis, $i\\omega_m \\to \\omega + i\\delta$, Eq.\\ (\\ref{sigmaattsubleading_1}) becomes the expression for the backscattering contribution to\nIm[$\\Sigma^R(\\omega)$]:\n\\beq\n\\text{Im}\\left[\\Sigma_{\\text{bs;a}}^R(\\omega)\\right] =\n- \\frac{\\omega^2}{Z \\pi {\\omega_{\\text{FL}}}} \n  f^*_\\text{\n  bs;a}(\\ell) \\ , \n\\label{sigmaattsubleading_2}\n\\eeq\nwhere on the real frequency axis $\\ell = {\\tilde \\lambda} \\ln({{\\omega_{\\text{FL}}}\/|\\omega|})$.  Eq.~(\\ref{sigmaattsubleading_2}) shows that the scattering rate increases when $\\omega$ decreases ($\\ell$ increases)  and\nbecomes singular at $\\ell =1$.\n\n\nThe  enhancement of $\\text{Im}[\\Sigma_{\\text{bs;a}}^R(\\omega)]$ near $\\ell =1$\ncan be interpreted as follows:\nto the lowest order in the interaction, the rate comes from the\n decay of a particle into two particles and one hole; $\\text{Im}[\\Sigma_{\\text{bs;a}}^R(\\omega)]$ measures the phase space for such a process. By energy conservation, the three outgoing states must  have energies smaller than $\\omega$.\nWhen Cooper pair formation sets in, the two electrons in the particle-particle ladder form a tightly bound Cooper pair, and there are only two decay products left. Therefore, the phase space restriction is less severe for small $\\omega$,   and the scattering rate increases.\n\nBefore concluding this section, we discuss the range of validity of Eqs.~(\\ref{sigmaattsubleading_1}) and (\\ref{sigmaattsubleading_2}).  The expression for the self-energy has been obtained with  logarithmic accuracy,\n i.e., at each loop order we neglected terms with the same power of ${\\tilde \\lambda}$, but a smaller power of $\\ln({\\omega_{\\text{FL}}}\/|\\omega|)$. This is valid when ${\\tilde \\lambda} \\ll 1$ and $|\\omega| \\ll {\\omega_{\\text{FL}}}$.\nThe condition on $\\omega$ does not pose a restriction as throughout the paper we assume that ${\\omega_{\\text{FL}}}$ is finite and focus on the self-energy at the smallest $\\omega$.  The condition ${\\tilde \\lambda} = \\lambda\/(1+\\lambda) \\ll 1$ implies  weak coupling, $\\lambda \\ll 1$.  Because $\\lambda = g\/(4\\pi v_F M) \\sim g\/(E_F \\epsilon)$, we need to keep $g\/E_F$ small to satisfy both $\\lambda \\ll 1$ and $\\epsilon \\ll 1$. \n  Weak coupling is advantageous to us because,  for ${\\tilde \\lambda} \\ll 1$, \nthere is a wide range of frequencies where on the one hand $|\\omega| \\ll {\\omega_{\\text{FL}}}$ and on the other  $\\ell  = {\\tilde \\lambda}  \\ln({\\omega_{\\text{FL}}}\/|\\omega|) <1$,  hence Eqs. (\\ref{sigmaattsubleading_1}) and (\\ref{sigmaattsubleading_2}) are applicable. This is indeed the consequence of the fact that superconductivity at weak coupling emerges at an exponentially small $\\omega \\sim \\Delta_0 ={\\omega_{\\text{FL}}} \\exp(-1\/{\\tilde \\lambda})$.   We emphasize that this only holds in a Fermi liquid regime, at small enough $g\/E_F$. Very close to a QCP, $\\lambda$ necessarily becomes\n large,\n and the whole Fermi liquid range $\\omega < {\\omega_{\\text{FL}}}$ falls into $\\omega < \\omega_{\\text{IN}} \\sim \\lambda^3 {\\omega_{\\text{FL}}}$.  The pairing instability in this case emerges at a \n   larger $\\omega \\leq \\omega_{\\text{IN}}$ (Refs. \\cite{PhysRevLett.77.3009,Moon2010,PhysRevB.91.115111,CHUBUKOV2020168142}), hence there is no range of applicability for our analysis.\n\n\n\n\\section{Self-energy in the superconducting state}\n\\label{scscatesec}\n\nWe now show that the singularity of the self-energy at $\\ell=1$ gets regularized once we take into consideration the fact that the ground state is actually an $s$-wave superconductor.\n In practical terms, superconductivity implies that for the calculation of the backscattering amplitude and the fermionic self-energy one has to use normal and anomalous Green's functions\n\\begin{align}\n\\label{Gmeanfieldmain}\nG(\\omega_m, {\\boldsymbol{k}}) &= -\n \\frac{i Z^{-1}\\omega_m + \\xi_{\\boldsymbol{k}}}{Z^{-2}\\omega^2_m + \\xi_{\\boldsymbol{k}}^2 + \\Delta_0^2}\\ ,  \\\\ \\notag    F(\\omega_m, {\\boldsymbol{k}}) &=  \\frac{\\Delta_0}{Z^{-2} \\omega^2_m + \\xi_{\\boldsymbol{k}}^2 + \\Delta_0^2} \\ .\n  \\end{align}\n\nThe  three key processes that determine the decay of a quasiparticle in a superconductor are \n(i)\n scattering by amplitude and phase fluctuations of the  superconducting order parameter,  \n (ii) scattering by a (potential) resonance mode in the nematic $D(\\omega, {{\\boldsymbol{q}}})$ at (real) $\\omega = \\omega_{\\text{res}} <3\\Delta_0$  and (iii) scattering due to a decay into particles and holes (this process starts at $\\omega =3\\Delta_0$).\n For a charged superconductor, scattering by phase fluctuations is affected by long-range Coulomb interaction, which converts a Goldstone phase fluctuation mode into a plasmon (still gapless in 2D).  However, we neglected the effects of long-range Coulomb interaction in the calculations assuming a normal state at $T=0$, and for consistency we also have to neglect this interaction in a superconductor.\n   \nIn general,  all three processes are important at $\\omega_m \\gtrsim \\Delta_0$,  and the full-fledged\n evaluation of $\\Sigma(\\omega_m \\gtrsim \\Delta_0)$\n is a complicated endeavor. \nSince our primary goal is to understand how superconductivity affects \n singular behavior of the self-energy that we found in the previous section, we limit ourselves to the scattering processes which cut off the singularity. \n \n \n We will directly focus on the imaginary part of the backscattering self-energy  in real frequencies, \n  $\\text{Im}[\\Sigma_\\text{bs;a}^R(\\omega)]$.  Consider first the contribution to  $\\text{Im}[\\Sigma_\\text{bs;a}^R(\\omega)]$ arising from the sum of all angular momentum channels,\n  Eq.~\\eqref{sigmaattleading}. \n  This expression shows a slope singularity\n   at $\\ell =1$, however,\n     since it was obtained with the logarithmic accuracy, it is only valid for $ \\ell \n     \\lesssim 1- \\tilde{\\lambda}$.  At the boundary of this regime, the self-energy \\eqref{sigmaattleading} is of the same order as\nthe one-loop result, Eq.\\ \\eqref{a_1}. Therefore, to estimate this part of the self-energy\n  at $1-\\ell <\\tilde{\\lambda}$,  it is sufficient to re-evaluate the one-loop diagram of Fig.\\ \\ref{diagtable}(b) in the superconducting state.\n     We find (see App.\\ \\ref{Imsigma1app} for details)\n\\begin{align}\\label{Imsigma3delta}\n&-\\text{Im}\\left[\\Sigma_\\text{bs;a}^{R}(\\omega)\\right] \\sim  \\\\ &   \\frac{\\Delta_0^2} {{\\omega_{\\text{FL}}}} \\tilde{\\lambda} \\ln\\left( \\frac{{\\omega_{\\text{FL}}}}{\\Delta_0}\\right) \\times \\sqrt{ \\frac{\\omega - 3\\Delta_0}{ \\Delta_0}} \\theta(\\omega - 3\\Delta_0) \\ .  \\notag\n\\end{align}  \nWe have suppressed the dependence on $Z \\simeq 1$. The scattering rate of Eq.\\ \\eqref{Imsigma3delta} starts at $\\omega = 3\\Delta_0$, which is transparent as the three scattering products, two quasiparticles and a quasihole, all have an energy gap $\\Delta_0$. At $\\omega - 3\\Delta_0 \\simeq \\Delta_0$, i.e., at  $1-\\ell \\sim \\tilde{\\lambda}$,  Eq.\\ \\eqref{Imsigma3delta} becomes of the same order as the normal state result, Eq.~\\eqref{sigmaattleading}.  \nWe see therefore that the portion of the self-energy that comes from all scattering channels smoothly evolves from\n Eq.~\\eqref{sigmaattleading} to Eq.~(\\ref{Imsigma3delta}).   We also computed the contribution to the self-energy \n  from the resonance in $D(\\omega, {\\boldsymbol{q}})$ in the superconducting state.  The resonance develops at $\\omega_{\\text{res}} = \\Delta_0 \\times \\left[1 + \\mathcal{O}(\\epsilon k_F\/(N \\rho g)^{1\/2})\\right]$.   Adding this contribution, we find that $\\text{Im}[\\Sigma_{\\text{bs;a}}^{R}(\\omega)]$ extends down to $\\omega_{\\text{res}}$, but near $\\omega \\sim 3\\Delta_0$ it is given by \n   Eq.\\ (\\ref{Imsigma3delta}) up to small corrections.   The form of  $\\text{Im}[\\Sigma_{\\text{bs;a}}^{R}(\\omega)]$ at $\\omega_{\\text{res}} \\leq \\omega \\leq 3\\Delta_0$ is rather involved and we will discuss it in a separate report. Ultimately, the complicated form of the self-energy is tied to the abundance of low-energy collective excitations in a superconductor with long-ranged interactions, as also recognized in Refs.\\ \\cite{PhysRevB.62.11778, KleinNpj2019}. \n \nTo find the correct cutoff of the $s$-wave pole, Eq.\\ \\eqref{sigmaattsubleading_1}, evaluation of the one-loop diagram is insufficient, and one \n needs to re-evaluate the  full backscattering amplitude $\\Gamma_{\\text{bs}}$ using  $\\Pi_\\text{pp}= GG + FF$. \n    The key new effect coming from this calculation is the scattering of a quasiparticle \n    by  massless (Goldstone) phase fluctuations of the superconducting order parameter. Evaluating the $s$-wave component of $\\Gamma_{\\text{bs}}$ and substituting the result into the self-energy, we obtain\n\\begin{align}\n\\label{Imsigmas}\n-\n \\text{Im}\\left[\\Sigma_\\text{bs;a}^{R}(\\omega)\\right] \\sim \n  \\frac{\\epsilon^2}{\\tilde{\\lambda}} \\frac{\\Delta_0^2}{{\\omega_{\\text{FL}}}}  \\times \\theta(\\omega - \\Delta_0)\\  .\n\\end{align}\nThis contribution to the self-energy starts at $\\omega = \\Delta_0$, since the phase mode is gapless. \n  For  $1- \\ell \\simeq \\tilde{\\lambda}$, i.e., $\\omega \\gtrsim \\Delta_0$, this self-energy \nmatches  with the $s$-wave piece  $\\omega^2\/{\\omega_{\\text{FL}}} \\times \\epsilon^2\/(1- \\ell)$, obtained in the normal state calculation,  Eq.\\ \\eqref{sigmaattsubleading_1}.\n  A similar expression for the self-energy at $T=0$  in a 3D superfluid has recently been computed in Ref.~\\cite{PhysRevLett.124.073404}, building on older work of Ref.\\ \\cite{haussmann1993crossover}. The contribution from the self-energy to the density of states of an $s$-wave superconductor has been obtained in Refs.\n  \\ \\cite{PhysRevB.42.10211, Larkin2008}.\n       \n\n\nCollecting  the results, we find the  full self-energy from backscattering in the form \n\\begin{widetext}\n\\begin{align} \\label{ImsigmafinalresultN}\n&0< \\  \\ell \\lesssim 1- \\mathcal{O}(\\tilde{\\lambda}): \\quad &&\\text{Im}\\left[\\Sigma_{\\text{bs;a}}^R(\\omega)\\right] =\n-\\frac{\\omega^2}{\\pi {\\omega_{\\text{FL}}}}   f^*_\\text{bs;a}(\\ell),  \\quad\n f^*_\\text{bs;a}(\\ell) \\simeq  \\frac{ (\\ell - 1)\\ln^2(1-\\ell)}{\\ell} + 2\\text{Li}_{2}(\\ell) + \\frac{\\epsilon^2}{4(1-\\ell)}, \\\\ &\n1- \\mathcal{O}(\\tilde{\\lambda})\\lesssim \\  \\ell < 1: \\quad &&\\text{Im}\\left[\\Sigma_{\\text{bs;a}}^R(\\omega)\\right] =  -  \\frac{\\Delta_0^2} {{\\omega_{\\text{FL}}}} \\tilde{\\lambda} \\ln\\left( \\frac{{\\omega_{\\text{FL}}}}{\\Delta_0}\\right) \\times \\sqrt{ \\frac{\\omega - 3\\Delta_0}{ \\Delta_0}} \\theta(\\omega - 3\\Delta_0) -   \\frac{\\epsilon^2}{\\tilde{\\lambda}} \\frac{\\Delta_0^2}{{\\omega_{\\text{FL}}}}  \\times \\theta(\\omega - \\Delta_0) .\n\\label{ImsigmafinalresultS}\n\\end{align}\n\\end{widetext}\n\n\nWe sketch $[\\Sigma_{\\text{bs;a}}^R(\\omega)]\/\\omega^2$ in Fig.\\ \\ref{interpolfig}. In panel (a) we set $\\epsilon < \\tilde{\\lambda}$.  In this situation,  the contribution from all angular channels dominates over the $s$-wave part  for all $\\ell$. In panel (b) we set  $\\epsilon > \\sqrt{\\tilde{\\lambda}}$. Here the contribution from the $s$-wave channel is the dominant one. We see that in this situation   Im$[\\Sigma_{\\text{bs;a}}^R(\\omega)]\/\\omega^2$ has a sharp peak at $\\ell \\leq 1$. This can be directly verified in ARPES experiments.\n\nNote that while  \n the imaginary part of the self-energy in real frequencies\n  %\n   vanishes at $|\\omega| < \\Delta_0$, the Matsubara self-energy (which can be obtained from Quantum Monte Carlo) is non-zero.\n     At $\\omega_m \\ll \\Delta_0$, $\\Sigma_{\\text{bs;a}}(\\omega_m)$ scales as \n    \n      $\\omega_m\/\\Delta_0$. At larger $\\omega_m \\sim \\Delta_0$, $\\Sigma_{\\text{bs;a}}(\\omega_m)$ passes through a maximum.\n     \n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=.95\\textwidth]{fig10.pdf}\\caption{Backscattering self-energy divided by $\\omega^2$ as a function of the logarithmic variable $\\ell$. Blue curves are the normal state result (Eq.\\ \\eqref{ImsigmafinalresultN} and Fig.\\ \\ref{fattractiveplot}), green curves correspond to the superconducting result, Eq.\\ \\eqref{ImsigmafinalresultS}.  The black line is a possible interpolation representing the correct physical backscattering rate. (a) Sum of angular momentum channels dominates, used parameters: $\\epsilon = 0.05,  \\tilde{\\lambda} = 0.1$. (b): $s$-wave channel dominates, used parameters: $\\epsilon = 0.3, \\tilde{\\lambda} = 0.05$.  }\n\\label{interpolfig}\n\\end{figure*}\n\n\n\n\n\\section{Conclusions}\n\\label{concsec}\n\nIn this work we computed the fermionic self-energy in a Fermi liquid near a nematic QCP.  We considered  the \n model of fermions interacting with soft fluctuations of a nematic order parameter, and extended it to $N$ fermionic flavors. The leading contributions to the self-energy at $N \\to \\infty$ come from planar diagrams, as was determined before at a QCP. In a Fermi liquid, the contributions from planar diagrams contain series of logarithms. We identified the \nleading logarithmic  contributions at all orders as coming from the fully dressed backscattering amplitude, i.e., \nthey describe one-dimensional scattering  processes. We further argued that the subleading contributions (the terms with a smaller power of $\\ln(\\omega)$ at each loop order) come from scattering with a finite momentum transfer counted from either backscattering or forward scattering.\n \nWe found that the logarithmic series are non-geometrical and summed them up using several computational tricks. \nWe first considered  a toy model with repulsive pairing interaction, and then a realistic model with attractive pairing interaction. For the latter case we found two contributions to the self-energ\n: one is the combined contribution from multiple pairing channels with near-equal attraction, and the second is the special contribution from an $s$-wave pairing channel, where the pairing interaction is a bit larger than in other channels. We first computed the two terms assuming the normal state and found that perturbation theory holds only above a certain frequency, comparable to superconducting gap $\\Delta_0$.  At the edge, both contributions become non-analytic and the one from $s$-wave channel diverges. We then extended the analysis to include superconductivity at $T=0$ and showed that the  would-be divergencies get regularized. In particular, the  divergence in the  $s$-wave channel is  regularized by scattering of a fermion by phase fluctuations of the superconducting order parameter. We obtained the full expression for the imaginary part of the retarded self-energy in real frequencies (the scattering rate) at $\\omega \\sim \\Delta_0$  and argued that, depending on parameters, $\\text{Im}[\\Sigma^R(\\omega)]\/\\omega^2$ either evolves smoothly at $\\omega \\geq \\Delta_0$ or has a peak.  This peak can be detected in ARPES measurements.\n\nThe  unconventional behavior that we found  stems from the competition between  different pairing channels. In our case this holds near a nematic QCP, but similar behavior is expected when there is a competition between multiple pairing channels for other reasons, like, e.g., a competition between $s$- and $d$-wave channels in iron-based superconductors \n    \\cite{PhysRevLett.109.087001, PhysRevLett.107.117001, HOSONO2015399}. Adapting our computations to a \n model that only allows for finite number of pairing channels, or, e.g., only selects pairing channels with  even angular momentum yielding singlet superconductivity could be an interesting future project. In addition, a recent Monte Carlo study of a quantum critical metal \\cite{grossman2020specific} observed a strong impact of pairing fluctuation on thermodynamic observables (e.g., specific heat), and it would be worthwhile to elucidate how the competition between different pairing  channels shows up in thermodynamics.\n \n{As a final remark, we emphasize that our analysis has been restricted to the isotropic case. On a lattice, the nematic form factor $f$ (which we set to one) becomes important, separating hot (where $f \\simeq 1$) and cold ($f \\ll 1$) regions on the Fermi surface. In the normal state $(\\omega \\gg \\Delta)$, our results carry over to the hot regions. Deep in the superconducting state ($\\omega < \\Delta$), the gap function will be maximal in the hot regions, and vanish in the cold ones, similar to Ref.\\ \\cite{PhysRevB.98.220501}. Accordingly, in this limit our results for the scattering rate have to be modified to account for the near-gapless regions. }\n\n\n\\section{Acknowledgments}\n\nWe thank  E.~Berg, M.~Hecker, Y.~B.~Kim, A.~Klein, S.-S. Lee and  Y.~Schattner for useful discussions. D.P.\\ is grateful to the Max-Planck-Institute for the Physics of Complex Systems Dresden (MPIPKS) for hospitality during the initial stage of this project. The work by A.V.C.\\ was supported by the Office of Basic Energy Sciences, U.S. Department of Energy, under award  DE-SC0014402. A.K. was supported by NSF grant DMR-2037654.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section*{List of the acronyms}\nWe introduce here the acronyms used throughout the document:\n\\begin{itemize}\n    \\setlength{\\itemsep}{0pt}%\n    \\setlength{\\parskip}{0pt}%\n\\item AD: Avalanche Diode\n\\item B\\&L: Box-and-Line dynode\n\\item BSM: Beyond the Standard Model\n\\item CC: charged currents\n\\item CCSNe: Core-Collapse Supernovae \n\\item CCQE: charge current quasi-elastic\n\\item CE: Collection Efficiency\n\\item CPL: Concrete Protective Liner\n\\item DAQ: Data Acquisition\n\\item DR: Design Report\n\\item DT: deuterium-tritium\n\\item EBU: Event Building Unit\n\\item ECal: ND280 Electromagnetic Calorimeter\n\\item FC: Fully Contained\n\\item FCFV: Fully Contained in Fiducial Volume \n\\item FGD: Fine Grained Detector\n\\item FRP: Fiber Reinforced Plastics\n\\item FV: Fiducial Volume\n\\item GUT: Grand Unified Theory\n\\item HDPE: High Density PolyEthylene\n\\item HK: Hyper-Kamiokande\n\\item HPD: Hybrid Photodetector\n\\item HPTPC: High Pressure Time Projection Chamber\n\\item HQE: High Quantum Efficiency\n\\item Hyper-K: Hyper-Kamiokande\n\\item IBC: International Board Representatives \n\\item IBD: Inverse Beta Decay\n\\item ID: Inner Detector\n\\item INGRID: Interactive Neutrino GRID\n\\item ISC: International Steering Committee\n\\item IWCD: Intermediate Water Cherenkov Detector\n\\item LAPPD: Large Area Picosecond PhotoDetector \n\\item LBNE: Long Baseline Neutrino Experiment\n\\item LAr: Liquid Argon calorimeter\n\\item LD: Laser Diode\n\\item LLDPE:Linear Low-Density PolyEthylene\n\\item LV: Lorentz Violation\n\\item MC: Monte Carlo\n\\item MLF: Material Science Facility\n\\item mPMT: Multi-channel Optical Module\n\\item MR: Main Ring synchrotron\n\\item NC: neutral currents\n\\item ND280: Near Detector 280m\n\\item NF: Nano Filter\n\\item OD: Outer Detector\n\\item PC: Partially Contained\n\\item PE: Photo Electron\n\\item PS: Power Supply\n\\item PTF: Photosensor Testing Facility\n\\item MH: neutrino mass hierarchy\n\\item QA: quality assurance\n\\item RBU: Readout Buffer Unit\n\\item RO: Reverse Osmosis\n\\item RCS: Rapid Cycling Synchrotron\n\\item SK: Super-Kamiokande\n\\item SM: Standard Model\n\\item Super-K: Super-Kamiokande\n\\item SUS: Stainless Steel (or Steel Use Stainless)\n\\item TITUS: Tokai Intermediate Tank for Unoscillated Spectrum\n\\item TPU: Trigger Processing Unit\n\\item TS: Target Station\n\\item UF: Ultra Filter\n\\item UPW: Ultra Purified Water\n\\item WAGASCI: Water Grid And SCIntillator detector\n\\item WC: Water Cherenkov\n\\end{itemize}\n\n\n\\subsection{Immersion test}\nSpecimens of HDPE lining sheet (GSE Gundle sheet, whose material is\nidentical to that for the CPL), with artificial extrusion-welded seam,\nwere immersed into the ultra-purified water (UPW) for certain periods\n(1, 2 to 7, and 8 to 31 days), and absorbance was compared to a\ncontrol sample without specimens. Amount of eluted materials into UPW,\ni.e. total organic carbon (TOC), anions and metals, were also\nmeasured.\nFigure \\ref{fig:liner-immersion} show the specimen and an example of\nmeasurements, where increase of the light absorbance were observed\nbetween the wavelength range of 200$\\sim$300\\,nm. Some amount of\nmaterial elution were observed, where eluted amounts per unit area and\ntime were significantly less for later periods. Although relation\nbetween the increase of light absorbance and the material elusion\nshould be studied, it is noted that range of PMT-sensitive wavelength\nis somewhat higher (300$\\sim$650\\,nm), so the effect to the\nexperiment may be limited. Similar results were obtained for\nGadolinium sulfate solutions.\n\n\\subsection{Measurements on material strength}\nTension tests were carried out for the CPL to estimate yield strength,\ntensile strength, and Young's modulus. Since HDPE has large elongation\nbefore breaking ($\\sim$500\\%), 1.0\\% proof stress was used as the\nyield strength (instead of 0.2\\% proof stress which is generally used\nfor other materials). Here, varying measurement conditions were\nexamined for tensioning velocity (0.05\\,mm\/min and 0.5\\,mm\/min) and for temperature (typical room temperature 23.5$^\\circ$\nCelsius and 15$^\\circ$ Celsius simulating water temperature).\nIt was found that measured yield strengths were smaller by a few to\nseveral tens of percent than the specification value (15.2\\,MPa as\nlisted in Table~\\ref{tab:liner-spec}). In general, lower tensioning\nspeed gives lower strength, due to large plasticity of HDPE. Strength\nalso depends significantly on temperature: HDPE becomes harder with\nlowering temperature, as common properties in high-polymer\nmaterials. The measurement in 15$^\\circ$ Celsius gave about 20\\%\nhigher strengths than those at 23.5$^\\circ$ Celsius.\nThe tension tests were also repeated on the samples with an\nextrusion-welded seam. For the most cases, none of peeling, fracture,\nnor other troubles were observed on welded seams, but the deformation\noccurred at the base material. The yield\/tensile strength were almost\nidentical to the values for base material.\n\n\\subsection{Creep test}\nCreep tests were performed with various tensile loads: 1\/4$\\times$M,\n3\/8$\\times$M, or 1\/2$\\times$M, where M = 18.2\\,MPa is the observed\nyield strength as 1.0\\% proof stress with tensioning speed of 5.0\\,cm\/min in 20$^\\circ$ Celsius. For the tests with load of 1\/4$\\times$M\n({\\it i.e.} 4.6 MPa), creep was not observed for about 30 days. Meanwhile, for tests with load of 3\/8$\\times$M (6.8\\,MPa) and\n1\/2$\\times$M (9.1\\,MPa), clear generation of creep was observed with 5\nand 10 days, respectively.\n\n\\subsection{Resistivity to localized water pressure}\n\\begin {figure}[htbp]\n  \\begin{center}\n    \\includegraphics[width=0.8\\textwidth]{liner-pressure-test.pdf}\n\\caption{Setup for the test applying localized water pressure to the lining material.}\n    \\label{fig:liner-pressure}\n  \\end{center}\n\\end {figure}\nIdeally, the CPL sheet should be closely attached to the surface of\nflatly-backfilled concrete walls and thus firmly supported by\nthem. However, it is probable that cracks or rough holes exist or\nhappen in the backfill concrete wall, and thus the liner should\nlocally stand for water pressure by itself. To simulate the situation,\ntests to apply localized water pressure for the lining were performed\nwith variety of slit widths (2 to 8\\,mm) and hole diameters (40\nto 120\\,mm-$\\phi$), as illustrated in\nFig.~\\ref{fig:liner-pressure}. It is found that the liner survived for\n0.8\\,MPa water pressure without breaking for all cases. Although the\ntest load was applied only in a short period and durability for longer\nperiod should be examined, it is probable that the expected water\npressure is enough lower than the critical pressure causing creep.\n\n\\subsection{Water permeability}\nGenerally, plastic materials have a property to pass water as moisture\nvapor through molecules. It is referred as moisture permeability, or\nwater vapor transmission rate (WVTR), being represented with\ntransferred mass through unit area and time (g\/m$^2$\/24-hours). The\npermeability was studied for the GSE geomembrane (Gundle sheet, whose\nmaterial is quite simmilar to that for CPL\nStudliner)~\\cite{liner:JAEA-Tech-2013-036}. For the sheet with\nthickness $t$= 1.5\\,mm, WVTR ($P_{a1}$) was obtained to be\n1\\,g\/m$^2$\/24-hours at most for standard testing temperature\n(40$^\\circ$\\,Celsius). The water permeability coefficient ($k$) was\nthen deduced to be\n\\[\nk = P_{a1} \\times t \\times \\frac{g}{\\Delta P_v} =  2.5 \\times 10^{-12} cm\/s,\n\\]\nwhere $g$ is standard gravitational acceleration (9.8\\,m$^2$\/s) and\n$\\Delta P_v$ is difference of the water-vapor pressures of both sides\nof the geomembrane (90\\% of saturated vapor pressure at\n40$^\\circ$\\,Celsius, 75.22\\,hPa). Since the CPL is 5\\,mm thick, time\nuntil water permeates to backside of the CPL is:\n\\[\n0.5 {\\rm [cm]} \\times \\frac{1}{2.5 \\times 10^{-12} {\\rm [cm\/s]}} = 2 \\times 10^{11} {\\rm [s]},\n\\]\n{\\it i.e.} about 6,300 years. Since the total inner surface area of\nthe tank is 18,259 (54,750)\\,m$^2$ for 1 (3) tank options, amount of\nwater permeation through the entire liner surface will be:\n\\begin{eqnarray}\n&~& 18,250 (54,750) \\times 10^4\\ {\\rm [cm^2]} \\times 2.5 \\times 10^{-12}\\ {\\rm [cm\/s]} \\nonumber \\\\\n&=& 4.6 \\times 10^{-4}(1.4 \\times 10^{-3}){\\rm [cm^3\/s]} = 40 (120) \\ {\\rm [cm^3\/day]}, \\nonumber\n\\end{eqnarray} \nthus being negligible amount. \n\n\n\\subsection{Penetration structure}\n\n\\begin {figure}[htbp]\n  \\begin{center}\n    \\includegraphics[width=0.8\\textwidth]{liner-penetration.pdf}\n\\caption{Schematic drawing for the penetration structure of a water pipe and \n         a photo of its prototype.}\n    \\label{fig:liner-penetration}\n  \\end{center}\n\\end {figure}\nThe leak can happen around the components which penetrate the water\ntank lining, such as anchors to support PMT framework columns, water\nsupply\/return pipes, and so on. A possible design of the penetration\nstructure for the water pipes is illustrated in\nFig.~\\ref{fig:liner-penetration}. A metal pipe with a flange is coated\ntogether with PE resin of about 1.5\\,mm thickness, which can be\nextrusion-welded to adjacent CPLs.  A prototype was made for the\ndesign as shown together in the figure, and tested with pressure up to\n1 MPa for 30 minutes. Tests with cyclic pressure (0.5\\,MPa, repeating\non\/off in a day for 5 days) and with continuous pressure (3 months\nwith applying 0.5\\,MPa were also performed. For all of the cases, no\nwater leak was observed.\n\n\n\\subsection{Rock Quality Information for Hakamagoshi and its Surroundings} \n\nThe Hakamagoshi area is characterized by Mount Hakamagoshi (elev. 1,159~m), Mount Mikata (elev. 1,142~m),\nand Mount Sarugayama (elev. 1,448~m), which form a mountain chain running from northeast to southwest. \nMountain streams along the north side of these peaks form part of the Oyabegawa river system, while \nsimilar streams on the southern side running to the east and south form the network of waterways feeding the Shogawa\nriver. \nIn short, the area around Hakamagoshi is the watershed for the Oyabegawa and Shogawa rivers, which is a \nstrong indicator of abundant underground water.\nIndeed, measurements have yielded spring water flows in excess of 100~tons per hour. \nHakamagoshi and its surroundings are formed from the \nprimary Futomiyama formation (from the Paloegene-Paleocene), sporadically covered with so-called Hida bedrock \n(a mixture of Shirakawa granite and Nohi ryolite). \nThe rock types in the Futomiyama formation as it is distributed throughout Hakamagoshi are primarily\nryholitic tuff, ryholitic welded tuff, and ryholitic lava. \nIn addition, quartz porphyry, porphyrite, dorelite and other rock intrusions of no known era are found \nwithin the formation. \nThe upper layers of the formation are inconsistently covered by Tori conglomerate from the Neogene-Miocene eras.\nIn particular this conglomerate is found above the Hakamagoshi tunnel at elevations around 900~m.\nThe upper most layers are composed of andesite and andesitic pyroclast from the same eras. \nOn site investigations of the lithic fragments from the Futomiyama formation indicate large amounts of CH$\\sim$CM class\nhardness rock.  \n\nFigure~\\ref{fig:hakamagoshi_layer} shows the rock quality distribution around Hakamagoshi in hardness classes.\nIn the lower layers of the Futomiyama formation Shogawa granodiorite is widespread, and in \naddition to this formation, at 1000~m depths other distributions of Shogawa granodiorite and \nKose diorite may be found.\nSince it is generally thought that diorite layers are higher quality in comparison to the Futomiyama formation,\nthe key to realizing the Hakamagoshi option lies in determining whether or not such layers exist at the site. \n\n\\begin {figure}[htbp]\n  \\begin{center}\n    \\includegraphics[width=0.8\\textwidth]{hakamagoshi_layer.pdf}\n\\caption{ Overhead and cross sectional views of the rock quality in the Hakamagoshi area appear in the upper and lower panels, respectively. \nThe dark pink color shows the diorite layer which is thought to extend to \nto areas where the rock overburden is 1000~m. \n}\n    \\label{fig:hakamagoshi_layer} \\end{center}\n\\end {figure}\n\n\n\\subsection{Physics sensitivities}\n\\subsubsection{Muon rate at Mt. Hakamagoshi} \n\nFor the moment we assume that a detector is placed directly beneath the peak of Mount Hakamagoshi at the same \nlevel as the expressway tunnel  and calculate the expected muon rate.\nIn order to study the reliability of these estimates the calculation is additionally performed for \nthe area inside of the tunnel and then compared with measurements made with a plastic scintillator-based detector.\nThe estimation uses 30~m elevation data from the ALOS database in the same way as calculations performed for the \nTochibora site. \nMUSIC and FLUKA are used for the muon simulation. \nWith a peak elevation of 1,159~m and a tunnel elevation of approximately 300~m, here the rock overburden is taken to be \nroughly 850~m. \nSimilarly the rock density is assumed to be the same as in Kamioka, 2.7~g\/$\\mbox{cm}^{3}$.\nFigure~\\ref{fig:hg_mu_rate} shows zenith, azimuthal, and energy distributions from the simulation.\nComparing to the Tochibora site, the muon rate at the Hakamagoshi site is roughly a factor of two times smaller \nand the mean muon energy is reduced by about 10\\%.\nThe following sensitivity studies have been performed based on these results.\n\\begin {figure}[htbp]\n  \\begin{center}\n    \\includegraphics[width=0.8\\textwidth]{hakamagoshi_mu.pdf}\n\\caption{ Muon zenith (left top) and azimuthal (left bottom) angular distributions as well as the energy distributions (right) for the Hakamagoshi site assuming an 850~m rock overburden are shown in thick red lines. Results for the Hyper-K Tochibora and \nMozumi sites, as well as the Super-K site are shown in thin red lines. \nBlue lines show measured values. \n}\n    \\label{fig:hg_mu_rate} \\end{center}\n\\end {figure}\n\n\nTo validate the results of these simulations the muon rate inside the Hakamagoshi tunnel was measured \nin cooperation with NEXCO Central Nippon in an air conditioning chamber from January 18-20, 2016. \nThis chamber is not located beneath the mountain peak and has an overburden of only 450~m.\nTwo plastic scintillators of dimension $1000\\times200\\times45\\mbox{mm}^{3}$ and separated by \n100~mm were used for coincident muon identification. \nFor comparison the same measurement was repeated at the Super-Kamiokande experimental site.\nUsing this apparatus the muon rates were measured to be $1.9\\pm 0.1 \\times 10^{-3}$~Hz \nand $1.9\\pm 0.2\\times 10^{-4}$~Hz at the Hakamagoshi and Super-K sites, respectively.\nThat is, the air conditioning chamber's muon rate is $10.0\\pm1.1$ times larger.\nBased on the simulations outlined above the expected difference is a factor of 8.9, indicating \nthat the data and simulation are consistent within errors.\nFor this reason the simulation of the Hakamagoshi site below the mountain peak is taken to be reasonable.\n\n\n\\subsubsection{Neutrino beam from J-PARC} \n\nIn the following the Hakamakoshi detector is assumed to be beneath the peak of the mountain, as in the simulations above, \nand the sensitivity neutrino oscillations using the J-PARC neutrino beam is estimated.\nHakamagoshi has an off-axis angle of $2.4^\\circ$ and a baseline of 335~km.\nThe corresponding flux and $\\nu_{\\mu} \\rightarrow \\nu_{e}$ oscillation probabilities are shown in Figure~\\ref{fig:hg_flux_osc}.\nSince the off-axis angle is sightly smaller than that for the Tochibora site, the flux peak has shifted to slightly \nhigher energies, but since the baseline is correspondingly larger the oscillation maximum is found at nearly the same energy.\nUsing this information the sensitivity to CP violation has been estimated as shown in Figure~\\ref{fig:hg_cpv}.\nHere the first tank is assumed to be located in Tochibora with the second tank beginning operations six years later.\nA comparison is made between a second tank in Tochibora and one in Hakamagoshi.\nDue to the larger baseline to Hakamagoshi there is a corresponding decrease in event rate, resulting in a slightly larger \nstatistical error, though the systematic errors are taken to be the same.\n\\begin {figure}[htbp]\n  \\begin{center}\n    \\includegraphics[width=0.8\\textwidth]{hakamagoshi_beamflux.pdf}\n\\caption{ The neutrino flux at the Tochibora (TB) and Hakamagoshi (HG) sites from the J-PARC beam is\nshown in the left figure.\nIn the right figure the electron neutrino appearance probability for both sites is shown \nas a function of energy.}\n    \\label{fig:hg_flux_osc} \\end{center}\n\\end {figure}\n\n\\begin {figure}[htbp]\n  \\begin{center}\n    \\includegraphics[width=0.8\\textwidth]{hakamagoshi_beamsens.pdf}\n\\caption{ Sensitivity to CP violation as a function of the true value of $\\delta_{cp}$ and for \nvarious assumed values of $\\mbox{sin}^{2}\\theta_{23}$.  \nBoth plots assume a single detector operating for five years with a second detector \nbeginning operations in the sixth year. \nThe left (right) plot shows the result assuming the second detector is placed in Hakamagoshi (Tochibora).\nSolid lines show the sensitivity assuming only statistical errors and dashed lines \ninclude both systematic and statistical uncertainties.\n}\n    \\label{fig:hg_cpv} \\end{center}\n\\end {figure}\n\n\n\n\n\\subsubsection{Low Energy Neutrino Observations} \n\nSpallation products from cosmic ray muons form the main background to low energy neutrino physics in Hyper-K, including \nsupernova and solar neutrino measurements.\nSince the muon background at Hakamagoshi is lower than that at Tochibora, the former is expected to have improved sensitivity \nto these neutrinos.\nUsing the muon simulation results presented above to derive the spallation backgrounds, Hyper-K's sensitivity \nto the observation of supernova relic neutrinos has been estimated.\nIn this analysis an analysis sample selected using neutron tagging (70\\% tagging efficiency) \nto identify the inverse beta decay signal in an energy window of 16 to 30~MeV has been assumed.\nFigure~\\ref{fig:hg_srn} shows the expected sensitivity assuming only one tank at at the Hakamagoshi site.\nAssuming a standard model of the relic neutrino flux and the first detector in Tochibora, the signal can \nbe observed with $3\\sigma$ ($5\\sigma$) significance after 4 (11)~years.  \n\n\\begin {figure}[htbp]\n  \\begin{center}\n    \\includegraphics[width=0.8\\textwidth]{hakamagoshi_srn.pdf}\n\\caption{ Sensitivity to supernova relic neutrinos as a function of operation time.\nThe left figure shows the number of relic neutrino candidate events and \nthe right figure shows the ability to discern this flux from the \nbackground in units of $\\sigma$.\n}\n    \\label{fig:hg_srn} \\end{center}\n\\end {figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\n  \n\n\n\n\n\n\n\\section{Background rate estimation} \\label{section:background}\n\n\\subsection{Background rate estimation for low energy neutrino study \\label{sec:lowe_bg}}\n\nIn this subsection, we will show the background rate estimation used\nfor the study of low energy neutrinos, such as solar neutrinos,\nsupernova neutrinos, and relic-supernova neutrinos. The most important\nbackground sources are the radioactive isotope of $^{222}$Rn contained\nin water, and radioactive spallation products created by cosmic-ray\nmuons. Other important radioactive isotopes\nare U and Th in the water, and $^{40}$K in the PMT glass; we need to\nsuppress the concentration of those isotopes to the similar level in\nSuper-K as well. On the other hand, the major background sources for\nthe anti-neutrino measurement is anti-neutrino backgrounds from\nnuclear power reactors, which limit the energy threshold at around\n10\\,MeV; the smaller contribution comes from the radioactive\nbackground of $(\\alpha, n)$ reactions. In the background calculation,\nthe most complicated task is the estimation of the muon spallation\nproductions and its background reduction by the analyses, because it\ndepends on the detector location and the detector performance. In the\nfollowing paragraphs, after discussing radon backgrounds,\nwe focus on the discussion of the muon spallation backgrounds.\n\n\\subsubsection{Radon background}\n\nRadon ($^{222}$Rn) is a radioactive noble gas, with a half-life of 3.8 days.  \n$^{222}$Rn occurs as a daughter nuclide in the $^{238}$U decay scheme,\nvia the decay of $^{226}$Ra ($\\tau_{1\/2} = 1599$ years).\nSmall but finite quantities of $^{226}$Ra exist in all materials and\ntherefore, every material can produce $^{222}$Rn.\nAs a gaseous isotope, $^{222}$Rn can easily escape from materials used\nin the construction of Hyper-K and the radioactivity content\nof construction materials must be carefully screened.\nThe decay of $^{222}$Rn produces several daughter isotopes,\nmost of which are not sufficiently energetic to produce Cherenkov\nlight in the Hyper-K detector.  \nThe most serious background for solar neutrino measurements\nis the radon daughter bismuth-214 ($^{214}$Bi)\nwhich decays via beta emission with a Q-value of 3.27 MeV.\nThis limits the energy threshold of the solar neutrino\nmeasurements in which a neutrino-electron elastic scattering reaction\nis used.\n\nIn the same energy region, $^{208}$Tl in thorium series could become\nanother serious source of the background.\nHowever, from the results of radon assay with special radon\ndetectors~\\cite{70Lradon,700Lradon} in Super-K, \nthe contamination of the radon in thorium series ($^{220}$Rn)\nlooks much smaller than that of $^{222}$Rn in Super-K water.\nSo, we discuss only $^{222}$Rn in this section. \n\nTypical radon concentrations in Mozumi mine and Super-K are\nsummarized in Table~\\ref{tab:radon_concentrations}.\n\\begin{table}[htbp]\n \\caption{\\label{tab:radon_concentrations}\n Typical radon concentrations in Mozumi mine and in\n Super-K.}\n\\begin{ruledtabular}\n\\begin{tabular}{lc}\nDetector site & Radon concentration \\\\\n\\hline\nMine air in Mozumi~\\cite{Takeuchi:1999}                      & $\\sim1200$ [Bq\/m$^3$]\\\\\nSK-I inner detector water (upper half)~\\cite{Takeuchi:1999}  & $<1.4$ [mBq\/m$^3$]\\\\\nSK-I inner detector water (bottom)~\\cite{Takeuchi:1999}      & $3 \\sim 5$ [mBq\/m$^3$]\\\\\nSK-IV supply water            & $< \\sim 1$ [mBq\/m$^3$]\\\\\nSK-IV return water            & $8 \\sim 10$ [mBq\/m$^3$]\\\\\n\\hline\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\nThe SK-IV return water is taken at $\\sim 18$ ton\/hour from inner detector\nand at $\\sim 42$ ton\/hour from outer detector~\\cite{Abe:2013gga}.\nSo, we think the radon concentration in the SK-IV outer detector\nis close to (or larger than) that in the SK-IV return water.\nFrom the event rate comparison between SK-IV and SK-I, the radon concentration\nin SK-IV inner detector is similar or less than that of SK-I.\nTherefore, the radon concentration would be different by several factors\nbetween inner and outer detectors in SK-IV.\nHowever, in Super-K detector, water flow is controlled well by temperature and flow\nrate balances~\\cite{Abe:2013gga}, and then the water condition in inner\ndetector has been stable.\nFigure~\\ref{fig:sk4-vertex} shows a vertex distribution of the final\ndata sample of the solar neutrino analysis in SK-IV.\n\\begin {figure}[htbp]\n  \\begin{center}\n   \\includegraphics[width=1.00\\textwidth]{background\/figs\/sk4-vertex.pdf}\n   \\caption{\n   Vertex distribution of the SK-IV 2645-day final data sample in\n   solar neutrino analysis in different electron kinetic energy regions.\n   The black lines indicate fiducial volume region in each energy range.\n   Whole area corresponds to 22.5 kton volume,\n   and the fiducial volume above 5.0 MeV is 22.5 kton.\n   The color shows the event rate (\/day\/bin). The R and Z correspond to\n   the detector horizontal and vertical axis, respectively.}\n   \\label{fig:sk4-vertex}\n  \\end{center}\n\\end {figure}\nThe remaining event rate in the central detector is kept low\nwhile the barrel and bottom of the detector are higher than that.\nThis shows the water flow control in Super-K detector works well.\nFor the solar neutrino measurements, a threshold of 4.5 MeV (electron\nkinetic energy) in 22.5 kton fiducial volume was achieved in Super-K I.\n\nIn this design report, we estimate the radon concentration in\nthe Hyper-K tank water would be about 1.6\\,${\\rm mBq}\/{\\rm m}^{3}$\nin average in whole detector, as described in section~\\ref{sec:radon-in-water}. \nWhen applying the same water flow control technique as Super-K,\nthe radon concentration in the central Hyper-K detector will be reduced.\n\nMoreover, we could suppress $^{214}$Bi events thanks to better\nphoton yield in Hyper-K. \nFigure~\\ref{fig:bi214} shows expected observed energy spectrum of\nbeta decay of $^{214}$Bi in Super-K and Hyper-K.\n\\begin {figure}[htbp]\n  \\begin{center}\n   \\includegraphics[width=0.70\\textwidth]{background\/figs\/bi214-kin.pdf}\n   \\caption{An estimation of expected Bi-214 energy spectrum.\n   The black, red and blue lines are the original beta decay spectrum,\n   the expected observed spectrum in SK-IV,\n   and expected observed spectrum in Hyper-K, respectively.\n   The black histogram is observed event rate of the SK-IV final data\n   sample in 22.5 kton fiducial volume.\n   The vertical axis is event rate in arbitrary unit.}\n   \\label{fig:bi214}\n  \\end{center}\n\\end {figure}\nAt 4.5 MeV (electron kinetic energy), a suppression about factor 10 is expected\nin Hyper-K detector comparing to Super-K detector. \nActually, we have observed an expected energy resolution difference\ndue to photon yield change between SK-I and SK-II.\n\nAnother possible difference among Super-K and Hyper-K detectors related\nto radon background would be the lining material of the detector.\nAs discussed in section~\\ref{section:tank-liner}, 5 mm thickness HDPE\nwill be used to line the Hyper-K detector, and radon could permeate a HDPE sheet.\nTypical radon permeability through a HDPE sheet \nis reported by various groups as $O(10^{-8}) \\sim O(10^{-7})$\ncm$^2$\/s~\\cite{radon1,radon2,radon3,radon4,radon5}. \n\nAs shown in Table~\\ref{tab:radon_concentrations}, the typical radon\nconcentrations are different by about 5 order of magnitude\nbetween mine air and Super-K OD water.\nFor Hyper-K, radon concentration in OD water through the\nradon permeation from outside detector into Hyper-K detector\ncould be estimated as $O(10)$ mBq\/m$^3$ in Hyper-K OD water\nunder several assumptions, like\n(1) a radon permeability of the Hyper-K HDPE sheet is $10^{-8}$ cm$^2$\/s,\n(2) radon concentration in mine water (spring water in the mine) is $10^{3}$ Bq\/m$^3$,\n(3) there is no water flow between Hyper-K ID and Hyper-K OD, and\n(4) the volumes of mine water and Hyper-K OD water contributing to this effect are similar.\nThis estimation gives a similar radon concentration in Super-K OD detector,\nthough the uncertainty is large.\n\nIn order to reduce the uncertainty of (1) in this estimation, we are planning to\nmeasure radon permeability of a Hyper-K HDPE sheet.\nFigure~\\ref{fig:radon-permeation} shows a device to assay radon\npermeability through a sheet in Kamioka.\n\\begin {figure}[htbp]\n  \\begin{center}\n   \\includegraphics[width=0.70\\textwidth]{background\/figs\/radon-permeation.pdf}\n   \\caption{A system to measure radon permeation of a sheet in Kamioka.}\n   \\label{fig:radon-permeation}\n  \\end{center}\n\\end {figure}\nThe performance of the device is still under tuning, but\nthe current sensitivity on radon permeation assay is about $O(10^{-9})$ cm$^2$\/s\nfor a 1 mm thickness sheet. Therefore, this device has enough\nsensitivity to test the Hyper-K HDPE sheet.\n\nIn order to achieve (3) in Hyper-K, we are designing more hermetic ID\ndetector to reduce water flow between ID and OD.\nWe also apply the same water flow control techniques realized in Super-K\nto keep radon around detector wall in Hyper-K ID.\n\nAs a summary, applying the same radon reduction techniques developed in\nSuper-K, a similar radon concentration in Hyper-K inner detector water is expected.\nThe radon permeation through lining material might increase, but an\ninitial estimation shows a similar level of the radon concentration in\nouter detector water in Hyper-K. In order to avoid increase of radon in inner\ndetector, we are planning more hermetic inner detector.\nWe are going to measure radon permeation of our liner candidates, too.\nMoreover, about factor 10 tolerance is expected at 4.5 MeV\nthanks to improvement of the energy resolution.\nTherefore, we think 4.5 MeV (electron kinetic energy)\nthreshold for solar neutrino measurements\nwould be feasible in the Hyper-K detector.\n\n\n\\subsubsection{Muon spallation}\n\t        \nRadioactive isotopes produced by cosmic-ray muon-induced spallation\nare potential backgrounds for low energy neutrinos. Generally, the\nproduction rate depends strongly on the muon flux and the average\nenergy, and the delayed radioactive decays cause the backgrounds in\nthe energy region below about 20\\,MeV. If the lifetime of radioactive\nisotope is relatively short on the order of a few seconds or less, the\nspallation backgrounds can be mitigated by time\/volume cuts based on\nthe reconstructed muon track. Therefore, the detailed estimation of\nthe cosmic-ray muon intensity and the spallation production rate are\nof great importance in demonstrating the sensitivity of Hyper-K to low\nenergy neutrinos.\n\nThe muon intensity at the planned site can be estimated using the\ncalculated surface muon flux and energy, the mountain profile, the\nrock density and compositions. The muon flux ($J_{\\mu}$) and average\nenergy ($\\overline{E}_{\\mu}$) at underground sites are estimated by\nthe muon simulation code (\\texttt{MUSIC})~\\cite{Antonioli:1997}, a\nthree-dimensional MC tool dedicated to muon transportation in\nmatter. In this MC, surface muons are generated according to\nthe \\textit{Modified Gaisser Parameterization}~\\cite{Tang:2006}\nsea-level muon flux distribution. A digital map of the topological\nprofile of Nijuugo-yama with a 5\\,m mesh\nresolution~\\cite{GeographicalSurvey:2010} is shown in\nFig.~\\ref{fig:profile_nijuugoyama}. The Hyper-K detector will be\nlocated around the basing point at the altitude of 508\\,m, referenced in\nSection~\\ref{section:location} corresponds to a position under the old mountain peak\nbefore the surface mining. Based on this elevation data, we calculate\nslant depths as a function of zenith and azimuth angle at an arbitrary\npoint of Hyper-K candidate sites, and estimate the survival probability of muons\nafter the muon transportation through the rock for each angle using\nthe \\texttt{MUSIC} simulation.\n\n\n\\begin {figure}[htbp]\n  \\begin{center}\n    \\includegraphics[width=0.70\\textwidth]{background\/figs\/Altitude-HyperK-figure2-PeakSearch-Center-WithoutLine.pdf}\n    \\caption{Topological profile of Nijuugo-yama~\\cite{GeographicalSurvey:2010}. The black point is the basing point for the Hyper-K site.}\n    \\label{fig:profile_nijuugoyama}\n  \\end{center}\n\\end {figure}\n\n\nPreviously, the value of $J_{\\mu}$ and $\\overline{E}_{\\mu}$ at KamLAND\nin Ikeno-yama are evaluated based on the \\texttt{MUSIC} simulation for\nvarious rock types~\\cite{Tang:2006,Abe:2010}. The value of $J_{\\mu}$\nis dependent on the type of rock. Varying the specific gravity of rock\nfrom 2.65 to 2.75\\,${\\rm g}\/{\\rm cm}^{3}$ in the \\texttt{MUSIC}\nsimulation yields values of $J_{\\mu}$ that agree with the KamLAND muon\nflux measurement~\\cite{Abe:2010}. As both Nijuugo-yama and Ikeno-yama\nare skarn deposit, which are common characteristic in the Kamioka\nmine, the simulation for Hyper-K assumes the same rock type used in\nRef.~\\cite{Abe:2010}. Figure~\\ref{fig:muon_flux_and_energy} shows the\ncalculated $J_{\\mu}$ and $\\overline{E}_{\\mu}$ for Hyper-K at the\naltitude of 508\\,m for 2.70\\,${\\rm g}\/{\\rm cm}^{3}$ specific gravity\nIkeno-yama rock. The values of $J_{\\mu}$ and $\\overline{E}_{\\mu}$ vary\ngreatly depending on the shallowest rock thickness on the west or\nsouth side, as indicated in Fig.~\\ref{fig:profile_nijuugoyama}.\n\n\n\\begin {figure}[htbp]\n  \\begin{center}\n    \\includegraphics[width=0.48\\textwidth]{background\/figs\/MuonFlux-HyperK-from-MountainDepth-xy-Fine-figure2-WithoutLine.pdf}\n    \\includegraphics[width=0.48\\textwidth]{background\/figs\/MuonFlux-HyperK-from-MountainDepth-xy-Fine-figure3-WithoutLine.pdf}\n    \\caption{Calculated muon flux ($J_{\\mu}$) and average energy ($\\overline{E}_{\\mu}$) at the altitude of the Hyper-K (508\\,m) detector. The black point is the basing point for the Hyper-K site.}\n    \\label{fig:muon_flux_and_energy}\n  \\end{center}\n\\end {figure}\n\n\nTable~\\ref{tab:muon_simulation} summarizes the calculated $J_{\\mu}$\nand $\\overline{E}_{\\mu}$ in Hyper-K and Super-K for 2.70\\,${\\rm\ng}\/{\\rm cm}^{3}$ specific gravity. Considering the variation of\n$J_{\\mu}$ for different rock types, we assume uncertainties of\n$\\pm$20\\% for $J_{\\mu}$. Because the Super-K site is deeper, the value\nof $J_{\\mu}$ for Hyper-K is higher than Super-K by a factor of 4.9. On\nthe other hand, the value of $\\overline{E}_{\\mu}$ for Hyper-K is\nsmaller than Super-K as indicated in Fig.~\\ref{fig:muon_flux_energy},\nbecause the relative contribution of lower energy muons becomes larger\nat a shallower site. Figure~\\ref{fig:muon_flux_angle} shows the muon\nflux as a function of zenith angle $\\theta$ (upper) and azimuth angle\n$\\phi$ (lower) for Super-K and Hyper-K at the basing point. We\nconfirmed that the \\texttt{MUSIC} Monte Carlo simulation has been shown to be in good agreement with the Super-K data, as shown in Figure~\\ref{fig:muon_flux_angle}. In Hyper-K, the major contribution of muon flux is introduced by\nthe flux in the west and the south.\n\n\n\\begin{table}[t]\n\\caption{\\label{tab:muon_simulation}Calculated muon flux ($J_{\\mu}$) and average energy ($\\overline{E}_{\\mu}$) in Hyper-K and Super-K for 2.70\\,${\\rm g}\/{\\rm cm}^{3}$ specific gravity Ikeno-yama rock based on the simulation method~\\cite{Tang:2006}. The basing point in Hyper-K is illustrated in Fig.~\\ref{fig:profile_nijuugoyama}.}\n\\begin{ruledtabular}\n\\begin{tabular}{lccc}\nDetector site & Vertical depth (m) & $J_{\\mu}$ ($10^{-7}\\,{\\rm cm}^{-2} {\\rm s}^{-1}$) & $\\overline{E}_{\\mu}$ (GeV) \\\\\n\\hline\nHyper-K (basing point) & 600 & 7.55 & 203 \\\\\nSuper-K & 1,000 & 1.54 & 258 \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\n\\begin {figure}[htbp]\n  \\begin{center}\n    \\includegraphics[width=0.60\\textwidth]{background\/figs\/MuonFlux-HyperK-Energy-figure1-WithSuperK-a.pdf}\n    \\caption{Calculated muon energy spectra for Super-K and Hyper-K at the basing point based on the \\texttt{MUSIC} simulation.}\n    \\label{fig:muon_flux_energy}\n  \\end{center}\n\\end {figure}\n\n\n\\begin {figure}[htbp]\n  \\begin{center}\n    \\includegraphics[angle=270,width=0.60\\textwidth]{background\/figs\/MuonFlux-HyperK-Theta-figure1-WithSuperK-a.pdf}\n    \\includegraphics[angle=270,width=0.60\\textwidth]{background\/figs\/MuonFlux-HyperK-Phi-figure1-WithSuperK-a.pdf}\n    \\caption{Muon flux as a function of zenith angle $\\theta$ (upper) and azimuth angle $\\phi$ (lower) for Super-K and Hyper-K at the basing point. The east corresponds to the azimuth angle of zero degree. The blue lines show the data for Super-K, and the red lines show the MC predictions for Super-K and Hyper-K based on the \\texttt{MUSIC} simulation. The absolute flux and the shape of the Super-K data, which are determined by slant depths for each angle, are well reproduced by MC.}\n    \\label{fig:muon_flux_angle}\n  \\end{center}\n\\end {figure}\n\n\nBased on the muon flux and energy spectrum calculated by\nthe \\texttt{MUSIC} simulation, we can estimate the isotope production\nrates by muon spallation in a planned detector. \\texttt{FLUKA} can be \nused to reliably model nuclear and particle physics processes\ninvolved in muon spallation. Previously, the measured isotope\nproduction rates in a underground detector were compared with\nthe \\texttt{FLUKA} simulation~\\cite{Abe:2010}. However, owing to\nlarge uncertainties on the isotope production cross sections by muons\nor their secondaries, the production rate between data and MC differ\nby up to a factor of two, as shown in Table V of\nRef.~\\cite{Abe:2010}. In order to minimize the uncertainties, we use\nthe isotope production rates observed in Super-K as a basis, and the\nvalues in Hyper-K are estimated based on the muon flux ratio\ncalculated by \\texttt{MUSIC} and the isotope yield ratio\nby \\texttt{FLUKA},\n\\begin{equation}\nR_{i} ({\\rm Hyper\\mathchar`-K}) = R_{i} ({\\rm Super\\mathchar`-K}) \\times \\frac{J_{\\mu} ({\\rm Hyper\\mathchar`-K})}{J_{\\mu} ({\\rm Super\\mathchar`-K})} \\times \\frac{Y_{i}({\\rm Hyper\\mathchar`-K})}{Y_{i} ({\\rm Super\\mathchar`-K})}\n\\end{equation}\nwhere $R_{i}$ is the production rate per unit volume for isotope $i$,\n$J_{\\mu}$ the muon flux (${\\rm cm}^{-2} {\\rm s}^{-1}$), $Y_{i}$ the\nyield per muon track length ($\/\\mu\/{\\rm m}$) for isotope\n$i$. We use \\texttt{FLUKA} version 2011.2b to estimate the isotope\nyields in Hyper-K and Super-K. A water-filled volume of 40-m square\nand 40-m length is used in the simulation, and the analysis of isotope\nproductions is limited within the inner volume of 40-m square and 20-m\nlength in order to avoid a boundary effect. To include the muon charge\nand energy dependence in isotope production yields, beams of both\n$\\mu^{+}$ and $\\mu^{-}$ with a calculated energy spectrum produced\nby \\texttt{MUSIC} were simulated, and the isotope yields by $\\mu^{+}$\nand $\\mu^{-}$ are combined based on their weighted average assuming\nthat a relative intensity of $\\mu^{+}$ to $\\mu^{-}$ is\n1.3. Table~\\ref{tab:isotope_yield_estimation} shows the estimation of\nisotope production yields for Hyper-K and Super-K, the ratio of\nisotope yields, $Y_{i} ({\\rm Hyper\\mathchar`-K}) \/ Y_{i} ({\\rm\nSuper\\mathchar`-K})$. The ratio of the production rates, $R_{i} ({\\rm\nHyper\\mathchar`-K}) \/ R_{i} ({\\rm Super\\mathchar`-K})$, are also\ncalculated by multiplying the isotope yield ratio by the muon flux\nratio, $J_{\\mu} ({\\rm Hyper\\mathchar`-K}) \/ J_{\\mu} ({\\rm\nSuper\\mathchar`-K}) = 4.9 \\pm 1.0$, which was evaluated from\nthe \\texttt{MUSIC} simulation. We assume uncertainties of $\\pm$20\\%\nfor the muon flux ratio considering the possibility of different rock\ntypes for the Hyper-K and Super-K sites. The resulting increase in\nisotope production rate per unit volume from Super-K is approximately\na factor of $4 \\pm 1$ in Hyper-K, which is used for studies of the\nHyper-K physics potential in the following sections.\n\n\n\\begin{table}[t]\n\\caption{\\label{tab:isotope_yield_estimation}Estimation of isotope production yields for Hyper-K and Super-K by muon spallation with \\texttt{FLUKA}. The ratio of the production yields for Hyper-K compared with Super-K are also listed. The ratio of the production rates are calculated by multiplying the isotope yield ratio by the muon flux ratio of $4.9 \\pm 1.0$, evaluated by the \\texttt{MUSIC} simulation.}\n\\begin{ruledtabular}\n\\begin{tabular}{lcccc}\n & \\multicolumn{2}{c}{Isotope yield by \\texttt{FLUKA} ($\\mu$\/m)} &Ratio of isotope yield & Ratio of production rate \\\\\n\\raisebox{1.5ex}[1.5ex][0.75ex]{Isotope} & Hyper-K & Super-K & (Hyper-K \/ Super-K) & (Hyper-K \/ Super-K) \\\\\n\\hline\n$^{12}$B & $8.05 \\times 10^{-5}$ & $9.93 \\times 10^{-5}$ & $0.811 \\pm 0.078$ & $3.98 \\pm 0.88$ \\\\\n$^{12}$N & $8.70 \\times 10^{-6}$ & $1.11 \\times 10^{-5}$ & $0.785 \\pm 0.075$ & $3.84 \\pm 0.85$ \\\\\n$^{9}$Li & $1.23 \\times 10^{-5}$ & $1.68 \\times 10^{-5}$ & $0.732 \\pm 0.070$ & $3.59 \\pm 0.80$ \\\\\n$^{8}$Li & $8.67 \\times 10^{-5}$ & $1.08 \\times 10^{-4}$ & $0.805 \\pm 0.077$ & $3.95 \\pm 0.87$ \\\\\n$^{15}$C & $5.12 \\times 10^{-6}$ & $6.68 \\times 10^{-6}$ & $0.768 \\pm 0.073$ & $3.76 \\pm 0.83$ \\\\\n$^{16}$N & $2.74 \\times 10^{-4}$ & $3.41 \\times 10^{-4}$ & $0.804 \\pm 0.077$ & $3.94 \\pm 0.87$ \\\\\n$^{11}$Be & $5.32 \\times 10^{-6}$ & $7.76 \\times 10^{-6}$ & $0.685 \\pm 0.065$ & $3.36 \\pm 0.74$ \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\\subsubsection{Muon spallation background reduction}\n\nThe spallation products as backgrounds have their origin in the\nspallation reaction of the cosmic muons.  Therefore, we can identify\nand remove these spallation products if we compare their\nfour-dimensional correlation with corresponding muons.  In this\nsection, we will discuss the muon spallation backgrounds using\nthese correlations.  The spallation reduction method is being used in\nsupernova relic neutrino searches at Super-K.\n\n\\paragraph{Method of muon spallation background reduction\\\\}\n\nWe apply a likelihood ratio test on low-energy events to reduce\nspallation backgrounds.  The detail is discussed below.\\\\ With water\nCherenkov detectors, such as Super-K or Hyper-K, we\ncan measure the times, positions and energies of the spallation\nproducts and their preceding muon tracks.  It is also possible to\nmeasure the energy deposit per unit length $dE\/dx$ along muon tracks,\nby deconvoluting Cherenkov ring hits.  The peak position of $dE\/dx$\ndistribution can be assumed as the position where muon spallation\noccurs.\\\\\n\nWe use following valiables for spallation background reduction. \n\\begin{itemize}\n\t\\item Time difference $\\delta t$, between the low-energy event and the preceding muon.\n\t\\item Transverse distance $l_{trans}$, which is defined as the perpendicular distance from the muon track to the low-energy event.\n\t\\item Longitudinal distance $l_{long}$, which is defined as the horizontal distance from the peak position of $dE\/dx$ on reconstructed muon track and the low-energy event.\n\t\\item Residual charge $Q_{peak}$, which is the amount of light seen in $the$ dE\/dx distribution in a width of 4.5 m centered on the peak.\n\\end{itemize}\n\n\\begin {figure}[htbp]\n\\begin{center}\n\t\\includegraphics[width=0.35\\textwidth]{background\/relic_spacut.pdf}\n\t\\caption{\n\t\tSchematic figure for showing spallation distance variables\\cite{KirkRyanDrThesis}.\n\t}\n\t\\label{fig:relic_spacut}\n\\end{center}\n\\end {figure}\n\nA schematic figure of spallation distance variables are shown in Figure~\\ref{fig:relic_spacut}.\nTheir actual distributions can be found in a reference \\cite{KirkRyanDrThesis}.\nThe probability density functions $PDF_{spa}$ and $PDF_{rand}$ of each valuables are given from the fitting results of actual spallation candidates and non-spallation (random) event samples, respectively.\nThe likelihood ratio $\\Lambda$ is defined using PDFs as follows:\n\\begin{equation}\n\\begin{split}\n\t\\Lambda = -2\\log{\\frac{PDF_{spa}(Q_{peak}) \\times PDF_{spa}(\\delta t) \\times PDF_{spa}(L_{trans}) \\times PDF_{spa}(L_{long})}\n\t{PDF_{rand}(Q_{peak}) \\times PDF_{rand}(\\delta t) \\times PDF_{rand}(L_{trans}) \\times PDF_{rand}(L_{long})}}\n\\end{split}.\n\\end{equation}\nBecause any preceding cosmic muon can be a cause of spallation backgrounds, we calculate the likelihood ratio $\\Lambda$ with all muons within 30\\,seconds before the low energy event.\nThe largest likelihood ratio $\\Lambda$ is adopted as the $\\Lambda$ value for the low energy event.\nWe can arbitrarily choose the cut value for the likelihood ratio, which defines the reduction efficiency and the signal efficiency.  \n\n\\paragraph{Estimated muon spallation background after reduction\\\\}\\label{par:spallation_reduction}\nWhen we have more cosmic muon flux, the number of preceding muons\nthat are randomly paired with a low-energy event will be increased.  As a result, we\nwill have more chance to have ``more spallation like'' $\\Lambda$ value\nfor a low-energy event even if it is a non-spallation event.  On the\nother hand the likelihood ratio is not changed for real spallation\nevents, because they will be paired to their mother muons regardless\nof the number of preceding muons.  Consequently we will have worse\nseparation between the likelihood distribution of spallation\nbackgrounds and that of non-spallation events with more cosmic muon flux.\nHere we studied the spallation reduction efficiency and the signal efficiency in Hyper-K, based on the data of SK-II.\nThe spallation reduction efficiency is defined as the rate of spallation events that survived the likelihood cut.\nThe likelihood distribution of\nnon-spallation event sample is also made from the data, pairing the\nlow energy events with muons which were detected 300$\\sim$330\\,s before these events.\nThe effect of shallower Hyper-K location on the spallation reduction method is studied by increasing the proceding muons with random muon data.\n\\\\\nIn following, we will show two cases of spallation cuts.\nOne is defined to keep the signal efficiency of 80\\%, which is applied for usual analysis, e.g. solar neutrino analysis.\nTo estimate the performances of this cut, those are signal efficiency and background reduction efficiency,\nwe use the real events of SK-II between 17.5 and 20\\,MeV.\nWe assume the same signal efficiency and the same reduction efficiency below 17.5\\,MeV.\n\nAs the result, the spallation reduction efficiency ($\\epsilon_{reduction}$) will be 1.2\\% and 3.9\\% for the same and 5 times larger amount of the\ncosmic muons for this criteria, respectively.\nFinally, the ratio of remaining spallation events in the solar neutrino analysis is calculated as follows:\n\\begin{equation}\n\tR_{spallation}({\\rm Hyper\\mathchar`-K\/}{\\rm Super\\mathchar`-K}) =  \\frac{R_{production}({\\rm Hyper\\mathchar`-K})}\n\t{R_{production}({\\rm Super\\mathchar`-K})} \\times\n\t\\frac{\\epsilon_{reduction}({\\rm Hyper\\mathchar`-K})}\n\t{\\epsilon_{reduction}({\\rm Super\\mathchar`-K})}.\n\t\\label{eq:spallation_production_rate}\n\\end{equation}\nHere, $\\epsilon_{reduction}({\\rm Hyper\\mathchar`-K})$ is found to be 3.9\\% for Hyper-K at Tochibora-site as discussed above.\n${\\epsilon_{reduction}({\\rm Super\\mathchar`-K})}$ of $\\sim$6\\% is taken form SK-II solar neutrino analysis.\n$R_{production}({\\rm Hyper\\mathchar`-K})$ and\n    $R_{production}({\\rm Super\\mathchar`-K})$ are the rate of spallation isotope production per unit volume in Hyper-K and Super-K respectively.\n\tReferring to the result of the former section, ${R_{production}({\\rm Hyper\\mathchar`-K})}\/{R_{production}({\\rm Super\\mathchar`-K})}$ is assumed to be $4\\pm1$.\n\tSince, we conclude the ratio of remaining spallation events of Hyper-K to Super-K is $R_{spallation}({\\rm Hyper\\mathchar`-K\/}{\\rm Super\\mathchar`-K})=2.7$.\\par\nMore strict spallation cut is applied for very low background analysis, e.g. supernova relic neutrino search.\nIn this case, the cut value is defined to remove the spallation backgrounds to the level of less than 1 event left between 17.5 and 20\\,MeV or between 20 and 26 MeV.\nSo, the signal efficiency will be affected by the increased amount of muons.\nThe signal efficiency will be 79\\% (29\\%) for the energy range of 17.5$\\sim$20~MeV and 90\\% (54\\%) for 20$\\sim$26~MeV, for the same (5 times larger) amount of the cosmic muons.\nBecause the amount of spallation backgrounds decreases exponentially at the higher energies, no spallation background is expected above 26\\,MeV.\nThe results are shown in Table~\\ref{tab:spacut_solar} and Table~\\ref{tab:spacut_relic}.\n\n\\begin{table}\n\\begin{center}\n\t\\caption{The expected spallation background reduction efficiency for solar neutrino analysis.\n\tThe signal efficiency of 80\\% is kept for the event selection.}\n\\label{tab:spacut_solar}\n\\begin{tabular}{lcccc}\n\\hline \\hline\nCosmic muons rate, comparing to Super-K   &&  $\\times 1$ & $\\times 5$ (Tochibora site) \\\\\n\\hline \nSignal Efficiency                                         &&  80\\%        &     80\\%                    \\\\\nSpallation Reduction Efficiency                           &&  1.2\\%       &     3.9\\%                   \\\\\n\\hline \\hline \n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\begin{table}\n\\begin{center}\n\t\\caption{The expected spallation background reduction efficiency for supernova relic neutrino searches.\n\tThe spallation background is reduced to the level of less than 1 event for the each energy range.}\n\\label{tab:spacut_relic}\n\\begin{tabular}{lcccc}\n\\hline \\hline\nCosmic muons rate, comparing to Super-K   &&  $\\times 1$ & $\\times 5$ (Tochibora site) \\\\\n\\hline \nSignal Efficiency ( - 20\\,MeV)                         &&  79\\%            &     29\\%                    \\\\\nSignal Efficiency (20   - 26\\,MeV)                         &&  90\\%            &     54\\%                    \\\\\n\\hline \\hline \n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\\subsection{Neutron background estimation for atmospheric neutrino\/proton decay study  \\label{sec:neutron_bg}}\n\nThis subsection will discuss the possible cosmic-ray\nbackgrounds for the atmospheric neutrino and proton decay analyses,\nwhich visible energy is greater than 30~MeV.  In this energy range the\nspallation background caused by cosmic muons, which is described in\nSection~\\ref{sec:lowe_bg}, can be neglected.\n\nIn the Super-K detector case, thanks to the double structure of the\ninner and outer detector, cosmic muons entering the detector can be\neasily rejected by looking at hit clusters around the entering and\nexiting points of muons.  According to Super-K's experience, the\nestimated background of the cosmic muons are negligible ($\\sim$0.1\\%\nin the final atmospheric neutrino fully-contained sample).\nConsidering that Hyper-K design is basically same structure, similar\nlevel of the background rejection performance for cosmic muons by the\nouter detector is expected even if the cosmic muon rate is increased\nby several factor due to the shallower site of Hyper-K.\n\n\nOne possible concern about the background due to neutral particle,\nsuch as neutrons and neutral kaons, which are produced by hadronic\ninteraction of cosmic muons near the detector, and enter the detector\nwithout being detected by the outer detector.  Such particles may\npenetrate deep into the detector and produce hadrons, such as $\\pi^0$,\nby interacting with water, which could become electron-like\nbackgrounds.  Figure~\\ref{fig:neutron_pi0_event} shows a Super-K event\ndisplay of the simulated neutron background events which produced\n$\\pi^0$ particle in the detector.\n\n\\begin{figure}[htb]\n\\includegraphics[width=0.6\\textwidth]{background\/neutron_bg_display-crop.pdf}\n\\caption{Event display of a neutron background simulation. $\\pi^0$ is produced by the interaction of $n + X \\to X' + \\pi^0$. A neutron is simulated with an energy of 1~GeV of the center of the detector.}\n\\label{fig:neutron_pi0_event}\n\\end{figure}\n\nFor the study of neutron backgrounds, the flux of cosmic neutron at the detector site are estimated \nbased on \\cite{PhysRevD.73.053004} and shown in Table~\\ref{tab:neutron_flux}. \nAccording to this table, the neutron flux of $E>100$~MeV at Hyper-K site will increase by a factor of $\\sim$8 \nthan that of Super-K site.\n\n\n\\begin{table}\n\\begin{center}\n\\caption{Comparison of various parameters related to neutron background estimation between Super-K and Hyper-K.\n         The estimation of these values are based on \\cite{PhysRevD.73.053004}. }\n\\label{tab:neutron_flux}\n\\begin{tabular}{lccc}\n\\hline \\hline\n                                          &&  Super-K site &  Hyper-K site \\\\\n\\hline \nSite depth (m.w.e.)                       &&  2700        &    1750  \\\\\nCosmic muon rate (10$^{-6}$\/cm$^2$\/sec)   &&  0.13$\\sim$0.14 &  1.0$\\sim$2.3      \\\\\nEffective depth (m.w.e.)                  &&  2050        &     1170  \\\\\n$<E_\\mu>$  (GeV)                          &&  219        &      146  \\\\\n$\\Phi_n$  (10$^{-9}$\/cm$^2$\/sec)          &&  12.3        &     101   \\\\\n~~~~~~~  ($>$100~MeV)                        &&  0.81        &      6.7  \\\\\n$<E_n>$  (MeV)                            &&  76        &        53   \\\\\n\\hline \\hline \n\\end{tabular}\n\\end{center}\n\\end{table}\n\nThough detecting neutron is difficult, their backgrounds can be reduced by two ways; \n\n\\begin{itemize}\n\\item Self-shielding effects due to surrounding water around fiducial volume. In Super-K case, \na water volume of $\\sim$ 4.6~m thick (2.0~m in the inner detector and\n2.6$\\sim$2.8~m in the outer detector) is surrounded around fiducial\nvolume.  Since the neutron is reduced by hadronic interactions in\nwater in a scale of several 10~cm, neutrons is expected to be reduced\nsignificantly before reaching the fiducial volume.\n\\item By detection of the accompanying cosmic muon. As seen in Fig~\\ref{fig:neutron_lateral},\ncosmic neutron and its parent muon are correlated spatially. This means that  neutrons are reduced after traveling in several meter from muon track in the rock. Considering the detector size of the Hyper-K, when neutrons comes into the detector, \nit is supposed that accompanying muons go through the detector also in most case, and rejected by the signal in\nthe outer detector.\n\\end{itemize}\n\n\n\\begin{figure}[htb]\n\\includegraphics[width=0.6\\textwidth]{background\/neutron_lateral_dist-crop.pdf}\n\\caption{Lateral distribution of cosmic neutron from parent muon track. Figure was taken from \\cite{PhysRevD.73.053004}  }\n\\label{fig:neutron_lateral}\n\\end{figure}\n\nIn order to estimate the neutron background in Hyper-K,\nneutron background simulations that took into account \nthe effect of the accompanying muons were performed.\nThe detector simulation of Super-K was used.  \nSince most of the neutrons are expected to be rejected by\ntaking the coincidence with muon signal, as described above, simple\ntoy Monte Carlo simulation considering the detector geometry are\nperformed, and then events in which only neutron is entering are\nsimulated with the Super-K simulator.  The detail procedure of the\nsimulation is described as follows:\n\n\\begin{enumerate}\n\\item Determine muon track. The starting position of muon track is in the plane about 20~meter above the top of Super-K detector \nand the vertex is randomly determined within the region of 200~meter from the detector center. \n\\item Determine the point at which neutron enters the detector according to the neutron lateral distribution from muon track. \n      If there is no neutron which track does not hit the detector, this event is not counted. \n\\item Rejection by muon track. If muon track goes through the detector region, this event is discarded. \n\\item For the events which pass the previous step, neutron vector information, such as vertex, energy, direction, are fed into Super-K detector simulator and simulate neutron interactions.\n\\item Apply simple fully-contained (FC) reduction cut to simulated neutron events. Criterion that the number of hits in the outer detector ($nhitac$) is less than 16 and the visible energy in the inner detector ($E_{vis}$) is greater than 30~MeV are required. \n\\end{enumerate}\n\nThe energy and directional angle distributions of neutrons are determined based on \\cite{PhysRevD.73.053004}. \nAccording to the toy simulation,  97\\% of events are rejected by the criteria of muon coincidence with neutron in step 3. \n\nFig~\\ref{fig:neutron_reduction} shows the distributions of the reduction parameters, $nhitac$ and $E_{vis}$. \nFig~\\ref{fig:neutron_energy} shows the distribution of neutron kinetic energy for all simulated events and the \nremaining events after FC and fiducial volume (FCFV) cut.\nFig~\\ref{fig:neutron_vertex} shows the vertex distribution in Z (height)  vs R (radius) of the detector, $D_{wall}$ distribution, which \nis the distance to the wall. \nIn the vertex distribution events are gathering around the side wall and fewer events around top, \nsuggesting the vertical muons passing nearby the detector produced neutrons entering the detector.  \n\n\\begin{figure}[htb]\n\\includegraphics[width=0.8\\textwidth]{background\/neutron_reduction_dist_new.pdf}\n\\caption{Distributions of number of hits in the outer detector ($nhitac$) (left) and visible energy in the inner detector ($E_{vis}$) (right) for simulated neutron background events. }\n\\label{fig:neutron_reduction}\n\\end{figure}\n\n\\begin{figure}[htb]\n\\includegraphics[width=0.6\\textwidth]{background\/neutron_energy-crop.pdf}\n\\caption{Distributions of true neutron kinetic energy for all simulated events (black) and remaining events after FCFV cut (red).}\n\\label{fig:neutron_energy}\n\\end{figure}\n \n\n\\begin{figure}[htb]\n\\includegraphics[width=0.8\\textwidth]{background\/neutron_vertex-crop.pdf}\n\\caption{Reconstructed vertex distributions of neutron background events which pass after fully-contained reduction (left),\nand $D_{wall}$ distribution ,which corresponds to the distance between reconstructed vertex to the detector wall, \nfor same event (right).}\n\\label{fig:neutron_vertex}\n\\end{figure}\n\n\nTable~\\ref{tab:neutron_MC_summary} shows the summary of the neutron\nbackground MC events in each reduction step.  Normalizing to the\nnumber of events per one year at Super-K detector condition, 2.1 and\n0.2 events are expected for fully-contained (FC) and fully-contained\nfiducial volume (FCFV) event, respectively.  Considering the event\nrate of $\\sim$3000 atmospheric neutrinos, this corresponds to\n$0.2\/3000=7\\times10^{-3}$\\% background rate in FCFV sample.  As for\nthe case of Hyper-K site, neutron flux is increased by about factor of\neight according to Table~\\ref{tab:neutron_flux} due to the shallower\noverburden of detector site condition.\nWhen the background rate is simply scaled by the factor of eight according to the increase of the muon flux,\n$7\\times10^{-2}$~$\\times$~8~$=$~$5\\times10^{-2}$\\% \nof the neutron background rate is estimated for Hyper-K case, which\nseems to be negligible level for physics study.\n\n\n\n\\begin{table}\n\\begin{center}\n\\caption{ Summary of the number of events in neutron background using Super-K detector simulation.}\n\\label{tab:neutron_MC_summary}\n\\begin{tabular}{lccc}\n\\hline \\hline\n                                        &&  All simulation    &  Event\/1year at Super-K \\\\\n\\hline \nentering neutrons                       &&  4.5$\\times$10$^8$   & 8.9$\\times$10$^6$     \\\\\nw\/o muon coincidence                    &&  1.1$\\times$10$^7$   & 2.1$\\times$10$^5$      \\\\\npassed fully-contained cut (FC)         &&   105       &    2.1   \\\\\nvertex is in FV (FCFV)                 &&   11       &     0.2   \\\\\n\\hline \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\nAnother possible neutral particle which could be the background is\nneutral kaon.  According to the calculation of neutral kaon flux in\nunderground~\\cite{JHEP04.041}, neutral flux is estimated to be\nsignificantly smaller; 0.3\\% of neutron flux at 3~km m.w.e..  \n\nThe neutral kaon background is also estimated by the same simulation \nmethod as in the neutron case with the estimated flux and energy spectrum \ndescribed in \\cite{JHEP04.041}. The simulation data corresponding to \n50~years livetime in Hyper-K are produced. After applying\nFCFV selection, no background events are remained in the fiducial volume,\nconcluding that the background from neutral kaon is negligible for the \natmospheric neutrino analysis.\n\nIt would plausible to consider that the impact of the neutron and \nkaon backgrounds on the proton decay analysis\nis negligible as in the atmospheric neutrino case since \nthe there is no reason that those backgrounds have the same event topologies\nas proton decay. These backgrounds will be also reduced similarly as \natmospheric neutrino background by the proton decay selection cuts. \n\n\n\n\n\n\\subsection{Detector calibrations \\label{sec:calibration}}\n\n\\input{design-calibration\/introduction.tex}\n\n\\subsubsection{Inner Detector Calibration}\n\n\\input{design-calibration\/ID_calib_preface.tex}\n\n\\input{design-calibration\/system.tex}\n\n\\input{design-calibration\/ptf.tex}\n\n\n\\input{design-calibration\/monitoring.tex}\n\n\\subsubsection{Calibrations dedicated for physics analyses}\n\n\\input{design-calibration\/LowE_phys.tex}\n\n\\input{design-calibration\/HighE_phys.tex}\n\n\\input{design-calibration\/OD_calib.tex}\n\n\n\n\n\n\n\n\n\\subsubsection{OD calibration system}\n\\label{sec:hk_od_calibration}\n\n\nThe major task of the OD part of the Hyper-K detector is not to\nobtain the exact energy deposited but to identify\nthe neutrino events out of the cosmic-ray muons.\nFor example, ``fully contained'' events are identified by requiring no\nenergy deposition in OD, and ``partially contained'' events and\n``upward-going muon'' events, which are important sub-samples in\natmospheric neutrino analyses, are identified with OD hits\ncoinciding with ID hits.\nFor these physics analyses, an `inter-calibrations' between OD and ID,\ne.g. timing calibrations between OD and ID, is also important in\naddition to the calibrations of OD itself.\n\nCompared the ID part, the OD has various disadvantages in having a\ncalibration system.  They are: 1) many light injection points are\nnecessary to illuminate all the light sensors in the OD area to an\nintensity level of a few 100 PE's, 2) there are sensor support\nstructures which can hinder the delivery of calibration light, and 3)\nthere is no easy way to deploy additional light injection points to\nreplace non-functional ones once the detector is filled with\nwater. The latter 2 points can be mitigated by having redundant light\ninjectors, but this will certainly increase the total cost of the\nsystem.\n\nIn the case of Super-Kamiokande experiment, the OD calibration system\nconsists of a $N_2$ and a dye laser, monitoring PMT's, a variable\nattenuation wheel, optical switches, and 52 fibers.  Each fiber is\nequipped with a light diffusing tip at the end. Of these fibers, 24\nare placed in wall section and 14 each are placed in top and bottom\nsections. They are 72\\,m long, except for those placed in the bottom\nsection which is 110\\,m long. In average, each wall fiber covers\n160\\,m$^2$ of OD sensor area and about 2.5\\,m away from the OD PMT\nplane. Top and bottom fibers cover 64\\,m$^2$ per fiber and about 1.6~m\naway. These fibers are reasonably redundant and a little over a half\nof them are actually used to calibrate all the OD PMT's.\n\nFor the SK OD calibration system to be adopted to the Hyper-K detector, 79\nfibers are required to achieve the same fiber density as the SK for\nthe Hyper-K wall section. For top and bottom, 61 each is necessary. In\ntotal, 201 fibers are needed for the entire Hyper-K detector.  In terms of\nlength, 200~m fibers for bottom and 120~m ones for other sections are\nnecessary to compensate the linear dimension difference between the SK\nand Hyper-K detectors.\n\n\n\n\n\n\\subsection{Cavern}\\label{section:cavern}\n\n\\subsubsection{Cavern shape}\n\nThe Hyper-K cavern has \na cylindrically shaped, barrel region 76\\,meters in diameter and\n62\\,meters in height with a 16\\,meter high dome above it.\nFigure~\\ref{fig:cavern_dimension} shows the cavern dimension.\n\\begin{figure\n  \\begin{center}\n  \\includegraphics[width=0.4\\textwidth]{cavern_dimension.pdf}\n  \\caption{Cavern shape and dimension. The dimensions in the figure are in meter.\n  The shape of the dome section (top portion of the cavern) is defined with \n  two different curvatures divided in 24.305\\,degree and 65.695\\,degree sections\n  denoted in top-right of the figure.}\n  \\label{fig:cavern_dimension}\n  \\end{center}\n\\end{figure}\nThe excavation volume of the cavern is approximately 0.34 Million m$^3$.\nIt should be noted that the dimension of the excavation volume will be\nslightly larger than the detector dimension since the water\ncontainment system, e.g.  a concrete lining, is constructed inside of\nthe excavated cavern surface.\n\n\\subsubsection{Cavern stability and support\\label{sec:cavern_stability_support}}\nThe excavated rock wall is supported by rock-bolts, pre-stressed (PS)\nanchors and shotcrete. A cavern structural stability analysis has been\ncarried out based on the geological condition obtained from the\ngeological surveys.\nThe vertical profile of rock quality is\nassumed to have the uniform distribution of the CH-class, which is the\nmajor component in the rock quality measurement.  The initial rock\nstress for this analysis is based on the measured stress at 553\\,m\na.s.l.  as shown in Fig.~\\ref{fig:ini-stress} and the rock stress at\neach depth is corrected by taking into account the depth, overburden.\nThe FLAC3D analysis software, which uses a finete difference method,\nis adopted to perform a three-dimensional stability analysis.  The\nHoek-Brown model \\cite{Hoek-1,Hoek-2,Hoek-3} is applied as a dynamic\nmodel.  The Hoek-Brown model is the method to estimate physical\nproperties of rock by using results obtained from examinations of\nsampled rock, and is widely used in the world.\n\nFigure~\\ref{fig:cavern_plastic_region_CH} shows the plastic region at\n45\\,degree and 105\\,degree slices in the case of no support (i.e., no\nrock-bolts, no PS-anchors, and no shotcrete).\n\\begin{figure\n  \\begin{center}\n  \\includegraphics[width=0.38\\textwidth]{cavern_CH_plastic_region_045deg.pdf}\n  \\includegraphics[width=0.38\\textwidth]{cavern_CH_plastic_region_105deg.pdf}\n  \\includegraphics[width=0.18\\textwidth]{view.pdf}\n  \\caption{The plastic region at 45\\,degree (left) and 105\\,degree (middle)\n  slices with assumption of uniform CH distribution. The right figure shows definition of the view angle.}\n  \\label{fig:cavern_plastic_region_CH}\n  \\end{center}\n\\end{figure}\nThe plastic region depth is estimated to be $\\sim$2.5\\,m to $12$\\,m.\nThe variation of the plastic region depth is due to the geological\ncondition, e.g. initial stress direction.\nBased on the plastic region obtained, the respective cavern support\n(PS-anchors) patterns are considered as shown in\nFig.~\\ref{fig:cavern_anchor_pattern_CH}.\n\\begin{figure\n  \\begin{center}\n  \\includegraphics[width=0.48\\textwidth]{cavern_CH_psanchors_045deg.pdf}\n  \\includegraphics[width=0.48\\textwidth]{cavern_CH_psanchors_105deg.pdf}\n  \\includegraphics[width=0.27\\textwidth]{cavern_CH_anchor_pattern_dorm.pdf}\n  \\includegraphics[width=0.68\\textwidth]{cavern_CH_anchor_pattern_barrel.pdf}\n  \\caption{PS-anchors pattern at 45\\,degree (top-left) and 105\\,degree (top-right)\n  slices with uniform CH distribution. \n  Colored lines indicate PS-anchors with different setting of initial force applied to\n  PS-anchors. Bottom figures show developed figures of PS-anchors pattern for dome (bottom-left)\n  and barrel (bottom-right) sections. Initial force and length of PS-anchors\n  are indicated by colored circles.}\n  \\label{fig:cavern_anchor_pattern_CH}\n  \\end{center}\n\\end{figure}\nThe number of PS-anchors and the total length for the cavern construction are\nsummarized in Table~\\ref{tab:PS_anchor_summary_CH}.\nThe total length of PS-anchors is estimated to be approximately\n45\\,km.\n\n\\begin{table\n\\caption{Summary of total number of PS-anchors and total length for the cavern\nexcavation in case of uniform CH distribution.\n\\label{tab:PS_anchor_summary_CH}}\n\\begin{tabular}{c|r|r}\n\\hline\\hline\nSection & \\# of anchors & Total length (m) \\\\ \\hline\\hline\nDome    & 1,537 &  26,539 \\\\\nBarrel  & 1,307 &  18,823  \\\\ \\hline\nTotal   & 2,844 &  45,362 \\\\\n\\hline\\hline\n\\end{tabular}\n\\end{table}\n\nAnother analysis is performed with a different vertical profile of\nrock quality, as shown in Fig.~\\ref{fig:rock_qual_assum}.\n\\begin{figure\n\\centering\n  \\includegraphics[width=0.5\\textwidth]{rock_qual_assum.pdf}\n  \\caption{Assumed rock quality distribution in vertical direction.\n  Rock quality in horizontal plane is assumed to be uniform.\n  Two classes of the rock quality, CH and CM classes, are used for this analysis.}\n  \\label{fig:rock_qual_assum}\n\\end{figure}\nIn Fig.~\\ref{fig:rock_qual_assum}, the fraction of rock quality is\nbased on the measurements of rock quality and CM-class is arranged to\nthe dome and bottom sections, which are structurally weaker due to its\nshape, so as to perform an analysis with a severe condition.\nFigure~\\ref{fig:cavern_plastic_region_CMCH} shows the plastic region at\n45\\,degree and 105\\,degree slices with the CH-CM mixed assumption in\nthe case of no support.\n\\begin{figure\n  \\begin{center}\n  \\includegraphics[width=0.38\\textwidth]{cavern_CMCH_plastic_region_045deg.pdf}\n  \\includegraphics[width=0.38\\textwidth]{cavern_CMCH_plastic_region_105deg.pdf}\n  \\caption{The plastic region at 45\\,degree (left) and 105\\,degree (right)\n  slices with assumption of CH-CM mixed distribution.}\n  \\label{fig:cavern_plastic_region_CMCH}\n  \\end{center}\n\\end{figure}\nThe plastic region depth is estimated to be $\\sim$10\\,m to $24$\\,m.\nIn the plastic region at 105\\,degree slice, two spikes in the plastic region\ncan be seen at the boundary between CH and CM classes. These sharp\nspikes are due to the discontinuity of rock quality, which corresponds\nto a discontinuous change in physical strength, and it is difficult to\ncorrectly analyze the plastic region in such a discontinuous\ncondition.  PS-anchor pattern is also considered for this case, as\nshown in Fig.~\\ref{fig:cavern_anchor_pattern_CMCH}.\n\\begin{figure\n  \\begin{center}\n  \\includegraphics[width=0.48\\textwidth]{cavern_CMCH_psanchors_045deg.pdf}\n  \\includegraphics[width=0.48\\textwidth]{cavern_CMCH_psanchors_105deg.pdf}\n  \\includegraphics[width=0.27\\textwidth]{cavern_CMCH_anchor_pattern_dorm.pdf}\n  \\includegraphics[width=0.68\\textwidth]{cavern_CMCH_anchor_pattern_barrel.pdf}\n  \\caption{PS-anchors pattern at 45\\,degree (top-left) and 105\\,degree (top-right)\n  slices with assumption of CH-CM mixed distribution.\n  Colored lines indicate PS-anchors with different setting of initial force applied to\n  PS-anchors. Bottom figures show developed figures of PS-anchors pattern for dome (bottom-left)\n  and barrel (bottom-right) sections. Initial force and length of PS-anchors\n  are indicated by colored circles.}\n  \\label{fig:cavern_anchor_pattern_CMCH}\n  \\end{center}\n\\end{figure}\nThe number of PS-anchors and the total length for the cavern excavation are\nsummarized in Table~\\ref{tab:PS_anchor_summary_CHCM}.\nThe total length of PS anchors is estimated to be approximately\n81\\,km \nin this assumption.  The difference in the total length\nbetween two cases can be considered as an uncertainty on the\nPS-anchors estimation.\n\n\\begin{table\n\\caption{Summary of total number of PS-anchors and total length for the cavern excavation\nin case of CH-CM mixed distribution.\n\\label{tab:PS_anchor_summary_CHCM}}\n\\begin{tabular}{c|r|r}\n\\hline\\hline\nSection & \\# of anchors & Total length (m) \\\\ \\hline\\hline\nDome    &  1,962 & 44,479 \\\\\nBarrel  &  2,020 & 36,392 \\\\\\hline\nTotal   &  3,982 & 80,871 \\\\\n\\hline\\hline\n\\end{tabular}\n\\end{table}\n\nWhile the geological surveys that have been completed already show the\nfeasibility of the required cavern construction, further detailed\nsurveys in the vicinity of the candidate site must be conducted for\nthe final determination of the cavern allocation and \nPS-anchors pattern before starting cavern excavation.\nIt should be stressed that structural stability of the detector cavern with the proposed\nshape can be achieved by using existing cavern construction technologies.\n\nA detailed plan for the additional geological surveys, which need to be done before actual\nconstruction begins, has been established.\nThe surveys are divided into three steps:\n{\\it Step-1} begins with drilling boreholes in the vertical direction from\nthe existing tunnels at 653\\,m a.s.l. to the cavern location where is defined\nin Fig.\\ref{fig:seismic_results},\n{\\it Step-2} carries out an `{\\it in-situ} testing' to measure the physical properties\nof the rock at around the cavern construction site using the existing tunnels,\n{\\it Step-3} is the final step of making the detailed cavern design (e.g. PS-anchor pattern)\nfor the actual cavern excavation, based on the geological survey results in {\\it Step-1} and\n{\\it Step-2}.\n\n\\subsubsection{Cavern construction\\label{sec:cavern_construction}}\n\nThis section describes the cavern construction\nmethod and procedure.\n\nThe cavern excavation begins with construction of access tunnels and\napproach tunnels.  The tunnels and cavern are excavated with a\nblasting technique.  \n\nThe tunnels leading from the mine entrance into the detector site\nvicinity are called ``access tunnels,'' and the tunnels leading from\nthe access tunnels into the group of tunnels connected to the cavern\nare collectively called ``approach tunnels.''\nFigure~\\ref{fig:tunnels_overview} shows overview of the access tunnels.\n\\begin{figure\n  \\begin{center}\n  \\includegraphics[width=0.98\\textwidth]{cavern_access_tunnel_layout_1cav.pdf}\n  \\caption{Overview of the access tunnels. For details of Hyper-K site, one can refer\n  to Fig.~\\ref{fig:approach_tunnels}.}\n  \\label{fig:tunnels_overview}\n  \\end{center}\n\\end{figure}\nThe access tunnel, named as `Wasabo' access tunnel, is also used to transport the excavated\nrock to the Wasabo site where excavated rock is temporarily stored (described\nin later section). \n\nThe cavern is excavated from top to bottom, and there are two\ngeneral phases in the cavern construction -- excavation of ``dome''\nand ``barrel'' sections.  The dome section is top portion of the\ncavern, and the barrel section is vertical straight wall section of the cavern\n(see Fig.~\\ref{fig:cavern_dimension}).  The barrel section is further\ndivided into four stages.\nEach stage has 15.5\\,m height, and the barrel section is excavated in\nstage by stage basis.  Top and bottom of each stage are connected to\napproach tunnels, and excavation of each stage proceed from the top\napproach tunnel to the bottom approach tunnel.\nFigures~\\ref{fig:approach_tunnels} and \\ref{fig:approach_tunnels2}\nillustrate the layout of the approach tunnels.\nAs shown in the figure, ``outer incline tunnel'' is helicoidally or spirally arranged around\nthe cavern and the outer incline tunnel works as an interface between access tunnels and\napproach tunnels.\n\\begin{figure\n  \\begin{center}\n  \\includegraphics[width=0.9\\textwidth]{cavern_approach_tunnel_layout1_3D_1cav.pdf}\n  \\includegraphics[width=0.9\\textwidth]{cavern_approach_tunnel_layout2_3D_1cav.pdf}\n  \\caption{Layout of approach tunnels (see also Fig.~\\ref{fig:approach_tunnels2}).\n  The figure shows the `water rooms' as well,\n  where the water purification systems are located.\n  The ``electronics huts,'' (a.k.a. counting room) which stores the readout electronics\n  and DAQ computers etc. (not shown in the figure), \n  will be built in the approach tunnel.}\n  \\label{fig:approach_tunnels}\n  \\end{center}\n\\end{figure}\n\\begin{figure\n  \\begin{center}\n  \\includegraphics[width=0.9\\textwidth]{cavern_approach_tunnel_layout2.pdf}\n  \\caption{Layout of approach tunnels.} \n  \\label{fig:approach_tunnels2}\n  \\end{center}\n\\end{figure}\n\\begin{figure\n  \\begin{center}\n  \\includegraphics[width=0.7\\textwidth]{cavern_excavation_step_dome.pdf}\n  \\includegraphics[width=0.7\\textwidth]{cavern_excavation_step_barrel.pdf}\n  \\caption{Illustration of the excavation steps for dome section (upper figure) and\n  barrel section (lower figure).}\n  \\label{fig:excavation_step}\n  \\end{center}\n\\end{figure}\nFigure~\\ref{fig:excavation_step} shows schematic of the excavation steps\nof the cavern construction.  The dome section is excavated with\nthirteen steps (from section ``A-1'' through section ``A-6'').  The barrel\nsection is divided into four stages and each stage has three\n``benches.''  The excavation of barrel section proceeds from ``bench\n1-1'' through ``bench 4-3''.\n\n\n\n\\subsubsubsection{Excavated rock handling and disposal}\n\nFor the excavated rock handling and disposal, two sites are used for different purposes:\n\\begin{itemize}\n\\item{\\bf Wasabo-site}\\\\\n          Wasabo-site is an intermediate (temporary) excavated rock deposition site.\n          All the excavated rock from cavern and tunnels excavation is transported and temporary\n          stored at Wasabo-site.\n\\item{\\bf Maruyama-site}\\\\\n          Maruyama-site is the main rock disposal site for all the excavated rock\n          from tunnels and cavern excavation.\n          Capacity of the site is more than two Million-m$^3$.\n          The total distance from Wasabo-site to Maruyama-site is about 14\\,km.\n\\end{itemize}\n\n\nThe excavated rock will be transported with dump-trucks\nfrom the detector site to Wasabo-site and from Wasabo-site to\nthe Maruyama-site.\nFigure~\\ref{fig:cavern_waste_rock_transportation} shows the route of\nexcavated rock transportation from the detector site vicinity through to\nMaruyama-site.\n\\begin{figure\n  \\begin{center}\n  \\includegraphics[width=0.8\\textwidth]{cavern_waste_rock_transport_routing.pdf}\n  \\caption{Overview of the excavated rock transportation routing from Hyper-K site to\n  Maruyama-site.\n  Magenta line denotes the transportation routing from Wasabo-site to Maruyama-site.\n  }\n  \\label{fig:cavern_waste_rock_transportation}\n  \\end{center}\n\\end{figure}\nThere are the existing roads from Wasabo-site to Maruyama-site, that\ncan be used for the transportation of the excavated rock.  Some part of the\nexisting roads, however, need to be improved, e.g. widening the roads\nor allocating turnouts (passing-places), in order to get a large\nnumber of dump-trucks pass through.\n\n\n\\begin{figure}\n    \\centering \n    \\includegraphics[width=1.0\\textwidth]{maruyama_photo_wide.pdf}\n    \\caption{The rock disposal site at Maruyama.\n    Its capacity is more than two Million-m$^3$.}\n    \\label{fig:maruyama_photo}\n\\end{figure}\nThe rock disposal site at Maruyama has a large sinkhole (see FIG.~\\ref{fig:maruyama_photo})\ninduced by the past underground block caving.\nThe excavated rock with a soil volume of 570,000\\,m$^3$ produced by the Hyper-K\ncavern construction will be piled up on top of this sinkhole.\nThe base of such a sinkhole induced by block caving is mainly filled with caved waste.\nTo investigate the geological condition of the rock disposal place,\ntwo vertical boring holes, No.1 and No.2, were excavated.\n\nThe No.1 borehole with a length of 24\\,m was excavated near the edge of the sinkhole\nto understand physical properties of the surface soil layer and its thickness.\nFirst, the standard penetration test has shown that the thickness of\nthe surface layer (i.e. the depth to bedrock) was about 21\\,m.\nThen, core samples were subjected to various laboratory testings,\nsuch as uniaxial compression test and consolidated-undrained triaxial compression test,\nto measure their physical properties,\nnamely the wet bulk density, the uniaxial compression strength,\nthe deformation coefficient, the cohesion, and the internal friction angle.\n\nThe No.2 borehole with a length of 100\\,m was excavated at the midpoint of the sinkhole\nto understand the geological condition of the caved waste.\nFirst, the core inspection has shown that\na gravel bed ($0.0\\sim9.4$\\,m) overlay a sandy clay layer ($9.4\\sim100$\\,m),\nand no cavity was found in the surveyed range.\nThe laboratory testings of the core samples obtained by the No.2 borehole drilling\nhave indicated that the filling state and relative density of the sandy clay\nwere higher at a deeper position.\nA suspension P-S velocity logging was also performed by using the No.2 borehole.\nA several meters long probe, containing a source and two receivers\nspaced 1\\,m apart, was lowered into the borehole to a specific depth,\nwhere the source generated seismic waves.\nThe elapsed time between arrivals of the waves at the receivers was used\nto determine the average velocity of a 1\\,m column of the ground around the borehole.\nIt was confirmed by the P-S logging that no cavity existed in the surveyed range\nand the filling state of the ground were higher at a deeper position.\n\n\\begin{figure}\n    \\centering \n    \\includegraphics[width=0.85\\textwidth]{maruyama_vertical_displacement2.pdf}\n    \\caption{Ground subsidence which might arise by piling up the excavated rock\n    on the Maruyama sinkhole. The figure shows a vertical section of the rock disposal place,\n    and the colored region above the gray line represents the piled rock produced by\n    the Hyper-K cavern excavation.\n    The white line show the boundary between the region filled with the caved waste\n    by past block caving and the surrounding bedrock.}\n    \\label{fig:maruyama_vertical_displacement2}\n\\end{figure}\nBy using geological information obtained by the boring survey,\nwe have performed two types of stability analyses.\nOne is a standard slope stability analysis considering circular slip surfaces.\nIn the analysis, safety factors during a normal period and those during earthquakes\nwere calculated both for the slope made of the piled excavated rock\nand for the existing slope around the sinkhole.\nThe design horizontal seismic coefficient was set at 0.15\naccording to the technical guideline established by METI\n(Ministry of Economy, Trade and Industry in Japan).\nThe safety factors were found to be above 1.20,\nwhich is the reference value described in the guideline,\nboth for the slope of the piled rock and for the existing slope.\n\nThe other is an elasto-plastic analysis using the finite element method (FEM) of the sinkhole ground.\nDistributions of stress, plastic region, and displacement were calculated\nfor both the situations before and after piling up the excavated rock\non top of the sinkhole, and were compared for estimating influences of the piling up.\nIn the analysis, physical properties of the excavated rock were set\naccording to results from the past boring survey at Tochibora\n(i.e. Hyper-K tank construction site), and those of the caved waste and surrounding bedrock\nwere set based on the boring survey results at the Maruyama sinkhole.\nAs a result of the analysis,\nFIG.~\\ref{fig:maruyama_vertical_displacement2} shows the distribution of\nthe expected ground subsidence which might arise by piling up the Hyper-K excavated rock\non the Maruyama sinkhole.\nThe size of possible subsidence is expected to be at most about 1.7\\,m,\nwhich would not be difficult to deal with.\nMonitoring subsidences during the rock piling work will be important for safety.\n\nIn the undergound of the sinkhole exist many mine tunnels, which were excavated\nto extract ores in the past block caving.\nDead ends of such tunnels are located near the boundary between\nthe caved waste region and the surrounding bedrock.\nCurrently the caved waste including broken ores stay at the dead ends by an arch effect\nand don't flow into the mine tunnels,\nbut it might be possible that they will start to move and flow in\ndue to the pressure from the excavated rock piled on the ground.\nAccording to the FEM stability analysis mentioned above,\nadditional horizontal compression stresses at dead ends of existing mine tunnels,\nwhich might cause such an inflow of rocks,\nwere calculated to be 0.8\\,MN at most.\nBy constructing concrete plugs with a length of about 3\\,m in front of dead ends,\nsuch an inflow of rocks into existing tunnels can be prevented.\n\nAll the stability analyses of the Maruyama rock disposal site described in this section\nwere performed by assuming that\nthe volume of the piled excavated rock was 2,600,000\\,m$^3$,\nwhich was an estimation from the old 1 mega-ton Hyper-K design\nand was much larger than 570,000\\,m$^3$ from the current design.\nTherefore, in the current rock disposal plan,\nthe safety factors of the slope stability during a normal period and during earthquakes\nmust be even higher,\nthe expected ground subsidence must be even smaller,\nand the length of concrete plugs necessary to prevent possible inflow of caved wastes\nto existing tunnels must be even shorter\nthan those described above.\n\n\n\n\\subsubsubsection{Cavern construction time}\n\nThe construction sequence has been established by making every effort\nto minimize the total construction time and construction cost, for\nexample, \nthe approach tunnels construction and cavern construction run in\nparallel at different elevations, {\\it etc}.\n\nAs described in the previous section, the cavern construction begins\nwith\nWasabo access tunnel constructions, and the\ncavern constructions will follow.\nThe construction of the access\ntunnels takes $\\sim$17\\,months, and the cavern\nexcavation takes $\\sim$30\\,months.\nThe total duration of cavern construction is estimated to be $\\sim$4 years.\n\nIt should be noted that additional $\\sim$10\\,months will be required in the cavern\nconstruction time if the cavern excavation volume\nand\/or PS-anchor supporting region is near a weak layer, such as a\nfracture zone, that requires additional construction work.\nFurther detailed surveys in the vicinity of the candidate site is important\nto minimize such slippage of the cavern construction,\n\n\n\n\\subsection{Computing}\nHyper-K adopts a Tiered computing model where Kamioka and KEK sites form the\nTier-0 due to the distributed nature of the experiment. A general overview of a future \nHyper-K tiered system is shown in Fig.~\\ref{fig:computing_tiers}.\n\n\\begin{figure}[htbp]\n  \\begin{center}\n    \\includegraphics[scale=0.4]{design-computation\/figures\/hk-comp.pdf}\n  \\end{center}\n\\caption {General overview of a possible Hyper-K tiered system.}\n  \\label{fig:computing_tiers}\n\\end{figure}\n\nThe Tier0 sites will hold the raw experiment data as well as the processed data. The KEK\nTier0 will also contain a  \nThe Tier1 centres (such as RAL, TRIUMF, ccin2p3) would hold portions of the raw, processed\nand simulated data and provide computational resources for the\nsimulation, processing and reprocessing. The Tier2 sites which\ntypically consist of universities will provide computational and\nstorage resources (the storage is usually used to hold specific\nsubsets of the data or simulation). The model makes efficient use the\navailable computing resources that exist at collaborating sites. The\nmodel will be regularly reviewed as changes to the computing landscape\ntake place.\n\n\n\nA current estimate of the rate of raw data to be stored is ~20\\,TB\/day. \nAbout 80\\,PB of the disk space\nis necessary to store the raw data for the 10 years operation.  \n\nReduction and reconstruction software will be\napplied to all the data in the Kamioka Tier0  as soon as the data are\ntaken to provide different samples for different energy regions or\ndifferent analysis groups, i.e. low energy region mainly for the study\nof solar neutrinos, higher energy for the study of nucleon decay and\natmospheric neutrinos, downward going muons for the background study\nof solar neutrinos, the data during the beam timing from the\naccelerator for the beam neutrino analyses. These data sets also\nprovide timely feedback to the experiment and beam operations groups\non the quality of the beam and performance of the detector.  Also,\nearly detection of a supernova burst is crucial and dedicated realtime\nanalysis has to be performed in the independent system.  The required\ncomputing power for data reduction and reconstruction at this level,\ntogether with the supernova detection system, is not so huge and 1000\ncores of the current Intel IA64 CPUs will be sufficient.\n\n\\subsubsection{Simulation production}\n\nMass production of the simulation data sets and their analyses are\nexpected to be performed in the Tier-1 centres because the required\nCPU resource for the huge amount of simulation data is expected to be\nat least a few tens of times larger than the ones necessary for the\nreal time data processing. On the other hand, the simulated data set\nis not extremely large and the cost of the storage could be less than\n10\\% of the storage for the real data sets.  All the data sets after\nreduction and the processed simulation data sets are shared among the\ngeographically distributed analysis working groups. The Tiered model\nensures results in a more scalable architecture capable of meeting the\ncomputational and storage demands of the experiment.\n\nThe Monte Carlo simulation production currently makes use of existing\nHEP computational Grid resources to produce sufficient quantities of\nphysics events necessary to optimise the detector design for maximum\nefficiency. The data are managed by the iRODS data management system\n(\\url{http:\/\/irods.org\/}) that enables distributed storage to be\nmanaged and accessed in an uniform manner. Collaborators access the\nstored data using the intuitive and simple iRODS client API.\n\n\n\n\n\n\\subsection{Data acquisition system }\\label{section:daq}\n\\subsubsection{Data acquisition and triggering}\n\nAll PMT hits from the detector (above a threshold of $\\sim\n0.25$\\,p.e.) will be delivered to the data readout and processing\nsystem where they will be formed into events and recorded on disk for\nfurther processing offline.  The overall rate of hits (mostly from\ndark noise) from the inner detector will be about\n460\\,MHz, \nleading to a total input data rate of 5GB\/s including additional data,\nin the absense of waveform information.\nThe OD adds less than 10\\% to this data load.  To reduce the data\nrecorded, trigger decisions will be made using real-time processing of\nthe hits in the detector.  Events will be formed from all hits within\na time-window surrounding the trigger and recorded to disk for offline\nstudy.\n\nThe main trigger will be the same as that used in SK-IV; a trigger\nwill be generated when the total number of hits seen (NHITS) in a\nsliding time-window exceeds a certain threshold (e.g. 27\\,hits).  This\ntrigger will accept all the necessary data for studies of proton\ndecay, atmospheric neutrinos, beam neutrinos and cosmic ray muon\nevents.  It is important that there is no dead time in the triggering\nor data collection so that delayed energy depositions following a\ntriggered event, such as from a Michel electron or neutron capture,\nare recorded, either as part of the same event or separately.  More\nsophisticated trigger algorithms, which can be added into the\narchitecture, are being studied to increase sensitivity to lower\nenergy events (by distinguishing events with fewer hits from random\ncombinations of dark-noise hits) and detection of supernova bursts (by\nobserving elevated trigger rates). As in Super-K, an additional\ntrigger input will be derived from the J-PARC beam-spill gate to\ndefine readout windows around the beam spill time, independent of the\nnumber of hits observed.  Triggers will be defined to receive\ncalibration events. Also, external trigger inputs are necessary to\ntake the calibration data with synchronized timing.  The estimated\nrate of events is shown in Table~\\ref{event_rates:daq} for readout\nwith the hit-only electronics, which require 12\\,bytes per hit (the\nwaveform option needs a factor four higher bandwidth, $\\sim 50$\\,bytes\nof information per hit).\n\n\n\n\\begin{table}[htbp]\n\\begin{center}\n \\caption{Estimates of data rates.  The ``pre-trigger'' data rates for each physics process\n    is given: (a) for events without dark noise and (b) for events including dark noise. \n    In these calculations, 12 Bytes per PMT hit is assumed. Event rates and event windows for background, muon, beam calibration and pedestal events are based on those from Super-Kamiokande.}\n \\label{event_rates:daq}\n \\begin{tabular}{l|l|r|r|r|r}\n\\hline\\hline\nData source & Event rate & Hits\/event         & Data rate  & Data rate & Data rate per\\\\\n                     &                  & (event window) & pre-trigger(a)& pre-trigger(b)& event window\\\\\n\\hline\nDark noise                             & 10\\,kHz each in 46,700\\,PMT & 1          & 5.6\\,GB\/s & -- & --  \\\\\nVery low energy background & 10\\,kHz (1.5\\,$\\mu$s)             & 25       & 3\\,MB\/s & 84\\,MB\/s   & 200\\,MB\/s \\\\\nLow energy background         & 35\\,Hz (40\\,$\\mu$s)                & 50        &21\\,kB\/s & 7.8\\,MB\/s   & 15\\,MB\/s\\\\\nCosmic muons                       & 100\\,Hz (40\\,$\\mu$s)             & 46,700 &56\\,MB\/s & 78\\,MB\/s & 15\\,GB\/s\\\\\nBeam events                         & 1\\,Hz (1\\,ms)                          &      0      & 0\\,MB\/s & 5.6\\,MB\/s &5.6\\,kB\\\\\nCalibration                             & 2\\,Hz                                      & 46,700  & 2\\,MB\/s & 2\\,MB\/s & -- \\\\\nPedestal                                & 1\\,Hz                                      & 46,700  & 2\\,MB\/s & 2MB\/s & -- \\\\\n\n\\hline\nTotal rate                &          &        &  5.6GB\/s & 180MB\/s &     \\\\\n\\hline\\hline\n \\end{tabular}\n\\end{center}\n\\end{table}\n\nA requirement of the data acquisition system is to reliably trigger\nand collect information from a supernova burst.  This is one of the\nmore challenging design aspects and benefits from the large memory\nbuffers available in modern commodity hardware, sufficient to retain\nall raw hit information for around 100\\,s of detector operation.  In \nthe event of a supernova, the detector will self-trigger on a supernova \nburst by searching for an elevated rate of\nindividual triggers in a sliding time-window e.g.~0.1\\,s.  If such a\ntrigger occurs, all raw hit information in a large time window around\nthe supernova trigger would be saved in the local hardware and then\nslowly written to disk.  Note that even a close supernova burst\nyielding 100,000~events per second of 50~PMT hits each (5~million\nhits\/s) will not overwhelm the readout which is designed to\ncontinuously read 460\\,MHz of dark noise hits.  A large storage buffer \non hard disk drives is used to save\nall the hit data, overwriting the oldest data.  An external\nobservation may reveal up to a few hours after the event that a\nsupernova signature was seen, perhaps in a neighbouring galaxy, in\nwhich only a few events are expected to be observed in Hyper-K.\n\n\\begin{figure}[htbp]\n\\begin{center}\n  \\includegraphics[width=0.8\\textwidth]{design-daq\/figures\/HKDAQ1.pdf}\n  \\caption{Simplified block diagram of the readout showing the\n    sequence of data transfers.}\n\\label{online_schematics:daq}\n\\end{center}\n\\end{figure}\nFigure~\\ref{online_schematics:daq} is a schematic diagram of the data\nreadout and processing system showing the five steps in the sequence\nof readout operations.  The main components are readout buffer units\n(RBU) that receive PMT hit (and possibly waveform) data from the\nfront-end boards, trigger processing units (TPU) and event building\nunits (EBU).  The sequence of operation is indicated by the numbers\n1-5 on Figure~\\ref{online_schematics:daq}: (1) Data on hits in a group\nof channels are streamed into buffers in the RBUs, where they are\nretained for the duration of the trigger decision.  (2) A summary\nblock of information for triggering is sent to the TPUs. (3) Trigger\ndecisions are delivered back to the RBUs to extract event data from\nthe buffer.  (4) The event data are sent to the EBUs.  (5) Built\nevents are written to disk for later processing offline. A description\nof each of the blocks shown is given in the following sections.  The\nDAQ system will be designed to be homogeneous, flexible and scalable.\n\nParameters of the readout system, including data rates at each point\nin the sequence of readout operation are shown in\nTable~\\ref{data_rates:daq}.  Since the architecture is flexible, the\nadditional data generated by the waveform option is easily\naccommodated, although a larger number of Readout buffer units (RBUs) would be required.\n\n\\begin{table}[htbp]\n\\begin{center}\n \\caption{Parameters of the readout design.}\n \\label{data_rates:daq}\n \\begin{tabular}{l|r|r}\n\\hline\\hline\n  Parameter &                              Hit-only option   & Waveform option \\\\\n\\hline\n  Pre-trigger input data rate &                  5,600 MB\/s  & 23,400 MB\/s \\\\\n  Number of RBUs &                                     38    & 122 \\\\\n  Input rate to each RBU  &                         150 MB\/s & 188 MB\/s \\\\\n  Latency provided by RBU (pre-trigger buffer length)& 109 s & 87 s \\\\\n  Trigger info output rate per RBU &                50 MB\/s  & 15 MB\/s \\\\\n  TPU data input rate (for 16 TPUs in detector) &  117 MB\/s  & 117 MB\/s \\\\\n\\hline\\hline\n \\end{tabular}\n\\end{center}\n\\end{table}\n \n\\subsubsection{Readout buffer unit (RBUs)}\n\nThe RBUs receive continuous streams of data from the front-end cards\nand perform three main tasks.  First they store incoming data in\nbuffer storage.  A short-term store of the most recent data is\nretained whilst trigger decisions are made and a long-term storage\narea is reserved in case of a supernova event until it can be read\nout.  Second, it forms a compressed block of data which is handed to\nthe TPU for trigger decision making.  Finally, the RBU handles trigger\nrequests to dispatch complete event data to the event builder unit,\nand supernova trigger requests to move data to the long term storage\narea.  Since the detector is large, and to allow scalability, there\nare many RBUs working in parallel, each responsible for processing\ndata from a designated region of the detector.  In the current design\nof 45,000~PMTs, between about 40~and 120~RBUs will be required\ndepending on the amount of data received per hit, and each will use\n16\\,GB of memory to provide the necessary storage.  The RBU design is\nscalable and thus largely independent of the detector size.\n\nThe final design of the RBUs depends on the layout and interface of\nthe upstream electronics. Possible implementations use either\ncommodity computers with commodity network switches to direct the\ndata, or hardware receiver cards in a communication-crate such as \nAdvanced Telecommunications Computing Architecture (ATCA)\nwith a commodity computer as a controller. The RBU may also include an\ninterface to the upstream link for monitoring information.\n\n\\subsubsection{Trigger processing unit}\n\nThe Trigger Processing Units (TPUs) will accept compressed trigger\ndata blocks from the RBUs and use these to form trigger decisions,\nsuch as the simple, robust NHITS trigger.  Hooks will be provided to\nallow for more sophisticated triggering.  The trigger will also search\nfor large collections of individual events, which may indicate a burst\nof neutrinos from a supernova explosion in the galaxy.\n\nThe trigger will operate using windows of fixed time duration\n(e.g.~60\\,per second) in order to allow the trigger processing to be\nparallelized.  The parallelisation allows for a scalable design that \ncan be extended easily if workloads change due to increased darknoise, \nas well as handling any reprocessing due to errors or data corruption.  \nOne TPU is\nallocated for a given time window and will process all the data in\nthat time window.  The TPUs are connected to the RBUs by a switched\nEthernet network to allow the data from the different RBUs (one for\neach section of the detector) to be routed to the correct TPU.  The\ndata is transferred asynchronously and the TPU starts processing when\nall data packets have arrived.\n\nThe trigger information can be compressed into a 32-bit word per hit\nto identify the channel number within the RBU (10\\,bits) and the time\nwithin the window (21\\,bits) to 10\\,ns accuracy (better time\nresolution is not needed in the trigger) and one spare status bit.\nThe trigger information can be truncated if it is clear that the NHITS\ntrigger will be satisfied from the data in that one RBU alone, as in\nthis case, the event is guaranteed to be collected without further\ntransfer or processing of trigger data.  For this level of packing,\nthe output rate per RBU will be about 30\\,MB\/s and if a farm of\n16\\,TPUs is used, the input rate to each will be around 100\\,MB\/s.\nThe final design of the TPUs is largely independent of the choice of \nfront-end electronics, because the RBUs provide the trigger data in the same \nformat regardless. The TPU design depends on the type of processing required \nfor the sophisticated triggers. The base line design is to use commodity \ncomputer componets, where data packets would be received by the computer \nand complex trigger processing would occur in FPGAs or GPUs that read the \ndata over PCIe links and deliver the trigger verdict for a given time \nblock back to the main memory. \n\n\\subsubsection{Event Building Unit}\n\nOnce the trigger decision has been made for a time-window, the\ndecision is reported back to a central trigger control process, from\nwhere it is delivered to the RBUs.  The request contains an event\nnumber, and the definition of the position and width of the trigger\nwindow.  The central trigger control process also looks for an\nincreased rate of positive triggers, which would be indicative of a\nsupernova burst, and in such case sends the RBUs an instruction to\nsave the relevant data in the long-term part of the buffer memory.  \n\nOn receipt of a normal trigger, the RBUs send all the hits in the\ntrigger window to a designated event-building node, which puts the\nevent together and writes it to the output file.  Once the file\nreaches a certain size, it is closed and released for offline\nprocessing, and new events are recorded in a new output file without\ninterruption.  The event builders also allow events to be read by\nmonitoring and event display programs.  Once a supernova trigger has\noccurred, a separate event-building stream is used to gather the data\nin the long-term part of the RBU memory and output it to a separate\nfile.  There will be one file per supernova trigger.  A\nstraightforward implementation of the event builders is to use\ncommodity computers.\n\n\\subsubsection{Triggering}\nA trigger will be issued if any event exceeds a pre-determined threshold \nof hits (NHITS) in a sliding time window.  This trigger will accept with \nhigh efficiency, all events for studies of proton decay, atmospheric \nneutrinos, beam neutrinos and cosmic ray interactions.  Aside from the \nNHITS requirement, the trigger must have no dead time as this could lead \nto the loss of information from associated delayed energy deposition events \nsuch as Michel electrons or neutron captures.\n\nThe low energy threshold limited by the dark noise rate of the PMTs, which \nprevents the use of such a simple NHITs trigger for low energy physics events \nsuch as solar neutrinos, supernova and neutron capture.  To allow sensitivity \nto low energy physics, the Hyper-K DAQ system must incorporate intelligent, \nfast trigger algorithms that can reject noise events but retain the low energy \nphysics events of interest.  Any event that fails the NHITS threshold will be \npassed to the low energy trigger.  \n\nThe base line low energy trigger design uses a grid of test vertex positions \nwith 5 m spacing inside the detector.  The distance between each test-vertex \nand PMT position is used to produce a look up table of time of flight \ninformation for each test-vertex PMT pair.  For an event, the ID and TDC \ninformation for each tube are recorded and \npassed to the test-vertices algortihm.  The algorithm loops over the test-vertices \nand corrects the measured TDC by the time of flight for each test-vertex PMT pair.  \nThis corrected value represents the time at which the photon was emitted by the \ntest vertex to produce the observed PMT hit. A one dimensional histogram of photon \nemission times is produced with a carefully chosen bin width corresponding to half \nof the time resolution.  The best vertex is found by selecting the histogram bin \nwith the highest number of entries and if this number of entries is greater than \na pre-determined threshold, the trigger is accepted. Due to the computational \nintensity of this process, it will be executed using high performance Graphical \nProcessing Units (GPUs).  Results from Monte Carlo simulations show high noise \nrejection and good low energy acceptance.\n\nA second trigger that uses convolution neural nets and GPUs to perform real-time \nimage recognition for triggering is also being investigated.  Initial studies have proven \nvery promising, with good performance even at very low energies.  Other specialist \ntriggers will be developed for calibration sources and supernova detection. \n\nAdaptations will be made to trigger algorithms for the use in the intermediate detector.  \nAs the low energy physics reach of the intermediate detector is less than that of \nthe far detector, the triggering scheme can be simplified.  However, low energy \ntriggers will be required to accept gadolinium gamma cascades.  Work in this area \nis continuing. \n\n\\subsection{Frontend electronics }\\label{section:electronics}\n\\subsubsection{ General concept of the baseline design }\\label{section:electronics-general}\n\n It is not possible to tell when and where a natural neutrino \ninteracts in the detector. Therefore, the front-end electronics \nmodules for the detectors, which are used to study neutrino from nature, \nare required to digitize all signals from photo-sensors that are above a certain\nthreshold -- i.e. the acquisition needs to be self-triggered. The digitized information \nis then either recorded or discarded, depending on the design of the detector-wide trigger system.\n\nCurrent design of the HK detector is quite similar to the SK\ndetector, in terms of the required specifications and the number of\nphoto-sensors in one detector. Therefore, it is reasonable to \nstart with the system used in the SK detector.\n\n The photo-sensor for the inner detector of HK is newly developed.\nBased on the baseline option, around 40,000 20inch PMT R12860-HQE \nis used. The R12860-HQE PMT has better timing and charge resolution \ncompared to the same diameter PMT (R3600), which has been used in SK.\nThe dark (noise) rate is required not to exceed 4~kHz, which is a \nsimilar requirement to the R3600 PMT. \nBased on these information, we have\nestimated the total data rate and concluded that it is possible to\ndesign the data acquisition system, which has similar to the concept \nof the SK-IV DAQ.\n\n As already realized in the SK-IV DAQ system, it is possible to \nread out all the hit information from the photo-sensors, including \nthe dark noise hits. There is no technical problem in selecting \nthe actual events to be recorded for the analyses by software.\n\n One difference is the size of the detector.  The total amount of\nphoto-sensors in one entire detector is expected to be up to $\\sim$\n47,000, including the sensors for OD.  If we locate the front-end \nelectronics modules on the top of the detector, it is necessary \nto run the cable from the PMT to the roof and the detector structure \nhas to support the weight of the cables, which is expected to be 800~tons. \nThus, it would be possible\nto simplify the detector structure if we can reduce the weight\nof the cables. Also, the maximum length of the cable is $\\sim$ \n30\\% longer than in the SK case. This not only reduces the signal amplitude,\nbut also degrades the quality of the signal -- the leading edge is smoothed out due\nto higher attenuation of the cable in the high frequency region.\nTherefore, we plan to place the modules with the front-end \nelectronics and power supplies for the photo-sensors\nin the water, close to the photo-sensors. This configuration\nmakes it possible to have shorter signal cables from the photo-sensors \nand also allows for significant reduction of the weight that needs to be supported by the photo-sensor \nsupport structure.  Of course, it is necessary to place the front-end module and\npower supply for the photo-sensors in a pressure tolerant enclosure, and also to \nuse water-tight connectors. This kind of ``water-tight'' casing \nhas been studied in other experiments and there are several \npossible options. One concern is the cost of the special cables\nand connectors -- consequently, we are developing special cables and connectors\ndedicated for HK.\n\n The other issue we have to keep in mind is an inability to do any repairs \n to a broken module that will be submerged in water.  Furthermore, a failure of\none module could affect the data transmission from other modules, resulting in a much\nlarger region of the detector that could be lost.  Therefore, the system\nmust be redundant and care must be taken to avoid a single\npoint of failure. Also, careful design of the data\ntransport connections and the timing distribution system are\nessential.\n \n The current baseline design of the front-end module is prepared\nconsidering these requirements. The schematic diagram of the front-end\nmodule is shown in Fig.~\\ref{schematic_frontend:daq}.\n\n\\begin{figure}[htbp]\n\\begin{center}\n  \\includegraphics[width=0.7\\textwidth]{design-electronics\/figures\/daq_schematics.pdf}\n  \\caption{Schematic diagram of the front-end module.}\n\\label{schematic_frontend:daq}\n\\end{center}\n\\end{figure}\n\n There are 4 main function blocks in the front-end board. The signal\ndigitization block, the photo-sensor power supply block, the slow\ncontrol block and the communication block. In the current baseline design, \none module accepts signals from 24 photo-sensors, digitizes them and \nsends out the data.\n\n In the following sections, details of each component of the\nfront-end module are described, along with the data readout and processing parts.\n\n\\subsubsection{ Signal digitization block }\n\n The signal digitization block accepts the signals from the photo-sensors \nand converts them to the digital timing and charge data.  As\nmentioned in the previous section, there is no way to tell when a\nneutrino or a nucleon decay event happens. Therefore, the front-end\nelectronics module is required to have self-triggered analog to\ndigital conversion mechanism and to be dead-time free.  The actual\nevent rate is expected to be smaller than a tens of kHz, even with the\nbackground events from gamma-rays from the surrounding wall or\nfrom cosmic-rays. Also, the number of photo-sensors, which detect\nsufficient charge in case of gamma-ray events is quite small, much \nless than 1\\% of the photo-sensors in the detector.\nTherefore, the time interval between photons hitting a single photo-sensor \nis rather long, much longer than the dark rate of the sensor.\nHowever, muons decay into electrons in the detector and photons from\nboth of them may hit a single photo-sensor. Therefore, it is necessary \nto have the capability to detect both photons, generated by the parent \nmuons and the decay electrons. The lifetime of muons are rather long, \n$\\sim$ 2\\,$\\mu$ sec and thus, it it not necessary to be completely \ndead-time free but the dead-time should be as short as possible.\n\n One possible way to satisfy these requirements is to employ the \ncharge-to-time conversion (QTC) chips. The QTC chip receives\nthe signal from the photo-sensor and produces the digital signal, \nwhose width is linearly dependent on the amount of the input charge. Also, \nthe leading edge of the output digital signal corresponds to the \ntime when the input signal exceeded the pre-defined threshold to produce the\noutput digital signal. The digital output signal from the QTC chip\nis read out by a TDC. Usually, the maximum width of the output \nsignal may be slightly longer than the charge integration gate \nwidth. Therefore, there is a small dead-time after the first \nsignal but it is no larger than several hundreds of ns and \nis acceptable for use in the water Cherenkov detector.\n\n The requirements of the charge and timing resolution are summarized\nin Table~\\ref{requirements1:daq}.\n\n\\begin{table}[htb]\n\\begin{tabular}{l|l}\n\\hline\\hline\nitems & required values \\\\\n\\hline\nBuilt-in discriminator threshold & 1\/4 p.e ( $\\sim$ 0.3 mV )\\\\\nProcessing speed & $\\sim$ 1$\\mu$sec. \/ hit \\\\\nCharge resolution & $\\sim$ 0.05 p.e. ( RMS ) for $<$ 5 p.e. \\\\\nCharge dynamic range & 0.2 $\\sim$ 2500 pC ( 0.1 $\\sim$ 1250 pe. )\\\\\nTiming response & 0.3 ns RMS ( 1 p.e. )\\\\\n                & 0.3 ns RMS ( $\\geq$ 5 p.e. )\\\\\nLeast time count & 0.52 ns \\\\\nTime resolution & 0.25 ns \\\\\nDynamic range   & $\\geq$ 15 bits \\\\\n\\hline\\hline\n\\end{tabular}\n\\caption{ Specification of the signal digitization block. }\n\\label{requirements1:daq}\n\\end{table}\n\n The QTC chips ( CLC101 ) used in the front-end module of SK-IV, \ncalled the QBEE, are a good reference and satisfy all the requirements.  \nThe design rule of these chips is 0.35$\\mu$m and it is possible to\nproduce them again. As for the TDC, the chips used in the QBEE, called AMT3,\nhave been discontinued. However, there are several implementations\nof a `TDC in an FPGA' and some of them seem to have sufficient performance.\nOne candidate is the `wave union TDC' developed at FNAL. The performance \nof this TDC is expected to be better than that of the AMT3 and we are currently\nevaluating this TDC design.\n\n Even though the current baseline design is to utilize the QTC-TDC \napproach, we are also investigating possibility of adopting Flash-ADC \n(FADC) type digitization. In this case, the FADC chip would run all \nthe time and digitize the input signal. Afterwards, FPGA-based \non-the-fly digital signal processing would be utilized to find the PMT \npulse and determine its charge and time of arrival. An advantage of \nthis approach is that it is completely dead-time free -- we would be \nable to detect photons both from prompt muons and from decay electrons, \neven if the latter happen only 100 ns after the initial interaction. \nWe may also be able to distinguish photons from direct and reflected light.\n Another potential advantage is an ability to remove deterministic \ninterference that may be present in the PMT signal. Example sources of \nsuch an interference are switching power supplies and high voltage \nsupplies. The disadvantage is  potentially larger power consumption and\nhigher cost.\n\n Since both the power consumption and the cost are highly dependent on \nthe speed and precision of the FADC ICs, it is advisable is to use the \nslowest possible configuration that will `do the job'. As such, a study \nhas been performed in order to understand the performance of the system \nas a function of both the resolution and the sampling frequency of the \nFADC.  Furthermore, models of the system were developed and validated, \nso that further studies can be streamlined. Finally, various signal \nprocessing methods were tested for determining timing of the pulse -- \namong them the digital constant fraction discriminator \\cite{Huber:2011dt}, \noptimal filters \\cite{Gatti:2004lms,Abbiati:2006tim} and matched filters. \nThe results of the study are presented in \nFig.~\\ref{electronics:fadc:shaper_study}. It is relatively easy to \nachieve the timing resolution that is below 10\\% of the sampling period \n($T_S$). With sufficiently high SNR it is also possible to reach even \nbetter timing accuracy, well below 1\\% of $T_S$. Based on these results,\nwe decided to use a 100 MSPS\/14 bit FADC as the current baseline design for \nthis digitization scheme. Two FADC channels will be used per single PMT (high-gain and low-gain channels), \nso that the dynamic range requirements are fulfilled. Current studies concentrate \non optimizing an anti-aliasing filter, in order to achieve best possible \ntiming resolution.\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=0.49\\textwidth,trim={12mm 74mm 22mm 74mm},clip]{design-electronics\/figures\/fadc_sigma_time_cfd.pdf}\n\\includegraphics[width=0.49\\textwidth,trim={12mm 74mm 22mm 74mm},clip]{design-electronics\/figures\/fadc_sigma_time_fir.pdf}\n\\caption{Results of the study of the timing performance of FADC-type digitization, with pulse timing using digital constant fraction algorithm (left) and optimal filter (right). The precision of the ADC is expressed as signal-to-noise ratio -- $SNR = 6.02N + 1.76$~[dB], with $N$ being the effective number of bits.}\n \\label{electronics:fadc:shaper_study}\n\\end{figure}\n \n Another solution, which allows for savings in power consumption and still \nmaintains high sampling frequency, is to use switched capacitor \narrays (SCA). The biggest advantage of this approach is that no \nbandwidth-limiting anti-aliasing filter is necessary. The price to pay \nis dead-time introduced due to `freezing' of the capacitor, which \nis necessary for its readout. One possibility of alleviating this problem is \nto utilize multiple SCA channels per photo-sensor. This way a dead-time free readout\ncan be provided up to a certain trigger frequency. A better option is to use SCAs\nwith segmented memory, so that one can avoid potentially expensive increase of the number\nof SCA chips. Unfortunately, we are currently not aware of an existence of such an IC, with sufficiently \nlarge memory buffer. Therefore, we are considering possibility of a development of a new IC. The goal would be\nto allow both a dead-time less readout, typical for pure FADC-type digitization,\n as well as high sampling speeds and low power consumption, provided by the SCAs.\n \n The basic assumption of the new design is that the chip would consist \nof both an SCA-type analog memory, a flash-ADC and a discriminator. The \nsampling speed of both the SCA and the flash-ADC would be equal. The\nflash-ADC, which is the most `power hungry' part, would be kept off \nmost of the time -- it would be activated only once a sufficiently high \npulse is detected at the input. Since some time is required for resuming\nthe flash-ADC operation, the analog memory would be used to store the \npulse -- sampled, but not yet quantized. Once the ADC is fully active, \nthe analog memory would act as a first-in first-out buffer, without any freezing \nof its content. Thus, the system would be able to work as long as there \nis any useful signal, without  any dead-time.\n\n In either case, it is necessary to be prepared for a failure of the\ndigitization component. We are therefore considering to have a set of spare\ndigitizer channels and to insert analog switches between the signal \ninputs and the digitization block. With this configuration, we have \nsome flexibility to change the assignment of input signals to \ndigitizer channels. This also provides us with an additional way to\ncalibrate the digitization blocks. Of course, analog switch is known\nto degrade the quality of the signal. Thus, we are going to study\ncarefully prior to implementing this solution. Furthermore, the photo-sensors, \nwhich are currently considered, produce faster pulses and, consequently, higher maximum\nvoltage, which is expected to exceed 6 volts. This is larger than\nthe maximum allowable voltage of the digitizer chips. Therefore, we need to\ndesign the protection circuit, which will\nnot degrade the timing and charge resolutions. \n\n Because the relative timing is used to reconstruct the event vertex \nin the detector, all the modules have to be synchronized.  Therefore, \nit is necessary to drive the TDC or FADC by a clock synchronized to \nthe reference clock fed externally.  Also, the system-wide counter \nis attached to the data to combine the data from different modules \nat a later stage.\n\n\\subsubsection{ The timing synchronization block }\n\n Synchronization of the timing of each TDC or FADC is crucial for precise \nmeasurement of the timing of photon arrival. In Hyper-Kamiokande,\ntiming resolution of the photo-sensor is expected to be largely\nimproved.  Therefore, we have to be careful with the\nsynchronization of the modules -- the design should minimize the clock jitter, so that\nthe timing resolution of the whole system is as good as possible. We are planning to distribute the\ncommon system clock and the reference counter to all the modules. We\nhave not yet started the actual design of this system, but there are\nseveral existing examples. The first method is to send the clock and\nserialized 32 bit counter information using special STP cable, called the\nnano-skew cable, whose skew is less than a few nanoseconds for a 100\\,m long\ncable.  This system has been used in current SK DAQ system and the\nskew is measured to be much smaller than 100\\,ps. It requires\nintermediate timing distributor and therefore needs to be modified for\nuse in Hyper-Kamiokande, but it is not difficult. The other\npossibility is to use the idea of White Rabbit. The White Rabbit\nsystem is designed for the synchronization in the accelerator complex\nand corrects the timing differences with measured delay in each node.\nIt is not necessary to implement entire functionality of White Rabbit\nfor our case but employ the main part of the timing synchronization\nand stabilization.\n\n\\subsubsection{ The photo-sensor power supply block }\n\n If HPDs are used as the photo-sensor, the high voltage supplies for the \n acceleration voltage and the avalanche photodiode bias voltage will\n be put on the HPD base. In this case, the front-end module will control \n both power supplies via control signals.\n\nIf the normal PMTs are used as the photo-sensors, we are considering to\nput the high voltage module in the same enclosure as the front-end\nmodule. In this case, the control signal would be fed internally between \nthe two modules.\n\nThe control signals to the HPD base or to the internal high \nvoltage modules will be controlled remotely through the communication \nblock.\n\n\\subsubsection{ The slow control and monitor block }\n\n It is important to control and monitor the status of the power supply\nfor the photo-sensors. Also, the voltage, the current and the\ntemperature of the front-end module has to be monitored.  This slow\ncontrol and monitor block is prepared for this purpose.  It\naccepts the commands from the communication block and also keeps the\ncurrent status, which is available for read-back. All communication with the\n`external world' is done through the communication block.\n\n\\subsubsection{ The communication block }\n\nIn order\nto reduce the amount of cables, we are planning to connect the modules\nin a mesh topology, with each module connected to its neighbors -- Fig.~\\ref{connections:daq}. \nOnly the top modules would be connected to the readout computers. Each module will have\nseveral communication ports, so that a single point of failure would be avoided. \nIn case of failure of one of the modules, the data would simply be re-routed to one of the neighbors, \nthus ensuring that communication path will be secured.\n\n\\begin{figure}[htbp]\n\\begin{center}\n  \\includegraphics[width=0.4\\textwidth]{design-electronics\/figures\/daq_connections.pdf}\n  \\caption{Schematic diagram of the connections between front-end modules.}\n\\label{connections:daq}\n\\end{center}\n\\end{figure}\n\n There are several possibilities for the connection, but one of\nthe promising ones is the SiTCP, an FPGA based TCP\/IP stack. This TCP\/IP\nstack does not require a CPU core in the FPGA and is accessible like a simple\nFIFO buffer. SiTCP acts as either a TCP\/IP server or a client, so it is \npossible both to receive data from the other module and to add own data\n and later to send everything to the next module. Also, SiTCP has\nregisters, which can be accessed via UDP commands. With this \nfunctionality, it is possible to realize the slow control and\nmonitor system, such as setting the high voltage or monitoring\nthe status, for example read-back of voltages of the power supply.\nRecently, CPU cores are embedded in the FPGA chip. With this kind\nof chips, TCP\/IP communication part is also possible to be handled\nwith the embedded CPU. We will investigate this possibility.\nApart from TCP\/IP, there are several other industry standard\ncommunication protocols available. One example is the Rocket-I\/O.\nRocket I\/O is the standard interface supported in Xilinx FPGA.\nThis allows us to transfer data at speeds exceeding gigabit per second.\nWe are also investigating this possibility for a faster \ncommunication between the modules.\n\n\\subsubsection{ Pressure tolerant cable and Water tight connectors }\n In order to connect the front-end electronics module with other modules, the photo-sensors\nand the clock modules, we need to have water tight connectors and\npressure tolerant cables. It is known that normal Ethernet\ncables are not capable of transmitting the data at full rate\nunder the pressure, because the characteristics of the cable\nare changed when the cable is squeezed under the pressure.\nTherefore, we have started the R\\&D of the pressure-tolerant,\nwater tight Ethernet cable. This cable will use the similar sheath\nmaterial to the one used as the photo-sensor signal cable\nnot to affect the water quality of the detector.\nWe have also started designing the water tight connectors\nfor the PMT connection and the Ethernet connection. Both of the \nconnectors are using screws and are easy to connect. This will \nreduce the time to connect cables during the construction.\nThe mock-up connectors have been designed and we are going\nto produce samples and evaluate them in the coming years.\n\n\\subsubsection{ Timeline }\n Current plan from the finalization of the design to the completion\nof the production and tests is shown in Table \\ref{timeline1:daq}\n\\begin{table}[h]\n\\begin{tabular}{l l}\n  Spring 2020 & Final design review of the system \\\\\n  Autumn 2020 & Start the design of the system based on the design review \\\\\n  Autumn 2021 & Start bidding procedure \\\\\n  Autumn 2022 & Start mass production \\\\\n  Autumn 2023 & Start final system test \\\\\n  Autumn 2024 & Complete mass production \\\\\n  Autumn 2025 & Complete system test and get ready for install\n\\end{tabular}\n\\caption{Timeline to complete the production for the installation.}\n\\label{timeline1:daq}\n\\end{table}\n\n In order to complete the design by Spring 2020, R\\&D and evaluation\nof each component have to be finished by then. Table \\ref{timeline2:daq}\nshows the deadlines for each component.\n\\begin{table}[h]\n\\begin{tabular}{l l}\n  Digitizer & Autumn 2018 based on the decision of the photo sensors \\\\\n  Timing and synchronization & Select technology by Autumn 2018 \\\\\n  Communication block & Fix specification by Autumn 2018 \\\\\n                      & Design by Spring 2019 \\\\\n  High voltage system & Product selection and design by Autumn 2019 \\\\\n  Water tight components & Technology choice by Spring 2019\n\\end{tabular}  \n\\caption{Deadlines for each components.}\n\\label{timeline2:daq}\n\\end{table}\n\n Considering the schedule, we need good coordination with the other\ngroups, including not only the photo-sensor groups but also the construction\ngroups. The allocated time for each item is not much but still achievable.\n\n\n\n\n\n\n\\subsection{Introduction of the Hyper-Kamiokande detector}\n\\graphicspath{{design-introduction\/figures\/}}\n\nThe 1TankHD{} design, i.e. one cylindrical vertical tank with\n40\\% phoocoverage as shown in Fig.\\ref{fig:hk-perspective-1TankHD}, corresponds to the optimized configuration studied\nby the proto-collaboration. It is referred as 1TankHD{} in this report.\n\nThe strategy considers also a staged second tank (2TankHK-staged)\ncurrenlty being investigated (see appendix~\\ref{sec:hakamagoshi}).\n\n\\begin{figure}[htbp]\n  \\begin{center}\n  \\includegraphics[width=0.69\\textwidth]{HK1TankHD_schematic.pdf}\n  \\caption{Schematic view for the configuration of single cylindrical tank instrumented with high density (40\\% photocoverage) PMTs.\nIt is referred as 1TankHD{} in this report.}\n  \\label{fig:hk-perspective-1TankHD}\n  \\end{center}\n\\end{figure}\n\nThe Hyper-K experiment employs a ring-imaging water Cherenkov detector\ntechnique to detect rare interactions of neutrinos and the possible\nspontaneous decay of protons and bound neutrons.\nTable~\\ref{Table:detectorparameters} summarizes the key parameters of\nthe Hyper-K detector compared with other previous and currently\noperating water Cherenkov detectors.  These types of detectors are\nlocated deep underground in order to be shielded from cosmic rays and\ntheir corresponding daughter particles and thereby to achieve a very\nlow background environment.\n\nThe detector mass -- or equivalently the underground detector cavern\nsize or water tank size -- is one of the key detector parameters that\ndetermines the event statistics in neutrino observations and nucleon\n(proton or bound neutron) decay searches.  The detector water plays\ntwo roles: a target material for incoming neutrinos and source of\nnucleons to decay.  We need a detector mass of at least $O(10^2)$\nkton.  in order to accumulate $O(10^3)$ electron neutrino signal\nevents (as shown in Table~\\ref{Tab:sens-selection-nue}) from the\nJ-PARC neutrino beam.  This is necessary to measure the $CP$ violation\neffect with a few \\% accuracy.  This mass of water contains\n$O(10^{35})$ nucleons (protons and nucleons) which would give an\nunprecedented sensitivity to nucleon lifetime at the level of\n$10^{35}$ years.  The location and detailed designs of the Hyper-K\ncavern and tank are presented in Section~\\ref{section:location},\n\\ref{section:cavern}, and \\ref{section:tank}.\n\nThe detector is filled with highly transparent purified water, as\nshown in Section~\\ref{section:water}. A light attenuation length above\n100\\,m can be achieved which allows us to detect a large fraction of\nthe emitted Cherenkov light around the periphery of the water volume.\nRadon concentration in the supplied water is kept below 1\\,mBq\/m$^3$\nto control the radioactive background event rate in solar neutrino and\nother low energy observation.  An option being investigated is the\nGd-doping of the water.  This option, in addition to the nominal water\none, is presented in Section~\\ref{section:water}.\n\n\\begin{table}[!tbp]\n  \\centering\n  \\caption{Parameters of past\n    (KAM~\\cite{Suzuki:1992as,Fukugita:1994wx}), running\n    (SK~\\cite{Fukuda:2002uc,Abe:2013gga}), and future\n    HK-1TankHD{}) water Cherenkov\n    detectors.\n  The KAM and SK have undergone several configuration changes\n  and parameters for KAM-II and SK-IV are referred \n  in the table.\n  The single-photon detection efficiencies are products of\n  the quantum efficiency at peak ($\\sim 400$\\,nm), \n  photo-electron collection efficiency,\n  and threshold efficiency.\n  }\\label{Table:detectorparameters}\n  \\begin{tabular}{lccc}\n   \\hline\n   \\hline\n                        & KAM & SK & HK-1TankHD{} \\ \\\\\n   \\hline\n   Depth   &    1,000 m    &  1,000 m  &    650 m  \\\\\n   Dimensions of water tank & & &  \\\\\n   ~~~~diameter &  15.6 m $\\phi$ &  39 m $\\phi$ & 74 m $\\phi$ \\\\\n   ~~~~height &  16 m &  42 m & 60 m \\\\\n   Total volume          &  4.5 kton & 50 kton & 258 kton \\\\\n   Fiducial volume        &    0.68 kton & 22.5 kton & 187 kton \\\\\n   Outer detector thickness  &  $\\sim$ 1.5 m & $\\sim$ 2 m & $1 \\sim 2$ m \\\\\n   Number of PMTs    &  &  & \\\\\n   ~~~~inner detector (ID)    &  ~~948 (50 cm $\\phi$)~~ & ~~11,129 (50 cm $\\phi$)~~ & ~~40,000 (50 cm $\\phi$)~~ \\\\\n   ~~~~outer detector (OD)   &  123 (50 cm $\\phi$) & 1,885 (20 cm $\\phi$) & 6,700 (20 cm $\\phi$) \\\\\n   Photo-sensitive coverage    &  20\\% & 40\\% & 40\\% \\\\\n   Single-photon detection & unknown  & 12\\% & 24\\%\\\\\n   efficiency of ID PMT  &  &  &  \\\\\n   Single-photon timing & $\\sim 4$ nsec  & 2-3 nsec & 1 nsec \\\\\n   resolution of ID PMT  &  & & \\\\\n   \\hline\n   \\hline\n  \\end{tabular}\n  \\label{table:example}\n \\end{table}\n\nThe detector is instrumented with an array of sensors with\nsingle-photon sensitivity in order to enable reconstruction of the\nspatial and timing distributions of the Cherenkov photons which are\nemitted by secondary particles from neutrino interactions and nucleon\ndecays.  The dimension of the photo-sensors and their density are\nsubject to an optimization that takes into account the required signal\nidentification efficiencies, background rejection power, and cost.  As\na reference, the Super-K detector shown in\nTable~\\ref{Table:detectorparameters} covers $40\\%$ of the detector\nwall with Hamamatsu R3600 50\\,cm diameter hemispherical\nphotomultiplier tubes (PMTs) with the original goal to measure the\nsolar neutrino energy spectrum above $\\sim$5\\,MeV.\n\nThe Hyper-K detector is designed to employ newly developed\nhigh-efficiency and high-resolution PMTs (Hamamatsu R12860) which\nwould amplify faint signatures such as neutron signatures associated\nwith neutrino interactions, nuclear de-excitation gammas and $\\pi^+$\nin proton decays into Kaons, and so on.  This increased sensitivity\ngreatly benefit the major goals of the Hyper-K experiment such as\nclean proton decay searches via $p\\rightarrow e^+ + \\pi^0$ and\n$p\\rightarrow \\bar{\\nu} + K^+$ decay modes and observation of\nsupernova electron anti-neutrinos.  The characteristics of the R12860\ntubes are shown in Section~\\ref{section:photosensors}.  The\nphoto-sensors have vacuum glass bulbs and will be located as much as\n60\\,m underwater in the Hyper-K cavern.  At this depth, the applied\npressure is close to the manufacturers upper specification of the\nSuper-K R3600 PMT (0.65\\,MPa).  Therefore, we need to develop a new\nbulb design and a quality controlled production method to ensure that\nthe sensors can withstand this pressure.  Furthermore, PMT cases\nwill envelop each photo-sensors to avoid a potential chain reaction\naccident due to the implosion of a glass bulb in the water.  The\ndesigns of the bulb and case are also described in\nSection~\\ref{section:photosensors}.\n\nThe detector is instrumented with front-end electronics and a readout\nnetwork\/computer system as shown in Section~\\ref{section:electronics}\nand \\ref{section:daq}.  The system is capable of high-efficient data\nacquisition for two successive events in which Michel electron events\nfollow muon events with a mean interval of 2\\,$\\mu$sec.  It is also\nable to collect the vast amount of neutrinos, which would come from\nnearby supernova in a nominal time period of 10\\,sec.\n\nSimilar to Super-K, an outer detector (OD) is being envisaged that, in\naddition to enabling additional physics, would help to constrain the\nexternal background.  Sparser photo-coverage and smaller PMTs than\nthat for the ID is also planned.\n\n\n\\subsection{Detector site}\\label{section:location}\n\n\\subsubsection{Detector location}\nThe Hyper-K detector candidate site, located 8\\,km south of Super-K,\nis in the Tochibora mine of the Kamioka Mining and Smelting Company,\nnear Kamioka town in Gifu Prefecture, Japan, as shown in\nFig.~\\ref{fig:map}.\n\\begin{figure}[tb]\n  \\includegraphics[width=1.0\\textwidth]{HKlocation.pdf} \n  \\caption{The candidate site map. Broad area map (left) and detailed map\n  (right).}  \\label{fig:map}\n\\end{figure}\nThe J-PARC neutrino beamline is designed so that the existing\nSuper-Kamiokande detector and the Hyper-K candidate site in Tochibora\nmine have the same off-axis angle.  The experiment site is accessible\nvia a drive-in, $\\sim$2.5\\,km long, (nominally) horizontal mine tunnel.\nThe detector will lie under the peak of Nijuugo-yama, with an\noverburden of 650\\,meters of rock or 1,750\\,meters-water-equivalent\n(m.w.e.), at geographic coordinates Lat. 36$^{\\circ}$21'20.105''N,\nLong. 137$^{\\circ}$18'49.137''E (world geographical coordinate\nsystem), and\nan altitude of 514\\,m above sea level (a.s.l.).  The candidate site is\nsurrounded by several faults as shown in Fig.~\\ref{fig:fault} and the\ncaverns and their support structure are placed to avoid a conflict\nwith the known faults.\n\\begin{figure}[tb]\n\\centering\n  \\includegraphics[width=0.95\\textwidth]{fault-side.pdf} \\caption{Location\n  of faults and existing tunnels around the candidate site.  The\n  existing tunnels are located at 423, 483, and 553\\,m\n  a.s.l.}  \\label{fig:fault}\n\\end{figure}\nThe site has a neighboring mountain, Maruyama, just 2.3\\,km away,\nwhose collapsed peak enables us to dispose of more than one million\nm$^3$ of the excavated rock from the detector cavern excavation.\n\n\\subsubsection{Geological condition at the site vicinity}\nRock quality is investigated in the existing tunnels and in sampled\nborehole cores near the candidate site.\nFig.~\\ref{fig:rock_qual_meas} summarizes the geological surveys.\n\\begin{figure}[tb]\n\\centering\n  \\includegraphics[width=1.0\\textwidth]{rock_qual_meas_1tank.pdf} \\caption{Location\n  of rock quality measurements in existing tunnels and bore-hole cores\n  at 423\\,m, 483\\,m, and 553\\,m a.s.l. The red rectangulars show the\n  surveyed regions in the measurements. \n  The black dashed circle indicates the Hyper-K cavern construction candidate site\n  and size of the cavern.\n  }  \\label{fig:rock_qual_meas}\n\\end{figure}\nThe rock wall in the existing tunnels and sampled borehole cores are\ndominated by Hornblende Biotite Gneiss and Migmatite in the state of\nsound, intact rock mass.  This is desirable for constructing such\nunprecedented large underground cavities.  A rock mass classification\nsytem developed by Central Research Institute of Electric Power\nIndustry (CRIEPI)~\\cite{rock_class}, which is widely used for dams and\nunderground cavities construction for the electric power plants in\nJapan,\nis utilized to classify rock quality. The CRIEPI system categorizes\nrock quality in six groups as A, B, CH, CM, CL, and D (in order of\ngood quality), among which the A, B, and CH classes are suitable for\ncavern construction.  Fraction of rock quality at the measured sites\nis summarized in Table~\\ref{tab:rock_qual_dist}.\n\\begin{table}[htbp]\n\\caption{Summary of measured rock quality fraction. Sum of rock quality fraction\nin some Bore-holes is not 100\\% since a small fraction of sampled rock cores was broken\nduring the survey due to a sampling failure.\n  \\label{tab:rock_qual_dist}}\n\\begin{tabular}{l|c|c|c|c|c|c}\n\\hline\\hline\nPlace           & \\multicolumn{6}{c}{Rock quality fraction (\\%)} \\\\ \\cline{2-7}\n                & A   & B    & CH          & CM  & CL  & D           \\\\\\hline\\hline\nTunnel No.1     & 0.0 & 51.6 & 43.6        & 3.0 & 1.8 & 0.0         \\\\\\cline{2-7}\n(553 m a.s.l.)  & \\multicolumn{3}{c|}{95.2} & \\multicolumn{3}{c}{4.8} \\\\\\hline\nBore-hole No.1  & 0.0 & 67.9 & 27.7        & 4.0 & 0.4 & 0.0         \\\\\\cline{2-7}\n(553 m a.s.l.)  & \\multicolumn{3}{c|}{95.6} & \\multicolumn{3}{c}{4.4} \\\\\\hline\nTunnel No.2     & 0.0 & 11.4 & 45.4        & 39.8 & 3.4 & 0.0        \\\\\\cline{2-7}\n(483 m a.s.l.)  & \\multicolumn{3}{c|}{56.8} & \\multicolumn{3}{c}{43.2}\\\\\\hline\nTunnel No.3     & 0.0 & 4.9 & 55.7         & 25.0 & 14.4 & 0.0       \\\\\\cline{2-7}\n(483 m a.s.l.)  & \\multicolumn{3}{c|}{60.6} & \\multicolumn{3}{c}{39.4}\\\\\\hline\nBore-hole No.2  & 2.4 & 10.5 & 49.2        & 29.7 & 5.7 & 0.2        \\\\\\cline{2-7}\n(483 m a.s.l.)  & \\multicolumn{3}{c|}{62.1} & \\multicolumn{3}{c}{35.6}\\\\\\hline\nBore-hole No.3  & 0.0 & 19.2 & 59.2        & 16.5 & 3.8 & 0.3        \\\\\\cline{2-7}\n(483 m a.s.l.)  & \\multicolumn{3}{c|}{78.4} & \\multicolumn{3}{c}{20.6}\\\\\\hline\nBore-hole No.4  & 6.6 & 20.5 & 36.4        & 22.6 & 7.1 & 3.1        \\\\\\cline{2-7}\n(483 m a.s.l.)  & \\multicolumn{3}{c|}{63.5} & \\multicolumn{3}{c}{32.8}\\\\\\hline\nTunnel No.4     & 0.0 & 18.1 & 39.0        & 38.1 & 1.9 & 2.9        \\\\\\cline{2-7}\n(423 m a.s.l.)  & \\multicolumn{3}{c|}{57.1} & \\multicolumn{3}{c}{42.9}\\\\\\hline\\hline\n\\end{tabular}\n\\end{table}\nThe geological surveys are performed at three different altitudes\n(423\\,m, 483\\,m and 553\\,m a.s.l.)  and better fraction of B and CH\nclasses is observed at higher altitude.  The measured fraction of rock\nquality is used for an assumption of rock quality distribution in\ncavern stability analyses.\n \nThe initial stress of the rock is also measured at three points, two\nof which are located at the bottom of the detector cavern (483\\,m\na.s.l.) and one at top (553\\,m a.s.l.).\nIt was found that the two measurements at 483\\,m a.s.l. are strongly\ninfluenced by existing faults.\nWe aim to build our detector at a place where there is no interference\nwith any faults or fracture zone, and the inputs, like initial stress,\nto the cavern stability analysis should not be influenced with faults.\nThus, the two measurements at 483\\,m a.s.l. are eliminated and the one at\n553\\,m a.s.l. is used for a cavern stability analysis described later.\nThe measured rock stress at 553\\,m a.s.l. is shown in\nFig~\\ref{fig:ini-stress}.\n\\begin{figure}[tb]\n\\centering\n  \\includegraphics[width=0.9\\textwidth]{initial_stress.pdf}\n  \\caption{Results of initial rock stress measurement at 553\\,m a.s.l.}\n  \\label{fig:ini-stress}\n\\end{figure}\nBased on the {\\it in-situ} measurements of the rock quality and the\nrock stress, it is confirmed that the Hyper-K caverns can be\nconstructed with the existing excavation techniques (described in\nsection~\\ref{section:cavern}).\n\n\n\n\\input{design-location\/seismic_prospecting.tex}\n\n\\subsubsection{Refining the cavern construction candidate site\\label{sec:seismic}}\n\nAs shown in previous section, the candidate site\nfor cavern construction has area of approximately 300\\,m$\\times$300\\,m\n(see Fig.~\\ref{fig:rock_qual_meas}).\nIn order to further refine and narrow the candidate area where has the best geological condition\nfor the cavern construction,\na geological survey in wide-range, called ``seismic prospecting,'' has been carried out.\n\nSeismic prospecting uses an artificially generated elastic wave that\ntransmits underground bedrock, and identifies physical properties\nof bedrock and geological structure underground, based on the classical physics\nprinciple of transmission, reflection, and refraction of the elastic wave.\nFor example, the speed of elastic wave transmission is varied\nif the elastic wave propagates in a bedrock with different physical\nproperties, e.g. elastic modulus.\n\n\nThe target area of seismic prospecting is defined as\n423$\\sim$703\\,m a.s.l., 400 meters from east to west and 400 meters\nfrom south to north, that covers the entire Hyper-K candidate site\nshown in Fig.~\\ref{fig:rock_qual_meas}.\nThere are six existing tunnels around the target area and they locate\nat different elevations between 423$\\sim$723\\,m a.s.l.\nFor the seismic prospecting, receivers or sensors, called `geophones,'\nwhich detect the elastic wave, were installed in all the six tunnels at interval of 20\\,m \n-- 111 locations in total and each location has three geophones to\ncapture triaxial components of the elastic wave.\nA seismic source is set in the tunnels and generated the elastic wave\nat all the six tunnels with interval of 2.5\\,m in order -- 738 seismic source points in total.\n\nFigure~\\ref{fig:seismic_waveform} shows a waveform data obtained with all geophones\nwhen an elastic wave generated at a location in the tunnel at 723\\,m a.s.l.\nOne can find that an elastic wave transmitted from the seismic source point\nthrough the tunnels in lower altitude.\n\\begin{figure\n  \\begin{center}\n  \\includegraphics[width=0.88\\textwidth]{seismic_waveform.pdf}\n  \\caption{Seismic prospecting waveform data obtained with geophones\n  when an elastic wave generated at a location in the tunnel at 723\\,m a.s.l., as an example.\n  Vertical axis is a time in millisecond and the origin of vertical axis is when\n  an elastic wave is generated.\n  Each line runs in vertical direction is a waveform data obtained with a geophone,\n  and figure shows waveform data for all the 333 geophones, which are arranged in the\n  lateral direction.\n  In the figure, pulse heights of the waveform are shown with different colors,\n  and the waveform in darker color corresponds to a time when geophones captured an elastic wave.}\n  \\label{fig:seismic_waveform}\n  \\end{center}\n\\end{figure}\n\nThe data obtained in the seismic survey were analyzed with two methods, seismic\ntomography and reflection imaging.\nSeismic tomography uses the speed of transmission of the elastic wave.\nThe speed of elastic wave transmission varies depending on the physical properties,\ne.g. elastic modulus, density,\nand seismic tomography identifies the physical properties of rock in the\ntarget region, the entire Hyper-K candidate site.\nReflection imaging identifies fault, fracture zone and open cracks in rock\nusing a nature that an elastic\nwave is reflected if there is a discontinuous or uneven structure, like a fault, in the bedrock.\nLeft figure in Fig.~\\ref{fig:seismic_results} shows the results of reflection imaging.\nIn the figure, blue dashed lines indicate the known faults location.\nAs shown in the figure, the reflection imaging identified the known faults, and\nconfirmed that there is no major fault, fracture zone nor open cracks in the\nHyper-K cavern construction candidate site.\nRight figure in Fig.~\\ref{fig:seismic_results} is rock class distribution obtained by\ncombining the results of seismic tomography, reflection imaging and by comparing\nthose results with the geological survey results shown in Fig.~\\ref{fig:rock_qual_meas}, \nwhich are obtained with borehole coring and investigation of the existing tunnels.\nThe red dashed rectangles in the figures denote a region where has the best rock\nquality and the least uneven rock over the entire Hyper-K candidate site.\nFrom the seismic prospecting results, the location for Hyper-K cavern\nconstruction is narrowed down to approximately $200$\\,m$\\times150$\\,m region\n(red dashed rectangle in Fig.~\\ref{fig:rock_qual_meas}).\n\n\nIn conclusion, the risk associated with insufficient geological information,\nespecially regarding geological discontinuities, e.g. faults, and low quality\nrock mass with CM or lower classes, has been largely reduced by the seismic surveys.\n\n\\begin{figure}[p\n  \\begin{center}\n  \\includegraphics[width=0.45\\textwidth]{seismic_reflection_imaging_Eng.pdf}\n  \\includegraphics[width=0.45\\textwidth]{seismic_rock_class_Eng.pdf}\n  \\includegraphics[width=0.45\\textwidth]{seismic_reflection_imaging_vertical_slice_Eng.pdf}\n  \\includegraphics[width=0.47\\textwidth]{seismic_rock_class_vertical_slice_Eng.pdf}\n  \\caption{Results of seismic prospecting at Hyper-K candidate site.\n  Top two figures are the results at altitude of 483\\,m a.s.l.\n  and lower two figures are results in vertical slice from south to north direction\n  as an example.\n  Top-left and lower-left plots show the results of reflection imaging,\n  and asterisk (*) indicate the identified locations where have fault,\n  fracture zone or open cracks in the bedrock.\n  In the top-left figure, blue dashed lines indicates the location of known faults which\n  are shown in Fig.~\\ref{fig:rock_qual_meas}.\n  Top-right and lower-right figures are rock class distribution obtained by combining the results of\n  seismic tomography, reflection imaging and the geological survey results\n  with borehole and the existing tunnels as shown in Fig.~\\ref{fig:rock_qual_meas}.\n  The red dashed rectangles in the figures denotes a region where has best rock\n  quality and least uneven rock structure over the entire Hyper-K candidate site.\n  Dashed circle indicates the size of Hyper-K cavern.}\n  \\label{fig:seismic_results}\n  \\end{center}\n\\end{figure}\n\n\\subsection{Photosensors}\\label{section:photosensors}\n\n        \\subsubsection{Introduction}\\label{section:photosensors:intro} \n\nThe Hyper-K photosensors detect the\nCherenkov ring pattern created in particle interactions. Photosensors\nview the ID, as well as the OD where they are used to tag particles\nentering or exiting the detector.   \n\nThe ID photosensor was newly developed for Hyper-K to meet the\nrequirements listed in Table~\\ref{photosensorrequirement}.  The new\nphotosensor is based on the well established and reliable design of\nthe 50\\,cm R3600 PMT by Hamamatsu Photonics K.K. with a Venetian blind\ndynode type, which is used for Super-K, and the 43\\,cm PMT with a\nbox-and-line dynode (Hamamatsu R7250), which is used for the KamLAND\nexperiment.  Further improvements include a higher quantum\nefficiency photocathode and an optimized box-and-line dynode, resulting in the new\nphotosensor, a Hamamatsu R12860-HQE PMT (Figure~\\ref{fig:R12860pic}).\n\n\n\\begin{figure}[htbp]\n     \\begin{center}\n \\includegraphics[width=0.4\\textwidth]{R12860HQEpic.jpg}%\n \\caption{Picture of the HQE 50\\,cm box-and-line R12860 PMT.}\n \\label{fig:R12860pic}\n     \\end{center}\n\\end{figure}\n\n\n\\begin{table}[htbp]\n \\begin{center}\n  \\begin{tabular}{l|l|l|l}\n   \\hline \\hline\n     Requirements                     & Value                &      & Conditions \\\\                                    \n   \\hline\n   Photon detection efficiency        & 26\\%                 & Typ. & Quantum Efficiency $\\times$ Collection Efficiency\\\\     \n                                      &                      &      & ~around 400\\,nm wavelength                        \\\\          %\n                                      & (10\\%)               &      & (including Photo-Coverage on the inner detection area)\\\\                \n   Timing resolution                  & 5.2\\,nsec            & FWHM, Typ. & Single Photoelectron (PE)              \\\\                          \n   Charge resolution                  & 50\\%                 & $\\sigma$, Typ. & Single PE \\\\                          \n   Signal window                      & 200\\,nsec            & Max. & Time window covering more than 95\\% \\\\           \n                                      &                      &      & ~of total integrated charge\\\\\n   Dynamic range                      & 2 photons\/cm$^{2}$   & Min. & Per detection area on wall\\\\                    \n   Gain                               & $10^{7} \\sim 10^{8}$ & Typ. & \\\\                                              \n   Afterpulse rate                    & 5\\%                  & Max. & For single PE, relative to the primary pulse\\\\   \n   Rate tolerance                     & 10\\,MHz               & Min. & Single PE pulse, within 10\\% change of gain \\\\   \n   Magnetic field tolerance           & 100\\,mG               & Min. & Within 10\\% degradation \\\\                       \n   Life time                          & 20\\,years             & Min. & Less than 10\\% dead rate \\\\                      \n   Pressure rating                    & 0.8\\,MPa              & Min. & Static, load in water \\\\                        \n   \\hline \\hline\n  \\end{tabular}\n  \\caption{Minimum requirements of the Hyper-K ID photosensors.\n           The dark rate is also an important parameter, but its\n           minimum required value depends on the photosensor\n           specification and is specific to each physics topic.\n           It will be explored in the future using the Hyper-K simulation.\n         }\n  \\label{photosensorrequirement}\n \\end{center}\n\\end{table}\n\nThe OD photosensor design is based on the Super-K OD photosensor, the\n20\\,cm Hamamatsu R5912 PMT, with an improved high quantum efficiency photocathod and a hard waterproofing cover to stand the 60\\,m deep water pressure in Hyper-K.\n\nThis section describes the characteristics of\nthese photosensors. In addition, we present prospects for alternative\noptions, which may be adopted in future.\n\n        \\subsubsection{Photosensor for Inner Detector}\\label{section:photosensors:ID} \n\n        \\subsubsubsection{Performance}\\label{section:photosensors:IDperformance} \nA newly developed 50 \\,cm R12860-HQE PMT (HQE, high quantum efficiency) for\nHyper-K by Hamamatsu (hereafter referred to as the HQE B\\&L PMT), has a faster time response, better\ncharge resolution and a higher detection efficiency with a stable\nmechanical structure, compared to the existing large aperture PMTs.\nThis section describes the specifications of the HQE B\\&L PMT, as well\nas the safety design that ensures longevity of operation.\n\n\n        \\subsubsubsubsection{Design and\nSpecifications}\\label{section:photosensors:IDperformance:design} \nFigure~\\ref{fig:R12860} shows a side view of the HQE B\\&L PMT, whose\nshape is similar to the PMT used in Super-K. Hence, the support structure\ndeveloped in Super-K to attach the PMT is also appropriate for\nthe HQE B\\&L PMT in Hyper-K. The dynode structure and the\nsurface curvature were improved.  A typical bias voltage of 2,000\\,V is\ndivided to each dynode by a PMT base circuit such the one shown in\nFigure~\\ref{fig:R12860bleeder}.  The specifications for a typical HQE\nB\\&L PMT is listed in Table~\\ref{pmtspec}.\n\n\\begin{figure}[htbp]\n     \\begin{center}\n \\includegraphics[width=0.8\\textwidth]{photosensor_R12860design.pdf}%\n \\caption{Outline of the HQE 50\\,cm box-and-line R12860 PMT.}\n \\label{fig:R12860}\n     \\end{center}\n\n\\end{figure}\n\n\\begin{figure}[htbp]\n     \\begin{center}\n \\includegraphics[width=0.8\\textwidth]{photosensor_VD12860-4.pdf}%\n \\caption{PMT base circuit of the HQE box-and-line R12860 PMT.}\n \\label{fig:R12860bleeder}\n     \\end{center}\n\\end{figure}\n\n\n\\begin{table}[htbp]\n \\begin{center}\n  \\begin{tabular}{l|l}\n   \\hline \\hline\n   Shape                  & Hemispherical \\\\\n   Photocathode area      & 50\\,cm diameter (20 inches)\\\\\n   Bulb material          & Borosilicate glass ($\\sim3$\\,mm) \\\\\n   Photocathode material  & Bialkali (Sb-K-Cs) \\\\\n   Quantum efficiency     & 30\\,\\% typical at $\\lambda=390$\\,nm \\\\\n   Collection efficiency  & 95\\,\\% at $10^7$ gain \\\\\n   Dynodes                & 10\\,stage box-and-line type\\\\\n   Gain                   & 10$^7$ at $\\sim2000$\\,V \\\\\n   Dark pulse rate        & $\\sim8$\\,kHz at $10^7$ gain (13 Celsius degrees, after stabilization for a long period) \\\\\n   Weight                 & 9\\,kg (without cable) \\\\ \n   Volume                 & 61,000\\,cm$^3$ \\\\\n   Pressure tolerance     & 1.25\\,MPa water proof \\\\\n   \\hline \\hline\n  \\end{tabular}\n  \\caption{Specifications of the 50\\,cm R12860-HQE PMT by Hamamatsu.}\n  \\label{pmtspec}\n \\end{center}\n\\end{table}\n\n\n\\subsubsubsubsection{Detection\n  Efficiency}\\label{section:photosensors:IDperformance:efficiency} \nThe single photon detection efficiency of the HQE B\\&L PMT is a factor of two\nbetter than the conventional R3600 in Super-K (Super-K PMT).\nFigure~\\ref{fig:HQE} shows the measured quantum efficiency (QE) of\nseveral HQE B\\&L PMTs as a function of wavelength compared with a\ntypical QE curve of the Super-K PMT (dotted line).  \nThe QE of the R12860-HQE PMT is typically 30\\% at peak wavelength around 390\\,nm, while the peak QE of the Super-K PMT is about 22\\%.\n\n \\begin{figure}[htbp]\n \\includegraphics[width=0.4\\textwidth]{HQEspectra-eps-converted-to.pdf}%\n \\caption{Measured QE for six high-QE R12860 (solid lines) and a R3600 (dashed line).\\label{fig:HQE}}\n \\end{figure}\n\n The HQE B\\&L PMT has a high collection efficiency and large sensitive photocathode area. The\n photocathode area with a collection efficiency (CE) of 50\\% or better\n is 49.2\\,cm for the HQE B\\&L PMT, compared to 46\\,cm in case of the\n Super-K PMT and 43.2\\,cm in the KamLAND PMT.  Compared with 73\\% CE\n of the Super-K PMT within the 46\\,cm area, the HQE B\\&L PMT reaches\n 95\\% in the same area and still keeps a high efficiency of 87\\% even\n in the full 50\\,cm area.  This high CE was achieved by optimizing the\n glass curvature and the focusing electrode, in addition to the use of a\n box-and-line dynode.  In the Super-K Venetian blind dynode, the\n photoelectron sometimes misses the first dynode while the wide first\n box dynode of the box-and-line accepts almost all the photoelectrons.\n This also helps improving the single photoelectron (PE) charge\n resolution, which then improves the hit efficiency at a single PE\n level.  By measuring the single PE level, we confirmed the CE\n improvement by a factor of 1.4 compared with the Super-K PMT, and 1.9\n in the total efficiency including HQE.\n Figure~\\ref{fig:EffUniformity} shows that the CE response is quite\n uniform over the whole PMT surface despite the asymmetric dynode\n structure.\n\nA relative CE loss in case of a 100\\,mG residual Earth magnetic field is at most 2\\% in the worst direction, or negligible when the PMT is aligned to avoid this direction on the tank wall.\nThe reduction of geomagnetism up to 100\\,mG can be achieved by active shielding by coils.\n\n \\begin{figure}[htbp]\n \\includegraphics[width=0.7\\textwidth]{photosensor_EffUniformityRel.pdf}%\n \\caption{Relative single photon detection efficiency as a function of the position in the photocathode, where a position angle is zero at the PMT center and $\\pm 90^{\\circ}$ at the edges.\n          The dashed line is the scan along the symmetric line of the box-and-line dynode whereas the solid line is along the perpendicular direction of the symmetric line.\n          The detection efficiency represents QE, CE and cut efficiency of the single photoelectron at 0.25 PE.\n          A HQE B\\&L PMT with a 31\\% QE sample shows a high detection efficiency by a factor of two compared with normal QE Super-K PMTs (QE = 22\\%, based on an average of four samples).\\label{fig:EffUniformity}}\n \\end{figure}\n\n\n        \\subsubsubsubsection{Performance of Single Photoelectron\nDetection}\\label{section:photosensors:IDperformance:1pe} \nThe single photoelectron pulse in a HQE B\\&L PMT has a 6.7\\,nsec rise\ntime (10\\% -- 90\\%) and 13.0\\,nsec FWHM without ringing, which is faster\nthan the 10.6\\,nsec rise time and 18.5\\,nsec FWHM in the Super-K PMT.  The\ntime resolution for single PE signals is 1.1\\,nsec in $\\sigma$ for the fast\nleft side of the transit time peak in Figure~\\ref{fig:PMTTTS} and 4.1\\,nsec\nat FWHM, which is about half of the Super-K PMTs.  This would be an\nimportant factor to improve the reconstruction performance of events\nin Hyper-K.\n\nThe nominal gain is $10^7$ and can be adjusted for several factors in\na range between 1500\\,V to 2200\\,V.\nFigure~\\ref{fig:PMT1PE} shows the charge distribution, where the 35\\%\nresolution in $\\sigma$ of the single PE is better for the HQE B\\&L PMT compared to the 50\\%\nof the Super-K PMT.  The peak-to-valley ratio is about 4, defined by\nthe ratio of the height of the single PE peak to that of the valley\nbetween peaks.\n\n\\begin{figure}[htbp]\n \\begin{tabular}{c}\n   \\begin{minipage}{0.45\\hsize}\n     \\begin{center}\n       \\includegraphics[width=0.9\\textwidth]{photosensor_1petts.pdf}\n       \\caption{Transit time distribution at single photoelectron, compared with the Super-K PMT in dotted line. \\label{fig:PMTTTS}}\n     \\end{center}\n   \\end{minipage}\n\n\\hspace{1cm}\n\n   \\begin{minipage}{0.45\\hsize}\n     \\begin{center}\n       \\includegraphics[width=0.9\\textwidth]{photosensor_1pecharge.pdf}\n       \\caption{Single photoelectron distribution with pedestal, compared with the Super-K PMT in dotted line. \\label{fig:PMT1PE}}\n     \\end{center}\n   \\end{minipage}\n \\end{tabular}\n\\end{figure}\n\n\n        \\subsubsubsubsection{Gain Stability}\\label{section:photosensors:IDperformance:gainstability} \nBecause the Hyper-K detector aims for various physics subjects in a wide energy range, the PMT is required to have a wide dynamic range. \nThe Super-K PMTs have an output linearity up to 250 PEs in\ncharge according to the specifications and up to about 700 PEs as measured in Super-K\n(with up to 5\\% distortion)~\\cite{Abe:2013gga}, while the linearity of\nthe HQE B\\&L PMT was measured to be within 5\\% up to 470 PEs as seen\nin Figure~\\ref{fig:pmtlinearity}.  Even with more than 1,000 PEs, the\noutput is not saturated and the number of PEs can be calculated by\ncorrecting the non-linear response.\nThe linearity range depends on the dynode current, and\ncan be optimized by changing the resistor values in the PMT base \ncircuit.  This result demonstrates sufficient detection capabilities\nin the wide MeV -- GeV region as in Super-K, as long as it is\ncorrected according to the response curve.\n\n\\begin{figure}\n  \\begin{center}\n  \\includegraphics[width=0.4\\textwidth]{photosensor_linearity_log.pdf}\n        \\caption{Output linearity of the HQE B\\&L PMT in charge, where a dotted line shows an ideal linear response.\n                 It is derived by measurements of a coincident emission by two light sources compared with an expectation by sum of individual detections. }\n        \\label{fig:pmtlinearity}\n  \\end{center}\n\\end{figure}\n\nA fast recovery of gain for high signal rate is needed for supernova\nobservation, decay electrons from muons, and any accidental pileup\nevents. On the other hand, separation of individual signals in time is limited by the charge integration range,\nthat is 200\\,nsec (equivalent to 5\\,MHz) or more depending on the electronics.\n\nThe rate dependence of the output charge was measured at several light\nintensities while varying the constant interval time of light pulses\n(as shown in Figure~\\ref{fig:pmtratetolerance}).  A 5\\% drop is\nobserved at the output current of 170 $\\mu$A.  It corresponds to 78\nMHz in the single PE intensity or 1\\,MHz in most of detected\nintensities like at the level of few tens of PE.  This is sufficient\nto detect possible burst physics events \n(Section~\\ref{sec:supernova}).\n\n\\begin{figure}[htbp]\n \\begin{tabular}{c}\n\n   \\begin{minipage}{0.46\\hsize}\n     \\begin{center}\n       \\includegraphics[width=0.9\\textwidth]{photosensor_rate_tolerance_log.pdf}\n       \\caption{Measured charge as a function of the pulse rate in three light intensities of 25, 50 and 100 photoelectrons, relative to outputs at 100\\,Hz.\n                Each charge is calculated using the baseline just before the pulse. \\label{fig:pmtratetolerance}}\n     \\end{center}\n   \\end{minipage}\n\n\\hspace{0.5cm}\n\n   \\begin{minipage}{0.46\\hsize}\n     \\begin{center}\n        \\includegraphics[width=0.9\\textwidth]{photosensor_gainrecovery.pdf}\n        \\caption{Output charge fidelity of a delayed pulse after a primary pulse, compared with no primary pulse.\n                 The charge set is about 150 PEs at 10$^{7}$ gain for both primary and delayed pulses in various delayed time. }\n        \\label{fig:pmtgainrecovery}\n     \\end{center}\n   \\end{minipage}\n \\end{tabular}\n\\end{figure}\n\n\nEven when two near continuous events are detected, like in the case of\nan event with a\ndecay particle, no loss of charge was observed for the\nsecond delayed pulse.  By measuring two continuous pulses of about 150\nPEs in both, the observed loss of gain is stable within 0.5\\% as shown\nin Figure~\\ref{fig:pmtgainrecovery}. \nTherefore, the output charge fidelity for a delayed signal is sufficient with the HQE B\\&L PMT.\n\nA long term stability test was performed on three HQE B\\&L PMTs\nin a 200\\,ton water Cherenkov detector at the Kamioka mine, which was\nconstructed to evaluate the feasibility of anti-neutrino tagging via gadolinium doping\nin water.  All the HQE B\\&L PMTs functioned over two years, and\nthe gain measured using the charge peak resulting from a pulsed calibration source was stable\nwithin 1\\% RMS (Figure~\\ref{fig:pmtstability}).\n\n\\begin{figure}\n  \\begin{center}\n  \\includegraphics[width=0.5\\textwidth]{photosensor_BLPMTpeakhistory.pdf}\n  \\caption{Relative charge of three HQE B\\&L PMTs for two months in a 200\\,ton water Cherenkov detector.\n           Signals of several tens photoelectrons from a xenon light pulse were monitored. }\n  \\label{fig:pmtstability}\n  \\end{center}\n\\end{figure}\n\n\n        \\subsubsubsubsection{Backgrounds}\\label{section:photosensors:IDperformance:BG} \nThe dark hit rate originates from a thermionic emission on the\nphotocathode, and depends on the environmental temperature, the bias\nhigh voltage and the accumulated operating time for stabilization.\nFor Hyper-K, the energy threshold for low energy physics studies depends largely on the PMT \ndark hit rate, because the sum of the PMT dark hits create fake event triggers.\n\nThe dark hit rates of several HQE B\\&L PMTs were measured to be\n8.3\\,kHz at a temperature of 15\\,$^\\circ$C in air after a month-long\nstabilization period. \nThe adequacy of this dark hit rate on the physics sensitivities will be discussed in Section \\ref{section:physics}.\nSince the detection efficiency is doubled for the HQE B\\&L PMTs, the obtained current dark hit rate is relatively high compared to the Super-K PMT (4.2\\,kHz). \nA lower dark rate results in a better sensitivity to low energy events, thus\nthe HQE B\\&L PMT production is being optimized to achieve a lower dark\nhit rate. We expect further improvements within the next year.\n\nThe afterpulse has a long delay of several microseconds order after\nthe primary PE, and can result in mis-reconstruction for tagging delayed\nparticles.  Afterpulsing is caused by a feedback of the ionized residual gas to\nthe photocathode, and several timing peaks appear due to the different gas\nmolecular masses.\nAs shown in Figure~\\ref{fig:pmtafterpulse}, hit timing distribution of the HQE B\\&L PMT has several peaks of afterpulses.\nThe afterpulse rate is, in total, less than 5\\% relative to the main\npulse at single PE observation.  It is comparable to the Super-K PMT.\n\n\\begin{figure}\n  \\begin{center}\n  \\includegraphics[width=0.85\\textwidth]{photosensor_afterpulse.pdf}\n  \\caption{Time distribution of hits, where the primary single PE signal comes at zero and others are afterpulses.\n           The dotted line represents the level of the dark hit which is set to zero.\n           The expected value of the number of afterpulses is measured to be 0.05 for one primary pulse in this sample. }\n  \\label{fig:pmtafterpulse}\n  \\end{center}\n\\end{figure}\n\nThe radioactive contamination in the surface glass was measured by a\ngermanium semiconductor detector.  It is listed\nin Table~\\ref{pmtglassri} by Uranium series, Thorium series and\nPotassium-40. The contamination of Potassium-40 was reduced by an\norder of magnitude compared to the glass used in the Super-K PMT. \n\n\\begin{table}[htbp]\n \\begin{center}\n  \\begin{tabular}{l|lll}\n   \\hline \\hline\n           & U-chain & Th-chain & K$^{40}$ \\\\\n   \\hline\n    Bq\/kg  & 5.4 & 1.8 & 1.6 \\\\\n    Bq\/PMT & 34.5  & 11.3 & 10.5 \\\\\n   \\hline \\hline\n  \\end{tabular}\n  \\caption{Radioactive contamination in glass for the HQE B\\&L PMT (Hamamatsu R12860-HQE).}\n  \\label{pmtglassri}\n \\end{center}\n\\end{table}\n\n\n        \\subsubsubsection{Mechanical Characteristics}\\label{section:photosensors:Mechanical}\nThe HQE B\\&L PMT bulb has been improved and proven to survive under\n60\\,meter water for Hyper-K as described in this section. It is\nsufficiently better than the Super-K PMT, which is only specified for 40\\,meter deep water. \n\nHowever, with the large number of photosensors in Hyper-K we expect\nthat even with a pre-selection (using a quick pressure test, etc.)\nbefore installation, it is difficult to\nensure that there is no glass failure.\nIn 2001, a chain implosion of 6,779 PMTs out of 11,146 took place at Super-K.  \nIt was triggered by an accidental implosion which was transmitted to other PMTs as pressure pulse.  \nIn order to avert a similar accident, a protective cover made of a ultraviolet (UV) transparent acrylic cover for the detection area and a Fiber Reinforced Plastics (FRP) for the rear was introduced in Super-K.\n\nSuch a protective cover is needed to avoid any cascade implosion of the photosensors, making up for the difficult control of the glass quality in the production.\nThe cover was re-designed to further reduce the impact of pressure pulse, because a 60\\,meter water depth boosts the peak pressure caused by the implosion by a factor of 1.6 from Super-K, corresponding to 6--7\\,MPa.\nThe new design of the ID photosensor cover and its validation are explained in this section.\n\n\n        \\subsubsubsubsection{Design and Confirmation Test}\\label{section:photosensors:Mechanical:Design} \nThe weakest point of the Super-K PMT, which is around the largest reverse curvature\n(Neck in Figure~\\ref{fig:pmtcurvature}), was improved for the HQE B\\&L\nPMT.\nBased on a stress analysis, the first bulb shape, R12860-A, was designed to reduce the stress concentration around the neck. \nFurther improvement was achieved in R12860-B by optimizing the\ncurvature because there was a crack observed on the photocathode\nsurface of R12860-A (in Figure~\\ref{fig:pmtcurvature}) after a high pressure water test. \n\n\\begin{figure}[h]\n    \\centering\n  \\includegraphics[width=4.8cm]{photosensor_R12860_curvature.pdf}\n    \\caption{Comparison of the glass bulb curvature between R3600 and R12860 PMTs. }\n    \\label{fig:pmtcurvature}\n\\end{figure}\n\nTo validate the degree of improvement, a dedicated test of the PMT in\nwater under high pressure was performed.  In a high pressure vessel\nfilled with water, one PMT was tested by increasing the pressure in\nsteps of 0.1\\,MPa above 0.5\\,MPa and waiting for 7 minutes in each\nstep.\n\nAt first, we tested 35 samples of the initial prototypes\n(R12860-A), and the depth range of implosion or crack was between 70\nand 155\\,meters pressure water (Figure~\\ref{fig:PMTdepth}).\nAccording to a survey of the glass thickness before the test, samples that\nimploded around a shallow depth of 70--100\\,meters were found to have a\nrelatively thin thickness of around 2.0--2.5\\,mm at the thinnest point as in\nFigure~\\ref{fig:PMTthickness}. \nTo mitigate this, a quality control step will be introduced whereby we\nmeasure the glass thickness and reject bulbs with thin thicknesses.\nThe glass quality, in regards to bubbles, foreign matter, cracks and\nthickness, is expected to be enhanced after improved training in bulb blowing\nover the year prior to mass production.\n\nThe R12860-B PMT improved with the new shape was also tested. 21\nR12860-B PMTs out of the total 25 did not implode up to 1.5\\,MPa (150\\,m equivalent) as shown in Figure~\\ref{fig:PMTdepth}.\nAll tested PMTs had sufficient high pressure resistance\nfor the 60\\,meter water depth of Hyper-K.\nIt should be noted that the test performed this time was in several\ndifferent conditions of PMT length or waterproofing, in order to find\nthe best design with high pressure bearing.\n\nOne of the two PMTs that imploded did so between 1.2 and 1.3\\,MPa\n(120\\,m and 130\\,m equivalent). This PMT had the smallest measured\nglass thickness around the neck out of all of the 25 tested R12860-B\nPMTs. Another PMT formed crack around the metal pins in the back,\nwhich the stress analysis determined to be the weakest part in the\nR12860.  This PMT had a waterproofed guard cover around the pins with\nthe same design as the R3600's, and its Polyethylene material is not\nsufficiently hard to guard the glass against the high pressure water\nthat exists above 1\\,MPa level.  Thus, the guard cover was improved\nwith a new hemisphere design made of PPS (Poly Phenylene Sulfide)\nresin and adopted for a subsequent test of fifty PMTs. \n\n\n\\begin{figure}[htbp]\n \\begin{tabular}{c}\n   \\begin{minipage}{0.45\\hsize}\n     \\begin{center}\n \\includegraphics[width=0.9\\textwidth]{photosensor_depth.pdf}\n \\caption{\nBroken pressure in tested R12860-A and R12860-B samples up to 1.5\\,MPa.\n\\label{fig:PMTdepth}}\n     \\end{center}\n   \\end{minipage}\n\n\\hspace{0.5cm}\n\n   \\begin{minipage}{0.5\\hsize}\n     \\begin{center}\n \\includegraphics[width=0.9\\textwidth]{photosensor_glassthickness.pdf}\n \\caption{\nRelation between a broken pressure in water depth and the minimum\nglass thickness of the R12860-A tested samples. The minimum glass\nthickness was estimated from several points around the neck of the\nbulb and the photocathode glass.\nSeveral PMTs that cracked, but did not implode, are shown in a blank\ncircle. \\label{fig:PMTthickness}}\n     \\end{center}\n   \\end{minipage}\n \\end{tabular}\n\\end{figure}\n\n        \\subsubsubsubsection{Quality Control of the PMT Glass Bulb}\n\\label{section:photosensors:Mechanical:Test} \nThe glass bulb is manufactured by hand; therefore it is difficult to expect uniform thickness throughout the mass production.  \nIn order to find an indication of a possible failure, the glass\nthickness was checked by an ultrasonic thickness gauge at various measurement points. \nAs indicated in Figure~\\ref{fig:PMTthickness} there exists a relation\nbetween the glass thickness and failure pressure, and therefore\nscreening PMTs based on glass thickness would be effective to minimize the failure. \nAlso the bulb was inspected by eye to find unexpected cracks, foam and foreign matter.  \n\nEventually after the production, individual PMTs will be tested before installation.\nIt is planned to load a high pressure in water over a few minutes before the installation to the Hyper-K tank, in order to reject a bad bulb.\nIn case of Super-K, 4,727 PMTs were checked with 0.65\\,MPa high pressure water to be safely used up to 40\\,m water depth for the reconstruction after the accident.\nSixteen PMTs of 4,727 were rejected using the test, that is 0.3\\% fraction.\n\nBefore the mass production of photosensors starts, we aim to establish\nthe quality control criteria.\nWe performed a screening test using fifty of the R12860 PMTs, in order\nto set the best criteria that we will use for the mass production. \n\n\\begin{itemize}\n \\item Remarkable failures such as bubbles, foreign matter and striae\n   were recorded and photographed during the quality check to measure\n   the size and to count the number of occurances.\n \\item The glass thickness was measured at 57 points.\n\\end{itemize}\n\nIt is noted that the photocathode and dynode are absent, but it does not matter because we only investigated the mechanical characteristics without measuring the detection performance.\nSimilar to the screening performed in Super-K, we tested all fifty PMTs in a high pressure water vessel. \nAt this time, the load pressure is assumed to be 0.95\\,MPa for the use in 60 meter water corresponding to 0.65\\,MPa for 40 meter in Super-K.\nSo far, fifty PMTs were tested after the production and there was no damage at 0.95\\,MPa.\nAll of the fifty PMTs were also tested up to 1.25\\,MPa for further\ninvestigation, but here we also found no damage in all PMTs.\n\nAccording to this test, the selection criteria and condition is optimized. \nIn the 2017 fiscal year, we will test 140 functional PMTs  \nand will apply the criteria that will be used in Hyper-K. \nAlthough the bulb design of the R12860 PMT is same as the previous test, the production will be improved as below.\n\\begin{itemize}\n \\item The metal mold is renewed. \n \\item The amount of glass is controlled by an automated machine to\n   reduce variations in glass thickness.\n \\item The automated measurement system of glass thickness is available for the precise and quick screening.\n\\end{itemize}\n\n\n        \\subsubsubsubsection{Degradation of the PMT Glass Bulb}\n\\label{section:photosensors:Mechanical:Degradation} \nAs for a degradation of the glass material, the HQE B\\&L PMT uses stable\nglass made of borosilicate like Super-K, which is highly resistant to\nan aqueous corrosion.\nA general pressure cooker test for the glass plate, in 100\\% humidity air at 121$^{\\circ}$C and\n0.2\\,MPa for 17 hours, showed almost no optical degradation, where the\nlargest degradation is about 1\\% only seen at around 350\\,nm wavelength\nand negligibly small.  A test to immerse a glass powder in\n98$^{\\circ}$C boiled pure water for an hour showed a small 0.03\\%\ndissolution in weight, which is very low compared to existing glass.\n\nMechanical characteristics were evaluated at\nfour different points on the PMT using PMT sampling glass taken from the bottom of the Super-K\ntank after 5-years of running. A composition ratio of material and\nbending strength are surveyed, and found to be comparable with other\nSuper-K PMTs stored in air at atmospheric pressure.  The high pressure\ntest was also performed on nine sample PMTs from Super-K -- three each\nfrom the top, bottom and barrel sections after 5-years in water -- and\nthere was no implosion with a 0.65\\,MPa load.\n\nTo examine the possibility of degradation by glass crystallization, the\nglass surface was surveyed using X-ray diffraction at the four\ndifferent points as well.  There was no diffraction peak originating\nfrom the crystallized glass, in either of the two samples;\none stored in air and the other from the bottom of\nSuper-K.\n\nThe number of dead channels of the ID photosensors is 0.4\\% after\nseven years of Super-K operation, including the ones with wrong\ncable connections.  It might also include PMTs with an unknown crack\nor implosion, and therefore we would expect a similar rate of glass\ndamage in Hyper-K, 1\\% at maximum for twenty years.\n\nIt is concluded that the borosilicate glass used in the HQE B\\&L PMT\nis expected to have stable material characteristics over twenty years, and\nseveral mechanical tests using the Super-K PMTs could\nfind no degradation in the long run.\nThe mass production and selection should be well managed such that the\nphysical damage rate is suppressed to the 1\\% level. This level of \nfailure is acceptable because the protective cover will avoid a chain implosion.\n\n\n        \\subsubsubsubsection{Shockwave Prevention Covers for PMTs}\\label{section:photosensors:cover} \n\n\\paragraph*{\\underline{Necessity of PMT covers}}\nAs described in previous sections, every effort has been and will be\nmade to avoid a PMT implosion inside the Hyper-K water tank.  Based on\nthe knowledge of the mechanical characteristics of the Super-K PMT,\nthe new 50\\,cm PMT has been designed to have enough strength for the\nsafe use at a water depth of 60\\,m, and its performance has been\ndemonstrated by hydrostatic pressure tests.  The production of the\nforty thousand PMTs for Hyper-K will be carried out under a strict\nquality control, and the total inspection of the products including a\npressure test will get rid of any individual PMTs having a higher risk\nof an implosion before their installation.\n\nHowever, the possibility of a single PMT implosion cannot be zero.  To\nprevent a chain reaction of imploding PMTs caused by the failure of a\nsingle one, all PMTs in Hyper-K are housed in the shockwave prevention\ncovers, similar to those installed on the 50\\,cm PMTs in Super-K after the catastrophic\naccident.\n\n\\paragraph*{\\underline{Basic design concept}}\nThe PMT cover for Hyper-K is designed on the same basic concept as\nthat for the Super-K PMT cover design.  In both detectors, the PMT\ncover has several small holes, and the gap between the PMT surface and\nthe cover is filled with the tank water.  Since the covers themselves\nare not usually exposed to the water pressure, there is no need to\ncare about any deformations caused by a long-term exposure to the high\nwater pressure.  On the other hand, the PMTs are constantly exposed to\nthe water pressure.  In the unlikely event of an imploding PMT, the\nwater pressure is immediately applied to the cover housing the broken PMT.  \nThe tank water slowly flows in through the small holes on the cover and fills up the vacuum region made by the PMT implosion.\nThe PMT cover is designed to have enough strength so that it can keep its shape even in such a case.  \nTherefore, the peak amplitude of the pressure shockwave is significantly\nreduced outside the PMT cover and thus cannot cause a chain reaction.\n\n\n\\paragraph*{\\underline{Super-K PMT covers}}\nIn developing the Super-K PMT cover, there were strict restrictions on\nits weight and shape, since the PMT supporting framework constructed\nin the tank had originally been designed to support the bare PMTs.\nAmong three major candidate designs, the PMT cover formed by combining\nan acrylic front window and a backside cover made of fiber-glass\nreinforced plastic (FRP) was selected in Super-K, since it had been\ndemonstrated by a hydrostatic test and a PMT implosion test that the\ncover would not be crushed even if the PMT inside would implode.\nThus, the Super-K group decided that the combination of the acrylic and FRP parts\nwith flange coupling bolts was suitable for a possible future PMT\nreplacement work and that the components can be readily produced.\n\nA cover formed by combining two half bodies of a molded acrylic\nproduct was another candidate. This full-acrylic cover also was found\nto have enough mechanical strength for withstanding the water pressure\nin the case of a PMT implosion, but it was not adopted due to its mass\nproduction difficulty, a higher manufacturing cost and uncertainty of\nthe strength when combining two bodies.\n\nA cover composed of an acrylic front cover and a stainless steel (SUS)\nbackside cover was also a candidate. \nThe reproducibility of its shape and thickness is better than those of FRP cases. \nThe SUS cover with a thickness of 2\\,mm could not pass a PMT implosion test, \nwhile PMT covers with a thicker SUS component were not possible \ndue to the restriction on the cover weight in Super-K, \n\n\\paragraph*{\\underline{Cover design selection for Hyper-K}}\nSince the depth of the Hyper-K water tank (60\\,m) is about 1.5 times\nlarger than that of the Super-K water tank (41.4\\,m), the Hyper-K PMT\ncovers have to withstand a higher pressure in the case of an implosion\nof the PMT inside.  The three cover designs which had been studied for\nSuper-K (i.e. acrylic+FRP, full acrylic, and acrylic+SUS covers) can\nbe also candidates for the Hyper-K PMT cover.\n\nHowever, it is now known that FRP does not just contain much more\nradioisotopes than those in the PMT glass but FRP itself also emits\nlight via chemiluminescence, as observed in Super-K.\nSince these unwelcome things can produce more background events for\nlow energy physics, we have decided not to use any FRP-made covers in\nHyper-K.  As for the PMT cover formed by a molded acrylic product,\nthere are still the problems such as a higher cost for an initial prototype.\nIf these problems could be solved\nwithin a reasonable time scale, the full-resin cover can be an\nalternative design for Hyper-K.\n\nThe cover made of the stainless steel contains less radioisotopes and\nis suitable for mass production.  Unlike the Super-K case, in which\nSUS-made PMT covers were not adopted due to the weight limit coming\nfrom the existing tank framework specification, the Hyper-K tank\nframework can be designed so that it can support the PMT system\nincluding the SUS covers of a sufficient strength.  Therefore, we have\nadopted the PMT cover design of a combination of acrylic and SUS\ncomponents for Hyper-K.\n\n\\paragraph*{\\underline{Hyper-K cover design}}\n\\begin{figure}\n  \\begin{center}\n  \\includegraphics[width=0.9\\textwidth]{protective_cover_shape.pdf}\n  \\caption{A schematic view of the shockwave prevention cover\nfor the Hyper-K ID PMTs.}\n  \\label{fig:protective_cover_shape}\n  \\end{center}\n\\end{figure}\n\nFigure~\\ref{fig:protective_cover_shape} shows the shape of the Hyper-K\nPMT cover.  The front-side cover with a partial spherical shape is\nmade of a UV transparent acrylic with thicknesses of 11\\,mm at the\ncenter position and 15\\,mm at the flange part\n(Figure~\\ref{fig:acrylic_cover_photo}), which is about 1.2 times\nthicker than the acrylic part of the Super-K PMT cover.  The light\ntransmittance of the acrylic cover measured in water is more than 95\\%\nfor a wavelength longer than 350\\,nm, which is reasonably good\nconsidering the quantum efficiency of the 50\\,cm Hyper-K PMT.  Since\nthe section modulus is proportional to the square of the thickness,\nthe Hyper-K acrylic cover could have about twice the strength of the\nSuper-K one, though this is just a crude estimation.  The backside\ncover with a combination of ring and circular truncated cone shapes is\nmade of stainless steel with a thickness of 3\\,mm.  The front acrylic\npart and the backside SUS part are connected to each other by flange\ncoupling bolts.\n\n\\begin{figure}\n  \\begin{center}\n  \\includegraphics[width=0.4\\textwidth]{acrylic_cover_photo.jpg}\n  \\caption{Acrylic window part of the PMT cover.}\n  \\label{fig:acrylic_cover_photo}\n  \\end{center}\n\\end{figure}\n\nThe detailed design of the cover, such as a thickness of each part,\nhas been determined based on a dynamic behavior analysis, simulating\nthe situation after a PMT implosion.  The analysis has shown that the\nPMT cover of the design mentioned above \nwill not be crushed even if the PMT inside would implode at a\nwater depth of 100\\,m.  The PMT implosion simulation should have some\nuncertainties, but we think they cannot change the conclusion that the\ncover will be functioning well at the depth of 60\\,m.  This was\nconfirmed by the performance demonstration tests described later. \n\nOn the acrylic cover, five holes with a diameter of 10\\,mm are formed;\none at the center and four near the flange.  These number, diameter\nand position of the holes on the acrylic cover are the same as those\nfor the Super-K PMT cover.  In developing the Super-K PMT cover, the\nsoundness of the holes on the acrylic cover against the water stream\ncaused by a PMT implosion had been checked. The test had shown that\nthe holes were not affected when exposed to a water stream with a\nhydraulic pressure of 0.65\\,MPa.  Therefore, the hole design should be\nsufficient for the Hyper-K PMT cover.\n\nIt is measured that acrylic is degraded to 77\\% by water absorption. \nTherefore, if it is confirmed by the PMT implosion test at 80\\,m that the covers have enough performance\nto prevent a chain implosion of PMTs, they are expected to be\nfunctioning for the duration of the Hyper-K lifetime.\n\n\n\n        \\subsubsubsubsection{Demonstration Test of Shockwave Prevention Covers}\n\\label{section:photosensors:implosiontest}\n\nThe Hyper-K PMT cover has been designed so that it will not be crushed in the unlikely event of a PMT implosion and will prevent the occurrence of the shockwave causing a chain reaction of imploding PMTs.\nTo measure the performance of our PMT cover, we carried out two tests.\nFirst, we carried out a hydrostatic pressure test of the PMT cover in Kamioka Mine to ensure the mechanical strength.\nIn this test, PMT cover was wrapped by a waterproof plastic bag and set in a high pressure vessel filled with water.\nWe pressurized the water surrounding the PMT cover, and the test showed that our PMT cover stood up to 1.1 to 1.5\\,MPa as it was designed.\n\nNext we carried out a prevention of the chain reaction of implosion with a mock-up of the PMT array of Hyper-K.\nWe utilized a test site formally used by Japan Microgravity Center (JAMIC), in Kami-Sunagawa, Hokkaido, Japan, for our chain implosion test.\nThe site has a vertical shaft depth of about 700\\,m with about 4\\,m diameter, filled with spring water.\n\nIn the test, nine PMTs were aligned 3 $\\times$ 3 and mounted on a framework with same 70\\,cm spacing in the Hyper-K inner detector.\nFigure~\\ref{fig:mock_implosion_test} shows the photograph of the mock-up just before sinking to deep water.\nThe central PMT was housed in the PMT cover, and implosion of the central PMT in the cover was induced by the apparatus which was designed to hit and crash the PMT remotely.\nThe others were bare to confirm there is no damage by a shockwave of the implosion at the center.\n\n\\begin{figure}\n  \\begin{center}\n   \\includegraphics[width=0.4\\textwidth]{cover_test.pdf}\n   \\caption{Mock of the chain implosion test. PMT encased in the cover was surrounded with bare PMTs to test the effect of the shockwave.\n            Four pressure sensors were set in front of the PMT array to measure the pressure of the shockwaves.\n            High-speed camera and lights are also set to investigate the crush visually.}\n  \\label{fig:mock_implosion_test}\n  \\end{center}\n\\end{figure}\n\nThis mockup was sunk into the deep water (60\\,m or 80\\,m) with pressure gauges and high-speed camera.\nThree trials for each depth were done, and we found that the pressure was reduced to less than 1\/100 and the cover was not crushed.\nFigure~\\ref{fig:shockwave_measurement} shows the measured pressure at the front of the central PMTs for each test at 60\\,m water depth.\nThe PMTs surrounding the center one were used in the second and third tests, but no damage was incurred due to a sufficient suppression of the shockwave by the central cover.\nWe conclude that our PMT cover works to prevent the chain implosion in both 60\\,m and 80\\,m water depths.\n\n\\begin{figure}\n  \\begin{center}\n   \\includegraphics[width=0.4\\textwidth]{woCover60mWater.pdf}\n   \\includegraphics[width=0.4\\textwidth]{wCover60mWater.pdf}\n   \\caption{Measured pressure at 70\\,cm ahead of the imploded PMT center at the 60\\,m water depth.\n            (Left panel) Shock wave without PMT cover. (Right panel) Shock wave with PMT cover. Three trials are shown in different colors.}\n  \\label{fig:shockwave_measurement}\n  \\end{center}\n\\end{figure}\n\nWe successfully established the cover design for Hyper-K using these tests.\nIn order to reduce cost and weight, improvement of the cover is on-going and several alternative ideas are under study.\n\n        \\subsubsubsubsection{Summary}\\label{section:photosensors:prospect} \n\\begin{figure}\n  \\begin{center}\n  \\includegraphics[width=1.\\textwidth]{photosensor_productionflow.pdf}\n  \\caption{Flow from the bulb manufacture to the operation in Hyper-K. }\n  \\label{fig:pmtproduction}\n  \\end{center}\n\\end{figure}\n\nThe strength of the PMT bulb was enhanced by the improved bulb design,\nquality check and pre-test in the high pressure vessel before the\ninstallation.  A schematic flow diagram of various measures to avoid a\nchain reaction of imploding PMTs is summarized in\nFigure~\\ref{fig:pmtproduction}.  In general, it is hard to expect\nthere will be no implosion in Hyper-K over the decades-long operation.\nThus, the protective cover is conservatively designed to avoid any\nchain implosion by suppressing the shockwave, and this design will be\ntested in advance of final production.\n\n        \\subsubsection{Photosensor for the Outer Detector}\\label{section:photosensors:OD} \n\n        The primary function of the Outer Detector is to reject the\n        incident cosmic ray muons that make up part of the background\n        in the measurement of nucleon decays and neutrino interactions\n        occurring in the Inner Detector.  The photosensor design for\n        the Hyper-K Outer Detector will be similar to that of the\n        successful Super-K Outer Detector using 20\\,cm Hamamatsu R5912\n        PMTs.  The OD PMT array is sparse relative to the ID PMT\n        array, resulting in a 1\\% photocathode coverage on the inner\n        wall of the OD.  To improve the light collection efficiency by\n        about a factor of 1.5, an acrylic wavelength shifting plate of\n        a 60\\,cm $\\times$ 60\\,cm square shape is placed around the\n        glass bulb of each of the 20\\,cm OD PMTs.\n\nThe pressure tolerance tests have demonstrated that R5912 PMTs could withstand the water pressure at a depth of 60\\,m.\nFor Hyper-K, the R5912 was improved using PPS resin for the guard cover like the R12860 PMT, as shown in Figures~\\ref{fig:R5912} and \\ref{fig:R5912pic}.\nIt also has improved high quantum efficiency of 30\\%\n\n\\begin{figure}[htbp]\n \\begin{tabular}{c}\n   \\begin{minipage}{0.5\\hsize}\n     \\begin{center}\n  \\includegraphics[width=1.0\\textwidth]{photosensor_8pmtassy2.png}\n  \\caption{Design of the HQE 20\\,cm box-and-line R5912 PMT.}\n  \\label{fig:R5912}\n     \\end{center}\n   \\end{minipage}\n   \\begin{minipage}{0.3\\hsize}\n     \\begin{center}\n \\includegraphics[width=1.0\\textwidth]{R5912HQEpic.jpg}\n  \\caption{Picture of the HQE 20\\,cm box-and-line R5912 PMT.}\n \\label{fig:R5912pic}\n     \\end{center}\n   \\end{minipage}\n \\end{tabular}\n\\end{figure}\n\nIn Super-K, there is no cover attached to the 20\\,cm OD PMTs. \nIt is noted that the volume of the 20\\,cm PMT bulb is 6\\% of the 50\\,cm one, with the similar glass thickness and far distance among OD PMTs. \nBecause of the 60\\,m deep water level compared with Super-K, we would reconsider a safety use of the PMT and cover with appropriate evaluation and tests such as implosion. \n\n        \\subsubsection{Alternative Designs}\\label{section:photosensors:Alternative} \nThere is still room to improve the Hyper-K performance with new possible\nphotosensors which are under development.  The key of the alternative\noptions is to show sufficient or superior physics sensitivities\nwhile demonstrating safe use in the water tank over a long period of\ntime at a reasonable cost.\nAll the listed alternative candidates are expected to be ready before\nthe Hyper-K construction period, but are currently shown as options\nbecause the product design is not finalized.\n\n        \\subsubsubsection{50\\,cm High-QE Hybrid Photodetector}\\label{section:photosensors:HPD} \nAnother new 50\\,cm photosensor with the better time and charge\nresolution than the\nexisting 50\\,cm photosensors is a combination semiconductor\ndevice, called a hybrid photodetector (HPD), and is made by Hamamatsu\n(R12850-HQE).\n\nThe HPD uses an avalanche diode (AD) instead of a metal dynode for the\nmultiplication of PEs emitted from a photocathode. \nA simple AD structure will have good quality\ncontrol in mass production, and a lower production cost than the\ncomplex of metal dynodes.  In order to collect PEs in a small 20\\,mm\ndiameter area of the AD, a high 8\\,kV is applied.  Related items such\nas the cable, connector and power supply were also developed.\n\nFigure~\\ref{fig:HPDamplification} shows that electrons are multiplied\nby a factor of $10^{5}$ with a combination of a bombardment gain and\nthen avalanche gain.  The gain is adjusted by the bias voltage applied\non the AD, around a few hundred volt, while the 8\\,kV is fixed.  The\nHPD is equipped with a pre-amplifier, so the resulting gain is\nequivalent to PMTs.\nThe size and surface material are almost the same as those of the 50\\,cm\nPMT as shown in Figure~\\ref{fig:HPDdesign}, thus the same support\nstructure and protective cover can be used.\n\n\\begin{figure}[htbp]\n \\begin{tabular}{cc}\n   \\begin{minipage}{0.5\\hsize}\n     \\begin{center}\n \\includegraphics[width=1.\\textwidth]{photosensor_HPDprinciple20inch.pdf}\n \\caption{Schematic view of amplification system on the HQE 50\\,cm HPD.\\label{fig:HPDamplification}}\n     \\end{center}\n   \\end{minipage}\n\n\\hspace{0.5cm}\n\n   \\begin{minipage}{0.45\\hsize}\n     \\begin{center}\n \\includegraphics[width=0.6\\textwidth]{photosensor_R12850HQE-10.pdf}\n \\caption{Design of the HQE 50\\,cm HPD (before waterproofing).\\label{fig:HPDdesign}}\n     \\end{center}\n   \\end{minipage}\n \\end{tabular}\n\\end{figure}\n\nThe single PE detection is significantly better, though it is still limited by the pre-amplifier because of a large junction capacitance of the AD (400\\,nF).\nThe transit time spread at single PE is 3.6\\,nsec in FWHM measured at the fixed threshold, as drawn by the red line in Figure~\\ref{fig:HPDTTS}.\nIt is superior to 7.3 and 4.1\\,nsec for the Super-K PMT (black) and B\\&L PMT (blue), respectively.\nWith the time walk correction using measured charge, the HPD resolution reached 3.2\\,nsec in FWHM as shown by the dotted magenta line.\n\nFigure~\\ref{fig:HPD1PE} shows the 15\\% charge resolution of the HPD as\nthe standard deviation at the single photoelectron peak as in the red line.\nIt is superior to both the 53\\% and 35\\% for the Super-K PMT (black) and B\\&L PMT (blue).\nIf the AD is segmented into two channels as a test, individual readout of HPD showed better resolution of 10\\% as shown by a dotted line due to a half junction capacitance.\nFurther five segmentation into a center channel and surrounding four channels brings a position sensitive detection with considering a hit probability in five channels\nby a slight focusing shift by different arrival points of photons on the glass.\nIt might help a good event reconstruction and background rejection.\n\n\n\\begin{figure}[htbp]\n \\begin{tabular}{cc}\n   \\begin{minipage}{0.45\\hsize}\n     \\begin{center}\n       \\includegraphics[width=0.9\\textwidth]{photosensor_HPDSPETTS.pdf}\n       \\caption{Transit time distribution at single photoelectron, compared with the Super-K PMT in dotted line. (HPD is added in Figure~\\ref{fig:PMTTTS}.) \\label{fig:HPDTTS}}\n     \\end{center}\n   \\end{minipage}\n\n\\hspace{1cm}\n\n   \\begin{minipage}{0.45\\hsize}\n     \\begin{center}\n       \\includegraphics[width=0.9\\textwidth]{photosensor_HPDPMTSPE_1ch.pdf}\n       \\caption{Single photoelectron distribution with pedestal, compared with the Super-K PMT in dotted line. (HPD is added in Figure~\\ref{fig:PMT1PE}.) \\label{fig:HPD1PE}}\n     \\end{center}\n   \\end{minipage}\n \\end{tabular}\n\\end{figure}\n\nThe waterproof HPD was successfully operated for 20 days in a dark box filled with water as shown in Figure~\\ref{fig:HPDwater}.\nIn very near future, the HPD would be a superior option to the PMTs, after successful long-term tests of all\nperformance and usability criteria.\n\n\\begin{figure}[htbp]\n   \\begin{center}\n     \\includegraphics[width=0.4\\textwidth]{photosensor_HPDwaterproof.pdf}\n     \\caption{The HQE 50\\,cm HPD (R12850) testing in water. \\label{fig:HPDwater}}\n   \\end{center}\n\\end{figure}\n\n\n        \\subsubsubsection{Smaller Photosensors}\\label{section:photosensors:SmallPD} \nPhotosensors with a 20--30\\,cm aperture\ncan also be an alternative option.\nThese are available with the HQE and by another manufacturer.\nUsing two small photosensors are comparable to a single 50\\,cm photosensor as for the detection efficiency, or a little larger aperture size than the 20\\,cm photosensor can be considered as the OD photosensor.\nA smaller 7.7\\,cm PMT with a large photo-collection plate is also a possible alternative option with a low cost for the OD photosensor system.\n\n\n\n\n        \\subsubsubsubsection{20--30\\,cm High-QE PMTs}\\label{section:photosensors:HQEODPMT} \nBased on the successful development of the 50\\,cm HQE B\\&L PMT, 20\\,cm\nor 30\\,cm PMTs can obtain superior performance compared to the existing small PMTs.\nBy applying the same techniques, the 20\\,cm and 30\\,cm PMTs with a high\nQE box-and-line dynode and improved performance will be available\neasily by scaling the 50\\,cm PMT down with a similar design.  The\nperformance is expected to be equivalent or better compared with the\n50\\,cm HQE B\\&L PMT.\n\nPrior to that, the HQE 30\\,cm PMT, R11780-HQE by Hamamatsu, was\ndeveloped aimed at a large water Cherenkov detector planned in US for\nthe LBNE project.  It reached a QE of 30\\% and pressure rating over 1\nMPa.  Further improvement was tried, and a new bulb design of the HQE\n30\\,cm PMT based on the R12860-HQE and R11780-HQE was made.  In order\nto validate the high pressure tolerance, a test in a high pressure\nwater was performed on three samples at Kamioka.  As a result, all the\nsamples got no implosion up to 150\\,meter water equivalently. \n\nUntil recently, photomultiplier tubes with an aperture over 25\\,cm have\nbeen almost exclusively supplied to the market by Hamamatsu.  It is\nimportant that additional vendors come in the marketplace for price\ncompetition and for additional supply capacity.\n\nET Enterprises Limited ADIT, now a US-based PMT manufacturer in Texas,\nhas been developing a large area PMT, financially supported by NSF.\nTesting of the operational first generation 28\\,cm HQE PMTs have been\nperformed at Pennsylvania, UC Davis, etc.  showing comparable\nefficiency and charge measurement performance to those of\nsimilarly-sized Hamamatsu HQE PMTs.\n\nIf successfully produced, the PMTs can be a cost-effective\nalternative to Hamamatsu for Hyper-K OD PMTs.\n\n\n        \\subsubsubsubsection{7.7\\,cm Photodetector Tube}\n\\label{section:photosensors:OD3inchPMT} \n\n\\par\nIn this section we will focus on an alternative design of the\nouter-detector using the 7.7\\,cm photosensors. It is expected that the\ncosts will be lower than the nominal configuration as well as also\nproviding an extended market and production capacity.  At the moment\nthe nominal configuration of the outer-detector consists of an array\nof 20\\,cm hemispherical photosensors placed on a grid to reach a\nphoto-coverage of 1\\%, as represented in\nFig.~\\ref{fig:pmt_arrangement}.  These photosensors are mounted on\nwavelength shifting plates, to improve the light collection.\n\nBy using 7.7\\,cm photosensors instead of the 20\\,cm, the number of\nphotosensors is multiplied by six to keep the same coverage.  The\nphotosensors are therefore closer to each other thus increasing the\nnumber of hits collected, and improving the coincidence accuracy.\nMoreover, the outer-detector water thickness is about 1\\,m, and\nparticles will produce less light and in a narrower region ---\ncompared to the Super-Kamiokande detector where the outer-detector\nwater thickness is about 2\\,m.  Thus, a setup with more photosensors\ncloser to each other should allow a better sensitivity, especially\nwith the configuration of Hyper-Kamiokande with a reduced water\nthickness.\n\nWe are currently testing candidate 7.7\\,cm photosensors to be used in\nthe outer-detector, see Fig.~\\ref{fig:QMSetup} for the teststand, as\nwell as implementing the outer detector in that simulation and\nreconstruction, which is needed for optimizing the configuration.\n\n\\begin{figure}[!htb]\n  \\centering \\includegraphics[width=0.5\\linewidth]{setup.png} \\caption{Setup\n    for photosensors testing at Queen Mary University of London.  This\n    picture was taken with a 20\\,cm ET9354KB (on the left) and a\n    7.7\\,cm ET9302B (on the right) photosensors.  One can see the XY\n    stage above the photosensors (blue) which moves the optical fibre\n    (in yellow) along the X-Y axis. The optical fibre guides the light\n    out from the LED driver to the blackbox.}  \\label{fig:QMSetup}\n\\end{figure}\n\nOur tests have been performed using the ET9302KB and other\nphotosensors.  The ET9302KB was extensively tested and many of its\nparameters measured.  It has a QE of 30\\% and a small dark current\nrate which has been measured at 400\\,Hz --- about ten times less than\ntypical rates for 20\\,cm photosensors --- and a small after-pulse rate\nwith respect to the gain.\n\n\n\nThe ET9302KB would fit perfectly in a configuration with several\nphotosensors like the outer detector, since the number of accidental\ncoincidences inherently raised by the amount of photosensors used will\nbe reduced thanks to its low-noises properties.\n\nFurthermore, these photosensors also showed excellent linearity and\nresolution, allowing an accurate reconstruction of the energy of the\nparticles inside the outer-detector.\nFigure~\\ref{fig:lin:ET9302KB} shows the linearity of the photosensor,\nwhere deviation was measured to be of the order of 1\\%.\nThe resolution at 1000 photoelectrons collected\n--- which is what is expected for a few-MeV gamma event for example ---\nis measured to be 2\\%.\n\n  \\begin{figure}[!htb]\n      \\begin{minipage}[t]{1.\\linewidth}\n       \\includegraphics[width=0.6\\linewidth]{lin}\n       \\caption{Linearity for the 7.7\\,cm photosensor ET9302KB,\n       measured with an LED driver ranged from few to several thousands\n       photons.}\n       \\label{fig:lin:ET9302KB}\n     \\end{minipage}\n  \\end{figure}\n\n\nThe newly improved photosensor from ET Enterprises Ltd., D793KFLB, with a transit time spread of 1.6\\,nsec, a nominal gain $6 \\times 10^{6}$ at 850--1100\\,V and nominal dark rate 1000\\,Hz at 800\\,V is currently under test. \nPreliminary results showed excellent linearity and resolution.\n\nThe photosensors are hemi-spherical and will also be mounted on\nwavelength shifting plates, which allow an easy way to enhance the\nlight collection.  Wavelength shifting plates will reemit the absorbed\nphotons in the direction of the photo-cathode of the photosensors at a\nwavelength around 400\\,nm, where the quantum efficiency of the\nET9302KB is maximum.\n\nOuter detector alternative configurations with 7.7\\,cm photosensors\nwill be evaluated in simulation and the best setting will be selected\naccordingly, using realistic cosmic flux files from the Super-K\ncollaboration to test the performance.\n\nThe chosen configuration should achieve at least the minimum detection\nefficiency to have sufficient ability to veto cosmic rays or catch\nthrough-going particles.\n\nFinally, also another alternative of using 12.7\\,cm photosensors,\ni.e. an intermediate solution between 7.7\\,cm and 20\\,cm photosensors,\nis planned to be addressed in both the optimization in simulation of\nthe configurations and in testing the photosensors.\n\n        \\subsubsubsection{Multi-PMT Optical Module }\\label{section:photosensors:MultiPMT}\n\\par The concept of an optical module with a 20 to 40\\,cm PMT housed in a glass pressure vessel\n     has been developed the past decades for neutrino telescopes in\n     water and ice (DUMAND~\\cite{DUMAND}, Baikal~\\cite{Baikal},\n     NESTOR@~\\cite{NESTOR,NESTOR-PMT}, ANTARES~\\cite{Antares,\n     Antares-PMT}, AMANDA~\\cite{AMANDA} and IceCube~\\cite{IceCube}).\n     For KM3NeT~\\cite{KM3NetDR}, the km$^3$ neutrino observatory under\n     construction in the Mediterranean, a single large area phototube\n     has been replaced by 31 7.7\\,cm PMTs packed in the same glass\n     pressure vessel. This new fly's eye photosensor concept, in which small PMTs replace a\n     single PMT, has been dubbed the ``multi-PMT Optical Module (mPMT)''.\n     The mPMT has several advantages as a photosensor module:\n     \\begin{itemize}\n     \\item Increased granularity\\\\\n         The increased granularity of mPMT provides enhanced event reconstruction, in particular for\n     multi-ring events, such as in proton decays and multi-GeV neutrinos which is important for mass hierarchy studies,\n     and for events near the wall where fiducial volume is defined.\n     Since each of the PMTs have different orientations with limited field of views,\n     the mPMT carries information on the direction of each detected photon.\n     This directional information effectively reduces the dark hit rate since the dark hits only applies to the field of view\n     of the PMT. As a result, it improves signal-to-noise separation for low energy events such as solar\/supernova\n     neutrinos and the neutron tagging as discussed in Section~\\ref{section:pdecay-coverage}.\n     \\item Mechanically safe pressure vessel with readout and calibration integrated\\\\\n     The pressure vessel protect against implosion. Since the vessel is filled with\n     small PMTs, electronics and the support structure without large void,\n     shock waves would be suppressed even when implosion takes place.\n     The pressure vessel contains digitization electronics and calibration sources,\n     providing natural solutions for readout and calibrations.\n    \\item Cost, geomagnetic field tolerance, and fast timing for 7.7\\,cm PMT's\\\\\n     Economical mass-production of 7.7\\,cm PMT's in particular for medical use\n     provides almost the same cost per photocathode area for 7.7\\,cm PMT compared\n     to larger PMTs. For example, KM3NeT claims the cost per photocathode\n     area is cheaper than for the 20 to 40\\,cm PMTs.\n     Additional advantages of small PMTs are their much lower sensitivity to the Earth's magnetic field,\n     which makes magnetic shielding unnecessary~\\cite{KM3Net:VLVnT15.1}, and potentially better\n     timing resolution, which could further reduce the dark hit background and event reconstruction.\n     The failure rate of small PMTs is of the order of 10$^{-4}$\/year. Any\n     loss of a single PMT would affect the detector performance\n     minimally compared to the loss of one large area PMT.\n     \\item Complex structure, an opportunity for international contributions\\\\\n     A potential draw back of mPMT compared to off-the-shelf 50\\,cm PMT is its complicated structure\n     with multiple components. This could be regarded as an advantage for international contributions\n     where each partner can take on mPMT module parts and\/or assembly and testing procedures,\n     as is successfully done for the KM3NeT mPMT and IceCube mDOM (Digital Optical Module) constructions.\n       \\end{itemize}\n\n\n\n\n     \n  \n\n\n\n\\begin{figure}[htbp]\n \\begin{tabular}{c}\n   \\begin{minipage}{0.55\\hsize}  \n     \\begin{center}\n \\includegraphics[width=0.8\\textwidth]{mPMT_NuPRISM.png}\n \\caption{Multi-PMT conceptual drawing with 19 7.7\\,cm PMTs as ID detectors and the OD detectors on the other half. Each small PMT has\n a reflector cone. An 50-cm acrylic covers on a cylindrical support is used as pressure vessel. Readout electronics and calibration\n sources are imbedded inside the vessel.\n \\label{fig:mPMT}}\n     \\end{center}\n   \\end{minipage}\n\\hspace{0.5cm}\n   \\begin{minipage}{0.38\\hsize}   \n     \\begin{center}\n \\includegraphics[width=0.9\\textwidth]{3inchPMT.jpg}\n \\caption{A Hamamatsu R12199-02 7.7\\,cm PMT that is currently used in KM3NeT and considered for\n IceCube-Gen2 modules. As this passed the Hyper-K PMT requirements, it is also a good candidate\nfor a Hyper-K mPMT. \\label{fig:3inchPMT}}      \n     \\end{center}\n   \\end{minipage}\n \\end{tabular}\n\\end{figure}\n\n\n\n\\subsubsubsubsection{The Multi-PMT Reference Design} \\label{multiPMT reference design}\nAs a reference design for Hyper-K, we consider 50\\,cm size vessel, the same size as the baseline\n50\\,cm PMT so that the same mechanical support structure can be used. This would allow part\nof the photocathode coverage to be replaced by an mPMT without major change in the support structure.\nFig.~\\ref{fig:mPMT} shows the 50\\,cm diameter mPMT design originally developed for the NuPRISM detector.\nThere are 19 7.7\\,cm PMTs looking inner detector side and 6 7.7\\,cm PMTs looking outer detector side.\nThe 7.7\\,cm PMTs will be supported by a 3D printed foam structure and\noptically and mechanically coupled by Silicon Gel to an acrylic pressure sphere.\nWhen reflector cones are added to each 7.7\\,cm PMT to increase the effective photocathode area\nby about 30\\%, we get about half of the effective photocathode area as a single 50\\,cm PMT.\nFor the low energy events such as neutron tagging, the number of detected photons will be reduced\nby a factor of two, which can be compensated by the reduction of the effective dark hit rate with the\nlimited field of view. A back-of-the-envelope calculation shows that the neutron tagging efficiency\nwould be similar or better than all 50\\,cm PMT, although more detailed simulation studies are required.\n\n\n\n\\par Currently, there are two main candidate 7.7\\,cm PMTs that have\n     been developed specifically for KM3NeT, the Hamamatsu R12199-02\n     (see Fig.~\\ref{fig:3inchPMT}) and the ET Enterprises\n     D792KFL\/9320KFL. They have been measured in\n     detail~\\cite{VLVnT13.1,VLVnT13.2,VLVnT13.3} and would be adequate\n     for Hyper-K. Both PMTs have a high peak QE of $\\sim$ 27\\% at\n     404\\,nm. Their collection efficiency is more than 90\\%.  The\n     transit time spread is about 4\\,nsec at FWHM and the dark rate is\n     200--300\\,Hz. \n     \nRecently, improved photosensors from both Hamamatsu and ET Enterprises have been made available, and are currently under test to fully characterize their performances.\nPreliminary measurements of the newly developed Hamamatsu R14374 7.7\\,cm PMT showed an improved TTS and dark rate respect to Hamamatsu R12199 PMT, finding a gain $\\sim 5 \\times 10^{6}$ at $\\sim$ 1.2\\,V with negative HV ($\\sim$ 1.1\\,V with positive HV) and a TTS of 1.35\\,ns with negative HV ($\\sim$ 1.58\\,ns with positive HV) at gain $\\sim 5 \\times 10^{6}$. \nThe dark hit rates were measured to be $\\sim$ 0.3\\,kHz at the 25\\,$^\\circ$C temperature in air. \nAn improved photosensor from ET Enterprises Ltd., D793KFLB, which has been recently made available, and is currently under test. \nPreliminary results are given in Section \\ref{section:photosensors:OD3inchPMT}.\nIn addition to Hamamatsu and ET Enterprises PMT's, HZC XP72B20 is currently reducing the TTS and dark rate and becomes a candidate.\nHZC has a mass production capacity and potential to provide significantly lower cost. \nMELZ PMT, that is considered for the JUNO experiment, is another potential candidate.\n\n\n\n\n\\par The price for $\\sim$19 7.7\\,cm PMTs is comparable or cheaper than\none large area 50\\,cm HQE B\\&L PMT. In addition, the cost could be reduced\ndue to the competition between several companies like\nHamamatsu, ET Enterprises and HZC in the next couple of years.\nThe front-end electronics\nwill be situated in modules in the water near the PMTs which need to\nbe pressure tolerant, water-tight and use water-tight connectors (see\nSection~\\ref{section:electronics-general}). This cost may be reduced\nby encasing the front-end electronics inside the same pressure vessel\nas the ID and OD PMTs. The HV generation for each ID PMT can be done\non a board attached to the PMT base. Only one water-proof cable for\nboth communication, LV and signal can then be connected to the whole\nmodule through a penetrator, as done in previous deep water neutrino\nexperiments.\n\n\n\\par A flexible implementation in the simulation software WCSim (see Section~\\ref{software:WCSim}) makes\n     optimization of the reference design possible and will be the\n     main topic for further study together with further improvement of\n     small photosensors. Based on the results of the optimization\n     studies \n     facilities a prototype mPMT will be built and tested.\n     Figure~\\ref{fig:mPMT_display} shows an event display of mPMT for NuPRISM\n     which assumes mPMT for all the photosensors. The event shows an improved\n     ring pattern recognition by mPMT for an event.\n\n\\begin{figure}[htbp]\n     \\begin{center}\n\\includegraphics[width=0.95\\textwidth]{Electron_Unrolled_Zoom.png}\n\\caption{An event display of mPMT for NuPRISM which assumes mPMT for all the photosensors. }\n\\label{fig:mPMT_display}\n     \\end{center}\n\\end{figure}\n\n\n\\subsubsubsubsection{The mPMT Prototype}\n\n\\par    The design of a KM3NeT mPMT module is restricted by the size of\n     commercially available transparent pressure vessels: borosilicate\n     glass spheres with a diameter of 33.3\\,cm and 43.5\\,cm.\n     The glass sphere also contains radioactive contaminants that emit Rn into the water,\n     which is not a problem for KM3NeT where the radioactive background rate is limited\n     by $^{40}$K in the sea water.\n     Pure glass could be used for the vessel in Hyper-K, but the cost\n     should increase, therefore a good alternative is given by\n     acrylics.  A first prototype of an mPMT of the future Hyper-K\n     experiment is under construction mainly to demonstrate the\n     effectiveness of a vessel system based on acrylic and to study a\n     better solution for the PMT Read-out system. Other prototypes\n     will be built and tested when the final design of the mPMT will\n     be defined on the basis of the optimization studies.  Several\n     comparative tests are ongoing to identify the best acrylic for\n     the experimental requirements, including optical tests, stress\n     and compression mechanical tests and thermal tests. The water\n     absorption tests are based on Nuclear Reaction Analysis (NRA),\n     whereas radioactivity contamination measurements are based on\n     gamma spectroscopy.  The UV transparency has\n     been measured for several commercial acrylics.  For some sample,\n     the light transmittance of the acrylic cover measured in water is\n     greater than 95\\% for a wavelength longer than 350\\,nm.  The\n     pressure vessel will be realized starting from two acrylic\n     hemispheres.  The hemisphere fabrication processes starts by blowing the\n     flat sheet onto a positive mold. A mold made from two parts\n     (i.e., positive and negative) could be used to have\n     a more uniform thickness. The final design of the vessel will be\n     defined on the basis of the simulation studies for the mPMT\n     optimization.\n\n\\par For the pressure vessel closure system, the two acrylic\nhemispheres might be glued by using a specific glue for\nacrylics. However, glue itself could emit light, producing more\nbackground events for low-energy physics. A mechanical system has been\nevaluated, both to avoid fluorescence emissions, and to guarantee a\nlonger endurance, and to simplify the anchorage to the tank frame and\nthe implementation of the cooling system of the mPMT. Thus a metallic\nring is being evaluated, modifying the spherical final shape by\nextending the equatorial zone with a cylinder.\nThe acrylic vessel in the mPMT is similar in price to the protective cover required for the single large PMTs.\n\n\\begin{figure}[htbp]\n     \\begin{center}\n \\includegraphics[width=0.7\\textwidth]{.\/design-photosensor\/figures\/VesselDesign.png}\n \\caption{Preliminary\n design of the mPMT vessel with its cooling system in the equator of\n the sphere.}\n \\label{fig:VesselDesign}\n     \\end{center}\n\\end{figure}\n\n\n\n\n\nThe first mPMT prototype will have an acrylic vessel with a diameter of 17 inches\n(432\\,mm, as in KM3NeT).  Several commercial acrylics have been studied\nand tested. EVONIK UV transmitting PLEXIGLAS GS \\footnote{\\href{http:\/\/www.plexiglas.net}{http:\/\/www.plexiglas.net}} has been\nchosen for the construction of the first prototype.\nThe thickness of the vessel has been studied on the basis of\nsimulation and three vessel prototype with thickness of 12, 15 and 18\\,mm\nwill undergo to preliminary pressure tests.  The sphere will house\n26 PMTs with a photocathode diameter of 7.7\\,cm.\n\n19 PMTs will view the inner detector side and 7 PMTs\nwill view the outer detector side.  The final number of PMTs in the\nmPMT will be defined on the basis of simulation studies.\nThe PMTs will be placed into a 3D printed structure and will be\noptically and mechanically coupled by Silicon Gel to an acrylic\npressure sphere.\nThe optical gel used for this prototype will be the same as in\nKM3NeT. The compatibility between optical gel and acrylic has been\nchecked and the transparency of acrylic+optical gel has been measured.\nFor the final mPMT design, other options for the optical gel are under\nstudy.\n\nFor the present prototype module, Hamamatsu R12199 PMTs are used. They\nare arranged in 3 rings of PMTs in the hemisphere looking at the inner\ndetector with zenith angles of 33$^\\circ$, 56$^\\circ$, 72$^\\circ$,\nrespectively. In each ring 6 PMTs are spaced at 60$^\\circ$ in azimuth\nand successive rings are staggered by 30$^\\circ$. The central PMT in\nthe hemisphere point at a zenith angle of 0$^\\circ$, looking at the\ninner detector axis.  Seven PMTs are arranged in the hemisphere looking at\nthe outer detector. Six of them are arranged in one ring which opens a\nhalf angle of 33$^\\circ$ with respect to the nadir.\n\nA basic Cockcroft-Walton voltage multiplier circuit design developed\nby KM3NeT Collaboration~\\cite{Timmer2010GF} is used to generate multiple\nvoltages to drive the dynodes of the photomultiplier tube. The system\ndraws less than 1.5\\,mA of supply current at a voltage of 3.3\\,V with\noutputs up to -1400\\,V$_{dc}$ cathode voltage.\nA passive cooling system, based on the heat conduction mechanism,\naimed at keeping the temperature of the electronic components as low\nas possible, thus maximizing their lifetime has been designed in order\nto optimize the transfer of the heat generated by the electronics.\nIn KM3NeT the time over threshold (ToT) strategy is exploited; this is\nnot a good solution for Hyper-K project in which charge measurement is\nimportant.\nTo fulfill the requirements of low consumption and charge and time\nresolution, a solution based on a Sample\\&Hold plus ADC has been\ninvestigated. Several commercial low power and highly versatile ASIC\nfrom Weeroc \\footnote{\\href{http:\/\/www.weeroc.com}{http:\/\/www.weeroc.com}} are under study.\n\nFor this prototype, the module was developed as a complete stand-alone\ndetector and it will fully test both in air and in water.\n\n        \\subsubsubsection{Alternative Cover Design}\n\\label{section:photosensors:coveralternatives}\n\nAn alternative design of the shockwave prevention cover is a cheap stainless steel tube with the acrylic window, instead of the conical shape cover, as shown in Figure~\\ref{fig:AcrSUStubecover}.\nBenefit of the mass production and installation is expected due to the simple outer shape.\n\nFigure~\\ref{fig:AcrPPScover} shows another possible idea to use a resin instead of stainless steel.\nIt can give a cheap and light weight option.\nOne of the possible material is PPS resin mixed with a reinforced filler.\n\n\\begin{figure}[h!]\n \\begin{tabular}{cc}\n   \\begin{minipage}{0.35\\hsize}\n     \\begin{center}\n \\includegraphics[width=1.\\textwidth]{photosensor_AcrSUStubeCover_outside.pdf}\n \\caption{Alternative cover design comprised of a stainless steel tube and the acrylic window.\\label{fig:AcrSUStubecover}}\n     \\end{center}\n   \\end{minipage}\n\n\\hspace{1.0cm}\n\n   \\begin{minipage}{0.4\\hsize}\n     \\begin{center}\n \\includegraphics[width=1.\\textwidth]{photosensor_AcrPPSCover_outside.pdf}\n \\caption{Alternative cover design comprised of a resin cover and the acrylic window.\\label{fig:AcrPPScover}}\n     \\end{center}\n   \\end{minipage}\n \\end{tabular}\n\\end{figure}\n\nThe light weight of the cover allows for a lower cost of the tank structure because the current cover weight is much heavier than the PMT and dominant in overall weight of the photosensor system.\nFast production and installation are also cost effective.\nIn both alternative cases, the design was evaluated by a simulation assuming a high pressure load outside, but it is still preliminary.\nThe hydrostatic test and demonstration test with implosion are necessary to ensure that the alternative covers are feasible for Hyper-K.\n\n\\subsubsection{Schedule}\\label{section:photosensors:schedule} \n\nBefore the photosensor mass production, it takes 0.5 years to complete\na design of a production line, and 1--1.5 years for the setup and\nstartup of the equipment in the factory.  The capacity of the factory\nproduction is expected to be 11k 50\\,cm and 4k 20\\,cm photosensors per\nyear.\n\n\nA test in Super-K using about a hundred of the B\\&L PMTs starts\nfrom 2018 to demonstrate the Hyper-K photosensor system.\nMoreover, criteria for the quality control including a\nselection with high pressure load will be established by the production.\n\nTable~\\ref{photosensoroption} summarizes the default design and\nalternatives for photosensors described in this section.\nEven without the alternatives, Hyper-K has sufficient\nperformance with a realistic time line using the default options.\nFurther improvements could be achieved and could be available in\ntime for Hyper-K tank construction.\n\n\\clearpage\n\n\\begin{table}[!h]\n \\begin{center}\n  \\begin{tabular}{ll|ll|l}\n   \\hline \\hline\n      Items & & Type & & Remaining studies \\\\\n   \\hline\n   \\hline\n    \n     \\multicolumn{2}{l|}{{\\bf ID photosensor}}  &\n     \\begin{minipage}{3.6cm}\n     \\begin{flushleft}\n     {\\bf HQE 50\\,cm\\\\\n          \\ B\\&L PMT}\\\\\n     {\\bf \\small (Hamamatsu\\\\\n                 \\ R12860-HQE)} \\\\\n     \\end{flushleft}\n     \\end{minipage}\n     &\n     \\begin{minipage}{3.cm}\n     \\includegraphics[width=\\textwidth]{photosensor_R12860design.pdf}\n     \\end{minipage}\n     &\n     \\begin{minipage}{7.cm}\n     \\begin{flushleft}\n     Mass test using about 100 PMTs \\\\\n     \\ with reduced dark hit rate\n     \\end{flushleft}\n     \\end{minipage}\n     \\\\\n   \\hline\n    \n     & (Alternative) &\n     \\begin{minipage}{3.6cm}\n     \\begin{flushleft}\n     HQE 50\\,cm HPD \\\\\n     (Hamamatsu R12850-HQE)\\\\\n     \\end{flushleft}\n     \\end{minipage}\n     &\n     \\begin{minipage}{2.4cm}\n     \\includegraphics[width=\\textwidth]{photosensor_HPDwaterproof.pdf}\n     \\end{minipage}\n     &\n     \\begin{minipage}{7.cm}\n     \\begin{flushleft}\n     Electronics tuning,\\\\\n     Confirmation of pressure resistance,\\\\\n     Long-term proof test in water,\\\\\n     Cost estimation\n     \\end{flushleft}\n     \\end{minipage}\n     \\\\\n   \\hline\n   \\hline\n    \n     \\multicolumn{2}{l|}{ {\\bf OD photosensor }} &\n     \\begin{minipage}{3.6cm}\n     \\begin{flushleft}\n     {\\bf HQE 20\\,cm\\\\\n          \\ B\\&L PMT}\\\\\n     {\\bf \\small (Hamamatsu\\\\\n                 \\ R5912)}\\\\\n     \\end{flushleft}\n     \\end{minipage}\n     &\n     \\begin{minipage}{2.5cm}\n     \\includegraphics[width=\\textwidth]{photosensor_8pmtassy2.png}\n     \\end{minipage}\n     &\n     Test in high pressure water\\\\\n   \\hline\n      & (Alternative)                      & HQE 20--30\\,cm PMT                & &  HQE study using prototype,            \\\\\n      &                                    &                                   & &  Trial manufacture and necessary test \\\\\n   \\hline\n      & (Alternative) &\n     \\begin{minipage}{3.6cm}\n     \\begin{flushleft}\n     7.7\\,cm PMT\n     \\end{flushleft}\n     \\end{minipage}\n     &\n     \\begin{minipage}{2.cm}\n     \\includegraphics[width=\\textwidth]{3inchPMT.jpg}\n     \\end{minipage}\n     &\n     \\begin{minipage}{7.cm}\n     \\begin{flushleft}\n     Performance simulation in Hyper-K,\\\\\n     Detection efficiency measured with prototype,\\\\\n     Selection between design\n     \\end{flushleft}\n     \\end{minipage}\n     \\\\\n   \\hline\n   \\hline\n     \\multicolumn{2}{l|}{\n     \\begin{minipage}{3.cm}\n     \\begin{flushleft}\n     {\\bf (ID and OD \\\\\n     \\ photosensor\\\\\n     \\ alternative) }\n     \\end{flushleft}\n     \\end{minipage}\n     }\n     &\n     \\begin{minipage}{3.6cm}\n     \\begin{flushleft}\n     Multi-PMT \\\\ optical module\\\\\n     \\end{flushleft}\n     \\end{minipage}\n     &\n     \\begin{minipage}{2.cm}\n     \\includegraphics[width=\\textwidth]{mPMT_NuPRISM.png}\n     \\end{minipage}\n     &\n     \\begin{minipage}{7.cm}\n     \\begin{flushleft}\n     Performance simulation in Hyper-K,\\\\\n     Cost estimation,\\\\\n     Selection of 7.7\\,cm photosensor,\\\\\n     Trial manufacture and necessary test\n     \\end{flushleft}\n     \\end{minipage}\n     \\\\\n   \\hline\n   \\hline\n     \\multicolumn{2}{l|}{ {\\bf ID cover }}      &\n     \\begin{minipage}{3.6cm}\n     \\begin{flushleft}\n     {\\bf Acrylic and \\\\\n      \\ stainless steel\\\\\n      \\ cone}\n     \\end{flushleft}\n     \\end{minipage}\n     &\n     \\begin{minipage}{2.8cm}\n     \\includegraphics[width=\\textwidth]{protective_cover_shape.pdf}\n     \\end{minipage}\n     &\n     \\begin{minipage}{7.cm}\n     \\begin{flushleft}\n     Improved with light weight,\\\\\n     Test of the improved design\n     \\end{flushleft}\n     \\end{minipage}\n     \\\\\n   \\hline\n      & (Alternative) &\n     \\begin{minipage}{3.6cm}\n     \\begin{flushleft}\n     Acrylic and \\\\\n     \\ stainless steel\\\\\n     \\ tube\n     \\end{flushleft}\n     \\end{minipage}\n     &\n     \\begin{minipage}{1.7cm}\n     \\includegraphics[width=\\textwidth]{photosensor_AcrSUStubeCover_outside.pdf}\n     \\end{minipage}\n     &\n     \\begin{minipage}{7.cm}\n     \\begin{flushleft}\n     Design and simulation,\\\\\n     Implosion test\n     \\end{flushleft}\n     \\end{minipage}\n     \\\\\n   \\hline\n      & (Alternative) &\n     \\begin{minipage}{3.6cm}\n     \\begin{flushleft}\n     Acrylic and \\\\\n     \\ full resin\n     \\end{flushleft}\n     \\end{minipage}\n     &\n     \\begin{minipage}{2.cm}\n     \\includegraphics[width=\\textwidth]{photosensor_AcrPPSCover_outside.pdf}\n     \\end{minipage}\n     &\n     \\begin{minipage}{7.cm}\n     \\begin{flushleft}\n     Design and simulation,\\\\\n     Prototype production,\\\\\n     Implosion test\n     \\end{flushleft}\n     \\end{minipage}\n     \\\\\n   \\hline \\hline\n  \\end{tabular}\n  \\caption{ Summary of the current base design in bold type and alternative options related with photosensors.\n            The remained studies of the baseline photosensors and cover will be finalized in fiscal year 2017.\n           }\n  \\label{photosensoroption}\n \\end{center}\n\\end{table}\n\n\n\\subsection{Summary of the Hyper-Kamiokande detector timeline}\n\\graphicspath{{design-summary\/figures\/}}\n\n\\begin{figure}[htbp]\n  \\begin{center}\n  \\includegraphics[width=1.0\\textwidth]{HK_timeline_NamatameCheck_v2_Eng.pdf}\n   \\caption{Construction period for the 1TankHD{}}\n  \\label{fig:construction-perod-1TankHD}\n  \\end{center}\n\\end{figure}\n\nFigure~\\ref{fig:construction-perod-1TankHD} shows the estimated construction\nperiod of the Hyper-K detector for 1TankHD{}.\nThe construction is estimated to take about 10 years \nincluding the geological survey and final design making.\n\n\nFor the operation of the 1TankHD{} detector, \nnecessary manpower is estimated to be\nabout 20 full-time equivalent (FTE) by taking into account\nmanagement and detector calibration\nin addition to the operation and maintenance of water system, electronics,\nDAQ system, and computer system. \nIn the case of three tank case, the necessary FTE would be as twice as\nthe single tank case.\n\n\n\n\n\n\\subsubsection{Tank Liner}\nThe lining covers inner surface of the Hyper-Kamiokande tank. It is to\ncontain ultra purified water (UPW) or gadolinium sulfate\n(Gd$_2$(SO$_4$)$_3$) water solution inside of the tank ideally without\nany leakage and without any dissolution of impurities into the\nmedium. Durability should be $\\sim$30 years.\nThe lining structure is to be constructed inside of the cavern bedrock\ncoated with shotcrete. Between the shotcrete and the lining, a\nbackfill concrete is to be employed. As a former example, a\n4\\,mm-thick stainless steel membrane, backfilled with a reinforced\nconcrete, was adopted as the lining material for the Super-Kamiokande\ntank~\\cite{Fukuda:2002uc}.\nIn designing a similar lining structure for Hyper-K, we assume the\nfollowing conditions:\n\\begin{itemize}\n\\item Physical properties of the bedrock surrounding the tank are the same as those used for the Super-Kamiokande designing. For example, elastic modulus of the bedrock is 51 (20)\\,GPa for non-damaged (damaged) region, respectively.\n\\item Physical properties of the backfill concrete are taken from {\\it Standard Specification for Concrete Structure}\\cite{liner:spec-concrete}.\n\\item The surrounding bedrock will not be displaced during\/after tank construction.\n\\item The backwater is controlled so that there is no water pressure on the lining structure from the bedrock side. To satisfy this critical condition, location of the entire detector cavern(s) is to be chosen at the -370\\,mL above the main water drainage level of the candidate mine site (-430\\,mL).\n\\end{itemize} \n\n\\paragraph{Liner sheet characteristics}  \nAs a lining material for the gigantic Hyper-K, firm adhesion to the\nbackfill concrete wall and enough elongation to follow possible\ndeficits and cracks of the concrete wall are both desirable\ncharacteristics. To fulfill these functionalities, concrete embedment\nliner, or the Concrete Protective Liner (CPL), made of High Density\nPolyEthylene (HDPE), has been chosen as the baseline candidate lining\nmaterial. Figure~\\ref{fig:liner-concept} shows schematic views of the\ncandidate CPL (Studliner, GSE Environmental). It has a\n2.0$\\sim$5.0\\,mm thick section of HDPE with a number of studs\nprotruding from one side, that lock the liner into the surface of\nconcrete to prolong the service life of concrete structures.\n\n\\begin {figure}[htbp]\n  \\begin{center}\n    \\includegraphics[width=0.8\\textwidth]{liner-concept.pdf}\n\\caption{Concrete Protective Liner made of High Density PolyEthylene, considered as the baseline tank lining material of Hyper-Kamiokande (Studliner, GSE Environmental).}\n    \\label{fig:liner-concept}\n  \\end{center}\n\\end {figure}\n\n\\begin {figure}[htbp]\n  \\begin{center}\n    \\includegraphics[width=0.8\\textwidth]{liner-examples.pdf}\n\\caption{Former examples of the HDPE lining. (a) IMB detector's 8\\,kt water tank lined with HDPE geomembrane sheets (product of Schlegel Lining Technology, a precursor of GSE Environmental). (b) The CPL (GSE Studliner) applied to water-sealing walls of a large industrial waste processing trench.}\n    \\label{fig:liner-examples}\n  \\end{center}\n\\end {figure}\n\nHDPE is a thermoplastic resin, a linear polymer prepared from ethylene\n(C$_2$H$_4$) by a catalytic process. The absence of branching results in\na more closely packed structure with a higher density (greater than\n0.94), and somewhat higher chemical resistance than Low Density\nPolyethylene (LDPE). HDPE is also harder and more opaque, and it can\nwithstand higher temperatures (120$^\\circ$\\,Celsius for short periods,\n110$^\\circ$\\,Celsius continuously). HDPE is known to maintain pure water\nquality as shown later. Advantages of HDPE as the lining\nmaterial are: impact\/wear resistance, flexibility (very high\nelongation before breaking), good chemical resistance, very low water\npermeability, good plasticity (particularly well to blow molding), and\nlow price. On the contrary, disadvantages of HDPE are: it may have\nvoids, bubbles or sink in the thick sections, poor dimensional\naccuracy, and low mechanical and thermal properties.\n\nA former example to apply HDPE liner to large water tank can be found\nin the IMB detector\\cite{BeckerSzendy:1992hr} as shown in\nFig.\\ref{fig:liner-examples}(a). The 8\\,kt water tank\n(22.5$\\times$17$\\times$18\\,m$^3$) utilized 2.5\\,mm -thick double\nlayered non-reflective black HDPE liners, separated by a plastic\ndrainage grid allowing water to flow between the liners. They were\nproduced and installed by Schlegel Lining Technology, one of the\nprecursors of GSE Environmental. Figure~\\ref{fig:liner-examples}(b)\nshows an application of the CPL as the water-sealing walls of a large\nindustrial waste processing trench. Table~\\ref{tab:liner-spec} shows\nmaterial parameters of the candidate CPL, GSE Studliner. It is also to\nbe noted that the original design for the far-site LBNE Water Cherenkov Detector (WCD) with\n200\\,kt volume adopted the water containment system option with\nuse of 1.5$\\sim$2.5\\,mm-thick Linear Low-Density PolyEthylene\n(LLDPE) geomembrane.\\cite{liner:LBNE-WCD-Golder}\n\n\\begin{table}[htbp]\n\\begin{center}\n\\caption{Material parameters, taken from specification of the candidate CPL (Studliner, GSE Environmental).} \n\\label{tab:liner-spec}\n\\footnotesize\n\\begin{tabular}{lll}\n\\hline\\hline\n\\multicolumn{2}{l}{Material property} & Nominal Value \\\\\n\\hline\nThickess\t& (mm)\t\t& 5.00 \\\\\nDensity\t& (g\/cm$^3$)\t& 0.94 \\\\\nYield strength & (MPa)\t& 15.2 \\\\\nElongation at break & (\\%)\t& 500 \\\\\nCarbon black content & (\\%) & 2$-$3 \\\\\nPigment content\t& (\\%) & 1.5$-$2.5 \\\\\nNotched constant tensile load & (hours) & 400 \\\\\nThermal Expansion Coefficient & (C$^\\circ$) & 1.20E-04 \\\\\nLow temperature brittleness    & (C$^\\circ$) & -77 \\\\\nDimensional stability in each direction & (\\%) & $\\pm$1.0 \\\\\nWater vapor transmission  & (g\/m$^2$\/day) & $<$ 0.01 \\\\\nTypical roll dimension & (m) & 2.44(W)$\\times$59.73(L) \\\\\n\\hline\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\begin {figure}[htbp]\n  \\begin{center}\n    \\includegraphics[width=0.8\\textwidth]{liner-install.pdf}\n\\caption{Schematics of the planned procedures for CPL installation into the cavern \n         with shotcrete surface.}\n    \\label{fig:liner-install}\n  \\end{center}\n\\end {figure}\n\\begin {figure}[htbp]\n  \\begin{center}\n    \\includegraphics[width=0.8\\textwidth]{liner-welding.pdf}\n\\caption{(a) The extrusion welding and a welded seam. (b) The high-voltage pin-hole test.}\n    \\label{fig:liner-welding}\n  \\end{center}\n\\end {figure}\n\nThe planned procedures of the CPL installation into cavern are\nillustrated in Fig.\\ref{fig:liner-install}: At first CPL is fastened\nto the inside of molds before concrete is poured to create a surface\nlined with HDPE. The backfill concrete flows around the studs\nanchoring CPL firmly in place, and fastens it securely to the surface\nof the concrete. The waterproof sheet between bedrock-covering\nshotcrete and backfill concrete aims conveyance of water, coming from\ntiny leakage of water through the CPL and backfill concrete structure,\nif any, and\/or penetrating underground backwater from bedrock.\n\nThe adjacent CPLs are welded by the extrusion welding of\nthermoplastics, which is used typically for assembly of large\nfabrications (such as chemical storage vessels and tanks) with wall\nthicknesses up to 50\\,mm. Figure~\\ref{fig:liner-welding}(a) shows\nan extrusion welding work and close-up to the welded seam. In this\nmethod, molten thermoplastic filler material is fed into the joint\npreparation from the barrel of a mini hand-held extruder based on an\nelectric drill. For the CPL welding the same HDPE is used as the\nfiller material. The molten material emerges from a PTFE shoe shaped\nto match the profile being welded. At the leading edge of the shoe a\nstream of hot gas is used to pre-heat the substrate prior to the\nmolten material being deposited, ensuring sufficient heat is available\nto form a weld.\n\nFor the quality control of the lining, the holes in the CPL sheets\nwith size of $>$0.5\\,mm, including those on the welded seams, can\nbe identified by a high-voltage pin-hole testing\nmethod~\\cite{liner:industrialwaste}, as illustrated in\nFig.~\\ref{fig:liner-welding}(b): It utilizes a charged metal or\nneoprene-rubber broom above the liner. The power source is grounded to\nthe conductive deck and creates a high potential difference ($\\sim$30\nkV at maximum) with tiny current. When the metallic broom head is\nswept over a breach or a hole in the insulating membrane surface,\ncurrent is detected by the test unit which turns off the power to the\nbroom and emits a beep sound to alert the test operator. The area is\nthen carefully swept again at $\\sim$90 degrees to the original\nsweeping direction to pinpoint the exact location of the\nbreach\/hole. This process is continued until all areas of the CPL have\nbeen tested. Occasionally negative pressure tests with a vacuum box\ncan be applied on the possible breaches and the welded seams. The\nleakage water through the holes less than 0.5\\,mm diameter, if\nany, can be collected and controlled by a leakage detection and drain\nsystem, as described later.\n\n\\paragraph{Liner sheet tests}\nVarious material tests were carried out for the candidate CPL, as are\ndescribed in Appendix~\\ref{sec:linertests}.\nTo see the change of light absorbance and elusion of impurities,\nspecimens of the lining sheet were soaked both into ultra-purified\nwater and into 1\\% gadolinium sulfate solution: increase of the light\nabsorbance were observed at the wavelength lower than 300\\,nm, and\ncertain amount of material elution, i.e. total organic carbon, anions\nand metals, were observed. The relation between material elusion and\nchange of light absorbance should be studied carefully. Meanwhile,\nsince PMT is sensitive for higher wavelength, the effect to the\nexperiment can be limited.\n\nMeasurements on material strength, i.e. tension test and creep\ntest, were performed: the candidate CPL sheet has basically enough\nstrength. If cracks or rough holes happen in the backfill concrete,\nthe liner should locally stand for water pressure. To simulate the\nsituation, tests to apply localized water pressure on the lining were\nperformed with variety of slits and holes: For all cases, the liner\nsurvived without breaking.\nAnother concern is that tensile or shear stress may be applied to the\nliner sheet if deformation in the backfill concrete occurs after installation.\nA crack elongation of the liner sheet was tested in the following way: \nA liner sheet was attached to a concrete block and a crack or a step\nwas created in the concrete as shown in Figure~\\ref{fig:liner_elongation_test_fig},\n\\begin{figure}[htbp]\n   \\begin{center}\n      \\includegraphics[width=0.8\\textwidth]{liner_elongation_test_fig.pdf}\n      \\caption{Two possible situations in deformation of backfill concrete, which\n               can introduce tensile or shear stress in the liner sheet. Left:\n               a crack is created. Right: a step is created.}\n      \\label{fig:liner_elongation_test_fig}\n   \\end{center}\n\\end{figure}\nthen strain on surface of the liner sheet was measured. The surface of the liner sheet\nwas pressurized with 0.8~MPa during the measurement to simulate a water pressure.\nFigure~\\ref{fig:liner_elongation_test_pic} shows the actual setup of the crack\nelongation test.\n\\begin{figure}[htbp]\n   \\begin{center}\n     \\includegraphics[width=0.8\\textwidth]{liner_elongation_test_pic.pdf}\n     \\caption{Picture of setup for crack elongation test. A HDPE sheet was\n              attached to a concrete block and a crack or a step was intentionally\n              produced in the concrete block. Strain on the liner sheet was then measured.}\n     \\label{fig:liner_elongation_test_pic}\n   \\end{center}\n\\end{figure} \nFrom the experience of Super-K operation\\footnote{During Super-K operation, \nobserved deformation was at most $O(0.01)$\\%. Cracks in concrete tend to be created\nin an even pitch. Assuming deformation of 0.1~\\% (with a safety factor of 10) and\ncrack pitch of 600~mm, a gap of crack can be estimated to be 0.6~mm.},\na gap or a step of 1~mm was applied to the concrete block as shown in\nFigure~\\ref{fig:liner_elongation_test_fig}. \nIn this test, the concrete was moved by 1~mm for 6 miniutes,\nheld for 30 minutes, and then moved back to the original position. This procedure was\nrepeated 10 times. \nThe measured strain with 1~mm gap\/step was at most 1.4\\%\/0.9\\%, respectively.\nThese values were consistent with the simulated ones based on the test condition.\nFrom the simulation results, it was found that strain of approximately 6\\% at maximum could be\napplied on the back (concrete) side of the liner sheet.\nAfter this test, no damage was observed on the liner sheet by visual inspection.\nFurthermore, water leak test with 0.8~MPa was also performed and no leakage was\nobserved. From the crack elongation test, it was concluded that the HDPE liner sheet\nhad enough strength against deformation and water tightness could be proved even in case that\ndeformation of the backfill concrete occured after installation.\n\nThe water leak can happen around components which penetrate the water\ntank lining, such as anchors and water pipes. A possible design of the\npenetration structure was developed. Its prototype was exposed to\nseries of pressure tests, which showed no leaks.\n\n\\paragraph{Long term stability}\nLong term stability of the liner sheet is one of big concerns on a water tank.\nPolyethylene materials have been used for sheeth of PMT cables and PMT endcap inside SK\nfor 20 years, and no obvious problem is observed. \nPossible sources of degradation of HDPE material are physical stress,\nand some other effects (materials attached to the liner, oxidization, temperature, \netc.)\\footnote{Generally, a UV light is one of the big concern on long term stability of \nHDPE material, but this is not the case for underground area.}.\nEffect of oxidization on lifetime of a HDPE material was reported in \\cite{liner:HDPE_lifetime}.\nLifetime of HDPE liner can be strongly affected by temperature. At 20~$^{\\circ}$C its lifetime\nis predicted to be 446 years, however it reduces to 69 years at 40~$^{\\circ}$C.\nSince water temperature in HK tank will be controlled below 15~$^{\\circ}$C,\nlifetime can be expected to be more than 500 years. To estimate long term stability, \nan accelerating test to measure strength of a HDPE liner sheet after long-term soak \nin ultra pure water is being considered.\n\n\\paragraph{Leakage detection and drain system}\n\n\\begin {figure}[htbp]\n  \\begin{center}\n    \\includegraphics[width=0.8\\textwidth]{liner-leakdrain.pdf}\n\\caption{Conceptual diagram of water leak detection\/drain system.}\n    \\label{fig:liner-leakdrain}\n  \\end{center}\n\\end {figure}\nThe water leakage, if it happens, will be not through the sheets\nthemselves, but through small holes, which are undetectable by the\npin-hole\/vacuum tests, or breaches, which are caused unavoidably by\nworks after the tests. To be prepared for these possible failures, a\nleak detection\/drainage system is to be\ndeveloped. Figure~\\ref{fig:liner-leakdrain} shows preliminary concept\nof the system. HDPE plastic moldings are embedded together with the\nCPL in the backfill concrete, to work as partition at a pitch of about\n10\\,m in the direction of circumference of the tank. Water leaks\nfrom the CPL(s) or seam(s) in each partition are to be collected\nindividually, so that leak detector installed at the bottom can\nidentify the partition with the problem.  Occasionally, water leak\nthrough the CPL can flow into bedrock side through cracks of the\nbackfill concrete. Water-proof sheets (high panel signal sheet),\ninstalled between bedrock and backfill concrete, can separate leakage\nof inside (tank) water and sump water coming from outside\nbedrock. These water will be drained separately, and tank water will\nbe treated with care especially for the case with gadolinium sulfate\nsolution is used in the tank.\n\n\n\n\\subsection{Water Tank}\\label{section:tank}\n\n\n\\input{design-tank\/overview.tex}\n\n\n\\subsubsection{Tank-Cavern Interface}  \n\\input{design-tank\/interface.tex}\n\n\n\\subsubsection{Tank Liner}  \n\\label{section:tank-liner}\n\\input{design-tank\/liner.tex}\n\n\n\\subsubsection{Photosensor Support Framework}  \n\\input{design-tank\/framework.tex}\n\n\n\\subsubsection{Geomagnetic Field Compensation Coils}  \n\\label{section:tank-coil}\n\\input{design-tank\/coil.tex}\n\n\n\\subsubsection{Construction}  \n\\input{design-tank\/construction.tex}\n\n\n\n\n\n\\subsection{Water purification and circulation system }\\label{section:water}\n\n\\subsubsection{Introduction}\nWater is the target material and signal-sensitive medium of the\ndetector, and thus its quality directly affects the sensitivity. In\norder to realize such a huge Cherenkov detector, achieving good water\ntransparency is the highest priority. In addition, as radon emanating\nfrom the photosensors and detector structure materials is the main\nbackground source for low energy neutrino studies, an efficient radon\nremoval system is indispensable.\n\nIn Super-Kamiokande the water purification system has been continually\nmodified and improved over the course of SK-I to SK-IV.  As a result,\nthe transparency is now kept above 100 m and is very stable, and the\nradon concentration in the tank is held below 1\\,mBq\/m$^3$.  Following\nthis success, the Hyper-Kamiokande water system design will be based\non the current Super-Kamiokande water system.\n\nNaturally, ever-faster water circulation is generally more effective\nwhen trying to keep huge amounts of water clean and clear, but\nincreasing costs limit this straightforward approach so a compromise\nbetween transparency and re-circulation rate must be found.  In\nSuper-Kamiokande, 50\\,ktons of water is processed at the rate of\n 60\\,tons\/hour in order to keep the water transparency (the attenuation\nlength for 400\\,nm-500\\,nm photons) above 100\\,m, and 20\\,Nm$^3$\/hour of\nradon free air is generated for use as a purge gas in degas modules,\nand as gas blankets for both buffer tanks and the Super-Kamiokande\ntank itself. For the 258\\,ktons of water in\nthe tank of Hyper-Kamiokande, these process speeds will need to be scaled-up to\n310\\,tons\/hour for water circulation and 50\\,Nm$^3$\/hour for radon free\nair generation.\n\n\\subsubsection{Source water}\nThe rate of initial water filling is restricted by the amount of available source water.\n In Mt.~Nijuugo-yama, the baseline location of Hyper-Kamiokande, the total\namount of the spring water is about 600\\,tons\/hour. (It varies\nseasonally between 300\\,tons\/hour and 800\\,tons\/hour and it is above\n600\\,tons\/hour except in Winter (December-March).) However, as the mine\ncompany uses all the water for their smelting factory, the available\nspring water for Hyper-Kamiokande is limited and cannot be allocated at this point. \nTherefore, the baseline plan is getting 105\\,tons\/hour of source water from the outside of the mine, making 78\\,tons\/hour of\nultra-pure water and filling the 258\\,ktons for 180\\,days.\nThe source water site is the well for snow-melting system in the Kamioka town at Oshima public hall which is\nabout 5\\,km away from the tank position. \nHida city is supportive in our use of the well and Gifu prefecture is also helping to\ndecide the route from the well to the entrance of the Tochibora mine.\nSerious investigations and negotiations are ongoing with these local governments.\n\nThe water quality of  the snow melt water and Tochibora spring\nwater are compared with that of Mozumi spring water in Table~\\ref{tab:source}.\nIn the Mozumi mine, the location of Super-Kamiokande, there is sufficient mine water and no mining\/smelting activities.\n\\begin{table}[htb]\n\\begin{center}\n\\scalebox{0.8}[0.8]{\n\\begin{tabular}{lr|rrr}\n\\hline\\hline\n   &  & The well for Kamioka snow melt      &  Tochibora spring water  & Mozumi spring water  \\\\\n   &  &  as of 21 Jun. 2016               &  as of 1 Mar. 2011       & as of 16 Mar. 2011   \\\\\n\\hline \\hline \nTemperature(Typical) & $^{\\circ}$C        &  11.9     &    11    &    12   \\\\\npH (25$^{\\circ}$C) &                             &  7.1    &   7.8    &   7.8   \\\\\nConductivity     & $\\mu$S\/cm               & 101    &   170    &   221   \\\\\nTurbidity        &degree(Kaolin)                & $<1$    &  $<1$   &   $<1$  \\\\\nAcid consumption (pH 4.8) & mg CaCO$_3$\/L &  27.9  &   40.0   &   75.8   \\\\\nTOC      & mg\/L                                   &  $<0.1$ &  $<1$   &   $<1$  \\\\\nPhosphate & mg\/L                               &  $<0.1$ &   $<0.1$   &  $<0.1$   \\\\\nNitrate   & mg\/L                                  &  3.0    &   1.0    &  1.6  \\\\\nSulfate   & mg\/L                                  &  4.4   &  36.4    &  30.2  \\\\\nFluoride  & mg\/L                                  &   $<0.1$    &   0.3    &  0.4  \\\\ \nChloride  & mg\/L                                  &  8.6    &   1.6    &  1.8  \\\\\nSodium    & mg\/L                                 &  4.6    &   4.9    &  6.2  \\\\\nPotassium & mg\/L                                &  0.8    &   0.5    &  0.5  \\\\   \nCalcium   & mg\/L                                  &  12.3   &  25.2    &  32.0 \\\\  \nMagnesium & mg\/L                                &  1.5    &   1.5    &  2.9  \\\\\nAmmonium  & mg\/L                              &  $<0.1$ &   $<0.1$   &  $<0.1$   \\\\\nIonic silicon dioxide  & mg\/L                   &  12.8   &   17.1    &  11.8  \\\\\nIron      & mg\/L                                    & $<0.01$ &  $<0.01$  &   $<0.01$  \\\\\nCopper    & mg\/L                                  & $<0.01$ &   $<0.01$  &   $<0.01$  \\\\\nZinc      & mg\/L                                    &   -    &   0.09     &    $<0.01$  \\\\\nLead      & mg\/L                                   & $<0.1$  &   $<0.1$  &   $<0.1$  \\\\\nAluminum & mg\/L                                 & $<0.01$ &    $<0.01$  &   $<0.01$  \\\\\nBoron     & mg\/L                                   & $<0.01$ &    $<0.01$  &  0.2  \\\\\nStrontium & mg\/L                                  & -   &   0.18      &  0.52 \\\\\nBarium    & mg\/L                                  & $<0.01$ &    $<0.01$  &  0.03  \\\\\n\\hline\\hline\n\\end{tabular}\n}\n\\caption{Source water quality.}\n\\label{tab:source}\n\\end{center}\n\\end{table}\n\n\n\\subsubsection{Main system flows and layouts}\nThe HK main water purification system consists of a 1st stage system\n(filling) and a 2nd stage (re-circulation) system for the 258,000\\,m$^3$ tank    \nas shown in Figure~\\ref{fig:1st2ndsys}.\nFigure~\\ref{fig:LO} shows their layouts.\n\\begin{figure}[htb]\n\\includegraphics[width=0.95\\textwidth]{design-water\/1stand2nd.pdf}\n\\caption{1st stage and 2nd stage water systems.}\n\\label{fig:1st2ndsys}\n\\end{figure}\n\\begin{figure}[htb]\n\\includegraphics[width=1.0\\textwidth]{design-water\/Layouts.pdf}\n\\caption{Necessary space for the main systems.}\n\\label{fig:LO}\n\\end{figure}\nThe process power of the 1st stage system is 78\\,m$^3$\/h, and accordingly, it takes\n138 days to fill the tank without consideration of any maintenance.\nIt may take about 180 days in the realistic case.\nPreferably, an additional, same amount, of 11 $^{\\circ}$C cooling water is required for the heat\nexchangers.\n\nThe process power of the 2nd stage system for the recirculation is \n310\\,m$^3$\/h.\n\n\\subsubsection{Water flow simulation in the tank}\nWater flow in the tank directly affects the water quality and the\nphysics results, therefore water flow simulations for the baseline\ndesign tank were conducted. Water flow is determined not only by the\ntotal water flow rate but also by detector geometry, the configuration\nof water inlets and outlets, supply water temperature, heat sources in\nthe tank, surrounding rock temperature and so on. The input parameters\nare summarized in Table~\\ref{tab:simpara}, and the main results are\nshown in Figure~\\ref{fig:flowsim}. \nWhen cold water is supplied from the bottom of the tank, convection in the tank is suppressed\nand the flow becomes laminar, resulting in effective water replacement. When cold\nwater is supplied from the top of the tank, large convection is evoked and the water quality in the\ntank becomes uniform, spoiling effective water replacement. Actually this behavior was\nconfirrmed in Super-Kamiokande's 50 kton tank and seem to be common to cylindrical tanks; \nthus the water flow in Hyper-Kamiokande should be controlled as in Super-Kamiokande.\n\n\n\\begin{table}[htb]\n\\begin{center}\n\\scalebox{1}[1]{\n\\begin{tabular}{l|r}\n\\hline\\hline\nID flow rate &    271.8 m$^3$\/h      \\\\ \nOD flow rate &    37.9 m$^3$\/h       \\\\  \nInlets\/Outlets & 65A$\\times$37\/65A$\\times$37 \\\\\nID boundary condition &  Inlet: 0.61 m\/s, 286K Outlet: 0Pa \\\\\nOD boundary condition &  Inlet: 0.67 m\/s, 286K Outlet: 0Pa \\\\\nSupply water temperature &  13.0 $^{\\circ}$C   \\\\\nTop level rock temperature &  16.7 $^{\\circ}$C  \\\\\nBottom level rock temperature &  17.7 $^{\\circ}$C \\\\\nHeat flux from the PMT\/electronics\/coil & 3.2W\/m$^2$\\\\\nTotal heat form ID top and bottom &  2100W and 2100W \\\\\nTotal heat from ID wall           & 6502W \\\\\nTotal heat from OD wall(rock)     & 5384W \\\\\nWater density               & 999.4 kg\/m$^2$ @286 K,  998.4 kg\/m$^2$ @292 K \\\\\nWater heat conductivity & 0.587 W\/m\/K  @286 K,  0.597 W\/m\/K  @292 K  \\\\\nWater viscosity             & 0.0012 kg\/m\/s  @286 K,  0.0010 kg\/m\/s  @292 K \\\\\n\\hline\\hline\n\\end{tabular}\n}\n\\caption{Input parameters for the water flow simulations.}\n\\label{tab:simpara}\n\\end{center}\n\\end{table}\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=0.85\\textwidth]{design-water\/Flowtemp.pdf}\n\\includegraphics[width=0.95\\textwidth]{design-water\/FlowSim.pdf}\n\\caption{Water temperature distributions (top 2 figures) \n  and water replacement efficiencies \n  as the result of water flow simulations. As the tank is in cylindrical shape and the \n water inlets and outlets are distributed symmetrically,  \n only 1\/6 of the tank was simulated with symmetric boundary condition and shown here.\n  (a) The case for supplying water from the bottom of the\n  tank and draining water from the top of the tank. (b) The case for\n  supplying water from the top of the tank and draining water from the\n  bottom of the tank. The elapsed days since the recirculation starts\n  are indicated. In this simulation, at first the tank was filled with\n  old water ($= 0$, blue), then new water ($= 1$, red) was supplied to the tank,\n  therefore the color scale in the figures corresponds to the water\n  replacement efficiency. After 40 days case (a) is more reddish, while case (b) is more uniform.}\n\\label{fig:flowsim}\n\\end{center}\n\\end{figure}\n\n\\subsubsection{Radon in the water}\n\\label{sec:radon-in-water}\nThe dominant low energy background is expected to be radon and \nthe dominant radon source in the tank is expected to be the PMTs\nthemselves. The radon emanation from Hyper-Kamiokande photon sensors have \nnot been measured yet, but each Super-Kamiokande ID PMT emanates about 10 mBq \nand the measured radon concentration in the Super-Kamiokande water is 2 mBq\/$m^3$.\nSuper-Kamiokande has 11129 ID PMTs and 50\\,ktons of water, therefore\nthe average radon concentration should be around\n10 mBq\/PMT $\\times$ 11129 PMTs \/ 50000 tons = 2.2 mBq\/$m^3$.\nAccordingly, the radon concentration expected in one Hyper-Kamiokande HD tank is\n10 mBq\/PMT $\\times$ 40000 PMTs \/ 258000 tons = 1.6 mBq\/$m^3$.\n\nRegarding radon suppression, the Hyper-Kamiokande water system includes Super-Kamiokande-based vacuum\ndegasifiers which reduce radon by about one order of magnitude as shown\nin Figure~\\ref{fig:1st2ndsys}. That being said, in the experience of Super-Kamiokande the best way to reduce\nradon is by inducing a gentle laminar flow in the fiducial volume, \nallowing the radon to primarily decay close to the PMTs (i.e., not in\nthe fiducial volume) where it can do the least harm.\n\n\n\\subsubsection{Gd option}\nIn order to realize the many physics benefits provided by efficient\ntagging of neutrons in water, it has been proposed (and recently approved) \nto add dissolved gadolinium sulfate to Super-Kamiokande.  As a result, \nover a period of\nyears much effort has gone into the design and demonstration of a\nspecialized water system capable of maintaining the exceptional water\ntransparency discussed above, while at the same time maintaining the\ndesired level of dissolved gadolinium in solution.  In other words,\nsomehow the water must be continuously recirculated and cleaned of\neverything {\\em except} gadolinium sulfate.\n\nBuilt in 2007, a 0.2 tons\/hour prototype selective filtration system at \nthe University of California, Irvine, led in 2010 to the 3 tons\/hour \nsystem at the heart of the Kamioka-based EGADS (Evaluating Gadolinium's \nAction on Detector Systems) project.  EGADS has now shown that this novel \nselective water filtration technology --- known as a \"molecular band-pass \nfilter\" --- is both feasible and scalable.\nIt continuously improves and then maintains the transparency of water loaded with\n\\mbox{Gd$_2$(SO$_4$)$_3$} to SK ultrapure water levels, removing\nunwanted impurities while simultaneously and indefinitely retaining\nthe desired levels of both the gadolinium and sulfate ions.\n\nSince EGADS was built specifically to show that gadolinium loading\nwould be feasible in Super-K, scalability was always an important\ndesign criterion. Therefore, from the beginning the EGADS band-pass\nsystem was conceived of as a modularized design.  It uses\ncost-effective, readily available components operating in parallel to\nachieve the desired throughput and assure serviceability.\n\nAs the band-pass design is modular and uses off-the-shelf equipment,\nalbeit in novel ways, scaling it up from the current 3~tons\/hour to\n60~tons\/hour for Super-Kamiokande, or 310\\,tons\/hour for \nHyper-Kamiokande, is straightforward.  Figure~\\ref{fig:rack} indicates\nhow one rack of filtration membrane housings, the modular unit around\nwhich the Hyper-K band-pass system is designed, is derived from the\noperating EGADS selective filtration system.\n\n\n\\begin{figure}\n    \\begin{center}\n      \\includegraphics[width=0.8\\textwidth]{design-water\/rack.pdf}\n      \\caption[Scaling the EGADS Band-Pass to Hyper-Kamiokande]\n              {Scaling the modular EGADS selective filtration\n                band-pass for Hyper-Kamiokande. One rack of filtration\n                membrane housings is shown here;\n                Figure~\\ref{fig:Gd_WS} shows many of them arranged\n                into a functional selective filtration system.}\n      \\label{fig:rack}\n    \\end{center}\n\\end{figure}\n\n\nFigure~\\ref{fig:Gd_WS} depicts how the modular rack from\nFigure~\\ref{fig:rack} may be duplicated and operated in parallel to\nprovide the needed throughput.  Further design simplification and cost\nsavings are achieved by using this standardized membrane housing array\nand filling the housings with a variety of filter membranes, each of\nwhich handles a different cleaning task.  These components include\nnanofilters (NF), ultrafilters (UF), and reverse osmosis (RO)\nmembranes; in each case there are two stages.  Note that the layout\nshown in Figure~\\ref{fig:Gd_WS} is schematic in nature.  Due to space\nconstraints underground the illustrated system would likely be split\ninto two levels, one atop the other.\n\n\\begin{figure}\n    \\begin{center}\n      \\includegraphics[width=0.8\\textwidth]{design-water\/HK_Gd_WS.pdf}\n      \\caption[Gadolinium-capable water system for Hyper-Kamiokande.]\n              {Gadolinium-capable water system for\n                Hyper-Kamiokande. Two stages each of nanofilters (NF),\n                ultrafilters (UF), and reverse osmosis (RO) membrane\n                racks are shown, sufficient to provide selectively\n                filtered water for Hyper-K.} \n      \\label{fig:Gd_WS}\n    \\end{center}\n\\end{figure}\n\nUsing the baseline Hyper-K design, the system shown in\nFigure~\\ref{fig:Gd_WS} represents what is needed for selective\nfiltration following the addition of\ngadolinium sulfate to the Hyper-K water. It is assumed that pure water\nfor filling the detector will be provided by the main, non-Gd-capable\nwater system described above.  The Gd-specific ``molecular band-pass''\nsystem described here will be augmented with additional Gd-capable water  \nhandling equipment -- also demonstrated by and scaled up from a working \nEGADS version -- known as a ``fast recirculation'' system. The Hyper-K \nfast recirculation system will be used in conjunction with HK's \nband-pass to maintain the Gd-loaded water's quality.\n\n\nDue to gadolinium sulfate's benign nature with regards to the usual\ndetector components (materials compatibility was another component of\nthe EGADS study), retaining the ability to add gadolinium to\nHyper-Kamiokande primarily means retaining the option of adding\ngadolinium filtration capability to the Hyper-K water system.  Indeed,\nif gadolinium works as well as expected in Super-Kamiokande over the\nnext few years, it is hard to imagine that a next-generation detector\nlike Hyper-K would not also want to enjoy the physics advantages a\ngadolinium-loaded Super-K would already have.  Therefore, we have been\ncareful to keep the possibility of gadolinium loading in mind when\ndesigning the overall Hyper-Kamiokande water system.\n\n\n\n\\part{Introduction}\n\\input{introduction\/introduction.tex}\n\n\n\\clearpage\n\\color{black}\n\\part{Experimental Configuration} \\label{part:experimentalconfiguration}\n\\input{jparc\/jparc.tex}\n\n\n\\clearpage\n\\color{black}\n\\section{Hyper-Kamiokande detector} \\label{section:design}\n\n\\input{design-introduction\/introduction.tex}\n\\newpage\n\\input{design-location\/location.tex}\n\\newpage\n\\input{design-cavern\/cavern.tex}\n\n\\newpage\n\\input{design-tank\/tank.tex}\n\n\\newpage\n\\input{design-water\/water.tex}\n\n\n\\clearpage\n\\input{design-photosensor\/photosensor.tex}\n\\clearpage\n\\input{design-electronics\/electronics.tex}\n\\newpage\n\\input{design-daq\/daq.tex}\n\n\\newpage\n\\input{design-calibration\/calibration.tex}\n\n\\newpage\n\\input{design-computation\/computation.tex}\n\n\\clearpage\n\\input{software\/software.tex}\n\n\\clearpage\n\\input{background\/background.tex}\n\n\n\\clearpage\n\\part{Physics Potential}\n\\label{section:physics}\n\n\\section{Neutrino Oscillation}\n\n\\input{physics-lbl\/lbl.tex}\n\\newpage\n\\input{physics-atmnu\/atmnu.tex}\n\\newpage\n\\input{physics-solarnu\/solarnu.tex}\n\n\\newpage\n\\section{Nucleon Decays}\n\\input{physics-pdecay\/pdecay.tex}\n\n\\section{Neutrino Astrophysics and Geophysics}\n\n\\input{physics-supernova\/supernova.tex}\n\\newpage\n\\input{physics-darkmatter\/dm.tex}\n\\newpage\n\\input{physics-astronu\/astronu.tex}\n\\newpage\n\\input{physics-geophys\/geophys.tex}\n\\newpage\n\n\\clearpage\n\\part{Second Detector in Korea}\n\\label{section:secon-detector-korea}\n\\section{Second Detector in Korea}\n\\input{second-tank-korea\/overview.tex}\n\\clearpage\n\n\\section{Introduction}\n\\label{section:intro}\n\nRecent advances in experimental particle physics have yielded\nfascinating insights into the inner workings of the smallest-scale\nphenomena.  In 2012, the last missing piece of the standard model (SM)\nof elementary particles, the Higgs boson, was finally observed by the\nATLAS and CMS experiments at the Large Hadron Collider (LHC) in\nCERN~\\cite{Aad:2012tfa,Chatrchyan:2012xdj}. The SM is highly\nsuccessful in explaining experimental data, however our current\nability to describe nature from a fundamental physics point of view is\nfar from satisfactory, most significantly the fact that neutrino mass\ncannot be incorporated, and so we need beyond the standard model (BSM)\nphysics. \n\nThe Nobel Prize in 2002 was awarded for the detection of cosmic\nneutrinos (in particular the ones coming from supernova) in Kamiokande\nand for the pioneering solar neutrino experiment at the Homestake\nmine. More recently, the 2015 Nobel Prize was awarded for the\ndiscovery of neutrino oscillations using data taken by the\nSuper-Kamiokande (Super-K) and the Sudbury Neutrino Observatory collaborations,\nwhich has the very profound implication that neutrinos have non-zero\nbut very tiny masses.\n\nBuilding on the expertise gained from the past and current\nexperiments, Kamiokande and Super-Kamiokande, Hyper-Kamiokande\n(Hyper-K) is a natural progression for the highly successful\nJapanese-hosted neutrino program.\n\n\\begin{figure}[htbp]\n  \\begin{center}\n  \\begin{tabular}{cc}\n    \\includegraphics[width=0.8\\textwidth]{introduction\/figs\/HKmain-170509-web.jpg}\n  \\end{tabular}\n  \\caption{Illustration of the Hyper-Kamiokande first cylindrical tank in Japan.}\n  \\label{fig:hk-perspective-JTank}\n  \\end{center}\n\\end{figure}\n\nHyper-Kamiokande is a next-generation, large-scale water Cherenkov\nneutrino detector. A dedicated task force determined the optimal tank\ndesign to be two cylindrical detectors that are 60\\,m in height and\n74\\,m in diameter with 40\\% photocoverage, where a staging between the\nfirst and second tank is considered. We first focus on building the\nfirst tank in Japan, see Fig.~\\ref{fig:hk-perspective-JTank} for the\ndrawing.\n\n\n\nCandidate sites for the Hyper-K experiment were selected such that\nneutrinos generated in the J-PARC accelerator facility in Tokai, Japan\ncan be measured in the detector. J-PARC will operate a 750\\,kW beam in\nthe near future, and has a long-term projection to operate with 1300\\,kW\nof beam power. Near detectors placed close to the J-PARC beam line\nwill determine the information about the neutrinos coming from the\nbeam, thus allowing for the extraction of oscillation parameters from\nthe Hyper-K detector. The ND280 detector suite, which has been used\nsuccessfully by the T2K experiment, could be upgraded to further\nimprove the measurement of neutrino cross section and flux. The\nWAGASCI detector is a new concept under development that would have a\nlarger angular acceptance and a larger mass ratio of water (and thus\nmaking the properties more similar to the Hyper-K detector) than the\nND280 design. Intermediate detectors, placed 1-2 km from the J-PARC\nbeam line, would measure the beam properties directly on a water\ntarget. Details of the beam, as well as the near and intermediate\ndetectors, can be found in Section~\\ref{section:jparc}.\n\nHyper-K is a truly international proto-collaboration with over 70\nparticipating institutions from Armenia, Brazil, Canada, France,\nItaly, Korea, Poland, Russia, Spain, Sweden, Switzerland, Ukraine, the United\nKingdom and the United States, in addition to Japan.\n\nHyper-K will be a multipurpose neutrino detector with a rich physics\nprogram that aims to address some of the most significant questions\nfacing particle physicists today. Oscillation studies from\naccelerator, atmospheric and solar neutrinos will refine the neutrino\nmixing angles and mass squared difference parameters and will aim to\nmake the first observation of asymmetries in neutrino and antineutrino\noscillations arising from a CP-violating phase, shedding light on\none of the most promising explanations for the matter-antimatter\nasymmetry in the Universe. The search for nucleon decays will probe\none of the key tenets of Grand Unified Theories. In the case of a\nnearby supernova, Hyper-K will observe an unprecedented number of\nneutrino events, providing much needed experimental results to\nresearchers seeking to understand the mechanism of the\nexplosion. Finally, the detection of astrophysical neutrinos from\nsources such as dark matter annihilation, gamma ray burst jets, and\npulsar winds could further our understanding of some of the most\nspectacular, and least understood, phenomena in the Universe. These\ntopics will be discussed further in Section~\\ref{section:physics}.\n\nThis design report is organized as follows. There are a total of five\nparts.  The remainder of this Part~\\ref{section:intro} outlines the\ntheoretical framework for the physics topics contained in this report\nand discusses the relationships between Hyper-K and other large-scale\nneutrino experiments. Part~\\ref{part:experimentalconfiguration}\ndescribes the experimental configuration where\nSection~\\ref{section:jparc} describes the J-PARC neutrino beam line\nand near detector facility; Section~\\ref{section:design} discusses the\ntechnical details of the experimental design and\nSection~\\ref{section:software} details the software packages that will\nbe utilized by the Hyper-K experiment. A discussion of pertinent\nradioactive backgrounds is contained in\nSection~\\ref{section:background}. Part~\\ref{section:physics} explains\nthe physics capabilities for\nHyper-K. Part~\\ref{section:secon-detector-korea} introduces a possible\nsecond tank in Korea.  \nThe last part, Part~\\ref{section:appendices}, is the Appendix with\ndetails on the liner sheet tests (A) and a\ndescription of a possible second tank in Japan at Hakamagoshi (B).\n\n\\subsection{Neutrino oscillations}\n\nNeutrino oscillations, discovered by the Super-Kamiokande (Super-K)\nexperiment in 1998~\\cite{Fukuda:1998mi}, implies that neutrinos have\nnonzero masses and flavor mixing, providing one of the most convincing\nexperimental proofs known today for the existence of physics beyond\nthe Standard Model (BSM).\nIndeed neutrino oscillation has been established as a very powerful\ntool to probe extremely small neutrino masses (or their differences)\nas well as lepton flavor mixing.\n\nThroughout this design report, unless stated otherwise, we consider\nthe standard three flavor neutrino framework.  The 3$\\times3$ unitary\nmatrix $U$ which describes the mixing of neutrinos~\\cite{Maki:1962mu}\n(that is often referred to as the Pontecorvo-Maki-Nakagawa-Sakata\n(PMNS) or Maki-Nakagawa-Sakata\n(MNS)~\\cite{Pontecorvo:1967fh,Maki:1962mu} matrix) relates the flavor\nand mass eigenstates of neutrinos as\n\\begin{eqnarray} \n\\nu_\\alpha  = \\sum_{i=1}^3 U_{\\alpha i} \\nu_i\n\\ \\ (\\alpha = e, \\mu, \\tau),\n\\end{eqnarray} \nwhere $\\nu_\\alpha  (\\alpha = e, \\mu, \\tau)$ \nand $\\nu_i (i = 1,2,3)$ \ndenote neutrino fields with definite flavor and mass, respectively. \n\nUsing the standard parameterization, found, e.g. in\nRef.~\\cite{Agashe:2014kda}, $U$ can be expressed as,\n\\begin{eqnarray}\n\\hskip -0.5cm\nU \n& = &\n\\left(\n\\begin{array}{ccc}\n1 & 0   & 0 \\\\\n0 & c_{23} & s_{23} \\\\\n0 & -s_{23} & c_{23} \\\\\n\\end{array}\n\\right)\n\\left(\n\\begin{array}{ccc}\nc_{13} & 0   & s_{13}e^{-i\\ensuremath{\\delta_{CP}} } \\\\\n0 & 1 & 0 \\\\\n-s_{13}e^{i\\ensuremath{\\delta_{CP}} } & 0 & c_{13} \\\\\n\\end{array}\n\\right)\n\\left(\n\\begin{array}{ccc}\nc_{12} & s_{12}  & 0 \\\\\n-s_{12} & c_{12} & 0 \\\\\n0 & 0 & 1 \\\\\n\\end{array}\n\\right) \n\\left(\n\\begin{array}{ccc}\n1 & 0 & 0 \\\\\n0 & e^{i\\frac{\\alpha_{21}}{2}} & 0 \\\\\n0 & 0 & e^{i\\frac{\\alpha_{31}}{2}} \\\\\n\\end{array}\n\\right)\n\\label{eq:mixing}\n\\end{eqnarray}\nwhere $c_{ij} \\equiv \\cos\\theta_{ij}$, $s_{ij} \\equiv \\sin\\theta_{ij}$, \nand $\\ensuremath{\\delta_{CP}} $ --- often called the Dirac $CP$ phase ---,  \nis the Kobayashi-Maskawa type $CP$ phase~\\cite{Kobayashi:1973fv} \nin the lepton sector. \nOn the other hand, the two phases, $\\alpha_{21}$ and $\\alpha_{31}$,\n--- often called Majorana $CP$ phases --- exist only if neutrinos are\nof Majorana type~\\cite{Schechter:1980gr, Bilenky:1980cx,Doi:1980yb}.\nWhile the Majorana $CP$ phases can not be observed in neutrino\noscillation, they can be probed by lepton number violating processes\nsuch as neutrinoless double beta ($0\\nu \\beta\\beta$) decay.\n\nIn the standard three neutrino flavor framework, only two mass squared\ndifferences, $\\Delta m^2_{21}$ and $\\Delta m^2_{31}$, for example, are\nindependent.  Here, the definition of mass squared differences is\n$\\Delta m^2_{ij}$ $\\equiv$ $m^2_i - m^2_j$. Therefore, for a given\nenergy and baseline, there are six independent parameters that\ndescribe neutrino oscillations: three mixing angles, one $CP$ phase,\nand two mass squared differences.\nAmong these six parameters, $\\theta_{12}$ and $\\Delta m^2_{21}$ have\nbeen measured by solar~\\cite{Ahmad:2002jz,Ahmad:2001an,Abe:2010hy} and\nreactor~\\cite{Eguchi:2002dm,Araki:2004mb,Abe:2008aa} neutrino\nexperiments.  The parameters $\\theta_{23}$ and $|\\Delta m^2_{32}|$\n(only its absolute value) have been measured by\natmospheric~\\cite{Ashie:2005ik,Ashie:2004mr} and\naccelerator~\\cite{Ahn:2006zza,Adamson:2011ig,Abe:2012gx,Abe:2014ugx}\nneutrino experiments.\nIn the last few years, \\(\\theta_{13}\\) has also been measured by\naccelerator~\\cite{Abe:2011sj,Adamson:2011qu,Abe:2013xua,Abe:2013hdq}\nand reactor experiments~\\cite{Abe:2011fz,Ahn:2012nd,An:2012eh,\n  An:2013zwz,Abe:2014lus}.\nRemarkably, the Super-K detector has successfully measured all of\nthese mixing parameters, apart from the $CP$ phase and the sign of\n$\\Delta m^2_{32}$.\nThe current best-measured values of the mixing parameters are listed\nin ~\\cite{Agashe:2014kda}, where the mass hierarchy and $CP$ phase are still unknown \nthough there are some weak preferences by the current \nneutrino data as will be mentioned later in this section. \n\nBy studying neutrino oscillation behaviour, Hyper-K is expected to\nimprove the current bounds obtained by Super-K for various\nnon-standard neutrino properties, such as the possible presence of\nsterile neutrinos~\\cite{Abe:2014gda}, non-standard interactions of\nneutrinos with matter~\\cite{Mitsuka:2011ty}, or violation of Lorentz\ninvariance~\\cite{Abe:2014wla}.\n\n\\subsubsection{Mass Hierarchy}\n\nThe positive or negative sign of $\\Delta m^2_{32}$ (or equivalently\nthat of $\\Delta m^2_{31}$) corresponds, respectively, to the case of\nnormal ($m_2 < m_3$) or inverted ($m_3 < m_2$) mass hierarchy\n(ordering).  From a theoretical point of view, it is of great interest\nto know the mass hierarchy to understand or obtain clues about how the\nneutrino masses and mixing are generated (see\ne.g. \\cite{Mohapatra:2006gs} for a review).  Also the mass hierarchy\nhas a significant impact on the observation of the $0\\nu \\beta \\beta $\ndecay for the case where neutrinos are Majorana particles.  If the\nmass hierarchy is inverted, a positive signal of $0\\nu\\beta\\beta$\nis expected in future experiments if the current sensitivity on the effective\nMajorana mass can be improved by about one order of magnitude beyond\nthe current limit.\n\nIn the $\\nu_\\mu \\to \\nu_e$ appearance channel, its oscillation\nprobability at around the first oscillation maximum, $O(L\/E_\\nu) \\sim\n1$, tends to be enhanced (suppressed) if the mass hierarchy is normal\n(inverted) due to the matter effect or the so called\nMikheev-Smirnov-Wolfenstein (MSW)\neffect~\\cite{Mikheev:1986gs,Wolfenstein:1977ue} as we will see in Part\n\\ref{section:physics}.  For the antineutrino channel, $\\bar{\\nu}_\\mu\n\\to \\bar{\\nu}_e$, the effect become opposite, namely, the normal\n(inverted) mass hierarchy tends to suppress (enhance) the appearance\nprobability.  The longer the baseline ($L$), larger the effect of such\nenhancement or suppression.  Therefore, in principle, the mass\nhierarchy can be determined by measuring the oscillation probability\nprovided that the matter effect is sufficiently large.  This is the\nmost familiar way to determine the mass hierarchy in neutrino\noscillation which can be done using accelerator or atmospheric\nneutrinos.\n\nIndependently from this method, it is also possible to determine the\nmass hierarchy by observing the small interference effects caused by\n$\\Delta m^2_{31}$ and $\\Delta m^2_{32}$ in the medium baseline ($L\n\\sim 50$ km) reactor neutrino oscillation experiment as first\ndiscussed in \\cite{Petcov:2001sy}.  The proposed projects such as\nJUNO~\\cite{An:2015jdp} and RENO-50~\\cite{Kim:2014rfa} aim to determine\nthe mass hierarchy by this method.\nFurthermore, in principle, it is possible to determine the mass\nhierarchy by comparing the absolute values of the effective mass\nsquared differences determined by reactor ($\\bar{\\nu}_e$\ndisappearance) and accelerator ($\\nu_\\mu$ disappearance) with high\nprecision ~\\cite{deGouvea:2005hk,Nunokawa:2005nx}.\n\nIt is expected by the time Hyper-K will start its operation,\naround the year 2025, the mass hierarchy could be determined at $\\sim$\n(3-4)$\\sigma$ or more by combining the future data coming from the\nongoing experiments such as NOvA, T2K and reactor experiments, Daya\nBay~\\cite{Guo:2007ug}, RENO~\\cite{Ahn:2010vy}, Double\nChooz~\\cite{Ardellier:2006mn}, and proposed future experiments such as\nJUNO~\\cite{An:2015jdp}, RENO-50~\\cite{Kim:2014rfa},\nICAL~\\cite{Ahmed:2015jtv}, PINGU~\\cite{Aartsen:2014oha}, and\nORCA~\\cite{Katz:2014tta} where the last three projects will use\natmospheric neutrinos to determine the mass hierarchy.\n\n\\subsubsection{CP Violation}\n\nThe magnitude of the charge-parity ($CP$) violation in neutrino\noscillation can be characterized by the difference of neutrino\noscillation probabilities between neutrino and anti-neutrino channels\n~\\cite{Barger:1980jm,Pakvasa:1980bz}.\n\nThe current data coming from T2K~\\cite{Abe:2015awa} and\nNOvA~\\cite{NOvA-talk-nufact15}, when combined with the result of the\nreactor $\\theta_{13}$ measurement, prefer the value around\n$\\delta_{CP} \\sim - \\pi\/2$ (or equivalently, $\\delta_{CP} \\sim\n3\\pi\/2$) for both mass hierarchies though the statistical significance\nis still small.\nInterestingly, the Super-K atmospheric neutrino data also prefers\nsimilar $\\delta_{CP}$ values with a similar statistical\nsignificance~\\cite{Wendell:2014dka}.\n\nIf $CP$ is maximally violated ($|\\sin \\delta_{CP} | \\sim 1$), $CP$\nviolation ($\\sin \\delta_{CP} \\ne 0$) could be established at $\\sim$\n(2-3)$\\sigma$ CL by combining the future data coming from T2K and NOvA\nas well as with data coming from the reactor $\\theta_{13}$\nmeasurements.\n\nIn Hyper-K the neutrino oscillation parameters will be measured using\ntwo neutrino sources which can provide complementary information.\nBoth atmospheric neutrinos, where neutrino oscillations were first\nconfirmed by Super-K, and a long baseline neutrino beam, where\nelectron neutrino appearance was first observed by T2K, will be\nemployed.\n\nWith a total exposure of 1.3~MW $\\times$ 10$^8$ sec integrated proton\nbeam power (corresponding to $2.7\\times10^{22}$ protons on target\nwith a 30~GeV proton beam) to a $2.5$-degree off-axis neutrino beam,\nit is expected that the leptonic $CP$ phase $\\ensuremath{\\delta_{CP}} $ can be\ndetermined to better than 23 degrees for all possible values of\n$\\ensuremath{\\delta_{CP}} $, and $CP$ violation can be established with a statistical\nsignificance of more than $3\\sigma$ ($5\\sigma$) for $76\\%$\n($57\\%$) of the $\\ensuremath{\\delta_{CP}} $ parameter space.\n\nFigure~\\ref{fig:hierarchy_CP} shows how both CP-violation and mass\nhierarchy affect the difference between $\\nu_\\mu \\to \\nu_e$ detection\nprobability relative to $\\bar{\\nu}_\\mu \\to \\bar{\\nu}_e$ detection\nprobability for a given set of\nneutrino parameters. \n\n\\begin{figure}\n   \\centering\n   \\includegraphics[width=0.9\\textwidth]\n                   {introduction\/figs\/Asym-Pmue-295km-test.pdf} \n   \\caption{The effect of neutrino mass hierarchy and $CP$-violation ($\\delta_{CP}$) on\n     the neutrino\/antineutrino detection probability, for a specific\n     set of neutrino mixing parameters, neutrino energy(E$_\\nu$), and\n     propagation length (L).}\n   \\label{fig:hierarchy_CP}\n\\end{figure}\n\n\\subsection{Astrophysical neutrino observations \\label{subsection:intro\/astrophysics}}\n\n Hyper-K is also capable of observing neutrinos from various\n astrophysical objects.  One main advantage of the detector is that\n its energy threshold can be set as low as several MeV; this enables\n us to reconstruct neutrinos from the Sun and supernovae on an\n event-by-event basis.\n\nThe Sun is an abundant and nearby source of neutrinos. Recently,\nSuper-K showed the first indication of the terrestrial matter effects\non $^8$B solar neutrino oscillations ~\\cite{sk4-daynight}. This was a\ndirect confirmation of the MSW model\n\\cite{Wolfenstein:1977ue,Mikheyev:1985zz,Mikheyev:1986zz} predictions\nfor neutrino interactions with matter, which is also used to describe\nneutrino behaviour as it travels through the Sun. Furthermore,\nterrestrial matter effects hint at an intriguing possibility of using\natmospheric and long baseline neutrinos to measure mass hierarchy and\n$CP$ phase as both these parameters affect how neutrinos interact with\nmatter. Hyper-K hopes to measure terrestrial matter effects with\nhigher precision to better understand neutrino oscillation behaviour\nin the presence of matter. This also might resolve the $\\sim 2 \\sigma$\ntension between the current best fit values of $\\Delta m^2_{21}$ from\nsolar and reactor neutrino experiments, which is thought to be due to\nsolar neutrino interactions in matter. Additionally, there are several\nphysics goals for the solar neutrino observations in Hyper-K, such as\nlong and short time variation of the $^8$B flux, the first measurement\nof $hep$ neutrinos, and precise measurement of solar neutrino energy\nspectrum.\n\nComputational simulations of core-collapse supernovae (CCSN) have\nfailed to successfully reproduce explosions for more than 40 years.\nHowever, thanks to the recent advances in modeling techniques and the\ngrowth of available computation power, multi-dimensional (2D and 3D)\nsimulations can now produce successful explosions \\cite{Bruenn:13,\n  Melson:15a, Lentz:15, Takiwaki:14}.  Nevertheless, there are still\nsome puzzles, such as the finding that the total explosion energy of\nthe available multi-dimensional models is small compared to the\nSN1987A observation.  Furthermore, the available 3D models are\ngenerally less energetic (or unsuccessful) compared with the more\nextensively simulated 2D models \\cite{Couch:14, Tamborra:13,\n  Lentz:15,Takiwaki:14}.  Clearly, details of the supernova explosion\nmechanism are still lacking.  High statistics observations of\nneutrinos from a CCSN (along with gravitational waves) are the only\nway to obtain precious inside information on the dynamics of the CCSN\ncentral engine and the explosion mechanism\n\\cite{OConnor:13,Tamborra:13}.  If a CCSN explosion were to take place\nnear the center of our Galaxy, Hyper-K would observe as many as\ntens of thousands of neutrino interactions (see Section~\\ref{sec:supernova}).\nFurthermore, Hyper-K will have the ability to precisely determine the\narrival time of supernova neutrinos, which will help contribute to the\nunderstandings of both neutrino and CCSN properties. For example, by\ncomparing of the number of $\\nu_e$ and $\\bar{\\nu}_e$ during the CCSN\nneutronization burst (first $\\sim$10\\,msec) we will be able to\ndetermine the neutrino mass hierarchy (see\nSection~\\ref{sec:supernova}). High frequency timing will also provide\nexperimental evidence of the multidimensional dynamics thought to be\ncrucial in the CCSN explosion mechanism \\cite{Tamborra:13}. A large\ntarget volume like that of Hyper-K is also required to observe\nneutrinos from CCSN explosions in nearby ($\\sim$ few Mpc) galaxies.\nIn this volume, CCSNe occur every few years \\cite{Ando:2005ka}.\nMeanwhile, while waiting for a nearby explosion to occur, the\ncontinuous flux of relic supernova neutrinos from all past CCSN\nexplosions in the observable universe will guarantee a steady\naccumulation of valuable astrophysical data.\n\nThanks to its good low energy performance for upward-going muons,\nHyper-K has a larger effective area for upward-going muons below 30\nGeV than do cubic kilometer-scale neutrino telescopes.  Additionally,\nfully contained events in Hyper-K have energy, direction, and flavor\nreconstruction and resolutions as good as those in Super-K.  This high\nperformance will be useful for further background suppression or\nstudies of source properties.  For example, the detector is extremely\nsensitive to the energy range of neutrinos from annihilations of light\n(below 100 GeV) WIMP dark matter, a region which is suggested by\nrecent direct dark matter search experiments. Hyper-K can search for\ndark matter WIMPs by looking for neutrinos created in pair\nannihilation from trapped dark matter in the Galactic centre or the\ncentre of the Sun. Atmospheric neutrinos are a background to this WIMP\nsearch, so spacial cuts are made to determine if there is an excess of\nneutrinos coming from the Galactic centre or the Sun. Hyper-K will\nhave the ability to detect both $\\nu_{e}$ and $\\nu_{\\mu}$ components\nof the signal, making it more sensitive to this type of analysis.\n\nThe detection of neutrinos from solar flares is another astrophysics\ngoal for Hyper-K.  This will give us important information about the\nmechanism of the particle acceleration at work in solar flares.  There\nhave been some estimations of the number of expected\nneutrinos. Although it has large uncertainties, about 20 neutrinos\nwill be observed at Hyper-K during a solar flare as large as the one\nin 20 January 2005.\nHyper-K also has the potential to see neutrinos from astrophysical\nsources such as magnetars, pulsar wind nebulae, active galactic\nnuclei, and gamma ray bursts. The large target volume of Hyper-K,\ncombined with the potential for these sources to emit neutrinos with\nenergy at the GeV-TeV scale, could make Hyper-K an interesting\nexperiment for observing these neutrinos. As with dark matter\nsearches, the most significant background for detecting neutrinos for\nthese astrophysical sources are atmospheric neutrinos. Spacial, and in\nsome cases temporal, cuts need to be utilized to disentangle the\nastrophysical neutrino signal from the atmospheric neutrino\nbackground.\n\n\\subsection{Nucleon decay searches\\label{subsection:intro\/nucleondecay}}\n\nThe stability of everyday matter motivated Weyl, Stueckelberg, Wigner,\nand other early quantum physicists to introduce a conserved quantity,\nbaryon number, to explain the observed and unobserved particle\nreactions. Baryon number violation is believed to have played an\nimportant role during the formation of the universe, and comprises one\nof the famous Sakharov Conditions to explain the baryon asymmetry of\nthe universe. Proton decay and the decay of bound neutrons are\nobservable consequences of the violation of baryon number.\n\nThe Standard Model Lagrangian explicitly conserves baryon number,\nalthough anomalous quantum effects do violate baryon number at an\nunobservably tiny level. Nevertheless, there are reasons to believe\nthat the Standard Model is part of a more expansive theory. Baryon\nnumber violation is a generic prediction of Grand Unified Theories\n(GUTs) that combine quarks and leptons and include interactions that\nallow their transition from one to the other. These theories are well\nmotivated by observations such as the equality of the sum of quark and\nlepton charges, the convergence of the running gauge couplings at an\nenergy scale of about $10^{16}$ GeV, and frequently have mechanisms to\ngenerate neutrino mass. If new forces carrying particles have masses\nat this GUT scale, the lifetime of the proton will be in excess of\n$10^{30}$ years, where past, present, and future proton decay\nexperiments must search.\n\nBaryon number violation has never been experimentally observed and\nlifetime limits, mainly by Super-Kamiokande, greatly restrict\nallowable Grand Unified theories and other interactions of interest to\nmodel building theorists. In Fig.~\\ref{fig:limits-compared-theory}, we\nshow 90\\% CL lifetime limits by Super-Kamiokande and earlier\nexperiments compared with representative lifetime ranges predicted by\nvarious GUTs. We also show the improvement in lifetime limits expected\nfor 10 years of Hyper-Kamiokande exposure. The complementary\nexperiment DUNE, assumed to be a 40 kt liquid argon time projection\nchamber (LArTPC) is also a sensitive to nucleon decay. Due to its\nsmaller mass compared to Hyper-K, it is competitive mainly in modes\nwith distinctive final state tracks such as those involving kaons.\n\n\\begin{figure}\n   \\centering\n   \\includegraphics[width=0.9\\textwidth]\n                   {introduction\/figs\/sklimits_compare_theory_2015_KamLAND.pdf} \n   \\caption{A comparison of historical experimental limits on the rate\n     of nucleon decay for several key modes to indicative ranges of\n     theoretical prediction. Included in the\n     figure are projected limits for Hyper-Kamiokande and DUNE based\n     on 10 years of exposure.}\n   \\label{fig:limits-compared-theory}\n\\end{figure}\n\nThe message the reader should conclude from this figure is that 10\nyears of Hyper-K exposure is sensitive to lifetimes that are commonly\npredicted by modern grand unified theories. The key decay channel $p\n\\rightarrow e^+\\pi^0$ has been emphasized, because it is dominant in a\nnumber of models, and represents a nearly model independent reaction\nmediated by the exchange of a new heavy gauge boson with a mass at the\nGUT scale. The other key channels involve kaons, wherein a final state\ncontaining second generation quarks are generic predictions of GUTs\nthat include supersymmetry. Example Feynman diagrams are shown in\nFig.~\\ref{fig:diagrams-ek}.\n\n\\begin{figure}\n   \\centering\n   \\includegraphics[width=0.9\\textwidth]\n                   {introduction\/figs\/diagrams-ek.pdf} \n   \\caption{Two sample Feynman diagrams that could be responsible for\n     proton decay. The left diagram is a $d = 6$ interaction mediated\n     by a heavy gauge boson, $X$, with mass at the GUT scale. The\n     right diagram contains superymmetric particles and a $d = 5$\n     operator that is predicted in many SUSY GUTs.}\n   \\label{fig:diagrams-ek}\n\\end{figure}\n\nGenerally, nucleon decay may occur through multiple channels and\nideally, experiments would reveal information about the underlying GUT\nby measuring branching ratios.  It is a strength of Hyper-K that it is\nsensitive to a wide range of nucleon decay channels, however the few\nshown here are sufficient to discuss the details of the search for\nnucleon decay by Hyper-Kamiokande later in this document.\n\nPractically, because of the stringent limits from more than\n300\\,kt$\\cdot$y of Super-K running, the next generation experiments\nwill have to concentrate on the discovery of nucleon decay, perhaps by\none or a small number of events. The predictions are uncertain to two\nor three orders of magnitude, and one should not expect a negative\nsearch to definitively rule out the idea of GUTs. To excel in the\nsearch for proton decay, Hyper-Kamiokande requires the largest mass\nthat is affordable in combination with sufficient instrumentation to\nminimize experimental background.\n\n\\subsection{Synergies between Hyper-K and other neutrino experiments}\n\nThis section will focus on understanding how Hyper-K will fit into the\ncontext of the global neutrino community.  This includes currently\noperating experiments such as T2K and Super-K, as well as future\nexperiments like DUNE.\n\n\\subsubsection{T2K}\n\nT2K \\cite{Abe:2011sj} is a currently-operating experiment which uses\nSuper-K to measure neutrinos produced in the J-PARC beam line. Hyper-K\nwill use much of the existing infrastructure used by T2K, particularly\nthe beam line and near detectors. Hyper-K will also benefit from any\nimproved data analysis techniques developed for T2K. Several important\nT2K upgrades and improvements are planned for the coming years, and\nthis will have a direct impact on improved Hyper-K performance.\n\n\\begin{itemize}\n\t\\item\n          \\textbf{Near detector improvements}: The T2K experiment uses\n          the ND280 near detector suite. Future analysis improvements\n          in the ND280 detector aim to reduce the cross section and\n          flux uncertainties. Hardware upgrades, particularly to the\n          time projection chamber component has also been\n          proposed. The reader should refer to\n          Section~\\ref{subsection:nd280} for the full details of the\n          current and future status of the ND280 detector.\n\t\\item\n          \\textbf{Increased beam power}: J-PARC is planning an upgrade\n          of the proton drivers in the neutrino beam. The near-term\n          goal is to improve the beam power from 365\\,kW to\n          750\\,kW. After the proton driver upgrade, beam power is\n          projected to reach 1300\\,kW. See Section~\\ref{section:jparc}\n          for more details.\n\t\\item\n          \\textbf{Better data analysis techniques}: T2K demonstrated\n          in its publications about $\\nu_e$ appearance that\n          aspects of the data analysis such $\\pi^0$ rejection can be\n          improved. Other improvements to the data analysis technique\n          are under development, including $\\nu_e$ detection\n          efficiency, precision of the vertex\n          determination (which could allow for an increased fiducial\n          volume), and an improvement in $\\pi\/\\mu$ separation.\n\\end{itemize}\n\nIn addition to benefiting directly from the upgraded hardware and\nanalysis techniques, Hyper-K will also benefit from the expertise\ngained through implementing these upgrades. Furthermore, these\nupgrades can serve as a test bed for new near detector designs that\nhave been proposed for Hyper-K (see Section~\\ref{section:jparc}).\n\n\\subsubsection{Super-K}\n\n\n\n\nIn June 2015, the Super-Kamiokande Collaboration approved the SK-Gd\nproject.  This project is an upgrade of the detector's capabilities,\nachieved by dissolving 0.2\\% gadolinium sulfate into Super-K's water\nin order to enhance detection efficiency of neutrons from neutrino\ninteractions. One of the main motivations of SK-Gd is to discover\nsupernova relic neutrinos (SRN), the diffuse flux of neutrinos emitted\nby all supernovae since the beginning of the universe.  SRN primarily\ninteract in Super-K via inverse beta decay (IBD). Therefore,\nfollowing the prompt detection of a positron, the accompanying IBD\nneutron can be identified in SK-Gd by a delayed gamma cascade, the\nresult of the neutron's capture on gadolinium.  As a result of this\npositive identification of true IBD events, a much improved separation\nbetween signal and background can be achieved.\n\nAs Super-K will be the first example of gadolinium loading in a\nlarge-scale water Cherenkov detector, this will be a template for any\nfuture possibility of loading gadolinium into Hyper-K. In addition to\ndetermining the physics performance of gadolinium-loaded water,\nHyper-K will also benefit from the extensive research done to optimize\nthe water purification system, as well as the tests for material\ncompatibility that was required to upgrade the Super-K detector.\n\n\\subsubsection{DUNE}\nThe Deep Underground Neutrino Experiment (DUNE), formerly LBNE\n\\cite{LBNE}, is a 40 kilotonne liquid argon neutrino experiment that\nis projected to begin taking data around the same time as\nHyper-K. Because DUNE will use a different target material than\nHyper-K (liquid argon rather than water), many complementary\nmeasurements can be made, including nucleon decay measurements (as\ndescribed in Section~\\ref{subsection:intro\/nucleondecay}) and\nsupernova neutrino detection.\n\nAs mentioned in Section~\\ref{subsection:intro\/astrophysics},\ninformation about the neutrino signature from supernovae is much\nsought after, and Hyper-K and DUNE will each add to the overall\npicture. The primary reaction channel for these neutrinos in Hyper-K\nis the inverse beta decay channel, in which only electron\nantineutrinos will take part. In DUNE, the reaction channel will be\nthe charged-current reaction on $^{40}$Ar, which measures electron\nneutrinos. Taken together, these measurements will be able to\ndetermine the relative abundance of neutrinos to\nantineutrinos. Furthermore, DUNE will be able to better determine some\nfeatures of the neutrino spectrum which are dominated by the electron neutrino\nsignal, such as the neutronization burst that occurs during early\ntimes, while Hyper-K will better measure features where there is an\nantineutrino signal, such as the accretion and cooling phases that\noccur at late times.\n\nDue to the fact that the baseline between the accelerator facility and Hyper-K will be\nshorter than the proposed baseline for the DUNE experiment, the two\nexperiments will have some complementarity in the information they can\nextract from their accelerator programs. The longer baseline to the\nDUNE experiment\nmeans their measurement will be more affected by matter effects, which\nwill give them more sensitivity to the mass\nhierarchy. The shorter baseline of Hyper-K experiment means less\nsensitivity to matter effects, which should lead to an increased\nsensitivity to the measurement of the CP-violation phase. This is further\ndescribed in Section~\\ref{sec:cp}.  \n\n\n\\subsubsection{Off-axis angle spanning configuration \\label{sec:nuprism}}\nThe intermediate WC detector can be oriented with the polar axis of\nthe cylinder in the vertical direction and the detector extending from\nthe ground level downward.  This configuration was originally \nproposed as the NuPRISM detector~\\cite{Bhadra:2014oma}, located at a\nbaseline of 1\\,km filling a 10\\,m diameter, 50\\,m deep pit.  \nFig.~\\ref{fig:nuprism_offaxis} shows the conceptual drawing for the\nNuPRISM detector and the $\\nu_{\\mu}$ spectra for the\n$1^{\\circ}-4^{\\circ}$ off-axis angle range spanned by the detector.\nThe baseline design for the detector is an instrumented structure with\na 10\\,m tall inner-detector containing 3215 8 inch inward facing\nphotomultiplier tubes to detect the Cherenkov light, giving 40\\%\nphoto-coverage. A crane system will move the detector structure\nvertically in the 50\\,m pit to make measurements at different off-axis\nangles. By orienting the long axis of the detector perpendicular to the beam\ndirection, the detector covers a range of angles relative to the beam\ndirection.  Hence, each vertical slice of the detector samples a\ndifferent neutrino spectrum due to the decay kinematics of the pions\nand kaons producing the neutrinos, the so-called off-axis beam effect.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.25\\textwidth]{jparc\/figs\/nuprism-middlecut.png}\n\\includegraphics[width=0.40\\textwidth]{jparc\/figs\/off_axis_spectrum.pdf}\n\\caption{Left: A conceptual drawing of the nuPRISM detector.  Right: the $\\nu_{\\mu}$ flux energy dependence for the\n$1^{\\circ}-4^{\\circ}$ off-axis angle range.\n\\label{fig:nuprism_offaxis}}\n\\end{figure}\n\nThere are three primary motivations for making neutrino measurements\nover a range of off-axis angles.  First, the change in the neutrino\nspectrum with off-axis angle is well known from the flux model, so the\npredicted off-axis spectra can be combined in a linear combination to\nproduce almost arbitrary neutrino spectra.  Measured distributions at\ndifferent off-axis angles can be combined in the linear combination to\nproduce the predicted measured quantity for the neutrino spectrum of\ninterest.  In this way, it is possible to measure the muon spectrum\nfor a nearly mono-chromatic neutrino spectrum, or a spectrum that\nclosely matches the oscillated spectrum that is expected at the far\ndetector.  This approach can nearly eliminate the main model dependent\nuncertainty in near to far extrapolations, which arises from the\ncombination of two factors: the near and far detector do not see the\nsame flux due to oscillations, and the relationship between the true\nneutrino energy and final state lepton kinematics strongly depends on\nnuclear effects, which are not well modelled~\\cite{Bhadra:2014oma}.\n\nThe second physics motivation is the measurement of the electron\nneutrino cross section relative to the muon neutrino cross section.\nAt further off-axis positions, the fraction of intrinsic\n$\\nu_{e},\\bar{\\nu}_{e}$ in the beam becomes large, making the\nselection of pure candidate samples possible.  By taking advantage of\nthe enhanced purity at large off-axis angles, a measurement of the\ncross section ratio, $\\sigma_{\\nu_{e}}\/\\sigma_{\\nu_{\\mu}}$ with 3\\%\nprecision or better may be possible.  A measurement of the\n$\\sigma_{\\bar{\\nu}_{e}}\/\\sigma_{\\bar{\\nu}_{\\mu}}$ ratio is also\npossible, although the precision is expected to be degraded due to the\nlarger neutral current background rate for electron antineutrino\ncandidates and the presence of a larger wrong-sign background for both\nmuon and electron antineutrino charge current interactions.\n\nThe third physics motivation is the search for sterile neutrino\ninduced oscillations that are consistent with the\nLSND~\\cite{Aguilar:2001ty} and\nMiniBooNE~\\cite{Aguilar-Arevalo:2013pmq} $\\bar{\\nu}_{e}$ and $\\nu_{e}$\nappearance anomalies.  At a 1\\,km baseline, the $L\/E$ of the neutrino\nspectrum peak varies between 1.1\\,km\/GeV at $1^{\\circ}$ off-axis to\n2.5\\,km\/GeV at $4^{\\circ}$ off-axis.  Since the neutrino spectrum\nvaries with off-axis angle, it is possible to search for the\noscillation pattern not only through the reconstructed energy of the\nneutrino candidate events, but also through the reconstructed off-axis\nangle.  This method provides a significant improvement in the electron\nneutrino appearance search sensitivity, and preliminary studies with a\nnon-optimal detector configuration already show that much of the LSND\nallowed region can be excluded at 5$\\sigma$~\\cite{NUPRISMProposal}.\n\n\\subsubsection{Gadolinium Loading}\n\nRecent developments in the addition of gadolinium\n(Gd)~\\cite{Watanabe:2008ru} and Water-based Liquid Scintillator (WbLS)\ncompounds~\\cite{Alonso:2014fwf} to water raise the possibility to\nseparate neutrino and antineutrino interactions by detecting the\npresence of neutrons or protons in the final state.\n\nFinal state proton tagging has been studied intensively for an\napplication for LArTPC detectors~\\cite{Acciarri:2014gev}, where final\nstate protons can be counted to further purify the sample to improve\nthe oscillation sensitivity~\\cite{Mosel:2013fxa}.  An analogous\napproach is possible for the larger WC detectors.\nNamely, Gd-doped WC detectors possess neutron tagging\nability on top of the 4$\\pi$ detector coverage~\\cite{Abe:2014oxa},\nwhich allows statistical separation of primary interaction modes,\notherwise impossible.  \nIn particular, the ability to tag neutrons provides charge separation\ninformation due to the enhanced presence of neutrons in the final\nstate for $\\bar{\\nu}$ charged current interactions. This will allow\nstudies of neutrino:anti-neutrino cross-section ratios on water and constraints on wrong-sign backgrounds, thus\nreducing a critical systematic uncertainty in both the beam\n$\\delta_{CP}$ and atmospheric neutrino oscillation analyses. Neutron\ntagging also allows more detailed studies of the interaction modes,\nand in particular final state interaction effects, for the main\nbackgrounds to proton decay. \n\nThe TITUS detector was originally proposed to provide an intermediate detector with neutron tagging capabilities for Hyper-K as described in Ref.~\\cite{TITUSpreprint}. \nFigure~\\ref{fig:neutrontaggingresolution} shows an example of a WC near detector simulation study for TITUS in which selecting ``neutron$\\ge$1'' increases the selection purity for $\\bar{\\nu}$CCQE\ninteractions and hence improves the energy resolution. This technique\nwill be also applied to the ANNIE experiment~\\cite{Anghel:2015xxt} in\nthe next years.\n\n\\begin{figure}[!tbp]\\centering\n\n\\includegraphics[width=0.32\\textwidth]{jparc\/figs\/resolution_titus_muon_rhc.pdf}\n\\includegraphics[width=0.32\\textwidth]{jparc\/figs\/resolution_titus_muon_rhc_noneutron.pdf}\n\\includegraphics[width=0.32\\textwidth]{jparc\/figs\/resolution_titus_muon_rhc_hasneutron.pdf}\n\\caption{The neutrino energy resolution due to the QE assumption in water Cherenkov near detector simulation \nfor the TITUS detector~\\cite{TITUSpreprint} during anti-neutrino mode running. \nThe effect of different neutron selections is shown. From left to right, no neutron tagging, neutron number =0, \nand neutron number $>$ 0.\\label{fig:neutrontaggingresolution}}\n\\end{figure}\n\nOne aspect of the intermediate detector's design that needs to be\ncarefully considered with Gd-doped water is how to veto incoming\nneutrons from beam-induced interactions in the material surrounding\nthe detector which will be the dominant contributor for the number of\nparticles entering the detector.  Vetoing most of these particles\nrequires at least 1\\,m of water to reduce the low energy tail, plus a\nfiducial cut on the reconstructed capture vertex. Preliminary studies\nusing spallation rates induced by muons~\\cite{Galbiati:2005ft} and\ninteractions in the material surrounding the detector show this veto\ncan reduce the number of neutrons entering the detector's ID to just\n10\\% of all the events entering the tank and the fiducial region\nfurther reduces this to approximately 7\\% of the entering particles.\n\nIn principle, Gd loading is compatible with the off-axis spanning\ndetector configuration described in the previous section.  The\noff-axis spanning detector should be as near as possible to the beam\norigin to reduce the depth of the excavated volume, while the Gd\nloaded detector should be far enough away to limit the beam induced\nentering neutron background to the necessary level.  Preliminary\nstudies suggest that the entering neutron rate is sufficiently low for\nthe off-axis spanning detector located at 1\\,km from the neutrino\nproduction point.\n\n    \n\n\n\\section{J-PARC neutrino beam facility} \\label{section:jparc}\nThe accelerator neutrinos detected by Hyper-K are produced at the\nJapan Proton Accelerator Research Complex (J-PARC)~\\cite{JPARCTDR}.\nThe proton accelerator chain, neutrino beamline and near detectors are\nlocated within J-PARC.  Proposed intermediate detectors would be\nlocated near the J-PARC site at a distance of 1-2\\,km from the\nproduction target. This section describes the proton accelerator\nchain, neutrino beamline, near detectors and proposed intermediate\ndetectors.  In each case, the current configuration and future\nupgrades are described.\n\n\\subsection{Neutrino beam and near detectors in long baseline oscillation measurements \\label{sec:beam_and_nd}}\nThe neutrino beam is produced by colliding 30\\,GeV protons\nextracted from the J-PARC accelerator chain with a 91\\,cm long graphite\ntarget.  Three magnetic horns focus secondary charged particles that\nare produced in the proton-target collisions.  The polarity of the\nhorns' currents determine which charge is focused and defocused,\nallowing for the creation of neutrino or antineutrino enhanced beams.\nThe secondary particles are allowed to decay in a 96\\,m long decay\nvolume.  The dominant source of neutrinos is the decay of $\\pi^{\\pm}$.\nMost $\\mu^{\\pm}$ are stopped in the absorber located at the end of the\ndecay volume before they decay, and $\\nu_{e}(\\bar{\\nu_{e}})$ from\n$\\mu^{\\pm}$ decays contribute less than 1\\% to the total neutrino flux\nat at the peak energy.\n\nThe J-PARC beam is aimed 2.5$^{\\circ}$ away from the Super-K and\nHyper-K detectors to take advantage of the pion decay kinematics to\nproduce a narrow band beam~\\cite{Beavis-BNL-52459} with a spectrum\npeaked at 600\\,MeV, at the first oscillation maximum for a baseline of\n295\\,km. Fig.~\\ref{fig:hk_fluxes} shows the calculated energy dependent\nneutrino fluxes in the absence of neutrino oscillations impinging on\nHyper-K for 320\\,kA horn currents in both horn polarities.  Neglecting\noscillations, neutrino detectors located near the neutrino source\nobserve a similar neutrino spectrum to the far detector spectrum, but\nthe peak of the spectrum is broader since the beam appears as a line\nsource for near detectors, compared to a point source for far\ndetectors.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{physics-lbl\/figures\/hk_numode_flux.pdf}\n\\includegraphics[width=0.45\\textwidth]{physics-lbl\/figures\/hk_anumode_flux.pdf}\n\\caption{The neutrino spectra at Hyper-K for the neutrino enhanced\n  (left) and antineutrino enhanced (right) horn current polarities\n  with the absolute horn current set to 320\\,kA.\n\\label{fig:hk_fluxes}}\n\\end{figure}\n\n\nThe neutrino flux is calculated using a data-driven simulation that\nemploys primary proton beam measurements, hadron production\nmeasurements, beamline element alignment measurements and horn current\nand field measurements~\\cite{Abe:2012av}.  The dominant uncertainty on\nthe flux calculation arises from modeling of hadron production in the\ngraphite target and surrounding material.  To minimize the hadron\nproduction uncertainties, the NA61\/SHINE\nexperiment~\\cite{Abgrall:2014xwa} has measured particle production\nwith 30\\,GeV protons incident on a thin (4\\% of an interaction length)\ntarget~\\cite{Abgrall:2011ae,Abgrall:2011ts}, and a replica T2K\ntarget~\\cite{Abgrall:2012pp}. The thin target data have been used in\nthe T2K flux calculation, and a 10\\% uncertainty on the flux\ncalculation has been achieved, as shown in\nFig.~\\ref{fig:t2k_flux_errors}.  Much of the remaining uncertainty\narises from the modeling of secondary particle re-interactions inside\nthe target.  Preliminary work suggests that the hadron production\nuncertainty can be reduced to $\\sim5\\%$ by using the NA61\/SHINE\nmeasurement of the particle multiplicities exiting the T2K replica\ntarget~\\cite{Hasler:2039148}.  In the context of Hyper-K, the thin\ntarget data from NA61\/SHINE are applicable to the flux calculation,\nand the replica target data may also be used if the target geometry\ndoes not change significantly.  If the target geometry or material \nare changed for Hyper-K, then new hadron production measurements will\nbe necessary.  \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{jparc\/figs\/total_err_sk_numode_numu.pdf}\n\\includegraphics[width=0.45\\textwidth]{jparc\/figs\/total_err_sk_anumode_numub.pdf}\n\\caption{The uncertainties on the T2K flux calculation at Super-K for neutrino enhanced (left) and \nantineutrino enhanced (right) beams.\n\\label{fig:t2k_flux_errors}}\n\\end{figure}\n\nThe near neutrino detectors of T2K are located 280\\,m from the pion\nproduction target and they include the INGRID on-axis\ndetector~\\cite{Otani:2010zza} and the ND280 off-axis\ndetector~\\cite{Assylbekov:2011sh,Allan:2013ofa,Aoki:2012mf,Amaudruz:2012agx,Abgrall:2010hi}.\nThe INGRID detector is used primarily to measure the beam direction and neutrino yield,\nwhile ND280 measurements provide constraints on the neutrino flux and\ninteraction models that are used to predicted the event rate at the\nfar detector after oscillations. Measurements with the ND280 detector\nare used for both dedicated neutrino cross-section measurements and\nevent rate constraints that are used directly in neutrino oscillation\nmeasurements.  For neutrino cross-section measurements, the neutrino\nflux is derived from the previously described flux calculation and the\nneutrino cross-section is inferred from the event rate and particle\nkinematics measured with the ND280 detector.  The cross-section\nmeasurements provide constraints on the building of models of\nneutrino-nucleus interactions that are ultimately used in oscillation\nmeasurements.  For the oscillation measurements themselves, nuisance\nparameters are introduced to describe the uncertainty on the neutrino\nflux and interaction models.  A fit to a subset of the ND280 data\nsimultaneously constrains the flux and interaction model nuisance\nparameters, and the predicted event rate and uncertainty at the far\ndetector are updated~\\cite{Abe:2015awa}.  The beam direction\nmeasurement, neutrino cross section measurements and direct\nconstraints on the neutrino event rate for oscillation measurements\nare all necessary for long baseline oscillations measurements at\nHyper-K.\n\n\n\\subsection{The J-PARC accelerator chain }\nThe J-PARC accelerator cascade~\\cite{JPARCTDR} \nconsists of a normal-conducting LINAC as an injection \nsystem, a Rapid Cycling Synchrotron (RCS), and a Main Ring synchrotron (MR). \nH$^-$ ion beams, with a peak current of 50 mA and pulse width of 500 $\\mu$s, \nare accelerated to 400 MeV by the LINAC. \nConversion into a proton beam is achieved by charge-stripping foils at\ninjection into the RCS ring, which accumulates and accelerates two\nproton beam bunches up to 3 GeV at a repetition rate of 25 Hz. Most of\nthe bunches are extracted to the Materials and Life science Facility\n(MLF) to generate intense neutron\/muon beams.  The beam power of RCS\nextraction is rated at 1\\,MW.  With a prescribed repetition cycle, four\nsuccessive beam pulses are injected from the RCS into the MR at 40 ms\n(= 25 Hz) intervals to form eight bunches in a cycle, and\naccelerated up to 30 GeV.  In fast extraction (FX) mode operation, the\ncirculating proton beam bunches are extracted within a single turn\ninto the neutrino primary beamline by a kicker\/septum magnet system.\n\n\n\\begin{table}[t]\n \\caption{Main Ring rated parameters for fast extraction, with numbers\n   achieved as of December 2017. The columns show (left to right): the\n   currently achieved operation parameters, the original design\n   parameters, the projected parameters after the MR RF and magnet\n   power supply upgrade, and the projected parameters for the maximum\n   beam power achievable after the upgrade. }\n \\label{jparc:MRFXpara}\n \\begin{center}\n   \\begin{tabular}{lcccc}\n     \\hline \\hline\n     Parameter   & Achieved & Original  & Doubled rep-rate & Long-term Projection \\\\\n     \\hline\n     Circumference          & \\multicolumn{4}{c}{1,567.5\\,m } \\\\\n     Beam kinetic energy    & 30\\,GeV  & 50\\,GeV & 30\\,GeV & 30\\,GeV \\\\\n     Beam intensity         & $2.45\\times 10^{14}$\\,ppp\n     & $3.3\\times 10^{14}$\\,ppp & $2.0\\times 10^{14}$\\,ppp & $3.2\\times 10^{14}$\\,ppp  \\\\\n     ~                      & $3.1\\times 10^{13}$\\,ppb\n     & $4.1\\times 10^{13}$\\,ppb & $2.5\\times 10^{13}$\\,ppb & $4.0\\times 10^{13}$\\,ppb \\\\\n     $[$ RCS equivalent power $]$  & $[$ 575\\,kW $]$\n     &   $[$ 1\\,MW $]$      & $[$ 610\\,kW $]$ &  $[$ 1\\,MW $]$  \\\\\n     Harmonic number        & \\multicolumn{4}{c}{9} \\\\\n     Bunch number           & \\multicolumn{4}{c}{8~\/~spill} \\\\\n     Spill width            & \\multicolumn{4}{c}{$\\sim$~5\\,$\\mu$s} \\\\\n     Bunch full width at extraction  & $\\sim$50\\,ns & -- & $\\sim$50\\,ns & $\\sim$50\\,ns  \\\\\n     Maximum RF voltage     & 280\\,kV  & 280\\,kV & 560\\,kV & 560\\,kV  \\\\\n     Repetition period      & 2.48\\,sec & 3.52\\,sec & 1.32\\,sec & 1.16\\,sec \\\\\n      \\hline\n     Beam power             & 485\\,kW\\footnote{As of 2018} & 750\\,kW & 750\\,kW & 1340\\,kW  \\\\\n     \\hline\n     \\hline\n   \\end{tabular}\n \\end{center}\n\\end{table}\n\nIn the MR FX mode operation,  a beam intensity of 2.45$\\times$10$^{14}$ proton-per-pulse (ppp) \nhas been achieved, corresponding to $\\sim$485\\,kW beam power (as of 2018).\nThe accelerator team is following \na concrete upgrade scenario~\\cite{jparc-midterm-new}\nto reach the design power of 750\\,kW in forthcoming years, \nwith a typical planned parameter set as listed in Table~\\ref{jparc:MRFXpara}.\nThis will double the current repetition rate by \n(i) replacing the magnet power supplies, \n(ii) replacing the RF system, and \n(iii) upgrading injection\/extraction devices. \nBased on high intensity studies of the current accelerator performance,\nit is expected that 1-1.3\\,MW beam power can be achieved after these upgrades~\\cite{jparc-status-upgrade,jparc-plan-2026}.\nThe projected beam performance up to 2028 is shown in Fig.~\\ref{fig:power_proj}.\nFor operation larger than 2\\,MW beam power, conceptual design studies are now\nunderway~\\cite{jparc-longterm-new}, and they include approaches such as raising the RCS top energy, enlarging the MR\naperture, or inserting an \"emittance-damping\" ring between the RCS and MR.\n\n\\begin {figure}[htbp]\n  \\begin{center}\n    \\includegraphics[width=0.8\\textwidth]\n     {jparc\/figs\/jparc_mr_power_projection_2017_labelled.pdf}\n    \\caption{The projected Main Ring fast extraction performance up to 2028, including the beam power, the protons per pulse, and the repitition rate.}\n    \\label{fig:power_proj}\n  \\end{center}\n\\end {figure}\n\n\n\\subsection{Neutrino beamline}\n\\begin {figure}[htbp]\n  \\begin{center}\n    \\includegraphics[width=0.8\\textwidth]\n     {jparc\/figs\/jparc-beamline-overview_v2.pdf}\n    \\caption{The neutrino experimental facility \n             (neutrino beamline) at J-PARC.}\n    \\label{fig:beamline}\n  \\end{center}\n\\end {figure}\nFig.~\\ref{fig:beamline} shows an overview of the neutrino experimental\nfacility~\\cite{Abe:2011ks}. The primary beamline guides the extracted\nproton beam to a production target\/pion-focusing horn system in a\ntarget station (TS). The pions decay into muons and neutrinos during\ntheir flight in a 96 m-long decay volume. A graphite beam dump is\ninstalled at the end of the decay volume, and muon monitors downstream\nof the beam dump monitor the muon profile.  The beam is aimed\n2.5$^{\\circ}$ off-axis~\\cite{OffAxisBeam} from the direction to\nSuper-K and the beamline has the capability to vary the off-axis angle\nbetween 2.0$^\\circ$ to 2.5$^\\circ$.\nThe centreline of the beamline extends 295 km to the west, \npassing midway between Tochibora and Mozumi, so that both \nsites have identical off-axis angles.  \n\\begin {figure}[htbp]\n  \\begin{center}\n    \\includegraphics[width=0.8\\textwidth]\n     {jparc\/figs\/jparc-nu-secondary-beamline-v2.pdf}\n    \\caption{(Left) Side view of the secondary beamline, \n             with a close up of the target station helium vessel.\n             (Right) A schematic view of a support module and shield blocks \n             for horn-3. If a horn fails, the horn together with its \n             support module is transferred remotely to a purpose-built \n             maintenance area, disconnected from the support module \n             and replaced. \n             }\n    \\label{fig:secondaryBL}\n  \\end{center}\n\\end {figure}\n\n\\subsubsection{Secondary beamline}\nThe secondary beamline consists of the beamline from the TS entrance\nto the muon monitors.  Fig.~\\ref{fig:secondaryBL} shows a cross\nsection of the secondary beamline, and a close-up of the TS helium\nvessel.  The secondary beamline components and their capability to\naccept high power beam are reviewed here.\n\nA helium cooled, double skin titanium alloy beam window separates the\nhelium environment in the TS vessel ($\\sim$1 atm pressure) from the\nvacuum of the primary beamline.  The proton beam collides with a\nhelium-cooled graphite production target that is inserted within the\nbore of the first of a three-horn pion-focusing system.  At 750\\,kW\noperation, a $\\sim$20\\,kW heat load is generated in the target.  The\nneutrino production target and the beam window are designed for 750\\,kW\noperation with 3.3$\\times$10$^{14}$ ppp (equivalent to RCS 1\\,MW\noperation) and 2.1\\,sec cycle. In the target, the pulsed beam generates\nan instantaneous temperature rise per pulse of 200 C$^\\circ$ and a\nthermal stress wave of magnitude 7 MPa.  Given the tensile strength,\nthe safety factor is $\\sim$5.  Although the tensile strength and\nsafety factor will be reduced by cyclic fatigue, radiation damage and\noxidization of the graphite, a lifetime of 2$-$5 years is expected.\n\nThe horns are suspended from the lid of the TS helium vessel.  Each\nhorn comprises two co-axial cylindrical conductors which carry up to a\n320 kA pulsed current. This generates a peak toroidal magnetic field\nof 2.1\\,Tesla which focuses one sign of pions.  The heat load generated\nin the inner conductors by secondary particles and by joule heating is\nremoved by water spray cooling.  So far the horns were operated with a\n250 kA pulsed current and a minimum repetition cycle of 2.48 sec.  To\noperate the horns at a doubled repetition rate of $\\sim$1 Hz requires\nnew individual power supplies for each horn utilizing an energy\nrecovery scheme and low inductance\/resistance striplines.  These\nupgrades will reduce the charging voltage\/risk of failure, and, as\nanother benefit, increase the pulsed current to 320 kA.  The horn-1\nwater-spray cooling system has sufficient capacity to keep the\nconductor below the required 80$^\\circ$C at up to 2\\,MW.\n\nAll secondary beamline components become highly radioactive during\noperation and replacements require handling by a remotely controlled\noverhead crane in the target station.  Failed targets can be replaced\nwithin horn-1 using a bespoke target installation and exchange\nmechanism.\n\nBoth the decay volume and the beam dump dissipate $\\sim$1\/3 of the\ntotal beam power, respectively.  The steel walls of the decay volume\nand the graphite blocks of the hadron absorber (core of the beam dump)\nare water cooled and both are designed to accept 3$\\sim$4\\,MW beam\npower since neither can be upgraded nor maintained after irradiation\nby the beam.\n\nConsiderable experience has been gained on the path to achieving\n475\\,kW beam power operation, and the beamline group is promoting\nupgrades to realize 750\\,kW operation and to expand the facilities for the\ntreatment of activated water.  Table~\\ref{jparc:BLupgrade} gives a\nsummary of acceptable beam power and\/or achievable parameters for each\nbeamline component~\\cite{IFW-nu750kW,IFW-numultiMW}, for both the\ncurrent configuration and after the proposed upgrades in forthcoming\nyears.\n\n\n\\begin{table}[t]\n \\caption{Acceptable beam power and achievable parameters\n   for each beamline component after proposed upgrades.\n   Limitations as of December 2017 are also given in\n   parentheses.}\n \\label{jparc:BLupgrade}\n \\begin{center}\n   \\begin{tabular}{lcc}\n     \\hline \\hline\n     Component   & \\multicolumn{2}{c}\n     {Acceptable beam power or achievable parameter}  \\\\\n     \\hline\n     Target      & \\multicolumn{2}{c}{3.3$\\times$10$^{14}$ ppp } \\\\\n     Beam window & \\multicolumn{2}{c}{3.3$\\times$10$^{14}$ ppp } \\\\\n     Horn        &   ~  & ~ \\\\\n     \\multicolumn{1}{c}{cooling for conductors} &\n     \\multicolumn{2}{c}{2 MW} \\\\\n     \\multicolumn{1}{c}{stripline cooling}\n     & ( 750 kW $\\rightarrow$) & $\\sim$3 MW \\\\\n     \\multicolumn{1}{c}{hydrogen production}\n     & ( 1 MW $\\rightarrow$) & $\\sim$2 MW \\\\\n     \\multicolumn{1}{c}{power supply} & ( 250 kA $\\rightarrow$) & 320\nkA  \\\\\n     ~              &  ( 0.4 Hz $\\rightarrow$)  &  1 Hz   \\\\\n     Decay volume   &  \\multicolumn{2}{c}{4 MW} \\\\\n     Hadron absorber (beam dump)  &  \\multicolumn{2}{c}{3 MW} \\\\\n     \\multicolumn{1}{c}{water-cooling facilities}\n      & ( 750 kW $\\rightarrow$) & $\\sim$2 MW \\\\\n     Radiation shielding  & ( 750 kW $\\rightarrow$) & 4 MW \\\\\n     Radioactive cooling water treatment\n     &  ( 600 kW $\\rightarrow$) & $\\sim$1.3 MW \\\\\n     \\hline \\hline\n   \\end{tabular}\n \\end{center}\n\\end{table}\n\\subsection{Near detector complex\\label{sec:NDcomplex}}\nThe accelerator neutrino event rate observed at Hyper-K depends on the\noscillation probability, neutrino flux, neutrino interaction\ncross-section, detection efficiency, and the detector fiducial mass of\nHyper-K.  To extract estimates of the oscillation parameters from\ndata, one must model the neutrino flux, cross-section and detection\nefficiency with sufficient precision.  In the case of the neutrino\ncross-section, the model must describe the exclusive differential\ncross-section that includes the dependence on the incident neutrino\nenergy, $E_{\\nu}$, the kinematics of the outgoing lepton, momentum\n$p_{l}$ and scattering angle $\\theta_{l}$, and the kinematics of final\nstate hadrons and photons.  In our case, the neutrino energy is\ninferred from the lepton kinematics, while the reconstruction\nefficiencies depend on the hadronic final state as well.\n\nThe near detectors measure the neutrino interaction rates close enough to the neutrino\nproduction point so that oscillation effects are negligible.  The prediction of \nevent rates at Hyper-K for a given set of oscillation parmeters will be precisely\nconstrained by event rates measured in the near detector and the flux simulation\nbased on hadron production data from NA61\/SHINE and other hadron production experiments.\nOur approach to using near detector data will build on the experience of\nT2K while considering new near detectors that address\nlimitations in reducing neutrino cross section modelling uncertainties\nwith the current T2K near detector suite.\n\nThe near detectors should be capable of measuring the signal and\nbackground processes relevant for neutrino oscillation measurements\nmade using the accelerator produced neutrinos.  The processes include:\n\\begin{itemize}\n\\item The charged current interactions with no detected final state\n  pion (CC0$\\pi$) that are the signal channel for the oscillation\n  measurements in Hyper-K.\n\\item The intrinsic electron neutrino component of the beam from muon\n  and kaon decays, which is a background for the electron\n  (anti)neutrino appearance signal.\n\\item The neutral current interactions with $\\pi^{0}$ production\n  (NC$\\pi^{0}$) that are a background for the electron (anti)neutrino\n  appearance signal.\n\\item The wrong-sign charged current processes (neutrinos in the\n  antineutrino beam and vise versa) which are a background in the CP\n  violation measurement.\n\\end{itemize}\nIn addition to measuring these processes, the near detectors should be\ndesigned to maximize the cancellation of systematic uncertainties when\nextrapolating from measured event rates in the near detector to\npredict the event rate at Hyper-K.  Hence, the near detector should be\nable to make measurements with the same angular acceptance (4$\\pi$)\nand target nuclei (H$_2$O) as Hyper-K.  Another source of uncertainty\nin the extrapolation is the difference between the near and far\ndetector neutrino spectra due to oscillations, which can amplify\nsystematic errors related to the modeling of the relationship between\nthe final state lepton kinematics and the incident neutrino\nenergy~\\cite{Martini:2012fa,Lalakulich:2012hs,Martini:2012uc}.  The\nnear detectors should be able to sufficiently constrain the modeling\nof the dependence of lepton kinematics on neutrino energy over the\nrelevant neutrino energy range.\n\nThe near detectors can also be used to constrain important neutrino\ninteraction modes for atmospheric neutrino and nucleon decay\nmeasurements at Hyper-K.  For example, Hyper-K may use neutron\ncaptures on Gd or H to statistically separate neutrinos and\nantineutrinos in the atmospheric measurements, or to reject\natmospheric neutrino backgrounds in the nucleon decay measurements.\nThe neutron multiplicities produced in the interactions of neutrinos\nand antineutrinos with energy of $\\mathcal{O}($1 GeV$)$ can be\nmeasured in the near detectors.  The dominant sources of uncertainty\nin the determination of the mass hierarchy and $\\theta_{23}$ quadrant\nwith atmospheric neutrinos are uncertainties in the neutrino to\nanti-neutrino cross section ratio for both CCQE and single pion\nproduction modes, the axial vector nucleon form factor, the\nneutrino-tau cross section, and the DIS cross section model.\nNear detector measurements that would constrain these uncertainties\nfor interactions on water, have the potential to significantly improve\nthe sensitivity of these atmospheric neutrino measurements.\nThe near detectors can also be used to measure the interaction modes for nucleon decay\nbackgrounds, including the CC$\\pi^{0}$ background to the $e^{+}(\\mu^{+})\\pi^{0}$ mode and the kaon production background to the $K^{+}\\nu$ mode.\n\nTo summarize, the near and intermediate detectors for Hyper-K should cover the full momentum and angular acceptance of the far detector, include homogenous H$_2$O targets to make precision measurements on H$_2$O, have sign selection capability to measure wrong sign backgrounds, be capable of directly measuring intrinsic $\\nu_e,\\bar{\\nu_e}$ and NC$\\pi^0$ backgrounds, be capable of reconstructing exclusive final states with low particle thresholds, be capable of measuring the $\\nu_e,\\bar{\\nu_e}$ cross sections, be capable of measuring the final state neutron multiplicities and be capable of measurements at multiple off-axis angles with varying peak neutrino energies.  We have not identified a single detector technology that can achieve all of these capabilities.  In the minimum configuration, it is necessary to have both a magnetized tracking detector and kiloton scale water Cherenkov detector.  The magnetized tracking detector may provide full momentum and angular acceptance, sign selection, reconstruction of exclusive final states with low particle thresholds, measurement of the $\\nu_e,\\bar{\\nu_e}$ rates and off-axis spanning measurements.  The water Cherenkov detector may provide full angular acceptance, a homogenous water target, direct measurement of the intrinsic $\\nu_e,\\bar{\\nu_e}$  and NC$\\pi^0$ backgrounds, measurement of the $\\nu_e,\\bar{\\nu_e}$ cross sections, measurements of final state neutron multiplicities and the off-axis spanning measurements.  Hence, in the following text we present the upgrade for the current ND280 magnetized tracking detector, new tracking detector technologies, and intermediate water Cherenkov detectors as potential components of the Hyper-K near and intermediate detector suite.\n\n\n    \\subsubsection{The ND280 Detector Suite}\n    \\input{jparc\/nd280.tex} \\label{subsection:nd280}\n\n     \n    \\subsubsection{Intermediate detector}\n    \\input{jparc\/intermediate.tex}\n\n\n\\subsection{Summary}\nThis section has outlined the performance and requirements of the\naccelerator complex, neutrino beamline and near detectors at J-PARC\nthat will be required for the Hyper-K physics program.  The J-PARC\naccelerator chain has achieved 475 kW beam power extracted to the\nneutrino beamline.  The accelerator upgrade plan, which includes the\nupgrade of the MR magnet powers supplies and RF, is expected to achieve\n1.3\\,MW beam operation with $3.2\\times10^{14}$ protons per pulse, as\nearly as 2026.\n\nThe neutrino beamline components require some upgrades to accept the\nrepetition rate, proton intensity and total beam power necessary to\nachieve 1.3\\,MW at $3.2\\times10^{14}$ protons per pulse.  To achieve\nthe 1.16 Hz operation, each magnetic horn requires an individual power\nsupply utilizing an energy recovery scheme and low\ninductance\/resistance striplines.  \nThe treatment facilities for activated cooling water will be expanded to accept up to 2 MW operation as well.  \nThe current beam window and\ntarget are rated to $3.3\\times10^{14}$ ppp, however their lifetime at\n$3.2\\times10^{14}$ ppp and 1.16 Hz will be studied and upgrades may be\nnecessary.\n\nThe current T2K near detectors, including ND280 and INGRID, are used\nto control neutrino flux and cross-section systematic errors at the\n$\\sim$5--6\\% level.  Further upgrades to the ND280 data analyses with\nwater target measurements and large angle tracks will reduce the\nsystematic error, although the ultimate performance may be limited by\nthe relatively low water fraction and low efficiency for large angle\ntrack reconstruction.  Upgrades to ND280 are being considered by T2K\nand these include a high pressure TPC, the WAGASCI water\/scintillator\n3D grid detector and emulsion detectors.  A reconfiguration of the TPC\ngeometry is also being considered to give better reconstruction at\nhigh angles.  It is expected that some of these upgrades may be\ncarried out during the T2K experiment and Hyper-K may benefit from\ntheir continued use.  If they are not implemented during T2K, these\nND280 upgrades as well as the continued operation of ND280 are an\nexpected area for international contributions to Hyper-K.\n\nAn intermediate water Cherenkov detector provides a necessary\ncomplement to the ND280 magnetized tracking detector in order to\nconstrain all the dominant systematics at the precision required.  The\nWC detector requires a new facility off of the J-PARC site and the\nexcavation of a new pit to house the detector.  \n\n\n\\subsubsection*{#1}}\n\n\\pagestyle{headings}\n\\markright{Reference sheet: \\texttt{natbib}}\n\n\\usepackage{shortvrb}\n\\MakeShortVerb{\\|}\n\n\\begin{document}\n\\thispagestyle{plain}\n\n\\newcommand{\\textsc{Bib}\\TeX}{\\textsc{Bib}\\TeX}\n\\newcommand{\\texttt{#1}\\def\\filedate{#2}\\def\\fileversion{#3}}}{\\texttt{#1}\\def\\filedate{#2}\\def\\fileversion{#3}}}\n\\begin{center}{\\bfseries\\Large\n    Reference sheet for \\texttt{#1}\\def\\filedate{#2}\\def\\fileversion{#3}}\\ usage}\\\\\n    \\large(Describing version \\fileversion\\ from \\filedate)\n\\end{center}\n\n\\begin{quote}\\slshape\nFor a more detailed description of the \\texttt{#1}\\def\\filedate{#2}\\def\\fileversion{#3}}\\ package, \\LaTeX\\ the\nsource file \\texttt{#1}\\def\\filedate{#2}\\def\\fileversion{#3}}\\texttt{.dtx}.\n\\end{quote}\n\\head{Overview}\nThe \\texttt{#1}\\def\\filedate{#2}\\def\\fileversion{#3}}\\ package is a reimplementation of the \\LaTeX\\ |\\cite| command,\nto work with both author--year and numerical citations. It is compatible with\nthe standard bibliographic style files, such as \\texttt{plain.bst}, as well as\nwith those for \\texttt{harvard}, \\texttt{apalike}, \\texttt{chicago},\n\\texttt{astron}, \\texttt{authordate}, and of course \\texttt{#1}\\def\\filedate{#2}\\def\\fileversion{#3}}.\n\n\\head{Loading}\nLoad with |\\usepackage[|\\emph{options}|]{|\\texttt{#1}\\def\\filedate{#2}\\def\\fileversion{#3}}|}|. See list of\n\\emph{options} at the end.\n\n\\head{Replacement bibliography styles}\nI provide three new \\texttt{.bst} files to replace the standard \\LaTeX\\\nnumerical ones:\n\\begin{quote}\\ttfamily\n plainnat.bst \\qquad abbrvnat.bst \\qquad unsrtnat.bst\n\\end{quote}\n\\head{Basic commands}\nThe \\texttt{#1}\\def\\filedate{#2}\\def\\fileversion{#3}}\\ package has two basic citation commands, |\\citet| and\n|\\citep| for \\emph{textual} and \\emph{parenthetical} citations, respectively.\nThere also exist the starred versions |\\citet*| and |\\citep*| that print\nthe full author list, and not just the abbreviated one.\nAll of these may take one or two optional arguments to add some text before\nand after the citation.\n\\begin{quote}\n\\begin{tabular}{l@{\\quad$\\Rightarrow$\\quad}l}\n  |\\citet{jon90}| & Jones et al. (1990)\\\\\n  |\\citet[chap.~2]{jon90}| & Jones et al. (1990, chap.~2)\\\\[0.5ex]\n  |\\citep{jon90}| & (Jones et al., 1990)\\\\\n  |\\citep[chap.~2]{jon90}| & (Jones et al., 1990, chap.~2)\\\\\n  |\\citep[see][]{jon90}| & (see Jones et al., 1990)\\\\\n  |\\citep[see][chap.~2]{jon90}| & (see Jones et al., 1990, chap.~2)\\\\[0.5ex]\n  |\\citet*{jon90}| & Jones, Baker, and Williams (1990)\\\\\n  |\\citep*{jon90}| & (Jones, Baker, and Williams, 1990)\n\\end{tabular}\n\\end{quote}\n\\head{Multiple citations}\nMultiple citations may be made by including more than one\ncitation key in the |\\cite| command argument.\n\\begin{quote}\n\\begin{tabular}{l@{\\quad$\\Rightarrow$\\quad}l}\n  |\\citet{jon90,jam91}| & Jones et al. (1990); James et al. (1991)\\\\\n  |\\citep{jon90,jam91}| & (Jones et al., 1990; James et al. 1991)\\\\\n  |\\citep{jon90,jon91}| & (Jones et al., 1990, 1991)\\\\\n  |\\citep{jon90a,jon90b}| & (Jones et al., 1990a,b)\n\\end{tabular}\n\\end{quote}\n\n\\head{Numerical mode}\nThese examples are for author--year citation mode. In numerical mode, the\nresults are different.\n\\begin{quote}\n\\begin{tabular}{l@{\\quad$\\Rightarrow$\\quad}l}\n  |\\citet{jon90}| & Jones et al. [21]\\\\\n  |\\citet[chap.~2]{jon90}| & Jones et al. [21, chap.~2]\\\\[0.5ex]\n  |\\citep{jon90}| & [21]\\\\\n  |\\citep[chap.~2]{jon90}| & [21, chap.~2]\\\\\n  |\\citep[see][]{jon90}| & [see 21]\\\\\n  |\\citep[see][chap.~2]{jon90}| & [see 21, chap.~2]\\\\[0.5ex]\n  |\\citep{jon90a,jon90b}| & [21, 32]\n\\end{tabular}\n\\end{quote}\n\\head{Suppressed parentheses}\nAs an alternative form of citation, |\\citealt| is the same as |\\citet| but\n\\emph{without parentheses}. Similarly, |\\citealp| is |\\citep| without\nparentheses.\n\nThe |\\citenum| command prints the citation number, without parentheses, even\nin author--year mode, and without raising it in superscript mode. This is\nintended to be able to refer to citation numbers without superscripting them.\n\n\\begin{quote}\n\\begin{tabular}{l@{\\quad$\\Rightarrow$\\quad}l}\n  |\\citealt{jon90}| & Jones et al.\\ 1990\\\\\n  |\\citealt*{jon90}| & Jones, Baker, and Williams 1990\\\\\n  |\\citealp{jon90}| & Jones et al., 1990\\\\\n  |\\citealp*{jon90}| & Jones, Baker, and Williams, 1990\\\\\n  |\\citealp{jon90,jam91}| & Jones et al., 1990; James et al., 1991\\\\\n  |\\citealp[pg.~32]{jon90}| & Jones et al., 1990, pg.~32\\\\\n  |\\citenum{jon90}| & 11\\\\\n  |\\citetext{priv.\\ comm.}| & (priv.\\ comm.)\n\\end{tabular}\n\\end{quote}\nThe |\\citetext| command\nallows arbitrary text to be placed in the current citation parentheses.\nThis may be used in combination with |\\citealp|.\n\\head{Partial citations}\nIn author--year schemes, it is sometimes desirable to be able to refer to\nthe authors without the year, or vice versa. This is provided with the\nextra commands\n\\begin{quote}\n\\begin{tabular}{l@{\\quad$\\Rightarrow$\\quad}l}\n  |\\citeauthor{jon90}| & Jones et al.\\\\\n  |\\citeauthor*{jon90}| & Jones, Baker, and Williams\\\\\n  |\\citeyear{jon90}|   & 1990\\\\\n  |\\citeyearpar{jon90}| & (1990)\n\\end{tabular}\n\\end{quote}\n\\head{Forcing upper cased names}\nIf the first author's name contains a \\textsl{von} part, such as ``della\nRobbia'', then |\\citet{dRob98}| produces ``della Robbia (1998)'', even at the\nbeginning of a sentence. One can force the first letter to be in upper case\nwith the command |\\Citet| instead. Other upper case commands also exist.\n\\begin{quote}\n\\begin{tabular}{rl@{\\quad$\\Rightarrow$\\quad}l}\n  when & |\\citet{dRob98}| & della Robbia (1998) \\\\\n  then & |\\Citet{dRob98}| & Della Robbia (1998) \\\\\n     &   |\\Citep{dRob98}| & (Della Robbia, 1998) \\\\\n     &   |\\Citealt{dRob98}| & Della Robbia 1998 \\\\\n     &   |\\Citealp{dRob98}| & Della Robbia, 1998 \\\\\n     &   |\\Citeauthor{dRob98}| & Della Robbia\n\\end{tabular}\n\\end{quote}\nThese commands also exist in starred versions for full author names.\n\n\\head{Citation aliasing}\nSometimes one wants to refer to a reference with a special designation,\nrather than by the authors, i.e. as Paper~I, Paper~II. Such aliases can be\ndefined and used, textual and\/or parenthetical with:\n\\begin{quote}\n\\begin{tabular}{lcl}\n  |\\defcitealias{jon90}{Paper~I}|\\\\\n  |\\citetalias{jon90}| & $\\Rightarrow$ & Paper~I\\\\\n  |\\citepalias{jon90}| & $\\Rightarrow$ & (Paper~I)\n\\end{tabular}\n\\end{quote}\nThese citation commands function much like |\\citet| and |\\citep|: they may\ntake multiple keys in the argument, may contain notes, and are marked as\nhyperlinks.\n\\head{Selecting citation style and punctuation}\nUse the command |\\setcitestyle| with a list of comma-separated\nkeywords (without spaces) as argument.\n\\begin{itemize}\n\\item\n  Citation mode: |authoryear| or |numbers| or |super|\n \n\\item\n  Braces: |round| or |square| or |open={|\\emph{char}|},close={|\\emph{char}|}|\n \n\\item\n  Between citations: |semicolon| or |comma| or |citesep={|\\emph{char}|}|\n \n\\item\n  Between author and year: |aysep={|\\emph{char}|}|\n \n\\item\n  Between years with common author: |yysep={|\\emph{char}|}|\n \n\\item\n  Text before post-note: |notesep={|\\emph{text}|}|\n \n\\end{itemize}\nDefaults are |authoryear|, |round|, |comma|, |aysep={;}|, |yysep={,}|,\n|notesep={, }|\n\nExample~1, |\\setcitestyle{square,aysep={},yysep={;}}| changes the author--year\noutput of\n\\begin{quote}\n  |\\citep{jon90,jon91,jam92}|\n\\end{quote}\ninto [Jones et al. 1990; 1991, James et al. 1992].\n\nExample~2, |\\setcitestyle{notesep={; },round,aysep={},yysep={;}}| changes the output of\n\\begin{quote}\n  |\\citep[and references therein]{jon90}|\n\\end{quote}\ninto (Jones et al. 1990; and references therein).\n\n\\head{Other formatting options}\nRedefine |\\bibsection| to the desired sectioning command for introducing\nthe list of references. This is normally |\\section*| or |\\chapter*|.\n\nRedefine |\\bibpreamble| to be any text that is to be printed after the heading but\nbefore the actual list of references.\n\nRedefine |\\bibfont| to be a font declaration, e.g.\\ |\\small| to apply to\nthe list of references.\n\nRedefine |\\citenumfont| to be a font declaration or command like |\\itshape|\nor |\\textit|.\n\nRedefine |\\bibnumfmt| as a command with an argument to format the numbers in\nthe list of references. The default definition is |[#1]|.\n\nThe indentation after the first line of each reference is given by\n|\\bibhang|; change this with the |\\setlength| command.\n\nThe vertical spacing between references is set by |\\bibsep|; change this with\nthe |\\setlength| command.\n\n\\head{Automatic indexing of citations}\nIf one wishes to have the citations entered in the \\texttt{.idx} indexing\nfile, it is only necessary to issue |\\citeindextrue| at any point in the\ndocument. All following |\\cite| commands, of all variations, then insert\nthe corresponding entry to that file. With |\\citeindexfalse|, these\nentries will no longer be made.\n\n\\head{Use with \\texttt{chapterbib} package}\n\nThe \\texttt{#1}\\def\\filedate{#2}\\def\\fileversion{#3}}\\ package is compatible with the \\texttt{chapterbib} package\nwhich makes it possible to have several bibliographies in one document.\n\nThe package makes use of the |\\include| command, and each |\\include|d file\nhas its own bibliography.\n\nThe order in which the \\texttt{chapterbib} and \\texttt{#1}\\def\\filedate{#2}\\def\\fileversion{#3}}\\ packages are loaded\nis unimportant.\n\nThe \\texttt{chapterbib} package provides an option \\texttt{sectionbib}\nthat puts the bibliography in a |\\section*| instead of |\\chapter*|,\nsomething that makes sense if there is a bibliography in each chapter.\nThis option will not work when \\texttt{#1}\\def\\filedate{#2}\\def\\fileversion{#3}}\\ is also loaded; instead, add\nthe option to \\texttt{#1}\\def\\filedate{#2}\\def\\fileversion{#3}}.\n\nEvery |\\include|d file must contain its own\n|\\bibliography| command where the bibliography is to appear. The database\nfiles listed as arguments to this command can be different in each file,\nof course. However, what is not so obvious, is that each file must also\ncontain a |\\bibliographystyle| command, with possibly differing arguments.\n\nAs of version~8.0, the citation style, including mode (author--year or\nnumerical) may also differ between chapters. The |\\setcitestyle| command\ncan be issued at any point in the document, in particular in different\nchapters.\n\\head{Sorting and compressing citations}\nDo not use the \\texttt{cite} package with \\texttt{#1}\\def\\filedate{#2}\\def\\fileversion{#3}}; rather use one of the\noptions \\texttt{sort}, \\texttt{compress}, or \\texttt{sort\\&compress}.\n\nThese also work with author--year citations, making multiple citations appear\nin their order in the reference list.\n\n\n\\head{Merged Numerical References}\nDo not use the \\texttt{mcite} package with \\texttt{#1}\\def\\filedate{#2}\\def\\fileversion{#3}}; rather use the package option \\texttt{merge}.\n\n\nWith this option in effect, citation keys within a multiple |\\citep| command\nmay contain a leading * that causes them to be merged in the bibliography\ntogether with the previous citation as a single entry with a single reference\nnumber. For example, |\\citep{feynmann,*salam,*epr}| produces a single number,\nand all three references are listed in the bibliography under one entry with\nthat number.\n\nThe \\texttt{elide} option also activates the merging features, but also sees\nto it that common parts of the merged references (e.g., authors) are not\nrepeated but are written only once in the single bibliography entry.\n\nThe \\texttt{mcite} option turns off the merging and eliding features, but\nallows the special syntax (the * and optional inserted texts) to be ignored.\n\nThese functions are available only to numerical-mode citations, and only when\nused parenthetically, similar to the restrictions on \\texttt{sort} and\n\\texttt{compress}.\n\nThey also require special \\texttt{.bst} files, as provided for example by the\nAmerican Physical Society for their REV\\TeX\\ class.\n\\head{Long author list on first citation}\nUse option \\texttt{longnamesfirst} to have first citation automatically give\nthe full list of authors.\n\nSuppress this for certain citations with |\\shortcites{|\\emph{key-list}|}|,\ngiven before the first citation.\n\n\\head{Local configuration}\nAny local recoding or definitions can be put in \\texttt{#1}\\def\\filedate{#2}\\def\\fileversion{#3}}\\texttt{.cfg} which\nis read in after the main package file.\n\n\\head{Options that can be added to \\texttt{\\char`\\\\ usepackage}}\n\\begin{description}\n\\item[\\ttfamily round] (default) for round parentheses;\n\\item[\\ttfamily square] for square brackets;\n\\item[\\ttfamily curly] for curly braces;\n\\item[\\ttfamily angle] for angle brackets;\n\\item[\\ttfamily semicolon] (default) to separate multiple citations with\n     semi-colons;\n\\item[\\ttfamily colon] the same as \\texttt{semicolon}, an earlier mistake in\n     terminology;\n\\item[\\ttfamily comma] to use commas as separators;\n\\item[\\ttfamily authoryear] (default) for author--year citations;\n\\item[\\ttfamily numbers] for numerical citations;\n\\item[\\ttfamily super] for superscripted numerical citations, as in\n     \\textsl{Nature};\n\\item[\\ttfamily sort] orders multiple citations into the sequence in\n     which they appear in the list of references;\n\\item[\\ttfamily sort\\&compress] as \\texttt{sort} but in addition multiple\n     numerical citations are compressed if possible (as 3--6, 15);\n\\item[\\ttfamily compress] to compress without sorting, so compression only\n     occurs when the given citations would produce an ascending sequence of\n     numbers;\n\\item[\\ttfamily longnamesfirst] makes the first citation of any reference\n     the equivalent of the starred variant (full author list) and subsequent\n     citations normal (abbreviated list);\n\\item[\\ttfamily sectionbib] redefines |\\thebibliography| to issue\n     |\\section*| instead of |\\chapter*|; valid only for classes with a\n     |\\chapter| command; to be used with the \\texttt{chapterbib} package;\n\\item[\\ttfamily nonamebreak] keeps all the authors' names in a citation on\n     one line; causes overfull hboxes but helps with some \\texttt{hyperref}\n     problems;\n\\item[\\ttfamily merge] to allow the * prefix to the citation key,\n     and to merge such a citation's reference with that of the previous\n     citation;\n\\item[\\ttfamily elide] to elide common elements of merged references, like\n     the authors or year;\n\\item[\\ttfamily mcite] to recognize (and ignore) the merging syntax.\n\\end{description}\n\\end{document}\n\n\\subsection{Other astrophysical neutrino sources}\\label{section:astro}\n   \n\\subsubsection{Solar flare}\nSolar flares are the most energetic bursts which occur in the solar\nsurface.  Explosive release of energy stored in solar magnetic fields\nis caused by magnetic reconnections, resulting in plasma heating,\nparticle accelerations, and emission of synchrotron X-rays or charged\nparticles from the solar surface.  In a large flare, an energy of\n10$^{33}$ ergs is emitted over 10's of minutes, and the accelerated\nprotons can reach energies greater than 10 GeV.  Such high energy\nprotons can produce pions by nuclear interactions in the solar\natmosphere. Evidence of such nuclear interactions in the solar\natmosphere are obtained via observations of solar neutrons, 2.2 MeV\ngamma rays from neutron captures on protons, nuclear de-excitation\ngamma rays, and possible $>100$ MeV gamma rays from neutral pion\ndecays.  Thus, it is likely that neutrinos are also emitted by the\ndecay of mesons following interactions of accelerated particles.\nDetection of neutrinos from a solar flare was first discussed in\n1970's by R.Davis~\\cite{dav, bacall_sf}, but no significant signal has\nyet been found~\\cite{aglietta,hirata}. There have been some estimates\nof the number of neutrinos which could be observed by large water\nCherenkov detectors~\\cite{fargion,kocharov}. According\nto \\cite{fargion}, about 6-7 neutrinos per tank will be observed at\nHyper-Kamiokande during a solar flare as large as the one in 20\nJanuary 2005, although the expected numbers have large uncertainties.\nTherefore, regarding solar flares our first astrophysics goal is to\ndiscover solar flare neutrinos with Hyper-K.  This will give us\nimportant information about the mechanism of the particle acceleration\nat work in solar flares.\n\n\n\\subsubsection{Gamma-Ray Burst Jets and Newborn Pulsar Winds}\nGamma-ray bursts (GRBs) are the most luminous astrophysical phenomena\nwith the isotropically-equivalent gamma-ray luminosity,\n$L_\\gamma\\sim{10}^{52}~{\\rm erg}~{\\rm s}^{-1}$, which typically occur\nat cosmological distance.  Prompt gamma rays are observed in the MeV\nrange, and their spectra can be fitted by a smoothened broken power\nlaw~\\cite{grbrev}.  The prompt emission comes from a relativistic jet\nwith the Lorentz factor of $\\Gamma\\sim{10}^{2}-{10}^{3}$, which is\npresumably caused by a blackhole with an accretion disk or\na fast-spinning, strongly-magnetized neutron star.  Observed gamma-ray\nlight curves are highly variable down to $\\sim1$~ms, suggesting\nunsteady outflows.  However, GRB central engines and their radiation\nmechanism are still unknown, and GRBs have been one of the biggest\nmysteries in high-energy astrophysics.\n\nInternal shocks are naturally expected for such unsteady jets, and the\njet kinetic energy can be converted into radiation via shock\ndissipation.  In the ``classical'' internal shock\nscenario~\\cite{rm94}, observed gamma rays are attributed to\nsynchrotron emission from non-thermal electrons accelerated at\ninternal shocks.  It has been suggested that GRBs may also be\nresponsible for ultrahigh-energy cosmic rays (CRs), and TeV-PeV\nneutrinos have been predicted as a smoking gun of CR acceleration in\nGRBs~\\cite{grbnu1}.  One of the key advantages in GRB neutrino\nsearches is that atmospheric backgrounds can significantly be reduced\nthanks to space- and time-coincidence, but no high-energy neutrino\nsignals correlated with GRBs have been found in any neutrino detector\nincluding Super-K~\\cite{skgrb} and IceCube~\\cite{icecubegrb}.\n\nOn the other hand, the recent theoretical and observational progress\nhas suggested that the above classical scenario has troubles in\nexplaining observational features such as the low-energy photon\nspectrum.  Alternatively, the photospheric scenario, where prompt\ngamma rays are generated around or under the ``photosphere'' (where\nthe Compton scattering optical depth is unity), has become more\npopular~\\cite{photosphere1,photosphere2,photosphere3}.  Indeed,\nobservations have indicated a thermal-like component in GRB\nspectra~\\cite{fermigrb,fermigrb2}.  Energy dissipation may be caused\nby inelastic nucleon-neutron\ncollisions~\\cite{neutron1,neutron2,neutron3}, and neutrons can\nnaturally be loaded by GRB central engines either\nblackhole-accretion-disk system or strongly-magnetized neutron star.\nThen, quasi-thermal GeV-TeV neutrino emission is an inevitable consequence of such inelastic nucleon-neutron collisions~\\cite{mur+13}.  Since neutrinos easily leave the flow, predictions for these neutrinos are insensitive to details of gamma-ray spectra.  \nHyper-K will enable us to search these quasi-thermal GeV-TeV neutrinos\nfrom GRB jets, and it also has an advantage over \nIceCube (that is suitable for higher-energy $>10$-$100$~GeV\nneutrinos).  The GeV-TeV neutrino detection is feasible if a GRB\nhappens at $\\lesssim100$~Mpc, and successful detections should allow\nus to discriminate among prompt emission mechanisms and probe the jet\ncomposition, leading to a breakthrough in understanding GRB physics.\nHowever, GRBs are rare astrophysical phenomena, so we have little\nchance to expect such a nearby bright burst in the next 10-100 years.\nMuch more promising targets as high-energy neutrino sources would be\nenergetic supernovae driven by relativistic\noutflows~\\cite{mw01,rmw04,ab05,mi13}.  A significant fraction of GRB\njets may fail to puncture their progenitor star, and photon emission\nfrom the jets can easily be hidden.  Indeed, theoretical studies\nrevealed the existence of conditions for a jet to make a successful\nGRB.  ``Choked jets'' or ``failed GRBs'' are naturally predicted when\nthe jet luminosity is not sufficient or the jet duration is too short\nor the progenitor is too big.  The choked jets can explain\ntrans-relativistic supernovae or low-luminosity GRBs, which show\nintermediate features between GRBs and supernovae~\\cite{sod+06}.\n\nThe neutrino event rate expected in Hyper-K depends on the\nisotropically-equivalent dissipation energy ${\\mathcal E}_{\\rm\ndiss}^{\\rm iso}$, Lorentz factor $\\Gamma$, and distance $d$.  For\n${\\mathcal E}_{\\rm diss}^{\\rm iso}={10}^{53}~{\\rm erg}~{\\rm s}^{-1}$,\n$\\Gamma=10$ and $d=10$~Mpc, the characteristic energy of quasi-thermal\nGeV-TeV neutrinos is $E_\\nu^{\\rm qt}\\sim3$~GeV~\\cite{mur+13}.  The\nneutrino-nucleon cross section for the charged-current interaction at\n1~GeV is $\\sim0.6\\times {10}^{-38}~{\\rm cm}^2$ (averaged over $\\nu$\nand $\\bar{\\nu}$), so the effective area of Hyper-K is\n$\\sim2\\times{10}^{-3}~{\\rm cm}^2$ at 1~GeV.  Then, Hyper-K will be\nable to detect $\\sim5$ signal events from such a jet-driven supernova.\nSuccessful detections enable us to probe jet physics that cannot be\ndirectly studied by electromagnetic observations.  Neutrinos enable us\nto understand how jets are accelerated and what the jet composition\nis, and will give us crucial keys to the mysterious connection between\nGRBs and energetic supernovae~\\cite{tho+04}.  Also, in principle,\nmatter effects in neutrino oscillation could be\ninvestigated~\\cite{fs08,rs10}.  Moreover, we will able to study how CR\nacceleration operates in dense radiation environments inside a GRB\nprogenitor star.  Whether CRs are accelerated or not depends on\nproperties of shocks. The conventional shock acceleration mechanism\ncan effectively operate only if the shock is radiation-unmediated\ncollisionless~\\cite{mi13}.  On the other hand, when the shock is\nmediated by radiation, the so-called neutron-proton-converter\nacceleration mechanism can work efficiently~\\cite{npc,kas+13}, which\nboosts the energy of quasi-thermal neutrinos produced by\nnucleon-neutron collisions~\\cite{mur+13}.\n\nAs discussed above, relativistic outflows containing neutrons should\nnaturally lead to GeV-TeV neutrino production, but the outflows do not\nhave to be jets.  Another interesting case may be realized when a\nsupernova explosion leaves a fast-spinning neutron star.  Neutrons are\nloaded in the proto-neutron star wind via neutrino heating.  Around\nthe base of the outflow, the particle density is so high that neutrons\nand ions are tightly coupled via elastic collisions.  Neutrons should\nbe accelerated together with ions as the Poynting-dominated pulsar\nwind is accelerated.\nOnce the outflow becomes relativistic enough to exceed the pion-production threshold, inelastic collisions naturally occur as the main dissipation process of relativistic neutrons.  \nThe neutrons then interact with the material decelerated by the shock\nand possibly with the overlying stellar material, producing 0.1-1~GeV\nneutrinos~\\cite{mur+14}.  Detecting this signal would probe the\notherwise completely obscured process of jet acceleration and the\nphysics of rotating and magnetized proto-neutron star birth during the\ncore collapse of massive stars.  Hyper-K may expect $\\sim20-30~({\\mathcal\nE}_{\\nu}^{\\rm iso}\/{10}^{48}~{\\rm erg})$ events for a core-collapse\nsupernova at 10~kpc.  In addition, Hyper-K could also allow us to see\n$\\sim10-100$~MeV neutrinos through the $\\bar{\\nu}_ep\\rightarrow e^+n$\nchannel.  However, detection of these lower energy neutrinos would be\nmore difficult because of the smaller cross sections at lower energies\nand because the signal may be buried in the exponential tail of\nthermal MeV neutrinos from the proto-neutron star.\n\nTo detect high-energy neutrino signals from hidden GRB jets or newborn\npulsar winds embedded in supernovae, it is crucial to reduce\natmospheric backgrounds using space and time coincidence, so\ninformation at other wavelengths is relevant.  The atmospheric\nneutrino background at GeV energies is $\\approx1.3\\times{10}^{-2}~{\\rm\nGeV}~{\\rm cm}^{-2}~{\\rm s}^{-1}~{\\rm sr}^{-1}$ for $\\nu_e+\\bar{\\nu}_e$\nand $\\approx2.6\\times{10}^{-2}~{\\rm GeV}~{\\rm cm}^{-2}~{\\rm\ns}^{-1}~{\\rm sr}^{-1}$ for $\\nu_\\mu+\\bar{\\nu}_\\mu$,\nrespectively~\\cite{hon+11}.  We may take the time window of $t_{\\rm\nthin}\\sim10-100$~s after the explosion time that is measurable with\nMeV neutrinos or possibly gravitational waves.  The localization is\npossible by follow-up observations at x-ray, optical, and infrared\nbands.  The atmospheric background flux in the typical angular and\ntime window is $\\sim2\\times{10}^{-3}~{\\rm erg}~{\\rm cm}^{-2}$, which\ncan be low enough for a nearby supernova.\n\nNote that it is critical to have large volume detectors for the\npurpose of detecting GeV-TeV neutrinos.  The present Super-K and\nliquid scintillator detectors such as JUNO and RENO-50 are too small\nto detect high-energy signals from astrophysical objects especially if\nextragalactic, and much bigger detectors such as Hyper-K and PINGU are\nnecessary to have a good chance to hunt high-energy neutrinos from\nGRBs and energetic supernovae.  Because of the atmospheric background,\nsensitivities above GeV energies are typically essential but searches\nfor neutrinos below $\\sim1$~GeV could also be useful for nearby\nevents.\n\n\\subsubsection{Neutrinos from gravitational-wave sources}\n\n100 years after the prediction by Einstein, gravitational waves (GWs)\nhave been detected by advanced-LIGO in 2015~\\cite{YS-ref1}. This\nhas allowed us to conduct multi-messenger observations of astrophysical\nobjects via multiple signals, i.e., electromagnetic (EM) waves (from\nradio to gamma-rays), neutrinos (from MeV to PeV) and GWs\n(kHz). Some strong GW emitters are also expected to be strong sources of\nneutrinos, e.g. supernovae and gamma-ray bursts. The searches\nfor neutrino counterparts to GW sources have been\nperformed~\\cite{YS-ref2}, including a search done by Super-Kamiokande\ncollaboration~\\cite{YS-ref3}, but there has been no clear counterpart\nfound.\nThis is consistent with the theoretical prediction because these GW\nsources (GW150914 and GW151226) are binary black-hole mergers.\nAlthough binary black-hole mergers as GW sources are not expected to\ngenerate other detectable signals, the mergers that contain at least\none neutron star (i.e. black hole-neutron star binary or neutron\nstar-neutron star binary) could emit other signals, includeing $\\sim\n10^{53}$ erg of neutrinos~\\cite{YS-ref4}. Indeed, a number of EM\ncounterparts associated with GW170817 (neutron-star binary merger)\nwere detected, including GRB170817A~\\cite{YS-ref5,YS-ref6}. A search\nfor neutrinos by Super-Kamiokande was also reported~\\cite{YS-ref7}, which did\nnot detect any coincident neutrino. Hyper-Kamiokande has the potential to\ndetect thermal neutrinos from nearby ($\\lesssim 10$Mpc) neutron\nstar meger events.\n\nDepending on the event rates, these objects would contribute to the\nrelic neutrino spectrum. The central engine of gamma-ray bursts are\nalso candidates of strong emitters of neutrinos and GWs.  It is\nevident that there are ultra-relativistic jets which are driven by the\ncentral engine. However, the mechanism that generates the jet is\nstill unclear. If this jet is driven by neutrino annihilation, which\nis one of the promising scenarios, concurrent observations of\nneutrinos and GWs will be important probe of the very central part of\nthe violent cosmic explosions at Hyper-Kamiokande era~\\cite{YS-ref8}.\n\n\\subsection{Atmospheric neutrinos}\\label{section:atmnu}\n\n  \\input{physics-atmnu\/intro.tex}\n\n  \\subsubsection{Neutrino oscillation studies (MH, \\(\\theta_{23}\\) octant, $CP$ phase)}\n      \n     \\input{physics-atmnu\/atmospheric_study.tex}\n\n  \\subsubsection{Combination with Beam Neutrinos}\n\n     \\input{physics-atmnu\/with_beam.tex}\n\n  \\subsubsection{Exotic Oscillations And Other Topics}\n\n     \\input{physics-atmnu\/exotic.tex}\n\n\n\n\n\n\n\\subsection{Dark matter searches}\\label{section:darkmatter}\n    \n       \\input{physics-darkmatter\/intro.tex}\n\n       \\input{physics-darkmatter\/wimp.tex}\n\n\n\n\n\\subsubsection{Search for WIMPs at the Galactic Center}\n\nDark matter trapped in the gravitational potential of galaxies is said\nto form a halo.  Halo models predict dark matter density distributions\nthat peak sharply near the center of a given galaxy and drop with\nradial distance.  For example, for the Milky Way galaxy the expected\ndensity distribution at the position of our solar system, $r =\n8.5$~kpc, is roughly 1000 times smaller than that at the galactic\ncenter in the NFW model~\\cite{Navarro:1995iw}.  When simulating the\nexpected signal distribution expected at Hyper-Kamiokande a diffuse\ndark matter profile following the full NFW density distribution is\nassumed and accordingly the signal is expected to arise primarily from\nthe galactic center.\n\n\\begin{figure}[thb]\n  \\begin{center}\n    \\includegraphics[scale=0.55]{physics-darkmatter\/figures\/gc_signal_demo_bbar_beta_1pc_5_20GeV_cos.pdf}\n  \\end{center}\n  \\caption{Signal and background (blue) distributions used in the Hyper-K sensitivity study of dark matter \n           annihilating via $\\chi\\chi \\rightarrow b \\bar b$ at the galactic center. \n           Analysis samples are binned in $\\mbox{cos}\\theta_{gc}$, the direction to the galactic \n           center.\n           Two WIMP hypotheses are shown: $m_{\\chi} = 5 \\mbox{GeV\/c}^{2}$ in green and $m_{\\chi} = 20 \\mbox{GeV\/c}^{2}$\n           in red. \n           All distributions have been area normalized with the WIMP normalization taken to 5\\% of the \n           background MC.\n           }\n\\label{fig:gc_signal_demo}\n\\end{figure}\n\n\nThe differential neutrino flux arising from WIMP annihilation into the\nstandard model particles listed above is simulated using the DarkSUSY\npackage~\\cite{Gondolo:2005we}, and the resulting spectrum adjusted to account for\noscillations on the way to the detector.  An independent set of\natmospheric neutrino MC is reweighted to this distribution to give a\nreconstructed signal MC at Hyper-Kamiokande.  In computing the\nsensitivity to an additional neutrino source, the analysis samples are\nrebinned in momentum and $\\mbox{cos}\\theta_{gc}$, where $\\theta_{gc}$\nis the angle between the galactic center position (RA = 266$^{\\circ}$,\nDec = -28$^{\\circ}$) and the reconstructed direction.\nFigure~\\ref{fig:gc_signal_demo} shows the $\\mbox{cos}\\theta_{gc}$\ndistributions of the atmospheric neutrino background and\ntwo WIMP hypotheses for each of the analysis samples.  During the fit\nMC data sets without a WIMP signal are fit against a PDF built from\nthe atmospheric background MC plus a WIMP signal modified by a\nnormalization parameter, $\\beta$.  Here the maximum value of $\\beta$\nthat is consistent with the background-only model within errors is\nused to compute the upper limit on the amount of additional neutrinos\nfrom the galactic center allowed after a 3.8~Mton$\\cdot$year exposure\nof Hyper-K.\n\n\n\\begin{figure}[thb]\n  \\begin{center}\n    \\includegraphics[width=0.90\\linewidth]{physics-darkmatter\/figures\/sigmaV-all-SKdata-HKsens-IceCube79-90CL-may2017.png}\n  \\end{center}\n  \\caption{Hyper-K's expected 90\\% C.L. limit on the WIMP velocity averaged \n           annihilation cross section for several annihilation modes after \n           a 3.8~Mton$\\cdot$year exposure overlaid with \n            limits from several experiments. \n            Limits are shown as a function of the dark matter mass. }\n\\label{fig:gc_wimps}\n\\end{figure}\n\nUnlike direct detection experiments this search method is insensitive\nto the WIMP-nucleon interaction cross section.  Instead limits can be\nplaced on the velocity averaged self-annihilation cross section,\n$< \\sigma \\times v > $, where $v$ is the assumed velocity distribution\nof WIMPs in the halo.  Figure~\\ref{fig:gc_wimps} shows the expected\nsensitivity of Hyper-K to WIMP annihilations at the galactic center.\nThe analysis makes use of potential signals\nin both $\\nu_{e}$- and $\\nu_{\\mu}$-enriched samples across the entire\nenergy range of atmospheric neutrinos and their energy and directional distributions\ncontribute to the sensitivity.  \nHyper-Kamiokande's sensitivity to the WIMP velocity\naveraged self-annihilation cross section is expected to exceed that of\nSuper-Kamiokande's limits by factors of three to ten, depending on \nthe assumed WIMP mass and annihilation channel.\nUnlike other experiments,\nHyper-K's ability to reconstruct down to $O(100)$~MeV neutrino\ninteractions gives it unparalleled sensitivity to WIMPs with masses\nless than $\\sim$ 100 GeV\/$c^{2}$.\n\n\\subsubsection{Search for WIMPs from the Earth}\n\n\\begin{figure}[thb]\n  \\begin{center}\n    \\includegraphics[width=0.9\\textwidth]{physics-darkmatter\/figures\/earth-wimp-spin-independent.png}\n  \\end{center}\n  \\caption{The 90\\% C.L. upper limits on the\n spin-independent WIMP-nucleon scattering cross section based on a search from\nWIMP-induced neutrinos coming from the center of the earth for several annihilation channels.\nLimits (lines) and allowed regions (hatched regions) from other experiments are also shown.\nResults from Super-Kamiokande assuming annihilations in the sun are taken from~\\cite{Choi:2015ara}.}\n\\label{fig:earth_wimps}\n\\end{figure}\n\nWIMPs bound in the halo of the galaxy may also become gravitationally\ntrapped within the Earth (or sun) after losing energy via scattering processes\nwith nuclei in its interior.  If these then pair annihilate and\nproduce neutrinos, they will escape the core of the Earth and be detectable at\nHyper-Kamiokande.  Assuming that the rate of WIMP capture within the\nsun is in equilibrium with the annihilation rate, measurements of the\nWIMP-induced neutrino flux can be directly translated into\nmeasurements of the WIMP-nucleon scattering cross section without the\nneed to measure the self-annihilation cross section.  \nSince the Earth is composed of heavy nuclei (relative to hydrogen) \nit is further possible to study WIMP interactions that are \nnot coupled to the nuclear spin (spin independent, SI).\n\n\n\nIn the analysis below the local dark matter density is assumed to be\n0.3 GeV\/$\\mbox{cm}^{3}$ with an RMS velocity of 270~km\/s.  The\nrotation of the solar system through the halo is taken to be 220~km\/s.\nSignal MC has been generated by reweighting atmospheric neutrino MC\nevents to spectra produced by the WIMPSIM package~\\cite{Blennow:2007tw}, which accounts for\nthe passage of particles through terrestrial matter.  Oscillation between\nflavors as the neutrinos travel from the Earth core to the detector are\nincluded.  An independent set of atmospheric neutrino MC is used to\nmodel the background.\n\nThe search for WIMPs bound and annihilating at the center of the Earth \nproceeds along similar lines as the search for events from the\ngalactic center, though events are now binned in momentum and\n$\\mbox{cos}\\theta_{zenith}$, the zenith angle of the reconstructed \nlepton direction relative to Hyper-K. \nIn these coordinates the atmospheric neutrino background takes its \ncharacteristic shape while the WIMP signal MC is peaked sharply in the direction \nof the Earth core; the most upward-going bin.\nLimits on the WIMP-induced neutrino flux are translated into\nlimits on the WIMP-nucleon SI cross sections using the DarkSUSY\nsimulation.  Hyper-K's sensitivity with a 1.9 Mton$\\cdot$year exposure\nshown in Figure~\\ref{fig:earth_wimps}.  \nThe plot shows the sensitivity to the WIMP-nucleon SI cross section\nfor masses $m_{\\chi} > 4 \\mbox{GeV\/c}^{2}$ compared to allowed regions\n(shown as hatched spaces) and limits (shown as lines) from current\nexperiments.  These limits have been produced assuming WIMPs have only\nSI interactions and have been estimated for\n$\\chi\\chi \\rightarrow W^{+}W^{-}, b \\bar b,$ and $\\tau^{+}\\tau^{-}.$\nHyper-K's is expected to produce limits a factor of $3\\sim 4$ times\nstringent than Super-K if no WIMP signal is seen.\nFurther, current hints for a positive SI\ndark matter signal~\\cite{Bernabei:2008yi,Savage:2009mk,Agnese:2013rvf,Angloher:2011uu,Aalseth:2010vx}, \ncan be probed completely by Hyper-K's $\\tau\\tau$ channel.\n\nIt is worth noting that in both the Earth and galactic center analyses\nno improvement in systematic errors beyond Super-K's current\nunderstanding has been assumed.  At Hyper-K the statistical\nuncertainty in the data is small enough that increases in sensitivity\nrelative to Super-K is limited by systematic errors in the atmospheric\nneutrino flux and cross section model.  Due to the relatively high energy\nof the signal events and the expected directional resolution of the\ndetector, systematic errors in the detector response, while currently\nless well known, are expected to be less significant.  While it is\nuncertain how the flux and cross section model will evolve in the\nfuture, improved modeling will translate directly into better\nsensitivity to WIMP-induced neutrinos at Hyper-K.\n\n\n\n\n\n\n\n\n\\subsection{Neutrino geophysics}\\label{section:radiography}\n\nThe chemical composition of the Earth's core is one of the most\nimportant properties of the planet's interior, because it is deeply\nconnected to not only the formation and evolution of the\nEarth~\\cite{Allegre1995} itself but also to the origin of the\ngeomagnetic field~\\cite{Fearn1981}.  While paleomagnetic evidence\nsuggests that the geomagnetic field has existed for roughly three\nbillion years, it is known that a core composed of iron alone could\nnot sustain this magnetic field for more than 20,000 years.\nExplaining the continued generation of the geomagnetic field as well\nas its other properties requires knowledge of composition of the core\nmatter.  Based on seismic wave velocity measurements and the\ncomposition of primordial meteorites the composition of the core is\npresumed to be an iron-nickel alloy that additionally includes light\nelements, such as oxygen, sulfur, or silicon~\\cite{McDonough1995}.\nHowever, since no sample of the Earth's mantle has even been acquired,\nlet alone a sample of the core, the composition of the latter,\nparticularly its light element a abundance, remains highly uncertain.\nSince the deepest wellbore to date has a depth of\n14~km~\\cite{Popov1999}, and the depth to the outer core is 2900~km it\nis unlikely that a core sample can be obtained within this century.\nAs a result, addition methods of determining the chemical composition\nof the core are essential to understanding the Earth and its magnetic\nfield.\n\n\\begin{figure}[thb]\n  \\begin{center}\n    \\includegraphics{physics-geophys\/geochemical_sensitivity.pdf}\n  \\end{center}\n  \\caption{Constraints on the proton to nucleon ratio of the Earth's outer core \n           for a 10 Mton year exposure of Hyper-K to atmospheric neutrinos. \n           Colored bands indicate the effect of present uncertainties in the \n           neutrino mixing parameters.}\n\\label{fig:geophys_za}\n\\end{figure}\n\n\nAs discussed in Section~\\ref{section:atmnu}, the oscillation\nprobability of atmospheric neutrinos depends on not only the various\nmixing angles, the neutrino mass differences, and CP-violating phase,\n$\\delta_{CP}$, but also on the electron density of the media they\ntraverse.  This last property makes atmospheric neutrinos an ideal\nprobe for measuring the electron density distribution of the Earth\npresuming the other oscillation parameters are well measured.  Since\naccelerator neutrino measurements at Hyper-K itself are expected to\ndramatically improve on the precision of these parameters\n(c.f. Section~\\ref{sec:cp}), Hyper-K may be able to make the first\nmeasurement of the core's chemical composition using its atmospheric\nneutrino sample.\n\nHyper-K's sensitivity in this regard has been studied in the context\nof atmospheric neutrino spectrum's dependence upon the ratio of the\nproton to nucleon ratio (Z\/A) of material in the outer core.\nConstraints from the combination of measurements of the Earth's\ngeodetic-astronomical parameters, such as its precession and nutation,\nwith its low frequency seismic oscillation modes (free oscillations),\nand seismic wave velocity measurements have yielded precise knowledge\nof the planet's density profile~\\cite{Dziewonski1981}.  Using this\ninformation the inner core and mantle layers of the Earth are fixed to\npure iron (Z\/A = 0.467) and pyrolite (Z\/A = 0.496) and the Z\/A value\nof the outer core is varied.  The analysis uses the same analysis\nsamples presented in Section~\\ref{section:atmnu} and focuses on\nupward-going events between 1 and 10 GeV.  Assuming the outer and\ninner core chemical compositions are the same,\nfigure~\\ref{fig:geophys_za} shows the expected constraint on the Z\/A\nparameter.  After a 10 Mton year exposure Hyper-K can exclude lead and\nwater (pyrolite) outer core hypotheses by approximately $\\sim 3\\sigma\n(1\\sigma)$.  \nWhile geophysics models will ultimately require even\ngreater precision in such measurements, Hyper-K has the potential to\nmake the spectroscopic measurements of the Earth's core.\nIt is worth noting that other proposed experiments with the ability \nto make similar geochemical measurements, such as the next generation of neutrino \ntelescopes, rely primarily on the muon disappearance channel. \nHyper-Kamiokande's, on the other hand, is unique in that its sensitivity \nis derived from the electron appearance channel. \n\n\n\n\n\n\\subsubsection{J-PARC to Hyper-Kamiokande long baseline experiment}\n\nThe neutrino energy spectrum of J-PARC neutrino beam is tuned to the\nfirst oscillation maximum with the off-axis technique, which enhances the\nflux at the peak energy while reducing the higher energy component\nthat produces background events.  The peak energy, around 600~MeV, is\nwell matched to the water Cherenkov detector technology, which has an excellent\n$e$\/$\\mu$ separation capability, high background rejection efficiency\nand high signal efficiency for sub-GeV neutrino events.\nDue to the relatively short baseline of 295~km and thus lower neutrino\nenergy at the oscillation maximum, the contribution of the matter\neffect is smaller for the J-PARC to Hyper-Kamiokande experiment\ncompared to other proposed experiments like DUNE \nin the United States~\\cite{Acciarri:2015uup}.\nThus the $CP$ asymmetry measurement with the J-PARC to Hyper-K long\nbaseline experiment has less uncertainty related to the matter effect,\nwhile other experiments with $>1000$~km baseline have much better\nsensitivity to the mass hierarchy (the sign of $\\Delta m^2_{32}$) \nwith accelerator neutrino beams.\nNevertheless, Hyper-K can determine the mass hierarchy using atmospheric \nneutrinos as described in Section~\\ref{section:atmnu}.  The sensitivities for\n$CP$ violation and mass hierarchy can be further enhanced by combining\naccelerator and atmospheric neutrino measurements.\n\nThe focus of the J-PARC to Hyper-K experiment is the measurements of $|\\Delta\nm^2_{32}|$, $\\sin^2\\theta_{23}$, $\\sin^2\\theta_{13}$ and $\\ensuremath{\\delta_{CP}} $.\nThe standard flavor mixing scenario is assumed in the following as a baseline study, although it is possible that new physics is involved in neutrino oscillation and will be revealed by Hyper-K.\nThe analysis presented in this report is based on \\cite{Abe:2015zbg} but with an updated treatment of systematic uncertainties.\n\n\\subsubsection{Oscillation probabilities and measurement channels}\nIn what follows, the oscillation probabilities and sensitivities to\noscillation parameters with $\\nue$ appearance and $\\numu$\ndisappearance measurements are discussed.  The analysis will be\nperformed by a combination of these two channels.\n\n\\paragraph{$\\numu \\to \\nue$ appearance channel}\nThe oscillation probability from $\\nu_\\mu$ to $\\nu_e$\nis expressed, to the first order of the matter effect, as follows~\\cite{Arafune:1997hd}:\n\\begin{eqnarray}\nP(\\numu \\to \\nue) & = & 4 c_{13}^2s_{13}^2s_{23}^2 \\cdot \\sin^2\\Delta_{31}  \\nonumber \\\\\n& & +8 c_{13}^2s_{12}s_{13}s_{23} (c_{12}c_{23}\\cos\\ensuremath{\\delta_{CP}}  - s_{12}s_{13}s_{23})\\cdot \\cos\\Delta_{32} \\cdot \\sin\\Delta_{31}\\cdot \\sin\\Delta_{21} \\nonumber \\\\\n& & -8 c_{13}^2c_{12}c_{23}s_{12}s_{13}s_{23}\\sin\\ensuremath{\\delta_{CP}}  \\cdot \\sin\\Delta_{32} \\cdot \\sin\\Delta_{31}\\cdot \\sin\\Delta_{21} \\nonumber \\\\\n& & +4s_{12}^2c_{13}^2(c_{12}^2c_{23}^2 + s_{12}^2s_{23}^2s_{13}^2-2c_{12}c_{23}s_{12}s_{23}s_{13}\\cos\\ensuremath{\\delta_{CP}} )\\cdot \\sin^2\\Delta_{21} \\nonumber \\\\\n& & -8c_{13}^2s_{13}^2s_{23}^2\\cdot \\frac{aL}{4E_\\nu} (1-2s_{13}^2)\\cdot \\cos\\Delta_{32}\\cdot \\sin\\Delta_{31} \\nonumber \\\\\n& & +8 c_{13}^2s_{13}^2s_{23}^2 \\frac{a}{\\Delta m^2_{31}}(1-2s_{13}^2)\\cdot\\sin^2\\Delta_{31}, \\label{Eq:cpv-oscprob}\n\\end{eqnarray}\n\\noindent where $s_{ij} = \\sin\\theta_{ij}$, $c_{ij}=\\cos\\theta_{ij}$, $\\Delta_{ij} = \\Delta m^2_{ij}\\, L\/4E_\\nu$, \nand $a =2\\sqrt{2}G_Fn_eE_\\nu= 7.56\\times 10^{-5}\\mathrm{[eV^2]} \\times \\rho \\mathrm{[g\/cm^3]} \\times E_\\nu[\\mathrm{GeV}].$\n$L$, $E_\\nu$, $G_F$ and $n_e$ are the baseline, the neutrino energy, the Fermi coupling constant and the electron density, respectively. \nThe corresponding probability for a $\\numubar \\to \\nuebar$ transition is obtained by replacing $\\ensuremath{\\delta_{CP}}  \\rightarrow -\\ensuremath{\\delta_{CP}} $\nand $a \\rightarrow -a$.\nThe third term, containing $\\sin\\ensuremath{\\delta_{CP}} $, is the $CP$ violating term\nwhich flips sign between $\\nu$ and $\\bar{\\nu}$ and thus introduces\n$CP$ asymmetry if $\\sin\\ensuremath{\\delta_{CP}} $ is non-zero.  The last two terms are\ndue to the matter effect.  Those terms which contain $a$ change their\nsign depending on the mass hierarchy.  As seen from the definition of\n$a$, the amount of asymmetry due to the matter effect is proportional\nto the neutrino energy at a fixed value of $L\/E_\\nu$.\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=0.48\\textwidth]{cpv-oscprob-nu.pdf}\n\\includegraphics[width=0.48\\textwidth]{cpv-oscprob-anti.pdf}\n\\caption{Oscillation probabilities as a function of the neutrino energy for $\\numu \\to \\nue$ (left) and $\\numubar \\to \\nuebar$ (right) transitions with L=295~km and $\\sin^22\\theta_{13}=0.1$. \nBlack, red, green, and blue lines correspond to $\\ensuremath{\\delta_{CP}}  = 0^\\circ$,\n90$^\\circ$, 180$^\\circ$ and $-90^\\circ$, respectively.  Solid (dashed)\nline represents the case for a normal (inverted) mass hierarchy.\n\\label{fig:cp-oscpob}}\n\\end{figure}\n\nFigure~\\ref{fig:cp-oscpob} shows the $\\numu \\to \\nue$ and\n$\\numubar \\to \\nuebar$ oscillation probabilities as a function of the\ntrue neutrino energy for a baseline of 295~km.  The Earth matter\ndensity is assumed to be 2.6\\,$g$\/cm$^3$.  The cases for $0^\\circ$,\n90$^\\circ$, 180$^\\circ$ and $-90^\\circ$, are shown together.  One can\nsee the effect of different $\\ensuremath{\\delta_{CP}} $ values on the oscillation\nprobabilities.  For example, if $\\ensuremath{\\delta_{CP}}  = -90^\\circ$, the appearance\nprobability will be enhanced for neutrino but suppressed for\nanti-neutrino.  By comparing the oscillation probabilities of\nneutrinos and anti-neutrinos, one can measure the $CP$ asymmetry. \nThe information on the $CP$ phase can be derived from not only the total\nnumber of events but also the energy spectrum of the oscillated events. \nFor example, for both $\\ensuremath{\\delta_{CP}}  = 0^\\circ$ and $180^\\circ$, $CP$ is conserved\n($\\sin\\ensuremath{\\delta_{CP}} =0$) and the oscillation probabilities in vacuum are the\nsame for neutrino and anti-neutrino, however those two cases can be\ndistinguished using spectrum information as seen in\nFig.~\\ref{fig:cp-oscpob}.\n\nAlso shown in Fig.~\\ref{fig:cp-oscpob} are the case of normal mass\nhierarchy ($\\Delta m^2_{32}>0$) with solid lines and inverted mass\nhierarchy ($\\Delta m^2_{32}<0$) with dashed lines.\nThere are sets of different mass hierarchy and values of $\\ensuremath{\\delta_{CP}} $\nwhich give similar oscillation probabilities, resulting in a potential\ndegeneracy if the mass hierarchy is unknown.  By combining information\nfrom experiments currently ongoing~\\cite{Abe:2011ks,Ayres:2004js,\nGuo:2007ug,Ahn:2010vy,Ardellier:2006mn} and\/or planned in the near\nfuture~\\cite{Aartsen:2014oha,Katz:2014tta,Ahmed:2015jtv,An:2015jdp,Kim:2014rfa},\nit is expected that the mass hierarchy will be determined by the time\nHyper-K starts to take data.  If not, Hyper-K itself has a sensitivity\nto the mass hierarchy by the atmospheric neutrino measurements as\ndescribed in the next section.  Thus, the mass hierarchy is assumed to\nbe known in this analysis, unless otherwise stated.\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{cpv-oscprob-breakdown-m90deg.pdf}\n\\includegraphics[width=0.45\\textwidth]{cpv-oscprob-breakdown-anti-m90deg.pdf}\n\\caption{Oscillation probabilities of $\\numu \\to \\nue$ (left) and  $\\numubar \\to \\nuebar$ (right) as a function of the neutrino energy with a baseline of 295~km.  $\\sin^22\\theta_{13}=0.1$,\n$\\ensuremath{\\delta_{CP}}  = -90^\\circ$, and normal hierarchy are assumed.\nContribution from each term of the oscillation probability formula is shown separately.\n\\label{fig:cp-oscpob-bd}}\n\\end{figure}\n\nFigure~\\ref{fig:cp-oscpob-bd} shows the contribution from each term of the $\\numu \\to \\nue$ and $\\numubar \\to \\nuebar$ oscillation probability formula, Eq.(\\ref{Eq:cpv-oscprob}),\nfor $L=295$\\,km, $\\sin^22\\theta_{13}=0.1$, $\\sin^22\\theta_{23}=1.0$, $\\ensuremath{\\delta_{CP}}  = -90^\\circ$, and normal mass hierarchy.\nFor $E_\\nu\\simeq 0.6$\\,GeV which gives $\\sin\\Delta_{32} \\simeq \\sin\\Delta_{31} \\simeq 1$,\n\\begin{eqnarray}\n\\frac{P(\\nu_\\mu \\to  \\nu_e) - P(\\bar\\nu_\\mu \\to  \\bar\\nu_e)}{P(\\nu_\\mu \\to  \\nu_e) + P(\\bar\\nu_\\mu \\to  \\bar\\nu_e)} &\\simeq& \n\\frac{-16J_{CP}\\sin \\Delta_{21} + 16 c_{13}^2s_{13}^2s_{23}^2 \\frac{a}{\\Delta m^2_{31}}(1-2s_{13}^2) }{8c_{13}^2s_{13}^2s_{23}^2}\\\\\n&\\simeq& -0.28 \\sin\\delta + 0.09,\n\\end{eqnarray}\nwhere $J_{CP} = c_{12}c_{13}^{2}c_{23}s_{12}s_{13}s_{23}\\sin\\delta$ is called Jarlskog invariant.\nThe effect of $CP$ violating term can be as large as 28\\%, while that of the matter effect is 9\\%.\nThe first term will be $-0.31 \\sin\\delta$ with $\\sin^{2}2\\theta_{13}=0.082$~\\cite{Olive:2016xmw}.\n\nThe uncertainty of the Earth's density between Tokai and Kamioka is estimated to be at most 6\\%~\\cite{Hagiwara:2011kw}. Because the matter effect contribution to the total $\\numu \\to \\nue$ appearance probability is less than 10\\% for 295km baseline, the uncertainty from the matter density is estimated to be less than 0.6\\% and neglected in the following analysis.\n\n\\paragraph{$\\numu$ disappearance channel}\nThe currently measured value of $\\theta_{23}$ is consistent with maximal mixing, $\\theta_{23} \\approx \\pi\/4$~\\cite{Abe:2014ugx, Adamson:2014vgd, Himmel:2013jva},\nwhile NOvA collaboration recently reported a possible hint of non-maximal mixing~\\cite{Adamson:2017qqn}.\nIt is of great interest to determine if $\\sin^22\\theta_{23}$ is maximal or not, and if not, whether $\\theta_{23}$ is less or greater than $\\pi\/4$, as \nit could constrain models of neutrino mass generation and quark-lepton unification~\\cite{King:2013eh,Albright:2010ap,Altarelli:2010gt,Ishimori:2010au,Albright:2006cw,Mohapatra:2006gs}.\nWhen we measure $\\theta_{23}$ with the survival probability $P(\\numu \\to \\numu)$ which is proportional to $\\sin^22\\theta_{23}$ to first order, \n\\begin{eqnarray}\nP(\\nu_\\mu \\rightarrow \\nu_\\mu) &\\simeq& 1-4c^2_{13}s^2_{23} [1-c^2_{13}s^2_{23}]\\sin^2(\\Delta m^2_{32}\\, L\/4E_\\nu) \\\\\n&\\simeq & 1-\\sin^22\\theta_{23}\\sin^2(\\Delta m^2_{32}\\, L\/4E_\\nu), \\hspace{2cm} \\textrm{(for $c_{13}\\simeq1$)}\n\\end{eqnarray}\nthere is an octant ambiguity, as for each value of $\\theta_{23} \\le\n45^\\circ $ (in the first octant), there is a value in the second\noctant ($\\theta_{23} > 45^\\circ$) that gives rise to the same\noscillation probability.  As seen from Eq.~\\ref{Eq:cpv-oscprob},\n$\\nu_e$ appearance measurement can determine\n$\\sin^2\\theta_{23}\\sin^22\\theta_{13}$.  In addition, the reactor\nexperiments provide an almost pure measurement of $\\sin^22\\theta_{13}$.\nThus, the combination of those complementary measurements will be able\nto resolve this degeneracy if $\\theta_{23}$ is sufficiently away from\n$\\frac{\\pi}{4}$~\\cite{Fogli:1996pv,Minakata:2002jv,Hiraide:2006vh}.\n\nMeasurement of $\\nuebar$ disappearance by reactor neutrino experiments\nprovides a constraint on the following combination of mass-squared\ndifferences,\n\\begin{equation}\n\\Delta m^2_{ee} = \\cos^2\\theta_{12}\\Delta m^2_{31}+\\sin^2\\theta_{12}\\Delta m^2_{32}.\n\\end{equation}\nwhile $\\numu$ disappearance measurement with Hyper-K provides a different combination~\\cite{Nunokawa:2005nx, deGouvea:2005hk}\n\\begin{equation}\n\\Delta m^2_{\\mu\\mu} = \\sin^2\\theta_{12}\\Delta m^2_{31}+\\cos^2\\theta_{12} \\Delta m^2_{32}\n+ \\cos\\ensuremath{\\delta_{CP}}  \\sin\\theta_{13} \\sin2\\theta_{12}\\tan\\theta_{23} \\Delta m^2_{21}.\n\\end{equation}\nBecause the mass squared difference measurements by Hyper-K and by\nreactor experiments give independent information, by comparing them\none can check the consistency of the mixing matrix framework, and\nobtain information on the neutrino mass hierarchy.  In order to have\nsensitivity to the mass hierarchy, uncertainties of both measurements\nmust be smaller than 1\\%.  Future medium baseline reactor experiments,\nJUNO~\\cite{An:2015jdp} and RENO-50~\\cite{Kim:2014rfa}, plan to measure\n$\\Delta m^2_{ee}$ with precision better than 1\\%.  Thus, precision\nmeasurement of $\\Delta m^2$ by Hyper-K will provide important\ninformation on the consistency of three generation mixing framework\nand mass hierarchy.\n\n\\subsubsection{Analysis method}\nAs described earlier, a binned likelihood analysis based on the\nreconstructed neutrino energy distribution is performed to extract the oscillation parameters. \nBoth \\nue\\ appearance and \\numu\\ disappearance samples, in both neutrino and\nantineutrino mode data, are simultaneously fitted.\n\nThe $\\chi^2$ used in this study is defined as \n\\begin{equation} \\label{eq:sens:chi2}\n\\chi^2 =  -2 \\ln \\mathcal{L}  + P,\n\\end{equation}\nwhere $\\ln \\mathcal{L}$ is the log likelihood for a Poisson distribution,\n\\begin{equation}\n-2\\ln \\mathcal{L} = \\sum_k \\left\\{ -{N_k^\\mathrm{test}(1+f_i)} + N_k^\\mathrm{hyp} \\ln \\left[ N_k^\\mathrm{test}(1+f_i) \\right] \\right\\}.\n\\end{equation}\nHere, $N_k^\\mathrm{hyp}$ and $N_k^\\mathrm{test}$ are the number of\nevents in $k$-th reconstructed energy bin for the hypothesis and test oscillation parameters, respectively.\nThe index $k$ runs over all reconstructed energy bins for muon and electron neutrino samples and for neutrino and anti-neutrino mode data.  \nThe parameters $f_i$ represent fractional variations of the bin entries due to systematic uncertainties.\n\nThe penalty term $P$ in Eq.~\\ref{eq:sens:chi2} constrains the systematic parameters $f_i$ with the normalized covariance matrix $C$,\n\\begin{equation}\nP = \\sum_{i,j} f_i (C^{-1})_{i,j} f_j.\n\\end{equation}\nIn order to reduce the number of the systematic parameters, several\nreconstructed energy bins that have similar covariance values are\nmerged for $f_i$.\n\nA robust estimate of the uncertainties is possible based on the T2K experience.\nFor each of three main categories of systematic uncertainties, we have made the following assumptions taking into account improvements expected with future T2K data and analysis improvements.\n\\begin{description}\n\\item[i) Flux and cross section uncertainties constrained by the fit to near detector data] \nData from near detectors will be used in conjunction with models for\nthe neutrino beam, neutrino interactions, and the detector performance\nto improve our predictions of the flux at SK and some cross-section\nparameters.  The understanding of the neutrino beam, interaction, and\ndetector is expected to improve in the future, which will result in\nreduction of uncertainties in this category.\nOn the other hand, the near detector analysis is expected to include\nmore samples to reduce the uncertainty for category ii), which will\nresult in migration of some errors into this category.  This category\nof uncertainties is assumed to stay at the same level as currently\nestimated by T2K.\n\\item[ii) Cross section uncertainties not constrained by the fit to near detector data ]\nThis category of error stems from the cross-section parameters which are independent between \nthe near and far detectors because of their different elemental composition and the cross-section \nparameters for which the near detector is insensitive.\nIn T2K, an intensive effort has been made to include more samples into analysis, such as data \nfrom FGD2 containing a water target~\\cite{Abe:2017uxa} and large scattering angle events, to \nprovide more constraints on the cross section models.\nFurther improvement is expected in future analysis as T2K accumulates and analyze more data.\nIn addition, the intermediate detector will significantly reduce the uncertainty due to \nthe neutrino interaction models.\n\\item[iii) Uncertainties on the far detector efficiency and reconstruction modeling]\nBecause most of the uncertainties related to far detector performance\nare estimated by using atmospheric neutrinos as a control sample and\nthe current error is limited by statistics, errors in this category\nare expected to decrease with much larger statistics available with \nHyper-K than currently used for T2K.\nUncertainties arising from the energy scale are kept the same because\nthey are not estimated by the atmospheric neutrino sample, although it\ncould be also reduced with a larger statistics control sample and\nbetter calibration of the detector.\n\\end{description}\nCompared to the systematic uncertainty used for the past \npublication~\\cite{Abe:2015zbg}, the uncertainties for anti-neutrino beam \nmode have been reduced to a similar level as those for neutrino beam mode, \nbased on the experience with T2K anti-neutrino oscillation analysis.\nThe flux and cross section uncertainties are assumed to be\nuncorrelated between the neutrino and anti-neutrino data, except for\nthe uncertainty of \\nue\/\\numu\\ cross section ratio which is treated to\nbe anti-correlated considering the theoretical uncertainties studied\nin~\\cite{Day:2012gb}.  Because some of the uncertainties, such as\nthose from the cross section modeling or near detector systematics,\nare expected to be correlated and give more of a constraint, this is a\nconservative assumption.  The far detector uncertainty is treated to\nbe fully correlated between the neutrino and anti-neutrino data.\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=0.48\\textwidth]{SystNue2.pdf}\n\\includegraphics[width=0.48\\textwidth]{SystNumu2.pdf}\n\\caption{\nFractional and total systematic error size for the appearance (left) and the disappearance\n(right) samples in the neutrino mode.  Black: total uncertainty, red:\nthe flux and cross-section constrained by the near detector, magenta:\nthe near detector non-constrained cross section, blue: the far\ndetector error.\n\\label{Fig:systerror}\n}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=0.48\\textwidth]{SystAntiNue2.pdf}\n\\includegraphics[width=0.48\\textwidth]{SystAntiNumu2.pdf}\n\\caption{\nFractional and total systematic error size for the appearance (left) and the disappearance\n(right) samples in the anti-neutrino mode.  Black: total uncertainty,\nred: the flux and cross-section constrained by the near detector,\nmagenta: the near detector non-constrained cross section, blue: the\nfar detector error.\n\\label{Fig:systerror-anti}\n}\n\\end{figure}\n\nFigures~\\ref{Fig:systerror} and \\ref{Fig:systerror-anti} show the\nfractional and total systematic uncertainties for the appearance and\ndisappearance reconstructed energy spectra in neutrino and\nanti-neutrino mode, respectively.  Black lines represent the prior\nuncertainties and bin widths of the systematic parameters $f_i$, while\ncolored lines show the contribution from each uncertainty source.  It\nshould be noted that because some uncertainties are correlated between\nbins, the uncertainty on the total number of events is not a simple\nflux-weighted sum of these errors.  For example, the energy scale\nuncertainty of the far detector has a large contribution around the\nflux peak, but it does not change the total number of events.\nFigure~\\ref{Fig:correlationmatrix} shows the correlation matrix of the\nsystematic uncertainties between the reconstructed neutrino energy\nbins of the four samples.  The systematic uncertainties of the number\nof expected events at the far detector are summarized in\nTable~\\ref{tab:sens:systsummary}.\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=0.6\\textwidth]{CorrelationMatrix2.pdf}\n\\caption{\nCorrelation matrix between reconstructed energy bins of the four samples due to the systematic uncertainties.\nBins 1--8, 9--20, 21--28, and 29--40 correspond to \nthe neutrino mode single ring $e$-like,\nthe neutrino mode single ring $\\mu$-like,\nthe anti-neutrino mode single ring $e$-like, and\nthe anti-neutrino mode single ring $\\mu$-like samples, respectively.\n\\label{Fig:correlationmatrix}\n}\n\\end{figure}\n\n\n\\begin{table}[htbp]\n\\caption{Uncertainties for the expected number of events at Hyper-K from the systematic uncertainties assumed in this study.}\n\\centering\n\\begin{tabular}{cccccc}  \\hline \\hline\n&  & ~~Flux \\& ND-constrained~~ & ~~ND-independent~~  & \\multirow{2}{*}{~~Far detector~~}  & \\multirow{2}{*}{Total} \\\\\n & &   cross section &  cross section \\\\ \\hline\n\\multirow{2}{*}{$\\nu$ mode~~} \t\t\t& Appearance \t& 3.0\\% & 0.5\\% & 0.7\\% & 3.2\\% \\\\\n\t\t\t\t\t\t\t\t\t& Disappearance \t& 3.3\\% & 0.9\\% & 1.0\\% & 3.6\\% \\\\ \\hline\n\\multirow{2}{*}{$\\overline{\\nu}$ mode~~}\t& Appearance \t& 3.2\\% & 1.5\\% & 1.5\\% & 3.9\\% \\\\\n\t\t\t\t\t\t\t\t\t& Disappearance \t& 3.3\\% & 0.9\\% & 1.1\\% & 3.6\\% \\\\\n\\hline \\hline\n\\end{tabular}\n\n\\label{tab:sens:systsummary}\n\\end{table}%\n\n\\subsubsection{Searches for new physics}\nIn addition to the study of standard neutrino oscillation, the\ncombination of intense beam and high performance detectors enables us\nto search for new physics in various ways.  Examples of possible\nsearches for new physics with Hyper-K and accelerator beam are listed\nbelow.\n\n\\paragraph{Search for sterile neutrinos}\nSterile neutrinos can be searched for in both disappearance and\nappearance channels in near and intermediate detectors.  With neutrino\nenergy of 0.1--few~GeV and baseline of 0.3~km and 1--2~km, it will be\nsensitive to $\\Delta m^2$ of $\\mathcal{O}(1)$~eV$^2$, which is\nthe interesting region in light of several anomalies reported by recent\nexperiments.  T2K has searched for sterile neutrino in $\\nue$\nappearance and $\\numu$ disappearance with near\ndetectors~\\cite{Abe:2014nuo, Dewhurst:2015aba}.  With more statistics,\nimproved detectors, and possible two detector configuration with near\nand intermediate detectors, Hyper-K near detectors will have chance to\nimprove the sensitivity for sterile neutrino searches.\n\nNeutral current measurements at the far detector will be also sensitive to the sterile neutrino,\nbecause the neutral current channel measures the total active flavor content.\nBy selecting two electron-like ring events with no decay electron and invariant mass consistent with $\\pi^0$, neutral current events with 96\\% purity can be obtained.\nWith $1.56\\times10^{22}$ protons on target and 560~kton fiducial mass, more than 4,000 \nNC $\\pi^0$ events are expected after selection.\nNormalization of the NC $\\pi^{0}$ production can be strongly\nconstrained by the high statistics data at the intermediate detector\nas shown in Table~\\ref{tab:lbl-ndcross}.\n\n\\paragraph{Test of Lorentz and CPT invariance}\nLorentz Violation arises when the behavior of a particle depends on\nits direction or boost velocity and is predicted to occur at the\nPlanck scale (10$^{19}$~GeV).  Searches for Lorentz Violation have\nbeen performed by various experiments, including T2K, by looking for a\nsidereal time dependence of the neutrino event rate.  Similar searches\ncan be carried out with larger statistics and improved detectors.\n\n\\paragraph{Heavy neutrino search}\nThe existence of heavy neutral leptons (heavy neutrinos) is predicted\nin many extensions of the Standard Model.  Such heavy neutrinos may be\nproduced in decays of kaons and pions from the target.  Then, decays\nof heavy neutrinos can be detected in the near detector.  The\nfeasibility of search for heavy neutrinos in accelerator neutrino\nexperiment, in particular with T2K, is studied in \\cite{Asaka:2012bb}\nand the sensitivity is expected to be better than previous searches.\nBecause interactions of ordinary neutrinos produce background to this\nsearch, having a low density detector such as a gas TPC inside a\nmagnetic field like ND280 is an advantage for this search.  The\nsensitivity will be further enhanced if a larger volume of gas\ndetector is employed.\n\n\n\\subsubsection{Analysis overview}\nThe analysis used in this report is based on a framework developed for\nthe sensitivity study by T2K presented in~\\cite{Abe:2014tzr}.  A\nbinned likelihood analysis based on the reconstructed neutrino energy\ndistribution is performed using both \\nue\\ (\\nuebar) appearance\nand \\numu\\ (\\numubar) disappearance samples simultaneously.  A full\noscillation probability formula, not the approximation shown in\nEq.~\\ref{Eq:cpv-oscprob}, is used in the analysis.\nTable~\\ref{Tab:oscparam} shows the nominal oscillation parameters used\nin the study presented in this report, and the treatment during the\nfitting.  Parameters to be determined with the fit are\n$\\sin^2\\theta_{13}$, $\\sin^2\\theta_{23}$, $\\Delta m^2_{32}$ and\n$\\ensuremath{\\delta_{CP}} $.\n\n\n\nAn integrated beam power of 13~MW$\\times$10$^7$~sec is assumed in\nthis study, corresponding to $2.7\\times10^{22}$ protons on target\nwith 30\\,GeV J-PARC beam.\nIt corresponds to about ten Snowmass years with 1.3~MW.\nWe have studied the sensitivity to $CP$\nviolation with various assumptions of neutrino mode and anti-neutrino\nmode beam running time ratio for both normal and inverted mass\nhierarchy cases.  The dependence of the sensitivity on the\n$\\nu$:$\\overline{\\nu}$ ratio is found not to be significant between\n$\\nu$:$\\overline{\\nu}$=1:1 to 1:5.  In this report,\n$\\nu$:$\\overline{\\nu}$ ratio is set to be 1:3 so that the expected\nnumber of events are approximately the same for neutrino and\nanti-neutrino modes.\n\n\n\\begin{table}[htbp]\n\\caption{Oscillation parameters used for the sensitivity analysis and treatment in the fitting. The \\textit{nominal} values are used for figures and numbers in this section, unless otherwise stated.}\n\\centering\n\\begin{tabular}{cccccccc} \\hline \\hline\nParameter & $\\sin^22\\theta_{13}$ & $\\ensuremath{\\delta_{CP}} $ & $\\sin^2\\theta_{23}$ &\n$\\Delta m^2_{32}$ & mass hierarchy & $\\sin^22\\theta_{12}$ & $\\Delta\nm^2_{21}$ \\\\ \\hline Nominal & 0.10 & 0 & 0.50 &\n$2.4\\times10^{-3}~\\mathrm{eV}^2$ & Normal & $0.8704$ &\n$7.6\\times10^{-5}~\\mathrm{eV}^2$ \\\\ Treatment & Fitted & Fitted &\nFitted & Fitted & Fixed & Fixed & Fixed \\\\ \\hline \\hline\n\\end{tabular}\n\\label{Tab:oscparam}\n\\end{table}%\n\nInteractions of neutrinos in the Hyper-K detector are simulated with\nthe NEUT program\nlibrary~\\cite{hayato:neut,Mitsuka:2007zz,Mitsuka:2008zz}, which is\nused in both Super-K and T2K.  The response of the detector is\nsimulated using the Super-K full Monte Carlo simulation based on the\nGEANT3 package~\\cite{Brun:1994zzo}, although some improvements are expected\nwith new photo-sensors with higher photon detection efficiency and timing\nresolution (see Section~\\ref{section:photosensors}). \nThe simulation is based on the SK-IV configuration, which has an upgraded electronics and DAQ system compared to SK-III.\nEvents are reconstructed with the Super-K reconstruction software, which\ngives a realistic estimate of the Hyper-K performance.\n\n\nBased on the experience with the SK-II period when the number of PMT\nwas about half compared to other periods (corresponding to 20\\%\nphotocoverage with the Super-K PMT R3600), the reconstruction\nperformance for beam neutrino events with around 1~GeV energy is known\nnot to degrade significantly with reduced photocathode coverage down to 20\\% (with\nR3600).  \nThus, the performance for the beam neutrino interactions is comparable within the range of \nthe photocathode coverage considered for Hyper-K.\nThere will be additional capabilities such as neutron tagging with higher coverage, \nbut they are not yet taken into account in the current study.\n\nIn what follows, results are presented assuming ten years of running with a single tank detector with 187\\,kton fiducial volume unless otherwise noticed.\nAlso shown for comparison are results from the staging approach with ten years of running, with a single tank for the first six years, and two tanks starting in the seventh year.\n\n\\subsubsection{Expected observables at the far detector}\nThe criteria to select \\nue\\ and \\numu\\ candidate events are based on\nthose developed for and established with the Super-K and T2K\nexperiments.  Fully contained (FC) events with a reconstructed vertex\ninside the fiducial volume (FV), which is defined as the region more than 1.5\\,m away from inner detector wall, and visible energy ($E_\\mathrm{vis}$)\ngreater than 30\\,MeV are selected as FCFV neutrino event candidates.\nIn order to enhance charged current quasielastic (CCQE, $\\nu_l +\nn \\rightarrow l^- + p$ or $\\overline{\\nu}_l + p \\rightarrow l^+ + n$)\ninteraction, a single Cherenkov ring is required.\n\nAssuming a CCQE interaction, the neutrino energy ($E_\\nu ^{\\rm rec}$)\nis reconstructed from the energy of the final state charged lepton\n($E_\\ell$) and the angle between the neutrino beam and the charged\nlepton directions ($\\theta_\\ell$) as\n\\begin{eqnarray}\nE_\\nu ^{\\rm rec}=\\frac {2(m_n-V) E_\\ell +m_p^2 - (m_n-V)^2 - m_\\ell^2} {2(m_n-V-E_\\ell+p_\\ell\\cos\\theta_\\ell)},\n\\label{eq:Enurec}\n\\end{eqnarray}\nwhere $m_n, m_p, m_\\ell$ are the mass of neutron, proton, and charged\nlepton, respectively, $p_\\ell$ is the charged lepton momentum, and $V$\nis the mean nuclear potential energy (27\\,MeV).\nIt was shown in T2K analysis that the sensitivity can be slightly improved\nby using two-dimentional information of $(p_\\ell, \\theta)$ in oscillation fit.\n\n\nThen, to select \\nue\/\\nuebar\\ candidate events the following criteria are applied;\nthe reconstructed ring is identified as electron-like ($e$-like),\n$E_\\mathrm{vis}$ is greater than 100 MeV, there is no decay electron\nassociated to the event, and $E_\\nu^\\mathrm{rec}$ is less than\n1.25~GeV.  Finally, in order to reduce the background from\nmis-reconstructed $\\pi^0$ events, additional criteria using the\nreconstructed $\\pi^0$ mass and the ratio of the best-fit likelihoods\nof the $\\pi^0$ and electron fits~\\cite{Abe:2013hdq} are applied.\n\n\\begin{figure}[tbp]%\n\\includegraphics[width=0.48\\textwidth]{Numode_Appearance_1tank.pdf}\n\\includegraphics[width=0.48\\textwidth]{AntiNumode_Appearance_1tank.pdf}\\\\\n\\caption{\nReconstructed neutrino energy distribution of the $\\nue$ candidate events.\nLeft: neutrino beam mode, right: anti-neutrino beam mode. Normal mass\nhierarchy with $\\sin^22\\theta_{13}=0.1$ and $\\ensuremath{\\delta_{CP}} =0^\\circ$ is\nassumed.\nCompositions of appearance signal, $\\numu \\to \\nue$ and $\\numubar \\to \\nuebar$,\nand background events originating from ($\\numu + \\numubar$) and ($\\nue + \\nuebar$) are shown separately.\n\\label{Fig:sens-enurec-nue}\n}\n\\end{figure}\n\n\\begin{table}[tbp]%\n\\caption{\\label{Tab:sens-selection-nue}%\nThe expected number of $\\nue\/\\nuebar$ candidate events and\nefficiencies with respect to FCFV events.\nNormal mass hierarchy with\n$\\sin^22\\theta_{13}=0.1$ and $\\ensuremath{\\delta_{CP}} =0$ are assumed.  Background is\ncategorized by the flavor before oscillation.}\n\\begin{center}%\n\\begin{tabular}{cc|cc|ccccc|c|c} \\hline \\hline\n&\t& \\multicolumn{2}{c|}{signal} & \\multicolumn{6}{c|}{BG} & \\multirow{2}{*}{Total} \\\\ \n&\t&~$\\numu \\to \\nue$~\t& ~$\\numubar \\to \\nuebar$~ \t&~$\\numu$ CC~\t&~$\\numubar$ CC~\t&~$\\nue$  CC~& ~$\\nuebar$ CC~ & ~NC~ & ~BG Total~\t&  \\\\ \\hline  \n\\multirow{2}{*}{$\\nu$ mode~~} & Events\t& 1643\t&\t15& 7 \t& 0\t & 248\t&11\t& 134\t&\t400 & 2058 \\\\ \n & Eff.(\\%)  & 63.6 & 47.3 & 0.1 & 0.0 & 24.5 & 12.6 & 1.4 & 1.6 & --- \\\\\n \\hline\n\\multirow{2}{*}{$\\bar{\\nu}$ mode~~} & Events\t& 206\t&\t1183& 2\t& 2\t& 101\t& 216\t& 196&\t517 & 1906 \\\\ \n & Eff. (\\%) & 45.0 & 70.8 & 0.03 & 0.02 & 13.5 & 30.8 & 1.6 & 1.6 & --- \\\\\n\\hline \\hline\n\\end{tabular}%\n\\end{center}\n\\end{table}%\n\nFigure~\\ref{Fig:sens-enurec-nue} shows the reconstructed neutrino\nenergy distributions of $\\nue\/\\nuebar$ events after all the\nselections. \nThe expected number of\n$\\nue\/\\nuebar$ candidate events is shown in\nTable~\\ref{Tab:sens-selection-nue} for each signal and background\ncomponent.  The efficiencies of selection with respect to FCFV events\nare also shown in Table~\\ref{Tab:sens-selection-nue}.  In the neutrino\nmode, the dominant background component is intrinsic $\\nue$\ncontamination in the beam.  The mis-identified neutral current $\\pi^0$\nproduction events are suppressed thanks to the improved $\\pi^0$\nreconstruction.  The total rejection factor, including FCFV selection,\nfor NC $\\pi^0$ interactions is $>99.5$\\%.  In the anti-neutrino mode,\nin addition to $\\nuebar$ and $\\numubar$, $\\nue$ and $\\numu$ components\nhave non-negligible contributions due to larger fluxes and\ncross-sections compared to their counterparts in the neutrino mode.\n\n\\begin{figure}[tbp]%\n\\includegraphics[width=0.48\\textwidth]{Numode_Disappearance_1tank.pdf}\n\\includegraphics[width=0.48\\textwidth]{AntiNumode_Disappearance_1tank.pdf}\n\\caption{%\nReconstructed neutrino energy distribution of the $\\numu\/\\numubar$ candidate events after oscillation.\nLeft: neutrino beam mode, right: anti-neutrino beam mode.\n\\label{Fig:sens-enurec-numu}\n}\n\\end{figure}\n\n\\begin{table}[tbp]%\n\\begin{center}%\n\\caption{\\label{Tab:sens-selection-numu}%\nThe expected number of $\\numu\/\\numubar$ candidate events and efficiencies (with respect to FCFV events) for each flavor and interaction type.\n}\n\\begin{tabular}{lcccccccccc} \\hline \\hline\n\t\t\t&\t&~$\\numu$CCQE\t& ~$\\numu$CC non-QE & ~$\\numubar$CCQE\t& ~$\\numubar$CC non-QE &~$\\nue+\\nuebar$ CC \t&~NC~ \t& ~$\\numu \\to \\nue$\t\t& ~total~ \t\t\\\\ \\hline \n\\multirow{2}{*}{$\\nu$ mode}\t& Events\t& 6043 & 2981\t &\t348 & \t\t194\t& 6\t\t\t\t& 480 \t\t& 29\t\t\t& 10080\t\t \\\\ \n & Eff. (\\%) & 91.0 & 20.7 & 95.6 &  53.5 &  0.5 & 8.8 & 1.1& --- \\\\ \\hline\n\\multirow{2}{*}{$\\bar{\\nu}$ mode} & Events\t& 2699 & \t2354\t&\t6099 &\t 1961 & 7\t\t& 603\t\t& 4  \t\t\t& 13726\t\t \\\\ \n & Eff. (\\%) & 88.0 & 20.1 & 95.4 & 54.8 & 0.4 & 8.8 & 0.7 & ---\\\\\n\\hline \\hline\n\\end{tabular}%\n\\end{center}\n\\end{table}%\n\nFor the \\numu\/\\numubar\\ candidate events the following criteria are applied;\nthe reconstructed ring is identified as muon-like ($\\mu$-like),\nthe reconstructed muon momentum is greater than 200 MeV\/$c$, and\nthere is at most one decay electron associated to the event.\n\nFigure~\\ref{Fig:sens-enurec-numu} shows the reconstructed neutrino\nenergy distributions of the selected $\\numu$\/$\\numubar$ events.\nTable~\\ref{Tab:sens-selection-numu} shows the number of\n$\\numu\/\\numubar$ candidate events for each signal and background\ncomponent.  In the neutrino beam mode, the purity of $\\numu$ CC\nevents, after oscillation and for $E_{rec}<1.5$~GeV, is 89\\%.  For the\nanti-neutrino mode data, the contribution of wrong-sign $\\numu$ CC\nevents is significant because the cross section for neutrino interactions is about\nthree times larger than anti-neutrino interactions in this energy range.  The\nfractions of $\\numubar$ and $\\numu$ CC events in anti-neutrino beam\nmode data after selection, for $E_{rec}<1.5$~GeV, are 66\\% and 26\\%,\nrespectively.\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=0.48\\textwidth]{DeltaCP_Energy_numode_Appearence_1tank.pdf}\n\\includegraphics[width=0.48\\textwidth]{DeltaCP_Energy_antinumode_Appearence_1tank.pdf} \\\\\n\\caption{\nTop: Reconstructed neutrino energy distribution for several values of\n$\\ensuremath{\\delta_{CP}} $.  $\\sin^22\\theta_{13}=0.1$ and normal hierarchy is assumed.\nBottom: Difference of the reconstructed neutrino energy distribution\nfrom the case with $\\ensuremath{\\delta_{CP}} =0^\\circ$.  The error bars represent the\nstatistical uncertainties of each bin.\n}\n\\label{enurecdiff-nue}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=0.48\\textwidth]{Numode_Disappearance_nonOsc_comp_total_1tank.pdf}\n\\includegraphics[width=0.48\\textwidth]{AntiNumode_Disappearance_nonOsc_comp_total_1tank.pdf}\n\\caption{\nReconstructed neutrino energy distributions of $\\numu$ candidates.\nDotted black lines are for no oscillation case,\nwhile solid red lines represent prediction with oscillation.\nLeft: for neutrino beam mode. Right: for anti-neutrino beam mode.\n}\n\\label{enurecdiff-numu}\n\\end{figure}\n\nThe reconstructed neutrino energy distributions of $\\nue$ events for\nseveral values of $\\ensuremath{\\delta_{CP}} $ are shown in the top plots of\nFig.~\\ref{enurecdiff-nue}.  The effect of $\\ensuremath{\\delta_{CP}} $ is clearly seen\nusing the reconstructed neutrino energy.  The bottom plots show the\ndifference of reconstructed energy spectrum from $\\ensuremath{\\delta_{CP}} =0^\\circ$\nfor the cases $\\ensuremath{\\delta_{CP}}  = 90^\\circ, -90^\\circ$ and $180^\\circ$.  The\nerror bars correspond to the statistical uncertainty.\nBy using not only the total number of events but also the reconstructed energy\ndistribution, the sensitivity to $\\ensuremath{\\delta_{CP}} $ can be improved and one\ncan discriminate all the values of $\\ensuremath{\\delta_{CP}} $, including the\ndifference between $\\ensuremath{\\delta_{CP}}  = 0^\\circ$ and $180^\\circ$ for which CP\nsymmetry is conserved.\n\nFigure~\\ref{enurecdiff-numu} shows the reconstructed neutrino energy\ndistributions of the $\\numu$ sample, for the cases with\n$\\sin^2\\theta_{23}=0.5$ and without oscillation.  Thanks to the narrow\nenergy spectrum tuned to the oscillation maximum with off-axis beam,\nthe effect of oscillation is clearly visible.\n\n\\subsubsection{Measurement of $CP$ asymmetry}\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=0.48\\textwidth]{2Dmap13_reactor_constraint_NH_1tank.pdf}\n\\includegraphics[width=0.48\\textwidth]{2Dmap13_reactor_constraint_IH_1tank.pdf}\n\\caption{The expected 90\\% CL allowed regions in the $\\sin^22\\theta_{13}$-$\\ensuremath{\\delta_{CP}} $ plane.\nThe results for the true values of $\\ensuremath{\\delta_{CP}}  = (-90^\\circ, 0, 90^\\circ, 180^\\circ)$ are shown.\nLeft: normal hierarchy case. Right: inverted hierarchy case.\nRed (blue) lines show the result with Hyper-K only (with $\\sin^22\\theta_{13}$ constraint from reactor experiments). \n\\label{fig:CP-contour}}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{CPresolvedSigmaVStruedCP_NH-1tank.pdf}\n\\caption{Expected significance to exclude $\\sin\\ensuremath{\\delta_{CP}}  = 0$ in case of normal hierarchy.  Mass hierarchy is assumed to be known.\n\\label{fig:CP-chi2}}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=0.6\\textwidth]{Delta3Sigma_POT-1tank.pdf}\n\\caption{Fraction of $\\ensuremath{\\delta_{CP}} $ for which $\\sin\\ensuremath{\\delta_{CP}} = 0$ can be excluded with  more than 3\\,$\\sigma$ (red) and 5\\,$\\sigma$ (blue) significance as a function of the running time. \nFor the normal hierarchy case, and mass hierarchy is assumed to be known.\nThe ratio of neutrino and anti-neutrino mode is fixed to 1:3. \n\\label{fig:delta-sens-time}}\n\\end{figure}\n\nFigure~\\ref{fig:CP-contour} shows examples of the 90\\% CL allowed\nregions on the $\\sin^22\\theta_{13}$--$\\ensuremath{\\delta_{CP}} $ plane resulting from\nthe true values of $\\ensuremath{\\delta_{CP}}  = (-90^\\circ, 0, 90^\\circ, 180^\\circ)$.\nThe left (right) plot shows the case for the normal (inverted) mass\nhierarchy.  Also shown are the allowed regions when we include a\nconstraint from the reactor experiments,\n$\\sin^22\\theta_{13}=0.100 \\pm0.005$.  With reactor constraints,\nalthough the contour becomes narrower in the direction of\n$\\sin^22\\theta_{13}$, the sensitivity to $\\ensuremath{\\delta_{CP}} $ does not\nsignificantly change because $\\delta_{CP}$ is constrained by the\ncomparison of neutrino and anti-neutrino oscillation probabilities by\nHyper-K and not limited by the uncertainty of $\\theta_{13}$.\n\nFigure~\\ref{fig:CP-chi2} shows the expected significance to exclude\n$\\sin\\ensuremath{\\delta_{CP}}  = 0$ (the $CP$ conserved case).  The significance is\ncalculated as $\\sqrt{\\Delta \\chi^2}$, where $\\Delta \\chi^2$ is the\ndifference of $\\chi^2$ for the \\textit{trial} value of \\ensuremath{\\delta_{CP}} \\ and\nfor $\\ensuremath{\\delta_{CP}}  = 0^\\circ$ or 180$^\\circ$ (the smaller value of\ndifference is taken).  We have also studied the case with a reactor\nconstraint, but the result changes only slightly.\nFigure~\\ref{fig:delta-sens-time} shows the fraction of $\\ensuremath{\\delta_{CP}} $ for\nwhich $\\sin\\ensuremath{\\delta_{CP}} = 0$ is excluded with more than 3\\,$\\sigma$ and\n5\\,$\\sigma$ of significance as a function of the integrated beam\npower.  The ratio of integrated beam power for the neutrino and\nanti-neutrino mode is fixed to 1:3.  The normal mass hierarchy is\nassumed.  The results for the inverted hierarchy are almost the same.\n$CP$ violation in the lepton sector can be observed with more than\n3(5)\\,$\\sigma$ significance for 76(57)\\% of the possible values of $\\ensuremath{\\delta_{CP}} $.\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=0.6\\textwidth]{DeltaErr_POT-1tank.pdf}\n\\caption{Expected 68\\% CL uncertainty of $\\ensuremath{\\delta_{CP}} $ as a function of running time.  \nFor the normal hierarchy case, and mass hierarchy is assumed to be known. \n\\label{fig:delta-error-time}}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=0.6\\textwidth]{Delta3Sigma_theta23_1tank.pdf}\n\\caption{Fraction of $\\ensuremath{\\delta_{CP}} $ for which $\\sin\\ensuremath{\\delta_{CP}} = 0$ can be excluded with more than 3\\,$\\sigma$ (red) and 5\\,$\\sigma$ (blue) significance as a function of the true value of $\\sin^2\\theta_{23}$, for the normal hierarchy case.\n\\label{fig:delta-theta23}}\n\\end{figure}\n\nFigure~\\ref{fig:delta-error-time} shows the 68\\% CL uncertainty of\n$\\ensuremath{\\delta_{CP}} $ as a function of the integrated beam power.\nThe value of $\\ensuremath{\\delta_{CP}} $ can be determined with an\nuncertainty of 7.2$^\\circ$ for $\\ensuremath{\\delta_{CP}} =0^\\circ$ or $180^\\circ$, and\n23$^\\circ$ for $\\ensuremath{\\delta_{CP}} =\\pm90^\\circ$.\n\nAs the nominal value we use $\\sin^2\\theta_{23}=0.5$, but the\nsensitivity to $CP$ violation depends on the value of $\\theta_{23}$.\nFigure~\\ref{fig:delta-theta23} shows the fraction of $\\ensuremath{\\delta_{CP}} $ for\nwhich $\\sin\\ensuremath{\\delta_{CP}} = 0$ is excluded with more than 3\\,$\\sigma$ and\n5\\,$\\sigma$ of significance as a function of the true value of\n$\\sin^2\\theta_{23}$.\nT2K collaboration reported $\\sin^{2}\\theta_{23}=0.55^{+0.05}_{-0.09}$ \nin case of the normal hierarchy~\\cite{Abe:2017vif}.\n\n\\begin{table}[htbp]\n\\caption{Comparison of CP sensitivity with different configurations. As a reference, the former results published in PTEP~\\cite{Abe:2015zbg} are also shown, where 560kton fiducial volume, 7.5~MW$\\times 10^7$s integrated beam power, and an old estimate of systematic uncertainty with larger anti-neutrino errors are assumed.}\n\\begin{center}\n\\begin{tabular}{l|cc|cc}\n                     & \\multicolumn{2}{c|}{($\\sin\\ensuremath{\\delta_{CP}} =0$) exclusion} & \\multicolumn{2}{c}{68\\% uncertainty of $\\ensuremath{\\delta_{CP}} $}\\\\\nConfiguration & ~~$>3\\sigma$~~ & ~~$>5\\sigma$~~  & ~~$\\ensuremath{\\delta_{CP}} =0^\\circ$~~ & ~~$\\ensuremath{\\delta_{CP}} =90^\\circ$~~\\\\ \\hline \\hline\n1 tank & 76\\% & 57\\% & 7.2$^\\circ$ & 23$^\\circ$\\\\ \\hline\nStaging & 78\\% & 62\\% & 7.2$^\\circ$ & 21$^\\circ$ \\\\ \\hline\nPTEP~\\cite{Abe:2015zbg} & 76\\% & 58\\% & 7$^\\circ$ & 19$^\\circ$\\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\label{tab:cp-tank-comparison}\n\\end{table}%\n\n\nTable~\\ref{tab:cp-tank-comparison} shows a comparison of several configurations for CP violation sensitivities.\n\n\\subsubsection{Precise measurements of $\\Delta m^2_{32}$ and $\\sin^2\\theta_{23}$}\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=0.6\\textwidth]{2Dmap_Dis_050_s2th23_err_1tank.pdf}\n\\caption{ The 90\\% CL allowed regions in the $\\sin^2\\theta_{23}$--$\\Delta m^2_{32}$ plane.\nThe true values are $\\sin^2\\theta_{23}=0.5$ and $\\Delta m^2_{32} = 2.4 \\times 10^{-3}$~eV$^2$.\nEffect of systematic uncertainties is included. The red (blue) line corresponds to the result with Hyper-K alone (with a reactor constraint on $\\sin^22\\theta_{13}$).\n\\label{fig:theta23-0.50}}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=0.48\\textwidth]{2Dmap_Dis_045_s2th23_err_1tank.pdf}\n\\includegraphics[width=0.48\\textwidth]{2Dmap_Dis_045_s2th23_withDB_err_1tank.pdf}\n\\caption{ 90\\% CL allowed regions in the $\\sin^2\\theta_{23}$--$\\Delta m^2_{32}$ plane.\nThe true values are $\\sin^2\\theta_{23}=0.45$ and $\\Delta m^2_{32} = 2.4 \\times 10^{-3}$~eV$^2$.\nEffect of systematic uncertainties is included.\nLeft: Hyper-K only. Right: With a reactor constraint. \n\\label{fig:theta23-0.45}}\n\\end{figure}\n\n\\begin{table}[tbp]\n\\caption{Expected 1$\\sigma$ uncertainty of $\\Delta m^2_{32}$ and $\\sin^2\\theta_{23}$  for true $\\sin^2\\theta_{23}=0.45, 0.50, 0.55$. \nReactor constraint on $\\sin^22\\theta_{13}=0.1\\pm 0.005$ is imposed.}\n\\begin{center}\n\\begin{tabular}{ccccccc} \\hline \\hline\nTrue $\\sin^2\\theta_{23}$\t& \\multicolumn{2}{c}{$0.45$}  \t\t& \\multicolumn{2}{c}{$0.50$} \t\t& \\multicolumn{2}{c}{$0.55$}\\\\ \nParameter  \t\t\t\t& $\\Delta m^2_{32}$ (eV$^2$) \t& $\\sin^2\\theta_{23}$  & $\\Delta m^2_{32}$\t(eV$^2$)\t& $\\sin^2\\theta_{23}$ \t& $\\Delta m^2_{32}$ (eV$^2$)\t& $\\sin^2\\theta_{23}$\\\\ \\hline\n NH\t& $1.4\\times10^{-5}$\t\t& 0.006 \t\t\t& $1.4\\times10^{-5}$\t\t& 0.017\t\t\t\t& $1.5\\times10^{-5}$\t\t\t& 0.009\\\\\n IH & $1.5\\times10^{-5}$\t\t& 0.006 \t\t\t& $1.4\\times10^{-5}$\t\t& 0.017\t\t\t\t& $1.5\\times10^{-5}$\t\t\t& 0.009\\\\\n\\hline \\hline\n\\end{tabular}\n\\end{center}\n\\label{tab:23sensitivity}\n\\end{table}%\n\nA joint fit of the $\\numu$ and $\\nue$ samples enables us to also precisely measure $\\sin^2\\theta_{23}$ and $\\Delta m^2_{32}$.\nFigure~\\ref{fig:theta23-0.50} shows the 90\\% CL allowed regions for the true value of $\\sin^2\\theta_{23}=0.5$.\nThe expected precision of $\\Delta m^2_{32}$ and $\\sin^2\\theta_{23}$ for true $\\sin^2\\theta_{23}=0.45, 0.50, 0.55$ with reactor constraint on $\\sin^22\\theta_{13}$ is summarized in Table~\\ref{tab:23sensitivity}.\n\n\\begin{figure}[tbp]\n\\centering\n\\includegraphics[width=0.6\\textwidth]{DeltaChiOCTANTvsSin2theta23_1tank.pdf}\n\\caption{The expected significance $(\\sigma \\equiv \\sqrt{\\Delta \\chi^2})$ for wrong octant rejection, by beam neutrino measurement with reactor constraint, as a function of true $\\sin^2\\theta_{23}$ in the normal hierarchy case.\n\\label{fig:lbl-octant}}\n\\end{figure}\n\nFigure~\\ref{fig:theta23-0.45} shows the 90\\% CL allowed regions on the\n$\\sin^2\\theta_{23}$--$\\Delta m^2_{32}$ plane, for the true values of\n$\\sin^2\\theta_{23}=0.45$ and $\\Delta m^2_{32} = 2.4 \\times\n10^{-3}$~eV$^2$.  With a constraint on $\\sin^22\\theta_{13}$ from the\nreactor experiments, Hyper-K measurements can resolve the octant\ndegeneracy and precisely determine $\\sin^2\\theta_{23}$.\nFigure~\\ref{fig:lbl-octant} shows the expected significance $(\\sigma \\equiv \\sqrt{\\Delta \\chi^2})$ for wrong octant rejection with beam neutrino measurement alone as a function of true value of $\\sin^2\\theta_{23}$ in the normal hierarchy case.\n\n\n\nAs discussed earlier, a precision measurement of $\\Delta m^2_{32}$,\ncompared with reactor measurements of $\\Delta m^2_{ee}$, will enable a\nconsistency check of the mixing matrix framework.  The difference\nexpected from the current knowledge of oscillation parameters is a\nfew \\%.  The uncertainty of $\\Delta m^2_{32}$ by Hyper-K is expected\nto reach $0.6\\%$, while measurements by future reactor experiments are\nexpected to achieve $<1\\%$ precision.  Thus, the comparison will yield\na significant consistency check.\n\n\\subsubsection{Summary}\nThe sensitivity to leptonic $CP$ asymmetry of a long baseline\nexperiment using a neutrino beam directed from J-PARC to the\nHyper-Kamiokande detector has been studied based on a full simulation\nof beamline and detector.  \nWith $1\\times$10$^8$~sec of running time with 1.3~MW beam power,\nthe value of $\\ensuremath{\\delta_{CP}} $ can be determined with 7.2$^\\circ$ for $\\ensuremath{\\delta_{CP}} =0^\\circ$ \nor $180^\\circ$, and 23$^\\circ$ for $\\ensuremath{\\delta_{CP}} =\\pm90^\\circ$.  \n$CP$ violation in the lepton sector can be observed with more than\n3~$\\sigma$(5~$\\sigma$) significance for 76\\%(57\\%) of the possible\nvalues of $\\ensuremath{\\delta_{CP}} $.\nUsing both $\\nu_e$ appearance and $\\nu_\\mu$ disappearance data,\nprecise measurements of $\\sin^2\\theta_{23}$ and $\\Delta m^2_{32}$ will\nbe possible.  The expected 1$\\sigma$ uncertainty of\n$\\sin^2\\theta_{23}$ is 0.017 (0.006) for $\\sin^2\\theta_{23}=0.5 (0.45)$.\nThe uncertainty of $\\Delta m^2_{32}$ is expected to reach $<1\\%$.\n\nThere will be also a variety of measurements possible with both near\nand far detectors, such as neutrino-nucleus interaction cross section\nmeasurements and search for exotic physics, using the well-understood\nneutrino beam.\n\n\\subsubsection{Neutrino cross section measurements}\n\nWith a set of highly capable neutrino detectors envisioned for Hyper-K\nproject, a variety of neutrino interaction cross section measurements\nwill become possible. The near detector suite offers a range of\ncapabilities to probe different theoretical models for neutrino\ninteractions: in particular data across different momenta ranges and a\nrange of lepton emission angles. Figure~\\ref{fig:lbl-ndcrosseff} shows\nthe efficiency of different detectors as a function of angle and muon\nmomentum. The ability to measure exclusive hadronic final states,\nusing techniques such as high pressure gas TPCs or emulsion detectors,\nprovides valuable additional information for exclusive\ncross-sections. In table\n\\ref{tab:lbl-ndcross} we estimate the sensitivity of each proposed near detector for key selections \nbased on a flux of $10^{21}$POT.\n\n \\begin{table}[htbp]\n\\begin{center}\n\\begin{tabular}{|c|c|c|p{5.5cm}|} \\hline\nDetector\t& Selection & Nevents & Selection Characteristics \\\\ \\hline\\hline\nND280 detector, 280m\t& $\\nu_{\\mu}$CC0$\\pi$ & 20k\t& FGD1 (1--3\\,GeV), $P\\approx $72\\%  \\cite{Abe:2015awa}\t \\\\ \\hline\nND280 detector, 280m\t& $\\nu_{\\mu}$CC1$\\pi$ & 6k\t& FGD1 (1--3\\,GeV), $P\\approx $50\\%\\ \\cite{Abe:2015awa}\t \\\\ \\hline\nND280 detector, 280m\t& $\\nu_{\\mu}$CC inclusive & 40k\t& FGD1 (1--3\\,GeV), $P\\approx $90\\%\\cite{Abe:2015awa}\t \\\\ \\hline\n\\hline\nINGRID &  $\\nu_{\\mu}$CC inclusive & 17.6$\\times 10^6$ & $\\epsilon>$70\\%  (1--3\\,GeV), $P = $  97\\% \\cite{Abe:2015biq}\\\\ \\hline\n\\hline\nHPTPC, 8\\,m$^3$, 10\\,bar Ne (CF$_4$) & $\\nu_{\\mu}$CC inclusive & 4.2k (18.4k) & $\\epsilon \\approx $70\\%, protons $>$ 5\\,MeV detected \\\\ \\hline\nHPTPC, 8\\,m$^3$, 10\\,bar Ne (CF$_4$) & $\\nu_{e}$CC inclusive & 80 (450)       & $\\epsilon \\approx $70\\%, protons $>$ 5\\,MeV detected \\\\ \\hline\n\\hline\nWAGASCI\t &  $\\nu_{\\mu}$CC0$\\pi$  & 63k &P=75\\%, proton reconstruction: $\\epsilon \\approx 15$\\% at  p=500\\,MeV\/c, water in; $\\epsilon \\approx 27$\\% at  p=250\\,MeV\/c, water out (15\\% @ 150MeV\/c) \\\\ \\hline\nWAGASCI\t &  $\\nu_{\\mu}$CC1$\\pi$  & 10k &P=50\\% (protons as above)\\\\ \\hline\nWAGASCI\t &  $\\nu_{\\mu}$CC inclusive  & 75k &P=96\\% (protons as above)\\\\ \\hline\n\n\\hline\n200kg Water target \t\t& $\\nu_\\mu$ CC+NC inclusive & 10k-20k & 4$\\pi$ automated readout \\\\\nemulsion off-axis, 280m    & & & proton $>$ 10-30\\,MeV detected \\\\ \\hline\n200kg Water target \t& $\\nu_e$ CC inclusive & 1k & 4$\\pi$ automated readout \\\\\nemulsion off-axis, 280m    & & & proton $>$ 10-30\\,MeV detected \\\\ \\hline\n\n\\hline\n1kton WC \\@ 1\\,km & $\\nu_{\\mu}$CC0$\\pi$ (1-2$^{\\circ}$,2-3$^{\\circ}$,3-4$^{\\circ}$) & 1682k,1060k,519k  &  $P\\approx$92\\%,95\\%,95\\% \\\\ \\hline \n1kton WC \\@ 1\\,km & $\\bar{\\nu}_{\\mu}$CC0$\\pi$ (1-2$^{\\circ}$,2-3$^{\\circ}$,3-4$^{\\circ}$) & 519k,331k,186k  &  $P\\approx$74\\%,77\\%,76\\% \\\\ \\hline \n1kton WC \\@ 1\\,km & $\\nu_{\\mu}$CC1$\\pi$ (1-2$^{\\circ}$,2-3$^{\\circ}$,3-4$^{\\circ}$) & 208k,65k,27k  &  $P\\approx$46\\%,44\\%,31\\% \\\\ \\hline \n1kton WC \\@ 1\\,km & $\\nu_{e}$CC0$\\pi$ (1-2$^{\\circ}$,2-3$^{\\circ}$,3-4$^{\\circ}$) & 11.2k,6.9k,4.6k  &  $P\\approx$54\\%,71\\%,80\\% \\\\ \\hline \n1kton WC \\@ 1\\,km & $\\nu$NC$\\pi^{0}$ (1-2$^{\\circ}$,2-3$^{\\circ}$,3-4$^{\\circ}$) & 300k,111k,45k  &  $P\\approx$58\\%,63\\%,60\\% \\\\ \\hline \n\\end{tabular}\n\n\\caption{\nSome of the primary cross-section measurements accessible with\ndifferent elements of the Near Detector Suite (see chapter 2 for\ndetails). The predicted number of events or measurement precision have\nbeen evaluated for $10^{21}$POT.  $\\epsilon$ = efficiency = number\nselected \/ total events for given topology, $P$ = purity = number of\ngiven topology \/ total events selected. For the ND280 measurements\nonly events for a single fine grained detector (FGD1) are projected,\nthe second FGD plus the use of other detector components as targets\nincreases the statistics. Numbers are obtained either from independent\nMonte Carlo studies, or extrapolated from the cited references.}\n\n\\label{tab:lbl-ndcross}\n\\end{center}\n\\end{table}%\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=0.85\\textwidth]{NDeffPlot.pdf}\n\\caption{\nExample detector efficiency in muon momentum and direction. a) A\nhorizontally oriented 2\\,kton cylindrical intermediate WC detector at\n2km, b) the same detector vertically oriented with respect to the\nbeam, c) the current ND280 near detector and d) the WAGASCI\ndetector. Good coverage of this phase space helps to constrain\nuncertainties in cross-section models. }\n\\label{fig:lbl-ndcrosseff}\n\\end{center}\n\\end{figure}\n\n\\subsection{Accelerator based neutrinos \\label{sec:cp}}\n\n\\input{physics-lbl\/lbl-intro.tex}\n\\input{physics-lbl\/lbl-observable.tex}\n\\input{physics-lbl\/lbl-method.tex}\n\\input{physics-lbl\/lbl-resuts.tex}\n\\input{physics-lbl\/lbl-xsec.tex}\n\\input{physics-lbl\/lbl-newphysics.tex}\n\\input{physics-lbl\/lbl-summary.tex}\n\n\\subsection{Impact of Photocathode Coverage and Improved Photosensors}\\label{section:pdecay-coverage}\n\\begin{figure}[htbp]\n  \\begin{center}\n    \\includegraphics[scale=0.35]{physics-pdecay\/figures\/hit_neutron_gamma_3tanks.pdf}\n  \\end{center}\n  \\caption{Number of hit PMTs for a toy MC simulation of the 2.2\\,MeV $\\gamma$ rays \n           emitted following neutron capture on hydrogen. \n           Cutting at more than nine hits in the Super-K distribution yields an \n           estimated 18\\% tagging efficiency.}\n  \\label{fig:n_dg_p_nhit}\n\\end{figure} \n\n\\begin{figure}[htbp]\n  \\begin{center}\n    \\includegraphics[scale=0.35]{physics-pdecay\/figures\/pdk_ntag_70pc.pdf}\n  \\end{center}\n  \\caption{Neutron multiplicity distribution for background events in the $p \\rightarrow e^{+}\\pi^{0}$\n           search at Hyper-K. The red histogram shows the distribution prior to \n           neutron tagging and the green histogram shows the result of \n           applying a tagging algorithm with 70\\% efficiency. }\n  \\label{fig:ep0_n_mult}\n\\end{figure} \n\nImproved photon collection with larger photocathode coverages, higher\nquantum efficiency photosensors, and their combination have\na dramatic effect on the physics sensitivity of Hyper-K.  \nNucleon decay searches, in particular, are expected to benefit significantly\nfrom enhanced ability to detect low levels of Cherenkov light.  With\nthe large exposures Hyper-K will provide, the atmospheric neutrino\nbackground to these searches becomes sizable and\ncan inhibit the discovery potential of the experiment.  However, these\nsame backgrounds are often expected to produce neutrons either\ndirectly through the CC interaction of antineutrinos or indirectly via\nthe secondary interaction of hadrons in the interaction.  Proton decay\nevents, in contrast, are only rarely expected to be accompanied by\nneutrons.  Though such neutrons are ordinarily transparent to water\nCherenkov detectors, Super-Kamiokande has demonstrated the ability to\ntag the 2.2\\,MeV photon emerging from neutron capture on hydrogen,\n$n(p,d)\\gamma$.  Naturally this channel will be available to\nHyper-K.\nIn this section we compare the performance of Hyper-K 1TankHD\\,   configuration with its 40\\% photocathode \ncoverage and high quantum efficiency photosensors\nagainst a design with 20\\% coverage and the same PMTs used in Super-K.\n\nSince 2.2\\,MeV is only barely visible in Super-K, their tagging algorithm\nmakes use of a neural network to isolate signal neutrons, which are\ncorrelated spatially and temporally with the primary neutrino\ninteraction, from background sources.  Though the method achieved\n20.5\\% tagging efficiency with a 1.8\\% false tag\nprobability~\\cite{Wendell:2014dka}  it is worth noting that so far it\nhas only been successful during the SK-IV phase of the experiment.\nDespite 40\\% photocathode coverage with Hamamatsu R3600 50~cm PMTs, on\naverage the neutron capture signal produces only 7~hits in the\ndetector~\\cite{Wendell:2014dka}.  Since the average photon travel\nlength in Hyper-K will be larger than that of SK-IV, in order to take\nadvantage of the neutron signal it is essential to augment Hyper-K's\nphoton yield wherever possible.\n\nWhile a full analysis of Hyper-K's neutron tagging capability is in\ndevelopment a rough estimation for the 1TankHD\\,   design \n(c.f. Section~\\ref{section:photosensors}) has been determined via\nsimulation.  Figure~\\ref{fig:n_dg_p_nhit} shows the number of hit\nphotosensors for a simulated sample of the 2.2\\,MeV $\\gamma$ ray\nemitted when a neutron captures on a proton for three water Cherenkov\ndetector configurations.  The green distribution shows the response of\na Super-K-sized detector with 40\\% photocathode coverage by Hamamatsu\nR3600 PMTs, while the blue line shows the effect of implementing the HQE\nBox and Line photosensors. \nUsing the same photocathode coverage and the\nlatter photosensors but with an enlarged tank representative of a\nthe Hyper-K design (60~m diameter and 74~m height results)\nresults in the red curve.  Using these distributions the number of hit\nphotosensors which reproduces the tagging efficiency realized in the\nSuper-K analysis is chosen as the metric to represent the expected\ntagging efficiency at Hyper-K.  For the Super-K distribution gamma\nevents with more than nine PMT hits result in an 18\\% tagging efficiency.\nAssuming the same 10 hit threshold can be used at Hyper-K, the\nequivalent distribution (red curve in the figure) suggests 73\\%\nefficiency can be achieved.  The sensitivity studies presented in this \ndocument assume this number for Hyper-K when neutron tagging is employed. \nIn what follows sensitivity estimates are\npresented assuming both this and the Super-K tagging efficiency for comparison.\n\n\\begin{figure}[htbp]\n\\begin{center}\n  \\includegraphics[scale=0.35]{physics-pdecay\/figures\/epi0_bkg_reduction_exposure.pdf}\n\\end{center}\n\\caption {Hyper-K sensitivity to proton decay for the $p \\rightarrow e^{+}\\pi^{0}$ mode \n          as a function of exposure. The black curve illustrates the sensitivity \n          assuming equivalent performance to Super-K.\n          Improved sensitivity curves as a result of a finer binned signal region \n          (red) in conjunction with a 50\\% reduction in background (green) \n          or a 70\\% reduction (blue) are also shown.\n          The latter is the analysis presented in Section~\\ref{section:pdecay}.  }\n\\label{fig:ep0_bkg_exp}\n\\end{figure} \n\n\nBased on MC studies, the fraction of atmospheric neutrino backgrounds\nto the $p \\rightarrow e^{+}\\pi^{0}$ search which are accompanied by at\nleast one final state neutron is $> 80\\%$.  At the same time, only 4\\%\nof signal MC events have such a neutron.  The multiplicity\ndistribution for the background sample is shown in\nFigure~\\ref{fig:ep0_n_mult}.  Due to the presence of high multiplicity\nevents, assuming 70\\% neutron tagging efficiency, the\nneutron background can be reduced by approximately 70\\% (green\nhistogram) by rejecting events with one or more neutron tags.\nReducing the background in this manner will have a large impact on the\nsensitivity to this decay mode, as shown in\nFigure~\\ref{fig:ep0_bkg_exp}.  In the figure the black curve represents\nthe sensitivity of the analysis without neutron tagging and cuts defining \nonly a single signal region.\nIf that signal region is divided into a piece where free proton\ndecays are enhanced ($p_{tot} < 100 \\mbox{MeV\/c}^{2}$) and a region\nwith predominantly bound decays ($p_{tot} > 100 \\mbox{MeV\/c}^{2}$), as is assumed \nin the analysis above, the\nresulting sensitivity is shown by the red line.  Further, if the total\nbackground is then reduced by 50\\% (70\\%), by the introduction of\nneutron tagging, the sensitivity improves as shown in\nthe green (blue) lines.  \nWith these background reductions Hyper-K will require exposures of \n3.0 and 2.4 Mton $\\cdot$years to reach lifetime limits of $10^{35}$ years if no signal is observed.\nWithout any background reduction 7.0~Mton$\\cdot$years are required.\n\n\nIt should be noted that $p \\rightarrow e^{+}\\pi^{0}$ is not the only\nmode that is expected to benefit from a higher photon yield; Most\nmodes are similarly expected to have reduced atmospheric neutrino\nbackgrounds with neutron tagging.  However, the $p\n\\rightarrow \\bar \\nu K^{+}$ search can also benefit from enhanced\nlight collection to improve the signal efficiency.  Its two lowest\nefficiency, but most sensitive search modes, one in which the decay is\naccompanied by a prompt 6.3 MeV de-excitation $\\gamma$ from the\nrecoiling nucleus and the other in which the $K^{+}$ decays into\n$\\pi^{+}\\pi^{0}$, both have components that are looking for small\namounts of light.  Higher photon yields are thus connected directly to\nefficiency improvements in these channels.\n\n\\begin{figure}[htbp]\n\\begin{center}\n  \\includegraphics[scale=0.35]{physics-pdecay\/figures\/ngam-bl.pdf}\n\\end{center}\n\\caption {Distribution of the number of hits within a 12~nsec timing window used to search for the 6.3 MeV $\\gamma$ \n          from $p \\rightarrow \\bar \\nu K^{+}$ events. \n          The upper panel corresponds to a Hyper-K design with 20\\% photocathode coverage \n          and the same PMTs used in SK (Hamamatsu R3600).\n          In the lower panel the result for the 1TankHD\\,   design is shown. \n          Red (blue) lines show the proton decay signal (atmospheric neutrino background) distribution. \n          The proton decay signal region is defined as $4 < N_{\\gamma} < 30$ hits and $8 < N_{\\gamma} < 120$ \n          for the 20\\% coverage and 1TankHD\\,   coverage designs, respectively.\n          Vertical lines in the plots show the lower bound of these signal regions.  }\n\\label{fig:ngam-bl}\n\\end{figure} \n\nAs discussed above, the search for the prompt $\\gamma$ ray is done\nusing the number of hit PMTs within a 12~nsec wide timing widow prior\nto the muon candidate from the $K^{+} \\rightarrow \\mu^{+} \\nu$\n(c.f. Figure~\\ref{fig:prompt-gamma}).  Figure~\\ref{fig:ngam-bl} shows\nthis distribution for the proton decay signal (red) and background\nfrom atmospheric neutrinos (blue) for a Hyper-K design with\n20\\% photocathode coverage (upper panel) and the 1TankHD\\,   design. \nIn the latter study the photon yield is assumed to be 1.9\ntimes greater than that of the PMT used in the 20\\% coverage design and\nwith half the intrinsic timing resolution of the photosensor\n(1~nsec at 1~p.e.).  Additionally, these sensors are assumed to have a\nhigher dark rate of 8.4~kHz.  For both detector configurations the peak\nnear zero corresponds to events in which a $\\gamma$ was not found; \nthese hits are dominated by dark noise.  Though\nthis peak is not well separated from the feature at higher hits in the\n20\\% coverage configuration, with improved photosensors and 40\\% coverage a clear distinction\nbetween the two can be seen.  Further, since the $\\gamma$ search is\ndesigned to avoid hit contamination from the $\\mu^{+}$, the narrower\ntiming resolution of the improved sensors means that the search can\noccur closer in time to the muon.  Practically speaking, this means $K^{+}$\nwith earlier decay times can be used in the analysis as shown in\nFigure~\\ref{fig:gam-eff}.  Both of these effects lead to an overall\nincrease in the signal efficiency of the $p \\rightarrow \\bar \\nu K^{+}$\nmode.  Based on this distribution the proton decay signal region is\ndefined as $4 < N_{\\gamma} < 30$ hits for the 20\\% coverage design \nand $8 < N_{\\gamma} < 120$ for the 1TankHD\\,   case.\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n  \\includegraphics[scale=0.35]{physics-pdecay\/figures\/eff-gam.pdf}\n\\end{center}\n\\caption {Prompt $\\gamma$ tagging efficiency as a function of the \n          $K^{+}$ decay time for the $p \\rightarrow \\bar \\nu K^{+}$ decay mode. \n          A Hyper-K design with only 20\\% photocathode coverage is shown in black and \n          the design with 40\\% photocathode coverage and photosensors with 1.9 times \n          higher photon yield (1TankHD\\,  ) is shown in red. \n          }\n\\label{fig:gam-eff}\n\\end{figure} \n\nAccordingly, the signal efficiency for the prompt $\\gamma$ tag method\nto search for $p \\rightarrow \\bar \\nu K^{+}$ increases from 6.7\\% in\nthe design with 20\\% photocathode coverage to 12.7\\% in the 1TankHD\\,   configuration.\nThough not detailed here, improved tagging of the $\\pi^{+}$ from the\n$K \\rightarrow \\pi^{+} \\pi^{0}$ component of this search yields 10.2\\%\nsignal efficiency in the 1TankHD\\,   design compared to 6.7\\% in the\n20\\% coverage case.\nAs in the $p \\rightarrow e^{+} \\pi^{0}$ study\nabove, if neutron tagging with 70\\% efficiency is assumed\n(c.f. Figure~\\ref{fig:ep0_bkg_exp}) the background to these $p\n\\rightarrow \\bar \\nu K^{+}$ search methods can be reduced to 0.87 and\n0.71~events\/Mton$\\cdot$year, respectively.  For comparison the\nbackground rates are 2.8 and\n3.4~events\/Mton$\\cdot$year without neutron tagging.  \nFigure~\\ref{fig:bl-sens} shows the\ncorresponding sensitivity curves for both the design with 20\\% coverage (black) and\nfor the 1TankHD\\,   design (red).  Two vertical lines at exposures of 1.9 and\n5.6~Mton$\\cdot$year in the figure denote positions of comparable\nsensitivity between the two designs.  \n\n\\begin{figure}[htbp]\n\\begin{center}\n  \\includegraphics[scale=0.35]{physics-pdecay\/figures\/nuk-sens-bl.pdf}\n\\end{center}\n\\caption {Sensitivity to $p \\rightarrow \\bar \\nu K^{+}$ at 90\\% C.L. for the 1TankHD\\,   design is shown in red. \n          The black curve shows the corresponding sensitivity for a design with 20\\% coverage and Super-K PMTs. }\n\\label{fig:bl-sens}\n\\end{figure}  \n\nIt should be noted that since backgrounds for the Super-K neutron\ntagging algorithm are taken from its data, it is difficult to provide\na realistic estimate of the potential of a similar algorithm at\nHyper-K at this time.  \nHowever assuming similar performance, Hyper-K's dramatic\nimprovement in proton decay sensitivity makes this topic among the\nmost fundamental to the development of its future program.  It is clear\nthat the potential for a discovery is connected to\nHyper-K's background reduction and efficiency enhancement\ncapabilities, both of which are realizable with higher photon yields.\n\n\n\n\n\\subsection{Nucleon decays }\\label{section:pdecay}\nOptimizing Hyper-Kamiokande for the observation and discovery of a\nnucleon decay signal is one if its primary design drivers.  In order\nto significantly extend sensitivity beyond existing limits, many of\nwhich have been set by Super-Kamiokande, Hyper-K needs both a much\nlarger number of nucleons than its predecessor and sufficient\nreconstruction ability to extract signals and suppress backgrounds.\nWhile it is possible to target specific decay channels, one of the\nstrengths of water Cherenkov technology is its sensitivity to a wide variety\nof modes.  Using MC and analysis techniques originally developed for\nSuper-Kamiokande, this section details Hyper-K's expected sensitivity\nto both the flagship proton decay modes, $p \\rightarrow e^{+} \\pi^{0}$\nand $p \\rightarrow \\overline{\\nu} K^{+}$, as well as other $\\Delta\n(B-L)$ conserving, $\\Delta B =2$ dinucleon, and $\\Delta (B-L)=2$ decays.  \n\\begin{figure}[htbp]\n  \\begin{center}\n    \\includegraphics[scale=0.4]{physics-pdecay\/figures\/epi0_massmom_pdk_corr.pdf}\n    \\includegraphics[scale=0.4]{physics-pdecay\/figures\/epi0_massmom_atmc.pdf}\n  \\end{center}\n\\caption {Reconstructed invariant mass and total momentum distributions for the $p \\rightarrow e^{+} \\pi^{0}$ \n          MC (left) and atmospheric neutrino MC (right) after all\n          event selections except the cuts on these variables.  The\n          final signal regions are shown by two black boxes in the\n          plane.  In the signal plot decays from bound and free\n          protons have been separated by color, dark blue and cyan\n          respectively.  Background events have been generated for a\n          45~Mton$\\cdot$year exposure and those falling in the signal\n          regions have been enlarged for\n          visibility. } \\label{fig:epi0_plane}\n\\end{figure} \n\n\n\n\n\n\n\\begin{figure}[htbp]\n  \\begin{center}\n     \\includegraphics[width=0.47\\textwidth,trim={0 4.5cm 0 4cm},clip]{physics-pdecay\/figures\/totmom-1hd.pdf}\n     \\includegraphics[width=0.45\\textwidth]{physics-pdecay\/figures\/totmass-1tank.pdf}\n  \\end{center}\n\\caption { Total momentum distribution of events passing all steps of the $p \\rightarrow e^{+} \\pi^{0}$ event selection \n          except the momentum cut after a 10 year exposure of a single Hyper-K tank (left).\n          Reconstructed invariant mass distribution of events passing all steps of the $p \\rightarrow e^{+} \\pi^{0}$ event selection \n          except the invariant mass cut after a 10 year exposure of a single Hyper-K tank (right).\n          The hatched histograms show\n          the atmospheric neutrino background and the solid crosses\n          denote the sum of the background and proton decay\n          signal. Here the proton lifetime is assumed to be,\n          $1.7 \\times 10^{34}$\\,years, just beyond current Super-K\n          limits.   \n          The free and bound proton-enhanced bins are shown by the lines in the left plot, \n          and are the upper and lower panels of the right plot. } \\label{fig:pmass_epi0}\n\\end{figure} \n\n\n\n\n\\subsubsection{Sensitivity to $p \\rightarrow e^{+} + \\pi^{0}$ Decay}\nProton decay into a positron and neutral pion is a favored mode of\nmany GUT models.\nExperimentally this decay has a very clean\nevent topology, with no invisible particles in the final state.  As a\nresult it is possible fully reconstruct the proton's mass from its\ndecay products and as a two body process the total momentum of\nthe recoiling system should be small.  The event selection focuses on\nidentifying fully contained events within the Hyper-K fiducial volume\nwith two or three electron-like Cherenkov rings.  Though the decay of\nthe pion is expected to produce two visible gamma rays, for\nforward-boosted decays the two photons may be close enough in space to\nbe reconstructed as a single ring.  Atmospheric neutrino events with\na muon below threshold are removed by requiring there are no Michel\nelectrons in the event.  For those events with three rings, the two\nrings whose invariant mass is closest to the $\\pi^{0}$ mass are\nlabeled the $\\pi^{0}$ candidate.  An additional cut on the mass of\nthose candidates, $ 85 < m_{\\pi} < 185$~MeV\/$c^{2}$, is applied.\nThe signal sample is selected by requiring the total invariant mass of\nthe event be near the proton mass, $800 < m_{inv} <\n1050$~MeV\/$c^{2}$ and that the total momentum, $p_{tot}$, be less\nthan 250~MeV\/$c$.  Water is an excellent molecule for studying proton\ndecay because in addition to providing 10 protons per molecule, two of\nthose are unbound free protons.  Decays from those protons are not\nsubject to nuclear effects and result in final state particles with\nvery low total momentum.  At the same time, very few atmospheric\nneutrino interactions are reconstructed with both a proton-like invariant\nmass and low total momentum.  To optimize the analysis sensitivity,\nproton-decay candidates are divided into two signal regions after the\ninvariant mass cut, a free proton decay enriched region ($0 < p_{tot}\n< 100$~MeV\/$c$) and a bound proton decay region ($100 < p_{tot}\n< 250$~MeV\/$c$).  \nFinally, neutron tagging (described in a later section)\nis used to reject background events by requiring events have no neutron candidates\nin the final state.\nFigure~\\ref{fig:epi0_plane} shows the signal\nand background MC in the invariant mass and total momentum plane\nbefore the final signal region is defined.  Signal efficiencies and\nbackground rates with corresponding systematic errors after all\nselections are listed in Table~\\ref{tbl:pdkepi0}.\n\n\\begin{table}[htb]\n  \\begin{center}\n  \\begin{tabular}{c|c||c|c}\n\\hline\n\\hline\n\\multicolumn{2}{c||}{ $0 < p_{tot} < 100$~MeV\/$c$ } &  \\multicolumn{2}{c}{ $100 < p_{tot} < 250$~MeV\/$c$ } \\\\\n$\\epsilon_{sig}$ [\\%] & Bkg [\/Mton$\\cdot$yr] & $\\epsilon_{sig}$ [\\%]   &  Bkg [\/Mton$\\cdot$yr] \\\\\n\\hline\n\\hline\n$18.7 \\pm 1.2$   &  $0.06 \\pm  0.02$    &     $19.4 \\pm 2.9$    &  $0.62 \\pm 0.20$           \\\\\n\\hline\n\\hline\n  \\end{tabular}\n  \\end{center}\n  \\caption{Signal efficiency and background rates as well as estimated systematic uncertainties \n           for the $p \\rightarrow e^{+} \\pi^{0}$ analysis at Hyper-K. }\n  \\label{tbl:pdkepi0}\n\\end{table}\n\n\n\n\\begin{figure}[p]\n  \\begin{center}\n    \\includegraphics[width=0.7\\textwidth,trim={0 0.5cm 0 0},clip]{physics-pdecay\/figures\/hk_epi0_disco_overlay_exposure_1and2HD_DUNE_JUNO_SK3s_C_year.pdf}\n  \\end{center}\n  \\caption{Comparison of the 3~$\\sigma$ $p \\rightarrow e^{+}\\pi^{0}$ discovery potential as a function of year\n           Hyper-K (red solid) assuming a single tank \n           as well as that of the 40~kton liquid argon detector DUNE (cyan solid) following~\\cite{Acciarri:2015uup}. \n           In the orange dashed line an additional Hyper-K tank is assumed to\n           come online six years after the start of the experiment.\n           Super-K's discovery potential in 2026 assuming 23 years of data is also shown.\n          }\n  \\label{fig:epi_discovery}\n  \\begin{center}\n    \\includegraphics[width=0.7\\textwidth,trim={0 0.5cm 0 0},clip]{physics-pdecay\/figures\/hk_epi0_limit_overlay_exposure_1and2HD_DUNE_JUNO_SK90pc_L_year.pdf}\n  \\end{center}\n\\caption {Hyper-K's sensitivity to the $p \\rightarrow e^{+} \\pi^{0}$ decay mode \n          at 90\\% C.L. as a function of run time appears in red assuming one detector in comparison with other experiments (see caption of Figure~\\ref{fig:epi_discovery}).\n          Super-K's current limit is shown by a horizontal line.\n         }\n  \\label{fig:sens_epi0}\n\\end{figure}\n\nMonte Carlo simulation of these decays includes the effects of the\nnucleon binding energy, Fermi motion, and interactions within the\n$^{16}O$ nucleus.  The latter represents a significant, but\nunavoidable, source of inefficiency as the signal $\\pi^{0}$ may be\nlost to absorption or charge exchange prior to exiting the nucleus.\nAlthough the signal efficiency of free proton decays is roughly 87\\% \nthese nuclear effects reduce the\nefficiency of bound proton decays such that the overall efficiency is\nonly $\\sim$ 40\\% when all decays are considered.\n\n\nAtmospheric neutrino interactions are the main background to proton\ndecay searches.  Not only can CC $\\nu_{e}$ single-pion production\nprocesses produce the same event topology expected in the\n$p \\rightarrow e^{+} \\pi^{0}$ decay, but recoiling nucleons from\nquasi-elastic processes can produce pion final states through hadronic\nscatters that mimic the signal.  After the event selection above the\nexpected background rate is 0.06 events in the free proton\nenhanced bin and 0.62 events in the bound proton enhanced bin\nper Mton$\\cdot$year.\nThese background expectations have been experimentally verified using\nbeam neutrino measurements with the K2K experiment's one kiloton water\nCherenkov detector, which found an expected atmospheric neutrino\nbackground contamination to $p \\rightarrow e^{+} \\pi^{0}$ searches of\n$1.63^{+0.42}_{-0.33}(stat)^{+0.45}_{-0.51}(sys)$ events per\nMegaton$\\cdot$year~\\cite{Mine:2008rt} without neutron tagging.\n\n\nThe ability to reconstruct the proton's invariant mass is a powerful\nfeature of this decay mode.  The left panel of Figure~\\ref{fig:pmass_epi0} shows the\nexpected distribution of this variable for both signal events and\natmospheric neutrino backgrounds after a 10 year exposure assuming the\nproton lifetime is $\\tau_{x} = 1.7 \\times 10^{34}$ years.  \nSimilarly, the left panel shows the total momentum distribution\nof candidate events prior to the momentum selection.\nBoth plots illustrate \nthe marked difference in the signal and background expectations \nfor the free proton and bound proton analysis bins.\n\nHyper-Kamiokande's proton decay discovery potential has been estimated \nbased on a likelihood ratio method. \nA likelihood function is constructed from a Poisson probability density \nfunction for the event rate in each total momentum bin of the $p \\rightarrow e^{+} \\pi^{0}$\nanalysis with systematic errors on the selection efficiency and background \nrate represented by Gaussian nuisance parameters.\nThe experiment's expected sensitivity to a proton decay signal at a given \nconfidence level, $\\alpha$, is calculated as the fastest proton decay rate, $\\Gamma$, \nwhose median likelihood ratio value assuming a proton decay signal\nyields a p-value not larger than $\\alpha$ from the likelihood ratio distribution\nunder the background-only hypothesis.\nThat is, \n\\begin{equation}\n\\Gamma_{disc} = \\max_{\\Gamma} \\left[ \\alpha = \\int_{ M(\\Gamma) }^{\\infty} f( q_{0} | b ) d q_{0}  \\right],  \n\\end{equation}\n\\noindent where $f$ is the distribution function of the likelihood ratio, $q_{\\Gamma}$, and \n$ M(\\Gamma) = \\mbox{median}[ f( q_{\\Gamma} | \\Gamma \\epsilon \\lambda + b ) ]$. \nHere $\\epsilon$ is the selection efficiency, $\\lambda$ the detector exposure, \nand $b$ is the expected number of background events.\nFor the calculated significance the corresponding proton lifetime is then $\\tau_{disc} = 1\/\\Gamma_{disc}$.\nUnder this definition Hyper-K's $3 \\sigma$ (one-sided) discovery potential as a function of run\ntime is shown in Figure~\\ref{fig:epi_discovery}.  Note that if the\nproton lifetime is as short as $\\tau_{x}$ its decay into\n$e^{+}\\pi^{0}$ will be seen at this significance within two years of\nHyper-K running. \n\nEven in the absence of a signal Hyper-K is expected\nto extend existing limits considerably.\nHyper-K's sensitivity to this decay mode is computed as\n\\begin{equation}\n\\tau_{limit} = \\sum_{n=0}^{\\infty} O(n|b) \/ \\Gamma_{n},\n\\end{equation}\n\\noindent where\n\\begin{equation}\n\\Gamma_{n} = \\left[ \\Gamma_{l} : 1 - \\alpha = \\int_{0}^{\\Gamma_{l}} P( \\Gamma | n ) d\\Gamma \\right]\n\\end{equation}\n\\noindent and $P( \\Gamma | n )$ is the probability that the proton decay rate is $\\Gamma$ given \nan observation of $n$ events.  \nSimilarly, $O(n|b)$ is the Poisson probability to observe $n$ events given a mean of $b$.\nThe function $P( \\Gamma | n )$ is obtained using Bayes' theorem and the likelihood function \noutlined above. \nHyper-K's expected sensitivity to $p \\rightarrow e^{+} \\pi^{0}$ using this metric is shown \nin Figure~\\ref{fig:sens_epi0}.\nIn both this and the discovery potential figure, systematic errors on the signal and background have been\nincluded based on the Super-K analysis; The errors are listed in Table~\\ref{tbl:pdkepi0}.  \nA detailed description of the systematic error estimation and the limit calculation may be found\nin~\\cite{Nishino:2012bnw}.\n\n\n\n\\subsubsection{Sensitivity study for the $p \\rightarrow \\overline{\\nu} K^{+}$ mode} \n\nProton decays into a lepton and a kaon are a feature of supersymmetric grand unified theories, of\nwhich $p \\rightarrow \\overline{\\nu} K^{+}$ is among the most\nprominent.  Searching for the $K^{+}$ in a water Cherenkov detector is\ncomplicated by the fact it emerges from the decay with a momentum of\nonly 340\\,MeV\/$c$, well below the threshold for light production,\n749\\,MeV\/$c$.  As a result the $K^{+}$ must be identified by its decay\nproducts: $K^+ \\to \\mu^{+}+\\nu$ (64\\% branching fraction) and\n$K^+ \\to \\pi^{+}+\\pi^{0}$ (21\\% branching fraction).  Since both of\nthese modes are two body decays the outgoing particles have\nmonochromatic momenta.  Furthermore, the 12~ns lifetime of the kaon\nmakes it possible to observe prompt $\\gamma$ ray emission produced\nwhen the proton hole leftover from a bound proton decay is filled by the\nde-excitation of another proton.  For $^{16}O$ nuclei the probability\nof producing a 6\\,MeV $\\gamma$ from such a hole is roughly 40\\%, making\nthis a powerful tool for identifying the $K^{+}$ decay products and rejecting \natmospheric neutrino backgrounds.\n\nThree methods, each targeting different aspects of the $K{+}$ decay,\nare used to search for $p \\rightarrow \\overline{\\nu} K^{+}$\nevents~\\cite{Abe:2014mwa}.  The ``prompt $\\gamma$'' method searches\nfor a prompt nuclear de-excitation $\\gamma$ ray (6.3\\,MeV) occurring\nprior to and separated in time from a 236\\,MeV\/$c$ muon.  A schematic\nof this process appears in Figure~\\ref{fig:prompt-gamma}.  In the\nsecond method the same monochromatic muon is searched for but without\nthe $\\gamma$ tag.  Finally, the third method, or ``$\\pi^{+}\\pi^{0}$\nmethod'', searches for a monochromatic $\\pi^{0}$ with light from a\nbackward $\\pi^{+}$.  Here the charged pion is only slightly above\nCherenkov threshold, making it difficult to reconstruct a full \nring.\n\n\\begin{figure}[htbp]\n  \\begin{center}\n    \\includegraphics[height=19pc]{physics-pdecay\/figures\/t-sample.pdf}\n  \\end{center}\n\\caption {Schematic view of the expected timing distribution of events in the \n          prompt $\\gamma$ search method for $p \\rightarrow \\overline{\\nu} K^{+}$ decays.\n          The upper panel shows the full event time window with the $\\mu$, Michel, \n          and $\\gamma$ candidate clusters. The bottom panel shows the time of flight \n          subtracted timing distribution around the $\\mu$ candidate.\n         }\n  \\label{fig:prompt-gamma}\n\\end{figure}\n\n\\begin{table}[htb]\n  \\begin{center}\n  \\begin{tabular}{c|c||c|c||c|c||c}\n\\hline\n\\hline\n\\multicolumn{2}{c||}{ Prompt $\\gamma$ } &  \\multicolumn{2}{c||}{ $\\pi^{+}\\pi^{0}$  } & \\multicolumn{3}{c}{ $p_{\\mu}$ Spectrum } \\\\\n$\\epsilon_{sig}$ [\\%] & Bkg  [\/Mton$\\cdot$yr] &  $\\epsilon_{sig}$ [\\%]   &  Bkg  [\/Mton$\\cdot$yr]  &  $\\epsilon_{sig}$ [\\%]   & Bkg  [\/Mton$\\cdot$yr] & $\\sigma_{fit}$ [\\%] \\\\\n\\hline\n\\hline\n$12.7 \\pm 2.4$  &  $0.9 \\pm 0.2$  &  $10.8 \\pm 1.1$  &  $0.7 \\pm 0.2$  & 31.0  &   1916.0  &  8.0             \\\\\n\\hline\n\\hline\n  \\end{tabular}\n  \\end{center}\n  \\caption{Signal efficiency and background rates as well as estimated systematic uncertainties \n           for the $p \\rightarrow \\bar \\nu K^{+}$ analysis at Hyper-K.}\n  \\label{tbl:pdknuk}\n\\end{table}\n\n\n\\begin{figure}[htbp]\n  \\begin{center}\n     \\includegraphics[width=0.48\\textwidth,trim={0 4.5cm 0 4cm},clip,trim={0 4.5cm 0 4cm},clip]{physics-pdecay\/figures\/mumom-1hd.pdf}\n     \\includegraphics[width=0.48\\textwidth,trim={0 4.5cm 0 4cm},clip]{physics-pdecay\/figures\/kmass-pipi-1hd.pdf}\n  \\end{center}\n\\caption { Reconstructed muon momentum distribution for muons found in the prompt $\\gamma$ search \n          of the $p \\rightarrow \\bar \\nu K^{+}$ analysis after a  10 year exposure of a single Hyper-K tank (left).\n          The right figure shows the reconstructed kaon mass based on the reconstructed final state in candidates from the  \n          $\\pi^{+}\\pi^{0}$. \n          The hatched histograms show the atmospheric neutrino background and the solid crosses\n          denote the sum of the background and proton decay signal. \n          Here the proton lifetime is assumed to be,\n          $6.6 \\times 10^{33}$ years, just beyond current Super-K limits.\n          All cuts except for the cut on visible energy opposite the $\\pi^{0}$ candidate have been applied in the right plot.  }\n  \\label{fig:nuk_pmu}\n\\end{figure} \n\n\n\nThe prompt $\\gamma$ search method proceeds by identifying fully\ncontained fiducial volume interactions with a single $\\mu$-like ring\naccompanied by a Michel electron.  In order to suppress atmospheric\nneutrino backgrounds, particularly those with an invisible muon, the\nmuon momentum is required to be $215< p_{\\mu} < 260$\\,MeV\/$c$ and the\ndistance between its vertex and that of the Michel electron cannot\nexceed 200~cm.  Events with high momentum protons that create a\n$\\mu$-like Cherenkov ring are rejected using a dedicated likelihood\ndesigned to select in favor of genuine muons based on the PMT hit\npattern and the Cherenkov opening angle.  Searching backward in time\nfrom the muon candidate, de-excitation $\\gamma$ ray candidates are\nidentified as the largest cluster of PMT hits within a 12~ns sliding\nwindow around the time-of-flight-subtracted time distribution.  \nThe time difference between the center of the time window\ncontaining the $\\gamma$ candidate, $t_{\\gamma}$, and the muon time,\n$t_{\\mu}$ is then required be consistent with decay of a kaon,\n$t_{\\mu}- t_{\\gamma} < 75$~ns ($\\sim 6\\tau_{K^+}$).  Finally, only\nevents whose number of hits in this window, $N_{\\gamma}$, is\nconsistent with 6\\,MeV of energy deposition ($8 < N_{\\gamma} < 120$) are\nkept.  The left panel of Figure~\\ref{fig:nuk_pmu} shows the expected $p_{\\mu}$\ndistribution after all selection cuts have been applied assuming a\nproton lifetime of $6.6\\times 10^{33}$ years, which is slightly less than the current \nSuper-K limit. \n\n\nThe second search ($p_{\\mu}$ spectrum) method also focuses on\nidentifying the monochromatic muon but with relaxed search criterion.\nOnly those cuts applied before the proton likelihood cut\nare used.  A fit is then applied to the resulting muon momentum\ndistribution to identify any proton decay-induced excess of muon\nevents over the considerable atmospheric neutrino background.\n\n\\begin{figure}[p]\n  \\begin{center}\n    \\includegraphics[width=0.7\\textwidth,trim={0 0.5cm 0 0},clip]{physics-pdecay\/figures\/hk_nuk_disco_overlay_exposure_1and2HD_DUNE_JUNO3s_C_year.pdf}\n  \\end{center}\n  \\caption{Comparison of the 3~$\\sigma$ $p \\rightarrow \\bar \\nu K^{+}$ discovery potential as a function of year for \n           the Hyper-K as well as that of the 40~kton DUNE detector (cyan solid) based on~\\cite{Acciarri:2015uup} and \n           the 20~kton JUNO detector based on~\\cite{An:2015jdp}. \n           The red line denotes a single Hyper-K tank, while the orange line shows the expectation when a second \n           tank comes online after six years.\n           The expected discovery potential for Super-K by 2026 assuming 23 years of data is also shown.\n          }\n  \\label{fig:nuK_discovery}\n  \\begin{center}\n    \\includegraphics[width=0.7\\textwidth,trim={0 0.5cm 0 0},clip]{physics-pdecay\/figures\/hk_nuk_limit_overlay_exposure_1and2HD_DUNE_JUNO_SK90pc_L_year.pdf}\n  \\end{center}\n\\caption {Hyper-K's sensitivity to the $p \\rightarrow \\bar \\nu K^{+}$ decay mode \n          at 90\\% C.L. as a function of run time is shown in red against other experiments (see caption of Figure~\\ref{fig:nuK_discovery}).\n          Super-K's current limit is also shown.\n         }\n  \\label{fig:sens_nuk}\n\\end{figure}\n\nLike the prompt $\\gamma$ search, the $\\pi^{+}\\pi^{0}$ search relies on\nmore sophisticated event selections to identify the signal.  While the\nmomentum of both the $\\pi^{+}$ and $\\pi^{0}$ from the $K^{+}$ decay\nwill be 205\\,MeV\/$c$, the former does not deposit enough light to be\nreconstructed fully.  To compensate for this the search method focuses\non PMT hits opposite the direction of a $\\pi^{0}$ at the correct\nmomentum.  Fully contained events with one or two reconstructed rings,\nall of which are $e$-like, are selected.  In addition there should be\none Michel electron from the charged pion decay chain.  For two-ring\nevents the invariant mass is required to be consistent with that of a\nneutral pion, $ 85 < m_{\\pi^{0}} < 185$\\,MeV\/$c^2$.  Further, the total\nmomentum should be between 175 and 250\\,MeV\/$c$.  At this stage the\n$\\pi^{+}$ light is identified as electron-equivalent visible energy between 7 and 17\\,MeV\nlocated at an angular separation between 140 and 180\\,degrees from the\n$\\pi^{0}$ candidate's direction.  Finally a likelihood method is used\nto evaluate the consistency of this light pattern with that produced\nby a $\\pi^{+}$.  Details of the full search method are documented\nelsewhere~\\cite{Abe:2014mwa}.\nIn addition, as in the $p\\rightarrow e^{+}\\pi^{0}$ search, the number \nof tagged neutron candidates is required to be zero.\n\n\nTable~\\ref{tbl:pdknuk} summarizes the expected signal efficiency and background rates \nfor each of the search methods. \nAccompanying systematic errors have been calculated based on Super-K methods~\\cite{Abe:2014mwa} and are included in \nthe table. \nAssuming a proton lifetime close to current Super-K limits, \nFigures~\\ref{fig:nuk_pmu} show the reconstructed muon momentum from \nthe prompt gamma search and the reconstructed kaon mass spectrum from the $\\pi^{+}\\pi^{0}$ search.\nNote that while the latter is unused in the analysis itself it provides a demonstration \nof the signal reconstruction since the proton mass for this decay mode cannot be reconstructed. \nThe expected discovery potential and sensitivity of Hyper-K in comparison with other experiments \nappears in Figures~\\ref{fig:nuK_discovery} and~\\ref{fig:sens_nuk}.\n\n\n\n\\subsubsection{Sensitivity study for other nucleon decay modes}\nAlthough the $p \\rightarrow e^{+} \\pi^{0}$ decay mode is predicted to be dominant \nin many GUT models, a variety of other decay modes are possible, each with a sizable \nbranching ratio.\nTable~\\ref{tab:branch} shows the branching ratio distribution and ratio of neutron to proton \nlifetimes as predicted by several GUT models.\nThe diversity in these predictions suggests that in order to make a discovery and to subsequently \nconstrain proton decay models, it is critical to probe as many nucleon decay modes as possible.\nFortunately, Hyper-Kamiokande is expected to be sensitive \nto many decay modes beyond the two standard decays discussed above. \n\n\\begin{table}[htb]\n\\caption{Branching ratios for various proton decay modes together with the\nratio of the neutron to proton lifetimes as predicted by SU(5) and SO(10) models.\n.\\label{tab:branch}}\n\\vspace{0.4cm}\n\\begin{center}\n\\begin{tabular}{l|rrrrr}\n\\hline\\hline\n        & \\multicolumn{5}{c}{Br.($\\%$)} \\\\ \n\\hline\n        & \\multicolumn{4}{c}{SU(5)}& SO(10)  \\\\ \n\\hline\nReferences& ~\\cite{mach} & ~\\cite{gav} & ~\\cite{dono} & ~\\cite{bucc} & ~\\cite{bucc}   \\\\ \n\\hline\n$p \\rightarrow e^{+} \\pi^{0}$ & 33   & 37  & 9  & 35 & 30 \\\\ \n\\hline\n$p \\rightarrow e^{+} \\eta^{0}$ & 12  & 7 & 3 & 15 & 13 \\\\\n\\hline \n$p \\rightarrow e^{+} \\rho^{0}$ & 17    & 2   & 21   & 2 & 2   \\\\ \n\\hline\n$p \\rightarrow e^{+} \\omega^{0}$ & 22  & 18 & 56 & 17 & 14 \\\\\n\\hline \nOthers & 17   & 35   & 11   & 31 & 31  \\\\ \n\\hline\n\\hline\n$\\tau_{p}\/\\tau_{n}$ & 0.8 &1.0 & 1.3 &  & \\\\\n\\hline  \\hline  \n\\end{tabular}\n\\end{center}\n\\end{table}\n\nHyper-Kamiokande's sensitivity to other nucleon decay modes has been \nestimated based in part on efficiencies and background rates\nfrom Super-Kamiokande~\\cite{:2009gd}. \nTable~\\ref{tab:other_mode} shows the 90~$\\%$ CL\nsensitivities with a 1.9\\,Megaton$\\cdot$year exposure in the 1TankHD\\ configuration. \nUnder these conditions Table~\\ref{tab:disc_other} shows the $3\\sigma$ discovery potential of Hyper-K \nfor a selected number of decay modes after a 1.9~Mton$\\cdot$year exposure.\n\n\n\\begin{table}[htp]\n\\caption{Summary of Hyper-K's sensitivity to various $|B-L|$ conserving nucleon decay modes after a 1.9\\,Megaton$\\cdot$year exposure of the 1TankHD\\  design compared in comparison with existing lifetime limits. The current limits for $p \\rightarrow e^{+} \\pi^{0}$, $p \\rightarrow \\mu^{+} \\pi^{0}$ are from a 0.316\\'Megaton$\\cdot$year exposure of Super-Kamiokande~\\cite{Miura:2016krn}, $p \\rightarrow \\overline{\\nu} K^{+}$ is from a 0.26\\,Megaton$\\cdot$year exposure~\\cite{Abe:2014mwa}, and the other modes are from a 0.316\\,Megaton$\\cdot$year exposure~\\cite{TheSuper-Kamiokande:2017tit}.\n\\label{tab:other_mode}}\n\\vspace{0.4cm}\n\\begin{center}\n\\begin{tabular}{l|c|c}\n\\hline\nMode        & Sensitivity (90$\\%$ CL) [years]  & Current limit [years] \\\\\n\\hline\n\\hline\n\\textcolor{blue}{$p \\rightarrow e^{+} \\pi^{0}$}        & 7.8 $\\times 10^{34}$ & 1.6$\\times 10^{34}$ \\\\\n\\textcolor{blue}{$p \\rightarrow \\overline{\\nu} K^{+}$} & 3.2 $\\times 10^{34}$   & 0.7$\\times 10^{34}$  \\\\\n\\hline\n\\hline\n$p \\rightarrow \\mu^{+} \\pi^{0}$ & 7.7$\\times 10^{34}$  & 0.77$\\times 10^{34}$  \\\\\n\\hline\n$p \\rightarrow e^{+} \\eta^{0}$& 4.3$\\times 10^{34}$  & 1.0$\\times 10^{34}$ \\\\\n\\hline\n$p \\rightarrow \\mu^{+} \\eta^{0}$& 4.9$\\times 10^{34}$  & 0.47$\\times 10^{34}$ \\\\\n\\hline\n$p \\rightarrow e^{+} \\rho^{0}$& 0.63$\\times 10^{34}$  & 0.07$\\times 10^{34}$ \\\\\n\\hline\n$p \\rightarrow \\mu^{+} \\rho^{0}$& 0.22$\\times 10^{34}$  & 0.06$\\times 10^{34}$ \\\\\n\\hline\n$p \\rightarrow e^{+} \\omega^{0}$& 0.86$\\times 10^{34}$  & 0.16$\\times 10^{34}$ \\\\\n\\hline\n$p \\rightarrow \\mu^{+} \\omega^{0}$& 1.3$\\times 10^{34}$  & 0.28$\\times 10^{34}$ \\\\\n\\hline\n$n \\rightarrow e^{+} \\pi^{-}$ & 2.0$\\times 10^{34}$   & 0.53$\\times 10^{34}$ \\\\\n\\hline\n$n \\rightarrow \\mu^{+} \\pi^{-}$ & 1.8$\\times 10^{34}$   & 0.35$\\times 10^{34}$ \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\nThe decay modes in Table~\\ref{tab:other_mode} all conserve baryon\nnumber minus lepton number, $(B-L)$.  \nHowever,  the $(B+L)$\nconserving mode, $n \\rightarrow e^{-} K^{+}$, was also given attention\nand searched for by Super-Kamiokande.  In $n \\rightarrow e^{-} K^{+}$,\nthe $K^{+}$ stops in the water and decays into $\\mu^{+} + \\nu$.  The\nfinal state particles observed in $n \\rightarrow e^{-} K^{+},K^+ \\to \\mu^+ \\nu$ are $e^{-}$ and $\\mu^{+}$.\n Both $e^{-}$ and\n$\\mu^{+}$ have monochromatic momenta as a result of originating from\ntwo-body decays.  Furthermore, the timing of the $\\mu^{+}$ rings are \ndelayed with respect to the $e^{-}$ rings because of the $K^{+}$\nlifetime.  In SK-II, the estimated efficiencies and the background\nrate are 8.4\\% and 1.1\\,events\/Megaton$\\cdot$year, respectively. From\nthose numbers, the sensitivity to the $n \\rightarrow e^{-} K^{+}$ mode\nwith a 1.9\\,Megaton$\\cdot$year exposure is estimated to be 1.0$\\times\n10^{34}$ years.\n\nThe possibility of $n \\overline{n}$ oscillation is another interesting\nphenomenon; it violates baryon number $(B)$ by $|\\Delta B|$ = 2.\nThese $n \\overline{n}$ oscillations have been searched for in\nSuper-Kamiokande with an exposure of 0.09\\,Megaton$\\cdot$year~\\cite{jang}. \nFurther improvement of the $n \\overline{n}$\noscillation search is expected in Hyper-Kamiokande.\n\n\\begin{table}[htp]\n\\caption{Summary of Hyper-K's $3 \\sigma$ discovery potential for several nucleon decay modes in the 1TankHD\\  configuration.\n         A 10 year exposure of a single detector  \n         has been assumed. Numbers in brackets denote the potential assuming a second detector comes online six years \n         after the start of the experiment. Current limits are summarized in Table~\\ref{tab:other_mode}. }\n\\label{tab:disc_other}\n\\vspace{0.4cm}\n\\begin{center}\n\\begin{minipage}[b][][b]{.45\\linewidth}\n\\begin{tabular}{>{\\raggedright}p{3cm}|>{\\centering}p{3cm}}\n\\hline\nMode  &  $\\tau_{disc}$ $3\\sigma$ [years]   \\tabularnewline\n\\hline\n\\hline\n\\textcolor{blue}{$p \\rightarrow e^{+} \\pi^{0}$}        & 6.3 (8.0)$\\times 10^{34}$  \\tabularnewline\n\\textcolor{blue}{$p \\rightarrow \\overline{\\nu} K^{+}$} & 2.0 (2.5)$\\times 10^{34}$  \\tabularnewline\n\\hline\n\\hline\n$p \\rightarrow \\mu^{+} \\pi^{0}$  & 6.9 (8.7)$\\times 10^{34}$ \\tabularnewline\n\\hline\n$p \\rightarrow e^{+} \\eta^{0}$   & 3.0 (3.9)$\\times 10^{34}$  \\tabularnewline \n\\hline\n$p \\rightarrow \\mu^{+} \\eta^{0}$ & 3.4 (4.7)$\\times 10^{34}$  \\tabularnewline \n\\hline\n$p \\rightarrow e^{+} \\rho^{0}   $   & 3.4 (5.0)$\\times 10^{33}$ \\tabularnewline \n\\hline\n$p \\rightarrow \\mu^{+} \\rho^{0}$     & 1.3 (1.6) $\\times 10^{33}$ \\tabularnewline \n\\hline\n$p \\rightarrow e^{+} \\omega    $ & 5.4 (6.9)$\\times 10^{33}$  \\tabularnewline\n\\hline\n$p \\rightarrow \\mu^{+} \\omega  $ & 0.78 (1.0)$\\times 10^{34}$  \\tabularnewline\n\\hline\n\\hline\n\\end{tabular}\n\\end{minipage}\n\\begin{minipage}[b][][b]{.45\\linewidth}\n\\begin{tabular}{>{\\raggedright}p{3cm}|>{\\centering}p{3cm}}\n\\hline\nMode  &  $\\tau_{disc}$ $3\\sigma$ [years]   \\tabularnewline\n\\hline\n\\hline\n$n \\rightarrow e^{+} \\pi^{-}$    & 1.3 (1.6)$\\times 10^{34}$   \\tabularnewline\n\\hline\n$n \\rightarrow \\mu^{+} \\pi^{-}$  & 1.1 (1.5)$\\times 10^{34}$  \\tabularnewline\n\\hline\n$n \\rightarrow e^{+} \\rho^{-}$    & 1.1 (1.5)$\\times 10^{33}$   \\tabularnewline\n\\hline\n$n \\rightarrow \\mu^{+} \\rho^{-}$  & 6.2 (8.1)$\\times 10^{32}$  \\tabularnewline\n\\hline\n\\hline\n\\end{tabular}\n\\end{minipage}\n\\end{center}\n\\end{table}\n\n\n\n\n\n\n\\begin{table}[htp]\n\\caption{Summary of Hyper-K's sensitivity to various $| \\Delta (B-L) | = 2 $ and $| \\Delta B | = 2$ proton and dinucleon decay modes after a 5.6\\,Megaton$\\cdot$year exposure compared with existing lifetime limits. \nThe current limits are taken from a 0.273~Mton$\\cdot$year exposure of Super-K.\nLimits on the dinucleon decay modes are reported per $^{16}O$ nucleus, whereas the single nucleon decays are displayed \nas limits per nucleon.\n}\n\\label{tab:dbl_eq_2}\n\\begin{center}\n\\begin{tabular}{l|c|c}\n\\hline\nMode                               & Sensitivity (90$\\%$ CL) [years]  & Current limit [years] \\\\\n\\hline\n$p \\rightarrow e^{+}   \\nu\\nu$       & 10.2 $\\times 10^{32}$             &  1.7  $\\times 10^{32}$  \\\\\n$p \\rightarrow \\mu^{+} \\nu\\nu$       & 10.7 $\\times 10^{32}$             &  2.2  $\\times 10^{32}$  \\\\\n\\hline\n\\hline\n$p \\rightarrow e{+} X        $       & 31.1 $\\times 10^{32}$             &  7.9  $\\times 10^{32}$  \\\\\n\\hline\n$p \\rightarrow \\mu^{+} X     $       & 33.8 $\\times 10^{32}$             &  4.1  $\\times 10^{32}$  \\\\\n\\hline\n$n \\rightarrow \\nu  \\gamma   $       & 23.4 $\\times 10^{32}$             &  5.5  $\\times 10^{32}$  \\\\\n\\hline\n\\hline\n$np \\rightarrow e^{+} \\nu    $       &  6.2 $\\times 10^{32}$             &  2.6  $\\times 10^{32}$  \\\\\n\\hline\n$np \\rightarrow \\mu^{+} \\nu  $       &  4.2 $\\times 10^{32}$             &  2.0  $\\times 10^{32}$  \\\\\n\\hline\n$np \\rightarrow \\tau^{+} \\nu $       &  6.0 $\\times 10^{32}$             &  3.0  $\\times 10^{32}$  \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nIn addition to $(B-L)$ conserving modes several other nucleon channels\nare available.  Unification schemes invoking left-right symmetry\n(c.f.~\\cite{Pati:1974yy}) predict trilepton decays such as\n$p \\rightarrow e^{+} (\\mu^{+}) \\nu\\nu$, which violate $(B-L)$ by two units\n($| \\Delta (B-L) |= 2 $).\nThough there are two particles in the final state that are invisible to Hyper-Kamiokande,\nthe presence of a positron or muon from such decays can, in principal, be detected.\nWhile this type of single charged lepton signature would naturally be subject to large \natmospheric backgrounds, with a sufficiently large decay rate, spectral information can be \nused to separate the two. \nGoing one step farther, then, spectral analysis makes it possible to search for generic decay \nmodes as well, such as $p \\rightarrow e^{+} (\\mu^{+}) X$, where $X$ is an unknown and unseen particle.\n\nThough only single nucleon decays have been considered up until this\npoint, it is worth noting that dinucleon processes, in which a\nneutron-proton (or proton-proton) pair decays into a pair of leptons,\ncan also be studied at Hyper-Kamiokande.\nModes where $\\Delta B =2$ , such as $np \\rightarrow l^{+}\\nu$,\nappear in models with extended Higgs sectors~\\cite{Perez:2013osa} and\nhave connections to Baryogenesis.  Interestingly, these models have the\nadditional property that the single nucleon decay modes are suppressed\nrelative to the dinucleon decays.  This wealth of predictions for\npossible channels further emphasizes the need to search for as many\nnucleon decay signatures as possible in the quest for grand\nunification.  Table~\\ref{tab:dbl_eq_2} lists Hyper-K's expected\nsensitivity to both $| \\Delta (B-L) | =2 $ and $\\Delta B =2$ decays\nfor a 5.6~Mton$\\cdot$year exposure.  While searches for these modes\nare dominated by the atmospheric neutrino background, Hyper-K can be\nexpected to extend existing limits by a factor of three to ten if \nno signal is observed.\n\n\\input{physics-pdecay\/coverage.tex}\n\n\n\\subsection{Solar neutrinos}\n\n\\label{section:solar}\n\nThe solar neutrino measurements are capable of determining the neutrino\noscillation parameters between mass eigenstate $\\nu_1$ and $\\nu_2$ in\nthe equation (\\ref{eq:mixing}).  Figure~\\ref{fig:sol-osc} shows the\nlatest combined results of the allowed neutrino oscillation\nparameters, $\\theta_{12}$ and $\\Delta m^2_{21}$ from all the solar\nneutrino experiments, as well as the reactor neutrino experiment\nKamLAND~\\cite{sol-neutrino2014}. The mixing angle is consistent between\nsolar and reactor neutrinos, while there is about 2$\\sigma$ tension in\n$\\Delta m^2_{21}$.  This mainly comes from the recent result of the solar\nneutrino day-night asymmetry and energy spectrum shape observed in Super-K.\nIn solar neutrino oscillations, regeneration of the electron neutrinos through the MSW\nmatter effect in the Earth is expected.  According to the MSW model,\nthe observed solar neutrino event rate in water Cherenkov detectors in\nthe nighttime is expected to be higher -- by about a few percent in\nthe current solar neutrino oscillation parameter region -- than that\nin the daytime as shown in the Figure~\\ref{fig:sol-osc}.  Super-K\nfound the first indication of this day-night flux asymmetry at the\n3$\\sigma$ level~\\cite{sk4-daynight}, but conclusive evidence is\nexpected in Hyper-K.  If the 2$\\sigma$ tension of $\\Delta m^2_{21}$\nbetween solar ($\\nu_e$) and reactor ($\\bar{\\nu_e}$) neutrinos is a\nreal effect, new physics must be introduced.\n\nIn addition to that, the observation of the upturn in the solar\nneutrino survival probability might be possible. The spectrum upturn is produced by\nthe transition of the survival probability in $\\nu_e$ from the matter\ndominant energy region to the vacuum dominant energy region in the\nsolar neutrino oscillation, and has been observed by the comparison\nbetween $^8$B solar neutrino flux in Super-K and SNO and $^7$Be solar\nneutrino flux in BOREXINO~\\cite{borexino-7be}. However, the precise\nmeasurement of the spectrum shape can distinguish the usual neutrino\noscillation scenario from several exotic models such as non standard\ninteraction~\\cite{sol-nsi}, MaVaN~\\cite{sol-mavan}, and sterile\nneutrino~\\cite{sol-sterile}, for example.  \nDue to the high photo-coverage of 40~\\%, the\nlower energy threshold required to measure the upturn is possible because of better energy\nresolution and reduction of the radio active background.~\\footnote{\nThough the energy of the radioactive events are lower than the energy\nthreshold, they can be misreconstructed such that they contaminate the\nregion above the energy threshold. \nIf the energy resolution is better, the background will be reduced more.\nSee more in Section~\\ref{sec:lowe_bg}.\n}\n\nIn the following sections, the sensitivity of the day-night flux\nasymmetry and spectrum upturn in Hyper-K are described.\n\n\\if 0\nIn solar neutrino oscillations, regeneration of the electron neutrinos\nthrough the Mikheyev-Smirnov-Wolfenstein (MSW) matter effect\n\\cite{Wolfenstein:1977ue,Mikheyev:1985zz,Mikheyev:1986zz} \nin the Earth is expected. \nRegeneration of the solar electron neutrinos in the Earth would constitute \nconcrete evidence of the MSW matter effect, and so it is important \nto experimentally observe this phenomenon.\nHowever, the matter effect acting on solar neutrinos passing through the Earth has not been \ndirectly confirmed yet, since the sensitivities of the current \nsolar neutrino experiments are not sufficient.\nAccording to the MSW model, the observed solar neutrino event rate \nin water Cherenkov detectors in the nighttime is expected to be higher \n-- by about a few percent in the current solar\nneutrino oscillation parameter region --\nthan that in the daytime. \nWe would like to measure this difference in Hyper-Kamiokande. \n\\fi\n\nHyper-K also could be used for variability analyses of the Sun.  For\nexample, the $^8$B solar neutrino flux highly depends on the Sun's\npresent core temperature~\\cite{1996PhRvD..53.4202B}.\nUnlike multiply scattered, random-walking\nphotons or slow-moving helioseismic waves, free streaming solar\nneutrinos are the only available messengers with which to precisely\ninvestigate ongoing conditions in the core region of the Sun.\nHyper-K, with its unprecedented statistical power, could measure the\nsolar neutrino flux over short time periods. Therefore, short time\nvariability of the temperature in the solar core could be monitored by\nthe solar neutrinos in Hyper-K.\n\n\\if 0\nIn order to achieve these precision measurements, background event\nlevels must be sufficiently small.  Here, we have estimated the basic\nperformance of Hyper-K for low energy events assuming some typical\nbackground levels.  In this study, the current analysis tools and the\ndetector simulation for the low energy analysis~\\cite{Abe:2011xx} in\nSuper-K were used.  The dark rate of the PMTs and the water\ntransparency were assumed to be similar to those in the current\nSuper-K detector.  A brief summary of the low energy event\nreconstruction performance in Hyper-K is listed in\nTable~\\ref{tab:performance}.\n\nThe analysis threshold of the total energy of the recoil electrons in\nHyper-K will be 7.0\\,MeV or lower, since a 7.0\\,MeV threshold was\npreviously achieved in the SK-II solar neutrino\nanalysis~\\cite{sk2-solar}.  The current analysis tools will work all\nthe way down to 4.5\\,MeV in Hyper-K with a vertex resolution of 3.0~m.\nNot surprisingly, higher energy events will be reconstructed with even\nbetter vertex resolution.\n\\fi\n\n\\begin{figure}[htb]\n \\begin{center} \\includegraphics[height=9.5cm]{physics-solarnu\/plot_skosc.pdf} \\\\ \\end{center} \\caption{Allowed\n  neutrino oscillation parameter region from all the solar neutrino\n  experiments (green), reactor neutrino from KamLAND (blue) and\n  combined (red) from one to five sigma lines and three sigma filled\n  area.  The star shows the best fit parameter from the solar\n  neutrinos.  The contour of the expected day-night asymmetry with\n  6.5\\,MeV (in kinetic energy) energy threshold is overlaid.}  \\label{fig:sol-osc}\n\\end{figure}\n\n\\subsubsection{Background estimation}\n\\label{section:solar_bg}\nThe major background sources for the $^8$B solar neutrino measurements\nare the radioactive spallation products created by cosmic-ray muons~\\cite{sk-spa} and\nthe radioactive daughter isotopes of ${}^{222}\\mbox{Rn}$ in water.\nThe spallation products is discussed in detail in the paragraph~\\ref{par:spallation_reduction},\nand the rate of spallation which result in relevant backgrounds is 2.7 times higher in Hyper-K compared to Super-K\nbecause of its shallow depth.\n\\if 0\n${}^{222}\\mbox{Rn}$  will be reduced to a similar (or lower) level as that currently\nin the Super-K detector, since Hyper-K will employ a similar water\npurification system and design improvements may well occur over the\nnext several years.  However, the spallation products will be\nincreased in Hyper-K.\n\\fi\nAs the radioactive daughter isotopes, ${}^{222}\\mbox{Rn}$  is an important background source for the spectrum upturn\nmeasurement.  First of all, the water purification system must achieve ${}^{222}\\mbox{Rn}$ levels similar to that achieved at Super-K.\nFurthermore, this background level must be achieved across the full fiducial volume, unlike at Super-K, where only a limited volume can be used for events with less than 5\\,MeV of energy.\nIt is a challenging task but we believe that this should be\npossible by design improvements over the next several years.\nTherefore, the same ${}^{222}\\mbox{Rn}$ background level as Super-K in full fiducial volume is assumed in the following calculation.\n\n\\if 0\n  While\nthe spallation products will be definitely increased in Hyper-K\nbecause of shallow cavern.  In the current design, the cosmic-ray muon\nrate is expected to be increased by a factor of 4.9 $\\pm$ 1.0 in equal\nvolumes, as discussed in Sec.~\\ref{sec:lowe_bg}.\n\nThe spallation products will not simply be increased by the same\nfactor.  This is because high energy cosmic-ray muons tend to produce\nthe spallation products, while the average energy of the cosmic-ray\nmuons at the shallower Hyper-K site is expected to be lower than that\nat the deeper Super-K site; greater overburden means less muons, but\nit also means those that do get through are more energetic.  We have\nestimated the average energies of the cosmic-ray muons to be $\\sim\n258$\\,GeV at the Super-K site and $\\sim 200$\\,GeV at the Hyper-K site.\nConsidering the discussion with FLUKA simulation in\nSec.~\\ref{sec:lowe_bg}, the density of spallation products will be\nincreased by a factor of 4.  We found the remaining spallation\nproducts will be decreased by a factor of 0.75 at the above condition,\ncomparing with current Super-K analysis, after the spallation\nreduction discussed in Sec.~\\ref{sec:lowe_bg}.\nIn summary, the density of the remaining spallation products will be increased \nby a factor of 2.7 in Hyper-K.\n\nIn Super-K, angular information is used to extract the solar neutrino\nsignal events~\\cite{full-solar}. \nWe have estimated the possible effect of the background level in the\nsignal extraction after considering angular information.\n\nIn this study, we used 9.0--9.5\\,MeV Super-K-I data as a reference.\nThe extracted solar neutrino signal events and background events in\nthis energy region over the entire run period\n(0.09~Megaton$\\cdot$years) were 1350 events and 7700 events,\nrespectively.  So, the Signal-to-Noise (S\/N) ratio is 18\\%.  We made\nartificial data samples with reduced S\/N ratios, then applied the\nsignal extraction.  As a result, we found the expected statistical\nerror is almost the square root of 2 $\\sim$ 15 times the number of\nsignal events for 1 $\\sim$ 20 times the Super-K-I background level,\nrespectively.  Table~\\ref{tab:sol-bgtest} shows a summary of the\nexpected statistical errors in a Super-K-I type detector with\nincreased backgrounds, as well as that of Hyper-K.\n\\begin{table}[htb]\n \\caption{Expected statistical uncertainties for 10000 signal events with increased background \n levels. The Super-K-I solar neutrino data sample between 9.0--9.5\\,MeV was used as a reference \n The 3rd column is the Hyper-K factor relative to Super-K given the same observation time.\n  To estimate the 3rd column, the same detector resolution\n  and 0.56~Mton fiducial volume are assumed in Hyper-K. }\n\\begin{center}\n\\begin{tabular}{cccc}\n\\hline \\hline\nBackground level & Stat. err. in SK &  & Stat. err. in HK\\\\\n\\hline\nSK-I BG $\\times 20$ & 3.6\\% &  & $\\times 1\/2.0$ \\\\ \nSK-I BG $\\times 10$ & 2.7\\% &  & $\\times 1\/2.7$ \\\\ \nSK-I BG $\\times 7$  & 2.4\\% &  & $\\times 1\/3.1$ \\\\ \nSK-I BG $\\times 5$  & 2.1\\% &  & $\\times 1\/3.5$ \\\\ \nSK-I BG             & 1.4\\% &  & $\\times 1\/5.2$ \\\\ \n\\hline \\hline\n\\end{tabular}\n\\end{center}\n\\label{tab:sol-bgtest}\n\\end{table}\nOnce the angular distribution is used to extract the solar signal, \nthe statistical error on this signal would be reduced by a factor of 2.0 in Hyper-K, \neven though the background level is increased by a factor of 20, \nfor the same observation time assuming both detectors have identical resolution.\n\nIn summary, Hyper-K will provide higher statistical measurements of\nsolar neutrinos than Super-K, even though there will be more spallation\nbackgrounds.\n\\fi\n\n\\subsubsection{Oscillation studies}\n\nIn order to calculate the neutrino oscillation sensitivity, the signal\nand background rates in each option have to be estimated.  As for the\nday-night asymmetry analysis, the energy threshold is set to 6.5\\,MeV\n(in kinetic energy)\nsince its effect is larger at higher energy region.  In this\nenergy region, only spallation backgrounds should be considered.  The\nremaining spallation background rate in Super-K phase IV (40\\%\nphoto-coverage) has been reduced by a factor three comparing to Super-K\nphase II (20\\% photo-coverage) because of the better energy resolution\nand better vertex resolution.\nFrom this experience in Super-K, the spallation background in Hyper-K\nwill be reduced by a factor three because of the higher photon detection\nefficiency than Super-K.\n\nFigure~\\ref{fig:sol-dn} shows the sensitivity of the day-night asymmetry\nas a function of the observation time.\nThe $\\Delta m^2_{21}$ separation ability between solar neutrino (HK) and\nreactor anti-electron neutrino (KamLAND) is expected 4$\\sim$5$\\sigma$ level in ten years observation, though it depends on the systematic uncertainty.\n\nIn the measurement of the spectrum upturn, the ${}^{222}\\mbox{Rn}$  background is\ncritical because the $^{214}$Bi beta decay events (3.27\\,MeV end point\nenergy) will come above the energy threshold due to the energy\nresolution.  The Hyper-K detector, which has better energy resolution\nbecause of the higher photon detection efficiency than Super-K,\nis strong to reduce such kind of radioactive background.\nFurthermore, the precise energy calibration has to be considered.\nHere, it is assumed that the same background level with full fiducial\nvolume and the same precision of the energy calibration as Super-K are\nachieved in Hyper-K.\nFigure~\\ref{fig:sol-upturn} shows the sensitivity of the spectrum upturn\ndiscovery as a function of the observation time.\nIt is about 3$\\sigma$ level in ten years observation with 4.5\\,MeV energy threshold.\n\n\\begin{figure}[htb]\n \\begin{center}\n  \\includegraphics[height=9.5cm]{physics-solarnu\/plot_dn_year.pdf} \\\\\n \\end{center}\n \\caption{Day-night asymmetry observation sensitivity as a function of observation time. The red line shows the sensitivity from the no asymmetry, while the blue line shows from the asymmetry expected by the reactor neutrino oscillation. The solid line shows that the systematic uncertainty which comes from the remaining background direction is 0.3\\%, while the dotted line shows the 0.1\\% case.}\n \\label{fig:sol-dn}\n\\end{figure}\n\n\\begin{figure}[htb]\n \\begin{center}\n  \\includegraphics[height=9.5cm]{physics-solarnu\/plot_upturn_year.pdf} \\\\\n \\end{center}\n \\caption{Spectrum upturn discovery sensitivity as a function of observation time. The solid line shows that the energy threshold is 4.5MeV, while the dotted line shows the 3.5MeV}\n \\label{fig:sol-upturn}\n\\end{figure}\n\n\\if 0\nIn solar neutrino oscillations, a difference in the solar neutrino\nevent rates during the daytime and the nighttime is expected from the\nMSW effect in the Earth.  This is called the day\/night asymmetry; it\nhas not yet been observed.  In Hyper-K, a precise measurement of the\nday\/night asymmetry will be performed using higher statistics than\nthose available in Super-K.\n\nThe upper plots in Fig.~\\ref{fig:sol-dn1} show the expected day\/night\nasymmetries with different lower energy thresholds. \n\\begin{figure}[htb]\n \\begin{center}\n  \\includegraphics[height=9.5cm]{physics-solarnu\/plot_map3-new.pdf} \\\\\n \\end{center}\n \\caption{Expected day\/night asymmetry in a megaton water Cherenkov\n  detector. 40\\% photo-coverage, 0.5~Megaton$\\cdot$years daytime data \n  and 0.5~Megaton$\\cdot$years nighttime data are assumed. \n  The effect of background events and reduction efficiencies are not considered.   \n  Upper left: expected day\/night asymmetry in the 5.0--20\\,MeV electron \n  total energy region. \n  Upper right: expected day\/night asymmetry in the 8.0--20\\,MeV region.\n  Lower left: expected day\/night asymmetry with uncertainties as a\n  function of the lower energy threshold at \n $(\\tan^2 \\theta_{12}, \\Delta \\rm{m}^2_{21}) = (0.40, 7.9 \\times 10^{-5} {\\rm eV}^2)$. \n The upper energy threshold is 20\\,MeV.\n The meaning of the different colors are defined in the lower right plot.\n Lower right: expected day\/night significance as a function of the\n energy threshold.}\n \\label{fig:sol-dn1}\n\\end{figure}\nThe expected day\/night asymmetry is at about the 1\\% level around the current\nsolar global oscillation parameters. \nIn order to observe the day\/night asymmetry in Hyper-K, \nwe must reduce the up-down systematic uncertainty below that level.\n\nThe expected day\/night asymmetry in the high energy region is larger \nthan that in the low energy region, as shown in the lower left plot in\nFig.~\\ref{fig:sol-dn1}.\nSo, high statistics data in this higher energy region would be desirable. \nWe have studied two typical values of systematic uncertainty, where the one at \n1.3\\% corresponds to the up-down systematic uncertainty of Super-K.\nFrom the lower right plot in Fig.~\\ref{fig:sol-dn1},  \nthe most sensitive lower energy threshold would be 6\\,MeV and 8\\,MeV for the\n0.5\\% and 1.3\\% up-down systematic uncertainties, respectively.\nFigure~\\ref{fig:sol-dn2} shows expected day\/night significance \nas a function of the observation time.\n\\begin{figure}[htb]\n \\begin{center}\n  \\includegraphics[height=6.5cm]{physics-solarnu\/plot_map3_all1.pdf} \\\\\n \\end{center}\n \\caption{Expected day\/night significance as a function of the\n observation time near the solar global oscillation best-fit parameters.\n The Super-K-I value of S\/N is assumed. \n The total electron energy region is 5.0--20\\,MeV.}\n \\label{fig:sol-dn2}\n\\end{figure}\nSince the expected day\/night asymmetry is small, it will be important to\nreduce the systematic uncertainties in order to observe the day\/night\nasymmetry with high precision.\nWe believe that this should be possible, especially if we design and prepare \nthe necessary calibration devices during detector construction.\n\\fi\n\n\\if 0\n\\subsubsection{Time variation}\n\nSolar neutrinos could be used as a direct probe of the nuclear \nreactions taking place in the solar core.\nIn particular, the $^8$B solar neutrino flux has a remarkable $T^{18}$ dependence \naccording to the Standard Solar Model (SSM)~\\cite{bahcall-textbook}. \nHere, $T$ is the solar core temperature, and with such a high-order dependence\nit is possible that even modest changes in the solar core temperature \ncould be amplified into something detectable via measurements of the $^8$B solar neutrino flux.\n\nAssuming the statistical uncertainties estimated in \nSec.~\\ref{section:solar_bg} can be used for Hyper-K,\nthe expected uncertainty on the solar core temperature  \nwhen the background level is increased by a factor of 20 \nwould be the following:\n\\[\n \\frac{\\sigma_T}{T} \n  = \\frac{1}{18} \\frac{\\sigma_N}{N} \n  = \\frac{\\sqrt{15 \\cdot N}}{18 \\cdot N}\n\\]\nHere $N$, $\\sigma_{\\rm T}$, and $\\sigma_{\\rm N}$ are the number of\nobserved $^8$B solar neutrinos, error in $T$, and error in $N$,\nrespectively.  The expected number of observed $^8$B solar neutrinos\nin Hyper-K is 200 events per day above 7.0\\,MeV, as shown in\nTable~\\ref{tab:targets}.  When $N$ is $200$, $\\sigma_T\/T$ will be\n$0.015$.  Therefore, the solar core temperature could be monitored\nwithin a few percent accuracy day by day.  Naturally, by integrating\nover longer periods, more subtle temperature changes - potentially\ndown to the 0.1\\% level - could be monitored.\n\\fi\n\n\\subsubsection{Hep solar neutrino}\n\nHep solar neutrino produced by the $^{3}$He + $p$ fusion reaction \nhas the highest energy in solar neutrinos.\nBut, most of the hep energy spectrum is overlapped with $^8$B solar neutrinos.\nThe expected $^8$B solar neutrino flux is more than 100 times larger than that of hep solar\nneutrino in Standard Solar Model (SSM).  \nSo far, only upper limits were reported from SNO and SK group~\\cite{hep-sno,hep-sk}, \nbut a recent improved analysis of the SNO charged-current data shows hints of \na hep solar neutrino signal~\\cite{hep-sno-2016}, and indicates a higher hep \nflux than the SSM prediction.\n \nThe measurement of the hep solar neutrino could provide new\ninformation on solar physics.  The production regions of the $^8$B and\nhep neutrinos are different in the Sun. \nThe energy production peak of hep neutrinos is located at the outermost \nradius in the solar core region\namong all the solar neutrinos in pp-chain~\\cite{bahcall-textbook}.\nSo, they could be used as a new probe of the solar interior around core region.\nNon-standard solar models, originally motivated by the solar neutrino problem, \npredict the potential enhancement of the hep neutrino flux~\\cite{Bahcall:1988}.\nThis is realized through the mixing of $^{3}$He into the inner core on a time scale \nshorter than the $^{3}$He burning time. Significant mixing is \nalready ruled out by helioseismology, however, the hep neutrino observation\ncan be a sensitive probe of the degree of the mixing in the solar core.\nThere is also the solar abundance problem. $^8$B and hep solar neutrino\nfluxes show different behavior with GS98 and AGSS09 chemical\ncompositions~\\cite{solar-abundance}.\nTheoretical calculation of hep solar neutrinos is a difficult \nchallenge~\\cite{solar-fusion}. The measurement of the hep solar neutrino\nflux will provide a better understanding of SSM.\nHep solar neutrinos could be also used to test non-standard neutrino physics\nin the energy range ($\\sim 18$~MeV)~\\cite{kubodera-hep}. \n\nIn Hyper-K, a high sensitivity measurements of hep solar neutrino flux\nwould be possible, since the detector has a good energy resolution.\nFigure~\\ref{fig:sol-hep} shows the expected solar neutrino fluxes\nin 1.9 Mton year in Hyper-K detector.\n\\begin{figure}[htb]\n \\begin{center}\n  \\includegraphics[height=9.5cm]{physics-solarnu\/plot_b8hep5.pdf} \\\\\n \\end{center}\n \\caption{Expected solar neutrino fluxes with neutrino oscillation \n  in Hyper-K. \n The horizontal axis is the energy threshold in electron total energy \n and the vertical axis is expected event rate in the energy range \n from the threshold up to 25\\,MeV in 10-year observation in Hyper-K.\n BP2004 SSM fluxes are assumed. \n The effect of background events, reduction efficiencies, systematic\n uncertainties are not considered.}\n \\label{fig:sol-hep}\n\\end{figure}\nThe separation between $^8$B and hep solar neutrinos highly depends\non the energy resolution of the detector. \nTable~\\ref{tab:sol-hep} shows a list of expected numbers of solar\nneutrino events in typical energy regions.\n\\begin{table}[htb]\n \\caption{Expected solar neutrino event rates in water Cherenkov detectors. \nThe assumptions are same as Fig.~\\ref{fig:sol-hep}.}\n\\begin{center}\n\\begin{tabular}{ccccc}\n\\hline \\hline\nEnergy resolution & Energy range & $^8$B        & hep          & hep \/ $^8$B \\\\\n                   & [MeV]        & [\/1.9 Mton\/year] & [\/1.9 Mton\/year] &  \\\\\n\\hline\n\\hline\n \n SK-III\/IV &  19.5--25.0 & 0.77 & 3.03 &  3.9 \\\\\n Hyper-K   &  18.0--25.0 & 0.56 & 6.04 & 10.6 \\\\\n\\hline \\hline\n\\end{tabular}\n\\end{center}\n\\label{tab:sol-hep}\n\\end{table}\nHyper-K has a better separation between hep and $^8$B solar neutrinos\ncomparing to SK-III\/IV.\\par\n\nFigure~\\ref{fig:sol-hep-sensitivity} shows an estimation of hep neutrino detection sensitivity.\nA spectrum fit analysis is performed here,\nconsidering the spallation background, detection efficiency and systematic uncertainties of the energy scale and resolution.\nThe statistical error due to remaining spallation background is the dominant source of ambiguity.\nWhen we simply scale the current remaining spallation background level in SK-IV solar analysis, with the cosmic muon rate at Tochibora,\nthe uncertainty of the hep neutrino flux will be $\\sim$60\\% ($\\sim$40\\%) and the non-zero significance will be 1.8$\\sigma$ (2.3$\\sigma$) in ten (twenty) years observation in Hyper-K.\nDue to the higher energy resolution of Hyper-K, there is still chance to improve the sensitivity.\nIf we can reduce the remaining spallation background to the SK-IV level, the uncertainty of hep neutrino flux will be $\\sim$40\\% ($\\sim$30\\%) and non-zero significance will be improved to 2.5$\\sigma$ (3.2$\\sigma$) in ten (twenty) years observation.\nHere the same systematic uncertainties of detector energy scale (0.5~\\%) and resolution (0.6~\\%) as SK-IV are considered.\n\n\\begin{figure}[htb]\n \\begin{center}\n  \\includegraphics[height=6.5cm]{physics-solarnu\/plot_hep_sensitivity.pdf} \\\\\n \\end{center}\n \\caption{\n\t Expected soler hep neutrino sensitivity with 187\\,kt fiducial volume.\n The horizontal axis is the observation time and the vertical axis is non-zero significance of hep neutrino signal expected from a energy spectrum analysis.\n BP2004 SSM fluxes are assumed. \n The effects of remaining spallation background, detection efficiency and systematic uncertainties of the energy scale and resolution estimated from SK-IV data are considered.\n The black solid line shows the expected sensitivity in Tochibora site.\n The red solid and dashed lines show the cases in Mozumi site and no spallation background, respectively.\n }\n \\label{fig:sol-hep-sensitivity}\n\\end{figure}\n\n\n\\subsubsection{Summary}\n\nIn this section, estimates of potential solar neutrino measurements\nare reported.  The solar neutrino analysis is sensitive to the\ndetector resolutions and background levels.  We have estimated\nexpected sensitivities in 10 years of Hyper-K observation based on the\ncurrent Super-K knowledge.\n\nAs a result of its shallower site, the increase of the spallation\nbackground level in Hyper-K will be up to a factor of 2.7 larger as compared\nto Super-K.  However -- due to its much greater size and high energy\nresolution-- the statistical uncertainties on solar\nneutrino measurements would actually be reduced in Hyper-K as compared\nto Super-K on an equal time basis.\n\nThe sensitivity to the difference in neutrino oscillation\nparameters between solar and reactor neutrinos due to the day-night asymmetry\nis estimated to be 4$\\sim$5$\\sigma$.\nThe possibility of spectrum upturn observation is estimated to be at\nthe 3$\\sigma$ level.\n\nThe solar hep neutrino could be measured in \nHyper-K \nwith a few Mton year data at the level of $2\\sim3 \\sigma$.\n\n\\if 0\nIn this section, rough estimates of potential solar neutrino\nmeasurements are reported.  The solar neutrino analysis is sensitive\nto the detector resolutions and background levels.  We have estimated\nthe expected sensitivities based on the current Super-K analysis tools.\n\nAs a result of its shallower site, the increase of the background\nlevel in Hyper-K will be up to a factor of 20 as compared to Super-K.\nHowever -- due to its much greater size -- the statistical\nuncertainties on solar neutrino measurements would actually be reduced\nby a factor of at least two in Hyper-K as compared to Super-K on an\nequal time basis, assuming similar detector resolutions.\n\nThe day\/night asymmetry of the solar neutrino flux -- concrete\nevidence of the matter effect on oscillations -- could be discovered\nand then precisely measured in Hyper-K, given that the detector\nup-down response is understood to better than about 1\\%.  Good\ncalibration tools will be a must for this physics.\n\nHyper-K will provide short time and high precision variability\nanalyses of the solar core activity.  The solar core temperature could\nbe monitored within a few percent accuracy day by day, and to a tenth\nof a percent over the period of several months.\n\nThe solar hep neutrino could be measured in a 1TankHD{} \ndetector with a few Mton year data.\n\\fi\n\n\n\\subsection{Supernova}\\label{sec:supernova}\n\\subsubsection{Supernova burst neutrinos}\\label{sec:supernova-burst}\n\\paragraph{Introduction}\n\nCore-collapse supernova explosions are the last process in the\nevolution of massive ($>8$M$_{\\rm sun}$) stars.  Working their way\nsuccessively through periods of predominantly hydrogen fusion, helium\nfusion, and so on, eventually silicon fusion starts making iron.  Once\nan iron core has formed, no more energy can be released via its fusion\ninto still-heavier elements, and the hydrodynamic balance between\ngravity and stellar burning is finally and catastrophically disrupted.\nThe sudden gravitational collapse of their iron cores --\nwhich will go on to form either a neutron star or a black hole -- is\nthe main source of energy from this type of supernova explosion.  The\nenergy released by a supernova is estimated to be $\\sim 3 \\times\n10^{53}$\\,ergs, making it one of the most energetic phenomena in the\nuniverse.  Since neutrinos interact weakly with matter, almost 99\\% of\nthe released energy from the exploding star is carried out by\nneutrinos.  As a result, the detection of supernova neutrinos gives\ndirect information of energy flow during the explosion.  The neutrino\nemission from a core collapse supernova starts with a short\n($\\sim$10\\,millisecond) burst phase of electron captures ($p +\ne^- \\rightarrow n + \\nu_e$) called the neutronization burst, which\nreleases about $10^{51}$\\,ergs.  Following that, the majority of the\nburst energy is released by an accretion phase ($< \\sim$1\\,second) and\na cooling phase (several seconds) in which all three flavours of\nneutrinos (as well as anti-neutrinos) are created.\\par\nThe observation of a handful (25 in total) of supernova burst\nneutrinos from SN1987a by the Kamiokande, IMB, and Baksan experiments\nproved that the basic scenario of the supernova explosion was correct.\nHowever, more than three decades later the detailed mechanism of\nexplosions is still not known.\nAchieving the necessary conditions\nfor a supernova explosion in computer simulations has been a\nlong-standing challenge.\nRecently, successful supernova explosions in two-dimensional and\nthree-dimensional simulations have been reported, the details of which will be\ndescribed later\\cite{Tamborra:13,SASIBlondin,SASIScheck,Takiwaki:2017tpe}.\nThough several models have produced successful\nsupernova explosions in simulations,\ndetecting supernova neutrinos will provide further input to improve the physics accuracy of the models.\nHyper-K can detect neutrinos with energy down to $\\sim$3\\,MeV and can point the supernova, due to its event-by-event directional sensitivity.\nBecause the supernova neutrinos are detected as a burst in a short time period, we can neglect the low energy radioactive backgrounds.\nThe localization in time also makes it possible to utilize most of full inner volume for our analysis, $i.e.$ 220\\,kt for each tanks.\nCompared with the current or planned experiments, $e.g.$ Super-K, the ice Cherenkov detector IceCube\/PINGU~\\cite{Aartsen:2014oha} and large liquid Ar TPC detectors like DUNE,\nHyper-K has several advantages for the supernova measurement.\nThe first advantage is the large volume and statistics.  \nHyper-K will have a FV of 8\\,times to 16\\,times larger than the Super-K detector, resulting in a commensurate increase in the number of detected supernova neutrinos and sensitivity to supernovae occurring in nearby galaxies.\nLikewise, Hyper-K will be significantly larger than DUNE\nwith mass in the tens of kt scale.\nFurthermore, Hyper-K primarily detects anti-electron-neutrino from the\nsupernova explosion using the inverse beta decay reaction\n($\\bar{\\nu}_e + p \\rightarrow e^+ + n$), unlike LArTPC detectors which\nprimarily detect electron neutrinos.\nLarge neutrino telescopes like IceCube\/PINGU has capability of huge\nstatistics detection, however, they would detect only single PMT hits from such low energy neutrino events, allowing them to separate supernova neutrinos from their dark noise only on a statistical basis.\nIn contrast, every single event will be reconstructable with Hyper-K down to an energy analysis threshold of $\\sim$3\\,MeV.\nOur precise event-by-event measurement will be essential for the comprehensive study of supernova neutrinos, especially with the detailed and time-dependent energy spectrum.\nWith these advantages, Hyper-K is able to perform unique measurements to reveal the mechanism of supernova explosions.\\\\\n\n\\paragraph{Expected observation in Hyper-Kamiokande}\n\nIn order to correctly estimate the expected number of neutrino events detected\nin Hyper-K, we must consider the neutrino oscillation due to the MSW matter\neffect through the stellar medium.\nThe flux of each neutrino type emitted from a supernova is related to the originally produced fluxes ($F^0_{\\nu_e}$, $F^0_{\\bar{\\nu}_e}$ and $F^0_{\\nu_x}$, where $\\nu_x$ is $\\nu_{\\mu,\\tau}$ and $\\bar{\\nu}_{\\mu,\\tau}$) by the following formulas~\\cite{Dighe:1999id, Fogli:2004ff} :\n\n\\noindent\nFor normal hierarchy,\n\\begin{eqnarray}\nF_{\\bar{\\nu}_e}  &\\simeq&  \\cos^2\\theta_{12} F^0_{\\bar{\\nu}_e} + \\sin^2\\theta_{12}F^0_{\\nu_x}\\ , \\nonumber \\\\\nF_{\\nu_e} &\\simeq& \\sin^2\\theta_{12} P_{H} F^0_{\\nu_e} + (1-\\sin^2\\theta_{12} P_{H}) F^0_{\\nu_x}, \\nonumber \\\\\nF_{\\nu_\\mu} + F_{\\nu_\\tau} &\\simeq&  (1-\\sin^2\\theta_{12} P_{H}) F^0_{\\nu_e} + (1 + \\sin^2\\theta_{12} P_{H}) F^0_{\\nu_x}, \\nonumber \\\\\nF_{\\bar{\\nu}_\\mu} + F_{\\bar{\\nu}_\\tau}  &\\simeq&  (1-\\cos^2\\theta_{12}) F^0_{\\bar{\\nu}_e} + (1 + \\cos^2\\theta_{12}) F^0_{\\nu_x} , \\nonumber\n\\end{eqnarray}\n\n\\noindent\nand, for inverted hierarchy,\n\\begin{eqnarray}\nF_{\\bar{\\nu}_e} &\\simeq& \\cos^2\\theta_{12} P_{H} F^0_{\\bar{\\nu}_e} + (1-\\cos^2\\theta_{12} P_{H}) F^0_{\\nu_x}\\ , \\nonumber \\\\\nF_{\\nu_e} &\\simeq& \\sin^2\\theta_{12} F^0_{\\nu_e}+ \\cos^2\\theta_{12}F^0_{\\nu_x} , \\nonumber \\\\\nF_{\\nu_\\mu} + F_{\\nu_\\tau} &\\simeq& (1-\\sin^2\\theta_{12}) F^0_{\\nu_e} + (1 + \\sin^2\\theta_{12}) F^0_{\\nu_x} , \\nonumber \\\\\nF_{\\bar{\\nu}_\\mu} + F_{\\bar{\\nu}_\\tau}  &\\simeq&  (1-\\cos^2\\theta_{12}P_H) F^0_{\\bar{\\nu}_e} + (1 + \\sin^2\\theta_{12} P_H) F^0_{\\nu_x} , \\nonumber\n\\end{eqnarray}\n\n\\noindent\nwhere $P_H$ is the crossing probability through the matter resonant\nlayer corresponding to $\\Delta m^2_{32}$. $P_H = 0$ ($P_H = 1$) for\nadiabatic (non-adiabatic) transition.\nRecent measurement of $\\theta_{13}$ indicates the adiabatic transition ($P_H=0$) for the matter transition in the supernova envelope.\nThe supernova neutrino spectrum is\naffected not only by stellar matter but also by other neutrinos and\nanti-neutrinos at the high density core (so-called collective\neffects).  These collective effects, which swap the $\\nu_{e}$ and\n$\\bar{\\nu}_e$ spectra with those of $\\nu_x$ in certain energy\nintervals bounded by sharp spectral splits, were first discussed\nin~\\cite{Duan:2006an,Duan:2006jv}.  This has become an active field of\nstudy whose recent investigations include taking into account the\npossibility of multiple splits~\\cite{Dasgupta:2009dd}, computation with\nthree neutrino flavors~\\cite{Friedland:2010sc}, and utilizing the full\nmulti-angle framework~\\cite{Duan:2010bf}.\nSo, in the following description of the performance of the\nHyper-Kamiokande detector, three cases are considered in order to\nfully cover the possible variation of expectations: (1)\nno~oscillations, (2) normal~hierarchy (N.H.) with $P_H=0$, and (3)\ninverted~hierarchy (I.H.) with $P_H=0$.  The process depends critically on\n$\\theta_{12}$; in what follows we assume $\\sin^2\\theta_{12}=0.31$.\nConcerning the neutrino fluxes and energy spectra at the production\nsite, we used results obtained by the Livermore\nsimulation~\\cite{Totani:1997vj}.\n\nFigure~\\ref{fig:sn-rate} shows time profiles for various interactions\nexpected at the Hyper-Kamiokande detector for a supernova at a\ndistance of 10\\,kiloparsecs~(kpc).  This distance is a bit farther than\nthe center of the Milky Way galaxy at 8.5\\,kpc; it is chosen as being\nrepresentative of what we might expect since a volume with a radius of\n10\\,kpc centered at Earth includes about half the stars in the galaxy.\n\\begin{figure}[tbp]\n  \\begin{center}\n    \\includegraphics[width=16cm]{physics-supernova\/pdf_1tank\/eventrate_persec_hk_1tank-crop_upto_100MeV.pdf}\n  \\end{center}\n\\vspace{-1cm}\n\\caption{Expected time profile of a supernova at 10\\,kpc with 1 tank. \nLeft, center, and right figures show profiles for no oscillation, normal \nhierarchy, and inverted hierarchy, respectively.\nBlack, red, purple, and light blue curves show event rates for\ninteractions of inverse beta decay ($\\bar{\\nu}_e + p \\rightarrow e^+ + n$), \n$\\nu e$-scattering ($\\nu + e^- \\rightarrow \\nu + e^-$), \n$\\nu_e~^{16}$O CC ($\\nu_e + {\\rm ^{16}O} \\rightarrow e^- + {\\rm^{16}F^{(*)}}$),\nand \n$\\bar{\\nu}_e~^{16}$O CC\n($\\bar{\\nu}_e + {\\rm ^{16}O} \\rightarrow e^+ + {\\rm ^{16}N^{(*)}}$), respectively.\nThe numbers in parentheses are integrated\nnumber of events over the burst. \nThe fluxes and energy spectra are from the Livermore \nsimulation~\\cite{Totani:1997vj}\n  \\label{fig:sn-rate}\n  }\n\\end{figure}\n\\begin{figure}[tbp]\n  \\begin{center}\n    \\includegraphics[width=8cm]{physics-supernova\/pdf_1tank\/sn-neutronization_1tank.pdf}\n  \\end{center}\n\\vspace{-1cm}\n  \\caption{Expected event rate at the time of neutronization burst for\n\t  a supernova at 10\\,kpc with 1 tank.  Red and green\n\t  show event rates for $\\nu e$-scattering events originated with neutronization neutrino and inverse beta\n\t  events, respectively.  Solid, dotted, and dashed curved\n\t  indicate the neutrino oscillation scenarios of no\n\t  oscillation, N.H., and I.H.,\n\t  respectively.\n  \\label{fig:sn-neutronization}\n  }\n\\end{figure}\n\\begin{figure}[tbp]\n  \\begin{center}\n    \\includegraphics[width=8cm]{physics-supernova\/pdf_1tank\/all_spectrum_hk_osc_resol.pdf}\n  \\end{center}\n\\vspace{-1cm}\n  \\caption{Total energy spectrum for each interaction\n\t  for a supernova at 10\\,kpc with 1 tank. \nBlack, red, purple, and light blue curves show event rates for\ninteractions of inverse beta decay, $\\nu e$-scattering, \n$\\nu_e+^{16}$O CC, and $\\bar{\\nu}_e+^{16}$O CC, respectively.\nSolid, dashed, and dotted curves correspond to no oscillation, N.H., and\nI.H., respectively.\n  \\label{fig:sn-spec}\n}\n\\end{figure}\nThe three graphs in the figure show the cases of no oscillation, normal hierarchy and inverted hierarchy, respectively.\nColored curves in the figure show event rates for inverse beta decay\n($\\bar{\\nu}_e + p \\rightarrow e^+ + n$), \n$\\nu e$-scattering($\\nu + e^- \\rightarrow \\nu + e^-$), \n$\\nu_e+^{16}$O CC($\\nu_e + {\\rm ^{16}O} \\rightarrow e^- + {\\rm ^{16}F^{(*)}}$),\nand \n$\\bar{\\nu}_e+^{16}$O CC\n($\\bar{\\nu}_e + {\\rm ^{16}O} \\rightarrow e^+ + {\\rm ^{16}N^{(*)}}$).\nThe burst time period is about 10\\,s and the peak event rate of inverse beta decay events reaches about 50\\,kHz at 10 kpc.\nThe DAQ and its buffering system of Hyper-K will be designed to accept the broad range of rates, for a galactic SN closer than 10 kpc.\nA sharp timing spike is expected for $\\nu e$-scattering events at the time of neutronization.\nFig.~\\ref{fig:sn-neutronization} shows the expanded plot around the neutronization burst peak region.\nWe expect $\\sim$9, $\\sim$23 and $\\sim$55 $\\nu e$-scattering events in this region for a supernova at 10\\,kpc, for N.H., I.H., and no oscillation respectively.\nAlthough the number of inverse beta events is $\\sim$100 (N.H.), $\\sim$210 (I.H.), and $\\sim$60 (no oscillation) in the 10\\,ms bin of the neutronization burst, \nthe number of events in the direction of the supernova is typically 1\/10 of the total events. \nSo, the ratio of signal events ($\\nu e$-scattering) to other events (inverse beta) is expected to be about 9\/10~(N.H.),\n23\/21~(I.H.) and 55\/6~(no oscillation).\n\n\nThe energy distributions of each interaction are shown in Fig.~\\ref{fig:sn-spec}, where the energy is the electron-equivalent total energy measured by a Cherenkov detector.\nThe energy spectrum of $\\bar{\\nu}_e$ can be extracted from the distribution.\n\nFigure~\\ref{fig:sn-evtvsdist} shows the expected number of supernova neutrino events at Hyper-K versus the distance to a supernova.\n\\begin{figure}[htbp]\n  \\begin{center}\n    \\includegraphics[width=8cm]{physics-supernova\/pdf_1tank\/evtvsdist-hk-oscrange-1tank-upto-100MeV-crop.pdf}\n  \\end{center}\n\\vspace{-1cm}\n  \\caption{Expected number of supernova burst events for each\n\t  interaction as a function of the distance to a supernova with 1 tank.\nThe band of each line shows the possible variation due to the assumption\nof neutrino oscillations.\n   \\label{fig:sn-evtvsdist}\n}\n\\end{figure}\nAt the Hyper-Kamiokande detector, we expect to see about 50,000 to 75,000 inverse beta decay events, 3,400 to 3,600 $\\nu e$-scattering events, 80 to 7,900 $\\nu_e+^{16}$O CC events, and 660 to 5,900 $\\bar{\\nu}_e+~^{16}$O CC events, in total 54,000 to 90,000 events, for a 10\\,kpc supernova.\nThe range of each of these numbers covers possible variations due to the neutrino oscillation scenario (no oscillation, N.H., or I.H.).\nEven for a supernova at M31 (Andromeda Galaxy), about 10 to 16 events are expected at Hyper-K.\nIn the case of the Large Magellanic Cloud (LMC) where SN1987a was located, about 2,200 to 3,600 events are expected.\n\nThe observation of supernova burst neutrino and the directional\ninformation can provide an early warning for electromagnetic observation\nexperiments, e.g. optical and x-ray telescopes. \nFigure~\\ref{fig:sn-cossn} shows expected angular\ndistributions with respect to the direction of the supernova for four\nvisible energy ranges.  The inverse beta decay events have a nearly\nisotropic angular distribution.  On the other hand, $\\nu e$-scattering\nevents have a strong peak in the direction coming from the supernova.\nSince the visible energy of $\\nu e$-scattering events are lower than\nthe inverse beta decay events, the angular distributions for lower energy\nevents show more enhanced peaks.  The direction of a supernova at\n10\\,kpc can be reconstructed with an accuracy of about 1 to 1.3 degrees with Hyper-K,\nassuming the performance of event direction reconstruction similar to Super-K~\\cite{Abe:2016waf}.\nThis pointing accuracy will be precise enough for the multi-messenger measurement of supernova at the center of our galaxy,\nwith the world's largest class telescopes, $i.e.$ Subaru HSC and future LSST telescope~\\cite{Nakamura:2016kkl}.\\\\\n\n\\begin{figure}[tbp]\n  \\begin{center}\n    \\includegraphics[width=10cm]{physics-supernova\/pdf_1tank\/sn-cossn_1tank.pdf}\n  \\end{center}\n\\vspace{-1cm}\n  \\caption{Angular distributions of a simulation of\n\t  a 10\\,kpc supernova with 1 tank. The plots show a visible energy range of \n5-10\\,MeV (left-top), \n10-20\\,MeV (right-top), 20-30\\,MeV (left-bottom), and 30-40\\,MeV (right-bottom).\nThe black dotted line and the red solid histogram (above the black dotted line)\nare fitted contributions of inverse beta decay and $\\nu e$-scattering events.\nConcerning the neutrino oscillation scenario, the $no~oscillation$ case is shown\nhere.\n  \\label{fig:sn-cossn}\n}\n\\end{figure}\n\n\\paragraph{Physics impacts}\n\nThe shape of the rising time of supernova neutrino flux and energy strongly depends on the model.\nFigure~\\ref{fig:sn-model} shows inverse beta decay event rates and mean $\\bar{\\nu}_e$ energy distributions predicted by various models\n\\cite{Totani:1997vj,Takiwaki:2013cqa,Tamborra:2014hga,Dolence:2014rwa,Pan:2015sga,Nakazato:2015rya,Bruenn:2014qea} for the first 0.3\\,sec after the onset of a burst.\nThe statistical error is much smaller than the difference between the models, and so Hyper-K should give crucial data for comparing model predictions.\n\\begin{figure}[tbp]\n  \\begin{center}\n    \\includegraphics[width=7cm]{physics-supernova\/pdf_1tank\/model_obsevents_nuebar_2016-crop.pdf}\n    \\includegraphics[width=7cm]{physics-supernova\/pdf_1tank\/model_obsenergy_nuebar_2016-crop.pdf}\n  \\end{center}\n\\vspace{-1cm}\n  \\caption{\n\t  Time profiles of the observed inverse beta decay event rate (left) and mean energy of these events, \npredicted by supernova simulations~\\cite{Totani:1997vj,Takiwaki:2013cqa,Tamborra:2014hga,Dolence:2014rwa,Pan:2015sga,Nakazato:2015rya,Bruenn:2014qea} for the first 0.3\\,seconds after the onset \nof a 10\\,kpc distant burst with Hyper-K 1 tank.\n  \\label{fig:sn-model} \n}\n\\end{figure}\nThe left plot in Fig.~\\ref{fig:sn-model} shows that about 150-500\nevents are expected in the first 20 millisecond bin.  This means that\nthe onset time can be determined with an accuracy of about 1 ms.  This\nis precise enough to allow examination of the infall of the core in\nconjunction with the signals of neutronization as well as\npossible data from future gravitational wave detectors.\nOur measurement will also provide an opportunity to observe black hole formation directly, as a sharp drop of the neutrino flux~\\cite{Sekiguchi:2010ja}.\n\nWe can use the sharp rise of the burst to make a measurement of\nthe absolute mass of neutrinos.  Because of the finite mass of\nneutrinos, their arrival times will depend on their energies.  This\nrelation is expressed as \\\\\n\\begin{eqnarray}\n\t\\Delta t = 5.15~{\\rm msec} (\\frac{D}{10~{\\rm kpc}})\n\t(\\frac{m}{1~{\\rm eV}})^2 (\\frac{E_\\nu}{10~{\\rm MeV}}) ^{-2}\n\\end{eqnarray}\nwhere $\\Delta t$ is the time delay with respect to that assuming zero\nneutrino mass, $D$ is the distance to the supernova, $m$ is the\nabsolute mass of a neutrino, and $E_\\nu$ is the neutrino energy.\nTotani~\\cite{Totani:1998nf} discussed Super-Kamiokande's sensitivity\nto neutrino mass using the energy dependence of the rise time; scaling\nthese results to the much larger statistics provided by Hyper-K, we\nexpect a sensitivity of 0.5 to 1.3\\,eV for the absolute neutrino\nmass~\\cite{Totani:2005pv}.  Note that this measurement of the absolute\nneutrino mass does not depend on whether the neutrino is a Dirac or\nMajorana particle.\n\nHyper-K can also statistically extract an energy distribution of\n$\\nu_e + \\nu_X (X = \\mu, \\tau)$ events using the angular distributions\nin much the same way as solar neutrino signals are separated from\nbackground in Super-K.  Although the effect of neutrino oscillations\nmust be taken into account, the $\\nu_e + \\nu_X$ spectrum gives another\nhandle on the temperature of neutrinos.\nHyper-K will be able to evaluate the temperature\ndifference between $\\bar{\\nu}_e$ and $\\nu_e + \\nu_X$.  This would be a\nvaluable input to model builders. For example, the prediction from\nmany of the \nmodels that the energy of $\\nu_e$ is less than $\\nu_X$ can be confirmed.\nThe temperature is also critical for the nucleosynthesis via supernova explosion~\\cite{2016inpc.confE.249H}.\n\nFrom recent computer simulation studies, new characteristic\nmodulations of the supernova neutrino flux are proposed.\nThese modulations are due to the dynamic motions in the supernovae such as convection.\nThe stall of shock wave after core bounce has been an issue in supernova computer simulations, which was not able to achieve successful explosions.\nThese dynamic motions enable the inner materials to be heated more efficiently by the neutrinos from collapsed core, and realize the shock wave revival.\nSuch modulations can be detected as a variance of the neutrino event rate in Hyper-K.\nIt will be the clear evidence that neutrino is the driver of supernova burst process.\nOne source of such modulation is the Standing Accretion Shock\nInstability (SASI)~\\cite{Tamborra:13,SASIBlondin,SASIScheck}.\nFigure~\\ref{fig:sn_tamborra_sasi} shows the detection rate modulation\nin Hyper-K, induced by SASI.\nThe modulation can be observed as a variance of $\\sim$10\\% of the number of supernova events in Hyper-K detector\n\\cite{Tamborra:13}.\nThis flux variance caused by SASI has a characteristic peak in the frequency space.\nWhen we assume a 3\\% flux modulation on the total supernova flux,\nthough the amount of the flux modulation depends on several variables, e.g. progenitor mass or equation of states,\nit is possible to see the existence of SASI for the supernovae within about 15\\,kpc distance.\\cite{Migenda:16}\nUnder this assumption of a 3\\% flux modulation, we will have chances to prove SASI effects for $\\sim$90\\% of galactic supernovae with Hyper-K, compared with only $\\sim$15\\% with Super-K.\n\\\\\nAnother source of modulation is the rotation of supernova~\\cite{Takiwaki2014,Takiwaki:2017tpe}.\nThe size of variation depends on the angle between the direction of\nearth and the rotational axis of supernova.  When the supernova\nrotational axis is orthogonal with the direction to the earth, the\ndetectable variance will be the maximum. In that case, Hyper-K will\ndetect a variance of $\\sim$50\\% in the number of observed signals\nas shown in Figure~\\ref{fig:sn_tamborra_sasi}.\nThe observation of these modulation with Hyper-K\nwill be a good test of such simulations and also provide important\ninformation for understanding the dynamics in supernovae.\n\n\\if0\n\\begin{figure}[tbp]\n  \\begin{center}\n    \\includegraphics[width=8cm]{physics-supernova\/sn_sasi_tamborra.pdf}\n  \\end{center}\n\\vspace{-1cm}\n  \\caption{\n\t  Detection rate modulation induced by SASI in 5\\,ms bins (0.56 Mt).\n\t  The SN progenitor mass is 27 solar mass.\n\t  The direction to the detector is chosen for strong signal modulation.\n\t  This figure is adopted from~\\cite{Tamborra:13}.\n  \\label{fig:sn_tamborra_sasi}\n  }\n\\end{figure}\n\\fi\n\\begin{figure}[tbp]\n  \\begin{center}\n    \\includegraphics[width=8cm]{physics-supernova\/pdf_1tank\/SASI_2msBin_220kt.pdf}\n  \\end{center}\n\\vspace{-1cm}\n  \\caption{\n\t  Detection rate modulation induced by SASI in Hyper-K 1 tank.\n\t  Red line shows the theoretical event rate estimation for the inverse beta decay reaction.\n\t  Gray line shows a simulated event rate taking into account statistical fluctuation.\n\t  The SN progenitor mass is 27 solar mass.\n\t  The direction to the detector is chosen for strong signal modulation.\n\t  This neutrino flux is adopted from~\\cite{Tamborra:13}.\n  \\label{fig:sn_tamborra_sasi}\n  }\n\\end{figure}\n\n\\begin{figure}[tbp]\n  \\begin{center}\n    \\includegraphics[width=10cm]{physics-supernova\/pdf_1tank\/time-Rev-HK1tank.pdf}\n  \\end{center}\n\\vspace{-1cm}\n  \\caption{\n\t  Detection rate modulation produced by a rotating SN model in Hyper-K 1 tank.\n\t  The SN progenitor mass is 27 solar mass.\n\t  The supernova rotational axis is orthogonal (red) and parallel (green) with the direction to the earth.\n\t  This figure is adopted from~\\cite{Takiwaki:2017tpe}.\n  \\label{fig:sn_tatewaki_rotation}\n  }\n\\end{figure}\n\nNeutrino oscillations could be studied using supernova neutrino events.\nThere are many papers which discuss the possibility of extracting signatures of neutrino oscillations free from uncertainties of supernova models \\cite{Totani:1997vj,Takiwaki:2013cqa,Tamborra:2014hga,Dolence:2014rwa,Pan:2015sga,Nakazato:2015rya,Bruenn:2014qea}.\nOne big advantage of supernova neutrinos over other neutrino sources (solar, atmospheric, accelerator neutrinos) is that they inevitably pass through very high density matter on their way to the detector.\nThis gives a sizeable effect in the time variation of the energy spectrum~\\cite{Schirato:2002tg, Fogli:2004ff, Tomas:2004gr}.\nThough the combination of MSW stellar matter effects and collective effects makes the prediction quite difficult~\\cite{Duan:2006an,Duan:2006jv,Dasgupta:2009dd,Friedland:2010sc,Duan:2010bf},\nwe will still have opportunities to determine the neutrino mass hierarchy from the supernova burst.\nThe first chance is the neutronization burst, where mostly pure $\\nu_e$ will be emitted from the proto-neutron star.\nSince, the collective effect through $\\nu_e\\bar{\\nu}_e\\rightarrow\\nu_X\\bar{\\nu}_X$ interaction can be negligible.\nThe multi-angle effect can also be ignored, because these neutrinos are emitted only from the very center of the supernova.\nThe flux is well predicted and hardly affected by the physics\nmodelling of the EOS or the progenitor mass~\\cite{Kachelriess:2004ds, Mirizzi:2015eza}.\nThe number of event will be about 50\\% larger in IH case comparing to NH, after 20\\,ms from the core bounce.\nIn the succeeding accretion phase, we will have another chance by observing the rise-time of neutrino event rate.\nThe mixing of $\\bar{\\nu}_X$ to $\\bar{\\nu}_e$, will result in a 100\\,ms faster rise time for the inverted hierarchy compared to the normal hierarchy case~\\cite{Serpico:2011ir}.\nWe will have fair chance to investigate it for a supernova at the galactic center, see Fig.~\\ref{fig:sn-model}.\n\n\\if0\nAn example is shown in Figure~\\ref{fig:sn-nuosc}~\\cite{Fogli:2004ff}.\nThe propagation of the supernova shock wave causes time variations in\nthe matter density profile through which the neutrinos must travel.\nBecause of neutrino conversion by matter, there may be a bump in the\ntime variation of the inverse beta event rate for a particular energy\nrange ($i.e.$, 45$\\pm$5\\,MeV as shown in\nFig.~\\ref{fig:sn-nuosc}(right)) while no change is observed in the\nevent rate near the spectrum peak ($i.e.$, 20$\\pm$5\\,MeV as shown in\nFig.~\\ref{fig:sn-nuosc}(left)).  This effect is observed only in the\ncase of inverted mass hierarchy; this is one way in which the mass\nhierarchy could be determined by a supernova burst.\n\\begin{figure}[tbp]\n  \\begin{center}\n    \\includegraphics[width=16cm]{physics-supernova\/sn-fp2-Fogli0412046.pdf}\n  \\end{center}\n\\vspace{-1cm}\n  \\caption{Time variation of the neutrino event rate affected by\n\t  neutrino conversion by matter due to shock wave propagation\n\t  (reproduced from~\\cite{Fogli:2004ff}).\n\t  Left (right) plot shows the time variation of inverse beta\n\t  events for the energy range of 20$\\pm$5\\,MeV (45$\\pm$5\\,MeV).\n\t  Solid black, dashed red, and blue dotted histograms show the\n\t  event rates for I.H. with shock wave propagation, I.H. with\n\t  static matter density profile, and N.H., respectively.  It\n\t  has been assumed that\n\t  $\\sin^2\\theta_{13}=10^{-2}$.  \\label{fig:sn-nuosc}}\n\\end{figure}\n\\fi\n\nIn Hyper-K, it could be possible to detect burst neutrinos from\nsupernovae in nearby galaxies.  As described above, we expect to\nobserve a very large number of neutrino events from a galactic\nsupernova.  However, galactic supernovae are expected to happen once\nper 30\\,-\\,50 years.  So, we cannot count on seeing many galactic\nsupernova bursts.  In order to examine a variety of supernova bursts,\nsupernovae from nearby galaxies are useful even though the expected\nnumber of detected events from any single such burst is small.\nFurthermore, the merged energy spectrum from these supernovae will be highly useful for understanding that of supernova relic neutrinos (see next sub-section) for the absence of red-shift.\nThe supernova events from nearby galaxies provide a reference energy spectrum for this purpose.\nThe supernovae rate in nearby galaxies was discussed in~\\cite{Ando:2005ka} and a figure from the\npaper is shown in Fig.~\\ref{fig:sn-nearby}.\nIt shows the cumulative supernova rate versus distance and indicates that if Hyper-K can see signals out to 4\\,Mpc then we could expect a supernova about every three years.\nIt should be noted that recent astronomical observations indicate about 3 times higher nearby supernova rate~\\cite{Horiuchi:2013}, compared to the conservative calculation.\nIt is also valuable to mention that two strange supernovae have been found at $\\sim$2\\,Mpc distance in the past 11 years observation, which are called dim supernovae~\\cite{Horiuchi:2013}.\nThe detections of supernova neutrinos from these dim supernovae will prove their explosion mechanism is core-collapse.\nFigure~\\ref{fig:sn-nearbyprob} shows detection probability versus distance for the Hyper-K detector 1 tank (left) and 2 tanks configuration (right).\nIn this estimate, we required the neutrino energy be greater than 10\nMeV in order to reduce background.\nRequiring the number of neutrino events to be more than or or equal to two\\,(one), the detection probability is 27 to 48\\% (64 to 80\\%) with 1 tank, for a supernova at 2\\,Mpc.\nThe probability will be 3 to 6\\% (22 to 33\\%) for the supernovae at 4\\,Mpc.\nWith 2 tank configuration and 375 kt fiducial volume, the detection probability will be increased to 10 to 18\\% (40 to 57\\%) for a supernova at 4\\,Mpc.\nIf we can use a tight timing coincidence with other types of supernova sensors (e.g. future gravitational wave detectors), we should be able to identify even single supernova neutrinos.\nWe expect to observe 2.4 to 4.6 or 0.6 to 1.4 supernovae with and without dim supernovae within 10\\,Mpc respectively, during 20 years of Hyper-K one tank operation.\nHere we required two or more neutrino events for each supernovae, and referred to the nearby galactic supernova rate given by CCSNe counting in ref.~\\cite{Horiuchi:2013}.\nThe number of observations can be increased to 5.0 to 8.2 and 1.6 to 3.3 supernovae with and without dim supernova respectively, with Hyper-K staging two tank scenario.\n\\begin{figure}[tbp]\n  \\begin{center}\n    \\includegraphics[width=7cm]{physics-supernova\/sn-nearbyrate.pdf}\n  \\end{center}\n\\vspace{-1cm}\n  \\caption{Cumulative calculated supernova rate versus distance for \n\t  supernovae in nearby galaxies. The dashed line is core-collapse supernova rate expectation, using the $z = 0$ limit of star formation rate measured by GALEX.  The figure is reproduced from ref.~\\cite{Ando:2005ka}.\n  \\label{fig:sn-nearby}\n}\n\\end{figure}\n\n\\begin{figure}[tbp]\n  \\begin{center}\n    \\includegraphics[width=7cm]{physics-supernova\/pdf_1tank\/sn-nearbyprob_1tank-crop.pdf}\n\\hspace{3mm}\n    \\includegraphics[width=7cm]{physics-supernova\/pdf_2tank\/sn-nearbyprob_2tank-crop.pdf}\n  \\end{center}\n\\vspace{-1cm}\n  \\caption{(Left) Detection probability of supernova neutrinos versus distance at Hyper-K assuming a 187\\,kiloton fiducial volume and 10\\,MeV threshold for this analysis.\nBlack, green, and blue curves show the detection efficiency resulting in requiring more than or equal to one, two, and three events per burst, respectively.\nSolid, dotted, and dashed curves are for neutrino oscillation scenarios of no oscillation, N.H., and I.H., respectively.\n\t(Right) Detection probability of supernova neutrinos with Hyper-K 2 tanks.\n  \\label{fig:sn-nearbyprob}\n}\n\\end{figure}\n\n\\paragraph{Summary}\n\nThe expected number of supernova neutrino events in the Hyper-Kamiokande detector is summarized in Table~\\ref{tab:phys_supernova_summary}.\nThese events will be enough to provide detailed information about the time profile and the energy spectrum for inspecting supernova explosion mechanism, including black hole formation.\nThey will also provide an opportunity for further physics topics, $i.e.$ the neutrino mass, the mass hierarchy and the neutrino oscillations in the supernova.\nPhysics beyond the standard model also can be tested, $e.g.$\nsterile neutrinos~\\cite{Hidaka:2006sg, Loveridge:2003fy, Hidaka:2007se, Tamborra:2011is},\nthe neutrino magnetic moment~\\cite{Lattimer:1988mf, Barbieri:1988nh}\nthe neutrino non-standard interaction~\\cite{EstebanPretel:2007yu, Ohlsson:2012kf} and\nthe axion~\\cite{Turner:1987by, Burrows:1988ah, Keil:1996ju, Raffelt:2006cw, Fischer:2016cyd}.\nThe better energy resolution from high photo-coverage will help these analyses.\nThe lower energy threshold will also help to understand the whole picture of supernova explosion.\nThe detection of supernovae in nearby galaxies will be possible even with a single tank, though the number of events will be small.\nThe energy spectrum of these events will still be useful to understand the nature of the supernova explosion.\n\\if0\nAxion \\cite{Turner:1987by, Burrows:1988ah, Keil:1996ju, Raffelt:2006cw, Fischer:2016cyd}\nSterile \\cite{Hidaka:2006sg, Loveridge:2003fy, Hidaka:2007se, Tamborra:2011is}\nNSI \\cite{EstebanPretel:2007yu, Ohlsson:2012kf}\nMagnetic Moment \\cite{Lattimer:1988mf, Barbieri:1988nh}\n\\fi\n\n\\if0\n\\begin{table}[t]\n\t\t\\caption{Summary table of expected supernova neutrino events in the Hyper-Kamiokande detector (E$_\\nu=$4 - 60\\,MeV). A supernova at Galactic center (10\\,kpc) is assumed. Our references for each cross-section are also shown.\n\t\t\\label{tab:phys_supernova_summary_old}\n\t\t}\n\t\\begin{ruledtabular}\n\t\t\\begin{tabular}{lccc}\n\t\t\tNeutrino source  & \\hkthreetank{} (660\\,kt Full Volume) & 1TankHD{} (220\\,kt Full Volume) & Ref. \\\\\n\t\t\t\\hline\n\t\t\t$\\bar{\\nu}_e + p$ & 147,000$\\sim$204,000\\,events& 49,000$\\sim$68,000\\,events &\\cite{Vogel:1999zy}\\\\\n\t\t\t${\\nu} + e^-$& 6,400$\\sim$7,500~events& 2,100$\\sim$2,500~events& \\cite{tHooft:1971ucy} \\\\\n\t\t\t${\\nu}_e + ^{16}O$ CC     & 250$\\sim$12,000~events& 80$\\sim$4,100~events& \\cite{Tomas:2003xn}\\\\\n\t\t\t$\\bar{\\nu}_e + ^{16}O$ CC    & 1,900$\\sim$12,000~events& 650$\\sim$3,900~events& \\cite{Kolbe:2002gk}\\\\\n\t\t\t${\\nu}_e + e^-$ (Neutronization) & 18$\\sim$120~events& 6$\\sim$40~events& \\cite{tHooft:1971ucy}\\\\\n\t\t\t\\hline\n\t\t\tTotal & 160,000$\\sim$240,000~events& 52,000$\\sim$79,000~events &\\\\\n\t\t\\end{tabular}\n\t\\end{ruledtabular}\n\\end{table}\n\\fi\n\\if0\n\\begin{table}[t]\n\t\t\\caption{Summary table of expected supernova neutrino events in the Hyper-Kamiokande detector. A supernova at Galactic center (10\\,kpc) is assumed.\n\t\t\\label{tab:phys_supernova_summary}\n\t\t}\n\t\\begin{ruledtabular}\n\t\t\\begin{tabular}{lcc}\n\t\t\tNeutrino source  & 2 Tanks (440\\,kt Full Volume) & Single Tank (220\\,kt Full Volume) \\\\\n\t\t\t\\hline\n\t\t\t$\\bar{\\nu}_e + p$ & 98,000$\\sim$136,000\\,events& 49,000$\\sim$68,000\\,events \\\\\n\t\t\t${\\nu} + e^-$     & 4,200$\\sim$5,000~events& 2,100$\\sim$2,500~events \\\\\n\t\t\t${\\nu}_e + ^{16}O$ CC     & 160$\\sim$8,200~events& 80$\\sim$4,100~events \\\\\n\t\t\t$\\bar{\\nu}_e + ^{16}O$ CC     & 1,300$\\sim$7,800~events& 650$\\sim$3,900~events \\\\\n\t\t\t${\\nu}_e + e^-$ (Neutronization) & 12$\\sim$80~events& 6$\\sim$40~events \\\\\n\t\t\t\\hline\n\t\t\tTotal & 104,000$\\sim$158,000~events& 52,000$\\sim$79,000~events \\\\\n\t\t\\end{tabular}\n\t\\end{ruledtabular}\n\\end{table}\n\\fi\n\\begin{table}[t]\n\t\t\\caption{Summary table of expected supernova neutrino events in the Hyper-Kamiokande detector, with E$_\\nu$ upto 100\\,MeV and the detection threshold of 3\\,MeV. A supernova at Galactic center (10\\,kpc) is assumed. Our references for each cross-section are also shown.\n\t\t\\label{tab:phys_supernova_summary}\n\t\t}\n\t\\begin{ruledtabular}\n\t\t\\begin{tabular}{lccc}\n\t\t\tNeutrino source  & Single Tank (220\\,kt Full Volume) & 2 Tanks (440\\,kt Full Volume) & Ref.\\\\\n\t\t\t\\hline\n\t\t\t$\\bar{\\nu}_e + p$ & 50,000 - 75,000\\,events& 100,000 - 150,000\\,events&\\cite{Vogel:1999zy}\\\\\n\t\t\t${\\nu} + e^-$ & 3,400 - 3,600~events& 6,800 - 7,200~events&\\cite{tHooft:1971ucy} \\\\\n\t\t\t${\\nu}_e + ^{16}O$ CC     & 80 - 7,900~events& 160 - 11,000~events&\\cite{Tomas:2003xn} \\\\\n\t\t\t$\\bar{\\nu}_e + ^{16}O$ CC     & 660 - 5,900~events& 1,300 - 12,000~events& \\cite{Kolbe:2002gk}\\\\\n\t\t\t${\\nu} + e^-$ (Neutronization) & 9 - 55~events& 17 - 110~events &\\cite{tHooft:1971ucy}\\\\\n\t\t\t\\hline\n\t\t\tTotal & 54,000 - 90,000~events& 109,000 - 180,000~events \\\\\n\t\t\\end{tabular}\n\t\\end{ruledtabular}\n\\end{table}\n\n\\subsubsection{High-energy neutrinos from supernovae with interactions with circumstellar material}\nCore-collapse supernovae are promising sources of high-energy ($\\gtrsim$ GeV) neutrinos as well as multi-MeV neutrinos.\nThe supernova shock\npropagates in the stellar material and experiences a shock breakout,\nwhich can be observed at ultraviolet or X-ray wavelengths.  Before the\nshock breakout, the supernova shock is mediated by radiation since the\nphoton diffusion time is longer than the expansion time.  During this\ntime, the conventional cosmic-ray (CR) acceleration is inefficient, so\nassociated neutrino production is not promising.  However, as the\nshock becomes collisionless after the breakout, the CR acceleration\nstarts to be effective~\\cite{wl01,mur+11}. The situation is expected\nto be analogous to supernova remnants, which are almost established as\nCR accelerators and widely believed as the origin of Galactic CRs.\n\nIn the early phase just after the breakout, the matter density is still high, so that accelerated CRs are efficiently used for neutrino production via inelastic $pp$ scatterings.\nFor type II supernovae, which are associated with red\nsuper-giants, the released energy of high-energy neutrinos is\ntypically ${\\mathcal E}_{\\nu}\\sim{10}^{47}$\\,erg~\\cite{wl01}.\nOne to two events of GeV neutrinos are expected in a timescale of hours after the core-collapse for a Galactic supernova at 10kpc in Hyper-K 1 tank.\n\nAbout 10\\% of core-collapse supernovae show strong interactions with\nambient circumstellar material, which are often called\ninteraction-powered supernovae.  If the circumstellar material mass is\n$\\sim0.1$-$1~M_\\odot$, the released high-energy neutrino energy\nreaches ${\\mathcal\nE}_{\\nu}\\sim{10}^{49}$-${10}^{50}$~erg~\\cite{mur+11}.  For example,\nEta Carinae at 2.3\\,kpc is an interesting candidate that showed violent\nmass eruptions in the past.  If a real supernova occurs and the ejecta\ncollides with the circumstellar material shell with $\\sim10\\,M_\\odot$,\none may expect $\\sim300$ events with Hyper-K.  However, because of\nthe long duration (from months to years), the signal can overwhelm the\nbackground only at early times and sufficiently high energies.\n\nHigh-energy neutrinos from supernovae are detectable hours to months\nafter the core-collapse, and detecting the signals will give us new\ninsights into supernova physics, such as how collisionless shocks are\nformed and CR acceleration starts, as well as the connection to\nsupernova remnants as the origin of Galactic CRs.  We may be able to\nsee the time evolution of multi-energy neutrino emission from the\ncore-collapse to shock breakout and following interactions with the\ncircumstellar material.\n       \n\\subsubsection{Supernova relic neutrinos}\\label{sec:supernova-relic}\nThe neutrinos produced by all of the\nsupernova explosions since the beginning of the universe are called\nsupernova relic neutrinos (SRN) or diffuse supernova neutrino background (DSNB).  They must fill the present universe\nand their flux is estimated to be a few tens\/cm$^2$\/sec.  If we can\ndetect these neutrinos, it is possible to explore the history of how\nheavy elements have been synthesized since the onset of stellar\nformation.\n\\par\n\\if0\n\\begin{figure}[tbp]\n  \\begin{center}\n    \\includegraphics[width=8cm]{physics-supernova\/srn-prediction.pdf}\n  \\end{center}\n\\vspace{-1cm}\n  \\caption{Predictions of the supernova relic neutrino (SRN) spectrum.\nFluxes of reactor neutrinos and atmospheric neutrinos are also shown.\n\t\\cite{Totani:1995rg,Totani:1995dw,Hartmann:1997qe,Malaney:1996ar,Kaplinghat:1999xi,Ando:2002ky,Lunardini:2006pd,Fukugita:2002qw}\n  \\label{fig:sn-srn-prediction} }\n\\end{figure}\n\\fi\n\\begin{figure}[tbp]\n  \\begin{center}\n    \\includegraphics[width=8cm]{physics-supernova\/relic_fluxplot_hbd-crop.pdf}\n  \\end{center}\n\\vspace{-1cm}\n  \\caption{Predictions of the supernova relic neutrino (SRN) spectrum.\nFluxes of reactor neutrinos and atmospheric neutrinos are also shown\n\t\\cite{Horiuchi:2008jz}.\n  \\label{fig:sn-srn-prediction} }\n\\end{figure}\nFigure~\\ref{fig:sn-srn-prediction} shows the SRN spectra predicted by\nvarious models.  Although searches for SRN have been conducted at\nlarge underground detectors, no evidence of SRN signals has yet been\nobtained because of the small flux of SRN.  The expected inverse beta\n($\\bar{\\nu}_e+p \\rightarrow e^++n$) event rate at Super-Kamiokande is\n0.8-5 events\/year above 10\\,MeV, but because of the large number of\nspallation products and the low energy atmospheric neutrino background\n(decay electrons from muons below Cherenkov threshold produced by\natmospheric muon neutrinos, the so-called invisible muon background),\nSRN signals have not yet been observed at Super-Kamiokande.  In order\nto reduce background, lower the energy threshold, individually\nidentify true inverse beta events by tagging their neutrons, and\nthereby positively detect SRN signals at Super-Kamiokande, a project\nto add 0.1\\% gadolinium (Gd) to tank (the SK-Gd project, called\nGADZOOKS! project previously) was proposed by J.F. Beacom and\nM.R. Vagins~\\cite{Beacom:2003nk}.\nAs the result of very active R\\&D works, the detector upgrade to SK-Gd is planned in 2018.\nThe first observation of the SRN could be made by the SK-Gd project.\nHowever, a megaton-scale detector is still desired to measure the spectrum of the SRN and to investigate the history of the universe because of its huge statistics as shown in Fig.~\\ref{fig:srn-comp}.\nFurthermore, Hyper-K could measure the SRN neutrinos at E =\n16-30\\,MeV, while the SK-Gd project concentrates on the energy of 10-20\\,MeV.\nThese observation at a different energy region can measure the contribution of extraordinary supernova bursts on the SRN, e.g. black hole formation~\\cite{Lunardini:2009,Horiuchi:2015HK}.\nFigure~\\ref{fig:srn-BH} shows the SRN signals in Hyper-K fiducial volume, with a different fraction of black hole formations.\nBecause the successful formation of black hole depends on the initial mass and metallicity of the progenitor,\nthe rate will provide information of the history about the formation of stars and their metallicity.\n\\begin{figure}[tbp]\n  \\begin{center}\n    \\includegraphics[width=8cm]{physics-supernova\/pdf_1tank\/Comparison_Fill_col2_v3.pdf}\n  \\end{center}\n\\vspace{-1cm}\n  \\caption{Expected number of inverse beta decay reactions due to supernova relic neutrinos in several experiments as a function of year.\nRed, gray and purple line shows Hyper-Kamiokande, SK-Gd, and JUNO, respectively.\nThe sizes of their fiducial volume and analysis energy thresholds were considered.\nThe neutrino temperature is assumed to be 6MeV.\nSolid line corresponds to the case, in which all the core-collapse supernovae emits neutrinos with the particular energy.\nDashed line corresponds to the case, in which 30\\% of the supernovae form black hole and emits higher energy neutrinos corresponding to the neutrino temperature of 8\\,MeV.  \\label{fig:srn-comp} }\n\\end{figure}\n\\begin{figure}[tbp]\n  \\begin{center}\n    \\includegraphics[width=8cm]{physics-supernova\/pdf_1tank\/calc_SRN-crop.pdf}\n  \\end{center}\n\\vspace{-1cm}\n  \\caption{\n\t  The SRN signal expectations in Hyper-K 1 tank fiducial volume and 10 years measurement.\n\t  Black line shows the case of neutrino temperature in supernova of 6\\,MeV, and red shows the case of 4\\,MeV~\\cite{Horiuchi:2008jz,Horiuchi:2017pv}.\n\t  Solid line corresponds to the case, in which all the core-collapse supernovae emits neutrinos with the particular energy.\n\t  Dashed line corresponds to the case, in which 30\\% of the supernovae form black hole and emits higher energy neutrinos corresponding to the neutrino temperature of 8\\,MeV.\n\t  Shaded energy region shows the range out of SRN search window at Hyper-K.\n  \\label{fig:srn-BH} }\n\\end{figure}\n\nFigure~\\ref{fig:srn-no-n-tag} shows expected SRN signals at Hyper-K with 10\\,years' livetime.\nBecause of the high background rate below 20\\,MeV from spallation products, the detection of SRN signals is limited to above $\\sim$16\\,MeV,\nwhile above 30\\,MeV the atmospheric neutrino backgrounds completely overwhelm the signal.\nConsidering the event selection efficiency after spallation product background reduction,\nthe expected number of SRN events in E = 16 to 30\\,MeV is about 70 (140) after 10 (20) years observation with Hyper-K 1 tank.\nThe statistical error will be 17 (25) events, corresponding to an observation of SRN in the energy range 16 to 30\\,MeV with 4.2 (5.7) $\\sigma$ significance (fig.~\\ref{fig:srn-det}). \nHere, we assumed the flux prediction described in ref.~\\cite{Ando:2003aa} and neutron tagging using $n + p \\rightarrow d + \\gamma\\,(2.2\\,$MeV$)$ with the tagging efficiency of 70\\%.\n\\begin{figure}[tbp]\n  \\begin{center}\n    \\includegraphics[width=7cm]{physics-supernova\/pdf_1tank\/srn-hyperk-10years-notag.pdf}\n\t  \\hspace{3mm}\n    \\includegraphics[width=7cm]{physics-supernova\/pdf_1tank\/srn-hyperk-10years.pdf}\n  \\end{center}\n\\vspace{-1cm}\n  \\caption{\n\t  Expected spectrum of SRN signals at Hyper-K with 10 years of livetime without tagging neutrons.\n\t  Left figure shows the case without tagging neutrons, assuming a signal selection efficiency of 90\\%.\n\t  Neutron tagging were applied for right figure, with the tagging efficiency of 67\\% and the pre-gamma cut for invisible muon background reduction.\n\tThe black dots show the sum of the signal and the total background, while the red shows the total background.\nGreen and blue show background contributions from the invisible muon and \n$\\nu_e$ components of atmospheric neutrinos.\nThe SRN flux prediction in~\\cite{Ando:2003aa} is applied.\n  \\label{fig:srn-no-n-tag} }\n\\end{figure}\n\n\\begin{figure}[tbp]\n\t  \\begin{center}\n\t\t      \\includegraphics[width=8cm]{physics-supernova\/pdf_1tank\/plot_SRN_candidates.pdf}\n\t\t\t      \\includegraphics[width=8cm]{physics-supernova\/pdf_1tank\/plot_SRN_sigma.pdf}\n\t\t\t\t    \\end{center}\n\t\t\t\t\t\\vspace{-1cm}\n\t\t\t\t\t  \\caption{The left (right) plot shows the number of observed SRN events (the discovery sensitivity) as a function of observation period.\n\t\t\t\t\t\t  Red solid line shows the continuous measurement with 1 tank and red dashed line shows the staging scenario, respectively.\n\t\t\t\t\t\t    \\label{fig:srn-det} }\n\t\t\t\t\t\t\\end{figure}\n\n\n\nIt is still important to measure the SRN spectrum down to $\\sim$ 10\\,MeV in order to explore the history of supernova bursts back to the epoch of red shift (z) $\\sim$1.\nTherefore, in the following discussion of the expected SRN signal with\ngadolinium neutron tagging, we assume that an\nanalysis with a lower energy threshold of $\\sim$ 10\\.MeV is possible.\nInverse beta reactions can be identified by coincident detection of both positron and delayed neutron signals, and requiring tight spatial and temporal correlations between them.\nWith 0.1\\% by mass of gadolinium dissolved in the water, neutrons are captured on gadolinium with about 90\\% capture efficiency; the excited Gd nuclei then de-excite by emitting 8\\,MeV gamma cascades.\nThe time correlation of about 30\\,$\\mu$sec between the positron and the Gd(n,$\\gamma$)Gd cascade signals, and the vertex correlation within about 50\\,cm are strong indicators of a real inverse beta event.\nRequiring both correlations (as well as requiring the prompt event to be Cherenkov-like and the delayed event to be isotropic) can be used to reduce background of spallation products by many orders of magnitude while also reducing invisible muon backgrounds by about a factor of 5.\nThe expected number of SRN events in the energy range of 10-30\\,MeV is about 280 (390) with 10 years of live time with Gd-loaded Hyper-K 1 tank (staging 2 tanks).\n\\if0\n\\begin{figure}[tbp]\n  \\begin{center}\n    \\includegraphics[width=8cm]{physics-supernova\/pdf_2tank\/srn-hyperk-10years_2tank.pdf}\n  \\end{center}\n\\vspace{-1cm}\n  \\caption{Expected spectrum of the SRN signals at Hyper-K with 10 years\nof livetime. The black dots show signal+background (red component).\nGreen and blue show background contributions from the invisible muon and \n$\\nu_e$ components of atmospheric neutrinos.\nA SRN flux prediction~\\cite{Ando:2003aa} was used, and \na 67\\% detection efficiency of 8\\,MeV gamma cascades and\na factor of 5 reduction in the invisible muon background were assumed.\n\\label{fig:srn-megaton}}\n\\end{figure}\n\\fi\nIn addition, by comparing the\ntotal SRN flux with optical supernova rate observations, a\ndetermination of the fraction of failed (optically dark) supernova\nexplosions, currently unknown but thought to occur in not less than\n5\\% and perhaps as many as 50\\% of all explosions, will be possible.\nPossible backgrounds to the SRN search down to $\\sim$ 10\\,MeV are (1) accidental\ncoincidences with the spallation event, (2) spallation products with accompanying neutrons, and\n(3) the resolution tail of the reactor neutrinos.  For (1) accidental\ncoincidences, the possible source of the prompt event is the\nspallation products. By requiring time coincidence, vertex correlation\nand energy and pattern of the delayed event, the accidental\ncoincidence rate can be reduced by a factor about 5 and can be below\nthe level of the expected SRN signal.\nFor (2) spallation products with accompanying neutrons, the only\npossible spallation product is $^9$Li and an estimation by a Geant4\nsimulation is shown in Fig.~\\ref{fig:srn-bg}.  Because of the short\nhalf-life of $^9$Li ($\\tau_{\\frac{1}{2}}=0.18$~sec), a high rejection\nefficiency of $\\sim$99.5\\% is expected.  With this expectation, the\n$^9$Li background is less than the signal level above 12~MeV; this\ncould be lowered by further development of the background reduction\ntechnique. For (3) the resolution tail of the reactor neutrinos, the\nestimated background rate is about 100(20)\/10\\,years above 10\\,MeV\n(11\\,MeV) as shown in Fig.~\\ref{fig:srn-bg} with full reactor\nintensity.\n\\begin{figure}[tbp]\n  \\begin{center}\n    \\includegraphics[width=8cm]{physics-supernova\/pdf_1tank\/srn-hk10yr-li9bg.pdf}\n  \\end{center}\n\\vspace{-1cm}\n  \\caption{Green (red) curve shows the estimated $^9$Li production rate\nbefore (after) applying cuts based on a correlation with cosmic ray muons.\nBlue shows estimated background from reactor neutrinos at full intensity.\nBlack data points show expected SRN signal based on the\nflux prediction in~\\cite{Ando:2003aa}.\n\\label{fig:srn-bg}}\n\\end{figure}\n\n\n\n\n\n\\subsection{Motivations}\\label{sec:secondtankkorea-motivations}\n\nStrategy of the future Hyper-K experiment is to build two identical water-Cherenkov detectors in stage with 260 kton of purified water per detector.\nThe first detector will be built at Tochibora mine in Japan, and this Hyper-K design report is dedicated to describe the design of the first detector in Japan.\nLocations in Korea are currently investigated for the second detector where the J-PARC neutrino beam passes through, and this section focuses on the benefits of building the second detector in Korea. \nFigure~\\ref{f:OAB} shows the J-PARC neutrino beam aiming at the Japanese site with 2.5 degree off-axis angle (OAA) and also passing through Korea\nwith an 1$\\sim$3 degree OAA range.\n\nIn fact, having a 2$^\\text{nd}$ detector regardless of its location, will improve sensitivities of all areas of physics covered by the Hyper-K 1$^\\text{st}$ detector.\nHowever, building a 2$^\\text{nd}$ detector in Korea rather than in Japan further enhances physics capabilities on neutrino oscillation physics\n(leptonic CP violation, determination of neutrino mass ordering, non-standard neutrino interaction etc.)\nusing beam neutrinos thanks to the longer baseline ($\\sim$1,100~km). \n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=1.0\\textwidth]{second-tank-korea\/figures\/Beam_Japan_to_Korea.pdf}\n\\end{center}\n\\caption{ Contour map of the J-PARC off-axis angle beam to Korea~\\cite{Hagiwara2006,Hagiwara2006e,Hagiwara2007}. }\n\\label{f:OAB}\n\\end{figure}\n\n\nThe benefits of Korean site for beam neutrino physics are well expressed by an appearance bi-probability plot: \n$P(\\nu_{\\mu}\\rightarrow\\nu_{e})$ vs. $P(\\bar{\\nu}_{\\mu}\\rightarrow\\bar{\\nu}_{e})$. \nFigure~\\ref{f:bp} shows appearance bi-probability plots at Hyper-K sites in Tochibora, Japan (left) and in Mt. Bisul, Korea (right). \nEach point in colored ellipses represents different $\\delta_{CP}$ value for normal (inverted) neutrino mass ordering \nin solid (dotted) lines.\nThree different colors correspond to three representing neutrino energies for each site.\nThe blue ellipses correspond to the peak energy at each OAA and the green and red ones represent median\nenergy after splitting the events into two parts below and above the peak. \nNote that there are high energy ($>$ 1.25 GeV) appearance events observed in Korean site due to the longer baseline,\nand this results in an important difference in physics sensitivities between Japan and Korean sites. \nThe gray ellipses indicates the sizes of the statistical\nuncertainties given by $\\sqrt N$ from the number of events around peak energy in $\\nu_{e}$ and $\\overline{\\nu_{e}}$ appearance signals. \nIt is clear from the bi-probability plots that the ellipses in Mt. Bisul site are well separated compare to those at Tochibora site\nalthough the statistics is lower with the longer distance. \nThis will allow us to resolve degeneracies (overlaps of ellipses) in neutrino mass ordering and $\\delta_{CP}$ parameters.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=1.0\\textwidth]{second-tank-korea\/figures\/bp_double_fig.pdf}\n\\caption{Appearance bi-probability plots at Hyper-K sites in Tochibora (left) at $2.5^{\\circ}$ OAA and in Mt. Bisul (right) at $1.3^{\\circ}$ OAA. \nThese plots show the principle by which Hyper-K determines the mass ordering and measures the CP phase in three representing energies\ndrawn with green, blue and red colors. \nNormal (inverted) ordering for different $\\delta_{CP}$ values is represented by a solid (dotted) curves. \nGrey ellipses represent the sizes of statistical error for a ten-year exposure on one Hyper-K detector. \nThe $\\Phi_{32}$ is defined as $\\frac{2}{\\pi}\\frac{\\left|\\Delta m^2_{32}\\right|L}{E}$ \nand is close to odd-integer values for oscillation maxima, and close to even-integer values for oscillation minima.\n}\n\\label{f:bp}\n\\end{figure}\n\nThanks to the longer baselines in Korean sites the 1$^\\text{st}$ and 2$^\\text{nd}$ oscillation maxima of the appearance probability are reachable\nwith higher ($>$ 1.25 GeV) neutrino energy, and this allows much better sensitivities especially on the neutrino mass ordering determination and non-standard neutrino interaction (NSI) in matter. \nBy having the 2$^\\text{nd}$ detector in Korea the fraction of $\\delta_{CP}$ coverage is also increased and more precise measurement of $\\delta_{CP}$ is achieved. \nAccording to the sensitivity studies performed, described later, smaller OAA in Korean site gives best sensitivities in most physics cases with beam neutrinos. \nReaching higher energy with smaller OAA, however, introduces more $\\pi^{\\circ}$ background but a good news is that T2K has recently reduced \nthe $\\pi^{\\circ}$ background and its systematic uncertainty~\\cite{Abe:2015awa}. \nThanks to the larger overburdens in Korean sites the sensitivities on solar neutrino and Supernova relic neutrinos (SRN) are also further improved. \n\n\n\\subsection{Location and Detector}\\label{sec:secondtankkorea-location}\n\n\nThere are several candidate sites to locate the 2$^\\text{nd}$ detector in Korea and they are listed in Table~\\ref{t:six_sites}.\nAmong the six candidate sites Mt. Bisul with the smallest OAA seems to be the most favorite ones according to our sensitivity study described later. \n\n\\begin{table}[hbt]\n\\captionsetup{justification=raggedright,singlelinecheck=false}\n\\small\n \\caption{Candidate sites with the off-axis angles between 1 and 2.5 degrees for the 2$^\\text{nd}$ Hyper-K detector in Korea. The baseline is the distance from the production point of the J-PARC\\\n neutrino beam~\\cite{Abe:2016t2hkk}.}\n \\centering\n \\begin{tabular*}{0.95\\textwidth}{@{\\extracolsep{\\fill}} l c c c l}\n\n  \\hline \\hline\nSite                 &  Height &  Baseline  &  Off-axis angle   &  Composition of rock \\\\[-1.4ex]\n     & (m) & (km) & (degree) & \\\\\n\\hline\n Mt. Bisul           &   1084    &  1088    &  1.3$^{\\circ}$      &   Granite porphyry,\\\\[-1.4ex]\n  & & & & andesitic breccia \\\\[0.4ex]\n Mt. Hwangmae  &   1113    &  1141    &  1.9$^{\\circ}$      &   Flake granite, \\\\[-1.4ex]\n& & & & porphyritic gneiss  \\\\[0.4ex]\n Mt. Sambong    &   1186    &  1169    &  2.1$^{\\circ}$     &   Porphyritic granite, \\\\[-1.4ex]\n  & & & & biotite gneiss \\\\[0.4ex]\n Mt. Bohyun       &   1124    &  1043    &  2.3$^{\\circ}$    &   Granite, volcanic rocks, \\\\[-1.4ex]\n  & & & & volcanic breccia \\\\[0.4ex]\n Mt. Minjuji         &   1242    &  1145    &  2.4$^{\\circ}$    &   Granite, biotite gneiss \\\\[0.4ex]\n Mt. Unjang        &   1125    &  1190    &  2.2$^{\\circ}$     &   Rhyolite, granite porphyry, \\\\[-1.4ex]\n  & & & & quartz porphyry  \\\\\n  \\hline \\hline\n  \\end{tabular*}\n \\label{t:six_sites}\n \\end{table}\n\n\nThe larger overburdens of the Korean candidate sites allow much less muon flux and spallation isotopes. \nWith a flat tunnel the overburden is $\\sim$820~m for both Mt. Bisul and Mt. Bohyun, the two favorable sites in a more favorable order.\nBy making a sloped tunnel the overburden becomes $\\sim$1000~m for both mountains, and this is the default tunnel construction plan.\nFigure~\\ref{fig:muon-direction} shows muon flux as a function of zenith (top) and azimuth (bottom) angles\nfor Mt. Bisul and Mt. Bohyun as well as Super-K and Hyper-K (Tochibora) sites for comparison.\nThe muon flux at the two Korean sites with $\\sim$1,000~m overburden are similar to that of Super-K site\nand about four times smaller than Hyper-K Tochibora site. \n\n\\begin{figure}[tb]\n\\centering\n\\includegraphics[width=0.7\\textwidth]{second-tank-korea\/figures\/muon-theta.pdf}\n\\includegraphics[width=0.7\\textwidth]{second-tank-korea\/figures\/muon-phi.pdf}\n\\caption{Muon flux as a function of cosine of zenith angle $\\cos\\theta$ (upper) and azimuth angle $\\phi$ (lower) for Hyper-K (Tochibora), \nMt.\\ Bisul (1,000~m overburden), Mt.\\ Bohyun (1,000~m overburden), and Super-K.\nThe east corresponds to the azimuth angle of zero degree.\nThe blue lines show the data for Super-K, and the red lines show the MC predictions based on the \\texttt{MUSIC} simulation. }\n\\label{fig:muon-direction}\n\\end{figure}\n\n\\subsection{Physics Sensitivities}\\label{sec:secondtankkorea-physics}\n\nPhysics sensitivity studies are performed \nfor three different off-axis angles (1.5$^{\\circ}$, 2.0$^{\\circ}$, and 2.5$^{\\circ}$) at a baseline distance of $L=1,000$~km.\nSensitivity study results are obtained for the determination of the mass ordering, the discovery of CP violation by excluding of the sin($\\delta_{cp}$) = 0\nhypothesis, the precision measurement of $\\delta_{cp}$, fraction of $\\delta_{cp}$ as a function of CPV significance and exposure, $\\theta_{23}$ octant, \natmospheric parameters and NSI.\nOnly part of these studies are shown here due to a limited space, but the full studies will be published in a separate paper. \nThe results are obtained by assuming 2.7$\\times$10$^{22}$ proton-on-target with $\\nu : \\bar{\\nu} = 1 : 3$\nwhich corresponds to 10-year operation with 1.3 MW beam power for 187 kton fiducial volume mass per detector. \nIn this section relatively simplistic systematic uncertainty model~\\cite{Abe:2016t2hkk} is used in the sensitivity studies \nwhile the size of the systematic uncertainties are mostly based on the current T2K analyses. \nNote that the same simplistic systematic uncertainty model is used for the Japanese detector, \nwhich is a little different from the systematic uncertainties used in other sections of this document. \nIn sensitivity studies other oscillation parameter values are from the Particle Data Group (PDG) 2015 Review of Particle Physics~\\cite{Olive:2016xmw}\nexcept for $\\theta_{23}$ and $\\Delta m^{2}_{32}$, where $\\sin^{2}\\theta_{23}=0.5$ and $\\Delta m^{2}_{32} = 2.5 \\times 10^{-3}$ eV$^{2}$ are used \ndue to their large uncertainties in their absolute values, and no CP violation ($\\delta$ = 0) is assumed unless it is specified.\nTwo detector configuration, either JD (Japanese Detector) + KD (Korean Detector) or JD $\\times$ 2, assumes no staging in the sensitivity studies shown in this section. \n\nFigure~\\ref{fig:cp_precision} shows the sensitivity on the 1$\\sigma$ precision of the $\\delta_{cp}$ measurement assuming no prior knowledge on neutrino mass ordering. \nFor most of the true $\\delta_{cp}$ the best sensitivity is from JD + KD with 1.5$^{\\circ}$ OAA. \nEspecially when the CP is maximally violated the sensitivity difference is largest between the JD + KD with 1.5$^{\\circ}$ OAA and JD $\\times$ 2 or JD $\\times$ 1. \n\\begin {figure}[htbp]\n\\captionsetup{justification=raggedright,singlelinecheck=false}\n  \\begin{center}\n    \\includegraphics[width=0.80\\textwidth]{second-tank-korea\/figures\/cp_precision_true_nh_jdx1.pdf}\\\\\n    \\includegraphics[width=0.80\\textwidth]{second-tank-korea\/figures\/cp_precision_true_ih_jdx1.pdf}\n    \\caption{The 1$\\sigma$ precision of the $\\delta_{cp}$ measurement as a function of the true $\\delta_{cp}$ value.  Here, it is assumed there is no prior knowledge of the\nmass ordering.}\n    \\label{fig:cp_precision}\n  \\end{center}\n\\end {figure}\nFigure~\\ref{fig:cp_prec_cp_frac} shows the sensitivity on the fraction of $\\delta_{cp}$ for the 1$\\sigma$ precision of the $\\delta_{cp}$ measurement. \nThe results with JD + KD with any off-axis angle are always better than JD $\\times$ 2 and JD $\\times$ 1. \nThe sensitivity with 2.0$^{\\circ}$ OAA is slightly better than that of 1.5$^{\\circ}$ OAA if the $\\delta_{cp}$ precision is less than 10 degree\nbut otherwise the 1.5$^{\\circ}$ OAA gives the best sensitivity. \nMore details on the physics sensitivity studies are found in Ref~\\cite{Abe:2016t2hkk}.\n\\begin {figure}[htbp]\n\\captionsetup{justification=raggedright,singlelinecheck=false}\n  \\begin{center}\n    \\includegraphics[width=0.8\\textwidth]{second-tank-korea\/figures\/cp_prec_frac_dcp_jdx1.pdf}\n    \\caption{The fraction of $\\delta_{cp}$ values (averaging over the true mass ordering) for which a given precision or better on $\\delta_{cp}$ can be achieved.}\n    \\label{fig:cp_prec_cp_frac}\n  \\end{center}\n\\end {figure}\nAccording to our sensitivity study CP violation sensitivity improves a little in JD + KD compared to JD $\\times$ 2. \nThe $\\theta_{23}$ octant sensitivity and atmospheric parameter sensitivities get slightly worse in JD + KD configuration compared to JD $\\times$ 2. \n\nThe sensitivity on NSI of neutrinos in matter (not in production nor in detection) \nis also greatly enhanced by having an additional  2$^\\text{nd}$ detector in Korea,\nand the details on NSI sensitivity studies are found in Ref~\\cite{Fukasawa:2016lew,Liao2017}. \n\n\nThanks to the similar overburden to Super-K in Korean candidate sites (see Fig.~\\ref{fig:muon-direction}) \nbut with a larger detection volume, \nlow energy astrophysics such as solar neutrinos and SRN sensitivities are expected to be good. \nAccording to our sensitivity study on the SRN we expect about 5 (4) sigma sensitivity for 10 years of data taking \nin Mt. Bisul or Mt. Bohyun (Tochibora) sites~\\cite{Yeom2017_SRN_sens}. \nMoreover we expect to observe spectral distribution of SRN due to large statistics from 8.4 times larger detection volume than Super-K,\nand this might lead to solving SN burst rate problem. \n\n\\subsection{Conclusion}\\label{sec:secondtankkorea-conclusion}\n\nHaving a 2$^\\text{nd}$ Hyper-K detector in addition to the 1$^\\text{st}$ one will enhance physics sensitivities\nfrom beam neutrino physics to astroparticle physics due to the increased detection volume.\nAccording to our sensitivity studies, physics capabilities of the Hyper-K project are further improved by locating the 2$^\\text{nd}$ detector in Korea, \nsuch as determination of neutrino mass ordering, precision of $\\delta_{CP}$ measurement and test of NSI.\nWith the longer baseline in Korean site both the 1$^\\text{st}$ and the 2$^\\text{nd}$ oscillation maximum of the appearance neutrino probability are reachable,\nand this is a unique opportunity since no other past and current experiment (MINOS, T2K, NOvA) can reach the 2$^\\text{nd}$ oscillation maxima. \nThe longer baseline to the detector in Korea allows to resolve the degeneracy between the mass ordering and the value of delta CP that would happen \nfor only certain values of the parameters with a detector(s) in Japan. The bi-probability plots can intuitively show this feature even before \nperforming any sensitivity studies.\nNSI sensitivities are also improved especially with the smaller OAA site in Korea. \nAccording to our sensitivity studies the best Korean candidate site seems to be the Mt. Bisul. \nWith $\\sim$1000~m overburden sensitivities on solar neutrino and SRN physics are further enhanced in the Korean candidate sites\nthan those in the Tochibora mine ($\\sim$650~m overburden). \n\n\\section{Hyper-Kamiokande software} \\label{section:software}\n\nThe Hyper-K software system is designed around the following\nprinciples:\n\\begin{itemize} \\itemsep0em\n\\item Adaptable. The Hyper-K experiment is expected to run for more\n  than a decade.  This period typically spans more than one\ngeneration of software and infrastructure. The Hyper-K offline system\nis being designed to be flexible enough to accommodate changes in\ntools or infrastructures.\n\\item Reliable. Each component needs to demonstrate it's reliability\nby exhibiting well defined behaviour on control samples.\n\\item Understandable. Documentation on what the component does, what\nit's dependencies are as well as test samples and outputs are\nessential in being able to use it successfully.\n\\item Low overhead. The management and maintenance should be as\nautomated as possible to free collaborators to focus on the challenge\nof extracting the high-quality physics measurements.\n\\end{itemize}\n\nThe software consists of a collection of loosely-coupled packages,\nsome of which are open-source and some of which are specific to\nHyper-K. The distributed code management system Git \\cite{git} is used to manage\nthe software. Each package is hosted on a third-party central\nrepository (\\url{https:\/\/github.com\/}) that provides distributed\naccess to the packages. The distributed nature of the code management\nallows researchers the possibility to develop the software independently without\nimpacting other researchers. The loose-coupling between packages\nallows those that reach their end of life to be replaced by better\nalternatives with minimal impact on the rest of the system.  Where\npossible standard particle physics software libraries are used to\nreduce the burden of support of experiment-specific code. The working\nlanguage for the Hyper-K software packages is C++, with the output\nfiles being written in ROOT \\cite{Brun97} format.\n\nThe flow for the simulation is as follows: The event topologies are\ngenerated by a neutrino interaction package(GENIE\n\\cite{Andreopoulos:2009rq} and NEUT \\cite{Hayato:2009zz}, for\nexample), and modeled by a Monte Carlo detector response code called\nWCSim \\cite{WCSim}. The event information is reconstructed using\neither BONSAI \\cite{icrc0213smy} (for low energy events) or fiTQun\n(for high energy events) \\cite{Tobayama_2016}. This is shown\nschematically in Figure~\\ref{fig:software_flow}. These packages will\nbe described in more detail in the next Sections.\n\nAn online workbook is also maintained to provide higher-level\ndocumentation on overall procedures and information for new users of\nthe software and developers.  An overall software control package\nallows for the fully automated download, compilation and running of\nthe software, based on user requests.\n\n\\begin{figure}[htbp]\n  \\begin{center}\n    \\includegraphics[scale=0.4]{software\/figures\/Software_flow_fig.pdf}\n  \\end{center}\n\\caption {Flowchart of the simulation process.}\n  \\label{fig:software_flow}\n\\end{figure}\n\n\\subsection{WCSim}\\label{software:WCSim}\n\nThe Water Cherenkov Simulation (WCSim) package is a flexible,\nGeant4-based code that is designed to simulate the geometry and\nphysics response of user-defined water Cherenkov detector\nconfigurations. WCSim is an open-source code and is available for\ndownload at \\url{https:\/\/github.com\/WCSim\/WCSim}.\n\nThe final performance of the Hyper-Kamiokande detector depends on the\ndetector geometry, the type of photodetectors, and the photocoverage\nthat will be used. WCSim takes these variables as inputs and simulates\nthe detector response, which can then be used to determine the physics\npotential. WCSim users specify the type of photodetectors, the number\nof photodetectors, the detector diameter and radius, and whether the\nwater should be doped with gadolinium. The outer detector volume is\ncurrently not implemented in WCSim, though it is actively being\ndeveloped for a future release.\n\nFor this report, the relevant photodetectors in WCSim are the R3600\n20'' diameter PMTs, as well as the R12850 20'' and 12'' diameter box\nand line photodetectors. Photodetector parameters in the simulation\ninclude the timing resolution, dark noise rate, and the overall\nefficiency for a photon to register a charge (including the quantum\nefficiency, collection efficiency, and hit efficiency as described in\nSection \\ref{section:photosensors}). For the R3600 PMTs, the\nparameters were taken from the Super-Kamiokande simulation code\nSKDETSIM. The parameters for the R12850 are taken from measurements as\ndescribed in Section \\ref{section:photosensors}. Some higher-level photodetector effects\nsuch as after-pulsing are not currently simulated in WCSim, though\nthis is a planned upgrade for a future releases.\n\nGeant4 \\cite{Agostinelli:2002hh} is used to track\nthe particles as they pass through the detector and compute the\nfinal deposited energy. Particles that reach the photodetector glass\nand pass the quantum efficiency and collection efficiency cuts are\nregistered as a hit. The hits are then digitized based on the SK-I\nelectronics scheme, though the code has the flexibility for users to\ninclude their own custom electronics configurations. \n\nThe output for the WCSim code includes both the raw hit and the\ndigitized information. The raw hit information includes which tubes\nwere hit and how many times each tube was hit. The digitized\ninformation includes the number of hits in a trigger window, as well\nas the charge and time of the hit tubes. WCSim output files can be\nused for event reconstruction by fiTQun or BONSAI, which are described in the\nfollowing subsections. Geant4 visualization tools can be used to display the\ndetector geometry and particle tracks. Figure~\\ref{fig:HK-WCSim} is a\nrendering of one of the proposed Hyper-K\ntanks. Figure~\\ref{fig:HK-WCSim-eventdisplay} shows an example of an\nevent display for an electron and for a muon, each with 1 GeV kinetic energy. \n\n\\begin{figure}[tbp]\n\n    \\includegraphics[scale=0.4]{software\/figures\/HK-WCSim.pdf}\n \n\\caption {Geant4 visualization of the Hyper-Kamiokande detector\nconfiguration. The top cap of the detector has\nbeen removed for visualization purposes. Phototubes are shown in\nblack, while the walls of the detector are shown in grey.}\n  \\label{fig:HK-WCSim}\n\\end{figure}\n\n \\begin{figure}[tbp]\n   \\centering\n   \\begin{tabular}{cc}\n    \\centering\n    \\includegraphics[scale=0.4]{software\/figures\/HK-WCSim-EDe-.pdf}\n   &\n    \\centering\n    \\includegraphics[scale=0.4]{software\/figures\/HK-WCSim-EDmu-.pdf}\n  \\end{tabular}\n  \\caption {Event displays in the HK detector for a 1 GeV electron\n    (left) and a 1 GeV muon (right).}\n  \\label{fig:HK-WCSim-eventdisplay}\n \\end{figure}\nFigure~\\ref{fig:total-charge} shows how the flexibility of WCSim can\nbe used to explore different detector configurations. Here, the total\ncharge distribution for electrons and muons at several momenta in\nthe Hyper-Kamiokande detector with two different photocoverage options\nare shown. RMS divided by mean charge\nis plotted in Figure~\\ref{fig:rms-mom} indicating better resolution\nwith 40\\% photocoverage than with\n14\\% photocoverage. For lower energy particles, the resolution can be\napproximated using nhits (the number of phototubes that register a\nhit). The nhit distribution for both 14\\% photocoverage and\n40 \\% photocoverage are shown in Figure~\\ref{fig:nhits}. \n\n \\begin{figure}[tbp]\n   \\begin{tabular}{cc}\n \n    \\centering\n    \\includegraphics[trim=0cm 7.0cm 2.5cm 7.0cm, clip=true, scale=0.4]{software\/figures\/QMean_60m_inner_e.pdf}\n  \n&\n    \\centering\n    \\includegraphics[trim=0cm 7.0cm 2.5cm 7.0cm, clip=true, scale=0.4]{software\/figures\/QMean_60m_inner_mu.pdf}\n\n  \\end{tabular}\n  \\caption {Total charge distributions for electrons (left) and muons (right)\n  with several momenta in the Hyper-K detector. The red line\n  corresponds to 14\\% photocoverage, while the blue line corresponds\n  to 40 \\% photocoverage.}\n  \\label{fig:total-charge}\n \\end{figure}\n \\begin{figure}[htbp]\n  \\begin{tabular}{cc}\n\n    \\centering\n    \\includegraphics[trim=0cm 7.0cm 2.5cm 7.0cm, clip=true, scale=0.4]{software\/figures\/ChargeRMS_inner_electron.pdf}\n\n&   \n    \\centering\n    \\includegraphics[trim=0cm 7.0cm 2.5cm 7.0cm, clip=true, scale=0.4]{software\/figures\/ChargeRMS_inner_muon.pdf}\n\n  \\end{tabular}\n  \\caption {RMS\/Total charge distributions for electrons (left) and muons (right)\n  with several momenta. The red line corresponds to the configuration\n  with 14\\% photocoverage, while the blue line corresponds to the\n  configuration with 40\\% photocoverage.}\n  \\label{fig:rms-mom}\n \\end{figure}\n \\begin{figure}[htbp]\n   \\begin{tabular}{cc}\n  \n    \\centering\n    \\includegraphics[scale=0.47]{software\/figures\/Nhits_3TankLD.pdf}\n  \n&\n \n    \\centering\n    \\includegraphics[scale=0.47]{software\/figures\/Resolution_3TankLD.pdf}\n  \n  \\end{tabular}\n  \\caption{ Expected number of PMT hits (N$_{PMT hits}$) and the RMS of\n    the N$_{PMT hit}$ distributions. \nWCsim is used for simulating the injection of electrons with several\nvalues of kinetic energy (E$_{kin}$).\nThe initial position is uniformly distributed inside the fiducial\nvolume ($>$2~m from inner detector wall). The red line corresponds to\nthe 14\\% photocoverage configuration, while the blue line corresponds to\nthe configuration with 40\\% photocoverage.}\n  \\label{fig:nhits}\n \\end{figure}\n\n\\subsection{FiTQun} FiTQun is an event reconstruction\npackage initially developed for the Super-K detector based on the\nformalism used by the MiniBooNE experiment\n\\cite{Patterson:2009ki}. The reconstruction algorithm allows for\nsingle- and multiple-ring event hypotheses to be tested against\nobserved data. For a given event hypothesis, a prediction is made for\nthe complete set of observables at each PMT. This includes whether or\n not the PMT was hit, and for hit PMTs, the hit time and integrated \ncharge. The hypothesis and associated kinematic parameters that best \ndescribe a given event are found by maximizing a likelihood function \nof the prediction with respect to the observed data.\n\nFiTQun has been shown to perform well on Super-K data, with\nsignificant improvements on vertex, angle and momentum resolutions, as\nwell as particle identification when compared to previous\nreconstruction algorithms. In particular, fiTQun was successfully\ndeployed to reject $\\pi^0$ events from the Super-K $\\nu_{e}$ sample in\nthe T2K $\\nu_{e}$ appearance analysis (Figure~\\ref{fig:pi0plot}) \\cite{Abe:2013hdq,Abe:2015awa}.\n\n\\begin{figure}[htbp]\n  \\begin{center}\n    \\includegraphics[width=0.6\\textwidth]{software\/figures\/sigbkg_pi0cut.pdf}\n  \\end{center}\n  \\caption {FiTQun used for $\\pi^0$ rejection in the $\\nu_{e}$\n    selection for the T2K $\\nu_{e}$ appearance\n    analysis\\cite{Abe:2013hdq}. The red line represents the cut, with\n    events above the line being rejected as $\\pi^0$ background.}\n  \\label{fig:pi0plot}\n\\end{figure}\n\n\\subsubsection{Reconstruction algorithm} \nAt the core of fiTQun lies a likelihood function, which is evaluated\nover all the PMTs in the detector:\n\\begin{equation} \nL \\left( {\\bf x} \\right)= \\prod_j^{unhit} P_j\\left(unhit | {\\bf x} \\right) \\prod_i^{ ihit} P_i \\left(hit | {\\bf x}\\right) f_q\\left(q_i|{\\bf x}\\right) f_t \\left(t_i | {\\bf x} \\right) \\,.\n\\end{equation} \n\nEvent hypotheses are characterized by ${\\bf x}$, which\nincludes the time and position of the interaction vertex, momentum and\ndirection of the charged particle tracks, and any other relevant\nkinematic parameters such as the distance or time interval between\ntracks, or the energy lost between track segments. For a given ${\\bf\nx}$, a prediction of the amount of charge at each PMT, $\\mu_i$, is\nmade and the time at which the light is expected to arrive each PMT is\ncalculated.\n\nThe detector response is folded into these predictions to give the\nprobabilities $P$ of a PMT being hit and the read-out time and charge\nprobability distribution functions $f_t$ and $f_q$, respectively. The\nnegative log-likelihood ($-log\\left( L \\right)$) is maximised to\nobtain the ${\\bf x}$ that best describe the event according to some\nevent topology (\\emph{e.g.}, single electron-like ring).\n\nOnce the best-fit parameters have been obtained for several\ntopologies, the ratio between their likelihoods is used to determine\nwhich topology gives the best match to the event. This can be used as\na particle identification tool if simple one-particle hypotheses are\nused, or as a more complex selection criterion if multiple final-state\nparticles are included in the hypothesis (\\emph{e.g.}, a nuclear\nde-excitation photon followed by a $K^+$ decay muon for the selection\nof $p\\rightarrow K^+ \\nu$ events).\n\n\n\n\n\n\n\\subsubsection{Integration with WCSim and tuning} FiTQun has been\nadapted to reconstruct events simulated with WCSim, in the various\ndetector configurations implemented in the simulation software. A\nC++ class was written (WCSimWrapper) that reads in both detector \ngeometry (positions and radius of PMTs) and event data from the WCSim\nROOT output files. A preprocessor flag allows fiTQun to be compiled \nagainst WCSim libraries, removing its dependence on Super-K software.\n\nIn the context of WCSim, events can be generated in an arbitrary\nnumber of detector configurations. For fiTQun to adequately\nreconstruct events in any given configuration, some of its components\nhave to be re-evaluated. For example, the charge and time response of\nPMTs must be accurately known in order to obtain unbiased estimates of\nparticle momentum and vertex position. The tuning procedure developed\nfor Super-K and SKDETSIM was adapted to be used with WCSim and with\ngeneralized cylindrical geometries.\n\nThe tunes produced for each simulated detector consist of ROOT files\nand run-time parameters. A configuration file (that contains the \ninformation needed by fiTQun to load the appropriate files and \nparameters) is given for each tune. These configuration files are \npackaged with fiTQun, such that any tune available can be selected\nin a single step.\n\n\\subsection{BONSAI} \nFor event reconstruction at low\nenergy, i.e. few MeV - few tens MeV, a reconstruction algorithm BONSAI\n(Branch Optimization Navigating Successive Annealing Iterations) is\nsupplied for Hyper-Kamiokande.  BONSAI was originally developed for\nSuper-Kamiokande \\cite{icrc0213smy} and written in C++. It has been used for the low\nenergy physics analysis of SK-I to SK-IV. In the low energy region,\nmost of the photosensor signals are single photon hits.  BONSAI uses\nthis relative hit time information to reconstruct the position of the\nCherenkov light source, i.e. the position of low energy event.\nFor Hyper-K analysis, a wrapper library (libWCsimBonsai) is supplied for ROOT environment.\n\n\\subsubsection{Vertex reconstruction} \nFor the vertex reconstruction, BONSAI performs a maximum likelihood fit using the\nphotosensor hit timing residuals.  This likelihood fit is done for the\nCherenkov signal as well as the dark noise background for each vertex\nhypothesis.  The likelihood of the selected hypothesis is compared\nto the likelihood of a hypothesis in an area nearby.  Highly ranked\nhypotheses and new points in the likelihood will survive this step.\nFinally, after several iterations, the hypothesis with the largest\nlikelihood is chosen as the reconstructed vertex.\n\nThe vertex goodness criterion testing the time residual distribution\nis defined as follows:\\\\\n\\begin{equation} g\n\t(\\vec{v})=\\sum_{i=1}^{N}w_{i}\\exp{-0.5(t_i-|\\vec{x_i}-\\vec{v}|\/c_{wat})\/\\sigma)^2}\n\\end{equation} where $t_i$ are the measured PMT hit times, $\\vec{x_i}$\nthe photosensor locations, $\\vec{v}$ is reconstructed event vertex,\n$\\sigma$ is the effective timing resolution expected for Cherenkov events (total of photosensor and DAQ resolution).\n$c_{wat}$ is the group speed of light in the water, i.e. $c\/n$ with the speed of light in vacuum $c$ and refractive index $n$.\n$\\omega_i$ are Gaussian hit weights also based on the hit time residuals, but with a much wider effective time resolution.\nThe weights reduces the dark noise contamination of the Cherenkov light.\nA result of vertex reconstruction performance study with BONSAI and WCSim can be found in the figure \\ref{fig:bonsaivtx}.\nMore Cherenkov photons could be detected with new photosensors for Hyper-K and it improves the reconstruction results, comparing to those of Super-K\nThough, at same time, the random photosensor signals caused by their dark pulse can spoil the merit.\nSo reducing dark pulse is a crucial factor to improve the low energy event detection.\nMany efforts for the dark pulse reduction (\\ref{section:photosensors:IDperformance:BG}) and improvements of the softwares are being continued.\n\\begin{figure}[htbp]\n  \\begin{center}\n    \\includegraphics[width=0.6\\textwidth]{software\/figures\/VtxResolution-crop.pdf}\n  \\end{center}\n  \\caption {\n\t  Vertex reconstruction resolution for electrons with BONSAI for Hyper-K and Super-K detectors.\n\t  Here, WCSim is used for Hyper-K detector simulation.\n\t  Red line shows the resolution with the PMT dark pulse rate of 8.4\\,kHz, as seen in \\ref{section:photosensors}.\n\t  Blue is for the case of PMT dark pulse rate of 4.2\\,kHz, which is same as the rate of Super-K photosensors.\n\t  Black line shows the performance with Super-K detector, simulated with SKDETSIM.\n  \\label{fig:bonsaivtx}\n    }\n\\end{figure}\n\n\\subsubsection{Energy and direction reconstruction}\nBONSAI and its related subroutines also determine the energy and the\nevent direction reconstruction.\nBecause most of the photosensor signals consist of single photon hits at low energy below few tens MeV,\nthe total number of photosensor hits is the leading parameter for\nreconstructing the energy of events.  First, time-of-flight values are\nsubtracted from each of the hit timing values based on the position of\neach photosensor and the result of the BONSAI vertex reconstruction.\nNext, the number of photosensor hits around the expected event timing\nis calculated, considering its cross-section and the local\nphotocoverage with neighboring photosensors.  Finally, the number of\nhits are scaled to energy using the information from detector\nsimulations and calibrations.\\\\\nThe direction reconstruction is also performed on the photosensor hit\npatterns using a circular KS test that checks the azimuthal symmetry\naround the Cherenkov cone.\nAs the result, the vertex position, direction and energy of low-energy\nevents are available after BONSAI reconstruction.\n\nSeveral likelihoods to test mis-reconstruction are also available\nduring the reconstruction.  Likelihoods calculated using photosensor\nhit patterns are also used in particle identification, e.g. to\ndifferentiate between electron and gamma events.\n\n\\section*{Executive summary}\n\nOn the strength of a double Nobel prize winning experiment\n(Super)Kamiokande and an extremely successful long baseline neutrino\nprogramme, the third generation Water Cherenkov detector,\nHyper-Kamiokande, is being developed by an international collaboration\nas a leading worldwide experiment based in Japan.\n\nIt will address the biggest unsolved questions in physics through a\nmulti-decade physics programme that will start in the middle of the\nnext decade.\n\nThe Hyper-Kamiokande detector will be hosted in the Tochibora mine,\nabout 295\\,km away from the J-PARC proton accelerator research complex\nin Tokai, Japan.\n\nThe currently existing accelerator will be steadily upgraded to reach\na MW beam by the start of the experiment.\nA suite of near detectors will be vital to constrain the beam for\nneutrino oscillation measurements. They will be a combination of\nupgraded and new detectors at a distance ranging from 280\\,m to\n1-2\\,km from the neutrino target.\n\nA new cavern will be excavated at the Tochibora mine to host the\ndetector. The corresponding infrastructure will be built.  The\nexperiment will be the largest underground water Cherenkov detector in\nthe world and will be instrumented with new technology photosensors,\nfaster and with higher quantum efficiency than the ones in\nSuper-Kamiokande. Pressure tests demonstrate that they will be able to\nsupport the pressure due to the massive tank.\n\nThe science that will be developed will be able to shape the future\ntheoretical framework and generations of experiments.\nHyper-Kamiokande will be able to measure with the highest precision\nthe leptonic CP violation that could explain the baryon asymmetry in\nthe Universe. The experiment also has a demonstrated excellent capability to\nsearch for proton decay, providing a significant improvement in\ndiscovery sensitivity over current searches for the proton lifetime.\nThe atmospheric neutrinos will allow to determine the neutrino mass\nordering and, together with the beam, able to precisely test the\nthree-flavour neutrino oscillation paradigm and search for new\nphenomena. A strong astrophysical programme will be carried out at the\nexperiment that will also allow to measure precisely solar neutrino\noscillation.  A set of other main physics searches is planned, like\nindirect dark matter.\n\nIn summary, a new experiment, based on the experience and facilities of\nthe already existing Super-Kamiokande and long baseline neutrino\nexperiment as T2K, is being developed by the international physics\ncommunity to provide a wide and groundbreaking multi-decade physics\nprogramme from the middle of the next decade (see Table~\\ref{tab:intro:phys}).\n\n\n\\begin{table}[hbtp]\n  \\caption{Expected sensitivities of the Hyper-Kamiokande experiment assuming 1 tank for 10 years.} \t\n  \\label{tab:intro:phys}\n  \\begin{center}\n    \\begin{tabular}{lll} \\hline \\hline\n      Physics Target & Sensitivity & Conditions \\\\\n      \\hline \\hline\n      Neutrino study w\/ J-PARC $\\nu$~~ && 1.3\\,MW $\\times$ $10^8$ sec\\\\\n      $-$ $CP$ phase precision & $<23^\\circ$ & @ $\\sin^22\\theta_{13}=0.1$, mass hierarchy known \\\\\n      $-$ $CPV$ discovery coverage & 76\\% (3\\,$\\sigma$), 57\\% ($5\\,\\sigma$) & @ $\\sin^22\\theta_{13}=0.1$, mass hierarchy known \\\\\n      $-$ $\\sin^2\\theta_{23}$ & $\\pm 0.017$ & 1$\\sigma$ @ $\\sin^2\\theta_{23}=0.5$ \\\\\n      \\hline\n      Atmospheric neutrino study && 10 years observation\\\\\n      $-$ MH determination & $> 2.2\\,\\sigma$ CL & @ $\\sin^2\\theta_{23}>0.4$ \\\\\n      $-$ $\\theta_{23}$ octant determination & $> 3\\,\\sigma$ CL & @  $|\\theta_{23} - 45^{\\circ}| > 4^{\\circ}$ \\\\\\hline\n      \\hline\n      Atmospheric and Beam Combination && 10 years observation\\\\\n      $-$ MH determination & $> 3.8\\,\\sigma$ CL & @ $\\sin^2\\theta_{23}>0.4$ \\\\\n      $-$ $\\theta_{23}$ octant determination & $> 3\\,\\sigma$ CL & @  $|\\theta_{23} - 45^{\\circ}| > 2.3^{\\circ}$ \\\\\\hline\n      %\n     Nucleon Decay Searches && 1.9 Mton$\\cdot$year exposure \\\\\n      $-$ $p\\rightarrow e^+ + \\pi^0$ & $7.8 \\times 10^{34}$ yrs (90\\% CL UL) &\\\\\n                     & $6.3 \\times 10^{34}$ yrs ($3\\,\\sigma$ discovery) &\\\\\n      $-$ $p\\rightarrow \\bar{\\nu} + K^+$ & $3.2 \\times 10^{34}$ yrs (90\\% CL UL) &\\\\\n                     & $2.0 \\times 10^{34}$ yrs ($3\\,\\sigma$ discovery) &\\\\ \n      \\hline\n      Astrophysical neutrino sources && \\\\\n      $-$ $^8$B $\\nu$ from Sun & 130 $\\nu$'s \/ day & 4.5\\,MeV threshold (visible\n                                                     energy) w\/ osc.\\\\\n      $-$ Supernova burst $\\nu$ & 54,000$-$90,000 $\\nu$'s & @ Galactic center (10 kpc)\\\\ \n                     & $\\sim$10 $\\nu$'s & @ M31 (Andromeda galaxy) \\\\ \n      $-$ Supernova relic $\\nu$ & 70 $\\nu$'s \/ 10 years & 10$-$30\\,MeV, 4.2$\\sigma$ non-zero significance  \\\\\n      $-$ WIMP annihilation in the Earth & & 10 years observation\\\\\n      ~~($\\sigma_{SD}$: WIMP-proton spin & $\\sigma_{SD}=10^{-40}$cm$^2$ & @ $M_{\\rm WIMP}=10$\\,GeV, $\\chi\\chi\\rightarrow b\\bar b$ dominant\\\\\n      ~~~~dependent cross section)& $\\sigma_{SD}=10^{-44}$cm$^2$ & @ $M_{\\rm WIMP}=50$\\,GeV, $\\chi\\chi\\rightarrow \\tau^+ \\tau^-$ dominant\\\\\n      \\hline \\hline\n    \\end{tabular}\n  \\end{center}\n\\end{table}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThere is strong observational evidence \\cite{observation_dm} that approximately $22\\%$ of the total energy density of the universe is in the form of dark matter. Up until now it is unclear what this dark matter should be made of. One of the favourite candidates are Weakly Interacting Massive Particles (WIMPs) which\narise in extensions of the Standard Model.\nRecently, new theoretical models of the dark matter sector have been proposed \\cite{dark}, in which\nthe Standard Model is coupled to the dark sector via an attractive interaction term.\nThese models have been motivated by new astrophysical observations \\cite{observation_ee} which show an excess \nin electronic production in the galaxy. Depending on the experiment, the energy of these\nexcess electrons is between a few GeV and a few TeV.\nOne possible explanation for these observations is the annihilation of dark matter \ninto electrons. Below the GeV scale, the interaction term in these models is basically of the form of a direct\ncoupling between the U(1) field strength tensor of the dark matter sector and the U(1) field strength tensor of\nelectromagnetism. The U(1) symmetry of the dark sector  has to be spontaneously broken, otherwise a ``dark photon'' background leading to observable consequences would exist.\n\nConsequently, it has been shown that the dark sector can have string-like solutions, denominated ``dark strings'' \nand the observational consequences of the interaction of these dark strings with the Standard Model\nhave been discussed \\cite{vachaspati}.\n\nTopological defects are believed to have formed in the numerous phase transitions in the early\nuniverse due to the Kibble mechanism \\cite{topological_defects}.\nWhile magnetic monopoles and domain walls, which result from the spontaneous\nsymmetry breaking of a spherical and parity symmetry, respectively, \nare catastrophic for the universe since they would overclose it, cosmic strings\nare an acceptable remnant from the early universe. These objects \nform whenever an axial symmetry gets spontaneously broken and (due to topological arguments)\nare either infinitely long or exist in the form of cosmic string loops. Numerical\nsimulations of the evolution of cosmic string networks have shown that\nthese reach a scaling solution, i.e. their contribution to the total energy density\nof the universe becomes constant at some stage. The main mechanism that allows\ncosmic string networks to reach this scaling solution is the formation\nof cosmic string loops due to self-intersection and the consequent decay of these loops\nunder the emission of gravitational radiation.\n\nFor some time, cosmic strings were believed to be responsible for the structure\nformation in the universe. New Cosmic Microwave background (CMB) data, however, clearly\nshows that the theoretical power spectrum associated to Cosmic strings\nis in stark contrast to the observed power spectrum. However, there has been\na recent revival of cosmic strings since it is now believed that cosmic  strings\nmight be linked to the fundamental strings of string theory \\cite{polchinski}.\n\nWhile perturbative fundamental strings were excluded to be observable on cosmic scales\nfor many reasons \\cite{witten}, there are now new theories containing\nextra dimensions, so-called brane world model, that allow to lower the fundamental\nPlanck scale down to the TeV scale. This and the observation that\ncosmic strings generically form at the end of inflation in inflationary models\nresulting from String Theory \\cite{braneinflation} and Supersymmetric Grand Unified Theories \\cite{susyguts}\nhas boosted the interest in comic string solutions again.\n\nDifferent field theoretical models describing cosmic strings have been investigated.\nThe $U(1)$ Abelian--Higgs model possesses string--like solutions \\cite{no}. This is a simple toy\nmodel that is frequently used to describe cosmic strings. However, the symmetry breaking\npattern $U(1)\\rightarrow 1$ has very likely never occurred in the evolution of the universe.\nConsequently, more realistic models with gauge group $SU(2)\\times U(1)$ and symmetry\n breaking $SU(2)\\times U(1)\\rightarrow U(1)$ have been considered and it has been\nshown that these models have string--like solutions \\cite{semilocal,gors}. Semilocal strings\nare solutions of a $SU(2)_{global}\\times U(1)_{local}$ model which -- in fact --\ncorresponds to the Standard Model of Particle physics in the limit $\\sin^2\\theta_{\\rm w}=1$, where\n$\\theta_{\\rm w}$ is the Weinberg angle. The simplest semilocal string solution is an\nembedded Abelian--Higgs solution \\cite{semilocal}. A detailed analysis of the stability\nof these embedded solutions has shown \\cite{hindmarsh} that they are unstable (stable)\nif the Higgs boson mass is larger (smaller) than the gauge boson mass. In the case\nof equality of the two masses, the solutions fulfill a Bogomolny--Prasad--Sommerfield (BPS) \\cite{bogo}\nbound such that their energy per unit length is directly proportional to the winding number.\nInterestingly, it has been observed \\cite{hindmarsh} that in this BPS limit, a one-parameter\nfamily of solutions exists: the Goldstone field can form a non-vanishing condensate\ninside the string core and the energy per unit length is independent of the value of this condensate.\nThese solutions are also sometimes denominated ``skyrmions'' and have been related to the zero-mode present\nin the BPS limit.\n\nIn this paper, we consider the interaction of dark strings with string--like solutions\nof the Standard Model in the specific limit $\\sin^2\\theta_{\\rm W}=1$. The two sectors\ninteract via an attractive interaction that couples the two $U(1)$ field strength tensors to each other.\nThis type of interaction has been studied before in \\cite{ha}, where the interaction\nbetween Abelian--Higgs strings and dark strings has been investigated. It has been \nfound that a BPS bound exists that depends on the interaction paramater and that Abelian--Higgs strings\nand dark strings can form bound states.\n\n\nOur paper is organized as follows: in Section 2, we give the model, the equations of motion, the boundary\nconditions and the asymptotics. In Section 3, we present our numerical results and Section 4 contains\nour conclusions.\n\n\n\n\\section{The model}\nWe study the interaction of a $SU(2)_{global}\\times U(1)_{local}$ model, which\nhas semilocal strings solutions \\cite{semilocal}\nwith the low energy dark sector, which is a $U(1)$ Abelian-Higgs model.\n\n\nThe matter Lagrangian\n${\\cal L}_{m}$ reads:\n\\begin{equation}\n{\\cal L}_{m}=(D_{\\mu} \\Phi)^{\\dagger} D^{\\mu} \\Phi-\\frac{1}{4} F_{\\mu\\nu} F^{\\mu\\nu}\n-\\frac{\\lambda_1}{2}(\\Phi^{\\dagger}\\Phi-\\eta_1^2)^2\n+(D_{\\mu} \\xi)^* D^{\\mu} \\xi-\\frac{1}{4} H_{\\mu\\nu} H^{\\mu\\nu}\n- \\frac{\\lambda_2}{2}(\\xi^*\\xi-\\eta_2^2)^2  + \\frac{\\varepsilon}{2} F_{\\mu\\nu}H^{\\mu\\nu}\n\\end{equation} \nwith the covariant derivatives $D_\\mu\\Phi=\\nabla_{\\mu}\\Phi-ie_1 A_{\\mu}\\Phi$,\n$D_\\mu\\xi=\\nabla_{\\mu}\\xi-ie_2 a_{\\mu}\\xi$\nand the\nfield strength tensors $F_{\\mu\\nu}=\\partial_\\mu A_\\nu-\\partial_\\nu A_\\mu$, \n$H_{\\mu\\nu}=\\partial_\\mu a_\\nu-\\partial_\\nu a_\\mu$  of the two U(1) gauge potential $A_{\\mu}$, $a_{\\mu}$ with coupling constants $e_1$\nand $e_2$.\n$\\Phi=(\\phi_1,\\phi_2)^T$ is a complex scalar doublet, while $\\xi$ is a complex scalar field.\nThe gauge fields have masses $M_{W,i}=\\sqrt{2}e_i \\eta_i$, $i=1,2$, while the Higgs fields\nhave masses $M_{H,i}=\\sqrt{2\\lambda_i} \\eta_i$, $i=1,2$.\nThe term proportional to $\\varepsilon$ is the interaction term \\cite{vachaspati}.\nTo be compatible with observations, $\\varepsilon$ should be on the order of $10^{-3}$. \n\n\\subsection{The Ansatz}\n\nFor the matter and gauge fields, we have \\cite{semilocal,hindmarsh,no}:\n\\begin{equation}\n\\phi_1(\\rho,\\varphi)=\\eta_1 h_1(\\rho)e^{i n\\varphi} \\ \\ , \\ \\  \\phi_2(\\rho)=\\eta_1 h_2(\\rho) \\ \\ , \\ \\\n\\xi(\\rho,\\varphi)=\\eta_2 f(\\rho)e^{i m\\varphi} \n\\end{equation}\n\\begin{equation}\nA_{\\mu}dx^{\\mu}=\\frac {1}{e_1}(n-P(\\rho)) d\\varphi \\ \\ , \\ \\ a_{\\mu}dx^{\\mu}=\\frac {1}{e_2}(m-R(\\rho)) d\\varphi \\ .\n\\end{equation}\n$n$ and $m$ are integers indexing the vorticity of the two Higgs fields  around the $z-$axis.\nIn the following, we will refer to solutions with $h_2(\\rho)\\equiv 0$ as ``embedded Abelian--Higgs solutions'', while solutions with $h_2(\\rho)\\neq 0$ will be referred to as ``semilocal solutions''.\nNote that in the case $\\varepsilon=0$, the solutions of the semilocal sector of\nour model are often also denominated ``skyrmions''.\n\n\n\\subsection{Equations of motion}\nWe define the following dimensionless variable $x=e_1\\eta_1 \\rho$, which measures the radial\ndistance in units of $M_{W,1}\/\\sqrt{2}$. \n\nThen, the total Lagrangian ${\\cal L}_m \\rightarrow {\\cal L}_m\/(\\eta_1^4 e_1^2)$ depends only on the following dimensionless coupling constants\n\n\\begin{equation}\n\\beta_i=\\frac{\\lambda_i}{e_1^2}=\\frac{M^2_{H,i}}{M^2_{W,1}}\\frac{\\eta_1^2}{\\eta_i^2} \\ , \\ i=1,2   \\ , \\ \\  g=\\frac{e_2}{e_1} \\ \\ , \\  \\ \\\nq=\\frac{\\eta_2}{\\eta_1} \\ \\ . \\ \\ \n\\end{equation}\n Varying the action with respect to the matter fields we\nobtain a system of five non-linear differential equations. The Euler-Lagrange equations for the matter field functions read:\n\\begin{equation}\n\\label{eq1}\n(xh_1')'=\\frac{P^2 h_1}{x}+\\beta_1x(h_1^2+h_2^2-1)h_1\n\\end{equation}\n\\begin{equation}\n\\label{eq2}\n(xh_2')'=\\frac{(n-P)^2 h_2}{x}+\\beta_1x(h_1^2+h_2^2-1)h_2\n\\end{equation}\n\\begin{equation}\n\\label{eq3}\n(xf')'=\\frac{R^2f}{x}+\\beta_2x(f^2-q^2)f\n\\end{equation}\n\\begin{equation}\n\\label{eq4}\n(1-\\varepsilon^2)\\left(\\frac{P'}{x}\\right)'=2 \\frac{h_1^2 P}{x} -2\\frac{(n-P)h_2^2}{x} + 2\\varepsilon g \\frac{R f^2}{x} \\ ,\n\\end{equation}\n\\begin{equation}\n\\label{eq5}\n(1-\\varepsilon^2)\\left(\\frac{R'}{x}\\right)'=2 g^2 \\frac{f^2 R}{x} + 2\\varepsilon g \\left(\\frac{P h_1^2}{x}-\\frac{(n-P)h_2^2}{x}\\right) \\ ,\n\\end{equation}\nwhere the prime now and in the following denotes the derivative with respect to $x$.\n\n\\subsection{Energy per unit length and magnetic fields}\n\nThe non-vanishing components of the energy-momentum tensor are (we use the notation\nof \\cite{clv}):\n\\begin{eqnarray}\nT_0^0 &=& e_s + e_v + e_w + u  \\ \\ , \\ \\ \nT_x^x = -e_s - e_v + e_w + u \\nonumber \\\\\nT_{\\varphi}^{\\varphi} &=&e_s - e_v - e_w + u \\ \\ , \\ \\ T_z^z =  T_0^0 \n\\end{eqnarray}\nwhere\n\\begin{equation}\n\\label{contributions}\ne_s= (h_1')^2 + (h_2')^2 + (f')^2   \\ \\ \\ , \\ \\ \\ e_v = \\frac{(P')^2}{2 x^2} + \\frac{(R')^2}{2 g^2 x^2} -\\frac{\\varepsilon}{g}\\frac{R'P'}{x^2} \\ \\ \\ , \\ \\ \\ e_w = \\frac{h_1^2 P^2}{x^2} + \\frac{h_2^2 (n-P)^2}{x^2} + \\frac{R^2 f^2}{x^2} \\end{equation}\nand\n\\begin{eqnarray}\nu & = & \\frac{\\beta_1}{2}\\left(h_1^2+h_2^2-1\\right)^2 + \\frac{\\beta_2}{2} \\left(f^2-q^2\\right)^2  \\ . \n\\end{eqnarray}\n\nWe define as inertial energy per unit length  of a solution describing the interaction of a semilocal\nstring with winding $n$ and a dark string with winding $m$:\n\\begin{equation}\n \\mu^{(n,m)}=\\int \\sqrt{-g_3} T^0_0 dx d\\varphi\n\\end{equation}\nwhere $g_3$ is the determinant of the $2+1$-dimensional space-time given by $(t,x,\\varphi)$.\nThis then reads:\n\\begin{equation}\n \\mu^{(n,m)}=2\\pi\\int_{0}^{\\infty} x \\left(\\varepsilon_s + \\varepsilon_v + \\varepsilon_w + u\\right) \\ dx\n\\end{equation}\nNote that the string tension $T=\\int \\sqrt{-g_3} \\ T^z_z dx d\\varphi$ is equal to the energy per unit length. There are a few special case, in which energy bounds can be given:\n\\begin{enumerate}\n \\item\nFor $h_2(x)\\equiv 0$, the energy per unit length of the solution is\ngiven by:\n\\begin{equation}\n\\mu^{(n,m)}=2\\pi n \\eta_1^2 g_1(\\beta_1) + 2\\pi m \\eta_1^2   g_2(\\beta_2) \n\\end{equation}\nwhere $g_1$ and $g_2$ are functions that depend only weakly on $\\beta_1$ and $\\beta_2$, respectively.\nThe energy bound is fulfilled, when the functions $g_1$ and $g_2$ become equal to unity.\nThis happens at $\\beta_1=\\beta_2=1\/(1-\\varepsilon)$  and $n=m$ \\cite{ha}.\n\n\\item For $\\varepsilon=0$ and $h_2(x)\\neq  0$, the energy per unit length of the solution is\ngiven by\n\\begin{equation}\n\\mu^{(n,m)}=2\\pi n \\eta_1^2 + 2\\pi m \\eta_2^2   g_2(\\beta_2) \n\\end{equation}\nwhere $g_2$ is a  function that depends only weakly on $\\beta_2$ with $g_2(1)=1$.\nNote that the solution of the semilocal sector exists only for $\\beta_1=1$ and fulfills the BPS bound\nfor all choices of $h_2(0)$.\n\n\\end{enumerate}\n\nThe magnetic fields associated to the solutions are given by \\cite{ha}~:\n\\begin{equation}\n\\label{magnetic}\n B_z(x)=\\frac{-P'(x)+\\frac{\\varepsilon}{g} R'(x)}{e_1 x} \\ \\ \\ {\\rm and} \\ \\ \\   b_z(x)=-\\sqrt{1-\\varepsilon^2}\\frac{R'(x)}{e_2 x}  \\ ,  \n\\end{equation}\nrespectively, where we have used the fact that the part of the Lagrangian containing\nthe field strength tensors can be rewritten as \\cite{vachaspati}~:\n\\begin{equation}\n -\\frac{1}{4} F_{\\mu\\nu} F^{\\mu\\nu}-\\frac{1}{4} H_{\\mu\\nu} H^{\\mu\\nu}+ \\frac{\\varepsilon}{2} F_{\\mu\\nu}H^{\\mu\\nu} \\Rightarrow -\\frac{1}{4} G_{\\mu\\nu} G^{\\mu\\nu} -\\frac{1}{4}(1-\\varepsilon^2) H_{\\mu\\nu} H^{\\mu\\nu}\n\\end{equation}\nwith $G_{\\mu\\nu}=\\partial_{\\mu} \\tilde{A}_{\\nu}- \\partial_{\\nu} \\tilde{A}_{\\mu}$\nwhere $\\tilde{A}_{\\mu}=A_{\\mu}-\\varepsilon a_{\\mu}$.\nThe corresponding magnetic fluxes $\\int d^2x \\ B$ are\n\\begin{equation}\n \\Psi= \\frac{2\\pi}{e_1}\\left(n-\\frac{\\varepsilon}{g} m\\right) \\ \\ {\\rm and} \\ \\ \n\\psi=\\sqrt{1-\\varepsilon^2} \\ \\frac{2\\pi m}{e_2} \\ ,\n\\end{equation}\nrespectively. Obviously, these magnetic fluxes are not quantized for generic $\\varepsilon$.\n\n\n\\subsection{Boundary conditions and asymptotics}\nThe requirement of regularity at the origin leads to the  following boundary \nconditions:\n\\begin{equation}\nh_1(0)=0 \\ , \\ h_2'(0)=0 \\ ,  \\ f(0)=0 \\ , \\ P(0)=n \\ , \\ R(0)=m\n\\label{bc1}\n\\end{equation}\nFor $h_2(0)=0$, the semilocal strings correspond to embedded Abelian--Higgs \nstrings. Here, we are mainly interested in constructing solutions that are truly semilocal, i.e. we require $h_2(0)\\neq 0$.\nThe finiteness of the energy per unit length requires:\n\\begin{equation}\nh_1(\\infty)=1 \\ , \\ h_2(\\infty)=0 \\ ,  \\ f(\\infty)=q \\ , \\ P(\\infty)=0 \\ , \\ R(\\infty)=0  \\ .\n\\end{equation}\n\nThe asymptotic behaviour for $x\\rightarrow \\infty$ depends crucially on whether the function $h_2(x)\\equiv 0$ or\n$h_2(x)\\neq  0$. \n\\begin{enumerate}\n \\item For $h_2(x)\\equiv 0$ we find:\n\\begin{eqnarray}\n P (x\\rightarrow \\infty) &=& - \\sqrt{x} \\  m_{12} \\ \\left[C_1  \\exp\\left(-x\\beta_+\\right) + C_2  \\exp\\left(-x \\beta_-\\right) \\right] + .... \\\\\n R(x\\rightarrow \\infty) &=& \\sqrt{x} \\ \n \\left[C_1 \\ m_{11}(\\beta_+) \\exp\\left(-x\\beta_+\\right) + C_2 \\ m_{11}(\\beta_-) \\exp\\left(-x \\beta_-\\right) \\right]+ ...\n\\end{eqnarray}\nwhere $C_1$ and $C_2$ are constants,  $m_{11}(\\beta_{\\pm})= (1-\\varepsilon^2)\\beta_{\\pm}^2 - 2$ and $m_{12} = -2 \\varepsilon q^2 g$. The $\\beta_{\\pm}$ are positive and are given by\n\\begin{equation}\n   \\beta^2_{\\pm} = \\frac{1+q^2 g^2 \\pm \\sqrt{(1-q^2 g^2)+ 4 \\varepsilon q^2 g^2}}{1-\\varepsilon^2}\n\\end{equation}\nThe numerical evaluation (see below) shows that for specific values of the coupling constants the constants $C_1$ and $C_2$\nhave opposite sign. Hence, the function $R(x)$ can possess a node asymptotically which we have confirmed\nnumerically. However, the numerics has shown that these type of solutions exist only\nfor values of $\\varepsilon$ of order one. Hence, we don't present them here since we believe that\nthey are unphysical.\n\nFor the scalar fields, we find \n\\begin{eqnarray}\nh_1 (x\\rightarrow \\infty) &=& 1 + \\frac{C_3}{\\sqrt{x}} \\exp\\left(-x\\sqrt{2\\beta_1}\\right) \n+ \\frac{c_+}{x} \\exp\\left(-2x \\beta_+\\right) + \\frac{c_-}{x} \\exp\\left(-2x \\beta_-\\right) + ...\\\\\nf(x\\rightarrow \\infty) &=& q + \\frac{C_4}{\\sqrt{x}} \\exp\\left(-x\\sqrt{2\\beta_2}\\right) \n+ \\frac{d_+}{x} \\exp\\left(-2x \\beta_+\\right) + \\frac{d_-}{x} \\exp\\left(-2x \\beta_-\\right) + ...\n\\end{eqnarray}\n$C_3$ and $C_4$ are two constants, while $c_{\\pm}$, $d_{\\pm}$ depend on \nthe constants $C_1, \\dots, C_4$ and on $\\beta_1$ and $\\beta_2$.\n\n\\item For $h_2(x)\\neq 0$ we find~:\n\\begin{equation}\n\\label{as_gauge}\nP(x\\rightarrow \\infty)= \\frac{n c^2}{x^{2n}} + ... \\ \\ , \\ \\ R(x\\rightarrow \\infty)= \\frac{c_R}{x^{2n+2}} + ...\n \\end{equation}\nfor the gauge field functions. Here $c$, $c_R$ are constants that depend on the values of the coupling constants.\nFor the scalar and Higgs field functions we have\n\\begin{equation}\n\\label{as_scalar}\nh_1(x\\rightarrow \\infty) = 1 - \\frac{c^2}{2} \\frac{1}{x^{2n}} + ... \\ \\ , \\ \\ h_2(x) = \\frac{c}{x^n} + ... \\ \\ , \\ \\ \nf(x\\rightarrow \\infty)= q - \\frac{c_R^2}{2 q \\beta_2} \\frac{1}{x^{4n+6}} + ....\n\\end{equation}\n\n\\end{enumerate}\nObviously, the presence of the scalar field $h_2(x)$ changes the asymptotics drastically.\nWhile for $h_2(x)\\equiv 0$, the gauge and Higgs fields decay exponentially, they have power-law \ndecay for $h_2(x)\\neq 0$. \n\n\\subsection{Stability}\n\nFollowing the investigation in the case $\\varepsilon=0$ \\cite{hindmarsh}, we are interested\nin the stability of the embedded Abelian--Higgs string coupled to a dark string.\nIn order to do that we will study\nthe normal mode along a very specific (but standard) direction\nin perturbation space about the embedded Abelian--Higgs string coupled to a dark string. We consider the perturbation\n\\begin{equation}\n      h_1(x)=\\tilde{h}_1(x) \\ , \\ h_2(x)= {\\rm e}^{i\\omega t}\\eta(x) \\ , \\ \n      P(x) = \\tilde{P}(x) \\ , \\ R(x) = \\tilde{R}(x) \\ , \\ f(x) = \\tilde{f}(x)\n\\end{equation}\nwhere the tilded functions denote the profiles of an embedded Abelian--Higgs string\ncoupled to a dark string, i.e. solutions to the equations (\\ref{eq1}),(\\ref{eq3}), (\\ref{eq4}) and (\\ref{eq5}) for $h_2(x)\\equiv 0$. The perturbation is denoted by $\\eta$\nand the parameter $\\omega$ is real. \nInserting this perturbation into (\\ref{eq2}) and keeping only the linear terms in $\\eta$ leads to the\n linear eigenvalue equation~:\n\\begin{equation}\n\\label{sta}\n       \\left(-\\frac{d^2}{dx^2} - \\frac{1}{x} \\frac{d}{dx} + V_{eff}\\right)\\eta(x) =  \\omega^2 \\eta(x) \\ \\ , \n       \\ \\ V_{eff} = \\frac{(n-\\tilde{P}(x))^2}{x^2} + \\beta_1 (\\tilde{h}_1(x)^2-1)\n\\end{equation}\nThe spectrum of the linear operator entering in (\\ref{sta}) consists of a continuum for $\\omega^2 > 0$ \nand of a finite number of  bound states (or normalisable solutions) for $\\omega^2 < 0$.\nIn the latter case, the solutions fulfill\n\\begin{equation}\n     \\eta(0) = 1 \\ , \\ \\eta'(0) = 0 \\ \\ \\ {\\rm with} \\ \\  \\ \\eta(x) \\to e^{-|\\omega| x} \\ {\\rm for} \\ x \\to \\infty\n\\end{equation} \nwhere we have fixed  the arbitrary normalisation by choosing $\\eta(0)=1$.\n\n\nOnly bound states are of interest to us since they signal the presence of an instability. \n  It should be pointed out that the functions $\\tilde{P}(x)$, $\\tilde{h}_1(x)$ entering in the\n effective potential feel the effect of the\ndark sector since the corresponding equations are directly coupled.\n\n\n\n\n\n\n\n\\section{Numerical results}\nFor all our numerical calculations, we have chosen $q=g=1$.\n\n\\subsection{Stability of the embedded Abelian--Higgs--dark strings}\nWe have first studied the stability of the embedded Abelian--Higgs strings coupled\nto dark strings by investigating the bound states of (\\ref{sta}) for different\nvalues of $\\varepsilon$. Our results for $n=m=1$ and $\\beta_2=1$ are shown in Fig.\\ref{fignew}.\n\n\\begin{figure}[!htb]\n\\centering\n\\leavevmode\\epsfxsize=12.0cm\n\\epsfbox{eigenvalue.eps}\\\\\n\\caption{\\label{fignew} We give the value of $\\omega^2$ (see (\\ref{sta}) in dependence\non $\\beta_1$ for three different choices of $\\varepsilon$ including the non--interacting\ncase $\\varepsilon=0$. Here $n=m=1$ and $\\beta_2=1$. }\n\\end{figure}\n\nFor $\\varepsilon=0$ we recover the result of \\cite{hindmarsh} that the\nembedded--Abelian-Higgs strings are unstable for $\\beta_1 >1$. \nFor $\\varepsilon \\neq 0$, we observe that the larger $\\varepsilon$, the larger the\nratio of Higgs to gauge boson mass $\\beta_1$ at which the embedded Abelian--Higgs strings\ncoupled to dark strings become unstable. In the following, we will denote the value of $\\beta_1$ at which\n$\\omega^2=0$ $\\beta_1^{cr}$.  \nWith view to the observations for the $\\varepsilon=0$ case, we would thus expect additional\nsolutions with $h_2(x)\\neq 0$ for $\\beta_1 > \\beta_1^{cr}$. \nIn section 3.2, we will discuss the properties of these solutions.\n \nLet us also remark that our analysis does not reveal the occurence of additional unstable modes in the sector explored.\n\n\n\n\\subsection{Properties of semilocal--dark strings}\nIn the case $\\varepsilon=0$, the two sector do not interact and for the semilocal sector\ntwo different types of solutions are possible: (a) embedded Abelian--Higgs solutions with\n$h_2(x)\\equiv 0$ which exist for generic choices of $\\beta_1$ \\cite{semilocal} and \n(b) semilocal strings (``skyrmions'') with $h_2(x)\\neq 0$ which exist only for $\\beta_1=1$ \\cite{hindmarsh}.\nIn the latter case, it was shown that \nthere is a zero mode associated to the fact that the energy of the ``skyrmions''\ndoes not depend on the value of $h_2(0)$. \n\nThe case with $\\varepsilon\\neq 0$ and $h_2(x)\\equiv 0$ corresponds hence to the case\nof an embedded Abelian--Higgs string interacting with a dark string. The equations\nof motion that describe this case are exactly those studied in \\cite{ha}.\nIn \\cite{ha}, the interaction of a dark string with an Abelian--Higgs string has been studied\nin detail. Since the only difference between an Abelian--Higgs string and\nan embedded Abelian--Higgs string is the stability  -- see section 3.1 -- we do not discuss this case in detail in this paper and focus on the\ncase of semilocal strings interacting with dark strings. We have solved the\ndifferential equations subject to the boundary conditions numerically using the\nODE solver COLSYS \\cite{colsys}.\n\n\\begin{figure}[!htb]\n\\centering\n\\leavevmode\\epsfxsize=12.0cm\n\\epsfbox{ener_dens.eps}\\\\\n\\caption{\\label{fig0} We give the profiles of the energy density $T_0^0$, the effective energy density $x\\cdot T^0_0$ as well as the\nmagnetic field $B_z$ (see (\\ref{magnetic})) for\n$\\varepsilon=1\/6$, $\\beta_1=3$ and $\\beta_2=(1-\\varepsilon)^{-1}=1.2$. We compare semilocal--dark string\nsolutions with $h_2(0) > 0$ (black) and embedded Abelian--Higgs--dark string solutions with $h_2(0)=0$ (red).}\n\\end{figure}\n\nTo see the difference between embedded Abelian--Higgs--dark string solutions\nand semilocal--dark string solutions, we present the energy density $T_0^0$, the effective\nenergy density $xT_0^0$ as well as the magnetic field $B_z$ (see (\\ref{magnetic}))\nin Fig.\\ref{fig0} for $\\varepsilon=1\/6$, $\\beta_1=3$ and $\\beta_2=(1-\\varepsilon)^{-1}=1.2$. Clearly, the effective energy density tends to zero very quickly for the\nembedded--Abelian--Higgs--dark string, while for the semilocal--dark string is has a long tail which\nresults from the power--law fall off of the functions. Moreover, the magnetic field $B_z$\ntends to zero exponentially for the embedded Abelian--Higgs--dark strings, while it falls\noff power--like for the semilocal--dark strings. Hence, the core of the\nmagnetic flux tube of the latter solution is not well defined.\n\n\n\nWhile for $\\varepsilon=0$ solutions with $h_2(x)\\neq 0$ exist only for\n$\\beta_1=1$, the situation is different here.\nFor $\\varepsilon\\neq 0$, we find solutions for generic values of $\\beta_1$, i.e. different from\nunity. In fact, the solutions exist only for $\\beta_1$ larger than a critical \nvalue, $\\beta_1^{cr}$,  which depends on the choice of the winding numbers\nand other coupling constants, in particular $\\varepsilon$. Moreover, we observe that\nthe $\\beta_1$ at which semilocal--dark strings exist is a function of $h_2(0)$.\nWhile for $\\varepsilon=0$, $\\beta_1=1$ for all choices of $h_2(0)$, we find that for $\\varepsilon\\neq 0$\nthe choice of $h_2(0)$ fixes the value of $\\beta_1$.\n\n\\begin{figure}[!htb]\n\\centering\n\\leavevmode\\epsfxsize=12.0cm\n\\epsfbox{data_eps_01.eps}\\\\\n\\caption{\\label{fig1} The energy per unit length $\\mu^{(1,1)}$ (in units of $2\\pi\\eta_1^2$) as well as the value of $h_2(0)$ and the asymptotic constants  $c$ and $c_R$ (see (\\ref{as_gauge}), \n(\\ref{as_scalar})) of the semilocal--dark  string solutions are shown in dependence on $\\beta_1$ for $\\varepsilon=0.1$, $\\beta_2=1$ and $n=m=1$ (dashed).\nFor comparison, we also give the energy per unit length of the embedded Abelian--Higgs--dark string solution (solid).}\n\\end{figure}\n\n\\begin{figure}[!htb]\n\\centering\n\\leavevmode\\epsfxsize=12.0cm\n\\epsfbox{data_eps_05.eps}\\\\\n\\caption{\\label{fig2}The energy per unit length $\\mu^{(1,1)}$ (in units of $2\\pi\\eta_1^2$) as well as the value of $h_2(0)$ and the asymptotic constants  $c$ and $c_R$ (see (\\ref{as_gauge}), (\\ref{as_scalar})) of the semilocal--dark string solution are shown in dependence on $\\beta_1$ for $\\varepsilon=0.5$, $\\beta_2=1$ and $n=m=1$ (dashed). For comparison, we also give the energy per unit length of the embedded Abelian--Higgs--dark string solution (solid). }\n\\end{figure}\n\nAt $\\beta_1^{cr}$ the branch of solutions describing a semilocal string in\ninteraction with a dark string bifurcates with the branch of solutions describing\nthe interaction of an embedded Abelian--Higgs string with a dark string. \nThis is shown in Fig.s \\ref{fig1},\\ref{fig2} for $\\varepsilon=0.1$ and $\\varepsilon=0.5$, respectively. Note that $\\beta_1^{cr}$ is exactly the value at which the\nembedded Abelian--Higgs--dark strings become unstable.\n\nHere, we give the value of $h_2(0)$ in dependence on $\\beta_1$ for $n=m=1$ and $\\beta_2=1.0$.\nClearly at some $\\beta_1^{cr}$, $h_2(0)$ tends to zero which means that $h_2(x)\\equiv 0$.\nHere the semilocal--dark string solutions bifurcate with the embedded Abelian--Higgs--dark string solutions.\nWe also compare the energy per unit length of the two types of solutions. Clearly,\nwhenever semilocal--dark string solutions exist, they have lower energy than \nthe corresponding embedded Abelian--Higgs--dark string solutions. Moreover, the larger $\\beta_1$, the bigger\nis the difference between the two energies per unit length.\nWe would thus expect the semilocal solutions to be stable with respect to the decay into the\nembedded Abelian solutions when coupled to dark strings. \nWe also present the values of the asymptotic constants $c$ and $c_R$ (see (\\ref{as_gauge}), (\\ref{as_scalar})). These vanish identically at $\\beta_1=\\beta_1^{cr}$.\n\n\n\\begin{figure}[!htb]\n\\centering\n\\leavevmode\\epsfxsize=14.0cm\n\\epsfbox{self_dual.eps}\\\\\n\\caption{\\label{fig4} The energy per unit length $\\mu^{(1,1)}$ (in units\nof $2\\pi\\eta_1^2$) is shown for semilocal strings interacting with dark strings as function of $\\beta_1$ for $\\beta_2=(1-\\varepsilon)^{-1}$ with $\\varepsilon=0.5$ and $\\varepsilon=1\/6$, respectively (dashed). \nFor comparison, we also give the energy per unit length of the corresponding embedded Abelian--Higgs\nsolutions interacting with dark strings (solid).}\n\\end{figure}\n\n\n\\begin{figure}[!htb]\n\\centering\n\\leavevmode\\epsfxsize=14.0cm\n\\epsfbox{ep_betacr.eps}\\\\\n\\caption{\\label{fig3} The value of $\\beta_1^{cr}$ at which the branch of semilocal\nsolutions bifurcates with the branch of embedded Abelian--Higgs solutions  is shown as function\nof $\\varepsilon$ for $m=1$, $m=2$, respectively and $\\beta_2=1.0$, $\\beta_2=2.0$, respectively.  }\n\\end{figure}\n\nIn general, $\\beta_1^{cr}$ will depend on the choice of $\\beta_2$, $n$ and $m$: $\\beta_1^{cr}(\\beta_2,n,m)$.\nAs shown in \\cite{ha} in the limit $h_2(x)\\equiv 0$ a BPS bound exists for $\\beta_1=\\beta_2=(1-\\varepsilon)^{-1}$ and $n=m$. In this limit, the energy per unit length (in units\nof $2\\pi\\eta_1^2$) is just $n+m=2n$. We have studied the dependence of the energy per unit length\non $\\beta_1$ for $\\beta_2=(1-\\varepsilon)^{-1}$ where $\\varepsilon=1\/6$\nand $\\varepsilon=0.5$, respectively. We have chosen $n=m=1$. Our results are given in Fig.\\ref{fig4}.\nInterestingly, we find that the branch of semilocal--dark string solutions\nbifurcates with the branch of embedded Abelian--Higgs--dark string solutions\nexactly at $\\beta_1=\\beta_2=(1-\\varepsilon)^{-1}$. For $\\beta_1 > (1-\\varepsilon)^{-1}$, the energy per unit length\nof the semilocal--dark string solutions is always smaller than that of the corresponding\nembedded Abelian--Higgs--dark string solutions, for $\\beta_1 < (1-\\varepsilon)^{-1}$ no semilocal--dark string\nsolutions exist at all. Hence, we find that \n\\begin{equation}\n \\beta_1^{cr}(\\beta_2=(1-\\varepsilon)^{-1},1,1)=(1-\\varepsilon)^{-1}\n\\end{equation}\n\n\nWe have also studied the dependence of $\\beta_1^{cr}$ on the winding of the dark string\nand the Higgs to gauge boson ratio $\\beta_2$ of the U(1) model describing the dark string in more detail.\nOur results are shown in Fig.\\ref{fig3}. Obviously, $\\beta_1^{cr}$ increases with increasing\n$\\varepsilon$. This is related to the fact that the core width of the strings decreases with increasing\n$\\varepsilon$. This means more gradient energy and hence we have to choose larger values\nof $\\beta_1$ to be able to compensate for this increase by decrease in potential energy.\n \nFor $\\beta_1=1.0$, which in fact corresponds to the BPS limit of the U(1) dark string model \nfor $\\varepsilon=0$, the value of $\\beta_1^{cr}$ increases for increasing winding $m$ of the\ndark string. Again increasing $m$ increases gradient energy such that we have to choose\nlarger value of $\\beta_1$ to compensate the increase by decrease in potential energy.\nThis is also true when increasing $\\beta_2$. Increasing $\\beta_2$ decreases the core size of\nthe dark string, this increases gradient energy and we again have to compensate by increasing\nthe value of $\\beta_1$.\n\n\n\n\n\\section{Conclusions}\nIn this paper we have shown that the interaction of semilocal strings with dark strings\nhas important effects on the properties of the former. While embedded Abelian--Higgs strings\nexist for all values of the Higgs to gauge boson ratio when interacting with dark strings, semilocal\nstrings with a condensate inside their core exist only above a critical value of the\nHiggs to gauge boson ratio. At this critical value, the embedded\nAbelian--Higgs--dark strings become unstable. The critical value of the ratio \ndepends on the choice of the Higgs to\ngauge boson ratio of the dark string and the windings. In the limit where the\nratio tends to the critical ratio, the condensate vanishes identically and the branch of\nsemilocal--dark string solutions bifurcates with the branch of embedded Abelian--Higgs--dark string\nsolutions. Apparently, the presence of the condensate lowers the energy in such a way\nthat whenever semilocal--dark strings exist, they are lower in energy than their\nembedded Abelian--Higgs-dark string counterparts. The value of the Higgs to gauge boson ratio\nfor which semilocal--dark strings exist depends on the value of the condensate\non the string axis and increases for increasing values of the condensate.\nAll these results are quite different from what is observed in the non--interacting case.\nIn the non--interacting case, semilocal strings exist only for Higgs to gauge boson\nratio equal to unity and in this limit, the energy per unit length is independent of the value\nof the condensate and in addition fulfills a BPS bound. To state it differently~: when not interacting\nwith dark strings, semilocal strings and embedded Abelian--Higgs strings are degenerate\nin energy, while the former are lower in energy as soon as they interact with dark strings.\nSince the branch of semilocal--dark string solutions bifurcates with the branch of\nembedded Abelian--Higgs--dark strings at the self--dual point of the embedded Abelian--Higgs-dark strings\n-- at which these fulfill an energy bound \\cite{ha} --  we expect that semilocal--dark strings\nare stable. Moreover, they are stable for all choices of the Higgs to gauge boson ratio for which they exist and not just -- as in the non--interacting case -- for Higgs to gauge boson ratio smaller\nor equal to unity. Since all current observations point to the fact that the Higgs boson\nmass is larger than the gauge boson masses, semilocal strings could still be stable\nwhen interacting with dark strings.\nInterestingly -- as mentioned above -- semilocal strings can lower their energy by \nforming a non--vanishing condensate inside their core. This could be important\nfor the evolution of cosmic string networks since next to the formation of bound states \\cite{wyman}\nthis would be a further mechanism for the network to loose energy.\n\nWe didn't study the gravitational properties of the solutions since we believe that the qualitative\nfeatures are similar to the case studied in \\cite{ha}. Since semilocal--dark strings\nhave lower energy per unit length than their embedded Ablian--Higgs--dark string\ncounterparts, we would expect the deficit angle created by the former to be smaller than that of the latter. Furthermore, the critical value of the gravitational coupling at which the solutions\nbecome singular is larger for the semilocal--dark strings than for the\nembedded Abelian--Higgs--dark strings.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\subsection{Introduction}\n\\ac{ML} is a set of models which can automatically identify the hidden patterns\nin the data and can then utilize hidden patterns to make decisions in condition\nof uncertainty.  \\ac{ML} has been progressively implemented in several areas\nincluding chemistry, biomedical science and robotics. \\ac{ML} falls into three\ncategories, i.e. supervised learning (e.g. classification), unsupervised\nlearning (e.g. clustering) and reinforcement learning. In this paper we focus on\nclassification, which is the way to represent and allocate objects into different\ncategories.\n\n\\ac{QT} is the probabilistic approach to representing and predicting properties\nof microscopic phenomena. Given an observable and an arbitrary state\nof a microscopic particle, \\ac{QT} computes a probability distribution of the\nvalues of the observable. The quantum formalism is explicitly acceptable to\nexplain distinct types of stochastic processes. Several nonstandard\nimplementations of the quantum formalism has emerged. For instance, the quantum\nformalism have been utilized vastly in the economic processes, game theory  and cognitive\nscience  as well.\n\nSince the data is growing exponentially, current state-of-the-art models are\nstill not effective.  In particular, \\emph{recall} is still unsatisfactory\nbecause most classification models aim to maximize precision especially when the\nitems of a class can be ranked by a certain measure of membership to the class;\na glaring example is the search of the Internet.  In contrast, recall is crucial\nin many daily tasks aiming to find all the pertinent items of a class such as patent search and biomedical image classification.\n\nOur approach is to develop a new theoretical approach inspired by \\ac{QM} in\norder to dig into the quantum world and come up with new and effective models\nwhich are capable of increasing recall.  Our hypothesis is that, since \\ac{QM} has\nalready shown its effectiveness in several fields, it may also be effective in\n\\ac{ML}.  To this end we will exploit Quantum Probability theory, which is the quantum\ngeneralization of classical probability theory and was developed by Von\nNeumann. While classical probability theory provides that a system can be in\neither state 0 or 1, quantum probability comes into existence to go beyond\nclassical theory and describes states which can be anything in-between 0 and 1. In\nthis paper, we propose the \\ac{BCIQT} model which is a step towards shifting\nfrom classical models to quantum models.\n\\section{Proposed Methodology}\n\n\\subsection{Classical and Quantum Signal Detection Theory}\n\\label{sec:quant-detect-fram}\n\n\\ac{BCIQT} is based on the overlap between \\ac{SDT} and \\ac{QM}. The main\ndifference between the classical framework and the quantum framework of signal\ndetection regards what encoders encode and what decoders decode\n\\cite{helstrom1969quantum}.\n\nIn the classical framework,\nthere is c-c (classical-classical) mapping from a symbol to the wave to the\ncorrupted channel; then the decoder produce c-c (classical-classical) mapping\nfrom the corrupted channel wave to a symbol.  In the quantum framework, there is a coder between the source and\nthe channel; the classical symbol is transmitted through the quantum\nstate. Initial encoding starts like c-q (classical-quantum) mapping from the\nsymbol to the quantum state selected from a finite set of possible states. More\ndetails about classical and quantum \\ac{SDT} can be found in\n\\cite{helstrom1969quantum}.\n\n\n\n\\subsection{Binary Classifier Inspired by Quantum Theory}\n\nA novel \\ac{BCIQT} that is inspired by quantum detection theory is described in\nthis section.  For each category we supposed that each training sample was about\nthe category or not.  For a given category and the set of training samples, we\nused the projector $\\Delta$ for each category to identify whether the test\nsample was about the category or not. To determine whether the test sample was\nabout the category, $\\Delta$ was examined against a vectorial representation of\nthe test samples.\n\nConsider a set of distinct features calculated from the whole sample collection.\nEach sample could be represented as a vector of features; each element in the\nfeature vector was a non-negative number such as frequency.  Each sample in the\ntraining set had a binary label in $\\{0,1\\}$.  The main goal of \\ac{BCIQT} was\nto obtain one binary label for each sample in the test set.\n\nThe \\ac{BCIQT} estimated two density operators $\\rho_{0}$ and $\\rho_{1}$, one\noperator for each category or class and its complement, by using the training\nsamples; in particular, for each class, the negative training samples were\nutilized to estimate $\\rho_{0}$ and the positive training samples were utilized\nto estimate $\\rho_{1}$.\n\nIn order to achieve these density operators $\\rho_{0}$ and $\\rho_{1}$, we first\ncalculated the total number of samples with non-zero values for each particular\nfeature.  In such a way, one vector $\\ket{v}$ was obtained for each class. Since\nwe were considering the binary case, two vectors $\\ket{v_{0}}$ and $\\ket{v_{1}}$\nwere obtained; the former referred to the negative training samples and the\nlatter referred to the positive training samples; these vectors may be\nconsidered as statistics of the features in a class.  We normalized the vectors\nto obtain $|\\braket{v|v}|^2=1$.  Then, we calculated the outer product in order\nto obtain the density operators $\\rho_{0}$ and $\\rho_{1}$ as follows:\n\\begin{equation}\n  \\label{eq:rhos}\n  \\rho_0 = \\frac{\\ket{v_0} {\\bra{v_0}} }{tr(\\ket{v_0} {\\bra{v_0}})}\n  \\qquad\n  \\rho_1 = \\frac{\\ket{v_1} {\\bra{v_1}} }{tr(\\ket{v_1} {\\bra{v_1}})}\n\\end{equation}\nWe computed the projection operator $\\Delta$ according to\n\\cite{melucci2016relevance}, that is,\n\\begin{equation}\n  \\label{eq:decomposition}\n  \\rho_1 - \\lambda \\rho_0 = \\eta\\,\\Delta + \\beta\\,\\Delta^\\perp \\qquad \\eta > 0\n  \\qquad \\beta < 0 \\qquad \\Delta\\,\\Delta^\\perp = 0\n\\end{equation}\nwhere $\\xi$ is the prior probability of the negative class and\n$ \\lambda = {\\xi}\\,\/\\,(1-\\xi) $; moreover, $\\eta$ is the positive eigenvalue\ncorresponding to $\\Delta$ which represents the subspaces of the vectors\nrepresenting the sample to be accepted in the target class.\n\nWe set $\\lambda = 1$ to simply mean that both classes had the same prior\nprobability ($\\xi=0.5$); moreover, there was no cost for wrong detection\n$C_{00}=C_{11}=0$; finally, the costs of false alarm and miss were constant\n($C_{01}=C_{10}$). Eventually, we determined the binary label for the given test\nsample $S_{j}$ by inspecting the value of\n$ \\langle w_{S_j} \\vert \\Delta \\vert w_{S_j} \\rangle$: If\n$\\langle w_{S_j} \\vert \\Delta \\vert w_{S_j}\\rangle \\geq 0.5$, then $C(S_j) = 1$;\notherwise $C(S_j) = 0$.\n\n\\section{Experiment}\n\nThe MNIST database \\footnote{\\url{http:\/\/yann.lecun.com\/exdb\/mnist\/}} of\nhandwritten digits has a training set of 60,000 examples, and a test set of\n10,000 examples. There are 9 categories from 0 to 9 but excluded 9. It is a subset of a larger set\navailable from the National Institute of Standards and Technology (NIST). The\ndigits have been size-normalized and centered in a fixed-size image.\n\nThe four models i.e. \\ac{NB}, \\ac{SVM}, \\ac{KNN}  and \\ac{DT}  were used as baselines. Prior to training the models,\nthe top 100 features were selected as the best features for all the models in terms of recall. The chi-square feature selection model was used.\n\nWe used one-vs-all strategy: for each category, the training samples labeled as\npertinent to the category are considered positive examples, while the rest are\nconsidered negative examples. While training the model, five fold cross\nvalidation was used. As it can be seen from Table \\ref{tab:1}, our proposed\nmodel performs better than any state-of-art-model in terms of recall for every\ncategory. By changing number of features, evaluation measures(i.e. accuracy, precision, recall and f-measure) also change and provide comparable results to the baselines.\n\n\n\\begin{table}[h!]\n\\large\n\\caption{Comparison of Recall among  k-nearest neighbors(KNN), Decision Tree(DT), Naive Bayes (NB), Support Vector Machine (SVM) and Binary Classifier Inspired by Quantum Theory (BCIQT)}\n\n\\begin{tabular}{cccccc}\n\t\\hline\n    Category\t&\tKNN\t& DT\t&\tNB\t&\tSVM\t&  BCIQT\t\\\\\n\t\\hline\n    \t\t0\t&0.959\t&0.884\t&0.889\t&0.292&\t\\textbf{1}\n\n\t\\\\\n    \t\t1\t& 0.699&\t0.710&\t0.582&\t0.390&\t\\textbf{0.996}\n\n\t\\\\\n    \t\t2\t&0.704&\t0.652&\t0.709&\t0.474&\t\\textbf{1}\n\n\t\\\\\n    \t\t3\t&0.623&\t0.508&\t0.792&\t0.346&\t\\textbf{0.997}\n\n\t\\\\\n    \t\t4\t&0.643&\t0.621&\t0.666&\t0.259&\t\\textbf{0.999}\n\n\t\\\\\n    \t\t5\t&0.892&\t0.855&\t0.872&\t0.621&\t\\textbf{1}\n\n\t\\\\\n    \t\t6\t&0.743&\t0.755&\t0.873&\t0.454&\t\\textbf{0.999}\n\n\t\\\\\n    \t\t7\t&0.753&\t0.728&\t0.779&\t0.332&\t\\textbf{1}\n\n\t\\\\\n    \t\t8\t&0.749&\t0.677&\t0.817&\t0.209&\t\\textbf{1}\n\n\t\\\\\n\t\\hline\n\\end{tabular}\n\\label{tab:1}\n\\end{table}\n\n\\section{Conclusion and Future Works}\n\nWe found out that our proposed model outperforms the state-of-the-art models in terms of\n\\emph{recall}; therefore, this model can be safely implemented if someone is\nlooking for high recall. We believe that this is an encouraging result and\nopens a gateway towards quantum inspired \\ac{ML} approaches. As for  future\nwork, we would like to develop multi-class classifiers (i.e. how to assign an\nitem to more than one class) and multi-label classifiers (i.e. how to deal with\nnon-binary labels), and re-rank the test items of a class by increasing precision as well.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{ Acknowledgments}\n\"This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 721321\".\n\n\\bigskip\n\\noindent \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nAll modern supersymmetric models are derived from a fundamental general scalar superfield. The application of the covariant superfield derivatives $D_\\alpha$ and $\\bar{D}_{\\dot{\\alpha}}$ allows then a systematic derivation of various chiral and anti-chiral superfields, see e.g. \\cite{sohnius85, dine07}. If the discussion is restricted to the supersymmetric description of bosonic fields with integer-valued mass dimension and fermionic fields with half-integer-valued mass dimension this approach is sufficient and leads to the Minimal Supersymmetric Standard Model. This changes, however, if the previous assumption on the mass dimensions of fermionic and bosonic fields is dropped.\n\nAn investigation of this matter is of special interest due to the recent proposal of fermionic fields with mass dimension one -- eigenspinors of the charge conjugation operator (ELKO) -- by Ahluwalia-Khalilova and Grumiller \\cite{ahluwaliakhalilova05a,ahluwaliakhalilova05b}. In their publications they use the field theory formalism to formulate a nonlocal theory of fermionic fields with mass dimension one. Ahluwalia et al. then modify this formalism by introducing a preferred direction along which the fermionic field with mass diemension one satisfies a local theory \\cite{ahluwalia10}. Subsequently da Rocha and Hoff da Silva construct a Lagrangian for ELKO spinors motivated from supergravity using mass dimension transmuting operators \\cite{darocha09}. Therfore, the question arises how to formulate a supersymmetric model from a fundamental superfield that is able to describe fermionic fields with mass dimension one.\n\nThe article is structured as follows. Section \\ref{CLSUSY} discusses the construction of a supersymmetric Lagrangian. It is shown that the straightforward approach using a general scalar superfield with redefined mass dimensions fails while the construction of a model based on a general spinor superfield is successful. Then, the corresponding supercurrent is calculated in Section \\ref{CJonshell}. In Section \\ref{CHposition} the Hamiltonian is derived using the supersymmetry algebra. This approach ensures that the resulting Hamiltonian is positive definite. Finally, the results are summarised in Section \\ref{Csummary}.\n\n\\section{A Supersymmetric Lagrangian}\n\\label{CLSUSY}\nIn this section a supersymmetric Lagrangian for fermionic fields with mass dimension one is derived. It is shown that a construction based on the general scalar superfield with redefined mass dimension of the component fields is impossible. This is due to problems generating a kinetic contribution for the fermionic fields with mass dimension one as well as constructing a consistent second quantisation. Afterwards the general spinor superfield is presented and all chiral and anti chiral superfields up to third order in covariant derivatives are derived systematically. This general spinor superfield is then used to construct a supersymmetric on-shell Lagrangian for fermionic fields with mass dimension one.\n\\subsection{Constructing a Model Based on the General Scalar Superfield}\n\\label{SLnot}\nThe most straightforward approach to formulate a supersymmetric model for fermionic fields with integer-valued mass dimension is to formulate a model in analogy to the commonly used formalism where fermionic fields have half-integer-valued mass dimension. This is done by starting from the general scalar superfield\n\\begin{align}\nV\n\t&= C\n\t- i \\theta \\chi\n\t+ i \\bar{\\chi}' \\bar{\\theta}\n\t- \\frac{i}{2} \\theta^2 \\left( M - i N \\right)\n\t+ \\frac{i}{2} \\left( M + i N \\right)\n\t- \\theta \\sigma^\\mu \\bar{\\theta} A_\\mu +\\notag \\\\\n\t&\\quad + i \\bar{\\theta}^2 \\theta \\left( \\lambda - \\frac{i}{2} \\dslash{\\partial} \\bar{\\chi}' \\right)\n\t- i \\theta^2 \\bar{\\theta} \\left( \\bar{\\lambda}' - \\frac{i}{2} \\bar{\\dslash{\\partial}} \\chi \\right)\n\t- \\frac{1}{2} \\theta^2 \\bar{\\theta}^2 \\left( D + \\frac{1}{2} \\Box C \\right) ,\n\\end{align}\nand redefining the mass dimensions of the component fields appropriately, e.g. $\\mathrm{dim}{\\left( C \\right)} = 1\/2$, $\\mathrm{dim}{\\left( \\chi \\right)} = 1$, etc. . The chiral superfields $X$ and $W_\\alpha$ are then defined as\n\\begin{align}\nX\n\t&= \\frac{i}{2} \\bar{D}^2 X \\, ,\\\\\nW_\\alpha\n\t&= \\frac{i}{4} \\bar{D}^2 D_\\alpha V \\, .\n\\end{align}\nwhere the covariant derivatives are given by\n\\begin{align}\nD_\\alpha\n\t&= \\partial_\\alpha - i \\dslash{\\partial}_{\\alpha \\dot{\\beta}} \\bar{\\theta}^{\\dot{\\beta}} \\, , \\label{covariantD}\\\\\n\\bar{D}_{\\dot{\\alpha}}\n\t&= - \\bar{\\partial}_{\\dot{\\alpha}} + i \\theta^\\beta \\dslash{\\partial}_{\\beta \\dot{\\alpha}} \\, . \\label{covariantDbar}\n\\end{align}\nThis choice of conventions differs by a factor of $- i$ from the conventions used in \\cite{wess82}.\n\nHowever, there are two fundamental problems that prevent a feasible theory using this approach. The first problem is that all possible contributions to the Lagrangian fail to produce a nonvanishing kinetic term for the fermionic fields. The second problem is encountered during second quantisation of the Lagrangian. It can be shown that already the simplest possible Lagrangian leads to negative energy solutions. In the following subsections these two problems will be discussed in detail.\n\n\\subsubsection{A Non-kinetic Supersymmetric Lagrangian}\n\\label{SSnonkinSUSYL}\n\\begin{TABLE}[t]{\n\\begin{tabular}{r|c|l}\nContribution & Mass Dimension & Possible Contributions \\\\\n\\hline\n$V V$ & $\\mathrm{dim}{\\left( V V \\right)} = 1$ & $\\left( m V V \\right)_D$ \\\\\n\\hline\n$X V$ & $\\mathrm{dim}{\\left( X V\\right)} = 2$ & $\\left( X V \\right)_D$ \\\\\n$D V D V$ & $\\mathrm{dim}{\\left( D V D V\\right)} = 2$ & $\\left( D V D V \\right)_D$  \\\\\n$V X$ & $\\mathrm{dim}{\\left( V X \\right)} = 2$ & $\\left( V X \\right)_D$ \\\\\n\\hline\n$D W V$ & $\\mathrm{dim}{\\left( D W V \\right)} = 3$ & mass dimension too big for D-component \\\\\n$W D V$ & $\\mathrm{dim}{\\left( W D V \\right)} = 3$ & mass dimension too big for D-component \\\\\n$X X$ & $\\mathrm{dim}{\\left( X X \\right)} = 3$ & $\\left( X X \\right)_F$ \\\\\n$D V W$ & $\\mathrm{dim}{\\left( D V W \\right)} = 3$ & mass dimension too big for D-component \\\\\n$V D W$ & $\\mathrm{dim}{\\left( V D W \\right)} = 3$ & mass dimension too big for D-component \n\\end{tabular}\n\\caption{Contributions to the Lagrangian based on the general scalar superfield if $\\chi$ is identified with the fermionic field of mass dimension one. In addition to the contributions built from products of unbarred superfields, the hermitian conjugates are permitted as well.}\n\\label{TChiDMnot}}\n\\end{TABLE}\nThe general scalar superfield has two possible candidates for a fermionic field with mass dimension one, $\\chi$ and $\\lambda$. For simplicity, the discussion is restricted to the case for $\\chi$ as fermionic field with mass dimension one. Similar calculations can be repeated for $\\lambda$. Due to the shift in mass dimension of the component fields the maximum number of covariant derivatives that needs to be considered is then increased by two and the discussion becomes more involved.\n\nIf $\\chi$ is identified with the fermionic field with mass dimension one it can be shown that the mass dimensions of the general superfield $V$ and the chiral superfields $X$ and $W_\\alpha$ are\n\\begin{align}\n\\mathrm{dim}{\\left( V \\right)}\n\t&= \\frac{1}{2} \\, , \\quad\n\\mathrm{dim}{\\left( X \\right)}\n\t= \\frac{3}{2} \\, , \\quad \n\\mathrm{dim}{\\left( W_\\alpha \\right)}\n\t= 2 \\, . \\quad\n\\end{align}\nThese results for the building blocks of the Lagrangian can be utilised to work out all possible contributions to the Lagrangian which have to satisfy three basic requirements. First, all contributions to the Lagrangian have to be Lorentz scalars and thus cannot contain any uncontracted indices. Second, all structure constants must have positive mass dimension for the theory to be renormalizable. Third, the contributions must have the appropriate mass dimension to contribute either via the $F$-component or the $D$-component.\n\nAll possible terms that satisfy the requirements are summarised in table \\ref{TChiDMnot}. It groups the contributions into three groups depending on the mass dimension of the superfield product without structure constants. It is possible to conceive terms with higher mass dimension, however, those terms cannot contribute to the Lagrangian and are irrelevant for the following discussion. For simplicity the discussion is restricted to the unbarred fields while the hermitian conjugated components have to be considered for the Lagrangian as well.\n\nThe first group of terms with mass dimension one consists of one single term which is the product of two general superfields. As the general superfield is neither chiral nor antichiral the only possible contribution to the Lagrangian is a mass term via the D-component.\n\nThe second group containing all terms with mass dimension two then encompasses all terms that can be constructed using two general superfields and two covariant derivatives. This results in three possible contributions to the kinetic term via the D-component. There can be no contributions to the mass term via the F-component as neither $V$ nor $D V$ are chiral or anti-chiral.\n\nFinally, the third group summarises all terms with mass dimension three which contain two general superfields as well as four covariant derivatives. Due to the mass dimension only contributions via the F-component are possible. The only term that satisfies the necessary symmetry requirements is $X X$ which contributes to the kinetic term.\n\nAltogether, there is one contribution to the mass term as well as four contributions to the kinetic term. On the first glance this seems to ensure the existence of a valid model. However, explicit calculations reveal that neither of the four kinetic terms in question is able to produce a kinetic term for $\\chi$ which was originally identified with the fermionic field with mass dimension one. A similar discussion can be repeated for the case where $\\lambda$ is identified with the fermionic field with mass dimension one. Although the discussion for $\\lambda$ produces an even larger number of potential contributions to the kinetic term neither of these terms produces a kinetic term for $\\lambda$. Therefore, it can be concluded that it is impossible to construct a Lagrangian -- other than the trivial solution for a constant background spinor field -- based on the general scalar superfield that is able to describe fermionic fields with mass dimension one.\n\n\\subsubsection{Problems with Second Quantisation}\n\\label{SSProblemSQ}\nThe second major problem arises from the second quantisation of the component fields. A simple way to demonstrate this is to start with the simplest possible Lagrangian for a fermionic field with mass dimension one\n\\begin{align}\n\\mathcal{L}\n\t&= \\partial_\\mu \\bar{\\psi} \\partial^\\mu \\psi\n\t- m^2 \\bar{\\psi} \\psi \\, .\n\\end{align}\nThe corresponding Hamiltonian is then found to be\n\\begin{align}\nH\n\t&= \\vec{\\mathbf{\\nabla}} \\mathbf{\\bar{\\psi}} \\vec{\\mathbf{\\nabla}} \\mathbf{\\psi}\n\t+ m^2 \\bar{\\psi} \\psi \\, .\n\\end{align}\nInserting the quantised Dirac field\n\\begin{align}\n\\psi\n\t&= \\int \\frac{\\mathrm{d}^3 \\mathbf{p}}{\\left( 2 \\pi \\right)^3} \\frac{1}{\\sqrt{2 E_\\mathbf{p}}} \\sum\\limits_s \\left( a^s_\\mathbf{p} u^s{\\left( \\mathbf{p} \\right)} e^{-i p \\cdot x} + {b^s_\\mathbf{p}}^\\dagger v^s{\\left( \\mathbf{p} \\right)} e^{i p \\cdot x} \\right) \\, , \\\\\n\\bar{\\psi}\n\t&= \\int \\frac{\\mathrm{d}^3 \\mathbf{p}}{\\left( 2 \\pi \\right)^3} \\frac{1}{\\sqrt{2 E_\\mathbf{p}}} \\sum\\limits_s \\left( b^s_\\mathbf{p} \\bar{v}^s{\\left( \\mathbf{p} \\right)} e^{- i p \\cdot x} + {a^s_\\mathbf{p}}^\\dagger \\bar{u}^s{\\left( \\mathbf{p} \\right)} e^{i p \\cdot x} \\right) \\, ,\n\\end{align}\ninto the Hamiltonian and removing the zero-point energy leads to\n\\begin{align}\nH\n\t&= \\int \\frac{\\mathrm{d}^3 \\mathbf{p}}{\\left( 2 \\pi \\right)^3} m E_\\mathbf{p} \\sum\\limits_s \\left( {a^s_\\mathbf{p}}^\\dagger a^s_\\mathbf{p} - {b^s_\\mathbf{p}}^\\dagger b^s_\\mathbf{p} \\right) .\n\\end{align}\nThe creation operator $b^\\dagger$ can be used to lower the energy arbitrarily and obtain negative energy solutions.\n\n\\subsection{The General Superfield with one Spinor Index}\n\\label{SVgeneral}\nIn the previous section it was shown that a theory based on the general scalar superfield cannot be viable. This motivated an ansatz based on the general spinor superfield. So far only few references to the general spinor superfield exist in the literature. One exception being the article by Gates \\cite{gates77} that contains an expansion of a spinor superfield in Grassmann variables. In addition, an expansion of the chiral spinor superfield was given by Siegel \\cite{siegel79}. Their results are also included in the book by Gates, Grisaru, Ro\\v{c}ek, and Siegel \\cite{gates01}. As our notation differs from previous publications and is based on spinor superfields with different mass dimension the spinor superfield is introduced in detail and all chiral and anti-chiral superfields up to third order in covariant derivatives are derived.\n\nIn analogy to the general scalar superfield , the general spinor superfield can immediately be written down as expansion in $\\theta$ and $\\bar{\\theta}$\n\\begin{align}\nV_\\alpha\n\t&= \\kappa_\\alpha\n\t+ \\theta^\\beta M_{\\beta \\alpha}\n\t+ \\bar{\\theta}^{\\dot{\\beta}} N_{\\dot{\\beta} \\alpha}\n\t+ \\theta^\\beta \\theta^\\gamma \\psi_{\\alpha \\beta \\gamma}\n\t+ \\bar{\\theta}^{\\dot{\\beta}} \\bar{\\theta}^{\\dot{\\gamma}} \\chi_{\\alpha \\dot{\\beta} \\dot{\\gamma}}\n\t+ \\theta^\\beta \\bar{\\theta}^{\\dot{\\gamma}} \\omega_{\\alpha \\beta \\dot{\\gamma}} + \\notag \\\\\n\t&\\quad + \\theta^\\beta \\theta^\\gamma \\bar{\\theta}^{\\dot{\\delta}} R_{\\dot{\\delta} \\alpha \\beta \\gamma}\n\t+ \\theta^\\beta \\bar{\\theta}^{\\dot{\\gamma}} \\bar{\\theta}^{\\dot{\\delta}} S_{\\alpha \\beta \\dot{\\gamma} \\dot{\\delta}}\n\t+ \\theta^\\beta \\theta^\\gamma \\bar{\\theta}^{\\dot{\\delta}} \\bar{\\theta}^{\\dot{\\epsilon}} \\lambda_{\\alpha \\beta \\gamma \\dot{\\delta} \\dot{\\epsilon}} \\, .\n\\end{align}\nTo bring this ansatz into a more convenient form the Grassmann variables need to be contracted over the respective indices. After absorbing some of the prefactors into the component fields, the general superfield is found to be\n\\begin{align}\nV_\\alpha\n\t&= \\kappa_\\alpha\n\t+ \\theta^\\beta M_{\\beta \\alpha}\n\t- \\bar{\\theta}^{\\dot{\\beta}} N_{\\dot{\\beta} \\alpha}\n\t+ \\theta^2 \\psi_\\alpha\n\t+ \\bar{\\theta}^2 \\chi_\\alpha\n\t+ \\theta \\sigma^\\mu \\bar{\\theta} \\left( \\sigma_\\mu \\right)^{\\beta \\dot{\\gamma}} \\omega_{\\alpha \\beta \\dot{\\gamma}} - \\notag \\\\\n\t&\\quad - \\theta^2 \\bar{\\theta}^{\\dot{\\delta}} R_{\\dot{\\delta} \\alpha}\n\t+ \\bar{\\theta}^2 \\theta^\\beta S_{\\beta \\alpha}\n\t+ \\theta^2 \\bar{\\theta}^2 \\lambda_\\alpha \\, ,\n\\end{align}\nwhere  $\\kappa$, $\\psi$, $\\chi$, and $\\lambda$ are Majorana spinors, $M$, $N$, $R$, and $S$ are complex second-rank spinors, and $\\omega$ is a complex  third-rank spinor. The four complex second-rank spinors contain 32 bosonic degrees of freedom while the four Majorana spinors contain 16 fermionic degrees of freedom. As the number of bosonic and fermionic degrees of freedom must be the same for a supersymmetric theory, the 3-rd rank spinor must also have 16 fermionic degrees of freedom. It is then tempting to rewrite the third-rank spinor as a vector of majorana spinors\n\\begin{align}\n\\left( \\sigma_\\mu \\right)^{\\beta \\dot{\\gamma}} \\omega_{\\alpha \\beta \\dot{\\gamma}}\n\t&= \\left( \\sigma_\\mu \\right)^{\\beta \\dot{\\gamma}} \\left( \\sigma^\\nu \\right)_{\\beta \\dot{\\gamma}} \\omega_{\\nu \\alpha}\n\t= 2 \\omega_{\\mu \\alpha} \\, ,\n\\end{align}\nwhich has 16 degrees of freedom as well. In the following discussion it will be referred to as a spinor-vector field. After appropriate rescaling of the component fields the most general spinor superfield is given by\n\\begin{align}\nV_\\alpha\n\t&= \\kappa_\\alpha\n\t+ \\theta^\\beta M_{\\beta \\alpha}\n\t- \\bar{\\theta}^{\\dot{\\beta}} N_{\\dot{\\beta}\\alpha}\n\t+ \\theta^2 \\psi_\\alpha\n\t+ \\bar{\\theta}^2 \\chi_\\alpha\n\t+ \\theta \\sigma^\\mu \\bar{\\theta} \\omega_{\\mu \\alpha} - \\notag \\\\\n\t&\\quad - \\theta^2 \\bar{\\theta}^{\\dot{\\beta}} R_{\\dot{\\beta} \\alpha}\n\t+ \\bar{\\theta}^2 \\theta^\\beta S_{\\beta \\alpha}\n\t+ \\theta^2 \\bar{\\theta}^2 \\lambda_\\alpha \\, .\n\\label{Valpha}\n\\end{align}\n\n\\subsubsection{The Chiral Superfields}\n\\label{SSChiralFields}\nFor the general scalar superfield the chiral and anti-chiral fields are derived by repeated operation of the covariant derivatives $D$ and $\\bar{D}$. By definition the chiral and anti-chiral superfields satisfy the following relations\n\\begin{align}\n\\bar{D}_{\\dot{\\alpha}} X\n\t&= 0 \\, , \\\\\nD_\\alpha Y\n\t&= 0 \\, ,\n\\end{align}\nwhere it is assumed that $X$ is a chiral superfield and $Y$ is an anti-chiral superfield which can have an arbitrary number of spinor indices. The chiral and anti-chiral superfields up to third order in covariant derivatives are then derived systematically by calculating $\\bar{D}^2 V$ and $D^2 V$, as well as $\\bar{D}^2 D V$ and $D^2 \\bar{D} V$.\n\nThe chiral spinor field is found by repeated operation of the covariant derivative $\\bar{D}$ onto the general superfield\n\\begin{align}\nX_\\alpha\n\t&= - \\frac{1}{4} \\bar{D}^2 V_\\alpha \\notag \\\\\n\t&= \\chi_\\alpha\n\t+ \\theta^\\beta \\left( S_{\\beta \\alpha} + \\frac{i}{2} \\dslash{\\partial}_\\beta{}^{\\dot{\\delta}} N_{\\dot{\\delta} \\alpha} \\right)\n\t+ \\theta^2 \\left( \\lambda_\\alpha + \\frac{i}{2} \\partial^\\mu \\omega_{\\mu \\alpha} - \\frac{1}{4} \\Box \\kappa_\\alpha \\right)\n\t- i \\theta \\dslash{\\partial} \\bar{\\theta} \\chi_\\alpha + \\notag \\\\\n\t&\\quad + \\frac{i}{2} \\theta^2 \\bar{\\theta}^{\\dot{\\gamma}} \\bar{\\dslash{\\partial}}_{\\dot{\\gamma}}{}^\\beta \\left( S_{\\beta \\alpha} + \\frac{i}{2} \\dslash{\\partial}_\\beta{}^{\\dot{\\delta}} N_{\\dot{\\delta} \\alpha} \\right)\n\t- \\frac{1}{4} \\theta^2 \\bar{\\theta}^2 \\Box \\chi_\\alpha \\, .\n\\label{Xchiral}\n\\end{align}\nComparison with the general expansion of a chiral field with one spinor index leads to the very elegant expression\n\\begin{align}\nX_\\alpha\n\t&= \\exp{\\left( - i \\theta \\dslash{\\partial} \\bar{\\theta} \\right)} \\left( \\chi_\\alpha + \\theta^\\beta \\left( S_{\\beta \\alpha} + \\frac{i}{2} \\dslash{\\partial}_\\beta{}^{\\dot{\\delta}} N_{\\dot{\\delta} \\alpha} \\right) + \\theta^2 \\left( \\lambda_\\alpha + \\frac{i}{2} \\partial^\\mu \\omega_{\\mu \\alpha} - \\frac{1}{4} \\Box \\kappa_\\alpha \\right) \\right) \\, .\n\\end{align}\nAs this notation is not used in the further discussion an explicit notation in exponential form is not given for $Y$, $Z$, $Z_0$, and $Z'$ but can be derived in a similar way.\n\nThe calculations for the anti-chiral spinor field can be performed in perfect analogy where the operation of the covariant derivatives $D$ on the general superfield replaces the operation of $\\bar{D}$\n\\begin{align}\nY_\\alpha\n\t&= - \\frac{1}{4} D^2 V_\\alpha \\notag \\\\\n\t&= \\psi_\\alpha\n\t- \\bar{\\theta}^{\\dot{\\beta}} \\left( R_{\\dot{\\beta} \\alpha} + \\frac{i}{2} \\bar{\\dslash{\\partial}}_{\\dot{\\beta}}{}^\\gamma M_{\\gamma \\alpha} \\right)\n\t+ \\bar{\\theta}^2 \\left( \\lambda_\\alpha + \\frac{i}{2} \\partial^\\mu \\omega_{\\mu \\alpha} - \\frac{1}{4}\\Box \\kappa_\\alpha \\right)\n\t- i \\theta \\dslash{\\partial} \\bar{\\theta} \\psi_\\alpha - \\notag \\\\\n\t&\\quad - \\frac{i}{2} \\theta^\\gamma \\bar{\\theta}^2 \\dslash{\\partial}_\\gamma{}^{\\dot{\\beta}} \\left( R_{\\dot{\\beta} \\alpha} + \\frac{i}{2} \\bar{\\dslash{\\partial}}_{\\dot{\\beta}}{}^\\delta M_{\\delta \\alpha} \\right)\n\t- \\frac{1}{4} \\theta^2 \\bar{\\theta}^2 \\Box \\psi_\\alpha \\, .\n\\label{Yantichiral}\n\\end{align}\n\nTo third order in covariant derivatives there is again one chiral and one anti-chiral superfield which are now second-rank spinor fields.  The chiral second-rank spinor field is found to be\n\\begin{align}\nZ_{\\gamma \\alpha}\n\t&= - \\frac{1}{4} \\bar{D}^2 D_\\gamma V_\\alpha \\notag \\\\\n\t&= \\left( S_{\\gamma \\alpha} - \\frac{i}{2} \\dslash{\\partial}_\\gamma{}^{\\dot{\\beta}} N_{\\dot{\\beta} \\alpha} \\right)\n\t+ \\theta^\\beta \\left( 2 \\epsilon_{\\beta \\gamma} \\lambda_\\alpha + \\left( \\sigma^{\\nu \\mu} \\right)_{\\beta \\gamma} \\partial_\\nu \\omega_{\\mu \\alpha} + \\frac{1}{2} \\epsilon_{\\beta \\gamma} \\Box \\kappa_\\alpha \\right) - \\notag \\\\\n\t&\\quad - i \\theta^2 \\left( \\dslash{\\partial}_\\gamma{}^{\\dot{\\beta}} R_{\\dot{\\beta} \\alpha} - \\frac{i}{2} \\Box M_{\\gamma \\alpha} \\right)\n\t- i \\theta \\dslash{\\partial} \\bar{\\theta} \\left( S_{\\gamma \\alpha} - \\frac{i}{2} \\dslash{\\partial}_\\gamma{}^{\\dot{\\beta}} N_{\\dot{\\beta} \\alpha} \\right) + \\notag \\\\\n\t&\\quad + \\frac{i}{2} \\theta^2 \\bar{\\theta}^{\\dot{\\delta}} \\dslash{\\partial}^\\beta{}_{\\dot{\\delta}} \\left( 2 \\epsilon_{\\beta \\gamma} \\lambda_\\alpha + \\left( \\sigma^{\\nu \\mu} \\right)_{\\beta \\gamma} \\partial_\\nu \\omega_{\\mu \\alpha} + \\frac{1}{2} \\epsilon_{\\beta \\gamma} \\Box \\kappa_\\alpha \\right) - \\notag \\\\\n\t&\\quad - \\frac{1}{4} \\theta^2 \\bar{\\theta}^2 \\Box \\left( S_{\\gamma \\alpha} - \\frac{i}{2} \\dslash{\\partial}_\\gamma{}^{\\dot{\\beta}} N_{\\dot{\\beta} \\alpha} \\right) \\, .\n\\label{Zchiral}\n\\end{align}\nA special case arises if the two undotted indices of the second-rank spinor field are contracted. It is then reduced to a scalar superfield\n\\begin{align}\nZ_0\n\t&= \\mathrm{Tr}{\\left( S + \\frac{i}{2} \\dslash{\\partial} N \\right)}\n\t- \\theta^\\beta \\left( 2 \\lambda_\\beta + \\sigma^{\\nu \\mu} \\partial_\\nu \\omega_\\mu + \\frac{1}{2} \\Box \\kappa_\\beta \\right) \n\t+ i \\theta^2 \\mathrm{Tr}{\\left( \\dslash{\\partial} R + \\frac{i}{2} \\Box M \\right)} - \\notag \\\\\n\t&\\quad - i \\theta \\dslash{\\partial} \\bar{\\theta} \\mathrm{Tr}{\\left( S + \\frac{i}{2} \\dslash{\\partial} N \\right)}\n\t- \\frac{i}{2} \\theta^2 \\bar{\\theta}^{\\dot{\\delta}} \\dslash{\\partial}^\\beta{}_{\\dot{\\delta}} \\left( 2 \\lambda_\\beta + \\sigma^{\\nu \\mu} \\partial_\\nu \\omega_\\mu + \\frac{1}{2} \\Box \\kappa_\\beta \\right) - \\notag \\\\\n\t&\\quad - \\frac{1}{4} \\theta^2 \\bar{\\theta}^2 \\Box \\mathrm{Tr}{\\left( S + \\frac{i}{2} \\dslash{\\partial} N \\right)} \\, .\n\\label{Z0chiral}\n\\end{align}\nThe calculations for the anti-chiral second-rank spinor field are nearly identical and it is found to be\n\\begin{align}\nZ'\n\t&= - \\frac{1}{4} D^2 \\bar{D}_{\\dot{\\gamma}} V_\\alpha \\notag \\\\\n\t&= \\left( R_{\\dot{\\gamma} \\alpha} + \\frac{i}{2} \\dslash{\\partial}^\\beta{}_{\\dot{\\gamma}} M_{\\beta \\alpha} \\right)\n\t- \\bar{\\theta}^{\\dot{\\beta}} \\left( 2 \\epsilon_{\\dot{\\beta} \\dot{\\gamma}} \\lambda_\\alpha - \\left( \\bar{\\sigma}^{\\nu \\mu} \\right)_{\\dot{\\beta} \\dot{\\gamma}} \\partial_\\nu \\omega_{\\mu \\alpha} + \\frac{1}{2} \\epsilon_{\\dot{\\beta} \\dot{\\gamma}} \\Box \\kappa_\\alpha \\right) + \\notag \\\\\n\t&\\quad + \\bar{\\theta}^2 \\left( i \\bar{\\dslash{\\partial}}_{\\dot{\\gamma}}{}^\\beta S_{\\beta \\alpha} - \\frac{1}{2} \\Box N_{\\dot{\\gamma} \\alpha} \\right)\n\t+ i \\theta \\dslash{\\partial} \\bar{\\theta} \\left( R_{\\dot{\\gamma} \\alpha} + \\frac{i}{2} \\dslash{\\partial}^\\beta{}_{\\dot{\\gamma}} M_{\\beta \\alpha} \\right) + \\notag \\\\\n\t&\\quad + \\frac{i}{2} \\theta^\\delta \\bar{\\theta}^2 \\dslash{\\partial}_\\delta{}^{\\dot{\\beta}} \\left( 2 \\epsilon_{ \\dot{\\beta} \\dot{\\gamma}}\\lambda_\\alpha - \\left( \\bar{\\sigma}^{\\nu \\mu} \\right)_{\\dot{\\beta} \\dot{\\gamma}} \\partial_\\nu \\omega_{\\mu \\alpha} + \\frac{1}{2} \\epsilon_{\\dot{\\beta} \\dot{\\gamma}} \\Box \\kappa_\\alpha \\right) - \\notag \\\\\n\t&\\quad - \\frac{1}{4} \\theta^2 \\bar{\\theta}^2 \\Box \\left( R_{\\dot{\\gamma} \\alpha} + \\frac{i}{2} \\dslash{\\partial}^\\beta{}_{\\dot{\\gamma}} M_{\\beta \\alpha} \\right) \\, .\n\\label{Zprimeantichiral}\n\\end{align}\nUnlike for the chiral second-rank spinor field, no special case exists for the anti-chiral second-rank spinor field. This is due to the odd number of dotted and undotted indices which makes it impossible to contract the indices to achieve a scalar superfield. At most it can be written as a product of a Pauli-matrix and a vector field.\n\n\\subsubsection{Unitary Supertranslations}\n\\label{SSSUSYtranslation}\nFor the later discussion of the supercurrent and the derivation of the second quantisation of the component fields the superfield variation of the general spinor superfield must be derived. The calculation follows the discussion for the general scalar superfield in \\cite{dick09} and was adapted accordingly to compensate for the additional spinor index.\n\nThe starting point for the derivation of the behaviour of a superfield under unitary supertranslations is the definition of a superspace eigenstate\n\\begin{align}\n\\left| x_0 , \\theta_0 , \\bar{\\theta}_0 \\right\\rangle \\, ,\n\\end{align}\nwhich has the eigenvalues\n\\begin{align}\nx^\\mu \\left| x_0 , \\theta_0 , \\bar{\\theta}_0 \\right\\rangle\n\t&= x_0 \\left| x_0 , \\theta_0 , \\bar{\\theta}_0 \\right\\rangle \\, , \\\\\n\\theta_\\alpha \\left| x_0 , \\theta_0 , \\bar{\\theta}_0 \\right\\rangle\n\t&= \\theta_{0 \\alpha} \\left| x_0 , \\theta_0 , \\bar{\\theta}_0 \\right\\rangle \\, , \\\\\n\\bar{\\theta}_{\\dot{\\alpha}} \\left| x_0 , \\theta_0 , \\bar{\\theta}_0 \\right\\rangle\n\t&= \\bar{\\theta}_{0 \\dot{\\alpha}} \\left| x_0 , \\theta_0 , \\bar{\\theta}_0 \\right\\rangle \\, .\n\\end{align}\nHere $\\theta_\\alpha$, $\\bar{\\theta}_{\\dot{\\alpha}}$, and $x^\\mu$ are operators acting on the superspace eigenstate while the eigenvalues are denoted by a subscript $0$ for the original eigenstate and with a prime for the translated state. Therefore, a state that is shifted under unitary supertranslations can be written as\n\\begin{align}\n\\left| x' , \\theta' , \\bar{\\theta}' \\right\\rangle\n\t&= \\exp{\\left( i a y \\cdot  P + i b \\zeta Q + i c \\bar{Q} \\bar{\\zeta} \\right)} \\left| x_0 , \\theta_0 , \\bar{\\theta}_0 \\right\\rangle \\, ,\n\\end{align}\nwhere the prefactors $a$, $b$, and $c$ still need to be determined. An arbitrary operator $\\mathcal{O}$ acting on the shifted state can be expressend as\n\\begin{align}\n\\mathcal{O} \\left| x' , \\theta' , \\bar{\\theta}' \\right\\rangle\n\t&= \\exp{\\left( i a y \\cdot P + i b \\zeta Q + i c \\bar{Q} \\bar{\\zeta} \\right)} \\exp{\\left( - i a y \\cdot  P - i b \\zeta Q - i c \\bar{Q} \\bar{\\zeta} \\right)} \\times \\notag \\\\\n\t& \\quad \\times \\mathcal{O} \\exp{\\left( i a y \\cdot P + i b \\zeta Q + i c \\bar{Q} \\bar{\\zeta} \\right)} \\left| x_0 , \\theta_0 , \\bar{\\theta}_0 \\right\\rangle \\, .\n\\end{align}\nUsing the Cambell-Baker-Hausdorff formula\n\\begin{align}\ne^{- i G \\lambda} \\mathcal{O} e^{i G \\lambda}\n\t&= \\sum\\limits_{j} \\left( - i \\lambda \\right)^j \\stackrel{j}{\\left[ \\right.} \\!\\!\\! \\left. G , \\mathcal{O} \\right]\n\\end{align}\nthis product of operators can be decomposed into an infinite sum of commutators\n\\begin{align}\n\\mathcal{O} \\left| x' , \\theta' , \\bar{\\theta}' \\right\\rangle\n\t&= \\exp{\\left( i a y \\cdot P + i b \\zeta Q + i c \\bar{Q} \\bar{\\zeta} \\right)} \\times \\notag \\\\\n\t&\\quad \\times \\sum\\limits_{j} \\left( - i \\lambda \\right)^j \\stackrel{j}{\\left[ \\right.} \\!\\!\\! \\left. a y \\cdot P + b \\zeta Q + c \\bar{Q} \\bar{\\zeta} , \\mathcal{O} \\right] \\left| x_0 , \\theta_0 , \\bar{\\theta}_0 \\right\\rangle \\, .\n\\end{align}\nTo evaluate the commutators it proves useful to utilise the following three commutators and anticommutators\n\\begin{align}\n\\left\\{ \\partial_\\beta , \\theta_\\alpha \\right\\}\n\t&= \\epsilon_{\\beta \\alpha} \\, , \\\\\n\\left\\{ \\partial_{\\dot{\\beta}} , \\bar{\\theta}_{\\dot{\\alpha}} \\right\\}\n\t&= \\epsilon_{\\dot{\\alpha} \\dot{\\beta}} \\, , \\\\\n\\left[ P_\\nu , x_\\mu \\right]\n\t&= i \\eta_{\\nu \\mu} \\, .\n\\end{align}\n\nFor the operator $\\theta$ acting on the translated eigenstate it is found that\n\\begin{align}\n\\theta_\\alpha \\left| x' , \\theta' , \\bar{\\theta}' \\right\\rangle\n\t&= \\exp{\\left( i a y \\cdot P + i b \\zeta Q + i c \\bar{Q} \\bar{\\zeta} \\right)} \\times \\notag \\\\\n\t&\\quad \\times \\sum\\limits_{j} \\left( - i \\lambda \\right)^j \\stackrel{j}{\\left[ \\right.} \\!\\!\\! \\left. a y \\cdot P + b \\zeta Q + c \\bar{Q} \\bar{\\zeta} , \\theta \\right] \\left| x , \\theta , \\bar{\\theta} \\right\\rangle \\, ,\n\\end{align}\nwhere the $j$-th commutator has to be derived recursively. Conveniently, the first commutator is given by\n\\begin{align}\n\\left[ a y^\\mu P_\\mu + b \\zeta^\\beta Q_\\beta + c \\bar{Q}_{\\dot{\\beta}} \\bar{\\zeta}^{\\dot{\\beta}} , \\theta_\\alpha \\right]\n\t&= i b \\zeta_\\alpha \\, .\n\\end{align}\nThis implies that the second commutator vanishes. Therefore, all higher order contributions to the infinite sum must vanish identically as well and the eigenvalue of the shifted state is\n\\begin{align}\n\\theta'_\\alpha \\left| x' , \\theta' , \\bar{\\theta}' \\right\\rangle\n\t&= \\left( \\theta_{0 \\alpha} + b \\zeta_\\alpha \\right) \\left| x' , \\theta' , \\bar{\\theta}' \\right\\rangle \\, .\n\\end{align}\n\nA similar calculation can be repeated for the operator $\\bar{\\theta}$. It is found that\n\\begin{align}\n\\bar{\\theta}_{\\dot{\\alpha}} \\left| x' , \\theta' , \\bar{\\theta}' \\right\\rangle\n\t&= \\exp{\\left( i a y \\cdot P + i b \\zeta Q + i c \\bar{Q} \\bar{\\zeta} \\right)} \\times \\notag \\\\\n\t&\\quad \\times \\sum\\limits_{j} \\left( - i \\lambda \\right)^j \\stackrel{j}{\\left[ \\right.} \\!\\!\\! \\left. a y \\cdot P + b \\zeta Q + c \\bar{Q} \\bar{\\zeta} , \\bar{\\theta}_{\\dot{\\alpha}} \\right] \\left| x_0 , \\theta_0 , \\bar{\\theta}_0 \\right\\rangle \\, .\n\\end{align}\nAgain, the $j$-th commutator must be calculated recursively starting with the first order commutator\n\\begin{align}\n\\left[ a y \\cdot P + b \\zeta^\\beta Q_\\beta + c \\bar{Q}_{\\dot{\\beta}} \\bar{\\zeta}^{\\dot{\\beta}} , \\bar{\\theta}_{\\dot{\\alpha}} \\right]\n\t&= i c \\bar{\\zeta}_{\\dot{\\alpha}} \\, .\n\\end{align}\nLike in the previous case this result implies that the second commutator vanishes identically and the eigenvalue of the shifted state is \n\\begin{align}\n\\bar{\\theta}'_{\\dot{\\alpha}} \\left| x' , \\theta' , \\bar{\\theta}' \\right\\rangle\n\t&= \\left( \\bar{\\theta}_{0 \\dot{\\alpha}} + c \\bar{\\zeta}_{\\dot{\\alpha}} \\right) \\left| x' , \\theta' , \\bar{\\theta}' \\right\\rangle \\, .\n\\end{align}\n\nFinally, the behaviour of the eigenvalue of the operator $x^\\mu$ is analysed\n\\begin{align}\nx^\\mu \\left| x' , \\theta' , \\bar{\\theta}' \\right\\rangle\n\t&= \\exp{\\left( i a y \\cdot P + i b \\zeta Q + i c \\bar{Q} \\bar{\\zeta} \\right)} \\times \\notag \\\\\n\t&\\quad \\times \\sum\\limits_{j} \\left( - i \\lambda \\right)^j \\stackrel{j}{\\left[ \\right.} \\!\\!\\! \\left. a y \\cdot P + i b \\zeta Q + i c \\bar{Q} \\bar{\\zeta} , x^\\mu \\right] \\left| x_0 , \\theta_0 , \\bar{\\theta}_0 \\right\\rangle \\, .\n\\end{align}\nThe first commutator is found to be\n\\begin{align}\n\\left[ a y^\\nu P_\\nu + b \\zeta^\\alpha Q_\\alpha + c \\bar{Q}_{\\dot{\\alpha}} \\bar{\\zeta}^{\\dot{\\alpha}} , x^\\mu \\right]\n\t&= i a y^\\mu\n\t- b \\zeta \\sigma^\\mu \\bar{\\theta}\n\t+ c \\theta \\sigma^\\mu \\bar{\\zeta} \\, .\n\\end{align}\nOn the first glance it seems as if the series expansion doesn't terminate after the first commutator. However, the explicit calculation of the second commutator reveals that it vanishes identically\n\\begin{align}\n\\stackrel{2}{\\left[ \\right.} \\!\\!\\! \\left. a y^\\nu P_\\nu + b \\zeta^\\alpha Q_\\alpha + c \\bar{Q}_{\\dot{\\alpha}} \\bar{\\zeta}^{\\dot{\\alpha}} , x^\\mu \\right]\n\t&= 0 \\, .\n\\end{align}\nThis terminates the infinite series and the eigenvalue for the operator $x^\\mu$ acting on the translated state is given by\n\\begin{align}\nx^\\mu \\left| x' , \\theta' , \\bar{\\theta}' \\right\\rangle\n\t&= \\left( x_0^\\mu + a y^\\mu + i \\left( b \\zeta \\sigma^\\mu \\bar{\\theta}_0 - c \\theta_0 \\sigma^\\mu \\bar{\\zeta} \\right) \\right) \\left| x' , \\theta' , \\bar{\\theta}' \\right\\rangle \\, .\n\\end{align}\n\nCombining the results for the operators $\\theta$, $\\bar{\\theta}$, and $x^\\mu$ yields a translated superspace eigenstate of\n\\begin{align}\n\\left| x' , \\theta' , \\bar{\\theta}' \\right\\rangle\n\t&= \\left| x_0 + a y_0 + i \\left( b \\zeta \\sigma \\bar{\\theta}_0 - c \\theta_0 \\sigma \\bar{\\zeta} \\right) , \\theta_0 + b \\zeta , \\bar{\\theta}_0 + c \\bar{\\zeta} \\right\\rangle \\, ,\n\\end{align}\nwhere the prefactors $a$, $b$, and $c$ are still arbitrary. As a convention it is assumed that the discussion is restricted to pure superspace translations for which the spatial translation vanishes and thus $a y_0 =0$. Furthermore, the translations of the superspace coordinates $\\theta$ and $\\bar{\\theta}$ are chosen to be positive which results in $b = c = 1$. This results in a relation between the original and shifted state of the following form\n\\begin{align}\n\\left| x' , \\theta' , \\bar{\\theta}' \\right\\rangle\n\t&= \\left| x + i \\left( \\zeta \\sigma \\bar{\\theta} - \\theta \\sigma \\bar{\\zeta} \\right) , \\theta + \\zeta , \\bar{\\theta} + \\bar{\\zeta} \\right\\rangle\n\t= \\exp{\\left( i \\zeta Q + i \\bar{Q} \\bar{\\zeta} \\right)} \\left| x , \\theta , \\bar{\\theta} \\right\\rangle \\, ,\n\\end{align}\nwhere the subscript $0$ was dropped as it is no longer necessary to distinguish between operators and eigenvalues. The eigenstate at the shifted superspace coordinates is expressed in terms of the superspace coordinates of the original superspace eigenstate. It can be seen that a superspace translation, unlike a translation of normal fields, not only induces a spatial translation, but also results in a shift of the superspace coordinates $\\theta$ and $\\bar{\\theta}$.\n\nNow that the behaviour of a superspace eigenstate under unitary supertranslation is known, the calculation of the translated general spinor superfield is straightforward\n\\begin{align}\nV'_\\alpha{\\left( x , \\theta , \\bar{\\theta} \\right)}\n\t&= \\left\\langle x , \\theta , \\bar{\\theta} \\right| \\exp{\\left( i \\zeta Q + i \\bar{Q} \\bar{\\zeta} \\right)} \\left| V_\\alpha \\right\\rangle \\notag \\\\\n\t&= \\left\\langle x - i \\left( \\zeta \\sigma \\bar{\\theta} - \\theta \\sigma \\bar{\\zeta} \\right) , \\theta - \\zeta , \\bar{\\theta} - \\bar{\\zeta} \\right| \\left. V_\\alpha \\right\\rangle \\notag \\\\\n\t&= V_\\alpha{\\left( x - i \\left( \\zeta \\sigma \\bar{\\theta} - \\theta \\sigma \\bar{\\zeta} \\right) , \\theta - \\zeta , \\bar{\\theta} - \\bar{\\zeta} \\right)} \\, .\n\\end{align}\nAs for the superspace eigenstate, a unitary supertranslation acting on a general superfield induces a spatial translation as well as a shift of superspace coordinates. In terms of the component fields the translated superfield can then be written as\n\\begin{align}\nV'_\\alpha{\\left( x , \\theta , \\bar{\\theta} \\right)}\n\t&= \\kappa_\\alpha{\\left( x - i \\left( \\zeta \\sigma \\bar{\\theta} - \\theta \\sigma \\bar{\\zeta} \\right) \\right)}\n\t+ \\left( \\theta^\\beta - \\zeta^\\beta \\right) M_{\\beta \\alpha}{\\left( x - i \\left( \\zeta \\sigma \\bar{\\theta} - \\theta \\sigma \\bar{\\zeta} \\right) \\right)} - \\notag \\\\\n\t&\\quad - \\left( \\bar{\\theta}^{\\dot{\\beta}} - \\bar{\\zeta}^{\\dot{\\beta}} \\right) N_{\\dot{\\beta}\\alpha}{\\left( x - i \\left( \\zeta \\sigma \\bar{\\theta} - \\theta \\sigma \\bar{\\zeta} \\right) \\right)}\n\t+ \\left( \\theta - \\zeta \\right)^2 \\psi_\\alpha{\\left( x - i \\left( \\zeta \\sigma \\bar{\\theta} - \\theta \\sigma \\bar{\\zeta} \\right) \\right)} + \\notag \\displaybreak[3] \\\\\n\t&\\quad + \\left( \\bar{\\theta} - \\bar{\\zeta} \\right)^2 \\chi_\\alpha{\\left( x - i \\left( \\zeta \\sigma \\bar{\\theta} - \\theta \\sigma \\bar{\\zeta} \\right) \\right)} + \\notag \\displaybreak[3] \\\\\n\t&\\quad + \\left( \\theta - \\zeta \\right) \\sigma^\\mu \\left( \\bar{\\theta} - \\bar{\\zeta} \\right) \\omega_{\\mu \\alpha}{\\left( x - i \\left( \\zeta \\sigma \\bar{\\theta} - \\theta \\sigma \\bar{\\zeta} \\right) \\right)} - \\notag \\displaybreak[3] \\\\\n\t&\\quad - \\left( \\theta - \\zeta \\right)^2 \\left( \\bar{\\theta}^{\\dot{\\beta}} - \\bar{\\zeta}^{\\dot{\\beta}} \\right) R_{\\dot{\\beta} \\alpha}{\\left( x - i \\left( \\zeta \\sigma \\bar{\\theta} - \\theta \\sigma \\bar{\\zeta} \\right) \\right)} + \\notag \\displaybreak[3] \\\\\n\t&\\quad + \\left( \\bar{\\theta} - \\bar{\\zeta} \\right)^2 \\left( \\theta^\\beta - \\zeta^\\beta \\right) S_{\\beta \\alpha}{\\left( x - i \\left( \\zeta \\sigma \\bar{\\theta} - \\theta \\sigma \\bar{\\zeta} \\right) \\right)} + \\notag \\\\\n\t&\\quad + \\left( \\theta - \\zeta \\right)^2 \\left( \\bar{\\theta} - \\bar{\\zeta} \\right)^2 \\lambda_\\alpha{\\left( x - i \\left( \\zeta \\sigma \\bar{\\theta} - \\theta \\sigma \\bar{\\zeta} \\right) \\right)} \\, .\n\\label{SUSYvariationTaylor}\n\\end{align}\nTo express the translated component fields in terms of the component fields at the original superspace coordinates a Taylor expansion of the component fields can be used. In the present case an expansion up to first order in the transformation parameters $\\zeta$ and $\\bar{\\zeta}$ of the form\n\\begin{align}\n\\kappa_\\alpha{\\left( x - i \\left( \\zeta \\sigma \\bar{\\theta} - \\theta \\sigma \\bar{\\zeta} \\right) \\right)}\n\t&\\approx \\kappa_\\alpha{\\left( x \\right)}\n\t- i \\left( \\zeta \\sigma^\\nu \\bar{\\theta} - \\theta \\sigma^\\nu \\bar{\\zeta} \\right) \\partial_\\nu \\kappa_\\alpha{\\left( x \\right)}\n\\end{align}\nis sufficient. After appropriately rewriting equation (\\ref{SUSYvariationTaylor}), neglecting all terms of second or higher order in the transformation parameters $\\zeta$ and $\\bar{\\zeta}$, and collecting the terms with corresponding orders in the Grassmann variables $\\theta$ and $\\bar{\\theta}$ the shifted superfield is given by\n\\begin{align}\nV'_\\alpha{\\left( x , \\theta , \\bar{\\theta} \\right)}\n\t&= \\kappa_\\alpha{\\left( x \\right)}\n\t- \\zeta^\\beta M_{\\beta \\alpha}{\\left( x \\right)}\n\t+ \\bar{\\zeta}^{\\dot{\\beta}} N_{\\dot{\\beta}\\alpha}{\\left( x \\right)} + \\notag \\\\\n\t&\\quad + \\theta^\\beta \\left( M_{\\beta \\alpha}{\\left( x \\right)}\n\t+ i \\left( \\sigma^\\mu \\right)_{\\beta \\dot{\\gamma}} \\bar{\\zeta}^{\\dot{\\gamma}} \\partial_\\mu \\kappa_\\alpha{\\left( x \\right)} \n\t- 2 \\zeta_\\beta \\psi_\\alpha{\\left( x \\right)}\n\t- \\left( \\sigma^\\mu \\right)_{\\beta \\dot{\\gamma}} \\bar{\\zeta}^{\\dot{\\gamma}} \\omega_{\\mu \\alpha}{\\left( x \\right)} \\right) - \\notag \\displaybreak[3] \\\\\n\t&\\quad - \\bar{\\theta}^{\\dot{\\beta}} \\left( N_{\\dot{\\beta}\\alpha}{\\left( x \\right)}\n\t- i \\left( \\bar{\\sigma}^\\mu \\right)_{\\dot{\\beta} \\gamma} \\zeta^\\gamma \\partial_\\mu \\kappa_\\alpha{\\left( x \\right)}\n\t- 2 \\bar{\\zeta}_{\\dot{\\beta}} \\chi_\\alpha{\\left( x \\right)}\n\t- \\left( \\bar{\\sigma}^\\mu \\right)_{\\dot{\\beta} \\gamma} \\zeta^\\gamma \\omega_{\\mu \\alpha}{\\left( x \\right)} \\right) + \\notag \\displaybreak[3] \\\\\n\t&\\quad + \\theta^2 \\left( \\psi_\\alpha{\\left( x \\right)}\n\t- \\frac{i}{2} \\bar{\\zeta}^{\\dot{\\delta}} \\left( \\bar{\\sigma}^\\mu \\right)_{\\dot{\\delta}}{}^\\beta \\partial_\\mu M_{\\beta \\alpha}{\\left( x \\right)} \n\t+ \\bar{\\zeta}^{\\dot{\\beta}} R_{\\dot{\\beta} \\alpha}{\\left( x \\right)} \\right) + \\notag \\displaybreak[3] \\\\\n\t&\\quad + \\bar{\\theta}^2 \\left( \\chi_\\alpha{\\left( x \\right)}\n\t- \\frac{i}{2} \\zeta^\\delta \\left( \\sigma^\\mu \\right)_\\delta{}^{\\dot{\\beta}} \\partial_\\mu N_{\\dot{\\beta}\\alpha}{\\left( x \\right)} \n\t- \\zeta^\\beta S_{\\beta \\alpha}{\\left( x \\right)} \\right) + \\notag \\displaybreak[3] \\\\\n\t&\\quad + \\theta \\sigma^\\mu \\bar{\\theta} \\left( \\omega_{\\mu \\alpha}{\\left( x \\right)}\n\t+ \\frac{i}{2} \\zeta^\\delta \\left( \\sigma^\\nu \\bar{\\sigma}_\\mu \\right)_\\delta{}^\\beta \\partial_\\nu M_{\\beta \\alpha}{\\left( x \\right)}\n\t- \\frac{i}{2} \\bar{\\zeta}^{\\dot{\\delta}} \\left( \\bar{\\sigma}^\\nu \\sigma_\\mu \\right)_{\\dot{\\delta}}{}^{\\dot{\\beta}} \\partial_\\nu N_{\\dot{\\beta}\\alpha}{\\left( x \\right)} - \\right. \\notag \\displaybreak[3] \\\\\n\t&\\qquad \\left. - \\zeta_\\gamma \\left( \\sigma_\\mu \\right)^{\\gamma \\dot{\\beta}} R_{\\dot{\\beta} \\alpha}{\\left( x \\right)}\n\t+ \\bar{\\zeta}_{\\dot{\\gamma}} \\left( \\bar{\\sigma}_\\mu \\right)^{\\dot{\\gamma} \\beta} S_{\\beta \\alpha}{\\left( x \\right)} \\right) - \\notag \\displaybreak[3] \\\\\n\t&\\quad - \\theta^2 \\bar{\\theta}^{\\dot{\\beta}} \\! \\left( \\! R_{\\dot{\\beta} \\alpha}{\\left( x \\right)} \n\t- i \\left( \\bar{\\sigma}^\\mu \\right)_{\\dot{\\beta} \\gamma} \\zeta^\\gamma \\partial_\\mu \\psi_\\alpha{\\left( x \\right)}\n\t+ \\frac{i}{2} \\left( \\bar{\\sigma}^\\mu \\sigma^\\nu \\right)_{\\dot{\\beta} \\dot{\\epsilon}} \\bar{\\zeta}^{\\dot{\\epsilon}} \\partial_\\nu \\omega_{\\mu \\alpha}{\\left( x \\right)}\n\t- 2 \\bar{\\zeta}_{\\dot{\\beta}} \\lambda_\\alpha{\\left( x \\right)} \\! \\right) \\! + \\notag \\displaybreak[3] \\\\\n\t&\\quad + \\bar{\\theta}^2 \\theta^\\beta \\! \\left( \\! S_{\\beta \\alpha}{\\left( x \\right)}\n\t+ i \\left( \\sigma^\\mu \\right)_{\\beta \\dot{\\gamma}} \\bar{\\zeta}^{\\dot{\\gamma}} \\partial_\\mu \\chi_\\alpha{\\left( x \\right)}\n\t+ \\frac{i}{2} \\left( \\sigma^\\mu \\bar{\\sigma}^\\nu \\right)_{\\beta \\delta} \\zeta^\\delta \\partial_\\nu \\omega_{\\mu \\alpha}{\\left( x \\right)}\n\t- 2 \\zeta_\\beta \\lambda_\\alpha{\\left( x \\right)} \\! \\right) \\! + \\notag \\\\\n\t&\\quad + \\theta^2 \\bar{\\theta}^2 \\left( \\lambda_\\alpha{\\left( x \\right)}\n\t- \\frac{i}{2} \\zeta^\\gamma \\left( \\sigma^\\mu \\right)_{\\gamma}{}^{\\dot{\\beta}} \\partial_\\mu R_{\\dot{\\beta} \\alpha}{\\left( x \\right)}\n\t- \\frac{i}{2} \\bar{\\zeta}^{\\dot{\\delta}} \\left( \\bar{\\sigma}^\\mu \\right)_{\\dot{\\delta}}{}^\\beta \\partial_\\mu S_{\\beta \\alpha}{\\left( x \\right)} \\right) \\, .\n\\label{SUSYvariation}\n\\end{align}\nThe variation of the general spinor superfield is then defined as the difference between the translated superfield and the superfield at the original superspace coordinates\n\\begin{align}\n\\delta V_\\alpha{\\left( x , \\theta , \\bar{ \\theta} \\right)}\n\t&= V'_\\alpha{\\left( x , \\theta , \\bar{ \\theta} \\right)} - V_\\alpha{\\left( x , \\theta , \\bar{ \\theta} \\right)} \\, .\n\\end{align}\nTherefore, the variation of the component fields can be extracted immediately from equation (\\ref{SUSYvariation})\n\\begin{align}\n\\delta \\kappa_\\alpha\n\t&= - \\zeta^\\beta M_{\\beta \\alpha}{\\left( x \\right)}\n\t+ \\bar{\\zeta}^{\\dot{\\beta}} N_{\\dot{\\beta}\\alpha}{\\left( x \\right)} \\, , \\label{deltakappa} \\displaybreak[3] \\\\\n\\delta M_{\\beta \\alpha}\n\t&= - 2 \\zeta_\\beta \\psi_\\alpha{\\left( x \\right)}\n\t+ i \\bar{\\zeta}^{\\dot{\\gamma}} \\left( \\bar{\\sigma}^\\mu \\right)_{\\dot{\\gamma} \\beta} \\partial_\\mu \\kappa_\\alpha{\\left( x \\right)} \n\t- \\bar{\\zeta}^{\\dot{\\gamma}} \\left( \\bar{\\sigma}^\\mu \\right)_{\\dot{\\gamma} \\beta} \\omega_{\\mu \\alpha}{\\left( x \\right)} \\, , \\label{deltaM} \\displaybreak[3] \\\\\n\\delta N_{\\dot{\\beta}\\alpha}\n\t&= - 2 \\bar{\\zeta}_{\\dot{\\beta}} \\chi_\\alpha{\\left( x \\right)}\n\t- i \\zeta^\\gamma \\left( \\sigma^\\mu \\right)_{\\gamma \\dot{\\beta}} \\partial_\\mu \\kappa_\\alpha{\\left( x \\right)}\n\t- \\zeta^\\gamma \\left( \\sigma^\\mu \\right)_{\\gamma \\dot{\\beta}} \\omega_{\\mu \\alpha}{\\left( x \\right)} \\, , \\label{deltaN} \\displaybreak[3] \\\\\n\\delta \\psi_\\alpha\n\t&= \\bar{\\zeta}^{\\dot{\\beta}} R_{\\dot{\\beta} \\alpha}{\\left( x \\right)} \n\t- \\frac{i}{2} \\bar{\\zeta}^{\\dot{\\beta}} \\left( \\bar{\\sigma}^\\mu \\right)_{\\dot{\\beta}}{}^\\gamma \\partial_\\mu M_{\\gamma \\alpha}{\\left( x \\right)} \\, , \\label{deltapsi} \\displaybreak[3] \\\\\n\\delta \\chi_\\alpha\n\t&= - \\zeta^\\beta S_{\\beta \\alpha}{\\left( x \\right)}\n\t- \\frac{i}{2} \\zeta^\\beta \\left( \\sigma^\\mu \\right)_\\beta{}^{\\dot{\\gamma}} \\partial_\\mu N_{\\dot{\\gamma} \\alpha}{\\left( x \\right)} \\, , \\label{deltachi} \\displaybreak[3] \\\\\n\\delta \\omega_{\\mu \\alpha}\n\t&= \\zeta^\\beta \\left( \\sigma_\\mu \\right)_\\beta{}^{\\dot{\\gamma}} R_{\\dot{\\gamma} \\alpha}{\\left( x \\right)}\n\t+ \\frac{i}{2} \\zeta^\\beta \\left( \\sigma^\\nu \\bar{\\sigma}_\\mu \\right)_\\beta{}^\\gamma \\partial_\\nu M_{\\gamma \\alpha}{\\left( x \\right)} - \\notag \\\\\n\t&\\quad - \\bar{\\zeta}^{\\dot{\\beta}} \\left( \\bar{\\sigma}_\\mu \\right)_{\\dot{\\beta}}{}^\\gamma S_{\\gamma \\alpha}{\\left( x \\right)} \n\t- \\frac{i}{2} \\bar{\\zeta}^{\\dot{\\beta}} \\left( \\bar{\\sigma}^\\nu \\sigma_\\mu \\right)_{\\dot{\\beta}}{}^{\\dot{\\gamma}} \\partial_\\nu N_{\\dot{\\gamma}\\alpha}{\\left( x \\right)} \\label{deltaomega} \\, , \\displaybreak[3] \\\\\n\\delta R_{\\dot{\\beta} \\alpha}\n\t&= - 2 \\bar{\\zeta}_{\\dot{\\beta}} \\lambda_\\alpha{\\left( x \\right)}\n\t- i \\zeta^\\gamma \\left( \\sigma^\\mu \\right)_{\\gamma \\dot{\\beta}} \\partial_\\mu \\psi_\\alpha{\\left( x \\right)}\n\t- \\frac{i}{2} \\bar{\\zeta}^{\\dot{\\gamma}} \\left( \\bar{\\sigma}^\\nu \\sigma^\\mu \\right)_{\\dot{\\gamma} \\dot{\\beta}} \\partial_\\nu \\omega_{\\mu \\alpha}{\\left( x \\right)} \\, , \\label{deltaR} \\displaybreak[3] \\\\\n\\delta S_{\\beta \\alpha}\n\t&= - 2 \\zeta_\\beta \\lambda_\\alpha{\\left( x \\right)} \n\t+ i \\bar{\\zeta}^{\\dot{\\gamma}} \\left( \\bar{\\sigma}^\\mu \\right)_{\\dot{\\gamma} \\beta} \\partial_\\mu \\chi_\\alpha{\\left( x \\right)}\n\t- \\frac{i}{2} \\zeta^\\gamma \\left( \\sigma^\\nu \\bar{\\sigma}^\\mu \\right)_{\\gamma \\beta} \\partial_\\nu \\omega_{\\mu \\alpha}{\\left( x \\right)} \\, , \\label{deltaS} \\displaybreak[3] \\\\\n\\delta \\lambda_\\alpha\n\t&= - \\frac{i}{2} \\zeta^\\beta \\left( \\sigma^\\mu \\right)_{\\beta}{}^{\\dot{\\gamma}} \\partial_\\mu R_{\\dot{\\gamma} \\alpha}{\\left( x \\right)}\n\t- \\frac{i}{2} \\bar{\\zeta}^{\\dot{\\beta}} \\left( \\bar{\\sigma}^\\mu \\right)_{\\dot{\\beta}}{}^\\gamma \\partial_\\mu S_{\\gamma \\alpha}{\\left( x \\right)} \\, . \\label{deltalambda}\n\\end{align}\nThese results then imply the variation of the on-shell component fields. After eliminating the auxiliary fields and using the definition of the component fields $\\tilde{R}$ and $\\tilde{S}$ from equations (\\ref{tildeS}) and (\\ref{tildeR}) in section \\ref{SSLmassiveonshell} the variation of the component field of the on-shell Lagrangian are found to be\n\\begin{align}\n\\delta \\psi_\\alpha\n\t&= \\bar{\\zeta}^{\\dot{\\beta}} \\tilde{R}_{\\dot{\\beta} \\alpha} \\, , \\displaybreak[3] \\label{deltapsionshell}\\\\\n\\delta \\chi_\\alpha\n\t&= - \\zeta^\\beta \\tilde{S}_{\\beta \\alpha} \\, , \\displaybreak[3] \\label{deltachionshell}\\\\\n\\delta \\tilde{R}_{\\dot{\\beta} \\alpha}\n\t&= m \\bar{\\zeta}_{\\dot{\\beta}} \\chi_\\alpha\n\t- 2 i \\zeta^\\gamma \\dslash{\\partial}_{\\gamma \\dot{\\beta}} \\psi_\\alpha \\, , \\displaybreak[3] \\label{deltaRonshell}\\\\\n\\delta \\tilde{S}_{\\beta \\alpha}\n\t&= m \\zeta_\\beta \\psi_\\alpha\n\t+ 2 i \\bar{\\zeta}^{\\dot{\\gamma}} \\bar{\\dslash{\\partial}}_{\\dot{\\gamma} \\beta} \\chi_\\alpha \\, . \\label{deltaSonshell}\n\\end{align}\n\n\\subsection{Constructing a Model Based on the General Spinor Superfield}\n\\label{SSLchi}\n\\begin{TABLE}[t]{\n\\begin{tabular}{r|c|l}\nProduct & Mass Dimension & Contributions \\\\\n\\hline\n$V V$ & $\\mathrm{dim}{\\left( V V \\right)} = 0$ & $\\left( m^2 V V \\right)_D$ \\\\\n\\hline\n$X V$ & $\\mathrm{dim}{\\left( X V \\right)} = 1$ & $\\left( m X V \\right)_D$ , $\\left( m Y V \\right)_D$\\\\\n$D V D V$ & $\\mathrm{dim}{\\left( D V D V \\right)} = 1$ & $\\left( m D V D V \\right)_D$ , $\\left( m \\bar{D} V \\bar{D} V \\right)_D$ \\\\\n$V X$ & $\\mathrm{dim}{\\left( V X \\right)} = 1$ & $\\left( m V X \\right)_D$ , $\\left( m V Y \\right)_D$ \\\\\n\\hline\n$D Z V$ & $\\mathrm{dim}{\\left( D Z V \\right)} = 2$ & $\\left( D Z V \\right)_D$ , $\\left( \\bar{D} Z' V \\right)_D$\\\\\n$Z D V$ & $\\mathrm{dim}{\\left( Z D V \\right)} = 2$ & $\\left( Z D V \\right)_D$ , $\\left( Z' \\bar{D} V \\right)_D$\\\\\n$X X$ & $\\mathrm{dim}{\\left( X X \\right)} = 2$ & $\\left( m X X \\right)_F$ , $\\left( m Y Y \\right)_F$ , $\\left( X Y \\right)_D$ , $\\left( Y X \\right)_D$ \\\\\n$D V Z$ & $\\mathrm{dim}{\\left( D V Z \\right)} = 2$ & $\\left( D V Z \\right)_D$ , $\\left( \\bar{D} V Z' \\right)_D$ \\\\\n$V D Z$ & $\\mathrm{dim}{\\left( V D Z \\right)} = 2$ & $\\left( V D Z \\right)_D$ , $\\left( V \\bar{D} Z' \\right)_D$ \\\\\n\\hline\n$D Z X$ & $\\mathrm{dim}{\\left( D Z X \\right)} = 3$ & mass dimension too big for D-component \\\\\n$Z Z$ & $\\mathrm{dim}{\\left( Z Z \\right)} = 3$ & $\\left( Z Z \\right)_F$ , $\\left( Z' Z' \\right)_F$ \\\\\n$X D Z$ & $\\mathrm{dim}{\\left( X D Z \\right)} = 3$ & mass dimension too big for D-component\n\\end{tabular}\n\\caption{Possible contributions to the Lagrangian for $\\chi$ as fermionic field with mass dimension one based on the general spinor superfield. The first two columns specify the product and mass dimensionality using the general superfield and chiral superfields only. The third column then summarises all possible contributions corresponding to the product outlined in the first column including the contributions that arise from the antichiral superfields.}\n\\label{TChiDM}}\n\\end{TABLE}\nIf $\\chi$ is identified with the fermionic field with mass dimension one it can be shown that\n\\begin{align}\n\\mathrm{dim}{\\left( V_\\alpha \\right)}\n\t&= 0 \\, , \\quad\n\\mathrm{dim}{\\left( X_\\alpha \\right)}\n\t= \\mathrm{dim}{\\left( Y_\\alpha \\right)}\n\t= 1 \\, , \\quad\n\\mathrm{dim}{\\left( Z_{\\gamma \\alpha} \\right)}\n\t= \\mathrm{dim}{\\left( Z'_{\\dot{\\gamma} \\alpha} \\right)}\n\t= \\frac{3}{2} \\, .\n\\end{align}\nIt is interesting to note that for $\\chi$ as fermionic field with mass dimension one the mass dimension of the general spinor superfield is $1\/2$ lower than for the previous approach based on the the general scalar superfield. This indicates that the structure of this model is richer as there are more allowed contributions to the Lagrangian. For convenience the discussion is resticted to the unbarred superfields while the hermitian conjugates contribute to the Lagrangian as well.\n\nThe contributions to the Lagrangian have to satisfy the same basic requirements as outlined in Section \\ref{SSnonkinSUSYL} -- no uncontracted spinor indices, positive mass dimension for structure constants, and appropriate mass dimension for contribution via $D$- or $F$-component. All conceivable terms that are in agreement with these conditions are then summarised in table \\ref{TChiDM}. It contains more possible contributions to the Lagrangian which are now divided into four groups. The additional group is due to the lower mass dimensionality of the general spinor superfield which allows a spectrum for the mass dimension ranging from 0 and 3.\n\nThe first group which contains only one term, the product of two general spinor superfields without additional covariant derivatives, has mass dimension 0. For symmetry reasons the only possible contribution to the Lagrangian is a mass term via the D-component.\n\nThe second group containing all terms with mass dimension 1 has 6 possible terms. As $V$ and $D V$ are neither chiral nor anti-chiral all six terms are contributions to the mass term via the D-component.\n\nIn the third group all terms with mass dimension 2 are grouped together. It contains 12 terms of which 10 are contributions to the kinetic term via the D-component while 2 are contributions to the mass term via the F-component. It is worth mentioning that this is the only group that contains contributions to the kinetic term as well as contributions to the mass term. Even more intriguing is the fact that a superfield product of the form $X_1 X_2$ where $X_1$ and $X_2$ can be either chiral or antichiral is able to produce both kind of contributions.\n\nFinally, the fourth group which contains all terms with mass dimension 3 has two entries. Due to the mass dimension only contributions via the F-component are possible which means that both terms can only contribute to the kinetic term.\n\nIt is interesting to note that some of the terms contained in table \\ref{TChiDM}, namely $D V D V$ and $X V$ were previously considered by Gates and Siegel \\cite{gates80,gates81}. However, in these articles the authors assume the commonly used mass dimensinos for fermionic and bosonic fields. This has two consequences. First, all kinetic terms in \\cite{gates80,gates81} become mass terms in the present scenario due to the change of mass dimensionality. Second, all contributions summarised in groups three and four of table \\ref{TChiDM}, and therefore the products of chiral superfields $X X$ and $Y Y$ do not exist without redefinition of mass dimensions to accommodate fermionic fields with mass dimension one and thus were not considered before.\n\n\\subsection{The On-shell Lagrangian}\n\\label{SSLmassiveonshell}\nA supersymmetric Lagrangian can be constructed by combining contributions that were found in the dimensional analysis of the previous section. It was mentioned earlier that the first two groups of table \\ref{TChiDM} with mass dimension 0 and 1 respectively contain only contributions to the mass term while the group with mass dimension 3 only produces contributions to the kinetic term. Therefore, the following discussion for the construction of a supersymmetric Lagrangian will be resticted to the third group which is the only one containing kinetic as well as mass terms. This limits the number of superfield products that need to be calculated to 12. Explicit calculations reveal that this number can be narrowed down even further. It can be shown that the terms $\\left( D Z V \\right)_D$, $\\left( Z D V \\right)_D$, $\\left( X Y \\right)_D$, $\\left( D V Z \\right)_D$, $\\left( V D Z \\right)_D$ are identical up to a prefactor. Therefore, only the D-component of the terms $X Y$ and $Y X$ will be considered for the kinetic term. The Lagrangian can then be written in a very compact form\n\\begin{align}\n\\mathcal{L}\n\t&= \\left( X Y \\right)_D + \\left( Y X \\right)_D + \\frac{m}{2} \\left( X X \\right)_F + \\frac{m}{2} \\left( Y Y \\right)_F + h.c. \\, .\n\\label{Lcompact}\n\\end{align}\nFrom the previous derivation of the chiral superfield $X$ in equation (\\ref{Xchiral}) and the anti-chiral superfield $Y$ in equation (\\ref{Yantichiral}) it can be seen that the component fields $N$, $M$, $S$, $R$, $\\lambda$, and $\\kappa$ are not independent. Therefore, it is convenient to introduce the new component fields\n\\begin{align}\n\\tilde{S}_{\\beta \\alpha}\n\t&= S_{\\beta \\alpha} + \\frac{i}{2} \\dslash{\\partial}_\\beta{}^{\\dot{\\gamma}} N_{\\dot{\\gamma} \\alpha} \\, , \\label{tildeS} \\\\\n\\tilde{R}_{\\dot{\\beta} \\alpha} \n\t&= R_{\\dot{\\beta} \\alpha} - \\frac{i}{2} \\bar{\\dslash{\\partial}}_{\\dot{\\beta}}{}^\\tau M_{\\tau \\alpha} \\, , \\label{tildeR} \\\\\n\\tilde{\\lambda}_\\alpha\n\t&= \\lambda_\\alpha - \\frac{1}{4} \\Box \\kappa_\\alpha \\, . \\label{tildelambda}\n\\end{align}\nFurthermore, it can be seen that the spinor-vector field $\\omega^\\mu_\\alpha$ is always contracted with a four derivative which is simplified by defining\n\\begin{align}\n\\tilde{\\omega}_\\alpha \n\t&= \\partial^\\mu \\omega_{\\mu \\alpha} \\, . \\label{tildeomega}\n\\end{align}\nThe chiral and anti-chiral superfields can then be written as\n\\begin{align}\nX_\\alpha\n\t&= \\chi_\\alpha\n\t+ \\theta^\\beta \\tilde{S}_{\\beta \\alpha}\n\t+ \\theta^2 \\left( \\tilde{\\lambda}_\\alpha + \\frac{i}{2} \\tilde{\\omega}_\\alpha \\right)\n\t- i \\theta \\dslash{\\partial} \\bar{\\theta} \\chi_\\alpha\n\t+ \\frac{i}{2} \\theta^2 \\bar{\\theta}^{\\dot{\\gamma}} \\bar{\\dslash{\\partial}}_{\\dot{\\gamma}}{}^\\beta \\tilde{S}_{\\beta \\alpha}\n\t- \\frac{1}{4} \\theta^2 \\bar{\\theta}^2 \\Box \\chi_\\alpha \\, , \\\\\nY_\\alpha\n\t&= \\psi_\\alpha\n\t- \\bar{\\theta}^{\\dot{\\beta}} \\tilde{R}_{\\dot{\\beta} \\alpha}\n\t+ \\bar{\\theta}^2 \\left( \\tilde{\\lambda}_\\alpha - \\frac{i}{2} \\tilde{\\omega}_\\alpha \\right)\n\t+ i \\theta \\dslash{\\partial} \\bar{\\theta} \\psi_\\alpha\n\t+ \\frac{i}{2} \\theta^\\gamma \\bar{\\theta}^2 \\dslash{\\partial}_\\gamma{}^{\\dot{\\beta}} \\tilde{R}_{\\dot{\\beta} \\alpha}\n\t- \\frac{1}{4} \\theta^2 \\bar{\\theta}^2 \\Box \\psi_\\alpha \\, .\n\\end{align}\nThis can be used to calculate the contributions to the Lagrangian outlined in equation (\\ref{Lcompact})\n\\begin{align}\n\\left( X^\\alpha X_\\alpha \\right)_F\n\t&= \\chi \\tilde{\\lambda}\n\t+ \\frac{i}{2} \\chi \\tilde{\\omega}\n\t- \\frac{1}{2} \\mathrm{Tr}{\\left( \\tilde{S}^T \\tilde{S} \\right)}\n\t+ \\tilde{\\lambda} \\chi\n\t+ \\frac{i}{2} \\tilde{\\omega} \\chi \\, , \\displaybreak[3] \\\\\n\\left( Y^\\alpha Y_\\alpha \\right)_F\n\t&= \\psi \\tilde{\\lambda}\n\t- \\frac{i}{2} \\psi \\tilde{\\omega}\n\t- \\frac{1}{2} \\mathrm{Tr}{\\left( \\tilde{R}^T \\tilde{R} \\right)}\n\t+ \\tilde{\\lambda} \\psi\n\t- \\frac{i}{2} \\tilde{\\omega} \\psi \\, , \\displaybreak[3] \\\\\n\\left( X^\\alpha Y_\\alpha \\right)_D\n\t&= \\partial_\\mu \\chi \\partial^\\mu \\psi\n\t+ \\tilde{\\lambda} \\tilde{\\lambda}\n\t- \\frac{i}{2} \\tilde{\\lambda} \\tilde{\\omega}\n\t+ \\frac{i}{2} \\tilde{\\omega} \\tilde{\\lambda}\n\t+ \\frac{1}{4} \\tilde{\\omega} \\tilde{\\omega}\n\t+ \\frac{i}{2} \\mathrm{Tr}{\\left( \\tilde{S}^T \\dslash{\\partial} \\tilde{R} \\right)} \\, , \\\\\n\\left( Y^\\alpha X_\\alpha \\right)_D\n\t&= \\partial_\\mu \\psi \\partial^\\mu \\chi\n\t+ \\tilde{\\lambda} \\tilde{\\lambda}\n\t+ \\frac{i}{2} \\tilde{\\lambda} \\tilde{\\omega}\n\t- \\frac{i}{2} \\tilde{\\omega} \\tilde{\\lambda}\n\t+ \\frac{1}{4} \\tilde{\\omega} \\tilde{\\omega}\n\t+ \\frac{i}{2} \\mathrm{Tr}{\\left( \\tilde{R}^T \\bar{\\dslash{\\partial}} \\tilde{S} \\right)} \\, .\n\\end{align}\nTherefore, the Lagrangian is given by\n\\begin{align}\n\\mathcal{L}\n\t&= \\partial_\\mu \\chi \\partial^\\mu \\psi\n\t+ \\partial_\\mu \\psi \\partial^\\mu \\chi\n\t+ 2 \\tilde{\\lambda} \\tilde{\\lambda}\n\t+ \\frac{1}{2} \\tilde{\\omega} \\tilde{\\omega}\n\t+ \\frac{m}{2} \\chi \\tilde{\\lambda}\n\t+ \\frac{i m}{4} \\chi \\tilde{\\omega}\n\t+ \\frac{m}{2} \\tilde{\\lambda} \\chi\n\t+ \\frac{i m}{4} \\tilde{\\omega} \\chi + \\notag \\\\\n\t&\\quad + \\frac{m}{2} \\psi \\tilde{\\lambda}\n\t- \\frac{i m}{4} \\psi \\tilde{\\omega}\n\t+ \\frac{m}{2} \\tilde{\\lambda} \\psi\n\t- \\frac{i m}{4} \\tilde{\\omega} \\psi\n\t+ \\frac{i}{2} \\mathrm{Tr}{\\left( \\tilde{S}^T \\dslash{\\partial} \\tilde{R} \\right)}\n\t+ \\frac{i}{2} \\mathrm{Tr}{\\left( \\tilde{R}^T \\bar{\\dslash{\\partial}} \\tilde{S} \\right)} - \\notag \\\\\n\t&\\quad - \\frac{m}{4} \\mathrm{Tr}{\\left( \\tilde{S}^T \\tilde{S} \\right)}\n\t- \\frac{m}{4} \\mathrm{Tr}{\\left( \\tilde{R}^T \\tilde{R} \\right)} \n\t+ h.c. \\, .\n\\label{Lmdimone}\n\\end{align}\nIt can be seen that this Lagrangian still contains the auxiliary fields $\\tilde{\\lambda}$ and $\\tilde{\\omega}$. They can be eliminated from the Lagrangian using their equations of motion\n\\begin{align}\n\\tilde{\\omega}_\\tau\n\t&= - \\frac{i m}{2} \\left( \\chi_\\tau - \\psi_\\tau \\right) \\, , \\label{Mdimoneeqmomega} \\\\\n\\tilde{\\lambda}_\\tau\n\t&= - \\frac{m}{4} \\left( \\chi_\\tau + \\psi_\\tau \\right)\\, . \\label{Mdimoneeqmlambda}\n\\end{align}\nThis process is also referred to as going \"on-shell\". The resulting on-shell Lagrangian is then found to be\n\\begin{align}\n\\mathcal{L}\n\t&= \\partial_\\mu \\chi \\partial^\\mu \\psi\n\t+ \\partial_\\mu \\psi \\partial^\\mu \\chi\n\t- \\frac{m^2}{4} \\psi \\chi\n\t- \\frac{m^2}{4} \\chi \\psi + \\notag \\\\\n\t&\\quad + \\frac{i}{2} \\mathrm{Tr}{\\left( \\tilde{S}^T \\dslash{\\partial} \\tilde{R} \\right)}\n\t+ \\frac{i}{2} \\mathrm{Tr}{\\left( \\tilde{R}^T \\bar{\\dslash{\\partial}} \\tilde{S} \\right)}\n\t- \\frac{m}{4} \\mathrm{Tr}{\\left( \\tilde{S}^T \\tilde{S} \\right)}\n\t- \\frac{m}{4} \\mathrm{Tr}{\\left( \\tilde{R}^T \\tilde{R} \\right)} \\, .\n\\label{Lonshell}\n\\end{align}\nIt is solely dependent on the on-shell component fields $\\chi$, $\\psi$, $\\tilde{S}$, and $\\tilde{R}$. On the first glance it seems that there are twice as many bosonic degrees of freedom as fermionic ones, because each of the second-rank spinor fields has in general 8 degrees of freedom while each of the complex spinor fields only encompasses four degrees of freedom. However, on-shell the bosonic second-rank spinor fields satisfy a Weyl type equation which reduces the number of bosonic on-shell degrees of freedom by a factor of 2. This means that the Lagrangian indeed has 8 fermionic and 8 bosonic degrees of freedom.\n\n\\section{The Supercurrent}\n\\label{CJonshell}\nIn classical field theory the Noether theorem describes the connection between symmetry transformations that leave the Lagrangian invariant and the corresponding conserved quantities. It states that every symmetry results in a conserved current which can alternatively be expressed as a conserved charge. Even though supersymmetry is not a symmetry in the classical sense the on-shell Lagrangian is invariant under the variation of the component fields as defined in equations (\\ref{deltapsionshell}) to (\\ref{deltaSonshell}). Therefore, according to Noether's theorem, a conserved supercurrent exists.\n\nThe general equation for the supercurrent is given by\n\\begin{align}\nJ_{\\mu \\kappa}\n\t&= \\frac{\\partial}{\\partial \\zeta^\\kappa} \\left( \\sum\\limits_\\phi \\delta \\phi \\frac{\\partial \\mathcal{L}}{\\partial \\partial^\\mu \\phi}\n\t- \\mathcal{K}_\\mu \\right) \\, ,\n\\end{align}\nwhere the summation runs over all component fields of the Lagrangian. It has to be emphasised that this compact general equation for the supercurrent suppresses any indices of the component fields and also includes all hermitian conjugate component fields as well. The term $\\mathcal{K}_\\mu$ in this equation is related to the variation of the Lagrangian by\n\\begin{align}\n\\partial^\\mu \\mathcal{K}_\\mu\n\t&= \\delta \\mathcal{L} \\, .\n\\end{align}\n\nAs the full supercurrent $J_\\mu$ is hermitian it is possible to restrict the discussion to the on-shell Lagrangian without hermitian conjugate part and to calculate both $J_\\mu^{1\/2}$ as well as $\\bar{J}_\\mu^{1\/2}$. The complete supercurrent can then be constructed from the two contributions $J_\\mu^{1\/2}$ and $\\bar{J}_\\mu^{1\/2}$.\n\nThe general equation for $J^{1\/2}_\\mu$ can be written as\n\\begin{align}\nJ^{1\/2}_{\\mu \\kappa}\n\t&= \\frac{\\partial}{\\partial \\zeta^\\kappa} \\left( \\delta \\chi^\\tau \\frac{\\partial \\mathcal{L}}{\\partial \\partial^\\mu \\chi^\\tau}\n\t+ \\delta \\psi^\\tau \\frac{\\partial \\mathcal{L}}{\\partial \\partial^\\mu \\psi^\\tau}\n\t+ \\delta \\tilde{S}^{\\tau \\omega} \\frac{\\partial \\mathcal{L}}{\\partial \\partial^\\mu \\tilde{S}^{\\tau \\omega}}\n\t+ \\delta \\tilde{R}^{\\dot{\\tau} \\omega} \\frac{\\partial \\mathcal{L}}{\\partial \\partial^\\mu \\tilde{R}^{\\dot{\\tau} \\omega}}\n\t- \\mathcal{K}_\\mu \\right) \\, .\n\\end{align}\nInserting the on-shell Lagrangian from equation (\\ref{Lonshell}) into the equation for the supercurrent yields\n\\begin{align}\nJ_{\\mu \\kappa}\n\t&= - 3 \\tilde{S}_\\kappa{}^\\alpha \\partial_\\mu \\psi_\\alpha\n\t- \\frac{i m}{2} \\psi^\\alpha \\left( \\sigma_\\mu \\right)_\\kappa{}^{\\dot{\\beta}} \\tilde{R}_{\\dot{\\beta} \\alpha}\n\t- i \\partial_\\nu \\psi^\\alpha \\left( \\sigma^\\nu{}_\\mu \\right)_\\kappa{}^\\beta \\tilde{S}_{\\beta \\alpha}\n\t- \\frac{\\partial}{\\partial \\zeta^\\kappa} \\mathcal{K}_\\mu \\, .\n\\end{align}\nThe explicit form of $\\mathcal{K}_\\mu$ is derived from the variation of the Lagrangian without hermitian conjugate part\n\\begin{align}\n\\delta \\mathcal{L}\n\t&= \\partial_\\mu \\delta \\chi \\partial^\\mu \\psi\n\t+ \\partial_\\mu \\chi \\partial^\\mu \\delta \\psi\n\t+ \\partial_\\mu \\delta \\psi \\partial^\\mu \\chi\n\t+ \\partial_\\mu \\psi \\partial^\\mu \\delta \\chi\n\t- \\frac{m^2}{4} \\delta \\psi \\chi\n\t- \\frac{m^2}{4} \\psi \\delta \\chi\n\t- \\frac{m^2}{4} \\delta \\chi \\psi - \\notag \\\\\n\t&\\quad - \\frac{m^2}{4} \\chi \\psi \\delta + \\frac{i}{2} \\mathrm{Tr}{\\left( \\delta \\tilde{S}^T \\dslash{\\partial} \\tilde{R} \\right)}\n\t+ \\frac{i}{2} \\mathrm{Tr}{\\left( \\tilde{S}^T \\dslash{\\partial} \\delta \\tilde{R} \\right)}\n\t+ \\frac{i}{2} \\mathrm{Tr}{\\left( \\delta \\tilde{R}^T \\bar{\\dslash{\\partial}} \\tilde{S} \\right)}\n\t+ \\frac{i}{2} \\mathrm{Tr}{\\left( \\tilde{R}^T \\bar{\\dslash{\\partial}} \\delta \\tilde{S} \\right)} - \\notag \\\\\n\t&\\quad - \\frac{m}{4} \\mathrm{Tr}{\\left( \\delta \\tilde{S}^T \\tilde{S} \\right)}\n\t- \\frac{m}{4} \\mathrm{Tr}{\\left( \\tilde{S}^T \\delta \\tilde{S} \\right)}\n\t- \\frac{m}{4} \\mathrm{Tr}{\\left( \\delta \\tilde{R}^T \\tilde{R} \\right)}\n\t- \\frac{m}{4} \\mathrm{Tr}{\\left( \\tilde{R}^T \\delta \\tilde{R} \\right)} \\, .\n\\end{align}\t\nIt can be shown that the variation of the Lagrangian is a four divergence as expected which implies that\n\\begin{align}\n\\mathcal{K}_\\mu\n\t&= \\zeta^\\beta \\tilde{S}_{\\beta \\alpha} \\partial_\\mu \\psi^\\alpha\n\t- \\bar{\\zeta}^{\\dot{\\beta}} \\tilde{R}_{\\dot{\\beta} \\alpha} \\partial_\\mu \\chi^\\alpha\n\t+ \\frac{i m}{2} \\left( \\bar{\\sigma}_\\mu \\right)_{\\dot{\\beta}}{}^\\gamma \\bar{\\zeta}^{\\dot{\\beta}} \\chi^\\alpha  \\tilde{S}_{\\gamma \\alpha}\n\t+ \\frac{i m}{2} \\left( \\sigma_\\mu \\right)_\\beta{}^{\\dot{\\gamma}} \\zeta^\\beta \\psi^\\alpha \\tilde{R}_{\\dot{\\gamma} \\alpha} + \\notag \\\\\n\t&\\quad + i \\bar{\\zeta}^{\\dot{\\delta}} \\left( \\bar{\\sigma}_\\mu{}^\\nu \\right)_{\\dot{\\delta}}{}^{\\dot{\\gamma}} \\chi^\\alpha \\partial_\\nu \\tilde{R}_{\\dot{\\gamma} \\alpha}\n\t+ i \\zeta^\\delta \\left( \\sigma_\\mu{}^\\nu \\right)_\\delta{}^\\gamma \\psi^\\alpha \\partial_\\nu \\tilde{S}_{\\gamma \\alpha} \\, .\n\\label{Kmu}\n\\end{align}\nThis result can then be inserted into the equation for the supercurrent. After differentiating with respect to the transformation parameter $\\zeta$ the supercurrent is found to be\n\\begin{align}\nJ^{1\/2}_{\\mu \\kappa}\n\t&= - i m \\left( \\sigma_\\mu \\right)_\\kappa{}^{\\dot{\\beta}} \\tilde{R}_{\\dot{\\beta} \\alpha} \\psi^\\alpha\n\t+ 2 \\left( \\sigma_\\mu \\right)^{\\beta \\dot{\\gamma}} \\bar{\\dslash{\\partial}}_{\\dot{\\gamma} \\kappa} \\psi^\\alpha \\tilde{S}_{\\beta \\alpha} \\, .\n\\label{Jhalf}\n\\end{align}\n\nThe contribution to the full supercurrent $\\bar{J}_\\mu^{1\/2}$ is defined in perfect analogy to $J_\\mu^{1\/2}$ by replacing the derivative with respect to the Grassmann variable $\\zeta$ with a derivative with respect to $\\bar{\\zeta}$. It has to be noted that the behaviour of the Grassmann derivative is rather subtle and depends on the conventions chosen. In the present scenario where by convention all derivatives are written as right derivatives the change from left to right derivative introduces an additional overall minus sign\n\\begin{align}\n\\bar{J}^{1\/2}_{\\mu \\dot{\\kappa}}\n\t&= - \\frac{\\partial}{\\partial \\bar{\\zeta}^{\\dot{\\kappa}}} \\left( \\delta \\chi^\\tau \\frac{\\partial \\mathcal{L}}{\\partial \\partial^\\mu \\chi^\\tau}\n\t+ \\delta \\psi^\\tau \\frac{\\partial \\mathcal{L}}{\\partial \\partial^\\mu \\psi^\\tau}\n\t+ \\delta \\tilde{S}^{\\tau \\omega} \\frac{\\partial \\mathcal{L}}{\\partial \\partial^\\mu \\tilde{S}^{\\tau \\omega}}\n\t+ \\delta \\tilde{R}^{\\dot{\\tau} \\omega} \\frac{\\partial \\mathcal{L}}{\\partial \\partial^\\mu \\tilde{R}^{\\dot{\\tau} \\omega}}\n\t- \\mathcal{K}_\\mu \\right) \\, .\n\\end{align}\nThe supercurrent $\\bar{J}_\\mu^{1\/2}$ for the Lagrangian without the complex conjugate part is then given by\n\\begin{align}\n\\bar{J}^{1\/2}_{\\mu \\dot{\\kappa}}\n\t&= 3 \\tilde{R}_{\\dot{\\kappa} \\alpha} \\partial_\\mu \\chi^\\alpha\n\t+ i \\left( \\sigma^\\nu{}_\\mu \\right)_{\\dot{\\kappa}}{}^{\\dot{\\beta}} \\partial_\\nu \\chi^\\alpha \\tilde{R}_{\\dot{\\beta} \\alpha}\n\t+ \\frac{i m}{2} \\left( \\bar{\\sigma}_\\mu \\right)_{\\dot{\\kappa}}{}^\\beta \\tilde{S}_{\\beta \\alpha} \\chi^\\alpha\n\t+ \\frac{\\partial}{\\partial \\bar{\\zeta}^{\\dot{\\kappa}}} \\mathcal{K}_\\mu \\, ,\n\\end{align}\nwhere the term $\\mathcal{K}_\\mu$ is already known from equation (\\ref{Kmu}). After differentiation with respect to the Grassmann variable the final result is\n\\begin{align}\n\\bar{J}^{1\/2}_{\\mu \\dot{\\kappa}}\n\t&= i m \\left( \\bar{\\sigma}_\\mu \\right)_{\\dot{\\kappa}}{}^\\beta \\tilde{S}_{\\beta \\alpha} \\chi^\\alpha\n\t+ 2 \\left( \\bar{\\sigma}_\\mu \\right)^{\\dot{\\beta} \\gamma} \\dslash{\\partial}_{\\gamma \\dot{\\kappa}} \\chi^\\alpha \\tilde{R}_{\\dot{\\beta} \\alpha} \\, .\n\\label{Jbarhalf}\n\\end{align}\nTogether with the previous result for $J^{1\/2}_\\mu$ from equation (\\ref{Jhalf}) the construction of the full supercurrent is straightforward\n\\begin{align}\nJ_{\\mu \\kappa}\n\t&= - i m \\left( \\sigma_\\mu \\right)_\\kappa{}^{\\dot{\\beta}} \\tilde{R}_{\\dot{\\beta} \\alpha} \\psi^\\alpha\n\t+ 2 \\left( \\sigma_\\mu \\right)^{\\beta \\dot{\\gamma}} \\bar{\\dslash{\\partial}}_{\\dot{\\gamma} \\kappa} \\psi^\\alpha \\tilde{S}_{\\beta \\alpha} - \\notag \\\\\n\t&\\quad - i m \\left( \\sigma_\\mu \\right)_{\\kappa}{}^{\\dot{\\beta}} \\bar{\\tilde{S}}_{\\dot{\\beta} \\dot{\\alpha}} \\bar{\\chi}^{\\dot{\\alpha}}\n\t+ 2 \\left( \\sigma_\\mu \\right)^{\\beta \\dot{\\gamma}} \\bar{\\dslash{\\partial}}_{\\dot{\\gamma} \\kappa} \\bar{\\chi}^{\\dot{\\alpha}} \\bar{\\tilde{R}}_{\\beta \\dot{\\alpha}} \\, .\n\\label{Jfull}\n\\end{align}\n\n\\section{The Hamiltonian in Position Space}\n\\label{CHposition}\nThe Hamiltonian in position space is usually derived from the Lagrangian by canonical quantisation. However, it is not immediately clear whether this approach is still valid for the present scenario that is based on a general spinor superfield instead of a scalar superfield. Therefore, a more conservative approach based on the supersymmetry algebra was chosen. It utilises the anticommutation relation between the barred and unbarred supersymmetry generator of the $N = 1$ supersymmetry algebra\n\\begin{align}\n2 \\left( \\sigma^\\mu \\right)_{\\alpha \\dot{\\beta}} P_\\mu\n\t&= \n\t\\left\\{ Q_\\alpha , \\bar{Q}_{\\dot{\\beta}} \\right\\} .\n\\label{PSUSYalgebra}\n\\end{align}\n\nAt this point it can already be seen that a successful derivation of the Hamiltonian from the supersymmetry algebra requires the knowledge of the commutation and anticommutation relations of the component fields in position space. Therefore, the second quantisation of the component fields in position space will be discussed in Section \\ref{SQuantCompFields}. Afterwards in Section \\ref{CHSUSYalgebra}, these results will be used to derive an expression for the Hamiltonian in position space which is founded in the supersymmetry algebra. Finally, in Section \\ref{CHcanonicalquant} it will be shown that canonical quantisation yields the same Hamiltonian in position space as the approach based on the supersymmetry algebra.\n\n\\subsection{Second Quantisation in Position Space}\n\\label{SQuantCompFields}\nA viable supersymmetric model of fermionic fields with mass dimension one requires a second quantisation that is in agreement with the superfield transformations of the component fields as derived in Section \\ref{SSSUSYtranslation}. This can be achieved by calculating the commutator between the component fields and the generators of the superspace translations\n\\begin{align}\n\\delta \\phi\n\t&= - i \\left[ \\phi , \\zeta^\\alpha Q_\\alpha + \\zeta_{\\dot{\\alpha}} Q^{\\dot{\\alpha}} \\right] \\, .\n\\label{compfieldvarcommutator}\n\\end{align}\nTo generalise the notation the spinor indices of the field $\\phi$ are suppressed and it can represent a scalar field as well as first, second, or higher rank spinor fields. Subsequently, the commutation and anticommutation relations of the barred component fields are derived from the results for the unbarred component fields.\n\nThe supersymmetry generators that appear in this equation are proportional to the supercurrent\n\\begin{align}\nQ_\\alpha\n\t&= \\int \\mathrm{d} \\mathbf{x} J_{0 \\alpha} \\, , \\label{QproptoJ}\\\\\n\\bar{Q}_{\\dot{\\alpha}} \n\t&= \\int \\mathrm{d} \\mathbf{x} \\bar{J}_{0 \\dot{\\alpha}} \\, . \\label{QbarproptoJbar}\n\\end{align}\nIn general the supersymmetry generators must contain the full supercurrent. However, the previous results for the superfield translations, equations (\\ref{deltapsionshell}) to (\\ref{deltaSonshell}), imply that no mixing between barred and unbarred component fields occurs. Therefore, it is sufficient to restrict the discussion in this section to the supercurrent arising from the Lagrangian without hermitian conjugate contribution, as any cross terms vanish identically and define the constrained generators\n\\begin{align}\nQ^{1\/2}_\\alpha\n\t&= \\int \\mathrm{d} \\mathbf{x} J^{1\/2}_{0 \\alpha} \\, , \\\\\n\\bar{Q}^{1\/2}_{\\dot{\\alpha}} \n\t&= \\int \\mathrm{d} \\mathbf{x} \\bar{J}^{1\/2}_{0 \\dot{\\alpha}} \\, .\n\\end{align}\nTo distinguish the constrained generators from the full generators as outlined in equations (\\ref{QproptoJ}) and (\\ref{QbarproptoJbar}) an additional superscript $1\/2$ was incorporated into the notation in analogy to the notation for the supercurrent in Chapter \\ref{CJonshell}. Inserting the results for the supercurrent from equations (\\ref{Jhalf}) and (\\ref{Jbarhalf}) then yields the following expression for the constrained supersymmetry generators\n\\begin{align}\nQ^{1\/2}_\\alpha\n\t&= \\int \\mathrm{d} \\mathbf{x} \\left( - i m \\left( \\sigma_\\mu \\right)_\\alpha{}^{\\dot{\\gamma}} \\tilde{R}_{\\dot{\\gamma} \\beta}{\\left( x \\right)} \\psi^\\beta{\\left( x \\right)}\n\t+ 2 \\left( \\sigma_\\mu \\right)^{\\gamma \\dot{\\delta}} \\tilde{S}_{\\gamma \\beta}{\\left( x \\right)} \\bar{\\dslash{\\partial}}_{\\dot{\\delta} \\alpha} \\psi^\\beta{\\left( x \\right)} \\right) \\, , \\label{generatorQ}\\\\\n\\bar{Q}^{1\/2}_{\\dot{\\alpha}}\n\t&= \\int \\mathrm{d} \\mathbf{x} \\left( i m \\left( \\bar{\\sigma}_\\mu \\right)_{\\dot{\\alpha}}{}^\\gamma \\tilde{S}_{\\gamma \\beta}{\\left( x \\right)} \\chi^\\beta{\\left( x \\right)}\n\t+ 2 \\left( \\bar{\\sigma}_\\mu \\right)^{\\dot{\\gamma} \\delta} \\tilde{R}_{\\dot{\\gamma} \\beta}{\\left( x \\right)} \\dslash{\\partial}_{\\delta \\dot{\\alpha}} \\chi^\\beta{\\left( x \\right)} \\right) \\, . \\label{generatorQbar}\n\\end{align}\n\n\\subsubsection{Superfield Transformation of the Fermionic Component Fields}\nInserting the constrained supersymmetry generators as defined in equations (\\ref{generatorQ}) and (\\ref{generatorQbar}) into equation (\\ref{compfieldvarcommutator}) for the commutator between component field $\\chi$ and the generators of superspace translations yields a variation of $\\chi$ of\n\\begin{align}\n\\delta \\chi_\\alpha{\\left( x \\right)}\n\t&= \\int \\mathrm{d} \\mathbf{x}' \\left( m \\zeta^\\beta \\left( \\sigma_0 \\right)_\\beta{}^{\\dot{\\gamma}} \\left\\{ \\chi_\\alpha{\\left( x \\right)} , \\tilde{R}_{\\dot{\\gamma} \\delta}{\\left( x' \\right)} \\psi^\\delta{\\left( x' \\right)} \\right\\} + \\right. \\notag \\\\\n\t&\\qquad + 2 i \\zeta^\\beta \\left( \\sigma_0 \\right)^{\\gamma \\dot{\\delta}} \\left\\{ \\chi_\\alpha{\\left( x \\right)} , \\tilde{S}_{\\gamma \\epsilon}{\\left( x' \\right)} \\bar{\\dslash{\\partial}}'_{\\dot{\\delta} \\beta} \\psi^\\epsilon{\\left( x' \\right)} \\right\\} - \\notag \\\\\n\t&\\qquad - m \\bar{\\zeta}_{\\dot{\\beta}} \\left( \\bar{\\sigma}_0 \\right)^{\\dot{\\beta} \\gamma} \\left\\{ \\chi_\\alpha{\\left( x \\right)} , \\tilde{S}_{\\gamma \\delta}{\\left( x' \\right)} \\chi^\\delta{\\left( x' \\right)} \\right\\} + \\notag \\\\\n\t&\\qquad \\left. + 2 i \\bar{\\zeta}_{\\dot{\\beta}} \\left( \\bar{\\sigma}_0 \\right)^{\\dot{\\gamma} \\delta} \\left\\{ \\chi_\\alpha{\\left( x \\right)} , \\tilde{R}_{\\dot{\\gamma} \\epsilon}{\\left( x' \\right)} \\dslash{\\partial}'_\\delta{}^{\\dot{\\beta}} \\chi^\\epsilon{\\left( x' \\right)} \\right\\} \\right) \\, .\n\\end{align}\nEach of the contributions to the variation of the component field $\\chi$ contains an anticommutator involving two fermionic fields and one bosonic field. They can be rewritten using the anticommutator relation\n\\begin{align}\n\\left\\{ F_1 , B_2 F_2 \\right\\}\n\t&= B_2 \\left\\{ F_1 , F_2 \\right\\} \\, .\n\\end{align}\nIn addition, it can be seen that the second and fourth term contain a four derivative $\\dslash{\\partial}$ acting on one of the component fields in the anticommutator. These terms can be rewritten by splitting the four derivative into its time and spatial components\n\\begin{align}\n\\dslash{\\partial}_{\\alpha \\dot{\\beta}}\n\t&= \\left( \\sigma^\\mu \\right)_{\\alpha \\dot{\\beta}} \\partial_\\mu\n\t= \\left( \\sigma^0 \\right)_{\\alpha \\dot{\\beta}} \\partial_0\n\t+ \\boldsymbol{\\sigma}_{\\alpha \\dot{\\beta}} \\cdot \\boldsymbol{\\nabla} \\, .\n\\end{align}\nIt is important to recall that there is a plus sign between the time and spatial components and not a minus sign as the derivative is a covariant three vector $\\boldsymbol{\\nabla} = \\left( \\partial_1 , \\partial_2 , \\partial_3 \\right)$ while all standard vectors, e.g. $\\mathbf{p} = \\left( p^1, p^2 , p^3 \\right)$, are contravariant three vectors. After partial integration over the spatial components each term involving a four-derivative is replaced by two terms -- one containing a time derivative acting on one of the fields in the commutator and one simply containing the commutator of component fields. Furthermore, the boundary terms from the partial integration which are 3-divergences vanish identically and are ignored. This results in\n\\begin{align}\n\\delta \\chi_\\alpha{\\left( x \\right)}\n\t&= \\int \\mathrm{d} \\mathbf{x}' \\left( m \\zeta^\\beta \\left( \\sigma_0 \\right)_\\beta{}^{\\dot{\\gamma}} \\tilde{R}_{\\dot{\\gamma} \\delta}{\\left( x' \\right)} \\left\\{ \\chi_\\alpha{\\left( x \\right)} , \\psi^\\delta{\\left( x' \\right)} \\right\\}\n\t+ 2 i \\zeta^\\beta \\tilde{S}_{\\beta \\epsilon}{\\left( x' \\right)} \\left\\{ \\chi_\\alpha{\\left( x \\right)} , \\dot{\\psi}^\\epsilon{\\left( x' \\right)} \\right\\} + \\right. \\notag \\\\\n\t&\\qquad + 2 i \\zeta^\\beta \\left( \\sigma_0 \\right)^{\\gamma \\dot{\\delta}} \\boldsymbol{\\sigma}_{\\dot{\\delta} \\beta} \\cdot \\boldsymbol{\\nabla}' \\tilde{S}_{\\gamma \\epsilon}{\\left( x' \\right)} \\left\\{ \\chi_\\alpha{\\left( x \\right)} , \\psi^\\epsilon{\\left( x' \\right)} \\right\\} - \\notag \\\\\n\t&\\qquad - m \\bar{\\zeta}_{\\dot{\\beta}} \\left( \\bar{\\sigma}_0 \\right)^{\\dot{\\beta} \\gamma} \\tilde{S}_{\\gamma \\delta}{\\left( x' \\right)} \\left\\{ \\chi_\\alpha{\\left( x \\right)} , \\chi^\\delta{\\left( x' \\right)} \\right\\} \n\t- 2 i \\bar{\\zeta}^{\\dot{\\beta}} \\tilde{R}_{\\dot{\\beta} \\epsilon}{\\left( x' \\right)} \\left\\{ \\chi_\\alpha{\\left( x \\right)} , \\dot{\\chi}^\\epsilon{\\left( x' \\right)} \\right\\} + \\notag \\\\\n\t&\\qquad \\left. + 2 i \\bar{\\zeta}_{\\dot{\\beta}} \\left( \\bar{\\sigma}_0 \\right)^{\\dot{\\gamma} \\delta} \\boldsymbol{\\sigma}_\\delta{}^{\\dot{\\beta}} \\cdot  \\boldsymbol{\\nabla}' \\tilde{R}_{\\dot{\\gamma} \\epsilon}{\\left( x' \\right)} \\left\\{ \\chi_\\alpha{\\left( x \\right)} , \\chi^\\epsilon{\\left( x' \\right)} \\right\\} \\right) \\, .\n\\label{deltachisecondquant}\n\\end{align}\nBy comparison with the previously derived superspace translation of $\\chi$ in equation (\\ref{deltachionshell}) it can be seen that the only nonvanishing contribution comes from the term proportional to $\\zeta \\tilde{S}$ while all other contributions have to vanish identically. This implies that three of the anticommutators vanish identically\n\\begin{align}\n\\left\\{ \\chi_\\alpha{\\left( x \\right)} , \\psi_\\beta{\\left( x' \\right)} \\right\\}\n\t&= 0 \\, ,\\\\\n\\left\\{ \\chi_\\alpha{\\left( x \\right)} , \\dot{\\chi}_\\beta{\\left( x' \\right)} \\right\\}\n\t&= 0 \\, , \\\\\n\\left\\{ \\chi_\\alpha{\\left( x \\right)} , \\chi_\\beta{\\left( x' \\right)} \\right\\}\n\t&= 0 \\, .\n\\end{align}\nThe only nonvanishing anticommutator satisfies\n\\begin{align}\n- \\zeta^\\beta \\tilde{S}_{\\beta \\alpha}{\\left( x \\right)}\n\t&= - 2 i \\zeta^\\beta \\int \\mathrm{d} \\mathbf{x}' \\tilde{S}_\\beta{}^\\gamma{\\left( x' \\right)} \\left\\{ \\chi_\\alpha{\\left( x \\right)} , \\dot{\\psi}_\\gamma{\\left( x' \\right)} \\right\\} \\, ,\n\\label{quantrelchi}\n\\end{align}\nwhich has the solution\n\\begin{align}\n\\left\\{ \\chi_\\alpha{\\left( x \\right)} , \\dot{\\psi}_\\gamma{\\left( x' \\right)} \\right\\}\n\t&= \\frac{i}{2} \\epsilon_{\\alpha \\gamma} \\delta{\\left( x - x' \\right)} \\, .\n\\end{align}\n\nAs the Lagrangian is symmetric with respect to the exchange of $\\chi$ and $\\psi$ there is no difference between the calculation of $\\delta \\chi$ and $\\delta \\psi$. Again, three of the anticommutators have to vanish identically\n\\begin{align}\n\\left\\{ \\psi_\\alpha{\\left( x \\right)} , \\dot{\\psi}_\\beta{\\left( x' \\right)} \\right\\}\n\t&= 0 \\, , \\\\\n\\left\\{ \\psi_\\alpha{\\left( x \\right)} , \\psi_\\beta{\\left( x' \\right)} \\right\\}\n\t&= 0 \\, , \\\\\n\\left\\{ \\psi_\\alpha{\\left( x \\right)} , \\chi_\\beta{\\left( x' \\right)} \\right\\}\n\t&= 0\\, ,\n\\end{align}\nwhile the only nonvanishing anticommutator is the one involving $\\psi$ and $\\dot{\\chi}$\n\\begin{align}\n\\left\\{ \\psi_\\alpha{\\left( x \\right)} , \\dot{\\chi}_\\gamma{\\left( x' \\right)} \\right\\}\n\t&= \\frac{i}{2} \\epsilon_{\\alpha \\gamma} \\delta{\\left( \\mathbf{x} - \\mathbf{x'} \\right)} \\, .\n\\end{align}\n\n\\subsubsection{Superfield Transformation of the Bosonic Component Fields}\nThe discussion for the superfield transformation of the bosonic component fields is in perfect analogy to those for the fermionic component fields. The change from a fermionic to a bosonic field results in an exchange of all anticommutators with commutators\n\\begin{align}\n\\delta \\tilde{S}_{\\beta \\alpha}{\\left( x \\right)}\n\t&= \\int \\mathrm{d} \\mathbf{x}' \\left( - m \\zeta^\\gamma \\left( \\sigma_0 \\right)_\\gamma{}^{\\dot{\\epsilon}} \\left[ \\tilde{S}_{\\beta \\alpha}{\\left( x \\right)} , \\tilde{R}_{\\dot{\\epsilon} \\delta}{\\left( x' \\right)} \\psi^\\delta{\\left( x' \\right)} \\right] - \\right. \\notag \\\\\n\t&\\qquad- 2 i \\zeta^\\gamma \\left( \\sigma_0 \\right)^{\\epsilon \\dot{\\delta}} \\left[ \\tilde{S}_{\\beta \\alpha}{\\left( x \\right)} , \\tilde{S}_{\\epsilon \\kappa}{\\left( x' \\right)} \\bar{\\dslash{\\partial}}_{\\dot{\\delta} \\gamma} \\psi^\\kappa{\\left( x' \\right)} \\right] +  \\notag \\\\\n\t&\\qquad + m \\bar{\\zeta}_{\\dot{\\gamma}} \\left( \\bar{\\sigma}_0 \\right)^{\\dot{\\gamma} \\epsilon} \\left[ \\tilde{S}_{\\beta \\alpha}{\\left( x \\right)} , \\tilde{S}_{\\epsilon \\delta}{\\left( x' \\right)} \\chi^\\delta{\\left( x' \\right)} \\right] - \\notag \\\\\n\t&\\qquad \\left. - 2 i \\bar{\\zeta}_{\\dot{\\gamma}} \\left( \\bar{\\sigma}_0 \\right)^{\\dot{\\epsilon} \\delta} \\left[ \\tilde{S}_{\\beta \\alpha}{\\left( x \\right)} , \\tilde{R}_{\\dot{\\epsilon} \\kappa}{\\left( x' \\right)} \\dslash{\\partial}_\\delta{}^{\\dot{\\gamma}} \\chi^\\kappa{\\left( x' \\right)} \\right] \\right) \\, .\n\\end{align}\nThe commutators involved in this expression each contain two bosonic and one fermionic component field and can be simplified using the commutator relation\n\\begin{align}\n\\left[ B_1 , B_2 F_2 \\right]\n\t&= F_2 \\left[ B_1 , B_2 \\right] \\, .\n\\end{align}\nTherefore, the variation of the bosonic second-rank spinor field $\\tilde{S}$ takes the form\n\\begin{align}\n\\delta \\tilde{S}_{\\beta \\alpha}{\\left( x \\right)}\n\t&= \\int \\mathrm{d} \\mathbf{x}' \\left( - m \\zeta^\\gamma \\left( \\sigma_0 \\right)_\\gamma{}^{\\dot{\\epsilon}} \\psi^\\delta{\\left( x' \\right)} \\left[ \\tilde{S}_{\\beta \\alpha}{\\left( x \\right)} , \\tilde{R}_{\\dot{\\epsilon} \\delta}{\\left( x' \\right)} \\right] - \\right. \\notag \\\\\n\t&\\qquad - 2 i \\zeta^\\gamma \\left( \\sigma_0 \\right)^{\\epsilon \\dot{\\delta}} \\bar{\\dslash{\\partial}}_{\\dot{\\delta} \\gamma} \\psi^\\kappa{\\left( x' \\right)} \\left[ \\tilde{S}_{\\beta \\alpha}{\\left( x \\right)} , \\tilde{S}_{\\epsilon \\kappa}{\\left( x' \\right)} \\right] + \\notag \\\\\n\t&\\qquad + m \\bar{\\zeta}_{\\dot{\\gamma}} \\left( \\bar{\\sigma}_0 \\right)^{\\dot{\\gamma} \\epsilon} \\chi^\\delta{\\left( x' \\right)} \\left[ \\tilde{S}_{\\beta \\alpha}{\\left( x \\right)} , \\tilde{S}_{\\epsilon \\delta}{\\left( x' \\right)} \\right] - \\notag \\\\\n\t&\\qquad \\left. - 2 i \\bar{\\zeta}_{\\dot{\\gamma}} \\left( \\bar{\\sigma}_0 \\right)^{\\dot{\\epsilon} \\delta} \\dslash{\\partial}_\\delta{}^{\\dot{\\gamma}} \\chi^\\kappa{\\left( x' \\right)} \\left[ \\tilde{S}_{\\beta \\alpha}{\\left( x \\right)} , \\tilde{R}_{\\dot{\\epsilon} \\kappa}{\\left( x' \\right)} \\right] \\right) \\, .\n\\label{deltaSsecondquant}\n\\end{align}\nIf this result is compared to the superspace translation of $\\tilde{S}$ in equation (\\ref{deltaSonshell}) it can be immediately read off that\n\\begin{align}\n\\left[ \\tilde{S}_{\\beta \\alpha}{\\left( x \\right)} , \\tilde{S}_{\\gamma \\delta}{\\left( x' \\right)} \\right]\n\t&= 0 \\, .\n\\end{align}\nThe remaining two relations for the commutator between $\\tilde{S}$ and $\\tilde{R}$ should yield the same result. Using the first relation \n\\begin{align}\nm \\zeta_\\beta \\psi_\\alpha\n\t&= - m \\zeta^\\gamma \\left( \\sigma_0 \\right)_\\gamma{}^{\\dot{\\epsilon}} \\int \\mathrm{d} \\mathbf{x}' \\psi^\\delta{\\left( x' \\right)} \\left[ \\tilde{S}_{\\beta \\alpha}{\\left( x \\right)} , \\tilde{R}_{\\dot{\\epsilon} \\delta}{\\left( x' \\right)} \\right]\n\\label{quantrelS}\n\\end{align}\nit is found that $\\tilde{S}$ and $\\tilde{R}$ satisfy the commutation relation\n\\begin{align}\n\\left[ \\tilde{S}_{\\beta \\alpha}{\\left( x \\right)} , \\tilde{R}_{\\dot{\\epsilon} \\delta}{\\left( x' \\right)} \\right]\n\t&= - \\epsilon_{\\alpha \\delta} \\delta{\\left( x - x' \\right)}\\left( \\sigma^0 \\right)_{\\beta \\dot{\\epsilon}} \\, .\n\\label{commutatorSR}\n\\end{align}\nInserting this result into the second relation then provides a consistency check as it satisfies the relation identically.\n\nAgain, the calculations for the superfield transformation of $\\tilde{R}$ are in perfect analogy to those for $\\tilde{S}$. It is found that the commutator of $\\tilde{R}$ with itself vanishes\n\\begin{align}\n\\left[ \\tilde{R}_{\\dot{\\beta} \\alpha}{\\left( x \\right)} , \\tilde{R}_{\\dot{\\gamma} \\delta}{\\left( x' \\right)} \\right]\n\t&= 0 \\, .\n\\end{align}\nFinally, the remaining commutator between $\\tilde{R}$ and $\\tilde{S}$ can be derived from equation (\\ref{commutatorSR}) by commuting the component fields and renaming the spinor indices appropriately\n\\begin{align}\n\\left[ R_{\\dot{\\beta} \\alpha}{\\left( x \\right)} , S'_{\\epsilon \\delta}{\\left( x' \\right)} \\right]\n\t&= - \\epsilon_{\\alpha \\delta} \\left( \\bar{\\sigma}^0 \\right)_{\\dot{\\beta} \\epsilon} \\delta{\\left( x - x' \\right)} \\, .\n\\label{commutatorRS}\n\\end{align}\n\n\\subsubsection{Transformation of the Conjugate Component Fields}\nGenerally it is possible to repeat the calculations outlined in the previous sections for the hermitian conjugate component fields. However, it is much easier to calculate the hermitian conjugate of the previously derived commutation and anticommutation relations.\n\nFor the anticommutation relations between the spinor fields hermitian conjugation is straightforward and the two nonvanishing anticommutation relations between the barred component fields are\n\\begin{align}\n\\left\\{ \\bar{\\chi}_{\\dot{\\alpha}}{\\left( x \\right)} , \\dot{\\bar{\\psi}}_{\\dot{\\gamma}}{\\left( x' \\right)} \\right\\}\n\t&= \\frac{i}{2} \\epsilon_{\\dot{\\alpha} \\dot{\\gamma}} \\delta{\\left( \\mathbf{x} - \\mathbf{x}' \\right)} \\, , \\\\\n\\left\\{ \\bar{\\psi}_{\\dot{\\alpha}}{\\left( x \\right)} , \\dot{\\bar{\\chi}}_{\\dot{\\gamma}}{\\left( x' \\right)} \\right\\}\n\t&= \\frac{i}{2} \\epsilon_{\\dot{\\alpha} \\dot{\\gamma}} \\delta{\\left( \\mathbf{x} - \\mathbf{x'} \\right)} \\, .\n\\end{align}\nThe only difficulty that arises is the sign change of the second-rank $\\epsilon$-tensor under hermitian conjugation.\n\nFor the commutation relations of the bosonic second-rank spinor fields the discussion is only slightly more involved as the hermitian conjugation inverts the ordering of the component fields which induces an additional sign flip for the commutators that didn't occur for the spinor fields. Therefore, the commutation relations for the barred component fields are given by\n\\begin{align}\n\\left[ \\tilde{\\bar{S}}_{\\dot{\\beta} \\dot{\\alpha}}{\\left( x \\right)} , \\tilde{\\bar{R}}_{\\epsilon \\dot{\\delta}}{\\left( x' \\right)} \\right]\n\t&= - \\epsilon_{\\dot{\\alpha} \\dot{\\delta}} \\delta{\\left( \\mathbf{x} - \\mathbf{x}' \\right)} \\left( \\bar{\\sigma}^0 \\right)_{\\dot{\\beta} \\epsilon} \\, , \\\\\n\\left[ \\tilde{\\bar{R}}_{\\beta \\dot{\\alpha}}{\\left( x \\right)} , \\tilde{\\bar{S}}_{\\dot{\\epsilon} \\dot{\\delta}}{\\left( x' \\right)} \\right]\n\t&= - \\epsilon_{\\dot{\\alpha} \\dot{\\delta}} \\left( \\sigma^0 \\right)_{\\beta \\dot{\\epsilon}} \\delta{\\left( \\mathbf{x} - \\mathbf{x}' \\right)} \\, .\n\\end{align}\n\n\\subsection{The Hamiltonian from the Supersymmetry Algebra}\n\\label{CHSUSYalgebra}\nTo derive an explicit equation for the Hamiltonian the supersymmetry generators in equation (\\ref{PSUSYalgebra}) have to be expressed in terms of the component fields. This can be achieved using the relations between the supersymmetry generators which are the conserved Noether charges of the system and the supercurrents which were defined in equations (\\ref{QproptoJ}) and (\\ref{QbarproptoJbar}). Inserting the result for the supercurrent from equations (\\ref{Jfull}) and its hermitian conjugate leads to the following expression of the supersymmetry generators in terms of the component fields\n\\begin{align}\nQ_\\alpha\n\t&= \\int \\mathrm{d} \\mathbf{x} \\left( - i m \\left( \\sigma_0 \\right)_\\alpha{}^{\\dot{\\gamma}} \\tilde{R}_{\\dot{\\gamma} \\beta}{\\left( x \\right)} \\psi^\\beta{\\left( x \\right)}\n\t+ 2 \\left( \\sigma_0 \\right)^{\\gamma \\dot{\\delta}} \\tilde{S}_{\\gamma \\beta}{\\left( x \\right)} \\bar{\\dslash{\\partial}}_{\\dot{\\delta} \\alpha} \\psi^\\beta{\\left( x \\right)} - \\right. \\notag \\\\\n\t&\\qquad \\left. - i m \\left( \\sigma_0 \\right)_\\alpha{}^{\\dot{\\gamma}} \\tilde{\\bar{S}}_{\\dot{\\gamma} \\dot{\\beta}}{\\left( x \\right)} \\bar{\\chi}^{\\dot{\\beta}}{\\left( x \\right)}\n\t+ 2 \\left( \\sigma_0 \\right)^{\\gamma \\dot{\\delta}} \\tilde{\\bar{R}}_{\\gamma \\dot{\\beta}}{\\left( x \\right)} \\bar{\\dslash{\\partial}}_{\\dot{\\delta} \\alpha} \\bar{\\chi}^{\\dot{\\beta}}{\\left( x \\right)} \\right) \\, , \\displaybreak[3] \\\\\n\\bar{Q}_{\\dot{\\alpha}}\n\t&= \\int \\mathrm{d} \\mathbf{x} \\left( i m \\left( \\bar{\\sigma}_0 \\right)_{\\dot{\\alpha}}{}^\\gamma \\tilde{S}_{\\gamma \\beta}{\\left( x \\right)} \\chi^\\beta{\\left( x \\right)}\n\t+ 2 \\left( \\bar{\\sigma}_0 \\right)^{\\dot{\\gamma} \\delta} \\tilde{R}_{\\dot{\\gamma} \\beta}{\\left( x \\right)} \\dslash{\\partial}_{\\delta \\dot{\\alpha}} \\chi^\\beta{\\left( x \\right)} + \\right. \\notag \\\\\n\t&\\qquad \\left. + i m \\left( \\bar{\\sigma}_0 \\right)_{\\dot{\\alpha}}{}^\\gamma \\tilde{\\bar{R}}_{\\gamma \\dot{\\beta}}{\\left( x \\right)} \\bar{\\psi}^{\\dot{\\beta}}{\\left( x \\right)}\n\t+ 2 \\left( \\bar{\\sigma}_0 \\right)^{\\dot{\\gamma} \\delta} \\tilde{\\bar{S}}_{\\dot{\\gamma} \\dot{\\beta}}{\\left( x \\right)} \\dslash{\\partial}_{\\delta \\dot{\\alpha}} \\bar{\\psi}^{\\dot{\\beta}}{\\left( x \\right)} \\right) \\, .\n\\end{align}\nTo streamline the notation it proves useful to introduce the short notation\n\\begin{align}\n\\dslash{P}_{\\alpha \\dot{\\beta}}\n\t&= \\left( \\sigma^\\mu \\right)_{\\alpha \\dot{\\beta}} P_\\mu \\, ,\n\\end{align}\nwhich is defined in analogy to the commonly used contraction with Dirac matrices. The momentum operator is then given by\n\\begin{align}\n2 \\dslash{P}_{\\alpha \\dot{\\beta}}\n\t&= \\left\\{ \\int \\mathrm{d} \\mathbf{x} \\left( - i m \\left( \\sigma_0 \\right)_\\alpha{}^{\\dot{\\gamma}} \\tilde{R}_{\\dot{\\gamma} \\omega}{\\left( x \\right)} \\psi^\\omega{\\left( x \\right)}\n\t+ 2 \\left( \\sigma_0 \\right)^{\\gamma \\dot{\\delta}} \\tilde{S}_{\\gamma \\omega}{\\left( x \\right)} \\bar{\\dslash{\\partial}}_{\\dot{\\delta} \\alpha} \\psi^\\omega{\\left( x \\right)} - \\right. \\right. \\notag \\\\\n\t&\\qquad \\left. - i m \\left( \\sigma_0 \\right)_\\alpha{}^{\\dot{\\gamma}} \\tilde{\\bar{S}}_{\\dot{\\gamma} \\dot{\\omega}}{\\left( x \\right)} \\bar{\\chi}^{\\dot{\\omega}}{\\left( x \\right)}\n\t+ 2 \\left( \\sigma_0 \\right)^{\\gamma \\dot{\\delta}} \\tilde{\\bar{R}}_{\\gamma \\dot{\\omega}}{\\left( x \\right)} \\bar{\\dslash{\\partial}}'_{\\dot{\\delta} \\alpha} \\bar{\\chi}^{\\dot{\\omega}}{\\left( x \\right)} \\right) , \\notag \\\\\n\t&\\quad \\int \\mathrm{d} \\mathbf{x}' \\left( i m \\left( \\bar{\\sigma}_0 \\right)_{\\dot{\\beta}}{}^\\kappa \\tilde{S}_{\\kappa \\epsilon}{\\left( x' \\right)} \\chi^\\epsilon{\\left( x' \\right)}\n\t+ 2 \\left( \\bar{\\sigma}_0 \\right)^{\\dot{\\kappa} \\tau} \\tilde{R}_{\\dot{\\kappa} \\epsilon}{\\left( x' \\right)} \\dslash{\\partial}'_{\\tau \\dot{\\beta}} \\chi^\\epsilon{\\left( x' \\right)} + \\right. \\notag \\\\\n\t&\\qquad \\left. \\left. + i m \\left( \\bar{\\sigma}_0 \\right)_{\\dot{\\beta}}{}^\\kappa \\tilde{\\bar{R}}_{\\kappa \\dot{\\epsilon}}{\\left( x' \\right)} \\bar{\\psi}^{\\dot{\\epsilon}}{\\left( x' \\right)}\n\t+ 2 \\left( \\bar{\\sigma}_0 \\right)^{\\dot{\\kappa} \\tau} \\tilde{\\bar{S}}_{\\dot{\\kappa} \\dot{\\epsilon}}{\\left( x' \\right)} \\dslash{\\partial}_{\\tau \\dot{\\beta}} \\bar{\\psi}^{\\dot{\\epsilon}}{\\left( x' \\right)} \\right) \\right\\} \\, .\n\\end{align}\nThe anticommutators containing two fermionic and two bosonic component fields can now  be rewritten using the commutator relation\n\\begin{align}\n\\left\\{ B_1 F_1 , B_2 F_2 \\right\\}\n\t&= \\left[ B_1 , B_2 \\right] F_1 F_2 + B_2 B_1 \\left\\{ F_1 , F_2 \\right\\} \\, ,\n\\end{align}\nwhere it was assumed that the fermionic and bosonic fields commute. This assumption is justified by the previous derivation of the commutation and anticommutation relations of the component fields as well as the results of the superfield translations.\n\nAfter separation of the time and spatial derivatives as well as partial spatial integration the momentum operator is given by\n\\begin{align}\n2 \\dslash{P}_{\\alpha \\dot{\\beta}}\n\t&= \\int \\mathrm{d} \\mathbf{x} \\mathrm{d} \\mathbf{x}' \\left(\n\tm^2 \\left( \\sigma_0 \\right)_\\alpha{}^{\\dot{\\gamma}} \\left( \\bar{\\sigma}_0 \\right)_{\\dot{\\beta}}{}^\\kappa \\psi^\\omega{\\left( x \\right)} \\chi^\\epsilon{\\left( x' \\right)} \\left[ \\tilde{R}_{\\dot{\\gamma} \\omega}{\\left( x \\right)} , \\tilde{S}_{\\kappa \\epsilon}{\\left( x' \\right)} \\right] - \\right. \\notag \\\\\n\t&\\qquad - 2 i m \\left( \\sigma_0 \\right)_\\alpha{}^{\\dot{\\gamma}} \\left( \\bar{\\sigma}_0 \\right)^{\\dot{\\kappa} \\tau} \\left( \\sigma^0 \\right)_{\\tau \\dot{\\beta}} \\tilde{R}_{\\dot{\\kappa} \\epsilon}{\\left( x' \\right)} \\tilde{R}_{\\dot{\\gamma} \\omega}{\\left( x \\right)} \\left\\{ \\psi^\\omega{\\left( x \\right)} , \\dot{\\chi}^\\epsilon{\\left( x' \\right)} \\right\\} + \\notag \\displaybreak[3] \\\\\n\t&\\qquad + 2 i m \\left( \\sigma_0 \\right)^{\\gamma \\dot{\\delta}} \\left( \\bar{\\sigma}_0 \\right)_{\\dot{\\beta}}{}^\\kappa \\left( \\bar{\\sigma}^0 \\right)_{\\dot{\\delta} \\alpha} \\tilde{S}_{\\kappa \\epsilon}{\\left( x' \\right)} \\tilde{S}_{\\gamma \\omega}{\\left( x \\right)} \\left\\{ \\chi^\\epsilon{\\left( x' \\right)} , \\dot{\\psi}^\\omega{\\left( x \\right)} \\right\\} + \\notag \\displaybreak[3] \\\\\n\t&\\qquad + 4 \\left( \\sigma_0 \\right)^{\\gamma \\dot{\\delta}} \\left( \\bar{\\sigma}_0 \\right)^{\\dot{\\kappa} \\tau} \\bar{\\dslash{\\partial}}_{\\dot{\\delta} \\alpha} \\psi^\\omega{\\left( x \\right)} \\dslash{\\partial}'_{\\tau \\dot{\\beta}} \\chi^\\epsilon{\\left( x' \\right)} \\left[ \\tilde{S}_{\\gamma \\omega}{\\left( x \\right)} , \\tilde{R}_{\\dot{\\kappa} \\epsilon}{\\left( x' \\right)} \\right] - \\notag \\displaybreak[3] \\\\\n\t&\\qquad - 4 \\left( \\sigma_0 \\right)^{\\gamma \\dot{\\delta}} \\left( \\bar{\\sigma}_0 \\right)^{\\dot{\\kappa} \\tau} \\left( \\bar{\\sigma}^0 \\right)_{\\dot{\\delta} \\alpha} \\boldsymbol{\\sigma}_{\\tau \\dot{\\beta}} \\cdot \\boldsymbol{\\nabla}' \\tilde{R}_{\\dot{\\kappa} \\epsilon}{\\left( x' \\right)} \\tilde{S}_{\\gamma \\omega}{\\left( x \\right)} \\left\\{ \\chi^\\epsilon{\\left( x' \\right)} , \\dot{\\psi}^\\omega{\\left( x \\right)} \\right\\} - \\notag \\\\\n\t&\\qquad - 4 \\left( \\sigma_0 \\right)^{\\gamma \\dot{\\delta}} \\left( \\bar{\\sigma}_0 \\right)^{\\dot{\\kappa} \\tau} \\left( \\sigma^0 \\right)_{\\tau \\dot{\\beta}} \\tilde{R}_{\\dot{\\kappa} \\epsilon}{\\left( x' \\right)} \\boldsymbol{\\bar{\\sigma}}_{\\dot{\\delta} \\alpha} \\cdot \\boldsymbol{\\nabla} \\tilde{S}_{\\gamma \\omega}{\\left( x \\right)} \\left\\{ \\psi^\\omega{\\left( x \\right)} , \\dot{\\chi}^\\epsilon{\\left( x' \\right)} \\right\\} + \\notag \\\\\n\t&\\qquad + m^2 \\left( \\sigma_0 \\right)_\\alpha{}^{\\dot{\\gamma}} \\left( \\bar{\\sigma}_0 \\right)_{\\dot{\\beta}}{}^\\kappa \\bar{\\chi}^{\\dot{\\omega}}{\\left( x \\right)} \\bar{\\psi}^{\\dot{\\epsilon}}{\\left( x' \\right)} \\left[ \\tilde{\\bar{S}}_{\\dot{\\gamma} \\dot{\\omega}}{\\left( x \\right)} , \\tilde{\\bar{R}}_{\\kappa \\dot{\\epsilon}}{\\left( x' \\right)}  \\right] - \\notag \\\\\n\t&\\qquad - 2 i m \\left( \\sigma_0 \\right)_\\alpha{}^{\\dot{\\gamma}} \\left( \\bar{\\sigma}_0 \\right)^{\\dot{\\kappa} \\tau} \\left( \\sigma^0 \\right)_{\\tau \\dot{\\beta}} \\tilde{\\bar{S}}_{\\dot{\\kappa} \\dot{\\epsilon}}{\\left( x' \\right)} \\tilde{\\bar{S}}_{\\dot{\\gamma} \\dot{\\omega}}{\\left( x \\right)} \\left\\{ \\bar{\\chi}^{\\dot{\\omega}}{\\left( x \\right)} , \\dot{\\bar{\\psi}}^{\\dot{\\epsilon}}{\\left( x' \\right)} \\right\\} + \\notag \\\\\n\t&\\qquad + 2 i m \\left( \\sigma_0 \\right)^{\\gamma \\dot{\\delta}} \\left( \\bar{\\sigma}_0 \\right)_{\\dot{\\beta}}{}^\\kappa \\left( \\bar{\\sigma}^0 \\right)_{\\dot{\\delta} \\alpha} \\tilde{\\bar{R}}_{\\kappa \\dot{\\epsilon}}{\\left( x' \\right)} \\tilde{\\bar{R}}_{\\gamma \\dot{\\omega}}{\\left( x \\right)} \\left\\{ \\bar{\\psi}^{\\dot{\\epsilon}}{\\left( x' \\right)} , \\dot{\\bar{\\chi}}^{\\dot{\\omega}}{\\left( x \\right)} \\right\\} + \\notag \\\\\n\t&\\qquad + 4 \\left( \\sigma_0 \\right)^{\\gamma \\dot{\\delta}} \\left( \\bar{\\sigma}_0 \\right)^{\\dot{\\kappa} \\tau} \\bar{\\dslash{\\partial}}_{\\dot{\\delta} \\alpha} \\bar{\\chi}^{\\dot{\\omega}}{\\left( x \\right)} \\dslash{\\partial}'_{\\tau \\dot{\\beta}} \\bar{\\psi}^{\\dot{\\epsilon}}{\\left( x' \\right)} \\left[ \\tilde{\\bar{R}}_{\\gamma \\dot{\\omega}}{\\left( x \\right)} , \\tilde{\\bar{S}}_{\\dot{\\kappa} \\dot{\\epsilon}}{\\left( x' \\right)} \\right] - \\notag \\\\\n\t&\\qquad - 4 \\left( \\sigma_0 \\right)^{\\gamma \\dot{\\delta}} \\left( \\bar{\\sigma}_0 \\right)^{\\dot{\\kappa} \\tau} \\left( \\bar{\\sigma}^0 \\right)_{\\dot{\\delta} \\alpha} \\boldsymbol{\\sigma}_{\\tau \\dot{\\beta}} \\cdot \\boldsymbol{\\nabla}' \\tilde{\\bar{S}}_{\\dot{\\kappa} \\dot{\\epsilon}}{\\left( x' \\right)} \\tilde{\\bar{R}}_{\\gamma \\dot{\\omega}}{\\left( x \\right)} \\left\\{ \\bar{\\psi}^{\\dot{\\epsilon}}{\\left( x' \\right)} , \\dot{\\bar{\\chi}}^{\\dot{\\omega}}{\\left( x \\right)} \\right\\} - \\notag \\\\\n\t&\\qquad \\left. - 4 \\left( \\sigma_0 \\right)^{\\gamma \\dot{\\delta}} \\left( \\bar{\\sigma}_0 \\right)^{\\dot{\\kappa} \\tau} \\left( \\sigma^0 \\right)_{\\tau \\dot{\\beta}} \\tilde{\\bar{S}}_{\\dot{\\kappa} \\dot{\\epsilon}}{\\left( x' \\right)} \\boldsymbol{\\bar{\\sigma}}_{\\dot{\\delta} \\alpha} \\cdot \\boldsymbol{\\nabla} \\tilde{\\bar{R}}_{\\gamma \\dot{\\omega}}{\\left( x \\right)} \\left\\{ \\bar{\\chi}^{\\dot{\\omega}}{\\left( x \\right)} , \\dot{\\bar{\\psi}}^{\\dot{\\epsilon}}{\\left( x' \\right)} \\right\\} \\right) \\, .\n\\end{align}\nInserting the previously derived results for the commutation and anticommutation relations between the component fields in position space then yields\n\\begin{align}\n2 \\dslash{P}_{\\alpha \\dot{\\beta}}\n\t&= \\int \\mathrm{d} \\mathbf{x} \\left(\n\t- m^2 \\left( \\sigma_0 \\right)_{\\alpha \\dot{\\beta}} \\psi_\\epsilon{\\left( x \\right)} \\chi^\\epsilon{\\left( x \\right)}\n\t- m \\left( \\sigma_0 \\right)_\\alpha{}^{\\dot{\\gamma}} \\tilde{R}_{\\dot{\\beta} \\epsilon}{\\left( x \\right)} \\tilde{R}_{\\dot{\\gamma}}{}^\\epsilon{\\left( x \\right)} + \\right. \\notag \\\\\n\t&\\qquad + m \\left( \\bar{\\sigma}_0 \\right)_{\\dot{\\beta}}{}^\\kappa \\tilde{S}_\\kappa{}^\\omega{\\left( x \\right)} \\tilde{S}_{\\alpha \\omega}{\\left( x \\right)}\n\t- 4 \\left( \\sigma_0 \\right)^{\\gamma \\dot{\\delta}} \\bar{\\dslash{\\partial}}_{\\dot{\\delta} \\alpha} \\psi_\\epsilon{\\left( x \\right)} \\dslash{\\partial}_{\\gamma \\dot{\\beta}} \\chi^\\epsilon{\\left( x \\right)} + \\notag \\\\\n\t&\\qquad + 2 i \\left( \\bar{\\sigma}_0 \\right)^{\\dot{\\kappa} \\tau} \\boldsymbol{\\sigma}_{\\tau \\dot{\\beta}} \\cdot \\boldsymbol{\\nabla} \\tilde{R}_{\\dot{\\kappa}}{}^\\omega{\\left( x \\right)} \\tilde{S}_{\\alpha \\omega}{\\left( x \\right)}\n\t+ 2 i \\left( \\sigma_0 \\right)^{\\gamma \\dot{\\delta}} \\tilde{R}_{\\dot{\\beta} \\epsilon}{\\left( x \\right)} \\boldsymbol{\\bar{\\sigma}}_{\\dot{\\delta} \\alpha} \\cdot \\boldsymbol{\\nabla} \\tilde{S}_\\gamma{}^\\epsilon{\\left( x \\right)} + \\notag \\\\\n\t&\\qquad + m^2 \\left( \\sigma_0 \\right)_{\\alpha \\dot{\\beta}} \\bar{\\chi}_{\\dot{\\epsilon}}{\\left( x \\right)} \\bar{\\psi}^{\\dot{\\epsilon}}{\\left( x \\right)}\n\t+ m \\left( \\sigma_0 \\right)_\\alpha{}^{\\dot{\\gamma}} \\tilde{\\bar{S}}_{\\dot{\\beta} \\dot{\\epsilon}}{\\left( x \\right)} \\tilde{\\bar{S}}_{\\dot{\\gamma}}{}^{\\dot{\\epsilon}}{\\left( x \\right)} - \\notag \\\\\n\t&\\qquad - m \\left( \\bar{\\sigma}_0 \\right)_{\\dot{\\beta}}{}^\\kappa \\tilde{\\bar{R}}_\\kappa{}^{\\dot{\\omega}}{\\left( x \\right)} \\tilde{\\bar{R}}_{\\alpha \\dot{\\omega}}{\\left( x \\right)}\n\t+ 4 \\left( \\sigma_0 \\right)^{\\gamma \\dot{\\delta}} \\bar{\\dslash{\\partial}}_{\\dot{\\delta} \\alpha} \\bar{\\chi}_{\\dot{\\epsilon}}{\\left( x \\right)} \\dslash{\\partial}_{\\gamma \\dot{\\beta}} \\bar{\\psi}^{\\dot{\\epsilon}}{\\left( x \\right)} - \\notag \\\\\n\t&\\qquad \\left. - 2 i \\left( \\bar{\\sigma}_0 \\right)^{\\dot{\\kappa} \\tau} \\boldsymbol{\\sigma}_{\\tau \\dot{\\beta}} \\cdot \\boldsymbol{\\nabla} \\tilde{\\bar{S}}_{\\dot{\\kappa}}{}^{\\dot{\\omega}}{\\left( x \\right)} \\tilde{\\bar{R}}_{\\alpha \\dot{\\omega}}{\\left( x \\right)}\n\t- 2 i \\left( \\sigma_0 \\right)^{\\gamma \\dot{\\delta}} \\tilde{\\bar{S}}_{\\dot{\\beta} \\dot{\\epsilon}}{\\left( x \\right)} \\boldsymbol{\\bar{\\sigma}}_{\\dot{\\delta} \\alpha} \\cdot \\boldsymbol{\\nabla} \\tilde{\\bar{R}}_\\gamma{}^{\\dot{\\epsilon}}{\\left( x \\right)} \\right) \\, .\n\\end{align}\nTo extract the Hamiltonian from the momentum operator it has to be contracted with the appropriate Pauli matrix\n\\begin{align}\n\\mathcal{H}\n\t&= \\frac{1}{2} \\left( \\sigma_0 \\right)^{\\alpha \\dot{\\beta}} \\dslash{P}_{\\alpha \\dot{\\beta}} \\, .\n\\end{align}\nThe Hamiltonian is therefore given by\n\\begin{align}\n\\mathcal{H}\n\t&= \\frac{1}{4} \\int \\mathrm{d} \\mathbf{x} \\left(\n\t2 m^2 \\psi{\\left( x \\right)} \\chi{\\left( x \\right)}\n\t+ m \\tilde{R}_{\\dot{\\beta} \\epsilon}{\\left( x \\right)} \\tilde{R}^{\\dot{\\beta} \\epsilon}{\\left( x \\right)}\n\t- m \\tilde{S}^{\\alpha \\omega}{\\left( x \\right)} \\tilde{S}_{\\alpha \\omega}{\\left( x \\right)} - \\right. \\notag \\\\\n\t&\\qquad - 4 \\left( \\sigma_0 \\bar{\\sigma}^\\mu \\sigma_0 \\right)^{\\gamma \\dot{\\beta}} \\partial_\\mu \\psi_\\epsilon{\\left( x \\right)} \\dslash{\\partial}_{\\gamma \\dot{\\beta}} \\chi^\\epsilon{\\left( x \\right)} \n\t+ 2 i \\left( \\bar{\\sigma}_0 \\sigma^i \\bar{\\sigma}_0 \\right)^{\\dot{\\kappa} \\alpha} \\partial_i \\tilde{R}_{\\dot{\\kappa}}{}^\\omega{\\left( x \\right)} \\tilde{S}_{\\alpha \\omega}{\\left( x \\right)} + \\notag \\\\\n\t&\\qquad + 2 i \\left( \\sigma_0 \\bar{\\sigma}^i \\sigma_0 \\right)^{\\gamma \\dot{\\beta}} \\tilde{R}_{\\dot{\\beta} \\epsilon}{\\left( x \\right)} \\partial_i \\tilde{S}_\\gamma{}^\\epsilon{\\left( x \\right)}\n\t+ 2 m^2 \\bar{\\chi}{\\left( x \\right)} \\bar{\\psi}{\\left( x \\right)}\n\t- m \\tilde{\\bar{S}}_{\\dot{\\beta} \\dot{\\epsilon}}{\\left( x \\right)} \\tilde{\\bar{S}}_{\\dot{\\gamma}}{}^{\\dot{\\beta} \\dot{\\epsilon}}{\\left( x \\right)} + \\notag \\\\\n\t&\\qquad + m \\tilde{\\bar{R}}^{\\alpha \\dot{\\omega}}{\\left( x \\right)} \\tilde{\\bar{R}}_{\\alpha \\dot{\\omega}}{\\left( x \\right)}\n\t+ 4 \\left( \\sigma_0 \\bar{\\sigma}^\\mu \\sigma_0 \\right)^{\\gamma \\dot{\\beta}} \\partial_\\mu \\bar{\\chi}_{\\dot{\\epsilon}}{\\left( x \\right)} \\dslash{\\partial}_{\\gamma \\dot{\\beta}} \\bar{\\psi}^{\\dot{\\epsilon}}{\\left( x \\right)} - \\notag \\\\\n\t&\\qquad \\left. - 2 i \\left( \\bar{\\sigma}_0 \\sigma^i \\bar{\\sigma}_0 \\right)^{\\dot{\\kappa} \\alpha} \\partial_i \\tilde{\\bar{S}}_{\\dot{\\kappa}}{}^{\\dot{\\omega}}{\\left( x \\right)} \\tilde{\\bar{R}}_{\\alpha \\dot{\\omega}}{\\left( x \\right)}\n\t- 2 i \\left( \\sigma_0 \\bar{\\sigma}^i \\sigma_0 \\right)^{\\gamma \\dot{\\beta}} \\tilde{\\bar{S}}_{\\dot{\\beta} \\dot{\\epsilon}}{\\left( x \\right)}  \\partial_i \\tilde{\\bar{R}}_\\gamma{}^{\\dot{\\epsilon}}{\\left( x \\right)} \\right) \\, .\n\\end{align}\nThis expression for the Hamiltonian can be further simplified using relations (\\ref{threesigplusthreesig}) and (\\ref{threebarsigplusthreebarsig}) in Appendix \\ref{Asigmarelations} for the special case where the first and last index are 0\n\\begin{align}\n\\sigma^0 \\bar{\\sigma}^\\mu \\sigma^0\n\t&= 2 \\eta^{\\mu 0} \\sigma^0 - \\sigma^\\mu \\, , \\\\\n\\bar{\\sigma}^0 \\sigma^\\mu \\bar{\\sigma}^0\n\t&= 2 \\eta^{\\mu 0} \\bar{\\sigma}^0 - \\bar{\\sigma}^\\mu \\, .\n\\end{align}\nThe Hamiltonian is then reduced to \n\\begin{align}\n\\mathcal{H}\n\t&= \\int \\mathrm{d} \\mathbf{x} \\left(\n\t2 \\dot{\\psi}{\\left( x \\right)} \\dot{\\chi}{\\left( x \\right)}\n\t+ 2 \\boldsymbol{\\nabla} \\psi{\\left( x \\right)} \\cdot \\boldsymbol{\\nabla} \\chi{\\left( x \\right)}\n\t+ \\frac{m^2}{2} \\psi{\\left( x \\right)} \\chi{\\left( x \\right)}\n\t+ 2 \\dot{\\bar{\\chi}}{\\left( x \\right)} \\dot{\\bar{\\psi}}{\\left( x \\right)} + \\right. \\notag \\\\\n\t&\\qquad + 2 \\boldsymbol{\\nabla} \\bar{\\chi}{\\left( x \\right)} \\cdot \\boldsymbol{\\nabla} \\bar{\\psi}{\\left( x \\right)}\n\t+ \\frac{m^2}{2} \\bar{\\chi}{\\left( x \\right)} \\bar{\\psi}{\\left( x \\right)}\n\t+ \\frac{m}{4} \\mathrm{Tr}{\\left( \\tilde{R}^T{\\left( x \\right)} \\tilde{R}{\\left( x \\right)} \\right)}\n\t+ \\frac{m}{4} \\mathrm{Tr}{\\left( \\tilde{S}^T{\\left( x \\right)} \\tilde{S}{\\left( x \\right)} \\right)} - \\notag \\\\\n\t&\\qquad - i \\mathrm{Tr}{\\left( \\tilde{R}^T{\\left( x \\right)} \\boldsymbol{\\bar{\\sigma}} \\cdot \\boldsymbol{\\nabla} \\tilde{S}{\\left( x \\right)} \\right)}\n\t+ \\frac{m}{4} \\mathrm{Tr}{\\left( \\tilde{\\bar{R}}^T{\\left( x \\right)} \\tilde{\\bar{R}}{\\left( x \\right)} \\right)}\n\t+ \\frac{m}{4} \\mathrm{Tr}{\\left( \\tilde{\\bar{S}}^T{\\left( x \\right)} \\tilde{\\bar{S}}{\\left( x \\right)} \\right)} - \\notag \\\\\n\t&\\qquad \\left. - i \\mathrm{Tr}{\\left( \\tilde{\\bar{S}}^T{\\left( x \\right)} \\boldsymbol{\\bar{\\sigma}} \\cdot \\boldsymbol{\\nabla} \\tilde{\\bar{R}}{\\left( x \\right)} \\right)} \\right) \\, .\n\\label{Hxspace}\n\\end{align}\nIt contains the sum of unbarred spinor products and their barred counterparts which is only restricted to be real but could, at least in principle, be either positive or negative. Therefore, on the first glance it seems that this Hamiltonian could have negative eigenvalues. However, as the Lagrangian is by construction supersymmetric and in addition the Hamiltonian was derived using the supersymmetry algebra the eigenvalues of the Hamiltonian must be positive definite. This can also be shown by deriving the momentum space expansion of the component fields in position space, calculating the commutation and anticommutation relations of the momentum space operators, and determining the normal ordered Hamiltonian in momentum space.\n\n\\subsection{The Hamiltonian from Canonical Quantisation}\n\\label{CHcanonicalquant}\nThe derivation of the Hamiltonian using the supersymmetry algebra is by construction positive definite and is founded in the fundamental properties of the algebra. However, it immediately raises the question whether this approach is equivalent to a construction of the Hamiltonian from canonical quantisation which doesn't require the Lagrangian to be supersymmetric.\n\nFor brevity the discussion is restricted to the Lagrangian without hermitian conjugate contribution. The Hamiltonian from canonical quantisation is then defined as \n\\begin{align}\n\\mathcal{H}_{c.q.}\n\t&= \\int \\mathrm{d}^3 \\mathbf{x} \\left( - \\frac{\\partial \\mathcal{L}}{\\partial \\dot{\\chi}^\\tau} \\dot{\\chi}^\\tau\n\t- \\frac{\\partial \\mathcal{L}}{\\partial \\dot{\\psi}^\\tau} \\dot{\\psi}^\\tau\n\t+ \\frac{\\partial \\mathcal{L}}{\\partial \\dot{\\tilde{S}}^{\\tau \\omega}} \\dot{\\tilde{S}}^{\\tau \\omega}\n\t+ \\frac{\\partial \\mathcal{L}}{\\partial \\dot{\\tilde{R}}^{\\dot{\\tau} \\omega}} \\dot{\\tilde{R}}^{\\dot{\\tau} \\omega}\n\t- \\mathcal{L} \\right) \\, .\n\\end{align}\nInserting the Lagrangian into this definition of the Hamiltonian results in\n\\begin{align}\n\\mathcal{H}_{c.q.}\n\t&= \\int \\mathrm{d}^3 \\mathbf{x} \\left( 2 \\dot{\\chi}{\\left( x \\right)} \\dot{\\psi}{\\left( x \\right)}\n\t+ 2 \\boldsymbol{\\nabla} \\chi{\\left( x \\right)} \\boldsymbol{\\nabla} \\psi{\\left( x \\right)}\n\t+ \\frac{m^2}{2} \\psi{\\left( x \\right)} \\chi{\\left( x \\right)} \n\t- i \\mathrm{Tr}{\\left( \\tilde{R}^T{\\left( x \\right)} \\boldsymbol{\\bar{\\sigma}} \\cdot \\boldsymbol{\\nabla} \\tilde{S}{\\left( x \\right)} \\right)} \\right. \\notag \\\\\n\t&\\qquad \\left. + \\frac{m}{4} \\mathrm{Tr}{\\left( \\tilde{S}^T{\\left( x \\right)} \\tilde{S}{\\left( x \\right)} \\right)}\n\t+ \\frac{m}{4} \\mathrm{Tr}{\\left( \\tilde{R}^T{\\left( x \\right)} \\tilde{R}{\\left( x \\right)} \\right)} \\right) \\, .\n\\end{align}\nIt turns out that the Hamiltonian derived from canonical quantisation after normal ordering is identical to the one derived using the supersymmetry algebra. This is intriguing as it paves the way for a significantly simplified derivation of the Hamiltonian in position space involving fermionic fields with mass dimension one. It represents an extension of the commonly used formalism of canonical quantisation to component fields with non-standard mass dimensions.\n\n\\section{Summary}\n\\label{Csummary}\nThe primary objective of this article was to construct a supersymmetric model for fermionic fields with mass dimension one.\n\nTo achieve this goal it was investigated whether it is possible to obtain a model based on the general scalar superfield commonly used in supersymmetric models. It has been shown that such a model cannot be formulated due to problems constructing a Lagrangian containing kinetic terms for the fermionic fields with mass dimension one. This eliminated all but the trivial solution which corresponds to a constant non-dynamic background spinor field and is not appealing. In addition no consistent second quantisation for the component fields can be constructed.\n\nThis motivated the formulation of a model for fermionic fields with mass dimension one based on a general spinor superfield. Up to now no explicit calculations for the general spinor superfield exist in the literature, therefore, necessitating the derivation of the model from the ground up. This included the calculation of all chiral and anti-chiral superfields up to third order in covariant derivatives. To second oder in covariant derivatives there is one chiral and one anti-chiral spinor field while to third order there is one chiral and one anti-chiral second rank spinor field. Interestingly, the chiral second-rank spinor field admits a special case that leads to a scalar superfield while the anti-chiral second-rank spinor field can at most be written as a vector superfield.\n\nDimensional analysis revealed that there is a large number of possible contributions to the mass and kintic terms. Therefore, the discussion was restricted to terms built from chiral and anti-chiral superfields. The resulting on-shell Lagrangian depends solely on two spinor fields and two second-rank spinor fields which corresponds to 8 fermionic and 8 bosonic degrees of freedom.\n\nAs it was not ad hoc clear that the Hamiltonian can be derived from the Lagrangian by canonical quantisation a conservative approach based on the supersymmetry algebra was utilised. It provides an anticommutation relation between the supersymmetry generators which is proportional to the momentum operator that contains the Hamiltonian as 0-th component. This is then related to the Lagrangin via the position space representation of the generators that are proportional to the spacetime integral of the supercurrent which itself can be derived from the Lagrangian. This process ensures a Hamiltonian that is consistent with the initial on-shell Lagrangian as well as the supersymmetry algebra. Therefore, the resulting Hamiltonian is positive definite. \n\nSubsequently it was shown that the Hamiltonian derived by canonical quantisation is identical to the one calculated using the supersymmetry algebra. This shows that it is possible to extend the commonly used formalism of canonical quantisation to component fields with non-standard mass dimensions.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nM87 is a nearby dominant elliptical galaxy in the Virgo Cluster with the first discovered extragalactic jet \\citep{Curtis1918}. It is one of the closest radio galaxies, with a distance of only about 16~Mpc \\citep{BlakesleeEtal2009}. It is generally believed that the observed relativistic jet is powered by a supermassive black hole with a mass of $(3-6)\\times10^9\\text{ M}_\\odot$ \\citep{MacchettoEtal1997,GebhardtEtal2011,WalshEtal2013}, which corresponds to an active galactic nucleus (AGN). The proximity of M87 facilitates detailed investigation of the activity, which manifests itself throughout the spectrum from radio to very-high-energy $\\gamma$-ray emission \\citep{WilsonYang2002,PerlmanWilson2005,MadridEtal2007,AcciariEtal2009,BaesEtal2010,PerlmanEtal2011,HadaEtal2014}, and makes it a good candidate for broadening our knowledge of physical processes occurring in the central engine of the AGN.\n\nAn interest in theoretical studies of relativistic jets, arisen after the first pioneering works on the energy extraction from a black hole and the jet origin \\citep{Penrose1969,Blandford1976,Lovelace1976,BlandfordZnajek1977,BlandfordPayne1982}, is yet more growing nowadays because of the necessity to firmly establish the nature of the jet and the exact mechanism of its launching, collimation, stabilization and propagation in the external medium \\citep{ChiuehEtal1991,ApplCamenzind1993,IstominPariev1996,Fendt1997,LyndenBell2003,BeskinNokhrina2009,porthFendt2010,PorthEtal2011,Cao2012,ColgateEtal2015,TchekhovskoyBromberg2016,FengEtal2016,YangReynolds2016,EnglishEtal2016,BritzenEtal2017,SobacchiEtal2017}. For the same reason, a significant role is given to high-resolution observational research \\citep{JunorEtal1999,KovalevEtal2007,HadaEtal2011,HadaEtal2012,HadaEtal2014,DoelemanEtal2012,AkiyamaEtal2015}, which can clarify some key features of the internal jet structure. The brightness and closeness of the M87 jet allowed one to reach an unprecedented ultra-high resolution down to $50\\text{ }\\mu$arcsec in Very Long Baseline Interferometer (VLBI) radio observations, which corresponds to only $6-10$ Schwarzschild radii \\citep{HadaEtal2016,KimM87Etal2016}.\n\nAll previous studies showed that the M87 jet is almost parabolic at relatively small ($<10^5$ Schwarzschild radii) distances from the base \\citep{AsadaNakamura2012}, characterized by a limb-brightened transverse profile of radio intensity at various frequencies, and then can be considered e.g. in the model of a magnetohydrodynamic nozzle \\citep{NakamuraAsada2013}. However, new radio observations of M87 at 15~GHz (2~cm) using the NRAO Very Long Baseline Array in concert with the phased Y27 Very Large Array clearly indicate the existence of a persistent triple-ridge structure across the jet \\citep{Hada2017}. Moreover, detection of an ultra-narrow central radio ridge in these observations sets up problems in explaining the effect with the standard spine-sheath jet model usually used when explaining the appearance of the limb brightness \\citep{GhiselliniEtal2005} and poses a question whether we really observe a single jet with some decaying radial velocity profile. In this paper, I study the possibility that observing the above structure in the radio image, we can in fact deal with a pure jet-in-jet structure in M87: the inner jet is placed inside the outer annular jet. This circumstance can be evidence of simultaneous operation of two different jet-launching mechanisms, one relating to the black hole and the other to the accretion disc.\n\n\\section{Jet in jet}\n\nThe whole jet is governed by the Maxwell equations\n\\begin{gather}\n\\label{divE}\n\\operatorname{div}\\mathbf{E}=4\\pi\\rho_e,\n\\\\\n\\label{divB}\n\\operatorname{div}\\mathbf{B}=0,\n\\\\\n\\label{rotE}\n\\operatorname{curl}\\mathbf{E}=-\\frac{\\partial\\mathbf{B}}{\\partial t},\n\\\\\n\\label{rotB}\n\\operatorname{curl}\\mathbf{B}=4\\pi\\mathbf{j}+\\frac{\\partial\\mathbf{E}}{\\partial t},\n\\end{gather}\nthe condition of infinite conductivity\n\\begin{equation}\n\\label{forceFree}\n\\mathbf{E}=-\\mathbf{v}\\times\\mathbf{B},\n\\end{equation}\nand the laws of conservation of matter\n\\begin{equation}\n\\label{matterConservation}\n\\frac{\\partial\\gamma\\rho}{\\partial t}+\\operatorname{div}\\gamma\\rho\\mathbf{v}=0,\n\\end{equation}\nenergy\n\\begin{equation}\n\\label{energyConservation}\n\\frac{\\partial}{\\partial t}\\biggl(\\gamma^2 \\rho h-p+\\frac{E^2+B^2}{8\\pi}\\biggr)+\\operatorname{div}\\biggl(\\gamma^2 \\rho h\\mathbf{v}+\\frac{\\mathbf{E}\\times\\mathbf{B}}{4\\pi}\\biggr)=0,\n\\end{equation}\nand momentum\n\\begin{align}\n\\label{momentumConservation}\n&\\frac{\\partial}{\\partial t}\\biggl(\\gamma^2 \\rho h\\mathbf{v}+\\frac{\\mathbf{E}\\times\\mathbf{B}}{4\\pi}\\biggr)\\nonumber\\\\\n&+\\operatorname{div}\\biggl[\\biggl(p+\\frac{E^2+B^2}{8\\pi}\\biggr)\\mathbf{I}+\\gamma^2 \\rho h\\mathbf{v}\\mathbf{v}-\\frac{\\mathbf{E}\\mathbf{E}+\\mathbf{B}\\mathbf{B}}{4\\pi}\\biggr]=0.\n\\end{align}\nIn the above equations, we put the speed of light $c=1$, and the electric and magnetic fields are denoted by $\\mathbf{E}$ and $\\mathbf{B}$, respectively, $\\rho_e$ and $\\mathbf{j}$ are the volume charge and current densities, respectively, $\\mathbf{v}$ is the velocity of the jet plasma at a given point, $\\gamma=(1-v^2)^{-1\/2}$ is the corresponding Lorentz factor, $h=1+\\varepsilon+p\/\\rho$ is the specific relativistic enthalpy, $\\varepsilon$ is the specific internal energy, $p$ is the pressure and $\\rho$ is the mass density in the comoving reference frame. We also use notations $\\mathbf{a}\\mathbf{b}=||a_ib_j||$ for a dyad, $\\mathbf{I}=||\\delta_{ij}||$ for the unit tensor, and $\\operatorname{div}\\mathbf{T}=\\nabla\\cdot\\mathbf{T}=||\\partial T_{ji}\/\\partial x_j||$ for the divergence of a tensor $\\mathbf{T}=||T_{ij}||$. The system is complemented by an equation of state $p=p(\\rho,\\varepsilon)$.\n\nWe will consider the case of stationarity and pure cylindrical symmetry. We may write in the cylindrical coordinates the velocity\n\\begin{equation}\n\\label{velocity}\n\\mathbf{v}=v_\\phi\\mathbf{e}_\\phi+v_z\\mathbf{e}_z\n\\end{equation}\nand electromagnetic fields\n\\begin{gather}\n\\label{electricField}\n\\mathbf{E}=E_r\\mathbf{e}_r=(v_z B_\\phi-v_\\phi B_z)\\,\\mathbf{e}_r,\n\\\\\n\\label{magneticField}\n\\mathbf{B}=B_\\phi\\mathbf{e}_\\phi+B_z\\mathbf{e}_z.\n\\end{gather}\nThe components $v_\\phi$, $v_z$, $B_\\phi$, $B_z$ as well as the other scalar quantities depend on $r$ only, the distance from the symmetry axis to a given point. Equations \\eqref{velocity}--\\eqref{magneticField} mean that the lines of the matter flow and the magnetic field lines are helices lying on a cylindrical tube, and the electric field lines are orthogonal to the lateral tube surface and radially diverge from (or converge to) the symmetry axis. In the stationary case, equations \\eqref{divB}, \\eqref{rotE} and \\eqref{forceFree}--\\eqref{energyConservation} are then automatically satisfied, while equations \\eqref{divE} and \\eqref{rotB} simply determine the charge and current densities\n\\begin{gather}\n\\label{charge}\n\\rho_e=\\frac{(r E_r)'}{4\\pi r},\n\\\\\n\\label{current}\n\\mathbf{j}=-\\frac{B'_z}{4\\pi}\\,\\mathbf{e}_\\phi+\\frac{(r B_\\phi)'}{4\\pi r}\\,\\mathbf{e}_z,\n\\end{gather}\nwhere prime denotes the $r$ derivative. Consequently, the momentum conservation law \\eqref{momentumConservation}, which now takes the form\n\\begin{equation}\n\\label{momentumConservation2}\n\\biggl(p+\\frac{B_z^2+B_\\phi^2-E_r^2}{8\\pi}\\biggr)'+\\frac{B_\\phi^2-E_r^2}{4\\pi r}-\\gamma^2 \\rho h\\frac{v_\\phi^2}{r}=0,\n\\end{equation}\nplays the major role in determining the jet equilibrium.\n\nAn equivalent approach to stationary axisymmetric plasma systems is based on the Grad-Shafranov equation \\citep{Fendt1997,BeskinNokhrina2009} and earlier allowed one to study jet collimation with the help of a steady-state trans-field force-balance equation \\citep{ChiuehEtal1991,ApplCamenzind1993}, consider differential rotation in the force-free approximation \\citep{Fendt1997}, and then extend the results to the full time-dependent two-dimensional solution for the collimated jet structure close to the disc and central object \\citep{porthFendt2010,PorthEtal2011}. The equivalence of the two approaches is evident as in the stationary axisymmetric case, when the radius of a magnetic tube may change with the distance from the jet base, we have the same integrals of motion as in the Grad-Shafranov approach that follow from conservation of the magnetic flux in the tube and, respectively, matter conservation \\eqref{matterConservation},\n\\begin{equation}\n\\label{GSeta}\n\\eta=\\frac{\\gamma\\rho v_\\text{p}}{B_\\text{p}},\n\\end{equation}\nenergy conservation \\eqref{energyConservation},\n\\begin{equation}\n\\label{GSE}\n\\mathcal{E}=\\gamma h\\eta-\\frac{\\Omega_\\text{F}I}{2\\pi},\n\\end{equation}\nand momentum conservation \\eqref{momentumConservation},\n\\begin{equation}\n\\label{GSL}\n\\mathcal{L}=\\gamma h\\eta r v_\\phi-\\frac{I}{2\\pi},\n\\end{equation}\nwhere $\\Omega_\\text{F}=(v_\\phi-v_\\text{p}B_\\phi\/B_\\text{p})\/r$ is the so-called Ferraro isorotation frequency, which is also conserved along the magnetic tube, $v_\\text{p}$ and $B_\\text{p}$ are the poloidal components of the velocity and magnetic field, and $I$ is the electric current in the tube.\n\nThe intensity of radio emission increases with the number of emitting particles, so the radio-emitting areas may simply reflect the areas with an active dense plasma. In this case, the observed three-peaked transverse radio profile (fig.~2 in \\cite{Hada2017}) can directly show up the intrinsic structure of the M87 jet and be naturally interpreted thus: the jet as a whole can represent a pinch-like inner jet that is placed in an outer jet, and both jets are coaxial. The inner jet is a solid plasma cylinder of a radius $r_0$, the outer jet is a hollow plasma cylinder of an inner radius $r_1>r_0$ and thickness $d$, so that the radius of the whole jet is $R=r_1+d$.\n\nEquation \\eqref{momentumConservation2} implies the following condition at an interface between plasmas with different pressures and electromagnetic fields in the absence of singular density distribution at the interface:\n\\begin{equation}\n\\label{boundaryRelation}\n\\Delta\\biggl(p+\\frac{B_z^2+B_\\phi^2-E_r^2}{8\\pi}\\biggr)=0,\n\\end{equation}\nwhere $\\Delta$ denotes the difference between the quantities on different sides from the interface.\n\nThe basic equations allow singular charge and current densities, such as current sheets, and if a jump in the electromagnetic fields takes place as one passes through the outer boundary of the inner jet or the inner or outer boundary of the outer jet, the corresponding surface charges and currents are non-zero and can readily be found in the standard way. However, the absence of discontinuities and surface charges and currents is beneficial to proper numerical simulations of jets \\citep{GourgouliatosEtal2012,KimEtal2017}. On this basis, we may consider surface charges and currents as a potential source of instability and assume their absence. This implies zero jump in electromagnetic fields and, hence, in pressure on the boundaries,\n\\begin{equation}\n\\label{boundaryContinuity}\n\\Delta p=\\Delta E_r=\\Delta B_\\phi=\\Delta B_z=0.\n\\end{equation}\nThus, the electromagnetic field and pressure are continuous everywhere in the case of zero surface charges and currents.\n\nThe inner jet bears a total axial electric current $I$ and charge per unit length $Q$. Both the jets may rotate and some toroidal currents are allowed in the plasma, so the electromagnetic field between the jets is\n\\begin{equation}\n\\label{electricFieldBetweenJets}\nE_r=\\frac{2Q}{r},\\text{ }B_\\phi=\\frac{2I}{r},\\text{ }B_z=\\frac{\\alpha}{r}+\\beta,\\text{ }r_0<r<r_1,\n\\end{equation}\nwhere $\\alpha=(B_{0z}-B_{1z})\/(r_0^{-1}-r_1^{-1})$, $\\beta=(B_{1z}r_1-B_{0z}r_0)\/(r_1-r_0)$, and $B_{0z}$ and $B_{1z}$ are generally different axial magnetic fields on the outer and inner boundaries of the inner and outer jets, respectively. The radial dependence of $B_z$ continuously connects the values of the magnetic fields on the boundaries and corresponds to a concentration of toroidal currents, which can provide a higher magnetization in the centre, towards the inner jet. The case of $\\alpha=0$ corresponds to the absence of toroidal currents and a uniform magnetization of the plasma between the jets, $B_z=B_{0z}=B_{1z}$.\n\nWe will further consider the case of total charge neutrality and complete current closure, which means that in the outer jet the linear charge density  is $-Q$ and the absolute value of the axial current is $I$, while its direction is opposite to the direction of the axial current in the inner jet. We also assume the absence of any external charges and currents. The total external field then vanishes, $E_r=B_\\phi=B_z=0$ for $r>R$.\n\nIn principle, the fields outside could have the same structure as those between the jets, so that the external field could be determined by some non-zero charge and current of the jet as a whole, while the axial magnetic field could be generated by some toroidal currents placed far away from the jet. In this case, the energy of the external electromagnetic field goes to infinity, but this fact itself cannot exclude the possibility of non-zero external field as it is largely an artefact of an idealized cylindrical consideration, seeing that the same situation is with an ordinary straight wire with current. Meanwhile, the absence of external field corresponds to full concentration of electromagnetic energy in the jet (which is in a certain sense `energetically profitable' when the internal field remains unchanged) and, as we will discuss below, favours the jet stability.\n\nThe quantities $Q$, $I$ and $B_z$ are not independent. We may relate the fields \\eqref{electricFieldBetweenJets} by equation \\eqref{electricField} to the toroidal and axial velocities $v_\\phi$ and $v_z$ of the intermediate plasma:\n\\begin{equation}\n\\label{chargeCurrentRelation}\nQ=v_zI-\\frac{r v_\\phi B_z}{2}.\n\\end{equation}\nWe may go to the outer boundary of the inner jet or to the inner boundary of the outer jet and relate the above three quantities to the corresponding boundary velocities $v_{0\\phi}$ and $v_{0z}$ at $r=r_0$ or $v_{1\\phi}$ and $v_{1z}$ at $r=r_1$. On the one hand, the axial magnetic field and velocities of the outer jet are related to the charge and current of the inner jet and, on the other hand, the axial magnetic field of the inner jet is determined by the toroidal current in the intermediate plasma and outer jet, so the jets are not independent but electromagnetically connected.\n\nEquation \\eqref{chargeCurrentRelation} implies the relation $v_{0z}-v_{1z}=(r_0 v_{0\\phi}B_{0z}-r_1 v_{1\\phi}B_{1z})\/2I$. When the axial velocities and magnetic fields coincide, $v_{0z}=v_{1z}$ and $B_{0z}=B_{1z}$, the ratio of the azimuthal velocities for the inner and outer jets is the inverse of the ratio for the corresponding radii, $v_{1\\phi}\/v_{0\\phi}=r_0\/r_1<1$. In this case, the outer jet rotates slower than the inner and the ratio of angular velocities of the boundaries is $\\Omega_1\/\\Omega_0=(r_0\/r_1)^2<1$.\n\nNow notice that for the electromagnetic fields of the form \\eqref{electricFieldBetweenJets} we have $(B_\\phi^2-E_r^2)'=-2(B_\\phi^2-E_r^2)\/r$. Substituting these in the momentum conservation law \\eqref{momentumConservation2} yields a balance of the relativistic centrifugal force  and the gradient of hydrodynamic plus axial magnetic pressure,\n\\begin{equation}\n\\label{intermediateMomentumConservation}\n\\biggl(p+\\frac{B_z^2}{8\\pi}\\biggr)'=\\gamma^2 \\rho h\\frac{v_\\phi^2}{r}.\n\\end{equation}\nFor a constant axial magnetic field, we would have a hydrodynamic balance of the pressure gradient and centrifugal force,\n\\begin{equation}\n\\label{hydrodynamicMomentumConservation}\np'=\\gamma^2 \\rho h\\frac{v_\\phi^2}{r}.\n\\end{equation}\nSince the total momentum conservation in fact signifies the balance of hydrodynamic and electromagnetic forces, the vanishing of electromagnetic fields from the force balance means that the fields so chosen generate zero Lorentz forces and hence the hydrodynamic forces have to balance themselves. Differently speaking, the plasma can bear significant electromagnetic fields but nevertheless flow in a purely hydrodynamic way.\n\nWe expect that the plasma density between the jets is significantly lower than in the jets, and hence neglect the centrifugal term in equation~\\eqref{intermediateMomentumConservation},\n\\begin{equation}\n\\label{intermediatePressureBalance}\np+\\frac{B^2_z}{8\\pi}=\\text{const},\\text{ }r_0<r<r_1.\n\\end{equation}\n\nUsing equations \\eqref{electricField} and \\eqref{electricFieldBetweenJets} and assuming a parabolic longitudinal velocity for the plasma between the jets that coincides with that for the outer and inner boundaries of the inner and outer jets,\n\\begin{equation}\n\\label{parabolicAxialVelocityBetweenJets}\nv_z=\\kappa-\\lambda r^2,\n\\end{equation}\nwhere $\\kappa=(v_{0z}r_1^2-v_{1z}r_0^2)\/(r_1^2-r_0^2)$ and $\\lambda=(v_{0z}-v_{1z})\/(r_1^2-r_0^2)$, leads us to the toroidal velocity distribution\n\\begin{equation}\n\\label{toroidalVelocityBetweenJets}\nv_\\phi=\\frac{\\xi-\\zeta r^2}{\\alpha+\\beta r},\n\\end{equation}\nwhere $\\xi=2(\\kappa I-Q)$ and $\\zeta=2\\lambda I$.\n\nInterestingly, for a constant axial magnetic field, when the plasma moves in a hydrodynamic way, equation \\eqref{toroidalVelocityBetweenJets} coincides with that for a viscous Couette flow between rotating cylinders, $v_\\phi=\\mu\/r-\\nu r$ with $\\mu=\\xi\/\\beta$ and $\\nu=\\zeta\/\\beta$, while equation \\eqref{parabolicAxialVelocityBetweenJets} coincides with a viscous Poiseuille flow in a pipe when $r_0\\ll r_1$ \\citep{LandauLifshitz1987}, and viscosity might take part in the formation of the described velocity distributions. When, in addition, we have a constant axial velocity between the jets, the toroidal velocity becomes $v_\\phi=\\mu\/r$ with $\\mu=r_0 v_{0\\phi}=r_1 v_{1\\phi}$.\n\nAfter the study of the fields and plasmas between the jets, let us devote our attention to the jets themselves. The simplest case for the inner jet corresponds to a uniform distribution of the electric charge and axial current, and equations \\eqref{charge} and \\eqref{current} give the resulting electromagnetic fields\n\\begin{equation}\n\\label{innerJetFields}\nE_r=\\frac{2Q r}{r_0^2},\\text{ }B_\\phi=\\frac{2I r}{r_0^2},\\text{ }B_z=B_{0z}.\n\\end{equation}\nWe have assumed here that the axial magnetic field is constant, which implies its generation by toroidal currents outside the inner jet. Assuming a constant axial velocity $v_{0z}$ throughout the inner jet, we get from equations \\eqref{electricField}, \\eqref{chargeCurrentRelation} and \\eqref{innerJetFields} a rigid-like radial distribution of the rotational velocity,\n\\begin{equation}\n\\label{innerJetToroidalVelocity}\nv_\\phi=\\frac{v_{0\\phi}r}{r_0},\n\\end{equation}\nwhich corresponds to a constant angular velocity $\\Omega_0=v_{0\\phi}\/r_0$.\n\nWhen the density~$\\rho_0\\approx\\text{const}$, relativistic enthalpy $h_0\\approx1$ and Lorentz factor $\\gamma_0\\approx\\text{const}$, we get from the full momentum conservation law~\\eqref{momentumConservation2} a parabolic pressure profile\n\\begin{equation}\n\\label{innerJetPressure}\np=p_0-\\biggl[\\frac{I^2-Q^2}{\\pi r_0^2}-\\frac{\\rho_0(\\gamma_0 v_{0\\phi})^2}{2}\\biggr]\\biggl(\\frac{r}{r_0}\\biggr)^2,\\text{ }r<r_0,\n\\end{equation}\nwhere $p_0$ is the pressure on the jet axis.\n\nIt remains to consider the distribution of electromagnetic fields and pressure in the outer jet. The external field is zero, so it is necessary to provide for $E_r$, $B_\\phi$ and $B_z$ some transition from their values at $r=r_1$ to zeroes at $r=R$. Let us assume that the width of the outer jet is sufficiently small compared with the width of the whole jet, $d\\ll R$. This condition suggests that the above field transition is sufficiently sharp in the sense that $E'_r\\sim E_r\/d\\gg E_r\/R$ and $B'_\\phi\\sim B_\\phi\/d\\gg B_\\phi\/R$, which gives the inequality $(B_\\phi^2-E_r^2)'\/8\\pi\\gg(B_\\phi^2-E_r^2)\/4\\pi r$ and thereby allows us to consider a simplified momentum conservation law\n\\begin{equation}\n\\label{outerJetMomentumConservation}\n\\biggl(p+\\frac{B_z^2+B_\\phi^2-E_r^2}{8\\pi}\\biggr)'=\\gamma^2 \\rho h\\frac{v_\\phi^2}{r}.\n\\end{equation}\nThe simplest linear field transition\n\\begin{equation}\n\\label{fieldTransition}\nE_r=\\frac{2Q}{r_1}\\frac{x}{d},\\text{ }B_\\phi=\\frac{2I}{r_1}\\frac{x}{d},\\text{ }B_z=B_{1z}\\frac{x}{d},\n\\end{equation}\nwhere $x=R-r$, gives a constant toroidal velocity $v_\\phi=v_{1\\phi}$ for a constant axial velocity $v_z=v_{1z}$.\n\nThus, the momentum conservation law is analogous to that taking place between the jets in the case of a uniform magnetization, equation \\eqref{hydrodynamicMomentumConservation}, with the difference that the ordinary pressure is replaced by the quantity $p+(B^2-E^2)\/8\\pi$, where the second term may be interpreted as the magnetic pressure in the comoving reference frame. In the laboratory frame, the term $B^2$ is responsible for the magnetic pressure while $-E^2$ for the radial tension of the electric field lines. We finally get the pressure\n\\begin{equation}\n\\label{outerJetPressure}\np=p_\\text{ex}-\\frac{B_z^2+B_\\phi^2-E_r^2}{8\\pi}-\\rho_1(\\gamma_1 v_{1\\phi})^2\\ln\\frac Rr,\\text{ }r_1<r<R,\n\\end{equation}\nwhere $\\rho_1$ is the plasma density in the outer jet, $\\gamma_1$ is the Lorentz factor, $p_\\text{ex}$ is the external pressure and the electromagnetic fields are given by equation~\\eqref{fieldTransition}. The external medium is subject to equation \\eqref{hydrodynamicMomentumConservation} and when it does not rotate, we have some uniform external pressure $p=p_\\text{ex}$.\n\n\\section{Discussion}\n\nThe whole jet-in-jet structure can be explained by the existence of a similar initial density distribution for the flowing plasma near the AGN central engine that plays the role of boundary conditions. In this case, the new radio observations of the innermost region in M87 \\citep{Hada2017} can be evidence for simultaneous activity of two different mechanisms of jet launching, one giving a plasma for the inner jet and relating to the central black hole and the other giving a plasma for the outer jet and relating to the accretion disc. In other words, the Blandford-Znajek and Blandford-Payne launching mechanisms can operate at the same time.\n\nIn general, the basic equations describing equilibrium and dynamics of jets admit infinitely many possible configurations of gas or plasma flows and concomitant electromagnetic fields, and the fact that we go on observing the same jet-in-jet configuration relatively far from the AGN indicates sufficient stability of the whole jet: the inner and outer jets are straight, gradually widen with distance from the central engine, and their coaxiality is not seen to be violated. How to qualitatively realize the possibility of stability? The electromagnetic field is an interlayer that maintains jet separation. The radial electric field attracts and the toroidal magnetic field repulses the jets, which corresponds to attraction of opposite charges and repulsion of opposite axial electric currents. If the M87 radio data indeed show that we deal with the jet-in-jet structure, then the net effect of the electromagnetic field should correspond to repulsion in order to provide mutual alignment and coaxiality of the jets. This fact requires that the jet necessarily bear significant axial currents because the electric field alone results in an instability with respect to displacement of the central jet from the axis of the outer jet.\n\nNote that if the internal magnetic field were absent, charge separation between the inner and outer jets would vanish due to neutralization of the radial electric field as a result of charge redistribution. The non-zero magnetic field plays a significant role in preventing such redistribution. In the first approximation, a charged particle moves along the magnetic field and simultaneously drifts in the perpendicular direction because of the crossed electric and magnetic fields, so that $\\mathbf{v}=v_\\parallel\\mathbf{B}\/B+\\mathbf{E}\\times\\mathbf{B}\/B^2$. Since the magnetic field lines are helices while the electric field is radial, the total motion of the charged particle is also helical though, in general, not coincident with a magnetic field line. We see that the electric drift prevents motion of charged particles in the radial direction and thereby maintains charge separation.\n\nThe current should stabilize the inner jet position, and the existence of the outer jet plays significant role not only as a way for the inverse current but also as an annular conducting wall that contains the internal toroidal magnetic field and does not allow it to go outside, thereby providing backward elastic forces due to an increase in the denseness of the magnetic field lines when the inner jet is displaced from the axis. Importantly, the longitudinal magnetic field also contributes to stabilization of the outer jet. The jet is not rigid, and if the inner jet will be of a curved form, the stabilizing effect of the outer jet will vanish if this jet takes the same form, seeing that the toroidal magnetic field will then not deform. The axial magnetic field helps the outer jet to maintain rigidity, thus playing the role of a resilient backbone: any bending in the outer jet results in the appearance of a returning force due to a string-like tension of the curved axial field. The same stabilizing effect of the axial field also takes place directly for the inner jet.\n\nIn the above stabilization the complete current closure, together with charge neutrality, is very significant because then we do not have a toroidal magnetic field in the external medium and the outside force is determined by the external pressure only. Were the current closure not complete, sufficiently bending the whole get would lead to increasing the denseness of toroidal magnetic field lines on the concave side and give a force that could potentially overcome the stabilizing force of the internal axial field and cause the kink instability of the whole jet. The absence of the external toroidal field and the existence of the internal longitudinal field is also a beneficial factor for stabilization of the sausage instability as we do not have a provocative accumulation of toroidal magnetic field lines around the waist formed in all-round squeezing the jet and at the same time we have a favourable increase in the internal magnetic pressure that resists such squeezing.\n\nThis circumstance might explain why astrophysical jets are more stable and lengthy than their laboratory hydrodynamic counterparts: even if the whole jet does not generate external electromagnetic fields and then in this aspect is formally similar to a hydrodynamic one as seen from outside, the existence of an internal electromagnetic structure characteristic of the jet in jet and absent in a hydrodynamic jet gives an extra stabilizing effect.\n\nWhat provides the current closure and the currents themselves? The jet as a whole is connected to the AGN central engine, a central supermassive black hole with a surrounding accretion disc. Under the condition of infinite conductivity the existence of a vertical magnetic field in the rotating disc results in the appearance of a radial electric field across the disc, so an induced potential difference between the inner and outer jets may be the source maintaining the circulating electric current in the whole jet. This effect is similar to the well-known generation of voltage by a unipolar inductor (Faraday's disc) \\citep{Shatskii2003,Okamoto2015}.\n\nThis conclusion is consistent with the existing observations. \\cite{Hada2017} has been observed that the width of the inner bright part is $r_0\\propto z^{0.5}$ while that of the whole jet is $R\\propto z^{0.56}$, where $z$ denotes the longitudinal distance. The voltage between the inner and outer jets is the same and does not depend on $z$. It is readily estimated from equation \\eqref{electricFieldBetweenJets} as\n\\begin{equation}\n\\label{voltage}\nV\\sim2Q\\ln\\frac{r_1}{r_0},\n\\end{equation}\nand for constant $V$ and $Q$ we have $r_1\\propto r_0$, which implies the same dependence of width on distance for both the jets irrespective of the nature of gradual widening when $R\\sim r_1$. Some difference might be assigned to a finite thickness of the outer jet.\n\nThe existence of charge separation can also be understood from the Grad-Shafranov approach. We can rewrite equation \\eqref{chargeCurrentRelation} as\n\\begin{equation}\n\\label{QWithIsorotation}\nQ=-\\Omega_\\text{F} \\Psi_\\text{eff},\n\\end{equation}\nwhere $\\Omega_\\text{F}=(v_\\phi-v_z B_\\phi\/B_z)\/r$ is the Ferraro isorotation frequency and $\\Psi_\\text{eff}=B_z\\pi r^2\/2\\pi$ is an effective normalized magnetic flux in the jet, which coincides with an actual normalized magnetic flux in the case of a constant axial magnetic field. Since we assume a constant field in the inner jet, we have $Q=-\\Omega_{\\text{F}0}\\Psi_0$, where $\\Omega_{\\text{F}0}$ and $\\Psi_0$ is the isorotation frequency and actual flux corresponding to the inner jet. Both the quantities are conserved along the magnetic tube, which may change its radius with the distance from the base, and then can be taken near the central engine. If we have a non-zero magnetic flux and rotation at the base, then the appearance of a charge is necessary and may be considered another reflection of the unipolar induction.\n\nFinally, let us estimate in the framework of the developed jet-in-jet model physical quantities related to the M87 jet. The total intensity $L$ is related to the power released in the jet via operation of the unipolar battery,\n\\begin{equation}\n\\label{totalIntensityViaVoltageAndCurrent}\nL\\sim V I.\n\\end{equation}\nUsing equation \\eqref{voltage} and estimating $Q\\sim I$ (we have considered relativistic motion with $v_z\\sim1$ and dropped the rightmost term in equation~\\eqref{chargeCurrentRelation}, the validity of which we will check below), we get an estimate for the jet charge and current\n\\begin{equation}\n\\label{chargeEstimate}\nQ\\sim I\\sim \\sqrt{\\frac{L}{2\\ln(r_1\/r_0)}}.\n\\end{equation}\n\nThe transverse sizes of the inner and outer jets are estimated from the radio intensity slice at $15$~marcsec (projected) from the core presented by \\cite{Hada2017} and from his statement that $R\/r_0\\sim8$. The inner radius of the outer jet is estimated as the semidistance between the peaks as the total length of segments of the line of sight lying in the emitting parts of the hollow cylinder is maximum when the line is tangent to the inner cylinder $r=r_1$. Adopting $1\\text{ marcsec}\\sim140\\text{ }R_\\text{Sch}\\sim2.5\\times10^{17}$~cm, we have the radius of the inner jet,\n\\begin{equation}\n\\label{M87InnerJetRadius}\nr_{0\\,\\text{M87}}\\sim8\\times10^{16}\\text{ cm},\n\\end{equation}\nand the inner and outer radii of the outer jet,\n\\begin{equation}\n\\label{M87OuterJetRadius}\nr_{1\\,\\text{M87}}\\sim4\\times10^{17}\\text{ cm},\\text{ }R_\\text{M87}\\sim6\\times10^{17}\\text{ cm}.\n\\end{equation}\n\nSince the intensity of the M87 jet is estimated from observations as \\citep{BroderickEtal2015}\n\\begin{equation}\n\\label{M87Intensity}\nL_\\text{M87}\\sim10^{44}\\text{ erg s}^{-1}\n\\end{equation}\n(with a factor of few uncertainty), we obtain the linear charge density and the electric current,\n\\begin{equation}\n\\label{M87ChargeCurrent}\nQ_\\text{M87}\\sim10^7\\text{ C cm}^{-1},\\text{ }I_\\text{M87}\\sim3\\times10^{17}\\text{ A}.\n\\end{equation}\nThe voltage between the inner and outer jets and the jet resistance become\n\\begin{equation}\n\\label{M87VoltageResistance}\nV_\\text{M87}\\sim3\\times10^{19}\\text{ V},\\text{ }R^\\Omega_\\text{M87}\\sim100\\text{ }\\Omega.\n\\end{equation}\nThe minimum volume particle densities (in the comoving frame; we adopt the Lorentz factor $\\gamma\\sim3$ from the observations of \\cite{MertensEtal2016}) in the inner and outer jets that are necessary for providing the above charges and currents are\n\\begin{equation}\n\\label{M87nE}\nn_{\\text{e}0\\,\\text{M87}}\\sim10^{-9}\\text{ cm}^{-3},\\text{ }n_{\\text{e}1\\,\\text{M87}}\\sim3\\times10^{-11}\\text{ cm}^{-3}\n\\end{equation}\nand, as we will see, are much less than the actual particle densities.\n\nAdopting the Kerr parameter $a\\sim0.6$, \\cite{MertensEtal2016} estimate the isorotation frequency $2.75\\times10^{-6}\\text{ s}^{-1}$ corresponding to the M87 black hole and conclude that the Blandford-Payne launching mechanism is more appropriate because observations of the jet rotation give the isorotation frequency $1.1\\times10^{-6}\\text{ s}^{-1}$, which corresponds to the Keplerian radius and velocity\n\\begin{equation}\n\\label{M87rBase}\nR^\\text{base}_\\text{M87}\\sim10^{16}\\text{ cm},\\text{ }v^\\text{base}_{\\phi\\,\\text{M87}}\\sim10^{10}\\text{ cm s}^{-1}.\n\\end{equation}\nSince in fact we have both mechanisms, we assign the first frequency to the inner jet and the second to the outer jet,\n\\begin{equation}\n\\label{M87OmegaF}\n\\Omega_{\\text{F}0\\,\\text{M87}}\\sim3\\times10^{-6}\\text{ s}^{-1},\\text{ }\\Omega_{\\text{F}1\\,\\text{M87}}\\sim10^{-6}\\text{ s}^{-1}.\n\\end{equation}\nThe actual angular velocities of the jets can be estimated from conservation of the quantity $l=\\gamma h(1-\\Omega_\\text{F}rv_\\phi)$ along the magnetic tube, which follows from a combination of the integrals \\eqref{GSeta}--\\eqref{GSL}. Putting $l\\approx1$ and $h\\approx1$, we arrive at\n\\begin{equation}\n\\label{M87OmegaGeneral}\n\\Omega\\sim\\frac{1-\\gamma^{-1}}{\\Omega_\\text{F}r^2}\n\\end{equation}\nand get the angular velocities\n\\begin{equation}\n\\label{M87Omega}\n\\Omega_{0\\,\\text{M87}}\\sim4\\times10^{-8}\\text{ s}^{-1},\\text{ }\\Omega_{1\\,\\text{M87}}\\sim3\\times10^{-9}\\text{ s}^{-1}\n\\end{equation}\nand ordinary toroidal velocities\n\\begin{equation}\n\\label{M87Vphi}\nv_{0\\phi\\,\\text{M87}}\\sim3\\times10^{9}\\text{ cm s}^{-1},\\text{ }v_{1\\phi\\,\\text{M87}}\\sim10^{9}\\text{ cm s}^{-1}.\n\\end{equation}\n\nEquation \\eqref{QWithIsorotation} allows us to express the axial magnetic field via charge, radius and isorotation frequency,\n\\begin{equation}\n\\label{BzGeneral}\nB_z\\sim\\frac{2Q}{\\Omega_\\text{F}r^2},\n\\end{equation}\nand thereby find the axial magnetic fields in the inner and outer jets,\n\\begin{equation}\n\\label{M87Bz}\nB_{0z\\,\\text{M87}}\\sim0.1\\text{ G},\\text{ }B_{1z\\,\\text{M87}}\\sim0.01\\text{ G}.\n\\end{equation}\nWe see that the inner jet is more magnetized than the outer, which can argue that the inner jet forms and collimates due to an effect analogous to the pinch effect: it can be squeezed by a toroidal magnetic field of the axial current (if we drop for clarity the inverse effect of the electric field), and a frozen-in axial magnetic field amplifies during such squeezing. Another possibility is a non-uniform magnetization in the disc, which can form due to some toroidal currents or accumulation of an accreting matter with a frozen-in field near the centre.\n\nThe other components of the electromagnetic field are calculated with equation~\\eqref{electricFieldBetweenJets}. The radial electric field\n\\begin{equation}\n\\label{M87Er}\nE_{0r\\,\\text{M87}}\\sim240\\text{ V cm}^{-1},\\text{ }E_{1r\\,\\text{M87}}\\sim50\\text{ V cm}^{-1}\n\\end{equation}\nand toroidal magnetic field\n\\begin{equation}\n\\label{M87Bphi}\nB_{0\\phi\\,\\text{M87}}\\sim0.8\\text{ G},\\text{ }B_{1\\phi\\,\\text{M87}}\\sim0.16\\text{ G}\n\\end{equation}\nfor the inner and outer jets exceed the above axial magnetic field. We can also verify here that the rightmost term in equation \\eqref{chargeCurrentRelation} can be dropped in our estimations: its ratio to the other term, $v_\\phi\/(B_\\phi\/B_z)$, is small because $B_\\phi\\gg B_z$ and $v_\\phi\\ll1$.\n\nLet us estimate the magnetic flux in the inner jet,\n\\begin{equation}\n\\label{M87Phi0}\n\\Phi_{0\\,\\text{M87}}=\\pi r_0^2 B_{0z}\\sim2\\times10^{33}\\text{ G cm}^{2},\n\\end{equation}\nin the outer jet,\n\\begin{equation}\n\\label{M87Phi1}\n\\Phi_{1\\,\\text{M87}}=\\pi d\\biggl(R-\\frac23\\,d\\biggr)B_{1z}\\sim4\\times10^{33}\\text{ G cm}^{2},\n\\end{equation}\nand between the jets,\n\\begin{equation}\n\\label{M87PhiIn}\n\\Phi_{\\text{in}\\,\\text{M87}}=\\pi(B_{0z}r_0+B_{1z}r_1)(r_1-r_0)\\sim1.3\\times10^{34}\\text{ G cm}^{2}.\n\\end{equation}\nThe total magnetic flux through the jet becomes\n\\begin{equation}\n\\label{M87Phi}\n\\Phi_\\text{M87}=\\Phi_0+\\Phi_1+\\Phi_\\text{in}\\sim2\\times10^{34}\\text{ G cm}^{2}\n\\end{equation}\nand allows us to find the magnetic field at the jet base under the condition of a uniform magnetization,\n\\begin{equation}\n\\label{M87Bbase}\nB^\\text{base}_\\text{M87}\\sim\\frac{\\Phi_\\text{M87}}{\\pi(R^\\text{base}_\\text{M87})^2}\\sim80\\text{ G}.\n\\end{equation}\nThus, the proposed magnetohydrodynamic model does not require the frequently discussed $B\\sim10^3-10^4$~G for explanation of the observed jet luminosity. This result is consistent with the estimate $50\\text{ G}<B<124$~G for the magnetic field at the base of the M87 jet derived from the observations of the radio core at $230$~GHz by applying the standard theory of synchrotron radiation \\citep{KinoEtal2015}.\n\nThe mass density in the disc is estimated from the balance of the magnetic and ram pressures,\n\\begin{equation}\n\\label{M87rhoDisc}\n\\rho^\\text{disc}_\\text{M87}\\sim\\frac1{8\\pi}\\biggl(\\frac{B^\\text{base}_\\text{M87}}{v^\\text{base}_{\\phi\\,\\text{M87}}}\\biggr)^2\\sim3\\times10^{-18}\\text{ g cm}^{-3},\n\\end{equation}\nwhich corresponds to the proton number density\n\\begin{equation}\n\\label{M87nDisc}\nn^\\text{disc}_\\text{M87}\\sim2\\times10^6\\text{ cm}^{-3}.\n\\end{equation}\n\nThe mass density in the inner jet is estimated from equation \\eqref{innerJetPressure} by assuming that the inner jet carries so much plasma as it can still carry, which corresponds to zero square bracket for zero pressure at the interface:\n\\begin{equation}\n\\label{M87rho0}\n\\rho_{0\\,\\text{M87}}\\sim\\frac2\\pi\\biggl(\\frac{I}{\\gamma_0^2 r_0 v_{0\\phi}}\\biggr)^2\\sim2\\times10^{-22}\\text{ g cm}^{-3}.\n\\end{equation}\nIn the outer jet, we have to use another approach because we do not know a priori the pressure of the external medium that enters equation~\\eqref{outerJetPressure}. Assuming that at the base $v_\\text{p}\\sim v_\\phi$ and using conservation of $\\eta$ \\eqref{GSeta}, we get for $\\rho^\\text{base}_\\text{M87}\\sim\\rho^\\text{disc}_\\text{M87}$\n\\begin{equation}\n\\label{M87rho1}\n\\rho_{1\\,\\text{M87}}\\sim\\frac{\\rho^\\text{disc}_\\text{M87}v^\\text{base}_{\\phi\\,\\text{M87}}}{\\gamma_1}\\frac{B_{1z\\,\\text{M87}}}{B^\\text{base}_\\text{M87}}\\sim4\\times10^{-23}\\text{ g cm}^{-3}.\n\\end{equation}\nWe then have the corresponding mass fluxes\n\\begin{equation}\n\\label{M87massFluxes}\n\\dot{M}_{0\\,\\text{M87}}\\sim5\\times10^{-3}\\text{ M}_\\odot\\text{ yr}^{-1},\\text{ }\\dot{M}_{1\\,\\text{M87}}\\sim0.05\\text{ M}_\\odot\\text{ yr}^{-1},\n\\end{equation}\nproton particle densities\n\\begin{equation}\n\\label{M87n}\nn_{0\\,\\text{M87}}\\sim100\\text{ cm}^{-3},\\text{ }n_{1\\,\\text{M87}}\\sim30\\text{ cm}^{-3},\n\\end{equation}\nand multiplicities $\\lambda=n\/n_\\text{e}$,\n\\begin{equation}\n\\label{M87lambda}\n\\lambda_{0\\,\\text{M87}}\\sim10^{11},\\text{ }\\lambda_{1\\,\\text{M87}}\\sim10^{12},\n\\end{equation}\nfor the inner and outer jets. The total mass flux through the jet,\n\\begin{equation}\n\\label{M87totalMassFlux}\n\\dot{M}_\\text{M87}\\sim0.05\\text{ M}_\\odot\\text{ yr}^{-1},\n\\end{equation}\nis very large, $\\dot{M}_\\text{M87}\\sim30L_\\text{M87}$, and comparable to the measured Bondi accretion rate $0.1-0.2\\text{ M}_\\odot\\text{ yr}^{-1}$ across the Bondi radius of M87 at $0.12-0.22$~kpc \\citep{RussellEtal2015}. This circumstance might favour a scenario when almost all initial accretion flow far from the jet goes to the outer jet so that the jet can substantially suppress accretion on to the black hole. Note that a limit of $10^{-3}\\text{ M}_\\odot\\text{ yr}^{-1}$ on the accretion flux is expected from the Faraday rotation measurements of \\cite{KuoEtal2014}, but the latest modelling of non-VLBI millimetre data admits much higher accretion rates \\citep{MoscibrodzkaEtal2017}.\n\nThe external pressure is estimated from equations \\eqref{intermediatePressureBalance} and \\eqref{outerJetPressure} by adopting zero pressure ar $r=r_0$,\n\\begin{align}\n\\label{M87pEx}\np_{\\text{ex}\\,\\text{M87}}&\\sim\\frac{B_{0z}^2}{8\\pi}+\\frac1{2\\pi}\\biggl(\\frac{I}{\\gamma_1 r_1}\\biggr)^2+\\rho_1(\\gamma_1v_{1\\phi})^2\\ln\\frac R{r_1}\n\\nonumber\\\\\n&\\sim10^{-3}\\text{ dyn cm}^{-2},\n\\end{align}\nand implies the jet temperature ($k_\\text{B}=1$)\n\\begin{equation}\n\\label{M87Temp}\nT_\\text{M87}\\sim\\frac{p_{\\text{ex}}}{2 n_1}\\sim10^{11}\\text{ K}.\n\\end{equation}\nThis temperature corresponds to non-relativistic protons, $T\\sim0.01m_\\text{p}$, and relativistic electrons, $T\\sim20m_\\text{e}$. Temperatures of the same order figured earlier in general relativistic magnetohydrodynamic simulations of the M87 jet \\citep{MoscibrodzkaEtal2016,MoscibrodzkaEtal2017}. Interestingly, the temperature slightly exceeds or is comparable to the brightness temperatures $10^{10}-10^{11}$~K observed at smaller distances from the base \\citep{KimM87Etal2016}.\n\n\\section{Conclusions}\n\nIn this paper, new high-resolution VLBI radio observations of the M87 jet at 15 GHz have been considered. A suggestion has been put forward that an unusual three-peaked structure of the transverse radio profile with a previously unobserved ultra-narrow central ridge can reflect the actual structure of the jet. A model in the framework of the relativistic ideal magnetohydrodynamics has been developed that the jet as a whole in fact consists of two coaxial embedded jets such that the outer jet is an annular hollow plasma cylinder that contains a narrow inner jet. The rotating relativistic inner and outer jets move in the same direction away from the central engine and have opposite charges and inversely directed electric currents. The charges, currents and axial magnetic fields are related to the velocities of the electromagnetically connected jets. A continuous distribution of electromagnetic fields and pressure without charge and current sheets is obtained and various physical quantities in relation to the existing observations of the M87 jet are calculated. In particular, the inferred magnetic field at the jet base is about $80$~G and appears sufficient to provide the observed jet luminosity.\n\nThe existence of the jet-in-jet structure can directly indicate that two different mechanisms of jet launching simultaneously operate in the central engine: one is responsible for the inner jet and related to the central supermassive black hole and the other is responsible for the outer jet and related to the accretion disc. The central engine plays the role of a battery operating as a unipolar inductor and generating voltage between the inner and outer jets, which provides the current closure in the whole jet. Qualitative consideration of the jet stability shows a beneficial role of the electromagnetic interlayer, charge neutrality and closure of the currents generated by the central engine in stabilization of the sausage and kink instabilities, which might explain why astrophysical jets are more stable than laboratory hydrodynamic jets. The constant voltage between the jets explains similarity between the widening laws for the inner and outer jets, which was found in the VLBI observations.\n\n\\section*{Acknowledgements}\n\nThe work is supported by the Russian Science Foundation, grant no. 16-12-10051.\n\n\n\n\n\\bibliographystyle{mnras}\n\\providecommand{\\noopsort}[1]{}\\providecommand{\\singleletter}[1]{#1}%\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{\\def\\@secnumfont{\\mdseries}\\@startsection{section}{1}%\n  \\z@{.7\\linespacing\\@plus\\linespacing}{.5\\linespacing}%\n  {\\normalfont\\scshape\\centering}}\n\\def\\subsection{\\def\\@secnumfont{\\bfseries}\\@startsection{subsection}{2}%\n  {\\parindent}{.5\\linespacing\\@plus.7\\linespacing}{-.5em}%\n  {\\normalfont\\bfseries}}\n\\makeatother\n\\def\\subl#1{\\subsection{}\\label{#1}}\n \\newcommand{\\Hom}{\\operatorname{Hom}}\n  \\newcommand{\\mode}{\\operatorname{mod}}\n\\newcommand{\\End}{\\operatorname{End}} \\newcommand{\\wh}[1]{\\widehat{#1}} \\newcommand{\\Ext}{\\operatorname{Ext}} \\newcommand{\\ch}{\\text{ch}} \\newcommand{\\ev}{\\operatorname{ev}}\n\\newcommand{\\Ob}{\\operatorname{Ob}} \\newcommand{\\soc}{\\operatorname{soc}} \\newcommand{\\rad}{\\operatorname{rad}} \\newcommand{\\head}{\\operatorname{head}}\n\\def\\operatorname{Im}{\\operatorname{Im}}\n\\def\\operatorname{gr}{\\operatorname{gr}}\n\\def\\operatorname{mult}{\\operatorname{mult}}\n\\def\\operatorname{Max}{\\operatorname{Max}}\n\\def\\operatorname{Ann}{\\operatorname{Ann}}\n\\def\\operatorname{sym}{\\operatorname{sym}}\n\\def\\operatorname{loc}{\\operatorname{loc}}\n\\def\\operatorname{\\br^\\lambda_A}{\\operatorname{\\br^\\lambda_A}}\n\\def\\underline{\\underline}\n\\def$A_k(\\lie{g})(\\bsigma,r)${$A_k(\\lie{g})(\\bsigma,r)$}\n\\def\\operatorname{res}{\\operatorname{res}}\n\n \\newcommand{\\Cal}{\\cal} \\newcommand{\\Xp}[1]{X^+(#1)} \\newcommand{\\Xm}[1]{X^-(#1)}\n\\newcommand{\\on}{\\operatorname} \\newcommand{\\Z}{{\\bold Z}} \\newcommand{\\J}{{\\cal J}} \\newcommand{\\C}{{\\bold C}} \\newcommand{\\Q}{{\\bold Q}}\n\\renewcommand{\\P}{{\\cal P}}\n\\newcommand{\\N}{{\\Bbb N}} \\newcommand\\boa{\\bold a} \\newcommand\\bob{\\bold b} \\newcommand\\boc{\\bold c} \\newcommand\\bod{\\bold d} \\newcommand\\boe{\\bold e} \\newcommand\\bof{\\bold f} \\newcommand\\bog{\\bold g}\n\\newcommand\\boh{\\bold h} \\newcommand\\boi{\\bold i} \\newcommand\\boj{\\bold j} \\newcommand\\bok{\\bold k} \\newcommand\\bol{\\bold l} \\newcommand\\bom{\\bold m} \\newcommand\\bon{\\bold n} \\newcommand\\boo{\\bold o}\n\\newcommand\\bop{\\bold p} \\newcommand\\boq{\\bold q} \\newcommand\\bor{\\bold r} \\newcommand\\bos{\\bold s} \\newcommand\\boT{\\bold t} \\newcommand\\boF{\\bold F} \\newcommand\\bou{\\bold u} \\newcommand\\bov{\\bold v}\n\\newcommand\\bow{\\bold w} \\newcommand\\boz{\\bold z} \\newcommand\\boy{\\bold y} \\newcommand\\ba{\\bold A} \\newcommand\\bb{\\bold B} \\newcommand\\bc{\\bold C} \\newcommand\\bd{\\bold D} \\newcommand\\be{\\bold E} \\newcommand\\bg{\\bold\nG} \\newcommand\\bh{\\bold H} \\newcommand\\bi{\\bold I} \\newcommand\\bj{\\bold J} \\newcommand\\bk{\\bold K} \\newcommand\\bl{\\bold L} \\newcommand\\bm{\\bold M} \\newcommand\\bn{\\bold N} \\newcommand\\bo{\\bold O} \\newcommand\\bp{\\bold\nP} \\newcommand\\bq{\\bold Q} \\newcommand\\br{\\bold R} \\newcommand\\bs{\\bold S} \\newcommand\\bt{\\bold T} \\newcommand\\bu{\\bold U} \\newcommand\\bv{\\bold V} \\newcommand\\bw{\\bold W} \\newcommand\\bz{\\bold Z} \\newcommand\\bx{\\bold\nx} \\newcommand\\KR{\\bold{KR}} \\newcommand\\rk{\\bold{rk}} \\newcommand\\het{\\text{ht }}\n\n\n\\newcommand\\toa{\\tilde a} \\newcommand\\tob{\\tilde b} \\newcommand\\toc{\\tilde c} \\newcommand\\tod{\\tilde d} \\newcommand\\toe{\\tilde e} \\newcommand\\tof{\\tilde f} \\newcommand\\tog{\\tilde g} \\newcommand\\toh{\\tilde h}\n\\newcommand\\toi{\\tilde i} \\newcommand\\toj{\\tilde j} \\newcommand\\tok{\\tilde k} \\newcommand\\tol{\\tilde l} \\newcommand\\tom{\\tilde m} \\newcommand\\ton{\\tilde n} \\newcommand\\too{\\tilde o} \\newcommand\\toq{\\tilde q}\n\\newcommand\\tor{\\tilde r} \\newcommand\\tos{\\tilde s} \\newcommand\\toT{\\tilde t} \\newcommand\\tou{\\tilde u} \\newcommand\\tov{\\tilde v} \\newcommand\\tow{\\tilde w} \\newcommand\\toz{\\tilde z} \\newcommand\\woi{w_{\\omega_i}}\n\\newcommand\\chara{\\operatorname{Char}}\n\\title{Conjecture $\\mathcal{O}$ holds for some Horospherical Varieties of Picard Rank 1}\n\\author[Bones, Fowler, Schneider, Shifler]{Lela Bones, Garrett Fowler, Lisa Schneider and Ryan M. Shifler}\\address{\\noindent Department of Mathematics, Salisbury University, MD 21801}\n\\email{lbones1@gulls.salisbury.edu, gfowler2@gulls.salisbury.edu, lmschneider@salisbury.edu, rmshifler@salisbury.edu}\n\\begin{abstract}\nProperty $\\mathcal{O}$ for an arbitrary complex, Fano manifold $X$, is a statement about the eigenvalues of the linear operator obtained from the quantum multiplication of the anticanonical class of $X$. Conjecture $\\mathcal{O}$ is a conjecture that Property $\\mathcal{O}$ holds for any Fano variety. Pasquier listed the smooth non-homogeneous horospherical varieties of Picard rank 1 into five classes. Conjecture $\\mathcal{O}$ has already been shown to hold for the odd symplectic Grassmannians which is one of these classes. We will show that Conjecture $\\mathcal{O}$ holds for two more classes and an example in a third class of Pasquier's list. The theory of Perron-Frobenius reduces our proofs to be graph-theoretic in nature.\n\\end{abstract}\n\\maketitle\n\\section{Introduction}\nThe purpose of this paper is to prove that Conjecture $\\mathcal{O}$ holds for some horospherical varieties of Picard rank 1. We recall the precise statement of Conjecture $\\mathcal{O}$ for varieties of Picard rank 1, following \\cite[Section 3]{GGI}. Let $F$ be a Fano variety, let $K:=K_F$ be the canonical line bundle of $F$, let $F_D$ be a fundamental divisor of $F$, and let $c_1(F):=c_1(-K) \\in H^2(F)$ be the anticanonical class. The Fano index of $F$ is $r$, where $r$ is the greatest integer such that $K_F \\cong -rF_D$. The small quantum cohomology ring $(QH^*(F), \\star)$ is a graded algebra over $\\mathbb{Z}[q]$, where $q$ is the quantum parameter. We define the small quantum cohomology in Section~\\ref{sec:Section 2.1}. Consider the specialization $H^{\\bullet}(F):=QH^*(F)|_{q=1}$ at $q=1$. The quantum multiplication by the first Chern class $c_1(F)$ induces an endomorphism $\\hat{c}_1$ of the finite-dimensional vector space $H^{\\bullet}(F)$: \\[y \\in H^{\\bullet}(F) \\mapsto \\hat{c}_1(y):=(c_1(F)\\star y)|_{q=1}. \\]\n\nDenote by $\\delta_0:=\\max \\{|\\delta| : \\delta \\mbox{ is an eigenvalue of } \\hat{c}_1 \\}.$ Then Property $\\mathcal{O}$ states the following:\n\\begin{enumerate}\n\\item The real number $\\delta_0$ is an eigenvalue of $\\hat{c}_1$ of multiplicity one.\n\\item If $\\delta$ is any eigenvalue of $\\hat{c}_1$ with $|\\delta|=\\delta_0$, then $\\delta=\\delta_0 \\gamma$ for some $r$-th root of unity $\\gamma \\in \\mathbb{C}$, where $r$ is the Fano index of $F$.\n\\end{enumerate}\nThe property $\\mathcal{O}$ was conjectured to hold for any Fano, complex manifold $F$ in \\cite{GGI}. If a Fano, complex, manifold has Property $\\mathcal{O}$ then we say that the space satisfies Conjecture $\\mathcal{O}$. Conjecture $\\mathcal{O}$ underlies Gamma Conjectures I and II of Galkin, Golyshev, and Iritani. The Gamma Conjectures refine earlier conjectures by Dubrovin on Frobenius manifolds and mirror symmetry. Conjecture $\\mathcal{O}$ has already been proved for the homogeneous $G\/P$ case in \\cite{CL}, the odd symplectic Grassmannians in \\cite{LMS}, del Pezzo surfaces in \\cite{delpezzo}, and projective complete intersections in \\cite{projinter}. The Perron-Frobenius theory of nonnegative matrices reduces the proofs that Conjecture $\\mathcal{O}$ holds for the homogeneous and the odd symplectic Grassmannian cases to be a graph-theoretic check. This is because Conjecture $\\mathcal{O}$ is largely reminiscent of Perron-Frobenius Theory. In this manuscript we will use the same graph-theoretic approach to prove that Conjecture $\\mathcal{O}$ holds for some smooth horospherical varieties of Picard rank 1.\n\n\nNext we recall the definition of a horospherical variety following \\cite{GPPS}. Let $G$ be a complex reductive group. A $G$-variety is a reduced scheme of finite type over the field of complex numbers $\\mathbb{C}$, equipped with an algebraic action of $G$. Let $B$ be a Borel subgroup of $G$. A $G$-variety $X$ is called spherical if $X$ has a dense $B$-orbit. Let $X$ be a $G$-spherical variety and let $H$ be the stabilizer of a point in the dense $G$-orbit in $X$. The variety $X$ is called {\\it horospherical} if $H$ contains a conjugate of the maximal unipotent subgroup of $G$ contained in the Borel subgroup $B$. \n\n\nSmooth horospherical varieties of Picard rank 1 were classified by Pasquier in \\cite{P}. These varieties are either homogeneous or can be constructed in a uniform way via a triple (Type($G$),$\\omega_Y$,$\\omega_Z$) of representation-theoretic data, where Type($G$) is the semisimple Lie type of the reductive group $G$ and $\\omega_Y, \\omega_Z$ are fundamental weights. See \\cite[Section 1.3]{P} for details. Pasquier classified the possible triples in five classes:\n\\begin{enumerate}\n\\item $(B_n, \\omega_{n-1},\\omega_n)$ with $n \\geq 3$;\n\\item $(B_3, \\omega_1,\\omega_3)$;\n\\item $(C_n,\\omega_m,\\omega_{m-1})$ with $n \\geq 2$ and $m \\in [2,n]$ (the odd symplectic Grassmannians);\n\\item $(F_4, \\omega_2, \\omega_3)$;\n\\item $(G_2,\\omega_1,\\omega_2)$.\n\\end{enumerate}\n\n In Proposition 3.6 of \\cite{Pthesis}, Pasquier showed the triples in the above list are Fano varieties. We are now able to state the main theorem:\n\n\\begin{thm}\\label{thm:main}\nIf $F$ belongs to the classes (1) for $n=3$, (2), (3), and (5) of Pasquier's list, then Conjecture $\\mathcal{O}$ holds for $F$.\n\\end{thm}\n{\\it Acknowledgements:} We would like to thank the anonymous referees for their useful comments and suggestions to improve the presentation of this paper.\n\\section{Preliminaries}\n\n\n\\subsection{Quantum Cohomology}\n\\label{sec:Section 2.1}\nThe small quantum cohomology is defined as follows. Let $(\\alpha_i)_i$ be a basis of $H^*(F)$, the classical cohomology ring, and let $(\\alpha_i^{\\vee})_i$ be the dual basis for the Poincar\\'e pairing. The multiplication is given by \\[ \\alpha_i \\star \\alpha_j= \\sum_{d\\geq 0, k} c_{i,j}^{k,d}q^d\\alpha_k \\] where $c_{i,j}^{k,d}$ are the 3-point, genus 0, Gromov-Witten invariants corresponding to the classes $\\alpha_i, \\alpha_j$, and $\\alpha_k^{\\vee}$. We will make use of the quantum Chevalley formula which is the multiplication of a hyperplane class $h$ with another class $\\alpha_j$. The result \\cite[Theorem 0.0.3]{GPPS} implies that if $F$ belongs to the classes (1) for $n=3$, (2), or (5) of Pasquier's list, then there is an explicit quantum Chevalley formula. The explicit quantum Chevalley formula is the key ingredient used to prove Property $\\mathcal{O}$ holds.\n\n\n\n\\subsection{Sufficient Criterion for Property $\\mathcal{O}$ to hold} We recall the notion of the (oriented) quantum Bruhat graph of a Fano variety $F$. The vertices of this graph are the basis elements $\\alpha_i \\in H^{\\bullet}(F):=QH^*(F)|_{q=1}$. There is an oriented edge $\\alpha_i \\rightarrow \\alpha_j$ if the class $\\alpha_j$ appears with positive coefficient (where we consider $q>0$) in the quantum Chevalley multiplication $h \\star \\alpha_i$ for some hyperplane class $h$. Using the Perron-Frobenius theory of non-negative matrices, Conjecture $\\mathcal{O}$ reduces to a graph-theoretic check of the quantum Bruhat graph. The techniques involving Perron-Frobenius theory used by Li, Mihalcea, and Shifler in \\cite{LMS} and Cheong and Li in \\cite{CL} imply the following lemma:\n\n\\begin{lemlab}\\label{lemma: propO}\nIf the following conditions hold for a Fano variety $F$:\n\\begin{enumerate}\n\\item the matrix representation of $\\hat{c}_1$ is nonnegative,\n\\item the quantum Bruhat graph of $F$ is strongly connected, and \n\\item there exists a cycle of length $r$, the Fano index, in the quantum Bruhat graph of $F$,\n\\end{enumerate}\nthen Property $\\mathcal{O}$ holds for $F$. We say the matrix representation of $\\hat{c}_1$ is nonnegative if all of the entries are nonnegative.\n\\end{lemlab}\n\nWe refer the reader to \\cite[section 4.3]{Minc} for further details on Perron-Frobenius theory.\n\n\\section{Checking Property $\\mathcal{O}$ Holds}\nLet $X$ be a horospherical variety. We will simplify our notation where the basis of $H^{\\bullet}(X)$ is $\\{1,h,\\alpha_i\\}_{i\\in I}$ for some finite index set $I$.  Observe by \\cite{GPPS} that  the anticanonical classes are \\[ c_1(X) = \\left\\{ \\begin{array}{ll}\n            5h & \\text{ when X is case (1) for }n=3 \\\\\n            7h & \\text{ when X is case (2)}\\\\\n            4h & \\text{ when X is case (5)}\n        \\end{array}\n    \\right.\\]\nand the Fano indices are\n \\[ r = \\left\\{ \\begin{array}{ll}\n            5 & \\text{ when X is case (1) for }n=3 \\\\\n            7 & \\text{ when X is case (2)}\\\\\n            4& \\text{ when X is case (5)}\n        \\end{array}.\n    \\right.\\]   \nThe endomorphism $\\hat{c}_1$ acting on the basis elements of $H^{\\bullet}(X)$ is determined by the Chevalley formula in the following way:\n\\begin{eqnarray*}\n            \\hat{c}_1(\\alpha_i)&=&5(h\\star \\alpha_i)|_{q=1}  \\text{ when X is case (1) for }n=3, \\\\\n            \\hat{c}_1(\\alpha_i)&=&7(h\\star \\alpha_i)|_{q=1}   \\text{ when X is case (2)}, \\text{and}\\\\\n            \\hat{c}_1(\\alpha_i)&=&4(h\\star \\alpha_i)|_{q=1}   \\text{ when X is case (5)}.\n\\end{eqnarray*}\n\nEach of the following three subsections will show that Conjecture $\\mathcal{O}$ holds for case (1) for $n=3$, case (2), and case (5) of Pasquier's list, respectively. In each subsection we will reformulate the quantum Chevalley formulas stated in \\cite{GPPS}, present the quantum Bruhat graph, and argue that each condition of Lemma \\ref{lemma: propO} is satisfied. For each case, we have kept the same format of the equations presented by \\cite{GPPS} with our prescribed basis for ease of identification for the reader. For example, line 3 in \\cite[Proposition 4.3]{GPPS} is \\[h*\\sigma'_{u_2}=\\sigma'_{u_3}+\\sigma'_{u'_3} \\mbox{ } and \\mbox{ } h* \\sigma'_{u'_2}=2\\sigma'_{u'_3}+\\tau_{v_0}. \\] In Proposition 1 below we identify this line with \\[\\hat{c}_1(\\alpha_1)=5\\alpha_3+5\\alpha_4 \\mbox{ } and \\mbox{ } \\hat{c}_1(\\alpha_2)=10\\alpha_3+5\\alpha_5. \\]\n\n\\subsection{Case (1) for $n=3$}  We will reformulate the quantum Chevalley formula stated in \\cite{GPPS} using the basis $\\{1,h, \\alpha_1, \\alpha_2,\\cdots, \\alpha_{18}\\}$.\n\n\\begin{proplab}\\label{Gon4.3} The following equalities hold by \\cite[Proposition 4.3]{GPPS}.\n\\begin{enumerate}\n\\item $\\hat{c}_1(1)=5h$\n\\item $\\hat{c}_1(h)=10\\alpha_1+5\\alpha_2$\n\\item $\\hat{c}_1(\\alpha_1)=5\\alpha_3+5\\alpha_4$ and $\\hat{c}_1(\\alpha_2)=10\\alpha_3+5\\alpha_5$\n\\item $\\hat{c}_1(\\alpha_3) = 10\\alpha_6+5\\alpha_7+5\\alpha_8, \\ \\ \\hat{c}_1(\\alpha_4)=5\\alpha_6 + 10\\alpha_7, $ and $\\hat{c}_1(\\alpha_5)=5\\alpha_8$\n\\item $\\hat{c}_1(\\alpha_6)=10\\alpha_9+5\\alpha_{10}+5\\alpha_{11},\\ \\ \\hat{c}_1(\\alpha_7) = 5\\alpha_{10}$ and $\\hat{c}_1(\\alpha_8)=5\\alpha_{11}+5\\cdot1$\n\\item $\\hat{c}_1(\\alpha_9)=5\\alpha_{12}+5\\alpha_{13}, \\ \\ \\hat{c}_1(\\alpha_{10})=10\\alpha_{13}+5\\alpha_{14}\\ \\ \\hat{c}_1(\\alpha_{11})=5\\alpha_{12}+5\\alpha_{14}+5h\n\\item $\\hat{c}_1(\\alpha_{12})=5\\alpha_{15}+5\\alpha_1, \\ \\ \\hat{c}_1(\\alpha_{13})=5\\alpha_{15} +5 \\alpha_{16},$ and $\\hat{c}_1(\\alpha_{14}) = 5\\alpha_{15}+5\\alpha_2\n\\item $\\hat{c}_1(\\alpha_{15})=5\\alpha_{17}+5\\alpha_3$ and $\\hat{c}_1(\\alpha_{16}) =5 \\alpha_{17}+5\\alpha_5\n\\item $\\hat{c}_1(\\alpha_{17})=5 \\alpha_{18}+5\\alpha_6+5\\alpha_8$\n\\item $\\hat{c}_1(\\alpha_{18}) =5 \\alpha_9+5\\alpha_{11}+10\\cdot1$\n\\end{enumerate}\n\\end{proplab}\n\n\\label{ex: case(1)}\nThe following figure is the quantum Bruhat graph of the Fano variety $X$ in case (1) for $n=3$. Colored edges are introduced in this figure to improve readability. The bold edges indicate a cycle of length $r=5$, the Fano index.\n\\begin{figure}[H]\n\\begin{center}\n\\caption{}\n\\label{qcbg:1-3}\n\\scalebox{.75}{\n\\begin{tikzpicture}\n\\tikzset{edge\/.style = {->,> = latex'}}\n    \\node (0) at (0,0) {$1$};\n    \\node (1) at (0,-1) {$h$};\n    \\node (2) at (-1,-2) {$\\alpha_1$};\n    \\node (3) at (1,-2) {$\\alpha_2$};\n    \\node (4) at (-2,-3) {$\\alpha_3$};\n    \\node (5) at (0,-3) {$\\alpha_4$};\n    \\node (6) at (2,-3) {$\\alpha_5$};\n    \\node (7) at (-2,-4) {$\\alpha_6$};\n    \\node (8) at (0,-4) {$\\alpha_7$};\n    \\node (9) at (2,-4) {$\\alpha_8$};\n    \\node (10) at (-2,-5) {$\\alpha_9$};\n    \\node (11) at (0,-5) {$\\alpha_{10}$};\n    \\node (12) at (2,-5) {$\\alpha_{11}$};\n    \\node (13) at (-2,-6) {$\\alpha_{12}$};\n    \\node (14) at (0,-6) {$\\alpha_{13}$};\n    \\node (15) at (2,-6) {$\\alpha_{14}$};\n    \\node (16) at (-1,-7) {$\\alpha_{15}$};\n    \\node (17) at (1,-7) {$\\alpha_{16}$};\n    \\node (18) at (0,-8) {$\\alpha_{17}$};\n    \\node (19) at (0,-9) {$\\alpha_{18}$};\n    \\draw [edge] (0) to (1);\n    \\draw [edge] (1) to (2);\n    \\draw [edge] (1) to (3);\n    \\draw [edge] (2) to (4);\n    \\draw [edge] (2) to (5);\n    \\draw [edge] (3) to (4);\n    \\draw [edge] (3) to (6);\n    \\draw [edge] (4) to (7);\n    \\draw [edge] (4) to (8);\n    \\draw [edge] (4) to (9);\n    \\draw [edge] (5) to (7);\n    \\draw [edge] (5) to (8);\n    \\draw [edge] (6) to (9);\n    \\draw [edge] (7) to (10);\n    \\draw [edge] (7) to (11);\n    \\draw [edge] (7) to (12);\n    \\draw [edge] (8) to (11);\n    \\draw [edge] (9) to (12);\n    \\draw [blue, edge] (9) to [bend right=100] (0);\n    \\draw [edge] (10) to (13);\n    \\draw [edge] (10) to (14);\n    \\draw [edge] (11) to (14);\n    \\draw [edge] (11) to (15);\n    \\draw [edge] (12) to (13);\n    \\draw [edge, ultra thick] (12) to (15);\n    \\draw [red, edge] (12) to [bend right=100] (1);\n    \\draw [edge] (13) to (16);\n    \\draw [green,edge] (13) to [bend left=100] (2);\n    \\draw [edge] (14) to (16);\n    \\draw [edge] (14) to (17);\n    \\draw [edge, ultra thick] (15) to (16);\n    \\draw [pink, edge] (15) to [bend right=100] (3);\n    \\draw [edge, ultra thick] (16) to (18);\n    \\draw [orange, edge] (16) to [bend left=100] (4);\n    \\draw [edge] (17) to (18);\n    \\draw [yellow, edge] (17) to [bend right=100] (6);\n    \\draw [edge, ultra thick] (18) to (19);\n    \\draw [purple, edge] (18) to [bend left =100] (7);\n    \\draw [teal, edge] (18) to [bend right=100] (9);\n    \\draw [lime, edge] (19) to [bend left=100] (10);\n    \\draw [edge, ultra thick] (19) to [bend right=100] (12);\n    \\draw [cyan, edge] (19) to [bend right=100] (0);\n\\end{tikzpicture}\n}\n\\end{center}\n\\end{figure}\n\n\\begin{lemlab}\\label{lemma:case1}\nProperty $\\mathcal{O}$ holds when $X$ is case (1) with $n=3$ of Pasquier's list.\n\\end{lemlab}\n\\begin{proof}\nThe coefficients that appear in the equations in Proposition \\ref{Gon4.3} are the entries of the matrix representation of $\\hat{c}_1$. Therefore, the matrix representation of $\\hat{c}_1$ is nonnegative. The quantum Bruhat graph is strongly connected by Figure \\ref{qcbg:1-3}, and the cycle $\\alpha_{18}\\alpha_{11}\\alpha_{14}\\alpha_{15}\\alpha_{17}\\alpha_{18}$ has length $r=5$.\n\\end{proof}\n\n\\subsection{Case (2)} Again, we reformulate the quantum Chevalley formula from \\cite{GPPS} using the basis $\\{1,h, \\alpha_1, \\alpha_2,\\cdots, \\alpha_{12}\\}$.\n\\begin{proplab}\\label{Gon4.4} The following equalities hold by \\cite[Proposition 4.4]{GPPS}.\n\\begin{enumerate}\n\\item $\\hat{c}_1(1)=7h$\n\\item $\\hat{c}_1(h)=7\\alpha_1$\n\\item $\\hat{c}_1(\\alpha_1)=14\\alpha_2+7\\alpha_3$\n\\item $\\hat{c}_1(\\alpha_2) = 7\\alpha_4+7\\alpha_5$ and $\\hat{c}_1(\\alpha_3)=7\\alpha_5$\n\\item $\\hat{c}_1(\\alpha_4)=7\\alpha_6+7\\alpha_7$ and $\\hat{c}_1(\\alpha_5)=7\\alpha_7$\n\\item $\\hat{c}_1(\\alpha_6)=7\\alpha_8$ and $\\hat{c}_1(\\alpha_7)=7\\alpha_8+7\\alpha_9\n\\item $\\hat{c}_1(\\alpha_8)=7\\alpha_{10}$ and $\\hat{c}_1(\\alpha_9) =7 \\alpha_{10}+7\\cdot1\n\\item $\\hat{c}_1(\\alpha_{10})=7\\alpha_{11}+7h\n\\item $\\hat{c}_1(\\alpha_{11})=7 \\alpha_{12}+7\\alpha_1$\n\\item $\\hat{c}_1(\\alpha_{12}) =7 \\alpha_2$\n\\end{enumerate}\n\\end{proplab}\n\nThe quantum Bruhat graph is \n\\begin{figure}[H]\n\\caption{}\n\\label{qcbg:2}\n\\begin{center}\n\\scalebox{.75}{\n\\begin{tikzpicture}\n\\tikzset{edge\/.style = {->,> = latex'}}\n    \\node (0) at (0,0) {$1$};\n    \\node (1) at (0,-1) {$h$};\n    \\node (2) at (0,-2) {$a_1$};\n    \\node (3) at (1,-3) {$a_3$};\n    \\node (4) at (-1,-3) {$a_2$};\n    \\node (5) at (1,-4) {$a_5$};\n    \\node (6) at (-1,-4) {$a_4$};\n    \\node (7) at (1,-5) {$a_7$};\n    \\node (8) at (-1,-5) {$a_6$};\n    \\node (9) at (2,-6) {$a_9$};\n    \\node (10) at (1,-6) {$a_8$};\n    \\node (11) at (1,-7) {$a_{10}$};\n    \\node (12) at (1,-8) {$a_{11}$};\n    \\node (13) at (1,-9) {$a_{12}$};\n    \\draw [edge] (0) to (1);\n    \\draw [edge] (1) to (2);\n    \\draw [edge] (2) to (3);\n    \\draw [edge] (2) to (4);\n    \\draw [edge] (3) to (5);\n    \\draw [edge] (4) to (5);\n    \\draw [edge, ultra thick] (4) to (6);\n    \\draw [edge] (5) to (7);\n    \\draw [edge] (6) to (7);\n    \\draw [edge, ultra thick] (6) to (8);\n    \\draw [edge] (7) to (9);\n    \\draw [edge] (7) to (10);\n    \\draw [edge, ultra thick] (8) to (10);\n    \\draw [edge] (9) to [bend right] (0);\n    \\draw [edge] (9) to (11);\n    \\draw [edge, ultra thick] (10) to (11);\n    \\draw [edge] (11) to [bend right=100] (1);\n    \\draw [edge, ultra thick] (11) to (12);\n    \\draw [edge] (12) to [bend right=100] (2);\n    \\draw [edge, ultra thick] (12) to (13);\n    \\draw [edge, ultra thick] (13) to [bend left=100] (4);\n\n\\end{tikzpicture}\n}\n\\end{center}\n\\end{figure}\n\n\n\\begin{lemlab} \\label{lemma:case2}\nProperty $\\mathcal{O}$ holds when $X$ is case (2) of Pasquier's list. \n\\end{lemlab}\n\\begin{proof}\nThe coefficients that appear in the equations in Proposition \\ref{Gon4.4} are the entries of the matrix representation of $\\hat{c}_1$. Therefore, the matrix representation of $\\hat{c}_1$ is nonnegative. The quantum Bruhat graph is strongly connected by Figure \\ref{qcbg:2}, and the cycle $\\alpha_{12}\\alpha_2\\alpha_4\\alpha_6\\alpha_8\\alpha_{10}\\alpha_{11}\\alpha_{12}$ has length $r=7$.\n\\end{proof}\n\n\n\\subsection{Case(5)} Again, we reformulate the quantum Chevalley formula from \\cite{GPPS} using the basis $\\{1,h, \\alpha_1, \\alpha_2,\\cdots, \\alpha_{10}\\}$.\n\\begin{proplab}\\label{Gon4.6} The following equalities hold by \\cite[Proposition 4.6]{GPPS}.\n\\begin{enumerate}\n\\item $\\hat{c}_1(1)=4h$\n\\item $\\hat{c}_1(h)=12\\alpha_1+4\\alpha_2$\n\\item $\\hat{c}_1(\\alpha_1)=8\\alpha_3+4\\alpha_4$ and $\\hat{c}_1(\\alpha_2)=4\\alpha_4$\n\\item $\\hat{c}_1(\\alpha_3) = 12\\alpha_5+4\\alpha_6$ and $\\hat{c}_1(\\alpha_4)=4\\alpha_6+4\\cdot1$\n\\item $\\hat{c}_1(\\alpha_5)=4\\alpha_7+4\\alpha_8$ and $\\hat{c}_1(\\alpha_6)=8\\alpha_7+4h$\n\\item $\\hat{c}_1(\\alpha_7)=4\\alpha_9+4\\alpha_1$ and $\\hat{c}_1(\\alpha_8)=4\\alpha_9+4\\alpha_2\n\\item $\\hat{c}_1(\\alpha_9)=4\\alpha_{10}+4\\alpha_3+4\\alpha_4\n\\item $\\hat{c}_1(\\alpha_{10})=4\\alpha_5+4\\alpha_6+8\\cdot 1\n\\end{enumerate}\n\\end{proplab}\n\n\n\nThe associated quantum Bruhat graph is \n\\begin{figure}[H]\n\\caption{}\n\\label{qcbg:5}\n\\begin{center}\n\\scalebox{.75}{\n\\begin{tikzpicture}\n\\tikzset{edge\/.style = {->,> = latex'}}\n    \\node (0) at (0,0) {$1$};\n    \\node (1) at (0,-1) {$h$};\n    \\node (2) at (-1,-2) {$\\alpha_1$};\n    \\node (3) at (1,-2) {$\\alpha_2$};\n    \\node (4) at (-1,-3) {$\\alpha_3$};\n    \\node (5) at (1,-3) {$\\alpha_4$};\n    \\node (6) at (-1,-4) {$\\alpha_5$};\n    \\node (7) at (1,-4) {$\\alpha_6$};\n    \\node (8) at (-1,-5) {$\\alpha_7$};\n    \\node (9) at (1,-5) {$\\alpha_8$};\n    \\node (10) at (-1,-6) {$\\alpha_9$};\n    \\node (11) at (-1,-7) {$\\alpha_{10}$};\n    \\draw [edge] (0) to (1);\n    \\draw [edge] (1) to (2);\n    \\draw [edge] (1) to (3);\n    \\draw [edge] (2) to (4);\n    \\draw [edge] (2) to (5);\n    \\draw [edge] (3) to (5);\n    \\draw [edge] (4) to (6);\n    \\draw [edge] (4) to (7);\n    \\draw [edge] (5) to (7);\n    \\draw [edge] (5) to [bend right=100] (0);\n    \\draw [edge] (6) to (8);\n    \\draw [edge] (6) to (9);\n    \\draw [edge, ultra thick] (7) to (8);\n    \\draw [edge] (7) to [bend right=100] (1);\n    \\draw [edge] (8) to [bend left=100] (2);\n    \\draw [edge, ultra thick] (8) to (10);\n    \\draw [edge] (9) to [bend right=100] (3);\n    \\draw [edge] (9) to (10);\n    \\draw [edge] (10) to [bend right=100] (5);\n    \\draw [edge, ultra thick] (10) to (11);\n    \\draw [edge] (10) to [bend left=100] (4);\n    \\draw [edge, ultra thick] (11) to [bend right=100] (7);\n    \\draw [edge] (11) to [bend left=100] (6);\n    \\draw [edge] (11) to [bend left=100] (0);\n\n\\end{tikzpicture}\n}\n\\end{center}\n\\end{figure}\n\n\\begin{lemlab}\\label{lemma:case5}\nProperty $\\mathcal{O}$ holds when $X$ is case (5) of Pasquier's list. \n\\end{lemlab}\n\\begin{proof}\nThe coefficients that appear in the equations in Proposition \\ref{Gon4.6} are the entries of the matrix representation of $\\hat{c}_1$. Therefore, the matrix representation of $\\hat{c}_1$ is nonnegative. The quantum Bruhat graph is strongly connected by Figure \\ref{qcbg:5}, and the cycle $\\alpha_{10}\\alpha_6\\alpha_7\\alpha_9\\alpha_{10}$ has length $r=4$.\n\\end{proof}\nTheorem \\ref{thm:main} follows from Lemmas \\ref{lemma:case1}, \\ref{lemma:case2}, \\ref{lemma:case5}, and the previously mentioned work done by Li, Mihalcea, Shifler for the odd symplectic Grassmannian case in \\cite{LMS}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe heavy ion collisions have the advantage of being able to study the hot medium in detail under controlled environments.\nIt has recently been reported that the strongest magnetic fields of the order of $10^{18}-10^{19}$ Gauss are produced in non-central heavy-ion collisions by the electric current from the positively charged spectators that travel at nearly the speed of light ~\\cite{Bzdak2012,Deng2012,Kharzeev2008}.\nIt is expected that such a huge magnetic field may have important consequences on the dynamics of the quark-gluon matter produced in heavy-ion collisions ~\\cite{Gyulassy2005}.\nIn particular, it has been proposed that the interplay of quantum anomalies with ultra-intense magnetic field results in several special transport phenomena\nthat are closely related to chiral anomaly and thus are called anomalous transports ~\\cite{Xu-Guang Huang2016,Kharzeev2016}.\nThe external magnetic fields may induce charge separation in a chirality-imbalanced medium, namely the chiral magnetic effect (CME) ~\\cite{Kharzeev2008,Fukushima2008}\nwhich has been observed at RHIC and the LHC and the measurements signals indeed consistent with the predictions of the CME\n~\\cite{STAR2010,STAR2009,ALICE2016}.\nAlong with CME, the chiral separation effect (CSE) ~\\cite{Metlitski2005,Son2004} represents the generation of the axial current along the external magnetic field in the presence of finite vector charge density. The duality between CME and CSE leads to the interesting collective effect,\ncalled ``chiral magnetic wave\"(CMW) ~\\cite{Kharzeev2011}, which induces a quadrupole deformation of the electric charge distribution\nthat might be responsible for breaking the degeneracy between the elliptic flows of $\\pi^{\\pm}$ ~\\cite{Burnier2011}.\n\nThe relativistic hydrodynamic models have so far nicely described the thermodynamic evolution of the produced matter and the\nexperimentally measured flow harmonics in heavy-ion collisions ~\\cite{Romatschke2007,Luzum2008,Songprl2008,Songprc2008,Schenke2012,\nRoy2012,Niemi2012}. The influence of strong magnetic fields on the hot and dense nuclear matter have been intensively investigated.\nIn principle, such studies can be accomplished by solving the relativistic magnetohydrodynamics (MHD) equations that takes into account\nthe dynamical coupling of the magnetic field to the fluid.\nAlthough the magnetic field generated in heavy-ion collisions rapidly decays in the vacuum,\nthe hot medium created in the heavy-ion collision as a conducting plasma might substantially\ndelay the decay of the magnetic field through the generation of an induction current due to\nLenz's law ~\\cite{Gursoy2014,Zakharov2014,Tuchin2013}.\n\nIn Refs.~\\cite{Roy2015}, one-dimensional magnetic fluid had been investigated by using the longitudinally boost-invariant Bjorken flow~\\cite{Bjorken1983} with a transverse and time-dependent homogeneous magnetic field.\nIn ideal MHD limits, with the infinite electrical conductivity and neglecting other dissipative effects such as viscosity and thermal conduction, it is extraordinary that the evolution of energy density is the same as the case without magnetic fields due to ``frozen-flux theorem\".\nLater, a nonzero magnetization effect is introduced to the Bjorken flow in MHD~\\cite{Shi2016}.\n\nQGP expanding in the beam direction can be described by considering a boost invariant Bjorken flow, that is known to be a good approximation at mid-rapidity. It leads to a flat rapidity distribution of final particle,\nwhich is inconsistent with observations at RHIC, except for a narrow region around mid-rapidity ~\\cite{Bjorken1983}.\nIt has been pointed out that in realistic situations the energy density at mid-rapidity decreases faster than in the Bjorken flow.\nAlthough the Bjorken-estimation for the initial energy density is widely used, the longitudinal expansion dynamics of hydrodynamics seems ~\\cite{Csorgo:2006ax,Csand:2016arx,Jiang2017,Jiang2018} to be able to offer a more realistic estimation for the initial energy density estimation and the final state description. Acceleration effects are important in the estimation of the initial energy density even at mid-rapidity, if the expanding system is finite: even the most central fluid element exert a force on the volume elements closer to the surface, and this work decreases the internal energy of cells even at mid-rapidity.\n\nThis paper is organized as follows. In Sec.\\ref{section2}, the ideal-MHD framework with acceleration effects is presented.\nIn Sec.\\ref{section3}, we present the evolution of the energy density in ideal transverse MHD with longitudinal expansion dynamics.\nWe consider the decay of the energy density within an external homogeneous magnetic field which decays with a power-law in proper time.\nIn subsection.~\\ref{CNC approximation}, an exact analytic solution under the CNC approximation is obtained.\nIn subsection.~\\ref{numerical solution}, we show the results obtained from numerical method for a realistic equation of state (EoS).\nFinally, we discuss and conclude in the last section.\nThroughout this work, $u^{\\mu}=\\gamma\\left(1,\\overrightarrow{\\boldsymbol{v}}\\right)$\nis the four-velocity field that satisfies $u^{\\mu}u_{\\mu}=1$ and\nthe spatial projection operator $\\Delta^{\\mu\\nu}=g^{\\mu\\nu}-u^{\\mu}u^{\\nu}$ is defined with\nthe Minkowski metric $g^{\\mu\\nu}={\\rm diag}\\left(1,-1,-1,-1\\right)$.\nIt is note-worthy that the orthogonality relation $\\Delta^{\\mu\\nu}u_{\\nu}=0$ is satisfied.\nWe adopt the standard convention for the summation over repeated indices.\n\n\\section{ideal MHD with acceleration}\n\\label{section2}\n\nRelativistic magnetohydrodynamics (RMHD for short) concerns the mutual interaction of fluid flow and magnetic fields.\nThe fluids in question must be electrically conducting and non-magnetic.\nThe RMHD evolution equations describe the dynamics of the overall system based on local conservation of this fluid current\n(associated to the net-baryon current or to any other conserved charge)\nand the total (matter and fields) energy-momentum as well as on the additional assumption of local thermal equilibrium.\n\nConsider an non-viscous fluid coupling with  a magnetic field.\nWe assume that the medium is perfectly conducting and the electric field in the co-moving frame vanishes\nto avoid the onset of huge currents in the plasma.\nThe total energy-momentum tensor of ideal fluid is given by ~\\cite{Huang2010,Giacomazzo2006,Giacomazzo2007}\n\\begin{eqnarray}\nT^{\\mu\\nu}=(e+p+B^{2})u^{\\mu}u^{\\nu}-\\left(p+\\frac{B^{2}}{2}\\right)g^{\\mu\\nu}-B^{\\mu}B^{\\nu},\n\\label{1}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nB^{2}=B^{\\mu}B_{\\mu},\\quad B^{\\mu}=\\frac{1}{2}\\epsilon^{\\mu\\nu\\alpha\\beta}u_{\\nu}F_{\\alpha\\beta},\n\\label{2}\n\\end{eqnarray}\nhere $e,~p$ and $F_{\\alpha\\beta}$ are the fluid energy density, pressure and the Faraday tensor.\nHere, $\\epsilon^{\\mu\\nu\\alpha\\beta}$ is the completely antisymmetric four tensor with $\\epsilon^{0123}=-\\epsilon_{0123}=1$.\nThe magnetic field four-vector $B^{\\mu}$ is a space-like vector with modulus $B^{\\mu}B_{\\mu}=-B^{2}$ and is orthogonal to $u^{\\mu}$, i.e.,~ $B^{\\mu}u_{\\mu}=0$, where $B=|\\vec{\\boldsymbol{B}}|$ and $\\vec{\\boldsymbol{B}}$\nis the magnetic field three-vector in the frame moving with four-velocity $u^{\\mu}$.\n\nIn the present paper we consider the special case of a fluid flow with the external magnetic field\n$\\vec{\\boldsymbol{B}}$ directed along the transverse plane.\nThis setup is consistent with the scenario in non-central heavy ion collision at top RHIC energy~\\cite{Deng2012}.\nThe system of ideal RMHD equations can be closed by choosing the rather general EoS\n\\begin{equation}\np={c}_{s}^{2}\\,e=\\frac{1}{\\kappa}\\,e\\,,\n\\label{3}\n\\end{equation}\nwhere $c_s$ stands for the local speed of sound which is assumed to be a constant.\nIn a fully realistic solution, we should use results form the lattice QCD, with the speed of sound being a function of temperature~\\cite{Gupta2004,Qin2015}.\nHowever, in current work we approximate $c_s(T)$ as a temperature independent constant $c_s$.\nWe postpone the analysis of the case of $c_s(T)$ for a later, more detailed investigation.\n\nWe decompose the covariant derivative as\n\\begin{eqnarray}\n\\partial_{\\mu}=u_{\\mu}D+\\nabla_{\\mu},\n\\label{4}\n\\end{eqnarray}\nwhere $D=u^{\\mu}\\partial_{\\mu}$ indicates the time derivative in the local rest frame,\nand $\\nabla^{\\mu}=\\Delta^{\\mu\\nu}\\partial_{\\nu}$ is the spatial gradient in the local rest frame.\nThe energy conservation equation is derived by projecting the conservation law $\\partial_{\\mu}T^{\\mu\\nu}=0$ along the fluid four-velocity $u^{\\mu}$,\n\\begin{eqnarray}\nu_{\\mu}\\partial_{\\nu}T^{\\mu\\nu}\t&=&\tu_{\\mu}u^{\\mu}u^{\\nu}\\partial_{\\nu}(e+p+B^{2})+(e+p+B^{2})u_{\\mu}\\partial_{\\nu}(u^{\\mu}u^{\\nu})-u_{\\mu}\\partial_{\\nu}\\left[\\left(p+\\frac{B^{2}}{2}\\right)g^{\\mu\\nu}\\right]-u_{\\mu}\\partial_{\\nu}(B^{\\mu}B^{\\nu})\\nonumber\\\\\n\t&=&\tu^{\\nu}\\partial_{\\nu}(e+p+B^{2})+(e+p+B^{2})\\partial_{\\nu}u^{\\nu}+0-u^{\\nu}\\partial_{\\nu}\\left(p+\\frac{B^{2}}{2}\\right)-\\partial_{\\nu}(u_{\\mu}B^{\\mu}B^{\\nu})+B^{\\mu}B^{\\nu}\\partial_{\\nu}u_{\\mu}\n\t\\nonumber\\\\\n\t&=&\t\tD(e+p+B^{2})+(e+p+B^{2})\\theta-D\\left(p+\\frac{B^{2}}{2}\\right)\n\t\\nonumber\\\\\n\t&=&\t\tD\\left(e+\\frac{B^{2}}{2}\\right)+(e+p+B^{2})\\theta\n\t\\nonumber\\\\\n\t&=&\t\t0,\n\\label{5}\n\\end{eqnarray}\nwhere $\\theta\\equiv\\partial_{\\mu}u^{\\mu}=\\nabla_{\\mu}u^{\\mu}$ is the expansion factor\nand we have used relation $u^{\\mu}B_{\\mu}=0$ and $B^{\\nu}\\partial_{\\nu}u_{\\mu}=0$,\nsince $u_{\\mu}=\\left(u_{0},0,0,u_{z}\\right)$ and $B_{\\mu}=\\left(0,B_{x},B_{y},0\\right)$ in our setup.\nThus one obtains the energy-conservation equation as follows,\n\\begin{eqnarray}\nD\\left(e+\\frac{B^{2}}{2}\\right)+(e+p+B^{2})\\theta=0 \\,.\n\\label{6}\n\\end{eqnarray}\n\nThe relativistic version of the MHD Euler equation is retrieved by projecting the energy-momentum conservation equation onto the direction orthogonal to $u^{\\mu}$,\n\\begin{eqnarray}\n\\triangle_{\\mu\\nu}\\partial_{\\alpha}T^{\\alpha\\nu}\t&=&\t\t(e+p+B^{2})\\triangle_{\\mu\\nu}\\partial_{\\alpha}(u^{\\alpha}u^{\\nu})+0-\\triangle_{\\mu\\nu}\\partial^{\\nu}\\left(p+\\frac{B^{2}}{2}\\right)-\\triangle_{\\mu\\nu}\\partial_{\\alpha}(B^{\\alpha}B^{\\nu})\\nonumber\\\\\n\t&=&\t\t(e+p+B^{2})(g_{\\mu\\nu}-u_{\\mu}u_{\\nu})\\partial_{\\alpha}(u^{\\alpha}u^{\\nu})-\\triangle_{\\mu\\nu}\\partial^{\\nu}\\left(p+\\frac{B^{2}}{2}\\right)-(g_{\\mu\\nu}-u_{\\mu}u_{\\nu})\\partial_{\\alpha}(B^{\\alpha}B^{\\nu})\\nonumber\\\\\n\t&=&\t\t(e+p+B^{2})\\left[u^{\\alpha}\\partial_{\\alpha}u_{\\mu}+u_{\\mu}\\partial_{\\alpha}u^{\\alpha}-u_{\\mu}u_{\\nu}u^{\\alpha}\\partial^{\\alpha}u^{\\nu}-u_{\\mu}u_{\\nu}u^{\\nu}\\partial_{\\alpha}u^{\\alpha}\\right]-\\triangle_{\\mu\\nu}\\partial^{\\nu}\\left(p+\\frac{B^{2}}{2}\\right)\\nonumber\\\\\n&&-B^{\\alpha}\\partial_{\\alpha}B_{\\mu}-B_{\\mu}\\partial_{\\alpha}B^{\\alpha}+u_{\\mu}u_{\\nu}B^{\\alpha}\\partial_{\\alpha}B^{\\nu}+u_{\\mu}u_{\\nu}B^{\\nu}\\partial_{\\alpha}B^{\\alpha}\n\\nonumber\\\\\n\t&=&\t\t(e+p+B^{2})Du_{\\mu}-\\triangle_{\\mu\\nu}\\partial^{\\nu}\\left(p+\\frac{B^{2}}{2}\\right)-B^{\\alpha}\\partial_{\\alpha}B_{\\mu}-B_{\\mu}\\partial_{\\alpha}B^{\\alpha}-u_{\\mu}B^{\\alpha}B^{\\nu}\\partial_{\\alpha}u_{\\nu}\\nonumber\\\\\n&=&\t\t0.\n\\label{7}\n\\end{eqnarray}\n\nThe last three terms vanish and leads to the Euler equation as follow,\n\\begin{equation}\n(e+p+B^{2})Du_{\\mu}-\\nabla_{\\mu}\\left(p+\\frac{B^{2}}{2}\\right)=0\\,.\n\\label{8}\n\\end{equation}\n\nWe use the well-known Rindler coordinates $\\tau=\\sqrt{t^{2}-r^{2}}$ and $\\eta_{s}=\\frac{1}{2}{\\rm log}\\left(\\left(t+r\\right)\/\\left(t-r\\right)\\right)$ as independent variables inside the forward lightcone and parametrize the fluid velocity as $v={\\rm tanh}\\Omega$, and the fluid rapidity $\\Omega$ depends only on $\\eta_s$ here. (For simplicity, we will use $\\Omega$ to denotes $\\Omega(\\eta_s)$, and $\\Omega^{\\prime}$ denotes $d\\Omega\/d\\eta_{s}$). One obtains\n\\begin{eqnarray}\nD&=&u^{\\mu}\\partial_{\\mu}=u^{0}\\partial_{t}+u^{z}\\partial_{z}\\nonumber\\\\\n&=&{\\rm cosh}\\Omega\\left({\\rm cosh}\\eta_{s}\\frac{\\partial}{\\partial\\tau}-\\frac{{\\rm sinh}\\eta_{s}}{\\tau}\\frac{\\partial}{\\partial\\eta_{s}}\\right)+{\\rm sinh}\\Omega\\left(-{\\rm sinh}\\eta_{s}\\frac{\\partial}{\\partial\\tau}+\\frac{{\\rm cosh}\\eta_{s}}{\\tau}\\frac{\\partial}{\\partial\\eta_{s}}\\right)\\nonumber\\\\\n&=&{\\rm cosh}(\\Omega-\\eta_{s})\\frac{\\partial}{\\partial\\tau}+\\frac{1}{\\tau}{\\rm sinh}(\\Omega-\\eta_{s})\\frac{\\partial}{\\partial\\eta_{s}}\n\\label{9}\n \\end{eqnarray}\nand\n\\begin{eqnarray}\n\\theta&=&\\partial_{\\mu}u^{\\mu}=\\partial_{t}u^{0}+\\partial_{z}u^{z}\\nonumber\\\\\n&=&\\left({\\rm cosh}\\eta_{s}\\frac{\\partial}{\\partial\\tau}-\\frac{{\\rm sinh}\\eta_{s}}{\\tau}\\frac{\\partial}{\\partial\\eta_{s}}\\right){\\rm cosh}\\Omega+\\left(-{\\rm sinh}\\eta_{s}\\frac{\\partial}{\\partial\\tau}+\\frac{{\\rm cosh}\\eta_{s}}{\\tau}\\frac{\\partial}{\\partial\\eta_{s}}\\right){\\rm sinh}\\Omega\\nonumber\\\\\n&=&{\\rm cosh}\\eta_{s}{\\rm sinh}\\Omega\\frac{\\partial\\Omega}{\\partial\\tau}-\\frac{{\\rm sinh}\\eta_{s}}{\\tau}{\\rm sinh}\\Omega\\frac{\\partial\\Omega}{\\partial\\eta_{s}}-{\\rm sinh}\\eta_{s}{\\rm cosh}\\Omega\\frac{\\partial\\Omega}{\\partial\\tau}+{\\rm \\frac{{\\rm cosh}\\eta_{s}}{\\tau}{\\rm cosh}\\Omega}\\frac{\\partial\\Omega}{\\partial\\eta_{s}}\\nonumber\\\\\n&=&{\\rm sinh}(\\Omega-\\eta_{s})\\frac{\\partial\\Omega}{\\partial\\tau}+\\frac{1}{\\tau}{\\rm cosh}(\\Omega-\\eta_{s})\\frac{\\partial\\Omega}{\\partial\\eta_{s}}.\n\\label{10}\n \\end{eqnarray}\n\nThe peak value of the magnetic field $\\vec{\\boldsymbol{B}}$ is well determined by using event-by-event simulations with in the Monte-Carlo Glauber model~\\cite{Bloczynski2013}.\nNevertheless, the lifetime of magnetic field is still an open question so far. We assume in this paper the homogeneous magnetic field obeys a power-law decay in proper time~\\cite{Roy2015},\n\\begin{equation}\n\\overrightarrow{B}(\\tau)=\\overrightarrow{B}_{0}\\left(\\frac{\\tau_{0}}{\\tau}\\right)^{a}.\n\\label{11}\n\\end{equation}\nHere $a=1$ correspond to the ideal-MHD case~\\cite{Davidson2017}, where $a>1$ corresponds to the case with the magnetic field decaying steeper than the ideal-MHD case,\nand $a<1$ corresponds to a decay slower than in the ideal-MHD limit.\n$\\tau_0$ is the initial proper time of the fluid expansion and $B_{0}\\equiv B(\\tau_{0})$ is the initial magnetic field strength.\n\nAbove assumptions allow one to rewrite the conservation equations in Rindler coordinate as follows\n\\begin{eqnarray}\n&&\\tau\\frac{\\partial\\widetilde{e}}{\\partial\\tau}+{\\rm tanh}(\\Omega-\\eta_{s})\\frac{\\partial\\widetilde{e}}{\\partial\\eta_{s}}+\n\\left(\\widetilde{e}(1+c_{s}^{2})+\\sigma_{0}\\left(\\frac{\\tau_{0}}{\\tau}\\right)^{2a}\\right)\\Omega'=\\sigma_{0}a\\left(\\frac{\\tau_{0}}{\\tau}\\right)^{2a},~\n\\label{12}\\\\\n&&\\frac{\\partial\\widetilde{e}}{\\partial\\eta_{s}}={\\rm tanh}(\\Omega-\\eta_{s})\\left[\\frac{\\sigma_{0}}{c_{s}^{2}}\n\\left(\\frac{\\tau_{0}}{\\tau}\\right)^{2a}(a-\\Omega')-\\frac{1+c_{s}^{2}}{c_{s}^{2}}\\widetilde{e}\\Omega'-\\tau\\frac{\\partial\\widetilde{e}}{\\partial\\tau}\\right],\n\\label{13}\n\\end{eqnarray}\nwith the dimensionless quantities $\\widetilde{e}\\equiv e\/e_{0},~\\sigma_{0}\\equiv B_{0}^{2}\/e_{0}$.\n\nThe combination of energy conservation equation Eq.~(\\ref{12}) and the Euler equation Eq.~(\\ref{13}) generates a partial differential equation,\n\n\\begin{eqnarray}\n\\tau\\frac{\\partial\\widetilde{e}}{\\partial\\tau}=\\left(\\frac{{\\rm sinh}^{2}(\\Omega-\\eta_{s})}{c_{s}^{2}}-{\\rm cosh}^{2}(\\Omega-\\eta_{s})\\right)\\left[\\left(\\sigma_{0}\\left(\\frac{\\tau_{0}}{\\tau}\\right)^{2a}+\\widetilde{e}(1+c_{s}^{2})\\right)\\Omega'-\\sigma_{0}a\\left(\\frac{\\tau_{0}}{\\tau}\\right)^{2a}\\right].\n\\label{14}\n\\end{eqnarray}\n\\section{energy-density evolution}\n\\label{section3}\nThe exact solution with CNC approximation (Sec.\\ref{CNC approximation}) and numerical solution (Sec.\\ref{numerical solution}) of energy density evolution in MHD are presented in this section step by step.\n\n\\subsection{Exact solution of MHD with CNC approximation}\n\\label{CNC approximation}\nFor a perfect fluid with longitudinal accelerating expansion, one finds $\\Omega\\neq\\eta_{s}$.\nThe exact solution for such longitudinal accelerating hydrodynamics is the well-known CNC solution with $\\Omega=\\lambda\\eta_{s}$ and $\\kappa=1$, $c_{s}^{2}=\\frac{1}{\\kappa}=1,~\\Omega'=\\lambda,~\\Omega''=0$. From Eq.(\\ref{14}), one gets\n\n\\begin{eqnarray}\n\\tau\\frac{\\partial\\widetilde{e}}{\\partial\\tau}=\n\\sigma_{0}a\\left(\\frac{\\tau_{0}}{\\tau}\\right)^{2a}-\\left(\\sigma_{0}\\left(\\frac{\\tau_{0}}{\\tau}\\right)^{2a}+2\\widetilde{e}\\right)\\lambda.\n\\label{15}\n\\end{eqnarray}\nThe solution $\\widetilde{e}(\\tau,\\eta_{s})$ is\n\\begin{eqnarray}\n\\widetilde{e}(\\tau,\\eta_{s})=-\\frac{1}{2}\\sigma_{0}\\left(\\frac{\\tau_{0}}{\\tau}\\right)^{2a}+\\tau^{-2\\lambda}C(\\eta_{s}),\n\\label{16}\n\\end{eqnarray}\nwhere $C(\\eta_{s})$ is an undetermined function related to the $\\eta_s$ part of the energy density $\\widetilde{e}(\\tau,\\eta_{s})$.\n\nPutting Eq.(\\ref{16}) into the Euler equation Eq.(\\ref{13}), one gets\n\\begin{eqnarray}\nC(\\eta_{s})=C.\n\\label{17}\n\\end{eqnarray}\nThen substituteing Eq.(\\ref{17}) to Eq.(\\ref{16}) and using the initial condition $\\widetilde{e}_{0}(\\tau_0,0)=1$, one obtains\n\\begin{eqnarray}\nC=\\frac{1}{2}(2+\\sigma_{0})\\tau_{0}^{2\\lambda}.\n\\label{18}\n\\end{eqnarray}\nFinally, inputting Eq.(\\ref{18}) and Eq.(\\ref{17}) into Eq.(\\ref{16}), an analytical solution of the fluid energy density with CNC approximation can be written as follow\n\\begin{eqnarray}\n\\widetilde{e}(\\tau,\\eta_{s})=\\left(\\frac{\\tau_{0}}{\\tau}\\right)^{2\\lambda}+\\frac{\\sigma_{0}}{2}\\left[\\left(\\frac{\\tau_{0}}{\\tau}\\right)^{2\\lambda}-\\left(\\frac{\\tau_{0}}{\\tau}\\right)^{2a}\\right].\n\\label{19}\n\\end{eqnarray}\n\n\\begin{table}[htb]\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n$$ & $a\\to\\lambda$ & $a\\gg\\lambda$ & $a\\ll\\lambda$\n\\\\\n\\hline\n$\\widetilde{e}(\\tau,\\eta_{s})$ & $\\left(\\frac{\\tau_{0}}{\\tau}\\right)^{2\\lambda}$ & $\\frac{2+\\sigma_{0}}{2}\\left(\\frac{\\tau_{0}}{\\tau}\\right)^{2\\lambda}$ & $-\\frac{\\sigma_{0}}{2}\\left(\\frac{\\tau_{0}}{\\tau}\\right)^{2a}$\n\\\\\n\\hline\n\\end{tabular}\n\\caption{The fluid energy density in the three kinds of limit conditions}\n\\end{table}\nOnce again, it is possible to see that in the limit of vanishing magnetization $\\sigma_{0}=0$ and $\\lambda=1$, Eq.(\\ref{19}) coincides with the solution for Bjorken flow.\nIf $\\sigma_{0}=0$ and $\\lambda \\neq 1$, the solution coincides with CNC solution.\nFurthermore, for $\\sigma_{0}\\neq 0$ and $\\lambda=1$, one obtains the same solution as the Bjorken-Victor type flow~\\cite{Roy2015}.\n\nOne can obtain the extreme value of the energy density from the following steps\n\\begin{eqnarray}\n\\frac{\\partial\\widetilde{e}(\\tau,\\eta_{s})}{\\partial\\tau}&&=\\frac{a\\sigma_{0}\\left(\\frac{\\tau_{0}}{\\tau}\\right)^{2a}-\\lambda(2+\\sigma_{0})\\left(\\frac{\\tau_{0}}{\\tau}\\right)^{2\\lambda}}{\\tau}=0\n\\label{20}\\\\\n&&\\Rightarrow\\tau=\\tau_{0}\\left(\\frac{a\\sigma_{0}}{\\lambda\\left(2+\\sigma_{0}\\right)}\\right)^{\\frac{1}{2(a-\\lambda)}}.\n\\label{21}\n\\end{eqnarray}\nUnfortunately, these CNC solutions have a shortcoming, namely the acceleration parameter $\\lambda$ becomes\na free fit parameter only for the superhard EoS of $\\kappa=1,~e=p$.\nIn this case, the speed of sound is equal to the speed of light $c$, so the investigation was thought to be rather academic.\n\n\\subsection{Numerical solution for MHD}\n\\label{numerical solution}\nTo get a realistic solution of the energy density, we consider the case in which $\\Omega\\equiv\\lambda\\eta_{s}\\equiv(1+\\lambda^{*})\\eta_{s}$ with $\\lambda^{*}$ being a very small constant acceleration parameter ($0<\\lambda^{*}\\ll1$) and $\\Omega'=1+\\lambda^{*},~\\Omega''=0$.\n\nThus, the energy equation and Euler equation can be expressed as\n\\begin{eqnarray}\n&&\\tau\\frac{\\partial\\widetilde{e}}{\\partial\\tau}+{\\rm tanh}(\\lambda^{*}\\eta_{s})\\frac{\\partial\\widetilde{e}}{\\partial\\eta_{s}}+\\left(\\widetilde{e}(1+c_{s}^{2})+\\sigma_{0}\\left(\\frac{\\tau_{0}}{\\tau}\\right)^{2a}\\right)(1+\\lambda^{*})=\\sigma_{0}a\\left(\\frac{\\tau_{0}}{\\tau}\\right)^{2a},\\label{22}\\\\\n&&\\frac{\\partial\\widetilde{e}}{\\partial\\eta_{s}}={\\rm tanh}(\\lambda^{*}\\eta_{s})\\left[\\frac{\\sigma_{0}}{c_{s}^{2}}\\left(\\frac{\\tau_{0}}{\\tau}\\right)^{2a}(a-1-\\lambda^{*})-\\frac{1+c_{s}^{2}}{c_{s}^{2}}\\widetilde{e}(1+\\lambda^{*})-\\tau\\frac{\\partial\\widetilde{e}}{\\partial\\tau}\\right].\n\\label{23}\n\\end{eqnarray}\n\nThe combination of energy equation Eq.(\\ref{22}) and Euler equation Eq. (\\ref{23}) can be rewritten as follows\n\\begin{eqnarray}\n\\tau\\frac{\\partial\\widetilde{e}}{\\partial\\tau}&=&\\left(\\kappa{\\rm sinh}^{2}(\\lambda^{*}\\eta_{s})-{\\rm cosh}^{2}(\\lambda^{*}\\eta_{s})\\right)\\left[\\left(\\sigma_{0}\\left(\\frac{\\tau_{0}}{\\tau}\\right)^{2a}+\\widetilde{e}(1+\\frac{1}{\\kappa})\\right)(1+\\lambda^{*})-\\sigma_{0}a\\left(\\frac{\\tau_{0}}{\\tau}\\right)^{2a}\\right],\n\\label{24}\\\\\n\\frac{\\partial\\widetilde{e}}{\\partial\\eta_{s}}&=&\\frac{1}{2}{\\rm sinh}(2\\lambda^{*}\\eta_{s})\\left[\\kappa\\sigma_{0}\\left(\\frac{\\tau_{0}}{\\tau}\\right)^{2a}(a-1-\\lambda^{*})-(1+\\kappa)\\widetilde{e}(1+\\lambda^{*})-\\sigma_{0}a\\left(\\frac{\\tau_{0}}{\\tau}\\right)^{2a}\\right.\\nonumber\\\\\n&&\\left.+\\left(\\widetilde{e}(1+\\frac{1}{\\kappa})+\\sigma_{0}\\left(\\frac{\\tau_{0}}{\\tau}\\right)^{2a}\\right)(1+\\lambda^{*})\\right].\n\\label{25}\n\\end{eqnarray}\nThe main idea of solving the above partial differential equations is to treat this two PDEs as two ordinary differential equations with a given initial condition $\\widetilde{e}(\\tau_{0},0)=1$. Then one can get the relation between $\\widetilde{e}$ and $\\tau$ from Eq. (\\ref{24}). The sets of data obtained above can be taken as the initial conditions for solving Eq.(\\ref{25}). For this purpose, one obtains the full profile of energy density right away and obtains the evolution of temperature by using relation $e\\propto T^{\\kappa+1}$.\n\nFig. \\ref{fig1} reports numerical solution of the fluid energy density and the temperature of accelerating fluid in one-dimensional relativistic magnetohydrodynamics with parameters $a=2,~\\sigma_{0}=1.0,~\\kappa=7,~\\lambda^{*}=0.03$. The profile of $\\widetilde{e}(\\tau,~\\eta_{s})$ is a (1+1) dimensional scaling solution, and it contains not only acceleration but also the magnetic field dependent terms now with the $\\eta_s$ dependence of the Gaussian form.\nNote that (1) if $\\lambda^{*}=0$ and $\\sigma_{0}=0$ one obtains the Bjorken solutions, (2) if $\\lambda^{*}=0$ and $\\sigma_0\\neq0$ one obtains the same solution as the Bjorken-Victor type flow, (3) if $\\lambda^{*}\\neq 0$ and $\\sigma_{0}=0$ one obtains the case of the well-known CNC solution.\n\\begin{figure}[htb]\n\\begin{minipage}[htb]{8.5cm}\n\\centerline{\\epsfig{figure=fig1.pdf,width=8cm}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[htb]{8.5cm}\n\\centerline{\\epsfig{figure=fig2.pdf,width=8cm}}\n\\end{minipage}\n\\caption{Left panel indicates the fluid energy density $e\/e_{0}$, while the right panel shows the temperature $T\/T_{0} $(right panel) profile, with parameters $a=2,\\sigma_{0}=1.0,\\kappa=7,\\lambda^{*}=0.03$.}\n\\label{fig1}\n\\end{figure}\n\nFor the sake of comparison with Bjorken-Victor type flow,\nwe take the space-time rapidity $\\eta_s=0$ in Fig.\\ref{fig2} and Fig.\\ref{fig3}. In Fig.\\ref{fig2}, we compare the evolution of fluid energy density $\\widetilde{e}$ for following different conditions: (\\textbf{a})~different longitudinal acceleration parameter $\\lambda^*$, (\\textbf{b})~different magnetic field decay parameter $a$, and (\\textbf{c})~different EoS parameter $\\kappa$. In Fig.\\ref{fig2} case (\\textbf{a}), different lines represent to different values of the longitudinal acceleration parameter $\\lambda^{*}$, ranging from $\\lambda^{*}=0$ (Bjorken-Victor type flow without longitudinal acceleration effect; black solid line) up to cases with $\\lambda^*=0.03$ (red dashed line), $\\lambda^*=0.06$ (blue dotted line) and $\\lambda^*=0.1$ (magenta dot-dashed line). In Fig.\\ref{fig2} case (\\textbf{b}), we show the evolution of the normalized energy density $\\widetilde{e}$ for $a=2$ (black solid line), $a=1$ (ideal-MHD limit; red dashed line), and $a=2\/3$ (blue dotted line). In Fig.\\ref{fig2} case (\\textbf{c}), different lines means different values of EoS $\\kappa=1$ (CNC approximation; black solid line), $\\kappa=3$ (red dashed line), $\\kappa=7$ (blue dotted), and $\\kappa=10$ (magenta dot-dashed line). As the graph illustrates, the longitudinal acceleration effect of the fluid increase the decay of the fluid energy density; $\\widetilde{e}$ decays faster for $a=2\/3$ than the ideal-MHD limit $a=1$ case, whereas for $a=2$ it initially decays more slowly and then decays asymptotically at the same rate as for the ideal-MHD $a=1$ case; the evolution of the fluid energy density $\\widetilde{e}$ decays more quickly with decreasing $\\kappa$.\n\n\\begin{figure}[htb]\n\\begin{minipage}[htb]{5cm}\n\\centerline{\\epsfig{figure=fig3.pdf,width=7.5cm}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[htb]{5cm}\n\\centerline{\\epsfig{figure=fig4.pdf,width=7.5cm}}\n\\end{minipage}\n\\hfill\n\\centering\n\\begin{minipage}[htb]{5cm}\n\\centerline{\\epsfig{figure=fig5.pdf,width=7.5cm}}\n\\end{minipage}\n\\caption{Evolution of fluid energy density $e\/e_0$ as a function of proper time $\\tau$ and we choose initial condition as $\\widetilde{e}_{0}\\equiv\\widetilde{e}(\\tau_{0})=1$. (a)Different lines refer to different levels of longitudinal acceleration parameter: $\\lambda^{*}=0$ (black solid line), $\\lambda^{*}=0.03$ (red dashed line), $\\lambda^{*}=0.06$ (blue dotted line), $\\lambda^{*}=0.1$ (magenta dot-dashed line). Clearly, it is gradually speed up the decay rate of fluid energy density with increasing acceleration parameter $\\lambda^*$. (b)\nDifferent lines refer to different levels of magnetic field decay parameter: $a=2$ (black solid line), $a=1$ (red dashed line), and $a=2\/3$ (blue dotted line). Clearly, the fluid energy density decreases more rapidly for $a=2\/3$ than in the case $a=1$, not to mention $a=2$. (c)Different lines refer to the evolution for $\\kappa=1$ (black solid line), $\\kappa=3$ (red dashed line), $\\kappa=7$ (blue dotted line), and $\\kappa=10$ (magenta dot-dashed line).\n}\n\\label{fig2}\n\\end{figure}\n\nIn Fig.\\ref{fig3}, we consider the evolution of the fluid energy density $\\widetilde{e}$ (upper panel) and the total energy density $e\/e_{0}+\\sigma_{0}(B\/B_{0})^{2}\/2$ (lower panel) in the different cases and when the parameters are set to $a=2\/3$ (left panel) and $a=2$ (right panel). Left panel report the evolution of fluid energy density and the total energy density $e\/e_{0}+\\sigma_{0}(B\/B_{0})^{2}\/2$ for $a=2\/3$ and where different lines refer to different levels of the initial magnetization: $\\sigma_0=0$ (black solid line), $\\sigma_0=0.5$ (red dashed line), $\\sigma_0=1.0$ (blue dotted line), and $\\sigma_0=2$ (magneto dot-dashed line). It is clear that larger values of initial magnetization $\\sigma_0$ will lead to a faster decrease in $\\widetilde{e}$ and $e\/e_{0}+\\sigma_{0}(B\/B_{0})^{2}\/2$. Right panel shows the evolution of fluid energy density $\\widetilde{e}$ and the total energy density $e\/e_{0}+\\sigma_{0}(B\/B_{0})^{2}\/2$ in the case $a=2$. In this case, different lines refer to different levels of the initial magnetization, $\\sigma_0=0.01$ (black solid), $\\sigma_0=1.0$ (red dashed), and $\\sigma_0=10$ (blue dotted). As shown in the Fig.~\\ref{fig3} (upper-right panel), it produces even a temporary increase in the fluid energy density evolution. This interesting phenomena, which can be associated with the resistive \"heating up\" of the fluid, and it depends on the values of the initial magnetization $\\sigma_0$ and the magnetic field decay parameter $a$. This increase in the fluid energy density evolution will be larger for larger magnetic field decay parameter $a$ due to the fact that the Lorentz force allows energy to transfer back and forth between the magnetic field and the fluid. The total energy density of this system decays quickly with increasing $\\sigma_0$ for $a<1$. Increasing $\\sigma_0$ only adds energy density to the system, but does not alter the temporal evolution of the total energy density for the case with $a>1$.\n\n\\begin{figure}[htb]\n\\begin{minipage}[htb]{8.8cm}\n\\centerline{\\epsfig{figure=fig6.pdf,width=8.5cm}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[htb]{8.8cm}\n\\centerline{\\epsfig{figure=fig7.pdf,width=8.5cm}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[htb]{8.8cm}\n\\centerline{\\epsfig{figure=fig8.pdf,width=8.5cm}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[htb]{8.8cm}\n\\centerline{\\epsfig{figure=fig9.pdf,width=8.5cm}}\n\\end{minipage}\n\\caption{The evolution of the fluid energy density $e\/e_0$ (upper panel) and the total energy density $e\/e_{0}+\\sigma_{0}(B\/B_{0})^{2}\/2$ (lower panel) in the different cases and when the parameters are set to $a=2\/3$ (left panel) and $a=2$ (right panel). (Left panel) Different lines refer to different levels of the initial magnetization: $\\sigma=0$ (black solid line), $\\sigma=0.5$ (red dashed line), $\\sigma=1.0$ (blue dotted line), and $\\sigma=2.0$ (magenta dot-dashed line). (Right panel) Different lines refer to different levels of the initial magnetization, ranging from $\\sigma_0=0.01$ (black solid line), $\\sigma_0=1.0$ (red dashed line), and $\\sigma_0=10.0$ (blue dotted line).}\n\\label{fig3}\n\\end{figure}\n\n\\section{discussion and conclusions}\n\\label{section4}\n\nWe have investigated the evolution of the energy density of the QGP generated by the non-central heavy ion collisions by one-dimensional MHD flow in the limit of infinite electrical conductivity with longitudinal acceleration parameter $\\lambda^*$ and got an exact solution under the CNC approximation. Compared with Bjorken-Victor type flow, the longitudinal acceleration effect accelerates the decay of the energy density. For larger $\\kappa$ of EoS, the energy density decays more slowly, thus the temperature dependent EoS should be calculated from lattice QCD simulations.\n\nBased on the definition of the acceleration coordinate (Rindler coordinate, Kottler-M\\o ller coordinates, and Radar coordinates), the \"acceleration parameter\" $\\lambda$ has following physics meaning: (1)$\\lambda<0$, for heavy ion collisions, it means that the fireball system's element flowing into the fireball's core and the system's thermodynamics quantities density is increasing with the time, or in other words, the fireball system does not swell but contracts, which means after enough long time, there will creating a black holw; (2)$\\lambda=0$ correspond to the rest fireball system; (3)$0<\\lambda<1$, the fireball system's expansion speed is decelerating. The energy density deposit to large $\\eta_s$; (4)$\\lambda=1$, the fireball system's expansion speed is average; (5)$\\lambda>1$, the fireball system's expansion is fast and many energy density deposit to the mid-rapidity $\\eta_s$, which is consistent with the experimental data. Thus, we only focus on the case that longitudinal acceleration parameter $\\lambda^*$ is greater than $0$ in the previous discussion.\n\nFor the case that the magnetic field evolution follows a power-law decay in proper time with exponent $a$, we find\nthe magnetic field decays more quickly than in the ideal-MHD case for $a>1$, while the magnetic field with $a<1$ correspond to a decay that is slower than in the ideal-MHD limit.\nIn heavy-ion collisions the remnants of colliding nuclei can give an additional contribution to the magnetic field to slow down its decay.\nThus, considering the case $a<1$ is reasonable in this paper.\nIt is clearly that larger values of initial magnetization $\\sigma_0$ leads to faster decreasing in $\\widetilde{e}$ for $a<1$. But it also results in a temporary increase in the fluid energy density for $a>1$. As we know, the magnetic field energy can be converted to fluid energy via Lorzent force, thus the evolution of the fluid energy density becomes more complex. For $a\\rightarrow0$, the magnetic field is constant in proper time and does not evolve with the fluid.\nThus, the fluid energy density must decay very rapidly to keep this constant magnetic field.\nFor $a\\rightarrow\\infty$, the magnetic field decays fast and the energy is transferred to the fluid-element according to the energy-conservation law.\nThus, one can expect a peak of the energy density near the initial time, which is associated with a \"reheating\" of the fluid with longitudinal acceleration effect.\n\nHowever, the recent estimates both from lattice QCD simulations ~\\cite{Alessandro2013,Amato2013,Greif2014} and fitting of experimental data point~\\cite{Yi Yin2014} toward high, but finite value for the electrical conductivity of the QGP. For a quantitative comparison with experimental data, the effects of the electrical resistivity has to be taken into account.\n\nAs a next step, we try to include the dissipative effects (shear and bulk viscosity and a finite electric conductivity), the rescatterings in the hadronic phase, the decays of hadronic resonance into stable hadrons and anomalous currents. Note it would be necessary to modify the Cooper-Frye formula by taking into account the presence of an electromagnetic field.\n\n\\begin{acknowledgements}\nWe specially thank Dirk H. Rischke for the useful suggestion about the MHD theory at the ATHIC2018. This work is in part supported by the Ministry of Science and Technology of China (MSTC) under the \"973\" Project No. 2015CB856904(4), by NSFC Grant Nos. 11735007, 11890711\nThis work was supported by the Sino-Hungarian bilateral cooperation program, under the Grand No.Te'T 12CN-1-2012-0016, by the financial supported from NNSF of China under grant No.11435004. Z-F. Jiang would like to thank T.~Cs\\\"org\\H{o}, M.~Csan\\'ad, L{\\'e}vai P{\\'e}ter and Gergely G{\\'a}bor Barnafoldi for kind hospitality during his stay at Winger RCP, Budapest, Hungary.\n\n\n\\end{acknowledgements}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nAt present, time invariance violation has only been observed\nindirectly,\nin the CP-violating decay\nof neutral K-mesons \\cite{CPv}. Parity and\ntime invariance violating nuclear and\natomic multipole moments, such\nas the magnetic monopole, electric dipole, magnetic quadrupole,\nand electric octupole moments, are interesting\nbecause, if discovered,\nthey would provide further proof of time invariance violation.\nEven the limits on these moments provide important tests of\ndifferent models of CP-violation.\nParity and time invariance violating nuclear moments induced by\nT- and P-odd nuclear forces have been discussed, e.g., in\nRefs. \\cite{Fein77,Cov83,Hax83,SFK84,FKS86,Khripl}.\n\nIn this paper we consider the electric octupole\nmoments of nuclei. We calculate the EOMs of odd $Z$ nuclei that\nare induced by a T- and P-odd interaction\nbetween the unpaired proton\nand the nuclear core. Note that nuclei with unpaired\nneutrons can\nhave an EOM of comparable magnitude due to the polarization of the\nnuclear core by the T-, P-odd field of the external neutron. A\nsimilar mechanism for the electric dipole and Schiff moments was\nconsidered in Ref. \\cite{FKS86}.\n\nWe give values of the collective electric\noctupole moments of nuclei having a static octupole \ndeformation, using the calculations done\nin Refs. \\cite{AFS96,AFS2}. We also present a\ncalculation of the atomic electric dipole moment (EDM) \nthat would be induced by a nuclear EOM. Finally, we discuss\na possible enhancement mechanism for\nmagnetic quadrupole moments (MQMs) in nuclei with\noctupole deformation.\n\nIn the appendix we present simple estimates of\nthe relative sizes of the contributions\nof various nuclear moments to the atomic EDM. The magnitude of\nthe nuclear EOM is comparable to that of the Schiff moment.\nHowever, the contribution of the EOM to the atomic EDM is\nsmaller than the contribution of the Schiff moment (and the\nmagnetic quadrupole moment) since the EOM interacts with\nhigher angular momentum electron states, whose wave functions\nare suppressed near the nucleus. \n\n\\section{The Hamiltonian of the T-, P-odd nucleon-nucleus\ninteraction and the resulting nucleon wave function}\n\\label{shptnw}\nFor a heavy nucleus, the T- and P-odd interaction between a\nnonrelativistic unpaired nucleon and the nuclear core can be \ndescribed\nby the following effective Hamiltonian (see, e.g.,\nRefs. \\cite{Henley,SFK84,Khripl}):\n\\begin{equation}\nH_{TP} = \\eta \\frac{G}{2 \\sqrt{2} m} \\bbox{\\sigma} \\cdot\n\\bbox{\\nabla} \\rho,\n\\label{ehpta}\n\\end{equation}\nwhere $\\bbox{\\sigma}$ is twice the spin operator for this nucleon,\n$\\rho$ is the density of the nuclear core,\n$G = 1.0 \\times 10^{-5}\/{m_p}^2$ is the Fermi constant,\n$m$ is the mass of the nucleon and $\\eta$ is a dimensionless\nconstant that describes the strength of the interaction.\nIn this paper we\ndeal with odd $Z$ nuclei, and so the nucleon involved is the\nunpaired proton.\n\nLet $U$ be the strong nuclear potential of the core that the \nunpaired proton moves in. The range of\nthe strong nucleon-nucleon interaction\nis small. This means that the potential $U({\\bf r})$ and \nthe nuclear density $\\rho ({\\bf r})$ will be similar in shape.\nIn fact, we assume\nthat they are approximately proportional: $U({\\bf r})\/U({\\bf 0})\n\\approx \\rho({\\bf r})\/\\rho({\\bf 0})$. This allows us to rewrite\nEq.\\ (\\ref{ehpta}) in a form that makes it\neasy to find the perturbed\nwave function due to $H_{TP}$ (we use the method of\nRef. \\cite{SFK84}). We obtain\n\\begin{equation}\nH_{TP} \\approx \\xi \\bbox{\\sigma} \\cdot \\bbox{\\nabla} U,\n\\label{ehptb}\n\\end{equation}\nwhere\n\\begin{equation}\n\\xi = \\eta \\frac{G}{2 \\sqrt{2} m_p} \\frac{\\rho ({\\bf 0})}\n{U ({\\bf 0})} = -2 \\times 10^{-21} \\eta \\mbox{ cm}.\n\\label{ehptc}\n\\end{equation}\nThe total potential that the unpaired proton experiences is\n\\begin{equation}\n\\widetilde{U} = U + H_{TP} \\approx U({\\bf r})\n+ \\xi \\bbox{\\sigma} \\cdot \\bbox{\\nabla} U \\approx\nU({\\bf r} + \\xi \\bbox{\\sigma}).\n\\label{ehptd}\n\\end{equation}\nAs a result, if $\\psi ({\\bf r})$ is the proton's wave function\nwhen only\n$U({\\bf r})$ is present, the perturbed wave function will be\n\\begin{equation}\n\\widetilde{\\psi} ({\\bf r}) \\approx \\psi ({\\bf r} + \\xi\n\\bbox{\\sigma})\n\\approx \\psi({\\bf r}) + \\xi \\bbox{\\sigma} \\cdot \\bbox{\\nabla}\n\\psi({\\bf r}).\n\\label{ehpte}\n\\end{equation}\n\n\\section{The electric octupole moment of a nucleus with\nan unpaired proton}\n\\label{sneom}\nIn this section we calculate the electric octupole moment\n(EOM) of a\nnucleus with an unpaired proton using\nthe perturbed wave function obtained above.\nThe electric octupole moment can be written as\n[see Eq. (\\ref{eneo8caa})]\n\\begin{equation}\nO_{ijk} = \\langle \\widetilde{\\psi} | \\hat{O}_{ijk} |\n\\widetilde{\\psi} \\rangle\n= e \\int \\widetilde{\\psi}^{\\dagger}({\\bf r})\n[r_i r_j r_k\n- {\\textstyle \\frac{1}{5}} r^2 (r_i \\delta_{jk} + r_j \\delta_{ik}\n+ r_k \\delta_{ij})] \\widetilde{\\psi} ({\\bf r}) \\, d^3 r.\n\\label{eaa}\n\\end{equation}\nNote that we use $e > 0$.\nSubstituting $\\widetilde{\\psi} ({\\bf r}) =\n\\psi ({\\bf r}) + \\xi \\sigma_m \n\\frac{\\partial \\psi}{\\partial r_m}$ (from Eq.\\ (\\ref{ehpte}))\ninto the above integral and expanding gives\n\\begin{equation}\nO_{ijk} = 2 e \\xi \\int \\frac{\\partial \\psi^{\\dagger}}{\\partial r_m}\n\\sigma_m [r_i r_j r_k - \n{\\textstyle \\frac{1}{5}} r^2 (r_i \\delta_{jk} + r_j \\delta_{ik}\n+ r_k \\delta_{ij})] \\psi \\, d^3 r.\n\\label{eneom1a}\n\\end{equation}\nNotice that we have discarded the $\\xi^2$ term\nand that the term not\ncontaining $\\xi$ vanishes, as it is an integral of an odd function\nof ${\\bf r}$. Applying integration by parts to the above\nintegral gives \n\\begin{eqnarray}\nO_{ijk} & =  - {\\textstyle \\frac{1}{5}} e \\xi \\langle \\psi\n| & \n5 (\\hat{r}_i \\hat{r}_j \\hat{\\sigma}_k +\n\\hat{r}_i \\hat{r}_k \\hat{\\sigma}_j +\n\\hat{r}_j \\hat{r}_k \\hat{\\sigma}_i)\n-{\\hat{r}}^2 (\\hat{\\sigma}_i \\delta_{jk} +\n\\hat{\\sigma}_j \\delta_{ik}\n+ \\hat{\\sigma}_k \\delta_{ij}) \\nonumber \\\\\n&& - 2 \\hat{r}_m \\hat{\\sigma}_m\n(\\hat{r}_i \\delta_{jk} + \\hat{r}_j \\delta_{ik} + \\hat{r}_k\n\\delta_{ij})\n|\n\\psi \\rangle.\n\\label{eac}\n\\end{eqnarray}\n\nWe can also write the octupole moment tensor in another form.\nSince it is a symmetric, irreducible (traceless) third rank tensor\nand the nuclear angular momentum ${\\bf I}$ is the only \nquantity which defines\na direction in the system, the EOM tensor must be in the form\nof the most general symmetric, irreducible third rank tensor that\ncan be formed from the components of $\\hat{\\bf I}$. That is\n\\begin{equation}\nO_{ijk} = \\langle \\widetilde{\\psi} | \\hat{O}_{ijk} |\n\\widetilde{\\psi} \\rangle ,\n\\label{ead}\n\\end{equation}\nwhere\n\\begin{eqnarray}\n\\hat{O}_{ijk} & = A [ & \\hat{I}_i \\hat{I}_j \\hat{I}_k +\n\\hat{I}_j \\hat{I}_k \\hat{I}_i + \\hat{I}_k \\hat{I}_i \\hat{I}_j\n+ \\hat{I}_k \\hat{I}_j \\hat{I}_i + \\hat{I}_j \\hat{I}_i \\hat{I}_k\n+ \\hat{I}_i \\hat{I}_k \\hat{I}_j \\nonumber \\\\\n&& - {\\textstyle \\frac{6I(I+1)-2}{5}}\n(\\hat{I}_i \\delta_{jk} + \\hat{I}_j \\delta_{ik}\n+ \\hat{I}_k \\delta_{ij} )]\n\\label{eae}\n\\end{eqnarray}\nand $A$ is some constant. The factor of $-[6I(I+1)-2]\/5$ follows\nfrom the requirement\nof tracelessness ($O_{iij}=O_{iji}=O_{jii}=0$).\nThe quantity which is usually referred to as the octupole moment\nis ${\\cal O}$, which is the $O_{zzz}$ component for the nuclear\nstate having\nangular momentum projection $I_z = I$. This is the quantity\nwhich we will calculate. We can write $A$ in terms of ${\\cal O}$\nusing Eqs.\\ (\\ref{ead}) and (\\ref{eae}), with $i=j=k=z$. This gives\n\\begin{eqnarray}\n\\hat{O}_{ijk} & =\n\\frac{5 {\\cal O}}{6I(I-1)(2I-1)}\n[ & \\hat{I}_i \\hat{I}_j \\hat{I}_k +\n\\hat{I}_j \\hat{I}_k \\hat{I}_i + \\hat{I}_k \\hat{I}_i \\hat{I}_j\n+ \\hat{I}_k \\hat{I}_j \\hat{I}_i + \\hat{I}_j \\hat{I}_i \\hat{I}_k\n+ \\hat{I}_i \\hat{I}_k \\hat{I}_j \\nonumber \\\\\n&& - {\\textstyle \\frac{6I(I+1)-2}{5}}\n(\\hat{I}_i \\delta_{jk} + \\hat{I}_j \\delta_{ik}\n+ \\hat{I}_k \\delta_{ij} )].\n\\label{eaf}\n\\end{eqnarray}\nWe now have two expressions for $O_{ijk}$: Eq.\\ (\\ref{eac}) and\nEq.\\ (\\ref{eaf}) (via Eq.\\ (\\ref{ead})). We calculate ${\\cal O}$ by\noperating on both of these equations with $\\hat{I}_i \\hat{I}_j$ on\nthe left and $\\hat{I}_k$ on the right and equating the results.\n\nTo evaluate the results of these operations, the following\ncommutation\nrelations are required: $[\\hat{r}_i,\\hat{\\sigma}_j] = 0$,\n$[\\hat{I}_i,\\hat{r}_j] = i \\varepsilon_{ijk} \\hat{r}_k$, and\n$[\\hat{I}_i,\\hat{\\sigma}_j] = i \\varepsilon_{ijk} \\hat{\\sigma}_k$.\n(Note that $\\hat{I}_i = \\hat{l}_i + \\hat{\\sigma}_i \/ 2$.)\nOther useful relations are\n$[\\hat{I}_i,\\hat{I}_j \\hat{r}_j] = [\\hat{I}_i,\\hat{I}_j\n\\hat{\\sigma}_j]\n= [\\hat{I}_i,\\hat{\\sigma}_j \\hat{r}_j] = 0$ and\n$\\varepsilon_{ijk} A_i A_j = {\\textstyle \\frac{1}{2}}\n\\varepsilon_{ijk} A_i A_j\n- {\\textstyle \\frac{1}{2}} \\varepsilon_{ijk} A_j A_i\n= {\\textstyle \\frac{1}{2}} \\varepsilon_{ijk} [A_i,A_j]$,\nfor an operator $A_i$.\nThe commutation relations\nare used to rearrange the operators so that pairs\nwith the same indices are adjacent.\n\nApplying the operators to\nEq. (\\ref{eac}) gives\n\\begin{eqnarray}\n\\langle \\widetilde{\\psi} | \\hat{I}_i \\hat{I}_j \\hat{O}_{ijk}\n\\hat{I}_k | \\widetilde{\\psi} \\rangle\n& = -{\\textstyle \\frac{1}{5}} e \\xi \\langle \\psi | &\n5 \\hat{I}_i \\hat{I}_j (\\hat{r}_i \\hat{r}_j \\hat{\\sigma}_k +\n\\hat{r}_i \\hat{r}_k \\hat{\\sigma}_j + \\hat{r}_j \\hat{r}_k\n\\hat{\\sigma}_i)\n\\hat{I}_k\n- \\hat{I}_i \\hat{I}_j \\hat{r}^2 (\\hat{\\sigma}_i \\delta_{jk} +\n\\hat{\\sigma}_j \\delta_{ik}\n+ \\hat{\\sigma}_k \\delta_{ij}) \\hat{I}_k \\nonumber \\\\\n&& - 2 \\hat{I}_i \\hat{I}_j \\hat{r}_m \\hat{\\sigma}_m \n(\\hat{r}_i \\delta_{jk} + \\hat{r}_j \\delta_{ik} + \\hat{r}_k\n\\delta_{ij})\n\\hat{I}_k | \\psi \\rangle \\nonumber \\\\\n& = -{\\textstyle \\frac{1}{5}} e \\xi \\langle r^2 \\rangle \\{ &\n\\langle I,I_z,l | (\\hat{\\bbox{\\sigma}} \\cdot {\\hat{\\bf n}})\n[ {\\textstyle \\frac{5}{4}} (\\hat{\\bf I} \\cdot \\hat{\\bbox{\\sigma}})\n- \\hat{I}^2 ] (\\hat{\\bbox{\\sigma}} \\cdot {\\hat{\\bf n}})\n| I,I_z,l \\rangle \\nonumber \\\\\n&&\n+ \\langle I,I_z,l | {\\textstyle \\frac{7}{2}} (\\hat{\\bf I} \\cdot\n\\hat{\\bbox{\\sigma}})\n- 3 \\hat{I}^2 (\\hat{\\bf I} \\cdot \\hat{\\bbox{\\sigma}}) + 1\n- 2 \\hat{I}^2 | I,I_z,l \\rangle \\},\n\\label{eal}\n\\end{eqnarray}\nwhere $\\hat{\\bf n} = \\hat{\\bf r} \/ r$.\nHere the facts that\n$\\hat{\\bf I} \\cdot \\hat{\\bf n} = (\\hat{\\bf l} +\n\\hat{\\bbox{\\sigma}}\/2)\n\\cdot \\hat{\\bf n} = (\\hat{\\bbox{\\sigma}} \\cdot \\hat{\\bf n})\/2$\nand $(\\hat{\\bbox{\\sigma}} \\cdot \\hat{\\bf n})^2 = 1$ \\cite{QMNRT}\nwere used. \nThe radial and angular parts of the\noperators have been separated in Eq.\\ (\\ref{eal});\n$\\langle r^2 \\rangle$\nis the expectation value of $r^2$ and $| I,I_z,l \\rangle$ is the\nangular part of $| \\psi \\rangle$.\nThe second matrix element in Eq.\\ (\\ref{eal}) is\n\\begin{equation}\n\\langle I,I_z,l | {\\textstyle \\frac{7}{2}}\n(\\hat{\\bf I} \\cdot \\hat{\\bbox{\\sigma}})\n- 3 \\hat{I}^2 (\\hat{\\bf I} \\cdot \\hat{\\bbox{\\sigma}}) + 1\n- 2 \\hat{I}^2 | I,I_z,l \\rangle\n= [{\\textstyle \\frac{7}{2}} - 3I(I+1)]\n({\\textstyle \\frac{1}{2}} - \\kappa) + 1 -2I(I+1),\n\\label{eam}\n\\end{equation}\nwhere we have used the fact that\n\\begin{eqnarray}\n(\\hat{\\bf I} \\cdot \\hat{\\bbox{\\sigma}}) | I,I_z,l \\rangle & =\n& [I(I+1) - l(l+1) + 3\/4] | I,I_z,l \\rangle \\nonumber \\\\\n& = & (1\/2 - \\kappa) | I,I_z,l \\rangle,\n\\label{ean}\n\\end{eqnarray}\nwith\n\\begin{equation}\n\\kappa = (I + 1\/2) (-1)^{I+1\/2-l}.\n\\label{eao}\n\\end{equation}\nTo evaluate the first matrix element in Eq.\\ (\\ref{eal}), we use\nthe following identity:\n\\begin{equation}\n(\\hat{\\bbox{\\sigma}} \\cdot \\hat{{\\bf n}}) | I,I_z,l \\rangle\n= - | I,I_z,\\widetilde{l} \\rangle,\n\\label{eap}\n\\end{equation}\nwhere $\\widetilde{l} = 2 I - l$.\nWe then have\n\\begin{eqnarray}\n\\langle I,I_z,l | (\\hat{\\bbox{\\sigma}} \\cdot {\\hat{\\bf n}})\n[ {\\textstyle \\frac{5}{4}} (\\hat{\\bf I} \\cdot \\hat{\\bbox{\\sigma}})\n- \\hat{I}^2 ] (\\hat{\\bbox{\\sigma}} \\cdot {\\hat{\\bf n}})\n| I,I_z,l \\rangle & = &\n\\langle I,I_z,\\widetilde{l} | {\\textstyle \\frac{5}{4}} (\\hat{\\bf I}\n\\cdot \\hat{\\bbox{\\sigma}})\n- \\hat{I}^2 | I,I_z,\\widetilde{l} \\rangle \\nonumber \\\\\n& = & {\\textstyle \\frac{5}{4}} ({\\textstyle \\frac{1}{2}}\n+ \\kappa) - I(I+1).\n\\label{eaq}\n\\end{eqnarray}\n(Note that $(\\hat{\\bf I} \\cdot \\hat{\\bbox{\\sigma}})\n| I,I_z,\\widetilde{l} \\rangle = (1\/2 + \\kappa)\n| I,I_z,\\widetilde{l} \\rangle$.)\nUsing these results in Eq.\\ (\\ref{eal}) gives\n\\begin{equation}\n\\langle \\widetilde{\\psi} | \\hat{I}_i \\hat{I}_j \\hat{O}_{ijk}\n\\hat{I}_k\n| \\widetilde{\\psi} \\rangle = -{\\textstyle \\frac{3}{5}} e \\xi\n\\langle r^2 \\rangle\n(\\kappa - {\\textstyle \\frac{3}{2}})\n(I-{\\textstyle \\frac{1}{2}})(I+{\\textstyle \\frac{3}{2}}).\n\\label{ear}\n\\end{equation}\nNow we find another form of the left hand side of the above\nequation by operating on Eq.\\ (\\ref{eaf}). We have\n\\begin{eqnarray}\n\\langle \\widetilde{\\psi} | \\hat{I}_i\n\\hat{I}_j \\hat{O}_{ijk} \\hat{I}_k\n| \\widetilde{\\psi} \\rangle\n& =\n\\frac{5 {\\cal O}}{6I(I-1)(2I-1)}\n\\langle \\widetilde{\\psi} |\n& \\hat{I}_i \\hat{I}_j (\\hat{I}_i \\hat{I}_j \\hat{I}_k +\n\\hat{I}_j \\hat{I}_k \\hat{I}_i + \\hat{I}_k \\hat{I}_i \\hat{I}_j\n+ \\hat{I}_k \\hat{I}_j \\hat{I}_i + \\hat{I}_j \\hat{I}_i \\hat{I}_k\n+ \\hat{I}_i \\hat{I}_k \\hat{I}_j) \\hat{I}_k \\nonumber \\\\\n&& - {\\textstyle \\frac{6I(I+1)-2}{5}} \\hat{I}_i \\hat{I}_j \n(\\hat{I}_i \\delta_{jk} + \\hat{I}_j \\delta_{ik}\n+ \\hat{I}_k \\delta_{ij} ) \\hat{I}_k | \\widetilde{\\psi} \\rangle.\n\\label{eas}\n\\end{eqnarray}\nThis can be evaluated using the equation\n$[\\hat{I}_i,\\hat{I}_j] = i \\varepsilon_{ijk} \\hat{I}_k$ to\nbring operators with the same indices together\nand $\\varepsilon_{ijk} \\hat{I}_i \\hat{I}_j =\n\\frac{i}{2}\n\\varepsilon_{ijk} \\varepsilon_{ijp} \\hat{I}_p = i \\hat{I}_k$.\nThe result is\n\\begin{equation}\n\\langle \\widetilde{\\psi} |\n\\hat{I}_i \\hat{I}_j \\hat{O}_{ijk} \\hat{I}_k\n| \\widetilde{\\psi} \\rangle = {\\cal O} (I+1)(I+3\/2)(I+2).\n\\label{eat}\n\\end{equation}\nEquating this result with Eq.\\ (\\ref{ear}) gives the following\nresult for the octupole moment:\n\\begin{equation}\n{\\cal O}_{\\rm sing} = \\frac{-3(\\kappa-3\/2)(I-1\/2)}\n{5(I+1)(I+2)} \\langle r^2 \\rangle e \\xi,\n\\label{eau}\n\\end{equation}\n(The subscript refers to the fact that this octupole moment is\ndue to a single particle, as opposed to a collective octupole\nmoment.)\nThe expectation value for $r^2$ can be\napproximated as \\cite{SFK84}\n\\begin{equation}\n\\langle r^2 \\rangle\n\\approx {\\textstyle \\frac{3}{5}} {r_0}^2 A^{2\/3},\n\\label{eaw}\n\\end{equation}\nto an accuracy of about 10\\%,\nwhere $r_0 = 1.1 \\mbox{ fm}$ and $A$ is the mass number of\nthe nucleus. Using the value of $r_0$ and Eqs.\\ (\\ref{ehptc}) and\n(\\ref{eao}) gives\n\\begin{equation}\n{\\cal O_{\\rm sing}} \\approx 8.7 \\times 10^{-9} A^{2\/3}\n\\eta e ({\\rm fm})^3 \\times\n\\left\\{\n\\begin{array}{ll}\n\\frac{-(I-1\/2)}{I+1} & \\mbox{for $I=l+1\/2$} \\\\\n\\frac{(I-1\/2)(I-1)}{(I+1)(I+2)} & \\mbox{for $I=l-1\/2$}\n\\end{array} \\right..\n\\label{eaw2}\n\\end{equation}\nObserve that ${\\cal O}_{\\rm sing}=0$ for $I=1\/2$. This\nis to be expected because\n$\\hat{O}_{ijk}$ is a third rank tensor, and so applying the\ntriangle rule for the addition of angular momenta\nto Eq.\\ ({\\ref{ead})\ngives the result that nuclei having angular momentum less\nthan $3\/2$ cannot have an octupole moment.\nValues of ${\\cal O}_{\\rm sing}$ for various nuclei\nare given in table \\ref{tsingp}, in terms of the parameter $\\eta$.\n  \n\\section{Collective electric octupole moments in nuclei with static\noctupole deformation}\n\\label{sceom}\nIn Refs.\\ \\cite{AFS96,AFS2}, a mechanism was suggested by which\nparity and time invariance\nviolating interactions can produce collective T- and P-odd\nmultipole moments in \neven-odd nuclei having a static octupole deformation\n[i.e. electric octupole moments in their intrinsic\n(or body-fixed) reference frames].\nSuch a deformation has been shown to exist for nuclei in the\nRa--Th and Ba--Sm regions (for a review see, e.g., \\cite{Ahmad93}).\nA similar\nmechanism, for the enhancement of the intrinsic electron EDM and\nother T-, P-odd interactions in polar molecules, was suggested in\nRef. \\cite{SF1978}.\nBelow, we explain the mechanism by which a collective nuclear\nEOM can be produced and provide values of this EOM for various\nnuclei.\n\n\\subsection{Parity and time invariance and octupole moments in\nthe laboratory frame}\nAn electric octupole moment can exist in the nucleus's intrinsic\nframe without parity or time invariance violation. Yet if parity\nand time invariance hold, the expectation value of the octupole\nmoment in the laboratory reference frame will be zero.\n\nConsider\n$|I M K \\rangle$ and $|I M -K \\rangle$, which are two almost\ndegenerate states of the nucleus in the laboratory\nframe. $I$ is the angular momentum of the nucleus,\n$M$ is its projection onto the $z$-axis, and $K$\nis its projection onto\nthe axis of symmetry of the deformed nuclear core\n(the $z^{\\prime}$-axis).\n(${\\bf I}$ is the sum of the unpaired nucleon's angular\nmomentum, ${\\bf j}$ and the nuclear core's orbital angular\nmomentum, ${\\bf R}$.)\nThese states can be written in terms of intrinsic states as\n\\begin{equation}\n|I M \\pm K \\rangle =\n\\sqrt{\\textstyle \\frac{2I+1}{4 \\pi}}\n D^I_{M \\pm K} (\\phi,\\theta,0) \\psi_{\\pm K}\n({\\bf r}^\\prime) \\chi_{\\rm core},\n\\label{ecn21a}\n\\end{equation}\nwhere $D^I_{M \\pm K} (\\phi,\\theta,0)$ is a Wigner $D$-function\n(see, e.g., \\cite{Varshalovich,RelQuant}),\n$\\chi_{\\rm core}$ is the wave function of the nuclear core\nin the intrinsic frame, and\n$\\psi_{\\pm K} ({\\bf r}^\\prime)$ is the wave function of the unpaired\nnucleon in the intrinsic frame, with\na $z^{\\prime}$ angular momentum projection of $\\pm K$. (In the\nintrinsic frame, the nuclear axis plays the role of the usual\n``$z$-axis'' in Quantum Mechanics.) Note that ${\\bf j}$ and\n${\\bf I}$ have the same $z^{\\prime}$ projection.\n\n$|I M K \\rangle$ and $|I M -K \\rangle$ do not\nhave good parity, as $K$\nchanges sign under a parity transformation. However, the following\nstates do, and they form a parity doublet:\n\\begin{equation}\n\\psi^{\\pm} = \\frac{1}{\\sqrt{2}} ( |I M K \\rangle\n\\pm |I M -K \\rangle).\n\\label{eceom1a}\n\\end{equation}\nFor these good parity states \n$\\langle \\psi^{\\pm} | {\\bf I} \\cdot {\\bf n} | \\psi^{\\pm}\n\\rangle = 0$\nbecause $K$ and $-K$ have equal probabilities\nand this means that there is no\naverage orientation of the nuclear axis in the laboratory frame\n($\\langle \\psi^{\\pm} | {\\bf n} | \\psi^{\\pm} \\rangle = {\\bf 0}$).\nThis is a consequence of time invariance and parity\nconservation since\nthe correlation ${\\bf I} \\cdot {\\bf n}$ is T-, P-odd. As a result\nof $\\langle \\psi^{\\pm} | {\\bf n} | \\psi^{\\pm} \\rangle = {\\bf 0}$,\nthe mean value of the octupole moment\n(whose orientation is determined by the direction of the\nnuclear axis)\nis zero in the laboratory frame.  \n\nNow, a T- and P-odd interaction, $H_{TP}$\nwill mix the members of the parity doublet ($\\psi^+$ and $\\psi^-$).\nThe admixed wave function of the predominantly positive parity\nmember of the doublet will be $\\psi = \\psi^+ + \\alpha \\psi^-$ or\n\\begin{equation}\n\\psi = \\frac{1}{\\sqrt{2}} [(1+\\alpha) |I M K \\rangle\n+ (1-\\alpha) |I M -K \\rangle],\n\\label{eceom2c}\n\\end{equation}\nwhere $\\alpha$ is a mixing coefficient:\n\\begin{equation}\n\\alpha = \\frac{\\langle \\psi^- | H_{TP} |\\psi^+ \\rangle}{E_+ - E_-}.\n\\label{enad32}\n\\end{equation}\n$E_+ - E_-$ is the energy splitting between the members of the\nparity doublet.\nThe interaction $H_{TP}$ is given by Eq.\\ (\\ref{ehpta}).\n(A similar expression can be obtained for the predominantly\nnegative parity member of the doublet.)\nThis mixing yields, on average,\nan orientation of the nuclear axis along the direction of the\nangular momentum:\n\\begin{equation}\n\\langle \\psi | {\\bf I} \\cdot {\\bf n} | \\psi \\rangle\n= \\langle \\psi | \\hat{K} | \\psi \\rangle\n= 2 \\alpha K,\n\\label{eceom2d}\n\\end{equation}\nand this means that the octupole moment need no longer vanish in\nthe laboratory frame.\n\n\\subsection{The magnitude of the collective octupole moment}\nThe collective EOM in the laboratory frame\nwas derived in \\cite{AFS96,AFS2},\nwith the following result:\n\\begin{equation}\n{\\cal O}_{\\rm coll} \\approx \\frac{4}{5}\n\\frac{I (I-1) (I - 1\/2)}{(I+1)\n(I+2) (I + 3\/2)}\n\\alpha O_{3,{\\rm intr}}.\n\\label{eceom1}\n\\end{equation}\nOnce again, observe that ${\\cal O}_{\\rm coll} = 0$ for $I=1\/2$.\n$O_{3,{\\rm intr}}$ refers to the octupole moment in the\nintrinsic frame ($ O_{zzz} \\equiv \\frac{2}{5} O_3$)\nand is given by \\cite{BohrMott,LeanderChen}:\n\\begin{equation}\nO_{3,{\\rm intr}} = e Z {R_0}^3 \\frac{3}{2 \\sqrt{7 \\pi}}\n(\\beta_3 + \\frac{2}{3} \\sqrt{\\frac{5}{\\pi}} \\beta_2 \\beta_3 + \n\\frac{15}{11 \\sqrt{\\pi}} \\beta_3 \\beta_4 + \\ldots),\n\\label{eceom2}\n\\end{equation}\nwhere $R_0 = r_0 A^{1\/3}$ ($r_0 = 1.1 \\mbox{ fm}$). \n$\\beta_2$, $\\beta_3$, and $\\beta_4$ are parameters\nthat describe the nuclear\ndeformation; the surface of a deformed nucleus is\n\\begin{equation}\nR = R_0 [1 + \\sum_{l=1}^{\\infty} \\beta_l Y_{l0} (\\theta,\\phi) ].\n\\label{eceom3}\n\\end{equation}\n\nWe will first present an order of magnitude estimate of\n$\\alpha$.\n$\\hat{K} = {\\bf I} \\cdot {\\bf n}$ and $H_{TP}$ are both\nT-, P-odd pseudoscalars. Therefore,\n$\\langle \\psi_{+K} | H_{TP} | \\psi_{+K} \\rangle \\propto K$ and\nso $\\langle \\psi_{-K} | H_{TP} | \\psi_{-K} \\rangle\n= -\\langle \\psi_{+K} | H_{TP} | \\psi_{+K} \\rangle$ (this fact can\nbe easily supported by model calculations). Using this fact\nand Eqs.\\ (\\ref{ecn21a}) and (\\ref{eceom1a}) we get\n$\\langle \\psi^- | H_{TP} | \\psi^+ \\rangle\n= \\langle \\psi_{+K} | H_{TP} | \\psi_{+K} \\rangle$. If\n$\\psi_{+K}$ were a good parity state this matrix element would\nbe zero. However, due to the perturbation caused by the static\noctupole deformation of the nucleus ($V_3$), it is a combination\nof the\nopposite parity spherical orbitals $\\psi_{1,+K}$\nand $\\psi_{2,+K}$ (e.g., $p_{3\/2}$ and $d_{3\/2}$):\n\\begin{equation}\n\\psi_{+K} = \\psi_{1,+K} + \\gamma \\psi_{2,+K},\n\\label{eae11a}\n\\end{equation}\n\\begin{equation}\n\\gamma = \\frac{\\langle \\psi_{2,+K} | V_3 | \\psi_{1,+K}\n\\rangle}{E_1 - E_2},\n\\label{eae11b}\n\\end{equation}\n\\begin{eqnarray}\n\\psi_{1,+K} & = & R_1(r^{\\prime}) \\Omega_{j,l,+K} (\\theta^{\\prime},\n\\phi^{\\prime}), \\nonumber \\\\\n\\psi_{2,+K} & = & R_2(r^{\\prime}) \\Omega_{j,\\widetilde{l},+K}\n(\\theta^{\\prime},\\phi^{\\prime})\n= - R_2(r^{\\prime}) (\\bbox{\\sigma}\n\\cdot {\\bf n}^{\\prime}) \\Omega_{j,l,+K}\n(\\theta^{\\prime},\\phi^{\\prime}),\n\\label{eae11c}\n\\end{eqnarray}\nwhere $\\widetilde{l} = 2j-l$.\n(Of course there will also be an admixture of\nother opposite parity states\nhaving different values of $j$. We neglect these states\nfor simplicity.)\nTherefore, we have\n\\begin{equation}\n\\alpha = \\frac{\\langle \\psi_{+K}|H_{TP}|\\psi_{+K} \\rangle}{E_+-E_-}\n= 2 \\gamma \\frac{\\langle\n\\psi_{1,+K} | H_{TP} | \\psi_{2,+K} \\rangle}{E_+\n- E_-}.\n\\label{ena42}\n\\end{equation}\n\nTo estimate $\\gamma$ we must first derive the form of $V_3$.\nAs in Sec. \\ref{shptnw}, let $U$ be\nthe strong nuclear potential that\nthe unpaired nucleon moves in. $V_3$\nis the difference between the potentials with the octupole\ndeformation present, $U_{\\rm pres}$ and\nabsent, $U_{\\rm abs}$. Once again we have\n$U({\\bf r}^{\\prime}) \\approx U({\\bf 0}) \\rho({\\bf r}^{\\prime}) \/\n\\rho ({\\bf 0})$. We also make the approximation that\n$\\rho({\\bf r}^{\\prime}) \/ \\rho ({\\bf 0}) \\approx\n\\theta (r^{\\prime} - R)$, where $R$ is the nuclear radius and\n$\\theta(x) = 1$ for $x<0$ and $\\theta(x) = 0$ for $x>0$. For an\noctupole deformed nucleus we have $R = R_0 (1 + \\beta_3 Y_{30})$\nand so\n\\begin{equation}\nV_3 = U_{\\rm pres} - U_{\\rm abs}\n\\approx U({\\bf 0}) [\\theta(r^{\\prime}- R_0 - \\beta_3 Y_{30} R_0)\n- \\theta(r^{\\prime} - R_0)]\n\\approx U({\\bf 0}) R_0 \\beta_3 \\delta(r^{\\prime}-R_0) Y_{30},\n\\label{eae21a}\n\\end{equation}\nwhere we have expanded $\\theta$ in a Taylor series, using\n$\\frac{d \\theta}{dx} = -\\delta(x)$.\nUsing Eq.\\ (\\ref{eae11b}) we then\nhave\n\\begin{equation}\n| \\gamma | \\approx \\left| U({\\bf 0})\n\\beta_3 R_1(R_0) R_2(R_0) {R_0}^3\n\\int \\Omega_2^{\\dagger} Y_{30} \\Omega_1 \\, d \\Omega^{\\prime}\n(E_1 - E_2)^{-1} \\right|\n\\sim \\beta_3,\n\\label{eae21b}\n\\end{equation}\nwhere we have used $R_1 (R_0) R_2 (R_0) \\approx 1.4 \/ {R_0}^3$\n\\cite{BohrMott}, $| U({\\bf 0}) | \\approx 50 \\mbox{ MeV}$,\n$|E_1-E_2| \\approx 5 \\mbox{ MeV}$, and\n$\\int \\Omega_2^{\\dagger} Y_{30} \\Omega_1 \\, d \\Omega^{\\prime} \\sim\n0.05$.\n\nFinally, we must estimate\nthe matrix element between the spherical orbitals,\n$\\langle \\psi_{1,+K} | H_{TP} | \\psi_{2,+K} \\rangle$.\nUsing Eq.\\ (\\ref{ehpta}) for\nthe form of $H_{TP}$ and $\\rho(r^{\\prime})\n= \\theta(r^{\\prime}-R_0) \/ ({\\textstyle \\frac{4}{3}} \\pi {r_0}^3)$\nwe get\n\\begin{equation}\nH_{TP} = -\\eta \\frac{3G}{8 \\pi \\sqrt{2} m {r_0}^3} (\\bbox{\\sigma}\n\\cdot {\\bf n}^{\\prime}) \\delta(r^{\\prime}-R_0).\n\\label{eae31c}\n\\end{equation}\nUsing Eq.\\ (\\ref{eae11c}) and $(\\bbox{\\sigma}\n\\cdot {\\bf n}^{\\prime})^2 = 1$\ngives\n\\begin{equation}\n\\langle \\psi_{1,+K} | H_{TP} | \\psi_{2,+K} \\rangle\n= \\eta \\frac{3G}{8 \\pi \\sqrt{2} m {r_0}^3}\nR_1(R_0) R_2(R_0) {R_0}^2\n\\approx \\frac{\\eta}{A^{1\/3}} 1 \\mbox{ eV}.\n\\label{eae31d}\n\\end{equation}\n\nUsing $|E_+ - E_-| \\sim 50 \\mbox{ keV}$\n(see, e.g., \\cite{AFS96,AFS2}),\n$\\beta_3 \\approx 0.1$ (see, e.g., \\cite{Cwiok91}),\nand Eqs.\\ (\\ref{ena42}), (\\ref{eae21b}), and (\\ref{eae31d})\ngives (for $A \\approx 225$)\n$|\\alpha| \\sim 2 \\beta_3 A^{-1\/3} \\eta\n\\, {\\rm eV} \/ |E_+ - E_-|\n\\sim 7 \\times 10^{-7} \\eta$.\nThis provides the following estimate for the collective EOM:\n\\begin{equation}\n| {\\cal O}_{\\rm coll} |\n\\sim 0.05 e {\\beta_3}^2 Z A^{2\/3} {r_0}^3 \\eta\n\\, {\\rm eV} \/ |E_+ - E_-|\n\\sim 4 \\times 10^{-5} \\eta e ({\\rm fm})^3.\n\\label{eeome30}\n\\end{equation}\nWe see that the collective EOM is two orders of magnitude\nlarger than the EOM due to unpaired protons.\n\nWe can do a more accurate calculation of the EOM\nusing Refs. \\cite{AFS96,AFS2},\nwhere the mixing coefficients, $\\alpha$ for various\nnuclei were calculated.\nWe use the values from \\cite{AFS2} that were calculated using\nthe Woods-Saxon potential.\nWe took the values of the $\\beta_i$ parameters from\n\\cite{Cwiok91} and the nuclei's angular momenta were taken\nfrom \\cite{Butler96,LeanderChen}.\nThe results are\nshown in table \\ref{tcoll} for various nuclei.\n\nNote the large value of $^{229}$Pa's collective octupole moment.\nThis is due to its large value of $\\alpha$, which is caused by\nthe small energy splitting between the members of\nits parity doublet \\cite{AFS96,AFS2}.\nThe possible existence of a static octupole deformation\nin $^{229}$Pa was stated in \\cite{Ahmad82}. However, more recent\npapers cast doubt on the existence of such a\ndeformation (see, e.g., \\cite{Grafen91,Levon94}).\nTherefore, the result given for $^{229}$Pa must be\nunderstood as being\nconditional on it having a static octupole deformation.\n\n\\section{The atomic electric dipole moment induced\nby a nuclear electric octupole moment}\n\\label{saedmi}\nIn this section we consider the electric dipole moment of an\natom that is induced by a nuclear electric octupole\nmoment.\nThe electric potential, $\\phi ({\\bf r})$ outside an arbitrary\ncharge distribution can be expanded in\nterms of spherical harmonics as (see, e.g., \\cite{Jackson})\n\\begin{equation}\n\\phi ({\\bf r}) = \\sum_{l=0}^{\\infty} \\sum_{m=-l}^l\n\\frac{4 \\pi}{2l+1} q_{lm} \\frac{Y_{lm} (\\theta,\\phi)}{r^{l+1}}.\n\\label{eedm1}\n\\end{equation}\nHere we neglect that part of the potential that comes\nfrom the screening of the nucleus's Coulomb field by\nthe atomic electrons as it\nis not important in the octupole potential (see the\nappendix).\nThe $q_{lm}$ are spherical electric multipole moments and they\ncan be written in terms of the charge distribution\n$\\rho_c ({\\bf r}^{\\prime})$ as follows \\cite{Jackson}:\n\\begin{equation}\nq_{lm} = \\int Y_{lm}^* (\\theta^\\prime,\\phi^\\prime) \n{r^\\prime}^l \\rho_c ({\\bf r}^{\\prime}) \\, d^3 r^\\prime.\n\\label{eedm2}\n\\end{equation}\nThe octupole term of the above electric potential is\n\\begin{equation}\n\\phi^{(3)} ({\\bf r}) = \\frac{4 \\pi}{7} q_{30} \\frac{1}{r^4}\nY_{30} (\\theta,\\phi),\n\\label{eedm3}\n\\end{equation}\nwhere we have only taken the $m=0$ term, as we will be\ncalculating the matrix element of the perturbation between\nstates with the same angular momentum projections. We can\nwrite this in terms of the octupole moment by using the\nrelation $q_{30} = \\int Y_{30} (\\theta,\\phi) r^3 \\rho_c\n({\\bf r}) \\, d^3 r\n= \\frac{5}{4} \\sqrt{\\frac{7}{\\pi}} O_{zzz}$.\nThe potential energy, $U_{\\rm oct}$ of an electron\n(of charge $-e$) in\n$\\phi^{(3)} ({\\bf r})$ will then be\n\\begin{equation}\nU_{\\rm oct} = -5 \\sqrt{\\frac{\\pi}{7}} e {\\cal O} \\frac{1}{r^4} \nY_{30} (\\theta,\\phi)\n\\label{eedm4}\n\\end{equation}\n(for the nuclear $I_z = I$ state).\n\nUsing perturbation theory, the electric dipole moment of,\nfor example,\nan atom with one electron above closed subshells induced\nby $U_{\\rm oct}$ can be written as\n\\begin{equation}\nd_z = -e \\langle \\widetilde{\\psi} | r_z | \\widetilde{\\psi} \\rangle\n= -2 e \\sum_{| k_2 \\rangle} \\frac{\\langle k_1 | r_z | k_2 \\rangle\n\\langle k_2 | U_{\\rm oct} | k_1 \\rangle}{E_{k_1} - E_{k_2}},\n\\label{eaedm3a}\n\\end{equation}\nwhere $\\widetilde{\\psi}$ denotes the perturbed atomic wave function,\n$| k_1 \\rangle = |n_1,j_1,l_1,m \\rangle$ is the unperturbed\nsingle-electron ground state,\nand $\\{| k_2 \\rangle \\}$ is the set of states with\nwhich $| k_1 \\rangle$ is mixed by the perturbation.\n\nAccording to the triangle rule for the addition of angular\nmomenta, $\\langle k_1 | r_z | k_2 \\rangle$ can only have a nonzero\nvalue if $|j_1 - j_2| \\le 1 \\le j_1 + j_2$. Similarly, for\n$\\langle k_2 | U_{\\rm oct} | k_1 \\rangle$ to be nonzero,\nwe must have\n$|j_1 - j_2| \\le 3 \\le j_1 + j_2$. This implies\nthat the following conditions need to be satisfied\nfor the dipole moment to be nonzero:\n\\begin{equation}\n|j_1 - j_2| \\le 1 \\mbox{ and } j_1 + j_2 \\ge 3.\n\\label{eaedm3b}\n\\end{equation}\nThe lowest pair of values that satisfies this condition\nis $j_1 = 3\/2$ and $j_2 = 3\/2$.\nTherefore, $s$ states cannot contribute\nto the dipole moment induced by the nuclear EOM. Also,\none of $l_1$ and $l_2$ must be even and the other odd, since the\nelectric dipole moment is a parity nonconserving effect.\n\nWe will carry out a relativistic calculation of the matrix\nelement of $U_{\\rm oct}$ between the single-electron states\n$|n_1,j_1,l_1,m \\rangle$ and $|n_2,j_2,l_2,m \\rangle$.\n(Note that although we wrote Eq.\\ (\\ref{eaedm3a}) for an atom with\none electron above closed subshells, this single-electron state\nmatrix element can be used for all atoms to compare various\nsources of atomic EDMs.)\nThe relativistic wave function of an electron\nis (see, e.g., \\cite{RelQuant})\n\\begin{equation}\n\\psi_{njlm} =\n\\left(\n\\begin{array}{c}\nf_{njl} (r) \\Omega_{jlm} (\\theta,\\phi) \\\\\ng_{njl} (r) i (-\\bbox{\\sigma} \\cdot {\\bf n})\n\\Omega_{jlm} (\\theta,\\phi)\n\\end{array}\n\\right),\n\\label{eedm5}\n\\end{equation}\nwhere $\\bbox{\\sigma}$ is twice the spin operator of this electron\nand ${\\bf n} = {\\bf r}\/r$.\nEvaluating the matrix element (using $(\\bbox{\\sigma}\n\\cdot {\\bf n})^2 = 1$) gives\n\\begin{equation}\n\\langle n_1 j_1 l_1 m | U_{\\rm oct} | n_2 j_2 l_2 m \\rangle\n= -5 \\sqrt{\\frac{\\pi}{7}} e {\\cal O}\n\\langle j_1 l_1 m | Y_{30} | j_2 l_2 m \\rangle T,\n\\label{eedm6}\n\\end{equation}\nwhere $T$ is a radial integral:\n\\begin{equation}\nT = \\int_0^{\\infty} \\frac{1}{r^4}\n(f_{n_1 j_1 l_1} f_{n_2 j_2 l_2}\n+ g_{n_1 j_1 l_1} g_{n_2 j_2 l_2}) r^2 \\, dr.\n\\label{eedm7}\n\\end{equation}\nBecause of the factor of $1\/r^4$ in the above integral,\nmost of the contribution to $T$ comes from small values\nof $r$. This allows us to use the following expressions\nfor $f_{njl}$ and $g_{njl}$, for\n$r \\ll a\/Z^{1\/3}$ \\cite{Khripl}:\n\\begin{eqnarray}\nf_{njl} (r) & = & \\frac{c_{njl}}{r}\n[(\\gamma+\\kappa) J_{2\\gamma}(x)\n- \\frac{x}{2} J_{2\\gamma-1}(x)], \\nonumber \\\\\ng_{njl} (r) & = & \\frac{c_{njl}}{r} Z\n\\alpha J_{2\\gamma}(x),\n\\label{eedm8}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nx & = & \\sqrt{\\frac{8 Z r}{a}}, \\nonumber \\\\\n\\gamma & = & \\sqrt{(j+{\\textstyle \\frac{1}{2}})^2\n- Z^2 \\alpha^2}, \\nonumber \\\\\n\\kappa & = & (-1)^{j+1\/2-l} (j+{\\textstyle\n\\frac{1}{2}}), \\nonumber \\\\\nc_{njl} & = & \\frac{\\kappa}{|\\kappa|}\n\\left( \\frac{1}{Z a \\nu^3} \\right)^{1\/2},\n\\label{eedm9}\n\\end{eqnarray}     \nwhere the $J$'s are Bessel functions, $a$ is the\nBohr radius, and $\\nu$ is the effective principal quantum\nnumber ($E_{nl} = -13.6 \\mbox{ eV} \/ \\nu^2$). \nTo avoid confusion, note that $l$ here is the orbital\nangular momentum of the electron, rather than the nucleus, as \nused in Sec. \\ref{sneom}. Also, the $\\kappa$ used\nhere is distinct from\nthe $\\kappa$ defined in Eq.\\ (\\ref{eao}).\nCarrying out the integration in Eq.\\ (\\ref{eedm7}), we obtain\n\\begin{eqnarray}\n\\label{eedm10}\nT & = & \\frac{192 (-1)^{j_2-j_1+1} Z^2}{a^4 \\nu_{1}^{3\/2}\n\\nu_{2}^{3\/2}} \\frac{\\Gamma (-3+\\gamma_1\n+\\gamma_2)}{\\Gamma(4+\\gamma_1+\\gamma_2)\n\\Gamma(4+\\gamma_1-\\gamma_2) \\Gamma(4-\\gamma_1+\\gamma_2)}\n\\\\ \\nonumber\n & \\times & \\{(3+\\gamma_1-\\gamma_2) (3-\\gamma_1+\\gamma_2)\n(3+\\gamma_1+\\gamma_2) (2+\\gamma_1+\\gamma_2) \\\\ \\nonumber\n&&-5 (\\gamma_1+\\kappa_1)(3-\\gamma_1+\\gamma_2)(3+\\gamma_1+\\gamma_2)\n-5 (\\gamma_2+\\kappa_2)(3+\\gamma_1-\\gamma_2)(3+\\gamma_1+\\gamma_2) \\\\\n\\nonumber\n&&+30 [(\\gamma_1+\\kappa_1)(\\gamma_2+\\kappa_2)+(Z \\alpha)^2] \\},\n\\end{eqnarray}\nwhere $\\Gamma$ is the gamma function.\n\nTo illustrate the numerical values involved, we will\nevaluate a value\nof the matrix element for an electron's interaction with the\nsingle particle octupole moment for $^{209}$Bi.\nThis atom has one $6 p_{3\/2}$ electron above closed subshells\n($(6 p_{1\/2})^2 (6 s_{1\/2})^2 \\ldots$). \nWe will consider\nmixing between $d_{5\/2}$ and $p_{3\/2}$ states, each with\nan angular momentum projection of $3\/2$. Using\ntable \\ref{tsingp} and Eqs.\\ (\\ref{eedm6}) and (\\ref{eedm10}) gives\n\\begin{equation}\n\\langle n_1 p_{3\/2},m=3\/2 | U_{\\rm oct} | n_2 d_{5\/2},m=3\/2 \\rangle\n\\approx 1.9 \\times\n10^{-13} (\\nu_1 \\nu_2)^{-3\/2} \\eta {\\rm cm}^{-1}.\n\\label{eaedmme1}\n\\end{equation}\nWe now compare this with the corresponding matrix element for the\ninteraction with a nuclear magnetic quadrupole\nmoment (MQM), which is\nanother possible source of an atomic EDM. The MQM induced by the T-\nand P-odd interaction (\\ref{ehpta}) was calculated\nin Ref. \\cite{SFK84},\nas well as the matrix element of the interaction between\nan electron\nand the MQM field.\nFor $^{209}$Bi the MQM is $\\approx 4.8 \\times 10^{-8} \\eta e\n(\\mbox{fm})^2$. \nWe get the following estimate:\n\\begin{equation}\n\\langle n_1 p_{3\/2},m=3\/2 | U_{\\rm quad} | n_2\nd_{5\/2},m=3\/2 \\rangle\n\\approx 1.3 \\times 10^{-11} (\\nu_1 \\nu_2)^{-3\/2}\n\\eta {\\rm cm}^{-1}.\n\\label{eaedmme2}\n\\end{equation}\nBy comparing the values of these matrix elements it\ncan be seen that the atomic EDM\ninduced by a nuclear EOM will be less than about 1\\% of\nthat induced by a nuclear magnetic quadrupole moment.\nThis same result\nheld for the other atoms and mixing states that we considered.\nAnother mechanism which could produce an atomic EDM is the nuclear\nSchiff moment, which for heavy atoms (which we are\nconsidering) gives a\ncontribution comparable to that of\nthe MQM \\cite{SFK84}. Because\nof this we can conclude that the atomic EDM induced by a\nsingle particle nuclear EOM\nis negligible in comparison with other possible mechanisms.\n(See the appendix for a discussion of the relative contributions\nof the octupole, magnetic quadrupole, and Schiff moments.)\n\nNow we will give example values of the matrix elements for those\nnuclei having a static octupole deformation.\nAs well as possibly having collective EOMs (as shown in Sec.\n\\ref{sceom}), these nuclei can also have collective MQMs that\nare an order of magnitude larger than the single particle MQMs\ndiscussed above \\cite{spinhh}. We will use $^{225}$Ac as an example\nand once again we will consider mixing between $d_{5\/2}$ and\n$p_{3\/2}$ states, each having an angular momentum projection of\n$3\/2$. This nucleus has an MQM of $\\sim 2 \\times 10^{-7} \\eta e\n(\\mbox{fm})^2$ \\cite{spinhh}.  \nWe obtain\n\\begin{equation}\n|\\langle n_1 d_{5\/2},m=3\/2 | U_{\\rm oct} | n_2 p_{3\/2},m=3\/2\n\\rangle|\n\\approx 1.4 \\times 10^{-11} (\\nu_1 \\nu_2)^{-3\/2} \\eta {\\rm cm}^{-1}\n\\label{ecme1}\n\\end{equation}\nand\n\\begin{equation}\n|\\langle n_1 d_{5\/2},m=3\/2 | U_{\\rm quad} | n_2\np_{3\/2},m=3\/2 \\rangle|\n\\sim 6 \\times 10^{-11} (\\nu_1 \\nu_2)^{-3\/2} \\eta {\\rm cm}^{-1}.\n\\label{ecme2}\n\\end{equation}\nThese results show that the contribution of the collective\nnuclear EOM to\nthe atomic EDM is smaller than that of the collective MQM.\nThis also applies to the other isotopes shown in table \\ref{tcoll},\nexcept for $^{229}$Pa. ($^{223}$Rn does have a fairly\nlarge collective\noctupole moment, but since it has a closed electron shell\nthe EOM does not contribute to the\natomic EDM.) For $^{229}$Pa the contribution of the\nEOM may be comparable to that of the MQM, but as stated above, it\nis not certain that this nucleus has a static octupole deformation.\n \n\\section{Possibly enhanced magnetic quadrupole moments in nuclei\nwith an octupole deformation}\n\nFinally, we discuss a mechanism by which single particle\nmagnetic quadrupole moments could be enhanced in even-odd nuclei\nwith an octupole deformation. We will first consider the MQM of\nsuch a nucleus in its intrinsic (body-fixed) frame and then\ntransform the MQM into its laboratory frame.\n\nAs in Sec. \\ref{sceom} the wave function of the external nucleon in\nthe intrinsic frame is $\\psi_{+K} ({\\bf r}^{\\prime})$, defined\nby Eqs. (\\ref{eae11a}), (\\ref{eae11b}), and (\\ref{eae11c}). The\nMQM in the intrinsic frame is then\n\\begin{eqnarray}\nM_{\\rm intr} & = & \\langle \\psi_{+K} | \\hat{M}_{z^{\\prime}\nz^{\\prime}} | \\psi_{+K} \\rangle\n\\label{emqn1a} \\\\\n& = & 2 \\gamma \\langle \\psi_{1,+K} | \\hat{M}_{z^{\\prime}\nz^{\\prime}} | \\psi_{2,+K} \\rangle,\n\\label{emqn1b}\n\\end{eqnarray}\nwhere $\\hat{M}$ is the operator for the MQM (see Ref. \\cite{SFK84}).\nNote that the MQM is defined for the maximum value of the\nprojection of the angular momentum onto the $z^{\\prime}$-axis,\nso $K=j=I$ (for the ground state of the rotational band $I=j$).\nUsing a result from Ref. \\cite{SFK84} we have\n\\begin{equation}\nM_{\\rm intr} = \\gamma \\frac{2I-1}{I+1} \\frac{e}{m}\n(\\mu - q) \\int R_1 R_2 r \\, r^2 dr,\n\\label{emqc}\n\\end{equation}\nwhere $\\mu$ is the magnetic moment of the external nucleon in\nnuclear magnetons and $q = 0$ ($1$) for a neutron (proton).\nIt should be noted that this (intrinsic frame)\nsingle particle MQM differs from\nthe spherical nucleus \nsingle particle EOMs and MQMs (considered in Sec. \\ref{sneom}\nand Ref. \\cite{SFK84}, respectively) in that the former is\ngenerated due to the interaction $V_3$, coming from the nucleus's\noctupole deformed shape [see Eqs. (\\ref{eae11b})\nand (\\ref{eae21a})], while the latter is due to the interaction\n$H_{TP}$ (\\ref{ehpta}). The $H_{TP}$ interaction will\ncome into the current\nsituation when we transform into the lab. frame.\n\nSince we are only working out a rough estimate of the MQM we take\nthe integral in the above equation to be\n\\begin{equation}\n\\int R_1 R_2 r \\, r^2 dr \\sim \\frac{1}{2} r_0 A^{1\/3},\n\\label{eqmd}\n\\end{equation}\nwhere $r_0 = 1.1 \\mbox{ fm}$. For both types of nucleons we\nhave $|\\mu - q| \\approx 1.8$ and so we can write\n\\begin{equation}\n| M_{\\rm intr}| \\sim 0.2 A^{1\/3} \\frac{2I-1}{I+1} |\\gamma| e\n(\\mbox{fm})^2.\n\\label{emqe}\n\\end{equation}\n\nNow we turn to the MQM in the laboratory frame. The wave function\nof the nucleus in the lab. frame is (as in Sec. \\ref{sceom})\n$\\psi = \\psi^+ + \\alpha \\psi^-$, where $\\psi^{\\pm}$ is defined\nby Eqs. (\\ref{eceom1a}) and (\\ref{ecn21a}) and $\\alpha$ is\ngiven by Eq. (\\ref{enad32}) --- this is where the interaction\n$H_{TP}$ comes into the present situation.\nThe MQM in the lab. frame is then\n\\begin{eqnarray}\nM_{\\rm lab} & = & \\langle \\psi | \\hat{M}_{zz} | \\psi\n\\rangle \\nonumber \\\\\n& = & 2 \\alpha \\langle \\psi^+ | \\hat{M}_{zz} | \\psi^-\n\\rangle \\nonumber \\\\\n& = & \\alpha ( \\langle I M K | \\hat{M}_{zz} | I M K \\rangle\n- \\langle I M -K | \\hat{M}_{zz} | I M -K \\rangle) \\nonumber \\\\\n& = & 2 \\alpha \\langle I M K | \\hat{M}_{zz} | I M K \\rangle.\n\\label{emqea2}\n\\end{eqnarray}\n(Once again, the MQM is defined for $M$, the projection of the\nangular momentum onto the $z$-axis, equal to $I$.)\nNote that $\\langle I M -K | \\hat{M}_{zz} | I M -K \\rangle\n= - \\langle I M K | \\hat{M}_{zz} | I M K \\rangle$, as\n$\\psi_{\\pm K} = \\psi_{1, \\pm K} \\pm \\gamma\n\\psi_{2, \\pm K}$ and so\n$\\langle \\psi_{-K} | \\hat{M}_{z^{\\prime} z^{\\prime}}\n| \\psi_{-K} \\rangle = -\\langle \\psi_{+K} |\n\\hat{M}_{z^{\\prime} z^{\\prime}}| \\psi_{+K} \\rangle$. \n\nNow consider $\\langle I M K | \\hat{M}_{zz} | I M K \\rangle$.\nWe have [from Eq. (\\ref{ecn21a})]\n\\begin{equation}\n\\langle I M K  | \\hat{M}_{zz} | I M K \\rangle\n= \\frac{2I+1}{4 \\pi} \\int {D^{I *}_{M K}} (\\phi,\n\\theta,0) \\psi_K^{*} ({\\bf r}^{\\prime}) \\hat{M}_{zz}\n(\\theta,\\phi,0) \\psi_K ({\\bf r}^{\\prime})\nD^I_{M K} (\\phi,\\theta,0) \\, d^3 r^{\\prime} \\, d \\Omega.\n\\label{emqf}\n\\end{equation}\nTransforming $\\hat{M}_{zz}$ from the lab. frame to the intrinsic\n($x^{\\prime},y^{\\prime},z^{\\prime}$) frame gives\n$\\hat{M}_{zz} (\\theta,\\phi,0) = {D^{2 *}_{0 0}} (\\phi,\n\\theta,0) \\hat{M}_{z^{\\prime} z^{\\prime}} (\\theta^{\\prime},\n\\phi^{\\prime},0)$ (see, e.g., \\cite{Varshalovich}). Substituting\nthis into Eq. (\\ref{emqf}) and using Eq. (\\ref{emqn1a}) gives\n(using a formula for the integral of $D$-functions from\n\\cite{Varshalovich})\n\\begin{eqnarray}\n\\langle I M K | \\hat{M}_{zz} | I M K \\rangle & = &\n\\frac{2I+1}{4 \\pi} M_{\\rm intr} \\int {D^{I *}_{M K}} (\\phi,\\theta,\n0) {D^{2 *}_{0 0}} (\\phi,\\theta,0) D^I_{M K} (\\phi,\\theta,0)\n\\, d \\Omega \\nonumber \\\\\n& = & M_{\\rm intr} \\langle I, M ; 2, 0|I, M \\rangle\n\\langle I, K; 2, 0| I, K \\rangle \\nonumber \\\\\n& = & M_{\\rm intr} {\\langle I, I ; 2, 0| I, I \\rangle}^2\n\\nonumber \\\\\n& = & \\frac{I (2I-1)}{(I+1) (2I+3)} M_{\\rm intr}.\n\\label{emqg}\n\\end{eqnarray}\n(Recall that $M=K=I$.) Using Eq. (\\ref{emqea2}) then gives\n\\begin{equation}\nM_{\\rm lab} = 2 \\alpha \\frac{I (2I-1)}{(I+1) (2I+3)}\nM_{\\rm intr}\n\\label{emq5a}\n\\end{equation}\nand so [using Eq. (\\ref{emqe})]\n\\begin{equation}\n|M_{\\rm lab}| \\sim 0.4 A^{1\/3} |\\alpha| |\\gamma|\n\\frac{I (2I-1)^2}{(I+1)^2 (2I+3)} e ({\\rm fm})^2.\n\\label{emq5b}\n\\end{equation}\nNow we have $|\\alpha| \\sim 7 \\times 10^{-7} \\eta$ and\n$|\\gamma| \\sim 0.1$ (see Sec. \\ref{sceom}). For\n$A \\approx 225$ we have\n\\begin{equation}\n|M_{\\rm lab}| \\sim 6 \\times 10^{-8} \\eta e ({\\rm fm})^2.\n\\label{emq5c}\n\\end{equation}\nThis is smaller than the collective MQM due to the spin\nhedgehog mechanism ---\n$\\sim 2 \\times 10^{-7} \\eta e ({\\rm fm})^2$ \\cite{spinhh} and\nso it is not the dominant mechanism. It is, in\nfact, of the same order of magnitude as the single particle\nMQM [e.g., for $^{209}$Bi, approximately\n$4.8 \\times 10^{-8} \\eta e ({\\rm fm})^2$ (see\nSec. \\ref{saedmi})].\nThis means that there is no enhancement.\n\nEnhancement was a possibility here due to the relatively large\nvalue of $\\alpha$ that comes from the smallness of the energy\nsplitting between members of the parity doublet ($E_+ - E_-$).\nHowever, the inclusion of the factors of $|\\gamma| \\sim 0.1$\nand $I (2I-1) \/ [(I+1) (2I+3)] \\sim 0.3$ (this enters on\ntransforming to the lab. frame) ensures that the possible\nenhancement is not realised.\n\nHowever, $^{229}$Pa may be an exception to this, as it has a\nlarge value of $\\alpha$, but it must be remembered that\nit is not certain that this nucleus has a static octupole\ndeformation (see Sec. \\ref{sceom}). If is does have such a\ndeformation then its MQM due to the present mechanism would\nbe $|M_{\\rm lab}| \\sim 3 \\times 10^{-7}\n\\eta e ({\\rm fm})^2$ [using Eq. (\\ref{emq5b}),\nthe value of $\\alpha$ given in Ref. \\cite{AFS2}, and\n$|\\gamma| \\sim 0.1$], which is of the same order of\nmagnitude as the collective MQM due to the spin hedgehog\nmechanism.\n\nNote that the enhancement of MQMs in\ndeformed nuclei with opposite\nparity levels close to each other has also been considered in\nRefs. \\cite{Hax83,SFK84}.\n\n\\begin{acknowledgements}\nOne of us (DWM) is grateful to G.F. Gribakin for\nhelpful discussions.\nThis work was supported by the Australian Research Council\nand by the National Science Foundation through a grant for the\nInstitute for Theoretical Atomic and Molecular Physics at\nHarvard University and the Smithsonian Astrophysical Observatory.\n\\end{acknowledgements}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nOver the last 10 or 12 years a great deal of outstanding observational\nwork has indicated that the best fit model of our universe is a nearly\nflat Friedmann-Lema\\^{i}tre-Robertson-Walker (FLRW) model with $\\Omega_M \n\\approx 0.27$ and $\\Omega_{\\Lambda} \\approx 0.73 $ (Riess {\\it et al.} 1998;\nPerlmutter {\\it et al.} 1999; Bennett {\\it et al} 2003 (WMAP results);\nPeacock {\\it et al.} 2001; Percival {\\it et al.} 2001; Efstathiou {\\it et al.}\n2002; Spergel {\\it et al.} 2003, and references therein),\nwhere $\\Omega_M$ and $\\Omega_{\\Lambda}$ are the usual density parameters for \nmatter, including nonbaryonic dark matter, and dark energy, modelled here\nas vacuum energy (the cosmological constant $\\Lambda$), respectively. Here\nand throughout this paper $\\Omega_M$ and $\\Omega_{\\Lambda}$ refer to these\nquantities as evaluated at our time now. This\nremarkable concordance is based on WMAP cosmic microwave background\n(CMB) anisotropy measurements, a large number of Supernovae Ia\ndata (see Riess {\\it et al.} 2004), and large scale structure studies, and has\nbeen confirmed by other more recent work. Riess and his collaborators (Riess\n{\\it et al.} 2004), for instance, have recently found a best-fit cosmology \nhaving $\\Omega_M = 0.29$ and $\\Omega_{\\Lambda} = 0.71$ for their sample of\n16 distant ($z > 1$) SN Ia, including 6 with $z > 1.25$, assuming the\nuniverse is exactly flat. Within\nthe errors this is consonant with the ``concordance'' model given above. \\\\\n\nDespite the strength of these results, they will obviously have to undergo\ngradual\nrevision and continual verification, as more precise data from higher redshifts\nare acquired. When $\\Lambda \\neq 0$, there are at present, from a strictly\nmathematical consideration of the Einstein field equations, not yet enough\ncompletely independent observables to constrain all the free parameters\nof the cosmological model (Hellaby, 2006; Stoeger \\& Hellaby, in\npreparation). \\\\  \n\nAssuming that the universe is spherically symmetric on the largest scales\n(FLRW or, more generally, Lema\\^{i}tre-Tolman-Bondi (LTB)), one generally needs\nredshifts, luminosity distances (or angular-diameter distances), and galaxy\nnumber counts, together with a reliable galaxy evolution model, or an \nequivalent set of measurements, to constrain the model fully (see Ellis, {\\it\net al.} 1985). If $\\Lambda \\neq 0$, however, or if there is\nsome other form of dark energy, these data are not enough. We need at least\none other independent parameter -- that is, independent of the observables we\nhave just mentioned and therefore of those which depend upon them. And,\nstrictly speaking, this is what we have not had. Thus, the impressive fittings\nthat have led to the concordance model are still model-dependent in some sense. \\\\\n\nThere is another pair of such independent observables. These would\nimprove and verify our cosmological fitting, when we are able to obtain an\nadequate number of precise luminosity distances -- or angular-diameter \ndistances -- and redshifts for SN Ia, or for other standard candles or standard\nrods , out to $z \\approx 1.8$. These observables are the maximum of the angular-diameter distance (or observer-area distance) $C_{max}$ and the\nredshift $z_{max}$ at which it occurs. It has been realized for many years\n(McCrea 1935, Hoyle 1961, Ellis \\& Tivon 1985) that this distance reaches\na maximum for relatively low redshifts in FLRW universes. For an\nEinstein-deSitter ($ \\Omega =1 $)universe filled with matter, for instance,\nthe observer area distance C has a maximum $C_{max}$ at $z_{max} = 1.25$. This\neffect is due to the global gravitational focusing of light rays\ncaused by the matter in the universe -- in effect the entire universe, filled\nwith homogeneously distributed matter, acts like a gravitational lens. \\\\\n\nKrauss and Schramm (1993) recognized that, for flat FLRW universes, \ndetermination of $z_{max}$ would give us $\\Omega_{\\Lambda}$. They plotted\nand provided a table giving this unique correspondence (see their Table 1),\nand proposed the possibility of using the measurement of compact parsec-scale\nradio jets to observationally exploit it, if the source-evolution problem\ncan be tamed. Since then, there has been little\ndevelopment or discussion of this potentially important connection -- except\nfor Hellaby's (2006) recent closely connected exploration of such\nmeasurements within the more general context of LTB universes (see below). \nCertainly, it is implicit in the Friedmann equation -- most clearly in \nRefsdal, {\\it et al.}'s (1967) numerical results of general cosmological\nmodels, in the brief treatment of cosmic distances by Carroll, {\\it et al.},\n1992 (see pages 510-512, and their Figure 5), and in Peeble's treatment of\nangular diameters in cosmology (Peebles 1993), but not pointed out or\ndiscussed further, until Hellaby's more general treatment. This may be\npartially due to the\ndifficulty of obtaining reliable data at the redshifts where we would expect to\nlocate $C_{max}$ (see below). Now, however, there is the very real prospect of\nobtaining angular diameter distances (indirectly, by measuring luminosity\ndistances of SN Ia) out to $z \\approx 1.8$ using telescopes in space. Thus, it\nis important to point out again and stress this promising connection, which could\neventually be incorporated in the Bayesian-Fisher matrix (see, for example,\nAlbrecht, {\\it et al.}, 2006) fitting of models to data, or be used as an\nindependent consistency check on such fittings. \\\\\n\nRecently, as already mentioned, Hellaby (2006) emphasized the importance\nof such a measurement within\na more general framework. He points out that in any LTB cosmology with\n$\\Lambda = 0$ (which includes all $\\Lambda = 0$ FLRW cosmologies as special\ncases) the measurement of $C_{max}$ is equivalent to a measurement of the\ntotal mass $M_{max}$ within the sphere defined by $C_{max}$. For\n$\\Lambda \\neq 0$ we have for any LTB model, instead, a simple relationship\nbetween the $\\Lambda$, $C_{max}$ and $M_{max}$ (see equation (11) below). So\na measurement of $M_{max}$, or its equivalent, and $C_{max}$ determines\n$\\Lambda$. What becomes apparent is that $C_{max}$ and the redshift $z_{max}$\nat which it occurs constitute independent cosmological observables -- directly\nconstraining $\\Lambda$ and $\\Omega_M$ (see Hellaby's Figure A1 in his Appendix,\nwhich shows how different cosmologcal parameters vary with $z_{max}$.)  \\\\\n\nApplying this directly to flat FLRW models, like those we have good evidence\nrepresent our universe, we quickly see that, since we implicitly have a \nrelation between the total mass-energy density and the matter density, or\nequivalently between the matter density and $\\Omega_{\\Lambda}$ --- i.e.\n$\\Omega_M = 1 - \\Omega_{\\Lambda}$ --- observational determination of \n$z_{max}$ will directly determine $\\Omega_{\\Lambda}$ in a very simple and\nstraightforward way, supporting Krauss and Schramm's results (1993). In\nthis paper we shall integrate and generalize these results, first of all\nverifying Krauss and Schramm's results for flat FLRW universes and writing\ndown that relationship as an algebraic equation in closed form (they\npresented their results numerically), and then generalizing those results\nto non-flat FLRW universes, using the relationship Hellaby (2006) noticed.\nIn this case, $C_{max}$ and $z_{max}$ directly determine both\n$\\Omega_{\\Lambda}$ and $\\Omega_M$, if we know $H_0$ independently.\nIn the course of doing this, we shall, as useful and important by-products,\nobtain the FLRW $C = C(z)$ and $\\hat{M}_0 = \\hat{M}_0(z)$ closed-form functional\nrelationships for $\\Lambda \\neq 0$ universes, parallel to those which\nare well-known for $\\Lambda = 0$ FLRW models (Ellis and Stoeger 1987;\nStoeger, {\\it et al.} 1992), as well as a very simple observational\ncriterion for flatness in terms of $C_{max}$. Here $C(z)$, of course, is\nsimply the angular-diameter distance as a function of the redshift $z$, and \n$\\hat{M}_0(z)$ is the mass density per source counted as a function of\n$z$, which is closely related to the differential galaxy number counts\n$dN\/dz$ (see Stoeger, {et al.} 1992). To our knowledge, these more\ngeneral results, along with the closed-form expressions and the flatness\ncriterion are new. \\\\\n\nWe have already indicated that these measurements will be able to be\nimplemented once we have luminosity distances and redshifts for SN Ia, or\nfor other standard candles or standard rods, in the interval $1.5 < z < 1.8$.\nAs we shall show, it is precisely in this region that a flat FLRW universe will\nhave a maximum in its angular-diameter distance, if $0.59 \\leq \\Omega_{\\Lambda}\n< 0.82$. For the best fit FLRW given by Riess {\\it et al.} (2004) with\n$\\Omega_M = 0.29$ and $\\Omega_{\\Lambda} = 0.71$, $z_{max} = 1.62$. Another \npotential way of obtaining such precise measurements is -- following Krauss\nand Schramm's (1993) idea -- the use of VLBI to determine the\nangular-size\/redshift relation for ultra-compact (milliarcsecond) radio\nsources. These have been argued to be standard rods (Jackson and Dodgson\n1997; Jackson 2004).  If we actually do \nfind the maximum angular-diameter distance near this value of the redshift,\nthis would be independent confirmation of the concordance model. If we\ndo not, but find the maximum angular-diameter distance $C$ at some other value\nof $z$, this will require further work at reconciling the models, and\npossibly modifying them. In that case, either the universe may still be flat,\nbut the relative amounts of matter and dark energy would be quite different\nfrom that given by the concordance, or there is a significant deviation from\nflatness that must be taken into account, or possibly there are significant\ndeviations from FLRW on the largest scales which must be included -- or all\nthree! At the very least, this would be a good consistency check on  \nour cosmological fitting so far. Alternatively, as we have already mentioned,\nwe could simply include both $C_{max}$ and $z_{max}$ data in our over-all \nfitting scheme -- which would further improve the relibility of our results. \\\\\n\nIt is important to point out that this redshift range is already attracting \nspecial attention. That is because there have been preliminary indications\n(Gilliland {\\it et al.} 1999) from an SN Ia at $z \\approx 1.7$ that the\nuniverse was decelerating at that time! Further studies (Riess {\\it et al.}\n2001; Mortsell {\\it et al.} 2001; Ben\\'{i}tez {\\it et al.} 2002) have confirmed\nthe plausibility of that conclusion, but were unable to strengthen it without\nfurther SN Ia measurements in that interval. Thus, we now have two strong\nmotivations for pursuing precise SN Ia searches and measurements in this\nredshift range. \\\\\n \nFinally, one might wonder how measurements of the luminosity distances\nof SN Ia can reveal maxima in the angular-diameter (or observer-area)\ndistances. The luminosity distances themselves will not have such maxima. The\nanswer to this question is simple, though rarely adverted to. According the\nreciprocity theorem of Etherington (1933; see also Ellis 1971), the luminosity\ndistance $d_L$ is very generally related to the angular-diameter, or\nobserver-area, distance by\n\n\\begin{equation}\nd_L = (1+z)^2 C.  \\label{recth}\n\\end{equation}\n\n\\noindent\nThis simple but important relationship holds for all cosmologies, even very\ninhomogenous ones. Thus, with observed luminosity distances and redshifts in the\nabove mentioned crucial redshift range, we can very quickly convert to\nangular-diameter distances, and determine whether the maximum for those \ndistances lies within that range. \\\\\n\nNow we shall go on to work out the simple relationship between $z_{max}$\nand $\\Omega_{\\Lambda}$ for flat FLRW. \n\n\\section{The Maximum Angular-Diameter Distance in Flat FLRW with $\\Lambda \\neq\n0$}\n\nThe basic equations relating $z_{max}$ and $\\Omega_{\\Lambda}$ in  flat FLRW\nwith $\\Lambda \\neq 0$ are not difficult, but require some effort to obtain\nand check, because they involve elliptic integrals. As we have already \nmentioned, this represents the simplest and clearest example of a more general\nrelationship between the redshift of the maximum of the angular-diameter\ndistance (in LTB models this is often referred to as the ``areal radius'') and\nthe matter and vacuum-energy content of the universe for all FLRW and LTB\nmodels (Hellaby 2006). Furthermore, neither Krauss and Schramm (1993) nor\nHellaby (2006) illustrate the actual calculation. Their results were obtained\nnumerically, and presented in plotted or table form. \\\\\n\nIn flat FLRW, the angular-diameter (or observer-area) distance $C(\\eta, y)$ is\ngiven by \n\n\\begin{equation}\nC(\\eta, y) = R(\\eta)y = \\frac{R_0 y}{1 + z}\\>, \\label{oadef}\n\\end{equation}\n\n\\noindent\nwhere $R(\\eta)$ is the scale factor, $\\eta$ is the conformal time, $R_0$ is\nthe scale factor now, $y$ is the comoving radial coordinate, and $z$ is the\nredshift of signals from distant sources. Here we have used the important\nFLRW relationship\n\n\\begin{equation}\n1 + z = \\frac{R_0}{R(\\eta)}\\>. \\label{red}\n\\end{equation}\n\nClearly, if we differentiate equation (\\ref{oadef}) with respect to $y$ and set\nthe result equal to zero, we\nshall have the equation for the maximum of $C(\\eta, y)$, subject to the \nusual condition that $d^2C\/dy^2 < 0$ for $dC\/dy = 0$. We have then from\nequation  (\\ref{oadef})\n\n\\begin{equation}\ndC\/dy = \\frac{R_0}{1+z} - \\frac{R_0 y}{(1+z)^2}dz\/dy = 0\\,, \\label{dcdy}\n\\end{equation}\nwhich becomes\n\n\\begin{equation}\n\\frac{R_0}{1+z} - \\frac{R_0 y}{(1 + z)^2} R_0 H_0 \\sqrt{\\Omega_{\\Lambda} \n+ (1 - \\Omega_{\\Lambda})(1+z)^3} = 0\\,, \\label{dcdy2}\n\\end{equation}\n\\noindent\nsince the Friedmann equation in this case yields\n\n\\begin{equation}\ndz\/dy = R_0 H_0 \\sqrt{\\Omega_{\\Lambda} + (1 - \\Omega_{\\Lambda})(1+z)^3}\\>.\n                                 \\label{dzdy}\n\\end{equation}\n\nThus, from solving equation (\\ref{dcdy2}) for $y$, we obtain the equation for\n$y_{max}$, the comoving radial coordinate distance to the point down\nthe observer's past light cone at which the angular-diameter distance is a\nmaximum, as a function of $z_{max}$, the redshift there, and of\n$\\Omega_{\\Lambda}$:\n\n\\begin{equation}\ny_{max} = \\frac{1 + z_{max}}{R_0H_0 \\sqrt{\\Omega_{\\Lambda} + (1+z_{max})^3\n                                    (1-\\Omega_{\\Lambda})}} \\> . \\label{ymax}\n\\end{equation}\n\nThis is the first and most essential step in finding the equation we are\nlooking for. \\\\\n\nThe second step involves finding the explicit solution to the Friedmann\nequation, essentially equation (\\ref{dzdy}), to give us another expression for\n$y_{max}$ at $z_{max}.$ Substituting this expression into left-hand-side of\nequation (\\ref{ymax}) gives a unique implicit equation for $\\Omega_{\\Lambda}$ as a\nfunction simply of $z_{max}$. This is the relationship we have been looking\nfor. \\\\\n\nSo, what is the solution of equation (\\ref{dzdy})? Normally, we might want to simply\ndo a numerical integration. However, this would not be very useful in\nour case. It turns out, as is well known (Byrd \\& Friedman (1954), pp.\n8-10 and formula 260.00 (p. 135); see also Jeffrey (1995), pp. 225-226), that,\nsince this equation involves the square \nroot of a cubic polynomial, it has an analytic solution in terms of elliptic\nintegrals. In our case the most useful form of the solution is:\n\n\\begin{equation}\ny = \\frac{g}{R_0 H_0 \\Omega_{\\Lambda}^{1\/2}}\\biggl[F(\\phi, k) \\mid_{(1+z)^{-1}=1} -\n                      F(\\phi, k)\\mid_{(1+z)^{-1}}\\biggr] \\>, \\label{yeliptic}\n\\end{equation}\n\n\\noindent\nwhere the $F(\\phi, k)$ are standard elliptic integrals of the first kind,\nfor the angle $\\phi$, which is a function of $1+z$, and $k$ is the modulus.\nMore explicitly \n\n\\begin{eqnarray}\n \\phi &=&cos^{-1}\\Biggl[\\frac{-m(1+z)+ (\\sqrt 3 -1)}{-m(1+z)-(\\sqrt 3 + 1)}\\Biggr]\\>, \\nonumber  \\\\\n m &= &\\Biggl[\\frac{1 - \\Omega_{\\Lambda}}{\\Omega_{\\Lambda}}\\Biggr]^{1\/3}, \\nonumber \\\\\nk^2 &=&\\frac{1}{2} + \\frac{\\sqrt 3}{4}\\>, \\nonumber  \\\\\ng &=&\\frac{1}{3^{1\/4}}\\Biggl[\\frac{\\Omega_{\\Lambda}}{1 - \\Omega_{\\Lambda}}\\Biggr]^{1\/3}. \\nonumber \n\\end{eqnarray}\n\n\\noindent\nThis solution was obtained and checked using elliptic integral tables\nin Byrd \\& Friedman (1954) (entry 260.00, p. 135) in conjunction with\nMAPLE.\n\nWith equation  (\\ref{yeliptic}) being substituted for $y$, equation (\\ref{oadef}) is the characteristic\nFLRW relationship for the angular-diameter distance $C = C(z)$ in terms of\n$z$. It turns out (see below) that this same form of the relationship\nholds in the general (non-flat) FLRW cases -- with the parameters $\\phi$,\n$k$, and $g$ being more complicated functions, involving $\\Omega_{\\Lambda}$,\neither $\\Omega_M$ or $C_{max}$, and $H_0$. We shall explicitly write these\ndown in the next section. Similarly, we quickly can write down the\ncomplementary characteristic $\\Lambda \\neq 0$ mass density per source counted\nas a function of $z$ (see Ellis and Stoeger 1987 and Stoeger, {\\it et al.}\n1992):\n\n\\begin{equation}\n \\hat{M}_0(z) = \\frac{\\mu_{m_{0}}(1+z)^2}{R_0H_0\\sqrt{\\Omega_{\\Lambda} +\\Omega_{M} (1+z)^3 - (\\Omega_0 - 1)(1+z)^2}}, \\label{Mz}\n\\end{equation}\nwhere $\\mu_{m_{0}}$ is the mass-energy density now and\n$\\Omega_0\\equiv\\Omega_{\\Lambda} +\\Omega_M$, and the last term under the\nradical sign in the denominator is zero when the universe is flat (see\nbelow). These characteristic FLRW relationships for $C(z)$ and for\n$\\hat{M}_0(z)$ are very useful to know (Ellis and Stoeger 1987;\nStoeger, {\\it et al.}(1992). If the universe is FLRW and $\\Lambda = 0$,\nthen these relationships inevitably follow. If, on the other hand, the data\ncan be put into these functional forms, then it can be shown by solving\nthe field equations with this data (Stoeger, {\\it et al.} 1992;\nAra\\'{u}jo, Stoeger, {\\it et al.}, in preparation) that the universe\nmust be FLRW. Thus, being able to fit the data to these forms, assures us\nthat the universe is FLRW. Not being able to do so, assures us that it is\nnot FLRW.\n \nReturning to the main object of our derivation, substituting equation (\\ref{yeliptic}) into\nthe left-hand-side of equation  (\\ref{ymax}),\nwe have simply:\n\n\\begin{eqnarray}\n\\frac{g}{\\Omega_{\\Lambda}^{1\/2}}\\biggl[F(\\phi, k)\\mid_{(1+z)^{-1} = 1} &-& F(\\phi, k)\n  \\mid_{(1+z_{max})^{-1}}\\biggr]    \\nonumber \\\\\n& & = \\frac{1+z_{max}}{\\sqrt{\\Omega_{\\Lambda} + (1+z_{max})^3(1 - \\Omega_{\\Lambda})}} \\>.  \\label{elzmax}\n\\end{eqnarray}\n\n\nThis is a transcendental relationship for $\\Omega_{\\Lambda}$ as a\nfunction of $z_{max}$. It is worth noticing that it does not involve\nany other parameters! This is the relationship which represents the\nnumerical results obtained by Krauss and Schramm (1993). \\\\\n\nThe solutions to this implicit algebraic equation were obtained using\nMAPLE, and were checked by hand for values of $\\Omega_{\\Lambda}$ near\nthe concordance model value of $\\Omega_{\\Lambda} = 0.73$. They are given in \nTable 1 and Figure 1 below.\\footnote{There are alternative sequences of\nsteps for obtaining these results -- for instance using the solution of (6) to\nwrite down a general formula for $C$ as a function of $z+1$ and then\ndifferentiating this, setting the result to zero, and solving for\n$\\Omega_{\\Lambda}$ in terms of $z_{max}$. But they all involve explicitly or\nimplicitly the steps we have indicated -- solving the Friedmann equation to\nobtain the relationship between $y$ and the observable $z$ (redshift), and\ndetermining the equation for $C_{max}$ in terms of $y_{max}$ or, from the first\nstep, its observational equivalent $z_{max}$. Because of the complication of\nincluding a non-zero $\\Omega_{\\Lambda}$, at some point a numerical solution\nwill always be needed. See, for instance Carroll, {\\it et al.} (1992), pp.\n510-512. We have chosen to keep the solution of Friedman\nequation analytic, in terms of elliptic integrals, in order to derive the\ncharacteristic FLRW closed-form expression for $C(z)$ and to solve the resulting\nalgebraic equation numerically.} We can immediately see, that for the\nconcordance model we should find $z_{max} = 1.64$. For the nearby best fit\nmodel of Riess,\n{et al.} (2004) we have already mentioned, $z_{max} = 1.62$. Values of\n$z_{max}$ for many other values of $\\Omega_{\\Lambda}$ are given, as well. These\nverify the values presented in Krauss and Schramm (1992), and those\nevident in the plots of Refsdal, {\\it et al.} (1967), Carroll, {\\it et al.}\n(1992), and Hellaby (2006). \\\\\n\n\\section{Non-Flat FLRW Universes}\n\nIf the universe is not flat, a slight\ngeneralization of these same equations obtains, with the solution for $y$\ntaking the same general form as given in equation  (\\ref{yeliptic}). The generalization\nof equation (\\ref{elzmax}) in this case will, however, include -- as is intuitively\nclear -- a dependence on $\\Omega_M$ as well as on $\\Omega_{\\Lambda}$. Using the\ngeneral relationship emphasized by Hellaby (2006)\n\n\\begin{equation}\n \\Lambda C_{max}^3 - 3C_{max} + 6 M_{max} = 0,  \\label{lcmmax}\n\\end{equation}\n\n\\noindent\nwe can determine $\\Omega_M$ through $M_{max}$ in terms of $C_{max}$ and \n$\\Lambda$. It is important to stress that equation (\\ref{lcmmax}) holds for these \nquantities as measured at $z_{max}$, or $y_{max}$, down the observer's past\nlight cone. From Hellaby's (2006) results, we easily find that, for FLRW,\n\n\\begin{equation}\nM_{max} = \\frac{4}{3} \\pi \\rho_M C_{max}^3, \\label{masseq}\n\\end{equation}\nwhere $\\rho_M = \\rho(t_{max}) = \\rho_0(1+z_{max})^3.$ Here $\\rho_0$ is\nthe density at our time now, $t_0$.\nUsing this together with the definition of $\\Omega_M \\equiv 8\\pi\\rho_0\/3{H_0}^2$\nand equation (\\ref{lcmmax}), we easily obtain\\footnote{As in Hellaby (2006), we also use units such that $G=c=1$.}\n\n\\begin{equation}\n\\Omega_M = \\frac{1}{H_0^2(1+z_{max})^3}[C_{max}^{-2} - \\Omega_{\\Lambda}H_0^2]. \\label{omegam}\n\\end{equation}\n\nThis can be substituted into the non-flat versions of equations (\\ref{dzdy}) and (\\ref{ymax}),\n\n\\begin{equation}\ndz\/dy = R_0H_0 \\sqrt{\\Omega_{\\Lambda} + \\Omega_M(1+z)^3 - (\\Omega_0 -1)\n   (1+z)^2}, \\label{dzdy2}\n\\end{equation}\nand\n\\begin{equation}\ny_{max} = \\frac{1+z_{max}}{R_0H_0\\sqrt{\\Omega_{\\Lambda}+\\Omega_M(1+z_{max})^3\n  - (\\Omega_0 - 1)(1+z_{max})^2}}, \\label{ymax2}\n\\end{equation}\n\nIn passing, we immediately see from equation (\\ref{omegam}) that we have a \nuseful observational criterion for flatness of an FLRW universe:\n\n\\begin{equation}\n\\Omega_0 = 1 \\Rightarrow (1+z_{max})^{-3}\\Biggl[\\frac{1}{H_0^2C_{max}^2} - \\Omega_{\\Lambda}\\Biggr] + \n  \\Omega_{\\Lambda} - 1 = 0, \\label{flatness}\n\\end{equation}\nThus, if already know that the universe is flat, or nearly so, and we know both $z_{max}$ and $C_{max}$, we\ncan directly determine $\\Omega_{\\Lambda}$, and therefore $\\Omega_M$ itself\nfrom equation (\\ref{flatness}).\n\nProceeding on, then, equation (\\ref{omegam}) can therefore be substituted into the\nnon-flat version of equation (\\ref{elzmax}), which\nis the same as equation (\\ref{elzmax}), except that its right-hand-side is identical\nto right-hand-side of equation (\\ref{ymax2}) without the $R_0H_0$ factors\nin the denominator (these have cancelled out, as before). Thus, we have, finally,\nthe resulting algebraic relationship involving  $C_{max}$, $z_{max}$,\n$H_0$ and $\\Omega_{\\Lambda}$ as the general FLRW relationship corresponding to the flat case\ngiven in equation (\\ref{elzmax}):\n\n\\begin{eqnarray}\n&  &\\frac{g}{\\Omega_{\\Lambda}^{1\/2}}\\biggl[ F(\\phi, k) \\mid_{(1+z)^{-1} = 1} -F(\\phi, k) \\mid_{(1+z_{\\max})^{-1}}\\biggr]  \\nonumber \\\\\n&  & \\hfill{\\qquad}= \\frac{1+z_{\\max}}{\\sqrt{\\Omega_{\\Lambda} + \\Omega_{M}(1+z_{\\max})^3 -(\\Omega_0-1)(1+z_{\\max})^2}}.  \\label{omegaleq}\n\\end{eqnarray}\nHere and in the solution of the Friedmann  equation for the general\nFLRW case, the parameters associated with that solution are now given by:\n\n\\begin{eqnarray}\n\\phi_{(1+z)^{-1}} &=&  \\cos^{-1}\\Biggl[\\frac{(A-B) - (\\bar{A}+\\bar{B})A(1+z)}\n{(A+B)-(\\bar{A}+\\bar{B})A(1+z)}\\Biggr], \\nonumber \\\\\nk^2 &=& \\frac{(A+B)^2 - (a - b)^2}{4AB}, \\nonumber \\\\\ng &=& \\frac{1}{\\sqrt{AB}}, \\nonumber\n\\end{eqnarray}\nwith $a \\equiv -\\frac{\\Omega_0-1}{\\Omega_{\\Lambda}}$,\n$b \\equiv \\frac{\\Omega_M}{\\Omega_{\\Lambda}}$,\nand\n\n\n\\begin{eqnarray}\nA^2 &=& \\bar{A}^2 + \\bar{B}^2 - \\bar{A}\\bar{B}, \\nonumber \\\\\nB^2 &=& 3(\\bar{A}^2 + \\bar{B}^2) + 3\\bar{A}\\bar{B}. \\nonumber \n\\end{eqnarray}\n\nHere, further, \n\n\\begin{eqnarray}\n\\bar{A} = \\Biggl\\{\\frac{\\Omega_M}{2\\Omega_{\\Lambda}} +\n\\Biggl[\\frac{{\\Omega_M}^2}{4\\Omega_{\\Lambda}^2} -\n\\frac{(\\Omega_0-1)^3}{27 \\Omega_{\\Lambda}^3}\\Biggr]^{1\/2}\\Biggr\\}^{1\/3},  \\nonumber \\\\\n\\bar{B} = \\Biggl\\{\\frac{\\Omega_M}{2\\Omega_{\\Lambda}} -\n\\Biggl[\\frac{{\\Omega_M}^2}{4\\Omega_{\\Lambda}^2} -\n\\frac{(\\Omega_0-1)^3}{27 \\Omega_{\\Lambda}^3}\\Biggr]^{1\/2}\\Biggr\\}^{1\/3}. \\nonumber \n\\end{eqnarray}\nIn these equations, remember that $\\Omega_M$ is given by equation (\\ref{omegam}), so that relationship given by\nequation (\\ref{omegaleq}) is indeed an algebraic relationship involving $C_{max}$,\n$z_{max}$, $H_0$ and $ \\Omega_{\\Lambda}$.\nThus, if both $C_{max}$ and $z_{max}$, together with $H_0$, are all known from data, then\nequation (\\ref{omegaleq}) will determine $\\Omega_{\\Lambda}$, the only unknown. Using \nthat result in equation (\\ref{omegam}) will also determine $\\Omega_M$. Thus,\nobservational determination of both $C_{max}$ and $z_{max}$,\nwill determine both $\\Omega_M$ and $\\Omega_{\\Lambda}$, as long as we also\nknow  $H_0$. \\\\\n\n\\clearpage\n\\begin{table}\n\\begin{tabular} {c c c c c c c c } \\hline\n${\\bf \\Omega_{\\Lambda}}$&${\\bf z_{max}}$&${\\bf \\Omega_{\\Lambda}}$&${\\bf z_{max}}$&${\\bf \\Omega_{\\Lambda}}$&${\\bf z_{max}}$&${\\bf \\Omega_{\\Lambda}}$&${\\bf z_{max}}$ \\\\ \n0.59&1.50&0.65&1.55&{\\bf0.71}&{\\bf1.62}&0.77&1.71 \\\\ \n0.60&1.51&0.66&1.56&0.72&1.63&0.78&1.72 \\\\ \n0.61&1.51&0.67&1.57&{\\bf0.73}&{\\bf1.64}&0.79&1.74 \\\\ \n0.62&1.52&0.68&1.58&0.74&1.66&0.80&1.76 \\\\ \n0.63&1.53&0.69&1.59&0.75&1.67&0.81&1.78 \\\\ \n0.64&1.54&0.70&1.61&0.76&1.69&0.82&1.81 \\\\ \\hline\n\\end{tabular}\n\\caption{List of pairs ($\\Omega_{\\Lambda}$,$z_{max}$) for $0.59 \\leq \\Omega_{\\Lambda}\n\\leq 0.82$ and $1.5 \\leq z_{max} \\leq 1.81$.}\n\\end{table}\n\\clearpage\n\\begin{figure}\n\\begin{center}\n\\includegraphics {asfigure1}\n\\end{center}\n\\caption{Plot of $\\Omega_{\\Lambda}$ -- $z_{max}$, given by equation (\\ref{elzmax}), which\nis for a flat FLRW universe. Here $z_{max}$ is the\nredshift at which the maximum of the angular diameter distance, $C_{max}$\noccurs.}\n\\label{araujo}\n\\end{figure}\n\\clearpage\n\n\\section{Observational Prospects and Conclusion}\n\nWhat are the prospects for actually determining $C_{max}$ and $z_{max}$ from\nobservations? We would\ncertainly need precise SN Ia luminosity-distance, or ultra-compact radio-source\nangular-diameter distance, and redshift data out to\n$z \\approx 1.8$ or so. In the SN Ia case this would require careful,\nlong-range programs\nusing space-telescopes. However, as already mentioned, we already have\ndetected and measured SN Ia out to $z \\approx 1.7$, and in a recent\nassessment (Davis, Schmidt and Kim 2006), precision SN Ia measurements\nto $z \\approx 1.8$ are considered attainable. This is already considered\nan important goal, in order to confirm at what redshift (and cosmic epoch)\nthe universe made the transition from deceleration to acceleration. It is\ncertainly fortuitous that the same redshift range promises to provide a\nstrong independent test of the concordance FLRW model we have derived from\nCMB, SN Ia, and large-scale structure measurements. \\\\\n\nHere we have provided a brief presentation of the straightforward relationship\n(first found in numerical form by Krauss and Schramm (1992)) between the\npresent value of $\\Omega_{\\Lambda}$ and the redshift $z_{max}$ at\nwhich the angular-diameter (or observer area) distance $C$ occurs in a flat\nFLRW cosmology, like that which apparently models our universe. Furthermore,\nwe have generalized this to non-flat FLRW cases, adding the $C_{\\max}$\nmeasurements themselves. In doing this we have derived the characteristic\nFLRW observational relationships in closed form for $C(z)$ and $\\hat{M}_0\n(z)$ in the $\\Lambda \\neq 0$ case, and found a very simple and potentially\nuseful observational criterion for flatness.  These results\npromise to provide improved determination of the\nbest fit cosmological model, or a strong consistency test of it, (depending on\nhow the relationship and the data supporting it are used), once we have\nenough precise high-redshift luminosity-distance (or angular-diameter distance)\ndata. That should be possible in the near future with the rapid progress being\nmade in SN Ia measurements from space. If the concordance model --\na nearly flat universe with $\\Omega_M = 0.27$ and $\\Omega_{\\Lambda} = 0.73$ --\nis approximately correct, we should find observationally that $z_{max}\n\\approx 1.64$. \\\\ \n\nOur thanks to George Ellis and Charles Hellaby for discussions and comments,\nand to an anonymous referees for several helpful suggestions for clarification \nand for checking our results, and to one of them for pointing out the\nmuch earlier 1993 Krauss and Schramm paper.  \\\\\n  \n\n\\noindent\n{\\Large \\bf References } \\\\\n\n\\noindent\nAlbrecht, A., {\\it et al.}, 2006, Report of the Dark Energy Task Force, \nastro-ph\/0609591. \\\\\nBen\\'{i}tez, N., Riess, A. G., Nugent, P., Dickinson, M., Chornook, R., \\& \nFilippenko, A. V. 2002, ApJ, 577, L1. \\\\\nBennett, C. L., {\\it et al.} 2003, ApJS, 148, 1. \\\\\nByrd, P. F. \\& Friedman, M. D. 1954, {\\it Handbook of Elliptic\nIntegrals for Engineers and Physicists}, Springer Verlag. \\\\\nCarroll, S. M.,  Press, W. H., \\& Turner, E. L., 1992, ``The Cosmological\nConstant,'' Ann. Rev. Astron. \\& Astrophys. 30, 499-542.  \\\\\nEfstathiou, G., {\\it et al.} 2002, MNRAS, 330, L29 \\\\\nDavis, T. M., Schmidt, B. P. \\& Kim, A. G. 2006, PASP, 118, 205. \\\\\nEllis, G. F. R. 1971, ``Relativistic Cosmology,'' in {\\it General Relativity\nand Cosmology}, Proc. Int. School Phys. ``Enrico Fermi,'' R. K. Sachs,\neditor (New York: Academic Press), pp. 104-182 (see especially pp. 153-1540.\\\\\nEllis, G. F. R., Nel, S. D., Maartens, R., Stoeger, W. R., \\& Whitman, A. P.\n1985, Phys. Reports, 124 (No. 5 and 6), 315. \\\\\nEllis, G. F. R. \\& Tivon, G. 1985, Observatory, 105, 189.\\\\\nEllis, G. F. R. \\& Stoeger, W. R. 1987. Class. Quantum Grav., 4, 1697. \\\\\nEtherington, I. M. H. 1933, Phil. Mag., 15, 761. \\\\\nGilliland, R. L., Nugent, P. E., \\& Phillips, M. M. 1999, ApJ, 521, 30. \\\\\nHellaby, C. W. 2006, MNRAS, 370, 239 (astro-ph\/0603637). \\\\\nHoyle, F., 1961, in Moller, C., ed., Proc. Enrico Fermi School of Physics,\nCourse XX, Varenna, {\\it Evidence for Gravitational Theories}, Academic\nPress, New York, p. 141. \\\\\nJackson, J. C. \\& Doddgson, M. 1997, Mon. Not. R. Astron. Soc., 285, 806. \\\\\nJackson, J. C. 2004, JCAP, 11, 007. \\\\\nJeffrey, A., 1995, {\\it Handbook of Mathematical Formulas and Integrals},\nAcademic Press, Inc., pp. 225-234. \\\\\nKrauss, L. M., and Schramm, D. N. 1993, ApJ, 405, L43. \\\\\nMcCrea, W. H., 1935, Z. Astrophys., 9, 290. \\\\\nMortsell, E., Gunnarson, C., \\& Goobar, A. 2001, ApJ 561, 106. \\\\\nPeacock, J. A., {\\it et al.} 2001, Nature, 410, 169. \\\\\nPeebles, P. J. E., 1993, {\\it Principles of Physical Cosmology}, Princeton\nUniversity Press, Princeton, NJ, pp. 325-329. \\\\\nPercival, W. J., {\\it et al.} 2001, MNRAS, 327, 1297. \\\\\nPerlmutter, S., {\\it et al.} 1999, ApJ, 517, 565. \\\\\nRefsdal, S., Stabell, R., \\& de Lange, F. G. 1967, Mem. R. Astron. Soc., 71,\n143. \\\\\nRiess, A. G., {\\it et al.} 1998, AJ, 116, 1009. \\\\\nRiess, A. G., {\\it et al.} 2001, ApJ, 560, 49. \\\\\nRiess, A. G., {\\it et al.} 2004, ApJ, 607, 665. \\\\\nSpergel, D. N., {\\it et al.} 2003, ApJS, 148, 175.\\\\\nStoeger, W. R., Ellis, G. F. R. \\& Nel, S. D. 1992, Class. Quantum Grav., 9,\n509.  \n\\end{document}  \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Preliminaries}\\label{sec:pre}\nIn this section we recall the definitions of Yang--Baxter operators and\nassociated link invariants.\nIf an invertible linear map $R:\\mathbb{C}^N\\otimes\\mathbb{C}^N\\to\\mathbb{C}^N\\otimes\\mathbb{C}^N$ satisfies the\nfollowing Yang--Baxter equation, it is called a Yang--Baxter operator.\n\\begin{equation*}\n  (R\\otimes{id})({id}\\otimes{R})(R\\otimes{id})\n  =\n  ({id}\\otimes{R})(R\\otimes{id})({id}\\otimes{R}),\n\\end{equation*}\nwhere $id:\\mathbb{C}^N\\to\\mathbb{C}^N$ is the identity.\nIf there exists a homomorphism $\\mu:\\mathbb{C}^N\\to\\mathbb{C}^N$ and scalars $\\alpha,\\beta$\nsatisfying the following two equations, the quadruple $S=(R,\\mu,\\alpha,\\beta)$\nis called an enhanced Yang-Baxer operator \\cite{Turaev:INVEM88}.\n\\begin{gather*}\n  (\\mu\\otimes\\mu)R=R(\\mu\\otimes\\mu),\n  \\\\\n  \\Sp_2\\left(R^{\\pm1}({id}\\otimes\\mu)\\right)=\\alpha^{\\pm1}\\beta\\,{id},\n\\end{gather*}\nwhere\n$\\Sp_k:\\operatorname{End}(\\mathbb{C}^{\\otimes{k}})\\to\\operatorname{End}(\\mathbb{C}^{\\otimes{k-1}})$ is the\noperator trace defined as\n\\begin{multline*}\n  \\Sp_k(f)(v_{i_1}\\otimes{v_{i_2}}\\otimes\\dots\\otimes{v_{i_{k-1}}})\n  \\\\=\n  \\sum_{j_1,j_2,\\dots,j_{k-1},j=0}^{N-1}\n  f_{i_1,i_2,\\dots,i_{k-1},j}^{j_1,j_2,\\dots,j_{k-1},j}\n  (v_{j_1}\\otimes{v_{j_2}}\\otimes\\dots\\otimes{v_{j_{k-1}}}\\otimes{v_{j}}),\n\\end{multline*}\nwhere\n\\begin{equation*}\n  f(v_{i_1}\\otimes{v_{i_2}}\\otimes\\dots\\otimes{v_{i_k}})\n  =\n  \\sum_{j_1,j_2,\\dots,j_k=0}^{N-1}\n  f_{i_1,i_2,\\dots,i_k}^{j_1,j_2,\\dots,j_k}\n  \\left({v_{j_1}}\\otimes{v_{j_2}}\\otimes\\dots\\otimes{v_{j_k}}\\right)\n\\end{equation*}\nfor a basis $\\{v_0,v_1,\\dots,v_{N-1}\\}$ of $\\mathbb{C}^N$.\n\\par\nFor an enhanced Yang--Baxter operator one can define a link invariant as follows\n\\cite{Turaev:INVEM88}.\nFirst we represent a given link $L$ as the closure of a braid $\\xi$ with\n$n$ strings.\nConsider the $n$-fold tensor product $\\left(\\mathbb{C}^N\\right)^{\\otimes{n}}$ and\nassociate the homomorphism\n$b_R(B):\\left(\\mathbb{C}^N\\right)^{\\otimes{n}}\\to\\left(\\mathbb{C}^N\\right)^{\\otimes{n}}$ by\nreplacing $\\sigma_i^{\\pm1}$\n(the usual $i$-th generator of the braid group) in $\\xi$ with\n\\begin{equation*}\n  \\underset{i-1}{\\underbrace{{id}\\otimes\\dots\\otimes{id}}}\n  \\otimes{R^{\\pm1}}\\otimes\n  \\underset{N-i-1}{\\underbrace{{id}\\otimes\\dots\\otimes{id}}}.\n\\end{equation*}\nThen taking the operator trace $n$ times we define\n\\begin{equation*}\n  T_S(\\xi)\n  =\n  \\alpha^{-w(\\xi)}\\beta^{-n}\n  \\Sp_1\\left(\\Sp_2\n         \\left(\\cdots\n           \\left(\\Sp_n\n             \\left(b_R(\\xi)\\mu^{\\otimes{n}}\n             \\right)\n           \\right)\n         \\right)\n       \\right),\n\\end{equation*}\nwhere $w(\\xi)$ is the sum of the exponents.\nThen $T_S(\\xi)$ defines a link invariant and denoted by $T_S(L)$.\n\\par\nTo define the (generalized) Alexander polynomial from an enhanced Yang--Baxter\noperator we have to be more careful, since $T_S$ always vanishes in this case.\nIf the following homomorphism\n\\begin{equation*}\n  T_{S,1}(\\xi)\n  =\n  \\alpha^{-w(\\xi)}\\beta^{-n}\n  \\Sp_2\\left(\\Sp_3\n         \\left(\\cdots\n           \\left(\\Sp_n\n             \\left(b_S(\\xi)({id}\\otimes\\mu^{\\otimes(n-1)})\n             \\right)\n           \\right)\n         \\right)\n       \\right)\n   \\in\\operatorname{End}(\\mathbb{C}^N)\n\\end{equation*}\nis a scalar multiple and\n\\begin{equation*}\n  \\Sp_1(\\mu)T_{S,1}(\\xi)=T_S(\\xi)\n\\end{equation*}\nfor any $\\xi$, then the scalar defined by $T_{S,1}(\\xi)$ becomes a link\ninvariant (even if $\\Sp_1(\\mu)=0$) and is denoted by $T_{S,1}(L)$.\nNote that this invariant can be regarded as an invariant for $(1,1)$-tangles,\nwhere a $(1,1)$-tangle is a link minus an open interval.\n\\section{The intersection of the generalized Alexander polynomials and the\ncolored Jones polynomials}\\label{sec:Alexander}\nIn \\cite{Akutsu\/Deguchi\/Ohtsuki:JKNOT92} Akutsu, Deguchi and Ohtsuki\ndefined a generalization of the multivariable Alexander polynomial for colored\nlinks.\nFirst we will briefly describe their construction only for the case where\nall the colors are the same according to \\cite{Deguchi:1994}.\n\\par\nFix an integer $N\\ge2$ and a complex number $p$.\nPut $s=\\exp(\\pi\\sqrt{-1}\/N)$ and $[k]=(s^k-s^{-k})\/(s-s^{-1})$ for a complex\nnumber $k$.\nNote that $[N]=0$ and $[N-k]=[k]$.\n\\par\nLet $U_q(sl(2,\\mathbb{C}))$ be the quantum group generated by $X,Y,K$ with the following\nrelations.\n\\begin{equation*}\n  KX=sXK,\\quad KY=s^{-1}YK,\\quad XY-YX=\\frac{K^2-K^{-2}}{s-s^{-1}}.\n\\end{equation*}\nLet $F(p)$ be the $N$-dimensional vector space over $\\mathbb{C}$ with basis\n$\\{f_0,f_1,\\dots,f_{N-1}\\}$.\nWe give an action of $U_q(sl(2,\\mathbb{C}))$ on $F(p)$ by\n\\begin{align*}\n  X(f_i)&=\\sqrt{[2p-i+1][i]}f_{i-1},\n  \\\\\n  Y(f_i)&=\\sqrt{[2p-i][i+1]}f_{i+1},\n  \\\\\n  K(f_i)&=s^{p-i}f_{i}.\n\\end{align*}\nUsing Drinfeld's universal $R$-matrix given in \\cite{Drinfeld:ICM86},\nwe can define a set of enhanced Yang--Baxter operators $S_A(p)$\nwith complex parameter $p$.\nThen Akutsu--Deguchi--Ohtsuki's generalized Alexander\npolynomial is defined to be $T_{S_A(p),1}$ by using the notation in the previous\nsection.\nWe denote it by $\\Phi_N(L,p)$ for a link $L$.\nNote that if $N=2$ the invariant $\\Phi_2(L,p)$ is the same as the\nmultivariable Alexander polynomial \\cite{Jun:PACJM93}.\n\\par\nNext we review the colored Jones polynomial at the $N$-th root of unity.\nThere is another $N$-dimensional representation of $U_q(sl(2,\\mathbb{C}))$,\ncorresponding to the usual $N$-dimensional irreducible representation of\n$sl(2,\\mathbb{C})$.\nLet $E$ be the $N$-dimensional complex vector space with basis\n$\\{e_0,e_1,\\dots,e_{N-1}\\}$ and we define the action of $U_q(sl(2,\\mathbb{C}))$\nby\n\\begin{align*}\n  X(e_i)&=[i+1]e_{i+1},\n  \\\\\n  Y(e_i)&=[i]e_{i-1},\n  \\\\\n  K(e_i)&=s^{i-(N-1)\/2}e_{i}.\n\\end{align*}\n(See for example \\cite[(2.8)]{Kirby\/Melvin:INVEM91}.)\nBy using Drinfeld's universal $R$-matrix again we have another enhanced\nYang--Baxter operator $S_J$.\nThen the invariant $T_{S_J,1}$ coincides with the colored Jones polynomial\nof a link each of whose component decorated by the $N$-dimensional irreducible\nrepresentation evaluated at $t=s^2=\\exp(2\\pi\\sqrt{-1}\/N)$.\nNote that before evaluating at $s^2$, we have to normalize the colored Jones\npolynomial so that its value of the trivial knot is one for otherwise\nthe invariant would be identically zero.\nThis is well-defined since the colored Jones polynomial defines a well-defined\n$(1,1)$-tangle invariant (\\cite[(3.9) Lemma]{Kirby\/Melvin:INVEM91}).\nWe will denote $T_{S_J,1}$ by $J_{N}$.\n\\par\nNow we put $p=(N-1)\/2$ in the Akutsu--Deguchi--Ohtsuki invariant.\nThen since $[N-k]=[k]$, we have\n\\begin{align*}\n  X(f_i)&=[i]f_{i-1},\n  \\\\\n  Y(f_i)&=[i+1]f_{i+1},\n  \\\\\n  K(f_i)&=s^{(N-1)\/2-i}f_{i}\n\\end{align*}\nand so the two representation $F((N-1)\/2)$ and $E$ are quite similar.\nIn fact if we exchange $X$ and $Y$, and replace $K$ with $K^{-1}$ then these two\ncoincide.\n(This automorphism is known as the Cartan automorphism\n\\cite[p.~123, Lemma VI.1.2]{Kassel:quantum_groups}.)\nTherefore they determine the same Yang--Baxter operator and the same\nlink invariant, that is, we have the following theorem.\n\\begin{thm}\\label{thm:Alexander=Jones}\n  The Akutsu--Deguchi--Ohtsuki invariant with all the colors $p=(N-1)\/2$\n  coincides with the colored Jones polynomial corresponding to the\n  $N$-dimensional irreducible representation evaluated at\n  $\\exp(2\\pi\\sqrt{-1}\/N)$.\n  More precisely, we have $\\Phi_{N}(L,(N-1)\/2)=J_{N}(L)$ for every link $L$.\n\\end{thm}\n\\begin{rem}\nAfter finishing this work we were informed by Deguchi that it has already\nobserved \\cite{Deguchi:JPHYS191} that the $R$-matrices given by $F((N-1)\/2)$\nand $E$ coincide.\n\\end{rem}\n\\section{$\\check{R}$-matrix for the colored Jones polynomial at the $N$th root of\n         unity}\\label{sec:Jones}\nLet $R_J$ be the $\\check{R}$-matrix shown in\n\\cite[Corollary~2.32]{Kirby\/Melvin:INVEM91}, which is the $N^2\\times N^2$ matrix\nwith\n$((i,j),(k,l))$th entry\n\\begin{align*}\n  \\left(R_J\\right)_{kl}^{ij}=\n  \\sum_{n=0}^{\\min{(N-1-i,j)}}\n  &\\delta_{l,i+n}\\delta_{k,j-n}\n  \\frac{(s-s^{-1})^n}{[n]!}\n  \\frac{[i+n]!}{[i]!}\\frac{[N-1+n-j]!}{[N-1-j]!}\n  \\\\\n  &\\times\n  s^{2(i-(N-1)\/2)(j-(N-1)\/2)-n(i-j)-n(n+1)\/2},\n\\end{align*}\nwhere $[k]!=[k][k-1]\\cdots[2][1]$.\nNote that our matrix $R_J$ corresponds to $\\check{R}$ in\n\\cite[Definition~2.35]{Kirby\/Melvin:INVEM91}.\nThis matrix is used to define an enhanced Yang--Baxter operator and\nthe link invariant $J_{N}$ described in the previous section.\n\\par\nThe aim of this section is to transform it to a matrix similar to Kashaev's\n$\\check{R}$-matrix.\nLet $W$ and $D$ be the $N\\times N$ matrices with $(i,j)$th entry\n$W_{j}^{i}=s^{2ij}$ and $D_{j}^{i}=\\delta_{i,j}s^{(N-1)i}$ respectively,\nwhere $\\delta_{i,j}$ is Kronecker's delta.\nWe will calculate the product\n$\\tilde{R}_J\n=(W\\otimes W)({id}\\otimes{D}){R_J}\n({id}\\otimes{D^{-1}})(W^{-1}\\otimes{W^{-1}})$\nwith $id$ the $N\\times{N}$ identity matrix and show the following proposition.\n\\begin{prop}\\label{prop:J}\n\\begin{equation*}\n  \\left(\\tilde{R}_J\\right)_{ab}^{cd}\n  =\n  \\begin{cases}\n    \\rho(a,b,c,d)(-1)^{a+b+1}\\dfrac{[d-c-1]![N-1+c-a]!}{[d-b]![b-a-1]!}\n    \\quad&\\text{if $d\\ge b>a\\ge c$},\n    \\\\[5mm]\n    \\rho(a,b,c,d)(-1)^{a+c}\\dfrac{[b-d-1]![N-1+c-a]!}{[c-d]![b-a-1]!}\n    \\quad&\\text{if $b>a\\ge c\\ge d$},\n    \\\\[5mm]\n    \\rho(a,b,c,d)(-1)^{b+d}\\dfrac{[N-1+b-d]![c-a-1]!}{[c-d]![b-a-1]!}\n    \\quad&\\text{if $c\\ge d\\ge b>a$},\n    \\\\[5mm]\n    \\rho(a,b,c,d)(-1)^{c+d}\\dfrac{[N-1+b-d]![a-b]!}{[c-d]![a-c]!}\n    \\quad&\\text{if $a\\ge c\\ge d\\ge b$},\n    \\\\[5mm]\n    0\\quad&\\text{otherwise},\n  \\end{cases}\n\\end{equation*}\nwhere\n$\\rho(a,b,c,d)=s^{-N^2\/2+1\/2+c+d-2b+(a-d)(c-b)}[N-1]!(s-s^{-1})^{2(N-1)}\/N^2$.\n\\end{prop}\n\\begin{proof}\nSince\n$(W\\otimes{W})_{ab}^{ef}=s^{2ae+2bf}$,\n$({id}\\otimes{D})_{ef}^{kl}=\\delta_{e,k}\\delta_{f,l}s^{(N-1)l}$,\n$\\left({id}\\otimes{D^{-1}}\\right)_{ij}^{gh}\n  =\\delta_{g,i}\\delta_{h,j}s^{-(N-1)j}$,\n$\\left(W^{-1}\\otimes{W^{-1}}\\right)_{gh}^{cd}=s^{-2cg-2dh}\/N^2$,\nwe have\n\\begin{align*}\n  &N^2\\left(\\tilde{R}_J\\right)_{ab}^{cd}\n  \\\\\n  &=\n  \\sum_{i,j,k,l,e,f,g,h=0}^{N-1}\\sum_{n=0}^{\\min{(N-1-i,j)}}\n  \\delta_{e,k}\\delta_{f,l}\\delta_{g,i}\\delta_{h,j}\\delta_{l,i+n}\\delta_{k,j-n}\n  s^{2ae+2bf-2cg-2dh+(N-1)(l-j)}\n  \\\\\n  &\\quad\\times\n  \\frac{(s-s^{-1})^n}{[n]!}\\frac{[i+n]!}{[i]!}\\frac{[N-1+n-j]!}{[N-1-j]!}\n  s^{2(i-(N-1)\/2)(j-(N-1)\/2)-n(i-j)-n(n+1)\/2}\n  \\\\\n  &=\n  \\sum_{i,j=0}^{N-1}\\sum_{n=0}^{\\min{(N-1-i,j)}}\n \n  s^{(N-1)^2\/2}s^{(2b-2a+N)n-n^2\/2-3n\/2+(2a-2d+n+2)j+(2b-2c+2j-n)i}\n  \\\\\n  &\\quad\\times\n  (s-s^{-1})^n\\frac{[N-1+n-j]!}{[N-1-j]!}\\qbinom{n+i}{i},\n\\end{align*}\nwhere\n\\begin{equation*}\n  \\qbinom{x}{y}=\\frac{[x]!}{[y]![x-y]!}.\n\\end{equation*}\nSince the summation $\\sum_{i,j=0}^{N-1}\\sum_{n=0}^{\\min{(N-1-i,j)}}$ is the same\nas $\\sum_{n=0}^{N-1}\\sum_{j=n}^{N-1}\\sum_{i=0}^{N-1-n}$, we have\n\\begin{align*}\n  N^2\\left(\\tilde{R}_J\\right)_{ab}^{cd}\n  &=\n  s^{(N-1)^2\/2}\\sum_{n=0}^{N-1}\\sum_{j=n}^{N-1}\n  s^{(2b-2a+N)n-n^2\/2-3n\/2+(2a-2d+n+2)j}(s-s^{-1})^n\n  \\\\\n  &\\quad\\times\n  \\frac{[N-1+n-j]!}{[N-1-j]!}S(n,2(b-c+j)-n)\n\\end{align*}\nwith $S(\\alpha,\\beta)=\\sum_{i=0}^{N-1}s^{\\beta i}\\qbinom{\\alpha+i}{i}$.\n\\par\nReplacing $j-n$ with $k$, the summation turns out be\n$\\sum_{k=0}^{N-1}\\sum_{n=0}^{N-1-k}$ and we have\n\\begin{equation*}\n  N^2\\left(\\tilde{R}_J\\right)_{ab}^{cd}\n  =\n  s^{(N-1)^2\/2}\\sum_{k=0}^{N-1}\n  s^{2(a-d+1)k}[N-1-k]!X(k),\n\\end{equation*}\nwhere\n\\begin{equation*}\n  X(k)=\n  \\sum_{n=0}^{N-1-k}\n  (-1)^n s^{2(b-d)n+kn+n(n+1)\/2}\n  \\frac{(s-s^{-1})^n}{[N-1-k-n]!}\n  S(n,2(b-c+k)+n).\n\\end{equation*}\nNote that from Lemma~\\ref{lem:ST}, we have\n\\begin{multline}\\label{eq:S}\n  S(n,2(b-c+k)+n)\n  =\n  \\\\\n  (1-s^{2(b-c+k-1)})(1-s^{2(b-c+k-2)})\\cdots(1-s^{2(b-c+k+n-N+1)}).\n\\end{multline}\n\\par\nAs easily seen from Lemma~\\ref{lem:ST}, $S(\\alpha,\\beta)$ and $T(\\alpha,\\beta)$\nvanishes if $(\\beta-\\alpha-2)\\ge0\\ge(\\beta+\\alpha-2N+2)$ and if\n$(\\beta+\\alpha-1)\\ge0\\ge(\\beta-\\alpha+1)$ respectively.\nWe will use this fact repeatedly from now on and divide the proof into\nsome cases according to the order of $a,b,c,d$.\n\\par\nFirst we divide the proof into two cases; $b>c$ and $c\\ge b$.\n\\medskip\\par\\noindent\n{\\bf Case 1 ($b>c$).}\\qquad\nIn this case $b-c+k-1\\ge 0$ and so from \\eqref{eq:S}, $S(n,2(b-c+k)+n)=0$ if\n$n\\le N-1-k-b+c$.\nIf $n>N-1-k-b+c$, we see that\n\\begin{multline*}\n  S(n,2(b-c+k)+n)=\n  \\\\\n  (-1)^{N-1-n}s^{(N-1-n)(2(b-c+k)+n-N)\/2}(s-s^{-1})^{N-1-n}\n  \\frac{[b-c+k-1]!}{[b-c+k+n-N]!}.\n\\end{multline*}\nTherefore\n\\begin{align*}\n  X(k)&=\n  (-1)^{N+1}(s-s^{-1})^{N-1}s^{(b-c+k)(N-1)-N(N-1)\/2}\n  \\frac{[b-c+k-1]!}{[b-c-1]!}\n  \\\\\n  &\\quad\\times\n  \\sum_{n=N-1-k-b+c}^{N-1-k}\n  (-1)^{n}s^{(b+c-2d)n}\n  \\qbinom{b-c-1}{N-1-k-n}\n  \\\\[5mm]\n  &\\text{(putting $i=N-1-k-n$)}\n  \\\\\n  &=\n  (-1)^{k}(s-s^{-1})^{N-1}s^{(2b-2d+k)(N-1)-(b+c-2d)k-N(N-1)\/2}\n  \\frac{[b-c+k-1]!}{[b-c-1]!}\n  \\\\\n  &\\quad\\times\n  \\sum_{i=0}^{b-c-1}\n  (-1)^{i}s^{(2d-b-c)i}\n  \\qbinom{b-c-1}{i}\n  \\\\\n  &=\n  (s-s^{-1})^{N-1}s^{(2d-2b)-(b+c-2d+1)k-N(N-1)\/2}\n  \\frac{[b-c+k-1]!}{[b-c-1]!}\n  \\\\\n  &\\quad\\times\n  T(b-c-1,2d-b-c),\n\\end{align*}\nwith $T(\\alpha,\\beta)=\\sum_{i=0}^{\\alpha}(-1)^{i}s^{\\beta i}\\qbinom{\\alpha}{i}$.\nFrom Lemma~\\ref{lem:ST} we have\n\\begin{align*}\\label{eq:X}\n  X(k)\n  &=(s-s^{-1})^{N-1}s^{2(d-b)-(b+c-2d+1)k-N(N-1)\/2}\n  \\frac{[b-c+k-1]!}{[b-c-1]!}\n  \\\\\n  &\\quad\\times\n  (1-s^{2(d-c-1)})(1-s^{2(d-c-2)})\\cdots(1-s^{2(d-b+1)}).\n\\end{align*}\nWe divide the case into two subcases; $d\\ge b$ and $b>d$.\n\\smallskip\\par\\noindent\n{\\bf Subcase 1.1 ($d\\ge b$).}\\quad\nSince\n\\begin{multline*}\n  (1-s^{2(d-c-1)})(1-s^{2(d-c-2)})\\cdots(1-s^{2(d-b+1)})\n  \\\\\n  =\n  (-1)^{b-c-1}s^{(2d-b-c)(b-c-1)\/2}(s-s^{-1})^{b-c-1}\n  \\frac{[d-c-1]!}{[d-b]!},\n\\end{multline*}\nwe have\n\\begin{align*}\n  X(k)\n  &=(-1)^{b+c+1}(s-s^{-1})^{N+b-c-2}\n  s^{2(d-b)-N(N-1)\/2+(2d-b-c)(b-c-1)\/2}\n  \\\\\n  &\\quad\\times\n  \\frac{[d-c-1]!}{[b-c-1]![d-b]!}\n  s^{(2d-b-c-1)k}[b-c+k-1]!\n\\end{align*}\nand so\n\\begin{align*}\n  &N^2\\left(\\tilde{R}_J\\right)_{ab}^{cd}\n  \\\\\n  &=\n  s^{(N^2+1)\/2}(-1)^{b+c}(s-s^{-1})^{N+b-c-2}\n  s^{2(d-b)-N(N-1)\/2+(2d-b-c)(b-c-1)\/2}\n  \\\\\n  &\\quad\\times\n  \\frac{[d-c-1]!}{[d-b]!}\n  \\sum_{k=0}^{N-1}s^{(2a-b-c+1)k}\\frac{[N-1-k]![b-c+k-1]!}{[b-c-1]!}\n  \\\\\n  &=\n  s^{(N^2+1)\/2}(-1)^{b+c}(s-s^{-1})^{N+b-c-2}\n  s^{2(d-b)-N(N-1)\/2+(2d-b-c)(b-c-1)\/2}\n  \\\\\n  &\\quad\\times\n  \\frac{[d-c-1]![N-1]!}{[d-b]!}\n  S(b-c-1,2a-b-c+1)\n  \\\\\n  &=\n  s^{(N^2+1)\/2}(-1)^{b+c}(s-s^{-1})^{N+b-c-2}\n  s^{2(d-b)-N(N-1)\/2+(2d-b-c)(b-c-1)\/2}\n  \\\\\n  &\\quad\\times\n  \\frac{[d-c-1]![N-1]!}{[d-b]!}\n  (1-s^{2(a-b)})(1-s^{2(a-b-1)})\\dots(1-s^{2(a-c-N+1)}),\n\\end{align*}\nwhere the second equality follows from $[N-1-k]!=[N-1]!\/[k]!$.\n\\par\nNoting that $a-c-N+1\\le 0$, we see that\n\\begin{multline*}\n  (1-s^{2(a-b)})(1-s^{2(a-b-1)})\\dots(1-s^{2(a-c-N+1)})\n  \\\\\n  =\n  \\begin{cases}\n    0 &\\quad\\text{if $a\\ge b$},\n    \\\\\n    (s-s^{-1})^{N+c-b}s^{(2a-b-c-N+1)(N+c-b)\/2}\\frac{[N-1+c-a]!}{[b-a-1]!}&\\quad\n    \\text{if $b>a$}.\n  \\end{cases}\n\\end{multline*}\nTherefore we have\n\\begin{equation*}\n  \\left(\\tilde{R}_J\\right)_{ab}^{cd}\n  =\\rho(a,b,c,d)(-1)^{a+b+1}\\frac{[d-c-1]![N-1+c-a]!}{[d-b]![b-a-1]!}\n\\end{equation*}\nif $b>a$ and zero otherwise.\nThus the proof is complete for the case where $d\\ge b>c$.\n(Note that $[N-1+c-a]=0$ if $c>a$.)\n\\smallskip\\par\\noindent\n{\\bf Subcase 1.2 ($b>d$).}\\quad\nIn this case we have\n\\begin{multline*}\n  (1-s^{2(d-c-1)})(1-s^{2(d-c-2)})\\cdots(1-s^{2(d-b+1)})\n  \\\\\n  =\n  s^{(2d-b-c)(b-c-1)\/2}(s-s^{-1})^{b-c-1}\n  \\frac{[b-d-1]!}{[c-d]!}\n\\end{multline*}\nif $c\\ge d$ and $0$ otherwise.\nTherefore $\\left(\\tilde{R}_J\\right)_{ab}^{cd}=0$ if $d>c$.\nIf $c\\ge d$, we have\n\\begin{align*}\n  &N^{2}\\left(\\tilde{R}_J\\right)_{ab}^{cd}\n  \\\\\n  &\\quad=\n  s^{(N-1)^2\/2}(s-s^{-1})^{N+b-c-2}\n  s^{2(d-b)-N(N-1)\/2+(2d-b-c)(b-c-1)\/2}\n  \\\\\n  &\\qquad\\times\n  \\frac{[N-1]![b-d-1]!}{[c-d]!}\n  S(b-c-1,2a-b-c+1)\n  \\\\\n  &\\quad=\n  s^{-(N-1)\/2}(s-s^{-1})^{N+b-c-2}\n  s^{2(d-b)+(2d-b-c)(b-c-1)\/2}\n  \\\\\n  &\\qquad\\times\n  \\frac{[N-1]![b-d-1]!}{[c-d]!}\n  (1-s^{2(a-b)})(1-s^{2(a-b-1)})\\cdots(1-s^{2(a-c-N+1)}).\n\\end{align*}\nThis vanishes if $a\\ge b$ and equals\n$N^2\\rho(a,b,c,d)(-1)^{a+c}\\frac{[b-d-1]![N-1+c-a]!}{[c-d]![b-a-1]!}$\notherwise, completing the proof for the case $b>d$ and $b>c$ since\n$[N-1+c-a]=0$ if $c>a$.\n\\medskip\\par\\noindent\n{\\bf Case 2 ($c\\ge b$)}.\\quad\nFirst note from \\eqref{eq:S}, $X(k)=0$ if $k\\ge c-b+1$.\nIf $k<c-b+1$, we have\n\\begin{multline*}\n  S(n,2(b-c+k)+n)\n  \\\\\n  =s^{(2(b-c+k)+n-N)(N-1-n)\/2}(s-s^{-1})^{N-1-n}\n  \\frac{[N-1-n+c-b-k]!}{[c-b-k]!}\n\\end{multline*}\nand so\n\\begin{align*}\n  X(k)\n  &=\n  (s-s^{-1})^{N-1}s^{(2(b-c+k)-N)(N-1)\/2}\\frac{[c-b]!}{[c-b-k]!}\n  \\\\\n  &\\quad\\times\n  \\sum_{n=0}^{N-1-k}s^{(b+c-2d)n}\\qbinom{N-1-k-n+c-b}{c-b}\n  \\\\[5mm]\n  &\\text{(putting $i=N-1-k-n$)}\n  \\\\\n  &=\n  (s-s^{-1})^{N-1}s^{(2(b-c+k)-N)(N-1)\/2+(b+c-2d)(N-1-k)}\\frac{[c-b]!}{[c-b-k]!}\n  \\\\\n  &\\quad\\times\n  S(c-b,2d-b-c)\n  \\\\\n  &=\n  (s-s^{-1})^{N-1}s^{(2(b-c+k)-N)(N-1)\/2+(b+c-2d)(N-1-k)}\\frac{[c-b]!}{[c-b-k]!}\n  \\\\\n  &\\quad\\times\n  (1-s^{2(d-c-1)})(1-s^{2(d-c-2)})\\cdots(1-s^{2(d-b-N+1)}).\n\\end{align*}\nSince $d-b-N+1\\le0$, $X(k)$ vanishes and so does\n$\\left(\\tilde{R}_J\\right)_{ab}^{cd}$ if $d>c$.\nTherefore we assume that $c\\ge d$.\n\\par\nIn this case since\n\\begin{multline*}\n  (1-s^{2(d-c-1)})(1-s^{2(d-c-2)})\\dots(1-s^{2(d-b-N+1)})\n  \\\\\n  =s^{(2d-b-c-N)(N-1+b-c)\/2}(s-s^{-1})^{N-1+b-c}\n  \\frac{[N-1+b-d]!}{[c-d]!},\n\\end{multline*}\nwe have\n\\begin{align*}\n  X(k)\n  &=\n  (s-s^{-1})^{2N-2+b-c}\n  \\\\\n  &\\quad\\times\n  s^{(2b-2c-N)(N-1)\/2+(b+c-2d)(N-1)+(2d-b-c-N)(N-1+b-c)\/2}\n  \\\\\n  &\\quad\\times\n  \\frac{[N-1+b-d]![c-b]!}{[c-d]!}\n  \\\\\n  &\\quad\\times\n  s^{(N-1+2d-b-c)k}\\frac{1}{[c-b-k]!}.\n\\end{align*}\nTherefore\n\\begin{align*}\n  N^{2}\\left(\\tilde{R}_J\\right)_{ab}^{cd}\n  &=\n  s^{(N-1)^2\/2}(s-s^{-1})^{2N-2+b-c}\n  \\\\\n  &\\quad\\times\n  s^{(2b-2c-N)(N-1)\/2+(b+c-2d)(N-1)+(2d-b-c-N)(N-1+b-c)\/2}\n  \\\\\n  &\\quad\\times\n  \\frac{[N-1]![N-1+b-d]!}{[c-d]!}T(c-b,2a-b-c+1)\n  \\\\\n  &=\n  s^{(N-1)^2\/2}(s-s^{-1})^{2N-2+b-c}\n  \\\\\n  &\\quad\\times\n  s^{(2b-2c-N)(N-1)\/2+(b+c-2d)(N-1)+(2d-b-c-N)(N-1+b-c)\/2}\n  \\\\\n  &\\quad\\times\n  (1-s^{2(a-b)})(1-s^{2(a-b-1)})\\cdots(1-s^{2(a-c+1)}).\n\\end{align*}\nThere are two subcases; $b>a$ and $a\\ge b$.\n\\smallskip\\par\\noindent\n{\\bf Subcase 2.1 ($b>a$)}.\\quad\nSince\n\\begin{multline*}\n  (1-s^{2(a-b)})(1-s^{2(a-b-1)})\\cdots(1-s^{2(a-c+1)})\n  =\n  \\\\\n  s^{(2a-b-c+1)(c-b)}(s-s^{-1})^{c-b}\\frac{[c-a-1]!}{[b-a-1]!},\n\\end{multline*}\nwe have\n\\begin{equation*}\n  \\left(\\tilde{R}_J\\right)_{ab}^{cd}\n  =\n  \\rho(a,b,c,d)(-1)^{b+d}\\frac{[N-1+b-d]![c-a-1]!}{[c-d]![b-a-1]!}\n\\end{equation*}\nand the proof for the case where $c\\ge b$ and $b>a$ is complete.\n(Note that $\\left(\\tilde{R}_J\\right)_{ab}^{cd}$ vanishes unless $c\\ge d\\ge b$.)\n\\smallskip\\par\\noindent\n{\\bf Subcase 2.2 ($a\\ge b$)}.\\quad\nIn this case we only have to consider the case where $a\\ge c$ for\notherwise $\\left(\\tilde{R}_J\\right)_{ab}^{cd}=0$.\nNow\n\\begin{multline*}\n  (1-s^{2(a-b)})(1-s^{2(a-b-1)})\\cdots(1-s^{2(a-c+1)})\n  =\n  \\\\\n  (-1)^{b+c}s^{(2a-b-c+1)(c-b)}(s-s^{-1})^{c-b}\\frac{[a-b]!}{[a-c]!}\n\\end{multline*}\nand so we have\n\\begin{equation*}\n  \\left(\\tilde{R}_J\\right)_{ab}^{cd}\n  =\n  \\rho(a,b,c,d)(-1)^{c+d}\\frac{[N-1+b-d]![a-b]!}{[c-d]![a-c]!}.\n\\end{equation*}\nThe proof for the case where $c\\ge b$ and $a\\ge b$ is now complete.\n(Note again that $\\left(\\tilde{R}_J\\right)_{ab}^{cd}=0$ unless $c\\ge d\\ge b$.)\n\\end{proof}\n\\section{Kashaev's R-matrix and his invariant}\\label{sec:Kashaev}\nIn this section we will calculate Kashaev's $\\check{R}$-matrix given in\n\\cite{Kashaev:MODPLA95} and prove that it coincides with the matrix $\\tilde{R}_J$\nup to a constant given in the previous section.\n\\par\nWe prepare notations following \\cite{Kashaev:MODPLA95}.\nFix an integer $N\\ge2$.\nPut $(x)_n=\\prod_{i=1}^{n}(1-x^i)$ for $n\\ge 0$.\nDefine $\\theta:\\mathbb{Z}\\to\\{0,1\\}$ by\n\\begin{equation*}\n  \\theta(n)\n  =\n  \\begin{cases}\n    1&\\quad\\text{if $N>n\\ge0$},\n    \\\\\n    0&\\quad\\text{otherwise}.\n  \\end{cases}\n\\end{equation*}\nFor an integer $x$, we denote by $\\operatorname{res}(x)\\in\\{0,1,2,\\dots,N-1\\}$ the\nresidue modulo $N$.\n\\par\nNow Kashaev's $\\check{R}$-matrix $R_K$ is given by\n\\begin{multline*}\n  {\\left(R_K\\right)}_{ab}^{cd}\n  \\\\\n  =\n  Nq^{1+c-b+(a-d)(c-b)}\n  \\frac{\\theta(\\operatorname{res}(b-a-1)+\\operatorname{res}(c-d))\\theta(\\operatorname{res}(a-c)+\\operatorname{res}(d-b))}\n  {(q)_{\\operatorname{res}(b-a-1)}(q^{-1})_{\\operatorname{res}(a-c)}(q)_{\\operatorname{res}(c-d)}(q^{-1})_{\\operatorname{res}(d-b)}}\n\\end{multline*}\nwith $q=s^2$.\nNote that we are using $P\\circ{R}$ with $R$ defined in\n\\cite[2.12]{Kashaev:MODPLA95} rather than $R$ itself where $P$ is the\nhomomorphism from $\\mathbb{C}^N\\otimes\\mathbb{C}^N$ to $\\mathbb{C}^N\\otimes\\mathbb{C}^N$ sending\n$x\\otimes{y}$ to $y\\otimes{x}$.\n\\par\nWe will show the following proposition.\n\\begin{prop}\\label{prop:K}\n\\begin{equation*}\n  {\\left({R}_K\\right)}_{ab}^{cd}\n    =\n    \\begin{cases}\n    \\lambda(a,b,c,d)(-1)^{a+b+1}\\dfrac{[d-c-1]![N-1+c-a]!}{[d-b]![b-a-1]!}\n    \\quad&\\text{if $d\\ge b>a\\ge c$},\n    \\\\[5mm]\n    \\lambda(a,b,c,d)(-1)^{a+c}\\dfrac{[b-d-1]![N-1+c-a]!}{[c-d]![b-a-1]!}\n    \\quad&\\text{if $b>a\\ge c\\ge d$},\n    \\\\[5mm]\n    \\lambda(a,b,c,d)(-1)^{b+d}\\dfrac{[N-1+b-d]![c-a-1]!}{[c-d]![b-a-1]!}\n    \\quad&\\text{if $c\\ge d\\ge b>a$},\n    \\\\[5mm]\n    \\lambda(a,b,c,d)(-1)^{c+d}\\dfrac{[N-1+b-d]![a-b]!}{[c-d]![a-c]!}\n    \\quad&\\text{if $a\\ge c\\ge d\\ge b$},\n    \\\\[5mm]\n    0\\quad&\\text{otherwise},\n  \\end{cases}\n\\end{equation*}\nwhere\n$\\lambda(a,b,c,d)=\ns^{-N^2\/2+N\/2+2+c+d-2b+(a-d)(c-b)}(s-s^{-1})^{1-N}N\/([N-1]!)^2$.\n\\end{prop}\n\\begin{proof}\nSince $N-1\\ge b-a-1\\ge-N$, $N-1\\ge c-d\\ge -N+1$, $N-1\\ge a-c\\ge -N+1$ and\n$N-1\\ge d-b\\ge -N+1$, we see that ${({R}_K)}_{ab}^{cd}$ vanishes except for the\nfollowing four cases, which have already appeared in Proposition~\\ref{prop:J}:\n(i) $d\\ge b>a\\ge c$, (ii) $b>a\\ge c\\ge d$, (iii) $c\\ge d\\ge b\\ge a$ and\n(iv) $a\\ge c\\ge d\\ge b$.\n\\par\nWe will only prove the first case because the other cases are similar.\nNoting that\n\\begin{align*}\n  (q     )_n&=(-1)^{n}s^{n(n+1)\/2}(s-s^{-1})^{n}[n]!,\n  \\\\\n  (q^{-1})_n&=s^{-n(n+1)\/2}(s-s^{-1})^{n}[n]!,\n\\end{align*}\nwe have\n\\begin{align*}\n  {\\left({R}_K\\right)}_{ab}^{cd}\n  &=\n  (-1)^{a+b+c+d}Ns^{2+2c-2b+2(a-d)(c-b)+a-d+N\/2+1\/2}(s-s^{-1})^{-N+1}\n  \\\\\n  &\\qquad\\times\n  \\frac{1}{[d-b]![b-a-1]![N+c-d]![a-c]!}\n\\end{align*}\nsince $\\operatorname{res}(b-a-1)=b-a-1$, $\\operatorname{res}(a-c)=a-c$, $\\operatorname{res}(c-d)=N+c-d$ and\n$\\operatorname{res}(d-b)=d-b$.\nNow since $[N-n]=[n]$, we see that\n\\begin{equation*}\n  \\frac{1}{[d-b]![b-a-1]![N+c-d]![a-c]!}\n  =\n  \\frac{1}{([N-1])^2}\\frac{[d-c-1]![N+c-a-1]!}{[b-a-1]![d-b]!}.\n\\end{equation*}\nTherefore\n\\begin{equation*}\n  {\\left({R}_K\\right)}_{ab}^{cd}\n  =\\lambda(a,b,c,d)(-1)^{a+b+1}\\frac{[d-c-1]![N-1+c-a]!}{[d-b]![b-a-1]!}\n\\end{equation*}\nas required.\n\\end{proof}\n\\par\nTherefore $R_K$ and $R_J$ are equal up to a constant depending\nonly on $N$.\nMore precisely we have\n\\begin{prop}\\label{prop:Kashaev=Jones}\nLet $R_K$ and $R_J$ be the $\\check{R}$-matrices defined by as above.\nThen we have\n\\begin{equation*}\n  {R}_K\n  =\n  s^{-(N+1)(N-3)\/2}(W\\otimes{W})({id}\\otimes{D})\n  R_J({id}\\otimes{D^{-1}})({W^{-1}}\\otimes{W^{-1}})\n\\end{equation*}\nfor any $N\\ge2$.\n\\end{prop}\n\\begin{proof}\nFrom Propositions~\\ref{prop:J} and \\ref{prop:K}, we only have to check that\n$\\rho(a,b,c,d)\/\\lambda(a,b,c,d)=s^{(N+1)(N-3)\/2}$.\nWe have\n\\begin{equation*}\n  \\rho(a,b,c,d)\/\\lambda(a,b,c,d)\n  =(-1)^{N}s^{(N-3)\/2}\\left(\\frac{(s-s^{-1})^{N-1}[N-1]!}{N}\\right)^3\n\\end{equation*}\nbut this coincides with $s^{(N+1)(N-3)\/2}$ as shown below.\n\\par\nWe have\n\\begin{align*}\n  (s-s^{-1})^{N-1}[N-1]!\n  &=\\prod_{k=1}^{N-1}\\left(2\\sqrt{-1}\\sin(k\\pi\/N)\\right)\n  \\\\\n  &=\\sqrt{-1}^{N-1}\\prod_{k=1}^{N-1}\\left(2\\sin(k\\pi\/N)\\right).\n\\end{align*}\nOn the other hand from \\cite[I.392-1, p.~33]{Gradshteyn\/Ryzhik:1980}, we have\n\\begin{equation*}\n  \\sin(Nx)=2^{N-1}\\prod_{k=0}^{N-1}\\sin(x+k\\pi\/N).\n\\end{equation*}\nDivided by $\\sin x$ and taking the limit $x\\to 0$, we have\n\\begin{equation*}\n  N=\\prod_{k=1}^{N-1}\\left(2\\sin(k\\pi\/N)\\right).\n\\end{equation*}\nTherefore we have\n\\begin{align*}\n  (-1)^{N}s^{(N-3)\/2}\\left(\\frac{(s-s^{-1})^{N-1}[N-1]!}{N}\\right)^3\n  &=\n  (-1)^{N}s^{(N-3)\/2}\\sqrt{-1}^{3(N-1)}\n  \\\\\n  &=\n  s^{N^2+(N-3)\/2+3(N-1)N\/2}\n  =\n  s^{(N+1)(N-3)\/2},\n\\end{align*}\ncompleting the proof.\n\\end{proof}\n\\par\nWe will show that the matrix ${R}_K$ also satisfies the Yang--Baxter\nequation.\nTo do that we prepare a lemma.\n\\begin{lem}\\label{lem:D_through_J}\nThe matrices $D$ and $D^{-1}$ can go through ${R_J}$ in pair, that is, the\nfollowing equality holds.\n\\begin{equation*}\n  ({id}\\otimes{D}){R_J}({id}\\otimes{D^{-1}})\n  =\n  (D^{-1}\\otimes{id}){R_J}({D}\\otimes{id}).\n\\end{equation*}\n\\end{lem}\n\\begin{proof}\nIt is sufficient to show that $(D\\otimes{D})R_J=R_J(D\\otimes{D})$.\nSince $D_{j}^{i}=\\delta_{i,j}s^{(N-1)j}$,\n\\begin{align*}\n  \\left((D\\otimes{D})R_J\\right)_{kl}^{ij}\n  &=\\sum_{a,b}\\delta_{a,k}\\delta_{b,l}s^{(N-1)k}s^{(N-1)l}{(R_J)}_{ab}^{ij}\n  \\\\\n  &=s^{(N-1)(k+l)}{(R_J)}_{kl}^{ij}\n\\end{align*}\nand\n\\begin{align*}\n  \\left(R_J(D\\otimes{D})\\right)_{kl}^{ij}\n  &=\\sum_{a,b}\\delta_{a,i}\\delta_{b,j}s^{(N-1)i}s^{(N-1)j}{(R_J)}_{kl}^{ab}\n  \\\\\n  &=s^{(N-1)(i+j)}{(R_J)}_{kl}^{ij}.\n\\end{align*}\nBut these two coincide since $(R_J)_{kl}^{ij}$ vanishes unless $k+l=i+j$\n(the charge conservation law), completing the proof.\n\\end{proof}\nUsing Lemma~\\ref{lem:D_through_J} we can give another proof of the following\nproposition.\n\\begin{prop}[Kashaev]\\label{prop:YBE_K}\nKashaev's $\\check{R}$-matrix ${R}_K$ satisfies the Yang--Baxter equation, that is,\n\\begin{equation*}\n  ({R}_K\\otimes{id})({id}\\otimes{{R}_K})({R}_K\\otimes{id})\n  =\n  ({id}\\otimes{{R}_K})({R}_K\\otimes{id})({id}\\otimes{{R}_K}).\n\\end{equation*}\n\\end{prop}\n\\begin{proof}\nFrom Proposition~\\ref{prop:Kashaev=Jones}, we have\n\\begin{align*}\n  &(R_K\\otimes{id})({id}\\otimes{R_K})({R}_K\\otimes{id})\n  \\\\\n  &\\quad=\n    (W\\otimes{W}\\otimes{id})({id}\\otimes{D}\\otimes{id})(R_J\\otimes{id})\n    ({id}\\otimes{D^{-1}}\\otimes{id})(W^{-1}\\otimes{W^{-1}}\\otimes{id})\n  \\\\\n  &\\qquad\\times\n   ({id}\\otimes{W}\\otimes{W})({id}\\otimes{id}\\otimes{D})({id}\\otimes{R_J})\n   ({id}\\otimes{id}\\otimes{D^{-1}})({id}\\otimes{W^{-1}}\\otimes{W^{-1}})\n  \\\\\n  &\\qquad\\times\n    (W\\otimes{W}\\otimes{id})({id}\\otimes{D}\\otimes{id})(R_J\\otimes{id})\n    ({id}\\otimes{D^{-1}}\\otimes{id})(W^{-1}\\otimes{W^{-1}}\\otimes{id})\n  \\\\\n  &\\quad=\n    (W\\otimes{W}\\otimes{W})({id}\\otimes{D}\\otimes{D})(R_J\\otimes{id})\n  \\\\\n  &\\qquad\\times\n    ({id}\\otimes{D^{-1}}\\otimes{id})({id}\\otimes{R_J})\n    ({id}\\otimes{D}\\otimes{id})\n  \\\\\n  &\\qquad\\times\n    (R_J\\otimes{id})({id}\\otimes{D^{-1}}\\otimes{D^{-1}})\n    (W^{-1}\\otimes{W^{-1}}\\otimes{W^{-1}})\n  \\\\\n  &\\quad=\n    (W\\otimes{W}\\otimes{W})({id}\\otimes{D}\\otimes{D})(R_J\\times{id})\n  \\\\\n  &\\qquad\\times\n    (id\\otimes{id}\\otimes{D})({id}\\otimes{R_J})\n    (id\\otimes{id}\\otimes{D^{-1}})\n  \\\\\n  &\\qquad\\times\n    (R_J\\otimes{id})({id}\\otimes{D^{-1}}\\otimes{D^{-1}})\n    (W^{-1}\\otimes{W^{-1}}\\otimes{W^{-1}})\n  \\\\\n  &\\quad=\n    (W\\otimes{W}\\otimes{W})(id\\otimes{D}\\otimes{D^2})\n  \\\\\n  &\\qquad\\times\n    (R_J\\otimes{id})(id\\otimes{R_J})(R_J\\otimes{id})\n  \\\\\n  &\\qquad\\times\n    (id\\otimes{D^{-1}}\\otimes{D^{-2}})(W^{-1}\\otimes{W^{-1}}\\otimes{W^{-1}}).\n  \\\\\n  \\intertext{Similar calculation shows}\n  &({id}\\otimes{R_K})(R_K\\otimes{id})({id}\\otimes{R_K})\n  \\\\\n  &\\quad=\n    (W\\otimes{W}\\otimes{W})(id\\otimes{D}\\otimes{D^2})\n  \\\\\n  &\\qquad\\times\n    ({id}\\otimes{R_J})(R_J\\otimes{id})(id\\otimes{R_J})\n  \\\\\n  &\\qquad\\times\n    (id\\otimes{D^{-1}}\\otimes{D^{-2}})(W^{-1}\\otimes{W^{-1}}\\otimes{W^{-1}}).\n\\end{align*}\nFrom the Yang--Baxter equation for $R_J$ these two coincide, completing\nthe proof.\n\\end{proof}\n\\par\nTo show that $R_J$ and ${R}_K$ define the same link invariant,\nwe will construct enhanced Yang--Baxter operators precisely by using them.\n\\par\nLet $\\mu_J$ be the $N\\times N$-matrix with $(i,j)$-entry\n$\\left(\\mu_J\\right)_{j}^{i}=\\delta_{i,j}s^{2i-N+1}$.\nThen the quadruple $S_J=(R_J,\\mu_{J},s^{N^2-1},1)$\nis a Yang--Baxter operator and the following lemma holds.\n\\begin{lem}\\label{lem:EYB_J}\n\\begin{gather*}\n  (\\mu_J\\otimes\\mu_J)R_J=R_J(\\mu_J\\otimes\\mu_J),\n  \\\\\n  \\sum_{j=0}^{N-1}\\left((R_J)^{\\pm1}(id\\otimes\\mu_J)\\right)_{kj}^{ij}\n  =(s^{N^2-1})^{\\pm1}id.\n\\end{gather*}\n\\end{lem}\n\\par\nNext we will give a Yang--Baxter operator using $R_K$.\nLet $\\mu_K$ be the $N\\times N$-matrix with $(i,j)$-entry\n$\\left(\\mu_K\\right)_{j}^{i}=-s\\delta_{i,j+1}$.\nThen we have\n\\begin{lem}\\label{lem:mu_nu}\n\\begin{equation*}\n  W\\,D\\,\\mu_J\\,D^{-1}\\,W^{-1}=\\mu_K.\n\\end{equation*}\n\\end{lem}\n\\begin{proof}\nSince $W_{j}^{a}=s^{2aj}$, $D_{a}^{b}=\\delta_{a,b}s^{(N-1)b}$,\n$\\left(\\mu_J\\right)_{b}^{c}=\\delta_{b,c}s^{2c-N+1}$,\n$\\left(D^{-1}\\right)_{c}^{d}=\\delta_{c,d}s^{-(N-1)d}$\nand $\\left(W^{-1}\\right)_{d}^{i}=\\delta_{d,i}s^{-2di}\/N$, we have\n\\begin{align*}\n  \\left(W\\,D\\,\\mu_J\\,D^{-1}\\,W^{-1}\\right)_{j}^{i}\n  &=\n  \\frac{1}{N}(-s)\\sum_{a=0}^{N-1}s^{2(j-i+1)a}\n  \\\\\n  &=\n  -s\\delta_{i,j+1},\n\\end{align*}\ncompleting the proof.\n\\end{proof}\nCombining Lemmas~\\ref{lem:EYB_J} and \\ref{lem:mu_nu}, we show that\n$S_K=({R}_K,\\mu_K,-s,1)$ is also an enhanced Yang--Baxter\noperator.\n\\begin{lem}\\label{lem:EYB_K}\n\\begin{gather*}\n  (\\mu_K\\otimes\\mu_K){R}_K={R}_K(\\mu_K\\otimes\\mu_K),\n  \\\\\n  \\sum_{j=0}^{N-1}\\left((R_K)^{\\pm1}(id\\otimes\\mu_K)\\right)_{kj}^{ij}\n  =(-s)^{\\pm1}id.\n\\end{gather*}\n\\end{lem}\n\\begin{proof}\nNoting that $\\mu_J$ and $D$ commutes since they are diagonal, the first equality\nfollows immediately from that in Lemma~\\ref{lem:EYB_J}.\n\\par\nThe second equality follows from\n\\begin{multline*}\n  ({R}_K)^{\\pm1}({id}\\otimes\\mu_K)\n  \\\\\n  =\n  s^{\\mp(N+1)(N-3)\/2}(W\\otimes{W})(id\\otimes{D})R_J(id\\otimes{\\mu_J})\n  (id\\otimes{D^{-1}})(W^{-1}\\otimes{W^{-1}}),\n\\end{multline*}\ncompleting the proof.\n\\par\nNote that the lemma can also be proved by using\n\\cite[(2.8) and (2.17)]{Kashaev:MODPLA95}.\n\\end{proof}\nNow we see that $S_J$ and $S_K$ define the same link invariant by using\nthe following lemma.\n\\begin{lem}\\label{lem:b_K}\nLet $\\xi$ be an $n$-braid.\nThen\n\\begin{multline*}\n  b_{R_K}(\\xi)\n  =\n  \\left(W^{\\otimes{n}}\\right)\n  \\left(D^{k_1}\\otimes{D^{k_2}}\\otimes\\dots\\otimes{D^{k_n}}\\right)\n  b_{R_J}(\\xi)\n  \\\\\n  \\times\n  \\left(D^{-k_1}\\otimes{D^{-k_2}}\\otimes\\dots\\otimes{D^{-k_n}}\\right)\n  \\left(\\left(W^{-1}\\right)^{\\otimes{n}}\\right)\n\\end{multline*}\nfor some non-negative integers $k_1,k_2,\\dots,k_n$.\n\\end{lem}\n\\begin{proof}\nIn fact we can show that $(k_1,k_2,\\dots,k_n)$ is of the form\n$$(0,1,2,\\dots,d_1,0,1,2,\\dots,d_2,\\dots,0,1,2,\\dots,d_h)$$\nby using Lemma~\\ref{lem:D_through_J} repeatedly\nto `push' $D$ and $D^{-1}$ from left to right\n(See the proof of Proposition~\\ref{prop:YBE_K}).\nDetails are omitted.\n\\end{proof}\n\\par\nSince we know that $J_N=T_{S_J,1}$ is well-defined as described in\n\\S\\ref{sec:pre}, from the previous lemma $T_{S_K,1}(L)$ is also a link\ninvariant, which we denote by $\\langle{L}\\rangle_{N}$.\nNote that it is implicitly stated in \\cite{Kashaev:MODPLA95} that the invariant\ncan be regarded as an invariant for $(1,1)$-tangles.\nNote also that though the invariant was defined only up to a multiple of $s$ in\n\\cite{Kashaev:MODPLA95}, we can now define it without ambiguity.\n\\par\nSince\n\\begin{align*}\n  &b_{R_K}(\\xi)({id}\\otimes{\\mu_K}^{\\otimes{(n-1)}})\n  \\\\\n  &\\quad=\n  \\left(W^{\\otimes{n}}\\right)\n  \\left(D^{k_1}\\otimes{D^{k_2}}\\otimes\\dots\\otimes{D^{k_n}}\\right)\n  b_{R_J}(\\xi)\n  \\\\\n  &\\qquad\\times\n  \\left({id}\\otimes{\\mu_J}^{\\otimes(n-1)}\\right)\n  \\left(D^{-k_1}\\otimes{D^{-k_2}}\\otimes\\dots\\otimes{D^{-k_n}}\\right)\n  \\left(\\left(W^{-1}\\right)^{\\otimes{n}}\\right)\n\\end{align*}\nfrom Lemma~\\ref{lem:b_K}\nwe conclude that $S_J$ and $S_K$ define the same link\ninvariant.\n\\begin{thm}\n  For any link $L$ and any integer $N\\ge2$, $\\langle{L}\\rangle_{N}$ and\n  $J_{N}(L)$ coincide.\n\\end{thm}\n\\section{Relation between the simplicial volume and the colored Jones\npolynomials.}\nLet $K$ be one of the three simplest hyperbolic knots $4_1$, $5_2$ and $6_1$.\nKashaev found in \\cite{Kashaev:LETMP97} that the hyperbolic volume of\n$S^3\\setminus K$, denoted by $\\operatorname{Vol}(K)$, coincides numerically with the growth\nrate of the absolute value of $\\langle K\\rangle_{N}$ with respect to $N$.\nMore precisely,\n\\begin{equation*}\n  \\operatorname{Vol}(K)\n  =2\\pi \\, \\lim_{N\\to \\infty}\\,\n  \\frac{\\log\\left|\\langle{K}\\rangle_{N}\\right|}{N}.\n\\end{equation*}\n\\par\nWe would like to modify his conjecture taking Gromov's simplicial\nvolume (or Gromov norm) \\cite{Gromov:INSHE82} into account.\nLet us consider the torus decomposition of the complement of a knot $K$\n\\cite{Jaco\/Shalen:MEMAM79,Johannson:1979}.\nThen the simplicial volume of $K$, denoted by $\\|K\\|$ is equal to the sum\nof the hyperbolic volumes of hyperbolic pieces of the decomposition divided\nby $v_3$, the volume of the ideal regular tetrahedron in $\\mathbb{H}^3$, the\nthree-dimensional hyperbolic space.\nRecall that it is additive under the connect sum \\cite{Soma:INVEM81}\n\\begin{equation*}\n  \\|K_1 \\sharp K_2\\|\n  =\n  \\|K_1\\| + \\|K_2\\|,\n\\end{equation*}\nand that it does not alter by mutation \\cite[Theorem~1.5]{Ruberman:INVEM87}.\n\\par\nNoting that $J_{N}$ is multiplicative under the connect sum, that is,\n\\begin{equation*}\n  J_{N}(K_1 \\sharp K_2)\n  =\n  J_{N}(K) \\,J_{N}(K_2)\n\\end{equation*}\nand that it does not alter by mutation \\cite[Corollary~6.2.5]{Jun:ORSJM89},\nwe propose the following conjecture.\n\\begin{conj}[Volume conjecture]\nFor any knot $K$,\n\\begin{equation}\\label{eq:volume}\n  \\|K\\|\n  =\n  \\frac{2\\pi}{v_3}\\lim_{N \\to \\infty}\n  \\frac{\\log\\left|J_{N}(K)\\right|}{N}.\n\\end{equation}\n\\end{conj}\n\\begin{rem}\nFirst note that if Kashaev's conjecture is true then our conjecture holds\nfor hyperbolic knots and their connect sums.\nIt is also true for torus knots since Kashaev and O.~Tirkkonen\n[private communication] showed that the right hand side of\n\\eqref{eq:volume} vanishes in this case by using H.~Morton's formula\n\\cite{Morton:MATPC95} (see also \\cite{Rosso\/Jones:JKNOT93}).\n\\end{rem}\n\\begin{rem}\nNote however that the volume conjecture does not hold for links since $J_{N}$ of\nthe split union of two links vanishes.\n\\end{rem}\n\\par\nAs a consequence of the volume conjecture, we anticipate the following\nsimplest case of V.~Vassiliev's conjecture\n\\cite[6.1 Stabilization conjecture]{Vassiliev:1990}\n(see also \\cite[Chapter~1, Part V (L), Conjecture]{Kirby:problems}).\n\\begin{conj}[V.~Vassiliev]\\label{conj:Vassiliev}\nAssume that every Vassiliev (finite-type) invariant of a knot is\nidentical to that of the trivial knot.\nThen it is unknotted.\n\\end{conj}\n\\par\nWe show that the volume conjecture implies Conjecture~\\ref{conj:Vassiliev} by\nusing the following two lemmas.\n\\begin{lem}[{\\cite[Corollary~4.2]{Gordon:TRAAM83}}]\\label{lem:zero_volume}\nIf $\\|K\\| = 0$  then $K$ is obtained from the trivial knot\nby applying a finite number (possibly zero) of the following two operations:\n\\begin{enumerate}\n\\item[1)] making a connect sum,\n\\item[2)] making a cable.\n\\end{enumerate}\n\\end{lem}\n\\begin{lem}\\label{lem:Alexander_of_cable}\nIf $\\|K\\| = 0$, then the Alexander polynomial $\\Delta(K)$ of $K$ is trivial\nif and only if $K$ is the trivial knot.\n\\end{lem}\n\\begin{proof}\nThis lemma comes from Lemma~\\ref{lem:zero_volume} and the following three facts\n\\cite[\\S2]{Burau:ABHMS32}\n(see also \\cite[8.23~Proposition]{Burde\/Zieschang:1985}).\n\\begin{enumerate}\n  \\item[i)]\n    the Alexander polynomial of a non-trivial torus knot is not trivial.\n  \\item[ii)]\n    the Alexander polynomial is multiplicative under the connect sum.\n    Therefore if $\\Delta(K_1)$ and $\\Delta(K_2)$ are non-trivial, then\n    $\\Delta(K_1\\sharp K_2)$  is also non-trivial.\n  \\item[iii)]\n    if $K^\\prime$ is a knot obtained from $K$ by a cabling operation, then\n    $\\Delta(K^\\prime)$ is $\\Delta(K)f(t)$ with some Laurent polynomial $f(t)$.\n    Hence if $\\Delta(K)$ is non-trivial, so is $\\Delta(K^\\prime)$ .\n\\end{enumerate}\n\\end{proof}\n\\par\n\\begin{proof}\n[Proof that the volume conjecture implies Conjecture~\\ref{conj:Vassiliev}]\nFirst note that every coefficient of both colored Jones polynomial and Alexander\npolynomial as a power series in $h=\\log t$ is a Vassiliev invariant.\nSo a knot $K$ with every Vassiliev invariant trivial has the trivial colored\nJones polynomial for any color and the trivial Alexander polynomial.\nIn particular $J_{N}(K)=1$ for any $N$.\nTherefore assuming the volume conjecture, $\\|K\\|$ vanishes.\nFrom Lemma~\\ref{lem:Alexander_of_cable}, $K$ should be trivial, completing the\nproof.\n\\end{proof}\n\\begin{rem}\nIt was pointed out by Vaintrob and Bar-Natan that using the\nMelvin--Morton--Rozansky conjecture\n\\cite{Melvin\/Morton:COMMP95,Rozansky:COMMP96} proved by Bar-Natan and\nS.~Garoufalidis \\cite{BarNatan\/Garoufalidis:INVEM96}, we\ncan also show that a knot is trivial if and only if all of\nits colored Jones polynomials are trivial since the Melvin--Morton--Rozansky\nconjecture says that the Alexander polynomial can be determined by the colored\nJones polynomials.\n\\end{rem}\n\\renewcommand{\\thesection}{\\Alph{section}}\n\\setcounter{section}{0}\n\\section{appendix}\nIn this appendix we prove some technical formulas used in the paper.\nPut\n\\begin{align*}\n  S(\\alpha,\\beta)&=\\sum_{i=0}^{N-1}s^{\\beta i}\\qbinom{\\alpha+i}{i},\n  \\\\\n  T(\\alpha,\\beta)&=\\sum_{i=0}^{\\alpha}(-1)^{i}s^{\\beta i}\\qbinom{\\alpha}{i}.\n\\end{align*}\nNote that the summation in $S(\\alpha,\\beta)$ is essentially from $0$ to\n$N-1-\\alpha$.\nThen we have\n\\begin{lem}\\label{lem:ST}\nThe following formulas hold.\n\\begin{align*}\n  S(\\alpha,\\beta)&=\n    \\prod_{j=1}^{N-\\alpha-1}(1-s^{\\beta-\\alpha-2j})\n   =(1-s^{\\beta-\\alpha-2})(1-s^{\\beta-\\alpha-4})\\cdots(1-s^{\\beta+\\alpha-2N+2}),\n  \\\\\n  T(\\alpha,\\beta)&=\n    \\prod_{j=1}^{\\alpha}(1-s^{\\beta+\\alpha+1-2j})\n    =(1-s^{\\beta+\\alpha-1})(1-s^{\\beta+\\alpha-3})\\cdots(1-s^{\\beta-\\alpha+1}).\n\\end{align*}\n\\end{lem}\n\\begin{proof}\nWe only prove the equality for $S(\\alpha,\\beta)$ since the other case is\nsimilar.\nWe use the following quantized Pascal relation.\n\\begin{equation*}\n  \\qbinom{\\alpha+i}{i}\n  =s^{-\\alpha}\\qbinom{\\alpha+i-1}{i-1}+s^{i}\\qbinom{\\alpha+i-1}{i}.\n\\end{equation*}\nThen since\n\\begin{align*}\n  S(\\alpha,\\beta)\n  &=\n  s^{-\\alpha}\\sum_{i=0}^{N-1}s^{\\beta i}\\qbinom{\\alpha+i-1}{i-1}\n  +\n  \\sum_{i=0}^{N-1}s^{(\\beta+1)i}\\qbinom{\\alpha+i-1}{i}\n  \\\\[5mm]\n  &\\text{(putting $k=i-1$ in the first term)}\n  \\\\\n  &=\n  s^{\\beta-\\alpha}\\sum_{k=-1}^{N-2}s^{k}\\qbinom{\\alpha+k}{k}\n  +\n  S(\\alpha-1,\\beta+1)\n  \\\\\n  &=\n  s^{\\beta-\\alpha}S(\\alpha,\\beta)+S(\\alpha-1,\\beta+1),\n\\end{align*}\nwe have the following recursive formula.\n\\begin{equation*}\n  S(\\alpha-1,\\beta+1)=(1-s^{\\beta-\\alpha})S(\\alpha,\\beta).\n\\end{equation*}\nNow the required formula follows since $S(N-1,\\gamma)=1$ for any integer\n$\\gamma$.\n\\end{proof}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction} \\label{sec1}\n\\IEEEPARstart{D}{irection} of arrival (DOA) estimation is a fundamental problem in the wireless communications, radar-based applications, and the future integrated sensing and communication (ISAC) systems~\\cite{zhang_overview_2021,xu_rate-splitting_2021}, and has been studied for decades. Typically, the DOA estimation is based on an ideal antenna array model, without considering any imperfect effect, including the mutual coupling effect, inconsistent gains\/phases, nonlinear effect, etc. In this ideal scenario, the DOAs can be estimated by traditional methods such as the monopulse angle estimation method~\\cite{zhu_combined_2018} and the fast Fourier transformation (FFT)-based methods. \n\nMoreover, the super-resolution estimation methods have also been proposed. On one hand, subspace-based methods such as the algorithms of multiple-signal classification (MUSIC)~\\cite{lin_fsf_2006,yan_low-complexity_2013,zhang_direction_2010} and estimation of signal parameters via rotational invariance techniques (ESPRIT)~\\cite{kim_joint_2015,lin_time-frequency_2016,xiaofei_zhang_multi-invariance_2009,fang-ming_han_esprit-like_2005} were proposed. On the other hand, sparse reconsturction-based methods have also been proposed by exploiting the signals' sparsity in the spatial domain. For example, the compressed sensing (CS)-based methods for the DOA estimation have been proposed, such as sparse Bayesian learning-based methods~\\cite{wan_deep_2021,p_chen_off-grid_2019,Dai_So_2021,mao_marginal_2021,wan_robust_2022,wang_alternative_2018}, the mixed  $\\ell _{2,0}$ norm-based methods~\\cite{5466152}, etc.\n \n\nHowever, the above works did not consider the effect of imperfect antenna arrays. As a result, the performance of these algorithms is significantly affected in practical DOA estimation systems. In the literature, some works have started to investigate the DOA estimation schemes under imperfect antenna array. For example, for an array with mutual coupling, gain or phase errors, and sensor location errors, a method estimating  both DOA and model errors  is proposed in~\\cite{lu_direction--arrival_2018}. A fourth-order parallel factor decomposition model using imperfect waveforms is given in~\\cite{ruan_parafac_2019} to estimate the DOA. Then, ref.~\\cite{liu_2-d_2021} proposes a two-dimensional (2D) DOA estimation method for an imperfect L-shaped array using active calibration. However, each of the above works only considered a subset of the imperfect array effects, because optimization over the complicated array model with all imperfect effects considered is challenging. This motivates us to use the deep learning (DL) technique for DOA estimation with all imperfect array effects taken into account because of its efficiency for training over difficult networks.  \n\n\nIn the literature, several works have been done for DL-based DOA estimation~\\cite{9173575,8400482,7497454,9178434}, and have the advantages of low computational complexity and high accuracy. There are some types of DL-based methods:\n\\begin{enumerate}\n\t\\item The input data is the raw sampled data from the array;\n\t\\item The input data is the covariance matrix of the received signal;\n\t\\item The outputs are the directly estimated DOAs;\n\t\\item The output is the spatial spectrum, and the DOAs are estimated from the spectrum.\n\\end{enumerate}\nFor example, in~\\cite{yuan_unsupervised_2021,wu_deep_2019}, a DL-based method is proposed for the DOA estimation, where the input is the covariance matrix and the output is the spectrum. A sparse loss function is used to train the network. Ref.~\\cite{papageorgiou_deep_2021} gives a CNN-based method for the DOA estimation using the estimated covariance matrix, and estimates the DOA by discretizing the spatial domain into grids. In~\\cite{akter_rfdoa-net_2021}, a synthetic dataset for angle classification is shown under the presence of additive noise, propagation attenuation and delay. Then, a CNN-based method is proposed for the DOA estimation. An angle separation learning method is proposed in~\\cite{wu_deep_2019} for the DOA estimation of the coherent signals, where the covariance matrix is formulated as the input features of the deep neural network (DNN). In~\\cite{8854868}, a deep convolution network (DCN) is given for the DOA estimation with the covariance matrix as the under-sampled linear measurements of the spatial spectrum, where the signal sparsity in the spatial domain is also exploited to improve the estimation performance. A MUSIC-based DOA estimation method is proposed in~\\cite{9328861} by using small antenna arrays, where DL is formulated to reconstruct the signals of a virtual large antenna array. Ref.~\\cite{8400482} gives an offline and online DNN method for the DOA estimation in the massive multiple-input multiple-output (MIMO) system, where DOA is the output of the network and can be estimated directly from the received signal. For the DOA estimation with low sjgnal-to-noise ratio (SNR), a convolutional neural network (CNN) is proposed in~\\cite{9457195}, where the covariance matrix is the network's input and shows enhanced robustness in the presence of noise. Moreover, a multiple deep CNN is designed in~\\cite{9034077}, where each CNN learns the MUSIC spectrum of the received signal, so a nonlinear relationship between the received sensor data and the angular spectrum is formulated in the network. For the imperfect array, ref.~\\cite{8485631} introduces a framework of the DNN to estimate the DOA using a multitask autoencoder and series of parallel multilayer classifiers. \n \nWe can find that the DL-based DOA estimation methods mainly use the CNN as a typical network structure~\\cite{chakrabarty_multi-speaker_2019}, and the input is the statistic results such as the covariance matrix. Since the estimation performance is limited by the information in the statistic data, the performance cannot be better than that use the raw sampled data. Furthermore, the network output is the estimated DOAs, the spatial spectrum cannot be obtained. Therefore,  the network structure should be adjusted with different targets numbers, and it is not suitable in the practical applications. There are some limitations with the existing DL-based methods:\n\\begin{itemize}\n\t\\item Since the classic DOA estimation algorithms such as MUSIC are just based on the covariance matrices of the received signals, most existing ML-based schemes use these covariance matrices as the input data to train the network. However, the covariance matrices are not sufficient for the optimal estimator design in general. As a result, the input data used in these works does not preserve all the useful information.  \n\t\\item In addition, the output of the existing ML-based DOA estimation schemes is usually the spatial spectrum of the targets. In this case, the training network depends on the number of targets, i.e., different networks should be trained given different number of targets. This is of high complexity in practice. \n\t\\item Furthermore, when the spatial spectrum is used, we must discretize the DOAs into grids, and the possible DOA must be on the discretized grids exactly. More girds as the output must be used for high accuracy, and the network will become more complexity and hardly to train.\n\\end{itemize}\n\n\nIn this paper, we propose a new type of DNN network based on CNN, i.e., super-resolution DOA network (SDOAnet), to overcome the above mentioned difficulties in the DOA estimation.  The proposed SDOAnet is used for performance evaluation of imperfect array under realistic conditions. Compared with the existing methods, the proposed SDOAnet can achieve better estimation performance with lower complexity. The contributions of this paper are given as follows:\n\\begin{enumerate}\n\t\\item \\textbf{A system model with imperfect array effects for the DOA estimation is formulated.} The imperfect effect includes the position perturbation, the inconsistent gains, the inconsistent phases, the mutual coupling effect, and the nonlinear effect, etc. \n\tAs a result, our results are directly applicable in a practical system.\n\t\\item \\textbf{A new DL architecture is proposed based on the imperfect array.} Different from existing methods, the input of the SDOAnet is the raw sampled signals, and the output is a vector, which can be used to estimate the spatial spectrum easily. Convolution layers are then used to get the signals' features and avoid the complexity due to high-dimension signals. The output of the SDOAnet is a vector for the spectrum estimation and can avoid the problem of discretizing the spatial domain. Compared with the existing CNN-based method, the proposed SDOAnet can be trained easily and achieve better estimation performance. \n\t\\item \\textbf{A loss function to train the SDOAnet is proposed,} where Gaussian functions are used to approximate the spatial spectrum. Inspired by the atomic norm minimization (ANM)-based DOA estimation method, the output of SDOAnet is used to formulate the spatial spectrum. Then, a loss function is used to measure the error between the refereed spectrum and the estimated one, and to train the network. \n\\end{enumerate}\n\nThe remainder of this paper is organized as follows. The system mode of practical DOA estimation is formulated in Section~\\ref{sec2}. The review of super-resolution DOA estimation method is given in Section~\\ref{sec3}. Then, the proposed SDOAnet for the DOA estimation is shown in Section~\\ref{sec4}. Simulation results are carried out in Section~\\ref{sec5}, and finally, Section~\\ref{sec6} concludes the paper. \n\n\\textit{Notations:} Upper-case and lower-case boldface letters denote matrices and column vectors, respectively. The  matrix transpose and the  Hermitian transpose are denoted as $(\\cdot)^\\text{T}$ and $(\\cdot)^\\text{H}$, respectively.  $\\mathcal{R}\\{\\cdot\\}$ and $\\mathcal{I}\\{\\cdot\\}$ denote the real and imaginary parts of a complex value, respecitvely.  $\\text{Tr}\\{\\cdot\\}$ is the trace of a matrix.  $\\|\\cdot\\|_2$ is  the $\\ell_2$ norm. \n\n\n\\section{The System Model for Practical DOA Estimation}\\label{sec2}\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=3in]{.\/Figures\/system.pdf}\n\t\\caption{The system model for the DOA estimation in a practical array.}\n\t\\label{sys}\n\\end{figure} \n\n\nIn this paper, we consider the DOA estimation problem in a practical system, and propose a DL-based estimation framework. As shown in Fig.~\\ref{sys}, we consider $K$ far-field signals, and the $k$-th ($k=0,1,\\dots,K-1$) signal is expressed as $s_{k}(t)\\in\\mathbb{C}$ with the DOA being $\\theta_k\\in\\left(-\\frac{\\pi}{2},\\frac{\\pi}{2}\\right]$. A linear array system with $N$  antennas is used to receive the signals and estimate the DOAs, where the wavelength is denoted as $\\lambda$. Considering an additive noise $w_n(t)\\in\\mathbb{C}$, the received signal at the $n$-th ($n=0,1,\\dots, N-1$) antenna can be expressed as\n\\begin{align}\\label{eq1}\n\tr_n(t) = g\\left(x_n(t)+\\sum_{n'\\neq n}B_{n,n'}x_{n'}(t)\\right)+w_n(t).\n\\end{align}\nThen, we have\n\\begin{align}\n\tx_n(t) = \\sum^{K-1}_{k=0}s_k(t)A_ne^{j\\phi_n}e^{j2\\pi\\frac{d_n}{\\lambda}\\sin\\theta_k},\n\\end{align}\nwhere taking the $0$-th antenna as the reference one, i.e., $d_0=0$, the position of the $n$-th antenna is $d_n$, and for a uniform linear array (ULA), the position of antenna is $d_n=n\\lambda\/2$. In the received signal (\\ref{eq1}), the following imperfect problems are considered:\n\\begin{enumerate}\n\t\\item \\emph{The mutual coupling effect:} The antennas cannot be ideally isolated and introduce the mutual coupling effect among the received signals. The mutual coupling coefficient between the $n$-th and $n'$-th ($n\\neq n'$) antenna is $B_{n,n'}\\in\\mathbb{C}$ with $|B_{n,n'}|<1$ in (\\ref{eq1});\n\t\\item \\emph{The position perturbations:} The antenna positions cannot be exactly at the desired positions, and will cause the phase errors of the received signals among antennas in the steering vector;\n\t\\item \\emph{The inconsistent gains: } The radio frequency (RF) channels usually cannot have exactly the same amplifiers, and will cause the amplitude differences among the received signals. The channel gain of the $n$-th antenna is denoted as $A_n>0$;\n\t\\item \\emph{The inconsistent phases: } The difference among the RF channels will also cause the delay and phases errors of the received signals, and The channel phase of the $n$-th antenna is denoted as $\\phi_n$;\n\t\\item \\emph{The nonlinear effect: } The nonlinear effect among RF channels and analog-to-digital converter (ADC) will introduce the nonlinear effect and degrade the DOA estimation performance.  We use a nonlinear function $g(\\cdot)$ to represent the nonlinear operation in the receiving channels.\n\\end{enumerate}\n \nHence, collect the received signals into a vector \n\\begin{align}\n\\boldsymbol{r}\\triangleq \\begin{bmatrix}\n\tr_0(t), r_1(t),\\dots, r_{N-1}(t)\n\\end{bmatrix}^{\\text{T}}.\t\n\\end{align}\nThe DOA estimation problem can be formulated as a  parameter estimation problem with the received signal $\\boldsymbol{r}$. Most existing works consider the methods in the scenario with perfect array, where we have the linear function $g(\\cdot)$, the mutual coupling coefficient $B_{n,n'}$ is $0$, the channel gains are the same ($A_n=1$ and $\\phi_0=0$), and the position $d_n$ of antenna is known. \n\nHowever, when an imperfect array is considered, the imperfect elements include the mutual coupling effect, the nonlinear effect, the inconsistent phases, the inconsistent gains and the position perturbations, etc. In the practical systems, most existing super-resolution methods cannot outperform the traditional methods, where the super-resolution methods must have perfect systems and high SNR. In this paper, we will focus on a robust super-resolution method for the DOA estimation with imperfect system effect. \n\n\\section{The Review of Super-Resolution DOA Estimation Methods}\\label{sec3}\n\\subsection{The Atomic Norm-Based Estimation Methods}\nIn recent years, the atomic norm-based methods have been proposed for the line-spectra estimation and achieved better performance by exploiting the sparsity of the spectrum in the frequency domain. Additionally, the DOA estimation problem can be easily described as a line-spectral estimation problem, so atomic norm-based methods have been proposed for the DOA estimation.\n\nUsually, in the atomic norm-based methods. the ideal ULA is assumed, and the received signal based on (\\ref{eq1}) in the $n$-th array can be expressed as\n\\begin{align}\n\tr_n=\\sum^{K-1}_{k=0}s_ke^{j2\\pi\\frac{n d}{\\lambda}\\sin\\theta_k} + w_n(t),\n\\end{align}\nwhere the distance between adjacent antennas is $d=\\frac{\\lambda}{2}$. Then, with the definition of a steering vector\n\\begin{align}\n\t\\boldsymbol{a}(\\theta)\\triangleq \\begin{bmatrix}\n\t1, e^{j\\frac{2\\pi d}{\\lambda}\\sin\\theta},\\dots, 1, e^{j\\frac{2\\pi (N-1)d}{\\lambda}\\sin\\theta}\n\\end{bmatrix}^{\\text{T}},\n\\end{align}\ncollect all the received signals into a vector, and we have\n\\begin{align}\n\t\\boldsymbol{r}&\\triangleq \\begin{bmatrix}\n\t\tr_0,r_1,\\dots,r_{N-1}\n\t\\end{bmatrix}^{\\text{T}}\\notag \\\\\n\t& = \\boldsymbol{As}+\\boldsymbol{w},\n\\end{align} \nwhere we define the steering matrix as\n\\begin{align}\n\t\\boldsymbol{A}&\\triangleq \\begin{bmatrix}\n\t\t\\boldsymbol{a}(\\theta_0),\\boldsymbol{a}(\\theta_1),\\dots, \\boldsymbol{a}(\\theta_{K-1})\n\t\\end{bmatrix},\n\\end{align}\nthe signal vector is defined as\n\\begin{align}\n\t\\boldsymbol{s}&\\triangleq \\begin{bmatrix}\n\t\ts_0,s_1,\\dots,s_{K-1}\n\t\\end{bmatrix}^{\\text{T}},\n\\end{align}\nand the noise vector is\n\\begin{align}\n\t\\boldsymbol{w}&\\triangleq \\begin{bmatrix}\n\t\tw_0,w_1,\\dots,w_{N-1}\n\t\\end{bmatrix}^{\\text{T}}.\n\\end{align}\n\n\nIn the ANM-based DOA estimation method, an atomic norm is defined as \n\\begin{align}\n\t\\|\\boldsymbol{x}\\|_{\\mathcal{A}}\\triangleq \\inf \\Big\\{ & \\sum_n \\alpha_{n'}:\\boldsymbol{x}=\\sum \\alpha_{n'} e^{j\\phi_{n'}} \\boldsymbol{a}(\\theta_{n'}), \\notag\\\\\n\t& \\phi_{n'}\\in[0,2\\pi), \\alpha_{n'}\\geq 0\n\t\\Big\\},  \n\\end{align}\nwhich describes a sparse representation  of $\\boldsymbol{x}$ with the sparse coefficients being $\\alpha_{n'}$ ($n'=0,1,\\dots, N'-1$). Then, with the received signal $\\boldsymbol{r}$, we denoise the signal with a sparse reconstruction signal $\\boldsymbol{x}$, which can be expressed as an ANM expression \n\\begin{align}\n\t\\min_{\\boldsymbol{x}} \\frac{1}{2}\\|\\boldsymbol{r}-\\boldsymbol{x}\\|^2_2+\\beta \\|\\boldsymbol{x}\\|_{\\mathcal{A}},\n\\end{align}\nwhere the parameter $\\beta$ is used to control the trade-off between the sparsity and the reconstruction accuracy. This ANM problem can be solved by introducing a semi-definite programming (SDP) method, which is\n\\begin{align}\\label{eq11}\n\t\\min_{\\boldsymbol{B},\\boldsymbol{h}}\\quad & \\|\\boldsymbol{r}-\\boldsymbol{h}\\|^2_2\\notag\\\\\n\t\\text{s.t}\\quad & \\begin{bmatrix}\n\t\t\\boldsymbol{B}& \\boldsymbol{h}\\\\\n\t\t\\boldsymbol{h}^{\\text{H}}& 1\n\t\\end{bmatrix}\\succeq 0\\\\\n\t& \\boldsymbol{B}\\text{ is Hermitian matrix}\\notag\\\\\n\t& \\text{Tr}\\{\\boldsymbol{B}\\} = \\beta^2\\notag\\\\\n\t& \\sum_n \\boldsymbol{B}_{n,n+n'} =0,\\text{ for } n'\\neq 0 \\notag \\\\ &\\qquad \\qquad\\qquad \\quad \\text{ and }n'=1-N,\\dots,N-1.\\notag\n\\end{align}\nBy solving the SDP problem (\\ref{eq11}), the sparse reconstruction signal $\\boldsymbol{h}$ can be obtained, and the DOA of the received signal can be estimated by finding the peak values of the following polynomial \n\\begin{align}\n\tf(\\theta) = |\\boldsymbol{a}^{\\text{H}}(\\theta)\\boldsymbol{h}|^2.\n\\end{align}\n\nThe ANM-based DOA estimation method is for the ideal array with perfect assumptions, but for the practical array, the ANM-based method must be extended. In~\\cite{p_chen_new_2020,wang_gridless_2019,govinda_raj_single_2019-1,gong_doa_2022}, the atomic norm-based methods are extended for the practical array. We can find that the much complex optimization problems are formulated, and a vector like $\\boldsymbol{h}$ denoted as $\\boldsymbol{h}'$ can be obtained. Then, the DOAs are estimated by the peak values of the following polynomial \n\\begin{align}\n\tf'(\\theta) = |\\boldsymbol{a}^{\\text{H}}(\\theta)\\boldsymbol{h}'|^2.\n\\end{align} \n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=7.2in]{.\/Figures\/sys.pdf}\n\t\\caption{The network architecture of the proposed SDOAnet.}\n\t\\label{sysnet}\n\\end{figure*} \n\n\\subsection{The MUSIC-Based Estimated Methods} \nIn the super-resolution estimation method, the MUSIC-based methods can achieve better performance by using the noise and signal subspaces. For the single-snapshot spectral estimation, ref~\\cite{liao_music_2016} proposes a MUSIC-based method. A Hankel matrix is obtained from the received signal $\\boldsymbol{r}$ as\n\\begin{align}\n\t\\boldsymbol{R}=\\text{Hankel}(\\boldsymbol{r})\\begin{bmatrix}\n\t\tr_0 & r_1 & \\dots & r_{N-L}\\\\\n\t\tr_1 & r_2 & \\dots & r_{N-L+1}\\\\\n\t\t\\vdots & \\vdots & \\ddots & \\vdots\\\\\n\t\tr_{L-1} & r_{L} & \\dots & r_{N-1}\n\t\\end{bmatrix},\n\\end{align}\nwhere the received signal $\\boldsymbol{r}$ is reshaped as a matrix $\\boldsymbol{R}\\in\\mathbb{C}^{L\\times N-L+1}$. Then, a singular value decomposition (SVD) is used as\n\\begin{align}\n\t[\\boldsymbol{U}_1, \\boldsymbol{U}_2] \\boldsymbol{\\Lambda}[\\boldsymbol{V}_1,\\boldsymbol{V}_2]=\\text{SVD}\\{\\boldsymbol{R}\\},\n\\end{align}\nwhere $\\boldsymbol{U}_2$ is corresponding to the small singular values, and $\\boldsymbol{\\Lambda}$ is a diagonal matrix with the entries from the singular values. Finally, the spatial spectrum can be estimated as\n\\begin{align}\n\tg(\\theta) = \\frac{1}{\\|\\boldsymbol{a}^{\\text{H}}(\\theta)\\boldsymbol{U}_2\\|^2_2}.\n\\end{align} \n\n\\section{The Proposed DOA Estimation Method}\\label{sec4}\n\n\nFrom the above sections about the exisiting DOA estimation methods, we can find that the DOAs are estimated by searching the peak values of the spatial spectrum. In this section, we will propose a new DL-based super-resolution method for the DOA estimation, and it is named as a super-resolution DOA network (SDOAnet), which contains more information and can be trained faster than the existing covariance matrix-based methods. \n\n\\subsection{The SDOAnet Architecture}\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=2in]{.\/Figures\/relu.pdf}\n\t\\caption{The ReLU function.}\n\t\\label{relu}\n\\end{figure} \n\nThe SDOAnet architecture is show in Fig.~\\ref{sysnet}. First, the received signal in (\\ref{eq1}) is rewritten as a vector with real and imaginary parts\n\\begin{align}\n\t\\boldsymbol{y}(t)\\triangleq [\\mathcal{R}^{\\text{T}}\\{\\boldsymbol{r}(t)\\},\\mathcal{I}^{\\text{T}}\\{\\boldsymbol{r}(t)\\}]^{\\text{T}}\\in\\mathbb{R}^{2N\\times 1},\n\\end{align}\nwhere we have the received signal vector \n\\begin{align}\n\t\\boldsymbol{r}(t)=[r_0(t),r_1(t),\\dots,r_{N-1}(t)]^\\text{T}\\in\\mathbb{C}^{N\\times 1}.\n\\end{align}\nWith the batch size being $M_B$, the  input signal is\n\\begin{align}\n\t\\boldsymbol{Y}\\triangleq [\\boldsymbol{y}(0), \\boldsymbol{y}(1),\\dots,\\boldsymbol{y}(M_B-1)]^{\\text{T}},\n\\end{align} \nand the size is $M_B\\times 2N$. \n\nThen, since the SDOAnet is based on the convolution network, we use a full connection (FC) as the input layer with the output dimension being $M_FM_I$, where $M_F$ denotes the number of filters in the convolution layers and $M_I$ denotes the inner dimension extension. After the input layer, the dimension of the signal is $M_B\\times M_FM_I$, and we reshape the signal as a tensor $f_1(\\boldsymbol{Y})$ with the dimension being $M_B \\times M_F\\times M_I$, where $f_1(\\cdot)$ is a input layer function.\n\nThe tensor is passed into the convolution layers and the number of convolution layers is $M_{C}$. In each convolution layer, a one-dimensional (1D) convolution operation is realized with the kernel size being $M_F\\times M_K$ and the padding operation is used to keep that the size of convolution output is the same with that of input. The output of convolution operation is $M_B\\times M_F \\times M_I$. Then, the batch normalization is applied on the convolution output, and the normalization output is denoted as $f_3(f_2(f_1(\\boldsymbol{Y})))$. The function $f_2(\\cdot)$ denotes the convolution operation and $f_3(\\cdot)$ is the batch normalization operation\n\\begin{align}\nf_3(x)=\\frac{x-\\mathcal{E}\\{x\\}}{\\sqrt{\\text{Var}\\{x\\}+\\epsilon}},\n\\end{align}\nwhere $\\mathcal{E}\\{x\\}$ and $\\text{Var}\\{x\\}$ are the mean and variance of $x$, respectively. $\\epsilon$ is a value added to the denominator for numerical stability and can be set as $\\epsilon=10^{-5}$. In each convolution layer, a ReLU function $f_4(\\cdot)$ as shown in Fig.~\\ref{relu} is applied on the output of the batch normalization, and is defined as \n\\begin{align}\n\tf_4(x)\\triangleq \\max(0,x).\n\\end{align}\nAfter the convolution layers, a FC layer is used as an output layer with the input and output sizes being $M_B\\times M_I M_F$ and $M_B\\times 2N$, respectively. The operation in the output layer is denoted as $f_5(\\cdot)$. \n\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=1.2in]{.\/Figures\/funcflow.pdf}\n\t\\caption{The flowchart of the function operations.}\n\t\\label{funcflow}\n\\end{figure} \n\n\n\nFinally, as shown in Fig.~\\ref{funcflow}, we can obtain the output of the SDOAnet as\n\\begin{align}\n\\boldsymbol{G}=f_5(f_4(f_3(f_2(f_4(\\dots f_4(\\dots f_2(f_1(\\boldsymbol{Y}))\\dots)\\dots ))))),\n\\end{align}\nwhere we have \n\\begin{align}\n\t\\boldsymbol{G}\\triangleq \\begin{bmatrix}\n\\boldsymbol{g}(0),\\boldsymbol{g}(1),\\dots,\\boldsymbol{g}(M_B-1)\n\\end{bmatrix}\\in\\mathbb{R}^{M_B\\times 2N}.\n\\end{align}\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=3.5in]{.\/Figures\/getsp.pdf}\n\t\\caption{The flowchart to obtain the spatial spectrum from the network output.}\n\t\\label{getsp}\n\\end{figure} \n\nAs shown in Fig.~\\ref{getsp}, the corresponding complex vector can be obtained from the network output $\\boldsymbol{g}(m)$ ($m=0,1,\\dots,M_B-1$) as\n\\begin{align}\n\\boldsymbol{z}(m)\\triangleq \\boldsymbol{g}_{0:N-1}(m)  +j \\boldsymbol{g}_{N:2N-1}(m),\n\\end{align}\nwhere $ \\boldsymbol{g}_{0:N-1}(m) $ denotes a sub-vector of $\\boldsymbol{g}(m)$ with the index  from $0$ to $N-1$, and $ \\boldsymbol{g}_{N:2N-1}(m) $ denotes that from $N$ to $2N-1$. With the output  $\\boldsymbol{z}$ of SDOAnet, the spatial spectrum can be estimated by \n\\begin{align}\\label{eq20}\nf_{\\text{sp}}(\\zeta) = |\\boldsymbol{a}^{\\text{H}}(\\zeta)\\boldsymbol{z}|^2,\n\\end{align}\nwhere $\\zeta$ is chosen based on the detection area, such as from $-\\pi\/2$ to $\\pi\/2$.\n\nIn the proposed SDOAnet, the network is different from existing methods. The novelty is that we use the raw sampled data as the network input, and the convolution layers are used to obtain the features of the raw data. The raw data contains all the information of the received signals. The network's output is a vector, which is neither the DOA results nor the spatial spectrum used by the existing methods. The output size is the same as the number of antennas in the array. Therefore, the dimension of the SDOAnet is lower than that using the spectrum as the output. Hence, the training time can be reduced significantly. Additionally, the DOAs can be obtained by finding the peak values of $f_{\\text{sp}}(\\zeta)$ in (\\ref{eq20}), which can avoid the problem of adopting the determined number of the received signals in the networks that using the DOA values as the output. \n\n\n\\subsection{The Training Approach}\nTo train the SDOAnet, the spatial spectrum $f_{\\text{sp}}(\\zeta)$ is obtained from (\\ref{eq20}) and the refereed spectrum is given as follows\n\\begin{align}\nf_{\\text{ref}}(\\zeta) = \\sum_{k=0}^{K-1} A_k e^{-\\frac{(\\zeta-\\theta_k)^2}{\\sigma_{\\text{G}}^2}},\n\\end{align}\nwhere we use Gaussian functions to approximate the spatial spectrum. $A_k$ denotes the spectrum value, and $\\sigma_{\\text{G}}$ is the standard deviation of the Gaussian function. In this paper, we set the value of $\\sigma_{\\text{G}}$ as \n\\begin{align}\n\t\\sigma_{\\text{G}}=\\bar{\\sigma}_{\\text{G}}\/N.\n\\end{align}\nAn example of the refereed spatial spectrum approximated by the Gaussian functions is shown in Fig.~\\ref{refsp}, where we use $16$ antennas, $\\bar{\\sigma}_{\\text{G}}=100$, and the ground-truth DOAs are $\\ang{-30}$, $\\ang{10}$, and $\\ang{20}$. The $3$ dB-spectrum width is about $\\ang{10.4}$.  \n\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=3.7in]{.\/Figures\/refsp.pdf}\n\t\\caption{The refereed spatial spectrum approximated by the Gaussian functions.}\n\t\\label{refsp}\n\\end{figure} \n\n\nWith the refereed spectrum the loss function is defined as\n\\begin{align} \\label{eq22}\nf_{\\text{loss}}(\\boldsymbol{\\zeta}) =\\frac{1}{\\Omega} \\left\\|f_{\\text{ref}}(\\boldsymbol{\\zeta})-f_{\\text{sp}}(\\boldsymbol{\\zeta})\\right\\|^2_2,\n\\end{align}\nwhere $f_{\\text{ref}}(\\boldsymbol{\\zeta})\\in\\mathbb{R}^{\\Omega\\times 1}$ and $f_{\\text{sp}}(\\boldsymbol{\\zeta})\\in\\mathbb{R}^{\\Omega\\times 1}$ are vectors with the $\\omega$-th ($\\omega=0,1,\\dots, \\Omega-1$) entry being $f_{\\text{ref}}(\\zeta_\\omega)$  and $f_{\\text{sp}}(\\zeta_\\omega)$, respectively. We define \n\\begin{align}\n\t\\boldsymbol{\\zeta}\\triangleq \\begin{bmatrix}\n\\zeta_0,\\dots, \\zeta_{\\Omega-1}\n\\end{bmatrix}^{\\text{T}},\n\\end{align}\nwhere $\\Omega$ is the number of the discretized spatial angles. The SDOAnet is trained to minimize the loss function $f_{\\text{loss}}(\\boldsymbol{\\zeta})$ in (\\ref{eq22}) by updating the network coefficients.\n\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=2.5in]{.\/Figures\/trainflow.pdf}\n\t\\caption{The training procedure for the SDOAnet.}\n\t\\label{trainflow}\n\\end{figure} \n\nFor the practical system, the mutual coupling effect, the nonlinear effect, the inconsistent phases, the inconsistent gains, and the position perturbations are considered in this paper. The training procedure is shown in Fig.~\\ref{trainflow}, and the following steps can be used to train the SDOAnet:\n\\begin{enumerate}\n\\item \\textbf{Perfect array step:} The received signals using perfect array without the imperfect effect are used during the training procedure;\n\\item \\textbf{Position perturbation step:} The received signals with position perturbation are used. The position perturbation is generated by a Gaussian distribution with the mean being $0$ and the standard deviation $\\sigma_{\\text{per}}$ selected by a uniform distribution $\\sigma_{\\text{per}}\\in [0,\\sigma_{\\text{max\\_per}}]$. The parameter $\\sigma_{\\text{max\\_per}}$ can be specified in the simulation;\n\\item \\textbf{Inconsistent gains step:} The inconsistent gains are considered in this step. Similarly, the inconsistent gains are generated by a zero-mean Gaussian distribution with the standard deviation $\\sigma_{\\text{gain}}$ being $\\sigma_{\\text{gain}}\\in [0, \\sigma_{\\text{max\\_gain}}]$, where $\\sigma_{\\text{max\\_gain}}$ is specified in the simulation;\n\\item \\textbf{Inconsistent phases step:} The inconsistent phases are also generated by a zero-mean Gaussian distribution with the  standard deviation $\\sigma_{\\text{phase}}$ being $\\sigma_{\\text{phase}}\\in [0, \\sigma_{\\text{max\\_phase}}]$, where $\\sigma_{\\text{max\\_phase}}$ is specified in the simulation;\n\\item \\textbf{Mutual coupling effect step:} The mutual coupling effect is described by a matrix $\\boldsymbol{B}$ with complex entries, and the diagonal entries are all ones. The entry at the $n$-th row and the $n'$-th column is denoted as \n\\begin{align}\n\tB_{n,n'}=|B_{n,n'}|e^{j\\psi_{n,n'}},\n\\end{align}\nand $|B_{n,n'}|$  is determined by a uniform distribution $|B_{n,n'}|\\in[0,\\sigma^{|n-n'|}_{\\text{mc}}]$ with $n\\neq n'$.  The phase $\\psi_{n,n'}$ follows a uniform distribution $\\psi_{n,n'}\\in[0,2\\pi)$. The parameter $\\sigma_{\\text{mc}}$ is specified in the simulation;\n\\item \\textbf{Nonlinear effect step:} The nonlinear effect is described by a nonlinear function\n\\begin{align}\n\tf_{\\text{nonlinear}}(x)=\\tanh(x\\sigma_{\\text{nonlinear}}),\n\\end{align}\nwhere $\\sigma_{\\text{nonlinear}}$ is specified in the simulation to control the nonlinear effect;\n\\item \\textbf{All the imperfect effect step:} We consider all the imperfect effect to train the network.\n\\end{enumerate}\nAfter training the SDOAnet in sequence according to the above steps, we start over from the first step to train the network again until the maximum number of the training procedures.    \n\n\n\n\\section{Simulation Results} \\label{sec5}\nIn this section, simulation results show the DOA estimation performance of the proposed SDOAnet using a practical array. The simulation results are carried out in a personal computer with MATLAB R2020b, Intel Core i5 @ 2.9 GHz processor, and 8 GB LPDDR3 @ 2133 MHz. The code about the SDOAnet is available online \\url{https:\/\/github.com\/chenpengseu\/SDOAnet.git}, where the training codes and a pre-trained network are also provided. The network is based on PyTorch~1.4 and Python~3.7. Simulation parameters are given in Table~\\ref{table1}.  We use $N=16$ antennas to receive the signals and the SDOAnet to estimate the DOA, where the number of signals is $K=3$. Moreover, the hyper-parameters for the imperfect array are also given in Table~\\ref{table1}.\n\n\n\\begin{table\n\t\\renewcommand{\\arraystretch}{1.3}\n\t\\caption{Simulation parameters}\n\t\\label{table1}\n\t\\centering\n\t\\begin{tabular}{cc}\n\t\t\\hline\n\t\t\\textbf{Parameter}                 & \\textbf{Value}                                                   \\\\\n\t\t\\hline\n\t\tThe standard deviation in the Gaussian function  & $\\bar{\\sigma}_{\\text{G}}=100$  \\\\\n\t\tThe batch size         & $64$                                                           \\\\\n\t\tThe number of convolution layers & $6$                                                    \\\\\n\t\tThe number of filters in the convolution layer               & $2$                                                            \\\\\n\t\tThe kernel size in the convolution layer                & $3$                                          \\\\\n\t\tThe learning rate          & $5\\times 10^{-4}$ \\\\\n\t\tThe number of antennas         & $N=16$ \\\\\n\t\tThe number of targets         & $K=3$ \\\\\n\t\tThe distance between adjacent antennas & $0.5$ wavelength \\\\\n\t\t\\tabincell{c}{The maximum standard deviation of\\\\ position perturbation} \n\t\t  & ${\\sigma}_{\\text{max\\_per}}=0.15$  \\\\\n\t\t\\tabincell{c}{The maximum standard deviation of\\\\ inconsistent gain}  & ${\\sigma}_{\\text{max\\_gain}}=0.5$  \\\\\n\t\t\\tabincell{c}{The maximum standard deviation of \\\\inconsistent phase}  & ${\\sigma}_{\\text{max\\_phase}}=0.2$  \\\\\n\t\tThe maximum mutual coupling effect  & ${\\sigma}_{\\text{mc}}=0.06$  \\\\\n\t\tThe nonlinear effect & ${\\sigma}_{\\text{nonlinear}}=1.0$  \\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table}\n    \nFirst, the proposed SDOAnet contains convolution layers, and each convolution layer has convolution, batch normalization, and ReLU active function operations. In the SDOAnet, some important hyperparameters must be considered for a better DOA estimation. The first hyperparameter is the number of 1D convolution layers. In Fig.~\\ref{layer}, we show the DOA estimation performance with different numbers of convolution layers. As shown in this figure, when the number of convolutions is $6$, a better estimation performance is achieved, so we use $6$ convolution layers in the following simulations.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=3.7in]{.\/Figures\/layers.pdf}\n\t\\caption{The DOA estimation performance with different numbers of layers.}\n\t\\label{layer}\n\\end{figure} \n\nThen, we compare the DOA estimation performance among the networks using different numbers $M_F$ of filters that used in the convolution layers. For the consideration of both the estimation performance and the network complexity, better performance is achieved with $M_F=2$, so we will use $2$ filters in the following simulations. \n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=3.7in]{.\/Figures\/filter.pdf}\n\t\\caption{The DOA estimation performance with different numbers of filters.}\n\t\\label{filter}\n\\end{figure} \n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=3.7in]{.\/Figures\/std.pdf}\n\t\\caption{The DOA estimation performance with different standard deviations $\\bar{\\sigma}_{\\text{G}}$.}\n\t\\label{std}\n\\end{figure} \n\nIn the procedure of training the SDOAnet, the referred spatial spectrum is used to measure the loss function, where we use the Gaussian functions to approximate the spatial spectrum. Hence, the standard deviation $\\bar{\\sigma}_{\\text{G}}$ in the Gaussian function is important in approximating the spatial spectrum. We show the DOA estimation performance with  different standard deviations $\\bar{\\sigma}_{\\text{G}}$ in Fig.~\\ref{std}. When the standard deviation $\\bar{\\sigma}_{\\text{G}}$ is $100$, a better DOA estimation performance is achieved, so we will use $\\bar{\\sigma}_{\\text{G}}=100$ in the following simulation. \n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=3.8in]{.\/Figures\/spectrum.pdf}\n\t\\caption{The spatial spectrum  compared with the existing methods.}\n\t\\label{spectrum}\n\\end{figure} \n\nNext, based on the above SDOAnet parameters, the estimated spatial spectrum is shown in Fig.~\\ref{spectrum} for the DOA estimation and is also compared with the following existing methods:\n\\begin{itemize}\n\t\\item \\textbf{MUSIC method~\\cite{liao_music_2016}:} In the traditional  MUSIC method, multiple snapshots are used to estimate the covariance matrix, which is used to obtain the DOA based on the eigenvalue decomposition. For a fair comparison, we use the MUSIC algorithm with only snapshot proposed in~\\cite{liao_music_2016}, where uses both a Hankel data matrix and the Vandermonde decomposition in the MUSIC method;\n\t\\item \\textbf{ANM method~\\cite{govinda_raj_single_2019,wei_gridless_2020,zai_yang_enhancing_2016}:} The  ANM-based methods have been proposed for the DOA estimation and can exploit the target's sparsity in the spatial domain. Unlike the current  CS-based methods discretizing the spatial domain into grids and using a dictionary matrix for the sparse reconstruction~\\cite{z_yang_orthonormal_2011,z_tan_joint_2014,g_yu_statistical_2011}, the ANM method estimates the DOA in the continuous domain. It can solve the \\emph{off-grid} problem caused by the discrete methods. \n\t\\item \\textbf{FFT method:} The  FFT  method is widely used in practical systems with low computational complexity. However, the resolution of the FFT method is unsatisfactory but robust to the imperfect array;\n\t\\item \\textbf{OMP method~\\cite{aghababaiyan_high-precision_2020,lin_single_2021,chen_source_2019}:} The \n\t  orthogonal matching pursuit (OMP) method is a CS-based method using the discretized spatial angles, and has relatively low computational complexity. Hence, it has been widely used in sparse reconstruction problems.\n\\end{itemize} \nAs shown in Fig.~\\ref{spectrum}, the spatial spectrum estimated by the proposed SDOAnet has better performance than the MUSIC, ANM, FFT, and OMP methods. Additionally, the proposed method is based on the convolution network and has low computational complexity than the ANM and MUSIC methods. Therefore, the proposed SDOAnet is efficient in the DOA estimation problem.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=3.7in]{.\/Figures\/snr.pdf}\n\t\\caption{The DOA estimation performance with different SNRs.}\n\t\\label{snr}\n\\end{figure} \n\nNext, the DOA estimation performance under different  SNRs is shown in Fig.~\\ref{snr}, where the SNR is from $0$ dB to $30$ dB. This figure shows that a better estimation performance is achieved by the proposed method  in the scenario with the imperfect array than that using  ANM, FFT, MUSIC, and OMP methods. The estimation performance is measured by the root mean square error\n\\begin{align}\n\t\\text{RMSE} = \\sqrt{\\frac{1}{N_{\\text{sim}}K}\\left\\|\n\t\\hat{\\boldsymbol{\\theta}}-\\boldsymbol{\\theta}\\right\\|^2_2},\n\\end{align}\nwhere $N_{\\text{sim}}$ is the number of simulations,  $\\hat{\\boldsymbol{\\theta}}$ is the estimated DOA vector, and $\\boldsymbol{\\theta}$ is the ground-truth DOA vector. For the SNR being $10$~dB, the RMSE of the proposed SDOAnet is about $\\ang{0.70}$ and that of ANM method is about $\\ang{1.15}$, so the RMSE improvement is about $39.13\\%$. Additionally, when the SNR is $7.5$~dB, the RMSE of the proposed SDOAnet method is the same as that of the ANM method with the SNR being $15$~dB, so the SNR improvement is about $7.5$~dB.\n\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=3.7in]{.\/Figures\/imperfect.pdf}\n\t\\caption{The DOA estimation performance with different imperfect factors.}\n\t\\label{imperfect}\n\\end{figure} \n\nWe use an imperfect factor to measure the imperfect effect, which is defined as $\\xi$. With the imperfect factor $\\xi$, the imperfect parameters for position perturbation, inconsistent gain, inconsistent phase, mutual coupling effect and nonlinear effect will be $\\xi\\sigma_{\\text{max\\_per}}$, $\\xi\\sigma_{\\text{max\\_gain}}$, $\\xi\\sigma_{\\text{max\\_phase}}$, $\\xi\\sigma_{\\text{mc}}$, and $\\xi\\sigma_{\\text{nonlinear}}$, respectively. In Fig.~\\ref{imperfect}, the DOA estimation performance with different imperfect factors is shown, where a better estimation performance is achieved by the proposed SDOAnet method than the compared methods. Additionally, the proposed method works better in the scenario with a higher imperfect factor, which means that the proposed method is robust to the imperfect effect.\n \n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=3.7in]{.\/Figures\/DLsp.pdf}\n\t\\caption{The spatial spectrum  compared with the existing deep learning-based method.}\n\t\\label{dlsp}\n\\end{figure} \n\nFurthermore, a DL-based method is also proposed in~\\cite{izacard2019data} for the DOA estimation, named by deep frequency network, where the output of the network is the spectrum. In Fig.~\\ref{dlsp}, the spatial spectrum of the proposed SDOAnet and that of the deep frequency network is given. Since the output of the deep frequency network is the spatial spectrum, the estimated spectrum is not smooth compared with the proposed SDOAnet. Hence, a better DOA estimation performance can be achieved by the SDOAnet than that by the deep frequency network.\n\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=3.7in]{.\/Figures\/DLsnr.pdf}\n\t\\caption{The DOA estimation performance compared with the existing deep learning-based method.}\n\t\\label{dlsnr}\n\\end{figure} \n \nFinally, the DOA estimation performance under different SNRs is given in Fig.~\\ref{dlsnr}, where the SNR is from $0$~dB to $30$~dB. The same training data set is used for the SDOAnet and the deep frequency network. Compared with the existing methods, including the FFT method, OMP method, and deep frequency network, the proposed SDOAnet can achieve better DOA estimation performance. Therefore, the proposed method is efficient in the DOA estimation problem using the imperfect array.\n\n\\section{Conclusions}\\label{sec6}\nThe DOA estimation problem in the imperfect array has been considered in this paper, and a general system model to describe the antenna position perturbations, the inconsistent gains\/phases, the mutual coupling effect, the nonlinear effect, etc., has also been formulated. Then, the novel SDOAnet has been proposed. Different from existing methods, the input of the SDOAnet is the raw sampled signals, and the output is a vector, which can be used to estimate the spatial spectrum. With the convolution layers, the convergence of training SDOAnet is much faster than existing DL-based methods. Simulation results show the advantages of the proposed SDOAnet in the DOA estimation problem with a practical array. Future work will focus on the theoretical analysis of the proposed SDOAnet in the DOA estimation performance.  \n  \n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nWheeled snake-like robots \\cite{tanaka2015control}\nare a class of hypermobile robots \\cite{granosik2014hypermobile}\nthat are able to navigate flexibly through rough terrains \nand restricted geometries. Movements may be generated either\nvia central pattern generators \\cite{hopkins2009survey}, or\nvia top-down commands \\cite{pfotzer2017autonomous}, with the\nlatter being a challenging task when a large number of actuators \nis involved. An alternative is autonomous decentralized\ncontrol, which has been studied for the case of serpentine robots \nin terms of a chain of locally coupled oscillators \n\\cite{sato2011applicability,sato2011decentralized}, and \nneurally-inspired generating schemes able of\nsensorless pathfinding \\cite{boyle2013adaptive}.\n\nA key rationale for developing biologically inspired robots \nis the drive for robust and highly adaptive designs\n\\cite{hirose2004biologically,liljeback2012review}. Similarly,\nthis is most of the time also the motive for studying how\nadaptive locomotion can be realized \\cite{aoi2017adaptive},\nf.i.\\ with soft robots \\cite{calisti2017fundamentals}.\nAbstracting from the direct engineering benefit, it is \nof particular interest to study generating mechanisms of \nlocomotion in general, an approach taken here. Our focus\nis on compliant locomotion generated by self-organizing \ndynamical systems, which may take the\nform of either limit-cycle \\cite{martin2016closed}\nor playful behavior \\cite{der2006rocking}. \n\nCompliance, which denotes the ability of a robot to \nreact elastically to environmental feedback \\cite{sprowitz2013towards},\nmay be achieved in several distinct ways, which include\nfrom the engineering perspective suitably designed actuators \\cite{van2009compliant} \nand control algorithms \\cite{calanca2016review}. Compliant \nbehavior can emerge on the other side also through the \nreciprocal dynamical coupling of control, body and environment\n\\cite{pfeifer2012challenges}, the sensorimotor loop. A particularly\ninteresting limit is here, from the perspective of complex system\ntheory, the limit of a fully reactive and hence embodied \ncontroller. In this limit, the controller is inactive in \nabsence of environmental feedback, with the consequence that\nthe sensorimotor feedback has not only a modulating effect on\nlocomotion, becoming instead essential. Locomotion then arises \nvia limit cycles and chaotic attractors that emerge within the \nsensorimotor loop, the telltale sign of self-organized locomotion. \nIt is hence important to ask, as we will do in this study, how \nlocomotion is generated in terms of a dynamical systems bifurcation \ndiagram.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[height=0.5\\textwidth]{control_schemes_v3.pdf}\n\\caption{Illustration of possible control schemes. Sensory\ninformation is processed, e.g.\\ by a neural network, and motor \ncommands sent to the actuators. The actuators may respond either \nrigidly (top), or elastically, viz compliant (bottom).\nCompliant actuators may be realized, as illustrated here, via \na direct feedback loop involving the state of the actuator \n(propriosensation). In the limiting case of an embodied\nactuator, as considered in this study, locomotion occurs also in\nthe absence of a modulatory top-down signal.\n}\n\\label{fig:control_topDownCompliant}\n\\end{figure}\n\nDecomposing complex behavior into a series (or into a \nsuperposition) of basic reusable building blocks, the\nmotor primitives \\cite{flash2005motor}, is a well studied \napproach for reducing the control problem of complex robots. \nMovement primitives may be modeled by nonlinear dynamical \nsystems \\cite{ijspeert2002movement} using, e.g., \nGaussian mixture models \\cite{khansari2011learning},\nwhere the parameters of the dynamical system are either\nuniquely defined or drawn from a suitable distribution\n\\cite{paraschos2013probabilistic,amor2014interaction}.\nMotor primitives can emerge also from embodied dynamics\nin terms of chaotic itinerancy \\cite{park2017chaotic},\nor, alternatively, as self-organized attracting states \nin the sensorimotor loop \\cite{sandor2015sensorimotor}, \nthat is within the state space comprising the controller, \nthe body of the robot and the environment. Here we propose \na new type of self-organized controller for wheeled robots\nthat leads to multiple fixpoint and limit-cycle \nattractor states and hence to self-organized motor \nprimitives in the sensorimotor loop. With the\nbehavior of the robot being self organized on the \nlevel of the individual wheels and with respect to \ninter-wheel coordination, the resulting dynamics reflects \nits affordances \\cite{chemero2007gibsonian}\nwhen placed in simple but structured environments.\n\n\\subsection{Control frameworks and the sensorimotor loop}\n\nSeveral in part non-exclusive routes for the \ngeneration of locomotion in robots and animats \ndo exist in generic terms. Standard\ntop-down control, as illustrated\nin Fig.~\\ref{fig:control_topDownCompliant},\nconsists of a central processor \ngenerating motor commands either reactively,\nin response to sensory inputs, or deliberately\non its own \\cite{nakhaeinia2011review}. The \nactuator may be in turn stiff, as for industrial\nrobots, or compliant, as for the muscles and tendons\nof animals, reacting either passively or actively \nto external forces \\cite{van2009compliant}. For\nthe latter case, as sketched in\n Fig.~\\ref{fig:control_topDownCompliant}, the\nactuator changes its stiffness upon sensing \nits own state. Compliance arises then\nin response to propriosensation.\n\nWe are interested in locomotion that arises through \nthe interaction of the degrees of freedom of the robot, \nincluding both internal variables and the body, with \nenvironmental feedback. The combined variables of the \nresulting sensorimotor loop constitute then the phase \nspace for dynamical attracting states, fixed points, \nlimit cycles and chaotic attractors, that correspond \nto self-organized behavioral primitives. The locomotion \ngenerated in this way is highly compliant in the sense \nthat the attracting states in the sensorimotor loop \nrespond elastically to additional top-down commands \nchanging internal parameters.\n\n\\section{Locomotive principles}\n\nStudies of real-world and simulated robots may focus \neither on performance, and its improvement, or on the \ngenerative capabilities of locomotive principles.\nThe latter approach is gaining in importance in view of\na recent study of the neural coding of leg dynamics \nin flies, which showed that the dynamics of the \nleg becomes dysfunction once the feedback loop between \nleg proprioception and motor commands is cut\n\\cite{mamiya2018neural}. These findings imply that\nself-organization plays a commanding role in fly\nlocomotion. Distributed computations has been found\nto be of relevance for the nematode C.~elegans\n\\cite{kaplan2018sensorimotor}. Here we concentrate \non generative principles that are time reversal \nsymmetric in the sense that a given set of internal \nparameters allows the robot to move both forwards and \nbackwards. The direction selected by the robot then \ndepends on the initial state, like a small positive \ninitial velocity or force.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=0.75\\textwidth]{wheel_controller_sketch}\n\\caption{Illustration of a one-neuron controller simulating \nthe transmission of classical steam engines. The actual\nposition $x^{(a)}=\\cos(\\varphi)$ of the wheel drives,\nas described by (\\ref{dot_x}), the neural\nactivity $y(x)$ setting the target position $x^{(t)}=y$. \nA simulated spring with spring constant $k$ between \n$x^{(a)}$ and $x^{(t)}$ generates subsequently the\ntorque $RF_\\mathrm{tan}$ acting on the wheel. Here \n$F_\\mathrm{tan}=F_k\\sin(\\varphi)$ denotes the tangential \nprojection of the spring force $F_k=k(x^{(t)}-x^{(a)})$.\n}\n\\label{fig:steamEngineController}\n\\end{figure}\n\n\\subsection{Locomotion via time reversal symmetry breaking}\n\nLocomotion is parametrized typically by a velocity vector\n$\\mathbf{v}=d\\mathbf{r}\/dt$ that incorporates both \nthe direction and the magnitude of the movement.\nReversing time $t\\leftrightarrow (-t)$ reverses then\nalso the velocity vector. Here we are interested in \nself-organized robots that break time reversal symmetry \nspontaneously, which in our case implies  that the \nattracting states in the sensorimotor loop come in pairs\nthat are related via time reversal symmetry. Whether\nthe robot moves for- or backwards depends then only\non the initial conditions. For this purpose we\nuse the one-neuron controller illustrated\nin Fig.~\\ref{fig:steamEngineController}.\n\nA wheel with a rotational angle $\\varphi$ \nis regulated individually via\n\\begin{equation}\n\\tau \\dot x = \\cos\\varphi -x, \n\\qquad\\quad y=\\tanh(ax)~,\n\\label{dot_x}\n\\end{equation}\nwhere $x$ is the membrane potential of the controlling\nneuron, $y=\\tanh(ax)$ the neural activity and $\\tau$ \nthe membrane time constant. The motor command is \nproportional to the spring force\n\\begin{equation}\nF_k=k\\big(x^{(t)}-x^{(a)}\\big),\n\\qquad\\quad\nx^{(t)}= y,\n\\qquad\\quad\nx^{(a)}=\\cos(\\varphi)\\,,\n\\label{F_k}\n\\end{equation}\nwhere $k$ is a spring constant and $x^{(a)}$ and\n$x^{(t)}$ respectively the actual and the target \nposition of the wheel in terms of a projection to \nthe ground \\cite{sandor2018kick}. Note that the \nangle $\\varphi$, which enters the right-hand side \nof Eqs.~(\\ref{dot_x}) and (\\ref{F_k}) as $\\cos(\\varphi)$, \nis the measured, the actual angle of the wheel. All \nforces, gravitational and mechanical, impact the \ncontroller hence exclusively via their influence on \nthe angle $\\varphi$.\n\nThe controller simulates the transmission rod of \na classical steam engine, as sketched in \nFig.~\\ref{fig:steamEngineController}, as it\ntranslates the bounded forth and back motion \nof the neural activity $y(t)$ into a rotational \nmotion. Alternatively, instead of using the\nangle $\\varphi$ as the determining variable, one \ncould postulate a discrete map $\\omega^{(t)}=\\tanh(a\\omega^{(a)})$\nbetween the actual and a target angular velocity\n\\cite{der2008predictive}, $\\omega^{(a)}$ and \n$\\omega^{(a)}$. This is not a problem for simulated\nrobots, for which $\\omega$ is a directly accessible \nvariable. To obtain a reliable estimate of the instantaneous\nangular velocity for real-world robots working with \nduty cycles of the order of $20\\,\\mathrm{Hz}$\nwould however be a challenge \\cite{sandor2018kick}. \n\nA controller enabling locomotive limit cycles to\nemerge in the sensorimotor loop \\cite{martin2016closed},\nas described here by (\\ref{dot_x}) and (\\ref{F_k}), \ndiffers qualitatively from controlling schemes employing\nlocal phase oscillators \\cite{ambe2018simple}, for which \na spontaneous reversal of the direction of motion would \nnot be possible.\n\n\\subsection{Isolated wheel}\n\nThe individual wheels of the simulated robots are \ncontrolled exclusively by (\\ref{dot_x}) and \n(\\ref{F_k}). There is no explicit inter-wheel\ncoupling present. It is illustrative to model,\nfor comparison, an idealized isolated wheel\nwith moment of inertia $I$, radius $R$, angle $\\varphi$ \nand angular velocity $\\omega$. The force $F_k$\ngenerated by the simulated transmission rod then\nenters the equations of motion as a torque \n$RF_k\\sin(\\varphi)$,\n\\begin{equation}\n\\tau \\dot x = \\cos\\varphi -x,\n\\qquad\\quad\n\\dot\\varphi= \\omega,\n\\qquad\\quad\nI\\dot\\omega = R(F_k\\sin\\varphi -f \\omega)~,\n\\label{eq_dot_x_phi_omega}\n\\end{equation}\nwhere $f>0$ is a friction coefficient. \nEq.~(\\ref{eq_dot_x_phi_omega}) is manifestly \ninvariant under $\\omega\\leftrightarrow(-\\omega)$,\n$\\varphi\\leftrightarrow(-\\varphi)$ and\n$x\\leftrightarrow x$, which implies\ntime-reversal symmetry in terms of an\ninvariance with respect to reversing \nthe direction of motion. We will investigate\n(\\ref{eq_dot_x_phi_omega}) further in\nSect.~\\ref{sec_theory}, noting here that\nsymmetry breaking may occur also in embodied \nrobots that incorporate forward world \nmodels \\cite{der2013behavior}.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[height=0.4\\textwidth]{screenshot_alone.png}\n\\hspace{4ex}\n\\includegraphics[height=0.4\\textwidth]{screenshot_slope.png}\n\\caption{Screenshots of the LPZRobots simulation\nenvironment. {\\it Left:} A snake-like train of cars composed \nof five passively coupled segments. Each segment contains\ntwo independent wheels that influence each other exclusively \nthrough the mechanics of the body and via the hinge joints\nconnecting the individual segments.\n{\\it Right:} A robot climbing intersecting slopes on its \nown, with the silver line illustrating the ground \ntrace of the last segment. No explicit control signal has \nbeen given. The wiggling observed when the robot moves fast \non straight stretches disappears at lower velocities, as it \nis the case when moving steeper up the slope. The train of \ncars reverses direction autonomously when hitting the intersecting\nslope. The wiggling amplitude on the last leg increases progressively \nwhile moving down, leading in the end to an upward curve\n(\\href{http:\/\/doi.org\/10.6084\/m9.figshare.7643123.v1}\n{click for movie}).\n}\n\\label{fig:screenshots}\n\\end{figure}\n\\section{Results}\n\nWe used the LPZRobots physics simulation\npackage \\cite{der2012playful} for the simulation\nof robots composed of chains of 1-5 two-wheeled\ncars linked passively through hinge joints, which\nare equipped with passively damped torsion springs. \nIn the absence of motor commands or external\nforces the equilibrium position of the hinge joints\ninduces a straight alignment of the connected body \nsegments. During locomotion the joints can store,\non the other hand, potential energy when bent.\n\nShown in Fig.~\\ref{fig:screenshots} is the trajectory \nof the train of cars climbing up a slope that is \nintersected orthogonally by two other slopes. One \nobserves wiggling and straight locomotion together \nwith direction reversal and large turns.\nIn order to develop an understanding we start \nby investigating the velocity profile of a robot on an \nextended slope, concentrating on the dependence of the \nself-regulated steady state velocity on the spring \nconstant $k$ of the actuator and on the inclination of \nthe slope. We note that the simulation cycle times of \nthe LPZRobots simulation package, which is based\non the Open Dynamics Engine \\cite{smith2005open},\nare of the order of 50\\,ms. \n\n\\subsection{Moving up and down an infinite slope}\n\nIn Fig.~\\ref{fig:velocities} we present the velocity\nprofile for a 5-segmented robot moving on a slope\nparallel to the gradient, that is straight up and down.\nThe downward velocity decreases in magnitude with \ndecreasing slope and spring constant $k$, as expected.\n\nFor the robot moving on a horizontal plane there\nexists a critical $k_c\\approx0.54$, such that\nthe limit-cycles corresponding to regular forward \nor backward movement disappear for $k<k_c$. For\na spring constant of $k=0.2$, which is below $k_c$,\nthe robot moves therefore only when the slope has a\nfinite downward inclination, as shown in \nFig.~\\ref{fig:velocities}.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=0.80\\textwidth]{slope_velocities.pdf}\n\\caption{The velocity profile of a 5-segmented robot on\na slope. The parameters are $a=1$ and \n$k=2.0\/1.2\/0.2$ (yellow\/red\/green). Shown are the \nsteady-state velocities for moving directly upwards \n(positive $v_{slope}$) and for moving down the slope \n(negative $v_{slope}$). A first order transition occurs\nfor $k=2.0$ (yellow dots) and $k=1.2$ (red dots) when \nincreasing the inclination slowly, that is adiabatically, \nas indicated by the arrows. The robot cannot move upwards \nfor $k=0.2$ (green dots), being subcritical.\n}\n\\label{fig:velocities}\n\\end{figure}\n\nThe torque exerted on the wheels is directly proportional \nto the spring constant, being generated otherwise through\nthe sensorimotor feedback. Moving upward the slope the\ntorque $R F_{tan}$, and hence also the sensorimotor feedback, \nneeds to counter the gravitational downhill force $F_G$. For \nan engine producing a constant torque, the balancing of the\ntangential and the gravitational force would lead to an uphill \nvelocity $v_{slope}\\propto F_{tan}-F_G$ that vanishes\nlinearly and hence continuously at a critical inclination.\nThis is however not the case when the motion is generated \nthrough sensorimotor feedback, as evident from the data \npresented in Fig.~\\ref{fig:velocities}. The sensorimotor\nfeedback involves a self-consistency condition that\nbreaks down discontinuously, at finite values of the \nuphill velocity, when the inclination of the slope becomes \ntoo large. It remains however the case that larger\nspring constants $k$ allow the robot to move up steeper\nslopes.\n  \n\\subsection{Autonomous direction reversal}\n\nThe trajectory of the robot presented in\nFig.~\\ref{fig:screenshots} hits twice an \nintersecting slope. A robot equipped with\nactuators producing constant torques would\nmove segment by segment onto the intersecting \nslope, up to the point where the gravitational \ndownward pull of the increasing number of segments \ncancels with the locomotive force. At this point\nthe robot would remain in place.\n\nThe locomotive force of the snake-like robot presented \nhere is however highly compliant, being generated \nwithin the sensorimotor loop. The velocity profile\npresented in Fig.~\\ref{fig:velocities} implies \nthat an equilibrium position resulting from the\nbalance of an upward locomotive force and the\ndownhill gravitational pull is not possible, namely \nthat the limit-cycle attractors present in the \nsensorimotor loop allow the robot to move only \nup- or downhill. We note that the stable fixpoint\nattractors corresponding to a non-moving state,\nthat exist in conjunction with the limit-cycle \nlocomotion for an isolated wheel, as discussed in \nSect.~\\ref{sec_theory}, possesses only a vanishing\nsmall basin of attraction for the case of a train\nof cars. A robot hitting a slope that is too steep \nis therefore likely to reverse direction, as observed in \nFig.~\\ref{fig:screenshots}, instead of being pulled\ninto a fixpoint attractor and coming to a stop.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[height=0.75\\textwidth]{timeSeries_allmodes.pdf}\n\\caption{As a function of time, the measured angular velocity \n$\\omega$ of one of the two wheels of the first and of the last\ncar. The membrane time constant is $\\tau=50\\,\\mbox{ms}$. \n{\\it Top:} For $a=0.5$ and $k=0.5$, a straight-moving mode\nLeft and right wheels of a given car are exactly synchronized.\nThe small difference in amplitude in the $\\omega(t)$ oscillations\nbetween cars is due to small oscillations in the respective\npitch angles. Note that $\\omega(t)$ is not exactly constant\nas resulting from (\\ref{dot_x}).\n{\\it Mid:} For $a=1.0$ and $k=1.0$, a regular meandering mode \n{\\it Bottom:} For $a=1.0$ and $k=5.0$, a chaotic mode resulting from wide\nsideway swings of the tail\n(\\href{https:\/\/doi.org\/10.6084\/m9.figshare.8143040.v1}{click for movies}).\n}\n\\label{fig:time_series}\n\\end{figure}\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[height=0.27\\textwidth]{wall_bounce_1.png}\n\\hfill\n\\includegraphics[height=0.27\\textwidth]{wall_bounce_2.png}\n\\hfill\n\\includegraphics[height=0.27\\textwidth]{wall_bounce_3.png}\n\\caption{A five-segmented robot bouncing off the wall, with\nthe silver line tracing the position of the darker segment.\nThe only knowledge the robot disposes of the outside world,\nand hence of the presence of a wall, is via propriosensation, \nthat is via the measurement of the angle of the wheels. The\nslight wiggling of the forward motion causes the robot to be \nreflected at various angles, even though it bumps into the wall \nperpendicularly (with respect to the average direction\nof locomotion).\nThe direction reversal results from the destruction of the\nforward limit cycle upon hitting the wall, with the flow\nin phase space evolving subsequently towards the backward\nattractor.\n}\n\\label{fig:train_wall_bounce}\n\\end{figure}\n\n\n\\subsection{Straight, meandering and chaotic modes}\n\nThe last leg of the trajectory shown in \nFig.~\\ref{fig:screenshots} shows growing\nleft- and rightward swings. This is a typical\nbehavior at larger velocities, here due to \nthe slight downhill direction, that results \nfrom a transversal mechanical instability of \nthe connected segments. The joints are elastic\nand therefore capable to store a certain amount\nof energy, akin to what happens when a string starts\nto vibrate. The final upturn of the robot\nmay occur at an angle, interestingly, that \nputs the robot below criticality. The robot will\nthen stop moving and reverses direction.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=0.75\\textwidth]{screenshot_boxes.pdf}\n\\caption{Superimposed screenshots from the LPZRobots simulation\nenvironment, with $a=1$ and $k=2$. The silver line represents \nthe ground trace of the darker of the two end segments. The \nsnake-like train of cars\nstarts from the lower center (shadowed), where it first hits \na movable box (shadowed), bending and pushing the box around in an \ninward spiral (as indicated by the shadowed box and robot in \nthe middle). As the angle of the spiral becomes steeper and the \nforward velocity smaller, the robot will reach the point at which\nthe forward limit cycle disappears, as seen also in\nFig.~\\ref{fig:velocities}. The robot then stops for a short\nperiod during which the dynamics flows in phase space \nautonomously towards the attractor corresponding to backward \nmotion. Once reached, the robot reverses direction, as one \ncan see from the positioning of the dark end segment\n(\\href{https:\/\/doi.org\/10.6084\/m9.figshare.8143040.v1}{click for movie}).\n}\n\\label{fig:train_boxes}\n\\end{figure}\n\nIn Fig.~\\ref{fig:time_series} we present the angular \nfrequency $\\omega(t)$ for one of the two wheels of the \nfirst and the last car, respectively, of a five-segmented\nsnake-like robot moving on a flat ground. Shown are the\ntimelines for two regular and for a chaotic mode. In the \nfirst limit-cycle mode the train of cars moves straight. \nThe ten wheels are in this case synchronized in the sense\nthat the small modulation of the respective angular \nvelocities, which occur because the controller (\\ref{dot_x}) \nis not rotationally invariant, appear all at exactly the \nsame time. Their respective amplitudes vary however from\ncar to car, which implies that the pitch angles of the \nindividual cars oscillates, even though only slightly.\n\nFor larger average velocities, the limit-cycle straight \nmode tends to be become unstable, making way to\nmeandering and chaotic modes, as illustrated in\nFig.~\\ref{fig:time_series}. We did not investigate\nthe exact nature of the respective transition to chaos, \nwhich may be due to a cascade of period-doubling within \nthe space of meandering modes \\cite{gros2015complex}.\nWe also note that our classification of the highly\nirregular mode as chaotic relies here only on a visual \ninspection, as we did not apply a formal test for the \npresence of deterministic chaos \\cite{wernecke2017test}.\n\n\\subsection{Interacting with a structured environment}\n\nThe robots exhibit interesting behavioral patterns when \nsituated in a structured environment. As a first example \nwe show in Fig.~\\ref{fig:train_wall_bounce} the interaction\nof a 5-segmented robot with a wall. Before hitting the\nwall the robot possesses, due to time-reversal symmetry,\nboth a forward- and a backward-moving limit cycle.\nApproaching the wall the sensorimotor state of the robot \nis in the forward limit cycle, which becomes however \ndestroyed upon hitting the wall. The flow in phase space,\nthat is the evolution of the membrane potentials\n$x$ of the individual wheels, is then attracted by\nthe remaining limit cycle, which is the one corresponding \nto moving backward. The robot hence reverses direction.\n\nThe observed direction reversal occurs autonomously\nin the absence of top-down control signals. As the robot possesses\nonly sensors measuring the angles of the wheels there \nare furthermore no external sensors present that would\ninform the robot about the distance to the obstacle. One \nobserves, as shown in Fig.~\\ref{fig:train_wall_bounce}, that \nthe angle at which the robot bounces varies considerably as a\nconsequence of the wiggling of the initial forward motion.\nIn a slow-velocity non-wiggling mode the bouncing occurs\nexactly perpendicularly.\n\nWe also studied the interaction of the multi-segmented robot with\nmovable boxes, as illustrated in Fig.~\\ref{fig:train_boxes}.\nDue to the mechanical feedback of the passive but elastic\nhinge joints, the body of the train of cars bends, such that \nthe robot continues to push the box in ever smaller circles.\nDuring the push, the robot slows down continuously, until\na critical velocity is reached and the forward limit cycle\ndisappears. At this point the robot stops,\nreversing direction autonomously.\n\nThe robot disposes of a single controlling neuron per actuator, \nwhich obtains in turn information about the external world only\nindirectly, namely via the measured angle of the respective\nwheel. The behavior illustrated in Fig.~\\ref{fig:train_boxes},\nsuch as pushing around boxes, is hence emergent and an example \nthat embodied controlling frameworks may lead robustly to novel \nbehavioral patterns. Functionally, the ability of the snake-like\nrobot to follow spiral-shaped trajectories, when pushing a box, \nis a direct consequence of the highly elastic working regime of \nthe actuators, viz of compliance. By themselves, the wheels on \nthe left-hand and the right-hand side of the body would acquire, \nbeing subject to identical controllers, identical angular velocities. \nThe fact, however, that the turning speed of the wheels is not determined \nby a top-down signal, but by the sensorimotor feedback, allows the \nemerging limit cycles to adapt autonomously to environmental\nforces. The wheels on opposite sides of the body react\nas a consequence distinctly when the box exerts a non-symmetric \nresistance onto the robot. The respective angular velocities then \ndiffer.\n\n\\subsection{Theory for an isolated single wheel}\\label{sec_theory}\n\nThe fixpoints of an isolated and non-moving single wheel, \nas described by Eq.~(\\ref{eq_dot_x_phi_omega}), are determined \nby $\\omega=0$, $x=\\cos \\varphi$ and\n\\begin{equation}\n\\sin\\varphi=0 \\qquad\\mbox{or}\\qquad y(x)=\\cos\\varphi\\,,\n\\label{omegaNull}\n\\end{equation}\ncompare Eqs.~(\\ref{eq_dot_x_phi_omega}) and \n(\\ref{F_k}). The Jacobian $J(\\varphi)$\nis in general\n\\begin{equation}\nJ(\\varphi)\\ =\\ \n\\left(\\begin{array}{ccc}\n-1\/\\tau & \\,-\\sin\\varphi\/\\tau\\, &0 \\\\[0.5ex]\n0 & 0 & 1 \\\\[0.5ex]\nka(1-y^2)\\sin\\varphi & A & -f\n\\end{array}\\right)\\,,\n\\label{Jacobian}\n\\end{equation}\nwhere $A=ky\\cos\\varphi+k(\\sin^2\\varphi-\\cos^2\\varphi)$.\nWe have set $R=1$, measuring in addition $k$ and\n$f$ relative to $I$.\n\n\\begin{itemize}\n\\item[--] For the first fixpoint $\\sin\\varphi=0$, that\n          is for $(x,\\omega,\\varphi)=(1,0,0)$ and\n          $(-1,0,\\pi)$, one has \n          $A(0,\\pi)=k(\\pm y(\\pm 1)-1)=k(\\tanh(a)-1)$, \n          which is always negative. The eigenvalues of \n          the Jacobian are then\n\\begin{equation}\n\\lambda_0(0,\\pi)=-\\frac{1}{\\tau},\\qquad\\quad\n\\lambda_\\pm(0,\\pi)=-\\frac{f}{2}\\pm\\frac{1}{2}\\sqrt{f^2+A(0,\\pi)}\\,,\n\\label{eq_fixpoint_0_pi}\n\\end{equation}\nSince $A(0,\\pi)<0$, the fixpoints at $\\varphi=0$ and \n$\\varphi=\\pi$ are always stable.\n\n\\item For the second set of fixpoints one has to solve the\nself-consistency condition $y(x)=\\tanh(ax)=x$,\nwith $x=\\cos\\varphi$. The trivial solution\n$x=0$, that is $(x,\\omega,\\varphi)=(0,0,\\pm\\pi\/2)$ \nsplits at $a_\\mathrm{c}=1$ via a pitchfork transition,\nallowing for three coexisting fixpoints for $a>a_c$.\nFor small $k$ the trivial fixpoint $x=0=\\cos\\varphi=y$ \nwith the Jacobian\n\\begin{equation}\nJ(\\varphi=\\pm\\pi\/2)\\ =\\ \n\\left(\\begin{array}{ccc}\n-1\/\\tau & \\,\\mp 1\/\\tau\\, &0 \\\\[0.5ex]\n0 & 0 & 1 \\\\[0.5ex]\n\\pm ka & k & -f\n\\end{array}\\right)\\,\n\\label{Jacobian_x_0}\n\\end{equation}\nis  a saddle for $0<a\\le1$, having two \nnegative and one positive eigenvalues, being\nstable however for $a>1$. For larger values of\nthe spring constant~$k$, the $x=0$ solution undergoes \na Hopf bifurcation leading to limit-cycle oscillations.\n\\end{itemize}\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=0.95\\textwidth]{kplot_a_smaller_1.pdf}\n\\caption{One-step heteroclinic route to locomotion for $a<1$.\nShown are stable limit cycles (red) and stable, unstable \nmanifolds   (black, light\/dark blue\/green) and \nselected sample trajectories (violet). The fixpoints at \n$\\varphi=0,\\,\\pi$ are stable nodes\/foci respectively for \nsmall and larger spring constants $k$ (top and upper middle panel), \nwith the saddles at $\\varphi=\\pm\\pi\/2$ remaining \nunchanged in character for all $k$. A symmetric heteroclinic \nconnection between the saddles is created when increasing \n$k$ further. For $k\\approx19.66$ (lower bottom panel)\na stable limit cycle (red) corresponding to limit-cycle \nlocomotion (bottom panel) is generated. \nThe parameters are $a=0.95$, $\\tau=0.2$, $I=0.25$ and $f=0.5$.\n}\n\\label{fig:bifurcations_a_smaller_1}\n\\end{figure}\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=0.95\\textwidth]{kplot_a_larger_1.pdf}\n\\caption{Multi-step heteroclinic route to locomotion for $a>1$.\nThe fixpoints at $\\varphi=0,\\pi$ are always stable nodes, \nwith the foci at $\\varphi=\\pm\\pi\/2$ undergoing a Hopf\nbifurcation upon increasing the spring constant $k$\n(top\/upper middle panel). The resulting limit\ncycle (red) corresponds to a periodic forth-and-back\nmotion characterized by $\\omega\\ne0$ and a vanishing\naverage $\\overline{\\omega}=0$. The forth-and-back\nmotion is destroyed by a symmetric heteroclinic transition\n(upper\/lower middle panel), leading to an intermediate\nphase without locomotion. A second heteroclinic transition\nthen generates stable limit-cycle locomotion \n(lower middle\/bottom panel). Parameters, besides $a=1.5$, \nand color-coding as for Fig.~\\ref{fig:bifurcations_a_smaller_1}.\n}\n\\label{fig:bifurcations_a_larger_1}\n\\end{figure}\n\n\\subsection{Routes to locomotion}\n\nIt is interesting to study how limit-cycle\nlocomotion arises from a configuration of\nindividual fixpoints upon increasing the\nforce acting on the wheel, that is the spring \nconstant $k$.\n\nIn Fig.~\\ref{fig:bifurcations_a_smaller_1} we illustrate\nthe case $a<1$, for which the saddles at \n$\\varphi=\\pm\\pi\/2$ do not undergo a pitchfork bifurcation \nyet.  Locomotion arises in this case via a one-step \nheteroclinic transition which allows for the generation \nof limit cycles of finite amplitudes. A pair of stable\nand unstable limit cycles is eventually produced\nwhen increasing the spring constant $k$, with the \nstable limit cycle corresponding to a locomotive \nbehavioral primitive. Note that the phase space\nof (\\ref{eq_dot_x_phi_omega}) is three dimensional \nand that the flow shown in \nFig.~\\ref{fig:bifurcations_a_smaller_1} corresponds\nto a projection onto the $(\\varphi,\\omega)$-plane.\nTrajectories may hence intersect.\n\nIn Fig.~\\ref{fig:bifurcations_a_larger_1} \na multi-step route to locomotion for $a>1$ is presented. \nOne observes first a Hopf-bifurcation (HB) at\n$\\varphi=\\pm\\pi\/2$, which leads to a first \nintermediate phase characterized by a closed \nlimit-cycle in the $(\\varphi,\\omega)$-plane. \nThis limit cycle, corresponding to \nsmall amplitude forth-and-back periodic motion, \nis destroyed when hitting in a symmetric heteroclinic \ntransition (SHE) the two saddles present additionally\nfor $a>a_\\mathrm{c}=1$. Motion ceases in the subsequent \nsecond intermediate phase, for which the \nunstable trajectories emerging from \n$\\varphi=\\pm\\pi\/2$ lead to the fixpoints~$\\varphi=0,\\pi$. \nA stable limit-cycle emerges however from a second \nheteroclinic transition (HE) when increasing the spring\nconstant $k$ further, namely when the unstable\nmanifold of one of the additional saddles\nhits another saddle.\n\nFor the parameters used for Fig.~\\ref{fig:bifurcations_a_larger_1} \nthere are hence for $a>1$ two phases without locomotion,\nviz for which the angular frequency $\\omega$ decays to \nzero. The average angular frequency $\\overline{\\omega}$ \nvanishing for forth-and-back motion, but not for limit-cycle\nlocomotion:\n$$\n\\fbox{$\\phantom{|}\\omega\\to0\\phantom{|}$} \n\\ \\ \\xrightarrow{\\mbox{\\small HB}} \\ \\\n\\fbox{$\\phantom{|}\\omega\\ne0,\\,\\overline{\\omega}=0\\phantom{|}$} \n\\ \\ \\xrightarrow{\\mbox{\\small SHE}} \\ \\\n\\fbox{$\\phantom{|}\\omega\\to0\\phantom{|}$} \n\\ \\ \\xrightarrow{\\mbox{\\small HE}} \\ \\\n\\fbox{$\\phantom{|}\\omega\\ne0,\\,\\overline{\\omega}\\ne0\\phantom{|}$} \n$$\nWith the motor command being proportional to the spring\nconstant $k$, it is somewhat intuitive that one needs a \ncritical $k$ for locomotion to emerge. Relatively large \nspring constants have been used in \nFig.~\\ref{fig:bifurcations_a_smaller_1} for \nillustrative purposes.\n\n\\section{Conclusions}\n\nThe gait of an animal corresponds to a coordinated\npattern of limb movements that repeats with a certain \nfrequency. Gaits are generated typically by a central \npattern generator \\cite{hopkins2009survey},\nthat is by a central processing unit that produces\ncoordinated motor signals. We have examined here an \nalternative framework for which the actuators of an \nanimat are self active, with the dynamics of the \nindividual actuators resulting from the presence \nof limit-cycle attractors within the sensorimotor loop.\nThe actuators, in our case the wheels of a snake-like\nrobot, are coupled in this framework only via the\nmechanics of the body and by the reaction of the environment.\nThe gaits of the animat result in our framework therefore \nfrom self-organizing principles. For a simulated \nwheeled snake-like animat, we find that the robot\ninteracts autonomously with the environment, e.g.\nby turning on its own on a slope. The robot will also\npush a movable box around for a while when colliding \nwith one.\n\nWe have tested in addition that locomotion modes also \narise for heterogeneous (not identical) body segments, \ne.g.\\ when the controllers have different spring constants \n$k_i$, or different wheel sizes. The multi-segmented robot \nis capable of generating locomotion in particular when\nseveral actuators are subcritical, i.e.\\ with wheels \nwhich on their own would not maintain oscillatory \ndynamics. Embodiment leads in our study robustly to \nemergent locomotion.\n\nThe here employed dynamical-system type approach to \nrobotic locomotion allows in addition to characterize \nthe motion primitives in terms of self-organized \nattractors formed in the extended phase space of the \nrobot and environment. Incorporating objects of the \nenvironment into the overarching dynamical system \nallows in consequence dynamical system approaches\nalso to classify computational models of affordance\n\\cite{zech2017computational}.\nIn this sense self-organizing attractors \nplay an important role in the generation of useful \nbehavior for the discovery of dynamic object \naffordances \\cite{der2017selforganizing}.\n\n\\section*{Acknowledgments}\nThe support of the German Science Foundation\n(DFG) is acknowledged.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nA Cyber-Physical System (CPS) forms a tight integration of cyber (computational) and physical components \\cite{Lee2008}.\nThe physical system is monitored and controlled by (networks of) embedded computers.\nUsually this occurs with a feedback loop: the computations affect the physical process and vice versa.\nExamples of CPSs are automobiles, medical devices, and manufacturing systems.\nIn this work we consider the supervisory control design of manufacturing systems.\nSupervisory control refers to the high-level (coordinated) monitoring and actuation of the system. \nIn a manufacturing system, the supervisory controller regulates the manufacturing processes and the movement of products through the system.\n\nThe design of supervisory controllers may be performed by applying Model-Based Systems Engineering (MBSE), in which models, rather than documents, are the primary means of information exchange, and engineering processes are applied to these models directly \\cite{Lee2008,Ramos2012}.\nMBSE (of supervisory control) contains many different disciplines, among which specification, variation management, controller synthesis, optimization, formal verification, and implementation.\nThese disciplines each use a specific set of methods, tools, and technologies that are loosely coupled both on a syntactic and semantic level. \nThis has a major impact on engineering efficiency: it hampers verification sufficiently early in the development process, especially concerning system-wide aspects such as throughput and collision avoidance. \nThe simultaneous use and the integration of heterogeneous models and tools to capture system-wide properties reliably and with firm guarantees is an open issue \\cite{Engell2015}.\nTo significantly improve engineering efficiency and enable rapid deployment of (new) systems and system features, seamless syntactic and semantic interoperability between engineering tools needs to be established \\cite{Seshia2017}.\n\n\n\nIn this work we address how to create a multi-disciplinary workflow that has seamless integration of mono-disciplinary MBSE technologies.\nWe show how interoperability of MBSE tools can be achieved through \\textit{Analytics as a Service (AaaS)}.\nIn AaaS, analytics functionality can be accessed over web-delivered technologies (i.e., the cloud).\nGenerally, AaaS is applied in the context of big data \\cite{Sun2012,Demirkan2013,Assunccao2015,Skourletopoulos2016}.\nIn our work, however, we apply the concept of AaaS to MBSE:\nkey functionalities from various MBSE tools are offered as separate services on a network and are made interoperable by automatically translating models to equivalent ones that are necessary for the required service.\nFurthermore, new functionality and tools can be added to the process in a modular manner.\nThe network is set up using the Arrowhead Framework, which is an service-oriented architecture for tool interoperability \\cite{Varga2017,Venanzi2020}. \n\nIn this paper we showcase and demonstrate the functionality of a number of state-of-the-art MBSE tools, and focus on the integration of these tools through AaaS to enable seamless synergistic multi-disciplinary MBSE.\n\n\nTo make the context more tangible, we first discuss a use case in Section \\ref{sec:usecase}.\nExamples from this use case are used throughout the paper.\nIn Section \\ref{sec:challenges}, we discuss some challenges that emerge during the design of the supervisory control of CPSs.\nSome state-of-the-art MBSE tools and technologies that address these challenges are discussed in Section \\ref{sec:tools}.\nThese tools are made interoperable through AaaS, and integration of their functionality into a toolchain is discussed in Section \\ref{sec:toolchain}.\nConclusions are provided in Section \\ref{sec:conclusion}.\n\n\n\n\n\\section{Use case: xCPS manufacturing system}\n\\label{sec:usecase}\nIn this section we discuss a demonstrative use case from which we use examples throughout the paper.\nOur method is demonstrated on the eXplore Cyber-Physical Systems (xCPS) manufacturing system.\nIt is a platform of industrial complexity for research and education on CPSs.\nThe xCPS system is elaborately discussed in \\cite{Adyanthaya2017} and \\cite{Basten2020}.\nFor simplicity, we only consider a part of the system, which is formed by the collection of components mentioned in Fig. \\ref{fig:overview}.\nThe realization of this (sub)system is displayed in Fig. \\ref{fig:realization}.\n\nThe xCPS system receives tops (red pieces in Fig. \\ref{fig:realization}) and bottoms (silver pieces) on the right side of conveyor belt 1 in an alternating manner.\nWhen a conveyor belt is moving, pieces can be held at a place by a stopper.\nPieces that are upside down can be turned with their right side up at the turner station.\nSwitch 1 can push pieces onto the indexing table, or allow pieces to keep running on conveyor belt 1.\nThe indexing table can hold up to six pieces (one on each arm), and turns counterclockwise.\nPieces that are not pushed on the indexing table transition to conveyor belt 2 where they reach the pick and place station.\nTo assemble a product, the pick and place robot picks up a top from conveyor belt 2, and places it onto a bottom on the indexing table.\nWhen the indexing table turns after assembly, it brings the product to switch 2. \nSwitch 2 can then push the assembled piece from the indexing table onto conveyor belt 3.\nFinally, the assembled pieces leave the system at the right side of conveyor belt 3.\n\n\\begin{figure}[t]\n\\centering\n\\subfigure[xCPS system schematic overview.]{\n\\includegraphics[width=0.46\\columnwidth]{images\/xCPS_overview.pdf}\n\\label{fig:overview}\n}\n\\subfigure[xCPS system realization.]{\n\\includegraphics[width=0.46\\columnwidth]{images\/screenshot_xCPS_edited.png}\n\\label{fig:realization}\n}\n\\caption{xCPS system layout.}\n\\label{fig:layout}\n\\end{figure}\n\n\\section{Challenges}\n\\label{sec:challenges}\nWhen controlling systems such as the xCPS system, several challenges may arise.\nWe mention some examples:\n\\begin{itemize}\n\\item How to manage variability in this system when there is for instance a system with, and a system without a turner station?\n\\item How to guarantee safe system operation, for instance, ensure a piece is never pushed to the indexing table when the spot on the table is already occupied by another piece?\n\\item How to control this system optimally to have the guaranteed highest throughput possible?\n\\item How to guarantee progress in the system, such that for every pair of top and bottom that enters the system, an assembled product will eventually leave the system?\n\\item After designing a controller for which the above guarantees can be made, how to deploy it on the system such that the guarantees are still made?\n\\end{itemize}\n\nGenerally, (theoretical) solutions have been found for such challenges.\nThese solutions originate from various disciplines.\nIf we consider the challenges mentioned above, in respective order some relevant disciplines are: \\textit{product line engineering} \\cite{Pohl2005}, \\textit{supervisory control synthesis} \\cite{Cassandras2008}, \\textit{timing analysis} \\cite{Cohen1989}, \\textit{formal verification} \\cite{Baier2008}, and \\textit{implementation} \\cite{Dietrich2002}. \nEven though the challenges can be separately addressed, it is still a single system that is being engineered.\nTherefore, a workflow is required that encompasses multiple disciplines.\nA workflow in which the above challenges can be addressed is shown in Fig. \\ref{fig:workflow_simplified}.\n\n\n\\begin{figure}[b!]\n\\centering\n\\includegraphics[width=1.00\\columnwidth]{images\/workflow_simplified.pdf}\n\\vspace*{-2\\baselineskip}\n\\caption{Overview of the demonstrative workflow.}\n\\label{fig:workflow_simplified}\n\\vspace*{0.5\\baselineskip}\n\\subfigure[In state-of-practice, manual processes are required to use functionality from different tools.\\label{fig:AaaS1}]{\n\\centering\n\\includegraphics[width=0.475\\columnwidth]{images\/AaaS_1.pdf}\n}\n\\hfill\n\\subfigure[By applying AaaS, tools are made interoperable resulting in readily available functionality from each tool.\\label{fig:AaaS2}]{\n\\centering\n\\includegraphics[width=0.475\\columnwidth]{images\/AaaS_2.pdf}\n}\n\\caption{From conventional MBSE (a) to AaaS (b).}\n\\end{figure}\n\n\nWe observe that for each step in the workflow specialized functionality is required, which is present in distinct tools that each use their own syntax and semantics.\nThis makes it challenging to (sequentially) apply each step in a unified engineering process (Fig. \\ref{fig:AaaS1}).\nThis brings us to the following major challenge that we address in this paper: \n\\begin{itemize}\n\\item How to create a multi-disciplinary workflow that has seamless integration of mono-disciplinary MBSE technologies?\n\\end{itemize}\nIn this paper we present AaaS as a solution to tackle this challenge.\nIn this solution multiple tools are used, each specialized in their own discipline and functionality.\nThey are integrated (i.e., made interoperable) over a service-oriented architecture.\nIn this way, their functionality is offered through services, and the engineer can access them from a single interface (Fig. \\ref{fig:AaaS2}).\nWhen other tools or services are required, they may be offered as additional services for integration into the workflow.\n\n\n\n\n\n\\section{State-of-the-art tools and technologies}\n\\label{sec:tools}\nIn the following, we discuss several state-of-the-art tools that each exist in separate disciplines of MBSE.\nThe tools are LSAT, PLE tool, CIF, SDF3, mCRL2, Activity Execution Engine, and model translation tool.\nThese tools are just a selection of tools that could be applied in an MBSE process.\nThe mentioned tools are applied in \\blind{the Arrowhead Tools}{\\xblackout{the Arrowhead Tools}} project and are made Arrowhead framework compatible as discussed in Section \\ref{sec:toolchain}.\n\n\n\n\\subsection{LSAT}\n\\label{sec:LSAT}\nLogistics Specification and Analysis Tool (LSAT) has been developed in a collaboration between \\blind{ESI (part of TNO, applied research center for high-tech systems design and engineering), ASML (manufacturer of lithography machines), and Eindhoven University of Technology (TU\/e)}{\\alttext{an applied research center, a manufacturing company, and a } \\alttext{ university}}\n\\cite{Sanden2021}\n\\footnote{\\blind{\\url{https:\/\/lsat.esi.nl}}{\\alttext{url 1}} , \\url{https:\/\/projects.eclipse.org\/projects\/technology.lsat}}.\nLSAT is being open sourced as an Eclipse project.\nIt is used for system specification through activity models \\cite{Sanden2016,Thuijsman2020LSAT}.\nA system is represented by a number of resources, where each resource consists of a number of peripherals.\nEach peripheral can execute a set of actions, to which a timing is prescribed. \nAn activity describes a cohesive piece of behavior in the system, and is modeled as a directed acyclic graph that captures the dependencies between actions performed on peripherals of the resources that it uses. \nAdditionally, LSAT has visualization techniques such as: movement trajectory plots for moving peripherals, graphical editing of activities, and Gantt charts to represent the timing of a sequence of activities.\n\nIn Listing \\ref{lst:LSATmachine}, an LSAT definition for three peripherals of the xCPS system is given: \\texttt{gripper}, \\texttt{turner}, and \\texttt{zMotor}.\nFor these peripherals, either actions are defined which they can perform, or axes are defined along which they can move.\nThe peripherals are instantiated in resource \\texttt{Turner}.\nThe instantiation of the \\texttt{zMotor} is parameterized with positions it can move to (\\texttt{Above\\_Belt}, \\texttt{At\\_Belt}), and which movements are possible between these positions (in this case in both directions between both positions), with a speed profile.\n\n\\begin{lstlisting}[frame=single,caption=LSAT machine specification.\\label{lst:LSATmachine}]\nPeripheralType gripper {\n\tActions {\n\t\tgrab\n\t\tungrab\n\t}\n}\n\nPeripheralType turner {\n\tActions {\n\t\tflip_left\n\t\tflip_right\n\t}\n}\n\nPeripheralType zMotor {\n  SetPoints {\n    Z [m]\n  }\n  Axes { \n    Z [m] moves Z\n  }\n}\n\nResource Turner {\n\tturner : turner \n\tgripper : gripper\n\tzMotor : zMotor {\n\t\tAxisPositions {  \n      Z (Above,At) \n\t\t}\n\t\tSymbolicPositions {\n\t\t\tAbove_Belt (Z.Above)\n\t\t\tAt_Belt (Z.At)\n\t\t}\n\t\tProfiles (normal)\n\t\tPaths {\n\t\t\tAbove_Belt <-> At_Belt profile normal \n\t\t}\n\t}\n}\n\\end{lstlisting}\n\nIn Listing \\ref{lst:LSATsetting}, part of the LSAT setting specification of the xCPS system is shown.\nThe timings of the actions of the \\texttt{gripper} and \\texttt{turner} peripherals in the \\texttt{Turner} resource are defined.\nThese timings are deterministic, but LSAT also supports specification of probability distributions for timing.\nFor the \\texttt{zMotor} of the \\texttt{Turner} a speed profile \\texttt{normal} is defined by specifying (maximal) velocity, acceleration, and jerk. \nCoordinates are defined for the symbolic positions of the peripheral.\nWhile specifying activities, the user can define actions that move the peripheral between these physical positions with a specific speed profile. \nThen, LSAT computes the timing of that action.\n\n\\begin{lstlisting}[frame=single,caption=LSAT setting specification.\\label{lst:LSATsetting}]\nTurner.gripper {\n\tTimings {\n\t\tgrab = 0.05\n\t\tungrab = 0.04\n\t}\n}\n\nTurner.turner {\n\tTimings {\n\t\tflip_left = 0.35\n\t\tflip_right = 0.35\n\t}\n}\n\nTurner.zMotor {\n\tAxis Z {\n\t\tProfiles {\n\t\t\tnormal (V = 5, A = 10, J = 10)\n\t\t}\n\tPositions {\n\t\tAbove = 0\n\t\tAt= 0.12\n    }\n  }\n}\n\\end{lstlisting}\n\n\nIn Listing \\ref{lst:LSATactivity}, an LSAT specification for the activity \\texttt{TurnerTurnTop} is given.\nThis is done by first giving arbitrary names for the actions that are used in the activity, and then specifying the directed acyclic graph of the activity which defines its action flow.\nAn arrow (\\texttt{->}) between two actions denotes that the succeeding action always starts after the preceding action has completed.\nSynchronization points are marked by a vertical bar: \\texttt{|s1} and \\texttt{|s2} are used in Listing \\ref{lst:LSATactivity} to denote multiple incoming or outgoing dependencies for an action.\nE.g., after \\texttt{Down}, both actions \\texttt{Release} and \\texttt{Up2} are allowed to take place (in any order\/at the same time).\nIn the activity framework, each resource needs to be claimed by the activity before it can perform actions, and all claimed resources need to be released after performing the actions \\cite{Sanden2016}.\nIn this way, activities can be deployed in a pipelined manner.\nI.e., when a resource is free, an activity can claim it and perform its actions on that resource, regardless of what previous activities might still be performing (on other resources). \n\n\\begin{lstlisting}[frame=single,caption=LSAT activity specification.\\label{lst:LSATactivity}]\nactivity TurnerTurnTop {\n\tprerequisites{\n\t\tTurner.zMotor at At_Belt\n\t}\n\tactions {  \n\t\tCT1     : claim Turner\n\t\tRT1     : release Turner\n\t\tCS1     : claim Stopper1\n\t\tRS1     : release Stopper1  \n\t\tCS2     : claim Stopper2\n\t\tRS2     : release Stopper2  \n\t\tLeft    : Turner.turner.flip_left \n\t\tRight   : Turner.turner.flip_right\n\t\tUp      : move Turner.zMotor to Above_Belt with speed profile normal \n\t\tUp2     : move Turner.zMotor to Above_Belt with speed profile normal\n\t\tDown    : move Turner.zMotor to At_Belt with speed profile normal\n\t\tGrab    : Turner.gripper.grab\n\t\tRelease : Turner.gripper.ungrab  \n\t}\n\taction flow {\n    CS2->CS1->CT1->Grab->Up->Left->Down->|s1->Release->|s2->Right->RT1->RS2->RS1\n                                         |s1->Up2    ->|s2\n\t} \n}\n\\end{lstlisting}\n\n\\subsection{PLE tool}\nThe Product Line Engineering (PLE) tool, developed at \\blind{TU\/e}{\\alttext{a university}}, is used to manage the variability of a system, and to automatically validate and derive product instances within a product line \n\\cite{Kahraman2021}.\nUsing the PLE tool, a product line is defined which consists of a feature model, a base LSAT model, delta modules, and a mapping model. \n\nThe feature model expresses the commonality and variability of the product line \\cite{Benavides2010}.\nFig. \\ref{fig:feature_model} shows the feature model of the xCPS product family that represents the variability in the resources, behavior, and assembly type of the system.\nConstraints expressed as propositional formulas are used to define dependencies between features. \nA configuration is valid if the combination of features is allowed by the feature model.\n\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.90\\columnwidth]{images\/FeatureModel.pdf}\n\\caption{xCPS feature model}\n\\label{fig:feature_model}\n\\end{figure}\n\nA base LSAT model, such as described in Section \\ref{sec:LSAT}, is input to the PLE tool and serves as source for deriving new product instances. \nDelta modules are used to define modifications that can be made to the base model.\nIn Listing \\ref{lst:PLEdelta} a delta module of the LSAT machine specification is provided for the absence of the turner station.\nNote that for example the gripper peripheral is not removed, since it is used in more resources next to the turner.\n\\begin{lstlisting}[frame=single,caption=PLE Tool delta module.\\label{lst:PLEdelta}]\ndelta \"machineDelta\"\n\tdialect <http:\/\/www.esi.lsat.nl\/machine>\n\tmodifies <..\/model\/xCPS.machine>\n{\tremoveResourceFromResourcesOfMachine(<Turner>, <xCPS>);\n\tremovePeripheralFromPeripheralTypesOfMachine(<turner>, <xCPS>);\n\t... }\n\\end{lstlisting}\nThe PLE tools uses mapping models to define what delta modules need to be applied when a particular configuration is selected.\nIn Listing \\ref{lst:PLEmapping} a mapping model is provided, where \\texttt{!Turner} states that the following deltas should only be applied if the Turner is not selected.\n\\begin{lstlisting}[frame=single,caption=PLE Tool mapping model.\\label{lst:PLEmapping}]\n!Turner:\n  <deltas\/machineDelta.decore>,\n  <deltas\/activityDelta.decore>,\n  ...\n\\end{lstlisting}\nA particular instance of the xCPS product line is defined in Listing \\ref{lst:PLEconfiguration}.\nNote that next to the absence or presence of components, the PLE tool can also be used to, e.g., instantiate settings (for the example, defined in \\texttt{FastMovement}) or assembly procedures.\nUsing the defined product line and the configuration, the PLE tool derives LSAT variant models.\nThese derived models can then be used to perform further analysis.\n\\begin{lstlisting}[frame=single,caption=PLE Tool configuration.\\label{lst:PLEconfiguration}]\nconfiguration <xCPS.defeaturemodel> {\n\t\"Resource\",\n\t\"PickPlace\",\n\t\"Turner\",\n\t\"Behavior\",\n\t\"FastMovement\",\n}\n\\end{lstlisting}\n\n\\subsection{CIF} \n\\label{sec:CIF}\nCIF is an automata-based tool and language, and is used to specify system behavior, formulate behavioral requirements, and perform supervisory controller synthesis \\cite{Cassandras2008} to obtain a correct-by-construction supervisory controller that adheres to the requirements\n\\cite{vanBeek2014}.\nCIF is part of the Eclipse Supervisory Control Engineering Toolkit (Eclipse ESCET\u2122) \\cite{ESCET}\\footnote{\\url{https:\/\/www.eclipse.org\/escet\/} , `Eclipse', 'Eclipse ESCET' and 'ESCET' are trademarks of Eclipse Foundation, Inc.}, that has become an Eclipse open source project since 2020.\nThis project builds upon research and tool development at \\blind{TU\/e, as well as collaboration with industry including ASML, and Rijkswaterstaat (part of the Dutch ministry of infrastructure and water management).}{\\alttext{a university}, as well as collaboration with industry including \\alttext{a manufacturing company}, and \\alttext{an infrastructural company}.}\n\n\n\nIn Listing \\ref{lst:CIFrequirement} a requirement specified in CIF is shown.\nThe requirement automaton specifies allowed behavior for the turner station.\nActivities \\texttt{TopGoTo}\\-\\texttt{Turner} and \\texttt{BottomGoToTurner} can only occur when there is no piece present at the turner station.\nA bottom can immediately pass through the turner station.\nIt is assumed that every top needs to be inverted.\nFirst, the \\texttt{TurnerGoDown} activity is executed.\nThis activity can be successful or fail.\nWhen it is successful, the turner will turn the top, and the top can continue after the turner.\nWhen the activity fails, the turner has to retry until it is successful.\n\n\\begin{lstlisting}[frame=single,caption=CIF requirement turner.\\label{lst:CIFrequirement}]\nrequirement automaton TurnerFlow:\n\tlocation NoPiece:\n\t\tinitial; marked;\n\t\tedge TopGoToTurner goto TurnTop;\n\t\tedge BottomGoToTurner goto Bottom;\n\tlocation TurnTop:\n\t\tedge TurnerGoDown goto TurningTop;\n\tlocation TurningTop:\n\t\tedge TURNERDOWNSUCC_TurnerTurnTop goto TurnedTop;\n\t\tedge TURNERDOWNFAIL_RetryTurnerGoDown;\n\tlocation TurnedTop:\n\t\tedge TopGoAfterTurner goto NoPiece; \n\tlocation Bottom:\n\t\tedge BottomGoAfterTurner goto NoPiece;\nend\n\\end{lstlisting}\n\n\nCIF can be used to perform supervisory controller synthesis. \nA supervisory controller is generated, that restricts the behavior of the system such that the requirements are always satisfied.\nEssentially, for each activity a predicate is computed that needs to hold for the activity to occur.\nFor example, Listing \\ref{lst:CIFsupervisor} shows the additional guard that is generated for the activity \\texttt{BottomGoAfterTurner}.\nIt makes sure that this activity can only occur when at the next station after the turner (\\texttt{Sensor3}) there are no inverted tops present, of which the amount is stored in \\texttt{Sensor3Location.nInvTops}, to avoid collisions.\nNote that when \\texttt{BottomGoAfterTurner} occurs, there can never be a non-inverted top or bottom at Sensor3, since it is assumed that tops and bottoms are input in alternating manner, cannot overtake, and every top is inverted at the turner station.\n\n\\begin{lstlisting}[frame=single,caption=CIF supervisor.\\label{lst:CIFsupervisor}]\nsupervisor automaton sup:\n  location:\n    initial; marked;\n    edge BottomGoAfterTurner when Sensor3Location.nInvTops = 0;\n    ...\nend\n\\end{lstlisting}\n\n\\subsection{SDF3} \n\\label{sec:SDF3}\nSDF3 is a toolset that has an extensive library of analysis and transformation algorithms for synchronous dataflow graphs, which are suitable for modeling both parallel and pipelined processing and cyclic dependencies \\cite{Stuijk2006}. \nAdditionally, it can be used to generate random synchronous dataflow graphs, if desirable with certain guaranteed properties.\nSDF3 is developed at \\blind{TU\/e \\footnote{\\url{https:\/\/www.es.ele.tue.nl\/sdf3\/}}}{\\alttext{a university} \\footnote{\\alttext{url}}}.\nIn this work we will use it for makespan optimization of activity models to find the optimal behavior that produces products as quickly as possible.\n\n\nFrom an Input\/Output (I\/O) automaton, SDF3 makes an internal conversion to a max-plus automaton and performs timing optimization using the methods of \\cite{Gaubert1995} to find the optimal order of activities to be dispatched to generate the lowest possible makespan.\nIn Listing \\ref{lst:SDF3MPA} an I\/O automaton is shown. \n\\texttt{loc1} is the initial location (denoted by \\texttt{i}).\nTransitions between the locations are defined, where in this case most input actions are empty (denoted by an empty string).\n\\texttt{loc4} is where the event outcome of \\texttt{TurnerGoDown} is processed. \nIf it succeeds, the top piece is turned, and if it fails, the turner retries to go down.\n\nFor the xCPS system model, the computed dispatching sequence is shown in Listing \\ref{lst:SDF3result}.\nThis is the optimal sequence of activities to execute when the system is empty to produce one product as quickly as possible.\nWhen there is no fail on the turner (or elsewhere), one product can be produced in 17.723 seconds.\n\n\\begin{lstlisting}[frame=single,caption=SDF3 I\/O automaton.\\label{lst:SDF3MPA}]\nioautomaton statespace {  \n  loc1 i -,InputTopInverted-> loc2 \n  loc2 -,TopGoToTurner-> loc3 \n  loc3 -,TurnerGoDown-> loc4 \n  loc3 -,InputBottom-> loc5  \n  loc4 -TURNERDOWNSUCC,TurnerTurnTop->  loc6 \n  loc4 -TURNERDOWNFAIL,RetryTurnerGoDown->  loc7\n  loc4 -,InputBottom-> loc8\n  ... }\n\\end{lstlisting}\n\\begin{lstlisting}[frame=single,caption=SDF3 makespan result.\\label{lst:SDF3result}]\nmakespan: 17.723\nsequence: InputTopInverted; TopGoToTurner; InputBottom; ; TurnerGoDown; TURNERDOWNSUCC,TurnerTurnTop; TopGoAfterTurner; BottomGoToTurner; TopGoToSensor4; ; BottomGoAfterTurner; TopGoToSwitch2; BottomGoToSensor4; BottomGofromSt3ToTable2; AlignTable2WithPickPlace; AlignTable2WithBelt; AlignTable2WithPickPlace; TopGoToPickPlace; TopPickUp; TopAssemble; AlignTable2WithBelt; AlignTable2WithPickPlace; ProdGoFromTable2ToBelt4\n\\end{lstlisting}\nTo study the sequence and evaluate potential bottlenecks, LSAT can generate a Gantt chart from the sequence, which is shown in Figure \\ref{fig:Gantt}.\nThe Gantt chart shows which activities are occupying which resources at what time, what actions are executed, and the dependencies between actions on different peripherals.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=1.00\\columnwidth]{images\/Gantt_edited.pdf}\n\\vspace*{-1.2\\baselineskip}\n\\caption{Gantt chart optimal makespan one product.}\n\\vspace*{-0.2\\baselineskip}\n\\label{fig:Gantt}\n\\end{figure}\n\n\\subsection{mCRL2}\nmCRL2 is a toolset designed to reason about concurrent and distributed systems.\nThe toolset consists of its own language as well as more than sixty tools supporting visualization, simulation, minimization, and model checking \\cite{Baier2008} of complex systems\n\\cite{Groote2014,Bunte2019}\n\\footnote{\\url{https:\/\/www.mcrl2.org\/}}.\nmCRL2 is open source and developed at \\blind{TU\/e}{\\alttext{a university}} in collaboration with \\blind{University of Twente}{\\alttext{another university}}. \nGiven an mCRL2 specification and a property in the modal mu calculus, mCRL2 can apply model checking to guarantee the (non-)existence of particular behaviors in a system.\n\nA portion of an mCRL2 model describing the behavior relevant to the turner station and the activity TurnerTurnTop is provided in Listing \\ref{lst:mCRL2model}.\nThe model is derived from an automata translation of the LSAT model using \\cite{Thuijsman2020LSAT}.\n\n\\begin{lstlisting}[frame=single,caption=mCRL2 model.\\label{lst:mCRL2model}]\nsort enum_LPE = struct enumlit_Grab | enumlit_Ungrab;\nsort enum_LPE2 = struct enumlit_Flip_left | enumlit_Flip_right;\n...\nact value_Turner_gripper : enum_LPE;\nact value_Turner_turner : enum_LPE2;\n...\nproc BehProc_M(Locvar_M : LocSort_M, Activity_TurnerTurnTop : enum_LPE4, Turner_gripper : enum_LPE, Turner_turner : enum_LPE2, Turner_zMotor : enum_LPE3) =\n  value_Activity_TurnerTurnTop(Activity_TurnerTurnTop) . BehProc_M(Locvar_M, Activity_TurnerTurnTop, Turner_gripper, Turner_turner, Turner_zMotor) +\n  value_Turner_gripper(Turner_gripper) . BehProc_M(Locvar_M, Activity_TurnerTurnTop, Turner_gripper, Turner_turner, Turner_zMotor) +\n  ...\n  ((Locvar_M == loc_M_L) && (Activity_TurnerTurnTop == enumlit_l0)) -> claim_Stopper2 . BehProc_M(Locvar_M, enumlit_l1, Turner_gripper, Turner_turner, Turner_zMotor) +\n  ((Locvar_M == loc_M_L) && (Activity_TurnerTurnTop == enumlit_l1)) -> claim_Stopper1 . BehProc_M(Locvar_M, enumlit_l2, Turner_gripper, Turner_turner, Turner_zMotor) +\n  ... ;\nact claim_Stopper1, renamed_claim_Stopper1, claim_Stopper2, ...;\ninit BehProc_M(loc_M_L, enumlit_l0, enumlit_Ungrab, enumlit_Flip_left, enumlit_Above_belt);\n\\end{lstlisting}\n\nIn Listing \\ref{lst:mCRL2requirement}, a modal mu calculus formula is provided (in mCRL2 syntax) that expresses that for every time the turner flips left, the turner must eventually flip right. \nmCRL2 can be used to verify whether this property holds for the model.\nWhen a property that is checked does not hold, a counter-example is given to aid in solving the problem.\n\\begin{lstlisting}[frame=single,caption=mCRL2 requirement.\\label{lst:mCRL2requirement}]\n[true*.flip_left]mu X.([!flip_right]X && <true>true)\n\\end{lstlisting}\n\n\\subsection{Activity Execution Engine}\nThe Activity Execution Engine (AEE) is an automated execution method for the activity framework developed at \\blind{TU\/e}{\\alttext{a university}}.\nIt receives an activity model with a supervisory controller in the form of an I\/O automaton as input and executes it on the system.\nThe AEE is time-preserving, meaning it adheres to (LSAT) model prescribed timing of actions within well defined bounds.\nThe AEE guarantees determinate behavior of the system despite timing variations that may happen in execution.\nThe supervisory control is directly performed by the activity execution engine, i.e., the activity execution engine directly connects to the low-level resource controllers.\n\nThe execution engine provides a generic solution for executing the LSAT model on the machine. \nIt consists of three layers: \nThe supervisory control layer deals with the high-level execution concerning the order of activities and decisions based on event outcomes that it receives from the lower layers.\nThe AEE layer is concerned with execution of individual activities and sequencing them together. \nThe action translation layer is responsible for translating action descriptions in LSAT specification to low-level function calls that execute the actions, and translates the sensor data back to communicate to the higher levels. \n\nThe generic solution provides all the mechanisms needed to guarantee timing and behavior-preserving execution of the model.\nThe only part of the engine that the system designers and engineers would need to implement based on their specific product is the action translation layer which is a relatively small library.\n\n\n\\subsection{Model translation tool}\nAs becomes apparent from the above listings, each tool has its own unique syntax.\nTo make the tools interoperable, it needs to be possible to translate a model in one tool to an equivalent model in another tool.\nIn this way it is for example possible to generate a behaviorally equivalent mCRL2 model of an LSAT model and perform formal verification on that model, which is a function of only mCRL2. \nThese translations are essential for the approach that we discuss next in Section \\ref{sec:toolchain}.\nIt allows to automate the use of functionality of tool \\emph{X} for models in (syntax of) tool \\emph{Y}.\nThe model translation tool is a toolset that offers a range of translations between models that are equivalent with respect to the process or computation that will be performed on the generated model.\nFor models with complex semantics, performing translation validation \\cite{Pnueli1998} may be desirable.\n\n\\section{Toolchain and interoperability}\n\\label{sec:toolchain}\nThe introduced tools each have their own specific functionality.\nWith their combined functionality, they can be used in a MBSE workflow for manufacturing systems.\nWe first discuss a workflow that uses all tools mentioned in Section \\ref{sec:tools}, and then present how the tools are integrated in an interoperable toolchain.\n\n\\subsection{Example workflow}\n\\label{sec:workflow}\nIn Section \\ref{sec:challenges}, we discussed a number of challenges that arise in MBSE of supervisory controllers.\nAn interdisciplinary workflow was introduced in Figure \\ref{fig:workflow_simplified}.\nIn that workflow, each of the steps tackled a challenge in the engineering process.\nAll steps of the workflow can individually be performed by the tools mentioned in Section \\ref{sec:tools}.\nIn this section we discuss integration of the tools into a toolchain, creating a MBSE process of supervisory control design for manufacturing systems that includes specification, variation management, supervisor synthesis, timing optimization, formal verification, and implementation.\n\nWe use the following workflow to show how the tools can be integrated together to form a toolchain:\n\\begin{enumerate}\n\\item A product line is specified using LSAT and the PLE tool.\n\\item Given a feature configuration, an LSAT product instance is derived with the PLE tool.\n\\item The LSAT product instance specification is converted to CIF.\n\\item Safety requirements are specified, and a maximally permissive supervisory controller automaton is synthesized using CIF.\n\\item The automaton supervisor is converted to an I\/O automaton.\n\\item Using the I\/O automaton and the timing information from the LSAT specification, SDF3 is used to perform makespan optimization to find the optimal dispatching sequence of activities to produce products as quickly as possible.\n\\item The LSAT specification and the optimal dispatching sequence are used to construct an mCRL2 model.\n\\item With mCRL2, progress properties are verified for the behavior that results from the obtained dispatching sequence.\n\\item The control strategy is deployed on the physical system using the AEE.\n\\end{enumerate}\nThe authors note that this is just a demonstrative workflow, the tools provide more services that could be used, or different tools can be used altogether.\n\nA graphical representation of the workflow is shown in Figure \\ref{fig:workflow}. \nEssentially, this is an elaboration of the workflow in Figure \\ref{fig:workflow_simplified}.\nAt the top of the diagram are the artifacts.\nSome artifacts are manually constructed (these do not have an incoming arrow), others are automatically generated during the process.\nIn the middle are services corresponding to processes that are performed during the workflow, with incoming and outgoing arrows linking them to the respective input and output artifacts. \nThe processes are executed as a service provided by one of the tools shown in the bottom of the diagram.\n\n\\begin{figure}[tbhp]\n\\centering\n\\includegraphics[width=1.00\\columnwidth]{images\/workflow.pdf}\n\\caption{Elaborated overview demonstrative workflow.}\n\\label{fig:workflow}\n\\end{figure}\n\nExcerpts from the artifacts are presented throughout Section \\ref{sec:tools} \\footnote{Complete models of the xCPS system for the various tools are available here: \\\\ \\url{https:\/\/github.com\/sbthuijsman\/FM_interoperability}}.\n\nThrough this workflow, the challenges mentioned in Section \\ref{sec:challenges} are addressed.\nUsing the PLE tool, variability of the system is managed by specifying a product line with deltas between particular configurations.\nIn this way, we avoid (manually) creating models of each unique configuration.\nBehavioral requirements are specified in a modular manner, and a minimally restrictive correct-by-construction supervisory controller is generated that adheres to these requirements by applying supervisory controller synthesis in CIF.\nTime-optimal control is guaranteed through application of timing optimization using SDF3, which selects the time-optimal activity sequence from the supervisory controller.\nProgress in the system can be guaranteed by specifying and verifying progress properties using mCRL2.\nFinally, the designed controller is deployed on the system with the AEE, which guarantees execution that adheres to the models.\nEven though these tools use their own semantics and syntax, their functionality can be applied sequentially on a single system through the use of model translations by using the model translation tool.\n\nThe authors note that even though behavioral requirements are already enforced by supervisor synthesis through CIF, formal verification in mCRL2 is still beneficial.\nA progress property such as listed in Listing \\ref{lst:mCRL2requirement} (eventually modality) can not be directly expressed as a requirement in CIF, and supervisory controller synthesis is in general not applicable to such requirements.\nAdditionally, in this case the CIF model only considers behavior on activity level, while the mCRL2 model includes action level behavior, which allows for more detailed inspection of the behavior.\n\n\n\n\\subsection{Toolchain integration over Arrowhead framework}\n\\label{sec:arrowhead}\nThe workflow described above spans from specification to realization.\nIn the steps along the way, several processes are performed using various tools.\nA seamless integration of these tools is established through AaaS: their functionalities are provided as services on an Arrowhead local cloud.\nThe Arrowhead cloud is constructed using the Arrowhead framework, which is an interoperability service-oriented architecture \\footnote{\\url{https:\/\/arrowhead.eu\/} , \\url{https:\/\/github.com\/eclipse-arrowhead} , the Arrowhead framework files used for the integration of the tools discussed in this paper can be found here: \\url{https:\/\/github.com\/sbthuijsman\/FM_interoperability}}.\nThe framework is elaborately discussed in \\cite{Varga2017,Venanzi2020}. \nNext to the MBSE tool services, the following three Arrowhead core services exist on the cloud:\n\\begin{enumerate}\n\\item Orchestration service to coordinate the connections between the consumer and provider services.\n\\item Authorization service to provide security such that services can only be accessed by authorized consumers.\n\\item Service registry system to keep track of all services within the network and ensure all systems can find each other.\n\\end{enumerate}\nBecause of the integration in the Arrowhead cloud, the workflow can be performed from a single interface and the functionality from each tool is readily accessible.\nFurthermore, additional tools and services can be added in a modular manner.\nAs long as there is a tool or service that can solve the problem, and model-to-model translation is possible from some existing artifact to the required syntax of the concerning service, the toolchain can be extended to include the service in the manner as the discussed services.\nDirect access to functionality of the MBSE tools enables rapid application of MBSE technologies from various disciplines, and allows seamless MBSE of the supervisory control of a CPS from start to finish.\nNext to this integration, the AaaS framework provides more benefits, such as provisioning of a centralized (model) database or management of computational resources.\n\n\\section{Conclusion}\n\\label{sec:conclusion}\nMany challenges can be addressed through MBSE of the supervisory control of CPSs.\nSolutions originate from several disciplines.\nEach discipline uses its own tools with its own semantics and syntax.\nIt is a major challenge to create a multi-disciplinary workflow that has seamless integration of mono-disciplinary MBSE technologies.\n\nIn this paper, several tools are discussed that are each state-of-the-art in their own discipline.\nEven though the tools use their own semantics and syntax, equivalent models can be generated for each tool by model-to-model translations.\nBy applying AaaS, in this case using the Arrowhead framework, the tools are made interoperable.\nThe translation steps are automated, and the services from each tool are readily accessible.\nA seamless integration of the tools is established: the engineer can easily access their functionality from a single interface.\nBecause of the modularity of the service-oriented architecture, the toolchain can without difficulty be extended to incorporate additional functionality as long as the required model-to-model translations can be established.\n\n \\bibliographystyle{splncs04}\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\nNakajima \\cite{N1} constructed a Heisenberg algebra using\ncorrespondence varieties which acts on the direct sum over all $n$\nof homology groups $H(X^{[n]})$ with complex coefficient of\nHilbert schemes $X^{[n]}$ of $n$ points on a quasi-projective\nsurface $X$. This representation is irreducible thanks to\nG\\\"ottsche's earlier work \\cite{G}. Similar results have been\nindependently obtained by Grojnowski \\cite{Gr}. We refer to\n\\cite{N2} for an excellent account of Hilbert schemes and related\nworks. In the case when $X$ is the minimal resolution $ \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} $ of\nthe simple singularity $\\Bbb C^2 \/ \\Gamma$ associated to a finite\nsubgroup $\\Gamma$ of $SL_2 (  {\\Bbb C} )$, this together with some additional\nsimple data provides a geometric realization of the\nFrenkel-Kac-Segal vertex construction of the basic representation\nof an affine Lie algebra \\cite{FK, S1}. We may view this as a\ngeometric McKay correspondence \\cite{Mc} which provides a\nbijection between finite subgroups of $SL_2 ( {\\Bbb C} )$ and affine Lie\nalgebras of ADE types.\n\nIn \\cite{W} we realized the important role of wreath products $\\G_n\n= \\Gamma \\sim S_n $ associated to a finite group $\\Gamma$ in equivariant\nK-theory. Various algebraic structures were constructed on the\ndirect sum over all $n$ of the topological $\\G_n$-equivariant\nK-theory $K^{ {\\footnotesize {top}} }_{\\G_n} (X^n) \\bigotimes  {\\Bbb C} $ of $X^n$ for a\n$\\Gamma$-space $X$. The results of \\cite{W} generalized the work of\nSegal \\cite{S2} (also see \\cite{VW, Gr}) which corresponds to our\nspecial case when $\\Gamma$ is trivial (i.e. $\\Gamma$ is the one-element\ngroup) and the wreath product $\\G_n$ reduces to the symmetric group\n$S_n$.\n\nThe wreath product approach obtains further significance in light\nof the conjectural equivalence of various algebraic structures in\nthe following three spaces:\n         %\n\\begin{eqnarray}   \\label{eq_master}\n \\begin{array}{ccc}\n   \\mbox{I} &  & \\mbox{II}    \\\\\n            \\bigoplus_{n \\geq 0} H( Y^{[n]})\n          & \\leftarrow  -\\rightarrow\n          &  \\bigoplus_{n \\geq 0} K^{ {\\footnotesize {top}} }_{S_n } (Y^n) \\bigotimes  {\\Bbb C}  \\\\\n             %\n           & \\;\\, \\nwarrow \\quad \\quad &\\uparrow  \\\\\n              %\n           & \\quad \\searrow & \\downarrow  \\\\\n          & &  \\mbox{III}  \\\\\n          & & \\bigoplus_{n \\geq 0} K^{ {\\footnotesize {top}} }_{\\G_n } (X^n) \\bigotimes  {\\Bbb C} \n \\end{array}\n\\end{eqnarray}\n         %\nHere one assumes that $X$ is a quasi-projective surface acted upon\nby a finite group $\\Gamma$ and $Y$ is a suitable resolution of\nsingularities of $X\/ \\Gamma$ such that there exists a canonical\nisomorphism between $K_{\\Gamma}(X)$ and $K( Y)$. For $\\Gamma$ trivial\nIII reduces to II.\nThe graded dimensions of the three spaces have been shown to\ncoincide \\cite{W}. The complexity of the geometry involved\ndecreases significantly from I to II, and then to III. In each of\nthe three setups various algebraic structures have been\nconstructed in \\cite{Gr, N2, S2, W} such as Hopf algebra, vertex\noperators, and Heisenberg algebra, etc. We remark that there has\nbeen a construction of an additive isomorphism between the spaces\nin I and II due to de Cataldo and Migliorini \\cite{CM}.\n\nIn a most important case when $X$ is $ {\\Bbb C} ^2$ acted upon by a finite\nsubgroup $\\Gamma \\subset SL_2( {\\Bbb C} )$ and $Y$ is the minimal resolution\n$ \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} $ of $ {\\Bbb C} ^2 \/\\Gamma$, the above diagram reduces to the following\none:\n         %\n\\begin{eqnarray}   \\label{eq_mckay}\n \\begin{array}{ccc}\n   \\mbox{I} &  & \\mbox{II}    \\\\\n\\bigoplus_{n \\geq 0} H(  \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^{[n]})\n          & \\leftarrow -\\rightarrow\n          &  \\bigoplus_{n \\geq 0} K^{ {\\footnotesize {top}} }_{S_n } ( \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^n) \\bigotimes  {\\Bbb C}  \\\\\n             %\n           & \\; \\,\\nwarrow \\quad \\quad &\\uparrow  \\\\\n              %\n           & \\quad \\searrow & \\downarrow  \\\\\n          & &  \\mbox{III}  \\\\\n          & & \\bigoplus_{n \\geq 0} R (\\G_n )\n \\end{array}\n\\end{eqnarray}\n     %\nby using the Thom isomorphism between $K^{ {\\footnotesize {top}} }_{\\G_n} ( {\\Bbb C} ^{2n})$\nand the representation ring $R_{ {\\Bbb Z} }(\\G_n)$ of the wreath product.\nHere $R (\\G_n ) = R_{ {\\Bbb Z} } (\\G_n) \\otimes  {\\Bbb C} $ in our notation. It was\npointed out in \\cite{W} that the Frenkel-Kac-Segal homogeneous\nvertex representation can be realized in terms of representation\nrings of such wreath products. Such a finite group theoretic\nconstruction, which can be viewed as a new form of McKay\ncorrespondence, has been firmly established recently in our work\n\\cite{FJW} jointly with I.~Frenkel and Jing. It remains a big\npuzzle however why there are many parallel algebraic structures in\nthese different setups.\n\nIn the present paper we propose a coherent approach to fill in the\ngap (at least in the setup of diagram (\\ref{eq_mckay}) above) and\npresent several canonical ingredients in our approach. More\nexplicitly, we provide direct links from wreath products to\nHilbert schemes. We find a natural interpretation of a main\ningredient (the so-called weighted bilinear form) in \\cite{FJW},\nwhich brings us one step closer to a {\\em direct} isomorphism of\nthe two forms of McKay correspondence respectively in terms of\nHilbert schemes and wreath products. We also establish\nisomorphisms of various algebraic structures in II and III. Let us\ndiscuss in more detail.\n\nGiven a finite subgroup $\\Gamma$ of $SL_2 ( {\\Bbb C} )$, we observe that there\nis a natural identification between $ {\\Bbb C} ^{2n} \/\\G_n$ and the $n$-th\nsymmetric product $( {\\Bbb C} ^2 \/\\Gamma)^{(n)}$ of the simple singularity\n$ {\\Bbb C} ^2 \/\\Gamma$. The following commutative diagram \\cite{W}\n         %\n\\begin{eqnarray*}\n \\begin{array}{ccc}\n    \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^{[n]}        & \\stackrel{\\pi_n}{\\longrightarrow}\n        & ( \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} )^n \/ S_n    \\\\\n   \\downarrow \\tau_n &                 & \\downarrow  \\tau_{(n)}      \\\\\n    {\\Bbb C} ^{2n} \/\\G_n      &   \\cong            & ( {\\Bbb C} ^2 \/ \\Gamma)^n \/S_n .\n \\end{array}\n\\end{eqnarray*}\n    %\ndefines a resolution of singularities $\\tau_n :  \\ale^{[n]}  \\rightarrow\n {\\Bbb C} ^{2n} \/\\G_n$, where $\\tau_{(n)}$ is naturally induced from the\nminimal resolution $ \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma}  \\rightarrow  {\\Bbb C} ^2 \/ \\Gamma$. We show that\n$\\tau_n$ is a semismall crepant resolution, which provides the\nfirst direct link between wreath products and Hilbert schemes. We\nshow that the fiber of $\\tau_n$ over $[0]\\in  {\\Bbb C} ^{2n} \/\\G_n$\n(associated to the origin of $ {\\Bbb C} ^{2n}$) is of pure dimension $n$\nand we give an explicit description of its irreducible components.\n\nWe conjecture that there exists a canonical isomorphism between\nthe Hilbert quotient $ \\C^{2n} \/\/ \\Gn $ of $ {\\Bbb C} ^{2n}$ by $\\G_n$ (see\n\\cite{Ka} or Subsection~\\ref{subsect_link} for the definition of\nHilbert quotient) and the Hilbert scheme $ \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^{[n]}$, and provide\nseveral nontrivial steps toward establishing this conjecture. More\nexplicitly, we first single out a distinguished nonsingular\nsubvariety $X_{\\G,n}$ of $( {\\Bbb C} ^2)^{[nN]}$ and construct a morphism\n$\\varphi$ from $ \\C^{2n} \/\/ \\Gn $ to $X_{\\G,n}$, where $N$ is the order of the\ngroup $\\Gamma$. We use here a description of a set of generators for\nthe algebra of $\\G_n$ invariant regular functions on $ {\\Bbb C} ^{2n}$\nwhich is a generalization of a theorem of Weyl \\cite{Wey}. It\nfollows by construction that our morphism from $ \\C^{2n} \/\/ \\Gn $ to\n$X_{\\G,n}$ when restricted to a certain Zariski open set is indeed an\nisomorphism. We give a quiver variety description of $X_{\\G,n}$ and\n$ {\\Bbb C} ^{2n} \/\\G_n$ in the sense of Nakajima \\cite{N, N3}. Such an\nidentification follows easily from Nakajima's quiver\nidentification of Hilbert scheme of points on $ {\\Bbb C} ^2$ (cf.\n\\cite{N2} and Varagnolo-Vasserot \\cite{VV}). According to Nakajima\n\\cite{N4}, it can be shown essentially by using\nKronheimer-Nakajima \\cite{KN} that the Hilbert scheme $ \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^{[n]}$\nis a quiver variety associated to the same quiver data as $X_{\\G,n}$\nbut with a different stability condition. It follows that $X_{\\G,n} $\nand $ \\ale^{[n]} $ are diffeomorphic by Corollary 4.2 in \\cite{N}. In\nthis way we have obtained a second direct link between $\\G_n$ and\n$ \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^{[n]}$. One plausible way to establish our main conjecture\nwill be to establish that $\\varphi$ is an isomorphism between\n$ \\C^{2n} \/\/ \\Gn $ and $X_{\\G,n}$ and that the diffeomorphism between $X_{\\G,n} $\nand $ \\ale^{[n]} $ is indeed an isomorphism as complex varieties.\n\nOur construction contains two distinguished cases which have been\nstudied by others. The morphism above for $n =1$ becomes an\nisomorphism due to Ginzburg-Kapranov (unpublished) and\nindependently Ito-Nakumura \\cite{INr}. Our morphism above also\ngeneralizes Haiman's construction \\cite{H} of a morphism from\n$ \\C^{2n} \/\/ S_n $ to $( {\\Bbb C} ^2)^{[n]}$ which corresponds to our special case\nfor $\\Gamma$ trivial (where no passage to the quiver variety\ndescription is needed). Haiman \\cite{H} has in addition shown that\nthe morphism being an isomorphism is equivalent to the validity of\nthe remarkable $n!$ conjecture due to Garsia and Haiman \\cite{GH}.\n(We remark that there has been also attempt by Bezrukavnikov and\nGinzburg \\cite{BG} in establishing this conjectural isomorphism\nfor $\\Gamma$ trivial.) Very recently a proof of the $n!$ conjeture\n(and this isomorphism conjecture) has been announced by Haiman in\nhis homepage by establishing a Cohen-Macaulay property of a\ncertain universal scheme which was conjectured in \\cite{H}. It is\nnatural for us to conjecture similarly a Cohen-Macaulay property\nof a certain universal scheme in our setup\n(Conjecture~\\ref{conj_cohen}), which is sufficient to imply that\n$\\varphi$ is an isomorphism.\n\nA distinguished virtual character of $\\G_n$ has been used to\nconstruct a semipositive definite symmetric bilinear form (called\na weighted bilinear form) on $R_{ {\\Bbb Z} }(\\G_n)$ which plays a\nfundamental role in the wreath product approach to McKay\ncorrespondence \\cite{FJW}. Indeed it is given by the $n$-th tensor\nof the McKay virtual character $\\lambda ( {\\Bbb C} ^2)$ of $\\Gamma$. On the other\nhand, the virtual character $\\lambda ( {\\Bbb C} ^{2n})$ of $\\G_n$ induced\nfrom the Koszul-Thom class defines a canonical bilinear form on\nthe Grothendieck group $K^0_{\\G_n} ( {\\Bbb C} ^{2n})$ of the bounded\nderived category $D^0_{\\G_n} ( {\\Bbb C} ^{2n})$ consisting of $\\G_n$\nequivariant coherent sheaves whose cohomology sheaves are\nconcentrated on the origin. Although they are defined very\ndifferently, these two virtual characters of $\\G_n$ are shown to\ncoincide. This establishes an isometry between $K^0_{\\G_n}\n( {\\Bbb C} ^{2n})$ and $R_{ {\\Bbb Z} }(\\G_n)$ endowed with the weighted bilinear\nform, and thus provides a natural explanation of the weighted\nbilinear form introduced from a purely group theoretic\nconsideration. (Actually we establish the coincidence of virtual\ncharacters for more general $\\Gamma \\subset GL_k ( {\\Bbb C} )$ and the induced\n$\\G_n$-action on $ {\\Bbb C} ^{kn}$.) We regard this isometry as an\nimportant ingredient toward a direct isomorphism of the two forms\nof McKay correspondence realized respectively in terms of Hilbert\nschemes \\cite{N2, Gr} and of wreath products \\cite{FJW}.\n\nWhile our motivation is quite different, our main conjecture fits\ninto the scheme of Reid \\cite{R} who asks for what finite subgroup\n$G \\subset SL_K( {\\Bbb C} )$ the Hilbert quotient $ {\\Bbb C} ^K \/\/ G$ (also called\n$G$-Hilbert scheme on $ {\\Bbb C} ^K$) is a crepant resolution of $ {\\Bbb C} ^K\/G$.\nNote that the notion of McKay correspondence is meant in the\nstrict sense in this paper while the McKay correspondence in\n\\cite{R} is in a generalized sense. Our work provides supporting\nevidence for an affirmative answer of the McKay correspondence in\nthe sense of \\cite{R} for $ {\\Bbb C} ^{2n}$ acted upon by $\\G_n$ which in\nturn is a key step to a direct isomorphism of the two form of\nMcKay correspondence mentioned above.\n\nBy applying a remarkable theorem of Bridgeland-King-Reid\n\\cite{BKR} to our situation, our main conjecture on the\nisomorphism between Hilbert quotients and Hilbert schemes implies\nthat the equivalence of bounded derived categories among $D_{\\G_n}\n( {\\Bbb C} ^{2n})$, $D_{S_n} ( \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^n ),$ and $D ( \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^{[n]})$. For $n =1$\nthis is a theorem due to Kapranov-Vasserot \\cite{KV}. Such an\nequivalence can be viewed as a direct connection among the objects\nin the diagram (\\ref{eq_mckay}), where K-groups of topological\nvector bundles are replaced by K-groups of sheaves and connection\nbetween K-group and homology is made via Chern character.\n\nIn the end we establish a direct isomorphism of various algebraic\nstructures in II and III in the diagram (\\ref{eq_master}). More\nexplicitly, we construct Schur bases of the equivariant K-groups\nin II and III and show that a canonical one-to-one correspondence\nbetween these two bases gives the desired isomorphism for various\nalgebraic structures such as Hopf algebras, $\\lambda$-rings, and\nHeisenberg algebra, etc.\n\nThe plan of the paper goes as follows. In\nSect.~\\ref{sect_morphism} we study the resolution of singularities\n$\\tau_n$ and provide various steps toward establishing our\nconjecture on the isomorphism between the Hilbert quotient\n$ \\C^{2n} \/\/ \\Gn $ and the Hilbert scheme $ \\ale^{[n]} $. In\nSect.~\\ref{sect_mckay} we show that two virtual characters of\n$\\G_n$ arising from different setups coincide with each other and\ndiscuss various implications of this and our main conjecture. In\nSect.~\\ref{sect_ktheory} we establish isomorphisms of various\nalgebraic structures in II and III of diagram (\\ref{eq_master}).\n\n\\noindent {\\bf Acknowledgment.}  Some ideas of the paper were\nfirst conceived when I was visiting the Max-Planck Institut f\\\"ur\nMathematik (MPI) at Bonn in 1998. I thank MPI for a stimulating\natmosphere. Hiraku Nakajima's papers and lecture notes are sources\nof inspiration. I am grateful to him for the influence of his\nwork. His comments on an earlier version of this paper helps much\nto clarify the exposition of Subsection~\\ref{subsect_quiver}. I\nthank Mark de Cataldo for helpful discussions and comments. I also\nthank Igor Frenkel for his comments and informing me that he has\nbeen recently thinking about some related subjects from somewhat\ndifferent viewpoint.\n\\section{Wreath products and Hilbert schemes}\n\\label{sect_morphism}\nIn this section, we establish some direct\nconnections between wreath products and Hilbert schemes, i.e.\nbetween I and III in the diagram (\\ref{eq_mckay}). In particular\nfor $\\Gamma$ trivial it reduces to relating I and II.\n\\subsection{The wreath products}\nGiven a finite group $\\Gamma$, we denote by $\\Gamma^*$ the set of all the\ninequivalent complex irreducible characters $\\{ \\gamma_0, \\gamma_1,\n\\ldots, \\gamma_r \\}$ and by $\\Gamma_*$ the set of conjugacy classes. We\ndenote by $\\gamma_0$ the trivial character and by $\\Gamma^*_0$ the set of\nnon-trivial characters $\\{\\gamma_1, \\ldots, \\gamma_r \\}$. The $ {\\Bbb C} $-span of\n$\\gamma \\in \\Gamma^*$, denoted by $ R(\\Gamma) $, can be identified with\nthe space of class functions on $\\Gamma$. We denote by $R_{\\Bbb Z} (\\G)$ the\nintegral span of irreducible characters of $ \\Gamma$.\n\nLet $\\Gamma^n = \\Gamma \\times \\cdots \\times \\Gamma$ be the $n$-th\ndirect product of $\\Gamma$. The symmetric group $S_n$ acts on\n$\\Gamma^n$ by permutations: $\\sigma (g_1, \\cdots, g_n)\n  = (g_{\\sigma^{ -1} (1)}, \\cdots, g_{\\sigma^{ -1} (n)}).\n$\nThe wreath product of $\\Gamma$ with $S_n$ is defined to be the\nsemi-direct product $$\n \\Gamma_n = \\{(g, \\sigma) | g=(g_1, \\cdots, g_n)\\in {\\Gamma}^n,\n\\sigma\\in S_n \\} $$\n with the multiplication given by\n $ (g, \\sigma)\\cdot (h, \\tau)=(g \\, {\\sigma} (h), \\sigma \\tau ) .\n$ Note that $\\Gamma^n$ is a normal subgroup of $\\G_n$.\n\nLet $\\lambda=(\\lambda_1, \\lambda_2, \\cdots, \\lambda_l)$ be a partition: $\\lambda_1\\geq\n\\dots \\geq \\lambda_l \\geq 1$. The integer $|\\lambda|=\\lambda_1+\\cdots+\\lambda_l$\nis called the {\\em weight}, and $l (\\lambda ) =l$ is called the {\\em\nlength} of the partition $\\lambda $. We will often make use of another\nnotation for partitions: $ \\lambda=(1^{m_1}2^{m_2}\\cdots) , $ where\n$m_i$ is the number of parts in $\\lambda$ equal to $i$.\n\nGiven a family of partitions $\\rho=(\\rho(x))_{x\\in S}$ indexed by\na finite set $S$, we define the {\\em weight} of $\\rho$ to be\n$$\\|\\rho\\|=\\sum_{x\\in S}|\\rho(x)|.$$ Sometimes it is convenient to\nregard $\\rho=(\\rho(x))_{x\\in S}$ as a partition-valued function on\n$S$. We denote by ${\\cal P}(S)$ the set of all partitions indexed\nby $S$ and by ${\\cal P}_n(S)$ the set of all partitions $\\rho$ in\n${\\cal P}(S)$ of weight $n$.\n\nThe conjugacy classes of ${\\Gamma}_n$ can be described in the\nfollowing way. Let $x=(g, \\sigma )\\in {\\Gamma}_n$, where $g=(g_1,\n\\cdots, g_n) \\in {\\Gamma}^n,$ $ \\sigma \\in S_n$. The permutation\n$\\sigma $ is written as a product of disjoint cycles. For each\nsuch cycle $y=(i_1 i_2 \\cdots i_k)$ the element $g_{i_k} g_{i_{k\n-1}} \\cdots g_{i_1} \\in \\Gamma$ is determined up to conjugacy in\n$\\Gamma$ by $g$ and $y$, and will be called the {\\em\ncycle-product} of $x$ corresponding to the cycle $y$. For any\nconjugacy class $c$ and each integer $i\\geq 1$, the number of\n$i$-cycles in $\\sigma$ whose cycle-product lies in $c$ will be\ndenoted by $m_i(c)$. Denote by $\\rho (c)$ the partition $(1^{m_1\n(c)} 2^{m_2 (c)} \\ldots )$. Then each element $x=(g, \\sigma)\\in\n{\\Gamma}_n$ gives rise to a partition-valued function $( \\rho\n(c))_{c \\in \\Gamma_*} \\in {\\mathcal P} ( \\Gamma_*)$ such that $\\sum_{i, c}\ni m_i(c) =n$. The partition-valued function $\\rho =( \\rho(c))_{ c\n\\in G_*} $ is called the {\\em type} of $x$. It is well known (cf.\n\\cite{M,Z}) that any two elements of ${\\Gamma}_n$ are conjugate in\n${\\Gamma}_n$ if and only if they have the same type.\n\\subsection{A resolution of singularities} \\label{subsect_fiber}\nLet $X$ be a smooth complex algebraic variety acted upon by a\nfinite group $\\Gamma$ of order $N$. We denote by $X^{[n]}$ the Hilbert\nscheme of $n $ points on $X$ and denote by $X^{(n)} = X^n \/S_n$\nthe $n$-th symmetric product. Both $X^{[n]}$ and $X^{(n)}$ carry\nan induced $\\Gamma$-action from $X$.\n\nNow assume $X$ is a quasi-projective surface. A beautiful theorem\nof Fogarty \\cite{Fo} states that $X^{[n]}$ is non-singular of\ndimension $2n$. It is well known (cf. e.g. \\cite{N2}) that the\nHilbert-Chow morphism $X^{[n]} \\rightarrow X^{(n)}$ is a\nresolution of singularities. Given a partition $\\nu$ of $n$ of\nlength $l$: $\\nu_1 \\geq \\nu_2 \\geq \\ldots \\geq \\nu_l\n>0$, we define\n\\[\nX^{(n)}_{\\nu} = \\left\\{ \\sum_{i =1}^l \\nu_i x_i \\in X^{(n)} |x_i\n\\neq x_i \\mbox{ for } i \\neq j \\right\\}.\n\\]\nA natural stratification of $X^{(n)}$ is given by\n\\begin{eqnarray*}\nX^{(n)} = \\bigsqcup_{\\nu} X^{(n)}_{\\nu}.\n\\end{eqnarray*}\n\nIn the remainder of this section, we let $\\Gamma$ be a finite subgroup\nof $ SL_2 ( {\\Bbb C}  )$ unless otherwise specified. The classification of\nfinite subgroups of $SL_2 ( {\\Bbb C} )$ is well known. The following is a\ncomplete list of them: the cyclic, binary dihedral, tetrahedral,\noctahedral and icosahedral groups. We denote by $\\tau :  \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma}  \\rightarrow  {\\Bbb C} ^2 \/\\Gamma$ the minimal resolution of the simple singularity.\n\nA canonical identification between $ {\\Bbb C} ^{2n} \/\\G_n$ and $( {\\Bbb C} ^2\n\/\\Gamma)^n \/S_n $ is given as follows: given a $\\G_n$-orbit, say $\\G_n .\n(x_1, \\ldots, x_n)$ for some $(x_1, \\ldots, x_n) \\in ( {\\Bbb C} ^2)^n =\n {\\Bbb C} ^{2n}$, we obtain a point $[\\Gamma.x_1] +\\ldots + [\\Gamma.x_n]$ in\n$( {\\Bbb C} ^2 \/ \\Gamma)^n \/S_n $, where $\\Gamma.x_i$ denotes the $\\Gamma$-orbit of\n$x_1$, i.e. a point in $ {\\Bbb C} ^2 \/\\Gamma$. It is easy to see that this map\nis independent of the choice of the representative\n$(x_1,\\ldots,x_n)$ in the $\\G_n$-orbit and it is one-to-one. The\nfollowing commutative diagram \\cite{W}\n         %\n\\begin{eqnarray}   \\label{eq_mine}\n \\begin{array}{ccc}\n    \\ale^{[n]}         & \\stackrel{\\pi_n}{\\longrightarrow}\n        & ( \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} )^n \/ S_n    \\\\\n   \\downarrow \\tau_n &                 & \\downarrow  \\tau_{(n)}      \\\\\n    {\\Bbb C} ^{2n} \/\\G_n      &   \\cong            & ( {\\Bbb C} ^2 \/ \\Gamma)^n \/S_n .\n \\end{array}\n\\end{eqnarray}\n    %\ndefines a morphism $\\tau_n :  \\ale^{[n]}  \\rightarrow   {\\Bbb C} ^{2n} \/\\G_n.$\n\n\\begin{proposition}  \\label{prop_resol}\nThe morphism $\\tau_n :  \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^{[n]} \\rightarrow  {\\Bbb C} ^{2n} \/\\G_n $ is a\nsemismall crepant resolution of singularities.\n\\end{proposition}\n   %\n\\begin{demo}{Proof}\nIt is clear by definition that $\\tau_n$ is a resolution of\nsingularities. We now describe a stratification of $ {\\Bbb C} ^{2n} \/\\G_n$.\n\nThe simple singularity $ {\\Bbb C} ^2 \/\\Gamma$ has a stratification given by\nthe singular point $o$ and its complement denoted by $ {\\Bbb C} _0^2 \/\\Gamma$.\nIt follows that a stratification of $( {\\Bbb C} ^2\/ \\Gamma)^n$ is given by $n\n+1 $ strata $( {\\Bbb C} ^2\/ \\Gamma)^n [i]$ $( 0 \\leq i \\leq n)$, where $( {\\Bbb C} ^2\/\n\\Gamma)^n [i]$ consisting of points in the Cartisian product $( {\\Bbb C} ^2\/\n\\Gamma)^n$ which has exactly $n - i$ components given by the singular\npoint. Then a stratification of $ {\\Bbb C} ^{2n} \/\\G_n = ( {\\Bbb C} ^2 \/ \\Gamma)^n \/S_n\n$ is given by\n\\begin{eqnarray*}\n( {\\Bbb C} ^2 \/ \\Gamma)^n \/S_n\n & =& \\bigsqcup_{i=0}^n ( {\\Bbb C} ^2\/ \\Gamma)^n [i] \/S_n \\\\\n & \\cong & \\bigsqcup_{i=0}^n ( {\\Bbb C} _0^2 \/\\Gamma)^i \/S_i \\times \\{ (n -i) o \\}  \\\\\n & \\cong & \\bigsqcup_{i=0}^n ( {\\Bbb C} _0^2 \/\\Gamma)^{(i)} \\\\\n & \\cong & \\bigsqcup_{i=0}^n \\bigsqcup_{|\\mu| =i} ( {\\Bbb C} _0^2 \/\\Gamma)^{(i)}_{\\mu}.\n\\end{eqnarray*}\nThe codimension of the strata $( {\\Bbb C} _0^2\/\\Gamma)^{(i)}_{\\mu}$ is $2n - 2\nl (\\mu)$. Clearly we also have\n\\begin{eqnarray*}\n\\tau_n^{-1} ( ( {\\Bbb C} _0^2 \/\\Gamma)^{(i)}_{\\mu} \\times \\{ (n -i) o \\} ) =\n ( {\\Bbb C} _0^2 \/\\Gamma)^{[i]}_{\\mu} \\times \\tau^{-1} (0)^{n -i}.\n\\end{eqnarray*}\nIt follows that the dimension of a fiber over a point in this\nstrata is equal to $(i - l (\\mu) ) + (n -i) = n - l (\\mu)$ which\nis half of the codimension of the strata above. Thus $\\tau_n$ is\nsemismall.\n\nThe canonical bundles over $ {\\Bbb C} ^{2n} \/\\G_n$ and $ \\ale^{[n]} $ is trivial\ndue to the existence of holomorphic symeplectic forms (note that\n$\\G_n$ preserves the symeplectic form on $ {\\Bbb C} ^{2n}$). Thus $\\tau_n$\nis crepant.\n\\end{demo}\n\n\\begin{remark} \\rm\n\\begin{enumerate}\n\\item $\\tau_n$ is a symplectic resolution in the sense that\nthe pullback of the holomorphic symplectic form on $ {\\Bbb C} ^{2n} \/ \\G_n$\noutside of the singularities can be extended to a holomorphic\nsymplectic form on $ \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^{[n]}$. When $n =1$ this becomes the\nminimal resolution $\\tau = \\tau_1 :  \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma}  \\rightarrow  {\\Bbb C} ^2 \/\\Gamma$.\n\\item  It follows from the semismallness of $\\tau_n$\nthat $\\tau_{(n)} $ is also semismall. Then the diagram\n(\\ref{eq_mine}) is remarkable in that all three maps $\\tau_n, \\pi_n$\nand $\\tau_{(n)} $ are semismall.\n\\item The resolution $\\tau_n :  \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^{[n]} \\rightarrow  {\\Bbb C} ^{2n} \/\\G_n $\nis one-to-one over the non-singular locus of the orbifold $ {\\Bbb C} ^{2n}\n\/ \\G_n$ corresponding to regular $\\G_n$-orbits in $ {\\Bbb C} ^{2n}$.\n\\end{enumerate}\n\\end{remark}\n\nWe denote by $ \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^{[n], 0}$ the fiber of $\\tau_n$ over $[0]$,\nwhere $[0]$ denotes the image in $ {\\Bbb C} ^{2n}\/\\G_n$ of the origin of\n$ {\\Bbb C} ^{2n}$. We assume that $\\Gamma$ is not trivial. By the diagram\n(\\ref{eq_mine}), we have\n  %\n\\begin{eqnarray*}\n \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^{[n], 0} = \\pi_n^{ -1} \\tau_{(n)}^{ -1} (0) = \\pi_n^{ -1}\n(D^{(n)}),\n\\end{eqnarray*}\nwhere $D = \\tau^{-1} (0)$ is the\nexceptional divisor in $ \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} $. It is well known that the\nirreducible components of $D$ are projective lines $\\Sigma_{\\gamma}$\nparameterized by the set of non-trivial characters $\\gamma \\in \\Gamma^*_0$\n(cf. e.g. \\cite{GSV}).\n\nRecall that given an irreducible curve $\\Sigma \\subset  \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} $, the\nvariety\n\\[\nL^n \\Sigma := \\left\\{ I \\in  \\ale^{[n]}  | \\mbox{Supp}( {\\cal O} \/I\n)\\subset \\Sigma \\right\\} = \\pi_n^{-1} (\\Sigma^{(n)}).\n\\]\nintroduced by Grojnowski \\cite{Gr} (also see \\cite{N2}) plays an\nimportant role in understanding the middle-dimensional homology\ngroups of Hilbert schemes and in connection with symmetric\nfunctions. One can show (cf. {\\em loc. cit.}) that the irreducible\ncomponents of $L^n \\Sigma$ are parameterized by partitions $\\nu$\nof $n$ and given by\n   %\n\\begin{eqnarray*}\nL^{\\nu} \\Sigma = \\overline{ \\pi_n^{-1} (\\Sigma^{(n)}_{\\nu} ) } ,\n\\end{eqnarray*}\nwhere $ \\Sigma^{(n)}_{\\nu} $  is the stratum of the symmetric\nproduct $\\Sigma^{(n)}$ associated to $\\nu$.\n\nIt is interesting to observe that the fiber $ \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^{[n], 0}$ is a\nnatural generalization of the above construction. Given $\\rho =\n(\\rho (\\gamma))_{\\gamma \\in \\Gamma^*_0} \\in {\\cal P}_n (\\Gamma^*_0)$, we set $n_i\n= |\\rho (\\gamma_i)|$ and define\n\\begin{eqnarray*}\nL^{\\rho}D = \\overline{ \\pi_n^{-1}\n\\left((\\Sigma_{\\gamma_1})^{(n_{1})}_{\\rho (\\gamma_1)} \\times \\ldots \\times\n(\\Sigma_{\\gamma_r})^{ (n_{r})}_{\\rho (\\gamma_r)} \\right)} .\n\\end{eqnarray*}\n\n\\begin{proposition} \\label{prop_fiber}\nLet $\\Gamma$ be non-trivial. The fiber $ \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^{[n], 0}$ is of pure\ndimension $n$, and its irreducible components are given by\n$L^{\\rho}D, \\rho \\in {\\cal P}_n (\\Gamma^*_0)$.\n\\end{proposition}\n\n\\begin{demo}{Proof}\nThe component $L^{\\rho}D$ is irreducible since the fiber of\n$\\pi_n$ is so. $L^{\\rho}D$ is of dimension $n$ since the dimension\nof $(\\Sigma_{\\gamma})^{ |\\rho (\\gamma)|}_{\\rho (\\gamma)}$ $(\\gamma \\in \\Gamma^*_0)$ is\n$l( \\rho (\\gamma))$ and the dimension of the fiber of $\\pi_n$ is equal\nto $$\\sum_{i =1}^r \\left( |\\rho (\\gamma_i) | - l (\\rho (\\gamma_i)) \\right)\n= n - \\sum_i l( \\rho (\\gamma_i)). $$ Therefore the dimension of\n$L^{\\rho}D$ is $( n - \\sum_i l( \\rho (\\gamma_i)) ) + \\sum_i l( \\rho\n(\\gamma_i)) = n$.\n\\end{demo}\n\\subsection{Hilbert quotient and a subvariety of $( {\\Bbb C} ^2)^{[nN]}$}\n\\label{subsect_link}\n  %\nLet $X$ be a smooth complex algebraic variety acted upon by a\nfinite group $\\Gamma$ of order $N$. A regular $\\Gamma$-orbit can be viewed\nas an element in the Hilbert scheme $X^{[N]}$ of $N$ points in\n$X$. The Hilbert quotient is the closure $X \/\/ \\G$ of the set of\nregular $\\Gamma$-orbits in $X^{[N]}$ (cf. \\cite{Ka}). It follows that\nthere exists a tautological vector bundle over $X \/\/ \\G$ of rank\n$N$. The group $\\Gamma$ acts on the tautological bundle fiberwise and\neach fiber is isomorphic to the regular representation of $\\Gamma$.\n\nNote that the wreath product $\\G_n $ acts faithfully on the affine\nspace $ {\\Bbb C} ^{2n} = ( {\\Bbb C} ^2)^n$. We make the following conjecture which\nprovides a key connection between I and III in the diagram\n(\\ref{eq_mckay}). We remark that a more general conjecture can be\neasily formulated in the setup of the diagram (\\ref{eq_master}).\n\n\\begin{conjecture} \\label{conj_main}\nThere exists a canonical isomorphism between the Hilbert quotient\n$ \\C^{2n} \/\/ \\Gn $ and the Hilbert scheme $ \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^{[n]}$ of $n$ points on\n$ \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} $.\n\\end{conjecture}\n   %\n\\begin{remark} \\rm \\label{rem_one}\n \\begin{enumerate}\n\\item\nConjecture~\\ref{conj_main} for $n =1$ reduces to a theorem due to\nGinzburg-Kapronov (unpublished) and independently Ito-Nakamura\n\\cite{INr} which says $ {\\Bbb C} ^2 \/\/ \\Gamma \\cong  \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} $.\n\\item When $\\Gamma$ is trivial and so $\\G_n $ is the\nsymmetric group $S_n$,  Haiman \\cite{H} has shown that\nConjecture~\\ref{conj_main} for $\\Gamma$ trivial is equivalent to a\nremarkable {\\em $n!$ conjecture} due to Garsia and Haiman\n\\cite{GH}. A proof of the $n!$ conjecture has been very recently\nposted by Haiman in his UCSD webpage.\n\\item Conjecture~\\ref{conj_main} implies an\nisomorphism of Hilbert quotients:\n\\[\n  \\C^{2n} \/\/ \\Gn  \\cong ( {\\Bbb C} ^2 \/\/ \\Gamma )^n \/\/S_n,\n\\]\nwhich seems to be in a more symmetric form. On the other hand,\nassuming the $n!$-conjecture, one can show that an isomorphism of\nHilbert quotients above implies Conjecture~\\ref{conj_main}.\n\\end{enumerate}\n\\end{remark}\n\nSince $\\tau_n :  \\ale^{[n]}  \\rightarrow  {\\Bbb C} ^{2n } \/\\G_n$ is a resolution\nof singularities ( and thus is proper) and $ {\\Bbb C} ^{2n } \/\\G_n$ is a\nnormal variety, the algebra of regular functions on $ \\ale^{[n]} $ is\nisomorphic to the algebra of $\\G_n$ invariants on the regular\nfunctions on $ {\\Bbb C} ^{2n}$. The following lemma gives a description of\nthe algebra of $\\G_n$ invariants in $ {\\Bbb C}  [ {\\bf x} ,  {\\bf y} ]$, where we denote\nby $ {\\bf x} $ (resp. $ {\\bf y} $) the $n$-tuple $x_1, \\ldots, x_n$ (resp. $y_1,\n\\ldots, y_n$). It generalizes Weyl's theorem \\cite{Wey} for\nsymmetric groups and the proof is similar.\n\n\\begin{lemma} \\label{lem_weyl}\nThe algebra of invariants $ {\\Bbb C}  [ {\\bf x} ,  {\\bf y} ]^{\\G_n}$ is generated by\n\\begin{eqnarray*}\n\\tilde{f} ( {\\bf x} ,  {\\bf y} ) = f(x_1, y_1) + f(x_2, y_2)\n                     + \\ldots + f(x_n, y_n),\n\\end{eqnarray*}\nwhere $f$ runs over an arbitrary linear basis $\\cal B$ for\nthe space of invariants $ {\\Bbb C}  [x,y]^{\\Gamma}$.\n\\end{lemma}\n\n\\begin{demo}{Proof}\nWe prove the lemma by induction on $n$. When $n =1$ it is evident.\nAssume now that we have established the lemma for $n-1$. We use\n$ {\\bf x}  '$ and $  {\\bf y} '$ to denote $x_2, \\ldots, x_n$ and respectively\n$y_2, \\ldots, y_n$. The space $ {\\Bbb C} [  {\\bf x}  ',  {\\bf y}  ' ]$ is acted on by\nthe wreath product subgroup $\\Gamma_{n -1} \\subset \\G_n$.\n\nGiven any $F ( {\\bf x} ,  {\\bf y} ) \\in  {\\Bbb C}  [ {\\bf x} ,  {\\bf y} ]^{\\G_n}$, we can write it as a\nlinear combination of $x_1^{\\alpha} y_1^{\\beta} F_{\\alpha\n\\beta}'( {\\bf x}  ',  {\\bf y}  '),$ where $ F_{\\alpha \\beta}'( {\\bf x}  ',  {\\bf y}  ')$ is\nsome $\\Gamma_{n -1}$-invariant polynomial. By induction assumption, we\ncan write $ F_{\\alpha \\beta}'( {\\bf x}  ',  {\\bf y}  ')$ as a polynomial in\nterms of $\\tilde{f} ( {\\bf x} ',  {\\bf y} ')$ where $f \\in {\\cal B}$, and in\nturn as a polynomial in terms of $\\tilde{f} ( {\\bf x} ,  {\\bf y} )$ and $x_1,\ny_1$. Therefore we can write $F ( {\\bf x} ,  {\\bf y} )$ as a linear combination\nof polynomials of the form $ x_1^{\\alpha} y_1^{\\beta} G_{\\alpha\n\\beta}( {\\bf x} ,  {\\bf y} ),$ where $ G_{\\alpha \\beta}( {\\bf x} ,  {\\bf y}  )$ is a\npolynomial in terms of $\\tilde{f} ( {\\bf x} ',  {\\bf y} ')$ where $f \\in \\cal\nB$. Since both $ F_{\\alpha \\beta}( {\\bf x} ,  {\\bf y} )$ and $G_{\\alpha\n\\beta}( {\\bf x} ,  {\\bf y} )$ are $\\G_n$-invariant and thus in particular\ninvariant with respect to the symmetric group $S_n$ and the first\nfactor $\\Gamma$ in $\\Gamma^n \\subset \\G_n$, $F( {\\bf x} ,  {\\bf y} )$ becomes a linear\ncombination of $p_{\\alpha \\beta}( {\\bf x} ,  {\\bf y} ) G_{\\alpha \\beta}( {\\bf x} , {\\bf y} ).$\nHere $p_{\\alpha \\beta}( {\\bf x} ,  {\\bf y} )$ denotes $\\frac1{nN} \\sum_{i =1}^n\n\\sum_{g \\in \\Gamma} (g. x_i)^{\\alpha} (g.y_i)^{\\beta}$, the average of\n$x_1^{\\alpha} y_1^{\\beta}$ over $\\Gamma \\times S_n$ (which is the same\nas the average over $\\G_n$). We complete the proof by noting that\n$p_{\\alpha \\beta}( {\\bf x} ,  {\\bf y} )$ is a linear combination of $\\tilde{f}\n( {\\bf x} ,  {\\bf y} )$ where $f \\in \\cal B$.\n\\end{demo}\n\nGiven $J \\in  \\C^{2n} \/\/ \\Gn $ we regard it as an ideal in $ {\\Bbb C}  [ {\\bf x} ,  {\\bf y} ]$ of\ncolength $N^n n!$ (which is the order of $\\G_n$). Then the quotient\n$ {\\Bbb C}  [ {\\bf x} ,  {\\bf y}  ] \/ J$ affords the regular representation $R$ of\n$\\G_n$, and its only $\\G_n$-invariants are constants. Thus we have\n$\\tilde{f} ( {\\bf x} ,  {\\bf y}  ) = c_f \\mbox{ mod }J$ for some constant $c_f$.\nRecall that $\\Gamma_{n -1}$ acts on $ {\\Bbb C}  [ {\\bf x}  ',  {\\bf y}  ']$. By\nLemma~\\ref{lem_weyl}, the space $ {\\Bbb C}  [ {\\bf x} ,  {\\bf y} ]^{\\Gamma_{n -1}}$ is\ngenerated by $x_1, y_1$ and $f(x_2, y_2) + \\ldots + f(x_n, y_n)$.\nThe latter is equal to $c_f - f(x_1, y_1) \\mbox{ mod } J$. Thus\n$( {\\Bbb C}  [ {\\bf x} ,  {\\bf y} ] \/ J)^{\\Gamma_{n -1}}$ is generated by $x_1, y_1$ and\n$c_f - f(x_1, y_1), f \\in  {\\Bbb C}  [x, y]^{\\Gamma}$. It follows that\n\\begin{eqnarray}  \\label{eq_link}\n   {\\Bbb C}  [x_1, y_1 ] \/ (J \\cap  {\\Bbb C}  [x_1, y_1 ])\n  \\equiv (  {\\Bbb C}  [ {\\bf x} ,  {\\bf y} ] \/ J)^{\\Gamma_{n -1}},\n\\end{eqnarray}\nwhich has dimension $nN = | \\G_n | \/ | \\Gamma_{n -1} |$ because\n$( {\\Bbb C} [ {\\bf x} ,  {\\bf y} ] \/J)^{\\Gamma_{n -1}}$ can be identified with the space of\n${\\Gamma_{n -1}}$-invariants in the regular representation of $\\Gamma$.\n\nThe first copy of $\\Gamma$ in the Cartesian product $\\Gamma^n \\subset \\G_n$\ncommutes with $\\Gamma_{n -1}$ above. It follows from (\\ref{eq_link})\nthat the quotient $ {\\Bbb C}  [x_1, y_1 ] \/ (J \\cap  {\\Bbb C}  [x_1, y_1 ])$ as a\n$\\Gamma$-module is isomorphic to $R^n$, a direct sum of $n$ copies of\nthe regular representation $R$ of $\\G_n$.\n\nThe $\\Gamma$-action on $ {\\Bbb C} ^2$ induces a $\\Gamma$-action on the Hilbert\nscheme $( {\\Bbb C} ^2 )^{[nN]}$ and the symmetric product $\n( {\\Bbb C} ^2)^{(nN)}$. The Hilbert-Chow morphism $( {\\Bbb C} ^2 )^{[nN]}\n\\rightarrow ( {\\Bbb C} ^2 )^{(nN)}$ induces one between the set of\n$\\Gamma$-fixed points $( {\\Bbb C} ^2 )^{[nN], \\Gamma} \\rightarrow ( {\\Bbb C} ^2\n)^{(nN),\\Gamma}$. As the fixed point set of a non-singular variety by\nthe action of a finite group, $( {\\Bbb C} ^2 )^{[nN], \\Gamma}$ is\nnon-singular. Denote by $X_{\\G,n}$ the set of $\\Gamma$-invariant ideals\n$I$ in the Hilbert scheme $ ( {\\Bbb C} ^2)^{[nN]}$ such that the quotient\n$ {\\Bbb C}  [x, y] \/ I$ is isomorphic to $R^n$ as a $\\Gamma$-module. Since the\nquotient $ {\\Bbb C}  [x, y]\/I$ are isomorphic as $\\Gamma$-modules for all $I$\nin a given connected component of $( {\\Bbb C} ^2 )^{[nN], \\Gamma}$, the\nvariety $X_{\\G,n}$ is a union of components of $( {\\Bbb C} ^2 )^{[nN], \\Gamma}$.\nIn particular $X_{\\G,n}$ is non-singular (we shall see that $X_{\\G,n}$\nis indeed connected of dimension $2n$).\n\nTherefore by sending the ideal $J$ to the ideal $J \\cap  {\\Bbb C}  [x_1,\ny_1]$, we have defined a map $\\varphi$ from $  \\C^{2n} \/\/ \\Gn $ to $ ( {\\Bbb C} ^2\n)^{[nN]}$, whose image lies in $X_{\\G,n}$. We also denote $\\varphi:\n \\C^{2n} \/\/ \\Gn  \\longrightarrow X_{\\G,n}$.\n\nThe map $\\varphi$ can be also understood as follows. Let\n${\\cal U}_{\\G, n} $ be the universal family over the Hilbert quotient\n$ \\C^{2n} \/\/ \\Gn $ which is a subvariety of the Hilbert scheme\n$( {\\Bbb C} ^{2n})^{[n!N^n]}$:\n\\begin{eqnarray}  \\label{eq_quotuniv}\n \\begin{array}{ccc}\n  {\\cal U}_{\\G, n}  & \\longrightarrow &  {\\Bbb C} ^{2n}    \\\\\n   \\downarrow   &                 &        \\\\\n     \\C^{2n} \/\/ \\Gn  &        &\n \\end{array}\n\\end{eqnarray}\n    %\nIt has a natural $\\G_n$-action fiberwise such that each fiber\ncarries the regular representation of $\\G_n$. Then ${\\cal U}_{\\G, n} \/\n\\Gamma_{n-1}$ is flat and finite of degree $nN$ over $ \\C^{2n} \/\/ \\Gn $, and\nthus can be identified with a family of subschemes of $ {\\Bbb C} ^2$ as\nabove. Then $\\varphi$ is the morphism given by the universal\nproperty of the Hilbert scheme $( {\\Bbb C} ^2 )^{[nN]}$ for the family\n${\\cal U}_{\\G, n} \/ \\Gamma_{n-1}$.\n\nThus we have established the following.\n\n\\begin{theorem} \\label{th_morph}\nWe have a natural morphism $\\varphi :  \\C^{2n} \/\/ \\Gn  \\longrightarrow\nX_{\\G,n}$ defined as above.\n\\end{theorem}\n\n\\begin{remark} \\rm\nFor $\\Gamma $ trivial, $\\G_n$ reduces to $S_n$ and $X_{\\G,n}$ becomes the\nHilbert scheme $( {\\Bbb C} ^2)^{[n]}$. In this case the above morphism\n$ \\C^{2n} \/\/ S_n  \\rightarrow ( {\\Bbb C} ^2)^{[n]}$ was earlier constructed by\nHaiman \\cite{H}. We plan to elaborate further on generalizations\nof \\cite{H} to our setup in the future.\n\\end{remark}\n\n\\begin{conjecture}\nThe morphism $\\varphi :  \\C^{2n} \/\/ \\Gn  \\longrightarrow X_{\\G,n}$ is an\nisomorphism.\n\\end{conjecture}\n\nObserve that $ ( {\\Bbb C} ^2 )^{(nN),\\Gamma}$ is the subset of $( {\\Bbb C} ^2)^{(nN)}$\nconsisting of points\n\\begin{eqnarray*}\n\\sum_{\\gamma \\in \\Gamma} [\\gamma \\cdot x_1] + \\ldots + \\sum_{\\gamma \\in \\Gamma} [\\gamma \\cdot\nx_a] + (n -a) N [0], \\quad 1 \\leq a \\leq n, x_1 , \\ldots , x_a \\in\n {\\Bbb C} ^2 \\backslash 0,\n\\end{eqnarray*}\nwhich can be thought as the $\\G_n$-orbit of $(x_1, \\ldots, x_a, 0,\n\\ldots, 0) \\in ( {\\Bbb C} ^2)^n =  {\\Bbb C} ^{2n}$. In this way $( {\\Bbb C} ^2)^{(nN),\\Gamma}$\nis identified with $ {\\Bbb C} ^{2n} \/ \\G_n$. Thus we have proved the\nfollowing proposition  by noting the inclusion $X_{\\G,n} \\subset\n( {\\Bbb C} ^2 )^{[nN], \\Gamma} $.\n\n\\begin{proposition}  \\label{prop_fix}\nThe $\\Gamma$-fixed point set $( {\\Bbb C} ^2)^{(nN),\\Gamma}$ of the symmetric\nproduct $( {\\Bbb C} ^2)^{(nN)}$ can be canonically identified with\n$ {\\Bbb C} ^{2n} \/ \\G_n$. The Hilbert-Chow morphism $( {\\Bbb C} ^2)^{[nN]}\n\\rightarrow ( {\\Bbb C} ^2)^{(nN)}$, when restricted to the $\\Gamma$-fixed\npoint set, induces a canonical morphism $X_{\\G,n} \\rightarrow  {\\Bbb C} ^{2n}\n\/ \\G_n$.\n\\end{proposition}\n\n\\begin{remark}  \\rm  \\label{rem_gener}\nTake an unordered $n$-tuple $T$ of distinct $\\Gamma$-orbits in $ {\\Bbb C} ^2\n\\backslash 0$. Such an $n$-tuple defines a set of $nN$ distinct\npoints in $ {\\Bbb C} ^2$, and thus can be regarded as an ideal $I(T)$ in\nthe Hilbert scheme $( {\\Bbb C} ^2)^{[nN]}$. This ideal is clearly\n$\\Gamma$-invariant and as a $\\Gamma$-module $ {\\Bbb C}  [x, y] \/ I(T)$ is\nisomorphic to $R^n$. On the other hand observe that such an\n$n$-tuple $T$ can be canonically identified with a regular\n$\\G_n$-orbit in $ {\\Bbb C} ^{2n}$. In this way the sets of generic points\nin $X_{\\G,n}$ and $ {\\Bbb C} ^{2n} \/\\G_n$ coincide. It is easy to see that the\nmorphism $X_{\\G,n} \\rightarrow  {\\Bbb C} ^{2n} \/ \\G_n$ above is surjective and\nit is one-to-one over the set of generic points in $ {\\Bbb C} ^{2n} \/ \\G_n$\nconsisting of regular $\\G_n$-orbits.\n\\end{remark}\n\nWe define the reduced universal scheme $W_{\\Gamma, n}$ as the reduced\nfibered product\n\\begin{eqnarray*}\n\\begin{array}{ccc}\n    W_{\\Gamma, n}  & {\\longrightarrow} &  {\\Bbb C} ^{2n} \\\\\n  \\downarrow   &                   & \\downarrow \\\\\n X_{\\G,n}     & \\stackrel{ \\tau_n}{\\longrightarrow} &  {\\Bbb C} ^{2n} \/\\G_n .\n\\end{array}\n\\end{eqnarray*}\nIt is known that a finite surjective morphism from $Z$ to a\nnon-singular variety is flat if and only if $Z$ is Cohen-Macaulay.\n\n\\begin{conjecture}  \\label{conj_cohen}\n$W_{\\Gamma, n}$ is Cohen-Macaulay.\n\\end{conjecture}\n\nUnder the assumption of the validity of\nConjecture~\\ref{conj_cohen}, the universal properties of the\nHilbert scheme $( {\\Bbb C} ^{2n})^{[n!N^n]}$ induces a morphism $\\psi:\nX_{\\G,n} \\rightarrow ( {\\Bbb C} ^{2n})^{[n!N^n]}$, whose image lies in\n$ \\C^{2n} \/\/ \\Gn $. By Remark~\\ref{rem_gener} and the fact that the set of\ngeneric points of $ \\C^{2n} \/\/ \\Gn $ and $ {\\Bbb C} ^{2n} \/ \\G_n$ coincide, the two\nmorphisms $\\varphi$ and $\\psi$ are mutually inverse to each other\nover generic points. Then it follows that they are inverse\neverywhere, establishing Conjecture~\\ref{conj_main}.\n\n\\begin{remark} \\rm\nThe above conjecture~\\ref{conj_cohen} for $\\Gamma$ trivial was first\nconjectured by Haiman \\cite{H}. The proof announced very recently\nby Haiman in his UCSD homepage of $n!$ conjecture is based on a\nproof of this conjecture of his.\n\\end{remark}\n\\subsection{A quiver variety description}  \\label{subsect_quiver}\nWe first recall (cf. \\cite{N2}) that the Hilbert scheme\n$( {\\Bbb C} ^2)^{[K]}$ of $K$ points in $ {\\Bbb C} ^2$ admits a description in\nterms of a quiver consisting of one vertex and one arrow starting\nfrom the vertex and returning to the same vertex itself. More\nexplicitly, we denote\n    %\n\\begin{eqnarray*}\n \\widetilde{H}(K) =\n\\left\\{\n\\begin{array}{lcl}\n              &|& i) [B_1, B_2 ] +ij =0  \\\\\n(B_1, B_2, i, j) &|& ii) \\mbox{ there exists no proper subspace\n}\\\\\n              &|&  S \\subset\n               {\\Bbb C} ^K \\mbox{ such that } B_{\\alpha} (S) \\subset S \\mbox{ and } \\\\\n               &|& \\mbox{ im } i \\subset S \\quad (\\alpha =1,2)\n\\end{array}\n              \\right\\}  ,\n\\end{eqnarray*}\n    %\nwhere $B_1, B_2 \\in  \\mbox{End}  ( {\\Bbb C} ^K ), i \\in  \\mbox{Hom}  ( {\\Bbb C} ,  {\\Bbb C} ^K ), j \\in\n \\mbox{Hom}  ( {\\Bbb C} ^K,  {\\Bbb C}  )$. Then we have an isomorphism\n    %\n\\begin{eqnarray}  \\label{eq_quot}\n( {\\Bbb C} ^2)^{[K]} \\cong \\widetilde{H}(K)\/ GL_K ( {\\Bbb C}  ) ,\n\\end{eqnarray}\n    %\nwhere the action of $GL_K ( {\\Bbb C}  )$ on $\\widetilde{H}(K)$ is given by\n   %\n\\begin{eqnarray*}\n  g \\cdot (B_1, B_2, i, j) = (g B_1 g^{ -1}, g B_2 g^{ -1}, g i, jg^{-1}).\n\\end{eqnarray*}\nIt is also often convenient to regard $(B_1, B_2)$ to be in $ \\mbox{Hom} \n( {\\Bbb C} ^K ,  {\\Bbb C} ^2 \\otimes  {\\Bbb C} ^K )$. We remark that one may drop $j$ in\nthe above formulation because one can show by using the stability\ncondition that $j =0$ (cf. \\cite{N2}).\n\nThe bijection in (\\ref{eq_quot}) is given as follows. For $I \\in\n( {\\Bbb C} ^2)^{[K]}$, i.e. an ideal in $ {\\Bbb C}  [x, y]$ of colength $K$, the\nmultiplication by $x, y$ induces endomorphisms $B_1, B_2$ on the\n$K$-dimensional quotient $ {\\Bbb C}  [x, y] \/I$, and the homomorphism $i\n\\in  \\mbox{Hom}  ( {\\Bbb C} ,  {\\Bbb C} ^K)$ is given by letting $i (1) =1 \\mbox{ mod }\nI$. Conversely, given $(B_1, B_2, i)$, we define a homomorphism\n$ {\\Bbb C}  [x, y] \\rightarrow  {\\Bbb C} ^K$ by $f \\mapsto f(B_1, B_2) i (1)$. One\ncan show by the stability condition that the kernel $I$ of this\nhomomorphism is an ideal of $ {\\Bbb C}  [x, y]$ of colength $K$. One\neasily checks that the two maps are inverse to each other.\n\nSet $K =nN$, where $N$ is the order of $\\Gamma$. We may identify\n$ {\\Bbb C} ^K$ with $R^n$, $ {\\Bbb C} ^2$ with the defining representation $Q$ of\n$\\Gamma$ by the embedding $\\Gamma \\subset SL_2 ( {\\Bbb C} )$, and $ {\\Bbb C} $ with the\ntrivial representation of $\\Gamma$. Denote by\n\\begin{eqnarray*}\nM (n) & =&  \\mbox{Hom}  (R^n , Q \\otimes R^n ) \\bigoplus  \\mbox{Hom}  ( {\\Bbb C} ,\nR^n)\\bigoplus  \\mbox{Hom}  (R^n,  {\\Bbb C} ).\n\\end{eqnarray*}\nBy definition $\\widetilde{H} (nN) \\subset M(n)$. Let $GL_{\\Gamma} (R)$\nbe the group of $\\Gamma$-equivariant automorphisms of $R$. Then the\ngroup $G \\equiv GL_{\\Gamma} (R^n) \\cong GL_n ( {\\Bbb C} ) \\times GL_{\\Gamma} (R)$\nacts on the $\\Gamma$-invariant subspace $M(n)^{\\Gamma} $. We have the\nfollowing description of $X_{\\G,n}$ as a quiver variety. This result\nis known to Nakajima \\cite{N4} (also cf. Theorem 1 of\nVaragnolo-Vasserot \\cite{VV})\\footnote{I.~Frenkel informed us that\nhe also noticed this recently.}.\n    %\n\\begin{theorem}  \\label{th_quiv}\nThe variety $X_{\\G,n} $ admits the following description:\n   %\n\\begin{eqnarray*}\n  X_{\\G,n} \\cong (\\widetilde{H} ( nN) \\cap M(n)^{\\Gamma} ) \/ GL_{\\Gamma} (R^n).\n\\end{eqnarray*}\nIt particular $X_{\\G,n} $ is non-singular of pure dimension $2n$.\n\\end{theorem}\n\n\\begin{remark}   \\rm \\label{rem_clear}\nConsider the $\\Gamma$-module decomposition $Q \\otimes V_{\\gamma_i}=\n\\bigoplus_j a_{ij} V_{\\gamma_j}$, where $a_{ij} \\in  {\\Bbb Z} _+$, and\n$V_{\\gamma_i} $ $(i =0, \\ldots, r)$ are irreducible representations\ncorresponding to the characters $\\gamma_i$ of $\\Gamma$. Set $\\dim V_{\\gamma_i}\n= n_i$. Then\n\\begin{eqnarray}\n M(n)^{\\Gamma}\n & =&  \\mbox{Hom} _{\\Gamma} (R^n ,Q \\otimes R^n ) \\bigoplus  \\mbox{Hom} _{\\Gamma} ( {\\Bbb C} , R^n)\n  \\bigoplus  \\mbox{Hom} _{\\Gamma} (R^n,  {\\Bbb C} )\n \\label{eq_invt} \\\\\n & =&  \\mbox{Hom} _{\\Gamma} (\\sum_i  {\\Bbb C} ^{n n_i} \\otimes V_{\\gamma_i},\n   {\\Bbb C} ^2 \\otimes \\sum_i  {\\Bbb C} ^{n n_i} \\otimes V_{\\gamma_i} )\\nonumber \\\\\n  && \\bigoplus  \\mbox{Hom} _{\\Gamma} ( {\\Bbb C} , R^n)  \\bigoplus  \\mbox{Hom} _{\\Gamma} (R^n,  {\\Bbb C} ) \\nonumber \\\\\n & =& \\sum_{i j} a_{ij}  \\mbox{Hom}  (  {\\Bbb C} ^{n n_i} , {\\Bbb C} ^{n n_j})\n  \\bigoplus  \\mbox{Hom}  ( {\\Bbb C} , V_{\\gamma_0}^n ) \\bigoplus  \\mbox{Hom}  (V_{\\gamma_0}^n,  {\\Bbb C} ).  \\nonumber\n\\end{eqnarray}\nwhere $ \\mbox{Hom} _{\\Gamma}$ stands for the $\\Gamma$-equivariant homomorphisms.\nIn the language of quiver varieties as formulated by Nakajima\n\\cite{N, N3}, the above desciption of $X_{\\G,n}$ identifies $X_{\\G,n}$\nwith a quiver variety associated to the following data: the graph\nconsists of the same vertices and edges as the McKay quiver which\nis an affine Dynkin diagram associated to a finite subgroup $\\Gamma$\nof $SL_2 ( {\\Bbb C} )$; the vector space $V_i$ associated to the vertex\n$i$ is isomorphic to the direct sum of $n$ copies of the $i$-th\nirreducible representation $V_{\\gamma_i}$; the vector space $W_i =0$\nfor nonzero $i$ and $W_0 = {\\Bbb C} $.\n\\end{remark}\n\n\\begin{demo}{Proof of Theorem~\\ref{th_quiv}}\nOur proof is modeled on the proof of Theorem 1.9 and Theorem 4.4\nin \\cite{N2} which are special cases of our isomorphism for $\\Gamma$\ntrivial and for $n =1$ respectively. We sketch below for the\nconvenience of the reader.\n\nOne shows $j =0$ by using the stability condition. The isomorphism\nstatement follows directly from the description of ${ {\\Bbb C} ^2}^{[nN]}$\ngiven by (\\ref{eq_quot}), the definition of $X_{\\G,n}$, and\nEq.~(\\ref{eq_invt}). We have seen earlier that $X_{\\G,n}$ is\nnonsingular by construction.\n\nOne shows by a direct check that $[B_1, B_2]$ is $\\Gamma$-invariant\nendomorphism in $R^n$ for $(B_1, B_2) \\in  \\mbox{Hom} _{\\Gamma} (R^n ,  {\\Bbb C} ^2\n\\otimes R^n )$. The cokernel of the differential of the map\n$(B_1,B_2, i) \\mapsto [B_1, B_2]$ from $M(n)^{\\Gamma}$ to\n$ \\mbox{End} _{\\Gamma}(R^n)$ consists of the $\\Gamma$-endomorphisms in $R^n$ which\ncommute with $B_1 $ and $B_2$. By sending $f \\mapsto f( i (1))$ we\ndefine a map from the cokernel to the $n$-dimensional space\n$ \\mbox{Hom} _{\\Gamma} ( {\\Bbb C} , R^n) \\cong V_{\\gamma_0}^n$. Conversely, given $v \\in\nV_{\\gamma_0}^n$, an endomorphism $f$ in $R^n$ is uniquely determined\nby the equation $f(B_1^a B_2^bi(1)) = B_1^a B_2^b v$ by the\nstability condition ii) in the definition of $\\widetilde{H} (nN)$.\nOne further checks that $f$ lies in the cokernel. These two maps\nare inverse to each other. Thus the cokernel has constant\ndimension $n$.\n\nThe dimension of $\\widetilde{H} ( nN) \\cap M(n)^{\\Gamma}$ is equal to\n$\\dim M^{\\Gamma}(n) +n - \\dim GL_{\\Gamma} (R^n) $ since the dimension of\nthe cokernel is $n$. Thus the quotient description of $X_{\\G,n}$\nimplies that the dimension of $X_{\\G,n} $ near $(B_1, B_2, i)$ is\ngiven by\n\\begin{eqnarray*}\n  & & \\dim M(n)^{\\Gamma} +n - 2 \\dim GL_{\\Gamma} (R^n) \\\\\n  & =& (n^2 \\dim  \\mbox{Hom} _{\\Gamma} (R,  {\\Bbb C} ^2 \\otimes R) +n)+ n -2 n^2 \\dim GL_{\\Gamma}\n  (R) \\\\\n  & =& n^2 (2 N) + n + n - 2 n^2 N \\\\\n  & =& 2n,\n\\end{eqnarray*}\nwhich is independent of which component a point $(B_1, B_2, i)$\nis in. Here we\nhave used the fact that the (complex) dimension of $  \\mbox{Hom} _{\\Gamma} (R,\n {\\Bbb C} ^2 \\otimes R)$ is equal to $N$ (cf. Kronheimer \\cite{Kr}).\n\\end{demo}\n\n\\begin{remark}  \\rm\nRecall that the minimal resolution $ \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} $ endowed with certain\nhyper-Kahler structures are called an ALE spaces \\cite{Kr}.\nAccording to Nakajima \\cite{N4}, one can show that the Hilbert\nscheme $ \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^{[n]}$ over an ALE space admits a quiver variety\ndescription in terms of the same quiver data as specified in\nRemark~\\ref{rem_clear} but with a different stability condition,\nby a modification of the proof for the description of the moduli\nspace of vector bundles over an ALE space \\cite{KN}. It follows by\nCorollary 4.2 of \\cite{N} that $ \\ale^{[n]} $ and $X_{\\G,n}$ is\ndiffeomorphic. We conjecture that they are indeed isomorphic as\ncomplex varieties. In this way we would have obtained a morphism\n$\\varphi:  \\C^{2n} \/\/ \\Gn  \\rightarrow  \\ale^{[n]} $ by combining with\nTheorem~\\ref{th_morph}.\n\\end{remark}\n\n\\begin{remark} \\rm\nIt follows readily from the affine algebro-geometric quotient\ndescription of the symmetric product $( {\\Bbb C} ^2)^{(nN)}$\n(Proposition~2.10, \\cite{N2}) that the $\\Gamma$-fixed-point set\n$( {\\Bbb C} ^2)^{(nN),\\Gamma}$ or rather the orbifold $ {\\Bbb C} ^{2n} \/ \\G_n$ (see\nProposition~\\ref{prop_fix}) has the following description (also\ncf. \\cite{VV}, Theorem~1):\n\\[\n {\\Bbb C} ^{2n} \/ \\G_n \\cong \\{(B_1, B_2, i, j) \\in M(n)^{\\Gamma} | [B_1, B_2]\n+ ij =0 \\}\n\/\/GL_{\\Gamma} (R^n).\n\\]\nIt follows from the general theory of quiver varieties that there\nis a natural projective morphism $ \\ale^{[n]}  \\rightarrow  {\\Bbb C} ^{2n} \/ \\G_n$\nwhich is a semismall resolution. We expect that this is the same\nas the semismall resolution $\\tau_n : \\ale^{[n]}  \\rightarrow\n {\\Bbb C} ^{2n}\/\\G_n$ explicitly constructed by the diagram\n(\\ref{eq_mine}). We also expect that the intermediate variety\n$( \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} )^n\/S_n$ (see (\\ref{eq_mine})) can also be identified with a\nquiver variety associated with the same quiver data (as specified\nin Remark~\\ref{rem_clear}) but with a new stability condition. In\nthis case it follows from Corollary 4.2 in \\cite{N} that the fiber\n$ \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^{[n], 0}$ is a lagrangian subvariety in $ \\ale^{[n]} $ and it is\nhomotopy equivalent to $ \\ale^{[n]} $ (compare with\nProposition~\\ref{prop_fiber}).\n\\end{remark}\n\n\\begin{remark} \\rm\nQuiver varieties are connected \\cite{N, N3}. Thus the variety\n$X_{\\G,n}$ can also be defined as the closure of the set of ideals\n$I(T)$ in $( {\\Bbb C} ^2)^{[nN]}$ associated to unordered $n$-tuples $T$\nof distinct $\\Gamma$-orbits in $ {\\Bbb C} ^2 \\backslash 0$.\n\\end{remark}\n\\subsection{Canonical vector bundles}\n  %\nSince $ \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} $ is isomorphic to the Hilbert quotient $ \\C^{2} \/\/ \\G $, there\nexists the tautological vector bundle $\\cal R$ on $ \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} $ of rank\n$N$, whose fiber affords the regular representation of $\\Gamma$ (cf.\n\\cite{GSV, N2}). It decomposes as follows:\n\\begin{eqnarray*}\n{\\cal V} \\cong \\sum_{\\gamma \\in \\Gamma^*} {\\cal R}_{\\gamma} \\bigotimes V_{\\gamma},\n\\end{eqnarray*}\nwhere $V_{\\gamma}$ is the irreducible representation of $\\Gamma$\nassociated to $\\gamma$ and ${\\cal R}_{\\gamma}$ is a vector bundle over\n$ \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} $ of rank equal to $ \\deg \\gamma$ (by definition $\\deg \\gamma = \\dim\nV_{\\gamma}$).\n\nOne can associate a vector bundle $E^{[n]}$ of rank $dn$ on the\nHilbert scheme $ \\ale^{[n]} $ to a rank $d$ vector bundle $E$ over $ \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} $\nas follows: let $U \\subset  \\ale^{[n]}  \\times  \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} $ be the universal\nfamily\n\\begin{eqnarray*}\n\\begin{array}{ccc}\n     U  &  \\stackrel{ p_1}{\\longrightarrow} &  \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma}  \\\\\n p_2 \\downarrow\\quad  &                  & \\\\\n  \\ale^{[n]}  &  &\n\\end{array}\n\\end{eqnarray*}\n$U$ is flat and finite of degree $n$ over $ \\ale^{[n]} $. Then $E^{[n]}$\nis defined to be $(p_2)_* p_1^* E$. In this way we obtain\ncanonical vector bundles ${\\cal R}^{[n]}$ and ${\\cal\nR}^{[n]}_{\\gamma}$ $(\\gamma \\in \\Gamma^*)$ over $ \\ale^{[n]} $ associated to $\\cal R$\nand ${\\cal R}_{\\gamma}$ above.\n\nThere exists a tautological vector bundle ${\\cal R}^{\\{n \\} }$ of\nrank $nN$ over $X_{\\G,n}$ (and thus over $ \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^{[n]}$) induced from\nthe inclusion $X_{\\G,n} \\subset ( {\\Bbb C} ^2)^{[nN]}$. The group $\\Gamma$ acts\non ${\\cal R}^{\\{n \\} }$ fiberwise such that each fiber as a\n$\\Gamma$-module is isomorphic to $R^n$. Then we have a decomposition:\n\\begin{eqnarray*}\n{\\cal R}^{\\{n \\} }\n = \\bigoplus_{\\gamma \\in \\Gamma^*} {\\cal R}^{\\{n \\} }_{\\gamma} \\bigotimes V_{\\gamma},\n\\end{eqnarray*}\nwhere ${\\cal R}^{\\{n \\} }_{\\gamma}$ is a vector bundle over\n$ \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^{[n]}$ of rank equal to $n \\deg \\gamma$.\n\nThe principle bundle\n\\begin{eqnarray*}\n\\widetilde{H} (K) \\cap M^{\\Gamma}(n) \\stackrel{\/ G}{\\longrightarrow}\nX_{\\G,n} \\end{eqnarray*}\n  %\n(which follows from Theorem~\\ref{th_quiv}) gives rise to various\ncanonical vector bundles associated to canonical representations\nof $G \\equiv GL_{\\Gamma} (R^n) \\cong GL_n ( {\\Bbb C} ) \\times GL_{\\Gamma} (R)$.\nFor example, the vector bundle associated to the representation\n$GL_{\\Gamma} (R^n) \\hookrightarrow GL (R^n)$ is exactly the above\ntautological vector bundle ${\\cal R}^{\\{n \\} }$ over $X_{\\G,n}$ (or\nrather over $ \\ale^{[n]} $); the one associated to $GL_{\\Gamma} (R^n)\n\\rightarrow G L ( \\gamma^n)$ is $ {\\cal R}^{\\{n \\} }_{\\gamma}$.\n\nIn the remainder of this section we assume the validity of\nConjecture~\\ref{conj_main} that $  \\C^{2n} \/\/ \\Gn  $ is isomoprhic to\n$ \\ale^{[n]} $. An immediate corollary is the existence of a tautological\nbundle $\\cal V$ on $ \\ale^{[n]} $ whose fiber affords the regular\nrepresentation of $\\G_n$. This comes from the tautological bundle\nover the Hilbert quotient $ \\C^{2n} \/\/ \\Gn $. It is well known (cf. e.g.\n\\cite{M, Z}) that the irreducible representations $S_{\\rho}  $ of\n$\\G_n$ are parameterized by the set ${\\cal P}_n (\\Gamma^*)$ of\npartition-valued functions $\\rho$ of weight $n$ on the set $\\Gamma^*$\nof irreducible characters of $\\Gamma$. It follows that one has a\ndecomposition\n\\begin{eqnarray} \\label{eq_wreathreg}\n {\\cal V} = \\bigoplus_{\\rho \\in {\\cal\nP}_n (\\Gamma^*)} S_{\\rho} \\bigotimes {\\cal V}_{\\rho},\n\\end{eqnarray}\n   %\nwhere ${\\cal V}_{\\rho}$ is a vector bundle on $ \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^{[n]}$ of rank\nequal to $\\dim S_{\\rho} $. The vector bundles ${\\cal V}_{\\rho}$\nare expected to be a basis for the K-group of $ \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^{[n]}$.\n\nWe denote by ${\\cal R}^{\\langle n \\rangle }$ the subbundle of\n$\\cal V$ which is given by the (fiberwise) $\\Gamma_{n -1}$ invariants\nof $\\cal V$. Since the first copy $\\Gamma$ in $\\Gamma^n \\subset \\G_n$\ncommutes with $\\Gamma_{n -1}$, ${\\cal R}^{\\langle n \\rangle }$ has a\n$\\Gamma$ action fiberwise such that each fiber affords the $\\Gamma$-module\n$R^n$ (cf. Subsect.~\\ref{subsect_link}). It decomposes as\n\\begin{eqnarray*}\n{\\cal R}^{\\langle n \\rangle } = \\bigoplus_{\\gamma \\in \\Gamma^*} {\\cal\nR}^{\\langle n \\rangle }_{\\gamma} \\bigotimes V_{\\gamma}.\n\\end{eqnarray*}\nwhere $ {\\cal R}^{\\langle n \\rangle }_{\\gamma}$ is a vector bundle\nover $ \\ale^{[n]} $ of rank $n \\deg \\gamma$.\n\nWe define the reduced universal scheme $ U_{\\G,n} $ as the reduced\nfibered product\n\\begin{eqnarray}  \\label{eq_univ}\n\\begin{array}{ccc}\n      U_{\\G,n}   & {\\longrightarrow} &  {\\Bbb C} ^{2n} \\\\\n  \\downarrow &                  & \\downarrow \\\\\n  \\ale^{[n]}  & \\stackrel{ \\tau_n}{\\longrightarrow} &  {\\Bbb C} ^{2n} \/\\G_n .\n\\end{array}\n\\end{eqnarray}\nUnder the isomorphism $\\varphi:  \\C^{2n} \/\/ \\Gn  \\rightarrow  \\ale^{[n]} $, the\nuniversal schemes ${\\cal U}_{\\G, n} ,  U_{\\G,n} $ defined respectively by\n(\\ref{eq_quotuniv}) and (\\ref{eq_univ}) can be identified. The\nfollowing proposition follows now from the way we define $\\varphi$\n(cf. Subsect.~\\ref{subsect_link}).\n\n\\begin{proposition}\nThere is a natural identification between the vector bundles\n${\\cal R}^{[n]}$ and ${\\cal R}^{\\langle n \\rangle }$, respectively\n${\\cal R}^{[n]}_{\\gamma}$ and ${\\cal R}^{\\langle n \\rangle }_{\\gamma}$.\n\\end{proposition}\n\\section{On the equivalence of two forms of McKay correspondence}\n\\label{sect_mckay}\n\\subsection{A weighted bilinear form}\nIn this subsection we recall the notion of a {\\em weighted}\nbilinear form on $R(\\G_n)$ introduced in \\cite{FJW}.\n\nThe standard bilinear form on $R(\\Gamma )$ is defined as follows:\n\n\\begin{eqnarray*}\n\\langle f, g \\rangle_{\\Gamma} = \\frac1{ | \\Gamma |}\\sum_{x \\in \\Gamma}\n          f(x) g(x^{ -1}).\n\\end{eqnarray*}\nWe will write $\\langle \\ , \\ \\rangle$ for $\\langle \\ , \\\n\\rangle_{\\Gamma }$ when no ambiguity may arise.\n\nLet us fix a virtual character $  \\xi  \\in R(\\Gamma)$. The\nmultiplication in $ R(\\Gamma)$ corresponding to the tensor product of\ntwo representations will be denoted by $* $. Recall that $\\gamma_0,\n\\gamma_1, \\ldots, \\gamma_r$ are all the inequivalent irreducible\ncharacters of $\\Gamma$ and $\\gamma_0$ denotes the trivial character. We\ndenote by $a_{ij} \\in  {\\Bbb Z} $ the (virtual) multiplicities of $\\gamma_j$\nin $  \\xi  * \\gamma_i $, i.e. $  \\xi   *  \\gamma_i = \\sum_{j =0}^r a_{ij}\n\\gamma_j.$ We denote by $A$ the $ (r +1) \\times (r +1)$ matrix $ (\na_{ij})_{0 \\leq i,j \\leq r}$.\n\nWe introduce the following {\\em weighted bilinear form} on\n$R(\\Gamma)$:\n $$\n  \\langle f, g \\rangle_{ \\xi  } = \\langle  \\xi  * f ,  g \\rangle_{\\Gamma },\n   \\quad f, g \\in R( \\Gamma).\n$$\nIt follows that $\\langle \\gamma_i, \\gamma_j \\rangle_{ \\xi  } = a_{ij}.$\n\nThroughout this paper we will always assume that $ \\xi  $ is a {\\em\nself-dual}, i.e. $ \\xi  (x) =  \\xi  (x^{-1}), x \\in \\Gamma$. The\nself-duality of $ \\xi $ implies that $ a_{ij} = a_{ji}, $ i.e. $A$\nis a symmetric matrix.\n\nGiven a representation $V$ of $\\Gamma$ with character $\\gamma \\in R(\\Gamma)$,\nthe $n$-th outer tensor product $V^{ \\otimes  n} $ of $V$ can be\nregarded naturally as a representation of the wreath product $\\G_n$\nwhose character will be denoted by $\\eta_n ( \\gamma )$: the direct\nproduct $\\Gamma^n$ acts on $\\gamma^{\\otimes n}$ factor by factor while\n$S_n$ by permuting the $n$ factors. Denote by $\\varepsilon_n$ the\n(1-dimensional) sign representation of $\\G_n$ on which $\\Gamma^n$ acts\ntrivially while $S_n$ acts as sign representation. We denote by\n$\\varepsilon_n ( \\gamma ) \\in R(\\G_n)$ the character of the tensor\nproduct of $\\varepsilon_n$ and $V^{\\otimes n}$.\n\nWe may extend naturally $\\eta_n$ to a map from $R(\\Gamma)$ to\n$R(\\G_n)$. In particular, if $\\beta$ and $\\gamma $ are characters of\n$\\Gamma$, then\n\\begin{eqnarray}  \\label{eq_virt}\n  \\eta_n (\\beta - \\gamma) =\n  \\sum_{m =0}^n ( -1)^m \\mbox{Ind}_{\\Gamma_{n -m} \\times \\Gamma_m }^{\\G_n}\n   [ \\eta_{n -m} (\\beta) \\otimes \\varepsilon_m (\\gamma ) ] .\n\\end{eqnarray}\n\nWe define a {\\em weighted bilinear form} on $R( \\G_n)$ by\nletting $$\n  \\langle  f, g\\rangle_{ \\xi , \\G_n } =\n   \\langle \\eta_n ( \\xi  ) * f, g \\rangle_{\\G_n} ,\n   \\quad f, g \\in R( \\G_n).\n$$ One can show that the bilinear form $\\langle \\ , \\\n\\rangle_{ \\xi , \\G_n}$ is symmetric.\nA symmetric bilinear form on $R_{\\G}  = \\bigoplus_{n} R(\\G_n)$ is\nthen given by\n\\[\n\\langle u, v \\rangle_{ \\xi }\n = \\sum_{ n \\geq 0} \\langle u_n, v_n \\rangle_{ \\xi , \\G_n } ,\n\\]\nwhere $u = \\sum_n u_n$ and $v = \\sum_n v_n$ with $u_n, v_n \\in\n\\G_n$.\n\nWe further specialize to the case when $\\Gamma$ is a finite subgroup\nof $SL_2 ( {\\Bbb C} )$ by an embedding $\\pi$, and fix the virtual\ncharacter $ \\xi $ of $\\Gamma$ to be $$\n \\lambda (\\pi) \\equiv \\sum_{i=0}^2 (-1)^i \\Lambda^i \\pi =2\\gamma_0 - \\pi.\n$$ where $\\Lambda^i$ denotes the $i$-th exterior power. We\nconstruct a diagram with vertices corresponding to elements $\\gamma_i$\nin $\\Gamma^*$ and we draw one edge (resp. two edges) between the\n$i$-th and $ j$-th vertices if $a_{ij} = -1$ (resp. $-2$).\nAccording to McKay \\cite{Mc}, the associated diagram can be\nidentified with affine Dynkin diagram of ADE type and the matrix\n$A$ is the corresponding affine Cartan matrix. It is shown in\n\\cite{FJW} that the weighted bilinear form on $R_{ {\\Bbb Z} }(\\G_n)$ is\nsemipositive definite symmetric.\n\\subsection{Identification of two virtual characters}\nIn this subsection we set $\\Gamma$ to be an arbitrary (not necessarily\nfinite) subgroup of $GL_k( {\\Bbb C} )$ unless otherwise specified. We\ndenote by $\\pi$ the $k$-dimensional defining representation of\n$\\Gamma$ for the embedding.\n\nThe wreath product $\\G_n$ acts naturally on $ {\\Bbb C} ^{kn} =( {\\Bbb C} ^k)^n$ by\nletting $\\Gamma^n$ act factor-wise and $S_n$ act as permutations of\n$n$-factors. We denote by $\\lambda ( {\\Bbb C} ^{kn})$ the virtual\ncharacter $\\sum_{i =0}^{kn} ( -1)^i \\Lambda^i  {\\Bbb C} ^{kn}$ of $\\G_n$,\nwhere the $i$-th exterior power $ \\Lambda^i  {\\Bbb C} ^{kn}$ carries an\ninduced $\\G_n$-action. The geometric significance of $\\lambda\n( {\\Bbb C} ^{kn})$ will become clear later. Let $\\eta_n (\\lambda (\\pi))$\nbe the virtual character of $\\G_n$ built on the $n$-th tensor of\nthe virtual character $\\lambda (\\pi ) = \\sum_{i =0}^k (-1)^i\n\\Lambda^i  {\\Bbb C} ^k$ of $\\Gamma$.\n  %\n\\begin{theorem} \\label{th_key}\n The virtual characters $\\lambda ( {\\Bbb C} ^{kn})$ and\n $\\eta_n (\\lambda (\\pi))$ of $\\G_n$ are equal.\n\\end{theorem}\n\n\\begin{demo}{Proof}\n Given $x \\times y \\in \\G_n$, where\n $x \\in \\Gamma_m$ and $y \\in \\Gamma_{n -m}$ for some $m$, we have by definition\n\\begin{eqnarray}   \\label{eq_mult}\n \\eta_n (\\lambda (\\pi) ) (x \\times y)\n   & =& \\eta_m (\\lambda (\\pi)) (x)\\; \\eta_{n-m} (\\lambda (\\pi))(y),\n            \\nonumber   \\\\\n  \\lambda ( {\\Bbb C} ^{kn}) (x \\times y)\n   & =& \\lambda ( {\\Bbb C} ^{km}) (x)\\; \\lambda ( {\\Bbb C} ^{k(n -m)})(y).\n\\end{eqnarray}\nThus it suffices to show that the character values $ \\eta_n\n(\\lambda (\\pi) ) (\\alpha, s)$ and $\\lambda ( {\\Bbb C} ^{kn}) (\\alpha, s)$\nare equal for $(\\alpha, s) \\in \\G_n,$ where $\\alpha =(g, 1, \\ldots,\n1) \\in \\Gamma^n$ and $s$ is an $n$-cycle, say $s = (12 \\ldots n)$. It\nis known (cf. \\cite{FJW}) that the character value of $\\eta_n\n( \\xi )(\\alpha, s)$ is $ \\xi  ( c)$, where $ \\xi $ is a class function\nof $\\Gamma$ and $c$ is the conjugacy class of $g$. In particular\n    %\n\\begin{eqnarray*}\n\\eta_n (\\lambda (\\pi) )(\\alpha, s) = \\lambda (\\pi) (c).\n\\end{eqnarray*}\n\nDenote by $x^i_a, a =1, \\ldots, n, i = 1, \\ldots, k$ the\ncoordinates in $ {\\Bbb C} ^{kn} = ( {\\Bbb C} ^k)^n.$ The $a$-th factor $\\Gamma$ in\n$\\Gamma^n \\subset \\G_n$ acts on $ {\\Bbb C} ^k$ with coordinates $x^i_a, i =1,\n\\ldots, k$.\n\nConsider the exterior monomial basis for $\\Lambda^i  {\\Bbb C} ^{kn}$.\nGiven such a monomial $X$, we first observe that the coefficient\nof $X$ in $(\\alpha, s) .X$ is $0$ unless there are equal numbers\nof lower subscripts $1, 2, \\ldots, n$ for $x^i_a$ appearing in\n$X$. It follows that\n\\begin{eqnarray}  \\label{eq_vanish}\n\\Lambda^i  {\\Bbb C} ^{kn} (\\alpha, s) = \\mbox{trace }(\\alpha, s)\n\\mid_{\\Lambda^i  {\\Bbb C} ^{kn}} = 0, \\quad \\mbox{if } i \\mbox{ is not\ndivisible by } n.\n\\end{eqnarray}\n\nFor $i =m n, 1\\leq m \\leq k$, we further observe that the\ncoefficient of $X$ in $(\\alpha, s) .X$ is $0$ unless the monomial\n$X$ is of the form\n\\begin{eqnarray*}\n X(i_1 \\ldots i_m) & =& x_1^{i_1} \\wedge x_2^{i_1} \\wedge \\ldots\n   \\wedge x_n^{i_1} \\wedge \\\\\n   & & x_1^{i_2} \\wedge x_2^{i_2} \\wedge \\ldots\n   \\wedge x_n^{i_2} \\wedge \\ldots \\wedge x_1^{i_m} \\wedge \\ldots\n   \\wedge x_n^{i_m},\n\\end{eqnarray*}\nwhere $\\{ i_1, \\ldots, i_m \\}$ is an unordered $m$-tuple of\ndistinct numbers among $1, 2, \\ldots, k$. Write $ \\pi( g) x^{i}_1\n= \\sum_j b_{ij} x^j_1 $ and denote by $B$ the $k \\times k$ matrix\n$(b_{ij})$. Then\n\\begin{eqnarray*}\n (\\alpha, s). X(i_1 \\ldots i_m)\n & =& x_2^{i_1} \\wedge x_3^{i_1} \\wedge \\ldots\n   \\wedge x_n^{i_1} \\wedge \\sum_j b_{i_1 j} x_1^j \\wedge \\\\\n   & &  x_2^{i_2} \\wedge x_3^{i_2} \\wedge \\ldots\n   \\wedge x_n^{i_2} \\wedge \\sum_j b_{i_2 j} x_1^j\n \\wedge \\ldots \\wedge \\\\\n & & x_2^{i_m} \\wedge x_3^{i_m} \\wedge \\ldots\n   \\wedge x_n^{i_m} \\wedge \\sum_j b_{i_m j} x_1^j .\n\\end{eqnarray*}\n\nIt follows that the coefficient of $X(i_1 \\ldots i_m)$ in\n$(\\alpha, s). X(i_1 \\ldots i_m)$ is equal to\n\\begin{eqnarray}  \\label{eq_det}\n && \\sum_{\\sigma} (-1)^{(n -1) m}\n(-1)^{l (\\sigma)} b_{i_1 \\sigma (i_1)}b_{i_2 \\sigma (i_2)} \\ldots\nb_{i_r \\sigma (i_m)}    \\nonumber \\\\\n & =&  (-1)^{(n -1) m} \\det B(i_1 \\ldots i_m),\n\\end{eqnarray}\nwhere the summation runs over all permutations $\\sigma$ of $i_1,\n\\ldots, i_m$, $l (\\sigma) $ is the length of $\\sigma$, and $\\det\nB(i_1 \\ldots i_m)$ denotes the determinant of the $m \\times m$\nminor of $A$ consisting of the rows and columns $i_1, \\ldots,\ni_m$.\n\nBy (\\ref{eq_vanish}) and (\\ref{eq_det}) we calculate that\n\\begin{eqnarray*}   \\label{eq_two}\n\\lambda ( {\\Bbb C} ^{kn}) (\\alpha, s)\n & =& \\sum_{m =0}^k (-1)^{mn}\\Lambda^{mn} ( {\\Bbb C} ^{kn})(\\alpha, s) \\nonumber \\\\\n & =& \\sum_{m =0}^k (-1)^{mn} (-1)^{m(n -1)} \\sum_{ \\{i_1, \\ldots , i_m\\} }\n  \\det B(i_1 \\ldots i_m) \\nonumber \\\\\n & =& \\sum_{m =0}^k (-1)^m e_m (t_1, \\ldots, t_k) \\nonumber   \\\\\n & =& \\sum_{m =0}^k (-1)^m (\\Lambda^m \\pi) (g) \\nonumber   \\\\\n & =& \\lambda (\\pi) (c ),\n \\end{eqnarray*}\nwhere the summation runs over all unordered $r$-tuples $ \\{i_1,\n\\ldots , i_m\\}$ of distinct numbers among $1, 2, \\ldots, k$, and\n$e_m$ denotes the $m$-th elementary symmetric polynomial of the\neigenvalues $t_1, \\ldots, t_k$ of the matrix $\\pi (g) \\in SL_k\n( {\\Bbb C} )$. The two identities involving $e_r$ used above are well\nknown.\n\nBy comparing the character values $\\eta_n (\\lambda (\\pi) )(\\alpha,\ns)$ and $\\lambda ( {\\Bbb C} ^{kn})(\\alpha, s) $ calculated above, we see\nthat\n\\[\n\\eta_n (\\lambda (\\pi) )(\\alpha, s) =\\lambda ( {\\Bbb C} ^{kn})(\\alpha, s) =\n\\lambda (\\pi) (c ).\n\\]\n\n\nTherefore if $x \\in \\G_n$ is of type $\\rho \\in {\\cal P}_n (\\Gamma_*)$\nthen by using (\\ref{eq_mult}) we obtain that\n \\begin{eqnarray*}\n  \\eta_n (\\lambda (\\pi)  ) ( x)\n    & =& \\prod_{c\\in \\Gamma_*} \\lambda (\\pi) (c)^{l (\\rho(c))} \\label{eq_term}\\\\\n    & =& \\lambda ( {\\Bbb C} ^{kn}) (x).\n \\end{eqnarray*}\n This completes the proof.\n\\end{demo}\n\n\\begin{remark}  \\rm\nThe identification of the two virtual characters can be also seen\nalternatively as follows:\n\\begin{eqnarray}\n\\lambda ( {\\Bbb C} ^{kn}) & =& \\sum_{i =0}^{kn} (-1)^i \\Lambda ( ( {\\Bbb C} ^k)^n )\n \\nonumber   \\\\\n & =&  \\sum_{i_1, \\ldots , i_n } (-1)^{i_1 + \\ldots + i_n}\n \\Lambda^{i_1} ( {\\Bbb C} ^k) \\otimes \\ldots \\otimes \\Lambda^{i_n} ( {\\Bbb C} ^k)\n  \\nonumber  \\\\\n & =& \\sum_{\\{n_0, \\ldots , n_k \\}} (-1)^{\\sum_i i n_i}\n  \\mbox{Ind}^{\\G_n}_{\\Gamma_{n_0} \\times \\ldots \\times \\Gamma_{n_k}}\n  \\Lambda^0 ( {\\Bbb C} ^k)^{\\boxtimes n_0} \\otimes \\ldots \\otimes\n  \\Lambda^k ( {\\Bbb C} ^k)^{\\boxtimes n_k} \\label{eq_def}  \\\\\n & =& (\\sum_i (-1)^i \\Lambda^i ( {\\Bbb C} ^k) )^{\\boxtimes n}   \\label{eq_work}\\\\\n & =& (\\lambda ( {\\Bbb C} ^k) )^{\\boxtimes n}  \\nonumber\n\\end{eqnarray}\n   %\nwhere $\\{n_0, \\ldots , n_k \\}$ ranges over the $(k+1)$-tuple of\nnon-negative integers such that $ \\sum_i n_i = n.$\nEq.~(\\ref{eq_def}) above basically follows from the definition of\nan induction functor. Eq.~(\\ref{eq_work}) is a generalization of\n(\\ref{eq_virt}) which can be established with some effort.\n\\end{remark}\n\nWhen $\\Gamma$ is trivial then $\\lambda (\\pi) = 0 \\in R(\\Gamma)$ and so $\\eta_n\n(\\lambda (\\pi)) =0$. We have an immediate corollary.\n    %\n\\begin{corollary}\nWhen $\\Gamma$ is trivial and $\\G_n$ becomes the symmetric group $S_n$,\nthe virtual $S_n$-character $\\lambda ( {\\Bbb C} ^{kn})$ is zero.\n\\end{corollary}\n\\subsection{Derived categories and Grothendieck Groups}\nIn this subsection we let $\\Gamma $ be a finite subgroup of $SL_2\n( {\\Bbb C} )$ unless otherwise specified.\n\nWe denote by $D_{\\G_n} ( {\\Bbb C} ^{2n})$ the bounded derived category of\n$\\G_n$-equivariant coherent sheaves on $ {\\Bbb C} ^{2n}$, and denote by\n$D( \\ale^{[n]} )$ the bounded derived category of coherent sheaves on\n$ \\ale^{[n]} $. Define two functors $\\Phi : D( \\ale^{[n]} )  \\rightarrow D_{\\G_n}\n( {\\Bbb C} ^{2n})$ and $\\Psi : D_{\\G_n} ( {\\Bbb C} ^{2n}) \\rightarrow D( \\ale^{[n]} ) $ by\n\\begin{eqnarray*}\n \\Phi (-)  & =& Rp_* ({\\cal O}_{ U_{\\G,n} } \\bigotimes q^*(-))\\\\\n \\Psi (-)  & =& (Rq_* RHom({\\cal O}_{ U_{\\G,n} }, p^* (-)) )^{\\G_n},\n\\end{eqnarray*}\nwhere $ U_{\\G,n} $ is the universal scheme defined in (\\ref{eq_univ}),\nand $p, q$ denote the projections $ \\ale^{[n]}  \\times  {\\Bbb C} ^{2n}$ to\n$ {\\Bbb C} ^{2n}$ and $ \\ale^{[n]} $ respectively.\n\nTake a basis of $R(\\G_n)$ given by the irreducible characters\n$s_{\\rho}, \\rho \\in {\\cal P}_n (\\Gamma^*)$ of $\\G_n$ (cf. \\cite{M, Z}).\nWe denote by ${\\cal O}_{ {\\Bbb C} ^{2n}}$ the structure sheaf over\n$ {\\Bbb C} ^{2n}$. Recall that the vector bundle (i.e. locally free sheaf)\n${\\cal V}_{\\rho}$ is defined in (\\ref{eq_wreathreg}). The\nfollowing theorem can be derived by using a general theorem due to\nBridgeland, King and Reid (Theorem~1.2, \\cite{BKR}) since $\\tau_n\n:  \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^{[n]} \\rightarrow  {\\Bbb C} ^{2n} \/\\G_n$ is a crepant resolution and\n$\\G_n$ preserves the symplectic structure of $ {\\Bbb C} ^{2n}$.\n\n\\begin{theorem}\nUnder the assumption of the validity of\nConjecture~\\ref{conj_main}, $\\Phi$ is an equivalence of categories\nand $\\Psi$ is its adjoint functor. In particular, $\\Psi$ sends\n${\\cal O}_{ {\\Bbb C} ^{2n}} \\otimes s_{\\rho}^{\\vee}$ to ${\\cal V}_{\\rho}$.\n\\end{theorem}\n\n\\begin{remark} \\rm\nWhen $n=1 $, Conjecture~\\ref{conj_main} is known to be true (cf.\nRemark~\\ref{rem_one}) and the above theorem was established by\nKapranov and Vasserot \\cite{KV}.\n\\end{remark}\n\n\\begin{remark} \\rm \\label{rem_alt}\nBy assuming the validity of the $n!$ conjecture (cf. \\cite{GH}),\nwe have by Remark~\\ref{rem_one} that $ \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^n\/\/S_n \\cong  \\ale^{[n]} $.\nThus we may replace $ \\ale^{[n]} $ by $ \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^n \/\/ S_n$ in the crepant\nHilbert-Chow resolution $ \\ale^{[n]}  \\stackrel{\\pi_n}{\\rightarrow}\n \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^n \/ S_n$. Clearly $S_n$ preserves the holomorphic symplectic\nstructure of the Cartisian product $ \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^n$. Then one can apply\nagain Theorem~1.2 in \\cite{BKR} to show that there is an\nequivalence of derived categories between $D( \\ale^{[n]} )$ and $D_{S_n}\n( \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^n)$ of $S_n$-equivariant coherent sheaves of $ \\ale^{[n]} $.\n\\end{remark}\n\nBelow we further specialize and apply the general results of\nBridgeland, King and Reid \\cite{BKR} to our setup. We denote by\n$D^0_{\\G_n} ( {\\Bbb C} ^{2n})$ the full subcategory of $D_{\\G_n} ( {\\Bbb C} ^{2n})$\nconsisting of objects whose cohomology sheaves are concentrated on\nthe origin of $ {\\Bbb C} ^{2n}$. We denote by $D^0_{S_n} ( \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^n)$ the\nfull subcategory of $D_{S_n} ( \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^n)$ consisting of objects whose\ncohomology sheaves are concentrated on the $n$-th Cartisian\nproduct of the exceptional divisor $D \\subset  \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} $. Denote by\n$D^0( \\ale^{[n]} )$ the full subcategory of $D( \\ale^{[n]} )$ consisting of\nobjects whose cohomology sheaves are concentrated on the fiber\n$ \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^{[n], 0}$ of $ \\ale^{[n]} $. Recall that $ \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^{[n], 0}$ is\ndescribed in Subsection~\\ref{subsect_fiber}.\n\nThe equivalence between the derived categories $D_{\\G_n} ( {\\Bbb C} ^{2n})$\nand $D( \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^{[n]})$ induces an equivalence between $D^0_{\\G_n}\n( {\\Bbb C} ^{2n})$ and $D^0( \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^{[n]})$. Thus we have the following\ncommutative diagram (under the assumption of the validity of\nConjecture~\\ref{conj_main}):\n     %\n\\begin{eqnarray}   \\label{eq_cat}\n \\begin{array}{ccc}\n   D^0_{\\G_n}( {\\Bbb C} ^{2n})   & \\stackrel{\\simeq}{\\longrightarrow}\n      & D^0( \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^{[n]})   \\\\\n   \\downarrow  &                 & \\downarrow   \\\\\n   D_{\\G_n}( {\\Bbb C} ^{2n})     & \\stackrel{\\simeq}{\\longrightarrow}\n      & D( \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^{[n]}) .\n \\end{array}\n\\end{eqnarray}\n    %\nGiven objects $E, F$ in $D_{\\G_n}( {\\Bbb C} ^{2n})$ and\n$D^0_{\\G_n}( {\\Bbb C} ^{2n})$ respectively, we define the Euler\ncharacteristic\n\\begin{eqnarray*}\n\\chi^{\\G_n} (E, F) = \\sum_i ( -1)^i \\dim Hom_{D_{\\G_n}( {\\Bbb C} ^{2n})} (E,\nF[i]).\n\\end{eqnarray*}\nThis gives a natural bilinear pairing between $D_{\\G_n} ( {\\Bbb C} ^{2n})$\nand $D^0_{\\G_n} ( {\\Bbb C} ^{2n})$. Similarly we can define the Euler\ncharacteristic $\\chi (A, B)$ for objects $A, B$ in $D( \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^{[n]})$\nand $D^0( \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^{[n]})$ respectively, which gives rise to a bilinear\npairing between $D( \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^{[n]})$ and $D^0( \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^{[n]})$. We further\nhave $\\chi^{\\G_n}(E, F) =\\chi (\\Psi (E), \\Psi (F)),$ cf.\n\\cite{BKR}.\n\nWe denote by $K_{\\G_n} ( {\\Bbb C} ^{2n}), K^0_{\\G_n} ( {\\Bbb C} ^{2n}), K\n( \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^{[n]})$, $ K^0 ( \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^{[n]})$, $K^0_{S_n} ( \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^n)$ and\n$K_{S_n} ( \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^n)$ the Grothendieck groups of the corresponding\nderived categories. It is well known that $K_{\\G_n}\n( {\\Bbb C} ^{2n})$ and $ K^0_{\\G_n} ( {\\Bbb C} ^{2n})$ are both isomorphic to the\nrepresentation ring $R_{ {\\Bbb Z} }( \\G_n)$. The bilinear pairings\nmentioned above together with the embeddings of categories induces\na bilinear form on $K^0_{\\G_n} ( {\\Bbb C} ^{2n})$ and respectively on\n$K^0( \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^{[n]})$.\n\nLet ${\\cal O}_0$ be the skyscraper sheaf at the origin $0$ on\n$ {\\Bbb C} ^{2n}$. The $\\G_n$-bundles ${\\cal O} \\otimes s_{\\rho}, \\rho \\in\n{\\cal P}_n (\\Gamma^*)$ form a basis for $K_{\\G_n} ( {\\Bbb C} ^{2n})$ while the\nmodules $s_{\\rho} \\otimes {\\cal O}_0, \\rho \\in {\\cal P}_n (\\Gamma^*)$\nform the dual basis for $K^0_{\\G_n} ( {\\Bbb C} ^{2n})$.\n    %\n\\begin{theorem} \\label{th_reform}\nThe map by sending $s_{\\rho}$ to ${\\cal O}_0 \\otimes s_{\\rho}$,\n$\\rho \\in {\\cal P}_n (\\Gamma^*)$, is an isometry between $R_{ {\\Bbb Z} }(\\G_n)$\nendowed with the weighted bilinear form and $K^0_{\\G_n} ( {\\Bbb C} ^{2n})$\nendowed with the bilinear form defined above. In particular the\nbilinear form on $K^0_{\\G_n} ( {\\Bbb C} ^{2n})$ is semipositive definite\nsymmetric.\n\\end{theorem}\n\n\\begin{demo}{Proof}\nFollowing a similar argument as in Gonzalez-Sprinberg and Verdier\n\\cite{GSV} (which is for $n =1$), we obtain the following\ncommutative diagram by using the Koszul resolution of ${\\cal O}_0$\non $ {\\Bbb C} ^{2n}$:\n\\begin{eqnarray*}\n \\begin{array}{ccc}\n   R_{ {\\Bbb Z} }(\\G_n)    & \\stackrel{\\simeq}{\\longrightarrow}\n     &  K^0_{\\G_n} ( {\\Bbb C} ^{2n} )     \\\\\n     \\jmath \\downarrow  &                 & \\downarrow  \\\\\n    R_{ {\\Bbb Z} }(\\G_n)    & \\stackrel{\\simeq}{\\longrightarrow}\n     &  K_{\\G_n} ( {\\Bbb C} ^{2n} ).\n \\end{array}\n\\end{eqnarray*}\n    %\nHere the horizontal maps are isomorphisms given by sending\n$s_{\\rho}$ to ${\\cal O}_0 \\otimes s_{\\rho}$ and respectively to\n${\\cal O}_{ {\\Bbb C} ^{2n}} \\otimes s_{\\rho}$, $\\rho \\in {\\cal P}_n\n(\\Gamma^*)$. The left vertical map $ \\jmath $ is given by\nmultiplication by the virtual character $\\lambda ( {\\Bbb C} ^{2n})$ of\n$\\G_n$, and the right vertical one is induced from the natural\nembedding of the corresponding categories. Now the theorem follows\nfrom the definition of the weighted bilinear form on $R(\\G_n)$,\nTheorem~\\ref{th_key}, and the fact that the basis ${\\cal O}_0\n\\otimes s_{\\rho}, \\rho \\in {\\cal P}_n (\\Gamma^*)$ for $K^0_{\\G_n}\n( {\\Bbb C} ^{2n})$ is dual to the basis ${\\cal O}_{ {\\Bbb C} ^{2n}} \\otimes\ns_{\\rho}, \\rho \\in {\\cal P}_n (\\Gamma^*)$ for $K_{\\G_n} ( {\\Bbb C} ^{2n})$.\n\\end{demo}\n\n\\begin{remark} \\rm\nThe main results in \\cite{FJW} (see Theorem 7.2 and Theorem 7.3 in\n{\\em loc. cit.}) can be now formulated by using the space\n\\[\n{\\cal F}_{\\Gamma} = \\bigoplus_{n \\geq 0}  K^0_{\\G_n} ( {\\Bbb C} ^{2n})\n\\bigotimes  {\\Bbb C}  [K^0_{\\Gamma}( {\\Bbb C} ^2 )]\n\\]\nwith its natural bilinear form induced from the Koszul-Thom class.\nHere $  {\\Bbb C}  [ - ]$ denotes the group algebra. Roughly speaking,\n${\\cal F}_{\\Gamma}$ affords a vertex representation of the toroidal\nLie algebra and a distinguished subspace of ${\\cal F}_{\\Gamma}$\naffords the basic representation of the affine Lie algebra\n$\\widehat{\\frak g}$ whose associated affine Dynkin diagram corresponds to\n$\\Gamma$ in the sense of McKay. This may be viewed as a form of McKay\ncorrespondence relating finite subgroups of $SL_2 ( {\\Bbb C} )$ to affine\nand toroidal Lie algebras.\n\\end{remark}\n\nNow we have the following commutative diagram (assuming the\nvalidity of Conjecture~\\ref{conj_main}):\n\\begin{eqnarray*}\n \\begin{array}{ccccc}\n   R_{ {\\Bbb Z} }(\\G_n)    & \\stackrel{\\simeq}{\\longrightarrow}\n     &  K^0_{\\G_n} ( {\\Bbb C} ^{2n} )   & \\stackrel{\\simeq}{\\longrightarrow}\n     &   K^0 ( \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^{[n]} ) \\\\\n     \\jmath \\downarrow  &   & \\downarrow &     & \\downarrow  \\\\\n    R_{ {\\Bbb Z} }(\\G_n)    & \\stackrel{\\simeq}{\\longrightarrow}\n     &  K_{\\G_n} ( {\\Bbb C} ^{2n} ) & \\stackrel{\\simeq}{\\longrightarrow}\n     & K ( \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^{[n]} ).\n \\end{array}\n\\end{eqnarray*}\n    %\nBy Remark~\\ref{rem_alt} and a similar argument as above, we have\nanother commutative diagram (assuming the validity of\nConjecture~\\ref{conj_main}):\n\\begin{eqnarray*}\n \\begin{array}{ccc}\n     K^0_{S_n} ( \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^n )   & \\stackrel{\\simeq}{\\longrightarrow}\n     &   K^0 ( \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^{[n]} ) \\\\\n      \\downarrow &     & \\downarrow  \\\\\n      K_{S_n} ( \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^n) & \\stackrel{\\simeq}{\\longrightarrow}\n     & K ( \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^{[n]} ).\n \\end{array}\n\\end{eqnarray*}\n\nCombining the two diagrams above, we have obtained the following\ntheorem.\n\n\\begin{theorem}\nUnder the assumption of the validity of\nConjecture~\\ref{conj_main}, the isomorphisms among $(R_{ {\\Bbb Z} }(\\G_n),\n\\langle -, - \\rangle_{\\lambda( {\\Bbb C} ^2)} )$, the K-groups $K^0_{S_n}\n( \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^{n} )$ and $K^0( \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^{[n]})$  are  isometries.\n\\end{theorem}\n\n\\begin{remark}  \\rm\nAll the algebraic structures on $\\bigoplus_n R (\\G_n)$ (cf.\n\\cite{W,FJW}) and thus on $\\bigoplus_n K^0_{\\G_n} ( {\\Bbb C} ^{2n})$ can\nnow be carried over to $\\bigoplus_n K^0( \\widetilde{  {\\Bbb C} ^2 \/ \\Gamma} ^{[n]})$. However it\nremains to match these with the Grojnowski-Nakajima construction\non $\\bigoplus_{n} H( \\ale^{[n]} )$ in terms of correspondence varieties.\n\\end{remark}\n\n\\begin{remark}  \\rm\nGiven a finite subgroup $G$ of $SL_K ( {\\Bbb C} )$, one asks whether there\nis a crepant resolution $Y$ of the affine orbifold $ {\\Bbb C} ^K \/ G$ so\nthat there exists a canonical isomorphism between $K_{G}( {\\Bbb C} ^K) =\nK(Y)$; one further asks whether or not the answer can be provided\nby the Hilbert quotient $ {\\Bbb C} ^K \/\/ G$, cf. Reid \\cite{R}. The answer\nis affirmative for $K =2$, known as the McKay correspondence\n\\cite{GSV} (also compare \\cite{IN, KV}). For $K=3$, there has been\nmuch work by various people, cf. \\cite{Ro, R, Nr, IN} and\nreferences therein, and it is settled by Bridgeland-King-Reid\n\\cite{BKR}. However not much is known in general (see however\n\\cite{BKR}) and there has been counterexamples. Our work provides\nstrong evidence for an affirmative answer in the case of $ {\\Bbb C} ^{2n}$\nacted upon by the wreath product $\\G_n$ associated to a finite\nsubgroup $\\Gamma \\subset SL_2 ( {\\Bbb C} )$.\n\\end{remark}\n\\section{A direct isomorphism of algebraic structures on equivariant K-theory}\n\\label{sect_ktheory}\n %\nIn this section we assume that the reader is familiar with\n\\cite{W}. For shortness of notations we will use $K^{ {\\footnotesize {top}} }(-)$\nand $K^{ {\\footnotesize {top}} }_G(-)$ to denote the {\\em complexified}\n($G$-equivariant) topological K-group. We further assume that $X$\nis a quasi-projective surface acted upon by a finite subgroup $\\Gamma$\nand $Y$ is a resolution of singularities of $X\/ \\Gamma$ such that\nthere exists a canonical isomorphism $\\theta$ between\n$K^{ {\\footnotesize {top}} }_{\\Gamma}(X)$ and $K^{ {\\footnotesize {top}} }( Y)$.\n\nLet ${\\cal C} = \\{ V_1, \\ldots, V_l \\}$ be a basis for $ {K}^{ {\\footnotesize {top}} }_{\\G}(X) $.\nWithout loss of generality we may assume they are genuine\n$\\Gamma$-vector bundles on $X$. We denote by $W_i = \\theta ( V_i)$.\nThe set $ \\{ W_1, \\ldots, W_l \\}$ is a basis for $ K(Y) $. We remark\nthat representatives of $W_i$'s can again be chosen as certain\ncanonical vector bundles in favorable cases, including the\nimportant case when $X=  {\\Bbb C} ^2$ and $\\Gamma \\subset SL_2 ( {\\Bbb C}  )$.\n\nLet $S_{\\lambda} $ be the irreducible representation of the symmetric\ngroup $S_n $ associated to the partition $\\lambda$ of $n$. Define\n$$\n  S_{\\lambda}  (V_i) = S_{\\lambda}  \\bigotimes V_i^{ \\boxtimes n}.\n$$ Endowed with the diagonal action of $S_n$ on the two tensor\nfactors and the action of $\\Gamma^n$ on the second factor, $S_{\\lambda}\n(V_i)$ is an $\\G_n$-equivariant vector bundle.\n\nGiven a partition-valued function $\\lambda = (\\lambda_i )_{ 1 \\leq i \\leq\nl} \\in {\\cal P}_n (\\cal C)$, we define the $\\G_n$-equivariant\nvector bundle $$ S^X_{\\lambda} =\n \\mbox{Ind} ^{\\G_n}_{ \\Gamma_{|\\lambda_1|} \\times \\ldots \\times \\Gamma_{|\\lambda_rl|} }\n  S_{\\lambda_1}(V_1 ) \\times \\ldots \\times S_{\\lambda_l}(V_l).\n$$\n   %\nIn a parallel way, we can define the $S_n$-equivariant bundle $\nS^Y_{\\lambda}$ associated to $ \\lambda \\in {\\cal P}_n (\\cal C) $ as $$\nS^X_{\\lambda}\n=\n \\mbox{Ind} ^{S_n}_{ S_{|\\lambda_1|} \\times \\ldots \\times S_{|\\lambda_l|} }\n  S_{\\lambda_1}(\\theta(V_1) ) \\times \\ldots \\times S_{\\lambda_l}(\\theta (V_l)).\n$$\n\nRecall that we constructed in \\cite{W} various algebraic\nstructures such as Hopf algebra, $\\lambda$-ring, Heisenberg\nalgebra on $\\bigoplus_{n \\geq 0}  K^{ {\\footnotesize {top}} }_{\\Gn}(X^n) $ for a $\\Gamma$-space $X$,\ngeneralizing the results of Segal \\cite{S2} for $\\Gamma$ trivial. The\nfollowing proposition can be proved using \\cite{W} in a way as\nMacdonald \\cite{M} did when $X$ is a point.\n\n\\begin{proposition}\nThe $\\G_n$-bundles $ S^X_{\\lambda}, \\lambda \\in {\\cal P}_n (\\cal C)$ form a\nbasis of $ K^{ {\\footnotesize {top}} }_{\\Gn}(X^n) $. The $S_n$-bundles $ S_{\\lambda}^Y, \\lambda \\in  {\\cal\nP}_n (\\cal C)$ form a basis of $ K_{S_n} (Y^n) $.\n\\end{proposition}\nThese bases will be referred to as Schur bases, generalizing the\nusual one for $R(\\G_n) = K^{ {\\footnotesize {top}} }_{\\G_n} (pt)$.\n\n\\begin{theorem}\nThe map $\\Theta$ from $\\bigoplus_{n \\geq 0}  K^{ {\\footnotesize {top}} }_{\\Gn}(X^n) $ to\n$\\bigoplus_{n \\geq 0}  K_{S_n} (Y^n) $ by sending $S_{\\lambda} ^X$ to $S_{\\lambda}\n^Y$ is an isomorphism of Hopf algebras, $\\lambda$-rings, and\nrepresentations over the Heisenberg algebra.\n\\end{theorem}\n\n\\begin{demo}{Proof}\nWe use \\cite{W} as a basic reference. We follow the notations\nthere with an additional use of $X, Y$ as subscripts to specify\nthe space we are referring to.\n\nAs graded vector spaces $\\bigoplus_{n \\geq 0}  K^{ {\\footnotesize {top}} }_{\\Gn}(X^n) $ and\n$\\bigoplus_{n \\geq 0}  K_{S_n} (Y^n) $ have the same graded dimension due to\nthe isomorphism between $K^{ {\\footnotesize {top}} }_{\\Gamma} (X)$ and $K^{ {\\footnotesize {top}} } (Y)$,\ncf. Theorem 3 in \\cite{W}. Thus the map $\\Theta$ given by matching\nthe Schur basis is an additive isomorphism.\n\nRecall that the Adam's operations $\\varphi^m_X$ on the space\n$\\bigoplus_{n \\geq 0}  K^{ {\\footnotesize {top}} }_{\\Gn}(X^n) $ and $\\varphi^m_Y$ on $\\bigoplus_{n \\geq\n0}  K_{S_n} (Y^n) $ satisfy the identities (where $q$ is a formal parameter),\ncf. Proposition 4 in \\cite{W}:\n\\begin{eqnarray*}\n\\bigoplus_{n \\geq 0} q^n V_i^{\\boxtimes n}\n  & =& \\exp \\left( \\sum_{m > 0} \\frac1m  \\varphi^m_X (V_i) q^m\n            \\right),   \\\\\n\\bigoplus_{n \\geq 0} q^n W_i^{\\boxtimes n}\n  & =& \\exp \\left( \\sum_{m > 0} \\frac1m \\varphi^m_Y (W_i) q^m\n            \\right).\n\\end{eqnarray*}\n\nIt follows that $\\varphi^m_X (V_i)$ and $\\varphi^m_Y (W_i)$ are\nuniquely determined by $V_i^{\\boxtimes n}$ $(n \\geq 0)$ and\nrespectively $W_i^{\\boxtimes n}$ in the same way (by taking\nlogarithms of the above identities). Since the isomorphism\n$\\Theta$ sends $V_i^{\\boxtimes n}$ to $W_i^{\\boxtimes n}$, the\nAdams operations $\\phi^r_X (V_i)$ and $\\phi^r_X (V_i)$ matches\nunder $\\Theta$ and so does the $\\lambda$-ring structures on\n$\\bigoplus_{n \\geq 0}  K^{ {\\footnotesize {top}} }_{\\Gn}(X^n) $ and $\\bigoplus_{n \\geq 0}  K_{S_n} (Y^n) $.\n\nRecall that Heisenberg algebras ${\\cal H}_X$ and ${\\cal H}_Y$ constructed\nin terms of K-theory maps act\nirreducibly on $\\bigoplus_{n \\geq 0}  K^{ {\\footnotesize {top}} }_{\\Gn}(X^n) $ and $\\bigoplus_{n \\geq\n0}  K_{S_n} (Y^n) $ respectively, cf. Theorem 4 in \\cite{W}. The Heisenberg\nalgebra generators are essentially defined in terms of Adams operations,\ninduction functors and restriction functors such as\n$\\mbox{Ind}_{\\Gamma_m \\times \\Gamma_{n -m}}^{\\G_n}$, $\\mbox{Res}_{\\Gamma_m\n\\times \\Gamma_{n -m}}^{\\Gamma_n}$, etc. Since the Adams operations,\ninduction and restriction functors are compatible with\nthe Schur bases and thus with the map\n$\\Theta$, the Heisenberg algebras acting on $\\bigoplus_{n \\geq 0}\n K^{ {\\footnotesize {top}} }_{\\Gn}(X^n) $ and $\\bigoplus_{n \\geq 0}  K_{S_n} (Y^n) $ also matches under\n$\\Theta$.\n\\end{demo}\n\n\\frenchspacing\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\nAbnormalities in medicine can be viewed as deviations from a normal feature in a healthy population. In medical images, such abnormalities appear as local or global deviations from an original normal anatomy. These changes should reflect a process where abnormalities occur under some initiation mechanisms from the normal anatomy and, with disease progression, existing normal tissues are eventually replaced by pathological ones. Indeed, when evaluating images of conditions to be diagnosed, physicians often need corresponding images without abnormal findings of interest or, conversely, images that contain similar abnormal findings regardless of normal anatomical context. This is called comparative diagnostic reading of medical images, which is essential for a correct diagnosis. For example, by comparing abnormal images of interest with normal images of the same anatomical site, physicians can diagnose the presence and extent of the disease. Meanwhile, images with similar abnormal features can also be useful as a reference because diseases with the same imaging phenotypes often have similar prognoses and treatment responses, somewhat regardless of the origin of the diseases. However, it is quite laborious to find similar images from a large database by focusing only on either normal or abnormal features of a query image by hands. Hence, we aimed to establish a content-based image retrieval (CBIR) system to support comparative diagnostic reading. \n\nCBIR is an important application to retrieve a set of the most similar images to a submitted patient's image from large databases to support clinical decision making. A CBIR system basically comprises two subsystems: At the first feature extraction stage, it converts images in a database to a set of features associated with image content, and at the next similarity search stage, similar images are retrieved from the database based on the feature similarity with respect to a query image. Traditionally, the feature extraction employed various types of handcrafted descriptors, such as shape, text, color, and texture; however, there was difficulty in reducing the ``semantic gap'' between low-level image features captured by the handcrafted descriptors and high-level visual concepts \\citep{Kumar2013, sift7935507}. Recently, with the success in many image processing tasks, the deep-learning-based approach has gained increasing attention in the field of CBIR. Since the deep neural network can automatically learn complex features in hierarchical levels of abstraction, it can simplify the feature extraction process. A common neural network architecture used in CBIR systems is the autoencoder, which enables to map each image into a latent representation that can store compressed image information \\citep{7727562, 7172448, 8310624, 8079843, 7351389}. Even though deep-learning-based approach often outperformed classical algorithm of CBIR systems \\citep{10.1117\/12.2251115, 10.1117\/12.2217587}, there is no specific method that can measure the distance between either normal or abnormal features of medical images as separated semantic components. To achieve our goals, we need to devise a novel feature extraction method that can decompose normal and abnormal features of medical images. We also aimed to establish a computationally effective method to retrieve similar images based on decomposed latent representations.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[]{.\/figures\/concept_of_decomp.jpg}\n  \\caption{\\textbf{Concept of the compositionality of medical images.} It indicates that the entire medical image can be decomposed into normal and abnormal anatomy codes relevant to normal and abnormal semantic components in the image, respectively.}\n  \\label{fig:concept_of_decomp}\n\\end{figure}\n\nIn this study, we define the two-tiered semantic nature of normal and abnormal anatomy as \\emph{compositionality} of medical images, as presented in {\\bf Fig. \\ref{fig:concept_of_decomp}}. Hereinafter, ``normal'' anatomy means counterfactual structures that should have existed if the sample is healthy, whereas ``abnormal'' anatomy attributes to disease changes that reflect the deviation from the normal baseline. When the sample is free from abnormality, ``normal'' anatomy corresponds to the entire sample image, and ``abnormal'' anatomy should not indicate any condition. Then, we consider how to decompose a given image into two low-dimensional representations in a manipulable manner, where the latent codes representing normal and abnormal anatomy should be mutually exclusive and collective in terms of the reconstruction of the original image. By measuring similarities based on decomposed normal or abnormal semantic components of medical images, it can be expected to retrieve images that have the same normal anatomical context or similar abnormal findings, respectively, while ignoring the other component.\n\nFactorizing compositionality into distinct and informative factors in data is a fundamental requirement of representation learning \\citep{Bengio2013}. The majority of studies has sought to capture purely independent factors of variation that contributes to generation of data, which are called \\emph{disentangled representations} \\citep{higgins2018definition}. To obtain disentanglement of features, two approaches have exploited implicit or explicit supervision to impose strong inductive biases on autoencoders \\citep{tschannen2018recent}. One approach is to collect a large amount of data to encompass sufficient variation for each factor and then apply appropriate regularization such that disentanglement can be implicitly performed \\citep{Higgins2017betaVAELB, burgess2018understanding, pmlr-v80-kim18b, Chen2018tcvae, zhao2017infovae, kumar2017variational, lopez2018information, esmaeili2018structured, Alex2017, Achille2018, Lample2017, louizos2015variational}. Another approach is to force a model to acquire separate representations by explicitly imposing modeling assumptions \\citep{cheung2014discovering, Kulkarni2015, Eslami2016attend, shanahan2019explicitly, charakorn2020explicit}. If distinct factors of variations explaining the characteristics of data can be separately represented using independent latent units, it can be useful in downstream tasks by providing interpretability and manipulability to the data. \n\nTherefore, disentangled representations can decompose normal and abnormal semantic components of medical images. Indeed, there has been an increasing number of studies focusing on feature disentanglement in medical imaging, including performing pseudo-healthy synthesis \\citep{XIA2020101719, ADN8788607, vorontsov2019semisupervised, Tang2021}, learning generalizable features across domains \\citep{meng2020learning, 9247170} or imaging modalities \\citep{Chen2019, 9250615, Chartsias2019}, and evaluating the reliability of individual annotators from true segmentation label distributions \\citep{HumanError2020}. However, with respect to the etiology of diseases in medical images, it is noteworthy that the assumption of disentanglement, where purely independent low-dimensional latent features can mimic the generation of high-dimensional data, can be too simple to be generalizable. For example, brain tumors originate as focal changes, and with their progression, cause compressional deformations and infiltration in adjacent structures, making it difficult to clearly define the boundary between normal and abnormal tissues. Hence, we consider \\emph{decomposition} of latent representation, which encompasses the notion of disentangled representation as its special case and allows much richer properties of latent spaces such as intricate structured relationships \\citep{pmlr-v97-mathieu19a}. \n\nIn this study, we first devised a neural network architecture, called \\emph{feature decomposing network}, to decompose normal and abnormal semantic components of medical images in a manipulable manner. Then, we demonstrated a novel CBIR framework by utilizing the decomposed latent codes to support comparative diagnostic reading. Given an input image, the feature decomposing network is trained to map it into two latent codes, {\\it normal anatomy code} and {\\it abnormal anatomy code}. It indicates that the original latent space for representing the whole image is divided into one portion as a normal anatomy code corresponding to normal anatomy and the remaining portion as an abnormal anatomy code corresponding to abnormal anatomy. After training the feature decomposing network, latent codes become representative of targeted semantic features in medical images. We also investigate the effectiveness of a method called \\emph{distribution matching} by utilizing Wasserstein generative adversarial networks (GANs) \\citep{pmlr-v70-arjovsky17a} with gradient penalty \\citep{NIPS2017_892c3b1c}. Distribution matching is imposed on the latent distribution to minimize the overlap in the semantic contents between normal and abnormal anatomy codes. Furthermore, by constructing the two latent codes to be discrete through vector quantization \\citep{oord2017neural}, we can reduce computational burden at the time of similarity search by binary hashing. The utility of these decomposed latent codes for CBIR applications is shown based on a large dataset containing brain magnetic resonance (MR) images of gliomas. By performing nearest neighbor search utilizing either normal or abnormal anatomy codes or the combination of the two codes, our CBIR system can retrieve images according to selected semantic components while ignoring the other, if necessary.\n\nThe main contributions of this study can be summarized into the following:\n\\begin{itemize}\n  \\item We propose a feature decomposing network that can explicitly decompose semantic features of medical images into normal and abnormal anatomy codes in a manipulable manner for downstream tasks. \n  \\item To enhance computational efficiency at the time of similarity search, latent spaces are configured using vector quantization to be discrete, rather than continuous.\n  \\item By employing the decomposed latent codes, we present a novel CBIR application that can search for similarities in images based on a selected latent code or the combination of the two latent codes, enabling retrieval of images viewed as any semantic components while ignoring the other, if necessary. \n\\end{itemize}\n\nThe proposed method is most closely related to our conference extended abstract \\citep{kobayashi2020decomposing}, a neural network architecture to decompose normal and abnormal features of medical images with its application to CBIR. However, the presented work has significantly improved the learning method with extensive experimental settings. We further validated the performance of the CBIR application with more appropriate metrics. \n\nThe rest of the manuscript is organized as follows: {\\bf Section \\ref{sec:related_work}} reviews the literature on disentangled representation, image-to-image translation in medical imaging, and CBIR. {\\bf Section \\ref{sec:methodology}} describes our proposed method with technical backgrounds. {\\bf Section \\ref{sec:experiments}} presents experimental settings and evaluation methods. {\\bf Section \\ref{sec:results}} provides the results. {\\bf Section \\ref{sec:discussion}} presents the discussion and conclusions. \n\n\\section{Related work}\n\\label{sec:related_work}\n\nHere, we briefly review literature related to disentangled representation learning, especially in the field of computer vision. Thereafter, as a research interest more related to our purpose, we review recent progress that mainly apply the image-to-image translation technique based on GANs for pseudo-healthy synthesis of medical images. We also introduce current progress in the CBIR system in medical imaging. \n\n\\subsection{Disentangled representation learning}\n\\label{sec:disentangleed_representation_learning}\nLearning disentangled representation attempts to separate the underlying factors of sample variations in a way that each factor exclusively represents one type of sample attributes. There are several benefits in learning disentangled representation from data because models that produce feature disentanglement can provide explainability of the model function and manipulability in the data generation process. One approach is to combine deep generative models, such as GANs \\citep{Goodfellow2014} and variational autoencoders (VAEs) \\citep{Kingma2013, Rezende2014} with regularization techniques to acquire disentanglement in an implicit manner \\citep{Higgins2017betaVAELB, pmlr-v80-kim18b, Chen2018tcvae, zhao2017infovae, kumar2017variational, lopez2018information, esmaeili2018structured, Alex2017, Achille2018, Lample2017, louizos2015variational}. For example, $\\beta$-VAE can automatically discover the independent latent factors of variation by imposing a limit on the capacity of latent information, which facilitates factorization of representations \\citep{Higgins2017betaVAELB}. However, it is still fundamentally difficult to learn disentangled features without any supervision. It is also argued that acquired disentangled representations sometimes mismatch human's predefined concepts \\citep{Locatello}. Another approach is to explicitly factorize representations into a component that is related to or independent of classes based on labeled data \\citep{cheung2014discovering, Kulkarni2015, Eslami2016attend, shanahan2019explicitly, charakorn2020explicit}. Label information or attribute annotation can serve as strong supervision for feature disentanglement; hence, the performance of disentanglement can be optimized and guaranteed. Therefore, by designing a network with an appropriate structure and exploiting segmentation labels indicating abnormal regions as supervision, we aim to obtain the desired decomposition of latent representations. \n\n\\subsection{Image-to-image translation}\n\\label{sec:image_to_image_translation}\nImage-to-image translation has been exploited in medical imaging to obtain disentangled representation. CycleGAN, which performs bidirectional translation between image domains, has been widely used in image-to-image translation \\citep{cyclegan8237506}. As an extended architecture of CycleGAN, the unsupervised image-to-image translation (UNIT) framework proposes a shared latent space assumption, where a pair of corresponding images in two different domains can be mapped \\citep{liu2017unsupervised}. More recently, multimodal UNIT decomposes an image into a content code that is domain invariant and a style code that represents domain-specific features \\citep{huang2018multimodal}. In line with studies focusing on medical imaging, Xia et al. demonstrated pseudo-healthy synthesis by creating a subject-specific healthy image from a pathological one by extending the learning framework of CycleGAN \\citep{XIA2020101719}. Similarly, Liao et al. proposed an artifact disentanglement network using the image-to-image translation architecture, achieving comparable performance in image restoration to existing supervised models \\citep{ADN8788607}. Vorontsov et al. also applied the same concept to improve semi-supervised training for semantic segmentation with autoencoding \\citep{vorontsov2019semisupervised}. To enhance realistic synthesis of chest radiographic images, Tang et al. proposed a disentangled generative model for disease decomposition and demonstrated that disease residual maps can indicate underlying abnormal regions \\citep{Tang2021}. Since one of our goals is to decompose medical images into normal and abnormal semantic components, the basic idea is somewhat similar to pseudo-healthy synthesis exploiting image-to-image translation techniques. However, previous approaches focused on transforming rather superficial appearance of images and did not evaluate the accessibility and validity of latent representations acquired inside the models. Thus, our feature decomposing network has a bottleneck where imaging features are compressed, enabling to handle latent representations of targeted semantic component for the downstream task. \n\n\\subsection{Content-based image retrieval}\n\\label{sec:content_based_image_retrieval}\nCBIR is an important application in retrieving a set of the most similar images from large databases, given a query image. Similarity measurements based on various information, such as shape, text, color and texture, and features acquired inside convolutional neural networks are used to resolve the semantic gap between imaging features and high-level visual concepts \\citep{Kumar2013, sift7935507}. Even though several studies utilizing state-of-the-art techniques of deep neural networks have been introduced to CBIR \\citep{Haq2021, MohdZin2018}, to the best of our knowledge, there are few studies that exploit disentangled representation from the viewpoint of image retrieval. Recently, Havaei et al. proposed a neural network architecture to ensure content-style disentanglement of medical images and demonstrated how the inferred style and content features are disentangled from each other by utilizing CBIR as an evaluation method \\citep{havaei2020conditional}. Since CBIR is one of essential technologies that assist physicians in diagnosis, a methodology that directly seeks to develop CBIR based on disentangled representation is worth considering. Therefore, in addition to the concept of decomposing normal and abnormal features of medical images, we particularly employ latent codes as to be discrete through vector quantization. By applying vector quantization for the latent space, subspaces can be fixed and transversed by the Hamming distance rearrangement, which is favorable in a large-scale CBIR by reducing the computational cost at the time of similarity search \\citep{9050998}. \n\n\\section{Material and methods}\n\\label{sec:methodology}\n\nIn this study, we propose a method to decompose two-tiered semantic components of medical images into normal and abnormal anatomical codes for the application of CBIR that can selectively focus on semantic components. The proposed method is presented in two stages: First, we describe a network architecture, which we call \\emph{feature decomposing network}, to decompose normal and abnormal features in medical images and its learning strategy. Then, we present how to establish a CBIR system that can support comparative diagnostic reading by utilizing these decomposed features. \n\n\\subsection{Feature decomposing network}\n\\label{sec:training_of_feature_decomposing_network}\n\n\\begin{figure*}\n  \\centering\n  \\includegraphics[]{.\/figures\/fdn_model_architecture.jpg}\n  \\caption{\\textbf{Shared architecture of the feature decomposing networks.} Input image $\\bm{x}$ is mapped to a pair of latent representations, $\\bm{z}_e^-$ and $\\bm{z}_e^+$. Vector quantization is performed based on two codebooks, $\\bm{e}^-$ and $\\bm{e}^+$, to produce normal anatomy code $\\bm{z}_q^-$ and abnormal anatomy code $\\bm{z}_q^+$, respectively. The segmentation decoder predicts the semantic segmentation of abnormality $\\hat{\\bm{y}}$ from $\\bm{z}_q^+$. Depending on the collateral input of the softmax logit of abnormality $\\tilde{\\bm{y}}$, the image decoder reconstructs either the entire input image $\\hat{\\bm{x}}^+$ or normal-appearing image ${\\hat{\\bm{x}}^-}$. Several loss functions in training the network, including latent, reconstruction and segmentation losses, are shown, except for those related to distribution matching.}\n  \\label{fig:fdn_model_architecture}\n\\end{figure*}\n\nThe feature decomposing network is used to decompose semantic components of medical images into normal and abnormal anatomy codes. It is trained based on a dataset containing input images $\\bm{x}$ and ground-truth segmentation labels $\\bm{y}$. The feature decomposing network consists of an encoder and two decoders, segmentation decoder, and image decoder. A pair of discrete latent spaces with latent codebooks exists at the bottom of the network, each of which produces normal and abnormal anatomy codes. The overview of the network architecture is shown in {\\bf Fig. \\ref{fig:fdn_model_architecture}}. It is noteworthy that the feature decomposing network has the latent space as its bottleneck, where no bypass connection between the encoder and decoders, such as skip connections \\citep{drozdzal2016importance}, is implemented. Therefore, we can expect that the information processed by the encoder can be compressed in latent spaces \\citep{razavi2019generating, 8969272}. \n\n\\subsubsection{Feature encoding}\n\\label{sec:encoder_network}\n\nThe encoder uses a two-dimensional (2D) medical image $\\bm{x} \\in \\mathbb{R}^{C \\times H \\times W}$ as an input, where $C$ is the number of channels and $H$ and $W$ represent the height and width of the images, respectively. Then, the encoder maps the image into two latent representations, $\\bm{z}_e^- \\in \\mathbb{R}^{D \\times H^{\\prime} \\times W^{\\prime}}$ and $\\bm{z}_e^+ \\in \\mathbb{R}^{D \\times H^{\\prime} \\times W^{\\prime}}$, where $\\bm{z}_e^-$ and $\\bm{z}_e^+$ correspond to the semantic features of normal and abnormal anatomies, respectively. We use $\\bm{z}^\\mp_e$ to represent both features. Subsequently, vector quantization is used to discretize $\\bm{z}^\\mp_e$. Namely, each elemental vector $z^\\mp_{e_i} \\in \\mathbb{R}^{D}$ is replaced with the closest code vector in each codebook $\\bm{e}^\\mp \\in \\mathbb{R}^{D \\times K}$ that comprises $D$-dimensional $K$ code vectors. Details in this vector quantization process are presented in the next subsection. We denote the quantized vector of $\\bm{z}_e^\\mp$ as $\\bm{z}_q^\\mp$. Hereinafter, $\\bm{z}_q^-$ is referred to as \\emph{normal anatomy code} and $\\bm{z}_q^+$ as \\emph{abnormal anatomy code}.\n\n\\subsubsection{Vector quantization}\n\\label{sec:vector_quantiser}\n\nThe latent space has two codebooks, $\\bm{e}^- = \\{e^-_k |k = 1, \\ldots, K\\}\\in \\mathbb{R}^{K \\times D}$ and $\\bm{e}^+ = \\{e^+_k |k = 1, \\ldots, K\\}\\in \\mathbb{R}^{K \\times D}$, corresponding to the normal and abnormal semantic features, respectively. The vector quantization process is similar to that of a vector-quantized (VQ) VAEs \\citep{oord2017neural}. An $i$-th elemental vector of $\\bm{z}^\\mp_e$, denoted as $z^\\mp_{e_i} \\in \\mathbb{R}^{D}$, is quantized by executing a nearest-neighbor lookup on the codebook as follows:\n\\begin{equation} \nk^\\mp = \\mathop{\\rm arg~min}\\limits_{k \\in \\{1, \\ldots,  K\\}} \\|z^\\mp_{e_i} - e_k\\|_2^2.\n\\end{equation} \nThereafter, an $i$-th elemental vector of $\\bm{z}^\\mp_e$ is replaced by the $k^\\mp$-th code vector in the codebook as follows: \n\\begin{equation} \nz^\\mp_{e_i} = e_{k^\\mp}.\n\\end{equation} \nThis replacement is performed for all ($H^\\prime \\times W^\\prime$) elemental vectors of $\\bm{z}^\\mp_e$ to collectively form a quantized vector $\\bm{z}^\\mp_q$.\n\nTo optimize this process, the encoder and codebooks are updated to minimize an objective, which is referred to as latent loss $L_{\\mathrm{lat}}$ as follows:\n\\begin{equation} \nL^\\mp_\\mathrm{lat} = \\|\\mathrm{sg}[\\bm{z}^\\mp_e] - \\bm{e}^\\mp\\|^2_2 + \\beta \\|\\bm{z}^\\mp_e - \\mathrm{sg}[\\bm{e}^\\mp]\\|^2_2,\n\\end{equation} \n\\begin{equation} \nL_\\mathrm{lat} = L_\\mathrm{lat}^- + L_\\mathrm{lat}^+,\n\\end{equation} \nwhere $\\mathrm{sg}$ indicates a stop-gradient operator, which serves as an identity function at the forward computation time and has zero partial derivatives, and $\\beta$ is a balancing hyperparameter. During training, the first term in the abovementioned equation updates the codebook variables by delivering the code vectors to the encoder output. Meanwhile, the second term encourages the encoder output to move closer to the targeted code vectors. We use the exponential moving average to train the codebook \\citep{kaiser2018fast}.\n\n\\subsubsection{Feature decoding}\n\\label{sec:feature_decoding}\n\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[]{.\/figures\/spade.jpg}\n  \\caption{\\textbf{Overview of spatially adaptive normalization (SPADE) module.} Softmax logit $\\tilde{\\bm{y}}$ multiplied by the segmentation mask $\\hat{\\bm{y}}$ is further downsampled to achieve resolutions corresponding to those of each layer in the image decoder. SPADE module propagates the semantic layout of abnormalities into the image generation process. Each convolution block comprises a convolution function and bias parameter.}\n  \\label{fig:spade}\n\\end{figure}\n\nThe segmentation decoder uses an abnormal anatomy code $\\bm{z}_q^+$ as an input and outputs a $S$-class segmentation label $\\hat{\\bm{y}} \\in \\mathbb{R}^{S \\times H \\times W}$ that corresponds to the ground-truth label $\\bm{y}$. A loss function for the segmentation output $L_{\\mathrm{seg}}$ is a composite of the generalized Dice \\citep{Sudre_2017} and focal \\citep{focal2017} losses as follows:\n\\begin{equation} \nL_\\mathrm{dice} = 1 - 2 \\frac{\\sum_{s \\in S} w_s |\\hat{\\bm{y}}_{s} \\cap \\bm{y}_{s}|}{\\sum_{s \\in S} w_s (|\\hat{\\bm{y}}_{s}| + |\\bm{y}_{s}|)},\n\\end{equation} \n\\begin{equation} \nL_\\mathrm{focal} = - \\frac{1}{N} \\sum_{s \\in S} \\bm{y}_s (1 - \\tilde{\\bm{y}}_s)^{\\gamma} \\log \\tilde{\\bm{y}}_s,\n\\end{equation} \n\\begin{equation} \nL_\\mathrm{seg} = L_\\mathrm{dice} + L_\\mathrm{focal},\n\\end{equation} \nwhere $\\tilde{\\bm{y}}$ indicates the softmax logits of the segmentation decoder, $N (= H \\times W)$ is the number of pixels, and $w_k$ is determined as $w_k = \\frac{1}{(\\sum_N |\\bm{y}_s|)^2}$ to mitigate the class imbalance problem \\citep{Sudre_2017}.\n\nMeanwhile, the image decoder $f$ performs conditional image generation using the \\emph{spatially adaptive normalization} (SPADE) \\citep{park2019SPADE}. SPADE is designed to propagate semantic layouts to the process of synthesizing images ({\\bf Fig. \\ref{fig:spade}}). The image decoder uses a normal anatomy code $\\bm{z}_q^-$ as its primary input. When the image decoder is used to reconstruct the entire input image $\\hat{\\bm{x}}^+$, the softmax logits of the segmentation decoder $\\tilde{\\bm{y}}$ is transmitted to each layer of the image decoder via the SPADE modules ($f(\\bm{z}_q^-, \\tilde{\\bm{y}}) = \\hat{\\bm{x}}^+$). When {\\it null} information, where $\\tilde{\\bm{y}}$ is filled with 0s, is propagated to the SPADE modules, a normal-appearing reconstruction $\\hat{\\bm{x}}^-$ is generated by the image decoder ($f(\\bm{z}_q^-, \\bm{0}) = \\hat{\\bm{x}}^-$).\n\nTo enforce different characteristics between the two types of generated images, $\\hat{\\bm{x}}^-$ and $\\hat{\\bm{x}}^+$, we apply a pixel-wise reconstruction loss depending on the region of abnormality. Suppose $\\bm{M}^+ \\in \\{0, 1\\}^{C \\times H \\times W}$ defines a mask, indicating that pixels with any abnormality labels are set to 1 and 0 otherwise, and $\\bm{M}^- = \\bm{1} - \\bm{M}^+$ is a complementary set of $\\bm{M}^+$. Briefly, $\\bm{M}^+$ presents the abnormal anatomy region, and $\\bm{M}^-$ indicates the normal anatomy region. Using these masks, image reconstruction loss $L_\\mathrm{rec}$ is defined as follows: \\begin{equation} \n\\begin{split}\nL^-_\\mathrm{rec} &= \\|\\bm{M}^- \\odot \\hat{\\bm{x}}^- - \\bm{M}^- \\odot \\bm{x}\\|^2_2 \\\\\n                            &+ (1 - \\mathrm{SSIM}(\\bm{M}^- \\odot \\hat{\\bm{x}}^-, \\bm{M}^- \\odot \\bm{x})),\n\\end{split}\n\\end{equation} \n\\begin{equation}\n\\begin{split}\nL^+_\\mathrm{rec} &= \\|\\hat{\\bm{x}}^+ - \\bm{x}\\|^2_2 \\\\\n                            &+ (1 - \\mathrm{SSIM}(\\hat{\\bm{x}}^+, \\bm{x})) \\\\\n                            &+ \\|{\\bm{M}^+} \\odot \\hat{\\bm{x}}^+ - {\\bm{M}^+} \\odot \\bm{x}\\|^2_2 \\\\\n                            &+ (1 - \\mathrm{SSIM}({\\bm{M}^+} \\odot \\hat{\\bm{x}}^+, {\\bm{M}^+} \\odot \\bm{x})),\n\\end{split}\n\\end{equation} \n\\begin{equation} \nL_\\mathrm{rec} = L^-_\\mathrm{rec} + L^+_\\mathrm{rec},\n\\end{equation} \nwhere SSIM indicates structural similarity \\citep{Wang2004}, which is added to the L2 loss as a constraint owing to its empirical effect to stabilize the image generation process. \n\n\\subsubsection{Distribution matching}\n\\label{sec:density_matching}\n\nIt is quite important to ensure that each decomposed feature, normal anatomy code $\\bm{z}_q^-$ and abnormal anatomy code $\\bm{z}_q^+$, corresponds to targeted semantic content in the images. For example, when some code vectors of normal anatomy codes convey not only features corresponding to normal anatomies but also those related to abnormal anatomies, the feature decomposition can be ``leaky,'' losing its reliability for downstream tasks. Particularly, this can occur at normal anatomy codes because, when a pathological image is provided, it is fundamentally impossible to obtain a \\emph{paired} normal counterpart that can be a ground-truth for the normal-appearing reconstructions $\\hat{\\bm{x}}^-$. Therefore, we utilize \\emph{unpaired} healthy images and employ distribution matching technique to minimize the discrepancy of the distributions between normal anatomy codes from healthy images and those from disease images. We consider this discrepancy as the Wasserstein distance $d_\\mathrm{W}$ and minimize it using Wasserstein GAN \\citep{pmlr-v70-arjovsky17a} with gradient penalty \\citep{NIPS2017_892c3b1c} that has a critic network $g$. \n\nHere, we consider that the set of input images $\\mathcal{X}$ can be divided into a set consisting of healthy images $\\mathcal{X}_h$ and a set consisting of diseased images $\\mathcal{X}_d$. Suppose the distribution of normal anatomy codes $\\mathcal{Z}^-_q$ can also be split into those originating from healthy images $\\mathcal{Z}^-_h$ and those originating from diseased images $\\mathcal{Z}^-_d$. When a batch $\\{\\bm{x} \\sim \\mathcal{X}\\}$ containing both healthy images $\\{\\bm{x}_h \\sim \\mathcal{X}_h\\}$ and diseased images $\\{\\bm{x}_d \\sim \\mathcal{X}_d\\}$ is fed into the encoder, a corresponding batch of normal anatomy codes $\\{\\bm{z}_q^- \\sim \\mathcal{Z}_q^- \\}$ can be split into those originating from healthy images $\\{\\bm{z}^-_h \\sim \\mathcal{Z}^-_h \\}$ and diseased images $\\{\\bm{z}^-_d \\sim \\mathcal{Z}^-_d \\}$. If there is no leakage of semantic content of abnormality into the normal anatomy codes, the two distributions, $\\mathcal{Z}^-_h$ and $\\mathcal{Z}^-_d$, should be identical with each other, irrespective of the presence of abnormality in the input images. Then, distribution matching imposes two types of loss functions, $L_\\mathrm{critic}$ and $L_\\mathrm{reg}$, as follows:\n\\begin{equation} \nL_\\mathrm{critic} = d_\\mathrm{W} + \\lambda_\\mathrm{gp} (\\| \\nabla_{\\bm{z}^-_m} g (\\bm{z}^-_m) \\|^2_2 - 1)^2,\n\\end{equation} \n\\begin{equation} \nL_\\mathrm{reg} = - g (\\bm{z}^-_d),\n\\end{equation} \nwhere $d_\\mathrm{W} = g (\\bm{z}^-_d) - g (\\bm{z}^-_h)$ is the Wasserstein distance, $\\lambda_\\mathrm{gp}$ is the balancing term for the gradient penalty, and $\\bm{z}^-_m = \\epsilon \\bm{z}^-_h + (1 - \\epsilon) \\bm{z}^-_d$ is to enforce the Lipschitz constraint by sampling a variable along straight lines between pairs of points sampled from the two latent distributions \\citep{NIPS2017_892c3b1c}. When optimizing $L_\\mathrm{critic}$, only the critic network is trained, not propagating gradients to the modules prior to $\\bm{z}_h^-$ and $\\bm{z}_d^-$. Both the encoder and codebook for normal anatomy codes are used to bring the two distributions closer together in the process of optimizing $L_\\mathrm{reg}$.\n\nNote that, unlike usual GANs for image synthesis, our goal is to achieve an alignment between two latent distributions, $\\mathcal{Z}^-_h$ and $\\mathcal{Z}^-_d$. Since distribution matching is applied to the batch containing quantized latent codes, it is expected that certain constraints will be added to the process of vector quantization. More specifically, it can be expected to regularize code vectors in the codebook for normal anatomy codes not to be too representative even for abnormal imaging features. \n\n\\subsubsection{Overall learning objectives}\n\\label{sec:overall_learning objectives}\n\nIn summary, we define several loss functions: latent loss $L_\\mathrm{lat}$ for optimization of the encoder and codebooks; segmentation loss $L_\\mathrm{seg}$ for the segmentation decoder, encoder, and codebook for abnormal anatomy codes; reconstruction loss $L_\\mathrm{rec}$ for the image decoder, encoder, and codebook for normal anatomy codes; and distribution matching loss for the critic network $L_\\mathrm{critic}$ and that for the encoder and codebook for normal anatomy codes $L_\\mathrm{reg}$. The overall learning algorithm is shown in {\\bf Algorithm \\ref{alg:learning_algorithm}}.\n\n\\begin{algorithm}[t]\n\\SetAlgoLined\n $e$: encoder\\\\\n $v^-$: vector quantiser for normal anatomy code\\\\\n $v^+$: vector quantiser for abnormal anatomy code\\\\\n $s$: segmentation decoder\\\\\n $f$: image decoder\\\\\n $g$: critic network\\\\\n $\\mathcal{D}$: training dataset\\\\\n sg: stop-gradient operator\\\\\n $m$: the number of inner iteration for the critic network\\\\\n $\\lambda_1, \\lambda_2, \\lambda_3, \\lambda_4, \\lambda_5$: balancing terms for each loss function\\\\\n \\While{not converge}{\n  Sample a batch of images $\\bm{x}$ and segmentation labels for abnormal regions $\\bm{y}$ from $\\mathcal{D}$.\\\\\n  $\\bm{z}_e^-, \\bm{z}_e^+ \\leftarrow e(\\bm{x})$\\\\\n  $\\bm{z}_q^- \\leftarrow v^- (\\bm{z}_e^-)$\\\\\n  $\\bm{z}_q^+ \\leftarrow v^+ (\\bm{z}_e^+)$\\\\\n  $\\hat{\\bm{y}}, \\tilde{\\bm{y}} \\leftarrow s (\\bm{z}_q^+)$\\\\\n  $\\hat{\\bm{x}}^+ \\leftarrow f (\\bm{z}_q^-, \\mathrm{sg}(\\tilde{\\bm{y}}))$\\\\\n  $\\hat{\\bm{x}}^- \\leftarrow f (\\bm{z}_q^-, \\bm{0})$\\\\\n  Compute $L_\\mathrm{lat}(\\bm{z}_e^-, \\bm{z}_e^+)$, $L_\\mathrm{seg} (\\hat{\\bm{y}}, \\bm{y})$, and $L_\\mathrm{rec} (\\hat{\\bm{x}}^-, \\hat{\\bm{x}}^+, \\bm{x})$.\\\\\n  Split the batch of $\\{\\bm{z}_q^-\\}$ into subbatches of $\\{\\bm{z}^-_h\\}$ and $\\{\\bm{z}^-_d\\}$ according to the presence of abnormal label in each input image.\\\\\n  \\For{i = 1, $\\dots$, m}{\n    $\\bm{z}^-_h \\leftarrow \\mathrm{sg} (\\bm{z}^-_h)$\\\\\n    $\\bm{z}^-_d \\leftarrow \\mathrm{sg} (\\bm{z}^-_d)$\\\\\n    Sample a random number $\\epsilon \\sim \\mathbb{U}[0, 1]$.\\\\\n    $\\bm{z}^-_m \\leftarrow \\epsilon \\bm{z}^-_h + (1 - \\epsilon) \\bm{z}^-_d$\\\\\n    Compute $L_\\mathrm{critic} (\\bm{z}^-_h, \\bm{z}^-_d, \\bm{z}^-_m)$.\\\\\n    Update parameters of $g$ to minimize $\\lambda_5 L_\\mathrm{critic}$ using stochastic gradient descent (e.g., Adam).\\\\\n  }\n  Compute $L_\\mathrm{reg} (\\bm{z}^-_d)$.\\\\\n  Update parameters of $e$, $v^\\mp$, $s$, and $f$ to minimize $\\lambda_1 L_\\mathrm{lat} + \\lambda_2 L_\\mathrm{seg} + \\lambda_3 L_\\mathrm{rec} + \\lambda_4 L_\\mathrm{reg}$ using stochastic gradient descent (e.g., Adam). \n }\n \\caption{Training of the Feature Decomposing Network}\n \\label{alg:learning_algorithm}\n\\end{algorithm}\n\n\\subsection{Modeling of content-based image retrieval}\n\\label{sec:modeling_of_content-based_image_retrieval}\n\nHere, we formulate the problem of CBIR to find the closest feature vector from a reference database containing $N$ $D$-dimensional database vectors $\\{ r_n \\}^N_{n = 1}$, given a query vector $q$, as follows:\n\\begin{equation} \n\\mathop{\\rm arg~min}\\limits_{n \\in {1, \\ldots, N}} D(q, r_n),\n\\label{eq:separating_hyperplane}\n\\end{equation} \nwhere $D$ is a distance function such as Euclidean distance. \n\nStarting with the abovementioned equation, our consideration into the CBIR system is twofold. First, to enhance computational efficiency in the distance calculation, we propose to binarize latent representations by leveraging the latent spaces that are constructed as to be discrete rather than continuous. Then, a novel CBIR framework by utilizing the decomposed latent codes to retrieve images with targeted semantic components is introduced. \n\n\\subsubsection{Binary hashing based on separating hyperplanes}\n\\label{sec:binary_hashing_based_on_separating_hyperplanes}\n\nHere, we consider how to binarize code vectors in a codebook $\\bm{e} = \\{e_k |k = 1, \\ldots, K\\}\\in \\mathbb{R}^{K \\times D}$, which is learned in the feature decomposing network. The goal is to find subspaces that can be transversed by Hamming distance rearrangement. Since each code vector $e_k \\in \\mathbb{R}^D$ has a fixed position in the latent space, the latent space can be divided by $K \\choose 2$ separating hyperplanes that perpendicularly bisect a line segment connecting any two code vectors. Given two code vectors, $e_i$ and $e_j$, points $x$ located on the separating hyperplane can be formulated as follows:\n\\begin{equation} \nH_{(i, j)} (x) = (e_i - e_j) x - \\frac{1}{2} (\\|e_i\\|^2_2 - \\|e_j\\|^2_2) = 0.\n\\label{eq:separating_hyperplane}\n\\end{equation} \nTherefore, the position of another code vector $e_k$ can be binarized according to the side of the separating hyperplanes it is on:\n\\begin{equation} \n\\mathrm{sgn} [H_{i, j} (e_k)], \n\\label{eq:separating_hyperplane}\n\\end{equation} \nwhere\n\\begin{equation} \n\\mathrm{sgn} (x) = \\begin{cases}\n  1 & (x \\geq 0) \\\\\n  -1 & (x < 0) \\\\\n\\end{cases}.\n\\end{equation} \nThen, by considering the positional relationship between a total of $K \\choose 2$ separating hyperplanes, the continuous codebook $\\bm{e} \\in \\mathbb{R}^{K \\times D}$ can be converted into a binarized codebook $\\bm{b} = \\{b_k | k = 1, \\dots, K\\} \\in \\mathbb{R}^{K \\times E}$, where $E = $ $K \\choose 2$. Experimentally, we confirmed that there was no code vector that is exactly located on any separating hyperplane, which would make Eq. (\\ref{eq:separating_hyperplane}) equal to zero. Since the binary representation corresponding to the location of each vector is uniquely determined, it can be regarded as binary hashing.\n\n\\subsubsection{Optimization of the binarized vector length}\n\\label{sec:optimization_of_binarized_vector_length}\n\nWhen naively performing the abovementioned binarization, the length of each binarized code vector is ${K \\choose 2} = {512 \\choose 2} = 130,816$, which is extremely long to obtain the computational benefit from the Hamming distance calculation. Therefore, we consider optimization for the length of the binarized code vectors. At the time of performing nearest neighbor search, a local sensitivity is primarily required; that is, the relationship between the closest code vectors should remain the same. Meanwhile, it does not necessarily guarantee the positional relationship between distance code vectors. Based on these ideas, we can optimize the length of the binarized code vector by removing each element one by one to investigate whether the proximity between the closest vectors changes or not. This optimization algorithm is shown in {\\bf \\ref{app:optimization_algorithm}}. Empirically, the length of the code vectors can be reduced to $<$ 1\\%, allowing the calculation of Hamming distance to be much faster than that of L2 distance using continuous code vectors. Hereinafter, we denote $\\bm{b}^* = \\{b_k^* | k = 1, \\dots, K\\} \\in \\mathbb{R}^{K \\times E^*}$ as a binarized codebook after optimization. \n\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[]{.\/figures\/cbir_overview.jpg}\n  \\caption{\\textbf{Overview of the content-based image retrieval framework.} First, a query image and reference images are decomposed into normal and abnormal anatomy codes. Then, by calculating the similarity of normal anatomy codes between the query image and reference images ($S_\\mathrm{normal}$), similar images when viewed as normal anatomies without any abnormality can be retrieved. In contrast, by calculating the similarity of abnormal anatomy codes, images with similar tumor regions can be retrieved ($S_\\mathrm{abnormal}$). Besides, the similarity retrieval based on the whole imaging features can be applied by calculating the similarity combining both normal and abnormal anatomy codes ($S_\\mathrm{sum}$). Note that these similarity measurements can be calculated based on different distance definitions, such as Euclidean distance, angular distance, and Hamming distance.} \n  \\label{fig:cbir_overview}\n\\end{figure}\n\n\\subsubsection{Image retrieval based on the decomposed latent codes}\n\\label{sec:query_by_image}\n\nWhen performing the CBIR framework, a query image is decomposed into a normal anatomy code and abnormal anatomy code through the feature decomposing network ({\\bf Fig. \\ref{fig:cbir_overview}}). Then, the similarity using either the latent codes or a combination of both is calculated between the query image and images in the reference dataset. Particularly, when the similarity is viewed as counterfactual normal images as they should be, the similarity measurement uses only normal anatomy codes to be compared $S_\\mathrm{normal} (q, r)$. In contrast, when viewed by focusing only on abnormal areas, the similarity measurement uses only abnormal anatomy codes $S_\\mathrm{abnormal} (q, r)$. Moreover, to calculate similarities of the whole imaging features between the query and reference images, we define $S_\\mathrm{sum} (q, r)$ for the summation of both measurements as follows:\n\\begin{equation} \n\\label{eq:dsum}\nS_\\mathrm{sum} (q, r) = S_\\mathrm{normal} (q, r) + S_\\mathrm{abnormal} (q, r).\n\\end{equation} \n\nThese similarity measurements can be calculated based on different distance definitions. Here, for comparison, three types of distances, that is, Euclidean distance $D_\\mathrm{E}$, angular distance $D_\\mathrm{A}$, and Hamming distance $D_\\mathrm{H}$, are calculated for each similarity measurement. Note that the continuous codebook $\\bm{e}$ is the basis for Euclidean distance $D_\\mathrm{E}$ and angular distance $D_\\mathrm{A}$, while the optimized binarized codebook $\\bm{b}^*$ is the basis for Hamming distance $D_\\mathrm{H}$.\n\n\\section{Experiments}\n\\label{sec:experiments}\n\n\\subsection{Dataset}\n\\label{sec:dataset}\n\nWe used brain MR images of gliomas from the 2019 BraTS Challenge \\citep{6975210brats, Bakas2017, TCGAGBM, TCGALGG}, containing a dataset of 355 patients for training (MICCAI\\_BraTS\\_Training), a dataset of 125 patients for validation (MICCAI\\_BraTS\\_Validation), and a dataset of 167 patients (MICCAI\\_BraTS\\_Testing) for testing. For each patient, T1, Gd-enhanced T1, T2, and fluid-attenuated inversion recovery (FLAIR) sequences were obtained. MICCAI\\_BraTS\\_Training contains three segmentation labels of abnormality: Gd-enhancing tumor (ET), peritumoral edema (ED), and necrotic and non-enhancing tumor core (NET). Conversely, MICCAI\\_BraTS\\_Validation and MICCAI\\_BraTS\\_Testing do not have any segmentation label. There is no ground-truth that represents normal anatomical structures. Therefore, we segmented all three datasets into six normal anatomical labels (left cerebrum, right cerebrum, left cerebellum, right cerebellum, left ventricle, and right ventricle). Moreover, the abnormal regions (ET, ED, and NET) in MICCAI\\_BraTS\\_Validation and MICCAI\\_BraTS\\_Testing were segmented in the same manner with those in MICCAI\\_BraTS\\_Training. The annotation process is described in detail in {\\bf \\ref{app:annotation_process}}.\n\nFor the training of the feature decomposing network, we concatenated both MICCAI\\_BraTS\\_Validation and MICCAI\\_BraTS\\_Testing as a \\emph{training dataset}. Then, the feature decomposing network was evaluated on MICCAI\\_BraTS\\_Training as a \\emph{test dataset}, which is also utilized in the demonstration of the CBIR system based on the decomposed latent codes. \n\n\\subsection{Preprocessing}\n\\label{sec:pre-processing}\n\nAll four sequences, T1, Gd-enhanced T1, T2, and FLAIR, were concatenated into a four-channel MR volume $\\bm{X} \\in \\mathbb{R}^{4 \\times 240 \\times 240 \\times 155}$. Then, a preprocessing pipeline, including axial image resizing to $256 \\times 256$ and Z-score normalization, was performed. Subsequently, each three-dimensional (3D) MR volume was decomposed into a collection of 2D axial slices $\\{\\bm{x}_1, \\bm{x}_2, \\dots, \\bm{x}_{155} \\in \\mathbb{R}^{4 \\times 256 \\times 256} \\}$. The training and test datasets underwent the same preprocessing process. Data augmentation, such as random rotation and random horizontal flip, was applied to each image in the training dataset to train the feature decomposing network.\n\n\\subsection{Implementation}\n\\label{sec:implementation}\n\nAll experiments were implemented in Python 3.7 with PyTorch library 1.2.0 \\citep{NEURIPS2019_9015} using an NVIDIA Tesla V100 graphics processing unit and CUDA 10.0. For all networks, Adam optimization \\citep{kingma2014adam} was used for the training. Network initialization was performed using the method described by He et al. \\citep{he2015delving}.\n\n\\subsubsection{Feature decomposing network}\n\\label{sec:implement_feature_decomposing_network}\n\nWhen using a quantized latent space, such as VQ-VAE, the size of latent representation (i.e., width and height of the feature maps) exerts a significant effect on the quality of image generation \\citep{razavi2019generating}. Therefore, for comparison, we configured several architectures of the feature decomposing network depending on the size of the latent space. Hereinafter, we denote a feature decomposing network with the latent representation of the size of $H^\\prime \\times W^\\prime$ as $\\mathrm{FDN}_{H^\\prime \\times W^\\prime}$. In this study, we compared $\\mathrm{FDN}_{4 \\times 4}$, $\\mathrm{FDN}_{8 \\times 8}$, $\\mathrm{FDN}_{16 \\times 16}$, and $\\mathrm{FDN}_{32 \\times 32}$ with or without the distribution matching for the normal anatomy codes. \n\nAs described in \\textbf{Section \\ref{sec:training_of_feature_decomposing_network}}, the feature decomposing network comprises the encoder, segmentation decoder, and image decoder. See {\\bf \\ref{app:detailed_architecture}} for the detail of the architectures of these neural networks. The input for the encoder is required to be a four-channel 2D image with the size of $4 \\times 256 \\times 256$ $(= \\mathrm{channel} \\times \\mathrm{height} \\times \\mathrm{width})$. The encoder has a common trunk for obtaining an input image and extracting low-level features of the image and then bifurcates into two branches with the same architecture, one of which is for the normal anatomy code and the other is for the abnormal anatomy code. The number of a repeated structure, consisting of residual blocks \\citep{He2015} with a strided convolution (ResBlock + StridedConv), was adaptively set for each size of the latent representation. For example, in $\\mathrm{FDN}_{8 \\times 8}$, the encoder utilized $32-64-128-128-128-128$ filter kernels in each layer. The encoder outputs two latent representations corresponding to normal and abnormal semantic components, $\\bm{z}_e^-$ and $\\bm{z}_e^+$, with the size of $64 \\times H^\\prime \\times W^\\prime$. These latent representations are subsequently quantized to $\\bm{z}_q^-$ and $\\bm{z}_q^+$ through the vector quantization. The dimension of code vectors $D$ and number of code vectors $K$ in the codebooks were fixed as follows: $D = 64$, and $K = 512$.\n\nThe image and segmentation decoders have almost the same architecture, except for the normalization layer, where the image decoder particularly utilizes SPADE module for the propagation of softmax logits $\\tilde{\\bm{y}}$ from the segmentation network. The number of a repeated structure, which comprises upsampling module with a bilinear interpolation function followed by residual block with or without SPADE [Upsample + (SPADE-)ResBlock], was adopted for the size of the latent representation. For example, in $\\mathrm{FDN}_{8 \\times 8}$, the image and segmentation decoders utilized $128-128-128-128-64-32$ filter kernels in each layer. The segmentation decoder takes $\\bm{z}_q^+$ as an input and outputs a segmentation map $\\hat{\\bm{y}}$ with the size of $4 \\times 256 \\times 256$ $(= \\mathrm{channel} \\times \\mathrm{height} \\times \\mathrm{width})$, the channel number of which corresponds to the number of abnormality labels (ET, ED, and TC) plus a background label. Besides, a softmax logit $\\tilde{\\bm{y}}$ with the size of $4 \\times 256 \\times 256$ is retained for the collateral input for the image decoder. The image decoder utilizes $\\bm{z}_q^-$ as primary input. Depending on the presence or absence of the softmax logit $\\tilde{\\bm{y}}$ through the SPADE modules, it generates either a normal-appearing reconstruction $\\hat{\\bm{x}}^-$ or entire reconstruction of the input image $\\hat{\\bm{x}}^+$, respectively. \n\nWhen using distribution matching to regularize normal anatomy codes (see \\textbf{Section \\ref{sec:density_matching}}), the critic network is trained to approximate the Wasserstein distance $d_\\mathrm{W}$ of distributions between the normal anatomy codes originating from healthy images $\\mathcal{Z}_h^-$ and those originating from diseased images $\\mathcal{Z}_d^-$. The detailed network architecture is described in {\\bf \\ref{app:detailed_architecture}}. The number of inner iterations in the training of the critic network $m$ was set to 5, and the balancing term for the gradient penalty $\\lambda_{\\mathrm{gp}}$ was 10.0. To balance the magnitudes of the loss functions, $\\lambda_4$ and $\\lambda_5$ were set to $1.0 \\times 10^{-4}$ for $\\mathrm{FDN}_{4 \\times 4}$ and $\\mathrm{FDN}_{8 \\times 8}$ and $5.0 \\times 10^{-4}$ for $\\mathrm{FDN}_{16 \\times 16}$ and $\\mathrm{FDN}_{32 \\times 32}$. The larger values of $\\lambda_4$ and $\\lambda_5$ for each configuration tended to fail in mode collapse, where the encoder learned only a few modes of data in the sample distribution. The other hyperparameters were shared across the configurations as follows: batch size = 112, number of training epochs = 400, learning late = $1.0 \\times 10^{-4}$, weight decay = $1.0 \\times 10^{-5}$, $\\lambda_1 = 0.25$, $\\lambda_2 = 5.0$, $\\lambda_3 = 5.0$, and $\\gamma = 2.0$.\n\n\\subsubsection{Preparation of codebooks}\n\\label{sec:content-based_image_retrieval_system}\n\nThe CBIR framework was constructed on a per-image basis; that is, each MR volume was separated into slices along the axial axis, which was in the same manner with the training of the feature decomposing network. By straightforwardly applying the trained feature decomposing network, every single image in the test dataset was decomposed into normal anatomy code $\\bm{z}_q^-$ and abnormal anatomy code $\\bm{z}_q^+$. These latent codes can be considered as a set of code vectors extracted from the continuous codebooks $\\bm{e}^\\mp$, which is subjective to the calculation of Euclidean distance $D_\\mathrm{E}$ and angular distance $D_\\mathrm{A}$. Then, according to the methods described in \\textbf{Section \\ref{sec:binary_hashing_based_on_separating_hyperplanes}} and \\textbf{Section \\ref{sec:optimization_of_binarized_vector_length}}, we prepared a set of latent codes in a manner of binarized code vectors. First, code vectors with L2 norm $<$ $1.0 \\times 10^{-5}$ were rounded to a zero vector. Then, each code vector was binarized with optimized length to obtain the optimized binarized codebooks $\\bm{b}^{\\mp*}$. Binarized latent codes were acquired by searching the optimized binarized codebooks based on the same indices of the corresponding latent codes in the continuous codebooks. The similarity between binarized code vectors was evaluated based on Hamming distance $D_\\mathrm{H}$. \n\n\\subsection{Evaluation}\n\\label{sec:evaluation}\n\nSince end-to-end learning of the whole modules cannot be performed, we evaluated individual component to find the optimal combination for the CBIR framework. The evaluation process comprised three stages. First, to find the optimal configuration of the feature decomposing network, we assessed the error of image reconstruction, performance of abnormality segmentation, ``leakiness'' of feature decomposition, and compactness of the codebooks. Second, based on the feature decomposing network with a selected configuration, we observed the extent to which the distance relationship between code vectors was changed through the binarization. Lastly, we demonstrated the effectiveness of the proposed CBIR framework based on the decomposed latent codes from qualitative and quantitative perspectives.\n\n\\subsubsection{Image reconstruction error}\n\\label{sec:image_reconstruction_error}\n\nThe image decoder performs conditional image generation while switching the entire reconstructions $\\hat{\\bm{x}}^+$ and normal-appearing reconstructions $\\hat{\\bm{x}}^-$ (see \\textbf{Section \\ref{sec:feature_decoding}}). At the training stage, the entire reconstructions are similar to the input images. Meanwhile, normal-appearing reconstructions learn to match the region of the input images that excludes the abnormal area, which can be indicated by the mask matrix $\\bm{M}^-$. Therefore, the reconstruction error of the entire reconstructions and that of normal-appearing reconstructions can be calculated as $\\sum \\| \\bm{x} - \\hat{\\bm{x}}^+ \\|_1$ and $\\sum \\| \\bm{M}^- \\odot \\bm{x} - \\bm{M}^- \\odot \\hat{\\bm{x}}^- \\|_1$, respectively, where $\\sum$ denotes pixel-wise summation of the residual errors.\n\n\\subsubsection{Segmentation performance}\n\\label{sec:segmentation_performance}\n\nThe segmentation decoder predicts a segmentation output $\\hat{\\bm{y}}$, which should be close to the ground-truth label $\\bm{y}$ (see \\textbf{Section \\ref{sec:feature_decoding}}). The segmentation performance was evaluated based on Dice score \\citep{dice1945} with respect to the abnormality labels (ET, ED, and NET). The segmentation outputs in each 2D axial slice $\\{\\bm{x}_1, \\bm{x}_2, \\dots, \\bm{x}_{155}\\}$ were concatenated into a volume to evaluate the Dice score based on the volume. \n\n\\subsubsection{Leakiness of feature decomposition}\n\\label{sec:leakiness_of_feature_decomposition}\n\nAs described in \\textbf{Section \\ref{sec:density_matching}}, distribution matching is introduced to ensure that each decomposed latent code corresponds to targeted semantic content of the input images without being ``leaky'' to other features. Especially, this is important for normal anatomy codes because there is no ground-truth for normal-appearing reconstructions originating from diseased images. However, observation of the latent space cannot provide information on whether the representations contained therein are decomposed in a desirable manner. Hence, we observe reconstructed images generated from these latent codes through the image decoder. In the evaluation, a batch containing only the normal-appearing reconstructions $\\{\\hat{\\bm{x}}^-\\}$ was fed into a classification network to predict whether each normal-appearing reconstruction was derived from healthy images $\\mathcal{X}_h$ or diseased images $\\mathcal{X}_d$. The architecture of the classification network and training settings based on the training dataset are described in {\\bf \\ref{app:detailed_architecture_classifier}}. When the normal-appearing reconstructions do not contain any abnormal characteristics, the classification network should not be possible to identify their origin, even if they are derived from diseased images. Otherwise, it indicates that there are ``leaked'' abnormal imaging features in the normal-appearing reconstructions, which can be distinguishable for the origin from the diseased images. Positive predictive value (PPV) of the classification network with respect to the origin from the diseased images was evaluated as an indicator of leakiness in the test dataset. \n\n\\subsubsection{Compactness of the codebooks}\n\\label{sec:compactness_of_codebooks}\n\nIn some configurations of the feature decomposing network, it was empirically observed that a portion of the $K$ code vectors stored in a codebook exhibited small norms close to zero (see \\textbf{\\ref{app:distribution_of_norm}}). It suggested that these code vectors with small norms did not encode distinguishable features with zero vector, allowing approximation as the zero vector. Therefore, in that case, the number of code vectors that are actually significant was $< K$. Hence, we define the compactness of the codebooks to be the ratio of insignificant code vectors that showed small L2 norms below a threshold values of $1.0 \\times 10^{-5}$ to the total number of code vectors $K$ in a codebook. When the value of the compactness is large, it implies that the codebook can be considered as compact with less than initial $K$ code vectors. This is preferential for the computational cost at the time of similarity search because we can approximately reduce the number of code vectors to be computed.\n\n\\subsubsection{Validity of the binarization process}\n\\label{sec:validity_of_binarization_process}\n\nThe validity of the binarization process, which is introduced in \\textbf{Section \\ref{sec:binary_hashing_based_on_separating_hyperplanes}} and \\textbf{Section \\ref{sec:optimization_of_binarized_vector_length}}, was evaluated from three perspectives: concordance of the distance relationship of the code vectors before and after the binarization process, compression ratio of the code vector length, and comparison of computational time. To demonstrate concordance, we investigated whether the nearest neighbor relationship among binarized code vectors changed with respect to the continuous ones. Since the fundamental element of the latent space composed a fixed set of code vectors, the distance relationship as a whole can be regarded as almost unchanged, given that the distance relationship between each code vector is consistent. Therefore, the distance relationship of each code vector with the other remaining $K - 1$ code vectors was assessed and compared between the two types of codebooks. In the evaluation, for each code vector, the indices of the top-$Q$ closest code vectors were obtained. The distance calculation was performed using Hamming distance $D_\\mathrm{H}$ for the optimized binarized codebook $\\bm{b}^*$ and Euclidean distance $D_\\mathrm{E}$ for the continuous codebook $\\bm{e}$. Then, the concordance was calculated with respect to the agreement by the Jaccard similarity coefficient between two sets of indices. The compression ratio of the code vectors owing to the optimization process was calculated as follows: $\\frac{E^*}{E}$. See {\\bf \\ref{app:comparison_of_computational_time}} for the methodological details in the comparison of computational time in the distance calculation.\n\n\\subsubsection{CBIR performance based on the decomposed latent codes}\n\\label{sec:eval_query_by_image}\n\nTo quantify the image retrieval performance using the decomposed latent codes (see \\textbf{Section \\ref{sec:query_by_image}}), images containing the largest area of abnormality were selected from each MR volume in the test dataset based on ground-truth labels. We refer to these representative images as \\emph{query images}. For each query image, a \\emph{reference dataset} that comprised the rest of the images in the test dataset except for the images in the same MR volume of the query image was constructed. The images in the reference dataset are called \\emph{reference images}. Then, for each MR volume in the reference dataset, the image with the closest latent code to that of the query image was obtained. The obtained images from each MR volume were sorted with respect to the similarity. Lastly, top-$Q$ closest images, each of which belonged to different MR volume, were provided as retrieved images. This MR volume-wise retrieval can be appropriate in evaluating variable retrieved images by a single query image because one MR volume usually contains several images with similar appearance in continuous axial slices. \n\nTo report retrieval performance, the mean Dice coefficient based on two types of ground-truth labels (see \\textbf{Section \\ref{sec:dataset}}), one is for six categories of normal anatomy and the other is for three categories of abnormality, was assessed between each query image and top-$Q$ closest images from the reference dataset. The mean Dice coefficient was calculated for all query images and then averaged. As shown in \\textbf{Fig. \\ref{fig:cbir_overview}}, when using $S_\\mathrm{normal}$, the mean Dice coefficients based on the categories of normal anatomy should be high because the similarity is evaluated based on the normal anatomy codes that correspond to normal-appearing reconstructions. Conversely, image retrieval using $S_\\mathrm{abnormal}$ should be accompanied by high mean Dice coefficients in the categories of the abnormalities because it evaluates the similarity based on the abnormal anatomy codes relevant to tumor segmentation labels. Therefore, the mean Dice coefficient based on the ground-truth labels of the normal or abnormal anatomical categories was reported for the image retrieval using $S_\\mathrm{normal}$ or $S_\\mathrm{abnormal}$, respectively. When it comes to the similarity retrieval using $S_\\mathrm{sum}$, the mean Dice coefficients based on the ground-truth labels of both normal and the abnormal anatomical categories were averaged because the similarity measurement should represent the features of the whole images. \n\nFor comparison, a \\emph{brutal search} to directly maximize Dice overlap was performed for each query image to retrieve top-$Q$ closest images. In comparing the retrieval performance using $S_\\mathrm{normal}$ or $S_\\mathrm{abnormal}$, the brutal search retrieved images by maximizing Dice coefficients between the ground-truth labels of the normal or abnormal anatomical categories of a query image and those of reference images, respectively. For the similarity measurement using $S_\\mathrm{sum}$, the brutal search maximized the simple summation of Dice coefficients of the ground-truth labels of the normal and abnormal anatomical categories. Similar image retrieval using the brutal search was conducted in the same MR volume-wise manner. While the brutal search requires a significant computational time and ground-truth labels for all query and reference images, the mean Dice coefficients obtained can be used as an oracle (technical upper bound). \n\n\\section{Results}\n\\label{sec:results}\n\n\\begin{figure*}[t]\n  \\centering\n  \\includegraphics[]{.\/figures\/example_training_results.jpg}\n  \\caption{\\textbf{Example results of model training based on $\\boldsymbol{\\mathrm{FDN}_{8 \\times 8}}$ with distribution matching.} The first row indicates the input images $\\bm{x}$. Entire input images were reconstructed $\\hat{\\bm{x}}^+$ (second row) based on both normal and abnormal anatomy codes, whereas normal-appearing reconstructions $\\hat{\\bm{x}}^-$ (third row) were generated on normal anatomy code only. A clear distinction can be observed between $\\hat{\\bm{x}}^+$ and $\\hat{\\bm{x}}^-$ at abnormal regions, which existed in both $\\bm{x}$ and $\\hat{\\bm{x}}^+$ but not in $\\hat{\\bm{x}}^-$. The fourth and fifth rows indicate ground-truth segmentation labels $\\bm{y}$ representing the tumor region and prediction for the labels $\\hat{\\bm{y}}$, respectively. The output of segmentation labels tended to be rounded and did not recover the detailed shape of each region. We consider this as a natural consequence since the compressed representation in the latent codes, which is advantageous for the computational cost of similarity search, did not have sufficient capacity to preserve the detailed feature in the input image as a tradeoff. ET, Gd-enhancing tumor; ED, peritumoral edema; NET, necrotic and non-enhancing tumor core.}\n  \\label{fig:example_training_results}\n\\end{figure*}\n\nHere, we first present an example training process of the feature decomposing network. Then, several configurations of the feature decomposing network, particularly focusing on the size of the latent representation and use of distribution matching, are compared to select a model to be exploited for the downstream task. Subsequently, using the feature decomposing network with the selected configuration, we confirm the validity of binary hashing. Lastly, the CBIR system utilizing decomposed latent codes is qualitatively and quantitatively evaluated. \n\n\\subsection{Example training results of the feature decomposing network}\n\\label{sec:training_results_of_the_feature_decomposing_network}\n\nAmong several configurations of the feature decomposing network, an example of the training results of $\\mathrm{FDN}_{8 \\times 8}$ with distribution matching at epoch 400 is shown in {\\bf Fig. \\ref{fig:example_training_results}}. The first, second, and third rows demonstrate the input images $\\bm{x}$, corresponding entire image reconstructions $\\hat{\\bm{x}}^+$, and normal-appearing reconstructions $\\hat{\\bm{x}}^-$, respectively. We can observe that there is a clear distinction between two types of reconstructions, $\\hat{\\bm{x}}^+$ and $\\hat{\\bm{x}}^-$, especially when evaluating abnormal regions, which existed in the input images $\\bm{x}$ and entire reconstructions $\\hat{\\bm{x}}^+$ but disappeared in the normal-appearing reconstructions $\\hat{\\bm{x}}^-$. Note that the regions where the abnormality had existed originally were replaced by imaging characteristics of normal neuroanatomy in the normal-appearing reconstructions $\\hat{\\bm{x}}^-$. This indicates the complementary capacity derived from normal anatomy codes. Besides, due to the shared normal anatomy codes between the two types of reconstructions, $\\hat{\\bm{x}}^+$ and $\\hat{\\bm{x}}^-$, the appearances outside the region of abnormality seems to be almost identical with each other. Furthermore, the fourth and fifth rows in {\\bf Fig. \\ref{fig:example_training_results}} indicate ground-truth segmentation labels for the abnormality (ET, TC, and NET) and prediction for the labels through the segmentation decoder, respectively. The output of segmentation labels tended to be rounded, without precisely recovering the detailed shape of each tumor region. This can be expected because the compressed representation of the latent space, where the spatial resolution is only $8 \\times 8$, is advantageous for computational efficiency at the time of similarity search. For the same reason, we did not pursue the generation quality of the reconstructed images as a primary purpose. Although the detailed part of the textual appearance as realistic MR images was not perfectly reproduced, we consider that it is sufficient for recognizing anatomical location and presence of abnormality in the reconstructed images. The learning process of this example model is demonstrated in {\\bf \\ref{app:example_learning_process}}.\n\n\\subsection{Comparison between several configurations of the feature decomposing network}\n\\label{sec:comparison_between_several_configuration}\n\nTo evaluate the effects of the size of the latent representation and use of the distribution matching, we compared several configurations of the feature decomposing network, such as $\\mathrm{FDN}_{4 \\times 4}$, $\\mathrm{FDN}_{8 \\times 8}$, $\\mathrm{FDN}_{16 \\times 16}$, and $\\mathrm{FDN}_{32 \\times 32}$, with or without distribution matching (see \\textbf{Section \\ref{sec:implement_feature_decomposing_network}}). See {\\bf Fig. \\ref{fig:comparison_reconstructions}} for visual results, where two types of reconstructed images, entire image reconstructions $\\hat{\\bm{x}}^+$ and normal-appearing reconstructions $\\hat{\\bm{x}}^-$, according to an input image are shown for each configuration. As the resolution of the latent space increased from $4 \\times 4$ to $32 \\times 32$, the quality of the reconstructed images was improved, showing that fine textures of the brain MR images were reproduced more realistically. As for the difference between entire reconstructions $\\hat{\\bm{x}}^+$ and normal-appearing reconstructions $\\hat{\\bm{x}}^-$, it is expected that there should be differences in the areas that correspond to the abnormal sites. Thus, the abnormal areas should appear only in the entire reconstructions $\\hat{\\bm{x}}^+$ and should be diminished in the normal-appearing reconstructions $\\hat{\\bm{x}}^-$. When the resolution of the latent representation is relatively low (i.e., $\\mathrm{FDN}_{4 \\times 4}$ and $\\mathrm{FDN}_{8 \\times 8}$), the difference was clear even without distribution matching. Conversely, particularly in $\\mathrm{FDN}_{16 \\times 16}$ and $\\mathrm{FDN}_{32 \\times 32}$, the abnormal regions, which exhibited as low-intensity area in Gd-enhanced T1 sequence and high-intensity area in FLAIR sequence, were partly reproduced even in the normal-appearing reconstructions $\\hat{\\bm{x}}^-$ especially when distribution matching was not utilized (see arrows in {\\bf Fig. \\ref{fig:comparison_reconstructions}}). As mentioned in \\textbf{Section \\ref{sec:leakiness_of_feature_decomposition}}, we call this failure in decomposing representations as ``leakiness'' of abnormal features into normal anatomy codes $\\bm{z}_q^-$. Note that the distribution matching imposed on the normal anatomy codes mitigated this leakiness and encouraged normal-appearing reconstructions to replace the region of abnormality with normal imaging features that would have existed therein if the sample is healthy (indicated by arrowheads in {\\bf Fig. \\ref{fig:comparison_reconstructions}}). \n\n\\begin{figure}[p]\n  \\centering\n  \\includegraphics[]{.\/figures\/comparison_reconstructions.jpg}\n  \\caption{\\textbf{Comparison of reconstructed images according to different configurations of the feature decomposing network.} As the spatial resolution of the latent representation increases from $\\mathrm{FDN}_{4 \\times 4}$ to $\\mathrm{FDN}_{32 \\times 32}$, visual fidelity of the reconstructed images tends to improve. Note that there are partially reconstructed abnormal regions (arrows) even in the normal-appearing images $\\hat{\\bm{x}}^-$ through the models with relatively high resolutions (i.e., $\\mathrm{FDN}_{16 \\times 16}$ and $\\mathrm{FDN}_{32 \\times 32}$). These ``leaky'' appearances can be alleviated by imposing the distribution matching (arrowheads). DM, distribution matching.}\n  \\label{fig:comparison_reconstructions}\n\\end{figure}\n\n\\begin{figure}[t!]\n  \\centering\n  \\includegraphics[]{.\/figures\/image_recon_results.jpg}\n  \\caption{\\textbf{Image reconstruction results.} Mean $\\pm$ standard deviation of the image reconstruction error (see \\textbf{Section \\ref{sec:image_reconstruction_error}}) is shown for each configuration of the feature decomposing network. As the resolution of the latent representation increased from $\\mathrm{FDN}_{4 \\times 4}$ to $\\mathrm{FDN}_{32 \\times 32}$, the quality of image reconstruction was improved with decreased reconstruction errors. Note that the distribution matching did not cause any negative effect. FDN, feature decomposing network; DM, distribution matching.}\n  \\label{fig:image_recon_results}\n\\end{figure}\n\n\\begin{figure}[t!]\n  \\centering\n  \\includegraphics[]{.\/figures\/segmentation_results.jpg}\n  \\caption{\\textbf{Segmentation results.} Mean $\\pm$ standard deviation of the averaged Dice scores for the abnormality labels (see \\textbf{Section \\ref{sec:segmentation_performance}}) is shown for each configuration of the feature decomposing network. As the resolution of the latent representation increased from $\\mathrm{FDN}_{4 \\times 4}$ to $\\mathrm{FDN}_{32 \\times 32}$, the segmentation performance was improved. Note that the distribution matching did not cause any negative effect. FDN, feature decomposing network; DM, distribution matching.}\n  \\label{fig:segmentation_results}\n\\end{figure}\n\n\\begin{figure}[t!]\n  \\centering\n  \\includegraphics[]{.\/figures\/leakiness_results.jpg}\n  \\caption{\\textbf{Evaluation of the leakiness in the normal-appearing reconstructions.} A classification network was trained based on four different reconstructions, such as entire reconstructions, entire reconstructions with distribution matching, normal-appearing reconstructions, and normal-appearing reconstructions with distribution matching, to classify whether the original images included the abnormality or not (see \\textbf{Section \\ref{sec:leakiness_of_feature_decomposition}}). Here, leakiness of feature decomposition means the tendency that the normal-appearing images unintentionally contained distinguishable features of abnormality. Mean $\\pm$ standard deviation of positive predictive values (PPVs) for the origin from diseased images measured in the test dataset are shown as an indicator of the leakiness. As the latent resolution increased, the PPVs increased (orange bars), especially without the distribution matching. This leakiness of abnormal features was alleviated by imposing the distribution matching (blue bars). FDN, feature decomposing network; DM, distribution matching.}\n  \\label{fig:leakiness_results}\n\\end{figure}\n\n\\begin{table}[t]\n\\resizebox{\\linewidth}{!}{\\begin{tabular}{@{}lccc@{}}\n\\toprule\n\\textbf{FDN model}                              & \\textbf{Normal anatomy code} & \\textbf{Abnormal anatomy code} \\\\ \\midrule\n$\\mathrm{FDN}_{4 \\times 4}$ w\/ DM               & 0.79                         & 0.75                           \\\\\n$\\mathrm{FDN}_{4 \\times 4}$ w\/o DM              & 0.75                         & 0.72                           \\\\\n$\\mathrm{FDN}_{8 \\times 8}$ w\/ DM               & 0.68                         & 0.69                           \\\\\n$\\mathrm{FDN}_{8 \\times 8}$ w\/o DM              & 0.63                         & 0.67                           \\\\\n$\\mathrm{FDN}_{16 \\times 16}$ w\/ DM             & 0.71                         & 0.52                           \\\\\n$\\mathrm{FDN}_{16 \\times 16}$ w\/o DM            & 0.25                         & 0.57                           \\\\\n$\\mathrm{FDN}_{32 \\times 32}$ w\/ DM             & 0.31                         & 0.0                            \\\\\n$\\mathrm{FDN}_{32 \\times 32}$ w\/o DM            & 0.0                          & 0.0                            \\\\ \\bottomrule\n\\end{tabular}}\n\\caption{\\textbf{Compactness of the codebooks.} The compactness, which is the ratio of insignificant code vectors showing small L2 norms below a threshold to the total number of the code vectors (see \\textbf{Section \\ref{sec:compactness_of_codebooks}}), is shown for each configuration of the feature decomposing network. FDN, feature decomposing network; DM, distribution matching.} \n\\label{tab:compactness_of_codebooks}\n\\end{table}\n\nSubsequently, we quantitatively compared several configurations based on the reconstruction error (see \\textbf{Section \\ref{sec:image_reconstruction_error}}), performance of abnormality segmentation (see \\textbf{Section \\ref{sec:segmentation_performance}}), leakiness of feature decomposition (see \\textbf{Section \\ref{sec:leakiness_of_feature_decomposition}}), and compactness of the codebooks (see \\textbf{Section \\ref{sec:compactness_of_codebooks}}). As shown in {\\bf Fig. \\ref{fig:image_recon_results}} and {\\bf Fig. \\ref{fig:segmentation_results}}, the performances of both image reconstruction and segmentation were improved as the resolution of the latent representation increased, which is demonstrated by the reduced reconstruction errors and increased Dice coefficients, respectively. Since the distribution matching is expected to impose some regularization effects on the codebooks, the performance of image reconstruction and segmentation could be degraded by training with distribution matching; however, the data also indicate that it did not cause any negative effect on both image reconstruction and segmentation performance. For the leakiness evaluated in {\\bf Fig. \\ref{fig:leakiness_results}}, without the distribution matching, the tendency that the normal-appearing reconstructions unintentionally contained distinguishable features of abnormality became apparent when the latent resolution increased. For example, this can be evident at $\\mathrm{FDN}_{32 \\times 32}$, where PPV for the origin of diseased images of the classification network trained based on normal-appearing reconstructions (see the orange bar in {\\bf Fig. \\ref{fig:leakiness_results}}) was as high as those of the classification network trained based on entire reconstructions (see the yellow and gray bars in {\\bf Fig. \\ref{fig:leakiness_results}}). However, the leakiness of abnormal features could be mitigated to some extent by introducing distribution matching. Note that PPVs for the origin of diseased images of the classification network trained on normal-appearing reconstructions (see the orange bars in {\\bf Fig. \\ref{fig:leakiness_results}}) consistently decreased by imposing distribution matching (see the blue bars in {\\bf Fig. \\ref{fig:leakiness_results}}). Regarding the compactness of the codebooks shown in {\\bf Table \\ref{tab:compactness_of_codebooks}}, models with lower latent resolutions showed higher value. Besides, it is noteworthy that distribution matching had the effect of rendering the codebooks more compact with the higher ratio of insignificant code vectors, which can be observed except for the abnormal anatomy codes of $\\mathrm{FDN}_{16 \\times 16}$ with distribution matching. See {\\bf \\ref{app:distribution_of_norm}} for the distribution of norms of code vectors in each configuration, which also supports that models with low latent space resolution had more code vectors with small norms.\n\nIn summary, there is a tradeoff between the performance of image reconstruction and segmentation and leakiness of feature decomposition and compactness of the codebooks according to the spatial resolution of latent representations. Note that, when the length and width of the latent space doubles (e.g., from $8 \\times 8$ to $16 \\times 16$), the computational complexity of the code vectors quadruples. Hereinafter, we selected $\\mathrm{FDN}_{8 \\times 8}$ with distribution matching for the following evaluation because the model exhibited intermediate characteristics, enabling good feature decomposition and simple latent representation with compact codebooks, which will be favorable at the time of similarity calculation. \n\n\\subsection{Assessment of the binarization process}\n\\label{sec:assessment_of_binarization}\n\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[]{.\/figures\/binarization.jpg}\n  \\caption{\\textbf{Relationship between Hamming distance and Euclidean distance.} The distances between one code vector and all other code vectors were calculated by Hamming distance (horizontal axis) and Euclidean distance (vertical axis). \\textbf{(a)} Distance relationship between the continuous codebook and binarized codebook before optimization. \\textbf{(b)} The same relationship after the optimization of the binarized codebook length. Note that not only the nearest relationship but also the global relationship is maintained through the optimization process.}\n  \\label{fig:binarization}\n\\end{figure}\n\nUsing $\\mathrm{FDN}_{8 \\times 8}$ with distribution matching as the feature decomposing network, we validated the process of the binarization of code vectors (see \\textbf{Section \\ref{sec:validity_of_binarization_process}}). Based on the techniques described in \\textbf{Section \\ref{sec:optimization_of_binarized_vector_length}}, the length of the optimized binarized vector was shortened from $13,816$ to 789 for the normal anatomy codes and 292 for the abnormal anatomy codes. The compression ratios $\\frac{E^*}{E}$ for the normal and abnormal anatomy codes were 0.60\\% and 0.22\\%, respectively. {\\bf Fig. \\ref{fig:binarization}} demonstrates an example result on the relationship between Hamming distance $D_\\mathrm{H}$ and Euclidean distance $D_\\mathrm{E}$ before ({\\bf Fig. \\ref{fig:binarization}a}) and after ({\\bf Fig. \\ref{fig:binarization}b}) optimization of the vector length. Note that, even though the optimization process considered only the nearest neighbor relationship, the global distance relationship was also maintained. The concordances of the top 1, 5, and 10 closest relationships between the Hamming distance calculation using the optimized binarized codebook $\\bm{b}^*$ and the Euclidean distance calculation using the continuous codebook $\\bm{e}$ were 0.81, 0.89, and 0.90 for normal anatomy codes and 0.81, 0.88, and 0.90 for abnormal anatomy codes, respectively. Notably, the relationships between the binarized codebook $\\bm{b}$ before the optimization and the continuous codebook $\\bm{e}$ were at similar levels, demonstrating that the concordances of top 1, 5, and 10 closest relationships were 0.82, 0.86, and 0.89 for normal anatomy codes and 0.84, 0.87, and 0.89 for abnormal anatomy codes, respectively. Therefore, the optimization process of the binarized vector length did not hinder the distance relationship with respect to the original Euclidean distance. Hamming distance calculation using the optimized binarized codebook $\\bm{b}^*$ reduced the computational time by 48.3\\% and 64.5\\% for normal and abnormal anatomy codes, respectively, compared to the Euclidean distance calculation using the continuous codebook $\\bm{e}$ (see \\textbf{\\ref{app:comparison_of_computational_time}}).\n\n\\subsection{Retrieval results based the decomposed latent codes}\n\\label{sec:result_query_by_image}\n\n\\begin{figure*}[t]\n  \\centering\n  \\includegraphics[]{.\/figures\/query_by_image.jpg}\n  \\caption{\\textbf{Example results of CBIR based on the decomposed latent codes.} Image retrieval was performed based on volume, comparing the closest images from each MR volume. Each similarity was calculated according to Euclidean distance $D_\\mathrm{E}$ based on the continuous codebooks $\\bm{e}$. Retrieved images are arranged from left to right, starting with the closest one to the query vector. ({\\bf a}) Similarity calculation based on normal anatomy codes $S_\\mathrm{normal}$ retrieved images with similar normal anatomical labels, irrespective of gross abnormalities. Query and retrieved images with ground-truth labels of the normal anatomical categories are shown. ({\\bf b}) Similarity calculation based on abnormal anatomy codes $S_\\mathrm{abnormal}$ retrieved images with similar abnormal anatomical labels. Query and retrieved images with ground-truth labels of the abnormal anatomical categories are shown. Note the variety of normal anatomical contexts of the retrieved images. ({\\bf c}) Similarity calculation using $S_\\mathrm{sum}$ retrieved images with combined features, as shown by the similar patterns of both normal and abnormal anatomical labels among the retrieved images. Query and retrieved images with ground-truth labels of the normal and abnormal anatomical categories are shown. ET, Gd-enhancing tumor; ED, peritumoral edema; NET, necrotic and non-enhancing tumor core.}\n  \\label{fig:query_by_image}\n\\end{figure*}\n\n\\begin{table}[t]\n\\centering\n\\resizebox{0.8\\linewidth}{!}{%\n\\begin{tabular}{@{}lccc@{}}\n\\toprule\n\\textbf{Distance measurements} & \\boldsymbol{$S_\\mathrm{normal}$} & \\boldsymbol{$S_\\mathrm{abnormal}$} & \\boldsymbol{$S_\\mathrm{sum}$}\\\\ \\midrule\n$D_\\mathrm{E}$                 & $0.50 \\pm 0.16$              & $0.28 \\pm 0.12$                        & $0.20 \\pm 0.16$\\\\\n$D_\\mathrm{A}$                 & $0.47 \\pm 0.19$              & $0.27 \\pm 0.11$                        & $0.30 \\pm 0.11$\\\\\n$D_\\mathrm{H}$                 & $0.48 \\pm 0.17$              & $0.28 \\pm 0.12$                        & $0.23 \\pm 0.15$\\\\\nBrutal search                  & $0.62 \\pm 0.16$              & $0.37 \\pm 0.08$                        & $0.42 \\pm 0.08$\\\\ \\bottomrule\n\\end{tabular}\n}\n\\caption{\\textbf{Quantitative evaluation of the image retrieval.} Mean $\\pm$ standard deviation of the averaged Dice coefficients for ground-truth labels with respect to the semantic components exploited by the similarity evaluation is shown.} \n\\label{tab:quantitative_evaluation}\n\\end{table}\n\nAn example of the CBIR results showing the top 5 images with the closest latent codes based on $S_\\mathrm{normal}$, $S_\\mathrm{abnormal}$, and $S_\\mathrm{sum}$ is presented in {\\bf Fig. \\ref{fig:query_by_image}}. Similarity calculation based on normal anatomy codes $S_\\mathrm{normal}$ retrieved images with similar normal anatomical labels irrespective of gross abnormalities ({\\bf Fig. \\ref{fig:query_by_image}a}). Similarity calculation based on abnormal anatomy codes $S_\\mathrm{abnormal}$ retrieved images with similar abnormal anatomical labels ({\\bf Fig. \\ref{fig:query_by_image}b}). Note that the variety of normal anatomical contexts of the retrieved images accompanied similar abnormal lesions. In the similarity retrieval using $S_\\mathrm{sum}$, as formulated in Eq. (\\ref{eq:dsum}), the similarity measurements between the normal and abnormal anatomy codes were summed. As shown in {\\bf Fig. \\ref{fig:query_by_image}c}, the whole imaging features of the retrieved images using $S_\\mathrm{sum}$ resemble those of the query image. In this example, Euclidean distance $D_\\mathrm{E}$ was employed based on the continuous codebooks $\\bm{e}$. \n\nAccording to the method described in \\textbf{Section \\ref{sec:eval_query_by_image}}, we performed top 10 nearest neighbor search between query images and reference dataset for quantitative evaluation. Here, the query images are the slices with the largest tumor area in each MR volume, and the reference dataset contains the rest of images in the test dataset, except for the images in the same MR volume of each query image. The similarity measurements utilized $S_\\mathrm{normal}$, $S_\\mathrm{abnormal}$, and $S_\\mathrm{sum}$. Mean Dice coefficients were assessed according to the semantics exploited in the similarity measurement: Dice coefficients between ground-truth labels of the normal anatomical categories were averaged when using $S_\\mathrm{normal}$, those between ground-truth labels of the abnormal anatomical categories were averaged when using $S_\\mathrm{abnormal}$, and those between ground-truth labels of the normal anatomical categories and those between ground-truth labels of the abnormal anatomical categories were averaged when using $S_\\mathrm{sum}$. The three types of distance calculations, Euclidean distance $D_\\mathrm{E}$, angular distance $D_\\mathrm{A}$, and Hamming distance $D_\\mathrm{H}$, were calculated for each similarity measurement. The results are summarized in {\\bf Table \\ref{tab:quantitative_evaluation}}. The technical upper bound was provided by the brutal search, which retrieved images to maximize Dice coefficients directly computing the overlap between ground-truth labels. Note that the brutal search is performed based on the ground-truth labels for all query and reference images, which is usually not given in real clinical practice. \n\n\\section{Discussion and conclusions}\n\\label{sec:discussion}\n\nComparative diagnostic reading is critical in correct diagnosis by comparing an image of the condition to be diagnosed with corresponding normal images without abnormal findings or images that contain similar abnormal findings. To the best of our knowledge, this is the first study that proposes a deep-learning-based algorithm specifically designed to support the comparative diagnostic reading. The fundamental contribution of our study is to extend the idea of disentangled representation into a CBIR application in medical imaging. By leveraging the feature decomposing network, a medical image can be decomposed into a normal and abnormal anatomy code, each of which represents the targeted semantic component in the image. Note that the codebooks located at the bottom of the network allows decomposed latent codes to be manipulable for the CBIR framework, which can switch the semantic components to be focused in the retrieval according to users' purposes. \n\nThe proposed CBIR framework demonstrated notable results in both qualitative and quantitative evaluation (see \\textbf{Section \\ref{sec:result_query_by_image}}). However, because of the two-staged approach to establish the CBIR framework, there are possible error origins for the final image retrieval performance, such as errors in image reconstruction, segmentation, and vector quantization in the codebooks. Moreover, the leakiness of abnormal imaging features into the normal anatomical codes can hinder retrieval performance such that the retrieval based on $D^\\mathrm{normal}$ unintentionally accompanies images with significant amount of abnormalities, which can be alleviated by distribution matching. Because it is quite overloaded as an experiment, directly comparing the image retrieval performance according to the different configurations of the feature decomposing network could not be performed. Instead, we precisely evaluated these possible origins of errors (see \\textbf{Section \\ref{sec:comparison_between_several_configuration}}) and selected $\\mathrm{FDN}_{8 \\times 8}$ trained with distribution matching for reporting the CBIR performance owing to its preferable features for the image retrieval.\n\nEven though there is no benchmark available for the image retrieval based on the dataset, we consider that the performance difference from the brutal search (see \\textbf{Table \\ref{tab:quantitative_evaluation}}) can be acceptable for clinical use. Under the condition that the leakiness of abnormal imaging features is controlled, it is quite natural to speculate that the performance of the image reconstruction and segmentation has a positive relationship between the image retrieval performance using the decomposed latent codes. Therefore, if higher retrieval performance is desired and computational resources are sufficient, a model with higher latent space resolution can be used. Because the brutal search is an ideal setting, where ground-truth labels are given to all query and reference images, our CBIR framework is more versatile and useful, and furthermore, different configurations can be used for different situations. \n\nDistance calculations using Euclidean distance $D_\\mathrm{E}$, angular distance $D_\\mathrm{A}$, and Hamming distance $D_\\mathrm{H}$ can be implemented according to different situations, taking into account the tradeoff between accuracy and computational efficiency at the time of similarity search. Note that, with respect to $S_\\mathrm{sum}$, angular distance calculation $D_\\mathrm{A}$ provided a higher value than others. This can be because the magnitude of the distance values is not normalized in the Euclidean distance calculation $D_\\mathrm{E}$ and Hamming distance calculation $D_\\mathrm{H}$, making it difficult to evenly weight the two terms, $S_\\mathrm{normal}(q, r)$ and $S_\\mathrm{abnormal}(q, r)$, in Eq. (\\ref{eq:dsum}). It is also noteworthy that, even though the proposed binarization technique is simple, Hamming distance calculation $D_\\mathrm{H}$ approximated the Euclidean distance calculation $D_\\mathrm{E}$ with notable accuracy (see \\textbf{Section \\ref{sec:assessment_of_binarization}}). Since the picture archiving and communication systems in hospitals usually contain a huge amount of medical images, our proposal of the binary hashing based on the discrete latent codes can also be effective even in the context of large-scale image search. \n\nOne may argue that the distribution matching should be applied in reconstructed images instead of latent codes. It seems to be one alternative method to decompose features of medical images, but there are some concerns. One is an issue called ``posterior collapse,'' which is caused by a powerful decoder ignoring latent codes, which can be observed in many VAE models \\citep{oord2017neural}. Because our primary purpose is to acquire good latent representation that can be faithfully representative for corresponding imaging features, we intentionally avoided imposing additional learning objective for decoders. Another concern is that the distribution matching for images with a resolution of $256 \\times 256$ was computationally expensive and required a long time for training of feature decomposing networks. We also encountered a more unstable training process, which is one of the intrinsic problems of GANs. As shown in \\textbf{Fig. \\ref{fig:comparison_reconstructions}}, where the constraints on the latent space are reflected in the differences in imaging features in the reconstructed images, we consider that distribution matching on the latent spaces should be more appropriate when it comes to CBIR utilizing latent codes. \n\nWe found that SPADE modules can effectively propagate the semantic layout obtained at the final layer of the segmentation decoder into the image reconstruction process of the image decoder. The success of the conditional image generation of the image decoder depending on the collateral input through the SPADE modules can also be observed in \\textbf{Fig. \\ref{fig:leakiness_results}}, where PPVs for the origin from diseased images of the classification network trained using the entire reconstructions were consistently $>$ 0.9 (see yellow and gray bars in \\textbf{Fig. \\ref{fig:leakiness_results}}). However, future technological challenges may lie in this regard. Ideally, the latent codes representing normal and abnormal semantic components of medical images should be distributed in a decomposable manner in a single space where they can be linearly computed with each other. Because the current study employed an architecture that holds two separated latent spaces for decomposed latent codes, the similarity calculation according to the whole imaging feature (Eq. (\\ref{eq:dsum})) might be deemed arbitrary. Even so, because simple autoencoders can be sufficient to calculate the similarity based on whole imaging features, we believe that our proposal is innovative to enable the CBIR to selectively utilize either normal or abnormal components of medical images to support comparative diagnostic reading.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section*{Introduction}\nThe theory of Lie groupoids can be viewed as a blend of geometry and Lie theory, and plays an \nimportant role in several branches of mathematics. Lie groupoids can be viewed as the global \nintegrations of geometrically defined Lie brackets and as such posses a natural deformation theory. \nIn \\cite{cms}, the authors wrote down an explicit cochain complex controlling such deformations. \nAs expected, this ``deformation cohomology'' is isomorphic to the cohomology of the adjoint representation (up to homotopy as in \\cite{ac}), but the point of \\cite{cms} is that the latter involves the choice of a connection, whereas the deformation complex is intrinsically defined. Any deformation\nof Lie groupoids defines a class in this deformation cohomology.\n\nLie groupoids also play an important role in noncommutative geometry where they give prime examples in the form of their associated convolution algebras. As is well-known, the deformation theory of an algebra is controlled by its Hochschild cohomology. Since a deformation of the underlying Lie groupoid induces a deformation of its convolution algebra, this strongly suggests a \nrelationship between Hochschild cohomology and the adjoint representation.\n\nThe aim of the present article is to shed light on this issue by exhibiting an explicit morphism of \ncochain complexes between the deformation complex of a Lie groupoid and the Hochschild complex \nof its convolution algebra which relates the deformation classes associated to a deformation of the \nunderlying Lie groupoid. We expect this morphism to be part of a larger picture computing the \nHochschild cohomology in terms of higher powers of the adjoint representation.\n\nA classical theme in the theory of Lie groupoids is its relation with the infinitesimal theory of Lie \nalgebroids. For the deformation complex this is highlighted by the ``van Est'' morphism to the \ndeformation complex of the Lie algebroid of Lie groupoid, a complex first considered in \\cite{cm}.  \n\nFrom the point of view of noncommutative geometry, the relation with the infinitesimal theory follows \nfrom ``quantization and the classical limit'': we show that our cochain morphism can be extended to the adiabatic groupoid \nof \\cite{connes} interpolating between a Lie groupoid and its Lie algebroid. The van Est-map is \nthen obtained by constructing a quantization map on the dual of the Lie algebroid using an exponential map on the Lie groupoid. The picture is completed by the relationship between \nthe deformation cohomology a Lie algebroid  and the Poisson cohomology of its dual. \n\nThis article is organized as follows: in \\cref{prelim} we recall the basic set-up of Lie groupoids and their convolution algebras. In \\cref{chainmap} we construct the morphism between the deformation complex of a Lie groupoid and the Hochschild complex of its convolution algebra, and explore some of its properties. Finally, \\cref{quant} is devoted to obtaining the van Est map using the adiabatic groupoi together with a quantization.\n\n\\section{Preliminaries}\\label{prelim}\n\\subsection{Densities along the fibers of a submersion}\n\\label{das}\nLet $V$ be an n-dimensional vector space. A \\textit{density} of $V$ is a map $a:\\Lambda^n V\\to\\mathbb{R}$ such that for every invertible map $A\\in\\text{GL}(n,\\mathbb{R})$ it holds that $a(Av_1,...,Av_n)=|\\det(A)|a(v_1,...,v_n)$.\n\nMore generally, from a vector bundle $E\\to M$, one constructs a bundle of densities $\\mathcal{D}_E$. Then if one has a vector bundle isomorphism $\\Psi: E\\to E$ covered by a diffeomorphism $\\Phi: M\\to M$, one obtains an action on the section of $\\mathcal{D}_E$, defined by\n\\begin{equation*}\n(\\Psi^\\ast a)_x(v_1,...,v_n)=a_{\\Phi(x)}(\\Psi v_1,...,\\Psi v_n)\n\\end{equation*}\nThe case where $E=TM$ is of particular interest because the integral $\\int_Ma$ is \ncanonically defined for a compactly supported density of $TM$. In this case one obtains an action of a vector field $X\\in\\mathfrak{X}(M)$ on the densities on $TM$, namely:\n\\begin{equation}\n\\label{apvf}\nXa=\\left.\\frac{d}{dt}\\right|_{t=0} (\\Phi^t_X)^\\ast a,\n\\end{equation}\nwhere $\\Phi^t_X$ denotes the flow of $X$. \n\nWe will be mostly interested in densities along the fibers of a submersion. For this, let $f: M\\to N$ be a submersion, and denote by $\\mathcal{D}_f$ the bundle of densities of the vector bundle $\\ker df$. In this case the fiber integral\n\\[\n\\int_f:\\Gamma_c(M,\\mathcal{D}_f)\\to C^\\infty_c(N)\n\\]\nis canonically defined. A vector field $X$ acts on sections of $\\mathcal{D}_f$ provided that the flow preserves the fibers of $f$. This is equivalent to there being a vector field $Y\\in\\mathfrak{X}(N)$ such that $df\\circ X=Y\\circ f$. In this case $X$ is called \\textit{$f$-projectable}, and since $\\Phi^t_X\\circ f=f\\circ\\Phi^t_Y$ the flow of $X$ preserves the fibers of $f$ and in turn acts on $\\ker df$, and we obtain an action of $X$ on $\\Gamma(\\mathcal{D}_f)$ by formula \\eqref{apvf}.\nWe denote by $\\text{Diff}_f(M)$ the diffeomorphisms of $M$ that preserve the fibers of $f$, and by $\\mathfrak{X}_f(M)$ the $f$-projectable vector fields of $M$.\n\nIn the following we shall consider $f$-projectable vector fields defined on only a single fiber of $f$ and let it act on densities to get a density on that one fiber. This is similar to the fact that the directional derivative $X(f)(p)$ of a function $f$ along a vector field $X$ in a point $p$ only depends\non $X(p)$. \n\\begin{lem}\n\\label{nl}\nLet $a\\in\\Gamma(\\mathcal{D}_f)$, let $y\\in N$, and let $X\\in\\mathfrak{X}_f(M)$ be an $f$-projective vector field. If $X$ vanishes along $f^{-1}(y)$, then $(Xa)_x=0$ for all $x\\in f^{-1}(y)$.\n\\end{lem}\n\\begin{proof}\nIf $X$ vanishes along $f^{-1}(y)$ we have $\\Phi^t_X(x)=x$ for all $t$ and all $x\\in f^{-1}(y)$. In particular $d(\\Phi^t_X)_x(v)=v$ for all $v\\in\\text{ker}(df)\\subset T_xM$. This means that $((\\Phi^t_X)^\\ast a)_x=a_x$ and hence $(Xa)_x=0$.\n\\end{proof}\n\\begin{rmk}\nThe previous Lemma allows us to define $(Xa)_x$ for $x\\in f^{-1}(y)$, $a\\in\\Gamma(\\mathcal{D}_f)$ and $X\\in\\mathfrak{X}_f(M)|_{f^{-1}(y)}$. Indeed, we can choose $Y\\in\\mathfrak{X}_f(M)$ to be an extension of $X$ to a global vector field and define $(Xa)_x=(Ya)_x$. The previous Lemma is then used to show that this definition is independent of the choice of $Y$.\n\\end{rmk}\n\\subsection{The convolution algebra of a Lie groupoid}\nLet $\\mathcal{G}\\rightrightarrows M$ be a Lie groupoid. For an introduction to the theory of Lie groupoids we refer to \\cite{mm}. Here we denote source and target maps by $s,t:\\mathcal{G}\\to M$ and will think of arrows $g\\in\\mathcal{G}$ as pointing from right to left, so that the product $g_1g_2$ is defined whenever $s(g_1)=t(g_2)$. The Lie algebroid  $A(\\mathcal{G})$ is defined as $A(\\mathcal{G})=\\ker(ds)|_M$. We will concern ourselves with the convolution algebra of $\\mathcal{G}$. To define the convolution product, we need entities which can be integrated, and this is where densities come into play. To this end we look at densities along the source-fibers, where we note that there is a canonical isomorphism between $\\ker ds$ and $t^\\ast A(\\mathcal{G})$ using right translations. In this way we can define the convolution product for two compactly supported densities $a_1, a_2\\in\\Gamma_c(\\mathcal{D}_s)$ by\n\\begin{equation*}\n(a_1\\ast a_2)_g(v_1,...,v_n)=\\int_{h\\in s^{-1}(s(g))}(a_1)_{gh^{-1}}(v_1,...,v_n)(a_2)_h\n\\end{equation*}\nIn this notation $v_1,...,v_n\\in A_{t(g)}=A_{t(gh^{-1})}$ so that the product in the integrand yields a well-defined compactly supported density along $s^{-1}(h)=s(g)$ that can be integrated. Colloquially this product will be written as:\n\\begin{equation*}\n(a_1\\ast a_2)(g)=\\int_{g_1g_2=g}a_1(g_1)a_2(g_2)=\\int_{h\\in s^{-1}(s(g))}a_1(gh^{-1})a_2(h)\n\\end{equation*}\nWe define the convolution algebra $\\mathcal{A}_\\mathcal{G}$ of $\\mathcal{G}$ to be $\\mathcal{A}_\\mathcal{G}=(\\Gamma_c(\\mathcal{D}_s),\\ast)$. This definition of the convolution algebra differs slightly (but is isomorphic as a complex algebra) from the more usual one in e.g. \\cite{connes} using $1\/2$-densities along source {\\em and} target fibers.\n\\subsection{The deformation complex of a Lie groupoid}\n Let $\\mathcal{G}\\rightrightarrows M$ be a Lie groupoid and write $\\overline{m}$ for the map $\\overline{m}(g,h)=gh^{-1}$. Note that $\\overline{m}$ has as domain $\\mathcal{G}\\hspace*{1mm}^s\\!\\times^s\\mathcal{G}:=\\{(g_1,g_2)\\in\\mathcal{G}\\times\\mathcal{G},~s(g_1)=s(g_2)\\}$. Furthermore we write $\\mathcal{G}^{(k)}$ for the $k$'th nerve of $\\mathcal{G}$:\n\\begin{equation*}\n\\mathcal{G}^{(k)}=\\{(g_1,...,g_k)\\in\\mathcal{G}^k|s(g_i)=t(g_{i+1})\\}\n\\end{equation*}\nIn \\cite{cms} the deformation complex is defined as follows:\n\\begin{defi}\nFor $k\\geq 1$ define \n$C^k_\\text{def}(\\mathcal{G})$ to be the set of smooth maps $c:\\mathcal{G}^{(k)}\\to T\\mathcal{G}$ such that $c(g_1,...,g_k)\\in T_{g_1}\\mathcal{G}$ and such that there is a section $s_c$ of tthe vector bundle $t^*TM$ over $\\mathcal{G}^{(k-1)}$ such that \n\\[\nds(c(g_1,...,g_k))=s_c(g_2,...,g_k).\n\\]\nThe differential $\\delta: C^k_\\text{def}(\\mathcal{G})\\to C^{k+1}_\\text{def}(\\mathcal{G})$ is defined by setting:\n\\begin{align*}\n(\\delta c)(g_1,...,g_{k+1})=&-d\\overline{m}(c(g_1g_2,g_3,...,g_{k+1}),c(g_2,...,g_{k+1}))\\\\\n&+\\sum_{i=2}^k(-1)^ic(g_1,...,g_ig_{i+1},...,g_{k+1})\n+(-1)^{k+1}c(g_1,...,g_k).\n\\end{align*}\nThe {\\em deformation complex} is defined by the graded vector space $C^\\bullet_\\text{def}(\\mathcal{G}):=\\bigoplus_{k\\geq 1} C^k_\\text{def}(\\mathcal{G})$ equipped with the differential $\\delta$, its cohomology is denoted $H^\\bullet_\\text{def}(\\mathcal{G})$. \n\\end{defi}\n\\begin{rmk}\n\\label{deg0}\nIt is possible, as in \\cite{cms}, to extend the deformation complex in degree zero by putting $C^0_\\text{def}(\\mathcal{G})=\\Gamma(M,A(\\mathcal{G}))$ \nwith differential defined for $\\alpha\\in\\Gamma(M,A(\\mathcal{G}))$ by\n\\begin{equation*}\n(\\delta \\alpha)(g)=(dr_g)(\\alpha(t(g))+(d(l_g\\circ\\iota))(\\alpha(s(g))\n\\end{equation*}\nWe exclude these elements in degree $0$ because, as we will see, these element cannot correspond to Hochschild $0$-cochains.\n\\end{rmk}\n\n\\begin{rmk}\n\\label{mv}\nIt follows from the definition above that the closed elements in degree $1$ are exactly the multiplicative vector fields, c.f.\\ \\cite[\\S 4.3]{cms}. These are vector fields $X\\in\\mathfrak{X}(\\mathcal{G})$ that are $s$ and $t$-projectable to the same image in $\\mathfrak{X}(M)$, satisfying the following equation:\n\\begin{equation*}\ndm_{(g,h)}(X(g),X(h))=X(gh)\n\\end{equation*}\n\\end{rmk}\n\nFor certain purposes, most importantly applying the Van Est map, it often necessary to impose more strict relations on elements $c\\in C^k_\\text{def}(\\mathcal{G})$ and their symbol $s_c$. To this end we also introduce the normalized deformation complex:\n\n\\begin{defi}\nThe \\textit{normalized deformation complex} is the subcomplex $\\hat{C}^\\bullet_\\text{def}(\\mathcal{G})$ of $C^\\bullet_\\text{def}(\\mathcal{G})$ consisting of those elements $c\\in C^k_\\text{def}(\\mathcal{G})$ which satisfy\n\\begin{equation*}\nc(1_x,g_2,...,g_k)=du(s_c(g_2,...,g_k))\n\\end{equation*}\nand\n\\begin{equation*}\ns_c(g_2,...,1_x,...,g_k)=0\n\\end{equation*}\nwhere the unit is put in any of the $k-1$ slots.\n\\end{defi}\nIt is shown in \\cite[Prop 11.8]{cms} that the inclusion of the normalized deformation complex into the whole deformation complex is a quasi-isomorphism.\n\\section{From deformation to Hochschild cohomology}\\label{chainmap}\n\\subsection{The cochain map}\n\\label{cmh}\nIn this section we define a cochain map from the deformation complex of $\\mathcal{G}$ to the Hochschild complex of the convolution algebra $\\mathcal{A}_\\mathcal{G}$. As a first hint for the existence of such a morphism, we make the following observation:\n\\begin{prop}\\label{multvfderiv}\nLet $\\mathcal{G}\\rightrightarrows M$ be a Lie groupoid. \nMultiplicative vector fields on $\\mathcal{G}$  act as derivations on the convolution algebra.\n\\end{prop}\n\\begin{proof}\nRecall the definition of a multiplicative vector field from \\cref{mv}. Since a multiplicative vector field on $\\mathcal{G}$ is by definition $s$-projectable to $M$, its action on an $s$-density is well-defined by \nthe discussion in \\cref{das}, c.f.\\ equation \\eqref{apvf}.\n\nThe key ingredient in the proof is the observation that the flow of a multiplicative vector field is a groupoid map, that is if $X\\in\\mathfrak{X}(\\mathcal{G})$ is a multiplicative vector field then $\\Phi^t_X(gh^{-1})=\\Phi^t_X(g)\\Phi^t_X(h)^{-1}$. A simple calculation then shows\n\\begin{align*}\nX(a_1 \\ast a_2)(g)&=\\left.\\frac{d}{dt}\\right|_{t=0}(a_1\\ast a_2)(\\Phi^t_Xg)\\\\\n&=\\left.\\frac{d}{dt}\\right|_{t=0}\\int_{h\\in s^{-1}(s(\\Phi^t_X g))}a_1((\\Phi^t_X g)h^{-1})a_2(h)\\\\\n&=\\left.\\frac{d}{dt}\\right|_{t=0}\\int_{h\\in s^{-1}(s(g))}a_1((\\Phi^t_X g)(\\Phi^t_X h)^{-1})a_2(\\Phi^t_Xh)\\\\\n&=\\left.\\frac{d}{dt}\\right|_{t=0}\\int_{h\\in s^{-1}(s(g))}a_1(\\Phi^t_X(gh^{-1}))a_2(\\Phi^t_Xh)\\\\\n&=\\left.\\frac{d}{dt}\\right|_{t=0}\\int_{h\\in s^{-1}(s(g))}a_1(\\Phi^t_X(gh^{-1}))a_2(h)\\\\ &\\hspace{2.5cm}+\\left.\\frac{d}{dt}\\right|_{t=0}\\int_{h\\in s^{-1}(s(g))}a_1(gh^{-1})a_2(\\Phi^t_Xh)\\\\\n&=(Xa_1\\ast a_2)(g)+(a_1\\ast Xa_2)(g)\n\\end{align*}\nwhich proves the proposition.\n\\end{proof}\nIn the following we write $C^\\bullet_\\text{Hoch}(\\mathcal{A}_\\mathcal{G},\\mathcal{A}_\\mathcal{G})$ for the Hochschild complex of the convolution algebra $\\mathcal{A}_\\mathcal{G}$ with values in the bimodule $\\mathcal{A}_\\mathcal{G}$ with differential $\\delta_{\\rm Hoch}$.\nWe now describe the cochain map $C^\\bullet_\\text{def}(\\mathcal{G})\\to C^\\bullet_\\text{Hoch}(\\mathcal{A}_\\mathcal{G},\\mathcal{A}_\\mathcal{G})$. \n\\begin{defi}\n\\label{chainmap}\nThe map $\\Phi:C^\\bullet_\\text{def}(\\mathcal{G})\\to C^\\bullet_\\text{Hoch}(\\mathcal{A}_\\mathcal{G},\\mathcal{A}_\\mathcal{G})$ is defined by \n\\begin{equation*}\n(\\Phi c)(a_1,...,a_k)(g)=\\int_{g_1\\cdots g_k=g}(c(-,g_2,...,g_k)a_1)(g_1)a_2(g_2)\\cdots a_k(g_k),\\qquad\\mbox{for}~c\\in C^k_\\text{def}(\\mathcal{G}).\n\\end{equation*}\nThis formula should be read as an inductive convolution (first over $g_1g_2=h_1$, then over $h_1g_3=h_2$, et cetera).\n\\end{defi}\n\\begin{rmk}\nThe formula for $\\Phi$ above is justified by \\cref{nl}: $c\\in C^k_\\text{def}(\\mathcal{G})$ is $s$-projectable and therefore the action of $c(-,g_2,...,g_k)$ on  $a\\in \\mathcal{A}_\\mathcal{G}$ along $s^{-1}(t(g_2))$ is well-defined. In particular $(c(-,g_2,...,g_k)a)(g_1)$ is a well-defined density at $g_1$ for $(g_1,...,g_k)\\in\\mathcal{G}^{(k)}$.\n\\end{rmk}\nShowing that $\\Phi$ is a chain map is done by a calculation similar to the one in \\cref{multvfderiv}. In particular we need to deal with the term $\\Phi^t_X(g)(\\Phi^t_X(h))^{-1}$ for divisible $g$ and $h$ when $t$ goes to $0$. In the mulitplicative case this is precisely $\\Phi^t_X(gh^{-1})$, but for general deformation elements we need a more general description.\n\nFor this we abbreviate the term in $\\delta c$ involving $d\\overline{m}$ by $\\overline{m}c$, that is:\n\\begin{equation*}\n(\\overline{m}c)(g_1,...,g_{k+1})=d\\overline{m}(c(g_1g_2,...,g_{k+1}),c(g_2,...,g_{k+1}))\n\\end{equation*}\nWe remark that this notation commutes with keeping the last all-but-two entries fixed, i.e.:\n\\begin{equation*}\n\\overline{m}(c(-,g_3,...,g_{k+1}))(g_1,g_2)=(\\overline{m}c)(g_1,...,g_{k+1})\n\\end{equation*}\nThe key Lemma is then as follows:\n\\begin{lem}\nLet $x\\in M$, $X\\in\\mathfrak{X}_s(\\mathcal{G})|_{s^{-1}(x)}$ and $a_1,a_2\\in \\mathcal{A}_\\mathcal{G}$. Then for all $h\\in s^{-1}(x)$ we have $\\overline{m}X(-,h)\\in\\mathfrak{X}_s(\\mathcal{G})|_{s^{-1}(t(h))}$ and for $g\\in s^{-1}(x)$:\n\\begin{equation*}\nX(a_1\\ast a_2)(g)=(a_1\\ast Xa_2)(g)+\\int_{h\\in s^{-1}(x)}((\\overline{m}X(-,h))a_1)(gh^{-1})a_2(h)\n\\end{equation*}\n\\end{lem}\n\\begin{proof}\nBy definition we have:\n\\begin{equation*}\n\\overline{m}X(gh^{-1},h)=d\\overline{m}(X(g),X(h))\\in T_{gh^{-1}}\\mathcal{G}\n\\end{equation*}\nwith $s$-projection\n\\begin{equation*}\nds(\\overline{m}X(gh^{-1},h))=dt(X(h))\n\\end{equation*}\nso indeed $\\overline{m}X(-,h)\\in\\mathfrak{X}_s(\\mathcal{G})|_{s^{-1}(t(h))}$.\n\nNext we assume that $X$ is a globally defined $s$-projectable vector field (otherwise, we choose an extension at this point). Then we know that $\\overline{m}X(-,h)$ is generated by the path $\\Phi_t$ through $\\text{Diff}_s(\\mathcal{G})$, which along $s^{-1}(t(h))$ looks like:\n\\begin{equation*}\n\\Phi_t(gh^{-1})=\\Phi^t_X(g)(\\Phi^t_X(h))^{-1}\n\\end{equation*}\nso that we see that:\n\\begin{equation*}\n\\int_{h\\in s^{-1}(x)}((\\overline{m}X(-,h))a_1)(gh^{-1})a_2(h)=\\left.\\frac{d}{dt}\\right|_{t=0}\\int_{h\\in s^{-1}(x)}a_1(\\Phi^t_X(g)(\\Phi^t_X(h))^{-1})a_2(h)\n\\end{equation*}\nUsing this we calculate $X(a_1\\ast a_2)(g)$:\n\\begin{align*}\nX(a_1\\ast a_2)(g)=&\\left.\\frac{d}{dt}\\right|_{t=0} (a_1\\ast a_2)(\\Phi^t_Xg)\\\\\n=&\\left.\\frac{d}{dt}\\right|_{t=0}\\int_{h\\in s^{-1}(s(\\Phi^t_Xg))}a_1((\\Phi^t_Xg)h^{-1})a_2(h)\\\\\n=&\\left.\\frac{d}{dt}\\right|_{t=0}\\int_{h\\in s^{-1}(x)}a_1((\\Phi^t_Xg)(\\Phi^t_Xh)^{-1})a_2(\\Phi^t_X(h))\\\\\n=&\\left.\\frac{d}{dt}\\right|_{t=0}\\int_{h\\in s^{-1}(x)}a_1((\\Phi^t_Xg)(\\Phi^t_Xh)^{-1})a_2(h)+\\left.\\frac{d}{dt}\\right|_{t=0}\\int_{h\\in s^{-1}(x)}a_1(gh^{-1})a_2(\\Phi^t_Xh)\\\\\n=&\\int_{h\\in s^{-1}(x)}((\\overline{m}X(-,h))a_1)(gh^{-1})a_2(h)\\\\\n&+(a_1\\ast Xa_2)(g)\n\\end{align*}\nwhich finishes the proof.\n\\end{proof}\n\\begin{prop}\nThe map $\\Phi: C^\\bullet_\\text{def}(\\mathcal{G})\\to C^\\bullet_\\text{Hoch}(\\mathcal{A}_\\mathcal{G},\\mathcal{A}_\\mathcal{G})$ is a morphism of cochain complexes.\n\\end{prop}\n\\begin{proof}\nThis proof is essentially writing out all the parts of the Hochschild differential and applying some bookkeeping. We start with $c\\in C^k_\\text{def}(\\mathcal{G})$ for $k\\geq 1$, and write down the definition of the various parts of $\\delta_\\text{Hoch}(\\Phi c)$.\n\\begin{equation*}\n(a_1\\ast(\\Phi c)(a_2,...,a_{k+1}))(g)=\\int_{g_1\\cdots g_{k+1}=g}a_1(g_1)(c(-,g_3,...,g_{k+1})a_2)(g_2)a_3(g_3)\\cdots a_{k+1}(g_{k+1})\n\\tag{$\\star$}\n\\end{equation*}\n\\begin{equation*}\n-(\\Phi c)(a_1\\ast a_2,a_3,...,a_{k+1})(g)=-\\int_{h\\cdot g_3\\cdots g_{k+1}=g}(c(-,g_3,...,g_{k+1})(a_1\\ast a_2))(h)a_3(g_3)\\cdots a_{k+1}(g_{k+1})\\tag{$\\star\\star$}\n\\end{equation*}\n\\begin{multline*}\n\\sum_{i=2}^k(-1)^i(\\Phi c)(a_1,...,a_i\\ast a_{i+1},...,a_{k+1})(g)=\\\\\n\\sum_{i=2}^{k}\\int_{g_1\\cdots g_{k+1}=g}(-1)^i (c(-,g_2,...,g_ig_{i+1},...,g_{k+1})a_1)(g_1)a_2(g_2)\\cdots a_{k+1}(g_{k+1})\n\\end{multline*}\n\\begin{equation*}\n(-1)^{k+1}((\\Phi c)(a_1,...,a_k)\\ast a_{k+1})(g)=(-1)^{k+1}\\int_{g_1\\cdots g_{k+1}=g}(c(-,g_2,...,g_k)a_1)(g_1)a_2(g_2)\\cdots a_{k+1}(g_{k+1})\n\\end{equation*}\nThe latter two terms we recognize from the differential of the deformation complex, while the first two terms can be rewritten to:\n\\begin{multline*}\n(\\star)+(\\star\\star)=\\int_{hg_3\\cdots g_{k+1}=g}\\left((a_1\\ast (c(-,g_3,...,g_{k+1}) a_2)-c(-,g_3,...,g_{k+1})(a_1\\ast a_2)\\right)(h)a_3(g_3)\\cdots a_{k+1}(g_{k+1})\n\\end{multline*}\nThen by the key Lemma we can rewrite this to\n\\begin{align*}\n(\\star)+(\\star\\star)=&-\\int_{g_1\\cdots g_{k+1}=g}((\\overline{m}c)(-,g_2,...,g_{k+1})a_1)(g_1)a_2(g_2)\\cdots a_{k+1}(g_{k+1})\n\\end{align*}\nPutting this all together we conclude that:\n\\begin{align*}\n(\\delta_\\text{Hoch}(\\Phi c))(a_1,...,a_{k+1})(g)\n&=(\\Phi(\\delta c))(a_1,...,a_{k+1})(g)\n\\end{align*}\nSo we see that $\\Phi$ is indeed a chain-map.\n\\end{proof}\n\\subsection{Comparing deformation classes}\nIn this section we compare the deformation classes in $H^2_\\text{def}(\\mathcal{G})$ and $H^2_{\\rm Hoch}(\\mathcal{A}_\\mathcal{G})$ coming from deformations of the Lie groupoid $\\mathcal{G}$.\nRecall from \\cite[\\S 5.2]{cms} that an $s$-constant deformation of $\\mathcal{G}$ is a smooth family $\\overline{m}_\\epsilon: \\mathcal{G}\\hspace*{1mm}^s\\!\\times^s\\mathcal{G}\\to\\mathcal{G}$ of division maps parameterized by $\\epsilon$ in an open interval in $\\mathbb{R}$ containing $0$, such that $\\overline{m}_0=\\overline{m}$. This induces a deformation cocycle $\\beta\\in C^2_\\text{Hoch}(\\mathcal{A}_\\mathcal{G},\\mathcal{A}_\\mathcal{G})$ by deforming the associative algebra that is the convolution algebra:\n\\begin{equation*}\n\\beta(a_1,a_2)(g)=\\left.\\frac{d}{d\\epsilon}\\right|_{\\epsilon=0}\\int_{h\\in s^{-1}(s(g))}a_1(\\overline{m}_\\epsilon(g,h))a_2(h)\n\\end{equation*}\nOn the other hand the deformation also induces a deformation element $\\xi\\in C^2_\\text{def}(\\mathcal{G})$ set for $(g,g')\\in\\mathcal{G}^{(2)}$ by:\n\\begin{equation*}\n\\xi(g,g'):=\\left.\\frac{d}{d\\epsilon}\\right|_{\\epsilon=0}\\overline{m}_\\epsilon(gg',g').\n\\end{equation*}\nBy \\cite[Lemma 5.3]{cms}, this cochain is is closed: $\\delta\\xi=0$.\n\\begin{prop}\nThe chain map $\\Phi$ sends $\\xi$ to $\\beta$.\n\\end{prop}\n\\begin{proof}\nThis follows from the observation that if $s(h)=s(g)$, then\n\\begin{equation*}\n\\xi(gh^{-1},h)=\\left.\\frac{d}{d\\epsilon}\\right|_{\\epsilon=0}\\overline{m}_e(g,h)\n\\end{equation*}\nWith this we see that\n\\begin{equation*}\n\\beta(a_1,a_2)(g)=\\int_{h\\in s^{-1}(s(g))}(\\xi(-,h)a_1)(gh^{-1})a_2(h)=\\Phi(\\xi)(a_1,a_2)(g),\n\\end{equation*}\nexactly as needed.\n\\end{proof}\n\\begin{rmk}\nIn \\cite[Prop 5.12]{cms} a deformation cocycle $\\xi\\in C^2_\\text{def}(\\mathcal{G})$ is assigned to any deformation (in particular those who are not $s$-constant), whose cohomology class is canonical. Then $\\Phi(\\xi)$ induces a Hochschild cohomology class of degree 2, which is not immediately linked to a deformation of the convolution product, since if the source map changes the underlying space of the convolution algebra also changes as it consists of densities along the $s$-fibers. Indeed, in \\cite{cms} the authors need an auxillary choice of a vector field on the larger deformation space to define the cocycle. This choice of an auxillary vector field is precisely what is needed to compare the various convolution algebras when the source map varies, and in this way $\\Phi$ maps $[\\xi]\\in H^2_\\text{def}(\\mathcal{G})$  to the Hochschild class of the deformation of the convolution product thus defined. \n\\end{rmk}\n\n\\subsection{Compatibility with the characteristic map to cyclic cochomology}\nDenote by $(C^\\bullet_{\\rm diff}(\\mathcal{G}),\\delta)$ the cochain complex of inhomogeneous groupoid cochains given by $C^k_{\\rm diff}(\\mathcal{G}):=C^\\infty(\\mathcal{G}^{(k)})$ with differential\n\\begin{align*}\n\\delta\\varphi(g_1,\\ldots,g_{k+1})&=\\varphi(g_2,\\ldots,g_{k+1})+\\sum_{i=1}^k(-1)^i\\varphi(g_1,\\ldots,g_ig_{i+1},\\ldots, g_k)+(-1)^{k+1}\\varphi(g_1,\\ldots, g_k).\n\\end{align*}\nWe can turn this cochain complex into a DGA by introducing the product $\\cup:C^k_{\\rm diff}(\\mathcal{G})\\times C^l_{\\rm diff}(\\mathcal{G})\\to C^{k+l}_{\\rm diff}(\\mathcal{G})$ given by\n\\[\n(\\varphi\\cup\\psi)(g_1,\\ldots,g_{k+l}):=\\varphi(g_1,\\ldots,g_k)\\psi(g_{l+1},\\ldots, g_{k+l}).\n\\]\nIn \\cite{cms} it is shown that by replacing $\\varphi$ by a deformation cochain $c\\in C^k_{\\rm def}(\\mathcal{G})$ in the above formula, $C^\\bullet_{\\rm def}(\\mathcal{G})$ becomes a right module over $C^\\bullet_{\\rm diff}(\\mathcal{G})$. On the other hand, in \\cite{ppt}, the smooth groupoid cohomology was used to construct cyclic cocycles. In this section we shall that these two structures are compatible with each others under the cochain map $\\Phi$ to Hochschild cohomology of \\cref{cmh}. We start by re-writing the map to cyclic cohomology of \\cite{ppt} in the following way.\n\nFirst recall that the Hochschild cochain complex $C^\\bullet(\\mathcal{A}_{\\mathcal{G}},\\mathcal{A}_{\\mathcal{G}})$ can be given a DGA structure by introducing the product $\\cup:C^k(\\mathcal{A}_{\\mathcal{G}},\\mathcal{A}_{\\mathcal{G}})\\times C^l(\\mathcal{A}_{\\mathcal{G}},\\mathcal{A}_{\\mathcal{G}})\\to C^{k+l}(\\mathcal{A}_{\\mathcal{G}},\\mathcal{A}_{\\mathcal{G}})$\n\\[\n(D\\cup E)(a_1,\\ldots,a_{k+l}):=D(a_1,\\ldots,a_k)*E(a_{k+1},\\ldots,a_{k+l}).\n\\]\nConstruct a map $\\Phi_0:C^\\bullet_{\\rm diff} (\\mathcal{G})\\to C^\\bullet(\\mathcal{A}_\\mathcal{G},\\mathcal{A}_\\mathcal{G})$ by\n\\begin{equation}\n\\label{dgm}\n\\Phi_0(\\varphi)(a_1,\\ldots,a_k)(g):=\\int_{g_1\\cdots g_k=g}\\varphi(g_1,\\ldots,g_k)a_1(g_1)\\cdots a_k(g_k).\n\\end{equation}\n\\begin{lem}\nThe map $\\Phi_0:(C^\\bullet_{\\rm diff} (\\mathcal{G}),\\delta,\\cup)\\to (C^\\bullet(\\mathcal{A}_\\mathcal{G},\\mathcal{A}_\\mathcal{G}),\\delta_{\\rm Hoch},\\cup)$ is a morphism of DGA's.\n\\end{lem}\n\\begin{proof}\nThis is a straightforward computation.\n\\end{proof}\nWith this Lemma we can also equip the Hochschild complex $C^\\bullet(\\mathcal{A}_\\mathcal{G},\\mathcal{A}_\\mathcal{G})$ with a module structure over $C^\\bullet_{\\rm diff}(\\mathcal{G})$ by using the cup-product on Hochschild cochains:\n\\[\n(D\\cup E)(a_1,\\ldots, a_{k+l}):=D(a_1,\\ldots,a_k)E(a_{k+1},\\ldots,a_{k+l}).\n\\] \nExplicitly, this module structure is given by\n\\[\nD\\cdot\\varphi:=D\\cup\\Phi_0(\\varphi).\n\\]\nWe then have:\n\\begin{prop}\nThe cochain map $\\Phi: C^\\bullet_\\text{def}(\\mathcal{G})\\to C^\\bullet_\\text{Hoch}(\\mathcal{A}_\\mathcal{G},\\mathcal{A}_\\mathcal{G})$ is a morphism of $C^\\bullet_{\\rm diff}(\\mathcal{G})$-modules.\n\\end{prop}\n\\begin{proof}\nLet us start with the following case: For $c\\in C^k_\\text{def}(\\mathcal{G})$ and $f \\in C^\\infty(\\mathcal{G})=C^1_{\\rm diff}(\\mathcal{G})$ we have\n\\begin{equation*}\n\\Phi(c\\cup f)(a_1,...,a_{k+1})=\\Phi(c)(a_1,...,a_k)\\ast (f\\cdot a_{k+1})\n\\end{equation*}\nThe claim follows by carefully writing out the definition\n\\begin{align*}\n\\Phi(c\\cup f)(a_1,...,a_{k+1})(g)&=\\int_{g_1\\cdots g_{k+1}=g}((c\\cup f)(-,g_2,...,g_{k+1})a_1)(g_1)a_2(g_2)\\cdots a_{k+1}(g_{k+1})\\\\\n&=\\int_{g_1\\cdots g_{k+1}=g}(f(g_{k+1})c(-,g_2,...,g_k)a_1)(g_1)a_2(g_2)\\cdots a_{k+1}(g_{k+1})\\\\\n&=\\int_{g_1\\cdots g_{k+1}=g}(c(-,g_2,...,g_k)a_1)(g_1)a_2(g_2)\\cdots a_k(g_k)\\left(f(g_{k+1})a_{k+1}(g_{k+1})\\right)\\\\\n&=\\int_{hg_{k+1}=g}\\int_{g_1\\cdots g_k=h}(c(-,g_2,...,g_k)a_1)(g_1)a_2(g_2)\\cdots a_k(g_k)\\left(f(g_{k+1})a_{k+1}(g_{k+1})\\right)\\\\\n&=\\int_{hg_{k+1}=g}\\Phi(c)(a_1,...,a_k)(h)\\left(f(g_{k+1})a_{k+1}(g_{k+1})\\right)\\\\\n&=\\left(\\Phi(c)(a_1,...,a_k)\\ast (f\\cdot a_{k+1})\\right)(g)\n\\end{align*}\nHence by induction we obtain\n\\begin{equation*}\n\\Phi(c\\cup (f_1\\otimes\\cdots\\otimes f_l))(a_1,...,a_{k+l})=\\Phi(c)(a_1,...,a_k)\\ast (f_1\\cdot a_{k+1})\\ast\\cdots\\ast (f_l\\cdot a_{k+l})\n\\end{equation*}\nWriting $f\\ast a=\\Phi_0(f)(a)$, we can rewrite this as\n\\[\n\\Phi(c\\cup (f_1\\otimes\\cdots\\otimes f_l))=\\Phi (c)\\cup\\Phi_0(f_1)\\cup\\ldots\\cup\\Phi_0(f_l).\n\\]\nFrom this the general statement of the proposition follows. \n\\end{proof}\n\n Now, analogous to the action of vector fields on differential forms in geometry, the Hochschild cochains act on Hochschild chains by contraction:\n\\[\nC^k(\\mathcal{A}_\\mathcal{G},\\mathcal{A}_\\mathcal{G})\\times C_l(\\mathcal{A}_\\mathcal{G})\\longrightarrow C_{l-k}(\\mathcal{A}_\\mathcal{G}),\\qquad (D,a)\\mapsto\\iota_Da,\n\\]\ngiven explicitly by\n\\[\n\\iota_D(a_0\\otimes\\ldots\\otimes a_{k+l}):=a_0D(a_1,\\ldots,a_k)\\otimes a_{k+1}\\otimes\\ldots\\otimes a_{k+l}.\n\\]\nThis action satisfies the properties\n\\begin{align*}\n\\iota_D\\circ\\iota_E&=\\iota_{D\\cup E}\\\\\n[b,\\iota_D]&=\\iota_{\\delta D}.\n\\end{align*}\nThe analogue of the Cartan formula for the ``Lie derivative''  $L_D:=B\\circ \\iota_D+\\iota_D\\circ B$ in noncommutative geometry also holds true on the level of Hochschild homology.\n\nNext, recall from \\cite{ppt} that when $\\mathcal{G}$ is unimodular we can define a trace on the convolution algebra $\\mathcal{A}_\\mathcal{G}$ by \n\\[\n\\tau(a):=\\int_Ma\\Omega,\n\\]\nwith on the right hand side $\\Omega$ a $\\mathcal{G}$-invariant section of the bundle $\\mathcal{D}_{A^*}\\otimes\\mathcal{D}_{TM}$, and we use the duality $\\mathcal{D}_A\\times\\mathcal{D}_{A^*}\\to\\mathbb{R}$ together with the isomorphism $\\mathcal{D}_s|_M=\\mathcal{D}_A$, to obtain a density on $M$ that can be integrated. With this trace (a degree $0$ cyclic cocycle), the cochain map \n\\begin{equation}\n\\label{chain-diff}\n\\Psi_\\tau:(C^\\bullet_{\\rm diff}(\\mathcal{G}),\\delta)\\longrightarrow (C^\\bullet(\\mathcal{A}_\\mathcal{G}),b_{\\rm Hoch}),\n\\end{equation}\nconstructed in \\cite{ppt} is simply given by $\\Psi_\\tau(c):=\\iota_{\\Phi_0(c)}\\tau$.\n\\begin{cor}\nLet $c\\in C^k_{\\rm def}(\\mathcal{G})$ and $f\\in C^l_{\\rm diff}(\\mathcal{G})$. Then the following\nidentity holds true:\n\\[\n\\iota_{\\Phi(c\\cup f)}\\tau=\\iota_{\\Phi(c)}\\Psi_\\tau(f).\n\\]\n\\end{cor}\nWith this Corollary, we can construct new cyclic cocycles on the convolution algebra. First of all, if we start with a smooth \ngroupoid cocycle $\\varphi\\in C^k_{\\rm diff}(\\mathcal{G})$, we obtain a Hochschild cocycle by applying $\\Psi_\\tau$ as in \\eqref{chain-diff}. A small computation shows that this cocycle is closed under the $B$-differential, i.e., $B\\Psi_\\tau(\\varphi)=0$, when $\\varphi$ is cyclic:\n\\[\n\\varphi(g_1,\\ldots,g_k)=(-1)^k\\varphi((g_1\\cdots g_k)^{-1},g_1,\\ldots,g_{k-1})\n\\]\nWe can work out similar conditions for elements $c\\in C^k_\\text{def}(\\mathcal{G})$, but they are more involved. For example, for $k=2$ we find\n\\begin{equation*}\n(d\\iota)(c(g,g^{-1}))=-c(g^{-1},g).\n\\end{equation*}\n\n\n\\begin{rmk}\nIt is proved in \\cite[\\S 9]{cms} that $H^\\bullet_\\text{def}(\\mathcal{G})\\cong H^\\bullet(\\mathcal{G},{\\rm Ad})$, where ${\\rm Ad}$ denotes the adjoint representation up to homotopy constructed in \\cite{ac}. Taking into account the morphism \\eqref{dgm}, this strongly suggests to relabel the morphism  of \\cref{chainmap} as $\\Phi_1$ and conjecture the existence of a map $\\Phi_p:H^\\bullet(\\mathcal{G},{\\rm Sym}^p({\\rm Ad}))\\to H^\\bullet_{\\rm Hoch}(\\mathcal{A}_\\mathcal{G},\\mathcal{A}_\\mathcal{G})$ extending the cases $p=0,1$ described in this paper. This would naturally fit with the infinitesimal theory (see also the next section) and the computation in \\cite{blom} of the Hochschild cohomology of the universal enveloping algebra $\\mathcal{U}(A)$ of the Lie algebroid $A$:\n\\[\nH^\\bullet_{\\rm Hoch}(\\mathcal{U}(A),\\mathcal{U}(A))\\cong \\bigoplus_{p\\geq 0} H^\\bullet_{CE}(A,{\\rm Sym}^pA).\n\\]\n\\end{rmk}\n\\subsection{The case $k=0$}\nFor the chain map between $C^\\bullet_\\text{def} (\\mathcal{G})$ and $C^\\bullet_\\text{Hoch}(\\mathcal{A}_\\mathcal{G},\\mathcal{A}_\\mathcal{G})$ we have just defined, a natural question is whether it can be extended to degree $k=0$, c.f. \\cref{deg0}. For this, one must find a  a map  $\\Phi^0: \\Gamma(A)\\to \\mathcal{A}_\\mathcal{G}$ which extends the chain map $\\Phi$. This is only possible if $\\Phi(\\delta(\\alpha))\\in\\text{Der}(\\mathcal{A}_\\mathcal{G})$ is an inner derivation for every $\\alpha\\in \\Gamma(A)$.\n\nIntuitively it is clear that is should not be always possible, since the derivation $\\Phi(\\delta(\\alpha))$ includes taking derivatives, while an inner derivation $\\partial_H(a)$ only includes integrations. The following example presents a concrete counterexample:\n\n\\begin{ex} Consider the pair groupoid $\\mathbb{R}\\times\\mathbb{R}\\rightrightarrows \\mathbb{R}$. For this groupoid, a bundle of densities is trivialized by $|dx|$, so that every compactly supported density is of the form $f|dx|$ for a compactly supported smooth function $f$.  Furthermore, a section of the algebroid is simply a vector field $X\\in\\mathfrak{X}(\\mathbb{R})$ and for this example we take $X=\\frac{\\partial}{\\partial x}$. We have\n\\begin{equation*}\n\\delta(X)(x,y)=(X(x),X(y))\n\\end{equation*}\nso that in this case $\\delta(X)=\\frac{\\partial}{\\partial x}+\\frac{\\partial}{\\partial y}$. This vector field has flow\n\\begin{equation*}\n\\Phi^t_{\\delta(X)}(x,y)=(x+t,y+t)\n\\end{equation*}\nNext we consider $\\Phi\\left(\\delta\\left(\\frac{\\partial}{\\partial x}\\right)\\right)$, so we look at the action of $\\frac{\\partial}{\\partial x}+\\frac{\\partial}{\\partial y}$ on a density $f(x,y)|dx|\\in \\mathcal{A}_{\\mathbb{R}\\times\\mathbb{R}}$. We see\n\\begin{equation*}\n(\\Phi^t_{\\delta(X)})^\\ast (f(x,y)|dx|)=f(x+t,y+t)|d(x+t)|=f(x+t,y+t)|dx|\n\\end{equation*}\nSo that:\n\\begin{equation*}\n\\Phi(\\delta(X))(f|dx|)=\\left(\\frac{\\partial f}{\\partial x}+\\frac{\\partial f}{\\partial y}\\right)|dx|\n\\end{equation*}\nNow suppose that there is some $g|dx|\\in \\mathcal{A}_{\\mathbb{R}\\times\\mathbb{R}}$, such that $\\Phi(\\delta(X))=\\partial_H(g|dx|)$. Then since always $\\partial_H(g|dx|)(g|dx|)=0$, we see that:\n\\begin{equation*}\n\\frac{\\partial g}{\\partial x}+\\frac{\\partial g}{\\partial y}=0\n\\end{equation*}\nso that\n\\begin{equation*}\ng(x+t,y+t)=g(x,y)\n\\end{equation*}\nSince $g$ has to be compactly supported, the only possibility is that $g=0$, which is obviously not a solution to $\\Phi(\\delta(X))=\\partial_H(g|dx|)$. We conclude that $\\Phi(\\delta(X))$ is not an inner derivation.\n\\end{ex}\nIn fact, using supports as an argument, we can deduce that $\\Phi(X)$ can never be an inner derivation for any $X\\in\\mathfrak{X}_s(\\mathcal{G})$.\n\\begin{prop}\nLet $D\\in\\text{Hom}(\\mathcal{A}_\\mathcal{G},\\mathcal{A}_\\mathcal{G})$ be a non-zero Hochschild-1-cochain. If $D$ satisfies $\\text{supp}(Da)\\subset\\text{supp}(a)$, then there is no $b\\in \\mathcal{A}_\\mathcal{G}$ such that $D=[-,b]$.\n\\end{prop}\n\\begin{proof}\nSuppose by contrary that there is a $b$ such that $D=[-,b]$. Let $g\\in\\mathcal{G}$ and let $a\\in \\mathcal{A}_\\mathcal{G}$ be supported arbitrarily close to $g$. For $h\\in t^{-1}(s(g))$ outside of the isotropy of $s(g)$ we obtain:\n\\begin{equation*}\n(a\\ast b)(gh)=\\int_k a(gk^{-1})b(kh)\\sim a(g)b(h)\n\\end{equation*}\nwhere we use that $a$ is only non-zero close enough to $g$. For the other part of the commutator we have\n\\begin{equation*}\n(b\\ast a)(gh)=\\int_k b(gk^{-1})a(kh)=0\n\\end{equation*}\nSince there is no way to let $kh$ come arbitrarily close to $g$ since $h$ is not in the isotropy of $s(g)$.\n\nSince $\\text{supp} (Da)\\subset \\text{supp}(a)$ we see that $(a\\ast b)(gh)$ also has to be supported arbitrarily close to $g$, so that $b$ is identically zero outside of the isotropy of $\\mathcal{G}$.\n\nIf we look at $h$ an isotropy element of $\\mathcal{G}$ we see that the second term acts like $b(ghg^{-1})a(g)$, so that we see that $b$ is invariant under conjugation. However, if $b$ is invariant under conjugation we conclude that $b\\in Z(\\mathcal{A}_\\mathcal{G})$, which is in contradiction to the fact that $D$ is non-zero. We conclude that there is no $b$ that solves $D=[-,b]$.\n\\end{proof}\n\\begin{rmk}\nIt is possible to define the map $\\Phi^0$ if one allows for distributions to be cochains of degree $0$, that is if one defines $C^0_\\text{Hoch}(\\mathcal{A}_\\mathcal{G},\\mathcal{A}_\\mathcal{G}):=\\Gamma^{-\\infty}_c(\\mathcal{D}_s)$.\n\\end{rmk}\n\\begin{cor} If $X\\in C^1_\\text{def}(\\mathcal{G})$ is non-zero, then $\\Phi(X)$ can never be an inner derivation.\n\\end{cor}\n\\begin{proof}\nThis follows from the previous proposition by the observation that $\\Phi(X)$ is local since it involves taking derivatives and the fact that $\\Phi$ is easily observed to be injective.\n\\end{proof}\n\\subsection{Examples}\nIn this section we discuss how the chain map $\\Phi$ links the deformation cohomology of $\\mathcal{G}$ and the Hochschild cohomology of $\\mathcal{A}_\\mathcal{G}$ in certain examples.\n\n\\begin{ex}[Trivial groupoid]\nWe consider the trivial groupoid $\\mathcal{G}=M\\rightrightarrows M$. On the density side we simply have $(\\mathcal{A}_\\mathcal{G},\\ast)=(C^\\infty_c(M),\\cdot)$, with $H^\\bullet_\\text{Hoch}(\\mathcal{A}_\\mathcal{G},\\mathcal{A}_\\mathcal{G})=\\Lambda^\\bullet\\mathfrak{X}(M)$. At the side of the deformation complex we note that the $k$-nerve of the trivial groupoid is $M$ for every $k$ and $s$-projectability is a void property, so that for $k>0$ we have $C^k_\\text{def}(\\mathcal{G})=\\mathfrak{X}(M)$, with differential alternating between the identity and the zero map:\n\\begin{equation*}\nC^\\bullet_\\text{def}(\\mathcal{G})=\\left[ 0\\to \\mathfrak{X}(M)\\xrightarrow{0}\\mathfrak{X}(M)\\xrightarrow{\\rm id}\\mathfrak{X}(M)\\to\\cdots\\right]\n\\end{equation*}\nSo the deformation cohomology equals:\n\\begin{equation*}\nH^k_\\text{def}(\\mathcal{G})\\cong\\left\\{\\begin{matrix}\n\\mathfrak{X}(M) &\\text{ if } k=1\\\\\n0 & \\text{ else}\n\\end{matrix}\\right.\n\\end{equation*}\nThe chain map $\\Phi: C^\\bullet_\\text{def}(\\mathcal{G})\\to C^\\bullet_\\text{Hoch}(\\mathcal{A}_\\mathcal{G},\\mathcal{A}_\\mathcal{G})$ simply becomes:\n\\begin{equation*}\n\\Phi(X)(f_1,...,f_k)=(Xf_1)\\cdot f_2\\cdots f_k\n\\end{equation*}\nand we simply see that:\n\\begin{equation*}\nH^k(\\Phi)=\\left\\{\\begin{matrix}\n\\text{id} &\\text{ if }k=1\\\\\n0 & \\text{ else}\n\\end{matrix}\\right.\n\\end{equation*}\nWe should also remark for this example that using the classical Hochschild--Kostant--Rosenberg theorem, we see that taking exterior powers of deformation elements we retrieve the whole Hochschild cohomology of $C^\\infty_c(M)$.\n\\end{ex}\n\\begin{ex}[\\'Etale groupoids]\nIn the case of an \\'Etale groupoid $\\mathcal{G}\\rightrightarrows M$, we have $\\mathcal{A}_\\mathcal{G}=C^\\infty_c(\\mathcal{G})$, since the distribution $\\ker(ds)$ is the trivial distribution. The convolution product in this case is commonly written as\n\\begin{equation*}\n(f_1\\ast f_2)(g)=\\sum_{g_1g_2=g}f_1(g_1)f_2(g_2)\n\\end{equation*}\nIn this case the action of vector fields on densities is just the normal action of vector fields on functions, and the map $\\Phi$ reduces to\n\\begin{equation*}\n\\Phi(c)(f_1,...,f_k)(g)=\\sum_{g_1\\cdots g_k=g}(c(g_1,...,g_k)f_1)\\cdot f_2(g_2)\\cdots f_k(g_k)\n\\end{equation*}\nSince the source map of $\\mathcal{G}$ is a local diffeomorphism, we see that there is a 1-1 correspondence between deformation elements $c\\in C^k_\\text{def}(\\mathcal{G})$ and their symbols $s_c\\in\\Gamma(t^\\ast TM\\to\\mathcal{G}^{(k-1)})$ since we have\n\\begin{equation*}\nc(g_1,...,g_k)=(ds_{g_1})^{-1}(s_c(g_2,...,g_k))\n\\end{equation*}\nIn fact, the correspondence establishes an isomorphism between $C^\\bullet_\\text{def}(\\mathcal{G})$ and $C^\\bullet(\\mathcal{G},TM)[-1]$ where we see $TM$ as a representation of $\\mathcal{G}$ where $g$ acts $T_{s(g)}M\\to T_{t(g)}M$ as\n\\begin{equation*}\ng\\cdot v=dt_g((ds_g)^{-1})(v))\n\\end{equation*}\nThe shift by 1 we see here also serves as a justification of why the case $k=0$ is a tricky thing (although for \\'Etale groupoids of course we have $C^0_\\text{def}(\\mathcal{G})=0$).\n\nIn the case that we have a proper \\'Etale groupoid (over a connected base $M$) we can calculate the cohomologies in both sides of the equation. On the side of the deformation complex we use \\cite[Thm 6.1]{cms} to obtain:\n\\begin{align*}\nH^0_\\text{def}(\\mathcal{G})&\\cong\\{0\\}\\\\\nH^1_\\text{def}(\\mathcal{G})&\\cong\\mathfrak{X}(M)_\\text{inv}\\\\\nH^k_\\text{def}(\\mathcal{G})&\\cong\\{0\\}\\hspace*{1cm}(k\\geq 2)\n\\end{align*}\nFor the Hochschild cohomology of the convolution algebra we refer to \\cite[Thm 3.11]{nppt} to obtain\n\\begin{align*}\nH^k(\\mathcal{A}_\\mathcal{G},\\mathcal{A}_\\mathcal{G})\\cong \\bigoplus_{\\mathcal{O}\\in\\text{Sec}(\\mathcal{G})}\\Gamma_\\text{inv}(\\Lambda^{k-\\text{codim}(\\mathcal{O})}T\\mathcal{O})\n\\end{align*}\nwhere the sum is over the sectors $\\mathcal{O}$ of $\\mathcal{G}$. The action of the chain map $\\Phi$ on the cohomology of degree $1$ is the inclusion of $\\mathfrak{X}(M)_\\text{inv}$ into this sum as the term for the sector $\\mathcal{O}=M$.\n\\end{ex}\n\\section{Deformation quantization and the van Est map}\\label{quant}\n\\subsection{The adiabatic groupoid}\\label{adiabatic}\n\nIn the theory of deformation quantizations and applications thereof, there is an inherent place for replacing a groupoid with its adiabatic groupoid, as first described in \\cite{connes}. In view of our discussion of the deformation complex, we describe it using the division map:\n\\begin{defi}\nLet $\\mathcal{G}\\rightrightarrows M$ be a Lie groupoid, with Lie algebroid $A\\xrightarrow{\\pi} M$. We define the \\textit{adiabatic groupoid} $\\mathcal{G}_{\\text{ad}}\\to M\\times\\mathbb{R}$ by:\n\\begin{equation*}\n\\mathcal{G}_{\\text{ad}}=A\\times\\{0\\}\\sqcup \\mathcal{G}\\times\\mathbb{R}^\\ast\n\\end{equation*}\nThe source and target are defined by:\n\\begin{equation*}\ns(v,0)=(\\pi(v),0)\n\\end{equation*}\n\\begin{equation*}\ns(g,\\tau)=(s(g),\\tau)\n\\end{equation*}\n\\begin{equation*}\nt(v,0)=(\\pi(v),0)\n\\end{equation*}\n\\begin{equation*}\nt(g,\\tau)=(t(g),\\tau)\n\\end{equation*}\nThen we define the inversion map by\n\\begin{equation*}\n\\iota(v,0)=(-v,0)\n\\end{equation*}\n\\begin{equation*}\n\\iota(g,\\tau)=(\\iota(g),\\tau)\n\\end{equation*}\nLastly, to define we division map, we note that pairs of divisible arrows come in 2 shapes, namely pairs $(v,0)$ and $(w,0)$ with $\\pi(v)=\\pi(w)$, and pairs $(g,\\tau)$ and $(h,\\tau)$ where $g$ and $h$ are divisible. We then define the division map by:\n\\begin{equation*}\n\\overline{m}((v,0),(w,0))=(v-w,0)\n\\end{equation*}\n\\begin{equation*}\n\\overline{m}((g,\\tau),(h,\\tau))=(\\overline{m}(g,h),\\tau)\n\\end{equation*}\n\\end{defi}\nThis is just the set-theoretical description, but the remarkable feature is that the adiabatic groupoid can be given a smooth\nstructure. Here we briefly recall this smooth structure and show how to extend normalized deformation elements to deformation elements of the adiabatic groupoid. Both will be done in the context of the procedure known as the {\\em deformation to the normal cone}.\n\\subsubsection{Deformation to the normal cone}\nThe part of the discussion below concerning the smooth structure and the smooth maps on the deformation to the normal cone is after \\cite[\\S 4]{higson} and \\cite[\\S 1.1]{ds}.\n\\begin{defi}\\label{nms}\nLet $S\\hookrightarrow M$ be a submanifold with normal bundle $N\\to S$. The \\textit{deformation to the normal cone} $N(M,S)$ is the manifold defined by:\n\\begin{equation*}\nN(M,S)=N\\times\\{0\\}\\sqcup M\\times\\mathbb{R}^\\ast\n\\end{equation*}\n\\end{defi}\nThe deformation to the normal cone can be given a topology and smooth structure in two ways. Either it is characterized by the fact that the following two types of maps\n\\begin{itemize}\n\\item The map $N(M,S)\\to M\\times\\mathbb{R}$ that sends $(x,\\tau)$ for $\\tau\\neq  0$ to $(x,\\tau)$ and sends $(v,0)$ with $v\\in N_x$ to $(x,0)$.\n\\item For every $f\\in C^\\infty(M)$ such that $f|_S=0$, the map $\\delta f: N(M,S)\\to\\mathbb{R}$ defined by\n\\begin{align*}\n(\\delta f)(x,\\tau)&=\\frac{f(x)}{\\tau}\\,\\,\\,(x\\in M,\\,\\tau\\neq 0),\\\\\n(\\delta f)(v,0)&=d_nf(v)\\,\\,\\,(v\\in N)\n\\end{align*}\n\\end{itemize}\nare smooth. Here by $d_nf$ we mean the smooth map on $N$ that for $v\\in TM|_S$ sends $[v]$ to $df(v)$ and which is well-defined since $f|_S=0$.\n\nEquivalently, one uses an exponential map, that is a map $\\theta: U\\to M$ from an open neighbourhood $U\\subset N$ of the zero-section, with the property that for all $p\\in S$ and $v\\in N_p$ it holds that\n\\begin{equation*}\n\\theta(0_p)=p,\\qquad \n\\left.\\frac{d}{d\\tau}\\right|_{\\tau=0}\\theta(\\tau v)=v ~\\text{ mod }T_pS\n\\end{equation*}\nThe smooth structure on $N(M,S)$ can then also be characterized by the fact that the maps\n\\begin{equation*}\ni_1: M\\times\\mathbb{R}^\\ast\\to N(M,S):\\,\\,(x,\\tau)\\mapsto (x,\\tau)\n\\end{equation*}\n\\begin{equation*}\ni_2: U'=\\{(v,\\tau)\\in N\\times\\mathbb{R}: \\tau v\\in U\\}\\to N(M,S):\\,\\,\\begin{matrix}(v,\\tau)&\\mapsto& (\\theta(\\tau v),\\tau)\\\\\n(v,0)&\\mapsto& (v,0)\\end{matrix}\n\\end{equation*}\nare open smooth embeddings.\n\nImportant in considering deformations to normal cones is the action of $\\mathbb{R}^\\ast$ on $N(M,S)$, which is given by:\n\\begin{align*}\n\\lambda\\cdot (x,\\tau)&=(x,\\lambda\\tau),\\\\\n\\lambda\\cdot (v,0)&=\\left(\\frac{v}{\\lambda},0\\right)\n\\end{align*}\nwhere $\\lambda,\\tau\\in\\mathbb{R}^\\ast$, $x\\in M$ and $v\\in N$.\n\nWe will describe how to extend a vector field on $M$, that is parallel to $S$, to a vector field on $N(M,S)$ that is invariant under the $\\mathbb{R}^\\ast$-action.\n\nThis will be done by writing down a vector field on the normal bundle and combining it with a vector field over $M\\times\\mathbb{R}^\\ast$ to a discrete vector field on $N(M,S)$, and using an explicit description of the smooth functions on $N(M,S)$ to show that this is in fact  a {\\em smooth} vector field.\n\\begin{defi} \\cite{higson}\nLet $X$ be a set and $\\mathcal{F}=\\{f_\\alpha: X\\to V_\\alpha\\}$ be a family of functions from $X$ into smooth manifolds. We say that a function $f: X\\to\\mathbb{R}$ is \\textit{smoothly composed from the family $\\mathcal{F}$} if there is a finite collection $(f_{\\alpha_1},...,f_{\\alpha_n})\\subset\\mathcal{F}$ and a smooth map $h: V_{\\alpha_1}\\times\\cdots V_{\\alpha_n}\\to\\mathbb{R}$ such that\n\\begin{equation*}\nf(x)=h(f_{\\alpha_1}(x),...,f_{\\alpha_n}(x))\n\\end{equation*}\n\\end{defi}\nThe smooth structure of $N(M,S)$ then means that all smooth functions on $N(M,S)$ are smoothly composed of type of functions as described after \\cref{nms}. If we then apply Taylors theorem we conclude the following.\n\\begin{lem}\\label{smoothvfnms}\nA discrete vector field $X$ on $N(M,S)$ is smooth if and only if for every $f\\in C^\\infty(M)$ with $f|_s=0$ and every $g\\in C^\\infty(M\\times\\mathbb{R})$ the maps $\\delta f$ and $\\tilde{g}\\in C^\\infty(N(M,S))$ defined by:\n\\begin{equation*}\n\\begin{matrix}\n(\\delta f)(x,\\tau)=\\frac{f(x)}{\\tau}&\\,\\,\\,(\\tau\\neq 0)\\\\\n(\\delta f)(v,0)=d_nf(v)&\\,\\,\\,(v\\in N)\\\\\n\\\\\n\\tilde{g}(x,\\tau)=g(x,\\tau)&\\,\\,\\,(\\tau\\neq 0)\\\\\n\\tilde{g}(v,0)=g(x,0)&\\,\\,\\,(v\\in N_x)\n\\end{matrix}\n\\end{equation*}\nsatisfy that $X(\\delta f),X(\\tilde{g})\\in C^\\infty(N(M,S))$.\n\\end{lem}\nWe start with writing down the vector field over $N$. This is the {\\em linearization}, as also in \\cite[\\S 4.1]{az}, that we describe in detail below:\n\\begin{prop}\\label{vectorfieldnormalbundle}\nLet $S\\hookrightarrow M$ be a submanifold with normal bundle $\\pi:N\\to S$ and $X\\in\\mathfrak{X}(M)$ a vector field that is parallel to $S$. Then:\n\\begin{itemize}\n\\item[\\textbf{a)}] The map that sends a smooth function $f\\in C^\\infty(M)$ satisfying $f|_S=0$ to the map $d_nf\\in C^\\infty_\\text{lin} (N)$ is a surjection onto $C^\\infty_\\text{lin}(N)$\n\\item[\\textbf{b)}] If $f\\in C^\\infty(M)$ satisfies that $f|_S=0$ and $d_nf=0$, then $Xf$ satisfies that $d_n(Xf)=0$.\n\\item[\\textbf{c)}] The maps $(X_N)_{\\text{lin}}: C^\\infty_\\text{lin}(N)\\to C^\\infty(N)$ and $(X_N)_{\\text{cst}}: C^\\infty(S)\\to C^\\infty(N)$ defined by\n\\begin{equation*}\n(X_N)_\\text{lin}(d_nf)=d_n(Xf)\n\\end{equation*}\n\\begin{equation*}\n(X_N)_\\text{cst}(g)=X|_S (g)\\circ \\pi\n\\end{equation*}\ndefine a smooth vector field $X_N\\in\\mathfrak{X}(N)$.\n\\end{itemize}\n\\end{prop}\n\\begin{proof} Working down the list:\n\\begin{itemize}\n\\item[\\textbf{a)}] By using a partition of unity this reduces to the local case $M=\\mathbb{R}^m\\times\\mathbb{R}^n$ with $S=\\mathbb{R}^m\\times\\{0\\}$. In this local case there is a canonical diffeomorphism between $M$ and $N$ and pushing a linear map on $N$ through this canonical diffeomorphism yields a smooth map on $M$ which normal derivative equals the linear map on $N$ we started with.\n\\item[\\textbf{b)}] This is again a computation in the local case $M=\\mathbb{R}^m\\times\\mathbb{R}^n$ with $S=\\mathbb{R}^m\\times\\{0\\}$. Write\n\\begin{equation*}\nX=\\sum_{i=1}^m \\alpha_i(x,y)\\frac{\\partial}{\\partial x_i}+\\sum_{j=1}^n\\beta_j(x,y)\\frac{\\partial}{\\partial y_j}\n\\end{equation*}\nThe fact that $X$ is parallel to $S$ means that $\\beta_j(x,0)=0$ for all $j=1,...,n$. The fact that $d_nf=0$ is equivalent to the fact $\\frac{\\partial f}{\\partial y_j}(x,0)=0$ for all $j=1,...,n$. Then we have\n\\begin{equation*}\nXf=\\sum_{i=1}^m \\alpha_i\\frac{\\partial f}{\\partial x_i}+\\sum_{j=1}^n \\beta_j\\frac{\\partial f}{\\partial y_j}\n\\end{equation*}\nSo that for $k=1,...,n$ we have\n\\begin{equation*}\n\\frac{\\partial (Xf)}{\\partial y_k}=\\sum_{i=1}^m\\frac{\\partial \\alpha_i}{\\partial y_k}\\frac{\\partial f}{\\partial x_i}+\\sum_{i=1}^m\\alpha_i\\frac{\\partial^2 f}{\\partial y_k\\partial x_i}+\\sum_{j=1}^n\\frac{\\partial\\beta_j}{\\partial y_k}\\frac{\\partial f}{\\partial y_j}+\\sum_{j=1}^n\\beta_j\\frac{\\partial^2 f}{\\partial y_k\\partial y_j}\n\\end{equation*}\nThen since respectively $\\frac{\\partial f}{\\partial x_i}(x,0)=0$ (since $f(x,0)=0$), $\\frac{\\partial^2 f}{\\partial y_k\\partial x_i}(x,0)=\\left(\\frac{\\partial}{\\partial x_i}\\frac{\\partial f}{\\partial y_k}\\right)(x,0)=0$ (since $\\frac{\\partial f}{\\partial y_k}(x,0)=0$), $\\frac{\\partial f}{\\partial y_j}(x,0)=0$ (by assumption) and $\\beta_j(x,0)=0$ (by assumption), we see that\n\\begin{equation*}\n\\frac{\\partial(Xf)}{\\partial y_k}(x,0)=0\n\\end{equation*}\nwhich implies that $d_n(Xf)=0$.\n\\item[\\textbf{c)}] First note that (by restriction) a smooth vector field $Y\\in\\mathfrak{X}(E)$ on a vector bundle $\\pi: E\\to M$ is the same as a pair of maps $Y_\\text{lin}: C^\\infty_\\text{lin}(E)\\to C^\\infty(E)$ and $Y_\\text{cst}: C^\\infty(M)\\to C^\\infty(E)$ such that for all $f,g\\in C^\\infty(M)$ and $h\\in C^\\infty_\\text{lin}$ it holds that\n\\begin{equation*}\nY_\\text{cst}(fg)=(f\\circ\\pi)\\cdot Y_\\text{cst}(g)+(g\\circ\\pi)\\cdot Y_\\text{cst}(f)\n\\end{equation*}\n\\begin{equation*}\nY_\\text{lin}((f\\circ\\pi)\\cdot h)=(f\\circ\\pi)\\cdot Y_\\text{lin}(h)+h\\cdot Y_\\text{cst}(f)\n\\end{equation*}\nWe show that these properties hold for the maps $(X_N)_\\text{cst}$ and $(X_N)_\\text{lin}$.\n\nFirst we note that $(X_N)_\\text{lin}$ is well-defined by parts a) and b). To show that they define a smooth vector field we check for $f,g\\in C^\\infty(S)$\n\\begin{align*}\n(X_N)_\\text{cst}(fg)&=(X|_S(fg))\\circ\\pi=(f\\cdot X|_S(g)+g\\cdot X|_S(f))\\circ\\pi\\\\\n&=(f\\circ\\pi)\\cdot (X|_S(g)\\circ\\pi)+(g\\circ\\pi)\\cdot (X|_S(f)\\circ\\pi)\\\\\n&=(f\\circ\\pi)X_\\text{cst}(g)+(g\\circ\\pi)X_\\text{cst}(f)\n\\end{align*}\nSecondly let $f\\in C^\\infty(S)$ and $h\\in C^\\infty_\\text{lin}(N)$ given by $h=d_ng$ with $g\\in C^\\infty(M)$ such that $g|_S=0$. Then first we need to find $g'\\in C^\\infty(M)$ with $g'|_S=0$ such that $fh=d_n(g')$. This can be done by choosing an extension of $f$ which is `constant in the normal direction', which is only well-defined locally or if we choose an exponential map.\n\nWe resort to the local case $M=\\mathbb{R}^m\\times\\mathbb{R}^n$ with $S=\\mathbb{R}^m\\times\\{0\\}$. Then the map $g'(x,y)=f(x)g(x,y)$ clearly satisfies that $d_ng'=fh$. Then writing $X$ in coordinates as\n\\begin{equation*}\nX=\\sum_{i=1}^m\\alpha_i\\frac{\\partial}{\\partial x_i}+\\sum_{j=1}^n\\beta_j\\frac{\\partial}{\\partial y_j}\n\\end{equation*}\nwe have\n\\begin{equation*}\n(Xg')(x,y)=\\sum_{i=1}^m\\alpha_i(x,y)\\frac{\\partial f}{\\partial x_i}(x)g(x,y)+f(x)(Xg)(x,y)\n\\end{equation*}\nso that we see\n\\begin{equation*}\n\\frac{\\partial (Xg')}{\\partial y_k}(x,0)=\\sum_{i=1}^m\\alpha_i(x,0)\\frac{\\partial f}{\\partial x_i}(x)\\frac{\\partial g}{\\partial y_k}(x,0)+\\sum_{i=1}^m\\frac{\\partial \\alpha_i}{\\partial y_k}(x,0)\\frac{\\partial f}{\\partial x_i}(x)g(x,0)+f(x)\\frac{\\partial (Xg)}{\\partial y_k}(x,0)\n\\end{equation*}\nThen $g(x,0)=0$ so that the middle term vanishes. Then recognizing terms we obtain\n\\begin{equation*}\nd(Xg')_{(x,0)}\\left(\\frac{\\partial}{\\partial y_k}\\right)=X|_S(f)(x)\\cdot(dg)_x\\left(\\frac{\\partial}{\\partial y_k}\\right)+f(x)\\cdot d(Xg)_{(x,0)}\\left(\\frac{\\partial}{\\partial y_k}\\right)\n\\end{equation*}\nso that globalizing we have\n\\begin{align*}\n(X_N)_\\text{lin}((f\\circ \\pi)\\cdot d_ng)&=(X_N)_\\text{lin}(d_ng')\\\\\n&=d_n(Xg')\\\\\n&=(X|_S(f)\\circ\\pi)d_ng+(f\\circ\\pi)d(Xg)\\\\\n&=(X_N)_\\text{cst}(f)d_ng+(f\\circ\\pi)(X_N)_\\text{lin}(d_ng)\n\\end{align*}\nSo we see that we obtain a smooth vector field $X_N\\in\\mathfrak{X}(N)$.\n\\end{itemize}\nThis completes the proof.\n\\end{proof}\nWe are now ready to define the $\\mathbb{R}^\\ast$-invariant extension of the vector field $X$.\n\\begin{prop}\\label{vfonnms}\nLet $S\\hookrightarrow M$ be a submanifold with normal bundle $N\\to S$. Let $X\\in\\mathfrak{X}(M)$ be a vector field that is parallel to $S$. Then the discrete vector field $X_\\text{inv}$ on $N(M,S)$ defined by\n\\begin{align*}\nX_\\text{inv}(x,\\tau)&=X(x),\\quad(\\tau\\neq 0)\\\\\nX_\\text{inv}|_{N\\times\\{0\\}}&=X_N\n\\end{align*}\nis a smooth vector field $X_\\text{inv}\\in\\mathfrak{X}(M,S)$ which is the unique vector field on $N(M,S)$ which equals $X$ on $M\\times\\mathbb{R}^\\ast$ and the unique $\\mathbb{R}^\\ast$-invariant vector field on $N(M,S)$ which equals $X$ along $M\\times\\{1\\}$.\n\\end{prop}\n\\begin{proof}\nThe invariance and uniqueness is clear assuming that $X_\\text{inv}$ is smooth. To show that it is smooth, by \\cref{smoothvfnms} the only thing we have to check is that $X_\\text{inv}(\\delta f)$ and $X_\\text{inv}(\\tilde{g})$ are smooth for $f\\in C^\\infty(M)$ with $f|_S=0$ and $g\\in C^\\infty(M\\times\\mathbb{R})$. The definition of $X_N$ makes sure that the result is\n\\begin{equation*}\nX_\\text{inv}(\\delta f)=\\delta(Xf)\n\\end{equation*}\n\\begin{equation*}\nX_\\text{inv}(\\tilde{g})=\\tilde{Xg}\n\\end{equation*}\nwhere in the second equation $X$ acts on $C^\\infty(M\\times\\mathbb{R})$ as the vector field $X(x,\\tau)=X(x)$ on $M\\times\\mathbb{R}$. By definition $\\delta(Xf)$ and $\\tilde{Xg}$ are smooth and so the result follows.\n\\end{proof}\n\\subsubsection{The adiabatic groupoid as a deformation to the normal cone}\nWe can now apply this to the case $M\\hookrightarrow\\mathcal{G}$ with normal bundle $A=\\ker ds|_M$. The fact that the source, target and division maps are smooth, follows from the fact that away from $\\tau=0$ they are just the respective maps of the original groupoid, while along $\\tau=0$ they are the normal derivatives of the respective maps. A general principle of deformations to normal cones then means they are smooth. We note that an exponential map can be obtained by choosing a connection on $A$, see \\cite{nwx} and \\cite{landsmanboek}.\n\nNext we want to describe the nerve of the adiabatic groupoid. As a set it equals $(\\mathcal{G}_{\\text{ad}})^{(k)}=\\mathcal{G}^{(k)}\\times\\mathbb{R}^\\ast\\sqcup A^{\\oplus k}\\times\\{0\\}$. From the view point of trying to define vector fields on the nerve of the adiabatic groupoid, this set-theoretic description leads to searching for a connection between $A^{\\oplus k}$ and the normal bundle of $M$ inside $\\mathcal{G}^{(k)}$ as the diagonal of units.\n\n\\begin{lem}\\label{normalbundenerve}\nLet $\\mathcal{G}\\rightrightarrows M$ be a Lie groupoid with $\\Delta: M\\to\\mathcal{G}^{(k)}$ the diagonal inclusion via the units. The vector bundle map $\\nu: A^{\\oplus k}\\to\\Delta^\\ast T\\mathcal{G}^{(k)}$ given by\n\\begin{equation*}\n\\nu(v_1,...,v_k)=(v_1+\\sum_{i=2}^k du(dt(v_i)),v_2+\\sum_{i=3}^k du(dt(v_i)),...,v_{k-1}+du(dt(v_k)),v_k)\n\\end{equation*}\ninduces an isomorphism between $A^{\\oplus k}$ and the normal bundle of $M$ inside $\\mathcal{G}^{(k)}$.\n\\end{lem}\n\\begin{proof}\nFirst one checks that $\\nu$ indeed maps into the tangent space of $\\mathcal{G}^{(k)}\\subset\\mathcal{G}^{\\times k}$, which is a simple calculation. Next to show that it induces an isomorphism to the normal bundle to $\\Delta$, we first use the decomposition $T_{1_x}M=A_x\\oplus T_xM$ to see that if $\\nu(v_1,...,v_k)\\in T_xM\\subset T_{\\Delta(x)}\\mathcal{G}^{(k)}$ then $(v_1,...,v_k)=0$, so that the map into the normal bundle is injective. A simple case of dimension counting then implies that it the induced map is an isomorphism.\n\\end{proof}\n\\begin{cor}\nThere is a natural isomorphism between $N(\\mathcal{G}^{(k)},M)$ and $\\mathcal{G}_{\\text{ad}}^{(k)}$ which away from $\\tau=0$ links $((g_1,...,g_k),\\tau)$ and $((g_1,\\tau),...,(g_k,\\tau))$.\n\\end{cor}\n\\subsubsection{Haar systems on the adiabatic groupoid}\nWe intend to link deformation quantizations of the Poisson manifold $A^\\ast$ with the Van Est map $\\mathcal{V}:\\tilde{C}^\\bullet_\\text{def}(\\mathcal{G})\\to C^\\bullet_\\text{def}(A)$. To make the syntax line up, we need to explicitely write down isomorphisms between smooth functions on $\\mathcal{G}$ and elements of the convolution algebra. This is done via Haar systems, which we will describe here in terms of densities.\n\n\\begin{defi}A Haar system on a groupoid $\\mathcal{G}\\rightrightarrows M$ is a collection $\\lambda=\\{\\lambda_x\\}_{x\\in M}$ of positive sections $\\lambda_x\\in\\Gamma(\\mathcal{D}_s|_{s^{-1}(x)})$ that are invariant under right translations $R_g: s^{-1}(t(g))\\to s^{-1}(s(g))$ and such that for every compactly supported function $f\\in C^\\infty_c(\\mathcal{G})$ the map $\\lambda(f): M\\to\\mathbb{R}$ given by\n\\begin{equation*}\n\\lambda(f)(x)=\\int_{s^{-1}(x)}f(g)\\lambda_x(g)\n\\end{equation*}\nis smooth.\n\\end{defi}\nWe know that every Lie groupoid admits a Haar system (\\cite[Prop 3.4]{landsman}) and if we have a Haar system $\\lambda$ on a Lie groupoid $\\mathcal{G}\\rightrightarrows M$ with $s$-fibers of dimension $d$, we can (\\cite[p.19]{landsman}) induce a Haar system $\\hat{\\lambda}$ on $\\mathcal{G}_{\\text{ad}}$ given by\n\\begin{equation*}\n\\hat{\\lambda}(g,\\tau)=|\\tau|^d\\lambda(g)\n\\end{equation*}\n\\begin{equation*}\n\\hat{\\lambda}(v,0)=\\lambda(\\pi(v))\n\\end{equation*}\nHere $\\pi: A\\to M$ is the projection and we take the canonical isomorphism $\\ker(d\\pi)\\cong\\pi^\\ast(A)$ as a given.\n\nNote that in particular we obtain a Haar system on the vector bundle $A\\to M$, seen as a groupoid in the canonical way.\n\nThe choice of a Haar system induces an isomorphism between the sheaf of smooth functions on $\\mathcal{G}$ and the sheaf of densities along the source fibers, and hence we can transport the convolution product over to the compactly supported functions where it is given by:\n\\begin{equation*}\n(f_1\\ast f_2)(g)=\\int_{s^{-1}(s(g))}f_1(gh^{-1})f_2(h)\\lambda_{s(g)}(h)\n\\end{equation*}\nIn particular on the adiabatic groupoid $\\mathcal{G}_{\\text{ad}}$ if we have two compactly supported functions $f_1,f_2$ we obtain:\n\\begin{equation*}\n(f_1\\ast f_2)(g,\\tau)=|\\tau|^{-d}\\int_{s^{-1}(s(g))}f_1(gh^{-1},\\tau)f_2(h,\\tau)\\lambda_{s(g)}(h)\\,\\,\\,\\,(\\tau\\neq 0)\n\\end{equation*}\n\\begin{equation*}\n(f_1\\ast f_2)(v,0)=\\int_{A_{\\pi(v)}}f_1(v-w,0)f_2(w,0)\\lambda_{\\pi(v)}(w)\n\\end{equation*}\nAt this point we notice that the convolution at $\\tau=0$ does not require the functions to be compactly supported on $A_x$, being Schwartz is enough (c.f. the usual theory of Fourier transform in $\\mathbb{R}^n$). This allows us, in the case of $\\mathcal{G}_{\\text{ad}}$, to enlarge the type of functions\/densities on which we let the deformation complex act.\n\nTo this end we refer to the work of \\cite{cr}, where a Fr\\'ech\\`et algebra $\\mathscr{S}_c(\\mathcal{G}_{\\text{ad}})$ is constructed with evaluations\n\\[\n\\mathscr{S}_c(\\mathcal{G}_{\\text{ad}})_t=\\begin{cases} \\mathscr{S}_c(A)& t=0\\\\ C_c^\\infty(\\mathcal{G})&t\\not = 0.\\end{cases}\n\\]\nHere $\\mathscr{S}_c(A)$ denotes the space of functions that are Schwartz along the fibers of the Lie algebroid and have compact support along $M$. This Schwartz type algebra should be thought of as a dense subalgebra the reduced $C^*$-algebra $C^*_r(\\mathcal{G}_{\\text{ad}})$.\n\nBy the discussion above, the convolution product is perfectly well-defined on $\\mathscr{S}_c(\\mathcal{G}_{\\text{ad}})$ and we can extend our viewpoint of the map $\\Phi: C^\\bullet_\\text{def}(\\mathcal{G}_{\\text{ad}})\\to C^\\bullet_\\text{Hoch}(\\mathcal{A}_{\\mathcal{G}_{\\text{ad}}})$ to let $\\Phi(c)$ (for $c\\in C^k_\\text{def}(\\mathcal{G}_{\\text{ad}})$) act on functions in $\\mathscr{S}_c(\\mathcal{G}_{\\text{ad}})$. At this point it should be remarked that the isomorphism between functions and densities induced by a Haar system does not preserve the action of vector fields (indeed on the level of densities one also needs to compare $\\mathcal{L}_X\\lambda$ with $\\lambda$!). So really we should introduce in parallel to $\\mathscr{S}_c(\\mathcal{G}_{\\text{ad}})$ the notion of densities with are of Schwartz-type along $\\tau=0$, but for the sake of not being overly pedantic we will not do this and just be careful when writing down the action of $\\Phi(c)$.\n\nIn what follows for a smooth family $\\{f_t\\}_{t\\neq 0}$ of compactly supported functions on $\\mathcal{G}$ and $f'\\in\\mathscr{S}_c(A)$ we will use the notation\n\\begin{equation*}\n\\lim_{t\\to 0} f_t=f'\n\\end{equation*}\nif the function $F:\\mathcal{G}_{\\text{ad}}\\to\\mathbb{R}$ given by\n\\begin{equation*}\nF(g,t)=f_t(g)\n\\end{equation*}\n\\begin{equation*}\nF(v,0)=f'(v)\n\\end{equation*}\nis an element of $\\mathscr{S}_c(\\mathcal{G}_{\\text{ad}})$.\n\\subsection{Fourier transform on vector bundles}\nWe briefly discuss the notion of Fourier transform on a vector bundle $E\\to M$ under the choice of a Haar system on $E$. This discussion follows the results of Landsman and Ramazan \\cite[\\S 7]{landsman}. Recall that a vector bundle $\\pi: E\\to M$ can be seen as a groupoid over $M$ where both the source and the target map are the projection $\\pi$ and the multiplication is the fiberwise addition. Since $\\ker(d\\pi)\\cong\\pi^\\ast E$ a choice of a Haar system is at every $v\\in E$ a choice of a density on $E_{\\pi(v)}$ that is invariant, where invariance in this case means that the choice is constant along the fiber.\n\nIf we choose such a Haar system $\\{\\mu_x\\}_{x\\in M}$, in \\cite{landsman} the Fourier transform $\\mathcal{F}_\\mu:\\mathscr{S}(E)\\to\\mathscr{S}(E^\\ast)$ was defined by\n\\begin{equation*}\n(\\mathcal{F}_\\mu f)(\\xi_x)=\\int_{E_x} f(v)e^{-i\\langle\\xi_x,v\\rangle}d\\mu_x(v)\n\\end{equation*}\nFurthermore, it was shown that this map is a linear isomorphism which intertwines the $\\mu$-convolution product on $E$ and the pointwise product on $E^\\ast$, and when $(x,v)$ are coordinates on $E$ induced by a frame with dual coordinates $(x,\\xi)$ we have for $f\\in\\mathscr{S}(E)$, $g\\in\\mathscr{S}(E^\\ast)$ and $a\\in C^\\infty(M)$ that\n\\begin{align*}\n\\mathcal{F}_\\mu((a\\circ\\pi)f)&=(a\\circ \\pi)\\mathcal{F}_\\mu\\\\\n\\frac{\\partial\\mathcal{F}_\\mu(f)}{\\partial x_j}&=\\mathcal{F}_\\mu\\left(\\frac{\\partial f}{\\partial x_j}\\right)+\\left(\\frac{\\partial\\text{log}(\\mu_e)}{\\partial x_j}\\circ\\pi\\right)\\mathcal{F}_\\mu(f)\\\\\n\\frac{\\partial\\mathcal{F}_\\mu(f)}{\\partial \\xi_j}&=-i\\mathcal{F}_\\mu(v_j f)\\\\\n\\frac{\\partial\\mathcal{F}_\\mu^{-1}(g)}{\\partial v_j}&=i\\mathcal{F}_\\mu^{-1}(\\xi_jg)\n\\end{align*}\nNote that after the choice of a Haar system $\\mu$ we obtain an isomorphism between the algebra of functions $C^\\infty_c(E)$ with the $\\mu$-convolution product and the convolution algebra $\\mathcal{A}_E$ of densities with the (intrinsic) convolution product. In particular if $X\\in\\mathfrak{X}(E)$, we can see $\\Phi(X)$ as defined on functions (which is, again, not equal to the usual action of vector fields on functions), and we can extend the action to Schwartz functions.\n\nNow using the Fourier transform, we can transport the action on the convolution algebra of $E$ to an action on the usual algebra with the pointwise product on $E^\\ast$.\n\\begin{prop}\nLet $X$ be a linear vector field on $E$. Then the map $\\hat{X}:\\mathscr{S}(E^\\ast)\\to\\mathscr{S}(E^\\ast)$ given by\n\\begin{equation*}\n\\hat{X}(f)=\\mathcal{F}_\\mu(\\Phi(X)(\\mathcal{F}_\\mu^{-1}(f)))\n\\end{equation*}\ndefines a linear vector field on $E^\\ast$. Here $\\Phi$ is the natural chain map we defined before, applied to the vector bundle $E$ seen as a groupoid.\n\\end{prop}\n\\begin{proof}\nFirst we show that $\\hat{X}$ is indeed a vector field, i.e. a derivation with respect to the pointwise product. Since $\\hat{X}$ is the conjugation of $\\Phi(X)$ with an isomorphism which intertwines the convolution product on $\\mathscr{S}(E)$ and the pointwise product on $\\mathscr{S}(E^\\ast)$ this is equivalent to showing that $\\Phi(X)$ is a derivation for the convolution product. When we see $E\\to M$ as a groupoid, this is equivalent to showing that $X$ is a multiplicative vector field, and it is easy to see that on a vector bundle the multiplicative vector fields are precisely the linear vector fields.\n\nTo see that $\\hat{X}$ is a linear vector field we do a local computation on a trivial vector bundle $E=\\mathbb{R}^m_x\\times\\mathbb{R}^n_v\\to\\mathbb{R}^m_x$ with Haar system $f(x)dv_1\\wedge\\cdots\\wedge dv_n$. Using the properties of the Fourier transform stated before it follows that if\n\\begin{equation*}\nX(x,v)=\\sum_{i=1}^mX_i(x)\\frac{\\partial}{\\partial x_i}+\\sum_{j=1}^n\\sum_{k=1}^n Y_{jk}(x)v_j\\frac{\\partial}{\\partial v_k}\n\\end{equation*}\nthen\n\\begin{equation*}\n\\hat{X}(x,\\xi)=\\sum_{i=1}^mX_i(x)\\frac{\\partial}{\\partial x_i}-\\sum_{j=1}^n\\sum_{k=1}^nY_{jk}(x)\\xi_k\\frac{\\partial}{\\partial\\xi_j}\n\\end{equation*}\nwhich indeed shows that $\\hat{X}$ is a linear vector field.\n\\end{proof}\nRecall that a linear vector field $X\\in\\mathfrak{X}(E)$ is the same as a linear map $X:\\Gamma(E^\\ast)\\to\\Gamma(E^\\ast)$ with a symbol $s_X\\in\\mathfrak{X}(M)$ such that\n\\begin{equation*}\nX(f\\alpha)=fX(\\alpha)+s_X(f)\\alpha\\qquad(f\\in C^\\infty(M),~\\alpha\\in\\Gamma(E)).\n\\end{equation*}\nFurthermore, recall the canonical pairing $\\langle-,-\\rangle:\\Gamma(E^\\ast)\\times\\Gamma(E)\\to C^\\infty(M)$. Then for a linear vector field $X$, the local calculation from the proof above generalizes to the following.\n\\begin{prop}\\label{xhat}\nLet $X\\in\\mathfrak{X}(E)$ be a linear vector field, then the linear vector field $\\hat{X}\\in\\mathfrak{X}(E^\\ast)$ is uniquely determined by the fact that for $\\beta\\in\\Gamma(E^\\ast)$ and $\\alpha\\in\\Gamma(E)$\n\\begin{equation*}\n\\langle\\beta,\\hat{X}(\\alpha)\\rangle+\\langle X(\\beta),\\alpha\\rangle=s_X(\\langle\\beta,\\alpha\\rangle)\n\\end{equation*}\n\\end{prop}\nWe can play a similar game, albeit slightly more involved in notation, for higher order deformation elements of the vector bundle. So consider an element $X\\in\\tilde{C}^k_\\text{def}(E)$ given by\n\\begin{equation*}\nX(v_1,...,v_n)=X_1(v_1)\\langle \\beta_2,v_2\\rangle\\cdots\\langle \\beta_k,v_k\\rangle\n\\end{equation*}\nwhere $X_1$ is a linear vector field on $E$ and $\\beta_2,...,\\beta_k\\in\\Gamma(E^\\ast)$. One immediately checks that this is a closed element of $\\tilde{C}^k_\\text{def}(E)$, so that the Fourier transform\n\\begin{equation*}\n\\hat{X}(f_1,...,f_k)=\\mathcal{F}_\\mu(\\Phi(X)(\\mathcal{F}_\\mu^{-1}(f_1),...,\\mathcal{F}_\\mu^{-1}(f_k)))\n\\end{equation*}\nis a closed element of the Hochschild complex of $C^\\infty(E^\\ast)$. By the specific form of $X$ is it easy to see that\n\\begin{equation*}\n\\Phi(X)(a_1,...,a_k)=\\Phi(X_1)(a_1)\\ast (\\beta_2a_2)\\ast\\cdots\\ast(\\beta_k a_k)\n\\end{equation*}\nwhere we see the $s_i$ as fiberwise linear maps on $E$. In particular we see that\n\\begin{equation*}\n\\hat{X}=\\hat{X_1}\\otimes\\hat{\\beta_2}\\otimes\\cdots\\otimes\\hat{\\beta_k}\n\\end{equation*}\nwhere for $\\beta\\in\\Gamma(E^\\ast)$, $\\hat{\\beta}$ is the vector field on $E^\\ast$ given by\n\\begin{equation*}\n\\hat{\\beta}(f)=\\mathcal{F}_\\mu(\\beta\\mathcal{F}_\\mu^{-1}(f))\n\\end{equation*}\nA local computation shows that $\\hat{\\beta}$ is identically zero on fiberwise constant maps and for the map induced by a section $\\alpha\\in\\Gamma(E)$ we have\n\\begin{equation*}\n\\hat{\\beta}(\\alpha)=\\frac{1}{i}\\langle \\beta,\\alpha\\rangle\n\\end{equation*}\nIn particular, we see that if we anti-symmetrize, we obtain the linear multivectorfield $\\hat{X_1}\\wedge\\hat{\\beta_2}\\wedge\\cdots\\wedge\\hat{\\beta_k}$ on $E^\\ast$.\n\\subsection{Deformation quantization of $A^\\ast$ and the Van Est map}\nNow, fix a choice of a Haar system of $\\mathcal{G}$, which by the discussion above induces a Haar system on $\\mathcal{G}_{\\text{ad}}$ and a Haar system $\\mu$ on $A\\to M$. The last one makes sure that we can talk about a Fourier transform $\\mathcal{F}_\\mu:\\mathscr{S}(A)\\to\\mathscr{S}(A^\\ast)$.\n\nSlightly tweaking the results of \\cite{landsman} we obtain {\\em quantization maps} $q_t:\\mathscr{S}_c(A^*)\\to C^\\infty_c(\\mathcal{G}),~t\\not = 0$ given by\n\\[\nq_t(f)(g):=\\chi(g)\\mathcal{F}_\\mu^{-1}(f)(\\frac{1}{t}\\exp^{-1}(g)),\n\\]\nwhich satisfy\n\\begin{equation}\n\\label{quant}\n\\lim_{t\\to 0}(q_t(f_1 f_2)-q_t(f_1)\\ast q_t(f_2))=0,\\quad \\lim_{t\\to 0}(\\frac{1}{it}[q_t(f_1),q_t(f_2)]-q_t(\\{f_1,f_2\\}))=0.\n\\end{equation}\nHere $\\chi\\in C^\\infty_c(\\mathcal{G})$ is a cut-off function that equals $1$ in a neighborhood of $M\\subset\\mathcal{G}$ \nwith support inside an open neighbourhood of the units onto which the exponential map is a diffeomorphism. The Poisson \nbracket $\\{~,~\\}$ is the bracket associated to the so-called Lie--Poisson structure on $A^*$.\n\nExplicitely, one of the differences with the results of \\cite{landsman} is that we do not need the property $q_t(f^\\ast)=q_t(f)^\\ast$ for which the Weyl exponential map $\\text{exp}^{\\text{W}}$ is used, and instead we can use the normal exponential map. Secondly, we do not need to restrict to Paley-Wiener functions, as we allow for Schwarz-type functions at $t=0$ and use the cut-off function on the level of $\\mathcal{G}$ instead of $A$, the deviation vanishing as $t$ approaches $0$. Lastly, as the relevant calculations on the local forms in $A$ and $A^\\ast$ are valid for all Schwarz functions and not just Paley-Wiener functions, the relevant propositions in \\cite{landsman} still hold in this situation. The variety of quantizations by using different types of exponential maps is also reflected on the more algebraic level in \\cite{nw} by using different orderings in the Fedossov construction of {\\em formal} deformation quantizations of $A^*$.\n\nWe now briefly recall the van Est-map as given in \\cite[\\S 10]{cms}. First the deformation complex of the algebroid $C^k_\\text{def}(A)$ is given by antisymmetric multilinear maps $D: \\Gamma(A)^k\\to\\Gamma(A)$ that have a symbol $s_D:\\Gamma(A)^{k-1}\\to\\mathfrak{X}(M)$ such that\n\\begin{equation*}\nD(\\alpha_1,...,f\\alpha_k)=fD(\\alpha_1,...,\\alpha_k)+s_D(\\alpha_1,...,\\alpha_{k-1})(f)\\alpha_k\n\\end{equation*}\nNote that we can, and will, see elements of $C^k_\\text{def}(A)$ as linear multivectorfields on $A^\\ast$ (by noting that sections of $A$ are the same as fiberwise linear maps on $A^\\ast$) and in turn see the deformation complex of $A$ as the linear Poisson complex of the Poisson manifold $A^\\ast$.\n\nThen for $\\alpha\\in\\Gamma(A)$ there are maps $R_\\alpha:\\tilde{C}^k_\\text{def}(\\mathcal{G})\\to\\tilde{C}^{k-1}_\\text{def}(\\mathcal{G})$ which are given for $k=1$ by\n\\begin{equation*}\nR_\\alpha(c)=[c,\\overrightarrow{\\alpha}]|_M\n\\end{equation*}\nand for $k>0$ by\n\\begin{equation*}\nR_\\alpha(c)(g_1,...,g_{k-1})=(-1)^{k-1} \\left.\\frac{d}{d\\epsilon}\\right|_{\\epsilon=0} c(g_1,...,g_{k-1},\\Phi^\\epsilon_{\\overrightarrow{\\alpha}}(s(g_{k-1}))^{-1})\n\\end{equation*}\nThe van Est-map $\\mathcal{V}:\\tilde{C}^k_\\text{def}(\\mathcal{G})\\to C^k_\\text{def}(A)$ is then given by\n\\begin{equation*}\n\\mathcal{V}(c)(\\alpha_1,...,\\alpha_k)=\\sum_{\\sigma\\in S_k}(-1)^\\sigma (R_{\\alpha_{\\sigma(k)}}\\circ\\cdots\\circ R_{\\alpha_{\\sigma(1)}})(c)\n\\end{equation*}\nThe connection between the van Est-map and the quantization maps is then as follows.\n\\begin{thm}\nLet $k\\geq 1$ and $c\\in \\tilde{C}^k_{\\text{def}}(\\mathcal{G})$ and suppose the choice of a Haar system on $\\mathcal{G}$ inducing a Haar system $\\mu$ on the algebroid $A$. Given $f_1,\\ldots,f_k\\in  \\mathscr{S}_c(A^*)$, the following \nequality holds true:\n\\[\n\\mathcal{V}(c)(f_1,\\ldots,f_k)=\\mathcal{F}_\\mu\\left(\\lim_{t\\to 0}\\left(\\sum_{\\sigma\\in S_k}(-1)^\\sigma\\frac{1}{(it)^{k-1}}\\Phi(c)(q_t(f_{\\sigma(1)}),\\ldots,q_t(f_{\\sigma(k)}))\\right)\\right)\n\\]\n\\end{thm}\n\\begin{rmk}\nNote that the right hand side of the equation above is well-defined, since the sum of which we take the limit is a function on the groupoid. The limit is then a Schwartz function on the algebroid and so if we take the Fourier transform we obtain a Schwartz function on the dual of the algebroid.\n\\end{rmk}\n\\begin{proof}\nWe start with the case $k=1$. First note that for $f\\in\\mathscr{S}_c(A^\\ast)$ the map $q(f):\\mathcal{G}_{\\text{ad}}\\to\\mathbb{R}$ given by\n\\begin{align*}\nq(f)(g,t)&=q_t(f)(g)\\\\\nq(f)(v,0)&=\\mathcal{F}_\\mu^{-1}(f)(v)\n\\end{align*}\nis an element of $\\mathscr{S}_c(\\mathcal{G}_{\\text{ad}})$. Then note that the family $\\{c\\}_{t\\neq 0}$ is a family of vector fields on $\\mathcal{G}$ which can be extended to a vector field on $\\mathcal{G}_{\\text{ad}}$, namely to the vector field $c_\\text{inv}$ obtaines by \\cref{vfonnms}. Then notice that $\\Phi(c_\\text{inv})(q(f))$ is an element of $\\mathscr{S}_c(\\mathcal{G}_{\\text{ad}})$ consisting of\n\\begin{align*}\n\\Phi(c_\\text{inv})(q(f))_t&=\\Phi(c)(q_t(f)),\\qquad(t\\neq 0)\\\\\n\\Phi(c_\\text{inv})(q(f))_0&=\\Phi(c_0)(\\mathcal{F}_\\mu^{-1}(f))\n\\end{align*}\nwhere $c_0$ is the linear vector field that is the restriction of $c_\\text{eqv}$ to $t=0$. Note that it is linear, since it is the application of \\cref{vectorfieldnormalbundle} to the vector field $c$ on $\\mathcal{G}$. In particular we see that\n\\begin{equation*}\n\\lim_{t\\to 0}\\Phi(c)(q_t(f))=\\Phi(c_0)(\\mathcal{F}_\\mu^{-1}(f))\n\\end{equation*}\nand so we need to show that $\\mathcal{V}(c)=\\hat{c_0}$.\n\nBy \\cref{xhat} this means that we need to show that for $\\beta\\in\\Gamma(A^\\ast)$ and $\\alpha\\in\\Gamma(A)$ we have\n\\begin{equation*}\n\\langle\\beta,\\mathcal{V}(c)(\\alpha)\\rangle+\\langle c_0(\\beta),\\alpha\\rangle=s_c(\\langle\\beta,\\alpha\\rangle)\n\\end{equation*}\nWe note two things. First that every $\\beta\\in\\Gamma(A^\\ast)=C^\\infty_\\text{lin}(A)$ can be written as $d_nh$ for $h\\in C^\\infty(\\mathcal{G})$ with $h|_M=0$. Second that since we have an explicit inclusion of $A$ into the tangent bundle of $\\mathcal{G}$ this means that:\n\\begin{equation*}\n\\langle d_nh,\\alpha\\rangle(x)=\\alpha(x)(h)\n\\end{equation*}\nWe are now ready to show the equality. First we have\n\\begin{align*}\n\\langle d_nh,\\mathcal{V}(c)(\\alpha)\\rangle(x)&=[c,\\overrightarrow{\\alpha}](1_x)(h)\\\\\n&=c(1_x)(\\overrightarrow{\\alpha}(h))-\\alpha(x)(c(h))\n\\end{align*}\n\\begin{equation*}\n\\langle c_0(d_n h),\\alpha\\rangle(x)=\\langle d_n(ch),\\alpha\\rangle(x)=\\alpha(x)(c(h))\n\\end{equation*}\nand since $c_{1_x}=du(s_c(x))$ combined with $\\overrightarrow{\\alpha}|_M=\\alpha$ we have\n\\begin{equation*}\ns_c(\\langle d_nh,\\alpha\\rangle)(x)=s_c(\\overrightarrow{\\alpha}(h)|_M)(x)=c(1_x)(\\overrightarrow{\\alpha}(h))\n\\end{equation*}\nFor $k>1$ we restrict to the case where $c=c_1\\otimes h_2\\otimes\\cdots\\otimes h_k$ with $c_1\\in\\mathfrak{X}(\\mathcal{G})$ and $h_2,...,h_k\\in C^\\infty(\\mathcal{G})$. For $c$ to be an element of $\\tilde{C}^k_\\text{def}(\\mathcal{G})$ it is necessairy and sufficient to have $c_1\\in\\tilde{C}^1_\\text{def}(\\mathcal{G})$ and $h_i|_M=0$. Similar to the case $k=1$ we note that\n\\begin{equation*}\n\\frac{1}{(it)^{k-1}}\\Phi(c)(q_t(f_1),...,q_t(f_k))=\\Phi(\\frac{1}{(it)^{k-1}}c)(q_t(f_1),...,q_t(f_k))\n\\end{equation*}\nwhich, as $t\\to 0$, converges\n\\begin{equation*}\n\\Phi(c_0)(\\mathcal{F}_\\mu^{-1}(f_1),...,\\mathcal{F}_\\mu^{-1}(f_k))\n\\end{equation*}\nif we find a vector field $c_0$ on $A$ that together with the family $\\{\\frac{1}{(it)^{k-1}}c\\}_{t\\neq 0}$ defines a smooth deformation element of $\\mathcal{G}_{\\text{ad}}$.\n\nTo calculate this localization we remark that we can do the calculation in $\\mathcal{G}^k$ using the cartesian product of the exponential map $A\\to\\mathcal{G}$, in stead of working in $\\mathcal{G}^{(k)}$ and using the machinery of the previous section. This is for two reasons: firstly our definition of $c$ extends to $\\mathcal{G}^k$. Secondly the difference of $(v_1,...,v_k)\\in A^{\\oplus k}$ seen as tangent vectors on $\\mathcal{G}^k$ and $(v_1,...,v_k)\\in A^{\\oplus k}$ seen as tangent vectors in $\\mathcal{G}^{(k)}$ which are normal to the units, using the isomorphism of \\cref{normalbundenerve}, are tangent vectors in $\\mathcal{G}^k$ which are along the units. Since $c$ vanishes along the units, we can neglect this.\n\nNow to do the actual calculation we consider the chart $\\theta: A^{\\oplus k}\\times\\mathbb{R}^\\ast\\to\\mathcal{G}^k\\times\\mathbb{R}^\\ast$ given by\n\\begin{equation*}\n\\theta(v_1,...,v_k,t)=(\\exp(tv_1),...,\\exp(tv_k),t)\n\\end{equation*}\nThen if we look at the family $\\{\\frac{1}{(it)^{k-1}}c\\}_{t\\neq 0}$, we see that if we take the pullback along $\\theta$ we obtain:\n\\begin{equation*}\n\\theta^\\ast(\\{\\frac{1}{(it)^{k-1}}c\\}_{t\\neq 0})(v_1,...,v_k,t)=\\frac{1}{(it)^k}c_1(\\exp(tv_1))h_2(\\exp(tv_2))\\cdots h_k(\\exp(tv_k))\n\\end{equation*}\nDistributing the $k$ powers of $\\frac{1}{t}$ over the $k$ different terms we see that\n\\begin{equation*}\nc_0(v_1,...,v_k)=\\frac{1}{i^{k-1}}(c_1)_0(v_1)d_nh_2(v_2)\\cdots d_nh_2(v_k)\n\\end{equation*}\nsince\n\\begin{equation*}\n\\frac{1}{t}c_1(\\exp(tv_1))\\to (c_1)_0(v_1)\n\\end{equation*}\n\\begin{equation*}\n\\frac{1}{t}h(\\exp(tv))\\to d_nh(v)\n\\end{equation*}\nas $t\\to 0$, so we see that $c_0=\\frac{1}{i^{k-1}}(c_1)_0\\otimes d_nh_2\\otimes\\cdots\\otimes d_nh_k$, which is a linear deformation element, and we want to show that $\\mathcal{V}(c)$ is the anti-symmetrization of the Fourier transform $\\hat{c_0}$. By the discussion at the end of the previous subsection we see that $\\hat{c_0}$ is determined for $\\alpha_1,...,\\alpha_k\\in\\Gamma(A)$ by\n\\begin{equation*}\n\\hat{c_0}(\\alpha_1,...,\\alpha_k)=\\frac{1}{i^{2(k-1)}}\\hat{(c_1)_0}(\\alpha_1)\\langle d_nh_2,\\alpha_2\\rangle\\cdots\\langle d_nh_k,\\alpha_k\\rangle\n\\end{equation*}\nNext we investigate $R_\\alpha(c)$, we obtain:\n\\begin{align*}\nR_\\alpha(c)(g_1,...,g_{k-1})&=(-1)^{k-1}\\frac{d}{d\\epsilon}|_{\\epsilon=0}c_1(g_1)h_2(g_2)\\cdots h_{k-1}(g_{k-1})h_k(\\Phi^\\epsilon_{\\overrightarrow{\\alpha}}(s(g_k))^{-1})\\\\\n&=(-1)^{k-1}c_1(g_1)h_2(g_2)\\cdots h_{k-1}(g_{k-1})dh_k(d\\iota(\\alpha(s(g_k)))\n\\end{align*}\nThen since $f_k|_M=0$ and for $v\\in A_x$ we have $d\\iota v=-v+d(u\\circ t)(v)$ we obtain\n\\begin{equation*}\nR_\\alpha(c)(g_1,...,g_{k-1})=(-1)^k c_1(g_1)h_2(g_2)\\cdots h_{k-1}(g_{k-1})d_nh_k(\\alpha(s(g_{k-1}))\n\\end{equation*}\nDoing this inductively, and using that the flow of $\\overrightarrow{\\alpha}$ preserves source fibers, we see\n\\begin{equation*}\n(R_{\\alpha_2}\\circ\\cdots\\circ R_{\\alpha_k})(c)(g)=(-1)^{\\frac{(k-1)(k-2)}{2}}c_1(g)d_nh_2(\\alpha_2(s(g))\\cdots d_nh_k(\\alpha_k(s(g))\n\\end{equation*}\nThen since this is simply $c_1$ multiplied with a function that is constant along the $s$-fibers, we obtain:\n\\begin{align*}\n(R_{\\alpha_1}\\circ\\cdots \\circ R_{\\alpha_k})(c)&=(-1)^{\\frac{(k-1)(k-2)}{2}}\\mathcal{V}(c_1)(\\alpha_1)\\langle d_nh_2,\\alpha_2\\rangle\\cdots\\langle d_nh_k,\\alpha_k\\rangle\\\\\n&=i^{(k-1)(k-2)}\\mathcal{V}(c_1)(\\alpha_1)\\langle d_nh_2,\\alpha_2\\rangle\\cdots\\langle d_nh_k,\\alpha_k\\rangle\n\\end{align*}\nSince already know by the calculation in the case $k=1$ that $\\mathcal{V}(c_1)(\\alpha_1)=\\hat{(c_1)_0}(\\alpha_1)$ we see that\n\\begin{equation*}\n(R_{\\alpha_1}\\circ\\cdots \\circ R_{\\alpha_k})(c)=i^{k(k-1)}\\hat{c_0}(\\alpha_1,...,\\alpha_k)\n\\end{equation*}\nThen note that there is a mismatch in the summation over $S_k$ in $\\mathcal{V}(c)$ and in the right hand side of the theorem. In particular the right hand side in the last equation corresponds to the identity permutation in the statement of the theorem, while the right hand side corresponds to the permutation in the definition of $\\mathcal{V}(c)$ that sends $j$ to $k-j$. This sign of this permutation is $(-1)^{\\frac{k(k-1)}{2}}$, for which we have to correct, so that we obtain\n\\begin{align*}\n\\mathcal{V}(c)(\\alpha_1,...,\\alpha_k)&=\\sum_{\\sigma\\in S_k}(-1)^\\sigma (R_{\\alpha_{\\sigma(k)}}\\circ\\cdots R_{\\alpha_{\\sigma(1)}})(c)\\\\\n&=\\sum_{\\sigma\\in S_k}(-1)^\\sigma i^{k(k-1)}(R_{\\alpha_{\\sigma(1)}}\\circ\\cdots\\circ R_{\\alpha_{\\sigma(k)}})(c)\\\\\n&=\\sum_{\\sigma\\in S_k}(-1)^\\sigma i^{2k(k-1)}\\hat{c_0}(\\alpha_{\\sigma(1)},...,\\alpha_{\\sigma(k)})\\\\\n&=\\sum_{\\sigma\\in S_k}(-1)^\\sigma \\hat{c_0}(\\alpha_{\\sigma(1)},...,\\alpha_{\\sigma(k)})\n\\end{align*}\nSo we see that $\\mathcal{V}(c)$ equals the linear multivector field that is the antisymmetrization of $\\hat{c_0}$. In particular this means that for $f_1,...,f_k\\in\\mathscr{S}_c(A^\\ast)$ we have\n\\begin{align*}\n\\mathcal{V}(c)(f_1,...,f_k)&=\\sum_{\\sigma\\in S_k}(-1)^\\sigma\\hat{c_0}(f_{\\sigma(1)},...,f_{\\sigma(k)})\\\\\n&=\\frac{1}{i^{k-1}}\\mathcal{F}_\\mu\\left(\\lim_{t\\to 0}\\left(\\sum_{\\sigma\\in S_k}(-1)^\\sigma\\frac{1}{(it)^{k-1}}\\Phi(c)(q_t(f_{\\sigma(1)}),\\ldots,q_t(f_{\\sigma(k)}))\\right)\\right)\n\\end{align*}\nThis completes the proof.\n\\end{proof}\n\n\\begin{rmk}\nThis theorem, restricted to multiplicative vector fields, can be viewed as a statement about the ``classical limit'' of certain derivations of the convolution algebra, and looks very similar to certain aspects of the proof of the Atiyah--Singer index theorem given in \\cite{enn}. Indeed, it would be interesting to investigate its use in index theory for Lie groupoids, as it \nexactly fits into the framework of relating the van Est map to the classical limit, as shown in the index theorem of \\cite{ppt} \nfor smooth groupoid cohomology $H^\\bullet_{\\rm diff}(\\mathcal{G})$.\n\\end{rmk}\n\nIn the previous proof we have only used the fact that $q_t(f)$ converges to $\\mathcal{F}_\\mu^{-1}(f)$ in $\\mathscr{S}_c(\\mathcal{G}_{\\text{ad}})$ as $t$ goes to $0$, we have not used the properties which makes the family $\\{q_t\\}_{t\\neq 0}$ a family of quantization maps, namely their compatibility with the Poisson bracket. However, we have not introduced these specific maps without reason, since we will use the fact that\n\\begin{equation*}\n\\lim_{t\\to 0}(\\frac{1}{it}[q_t(f_1),q_t(f_2)])=\\lim_{t\\to 0}q_t(\\{f_1,f_2\\}))\n\\end{equation*}\nto give an alternative proof of the fact that the Van Est map is a {\\em chain map}, i.e, compatible with the differentials:\n\\begin{cor}\nThe van Est map $\\mathcal{V}:\\tilde{C}_\\text{def}^\\bullet(\\mathcal{G})\\to C^\\bullet_{\\text{Pois,lin}}(A^\\ast)$ is a chain map.\n\\end{cor}\n\\begin{proof}\nLet $c\\in\\tilde{C}^k_\\text{def}(\\mathcal{G})$ for $k\\geq 1$ and we start by dissecting $\\mathcal{V}(\\partial c)$. Using the previous theorem we obtain\n\\small\n\\begin{align*}\n\\mathcal{V}(\\delta c)(f_1,...,f_{k+1})=&\\mathcal{F}_\\mu\\left(\\lim_{t\\to 0}\\left(\\sum_{\\sigma\\in S_{k+1}}(-1)^\\sigma\\frac{1}{(it)^{k}}\\Phi(\\delta c)(q_t(f_{\\sigma(1)}),\\ldots,q_t(f_{\\sigma(k+1)}))\\right)\\right)\\\\\n=&\\mathcal{F}_\\mu\\left(\\lim_{t\\to 0}\\left(\\sum_{\\sigma\\in S_{k+1}}(-1)^\\sigma\\frac{1}{(it)^{k}}(\\delta_\\text{Hoch}\\Phi(c))(q_t(f_{\\sigma(1)}),\\ldots,q_t(f_{\\sigma(k+1)}))\\right)\\right)\\\\\n=&\\mathcal{F}_\\mu\\left(\\lim_{t\\to 0}\\left(\\sum_{\\sigma\\in S_{k+1}}(-1)^\\sigma\\frac{1}{(it)^k}[q_t(f_{\\sigma(1)}),\\Phi(c)(q_t(f_{\\sigma(2)}),...,q_t(f_{\\sigma(k+1)}))]\\right)\\right)\\\\\n&+\\mathcal{F}_\\mu\\left(\\lim_{t\\to 0}\\left(\\sum_{j=1}^k\\sum_{\\substack{\\sigma\\in S_{k+1}\\\\\\sigma^{-1}(j)<\\sigma^{-1}(j+1)}}(-1)^\\sigma (-1)^j\\frac{1}{(it)^k}\\Phi(c)(q_t(f_{\\sigma(1)}),...,[q_t(f_{\\sigma(j)}),q_t(f_{\\sigma(j+1)})],...,q_t(f_{\\sigma(k)}))\\right)\\right)\n\\end{align*}\n\\normalsize\nNow the relation between the commutator, the Poisson bracket and the quantization maps, we can use 1 power of $\\frac{1}{it} $ to turn the commutators into Poisson brackets. Also using the fact that $q_t(f)\\to \\mathcal{F}_\\mu^{-1}(f)$ as $t\\to 0$ this results in\n\\small\n\\begin{align*}\n\\mathcal{V}(\\delta c)(f_1,...,f_{k+1})=&\\sum_{\\sigma\\in S_{k+1}}(-1)^\\sigma\\left\\{f_{\\sigma(1)},\\mathcal{F}_\\mu\\left(\\lim_{t\\to 0}\\left(\\frac{1}{(it)^{k-1}}\\Phi(c)(q_t(f_{\\sigma(2)}),...,q_t(f_{\\sigma(k+1)}))\\right)\\right)\\right\\}\\\\\n&+\\mathcal{F}_\\mu\\left(\\lim_{t\\to 0}\\left(\\sum_{j=1}^k\\sum_{\\substack{\\sigma\\in S_{k+1}\\\\\\sigma^{-1}(j)<\\sigma^{-1}(j+1)}}(-1)^\\sigma (-1)^j\\frac{1}{(it)^{k-1}}\\Phi(c)(q_t(f_{\\sigma(1)}),...,q_t(\\{f_{\\sigma(j)},f_{\\sigma(j+1)}\\}),...,q_t(f_{\\sigma(k)}))\\right)\\right)\n\\end{align*}\n\\normalsize\nThen using the previous Theorem in reverse order we see that this leads to\n\\begin{align*}\n\\mathcal{V}(\\delta c)(f_1,...,f_{k+1})&=\\sum_{j=1}^{k+1}(-1)^{j+1}\\left\\{f_j,\\mathcal{V}(c)(f_1,...,\\hat{f_j},...,f_{k+1})\\right\\}\\\\\n&+\\sum_{j_1<j_2}(-1)^{j_1+j_2}\\mathcal{V}(c)(\\{f_{j_1},f_{j_2}\\},f_1,...,\\hat{f_{j_1}},\\hat{f_{j_2}},...,f_{k+1})\n\\end{align*}\nwhich shows that the Van Est map is a chain map.\n\\end{proof}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{introduction}\nRecent advances in deep reinforcement learning\\cite{mnih2013playing, van2016deep, wang2015dueling} have triggered subgoal and option research within reinforcement learning using deep neural networks. Here we investigate Micro-Objective Learning, a new technique to discover important states for a complex task and exploit them to accelerate learning.\n\nMany techniques have been investigated with respect to subgoals. \\cite{kulkarni2016deep} uses a Successor map, which approximates the count of the successive states given (state, action) pairs. It successfully extracted subgoals in the simple grid world and 3D Doom task. However, it did not exploit the subgoals while discovering them online to accelerate learning. \\cite{kulkarni2016hierarchical} used heuristically defined subgoals and two neural networks: One to maximize the external reward, and the other to maximize the internal reward to achieve subgoals. This \\cite{kulkarni2016hierarchical} paper has had promising results in Montezuma's Revenge, an Atari game notorious for its difficult exploration. \\cite{bacon2016option} has incorporated option learning into policy gradient theorem to directly do end-to-end learning of the options for semi-MDP (Markov Decision Process) and has been tested on four Atari games\\cite{bellemare2013arcade} but left the number of options as a hyper-parameter.\n\nDespite the promising results by \\cite{kulkarni2016hierarchical, bacon2016option}, several drawbacks still exist. 1) Although subgoals and options are inseparable, discovering the subgoals online while learning options has been extremely difficult as they need to be learned in addition to original reinforcement learning algorithms. As far as we know, online learning of subgoals while exploiting them to accelerate learning in large state-space, such as the Atari domain, has not been accomplished. In \\cite{bacon2016option}, they did not use the idea of subgoals to learn options.\n\nAdditionally, 2) the definition of subgoal has been somewhat ambiguously defined. In \\cite{murata2007introduction}, a subgoal was defined as a set of states that are known to be visited when reaching the goal. However, every state can be considered a subgoal. For example, when a robot is learning to move from one room to another and a door is in between, the door can be viewed as a subgoal in a cognitive way. Also, we can consider a desk or a chair that one can pass (but does not need to pass) to get to the other room as a subgoal with lower importance than the door.\n\nIn this paper, we present a hierarchical reinforcement learning algorithm, Micro-Objective Learning(MOL). We first define micro-objectives, a set of states with a continuous measurement of importance assigned to each element, which is a micro-version of the subgoal. We induce the logic that frequently visited states can be considered as subgoals, from \\cite{mcgovern2001accelerating}, however only successful trajectories are used to estimate the importance of micro-objectives. Similar to the empirical results in \\cite{mcgovern2001accelerating}, simply counting all states results in distracted subgoals, while counting the states only when visited for the first time gives clearer results.\n\nThere are drawbacks of first-visit counting as mentioned in \\cite{mcgovern2001accelerating}: It is noisy, and impractical in large or continuous state-space domains. Though it appears noisy from the traditional subgoal point of view, it does help to approximate the importance of micro-objective states. Also, noisy subgoals are normally discouraged because learning option policy takes time. However, since we give an additional reward to the original MDP rather than explicitly creating macro-actions, estimating the importance does not need to be precise. Additionally, we use recent pixel differences to sample dissimilar states and count them from successful trajectories instead of using first-visit counts in non-tabular domains. For justification of first-visit sampling, we analyzed the difference between theoretical importance, approximated importance achieved with first-visit counting, and approximated importance achieved with every-state counting of successful trajectories. In MOL, we use pseudo-count from \\cite{bellemare2016unifying} for fast learning of importance.\n\nBy using first-visit pseudo-counts for approximating the importance of micro-objectives and sampling dissimilar states, we achieved significantly better results than the baselines in two Atari games: Montezuma's Revenge and Seaquest, where we used \\cite{bellemare2016unifying} as a baseline for Montezuma's Revenge, the state-of-the-art model as far as we know, and Deep Q-learning for Seaquest.\n\nOverall, our main contributions are:\n\\begin{itemize}\n\\item Defined micro-objectives with precise measurement of their importance using visited counts combining the notions of the subgoal and the option.\n\\item Automatic discovery and utilization of the micro-objectives at the same time resulting in accelerated learning.\n\\item Micro-Objective Learning scales up to large and continuous states, such as the Atari domain.\n\\end{itemize}\n\n\\section{Background}\n\\subsection{Double Deep Q-learning}\nIn Double Deep Q-learning (DDQN)\\cite{van2016deep}, one operator calculates the action-value and chooses the action simultaneously. This results in a positive bias. Double Deep Q-learning uses two parameters for the action-value function to solve this problem.\n\\begin{equation}\ny_i^{DDQN} = r + \\gamma \\, Q(s', \\arg\\max_{a'} Q(s', a';\\theta_i);\\theta^-)\n\\end{equation}\n\n\\subsection{Pseudo-Count Exploration}\nPseudo-Count Exploration (PSC)\\cite{bellemare2016unifying} extends traditional count-based approach for efficient exploration. PSC uses a sequential density model to approximate the count of the state in large and continuous state-space. Specifically, it uses a CTS density model for each pixel to get the probability $\\rho_{n}(x)$ of the state and define the recording probability $\\rho'_{n}(x)$ to get the pseudo-count $\\hat{N}_{n}(x)$ and pseudo-count total $\\hat{n}$.\n\\begin{equation}\n\\rho_{n}(x) = \\frac{\\hat{N}_{n}(x)}{\\hat{n}}, \\;\\;\n\\rho'_{n}(x) = \\frac{\\hat{N}_{n}(x)+1}{\\hat{n}+1}\n\\end{equation}\nPSC calculates the pseudo-count by solving the equations between $\\rho'_{n}(x)$ and $\\rho_n$.\n\\begin{equation}\n\\hat{N}_n(x) = \\frac{\\rho_n(x)(1-\\rho'_n(x))}{\\rho'_n(x)-\\rho_n(x)}\n\\end{equation}\nBy pseudo-counting the states the agent has visited, it gives an additional reward to the states the agent has not seen before. \n\\begin{equation}\nR_{new} = R + \\beta \\ (\\hat{N}_n(x)+0.01)^{-1\/2}\n\\end{equation}\nDirect usage of an exploration bonus results in a destabilized Q function. PSC mixed Double Q target with Monte Carlo return as below:\n\\begin{equation}\n\\begin{aligned}\n\\Delta Q(x_t, a_t) :=\n(1-\\eta)\\Delta Q_{DOUBLE}(x_t, a_t) + \\\\ \\eta[\\sum_{s=0}^\\infty \\gamma^s((R+R_n^+)(x_{t+s}, a_{t+s}))-Q(x_t, a_t)]\n\\end{aligned}\n\\end{equation}\n\n\\section{Micro-Objective Learning}\nIn Micro-Objective Learning (MOL), we approximate how important a state is given the current policy and give an additional reward to the original MDP that is proportional to the importance, as below.\n\\begin{equation}\n    R_{obj} = \\alpha \\cdot min(R_{max}, \\frac{(1-R_{exp})}{R_{c-max}})\n\\end{equation}\n\\begin{equation}\n    R_{new} = R + R_{obj}\n\\end{equation}\nwhere $R_{max}$ is a constant value to limit the maximum of $R_{obj}$, $\\alpha$ is a coefficient for $R_{obj}$, and $R_{c-max}$ is the current maximum value of (1-$R_{exp}$).\n\\begin{equation}\n    R_{exp}=\\frac{0.1}{\\sqrt{N(s)+0.01}}\n\\end{equation}\n\n$R_{exp}$ is an additional reward function used in  \\cite{bellemare2016unifying} for exploration.\n\nLet us define a successful trajectory as a trajectory that has its last state as the only goal state in the trajectory which is given in MDP. A goal state is dependent on the task, but typically we can define a goal state as a positive rewarding state. The visit-counts of the states in successful trajectories were used to estimate the importance of the states. However, counting all of the states in the successful trajectories results in poor estimation of the importance. We use dissimilar sampling, which is an extension of first-visit sampling, to solve this issue. First-visit sampling is a simple method that samples the states that have been visited for the first time when moving along the trajectory.\n\nGiving an additional reward to the original MDP has several benefits over the option framework. By giving an additional reward, we do not need to learn the option policies, a task which is as expensive as the original RL. Also, the options need to find an initiation set and learn the termination condition to decide where to start and terminate the options, which MOL does not require. Because micro-objectives are smaller units than the subgoal or initiation set in the option framework, MOL can be viewed as a process that learns a micro-version of the option policies, but in a flexible manner.\n\n\\subsection{Empirical explanation of Micro-Objective Learning}\nIntuitively, giving an additional reward to frequently visited states in successful trajectories accelerates the learning process. One possible approach to calculating importance is to count all of the states from the successful trajectories and use this count as an importance of the states. Giving an additional reward, proportional to the importance, to every state when the agent visits can encourage the agent to go to important states. However, this simple approach has two critical flaws: It creates overly-important states and it encourages the agent to revisit a certain state that gives a substantial additional reward.\n\\begin{figure}[t]\n\\vskip 0.2in\n\\begin{center}\n\\centerline{\\includegraphics[width=0.7\\columnwidth]{grid_with_trajectory}}\n\\caption{A simple Grid World MDP with $s_0$ as an initial state and $s_8$ as a goal, the terminal state. Figures (a) and (b) show two different successful trajectories. Figures below show the estimated importance of states using 1) every-visit counting ((c)), and 2) first-visit counting ((d)). Redder areas mean increased importance.}\n\\label{simple-mdp}\n\\end{center}\n\\vskip -0.2in\n\\end{figure}\n\\begin{itemize}\n \\item{\\bf Overly-important states}\\\\\nLook at the simple MDP in figure \\ref{simple-mdp}. Assume that the agent has succeeded in achieving the goal while exploring, as in figure \\ref{simple-mdp}(c). Because the successful trajectory has a loop, states $s_1, s_3, s_6$, the states that have no relationship with achieving the goal, receive the same importance as the states $s_0, s_2, s_7$ which actually contributed to achieving the goal. If the successful trajectory has many loops, which is the normal case in early stages of exploration, the importance becomes high in the states that are included in the loops. Since we are going to give an additional reward proportional to the importance, this will result in encouraging the agent to follow the loop many times, which is definitely not what we desire.\n \\item{\\bf Revisiting states}\\\\\nLet us say we have a good estimate of the importance. In the case of figure \\ref{simple-mdp}(c), huge importance would be given to states $s_0, s_2, s_4, s_7, s_8$ which are the states that have contributed to earning the goal. However, if we give an additional reward to every state the agent has visited, it results in the agent going back to the states that have large importance. It is because the estimated Q value of (state, action) pairs that have states with large importance as the next state, will explode as substantial reward is continuously given. This is also not what we desire.\n\\end{itemize}\n\nWe solve these two problems by sampling the first-visit states from the trajectories. By using first-visit sampling of the successful trajectories before counting, we can avoid giving too much importance to any specific state because it limits the maximum count for each state in a single successful trajectory to 1. An example is shown in figure \\ref{simple-mdp}. When using every-visit counting on two successful trajectories in figure \\ref{simple-mdp}(c), and \\ref{simple-mdp}(d), we get more importance on the states that are near the initial state because the agent has explored them frequently. However, with first-visit counting, we get a clear path to the goal state. Also, if we use first-visit sampling when we give an additional reward, the additional reward will only be given once to the state. This prevents the agent from revisiting a certain state because it will only get the reward when it first visits that state. Using first-visit sampling when giving an additional reward has one additional benefit; because we give the reward only once even if the agent visits that state several times, the expected additional reward given by visiting that state becomes $1\/n$ (n is the number of times the agent has visited that certain state) times the additional reward it gets when it visits that state for the first time. We can assume the current policy of the agent has a bias for a certain state if the agent visits that state often. However, because the additional reward decreases on subsequent visits, it naturally forces the agent to go to the next state rather than forming unnecessary loops.\n\nDefining what first-visit is in a large and continuous space is difficult because every state is different. To create the same effect as first-visit sampling, we use dissimilar sampling. Dissimilar sampling is a method that samples dissimilar states from the trajectories. This is a natural extension of first-visit sampling because the main idea of first-visit sampling is to avoid reusing the same states when calculating the importance and giving an additional rewards, and, the 'same states' can be extended to 'similar states' in large and continuous domains. To define similar states, we use the notion of pixel difference between sequential states, which has been successfully used in a 3D domain \\cite{jaderberg2016reinforcement} as a criteria to find the occurrence of an event. In MOL, we use this notion as a similarity between states. While using the mean of the pixel difference in the whole trajectory seems natural, it gets awkward when the environment changes dramatically. This pixel difference $\\delta$ can be viewed as\n\\begin{equation}\n\\delta = \\delta_{agent} + \\delta_{env}\n\\end{equation}\n\nThough we need to take the environment into account, the environment alone may have too great an effect to be a good criterion for sampling. This can result in sampling only the states that have huge environmental changes, even if they are not related to the task. To address this issue, we use the mean of recent pixel differences as a criteria because recent pixel difference will exclude meaningless changes in the current environment.\n\nWhile algorithm \\ref{alg:dissimilar sampling} takes the trajectory as an input which assumes using an off-policy algorithm, we can still do dissimilar sampling when using an on-policy algorithm because we sample the states in sequential order. At each step, we can update the recent history of the states and decide whether the current state should be chosen or not. If the state is chosen, we add the state into the sampled trajectory and give an additional reward to it. When the reward is given, we update the pseudo-count density model with the sampled trajectory and re-initialize the sampled trajectory. Remember, a successful trajectory is defined as a trajectory that ends with the only goal state in the trajectory. This means that there can be several successful trajectories in a single episode. Splitting an episode trajectory into several successful trajectories is reasonable because for each reward, micro-objectives should be counted separately. In other words, if a single state is important in acquiring several goals, we need to count it several times to represent its importance.\n\n\\begin{algorithm}[tb]\n   \\caption{Dissimilar Sampling}\n   \\label{alg:dissimilar sampling}\n\\begin{algorithmic}\n   \\STATE {\\bfseries Input:} A trajectory $L =$ ($s_0, s_1, \u2026 , s_n$), RecentHistorySize = $h$\n   \\STATE {\\bfseries Output:} sampled trajectory $L^*$\n   \\STATE $L^*$ = [$s_0$]\n   \\FOR{$i=1$ {\\bfseries to} $n$}\n   \\STATE $\\delta$ = $||((s_{i-h}:s_i)-(s_{i-h+1}:s_{i+1}))||$\n   \\IF{all($||(L^*[j]-s_i)||$ $\\geq$ $\\delta$) (for $j = 1$ to the length\\\\ of ($L^*$)))}\n   \\STATE Append $s_i$ to $L^*$\n   \\ENDIF\n   \\ENDFOR\n\\end{algorithmic}\n\\end{algorithm}\n\nWhen designing an additional reward function, we have considered several requirements.\n\\begin{itemize}\n  \\item {\\bf Convergence}\nIt must have limits as the pseudo-count grows larger. Though we are going to clip the reward, an expanding reward is undesirable because it makes the Q function unstable.\n \\item {\\bf Early stage exploitation}\n It must be significant, even in the early stages of learning, because exploiting micro-objectives is most effective when the learning is more imperfect.\n \\item {\\bf Distinguish micro-objectives and non micro-objectives}\n It must be able to distinguish between micro-objectives and non micro-objectives, which have an importance of nearly 0.\n\\end{itemize}\n\nScaling the reward with a maximum value of (1-$R_{exp}$) for steps leading to the current learning step, will result in giving substantial reward values in the early stages of learning. Also, we clip the reward with $R_{max}$ to limit the maximum additional reward.\n\n\\begin{algorithm}[tb]\n   \\caption{Micro-Objective Learning in Deep Q-Learning}\n   \\label{alg:micro-objective learning}\n\\begin{algorithmic}\n   \\STATE Initialize replay memory $D$ and action-value function $Q$\n   \\STATE Initialize trajectory $L$ and pseudo-count model $M$ for $R_{obj}$\n   \\FOR{episode=$1$ {\\bfseries to} $n$}\n   \\REPEAT\n   \\STATE Select an action $a$ with epsilon-greedy policy\n   \\STATE Do action $a$ in current state $s$\n   \\STATE Update parameters in function $Q$ using replay memory $D$\n   \\STATE Use dissimilar sampling($L+s', h$) to get sampled trajectory $L^*$ where $s'$ is the next state\n   \\IF{$s'$ $\\in$ $L^*$}\n   \\STATE $R$ = $R$ + $R_{obj}(s')$\n   \\STATE Append $s'$ to $L$\n   \\ENDIF\n   \\STATE Insert $(s, a, s', R, terminal)$ in replay memory $D$\n   \\IF{next state is a goal state(external reward $> 0$)}\n   \\STATE Update $M$ with $L$\n   \\STATE Re-initialize trajectory $L$\n   \\ENDIF\n   \\UNTIL{Terminal is True}\n   \\ENDFOR\n\\end{algorithmic}\n\\end{algorithm}\n\n\\subsection{Analysis on Micro-Objective Learning}\nConsider a Markov Decision Process (MDP) which is defined with $(S, A, R, \\rho_0, \\gamma, P)$. $S$ is a set of states, $A$ is a set of possible actions, $R$ is an external reward from the environment, $\\rho_0$ is an initial state distribution, gamma is a discount factor, and $P : S \\times A \\times S \\rightarrow \\mathbb{R}$, is the transition probability distribution.\n\nWe define the importance of a state in a given trajectory.\n\n\\theoremstyle{definition}\n\\begin{definition}{Importance Count}\\\\\nLet there be a successful trajectory $L$ with visited states $s_0$ to $s_n$ in sequential order. We define a {\\bf importance count} $I^L(s_i)$ for every state $s_i$ in a MDP as follows, where $L^*$ is the optimal path from $s_0$ to $s_n$:\n\\end{definition}\n\n\\begin{equation*}\nI^L (s_i) = \n\\begin{cases}\n & 1 \\text{ if } s_i \\in L^*  \\\\ \n & 0 \\text{ otherwise}\n\\end{cases}\n\\end{equation*}\n\nFor example, in the MDP shown in figure \\ref{simple-mdp2}, assume the successful trajectory $L$ has all (state, next state) pairs except for $(s_6, s_7)$. By definition, we get importance of 1 except for the states $s_4$, and $s_6$. These two states did not contribute to reaching the goal state. This definition naturally arises from what humans do when searching for valuable steps in a complex task. Humans try to figure out why a trial was successful by tracing back the cause of the success, which is a similar process to finding the optimal path given a successful trajectory. We use the importance count to define the importance of a state given the policy.\n\n\\begin{definition}{Micro-objective} \\\\\nA micro-objective is a state $s_i$ that has importance $M_{\\pi}(s_i)$ $> 0$ given the current policy $\\pi$. Let $H$ be the set of possible successful trajectories in the given MDP.\n\\begin{equation}\nM_{\\pi}(s_i) = \\sum_{L \\in H} I^L(s_i) \\cdot p_{\\pi}(L)\n\\end{equation}\n, where $p_{\\pi}(L)$ is the probability of following the trajectory $L\n$ when using the policy $\\pi$.\n\\end{definition}\n\\begin{figure}[t]\n\\vskip 0.2in\n\\begin{center}\n\\centerline{\\includegraphics[width=\\columnwidth]{simple_mdp}}\n\\caption{A simple MDP with $s_0$ as an initial state and $s_8$ as a goal state.}\n\\label{simple-mdp2}\n\\end{center}\n\\vskip -0.2in\n\\end{figure}\nThe state importance should be dependent on the current policy as the value function $V(s)$. For example, in the MDP in figure \\ref{simple-mdp}, if we have a policy of going up, then the important states would be the states from below because in the current policy, getting to the states in the bottom row makes the probability of getting to the goal state higher. However, if we have a policy of going down, the states that are above are more important for reaching the goal state.\n\nConsidering that the definition of importance count is whether or not the state has contributed to achieving the goal, the importance of the micro-objective defines how likely we succeed if we are in that state using the current policy. Because we update the policy at every step, using recent successful trajectories is a reasonable approximation. For convenience, we used all successful trajectories to estimate the importance of micro-objectives.\n\nWe argue that giving an additional reward proportional to the estimated importance accelerates learning to reach the goal. Because of the discount factor $\\gamma$, states near the initial states have small $V(s)$. Also, to update the $V(s)$ of those states, updating $V(s)$ of the states between those states and the goal state is required. However, if we give an additional reward to each state, $V(s)$ will be updated quickly and the requirements mentioned above are eliminated. For example, in the figure \\ref{simple-mdp2} MDP, to update the value of state $s_1$, the values of state $s_k$ $(k \\geq 2)$ need to be updated. However, when giving an additional reward, a direct update is possible with $(s_1, a, s_j)$ $(j = 2, 3, 4)$ in the replay pool $D$.\n\\begin{figure}[t]\n\\vskip 0.2in\n\\begin{center}\n\\centerline{\\includegraphics[width=\\columnwidth]{montezuma-frames}}\n\\caption{The states that have the largest, medium, and smallest $R_{obj}$ in Montezuma's Revenge: First, second, and third row each. For Montezuma's Revenge, we used 0.2 and 0.4 as the criteria. The states with the largest $R_{obj}$ are traditional subgoal states or actual goal states (key, rope, and the door). The states with the medium $R_{obj}$ are not necessary to reach the goal states, but are helpful. The states with the smallest $R_{obj}$ do not have much relationship with the goal states.}\n\\label{montezuma-frames}\n\\end{center}\n\\vskip -0.2in\n\\end{figure}\nWhile the benefits of giving an additional reward to the appropriate states are obvious, estimating the appropriate states is not. When using first-visit sampling, we can effectively approximate the state importance. Assuming that we are using a substantial number of successful trajectories obtained from the current policy, with an estimated importance count $I^L_{est}$, actual importance count $I^L$, the set of states $S$, and the difference between the estimated importance $M_{est}$ and the actual importance $M$, $L_{M}$,\n\\begin{equation}\n\\label{equation_loss}\nL_{M_{est}} = \\sum_{s_i \\in S}\\sum_{L \\in H} (I^L(s_i)-I^L_{est}(s_i)) \\cdot p_{\\pi}(L)\n\\end{equation}\nAs in equation \\ref{equation_loss}, the loss comes from the difference of importance count, which is caused by states that are not included in the optimal path of a successful trajectory.\n\nTo approximate the loss of importance count $(I^L(s_i)-I^L_{est}(s_i)$, we analyze how the agent is trained. When giving rewards to every state that is visited in a successful trajectory, there can be states that distract the agent from reaching the goal.  With appropriate exploration methods, the count of these states will be low. Also, though the agent may not follow the optimal trajectory, since the estimated importance is as follows,\n\n\\begin{equation}\nM_{est}(s_i) = \\sum_{L \\in H} I^L_{est}(s_i) \\cdot p_{\\pi}(L)\n\\end{equation}\nthe agent is encouraged to follow the successful trajectory that is the most likely in the current policy, resulting in fast convergence of the policy for getting to the goal state. \n\\begin{figure}[t]\n\\vskip 0.2in\n\\begin{center}\n\\centerline{\\includegraphics[width=\\columnwidth]{seaquest-frames}}\n\\caption{The states that have the largest, medium, and smallest $R_{obj}$ in Seaquest: First, second, and third row each. For Seaquest, we used $0.01, 0.1$, and $0.5$ as the criteria. The states with the largest $R_{obj}$ are states that give an external reward. The states with the medium $R_{obj}$ are the states where the submarine is firing its weapon, which is a way to get an external reward. The states with the smallest $R_{obj}$ do not have much relationship with the goal states.}\n\\label{seaquest-frames}\n\\end{center}\n\\vskip -0.2in\n\\end{figure}\nTherefore, though estimated convergence by first-visit sampling does not converge to the actual importance, $L_{M_{est}}$ converges. Though MOL does not guarantee an optimal policy, it helps for an agent to learn the policy that succeeds in reaching the goal state and this makes it possible to get to the next reward, resulting in a higher average score.\n\n\n\n\\section{Experiments}\nIn the experiments, we focused on 1) analyzing which states are chosen as micro-objectives and which states have large or small importance, and 2) evaluating how much MOL accelerates learning using the discovered micro-objectives. We compared our agent to existing methods in two Atari games: Montezuma's Revenge and Seaquest. Montezuma's Revenge is notorious for its difficult exploration while Seaquest is one of the most dense reward games in Atari.\n\nIn Montezuma's Revenge, we compared a pseudo-count exploration model from \\cite{bellemare2016unifying}, which is the state-of-the-art model in this domain, with and without MOL. This was because Deep Q-learning has a difficult time obtaining successful trajectories which are needed for MOL. We tested our agent in a stochastic ALE setting with a probability to repeat previous actions of 0.25, the same setting as in \\cite{bellemare2016unifying}. \n\\begin{figure}[ht]\n\\vskip 0.2in\n\\begin{center}\n\\centerline{\\includegraphics[width=\\columnwidth]{montezuma_result}}\n\\caption{Average Training Episode Score of Montezuma's Revenge for 3 million training frames with three random seeds - Pseudo count exploration model from \\cite{bellemare2016unifying} versus MOL with Pseudo-count exploration model.}\n\\label{montezuma-result}\n\\end{center}\n\\vskip -0.2in\n\\end{figure} \nIn Seaquest, we compared the Double Q-learning with Monte-Carlo return, with and without MOL. Gym environment was used as an emulator\\cite{1606.01540}. Parameters used are set the same as \\cite{van2016deep}.\n\nFor dissimilar sampling, we used recent history size h = 5, and 2,500 as a minimum pixel difference for clear sampling. We reset the emulator every 250,000 frames to lower the memory usage as extremely long episodes take too much memory. Also, we used $\\alpha = 1$ and $R_{max} = 0.9$ for $R_{obj}$.\n\nWe trained the agents for 3 million frames in both games with three different random seeds. In Montezuma's Revenge, after getting the reward of 300 by reaching the door, the rewards following that come from exploring additional rooms. 3 million frames were chosen to see how the agent is trained in diverse rooms. Also, in Seaquest, it is sufficient to observe the effect of MOL.\n\n\\subsection{Analysis on Micro-Objectives}\n\nTo analyze the pseudo-count used for $R_{obj}$, we took one successful trajectory of the agent trained with 3 million frames. We sampled with dissimilar sampling and compared $R_{obj}$ of the sampled frames. Figures \\ref{montezuma-frames} and \\ref{seaquest-frames} show three sample states with the largest, medium, and the smallest $R_{obj}$ for each game when training on 3 million frames.\n\nAs we can see in the figure \\ref{montezuma-frames}, in Montezuma's Revenge, the states where the character is reaching the key, rope, and the door have the largest counts, which are traditional subgoal states. \n\\begin{figure}[ht]\n\\vskip 0.2in\n\\begin{center}\n\\centerline{\\includegraphics[width=\\columnwidth]{seaquest_result}}\n\\caption{Average Training Episode Score of Seaquest for 3 million training frames with three random seeds - Double Q-learning with Monte Carlo return(a model from \\cite{bellemare2016unifying} without exploration bonus) versus MOL with Double Q-learning with Monte Carlo return model.}\n\\label{seaquest-result}\n\\end{center}\n\\vskip -0.2in\n\\end{figure}\nHowever, the states which have medium counts are not traditional subgoal states, but are important in reaching the key or the door. In the first sample, the character is reaching the rope while he does not need to use that path to reach it. Also in the third sample, the character does not need to go left since he can also jump to the left. The states which have the smallest counts appear to have weak relationships with the goal states.\n\nIn Seaquest, the initial game states and goal states have the largest counts as in figure \\ref{seaquest-frames}. In the states with medium counts, the submarine appears to do the \"fire\" action which is needed to get additional points. As in Montezuma's Revenge, the states which have the smallest counts seem to have weak relationships with the goal states.\n\n\n\n\n\\subsection{Accelerating Learning}\nThe learning curve of both games are shown in figure \\ref{montezuma-result} and \\ref{seaquest-result}. We averaged the training episode scores of three experiments. In Montezuma's Revenge, because the reward is sparse and the subgoal state is clear, the gap dramatically increases as expected. After three million frames of training, the average training episode score of the agent with MOL exceeded 100, which means the agents are constantly getting to the door after obtaining the key. Meanwhile, Seaquest is a dense reward game which is rather hard to interpret the subgoal states. However, even in Seaquest, the gap between the baseline and MOL increased as training proceeds. This suggests MOL can be applied to games with unclear subgoal states. After training 3 million frames, using MOL resulted in 120.34\\%, and 18.25\\% increase in Montezuma's Revenge and Seaquest scores, respectively.\n\\begin{table}[t]\n\\label{ratio-table}\n\\vskip 0.15in\n\\begin{center}\n\\begin{small}\n\\begin{sc}\n\\begin{tabular}{lcc}\n\\hline\n\\abovespace\\belowspace\nAlgorithm & Seaquest & Montezuma's\\ Revenge \\\\\n\\hline\n\\abovespace\nBaseline       & 267.10 $\\pm$\\ 11.88 & 51.40 $\\pm$\\ 10.73 \\\\\nWith MOL       & 315.84 $\\pm$\\ 15.01 & 113.26 $\\pm$\\ 40.62\\\\\n\\hline\n\\abovespace\nRatio(\\%) & 18.25 & 120.34 \\\\\n\\hline\n\\belowspace\n\\end{tabular}\n\\end{sc}\n\\end{small}\n\\end{center}\n\\vskip -0.1in\n\\caption{Comparison between the baseline and with MOL after training on 3 million frames.}\n\\end{table}\n\n\\begin{figure}[ht]\n\\vskip 0.2in\n\\begin{center}\n\\centerline{\\includegraphics[width=\\columnwidth]{montezumaroom}}\n\\caption{Explored rooms in Montezuma's Revenge after 1.5 million training frames - PSC only explored room 1 and 2 in all 3 experiments with room 0 explored in 2 experiments and room 6 and 7 explored in only 1 experiment. PSC+MOL explored room 0, 1, 2, 6, and 7 in all 3 experiments with room 5 explored in 2 experiments.}\n\\label{montezuma-room}\n\\end{center}\n\\vskip -0.2in\n\\end{figure}\n\n\nUsing MOL can be viewed as an exploitation of the successful trajectories which might distract exploration. However, the agent can explore better with improved exploitation methods. As in figure \\ref{montezuma-room}, with MOL, the agent normally explores 6 rooms (one experiment fails to explore room 5), before 1.5 million training frames. The fastest room search was exploring 6 rooms in 0.4 million training frames while the baseline searches 6 rooms after 5 million training frames according to \\cite{bellemare2016unifying}. Also, in one experiment, the agent with MOL started to get the external reward of 2,500 before training 1.5 million frames, while the agent needs to collect three items (a key, a door, and a knife) to reach a score of 2,500.\n\n\\section{Conclusion}\nIn this paper, we proposed an autonomous and effective hierarchical reinforcement learning method, Micro-Objective Learning, which accelerates learning by setting micro-objectives with pseudo-counts. Using dissimilar sampling, we avoided counting and giving rewards to similar states, which was critical to discovering precise micro-objectives and learning. Experimental results in Montezuma's Revenge and Seaquest show that micro-objectives embrace the subgoals which were heuristically designed previously and are effective in both sparse and dense reward settings.\n\nIn this work, we have successfully applied the notion of pixel difference for dissimilar sampling. However, for generalization, we have to use higher level features instead of pixel difference. Currently, we need to pre-train the networks to get higher level features, but it does not give sufficiently good features. With additional exploration in unsupervised learning, we could generalize further. In addition, although giving an additional reward directly to an original MDP is effective in accelerating the learning process, it does not guarantee convergence to the optimal policy. Therefore, finding a way to guarantee convergence while still getting the advantages of directly giving an additional reward will be our next step.\n\n\\section*{Acknowledgements} \nThe authors would like to thank Heidi Lynn Tessmer for discussion and helpful comments.\n\\nocite{silver2016mastering}\n\\nocite{sutton1999between}\n\\nocite{konidaris2009skill}\n\\nocite{bellemare2012investigating}\n\\nocite{stolle2002learning}\n\\nocite{stadie2015incentivizing}\n\\nocite{hasselt2010double}\n\\nocite{jaderberg2016reinforcement}\n\\nocite{chiu2011subgoal}\n\\nocite{csimcsek2005identifying}\n\\nocite{goel2003subgoal}\n\\nocite{bakker2004hierarchical}\n\\nocite{oh2015action}\n\\nocite{houthooft2016vime}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nOnline learning has received little attention from physics education researchers relative to topics such as conceptual understanding or student discussions in the classroom \\citep{docktor_synthesis_2014}. Physics courses are comparatively rare in online offerings, in part because of the hands-on laboratory courses required by the introductory sequence. However, many instructors are interested in promoting more student discussion in their classes, and web-based forums are a readily available tool for this purpose \\citep{howard_discussion_2015}. Some work in physics has analyzed student discussion posts about homework problems \\citep{kortemeyer_analysis_2006} or in textbook annotation \\citep{miller_analysis_2016}, but more general-purpose forums of the type commonly discussed in distance learning literature are only beginning to be studied \\citep{traxler_coursenetworking_2016,gavrin_connecting_2017}. \n\nForums are included with learning management systems at universities, and are freely available on various stand-alone platforms. Thus, they represent a tool that is available to instructors regardless of their choice of homework system or textbook. \nTo better understand these tools, this paper turns to network methods, which are a natural framework for analyzing the intricate record of transactions produced by discussion forums \\citep{garton_studying_1997}.\nWe consider data from three semesters of an introductory physics course, taking the entire forum transcript (on- or off-topic) as our data. We use network analysis to explore and compare the structure of the discussions between semesters, drawing on the ``map'' of student connectivity that electronic records preserve. Since online environments have not been extensively studied in physics, we will summarize key results and questions of interest from \neducational technology and distance learning. \nThis study applies network analysis in a new context for physics education research and aims to begin building a physics-specific understanding of how students form asynchronous discussion communities that help their learning.\n\n\n\\subsection{Computer-mediated communication}\n\nResearch about how students talk online is typically published \nunder keywords such as computer-mediated communication (CMC) and computer-supported collaborative learning (CSCL). \nNumerous CSCL studies compare online to offline classes in terms of student achievement or satisfaction, and in many cases find that the online environment does at least as well as face-to-face classes \\citep{johnson_synchronous_2006}. \nPotential strengths of asynchronous forums include longer ``think time'' and the ability to easily reference comments from previous weeks, while drawbacks include reluctance to participate and high variability in comment quality \\citep{guzdial_effective_2000,howard_discussion_2015}. The reduced-social-cues nature of text communication can lead to an unpredictable social gestalt in CMC. Researchers have observed both impersonal, highly task-focused environments, and equally strong interpersonal groups where a sense of community can even interfere with ``on-task'' discussions\nif members hesitate to disagree with each other \\citep{walther_computer-mediated_1996}. A review by \\citet{walther_computer-mediated_1996} synthesizes early results to suggest that the speed and quality of community development are shaped by a sense of shared purpose among users, longevity of\nthe group, and outside cues or facilitation.\n\nEducational settings vary in formality from technical, highly-focused project work to free-for-all socializing, producing a range of conversation styles from \nexpository to epistolary \\citep{fahy_epistolary_2002}. \nShared purpose might be expected as a given in course forums, but in practice is often missing, and this is one area where instructor guidance can be very influential \\citep{guzdial_effective_2000,howard_discussion_2015}. \nTo analyze the cognitive level of discussions, many researchers have turned to content analysis. Key results from this area are summarized by \\citet{de_wever_content_2006} in their review of 15 \ncontent analysis schemes for asynchronous discussion groups. They find that content analysis tools vary widely in how clearly they connect learning theory to content codes and how (or if) they report inter-rater reliability measures. Few schemes were used in more than one study, and there is no wide consensus about how to break online conversations into an appropriate ``length scale'' (post, sentence, etc.) for analysis \\citep{rourke_methodological_2001}.\n\nMany researchers instead seek purely quantitative ways to study online talk, including social network analysis. \\citet{garton_studying_1997} argue that social network analysis can effectively describe online interactions with concepts like tie strength, multiplexity (different channels or purposes of communication), or structural roles of nodes in the network. \\citet{wortham_nodal_1999} notes that different network topologies could be productively linked to claims about communities of practice or cognitive apprenticeship. Though network analysis does not speak to the details of messages between students, it can show who talks to whom, the density and frequency of those ties, and how they evolve over time. For instructors trying to build a useful community for an online or online-supplemented course, there are many open questions, some of them first posed decades ago \\citep{rice_electronic_1987,guzdial_effective_2000}: What time scales are appropriate to characterize discussions? What does reciprocity in relationships mean online, where many students might read a post but few respond? How much instructor involvement is needed to promote useful conversation?\n\nIn this study, we include data from the entire semester, to eliminate possible selection effects from only sampling a slice of weeks. The question of reciprocity is taken up again in Section \\ref{sec:meth-nw} where our network model is discussed. We found no obvious link between the instructor's posting frequency and the discussion network that develops, but a future content analysis of the data may better address this question. \nA final caution in generalizing from the CSCL literature is that most results come from fully online courses, and graduate-level courses are overrepresented. \nIt may be possible to draw on the discussion strengths of forums without the isolating effects of a distance course by using a web-based forum to supplement a traditional live class. Studies of this type of forum use are still relatively rare \\citep{guzdial_effective_2000,yang_effects_2003}, especially at the introductory undergraduate level. This adjunct or ``anchored'' mode may be of the most interest to physics educators, whose courses are typically offered face-to-face \nand who increasingly want to build community as part of active learning.\n\n\\subsection{Network analyses of online learning}\n\nIn a recent review of social network analyses in educational technology, \\citet{sie_social_2012} classify study goals as visualization, analysis, simulation, or intervention. Work reviewed here fits in the first two types, and \ncan be grouped into two broad categories: descriptive studies of the structure of student networks in online education, and research connecting students' network positions with performance measures. \nIn the first category, work appearing in distance learning or online education literature has used network methods to understand online community structures (or lack thereof). \nResearchers use network analysis to show power relations in the group or the engagement level of learners \\citep{wortham_nodal_1999,de_laat_investigating_2007}. Other work contrasts between semesters or between student groups within a semester \\citep{reffay_social_2002,aviv_network_2003}, and uses visual displays or clustering analysis to show differences in the community structure. \nThese studies all function as proofs of concept for analyzing online talk via networks, and some suggest best practices for constructing learning environments, but they are primarily exploratory.\nThey also span a range of communication channels, from synchronous text chat to asynchronous forums or email lists. \nOne larger pattern that emerges from this literature review is that \nthe communication medium affects network models---for example, using emails to link the network may produce many one-way but few reciprocal connections.\nWe will return to this issue in Section \\ref{sec:methods}.\n\nA second category of studies chooses one or more markers of course success and tries to link students' network centrality with those outcomes. \\citet{yang_effects_2003} correlated centrality in friendship, advice, and adversarial networks with several components of course grade in an undergraduate business course that used a forum to supplement the face-to-face class. They found that centrality in the advice network was positively correlated with performance in both online and offline class activities. Centrality in the adversarial network (\\textit{e.g.}, ``Which of the following individuals are difficult to keep a good relationship with?'') was negatively correlated with final exam and overall course grade. \\citet{cho_social_2007} collected survey-based networks at the beginning and end of a two-semester online course sequence on aerospace system design. They looked for links between centrality and final grade and between a Willingness-to-Communicate (WTC) construct and network growth. They found that post-course (but not pre-course) degree and closeness centrality were positively correlated with final grade, and that students with higher WTC were more likely to form new ties during the two semesters. \n\nOther approaches use different positive outcomes or look for network characteristics of successful students rather than course-wide correlations. \n\\citet{dawson_study_2008} correlated students' centrality in course forum networks and their sense of course community as measured by Rovai's Classroom Community Scale \\citep{rovai_development_2002}. He found that degree and closeness centrality were positively correlated and betweenness centrality was negatively correlated with greater feelings of classroom community. \nHowever, the data pools 25 courses at undergraduate and graduate levels, different amounts of online integration, and different communication channels, so direct comparisons with these results are difficult. \nIn a second study \\citep{dawson_seeing_2010}, the same author examined student participation in an optional (but encouraged) discussion forum used as a supplement to a large introductory chemistry course. Focusing on the ``ego networks'' (immediate connections, see \\citep{hanneman_introduction_2005}) of individual students in the top and bottom 10\\% of the grade distribution, he found that students in the high-performing group had larger ego networks, and the members of those networks had higher average grades. Additionally, there was a higher percentage of instructor presence in the networks of high-scoring students, who tended to ask a larger number of conceptual questions. Students in the lower-performing group often asked more fact-based questions, which were typically answered by other students, leading to an unintended ``rich get richer'' effect of the higher-performing students receiving a larger share of instructor attention. \n\nThere is thus evidence to support networks' ability to distinguish between at least some types of online dialog structure, and to support links between network centrality and final grade. The latter point has been observed in some physics classrooms \\citep{bruun_talking_2013}, but not previously sought in electronic forums. \nWith some exceptions \\citep{aviv_network_2003}, most of the online network studies either give results for a single course offering or pool multiple courses together. They thus provide interesting cases, but it is unclear how stable their results may be from one semester to the next. Since network analysis requires start-up time for data cleaning and analysis, it is also reasonable to ask if it shows anything not already evident from the participation statistics reported by most forum software. \nBuilding on the literature above, we consider three research questions:\n\\begin{enumerate}\n\\item How do discussion forum networks differ among multiple semesters of an introductory physics course, and can this information be extracted more easily from participation statistics?\n\\item How much are student final grades correlated with their centrality in the discussion forum network?\n\\item Do centrality\/grade correlations, if present, strengthen when reducing the network to a more simplified ``backbone?''\n\\end{enumerate}\nThe third question has not been considered in any prior educational network studies we could find, but emerged from the high density of our discussion networks (Sec.\\ \\ref{sec:results}) and recent work piloting network sparsification in physics education research \\citep{brewe_using_2016}. \n\n\\section{Methods\\label{sec:methods}}\n\nBelow, we describe data collection, the process of building forum networks, comparison of network measures with final course grades, and how we simplified the network using backbone extraction. Further details on the backbone process, including source code, are in the Supplemental Material. \n\n\\subsection{Data collection}\n\nWe adopted the CourseNetworking (CN) platform \\citep{theCN}, which combines a robust forum tool with features typical of learning management systems. CN is a cloud-based platform, accessible either through a web browser or through apps on IOS and Android mobile devices. We selected CN primarily because the interface is ``student-centric,'' that is, student work occupies the majority of the view, and faculty focused tools are secondary. Although it is possible to use CN as a standalone LMS, the instructor coupled it with another system (Canvas) and used CN exclusively as a forum. The CN forum has a look and feel similar to other popular social media, so students pick it up with minimal introduction. The forum supports starting threads as either posts or polls and allows hyperlinks, embedded images and videos, and downloadable files.\nPolls may be structured as multiple choice, ranking, free response, and other formats, allowing students to create and post ``sample questions'' for one another. Students may also post Reflections (comments) beneath Posts and Polls, and rate Posts and Polls using a 1--3 star system. Our network analysis is based on which students post comments on one another's work, as detailed in the next subsection.\n\nOne of us (AG) used the CN forum in three full semesters of a calculus-based introductory mechanics class. The initial enrollment was over 160 students each semester, with the majority of the students being engineering and computer science majors. The institutional context is an urban, public university enrolling approximately 30,000 students. \nIn all three semesters, the university had undergraduate racial\/ethnic demographics of 71--72\\% white, 10\\% African American, 6--7\\% Hispanic\/Latino, other groups (including international students) 4\\% or less. \nThe majority of students commute, and most work part- or full-time in off campus jobs. \n\nThe course is heavily interactive, using Peer Instruction \\citep{mazur_peer_1997} and Just-in-Time Teaching (JiTT) \\citep{novak_just--time_1999} in the lectures, and group problem solving in the recitations. Students received extra credit (maximum 5\\% of the course grade) for use of the forum. (All calculations below involving student grades exclude forum bonus points.)\nFurther course details are described by \\citet{gavrin_students_2016}. In all semesters, CN was introduced on the first day of class with a brief demonstration. In Semesters 1 and 3, the instructor used the CN ``Tasks'' feature to provide an optional weekly discussion topic, which took place in the forum and did not involve extra class time. Finally, in Semester 3, the first-day introduction included mention of a new ability in the software to tag posts with instructor- or user-created ``hashtags.'' In all other respects, the CN implementation was identical across terms.\n \n\n\\subsection{Casting forum data as networks\\label{sec:meth-nw}}\n\nThe forum transcript contains the following data: Content ID (Post, Poll, or Reflection); a unique student identifier code; the date, time, and text of the post; the number of attachments (pictures or ``other''); and the star rating (pre-2016, number of ``likes'') accumulated by the post or comment. \nIn this analysis, the fields for text, number of attachments, and stars or Likes are not used; content ID, student code, date, and time are retained. The transcript also groups all reflections below their parent post or poll, showing a threaded view that corresponds to the student view of the forum. The CN software does track the ``nesting'' level of a reply (whether a student hit the reply button for the original post, or for another reply to that post). In practice, most students did not organize their replies in a multi-layer fashion, using a single reply layer even when the content was clearly a response to another comment. For this analysis, we treat each thread as consisting of a root plus single reply level (Fig.\\ \\ref{fig:nesting}, left). This has consequences for constructing a network---in \nsome other studies using transcript data \\citep{aviv_network_2003,de_laat_investigating_2007}, clear\nnested structure in the electronic logs \nled the authors to draw links only between a poster and the person to whom they were immediately replying. In our data, accurate nesting information is largely unavailable, requiring a different model for drawing connections between participants in a thread. \n\n\\begin{figure}[bthp]\n\\begin{center}\n\\includegraphics[width=0.9\\linewidth]{fig1}\n\\caption{Structure of forum transcript data. The CN data largely shows a post with a single reply layer (left), in contrast to studies where more nested structure is retained and informs the network construction (right). \\label{fig:nesting}}\n\\end{center}\n\\end{figure}\n\nThough it is intuitive that students talking in a forum are interacting with each other somehow, some set of assumptions must be chosen to map the logs into a network object. Prior studies of forum networks have used several different approaches: adding a link between a student commenter and the poster they were directly replying to~\\citep{aviv_network_2003,de_laat_investigating_2007}, surveying students at the beginning and end of the semester \\citep{yang_effects_2003,cho_social_2007}, or unspecified methods \\citep{dawson_study_2008,fahy_patterns_2001}. \nWe used a bipartite network model, often used to model situations where both people (``actors'') and some set of shared activities (\"events\") are of interest \\citep{borgatti_network_1997}. This approach has been used to model scientific collaboration networks and is starting to see use in online education studies \\citep{ouyang_influences_2017,rodriguez_exploring_2011}.\nAfter constructing a bipartite network, the analysis presented here focuses on the actor projection (see Fig.\\ \\ref{fig:bipartite}), which links together all students who posted together in the same discussion thread \\citep{borgatti_analyzing_2011,traxler_coursenetworking_2016}. \nFor the full-semester forum network, this process creates a dense, heavily-interlinked representation of student nodes, including the instructor's place in this web. People who post in many threads in common with each other will be connected by high-weight links (``edges''), while those who post in only one or two threads can only have low-weight links to others. \n\n\\begin{figure}[tbph]\n\\begin{center}\n\\includegraphics[width=0.9\\linewidth]{fig2}\n\\caption{Bipartite network model for transforming forum transcript into a network object. Students are ``actor'' nodes, who post to thread or ``event'' nodes. The actor network projection links together student nodes who posted to the same thread.\\label{fig:bipartite}}\n\\end{center}\n\\end{figure}\n\n\n\\subsection{Network measures}\n\nEven a small network quickly becomes unwieldy to describe by naming all actors and listing their connections. (But for very small classes this kind of description can be very illuminating, see \\citep{de_laat_investigating_2007}.)\nStructural measures condense broad features of network objects, and centrality measures quantify the position and importance of a particular node. Basic structure descriptors include the number of nodes ($N$) and edges ($N_E$) and the network density, defined as the ratio of total to possible edges: \n\\begin{equation}\n\\rho = \\frac{N_E}{N(N-1)\/2} \\label{eq:dens}\n\\end{equation}\nfor an undirected network. \nLarger social networks tend to be less dense---mathematically, because the denominator of (\\ref{eq:dens}) scales as $N^2$, and practically because any individual can only sustain relationships with so many other people \\citep{dunbar_neocortex_1992}. In a forum environment, where both rare and frequent interactions are recorded, higher density values may be expected unless some thresholding process is used (see Sec. \\ref{sec:meth-bb}). \n\nUncertainty in the network density can be estimated using boostrap methods~\\citep{snijders_non-parametric_1999}. Using this technique, a new sample of $N$ nodes is drawn from the network and an artificial network is constructed using the connections belonging to those nodes. The density of this artificial network represents a new possible value, and the process is repeated many times, generating a distribution of artificial density values. This distribution can be used to calculate a standard error for the observed statistic. We use the bootstrap method of~\\citet{snijders_non-parametric_1999} with 5000 samples.\n\nCentrality measures describe the position or importance of a node in a network. ``Position'' does not refer to physical location on a network diagram, as plotting algorithms use randomized processes to find reasonable display configurations (for example, minimizing overlap of edges). The number and strength of a node's connections to others, and the extent to which that person is at the core or periphery of the whole network, form the basis of centrality. \nThe most basic measure is degree centrality, which counts the number of edges (connections) attached to a node. In weighted networks, this concept is often expanded to node strength \\citep{hanneman_introduction_2005}, which is the sum of the node's weighted degree. In directed networks such as the reduced backbones described below, directionality of links can be tracked using in- and out-degree or in- and out-strength. All of these values account for only the direct neighbors of a node, but in many networks the \nwider set of the neighbors' connections can also constrain or boost a node's access to information or resources (for example, study group invitations).\n\nA later generation of centrality measures accounts for both the number of neighbors of a node and the importance of each of those neighbors. Their importance, in turn, depends on that of their own neighbors, requiring simultaneous solution over the whole network. Measures of this type can be computed as eigenvalue problems \\citep{bonacich_power_1987}. One of the most popular measures of this type is PageRank, the same base algorithm used by the Google search engine to rank the importance of pages on the internet \\citep{brin_anatomy_1998}. PageRank designates a node as being important if a large number of important pages point to it. It was developed for directed networks (on the internet, linking to another page makes a directed network edge), but can be used in undirected networks as well. PageRank is one of the three centrality measures used below. \n\nThe other two centrality values we will test, Target Entropy (TE) and Hide \\citep{sneppen_hide-and-seek_2005}, have been used in network analyses of classroom interactions between physics students over a university term \\citep{bruun_talking_2013}. Target Entropy is a measure of the diversity of a node's information sources; high TE nodes will have many neighbors who themselves talk to a wide array of other students. Conversely, Hide quantifies how difficult it is to ``find'' a node in the network. High-Hide nodes will have few neighbors, who may themselves be more sparsely connected than average. \n\nFor each semester, we calculate PageRank, Target Entropy, and Hide for all nodes, with PageRank using the \\texttt{igraph} package in R~\\citep{R_software,igraph} and the other two measures using code from \\citet[][Supplemental Material]{bruun_talking_2013}. \nWe then calculate Pearson correlations between each of these three centrality measures and final course grade.\nNetwork centralities inherently violate the assumption of independence that underlies typical correlation calculations. To correct for this issue, permutation tests can be used, where the data set is repeatedly resampled and the correlation re-calculated, typically thousands or tens of thousands of times~\\citep{grunspan_understanding_2014}. The resulting distribution of correlation coefficients gives an estimate of how likely the observed correlation was to occur by chance in a network of the same size and density---in other words, an empirical $p$-value. \nThough network measures are our primary interest, for comparison we also report Pearson correlations between final grade and a student's total contributions to the forum (their combined number of Posts, Polls, and Reflections). \n\n\\subsection{Backbone extraction\\label{sec:meth-bb}}\n\nThe forum networks generated by the process described above are much more dense than typical survey-based networks in a physics class of comparable size~\\citep{brewe_changing_2010,traxler_community_2015,sandt_TNT_2016}. Since they are built from thousands of posts, with content ranging from physics-based conversations to ``post count'' boosting, it seems reasonable that not all interactions are equally important. The most active individuals might be connected by some core structure underlying the ``noisy'' full network, and it is these types of structures that backbone extraction is designed to uncover \\citep{serrano_extracting_2009}. \n\nVarious methods exist for extracting backbones, and for this work we used the Locally Adaptive Network Sparsification (LANS) algorithm of~\\citet{foti_nonparametric_2011}, which has been tested on several real-world dense networks including answer distributions from the Force Concept Inventory~\\citep{brewe_using_2016}.\n LANS is tuned through a parameter $\\alpha$: for each node in the network, all edges below the $(1-\\alpha)$ percentile of edge weight are discarded. Thus, an alpha value of 0.05 would correspond to keeping only the 95th percentile and above of a node's strongest links. For a node with edges of weights 1, 5, and 10, a threshold of $\\alpha=0.05$ would remove all but the weight-10 edge(s). There is no single value of alpha which will suit for all network problems; rather, each analysis should test several values and select one that simplifies to the desired density while still preserving necessary information. Here, we test several values of $\\alpha$ and re-run permutation correlation tests between centrality \n and final grade, investigating whether backbone extraction strengthens the correlations by removing the effect of extraneous low-weight connections.\n\n\\section{Results\\label{sec:results}}\n\n\\subsection{Participation and network statistics}\n\n\\begin{table*}\n\\begin{ruledtabular}\n\\caption{Forum participation and network statistics by semester. Participation includes students enrolled ($N_{class}$), percent who posted in the forum, total number of threads and replies, and average replies per thread and posts per student plus or minus standard deviation. Network statistics include number of nodes ($N$), isolates, average degree ($\\pm SD$), and network density ($\\pm$ boostrapped standard error).\\label{tab:part-nw}}\n\\begin{center}\n\\begin{tabular}{c|cccccc|cccc}\nSemester\t& $N_{class}$\t& Part.\\ (\\%)\t& Threads\t& Replies\t& Replies\/thread\t& Posts\/student\t& $N$\t& Isolates\t& Degree\t& Density \\\\ \\hline\n1 \t& 173\t& 90\t& 936\t& 2376\t& $2.5\\pm\t3.6$\t& $21\\pm16$ & 156\t& 12\t& $53\\pm30$\t& $0.32\\pm0.03$ \\\\\n2\t& 152\t& 86\t& 912\t& 2253\t& $2.5\\pm2.4$\t& $23\\pm24$ & 131\t& 5\t& $29\\pm22$\t& $0.22\\pm0.03$ \\\\\n3\t& 166\t& 87\t& 762\t& 2508\t& $3.3\\pm3.3$\t& $22\\pm22$ & 145\t& 6\t& $42\\pm29$\t& $0.28\\pm0.03$ \n\\end{tabular}\n\\end{center}\n\\end{ruledtabular}\n\\end{table*}\n\nTable \\ref{tab:part-nw} shows summary participation statistics for the forum. Each semester, 85--90\\% of the enrolled students posted at least once. The number of threads was similar between the first two semesters and lower in the third, when the average number of replies per thread increased. We compared thread length and posts per student between semesters using pairwise Wilcoxan tests, which account for the non-normal distribution and presence of outliers in the data. Only Semester 3 had a significantly different ($p < 10^{-5}$) average number of replies per thread. \nThere were no significant differences in the number of posts per student between semesters.\n\nAverage posts per student can mask very different posting patterns, if some semesters have a few high-volume participants and others have a lower but more widespread posting rate.\nFigure \\ref{fig:par} shows the distribution of forum contributions among students. \nTo control for varying class size, the figure shows the density, essentially a smoothed histogram normalized to integrate to 1 for each semester. \nAll three semesters have a peak at low activity (0--15 contributions), a few very active members around 75--100 contributions, and a high-activity ``tail.'' Semester 1 has its largest peak around 25 contributions, while the other two semesters had a less prominent ``shoulder'' there. \n\n\\begin{figure}[ptbh]\n\\begin{center}\n\\includegraphics[width=0.95\\linewidth]{fig3}\n\\caption{Density distribution of forum activity (combined posts, polls, and comments) for class members by semester. The instructor's contribution totals are included and are 94, 182, and 141 by semester.\\label{fig:par}}\n\\end{center}\n\\end{figure}\n\n\nTable \\ref{tab:part-nw} also shows descriptive statistics for the forum discussion networks. Nodes are all students who posted at least once, and isolates are students who only posted one thread, which received no replies (see  student D in Fig.\\ \\ref{fig:bipartite}). Though there are fewer isolates in the second semester compared to the first, the average degree of nodes is lower, as is the network density. Because larger networks will tend to have lower density, \nthe ``natural'' ranking of density values in the three semesters would be (2, 3, 1) for a comparable level of network structure. The observed ranking reverses this. \n\nThe aggregate forum network for the whole semester is too dense to be visually useful without extensive filtering of low weight edges (see Fig.\\ 1 in \\citet{traxler_coursenetworking_2016}). Fig.\\ \\ref{fig:networks} shows the week 7--8 subset of Semesters 1 and 2, \na time of similar activity in the middle of the semester. Each circle shows a student, \nsized by total contributions over the semester and colored by final grade. \nDarker connecting lines indicate higher-weight edges, resulting from more common posting by a pair of students.\n Though total forum activity was similar between the two semesters, the Semester 2 network is less dense and more dominated by a few high-participation, high-grade students during the time shown.\nSemester 3 (not shown) is visually ``between'' the two pictures, with fewer students posting than Semester 1 during this time slice but notably higher density than Semester 2.\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=3.3in]{fig4a}\n\\includegraphics[width=3.3in]{fig4b}\n\\caption{Forum networks from weeks 7--8 in Semester 1 (left) and Semester 2 (right). Line opacity is scaled by edge weight, so darker lines indicate more threads in common for a student pair. Nodes are sized by total contributions over the semester and colored by grade (red low, yellow medium, blue high). Withdrawals and instructor or CN staff accounts are white, and the instructor's node is labeled ``I.''\\label{fig:networks}}\n\\end{center}\n\\end{figure*}\n\n\n\\subsection{Centrality\/grade correlations}\n\nTable \\ref{tab:corr} shows the results of the bootstrap correlations between final grade and centrality in the discussion forum network. In the first semester, PageRank and Target Entropy are positively correlated with final grade and Hide is negatively correlated, all at small effect sizes (using Cohen's suggested thresholds of (0.1, 0.3, 0.5) for size of effect \\citep{cohen_power_1992}). In the second semester, no correlations are significant. The third semester repeats the pattern of semester 1, with the PageRank and Hide correlations now above the threshold for medium effect size. \nThe table also gives the Pearson correlation between total number of forum contributions and final grade for each semester. This correlation is only significant in Semester 3, at a medium effect size.\n\n\\begin{table}[htbp]\n\\begin{ruledtabular}\n\\caption{Correlation coefficients between final grade, the network centrality measures PageRank (PR), Target Entropy (TE), and Hide, and forum participation (total threads$+$comments). Asterisks show the level of statistical significance ($^* p < 0.05$, $^{**} p < 0.01$, and $^{***} p < 0.001$).\\label{tab:corr}}\n\\begin{center}\n\\begin{tabular}{cllll}\nSemester\t& PR\t\t\t& TE \t\t\t& Hide\t\t\t\t& Participation \t\\\\ \\hline\n1 \t\t& 0.18 $^*$\t& 0.29 $^{**}$\t& -0.27 $^{**}$\t\t\t& 0.091 \t\t\\\\\n2\t\t& 0.13 \t\t& 0.17 \t\t& -0.18 \t\t\t\t& 0.12\t\t\\\\\n3\t\t& 0.34 $^{***}$\t& 0.28 $^{**}$\t& -0.31 $^{**}$\t\t& 0.33 $^{***}$\t\t\n\\end{tabular}\n\\end{center}\n\\end{ruledtabular}\n\\end{table}\n\n\n\\subsection{Backbone extraction}\n\nThe goal of backbone extraction is to simplify a network to its essential structure, so high-density forum networks are ideal candidates for this technique. \nFor each semester of data, we calculated the LANS backbone extraction at values of $\\alpha=(0.5, 0.1, 0.05, 0.01)$. Table~\\ref{tab:bbstats} shows the number of edges and the fraction of the original total edge weight remaining~\\citep{serrano_extracting_2009} for each reduction of the three semesters. \n\n\\begin{table}[htbp]\n\\begin{ruledtabular}\n\\caption{Edges ($N_E$) and fraction of total original weight (\\%$W_T$) at each level of backbone extraction; $\\alpha=1$ is the original network.\\label{tab:bbstats}}\n\\begin{center}\n\\begin{tabular}{lrcrcrc}\n\t& \\multicolumn{2}{c}{Semester 1}\t& \\multicolumn{2}{c}{Semester 2}\t& \\multicolumn{2}{c}{Semester 3}\t\\\\ \n$\\alpha$\t& $N_E$\t& \\%$W_T$\t& $N_E$\t& \\%$W_T$\t& $N_E$\t& \\%$W_T$ \\\\ \\hline \n1\t& 7628\t& 1.00\t& 3704\t& 1.00\t& 5858\t& 1.00 \\\\\n0.5\t& 5635\t& 0.88\t& 2476\t& 0.88\t& 4042\t& 0.88 \\\\\n0.1\t& 1173\t& 0.36\t& 572\t& 0.39\t& 1000\t& 0.39 \\\\\n0.05\t& 661\t& 0.24\t& 334\t& 0.26\t& 530\t& 0.25 \\\\\n0.01\t& 194\t& 0.09\t& 186\t& 0.12\t& 221\t& 0.10 \n\\end{tabular}\n\\end{center}\n\\end{ruledtabular}\n\\end{table}\n\nThere are competing criteria for judging a backbone extraction to be appropriate or a value of alpha to be suitably small. One heuristic is that a large portion of the original network weight (the sum of its weighted degree) should remain~\\citep{serrano_extracting_2009}. \nAnother possible metric is to lower $\\alpha$ until the forum network reaches a comparable density or average degree to a classroom survey-based network of similar size~\\citep{traxler_community_2015}. By the first measure, values of $\\alpha=0.05$ or lower may be cause for concern in this data, since they hold only a quarter of the original network weight (a small amount in comparison to the example backbones of\\citet{serrano_extracting_2009}). By the second measure, values of $\\alpha=0.05$ or 0.01 might be most appropriate. \n\nTo resolve this possible contradiction, the ultimate arbiter is what happens to the centrality values of the nodes: their relative distribution and their correlations with students' final grades. For all three semesters, backbone reduction appears to destroy rather than strengthen correlations between network centrality and final grade. \nThe negative Hide\/grade correlation vanishes immediately, \nwith similar though less severe effects on PageRank and Target Entropy (see Supplemental Material for details). In the third semester, there is some suggestion that backbone reduction does not hurt and may even help the PageRank and Target Entropy correlations down to $\\alpha=0.1$. However, the overall effect of the technique is to reduce rather than highlight the useful information.\n\nFigure \\ref{fig:boxplot-PR} shows boxplots of the PageRank scores of nodes for the original ($\\alpha=1$) and reduced networks for Semester 1. These distributions help to explain why correlations with final grade decrease as supposedly ``extraneous'' links are removed. Backbone extraction flattens calculated centrality values for most nodes in the network as $\\alpha$ decreases, with the distribution skewing lower and many nodes eventually occupying the minimum possible PageRank value. \nPlots for the other semesters and the other two centrality measures show a similar effect.\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.93\\linewidth]{fig5.eps}\n\\caption{Boxplot of PageRank centrality values for the Semester 1 network backbones. \nThe bottom, middle, and top line of the boxes show first quartile, median, and third quartile. \nThe upper and lower ``whiskers'' extend to the maximum\/minimum values or 1.5 times the inter-quartile range, whichever is larger.\nAs alpha decreases, more node centralities cluster at the minimum value.\\label{fig:boxplot-PR}}\n\\end{center}\n\\end{figure}\n\n\n\\section{Discussion}\n\n\\subsection{Network analysis reveals important differences in forum use between semesters}\n\nOur first research question was: {\\em How do discussion forum networks differ among multiple semesters of an introductory physics course, and can this information be extracted more easily from participation statistics?} From the analyses summarized in Table \\ref{tab:part-nw} and Fig.\\ \\ref{fig:networks}, we find that the forum networks have different densities, average degree, and breadth of participation between semesters. In particular, Semesters 1 and 3 show a higher level of connectivity that is not easily explained by fluctuations in class size or numbers of discussion threads and comments.\nIn contrast, non-network participation statistics show few significant differences between the classes, with only Semester 3 having longer discussion threads (but not more activity overall).\nSome essential structure of discussions is captured by network analysis, beyond that available by participation tracking but without time-consuming content analysis of posts.\n\nOur second research question was: {\\em How much are student final grades correlated with their centrality in the discussion forum network?}\nStudents who are more central in the forum network tend to score higher in the course, but not in every semester---in particular, the higher-density networks are those in which centrality is correlated with grade. \nTarget Entropy and Hide seem to be the most reliable predictors, \nwith PageRank somewhat less consistent. Exploratory analysis shows that in this data set, Target Entropy and Hide are highly correlated, so we focus our discussion below on Target Entropy. \nThis result builds on the findings of question 1: not only do networks better capture the discussion connectivity, but they track a kind of interaction that benefits students in the course.\n\nOur third research question was: {\\em Do centrality\/grade correlations, if present, strengthen when reducing the network to a more simplified ``backbone?''}\nWe predicted that using backbone extraction on the forum networks would clarify correlations between centrality and final grade, by streamlining low-weight links that proliferate in long ``chat'' threads and leaving the most important connections between students. We found that instead, this ``noise'' is part of the signal, and that reducing the forum networks to backbone representations destroys correlations between centrality and grade.\nIt is possible that\na backbone extraction method developed specifically for bipartite networks \\citep{neal_backbone_2014} might improve this result. However, plotting PageRank, Target Entropy, and Hide at successive alpha levels shows that backbone extraction flattens these centrality distributions and pushes more and more nodes to the minimum value. This issue seems likely to recur even with a change in algorithm.\n\n\\subsection{Implications for network research}\n\nOne recommendation that emerges from the literature review of this paper is for researchers to carefully document their choices in using network models to describe online learning. Some past studies have used survey methods to gather network data \\citep{yang_effects_2003,cho_social_2007}, while others draw from electronic logs \\citep{wortham_nodal_1999,reffay_social_2002,aviv_network_2003,de_laat_investigating_2007,dawson_seeing_2010}. Studies in the first category base their approach on earlier social network analysis studies of business organizations, though physics education researchers have tied similar data collection to theoretical frameworks of transformation of participation or communities of practice \\citep{brewe_investigating_2012,bruun_networks_2012}. \n\nStudies that derive their data from electronic logs are more common in the CSCL literature, and this is a promising direction given the growing amount of data that is available from learning management systems.\n\\citet{kortemeyer_analysis_2006} argues that these data open a more natural window onto students' thought processes than think-aloud interviews, where students may be trying to perform to the interviewer's expectations.  \nFor instructors, detecting differences in student participation early in the semester, based on their use of resources like forums, can give early warnings about at-risk students in live or online courses \\citep{dawson_seeing_2010,reffay_social_2002}. \n\nA few studies do not specify how they constructed their networks. \nBoth the data source (survey or logs) and the assumptions made about how to connect the network have consequences for the density and structures that result. In other words, the network model---what constitutes a link between students?---is an interaction model \\citep{freeman_centrality_1978}, which makes a statement about what communication processes the researcher thinks are important to learning in a given environment. Our bipartite model generated far denser networks than survey-based classroom studies, even those drawn from weekly sampling (see \\citep{bruun_talking_2013}, Supplemental Material Fig.\\ 5 for link weight distribution of their densest network). We chose an expansive definition of interaction, and find that centrality in the resulting network is an equally strong predictor of grades as a sparser survey approach. \nOur measured correlations between network centrality and grades are also comparable to those found between annotation quality and exam grade in a physics content analysis study \\citep{miller_analysis_2016}.\nDifferent online learning studies have used a variety of centrality measures, and it is not at all clear that a ``best'' set will emerge. Only by documenting their assumptions can researchers allow for any hope of comparing between or replicating results. \n\n\n\\subsection{Implications for online learning research}\n\nAs outlined above, the range of data sources, network statistics, and outcome measures makes it challenging to check results between CSCL network analyses. However, we can look for alignment in trends or effect sizes of results. \\citet{dawson_seeing_2010} \nfound that high-performing students had more connections and were more likely to be linked to the instructor. High Target Entropy students in our Semesters 1 and 3, who were more likely to do well in the course, would tend to have a large number of connections like the high-scoring students in Dawson's study. Similarly, low Target Entropy---signaling limited sources of information---would generally correspond to student ego networks with only a few connections. \n\nThough the instructor in our data was not intentionally making an anchored forum with the traits recommended by \\citet{guzdial_effective_2000}, the CN interface builds in two of those authors' recommendations: a thread-grouped view with always-visible archives and the ability to choose a post category (through instructor- or user-created ``hashtags''). The authors make a third recommendation of ``anchor'' threads that prompt students with a few key discussion topics and include a link to post their contributions. In Semesters 1 and 3, the instructor created anchor threads via the Tasks feature on CN. Tasks show at the top of the forum page, and were updated once a week in those two semesters. \nThe instructor did not use these weekly tasks in Semester 2, and this change came with (though we cannot say it was the sole cause of) a loss of network connectivity.\n\n\\citet{aviv_network_2003} compared two semesters and found that the level of integration between the forum and class assignments was linked to substantial differences in the amount and cognitive level of discussion by students. Our results match theirs in part: the raw amount of discussion was not necessarily tied to facilitation, but the resulting network between students was more dense and appears to be more educationally useful in the more-structured semesters. \nThe work by Aviv and collaborators is one of a small but growing number of studies that combine network measures with content analysis of posts \\citep{rice_electronic_1987,fahy_epistolary_2002,de_laat_investigating_2007}. Work in physics has shown links between the cognitive level of student comments on homework problems \\citep{kortemeyer_analysis_2006} or textbook annotation \\citep{miller_analysis_2016} with their grades \\citep{kortemeyer_analysis_2006,miller_analysis_2016} or conceptual gains \\citep{miller_analysis_2016}.\nContent analysis of the CN data, currently in progress, will let us look for interplay between the quantitative network structures and qualitative content of discussions. \n\n\\citet{cho_social_2007} and \\citet{yang_effects_2003} found that degree centrality positively correlated with final grade in survey-based classroom networks, though in the first study, the correlation was only marginally significant and a significant correlation instead appeared with closeness centrality. Though their network construction methods were different, the correlations found ($r=0.442$ for \\citep{cho_social_2007}, $r=0.4$ or 0.46 for \\citep{yang_effects_2003}) are similar to the results of this study as well as the correlations with PageRank, Target Entropy, and Hide found by \\citet{bruun_talking_2013}. \n\nThe closest comparison study in physics is \\citet{bruun_talking_2013}, who used weekly surveys to build an aggregate network for an introductory mechanics course. We found that the three centrality measures that emerge as most important in their study---PageRank, Target Entropy, and Hide---are also useful for exploring position\/grade correlations in the forum data. Of these, Target Entropy and Hide seem to show the most consistent signal; these measures originate from a theoretical perspective of quantifying information flow \\citep{bruun_talking_2013,sneppen_hide-and-seek_2005}, which may be especially suited for describing long post chains in forum networks.\n\n\n\\section{Limitations and future work}\n\nLike most CSCL studies \\citep{johnson_synchronous_2006}, this is not a control-group experimental study. One possible reading of our results is that more engaged students tend to participate in the forum, and that high-centrality nodes are merely the ``good'' students (however a reader might define that) who would succeed regardless of the presence of a forum or discussion prompts. Certainly, there is evidence that students who are inclined to talk to others are more likely to benefit from forums \\citep{cho_social_2007}. However, the lack of forum\/grade correlations in Semester 2 suggests that this explanation is incomplete. First, and as a general argument for forum use, even students who are predisposed to talk about class material can benefit from tools for doing so outside of class hours at commuter schools. Furthermore, the differences in Semester 2 show that even a similarly-active forum may not be equally useful. There is no reason to believe that the fraction of engaged, self-starting students was substantially different between our three semesters, but there are significant differences in network structure and in correlations between forum position and grade. Taken together, these points suggest that not only does instructor facilitation matter, but that network analysis can detect this difference even when participation tracking does not. \n\nFinally, a detailed content analysis is beyond the scope of this paper, but spot-checking suggests that the most active threads (which contribute to higher network density) are a mixture of physics-based and social topics. This further weakens the idea that the correlations we found only show the ``best'' students using the forum for strictly studious purposes. The nature of the conversations and community that arise are more complicated than an on\/off-topic dichotomy \\citep{rourke_assessing_1999}, so the \nnext stage of this project will use post text to analyze the discussion differences between semesters and the effect of anchoring by weekly tasks.\nUltimately, content analysis results can be combined with a time-developing picture of the network characteristics \\citep{ouyang_influences_2017} to better understand instructor facilitation, the student discussion culture that emerges, and the benefits that both have for learning in physics forums.\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn insurance and risk, a pivotal role is played by the classical {\\it Cram\\'er-Lundberg model} (also known as the {\\it compound Poisson model}). In this model independent and identically distributed claims arrive according to a Poisson process, whereas premiums are earned at a constant rate. This means that if the initial reserve is given by $u\\geqslant 0$, then the reserve level at time $t\\geqslant 0$ is given by \n\\begin{equation}X_t:= u + rt - \\sum_{i=1}^{N_t} L_i,\\label{CP}\\end{equation}\nwith $r>0$ the premium rate, $(N_t)_{t\\geqslant 0}$ a Poisson process with intensity $\\lambda>0$, and $(L_i)_{i\\in{\\mathbb N}}$ a sequence of i.i.d.\\ random variables. The key quantity of interest is the (finite-horizon) ruin probability\n${\\mathbb P}(\\exists s\\in[0,t]: X_s<0)$ and its infinite-horizon counterpart ${\\mathbb P}(\\exists s\\geqslant 0: X_s<0)$. A broad set of techniques has been developed to analyze this quantity, for the Cram\\'er-Lundberg model itself as well as for more advanced variants; we refer to \\cite{AsmussenAlbrecher} for an exhaustive overview. With the random variable $L$ denoting a generic claim, often the {\\it net profit condition} ${\\mathbb E}\\,(X_t-X_0)= rt - {\\mathbb E}N_t\\,{\\mathbb E}L>0$ is imposed. Under this condition, which effectively means that $r> \\lambda\\,{\\mathbb E}L$, it is guaranteed that ruin is rare. A practically relevant objective is to select the initial reserve $u$ such that the (finite or infinite-horizon) ruin probability is below some threshold $\\varepsilon.$\n\nEssentially the same modeling framework can be applied in the context of credit as well. Then the claim arrival process describes the default epochs, the premiums correspond to the interest received from the obligors, and the claims are the corresponding losses. One may wonder, however, whether in this setting the assumption of Poisson arrivals is any realistic: {whereas in the insurance context the number of claims issued can in principle exceed any bound, it is obvious that in the credit context the number of defaults cannot exceed the number of obligors. More concretely, as soon as an obligor goes into default, it effectively leaves the system.} Motivated by this observation, we study in this paper the ruin probability in a transient variant of the Cram\\'er-Lundberg model. We do so by defining for each obligor a random variable (e.g.\\ exponentially distributed) corresponding to the time-to-default, where after the default the obligor can neither cause any new default nor generates any interest anymore. \n\n\\vspace{3mm}\n\n{\\it Model.} We proceed by providing a more formal description of our transient variant of the classical Cram\\'er-Lundberg model. Here we state the main model, which we will generalize in various directions later in the paper.\n\nWe consider a setting in which there are initially $n\\in{\\mathbb N}$ obligors, each of which goes into default after some random amount of time. The corresponding $n$ times-to-default are assumed to be i.i.d.\\ non-negative random variables, characterized by the density $f(\\cdot)$. {In the credit context, risk is quantified over a finite time horizon justifying the use of a model in which clients eventually all go into default.} Let the loss-at-default, per obligor, be distributed as a non-negative random variable $L$, and let these losses be i.i.d., each with Laplace transform $\\ell(\\cdot)$. It is natural to assume that the income per unit of time is proportional to the number of obligors that have not gone into default yet. In other words, the surplus process increases at a rate $ri$ per unit of time, for some $r>0$, when there are $i$ obligors that have not defaulted yet, for $i\\in\\{0,\\ldots,n\\}.$ \nThe company has an initial reserve level $u>0$. Because of the similarity with insurance and risk models, throughout this paper we sometimes refer to losses as claims. \n\nThe primary objective of this paper is to evaluate $p_n(u,t)$, defined as the ruin probability of the company before time $t$, given there are $n$ obligors at time $0$ and that the initial reserve is $u$. Being able to compute $p_n(u,t)$, one can pick $u$ such that this ruin probability remains below an acceptable level $\\varepsilon>0$. In addition, when a new obligor wishes to get a loan, knowledge of $p_n(u,t)$ allows one to decide if (and, if yes, by how much) the initial level should be adjusted. \n\n\n\n\\vspace{3mm}\n\n{\\it Contributions.} For the main model, we provide a procedure by which, for any $n$, the double transform (in space and time, that is) of $p_n(u,t)$ can be determined. More specifically, we develop a recursive relation by which these transforms can be determined. While this means that one can evaluate the finite-horizon ruin probability $p_n(u,t)$ by numerical inversion, we in addition point out how to efficiently estimate this rare-event probability relying on importance sampling simulations; the procedure proposed has provable optimality properties. In addition we provide the logarithmic asymptotics of $p_n(nu,t)$ as $n$ grows large (i.e., in this setting the initial reserve $u$ is scaled by the number of obligors $n$). \n\nBesides the base model, four generalizations are dealt with in this paper.\nOne could argue that the assumption of the times-to-default being independent is not realistic, as in reality defaults tend to cluster. To resolve this issue, in one of the generalizations we allow a {\\it regime switching} mechanism (also frequently referred to as {\\it Markov modulation}) that induces dependence between the obligors. The regime could be thought of as the `state of the economy', wherein every state of the economy the dynamics of the reserve level are described by a specific Cram\\'er-Lundberg model. \nIn a second generalization, we consider a model in which some loss events correspond to defaults (reducing the number of obligors by one) while others do not (leaving the number of obligors unchanged). Another unrealistic feature of the main model is that the obligors are homogeneous: their times-to-default (losses, respectively) stem from the same distribution. To remedy this, we\nalso analyze a model variant corresponding to heterogenous obligors: there are multiple groups, each of them consisting of statistically identical obligors. {This extension offers an important additional flexibility as one can cluster obligors based on the loss distribution, which is often deterministic in the credit context, and consider classes of obligors that do not go into default or have a class-specific income rate.}\nA last extension that we discuss in this paper concerns a model in which between loss events the reserve level behaves as a Brownian motion (rather than as a deterministic drift).\n\n\\vspace{3mm}\n\n{\\it Related literature.} \nStarting from the pioneering papers by Cram\\'er \\cite{C1} and Lundberg \\cite{L1,L2}, focusing on the classical compound Poisson model \\eqref{CP},\na broad range of risk models has been analyzed. Without attempting to provide a complete overview, we proceed by discussing a few important branches; we refer to \\cite{MIK,KGS, TEU} for general accounts of risk theory.\nIn the first place, the assumption of the cumulative claim process being of compound Poisson type has been lifted, thus allowing a compound Poisson claim process perturbed by a diffusion \\cite{G2, G1}, and even a (spectrally one-sided) L\\'evy claim process; see e.g.\\ \\cite[Ch.\\ X and XI]{AsmussenAlbrecher} and \\cite{DM,KYP}. In addition, some models incorporate returns on investment, while in other models the dynamics of the reserve process are level-dependent; see e.g.\\ \\cite[Ch.\\ VIII]{AsmussenAlbrecher} and \\cite{AC, BM}. Finally, there is a substantial body of papers exploring the effect of specific dependence structures; see e.g.\\ \\cite{CKM} and, for an overview, \\cite[Ch.\\ XIII]{AsmussenAlbrecher}. More specifically, the effect of parameter uncertainty can be analyzed through the resampling model recently proposed in \\cite{CDMR}.\n\n\\vspace{3mm}\n\n{\\it Organization.} Section \\ref{S3} provides an explicit analysis, in terms of transforms, for the base model introduced above. A large deviations analysis of the tail probability is presented in Section \\ref{asy}, together with an importance-sampling based simulation approach and a uniform upper bound. The four extensions of the base model are presented by Section \\ref{ext}. {The final section contains a series of numerical experiments.}\n\n\\section{Exact analysis}\\label{S3}\nIn this section we analyze the base model that was described in the introduction. We start by defining the key quantities of this base model, pertaining to the case that each of the obligors has a time-to-default that is exponentially distributed.\nWe then present our analysis yielding a recursion for the double transform of the ruin probability. \n\n\\subsection{Notation and preliminaries}\nPer obligor the rate of going into default is $\\lambda>0$. This means that if there are still $i$ obligors left (i.e., being not in default), the time till the next default is exponentially distributed with mean $(\\lambda i)^{-1}$.\n\nRecall that $p_n(u,t)$ is the probability of ruin before time $t$, starting with $n$ obligors at time $0$, given the initial reserve level is $u$.\nIn our approach we (uniquely) characterize $p_n(u,t)$ through its double transform\n\\[\\psi_n(\\gamma) := \\int_0^\\infty e^{-\\gamma u} \\int_0^\\infty \\vartheta e^{-\\vartheta t} p_n(u,t)\\,{\\rm d}t\\,{\\rm d}u\n=\\int_0^\\infty e^{-\\gamma u}  p_n(u)\\,{\\rm d}u,\\]\nwhere $p_n(u)$ can be interpreted as the probability of ruin before an exponentially distributed clock with mean $\\vartheta^{-1}$ (which is sampled independently from anything else). The case of $t=\\infty$ corresponds with $\\vartheta\\downarrow 0$.\nThe main result of this section is an expression (recursive in $n$) for $\\psi_n(\\gamma)$: we express $\\psi_n(\\cdot)$ in terms of $\\psi_{n-1}(\\cdot)$. Observe that we can equivalently write $p_n(u)$ as ${\\mathbb P}(Z_n\\geqslant u)$, where $Z_n$ is the maximum of the {\\it net} cumulative {loss} process (the net cumulative {claim} process, in the insurance context) over the above-mentioned exponentially distributed amount of time (with mean $\\vartheta^{-1}$, that is). \n\nIn practical settings, one typically has that $r>-\\lambda\\ell'(0)=\\lambda\\,{\\mathbb E}L$, so that at any point in time ruin is rare, in the sense that the expected reserve increases as a function of time; to this end, realize that when there are $i\\in\\{0,\\ldots,n\\}$ obligors left, the `local drift' of the reserve process is $ri + \\lambda i\\,\\ell'(0)>0.$\n\n\\subsection{Analysis}\\label{subsec_analysis}\nIn this subsection we present a recursive scheme to evaluate $\\psi_n(\\gamma)$.\nThe main idea is to condition on the first event, being either the first default (which happens after an exponentially distributed time with mean $(\\lambda n)^{-1})$ or the expiration of the exponential clock (which happens after an exponentially distributed time with mean $\\vartheta^{-1})$. If the former event happens to apply first, then we can still reach ruin, but now with $n-1$ obligors and an adapted initial reserve. If the latter events occurs first, then we won't be facing ruin before the exponential clock expires.\nThese ideas can be translated into mathematical terms as\n\\begin{equation}\n\\label{RECU}p_n(u) = \\int_0^\\infty \\lambda n\\, e^{- (\\lambda n+\\vartheta) t} \\,{\\mathbb P}(Z_{n-1}+L\\geqslant u+rnt)\\,{\\rm d}t;\\end{equation}\nuse that the time till the first event is exponentially distributed with mean $(\\lambda n+\\vartheta)^{-1}$, and that the first event is a default with probability $\\lambda n\/(\\lambda n+\\vartheta)$. \n\nWe proceed by analyzing $\\psi_n(\\gamma)$ using the relation \\eqref{RECU}, with the objective to express it in terms of $\\psi_{n-1}(\\cdot)$. By a change-of-variable $v:=u+rnt$, we obtain\n\\begin{align*}\\psi_n(\\gamma) &=\\int_0^\\infty e^{-\\gamma u}  \\int_0^\\infty \\lambda n\\, e^{- (\\lambda n+\\vartheta) t} \\,{\\mathbb P}(Z_{n-1}+L\\geqslant u+rnt)\\,{\\rm d}t\\,{\\rm d}u\\\\\n&=\\frac{1}{rn} \\int_0^\\infty e^{-\\gamma u}  \\int_u^\\infty \\lambda n\\, e^{- (\\lambda n+\\vartheta) (v-u)\/(rn)}\\, {\\mathbb P}(Z_{n-1}+L\\geqslant v)\\,{\\rm d}v\\,{\\rm d}u.\n\\end{align*}\nThe next step is to swap the order of the integrals, exploiting the fact that the integral over $u$ allows an elementary solution:\n\\begin{align*}\\frac{1}{rn}& \\int_0^\\infty  \\left(\\int_0^v e^{-\\gamma u} e^{ (\\lambda n+\\vartheta) u\/(rn)} \\,{\\rm d}u\\right)\n \\lambda n\\, e^{- (\\lambda n+\\vartheta) v\/(rn)}\\, {\\mathbb P}(Z_{n-1}+L\\geqslant v)\\,{\\rm d}v\\\\\n &=\\frac{\\lambda n}{\\gamma rn -\\lambda n-\\vartheta}\\int_0^\\infty \\big(e^{- (\\lambda n+\\vartheta) v\/(rn)}-e^{-\\gamma v}\\big) {\\mathbb P}(Z_{n-1}+L\\geqslant v)\\,{\\rm d}v.\\end{align*}\nIn the last expression, we see an object that resembles a Laplace transform, but observe that it features a complementary cumulative distribution function rather than a density. Recall however the standard identity\n\\begin{equation}\\label{e1}\\int_0^\\infty e^{-\\gamma u} {\\mathbb P}(X\\geqslant u){\\rm d}u =\\frac{1}{\\gamma}-\\frac{1}{\\gamma}\\int_0^\\infty e^{-\\gamma u} {\\mathbb P}(X\\in{\\rm d}u)= \\frac{1-{\\mathbb E}\\,e^{-\\gamma X}}{\\gamma}.\\end{equation}\nIn addition, using integration by parts, for the non-negative random variable $Z_{n-1}$,\n\\begin{equation}\n\\label{e2}{\\mathbb E}\\,e^{-\\gamma Z_{n-1}} = \\int_0^\\infty e^{-\\gamma x} {\\mathbb P}(Z_{n-1}\\in {\\rm d}x) = 1-\\gamma  \\int_0^\\infty {\\mathbb P}(Z_{n-1}>x)e^{-\\gamma x}{\\rm d}x= 1-\\gamma\\psi_{n-1}(\\gamma).\\end{equation}\nBy the identity (\\ref{e1}), and using the independence between the random variables $Z_{n-1}$ and $L$, we obtain, for any $\\gamma\\geqslant 0$, with $d_n:=(\\lambda n+\\vartheta)\/(rn)$,\n\\begin{align*}\\psi_n(\\gamma) &=\\frac{\\lambda n}{\\gamma rn -\\lambda n-\\vartheta}\\Big(\\frac{rn}{\\lambda n+\\vartheta}\\left(1- {\\mathbb E}\\,e^{-(\\lambda n+\\vartheta)\/(rn)\\,(Z_{n-1}+L)}\\right)\\,-\\frac{1}{\\gamma}\\left(1- {\\mathbb E}\\,e^{-\\gamma\\,(Z_{n-1}+L)}\\right)\\Big)\\\\\n&=\\frac{\\lambda n}{\\lambda n+\\vartheta}\\frac{1}{\\gamma}+\\frac{\\lambda n}{\\gamma rn -\\lambda n-\\vartheta}\\left(\\frac{{\\mathbb E}\\,e^{-\\gamma Z_{n-1}}\\ell(\\gamma)}{\\gamma}-\\frac{{\\mathbb E}\\,e^{-d_n Z_{n-1}}\\ell(d_n)}{d_n}\\right),\n\\end{align*}\nwhich, by applying (\\ref{e2}) and a few elementary algebraic steps, equals\n\\[\\frac{\\lambda n}{\\lambda n+\\vartheta}\\frac{1}{\\gamma}+\\frac{\\lambda n}{\\lambda n +\\vartheta-\\gamma r n}\\left(B\\left(\\frac{\\lambda n +\\vartheta}{rn},\\psi_{n-1}\\left(\\frac{\\lambda n +\\vartheta}{rn}\\right)\\right)-B\\left(\\gamma,\\psi_{n-1}(\\gamma)\\right)\\right),\\]\nwhere we define\n\\[B(x,y):= \\ell(x)\\left(\\frac{1}{x}-y\\right).\\]\n\nConclude that we have expressed $\\psi_n(\\cdot)$ in terms of $\\psi_{n-1}(\\cdot)$, so that we would obtain a recursion if we would have an explicit expression for $\\psi_0(\\cdot)$. Recall that $\\psi_0(\\cdot)$ corresponds to ruin in the scenario without any obligor left. Obviously $p_0(u,t)\\equiv 0$ for any $u$ and $t$, entailing that $\\psi_0(\\gamma)\\equiv 0$ for any value of $\\gamma$.\nIt means that we can thus recursively compute $\\psi_n(\\gamma)$.\nThe theorem below summarizes the findings so far.\n\n\\begin{theorem} \\label{TH1} For any $\\gamma\\geqslant 0$ and $n\\in{\\mathbb N}$, we have the recursion\n\\[\\psi_{n}(\\gamma)= \\frac{\\lambda n}{\\lambda n+\\vartheta}\\frac{1}{\\gamma}+\\frac{\\lambda n}{\\lambda n +\\vartheta-\\gamma r n}\\left(B\\left(\\frac{\\lambda n +\\vartheta}{r n},\\psi_{n-1}\\left(\\frac{\\lambda n +\\vartheta}{rn}\\right)\\right)-B\\left(\\gamma,\\psi_{n-1}(\\gamma)\\right)\\right),\\]\nwhere $\\psi_0(\\gamma)\\equiv 0$.\n\\end{theorem}\n\n\n\\begin{remark}\\label{R1a}{\\em \nInterestingly, one could interpret the departure of an obligor as a time change: \nthe default arrival rate drops from $\\lambda n$ to $\\lambda (n-1)$, and simultaneously the aggregate income per time unit drops from $rn$ to $r(n-1)$. As a consequence, in the infinite-horizon setting ($\\vartheta=0$, that is) the recursion in Theorem \\ref{TH1} greatly simplifies. \n}$\\hfill\\Diamond$\\end{remark}\n\n\\begin{remark}\\label{R1}{\\em \nUpon inspecting the above proof, it is readily checked that it has not been used that the income rate is proportional to the number of obligors present; similarly, it is not crucial that the time till the next default when there are still $i$ obligors is exponential with parameter $\\lambda i$. This effectively means that we can work with an income rate $r_i$ (rather than $ri$) and a default rate $\\lambda_i$ (rather than $\\lambda i$) during times that there are $i$ obligors left. We thus obtain the recursion\n\\[\\psi_{n}(\\gamma)= \\frac{\\lambda_n}{\\lambda_n+\\vartheta}\\frac{1}{\\gamma}+\\frac{\\lambda_n}{\\lambda_n +\\vartheta-\\gamma r_n}\\left(B\\left(\\frac{\\lambda_n +\\vartheta}{r_n},\\psi_{n-1}\\left(\\frac{\\lambda_n +\\vartheta}{r_n}\\right)\\right)-B\\left(\\gamma,\\psi_{n-1}(\\gamma)\\right)\\right),\\]\nwhere $\\psi_0(\\gamma)\\equiv 0$. It is also remarked that one can make the loss distribution dependent on the number of obligors in the system, by working with the transform $\\beta_i(\\cdot)$ when there are still $i$ obligors that have not gone into default yet.\n}$\\hfill\\Diamond$\\end{remark}\n\n\\begin{remark}\\label{rem2}{\\em\nAn interesting special case relates to the situation in which $r_n = r$ and $\\lambda_n=\\lambda$, i.e., the conventional Cram\\'er-Lundberg model. Sending $n\\to\\infty$, one should recover the (transient version of the) Pollaczek-Khinchine formula. As an illustration, we show this computation for $\\vartheta=0$, writing $a$ for $\\lambda\/r$ and assuming that $-a\\ell'(0)<1$. We obtain the relation, with the limit of $\\psi_n(\\cdot)$ being denoted by $\\psi(\\cdot)$,\n\\[\\psi(\\gamma) = \\frac{1}{\\gamma}+\\frac{a}{a-\\gamma}\\big(B(a,\\psi(a))-B(\\gamma,\\psi(\\gamma)\\big).\\]\nIt yields, after some elementary algebra, that \n\\[1-\\gamma\\psi(\\gamma) = \\frac{\\gamma}{\\gamma-a+a\\ell(\\gamma)}\\ell(a)(1-a\\psi(a)).\\]\nThe constant $\\ell(a)(1-a\\psi(a))$ can be identified by observing that the left-hand side goes to $1$ as $\\gamma\\downarrow 0$; hence, an application of H\\^opital's rule yields that\n\\[\\ell(a)(1-a\\psi(a)) =\\lim_{\\gamma\\downarrow 0}\\frac{\\gamma-a+a\\ell(\\gamma)}{\\gamma} = 1+a\\ell'(0).\\]\nWe conclude\n\\[\\psi(\\gamma) =\\frac{1}{\\gamma}-\\frac{1+a\\ell'(0)}{\\gamma-a+a\\ell(\\gamma)},\\]\nwhich directly corresponds to the Pollaczek-Khinchine formula \\cite{AsmussenAlbrecher,DM}. Our new results can be thus be seen as a true generalization of the classical results from ruin theory. $\\hfill\\Diamond$\n}\\end{remark}\n\n\\begin{remark}{\\em\nThe recursion featuring in Thm.\\ \\ref{TH1} can be made more explicit when working with its generating function. To demonstrate this, we focus on the case of $\\vartheta=0$, $r_n=rn$, and $\\lambda_n=\\lambda n$. We have, again with $a=\\lambda\/r$,\n\\[\\psi_n(\\gamma) = \\frac{1}{\\gamma} +\\frac{a}{a-\\gamma}\\left(\\ell(a)\\left(\\frac{1}{a}-\\psi_{n-1}(a)\\right)-\\ell(\\gamma)\\left(\\frac{1}{\\gamma}-\\psi_{n-1}(\\gamma)\\right)\\right).\\]\nWe thus obtain that, using that $\\psi_0(\\gamma)=0$, \n\\begin{align*}\n\\Psi(z,\\gamma)&:= \\sum_{n=1}^\\infty z^n\\psi_n(\\gamma)\\\\\n&=\\sum_{n=1}^\\infty z^n\\frac{1}{\\gamma}+z\\frac{a}{a-\\gamma} \\sum_{n=1}^\\infty z^{n-1}\\left(\\ell(a)\\left(\\frac{1}{a}-\\psi_{n-1}(a)\\right)-\\ell(\\gamma)\\left(\\frac{1}{\\gamma}\\psi_{n-1}(\\gamma)\\right)\\right)\\\\\n&=\\frac{z}{1-z}\\frac{1}{\\gamma}+z\\frac{a}{a-\\gamma}\\left(\\ell(a)\\left(\\frac{1}{a{(1-z)}}-\\Psi(z,a)\\right)-\\ell(\\gamma)\\left(\\frac{1}{\\gamma{(1-z)}}-\\Psi(z,\\gamma)\\right)\\right) .\\end{align*} \nWe conclude that \\[\\Psi(z,\\gamma)=\\frac{1}{a-\\gamma-za\\,\\ell(\\gamma)}\\,\\left( \\frac{z}{1-z} \\frac{a-\\gamma}{\\gamma} + za\\, \\ell(a) \\left(\\frac{1}{a{(1-z)}}-\\Psi(z,a)\\right)-\\frac{z}{1-z} \\frac{ a\\,\\ell(\\gamma)}{\\gamma}\\right).\\]\nWe are thus left with determining $\\Psi(z,a)$. \nFor $a$ and $z$ fixed there is a unique positive $\\gamma\\equiv \\gamma(z,a)$ for which the denominator equals 0 (as follows from the fact that $\\nu(\\gamma):=a-\\gamma-za\\,\\ell(\\gamma)$ is concave with $\\nu(0)=a(1-z)>0$ and $\\nu(\\gamma)\\to-\\infty$ as $\\gamma\\to\\infty$). We therefore have that in $\\gamma\\equiv \\gamma(z,a)$ the numerator should equal $0$ as well. \nThis leads to\n\\begin{align*}\\Psi(z,a) &= \\frac{1}{a{(1-z)}}+\\frac{1}{1-z}\\frac{1}{\\gamma(z,a)\\,\\ell(a)}\\left(\\frac{a-\\gamma(z,a)}{a} -\\ell(\\gamma(z,a))\\right)\\\\\n&=\\frac{1}{a{(1-z)}}+{\\frac{a-\\gamma(z,a)-a\\,\\ell(\\gamma(z,a))}{(1-z)\\,a\\gamma(z,a)\\,\\ell(a)}}.\\end{align*}\nCombining the above, we have thus identified \\[\\Psi(z,\\gamma)=\\frac{z}{1-z}\\frac{1}{a-\\gamma-za\\,\\ell(\\gamma)}\\left(\\frac{a-\\gamma -a\\,\\ell(\\gamma)}{\\gamma}-\\frac{a-\\gamma(z,a)-a\\,\\ell(\\gamma(z,a))}{\\gamma(z,a)}\\right).\\] \nBy multiplying with $(1-z)$, we obtain the transform at a geometrically distributed time with success probability $z$. Sending $z\\uparrow 1$, and realizing that $\\gamma(1,a)=0$, we recover the stationary result discussed in Remark \\ref{rem2}. \n $\\hfill\\Diamond$ }                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     \n\\end{remark}\n\n\n \n\\section{Asymptotics, efficient simulation, and uniform bound}\\label{asy}\nThe previous section provides us with a way of computing $p_n(u,t).$ Here one should realize that $\\psi_n(\\gamma)$ is a (double) transform, so that numerical Laplace inversion needs to be applied in order to evaluate $p_n(u,t)$. Over the past decades significant progress has been made in the domain of Laplace inversion; see for instance the fast, accurate, and generally applicable algorithms described in\\ \\cite{AW,dI}. If one wishes to avoid numerical inversion, two frequently used alternatives are  (i)~asymptotic techniques, and (ii)~simulated-based estimation. In approach (i), one scales one or more of the model parameters, and aims at finding an explicit expression for the quantity under study (in our case the ruin probability) in the regime that this scaling parameter grows large. Approach (ii) has the intrinsic drawback that, in order to obtain reliable estimates in the domain of small ruin probabilities, many runs are needed. These issues can be remedied by simulating under a suitably chosen alternative measure rather than the actual one, and correcting the simulation output by likelihood ratios; this method is known as importance sampling. \n\nIn this section we present a series of results that help to quantify the ruin probability $p_n(u,t)$ without the need to resort to numerical inversion. Our findings come in three flavors. In the first place we find, for a given $u$ and $t$, the asymptotics of $p_n(nu,t)$ as $n$ grows large; i.e., we scale the initial capital level by the initial number of obligors.\nSecondly, we derive a uniform upper bound on $p_n(u,t)$, comparable to the well-known Lundberg inequality for the conventional Cram\\'er-Lundberg model. Finally, we develop a provably efficient importance-sampling based simulation algorithm. \nImportantly, in this section we can lift the assumption of exponentially distributed time-to-defaults. \n\n\n\\subsection{Notation and preliminaries}\nThroughout this entire section we let the times-to-default $T_1,\\ldots,T_n$ be non-negative i.i.d.\\ random variables, with density $f(\\cdot)$ and distribution function $F(\\cdot)$, distributed as a generic random variable $T$.\nLet $Z_n(t)$ be the net {cumulative loss amount} at time $t\\geqslant 0$, given that at time $0$ there are $n\\in{\\mathbb N}$ obligors present. We denote, for $i=1,\\ldots,n$ and $t\\geqslant 0$, by $W_i(t)$ the net {cumulative loss amount} of the $i$-th obligor at time $t$. By distinguishing between the scenario that obligor $i$ has gone into default at time $t$ and its complement, we can write $W_i(t)$ as\n\\begin{equation}\\label{weetje}W_i(t):=1_{\\{T_i\\leqslant t\\}}L_i-r\\min\\{T_i,t\\}.\\end{equation}\nWe define the moment generating function ${\\mathbb E}\\, e^{\\alpha L}$ of the loss $L$ by $\\bar\\ell(\\alpha):=\\ell(-\\alpha)$. Then, due to fact that the obligors are statistically identical,\n\\[{\\mathbb E}\\,e^{\\alpha Z_n(t)} =\\left({\\mathbb E}\\,e^{\\alpha W_1(t)} \\right)^n.\\]\nIn addition, we can compute the moment generating function of the net {loss} amount of a single obligor at time $t$. By conditioning on the time-to-default, using \\eqref{weetje},\n\\begin{align*}\n\\omega_t(\\alpha):={\\mathbb E}\\,e^{\\alpha W_1(t)}&=\\bar\\ell(\\alpha)\\int_0^t f(s) e^{-r\\alpha s}{\\rm d}s  + \ne^{-r\\alpha t}\\int_t^\\infty f(s) {\\rm d}s\\\\\n&=\\bar\\ell(\\alpha)\\int_0^t f(s) e^{-r\\alpha s}{\\rm d}s  + \ne^{-r\\alpha t} (1-F( t)).\n\\end{align*}\nFor instance, in the special case that the time-to-defaults are exponentially distributed with mean $\\lambda^{-1}$, we have\n\\[\\omega_t(\\alpha)=\\left(1-e^{-(\\lambda+r\\alpha)t}\\right) \\frac{\\lambda}{\\lambda+r\\alpha}\\bar\\ell(\\alpha) + e^{-(\\lambda+r\\alpha)t}.\\]\n\n\\subsection{Large-deviations asymptotics}\\label{subsec_LD}\nThe goal of this subsection is to establish a limit theorem for our ruin probability, given that we start with $n$ obligors and an initial capital reserve level $nu>0$, as $n$ grows large. In other words, we analyze how the probability\n\\begin{equation}\\label{qn}q_n(t):=  p_n(nu,t)={\\mathbb P}\\left(\\exists s\\in[0,t]: Z_n(s) \\geqslant nu\\right)= {\\mathbb P}\\left(\\exists s\\in[0,t]:\\sum_{i=1}^n W_i(s) \\geqslant nu\\right)\\end{equation}\nbehaves as $n\\to\\infty.$\nWe do so under the evident `rarity condition' that, for all $t\\geqslant 0$, ${\\mathbb E}Z_n(t)$ is smaller than $nu$, or, equivalently, \n\\[\\sup_{t\\geqslant 0}\\big({\\mathbb P}(T\\leqslant t) \\,{\\mathbb E}L - r\\,{\\mathbb E}\\min \\{T,t\\} \\big) < u,\\]\nwhere we use that ${\\mathbb E}W_1(t)=\\omega'_t(0) = {\\mathbb P}(T\\leqslant t) \\,{\\mathbb E}L - r\\,{\\mathbb E}\\min \\{T,t\\}.$\nWe start by establishing a lower bound. The underlying principle is that the probability of a union of events is bounded from below by the probability of the most likely event among them. This entails that, for any $s\\in[0,t]$ we have that $q_n(t)\\geqslant \\check q_n(s)$, where\n\\[\\check q_n(s):= {\\mathbb P}\\left(\\sum_{i=1}^n W_i(s) \\geqslant nu\\right).\\]\nDefine the Legendre transform pertaining to $W_1(s)$:\n\\[I(s):=\\sup_{\\alpha}\\left(\\alpha u - \\log \\omega_s(\\alpha)\\right).\\]\nBecause of the rarity condition $\\omega'_s(0) <u$ for all $s\\geqslant 0$, we can restrict ourselves to maximizing over $\\alpha>0$ only;\nwe define $\\alpha^\\star(s):=  \\arg\\sup_{\\alpha}\\left(\\alpha u - \\log \\omega_s(\\alpha)\\right).$\nBy Cram\\'er's theorem \\cite{DZ}, we immediately have that, for any $s\\in[0,t]$,\n\\begin{equation}\\label{LBS}\\liminf_{n\\to\\infty}\\frac{1}{n}\\log q_n(t) \\geqslant\\liminf_{n\\to\\infty}\\frac{1}{n}\\log \\check q_n(s) =\n-\\alpha^\\star(s) u + \\log \\omega_s(\\alpha^\\star(s))=-I(s).\\end{equation}\nWe also define \n\\[t^\\star:= \\arg\\inf_{s\\in[0,t]}I(s),\\]\nwhich has the informal interpretation of the most likely time $Z_n(\\cdot)$ exceeds $nu.$\nFrom the fact that the lower bound \\eqref{LBS} applies for any $s\\in[0,t]$, we thus obtain that\n\\[\\liminf_{n\\to\\infty}\\frac{1}{n}\\log q_n(t) \\geqslant-\\inf_{s\\in[0,t]}I(s) = -I(t^\\star).\\]\n\nWe proceed by proving that $-I(t^\\star)$ is also an upper bound on the decay rate of $q_n(t)$. The first step is to realize that ruin occurs at the default time of one of the $n$ obligors. As a consequence, we can rewrite $q_n$ in terms of the union of $n$ events:\n\\[q_n(t)={\\mathbb P}\\left(\\exists j\\in\\{1,\\ldots,n\\}: T_j\\in[0,t],  \\sum_{i=1}^n W_i(T_j) \\geqslant nu\\right),\\]\ninstead of the union of uncountable many events featuring in the representation (\\ref{qn}).\nBy the union bound, we obtain that this probability $q_n(t)$ is majorized by $n\\hat q_n(t)$, where\n\\[\\hat q_n(t):={\\mathbb P}\\left(T_1\\in[0,t],  \\sum_{i=1}^n W_i(T_1) \\geqslant nu\\right).\\]\nAs $n^{-1} \\log n \\to 0$, it suffices to prove that $\\limsup_{n\\to\\infty} n^{-1} \\log \\hat q_n(t)\\leqslant -I(t^\\star).$ To this end,  by conditioning on $T_1$,\n\\[\\hat q_n(t) = \\int_0^t f(s) \\,{\\mathbb P}\\left( \\sum_{i=2}^{n} W_i(s) +L_1 -rs \\geqslant nu\\right) {\\rm d}s.\\]\nThen observe that the $W_i(T_1)$ are dependent, but once conditioned on $T_1=s$ they have become independent.  The next step is to apply the Markov inequality: for any $\\alpha\\geqslant 0$, with $L_1$ being independent from $W_2(s),\\ldots,W_{n}(s)$,\n\\begin{align*}\n{\\mathbb P}\\left(\n \\sum_{i=2}^{n} W_i(s) +L_1 -rs \\geqslant nu\n \\right)&=\n{\\mathbb P}\\left( \\exp\\left({\\alpha \\sum_{i=2}^{n} W_i(s) +\\alpha L_1}\\right) \\geqslant \\exp({\\alpha (nu+rs)})\\right)  \\\\\n&\\leqslant (w_s(\\alpha))^{n-1}\\bar\\ell(\\alpha) \\,e^{-\\alpha(nu+rs)}\n\\leqslant (w_s(\\alpha))^{n-1}\\bar\\ell(\\alpha) \\,e^{-\\alpha(n-1)u}\n.\\end{align*}\nUpon combining the above, we have thus found that for any $\\alpha(\\cdot)\\geqslant 0$,\n\\[\\limsup_{n\\to\\infty}\\frac{1}{n}\\log \\hat q_n(t) \\leqslant \\limsup_{n\\to\\infty}\\frac{1}{n}\\log \n\\int_0^t f(s) \\,(w_s(\\alpha(s)))^{n-1} \\bar\\ell(\\alpha(s))\\,e^{-\\alpha(s)\\,(n-1)u} {\\rm d}s.\\]\nRecall that, for any $t\\geqslant 0$,  $I(t)=\\alpha^\\star(t) u - \\log \\omega_t(\\alpha^\\star(t)).$\nPlugging in $\\alpha(\\cdot)=\\alpha^\\star(\\cdot),$ we thus obtain, in the second inequality using the definition of $t^\\star$,\n\\begin{align}\\nonumber\\limsup_{n\\to\\infty}\\frac{1}{n}\\log \\hat q_n (t)&\\leqslant \\limsup_{n\\to\\infty}\\frac{1}{n}\\log \n\\int_0^t f(s)\\, \\bar\\ell(\\alpha^\\star(s))\\,e^{-(n-1)I(s)} {\\rm d}s\\\\ \\nonumber\n&\\leqslant \\limsup_{n\\to\\infty}\\frac{1}{n}\\log \n\\int_0^t f(s)\\, \\bar\\ell(\\alpha^\\star(s))\\,e^{-(n-1)I(t^\\star)} {\\rm d}s\\\\\n&=-I(t^\\star)+ \\limsup_{n\\to\\infty}\\frac{1}{n}\\log \n\\int_0^t f(s)\\, \\bar\\ell(\\alpha^\\star(s))\\, {\\rm d}s.\\label{laatste}\n\\end{align}\nObserve that we are done if we succeed in proving that the second term in \\eqref{laatste} is $0$, for which it suffices to prove that the integral appearing in this term is finite.\nTo this end, first observe that, with $\\tau(\\alpha):= {\\mathbb E}\\,e^{\\alpha T}$, \n\\[\\lim_{t\\to\\infty} \\omega_t(\\alpha) = \\bar\\ell(\\alpha) \\tau(-r\\alpha)=:\\Delta(\\alpha),\\]\nso that $\\alpha^\\star(\\infty)$ solves $\\Delta'(\\alpha)\/\\Delta(\\alpha) = u$. \n\n\\begin{assumption} \\label{ASS1} The function $\\alpha^\\star(\\cdot)$ is bounded on $[0,t]$.\n\\end{assumption}\n\nUnder Assumption \\ref{ASS1}, we have $\\sup_{s\\in[ 0,t]}\\alpha^\\star(s)\\leqslant M$ for some finite $M$. Note that this holds whenever the function $\\alpha^\\star(\\cdot)$ is continuous, whereas in case $t=\\infty$ we additionally require $\\alpha^\\star(\\infty)<\\infty$. \nWith this assumption in place and using that $\\alpha\\mapsto \\bar\\ell(\\alpha)$ is increasing, we conclude that \n\\[\\int_0^t f(s)\\, \\bar\\ell(\\alpha^\\star(s))\\, {\\rm d}s \\leqslant \\bar\\ell(M) \\int_0^t f(s)\\, {\\rm d}s \\leqslant \\bar\\ell(M)<\\infty.\\]\nSummarizing, we have shown\n\\[\\limsup_{n\\to\\infty}\\frac{1}{n}\\log q_n(t) \\leqslant -I(t^\\star).\\]\nWe have arrived at the following result.\n\\begin{theorem} \\label{TH31} As $n\\to\\infty$, under Assumption $\\ref{ASS1}$, \n\\[\\frac{1}{n}\\log q_n(t) \\to -I(t^\\star).\\]\n\\end{theorem}\n\n\\subsection{Efficient simulation}\\label{subsec_is}\nAs the above theorem only provides us with the logarithmic asymptotics of $q_n$, it is inherently imprecise. For instance, if the true asymptotic shape of $q_n$ is $n^\\alpha \\,\\exp({-nI(t^\\star)})$ for some $\\alpha\\in{\\mathbb R}$, or $\\exp(n^\\eta) \\exp({-nI(t^\\star)})$\nfor some $\\eta\\in (0,1)$, the effect of the $\\alpha$ and $\\eta$ is not visible. \nOne can get accurate estimates in an efficient way, however, applying importance sampling.  Below we present an importance sampling algorithm, which we prove to be logarithmically efficient. \n\nThe key idea is that we decompose our rare-event probability $q_n$ into $n$ rare-event probabilities, which we will be dealing with separately. We write\n\\begin{equation}\\label{sum}q_n(t) = \\sum_{j=1}^n q_{nj}(t),\\end{equation}\nwhere \n\\[q_{nj}(t):={\\mathbb P}\\left({\\mathscr F}_j\\right),\\:\\:\\:\\:\n{\\mathscr F}_j:=\n{\\mathscr E}_j\\cap \\bigcap_{i=1}^{j-1} {\\mathscr E}_i^{\\rm c},\\:\\:\\:\\:\n{\\mathscr E}_j:= \\left\\{T_j\\in[0,t],\n \\sum_{i\\not=j} W_i(T_j) +L_j -rT_j \\geqslant nu\n\\right\\};\\]\nthe validity of \\eqref{sum} is due to the events ${\\mathscr F}_j$ being (by construction) disjoint, while the union of the ${\\mathscr E}_j$ equals the union of the ${\\mathscr F}_j$. The problem of efficiently estimating $q_n(t)$ thus reduces to the problem of efficiently estimating each of the $q_{nj}(t)$ (and adding up all the resulting estimates). \n\nFix a $j\\in\\{1,\\ldots,n\\}$ and focus on the estimation of $q_{nj}$. We now define an importance sampling probability measure ${\\mathbb Q}$.\n\\begin{itemize}\n\\item[$\\circ$]\nUnder ${\\mathbb Q}$ the density of $T_j$ remains $f(\\cdot)$. \n\\item[$\\circ$] Conditionally on $T_j=s$, the moment generating function of $L_j$ becomes\n\\[\\bar\\ell^{\\mathbb Q}(\\alpha) = \\frac{\\bar\\ell(\\alpha+\\alpha^\\star(s)}{\\bar\\ell(\\alpha^\\star(s))}.\n\\]\nSampling $L_j$ from ${\\mathbb Q}$ amounts to sampling from an exponentially twisted version of the actual distribution. This is a standard procedure in applied probability; for many frequently used distributions the twisted distribution remains within the same class of distributions, but with different parameters. For instance, the $\\alpha$-twisted version of an exponentially distributed random variable with parameter $\\mu$ corresponds to an exponentially distributed random variable with parameter $\\mu-\\alpha$ (requiring that $\\alpha\\in[0,\\mu)$).\n\\item[$\\circ$] Conditionally on $T_j=s$, the moment generating function of $W_i(s)$ (for $i\\not= j$) becomes\n\\begin{equation}\\label{alp}\\omega_s^{\\mathbb Q}(\\alpha):= \\frac{\\omega_s(\\alpha+\\alpha^\\star(s))}{\\omega_s(\\alpha^\\star(s))}.\\end{equation}\nTo decide whether the event ${\\mathscr F}_j$ applies, we have to sample the default times $T_i$ and (if $T_i<t$) the losses $L_i$, for $i\\not=j$, in accordance with \\eqref{alp}. This can be done as follows. By (\\ref{alp}), the exponentially twisted version of $W_i(s)$ has the moment generating function\n\\begin{align*}\\omega_s^{\\mathbb Q}(\\alpha)=\n\\frac{1}{\\omega_s(\\alpha^\\star(s))}&\\left(\\int_0^s f(v)\\, e^{-(\\alpha+\\alpha^\\star(s))\\, rv} \\bar\\ell(\\alpha+\\alpha^\\star(s)){\\rm d}v \\:+\\right.\\\\ &\\hspace{3mm}\\left.\\int_s^\\infty f(v)\\, e^{-(\\alpha+\\alpha^\\star(s))\\, rs} {\\rm d}v\\right).\\end{align*}\nFrom this identity we observe that the $L_i$ can be sampled from a distribution with moment generating function $\\bar\\ell^{\\mathbb Q}(\\cdot)$, as defined above, whereas the density $f^{\\mathbb Q}(\\cdot)$ of the $T_i$ (for $i\\not=j$) becomes\n\\[f^{\\mathbb Q}(v) =\\frac{1}{\\omega_s(\\alpha^\\star(s))}f(v)\\left( e^{-\\alpha^\\star(s)\\, rv} \\bar\\ell(\\alpha^\\star(s))1_{\\{v\\leqslant s\\}} +  e^{-\\alpha^\\star(s)\\, rs} 1_{\\{v>s\\}}\\right).\\]\n\\end{itemize}\nWe proceed by detailing the importance-sampling based simulation procedure, and establishing its asymptotic efficiency. \nTo this end, we first observe that a generic sample of the likelihood ratio, say ${\\mathscr L}_j$, has the form\n\\[\n{e^{-\\alpha^\\star(T_j)\\,L_j}}\\cdot{\\bar\\ell(\\alpha^\\star(T_j))}\n\\prod_{i\\not=j} \\left({e^{-\\alpha^\\star(T_j)\\,W_i(T_j)}}\\cdot{\\omega_{T_j}(\\alpha^\\star(T_j))}\\right).\\]\nRecall that on the event ${\\mathscr F}_j$ we have $ \\sum_{i\\not= j} W_i(T_j) +L_j -rT_j \\geqslant nu$. As a consequence, on the event ${\\mathscr F}_j$ the likelihood ratio ${\\mathscr L}_j$ is majorized by\n\\begin{align*}e^{-\\alpha^\\star(T_j)(nu + r T_j)}\\cdot{\\bar\\ell(\\alpha^\\star(T_j))}&\\cdot\\big(\\omega_{T_j}(\\alpha^\\star(T_j))\\big)^{n-1}\\\\\n&\\leqslant e^{-\\alpha^\\star(T_j)(n-1)u}\\cdot{\\bar\\ell(\\alpha^\\star(T_j))}\\cdot\\big(\\omega_{T_j}(\\alpha^\\star(T_j))\\big)^{n-1}\\\\\n&=\\bar\\ell(\\alpha^\\star(T_j))\\,e^{-(n-1)\\,I({T_j})} \\leqslant \\bar\\ell(M)\\,e^{-(n-1)\\,I({T_j})}\\\\\n&\\leqslant  \\bar\\ell(M)\\,e^{-(n-1)\\,I({t^\\star})},\n\\end{align*}\nwith $M$ as defined in Section \\ref{subsec_LD} (where we let Assumption \\ref{ASS1} be in force).\nWe thus find that, with ${\\mathscr I}_j$ denoting the indicator function of ${\\mathscr F}_j$, the almost-sure inequality\n${\\mathscr L}_j\\,{\\mathscr I}_j \\leqslant \\bar\\ell(M)\\,e^{-(n-1)\\,I({t^\\star})}$, and therefore\n\\[\\sum_{j=1}^n {\\mathscr L}_j\\,{\\mathscr I}_j \\leqslant n \\,\\bar\\ell(M)\\,e^{-(n-1)\\,I({t^\\star})}.\\]\nEvidently, to obtain an estimator with good precision, we have to repeat the above experiment sufficiently often. Suppose, for each $j\\in\\{1,\\ldots,n\\}$, we perform $N\\in{\\mathbb N}$ independent trials. The corresponding likelihood ratios are denoted by \n${\\mathscr L}_{j,k}$, and the indicator functions are ${\\mathscr I}_{j,k}$, with $j\\in\\{1,\\ldots,n\\}$ and $k\\in\\{1,\\ldots,N\\}$. \nOur estimator thus becomes  \n\\[\\xi_N:=\\frac{1}{N} \\sum_{k=1}^N \\sum_{j=1}^n {\\mathscr L}_{j,k}\\,{\\mathscr I}_{j,k},\\]\nwhich is (by construction) unbiased.\nThe next step is to analyze the performance of this estimator. To this end, we observe in relation to its second moment that\n\\[{\\mathbb E}_{\\mathbb Q}\\left(\\left(\\sum_{j=1}^n {\\mathscr L}_{j}\\,{\\mathscr I}_{j}\\right)^2\\right) \\leqslant n^2 \\,(\\bar\\ell(M))^2\\,e^{-2(n-1)\\,I({t^\\star})},\\]\nwith ${\\mathbb E}_{\\mathbb Q}(\\cdot)$ denoting expectation under ${\\mathbb Q}$.\nWe find the following upper bound for the second moment: \n\\[\\limsup_{n\\to\\infty}\\frac{1}{n}\\log {\\mathbb E}_{\\mathbb Q}\\left(\\left(\\sum_{j=1}^n {\\mathscr L}_{j}\\,{\\mathscr I}_{j}\\right)^2\\right) \\leqslant -2 I(t^\\star).\\]\nBy Theorem \\ref{TH31}, and in addition using that variances are non-negative, we also have the corresponding lower bound:\n\\begin{align*}\n\\liminf_{n\\to\\infty}\\frac{1}{n}\\log {\\mathbb E}_{\\mathbb Q}\\left(\\left(\\sum_{j=1}^n {\\mathscr L}_{j}\\,{\\mathscr I}_{j}\\right)^2\\right)& \\geqslant \\liminf_{n\\to\\infty}\\frac{2}{n}\\log {\\mathbb E}_{\\mathbb Q}\\left(\\sum_{j=1}^n {\\mathscr L}_{j}\\,{\\mathscr I}_{j}\\right) \\\\&= \\liminf_{n\\to\\infty}\\frac{2}{n}\\log q_n(t) = -2 I(t^\\star).\\end{align*}\nThe above bounds lead to the following conclusion, which in practical terms entails that the number of runs needed to obtain an estimate of a given relative precision, grows sub-exponentially in $n$. For the definition of logarithmic efficiency, and related performance notions in rare-event simulation, we refer to \\cite[Ch. VI]{AG}.\n\\begin{theorem} Under Assumption $\\ref{ASS1}$, the estimator $\\xi_N$ is logarithmically efficient as $N\\to\\infty$.\n\\end{theorem}\n\n\\subsection{Uniform bound} \\label{Unif}\nIntrinsic drawbacks of the large-deviations asymptotics is that they only kick in for large $n$, and they provide us with the decay rate only. This motivates the search for a uniform upper bound on the ruin probability $p_n(u,t)$. The result is a Lundberg-type inequality derived along the same lines was done in \\cite[Section XIII.5a]{AsmussenAlbrecher} for the conventional Cram\\'er-Lundberg model in which claims (or losses in the credit context) arrive according to a fixed-intensity Poisson process. We focus on the situation that when there are $n$ obligors the time to the first default is exponentially distributed with mean $\\lambda_n^{-1}$ and the income rate is $r_n.$ Let $\\gamma_n$ be the positive solution for $\\gamma$ in \n\\[\\bar\\ell(\\gamma) \\frac{\\lambda_n}{\\lambda_n+\\gamma r_n}=1.\\]\n\n\\vspace{3mm}\n\n\\begin{theorem}\\label{th_upper}\nSuppose that $\\gamma_n$ is non-increasing in $n$. Then \n\\[p_n(u,t)\\leqslant p_n(u,\\infty)\\leqslant e^{-\\gamma_n u}.\\]\n\\end{theorem}\n\\begin{proof} It is evident that $p_n(u,t)\\leqslant p_n(u,\\infty)$.\nLet $Y_n$ be distributed as $L-r_n\\,T_1$, where $T_1$ is assumed exponentially distributed with mean $\\lambda_n^{-1}$ (independent of $L$).\nConditioning on $Y_n$ immediately yields\n\\[p_n(u,\\infty)=\\mathbb{P}(Y_{n}>u)+\\int_{-\\infty}^u p_{n-1}(u-y,\\infty){\\mathbb P}({Y_{n}}\\in{\\rm d}y).\\]\nWe claim that this implies $p_n(u,\\infty)\\leqslant e^{-\\gamma_n u}$. The proof is by induction. First note that the claim holds true for $n=0$ as $p_0(u,\\infty)=0$ for all $u> 0$. Assuming the inequality holds true for $n-1$,\\begin{align*}\np_{n}(u,\\infty)&\\leqslant\\mathbb{P}(Y_{n}>u)+\\int_{-\\infty}^u e^{-\\gamma_{n-1} (u-y)}\\,{\\mathbb P}({Y_{n}}\\in{\\rm d}y)\\\\\n&\\leqslant\\mathbb{P}(Y_{n}>u)+\\int_{-\\infty}^u e^{-\\gamma_{n} (u-y)}\\,{\\mathbb P}({Y_{n}}\\in{\\rm d}y)\\\\\n&\\leqslant e^{-\\gamma_{n} u}\\int^{\\infty}_u e^{\\gamma_{n} y}\\,{\\mathbb P}({Y_{n}}\\in{\\rm d}y)+\\int_{-\\infty}^u e^{-\\gamma_{n} (u-x)}\\,{\\mathbb P}({Y_{n}}\\in{\\rm d}y)\\\\\n&=e^{-\\gamma_{n} u}\\,{\\mathbb E}\\,e^{\\gamma_{n} Y_n}=e^{-\\gamma_{n} u}\\,\\bar\\ell(\\gamma_n) \\frac{\\lambda_n}{\\lambda_n+\\gamma_n r_n}=e^{-\\gamma_n u},\n\\end{align*}\nwhere in the second inequality it has been used that that $\\gamma_n$ is non-increasing in $n$.\n\\end{proof}\n\n\\begin{remark}{\\em \nIn the special case the default arrival intensity $\\lambda_n$ and the income rates $r_n$ are linear in the number of obligors $n$, it is readily checked that $\\gamma_n$ does not depend on $n$. As a consequence, also the upper bound derived above does not depend on $n$. $\\hfill\\Diamond$\n}\\end{remark}\n\n\n\n\\section{Non-default losses, Markov modulation, \\\\Brownian perturbations, and multiple groups} \\label{ext}\n\nIn this section we consider four important extensions of our base model.\n\\begin{itemize}\n\\item[$\\circ$] In the first extension there are both losses due to defaults (reducing the number of obligors by one) and losses that do not correspond to defaults (leaving the number of obligors unchanged).\n\\item[$\\circ$] Then we consider a model in which the dynamics are affected by a Markovian background process, thus creating dependence between the individual obligors.\n\\item[$\\circ$] We proceed by analyzing a model in which the cumulative process between jumps behaves as a Brownian motion (rather than being linear).\n\\item[$\\circ$] Finally we discuss an extension that allows heterogeneous obligors (by working with multiple groups). \n\\end{itemize}\nNote that, as opposed to the analysis presented in the previous section, in this section we let the default times be exponentially distributed. In principle, the four generalizations introduced above can be combined, but to keep the presentation as transparent as possible we have decided to discuss them separately.\n\n\n\\subsection{Non-default losses} In this subsection we consider the following extension of the model analyzed in Section \\ref{S3} (or, actually, the more general one featured in Remark \\ref{R1}). Next to losses due to defaults (happening at a Poisson rate $\\lambda_n$ with the losses having Laplace transform $\\ell(\\cdot)$ when $n$ obligors are present) there are losses that do {\\it not} correspond to defaults (happening at a Poisson rate $\\lambda^\\circ_n$ with the losses having Laplace transform $\\ell^\\circ(\\cdot)$ when $n$ obligors are present).\n\n\nWe again start our derivations by conditioning on the first event, being the first default, the first loss (not leading to default), or the expiration of the exponential clock. If a default happens first, then we can still reach ruin, but now with $n-1$ obligors and an adapted initial reserve. In case the first event is a loss which does not correspond to a default, then we can still reach ruin with $n$ obligors but an adapted initial reserve. If the exponential clock expires, then we will not be facing ruin.\n\nThis idea can be formalized as follows. With $L^\\circ$ denoting a generic random variable corresponding with a non-default loss, we obtain the relation\n\\begin{align*}p_n(u) &= \\int_0^\\infty e^{- (\\bar \\lambda_n+\\vartheta) t}\\Big(\n\\lambda_n\\, {\\mathbb P}(Z_{n-1}+L\\geqslant u+r_nt)+ \\lambda^\\circ_n\\,  {\\mathbb P}(Z_n+L^\\circ\\geqslant u+r_nt)\\Big){\\rm d}t.\n\\end{align*}\nGoing through the same type of computations as those relied on in Section \\ref{S3}, we end up with a relation between $\\psi_n(\\cdot)$ and $\\psi_{n-1}(\\cdot)$. More specifically, \nfor any $\\gamma\\geqslant 0$, using the notation $\\bar{\\lambda}_n=\\lambda_n+\\lambda_n^\\circ$, we find that\n\\begin{align}\\nonumber\\psi&_{n}(\\gamma)= \\frac{\\bar\\lambda_n}{\\bar\\lambda_n+\\vartheta}\\frac{1}{\\gamma}\\:+\\\\\\nonumber&\\:\\:\\:\\frac{\\lambda_n}{\\bar\\lambda_n+\\vartheta-\\gamma r_n}\\left(B\\left(\\frac{\\bar\\lambda_n+\\vartheta}{r_n},\\psi_{n-1}\\left(\\frac{\\bar\\lambda_n+\\vartheta}{r_n}\\right)\\right)-B\\left(\\gamma,\\psi_{n-1}(\\gamma)\\right)\\right)\\:+\\\\&\\:\\:\\:\\frac{\\lambda_n^\\circ}{\\bar\\lambda_n+\\vartheta-\\gamma r_n}\\left(B^\\circ\\left(\\frac{\\bar\\lambda_n+\\vartheta}{r_n},\\psi_{n}\\left(\\frac{\\bar\\lambda_n+\\vartheta}{r_n}\\right)\\right)-B^\\circ\\left(\\gamma,\\psi_{n}(\\gamma)\\right)\\right),\\label{psin}\\end{align}\nwhere $B^\\circ(\\cdot,\\cdot)$ is defined as $B(\\cdot,\\cdot)$ but with $\\ell(\\cdot)$ replaced by $\\ell^\\circ(\\cdot)$. \nUnfortunately, this relation between $\\psi_n(\\cdot)$ and $\\psi_{n-1}(\\cdot)$ cannot be directly written in terms of an explicit recursion (as opposed to the model without non-default losses; see Theorem \\ref{TH1}). The $\\psi_n(\\cdot)$, however, can still be found recursively, using the following procedure.\n\nTo this end, we start by defining the (yet unknown) constants\n\\[A_n:=B^\\circ\\left(\\frac{\\bar\\lambda_n+\\vartheta}{r_n},\\psi_{n}\\left(\\frac{\\bar\\lambda_n+\\vartheta}{r_n}\\right)\\right).\\]\nThen, using that $\\psi_0(\\cdot)\\equiv 0$, observe that $\\psi_1(\\gamma)$ obeys \n\\begin{align}\\nonumber\\psi_1(\\gamma) = \\:&\\frac{\\bar\\lambda_1}{\\bar\\lambda_1+\\vartheta}\\frac{1}{\\gamma}+\\frac{\\lambda_1}{\\bar\\lambda_1+\\vartheta-\\gamma r_1}\\left(\\frac{r_1}{\\bar\\lambda_1+\\vartheta}\\ell\\left(\\frac{\\bar\\lambda_1+\\vartheta}{r_1}\\right)-\\frac{\\ell(\\gamma)}{\\gamma}\\right)\\:+\\\\\n&\\:\\:\\:\\frac{\\lambda_1^\\circ}{\\bar\\lambda_1+\\vartheta-\\gamma r_1}\\left(B^\\circ\\left(\\frac{\\bar\\lambda_1+\\vartheta}{r_1},\\psi_{1}\\left(\\frac{\\bar\\lambda_1+\\vartheta}{r_1}\\right)\\right)-B^\\circ\\left(\\gamma,\\psi_{1}(\\gamma)\\right)\\right).\\label{psi1_nd}\\end{align}\nWe can rewrite (\\ref{psi1_nd}), for a known function $F(\\cdot)$, as\n\\[\\psi_1(\\gamma) = F(\\gamma) +\\frac{\\lambda_1^\\circ}{\\bar\\lambda_1+\\vartheta-\\gamma r_1}\\left(A_1-\\ell^\\circ(\\gamma)\\left(\\frac{1}{\\gamma}-\\psi_1(\\gamma)\\right)\\right),\\]\nwhich can be rearranged to\n\\[1-\\gamma \\psi_1(\\gamma) = 1 -\\frac{\\gamma F(\\gamma)(\\bar\\lambda_1+\\vartheta-\\gamma r_1)+\\gamma\\lambda_1^\\circ A_1-\\lambda_1^\\circ\\ell^\\circ(\\gamma)}{\\bar\\lambda_1-\\lambda_1^\\circ\\ell^\\circ(\\gamma)+\\vartheta-\\gamma r_1}.\\]\nAs we know that $1-\\gamma \\psi_1(\\gamma)$ is a Laplace transform, its value should be between 0 and 1 for any $\\gamma\\geqslant 0$.\nHence, any zero of the denominator is necessarily also a zero of the numerator. It is standard to verify that the numerator has a single positive zero, say $\\bar\\gamma$. Then it follows that \n\\[A_1 = \\frac{\\ell^\\circ(\\bar\\gamma)}{\\bar{\\gamma}}-F(\\bar\\gamma)\\frac{\\bar\\lambda_1+\\vartheta-\\bar\\gamma r_1}{\\lambda_1^\\circ}.\\]\nNow that we have found $A_1$ and hence $\\psi_1(\\gamma)$, we can identify $A_2$ and $\\psi_2(\\gamma)$ along the same lines:\nwe first express $\\psi_2(\\gamma)$ in terms of $A_2$ using \\eqref{psin}, and then identify $A_2$ using that the zero of the denominator (which we know to equal $\\bar\\lambda_2 -\\lambda_2^\\circ\\ell^\\circ(\\gamma)+\\vartheta-\\gamma r_2$) is a zero of the numerator as well. Continuing this procedure, all $\\psi_n(\\gamma)$ (and constants $A_n$) can be found. \n\n\n\\subsection{Markov modulation}\nIn the models discussed so far the individual obligors are independent. In reality they may be affected by common external factors, to be thought of as the `state of the economy', and hence behave dependently. In this subsection we consider a model in which a particular dependence structure is incorporated, through the mechanism of Markov modulation (also known as regime-switching). \n\nWe start by describing the model. Let $(J(t))_{t\\geqslant 0}$ be an irreducible continuous-time Markov process living on $\\{1,\\ldots,d\\}$. We denote by $q_{jk}\\geqslant 0$ (for $j\\not=k$) the transition rate from state $j$ to state $k$, and $q_j:=-q_{jj}=\\sum_{k\\not=j} q_{jk}$. Let $r_{nj}$ be the rate at which the surplus process increases when there are $n$ obligors and the background process is in state $j$, let $\\lambda_{nj}$ be the corresponding hazard rate of the time to the next default, and let $\\ell_j(\\cdot)$ be the Laplace transform of the loss (with the associated generic random variable being denoted by $L_j$). \n\nLet $T_n$ be the minimum of the time of the first default and the expiration of an exponential clock of rate $\\vartheta$. Denote by\n\\[R(T_n):=\\int_0^{T_n} r_{nJ(t)}{\\rm d}t\\]\nthe increase of the surplus process till $T_n$. We start by analyzing the distribution of $R(T_n)$ through the object\n\\[F_{i,j,n}(x):= {\\mathbb P}_i(R(T_n)\\geqslant x, J(T_n)=j) := {\\mathbb P}(R(T_n)\\geqslant x, J(T_n)=j\\,|\\, J(0)=i).\\]\nUsing the standard `Markovian reasoning', i.e., by distinguishing between all possible events in a (small) time interval of length $\\Delta$ and using the memory-less property, we obtain the relation, as $\\Delta\\downarrow 0$,\n\\[F_{i,j,n}(x) = \\sum_{k\\not = j}F_{i,k,n}(x)\\,q_{kj}\\Delta + F_{i,j,n}(x-r_j\\Delta)\\big(1- (q_j+\\lambda_{nj}+\\vartheta)\\big)+o(\\Delta).\\]\nSubsequently subtracting $F_{i,j,n}(x-r_j\\Delta)$ from both sides, dividing by $\\Delta$ and taking the limit $\\Delta\\downarrow 0$, we end up with a system of linear differential equations:\n\\[F'_{i,j,n}(x) = \\sum_{k=1}^d F_{i,k,n}(x)\\,q_{kj} + F_{i,j,n}(x)\\,(\\lambda_{nj}+\\vartheta).\\]\nFor given $i$ and $n$, this is a system of $d$ coupled linear differential equations, that can be solved in the standard manner; the resulting structure depends on the multiplicities of the eigenvalues. In the sequel we assume that its solution is such that the corresponding density obeys\n\\[{\\mathbb P}_i(R(T_n)\\in {\\rm d} x, J(T_n)=j) = \\sum_{k=1}^d \\xi_{i,j,k,n} e^{-\\zeta_{k,n}x},\\]\nbut a similar analysis can be done if the terms in the right-hand side of the previous display also involve polynomial factors (as a consequence of the multiplicities of some of the eigenvalues being larger than one). \n\nThe key observation is the identity\n\\begin{align*}{\\mathbb P}_i(Z_n\\geqslant u) &= \\sum_{j=1}^d \\frac{\\lambda_{nj}}{\\lambda_{nj}+\\vartheta}\\int_0^\\infty {\\mathbb P}_j(Z_{n-1}\\in {\\rm d}z) {\\mathbb P}_i(L_j\\geqslant R(T_n)+u-z, J(T_n)=j)\\\\\n&= \\sum_{j=1}^d \\frac{\\lambda_{nj}}{\\lambda_{nj}+\\vartheta}\\int_0^\\infty\\int_0^\\infty {\\mathbb P}_j(Z_{n-1}\\in {\\rm d}z) {\\mathbb P}(L_j\\geqslant x+u-z) \\sum_{k=1}^d \\xi_{i,j,k,n} e^{-\\zeta_{k,n}x}\\,{\\rm d}x\\\\\n&= \\sum_{j=1}^d \\frac{\\lambda_{nj}}{\\lambda_{nj}+\\vartheta}\\int_0^\\infty {\\mathbb P}_j(Z_{n-1}+L_j\\geqslant x+u)\\sum_{k=1}^d \\xi_{i,j,k,n} e^{-\\zeta_{k,n}x}\\,{\\rm d}x\n\\end{align*}\nTherefore, using the by now familiar steps concerning a change-of-variables and swapping the order of integration,\n\\begin{align*}\n\\psi_{ni}(\\gamma)&:=\\int_0^\\infty e^{-\\gamma u} {\\mathbb P}_i(Z_n\\geqslant u)\\,{\\rm d}u\\\\\n&=\\sum_{j=1}^d \\frac{\\lambda_{nj}}{\\lambda_{nj}+\\vartheta}\\int_0^\\infty\\int_0^\\infty e^{-\\gamma u} {\\mathbb P}_j(Z_{n-1}+L_j\\geqslant x+u)\\sum_{k=1}^d \\xi_{i,j,k,n} e^{-\\zeta_{k,n}x}\\,{\\rm d}x\\,{\\rm d}u\\\\\n&=\\sum_{j=1}^d \\frac{\\lambda_{nj}}{\\lambda_{nj}+\\vartheta}\\int_0^\\infty\\int_u^\\infty e^{-\\gamma u} {\\mathbb P}_j(Z_{n-1}+L_j\\geqslant v)\\sum_{k=1}^d \\xi_{i,j,k,n} e^{-\\zeta_{k,n}(v-u)}\\,{\\rm d}v\\,{\\rm d}u\\\\\n&=\\sum_{j=1}^d \\frac{\\lambda_{nj}}{\\lambda_{nj}+\\vartheta}\\int_0^\\infty\\sum_{k=1}^d \\xi_{i,j,k,n}\n\\left(\\int_0^v e^{-\\gamma u} e^{\\zeta_{k,n}u}\n\\,{\\rm d}u\\right){\\mathbb P}_j(Z_{n-1}+L_j\\geqslant v) e^{-\\zeta_{k,n}v}\\,{\\rm d}v\\\\\n&=\\sum_{j=1}^d \\frac{\\lambda_{nj}}{\\lambda_{nj}+\\vartheta}\\int_0^\\infty\\sum_{k=1}^d \\xi_{i,j,k,n}\n\\frac{e^{-\\zeta_{k,n}v}-e^{-\\gamma v}}{\\gamma- \\zeta_{k,n}}\n{\\mathbb P}_j(Z_{n-1}+L_j\\geqslant v)\\,{\\rm d}v.\n\\end{align*}\nFrom now on we can follow the approach presented in Section \\ref{S3}: the last expression in the previous display can be expressed in terms of $\\psi_{n-1,j}(\\cdot)$, for $j=1,\\ldots,d.$ We thus end up with a vector-valued recursion. As the derivation is fully analogous to the one corresponding to the non-modulated case, we omit the details. \n\n\\subsection{Brownian perturbations}\n\nWe proceed by making the model more realistic by allowing the process to evolve, between defaults, as Brownian motion rather than a deterministic drift. The parameters of this Brownian motion depend on the number of obligors that have not gone in default yet, say with drift coefficient $r_i$ and variance coefficient $\\sigma_i^2$ when there are $i$ obligors left. In this section the time between the $i$-th and $(i+1)$-st default is exponentially distributed with mean $\\lambda_i^{-1}$.\n\n\nConsidering a Brownian motion with parameters $r$ and $\\sigma^2$ over an interval with exponentially distributed length with mean $\\lambda^{-1}$, it is known from Wiener-Hopf theory, that \n\\begin{itemize}\n\\item[$\\circ$]\nthe maximum value $M^+$ achieved is exponentially distributed with the\nparameter \n\\[\\nu^+\\equiv \\nu^+(r,\\sigma^2,\\lambda):= \\frac{\\sqrt{r^2+2\\lambda\\sigma^2}}{\\sigma^2}-\\frac{r}{\\sigma^2}.\\]\n\\item[$\\circ$] the (absolute value of the) amount by which the process goes down after the maximum is achieved until the end of the exponentially distributed interval, say $M^-$, is exponentially distributed with the parameter\n\\[\\nu^-\\equiv \\nu^-(r,\\sigma^2,\\lambda):= \\frac{\\sqrt{r^2+2\\lambda\\sigma^2}}{\\sigma^2}+\\frac{r}{\\sigma^2}.\\]\n\\item[$\\circ$] the random variables $M^+$ and $M^-$ are independent. The rates $\\nu^+$ and $\\nu^-$ are the roots of the equation\n$\\lambda+r\\alpha-\\frac{1}{2}\\alpha^2\\sigma^2=0$. \n\\end{itemize}\nNow define $\\nu_n^\\pm:= \\nu^\\pm(-r_n,\\sigma_n^2,\\lambda_n+\\vartheta)$; note that the first parameter is $-r_n$ rather than $r_n$, as we consider the event of the cumulative claim process exceeding the value $u$ (i.e., the reserve level dropping below $0$). \nAs before, we set up a relation between $\\psi_n(\\cdot)$ and $\\psi_{n-1}(\\cdot)$. Realize that, due to the Brownian term, ruin can occur before the exponential clock (with parameter $\\vartheta$) expires; this happens with probability $ e^{-\\nu_n^+ u}$. \nFollowing the approach we have been using in the case without the Brownian term, we thus obtain the relation\n\\[p_n(u) =  e^{-\\nu_n^+ u}  + I_n(u,\\vartheta),\\]\nwhere\n\\begin{align*}I_n(u,\\vartheta)&:=\\int_0^u \\int_0^\\infty\\nu_n^+ e^{-\\nu_n^+ v}\\nu_n^- e^{-\\nu_n^- w}\n\\frac{\\lambda_n}{\\lambda_n+\\vartheta}\\,{\\mathbb P}(Z_{n-1}+L\\geqslant u-v+w)\\,{\\rm d}w\\,{\\rm d}v\\\\\n&=\\frac{\\lambda_n}{\\lambda_n+\\vartheta}\\int_0^u \\int_{u-v}^\\infty\\nu_n^+ e^{-\\nu_n^+ v}\\nu_n^- e^{-\\nu_n^- (z-u+v)}\\,\n{\\mathbb P}(Z_{n-1}+L\\geqslant z)\\,{\\rm d}z\\,{\\rm d}v\n.\\end{align*}\n\nThe next step is to evaluate $\\psi_n(\\gamma)$, by multiplying $p_n(u)$ by $e^{-\\gamma u}$ and integrating over $u\\in[0,\\infty).$ We obtain that, interchanging the order of the integrals such that the `easy' integration (over $u$, that is) can be done first,\n\\begin{align*}\\int_0^\\infty &e^{-\\gamma u} I_n(u,\\vartheta){\\rm d}u \\\\&= \\frac{\\lambda_n}{\\lambda_n+\\vartheta}\n\\int_0^\\infty\\int_0^\\infty\\int_v^{z+v} e^{-\\gamma u} \\,\\nu_n^+ e^{-\\nu_n^+ v}\\nu_n^- e^{-\\nu_n^- (z-u+v)}\n\\,{\\mathbb P}(Z_{n-1}+L\\geqslant z)\\,{\\rm d}u\\,{\\rm d}v\\,{\\rm d}z\\\\\n&=\n \\frac{\\lambda_n}{\\lambda_n+\\vartheta}\n\\int_0^\\infty\\int_0^\\infty {\\nu_n^-}e^{-\\gamma v}\\frac{e^{-\\nu_n^- z}-e^{-\\gamma z}}{\\gamma-\\nu_n^-}\\nu_n^+e^{-\\nu_n^+ v}\n\\,{\\mathbb P}(Z_{n-1}+L\\geqslant z)\\,{\\rm d}v\\,{\\rm d}z\\\\\n&= \\frac{\\lambda_n}{\\lambda_n+\\vartheta} \\frac{\\nu_n^-\\nu_n^+}{(\\gamma-\\nu_n^-)(\\gamma+\\nu_n^+)} \\int_0^\\infty \n\\big(e^{-\\nu_n^- z}-e^{-\\gamma z}\\big) \\,{\\mathbb P}(Z_{n-1}+L\\geqslant z)\\,{\\rm d}z.\n\\end{align*}\nPerforming the same steps as in the proof of Theorem \\ref{TH1}, as before relying on the identities \\eqref{e1} and \\eqref{e2} in combination with the independence of $L$ and $Z_{n-1}$, we find\nafter some standard algebra the following result.\n\n\\begin{theorem} \\label{TH41} For any $\\gamma\\geqslant 0$ and $n\\in{\\mathbb N}$, we have the recursion,\n\\begin{align*}\\psi_n(\\gamma)&=\\frac{1}{\\gamma+\\nu_n^+}+\\frac{\\lambda_n}{\\lambda_n+\\vartheta}\n\\frac{1}{\\gamma+\\nu_n^+}{\\frac{\\nu_n^+}{\\gamma}}\\\\\n&\\hspace{20mm}-\\frac{\\lambda_n}{\\lambda_n+\\vartheta} \\frac{\\nu_n^-\\nu_n^+}{(\\gamma-\\nu_n^-)(\\gamma+\\nu_n^+)}\\big( B(\\nu_n^-,\\psi_{n-1}(\\nu_n^-))-B(\\gamma,\\psi_{n-1}(\\gamma))\\big),\n\\end{align*}\nwhere $\\psi_0(\\gamma)\\equiv 0.$\n\\end{theorem}\n\n\\begin{remark}{\\em\nIn Theorem \\ref{TH41} we can simplify\n\\[\\frac{\\lambda_n}{\\lambda_n+\\vartheta} \\frac{\\nu_n^-\\nu_n^+}{(\\gamma-\\nu_n^-)(\\gamma+\\nu_n^+)}=\\frac{\\lambda_n}{\\lambda_n+\\vartheta+r_n\\gamma-\\frac{1}{2}\\gamma^2\\sigma_n^2},\\]\nusing that $\\nu_n^+$ and $\\nu_n^-$ solve\n$(\\lambda_n+\\vartheta)+r_n\\alpha-\\frac{1}{2}\\alpha^2\\sigma_n^2=0$. $\\hfill\\Diamond$\n}\\end{remark}\n\n\\subsection{Multiple groups}\\label{MG}\n\nTo make the model more realistic, one could work with multiple (heterogeneous) groups of obligors. Suppose there are $G\\in\\mathbb{N}$ groups of obligors with initially $n_j$ obligors in group $j\\in\\{1,\\ldots,G\\}$; write ${\\boldsymbol n}=(n_1,\\ldots, n_G).$ We consider the multi-group counterpart of the base model of Section \\ref{S3}:\neach obligor in group $j$ has a time-to-default that is exponentially distributed with rate $\\lambda_j$. The losses at default per obligor in group $j$ are i.i.d.\\ random variables with Laplace transform $\\ell_j(\\cdot)$; in addition these per-group sequences are assumed independent. The income per unit time for this group is $r_j i$ when there are $i\\in\\{1,\\ldots, n_j\\}$ obligors that have not gone into default yet. \n\nThe company's capital reserve is given by the sum of the reserves of the individual groups; its initial level is $u>0$. Let $\\psi_{{\\boldsymbol n}}(\\gamma)$ denote the double transform of the probability of ruin over an exponentially distributed interval (with, as usual, mean $\\vartheta^{-1}$), given there $n^j$ obligors in group $j$ that have not gone into default yet. Then by the same argumentation as before we find, for ${\\boldsymbol n}$ component-wise at least equal to 1, and with ${\\boldsymbol e}_j$ the $j$-th unit vector,\n\n\\begin{align*}\n\\psi_{\\boldsymbol n}(\\gamma)=&\\sum_{j=1}^G\\frac{\\lambda_j n_j}{\\sum_{k=1}^G\\lambda_k n_k+\\vartheta}\\frac{1}{\\gamma}+\\sum_{j=1}^G\\frac{\\lambda_j n_j}{\\sum_{k=1}^G\\lambda_k n_k +\\vartheta-\\gamma r_j n_j}\\\\\n&\\times\\Bigg(B_j\\left(\\frac{\\lambda_j +\\vartheta\/n_j}{r_j},\\psi_{{\\boldsymbol n}-{\\boldsymbol e}_j}\\left(\\frac{\\lambda_j +\\vartheta\/n_j}{r_j}\\right)\\right)-B_j\\left(\\gamma,\\psi_{{\\boldsymbol n}-{\\boldsymbol e}_j}(\\gamma\\right)\\Bigg),\n\\end{align*}\n\\normalsize\nwhere \\[B_j(x,y):= \\ell_j(x)\\left(\\frac{1}{x}-y\\right).\\] We have thus expressed $\\psi_{\\boldsymbol n}(\\gamma)$ as a linear function of $\\psi_{{\\boldsymbol n}-{\\boldsymbol e}_1}(\\gamma)$ up to $\\psi_{{\\boldsymbol n}-{\\boldsymbol e}_G}(\\gamma)$.\nA similar recursive relation be found if some of the entries of ${\\boldsymbol n}$ equal 0. Given that $\\psi_{{\\boldsymbol 0}}(\\gamma)=0$, with ${\\boldsymbol 0}$ denoting the $G$-dimensional all-zeroes vector, we have thus devised a procedure to identify $\\psi_{\\boldsymbol n}(\\gamma)$.\n\n\n\\begin{remark}{\\em \nThe above model extension with multiple classes offers an important additional flexibility. In the first place, one could cluster the obligors in terms of the loss distributions. Per class this loss can even be deterministic; this is a useful property, as in the credit context the losses of some obligors may be a priori known. In addition, we could work with some classes in which the obligors do not go bankrupt and some classes in which they do. Also, one could work with a class-specific income rate.  $\\hfill\\Diamond$\n}\\end{remark}\n\n\n\n\\section{Numerical experiments}\\label{num}\nIn this section we focus on issues concerning the numerical evaluation of the ruin probability. \nIn the first subsection, we specialize to the case that the losses are exponentially distributed, where some of the quantities that feature in the numerical analysis allow closed-form analysis. In the second subsection, we present a couple of illustrative examples. These in particular quantify the effect of the size of the obligor population.\n\n\\subsection{Exponentially distributed losses}\\label{subsec_exp}\nIn Section \\ref{subsec_analysis} the focus was on finding an expression for the double transform $\\psi_n(\\gamma)$,\nwhich can then be inverted numerically. In Section~\\ref{asy} we presented a couple of other approaches: asymptotics, an efficient importance sampling algorithm, and bounds. In this section we present an alternative technique, namely an iterative procedure that directly provides the ruin probabilities $p_n(u,t)$ themselves.  We consider the model variant in which the default rate and the income rate are $\\lambda_i$ and $r_i$, respectively, during time periods in which there are $i$ obligors left. \n\nAs in Section \\ref{subsec_analysis}, the idea is to condition on the first default.\nWe thus obtain, with $W(\\cdot)$ as introduced in Section \\ref{asy}, the following recursive relation:\n\\begin{align}\\nonumber\np_n(u,t)&=\\int_0^t  \\lambda_n e^{- \\lambda_n s} {\\mathbb P}\\left(\\sup_{0\\leqslant v\\leqslant t-s} \\sum_{i=1}^{n-1}W_i(v)+L\\geqslant u+r_ns\\right)\\,{\\rm d}s\\\\ \\nonumber\n&=\\int_0^t \\lambda_n e^{- \\lambda_ns} \\,{\\rm d}s-\\int_0^t  \\lambda_n e^{- \\lambda_n s} {\\mathbb P}\\left(\\sup_{0\\leqslant v\\leqslant t-s} \\sum_{i=1}^{n-1}W_i(v)+L\\leqslant u+r_ns\\right)\\,{\\rm d}s\\\\\n&=1-e^{-\\lambda_n t}-\\int_0^t\\int_0^{u+r_n s}  \\lambda_n e^{- \\lambda_n s}\\left(1-p_{n-1}(u+r_n s-x,t-s)\\right)\\mathbb{P}(L\\in {\\rm d}x)\\,{\\rm d}s. \\label{ECURS}\n\\end{align}\n\nWhen there is only one obligor left, there is only one scenario leading to ruin: default should take place before the exponential clock (with mean $\\vartheta^{-1}$)  expires and the loss should be sufficiently large. In other words, \n\\begin{align}\n\\nonumber p_1(u,t)&= \\int_0^t\\int^\\infty_{u+r_1s}  \\lambda_1 e^{- \\lambda_1 s} \\mathbb{P}(L\\in {\\rm d}x)\\,{\\rm d}s= \\int_0^t \\lambda_1 e^{- \\lambda_1 s} \\mathbb{P}(L\\geqslant u+r_1s)\\,{\\rm d}s\n\\end{align}\nFrom this point on we focus on the case of exponentially distributed claims with mean $\\mu^{-1}$, i.e., ${\\mathbb P}(L\\geqslant x)=e^{-\\mu x}$. We readily obtain\n\\[p_1(u,t)=\\int_0^t \\lambda_1 e^{- \\lambda_1 s} e^{-\\mu (u+r_1s)}\\,{\\rm d}s=\\frac{\\lambda_1 e^{-\\mu u}}{\\lambda_1+\\mu r_1}\\left(1-e^{-(\\lambda_1+\\mu r_1) t}\\right).\\]\nWe can thus obtain $p_2(u,t)$ applying numerical integration to \\eqref{ECURS} with $n=2$. Continuing along these lines,  $p_n(u,t)$ can be numerically evaluated for higher values of $n$. \n\n\\vspace{3mm}\n\nWe now point out how to evaluate the large-deviations asymptotics that were presented in Section~\\ref{subsec_LD}, in the case of exponentially distributed claims.\nThe moment generating function of $W_1(s)$ is for $\\alpha<\\mu$ given by\n\\[\\omega_s(\\alpha)=\\left(1-e^{-(\\lambda+ r\\alpha)s}\\right) \\frac{\\lambda}{\\lambda+r\\alpha}\\frac{\\mu}{\\mu-\\alpha} + e^{-(\\lambda+r\\alpha)s},\\]\nwhereas for $\\alpha\\geqslant \\mu$ the moment generating function is infinite. We continue by computing the mean net loss corresponding to a single obligor (as a function of time):\n\\begin{align*}m(s)&:={\\mathbb E}W_1(s) = \\frac{1}{\\mu}(1-e^{-\\lambda s}) - r\\int_0^s u\\, \\lambda e^{-\\lambda v}{\\rm d}v\n- rs\\int_s^\\infty  \\lambda e^{-\\lambda v}{\\rm d}v\\\\\n&=\\left(\\frac{1}{\\mu}-\\frac{r}{\\lambda}\\right)(1-e^{-\\lambda s}).\n\\end{align*}\n{In the sequel we will assume $u>m(\\infty)$, or equivalently $\\lambda-r\\mu<\\lambda\\mu u$, to make sure the event under consideration is rare.}\n\nThe Legendre transform pertaining to $W_1(s)$ reads\n\\[I(s):=\\sup_{0<\\alpha<\\mu}\\left(\\alpha u - \\log \\omega_s(\\alpha)\\right);\\]\nwe can rule out $\\alpha\\geqslant \\mu$ as  $\\omega_s(\\alpha)=\\infty$ for these $\\alpha$.\nBecause the first-order condition does not allow an explicit solution, one cannot write $I(s)$ in closed form. \nTwo boundary cases can be dealt with explicitly, though. It is first observed that, denoting by $\\omega'_{s,1}(\\alpha)$ the derivative of $\\omega_s(\\alpha)$ with respect to $\\alpha$, and by $\\omega'_{s,2}(\\alpha)$ the derivative of $\\omega_s(\\alpha)$ with respect to $s$,\n\\begin{align}\\nonumber I'(s)& = \\frac{\\rm d}{{\\rm d}s} \\left(\\alpha^\\star(s) u - \\log \\omega_s(\\alpha^\\star(s))\\right)\\\\&=\\frac{{\\rm d}\\alpha^\\star(s)}{{\\rm d}s}\\left(u- \\frac{\\omega'_{s,1}(\\alpha^\\star(s))}{\\omega_s(\\alpha^\\star(s))}\\right)- \\frac{\\omega'_{s,2}(\\alpha^\\star(s))}{\\omega_s(\\alpha^\\star(s))}=- \\frac{\\omega'_{s,2}(\\alpha^\\star(s))}{\\omega_s(\\alpha^\\star(s))},\\label{IS}\\end{align}\nwhere the last equality is due to the definition of $\\alpha^\\star(s)$. By an elementary computation,\n\\begin{equation}\\label{afge}\\omega'_{s,2}(\\alpha)= \\left(\\frac{\\lambda\\mu}{\\mu-\\alpha}-(\\lambda+r\\alpha)\\right)e^{-(\\lambda +r\\alpha)s}=\n\\frac{r\\alpha^2+\\lambda\\alpha -r\\mu\\alpha}{\\mu-\\alpha}\\,e^{-(\\lambda +r\\alpha)s}.\\end{equation}\nWe observe that the Legendre transform $I(s)$ is decreasing in $s$ whenever $\\alpha^*(s)>\\mu-{\\lambda}\/{r}$.\n\\begin{itemize}\n\\item[$\\circ$] For $s=0$, we immediately see that $\\omega_0(\\alpha) = 1$ for all $\\alpha$, so that $\\alpha^\\star(0)=\\mu$ and $I(0) = \\mu u.$ In addition, we obtain by some straightforward algebra that\n\\[I'(0) = - \\lim_{\\alpha\\uparrow \\mu} \\frac{\\omega'_{0,2}(\\alpha)}{\\omega_0(\\alpha)}=-\\infty.\\]\n\\item[$\\circ$] For $s=\\infty$,\n\\[I(s) = \\sup_{0<\\alpha<\\mu} \\kappa(\\alpha),\\:\\:\\:\\:\\kappa(\\alpha):=\\alpha u - \\log (\\lambda\\mu) +\\log(\\lambda+r\\alpha)+\\log(\\mu-\\alpha).\\]\nObserve that $\\kappa(\\cdot)$ is concave, with $\\kappa'(0)>0$ (under the assumption $u>m(\\infty)$) and $\\kappa(\\alpha)\\to-\\infty$ as $\\alpha\\uparrow\\mu.$ In other words, $\\kappa(\\cdot)$ attains a maximum in $(0,\\mu).$\nThe first order condition, determining $\\alpha^\\star(\\infty)$, is\n\\[u=\\frac{1}{\\mu-\\alpha}-\\frac{r}{\\lambda+r\\alpha},\\]\nor equivalently\n\\[ru\\alpha^2 +\\big((\\lambda-r\\mu )u+2r\\big)\\alpha - \\lambda\\mu\\big(u-m(\\infty)\\big)=0.\\]\nAs $\\lambda\\mu (u-m(\\infty))>0$, this equation has a positive and negative root. \nConsequently, $\\alpha^\\star(\\infty)$ is the positive root, i.e.,\n\\[\\alpha^\\star(\\infty) = \\frac{-2r-\\lambda u+r\\mu  u +\\sqrt{4r^2+\\lambda^2 u^2+2r\\lambda\\mu  u^2 + r^2\\mu^2  u^2}}{2ru},\\]\nso that $I(\\infty) = \\kappa(\\alpha^\\star(\\infty)).$\nNext, we want to find the sign  of $I(s)$ in the regime that $s\\to\\infty$. Based on \\eqref{IS} and \\eqref{afge}, this is the sign of \n$-r\\alpha^\\star(\\infty)-\\lambda+r\\mu$. Using the explicit solution of $\\alpha^\\star(\\infty)$, it requires some straightforward calculus to verify that this leads to a negative sign, i.e. $I(s)$ is decreasing in the regime that $s\\to\\infty$, if and only if $ \\lambda- r\\mu>-\\lambda\\mu u$. \n\\end{itemize}\n\n\\subsection{Numerical example}\nFor the numerical results we have used a setup that aligns with the one considered in \\cite{Asmussen1984}. \n\\begin{enumerate}\n\\item[$\\circ$] We consider the case that both the income rates $r_i$ and the default intensity $\\lambda_i$ are linear in the number of obligors $i$ that have not gone into default yet. We let the proportionality constants be $r=  1$ and $\\lambda=0.9$, respectively. In other words, when there are $i$ obligors in the system that have not gone into default yet, the income rate is given by $i$ and the default intensity rate by $0.9\\,i$.\n\\item[$\\circ$] The losses are exponentially distributed with parameter $\\mu=1$.\n\\end{enumerate}\nWith these parameter settings the rarity condition $m(\\infty)<u$ is satisfied for all $u>0$, as we have that $0.9-1=-0.1<0<0.9\\,u$.\n\nFirst, we focus on the evaluation of the large-deviation asymptotics. For $s\\rightarrow \\infty$ we have that the Legendre transform $I(s)$ is decreasing (increasing) if $u>\\frac{1}{9}$ (if $u<\\frac{1}{9}$, respectively). For illustrational purposes we have plotted the functions $\\alpha^\\star(s)$ and $I(s)$ in Figure \\ref{fig1}, as a function of time $s$, for $u=5$ as well as $u=0.1$.\n In the first instance, with $u=5$, the function $I(\\cdot)$ is decreasing, so that the optimal $t^\\star=\\infty$, whereas for $u=0.1$ we see that $I(\\cdot)$ attains a minimal value at $t^\\star=2.3$. \n\n\\begin{figure}\n  \\centering\n \n\\resizebox{8cm}{6cm}{\n \\pgfplotstableread{Data.txt}{\\table}\n    \\begin{tikzpicture}\n        \\begin{axis}[\n            xmin = 0, xmax = 5,\n            ymin = 2.75, ymax = 4.75,\n            xtick distance = 1,\n            ytick distance = 0.25,\n            grid = both,\n            minor tick num = 1,\n            major grid style = {lightgray},\n            minor grid style = {lightgray!25},\n            width = \\textwidth,\n            height = 0.75\\textwidth,\n            legend cell align = {left},\n            legend pos = north west\n        ]\n            \\addplot[blue, mark = *, mark size = 0.3pt] table [x = {x}, y = {b}] {\\table};\n                       \\legend{$I(s)$}\n        \\end{axis}\n    \\end{tikzpicture}}\n  \\resizebox{8cm}{6cm}{\n \\pgfplotstableread{Data.txt}{\\table}\n    \\begin{tikzpicture}\n        \\begin{axis}[\n            xmin = 0, xmax = 5,\n            ymin = 0.75, ymax = 1,\n            xtick distance = 1,\n            ytick distance = 0.05,\n            grid = both,\n            minor tick num = 1,\n            major grid style = {lightgray},\n            minor grid style = {lightgray!25},\n            width = \\textwidth,\n            height = 0.75\\textwidth,\n            legend cell align = {left},\n            legend pos = north west\n        ]\n            \\addplot[blue, mark = *,mark size = 0.3pt] table [x = {x}, y = {a}] {\\table};\n         \n                     \\legend{\n                $\\alpha^\\star(s)$}\n        \\end{axis}\n    \\end{tikzpicture}\n}\n\n\n\\resizebox{8cm}{6cm}{\n \\pgfplotstableread{Data.txt}{\\table}\n    \\begin{tikzpicture}\n        \\begin{axis}[\n            xmin = 0, xmax = 5,\n            ymin = 0.0098, ymax = 0.0107,\n            xtick distance = 1,\n            ytick distance = 0.00025,\n            grid = both,\n            minor tick num = 1,\n            major grid style = {lightgray},\n            minor grid style = {lightgray!25},\n            width = \\textwidth,\n            height = 0.75\\textwidth,\n            legend cell align = {left},\n            legend pos = north west\n        ]\n            \\addplot[blue, mark = *, mark size = 0.3pt] table [x = {x}, y = {d}] {\\table};\n          \n            \\legend{$I(s)$}\n        \\end{axis}\n    \\end{tikzpicture}}\n    \\resizebox{8cm}{6cm}{\n \\pgfplotstableread{Data.txt}{\\table}\n    \\begin{tikzpicture}\n        \\begin{axis}[\n            xmin = 0, xmax = 5,\n            ymin = 0, ymax = 0.75,\n            xtick distance = 1,\n            ytick distance = 0.25,\n            grid = both,\n            minor tick num = 1,\n            major grid style = {lightgray},\n            minor grid style = {lightgray!25},\n            width = \\textwidth,\n            height = 0.75\\textwidth,\n            legend cell align = {left},\n            legend pos = north west\n        ]\n            \\addplot[blue, mark = *, mark size = 0.3pt] table [x = {x}, y = {c}] {\\table};\n          \n            \\legend{\n                $\\alpha^\\star(s)$}\n        \\end{axis}\n    \\end{tikzpicture}}\n\n\n  \\caption{The Legendre transform $I(s)$ and the underlying optimal $\\alpha^\\star(s)$ parameter as a function of time $s$ (for $s\\in[0,5]$). In the top panels we took  for $u=5$, whereas in the bottom panels we took $u=0.1$.}\\label{fig1}\n\\end{figure}\n\n\\vspace{3mm}\n\nIn Figure \\ref{fig_Exact_n10} we present, for different values of the initial number of obligors $n$ and $u=5$,  the ruin probabilities as a function of time. This has been done relying on the iterative approach presented of Section \\ref{subsec_exp}. The double integral involved has been evaluated analytically for $n=1,2$ while numerical integration methods have been employed for $n>2$. We do observe that the ruin probability increases in the length of the time interval, as desired. \nThe upper bound (as derived in Section \\ref{Unif}) in this instance is given by 0.6065, and is independent of the number of obligors $n$. As can be observed, this upper bound is rather conservative, in particular  when there are only a few obligors in the system.\n\n\n\\begin{figure}\\resizebox{12.8cm}{9cm}{\n \\pgfplotstableread{Data2.txt}{\\table}\\pgfplotsset{scaled y ticks=false}\n    \\begin{tikzpicture}\n        \\begin{axis}[\n            xmin = 0, xmax = 10,\n            ymin = 0, ymax = 0.15,\n            xtick distance = 1,\n            ytick distance = 0.01,\n            grid = both,\n            minor tick num = 1,\n            major grid style = {lightgray},\n            minor grid style = {lightgray!25},\n            width = \\textwidth,\n            height = 0.75\\textwidth,\n            legend cell align = {left},\n            yticklabel style={\n        \/pgf\/number format\/fixed,\n        \/pgf\/number format\/precision=5\n},\nscaled y ticks=false,\n            legend pos = north west]\n                       \\addplot[blue, mark = *, mark size = 0.3pt] table [x = {x}, y = {a}] {\\table};\n             \\addplot[blue, mark = *, mark size = 0.3pt] table [x = {x}, y = {b}] {\\table};\n              \\addplot[blue, mark = *, mark size = 0.3pt] table [x = {x}, y = {c}] {\\table};\n               \\addplot[blue, mark = *, mark size = 0.3pt] table [x = {x}, y = {d}] {\\table};\n                \\addplot[blue, mark = *, mark size = 0.3pt] table [x = {x}, y = {e}] {\\table};\n                 \\addplot[blue, mark = *, mark size = 0.3pt] table [x = {x}, y = {f}] {\\table};\n                  \\addplot[blue, mark = *, mark size = 0.3pt] table [x = {x}, y = {g}] {\\table};\n                   \\addplot[blue, mark = *, mark size = 0.3pt] table [x = {x}, y = {h}] {\\table};\n                    \\addplot[blue, mark = *, mark size = 0.3pt] table [x = {x}, y = {i}] {\\table};\n                     \\addplot[blue, mark = *, mark size = 0.3pt] table [x = {x}, y = {j}] {\\table};\n          \n           \n        \\end{axis}\n         \\end{tikzpicture}}\n\\caption{Ruin probabilities over time: $p_n(u,t)$ as a function of $t$, for $n=1$ (bottom line) to $n=10$ (top line), with $u=5$.}\n\\label{fig_Exact_n10}\n\\end{figure}\n\nIn a next experiment we study the performance of the importance sampling technique that was presented Section \\ref{subsec_is}. The top panel of Figure \\ref{fig3}  shows, for the initial capital reserve $u$ being equal to $5$, the estimates of the ruin probability as a function of time, obtained by \nsimulation, using our importance sampling algorithm. The values nearly coincide with what is obtained \nby applying the na\\\"{i}ve, direct simulation approach (i.e., without a change of measure); \nfrom Figure \\ref{fig_Exact_n10} we in addition observe that there is a highly accurate match with the values computed using the iterative approach of Section \\ref{subsec_exp}. \nRegarding the  importance sampling simulations  it is noted that we let the events ${\\mathscr E}_j$ correspond to the event where the net cumulative loss process exceeds the initial level $u$ (instead of $nu$), as $u$ in this example corresponds to the {\\it unscaled} initial capital level. The fact that we have used as many as $10^6$ runs guarantees estimates with a high precision. The importance sampling based approach substantially outperforms direct simulation, in that it greatly reduces the variance of the estimator, as can be observed in the bottom panels of Figure \\ref{fig3}.\n\n\n\\begin{figure}\n  \\centering\n\\begin{center}\n\\resizebox{11cm}{7cm}{\n     \\pgfplotstableread{Data3.txt}{\\table}\n    \\begin{tikzpicture}    \n        \\begin{axis}[\n            xmin = 0, xmax = 5,\n            ymin = 0, ymax = 0.045,\n            xtick distance = 1,\n            ytick distance = 0.005,\n            grid = both,\n            minor tick num = 1,\n            major grid style = {lightgray},\n            minor grid style = {lightgray!25},\n            width = \\textwidth,\n            height = 0.75\\textwidth,\n            legend cell align = {left},\n            legend pos = north west\n        ]\n            \\addplot[blue, mark = *, mark size = 0.3pt] table [x = {x}, y = {regular simulation n=1}] {\\table};\n             \\addplot[purple, mark = *, mark size = 0.3pt] table [x = {x}, y = {regular simulation n=2}] {\\table};\n              \\addplot[black, mark = *, mark size = 0.3pt] table [x = {x}, y = {regular simulation n=3}] {\\table};\n               \\addplot[red, mark = *, mark size = 0.3pt] table [x = {x}, y = {regular simulation n=4}] {\\table};\n                      \n                       \\legend{\n                $n=1$,\n                $n=2$,\n                $n=3$,\n                $n=4$}   \n           \n        \\end{axis}\n         \\end{tikzpicture}}\\end{center}\n\n\n\\resizebox{8cm}{4.99cm}{\n     \\pgfplotstableread{Data3.txt}{\\table}\n    \\begin{tikzpicture}\n        \\begin{axis}[\n            xmin = 0, xmax = 5,\n            ymin = 0, ymax = 0.175,\n            xtick distance = 1,\n            ytick distance = 0.025,\n            grid = both,\n            minor tick num = 1,\n            major grid style = {lightgray},\n            minor grid style = {lightgray!25},\n            width = \\textwidth,\n            height = 0.75\\textwidth,\n            legend cell align = {left},\n            legend pos = north west\n        ]\n            \\addplot[blue, mark = *, mark size = 0.3pt] table [x = {x}, y = {regular variance n=1}] {\\table};\n             \\addplot[purple, mark = *, mark size = 0.3pt] table [x = {x}, y = {regular variance n=2}] {\\table};\n              \\addplot[black, mark = *, mark size = 0.3pt] table [x = {x}, y = {regular variance n=3}] {\\table};\n               \\addplot[red, mark = *, mark size = 0.3pt] table [x = {x}, y = {regular variance n=4}] {\\table};\n \n\n \n   \n                      \n                       \\legend{\n                $n=1$,\n                $n=2$,\n                $n=3$,\n                $n=4$}   \n           \n        \\end{axis}\n         \\end{tikzpicture}}\n         \\resizebox{8cm}{4.99cm}{\n     \\pgfplotstableread{Data3.txt}{\\table}\n    \\begin{tikzpicture}\n        \\begin{axis}[\n            xmin = 0, xmax = 5,\n            ymin = 0, ymax = 0.175,\n            xtick distance = 1,\n            ytick distance = 0.025,\n            grid = both,\n            minor tick num = 1,\n            major grid style = {lightgray},\n            minor grid style = {lightgray!25},\n            width = \\textwidth,\n            height = 0.75\\textwidth,\n            legend cell align = {left},\n            legend pos = north west\n        ]\n     \n       \n         \n            \n                \\addplot[blue, mark = *, mark size = 0.3pt] table [x = {x}, y = {is variance n=1}] {\\table};\n             \\addplot[purple, mark = *, mark size = 0.3pt] table [x = {x}, y = {is variance n=2}] {\\table};\n              \\addplot[black, mark = *, mark size = 0.3pt] table [x = {x}, y = {is variance n=3}] {\\table};\n               \\addplot[red, mark = *, mark size = 0.3pt] table [x = {x}, y = {is variance n=4}] {\\table};\n                      \n                       \\legend{\n                $n=1$,\n                $n=2$,\n                $n=3$,\n                $n=4$}   \n           \n        \\end{axis}\n         \\end{tikzpicture}}\n\n\\caption{Top panel: ruin probabilities, as simulated by importance sampling: $p_n(u,t)$ as a function of time $t$. Bottom left panel: variance of the estimator under direct simulation as a function of $t$. Bottom right panel: variance of the estimator under importance sampling as a function of $t$. In all experiments we took $u=5.$}\\label{fig3}\n\\end{figure}\n\n\\section{Concluding remarks}\nMotivated by applications in credit risk,  we have analyzed in this paper a transient counterpart of the classical Cram\\'er-Lundberg model. We have presented a broad range of results: exact analysis in terms of transforms, asymptotic analysis including an efficient rare-event simulation algorithm, and four model variants (viz.\\ a setup that also includes non-default losses, one with Markov modulation to make the obligors dependent, one in which the linear drifts are replaced by Brownian motions, and a last one in which there are multiple groups of obligors). \n\nFollow-up research could relate to the next steps to make this model operational. A main challenge concerns dealing with the heterogeneity between the obligors. When there are relatively few groups (with homogeneity within these groups) the approach of Section \\ref{MG} can be relied upon, but when effectively all obligors have a specific time-to-default and loss distribution, an alternative approach needs to be developed. Another topic for future research could concern procedures to on-the-fly adjust the capital level given realizations of the defaults; cf.\\ e.g.\\ the approach proposed in\\cite{DMSW}. \n\n\n\\bibliographystyle{plain}\n{\\small ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nPolaritons in semiconductor microcavities are light-matter bosonic quasi-particles formed by strong coupling of cavity photons and intra-cavity excitons ~\\cite{deng2010}. Their excitonic part gives rise to strong interactions essential for fast thermalization and condensation, while their photonic part contributes to their very low effective mass ($~5 \\times 10^{-5}m_{e}$) allowing for high temperature condensation ~\\cite{roomtemperature}. Polariton condensates have been observed both under non-resonant optical excitation~\\cite{BEC2006} and more recently under electrical injection of carriers~\\cite{schneider_2013,solid_2013}. However, polaritons populate a two dimensional plane where a true Bose phase transition is theoretically possible only in the presence of a confining potential~\\cite{berman_theory_2008} and this was first demonstrated with a stress induced trap~\\cite{balili_bose-einstein_2007}. Unlike the weak atom-atom interactions in cold atomic Bose Einstein condensates (BEC), inter-particle interactions in a semiconductor microcavity are strong enough to substantially renormalise polariton self-energy, experimentally observed as a local blue-shift of the polariton spectrum. Variations of the polariton density in the plane of the cavity result in a potential landscape that can be externally controlled through real space modulation of the optical excitation beam. The malleability of the potential landscape can be used to imprint scattering centres~\\cite{sanvitto_all-optical_2011} and devise polariton traps~\\cite{Ring,chiral2014} and gates~\\cite{gao_polariton_2012}. The dynamics of polariton condensates in externally modulated potential landscapes can lead to trapped states, standing polariton waves and phase-locking of remote condensates in non-trivial configurations~\\cite{Ring,manni_spontaneous_2011,opticalSPT2013,rotating_2014,kalevich,ohadi_dissipative_2014}. Extensive control over mesoscopic polariton wavefunctions and their transitions between quantum states, coupled with the extensive propagation of polaritonic flows~\\cite{sanvitto_all-optical_2011,schmutzler_all-optical_2015}, bares applications in quantum control, quantum circuits and on-chip quantum information processing~\\cite{YamamotoReview_2014,qubits_2014}.\n\n\nIn this letter, we investigate the dynamics of pure quantum state transitions of polariton condensates under optical confinement. We utilise a ring-shaped, non-resonant optical excitation scheme to create a size-tunable annular potential trap. Under continuous wave excitation, we study the steady-state regime of trapping and condensate formation. We control the height of the potential trap by tuning the optical excitation density and observe that at coherence threshold, polaritons coalesce preferentially at the uppermost confined energy state that has the largest wavefunction overlap to the exciton reservoir that forms the trap barriers. To confirm that excited state polariton condensates are realised predominantly by polariton confinement in the optically induced potential trap, we study the transient dynamics of the formation mechanism. For this purpose, we change from continuous wave to pulsed excitation, while keeping all other parameters unaltered, and time-resolve the evolution of the spatial polariton state. Under pulsed excitation, the height of the potential barrier is transiently diminishing following the decay of the exciton reservoir. We observe that the mesoscopic polariton condensate switches between states, progressively transforming to the highest available confined energy state. The experimental observations are accurately reproduced using the extended Gross-Pitaevski equation. \n\n\n\\begin{figure*}\n\t\\includegraphics[scale=0.35]{fig1.pdf}\n\t\\centering\n\t\\caption{False color-scale experimental \\textbf{(a)-(e)} and theoretical \\textbf{(f)-(j)} states of polariton condensates.\n\t\\textbf{(a),(f)} $\\Psi_{00}$, \\textbf{(b),(g)} $\\Psi_{11}$, \\textbf{(c),(h)} $\\Psi_{01}$, \n\t\\textbf{(d),(i)} $\\Psi_{02}$, \\textbf{(e),(j)} $\\Psi_{03}$. \n\t$\\epsilon$ denotes the ellipticity of individual configurations.}\n\t\\label{fig:modes}\n\t\\vspace{-0.4cm}\n\\end{figure*}\n\nNon-ground state condensates of spatially-confined polaritons were previously observed in optical defect sites and in pillar microcavities, under Gaussian-shaped non-resonant optical excitation incident to the confinement area ~\\cite{sanvitto2009,maragkou,nardin_2010}. While gain competition in thermodynamic equilibrium has been predicted to give rise to occupation of a single or several excited states~\\cite{eastham2008,portolan2008}, in both cases, excited state condensates were shown to be driven by the dynamics of energy relaxation across the confined energy states resulting in multi-state condensation. In the case of ring-shaped excitation, two characteristically different regimes of polariton condensates have been realised. For ring radii comparable to the thermal de-Broglie wavelength a phase-locked standing-wave condensate co-localised to the excitation area was observed ~\\cite{manni_spontaneous_2011}. For ring radii comparable to the polariton propagation length in the plane of the cavity, the excitation ring acted as a potential barrier and a Gaussian-shaped ground state polariton condensate was realized ~\\cite{Ring}. \nChristofolini and co-workers examined the transition between phase-locked and trapped condensates using multiple-excitation spots and a ring-shaped excitation pattern~\\cite{opticalSPT2013}. Despite earlier work by Manni et al~\\cite{manni_spontaneous_2011}, the authors claimed that for ring-shaped pumps, no phase-locked state is geometrically possible, and that when the spacing between the pumps reduces, the trapped condensate collapses into a Gaussian-shaped ground state. Here, we show that under ring-shaped excitation, the formation of excited state condensates is driven by polariton confinement in the linear potentials and that the presence of non-ground polariton condensates does not necessitate asymmetries in the shape and\/or power distribution of the ring excitation. The dependence of the state selection on the height of the trap's barrier and shape at threshold, provides a robust platform for engineering switches of mesoscopic multi-particle coherent states.\n\n\n\nThe experimental configuration that produces an annular beam of zero angular momenta consists of a double axicon arrangement. A variable telescope is used to control the radii of the excitation beam that we project on the sample. The excitation and detection configuration and the microcavity sample is described in ref.~\\cite{Ring}. The microcavity is held in a cold finger cryostat operating at 6 K. We study the steady-state dynamics under non-resonant excitation at \\unit[752]{nm} using a single mode quasi-continuous wave (CW) laser (2\\% on-off ratio at 10kHz). The microcavity used in these experiments is a high Q factor ($>$ 15000) $5\\lambda\/2$ GaAs\/AlGaAs microcavity with 4 triplets of \\unit[10]{nm} GaAs quantum wells, with a Rabi splitting of \\unit[9]{meV} and a cavity lifetime of \\unit[7]{ps}, as described in ref.~\\cite{tsotsis_lasing_2012}. All experiments were performed for a small negative detuning range of $\\unit[-7]{meV}\\leq$ d $\\leq\\unit[-5]{meV}$.\n\nFigures \\fig{fig:modes}a-e, show the spatial profile of mesoscopic wavefunctions for a range of excitation radii and asymmetries, characterised by the ellipticity and radius of the excitation ring, at the coherence threshold that defines the depth of the trap via the interactions in the reservoir. Theses states resemble the TEM modes of a harmonic oscillator and in what follows we will adapt their symbolism to annotate the state of the polariton wavefunction. For an excitation ring with a radius of $\\sim$\\unit[10]{$\\mu$m} we observe a ground-state polariton condensate (\\Fig{fig:modes}a), as in ref.~\\cite{Ring}, which remains in the ground-state as long as the long axis of the asymmetric excitation does not exceed $\\sim$\\unit[10]{$\\mu$m}. For larger excitation ring radius ($\\sim$\\unit[17]{$\\mu$m}) and similar ellipticity as in \\Fig{fig:modes}a ($\\epsilon=0.22$) at coherence threshold we observe that polaritons coalesce at a higher excited state ($\\psi_{11}$) as shown in \\Fig{fig:modes}b. We note that the symmetry of the excited state wavefunction is robust to small asymmetries in the excitation ring ($0<\\epsilon<0.23$) and the transition from ground to non-ground polariton condensates is predominantly dependent on the radius of the ring. Increasing the ring radius and the asymmetry of the excitation it is possible to observe excited state polariton condensates as shown in \\Fig{fig:modes}c-e. On top of each panel we have annotated the ellipticity of the excitation ring. Interferometric measurements of excited states $\\psi_{01},\\psi_{02},\\psi_{03}$ confirm that these are coherent mesoscopic wavefunctions of extended condensates (\\Fig{int}a-c). \n\n\\begin{figure}[ht]\n\t\\includegraphics[scale=0.45]{fig2.pdf}\n\t\\centering \\vspace{-0.25cm}\n\t\\caption{\\textbf{Interference patterns of trapped polariton condensates: (a)} $\\Psi_{01}$, \\textbf{(b)} $\\Psi_{02}$, \\textbf{(c)} $\\Psi_{03}$. The interference patterns where obtained with a retro-reflector configuration.}\n\t\\vspace{-0.2cm}\n\t\\label{int}\n\\end{figure}\n\n\n\nWe investigate the dependence of the quantum state selectivity on the barrier height by varying the non-resonant excitation density of a geometrically fixed, ring-shaped, asymmetric excitation profile. We use an excitation ring of radius $\\sim$16 $\\mu m$ and $\\epsilon=0.27$ that at coherence threshold produces the $\\Psi_{04}$ polariton state as shown in Fig.\\ref{Ptran}a. By increasing the excitation density above the coherence threshold, while keeping all other parameters the same, we observe the transition from $\\Psi_{04}$ to $\\Psi_{05}$ (Fig. \\ref{Ptran}b). The order of the latter state is clearly revealed in Fig.\\ref{Ptran}c, where we plot the normalised spatial profiles along the white dashed lines of the real space intensity images of Fig.\\ref{Ptran}a,b. Fig.\\ref{Ptran}c shows the presence of an extra node at the higher excitation density indicative of $\\Psi_{05}$. In Fig. \\ref{Ptran}d we plot the energy shift of the condensate in the transition from $\\Psi_{04}$ to $\\Psi_{05}$ with respect to its energy at the coherence threshold ($\\Delta(E_P-E_{P_{th}})$). A sharp increase of the energy shift ($\\sim 45\\mu eV$) is observed in Fig.\\ref{Ptran}d at $P\\sim 1.12 P_{th}$. Within the grey stripe intensity fluctuations of the excitation beam artificially blur the two states. The top panels in Fig.\\ref{Ptran}a,b depict the calculated energy levels for the trap shape and the corresponding probability density of the confined states. In both panels, the red-filled probability density corresponds to the occupied state. It is worth noting here the greater overlap of the probability density of the highest energy level ($\\Psi_{04}$ in \\ref{Ptran}(a) and $\\Psi_{05}$ \\ref{Ptran}(b)) with the reservoir compared to the lowest energy levels. Evidently, with increasing the barrier height a polariton condensate is realised at the next confined energy level as a pure-quantum-state that can be singularly described by the principal quantum number $n$ $(\\Psi_{0,n+1})$.  \n\n\\begin{figure}[ht]\n\t\\includegraphics[scale=0.25]{fig3.pdf}\n\t\\centering \n\t\\caption{\\textbf{Evolution of $\\Psi_{04}$ for increasing excitation density. Bottom panel: (a)}  $\\Psi_{04}$ at P=1.03$P_{th}$. \\textbf{(b)} Subsequent increase of the power results in the appearance of $\\Psi_{05}$. Top panel: Schematic representation of the confined energy states for two different barrier heights. \\textbf{(c)} Profiles of the wavefunction for different excitation densities extracted along the dashed white lines at \\textbf{(a)} and \\textbf{(b)}. \\textbf{(d)} Corresponding energy difference with respect to the energy at coherence threshold for increasing excitation power normalised at the coherence threshold power $P_{th}$. Inset in \\textbf{(d)} shows the spectra of the points denoted by the arrows}\n\t\\label{Ptran}\n\t\\vspace{-0.25cm}\n\\end{figure}\n\nWe explore the robustness of the formation of pure-quantum-states on density fluctuations in the exciton reservoir, by extending our study from the excitation density dependent switching between successive states in the dynamic equilibrium regime, to transitions in the time domain under non-resonant pulsed excitation. We use a ring-shaped non-resonant $200$ femtosecond pulse at \\unit[755]{nm} with $\\sim$\\unit[11]{$\\mu$m} radius of the major axis and $\\epsilon= 0.3$ at $\\sim$1.6$P_{th}$. We record the spatio-temporal dynamics of the emission and observe the formation of the $\\Psi_{01}$ polariton state and its transition to $\\Psi_{00}$~\\cite{supp}. We set the transition point to define the zero time frame for the rest of our analysis. Figure \\ref{Ttran}a shows a snapshot of the $\\Psi_{01}$ state at \\unit[-30]{ps}. At later times, the two lobes of the $\\Psi_{01}$ state appear to move closer together and the condensate rapidly transforms to the ground polariton state ($\\Psi_{00}$) of Fig.\\ref{Ttran}b. The decrease of the density in the  barriers in the time-domain results in a shallower trap in which the $\\Psi_{01}$ state is no longer confined, leading to a polariton condensate at the next available state, here the ground state $\\Psi_{00}$. We spectrally and time resolve the decay of emission at normal incidence with an angular width corresponding to $|k|\\leq 1.4 \\mu$m and observe a sharp energy shift from $\\Psi_{01}$ to $\\Psi_{00}$ as shown in Figure \\ref{Ttran}c. This dynamic transition further illustrates that under optical confinement a polariton condensate spontaneously occurs at a higher confined state as defined by the barrier height of the trap and that the transition to the ground state is hindered solely by the existence of higher energy levels. \n\\begin{figure}[ht]\n\t\\includegraphics[scale=0.29]{fig4.pdf}\n\t\\centering \\vspace{-0.35cm}\n\t\\caption{\\textbf{False colour-scale real space tomographic frames in the time domain. (a)} $\\Psi{01}$ state at -30ps and \\textbf{(b)} subsequent \n\t\ttransition to $\\Psi_{00}$ at 30ps. \\textbf{(c)} Intensity normalised time evolution of the emission energy showing the characteristic energy jump upon the transition threshold.}\n\t\\label{Ttran}\n\t\\vspace{-0.25cm}\n\\end{figure}\n\n\n\\begin{figure*}\n\t\\vspace{-0.1cm}\n\t\\includegraphics[scale=0.29]{fig5.pdf}\n\t\\centering \\vspace{-0.15cm}\n\t\\caption{ Polariton dispersion at -30ps a), 0ps, b) and 30ps c). White dotted lines in a) denote the integrated area from which Fig.\\fig{Ttran}c was extracted. d) Schematic representation of the momentum acquired by polaritons tunnelling outside of the potential trap (potential and dispersion energy not in scale). e) Intensity normalised time evolution of $k_x$ showing the characteristic $\\Delta k_x$ jump of the tunnelling mode.}\n\t\\label{tunel}\n\t\\vspace{-0.25cm}\n\\end{figure*}\n\nThe time resolved dispersion images from which the energy evolution of the system was extracted (Fig.\\fig{Ttran}c) are presented in Fig.\\fig{tunel}a-c.\nThe appearance of the $\\Psi_{01}$ mode is accompanied by a distinct doublet mode in the dispersion (Fig.~\\fig{tunel}a), which corresponds to the counter-propagating components of the standing wave~\\cite{rotating_2014}. As the barrier dynamically decays and the $\\Psi_{00}$ mode is switched on as previously discussed, it quickly overtakes $\\Psi_{01}$ in intensity at $\\sim$\\unit{0}{ps}. The first excited state quickly dissipates after this point with the polariton lifetime and the dispersion is dominated by the emission of the trap ground state. Interestingly, Fig.\\fig{tunel}a-c also reveal distinct satellite modes at the same energy of the confined modes but for greater in plane wave-vector. For quantum states in traps with a finite barrier width, coherent tunnelling modes are a characteristic feature. Moreover, in our system these modes will be accelerated by the potential landscape outside the trap eventually acquiring momentum characteristic of the difference between the energy level in the trap and of the low-density polariton dispersion of the system outside the excitation region (Fig.\\fig{tunel}d). From this description it becomes clear that the tunnelling modes are expected to be at the same energy but with higher momentum, as observed in Fig.\\fig{tunel}a-c. \n\nIntegrating the time-dispersion images over energy, while intensity normalising for every time frame, we compile the time evolution of k$_x$ (Fig.\\fig{tunel}e). This analysis reveals the expected $\\Delta$k$_x$ difference of the tunnelling modes of the two states. Intuitively, the relative (to the trapped state) intensity of the $\\Psi_{01}$ tunnelling mode at the transition is substantial, as the width of the barrier goes to zero at this energy level. In contrast to the tunnelling amplitude of the ground state that is effectively suppressed as the potential width at the $\\Psi_{01}$ energy level is still significant. Nevertheless, following the dynamic dissipation of the barrier, due to the decay of particles as well as draining of the reservoir by the condensate, we observe a continuous increase of the relative intensity of the tunnelling amplitude of the ground state at k$_x\\sim \\unit{1.4}{\\mu m ^{-1}}$. The observation of a strong tunnelling component from the E$_{01}$ energy just before the transition, verifies that the barrier width for this level is indeed minimal and that $\\Psi_{01}$ is close to the rim of the trap barrier, further corroborating our interpretation.\n\n\nThe system can be theoretically modelled with a non-linear Schr\\\"{o}dinger equation, namely the Gross-Pitaevski equation.\nSimulations with the Gross-Pitaevski equation with a potential similar to the one from the experimental measurements in our system \nqualitatively reproduce the states recorded experimentally. Using a potential $V(r)$ that consists of the exciton-exciton interactions in \nthe reservoir, that blue-shift the polariton energy levels, and of the polariton-polariton interactions in the condensate,the Hamiltonian of the system is: \n\\begin{IEEEeqnarray}{l}\nH(r) = T+V(r) \\\\\nV(r)  =  V_{r}(r)+V_{c}(r)  \\\\\n V_{r}(r)=N_{r}U_{ex-ex} f_{r}(r)\n\\end{IEEEeqnarray}\nwhere $N_{r}$ is the density of excitons in the reservoir, $U_{ex-ex}$ the exciton-exciton interaction \nstrength, $f_r(\\mathbf{r})$ is the spatial distribution of the exciton \nreservoir taking into account exciton diffusion beyond the pump spot and $V_c=U_{pol-pol}|\\psi_n(\\mathbf{r})|^2$ where $U_{pol-pol}$ the polariton-polariton interaction strength and $\\psi(\\mathbf{r})$ the condensate wavefunction. In addition to kinetic and potential energy terms in the above Hamiltonian, to account for \npolariton spatial dynamics, a generalization of the extended Gross-Pitaevskii equation is required to include incoherent pumping and \ndecay~\\cite{Wouters2007}. In continuous wave experiments one expects the excitation of a steady state of hot excitons with the spatial profile set by the optical pumping extended by exciton diffusion. One can then make use of the Landau-Ginzburg approach for describing the dynamics of the 2D polariton wavefunction~\\cite{Keeling2008}:\n\\begin{align}\ni\\hbar\\frac{d\\psi(\\mathbf{r},t)}{dt}&=\\left[-\\frac{\\hbar^2\\hat{\\nabla}^2}{2m_P}+\\left(U_{pol-pol}-i\\Gamma_\\mathrm{NL}\\right)|\\psi(\\mathbf{r},t)|^2\\right.\\notag\\\\\n&\\hspace{10mm}\\left.+\\left(U_{pol-ex}+ir\\right)N_rf_r(\\mathbf{r})-\\frac{i\\Gamma}{2}\\right]\\psi(\\mathbf{r},t)\\notag\\\\\n&\\hspace{10mm}+i\\hbar\\mathfrak{R}\\left[\\psi(\\mathbf{r},t)\\right].\\label{eq:GP}\n\\end{align}\nHere $m_P$ is the polariton effective mass and $f_r(\\mathbf{r})$ describes the 2D spatial distribution of $N_r$ excitons. The condensation rate $r$ describes the gain of polaritons in the presence of the exciton reservoir. The polaritons experience both a linear decay $\\Gamma$ and non-linear loss $\\Gamma_{NL}$, which represents the scattering of polaritons out of the condensate when its density is high~\\cite{Keeling2008}. The final term in Eq.~\\ref{eq:GP} represents a phenomenological energy relaxation~\\cite{Wouters2012} in the system, which can play an important role when non-ground state polaritons interact with a potential gradient~\\cite{Wouters2010,Wertz2012,Anton2012}:\n\\begin{equation}\n\\mathfrak{R}[\\psi(x,t)]=-\\lambda N_rf_r(\\mathbf{r})\\left(\\hat{E}_\\mathrm{LP}-\\mu(\\mathbf{r},t)\\right)\\psi(x,t).\\label{eq:relax}\n\\end{equation}\nwhere $\\lambda$ determines the strength of energy relaxation~\\cite{Wouters2012,Wertz2012} and $\\mu(\\mathbf{r},t)$ is a local effective chemical potential that conserves the polariton population~\\cite{Wouters2012}. Kinetic energy relaxation of this form was derived with a variety of methods~\\cite{Solnyshkov2014,Sieberer2014} and offers a simple model for the qualitative description of our experiment. We note however that this model does not distinguish between different mechanisms of energy relaxation, which may have different power dependences~\\cite{Haug2014}.\n\n\n\nFixing $N_rf_r(\\mathbf{r})$ to represent a ring shaped excitation (with slight asymmetry), the numerical solution of Eq.~\\ref{eq:GP} gives the steady state intensity profiles shown in \\Fig{fig:modes}f-j. Different configurations are accessed by varying the spatial distribution ($f_r(\\mathbf{r})$) and population ($N_r$) of hot excitons, as in the experiment~\\cite{TimV}. The simulations support that excited state condensation occurs preferentially at the uppermost confined energy state. \n\n\nAlthough it cannot be explicitly verified that there is no available state in the trap above the condensate energy level, since the polariton potential landscape is not directly measurable, the evidence presented from the steady state switching, the transient study including the dynamic behaviour of the tunnelling components of the system, as well as the theoretical simulations and the calculations for the condensate reservoir overlap~\\cite{supp} strongly supports our interpretation that polaritons condense in the highest available energy state within the optical trap.\n\nIn conclusion, we have investigated the dynamics of polariton condensates under optical confinement and observed that, in contrast to previously reported excited state condensation in defect traps and pillar structures, injection of polaritons from the trap barriers leads to the formation of a pure quantum-confined state with a mesoscopic coherent wavefunction above condensation threshold. This behaviour is in agreement with theoretical expectations for a true Bose condensate that is anticipated to resist multi-mode behaviour~\\cite{combescot_stability_2008,nelsen_dissipationless_2013} in the presence of inter-particle interactions.  Moreover we revealed that the state selectivity of this system strongly depends on the geometric properties of the trap and have demonstrated a highly controllable switching between successive mesoscopic coherent quantum confined states, in the dynamic equilibrium regime and in the time domain. These results highlight the capability of tailoring and manipulating on-chip pure-quantum-states in semiconductor microcavities that can facilitate the implementation of polariton bosonic cascade lasers~\\cite{liew_proposal_2013}. Taking into account that the extensive propagation~\\cite{nelsen_dissipationless_2013} as well as the susceptibility of the polaritonic flow to the potential landscape~\\cite{nguyen_realization_2013} has been widely demonstrated, these results also indicate the potential of engineering confined condensate lattices, coupled by their respective tunnelling amplitudes. Moreover the coupling strength in this architecture can be finely tuned by controlling the barrier height enabling the emergence of applications such as many-body quantum circuitry and quantum simulators. \n\n\n\n\nP. S. acknowledges funding from Greek GSRT programm APOLLO. A. A. acknowledges useful discussions with W. Langbein and S. Portolan.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\tIn recent years, the hope of cheap, solution-based solar cells with efficiencies above 20\\% has fueled the scientific research in halide perovskites (HaPs).\n    \\cite{David1_1, David1_2, David1_3, David1_4, David1_5, David1_6, David1_7}    \n    HaPs exhibit many interesting optoelectronic properties that make them ideal candidates for future semiconductor based applications, yet some of their structural and dynamic properties remain of central scientific interest since they also present major stumbling blocks in the development of stable devices. Fundamentally, HaPs can crystallize in the typical perovskite structure $ABX_3$, where the $B$-site is a heavy metallic element such as lead, and halide ions ($X$) form octahedra centered around the $B$-site. \n    \\cite{David2}\n    In hybrid organic-inorganic HaPs, the voids between the inorganic scaffolds are occupied by the organic $A$-site (in this work methylammonium, MA). As is typical for perovskites, HaPs undergo phase transitions with changes in temperature. The prototypical variant $\\mathrm{MAPbI_3}${} has an orthorhombic symmetry up to 162~K, where it changes to a tetragonal structure. The second phase transition occurs at 327~K, where it switches to a cubic crystal structure.\n    \\cite{Poglitsch87, Stoumpos13}\n    Importantly, HaPs are compared to other well established semiconductors\n    \\cite{David4}\n    and perovskite materials very soft mechanically,\n    \\cite{Feng14, David5_2, Rakita15, David5_4, David5_5, David5_6}\n    which is interesting fundamentally and could become problematic when they are exposed to the operating conditions of commercially used solar cells. Furthermore, experimental and theoretical studies have also shown that the structure and binding in HaPs can be modified already by applying relatively low pressure. \n    \\cite{Capitani16, David6_2, David6_3}\nTherefore, it is interesting to study the interrelation of binding and structural modifications in HaPs, which can be induced by a change in temperature or applied pressure, and may have important consequences for their optoelectronic properties.\n    \\newline\n    Such insight can be generated from first-principles based computations. However, from a theoretical point of view there are a number of physical effects and properties, all of which could play a role regarding the structure of and binding in HaPs. First, hybrid HaPs consist of highly polarizable atoms such as iodine and lead and contain hydrogen atoms bound to electronegative species (e.g. nitrogen); hence, from simple chemical arguments dispersive interactions, such as van-der-Waals (vdW) and hydrogen bonding, can be expected to play an important role in HaPs. {Several computational studies have shown that dispersive interactions are crucial in static calculations of the lattice constants and mechanical properties of HaPs.} \n    \\cite{Feng14, David7_2, Egger14, David7_4, David7_5,Faghihnasiri17, David5_5, Motta16, Egger18}\n    However, as to how their treatment for HaPs should best be implemented in density functional theory (DFT), the most widely used method for HaPs, is still an open question.\\cite{Kronik14, Hermann17} {In particular, a recent study has concluded that correcting for dispersive interactions in calculations of HaPs does not improve their description.}\n    \\cite{Bokdam17}\nSecond, Kohn-Sham DFT functionals, such as the frequently used generalized gradient approximation (GGA), cannot accurately describe the band gap of semiconductors even in principle,\n    \\cite{David8_1, David8_2}\n    which can be improved by applying computationally more costly hybrid functionals in the generalized Kohn-Sham framework.\n    \\cite{David9,Zhang11}\nFor classical inorganic semiconductors, screened hybrid functionals were shown to improve also the description of lattice constants compared to GGA-based approaches.\n\t\\cite{David10_1, David10_2, David10_3}\n    The question that naturally arises then is whether a similar improvement is found when using a hybrid functional for calculating the structure of HaPs, {as has been suggested recently.}\\cite{Bokdam17}\n    Third, it is well established that spin-orbit coupling (SOC) due to the presence of the lead atom in $\\mathrm{MAPbI_3}${} leads to strong modifications in the electronic structures, i.e., it lifts the degeneracy of the conduction and valence band and lowers the band gap.\n    \\cite{David11,Whalley17}\n    Lastly, since HaPs are mechanically soft, they exhibit large structural dynamical effects, such as massive ionic displacements and molecular rotation around room temperature,\n    \\cite{David12_1, Egger16}\n    which is the relevant scenario for device applications. Therefore, it is important to investigate the consequences of these unusual structural dynamical effects on the pertinent structural properties of HaPs. {To the best of our knowledge, the impact of finite temperature on the lattice constants and mechanical properties, and the role of dispersive interactions in calculating them, has not been addressed by means of DFT-based molecular dynamics (MD).}\n    \\newline\n    In this study, we first investigate how the choice of vdW correction and DFT functional affects the results of calculations for structure and binding in the prototype $\\mathrm{MAPbI_3}$. To this end, we compare data obtained in static DFT calculations for a primitive cubic unit cell to experimental ones, using various computational approaches. The most promising choice of method is then used to explore how different MA orientations impact structural properties in static and {finite-temperature MD calculations} of $\\mathrm{MAPbI_3}$.    \n    \n    \n\\section{Methods and Computational Setup}\n\n\tA satisfactory treatment of dispersion forces within conventional approximate Kohn-Sham DFT functionals requires dispersive correction schemes. In our calculations, we considered the Tkatchenko-Scheffler method with regular Hirshfeld partitioning (TS)\n    \\cite{Tkatchenko09}\n    and with an iterative Hirshfeld partitioning (HI)\n    \\cite{Bucko14, Bucko13}\nas well as the many-body dispersion (MBD) method.\n\t\\cite{David15, Ambrosetti14, Bucko16}\nThe TS and HI methods include pairwise dispersive interactions between atoms in the crystal. The HI scheme expands the Hirshfeld partitioning with an iterative process that results in a more realistic allocation of charges for strongly ionic systems.\n\\cite{Bucko14}\nTo include the screening of dispersive interactions that occurs in a many-body system, we apply the MBD method based on the random phase expression of the correlation energy, as developed by Tkatchenko et al.\n\t\\cite{David15}\n\\\\    \n    For the comparison of results obtained by using a GGA functional and a hybrid functional, we apply two very common variants in the context of solid-state calculations, namely the PBE\n    \\cite{Perdew96}\n    and HSE functionals.\n    \\cite{Heyd03, David18_2}\n    In the HSE functional, the short-range exchange energy is calculated as a mix of PBE and exact exchange,\n    \\cite{Heyd03, David18_2}\n   \n    which was often found to improve the description of electronic-structure and structural properties of semiconductors. \n    \\cite{David20_1, David10_1, David10_2, David10_3}\n    SOC describes the interaction of the spin angular momentum with the orbital momentum, and it plays a crucial role in systems with heavy elements such as lead. In our calculations, it was described {fully self-consistently} within the framework of non-collinear magnetism,\n    \\cite{David21}\n    in conjunction with the various DFT methods applied here.    \n \n    In order to obtain structural and mechanical parameters, we use an equation of state that accurately describes the macroscopic properties of the crystal. One of the most frequently used ones is the Birch-Murnaghan equation of state (BMEOS).\n    \\cite{Birch,Murnaghan,Fu83}\n    It describes the free energy of a solid as a function of the volume of the unit cell, $E(V)$, at isothermal conditions as:\n    \\begin{eqnarray}\n    \tE(V) = E_0 + \\frac{B_0V}{B^\\prime_0} \\left( \\frac{(V_0\/V)^{B^\\prime_0}}{B^\\prime_0 - 1} +1 \\right) - \\frac{V_0B_0}{B^\\prime_0 -1}.\\label{BMEOS}\n        \\end{eqnarray}\n    $E_0$ is the free energy at the equilibrium volume, $V_0$, and $B_0$ and $B^\\prime_0$ are the bulk modulus and its derivative, respectively. In order to obtain these parameters, we first calculate the free energy of the static system at eight equidistant volumes between 90\\% and 111\\% of the experimental value\n    \\cite{Stoumpos13}\n \n    and fit Eq.~\\ref{BMEOS} using these eight values (tests with more data points did not show any significant improvement) with the method of least squares. Note that while the $E_0$ and  $B^\\prime_0$ are required to fit Eq.~\\ref{BMEOS}, they are not relevant for our discussion.\n  \n  \\begin{figure}\n\t\\includegraphics[width=\\linewidth]{combined}\n    \\caption{\\label{combined} Schematic structural representations of the configurations of $\\mathrm{MAPbI_3}${} that were used in the calculations: a) Primitive unit cell (red solid line) and a 2x2x2 supercell (black dashed line) used in static calculations; note that we have considered various orientations of MA, see text for details. b) Visualization of the structural changes in molecular dynamics (MD) calculation of a 2x2x2 supercell, which was obtained as an overlay of five structures separated by 200~fs along the 20~ps DFT-MD trajectory. Shown are carbon (brown), nitrogen (light blue), hydrogen (white), iodine (violet), and lead (gray) atoms. Atoms belonging to more than the computational cell are displayed for visual clarity.}\n\\end{figure}\n  \n    In our calculations, we first considered the cubic primitive unit cell of $\\mathrm{MAPbI_3}${} as has been reported experimentally, see Fig.~\\ref{combined}a.\n    \\cite{Stoumpos13}\n    To be able to also investigate the effect of MA orientation, we used a supercell that consists of eight unit-cells stacked together in a 2x2x2 pattern (see Fig.~\\ref{combined}a), with the MA ions orientated in parallel or anti-parallel to each other. {In this way, we can test the effect of the assumption that MA molecules are oriented perfectly in parallel throughout the material on computing the structural and binding properties of $\\mathrm{MAPbI_3}${}.} Note that in these calculations, the angle between the MA molecules was fixed during the atomic relaxation. To fit Eq.~\\ref{BMEOS} with the data obtained in the MD calculations, the average free energy along a trajectory of 20~ps was calculated for each volume point, which was used, together with the standard deviation as the uncertainty, in the fit of Eq. \\ref{BMEOS}. We note that since the fit errors of $B^\\prime$ were quite large in the case of the MD calculations, in fitting Eq.~\\ref{BMEOS} we set $B^\\prime$ to the value obtained in the static calculation of the unit cell.\n    \n    \n\t\\begin{table*}\n    \t\\caption{\\label{static_unitcell} Volume of the primitive unit cell, $V_0$, and bulk modulus, $B_0$, obtained by fitting the DFT-calculated data with Eq.~\\ref{BMEOS}, using various methods applied to the primitive unit-cell of $\\mathrm{MAPbI_3}$.}\n    \t\\begin{ruledtabular}\n    \t\t\\begin{tabular}{lccccccccccc}\n\t\t\t\t& PBE & PBE+TS & PBE+MBD & PBE+HI & HSE & HSE+TS & HSE+MBD & HSE+HI & PBE+SOC & PBE+TS+SOC & Experiment \\\\\n                ine\n                $V_0$ [\\AA$^3$] \\footnote{Errors are between 0.1 and 0.4~\\AA$^3$} & 272.9 & 256.2 & 257.1 & 262.0 & 266.6 & 252.9 & 252.1 & 258.1 & 274.4 & 256.3 & 247-253 \\cite{Stoumpos13,Baikie13,Feng14}\\\\\n                $B_0$ [GPa] \\footnote{Errors are between 0.2 and 0.4~GPa}& 10.6 & 15.7 & 13.6 & 14.3 & 11.1 & 16.4 & 14.7 & 14.2 & 9.6 & 15.0 & 12-16 \\cite{Ferreira18,Rakita15}\\\\\n           \t\\end{tabular}\n    \t\\end{ruledtabular}\n    \\end{table*}\n    \n    The DFT calculations were performed with a plane-wave basis (cutoff energy: 400 eV) and the projector-augmented wave (PAW) method,\n    \\cite{David24}\n    as implemented in the VASP code.\n    \\cite{David25}\n    For unit cell calculations, a 6x6x6 grid of k-points centered around the $\\Gamma$-point was used, for static and dynamic calculations that considered the supercell, the grid was reduced to a sampling of 3x3x3. For each static calculation, the ionic coordinates were relaxed, using the PBE functional and the respective dispersive correction scheme, with the Gadget tool in internal coordinates,\n    \\cite{David26}\n    before calculating the energies to fit Eq.~\\ref{BMEOS}. In the calculations applying the HSE functional or SOC, the respective PBE-based geometry was used to reduce computational costs. For all static calculations, the geometry was considered relaxed when the forces acting  one the atoms were below 10~meV per \\AA. In the canonical (NVT) MD simulations, a $\\mathrm{MAPbI_3}${} 2x2x2 supercell with randomly orientated MA was used as a starting point for the structural geometry, {in line with the recommendations provided by Lahnsteiner et al.} \\cite{Lahnsteiner16\n    It was then equilibrated for 5~ps and computed for 20~ps in time steps of 1~fs at a temperature of 400~K. {This protocol is sufficient to compute the impact of the dynamic nuclear effects on the structure and binding despite the limited supercell size} \\cite{Lahnsteiner16}{ and trajectory length. The latter was checked explicitly by adding another 10~ps to the simulation, which resulted in only insignificant changes of the calculated observables.} Schematic representations of the structures were generated with VESTA. \\cite{vesta}\n            \n   \n    \n\\section{Results}\n\n    \nFig.~\\ref{vdw_comp} shows the energy change due to volume change, i.e., $\\Delta E(V) = E(V)- E_0$, for the static calculations of the $\\mathrm{MAPbI_3}${} unit cell, calculated by using the PBE and HSE functionals augmented by different dispersion corrections. The optimized volume ($V_0$) and bulk modulus ($B_0$), obtained from the fit, are listed in Table~\\ref{static_unitcell}. Considering the PBE data, it can readily be seen that using dispersive corrections has a large impact on the result. Furthermore, it was found that the results from the TS and MBD methods are very similar, and provide a $V_0$ of 256.2~\\AA$^3$ (PBE+TS) and 257.1~\\AA $^3$ (PBE+MBD) which are close to the experimental range of 247-253~\\AA$^3$.\nIn contrast, the bare PBE calculations are far off (272.9~\\AA$^3$) the experimental range, in agreement with literature findings,\n \\cite{Feng14, David7_2, Egger14, David7_4, David7_5,Faghihnasiri17}\nand the PBE+HI (262.0~\\AA$^3$) results basically lie in between the results of bare PBE and PBE+TS\/PBE+MBD. The calculated $B_0$ data show similar trends, i.e., the experimentally reported values (12-16~GPa)\nagree well with the PBE+TS, PBE+MBD, and PBE+HI results, but deviate more from the bare PBE value. From these data, one can see that the TS dispersive correction scheme with a regular Hirshfeld partitioning and the MBD approach perform best in reproducing structural and binding properties measured in experiments. {It should be noted that the experimental data were recorded at elevated temperatures, at least at $T\\approx$330~K, while the calculations were performed at 0~K and do not address thermal expansion, an effect we discuss below based on our results obtained from finite-temperature MD calculations.}\n\n    Fig.~\\ref{vdw_comp} also shows the results for regular and TS-corrected calculations using the HSE functional (see Table~\\ref{static_unitcell} for full dataset using the other correction schemes). While there is some visible difference between the PBE and HSE calculations without dispersive corrections, the bare HSE value for $V_0$ is still quite off the experimental result, in stark contrast to findings reported for conventional inorganic semiconductors.\n\\cite{David10_1, David10_2, David10_3}    \n    Considering the dispersion-corrected HSE data, it is found that these compare almost equally well to experiment as their PBE counterparts. Importantly, the optimized $V_0$ value using the PBE+TS, PBE+MBD, HSE+TS, and HSE+MBD are all within $< 5$~\\AA$^3$, which is smaller than the range of experimentally reported values. Hence, we find that the improvement due to using the HSE functional is minor compared to using dispersive corrections, which is thus found to be the essential computational ingredient for obtaining accurate structural and binding properties of $\\mathrm{MAPbI_3}$. From these findings, we argue that using PBE+TS provides a reasonable choice for calculating structural and binding properties of HaPs such as $\\mathrm{MAPbI_3}$. For completeness, we note that calculations including SOC showed very little difference to those without, as can be seen in Table~\\ref{static_unitcell}.\n  \n  \n    \\begin{figure}\n    \t\\includegraphics[width=1.0\\linewidth]{unit_cell_comp}\n        \\caption{\\label{vdw_comp} Energy change as a function of unit-cell volume, $\\Delta E(V) = E(V)- E_0$, for the static calculations of the primitive $\\mathrm{MAPbI_3}${} unit cell, using the PBE and HSE functionals augmented by different dispersion correction schemes. Also shown are fits to Eq.~\\ref{BMEOS} and the range of experimental results for the volumes (green-shaded area). It is noted that a larger slope in the fit corresponds to a larger value of $B_0$.}\n    \\end{figure}\n  \n    It is well-known that the electronic interaction between MA and the inorganic ions in $\\mathrm{MAPbI_3}${} is minor, since the electronic states of MA are energetically not close to the band edges. Nevertheless, electrostatic and dispersive interactions\nbetween the inorganic ions and MA could still be important for the structure and binding of $\\mathrm{MAPbI_3}$. In order to test this, we varied the MA orientation in a 2x2x2 supercell, considering the extreme cases of either perfectly parallel or antiparallel MA molecules. {It is noted that the former scenario is equivalent to considering a primitive cubic unit cell of $\\mathrm{MAPbI_3}${} containing one MA unit.} When calculating the structural parameters of the 2x2x2 supercell with different MA orientations, we limit the calculations to the bare PBE and the PBE+TS approach, since neither using HSE nor including SOC has shown a significant improvement that would justify the increase in computational cost {(see above and Table}~\\ref{static_unitcell}). Furthermore, using the TS and MBD dispersive correction scheme provided essentially equal results, but the computational costs associated with the MBD method increase more rapidly when the number of atoms becomes larger compared to the TS scheme.\n   \n   The first relevant observation in the data shown in Table~\\ref{static_supercell} is that the results of the supercell calculations considering the parallel MA orientation are, within the fitting error, identical to the results obtained with the calculations for the primitive unit cell, as expected. However, a noticeable difference occurs in both the structure (see Fig.~\\ref{supercell_MA}) and the optimized parameters (see Table~\\ref{static_supercell}) when the MA orientation is changed to the other extreme of perfectly antiparallel MA molecules. In the case of the parallel MA orientation, at all volumes the octahdra retain cubic symmetry, whereas in the antiparallel orientation they tilt strongly (see Fig.~\\ref{supercell_MA}). Indeed, this effect is important, since for the fit of the supercell data calculated with bare PBE we find that $V_0\/8$ changes from 272.7\\AA$^3$, for the parallel orientation, to 265.8\\AA$^3$, for the antiparallel one. The same trend is confirmed in the PBE+TS calculations, although the differences are smaller. Hence, the interaction between the inorganic cage and the MA molecules is important for the structure and binding in $\\mathrm{MAPbI_3}$. Since our DFT calculations show that the antiparallel orientation is preferred in ~$\\mathrm{MAPbI_3}$, i.e., the free energy is consistently lower for the antiparallel MA orientation at all eight volumes, this requires further investigations.\n    \n    \\begin{table}\n    \t\\caption{\\label{static_supercell}Cell volume, $V_0$, and bulk modulus, $B_0$, obtained by fitting the DFT-calculated data with Eq.~\\ref{BMEOS}, using PBE and PBE+TS applied to a 2x2x2 supercell of $\\mathrm{MAPbI_3}${} with differently orientated MA molecules. Note that in order to improve the comparison, we here report $V_0\/8$.}\n        \\begin{ruledtabular}\n            \\begin{tabular}{lcc}\n            \t& parallel MA & anti-parallel MA \\\\\n                ine                \n                & \\multicolumn{2}{c}{\\textbf{PBE}} \\\\\n                $V_0$ [\\AA$^3$] \\footnotemark[1]& 272.9 & 267.5  \\\\ \n                $B_0$ [GPa] \\footnotemark[2]& 10.8 & 11.2 \\\\\n               \n                & \\multicolumn{2}{c}{\\textbf{PBE+TS}} \\\\\n                $V_0$ [\\AA$^3$] \\footnotemark[1] & 256.3 & 253.8 \\\\ \n                $B_0$ [GPa] \\footnotemark[2]& 15.7 & 14.7 \\\\\n            \\end{tabular}\n            \\footnotetext[1]{Errors are between 0.1 and 0.3~\\AA$^3$}\n            \\footnotetext[2]{Errors are between 0.4 and 0.4~GPa}\n        \\end{ruledtabular}\n    \\end{table}\n    \n    \n    \\begin{figure}\n    \t\\includegraphics[width=.49\\linewidth]{CONTCAR_parallel}\n        \\includegraphics[width=.49\\linewidth]{CONTCAR_antiparallel}\n        \\caption{\\label{supercell_MA}Schematic representations of the supercells with optimized atomic geometries obtained with PBE+TS at a volume of $256.6$~\\AA$^3$, for the case of parallel (a) and antiparallel MA orientation (b). The x-z plane is shown, and atoms belonging to more than the computational cell are displayed for visual clarity}\n    \\end{figure}    \n    \n   \n    \n    \\begin{table}\n    \t\\caption{\\label{MD_results}\nCell volume, $V_0$, and bulk modulus, $B_0$, obtained by fitting the DFT-calculated data with Eq.~\\ref{BMEOS}, using PBE and PBE+TS applied to a 2x2x2 supercell of $\\mathrm{MAPbI_3}${} computed along an MD trajectory of 20~ps. Note that in order to improve the comparison, we here report $V_0\/8$.}\n       \t\\begin{ruledtabular}\n          \t\\begin{tabular}{lcc}\n            \t& PBE & PBE+TS \\\\\n                ine\n            \t$V_0$ [\\AA$^3$] \\footnotemark[1] & 267.3 & 248.1 \\\\ \n            \t$B_0$ [GPa] \\footnotemark[1] & 8.8 & 13.0 \\\\\n            \\end{tabular}\n            \\footnotetext[1]{Errors are 0.1~\\AA$^3$}\n            \\footnotetext[2]{Errors are between 0.6 and 0.7~GPa}\n        \\end{ruledtabular}\n\t\\end{table}\n    \n    \n    \\begin{figure}\n    \t\\includegraphics[width=\\linewidth]{Figure_md_pbe}\n        \\includegraphics[width=\\linewidth]{Figure_md_ts}\n        \\caption{\\label{md_bmeos}\n        Energy change as a function of unit-cell volume, $\\Delta E(V) = E(V)- E_0$, calculated by using MD-DFT with PBE (top) and PBE+TS (bottom) along a trajectory of 20~ps, shown as yellow dots, where the symbol size is given by number of occurrences in the MD run. The blue cross denotes the average $\\Delta E(V)$, and the black dashed line is the fit according to Eq.~\\ref{BMEOS}. Note that in order to improve the comparison, we here report $\\Delta E\/8$ as a function of $V\/8$.}\n    \\end{figure}\n    \n\n    Hence, motivated by the finding that MA orientation impacts structure and binding in $\\mathrm{MAPbI_3}$, we performed fully unconstrained NVT MD calculations at 400~K. {This is relevant, since the MA unit was found to be only weakly bound in the cubic phase,} \\cite{Chen15, Jingrui18}\n    {and hence undergoes rotational and translational motion at elevated temperatures, which could impact the energetics in the material dynamically. Furthermore,} the MD calculations allow for testing whether the effect of dispersive corrections seen for the static structures is still visible at elevated temperatures, {a comparison that has not been attempted previously but is} important for investigating dynamical effects in the structural interaction of $\\mathrm{MAPbI_3}$. Fig.~\\ref{md_bmeos} shows the entire distribution as well as the mean value of the PBE- and PBE+TS-calculated change in free energy of $\\mathrm{MAPbI_3}$, $\\Delta E$, determined for a 20~ps MD simulation, again fitted by Eq.~\\ref{BMEOS}. The most apparent finding from Fig.~\\ref{md_bmeos} is yet again the sizable difference between the PBE and PBE+TS curves, as also quantified by the $V_0$ parameters provided in Table~\\ref{MD_results}. Furthermore, a slight decrease of $V_0$ is found in the MD calculations at 400~K compared to the static 0~K calculation, contrary to the expected thermal expansion. In regard to $B_0$, while in the case of PBE it is similar to the 0~K calculation of the primitive unit cell, for PBE+TS $B_0$ it is slightly lower than what was obtained in the static calculations. \n    \n    In order to better understand the origin of these findings, the average atomic positions along the {PBE+TS} MD trajectories were calculated. Fig.~\\ref{md_poscars} shows that depending on the volume of the supercell, two different types of structures emerged. For the three smallest volumes considered in the MD, the octahedra were found to be tilted and the C-N bond of MA is still clearly visible in the average structure: successive MA molecules are oriented in parallel to each other along one direction and orthogonal to each other in the other two directions, aligning\n    with the long axis of the rhombus created by the tilted octahedra. For the five larger considered volumes, on the other hand, the octahedra form an on average near perfect cubic symmetry, and the average carbon and nitrogen atoms are almost conjoined. The latter could either mean that there is absolutely no preferred direction of the MA molecules, i.e., it {rotates such that it is entirely disordered over the course of the 20~ps trajectory}, or that there are preferred directions which are equally occupied thermally.\n    \n    \\begin{figure}\n    \t\\includegraphics[width=.49\\linewidth]{POSCAR_md_090}\n        \\includegraphics[width=.49\\linewidth]{POSCAR_md_102}\n        \\caption{\\label{md_poscars}Schematic representation of the time-averaged atomic positions along the MD trajectory, calculated with PBE+TS, for $V_0=231.0$\\AA$^3$ (a) and $256.6$\\AA$^3$ (b). Note that hydrogen atoms were omitted for clarity.}\n    \\end{figure}   \n  \n\\FloatBarrier    \n\\section{Discussion}\n\nThe results from the static unit cell calculations confirmed previous findings that showed the importance of including dispersive interaction in DFT calculations {for the structure and binding in $\\mathrm{MAPbI_3}${}}.\\cite{Feng14, David7_2, Egger14, David7_4, David7_5,Faghihnasiri17} {Here, we went beyond in testing a range of dispersive-correction schemes}: PBE calculations corrected by the TS scheme with the regular Hirshfeld partitioning and the MBD scheme showed good agreement with experimental data. The TS method with iterative Hirshfeld partitioning performed slightly worse than the regular TS and MBD schemes. {While the iterative Hirshfeld partitioning has indeed been shown to improve the description of ionic materials,}\\cite{Bucko14, Bucko13} {$\\mathrm{MAPbI_3}${} is a more complex case, since it is a hybrid organic-inorganic system that contains covalent bonds (in the MA molecule) as well as partially covalent-ionic bonds (in the inorganic framework)}. One perhaps surprising result is that the MBD method did not improve the results significantly compared to the TS method, even though one would expect that the iodine atoms, which account for most of the vdW-interactions,\\cite{Egger14} are screened by the surrounding dielectric environment. {We considered a comparison of the $C_6$ parameters calculated from PBE+TS and PBE+MBD, which determine the dispersive correction in either case,}\\cite{Kronik14,Hermann17}{ to further understand this result. We find that the changes are minor, on the order of $1~\\%$ or smaller, and furthermore do not depend on the volume of the unit cell.} This implies that the screening is barely affected by the changes in distance in the calculations at different volumes, and thereby only leads to a constant energy shift. {Furthermore, we considered the higher-order contributions to the dispersive energy calculated with PBE+MBD, to find that indeed the second-order term is dominating. This could imply that the dispersive interactions in $\\mathrm{MAPbI_3}${} are such that higher-order interactions are indeed negligible compared to the pairwise contribution, and also that the polarizability is largely isotropic (see ref. }\\cite{Hermann17}\n{for further discussion).}\n    \nFurthermore, we found that using the hybrid functional HSE, which improves the description of the electronic structure, did not result in improvements for the optimized unit-cell volume and bulk modulus, once it was combined with dispersive corrections. Indeed, the results from the PBE+TS, PBE+MBD, HSE+TS, and HSE+MBD calculations are all within experimentally reported range for these two important structural parameters. Since {our data, presented in Table~}\\ref{static_unitcell}{, also confirm the effect of SOC for the structure and binding to be minor, and since the PBE+TS approach was shown to be very accurate for structural and mechanical properties of multiple HaP crystals,} \\cite{Egger14, David7_5, David5_5,David7_4, Motta16, Egger18}\nwe conclude that using PBE+TS for investigating the structure and binding in static and dynamical calculations of HaPs is a reasonable choice.\n   \n{In this context, it is worth noting that} \\citealt{Bokdam17} suggested the choice of DFT functional to have a bigger impact on the structural properties of $\\mathrm{MAPbI_3}${} than the addition of dispersive corrections. While this is certainly true for electronic properties, and while our calculations showed some minor differences between the PBE and HSE results, {we have shown that} this clearly cannot diminish the important role of dispersive interactions in $\\mathrm{MAPbI_3}$. {We further note that the different conclusions of our work and} \\citealt{Bokdam17} {could be related to the different points of reference used in either case: while we have chosen to consider experimental data on the lattice constants and bulk modulus,} \\citealt{Bokdam17} {considered energies calculated in the random-phase approximation (RPA) as a reference. Indeed, RPA energies can be very accurate for solids and are broadly applicable, but it is worth noting that ``standard'' RPA suffers from known deficiencies, such as potentially incorrect descriptions of short-range correlation.} \\cite{RPA_range,Hermann17}\n    \n    \\begin{figure}\n    \t\\includegraphics[width=\\linewidth]{green_iodines}\n        \\caption{\\label{antiparallel_mixed} Schematic representation of the interaction between the ammonium-end of MA and the surrounding iodine atoms; the relevant atoms are highlighted in green, and the z-x plane (left) and y-x plane (right) are shown. The structure is taken from the PBE+TS calculation of $V= 256.6$~\\AA$^3$ as visualized in Fig.~\\ref{supercell_MA}.}\n    \\end{figure}\n    \nUsing the PBE+TS approach allowed for studying more complicated static and dynamic structural phenomena: First, we investigated the impact of MA orientation on the structural parameters of $\\mathrm{MAPbI_3}$, studying the extreme cases of either perfectly parallel or antiparallel MA molecules contained in a supercell. We found that only for the parallel orientation of MA do the octahedra retain cubic symmetry, while for antiparallel MA molecules they strongly tilt. Note that the lattice vectors were still constrained to the primitive unit cell, i.e., the volume change corresponds to a hydrostatic pressure in the system. Since the relative energy gain\/cost due to these structural distortions depends on the unit-cell volume, this effect modified the obtained volume and bulk modulus. Hence, the interaction between MA and the inorganic atoms is important, especially because the calculations showed that the antiparallel orientations are actually energetically preferred.\n\nThe origin of these distortions is related to the interactions between the MA molecules and iodine atoms, since the partially positive ammonium-part of the MA interacts stronger with the partially negatively iodide ion than the methyl-side. In Fig.~\\ref{antiparallel_mixed} we illustrate that the nitrogen atoms of MA interact mainly with five of the neighboring iodines, which results in a distortion of the octahedra such that the distances between the nitrogen and iodine atoms are maintained between 3.64 and 3.81~\\AA. These interactions seem to be the main driving force behind the MA-induced distortions of the octahedra in the antiparallel case, which are energetically favorable for the system. Due to symmetry, these interactions are cancelled in the case of parallel MA orientation, and our geometry relaxations did not automatically adapt to the more favorable antiparallel szenario. This finding is also quite interesting in view of the fact that the experimentally-determined lattice symmetry of ~$\\mathrm{MAPbI_3}${} above 327~K is almost perfectly cubic,\\cite{Poglitsch87,Stoumpos13} despite the fact that different MA orientations can induce lower energy structures.\n\nThe implications of these findings can be fully understood by means of fully-unconstrained MD calculations at 400~K, since in these the MA molecules, as well as all other atoms, are allowed to move freely. The first important finding from these data was that inclusion of dispersive corrections is equally important to obtain reasonable structural parameters in MD calculations. Second, the results obtained from the MD simulations are quite surprising at first sight, since the unit-cell volume was found to be smaller than the one obtained from the 0~K static calculation. To understand this finding, consider that the time-scales associated with MA motion are much faster than the ones corresponding to the octahedral distortions of the heavy Pb-I cage, which is included in the MD but absent in the static calculations. Our analysis further showed that the average nuclear positions of MA are such that the carbon and nitrogen atoms are essentially conjoined at this temperature, meaning that MA is disordered, which is well known.\\cite{Poglitsch87} Hence, the inorganic atoms essentialy respond to a time-averaged volume corresponding to the moving MA molecule. Due to the MA disorder, this volume is effectively smaller at 400~K than for a fixed pattern of MA orientations, allowing for shorter I-Pb-I bonds and hence smaller crystal volumes. Last but not least, this rationale also explains the finding from the MD data showing that the time-averaged octahedral symmetry is almost perfectly cubic at 400~K in agreement with experiment: MA disorder implies that all possible octahedral distortions induced by MA-iodine interactions are statistically equally likely. Therefore, the cubic perovskite structure is maintained as the most energetically favorable average crystal structure, exhibiting all the electronic and optical properties that render $\\mathrm{MAPbI_3}${} so favorable for device applications.\n\nFinally, the data contained in Fig.~\\ref{md_poscars} showed that at smaller volumes the MD-averaged nuclear positions exhibit octahedral distortions together with a more preferred orientational order of MA. This makes sense, considering that smaller unit-cell volumes correspond to smaller voids between the octahedra, which increases the organic-inorganic interactions,\\cite{David7_5} and partially hinders free MA motion.\nSuch tilting of the octahedra into an orthorhombic-like structure was also observed experimentally in pressure experiments\\cite{Capitani16} for $\\mathrm{MAPbI_3}$, and the variation in the preferred direction for MA at different volumes was discussed also in a recent study reporting MD simulations.\\cite{Lahnsteiner18} Therefore, in agreement with previous findings our study shows that even relatively mild changes in external pressure can result in large changes of the structure and binding in $\\mathrm{MAPbI_3}$.\n\n\\section{Conclusion}\n\n In summary, we investigated the impact of various levels of DFT-related approximations for calculations of the structural and binding properties of the prototypical HaP material $\\mathrm{MAPbI_3}$. Our tests considered the effects of including different dispersive correction schemes, applying a hybrid functional, including SOC, and also addressed the role of dynamic effects in MD calculations. The data confirmed previous theoretical work showing that dispersive corrections are important for accurate calculations of $\\mathrm{MAPbI_3}$, and also highlight that applying a computationally much more expensive hybrid functional improves the description of structural and mechanical properties by only a small amount. From this, we conclude that the use of a semilocal functional, augmented by pairwise dispersive interactions, is a suitable choice when computing more complicated static as well as structural dynamical phenomena in HaPs. Applying this methodology to DFT-based MD calculations of $\\mathrm{MAPbI_3}$, we analyzed the dynamic effect of molecular motion and its interplay with the structure of and binding in $\\mathrm{MAPbI_3}$. From this analysis, we could rationalize microscopically the simultaneous occurrence of a preferred cubic octahedral symmetry and MA disorder.\n\t\n\\section{Acknowledgements}\n\nFunding provided by the Alexander von Humboldt Foundation in the framework of the Sofja Kovalevskaja Award endowed by the German Federal Ministry of Education and Research is acknowledged. The authors gratefully acknowledge the Gauss Centre for Supercomputing e.V. (www.gauss-centre.eu) for funding this project by providing computing time through the John von Neumann Institute for Computing (NIC) on the GCS Supercomputer JUWELS at J\u00fclich Supercomputing Centre (JSC).\n\n\\section{References}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nFictitious Play (FP), introduced in \\cite{Brown51}, is one of the oldest and best-known game theoretic learning algorithms. FP has been shown to be an effective algorithm for distributed learning of Nash equilibria in various classes of games including two-player zero-sum games \\cite{robinson1951iterative}, generic $2\\times m$ games \\cite{berger2005fictitious}, supermodular games\n\\cite{milgrom1990rationalizability,berger2008learning}, one-against-all games \\cite{sela1999fictitious}, and potential games \\cite{Mond96,benaim2005stochastic}. However, the manner in which players \\emph{learn} in FP is often unsatisfactory, especially in the context of distributed control.\n\nIn FP, players learn equilibrium strategies in the sense that the time-averaged empirical distribution of players' actions converges to the set of Nash equilibria ---a form of learning known as \\emph{convergence in empirical distribution}. This notion of learning tends to be problematic when the limit set of a learning algorithm contains mixed-strategy equilibria. In particular, convergence of the time-averaged empirical distribution to a mixed-strategy equilibrium does not imply any form of convergence in players' period-by-period strategies or actions. In practice, players' period-by-period strategies tend to move in progressively longer and longer cycles around an equilibrium set---the time-averaged empirical distribution is driven to equilibrium, but the period-by-period strategies never approach the equilibrium set themselves.\n\nIn the context of repeated-play algorithms, we refer to convergence of the empirical distribution (or some function thereof) to an equilibrium set as weak convergence, and we refer to any form of learning involving weak convergence as weak learning. We refer to the convergence of players' period-by-period strategies to an equilibrium set as strong convergence, and we refer to any form of learning involving strong convergence as strong learning. Intuitively speaking, weak learning means that players learn an equilibrium strategy in some abstract sense (i.e., convergence in empirical distribution) but may never actually implement the strategy they are learning. In strong learning, not only do players \\emph{learn} an equilibrium strategy, but they also implement it.\n\nFP is proven to achieve learning only in the weak sense, and thus no guarantees can be made regarding the convergence nor optimality of players period-by-period strategies. For example, Jordan \\cite{jordan1993} presents a continuum of games for which FP achieves weak learning, yet in all but a countable subset of games, the period-by-period strategies produced by FP never approach the game's unique equilibrium. As another example, Young \\cite{young2004strategic} presents a $2\\times 2$ game in which FP achieves weak learning, but the period-by-period actions produced by FP achieve the lowest possible utility in every stage of the repeated play (see also Section \\ref{sec_weak_convergence}).\n\nOur first main contribution is the presentation of a simple variant of FP that converges strongly to equilibrium. In our strongly convergent variant of FP, players gradually and independently transition from using the FP best response rule to determine the next-iteration action, to using their current empirical distribution as a probability mass function from which they sample to determine the next-iteration action. We show that, for any game in which FP can be shown to converge weakly to equilibrium (and for which a certain robustness assumption holds---see \\textbf{A.\\ref{a_robustness}}), our variant of FP will converge strongly to equilibrium.\n\nOne advantage of this approach is that it is readily applicable to more general FP-type learning algorithms. Our second (and more general) main contribution is a method for taking a weakly convergent FP-type learning algorithm, and constructing from it, a strongly convergent variant.\nWe study a general class of FP-type algorithms and show that, so long as an algorithm achieves weak learning in a sufficiently robust sense (see \\textbf{A.\\ref{a_robustness}}), then a strongly convergent variant of the algorithm can be constructed. As an example of how the general result may be applied, we consider three weakly convergent FP-type algorithms---classical FP, Generalized Weakened FP \\cite{leslie2006generalised}, and Empirical Centroid FP \\cite{swenson2012ECFP,Swenson-MFP-Asilomar-2012}---and construct the strongly convergent variant of each.\n\n\n\n\\subsection{Related Work}\nAn overview of the topic of learning in games can be found in \\cite{fudenberg1998theory,young2004strategic}. Various problems associated with learning mixed-strategy equilibria in best-response-type learning algorithms (including FP-type algorithms) are discussed in \\cite{jordan1993}. In particular, the issue of weak convergence is considered, along with a discussion of some of the underlying mechanics that lead to weak convergence.\n\nMany learning algorithms are designed to ensure that their limit points are pure-strategy equilibria \\cite{marden06,marden-payoff,chasparis2010aspiration,pradelski2012learning,marden-shamma-08}. Ensuring convergence to a pure strategy is a natural way of ensuring strong learning, since weak learning can generally only occur when the limit set contains mixed strategies.\n\nIn contrast, this paper studies a method of ensuring strong convergence when the limit set of the algorithm contains mixed strategies. The ability to (strongly) learn mixed equilibria is important for many reasons, the foremost being that, in finite games, the set of Nash equilibria (NE) is only guaranteed to be non-empty if mixed equilibria are considered. Mixed strategies play an important role when the learned strategy needs to be robust to uncertainty in opponent behavior or game structure, or secure against the actions of malicious players \\cite{rass2014numerical,voorneveld1999pareto,alpcan2010network,sela1999fictitious,dabcevic2014fictitious}. With regards to FP in particular, it was recently shown in \\cite{candogan2013dynamics} that, for the class of near-potential games, the limit set of the FP dynamics (weakly speaking) is a neighborhood of a mixed equilibrium.\n\nRegret-testing algorithms \\cite{foster2003regret},\\cite{germano2007global} achieve strong convergence to mixed-strategy equilibria in generic finite games. However, such algorithms operate on fundamentally different principles from FP-type algorithms---players implement a form of exhaustive search to coordinate on a NE strategy. Such algorithms tend to have slow convergence rates, especially when the number of players or available actions is large.\n\nStochastic FP (SFP)---introduced in \\cite{Fud92}---was proposed as a learning mechanism that could (i) mitigate the problem of weak convergence to mixed equilibria in FP and (ii) provide a reasonable explanation for why real-world players might learn mixed-strategy equilibria. In SFP, the issue of weak convergence is addressed by smoothing each player's best response correspondence with the addition of small random shocks or perturbations. The stable points of SFP are not Nash equilibria, but rather Nash distributions. The set of Nash distributions converges to the set of Nash equilibria as the size of the perturbations goes to zero \\cite{Fud92}. SFP has been shown to obtain strong convergence to the set of Nash distributions in various classes of games \\cite{hofbauer2002global,benaim2005stochastic,fudenberg1998theory}. Moreover, if the perturbations are permitted to gradually decay throughout the course of the repeated play, then SFP converges to the set of NE \\cite{leslie2006generalised}.\n\nIn contrast to SFP, the present work does not consider the descriptive agenda of providing an explanation for why real-world learners might act according to a given behavior rule. Furthermore, we present a simple and intuitive procedure for modifying a variety of weakly convergent learning algorithms in order to obtain a strong convergent variant. From a technical perspective, the current work differs from SFP in that the best response correspondence is not directly smoothed in any way.\n\nThe work \\cite{leslie2006generalised} by Leslie  et al. studies a useful generalization of FP termed Generalized Weakened FP (GWFP). Among other contributions, the paper demonstrates that the convergence of FP is not affected by asymptotically decaying perturbations to players' best response sets. This result provides a cornerstone for our proofs by ensuring that FP (and GWFP) meet the critical robustness assumption \\textbf{A.\\ref{a_robustness}}. We study a strongly convergent variant of GWFP in Section \\ref{sec_apps2}. Furthermore,  \\cite{leslie2006generalised} also presents a payoff-based, actor-critic learning algorithm based on GWFP that achieves strong learning. Our work differs from this in that we provide a general method for constructing a strongly convergent algorithm from a weakly convergent one in a setting where instantaneous payoffs information may or may not be available.\n\nOur preliminary results on strong convergence in FP is found in \\cite{swenson2014strong}. The present work expands on \\cite{swenson2014strong} by considering algorithms beyond classical FP and establishing more general conditions under which convergence can be attained (in particular, see \\textbf{A.\\ref{rho_a1}}--\\textbf{A.\\ref{rho_a3}}). Furthermore, \\cite{swenson2014strong} contains a gap in reasoning in the proof of Lemma 2 which the present paper fills in.\n\nThe remainder of the paper is organized as follows. Section \\ref{sec_prelims} sets up notation to be used in the subsequent development. Section \\ref{sec_FP} introduces classical FP and discusses the problem of weak convergence in classical FP. Section \\ref{sec_strong_fp} presents the strongly convergent variant of classical FP and states the strong convergence theorem for classical FP. Section \\ref{sec_general_setup} presents the general notion of an FP-type algorithm, then presents the strongly convergent variant of an FP-type algorithm, states the general strong convergence result in the context of an FP-type algorithm, and presents the proof of the result. In Section \\ref{sec_apps}, the general result is applied to prove strong convergence in classical FP, Generalized Weakened FP, and Empirical Centroid FP. Section \\ref{sec_conclusion} concludes the paper.\n\n\\section{Preliminaries}\n\\label{sec_prelims}\n\\subsection{Setup and Notation}\nA game in normal form is represented by the triple $\\Gamma := (N,(Y_i,u_i)_{i\\in N})$, where $N = \\{1,\\ldots,n\\}$ denotes the set of players, $Y_i$ denotes the finite set of actions available to player $i$, and $u_i:\\prod_{i\\in N}Y_i \\rightarrow \\mathbb{R}$ denotes the utility function of player $i$. Denote by $Y:= \\prod_{i\\in N} Y_i$ the joint action space.\n\nIn order to guarantee the existence of Nash equilibria it is necessary to consider the mixed extension of $\\Gamma$ in which players are permitted to play probabilistic strategies. Let $m_i := |Y_i|$ be the cardinality of the action space of player $i$, and let $\\Delta_i := \\{p\\in \\mathbb{R}^{m_i}:\\sum_{k=1}^{m_i}p(k) = 1,~p(k)\\geq 0 ~\\forall k\\}$ denote the set of mixed strategies available to player $i$---note that a mixed strategy is probability distribution over the action space of player $i$. Denote by $\\Delta^n := \\prod_{i\\in N} \\Delta_i$, the set of joint mixed strategies.\n\nIn this context, we often wish to retain the notion of playing a deterministic action. For this purpose, let $A_i := \\{e_1,\\ldots,e_{m_i}\\}$ denote the set of ``pure strategies'' of player $i$, where $e_j$ is the $j$-th cannonical vector containing a $1$ at position $j$ and zeros otherwise.\n\nThe mixed utility function of player $i$ is given by $U_i(p) := \\sum_{y \\in Y} u_i(y) p_1(y)\\ldots p_n(y)$, where $U_i:\\Delta^n \\rightarrow \\mathbb{R}$. When convenient we sometimes write $U_i(p)$ as $U_i(p_i,p_{-i})$, where $p_i$ denotes the mixed strategy of player $i$ and $p_{-i}$ denotes the mixed strategies of all other players.\nThe set of Nash equilibria is given by $NE := \\{p\\in \\Delta^n: U_i(p_i,p_{-i}) \\geq U_i( p_i',p_{-i}), ~\\forall p_i' \\in \\Delta_i,~\\forall i\\in N\\}$.\nLet\n\\vskip-15pt\n\\begin{equation}\nBR_i^{\\epsilon}(p_{-i}) := \\{a_i \\in A_i: U(a_i,p_{-i}) \\geq \\max_{\\alpha_i \\in A_i} U(\\alpha_i,p_{-i})-\\epsilon\\}\n\\label{BR_epsilon_set}\n\\end{equation}\n\\vskip-5pt\n\\noindent\nbe the $i$-th players set of $\\epsilon$-best responses to a strategy profile $p_{-i}$ adopted by the other players. Note that in this definition we only consider pure-strategy $\\epsilon$-best responses.\nDenote by $v_i(p_{-i}) := \\max_{p_i\\in \\Delta_i} U_i(p_i,p_{-i}),$ the value obtained by playing a best response.\n\nThroughout, we assume there exists a probability space $(\\Omega,\\mathcal{F},\\mathbb{P})$ rich enough to carry out the construction of the various random variables required in this paper. For a random object $X$ defined on a measurable space $(\\Omega,\\mathcal{F})$, let $\\sigma(X)$ denote the $\\sigma$-algebra generated by $X$ \\cite{williams_book}. As a matter of convention, all equalities and inequalities involving random objects are to be interpreted almost surely (a.s.) with respect to the underlying probability measure, unless otherwise stated.\n\n\\subsection{Repeated Play}\nSuppose players repeatedly face off in the game $\\Gamma$. Denote by $t\\in \\{1,2,\\ldots\\}$ a round of the repeated play. Let $\\{a_i(t)\\}_{t\\geq 1}$ denote the sequence of actions taken by player $i$, where $a_i(t) \\in A_i$, and let $\\{a(t)\\}_{t\\geq 1}$, $a(t) = (a_1(t),\\ldots,a_n(t))$ denote the sequence of joint actions.\n\nLet $\\{\\mathcal{F}_t\\}_{t\\geq 1}$ be a filtration (sequence of $\\sigma$-algebras) that contains the information available to players in round $t$ of the repeated play.\nFor $t\\geq 1$ and $\\alpha_i \\in A_i$, let $g(\\alpha_i,~t)\\in\\mathbb{R}$ be an $\\mathcal{F}_{t-1}$-measurable random variable with $g_i(\\alpha_i,~t) := \\mathbb{P}(a_i(t) = \\alpha_i\\vert \\mathcal{F}_{t-1})$, and let $g_i(t)\\in \\Delta_i$ be the vector with components $g_i(t) := (g_i(\\alpha_1,~t),\\ldots,g_i(\\alpha_{m_i},~t))$, where $m_i$ is the cardinality of $A_i$.\nWe say $g_i(t)$ is the mixed strategy used by player $i$ in round $t$, and we say $\\{g_i(t)\\}_{t\\geq}$ is the sequence of period-by-period (mixed) strategies used by player $i$. The sequence of joint period-by-period strategies is given by $\\{g(t)\\}_{t\\geq 1}$, $g(t) := (g_1(t),\\ldots,g_n(t))$.\n\nDenote by $q_i(t) \\in \\Delta_i$, the empirical distribution of player $i$. The precise manner in which the empirical distribution\\footnote{The term \\emph{empirical distribution} is often used to refer explicitly to the time-averaged histogram of the action choices of some player $i$; i.e., $q_i(t) = \\frac{1}{t}\\sum_{s=1}^t a_i(s)$. Here, we allow for a broader definition that will permit interesting and useful algorithmic generalizations.} is formed will depend on the algorithm at hand. In general, $q_i(t)$ is formed as a function of the action history $\\{a_i(s)\\}_{s=1}^t$ and serves as a compact representation of the action history of player $i$ up to and including the round $t$. The joint empirical distribution is given by $q(t) := (q_1(t),\\ldots,q_n(t))$.\n\nUnless otherwise stated, $d(\\cdot,~\\cdot)$ denotes the standard Euclidean norm. For $m\\geq 1$ and $S \\subset \\mathbb{R}^m$ define the distance from $p\\in \\mathbb{R}^m$ to $S\\subset\\mathbb{R}^m$ by $d(p,~S) := \\inf\\{d(p,~p'):~p'\\in S\\}$.  We say a repeated-play learning process converges \\emph{weakly} to equilibrium if for some map $f:\\Delta^n\\rightarrow\\Delta^n$ there holds $d(f(q(t)),~NE)\\rightarrow 0$ as $t\\rightarrow \\infty$. In most cases in this paper, $f$ will simply be the identity function. We say a repeated-play learning process converges \\emph{strongly}\\footnote{The notion of strong convergence presented in this paper is comparable to the notions of ``convergence in intended behavior'' presented in \\cite{Fud92} and ``convergence in strategic intentions'' given in \\cite{young2004strategic}.} to equilibrium if $d(g(t),~NE)\\rightarrow 0$ as $t\\rightarrow \\infty$. Note that weak learning implies that players \\emph{learn} an equilibrium strategy, but may never actually begin to implement the strategy that is being learned. On the other hand, in strong learning players both \\emph{learn} an equilibrium strategy, and implement the strategy that is being learned (see Section \\ref{sec_weak_convergence} for more details).\n\n\\section{Fictitious Play}\n\\label{sec_FP}\n\\subsection{Fictitious Play}\n\\label{sec_FP_subsection}\nLet\n\\vskip-20pt\n\\begin{equation}\n\\label{q_FP}\nq_i(t) := \\frac{1}{t}\\sum_{s=1}^t a_i(s),\n\\end{equation}\n\\vskip-5pt\n\\noindent be the normalized histogram\\footnote{Recall that the actions $a_i(t)\\in A_i$ are dirac distributions in the mixed-strategy space $\\Delta_i$.} of the actions of player $i$.\n\nFP may be intuitively understood as follows. Players repeatedly face off in a stage game $\\Gamma$. In any given stage of the game, players choose a next-stage action by assuming (perhaps incorrectly) that opponents are using stationary and independent strategies. Thus, in FP, players use the marginal empirical distribution of each opponent's past play, $q_i(t)$, as a prediction of the opponent's behavior in the upcoming round and choose a next-round strategy which is a best response against this prediction.\n\nA sequence of actions $\\{a(t)\\}_{t\\geq 1}$ such that\\footnote{In all variants of FP discussed in this paper, the initial action $a_i(1)$ may be chosen arbitrarily for all $i$.\\label{footnote_initial_cond}}\n\\vskip-15pt\n\\begin{equation}\na_i(t+1) \\in BR_i(q_{-i}(t)),~\\forall i,\n\\label{FP_BR}\n\\end{equation}\n\\vskip-5pt\n\\noindent for all $t\\geq 1$, is referred to as a \\emph{fictitious play process}. FP has been studied extensively to determine the classes of games for which it can be said to converge (weakly) to the set of Nash equilibria.\nAmong other results, it has been shown that FP leads to weak learning in two-player zero-sum games \\cite{robinson1951iterative}, potential games \\cite{Mond96}, and generic $2\\times m$ games \\cite{berger2005fictitious}. We summarize these results in the following theorem.\n\\begin{theorem}\nLet $\\Gamma = (N,\\{u_i(\\cdot)\\}_{i\\in N},Y^n)$ be a two-player zero-sum game, potential game, or generic $2\\times m$ game, and let $\\{a(t)\\}_{t\\geq 1}$ be a fictitious play process on $\\Gamma$. Then $d(q(t),~NE)\\rightarrow 0$ as $t\\rightarrow \\infty$.\n\\label{theorem_classical_fp}\n\\end{theorem}\n\n\\subsection{Weak Convergence in Fictitious Play}\n\\label{sec_weak_convergence}\nThe following example (see \\cite{young2004strategic}, p. 78), while fairly simple, clearly illustrates the phenomenon of weak convergence in FP, and demonstrates why weak convergence can be a deeply unsatisfactory notion of learning.\n\n\n\\begin{wrapfigure}{l}{0.35\\textwidth}\n  \\begin{center}\n    \\includegraphics[width=0.28\\textwidth]{figures\/miscoord_game.eps}\n  \\end{center}\n  \\caption{}\\label{fig:miscoord_game}\n\\end{wrapfigure}\n\nConsider the two-player asymmetric coordination game shown in Figure \\ref{fig:miscoord_game}. The game has three Nash equilibria: both players play A, both players play B, and an asymmetric mixed-strategy Nash equilibrium. The game is a potential game \\cite{Mond96} (in fact, an identical interests game \\cite{Mond01}) and hence falls within the purview of Theorem \\ref{theorem_classical_fp}---regardless of the initial conditions, players engaged in an FP process will learn an equilibrium in the weak sense that $d(q(t),~NE)\\rightarrow 0$ as $t\\rightarrow\\infty$.\n\nSuppose that the players are engaged in an FP process on this game, and in the first round they miscoordinate their actions (e.g., one chooses A, and the other chooses B). Young \\cite{young2004strategic} shows the somewhat counterintuitive result that the FP dynamics will in fact lead players to miscoordinate their action choices in every subsequent round of the learning process. Thus, despite the fact that $\\lim_{t\\rightarrow\\infty} d(q(t),~NE) = 0$, the players' realized action choices are extremely suboptimal---yielding the lowest possible utility in each round of play. Intuitively speaking, this phenomenon occurs when players' actions cycle in such a way as to drive the time-averaged empirical distribution to a mixed-strategy Nash equilibrium, yet player's period-by-period strategies never constitute (nor even approach) a Nash equilibrium themselves.\n\nIt may be said that in weak learning players ``learn'' a NE strategy in some abstract sense, but never actually implement the strategy they are learning. In strong learning, players not only learn a NE strategy, but they also physically implement the strategy that is being learned.\n\nThe following section presents a simple modification of FP that achieves strong learning; i.e., players' period-by-period strategies converge to equilibrium in addition to convergence of the empirical distributions.\n\n\n\n\n\\section{Strong Convergence in Classical Fictitious Play}\n\\label{sec_strong_fp}\n\nConsider a variant of FP in which the action for player $i$ at time $t$ is chosen by drawing a random sample from the mixed strategy (i.e., probability distribution) $g_i(t)$, where\n\\begin{equation}\ng_i(t) \\in BR_i(q_{-i}(t-1))\\rho_i(t) + q_i(t-1)(1-\\rho_i(t)),\n\\label{strong_FP_informal}\n\\end{equation}\n$\\rho_i(t) \\in [0,1]$, and $\\lim_{t\\rightarrow\\infty} \\rho_i(t) = 0$. Intuitively, this is similar to the classical FP process \\eqref{FP_BR}, but rather than playing a deliberate best response each round, players gradually transition toward drawing their stage $t$ action as a random sample from their own empirical distribution, $q_i(t)$.\n\nThe idea is that players will play a best response sufficiently often so that, per FP, the empirical distribution $q(t)$ will be driven toward equilibrium, as in Theorem \\ref{theorem_classical_fp}. Then, since $\\rho_i(t) \\rightarrow 0$ as $t\\rightarrow \\infty$, the mixed strategy $g_i(t)$ tends towards $q_i(t)$, which is itself tending towards equilibrium. Informally, \\eqref{strong_FP_informal} captures the main idea of strongly convergent FP. A formal presentation of the algorithm is given below.\n\n\n\\subsection{Strongly Convergent Variant of Classical FP}\n\\label{sec_strong_FP_construction}\nConsider a variant of FP in which the action for player $i$ at time $t$ is chosen according to the following randomized rule:\n\\vskip-20pt\n\\begin{equation}\na_i(t) \\sim g'_i(t) :=\n\\begin{cases}\nb_i(t-1), & \\mbox{ if } X_i(t) = 1,\\\\\nq_i(t-1), & \\mbox{otherwise},\n\\end{cases}\n\\label{action_rule0}\n\\end{equation}\nwhere\n$b_i(t-1) \\in BR_i(q_{-i}(t-1)),$ the notation\n$a_i(t) \\sim g'_i(t)$ indicates that the action $a_i(t)$ is drawn as a random sample\\footnote{The action $a_i(t) \\in A_i$ is technically a dirac distribution over the finite action space $Y_i$ (see Section \\ref{sec_prelims}), and the mixed strategy $g_i'(t)$ is a probability distribution over $Y_i$. More precisely, the notation $a_i(t) \\sim g_i'(t)$ means that an action $y_i(t)$ is drawn as a random sample from $g_i'(t)$ with $a_i(t) := \\delta_{y_i(t)}(y_i)$, where $\\delta_{y_i(t)}(y_i) = 1$ if $y_i = y_i(t)$ and $\\delta_{y_i(t)}(y_i) = 0$ otherwise.} from the probability mass function $g'_i(t)$, $X_i(t) \\in \\{0,1\\}$ is a random variable, and $q_i(t)$ is the player's empirical distribution as defined in \\eqref{qt_update2} below.\nLet $\\mathcal{F}_t := \\sigma(\\{a(s),X_1(s),\\ldots,X_n(s),b_1(s),\\ldots,b_n(s)\\}_{s\\leq t}),$\nand note that $g'_{i}(t)$ is $\\mathcal{F}_{t}$-measurable.\nLet\n\\vskip-20pt\n\\begin{equation}\n\\rho_i(t) := \\mathbb{P}(X_i(t) = 1\\vert~\\mathcal{F}_{t-1}),\n\\end{equation}\n\\vskip-5pt\n\\noindent and note that $\\rho_i(t)$ is $\\mathcal{F}_{t-1}$-measurable.\nIntuitively speaking, $\\rho_i(t)$ represents the probability that player $i$ deliberately chooses to play a best response strategy in round $t$ given the history of play up through the previous round.\nWe make the following assumptions regarding each player's probability of deliberately choosing a best response:\n\\begin{assumption}\n$\\lim\\limits_{t\\rightarrow\\infty} \\rho_i(t) = 0, ~\\forall i\\in N$, a.s.,\n\\label{rho_a1}\n\\end{assumption}\n\\begin{assumption}\n$\\sum\\limits_{t\\geq 1} \\rho_i(t) = \\infty, ~\\forall i\\in N$, a.s.,\n\\label{rho_a2}\n\\end{assumption}\n\\begin{assumption}\n$\\lim\\limits_{t\\rightarrow\\infty} \\frac{\\sum_{k=1}^t\\rho_i(k)}{\\sum_{k=1}^t\\rho_j(k)} = 1, ~\\forall i,j \\in N,$ a.s.\n\\label{rho_a3}\n\\end{assumption}\n\nThe first assumption ensures that players eventually transition towards playing their next-stage action as a sample from their empirical distribution rather than playing a deliberate best response. The second assumption ensures that, for each player, a deliberate best response is played infinitely often. The third assumption ensures that the number of deliberate best responses taken by each player remain relatively in sync.\\footnote{Note that since $\\rho_i(t)$ is only required to be $\\mathcal{F}_{t-1}$-measurable, this parameter is in fact adaptively tunable. This is a feature of practical interest since it allows players to adjust their deliberate best response rates on the fly---possibly adapting to the (initially unknown) deliberate best response rates of others and to underlying process dynamics---in order to satisfy \\textbf{A.\\ref{rho_a1}}--\\textbf{A.\\ref{rho_a3}}.}\nIn practice, players may choose their deliberate best responses completely asynchronously; for example, setting $\\rho_i(t) = 1\/t^{r},~\\forall i$, with $r\\in (0,1]$, results in (purely) independent sampling of deliberate best response rounds and secures \\textbf{A.\\ref{rho_a1}}--\\textbf{A.\\ref{rho_a3}}.\n\n\nLet\n\\vskip-25pt\n\\begin{equation}\n\\ell_i(t) := \\sum\\limits_{k=1}^{t} X_i(k)\n\\label{ell_def}\n\\end{equation}\n\\vskip-5pt\n\\noindent count the number of times player $i$ has deliberately played a best response until and including round $t$. Note that $\\ell_i(t)$ is $\\mathcal{F}_t$-measurable.  The empirical distribution $q_i(t)$ is defined recursively as\\footnote{To initialize the process, let the action $a_i(1)$ be chosen arbitrarily, let $q_i(1) = a_i(1)$, and let $X_i(1) = 1$ for all $i$.}\n\\vskip-15pt\n\\begin{equation}\nq_i(t+1) = q_i(t) + \\frac{1}{\\ell_i(t+1)}\\left(a_i(t+1) - q_i(t)\\right)X_i(t+1).\n\\label{qt_update2}\n\\end{equation}\n\\vskip-5pt\n\\noindent Intuitively speaking, the empirical distribution \\eqref{qt_update2} is updated only over rounds when a deliberate best response was played. Note that $q_i(t)$ is $\\mathcal{F}_t$-measurable.\\footnote{Note that, \\eqref{action_rule0} implicitly assumes that players have knowledge of the empirical distributions of opponents when computing a best response. This may be accomplished by assuming that players actions are accompanied with a ``tag'' indicating whether or not the played action was a deliberate best response. Alternatively, the information regarding $q_i(t)$ may tracked by the individual player $i$ and disseminated by a gossip-type algorithm \\cite{swenson2012ECFP} or implicitly disseminated through a payoff-based scheme.}\n\nFinally, let\n\\vskip-20pt\n\\begin{equation}\ng_i(t) := b_i(t-1)\\rho_i(t) + q_i(t-1)(1-\\rho_i(t)),\n\\label{g_def}\n\\end{equation}\n\\vskip-5pt\n\\noindent and note that $g_i(t)$ is $\\mathcal{F}_{t-1}$ measurable.\\footnote{To see this, note first that $q_i(t-1)$ and $\\rho_i(t)$ have been shown to be $\\mathcal{F}_{t-1}$ measurable. Furthermore, this implies that $BR_i(q_i(t-1))$ is $\\mathcal{F}_{t-1}$-measurable. Lastly, by construction, $b_i(t) \\in BR_i(q_i(t-1))$ is $\\mathcal{F}_{t-1}$-measurable.} More importantly, note that for every $\\alpha_i \\in A_i$,\n$g_i(\\alpha_i,t) = \\mathbb{P}(a_i(t) = \\alpha_i\\vert~\\mathcal{F}_{t-1}),$\nand thus $g_i(t)$ represents the mixed strategy (conditioned on past play) used by player $i$ in round $t$. The joint mixed strategy used in round $t$ is given by $g(t) := (g_1(t),\\ldots,g_n(t))$.\n\nWe refer to a process where, for each player $i$, $a_i(t)$ is updated according to \\eqref{action_rule0}, $q_i(t)$ is updated according to \\eqref{qt_update2}, and $g_i(t)$ is updated according to \\eqref{g_def} as the strongly convergent variant of (classical) FP (for reasons to be clear soon).\n\n\\subsection{Strong Convergence in Classical FP: Main Result}\nThe following result states that in the strongly convergent variant of FP, players' period-by-period mixed strategies converge to the set of Nash equilibria---i.e., strong learning is achieved.\n\\begin{cor}\nLet $\\Gamma$ be a two-player zero-sum game, potential game, or generic $2\\times m$ game. Assume \\textbf{A.\\ref{rho_a1}}--\\textbf{A.\\ref{rho_a3}} hold. Then the strongly convergent variant of FP achieves strong learning in the sense that $\\lim_{t\\rightarrow\\infty} d(g(t),~NE) = 0$ almost surely.\n\\label{theorem_main_result}\n\\end{cor}\n\nIn order to prove the above result, we first study a more general notion of fictitious play and then prove the result as a corollary of the general theorem (see Theorem \\ref{theorem_general_result}). Taking this general approach allows our strong convergence results to be be applied to other FP-type algorithms, e.g., Generalized Weakened FP (Section \\ref{sec_apps2}) and Empirical Centroid FP (Section \\ref{sec_apps3}). The proof of Corollary \\ref{theorem_main_result} is given in Section \\ref{sec_apps1}.\n\n\\subsection{Simulation Example}\nIn order to demonstrate the learning properties of strongly convergent FP, we simulated classical FP and strongly convergent FP in a simple two-player matching pennies game with utility functions as shown in Figure \\ref{fig:payoff_matrix}. The game has a unique (symmetric) mixed-strategy equilibrium in which both players choose either action with probability $1\/2$. Figure \\ref{fig:FP_strategies} shows the period-by-period strategies generated by classical FP. Players' strategies are always pure and progress in continuously lengthening cycles. While the time-averaged empirical distribution is being driven to equilibrium, the period-by-period strategies clearly are not.\n\nFigure \\ref{fig:strong_FP_strategies} shows the period-by-period strategies generated by strongly convergent FP with $\\rho(t) = t^{-.35}$. Players' period-by-period strategies are converging to the unique Nash equilibrium of the game.\n\nFigure \\ref{fig:received_utility} shows the utility received by the realized joint action $a(t)$ in each round of repeated play for both learning algorithms. The received payoffs in classical FP cycle around the value of the game, while the received payoffs in strongly convergent FP converge to the value of the game.\n\nOne possible tradeoff in strongly convergent FP is that less frequent deliberate best response actions and less frequent updating of the empirical distribution (see \\eqref{qt_update2}) may lead to a slow-down in convergence rate. The empirical distribution processes for player $1$ in each algorithm is shown in Figure \\ref{fig:empirical_dist} with $\\rho(t) = t^{-.35}$.\n{\\small\n\\begin{figure}\n        \\centering\n        \\begin{subfigure}[b]{0.27\\textwidth}\n                \\includegraphics[width=\\textwidth]{figures\/match_pennies_utility_matrix.eps}\n                \\caption{}\n                \\label{fig:payoff_matrix}\n        \\end{subfigure\n        ~\n         \n        \\begin{subfigure}[b]{0.3\\textwidth}\n                \\includegraphics[width=\\textwidth]{figures\/fp_strategies.eps}\n                \\caption{}\n                \\label{fig:FP_strategies}\n        \\end{subfigure}\n        ~\n         \n        \\begin{subfigure}[b]{0.3\\textwidth}\n                \\includegraphics[width=\\textwidth]{figures\/strong_fp_strategies.eps}\n                \\caption{}\n                \\label{fig:strong_FP_strategies}\n        \\end{subfigure}\n        \\begin{subfigure}[b]{0.3\\textwidth}\n                \\includegraphics[width=\\textwidth]{figures\/match_pennies_utilities.eps}\n                \\caption{}\n                \\label{fig:received_utility}\n        \\end{subfigure}\n        \\begin{subfigure}[b]{0.3\\textwidth}\n                \\includegraphics[width=\\textwidth]{figures\/emp_distributions.eps}\n                \\caption{}\n                \\label{fig:empirical_dist}\n        \\end{subfigure}\n        \\caption{\\small \\ref{fig:payoff_matrix}: Matching pennies payoff matrix, \\ref{fig:FP_strategies}: The probability of each player playing heads in round $t$ using the classical FP algorithm, \\ref{fig:strong_FP_strategies}: The probability of each player playing heads in round $t$ using the strongly convergent FP algorithm, \\ref{fig:received_utility}: The received utility in round $t$ given the realized action $a(t)$, \\ref{fig:empirical_dist}: The empirical distribution process of the action H (heads) for player $1$ in both FP and strongly convergent FP.}\\label{fig:simulation}\n\\end{figure}\n}\n\\section{General Setup}\n\\label{sec_general_setup}\nIn this section we study strong learning in FP-type algorithms ---a class of algorithms that generalizes FP and includes many learning processes based on best-response dynamics.\\footnote{The class of FP-type algorithms proposed here is similar in spirit to the class of best-response-based algorithms considered in \\cite{jordan1993}.} In Section \\ref{sec_FP_type}, we define the notion of an FP-type algorithm. In Section \\ref{sec_FP_type_examples} we present some examples of an FP-type algorithm. In Section \\ref{sec_FP_type_strong} we define the strongly convergent variant of an FP-type algorithm. In Section \\ref{sec_FP_type_main_result} we provide the general strong convergence result for an FP-type algorithm (see Theorem \\ref{theorem_general_result}), and in Sections \\ref{sec_additional_defs}--\\ref{sec_proof_general_result} we prove the general result.\n\n\\subsection{FP-Type Algorithm}\n\\label{sec_FP_type}\nAn FP-type algorithm generalizes classical FP in the following ways: (i) the notion of a player's empirical distribution is generalized, (ii) players are permitted to use a function of the empirical distribution (rather than use the empirical distribution itself) as a predictor of the next-round strategy of opponents, (iii) convergence to equilibrium may occur in terms of a function of the empirical distribution (rather than convergence to equilibrium of the empirical distribution itself), and (iv) limit sets other than the set of NE are permitted.\n\nWe define an FP-type algorithm as follows. Let players be engaged in repeated play of a stage game $\\Gamma$. Let $a_i(t)$ represent the action of player $i$ in round $t\\in\\{1,~2,\\ldots\\}$, and let\n$H_i(t) := \\{a_i(s)\\}_{s=1}^t$\nrepresent the action history of player $i$ up to and including round $t$.\n\nIn classical FP, for each player $i$, the normalized histogram of the player's action choices \\eqref{q_FP} is used as a compact representation of the player's action history. In the general formulation of an FP-type algorithm, we still suppose that players track a compact representation of the action history, but we allow the compact representation to take on a fairly general form,\\footnote{In most literature, the notion of an \\emph{empirical distribution} refers strictly to the time-averaged empirical histogram of a player's action choices, as in classical FP \\eqref{q_FP}. However, as discussed in Section \\ref{sec_prelims}, we use the term empirical distribution more generally to refer to an arbitrarily formed (see \\textbf{A.\\ref{a_general_q}}) distribution that a player uses to track information regarding opponents' empirical action histories. This abuse of terminology allows us to more naturally extend concepts to the general FP-type setting.}  as stated in the following assumption:\n\\begin{assumption}\n\\label{a_general_q}\nThe empirical distribution of player $i$ is of the form $q_i(t) := f^q_i(H_i(t),~t)$, where $f^q_i(\\cdot,~t):\\prod_{s=1}^t A_i \\rightarrow \\Delta_{i}$.\n\\end{assumption}\nWe make the following assumption regarding the sequence of functions $\\{f^q_i(\\cdot,~t)\\}_{t\\geq 1}$ used to form the empirical distribution sequence of player $i$:\n\\begin{assumption}\n\\label{a_step_size_bound}\nFor any history sequence $\\{H_i(t)\\}_{t\\geq 1}$ for player $i$, there holds $\\lim_{t\\rightarrow\\infty} \\|f^q_i(H_i(t+1),~t+1) - f^q_i(H_i(t),~t)\\| = 0$.\n\\end{assumption}\n\nIn particular, this implies that---regardless of the action history---there holds\\\\ $\\lim_{t\\rightarrow \\infty} \\|q_i(t+1) - q_i(t)\\| = 0$ for each player $i$. This fairly mild assumption captures the essential characteristics required for our asymptotic analysis, and may be seen as a generalization of classical FP where exact averaging of actions over time yields $\\|q_i(t+1) - q_i(t)\\| \\leq \\frac{1}{t}$ (see Section \\ref{sec_FP_example}). Together, assumptions \\textbf{A.\\ref{a_general_q}}--\\textbf{A.\\ref{a_step_size_bound}} allow us to consider a variety of FP inspired algorithms, including those with general step sizes \\cite{leslie2006generalised} and those with more intricate history dependent rules such as derivative action \\cite{Arslan04}.\n\nIn an FP-type algorithm, players form a prediction of the future behavior of opponents as a function of the current empirical distribution. Let $p_i(t)$ be player $i$'s prediction of opponent strategies for the upcoming round $(t+1)$. We assume,\n\\begin{assumption}\nPlayer $i$'s prediction $p_i(t)$ of opponent behavior is of the form $p_i(t) = f_i^p(q(t))$, where $f_i^p:\\Delta^n \\rightarrow \\Delta_{-i}$ is a Lipschitz continuous, time-invariant function.\n\\label{a_prediction}\n\\end{assumption}\n\nWe say a sequence of actions $\\{a(t)\\}_{t\\geq 1}$ is an FP-type process if for all $i\\in N$ and all $t\\geq 1$,\n$a_i(t+1) \\in BR_i^{\\epsilon_t}(p_i(t)),$\nwhere $BR^{\\varepsilon_t}_i(\\cdot)$ is the $\\varepsilon_t$-best response set (recall \\eqref{BR_epsilon_set}), and $\\{\\epsilon_t\\}_{t\\geq 1}$ is a sequence satisfying $\\lim_{t\\rightarrow\\infty} \\epsilon_t = 0$.\n\nIn many variants of FP, including classical FP, learning occurs in the sense that $d(q(t),~NE)\\rightarrow 0$. We generalize this notion of learning by allowing for limit sets other than the set of NE and allowing for convergence in terms of a function of $q(t)$ rather than permitting convergence only in terms of $q(t)$ itself.\n\nLet $E$ be some target equilibrium set (not necessarily the set of NE). An FP-type process is said to learn elements of $E$ if for each $i$ there exists a function $f^\\xi_i$ satisfying:\n\\begin{assumption}\nThe function $f^\\xi_i:\\Delta^n\\rightarrow\\Delta_i$ is Lipschitz continuous and time invariant,\n\\label{a_xi}\n\\end{assumption}\nand such that, for $\\xi_i(t) := f^\\xi_i(q(t))$ and $\\xi(t) := (\\xi_1(t),\\ldots,\\xi_n(t))$ there holds\n$\\lim_{t\\rightarrow 0} d(\\xi(t),~E) =0.$\nWe refer to $\\xi(t)$ as the asymptotic learning distribution, and $f^\\xi_i$ as the convergence map of player $i$.\n\nIn general, we will denote an instance of an FP-type learning algorithm by $\\Psi = (\\{f^q_i(\\cdot,~t)\\}_{t\\geq 1},f_i^p,f^\\xi_i)_{i\\in N}$.\nIn order to construct a strongly convergent variant of $\\Psi$ we will require that $\\Psi$ obtain weak convergence in sufficiently robust sense as stated in the following assumption.\n\n\\begin{assumption}\nFor the stage game $\\Gamma$ and equilibrium set $E$, the FP-type algorithm $\\Psi$ is such that for any sequence $(\\epsilon_t)_{t\\geq 1}$ satisfying $\\lim_{t\\rightarrow\\infty} \\epsilon_t = 0$, the FP-type algorithm $\\Psi$ obtains weak convergence in the sense that $\\lim_{t\\rightarrow 0} d(\\xi(t),~E)= 0$.\n\\label{a_robustness}\n\\end{assumption}\n\nThe above assumption ensures that the FP-type algorithm is robust to asymptotically decaying perturbations in a player's best response set. When studying the strongly convergent variant of $\\Psi$ in the following section, the assumption \\textbf{A.\\ref{a_robustness}} will serve to ensure that convergence of the process is not disrupted by minor asynchronies in the number of deliberate best responses taken by each player (i.e., minor disparities in \\eqref{ell_def}).\n\n\\subsection{Examples}\n\\label{sec_FP_type_examples}\n\\subsubsection{Classical Fictitious Play}\n\\label{sec_FP_example}\nClassical FP (Section \\ref{sec_FP_subsection}) fits the template of an FP-type algorithm with $q_i(t) = \\frac{1}{t}\\sum_{s=1}^t a_i(s)$. Note that $q_i(t)$ may be written in recursive form as: $q_i(t+1) = q_i(t) + 1\/(t+1)\\left(a_i(t+1) - q_i(t) \\right)$. Thus, $\\|q_i(t+1) - q_i(t)\\| \\leq \\frac{M_i}{t+1}$, where $M_i := \\sup_{p_i',p_i''\\in \\Delta_i} \\|p_i' - p_i''\\|$, and \\textbf{A.\\ref{a_step_size_bound}} is satisfied. The prediction map $f_i^p$ is given by the identity function, and convergence map $f^\\xi_i$ also given by the identity function. The target equilibrium set is given by $E := NE$, the set of Nash equilibria.\n\n\\subsubsection{Generalized Weakened Fictitious Play}\n\\label{sec_GWFP_intro}\nLeslie et al. \\cite{leslie2006generalised} study a useful generalization of FP, termed Generalized Weakened FP (GWFP), in which players are permitted to choose a suboptimal best response each round, so long as the degree of suboptimality decays asymptotically to zero, and in which step-size sequences other than $\\{1\/t\\}_{t\\geq 1}$ are permitted.\n\nFormally, for $p_{-i} \\in \\Delta_{-i}$ and $\\epsilon\\geq 0$, let\\footnote{The set $\\bar{BR^{\\epsilon}_i}(p_{-i})$ defined below differs from the set $BR_i^\\epsilon(p_{-i})$ defined in the preliminaries in that $\\bar{BR^{\\epsilon}_i}(p_{-i})$ includes all \\emph{mixed} strategy best responses, whereas $BR_i^\\epsilon(p_{-i})$ contains only the pure strategy best responses. The set $\\bar{BR^{\\epsilon}_i}(p_{-i})$ is used here in order to precisely define a GWFP process as given in \\cite{leslie2006generalised}, but the remainder of the paper focuses on the set $BR_i^\\epsilon(p_{-i})$.}\n$\\bar{BR^{\\epsilon}_i}(p_{-i}) := \\{p_i\\in \\Delta_i: U_i(p_i,p_{-i}) \\geq \\max_{\\alpha_i \\in A_i}U_i(\\alpha_i,p_{-i})-\\epsilon\\},$\nand for $p\\in \\Delta^n$, let $\\bar{BR^{\\epsilon}}(p) := (\\bar{BR^{\\epsilon}_1}(p_{-1}),\\ldots,\\bar{BR^{\\epsilon}_n}(p_{-n}))$. A sequence $\\{q(t)\\}_{t\\geq 1}$ is said to be a GWFP process if\n$q(t+1) \\in (1-\\gamma(t+1))q(t) + \\gamma(t+1)(\\BRtildet(q(t)) + M_{t+1})$\nwith $\\gamma(t) \\rightarrow 0$ and $\\epsilon_t \\rightarrow 0$ as $t\\rightarrow \\infty$, $\\sum_{t\\geq 1} \\gamma(t) = \\infty$, and $\\{M_t\\}_{t\\geq 1}$ is a deterministic (or stochastic) perturbation sequence satisfying\n$\\lim\\limits_{t\\rightarrow\\infty}\\sup_{k}\\{\\|\\sum_{i=t}^{k-1} \\gamma_{i+1} M_{i+1} \\|:\\sum_{i=t}^{k-1}\\gamma_{i+1} \\leq T\\} = 0$ (a.s.).\n\nWe consider a special case of GWFP in which $M_t = 0, ~\\forall t$ and the $\\epsilon$-best response set is restricted to the set of pure strategy $\\epsilon$-best responses. That is, we consider the subset of GWFP process such that\n$a(t+1) \\in BR^{\\epsilon_t}(q_{-i}(t)),$\nand,\n\\begin{equation}\n\\label{q_gwfp_def}\nq(t+1) = q(t) + \\gamma(t+1)\\left(a(t+1) - q(t) \\right),\n\\end{equation}\nwith $\\epsilon_t \\rightarrow 0$,\nand in a slight variation of terminology we refer to the sequence of actions $\\{a(t)\\}_{t\\geq 1}$ satisfying the above as a GWFP process.\n\nIn the terminology of Section \\ref{sec_FP_type}, GWFP fits the template of an FP-type algorithm with the empirical distribution $q_i(t)$ defined recursively as in \\eqref{q_gwfp_def} (where it is assumed that $\\lim_{t\\rightarrow\\infty} \\gamma(t) = 0$), the prediction map $f_i^p$ given by the identity function for all $i$, and the convergence map $f^{\\xi}_i$ given by the identity function for all $i$, and the target equilibrium set is given by $E := NE$---the set of Nash equilibria.\n\n\\subsubsection{Empirical Centroid Fictitious Play---Learning Consensus Equilibria}\n\\label{sec_ECFP_intro}\nEmpirical Centroid FP (ECFP) was conceived as a variant of FP suited to implementation in large-scale games \\cite{swenson2012ECFP,Swenson-MFP-Asilomar-2012}. In ECFP, rather than tracking the empirical distribution of each individual opponent (as in FP), players track and respond to only the centroid of the empirical distributions. In order to ensure the process is well defined the following assumption is made:\n\\begin{assumption}\n\\label{a_ident_strat}\nAll players use the same strategy space.\n\\end{assumption}\nUnder this assumption, let the empirical distribution be defined by\n\\vskip-20pt\n\\begin{equation}\n\\label{q_ecfp_def}\nq_i(t) := \\frac{1}{t}\\sum_{s=1}^t a_i(s),\n\\end{equation}\n\\vskip-5pt\n\\noindent and let the empirical centroid distribution be defined by\n$\\bar q(t) := \\frac{1}{n}\\sum_{i\\in N} q_i(t).$\nWe say a sequence of actions $\\{a(t)\\}_{t\\geq 1}$ is an ECFP process if for all $i$ and all $t\\geq 1$,\n\\begin{equation}\na_i(t+1) \\in BR_i^{\\epsilon_t}(\\bar q_{-i}(t)),\n\\label{ecfp_process}\n\\end{equation}\nwhere $\\bar q_{-i}(t) = (\\bar q(t),\\ldots,\\bar q(t)) \\in \\prod_{j\\not= i} \\Delta_j$ is the $(n-1)$-tuple containing $(n-1)$ repeated copies of $\\bar q(t)$, and $\\{\\epsilon_t\\}_{t\\geq 1}$ is a sequence satisfying $\\lim_{t\\rightarrow\\infty} \\epsilon_t = 0$.\n\nIn ECFP, players learn elements of the set of consensus Nash equilibria\\footnote{We assume here that the set of consensus Nash equilibria is non-empty. When revisiting ECFP in Section \\ref{sec_apps3}, we provide an assumption on the utility structure that ensures that the set is indeed non-empty.}, defined by\n$C:= \\{p = (p_1,~\\ldots~,p_n)\\in NE:~ p_1 = p_2 = \\ldots = p_n\\},$\nthe subset of Nash equilibria in which all players use identical strategies (see \\cite{swenson2012ECFP} for more details).\nDefine $\\bar q^n(t) := (\\bar q(t),\\ldots,\\bar q(t)) \\in \\Delta^n$ to be the $n$-tuple containing repeated copies of $\\bar q(t)$; learning in ECFP takes place in the sense that $\\lim_{t\\rightarrow\\infty} d(\\bar q^n(t),~C) = 0$.\n\nIn the terminology of Section \\ref{sec_FP_type}, ECFP fits the template of an FP-type algorithm with the empirical distribution given by \\eqref{q_ecfp_def}, the prediction map $f_i^p$ given by\n$f_i^p(q(t))  := \\left(\\frac{1}{n}\\sum_{j\\in N}q_j(t), \\ldots, \\frac{1}{n}\\sum_{j\\in N}q_j(t)\\right),~ \\forall i,$\nwhere the right-hand side is a $(n-1)$-tuple containing repeated copies of $\\bar q(t)$,\nand the convergence map given by\n$f^\\xi_i(q(t)) := \\frac{1}{n}\\sum_{j=1}^n q_j(t),~\\forall i.$\nThe target equilibrium set is given by $E:= C$, the set of consensus Nash equilibria.\n\n\\subsubsection{Empirical Centroid Fictitious Play---Learning Mean-Centric Equilibria}\n\\label{sec_ecfp_mce_intro}\nIn this section we consider a slight modification of the ECFP algorithm presented in Section \\ref{sec_ECFP_intro} that enables players to learn elements of an alternate (non-Nash) equilibrium set.\n\nLet an ECFP action process be defined as in \\eqref{ecfp_process}. Define the set of mean-centric equilibria by\n$MCE := \\{p \\in \\Delta^n: ~U_i(p_i,~\\bar p_{-i}) \\geq U_i(p_i',~\\bar p_{-i})~\\forall p_i' \\in \\Delta_i,~\\forall i\\}.$\nThe set of MCE is neither a superset nor a subset of the NE---rather, it is a set of natural equilibrium points tailored to the ECFP dynamics \\cite{swenson2013MCE}. The set of consensus Nash equilibria $C$ (see Section \\ref{sec_ECFP_intro}) however, is contained in the set of MCE.\n\nIn ECFP, players learn elements of MCE in the sense that $\\lim_{t\\rightarrow\\infty}d(q(t),~MCE)= 0$. In the terminology of Section \\ref{sec_FP_type}, this fits the template of an FP-type algorithm with $q_i(t)$ given by \\eqref{q_ecfp_def}, $f_i^p$ defined in the same way as in Section \\ref{sec_ECFP_intro}, the convergence map $f^\\xi_i$ given by the identity for all $i$, and the target equilibrium set given by $E := MCE$.\n\nNote that the only difference between the ECFP algorithm discussed in the Section \\ref{sec_ECFP_intro} and the ECFP algorithm discussed here is the choice of target equilibrium set $E$ and convergence maps $f^\\xi_i$.\n\n\\subsection{Strongly Convergent Variant of an FP-type Algorithm}\n\\label{sec_FP_type_strong}\nIn this section we construct the strongly convergent variant of an FP-type learning algorithm. The construction here is a generalization of that of Section \\ref{sec_strong_FP_construction} where we constructed the strongly convergent variant of classical FP.\n\nLet $\\Psi = (\\{f^q_i(\\cdot,~t)\\}_{t\\geq 1},~f_i^p,~f^\\xi_i)_{i\\in N}$ be an FP-type learning algorithm.\nFor each $i\\in N$, let $\\{X_i(t)\\}_{t\\geq 1}$ be a sequence of random variables with $X_i(t) \\in \\{0,1\\}$. Analogous to Section \\ref{sec_strong_fp}, $X_i(t)=1$ will serve to indicate that player $i$ took a deliberate best response in round $t$. Let\n\\vskip-20pt\n\\begin{equation}\n\\label{ell_def2}\n\\ell_i(t) := \\sum_{s=1}^t X_i(s)\n\\end{equation}\n\\vskip-5pt\n\\noindent count the number of deliberate best responses taken by player $i$ through $t$.\n\nIn Section \\ref{sec_strong_FP_construction} the empirical distribution of player $i$, \\eqref{qt_update2}, is a time average taken only over rounds when player $i$ took a deliberate best response. In order to generalize this notion to an FP-type algorithm, define the term\n\\begin{equation}\n\\label{tau_def}\n\\tau_i(s) := \\inf\\{t:\\ell_i(t)=s\\}.\n\\end{equation}\nFor $s\\geq 1$, $\\tau_i(s)$ indicates the round when player $i$ took their $s$-th deliberate best response,\\footnote{Note that by \\eqref{tau_exist}, $\\tau_i(s)$ is finite valued a.s. for any $s\\in \\{1,~2,\\ldots\\}$.}\nand the sequence $\\{\\tau_i(s)\\}_{s\\geq 1}$ gives the subsequence of rounds when player $i$ took a deliberate best response. For $t\\in \\{1,~2,\\ldots\\}$ let\n$\\bar H_i(t) := \\{a_i(\\tau_i(s)):~\\tau_i(s) \\leq t\\}$\ndenote the action history of player $i$. Note that $\\bar H(t)$ records only the history of actions that were taken as deliberate best responses.\nLet the empirical distribution of player $i$ at time $t$ be formed as\n\\vskip-18pt\n\\begin{equation}\n\\label{qt_general_strong_update}\nq_i(t) := f_i^q(\\bar H_i(t),~\\ell_i(t)).\n\\end{equation}\n\\vskip-5pt\n\\noindent Let the asymptotic learning distribution (see \\textbf{A.\\ref{a_xi}} and subsequent discussion) be given by $\\xi_i(t) := f_i^\\xi(q(t))$ and  $\\xi(t) := (\\xi_1(t),\\ldots,~\\xi_i(t))$.\n\nLet the action for player $i$ in round $t\\geq 2$ be chosen according to the random rule\\footnote{To initialize the process, let the action $a_i(1)$ be chosen arbitrarily, let $X_i(1) = 1$, and let $\\bar H(1) = a_i(1)$ for all $i$.}\n\\begin{equation}\na_i(t) \\sim g'_i(t) :=\n\\begin{cases}\nb_i(t-1), & \\mbox{ if } X_i(t) = 1,\\\\\n\\xi_i(t-1), & \\mbox{otherwise},\n\\end{cases}\n\\label{action_rule1}\n\\end{equation}\nwhere $p_i(t-1) = f_i^p(q(t-1))$, and $b_i(t-1) \\in BR^{\\eta_t}_i(p_i(t-1)),$\nand assume:\\footnote{Note that this assumption subsumes the more typical assumption that $\\eta_t = 0,~\\forall t$. By making this more general assumption we are able to handle interesting scenarios that may arise in a practical implementation of the algorithm; e.g., players have some asymptotically decaying error in their knowledge of their utility function or knowledge of opponent's empirical distributions.}\n\\begin{assumption}\n\\label{a_eta}\nThe sequence $(\\eta_t)_{t\\geq 1}$ associated with $b_i(t)$ of \\eqref{action_rule1} is such that $\\lim\\limits_{t\\rightarrow\\infty} \\eta_t = 0$.\n\\end{assumption}\nLet $\\mathcal{F}_t := \\sigma(\\{a(s),X_i(s),\\ldots,X_n(s),b_1(s),\\ldots,b_n(s)\\}_{s\\leq t}).$\nLet the probability that player $i$ chooses a deliberate best response in round $t$ conditioned on past events be given by\n$\\rho_i(t) := \\mathbb{P}(X_i(t) = 1\\vert \\mathcal{F}_{t-1}),$\nand assume \\textbf{A.\\ref{rho_a1}}--\\textbf{A.\\ref{rho_a3}} hold.\nNote that $q_i(t)$, $p_i(t)$, $\\xi_i(t)$, and $g_i'(t)$ are $\\mathcal{F}_t$--measurable and that by definition, $\\rho_i(t)$ is $\\mathcal{F}_{t-1}$--measurable.\n\nFinally, let\n\\vskip-15pt\n\\begin{equation}\n\\label{g_def_general}\ng_i(t) := b_i(t-1)\\rho_i(t) + \\xi_i(t)(1-\\rho_i(t)).\n\\end{equation}\n\\vskip-5pt\n\\noindent Note that $g_i(t)$ is $\\mathcal{F}_{t-1}$--measurable and that $g(\\alpha_i,t) = \\mathbb{P}(a_i(t) = \\alpha_i\\vert \\mathcal{F}_{t-1})$; that is, $g_i(t)$ represents the mixed strategy in use by player $i$ in round $t$ (compare with \\eqref{g_def}). Let $g(t) := (g_1(t),\\ldots,g_n(t))$ denote the joint mixed strategy in use at time $t$.\n\nWe refer to a process where, for each player $i$, $q_i(t)$ is updated according to \\eqref{qt_general_strong_update}, $a_i(t)$ is updated according to \\eqref{action_rule1}, and $g_i(t)$ is updated according to \\eqref{g_def_general} as the strongly convergent variant of $\\Psi$ (for reasons to be clear soon---see Theorem \\ref{theorem_general_result}). In Section \\ref{sec_apps} we will demonstrate applications of this in the context of the previous examples.\n\n\\subsection{General Result}\n\\label{sec_FP_type_main_result}\nThe following theorem provides the general result from which the strong convergence of various FP-type algorithms can be derived.\n\\begin{theorem}\nLet $\\Gamma$ be a finite normal form game, let $E$ be an equilibrium set, and let $\\Psi$ be an FP-type algorithm satisfying \\textbf{A.\\ref{a_general_q}}--\\textbf{A.\\ref{a_robustness}}. If the strongly convergent variant of $\\Psi$ satisfies \\textbf{A.\\ref{rho_a1}}--\\textbf{A.\\ref{rho_a3}} and \\textbf{A.\\ref{a_eta}} then it achieves strong learning in the sense that $\\lim_{t\\rightarrow\\infty} d(g(t),~E) = 0$, almost surely.\n\\label{theorem_general_result}\n\\end{theorem}\n\nWe emphasize that in the above result players' period-by-period mixed strategies $g(t)$ are converging to equilibrium. In general,\nwhen seeking to construct the strongly convergent variant of some FP-type algorithm $\\Psi$, the most challenging aspect of applying Theorem \\ref{theorem_general_result} is the verification that $\\Psi$ satisfies \\textbf{A.\\ref{a_robustness}}. The remaining assumptions \\textbf{A.\\ref{a_general_q}}--\\textbf{A.\\ref{a_xi}} are generally fairly trivial to verify. Assumptions \\textbf{A.\\ref{rho_a1}}--\\textbf{A.\\ref{rho_a3}} and \\textbf{A.\\ref{a_eta}} pertain to the manner in which the strongly convergent variant of $\\Psi$ is constructed and are not related to intrinsic properties of $\\Psi$ itself.\n\n\\subsection{Some Additional Definitions}\n\\label{sec_additional_defs}\nIn order to prove Theorem \\ref{theorem_general_result} we will study the behavior of an underlying FP-type process that is embedded in the action, history, and empirical distribution processes produced by the strongly convergent variant of $\\Psi$. In particular, for $i\\in N$ and $s\\in \\{1,2,\\ldots\\}$, let $\\tau_i(s)$ be defined as in \\eqref{tau_def}, and define the following terms:\n$\\tilde a_i(s) := a_i(\\tau_i(s)),~\n \\tilde a(s) := (\\tilde a_1(s),\\ldots,\\tilde a_n(s)),~\n \\tilde H_i(s) := \\bar H_i(\\tau_i(s)),~\n \\tilde q_i(s) := q_i(\\tau_i(s)),~\n \\tilde q(s) := (\\tilde q_1(s),\\ldots,\\tilde q_n(s)),~\n \\tilde p_i(s) := f^p_i(\\tilde q(s)),~\n \\tilde \\xi(s) := (f^\\xi_1(\\tilde q(s)),\\ldots,f^\\xi_n(\\tilde q(s))).$\nThe aforementioned terms (marked with a tilde) correspond to to the embedded FP-type process that we will study in the proof of Theorem \\ref{theorem_general_result}. In particular, for each player $i$, the sequence $\\{\\tau_i(s)\\}_{s\\geq 1}$ denotes the subsequence of rounds when the player chose to play a deliberate best response. The sequence ${\\tilde a_i(s)}_{s\\geq 1}$ is the action sequence occurring along the subsequence of rounds when player $i$ chose to play a deliberate best response. The sequence $\\{\\tilde H_i(s)\\}_{s\\geq 1}$ corresponds to the action history of player $i$ along the same subsequence. The sequence $\\{\\tilde q_i(s)\\}_{s\\geq 1}$ corresponds to the empirical distribution of player $i$ along the same subsequence; in particular, note that by Lemma \\ref{q_tilde_lemma} (see appendix), $\\{\\tilde q_i(s)\\}_{s\\geq 1}$ fits the format prescribed by \\textbf{A.\\ref{a_general_q}} for the embedded FP-type process: $\\tilde q_i(s) = f_i^q(\\tilde H(s),s).$\nFinally, the term $\\tilde \\xi(s)$ is the asymptotic learning distribution associated with the embedded FP-type process.\n\nIn studying the embedded FP-type process, it will be important to characterize the terms to which players are best responding. With this in mind, note that per \\eqref{action_rule1}, the action at time $\\tau_i(s+1)$ (in the strongly convergent variant of $\\Psi$) is chosen as $a_i(\\tau_i(s+1)) \\in BR_i^{\\eta_{\\tau_i(s+1)}}(p_i(\\tau_i(s+1)-1))$. In order to translate this to the embedded FP-type process, define the following terms:\n$\\hat q^i_j(s) := q_j(\\tau_i(s+1)-1),~\n\\hat q^i(s) := (q_1(\\tau_i(s+1)-1),\\ldots,q_n(\\tau_i(s+1)-1))~\n\\hat p_i(s) := f_i^p(\\hat q^i(s)),$\nBy construction, the $(s+1)$-th action of player $i$ in the embedded FP-type process is chosen as,\n\\begin{equation}\n\\label{embedded_BR}\n\\tilde a_i(s+1) \\in BR_i^{\\eta_{\\tau_i(s+1)}}(\\hat p_i(s)).\n\\end{equation}\nIn the embedded FP-type process, the term $\\tilde q_j(s)$ may be thought of as the `true' empirical distribution of player $j$. The term $\\hat q_j^i(s)$ may be thought of as the estimate which player $i$ maintains of $\\tilde q_j(s)$, and the term $\\hat q^i(s)$ (note the superscript) may be thought of as player $i$'s estimate of the joint empirical distribution $\\tilde q(s)$ at the time of player $i$'s $(s+1)$-th best response. Finally, the term $\\hat p_i(s)$ may be thought of as player $i$'s prediction of opponents next-stage strategy given $\\hat q^i(s)$; in particular, note that---in the embedded FP-type process---player $i$ chooses their stage $(s+1)$ action \\eqref{embedded_BR} as an asymptotic best response to $\\hat p_i(s)$.\n\n\n\\subsection{Some Useful Properties}\n\\label{sec_useful_props}\nLet\n\\begin{equation}\n\\Omega' := \\{\\omega: \\lim_{t\\rightarrow\\infty} \\frac{\\ell_i(t)}{\\sum_{k=1}^t \\rho_i(t)} = 1,~ \\forall i \\}.\n\\end{equation}\nBy Lemma \\ref{IR3} (see appendix), there holds $\\mathbb{P}(\\Omega')=1$. In proving Theorem \\ref{theorem_general_result} we will restrict attention to (sample path) realizations in $\\Omega'$.\n\nNote that under assumption \\textbf{A.\\ref{rho_a2}}, there holds $\\{\\omega: \\lim_{t\\rightarrow\\infty}\\ell_i(t) = \\infty,~ \\forall i\\}\\supset \\Omega'.$ By the equivalence $\\{\\omega: \\lim_{t\\rightarrow\\infty}\\ell_i(t) = \\infty,~ \\forall i\\}=\\{\\omega:X_i(t)=1 \\mbox{ infinitely often } \\forall i\\}$, there holds $\\{\\omega:X_i(t)=1 \\mbox{ infinitely often } \\forall i\\}\\supset \\Omega'.$\nTherefore, by the definitions of $\\ell_i$ and $\\tau_i$, there holds for any realization in $\\Omega'$, $\\lim_{t\\rightarrow\\infty} \\ell_i(t) = \\infty$, and\n\\vskip-15pt\n\\begin{align}\n\\label{tau_exist}\n&\\tau_i(s) <\\infty, ~\\forall s\\in \\mathbb{N},\\\\\n\\label{tau_lim}\n&\\lim\\limits_{s\\rightarrow\\infty} \\tau_i(s) = \\infty.\n\\end{align}\n\\vskip-5pt\n\nThese properties will be useful in the proof of Theorem \\ref{theorem_main_result}. In particular, the proof will frequently make reference to $\\tilde q_i(s)$, or $\\tilde a_i(s)$ for arbitrary $s\\in \\mathbb{N}$---the property \\eqref{tau_exist} ensures that such terms are well defined for any $\\omega \\in \\Omega'$.\n\nNote also that for any realization in $\\Omega'$, for $i\\in N$ and $s\\in \\{1,2,\\ldots\\}$,\n\\vskip-15pt\n\\begin{equation}\n\\label{ell_tau_eq}\n\\ell_i(\\tau_i(s)) = s,\n\\end{equation}\n\\vskip-15pt\n\\noindent and for $i\\in N$ and $t\\in \\{1,2,\\ldots\\}$\n\\vskip-15pt\n\\begin{equation}\n\\label{X_i_implication}\nX_i(t) = 1 \\implies \\tau_i(\\ell_i(t)) = t.\n\\end{equation}\n\\vskip-5pt\n\\noindent Furthermore, note that $X_i(t) = 0$ implies that $\\ell_i(t) = \\ell_i(t-1)$ and $\\bar H_i(t) = \\bar H_i(t-1)$, and in particular,\n\\vskip-20pt\n\\begin{align}\n\\label{q_step_equality}\nX_i(t) = 0 \\implies q_i(t) = q_i(t-1).\n\\end{align}\n\\vskip-10pt\n\\noindent These facts are readily verified by conferring with the definitions of $\\tau_i$, $\\ell_i$, and $X_i$.\n\n\\subsection{Proof of Theorem \\ref{theorem_general_result}}\n\\label{sec_proof_general_result}\n\\begin{proof}\nSince $\\mathbb{P}(\\Omega') = 1$ it is sufficient to show that the desired result holds for any $\\omega \\in \\Omega'$. Henceforth, we restrict attention to realizations $\\omega \\in \\Omega'$, and for ease of notation suppress the term $\\omega$ when referring to random variables.\n\nAs a first step, we wish to show that $\\lim_{s\\rightarrow\\infty} d(\\tilde \\xi(s),~E) = 0$. We accomplish this by showing that there exists a sequence $\\{\\epsilon_s\\}_{s\\geq 1}$ such that $\\lim_{s\\rightarrow\\infty}\\epsilon_s = 0$ and $\\tilde a_i(s+1) \\in BR_i^{\\epsilon_s}(\\tilde p_i(s))$. By assumption \\textbf{A.\\ref{a_robustness}}, it will then follow that $\\lim_{s\\rightarrow\\infty} d(\\tilde \\xi(s),~E) = 0$.\n\nTo that end, note that by Lemma \\ref{lemma_BR_limit} (see appendix),\n$\\lim\\limits_{s\\rightarrow\\infty} |U_i(a_i(\\tau_i(s+1)),p_i(\\tau_i(s+1)-1)) - v_i(p_i(\\tau_i(s+1)-1))| = 0,~\\forall i,$\nor equivalently by the definitions of $\\tilde a(s)$ and $\\hat p_i(s)$ (see Section \\ref{sec_additional_defs}),\n\\vskip-20pt\n\\begin{align}\n\\lim\\limits_{s\\rightarrow\\infty} |U_i(\\tilde a_i(s+1)),\\hat p_i(s)) - v_i(\\hat p_i(s))| = 0,~\\forall i.\n\\label{thrm1_eq1}\n\\end{align}\n\\vskip-5pt\n\\noindent By Lemma \\ref{IR0} (see appendix), $\\lim_{s\\rightarrow\\infty} \\|\\hat q^i(s) - \\tilde q(s)\\| = 0$. By \\textbf{A.\\ref{a_prediction}}, it follows that $\\lim_{s\\rightarrow\\infty} \\|\\hat p_i(s) - \\tilde p_i(s)\\| = 0$, which by the Lipschitz continuity of $U_i(\\cdot)$ implies that\n$\\lim_{s\\rightarrow\\infty} | U_i(\\alpha_i,\\hat p_i(s)) - U_i(\\alpha_i,\\tilde p_i(s))| = 0,~ \\forall \\alpha_i \\in A_i, \\forall i,$\nand\n$\\lim_{s\\rightarrow\\infty} |v_i(\\hat p_i(s)) - v_i(\\tilde p_i(s))| = 0, \\forall i.$\nReturning to \\eqref{thrm1_eq1} we see that\n$\\lim\\limits_{s\\rightarrow\\infty} |U_i(\\tilde a_i(s+1)),\\tilde p_i(s)) - v_i(\\tilde p_i(s))| = 0, ~\\forall i,$\ni.e., there exists a sequence $\\{\\epsilon_s\\}_{s\\geq 1}$ such that $\\epsilon_s \\rightarrow 0$ and $\\tilde a_i(s+1) \\in BR_i^{\\epsilon_s}(\\tilde p_i(s))$. It follows by \\textbf{A.\\ref{a_robustness}} that\n\\vskip-20pt\n\\begin{equation}\n\\lim\\limits_{s\\rightarrow\\infty} d(\\tilde \\xi(s),~E) = 0.\n\\label{theorem_main_result_eq3}\n\\end{equation}\n\\vskip-5pt\n\nWe now proceed to show that $\\lim_{t\\rightarrow\\infty} d(\\xi(t),~E) =0$. Let $\\varepsilon > 0$ be given. By Lemma \\ref{IR1} (see appendix) and assumption \\textbf{A.\\ref{a_xi}}, for each $i\\in N$, there exists a random time $S_i > 0$ such that $\\forall s\\geq S_i$, $\\|\\xi(\\tau_i(s)) - \\tilde\\xi(s)\\| < \\frac{\\varepsilon}{2}$. Let $S^{'} = \\max_i\\{S_i\\}$. By \\eqref{theorem_main_result_eq3} there exists a random time $S^{''}$ such that $\\forall s\\geq S^{''}$, $d(\\tilde \\xi(s), ~E) < \\frac{\\varepsilon}{2}$. Let $S=\\max\\{S^{'},S^{''}\\}$. Then\n\\vskip-15pt\n\\begin{equation}\nd(\\xi(\\tau_i(s)),~E) < \\varepsilon, ~\\forall i,~ \\forall s\\geq S.\n\\label{thrm1_eq6}\n\\end{equation}\n\\vskip-5pt\n\nLet $T = \\max_{i}\\{\\tau_i(S)\\}$. Note that for some $i$, $\\xi(T) = \\xi(\\tau_i(S))$, and therefore by \\eqref{thrm1_eq6},\n\\vskip-15pt\n\\begin{equation}\nd(\\xi(T),~E) < \\varepsilon.\n\\label{thrm1_eq4}\n\\end{equation}\n\\vskip-5pt\nAlso note that for any $t_0>T$, it holds that $\\ell_i(t_0) \\geq S$ (since $\\ell_i(\\tau_i(S)) = S$, and $\\ell_i(t)$ is non-decreasing in $t$), and moreover\n\\vskip-15pt\n\\begin{align}\nX_i(t_0) = 1 \\mbox{ for some } i & ~\\implies ~q(t_0) = q(\\tau_i(\\ell_i(t_0))) \\implies \\xi(t_0) = \\xi(\\tau_i(\\ell_i(t_0))),\\\\\nX_i(t_0) = 0 \\mbox{ for all $i$ } & ~\\implies ~q(t_0) = q(t_0-1) \\implies \\xi(t_0) = \\xi(t_0-1) ,\n\\label{thrm1_eq5}\n\\end{align}\n\\vskip-5pt\nwhere the first implication holds with $ ~\\mbox{ with } \\ell_i(t_0) \\geq S$. In the above, the first line follows from \\eqref{X_i_implication}, and the second line follows from \\eqref{q_step_equality}.\nConsider $t\\geq T$. If for some $i$, $X_i(t) = 1$, then by \\eqref{thrm1_eq5} and \\eqref{thrm1_eq6},\n$d(\\xi(t),~E) = d(\\xi(\\tau_i(\\ell_i(t))),~E) < \\varepsilon.$\nOtherwise, if $X_i(t) = 0 ~\\forall i$, then $\\xi(t) = \\xi(t-1)$.\n\nIterate this argument $m$ times until either (i) $X_i(t-m) = 1$ for some $i$, or (ii), $t-m = T$. In the case of (i),\n$d(\\xi(t),~E) = d(\\xi(t-m),~E) = d(\\xi(\\tau_i(\\ell_i(t-m))),~E) < \\varepsilon,$\nwhere the inequality again follows from \\eqref{thrm1_eq6} and the fact that $t-m>T \\implies \\ell_i(t-m)\\geq S$.\nIn the case of (ii),\n$d(\\xi(t),~E) = d(\\xi(T),~E) < \\varepsilon,$\nwhere the inequality  follows from \\eqref{thrm1_eq4}. Since $\\varepsilon >0$ was chosen arbitrarily, it follows that\n$\\lim\\limits_{t\\rightarrow\\infty} d(\\xi(t),~E)=0.$\n\nFinally, we show that $\\lim_{t\\rightarrow\\infty} d(g(t),~E)=0$. Note that by \\eqref{g_def_general}, $\\|g_i(t) - \\xi_i(t-1)\\| \\leq M_i\\rho_i(t), ~\\forall i,$ where $M_i:=\\max_{p',p'' \\in \\Delta_i} \\|p' - p''\\|$ is a constant. Invoking assumption \\textbf{A.\\ref{rho_a1}} gives, $\\lim\\limits_{t\\rightarrow\\infty}\\|g_i(t) - \\xi_i(t-1)\\|=0, ~\\forall i.$\nCombining this with the fact that $\\lim\\limits_{t\\rightarrow\\infty} d(\\xi(t),~E)=0$ yields the desired result, $\\lim_{t\\rightarrow\\infty}d(g(t),~E) =0$.\n\\end{proof}\n\n\n\\section{Applications of the General Result}\n\\label{sec_apps}\nIn this section we consider three different FP-type algorithms and study the strongly convergent variant of each. In each case, we prove strong convergence by showing that the FP-type algorithm fits the template of Theorem \\ref{theorem_general_result}. Generally, the only non-trivial aspect of applying Theorem \\ref{theorem_general_result} will be to show that \\textbf{A.\\ref{a_robustness}} is satisfied.\n\nIn Section \\ref{sec_apps1} we consider classical FP. The fact that classical FP satisfies \\textbf{A.\\ref{a_robustness}} was shown by Leslie et al. \\cite{leslie2006generalised}. In Section \\ref{sec_apps2} we consider GWFP---a generalization of FP proposed in \\cite{leslie2006generalised}. Again, the crucial step of showing that GWFP satisfies \\textbf{A.\\ref{a_robustness}} was shown in \\cite{leslie2006generalised}. In Section \\ref{sec_apps3} we consider a variant of FP termed ECFP. That ECFP satisfies \\textbf{A.\\ref{a_robustness}} was shown in \\cite{swenson2015weakECFP}. We emphasize that each of these algorithms is known to achieve weak learning in the sense that $d(\\xi(t),~E) \\rightarrow 0$ as $t\\rightarrow \\infty$. Our contribution is to construct a variant where players also achieve learning in the strong sense that period-by-period mixed strategies also converge to equilibrium.\n\n\\subsection{Strong Convergence in Classical FP}\n\\label{sec_apps1}\nWe now prove Corollary \\ref{theorem_main_result} using the general convergence result of Theorem \\ref{theorem_general_result}.\n\n\\begin{proof}\nClassical FP fits the template of an FP-type algorithm with the empirical distribution given by $q_i(t) = \\frac{1}{t}\\sum_{s=1}^ta_i(s)$, the functions $f_i^p$ and $f_i^\\xi$ given by the identity function for each $i$, and the best response perturbation given by $\\epsilon_t = 0,~\\forall t$. To show that the strongly convergent variant of classical FP attains strong learning, it suffices to show that the assumptions of Theorem \\ref{theorem_general_result} are met.\n\nTo that end, note that \\textbf{A.\\ref{rho_a1}}--\\textbf{A.\\ref{rho_a3}} are satisfied by assumption, and \\textbf{A.\\ref{a_eta}} is trivially satisfied (with $\\eta_t = 0,~\\forall t$). Furthermore, the empirical distribution sequence satisfies $\\lim_{t\\rightarrow\\infty}\\|q_i(t) - q_i(t-1)\\| = 0$ (see Section \\ref{sec_FP_example}), and hence \\textbf{A.\\ref{a_step_size_bound}} is satisfied. The functions $f_i^p$ and $f_i^\\xi$ (each being the identity function) satisfy \\textbf{A.\\ref{a_prediction}}--\\textbf{A.\\ref{a_xi}}.\nTherefore, it is sufficient to show that \\textbf{A.\\ref{a_robustness}} is satisfied. But, for zero-sum games, potential games, and generic $2\\times m$ games this holds by \\cite{leslie2006generalised}, Corollary 5.\n\\end{proof}\n\n\\subsection{Strong Convergence in Generalized Weakened FP}\n\\label{sec_apps2}\n\nGWFP was introduced in Section \\ref{sec_GWFP_intro}, where it was shown to fit the template of an FP-type algorithm.\n\nSince, by definition, a GWFP process allows players to choose an $\\epsilon_t$ sub-optimal best response with $\\epsilon_t \\rightarrow 0$, the following result (\\cite{leslie2006generalised}, Corollary 5) guarantees a GWFP process satisfies \\textbf{A.\\ref{a_robustness}} in the noted classes of games.\n\\begin{theorem}\nAny generalized weakened fictitious play process will converge to the set of Nash equilibria in two-player zero-sum games, potential games, and generic $2\\times m$ games.\n\\label{thrm_wfp}\n\\end{theorem}\n\nTo clarify the precise meaning of the convergence stated above as it relates to the present work, we emphasize that Theorem \\ref{thrm_wfp} implies that $\\lim_{t\\rightarrow\\infty} d(q(t),~NE)=0$; i.e., the process converges weakly to equilibrium.\n\nLet the strongly convergent variant of GWFP be constructed using the approach laid out in Section \\ref{sec_FP_type_strong}. The following Corollary to Theorem \\ref{theorem_general_result} states that the strongly convergent variant of a GWFP process will achieve strong learning.\\footnote{It should be noted that classical FP may be seen as an instance of GWFP, and thus Corollary \\ref{theorem_main_result} may in fact be deduced as a corollary to Corollary \\ref{cor_gwfp}. However, for clarity and continuity of presentation, the results regarding classical FP have been presented separately.}\n\\begin{cor}\n\\label{cor_gwfp}\nLet $\\Gamma$ be a two-player zero-sum game, potential game, or generic $2\\times m$ game. Let $\\Psi$ be an instance of GWFP. If the strongly convergent variant of $\\Psi$ satisfies \\textbf{A.\\ref{rho_a1}}--\\textbf{A\\ref{rho_a3}} and \\textbf{A.\\ref{a_eta}}, then it achieves strong learning in the sense that $\\lim_{t\\rightarrow\\infty} d(g(t),~NE) = 0$.\n\\end{cor}\n\\begin{proof}\nIt is sufficient to show that the conditions of Theorem \\ref{theorem_general_result} are met. Note that \\textbf{A.\\ref{rho_a1}}--\\textbf{A.\\ref{rho_a3}}, \\textbf{A.\\ref{a_eta}} hold by assumption. Furthermore, by definition, any GWFP process satisfies $\\lim_{t\\rightarrow\\infty} \\gamma(t)=0$, and hence satisfies \\textbf{A.\\ref{a_step_size_bound}}. The functions $f_i^p$ and $f_i^\\xi$ are given by the identity function for each $i$, and hence \\textbf{A.\\ref{a_prediction}} and \\textbf{A.\\ref{a_xi}} hold.  Thus, it suffices to show that \\textbf{A.\\ref{a_robustness}} holds for the specified class of games---but, this follows from Theorem \\ref{thrm_wfp}.\n\\end{proof}\n\n\n\\subsection{Strong Convergence in Empirical Centroid FP}\n\\label{sec_apps3}\nECFP was introduced in Sections \\ref{sec_ECFP_intro} and \\ref{sec_ecfp_mce_intro}. It\nIn order to study the asymptotic behavior of ECFP (in either of the above formats introduced in Sections \\ref{sec_ECFP_intro} and \\ref{sec_ecfp_mce_intro}) we make the following assumption regarding the structure of players' utility functions:\n\\begin{assumption}\nThe players' utility functions are identical and permutation invariant. That is, for any $i,j\\in N$, $u_i(y) = u_j(y)$, and $u([y']_i,[y'']_j,y_{-(i,j)}) = u([y'']_i,[y']_j,y_{-(i,j)}),$\nwhere, for any player $k\\in N$, the notation $[y']_k$ indicates the action $y'\\in Y_k$ being played by player $k$, and $y_{-(i,j)}$ denotes the set of actions being played by all players other than $i$ and $j$.\n\\label{a_perm_inv}\n\\end{assumption}\n\nWe note that, under this assumption, the sets $C$ and $MCE$ are nonempty \\cite{swenson2012ECFP,swenson2013MCE}.\nThe following theorem (\\cite{swenson2015weakECFP}, Theorem 1) specifies the manner in which players engaged in an ECFP process (weakly) learn elements of the sets $C$ and $MCE$.\n\\begin{theorem}\nLet $\\{a(t)\\}_{t\\geq 1}$ be an ECFP process.  \\\\\nAssume $\\Gamma$ is such that \\textbf{A.\\ref{a_ident_strat}} and \\textbf{A.\\ref{a_perm_inv}} hold. Then players learn equilibrium strategies in the sense that\n(i) $\\lim_{t\\rightarrow \\infty} d(\\bar q^n(t),~C) = 0$, and (ii) $\\lim_{t\\rightarrow \\infty} d(q(t),~MCE) = 0$.\n\\label{theorem_ecfp_weak}\n\\end{theorem}\n\nNote that case (i) above corresponds to ECFP with the convergence map $f_i^\\xi$ as given in Section \\ref{sec_ECFP_intro}, and case (ii) corresponds to the convergence map $f_i^\\xi$ given by the identity function (as in Section \\ref{sec_ecfp_mce_intro}). Since, by definition, an ECFP process \\eqref{ecfp_process} allows players to choose actions from the $\\epsilon_t$-sub-optimal best response set with $\\epsilon_t \\rightarrow 0$, Theorem \\ref{theorem_ecfp_weak} ensures that ECFP satisfies \\textbf{A.\\ref{a_robustness}}.\n\nLet $\\Psi$ be an instance of ECFP as presented in either Section \\ref{sec_ECFP_intro} or Section \\ref{sec_ecfp_mce_intro}, and let the strongly convergent variant of $\\Psi$ be constructed using the approach laid out in Section \\ref{sec_FP_type_strong}.\nThe following corollary to Theorem \\ref{theorem_general_result} states that players engaged in the strongly convergent variant of an ECFP process learn elements of $C$ and $MCE$ in the strong sense that players' period-by-period strategies converge to equilibrium.\n\\begin{cor}\n(i) Let $\\Psi$ be an instance of ECFP with $f^{\\xi}_i(q) = \\frac{1}{n}\\sum_jq_j,~\\forall i$ and assume $\\Gamma$ is such that \\textbf{A.\\ref{a_ident_strat}} and \\textbf{A.\\ref{a_perm_inv}} hold. If the strongly convergent variant of $\\Psi$ satisfies \\textbf{A.\\ref{rho_a1}}--\\textbf{A.\\ref{rho_a3}} and \\textbf{A.\\ref{a_eta}}, then it achieves strong learning in the sense that $\\lim_{t\\rightarrow 0} d(g(t),~C)=0$.\\\\\n(ii) Let $\\Psi$ be an instance of ECFP with $f^{\\xi}_i(q)$ given by the identity function for all $i$ and assume $\\Gamma$ is such that \\textbf{A.\\ref{a_ident_strat}} and \\textbf{A.\\ref{a_perm_inv}} hold. If the strongly convergent variant of $\\Psi$ satisfies \\textbf{A.\\ref{rho_a1}}--\\textbf{A.\\ref{rho_a3}} and \\textbf{A.\\ref{a_eta}}, then it achieves strong learning in the sense that $\\lim_{t\\rightarrow 0} d(g(t),~MCE)=0$.\n\\end{cor}\n\\begin{proof}\nCases (i) and (ii) differ only in terms of the function $f^{\\xi}_i(t)$ and target equilibrium set $E$. However, in both cases the function $f^{\\xi}_i$ satisfies \\textbf{A.\\ref{a_xi}}. It suffices to show the remaining conditions of Theorem \\ref{theorem_general_result} are satisfied. Henceforth we treat cases (i) and (ii) equivalently.\n\nNote that \\textbf{A.\\ref{rho_a1}}--\\textbf{A.\\ref{rho_a3}} and \\textbf{A.\\ref{a_eta}} hold by assumption. The empirical distribution sequence satisfies $\\|q_i(t) - q_i(t-1)\\| \\leq \\frac{M_i}{t} \\rightarrow 0 \\mbox{ as } t\\rightarrow \\infty$, where $M_i := \\sup_{p',p'' \\in \\Delta_i} \\|p' - p''\\|$, and hence \\textbf{A.\\ref{a_step_size_bound}} is satisfied. Note that the function $f_i^p(q) = \\frac{1}{n}\\sum_j q_j$ satisfies \\textbf{A.\\ref{a_prediction}}.  Finally, Theorem \\ref{theorem_ecfp_weak} shows that \\textbf{A.\\ref{a_robustness}} is satisfied.\n\\end{proof}\n\n\n\\section{Conclusions}\n\\label{sec_conclusion}\nAn algorithm is said to achieve weak learning if players learn an equilibrium strategy in an abstract sense (see Section \\ref{sec_prelims}), but period-by-period strategies do not necessarily converge to equilibrium. An algorithm is said to achieve strong learning if (additionally) players' period-by-period strategies converge to equilibrium. Weak learning may be thought of as a form of learning where players \\emph{learn} a strategy in some abstract sense, but never begin to implement the strategy they are learning. On the other hand, in strong learning, not only do players \\emph{learn} a strategy, but they also physically implement the learned strategy through the course of the learning process.\n\nFictitious Play (FP) and its variants are known to exhibit weak learning but not necessarily strong learning. An approach was presented for taking a general FP-type algorithm that achieves weak learning, and constructing from it a strongly convergent variant of the algorithm. General convergence results were proved and used to construct a strongly convergent variant of several example FP-type processes.\n\nIn order to apply the convergence results proved in this paper, it is necessary to ensure a candidate algorithm meets \\textbf{A.\\ref{a_robustness}} (the other necessary assumptions are relatively trivial to verify). An interesting future research direction might be to investigate other FP-type algorithms (e.g., \\cite{Arslan04,Shamma03}) and verify whether they meet the assumptions sufficient for construction of a strongly convergent variant.\n\n\\section*{Appendix}\n{\\small\n\\subsection{Some Useful Inequalities}\n\\label{sec_IR}\nWe consider some useful inequalities related to the strongly convergent variant of an FP-type algorithm. We restrict attention to realizations $\\omega \\in \\Omega'$. Let $\\{q_i(t)\\}_{t\\geq 1}$ be given by \\eqref{qt_general_strong_update}.\nBy \\textbf{A.\\ref{a_step_size_bound}} there exists a sequence $\\gamma(t)$ such that\n$\\lim\\limits_{t\\rightarrow\\infty} \\gamma(t) = 0,$\nand for each $i\\in N$,\n\\vskip-5pt\n\\begin{equation}\n\\|q_i(t+1) - q_i(t)\\| \\leq M_i\\gamma(\\ell_i(t)),\n\\label{qt_bound}\n\\end{equation}\nwhere $M_i:= \\sup_{q',q'' \\in \\Delta_i}\\|q' - q''\\|$.\nSimilarly, there holds for any integer $s>0$,\n\\begin{equation}\n\\|\\tilde q(s+1) - \\tilde q(s)\\| \\leq M\\gamma(s),\n\\label{qs_bound1}\n\\end{equation}\nwhere $M:= \\sup_{q',q'' \\in \\Delta^n}\\|q' - q''\\|$.\nMore generally, for any integers $s_1,s_2>0$, if \\textbf{A.\\ref{a_step_size_bound}} holds then,\n\\vskip-20pt\n\\begin{align}\n\\|\\tilde q(s_1) - \\tilde q(s_2)\\| & \\leq M\\sum\\limits_{s=\\min\\{s_1,s_2\\}}^{\\max\\{s_1,s_2\\}-1} \\gamma(s)\\leq |s_1 - s_2|B,\n\\label{qs_bound2}\n\\end{align}\n\\vskip-5pt\n\\noindent where $0<B<\\infty$ is such that $\\sup_t \\gamma(t) \\leq B\/M$.\n\n\\subsection{Intermediate Results}\n$~$\\\\\n\\vskip-5pt\n\\begin{lemma}\n\\label{lemma_BR_limit}\nLet $\\tau_i(s)$ be defined as in section \\ref{sec_additional_defs}, and assume \\textbf{A.\\ref{a_eta}} holds. Then for any realization in $\\Omega'$ there holds,\n$\\lim\\limits_{s\\rightarrow\\infty} |U_i(a_i(\\tau_i(s)),p_i(\\tau_i(s)-1)) - v_i(p_i(\\tau_i(s)-1))| = 0,~\\forall i.$\n\\end{lemma}\n\\begin{proof}\nLet $s\\in \\mathbb{N}$. Note that by definition $\\tau_i(s) := \\inf\\{t:\\ell_i(t) = s\\}$ and $\\ell_i(t) := \\sum_{k=1}^t X_i(k)$, thus $X_i(\\tau_i(s)) = 1$. By \\eqref{action_rule1} this implies\n$a_i(\\tau_i(s)) = b_i(\\tau_i(s)) \\in BR_i^{\\eta_{\\tau_i(s)}}(p_i(\\tau_i(s)-1)),$\nwhich implies\n$|U_i(a_i(\\tau_i(s)),p_i(\\tau_i(s)-1)) - v_i(p_i(\\tau_i(s)-1))| \\leq \\eta_{\\tau_i(s)}.$\nBy \\textbf{A.\\ref{a_eta}}, $\\eta_t \\rightarrow 0$ as $t\\rightarrow \\infty$, and moreover, by \\eqref{tau_lim}, $\\tau_i(s) \\rightarrow \\infty$ as $s\\rightarrow \\infty$. Thus $\\eta_{\\tau_i(s)} \\rightarrow 0$ as $s\\rightarrow \\infty$, and the claim holds.\n\\end{proof}\n\n\\begin{lemma}\nLet $i,j\\in N$, let $\\tau_i(s)$ and $\\tilde q_j(s)$ be defined as in Section \\ref{sec_additional_defs}, and assume \\textbf{A.\\ref{rho_a2}}--\\textbf{A.\\ref{rho_a3}} hold. Then for any realization in $\\Omega'$,\n$\\lim_{s\\rightarrow\\infty} \\|  q_j(\\tau_i(s)) - \\tilde q_j(s)\\| = 0.$\n\\label{IR1}\n\\end{lemma}\n\\begin{proof}\nNote that for any $t\\in \\mathbb{N}$,\n$q_j(t) = q_j(\\tau_j(\\ell_j(t))) =  \\tilde q_j(\\ell_j(t)),$\nwhere the first equality follows from Lemma \\ref{IR6}, and the second equality follows from the definition of $\\tilde q_i(s)$. Hence,\n\\vskip-15pt\n\\begin{align}\n\\|q_j(\\tau_i(s)) - \\tilde q_j(s)\\| = \\|\\tilde q_j(\\ell_j(\\tau_i(s))) - \\tilde q_j(s)\\|\n& = \\|\\tilde q_j(\\ell_j(\\tau_i(s))) - \\tilde q_j(\\ell_i(\\tau_i(s)))\\|\\\\\n& \\leq \\vert\\ell_j(\\tau_i(s)) - \\ell_i(\\tau_i(s))\\vert B,\n\\label{IR1_eq1}\n\\end{align}\n\\vskip-5pt\nwhere the first equality follows from the previous statement, and the second equality follows from the fact that $\\ell_i(\\tau_i(s)) = s$ (see \\eqref{ell_tau_eq}), and the final inequality follows from \\eqref{qs_bound2}. Thus, it suffices to show that\n\\vskip-10pt\n\\begin{equation}\n\\lim\\limits_{s\\rightarrow\\infty} \\vert\\ell_j(\\tau_i(s)) - \\ell_i(\\tau_i(s))\\vert = 0.\n\\label{IR1_eq2}\n\\end{equation}\nFor convenience in notation let $h_i(t) := \\sum_{m=1}^t \\rho_i(m)$. By Lemma \\ref{IR3} and the definition of $\\Omega'$ there holds for any $k\\in N$,\n$\\lim_{t\\rightarrow\\infty} \\frac{ \\ell_k(t) }{h_k(t)} = 1.$\nBy assumption \\textbf{A.\\ref{rho_a3}}, for any $k\\in N$, $\\lim_{t\\rightarrow\\infty} \\left(h_k(t) \/ (h_i(t)\\right) = 1.$ Hence, for any $k\\in N$,\n\\begin{align}\n\\lim\\limits_{t\\rightarrow\\infty} \\frac{\\ell_k(t)}{h_i(t)} = \\lim\\limits_{t\\rightarrow\\infty} \\frac{\\ell_k(t)}{h_k(t)} \\frac{h_k(t)}{h_i(t)}\n=1.\n\\label{IR1_eq4}\n\\end{align}\nReturning attention to \\eqref{IR1_eq2} and recalling that by \\eqref{tau_lim}, $\\lim_{s\\rightarrow\\infty}\\tau_i(s)=\\infty$ on $\\Omega'$, we have,\n\\vskip-10pt\n\\begin{align}\n\\limsup\\limits_{s\\rightarrow\\infty}\\left| \\ell_j(\\tau_i(s)) - \\ell_i(\\tau_i(s))\\right|\n\\leq \\limsup\\limits_{t\\rightarrow\\infty} \\left|\\ell_j(t) - \\ell_i(t) \\right|\n& = \\limsup\\limits_{t\\rightarrow\\infty} \\left| \\frac{\\ell_j(t)}{h_i(t)} h_i(t) - \\frac{\\ell_i(t)}{h_i(t)}h_i(t)\\right|\\\\\n& = \\limsup\\limits_{t\\rightarrow\\infty} \\left|h_i(t) - h_i(t)\\right| = 0,\n\\label{IR1_eq3}\n\\end{align}\n\\vskip-5pt\n\\noindent where the transition to the last line follows from application of \\eqref{IR1_eq4}.\nThus, \\eqref{IR1_eq2} is verified, and the desired result holds.\n\\end{proof}\n\n\\begin{lemma}\nLet $i,j\\in N$, let $\\hat q_j^i(s)$ and $\\tilde q_j(s)$ be defined as in Section \\ref{sec_additional_defs}, and assume \\textbf{A.\\ref{rho_a2}}--\\textbf{A.\\ref{rho_a3}} hold. Then for any realization in $\\Omega'$ there holds\n$\\lim_{s\\rightarrow\\infty} \\|  \\hat q_j^i (s) - \\tilde q_j(s)\\| = 0.$\n\\label{IR0}\n\\end{lemma}\n\\begin{proof}\nRecall that by definition, $\\hat q_j^i(s) = q_j(\\tau_i(s+1)-1)$; our objective then is to show that\n$\\lim_{s\\rightarrow\\infty} \\|  q_j (\\tau_i(s+1)-1) - \\tilde q_j(s)\\| = 0.$\nBy Lemma \\ref{IR1},\n$\\lim\\limits_{s\\rightarrow\\infty} \\|q_j(\\tau_i(s)) - \\tilde q_j(s)\\| = 0.$\nBy \\eqref{qs_bound1} and \\textbf{A.\\ref{a_step_size_bound}} there holds,\n$\\lim\\limits_{s\\rightarrow\\infty} \\|\\tilde q_j(s+1) - \\tilde q_j(s)\\| = 0.$\nCombining this with the previous statement,\n\\vskip-10pt\n\\begin{equation}\n\\lim\\limits_{s\\rightarrow\\infty} \\|q_j(\\tau_i(s+1)) - \\tilde q_j(s)\\|=0.\n\\label{IR0_eq2}\n\\end{equation}\n\\vskip-5pt\n\\noindent Recalling \\eqref{qt_bound}, there holds,\n\\vskip-15pt\n\\begin{align}\n\\limsup\\limits_{s\\rightarrow\\infty} \\|q_j(\\tau_i(s+1)-1) - q_j(\\tau_i(s+1))\\| \\leq \\limsup\\limits_{s\\rightarrow\\infty}M_{j} \\gamma(\\ell_j(\\tau_i(s+1))) = 0,\n\\label{IR0_eq3}\n\\end{align}\n\\vskip-10pt\n\\noindent where the equality holds since $\\lim_{s\\rightarrow\\infty} \\ell_j(\\tau_i(s)) = \\infty$ on $\\Omega'$, and by \\textbf{A.\\ref{a_step_size_bound}}, $\\lim_{s\\rightarrow\\infty} \\gamma(s) = 0$.\n\nConsider now the quantity of interest,{\\small\n\\vskip-15pt\n\\begin{align}\n\\|q_j(\\tau_i(s+1)-1) - \\tilde q_j(s)\\| \\leq & \\|q_j(\\tau_i(s+1)-1) - q_j(\\tau_i(s+1))\\| + \\|q_j(\\tau_i(s+1)) - \\tilde q_j(s)\\|.\n\\end{align}\n}\n\\vskip-15pt\n\\noindent The first term on the right hand side (RHS) goes to zero by \\eqref{IR0_eq3} and the second term on the RHS goes to zero by \\eqref{IR0_eq2}. Thus,\n$\\lim\\limits_{s\\rightarrow\\infty} \\|q_j(\\tau_i(s+1)-1) - \\tilde q_j(s)\\| = 0,$ and the claim holds.\n\\end{proof}\n\n\\begin{lemma}\nLet $i\\in N$, let $q_i(\\cdot)$ be as defined in \\eqref{qt_general_strong_update}, let $\\ell_i(\\cdot)$ be as defined in \\eqref{ell_def2}, and let $\\tau_i(\\cdot)$ be as defined in \\eqref{tau_def}. Then for every realization in $\\Omega'$ and any $t\\in\\{1,2,\\ldots\\}$ there holds $q_i(\\tau_i(\\ell_i(t))) = q_i(t).$\n\\label{IR6}\n\\end{lemma}\n\\begin{proof}\nLet\n$t_0 := \\tau_i(\\ell_i(t)) = \\inf\\{t':\\ell_i(t') = \\ell_i(t)\\},$ where the second equality follows from the definition of $\\tau_i(\\cdot)$. Note that $t_0 \\leq t$ and by definition of $t_0$, there holds $\\tau_i(\\ell_i(t_0)) = t_0$, and hence\n$ q_i(\\tau_i(\\ell_i(t_0)))=q_i(t_0).$\nFurthermore, by the definition of $t_0$, for $t_0  \\leq t' \\leq t$, there holds $\\ell_i(t) = \\ell_i(t') = \\ell_i(t_0)$, and hence\n$\\tau_i(\\ell_i(t)) = \\tau_i(\\ell_i(t_0)).$\nMoreover, the fact that $\\ell_i(t) = \\ell_i(t') = \\ell_i(t_0)$ implies by definition of $\\ell_i(\\cdot)$ that $X_i(t') = 0$ for $t_0 < t' \\leq t$ (if such a $t'$ exists). Thus, by \\eqref{q_step_equality} there holds $q_i(t) = q_i(t') = q_i(t_0)$ for $t_0 \\leq t'\\leq t$, and in particular $q_i(t) = q_i(t_0).$\nCombining this with the facts that $q_i(\\tau_i(\\ell_i(t_0)))=q_i(t_0)$ and $\\tau_i(\\ell_i(t)) = \\tau_i(\\ell_i(t_0))$ yields the desired result.\n\\end{proof}\n\n\\begin{lemma}\n\\label{q_tilde_lemma}\nLet $\\Psi = (\\{f^q_i(\\cdot,t)\\}_{t\\geq 1},f_i^p,f^\\xi_i)_{i\\in N}$ be an FP-type algorithm, and let the strongly convergent variant of $\\Psi$ be constructed as in Section \\ref{sec_FP_type_strong}.\nLet $\\tilde a(s)$, $\\tilde H(s)$, and $\\tilde q_i(s)$ be as defined in Section \\ref{sec_additional_defs}. Then for every realization in $\\Omega'$, and for $s\\geq 1$,\n$\\tilde q_i(s) = f_i^q(\\tilde H(s),s).$\n\\end{lemma}\n\\begin{proof}\nFor $s\\geq 1$, note that\n$\\tilde q_i(s) = q_i(\\tau_i(s)) = f_i^q(\\bar H_i(\\tau_i(s)),\\ell_i(\\tau_i(s))) = f_i^q(\\tilde H(s),s)$, where the first equality follows from the definition of $\\tilde q_i(s)$ in Section \\ref{sec_additional_defs}, the second follows from \\textbf{A.\\ref{a_general_q}}, and the third follows from the definition of $\\tilde H_i(s)$ in Section \\ref{sec_additional_defs} and \\eqref{ell_tau_eq}.\n\\end{proof}\n\n\\begin{lemma}\nLet $\\{X(t)\\}_{t\\geq 1}$ be $0 - 1$ Bernoulli random variables, let $\\ell(t) := \\sum_{k=1}^t X(k)$ be the associated counting process, let $\\mathcal{G}_t := \\sigma(\\{X(k)\\}_{k=1}^t)$, and let $\\rho(t) = \\mathbb{P}(X(t) = 1|\\mathcal{G}_{t-1})$.  Assume $\\sum_{t\\geq 1} \\rho(t) = \\infty$. Then there holds, $\\lim\\limits_{t\\rightarrow\\infty} \\left(\\ell(t)\\right)\/\\left(\\sum_{k=1}^t \\rho(t)\\right) = 1, ~\\mbox{ a.s.}$\n\\label{IR3}\n\\end{lemma}\n\\begin{proof}\nThe result follows via Levi's extension of the Borel-Cantelli Lemmas, \\cite{williams_book} p.124.\n\\end{proof}\n}\n\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nQuantum entanglement~\\cite{SSS} plays an important role in both\nthe foundations of quantum physics and quantum-information\nprocessing~\\cite{Nielsen}. Moreover, various entangled states have\nbeen produced in many experiments for different goals when\nstudying various nonclassical phenomena and their\napplications~\\cite{Shiyh,Polzik,Lukin,Leibfried,Guogc,Panjw,Dujf,Appel,Andre}.\nThus, entanglement is a quantum resource, and how to measure and\ndetect entanglement is very crucial for both theoretical\ninvestigations and potential practical applications.\n\nFor a system of two spin-1\/2 particles or a composite system of a\nspin-1\/2 and a spin-1, there are operationally computable\nentanglement measures such as concurrence~\\cite{Conc} and\nnegativity~\\cite{Neg,Horodecki}, but no universal measures have\nbeen found for general many-body systems. To overcome this\ndifficulty, entanglement witnesses are presented to detect some\nkinds of entanglement in many-body\nsystems~\\cite{Horodecki,Witness}. Now it is believed that spin\nsqueezing~\\cite{KU, Wineland} may be useful for this\ntask~\\cite{Sorensen,Toth,Korbicz}. In a general sense,\nspin-squeezing parameters are multipartite entanglement witnesses.\nFor a class of many-particle states, it has been proved that the\nconcurrence is linearly related to some squeezing\nparameters~\\cite{WangSanders}. In fact, spin-squeezing\nparameters~\\cite{KU,Wineland,Sorensen,Toth} could be calculated\nalso in a simple operational fashion, which characterizes\nmultipartite quantum correlations beyond the pairwise\nentanglement. Another important reason for choosing spin-squeezing\nparameters as indicators of multipartite correlations is that spin\nsqueezing is relatively easy to generate~\\cite{Wineland,Berman}\nand measure experimentally~\\cite{Fernholz,Takano}.\n\nBesides being a parameter characterizing multipartite\ncorrelations, spin squeezing is physically natural for controlling\nmany-body systems. It is difficult to control a quantum many-body\nsystem since its constituents cannot be individually addressed. In\nthis sense, one needs to use collective operations, and spin\nsqueezing is one of the most successful approaches for controlling\nsuch systems. For example, creating spin squeezing of an atomic\nensemble could result in precision measurements based on many-atom\nspectroscopy~\\cite{Wineland}. Therefore, we can also regard spin\nsqueezing as a quantum resource since for more than two particles\nit behaves as two-particle entanglement in controlling and\ndetecting quantum correlations. On this quantum resource, we need\nto further consider the effects of\ndecoherence~\\cite{Deco,Ozdemir}. Thus, it is important to study\nthe environment-induced decoherence effects on both spin squeezing\nand multipartite\nentanglement~\\cite{SimonKempe,Dur,Carvalho,JangNonlocality,Aolita,Lopez,Xiayj,Ficek,Guhne,Vedral,Werlang}.\nA decaying time evolution of the spin squeezing under\ndecoherence~\\cite{SimonKempe,Stockton,Laurat,Liyun} can be used to\nanalyze whether this quantum resource is robust.\n\nIn this article we address this problem by calculating three\nspin-squeezing parameters  for a spin ensemble in a collective\nexcited state. We study the time evolution of spin squeezing under\nlocal decoherence, acting independently and equally on each spin.\nHere, the irreversible processes are modelled as three decoherence\nchannels: the amplitude damping, pure dephasing and depolarizing\nchannels. We find that, similar to the sudden death of pairwise\nentanglement~\\cite{Yuting}, spin squeezing can also suddenly\nvanish with different lifetimes for some decoherence channels,\nshowing in general different vanishing times in multipartite\ncorrelations in quantum many-body systems. Thus, similar to the\ndiscovery of pairwise entanglement sudden death\n(ESD)~\\cite{Yuting}, the spin-squeezing sudden death (SSSD) occurs\ndue to decoherence. We will see that for some initial states, the\nSSSD never occurs under the amplitude-damping channel. We also\ngive analytical expressions for the vanishing time of spin\nsqueezing and pairwise entanglement. The ESD has been tested\nexperimentally~\\cite{Almeida,Laurat} and we also expect that the\nSSSD can also be realized experimentally.\n\nThis article is organized as follows. In Sec.~II, we introduce the\ninitial state from the one-axis twisting Hamiltonian and then, in\nSec.~III, the decoherence channels. In Sec.~IV, we list three\nparameters of spin squeezing and discuss the relations among them.\nFor a necessary comparison, the concurrence is also calculated. We\nalso study  initial-state squeezing. In Sec.~V, we study three\ndifferent types of spin squeezing and concurrence under three\ndifferent decoherence channels. Both analytical and numerical\nresults are given. We conclude in Sec.~VI.\n\n\\section{Initial state}\n\nWe consider an ensemble of $N$ spin-1\/2 particles with ground\nstate $|1\\rangle $ and excited state $|0\\rangle.$ This system has\nexchange symmetry, and its dynamical properties can be described\nby the collective operators\n\\begin{equation}\nJ_{\\alpha }=\\sum_{k=1}^N\nj_{k\\alpha}=\\frac{1}{2}\\sum_{k=1}^{N}\\sigma _{k\\alpha }\n\\label{qqq}\n\\end{equation}\nfor $\\alpha =x,y,z.$ Here, $\\sigma _{k\\alpha }$ are the Pauli\nmatrices for the $k$th qubit. To study the decoherence of spin\nsqueezing, we choose a state which is initially squeezed. One\ntypical class of such spin-squeezed states is the one-axis\ntwisting collective spin state~\\cite{KU},\n\\begin{equation}\n|\\Psi (\\theta_0 )\\rangle _{0}=e^{-i\\theta_0\nJ_{x}^{2}\/2}|1\\rangle^{\\otimes N}=e^{-i\\theta_0 J_{x}^{2}\/2}|{\\bf\n1}\\rangle , \\label{initial}\n\\end{equation}\nwhich could be prepared by the one-axis twisting Hamiltonian\n\\begin{equation}\nH=\\chi J_{x}^{2},\n\\end{equation}\nwhere\n\\begin{equation}\\label{angle}\n\\theta_0=2\\chi t\n\\end{equation}\nis the {\\it one-axis twist angle} and $\\chi$ is the coupling\nconstant. For this state, it was proved~\\cite{WangSanders} that\nthe spin squeezing $\\xi _{1}^{2}$~\\cite{KU} and the concurrence\n$C_{0}$~\\cite{Conc} are equivalent since there exists a linear\nrelation $$\\xi_1^2=1-(N-1)C_{0}$$ between them. Physically, they\noccur and disappear simultaneously.  The spin squeezing of this\nstate can be generated and stored in,  e.g., a two-component\nBose-Einstein condensate~\\cite{Jingr}.\n\n\\subsection{Initial-state symmetry}\n\nThe initial state has an obvious symmetry resulting from\nEq.~(\\ref{initial}), the so-called even-parity symmetry, which\nmeans that only even excitations of spins occur in the state.\nSince $J_{\\alpha }$ define an angular-momentum spinor\nrepresentation of SO(3), the general definitions of spin squeezing\nfor abstract operators $J_{x},J_{y},$ and $J_{z}$ can work well by\nidentifying $N\/2$ with the highest weight $J$, which corresponds\nto the collective ground state\n\\begin{equation}\n|J,-J\\rangle =|1\\rangle^{\\otimes N}\\equiv |{\\bf 1}\\rangle\n\\end{equation}\nindicating that all spins are in the ground state. The symmetric\nspace is generated by the collective operator $$J_{+}=\n\\frac12\\sum_{k=1}^{N}\\sigma _{k+} $$ acting on the collective\nground state. Here, $$\\sigma_{k\\pm}=\\frac12(\\sigma_{kx}\\pm\ni\\sigma_{ky})$$. In others words, the state is in the maximally\nsymmetric space spanned by the Dicke states. So, the $N$ spin-1\/2\nsystem behaves like a larger spin-$N\/2$ system. It can be proved\nthat any pure state with exchange symmetry belongs to the\nabove-mentioned symmetric space, but for mixed states the state\nspace can be extended to include a space beyond the symmetric\none~\\cite{Molmer}. In the following discussions, we focus on such\nan extended space.\n\nIn fact, after decoherence, not only the symmetric Dicke states\nare populated, but also states with lower symmetry. So, it is not\nsufficient to describe the system in only ($N+1)$-dimensional\nspace. Although the maximal symmetry is broken, the exchange\nsymmetry is not affected by the decoherence as each local\ndecoherence equally acts on each spin. In other words, a state\nwith exchange symmetry does not necessarily belong to the\nmaximally symmetric space.\n\nWith only the exchange symmetry, from Eq.~(\\ref{qqq}),\\ the global\nexpectations or correlations of collective operators are obtained as\n\\begin{eqnarray}\n \\langle J_{\\alpha }^{2}\\rangle\n&=&\\frac{N}{4}+\\frac{N(N-1)}{4}\\langle \\sigma\n_{1\\alpha }\\sigma _{2\\alpha }\\rangle ,  \\label{square4} \\\\\n \\langle J_{-}^{2}\\rangle &=&N(N-1)\\langle \\sigma _{1-}\\sigma _{2-}\\rangle ,\n\\label{s6} \\\\\n \\langle \\lbrack J_{x},J_{y}]_{+}\\rangle &=&\n\\frac{N(N-1)}{4}\\langle \\lbrack \\sigma _{1x},\\sigma\n_{2y}]_{+}\\rangle.  \\label{s5}\n\\end{eqnarray}\nFurthermore, it follows from Eq.~(\\ref{square4}) that\n\\begin{align}\n& \\langle J_{x}^{2}+J_{y}^{2}\\rangle =\\frac{N}{2}+\\frac{N(N-1)}{2}\\langle\n\\sigma _{1+}\\sigma _{2-}+\\sigma _{1-}\\sigma _{2+}\\rangle ,  \\label{square2}\n\\\\\n& \\langle J_{x}^{2}+J_{y}^{2}+J_{z}^{2}\\rangle\n=\\frac{N^{2}}{4}\\left[ \\frac{3}{N}+\\left( 1-\\frac{1}{N}\\right)\n\\langle \\vec{\\sigma}_{1}\\cdot \\vec{\\sigma}_{2}\\rangle \\right] .\n\\label{square3}\n\\end{align}\nThese equations show the relations between the global and local\nexpectations and correlations, which are useful in the following\ncalculations.\n\n\\section{Decoherence channels and examples of their implementations}\n\nHaving introduced the initial state, now we discuss three typical\ndecoherence channels: the amplitude-damping channel (ADC), the\nphase-damping channel (PDC), and the depolarizing channel (DPC).\n\nThese channels are prototype models of dissipation relevant in\nvarious experimental systems. They provide ``a revealing\ncaricature of decoherence in realistic physical situations, with\nall inessential mathematical details stripped\naway''~\\cite{Preskill}. But yet this ``caricature of decoherence''\nleads to theoretical predictions being often in good agreement\nwith experimental data. Examples include multiphoton systems, ion\ntraps, atomic ensembles, or a solid-state spin systems such as\nquantum dots or NV diamonds, where qubits are encoded in electron\nor nuclear spins.\n\nHere, we briefly describe only a few of such implementations.\n\n\\subsection{Amplitude-damping channel}\n\nThe ADC is defined as\n\\begin{equation}\n{\\cal E}_\\text{ADC}(\\rho)=E_0\\rho E_0^\\dagger + E_1\\rho\nE_1^\\dagger, \\label{ADC}\n\\end{equation}\nwhere\n\\begin{equation}\nE_{0}=\\sqrt{s}\\,|0\\rangle \\langle 0|+|1\\rangle \\langle 1|,\\quad\nE_{1}=\\sqrt{p}\\,|1\\rangle \\langle 0|  \\label{kraus1}\n\\end{equation}\nare the Kraus operators, $p=1-s$, $s=\\exp (-\\gamma t\/2)$, and\n$\\gamma $ is the damping rate. In the Bloch representation, the\nADC squeezes the Bloch sphere into an ellipsoid and shifts it\ntoward the north pole. The radius in the $xy$ plane is reduced by\na factor $\\sqrt{s}$, while in the $z$ direction it is reduced by a\nfactor $s$.\n\nThe ADC is a prototype model of a dissipative interaction between\na qubit and its environment. For example, the ADC model can be\napplied to describe the spontaneous emission of a photon by a\ntwo-level system into an environment of photon or phonon modes at\nzero (or very low) temperature in (usually) the weak Born-Markov\napproximation. The ADC can also describe processes contributing to\n$T_1$ relaxation in spin resonance at zero temperature. Note that\nby introducing an ``upward'' decay (i.e, a decay toward the south\npole of the Bloch sphere), in addition to the standard\n``downward'' decay, the ADC can be used to describe dissipation\ninto the environment also at finite temperature.\n\nThe ADC acting on a system qubit in an unknown state $\\rho$ can be\nimplemented in a two-qubit circuit performing a rotation\n$R_y(\\theta)$ of an ancilla qubit (initially in the ground state)\ncontrolled by the system qubit and followed by a controlled-NOT\n(CNOT) gate on the system qubit controlled by the ancilla\nqubit~\\cite{Nielsen}. The parameter $\\theta$ is simply related to\nthe probability $p$ in Eq.~(\\ref{ADC}). The ancilla qubit, which\nmodels the environment, is measured after the gate operation.\n\nThe ADC-induced sudden vanishing of entanglement was first\nexperimentally demonstrated for polarization-encoded\nqubits~\\cite{Almeida}. For this reason let us shortly describe\nthis optical implementation of the ADC. It is based on a\nSagnac-type ring interferometer composed of a polarizing beam\nsplitter and a half-wave plate at an angle corresponding to the\nparameter $p$ in Eq.~(\\ref{ADC}). The beam splitter separates an\nincident beam (being in a superposition of states with horizontal,\n$|H\\>$, and vertical, $|V\\>$, polarizations) into spatially\ndistinct counter propagating light beams. The $H$ component leaves\nthe interferometer unchanged. But the $V$ component is rotated in\nthe wave plate, which corresponds to probabilistic damping into\nthe $H$ component. Then, at the exit from the interferometer, this\ncomponent is probabilistically transmitted or reflected from the\nbeam splitter. So it is cast into two orthogonal spatial modes\ncorresponding the reservoir states with and without excitation.\n\nThe action of the ADC can be represented by an interaction\nHamiltonian~\\cite{Nielsen}: $H\\sim  a b^\\dagger + a^\\dagger b$,\nwhere $a$ ($a^\\dagger$) and $b$ ($b^\\dagger$) are annihilation\n(creation) operators of the system and environment oscillators,\nrespectively. In more general models of damping, a single\noscillator $b$ of the reservoir is replaced by a finite or\ninfinite collection of oscillators $\\{b_n\\}$ coupled to the system\noscillator with different strengths (see, e.g.,\nRef.~\\cite{Louisell,Leibfried03}). For the example of quantum\nstates of motion of ions trapped in a radio-frequency (Paul) trap,\nthe amplitude damping can be modeled by coupling an ion to the\nmotional amplitude reservoir described by the above\nmultioscillator Hamiltonian~\\cite{Leibfried03}. The\nhigh-temperature reservoir is possible to simulate by applying (on\ntrap electrodes) a random uniform electric field with spectral\namplitude at the ion motional\nfrequency~\\cite{Myatt00,Turchette00}.  The zero-temperature\nreservoir can be simulated by laser cooling combined with\nspontaneous Raman scattering~\\cite{Poyatos}.\n\n\\subsection{Phase-damping channel}\n\nThe PDC is a prototype model of dephasing or pure decoherence,\ni.e., loss of coherence of a two-level state without any loss of\nsystem's energy. The PDC is described by the map\n\\begin{equation}\n{\\cal E} _{\\text{PDC}}(\\rho )=s\\rho +p\\left( \\rho _{00}|0\\rangle\n\\langle 0|+\\rho _{11}|1\\rangle \\langle 1|\\right) \\label{PDC},\n\\end{equation}\nand obviously the three Kraus operators are given by\n\\begin{equation}\nE_{0}=\\sqrt{s}\\,\\openone,\\; E_{1}=\\sqrt{p}\\,|0\\rangle \\langle\n0|,\\; E_{2}=\\sqrt{p}\\,|1\\rangle \\langle 1|,  \\label{kraus2}\n\\end{equation}\nwhere $\\openone$ is the identity operator. For the PDC, there is\nno energy change and a loss of decoherence occurs with probability\n$p.$ As a result of the action of the PDC, the Bloch sphere is\ncompressed by a factor $(1-2p)$ in the $xy$ plane.\n\nIn analogy to the ADC, the PDC can be considered as an interaction\nbetween two oscillators (modes) representing system and\nenvironment as described by the interaction Hamiltonian: $H\\sim\na^\\dagger a(b^\\dagger + b)$~\\cite{Nielsen}. In more general\nphase-damping models, a single environmental mode $b$ is usually\nreplaced by an infinite collection of modes $b_n$ coupled, with\nvarious strengths, to mode $a$.\n\nIt is evident that the action of the PDC is nondissipative. It\nmeans that, in the standard computational basis $|0\\>$ and $|1\\>$,\nthe diagonal elements of the density matrix $\\rho$ remain\nunchanged, while the off-diagonal elements are suppressed.\nMoreover, the qubit states $|0\\>$ and $|1\\>$ are also unchanged\nunder the action of the PDC, although any superposition of them\n(i.e., any point in the Bloch sphere, except the poles) becomes\nentangled with the environment.\n\nThe PDC can be interpreted as elastic scattering between a\n(two-level) system and a reservoir. It is also a model of coupling\na system with a noisy environment via a quantum nondemolition\n(QND) interaction. Note that spin squeezing of atomic ensembles\ncan be generated via QND\nmeasurements~\\cite{Kuzmich99,Takahashi99,Kuzmich00,Julsgaard01,Schleier,Appel,Takano}.\nSo modeling the spin-squeezing decoherence via the PDC can be\nrelevant in this context.\n\nThe PDC is also a suitable model to describe $T_2$ relaxation in\nspin resonance. This in contrast to modeling $T_1$ relaxation via\nthe ADC.\n\nA circuit modeling the PDC can be realized as a simplified version\nof the circuit for the ADC, discussed in the previous subsection,\nobtained by removing the CNOT gate~\\cite{Nielsen}. Then, the angle\n$\\theta$ in the controlled rotation gate $R_y(\\theta)$ is related\nto the probability $p$ in Eq.~(\\ref{PDC}).\n\nThe sudden vanishing of entanglement under the PDC was first\nexperimentally observed in Ref.~\\cite{Almeida}. This optical\nimplementation of the PDC was based on the same system as the\nabove-mentioned Sagnac interferometer for the ADC but with an\nadditional half-wave plate at a $\\pi\/4$ angle in one of the\noutgoing modes.\n\nSome specific kinds of PDCs can be realized in a more\nstraightforward manner. For example, in experiments with trapped\nions, the motional PDC can be implemented just by modulating the\ntrap frequency, which changes the phase of the harmonic motion of\nions~\\cite{Myatt00,Turchette00} (for a review see\nRef.~\\cite{Leibfried03} and references therein).\n\n\\subsection{Depolarizing channel}\n\nThe definition of the DPC is given via the map\n\\begin{align}\n{\\cal E }_{\\text{DPC}}(\\rho )& =\\sum_{i=0}^{3}E_{k}\\rho\nE_{k}^{\\dagger },\n\\\\\n& =(1-p^{\\prime })\\rho +\\frac{p^{\\prime }}{3}(\\sigma _{x}\\rho\n\\sigma _{x}+\\sigma _{y}\\rho \\sigma _{y}+\\sigma _{z}\\rho \\sigma\n_{z}),  \\notag\\label{DPC},\n\\end{align}\nwhere\n\\begin{eqnarray}\nE_{0} &=&\\sqrt{1-p^{\\prime }}\\openone,\n\\quad E_{1}=\\sqrt{\\frac{p^{\\prime }}{3}}\\sigma _{x},  \\notag \\\\\nE_{2} &=&\\sqrt{\\frac{p^{\\prime }}{3}}\\sigma _{y},\\quad \\quad\nE_{3}=\\sqrt{\\frac{p^{\\prime }}{3}}\\sigma _{z}, \\label{kraus3}\n\\end{eqnarray}\nare the Kraus operators. By using the following identity\n\\begin{equation*}\n\\sigma _{x}\\rho \\,\\sigma _{x}+\\sigma _{y}\\rho \\,\\sigma _{y}+\\sigma\n_{z}\\rho \\,\\sigma_{z}+\\rho =2\\openone,\n\\end{equation*}%\nwe obtain\n\\begin{equation}\n{\\cal E}_{\\text{DPC}}(\\rho )=s\\rho +p\\frac{\\openone}{2},\n\\end{equation}\nwhere $p ={4p^{\\prime }}\/{3}$. We see that for the DPC, the spin\nis unchanged with probability $s=1-p$ or it is depolarized to the\nmaximally mixed state $\\openone\/2\\,\\ $with probability $p.$ It is\nseen that due to the action of the DPC, the radius of the Bloch\nsphere is reduced by a factor $s$, but its shape remains\nunchanged.\n\nFormally, the action of the DPC on a qubit in an unknown state\n$\\rho$ can be implemented in a three-qubit circuit composed of two\nCNOT gates with two auxiliary qubits initially in mixed states\n\\begin{eqnarray}\n  \\rho_1=\\openone\/2,\\quad \\rho_2=(1-p)|00\\rangle\\langle\n00|+p|11\\rangle\\langle 11|, \\label{N1}\n\\end{eqnarray}\nwhich model the environment. Qubit $\\rho_2$ controls the other\nqubits via the CNOT gates~\\cite{Nielsen}.\n\nThe DPC map can also be implemented by applying each of the Pauli\noperators $[\\openone,\\sigma_x,\\sigma_y,\\sigma_z]$ at random with\nthe same probability. Using this approach, optical DPCs have been\nrealized experimentally both in free space~\\cite{Ricci04} and in\nfibers~\\cite{Karpinski08}, where qubits are associated with\npolarization states of single photons. In Ref.~\\cite{Ricci04}, the\nDPC was implemented by using a pair of equal electro-optical\nPockels cells. One of them was performing a $\\sigma_x$ gate and\nthe other a $\\sigma_y$ gate. The simultaneous action of both\n$\\sigma_x$ and $\\sigma_y$ corresponds to a $\\sigma_y$ gate. The\ncells were driven (with a mutual delay of $\\tau\/2$) by a\ncontinuous-wave periodic square-wave electric field with a\nvariable pulse duration $\\tau$, so the total depolarizing process\nlasted 2$\\tau$ for each period.\n\nAnalogous procedures can be implemented in other systems,\nincluding collective spin states of atomic ensembles. The coherent\nmanipulation of atomic spin states by applying off-resonantly\ncoherent pulses of light is a basic operation used in many\napplications~\\cite{Julsgaard04}. We must admit that the standard\nmethods enable rotations in the Bloch sphere of only classical\nspin states (i.e., coherent spin states). Nevertheless,\nrecently~\\cite{Takano} an experimental method has been developed\nto rotate also spin-squeezed states.\n\nIt is worth noting that in experimental realizations of\ndecoherence channels (e.g, in ion-trap\nsystems~\\cite{Hannemann09}), sufficient resources for complete\nquantum tomography are provided even for imperfect preparation of\ninput states and the imperfect measurements of output states from\nthe channels.\n\n\\section{Spin-squeezing definitions and concurrence}\n\nNow, we discuss several parameters of spin squeezing and give\nseveral relations among them. To compare spin squeezing with\npairwise entanglement, we also give the definition of concurrence.\nWe notice that most previous investigations on ESD of concurrence\nwere only carried out for two-particle system rather than for\ntwo-particle subsystem embedded in a larger system. For the\ninitial states, spin-squeezing parameters and concurrence are also\ngiven below.\n\n\\subsection{Spin-squeezing parameters and their relations}\n\\subsubsection{Definitions of spin squeezing}\n\nThere are several spin-squeezing parameters, but we list only\nthree typical and related ones as\nfollows~\\cite{KU,Wineland,Sorensen,Toth}:\n\\begin{eqnarray}\n\\xi _{1}^{2} &=&\\frac{4(\\Delta J_{\\vec{n}_\\perp })_{\\min }^{2}}{N},~~  \\label{x1} \\\\\n\\xi _{2}^{2} &=&\\frac{N^{2}}{4\\langle \\vec{J}\\rangle ^{2}}\\xi _{1}^{2},~~\n\\label{x2} \\\\\n\\xi _{3}^{2} &=&\\frac{\\lambda _{\\min }}{\\langle \\vec{J}^{2}\\rangle\n-\\frac{N}{2}}.  \\label{x3}\n\\end{eqnarray}\nHere, the minimization in the first equation is over all\ndirections denoted by $\\vec{n}_\\perp,$ perpendicular to the mean\nspin direction $\\langle \\vec{J}\\rangle \/\\langle \\vec{J}^{2}\\rangle\n$; $\\lambda _{\\min }$ is the minimum eigenvalue of the\nmatrix~\\cite{Toth}\n\\begin{equation}\n\\Gamma =(N-1)\\gamma +\\mathbf{C},  \\label{gamma}\n\\end{equation}\nwhere\n\\begin{equation}\n\\gamma _{kl}={C}_{kl}-\\langle J_{k}\\rangle \\langle J_{l}\\rangle\n\\;\\;\\text{for}\\;\\; k,l\\in \\{x,y,z\\}=\\{1,2,3\\}, \\label{comatrix}\n\\end{equation}\nis the covariance matrix and ${\\bf C}=[C_{kl}]$ with\n\\begin{equation}\n{C}_{kl}=\\frac{1}{2}\\langle J_{l}J_{k}+J_{k}J_{l}\\rangle\n\\label{cmatrix}\n\\end{equation}\nis the global correlation matrix. The parameters $\\xi _{1}^{2},\n\\xi _{2}^{2},$ and $\\xi _{3}^{2}$ were defined by Kitagawa and\nUeda \\cite{KU}, Wineland {\\em et al.}~\\cite{Wineland}, and\nT\\'{o}th {\\it et al.}~\\cite{Toth}, respectively. If $\\xi\n_{2}^{2}<1$ $(\\xi _{3}^{2}<1),$ spin squeezing occurs, and we can\nsafely say that the multipartite state is\nentangled~\\cite{Sorensen,Toth}. Although we cannot say that the\nsqueezed state via the parameter $\\xi_1^2$ is entangled, it is\nindeed closely related to quantum entanglement~\\cite{WangSanders}.\n\n\\subsubsection{Squeezing parameters for states with parity}\n\nWe know from Sec.~II.A that the initial state has an even parity\nand that the mean spin direction is along the $z$ direction.\nDuring the transmission through all the three decoherence channels\ndiscussed here, the mean spin direction does not change. For\nstates with a well-defined parity (even or odd), the\nspin-squeezing parameter $\\xi _{1}^{2}$ was found to be\n\\cite{WangSanders}\n\\begin{equation}\n\\xi _{1}^{2}=\\frac{2}{N}\\left( \\langle J_{x}^{2}+J_{y}^{2}\\rangle\n-|\\langle J_{-}^{2}\\rangle |\\right).  \\label{xixi1}\n\\end{equation}\nThen, the parameter $\\xi _{2}^{2}$ given by Eq.~(\\ref{x2}) becomes\n\\begin{equation}\n\\xi _{2}^{2}=\\frac{N^{2}\\xi _{1}^{2}}{4\\langle J_{z}\\rangle\n^{2}}=\\frac{N\\left( \\langle J_{x}^{2}+J_{y}^{2}\\rangle -|\\langle\nJ_{-}^{2}\\rangle |\\right) }{2\\langle J_{z}\\rangle ^{2}}.\n\\end{equation}\nFor the third squeezing parameter (see Appendix A for the\nderivation), we have\n\\begin{equation}\n\\xi _{3}^{2}=\\frac{\\min \\left\\{ \\xi _{1}^{2},\\varsigma\n^{2}\\right\\} }{{4}{N^{-2}}\\langle \\vec{J}^{2}\\rangle\n-{2}{N^{-1}}},  \\label{xixixi}\n\\end{equation}\nwhere\n\\begin{equation}\n\\varsigma ^{2}=\\frac{4}{N^{2}}\\left[ N(\\Delta J_{z})^{2}+\\langle\nJ_{z}\\rangle ^{2}\\right] .  \\label{zzz}\n\\end{equation}\nNote that the first parameter $\\xi _{1}^{2}$ becomes a key\ningredient for the latter two squeezing parameters ($\\xi_2^2$ and\n$\\xi_3^2$).\n\n\\subsubsection{Spin-squeezing parameters in terms of local expectations}\n\nFor later applications, we now express the squeezing parameters in\nterms of local expectations and correlations, and also examine the\nmeaning of $\\varsigma ^{2}$, which will be clear by substituting\nEqs.~(\\ref{qqq}) and (\\ref{square4}) into Eq.~(\\ref{zzz}),\n\\begin{eqnarray}\n\\varsigma ^{2} &=&1+\\mathcal{C}_{zz}  \\notag \\\\\n&=&1+(N-1)\\left( \\langle \\sigma _{1z}\\sigma _{2z}\\rangle -\\langle \\sigma\n_{1z}\\rangle \\langle \\sigma _{2z}\\rangle \\right) .  \\label{zzzz}\n\\end{eqnarray}\nThus, the parameter $\\varsigma ^{2}$ is simply related to the\ncorrelation $\\mathcal{C}_{zz}$ along the $z$ direction. A negative\ncorrelation ${\\cal C}_{zz}<0$ is equivalent to $\\varsigma ^{2}<1.$\nIt is already known that the spin-squeezing parameter $\\xi\n_{1}^{2}$ can be written as \\cite{Kitagawa}\n\\begin{equation}\\label{perp}\n\\xi _{1}^{2}=1+(N-1)\\mathcal{C}_{\\vec{n}_{\\perp }\\vec{n}_{\\perp\n}},\n\\end{equation}\nwhere $\\mathcal{C}_{\\vec{n}_{\\perp }\\vec{n}_{\\perp }}$ is the\ncorrelation function in the direction perpendicular to the mean\nspin direction. So, the spin squeezing $\\xi_1^2<1$ is equivalent\nto the negative pairwise correlations $\\mathcal{C}_{\\vec{n}_{\\perp\n}\\vec{n}_{\\perp }}<0$~\\cite{Kitagawa}.\n\nThus, from the above analysis, spin squeezing and negative\ncorrelations are closely connected to each other. The parameter\n$\\varsigma ^{2}<1$ indicates that spin squeezing occurs along the\n$z$ direction, and $\\xi_1^{2}<1$ implies spin squeezing along the\ndirection perpendicular to the mean spin direction. Furthermore,\nfrom Eq.~(\\ref{xixixi}), a competition between the transverse and\nlongitudinal correlations is evident.\n\nBy substituting Eqs.~(\\ref{s6}) and (\\ref{square2}) to\nEq.~(\\ref{xixi1}), one can obtain the expression of $\\xi _{1}^{2}$\nin terms of local correlations $\\langle \\sigma _{1+}\\sigma\n_{2-}\\rangle $ and $\\langle \\sigma _{1-}\\sigma _{2-}\\rangle $ as\nfollows:\n\\begin{eqnarray}\n\\xi _{1}^{2} &=&1+(N-1)\\langle \\sigma _{1+}\\sigma _{2-}+\\sigma _{1-}\\sigma\n_{2+}\\rangle  \\notag \\\\\n&&-2(N-1)|\\langle \\sigma _{1-}\\sigma _{2-}\\rangle |  \\notag \\\\\n&=&1+2(N-1)\\langle \\sigma _{1+}\\sigma _{2-}\\rangle -|\\langle\n\\sigma _{1-}\\sigma _{2-}\\rangle |).  \\label{xixixi1}\n\\end{eqnarray}\nThe second equality in Eq.~(\\ref{xixixi1}) results from the\nexchange symmetry. From Eqs.~(\\ref{qqq}), (\\ref{square3}), and\n(\\ref{zzzz}), one finds\n\\begin{eqnarray}\n\\xi _{2}^{2} &=&\\frac{\\xi _{1}^{2}}{\\langle \\sigma _{1z}\\rangle ^{2}},~\n\\label{xixixi2} \\\\\n\\xi _{3}^{2} &=&\\frac{\\min \\left\\{ \\xi\n_{1}^{2},1+\\mathcal{C}_{zz}\\right\\} }{(1-N^{-1})\\langle\n\\vec{\\sigma}_{1}\\cdot \\vec{\\sigma}_{2}\\rangle +{N^{-1}}}.\n\\label{third}\n\\end{eqnarray}\nThus, we have reexpressed the squeezing parameters in terms of\nlocal correlations and expectations.\n\n\\subsubsection{New spin-squeezing parameters}\n\nIn order to characterize spin squeezing more conveniently, we\ndefine the following squeezing parameters:\n\\begin{equation}\n\\zeta _{k}^{2}=\\max (0,1-\\xi _{k}^{2}), \\; k\\in \\{1,2,3\\}.\n\\label{zeta}\n\\end{equation}\nThis definition is similar to the expression of the concurrence\ngiven below. Spin squeezing appears when $\\zeta _{k}^{2}>0$, and\nthere is no squeezing when $\\zeta _{k}^{2}$ vanishes. Thus, the\ndefinition of the first parameter $\\zeta_{1}^{2}$ has a clear\nmeaning, namely, it is the \\emph{strength} of the negative\ncorrelations as seen from Eq.~(\\ref{perp}). The larger is\n$\\zeta_1^2$, the larger is the strength of the negative\ncorrelation, and the larger of is the squeezing. More explicitly,\nfor the initial state, we have $\\xi _{1}^{2}=1-(N-1)C_{0}$\n\\cite{WangSanders}, so $\\zeta _{1}^{2}$ is just the rescaled\nconcurrence $\\zeta_1^2=C_{r}(0)=(N-1)C_{0}$~\\cite{Vidal}.\n\nHere, we give a few comments on the spin-squeezing parameter $\\xi\n_{2}^{2}$, which represents a competition between $\\xi _{1}^{2}$\nand $\\langle \\sigma _{1z}\\rangle ^{2}$: the state is squeezed\naccording to the definition of $\\ \\xi _{2}^{2}$ if $\\xi\n_{1}^{2}<\\langle \\sigma _{1z}\\rangle ^{2}$. We further note\nthat~\\cite{CPL}\n\\begin{equation}\n\\langle \\sigma _{1z}\\rangle ^{2}=1-2E_{L},\n\\end{equation}\nwhere $E_{L}$ is the linear entropy of one spin and it can be used\nto quantify the entanglement of pure states~\\cite{Horodecki}. So,\nthere is a competition between the strength of negative\ncorrelations and the linear entropy $2E_L$ in the parameter $\\xi\n_{2}^{2},$ and $\\zeta _{1}^{2}>2E_{L}$ implies the appearance of\nsqueezing.\n\n\n\\subsection{Concurrence for pairwise entanglement}\n\nIt has been found that the concurrence is closely related to spin\nsqueezing~\\cite{WangSanders}. Here, we consider its behavior under\nvarious decoherence channels. The concurrence quantifying the\nentanglement of a pair of spin-1\/2 can be calculated from the\nreduced density matrix. It is defined as~\\cite{Conc}\n\\begin{equation}\n{C}=\\max(0,\\lambda _1-\\lambda _2-\\lambda _3-\\lambda _4),\n\\label{Cdef}\n\\end{equation}\nwhere the quantities $\\lambda _i$ are the square roots of the\neigenvalues in descending order of the matrix product\n\\begin{equation}\n\\varrho _{12}=\\rho _{12}(\\sigma _{1y}\\otimes \\sigma _{2y})\\rho\n_{12}^{*}(\\sigma _{1y}\\otimes \\sigma _{2y}).  \\label{varrho}\n\\end{equation}\nIn (\\ref{varrho}), $\\rho _{12}^{*}$ denotes the complex conjugate\nof $\\rho _{12}$.\n\nThe two-spin reduced density matrix for a parity state with the\nexchange symmetry can be written in a block-diagonal\nform~\\cite{WangMolmer}\n\\begin{equation}\n\\rho _{12}=\\left(\n\\begin{array}{cc}\nv_{+} & u^{\\ast } \\\\\nu & v_{-}%\n\\end{array}%\n\\right) \\oplus \\left(\n\\begin{array}{cc}\nw & y \\\\\ny & w%\n\\end{array}%\n\\right) ,  \\label{re}\n\\end{equation}\nin the basis \\{$|00\\rangle ,|11\\rangle ,|01\\rangle ,|10\\rangle $\\}, where\n\\begin{eqnarray}\nv_{\\pm } &=&\\frac{1}{4}\\left( 1\\pm 2\\langle \\sigma _{1z}\\rangle +\\langle\n\\sigma _{1z}\\sigma _{2z}\\rangle \\right) ,  \\label{r1} \\\\\nw &=&\\frac{1}{4}\\left( 1-\\langle \\sigma _{1z}\\sigma _{2z}\\rangle \\right) ,\n\\label{r3} \\\\\nu &=&\\langle \\sigma _{1+}\\sigma _{2+}\\rangle ,  \\label{rrr} \\\\\ny &=&\\langle \\sigma _{1+}\\sigma _{2-}\\rangle .  \\label{rr}\n\\end{eqnarray}\nThe concurrence is then given by \\cite{Wootters2}\n\\begin{equation}\nC=\\max \\left\\{ 0,2\\left( |u|-w\\right) ,2(y-\\sqrt{v_{+}v_{-}})\\right\\} .\n\\label{conc}\n\\end{equation}\nFrom the above expressions of the spin-squeezing parameters and\nconcurrence, we notice that if we know the expectation $\\langle\n\\sigma _{1z}\\rangle$, and the correlations $\\langle \\sigma\n_{1+}\\sigma _{2-}\\rangle ,$ $\\langle \\sigma _{1-}\\sigma\n_{2-}\\rangle$, and $\\langle \\sigma _{1z}\\sigma _{2z}\\rangle,$ all\nthe squeezing parameters and concurrence can be determined. Below,\nwe will give explicit analytical expressions for them subject to\nthree decoherence channels.\n\n\\subsection{Initial-state squeezing and concurrence}\n\nWe will now investigate initial spin squeezing and pairwise\nentanglement by using our results for the spin-squeezing\nparameters and concurrence obtained in the last subsections. We\nfind that the third squeezing parameter $\\xi_3^2$ is equal to the\nfirst one $\\xi_1^2$. The squeezing parameter $\\xi _{1}^{2}$ is\ngiven by (see Appendix B):\n\\begin{align}\n\\xi _{1}^{2}(0)&=1-C_{r}(0)  \\notag \\\\\n& =1-(N-1)C_{0},  \\notag \\\\\n& =1-2(N-1)(|u_{0}|-y_{0}),  \\label{ccc1}\n\\end{align}\nwhere\n\\begin{align}\nC_{0}=&\\frac{1}{4}{\\LARGE \\{}[(1-\\cos ^{N-2}\\theta_0 )^{2}+16\\sin\n^{2}{(\\theta_0\/2)} \\cos ^{2N-4}{(\\theta_0\/2)}]^{\\frac{1}2}\n\\notag \\\\\n& -1+\\cos ^{N-2}\\theta_0 {\\Large \\}}\n\\end{align}\nis the concurrence \\cite{WangSanders}.\n\nThe parameter $\\xi _{2}^{2}(0)$ is easily obtained, as we know\nboth $\\xi _{1}^{2}(0)$ and $\\langle \\sigma _{1z}\\rangle_0^{2}$\n(\\ref{sigmaz})$.$ For this state, following from\nEq.~(\\ref{square3}), $\\langle \\vec{\\sigma}_{1}\\cdot\n\\vec{\\sigma}_{2}\\rangle_0 =1,$ and thus the third parameter given\nby Eq.~(\\ref{third}) becomes\n\\begin{align}\n\\xi _{3}^{2}(0)&=\\min [\\xi _{1}^{2}(0),\\varsigma ^{2}(0)]\\notag\\\\\n&=\\min [\\{ 1-C_{r}(0),1+\\mathcal{C}_{zz}(0)] ,  \\label{thirdd}\n\\end{align}\nwhere the correlation function is\n\\begin{equation}\n\\mathcal{C}_{zz}(0)=\\frac{1}{2}\\left( 1+\\cos ^{N-2}\\theta_0\n\\right) -\\cos ^{2N-2}{(\\theta_0\/2)}\\geq 0.  \\label{c5}\n\\end{equation}\nThe proof of the above inequality is given in Appendix C.\n\nAs the correlation function $\\mathcal{C}_{zz}(0)$ and the\nconcurrence $C_{r}(0)$ are always $\\ge 0$, Eq.~(\\ref{thirdd})\nreduces to\n\\begin{equation}\n\\xi _{3}^{2}(0)=\\xi _{1}^{2}(0)=1-C_{r}(0).\n\\end{equation}\nSo, for the initial state, the spin-squeezing parameters $\\xi\n_{3}^{2}(0)$ and $\\xi _{1}^{2}(0)$ are equal or equivalently, we\ncan write $\\zeta _{1}^{2}(0)=\\zeta _{3}^{2}(0)=C_{r}(0)$ according\nto the definition of parameter $\\zeta _{k}^{2}$ given by\nEq.~(\\ref{zeta}). Below we made a summary of results of this\nsection in Table I.\n\n\\begin{table*}[tbp]\n\\caption{Spin-squeezing parameters $\\xi_1^2$~\\cite{KU},\n$\\xi_2^2$~\\cite{Wineland}, $\\xi_3^2$~\\cite{Toth} and concurrence\n$C$~\\cite{Conc} for arbitrary states (first two columns), states\nwith parity (third column). The squeezing parameters are also\nexpressed in terms of local expectations (fourth column) and in\nterms of the initial rescaled concurrence $C_r(0)$ for initial\nstates (last column). Also, $C_0$ is the initial concurrence, and\nother parameters are defined in the text.}\n\\begin{center}\n\\begin{tabular}{c||c|c|c|c}\n\\hline\\hline Squeezing parameters& Definitions & States with\nparity & In terms of local expectations &Initial state\n\\\\ \\hline\\hline\n\\parbox{2 cm} {$\\xi_{1}^{2}$} &\n\\parbox{3.7 cm} {\\vspace{2mm} $\\displaystyle\\frac{4(\\Delta J_{\\vec{n}_\\perp })_{\\min\n}^{2}}{N}$ \\vspace{2mm}} &\n\\parbox{4 cm}{$\\displaystyle\\frac{2}{N}\\left( \\langle J_{x}^{2}+J_{y}^{2}\\rangle\n-|\\langle J_{-}^{2}\\rangle |\\right)$}&\n\\parbox{4 cm}{$1+2(N-1)(y -|u|)$}&\n\\parbox{1.5 cm}{$1-C_r(0)$}\n\n\\\\ \\hline\n\\parbox{2 cm} {$\\xi_{2}^{2}$} &\n\\parbox{3.7cm} {\\vspace{2mm} $\\displaystyle\\frac{N^{2}}{4\\langle\n\\vec{J}\\rangle ^{2}}\\xi _{1}^{2}$ \\vspace{2mm}} &\n\\parbox{4 cm} {$\\displaystyle\\frac{N^{2}\\xi _{1}^{2}}{4\\langle J_{z}\\rangle ^{2}}$} &\n\\parbox{4 cm}{$\\displaystyle\\frac{\\xi _{1}^{2}}{\\langle \\sigma _{1z}\\rangle ^{2}}$}&\n\\parbox{1.5 cm}{$\\displaystyle\\frac{1-C_r(0)}{\\langle\\sigma_{1z}\\rangle_0^2}$}\n\\\\ \\hline\n\n\\parbox{2 cm} { $\\xi_{3}^{2}$} &\n\\parbox{3.9cm} {\\vspace{2mm} \\ $\\displaystyle\n\\frac{\\lambda _{\\min }}{\\langle \\vec{J}^{2}\\rangle\n-\\displaystyle\\frac{N}{2}} $ \\vspace{2mm} } &\n\\parbox{4.2 cm} { $\\displaystyle\\frac{\\min\n\\left\\{ \\xi _{1}^{2},\\varsigma ^{2}\\right\\} }{{4}{N^{-2}}\\langle\n\\vec{J}^{2}\\rangle -{2}{N^{-1}}}$}&\n\\parbox{4 cm}{$\\displaystyle\\frac{\\min \\left\\{ \\xi _{1}^{2},1+\\mathcal{C}_{zz}\\right\\} }{(1-N^{-1})\\langle \\vec{\\sigma}_{1}\\cdot \\vec{\\sigma}_{2}\\rangle +{N^{-1}}} $}&\n\\parbox{1.5 cm}{$1-C_r(0)$}\n\\\\ \\hline\n\n\\parbox{2.5 cm} {\\vspace{0.3cm}Concurrence $C$\\vspace{0.3cm}} &\n\\parbox{3.7 cm} {$\\max(0,\\lambda_1-\\lambda_2-\\lambda_3-\\lambda_4)$} &\n\\parbox{4.0 cm} {$2\\max (0,|u|-w,y-\\sqrt{v_{+}v_{-}})$}&\n\\parbox{4.0 cm}{ $2\\max (0,|u|-w,y-\\sqrt{v_{+}v_{-}})$} &\n\\parbox{1.5 cm}{$C_0$}\n\\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\n\\section{Spin squeezing under decoherence}\n\nNow we begin to study spin squeezing under three different\ndecoherence channels. From the previous analysis, all the\nspin-squeezing parameters and the concurrence are determined by\nsome correlation functions and expectations. So, if we know the\nevolution of them under decoherence, the evolution of any\nsqueezing parameters and pairwise entanglement can be calculated.\n\n\\subsection{Heisenberg approach}\n\nWe now use the Heisenberg picture to calculate the correlation\nfunctions and the relevant expectations. A decoherence channel\nwith Kraus operators $K_{\\mu }$ is defined via the map\n\\begin{equation}\n{\\cal E}(\\rho )=\\sum_{\\mu }K_{\\mu }\\rho K_{\\mu\n}^{\\dagger }.\n\\end{equation}\nThen, an expectation value of the operator $A$ can be calculated\nas $\\langle A\\rangle =$Tr$\\left[A{\\cal E}(\\rho )\\right] .$\nAlternatively, we can define the following map,\n\\begin{equation}\n{\\cal E}^{\\dagger }(\\rho )=\\sum_{\\mu }K_{\\mu }^{\\dagger }\\rho\nK_{\\mu }.\n\\end{equation}\nIt is easy to check that%\n\\begin{equation}\\label{four}\n\\langle A\\rangle =\\text{Tr}\\left[ A{\\cal E} (\\rho) \\right]\n=\\text{Tr}\\left[{\\cal E}^{\\dagger }(A)\\rho \\right] .\n\\end{equation}\nSo, one can calculate the expectation value via the above equation\n(\\ref{four}). This is very similar to the standard Heisenberg\npicture.\n\n\\subsection{Amplitude-damping channel}\n\n\\begin{figure}[tbp]\n\\includegraphics[width=9cm,clip]{fig1.eps}\n\\caption{(Color online) Spin-squeezing parameters\n$\\protect\\zeta_2^2$ (red curve with squares), $\\protect\\zeta_3^2$\n(top green curve with circles), and the concurrence $C_r$ (blue\nsolid curve) versus the decoherence strength $p=1-\\exp(-\\gamma t)$\nfor the amplitude-damping channel, where $\\gamma$ is the damping\nrate. Here, $\\theta_0$ is the initial twist angle given by\nEq.~(\\ref{angle}). In all figures, we consider an ensemble of\n$N=12$ spins. Note that for a small initial twist angle $\\theta_0$\n(e.g., $\\theta_0=0.1\\pi$), the two squeezing parameters and the\nconcurrence all concur. For larger values of $\\theta_0$, the\nparameters $\\zeta_2^2$, $\\zeta_3^2$, and $C$ become quite\ndifferent and all vanish for sufficiently large values of the\ndecoherence strength.}\n\\end{figure}\n\n\\subsubsection{\\protect\\bigskip Squeezing parameters}\nBased on the above approach and the Kraus operators for the ADC\ngiven by Eq.~(\\ref{kraus1}), we now find the evolutions of the\nfollowing expectations under decoherence (see Appendix D for\ndetails)\n\\begin{subequations}\n\\begin{align}\n\\langle \\sigma _{1z}\\rangle =& \\; s\\langle \\sigma _{1z}\\rangle _{0}-p, \\\\\n\\langle \\sigma _{1-}\\sigma _{2-}\\rangle =&\\; s\\langle \\sigma\n_{1-}\\sigma\n_{2-}\\rangle_0 ,  \\label{c2} \\\\\n\\langle \\sigma _{1+}\\sigma _{2-}\\rangle =&\\; s\\langle \\sigma\n_{1+}\\sigma\n_{2-}\\rangle_0 ,  \\label{c3} \\\\\n\\langle \\sigma _{1z}\\sigma _{2z}\\rangle =&\\; s^{2}\\langle \\sigma\n_{1z}\\sigma _{2z}\\rangle _{0}-2sp\\langle \\sigma _{1z}\\rangle\n_{0}+p^{2}.  \\label{c44}\n\\end{align}\nTo determine the squeezing parameters and the concurrence, it is\nconvenient to know the correlation function $\\mathcal{C}_{zz}$ and\nthe expectation $\\langle \\vec{\\sigma}_{1}\\cdot\n\\vec{\\sigma}_{2}\\rangle ,$ which can be determined from the above\nexpectations as follows:\n\\end{subequations}\n\\begin{align}\n\\langle \\vec{\\sigma}_{1}\\cdot \\vec{\\sigma}_{2}\\rangle =&1-s\\, p\\, x_{0}, \\\\\n\\mathcal{C}_{zz}=&s^{2}\\left( \\langle \\sigma _{1z}\\sigma _{2z}\\rangle\n_{0}-\\langle \\sigma _{1z}\\rangle _{0}\\langle \\sigma _{2z}\\rangle _{0}\\right)\n\\notag  \\label{c4} \\\\\n=&s^{2}\\mathcal{C}_{zz}(0),\n\\end{align}\nwhere%\n\\begin{equation}\nx_{0}=1+2\\langle \\sigma _{z}\\rangle _{0}+\\langle \\sigma _{1z}\\sigma\n_{2z}\\rangle _{0}.\n\\end{equation}\nSubstituting the relevant expectation values and the correlation\nfunction into Eqs.~(\\ref{xixixi1}), (\\ref{xixixi2}), and\n(\\ref{third}) leads to the explicit expression of the\nspin-squeezing parameters\n\\begin{eqnarray}\n\\xi _{1}^{2} &=&1-sC_{r}(0),  \\label{xi1} \\\\\n\\xi _{2}^{2} &=&\\frac{\\xi _{1}^{2}}{\\left( s\\langle \\sigma\n_{1z}\\rangle\n_{0}-p\\right) ^{2}},  \\label{xix2} \\\\\n\\xi _{3}^{2} &=&\\frac{\\min \\left\\{\\xi _{1}^{2},1+s^{2}\\mathcal{C}%\n_{zz}(0)\\right\\} }{1+({N}^{-1}-1)s\\, p\\, x_{0}}.  \\label{xix3}\n\\end{eqnarray}\nAs the correlation function $\\mathcal{C}_{zz}(0)\\geq 0$, given by\nEq.~(\\ref{c5})$,$ the third parameter can be simplified as\n\\begin{equation}\n\\xi _{3}^{2}=\\frac{1-sC_{r}(0)}{1+({N}^{-1}-1)s\\, p\\, x_{0}}.\n\\end{equation}\n\nInitially, the state is spin squeezed, i.e., $\\xi _{1}^{2}(0)<1$\nor $C_r(0)>0$. From Eq.~(\\ref{xi1}), one can find that $\\xi\n_{1}^{2}<1$, except in the asymptotic limit of $p=1$. As we will\nsee below, for the PDC and DPC, $$\\xi _{1}^{2}=1-s^{2}C_{r}(0).$$\nThus, we conclude that according to $\\xi _{1}^{2}$, the initially\nspin-squeezed state is always squeezed for $p\\neq 1$, irrespective\nof both the decoherence strength and decoherence models. In other\nwords, there exists no SSSD if we quantify spin squeezing by the\nfirst parameter $\\xi_1^2$.\n\n\\subsubsection{Concurrence}\n\nIn the expression (\\ref{conc}) of the concurrence, there are three\nterms inside the max function. The expression can be simplified to\n(see Appendix E for details):\n\\begin{equation}\nC_{r}=2(N-1)\\max (0,|u|-w).  \\label{sim}\n\\end{equation}\nBy using Eqs.~(\\ref{r3}) and (\\ref{c3}), one finds\n\\begin{align}\n&2(|u|-w) \\\\\n&\\hspace{5mm}=2s|u_{0}|+\\frac{s}{2}\\left[ s-2+s\\langle \\sigma\n_{1z}\\sigma _{2z}\\rangle\n_{0}-2p\\langle \\sigma _{1z}\\rangle _{0}\\right] ) \\notag\\\\\n&\\hspace{5mm}=sC_{0}-\\frac{s\\,p\\,x_{0}}{2}.\n\\end{align}\nSo, we obtain the evolution of the rescaled concurrence as\n\\begin{equation}\nC_{r}=\\max \\left[ 0,sC_{r}(0)-2^{-1}{(N-1)s\\,p}\\,x_{0}\\right],\n\\label{cccc}\n\\end{equation}\nwhich depends on the initial concurrence, expectation\n$\\langle\\sigma_{1z}\\rangle_0$, and correlation\n$\\langle\\sigma_{1z}\\sigma_{2z}\\rangle_0$.\n\n\\subsubsection{Numerical results}\n\nThe numerical results for the squeezing parameters and concurrence\nare shown in Fig.~1 for different initial values of  the twist\nangle $\\theta_0$, defined in Eq.~(\\ref{angle}). For the smaller\nvalue of $\\theta_0$, e.g., $\\theta_0=\\pi\/10$, we see that there is\nno ESD and SSSD. All the spin squeezing and the pairwise\nentanglement are completely robust against decoherence.\nIntuitively, the larger is the squeezing, the larger is the\nvanishing time. However, here, in contrast to this, no matter how\nsmall are the squeezing parameters and concurrence, they vanish\nonly in the asymptotic limit. This results from the complex\ncorrelations in the initial state and the special characteristics\nof the ADC.\n\nFor larger values of $\\theta_0$, as the decoherence strength $p$\nincreases, the spin squeezing decreases until it suddenly\nvanishes, so the phenomenon of SSSD occurs. There exists a\ncritical value $p_{c},$ after which there is no spin squeezing.\nThe vanishing time of $\\xi _{3}^{2}$ is always larger than those\nof $\\xi _{2}^{2}$ and the concurrence. We note that depending on\nthe initial state, the concurrence can vanish before or after $\\xi\n_{2}^{2}$. This means that in our model, the parameter $\\xi\n_{3}^{2}<1$ implies the existence of pairwise entanglement, while\n$\\xi _{2}^{2}$ does not.\n\n\n\n\\subsubsection{Decoherence strength $p_c$ corresponding to the SSSD}\n\nFrom Eqs.~(\\ref{xix2}), (\\ref{xix3}), and (\\ref{cccc}), the\ncritical value $p_{c}$ can be analytically obtained as\n\\begin{eqnarray}\np_{c}^{(k)} &=&\\frac{x_kC_{r}(0)}{\\left( N-1\\right) x_{0}}, \\quad (k=1,3) \\label{eq1} \\\\\np_{c}^{(2)} &=&\\frac{\\langle \\sigma_{1z}\\rangle _{0}^{2}+C_{r}(0)-1}{%\n1+2\\langle \\sigma_{1z}\\rangle _{0}+\\langle \\sigma _{z}\\rangle\n_{0}^{2}},\n\\end{eqnarray}\nwhere $x_1=2$ for the concurrence and $x_3=N$ for the squeezing\nparameter $\\zeta_{3}^{2}$.  The critical value $p_{c}^{(2)}$ is\nfor the second squeezing parameter $\\zeta_{2}^{2}$. Here, $p_c$ is\nrelated to the vanishing time $t_v$ via $p_c=1-\\exp(-\\gamma t_v)$.\n\nIn Fig.~2, we plot the critical values $p_c$ of the decoherence\nstrength versus $\\theta_0$. The initial-state squeezing parameter\n$\\zeta_{1}^{2}$ is also plotted for comparison. For a range of\nsmall values of $\\theta_0$, the entanglement and squeezing are\nrobust to decoherence. The concurrence and parameter $\\zeta\n_{2}^{2}$ intersect. However, we do not see the intersections\nbetween $\\zeta _{3}^{2}$ and $\\zeta _{2}^{2}$ or between\n$\\zeta_3^2$ and the concurrence. We also see that for the same\ndegree of squeezing, the vanishing times are quite different,\nwhich implies that except for the spin-squeezing correlations,\nother type of correlations exist. For large enough initial twist\nangles $\\pi\\le \\theta_0\\le 2\\pi$, the behavior of the squeezing\nparameter $\\xi_1^2$ is similar to those corresponding to\n$p_{c}^{(1)}$ and $p_{c}^{(3)}$.\n\n\\begin{figure}[tbp]\n\\includegraphics[width=9cm,clip]{fig2.eps}\n\\caption{(Color online) Critical values of the decoherence\nstrength $p_{c}^{(1)}$ (blue solid curve), $p_{c}^{(2)}$ (red\ncurve with squares), $p_{c}^{(3)}$ (green curve with circles), and\nthe squeezing parameter $\\protect\\zeta_1^2$ (black dashed curve)\nversus the initial twist angle $\\protect\\theta_0$ given by\nEq.~(\\ref{angle}) for the amplitude-damping channel, ADC. Here,\n$p_c$ is related to the vanishing time $t_v$ via\n$p_c=1-\\exp(-\\gamma t_v)$. At vanishing times, SSSD occurs. The\ncritical values $p_{c}^{(1)}$ , $p_{c}^{(2)}$, and $p_{c}^{(3)}$\ncorrespond to the concurrence, squeezing parameter $\\zeta_2^2$,\nand $\\zeta_3^2$, respectively.}\n\\end{figure}\n\n\\subsection{Phase-damping channel}\n\n\\subsubsection{Squeezing parameters and concurrence}\nNow, we study the spin squeezing and pairwise entanglement under\nthe PDC. For this channel, the expectation values $\\langle \\sigma\n_{z}^{\\otimes n}\\rangle $ are unchanged and the two correlations\n$\\langle \\sigma _{1-}\\sigma _{2-}\\rangle $ and $\\langle \\sigma\n_{1+}\\sigma _{2-}\\rangle $ evolve as (see Appendix D for\ndetails)%\n\\begin{eqnarray}\n\\langle \\sigma _{1-}\\sigma _{2-}\\rangle &=& s^{2}\\langle \\sigma\n_{1-}\\sigma _{2-}\\rangle , \\notag \\\\ \\langle \\sigma _{1+}\\sigma\n_{2-}\\rangle &=& s^{2}\\langle \\sigma _{1+}\\sigma _{2-}\\rangle.\n\\label{evolve}\n\\end{eqnarray}\nFrom the above equations and the fact $\\langle \\vec{\\sigma}_{1}\\cdot \\vec{%\n\\sigma}_{2}\\rangle _{0}=1$, one finds\n\\begin{align}\n\\langle \\vec{\\sigma}_{1}\\cdot \\vec{\\sigma}_{2}\\rangle\n&=s^{2}\\langle \\sigma _{1x}\\sigma _{2x}+\\sigma _{1y}\\sigma\n_{2y}\\rangle _{0}+\\langle \\sigma\n_{1z}\\sigma _{2z}\\rangle _{0}  \\notag \\\\\n &=s^{2}(1-\\langle \\sigma _{1z}\\sigma _{2z}\\rangle _{0})+\\langle \\sigma\n_{1z}\\sigma _{2z}\\rangle _{0},  \\label{sigmasigma} \\\\\n\\mathcal{C}_{zz}(p)&=\\mathcal{C}_{zz}(0).  \\label{sigmasigma2}\n\\end{align}\nTherefore, from the above properties, we obtain the evolution of the\nsqueezing parameters,\n\\begin{eqnarray}\n\\xi _{1}^{2} &=&1-s^{2}C_{r}(0), \\label{eee}\\\\\n\\xi _{2}^{2} &=&\\frac{\\xi _{1}^{2}}{\\langle \\sigma _{1z}\\rangle _{0}^{2}}%\n,~  \\label{ee2}\n\\end{eqnarray}\nand the third parameter becomes\n\\begin{eqnarray}\n\\xi _{3}^{2} &=&\\frac{N\\min \\left[\\xi _{1}^{2},1+\\mathcal{C}%\n_{zz}(0)\\right] }{(N-{1})[ s^{2}+(1-s^{2})\\langle \\sigma_{1z}\\sigma _{2z}\\rangle_{0}] +1}  \\label{ee3} \\\\\n&=&\\frac{N\\xi _{1}^{2}}{(N-{1})[s^{2}+(1-s^{2})\\langle \\sigma\n_{1z}\\sigma _{2z}\\rangle_{0}] +{1}}.\n\\end{eqnarray}\nwhere we have used Eqs.~(\\ref{sigmasigma}) and\n(\\ref{sigmasigma2}), and the property $\\mathcal{C}_{zz}(0)\\geq 0.$\n\nFrom Eq.~(\\ref{evolve}) and the simplified form of the concurrence\ngiven by Eq.~(\\ref{sim}), the concurrence is found to be\n\\begin{eqnarray}\nC_{r} &=&\\max \\Big\\{0,2(N-1)  \\notag \\\\\n&&\\times \\left[ s^{2}|u_{0}|-{4}^{-1}(1-\\langle \\sigma _{1z}\\sigma\n_{2z}\\rangle _{0}\\rangle )\\right] \\Big\\}  \\notag \\\\\n&=&\\max \\left[0,s^{2}C_{r}(0)+\\frac{a_{0}(s^{2}-1)}{2}\\right].\n\\label{ee4}\n\\end{eqnarray}\nwhere\n\\begin{align}\na_{0}=\\left( N-1\\right) (1-\\langle \\sigma _{1z}\\sigma _{2z}\\rangle\n_{0}).\n\\end{align}\nThus, we obtained all time evolutions of the spin-squeezing\nparameters and the concurrence. To study the phenomenon of SSSD,\nwe below examine the vanishing times.\n\n\\subsubsection{Decoherence strength $p_c$ corresponding to the SSSD}\n\nThe critical decoherence strengths $p_c$ can be obtained from Eqs.~(\\ref{ee2}), (\\ref{ee3}), and (%\n\\ref{ee4}) as follows:\n\\begin{eqnarray}\np_{c}^{(k)} &=&1-\\left[ \\frac{a_{0}}{x_kC_{r}(0)+a_{0}}\\right]\n^{\\frac{1}{2}},\n\\label{eq2} \\\\\np_{c}^{(2)} &=&1-\\left[ \\frac{1-\\langle \\sigma _{1z}\\rangle _{0}^{2}}{C_{r}(0)%\n}\\right]^{\\frac{1}{2}},\n\\end{eqnarray}\nwhere $k=1,3$ and $x_1=2, x_3=N$.\n\\begin{figure}[tbp]\n\\includegraphics[width=9cm,clip]{fig3.eps}\n\\caption{(Color online) Same as in Fig.~2 but for the\nphase-damping channel, PDC, instead of ADC. }\n\\end{figure}\n\nIn Fig. 3, we plot the decoherence strength $p_c$ versus the twist\nangle $\\theta_0$  of the initial state for the PDC. For this\ndecoherence channel, the critical value $p_c's$ first decrease\nuntil they reach zero. Also, it is symmetric with respect to\n$\\theta_0 =\\pi ,$ which is in contrast to the ADC. There are also\nintersections between the concurrence and parameter $\\xi\n_{2}^{2},$ and the critical value $p_{c}^{(3)}$ is always larger\nthan $p_{c}^{(1)}$ and $p_{c}^{(2)}.$\n\n\\begin{table*}[tbp]\n\\caption{Analytical results for the time evolutions of all\nrelevant expectations, correlations, spin-squeezing parameters,\nand concurrence, as well as the critical values $p_c$ of the\ndecoherence strength $p$. This is done for the three decoherence\nchannels considered in this work. For the concurrence $C$, we give\nthe expression for $C_r'$, which is related to the rescaled\nconcurrence $C_r$ via $C_r=\\max(0,C_r')$.}\n\\begin{center}\n\\begin{tabular}{c||c|c|c}\n\\hline\\hline &  Amplitude-damping channel & Phase-damping channel\n& Depolarizing channel\n\\\\\n& (ADC) & (PDC) & (DPC)\n\\\\ \\hline\\hline\n\\parbox{1.5 cm} {\\vspace{0.3cm} $\\langle\\sigma_{1z}\\rangle$\\vspace{0.2cm} } &\n\\parbox{4 cm} {$s\\langle\\sigma_{1z}\\rangle_0-p$} &\n\\parbox{4 cm}{$\\langle\\sigma_{1z}\\rangle_0$}&\n\\parbox{4 cm}{$s\\langle\\sigma_{1z}\\rangle_0$}\n\\\\ \\hline\n\n\\parbox{1.5 cm} {\\vspace{0.3cm}$\\langle\\sigma_{1z}\\sigma_{2z}\\rangle$\\vspace{0.3cm}} &\n\\parbox{4.3cm} {$s^2\\langle\\sigma_{1z}\\sigma_{2z}\\rangle_0-2sp\\langle\\sigma_{1z}\\rangle_0+p^2$} &\n\\parbox{4 cm} {$\\langle\\sigma_{1z}\\sigma_{2z}\\rangle_0$}&\n\\parbox{4 cm}{$s^2\\langle\\sigma_{1z}\\sigma_{2z}\\rangle_0$}\n\\\\ \\hline\n\n\\parbox{1.5 cm} {\\vspace{0.3cm}$\\langle\\sigma_{1+}\\sigma_{2-}\\rangle$\\vspace{0.3cm}} &\n\\parbox{4cm} {$s\\langle\\sigma_{1+}\\sigma_{2-}\\rangle_0$} &\n\\parbox{4 cm} {$s^2\\langle\\sigma_{1+}\\sigma_{2-}\\rangle_0$}&\n\\parbox{4 cm}{$s^2\\langle\\sigma_{1+}\\sigma_{2-}\\rangle_0$}\n\\\\ \\hline\n\n\n\\parbox{1.5 cm} {\\vspace{0.3cm}$\\langle\\sigma_{1-}\\sigma_{2-}\\rangle$\\vspace{0.3cm}} &\n\\parbox{4cm} {$s\\langle\\sigma_{1-}\\sigma_{2-}\\rangle_0$} &\n\\parbox{4 cm} {$s^2\\langle\\sigma_{1-}\\sigma_{2-}\\rangle_0$}&\n\\parbox{4 cm}{$s^2\\langle\\sigma_{1-}\\sigma_{2-}\\rangle_0$}\n\\\\ \\hline\n\n\\parbox{1.5 cm} {\\vspace{0.3cm}$\\langle\\vec{\\sigma}_{1}\\cdot\\vec{\\sigma}_{2}\\rangle$\\vspace{0.3cm}} &\n\\parbox{4cm} {$1-s\\,p\\,x_0$} &\n\\parbox{4 cm} {$s^2(1-\\langle\\sigma_{1z}\\sigma_{2z}\\rangle_0)+\\langle\\sigma_{1z}\\sigma_{2z}\\rangle_0$}&\n\\parbox{4 cm}{$s^2$}\n\\\\ \\hline\n\n\\parbox{1.5 cm} {\\vspace{0.3cm}${\\cal C}_{zz}$\\vspace{0.3cm}} &\n\\parbox{4 cm} {$s^2{\\cal C}_{zz}(0)$} &\n\\parbox{4 cm} {${\\cal C}_{zz}(0)$}&\n\\parbox{4 cm}{$s^2{\\cal C}_{zz}(0)$}\n\\\\ \\hline\n\n\n\n\\parbox{1.5 cm} {\\vspace{0.3cm}$\\xi_1^2$\\vspace{0.3cm}} &\n\\parbox{4cm} {$1-sC_r(0)$} &\n\\parbox{4 cm} {$1-s^2C_r(0)$}&\n\\parbox{4 cm}{$1-s^2C_r(0)$}\n\\\\ \\hline\n\n\\parbox{1.5 cm} {\\vspace{0.3cm}$\\xi_2^2$\\vspace{0.3cm}} &\n\\parbox{4cm} {$\\displaystyle\\frac{1-sC_r(0)}{(s\\langle\\sigma_{1z}\\rangle_0-p)^2}$} &\n\\parbox{4 cm} {\\vspace{0.15cm}$\\displaystyle\\frac{1-s^2C_r(0)}{\\langle\\sigma_{1z}\\rangle_0^2}$\\vspace{0.15cm}}&\n\\parbox{4 cm}{$\\displaystyle\\frac{1-s^2C_r(0)}{s^2\\langle\\sigma_{1z}\\rangle_0^2}$}\n\\\\ \\hline\n\n\n\\parbox{1.5 cm} {\\vspace{0.3cm}$\\xi_3^2$\\vspace{0.3cm}} &\n\\parbox{4cm} {$\\displaystyle\\frac{1-sC_r(0)}{1+(N^{-1}-1)s\\,p\\,x_0}$} &\n\\parbox{6 cm} {\\vspace{0.15cm}$\\displaystyle\\frac{1-s^2C_r(0)}{(1-N^{-1})[s^2+(1-s^2)\\langle\\sigma_{1z}\\sigma_{2z}\\rangle_0]+N^{-1}}$\\vspace{0.2cm}}&\n\\parbox{4 cm}{$\\displaystyle\\frac{1-s^2C_r(0)}{(1-N^{-1})s^2+N^{-1}}$}\n\\\\ \\hline\n\n\n\\parbox{1.5 cm} {\\vspace{0.3cm}$C_r'$\\vspace{0.3cm}} &\n\\parbox{4cm} {$sC_{r}(0)-(N-1)s\\,p\\,x_{0}\/2$} &\n\\parbox{4 cm} {$s^{2}C_{r}(0)+{a_{0}(s^{2}-1)}\/{2}$}&\n\\parbox{4.3 cm}{$s^{2}C_{r}(0)+(N-1)(s^{2}-1)\/2$}\n\\\\ \\hline\n\n\\parbox{1.5 cm} {\\vspace{0.3cm}$p_{c}^{(1)}$\\vspace{0.3cm}} &\n\\parbox{4cm} {$\\displaystyle\\frac{2C_{r}(0)}{\\left( N-1\\right) x_{0}}$} &\n\\parbox{4 cm} {$\\displaystyle 1-\\left(\n\\frac{a_{0}}{2C_{r}(0)+a_{0}}\\right)\n^{\\frac{1}{2}}$}&\n\\parbox{4 cm}{$\\displaystyle 1-\\left( \\frac{N-1}{2 C_{r}(0)+N-1}\\right)\n^{\\frac{1}2}$}\n\\\\ \\hline\n\n\n\\parbox{1.5 cm} {\\vspace{0.3cm}$p_{c}^{(2)}$\\vspace{0.3cm}} &\n\\parbox{4cm} {$\\displaystyle\\frac{\\langle \\sigma_{1z}\\rangle _{0}^{2}+C_{r}(0)-1}{%\n1+2\\langle \\sigma_{1z}\\rangle _{0}+\\langle \\sigma _{z}\\rangle\n_{0}^{2}}$} &\n\\parbox{4 cm} {$\\displaystyle 1-\\left( \\frac{1-\\langle \\sigma _{1z}\\rangle _{0}^{2}}{C_{r}(0)%\n}\\right) ^{\\frac{1}2}$}&\n\\parbox{4 cm}{$\\displaystyle 1-\\left( \\frac{1}{C_{r}(0)+\\langle \\sigma _{1z}\\rangle _{0}^{2}%\n}\\right) ^{\\frac{1}2}$}\n\\\\ \\hline\n\n\\parbox{1.5 cm} {\\vspace{0.3cm}$p_{c}^{(3)}\\vspace{0.3cm}$} &\n\\parbox{4cm} {$\\displaystyle\\frac{NC_{r}(0)}{\\left( N-1\\right) x_{0}}$} &\n\\parbox{4 cm} {$\\displaystyle 1-\\left(\n\\frac{a_{0}}{NC_{r}(0)+a_{0}}\\right)\n^{\\frac{1}{2}}$}&\n\\parbox{4 cm}{$\\displaystyle 1-\\left( \\frac{N-1}{N C_{r}(0)+N-1}\\right)\n^{\\frac12}$}\n\\\\ \\hline\n\n\\end{tabular}%\n\\end{center}\n\\end{table*}\n\n\\subsection{Depolarizing channel}\n\n\\begin{figure}[tbp]\n\\includegraphics[width=9cm,clip]{fig4.eps}\n\\caption{(Color online) Same as in Fig.~2 but for the depolarizing\nchannel, DPC, instead of ADC.}\n\\end{figure}\n\n\\subsubsection{Squeezing parameters and concurrence}\n\nThe decoherence of the squeezing parameter defined by S\\o rensen\n{\\it et al.}~\\cite{Sorensen} has been studied in\nRef.~\\cite{SimonKempe} for the DPC. It is intimately related to\nthe second squeezing parameter $\\xi_2^2$. For the DPC, the\nevolution of correlations $\\langle \\sigma _{1-}\\sigma _{2-}\\rangle\n$ and $\\langle \\sigma _{1+}\\sigma _{2-}\\rangle $ are the same as\nthose of the DPC given by Eq.~(\\ref{evolve}), and the expectations\n$\\langle \\sigma _{1z}\\rangle $ and $\\langle \\sigma _{1z}\\sigma\n_{2z}\\rangle $ change as (see Appendix D).\n\\begin{eqnarray}\n\\langle \\sigma _{1z}\\rangle  &=&s\\langle \\sigma _{1z}\\rangle _{0}, \\\\\n\\langle \\sigma _{1z}\\sigma _{2z}\\rangle  &=&s^{2}\\langle \\sigma\n_{1z}\\sigma _{2z}\\rangle_{0}.  \\label{cccccc}\n\\end{eqnarray}\nFrom these equations, we further have\n\\begin{align}\n& \\langle \\vec{\\sigma}_{1}\\cdot \\vec{\\sigma}_{2}\\rangle =s^{2}\\langle \\vec{%\n\\sigma}_{1}\\cdot \\vec{\\sigma}_{2}\\rangle _{0}=s^{2}, \\\\\n& \\mathcal{C}_{zz}=s^{2}\\left( \\langle \\sigma _{1z}\\sigma\n_{2z}\\rangle _{0}-\\langle \\sigma _{1z}\\rangle _{0}\\langle \\sigma\n_{2z}\\rangle _{0}\\right) =s^{2}\\mathcal{C}_{zz}(0).\n\\end{align}\n The squeezing parameter $\\xi_1^2$ is\ngiven by Eq.~(\\ref{eee}), and the other two squeezing parameters\nare obtained as\n\\begin{eqnarray}\n\\xi _{2}^{2} &=&\\frac{\\xi _{1}^{2} }{s^{2}\\langle \\sigma\n_{1z}\\rangle\n_{0}^{2}},~  \\label{k1} \\\\\n\\xi _{3}^{2} &=&\\frac{N\\min \\left\\{\\xi_{1}^{2} ,1+s^{2}\\mathcal{C}%\n_{zz}(0)\\right\\} }{(N-{1})s^{2}+{1}}  \\notag \\\\\n&=&\\frac{N\\xi _{1}^{2} }{(N-{1})s^{2}+{1}}. \\label{k2}\n\\end{eqnarray}\nBy making use of Eqs.~( \\ref{evolve}) and (\\ref{cccccc}) and\nstarting from the simplified form of the concurrence (\\ref{sim}),\nwe obtain\n\\begin{eqnarray}\nC_{r} &=&\\max\n\\left\\{0,2(N-1)\\left[s^{2}|u_{0}|-\\textstyle{\\frac14}(1-s^{2}\\langle\n\\sigma\n_{1z}\\sigma _{2z}\\rangle _{0})\\right]\\right\\}   \\notag \\\\\n&=&\\max \\left[ 0,s^{2}C_{r}(0)+{2}^{-1}(N-1)(s^{2}-1)\\right] .\n\\label{cb}\n\\end{eqnarray}\nWe observe that the concurrence is dependent only on the initial\nvalue itself, not other ones.\n\n\n\\subsubsection{Decoherence strength $p_c$ corresponding to the SSSD}\n\nFrom Eqs.~(\\ref{cb}), (\\ref{k1}), and (\\ref{k2}), the vanishing\ntimes are analytically calculated as\n\\begin{eqnarray}\np_{v}^{(k)} &=&1-\\left[ \\frac{N-1}{x_k C_{r}(0)+N-1}\\right]\n^{\\frac 12}, \\label{eq3}\n\\\\\np_{v}^{(2)} &=&1-\\left[ \\frac{1}{C_{r}(0)+\\langle \\sigma _{1z}\\rangle _{0}^{2}%\n}\\right]^{\\frac 12},\n\\end{eqnarray}\nwhere $k=1,3$ and $x_1=2, x_3=N$.\n\nIn Fig.~3, we plot the critical values $p_c$ versus the initial\ntwist angle $\\theta_0$ for the DPC. For the DPC, the $p_c's$ first\nincrease until they reach their maxima and then decrease to zero.\nAlso, it is symmetric with respect to $\\theta_0=\\pi$, which is the\nsame as for the PDC. There are also intersections between the\nconcurrence and the parameter $\\xi _{2}^{2}.$ Qualitatively, the\nbehaviors of $p_{c}^{(1)}$ and $p_{c}^{(3)}$ are the same as that\nof the squeezing parameter $\\zeta _{1}^{2}$. This implies that the\nlarger the squeezing, the larger is the critical value $p_c$.\n\nThe common features of these three decoherence channels are: (i)\nThe critical value $p_{v3}$ is always larger or equal than the\nother two, namely, the spin-squeezing correlations according to\n$\\xi _{3}^{2}$ are more robust; (ii) there always exist two\nintersections between the concurrence and the parameter $\\xi\n_{2}^{2},$ for $\\theta_0$ from 0 to $2\\pi $, irrespective of the\ndecoherence channels; (iii) when there is no squeezing (central\narea of Figs. 2, 3, and 4), all vanishing times are zero. Table II\nconveniently lists all the analytical results obtained in this\nsection.\n\n\\section{Conclusions and remarks}\n\nTo summarize, for a spin ensemble in a typical spin-squeezing\ninitial state under three different decoherence channels, we have\nstudied spin squeezing with three different parameters in\ncomparison with the pairwise entanglement quantified by the\nconcurrence. When the subsystems of the correlated system decay\nasymptotically in time, the spin-squeezing parameter $\\zeta\n_{1}^{2}$ also decays asymptotically in time for all three types\nof decoherence. However, for the other two squeezing parameters\n$\\zeta_2^2$ and $\\zeta_3^2$, we find the appearance of\nspin-squeezing sudden death and entanglement sudden death. The\nglobal behaviors of the correlated state are markedly different\nfrom the local ones. The spin-squeezing parameter $\\zeta _{2}^{2}$\ncan vanish before, simultaneously, or after the concurrence, while\nthe squeezing parameter $\\zeta _{3}^{2}$ is always the last to\nvanish. This means that this parameter is more robust to\ndecoherence, and it can detect more entanglement than~$\\xi_2^2$.\n\nOur analytical approach for the vanishing times can be applied to\nany initial quantum correlated states, not restricted to the\npresent one-axis twisted state. Moreover, for more complicated\nchannels, such as the amplitude-damping channel at finite\ntemperatures~\\cite{Aolita} or the channel discussed in\nRef.~\\cite{Lidar}, the method developed in this article can be\nreadily applied to study spin squeezing under these decoherence\nchannels.\n\nOur investigations show the widespread occurrence of sudden death\nphenomena in many-body quantum correlations. Since there exists\ndifferent vanishing times for different squeezing parameters, spin\nsqueezing offers a possible way to detect the total spin\ncorrelation and their quantum fluctuations with distinguishable\ntime scales. The discovery of different lifetimes for various\nspin-squeezing parameters means that, in some time region, there\nstill exists another quantum correlation when other quantum\ncorrelations suddenly vanish. However, to determine which kind of\ncorrelations will vanish, one possible approach is to further\ninvoke irreducible multiparty correlations~\\cite{Zhou}, where the\nmultipartite correlations are classified in a series of\nirreducible $k$ party ones. If we could obtain the time evolution\nbehaviors of such irreducible multipartite correlations in various\ndecoherence channels, we could classify lifetimes for the\nspin-squeezing sudden death of various multipartite correlations\norder by order.\n\n\\begin{acknowledgments}\nWe gratefully acknowledge partial support from the National\nSecurity Agency, Laboratory of Physical Sciences, Army Research\nOffice, National Science Foundation under Grants Nos. 0726909, and\nJSPS-RFBR 06-02-91200. X. Wang acknowledges support from the\nNational Natural Science Foundation of China under No. 10874151,\nthe National Fundamental Research Programs of China under Grant\nNo. 2006CB921205, and the Program for New Century Excellent\nTalents in University (NCET). A.~M. acknowledges support from the\nPolish Ministry of Science and Higher Education under Grant No. N\nN202 261938.\n\\end{acknowledgments}\n\n\\begin{appendix}\n\n\\section{Spin-squeezing parameter $\\protect\\xi _{3}^{2}$ for states with\nparity symmetry}\n\nHere, we calculate the spin-squeezing parameter $\\xi_3^2$ for\ncollective states with either even or odd parity symmetry. For\nsuch states, we immediately have\n\\begin{equation}\n\\langle J_{x}\\rangle =\\langle J_{y}\\rangle =\\langle\nJ_{x}J_{z}\\rangle =\\langle J_{y}J_{z}\\rangle =0\n\\end{equation}\nas the operators change the parity of the state. Then, the mean\nspin direction is along the $z$ direction and the correlation\nmatrix given by Eq.~(\\ref{cmatrix}) is simplified to\n\\begin{equation}\n\\mathbf{C}=\\left(\n\\begin{array}{ccc}\n\\langle J_{x}^{2}\\rangle & C_{xy} & 0 \\\\\nC_{xy} & \\langle J_{y}^{2}\\rangle & 0 \\\\\n0 & 0 & \\langle J_{z}^{2}\\rangle%\n\\end{array}%\n\\right),\n\\end{equation}\nwhere $C_{xy}=\\langle \\lbrack J_{x},J_{y}]_{+}\\rangle\/2$. From the\ncorrelation matrix $\\mathbf{C}$ and the definition of covariance\nmatrix $\\gamma $ given by Eq.~(\\ref{comatrix}), one finds\n\\begin{equation}\n\\Gamma =\\left(\n\\begin{array}{ccc}\nN\\langle J_{x}^{2}\\rangle & {N}C_{xy} & 0 \\\\\n{N}C_{xy} & N\\langle\nJ_{y}^{2}\\rangle & 0 \\\\\n0 & 0 & N(\\Delta J_{z})^{2}+\\langle J_{z}^{2}\\rangle%\n\\end{array}%\n\\right).\n\\end{equation}\nThis matrix has a block-diagonal form and the eigenvalues of the\n$2\\times 2$ block are obtained as\n\\begin{equation}\n\\lambda _{\\pm }=\\frac{N}{2}\\left( \\langle J_{x}^{2}+J_{y}^{2}\\rangle\n\\pm |\\langle J_{-}^{2}\\rangle |\\right) .\n\\end{equation}\nIn deriving the above equation, we have used the relation%\n\\begin{equation}\nJ_{-}^{2}=J_{x}^{2}-J_{y}^{2}-i[J_{x},J_{y}]_{+}.\n\\end{equation}\nTherefore, the smallest eigenvalue $\\lambda _{\\min }$ of $\\Gamma $\nis obtained as\n\\begin{equation}\n\\lambda _{\\min }=\\min \\left(\\lambda _{-},N(\\Delta\nJ_{z})^{2}+\\langle J_{z}^{2}\\rangle \\right),  \\label{xixi2}\n\\end{equation}\nwhere $\\lambda _{-}$ differs from the squeezing parameter\n$\\xi_1^2$ given by Eq.~(\\ref{xixi1}) by only a multiplicative\nconstant, as seen by comparing Eqs.~(\\ref{xixi1}) and\n(\\ref{xixi2}). From Eqs.~(\\ref{xixi2}) and (\\ref{x3}), one finds\nthat the squeezing parameter $\\xi_3^2$ is given by\nEq.~(\\ref{xixixi}).\n\n\\section{Spin-squeezing parameters for the one-axis twisted state}\n\nHere, we will use the Heisenberg picture to derive the relevant\nexpectations and spin-squeezing parameters for the initial\nstate~\\cite{Molmer2,WangMolmer2}. To determine the spin-squeezing\nparameter $\\xi _{1}^{2}$ given by Eq.~(\\ref{xixixi1}), one needs\nto know the expectation $\\langle \\sigma_{1z}\\rangle_0$, and\ncorrelations $\\langle \\sigma _{1+}\\sigma _{2-}\\rangle_0$ and\n$\\langle \\sigma _{1-}\\sigma _{2-}\\rangle_0$. We first consider the\nexpectation $\\langle \\sigma_{1z}\\rangle_0$. For simplicity, we\nomit the subscript $0$ in the following formulas.\n\n\\subsection{Expectation $\\langle\\sigma_{1z}\\rangle$}\n\nThe evolution operator can be written as,\n\\begin{equation}\nU=\\exp({-i\\chi tJ_{x}^{2}})=\\exp\\left({-i\\theta\n\\sum_{k>l}j_{kx}j_{lx}}\\right)\n\\end{equation}\nup to a trivial phase, where $\\theta=2\\chi t$ given by\nEq.~(\\ref{angle}). From this form, the evolution of $j_{1z}$ can\nbe obtained as\n\\begin{eqnarray}\nU^{\\dagger }j_{1z}U =j_{1z}\\cos [ \\theta j_x^{(2)}] +j_{1y}\\sin [\n\\theta j_x^{(2)}],\n\\end{eqnarray}\nwhere\n\\begin{equation}\nj_{x}^{(k)}=\\sum_{l=k}^N j_{lx}.\n\\end{equation}\nTherefore, the expectations are\n\\begin{equation} \\langle j_{1z}\\rangle\n=-{2}^{-1}\\langle {\\bf 1'}| \\cos [ \\theta j_x^{(2)}] |{\\bf\n1'}\\rangle \\label{jz1}\n\\end{equation}\nsince $\\langle 1|j_{1y}|1\\rangle =0.$ Here, $|{\\bf 1'}\\rangle\n=|1\\rangle _{2}\\otimes ...\\otimes |1\\rangle _{N}.$ So, one can\nfind the following form for the expectation values\n\\begin{eqnarray}\n\\langle {\\bf 1}|\\cos \\left[ \\theta J_{x}\\right] |{\\bf 1}\\rangle\n&=&\\left( \\langle {\\bf 1}|e^{i\\theta J_{x}}|{\\bf 1}\\rangle\n+ {\\rm c.c.}\\right) \/2  \\notag \\\\\n&=&\\left( \\Pi _{k=1}^{N}\\langle 1|e^{i\\theta j_{kx}}|1\\rangle + {\\rm c.c.} \\right) \/2  \\notag \\\\\n&=&\\cos ^{N}({\\theta'}),  \\label{jz2}\n\\end{eqnarray}\nwhere $\\theta'=\\theta\/2$ and $|{\\bf 1}\\rangle=|1\\rangle^{\\otimes\nN}$.\n\nBy using Eqs.~(\\ref{jz1}) and (\\ref{jz2}), one gets\n\\begin{equation}\\label{sigmaz}\n\\langle \\sigma _{z}\\rangle =-\\cos ^{N-1}\\left( {\\theta' }\\right).\n\\end{equation}\n\n\n\\subsection{\\protect\\bigskip Correlation $\\langle \\protect\\sigma _{1+}%\n\\protect\\sigma _{2-}\\rangle $}\n\nSince the operator $\\sigma _{1x}\\sigma _{2x}$ commutes with the\nunitary operator $U,$ we easily obtain\n\\begin{equation}\n\\langle \\sigma _{1x}\\sigma _{2x}\\rangle =0.  \\label{xx}\n\\end{equation}\nWe now compute the correlations $\\langle \\sigma _{1z}\\sigma\n_{2z}\\rangle .$ From the unitary operator,\n\\begin{eqnarray*}\n&&\\hspace*{-7mm} U^{\\dagger }j_{1z}j_{2z}U\\nonumber\\\\\n&=&\\left[j_{1z}\\cos ( \\theta\nj_x^{(2)})+j_{1y}\\sin ( \\theta j_x^{(2)}) \\right] \\\\\n&&\\times \\left[j_{2z}\\cos [ \\theta (j_{1x}+j_{x}^{(3)})]\n+j_{2y}\\sin [ \\theta (j_{1x}+j_{x}^{(3)})] \\right] \\\\\n&=&\\left[j_{1z}\\cos (\\theta j_{2x})\\cos(\\theta\nj_{x}^{(3)})-j_{1z}\\sin\n(\\theta j_{2x})\\sin (\\theta j_{x}^{(3)})\\right. \\\\\n&&\\left. +j_{1y}\\sin (\\theta j_{2x})\\cos (\\theta\nj_{x}^{(3)})+j_{1y}\\cos (\\theta j_{2x})\\sin (\\theta j_{x}^{(3)})\n\\right] \\\\\n&&\\times \\left[j_{2z}\\cos (\\theta j_{1x})\\cos (\\theta j_{x}^{(3)})\n-j_{2z}\\sin (\\theta j_{1x})\\sin (\\theta j_{x}^{(3)})\\right.\n\\\\\n&&\\left. +j_{2y}\\sin (\\theta j_{1x})\\cos (\\theta j_{x}^{(3)})\n+j_{2y}\\cos (\\theta j_{1x})\\sin (\\theta j_{x}^{(3)}) \\right].\n\\end{eqnarray*}%\nAlthough there are 16 terms after expanding the above equation,\nonly 4 terms survive when calculating $\\langle s_{1z}s_{2z}\\rangle\n.$ We then have\n\\begin{eqnarray}\\label{eef}\n\\langle j_{1z}j_{2z}\\rangle\n&=&\\langle {\\bf 1}|j_{1z}j_{2z}\\cos ^{2}(\\theta\/2)\\cos ^{2}(\\theta j_{x}^{(3)})\n\\notag \\\\\n&&-j_{1z}j_{2x}j_{2y}\\sin (\\theta )\\sin ^{2}(\\theta j_{x}^{(3)})\n\\notag \\\\\n&&+4j_{1y}j_{1x}j_{2x}j_{2y}\\sin ^{2}(\\theta \/2)\\cos ^{2}(\\theta j_{x}^{(3)})\n\\notag \\\\\n&&-j_{1y}j_{1x}j_{2z}\\sin (\\theta )\\sin ^{2}(\\theta j_{x}^{(3)}) |{\\bf 1}\\rangle\n\\notag \\\\\n&=&{4}^{-1}\\langle {\\bf 1}'|\\cos ^{2}(\\theta j_{x}^{(3)}) |{\\bf 1}'\\rangle\n\\notag \\\\\n&=&{8}^{-1}\\langle {\\bf 1}'|\\left[ 1+\\cos (2\\theta j_{x}^{(3)}\n) \\right]|{\\bf 1}'\\rangle\n\\notag \\\\\n&=&{8}^{-1}\\left[ 1+\\cos ^{N-2}(\\theta )\\right],\n\\end{eqnarray}\nwhere $|{\\bf 1}'\\rangle=|1\\rangle_3\\otimes...\\otimes|1\\rangle_N$.\nThe second equality in Eq.~(\\ref{eef}) is due to the property\n$j_{x}j_{y}=-j_{y}j_{x}={ij_z}\/{2}$, and the last equality from\nEq.~(\\ref{jz2}). Finally, from the above equation, one finds\n\\begin{equation}\n\\langle \\sigma _{1z}\\sigma _{2z}\\rangle ={2}^{-1}\\left( 1+\\cos\n^{N-2}\\theta \\right).  \\label{zz}\n\\end{equation}\nDue to the relation $\\langle \\sigma _{1x}\\sigma _{2x}+\\sigma\n_{1y}\\sigma _{2y}+\\sigma _{1z}\\sigma _{2z}\\rangle =1$ for the\ninitial state, the correlation $\\langle \\sigma _{1y}\\sigma\n_{2y}\\rangle $ is obtained from Eqs.~(\\ref{xx}) and (\\ref{zz}) as\n\\begin{equation}\n\\langle \\sigma _{1y}\\sigma _{2y}\\rangle ={2}^{-1}\\left( 1-\\cos\n^{N-2}\\theta \\right) .  \\label{yy}\n\\end{equation}\nSubstituting Eqs.~(\\ref{xx}) and (\\ref{yy}) into the following\nrelations\n\\begin{equation*}\n\\sigma _{1x}\\sigma _{2x}+\\sigma _{1y}\\sigma _{2y}=2\\left( \\sigma\n_{1+}\\sigma _{2-}+\\sigma _{1-}\\sigma _{2+}\\right)\n\\end{equation*}\nleads to one element of the two-spin reduced density matrix,\n\\begin{equation}\\label{y0}\ny_{0}=\\langle \\sigma _{1+}\\sigma _{2-}\\rangle ={8}^{-1}\\left(\n1-\\cos ^{N-2}\\theta \\right) ,\n\\end{equation}\nwhere the relation $\\langle \\sigma _{1+}\\sigma _{2-}\\rangle\n=\\langle \\sigma _{1-}\\sigma _{2+}\\rangle $ is used due to the\nexchange symmetry.\n\n\\subsection{Correlation $\\langle \\protect\\sigma _{1-}\\protect\\sigma %\n_{2-}\\rangle $}\n\nTo calculate the correlation $\\langle \\sigma _{1-}\\sigma\n_{2-}\\rangle ,$ due to the following relations%\n\\begin{eqnarray}\n\\sigma _{1x}\\sigma _{2x}-\\sigma _{1y}\\sigma _{2y} &=&2\\left( \\sigma\n_{1+}\\sigma _{2+}+\\sigma _{1-}\\sigma _{2-}\\right) ,  \\label{sigma1} \\\\\ni\\left( \\sigma _{1x}\\sigma _{2y}+\\sigma _{1y}\\sigma _{2x}\\right)\n&=&2\\left( \\sigma _{1+}\\sigma _{2+}-\\sigma _{1-}\\sigma\n_{2-}\\right) ,\\quad  \\label{sigma2}\n\\end{eqnarray}\nwe need to know the expectations $\\langle j_{1x}j_{2y}\\rangle .$ The\nevolution of $j_{1x}j_{2y}$ is given by\n\\begin{eqnarray*}\nU^{\\dagger }s_{1x}s_{2y}U &=&j_{1x}\\left\\{j_{2y}\\cos \\left[ \\theta\n(j_{1x}+j_{x}^{(3)})\\right] \\right. \\\\\n&&\\left.\\quad~~ -j_{2z}\\sin \\left[ \\theta\n(j_{1x}+j_{x}^{(3)})\\right] \\right\\},\n\\end{eqnarray*}%\nand the expectation is obtained as\n\\begin{eqnarray*}\n\\langle j_{1x}j_{2y}\\rangle &=&{2}^{-1}\\langle {\\bf 1'}|j_{1x}\\sin\n\\left[ \\theta\n(j_{1x}+j_{x}^{(3)})\\right] |{\\bf 1'}\\rangle  \\\\\n&=&{(4i)}^{-1}\\langle {\\bf 1'}|j_{1x}e^{i\\theta j_{1x}}\\Pi\n_{k=3}^{N}e^{i\\theta j_{kx}} \\\\\n&&-j_{1x}e^{-i\\theta j_{1x}}\\Pi _{k=3}^{N}e^{-i\\theta j_{kx}}|{\\bf\n1'}\\rangle \\\\\n&=&{(4i)}^{-1}{\\cos ^{N-2}\\left( {\\theta'}{}\\right) }\\langle\n1|j_{1x}e^{i\\theta j_{1x}}-j_{1x}e^{-i\\theta j_{1x}}|1\\rangle  \\\\\n&=&{2}^{-1}{\\cos ^{N-2}\\left( {\\theta'}\\right) }\\langle 1|\nj_{1x}\\sin\n(\\theta j_{1x})|1\\rangle  \\\\\n&=&{4}^{-1}{\\sin \\left({\\theta'}{}\\right) \\cos ^{N-2}\\left(\n\\theta'\\right) }\n\\end{eqnarray*}%\nHere, $|{\\bf 1'}\\rangle=|1\\rangle _{1}\\otimes |1\\rangle\n_{3}\\otimes ...\\otimes |1\\rangle _{N}$, where $|1\\rangle_2$ is\nabsent. Moreover, $\\langle j_{1y}j_{2x}\\rangle =\\langle\nj_{1x}j_{2y}\\rangle $ due to the exchange symmetry, and thus,\n\\begin{equation*}\n\\langle j_{1x}j_{2y}+j_{1y}j_{2x}\\rangle =2^{-1}{\\sin \\left({\\theta'}{}%\n\\right) \\cos ^{N-2}\\left({\\theta'}{}\\right) }.\n\\end{equation*}\nFor the initial state (\\ref{initial}), we obtain the following\nexpectations \\cite{KU,WangMolmer}\n\\begin{equation}\\label{b12}\n\\langle \\sigma _{1x}\\sigma _{2y}+\\sigma _{1y}\\sigma _{2x}\\rangle\n=2\\sin \\left( {\\theta'}{}\\right) \\cos ^{N-2}\\left( {\\theta\n'}{}\\right).\n\\end{equation}\nThe combination of Eqs.~(\\ref{xx}), (\\ref{yy}),  (\\ref{sigma1}), (\\ref{sigma2}%\n), and (\\ref{b12}) leads to the correlation\n\\begin{eqnarray}\\label{u0}\nu_{0} &=&\\langle \\sigma _{1-}\\sigma _{2-}\\rangle =-{8}^{-1}\\left(\n1-\\cos\n^{N-2}\\theta \\right)   \\notag \\\\\n&&-{i}{2}^{-1}\\sin \\left({\\theta'}{}\\right) \\cos ^{N-2}\\left({%\n\\theta' }{}\\right).  \\label{cr}\n\\end{eqnarray}\nSubstituting Eqs.~(\\ref{y0}) and (\\ref{u0}) to Eq.~(\\ref{xixixi1})\nleads to the expression of the squeezing parameter $\\xi_1^2$ given\nby Eq.~(\\ref{ccc1}).\n\n\\section{Proof of ${\\cal C}_{zz}(0)\\ge 0 $}\n\nTo prove this, we will not use this specific function of the\ninitial twist angle $\\theta$ as given by Eq.~(\\ref{c5}), but use\nthe positivity of the reduced density matrix (\\ref{re})$.$ We\nfirst notice an identity\n\\begin{equation*}\n\\mathcal{C}_{zz}=4(v_{+}v_{-}-w^{2}),\n\\end{equation*}\nwhich results from Eqs.~(\\ref{r1}) and (\\ref{r3}).  This is a key\nstep. Also there exists another identity\n\\begin{equation}\n\\label{e1} w_0=y_0\n\\end{equation}\nas $\\langle \\vec{\\sigma}_{1}\\cdot \\vec{\\sigma}_{2}\\rangle _{0}=1.$\nFrom the positivity of the reduced density matrix (\\ref{re}), one\nhas\n\\begin{equation*}\nv_{0+}v_{0-}\\geq |u_0|^{2}\\geq y_0^{2}=w_0^{2},\n\\end{equation*}\nwhere the second inequality follows from Eq.~(\\ref{r3}) and the\nlast equality results from Eq.~(\\ref{e1}). This completes the\nproof.\n\n\\section{Derivation of the evolution of the correlations and expectations under decoherence}\n\nFor an arbitrary matrix\n\\begin{equation*}\nA=\\left(\n\\begin{array}{cc}\na & b \\\\\nc & d%\n\\end{array}%\n\\right) ,\n\\end{equation*}\nfrom the Kraus operators (\\ref{kraus1}) for the ADC, it is\nstraightforward to find\n\\begin{eqnarray*}\n{\\cal E} (A) &=&\\left(\n\\begin{array}{cc}\nsa & \\sqrt{s}b \\\\\n\\sqrt{s}c & d+pa%\n\\end{array}%\n\\right) , \\\\\n{\\cal E}^{\\dagger }(A) &=&\\left(\n\\begin{array}{cc}\nsa+pd & \\sqrt{s}b \\\\\n\\sqrt{s}c & d%\n\\end{array}%\n\\right) .\n\\end{eqnarray*}%\nThe above equations imply that\n\\begin{eqnarray*}\n{\\cal E} ^{\\dagger }(\\sigma _{\\mu}) &=&\\sqrt{s}\\sigma _{\\mu} \\; \\text{for}\\; \\mu=x,y, \\\\\n{\\cal E} ^{\\dagger }(\\sigma _{z}) &=&s \\sigma _{z}-p.\n\\end{eqnarray*}%\nAs we considered independent and identical decoherence channels\nacting separately on each spin, the evolution correlations and expectations in Eqs.~(%\n\\ref{c2}), (\\ref{c3}), and (\\ref{c44}) are obtained directly from\nthe above equations.\n\nFrom the Kraus operators (\\ref{kraus2}), the evolution of the\nmatrix $A$ under the PDC is obtained as\n\\begin{equation*}\n{\\cal E} (A)={\\cal E} ^{\\dagger }(A)=\\left(\n\\begin{array}{cc}\na & sb \\\\\nsc & d\n\\end{array}\n\\right) ,\n\\end{equation*}\nfrom which one finds\n\\begin{eqnarray*}\n{\\cal E} ^{\\dagger }(\\sigma _{\\mu}) &=&s \\sigma _{\\mu} \\quad \\text{for} \\; \\mu=x,y \\\\\n{\\cal E} ^{\\dagger }(\\sigma _{z}) &=&\\sigma _{z}.\n\\end{eqnarray*}%\nSo expectations $\\langle \\sigma _{z}^{\\otimes n}\\rangle $ are\nunchanged and Eq.~(\\ref{evolve}) is obtained.\n\nFrom the Kraus operators (\\ref{kraus3}) of the DPC, the evolution\nof the matrix $A$ is given by\n\\begin{eqnarray*}\n{\\cal E} (A) &=&{\\cal E} ^{\\dagger }(A) \\\\\n&=&\\left(\n\\begin{array}{cc}\nas +\\frac{p}{2}(a+d) & sb \\\\\nsc & ds +\\frac{p}{2}(a+d)%\n\\end{array}%\n\\right)\n\\end{eqnarray*}%\nfrom which one finds%\n\\begin{eqnarray*}\n{\\cal E} ^{\\dagger }(\\sigma _{\\alpha}) =s \\sigma _{\\alpha}\\quad\n\\text{for}\\; \\alpha\\in\\{x,y,z\\}.\n\\end{eqnarray*}%\nThen, Eq.~(\\ref{cccccc}) is obtained.\n\n\\section{Simplified form of the concurrence}\n\nFor our three kinds of decoherence channels, the concurrence\n(\\ref{conc}) can be simplified and given by\n\\begin{eqnarray}\nC &=&\\max \\left\\{ 0,2\\left( |u|-w\\right)\n,2(y-\\sqrt{v_{+}v_{-}})\\right\\}\n\\notag\\\\\n&=&\\max \\left\\{ 0,2\\left( |u|-w\\right) \\right\\} . \\label{E1}\n\\end{eqnarray}\nIf one can prove\n\\begin{eqnarray}\n|u|-y &\\geq &0, \\\\\nw-\\sqrt{v_{+}v_{-}} &\\leq &0,\n\\end{eqnarray}\nthen we obtain the simplified form shown in Eq.~(\\ref{E1}). The\nlast inequality can be replaced by\n\\begin{equation}\\label{c22}\nw^{2}-v_{+}v_{-}\\leq 0\n\\end{equation}\nas $w$ and $v_{+}v_{-}$ are real.\n\nWe first consider the ADC channel. From Eqs.~(\\ref{c2}), (\\ref{c3}), and (\\ref{c4}%\n), one obtains\n\\begin{eqnarray}\n|u|-y &=&s(|u_{0}|-y_{0})\\geq 0, \\\\\nw^{2}-v_{+}v_{-} &=&-\\frac{1}{4}\\mathcal{C}_{zz}=-\\frac{s^{2}}{4}\\mathcal{C}%\n_{zz}(0)\\leq 0.\n\\end{eqnarray}\nwhere the inequalities result from Eqs.~(\\ref{ccc1}) and\n(\\ref{c5}), respectively. So, the inequality (\\ref{c22}) follows.\n\nFor the PDC, from Eq.~(\\ref{evolve}) and fact that\n$\\langle\\sigma_z^{\\otimes n}\\rangle$ is unchanged under\ndecoherence, the concurrence can also be simplified due to the\nfollowing properties:\n\\begin{eqnarray*}\n|u|-y &=&s^{2}(|u_{0}|-y_{0})\\geq 0, \\\\\nw^{2}-v_{+}v_{-} &=&-\\frac{1}{4}\\mathcal{C}_{zz}(0)\\leq 0.\n\\end{eqnarray*}\nFor the DPC, from Eqs.~(\\ref{evolve}) and (\\ref{cccccc}), one has\n\\begin{eqnarray}\n|u|-y &=&s^{2}(|u_{0}|-y_{0})\\geq 0, \\\\\nw^{2}-v_{+}v_{-} &=&-\\frac{s^{2}}{4}\\mathcal{C}_{zz}(0)\\leq 0.\n\\end{eqnarray}\nSo, again, the concurrence can be simplified to the form shown in\nEq.~(\\ref{E1}). This completes the proof.\n\n\\end{appendix}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec_intro}\nOver the last decades nonlocal models have attracted much attention owing to its potentially promising application in various disciplines of science and engineering, such as the preridynamic (PD) theory of continuum mechanics, and the modeling of nonlocal diffusion process, see \\cite{bobaru2010the,du2012analysis,silling2000reformulation,weckner2005the,zhou2010mathematical}. Peridynamics originally introduced in \\cite{silling2000reformulation} is a nonlocal formulation of elastodynamics which can more easily incorporate discontinuities such as cracks and damage, and has been extended past its original formulation including micropolar, nanofiber networks and so on \\cite{bobaru2007influence,foster2010viscoplasticity,gerstle2007micropolar,weckner2009green}. While most nonlocal models are formulated on bounded domains with volume constraints, there are indeed applications in which the simulation in an infinite medium may be useful, such as wave or crack propagation in the whole space.\n\nIn this paper, we consider constructing perfectly matched layers (PMLs) to numerically solve the following nonlocal wave equation\n\\begin{align}\n& (\\partial_t^2+\\mathcal{L}) q(x,t) = f(x,t), \\quad x\\in\\mathbb{R},\\ t>0,\\label{eq:nonlocalwave}\\\\\n& q(x,0) = \\psi_0(x), \\quad \\partial_t q(x,0) = \\psi_1(x),\\quad x\\in\\mathbb{R}, \\label{eq:nonlocalwavecon}\n\\end{align}\nwhere $q(x,t)$ represents the displacement field, $\\psi_0(x)$ and $\\psi_1(x)$ are the initial values, $f(x,t)$ is the body force. The nonlocal operator $\\mathcal{L}$ acting on $q$ is defined by\n\\begin{align} \n\t\\mathcal{L} q(x,t) = \\int_{\\mathbb{R}} \\big( q(x,t)-q(y,t) \\big) \\gamma\\Big(y-x,\\frac{x+y}{2}\\Big) \\mathrm{d}y,\t\\label{eq:nonlocalOperator}\n\\end{align}\nwhere the nonnegative kernel function $\\gamma(\\alpha,\\beta)$ satisfies\n\\begin{align}\n\\gamma(-\\alpha,\\beta)=\\gamma(\\alpha,\\beta),\\ \\forall \\alpha,\\beta\\in\\mathbb{R},\\quad \\mathrm{and}\\quad \\gamma(\\alpha,\\beta)=0,\\ \\mbox{if}\\ |\\alpha|>\\delta>0. \\label{eq:symkernel}\n\\end{align}\nWe assume the initial values $\\psi_k(x)\\ (k=0,1)$ and the source $f(x,t)$ are compactly supported over a bounded domain $\\Omega_f$ for all $t$.\n\n\nThe aim of this paper is to develop an efficient numerical scheme to compute the solution of problem \\eqref{eq:nonlocalwave}-\\eqref{eq:nonlocalwavecon} on the whole real axis.  We are facing two difficulties: \n\\begin{itemize}\n\\item The definition domain is unbounded. This requires us to construct artificial\/absorbing boundary conditions (ABCs) which artificially bounds the computational domain without changing the solution of a PDE or nonlocal model. Here we consider perfectly matched layers (PMLs) of nonlocal models to overcome the unboundedness of spatial domain;\n\\item The kernel in the proposed nonlocal PML equation is complex-valued and depends on the time $t$. As a result, the modified nonlocal operator in the PML equation is given by a convolution in time,  which differs from the original nonlocal operator \\eqref{eq:nonlocalOperator}. In addition, the simulations are implemented in multi-scale media. These require us to develop an asymptotically compatible (AC) scheme which should be consistent with both its local limiting model (i.e., taking $\\delta \\to 0$) and the nonlocal model itself (i.e., taking $\\delta =\\mathcal{O}(1)$). \n\\end{itemize}\n\n\nTo overcome the first difficulty of the unboundedness of definition domain, the accurate ABCs is a successful approach by absorbing any impinging waves on the artificial boundaries\/layers. The great progress has been made for the construction of ABCs for various nonlocal models, see \\cite{DuHanZhangZheng,DuZhangZheng,ZYZD16,ZhengHuDuZhang,zheng2020stability}. In this paper, we will apply the perfectly matched layer (PML) to confine a bounded domain of physical interest.\nThe PML has two important properties: (i) waves in the PML regions decay exponentially and; (ii) the returning waves after one round trip through the absorbing layer are very tiny if the wave reflects off the truncated boundary. These properties make it useful to simulate wave propagations in various media and fields, e.g., \\cite{ber94,becache2004perfectly,berenger1996three,bermudez2007an,cw,cl03,collino1998the,turkel1998absorbing}. While the PML has been well developed for local problems, there are few works on PMLs for nonlocal problems \\cite{antoine2019towards,DuZhangNonlocal1,DuZhangNonlocal2,wildman2011,wildman2012a,wang2013matching,ji2020artificial}. The main reason is that,  due to the nonlocal horizon, the design of PMLs for nonlocal models poses challenges not faced in the PDEs setting. For example, when constructing local PMLs, one replaces derivatives with respect to real numbers by the corresponding complex derivatives. However, this process cannot be applied to the nonlocal operator which is in the form of integral. \n\nIn this paper, we provide a way of constructing an efficient PML for nonlocal wave problem \\eqref{eq:nonlocalwave}--\\eqref{eq:nonlocalwavecon}. To do so, we first reformulate the wave equation into a nonlocal Helmholtz equation by using Laplace transform. The Laplace transform introduces a complex variable $s$. After that, we apply the PML modifications, recently developed in \\cite{DuZhangNonlocal1,DuZhangNonlocal2} for nonlocal Helmholtz equations, to derive PMLs for the resulting nonlocal Helmholtz equation with $s$. In this situation, the kernel is still analytically continued into the complex coordinates and consequently, the modified equation has a complex-valued kernel depending on complex value $s$. Finally, we transform the modified nonlocal equation into its time-domain form by inverse Laplace transform. As a result, we obtain the nonlocal wave equation with PML modifications.\n\nIn term of the discretization of the nonlocal PML equation, asymptotic compatibility (AC) schemes, a concept developed in \\cite{TianDu,TianDu2}, is needed to discretize the nonlocal operator \\cite{Du2016handbook}. \nIn this paper, the kernel is taken by the following heterogeneous diffusion coefficient\n\\begin{equation}\\label{sm}\n0< \\sigma(x)=\\frac{1}{2}\\int_\\mathbb{R} s^2\\gamma(s,x)ds < \\infty,\n\\end{equation}\nwhich implies that the nonlocal model is in multi-scale media. Under the assumption \\eqref{sm}, the nonlocal operator \\eqref{eq:nonlocalOperator} will converge\nto a local operator \\cite{DuZhangZheng} in the form of \n\\begin{equation} \\label{LO}\n\\lim_{\\delta\\to 0^+} \\mathcal{L} q(x)=-\\partial_x\\left[\\sigma(x)\\partial_xq(x)\\right].\n\\end{equation}\nAs $\\delta\\rightarrow 0$, the solution of problem \\eqref{eq:nonlocalwave} will converge to the solution of local wave equation\n\\begin{align}\n \\partial_t^2 q(x,t)-\\partial_x\\left[\\sigma(x)\\partial_x  \\right] q(x,t)= f(x,t), \\quad x\\in\\mathbb{R},\\ t>0. \n\\end{align}\n\nThe AC scheme can ensure that numerical solutions of nonlocal models converge to the correct local limiting solution, as both the mesh size $h$ and the nonlocal effect $\\delta$ tend to zero. One can refer to \\cite{du2019asymptotically,tian2017a,TianDu,tian2014asymptotically} for more details of AC schemes. Noting that our nonlocal PML problem involves a new complex-valued kernel arising from the inverse Laplace transform, we here present the analogous ideas given in \\cite{DuZhangZheng} to discretize the one-dimensional nonlocal operator with the general complex and time-dependent kernels and complex functions. For practical multi-scale simulations, we apply Talbot's contour \\cite{weideman2006optimizing} to the inverse Laplace transform and obtain its approximation consisting of several sub-kernels. For each sub-kernel, we employ an AC scheme, developed for complex functions in \\cite{DuZhangNonlocal1,DuZhangNonlocal2}, to discretize the resulting nonlocal operator. After that, we introduce some new auxiliary functions and reformulate the semi-discrete problem into a second-order ODE system, which is finally solved by a Verlet-type scheme.\n\n\n \n\n\n\nThe outline of this paper is organized as follows. In section~\\ref{sec:NPML}, we design the nonlocal PMLs and obtain a truncated nonlocal PML problem on a bounded domain. In section~\\ref{sec_dis}, we first spatially discretize the truncated nonlocal PML problem into an ODE system with the variable $t$ and solve it by a Verlet-type scheme. In section~\\ref{sec_num} we introduce the basic setting of parameters for the discretization, and present numerical examples to verify the efficiency of the nonlocal PMLs and the convergence order of our numerical scheme. \n\n\n\\section{Nonlocal Perfectly Matched Layers} \\label{sec:NPML}\n\nWe now consider the construction of nonlocal PMLs by using complex-coordinate approach. The complex-coordinate approach is essentially based on analytic continuation of the wave equation into complex spatial coordinates where the fields are exponentially decaying. To do so, we assume the initial data functions and the kernel functions satisfy the following properties:\n\\begin{itemize}\n\\item[A1:] $\\psi_1$ and $f$ are compactly supported into a finite interval $\\mathcal{D}=(x_l,x_r)$, and $\\psi_0$ is compactly supported into $(x_l+\\delta,x_r-\\delta)$; \n\\item[A2:] $\\gamma$ is compactly supported over a strip $[-\\delta,\\delta]\\times\\mathbb{R}$ with $\\delta\\leq x_r-x_l$;\n\\item[A3:] $\\gamma$ is homogeneous in both $[x_r,+\\infty)$ and $(-\\infty,x_l]$, namely,\n\\begin{align}\n\\gamma(\\alpha,\\beta) =& \\gamma_L(\\alpha),\\quad \\beta\\in (-\\infty, x_l+\\delta\/2],\\\\\n\\gamma(\\alpha,\\beta) =& \\gamma_R(\\alpha),\\quad \\beta\\in (x_r-\\delta\/2,+\\infty].\n\\end{align}\n\\end{itemize}\nIn the sequel we take $\\gamma_L=\\gamma_R=\\gamma_\\infty$ for brevity. \n\nPerforming the Laplace transform on \\eqref{eq:nonlocalwave}, we have \n\\begin{align}\ns^2\\hat q(x,s) +\\mathcal{L} \\hat q(x,s) = \\hat f(x,s)+s\\psi_0(x)+\\psi_1(x), \\quad x\\in\\mathbb{R}, \\label{eq:Hel}\n\\end{align}\nwhere $\\hat{q}(x,s) = \\mathscr{L}(q(x,t);s)$ with $\\mathscr{L}$ representing the Laplace transform in time with $\\Re \\{s\\} >0$.\n\nNoting the nonlocal operator $\\mathcal{L}$ is self-adjoint, we can rewrite \\eqref{eq:Hel} into the weak form of \n\\begin{align*}\n\\int_{\\mathbb{R}} s^2\\hat q(x,s) v(x)\\,\\mathrm{d}x -& \\frac12\\int_{\\mathbb{R}}\\int_{\\mathbb{R}} \\big[\\hat q(x,s) - \\hat q(y,s) \\big]\\big[ v(x)-v(y) \\big]\\\\\n& \\gamma\\Big(y-x,\\frac{x+y}{2}\\Big)\\,\\mathrm{d}x\\,\\mathrm{d}y = \\int_\\mathbb{R} \\big(\\hat f(x,s)+s\\psi_0(x)+\\psi_1(x)\\big)v(x)\\mathrm dx,\\quad \\forall v\\in C_0^\\infty(\\mathbb{R}).\n\\end{align*}\nThe PML modifications can be viewed as a complex coordinate stretching of the original problem by constructing an analytic continuation to the complex plane \\cite{DuZhangNonlocal1,DuZhangNonlocal2}. In this paper, we take \n\\begin{align}\n\t\\tilde x:=\\int_0^x \\alpha(\\eta,s)\\mathrm{d}\\eta= \\int_0^x \\Big(1 + \\frac{z}{s}\\sigma(\\eta)\\Big) \\mathrm{d}\\eta,\\qquad \\tilde y:=\\int_0^y \\alpha(\\eta,s)\\mathrm{d}\\eta= \\int_0^y \\Big(1 + \\frac{z}{s}\\sigma(\\eta)\\Big) \\mathrm{d}\\eta, \\label{eq:complexStreching}\n\\end{align}\nwhere the absorption function $\\sigma(\\eta)\\leq1$ is positive in $\\mathbb{R}\\setminus\\mathcal{D}$ and is zero in $\\mathcal{D}$. The PML coefficient $z$ is a real or complex constant, such as $z= 10$ or $ z = 10+\\i$. By replacing\n\\begin{equation*}\nx\\to\\tilde x(x,s),\\quad y\\to \\tilde y(y,s),\\quad\n\\,\\mathrm{d}x\\to\\frac{\\partial \\tilde x}{\\partial x}\\,\\mathrm{d}x=\\alpha(x,s)\\,\\mathrm{d}x,\\quad\n\\,\\mathrm{d}y\\to\\frac{\\partial \\tilde y}{\\partial y}\\,\\mathrm{d}y=\\alpha(y,s)\\,\\mathrm{d}y,\n\\end{equation*}\nwe can transform Eq. \\eqref{eq:Hel} into the following nonlocal equation with PML modifications\n\\begin{align*}\n\\int_{\\mathbb{R}} s^2\\hat q(\\tilde x,s) v(\\tilde x)\\,\\mathrm{d}x &- \\frac12\\int_{\\mathbb{R}}\\int_{\\mathbb{R}} \\big[\\hat q(\\tilde x,s) - \\hat q(\\tilde y,s) \\big]\\big[ v(\\tilde x)-v(\\tilde y) \\big]\\gamma\\Big(\\tilde y-\\tilde x,\\frac{\\tilde x+\\tilde y}{2}\\Big)\\alpha(x,s)\\alpha(y,s)\\,\\mathrm{d}x\\,\\mathrm{d}y\\\\\n& = \\int_\\mathbb{R}\\big(\\hat f(\\tilde x,s)+s\\psi_0(\\tilde x)+\\psi_1(\\tilde x)\\big) v(\\tilde x)\\alpha(x,s)\\mathrm dx,\\quad \\forall v\\in C_0^\\infty(\\mathbb{R}),\n\\end{align*}\nwhich implies that \n\\begin{align}\ns^2\\alpha(x,s)\\hat q(\\tilde x,s) + \\int_{\\mathbb{R}} \\big[\\hat q(\\tilde x,s) - \\hat q(\\tilde y,s) \\big] \\gamma\\Big(\\tilde y-\\tilde x,\\frac{\\tilde x+\\tilde y}{2}\\Big)\\alpha(x,s)\\alpha(y,s)\\mathrm dy \\notag\\\\\n= \\hat f(x,s)+s\\psi_0(x)+\\psi_1(x). \\label{eq_HelPML}\n\\end{align}\nNoting that to derive the right hand side of the above equation,  we have used the facts that $\\tilde x=x,\\alpha(x,s)=1$ for $x\\in\\mathcal{D}$, the initial data $\\psi_k(k=0,1)$ and the source function $f$ are compactly supported in $\\mathcal{D}$. Thus, we continue the equation~\\eqref{eq:Hel} into \\eqref{eq_HelPML} in complex coordinates. One can see that the solutions $\\hat q(\\tilde x,s)$ will not change in the interior domain $\\mathcal{D}$ and exponentially decay in the absorbing region $\\sigma(x)>0$ by choosing an appropriate PML coefficient $z$. \n\nWe now perform the inverse Laplace transform to turn the equation back into the time-domain form. To do so, we set  \n\\begin{align}\n\\tilde q(x,t) = \\mathscr{L}_s^{-1}[\\hat q(\\tilde x,s)],\\qquad \\tilde \\gamma(x,y,t)=\\mathscr{L}_s^{-1}\\Big[\\frac{1}{s}\\gamma\\Big(\\tilde y-\\tilde x,\\frac{\\tilde x+\\tilde y}{2}\\Big)\\alpha(x,s)\\alpha(y,s)\\Big]. \\label{eq_invkernel}\n\\end{align}\nSince $\\tilde x=x$ for $x\\in\\mathcal{D}$, we have $\\tilde q(x,t)=q(x,t)$ for $x\\in\\mathcal{D}$ and all time $t$, which implies that $\\tilde q(x,0)=q(x,0)$ and $\\partial_t\\tilde q(x,t)|_{t=0}=\\partial_t q(x,t)|_{t=0}$ for $x\\in \\mathcal{D}$. Therefore, we can naturally assume that $\\tilde q(x,0)=\\psi_0(x)$ and $\\partial_t\\tilde q(x,0)=\\psi_1(x)$. Then, we have the following inverse Laplace transforms \n\\begin{align}\n\\mathscr{L}_s^{-1}[ s^2\\alpha(x,s)\\hat q(\\tilde x,s)-s\\psi_0(x)-\\psi_1(x) ] &= \\mathscr{L}_s^{-1}[ (s^2+zs\\sigma(x))\\hat q(\\tilde x,s) -s\\psi_0(x)-\\psi_1(x)]\\notag\\\\\n& = \\partial_t^2 \\tilde q(x,t) + z\\sigma(x) \\partial_t \\tilde q(x,t), \\label{eq_wavePMLp1}\n\\end{align}\nand\n\\begin{align}\n& \\mathscr{L}_s^{-1}\\Big[\\big[\\hat q(\\tilde x,s) - \\hat q(\\tilde y,s) \\big]\\gamma\\Big(\\tilde y-\\tilde x,\\frac{\\tilde x+\\tilde y}{2}\\Big)\\alpha(x,s)\\alpha(y,s)\\Big] \\notag\\\\\n=& \\mathscr{L}_s^{-1}\\Big[\\big[\\big(s\\hat q(\\tilde x,s)-q(x,0)\\big) -\\big(s \\hat q(\\tilde y,s)-q(y,0)\\big) \\big] \\cdot \\frac{1}{s} \\gamma\\Big(\\tilde y-\\tilde x,\\frac{\\tilde x+\\tilde y}{2}\\Big)\\alpha(x,s)\\alpha(y,s)\\Big] \\notag\\\\\n& + \\mathscr{L}_s^{-1}\\Big[\\big[q(x,0)-q(y,0) \\big] \\cdot \\frac{1}{s} \\gamma\\Big(\\tilde y-\\tilde x,\\frac{\\tilde x+\\tilde y}{2}\\Big) \\alpha(x,s)\\alpha(y,s)\\Big] \\notag\\\\\n=& \\big[ \\partial_t \\tilde q(x,t)- \\partial_t \\tilde q(y,t) \\big] \\ast \\tilde \\gamma(x,y,t) +\\big[q(x,0)-q(y,0) \\big] \\tilde \\gamma(x,y,t),\\label{eq_wavePMLp2}\n\\end{align}\nwhere $*$ indicates the convolution of two functions in time. Combining \\eqref{eq_wavePMLp1} and \\eqref{eq_wavePMLp2} with \\eqref{eq_HelPML} yields the nonlocal wave equation with PML modifications as\n\\begin{align}\n\\big(\\partial_t^2 +z\\sigma(x)\\partial_t \\big) \\tilde q(x,t) + &\\int_\\mathbb{R} \\big[ \\partial_t \\tilde q(x,t)- \\partial_t \\tilde q(y,t) \\big] \\ast \\tilde \\gamma(x,y,t) \\mathrm dy \\notag\\\\\n&= f(x,t) - \\int_\\mathbb{R}\\big[\\psi_0(x)-\\psi_0(y) \\big] \\tilde \\gamma(x,y,t)\\mathrm{d}y,\\quad x\\in\\mathbb{R}.\n\\end{align}\n\nNoting $ \\tilde x =x, \\tilde y = y \\; (\\forall x,y\\in\\mathcal{D})$ and $\\mathrm{supp}\\;\\psi_0(x)\\subset(x_l+\\delta,x_r-\\delta)$ (see A1), we have $\\tilde \\gamma(x,y,t)=\\gamma\\Big( y- x,\\frac{ x+ y}{2}\\Big)$, which implies that for all $x$,\n\\begin{align*}\n\\int_\\mathbb{R}\\big[\\psi_0(x)-\\psi_0(y) \\big] \\tilde \\gamma(x,y,t)\\mathrm{d}y = \\int_{\\mathcal{D}}\\big[\\psi_0(x)-\\psi_0(y) \\big]  \\gamma\\Big( y- x,\\frac{ x+ y}{2}\\Big)\\mathrm dy.\n\\end{align*}\n\nWe finally have the nonlocal PML wave equations\n\\begin{align} \\label{PMLw}\n\\big(\\partial_t^2 +z\\sigma(x)\\partial_t \\big) \\tilde q(x,t) + \\mathcal{L}_{pml} \\partial_t \\tilde q(x,t) = f(x,t) - \\int_{\\mathcal{D}}\\big[\\psi_0(x)-\\psi_0(y) \\big]  \\gamma\\Big( y- x,\\frac{ x+ y}{2}\\Big)\\mathrm dy,\n\\end{align}\nwhere the nonlocal operator $\\mathcal{L}_{pml}$ for the PML is given by\n\\begin{align}\n \\mathcal{L}_{pml}  \\partial_t \\tilde q(x,t)=\\int_\\mathbb{R} \\big[ \\partial_t \\tilde q(x,t)- \\partial_t \\tilde q(y,t) \\big] \\ast \\tilde \\gamma(x,y,t) \\mathrm dy.\n\\end{align}\nWe point out that the nonlocal PML operator $\\mathcal{L}_{pml}$ involves a convolution in time, which differs from the original nonlocal operator $\\mathcal{L}$.\n\nNoting the PML equation \\eqref{PMLw} is still defined on the whole space, we need to truncate the computational region at some sufficiently large $x$ by putting homogeneous Dirichlet boundary conditions. To do so, we define the PML layer $\\mathcal{D}_p=(x_l-d_p,x_l]\\cup[x_r,x_r+d_p)$ with the thickness $d_p$ of the absorbing layer, and define the boundary layer $\\mathcal{D}_b$ of width $\\delta$ which surrounds $\\mathcal{D}\\cup\\mathcal{D}_p$.\n\nThus, we derive the following truncated nonlocal wave problem with PML modifications:\n\\begin{align}\n&\\big(\\partial_t^2 +z\\sigma(x)\\partial_t \\big) \\hat{\\tilde q}(x,t) +  \\mathcal{L}_{pml}  \\partial_t \\hat{\\tilde q}(x,t)\\notag\\\\\n&\\qquad\\qquad\\qquad= f(x,t) - \\int_{\\mathcal{D}}\\big[\\psi_0(x)-\\psi_0(y) \\big]  \\gamma\\Big( y- x,\\frac{ x+ y}{2}\\Big)\\mathrm dy,\\quad x\\in\\mathcal{D}\\cup\\mathcal{D}_p, \\label{eq_truPML1}\\\\\n& \\hat{\\tilde q}(x,0) = \\psi_0(x),\\quad \\partial_t \\hat{\\tilde q}(x,0) = \\psi_1(x),\\quad x\\in\\mathcal{D}_b\\cup\\mathcal{D}_p\\cup\\mathcal{D},\\label{eq_truPML2}\\\\\n& \\hat{\\tilde q}(x,t)=0,\\quad x\\in \\mathcal{D}_b,\\ 0<t\\leq T \\label{eq_truPML3}.\n\\end{align}\nThe solution $\\hat{\\tilde q}(x,t)$ above will use to approximate the solution $q(x,t)$ of problem \\eqref{eq:nonlocalwave}--\\eqref{eq:nonlocalwavecon} in $\\mathcal{D}$.\n\n\\section{Discretization of the truncated nonlocal wave problem}\\label{sec_dis}\nWe now consider the discrete scheme of problem~\\eqref{eq_truPML1}--\\eqref{eq_truPML3} by using the AC scheme, developed in \\cite{TianDu,du2019asymptotically,tian2014asymptotically,tian2017a}, to discretize the nonlocal PML operator, and using the Verlet-type scheme to solve the discrete ODE system obtained from the spatial discretization.\n\nWe first introduce a spatial uniform grid $\\{x_i\\}_{i\\in\\mathbb{Z}}$ with mesh size $h$. For simplicity, we take $x_0=x_l-d_p$ and $x_{N+1}=x_r+d_p$, and set \n\\begin{align}\n&\\mathcal{I}_p = \\{i|1\\leq i\\leq N, x_i\\in{\\mathcal{D}}_p\\},\\quad\\quad \\mathcal{I} = \\{i|1\\leq i\\leq N, x_i\\in{\\mathcal{D}}\\}.\n\\end{align}\n\\subsection{The approximation of nonlocal PML operator $\\mathcal{L}_{pml}$}\nHere we consider the numerical approximation of the complex-valued kernel $\\tilde \\gamma(x,y,t)$ \\eqref{eq_invkernel} given by\n\\begin{align}\n\\tilde \\gamma(x,y,t)=\\mathscr{L}_s^{-1}[\\mathcal{K}(x,y,s) ]=\\frac{1}{2\\pi\\i}\\int_{\\Gamma} \\mathcal{K}(x,y,s)e^{st} \\mathrm ds,\\label{eq_invkernel2}\n\\end{align}\nwhere $\\mathcal{K}(x,y,s)$ represents\n\\begin{align}\n&\\mathcal{K}(x,y,s) = \\frac{1}{s}\\gamma\\Big(\\tilde y-\\tilde x,\\frac{\\tilde x+\\tilde y}{2}\\Big)\\alpha(x,s)\\alpha(y,s). \\label{eq:mk}\n\\end{align}\n\nDenote by $\\Omega_\\mathcal{K}$ the $s$-complex domain where $\\mathcal{K}(x,y,s)$ viewed as a function of the variable $s$ is analytic for any given $x,y\\in \\mathcal{D}_p$. The notation $\\Gamma$ denotes the Bromwich line $\\Re(s)=\\eta$ initially, where the parameter $\\eta$ is taken large enough such that the complement of $\\Omega_\\mathcal{K}$ lies in the half-plane $\\Re(s)<\\eta$. A typical approach of numerically approximating the inverse Laplace transform is to deform the Bromwich line into Talbot's contour \\cite{weideman2006optimizing}\n\\begin{align}\n\\Gamma:\\ s(\\theta) = \\omega +\\mu(\\theta\\cot\\theta+\\nu\\i\\theta),\\quad -\\pi\\leq\\theta\\leq\\pi,\n\\end{align}\nwhere $\\omega,\\mu$, and $\\nu$ are real parameters such that $\\Gamma$ encloses $\\mathbb{C}\\setminus\\Omega_\\mathcal{K}$.\n\nWe define the grid \n\\begin{align*}\n\\theta_j=-\\pi+\\frac{\\pi}{m}(2j-1),\\ j=1,2,\\cdots,m,\n\\end{align*}\nand approximate Eq.~\\eqref{eq_invkernel2} by the trapezoidal rule\n\\begin{align}\n\\tilde \\gamma(x,y,t)=\\frac{1}{2\\pi\\i}\\int_{-\\pi}^\\pi \\mathcal{K}(x,y,s(\\theta))e^{s(\\theta)t} s'(\\theta) \\mathrm d\\theta \\approx \\sum_{j=1}^m \\varpi_j\\mathcal{K}(x,y,\\xi_j)e^{\\xi_j t}, \\label{eq_tabapp}\n\\end{align}\nwhere $\\xi_j=s(\\theta_j)$ are the sampling points on $\\Gamma$ and $\\varpi_j=\\frac{s'(\\theta_j)}{m\\i}$ are associated quadrature weights.\n\nBy \\eqref{eq_tabapp}, we derive the approximation of the nonlocal PML operator $\\mathcal{L}_{pml}$\n\\begin{align}\n\\mathcal{L}_{pml} \\partial_t \\hat{\\tilde q}(x,t) \\approx& \\int_{\\mathbb{R}} \\big[ \\partial_t \\hat{\\tilde q}(x,t)- \\partial_t \\hat{\\tilde q}(y,t) \\big]   \\ast \\sum_{j=1}^m \\varpi_j  \\mathcal{K}(x,y,\\xi_j)e^{\\xi_jt} \\mathrm dy \\notag\\\\\n=& \\sum_{j=1}^m \\varpi_j \\Big(\\int_{\\mathbb{R}} \\big[ \\partial_t \\hat{\\tilde q}(x,t)- \\partial_t \\hat{\\tilde q}(y,t) \\big] \\mathcal{K}(x,y,\\xi_j) \\mathrm dy\\Big) \\ast e^{\\xi_jt} \\notag\\\\\n=:&  \\sum_{j=1}^m  \\mathcal{L}_{pml}^j \\partial_t  \\hat{\\tilde q}(x,t)  \\ast e^{\\xi_jt},\n\\end{align}\nwhich yields the approximation of Eq.~\\eqref{eq_truPML1}\n\\begin{align}\n\\big(\\partial_t^2 +z\\sigma(x)\\partial_t \\big) \\hat{\\tilde q}(x,t) +& \\sum_{j=1}^m \\mathcal{L}_{pml}^j \\hat{\\tilde q}(x,t)\\ast e^{\\xi_jt}\\notag\\\\\n =& f(x,t) -\\int_{\\mathcal{D}}\\big[\\psi_0(x)-\\psi_0(y) \\big]  \\gamma\\Big( y- x,\\frac{ x+ y}{2}\\Big)\\mathrm dy,\\quad x\\in\\mathcal{D}.\n\\end{align}\n\n\n\\subsection{The spatial discretization and semi-discrete problem}\nWe now consider the AC scheme, originally given in \\cite{DuZhangZheng} and further developed for complex functions in \\cite{DuZhangNonlocal1,DuZhangNonlocal2}, to discretize the nonlocal operators $\\mathcal{L}_{pml}^j$. The approximation of $\\mathcal{L}_{pml}^j $ is given as \n\\begin{align}\n\\mathcal{L}_{pml}^{j,h} \\partial_t\\hat{\\tilde q}(x_n,t) =& \\varpi_j\\int_{\\mathbb{R}} \\sum_{k\\neq n}\\phi_{x_k}(y)\\frac{ \\partial_t \\hat{\\tilde q}(x_n,t)- \\partial_t \\hat{\\tilde q}(x_k,t) }{x_k-x_n}(y-x_n) \\notag\\\\\n&\\qquad \\cdot\\mathcal{K}(\\frac{x_n+x_k}{2}-\\frac{y-x_n}{2},\\frac{x_n+x_k}{2}+\\frac{y-x_n}{2},\\xi_j) \\mathrm dy\\label{eq_disker}\\\\\n=& \\sum_{k} \\tilde a_{n,k}^j \\partial_t \\hat{\\tilde q}(x_k,t),\\qquad \\forall n\\in\\mathcal{I}_p\\cup\\mathcal{I},  \\quad j=1,\\cdots,m, \\notag\n\\end{align}\nwhere $\\tilde a_{n,n}^j = -\\sum_{ k\\neq n} \\tilde a_{n,k}^j $ with \n\\begin{align*}\n\t\\tilde a_{n,k}^j = - \\frac{\\varpi_j}{(k-n)h} \\int_{\\mathbb{R}}\\phi_{x_k}(y) (y-x_n) \\mathcal{K}(x_n+\\frac{x_k-y}{2},\\frac{x_k+y}{2},\\xi_j)\\,\\mathrm{d}y,\\ k\\neq n.\n\\end{align*}\n\nUsing the spatial discretizations \\eqref{eq_disker}, we derive the following semi-discrete problem:\n\\begin{align}\n&\\big(\\partial_t^2 +z\\sigma(x_n)\\partial_t \\big)\\hat{\\tilde q}_n(t) + \\sum_{j=1}^m\\sum_{k}\\tilde a_{n,k}^j \\partial_t \\hat{\\tilde q}_k(t) \\ast e^{\\xi_jt}\\notag\\\\\n&\\quad= f(x_n,t)-\\int_{\\mathcal{D}}\\big[\\psi_0(x_n)-\\psi_0(y) \\big]  \\gamma\\Big( y- x_n,\\frac{ x_n+ y}{2}\\Big)\\mathrm dy,\\quad 1\\leq n\\leq N,\\ 0<t, \\label{eq_sdp1}\\\\\n& \\hat{\\tilde q}_n(0) = \\psi_0(x_n),\\quad \\partial_t \\hat{\\tilde q}_n(0) = \\psi_1(x_n),\\quad n\\in\\mathbb{Z},\\label{eq_sdp2}\\\\\n& \\hat{\\tilde q}_n(t) = 0,\\quad n<1\\ \\mathrm{or}\\ n>N,\\ 0< t\\leq T,\\label{eq_sdp3}\n\\end{align}\nwhere $\\hat{\\tilde q}_n(t)\\approx\\hat{\\tilde q}(x_n,t)$.\n\nIn the remainder, we consider the convolutions of functions $\\partial_t \\hat{\\tilde q}_k(t)$ and $e^{\\xi_jt}$ over the range $[0,t]$ by introducing the auxiliary functions\n\\begin{align}\np_{k,j}(t) = \\partial_t \\hat{\\tilde q}_k(t) \\ast e^{\\xi_jt},\\quad k\\in\\mathcal{I}_p\\cup\\mathcal{I},\\ j=1,\\cdots,m.\n\\end{align}\nIt's clear that functions $p_{k,j}$ satisfy the following ODEs\n\\begin{align}\n\\partial_t p_{k,j}(t) = \\xi_j p_{k,j}(t) + \\partial_t \\hat{\\tilde q}_k(t) \\label{eq_pjode}\n\\end{align}\nwith the initial conditions $p_{k,j}(0) = 0$.\nThen we reformulate the the semi-discrete problem~\\eqref{eq_sdp1}--\\eqref{eq_sdp3} into the following ODEs\n\\begin{align}\n&\\big(\\partial_t^2 +z\\sigma(x_n)\\partial_t \\big)\\hat{\\tilde q}_n(t) + \\sum_{j=1}^m\\sum_{k}\\tilde a_{n,k}^j p_{k,j}(t) \\notag\\\\\n&\\qquad = f(x_n,t)-\\int_{\\mathcal{D}}\\big[\\psi_0(x_n)-\\psi_0(y) \\big]  \\gamma\\Big( y- x_n,\\frac{ x_n+ y}{2}\\Big)\\mathrm dy,\\quad 1\\leq n\\leq N,\\ 0<t,\\label{eq_disPML1}\\\\\n& \\partial_t p_{n,j}(t) = \\xi_j p_{n,j}(t) + \\partial_t \\hat{\\tilde q}_n(t),\\quad 1\\leq n\\leq N,\\ 0<t\\leq T,\\label{eq_disPML2}\\\\\n& \\hat{\\tilde q}_n(0) = \\psi_0(x_n),\\quad \\partial_t \\hat{\\tilde q}_n(0) = \\psi_1(x_n),\\quad p_{n,j}(0)=0,\\quad n\\in\\mathbb{Z},\\label{eq_disPML3}\\\\\n& \\hat{\\tilde q}_n(t) = 0,\\quad p_{n,j}(t)=0,\\quad n<1\\ \\mathrm{or}\\ n>N,\\ 0< t\\leq T.\\label{eq_disPML4}\n\\end{align}\n\n\n\\subsection{The Verlet-type ODE solver}\n\nWe here introduce the Verlet-type algorithm to numerically solve the ODE system \\eqref{eq_disPML1}--\\eqref{eq_disPML4}. Denote by $D_\\sigma$ the $N\\times N$  the diagonal matrix with entries $\\sigma(x_1),\\sigma(x_2),\\cdots,\\sigma(x_N)$, and by $\\tilde A_j\\ (j=1,2,\\cdots,m)$ the $N\\times N$ matrices with entries $\\tilde a_{n,k}^j\\ (n,k=1,2,\\cdots,N)$. The the ODE system \\eqref{eq_disPML1}--\\eqref{eq_disPML4} can be rewritten into the following form\n\\begin{align}\n\\mathbf{w}(t)-\\mathbf{q}'(t) & = 0,\\\\ \n\\mathbf{w}'(t) + zD_\\sigma \\mathbf{w}(t) + \\sum_{j=1}^m \\tilde A_j \\mathbf{p}_j(t) &= \\mathbf{f}(t),\\\\\n\\mathbf{p}_j'(t) - \\xi_j \\mathbf{p}_j(t) - \\mathbf{w}(t) &= 0,\\quad j=1,2,\\cdots,m,\n\\end{align}\nwhere $\\mathbf{q}=(\\hat{\\tilde q}_1,\\hat{\\tilde q}_2,\\cdots,\\hat{\\tilde q}_N)^T$, $\\mathbf{p}_j=(p_{1,j},p_{2,j},\\cdots,p_{N,j})^T$ and\n\\begin{align*}\n\\mathbf{f} = \\begin{pmatrix}\nf(x_1,t)-\\int_{\\mathcal{D}}\\big[\\psi_0(x_1)-\\psi_0(y) \\big]  \\gamma\\Big( y- x_1,\\frac{ x_1+ y}{2}\\Big)\\mathrm dy\\\\\nf(x_1,t)-\\int_{\\mathcal{D}}\\big[\\psi_0(x_2)-\\psi_0(y) \\big]  \\gamma\\Big( y- x_2,\\frac{ x_2+ y}{2}\\Big)\\mathrm dy\\\\\n\\vdots\\\\\nf(x_1,t)-\\int_{\\mathcal{D}}\\big[\\psi_0(x_N)-\\psi_0(y) \\big]  \\gamma\\Big( y- x_N,\\frac{ x_N+ y}{2}\\Big)\\mathrm dy\n\\end{pmatrix}.\n\\end{align*}\nLet $\\tau$ be the temporal stepsize and $t_k=k\\tau$ be the $k$-th time point. Denote by $\n\\mathbf{w}^{k+1\/2} \\approx \\mathbf{w}(t_{k+1\/2}),\\; \\mathbf{q}^k \\approx \\mathbf{q}(t_k),\\; \\mathbf{p}_j^k \\approx \\mathbf{p}(t_k)\\; (k=0,1,\\cdots).$ \nLet $\\mathbf{f}^j=\\mathbf{f}(t_j), \\mathbf{\\Psi}_0 =  (\\psi_0(x_1),\\psi_0(x_2),\\cdots,\\psi_0(x_N))^T$ and $\\mathbf{\\Psi}_1 =  (\\psi_1(x_1),\\psi_1(x_2),\\cdots,\\psi_1(x_N))^T$.\nThe initial values can be written as\n\\begin{align} \\label{1s}\n\\mathbf{q}^0 &= \\mathbf{\\Psi}_0,\\\\\n\\mathbf{p}_j^0 &=0,\\quad j=1,2,\\cdots,m,\\\\\n\\mathbf{w}^{1\/2} &= \\mathbf{\\Psi}_1 + \\frac{\\tau}{2}\\Big[\\mathbf{f}^0-\\sum_{j=1}^m\\tilde A_j \\mathbf{p}_j^0-zD_\\sigma\\mathbf{\\Psi}_1\\Big].\n\\end{align}\nFor $k\\geq0$, we apply the following second-order central difference to calculate $\\mathbf{q}^{k+1}$ and $\\mathbf{p}_j^{k+1}$ by \n\\begin{align}\n\\mathbf{w}^{k+\\frac12} - \\frac{\\mathbf{q}^{k+1}-\\mathbf{q}^{k}}{\\tau} & =0,\\\\\n\\frac{\\mathbf{p}_j^{k+1}-\\mathbf{p}_j^{k}}{\\tau} - \\xi_j\\frac{\\mathbf{p}_j^{k+1}+\\mathbf{p}_j^{k}}{2} - \\mathbf{w}^{k+\\frac12} &=0.\n\\end{align}\nAfter that, we still update $\\mathbf{w}^{k+3\/2}$ by using the above $\\mathbf{q}^{k+1}$ and $\\mathbf{p}_j^{k+1}$ via the second-order scheme: \n\\begin{align}\n& \\frac{\\mathbf{w}^{k+\\frac32}-\\mathbf{w}^{k+\\frac12}}{\\tau} + zD_\\sigma \\frac{\\mathbf{w}^{k+\\frac32}+\\mathbf{w}^{k+\\frac12}}{2} + \\sum_{j=1}^m \\tilde A_j \\mathbf{p}_j^{k+1}= \\mathbf{f}^{k+1}. \\label{3s}\n\\end{align}\n\n\\section{Numerical examples}\\label{sec_num}\nIn this section, three examples are provided to verify the effectiveness of our PML strategy,  the convergence and asymptotic compatibility of scheme \\eqref{1s}--\\eqref{3s}. Define the $L^2$-error at $t=t_k$ by \n\\begin{align}\ne_h &= \\sqrt{\\frac{1}{|\\mathcal{I}|}\\sum_{n\\in\\mathcal{I}} |\\hat{\\tilde q}^k(x_n)-q(x_n,t_k)|^2 },\\end{align}\nand the error to study the AC property, i.e., the so-called ``$\\delta$-convergence'' in \\cite{silling2000reformulation,DuZhangZheng,bobaru2009convergence}, by \n\\begin{align}\ne_\\delta &= \\sqrt{\\frac{1}{|{\\mathcal{I}\\cup\\mathcal{I}_p}|}\\sum_{n\\in{\\mathcal{I}\\cup\\mathcal{I}_p}} |\\hat{\\tilde q}^k(x_n)-u(x_n,t_k)|^2 },\n\\end{align}\nwhere $u(x,t)$ is the corresponding local PML soltion.\n\nIn the simulations, we choose the interior domain $\\mathcal{D}$ such that $x_l=-l,\\ x_r=l$ for some constant $l$ and set the PML absorbing function as the piecewise linear function\n\\begin{align}\n\\sigma(\\eta) = \n\\begin{cases}\n0, & -l<\\eta<l,\\\\\n\\frac{1}{d_{p}} (|\\eta|-l), & l\\leq|\\eta|<d_{p}+l,\\\\\n1, &  d_{p}+l\\leq |\\eta|.\n\\end{cases}\n\\end{align}\n\n\\noindent\\textbf{Example 5.1.} In this example we take the source $f(x,t)\\equiv0$ and the initial values as \n\\begin{align*} \n\\psi_0(x) = e^{-20(x-0.2)^2} + e^{-20(x+0.2)^2}, \\quad \\psi_1(x) = 100x^2e^{-20x^2},\n\\end{align*}\nand consider the kernel\n\\begin{align}\n\\gamma(y-x,\\frac{x+y}{2})=\\frac{1}{\\delta^3}\\gamma_0(\\frac{|x-y|}{\\delta}),\\label{eq_ker1}\n\\end{align}\nwhere $\\gamma_0(s) = \\frac{1}{2c_0^3} e^{-\\frac{|s|}{c_0}}$ and $c_0$ can be taken as a positive constant measuring the nonlocal horizon. \n\nIn the simulations, we take $l=1.5$, $d_p=1$, the PML coefficient $z=20$, and set the parameters in the numerical implementation of Laplace transform as \n$\\mu = \\frac12 |z|, \\nu = 1.$\nWe leave the discussion about the domain $\\Omega_\\mathcal{K}$ for the kernel~\\eqref{eq_ker1} in the appendix~\\ref{sec_Ap}.\n\n Figure~\\ref{fig_ex5p1solutions} shows the evolution of the numerical and reference solutions by taking $T=3, h=2^{-8}, \\tau=\\frac{1}{12000}$ and $m=400$ for different $\\delta=0.5,0.2,0.1$. The reference solutions are obtained by the pseudo-spectral method on a sufficiently large truncated domain. Figure~\\ref{fig_ex5p1solutionsT2} shows the PML solution and reference solution at time $t=2$, which indicates the waves decay exponentially in the PML media.\n\n\n\n\n\n\\begin{figure}[htbp]\n\\centering\n\\begin{subfigure}{.32\\textwidth}\n \\centering\n\t\\includegraphics[width=1\\textwidth]{ex1_d0p5_z20Nxi400}\n \\caption{numerical solution $\\delta=0.5$}\n \\label{fig:ex5p1solutionsa}\n\\end{subfigure}\n\\begin{subfigure}{.32\\textwidth}\n \\centering\n\t\\includegraphics[width=1\\textwidth]{ex1_d0p2_z20Nxi400}\n \\caption{numerical solution $\\delta=0.2$}\n \\label{fig:ex5p1solutionsb}\n\\end{subfigure}\n\\begin{subfigure}{.32\\textwidth}\n \\centering\n\t\\includegraphics[width=1\\textwidth]{ex1_d0p1_z20Nxi400}\n \\caption{numerical solution $\\delta=0.1$}\n \\label{fig:ex5p1solutionsc}\n \\end{subfigure}\n \n \\begin{subfigure}{.32\\textwidth}\n \\centering\n\t\\includegraphics[width=1\\textwidth]{ex1_d0p5_z20Nxi400_ref}\n \\caption{reference solution $\\delta=0.5$}\n \\label{fig:ex5p1solutionsd}\n\\end{subfigure}\n\\begin{subfigure}{.32\\textwidth}\n \\centering\n\t\\includegraphics[width=1\\textwidth]{ex1_d0p2_z20Nxi400_ref}\n \\caption{reference solution $\\delta=0.2$}\n \\label{fig:ex5p1solutionse}\n\\end{subfigure}\n\\begin{subfigure}{.32\\textwidth}\n \\centering\n\t\\includegraphics[width=1\\textwidth]{ex1_d0p1_z20Nxi400_ref}\n \\caption{reference solution $\\delta=0.1$}\n \\label{fig:ex5p1solutionsf}\n \\end{subfigure}\n\t\\caption{(Example 5.1) Numerical solutions for $\\delta=0.5,0.2,0.1$ and reference solutions up to $T=3$. The numerical solutions are obtained by taking $h=2^{-8}$ and $m=400$..}\n\t\\label{fig_ex5p1solutions}\n\\end{figure}\n\n\n\n\\begin{figure}[htbp]\n\\centering\n\\begin{subfigure}{.32\\textwidth}\n \\centering\n\t\\includegraphics[width=1\\textwidth]{ex1_d0p5_solution_T2}\n \\caption{$\\delta=0.5$}\n \\label{fig:ex5p1solutionsT2a}\n\\end{subfigure}\n\\begin{subfigure}{.32\\textwidth}\n \\centering\n\t\\includegraphics[width=1\\textwidth]{ex1_d0p2_solution_T2}\n \\caption{$\\delta=0.2$}\n \\label{fig:ex5p1solutionsT2b}\n\\end{subfigure}\n\\begin{subfigure}{.32\\textwidth}\n \\centering\n\t\\includegraphics[width=1\\textwidth]{ex1_d0p1_solution_T2}\n \\caption{$\\delta=0.1$}\n \\label{fig:ex5p1solutionsT2c}\n \\end{subfigure}\n\t\\caption{(Example 5.1) The comparison of the numerical solution and reference solution at time $t=2$. The PML layers are shaded in light grey.}\n\t\\label{fig_ex5p1solutionsT2}\n\\end{figure}\n\n\n\n\n\n\\begin{table}\n\\centering\n\\begin{tabular}{c|cc|cc|cc}\n\\hline\n$h$\t&\t$\\delta=0.5$  & order & $\\delta=0.2$ & order &  $\\delta=0.1$ & order\\\\ \\hline\n$2^{-4}$\t&  8.39e-03   & -- & 7.62e-03 & -- & 7.91e-03 & --\\\\\n$2^{-5}$\t& 1.36e-03 &  2.62 & 1.19e-03 & 2.67 & 1.31e-03 & 2.59\\\\\n$2^{-6}$ &  3.16e-04 & 2.11 & 2.68e-04 & 2.15 & 2.87e-04 & 2.19\\\\\n$2^{-7}$ & 7.79e-05 & 2.02 & 6.32e-05 & 2.09 & 6.73e-05 & 2.09\\\\\\hline\n\\end{tabular}\n\\caption{(Example 5.1) Errors $e_h$ and convergence orders at $t=2$.}\\label{tab_5p1_l2}\n\\end{table}\n\nTable~\\ref{tab_5p1_l2} shows the errors $e_h$ at time $t=2$ and the spatial convergence rates for various horizons $\\delta=0.5,0.2,0.1$.\nWe now investigate if numerical solutions ${\\tilde q}$ of \\eqref{eq_truPML1}--\\eqref{eq_truPML3} converge to the correct solution $u$ of the corresponding local problem \\eqref{eq_localPML1}--\\eqref{eq_localPML2} as the horizon $\\delta$ goes to zero. The local wave equations with PML modifications in \\cite{basu2008perfectly} are given by\n\\begin{align}\n\\frac{\\partial^2 u}{\\partial t^2} + z\\sigma(x)\\frac{\\partial u}{\\partial t} - \\frac{\\partial v}{\\partial x}&= 0,\\label{eq_localPML1}\\\\\n\\frac{\\partial v}{\\partial t} - \\frac{\\partial^2 u}{\\partial t\\partial x} + z\\sigma(x)v & = 0.\\label{eq_localPML2}\n\\end{align}\nIn the simulation, we fix the ratio $\\delta\/h=M$ with $M=1,2,3$. Table~\\ref{tab_5p1_dh_l2} shows the errors $e_\\delta$ at $t=2$ and the second-order convergence rates, which is consistent to the analysis in \\cite{TianDu}.\n\n\n\\begin{table}\n\\centering\n\\begin{tabular}{c|cc|cc|cc}\n\\hline\n$h$\t&\t$\\delta=h$  & order & $\\delta=2h$ & order &  $\\delta=3h$ & order\\\\ \\hline\n$2^{-4}$\t&  2.63e-02   &-- & 2.62e-02 & -- & 2.62e-03 & --\\\\\n$2^{-5}$\t&  5.72e-03 &  2.20 & 5.71e-03 & 2.20 & 5.70e-03 & 2.20\\\\\n$2^{-6}$ &  1.35e-03  & 2.08 & 1.34e-03 & 2.09 & 1.35e-04 & 2.09\\\\\n$2^{-7}$ & 3.15e-04 & 2.09 & 3.14e-04 & 2.10 & 3.13e-05 & 2.09\\\\\\hline\n\\end{tabular}\n\\caption{(Example 5.1) Errors $e_\\delta$ and $\\delta$-convergence orders between the numerical solutions and exact solutions of local problem \\eqref{eq_localPML1}--\\eqref{eq_localPML2} by vanishing $\\delta$ and $h$ simultaneously at $t=2$.}\\label{tab_5p1_dh_l2}\n\\end{table}\n\n\\bigskip \n\n\\noindent\\textbf{Example 5.2.} In this example, we take the source $f(x,t)\\equiv0$ and the initial values\n$$\\psi_0(x) = e^{-25(x-0.2)^2} + e^{-25(x+0.2)^2}, \\quad \\psi_1(x)  = 50xe^{-25x^2},$$ \nand consider the Gaussian kernel in the form of \n\\begin{align}\n\\gamma(y-x,\\frac{x+y}{2}) = \\frac{4}{\\delta^3}\\sqrt{\\frac{10^3}{\\pi}} e^{-10\\frac{(x-y)^2}{\\delta^2}}. \\label{eq_gaukernel}\n\\end{align}\nIn the simulations, we set $l=2,d_p=2$, the PML coefficient $z=10$, and take Talbot's contour parameters $\\mu = |z|$ and $\\nu = 1$. The reason for choosing these parameters is discussed in Appendix~\\ref{sec_Ap}.\n\nFigure~\\ref{fig_ex5p2solutions} plots the evolution of numerical and the reference solutions by taking $T=4$, $h=2^{-8}$, $\\tau=\\frac{1}{12000}$ and $m=800$ for different $\\delta=0.5,0.2,0.1$. Figure~\\ref{fig_ex5p2solutionsT2} plots the PML solution and the reference solution at time $t=2$, which indicates the waves decay exponentially in the PML media.\n\n\n\n\\begin{figure}[htbp]\n\\centering\n\\begin{subfigure}{.32\\textwidth}\n \\centering\n\t\\includegraphics[width=1\\textwidth]{ex2_d0p5_z20Nxi400}\n \\caption{numerical solution $\\delta=0.5$}\n \\label{fig:ex5p2solutionsa}\n\\end{subfigure}\n\\begin{subfigure}{.32\\textwidth}\n \\centering\n\t\\includegraphics[width=1\\textwidth]{ex2_d0p2_z20Nxi400}\n \\caption{numerical solution $\\delta=0.2$}\n \\label{fig:ex5p2solutionsb}\n\\end{subfigure}\n\\begin{subfigure}{.32\\textwidth}\n \\centering\n\t\\includegraphics[width=1\\textwidth]{ex2_d0p1_z20Nxi400}\n \\caption{numerical solution $\\delta=0.1$}\n \\label{fig:ex5p2solutionsc}\n \\end{subfigure}\n \n \\begin{subfigure}{.32\\textwidth}\n \\centering\n\t\\includegraphics[width=1\\textwidth]{ex2_d0p5_z20Nxi400_ref}\n \\caption{reference solution $\\delta=0.5$}\n \\label{fig:ex5p2solutionsd}\n\\end{subfigure}\n\\begin{subfigure}{.32\\textwidth}\n \\centering\n\t\\includegraphics[width=1\\textwidth]{ex2_d0p2_z20Nxi400_ref}\n \\caption{reference solution $\\delta=0.2$}\n \\label{fig:ex5p2solutionse}\n\\end{subfigure}\n\\begin{subfigure}{.32\\textwidth}\n \\centering\n\t\\includegraphics[width=1\\textwidth]{ex2_d0p1_z20Nxi400_ref}\n \\caption{reference solution $\\delta=0.1$}\n \\label{fig:ex5p2solutionsf}\n \\end{subfigure}\n\t\\caption{(Example 5.2) Numerical solutions for $\\delta=0.5,0.2,0.1$ and reference solutions up to $T=4$. The numerical solutions are obtained by taking $h=2^{-8}$ and $m=400$.}\n\t\\label{fig_ex5p2solutions}\n\\end{figure}\n\n\n\n\\begin{figure}[htbp]\n\\centering\n\\begin{subfigure}{.32\\textwidth}\n \\centering\n\t\\includegraphics[width=1\\textwidth]{ex2_d0p5_solution_T4}\n \\caption{$\\delta=0.5$}\n \\label{fig:ex5p2solutionsT2a}\n\\end{subfigure}\n\\begin{subfigure}{.32\\textwidth}\n \\centering\n\t\\includegraphics[width=1\\textwidth]{ex2_d0p2_solution_T4}\n \\caption{$\\delta=0.2$}\n \\label{fig:ex5p2solutionsT2b}\n\\end{subfigure}\n\\begin{subfigure}{.32\\textwidth}\n \\centering\n\t\\includegraphics[width=1\\textwidth]{ex2_d0p1_solution_T4}\n \\caption{$\\delta=0.1$}\n \\label{fig:ex5p2solutionsT2c}\n \\end{subfigure}\n\t\\caption{(Example 5.2) The comparison of the numerical solution and reference solution at time $t=4$. The PML layers for numerical solutions are shaded in light grey.}\n\t\\label{fig_ex5p2solutionsT2}\n\\end{figure}\n\nTable~\\ref{tab_5p2_l2} shows the errors $e_h$ and the spatial convergence order at $t=4$ by refining $h$ for $\\tau=\\frac{1}{12000}$ and $m=400$. Table~\\ref{tab_5p2_dh_l2} shows the errors $e_\\delta$ and $\\delta$-convergence rates of second-order with the corresponding local PML equations~\\eqref{eq_localPML1}--\\eqref{eq_localPML2}.\n\\begin{table}\n\\centering\n\\begin{tabular}{c|cc|cc|cc}\n\\hline\n$h$\t&\t$\\delta=0.5$  & order & $\\delta=0.2$ & order &  $\\delta=0.1$ & order\\\\ \\hline\n$2^{-4}$\t&  4.07e-02   & -- & 3.18e-02 & -- & 1.50e-02 & --\\\\\n$2^{-5}$\t& 1.04e-02 &  1.96 & 7.19e-03 & 2.15 & 2.38e-03 & 2.65\\\\\n$2^{-6}$ &  2.63e-03 & 1.99 & 1.74e-03 & 2.05 & 5.62e-04 & 2.08\\\\\n$2^{-7}$ & 6.57e-04 & 2.00 & 4.30e-04 & 2.02 & 1.38e-05 & 2.02\\\\\\hline\n\\end{tabular}\n\\caption{(Example 5.2) Errors $e_h$ and convergence orders at $t=4$.}\\label{tab_5p2_l2}\n\\end{table}\n\n\\begin{table}\n\\centering\n\\begin{tabular}{c|cc|cc|cc}\n\\hline\n$h$\t&\t$\\delta=h$  & order & $\\delta=2h$ & order &  $\\delta=3h$ & order\\\\ \\hline\n$2^{-4}$\t& 4.47e-03   & -- & 5.73e-03  & -- & 1.30e-02  & --\\\\\n$2^{-5}$\t&  8.29e-04 &  2.43 & 9.89e-04  & 2.53 & 1.82e-03 & 2.83\\\\\n$2^{-6}$ &  1.92e-04  & 2.11 & 2.24e-04 & 2.14 & 3.91e-04  & 2.22\\\\\n$2^{-7}$ & 4.43e-05 & 2.11 & 5.17e-05 & 2.11 & 9.16e-05& 2.09\\\\\\hline\n\\end{tabular}\n\\caption{(Example 5.2) Errors $e_\\delta$ and $\\delta$-convergence orders between the numerical solutions and exact solutions of local problem \\eqref{eq_localPML1}--\\eqref{eq_localPML2} by vanishing $\\delta$ and $h$ simultaneously at $t=4$.}\\label{tab_5p2_dh_l2}\n\\end{table}\n\n\\bigskip \n\\noindent\\textbf{Example 5.3.} Here we use the same source term and the initial values as Example 5.2, and consider the following spatially inhomogeneous kernel\n\\begin{align}\n\\gamma(y-x,\\frac{x+y}{2}) = \\frac{\\omega(\\frac{x+y}{2})}{\\zeta^3(\\frac{x+y}{2})} H\\Big(\\frac{y-x}{\\zeta(\\frac{x+y}{2})}\\Big),\n\\end{align}\nwhere \n\\begin{align}\n\\omega(\\beta) = 1+e^{-3\\beta^2},\\ \\zeta(\\beta) = \\delta(2+\\tanh(-1.5\\beta)),\\ H(s) = 4\\sqrt{\\frac{10^3}{\\pi}} e^{-10s^2}.\n\\end{align}\nIn the simulations, we take $l=2, d_p=2$, the PML coefficient $z=10$ and $\\mu = |z|, \\nu = 1$. \n\n\n\n\\begin{figure}[htbp]\n\\centering\n\\begin{subfigure}{.4\\textwidth}\n \\centering\n\t\\includegraphics[width=1\\textwidth]{inhpar}\n\\end{subfigure}\n\\begin{subfigure}{.4\\textwidth}\n \\centering\n\t\\includegraphics[width=1\\textwidth]{ex3_kernel}\n\\end{subfigure}\n\t\\caption{(Example 5.3) Left: the functions $\\omega(\\beta)$ and $\\zeta(\\beta)$. Right: the kernel $\\gamma(\\alpha,\\beta)$ for $\\delta=1$.}\n\t\\label{fig_ex5p4ker}\n\\end{figure}\n\nThe limiting local wave equation with PML modifications is given by\n\\begin{align}\n\\frac{\\partial^2 u}{\\partial t^2} + z\\sigma(x)\\frac{\\partial u}{\\partial t} - \\frac{\\partial v}{\\partial x}&= 0,\\label{eq_localinPML1}\\\\\n\\frac{\\partial v}{\\partial t} - \\mu(x)\\frac{\\partial^2 u}{\\partial t\\partial x} + z\\sigma(x)v & = 0.\\label{eq_localinPML2}\n\\end{align}\n\nFigure~\\ref{fig:ex5p3solutions} shows the evolution of numerical and reference solutions by taking $T=4$, $h=2^{-8}$, $\\tau=1\/12000$ and $m=400$ for different $\\delta=0.3,0.2,0.1$. Figure~\\ref{fig_ex5p3solutionsT4} shows numerical and reference solutions at $t=4$, which again indicates the numerical solutions decay exponentially in PML layers.\n\n\n\\begin{figure}[htbp]\n\\centering\n\\begin{subfigure}{.32\\textwidth}\n \\centering\n\t\\includegraphics[width=1\\textwidth]{ex3_d0p3_z5Nxi400}\n \\caption{numerical solution $\\delta=0.3$}\n \\label{fig:ex5p3solutionsa}\n\\end{subfigure}\n\\begin{subfigure}{.32\\textwidth}\n \\centering\n\t\\includegraphics[width=1\\textwidth]{ex3_d0p2_z5Nxi400}\n \\caption{numerical solution $\\delta=0.2$}\n \\label{fig:ex5p3solutionsb}\n\\end{subfigure}\n\\begin{subfigure}{.32\\textwidth}\n \\centering\n\t\\includegraphics[width=1\\textwidth]{ex3_d0p1_z5Nxi400}\n \\caption{numerical solution $\\delta=0.1$}\n \\label{fig:ex5p3solutionsc}\n \\end{subfigure}\n \n \\begin{subfigure}{.32\\textwidth}\n \\centering\n\t\\includegraphics[width=1\\textwidth]{ex3_d0p3_z5Nxi400_ref}\n \\caption{reference solution $\\delta=0.3$}\n \\label{fig:ex5p3solutionsd}\n\\end{subfigure}\n\\begin{subfigure}{.32\\textwidth}\n \\centering\n\t\\includegraphics[width=1\\textwidth]{ex3_d0p2_z5Nxi400_ref}\n \\caption{reference solution $\\delta=0.2$}\n \\label{fig:ex5p3solutionse}\n\\end{subfigure}\n\\begin{subfigure}{.32\\textwidth}\n \\centering\n\t\\includegraphics[width=1\\textwidth]{ex3_d0p1_z5Nxi400_ref}\n \\caption{reference solution $\\delta=0.1$}\n \\label{fig:ex5p3solutionsf}\n \\end{subfigure}\n\t\\caption{(Example 5.3) Numerical solutions for $\\delta=0.3,0.2,0.1$ and reference solutions up to $T=4$. The numerical solutions are obtained by taking $h=2^{-8}$ and $m=400$.}\n\t\\label{fig:ex5p3solutions}\n\\end{figure}\n\n\\begin{figure}[htbp]\n\\centering\n\\begin{subfigure}{.32\\textwidth}\n \\centering\n\t\\includegraphics[width=1\\textwidth]{ex3_d0p3_solution_T4}\n \\caption{$\\delta=0.3$}\n \\label{fig:ex5p3solutionsT2a}\n\\end{subfigure}\n\\begin{subfigure}{.32\\textwidth}\n \\centering\n\t\\includegraphics[width=1\\textwidth]{ex3_d0p2_solution_T4}\n \\caption{$\\delta=0.2$}\n \\label{fig:ex5p3solutionsT2b}\n\\end{subfigure}\n\\begin{subfigure}{.32\\textwidth}\n \\centering\n\t\\includegraphics[width=1\\textwidth]{ex3_d0p1_solution_T4}\n \\caption{$\\delta=0.1$}\n \\label{fig:ex5p3solutionsT2c}\n \\end{subfigure}\n\t\\caption{(Example 5.3) The comparison of the numerical solution and reference solution at time $t=4$. The PML layers for numerical solutions are shaded in light grey.}\n\t\\label{fig_ex5p3solutionsT4}\n\\end{figure}\n\nTable~\\ref{tab_5p3_l2} lists the errors $e_h$ at $t=4$ and the spatial convergence order for numerical solutions, which verify the second-order accuracy of the spatial discretization by refining $h$. Table~\\ref{tab_5p3_dh_l2} lists the errors $e_\\delta$ and $\\delta$-convergence orders between the numerical solutions and exact solutions of local problem \\eqref{eq_localinPML1}--\\eqref{eq_localinPML1} by vanishing $\\delta$ and $h$ simultaneously at $t=2$. The behavior of the errors is similar to that shown in the examples above for sufficiently small $\\delta$.\n\n\n\n\\begin{table}\n\\centering\n\\begin{tabular}{c|cc|cc|cc}\n\\hline\n$h$\t&\t$\\delta=0.3$  & order & $\\delta=0.2$ & order &  $\\delta=0.1$ & order\\\\ \\hline\n$2^{-5}$\t& 3.74e-02 &  --& 4.7483e-02 & -- &  4.35e-02  & --\\\\\n$2^{-6}$ &  9.96e-03  & 1.91 &  1.3050e-02  & 1.86 &  1.19e-02 & 1.87\\\\\n$2^{-7}$ & 2.52e-03 & 1.98 &  3.3125e-03& 1.98 &  2.99e-03 & 2.00\\\\\n$2^{-8}$ & 6.36e-04 & 1.98 & 8.25e-04  & 2.01 & 7.21e-04  & 2.05 \\\\\\hline\n\\end{tabular}\n\\caption{(Example 5.3) errors $e_h$ and convergence orders at $t=4$.}\\label{tab_5p3_l2}\n\\end{table}\n\n\\begin{table}\n\\centering\n\\begin{tabular}{c|cc|cc|cc}\n\\hline\n$h$\t&\t$\\delta=h$  & order & $\\delta=2h$ & order &  $\\delta=3h$ & order\\\\ \\hline\n$2^{-5}$\t&  5.3521e-02 &  -- & 1.1867e-01 & -- & 1.4328e-01 & --\\\\\n$2^{-6}$ &  1.3937e-02& 1.94 &  4.2328e-02 & 1.49 & 8.1629e-02 & 0.81\\\\\n$2^{-7}$ &  3.4432e-03  & 2.02 &  1.0714e-02 & 1.98 &  2.2952e-02 & 1.83\\\\\n$2^{-8}$ &  8.3835e-04  & 2.03 &  2.6479e-03& 2.01&  5.6996e-03 & 2.01\\\\\\hline\n\\end{tabular}\n\\caption{(Example 5.3) The errors $e_\\delta$ and $\\delta$-convergence orders between numerical solutions and exact solutions of local PML problem \\eqref{eq_localinPML1}--\\eqref{eq_localinPML2} by vanishing $\\delta$ and $h$ simultaneously at $t=2$.}\\label{tab_5p3_dh_l2}\n\\end{table}\n\n\n\n\n\n\n \n\\section{Conclusion}\nThe nonlocal PML and its numerical discretization of a nonlocal wave equation in unbounded spatial domains are studied in this paper. We first propose the nonlocal PML equation with a time-dependent nonlocal operator which has a complex-valued kernel and is defined by a convolution. This feature differs from the results in \\cite{wildman2011,wildman2012a}. After discretizing the contour integrals arising from the inverse Laplace transform, applying the AC scheme and introducing auxiliary functions, we get a truncated semidiscrete nonlocal PML problem which only involves a finite number of degrees of freedom. A Verlet-type ODE solver is used for the time integration. Numerical experiments demonstrate the effectiveness and accuracy of our proposed PML.\n\n\n\\section*{Appendix A.} \\label{sec_Ap}\n\\textbf{We first consider $\\Omega_{\\mathcal{K}}$ for the kernel~\\eqref{eq_ker1}.} First, we give the analytic continuation of the reference kernel $\\gamma_0(s) = \\frac{1}{2c_0^3} e^{-\\frac{|s|}{c_0}}$ by\n$$\\gamma_0(\\tilde x-\\tilde y) = \\frac{1}{2c_0^3} e^{-\\frac{\\rho(\\tilde x,\\tilde y)}{c_0}},$$\nwhere $\\rho(\\tilde x,\\tilde y)=\\sqrt{(\\tilde x-\\tilde y)^2}$ is its analytic branch $\\Re \\rho(\\tilde x,\\tilde y)\\geq0$. Let $z=z_1+\\i z_2$ and $s = s_1 + \\i s_2$ where $z_1,z_2,s_1,s_2\\in\\mathbb{R}$. We have\n\\begin{align*}\n\\tilde x-\\tilde y &= (x-y) + \\frac{z}{s}\\Big(\\int_0^x\\sigma(t)\\mathrm dt-\\int_0^y\\sigma(t)\\mathrm dt\\Big)\\\\\n&= (x-y) \\Big[\\Big( 1 + \\frac{z_1s_1+z_2s_2}{|s|^2}\\cdot \\frac{\\int_0^x\\sigma(t)\\mathrm dt-\\int_0^y\\sigma(t)\\mathrm dt}{x-y}\\Big) x+\\i\\frac{-z_1s_2+z_2s_1}{|s|^2} \\cdot\\frac{\\int_0^x\\sigma(t)\\mathrm dt-\\int_0^y\\sigma(t)\\mathrm dt}{x-y}\\Big].\n\\end{align*}\nSet\n\\begin{equation}\\label{Ge}\ng=\\frac{\\int_0^x\\sigma(t)\\mathrm dt-\\int_0^y\\sigma(t)\\mathrm dt}{x-y}.\n\\end{equation}\n It is clear that $0\\leq g\\leq1$ since $\\sigma\\leq 1$. Therefore, the real part of $\\tilde x-\\tilde y$ is\n\\begin{align*}\n (x-y)\\Big(1 + \\frac{(z_1s_1+z_2s_2)g}{|s|^2} \\Big) &=  (x-y)\\Big(\\frac{|s|^2 + (z_1s_1+z_2s_2)g}{|s|^2} \\Big)\\\\\n &= \\frac{x-y}{|s|^2}\\Big[ \\Big( s_1+\\frac{z_1g}{2} \\Big)^2 +  \\Big( s_2+\\frac{z_2g}{2} \\Big)^2 - \\frac14|z|^2g^2\\Big],\n\\end{align*}\nwhich implies that $\\rho(\\tilde x,\\tilde y)=\\sqrt{(\\tilde x-\\tilde y)^2}$ as a function of $s$ is analytic in the $s$-domain\n\\begin{align*}\n\\Omega_{\\mathcal{K}}:\\ \\Big( s_1+\\frac{z_1}{2} \\Big)^2 +  \\Big( s_2+\\frac{z_2}{2} \\Big)^2 > \\frac14|z|^2, \\quad \\forall x,y. \n\\end{align*}\n\n\n\\noindent\\textbf{We then consider Talbot's contour parameters $\\mu$ and $\\nu$ for the Gaussian kernel~\\eqref{eq_gaukernel}.} The analytic continuation of the kernel $\\gamma(y-x,\\frac{x+y}{2}) =  \\frac{4}{\\delta^3}\\sqrt{\\frac{10^3}{\\pi}} e^{-10\\frac{(x-y)^2}{\\delta^2}}$ is given by\n\\begin{align}\n\\gamma(\\tilde y-\\tilde x,\\frac{\\tilde x+\\tilde y}{2}) =  \\frac{4}{\\delta^3}\\sqrt{\\frac{10^3}{\\pi}} e^{-10\\frac{(\\tilde x-\\tilde y)^2}{\\delta^2}}.\n\\end{align}\nNote the $\\Omega_\\mathcal{K}$ is the whole complex plane for any given $z\\in\\mathbb{C}$ and $x,y\\in\\mathbb{R}$. However, to ensure the stability, we have to choose the Talbot's contour parameters $\\mu$ and $\\nu$ such that $\\Re[(\\tilde x-\\tilde y)^2] \\geq 0$. Denote by $\\zeta=\\zeta_1+\\i \\zeta_2=\\frac{z}{\\xi_j}$. We have\n\\begin{align*}\n\\Re\\Big[(\\tilde x-\\tilde y)^2\\Big] =& \\Re\\Big[\\Big( (x+\\zeta_1\\int_0^x\\sigma(t)\\mathrm dt+\\i\\zeta_2\\int_0^x\\sigma(t)\\mathrm dt) - (y+\\zeta_1\\int_0^y\\sigma(t)\\mathrm dt+\\i\\zeta_2\\int_0^y\\sigma(t)\\mathrm dt) \\Big)^2\\Big] \\\\\n=& \\Big( (x+\\zeta_1\\int_0^x\\sigma(t)\\mathrm dt)- (y+\\zeta_1\\int_0^y\\sigma(t)\\mathrm dt)\\Big)^2 -\\Big (\\zeta_2\\int_0^x\\sigma(t)\\mathrm dt-\\zeta_2\\int_0^y\\sigma(t)\\mathrm dt\\Big)^2\\\\\n=&(x-y)^2\\big[ (1+\\zeta_1 g)^2 - \\zeta_2^2 g^2\\big]\\\\\n=&(x-y)^2\\big[ \\big(1+(\\zeta_1-\\zeta_2)g\\big) \\big(1+(\\zeta_1+\\zeta_2)g\\big)\\big],\n\\end{align*}\nwhere $g$ is defined in \\eqref{Ge} with $0\\leq g\\leq1$ as $0\\leq\\sigma\\leq 1$. \n\nTo ensure $\\Re[(\\tilde x-\\tilde y)^2] \\geq 0$, we have $\\big(1+(\\zeta_1-\\zeta_2)g\\big) \\big(1+(\\zeta_1+\\zeta_2)g\\big)\\geq0$ for any $g\\in[0,1]$, which implies that $\\zeta_1-\\zeta_2\\geq -1$ and $\\zeta_1+\\zeta_2\\geq-1$. Therefore, we may simply choose Talbot's contour parameters $\\mu$ and $\\nu$ such that\n\\begin{align}\n\\sqrt{2}|z|\\leq |\\xi_j|,\\quad \\forall\\ j=1,2,\\cdots,m.\n\\end{align}\n\n\n\n\n\\bibliographystyle{siam}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\\section{Related Work}\n\\label{sec:related}\nIn this section, we review the recent studies that are most relevant to our work, including data-driven storytelling, automatic data visualization, and natural language generation.\n\n\\subsection{Data-Driven Storytelling} \nData-driven storytelling is a rapidly developing research direction that focuses on techniques for enhancing data understanding, information expression, and communication. Narrative visualization is one promising approach frequently used for data-driven story telling~\\cite{tong2018storytelling}. Recently, the visualization community has extensively investigated storytelling and narrative visualization techniques~\\cite{segel2010narrative, tong2018storytelling}. According to Segel and Heer~\\cite{segel2010narrative}, narrative visualization can be largely classified into seven genres, including magazine style, annotated chart, partitioned poster, flow chart, comic strip, slideshow, and video. Evidence shows that an effective composition and visual narrative of the story can guide readers through the data and improve the comprehension and memory~\\cite{borkin2015beyond, hullman2013deeper}. \nTo compose effective data-driven stories, Hullman~{\\it et~al.}\\xspace~\\cite{hullman2013deeper} identified several key design actions in the sequential story creation process, including context definition, facts selection, modality selection, and order selection.\nLee~{\\it et~al.}\\xspace also decomposed the creation process of a data-driven story into three major steps: insights finding, scripting, and communicating to the audience~\\cite{lee2015more}. These valuable study results guide the designs of many data story creation systems including Calliope\\xspace, which is introduced in this paper.\n\nUsers commonly experience difficulty in creating a data-driven story due to technical barriers, which motivate the design and development of various authoring tools. General tools such as Ellipsis~\\cite{satyanarayan2014authoring} allows a user to directly integrate visualizations to an illustrative story. Recent studies focus on designing tools to generate a specific type of visual narrative. For example, ChartAccent~\\cite{ren2017chartaccent} and InfoNice~\\cite{wang2018infonice} are respectively designed for creating annotated charts and infographics. Narvis~\\cite{wang2018narvis} is introduced to extract the combination of visual elements of a visualization and organize them as a slideshow to help with the narrative interpretation of a visualization design. DataClips~\\cite{amini2016authoring} is designed to help users generate data videos. Various authoring tools that are specifically designed to create narrative visualization for certain types of data. For example, Timeline Storyteller~\\cite{brehmer2019timeline}, is a visual storytelling tool designed specifically for time-oriented data. Several visualization tools are also introduced to bridge the gap between visual analysis and storytelling~\\cite{chen2018supporting, gratzl2016visual}, but they still target on expert users.\n\nThe above tools assume that the story content is manually created, resulting in inefficiency. By contrast, Calliope\\xspace supports automatic story generation and flexible story editing functions, which ensure the quality, lower the barrier, and improve the efficiency of visual narration.\n\n\\subsection{Automatic Data Visualization}\nStudies on automatic visualization have experienced three stages, including visualization chart recommendation, automatic data mapping generation, and auto-insights. To recommend a chart given an input data, early studies employed a rule-based method, which checks the data types to make a suggestion~\\cite{mackinlay2007show,gotz2010harvest}. A recent study~\\cite{hu2019vizml} trained a classification model based on a collection of ``data feature - chart type\" pairs extracted from a visualization design corpus. As a result, given data features, a proper chart type can be selected. In Calliope\\xspace, we select a chart for each data fact based on its fact type and data fields.\n\nTo visualize data in a chart, one must determine the detailed data mapping strategy. To this end, various techniques are introduced. Draco~\\cite{moritz2018formalizing} uses an optimization model to find the best data mapping strategy under a set of constraints formulated by several common visual design guidelines. DeepEye~\\cite{luo2018deepeye} enumerates all possible data mapping strategies and uses a decision tree to select the good ones. Data2Vis~\\cite{dibia2019data2vis} ``translates\" the data into a visual encoding scheme based on a sequence-to-sequence deep model. Text-to-Viz~\\cite{cui2019text} employs natural language processing techniques to identify and parse data entities, such as numbers, portions, and data scopes from an input text, and convert them into a statistic diagram. Shi~{\\it et~al.}\\xspace\\cite{Shi2019TaskOrientedOS} explored the chart design space via a reinforcement learning model and generated a sequence of data mapping approaches regarding an analytical task. Calliope\\xspace encodes different data fact fields in a chart following a rule-based method.\n\nRecent studies focused on extracting data patterns and representing them in charts to reveal data insights, i.e., auto-insights~\\cite{tang2017extracting,ding2019quickinsights}. The extracted insights can be quantitatively measured based on their statistical significance~\\cite{ding2019quickinsights}. Visual analysis techniques were also developed to support auto-insights. For example, SeeDB~\\cite{vartak2015seedb} finds and illustrates the most interesting trend in the data by exploring various data mapping strategies. Foresight~\\cite{demiralp2017foresight} extracts and visualizes insights from multidimensional data using rule-based methods. DataShot~\\cite{wang2019datashot} randomly visualizes a set of automatically generated data facts as a factsheet. \\rv{Inspired by DataShot, Calliope\\xspace also borrowed the auto-insights techniques to generate data facts, but made a step further by organizing the facts in a logical order to generate a meaningful data story.}\n\n\\subsection{Natural Language Generation}\nRecently, studies on natural language generation (NLG) demonstrate the capability of producing descriptive text from various types of data~\\cite{mishra2019storytelling,turner2009generating,galanis2007generating}. Many methods use templates to generate sentences~\\cite{swanson2012say,li2013story}, and techniques that automatically enrich a template were developed~\\cite{dou2018data2text,ye2020variational}. Recent studies leveraged deep learning models to generate textual content from scratch~\\cite{fan2018hierarchical,fan2019strategies}. Several methods create an intermediate structure based on a recurrent neural network~\\cite{martin2018event,xu2018skeleton,yao2019plan}, and others use the auto-encoder architecture to generate diverse sentences from a latent space~\\cite{liu2019transformer,li2019learning}. Among various techniques, those aim to generate text content based on structured data are the most relevant to our work~\\cite{mahapatra2016statistical,belz2008automatic,jain2018mixed}. For example, commercial software, such as PowerBI\\footnote{\\url{https:\/\/powerbi.microsoft.com}} and Quill\\footnote{\\url{https:\/\/narrativescience.com}} describe important data facts based on a set of templates to help interpret the data and the corresponding visualization. A number of visual auto-insights systems, such as DataSite~\\cite{cui2019datasite}, DataShot~\\cite{wang2019datashot}, and Voder~\\cite{srinivasan2018augmenting}, use the template-based NLG to generate captions for visualization charts. In Calliope\\xspace, we also employ the template-based method to generate captions for each chart, but to ensure the readability and avoid ambiguity, we define a syntax for each fact type that regulates the generation results.\n\\section{Design of the Calliope System}\n\\label{sec:system}\nThis section introduces the design of Calliope\\xspace system. We first introduce the formal definition of a data story, then survey on a collection of data videos to help us understand how a story is generated by human designers. After that, we summarize the design requirements and introduce the architecture design of the system. \n\n\n\\vspace{-0.5em}\n\\input{tables\/1-factvischarts}\n\\setlength{\\floatsep}{5pt}\n\\input{tables\/2-narrativelogic}\n\\setlength{\\textfloatsep}{5pt}\n\n\\subsection{Data Story}\nData story is a set of story pieces that are meaningfully connected to support the author's communication goal~\\cite{lee2015more}. A story piece is a fact backed up by data, and it is usually visualized by succinct but expressive charts, accompanied with annotations (labels, pointers, text, etc.) and narrations to express the message and avoid the ambiguity. We design Calliope\\xspace to automatically generate visual data stories by following this definition. Formally, a data story $\\mathcal{S}$ consists of a sequence of data facts that are connected by coherent relations (denoted as $r_i \\in \\mathcal{R}$) $\\{f_1,r_1, \\cdots, f_{n-1}, r_{n-1}, f_n\\}$ with each fact $f_i \\in F$. We will use these notations throughout the paper. \n\n\\subsection{Preliminary Survey}\nBefore designing the system, it is necessary to understand how a data story is created by a human designer. To this end, as data video is a frequently used narrative visualization form, we collected a set of 602 data videos from YouTube and Vimeo by searching keywords, such as ``animated infographic'', ``data video'', and ``motion infographic''. A total of 230 high-quality videos were selected and manually segmented into 4186 story pieces. The fact and chart types of 2583 data-related story pieces and the coherence relations used for connecting two succeeding pieces were labeled for analysis. \\rv{Here, we borrowed the definition of fact types introduced in DataShot~\\cite{wang2019datashot} and coherence relations introduced in~\\cite{wellner2006classification, wolf2005representing} to label our data}. \n\nAs a result, even some simple statistics are able to help us answer a number of questions that are critical to the design of an automatic story generation system. For example, Fig.~\\ref{fig:statistic} suggests the frequently used fact types, the fact types frequently used as start points, and the frequently used coherence relations regarding our data video corpus. Table~\\ref{tab:charts} suggests the preferred visualization charts given a fact type. Table~\\ref{tab:logic} suggests the narrative logic probably used following each fact type. These results guide the design of the story generation algorithm and will be further described later. \n\n\\subsection{System Design}\nOur goal is to design a system that can automatically generate high-quality initial data stories directly from an input spreadsheet and support flexible story editing functions to lower the technical barriers of creating a data story. To achieve this goal, a number of key requirements need to be fulfilled:\n\n\n\\begin{enumerate\n\\itemsep -1mm\n\\item[{\\bf R1}] {\\bf Generating ``successful\" stories.} The most important thing for the system is to ensure the quality of story generation. Among various factors that contribute to a successful narrative artifact, the key is understandability~\\cite{riedl2010narrative}, which is usually determined by the narrative logic, and the meaningful and believable content~\\cite{lee2015more}. Therefore, the system should be intelligent enough to automatically generate meaningful stories logically with correct, i.e., believable, data backups.\n\n\\item[{\\bf R2}] {\\bf Efficient story generation.} The system should be able to efficiently generate a data story within a reasonable period of time that is affordable to the users. Therefore, the generation time should be controllable to grantee the efficiency of the system while keeping the quality of the story.\n\n\\item[{\\bf R3}] {\\bf Expressive story representation.} As suggested in~\\cite{lee2015more}, the generated visual data story should be expressively represented in both visual and textual forms to precisely express the message and avoid ambiguity. Here, simple but intuitive charts~\\cite{wang2019datashot}, as well as precise and meaningful narratives~\\cite{lee2015more} should be guaranteed to reduce users' learning efforts.\n\n\\item[{\\bf R4}] {\\bf Easy story editing.} The system should provide flexible interactions to support comprehensive editing of the generated storyline, text narration, visual representation, and the corresponding data facts, so that a user can further refine and adjust an automatically generated story based on their own requirements.\n\n\\item[{\\bf R5}] {\\bf Easy communication and sharing.} The visual and textual representations of a data story should be probably aligned and adaptive laid out to fit into different devices such as a laptop, tablets, and smartphones to facilitate an easy story exploration, communication, and sharing.\n\\end{enumerate}\n\nTo fulfill these requirements, the design of Calliope\\xspace system consists of two modules (Fig.~\\ref{fig:system}): (1) the story generation engine and (2) the story editor. The story generation engine is designed based on a logic-oriented Monte Carlo tree search process, in which a story is gradually generated fact by fact while searching through the data space defined by an input spreadsheet. The whole search process is guided by narrative logic and a reward function that measures the importance of facts to ensure the quality of the generated story (\\textbf{R1}). In addition, the time spent on each searching step is configurable, which guarantees the generation efficiency (\\textbf{R2}). The generated story is visualized in the story editor as a series of captioned visualization charts (\\textbf{R3}), whose data facts, caption, chart type, and logic orders can be revised according to user preferences (\\textbf{R4}). The final visual data story can be represented in three modes to fit different devices (\\textbf{R5}).\n\n\\setlength{\\textfloatsep}{20pt}\n\\begin{figure}[tb]\n  \\centering\n  \\includegraphics[width=\\linewidth]{figures\/pipeline.png}\n  \\vspace{-2em}\n  \\caption{Calliope\\xspace system consists of two modules : the story generation engine and the story editor.} \\label{fig:system}\n  \\vspace{-1.5em}\n\\end{figure}\n\\setlength{\\textfloatsep}{20pt}\n\\subsection{Story Generation Algorithm}\nIn Calliope\\xspace, we introduce an intelligent algorithm that generates data facts from an input spreadsheet and threads them logically to create a meaningful data story. However designing such an algorithm is challenging. The story design space, formulated by a collection of data facts generated from the input spreadsheet, could be extremely large due to the huge number of possible combinations of the fact fields even based on a small dataset. The algorithm cannot generate all facts first and then pick up those important ones to build the story as users cannot afford a long waiting time. We addressed this issue by introducing an efficient logic-oriented searching algorithm based on the Monte Carlo tree search (MCTS)~\\cite{browne2012survey,silver2016mastering}. The algorithm can efficiently explore a large space that contains a huge number of states via a searching process organized by a tree structure and guided by a reward function towards a logic direction.\n\n\\begin{figure}[!b]\n  \\centering\n  \\vspace{-0.5em}\n  \\includegraphics[width=\\linewidth]{figures\/algorithm.png}\n  \\vspace{-1em}\n  \\caption{An iteration of the logic-oriented Monte Carlo tree search consists of four steps, including (a) selection, (b) expansion, (c) simulation, and (d) back-propagation.} \n  \\label{fig:algorithm}\n  \\vspace{-0.5em}\n\\end{figure}\n\n\\paragraph{\\bf Algorithm Overview} In general, the algorithm explores the design space by dynamically constructing a searching tree $\\mathcal{T}$. As shown in Fig.~\\ref{fig:algorithm}(a), each node in the tree is a data fact $f_i$ and each directed edge indicates a logic relation $r_i$.  A data story $\\mathcal{S} = \\{f_1, r_1, \\cdots,f_{n-1}, r_{n-1}, f_n\\}$ is thus represented by a path starting from the root. A reward function is designed to estimate the quality of each path in the tree. The reward scores are marked on the last node in paths. A node shared by multiple paths is weighted by the maximum reward. These scores are used to guide the exploration of the design space.\n\n\nThe tree $\\mathcal{T}$ is gradually generated through a searching process as described in Algorithm~\\ref{alg:generation}. In particular, the algorithm takes a spreadsheet $\\mathcal{D}$, i.e., a data table, and a goal $\\mathcal{G}$, such as generating a story with a desired information quantity or length as the inputs and automatically generates a story $\\mathcal{S}$ that fulfills the goal. \\rv{Initially, it randomly generates a set of facts in types that are frequently used as the starting point in a data story (Fig.~\\ref{fig:statistic}(b)). These facts usually reveal general and common data patterns which may already known by the audience as the background of the story. Among these facts, the most important one, denoted as $f_0$, is used as the root of $\\mathcal{T}$. In the next, the algorithm generates a story by iteratively searching more informative and significant data facts to elaborate the story via four major steps: \\textit{\\textbf{selection}}\\xspace, \\textit{\\textbf{expansion}}\\xspace, \\textit{\\textbf{simulation}}\\xspace, and \\textit{\\textbf{back-propagation}}\\xspace.}\nThe first step finds a node $f_i$ with the largest reward in $\\mathcal{T}$, from which the next searching step will be performed (\\textit{line 3}, Fig.~\\ref{fig:algorithm}(a)). \nThe second step searches the design space by creating a set of data facts (denoted as $F_i$), that is logically relevant to $f_i$ (\\textit{line 4}, Fig.~\\ref{fig:algorithm}(b)).\nThe third step finds the best searching direction $f_i \\rightarrow f^*, f^* \\in F_i$ with the largest reward $\\Delta^*$ through a simulation process (\\textit{lines 5 - 11}, Fig.~\\ref{fig:algorithm}(c)). This process simulates the cases in which each $f \\in F_i$ is expanded in a simulation tree $\\mathcal{T}_s$ rooted at $f_i$ to help the algorithm explore the space a few steps further, so that the different searching directions can be estimated in advance. The simulation runs within a time limit to ensure the efficiency of the algorithm (\\textbf{R2}).\nIn the last step, the tree $\\mathcal{T}$ is updated via a back-propagation process, in which the weights of the relevant nodes are updated based on $\\Delta^*$ and $f^*$ is formally added into $\\mathcal{T}$ as a child of $f_i$ (\\textit{line 11}, Fig.~\\ref{fig:algorithm}(d)). Finally the path with the highest reward in $\\mathcal{T}$, $\\mathcal{P}^*$, is identified as the best story generated at the current iteration (\\textit{line 12}). The algorithm stops when the goal $\\mathcal{G}$ is fulfilled.\n\n\n\\setlength{\\textfloatsep}{0pt}\n\\begin{algorithm}[tb]\n\\label{alg:generation}\n\\SetAlgoLined\n\\SetKwInOut{Input}{Input}\n\\SetKwInOut{Output}{Output}\n\\Input{$\\mathcal{D}, \\mathcal{G}$}\n\\Output{$\\mathcal{S} = \\{f_1,r_1 \\cdots, f_{n-1}, r_{n-1}, f_n\\}$}\n$f_0$ $\\leftarrow$ Initialize($\\mathcal{D}$);\n$\\mathcal{T}$ $\\leftarrow$ \\{$f_0$\\};\n$\\mathcal{S}$ $\\leftarrow$ \\{\\}\\;\n\\While{$\\mathcal{G}$ is not fulfilled}{\n    \\tcc{1.\\textit{\\textbf{selection}}\\xspace}\n    $f_i \\leftarrow select(\\mathcal{T})$\\; \n    \n    \\tcc{2.\\textit{\\textbf{expansion}}\\xspace}\n    $F_i$ $\\leftarrow$ Expand($f_i$)\\;\n    \n    \\tcc{3.\\textit{\\textbf{simulation}}\\xspace}\n    $\\mathcal{T}_s \\leftarrow \\{f_i\\}$;\n    $F \\leftarrow F_i$;\n    $f_p \\leftarrow f_i$\\;\n    \\While{within time limitation}{\n    \n        \\tcp{Calculate the reward of each node in $F$ in context of $\\mathcal{T}_s$ and find the node $f \\in F$ with the highest reward $\\Delta$. The design space will be explored in direction of $f_p \\rightarrow f$ in the simulation process.}\n        $f, \\Delta \\leftarrow$ Reward($\\mathcal{T}_s, F$)\\;\n        \n        \\tcp{Add $f$ in the simulation tree $\\mathcal{T}_s$ as a child of $f_p$ and update reward of the relevant nodes in $\\mathcal{T}_s$ based on $\\Delta$. After that, find the node $f^* \\in F_i$ with the highest reward in $\\mathcal{T}_s$, where $f_i \\rightarrow f^*$ determines the best searching direction found so far.}\n        $f^*,\\Delta^* \\leftarrow$ BackPropagation($\\mathcal{T}_s, f, \\Delta$)\\;\n        \n        \\tcp{Select the next node $f_p$ and expand it in the simulation tree for a further exploration}\n        $f_p \\leftarrow select(\\mathcal{T}_s)$;\n        $F$ $\\leftarrow$ Expand($f_p$)\\;\n    }\n    \\tcc{4.\\textit{\\textbf{back-propagation}}\\xspace}\n    BackPropagation($\\mathcal{T},f^*, \\Delta^*$)\\;\n    \n    $\\mathcal{S} \\leftarrow \\mathcal{P}^* = \\{f_1, r_1, \\cdots, f_{n-1}, r_{n-1}, f_n\\}$\\;\n}\n\\Return $\\mathcal{S}$\\;\n\\caption{Logic-Oriented Monte Carlo Tree Search}\n\\end{algorithm}\n\n\\paragraph{\\bf Logic-Oriented Node Expansion} Expanding a selected node $f_i$ in the search tree $\\mathcal{T}$ to elaborate the story design space is a critical step in the aforementioned searching algorithm. The expansion should generate a set of nodes that is logically relevant to $f_i$ to gradually generate a meaningful data story through the searching process. To this end, we investigated how a set of commonly used coherence relations (denoted as $\\mathcal{R}$)~\\cite{wellner2006classification, wolf2005representing} was used in data stories during our preliminary survey. As a result, Table~\\ref{tab:logic} summarizes the likelihood, $P(r_i|f_i)$, of each relation $r_i$ occurring after a fact $f_i$ regarding to their fact types, which guides the node expansion process. In particular, during the expansion, we create a set of data facts regarding each coherence relation. The proportion of the newly generated facts is given by $P(r_i|f_i)$, and each new fact, $f_{i+1}$, is generated by the following rules:\n\\begin{itemize}[leftmargin=10pt,topsep=2pt]\n\\itemsep -.5mm\n    \\item {\\textit{\\textbf{Similarity}}} indicates two succeeding facts are logically parallel to each other. Therefore, $f_{i+1}$ can be generated by a variety of methods, such as modifying the measure \/ breakdown \/ focus field without changing the subspace. \n    \\item {\\textit{\\textbf{Temporal}}} relation communicates the ordering in time of events or states. In this case, we generate $f_{i+1}$ by changing the value of the temporal filter in $f_{i}$'s subspace to a succeeding time.\n    \\item {\\textit{\\textbf{Contrast}}} indicates a contradiction between two facts. For simplicity, we only check the contradictions in two types of facts, i.e., trend and association. $f_{i+1}$ is generated by modifying the subspace of $f_i$ to form a data contradiction in measures. For example, the sales trends of a product increases, but that of another product decreases. The sales number of a product is positively associated with its price, but the association is negative in case of another product. In these examples, the subspace is determined by different products.\n    \\item {\\textit{\\textbf{Cause-Effect}}} indicates the later event is caused by the former one. \\rv{In multidimensional data, a causal relation can be determined between dimensions based on the data distribution}. In this way, $f_{i+1}$ can be generated by changing the measure field $m_i$ of $f_i$ to another numerical field in the spreadsheet that is most likely caused by $m_i$ in accordance with causal analysis~\\cite{schaechtle2013multi}.\n    \\item {\\textit{\\textbf{Elaboration}}} indicates a relation in which a latter fact $f_{i+1}$ adds more details to the previous one $f_i$. In this way, we create $f_{i+1}$ by shrinking the scope of $f_i$'s subspace via adding more constraints (i.e., filters) or setting a focus to ``zoom\" $f_i$ into a more specific scope. \n    \\item {\\textit{\\textbf{Generalization}}} indicates $f_{i+1}$ is an abstraction of the previous $f_i$, which is in opposite to elaboration. Therefore, we create $f_{i+1}$ by enlarging the scope of $f_i$'s subspace via removing constraints.\n\\end{itemize}\n\n\\paragraph{\\bf Reward Function} We propose a reward function that estimates the quality of each generated story $\\mathcal{S}$ via three criteria, i.e., diversity $D(\\mathcal{S})$, logicality $L(\\mathcal{S})$, and integrity (i.e, data coverage) $C(\\mathcal{S})$ based on the story's information entropy $H(\\mathcal{S})$:\n\\begin{equation}\nreward(\\mathcal{S}) = \\{\\gamma_1 \\cdot D(\\mathcal{S}) + \\gamma_2 \\cdot L(\\mathcal{S}) + \\gamma_3 \\cdot C(\\mathcal{S})\\} \\cdot H(\\mathcal{S})\n\\label{eq:reward}\n\\end{equation}\nwhere $\\gamma_i \\in [0,1], \\sum_i \\gamma_i = 1$ are the weighting parameters given by users to balance different criteria. All the criteria are also normalized to $[0,1]$. $H(\\mathcal{S})$ is the story's information entropy that indicates the expected self-information of the data facts in the story, which is used as the basis of the reward and formally defined as follows:\n\\begin{equation}\n    H(\\mathcal{S})= \\sum_{i=1}^{n} { P(f_i) \\cdot I_s(f_i) } = -\\sum_{i=1}^{m} P(f_i) \\cdot S(f_i) \\cdot \\log _{2}\\left(P(f_i)\\right)\n\\end{equation}\nwhere, $I_s(f_i)$ is the fact's importance score defined in Formula (\\ref{eq:importance}). The definition of each story estimation criterion is described as follows:\n\n\\begin{itemize}[leftmargin= 10pt]\n\\itemsep -.5mm\n\\item \\textbf{\\textit{Diversity}} estimates the variance of the fact types in $\\mathcal{S}$. Rich fact types will make the story vivid and attractive. Diversity is given by two terms as follows:\n\\begin{equation}\nD(\\mathcal{S}) = \\frac{n}{\\min(|\\mathcal{S}|,10)} \\cdot \\frac{-\\sum_{i=1}^{n} {p}_{i} \\cdot \\ln \\left({p}_{i}\\right)}{\\ln (n)} \n\\end{equation}\nwhere $n$ indicates the total number of fact types used in $\\mathcal{S}$ whose maximum value is known as 10; $p_i$ is the proportion of the $i$-th fact type in the story. When $D(\\mathcal{S})$ is maximized, the first term encourages to use more fact types in a story, and the second term, a normalized Shannon diversity index~\\cite{ramezani2012note}, ensures that different fact types can be evenly used in the story.\n\n\\item \\textbf{\\textit{Logicality}} estimates the logical coherence of a story. A higher logicality score indicates the story is more coherent and easier to follow. Logicality is defined by the averaged likelihood of each coherent relation $r_i$ occurred after each fact $f_i$ in the story: \n\n\\begin{equation}\nL(S) = \\frac{1}{n-1}\\sum_{r_i,f_i \\in S}P\\left(r_{i} | f_{i}\\right)\n\\end{equation}\n\n\\item \\textbf{\\textit{Integrity}} is the data coverage rate of $\\mathcal{S}$. A larger integrity, indicates the story more comprehensively represents the input data. Integrity is defined as:\n\\begin{equation}\nC(\\mathcal{S}) = \\frac{count(\\bigcup\\limits_{i=0}^{n-1} f_{i})}{\\mathcal{N}}\n\\end{equation}\nwhere the molecule is the total number of data items in the spreadsheet used in the story, and $\\mathcal{N}$ is the total number of data items.\n\\end{itemize}\n\\section{Story Generation Engine}\n\\label{sec:engine}\nIn this section, we first formally define a data fact and its importance measurement. We then describe the details of the proposed automatic story generation algorithm.\n\n\\input{tables\/3-datafact}\n\n\\subsection{Data Facts}\n\\label{sec:fact}\nData facts are the elementary building blocks of a data story. Each of these facts represents a piece of information extracted from the data. \\rv{We first give a formal definition of data facts by simplifying the concepts introduced in~\\cite{wang2019datashot,chen2009toward} to guarantee a clear semantic and then introduce a novel method used to estimate the importance of a given data fact.}\n\n\\paragraph{\\bf Definition} A data fact is designed to measure a collection of data items in a subspace of an input dataset based on a measurable data field. The data items can be further divided into groups via a breakdown method. Formally, a fact, $f_i \\in F$, is defined by a 5-tuple:\n\\[\n\\begin{aligned}\n    f_i &= \\{type, subspace, breakdown, measure, focus\\}\\\\\n    &= \\{t_i, s_i, b_i, m_i, x_i\\}\n\\end{aligned}\n\\]\n\\rv{where \\textit{\\textbf{type}}\\xspace (denoted as $t_i$) indicates the type of information described by the fact. As summarized in Table~\\ref{tab:fact}, Calliope\\xspace includes 10 fact types;}\n\\textit{\\textbf{subspace}}\\xspace (denoted as $s_i$) describes the data scope of the fact, which is defined by a set of data filters in the following form: \n\\[\n\\{\\{\\mathcal{F}_1 = \\mathcal{V}_1\\},\\cdots, \\{\\mathcal{F}_k = \\mathcal{V}_k\\}\\}\n\\]\nwhere $\\mathcal{F}_i$ and $\\mathcal{V}_i$ respectively indicate a data field and its corresponding value selected to filter the data. By default, the subspace is the entire dataset. \\textit{\\textbf{breakdown}}\\xspace (denote as $b_i$) is a set of temporal or categorical data fields based on which the data items in the subspace are further divided into groups; \\textit{\\textbf{measure}}\\xspace (denote as $m_i$) is a numerical data field based on which we can retrieve a data value or calculate a derived value, such as count, sum, average, minimum, or maximum, by aggregating the subspace or each data group; \\textit{\\textbf{focus}}\\xspace (denote as $x_i$) indicates a set of specific data items in the subspace that require attention. Despite the above five fields, certain facts may also have a \\textit{\\textbf{derived value}} (denoted as $V_d$) such as a textual summary of the trend (i.e., ``increasing\" or ``decreasing\") or the specific difference value between two cases described by a difference fact, or the correlation coefficient computed for an association fact as shown in Table~\\ref{tab:fact}. These values help with a more insightful description of the fact.\n\n\\rv{When compared to the concepts introduced in~\\cite{wang2019datashot,chen2009toward}, the above definition simplified and restricted the fact fields to ensure a clear semantic expression of the data that avoids redundancy, overlaps, and ambiguity. Specifically, when compared to \\cite{wang2019datashot}, we removed the fact fields that are irrelevant to the fact semantics  and treated ``aggregation\" as an operation on ``measures\" instead of a fact type to avoid duplicated fact definitions. In addition, as summarized in Table~\\ref{tab:fact}, we add constraints on each fact field to ensure a clear semantics. For example, the facts in ``distribution\" and ``trend\" types are designed to capture the data patterns given by the measures of different data groups in the subspace. Both fact types can be differentiated by their ways of breaking down a subspace: the subspace in a ``trend\" fact must be divided by a temporal field, whereas the subspace in a ``distribution\" fact can only be divided by a categorical field. Thus, each fact can be described by a syntax which is used for generating a textual description of the fact.}\n\nTo understand the above concepts, let's consider the following examples. Given a dataset about the COVID-19 virus outbreak in China, the data fact, {\\it \\{``distribution\", \\{\\{Country =``China\"\\}\\}, \\{Province\\}, \\{sum(Infections)\\}, \\{Province=``Hubei\"\\}\\}}, describes ``the distribution of the \\underline{total number of} \\underline{infections} over all \\underline{provinces} when \\underline{the country is China} (subspace) and \\underline{Hubei} needs to pay attention\" regarding to the syntax of the distribution fact. Similarly, the data fact, {\\it \\{``trend\", \\{\\{Province =``Hubei\"\\}\\}, \\{Date\\}, \\{sum(Infections)\\}, \\{Date=``2020-1-24\"\\}\\}}, indicates ``the changing trend of the \\underline{total number of} \\underline{infections} over different \\underline{dates} when \\underline{province is Hubei} and the values of \\underline{2020-1-24} need to pay attention\".\n\n\n\n\n\n\\paragraph{\\bf Importance Score} We estimate the importance of a data fact $f_i = \\{t_i,s_i,b_i,m_i,x_i\\} \\in F$ based on its \\textit{self-information} (denoted as $I(f_i)$) weighted by its \\textit{pattern significance} (i.e., $S(f_i)\\in [0,1]$) as follows:\n\\begin{equation}\n    I_s(f_i) = S(f_i) \\cdot I(f_i)\n\\label{eq:importance}\n\\end{equation}\n\nIn particular, $I(f_i) \\in [0, \\infty]$ is defined based on the information theory and can be measured in ``bit\" using the following formula:\n\\begin{equation}\nI(f_i) = -log_2(P(f_i))\n\\label{eq:info}\n\\end{equation}\n\\rv{where $P(f_i)$ indicates the occurrence probability of the fact given the input data. A data fact with a lower occurrence probability in the data space has a higher self-information value as it reveals uncommon patterns which are usually more meaningful and interesting.} $P(f_i)$ is formally determined by the occurrence probability of the fact's  \\textit{\\textbf{subspace}}\\xspace $s_i$, the probabilities of selecting $x_i$ as the \\textit{\\textbf{focus}}\\xspace in $s_i$, and the probabilities of choosing $m_i$ ($P(m_i|t_i)$) and $b_i$ ($P(b_i|t_i)$) to measure and break down $s_i$ given a fact type $t_i$: \n\\begin{equation}\nP(f_i) = P(m_i|t_i) \\cdot P(b_i|t_i) \\cdot P(s_i) \\cdot P(x_i | s_i)\n\\label{eq:pfi}\n\\end{equation}\nwhere $P(m_i|t_i)$ and $P(b_i|t_i)$ is defined regarding the data type constraints of $m_i$ and $b_i$ as summarized in Table~\\ref{tab:fact}. For example, when the fact type is ``Value\", $P(m_i|Value)$ is $1\/N$, where $N$ is the total number of numerical fields in the data. Similarly, $P(b_i|Difference)$ is $1\/(C + T)$ where $C,T$ are the total number of categorical and temporal fields in the data. Moreover, in Formula (\\ref{eq:pfi}), $P(x_i | s_i)$ is defined as the proportion of the focused data items in the subspace, i.e., $P(x_i | s_i) = count(x_i) \/ count(s_i)$, with the assumption that the probability of selecting each data item as a focus in the subspace is equivalent. In our design, all the data items in $s_i$ are focused by default, i.e., $P(x_i | s_i) = 1$ when the \\textit{\\textbf{focus}}\\xspace field is unspecified. To calculate $P(s_i)$, we first assume $s_i$ consists of $k$ data filters, i.e., $\\{\\{\\mathcal{F}_1 = \\mathcal{V}_1\\},\\cdots, \\{\\mathcal{F}_k = \\mathcal{V}_k\\}\\}$ and there are $m$ independent data fields that can be used for formulating a subspace. In this way, $P(s_i)$ is defined as follows:\n\\begin{equation}\nP(s_i) = \\frac{1}{\\sum_{i=0}^{m} C(m, i)}\\cdot\\prod_{j=1}^{k}P(\\mathcal{F}_j = \\mathcal{V}_j)\n\\label{eq:p_si}\n\\end{equation}\nwhere the first term indicates the probability of choosing the fields $\\mathcal{F}_1,\\cdots, \\mathcal{F}_k$ from the input data to formulate the subspace $s_i$. $C(m, i)$ is an $i$-combination over a set of $m$ possible data fields. $\\sum_{i=0}^{m} C(m, i)$ summarizes the number of all possible cases for formulating a subspace. In this way, the first term in Formula (\\ref{eq:p_si}) indicates the method we use for formulating the current subspace $s_i$ is just one possible case. The second term in Formula (\\ref{eq:p_si}) indicates the probability of using the corresponding values $\\mathcal{V}_1,\\cdots, \\mathcal{V}_k$ on the selected fields to filter the data. This probability is directly given by the products of the proportions of the data that satisfy each filter conditions, i.e.,  $\\{\\mathcal{F}_j = \\mathcal{V}_j\\}$.\n\nIn Formula (\\ref{eq:importance}), $S(f_i) \\in [0, 1]$ estimates the significance of the data patterns described by the fact $f_i$, which is calculated based on auto-insight techniques~\\cite{ding2019quickinsights, tang2017extracting, wang2019datashot}. The detailed methods are described in the supplemental material. \\rv{It worth mentioning that a significant pattern may not necessary have a high self-information value. Only using both measurements as shown in Formula (\\ref{eq:importance}) will guarantee a comprehensive estimation. Under this definition, the importance of a fact is only determined by its data content and irrelevant to how frequently a type of fact is used in data stories (Fig.~\\ref{fig:statistic}(a)).}\n\n\n\\input{sections\/4-2-algorithm}\n\\section{Data Story Editor}\n\\label{sec:editor}\n\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=\\linewidth]{figures\/UI.png}\n  \\vspace{-1.5em}\n  \\caption{The story editor of Calliope\\xspace system consists of three major views: (1) the storyline view for story configuration, generation, and storyline editing, (2) the fact view for fact editing, and (3) the story visualization view for the visual data story preview and sharing \\rv{(\\textit{\\textbf{factsheet}}\\xspace mode)}. \n    } \n  \\label{fig:ui}\n  \\vspace{-1em}\n\\end{figure}\n\\setlength{\\textfloatsep}{20pt}\n\n\nIn this section, we introduce the design of the story editor and the methods used for visualizing a data story.\n\n\\subsection{User Interface}\nThe story editor, as shown in Fig.~\\ref{fig:ui}, consists of three major views: the storyline view (Fig.~\\ref{fig:ui}-1), fact view (Fig.~\\ref{fig:ui}-2), and story visualization view (Fig.~\\ref{fig:ui}-3). In the storyline view, a user can upload a spreadsheet, set the story generation goal, and adjust the reward function in a group of configuration panels (Fig.~\\ref{fig:ui}-1(a)). The generated data facts are shown in a row (Fig.~\\ref{fig:ui}-1(b)), in which a user can remove a fact or change the generated narrative order based on his\/her own preferences. Each fact is visualized by a chart and captioned by a generated text description (\\textbf{R3}). When a fact is selected, the data details on each of its fields, importance scores, and visual and textual representations, will be shown in the fact view (\\textbf{R4}). \\rv{The generated data story can be visualized in the story visualization view through three visualization modes: (1) \\textit{\\textbf{storyline}}\\xspace mode (Fig.~\\ref{fig:teaser}), (2) \\textit{\\textbf{swiper}}\\xspace mode (Fig.~\\ref{fig:casestudy}(a)), and (3) \\textit{\\textbf{factsheet}}\\xspace mode (Fig.~\\ref{fig:casestudy}(b)). These modes are respectively designed for representing the story on laptops\/tablets, smartphones, and printouts to facilitate a flexible story communication and sharing (\\textbf{R5}). A user can easily switch between different \nmodes in the story visualization view \nvia a drop down menu.}\n\n\n\\subsection{Visualizing a Data Story}\nA data story generated by the engine is visualized as a sequence of charts through two steps: \\textit{showing a data fact} and \\textit{showing a story}. The first step maps a data fact to a chart while the second step organizes a sequence of charts in an expressive layout as a story.\n\n\\paragraph{\\bf Showing a Data Fact} Benefiting from the simple and clear definition of each fact type introduced in Section~\\ref{sec:fact}, Calliope\\xspace is able to directly convert a data fact into a captioned chart that incorporates both the visual and textual representations. Specifically, the caption is generated based on the syntax defined in Table~\\ref{tab:fact}, and the fact is automatically visualized in two steps by following a rule-based approach. In particular, the system first selects the most frequently used chart regarding the fact type in Table~\\ref{tab:charts}. After that, it selects a subset of data from the input spreadsheet regarding the filters given by the \\textit{\\textbf{subspace}}\\xspace field and then maps the \\textit{\\textbf{breakdown}}\\xspace field(s) to the categorical channel(s) and the \\textit{\\textbf{measure}}\\xspace field(s) to the numerical channel(s) in the chart. Finally, the data values indicated by the \\textit{\\textbf{focus}}\\xspace field are highlighted in the chart. Fig.~\\ref{fig:ui}(2) illustrates an example of showing a difference fact in a captioned bar chart.\n\n\n\\begin{figure*}[htb]\n  \\centering\n  \\includegraphics[width=0.86\\linewidth]{figures\/case23.png}\n  \\vspace{-0.5em}\n  \\caption{Two data story examples generated by Calliope\\xspace: (a) a story about car selling records around economic crisis in 2008 shown in the \\textit{\\textbf{swiper}}\\xspace mode and (b) a story about startup failures after the tide of ``new economics\" in China shown in the \\textit{\\textbf{factsheet}}\\xspace mode.} \n  \\label{fig:casestudy}\n  \\vspace{-1em}\n\\end{figure*}\n\n\\paragraph{\\bf Showing a Story} The story visualization view provides a variety of visualization modes to represent the generated data story for different application scenarios. In particular, a summarization is first provided to give a textual briefing of the story to help users obtain data insights at a glance. This step shows the data coverage rate, the total number of data facts in the story, and the generated textual narrative of the story. The \\textit{\\textbf{storyline}}\\xspace and \\textit{\\textbf{swiper}}\\xspace visualization modes are respectively designed to facilitate an efficient story exploration on tablets and smartphones. \\rv{In particular, the captioned charts showing data facts in the story are horizontally aligned in a row to represent the narrative order in the \\textit{\\textbf{storyline}}\\xspace mode} and are shown one at a time in the \\textit{\\textbf{swiper}}\\xspace mode. A user can swipe the touch screen to explore the story through these representations. Finally, the \\textit{\\textbf{factsheet}}\\xspace mode is designed to show the story in the form of a poster that could be easily printed out.\n\n\n\\section{Evaluation}\nWe evaluate Calliope\\xspace system via (1) three examples to showcase the quality of the generated story, (2) two controlled experiments to estimate the usefulness and quality of a generated logic, and (3) domain expert interviews to estimate the usability of the system.\n\n\\subsection{Example Stories}\nWe demonstrate three visual data stories generated by Calliope\\xspace respectively based on three real-world datasets as shown in Fig.~\\ref{fig:teaser} and Fig.~\\ref{fig:casestudy}, which are described as follows:\n\nFig.~\\ref{fig:teaser} shows a story generated based on a \\textit{\\textbf{COVID-19}}\\xspace dataset (903 rows, 5 columns). The data record the recent numbers of daily infections, deaths, and healings of COVID-19 in China from March 1st to March 21st. The generated story illustrates the daily mortality in China decreased in March ({\\it Fact 1}), and the largest number was 42 occurred on March 2nd ({\\it Fact 2}). Hubei was the most affected province ({\\it Fact 3}). The total death in Hubei accounted for 97.4\\% of that in China ({\\it Fact 4}), which was 423 ({\\it Fact 5}). A large number of patients recovered in March ({\\it Fact 6}), showing the improving situation in China.\n\nFig.~\\ref{fig:casestudy}(a) shows a story generated based on a \\textit{\\textbf{Car Sales}}\\xspace dataset (275 rows, 4 columns), including the sales records of different automobile brands around the financial crisis in 2008. The story shows that in 2007-2011, 21,921,768 cars were sold in total ({\\it Fact 1}). The top three sellers were Ford, Toyota, and Honda ({\\it Fact 2}). The difference was huge when comparing the best and worst sellers ({\\it Fact 3}). Specifically, SUV was the best selling model ({\\it Fact 4}) which sold 6,764,065 more than MPV ({\\it Fact 5}). Generally, the sales records decreased during the final financial crisis ({\\it Fact 6}).\n\nFig.~\\ref{fig:casestudy}(b) presents a dataset about \\textit{\\textbf{Startup Failures}}\\xspace . The data (1234 rows, 6 columns) record a set of companies closed during or after the raising tide of ``new economics\" in China from 2010 to 2019. Each startup company is described from six criteria, including its broken year, location, industry, funded status, survival time, and the main cause of failure. The story shows that numerous companies were closed in recent years ({\\it Fact 1}), and most of them were located in Eastern China ({\\it Fact 2}). The most dangerous fields were e-commerce, social media, and local business ({\\it Fact 3}). Most companies closed in these fields were still in the early stages before the series A+ round ({\\it Fact 4}), and some even closed without receiving any investment ({\\it Fact 5}). Regarding the reasons, ``no business model\" and ``no market need\" were the most frequently occurring problems in these startup companies ({\\it Fact 6}).\n\n\n\\subsection{Evaluation of the Generated Logic}\nWe verified the usefulness and the quality of the generated logic in a story via two controlled experiments.\n\n\\paragraph{\\bf Experiment I: Usefulness}\nWe first estimated whether the logic generated by Calliope\\xspace helps with the understanding of a data story. To this end, we compared our generation results with the factsheets generated by DataShot~\\cite{wang2019datashot}, in which a set of selected data facts was randomly organized. \n\n{\\underline{\\textit{Data.}}}\n\\rv{We collected the same datasets illustrated in Fig. 4 (C, D) in DataShot, including CarSales and SummerOlympics. Based on these two datasets, we generated two factsheets using Calliope automatically and picked two cases from the DataShot paper directly as the baseline. To ensure a more fair study setting, we made sure these generated stories from different systems contained similar data facts, and we also revised the design of each factsheet with the same style.}\n\n{\\underline{\\textit{Procedure and Tasks.}}}\n\\rv{We recruited 16 college students (12 females) ages 22 - 26 years old (M=23.94, SD=1.43) as participants. Each participant was presented with two factsheets of different topic from our data, one by Calliope and one by DataShot.\nWe counterbalanced the presentation order of the factsheets for a fair comparison.\nThe participants were asked to read the factsheets carefully and compare two factsheets from five specific aspects including logicality, comprehension, memorability, engagement, and dissemination. The experiment lasted approximately 40 minutes for each participant.}\n\n{\\underline{\\textit{Results.}}}\n\\rv{In terms of \\textit{Logicality} and \\textit{Memorability}, Calliope received more positive feedback than DataShot. One participant commented that ``\\textit{I can smoothly follow it's (Calliope) logic from whole to part, as it first introduces the overall information about Olympic golds and then zooms in on specific sports and countries.}\" Another participant said, ``\\textit{it's much easier to remember the story generated by Calliope, as the annual car sales in different brands is presented step by step in a proper order}\". \nRegarding \\textit{Comprehension}, \\textit{Engagement}, and \\textit{Dissemination}, Calliope performs comparably to Datashot. One participant said, ``\\textit{I enjoy the \nsimple and beautiful visualization of both factsheets and would love to share them on social media if the data is relevant.}\"} \n\n\n\n\n\n\\paragraph{\\bf Experiment II : Quality} \nThe second experiment was designed to evaluate the quality of the generated logic. To this end, we objectively estimated consistency of the logic orders respectively given by users and generated by Calliope\\xspace based on the same set of data facts.\n\n\\underline{\\textit{Procedure and Tasks.}} We first shuffled the order of a set of data facts in a visual data story generated by Calliope\\xspace and then asked a group of users to restore the logic order by reading the chart and description of each fact. Finally, we checked the consistency between the human-generated logic orders and those produced by Calliope\\xspace based on Kendall's $\\tau_b$ correlation~\\cite{kendall1945treatment}. \n\\rv{This measure was introduced to estimate the consistency of the element orders between two sequences, whose value lies in $[-1, 1]$ with ``-1\" indicating a completely reverse order and ``1\" indicating the orders are identical}. To ensure a fair and comprehensive comparison, we generated 12 data stories based on the aforementioned three datasets, four stories per dataset. Each story contained six data facts whose orders were shuffled for the experiment. \n\nA group of 20 participants (17 female) aged 22-30 years old ($M=26, SD=2.63$) were recruited for Experiment II. All of the participants reported that they have fundamental knowledge about data visualization or experience in data-oriented storytelling. The experiments started by a brief introduction about the data, and the participants were asked to reorder the data facts of all the aforementioned 12 shuffled stories via an interactive user interface. \nWe also encouraged the participants to fully explore the data and try their best to understand the data insights represented by each data fact.\n\n{\\underline{\\textit{Results.}}}\n\\rv{We calculated the average Kendall's $\\tau_b$ value on each dataset, and the result showed that the logic orders generated by Calliope\\xspace were consistent with those generated by our participants (\\textit{\\textbf{Car Sales}}\\xspace: $\\mu=0.487, \\sigma=0.29$; \\textit{\\textbf{COVID-19}}\\xspace: $\\mu=0.648, \\sigma=0.327$; \\textit{\\textbf{Startup Failures}}\\xspace: $\\mu=0.63, \\sigma=0.295$). \nWe also leveraged the sequences generated by humans as a ground truth and calculated Kendall's $\\tau_b$ values between the random results and human generated results as the baseline. As a result, the baseline is 0.015, which indicates these two orders are relatively irrelevant (as the value is close to 0). \nBy comparing the $\\tau_b$ values, we found that the logic orders generated by Calliope are much more consistent to humans than the baseline.}\n\n\\begin{figure}[b]\n  \\centering\n  \\includegraphics[width=0.9\\linewidth]{figures\/interview.png}\n  \\vspace{-0.5em}\n  \\caption{The ratings of generated story, visualization design, and system from different criteria based on a 5-point Likert scale given by 10 expert users, where 5 is the best and 1 is the worst.}\n  \\label{fig:questionare}\n  \\vspace{-0.5em}\n\\end{figure}\n\n\\subsection{Expert Interview}\nTo further evaluate the usability of Calliope\\xspace system, a series of interviews with three groups of expert users from different areas were performed. The first group included four data journalists (denoted as J1-J4) from different news medias in China. They had over 3.5 years' working experiences on average, and were very familiar with the data-oriented storytelling and had technical skills for creating a visual data story. The second group comprised three data analysts (denoted as D1-D3) from an international IT consultant company. Their major job was to analyze customers' data and write analysis reports. BI tools such as Tableau and PowerBI were frequently used in their daily work. The third group consisted of three senior visualization researchers (denoted as V1-V3), all of whom had experiences on publishing papers in major visualization conferences such as IEEE VIS and EuroVis. \n\n\\paragraph{\\bf Procedure} The interviews were performed via an online meeting system. Each interview started by a 10-minutes introduction about the system. After that, the experts were encouraged to use the online Calliope\\xspace system on their own. After fully explored the functions of the system, the experts were asked to generate a data story based on one of the three datasets introduced in Experiment II. To arouse their interests, we let the journalists explore the \\textit{\\textbf{COVID-19}}\\xspace dataset as it is a recent and important news topic. We let the data analysts use the \\textit{\\textbf{Car Sales}}\\xspace data given its similarity to the business data analyzed in their work. We let the visualization researchers explore the \\textit{\\textbf{Startup Failures}}\\xspace data as it is the most complex one. They were also encouraged to edit the generated story and share their findings with us. The experts were asked to finish a questionnaire, followed by an interview after they fully explored the functionalities of Calliope\\xspace system. The whole process lasted for about one hour and the interviews were recorded for later analysis.\n\n\\paragraph{\\bf Results} \nFig.~\\ref{fig:questionare} shows that questionnaire results, Calliope\\xspace was rated relatively high in terms of the generated story, visualization, and system.\nA large number of positive comments were recorded during the interview. We summarized the interview results due to the page limitation.\n\n\\textit{\\underline{Data Story.}} \nAll the experts agreed that the generated data story was able to express useful data insights. The visualization researcher V2 mentioned that ``\\textit{the data facts in the story are quite clear and are well-organized}\". J1 commented that ``\\textit{the story starts from an overview, followed by a series of data details, ..., It's the way we frequently adopted when writing a news story. It's amazing that now it can be automatically generated}\". J3 also noted that \\textit{``the logical order [of the story] can help readers get into the points\"}. D1 observed that the ordered data facts can help with the efficient exploration of the data,  \\textit{``[Automatically] showing the appropriate dimensions and measures in a sequence of visualizations can definitely guide the data exploration, ..., it's helpful when you have no idea about how to get started\"}.\n\n\n\\textit{\\underline{Visualization.}}\nIn terms of visualization, all the experts were satisfied with the design of three visualization modes in Calliope\\xspace, \\textit{``it's a very nice and thoughtful design\"} (J1-J4). Especially, they felt that the \\textit{\\textbf{storyline}}\\xspace mode provides a good overview (J1-J3, D1, D3, V1, V2). All the experts believed that the \\textit{\\textbf{swiper}}\\xspace mode was neat and helpful when viewing the story on a smartphone while the \\textit{\\textbf{factsheet}}\\xspace mode supports easy printing of a story. \nAll the journalists felt that editing was an important function, \\textit{``it (editing function) allows us to create a high-quality story quickly based on the generation results\"} (J1). They also felt generating stories by interactively changing the reward was \\textit{``interesting\"} and \\textit{``inspiring\"}. J2 suggested that \\textit{``we usually write stories from different perspectives, and it can facilitate my ideation process\"}.\n\n\\textit{\\underline{System.}}\nMost experts (J1-J3, D1, D3, V2, V3) mentioned that the system was useful for users who are not skilled at data analysis or visualization. The data journalists J1-J4 especially appreciated the efficient story generation and editing function of Calliope\\xspace. J4 said, \\textit{``with this tool I can quickly create a story by first generating a draft and then revising it accordingly\"}. The data analysts felt the system is powerful to help them efficiently preview an unknown data, \\textit{``with this tool, I can quickly find where to start when getting a [new] spreadsheet\"} (D1). The visualization researchers believed that our system lowers the barriers of creating a data story. V3 said, ``when compared with other data story authoring tools, this system is much more smart as it requires limited knowledge about data analysis and visualization design\". \n\n\\section{\\bf Limitations and Future Work}\n\\rv{Despite the above positive results from the evaluation, we would also like to summarize and discuss several limitations that was found during the design and implementation process and mentioned by our expert users during the interview. \nWe hope highlight these limitations will help point out several potential future research directions and inspire new studies by following our work.\n}\n\n\\rv{\\underline{\\textit{Supporting a Better Textual Narrative.}} During the interview, many data journalists (J1,J2,J4) felt the generated captions were too rigid to be used especially in a data news. More diverse and insightful descriptions were desired. Moreover, the current results also contained some grammar errors, which also need to be addressed (J1,2 D1, V2,3)). However, all of them acknowledged that the current results, although unsatisfactory, were still useful for a rapid preview and briefing of the input data. In the future, it is necessary to leverage more advanced techniques introduced in the field of nature language processing to generate text narratives in the data story in a higher quality.}\n\n\\underline{\\textit{Understanding Data Semantics.}} After using Calliope\\xspace, although impressed, some experts (J1-J3, V2) expect a more intelligent tool that can even understand the semantics of the data to better generate the story content and logic. \\rv{We acknowledge this is a key limitation of the current system and understanding the underlying semantics of the data is critical for generating a meaningful and insightful data story. This is a promising research direction that is worth a further exploration. To address the issue, one could leverage or develop more advanced AI techniques or could also introduce sophisticated interactive feedback mechanism to keep user in the generation loop and leverage human intelligence to \nsteer data quality~\\cite{liu2018steering} and guide the underlying generation algorithm\/model to better understand data semantics.}\n\n\\underline{\\textit{Enriching Visualization.}} Several experts (J3, D1, D2) would like to have a slides mode and dashboard mode to support more application cases. J1 and V3 also pointed out that some of the current generated visual encoding are notably simple, and a chart should encode more information at a time. For example, when showing a line chart, the size of point could also be used to encode data, and a stack bar chart could be used to show an additional categorical field in the data. \\rv{In addition, Calliope\\xspace cannot deal with hierarchical or relational datasets, which are also desired functions (V1,V2). Providing more advanced visualization representations for a story is also a valuable future work.}\n\n\\underline{\\textit{Performance Issues.}}\nThe current system design and implementation have some performance bottlenecks that worth a future study.  \\rv{In particular, the calculation of data fact significance, i.e., $S(f_i)$ in Formula (\\ref{eq:importance}) consists of statistical computations, which is usually slow and thus limits the number of facts that can be explored in each searching iteration, thus affecting the generation quality.}\n\\section{Conclusion}\n\nWe have presented Calliope\\xspace, a novel system designed for automatically generating visual data stories from a spreadsheet. The system incorporates a novel logic-oriented Monte Carlo tree search algorithm to create a data story by gradually generating a sequence of data facts in a logical order while exploring the data space. The importance of a fact is measured by its information quantity and its statistical significance. Each fact in the story is visualized in a chart with an automatically generated caption. A story editor is introduced in the system to facilitate the easy and efficient editing of a generated story. The proposed technique was evaluated via three example cases, two controlled experiments, and a series of interviews with 10 expert users from different domains. The evaluation showed the power of Calliope\\xspace system and revealed several limitations of the current system, which will be addressed in the future.\n\n\n\\section{Fact Significance}\n\nFact significance is calculated based on the former auto-insights techniques \\cite{tang2017extracting, ding2019quickinsights, wang2019datashot}. \nAs the definition in the references, the significance measure reveals the uncommonness of an observed insight in the result set \\cite{tang2017extracting}.\nA meaningful fact should exhibit significant differences against a baseline which reflects common situations formed up by majority of non-insights \\cite{ding2019quickinsights}.\nIn the following section, we state the calculation methods for each fact type using the same statements in the these references. \n\n\n\\paragraph{\\bf Value} DataShot states ``Some data facts (e.g., Value, Aggregation) only derive values from the data simply assign it to zero. We simply assign them to zero.\"\\cite{wang2019datashot}. In our implementation, we use the probability of the fact to calculate the significant score.\n\n\\paragraph{\\bf Difference}\nA significant difference between two $focus$ corresponds to a high score for Difference. \nThe significance is equal to 1 when the difference is maximum.\n\n\\paragraph{\\bf Proportion}\nWe follow the definition of \\underline{Proportion} in DataShot\\cite{wang2019datashot} and \\underline{Attribution} in QuickInsights\\cite{ding2019quickinsights}.\nDataShot states ``A high proportion corresponds to a high score for Proportion\".\nQuickInsights states ``Attribution shows the fact that the leading value dominates (accounting for >= 50\\% of) the group\".\nThus we directly use the proportion as the significance of proportion. \nIn addition, if the proportion is larger than 50\\%, we set the significance as 1 because the focus part dominates the group.\n\n\\paragraph{\\bf Trend}\nWe follow the definition of \\underline{Trend} in DataShot and \\underline{Shape Insight} in Top-K Insights\\cite{tang2017extracting}.\nDataShot states ``A sharp increase corresponds to a high score for Trend\".\nTop-K Insights states ``In business intelligence applications, data analysts are attracted to a clear rising\/falling trend, whose slope is very different from 0\".\nThe following is the method mentioned in the Top-K Insights:\n\nWe set the null hypothesis as $X$ forms a shape with a slope near 0.\nThus, the p-value should measure how surprisingly the slope differs from 0.\n\n1. First, we fit $X$ to a line by linear regression analysis, and then compute its slope $slope$ and the goodness-of-fit value $r^{2}$.\n\n2. we use logistic distribution to model the distributions of slopes.\n\n3. The p-value is the probability of the slope values equal to or larger than the observed slope of the rising trend.\n\n4. Finally, we define the significance as $S(f)=r^2\\cdot (1-p)$, where the goodness-of-fit value $r^2$ is used as a weight.\n\n\\paragraph{\\bf Categorization}\nWe follow the definition of \\underline{Evenness} in QuickInsights\\cite{ding2019quickinsights}.\nQuickInsights states that Evenness shows that the cases where all value of a measure for a given category should be close to each other. The following is the method:\n\n1. Perform the CHI square test for the hypothesis: the counts in each category are equal.\n\n2. We obtain the significance as $S(f)=1-p$.\n\n\\paragraph{\\bf Distribution}\nThe significance of distribution should reveal how surprisingly $X$ differs from the Gaussian distribution, which is the common distribution in the natuaral world.\nThe following is the method:\n\nWe set the null hypothesis $X$ as a power-law distribution with Gaussian noises.\n\n1. Perform the Shapiro-Wilk test for $X$.\n\n2. We obtain the significance as $S(f)=1-p$.\n\n\\paragraph{\\bf Rank}\nWe follow the definition of \\underline{Point Insight} in Top-K Insights\\cite{tang2017extracting}.\nTop-K Insights states ``In the business domain, the sale of products often follows a power-law distribution\".\nThe following is the method mentioned in Top-K Insights:\n\nWe set the null hypothesis $X$ as a power-law distribution with Gaussian noises.\n\n1. First, we sort $X$ in the descending order and obtain the maximum value $x_{max}$.\n\n2. Then, we fit the values in $X$ to a power-law distribution if it is good fit, where the prediction errors (i.e., subtracting observed value $\\hat{x_i}$ from estimated value $x_i$ , also called residuals) approximately follow Gaussian distribution.\n\n3. Next, we determine how surprising it is that $X$ observed against the hypothesis and calculate the p-value $p$.\n\n4. Finally, we obtain the significance as $S(f)=1-p$.\n\n\\paragraph{\\bf Association}\nWe follow the definition of \\underline{Correlation} in QuickInsights\\cite{ding2019quickinsights}.\nThe following is the method mentioned in the QuickInsights:\n\nThe significance of two $measure$ is defined based on testing using Student's t-distribution with Pearson's correlation coefficient $r$.\n\n1. Specify the null and alternative hypotheses.\n\n2. Calculate the value of test statistic: $t=r \\sqrt{\\frac{n-2}{1-r^{2}}}$.\n\n3. Use the resulting test statistic $t$ to calculate the p-value, which is determined by referring to a t-distribution with n-2 degrees of freedom.\n\n4. The p-value is translated into significance. The lower the p-value, the higher the significance.\n\n\\paragraph{\\bf Extreme}\nWe follow the definition of \\underline{Outstanding No.1} and \\underline{Outstanding Last} in QuickInsights\\cite{ding2019quickinsights}.\nThe following is the method mentioned in the QuickInsights:\n\nTake Outstanding No. 1 as an example.\nGiven a group of non-negative numerical values {x} and their biggest value $x_{max}$ , the significance of $x_{max}$ being Outstanding No.1 of $X$ is defined based on the p-value against the null hypothesis of $X$ obeys an ordinary long-tail distribution.\n\n1. We sort $X$ in descending order;\n\n2. We assume the long-tail shape obeys a power-law function. Then we conduct regression analysis for the values in $X$. $x_{max}$ using power-law functions, where i is an order index and in our current implementation we fix $\\beta = 0.7$ in the power-law fitting;\n\n3. We assume the regression residuals obey a Gaussian distribution. Then we use the residuals in the preceding regression analysis to train a Gaussian model $H$;\n\n4. We use the regression model to predict $x_{max}$ and get the corresponding residual $R$;\n\n5. The p-value will be calculated via $P(R|H)$.\n\n\\paragraph{\\bf Outlier} We follow the definition of \\underline{Outlier} in DataShot and QuickInsights\\cite{wang2019datashot, ding2019quickinsights}.\nBecause of no exact method mentioned in these papers, we use the a statistic method Grubbs' test, which is a commonly used way to detect outliers.\n\n1. The first step is to quantify how far the outlier is from the others. We calculate The Grubbs test statistic $G$ as the largest absolute deviation from the sample mean in units of the sample standard deviation.\n\n2. The hypothesis of no outliers at a certain significance level is rejected if the calculated $G$ is greater than the corresponding critical value in the table, which have been tabulated for the Grubbs test statistic. \n\n3. If there exist at least an outlier and the p-value is small, we can conclude that the deviation of the outlier from other values is statistically significant. The lower the p-value is, the higher the significance is. Hence We obtain the significance as $S(f)=1-p$. \n\n4. Otherwise, If the hypothesis of no outliers is accepted which means there is no outlier in the sample, the significance is equal to 0.\n\n\\section{An Example of the Algorithm}\n\nTo explain the algorithm, we provide a concrete example in which the process of creating a 3-facts story is explained in detail. Here we use a simplified spreadsheet about the deaths of COVID-19 in China from March 1st to March 21st as a running example. The spreadsheet contains 3 columns including Date(temporal), Province(categorical), and Deaths(numerical). \n\nInitially, the algorithm generates a set of random facts according to the preliminary survey as the first fact, such as the Value fact \"The value of the \\underline{total deaths} is 423\" (\\textbf{F1}), the Trend fact \"The trend of the \\underline{total deaths} over \\underline{dates} is decreasing\" (\\textbf{F2}), and the Categorization fact \"The data contains 32 \\underline{provinces}\"(\\textbf{F3}). According to the importance score, the \\textbf{F2} is chosen as the root of the tree.\n\n\\begin{figure}[htb]\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/example-step-1.png}\n   \n\\end{figure}\n\nIn the next, the algorithm begins an iterative process with 4 major steps: selection, expansion, simulation, and back-propagation. \n\n\\begin{itemize}\n    \\item {\\bf Selection}: At first, it finds a fact with the largest reward in the tree. In the current stage, \\textbf{F2} is picked because it is the only fact in the tree. \n\n    \\item {\\bf Expansion}: Secondly, the algorithm expands a set of data facts from the Trend fact according to the different logical relation. For example, a contrast relation may lead to the Trend fact \"The \\underline{total deaths} over \\underline{dates} shows an increasing trend when \\underline{the province is Hong Kong}\" (\\textbf{F4}), an elaboration relation can lead to the Extreme fact \"The maximum value of the \\underline{total deaths} is 42 when \\underline{the date is 2020\/3\/2}\" (\\textbf{F5}), and a similarity relation can trigger the Distribution fact \"The distribution of the \\underline{total deaths} over different \\underline{provinces} shows an overview\" (\\textbf{F6}).\n\n    \\item {\\bf Simulation}: Thirdly, the algorithm starts to simulate from all the expanded facts (\\textbf{F4}, \\textbf{F5} and \\textbf{F6}). In each simulation process, the algorithm tries to explore the design space and find a path with the largest reward. \n\n    \\item {\\bf Back-propagation}: The rewards of the facts in the tree are updated after each simulation. In this example, the algorithm finds that choosing \\textbf{F5} will lead to a path with the largest reward. Thus, it updates the reward $\\Delta$ in the path (\\textbf{F2}, \\textbf{F5}) during back-propagation.\n\\end{itemize}\n\n\\begin{figure}[htb]\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/example-step-2.png}\n   \n\\end{figure}\n\n\nNow \\textbf{F5} is the node with the largest reward in the tree. Thus, it will be selected as the beginning node in the next iteration. A new Distribution fact \\textbf{F7} is expanded during the similar search process. The algorithm stops when the goal is fulfilled. In the end, the path (\\textbf{F2}, \\textbf{F5}, \\textbf{F7}) with the highest reward is the best story in the tree.\n\n\\begin{figure}[htb]\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/example-step-3.png}\n   \n\\end{figure}\n\n\nThe following figure is the storyline of the final 3-facts data stories. It illustrates that the trend of daily mortality in China was decreasing in March (Fact 1). In particular, the largest number was 42 occurred on March 2nd (Fact 2). Finally, the distribution shows that Hubei was the most dangerous province (Fact 3).\n\n\\begin{figure}[htb]\n    \\centering\n    \\includegraphics[width=0.82\\linewidth]{figures\/Example-Result.png}\n   \n\\end{figure}\n\\subsection{Story Generation Algorithm}\nIn Calliope\\xspace, we introduce an intelligent algorithm that generates data facts from an input spreadsheet and threads them in logic to create a meaningful data story. However design such an algorithm is quite challenge. The story design space, formulated by a collection of data facts generated from the input spreadsheet, could be extremely large due to the huge number of possible combinations of the fact fields event based on a small dataset. It is almost impossible for the algorithm to generate all facts first and then pick up those important ones to build the story as users cannot afford a long waiting time ({\\bf R2}). We addressed this issue by introducing an efficient logic-oriented searching algorithm based on the Monte Carlo tree search~\\cite{browne2012survey,silver2016mastering}. It can efficiently explore a large space that contains numerous status via a searching process organized by a tree structure and guided by a reward function.\n\nAlgorithm~\\ref{alg:generation} provides an overview of the proposed technique. It takes a spreadsheet $\\mathcal{D}$, the desired story length $\\mathcal{L}$, and a set of parameters \\{$w_d, w_i, w_l$\\} as the inputs and outputs a story  $\\mathcal{S} = \\{f_1,\\cdots,f_n\\}$. Here, the parameters are given by users that indicate their personal preferences of the generated story in terms of its fact diversity ($w_d$), the data coverage rate ($w_d$), and logicality ($w_l$). Specifically, the algorithm first generates a set of initial facts in three fact types, i.e., value, trend, and categorization, that are most frequently used as the starting points in a data story according to our preliminary survey as shown in Fig.~\\ref{fig:statistic}(b). The most important one is selected as the initial fact in $\\mathcal{S}$, denoted as $f_0$, and used as the root of a search tree. In the next, the algorithm expands $f_0$ in the tree by creating a set of data facts as its children that are logically relevant to $f_0$. \n\nwhich is estimated by as reward function  estimated by a reward function,  \n\nbased on which the  a search tree is gradually generated. \n\n\n\\begin{algorithm}[htb]\n    \\label{alg:generation}\n    \\SetAlgoLined\n    \\SetKwInOut{Input}{Input}\n    \\SetKwInOut{Output}{Output}\n    \\Input{$data$}\n    \\Output{$S$}\n   \n    $f_0$ $\\leftarrow$ InitalFact($data$)\\;\n    $S$ $\\leftarrow$ [$f_0$]\\;\n    \\While{length(S) \\textless max}{\n        \\While{within computational time}{\n            ($f_l$,$r_l$) $\\leftarrow$ TreePolicy($S$)\\;\n            $reward$ $\\leftarrow$ DefaultPolicy($story(f_l, r_l)$)\\;\n            Backpropagation(($f_l$,$r_l$), $reward$)\\;\n        }\n        $S$.insert(BestChild($S$))\\;\n    }\n    $S$ $\\leftarrow$ Aggregation($S$)\\;\n    \\Return $S$\\;\n    \\caption{Data Story Generation}\n\\end{algorithm}\n\n\\subsubsection{Logic Calculation}\n\n\nA $tree$ $policy$ determines the most possible node to reach in each iteration of MCTS. In data story generation, the $tree policy$ is used to find the next fact to expand the story. To ensure the facts in story are meaningful connected and reduce the search space, we introduce coherence relations as the latent relation between the adjacent facts to guide a logic-oriented tree policy in MCTS. In the discourse, coherence relations refers to informational relations that hold between natural language sentences \\cite{hobbs1985coherence}. According to preliminary data videos survey and related corpus studies \\cite{wolf2005representing, wellner2006classification}, we summarize 6 discourse relations in data stories in Table \\ref{table:relation}. A pair of facts can be described by three components $(f_a, r_{ab}, f_b)$, where $f_a$ and $f_b$ denote two adjacent facts connected by relation $r_{ab}$.\n\n\\input{tables\/4-relation}\n\nA $tree$ $policy$ determines the most possible node to reach in each iteration of MCTS. In data story generation, the $tree policy$ is used to find the next fact to expand the story. To ensure the facts in story are meaningful connected and reduce the search space, we introduce coherence relations as the latent relation between the adjacent facts to guide a logic-oriented tree policy in MCTS. In the discourse, coherence relations refers to informational relations that hold between natural language sentences \\cite{hobbs1985coherence}. According to preliminary data videos survey and related corpus studies \\cite{wolf2005representing, wellner2006classification}, we summarize 6 discourse relations in data stories in Table \\ref{table:relation}. A pair of facts can be described by three components $(f_a, r_{ab}, f_b)$, where $f_a$ and $f_b$ denote two adjacent facts connected by relation $r_{ab}$.\n\n\\input{tables\/4-relation}\n\nWe regard these relations as the constraints in tree policy. Calliope\\xspace applies the rules in the defined relations to keep the connection between $f_a$ and $f_b$. As follow, when the current fact is $f_a$ and relation is $r_{ab}$, the next fact $f_b$ can be derived after a set of edit operations from $f_a$:\n\n\\begin{itemize}\n    \\item {\\bf Similarity} modifies ($meausre$\/$breakdown$\/categorical filter in $subspace$) in $f_a$ to construct the $f_b$. It juxtaposes two facts with same measurement or same scope of data, such as ``the confirmed count of COVID-19 in different provinces\".\n    \\item {\\bf Temporal} adds\/removes\/modifies the temporal filter of $subspace$ in $f_a$ to construct the $f_b$. It statement two facts based on time, such as ``the confirmed count of COVID-19 in February and March\".\n    \\item {\\bf Contrast} modifies ($meausre$\/filter in $subspace$) in $f_a$. In addition, the result should be the opposite fact to $f_a$, thus we take it as $f_b$. In this relation, $f_a$ and $f_b$ are reverse to each other, such as ``the decreasing trend of new confirmed count of COVID-19 in China and the increasing trend in other countries\". \n    \\item {\\bf Causality} modifies $meausre$ that has the cause-effect relation to the $f_a$. $f_b$ should be the reason of $f_a$, such as ``the confirmed count leads to the death count of COVID-19\". The causality discover technique can be used to build a causal graph for the multi-dimensional data \\cite{schaechtle2013multi}. The directed link in the causal graph means the causal relation from one numerical data to another. Thus we use the causal graph to modify the $measure$.\n    \\item {\\bf Elaboration} add filter in $subspace$ or add $focus$ from $f_a$ to $f_b$.  It means $f_b$ is the specific condition to $f_a$, such as ``the confirmed count of COVID-19 in China and details in each provinces\".\n    \\item {\\bf Generalization} remove filter in $subspace$ or remove $focus$ from $f_a$ to $f_b$. It means $f_b$ is the general condition to $f_a$, such as ``the confirmed count of COVID-19 in China and overall information in the world\".\n\\end{itemize}\n\nIn addition, we follow the statistic of narrative logic frequently used behind each fact type (Table~\\ref{tab:logic}) to determine the probability of each relation during tree search. \n\n\\label{4.2.1}\n\n\\subsubsection{Reward Function}\n\nThe reward function is to ensure the quality of the story generation. As we mentioned before, our system should generate a meaningful story in a clear logic with correct data. The meaningful story refers to a data story with a high importance score, which can be measured by the information entropy and significance.\n\nBesides the story importance, the story should also provide the narrative logic, and believable content. Furthermore, the content in story should should be diverse to avoid a boring presentation. Therefore, the final reward of story also consider narrative logic, data integrity, and diversity. We put these three parameters into the coefficient of preference which can be adjusted by the user.\n\nThe total reward of story is the product of the coefficient of user preference and the importance score of the story. Bringing each parameters together, the final $reward$ is:\n\n\\begin{equation}\nreward =  (\\alpha \\cdot Diversity  + \\beta \\cdot Integrity + \\gamma \\cdot Logicality) \\cdot score(S)\n\\end{equation}\nwhere $\\alpha$, $\\beta$ and $\\gamma$ is the weight in [0,1] and the sum of them is equal to 1. Users can determine the value by interaction in the storyline view. The interaction detail will be introduced in Section~\\ref{5.4}.\n\nBased on the information theory, we define the information entropy of story $S$ as $H(S)$ by the formula of information entropy:\n\n\\begin{equation}\n    H(S)=-\\sum_{i=1}^{m} P(f_i) \\log _{2}\\left(P(f_i)\\right) = \\sum_{i=1}^{n} { P(f_i) \\cdot I(f_i) }\n\\end{equation}\n\nWe add significance importance of each fact into this formula to calculate the importance score of the story $score(S)$. Then, we put formula of fact importance into $score(S)$.\n\n\\begin{equation}\nscore(S) = \\sum_{i=1}^{n} { P(f_i) \\cdot S(f_i) \\cdot I(f_i) } = \\sum_{i=1}^{n} { P(f_i) \\cdot score(f_i) }\n\\end{equation}\n\n\\textbf{Logicality} represents the logic of the story.\nIn the logic tree search, we only consider the logic from current fact to next relation. By contrast, we consider the logic from fact to last relation in reward term. Logicality is the average of each conditional probability of last relation $r_{i-1}$ given fact $f_{i}$:\n\n\\begin{equation}\nLogicality = \\frac{1}{n-1}\\sum_{i=1}^{n-1}P\\left(r_{i-1} | f_{i}\\right)\n\\end{equation}\n\n\\textbf{Diversity} represents the diversity of the story.\nWhen there are more various fact types in a story, the story is considered more diverse. We define $m$ as the number of fact types in the story and ${p}_{j}$ as the proportion of $j$th fact type in the story. The diversity is calculated as the follow, where the former item is to make sure more unique fact types in the story and the latter item is the normalized Shannon diversity index to keep these fact types have roughly equal proportion:\n\n\\begin{equation}\nDiversity = \\frac{m}{\\max(n,10)} \\cdot -\\frac{\\sum_{j=1}^{m} {p}_{j} \\cdot \\ln \\left({p}_{j}\\right)}{\\ln (m)} \n\\end{equation}\n\n\\textbf{Integrity} represents the data integrity of the story.\nThe more complete the story cover the spreadsheet, the better the data integrity. The calculation method is to take the union of the table cells covered by each fact and divide by the count of total table cells:\n\n\\begin{equation}\nIntegrity = \\frac{count_{cell}(\\bigcup\\limits_{i=0}^{n-1} f_{i})}{count_{cell}(data)}\n\\end{equation}\n\n\\label{4.2.2}\n\n\\subsection{Story Aggregation}\n\nFor now, the story pieces in our generated data story only convey single fact. However, the compound fact is the common used fact that consists of multiple primitive facts \\cite{amar2005low, chen2009toward}. For instance, \"The trend of sales is decreasing and the maximum of sales is in 2007\" is a compound fact that involves a trend fact and an extreme fact. To keep the story concise, Calliope\\xspace provides an option for users to aggregate the story by merging single facts into a compound one automatically.\n\nThe story aggregation process consists of three major steps: (1) similarity measuring, (2) hierarchical clustering, and (3) fact merging.\n\nFirst, we measures the similarity between data facts using the overlap similarity for categorical data \\cite{boriah2008similarity}. The overlap similarity of the $type$ is 0 when types of two facts are no match, and 1 when the attribute values match. For the $measure$ and the $breakdown$, the similarity is the proportion of the fields in common. For the $subspace$ and the $focus$, we use the the fraction of union covered by intersection for similarity calculation. The similarity in total is the sum of these five results.\n\nThen, we aggregate the data facts into a hierarchical structure based on the similarity using hierarchical clustering technique. After clustering, the pairs of leaf nodes, such as the dashed box in fig, are the potential facts to be merged. These pairs of facts are ranked according to the similarity.The more similar facts have more tendency to be merged.Calliope\\xspace provides a slider bar that allows users to control the level of story aggregation in percentage, with a value of 0\\% to present single facts, and a value of 100\\% to merge all pairs of data facts.\n\nTo present the compound fact, Calliope\\xspace merge facts in two aspects, including visualization and natural language. For simplicity, we define a rule table for each combinations of two fact types. In the table, we choose chart type for compound fact according to the intersection of chart candidates for each primitive fact type. We also merge the fact tuples that have no conflict between each other. If there is no common chart type between two facts or conflict in the fact tuples, we simply place the two visualization by juxtaposition. For example, there is a trend fact and a extreme fact share same $measure$, $subspace$, and $breakdown$ shown in Fig.\\ref{fig:aggregation}. The difference is that the extreme fact has an additional \\textit{focus} which highlight the maximum value. Based on our rule table, both of these two facts can use the line chart to present the information. As a result, the compound fact use line chart  which is shown in Fig.\\ref{fig:aggregation}. For the description of compound fact, we choose to join together two sentences from both primitive facts.\n\n\\begin{figure}[htb]\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/aggregation.png}\n    \\caption{Compound fact} \n    \\label{fig:aggregation}\n\\end{figure}\n\nUp to this point, Calliope\\xspace does not consider the perception cost during the fact merging \\cite{hullman2013deeper}. Besides, the natural language in the compound fact may has redundancy, which can be solved by NLP techniques such as coreference resolution \\cite{ng2002improving}. We leave these questions in the future work.\n\n\\label{4.3}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Story Generation Engine}\n\\label{sec:engine}\nIn this section, we describe the technique details of the visual data story generation engine introduced in Calliope\\xspace. Before describing the algorithm, we first introduce different types of data facts and their importance measures. \n\n\\subsection{Data Facts}\nData facts are the elementary pieces that build up a data story. In Calliope\\xspace, we formally define fact as a 5-tuple \n\nBased on preliminary study and formative work \\cite{wang2019datashot}, Calliope provides a formal model for representing data facts.\nIn our definition, a data fact consist of a five-tuple, including \\textit{type}, \\textit{measure}, \\textit{subspace}, \\textit{group-by} and \\textit{focus}.\n$$\nfact := (type, measure, subspace, groupby, focus)\n$$\n\nThe \\textit{type} definition identifies the fact type from 10 pre-defined taxonomies, including \\textit{value}, \\textit{difference}, \\textit{proportion}, \\textit{trend}, \\textit{categorization}, \\textit{distribution}, \\textit{rank}, \\textit{association}, \\textit{extreme} and \\textit{outlier}.\nThese 10 types derive from our preliminary survey on data videos and several formative works \\cite{wang2019datashot, chen2009toward}. \nThe \\textit{measure} determines which numerical field and aggregation function to use as the measurement in the fact.\nThe \\textit{subspace} specifies the range of data that fulfill a filter condition.\nThe \\textit{group-by} groups records that have same value in categorical or temporal fields into summary records.\nThe \\textit{focus} highlights the specific information in the subspace.\n\nTable \\ref{tab:fact_table} summarizes the rules to construct the data fact for each fact type.\nAs shown in the table, \\textit{measure}, \\textit{group-by} and \\textit{focus} are not necessary for all fact types.\nThe definition of each fact type are depicted in the following list.\nTo explain the meaning of each fact type, we use a sample spreadsheet about automatic sales.\nThe columns' information is shown in Table \\ref{table:tabledetail}.\n\n\\begin{table}[htb]\n    \\label{table:tabledetail}\n    \\caption{The information of columns in the sample}\n    \\centering\n    \\small\n    \\begin{tabular}{ccl}\n    \\hline\n    field name & type & description \\\\ \\hline\n    Sales & numerical & automobile sales \\\\ \n    Income & numerical & automobile income \\\\ \n    Category & categorical & types of automobile (e.g. SUV, Compact) \\\\ \n    Brand & categorical & brands of automobile (e.g. BMW, Ford) \\\\ \n    Year & temporal & year of Sales (2007-2011) \\\\ \\hline\n    \\end{tabular}\n\\end{table}\n\n\\begin{itemize}\n\n    \\item \\textbf{Value} is defined on (\\textit{measure}, \\textit{subspace}).\n    It derives the value according to the \\textit{measure} in \\textit{subspace}. \n    E.g. Show total sales in Brand Ford.\n    \n    \\item \\textbf{Difference} is defined on (\\textit{measure}, \\textit{subspace}, \\textit{group-by}, \\textit{focus}).\n    It compares the \\textit{measure} of two \\textit{focus} on \\textit{group-by} in \\textit{subspace}.\n    E.g. Compare difference between sales of SUV and Compact in Brand BMW.\n    \n    \\item \\textbf{Proportion} is defined on (\\textit{measure}, \\textit{subspace}, \\textit{group-by}, \\textit{focus}).\n    It shows the proportion of the \\textit{measure} in the \\textit{focus} from all \\textit{group-by}s in \\textit{subspace}.\n    E.g. Show proportion of SUV in total sales of Brand Ford.\n    \n    \\item \\textbf{Trend} is defined on (\\textit{measure}, \\textit{subspace}, \\textit{group-by}).\n    It shows the \\textit{measure} order by the temporal \\textit{group-by} in \\textit{subspace}.\n    E.g. Show the trend of sales in Brand Ford order by year.\n    \n    \\item \\textbf{Categorization} is defined on (\\textit{subspace}, \\textit{group-by}).\n    It lists all \\textit{group-by}s in \\textit{subspace}.\n    E.g. Show all categories contained in Brand Ford.\n    \n    \\item \\textbf{Distribution} is defined on (\\textit{measure}, \\textit{subspace}, \\textit{group-by}).\n    It shows the \\textit{measure} of each \\textit{group-by} in \\textit{subspace}. \n    E.g. Show the sales of each category in Brand Ford.\n    \n    \\item \\textbf{Rank} is defined on (\\textit{measure}, \\textit{subspace}, \\textit{group-by}, \\textit{focus}).\n    It lists all \\textit{group-by} order by \\textit{measure} in \\textit{subspace} and \\textit{focus} on the top rank.\n    E.g. List the sales rank of brand in 2011.\n    \n    \\item \\textbf{Association} is defined on (\\textit{measure}, \\textit{subspace}, \\textit{group-by}).\n    It shows the relation between two \\textit{measure}s by \\textit{group-by} in \\textit{subspace}.\n    E.g. Show the relationship between sales and income in each brand in 2011.\n    \n    \\item \\textbf{Extreme} is defined on (\\textit{measure}, \\textit{subspace}, \\textit{group-by}, \\textit{focus}).\n    It shows the \\textit{measure} of each \\textit{group-by} in \\textit{subspace} and \\textit{focus} on the extreme.\n    E.g. Show the best-selling category in Brand Ford.\n    \n    \\item \\textbf{Outlier} is defined on (\\textit{measure}, \\textit{subspace}, \\textit{group-by}, \\textit{focus}).\n    It shows the \\textit{measure} of each \\textit{group-by} in \\textit{subspace} and \\textit{focus} on the outlier.\n    E.g. Show the outlier of sales by year in Brand Ford.\n    \n\\end{itemize}\n\nAccording to this fact definition, We describe a single data fact using a JSON specification in Calliope.\nBased on the automobile spreadsheet, we demonstrate a \\textit{Proportion} fact in Listing \\ref{lst:json}.\nThe \\textit{measure} of the fact is total sum of ``Sales\".\nThe \\textit{subspace} is the records when ``Brand\" is equal to ``Ford\".\nThe \\textit{group-by} divide the subspace by ``Category\", and the \\textit{focus} is the records when when ``Category\" is equal to ``SUV\".\nThe meaning of this specification is to show the proportion fact about total Sales of SUV in the Brand Ford.\n\n\\begin{lstlisting}[caption=JSON specification of a data fact,\nbasicstyle=\\small, label={lst:json}, language=json,firstnumber=1]\n{\n    \"type\": \"proportion\",\n    \"measure\": [{\"field\": \"Sales\", \"aggregate\": \"sum\"}],\n    \"subspace\": [{\"field\": \"Brand\", \"value\": \"Ford\"}],\n    \"group-by\": [{\"field\": \"Category\"}],\n    \"focus\": [{\"field\": \"Category\", \"value\": \"SUV\"}]\n}\n\\end{lstlisting}\n\nIn addition the basic fact defination, data facts also have \\textit{parameter} to explain the status of the fact, such as increasing or decreasing in the \\textit{trend} fact.\nTable \\ref{tab:fact_table} displays all \\textit{parameter} in 10 fact types.\nThese \\textit{parameter}s can be derived from the data and fact specification directly.\nWith the \\textit{parameter}, we can generate natural language for each fact. \nThe detail of facts to natural language will be descripted in Section 5.2.1.\n\n\\begin{table*}[t]\n    \\caption{Data Facts}\n    \\label{tab:fact_table}\n    \\small\n    \\def1.5{1.2}\n    \\begin{tabular}{|l|p{0.8cm}|p{0.8cm}|p{0.8cm}|p{0.8cm}|l|p{7.7cm}|}\n    \\hline\n    fact type & \\multicolumn{1}{l|}{measure} & \\multicolumn{1}{l|}{subspace} & \\multicolumn{1}{l|}{group-by} & \\multicolumn{1}{l|}{ focus } & parameter & sentence template \\\\ \\hline\n    Value & O & O & X & X & derived value & The \\{\\{aggregate\\}\\} \\{\\{measure\\}\\} is \\{\\{parameter\\}\\} when \\{\\{subspace\\}\\}. \\\\ \\hline\n    Difference & O & O & O & O & difference value & \\begin{tabular}[c]{@{}l@{}}The difference between \\{\\{focus{[}1{]}\\}\\} and \\{\\{focus{[}2{]}\\}\\} regarding to their \\\\ \\{\\{aggregate\\}\\} \\{\\{measure\\}\\} is \\{\\{parameter\\}\\} when \\{\\{subspace\\}\\}.\\end{tabular} \\\\ \\hline\n    Proportion & O & O & O & O & percentage & \\begin{tabular}[c]{@{}l@{}}The \\{\\{focus\\}\\} accounts for \\{\\{parameter\\}\\} of the \\{\\{aggregate\\}\\} \\\\ \\{\\{measure\\}\\} when \\{\\{subspace\\}\\}.\\end{tabular} \\\\ \\hline\n    Trend & O & O & O & X & increasing\/decreasing & \\begin{tabular}[c]{@{}l@{}}The trend of \\{\\{aggregate\\}\\} \\{\\{measure\\}\\} over \\{\\{group-by\\}\\} is \\\\ \\{\\{parameter\\}\\} when \\{\\{subspace\\}\\}.\\end{tabular} \\\\ \\hline\n    Categorization & X & O & O & X & number of categories & \\begin{tabular}[c]{@{}l@{}}Given \\{\\{subspace\\}\\}, there are \\{\\{parameter\\}\\} \\{\\{group-by\\}\\}(s) \\\\ which are \\{\\{C\\_1\\}\\}, \\{\\{C\\_2\\}\\}, \\{\\{...\\}\\}, and \\{\\{C\\_n\\}\\}.\\end{tabular} \\\\ \\hline\n    Distribution & O & O & O & X & N\/A & \\begin{tabular}[c]{@{}l@{}}The distribution of the \\{\\{aggregate\\}\\} \\{\\{measure\\}\\} over different \\\\ \\{\\{group-by\\}\\}(s) shows an overview of the data when \\{\\{subspace\\}\\}.\\end{tabular} \\\\ \\hline\n    Rank & O & O & O & O & N\/A & \\begin{tabular}[c]{@{}l@{}}In the \\{\\{aggregate\\}\\} \\{\\{measure\\}\\} ranking of different \\{\\{group-by\\}\\}(s), \\\\ the top three \\{\\{group-by\\}\\}(s) are \\{\\{focus{[}0{]}\\}\\}, \\{\\{focus{[}1{]}\\}, \\{\\{focus{[}2{]}\\}\\},\\\\ when \\{\\{subspace\\}\\}.\\end{tabular} \\\\ \\hline\n    Association & O & O & O & X & correlation value & \\begin{tabular}[c]{@{}l@{}}The Pearson correlation between the \\{\\{measure{[}1{]}\\}\\} and the \\\\ \\{\\{measure{[}2{]}\\}\\} is \\{\\{parameter\\}\\} in case of \\{\\{subspace\\}\\}.\\end{tabular} \\\\ \\hline\n    Extreme & O & O & O & O & \\begin{tabular}[c]{@{}l@{}}0: max\/min\\\\ 1: extreme value\\end{tabular} & \\begin{tabular}[c]{@{}l@{}}Given \\{\\{subspace\\}\\}, the \\{\\{parameter{[}0{]}\\}\\} value of the \\{\\{aggregate\\}\\} \\\\ \\{\\{measure\\}\\}is \\{\\{parameter{[}1{]}\\}\\} when \\{\\{group-by\\}\\} is \\{\\{focus\\}\\}.\\end{tabular} \\\\ \\hline\n    Outlier & O & O & O & O & outlier value & \\begin{tabular}[c]{@{}l@{}}The\\{\\{aggregate\\}\\} \\{\\{measure\\}\\} of \\{\\{focus\\}\\} is an outlier when compare \\\\ with that of other \\{\\{group-by\\}\\}(s) when \\{\\{subspace\\}\\}.\\end{tabular} \\\\ \\hline\n    \\end{tabular}\n\\end{table*}\n\nCalliope calculate the importance score $I(f)$ for a data fact $f$ to measure the value of fact.\nThe score contains two aspects: information importance ${\\rm Info}(f)$ and significance importance ${\\rm Sig}(f)$.\nThe importance score for each fact is:\n\n\\begin{equation}\n    I(f) = {\\rm Sig}(f) * {\\rm Info}(f)\n\\end{equation}\n\n\\textbf{Information importance} is measured by Shannon entropy \\cite{shannonentropy}, which is also called the self-information of a data fact.\nThe information importance should be high when an unlikely data fact is observed in spreadsheet.\nIt is calculated as follows, where $X$ is the scope of the data:\n\n\\begin{equation}\n{\\rm Info}(f) = -log_2(P(X))\n\\end{equation}\n\nAccording our fact definition, the \\textit{subspace} and the \\textit{focus} are the only two elements that related to the scope of the data $X$.\nSo the probability of $X$ is then determined by the product of the probability of \\textit{subspace} and conditional probability of \\textit{focus} given \\textit{subspace}:\n\n\\begin{equation}\nP(X) = P(subspace) * P(focus | subspace)\n\\end{equation}\n\nFor now, we only support at most 2 fields in the subspace of the fact.\nThe following formulation is to calculate the probability of the subspace:\n\n\\begin{equation}\nP(subspace)=\\frac{1}{\\sum_{i=0}^{2} C(n, i)}*\\prod_{i=1}^{k}\\frac{count({subspace}_i)}{count(all)}\n\\end{equation}\nThe former item is the probability of the occurence of the \\textit{subspace}, where $n$ is the count of categorical and temporal fields in the spreadsheet, and $C(n,i)$ is the number of $i$-combinations from $n$ fields. \nThe latter item is the probability of the \\textit{subspace}, where $count({subspace}_i)$ is the count of records which fulfills the $i$th condition in \\textit{subspace} and $count(all)$ is the total count of the spreadsheet.\nWhen the \\textit{subspace} includes two fields, we assume they are independent to each other.\nSo we calculate the product of the probabilities of two \\textit{subspace}s.\n\nThe conditional probability of \\textit{focus} given \\textit{subspace} is determined by the fraction of the count of \\textit{subspace} covered by the count of \\textit{focus}:\n\n\\begin{equation}\nP(focus|subspace)= \\frac{count(focus)}{count(subspace)}\n\\end{equation}\n\n\\textbf{Significance importance} $Sig(f)$ is the measurement from the statistical aspect. \nTo determine the importance of fact, significance is the weight for the information importance in [0, 1].\nFor instance, if a trend fact is steady with no obvious increasing or decreasing, it should have a low significance close to 0.\nIn contrast, a fact shows a clear rising trend line should get a high significance close to 1.\nWe calculate the significance importance of the fact using the methods in formative works \\cite{ding2019quickinsights, tang2017extracting, wang2019datashot}.\n\n\\subsection{Story Generation Algorithm}\nFirst describe the algorithm overview via pseudocode and explain the algorithm pipeline by important steps.\n\n\\subsubsection{Logic Calculation}\nDescribing how did we ensures the story logic and reduce the search space. Describe the statistics that we summarized based on the preliminary survey.\n\n\\subsubsection{Reward Function}\nDescribing the reward function followed by the details of each terms in the function. The key is to clearly describe the design rationale of each term.\n\nWe define the information entropy of the data story as $H(story)$.\n\n\\begin{equation}\nH(story) = E(I(F)) = \\sum_{i=1}^{n} { P(f_i) * I(f_i) }\n\\end{equation}\n$I(f_i)$ is the quantity of information for the $i$th fact and the $X$ is the data of the fact.\n\nBased on the information entropy, we add significance into the formulation to calculate the importance score of the story $I(story)$.\n\n\\begin{equation}\nI(S)  = E(S(F) * I(F)) =  \\sum_{i=1}^{n} { P(f_i) * S(f_i) * I(f_i) }\n\\end{equation}\n\nThe total story reward is the importance score of the story with the weight of user preference.\n\n\\begin{equation}\nReward = w_{S} * Importance(S)\n\\end{equation}\n\nThe weight of user preference includes: logicality, diversity and integrity.\n\n\\begin{equation}\nw_{story} =  \\alpha * Diversity  + \\beta * Integrity + (1- \\alpha - \\beta) * Logicality\n\\end{equation}\n\n\\subsection{Story Aggregation}\n\nFor now, the story pieces in our generated data story only convey single fact.\nHowever, the compound fact is the common used fact that consists of multiple primitive facts \\cite{amar2005low, chen2009toward}.\nTo keep the story concise, we aggregate the story by merging single facts into a compound one.\nThe story aggregation process consists of three major steps: (1)similarity measuring, (2)hierarchical clustering, and (3) fact merging.\n\nFirst, we measures the similarity between data facts using the overlap similarity for categorical data \\cite{boriah2008similarity}.\nThe overlap similarity of the \\textit{type} is 0 when types of two facts are no match, and 1 when the attribute values match.\nFor the \\textit{measure} and the \\textit{group-by}, the similarity is the proportion of the fields in common.\nFor the \\textit{subspace} and the \\textit{focus}, we use the the fraction of union covered by intersection for similarity calculation.\nThe similarity in total is the sum of these five results.\n\nThen, we aggregate the data facts into a hierarchical structure based on the similarity using hierarchical clustering technique.\nAfter clustering, the pairs of leaf nodes, such as the dashed box in fig, are the potential facts to be merged.\nThese pairs of facts are ranked according to the similarity. \nThe more similar facts have more tendency to be merged.\nCalliope provides a slider bar that allows users to control the level of story aggregation in percentage, with a value of 0\\% to present single facts, and a value of 100\\% to merge all pairs of data facts.\n\nTo present the compound fact, Calliope merge facts in two aspects, including visualization and natural language.\nFor simplicity, we define a rule table for each combinations of two fact types.\nIn the table, we choose chart type for compound fact according to the intersection of chart candidates for each primitive fact type.\nWe also merge the fact tuples that have no conflict between each other.\nIf there is no common chart type between two facts or conflict in the fact tuples, we simply place the two visualization by juxtaposition.\nFor example, there is a trend fact and a extreme fact share same \\textit{measure}, \\textit{subspace}, and \\textit{group-by} shown in Fig.\\ref{fig:aggregation}.\nThe difference is that the extreme fact has an additional \\textit{focus} which highlight the maximum value.\nBased on our rule table, both of these two facts can use the line chart to present the information.\nAs a result, the compound fact use line chart  which is shown in Fig.\\ref{fig:aggregation}.\nFor the description of compound fact, we choose to join together two sentences from both primitive facts.\n\n\\begin{figure}[htb]\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/aggregation.png}\n    \\caption{Compound fact} \n    \\label{fig:aggregation}\n\\end{figure}\n\nUp to this point, Calliope does not consider the perception cost during the fact merging \\cite{hullman2013deeper}.\nBesides, the natural language in the compound fact may has redundancy, which can be solved by NLP techniques such as coreference resolution \\cite{ng2002improving}.\nWe leave these questions in the future work.\n\\section{Story Generation Engine}\n\\label{sec:engine}\n\nIn this section, we describe the technique details of the data story generation engine in Calliope system. We first introduce the definitions of different types of data facts and their importance measures followed by the details of the proposed story generation algorithm.\n\n\\subsection{Data Facts}\n\nBased on our preliminary study and formative work \\cite{wang2019datashot}, Calliope provides a formal model for representing data facts.\nIn definition, a data fact consist of a five-tuple, including \\textit{type}, \\textit{measure}, \\textit{subspace}, \\textit{group-by} and \\textit{focus}.\n$$\nfact := (type, measure, subspace, groupby, focus)\n$$\n\nThe \\textit{type} definition identifies the fact type from 10 pre-defined taxonomies, including value, difference, proportion, trend, categorization, distribution, rank, association, extreme and outlier.\nThese 10 types derive from our preliminary survey on data videos and several formative works \\cite{wang2019datashot, chen2009toward}. \nThe \\textit{measure} determines which numerical field and aggregation function to use as the measurement in the fact.\nThe \\textit{subspace} specifies the range of data that fulfill a filter condition.\nThe \\textit{group-by} groups records that have same value in categorical or temporal fields into summary records.\nThe \\textit{focus} highlights the specific information in the subspace.\n\nTable \\ref{tab:fact_table} summarizes the rules to construct the data fact for each fact type.\nAs shown in the table, \\textit{measure}, \\textit{group-by} and \\textit{focus} are not necessary for all fact types.\nThe definition of each fact type are depicted in the following list.\nTo explain the meaning of each fact type, we use a sample spreadsheet about automatic sales.\nThe columns' information is shown in Table \\ref{table:tabledetail}.\n\n\\begin{table}[htb]\n    \\label{table:tabledetail}\n    \\caption{The information of columns in the sample}\n    \\centering\n    \\small\n    \\begin{tabular}{ccl}\n    \\hline\n    field name & type & description \\\\ \\hline\n    Sales & numerical & automobile sales \\\\ \n    Income & numerical & automobile income \\\\ \n    Category & categorical & types of automobile (e.g. SUV, Compact) \\\\ \n    Brand & categorical & brands of automobile (e.g. BMW, Ford) \\\\ \n    Year & temporal & year of Sales (2007-2011) \\\\ \\hline\n    \\end{tabular}\n\\end{table}\n\n\\begin{itemize}\n\n    \\item \\textbf{Value} is defined on (\\textit{measure}, \\textit{subspace}).\n    It derives the value according to the \\textit{measure} in \\textit{subspace}. \n    E.g. Show total sales in Brand Ford.\n    \n    \\item \\textbf{Difference} is defined on (\\textit{measure}, \\textit{subspace}, \\textit{group-by}, \\textit{focus}).\n    It compares the \\textit{measure} of two \\textit{focus} on \\textit{group-by} in \\textit{subspace}.\n    E.g. Compare difference between sales of SUV and Compact in Brand BMW.\n    \n    \\item \\textbf{Proportion} is defined on (\\textit{measure}, \\textit{subspace}, \\textit{group-by}, \\textit{focus}).\n    It shows the proportion of the \\textit{measure} in the \\textit{focus} from all \\textit{group-by}s in \\textit{subspace}.\n    E.g. Show proportion of SUV in total sales of Brand Ford.\n    \n    \\item \\textbf{Trend} is defined on (\\textit{measure}, \\textit{subspace}, \\textit{group-by}).\n    It shows the \\textit{measure} order by the temporal \\textit{group-by} in \\textit{subspace}.\n    E.g. Show the trend of sales in Brand Ford order by year.\n    \n    \\item \\textbf{Categorization} is defined on (\\textit{subspace}, \\textit{group-by}).\n    It lists all \\textit{group-by}s in \\textit{subspace}.\n    E.g. Show all categories contained in Brand Ford.\n    \n    \\item \\textbf{Distribution} is defined on (\\textit{measure}, \\textit{subspace}, \\textit{group-by}).\n    It shows the \\textit{measure} of each \\textit{group-by} in \\textit{subspace}. \n    E.g. Show the sales of each category in Brand Ford.\n    \n    \\item \\textbf{Rank} is defined on (\\textit{measure}, \\textit{subspace}, \\textit{group-by}, \\textit{focus}).\n    It lists all \\textit{group-by} order by \\textit{measure} in \\textit{subspace} and \\textit{focus} on the top rank.\n    E.g. List the sales rank of brand in 2011.\n    \n    \\item \\textbf{Association} is defined on (\\textit{measure}, \\textit{subspace}, \\textit{group-by}).\n    It shows the relation between two \\textit{measure}s by \\textit{group-by} in \\textit{subspace}.\n    E.g. Show the relationship between sales and income in each brand in 2011.\n    \n    \\item \\textbf{Extreme} is defined on (\\textit{measure}, \\textit{subspace}, \\textit{group-by}, \\textit{focus}).\n    It shows the \\textit{measure} of each \\textit{group-by} in \\textit{subspace} and \\textit{focus} on the extreme.\n    E.g. Show the best-selling category in Brand Ford.\n    \n    \\item \\textbf{Outlier} is defined on (\\textit{measure}, \\textit{subspace}, \\textit{group-by}, \\textit{focus}).\n    It shows the \\textit{measure} of each \\textit{group-by} in \\textit{subspace} and \\textit{focus} on the outlier.\n    E.g. Show the outlier of sales by year in Brand Ford.\n    \n\\end{itemize}\n\nAccording to this fact definition, We describe a single data fact using a JSON specification in Calliope.\nBased on the automobile spreadsheet, we demonstrate a \\textit{Proportion} fact in Listing \\ref{lst:json}.\nThe \\textit{measure} of the fact is total sum of ``Sales\".\nThe \\textit{subspace} is the records when ``Brand\" is equal to ``Ford\".\nThe \\textit{group-by} divide the subspace by ``Category\", and the \\textit{focus} is the records when when ``Category\" is equal to ``SUV\".\nThe meaning of this specification is to show the proportion fact about total Sales of SUV in the Brand Ford.\n\n\\begin{lstlisting}[caption=JSON specification of a data fact,\nbasicstyle=\\small, label={lst:json}, language=json,firstnumber=1]\n{\n    \"type\": \"proportion\",\n    \"measure\": [{\"field\": \"Sales\", \"aggregate\": \"sum\"}],\n    \"subspace\": [{\"field\": \"Brand\", \"value\": \"Ford\"}],\n    \"group-by\": [{\"field\": \"Category\"}],\n    \"focus\": [{\"field\": \"Category\", \"value\": \"SUV\"}]\n}\n\\end{lstlisting}\n\nIn addition the basic fact defination, data facts also have \\textit{parameter} to explain the status of the fact, such as increasing or decreasing in the \\textit{trend} fact.\nTable \\ref{tab:fact_table} displays all \\textit{parameter} in 10 fact types.\nThese \\textit{parameter}s can be derived from the data and fact specification directly.\nWith the \\textit{parameter}, we can generate natural language for each fact. \nThe detail of facts to natural language will be descripted in Section 5.2.1.\n\n\\begin{table*}[t]\n    \\caption{Data Facts}\n    \\label{tab:fact_table}\n    \\small\n    \\def1.5{1.2}\n    \\begin{tabular}{|l|p{0.8cm}|p{0.8cm}|p{0.8cm}|p{0.8cm}|l|p{7.7cm}|}\n    \\hline\n    fact type & \\multicolumn{1}{l|}{measure} & \\multicolumn{1}{l|}{subspace} & \\multicolumn{1}{l|}{group-by} & \\multicolumn{1}{l|}{ focus } & parameter & sentence template \\\\ \\hline\n    Value & O & O & X & X & derived value & The \\{\\{aggregate\\}\\} \\{\\{measure\\}\\} is \\{\\{parameter\\}\\} when \\{\\{subspace\\}\\}. \\\\ \\hline\n    Difference & O & O & O & O & difference value & \\begin{tabular}[c]{@{}l@{}}The difference between \\{\\{focus{[}1{]}\\}\\} and \\{\\{focus{[}2{]}\\}\\} regarding to their \\\\ \\{\\{aggregate\\}\\} \\{\\{measure\\}\\} is \\{\\{parameter\\}\\} when \\{\\{subspace\\}\\}.\\end{tabular} \\\\ \\hline\n    Proportion & O & O & O & O & percentage & \\begin{tabular}[c]{@{}l@{}}The \\{\\{focus\\}\\} accounts for \\{\\{parameter\\}\\} of the \\{\\{aggregate\\}\\} \\\\ \\{\\{measure\\}\\} when \\{\\{subspace\\}\\}.\\end{tabular} \\\\ \\hline\n    Trend & O & O & O & X & increasing\/decreasing & \\begin{tabular}[c]{@{}l@{}}The trend of \\{\\{aggregate\\}\\} \\{\\{measure\\}\\} over \\{\\{group-by\\}\\} is \\\\ \\{\\{parameter\\}\\} when \\{\\{subspace\\}\\}.\\end{tabular} \\\\ \\hline\n    Categorization & X & O & O & X & number of categories & \\begin{tabular}[c]{@{}l@{}}Given \\{\\{subspace\\}\\}, there are \\{\\{parameter\\}\\} \\{\\{group-by\\}\\}(s) \\\\ which are \\{\\{C\\_1\\}\\}, \\{\\{C\\_2\\}\\}, \\{\\{...\\}\\}, and \\{\\{C\\_n\\}\\}.\\end{tabular} \\\\ \\hline\n    Distribution & O & O & O & X & N\/A & \\begin{tabular}[c]{@{}l@{}}The distribution of the \\{\\{aggregate\\}\\} \\{\\{measure\\}\\} over different \\\\ \\{\\{group-by\\}\\}(s) shows an overview of the data when \\{\\{subspace\\}\\}.\\end{tabular} \\\\ \\hline\n    Rank & O & O & O & O & N\/A & \\begin{tabular}[c]{@{}l@{}}In the \\{\\{aggregate\\}\\} \\{\\{measure\\}\\} ranking of different \\{\\{group-by\\}\\}(s), \\\\ the top three \\{\\{group-by\\}\\}(s) are \\{\\{focus{[}0{]}\\}\\}, \\{\\{focus{[}1{]}\\}, \\{\\{focus{[}2{]}\\}\\},\\\\ when \\{\\{subspace\\}\\}.\\end{tabular} \\\\ \\hline\n    Association & O & O & O & X & correlation value & \\begin{tabular}[c]{@{}l@{}}The Pearson correlation between the \\{\\{measure{[}1{]}\\}\\} and the \\\\ \\{\\{measure{[}2{]}\\}\\} is \\{\\{parameter\\}\\} in case of \\{\\{subspace\\}\\}.\\end{tabular} \\\\ \\hline\n    Extreme & O & O & O & O & \\begin{tabular}[c]{@{}l@{}}0: max\/min\\\\ 1: extreme value\\end{tabular} & \\begin{tabular}[c]{@{}l@{}}Given \\{\\{subspace\\}\\}, the \\{\\{parameter{[}0{]}\\}\\} value of the \\{\\{aggregate\\}\\} \\\\ \\{\\{measure\\}\\}is \\{\\{parameter{[}1{]}\\}\\} when \\{\\{group-by\\}\\} is \\{\\{focus\\}\\}.\\end{tabular} \\\\ \\hline\n    Outlier & O & O & O & O & outlier value & \\begin{tabular}[c]{@{}l@{}}The\\{\\{aggregate\\}\\} \\{\\{measure\\}\\} of \\{\\{focus\\}\\} is an outlier when compare \\\\ with that of other \\{\\{group-by\\}\\}(s) when \\{\\{subspace\\}\\}.\\end{tabular} \\\\ \\hline\n    \\end{tabular}\n\\end{table*}\n\nCalliope calculate the importance score $I(f)$ for a data fact $f$ to measure the value of fact.\nThe score contains two aspects: information importance ${\\rm Info}(f)$ and significance importance ${\\rm Sig}(f)$.\nThe importance score for each fact is:\n\n\\begin{equation}\n    I(f) = {\\rm Sig}(f) \\cdot {\\rm Info}(f)\n\\end{equation}\n\n\\textbf{Information importance} ${\\rm Info}(f)$ is measured by Shannon entropy \\cite{shannonentropy}, which is also called the self-information of a data fact.\nThe information importance should be high when an unlikely data fact is observed in spreadsheet.\nIt is calculated as follows, where $X$ is the scope of the data:\n\n\\begin{equation}\n{\\rm Info}(f) = -log_2(P(X))\n\\end{equation}\n\nAccording our fact definition, the \\textit{subspace} and the \\textit{focus} are the only two elements that related to the scope of the data $X$.\nSo the probability of $X$ is then determined by the product of the probability of \\textit{subspace} and conditional probability of \\textit{focus} given \\textit{subspace}:\n\n\\begin{equation}\nP(X) = P(subspace) \\cdot P(focus | subspace)\n\\end{equation}\n\nFor now, we only support at most 2 fields in the subspace of the fact.\nThe following formulation is to calculate the probability of the subspace:\n\n\\begin{equation}\nP(subspace)=\\frac{1}{\\sum_{i=0}^{2} C(n, i)}\\cdot\\prod_{i=1}^{k}\\frac{count({subspace}_i)}{count(all)}\n\\end{equation}\nThe former item is the probability of the occurence of the \\textit{subspace}, where $n$ is the count of categorical and temporal fields in the spreadsheet, and $C(n,i)$ is the number of $i$-combinations from $n$ fields. \nThe latter item is the probability of the \\textit{subspace}, where $count({subspace}_i)$ is the count of records which fulfills the $i$th condition in \\textit{subspace} and $count(all)$ is the total count of the spreadsheet.\nWhen the \\textit{subspace} includes two fields, we assume they are independent to each other.\nSo we calculate the product of the probabilities of two \\textit{subspace}s.\n\nThe conditional probability of \\textit{focus} given \\textit{subspace} is determined by the fraction of the count of \\textit{subspace} covered by the count of \\textit{focus}:\n\n\\begin{equation}\nP(focus|subspace)= \\frac{count(focus)}{count(subspace)}\n\\end{equation}\n\n\\textbf{Significance importance} ${\\rm Sig}(f)$ is the measurement from the statistical aspect. \nTo determine the importance of fact, significance is the weight for the information importance in [0, 1].\nFor instance, if a trend fact is steady with no obvious increasing or decreasing, it should have a low significance close to 0.\nIn contrast, a fact shows a clear rising trend line should get a high significance close to 1.\nWe calculate the significance importance of the fact using the methods in formative works \\cite{ding2019quickinsights, tang2017extracting, wang2019datashot}.\n\n\\subsection{Story Generation Algorithm}\n\nWe introduce Monte Carlo Tree Search\n\nThere are 4 steps in each search iteration: (1) Selection, (2) Expansion, (3) Simulation, (4) Backpropagation.\nThese steps are grouped into two policies, including logic tree policy and default policy.\nThe total generation algorithm is summarized in pseudocode in Algorithm \\ref{alg:generation}.\n\n\\begin{algorithm}[htb]\n    \\label{alg:generation}\n    \\SetAlgoLined\n    \\SetKwInOut{Input}{Input}\n    \\SetKwInOut{Output}{Output}\n    \\Input{Data from spreadsheet}\n    \\Output{Story with a sequence of data facts}\n   \n    $f_0$ $\\leftarrow$ InitalFact($data$)\\;\n    $story$ $\\leftarrow$ [$f_0$]\\;\n    \\While{length(story) < max}{\n        \\While{within computational time}{\n            ($f_l$,$r_l$) $\\leftarrow$ LogicTreePolicy($story$)\\;\n            $\\Delta$ $\\leftarrow$ DefaultPolicy($story(f_l, r_l)$)\\;\n            Backup(($f_l$,$r_l$), $\\Delta$)\\;\n        }\n        $story$.insert(BestChild($story$))\\;\n    }\n    $story$ $\\leftarrow$ Aggregation($story$)\\;\n    \\Return $story$\\;\n    \\caption{Data Story Generation}\n\\end{algorithm}\n$f_0$ is the first fact and the root node corresponding to the $story$.\n$f_l$ and $r_l$ is the last fact and relation reached during the logic tree policy, which will be introduced in Section 4.2.1.\n$\\Delta$ is the reward for terminal $story$ by running the default policy.\nThe result of the overall search BestChild($story$) is the fact and relation that leads to the best reward.\nSo we take the result to append the current $story$.\nAfter all iterations are finished, we get the final generated $story$ and aggregate it using the method mentioned in Section 4.3.\n\n\\subsubsection{Logic Calculation}\n\nAccording to preliminary data videos survey and formative corpus study \\cite{wolf2005representing}, we summarize 6 discourse relations in data stories.\nWe list them in Table \\ref{table:relation}.\n\n\\begin{table}[]\n    \\centering\n    \\label{table:relation}\n    \\caption{6 discourse relations and their example templates in data stories.}\n    \\begin{tabular}{ll}\n    \\hline\n    relation & example template \\\\ \\hline\n    similarity & \\{\\{$Fact_a$\\}\\}. \\{\\{$Fact_b$\\}\\}. \\\\ \n    temporal & \\{\\{$Fact_a$\\}\\}. After that, \\{\\{$Fact_b$\\}\\}. \\\\ \n    contrast & \\{\\{$Fact_a$\\}\\}. However, \\{\\{$Fact_b$\\}\\}. \\\\ \n    cause-effect & \\{\\{$Fact_a$\\}\\}. The reason is \\{\\{$Fact_b$\\}\\}. \\\\ \n    elaboration & \\{\\{$Fact_a$\\}\\}. Furthermore, \\{\\{$Fact_b$\\}\\}. \\\\ \n    generalization & \\{\\{$Fact_a$\\}\\}. In general, \\{\\{$Fact_b$\\}\\}. \\\\ \\hline\n    \\end{tabular}\n\\end{table}\n\nInitial fact:\n\n\\subsubsection{Reward Function}\nDescribing the reward function followed by the details of each terms in the function. The key is to clearly describe the design rationale of each term.\n\nWe define the information entropy of the data story as $H(story)$.\n\n\\begin{equation}\nH(story) = E(I(F)) = \\sum_{i=1}^{n} { P(f_i) \\cdot I(f_i) }\n\\end{equation}\n$I(f_i)$ is the quantity of information for the $i$th fact and the $X$ is the data of the fact.\n\nBased on the information entropy, we add significance into the formulation to calculate the importance score of the story $I(S)$.\n\n\\begin{equation}\nI(S)  = E(S(F) \\cdot I(F)) =  \\sum_{i=1}^{n} { P(f_i) \\cdot S(f_i) \\cdot I(f_i) }\n\\end{equation}\n\nThe coefficient of user preference $w_{S}$ includes three parameters: logicality, diversity and integrity.\n\n\\textbf{Logicality}\n\n\\begin{equation}\nLogicality = \\frac{1}{n}\\sum_{i=1}^{n-1}P\\left(r_{i-1} | f_{i}\\right)\n\\end{equation}\n\n\\textbf{Diversity}\n\n\\begin{equation}\nDiversity = \\frac{count(type)}{\\max (length(S),10)} *-\\frac{\\sum_{j=1}^{i} \\hat{p}_{j} \\cdot \\ln \\left(\\hat{p}_{j}\\right)}{\\ln (s)} \n\\end{equation}\n\n\\textbf{Integrity}\n\n\\begin{equation}\nIntegrity = \\frac{count_{cell}(\\bigcup\\limits_{i=0}^{n-1} f_{i})}{count_{cell}(data)}\n\\end{equation}\n\nThe total story reward is the importance score of the story with the coefficient of user preference.\nBringing each parameters together, the final $reward$ is:\n\n\\begin{equation}\nreward =  (\\alpha \\cdot Diversity  + \\beta \\cdot Integrity + \\gamma \\cdot Logicality) \\cdot I(S)\n\\end{equation}\nwhere $\\alpha$, $\\beta$ and $\\gamma$ is the weight in [0,1] and the sum of them is equal to 1.\nUsers can determine the value by interaction in the storyline view.\nThe interaction detail will be introduced in Section 5.1.\n\n\\subsection{Story Aggregation}\n\nFor now, the story pieces in our generated data story only convey single fact.\nHowever, the compound fact is the common used fact that consists of multiple primitive facts \\cite{amar2005low, chen2009toward}.\nFor instance, \"The trend of sales is decreasing and the maximum of sales is in 2007\" is a compound fact that involves a trend fact and an extreme fact.\nTo keep the story concise, Calliope provides an option for users to aggregate the story by merging single facts into a compound one automatically.\n\nThe story aggregation process consists of three major steps: (1) similarity measuring, (2) hierarchical clustering, and (3) fact merging.\n\nFirst, we measures the similarity between data facts using the overlap similarity for categorical data \\cite{boriah2008similarity}.\nThe overlap similarity of the \\textit{type} is 0 when types of two facts are no match, and 1 when the attribute values match.\nFor the \\textit{measure} and the \\textit{group-by}, the similarity is the proportion of the fields in common.\nFor the \\textit{subspace} and the \\textit{focus}, we use the the fraction of union covered by intersection for similarity calculation.\nThe similarity in total is the sum of these five results.\n\nThen, we aggregate the data facts into a hierarchical structure based on the similarity using hierarchical clustering technique.\nAfter clustering, the pairs of leaf nodes, such as the dashed box in fig, are the potential facts to be merged.\nThese pairs of facts are ranked according to the similarity. \nThe more similar facts have more tendency to be merged.\nCalliope provides a slider bar that allows users to control the level of story aggregation in percentage, with a value of 0\\% to present single facts, and a value of 100\\% to merge all pairs of data facts.\n\nTo present the compound fact, Calliope merge facts in two aspects, including visualization and natural language.\nFor simplicity, we define a rule table for each combinations of two fact types.\nIn the table, we choose chart type for compound fact according to the intersection of chart candidates for each primitive fact type.\nWe also merge the fact tuples that have no conflict between each other.\nIf there is no common chart type between two facts or conflict in the fact tuples, we simply place the two visualization by juxtaposition.\nFor example, there is a trend fact and a extreme fact share same \\textit{measure}, \\textit{subspace}, and \\textit{group-by} shown in Fig.\\ref{fig:aggregation}.\nThe difference is that the extreme fact has an additional \\textit{focus} which highlight the maximum value.\nBased on our rule table, both of these two facts can use the line chart to present the information.\nAs a result, the compound fact use line chart  which is shown in Fig.\\ref{fig:aggregation}.\nFor the description of compound fact, we choose to join together two sentences from both primitive facts.\n\n\\begin{figure}[htb]\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/aggregation.png}\n    \\caption{Compound fact} \n    \\label{fig:aggregation}\n\\end{figure}\n\nUp to this point, Calliope does not consider the perception cost during the fact merging \\cite{hullman2013deeper}.\nBesides, the natural language in the compound fact may has redundancy, which can be solved by NLP techniques such as coreference resolution \\cite{ng2002improving}.\nWe leave these questions in the future work.\n\\subsection{Storyline View}\nThe storyline view(Fig.~\\ref{fig:ui}-1) contains two panels, the story configuration panel(Fig.~\\ref{fig:ui}-1(a)) and the story pieces panel(Fig.~\\ref{fig:ui}-1(b)). The story configuration panel is where users upload data, tweak parameters for the generated story, such as the story length, the chart diversity and the tolerable time limitation of the generation. The chart diversity represents chart type diversities of each fact type in the generated story ranging from zero to one. Particularly, the value of zero means that each type of facts maps to only one chart type in the generated result. When users increase the value with the slider, more chart types will show up for each fact type in the story. \n\nIn the reward view within the story configuration panel, users can also set up their preference for the reward of the story generation algorithm, including three key parameters: logicality, diversity and integrity. A chart is implemented with one movable circle and three fixed circles to encode the relation among the generated story and the three key parameters.\nHere, the circle $S$ represents the story, and three fixed circles representing three parameters shape a circular area, and the distances between $S$ and three parameter circles capture the weights of each parameter. While moving the circle $S$, the parameter circles that are closer to $S$ will obtain higher weights and more attention when generating.\n\nAfter clicking the ``Generate Story\" button in the configuration panel, Calliope\\xspace will yield a sequence of data facts in a logic order and line them in the story pieces panel successively. \nEach piece in this panel shows a fact of data with visualization and description. \nFor easy revision, Calliope\\xspace allows users to add, remove or re-arrange these fact pieces by simple interaction, as well as edit the statement of facts. When a specific fact piece is selected, its details will show up in the fact view, where users can modify it to meet their needs. These actions will apply in all the other views accordingly to reduce repetitive modification. \n\n\\subsection{Fact View}\nThe fact view (Fig.~\\ref{fig:ui}-2) is a detailed view that not only represent the selected story piece but also enables users to customize this fact by changing the configuration. \nFact's configuration items include the following: \\textit{fact type}, \\textit{visualization}, \\textit{measure},\\textit{subspace}, \\textit{breakdown}, and \\textit{focus}. Different fact types may have specific combination of the configuration items mentioned above according to table~\\ref{tab:fact}.\nThrough the adjustment of these configuration items, the nature language(\\ref{5.2.1}) and the visualization(\\ref{5.2.2}) of the fact will be generated in real time. Also, the information quantity and significance score of the selected fact will be updated accordingly. \n\n\\subsubsection{Facts to Nature Language}\nEach facts is equivalent to a context free grammar which can be used to generate nature language. Describe how the description text are generated for each fact type.\n\\label{5.2.1}\n\n\\subsubsection{Facts to Visualization}\n\nBased on our preliminary survey, we design a rule based method to match fact types with different chart types. For ensuring the visual diversity, we add some alternative designs of each basic chart based on statistical results. For example, donut chart and half donut chart can be  the variants of the basic pie chart for users to choose.\n\nThe visualization changes accordingly with the adjustment of fact configuration. Among the configuration items, \\textit{subspace} subspace is used to specify the range of data to be visualized, \\textit{measure} and \\textit{groupby} respectively correspond to the measure and dimension that make up the chart, and \\textit{focus} is used as the highlighted part in the chart.\n\\label{5.2.2}\n\n\\subsection{Story Visualization View}\nThe story visualization view provides a variety of visualization forms to represent the generated data story line for different application scenarios. In particular, a summarization view is first provided to give a textual briefing of the story to help users obtain data insights at a glance. It shows the data coverage rate, total number of data facts in the story, and a textual description of the story. \n\nTo facilitate an efficient story exploration on tablets and smart phones, a storyline view and a mobile view are respectively developed. The visualizations and textual descriptions of the facts in the story are grouped together and horizontally aligned in a row in the storyline view and are showed one at a time in the mobile view. Users can swipe on the touch screen to scroll the horizontal storyline in the storyline view or switch to another fact in the mobile view to navigate through the story. \n\nFinally, a fact-sheet view is also introduced, in which the visual and textual representation of all the facts are organized into a poster that can be directly downloaded and printed. This view supports an easy offline usage of the generation results. A layout algorithm is designed to enable an efficient and aesthetic representation of the story facts. Formally, the algorithm is designed to optimize the following objective: \n\n\\begin{equation}\n    f = f_s + f_d\n    \\label{eq:optimize}\n\\end{equation}\nwhere $f_s$ indicates how the areas of each fact take up in the fact sheet in proportion of their importance scores, shown in Eq.(\\ref{eq:opti_area}); $f_d$ measures whether the layout of the fact sheet matches the guideline of low intra-row distances and high inter-row distances defined in Eq.(\\ref{eq:opti_layout}).\n\\begin{equation}\n    f_s = \\frac{\\sum_{i=1}^{n} s_i \\times a_i }{\\sum_{i=1}^{n} a_i }\n    \\label{eq:opti_area}\n\\end{equation}\nwhere $n$ is the number of facts in the generated story; $s_i$ is the importance score of $i$-th fact derived from the story generation algorithm, and $a_i$ is the area $i$-th fact occupied. With $f_s$, we expect that facts with higher scores would involve larger areas to attract enough attention from readers for they bring more information.\n\nHere we denote the layout of fact sheet with $n$ facts as $C = \\{C_1, C_2, \\cdots, C_k\\}$, where $k$ is the number of rows in the fact sheet. Formally, $f_d$ is calculated as follows:\n\\begin{equation}\nf_d = f_d^{inter} - f_d^{intra}\n\\label{eq:opti_layout}\n\\end{equation}\n\\begin{equation}\nf_d^{inter} = \\frac{\\sum_{1 \\leq j < |C|} d(C_j, C_j+1)}{|C|-1}\n\\label{eq:opti_inter}\n\\end{equation}\n\\begin{equation}\nf_d^{intra} = \\frac{\\sum_{C_j \\in C} \\sum_{1 \\leq i < |C_i|} d(x_i, x_{i+1})} {n - |C|}\n\\label{eq:opti_intra}\n\\end{equation}\nwhere $d$ measures the distance between two facts. To obtain it, we first calculate the similarity between two facts as described in Sec.~\\ref{4.3} and then minus it from one, indicating low similarities lead to high distances. As for the distance between two rows, we identify it as the distance between two adjacent facts connecting rows. Finally, $f_d$ is derived by subtracting $f_d^{intra}$, the average distance inside rows, from $f_d^{inter}$, the average distance between rows. With higher value, $f_d$ represents a more compact layout of a fact sheet in terms of the logic relations among facts through the generated story.\n\\label{StoryView}\n\n\n\\subsection{Interactions}\n\nCalliope\\xspace provides a set of interaction methods to support the data story generation, fact exploration, story visualization and publishing to the public web.\n\n{\\bf Data story generation.} Calliope\\xspace allows users to fine-tune several details of the story in the story configuration panel within the storyline view. For the length of the story generated by Calliope\\xspace, the chart diversity for each fact type and the time limitation of generation, users can drag the sliders of each parameter to pick a specific value from the given range. As for the reward preference for the generation, user can drag the circle $S$ which presents the story in the reward panel, to a position where distances between the story circle and three parameter nodes capture the relevant weights accordingly. After all parameters settled, users can click the ``Generate Story\" button and wait for the story from Calliope\\xspace in a few seconds. With the generated story, Calliope\\xspace allows users to add a new fact piece or remove current fact pieces by clicking buttons, or re-order the fact pieces by simple drag-and-drop interaction in the story pieces panel. Editing the fact description is also enabled in the story pieces panel. All actions will apply in all the other views accordingly to reduce repetitive modification. \n\n{\\bf Modifying data fact.} The fact view enables users to customize the Currently selected fact by changing the fact's configuration. \nFact view provides selectors for \\textit{fact type}, \\textit{visualization}, \\textit{measure},\\textit{subspace}, \\textit{breakdown}, and \\textit{focus}. When adjusting these configuration items, visual and textual representations of the current fact will be modified. \n\n{\\bf Choosing visualization form and publishing.} \nCalliope\\xspace allows user to switch between all the visualization forms mentioned in~\\ref{StoryView} in the story visualization view.\nBy clicking the ``share\" button, Calliope\\xspace can automatically generate an online access link and a code snippet that can be embedded in other web pages for the generated data story represented in its current visualization form. When the user clicks the copy button, the online link or the code snippet will be copied accordingly. If others visit the link, they can access this data story. As for the generated \ncode snippet, user can easily embed this data story into his own web page by inserting the snippet into it.\n\n\\label{5.4}\n\\section{Data Story Editor}\n\\label{sec:editor}\n\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=\\linewidth]{figures\/UI.png}\n  \\caption{The story editor of Calliope\\xspace system contests of three major views: (1) the storyline view for story configuration, generation and storyline editing, (2) the fact view for fact editing, and (3) the story visualization view for the visual data story preview and sharing.} \\label{fig:ui}\n\\end{figure}\n\nIn this section, we describe the design details of each views in the data story editor and the corresponding interactions of the system. \n\n\\subsection{User Interface}\nThe story editor, as shown in Fig.~\\ref{fig:ui}, consists of three major views: the storyline view (Fig.~\\ref{fig:ui}-1), the fact view (Fig.~\\ref{fig:ui}-2), and the story visualization view (Fig.~\\ref{fig:ui}-3). In the storyline view, a user can upload a spreadsheet, set the story generation goal, and adjust the reward function in a group of configuration panels (Fig.~\\ref{fig:ui}-1(a)). The generated data facts are shown in a row (Fig.~\\ref{fig:ui}-1(b)), in which a user can remove a fact or change the generated narrative order based on his\/her own preferences. Each fact is visualized by a chart and captioned by a generated text description (\\textbf{R3}). When a fact is selected, its data details, importance measures, and visual and textual representations will be shown in the fact view, which are also editable (\\textbf{R4}). The generated data story can be visualized in the story visualization view through three visualization modes: (1) the storyline mode, (2) the swiper mode, and (3) the fact sheet mode, respectively designed for representing the story on laptops\/tablets, smart phones, and in printouts to facilitate a flexible story communication and sharing (\\textbf{R5}).\n\nLet's consider the following scenario for using Calliope\\xspace system. Jean is a data journalist. Her primary job is to tracing important news events, collect, analyze, and visualize the corresponding data to write news stories with the data support. Her job frequently requires a quick response to emergencies, but data analysis and visualization usually take days to finish. She uses Calliope\\xspace to help her on these tasks. Recently, she is monitoring the public health emergency on the propagation of CoVID-19 virus in China. A data containing everyday infections and deaths in different provinces and cities are collected. She uploads the data into Calliope\\xspace system, sets the story length as 6 (i.e., contains at most 6 data facts) in the story configuration panel, and clicks the generate button. Only after a few seconds, the system automatically outputs a sequence of related data facts represented by charts and captioned by text in a logic order in the storyline view. She clicks a generated fact to edit it in the fact view, where she can change the generated description, chart and the fact types, and the data details based on her own observations and preferences. After a few edits on the story facts, Jean switched the visual representation style to the swiper mode in the story visualization view and shared the results with her colleague via a link by clicking the share button for a future discussion.\n\n\n\n\n\\subsection{Fact View}\nThe fact view (Fig.~\\ref{fig:ui}-2) is a detailed view that not only represent the selected story piece but also enables users to customize this fact by changing the configuration. \nFact's configuration items include the following: \\textit{fact type}, \\textit{visualization}, \\textit{measure},\\textit{subspace}, \\textit{breakdown}, and \\textit{focus}. Different fact types may have specific combination of the configuration items mentioned above according to table~\\ref{tab:fact}.\nThrough the adjustment of these configuration items, the nature language(\\ref{5.2.1}) and the visualization(\\ref{5.2.2}) of the fact will be generated in real time. Also, the information quantity and significance score of the selected fact will be updated accordingly. \n\n\\subsubsection{Facts to Nature Language}\nEach facts is equivalent to a context free grammar which can be used to generate nature language. Describe how the description text are generated for each fact type.\n\\label{5.2.1}\n\n\\subsubsection{Facts to Visualization}\n\nBased on our preliminary survey, we design a rule based method to match fact types with different chart types. For ensuring the visual diversity, we add some alternative designs of each basic chart based on statistical results. For example, donut chart and half donut chart can be  the variants of the basic pie chart for users to choose.\n\nThe visualization changes accordingly with the adjustment of fact configuration. Among the configuration items, \\textit{subspace} subspace is used to specify the range of data to be visualized, \\textit{measure} and \\textit{groupby} respectively correspond to the measure and dimension that make up the chart, and \\textit{focus} is used as the highlighted part in the chart.\n\\label{5.2.2}\n\n\\subsection{Story Visualization View}\nThe story visualization view provides a variety of visualization forms to represent the generated data story line for different application scenarios. In particular, a summarization view is first provided to give a textual briefing of the story to help users obtain data insights at a glance. It shows the data coverage rate, total number of data facts in the story, and a textual description of the story. \n\nTo facilitate an efficient story exploration on tablets and smart phones, a storyline view and a mobile view are respectively developed. The visualizations and textual descriptions of the facts in the story are grouped together and horizontally aligned in a row in the storyline view and are showed one at a time in the mobile view. Users can swipe on the touch screen to scroll the horizontal storyline in the storyline view or switch to another fact in the mobile view to navigate through the story. \n\nFinally, a fact-sheet view is also introduced, in which the visual and textual representation of all the facts are organized into a poster that can be directly downloaded and printed. This view supports an easy offline usage of the generation results. A layout algorithm is designed to enable an efficient and aesthetic representation of the story facts. Formally, the algorithm is designed to optimize the following objective: \n\n\\begin{equation}\n    f = f_s + f_d\n    \\label{eq:optimize}\n\\end{equation}\nwhere $f_s$ indicates how the areas of each fact take up in the fact sheet in proportion of their importance scores, shown in Eq.(\\ref{eq:opti_area}); $f_d$ measures whether the layout of the fact sheet matches the guideline of low intra-row distances and high inter-row distances defined in Eq.(\\ref{eq:opti_layout}).\n\\begin{equation}\n    f_s = \\frac{\\sum_{i=1}^{n} s_i \\times a_i }{\\sum_{i=1}^{n} a_i }\n    \\label{eq:opti_area}\n\\end{equation}\nwhere $n$ is the number of facts in the generated story; $s_i$ is the importance score of $i$-th fact derived from the story generation algorithm, and $a_i$ is the area $i$-th fact occupied. With $f_s$, we expect that facts with higher scores would involve larger areas to attract enough attention from readers for they bring more information.\n\nHere we denote the layout of fact sheet with $n$ facts as $C = \\{C_1, C_2, \\cdots, C_k\\}$, where $k$ is the number of rows in the fact sheet. Formally, $f_d$ is calculated as follows:\n\\begin{equation}\nf_d = f_d^{inter} - f_d^{intra}\n\\label{eq:opti_layout}\n\\end{equation}\n\\begin{equation}\nf_d^{inter} = \\frac{\\sum_{1 \\leq j < |C|} d(C_j, C_j+1)}{|C|-1}\n\\label{eq:opti_inter}\n\\end{equation}\n\\begin{equation}\nf_d^{intra} = \\frac{\\sum_{C_j \\in C} \\sum_{1 \\leq i < |C_i|} d(x_i, x_{i+1})} {n - |C|}\n\\label{eq:opti_intra}\n\\end{equation}\nwhere $d$ measures the distance between two facts. To obtain it, we first calculate the similarity between two facts as described in Sec.~\\ref{4.3} and then minus it from one, indicating low similarities lead to high distances. As for the distance between two rows, we identify it as the distance between two adjacent facts connecting rows. Finally, $f_d$ is derived by subtracting $f_d^{intra}$, the average distance inside rows, from $f_d^{inter}$, the average distance between rows. With higher value, $f_d$ represents a more compact layout of a fact sheet in terms of the logic relations among facts through the generated story.\n\\label{StoryView}\n\n\n\\subsection{Interactions}\n\nCalliope\\xspace provides a set of interaction methods to support the data story generation, fact exploration, story visualization and publishing to the public web.\n\n{\\bf Data story generation.} Calliope\\xspace allows users to fine-tune several details of the story in the story configuration panel within the storyline view. For the length of the story generated by Calliope\\xspace, the chart diversity for each fact type and the time limitation of generation, users can drag the sliders of each parameter to pick a specific value from the given range. As for the reward preference for the generation, user can drag the circle $S$ which presents the story in the reward panel, to a position where distances between the story circle and three parameter nodes capture the relevant weights accordingly. After all parameters settled, users can click the ``Generate Story\" button and wait for the story from Calliope\\xspace in a few seconds. With the generated story, Calliope\\xspace allows users to add a new fact piece or remove current fact pieces by clicking buttons, or re-order the fact pieces by simple drag-and-drop interaction in the story pieces panel. Editing the fact description is also enabled in the story pieces panel. All actions will apply in all the other views accordingly to reduce repetitive modification. \n\n{\\bf Modifying data fact.} The fact view enables users to customize the Currently selected fact by changing the fact's configuration. \nFact view provides selectors for \\textit{fact type}, \\textit{visualization}, \\textit{measure},\\textit{subspace}, \\textit{breakdown}, and \\textit{focus}. When adjusting these configuration items, visual and textual representations of the current fact will be modified. \n\n{\\bf Choosing visualization form and publishing.} \nCalliope\\xspace allows user to switch between all the visualization forms mentioned in~\\ref{StoryView} in the story visualization view.\nBy clicking the ``share\" button, Calliope\\xspace can automatically generate an online access link and a code snippet that can be embedded in other web pages for the generated data story represented in its current visualization form. When the user clicks the copy button, the online link or the code snippet will be copied accordingly. If others visit the link, they can access this data story. As for the generated \ncode snippet, user can easily embed this data story into his own web page by inserting the snippet into it.\n\n\\label{5.4}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Evaluation}\nThe evaluation of Calliope\\xspace system consists of two parts, which respectively estimate the quality of the generated story logic and the visual data story content.\n\n\nWe evaluate Calliope\\xspace via two controlled experiments to evaluate the generate logic and case study with 10 expert users followed by a series of expert interviews to collect their feedback.\n\n\\begin{figure*}[tb]\n  \\centering\n  \\includegraphics[width=\\linewidth]{figures\/case23.png}\n  \\vspace{-1.5em}\n  \\caption{Two alternative visualization modes, (a)\\textit{\\textbf{swiper}}\\xspace mode and (b) \\textit{\\textbf{factsheet}}\\xspace mode, that are developed to facilitate the story exploration, sharing and communication via mobile phones and printouts.} \n  \\label{fig:vmodes}\n  \\vspace{-1em}\n\\end{figure*}\n\n\\subsection{Evaluation of the Generated Logic}\nTwo controlled experiments were performed to evaluate the generated story logic. The first one is designed to compares the logic generated by users and \n\nIn the first experiment, we first randomized the orders of a set of data facts in a visual data story generated by Calliope\\xspace and then asked a group of users to restore the order of the data facts based on their own observations and understandings of the data facts. Finally, we checked the consistency of the human generated logic orders and the logic order produced by Calliope\\xspace. In this experiment, we recruited 20 participants (17 female) aged from 22 to 30 (mean 26). 11 of them had data analysis experiences and 14 of them had some basic knowledge about data visualization. 3 participants have experience in storytelling and 16 are with the skill of design. Participants took 19.5 minutes in average ($SD=10$) to finish the study.\n\n\n\\subsubsection{User Study \\uppercase\\expandafter{\\romannumeral1}}\nThe first study evaluates the logicality of the data stories of Calliope\\xspace by comparing the orders of facts from Calliope\\xspace and human.\n\n\\textbf{Hypotheses} \nAs we apply a logic-oriented MCTS to explore the space of the input data and generate data facts as a data story, we hypothesize that:\n\\begin{enumerate}\n\\itemsep -1mm\n\\item[{\\bf H1}] The logic order of facts in the story generated by Calliope\\xspace would strongly correlated with users' logical reasoning.\n\\end{enumerate}\n\n\\textbf{Procedure and Task} We collected three datasets in different topics to generate data stories for the study. \\textit{CarSales} dataset(275 rows, 4 columns) describes the sales of different automobile brands from 2007 to 2011; \\textit{COVID-19} dataset(903 rows, 6 columns) provides epidemic data around the world from Mar. 1, 2020 to Mar. 21, 2020, recording the number of infections, deaths and cures everyday; \\textit{Startup Failures} dataset(1234 rows, 6 columns) reports the situation of companies that died through the tide of the new economy in China from 2010 to 2019, showing these companies' industry information, fund status they ended up with, reasons for failure, etc. For each dataset, we uploaded it to Calliope\\xspace and generated 4 stories with length 6, then shuffled the order of each story. \n\nIn the study, we presented brief introduction of each dataset before the questions. For each question, participants were asked to re-arrange the facts from an out-of-order story based on the information from charts and descriptions, to form a logical and reasonable story. After answering all 12 questions of re-ordering, participants were asked to write down their judgement and their personal information. \n\n\\begin{figure}[htb]\n  \\centering\n  \\includegraphics[width=\\linewidth]{figures\/userstudy1.png}\n  \\vspace{-2em}\n  \\caption{The evaluation results of the story logic: (a) the average Kendall's $\\tau_b$ values and standard errors on each dataset; (b) the average Kendall's $\\tau_b$ values and standard errors of Human and Calliope\\xspace; (c) the means and standard errors of rates on logic of DataShot and Calliope\\xspace by participants using a 5-point Likert scale.} \n  \\label{fig:userstudy_tau}\n  \\vspace{-1.7em}\n\\end{figure}\n\n\\textbf{Results}\nTo measure the correspondence between the rankings of Calliope\\xspace and of participants, we used Kendall's $\\tau_b$ correlation~\\cite{kendall1945treatment}  for evaluation, with value lies between -1(reversed order) and 1(original order). First we collected the participant scores of the Kendall's $\\tau_b$ on the orderings between participants and Calliope\\xspace and then averaged them on each dataset, as shown in Fig.~\\ref{fig:userstudy_tau}(a). The results indicate that there exists some correlation between two rankings, especially in the latter two datasets, which are more familiar to the participants.   \n\nTo understand whether Calliope\\xspace really perceives logic and generate meaningful stories, we invited an expert who knows much about the three datasets to assign reasonable orders of the facts of each story. Then we obtained the Kendall's $\\tau_b$ correlation between participants' rankings and expert's rankings on 12 questions, marked as \\textit{Human}, and the Kendall's $\\tau_b$ correlation between Calliope\\xspace's rankings and expert's rankings on 12 questions, marked as \\textit{Calliope\\xspace}. We conducted an unpaired t-test to examine the differences on the $\\tau_b$ scores of two groups. The result showed in Fig.~\\ref{fig:userstudy_tau}(b) indicates that there is no significant difference ($t(22)=.097, p=.92$) of participants($M=.50, SD=.13$) and Calliope\\xspace($M=.55, SD=.17$) when comparing both of their rankings with that of the expert. Hence \\textbf{H1} is accepted.\n\nWhen asked about the criteria of a logical story, the answers can be summarized into four aspects. \nFirst, most of the participants valued the consistency of stories. \\textit{``I would pay attention to the key words in the facts, and place those with similar key words together\"}, one participants said. Since our stories were backed up by data, some participants arranged the facts by tracking the relevant data scopes as well. \nSecondly, participants would try to organize data facts using common discourse relations. Among the answers(9 of 20), the overall-detail structure of stories was mentioned most frequently, with which the story gives the whole picture in the beginning, followed by a series of details. \nAlso, participants would interpret the semantic contents of each fact. one commented that \\textit{``I assigned the orders through the emotions from negative to positive for the COVID-19 dataset, first reporting the number of infections, then deaths and cures.\"} while another said that \\textit{``I followed the chronological order and put facts depicting infections of COVID-19 ahead of those with deaths situation.\"} \nParticularly, since Calliope\\xspace presents data stories with visualizations, some participants(5 of 20) mentioned that they would take the visualizations into account. One commented that one of his judgements was \\textit{``the correlation of types of charts in each fact\"}.\n\n\\subsubsection{User Study \\uppercase\\expandafter{\\romannumeral2}}\nTo further evaluate the quality of the output of Calliope\\xspace, we conducted the second user study to validate whether the data stories generated by our approach would be more logical and reasonable than existing work. For comparison, we choose DataShot\\cite{wang2019datashot} as a baseline, which automatically generates ``random facts\" sheet from the tabular data. \n\n\\textbf{Hypotheses} \nAs the facts in our stories follow a logical order and are displayed sequentially in all visualization modes, we hypothesize that:\n\\begin{enumerate}\n\\itemsep -1mm\n\\item[{\\bf H2}] The visual data story generated by Calliope\\xspace would be better for reading than the story with facts in random order under the same topic significantly.\n\\end{enumerate}\n\n\\textbf{Participants} In this study, we invited 20 participants (14 female) whose ages range from 23 to 30 (mean: 25). All the participants are students majoring in design. 13 of them have the experience in designing infographic posters. They took an average of 13 minutes ($SD=12$) to complete this study.\n\n\\textbf{Procedure and Task} Our study followed a 2 (visualization tool) $\\times$ 3 (dataset) mixed design. We collected the three datasets presented in DataShot and uploaded them to Calliope\\xspace to generate stories in the \\textit{\\textbf{factsheet}}\\xspace mode. After the processing, we got 6 factsheets, including 3 of them collected from the paper of DataShot and the other 3 generated from Calliope\\xspace. To avoid other factors that interfere with the study, we unified the design style for the factsheets from both visualization tools. We also marked the sequential number for all facts in each factsheet to guide participants. Finally, we randomized the order of the 6 factsheets in each questionnaire.\n\nEach participant performed 6 trials in the questionnaire. In each trail, participants were shown one factsheet and asked to read it following the sequential order from left to right, top to bottom. After each trail, they rated the logic of the factsheet in a 5-point Likert scale ranging from ``Very Unclear\" to ``Very Clear\".\n\n\\textbf{Results} We analyzed a total of 120 user ratings and conducted unpaired t test between DataShot and Calliope\\xspace. The result (Fig.~\\ref{fig:userstudy_tau}(c)) shows that there is a significant difference ($t(118)=3.99, p<.01$) between the rating of DataShot($M=3.02, SD=.95$) and Calliope\\xspace($M=3.72, SD=.98$), which means that stories generated by Calliope\\xspace have clearer logic than random facts stories for reading and comprehension. Therefore \\textbf{H2} is accepted. \n\n\\subsection{Case Studies and Expert Interview}\n\nTo evaluate the utility of our system, we conducted in-depth interviews with ten experts. Due to the application fields of the system, we chose experts from three relevant areas, including 4 professional data journalists, 3 data analysts in BI department, and 3 visualization researchers. In the following section, these 10 participants are denoted as \\textbf{P1}, \\textbf{P2}, ..., \\textbf{P10}, where \\textbf{P1-P4} are data journalists, \\textbf{P5-P7} are data analysts, and \\textbf{P8-P10} are visualization researchers.\n\n\\textbf{Interview methodology} \nThe interview was started with a 10-minute system introduction. We introduced the system interface and each module to the interviewees. \nThen all the participants were asked to explore our system freely and generate a high quality data story with a provided dataset in 15-20 minutes. During the exploration process, if participants have any questions with Calliope\\xspace, we'll provide hints to them promptly. After finishing exploring and completing a data story, the user was asked to use the sharing feature to generate a link as a delivered work.\nThen we conducted a brief interview with the user, including the following three aspects: the generated data story, visual and interaction design and the overall system. \nAt the end of the interview, the participants were asked to fill out a questionnaire to assess these three aspects using a 5-point Likert scale.\nOn average, each of the interviews last approximately one hour.\n\n\\textbf{Case Studies}\nWe collected all visual data stories authored by participants. Here we present three of them with different spreadsheets, including CarSales, COVID-19, and Startup Failures.\n\n\\textit{CarSales.} \\textbf{P7} (data analyst, male, 34) generated and authored this story about CarSales, which is shown in Fig.~\\ref{fig:vmodes}(a). During the whole authoring process, he acknowledged that the generated story is well structured which can be easily understood and provided an overview of the dataset. Thus, he used the generated story as backbone and edited facts to convey his insights about this data. The final story is start with the sales trend from 2007 to 2011. He thought the financial crisis is to blame for this decreasing trend. Then the story shows the rank of sales for each brand and the top 1 Ford accounts for 25.04\\% of the total sales. After that, it displays the rank of sales for different vehicle models and the comparison between top 1 (SUV) and top 3 (Subcompact). In the end of the story, P7 indicated that the sales of Subcompact model is the only one not recovered after the financial crisis. After finishing the story, he shared it in \\textit{\\textbf{swiper}}\\xspace mode which he thought is the best way to spread via phones quickly. \n\n\\textit{COVID-19.} This story is generated and authored by \\textbf{P4} (data journalist, female, 25), which is shown in Fig.~\\ref{fig:teaser}. P4 is an experienced data journalist. Because of her daily work needing the cooperation with designers and engineers, she appreciated the effectiveness of this system that can let her create data news on her own. After the story generation, she use the data story editor to reorder facts to ensure the attractiveness of the story based on her background knowledge. The story first presents the distribution of death cases in the provinces of China during March and points out that the maximum is 42 on March 2. Then it displays the distribution of death cases geographically, followed by the detail data in Hubei. After that, the story is concluded with the total death and recovered cases in March, indicating that China has turned for the better. She decided to represent the story in the \\textit{\\textbf{storyline}}\\xspace mode, which can be embedded into websites for news reports easily and read in web browsers.\n\n\\textit{Startup Failures.} \\textbf{P10} (visualization researcher, female, 29) was interested in the spreadsheet about startup failures in the past 10 years. She has no background information about this data before. Instead of the data exploration, she started with the story generation in the Calliope\\xspace system. \nThis generated story begins with the trend and distribution of failed startups in terms of time and space respectively, then represents the survival time ranking of failed startups in each industry and the situation of failed startups in each fund status. An outlier is also detected that no-funding is abnormal among all fund statuses. At the end of the story it shows the distribution of failure reasons of startups. After viewing the initial generated story, she pointed out that the logic and statement of generated story is very clear. Then she checked the schema of the data table and she found the generated story covers all aspects of the data. Thus she felt satisfied with the story without making any significant changes and only tried to adjust the visualization of the facts. Finally, she chose to print the story in the \\textit{\\textbf{factsheet}}\\xspace mode as shown in Fig.~\\ref{fig:vmodes}(b). \n\n\\textbf{Participant Feedback} \nAfter the interview, we summarized all the participants' feedback and collated the results of user ratings of Calliope\\xspace.\nFig.~\\ref{fig:interview} shows the results of a 5-point Likert scale, which express how much the users agree or disagree with 8 standards on 3 aspects of our system. Generally speaking, all the expert users gave promising and positive feedback on Calliope\\xspace.\n\n\\begin{figure}[tbh]\n  \\centering\n  \\includegraphics[width=\\linewidth]{figures\/interview.png}\n  \\vspace{-1.5em}\n  \\caption{The interview ratings of Calliope\\xspace on a 5-point Likert scale.} \n  \\label{fig:interview}\n  \\vspace{-0.5em}\n\\end{figure}\n\n\\textit{\\textbf{The generated data story.}} \nAll the experts agreed that the data stories generated by Calliope\\xspace can effectively convey the necessary information in the data. \n\\textbf{P10} commented, \\textit{``Each part of the story has the information that I want to present, and the story has both the overall content and the details. It's amazing!\"}. \nExperts commented regarding the data exploration that, \\textit{``This system can effectively help users explore data in order to discover intriguing data facts\"}(\\textbf{P3}), and \\textit{``The system automatically generates structured framework of data stories about the dataset, helping users to explore and understand the data easily\"}(\\textbf{P7}).\nMeanwhile, participants commented on the logic of the generated story, \\textit{``The logic of the generated story is very clear, especially when the logicality attribute in the reward panel was adjusted to a larger value\"} (\\textbf{P9}), and\n\\textit{``The logical order of the generated story can help the readers get to the point\"}(\\textbf{P1}).\nMoreover, experts also reported that the current presentation of data facts is meaningful and easy to understand.\n\\textbf{P5} commented, \\textit{``Instead of presenting multi-dimensional data in complex visualizations, the system arranges the appropriate dimensions and measures, and uses a series of simple visualizations to make the generated story easier to understand.\" }\n\n\\textit{\\textbf{Visual and interaction design.}}\nAll experts provided positive comments on the visual and interaction design of Calliope\\xspace.\nRegarding the 3 visual forms of the data story, expert users acknowledged that these 3 visual forms are very useful and can be selected for different scenarios. \nExperts commented, \\textit{``Each fact in the \\textit{\\textbf{factsheet}}\\xspace mode is sized according to how important it is. Therefore, it makes sense that this layout allows people to see relatively important information of the data story at a glance. It's also very nice that the \\textit{\\textbf{factsheet}}\\xspace can be printed as a PDF\"}(\\textbf{P3}); \n\\textit{``The \\textit{\\textbf{swiper}}\\xspace mode shows a fact per page, which is suitable for mobile devices\"}(\\textbf{P9});\n\\textit{``The \\textit{\\textbf{storyline}}\\xspace mode is a great form for presenting stories in web pages\"}(\\textbf{P7}).\nAs for the storyline view and the fact view, \\textbf{P9} appreciated the interaction between these two views. He commented, \\textit{``I can roughly explore the whole data story through the storyline view, and then further explore the data details on each of its fields of the selected fact through the fact view.\"}\n\\textbf{P3} also expressed her thoughts in that, `\\textit{`The storyline view enable me to add, delete, reorder and modify generated data facts, which is really convenient for me to make a data story on my preference\"}\nAt the same time, two experts(\\textbf{P3} and \\textbf{P9}) mentioned that the adjustment of reward configuration can help users with different needs to generate stories with different emphasis, \\textit{``For example, data journalists like me may focus more on the logicality of a data story, while data analysts may prefer integrity because they may want to cover as much data as possible\"}(\\textbf{P3}).\n\n\\textit{\\textbf{The overall system.}}\nThese 10 experts all gave an overall assessment of Calliope\\xspace from the perspective of their areas of expertise.\n\\textbf{P1}, a data journalist with four years' experience, commented that this system can help users generate stories quickly and intelligently, compared to traditional journalism, which uses templates to draw charts. \n\\textbf{P2} also appreciated the effectiveness of Calliope\\xspace. she commented, \\textit{``This system is very useful and gives a lot of effective advice, especially for those who have no experience in data visualization or data journalism.\"}\nAnother data journalist \\textbf{P4} said the system enable users to quickly create graphics for some easy-to-read articles with little interaction, thus there is no need for the designers to typesetting. She also commented that charts provided in Calliope\\xspace are commonly used in journalism, such as line charts, bar charts, pie charts, etc.\nThe data journalist offered insights of our system from a news perspective, while data analysts came up with insights from the perspective of data analysis.\n\\textbf{P7} commented, \\textit{``With this system, analysts can quickly generate stories that help quickly summarize the information of the data without much exploration or insight. It is really convenient in simple application scenarios.\"}\nIn addition, data visualization experts also put forward their own views.\n\\textbf{P9} commented, \\textit{``The system lays out the overall story frame of a dataset, giving us a direction on how to create a data story. It is worth mentioning that the visual design is also very good.\"}\n\\textbf{P10} said, \\textit{``The overall system is very useful, especially for people who are not professional in data visualization.\"}\n\n\n\\textbf{Discussions}\nIn spite of the positive feedback mentioned above, some suggestions for our system were also put forward, which can be summarized in the following aspects:\n\\begin{itemize}[leftmargin=10pt,topsep=2pt]\n\\itemsep -.5mm\n    \\item {\\textbf{Story optimization.}} \n    Both \\textbf{P2} and \\textbf{P9} said that the automatic-generated text may be suitable for Simple application scenarios or initial story exploration. However, the formal scenarios, such as professional news, have high requirements of expression. They suggested the statement of the story can be more diverse and vivid. \n    Journalists are more concerned about the specific cases behind the data, hoping to dig deeper into the information behind the data.\n    Meanwhile, data analysts also pointed out that the data story just describe the data fact without further insight which can guide users on how to take actions.\n    These concerns indicate the gap between generated and real-world data stories. Since Calliope\\xspace provides editing functions, users can further optimize the generated stories to meet their needs on various aspects. \n    \n    \\item {\\textbf{Semantic understanding.}} Three experts (\\textbf{P3}, \\textbf{P6}, \\textbf{P7}) pointed out that the generated data stories are lack of semantic understanding and common sense to build bridges between information. For example, in the news about COVID-19, the number of confirmed cases is usually depicted before the number of deaths and cures. However, our system does not have this background knowledge. We acknowledge it as the limitation of the current system. Spreadsheets in the real world are usually relevant to some domain knowledge, which has significant impact on the story structure. We think it is a promising direction for future work.\n    \n    \\item {\\textbf{Visualization.}}\n    \\textbf{P2}, \\textbf{P3} and \\textbf{P4} all suggested that the system should provide the ability to customize charts(such as changing color, transparency, axis labels, etc.) and provide more visual designs for users to choose. Regarding the \\textit{\\textbf{factsheet}}\\xspace mode, 3 of the experts suggested that the reading order of the data facts should be more clear. Moreover, \\textbf{P10} suggested, \\textit{``I'd appreciate it if there was a visualization form of slides, which can be helpful when I need to prepare a presentation.\"} As future work, we plan to integrate more charts and layouts to enrich our system.\n\\end{itemize}\n\nOverall, Calliope\\xspace is appreciated by these ten experts. These interviews also pointed out the next steps of the system.\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nThe minimum dominating set (MDS) problem is  notoriously difficult and yet extremely important because of its numerous applications. Recall that for a graph $G=(V,E)$, set $D\\subseteq V$ is called a {\\it dominating set} if every vertex in $V\\setminus D$ has a neighbor in $D$. For general graphs $G$ even finding a  $C\\log{n}$-approximation for some constant $C$ where $n$ is the order of $G$  is NP-hard \\cite{RS}. At the same time, the problem becomes much more tractable when restricted to certain classes of graphs. In particular, assumptions about the sparsity of graphs, measured in various ways, can make the MDS problem easier to approximate. The situation is not much different in the distributed setting, where on one hand, only a $O(\\log{\\Delta})$-approximation for general graphs is known  \\cite{KMW}, and on the other, the problem becomes significantly easier for special cases of graphs, like for example planar graphs \\cite{LOW}.\n\nIn this paper, we shall consider an important generalization of the MDS problem, the distance-$k$ minimum dominating set problem, and we shall give fast deterministic distributed approximations in the {\\it Local} model in the case the underlying network satisfies certain sparsity conditions.  \n\nThe term distance-$k$ dominating set was given by Henning et al. \\cite{henning}. For a graph $G=(V,E)$ and $k\\in \\mathbb{Z}^+$, set $D\\subseteq V$ is called a {\\it distance-$k$ dominating set} if every vertex $v\\in V$ is within distance $k$ of a vertex from $D$. In particular, $1$-distance dominating set is a dominating set. The problem has many applications in networking and other areas of computer science. Maybe the most natural applications of  distance-$k$ dominating sets arise when considering the problem of allocating centers in a network that can share resources with the remaining vertices of the graph when needed \\cite{alloc}.\n\\subsection{Related Work}\nIn the distributed setting, the MDS problem has been extensively studied for many different classes of sparse networks. Lenzen et al. \\cite{LOW} gave a constant-factor distributed approximation of a minimum dominating set that runs in a constant number of rounds in planar graphs in the {\\it Local} model of computations. Using more careful analysis, Wawrzyniak \\cite{ww-ipl} improved the approximation ratio and showed that this algorithm gives in fact a 52-approximation.   Amiri et al. \\cite{amiri} showed that a small modification of the algorithm from \\cite{LOW} also gives a constant-factor approximation of a minimum dominating set in graphs of the bounded genus, and even more generally in graphs with no $K_{3,t}$-minor for some constant $t$. In fact, a further generalization is given in \\cite{CHWW} where the authors give a constant-time distributed algorithm for $K_t$-minor-free graphs.  In addition, using the methods from \\cite{CHW}, it is possible to improve the approximation factor in these classes of graphs at the expense of the time complexity. Specifically, it can be proved that there is a distributed algorithm which given $\\epsilon>0$ finds a $(1+\\epsilon)$-approximation of a MDS in a graph $G=(V,E)$ that is $K_t$-minor-free in $O(\\log^*{|V|})$ rounds.\nFor graphs of a constant arboricity, a much more general class of graphs, there is a randomized algorithm of Lenzen and Wattenhofer \\cite{LW-arb} that finds a constant approximation in time which is $O(\\log{|V|})$ rounds with high probability.\nIn addition, tight results are known for outerplanar graphs. Recently, using an analysis of a maximal counterexample, Bonamy, Cook, Groenland, and Wesolek \\cite{B} manged to prove very tight bounds for the approximation ratio for MDS in the case of outerplanar graphs. Specifically they showed the following two facts.\n\\begin{itemize}\n    \\item There is a deterministic $5$-approximation of the MDS in outerplanar graphs.\n    \\item There is no $(5-\\epsilon)$-approximation for outerplanar graphs for any $\\epsilon>0$.\n\\end{itemize}\nVery little is known about distributed algorithms for distance-$k$ dominating sets when $k>1$ as the problem becomes significantly different when $k$ increases making it impossible to adapt solutions for $k=1$. Amiri et al. \\cite{amiri-ossona} gave a constant-factor approximation for the minimum distance-$k$ dominating set problem in graphs $G=(V,E)$ of bounded expansion in $O(\\log{|V|})$ rounds (for a fixed $k$) in the more restrictive {\\it Congest$_{BC}$} model. \n\nThe main motivation for our work  comes from the recent paper by Amiri and Wiederhake \\cite{AmiriWieder} who managed to provide a first constant approximation algorithm in a constant number of rounds for distance-$k$ domination in graphs of bounded expansion of high girth (i.e. graphs that are sparse and are trees locally).  In fact, we will use the very same procedure from \\cite{AmiriWieder}, but give a different argument in the first part of the paper as we will examine a different class of graphs. Note that the girth assumption in \\cite{AmiriWieder} was related to a previous work on lower bounds that were established for graphs of high girths and \nis absolutely critical to their analysis. It is this assumption that we will get rid of in the current paper (at the expense of dealing with graphs with no $K_{2,t}$-minor rather than a much more general class of graphs of bounded expansion). Therefore, our paper is a step towards constant time, constant approximation algorithms for minimum distance-$k$ domination in graphs of arbitrarily small girth for which constant time approximation are known.\n\nIt is worth mentioning that the problem for $k>1$ seems to be genuinely different than the classical MDS problem, especially in the realm of sparse graphs. For example, the probabilistic algorithm from \\cite{LW-arb} is specific to the case $k=1$, and the methods from \\cite{CHW} used to ameliorate the approximation ratio when a constant approximation is furnished are again applicable only to the regular distance-$1$ domination. \n\n\\subsection{Summary of results}\nWe will work in the {\\it Local} model of computations and assume throught the paper that $k\\geq 2$. Although the first algorithm is identical to the algorithm from \\cite{AmiriWieder} which works in the {\\it Congest} model, the algorithm of Amiri et al.  exploits the fact that graphs are locally trees to allow for a {\\it Congest} model implementation. Since the graphs considered in this paper can have many short cycles, the algorithm works only in the {\\it Local} model. In addition, our algorithm for the $(1+\\epsilon)$-factor approximation heavily relies on the assumptions of the {\\it Local} model. In this model, vertices correspond to computational units, and computations are synchronized. In each round, a vertex can send, receive messages from its neighbors, and can perform individual computations. In addition, we assume that vertices have unique identifiers and denote the identifier of $v$ by $ID(v)$.\n\n\nAlthough our results are stated for graphs with no $K_{2,t}$-minor, an important subclass of this class  is outerplanar graphs that have no $K_{2,3}$-minor and no $K_4$-minor, that is graphs that admit a planar embedding in $\\mathbb{R}^2$ such that all vertices lie on the boundary of the outer face. It would be possible to phrase the main result of the first part of the paper in a more general language of graphs of bounded expansion that are locally $K_{2,t}$-minor-free, but this would require additional terminology and the benefit seems quite minuscule. \n\nWe will prove the following results. First, we will show that there is a distributed algorithm which finds a constant-approximation of a minimum distance-$k$ dominating set in graphs with no $K_{2,t}$-minor in a constant time which depends on $t$ and $k$ (Theorem \\ref{const-approx}). Second, we will show that a suitable modification of methods from \\cite{CHW} gives a $(1+\\epsilon)$-approximation of the $k$-MDS problem in $O(\\log^*{|V|})$ rounds in graphs $G=(V,E)$ that are $K_{2,t}$-minor-free (Theorem \\ref{main-approx-thm}).\n\nFinally we show that it is possible to find a $(1+\\epsilon)$-factor approximation that runs in  $O(\\log^*{|V|})$ rounds in  $K_t$-minor-free graphs $G=(V,E)$ of a constant maximum degree (Theorem \\ref{const-deg-thm}).\n\nThe rest of the paper is structured as follows. In the next section, we shall fix some terminology and prove a fact about $K_{2,t}$-minor-free graphs that will be useful in the main part of the paper. Section \\ref{sec-const} contains the analysis of the constant approximation algorithm and Section \\ref{sec-eps} discusses the $(1+\\epsilon)$-approximation.  \n\n\\section{Preliminaries}\nLet $G=(V,E)$ and $H=(W,F)$ be graphs. We say that $G$ contains an $H$-minor if $H$ can be obtained from a subgraph of $G$ by a sequence of edge contractions. More formally, $H$ is a minor of $G$ if for some subgraph $G'=(V',E')$ of $G$ we can partition $V'$ into sets $V_1, \\dots, V_l$ so that each $G'[V_i]$ is connected and the graph obtained from $G'$ by contracting every $V_i$ to a vertex is isomorphic to $H$. (Note that we discard all parallel edges or loops if they appear when contracting connected subgraphs.) We will be mainly interested in graphs $G$ that are $H$-minor-free (i.e. have no $H$-minor) for $H=K_{2,t}$ where $t\\in \\mathbb{Z}^+$ is a constant. Clearly, if a graph has no $K_{2,t}$-minor then it has no $K_{2,t+1}$-minor and  so assuming there is no $K_{2,t+1}$-minor is weaker than supposing no $K_{2,t}$-minor. Recall that if $G$ is planar, then $G$  has not $K_{3,3}$-minor and if it is outerplanar, then it has no $K_{2,3}$-minor. Consequently, our results apply to outerplanar graphs as $t$ can be a large but fixed positive integer.\n\nA subdivision of a graph $H$, denoted $TH$, is obtained from $H$ by replacing its edges with internally disjoint paths of length at least one.\n\nWe will follow terminology from \\cite{diestel} but will recall main concepts used throughout the paper. In particular, a path between two vertices does not contain a vertex more than once, a walk can contain repeated vertices or edges.\n\nFor two distinct vertices $u,v\\in V$, a $u,v$-path is a path which ends in $u$ and $v$.  \nWe use $d_G(u,v)$ to denote the distance between $u$ and $v$ in $G$, that is, the length of a shortest $u,v$-path (allowing for $u=v$). \nFor a subset $Q\\subseteq V$, a $Q$-path is a path $P$ such that $V(P)\\cap Q$ contains only the endpoints of $P$. In particular, every vertex of $Q$ is a trivial $Q$-path. For two disjoint sets $Q_1,Q_2$, a $Q_1, Q_2$-path is a path that has one endpoint in each of the $Q_i$'s and no other vertices in $Q_1\\cup Q_2$. In the case $Q_1=\\{u\\}$, we will use $u,Q_2$-paths for $\\{u\\}, Q_2$-paths. We denote by $uPv$ a the subpath of $P$ between vertices $u$ and $v$.\n\nFor a vertex $v\\in V$, $N(v), N[v]$ denote the neighborhood of $v$ and the closed neighborhood of $v$ respectively, that is $N[v]=\\{v\\} \\cup N(v)$. In addition, for $l\\in \\mathbb{Z}^+$, let $N^l[v]$ denote the set of vertices within distance $l$ of $v$ and we set $N^l(v)=N^l[v]\\setminus \\{v\\}$.\nSimilarly, for a set of vertices $X\\subseteq V$, we let $N^l[X]=\\bigcup_{v\\in X}N^l[v]$.\n\nFinally, we will use $\\gamma_k(G)$ to denote the size of a smallest distance-$k$ dominating set in $G$.\n\nGraphs with no $K_{2,t}$-minor are sparse. To be precise, we have the following fact \\cite{CRS}.\n\\begin{lemma}\\label{crs-lem}\nLet $t\\geq 2$ and let $H$ be a graph of order at least one with no $K_{2,t}$-minor, then $|E(H)|\\leq \\frac{1}{2}(t+1)(|V(H)|-1).$\n\\end{lemma}\nAlthough we are not going to attempt to optimize the constants and do not really need the full power of the previous lemma, we will use it to obtain a bound for the number of vertices on $Q$-paths of length at most $h$.\n\n\\begin{lemma}\\label{main-lem-estimate}\nLet $t\\geq 2$, $h\\in \\mathbb{N}$ and let $H=(V,E)$ be a graph with no $K_{2,t}$-minor. Let $Q\\subseteq V$ and let $Q_h$ denote the set of vertices which are on $Q$-paths in $H$ of length at most $h$. Then $|Q_h|\\leq \\alpha_{h,t} |Q|$ for some $\\alpha_{h,t}$ that depends on $h$ and $t$ only. \n\\end{lemma}\n\\begin{proof} We will induct on $h$. If $h=0$ then $Q_0=Q$, and $\\alpha_{0,t} = 1$.  For the inductive step, let $\\mathcal{P}$ be a maximal set of $Q$-paths of length at most $h$ which are internally disjoint. For $u,v\\in Q$ let $P_{u,v}$ denote $u,v$-paths in $\\mathcal{P}$ and note that $|P_{u,v}|\\leq t+1$ because otherwise, graph $H$ has a subdivision of $K_{2,t}$ (as there can be only one edge $uv$, and every other path has length at least two), and so a $K_{2,t}$-minor. Contract paths from $P_{u,v}$ to edge $uv$ and apply Lemma \\ref{crs-lem} to conclude that the number of edges in the contracted graph  is less than $(t+1)|Q|\/2$. Consequently, since each path has length at most $h$, the number of vertices on paths from $\\mathcal{P}$ is less than $(t+1)^2(h+1)|Q|\/2$. Let $Q'$ denote the set of vertices on paths from $\\mathcal{P}$ (including paths of length $0$, or $1$, so notice that it does contain every vertex from set $Q$). \n\nIf $S$ is a $Q$-path of length at most $h$ which does not belong to $\\mathcal{P}$, then $S$ contains an internal vertex of a path from $\\mathcal{P}$ and so, all vertices on $S$ that have not been already counted belong to $Q'$-paths of length at most $h-1$ (see Figure~\\ref{fig1}). Thus, by induction, the number of vertices on such paths is at most $\\alpha_{h-1, t}|Q'|$. Consequently, by (\\ref{eqQ}), the number of vertices on all paths is at most \n\\begin{equation}\\label{eqQ2}\n\\begin{split}\n|Q_h| & \\leq (1+\\alpha_{h-1,t})|Q'| \\\\\n& <(1+\\alpha_{h-1,t})(t+1)^2(h+1)|Q|\/2 \\\\\n& \\leq  (t+1)^2(h+1)\\alpha_{h-1,t}|Q|,\n\\end{split}\n\\end{equation}\nwhich gives us rough estimate on $\\alpha_{h,t} \\leq (t+1)^{2h}(h+1)!$, and so\n\\end{proof}\n\\begin{equation}\\label{eqQ}\n    |Q'|\\leq (t+1)^2(h+1)|Q|\/2.\n\\end{equation}\n\\begin{figure}\n\\begin{center}\n \\scalebox{0.7}{\\input{picture1.pdf_t}}\n\\caption{\\normalsize Example for Lemma 2. The set $Q$ consists of the five black vertices, $t = 3$, $h = 5$. Every $Q$-path not in $P$ has its vertices covered by $P$ and $Q'$-path.}\n\\label{fig1}\n\\end{center}\n\\end{figure}\n\\section{Constant factor approximation}\\label{sec-const}\nIn this section, we will show that the simple algorithm (Algorithm 1 from \\cite{AmiriWieder}) finds a distance-$k$ dominating set of size  $O(\\gamma_k(G))$ in graphs $G$ with no $K_{2,t}$-minor. \n\nWe work in the {\\it Local} model, and as a result we may assume that each connected component of $G$ has diameter at least $4k$. Indeed, if component $C$ has diameter less than $4k$, then a simple $O(k)$-round algorithm can test that $C$ is a component and will compute an optimal distance-$k$ dominating set in $C$. In particular, $t\\geq 2$.\nIn the procedure from \\cite{AmiriWieder},  every vertex $v\\in V$ selects vertex $w\\in N^k[v]$ with $|N^k[w]|$  maximum and resolves ties using $ID(w)$. \nMore formally, the algorithm can be described as follows:\\\\\n\\begin{algorithm}[H]\n \\KwData{Graph $G=(V,E)$}\n \\KwResult{Set $D$ }\n \\label{alg1}\n \\caption{{\\sc DomSet}}\n\\begin{enumerate}\n    \\item For every $v\\in V$, in parallel, find $q_v=|N^k[v]|.$\n    \\item For every $v\\in V$ let $w:=w_v$ be the vertex in $N^k[v]$ such that \n    \\begin{itemize}\\item $q_w$ is maximum,\n    \\item and subject to this, $ID(w)$ is maximum.\n    \\end{itemize}\n    \\item Return $D:=\\bigcup \\{w_v\\}$.\n\\end{enumerate}\n\\end{algorithm}\nThe algorithm clearly runs in $O(k)$ rounds and outputs a distance-$k$ dominating set. The only difficulty is to show that it indeed finds a distance-$k$ dominating set $D$ such that $|D|=O(\\gamma_k(G))$, which we will do in the remainder of this section.\n\nIn our analysis we may assume that $G$ is connected because the same argument can be applied to each connected component.\n\nLet $M$ be an optimal distance-$k$ dominating set in $G=(V,E)$. Create Voronoi cells (also called clusters) centered at $M = \\{u_1,...,u_m\\}$ with a vertex $v$ joining cell $C_{i}$ if $d_G(u_i,v)$ is the smallest and with ties resolved by selecting $u_i$ with the maximum ID. This gives a set of cells $\\mathcal{C}=\\{C_1, \\dots, C_m\\}$ such that each $G[C_i]$ is connected. For a cell $C\\in \\mathcal{C}$ let $v_C$ be the vertex in $C$ such that $d_G(v_C,w)\\leq k$ for every $w\\in C$, and subject to this, $|N^k_G(v_C)|$ is maximum, and subject to that, $ID(v_C)$ is maximum. If $C=C_i$, then $v_C$ might be the same as $u_i$ or it can be a different vertex but $u_i$ is always an option.\n\n\\begin{definition}\nLet $C\\in \\mathcal{C}$. A vertex $v\\in C$ is called a border vertex if  $v$ has a neighbor in $V\\setminus C$.\n\\end{definition}\nLet $C^*$ denote the set of border vertices in $C$.\nWe have the following simple observation.\n\\begin{lemma}\\label{simple-lem}\nLet $G=(V,E)$ be a connected graph of diameter at least $4k-1$ and let $C$ be a Voronoi cell. If $y,y^*\\in C$ are such that $N^k[y]\\subseteq N^k[y^*]$ and $d_G(y^*,w)<d_G(y,w)\\leq k$ for every $w\\in C^*$, then $N^k[y^*]\\setminus N^k[y]\\neq \\emptyset.$\n\\end{lemma}\n\\begin{proof} We have $N^{k}[C]\\setminus N^{k-1}[C]\\neq \\emptyset$ because otherwise $N^{2k-1}[v_C]=V$. Let $v \\in N^{k}[C]\\setminus N^{k-1}[C]$ and let $Q$ be a shortest $v,C$-path in $G$. Let $\\{w\\}=Q\\cap C$. Then for any $i=1,\\dots, k$ there is a vertex $x_i$ on $Q$ such that $d_G(x_i, C) =d_G(x_i,w) = i$. Since $d_G(y^*,w)< d_G(y,w)\\leq k$ there is a vertex on $Q$ at distance $k$ from $y^*$ and at distance $k+1$ from $y$. \\end{proof}\n\nWe will now prove our main lemma which shows that the number of vertices from $C$ that can be added by the Algorithm \\ref{alg1} is $O(|C^*|)$. The idea is to show that the number of potential choices for vertices selected by {\\sc DomSet} can be bounded from above by repeated application of Lemma \\ref{main-lem-estimate}.\n\\begin{definition}\n Let $U_0:=C^*\\cup \\{v_C\\}$ and for $i>0$ let $U_i$ be the set of vertices on $U_{i-1}$-paths of length at most $3k$.\n\\end{definition}\nNote that we have $U_{i-1}\\subseteq U_{i}$ because of the trivial paths.\n\\begin{lemma}\n\\label{lemmaU_k}\nOutput of the {\\sc DomSet} contains only vertices of $U_k$.\n\\end{lemma}\n\\begin{proof}\nAssume towards contradiction that for some $C$ there is a vertex  $y\\in C$ selected by {\\sc DomSet} such that $y \\notin U_k$. \nWe have $d_{G}(y,v_C)\\leq k$. Fix one $y,v_C$-path of the shortest length in $G$ and call it $P$. Let $l \\leq k$ denote the length of $P$. We will now select a vertex which is closest to $y$ on $P$ and belongs to the sets $U_{l-i}$ for some $i\\geq 0$. \nLet $i$ be the smallest non-negative integer such that there is a vertex $y^*$ in $U_{l-i}\\cap V(P)$ which satisfies $d_P(y,y^*)\\leq i$. We choose a vertex $y^*$ for which $d_P(y,y^*)$ is the smallest. Since $v_C\\in U_0$ and $d_P(v_C,y)=l$, we can deduce that such $y^*$ always exists and $i\\leq l \\leq k$. \n\n\\begin{figure}\n\\label{fig2}\n\\begin{center}\n \\scalebox{0.8}{\\input{picture2.pdf_t}}\n\\caption{\\normalsize Example of Lemma \\ref{lemmaU_k}. a) Path $S$ connecting $y$ and $y^*$. b) Path $R_1$ and $R_2$ having no intersection with $S$. c) Path $R_1$ having an intersection with $S$ in vertex $u$.\n\\end{center}\n\\end{figure}\n\nNote that for the case $i=0$, we have $y=y^*\\in U_l\\subseteq U_k$. Thus,  consider cases when $y^*\\in U_{l-i}$ where $i\\geq 1$, and let $S=yPy^*$.\nWe will now analyze possibilities of the placement of $y^*$ in relation to $y$ (see Figure \\ref{fig2}) and prove the following three claims.\n\\begin{enumerate}\n\\item \\begin{clm}\\label{cl1}\nIf $w\\in C^*$ and $d_{G}(y,w)\\leq k$ then $d_G(y^*,w) < d_G(y,w)$.\n\\end{clm}\n\\begin{proof}\nLet $Q$ be a shortest $y,w$-path of length at most $k$ and assume towards contradiction that $Q$ does not contain $y^*$  (otherwise the condition is trivially true). If $Q\\cap S=\\{y\\}$, then $y\\in U_{(l-i)+1}\\subseteq U_k$ because the length of $Q\\cup S$ is at most $2k$ and both $w,y^*\\in U_{l-i}$. \n\nIf $Q\\cap S\\neq\\{y\\}$ then there is another vertex $z\\in Q\\cap S$ with $z\\neq y$. If $d_G(z,y)>d_G(z,y^*)$ then $d_G(y^*,w) < d_G(y,w)$. Thus assume  $d_G(z,y)\\leq d_G(z,y^*)$ and $zSy^*$ does not contain any vertices of $Q$. By the choice of path $P$ we have that $d_P(z,y)<d_P(y^*,y)$. Then, by definition of $y^*$, we have $z\\neq w$, but, as above, since $zSy^* \\cap zQw=\\{z\\}$, we have $z\\in U_{(l-i)+1}$, which gives a contradiction with the choice of $y^*$. \n\n\nThus for every $w\\in C^*$ such that $d_G(y, w)\\leq k$ every  shortest $y,w$-path $Q$ in $G$ contains $y^*$, and so $d_{G}(y^*,w)< d_G(y,w)$. \n\\end{proof}\n\\item \\begin{clm}\\label{cl2}\n$N^k[y]\\subseteq N^k[y^*]$\n\\end{clm}\n\\begin{proof}\nLet $x\\in N^k[y]$, then we have $d_G(x,y)\\leq k$ and we show that $d_G(x, y^*)\\leq k$.  Clearly $d_G(y^*,y)\\leq k$ and so we may assume that $x\\neq y$. In addition, if $x\\notin C$ then by the previous argument $d_G(x,y^*)<d_G(x,y)$, because $d_G(w, y^*)<d_G(w,y)$ for every $w\\in C^*$. Therefore, we may assume that $x\\in C$. \n\nLet $R_1$ be a $y,x$-path of length at most $k$. Since $x$ is in $C$, there is a $v_C,x$-path $R_2$ of length at most $k$. If $R_1\\cap S=\\{y\\}$ and $R_2\\cap S=\\emptyset$, then the walk $R_1\\cup R_2$ contains a $v_C,y$-path $R$ such that $R\\cap S=\\{y\\}$. The length of $R$ is at most $2k$ and the length of $S$ is at most $k$. Consequently $y\\in U_{k}$, which is a contradiction.\n\nIn addition, we either have $y^*\\in R_2$ or $R_2\\cap S=\\emptyset$ (again if $y^*\\notin R_2$ and there is some $u\\in R_2\\cap S$ that is closest to $y^*$ on $P$, then $u\\in U_{(l-i)+1}$, contradicting the choice of $y^*$). In the former case $d_G(x,y^*)\\leq d_{R_2}(x,v_C)\\leq k$ and so we may assume that $R_2\\cap S=\\emptyset$. By the previous discussion, the case that is left to analyze is when $R_1\\cap S$ contains some vertex $u\\neq y$. If $R_1\\cap S$ contains $y^*$, then $d_G(y^*,x)\\leq k$. Otherwise selecting $u$ to be closest to $y^*$ gives a $u,x$-path  of length at most $k$ which in connection with the $v_C,x$-path $R_2$ gives a $u,v_C$-walk of length at most $2k$. This walk intersects $S$ in $u$ only. Consequently $u\\in U_{(l-i)+1}$ because $y^*\\in U_{l-i}$ and $u$ belongs to a $y^*,v_C$-path of length at most $3k$. This again contradicts the choice of $y^*$. \nTherefore, $N^k[y]\\subseteq N^k[y^*]$. \n\\end{proof}\n\\item \\begin{clm}\\label{cl3}\n$N^k[y]$ is a proper subset of $N^k[y^*]$.\n\\end{clm}\n\\begin{proof}\nBy Claim \\ref{cl2}, $N^k[y]\\subseteq N^k[y^*]$. Suppose there exists $w\\in C^*$ such that $d_G(y,w)\\geq k+1$ and let $Q$ be a shortest $v_C,w$-path in $G$. Note that  the length of $Q$ is at most $k$ and so $d_G(y,w)\\leq 2k$ as witnessed by $P\\cup Q$.\nIf there is a $y,w$-path $R$ of length at most $2k$ such that $S\\cap R$ contains $z\\neq y^*$, and $y^*\\notin R$ then choosing $z$ closest to $y^*$ on $S\\cap R$ gives $z\\in U_{(l-i)+1}$ contradicting the choice of $y^*$.  Thus every such path $R$ contains $y^*$ because otherwise $S\\cap R=\\{y\\}$ implying $y\\in U_k$. Considering a $y,w$-path of  the shortest length gives a vertex $z$ such that $d_G(y^*,z)\\leq k$  and $d_G(y,z)>k$. Thus $z\\in N^k[y^*]\\setminus N^k[y]$.\nIf for every $w\\in C^*$ $d_G(y,w)\\leq k$ then by Claim \\ref{cl1} for every $w\\in C^*$ $d_G(y^*,w)< d_G(y,w)\\leq k$ and so, by Lemma \\ref{simple-lem},  $N^k[y^*]$ is a proper subset of $N^k[y]$.\n\\end{proof}\n\\end{enumerate}\nTherefore, by Claim \\ref{cl3}, $N^k[y]$ is a proper subset of $N^k[y^*]$ and so {\\sc DomSet} chooses $y^*\\in U_k$. \\end{proof}\n\n\\begin{lemma}\\label{main-const-approx}\nFor every $t,k\\in \\mathbb{Z}^+$ there is $\\beta_{k,t}$ such that the number of vertices in $C$ selected by {\\sc DomSet} is at most $\\beta_{k,t}|C^*|.$\n\\end{lemma}\n\\begin{proof} Since output of {\\sc DomSet} is a subset of $U_k$ and in view of Lemma \\ref{main-lem-estimate}, $|U_i|\\leq \\alpha_{3k,t}|U_{i-1}|$.\n\\end{proof}\n\n\\begin{definition}\nLet $V^*=\\bigcup_{C\\in \\mathcal{C}} C^*$.\n\\end{definition}\nUsing a few relatively easy lemmas, we can conclude the analysis of {\\sc DomSet}.\n\\begin{lemma}\\label{lem-simple-W}\nLet $C\\subseteq V$ be such that $G[C]$ is connected, and such that for some vertex $v_C$, $d_{G[C]}(v_C, w)\\leq k$ for every $w\\in C$.\nLet  $W\\subseteq C$ be a set which satisfies $|W|> ks^{k}$.  Then $G[C]$ contains a subdivision of $K_{1,s}$ with all leaf vertices in $W$. \n\\end{lemma}\n\\begin{proof}  Let $T$ denote a spanning BFS tree in $G[C]$ rooted at $v_C$ and let $W_i$ denote the set of vertices in $W$ that are at distance $i$ from $v_C$ in $T$. We have $\\sum_{i=0}^k|W_i| =|W|$, and so there is an $i$ such that $|W_i|\\geq s^{k}$. For $w\\in W_i$, let $P_w$ denote the path $wTv_C$ (path from $w$ to $v_C$ using the edges of the spanning tree $T$), and let $T'$ be the union $\\bigcup_{w\\in W_i} P_w$. Then $T'$ is a tree with leaves in $W_i$ and for every $w\\in W_i$, $d_{T'}(w,v_C)\\leq k$. If there is a vertex $z\\in T'$ such that $deg_{T'}(z)\\geq s$, then $T'$, and so $G[C]$, contains a subdivision of $K_{1,s}$ with all leaf vertices in $W$. Otherwise, the number of vertices in $W_i$ is less\nthan $s^{k}$. \\end{proof}\n\\begin{lemma}\\label{lem-two-clusters-edges}\nLet $C,C'$ be two Voronoi cells as in Lemma \\ref{lem-simple-W}. Then the number of edges between $C$ and $C'$ is at most $k^2t^{2kt^k}$.\n\\end{lemma}\n\\begin{proof} If there is a vertex $z\\in C$ which has more than $kt^{k}$ neighbors in $C'$, then, by Lemma \\ref{lem-simple-W} applied to $W=N(z)\\cap C'$, $G$ contains a subdivision of $K_{2,t}$. If there is a matching $Q$ between $C$ and $C'$ of size larger than $kt^{kt^k+1}$, then we can apply Lemma \\ref{lem-simple-W} twice. Apply it first with $s=kt^k+1$  and $W= V(Q)\\cap C'$ to get a subdivision of $K_{1,s}$, $T$, in $C'$. Then apply it again with $s=t$  and $W\\subseteq V(Q)\\cap C$ which contains vertices matched by $Q$ with the leaves of $T$ to get a subdivision of $K_{1,t}$ in $C$. Therefore the number of edges between $C$ and $C'$ is at most $k^2t^{2kt^k}.$ \\end{proof}\n\nFinally, we have with the following observation (we recall that $M$ is an optimal distance-$k$ dominating set in $G$).\n\n\\begin{lemma}\\label{bound-lemma}\n$|V^*|\\leq  k^2t^{2kt^k}(t+1)|M|.$\n\\end{lemma}\n\\begin{proof} Contracting each $C\\in \\mathcal{C}$ to a vertex gives a minor of $G$ which by Lemma \\ref{crs-lem} has at most $(t+1)|M|\/2$ edges. By Lemma \\ref{lem-two-clusters-edges}, the number of edges with endpoints in two different Voronoi cells is at most $k^2t^{2kt^k}(t+1)|M|\/2$. Consequently, the  number of vertices that belong to these edges is at most $k^2t^{2kt^k}(t+1)|M|$. \\end{proof}\n\nWe will now combine the previous facts to prove the main result of this section.\n\n\\begin{theorem}\\label{const-approx}\nLet $t, k\\in Z^+$. Then there exists $\\delta=\\delta(t,k)$ such that given a connected graph $G$ with no $K_{2,t}$-minor and such that $diam(G)\\geq 4k$, algorithm {\\sc DomSet} finds in $O(k)$ rounds a distance-$k$ dominating set $D$ in $G$ such that $|D|\\leq \\delta \\cdot \\gamma_k(G).$   \n\\end{theorem}\n\\begin{proof}\nLet $M$ be an optimal distance-$k$ dominating set in $G$.\nBy Lemma \\ref{main-const-approx}, the set $D$ obtained by {\\sc DomSet} satisfies $|D|\\leq \\beta_{k,t}|V^*|$ for some $\\beta_{k,t}$. In view of Lemma \\ref{bound-lemma}, $|V^*|\\leq k^2t^{2kt^k}(t+1)|M|$ and so, $|D|\\leq  \\delta |M|$ for $\\delta =\\beta_{k,t} k^2t^{2kt^k}(t+1)$. \\end{proof}\n\n\\section{The $(1+\\epsilon)$-factor approximation}\\label{sec-eps}\nIn this section, we will first give a distributed $(1+\\epsilon)$-factor approximation of an optimal $k$-MDS in the case when $G$ is $K_{2,t}$-minor-free. This algorithm runs in $O(\\log^*{|V|})$ rounds in the {\\it Local} model.\nAs noted in the introduction, adapting methods from \\cite{CHW} is not automatic. However, there are some instances when this can be accomplished with relatively little effort.  In the second part of this section, we give one example of such a situation when a graph $G$ is $K_t$-minor-free and satisfies  $\\gamma_k(G)\\leq C \\gamma_1(G)$ for some constant $C$. For example, $K_t$-minor-free graphs of a bounded maximum degree satisfy this condition.\n\\subsection{Graphs with no $K_{2,t}$-minor}\n\nLet $H=(W,F)$ be a graph and let $P=(W_1, \\dots, W_l)$ be an ordered partition of $W$. We define $\\partial(P)$ to be the set of vertices $v\\in W$ such that $v \\in W_i$ and $N(v)\\cap W_j\\neq \\emptyset$ for some $i\\neq j$.\nWe have the following theorem which can be proved by applying methods from \\cite{CHW} and is a special case of the corresponding theorem in \\cite{CHWW1}.\n\\begin{theorem}\\label{waw-thm} Let $s\\in \\mathbb{Z}^+$ and let $\\epsilon>0$. There exists $L$ such that the following holds. Let $H=(W,E)$ be a graph on $n$ vertices with no $K_{s}$-minor. There is a distributed algorithm which finds a partition $P=(W_1, \\dots, W_l)$ such that: \n\\begin{itemize}\n    \\item For every $i$, $H[W_i]$ has diameter $O(L)$ and\n    \\item $|\\partial(P)|\\leq \\epsilon |W|$.\n    \\end{itemize}\nThe algorithm runs in $L\\log^*{n}$ rounds.\n\\end{theorem}\nWe will use the algorithm from  Theorem \\ref{waw-thm} to improve the approximation ratio of the algorithm from the previous section. Although the general idea is the same as in \\cite{CHW}, there are a few changes in the analysis that must be made to account for the fact that we are dealing with a distance-$k$ dominating set with $k\\geq 2$. In particular, the assumption that there is no $K_{2,t}$-minor (rather than a more relaxed assumption that there is no $K_{s}$-minor for $s\\geq t+2$) will play a critical role in the analysis via Lemma \\ref{two-cluster-lem}.\n \nLet $\\alpha \\in (0,1)$ be given and let $D$ be the set obtained by {\\sc DomSet}. Then, by Theorem \\ref{const-approx}, \\begin{equation}\\label{eq-C}|D|\\leq \\delta \\cdot \\gamma_k(G)\\end{equation} for some $\\delta$ that depends on $t,k$ only.\nConsider Voronoi cells with centers in vertices from $D$, that is, Voronoi cells $C_v$ for $v\\in D$ with $w$ joining $C_v$ if $d_G(v,w)$ is minimum over all $v\\in D$ and ties resolved by selecting $v$ with maximum $ID(v).$ Let $H=(W,F)$ be obtained from $G$ by contracting each $C_v$ to a vertex. \nSet $\\epsilon :=\\frac{\\alpha}{2\\delta k^2t^{2kt^k}}$ and let $P = (W_1, \\dots, W_l)$ be the partition of $W$ from Theorem \\ref{waw-thm}. We have \\begin{equation}\\label{partial1}|\\partial(P)|\\leq \\epsilon |W| =\\epsilon |D|.\\end{equation}\nPartition $P$ yields partition $P'=(V_1, \\dots, V_l)$ of $V(G)$ by setting $V_i:=\\bigcup_{u\\in W_i} C_u$.\n\\begin{lemma}\\label{two-cluster-lem}\nLet $u,w\\in V(H)$. Then the number of edges in $G$ between $C_u$ and $C_v$ satisfies $|E_{G}(C_u, C_v)|\\leq k^2 t^{2kt^k}.$\n\\end{lemma}\n\\begin{proof} By construction every vertex $w\\in C_u$ is within distance $k$ of $u$. Consequently, by Lemma \\ref{lem-two-clusters-edges}, $|E_G(C_u,C_v)|\\leq k^2t^{2kt^k}.$ \\end{proof}\n\nNow combining Lemma \\ref{two-cluster-lem} and (\\ref{partial1}) we have that $\\epsilon |D|$ Voronoi cells can have $k^2t^{2kt^k}$ edges between them, each connecting a pair of vertices. Thus\n\\begin{equation}\\label{partial2}\n|\\partial(P')|\\leq 2\\epsilon k^2t^{2kt^k}|D|.\n\\end{equation}\nInformally speaking, we use Algorithm 1 to find a seed dominating set. We define Voronoi cells and construct groups of Voronoi cells using Theorem \\ref{waw-thm}, and solve the subgraphs inside these groups optimally. Specifically, we consider the following procedure.\\\\\n\\begin{algorithm}[H]\n \\KwData{Graph $G$ with no $K_{2,t}$-minor, $k\\in Z^+$, $0<\\alpha<1$}\n \\KwResult{Set $Q$}\n\\caption{{\\sc $k$-DomSet Approximation}}\n\\begin{enumerate}\n    \\item Find $D$ using {\\sc DomSet}.\n    \\item Construct graph $H$ as above and set $\\epsilon:=\\frac{\\alpha}{2\\delta k^2t^{2kt^k}}$.\n    \\item Use the algorithm from Theorem \\ref{waw-thm} to find $P$. Let $Q:= \\partial(P')$.\n    \\item For every $i=1, \\dots, l$ find a set $Q_i$ in $G[V_i]$ such that $|Q_i|$ is the smallest and $Q_i \\cup (\\partial(P')\\cap V_i)$ distance-$k$ dominates $V_i$ in $G$.\n    \\item Return $Q:=Q\\cup \\bigcup_i Q_i.$\n\\end{enumerate}\n\\end{algorithm}\nUsing the above discussion we can now prove the main theorem.\n\\begin{theorem}\\label{main-approx-thm}\nLet $\\alpha\\in (0,1)$ and let $t,k\\in Z^+$. Given a connected graph $G=(V,E)$ with no $K_{2,t}$-minor of diameter at least $4k$, procedure {\\sc $k$-DomSet Approximation} finds in $O(\\log^*{|V|})$ rounds set $Q$ such that $|Q|\\leq (1+\\alpha)\\gamma_k(G).$   \n\\end{theorem}\n\\begin{proof} The algorithm runs in $O(kL \\log^*{|V|})=O(\\log^*{|V|})$ rounds (where $L$ is the constant from Theorem \\ref{waw-thm}) because $diam(G[V_i])=O(kL)$ by Theorem \\ref{waw-thm} and the construction, and so step 4 requires $O(kL)$ rounds in the {\\it Local} model.\n\nLet $M$ be an optimal distance-$k$ dominating set in $G$, $i\\in \\{1, \\dots, l\\}$, and $M_i:=M\\cap V_i$.\nLet $V_i^O$ denote the set of vertices $w\\in V_i$ such that $d_G(w, \\partial(P')\\cap V_i)\\leq k$, and let $V_i^I:=V_i\\setminus V_i^O$. \nClearly every vertex from $V_i^O$ is distance-$k$ dominated by $\\partial(P')\\cap V_i$. In addition, if $w\\in V_i^I$, then $w$ must be distance-$k$ dominated by a vertex from $M_i$. Thus by step 4 of {\\sc $k$-DomSet Approximation}, we have $|Q_i|\\leq |M_i|$, and $|Q|\\leq |\\partial(P')|+\\sum_{i=1}^l |M_i|$, \nwhich in view of (\\ref{partial2}) gives $|Q|\\leq 2\\epsilon k^2t^{2kt^k} |D|+|M|.$\nFurther by (\\ref{eq-C}) and the definition of $\\epsilon$ we have\n$$\n|Q|\\leq (2\\epsilon k^2t^{2kt^k}\\delta+1)\\gamma_{k}(G)=(1+\\alpha)\\gamma_k(G).\n$$\n\n\\end{proof}\n\\subsection{$K_t$-minor free graphs of a constant maximum degree}\nIn this last short section, we show a simple method to find a $(1+\\epsilon)$-factor approximation of the minimum distance-$k$ dominating set if $G$ is $K_t$-minor-free and the maximum degree of $G$ satisfies $\\Delta(G)\\leq L$ for some $L$ independent of $G$. \n\nIn fact, we will give an algorithm for a somewhat more general class of $K_t$-minor-free graphs that we call $(C, \\gamma_k)$-bounded. \n\nNote that obviously, for every graph $G$ and every $i\\in Z^+$, we have $\\gamma_{i}(G)\\geq \\gamma_{i+1}(G)$.\n\nFix $k\\in Z^+$. We say that a graph $G$ is $(C,\\gamma_k)$-bounded if  $\\gamma_1(G)\\leq C \\gamma_k(G)$. For example, if for some $L\\geq 3$, graph $G=(V,E)$ is such that $\\Delta(G)\\leq L$, then $\\gamma_{k}(G)> |V|\/L(L-1)^k\\geq  \\gamma_1(G)\/L(L-1)^k$ and so $G$ is $(C,\\gamma_k)$-bounded with $C=L(L-1)^k$.\n\nThe algorithm is very simple and we will only outline the main idea.\nFix $C,k, t$ which are known to the algorithm and let $G$ be a graph that is $K_t$-minor-free and $(C,\\gamma_k)$-bounded. Find a constant approximation $S$ of a minimum dominating set in $G$ (i.e. distance-$k$ dominating set  with $k=1$) by using the algorithm from \\cite{CHWW}. Partition $V(G)$ into $\\{S_v|v\\in S\\} $ by setting $S_v=\\{v\\}$ and adding $u$ to $S_v$  if $uv\\in E$ and $v$ has the maximum ID over vertices from $S$. Construct $H$ by contracting each $S_v$ to a vertex. Set $\\epsilon$ appropriately and use Theorem \\ref{waw-thm} to find a partition $(W_1,\\dots, W_l)$ of $V(H)$. If a vertex in $S_v\\in W_i$ has a neighbor in $W_j$ for some $j\\neq i$ in $G$ (border set), then add the center of $S_v$, namely $v$, to $D$. Finally, for each $i=1 ,\\dots, l$ find an optimal set $D_i$ in $G$ such that $D_i\\cup D$ distance-$k$ dominates $V_i:=\\bigcup_{S_u\\in W_i} S_u$ in $G$. \n\\begin{theorem}\\label{const-deg-thm}\nLet $C,k,t\\in Z^+$ and let $0<\\alpha <1$. There is a distributed algorithm which given a $K_t$-minor-free graph $G=(V,E)$ which is $(C,\\gamma_k)$-bounded, finds in $O(\\log^*{|V|})$ rounds a set $D\\subseteq V(G)$ such that $|D|\\leq (1+\\alpha)\\gamma_k(G).$  \n\\end{theorem}\n\\begin{proof} (Sketch) The argument is analogous to the proof of Theorem \\ref{main-approx-thm}. In particular, it is easy to see that the number of vertices added to $D$ from border sets $S_v$ is  $O(\\epsilon \\gamma_1(G))$ which can be made smaller than $\\alpha \\gamma_k(G)$ using appropriately defined $\\epsilon$ and the fact that $G$ is $(C,\\gamma_k)$-bounded. Now assume $w \\in V_i$ and $w$ is distance-$k$ dominated by a vertex $u\\in V_j$ for some $j\\neq i$. Then a shortest $u,w$-path $P$ in $G$ contains a vertex $x$ from some border set $S_v\\in W_i$. Then however $vxPu$ has length which is less than or equal to the length of $P$ and so the center of $S_v$,  which is added to $D$, distance-$k$ dominates $w$. Finally, vertices $w\\in V_i$ that are not distance-$k$ dominated by vertices from other Voronoi cells than $V_i$ are distance-$k$ dominated by $D_i$ and $|D|=\\sum_{i}|D_i|\\leq \\gamma_k(G)$. \\end{proof} \n\n\\section{Conclusions}\nWe finish  with a short summary. In this paper we considered the distance-$k$ dominating set problem in a special class of graphs and showed three facts. \n\\begin{itemize}\n    \\item There is a (simple) distributed (LOCAL) constant-factor approximation of an optimal distance-$k$ dominating set in graphs $G$ that have no $K_{2,t}$-minor. The algorithm runs in $O_t(k)$ rounds.\n    \\item There is a distributed (LOCAL) algorithm which given $\\epsilon>0$ finds in a $K_{2,t}$-minor free graph $G$ of order $n$ a distance-$k$ dominating set of size at most $(1+\\epsilon)\\gamma_k(G).$ The algorithm runs in $O_{\\epsilon, k, t}(\\log^*{n})$ rounds. \n    \\item There is  a distributed (LOCAL) algorithm which given $\\epsilon>0$ finds in a $K_{t}$-minor free graph $G$ of a constant maximum degree and order $n$ and a distance-$k$ dominating set of size at most $(1+\\epsilon)\\gamma_k(G).$ The algorithm runs in $O_{\\epsilon, k, t,\\Delta(G)}(\\log^*{n})$ rounds. \n    \\end{itemize}\n    The proofs of the first two statements critically rely on the fact that graph $G$ is $K_{2,t}$-minor free and the third one applies only to very restrictive class of graphs, the class of $K_t$-minor-free graphs of a constant maximum degree. It would be interesting to see if similar facts can be obtained for graphs with no $K_{3,t}$-minor, or in general for graphs which are $K_t$-minor free.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nWith its first detections of gravitational waves~\\cite{Detection1,Detection2,\nDetection3,Detection4,Detection5,Detection6}, the Advanced Laser\nInterferometer Gravitational Wave Observatory (Advanced LIGO) has provided a\nfundamentally new means of observing the Universe. At the heart of\neach of these detections was a merger of compact binaries.\nIn such binaries, each compact object possesses four intrinsic parameters:\nmass, and the three components of the spin vector. Inferring all eight\nintrinsic parameters\\footnote{\\textit{Intrinsic} parameters are fundamental to\nthe underlying physics of the system. In contrast, \\textit{extrinsic}\nparameters are related to the observer (e.g.~polarization, sky location, and\ndistance) and are not considered in this paper. Some authors refer to seven\nintrinsic parameters in the full-dimensional space, which include each spin\ncomponent and the mass ratio of the system. This is because the total mass of\nthe system is simply a scaling factor; we choose to refer to eight parameters\nsince the total mass sets the time and frequency scales and therefore must be\nconsidered in PE.} from a gravitational wave observation, which analysis is\npart of the more general \\textit{parameter estimation} (PE), remains a\nchallenging and computationally expensive enterprise.\n\nThe LIGO\/Virgo Scientific Collaboration (LVC) performs PE in a Bayesian\nframework, implemented within the \\textsc{lalinference} software package that\nis part of the larger open-source software framework\n\\texttt{LALSuite}~\\cite{lalsuite}. In such a framework, we sample the\nposterior distribution by repeatedly calculating the likelihood that a\nparticular waveform matches the data and applying Bayes' theorem. Evaluating\nthe likelihood requires the rapid, sequential generation of as many as \n$\\mysim10^8$ theoretical gravitational wave predictions~\\cite{Veitch2015}.\nGenerating so many predictions via a full solution of the general relativistic\nfield equations (using the tools of numerical relativity) would be far too\ncomputationally expensive. Thus theoretical models adopted for PE generally\nemploy approximate solutions called \\emph{approximants}. State-of-the-art\napproximants adopt post-Newtonian techniques for evaluating the gravitational\nwaveform throughout most of the inspiral and ringdown, and inject information\nfrom numerical relativity calculations for the late inspiral and merger.\n\nOne such gravitational wave approximant is the Spinning Effective One\nBody--Numerical Relativity (SEOBNR) algorithm. This algorithm marries an\neffective-one body inspiral gravitational waveform approximation---with\nunknown higher-order terms fit to numerical relativity-generated gravitational\nwave predictions---to a black hole ringdown model~\\cite{Pan2014}. In\nparticular, SEOBNR starts with the Effective One Body (EOB) approach to\nnon-spinning binary modeling~\\cite{Buonanno1999} by mapping the dynamics of\nthe two-body system to the dynamics of an effective particle moving in a\ndeformed Schwarzschild metric. This work was then extended to include the\neffects of spinning, precessing binaries~\\cite{Buonanno2006}. Implemented\nnumerically, this Spinning EOB procedure adopts a precessing source frame in\nwhich precession-induced variations in amplitude and phase are minimized\nduring inspiral, and a source frame aligned with the spin of the final body\nfor matching the inspiral to the merger-ringdown~\\cite{Pan2014}. \n\nThe other widely adopted approximant within the LVC for PE is the Phenom\nseries of phenomenological waveform models. These waveform models are based on\nthe combination of accurate post-Newtonian inspiral models with late-inspiral\nand merger phenomenological fits to suites of numerical relativity\nsimulations~\\cite{Santamaria2010}. More recently, Phenom models have been \nbuilt to include the effects of precession~\\cite{Hannam2014}. In particular,\nprecession effects are included by using post-Newtonian methods to compute\nprecession angles and then ``twisting'' the underlying non-precessing\nmodel~\\cite{Hannam2014,Husa2016,Khan2016}. Phenom models are simulated\ncompletely in the frequency domain, and therefore simplify some aspects of\nanalysis. The only Phenom model designed to generate gravitational waveform\npredictions across all eight dimensions of parameter space is\nPhenomP~\\cite{Hannam2014}, which was extensively used in the first six\ndetection papers. We remark that Phenom is limited to a relatively small\nnumber of numerical relativity simulations against which it has been\ncalibrated, and it is difficult to determine the degree of systematic\nuncertainty in the model without appealing to another model for comparison. \n\nEvaluating the systematic uncertainties of the Phenom model requires\nconstruction of an independent gravitational waveform model with independent\nsystematics, and the SEOBNR family of models is a good candidate for this\ntask. The only SEOBNR model capable of generating theoretical gravitational\nwaveform predictions in all 8 intrinsic dimensions of parameter space is the\nthird version of the model, v3; the first and second versions were restricted\nto aligned-spin cases. In particular, v3 was built to accommodate arbitrary\nmass ratios, spin magnitudes, and spin orientations and has been calibrated and\nvalidated against a variety of numerical relativity\nsimulations~\\cite{Babak2017}. Thus v3 is vital for precessing compact binary\nmerger PE.\n\nUnfortunately, v3 is too currently too slow for PE. A single waveform\ngeneration across the LIGO band for, say, a black hole-neutron star system\nusing v3 can take as long as an hour on a modern desktop computer. If LIGO\nobserved a black hole-neutron star system merge, a\nsequential-gravitational-wave-generation PE would take thousands of years.\nAttempts to overcome the computational challenge of generating such\ntime-consuming gravitational waveforms include the construction of Reduced\nOrder Model (ROM) approximants. ROMs make use of multidimensional\ninterpolations between sampled points in another underlying approximant. For\nexample, a ROM based on the aligned-spin SEOBNR version 2 (v2)\napproximant~\\cite{Purrer2016,Field2014} is constructed by first generating an\nextensive collection of waveform predictions using v2 that adequately samples\nthe 4D parameter space reliably covered by v2. Then to obtain the\ngravitational waveform at any desired point in parameter space, the ROM simply\ninterpolates within the four dimensions of sampled parameter space. A ROM\nversion of v2 can generate waveforms up to $\\mysim3000$x faster than v2\ndirectly~\\cite{Purrer2016}, which explains in part why ROMs enjoy such\nwidespread use within the LVC for data analysis applications. \n\nWhile ROMs have been constructed with favorable performance characteristics in\naligned-spin situations, the cost of generating a ROM grows exponentially with\nthe dimension of the ROM (though see~\\cite{Field2012} for ideas on combating\nthis using a reduced basis approach). No strategy yet exists that can perform\nthe 8-dimensional (8D) interpolations faster than the 8D approximant; until\nsuch a strategy is invented, the most promising way to improve the performance\nof theoretical waveform generation in the full 8D parameter space will be to\noptimize the approximant directly. As a proof-of-principle, we demonstrated\nthat such an approach is capable of improving the performance of the\naligned-spin v2 approximant by a typical factor of $\\mysim280$x~\\cite{v2opt}.\nWe call our optimized v2 approximant v2\\_opt. The precessing (8D) v3\napproximant was in development as we independently prepared v2\\_opt, and thus\noriginally contained all the same inefficiencies as v2. This suggests that if\nthe full suite of optimizations we implemented in v2 were incorporated into\nv3, v3-based PE timescales might drop by two orders of magnitude at least.\n\nThis paper documents our incorporation of applicable v2 optimizations into v3,\nas well as our implementation of innovative new optimization ideas, which\ntogether act to speed up v3 by $\\mysim340$x. Optimization strategies are\nsummarized in Sec.~\\ref{OptStrats}. Section~\\ref{Results}  presents code\nvalidation tests that demonstrate roundoff-level agreement between v3 and our\nlatest optimized version of v3, designated v3\\_Opt, along with benchmarks\nproviding an overview of performance gains across parameter space in v3\\_Opt.\n{\\bf For convenience, Table~\\ref{approx_conv} defines all SEOBNR approximants\nreferenced in this paper.}\n\n  \\begin{table}\n    \\begin{adjustbox}{max width=\\textwidth}\n    \\centering\n      \\begin{tabular}{|c|c|l|}\n        \\hline\n        \\textbf{Base} & \\textbf{Approx.} & \\textbf{Description} \\\\\n        \\textbf{Approximant}  & \\textbf{Name} & \\\\\n        \\hline\\hline\n        SEOBNRv2 & v2 & Initial SEOBNRv2 implementation\\tablefootnote{\\label{2cce415}As of publication, the most recent updates to v2\/v2\\_opt are found on commit ID \\texttt{2cce415} in the \\texttt{LALSuite} \\texttt{master} branch.}; see \\cite{Taracchini2014}. \\\\\\cline{2-3}\n        (spin-aligned) & v2\\_opt & Optimized v2\\cref{2cce415}; see \\cite{v2opt}. \\\\\n        \\hline\n        SEOBNRv3 & v3\\_preopt & Initial SEOBNRv3 implementation\\tablefootnote{To generate a waveform with v3\\_preopt, download \\texttt{LALSuite} from the archived repository page \\url{https:\/\/git.ligo.org\/lscsoft\/lalsuite-archive\/tree\/14414694698a2f18c9135445003cade805ad2096} and use approximant tag SEOBNRv3.}; see \\cite{Pan2014}. \\\\\\cline{2-3}\n        (precessing) & v3 & Partially optimized v3\\_preopt with bug fixes\\tablefootnote{\\label{19e95b4}As of publication, the most recent updates to v3 and v3\\_opt are found on commit ID \\texttt{19e95b4} in the \\texttt{LALSuite} \\texttt{master} branch.}. \\\\\\cline{2-3}\n        & v3\\_pert & v3 with machine-$\\epsilon$ mass perturbation \\cref{19e95b4}. \\\\\\cline{2-3}\n        & v3\\_opt &v3 optimized similarly to v2\\_opt \\cref{19e95b4}. \\\\\\cline{2-3}\n        & v3\\_Opt & v3\\_opt with new optimization strategies\\tablefootnote{\\label{Opt}Approximants v3\\_opt and v3\\_opt\\_rk4 were updated to run v3\\_Opt and v3\\_Opt\\_rk4, respectively, on commit ID \\texttt{1391f77} in the \\texttt{LALSuite master} branch.}. \\\\\\cline{2-3}        \n        & v3\\_Opt\\_rk4 & v3\\_Opt implementing RK4 rather than RK8\\cref{Opt}. \\\\\n        \\hline\n      \\end{tabular}\n      \\caption{Approximant naming conventions. These conventions apply\n        throughout this paper.}\\label{approx_conv}\n    \\end{adjustbox}\n  \\end{table}\n\n\\section{SEOBNRv3\\_opt: Optimizations migrated from v2\\_opt}\n\\label{OptStrats}\nOptimizations to v3 were performed in two phases. In the first phase,\ndescribed in Sec.~\\ref{v2opts}, we migrated to v3 all applicable optimizations\ndeveloped during the preparation of v2\\_opt. Sections~\\ref{v3opts:gad} and~\\ref{v3opts:do} detail the second phase of optimization, outlining new strategies\nincorporated into v3\\_Opt.\n\n\\subsection{Migrated Optimizations}\n\\label{v2opts}\nHere we summarize the optimizations to v2 which were migrated to v3 and thus\nimplemented in v3\\_opt.\n\n\\begin{itemize}\n  \\item \\emph{Switching compilers}. Switching from the \\texttt{GNU Compiler\n    Collection} (\\texttt{gcc})~\\cite{gcc} \\texttt{C} compiler to the\n    \\texttt{Intel Compiler Suite} (\\texttt{icc})~\\cite{icc} \\texttt{C}\n    compiler improves performance by roughly a factor of 2x. It is well-known\n    that the \\texttt{icc} compiler often produces more efficient executables\n    than the \\texttt{gcc} compiler\\footnote{We used the following compiler\n    flags when compiling with \\texttt{icc}: \\texttt{-xHost}, \\texttt{-O2}, and\n    \\texttt{-fno-strict-aliasing}.}.\n\n  \\item \\emph{Minimize transcendental function evaluations}. The EOB\n    Hamiltonian equations of motion were hand-optimized by minimizing calls to\n    some expensive transcendental functions such as \\texttt{exp()},\n    \\texttt{log()}, and \\texttt{pow()}.\n\n  \\item \\emph{Replacing finite difference with exact derivatives}. When\n    solving the EOB Hamiltonian equations of motion, v3 computes partial\n    derivatives of the Hamiltonian using finite difference approximations. We\n    replaced these with exact, Mathematica-generated expressions for the\n    derivatives, using Mathematica's code generation facilities---which\n    includes common subexpression elimination (CSE)---to generate the\n    \\texttt{C} code~\\cite{Mathematica}. Although this alone acts to\n    significantly speed up v3, in this work we further optimize these\n    Mathematica-generated derivatives.\n\n  \\item \\emph{Increasing the order of the ODE solver}. v3 solves the EOB\n    Hamiltonian equations of motion via a Runge-Kutta fourth order (RK4) ODE\n    solver. After implementing exact derivatives, we noticed that the number\n    of RK4 steps needed dropped significantly---presumably due to the\n    effective removal of high numerical noise intrinsic to finite-difference\n    derivatives. We then found that adopting a Runge-Kutta eighth order (RK8)\n    ODE solver resulted in 2x larger timesteps, so an even larger speed-up was\n    observed.\n\n  \\item \\emph{Reducing orbital angular velocity calculations}. The orbital\n    angular velocity $\\omega$ was calculated for each ($\\ell,m$) mode (as\n    defined in \\cite{Pan2014}) inside the ODE solver. As $\\omega$ exhibits no\n    dependence on $\\ell$ or $m$, this expensive recalculation was unnecessary\n    and needs only be performed once.\n\\end{itemize}\n\nFor more details about these optimizations see our v2 optimization\npaper~\\cite{v2opt}. \n\n\\subsection{Guided Automatic Differentiation: A more efficient way of\n  generating symbolic derivatives of the Hamiltonian}\\label{v3opts:gad}\n\nAfter migrating the v2 optimizations described in Sec.~\\ref{v2opts} to v3,\nprofiling analyses indicated that approximately $75\\%$ of v3\\_opt's total\nruntime was spent computing the v3 Hamiltonian~\\cite{Pan2014} and its partial\nderivatives with respect to the twelve degrees of freedom (consisting of three\nspatial degrees $\\{x,y,z\\}$, three momentum degrees $\\{p_x,p_y,p_z\\}$, and\nthree spin degrees for each of the two binary components $i\\in\\{1,2\\}$:\n$\\{s_{i}^{x},s_{i}^{y},s_{i}^{z}\\}$).\n\nIn v3, the ODE solver computes these partial derivatives by direct evaluations\nof the Hamiltonian itself via finite difference\ntechniques~\\cite{Taracchini2014}. In v3\\_opt, these numerical derivatives were\nreplaced with Mathematica-generated exact derivatives. Although these exact\nderivatives unlock significant performance gains, the Mathematica-generated\n\\texttt{C} code was neither particularly human-readable (comprising thousands\nof lines of code output by Mathematica's CSE routines) nor particularly\nwell-optimized (common patterns were still visible and recomputed in the\n\\texttt{C} code). Attempts to gain performance through consolidation of all\nderivatives---as was possible in our optimizations of v2---proved beyond\nMathematica's capabilities when differentiating the v3 Hamiltonian on our\nhigh-performance workstations. Therefore, \\texttt{C} codes for all twelve\nexact derivatives needed to be output separately, resulting in a significant\nnumber of unnecessary re-computations.\n\nWe present here our new strategy for computing partial derivatives of the\nHamiltonian, called \\textit{guided automatic differentiation} (GAD), which\nresults in a significant reduction in computational cost while ensuring the\nresulting code is highly human-readable.  GAD is based on forward accumulation\nautomatic differentiation, with the advantage of the subexpressions being\nchosen by hand to minimize the overall number of floating point operations.\n\nThe following describes the process of computing a partial derivative of the\nv3 Hamiltonian $H$ with respect to an \\textit{arbitrary} independent variable\n$x_{1}$ using GAD. We may write $H$ in the following form, where $I$ is a set\nof input quantities:\n\\begin{align*}\nv_{1} &= f_{1}(I) \\\\\nv_{2} &= f_{2}(v_1,I) \\\\\nv_{3} &= f_{3}(v_1,v_2,I) \\\\\n &\\ \\ \\vdots  \\\\\nH &= f_{N}(v_{1},v_{2},v_{3},...,I).\n\\end{align*}\nHere $f_\\ell$ is the $\\ell$th function of the set of input quantities $I$ and\npreviously computed subexpressions\n$\\left\\{v_{0},v_{1},\\hdots,v_{\\ell-1}\\right\\}$. Although $N\\approx 200$ for\nv3, \\textit{for the sake of example} we suppose $N=3$, $I=\\{x_{1},x_{2}\\}$,\nand\n\\begin{align*}\nv_1 &= \\sqrt{x_{1}} + ax_{1} \\\\\nv_2 &= \\sqrt{x_{2}} + ax_{2} \\\\\nv_3 &= (v_1 + v_2)\/(v_1v_2)  \\\\\nH  &= v_3^2.\n\\end{align*}\nWe demonstrate GAD by taking a partial derivative of $H$ with respect to the\nindependent input variable $x_{1}$. Table \\ref{doGAD} displays the evolution\nof this example code under the GAD scheme, which proceeds as follows:\n\\begin{enumerate}[1.]\n  \\item We begin with a list of variables and subexpression computations for\n    the Hamiltonian, and translate this \\texttt{C} code into the Mathematica\n    language.\n  \\item We parameterize the terms of each subexpression according to their\n    dependence on $x_{1}$.\n  \\item Mathematica computes derivatives of each subexpression.\n  \\item We convert the Mathematica output into \\texttt{C} code.\n  \\item We replace each occurrence of $x_{1}'$ with 1 and remove terms equal to 0.\n\\end{enumerate}\n\nThe resulting \\texttt{C} code is short, optimized, and human readable.\nFurthermore, any terms that are common to all derivative expressions are\ncomputed and saved before computing the partial derivatives, further reducing\nthe computational cost.\n\n\\begin{table}\n  \\centering\n  \\begin{adjustbox}{max width=\\textwidth}\n    \\begin{tabular}{|l|l|}\n      \\hline\n      \\textbf{Step 1: Convert \\texttt{C} to Mathematica.} & \\textbf{Step 2: Parameterize subexpressions.} \\\\\n        \\texttt{v1 = Sqrt[x1] + a*x1}                     & \\texttt{v1 = Sqrt[x1[x]] + a*x1[x]}           \\\\\n        \\texttt{v2 = Sqrt[x2] + a*x2}                     & \\texttt{v2 = Sqrt[x2] + a*x2}                 \\\\\n        \\texttt{v3 = (v1 + v2)\/(v1*v2)}                   & \\texttt{v3 = (v1[x] + v2[x])\/(v1[x]*v2[x])}   \\\\\n        \\texttt{H~ = v3*v3}                               & \\texttt{H = v3[x]*v3[x]}                      \\\\\n      \\hline\n        \\multicolumn{2}{|l|}{\\textbf{Step 3: Utilize Mathematica to compute derivatives.}}                 \\\\\n        \\multicolumn{2}{|l|}{\\texttt{v1' = x1'[x]\/(2*Sqrt[x1[x]]) + a*x1'[x]}}                             \\\\\n        \\multicolumn{2}{|l|}{\\texttt{v2' = 0}}                                                             \\\\\n        \\multicolumn{2}{|l|}{\\texttt{v3' = (v1[x]*v2[x]*(v1'[x] + v2'[x])-((v1[x] + v2[x])*(v1'[x]*v2[x]}} \\\\\n        \\multicolumn{2}{|l|}{\\texttt{~~~~~ + v1[x]*v2'[x]))\/(v1[x]*v1[x]*v2[x]*v2[x])}}                    \\\\\n        \\multicolumn{2}{|l|}{\\texttt{H'~ = 2*v3'[x]*v3[x]}}                                                \\\\\n      \\hline\n        \\multicolumn{2}{|l|}{\\textbf{Step 4: Convert Mathematica to \\texttt{C}; prime notation becomes a protected \\texttt{prm} suffix.}} \\\\\n        \\multicolumn{2}{|l|}{\\texttt{v1prm = x1prm\/(2*sqrt(xi)) + a*x1prm}}                                                               \\\\\n        \\multicolumn{2}{|l|}{\\texttt{v2prm = 0}}                                                                                          \\\\\n        \\multicolumn{2}{|l|}{\\texttt{v3prm = (v1*v2*(v1prm + v2prm)-((v1 + v2)*(v1prm*v2 + v1*v2prm))\/(v1*v1*v2*v2)}}                     \\\\\n        \\multicolumn{2}{|l|}{\\texttt{Hprm~ = 2*v3prm*v3}}                                                                                 \\\\\n      \\hline\n        \\multicolumn{2}{|l|}{\\textbf{Step 5: Replace \\texttt{x1prm} with 1 and remove terms equaling 0.}} \\\\\n        \\multicolumn{2}{|l|}{\\texttt{v1prm = 1.\/(2*sqrt(x1)) + a}}                                        \\\\\n        \\multicolumn{2}{|l|}{\\texttt{v3prm = (v1*v2*v1prm-(v1 + v2)*v1prm*v2)\/(v1*v1*v2*v2)}}             \\\\\n        \\multicolumn{2}{|l|}{\\texttt{Hprm~ = 2*v3prm*v3}}                                                 \\\\\n      \\hline\n    \\end{tabular}\n  \\end{adjustbox}\n  \\caption{Step-by-step GAD code evolution.}\\label{doGAD}\n\\end{table}\n\nSince each $v_{\\ell}$ is merely an intermediate of $H$, there is significant\nfreedom in our choice of the set of subexpressions\n$\\mathcal{V}\\equiv\\left\\{v_{1},v_{2},\\hdots,v_{N-1}\\right\\}$. Our choices do,\nhowever, have a direct effect on the number of calculations necessary to\ncompute $\\partial_{x_{1}}H$, which we measure in floating point operations\n(FLOPs\\footnote{Not to be confused with ``FLOPs per second'' (FLOPS).}.) Our\ngoal in GAD, therefore, is to choose $\\mathcal{V}$ to minimize the number of\nFLOPs needed to compute $\\partial_{x_{1}}H$.\n\nIn general, the largest contributor to FLOPs is the product rule. If there are\n$M$ different subexpressions multiplied together in a given expression,\ncomputing the derivative will require $\\mathcal{O}(M^2)$ FLOPs. If we\ntherefore choose $\\mathcal{V}$ such that each $v_{\\ell}$ contains no more than\ntwo previous subexpressions multiplied together, we should minimize the\noverall cost. We expect a significant reduction in FLOPs to correspond to a\nsignificant reduction in the time to generate a waveform.\n\nWe estimated the number of FLOPs based on benchmarks provided\nin~\\cite{FLOPsite} for CPUs corresponding to the CPU family in our\nworkstations (Intel Core i7-6700) and generated Table \\ref{countratio}. We\nemphasize that the values listed in Table~\\ref{countratio} are truly rough\nestimates, used only to provide us general direction as we seek an optimal\n$\\mathcal{V}$.\n\n\\begin{table}\n  \\centering\n  \\begin{adjustbox}{max width=\\textwidth}\n    \\begin{tabular}{|c|c|c|c|c|c|c|c|}\n      \\hline\n      $a+b$ & $a-b$ & $a*b$ & $a=b$ & $a\/b$ & ${\\tt sqrt}(a)$ & ${\\tt log}(a)$ & ${\\tt pow}(a,b)$  \\\\ \\cline{3-5}\n      \\hline\n        1   &   1   &   1   &   1   &   3   &     3     &    24    &     24      \\\\ \\cline{3-5}\n      \\hline\n    \\end{tabular}\n  \\end{adjustbox}\n  \\caption{Relative FLOPs count of the mathematical\n    operations.}\\label{countratio}\n\\end{table}\n\nTable~\\ref{FLOPs} compares the number of Hamiltonian derivative FLOPs under\nGAD to the number in the exact derivatives (EDs) generated by Mathematica's\nCSE code generation algorithm.  In principle, the difference in FLOPs between\nED and GAD schemes may be used to predict the waveform generation speedup\nfactor. A direct comparison from Table~\\ref{FLOPs} indicates a 3.6x reduction\nin FLOPs when using GAD. For a double neutron star coalescence, Hamiltonian\nderivative computations constitute about 80\\% of waveform generation time.\nThis suggests a speedup factor of 2.3x. Waveform generation times for three\nscenarios comparing ED and GAD implemented in v3\\_opt are shown in\nTable~\\ref{previewbench}, and demonstrates a speedup factor of about 1.7x. We\nemphasize again that counting FLOPs using the relative values of\nTable~\\ref{countratio} only provides a rough estimate of the reduction in\nFLOPs, and the compiler itself rearranges arithmetic expressions to minimize\nFLOPs as well so the gap between our estimated and observed speed-ups is not\nsurprising.\n\n  \\begin{table}\n    \\centering\n    \\begin{adjustbox}{max width=\\textwidth}\n      \\begin{tabular}{|c|ccc|c|}\n        \\hline\n        Derivative scheme & Space derivative &  Momentum derivative &  Spin derivative &  Total   \\\\\n                          & (FLOPs)          & (FLOPs)              & (FLOPs)          &  (FLOPs) \\\\\n        \\hline\n        ED                & 3 x 5073         & 3 x 2319             &  6 x 4333        &  48174   \\\\\n        GAD               & 3 x 1418         & 3 x 527              &  6 x 1264        &  13419   \\\\\n        \\hline\n      \\end{tabular}\n    \\end{adjustbox}\n    \\caption{Number of FLOPs using ED versus GAD methods.}\\label{FLOPs}\n  \\end{table}\n\n  \\begin{table}\n    \\centering\n    \\begin{adjustbox}{max width=\\textwidth}\n      \\begin{tabular}{|c|c|c|c|}\n        \\hline\n                                                                            & v3\\_opt (s)       & v3\\_opt (s)       \\\\\n        \\textbf{Parameters}                                                 & ED                & GAD               \\\\\n        \\hline\n        \\textbf{Neutron Star Binary}                                        & 36.75             & 20.49             \\\\\n        $1.4M_{\\odot} + 1.4M_{\\odot}$, $s_{1}^{y} = 0.05$                     &                   & x(\\textbf{1.79})  \\\\ \n        \\hline\n        \\textbf{Black Hole + Neutron Star}                                  & 8.07              & 4.69              \\\\\n        $10M_{\\odot} + 1.4 M_{\\odot}$, $s_{1}^{y} = 0.4$                      &                   & x(\\textbf{1.72})  \\\\\n        \\hline\n        \\textbf{Black Hole Binary} (GW150914-like)                          &                   &                   \\\\\n        $36M_{\\odot} + 29M_{\\odot}$                                           & 0.64              & 0.38              \\\\\n        $s_{1}^{y}=0.05,\\ s_{1}^{z}=0.5,\\ s_{2}^{y}=-0.01,\\ s_{2}^{z}=-0.2$ &                   & x(\\textbf{1.68})  \\\\\n        \\hline\n      \\end{tabular}\n    \\end{adjustbox}\n    \\caption{Benchmark comparison of ED to GAD strategies. In each\n      scenario, we adopt a 10Hz start frequency.}\\label{previewbench}\n  \\end{table}\n\n\\subsection{Dense Output: A more efficient way of interpolating\n  sparsely-sampled data}\n\\label{v3opts:do}\n\nAn RK4 ODE solver with adaptive timestep control solves the EOB Hamiltonian\nequations of motion in v3; thus solutions are unevenly sampled in time.\nSubsequent analyses require mapping these data into the frequency domain via\nthe fast Fourier transform (FFT), which expects evenly-sampled data. Rather\nthan restricting the integration timestep, v3 uses cubic splines to\ninterpolate the Hamiltonian solutions after RK4 runs to completion. During\noptimization of v2, the GSL cubic spline interpolation routine was optimized\nand gave significant performance gains. During optimization of v3, it was\ndiscovered that third-order Hermite interpolation made v3\\_Opt more faithful\nto v3 (see Section \\ref{Results}). Hermite interpolation requires only two\nfunction values and the derivatives at those values, which are available at\neach step of RK8. Thus we may interpolate the sparsely-sampled data to the\ndesired evenly-sampled data ``on the fly'' during integration. Such an\nintegration routine is called a \\textit{dense output} method~\\cite{NR}. In\nparticular, suppose the RK8 integrator computes the solution $y(t_{0})$ and\n$y(t_{1})$ at times $t_{0}$ and $t_{1}$ with timestep $h$ and derivative\nvalues $y'_{0}$ $y'_{1}$. Then for any $0\\le\\theta\\le1$, we have\n  \\begin{equation*}\n  \\begin{adjustbox}{max width=\\textwidth}\n      $y(t_{0} + \\theta h) = (1-\\theta)y(t_{0}) + \\theta y(t_{1}) + \\theta(\\theta-1)\\left[(1-2\\theta)(y(t_{1}) - y(t_{0}) + (\\theta-1)hy'(t_{0}) + \\theta hy'(t_{1})\\right]$.\n  \\end{adjustbox}\n  \\end{equation*}\nAs this cubic Hermite interpolation routine uses both the solution data and\nderivative values at each point, it therefore requires only the output of the\nRK8 integration and no further data storage or function evaluations.\n\n\\section{Results}\\label{Results}\nIn Sec.~\\ref{subsec:faith} we establish that v3\\_Opt produces waveforms which\nagree with v3 at the level of roundoff error. Section \\ref{subsec:bench} then\ndescribes the process of measuring speedup and demonstrates the speedup factor\nachieved.\n\n\\subsection{Determining Faithfulness}\\label{subsec:faith}\nGiven two waveforms $h_1(t)$ and $h_2(t)$ (in the time domain), we determine\nif $h_{1}(t)$ is \\emph{faithful} to $h_{2}(t)$ using the LVC's open-source\n\\textsc{PyCBC} software~\\cite{pycbc-software,Canton2014,Usman2015}. This\ncomputation depends on the following definitions, which we write in the same\nform as~\\cite{v4}. The \\emph{noise-weighted overlap} between $h_{1}$ and\n$h_{2}$ is defined as\n  \\[\n    (h_1 | h_2) \\equiv 4{\\rm Re} \\int_{f_{l}}^{f_{h}} \\frac{ \\tilde{h}_{1}(f)\\tilde{h}^{*}_{2}(f) }{ S_{n}(f) } {\\rm d}f\n  \\]\nwith $\\tilde{h}_{i}(f)$ denoting the Fourier transform of the waveform\n$h_{i}(t)$, $h_{i}^{*}$ denoting the complex conjugate of $h_{i}$, $f_{l}$ and\n$f_{h}$ denoting the endpoints of the range of frequencies of interest, and\n$S_n(f)$ denoting the one-sided power spectral density (PSD) of the LIGO\ndetector noise. We chose $f_{l}=20$ Hz and $S_{n}(f)$ to be Advanced LIGO's \ndesign zero-detuned high-power noise PSD~\\cite{noisecurve}. For each waveform,\n$f_{h}$ is the Nyquist critical frequency~\\cite{NR}. We then define the\n\\emph{faithfulness} between $h_{1}$ and $h_{2}$ to  be the overlap between the\nnormalized waveforms maximized over relative time and phase shifts:\n  \\[\n    \\braket{h_{1} | h_{2}} \\equiv \\max_{\\phi_c, t_c} \\frac{ (h_1(\\phi_c,t_c) | h_2) }{ \\sqrt{(h_1 | h_1)(h_2 | h_2)} }.\n  \\]\nHere $t_{c}$ and $\\phi_c$ denote the coalescence time and phase, respectively.\nNote that normalization forces $\\braket{h_{1} | h_{2}} \\in [0,1]$, with\n$\\braket{h_{1} | h_{2}} = 1$ indicating complete overlap (and therefore a\nperfectly faithful waveform) while $\\braket{h_{1} | h_{2}} = 0$ indicates no\noverlap (an \\emph{unfaithful} waveform\\footnote{Another common measure in\nfaithfulness tests is \\textit{mismatch}, defined as\n$1-\\braket{h_{1} | h_{2}}$.}). For each faithfulness test conducted, we\ngenerate a waveform with two different approximants and the same set of input\nparameters.\n\nWe ran 100,000 faithfulness tests for each set of waveform approximants we\nwished to compare. The input parameters for each test are randomly chosen by\n\\textsc{PyCBC} with bounds as outlined in Table \\ref{faithparams}; these\nbounds are chosen to capture the relevant parameter space for v3.  Note that\neach of the spin parameters $s_{i}^{x}, s_{i}^{y}, s_{i}^{z}$ are chosen\nrandomly in $(-1,1)$ with the constraint\n\\[\n  \\sqrt{ (s_{i}^{x})^{2} + (s_{i}^{y})^{2} + (s_{i}^{z})^{2} } \\le 0.99, \\ \\ i \\in \\{ 1,2\\}.\n\\]\n\n\\begin{table}\n  \\begin{tabular}{|c|c|}\n    \\hline\n    Mass of Object 1 (solar masses)            & $m_{1}\\in[1,100]$               \\\\\n    \\hline\n    Mass of Object 2 (solar masses)            & $m_{2}\\in[1,100]$               \\\\\n    \\hline\n    Spin magnitude of Object 1 (dimensionless) & $\\lvert a_{1}\\rvert\\in[0,0.99]$ \\\\\n    \\hline\n    Spin magnitude of Object 2 (dimensionless) & $\\lvert a_{2}\\rvert\\in[0,0.99]$ \\\\\n    \\hline\n    Binary total mass (solar masses)           & $m_{\\rm total}\\in[4,100]$       \\\\\n    \\hline\n    Starting orbital frequency (Hz)            & $f=19$                          \\\\\n    \\hline\n  \\end{tabular}\n\\caption{Ranges of values for random input parameters in our faithfulness\n  tests.}\\label{faithparams}\n\\end{table}\n\nThe specific faithfulness runs we conducted were organized as follows. The\napproximant v3\\_pert is identical to v3 except $m_{1}$ is replaced with\n$m_{1}\\left( 1 + 10^{-16} \\right)$; such a perturbation should result\nin waveforms that are nearly identical and provides a measure of how sensitive\nv3 is to roundoff error. Thus faithfulness tests comparing v3 and v3\\_pert\nprovide a ``control'' against which we compare the faithfulness of v3\\_Opt to\nv3. As another point of comparison, we also test v3 (which is RK4-based)\nagainst the RK4-based v3\\_Opt\\_rk4. For each approximant comparison we\ncompare the effect of increasingly stricter ODE solver tolerance. By default,\nv3 sets the ODE solver's absolute and relative error tolerances to\n$\\varepsilon\\equiv1\\times10^{-8}$; we compare faithfulness at tolerances of\n $\\varepsilon$, $\\varepsilon\\times10^{-1}$, $\\varepsilon\\times10^{-2}$,\n$\\varepsilon\\times10^{-3}$, and $2\\varepsilon\\times10^{-4}$. Finally, we also\nconsider the effect of compiler choice on faithfulness and so conduct\nfaithfulness runs using both \\texttt{gcc} and \\texttt{icc}. Table\n\\ref{faithresults} summarizes the faithfulness tests conducted and their\nresults; the rightmost column displays the counting error $\\sqrt{n}$ for the\nnumber of waveforms $n$ with $\\braket{\\cdot|\\cdot} < 0.999$.\n\n\\begin{table}\n\\begin{adjustbox}{max width=\\textwidth}\n\\begin{tabularx}{1.3\\textwidth}{c c c *{5}{Y} c}\n\\toprule\n                                  &               & \\bf{ODE}                   & \\multicolumn{5}{c}{\\textbf{Number of waveforms with faithfulness}} & \\textbf{Counting} \\\\\n\\cmidrule(lr){4-8}\n\\bf{Comparison}                   & \\bf{Compiler} & \\bf{tolerance}             & $<0.8$ & $<0.9$ & $<0.95$ & $<0.99$ & $<0.999$                     & \\textbf{Error} \\\\\n\\midrule\nv3 vs.~v3\\_pert                   & \\texttt{gcc}  & $\\varepsilon$              & 1      & 5      & 13      & 104     & 399                          & $\\pm20$ \\\\\n(per $10^{5}$ for $10^{6}$ tests) & \\texttt{icc}  & $\\varepsilon$              & 1.0    & 4.2    & 11.5    & 109.0   & 398.2                        & $\\pm6.3$ \\\\\n\\hline\nv3 vs.~v3\\_Opt                    & \\texttt{gcc}  & $\\varepsilon$              & 5      & 28     & 136     & 1184    & 5466                         & $\\pm74$ \\\\\n                                  & \\texttt{icc}  & $\\varepsilon$              & 5      & 28     & 135     & 1174    & 5509                         & $\\pm74$ \\\\\n                                  & \\texttt{icc}  & $\\varepsilon\\times10^{-1}$ & 2      & 16     & 44      & 327     & 1510                         & $\\pm39$ \\\\\n                                  & \\texttt{icc}  & $\\varepsilon\\times10^{-2}$ & 0      & 2      & 12      & 143     & 727                          & $\\pm27$ \\\\\n                                  & \\texttt{icc}  & $\\varepsilon\\times10^{-3}$ & 1      & 3      & 8       & 80      & 511                          & $\\pm23$ \\\\\n                                  & \\texttt{icc}  & $2\\varepsilon\\times10^{-4}$& 1      & 1      & 2       & 60      & 457                          & $\\pm21$ \\\\\n\\hline\nv3 vs.~v3\\_Opt\\_rk4               & \\texttt{gcc}  & $\\varepsilon$              & 1      & 9      & 35      & 427     & 1529                         & $\\pm39$ \\\\\n                                  & \\texttt{icc}  & $\\varepsilon$              & 0      & 9      & 35      & 420     & 1510                         & $\\pm39$ \\\\\n                                  & \\texttt{icc}  & $\\varepsilon\\times10^{-1}$ & 1      & 7      & 24      & 223     & 926                          & $\\pm30$ \\\\\n                                  & \\texttt{icc}  & $\\varepsilon\\times10^{-2}$ & 0      & 0      & 8       & 114     & 585                          & $\\pm24$ \\\\\n                                  & \\texttt{icc}  & $\\varepsilon\\times10^{-3}$ & 1      & 3      & 8       & 77      & 483                          & $\\pm22$ \\\\\n                                  & \\texttt{icc}  & $2\\varepsilon\\times10^{-4}$& 1      & 2      & 3       & 52      & 423                          & $\\pm21$ \\\\\n\\bottomrule\n\\end{tabularx}\n\\end{adjustbox}\n\\caption{Summary of \\texttt{PyCBC} faithfulness results. Here\n$\\varepsilon=1\\times10^{-8}$ and each row reports the results of a run of\n\\textbf{100,000 faithfulness tests}. \\texttt{icc} refers to Intel compiler\nversion 15.5.223, while \\texttt{gcc} refers to GNU compiler version\n4.9.}\\label{faithresults}\n\\end{table}\n\nWe comment on the values in Table \\ref{faithresults}. For a couple of\nparameters for which $\\braket{\\cdot | \\cdot} < 0.8$ when comparing v3 to\nv3\\_Opt compiled with \\texttt{gcc}, one author back-traced a significant\ndifference between v3 and v3\\_Opt to the ODE stopping condition or the time of\nmaximum amplitude being clearly wrong in v3 but not v3\\_Opt. In particular,\nthere are some algorithms within v3 that are fundamentally non-robust, and\nv3\\_Opt inherits most of these functions. The RK8 integration of v3\\_Opt\nshould be just as accurate as the RK4 integration of v3\\_Opt\\_rk4 when the\ntolerances are equal, but the output from RK8 should be much sparser (by more\nthan a factor of 2) than RK4. Since we observe worse faithfulness with v3\\_Opt\nthan v3\\_Opt\\_rk4, we conclude that most of the truncation error stems from\nthe interpolation of the sparsely-sampled ODE solution to a uniform timestep.\n\nMost importantly, notice that as we make the ODE solver's tolerance $\\varepsilon$\nstricter (resulting in smaller errors and more finely sampled output data from\nthe ODE solver), the faithfulness between v3 and v3\\_Opt improves to the level\nof agreement between v3 and v3\\_pert.  Thus we conclude that v3\\_opt generates\nroundoff-level agreement in the limit of $\\varepsilon\\to0$ with errors dominated\nby interpolation otherwise.\n\n\\subsection{Performance Benchmarks}\\label{subsec:bench}\nIn order to capture the full effect of our optimizations to v3, we compared\nwaveform generation times of v3\\_Opt with waveform generation times of\nv3\\_preopt. In particular, v3\\_preopt lacks by-hand optimizations of the EOB\nHamiltonian implemented in the development of v2\\_opt; thus unnecessary\ncomputations of transcendental functions \\texttt{pow()}, \\texttt{log()}, and\n\\texttt{exp()} remain therein. All reported benchmarks were completed on a\nsingle core of a modern desktop computer with an Intel Core i7-7700 CPU and 64\nGB RAM.\n\nTo highlight cases of interest, Table \\ref{benchmarkresults} summarizes\nbenchmarks of v3\\_Opt and v3\\_Opt\\_rk4 in comparison to v3\\_preopt for a\nhandful of scenarios of interest to LIGO. The speedup factors are also\nincluded, with speedup simply defined to be the ratio of time to generate a\nwaveform with v3\\_preopt to the time to generate the same waveform with\nv3\\_Opt or v3\\_Opt\\_rk4.\n\n  \\begin{table}\n    \\centering\n    \\begin{adjustbox}{max width=\\textwidth}\n      \\begin{tabular}{|c|c|c|c|c|}\n        \\hline\n        Physical scenario                   & v3\\_preopt                    & v3\\_Opt\\_rk4     & v3\\_Opt                   & v3\\_Opt                  \\\\\n                                            & \\texttt{gcc}, (s)                  & \\texttt{gcc}, (s)               & \\texttt{gcc}, (s)               & \\texttt{icc}, (s)                \\\\\n        \\hline\n        \\hline\n        DNS, $s_{2}^{y}=0.05$               & \\multirow{2}{*}{8618.60}           & 98.51                           & 42.85                           & 21.22                            \\\\ \\cline{3-5}\n        $1.3M_{\\odot}+1.3M_{\\odot}$           &                                    & x(\\textbf{87.49})               & x(\\textbf{201.1})               & x(\\textbf{406.2})                \\\\\n        \\hline\n        BHNS, $s_{\\rm NS}^{y}=0.05$         & \\multirow{2}{*}{2760.77}           & 20.75                           & 8.84                            & 4.37                             \\\\ \\cline{3-5}\n        $10M_{\\odot}+1.3M_{\\odot}$            &                                    & x(\\textbf{133.0})               & x(\\textbf{312})                 & x(\\textbf{632})                  \\\\\n        \\hline\n        BHB, $s_{2}^{y}=0.05$               & \\multirow{2}{*}{127.71}            & 1.70                            & 0.90                            & 0.46                             \\\\ \\cline{3-5}\n        $16M_{\\odot}+16M_{\\odot}$             &                                    & x(\\textbf{75.1})                & x(\\textbf{140})                 & x(\\textbf{280})                  \\\\\n        \\hline\n        BHB, $s_{1}^{y}=s_{2}^{y}=0.9$      & \\multirow{2}{*}{168.13}            & 1.75                            & 0.91                            & 0.46                             \\\\ \\cline{3-5}\n        $16M_{\\odot}+16M_{\\odot}$             &                                    & x(\\textbf{96.1})                & x(\\textbf{180})                 & x(\\textbf{370})                  \\\\\n        \\hline\n        BHB, $s_{1}^{y}=s_{2}^{z}=0.9$      & \\multirow{2}{*}{235.53}            & 3.48                            & 1.55                            & 0.76                             \\\\ \\cline{3-5}\n        $10M_{\\odot}+10M_{\\odot}$             &                                    & x(\\textbf{67.7})                & x(\\textbf{152})                 & x(\\textbf{310})                  \\\\\n        \\hline\n        BHB, GW150914-like                  & \\multirow{4}{*}{31.48}             & \\multirow{2}{*}{0.75}           & \\multirow{2}{*}{0.51}           & \\multirow{2}{*}{0.27}            \\\\\n        $36M_{\\odot}+29M_{\\odot}$             &                                    &                                 &                                 &                                  \\\\ \\cline{3-5}\n        $s_{1}^{y}=0.05$, $s_{1}^{z} = 0.5$ &                                    & \\multirow{2}{*}{x(\\textbf{42})} & \\multirow{2}{*}{x(\\textbf{60})} & \\multirow{2}{*}{x(\\textbf{120})} \\\\\n        $s_{2}^{y}=-0.01$, $s_{2}^{z}=-0.2$ &                                    &                                 &                                 &                                  \\\\\n        \\hline\n      \\end{tabular}\n    \\end{adjustbox}\n    \\caption{Benchmarks and speedups of v3\\_Opt and v3\\_Opt\\_rk4 compared to\n      v3.}\\label{benchmarkresults}\n  \\end{table}\n\nTo demonstrate that the advertised speedup factors of Table\n\\ref{benchmarkresults} apply across the parameter space of binaries of\ninterest to the LVC, we completed four benchmark surveys. The first two\nconcern binary black hole systems, one with varying masses and the other with\nvarying spins. The third survey considers mixed binaries (one black hole and\none neutron star), and the fourth binary neutron stars. The parameters tested\nin each run are included in Table \\ref{parambench}. The results of these\nsurveys are plotted in Figure \\ref{speed} and summarized in Table\n\\ref{benchmarkresults}.\n\n  \\begin{table}\n    \\begin{adjustbox}{max width=\\textwidth}\n    \\begin{tabular}{lcccc}\n      \\hline\n      Ranges        & $m_{1}$ ($M_{\\odot}$) & $q$ (dimensionless)       & $a_{1}$ (dimensionless) & $a_{2}$ (dimensionless) \\\\\n      \\hline\\hline\n      BHB$_{\\rm M}$ & $[16.7,100.3]$       & $[1,10]$                  & 0.0500001               & 0                       \\\\\n      BHB$_{\\rm S}$ & 10                   & 1                         & $[-0.95,0.95]$          & $[-0.95,0.95]$          \\\\\n      BHNS          & $[7,100]$            & $\\frac{M}{1.4}$           & $[-0.95,0.95]$          & 0                       \\\\\n      DNS           & $[1.2,2.3]$          & $\\frac{M}{m\\in[1.2,2.3]}$ & 0.0500001               & 0                       \\\\\n      \\hline\n    \\end{tabular}\n    \\end{adjustbox}\n    \\caption{Surveyed parameters: each survey tested 400 parameter\n      combinations, with 20 evenly-spaced values taken in each range\n      indicated. Here BHB$_{\\rm M}$ indicates the black hole binary mass\n      survey, BHB$_{\\rm S}$ the black hole binary spin survey, BHNS the black\n      hole neutron star survey, and DNS the double neutron star survey. We\n      define $q \\equiv \\frac{m_{1}}{m_{2}}$, the ratio of the mass of object 1\n      to the mass of object 2. The dimensionless Kerr spins of each object are\n      denoted $a_{1}$ and $a_{2}$, respectively. Each waveform generation\n      started with a frequency of 10 Hz used a sample rate of 16,384\n      Hz.}\\label{parambench}\n  \\end{table}\n\n\\begin{figure}\n  \\centering\n  \\begin{subfigure}[t]{0.5\\textwidth}\n    \\centering\n    \\includegraphics[height=5cm]{speedup_plot.eps}\n  \\end{subfigure}%\n~\n  \\begin{subfigure}[t]{0.5\\textwidth}\n    \\centering\n    \\includegraphics[height=5cm]{time_plot.eps}\n  \\end{subfigure}\n  \\caption{\\textbf{Performance benchmarks:} \\textbf{Left panel:} plots speedup\n    factor versus number of wavecycles in the binary inspiral. Measuring the\n    number of wavecycles allows us to compactly display the results of the\n    benchmark tests without explicit reference to mass or spin. \\textbf{Right\n      panel:} plots the number of wavecycles versus the time taken to output\n    the waveform. Note that the speedup factor in the left panel is simply the\n    ratio of the curves in the right panel.}\\label{speed}\n\\end{figure}\n\nWe would like to measure an average speedup based on the four surveys. As\nin~\\cite{v2opt}, we define an overall speedup factor as a waveform\ncycle-weighted average\n  \\[\n    \\mathcal{S} = \\frac{\\sum_{i}\\mathcal{S}_{i}N_{i}}{\\sum_{i} N_{i}}\n  \\]\nwhere $\\mathcal{S}_{i}$ is the speedup factor for generating the\n$i^{\\text{th}}$ waveform and $N_{i}$ is the number of\nwavecycles in the $i^{\\text{th}}$ waveform. We found $\\mathcal{S}\\sim340$.\nThis reduces the time necessary for a black hole binary PE run from\n$\\mathord{\\sim}$100 years (with v3\\_preopt) to $\\mathord{\\sim}$8 months (with v3\\_Opt). We\nexpect lower mass PE runs will be possible on similar timescales with\nadditional optimizations.\n\n\\section{Conclusions and Future Work}\n\\label{Conclusions}\nAnticipating the potential detection by Advanced LIGO of significantly\nprecessing compact binaries, we have optimized v3 to make costly\nprecessing-waveform-approximant-based data analysis applications like PE\npossible in a reasonable amount of time. If an efficient 8D ROM is found, such\noptimizations will make the construction of this ROM faster. After migrating\nv2\/v4 optimizations to v3, we further optimized partial derivatives of the\nHamiltonian using a GAD scheme. This resulted in waveforms  that are faithful\nto v3, as evidenced by faithfulness increasing to 1 as ODE tolerance\ndecreases. We achieved an average overall speedup of $\\mysim340$x, ranging\nfrom $\\mysim120$x for GW150914-like black hole binaries to $\\mysim630$x for\nblack hole-neutron star binaries. We expect that further optimizations are\npossible, achieving an additional speedup factor of at least $\\mathord{\\sim}$3x.\nFuture work will focus on transforming Cartesian coordinates to spherical\ncoordinates to lower sampling rates even more during ODE solving and\nintegration.\n\n\\ack\nWe thank O.~Birnholtz, N.~Johnson-McDaniel, R.~Sturani, A.~Taracchini, and\nC.~Haster for helpful comments and discussion during a review of the v3\\_opt\ncode. A.~Taracchini is especially thanked for introducing us to faithfulness\ntesting via \\texttt{PyCBC}, as is S.~Teukolsky for suggesting dense output\nmethods. We also thank R.~Haas for his work optimizing v3 during its\ndevelopment, and I.~Ruchlin for numerous helpful discussions. This work was\nsupported primarily by NSF LIGO Research Support Grant PHY-1607405. Early work\non this project was supported by NSF EPSCoR Grant 1458952 and NASA Grant\n13-ATP13-0077. We are grateful for the computational resources provided by the\nLeonard E.~Parker Center for Gravitation, Cosmology and Astrophysics at\nUniversity of Wisconsin-Milwaukee (NSF Grant 0923409), LIGO Livingston\nObservatory, LIGO Hanford Observatory, and the LIGO-Caltech Computing Cluster.\n\n\\section*{References}\n\n\\sloppy\n\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe classical theories as well as most of the recent theoretical studies and experiments regarding the efficiency of plasmonic resonances \nreported in the literature are concerned with metal nanoparticles where the exterior domain is lossless, \nsee \\textit{e.g.}\\\/,  \\cite{Bohren+Huffman1983,Link+etal2000,Maier2007,Tretyakov2014,Miller+etal2016,Tzarouchis+etal2016}. \nAs \\textit{e.g.}\\\/,  in \\cite{Miller+etal2016}, a variational approach is employed in connection with a generalized optical theorem for scattering, absorption and extinction \n\\cite{Lytle+etal2005}, to obtain an upper bound on the absorption that can be achieved inside a scatterer with arbitrary geometry and with a given volume and material property.\nFurthermore, an upper bound on the dipole absorption of an electrically small particle with arbitrary geometry and structural parameters is given in \\cite{Tretyakov2014}.\nThe bound \\cite[Eq.~(16) on p.~937]{Tretyakov2014} is based on the \noptical theorem \\cite{Bohren+Huffman1983} and obtained by optimizing absorption \ndirectly in terms of the complex valued polarizability of the particle. \nBoth results in \\cite{Tretyakov2014} and \\cite{Miller+etal2016} are valid for a lossless surrounding medium only. \n\nHowever, there are many application areas of plasmonics where the exterior losses must be taken into account.\nOne potential application in medicine is the localized electrophoretic heating of a bio-targeted and electrically \ncharged gold nanoparticle (GNP) suspension as a radiotherapeutic hyperthermia based method to treat cancer, \n\\textit{cf.}\\\/,  \\cite{Corr+etal2012,Sassaroli+etal2012,Collins+etal2014,Nordebo+etal2017a,Dalarsson+etal2017a}\nor with the related plasmonic photothermal therapy as proposed in \\cite{Huang+etal2008}.\nOther potential application areas include plasmon waveguides, aperture arrays, extraordinary transmission, superlenses, artificial magnetism, negative refractive index,\nand surface-enhanced biological sensing with molecular monolayer spectroscopy, etc., see \\textit{e.g.}\\\/,  \\cite{Maier2007}. \n\nThere has been a number of investigations devoted to the scattering, absorption and extinction of small particles embedded \nin a lossy medium, see \\textit{e.g.}\\\/,  \\cite{Mundy+etal1974,Chylek1977,Bohren+Gilra1979,Lebedev+etal1999,Sudiarta+Chylek2001,Durant+etal2007a}. \nThis topic has even been subjected to some controversy due to the difficulties to define a general theory\nencompassing the notion of a cross section when the surrounding medium is lossy, see \\textit{e.g.}\\\/, \\cite{Bohren+Gilra1979,Lebedev+etal1999,Sudiarta+Chylek2001,Durant+etal2007a}. \nIn contrast, for a lossless surrounding medium the absorption cross section can be defined\nfrom the power flowing into a conceptual sphere surrounding the particle at an arbitrary radius, and which enables the derivation of an optical theorem valid for arbitrary geometries \\cite[pp.~71 and 140]{Bohren+Huffman1983}. For a lossy medium this theory is no longer valid, which is due to the fact\nthat the absorption in the surrounding medium depends on the geometry of the scatterer.\nHence, the optical theorems for lossy media are typically given only for spheres. As \\textit{e.g.}\\\/,  in \\cite{Nordebo+etal2018f} is given \nnew fundamental upper bounds on the multipole absorption and scattering of a rotationally invariant\nsphere embedded in a lossy surrounding medium and which are derived based on the corresponding generalized optical theorem \nas given in \\textit{e.g.}\\\/,  \\cite[Eq.~(7) on p.~1276]{Sudiarta+Chylek2001}.\n\nWe are concerned here with the optimal absorption of an electrically small spherical object \nembedded in a lossy surrounding medium when the near-field distribution can be found using the quasistatic approximation. \nFor simplicity, we are considering only the important special case of non-magnetic, dielectric materials which are common in plasmonic applications. \nMagnetic materials can be treated similarly.\nWe are also considering only a single electric dipole resonance, and we are\ndiscarding any possibilities of having an unbounded absorption due to multiple mode super resolution effects, etc.,\nsee \\textit{e.g.}\\\/,  \\cite{Valagiannopoulos+etal2015,Maslovski+etal2016,Valagiannopoulos+Tretyakov2016}.\nA quasistatic theory has been developed in \\cite{Nordebo+etal2017a,Dalarsson+etal2017a} giving the optimal plasmonic dipole resonance of small dielectric \nellipsoids in terms of an optimal conjugate match with respect to the background loss. \nHowever, an important limitation of this theory is that it does not give the correct physical answers when the background becomes lossless or has very small losses.\nThis limitation is obvious since the quasistatic optimal absorption becomes unbounded in the case when the external losses vanish\n\\textit{cf.}\\\/,  \\textit{e.g.}\\\/,  \\cite[Eq.~(33) on p.~5]{Dalarsson+etal2017a}, \nin contrast to the well known fact that the absorption cross section of a small dipole scatterer in a lossless exterior medium is bounded \nby $3\\lambda^2\/8\\pi$, \\textit{cf.}\\\/,  \\cite[Eq.~(16) on p.~937]{Tretyakov2014}.\n\nThis limitation, which may even appear as a contradiction, can be understood simply by realizing that the quasistatic model disregards radiation damping. \nMoreover,  the plasmonic singularity of the sphere ($\\epsilon=-2$ in vacuum) \nexists only in the sense of a limit as the size of the particle approaches zero \\cite{Tzarouchis+etal2016}.\nIn fact, it turns out that the dipole resonance of a small sphere has a very subtle limiting behavior as the electrical size approaches zero,\nand as \\textit{e.g.}\\\/,  in \\cite{Tzarouchis+etal2016} Pad\\'{e} approximants are used to reveal new scattering aspects of small spherical particles.\nIn this paper, an asymptotic analysis based on the Mie theory is employed to study the limiting behavior of the quasistatic optimal resonance \\cite{Nordebo+etal2017a},\nas the electrical size of the sphere as well as the external losses tend to zero. The limitation of the quasistatic theory is then finally assessed by providing explicit\nasymptotic formulas for the validity of the quasistatic model of the optimal resonance.  \nWe explicitly find the validity region of the quasistatic model, which is determined by the scattering loss factor.\n\nThe rest of the paper is organized as follows. In Section \\ref{sect:paradox} is given a description of\nthe quasistatic optimal plasmonic resonance of an ellipsoid embedded in a lossy background medium. \nIn Section \\ref{sect:Miesolution} we develop a detailed full electrodynamic analysis with explicit asymptotic results\nconcerning the special case with a sphere.\nNumerical examples are given in Section \\ref{sect:numexamples} and the vector spherical \nwaves are defined in Appendix \\ref{sect:spherical}.\n\n\\section{The quasistatic optimal plasmonic resonance of an ellipsoid in a lossy background medium}\\label{sect:paradox}\n\\subsection{Notation and conventions}\nThe following notations and conventions are used in this paper.\nClassical electrodynamics is considered where the electric and magnetic field intensities $\\bm{E}$ and $\\bm{H}$\nare given in SI-units \\cite{Jackson1999}.  \nThe time convention for time harmonic fields (phasors) is given by $\\eu^{-\\iu\\omega t}$\nwhere $\\omega$ is the angular frequency and $t$ the time. \nConsequently,  the relative permittivity $\\epsilon$ of a passive isotropic dielectric material has positive imaginary part.\nLet $\\mu_0$, $\\epsilon_0$, $\\eta_0$ and $\\mrm{c}_0$ denote the permeability, the permittivity, the wave impedance and\nthe speed of light in vacuum, respectively, and where $\\eta_0=\\sqrt{\\mu_0\/\\epsilon_0}$ and $\\mrm{c}_0=1\/\\sqrt{\\mu_0\\epsilon_0}$.\nThe wavenumber of vacuum is given by $k_0=\\omega\\sqrt{\\mu_0\\epsilon_0}$.\nThe wavenumber of a homogeneous and isotropic medium with relative permeability $\\mu$ and permittivity $\\epsilon$ is given by $k=k_0\\sqrt{\\mu\\epsilon}$\nand the wavelength $\\lambda$ is defined by $k\\lambda=2\\pi$.\nThe wave impedance of the same medium is given by $\\eta_0\\eta$ where $\\eta=\\sqrt{\\mu\/\\epsilon}$ is the relative wave impedance.\nIn the following, we will consider only non-magnetic, homogeneous and isotropic dielectric or conducting materials, and hence $\\mu=1$ from now on. \nThe spherical coordinates are denoted by $(r,\\theta,\\phi)$, the corresponding unit vectors $(\\hat{\\bm{r}},\\hat{\\bm{\\theta}},\\hat{\\bm{\\phi}})$,\nand the radius vector $\\bm{r}=r\\hat{\\bm{r}}$.\nFinally, the real and the imaginary parts, and the complex conjugate of a complex number $\\zeta$ are denoted \n$\\Re\\left\\{\\zeta\\right\\}$, $\\Im\\left\\{\\zeta\\right\\}$ and $\\zeta^*$, respectively.\n\n\n\\subsection{Optimization under the quasistatic approximation}\nThe maximal absorption of a small dielectric ellipsoid under the quasistatic approximation can readily be calculated as follows, see also \\cite{Nordebo+etal2017a,Dalarsson+etal2017a}.\nConsider a small, homogeneous and isotropic dielectric ellipsoid with relative permittivity $\\epsilon$ which is embedded in \na lossy dielectric background medium with relative permittivity $\\epsilon_\\mrm{b}$. In the quasistatic approximation, \nthe polarizability of the ellipsoid with a uniform excitation $\\bm{E}_\\mrm{i}=E_0\\hat{\\bm{e}}$ along one of its axes is given by the expression\n\\begin{equation}\\label{eq:alphaellipsoiddef}\n\\alpha=V\\frac{\\epsilon-\\epsilon_\\mrm{b}}{\\epsilon_\\mrm{b}+L(\\epsilon-\\epsilon_\\mrm{b})},\n\\end{equation}\nwhere $0<L<1$ is the corresponding depolarizing factor and $V$ the volume of the ellipsoid, \\textit{cf.}\\\/,  \\textit{e.g.}\\\/,  \\cite{Bohren+Huffman1983}.\nNote that $L<1\/3$, $L=1\/3$ and $L>1\/3$ for a prolate spheroid, the sphere and an oblate spheroid, respectively.\nThe dipole moment of the small ellipsoid is given by $\\bm{p}=\\epsilon_0\\epsilon_\\mrm{b}\\alpha\\bm{E}_\\mrm{i}=\\int_\\Omega\\epsilon_0(\\epsilon-\\epsilon_\\mrm{b})\\bm{E}\\mrm{d}v$\nwhere $\\Omega$ denotes the ellipsoidal domain, and since the resulting internal field $\\bm{E}$ of the ellipsoid is a constant vector parallel to \n$\\bm{E}_\\mrm{i}$ \\cite{Bohren+Huffman1983}, it follows readily that\n\\begin{equation}\n\\bm{E}=\\frac{\\epsilon_\\mrm{b}\\alpha}{V(\\epsilon-\\epsilon_\\mrm{b})}\\bm{E}_\\mrm{i}=\\frac{\\epsilon_\\mrm{b}}{L\\left(\\epsilon+\\epsilon_\\mrm{b}\\frac{(1-L)}{L} \\right)}\\bm{E}_\\mrm{i}.\n\\end{equation}\nThe power absorbed in the ellipsoid can now be calculated from Poynting's theorem as\n\\begin{multline}\\label{eq:Pabsellipsoid1}\nP_\\mrm{abs}=\\frac{\\omega\\epsilon_0}{2}\\Im\\{\\epsilon\\}\\int_\\Omega\\left|\\bm{E} \\right|^2\\mrm{d}v= \\\\\n\\frac{\\omega\\epsilon_0}{2}\\frac{\\left|\\epsilon_\\mrm{b}\\right|^2}{L^2}\\frac{\\Im\\{\\epsilon\\}}{\\left|\\epsilon+\\epsilon_\\mrm{b}\\frac{(1-L)}{L} \\right|^2}\\left| E_0\\right|^2V,\n\\end{multline}\nor \n\\begin{equation}\\label{eq:Pabsellipsoid2}\nP_\\mrm{abs}=\n\\frac{\\omega\\epsilon_0}{2}\\frac{\\left|\\epsilon_\\mrm{b}\\right|^2\\Im\\{\\epsilon\\}}{\\left|\\epsilon_\\mrm{b}+L(\\epsilon-\\epsilon_\\mrm{b})\\right|^2}\\left| E_0\\right|^2V.\n\\end{equation}\nAs \\textit{e.g.}\\\/, , for the sphere ($L=1\/3$), the absorption cross section $C_\\mrm{abs}$ is obtained by normalizing with the power intensity \nof the incoming plane wave, yielding\n\\begin{equation}\\label{eq:Cabssphere1}\nC_\\mrm{abs}=\\frac{P_\\mrm{abs}}{I_\\mrm{i}}\n=12\\pi k_0 a^3\\frac{\\left|\\epsilon_\\mrm{b}\\right|^2}{\\Re\\{\\sqrt{\\epsilon_\\mrm{b}}\\}}\\frac{\\Im\\{\\epsilon\\}}{\\left|\\epsilon+2\\epsilon_\\mrm{b} \\right|^2},\n\\end{equation}\nwhere $I_\\mrm{i}=\\frac{1}{2}\\Re\\{E_0H_0^*\\}=\\left| E_0\\right|^2\\Re\\{\\sqrt{\\epsilon_\\mrm{b}}\\}\/2\\eta_0$ and where $H_0=E_0\/\\eta$.\n\nWhen the background medium as well as the shape of the ellipsoid are fixed, the expression \\eqref{eq:Pabsellipsoid1} can be maximized as follows.\nLet $\\epsilon_\\mrm{opt}$ denote a fixed complex-valued constant with $\\Im\\{\\epsilon_\\mrm{opt}\\}>0$ and consider the real-valued function \n\\begin{equation}\\label{eq:fofepsilondef}\nf(\\epsilon)=\\frac{\\Im\\{\\epsilon\\}}{\\left|\\epsilon-\\epsilon_\\mrm{opt}^* \\right|^2},\n\\end{equation}\nwhere $\\epsilon$ is a complex-valued variable with $\\Im\\{\\epsilon\\}>0$.\nIt can be shown that the function $f(\\epsilon)$ has a local maximum at $\\epsilon_\\mrm{opt}$ \\textit{cf.}\\\/,  \\cite[Sect.~2.5, Eqs.~(15) through (17)]{Nordebo+etal2017a}.\nHence, for the ellipsoid the maximizer of $P_\\mrm{abs}$ is given by\n\\begin{equation}\\label{eq:epsilonodef}\n\\epsilon_\\mrm{opt}=-\\epsilon_\\mrm{b}^*\\frac{(1-L)}{L},\n\\end{equation}\nand in particular for the sphere $\\epsilon_\\mrm{opt}=-2\\epsilon_\\mrm{b}^*$, \n\\textit{cf.}\\\/,  \\cite{Nordebo+etal2017a,Dalarsson+etal2017a}.\nThe corresponding maximal absorption is given by\n\\begin{equation}\\label{eq:Pabsellipsoid1o}\nP_\\mrm{abs}^\\mrm{qs,opt}(\\epsilon_\\mrm{b},L)=\\frac{\\omega\\epsilon_0}{2}\\frac{1}{4L(1-L)}\\frac{\\left|\\epsilon_\\mrm{b}\\right|^2}{\\Im\\{\\epsilon_\\mrm{b}\\}}\\left| E_0\\right|^2V,\n\\end{equation}\nand for the sphere we obtain the optimal absorption cross section\n\\begin{equation}\\label{eq:Cabssphere1o}\nC_\\mrm{abs}^\\mrm{qs,opt}\n=\\frac{3\\pi}{2}k_0 a^3\\frac{\\left|\\epsilon_\\mrm{b}\\right|^2}{\\Re\\{\\sqrt{\\epsilon_\\mrm{b}}\\}}\\frac{1}{\\Im\\{\\epsilon_\\mrm{b}\\}}.\n\\end{equation}\nThe solution \\eqref{eq:epsilonodef} is referred to as an optimal conjugate match and can be\ninterpreted in terms of an optimal plasmonic resonance for the ellipsoid \\cite{Nordebo+etal2017a,Dalarsson+etal2017a}.\nNote that \\eqref{eq:Pabsellipsoid1o} is unbounded in the cases when the exterior domain becomes lossless and $\\Im\\{\\epsilon_\\mrm{b}\\}\\rightarrow 0$,\nas well as when the ellipsoid collapses and $L\\rightarrow 0$ or $L\\rightarrow 1$.\n\nWhen both media parameters $\\epsilon$ and $\\epsilon_\\mrm{b}$ are fixed, the expression \\eqref{eq:Pabsellipsoid2} can be maximized\nwith respect to the shape parameter $L$. By straightforward differentiation it is found that the maximizing shape parameter $L_\\mrm{opt}$ is given by\n\\begin{equation}\\label{eq:Lodef}\nL_\\mrm{opt}=\\Re\\left\\{\\frac{\\epsilon_\\mrm{b}}{\\epsilon_\\mrm{b}-\\epsilon} \\right\\},\n\\end{equation}\nand the corresponding maximal absorption is given by\n\\begin{equation}\\label{eq:Pabsellipsoid2o}\nP_\\mrm{abs}^\\mrm{qs,opt}(\\epsilon_\\mrm{b},\\epsilon)\n=\\frac{\\omega\\epsilon_0}{2}\\frac{\\left|\\epsilon_\\mrm{b}\\right|^2\\Im\\{\\epsilon\\}\\left|\\epsilon-\\epsilon_\\mrm{b}\\right|^2}{\\left(\\Im\\{\\epsilon_\\mrm{b}^*\\epsilon\\} \\right)^2}\n\\left| E_0\\right|^2V.\n\\end{equation}\nIn the limiting case when the exterior region becomes lossless and $\\Im\\{\\epsilon_\\mrm{b}\\}\\rightarrow 0$, the expression \\eqref{eq:Pabsellipsoid2o} simplifies to\n\\begin{equation}\\label{eq:Pabsellipsoid3o}\nP_\\mrm{abs}^\\mrm{qs,opt}(\\epsilon_\\mrm{b},\\epsilon)\n=\\frac{\\omega\\epsilon_0}{2}\\frac{\\left|\\epsilon-\\epsilon_\\mrm{b}\\right|^2}{\\Im\\{\\epsilon\\}}\n\\left| E_0\\right|^2V,\n\\end{equation}\nand which agrees with the upper bound given in \\cite[Eq.~(32b) on p.~3345 and Eq.~(41) on p.~3349]{Miller+etal2016} under the same quasistatic assumption (incident field is uniform).\n\nWhat is important to note at this point is the unboundedness of the quasistatic maximal absorption \\eqref{eq:Pabsellipsoid1o} as the external loss factor\n$\\Im\\{\\epsilon_\\mrm{b}\\}$ tends to zero. Obviously, this is in contradiction to the well known bound on the \nabsorption of an arbitrary electric dipole scatterer in a lossless background medium given by\n\\begin{equation}\\label{eq:Cabsdip}\nC_\\mrm{abs}^\\mrm{dip}=\\frac{3\\pi}{2}\\frac{1}{k_\\mrm{b}^2},\n\\end{equation}\nwhere $k_\\mrm{b}=k_0\\sqrt{\\epsilon_\\mrm{b}}$, \\textit{cf.}\\\/,  \\cite[Eq.~(16) on p.~937]{Tretyakov2014}. As we will see later, this apparent contradiction is due simply\nto the fact that the quasistatic approximation does not take the scattering loss into account. In particular, in the next section we will see\nthat the fully dynamical model for the normalized absorption cross section area of a small dielectric spherical dipole has a very subtle limiting behavior, \nand is in fact unbounded when $\\epsilon=-2\\epsilon_\\mrm{b}^*$ and both the electrical size as well as the exterior losses tend to zero at the same time. \nTo this end, it is also interesting to observe that the limit of \\eqref{eq:Pabsellipsoid2o} as $(\\Im\\{\\epsilon\\},\\Im\\{\\epsilon_\\mrm{b}\\})\\rightarrow (0,0)$ will depend on how\nthis limit is taken. In particular, \\eqref{eq:Pabsellipsoid3o} is obtained for fixed $\\epsilon$ ($\\Im\\{\\epsilon\\}>0$) as $\\Im\\{\\epsilon_\\mrm{b}\\}\\rightarrow 0$ and \\eqref{eq:Pabsellipsoid3o} is\nthen unbounded as $\\Im\\{\\epsilon\\}$ approaches zero. On the other hand, \\eqref{eq:Pabsellipsoid2o} approaches zero for fixed  $\\Im\\{\\epsilon_\\mrm{b}\\}>0$ as  $\\Im\\{\\epsilon\\}\\rightarrow 0$ \n(and $\\Re\\{\\epsilon\\}\\neq 0$).\n\nThe conclusion of this discussion is that one can not optimize (with respect to $\\epsilon$) the absorption of an ellipsoid under the quasistatic assumption\nwhen the exterior domain is lossless. The natural question that arises is then under which circumstances this optimization model is valid for a lossy exterior domain.\nThis is the topic of the next section, where we restrict the analysis to the absorption of a dielectric sphere in a lossy medium.\n\n\n\\section{The absorption of a small dielectric sphere in a lossy background medium}\\label{sect:Miesolution}\n\\subsection{Electrodynamic solution}\nThe complete electrodynamic solution for the internal absorption of a small dielectric sphere in a lossy background medium is analyzed below.\nThe definition of the spherical vector waves, the spherical Bessel and Hankel functions and the related Lommel integrals are given in Appendix \\ref{sect:spherical}, see also  \\cite{Nordebo+etal2017a}.\n\nConsider the scattering of the electromagnetic field due to a homogeneous dielectric sphere of radius $a$, complex-valued permittivity $\\epsilon$, \nand wavenumber $k=k_0\\sqrt{\\epsilon}$.\nThe medium surrounding the sphere is characterized by the permittivity $\\epsilon_\\mrm{b}$ and the wavenumber $k_\\mrm{b}=k_0\\sqrt{\\epsilon_\\mrm{b}}$.\nThe incident and the scattered fields for $r>a$ are expressed as in \\eqref{eq:Esphdef} with multipole coefficients $a_{\\tau ml}$ and $b_{\\tau ml}$, respectively,\nand the interior field is similarly expressed using regular spherical vector waves for $r<a$ with multipole coefficients $a_{\\tau ml}^\\mrm{int}$.\nBy matching the tangential electric and magnetic fields at the boundary of radius $a$, it can be shown that\n\\begin{eqnarray}\\label{eq:btaumldef}\nb_{\\tau ml}=t_{\\tau l}a_{\\tau ml}, \\\\\na_{\\tau ml}^\\mrm{int}=r_{\\tau l}a_{\\tau ml},\\label{eq:ataumldef}\n\\end{eqnarray}\nfor $\\tau=1,2$, $l=1,2,\\ldots$, and $m=-l,\\ldots,l$, and where $t_{\\tau l}$ and $r_{\\tau l}$ are transition matrices for scattering and absorption, \nrespectively, see \\textit{e.g.}\\\/,  \\cite[Eqs.~(4.52) and (4.53) on p.~100]{Bohren+Huffman1983}.\nIn particular, for the internal fields of the sphere the corresponding electric multipole coefficients are given by\n\\begin{multline} \nr_{2l} = \\\\\n \\displaystyle  \\frac{ (\\mrm{j}_l(k_\\mrm{b}a) (k_\\mrm{b}a\\mrm{h}_l^{(1)}(k_\\mrm{b}a))^\\prime-\\mrm{h}_l^{(1)}(k_\\mrm{b}a) (k_\\mrm{b}a\\mrm{j}_l(k_\\mrm{b}a))^\\prime)\\sqrt{\\epsilon_\\mrm{b}\\epsilon} }\n   { -\\mrm{h}_l^{(1)}(k_\\mrm{b}a) (ka\\mrm{j}_l(ka))^\\prime\\epsilon_\\mrm{b} + \\mrm{j}_l(ka) (k_\\mrm{b}a\\mrm{h}_l^{(1)}(k_\\mrm{b}a))^\\prime\\epsilon}.  \\label{eq:r2l}\n\\end{multline}\n\nIt can be shown that the multipole expansion coefficients for a plane\nwave $\\bm{E}_\\mrm{i}(\\bm r)=\\bm{E}_0\\eu^{i k_\\mrm{b}\\hat{\\bm{k}}\\cdot\\bm{r}}$ are given by\n\\begin{equation}\na_{\\tau ml}=4\\pi\\iu^{l-\\tau+1}\\bm{E}_0\\cdot\\bm{A}_{\\tau ml}^*(\\hat{\\bm{k}}),\n\\end{equation}\nfor $\\tau=1,2$, $l=1,2,\\ldots$, and $m=-l,\\ldots,l$, and where the vector spherical harmonics $\\bm{A}_{\\tau ml}(\\hat{\\bm{k}})$\nare defined as in Appendix \\ref{sect:sphericaldef}, \\textit{cf.}\\\/,  \\textit{e.g.}\\\/,  \\cite[Eq.~(7.27) on p.~374]{Kristensson2016}. Without loss of generality, \nfor a homogeneous and isotropic sphere we may assume that $\\hat{\\bm{k}}=\\hat{\\bm{z}}$, and hence from \\cite[Eq.~(7.29) on p.~375]{Kristensson2016} it follows that \n\\begin{equation}\\label{eq:sumoverm}\n\\sum_{m=-l}^l \\left| a_{\\tau ml} \\right|^2=2\\pi (2l+1)\\left| \\bm{E}_0\\right|^2.\n\\end{equation}\n\nBased on Poynting's theorem and the orthogonality of the spherical vector waves\nthe absorbed power can now be calculated as\n\\begin{multline}\\label{eq:Pabssphexpdef}\nP_\\mrm{abs}=\\frac{1}{2}\\omega\\epsilon_0\\Im\\{\\epsilon\\}\\int_{V_{a}} \\left| \\bm{E} \\right|^2\\mrm{d}v  \\\\\n =\\pi\\left| \\bm{E}_0\\right|^2\\omega\\epsilon_0\\Im\\{\\epsilon\\}\\sum_{l=1}^{\\infty}\\sum_{\\tau=1}^2  (2l+1) W_{\\tau l}(k,a)\\left| r_{\\tau l} \\right|^2, \n\\end{multline}\nwhere $V_{a}$ denotes the spherical volume of radius $a$ and where \n$W_{\\tau l}(k,a)=\\int_{V_{a}}\\left|{\\bf v}_{\\tau ml}(k\\bm{r})\\right|^2\\mrm{d} v$ is the volume integral of the regular spherical vector waves, \n\\textit{cf.}\\\/,  \\eqref{eq:vorthogonal2} and \\eqref{eq:Wtauldef}. \nIn \\eqref{eq:Pabssphexpdef}, the relations \\eqref{eq:ataumldef} and \\eqref{eq:sumoverm} have also been employed.\n\nThe absorption cross section ($C_\\mrm{abs}=P_\\mrm{abs}\/I_\\mrm{i}$)\nbased solely on the dominating Transverse Magnetic (TM) dipole fields ($\\tau=2$ and $l=1$) can now be expressed as\n\\begin{equation}\\label{eq:CabssphTMdip}\nC_\\mrm{abs}^\\mrm{dyn}=6\\pi k_0 \\frac{\\Im\\{\\epsilon\\}}{\\Re\\{\\sqrt{\\epsilon_\\mrm{b}}\\}}W_{21}\\left| r_{21} \\right|^2,\n\\end{equation}\nwhere \n\\begin{multline}\nr_{21} =   \\\\\n \\displaystyle  \\frac{ (\\mrm{j}_1(k_\\mrm{b}a) (k_\\mrm{b}a\\mrm{h}_1^{(1)}(k_\\mrm{b}a))^\\prime-\\mrm{h}_1^{(1)}(k_\\mrm{b}a) (k_\\mrm{b}a\\mrm{j}_1(k_\\mrm{b}a))^\\prime)\\sqrt{\\epsilon_\\mrm{b}\\epsilon} }\n   { -\\mrm{h}_1^{(1)}(k_\\mrm{b}a) (ka\\mrm{j}_1(ka))^\\prime\\epsilon_\\mrm{b} + \\mrm{j}_1(ka) (k_\\mrm{b}a\\mrm{h}_1^{(1)}(k_\\mrm{b}a))^\\prime\\epsilon}, \\label{eq:r21}\n \\end{multline}\n is given by \\eqref{eq:r2l}, and\n\\begin{equation}\\label{eq:W21expr}\nW_{21}=\\frac{a^2}{3}\\frac{\\Im\\left\\{k\\left( 2\\mrm{j}_1(ka)\\mrm{j}_0^*(ka)+ \\mrm{j}_3(ka)\\mrm{j}_2^*(ka) \\right)\\right\\}}{\\Im\\{k^2\\}},\n\\end{equation}\nis obtained from \\eqref{eq:W1ldef} and \\eqref{eq:W2ldef}.\n\n\n\\subsection{Asymptotic analysis}\\label{sect:asymptoticanalys}\nTo analyze the asymptotic properties of the dynamic solution $C_\\mrm{abs}^\\mrm{dyn}$, the expression \\eqref{eq:CabssphTMdip}\nis first rewritten in the following area normalized form\n\\begin{equation}\\label{eq:CabssphTMdipnorm}\nQ_\\mrm{abs}^\\mrm{dyn}=\\frac{C_\\mrm{abs}^\\mrm{dyn}}{\\pi a^2}=6 k_0a \\frac{\\Im\\{\\epsilon\\}}{\\Re\\{\\sqrt{\\epsilon_\\mrm{b}}\\}}\\frac{W_{21}}{a^3}\\left| r_{21} \\right|^2.\n\\end{equation}\nAn asymptotic analysis can now be carried out based on the asymptotic expansions of the spherical Bessel and Hankel functions for small arguments, see \\textit{e.g.}\\\/,  \\cite{Olver+etal2010}.\nFurthermore, with $\\epsilon=\\epsilon^\\prime+\\iu\\epsilon^{\\prime\\prime}$ and \n$\\epsilon_\\mrm{b}=\\epsilon_\\mrm{b}^\\prime+\\iu\\epsilon_\\mrm{b}^{\\prime\\prime}$ and small loss factors $\\epsilon^{\\prime\\prime}$\nand $\\epsilon_\\mrm{b}^{\\prime\\prime}$, it is readily seen that\n\\begin{equation}\n\\left\\{\\begin{array}{l}\nka=k_0a\\sqrt{\\epsilon}=k_0a\\left(\\sqrt{\\epsilon^{\\prime}}+{\\cal O}\\{\\epsilon^{\\prime\\prime}\\} \\right), \\vspace{0.2cm} \\\\\nk_\\mrm{b}a=k_0a\\sqrt{\\epsilon_\\mrm{b}}=k_0a\\left(\\sqrt{\\epsilon_\\mrm{b}^{\\prime}}+{\\cal O}\\{\\epsilon_\\mrm{b}^{\\prime\\prime}\\} \\right),\n\\end{array}\\right.\n\\end{equation}\nwhere ${\\cal O}\\{\\cdot\\}$ denotes the big ordo \\cite[p.~4]{Olver1997},\nand where it is assumed that $\\epsilon^\\prime\\neq 0$ and $\\epsilon_\\mrm{b}^\\prime\\neq 0$ are fixed. \nFrom this observation, it is also noted that \nfor integer $m$\n\\begin{equation}\n\\left\\{\\begin{array}{l}\n{\\cal O}\\{(ka)^m\\}={\\cal O}\\{(k_0a)^m\\}, \\vspace{0.2cm} \\\\\n {\\cal O}\\{(k_\\mrm{b}a)^m\\}={\\cal O}\\{(k_0a)^m\\},\n\\end{array}\\right.\n\\end{equation}\nindependent of the (small) values of $\\epsilon^{\\prime\\prime}$ and $\\epsilon_\\mrm{b}^{\\prime\\prime}$.\n\nBased on the asymptotic tools mentioned above, the different factors of \\eqref{eq:CabssphTMdipnorm} can now be analyzed in detail.\nWe start by studying the normalized factor\n\\begin{equation}\\label{eq:W21normexpr}\n\\frac{W_{21}}{a^3}=\\frac{1}{3k_0a}\\frac{\\Im\\left\\{ F(k_0a,\\epsilon)\\right\\}}{\\Im\\{\\epsilon\\}},\n\\end{equation}\nwhere\n\\begin{multline}\nF(k_0a,\\epsilon) \n=\\sqrt{\\epsilon}\\left( 2\\mrm{j}_1(k_0a\\sqrt{\\epsilon})\\mrm{j}_0^*(k_0a\\sqrt{\\epsilon}) \\right.  \\\\\n + \\left. \\mrm{j}_3(k_0a\\sqrt{\\epsilon})\\mrm{j}_2^*(k_0a\\sqrt{\\epsilon}) \\right)\n\\end{multline}\nis given by \\eqref{eq:W21expr}.\nNote that the spherical Bessel functions $\\mrm{j}_n(x)$ of order $n$ are analytic functions with a Taylor series expansion starting at the polynomial order $n$.\nThe functions $\\mrm{j}_n(x)$ are furthermore odd for odd orders and even for even orders. Hence, it is readily seen that the function\n$F(k_0a,\\epsilon)$ has a convergent power series expansion in terms of $\\epsilon$ and $\\epsilon^*$ with real valued coefficients, and that\n\\begin{eqnarray}\nF(k_0a,\\epsilon) =k_0a\\frac{2}{3}\\epsilon+{\\cal O}\\{\\left(k_0a\\right)^3\\}{\\cal O}\\{\\epsilon\\},\n\\end{eqnarray}\nand \n\\begin{eqnarray}\n\\Im\\{F(k_0a,\\epsilon)\\} =k_0a\\frac{2}{3}\\epsilon^{\\prime\\prime}+{\\cal O}\\{\\left(k_0a\\right)^3\\}{\\cal O}\\{\\epsilon^{\\prime\\prime}\\}.\n\\end{eqnarray}\nThis means that the expression in \\eqref{eq:W21normexpr} is well behaved when $k_0a$ as well as $\\epsilon^{\\prime\\prime}$ approaches zero,\nand that\n\\begin{equation}\\label{eq:W21normas}\n\\frac{W_{21}}{a^3}=\\frac{2}{9}+{\\cal O}\\{(k_0a)^2\\},\n\\end{equation}\nindependent of $\\epsilon^{\\prime\\prime}$. This observation can also be derived directly from the definition \\eqref{eq:W2ldef} and the regularity\nof the spherical Bessel function $\\mrm{j}_1(x)$.\n\nA detailed asymptotic study of \\eqref{eq:r21} shows that\n\\begin{equation}\\label{eq:r21as}\nr_{21}=\\frac{3\\epsilon_\\mrm{b}+{\\cal O}\\{(k_0a)^2\\}}{\\epsilon+2\\epsilon_\\mrm{b}+(k_0a)^2C+(k_0a)^3D+ {\\cal O}\\{(k_0a)^4\\}},\n\\end{equation}\nwhere \n\\begin{equation}\\label{eq:CDdef}\n\\left\\{\\begin{array}{l}\nC=(\\epsilon_\\mrm{b}-\\epsilon)(\\epsilon_\\mrm{b}+\\epsilon\/10), \\vspace{0.2cm} \\\\\nD=\\iu\\frac{2}{3}\\epsilon_\\mrm{b}\\sqrt{\\epsilon_\\mrm{b}}(\\epsilon_\\mrm{b}-\\epsilon). \n\\end{array}\\right.\n\\end{equation}\nIt can safely be assumed that $C\\neq 0$ and $D\\neq 0$ since\n$\\epsilon_\\mrm{b}=\\epsilon$ is not an interesting case and $\\epsilon_\\mrm{b}+\\epsilon\/10\\neq 0$ for lossy and passive media.\nFor fixed $\\epsilon$ and $\\epsilon_\\mrm{b}$ with $\\epsilon+2\\epsilon_\\mrm{b}\\neq 0$, and by using \\eqref{eq:W21normas} and \\eqref{eq:r21as}, \nit is now concluded that \\eqref{eq:CabssphTMdip} is given by\n\\begin{equation}\\label{eq:CabssphTMdipas}\nC_\\mrm{abs}^\\mrm{dyn}=12\\pi k_0 a^3\\frac{\\left|\\epsilon_\\mrm{b}\\right|^2}{\\Re\\{\\sqrt{\\epsilon_\\mrm{b}}\\}}\n\\frac{\\Im\\{\\epsilon\\}}{\\left|\\epsilon+2\\epsilon_\\mrm{b} \\right|^2}+{\\cal O}\\{(k_0a)^2\\},\n\\end{equation}\nwhich is in agreement with the quasistatic approximation given in \\eqref{eq:Cabssphere1}. \n\nHowever, when $\\epsilon^{\\prime}$ and $\\epsilon_\\mrm{b}^{\\prime}$ are fixed at $\\epsilon^{\\prime}=-2\\epsilon_\\mrm{b}^{\\prime}$,\nand at the same time $k_0a$ as well as $\\epsilon^{\\prime\\prime}$ and $\\epsilon_\\mrm{b}^{\\prime\\prime}$ can be chosen freely, \nthe normalized  absorption cross section area \\eqref{eq:CabssphTMdipnorm} turns out to be unbounded. To study this behavior in detail, it is noted that to the lowest order in\n$(k_0a,\\epsilon^{\\prime\\prime},\\epsilon_\\mrm{b}^{\\prime\\prime})$, the expression \\eqref{eq:CabssphTMdipnorm} can be approximated by\n\\begin{equation}\\label{eq:CabssphTMdipnormapprox1}\nQ_\\mrm{abs}^\\mrm{dyn}\\sim 12 k_0a \\frac{\\left|\\epsilon_\\mrm{b}\\right|^2}{\\Re\\{\\sqrt{\\epsilon_\\mrm{b}}\\}}\n\\frac{\\Im\\{\\epsilon\\}}{\\left|\\epsilon+2\\epsilon_\\mrm{b}+(k_0a)^2C_0\\right|^2},\n\\end{equation}\nwhere $C_0=(\\epsilon_\\mrm{b}^\\prime-\\epsilon^\\prime)(\\epsilon_\\mrm{b}^\\prime+\\epsilon^\\prime\/10)$ and the symbol $\\sim$ indicates\nan asymptotic approximation in the sense of \\cite[p.~4]{Olver1997}.\nObviously, for any small but finite value of $k_0a\\neq 0$ the real part of the expression under the absolute-value sign can be brought to zero, \nand we see that in this approximation the absorption cross section is unbounded at $\\epsilon^{\\prime\\prime}$ and \n$\\epsilon_b^{\\prime\\prime}$ tending to zero. We again confirm that the second-order correction is not enough to reveal the dynamic absorption bound.\nInterestingly, the absorption cross section can be unbounded even without tuning to the dynamically-corrected resonant value of the permittivity. \nIn particular, by considering the special case of interest, \\textit{i.e.}\\\/,  the quasistatic optimal conjugate match where \n$\\epsilon=-2\\epsilon_\\mrm{b}^*=-2\\epsilon_\\mrm{b}^\\prime+\\iu 2\\epsilon_\\mrm{b}^{\\prime\\prime}$\nand $\\epsilon+2\\epsilon_\\mrm{b}=\\iu 4 \\epsilon_\\mrm{b}^{\\prime\\prime}$, it follows that\n\\begin{equation}\\label{eq:CabssphTMdipnormapprox2}\nQ_\\mrm{abs}^\\mrm{dyn}\\sim \\frac{3}{2} \\frac{\\left|\\epsilon_\\mrm{b}\\right|^2}{\\Re\\{\\sqrt{\\epsilon_\\mrm{b}}\\}}\n\\frac{k_0a\\epsilon_\\mrm{b}^{\\prime\\prime}}{{\\epsilon_\\mrm{b}^{\\prime\\prime}}^2+(k_0a)^4C_0^2\/16},\n\\end{equation}\nwhere $C_0=12{\\epsilon_\\mrm{b}^\\prime}^2\/5$ and where it is assumed that $\\epsilon_\\mrm{b}^\\prime\\neq 0$. Note that the assumption\n$\\epsilon_\\mrm{b}^{\\prime\\prime} \\gg (k_0a)^2C_0\/4$ will restore the expression for the maximal absorption given by \\eqref{eq:Cabssphere1o}.\nHence, it is observed that the quasistatic theory \\eqref{eq:Cabssphere1o} is valid provided that $k_0a$ is sufficiently small\nat the same time as the background loss factor $\\epsilon_\\mrm{b}^{\\prime\\prime}$ is sufficiently large.\n\nThe factor governing the convergence of \\eqref{eq:CabssphTMdipnormapprox2} and \\eqref{eq:CabssphTMdipnorm} is given by\n\\begin{equation}\\label{eq:factorgovconv1}\nF_1=\\frac{k_0a\\epsilon_\\mrm{b}^{\\prime\\prime}}{{\\epsilon_\\mrm{b}^{\\prime\\prime}}^2+(k_0a)^4C_0^2\/16}.\n\\end{equation}\nBy choosing \n\\begin{equation}\\label{eq:k0aasymptotics}\nk_0a=A{\\epsilon_\\mrm{b}^{\\prime\\prime}}^\\alpha,\n\\end{equation}\nwhere $A$ is an arbitrary constant and $\\alpha>0$, this factor becomes\n\\begin{equation}\\label{eq:factorgovconv2}\nF_2=\\frac{A{\\epsilon_\\mrm{b}^{\\prime\\prime}}^{\\alpha+1}}{{\\epsilon_\\mrm{b}^{\\prime\\prime}}^2+A^4{\\epsilon_\\mrm{b}^{\\prime\\prime}}^{4\\alpha}C_0^2\/16}.\n\\end{equation}\nA detailed study of the expression \\eqref{eq:factorgovconv2} for small $\\epsilon_\\mrm{b}^{\\prime\\prime}>0$ \nreveals the condition for convergence of \\eqref{eq:CabssphTMdipnorm}, which can be summarized as\n\\begin{equation}\\label{eq:convdiv}\n\\left\\{\\begin{array}{ll}\n\\mrm{Convergence} & 0<\\alpha<\\frac{1}{3}, \\vspace{0.2cm} \\\\\n\\mrm{Divergence} & \\frac{1}{3}\\leq \\alpha\\leq 1, \\vspace{0.2cm} \\\\\n\\mrm{Convergence} & \\alpha>1, \n\\end{array}\\right.\n\\end{equation}\nwhere $k_0a=A{ \\epsilon_\\mrm{b}^{\\prime\\prime}}^\\alpha$, $\\epsilon_\\mrm{b}^\\prime$ is fixed and $\\epsilon=-2\\epsilon_\\mrm{b}^*$ (the quasistatic optimal conjugate match).\n\nFor a fixed background loss parameter $\\epsilon_\\mrm{b}^{\\prime\\prime}>0$, the factor \\eqref{eq:factorgovconv1} can furthermore be maximized\nwith respect to the electrical size $k_0a$, yielding\n\\begin{equation}\\label{eq:k0aoptasymptotics}\nk_0a=\\frac{2}{3^{1\/4}C_0^{1\/2}}{\\epsilon_\\mrm{b}^{\\prime\\prime}}^{1\/2}.\n\\end{equation}\nFor fixed $\\epsilon_\\mrm{b}^{\\prime\\prime}$, the relation \\eqref{eq:k0aoptasymptotics} expresses a stationary point\nfor \\eqref{eq:CabssphTMdipnormapprox2} regarded as a function of $k_0a$, and hence an indicator of the domain of validity of \nthe quasistatic approximation \\eqref{eq:Cabssphere1o}.\n\nIn conclusion, it has been shown that the  normalized absorption cross section area\n\\eqref{eq:CabssphTMdipnorm} has a subtle limiting behavior and is in fact unbounded as \n$(k_0a,\\epsilon^{\\prime\\prime},\\epsilon_\\mrm{b}^{\\prime\\prime})\\rightarrow (0,0,0)$ for fixed $\\epsilon^{\\prime}$ and $\\epsilon_\\mrm{b}^{\\prime}$.\nIt is also shown that if $k_0a$ is sufficiently small, the optimal conjugate match $\\epsilon=-2\\epsilon_\\mrm{b}^*$ defined in \\eqref{eq:epsilonodef} yields\nan accurate quasistatic approximation \\eqref{eq:Cabssphere1o} provided that \n\\begin{equation}\\label{eq:k0aoptasymptotics2}\n\\epsilon_\\mrm{b}^{\\prime\\prime}>\\frac{3\\sqrt{3}}{5}\\left(k_0a\\right)^2{\\epsilon_\\mrm{b}^\\prime}^2,\n\\end{equation}\nwhere \\eqref{eq:k0aoptasymptotics} and $C_0=12{\\epsilon_\\mrm{b}^\\prime}^2\/5$ have been used.\nIt should be noted that a direct maximization of \\eqref{eq:CabssphTMdipnorm} with respect to $\\epsilon$ would be possible by numerical optimization techniques,\nbut is not necessary if the quasistatic approximation is valid. The validity of the quasistatic approximation can be assessed by checking\nthe criteria \\eqref{eq:k0aoptasymptotics2} as well as by a direct comparison of \\eqref{eq:Cabssphere1} and \\eqref{eq:Cabssphere1o} with\nthe electrodynamic solution \\eqref{eq:CabssphTMdipnorm}.\n\n\\subsection{Optimal absorption of the small dielectric sphere in a lossy media}\\label{sect:optimalabsorptionepsilonsol}\nFinally, the asymptotic analysis above with \\eqref{eq:r21as} and \\eqref{eq:CDdef} makes it possible to analyze the pole structure of\n\\eqref{eq:CabssphTMdipnorm} in full dynamics. By making the following Ansatz for the pole\n$\\epsilon_\\mrm{p}=a_0+a_1 k_0a +a_2(k_0a)^2+a_3(k_0a)^3$,\nand identifying terms up to the third order in the denominator of \\eqref{eq:r21as},\nit is found that \\eqref{eq:CabssphTMdipnorm} can be approximated for small $k_0a$ as\n\\begin{equation}\\label{eq:Qabsdynpoleapprox}\nQ_\\mrm{abs}^\\mrm{dyn}\\sim 12 k_0a \\frac{\\left| \\epsilon_\\mrm{b} \\right|^2}{\\Re\\{\\sqrt{\\epsilon_\\mrm{b}}\\}}  \\frac{\\Im\\{\\epsilon\\}}{\\left|\\epsilon-\\epsilon_\\mrm{p} \\right|^2},\n\\end{equation}\nand where the pole is given by \n\\begin{equation}\\label{eq:epsilonpolespherethirdorder}\n\\epsilon_\\mrm{p}=-2\\epsilon_\\mrm{b}-\\frac{12}{5}\\epsilon_\\mrm{b}(k_\\mrm{b}a)^2-\\iu 2 \\epsilon_\\mrm{b}(k_\\mrm{b}a)^3+ {\\cal O}\\{(k_\\mrm{b}a)^4\\},\n\\end{equation}\nsee also \\cite{Tretyakov2014}, and \\cite[Eq.~(11) on p.~3]{Tzarouchis+etal2016}.\nThe expression \\eqref{eq:Qabsdynpoleapprox} is of the form \\eqref{eq:fofepsilondef} where $\\Im\\{\\epsilon_\\mrm{p}\\}<0$ for small $\\epsilon_\\mrm{b}^{\\prime\\prime}$, \nand hence it can be concluded that the maximal absorption is approximately (asymptotically) achieved at $\\epsilon_\\mrm{opt}=\\epsilon_\\mrm{p}^*$, yielding\n\\begin{equation}\\label{eq:epsilonospherethirdorder}\n\\epsilon_\\mrm{opt}=-2\\epsilon_\\mrm{b}^*-\\frac{12}{5}\\epsilon_\\mrm{b}^*(k_\\mrm{b}^*a)^2+\\iu 2 \\epsilon_\\mrm{b}^*(k_\\mrm{b}^*a)^3+ {\\cal O}\\{(k_\\mrm{b}^*a)^4\\}.\n\\end{equation}\nNote that the result \\eqref{eq:epsilonospherethirdorder} generalizes the previous quasistatic result $\\epsilon_\\mrm{opt}=-2\\epsilon_\\mrm{b}^*$ given by \\eqref{eq:epsilonodef}.\nThe expression is valid for small $k_0a$ as well as for small loss factors $\\epsilon_\\mrm{b}^{\\prime\\prime}$.\nIn particular, the optimal absorption of the sphere is given by\n\\begin{equation}\\label{eq:Qabsoptasymptotic}\nQ_\\mrm{abs}^\\mrm{dyn,opt}\\sim 3k_0a\\frac{\\left| \\epsilon_\\mrm{b} \\right|^2}{\\Re\\{\\sqrt{\\epsilon_\\mrm{b}}\\}}\\frac{1}{\\Im\\{\\epsilon_\\mrm{opt}\\}},\n\\end{equation}\nand in the limit as the exterior losses vanish, \nwe have that $\\Im\\{\\epsilon_\\mrm{opt}\\}\\rightarrow 2{\\epsilon_\\mrm{b}^\\prime}^2\\sqrt{\\epsilon_\\mrm{b}^\\prime}(k_0a)^3$ and\n\\begin{equation}\\label{eq:Qabsoptasymptotic2}\n\\lim_{\\epsilon_\\mrm{b}^{\\prime\\prime}\\rightarrow 0}Q_\\mrm{abs}^\\mrm{dyn,opt}=\\frac{3}{2}\\frac{1}{(k_\\mrm{b} a)^2},\n\\end{equation}\nwhere $k_\\mrm{b}=k_0\\sqrt{\\epsilon_\\mrm{b}^\\prime}$, and which is in agreement with the upper bound \\eqref{eq:Cabsdip} given in \\cite{Tretyakov2014}.\n\nFinally, it should be noted that there are major discrepancies in the asymptotic analysis \nof the full dynamic solution performed in this section in comparison to the analysis of the quasistatic\napproximation as in \\eqref{eq:CabssphTMdipnormapprox1} and \\eqref{eq:CabssphTMdipnormapprox2} where $\\epsilon^{\\prime}$ and $\\epsilon_\\mrm{b}^{\\prime}$ are fixed \nand $\\epsilon^{\\prime}=-2\\epsilon_\\mrm{b}^{\\prime}$.\nIn essence, the quasistatic approximation \\eqref{eq:epsilonodef} is missing the second-order term giving a shift in the resonance frequency as well as the\nthird-order term taking the scattering loss into account. \nThese factors are negligible only when the external losses are large enough  to make the second-order term redundant as expressed in \\eqref{eq:k0aoptasymptotics2},\nat the same time as the electrical size of the sphere is small enough to make the scattering loss insignificant. \nHence, when the external losses are too small in relation to the electrical size of the sphere, \nthe quasistatic solution $\\epsilon_\\mrm{opt}=-2\\epsilon_\\mrm{b}^*$ is not the correct choice to maximize the absorption and\none should instead use \\eqref{eq:epsilonospherethirdorder}.\n\n\n\\section{Numerical examples}\\label{sect:numexamples}\n\nIn Figures \\ref{fig:matfig3} through \\ref{fig:matfig4b} we show the  normalized absorption cross section area $Q_\\mrm{abs}$\nand its behavior for $(k_0a,\\epsilon_\\mrm{b}^{\\prime\\prime})$ close to $(0,0)$.\nHere, $Q_\\mrm{abs}^\\mrm{dyn}$ denotes the electrodynamic solution given by \\eqref{eq:CabssphTMdipnorm} and $Q_\\mrm{abs}^\\mrm{qs,opt}$ corresponds to the optimal quasistatic solution\ngiven by \\eqref{eq:Cabssphere1o}, both of which are calculated for the background permittivity $\\epsilon_\\mrm{b}=\\epsilon_\\mrm{b}^{\\prime}+\\iu\\epsilon_\\mrm{b}^{\\prime\\prime}$\nand the quasistatic optimal conjugate match $\\epsilon=-2\\epsilon_\\mrm{b}^*$, yielding $\\epsilon+2\\epsilon_\\mrm{b}=\\iu 4\\epsilon_\\mrm{b}^{\\prime\\prime}$. \nIn Figure \\ref{fig:matfig3} we show also the normalized absorption cross section $Q_\\mrm{abs}^\\mrm{dyn,opt}$ corresponding\nto the dynamic optimal solution \\eqref{eq:epsilonospherethirdorder}, as well as the break points for the quasistatic approximation given by \\eqref{eq:k0aoptasymptotics}.\nAs a comparison with the case with a lossless background $(\\epsilon_\\mrm{b}^{\\prime\\prime}=0)$, the optimal dipole absorption cross section \n$Q_\\mrm{abs}^\\mrm{dip}$ corresponding to \\eqref{eq:Cabsdip} is also shown in Figure  \\ref{fig:matfig3}.\nIn Figures \\ref{fig:matfig3} through \\ref{fig:matfig8} the background\nis defined by $\\epsilon_\\mrm{b}^{\\prime}=1$, and in Figure \\ref{fig:matfig4b} the background corresponds to a saline water with $\\epsilon_\\mrm{b}^{\\prime}=80$.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=0.48\\textwidth]{matfig3b.pdf}\n\\end{center}\n\\vspace{-5mm}\n\\caption{The quasistatic optimal normalized absorption cross section $Q_\\mrm{abs}^\\mrm{qs,opt}$ and the corresponding dynamic cross section $Q_\\mrm{abs}^\\mrm{dyn}$ \nas functions of the electrical size $k_0a$. The break points are given by the asymptotic analysis\n\\eqref{eq:k0aoptasymptotics}. For comparison, the optimal dynamic $Q_\\mrm{abs}^\\mrm{dyn,opt}$ is calculated based on \\eqref{eq:epsilonospherethirdorder}.\nHere, $\\epsilon_\\mrm{b}^{\\prime}=1$.}\n\\label{fig:matfig3}\n\\end{figure}\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=0.48\\textwidth]{matfig4.pdf}\n\\end{center}\n\\vspace{-5mm}\n\\caption{The quasistatic optimal normalized absorption cross section $Q_\\mrm{abs}^\\mrm{qs,opt}$ and the corresponding dynamic cross section $Q_\\mrm{abs}^\\mrm{dyn}$  \nas functions of the background loss $\\epsilon_\\mrm{b}^{\\prime\\prime}$. Here, $\\epsilon_\\mrm{b}^{\\prime}=1$.}\n\\label{fig:matfig4}\n\\end{figure}\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=0.48\\textwidth]{matfig8.pdf}\n\\end{center}\n\\vspace{-5mm}\n\\caption{The normalized dynamic absorption cross section $Q_\\mrm{abs}^\\mrm{dyn}$ as a function of the electrical size $k_0a$ \nand the background loss $\\epsilon_\\mrm{b}^{\\prime\\prime}$, and where $\\epsilon_\\mrm{b}^{\\prime}=1$. \nThe blue dashed line shows the stationary (break) points given by the asymptotic analysis\n\\eqref{eq:k0aoptasymptotics} indicating the region of validity \\eqref{eq:k0aoptasymptotics2} of the quasistatic approximation.\nNote also the asymptotes for divergence given by $\\log k_0a=\\log A+\\alpha\\log { \\epsilon_\\mrm{b}^{\\prime\\prime}}$ with $\\alpha=1\/3$ and $\\alpha=1$ as in \\eqref{eq:convdiv}.}\n\\label{fig:matfig8}\n\\end{figure}\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=0.48\\textwidth]{matfig4b.pdf}\n\\end{center}\n\\vspace{-5mm}\n\\caption{The quasistatic optimal normalized absorption cross section $Q_\\mrm{abs}^\\mrm{qs,opt}$ and the corresponding dynamic cross section $Q_\\mrm{abs}^\\mrm{dyn}$  \nas functions of the background loss $\\epsilon_\\mrm{b}^{\\prime\\prime}$. Here, $\\epsilon_\\mrm{b}^{\\prime}=80$, as with a biological tissue in the lower GHz region.}\n\\label{fig:matfig4b}\n\\end{figure}\n\nAs can be seen in the Figures \\ref{fig:matfig3} through \\ref{fig:matfig8}, the normalized absorption cross section $Q_\\mrm{abs}^\\mrm{dyn}$\nhas a very subtle behavior for $(k_0a,\\epsilon_\\mrm{b}^{\\prime\\prime})$ close to the plasmonic singularity at $(0,0)$ where $\\epsilon=-2$.\nNote that $Q_\\mrm{abs}^\\mrm{dyn}\\rightarrow 0$ as $k_0a\\rightarrow 0$\n ($\\epsilon_\\mrm{b}^{\\prime\\prime}>0$ fixed) and $Q_\\mrm{abs}^\\mrm{dyn}\\rightarrow 0$ as $\\epsilon_\\mrm{b}^{\\prime\\prime}\\rightarrow 0$\n ($k_0a>0$ fixed), \\textit{cf.}\\\/,  also the approximate expression in \\eqref{eq:CabssphTMdipnormapprox2}.\n However, as illustrated in Figure \\ref{fig:matfig8},  $Q_\\mrm{abs}^\\mrm{dyn}$ is unbounded in\n any neighborhood where $(k_0a,\\epsilon_\\mrm{b}^{\\prime\\prime})$ is close to $(0,0)$ in accordance with the analysis given in Section \\ref{sect:asymptoticanalys}.\n It should be noted that this behavior is perfectly consistent with the bound for a lossless background $C_\\mrm{abs}\\leq 3\\lambda^2\/8\\pi$ given in \\cite{Tretyakov2014}\n since $C_\\mrm{abs}\/a^2\\leq (3\/8\\pi)\/(a\/\\lambda)^2$ is unbounded as $a\/\\lambda$ approaches zero.\n \nFigures  \\ref{fig:matfig3}  through \\ref{fig:matfig8} illustrate how the approximation $Q_\\mrm{abs}^\\mrm{qs,opt}\\sim Q_\\mrm{abs}^\\mrm{dyn}$\nbecomes valid when $k_0a$ is sufficiently small  \nat the same time as $\\epsilon_\\mrm{b}^{\\prime\\prime}$ is sufficiently large. \nIn Figure \\ref{fig:matfig3} is also illustrated that the break points given by \\eqref{eq:k0aoptasymptotics} (or \\eqref{eq:k0aoptasymptotics2})\ngive a very accurate estimate for the validity of the quasistatic approximation \\eqref{eq:Cabssphere1o}.\n\nThe whole feasibility investigation regarding the quasistatic approximation above can \nreadily be executed similarly for any other background permittivity $\\epsilon_\\mrm{b}^{\\prime}$.\nAs \\textit{e.g.}\\\/,  Figure \\ref{fig:matfig4b} shows a comparison between  $Q_\\mrm{abs}^\\mrm{dyn}$ and $Q_\\mrm{abs}^\\mrm{qs,opt}$ \nwith $\\epsilon_\\mrm{b}^{\\prime}=80$ corresponding to the permittivity of biological tissue for frequencies \nin the lower GHz region, \\textit{cf.}\\\/,  \\cite{Gabriel+etal1996b}. In this frequency region the corresponding\ndielectric losses will be at least $\\epsilon_\\mrm{b}^{\\prime\\prime}>10$. Hence, the evaluation shown in Figure \\ref{fig:matfig4b}\nverifies that the previous investigations made in \\cite{Nordebo+etal2017a,Dalarsson+etal2017a} are safely in the quasistatic regime\nif \\textit{e.g.}\\\/,  $k_0a<10^{-2}$, and which is certainly the case with cellular ($\\mu$m) structures in the GHz frequency range.\n\nFinally, to illustrate the theory based on an application in plasmonics, we investigate the absorption in a silver \nnanosphere embedded in a slightly lossy medium.\nFigure \\ref{fig:matfig21} shows the dielectric function of silver according to the Brendel-Bormann (BB) model fitted to experimental data as presented in \n\\cite[the dielectric model in Eq. (11) with parameter values from Table 1 and Table 3]{Rakic+etal1998}.\nThe frequency axis is given here in terms of the photon energy $h\\nu$ in units of electron volts\\unit{(eV)} where $h$ is Planck's constant and $\\nu$ the frequency.\nFigures \\ref{fig:matfig24} and \\ref{fig:matfig24b} present the normalized absorption cross section areas of a sphere with radius $a=10$\\unit{nm} \nand where the quasistatic optimal $Q_\\mrm{abs}^\\mrm{qs,opt}$ is given by \\eqref{eq:Cabssphere1o},  \nthe dynamic optimal $Q_\\mrm{abs}^\\mrm{dyn,opt}$ is given by \\eqref{eq:CabssphTMdipnorm} \ntogether with \\eqref{eq:epsilonospherethirdorder} and $Q_\\mrm{abs}^\\mrm{Ag}$ is given by \\eqref{eq:CabssphTMdipnorm} together with the dielectric model of silver as illustrated\nin Figure \\ref{fig:matfig21}. The background medium is defined by $\\epsilon_\\mrm{b}=1+\\iu\\epsilon_\\mrm{b}^{\\prime\\prime}$, where \n$\\epsilon_\\mrm{b}^{\\prime\\prime}=10^{-1}$ and $\\epsilon_\\mrm{b}^{\\prime\\prime}=10^{-3}$, respectively.\nWhat is interesting to observe here is that the absorption of silver is in fact rather close to being optimal (the peak at $h\\nu= 3.4$\\unit{eV}) \nwith the larger loss factor $\\epsilon_\\mrm{b}^{\\prime\\prime}=10^{-1}$,\nand where the quasistatic and dynamic optimal solutions also agree rather well. With the smaller loss factor $\\epsilon_\\mrm{b}^{\\prime\\prime}=10^{-3}$,\nthe absorption is no longer close to being optimal and the quasistatic and dynamic optimal solutions diverge. \nHere, we have chosen the loss factors so that  $10^{-3}<\\epsilon_\\mrm{b,break}^{\\prime\\prime}<10^{-1}$ where\n$\\epsilon_\\mrm{b,break}^{\\prime\\prime}=0.03$ corresponds to the break point defined by \\eqref{eq:k0aoptasymptotics2} for resonance at $h\\nu=3.4$\\unit{eV}. \n\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=0.48\\textwidth]{matfig21.pdf}\n\\end{center}\n\\vspace{-5mm}\n\\caption{Dielectric function of silver according to the Brendel-Bormann model fitted to experimental data \\cite{Rakic+etal1998}.}\n\\label{fig:matfig21}\n\\end{figure}\n\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=0.48\\textwidth]{matfig24.pdf}\n\\end{center}\n\\vspace{-5mm}\n\\caption{Normalized absorption cross sections for a sphere with radius  $a=10$\\unit{nm} \nwith $Q_\\mrm{abs}^\\mrm{dyn,opt}$ denoting the dynamic optimal absorption,  $Q_\\mrm{abs}^\\mrm{qs,opt}$ the quasistatic optimal absorption and \n$Q_{\\mrm{abs}}^\\mrm{Ag}$ the full dynamic dipole absorption corresponding to the Brendel-Bormann \nmodel for the dielectric function of silver \\cite{Rakic+etal1998}, \\textit{cf.}\\\/,  also Fig.~\\ref{fig:matfig21}.\nHere, the background loss is given by $\\epsilon_\\mrm{b}=1+\\iu 10^{-1}$.\n}\n\\label{fig:matfig24}\n\\end{figure}\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=0.48\\textwidth]{matfig24b.pdf}\n\\end{center}\n\\vspace{-5mm}\n\\caption{Same plot as in Fig.~\\ref{fig:matfig24}, but here with the background loss given by $\\epsilon_\\mrm{b}=1+\\iu 10^{-3}$.}\n\\label{fig:matfig24b}\n\\end{figure}\n\n\n\n\n\n\\section{Summary and conclusions}\n\nIt has been demonstrated that the maximal dipole absorption of a small dielectric or conducting sphere is \nunbounded under the quasistatic approximation if the losses in the surrounding medium can be made arbitrarily small. \nThis deviation has been rectified by using the general Mie theory and an asymptotic analysis to give explicit formulas to assess the validity of the quasistatic approximation.\nIn particular, it turns out that the quasistatic theory is valid provided that the electrical size of the sphere is sufficiently small\nat the same time as the exterior losses are sufficiently large. Moreover, it has been shown that the optimal normalized absorption cross section area of \nthe small dielectric sphere has a very subtle limiting behavior, and is in fact unbounded when both the electrical size as well as the exterior losses tend to zero.\nFinally,  an improved asymptotic formula based on full dynamics has been given for the optimal plasmonic dipole absorption of the sphere, which is \nvalid for small spheres as well as for small losses. Numerical examples have been included to illustrate the asymptotic results.\n\n\n\\begin{acknowledgments}\nThis work has been partly supported by the Swedish Foundation for Strategic Research (SSF) under the programme \nApplied Mathematics and the project Complex Analysis and Convex Optimization for EM Design.\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\n\\begin{figure*}[ht]\n  \\centering\n  \\includegraphics[scale=0.67]{figures\/new_main_figure_3.pdf}\n    \\caption{An example from the SciERC dataset~\\cite{luan2018multi}, where a system is expected to identify that \\ts{MORPA} and \\ts{parser} are entities of type \\ts{Method}, \\textsc{text-to-speech} is a \\ts{Task}, as well as \\textsc{MORPA} is a \\ti{hyponym of} \\textsc{parser} and \\textsc{MORPA} is \\ti{used for} \\textsc{text-to-speech}. Our entity model (a) predicts all the entities at once and our relation model (b) considers every pair of entities independently by inserting typed entity markers (e.g., \\textsc{[S:Md]} = the subject is a \\ts{Method}, \\textsc{[O:Tk]} = the object is a \\ts{Task}). We also proposed an approximation relation model (c) which supports batch computations. The tokens of the same color in (c) share the positional embeddings. See text for more details. }\n    \\label{fig:mainfig}\n\\end{figure*}\n\nExtracting entities and their relations from unstructured text is a fundamental problem in information extraction. This problem can be decomposed into two subtasks: named entity recognition~\\cite{sang2003introduction,ratinov2009design} and relation extraction~\\cite{zelenko2002kernel,bunescu2005shortest}. Early work employed a pipelined approach, training one model to extract entities and another to classify relations between them. More recently, however, end-to-end evaluations have been dominated  by systems that model these two tasks jointly~\\cite{li2014incremental,miwa2016end,katiyar2017going,zhang2017end,li2019entity,luan2018multi,luan2019general,wadden2019entity,lin2020joint,wang2020two}. It is commonly thought that joint models can better capture the interactions between entities and relations and help mitigate error propagation issues.\n\nIn this work, we review this problem and present a very simple approach which learns \\ti{two} encoders built on top of deep pre-trained language models~\\cite{devlin2019bert,beltagy2019scibert,lan2020albert}. The two models --- which we refer them as to an \\ti{entity model} and a \\ti{relation model} throughout the paper --- are trained independently and the relation model only relies on the entity model as input features. Our entity model builds on span-level representations and our relation model builds on contextual representations specific to a given pair of spans. Despite its simplicity, we find this pipelined approach to be extremely effective: using the same pre-trained encoders, our model outperforms all previous joint models on three standard benchmarks (ACE04, ACE05, SciERC).\n\nTo understand why this approach performs so well, we carry out a series of careful analyses. We observe that, (1) the contextual representations for the entity and relation models essentially capture distinct information, so sharing their representations hurts performance; (2) it is crucial to fuse the entity information (both boundary and type) at the \\ti{input layer} of the relation model; (3) leveraging cross-sentence information is useful in both tasks; (4) stronger pre-trained language models can bring further gains.   Hence, we expect that this simple model will serve as a very strong baseline and make us rethink the value of joint training in end-to-end relation extraction.\n\nFinally, one possible shortcoming of our approach is that we need to run our relation model once for every pair of entities. To alleviate this issue, we present a novel and efficient alternative by approximating and batching the computations of different groups of entity pairs at inference time. This approximation achieves a 8-16$\\times$ speedup with only a very small accuracy drop (e.g., 0.5-0.9\\% F1 drop on ACE05), which makes our model fast and accurate to use in practice.\n\nWe summarize our contributions as follows:\n\\begin{itemize}\n    \\item We present a very simple and effective approach for end-to-end relation extraction, which learns two independent encoders for entity recognition and relation extraction. Our model establishes the new state-of-the-art on three standard benchmarks and surpasses all previous joint models using the same pre-trained models.\n    \\item We conduct careful analyses to understand why our approach performs so well and how different factors affect final performance. We conclude that it is more effective to learn distinct contextual representations for entities and relations than to learn them jointly.\n    \\item To speed up the inference time of our model, we also propose a novel and efficient approximation, which achieves a 8-16$\\times$ runtime improvement with only a small accuracy drop.\n\\end{itemize}\n\n\\label{sec:intro}\n\n\\section{Related Work}\n\\label{sec:related}\n\nTraditionally, extracting relations between entities in text has been studied as two separate tasks: named entity recognition and relation extraction. In the last several years, there has been a surge of interest in developing models for joint extraction of entities and relations~\\cite{li2014incremental,miwa2016end}.  We group existing work of end-to-end relation extraction into two main categories: \\ti{structured prediction} and \\ti{multi-task learning}.\n\n\n\n\n\\vspace{-0.4em} \\paragraph{Structured prediction} The first category casts the two tasks into one structured prediction framework, although it can be formulated in various ways. \\newcite{li2014incremental} propose an action-based system which identifies new entities as well as links to previous entities, \\newcite{zhang2017end,wang2020two} adopt a table-filling approach proposed in \\cite{miwa2014modeling};\n\\newcite{katiyar2017going} and \\newcite{zheng2017joint} employ sequence tagging-based approaches; \\newcite{sun2019joint} and \\newcite{fu2019graphrel} propose graph-based approaches to jointly predict entity and relation types;\nand, \\newcite{li2019entity} convert the task into a multi-turn question answering problem. All of these approaches need to tackle a global optimization problem and perform joint decoding at inference time, using beam search or reinforcement learning.\n\n\n\\vspace{-0.4em} \\paragraph{Multi-task learning} This family of models essentially builds two separate models for entity recognition and relation extraction and optimizes them together through parameter sharing. \\newcite{miwa2016end} propose to use a sequence tagging model for entity prediction and a tree-based LSTM model for relation extraction.\nThe two models share one LSTM layer for contextualized word representations and they find sharing parameters improves performance for both models. The approach of \\newcite{bekoulis2018adversarial} is similar except that they model relation classification as a multi-label head selection problem. Note that these approaches still perform pipelined decoding: entities are first extracted and the relation model is applied on the predicted entities.\n\n\nThe closest work to ours is probably DYGIE and DYGIE++~\\cite{luan2018multi,luan2019general,wadden2019entity} which builds on recent span-based models for coreference resolution~\\cite{lee2017end} and semantic role labeling~\\cite{he2018jointly}. The key idea is to share span representations between the two tasks with joint optimization. It is later improved by incorporating relation and coreference propagation layers to update span representations~\\cite{luan2019general} and replacing LSTM encodings with BERT-based representations~\\cite{wadden2019entity}. A more recent work~\\cite{lin2020joint} further extends~\\newcite{wadden2019entity} by incorporating global features based on cross-substask and cross-instance constraints.\\footnote{This is an orthogonal contribution to ours and we will explore it for future work.} Compared to DYGIE++, our approach is much simpler: we use two independent encoders and do not use beam search or graph propagation layers. We will detail the differences in Section~\\ref{sec:our-model} and argue why our simpler model performs better.\n\n\n\n\n\n\\paragraph{BERT for relation extraction} Earlier work explored a variety of neural network architectures for relation extraction, including convolutional neural networks~\\cite{zeng2014relation}, recurrent neural networks~\\cite{xu2015classifying,zhang2017tacred}, and graph neural networks~\\cite{zhang2018graph}. More recent work uses pre-trained language models (LMs) such as BERT, built on deep Transformer encoders~\\cite{shi2019simple,soares2019matching}. We follow this trend and also study the impact of different pre-trained LMs on final performance.\n\n\\section{Method}\n\\label{sec:method}\n\nIn this section, we first formally define the problem of end-to-end relation extraction in Section~\\ref{sec:problem-definition} and then detail our approach in Section~\\ref{sec:our-model}.  Finally, we present our approximation solution in Section~\\ref{sec:method-approx} which improves the efficiency of our approach during inference considerably.\n\n\\subsection{Problem Definition}\n\\label{sec:problem-definition}\nThe input of the problem is a sentence $X$ consisting of $n$ tokens $x_1, \\dots, x_n$. Let $S=\\{s_1, \\dots, s_m\\}$ be all the possible spans in $X$ of up to length $L$ and $\\ts{START}(i)$ and $\\ts{END}(i)$ denote start and end indices of $s_i$. Optionally, we can incorporate cross-sentence context to help improve predictions, which we will elaborate in the next section. The problem can be decomposed into two sub-tasks:\n\\paragraph{Named entity recognition} Let $\\mathcal{E}$ denote a set of pre-defined entity types. The named entity recognition task is, for each span $s_i \\in S$, to predict an entity type $y_e(s_i) \\in \\mathcal{E}$ or $y_e(s_i) = \\epsilon$ representing span $s_i$ is not an entity. The output of the task is $Y_e = \\{(s_i, e), s_i \\in S, e \\in \\mathcal{E}\\}$.\n\\paragraph{Relation extraction} Let $\\mathcal{R}$ denote a set of pre-defined relation types. The task is, for every pair of spans $s_i \\in S, s_j\\in S$, to predict a relation type $y_r(s_i, s_j) \\in \\mathcal{R}$, or there is no relation between them: $y_r(s_i, s_j) = \\epsilon$. The output of the task is $Y_r = \\{(s_i, s_j, r), s_i, s_j \\in S, r \\in \\mathcal{R}\\}$.\n\nWe aim to build a model which takes $X$ as input and outputs $Y_e$ and $Y_r$ at the same time. During evaluation, $Y_e$ and $Y_r$ are compared against the ground truth $Y^*_e$ and $Y^*_r$ and entity and relation F1 will be reported respectively.\\footnote{The strict evaluation of relation F1 also takes the entity type into account.  See Section~\\ref{sec:exp-setup} for more details.}\n\n\\subsection{Our Model}\n\\label{sec:our-model}\nIn the following, we will describe our full model which consists of an entity and a relation model. As shown in Figure~\\ref{fig:mainfig}, our entity model first takes the input sentence and predict an entity type (or $\\epsilon$) for each single span. We then process every pair of candidate entities independently in the relation model by inserting extra marker tokens to highlight the subject and object and their types. We will detail each component below. At the end of this section, we will also summarize the main differences of our approach and DYGIE++~\\cite{wadden2019entity}, which is the closest work to ours and serves as a strong baseline in the literature.\n\n\\paragraph{Entity model} Our entity model is a standard span-based model following prior work~\\cite{lee2017end,luan2018multi,luan2019general,wadden2019entity}. We first use a pre-trained language model (e.g., BERT) to obtain contextualized representations $\\mf{x}_t$ for each input token $x_t$. Given a span $s_i \\in S$, the span representation $\\mathbf{h}_e(s_i)$ is defined as:\n\\vspace{-0.3em}\n\\begin{align*}\n  \\mathbf{h}_e(s_i) = [\\mathbf{x}_{\\textsc{START}(i)}; \\mathbf{x}_{\\textsc{END}(i)}; \\phi(s_i)],\n\\end{align*}\nwhere $\\phi(s_i) \\in \\mathbb{R}^{d_W}$ represents the learned embeddings of span width features. The span representation $\\mathbf{h}_e(s_i)$ is then used to predict entity types $e \\in \\mathcal{E} \\cup \\{\\epsilon\\}$:\n\\vspace{-0.3em}\n\\begin{align*}\n    P_e(e \\mid s_i) = \\mathrm{softmax}(\\mathbf{W}_e \\text{FFNN}(\\mathbf{h}_e(s_i)).\n\\end{align*}\nWe follow~\\newcite{wadden2019entity} and use a 2-layer feedforward neural network with ReLU activations.\n\n\\paragraph{Relation model} The relation model aims to take a pair of spans $s_i, s_j$ (a subject and an object) as input and predicts a relation type or $\\epsilon$, between the two spans. Previous approaches~\\cite{luan2018multi,luan2019general,wadden2019entity} re-use span representations $\\mf{h}_e(s_i), \\mf{h}_e(s_j)$ to predict their relation. We observe that these representations only capture contextual information around each individual entity and might fail to capture the dependencies between a specific pair of spans. We also hypothesize that sharing the contextual representations for different pairs of spans may be suboptimal. For example, the words \\ti{is a} in ~Figure~\\ref{fig:mainfig} are crucial in classifiying the relationship between \\ts{MORPA} and \\ts{parser} but not for \\ts{MORPA} and \\ts{text-to-speech}.\n\n\nOur relation model instead processes each pair of spans independently and inserts typed markers at the input layer to highlight the subject and object and their types. Specifically, given an input sentence $X$ and a pair of spans $s_i, s_j$, where $s_i$, $s_j$ have a type of $e_i, e_j \\in \\mathcal{E} \\cup \\{\\epsilon\\}$ respectively.\nWe define text markers as $\\langle \\textsc{S:}e_i\\rangle$, $\\langle \\textsc{\/S:}e_i\\rangle$, $\\langle \\textsc{O:}e_j\\rangle$, and $\\langle \\textsc{\/O:}e_j\\rangle$,\nand insert them into the input sentence before and after the subject and object spans (Figure~\\ref{fig:mainfig} (b)).\\footnote{Our final model indeed only considers $e_i, e_j \\neq \\epsilon$. We have explored using spans which are predicted as $\\epsilon$ for the relation model but didn't find improvement. See Section~\\ref{sec:error-prop} for more discussion.}\nLet $\\widehat{X}$ denote this modified sequence with text markers inserted:\n\\vspace{-0.3em}\n\\begin{align*}\n    &\\widehat{X} = \\dots \\langle \\textsc{S:}e_i\\rangle, x_{\\textsc{START}(i)}, \\dots, x_{\\textsc{END}(i)}, \\langle \\textsc{\/S:}e_i\\rangle, \\\\\n    &\\dots \\langle \\textsc{O:}e_j\\rangle, x_{\\textsc{START}(j)}, \\dots, x_{\\textsc{END}(j)}, \\langle \\textsc{\/O:}e_j\\rangle, \\dots.\n\\end{align*}\n\nWe then apply another pre-trained encoder on $\\widehat{X}$ and denote the output representations by $\\mathbf{\\widehat{x}}_t$. We concatenate the output representations  of two start positions and obtain the span pair representation:\n\\begin{align*}\n    \\mathbf{h}_r(s_i, s_j)=[\\mathbf{\\widehat{x}}_{\\widehat{\\textsc{START}}(i)}; \\mathbf{\\widehat{x}}_{\\widehat{\\textsc{START}}(j)}],\n\\end{align*}\nwhere $\\widehat{\\textsc{START}(i)}$ and $\\widehat{\\textsc{START}(j)}$ are the indices of  $\\langle \\textsc{S:}e_i\\rangle$ and $\\langle \\textsc{O:}e_j\\rangle$ in $\\widehat{X}$. Finally, the representation $\\mathbf{h}_r(s_i, s_j)$ will be used to predict the relation type $r \\in \\mathcal{R} \\cup \\{\\epsilon\\}$:\n\\begin{align*}\n    P_r(r|s_i, s_j) = \\mathrm{softmax}(\\mathbf{W}_r \\mathbf{h}_r(s_i, s_j)),\n\\end{align*}\n\n\n\n\nThis idea of using additional markers to highlight the subject and object is not entirely new as it  has been studied recently in relation classification tasks~\\cite{zhang2019ernie,soares2019matching}. However, most relation classification tasks (e.g., TACRED~\\cite{zhang2017tacred}) only focus on a given pair of subject and object in an input sentence and its effectiveness has not been evaluated in the end-to-end setting in which we need to classify the relationships between multiple entity mentions. We observed a large improvement in our experiments (Section~\\ref{sec:input-features})  and this strengthens our hypothesis that modeling the relationship between different entity pairs in one sentence require different contextual representations. Furthermore, \\newcite{zhang2019ernie,soares2019matching} only considered untyped markers (e.g., $\\langle\\textsc{S}\\rangle$, $\\langle\\textsc{\/S}\\rangle$) and previous end-to-end models e.g., \\cite{wadden2019entity} inject the entity type information into the relation model through auxiliary losses.  Using \\ti{typed} entity markers hasn't been explored before. We find that injecting type information at the input layer is very helpful in distinguishing entity types --- for example, whether ``Disney'' refers to a \\ti{person} or an \\ti{organization}--- before trying to understand the relations between them.\n\n\n\\paragraph{Cross-sentence context}\nCross-sentence information can be used to help predict entity types and relations, especially for pronominal mentions.\n\\newcite{luan2019general,wadden2019entity} employ a propagation mechanism through coreference and relation links to incorporate cross-sentence context. \\newcite{wadden2019entity} also add a 3-sentence context window which is shown to improve performance.\nWe also evaluate the importance of leveraging cross-sentence context in end-to-end relation extraction. As we expect that pre-trained language models to be able to capture long-range dependencies already, we simply incorporate cross-sentence context by extending the sentence to a fixed window size $W$ for both the  entity and relation model. Specifically, given an input sentence with $n$ words, we augment the input with $(W-n) \/ 2$ words from the left context and right context respectively ($W = 100$ in our default model).\n\n\\paragraph{Training \\& inference} For both entity model and relation model, we employ two pre-trained language models and fine-tune them using task-specific losses. We use cross-entropy loss for both models:\n\\begin{align*}\n    \\mathcal{L}_e &= -\\sum_{s_i \\in S} \\log P_e(e_i^* | s_i) \\\\\n    \\mathcal{L}_r &= -\\sum_{s_i, s_j \\in S_G, s_i \\neq s_j} \\log P_r(r_{i,j}^* \\mid s_i, s_j),\n\\end{align*}\nwhere $e_i^*$ represents the gold entity type of $s_i$ and $r_{i,j}^*$ represents the gold relation type of span pair $s_i, s_j$ in the training data. For training the relation model, we only consider the gold entities $S_G \\subset S$ in the training set and use the gold entity labels as the input of the relation model. We considered training on all spans $S$ (with pruning) as well as predicted entity types but none of them led to meaningful improvements compared to this simple pipeline training (see more details in Section~\\ref{sec:error-prop}).\n\nDuring inference, we first predict the entities by taking $y_e(s_i) = \\argmax_{e\\in \\mathcal{E} \\cup \\{\\epsilon\\}}P_e(e | s_i)$. Denote $S_{\\text{pred}} = \\{s_i: y_e(s_i) \\neq \\epsilon\\}$, we enumerate all the spans $s_i, s_j \\in S_{\\text{pred}}$ and use $y_e(s_i), y_e(s_j)$ as the inputs for the relation model $P_r(r \\mid s_i, s_j)$.\n\n\n\\paragraph{Differences from DYGIE++} Our model differs from DYGIE++ in the following ways: (1) We use separate encoders for the entity and relation model, without any multi-task learning; the predicted entity labels are used directly as the input features of the relation model. (2) The contextual representations in the relation model are specific to each pair of spans. (3) We incorporate cross-sentence information by extending the input with additional context. (4) We do not use beam search or graph propagation layers. As a result, our model is much simpler. Moreover, we will show  that it also achieves large gains in all the benchmarks, using the same pre-trained encoders.\n\n\\subsection{Efficient Batch Computations}\n\\label{sec:method-approx}\nDespite the simplicity and effectiveness of our approach (which we will demonstrate in our experiments), one possible shortcoming is that we need to run our relation model once for every pair of entities.\nTo alleviate this issue, we propose a novel and efficient alternative for our relation model. The key problem is that we would like to re-use computations for different span pairs in the same sentence. This is impossible in our original model because we must insert special entity markers for each pair of spans independently. Thus we propose an approximation model by making two major changes to the original relation model.  First, instead of directly inserting entity markers into the original sentence, we tie the position embeddings of the markers with the start and end tokens of the corresponding span:\n\\begin{align*}\n    & \\textsc{pos}(\\langle \\textsc{S:}e_i\\rangle) = \\textsc{pos}(x_{\\textsc{START}(i)}) \\\\\n    & \\textsc{pos}(\\langle \\textsc{\/S:}e_i\\rangle) = \\textsc{pos}(x_{\\textsc{END}(i)}) \\\\\n    & \\textsc{pos}(\\langle \\textsc{O:}e_j\\rangle) = \\textsc{pos}(x_{\\textsc{START}(j)}) \\\\\n    & \\textsc{pos}(\\langle \\textsc{\/O:}e_j\\rangle) = \\textsc{pos}(x_{\\textsc{END}(j)}),\n\\end{align*}\nwhere $\\textsc{pos}(\\cdot)$ denotes the position id of a token. As the example shown in Figure~\\ref{fig:mainfig}, if we want to classify the relationship between \\ts{MORPA} and \\ts{parser}, the first entity marker \\ts{$\\langle S$$:\\ts{Method}\\rangle$} will share the positional embedding with the token \\ts{mor}. To do this, the positional embeddings of the original tokens will not be changed.\n\nSecond, we add a constraint to the attention layers: We enforce the text tokens to only attend to text tokens and not attend to the marker tokens while an entity marker token can attend to all the text tokens and all the 4 marker tokens  associated with the same span pair. These two modifications allow us to re-use the computations of all text tokens, because text tokens are independent of the entity marker tokens.\nThus, we can batch multiple pairs of spans from the same sentence in one run of the relation model. In practice, we add all marker tokens to the end of the sentence to form an input that batches a set of span pairs (Figure~\\ref{fig:mainfig} (c)). This leads to a large speedup at inference time and only a small drop in performance (Section~\\ref{sec:approx}).\n\n\n\\section{Experiments}\n\\label{sec:exp}\n\n\\subsection{Experimental Setup}\n\\label{sec:exp-setup}\n\n\\paragraph{Datasets} We evaluate our approach on three end-to-end relation extraction datasets: ACE04, ACE05\\footnote{\\url{catalog.ldc.upenn.edu\/LDC2005T09} \\url{catalog.ldc.upenn.edu\/LDC2006T06}}, and SciERC~\\cite{luan2018multi}.\nWe follow \\newcite{luan2019general}'s preprocessing steps and split ACE04 into 5 folds, and split ACE05 and SciERC into train, development, and test sets.\nThe detailed data statistics are given in Table~\\ref{tab:data_stats}.\n\n\n\\input{tables\/data_stat.tex}\n\\input{tables\/main_result}\n\\input{tables\/approx.tex}\n\n\\paragraph{Evaluation metrics}\nWe follow the standard evaluation protocol and use F1 scores as the evaluation metric. For the named entity recognition task, a predicted entity is considered as a correct prediction if its span boundaries and the predicted entity type are both correct.\nFor the relation extraction task, a predicted relation is considered as a correct prediction if the boundaries of two spans are correct and the predicted relation type is correct.\nWe also report the strict relation F1 score (denoted by Rel+), where a predicted relation is considered as a correct prediction if the entity types and boundaries of two spans are correct, as well as the predicted relation type is correct. More discussion of the evaluation settings can be found in \\newcite{bekoulis2018adversarial,taille2020sincere}.\n\n\\paragraph{Implementation details}\nFor the entity model, we follow \\newcite{wadden2019entity} and set the width embedding size as $d_W = 150$ and use a 2-layer FFNN with $150$ hidden units. For our approximation model (Section~\\ref{sec:approx}), we batch candidate pairs by adding $4$ markers for each pair to the end of the sentence, until the total number of tokens exceeds $250$.\nWe use a context window size of $W=100$ in our default setting using cross-sentence context and we will study the effect of different context sizes in Section~\\ref{sec:context}. We consider spans up to $L = 8$ words. For all the experiments, we report the averaged F1 scores of 5 runs.\n\nWe implement our models based on HuggingFace's \\ti{Transformers} library~\\cite{Wolf2019HuggingFacesTS}. We use \\ttt{bert-base-uncased}~\\cite{devlin2019bert} and \\ttt{albert-xxlarge-v1}~\\cite{lan2020albert} as the base encoders for ACE04 and ACE05, for a fair comparison with previous work and an investigation of the impact of small vs large pre-trained models.\\footnote{ As detailed in Table~\\ref{tab:main-results}, some previous work used BERT-large models. We are not able to do a comprehensive study of all the pre-trained models and our BERT-base results are generally higher than most published results using larger models.}  We also use \\ttt{scibert-scivocab-uncased}~\\cite{beltagy2019scibert} as the base encoder for SciERC, as this in-domain pre-trained model is shown to be more effective than BERT~\\cite{wadden2019entity}.\nWe train our models with Adam optimizer of a linear scheduler with a warmup ratio of $0.1$. For all the experiments, we train the entity model for $100$ epochs, and a learning rate of 1e-5 for weights in pre-trained LMs, 5e-4 for others and a batch size of 16. We train the relation model for $10$ epochs with a learning rate of 2e-5 and a batch size of 32.\n\n\\subsection{Main Results}\n\nTable~\\ref{tab:main-results} compares our approach to all the previous results.\nWe report the F1 scores in both single-sentence (no cross-sentence context) and cross-sentence (a context window size of $W=100$) settings for a fair comparison with previous work. As is shown, our single-sentence models achieve strong performance and incorporating cross-sentence context further improves the results consistently.\nOur BERT-base (or SciBERT) models achieve similar or better results compared to all the previous work including models built on top of larger pre-trained LMs, and the performance is further improved by using a larger encoder, i.e., ALBERT.\n\n\nFor entity recognition, our best model achieves an absolute F1 improvement of $+1.4\\%$, $+1.7\\%$, $+0.7\\%$ on ACE05, ACE04, and SciERC respectively.\nThis shows that cross-sentence information is useful for the entity model and that pre-trained Transformer encoders are able to capture long-range dependencies from a large context. For relation extraction, our approach outperforms the best previous methods by an absolute F1 of $+2.6\\%$, $+2.8\\%$, $+1.7\\%$ on ACE05, ACE04, and SciERC respectively. We also obtained a $4.3\\%$ higher relation F1 on ACE05 compared to DYGIE++~\\cite{wadden2019entity} using the same BERT-base pre-trained model. All these large improvements demonstrate the effectiveness of learning distinct representations for entities and relations of different entity pairs, as well as fusing entity information at the input layer of the relation model.\n\nWe also noticed that compared to the previous state-of-the-art model~\\cite{wang2020two} based on ALBERT, our model achieves a similar entity F1 (89.5 vs 89.7) but a substantially better relation F1 (67.6 vs 69.0) without using cross-sentence context. This clearly demonstrates the superiority of our relation model.\n\n\\subsection{Batch Computations and Speedup}\n\\label{sec:approx}\n\nIn Section~\\ref{sec:method-approx}, we proposed an efficient approximation solution for the relation model, which enables us to re-use the computations of text tokens and batch multiple span pairs in one input sentence.\nWe evaluate this approximation model on ACE05 and SciERC.\nTable~\\ref{tab:approx} shows the relation F1 scores and the inference speed of the full relation model and the approximation model.\nOn both datasets, our approximation model significantly improves the efficiency of the inference process. For example, in the single-sentence setting, we obtain a $11.9\\times$ speedup on ACE05 and a $8.7\\times$ speedup on SciERC.\nBy re-using a large part of computations, we are able to make predictions on the full ACE05 test set (2k sentences) in less than $10$ seconds on a single NVIDIA GeForce 2080 Ti GPU. On the other hand, this approximation only brings a small performance drop --- compared to the full model, the F1 score drops $0.5\\%$ and $1.0\\%$ on ACE05 and SciERC respectively in the single-sentence setting. Considering the accuracy and efficiency of this approximation model, we expect it to be very effective to use in practice.\n\n\n\\section{Analysis}\n\\label{sec:analysis}\n\nDespite its simple design and training paradigm, we have shown that our approach outperforms all previous joint models. In this section, we aim to take a deeper look and understand why this model performs so well and what contributes to its final performance.\n\n\\subsection{Importance of Typed Text Markers}\n\\label{sec:input-features}\n\nOur first argument is that it is important to build different contextual representations for different pairs of spans and an early fusion of entity type information can further improve performance. To validate the importance of typed text markers, we experiment the following variants on both ACE05 and SciERC when the gold entities are given:\n\n\\vspace{0.5em}\n\\noindent\\textbf{\\textsc{Text}}: We use the span representations defined in the entity model (Section~\\ref{sec:our-model}) and concatenate the hidden representations for the subject and the object, as well as their element-wise multiplication: $[\\mathbf{h}_e(s_i), \\mathbf{h}_e(s_j), \\mathbf{h}_e(s_i) \\odot \\mathbf{h}_e(s_j)]$. This is similar to the relation model in \\newcite{luan2018multi,luan2019general}.\n\n\\noindent\\textbf{\\textsc{TextEType}}: In addition to \\ts{Text}, we concatenate the span-pair representations with entity type embeddings $\\psi(e_i), \\psi(e_j) \\in \\mathbb{R}^{d_E}$ ($d_E$ = 150).\n\n\\noindent\\textbf{\\textsc{Markers}}: We use untyped entity types ($\\langle\\ts{S}\\rangle$, $\\langle\\ts{\/S}\\rangle$, $\\langle\\ts{O}\\rangle$, $\\langle\\ts{\/O}\\rangle$) at the input layer and concatenate the representations of two spans' starting points.\n\n\\noindent\\textbf{\\textsc{MarkersEType}}: In addition to \\ts{Markers},  we concatenate the span-pair representations with entity type embeddings $\\psi(e_i), \\psi(e_j) \\in \\mathbb{R}^{d_E}$ ($d_E$ = 150).\n\n\n\\noindent\\textbf{\\textsc{MarkersELoss}}: We also consider a variant which uses untyped markers but add another FFNN to predict the entity types of subject and object through auxiliary losses. This is similar to how the entity information is used in multi-task learning~\\cite{luan2019general,wadden2019entity}.\n\n\\noindent\\textbf{\\textsc{TypedMarkers}}: This is our final model described in Section~\\ref{sec:our-model}. We use typed markers at the input layer.\n\n\\input{tables\/input_features}\n\nTable~\\ref{tab:input} shows the performance of all the variants and it clearly indicates that different input representations make a real difference in the relation accuracy. Compared to \\ts{Text}, \\ts{TypedMarkers} improved the F1 scores largely by $+5.5\\%$ and $+7.4\\%$ absolute. All the variants of using marker tokens are significantly better than the standard text representations and this suggests the importance of learning different representations with respect to different subject-object pairs. Finally, entity type is useful in improving the relation performance and an early fusion of entity information is particularly effective (\\ts{TypedMarkers} vs \\ts{MarkersEType} and \\ts{MarkersELoss}). We also find that \\ts{MarkersEType} to perform even better than \\ts{MarkersEloss} which suggests that using entity types directly as features is better than using them to provide training signals through auxiliary losses.\n\n\\subsection{Modeling Interactions between Entities and Relations}\nOne main argument for joint models is that modeling the interactions between the two tasks can contribute to each other. In this section, we aim to validate if it is the case in our approach. We first study whether sharing the two representation encoders can improve the performance or not. We train the entity and relation models together by jointly optimizing $\\mathcal{L}_e + \\mathcal{L}_r$.  As shown in Table~\\ref{tab:shared-encoder}, we find that simply sharing the encoders hurts both the entity and relation F1. We think this is because the two tasks have different input formats and require different features for predicting entity types and relations, thus using separate encoders indeed learns better task-specific features.\n\\input{tables\/shared_encoder}\n\nIn the previous section, we have already shown that the entity information is useful in the relation model (either entity embeddings, auxiliary loss or input features) and the best way to use it is through typed markers.  Next, we aim to investigate whether the relation information can improve the entity performance. To do so, we add an auxiliary loss to our entity model, which concatenates the two span representations as well as their element-wise multiplication (see the \\textsc{Text} variant in Section~\\ref{sec:input-features}) and predicts the relation type between the two spans ($r \\in \\mathcal{R}$ or $\\epsilon$).  Through joint training with this auxiliary relation loss, we observe a negligible improvement ($<0.1\\%$) on averaged entity F1 over $5$ runs on the ACE05 development set. Hence, we conclude that relation information does not improve the entity model substantially.\n\nTo summarize our findings, (1) entity information is clearly helpful in predicting relations. However, we don't find enough evidence in our experiments that relation information can improve the entity performance substantially.\\footnote{\\newcite{miwa2016end} observed a  slight improvement on entity F1 by sharing the parameters (80.8 $\\rightarrow$ 81.8 F1) on the ACE05 development data. \\newcite{wadden2019entity} observed that their relation propagation layers improved the entity F1 slightly on SciERC but it hurts performance on ACE05.} (2) Simply sharing the encoders does not provide benefits to our approach.\n\n\\input{tables\/error_propagation}\n\n\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[scale=0.47]{figures\/context_size.pdf}\n    \\vspace{-0.6em}\n    \\caption{Effect of different context window sizes, measured on the ACE05 development set with the BERT-base model. We use the same entity model (an entity model with $W=100$) to report the relation F1 scores.}\n    \\label{fig:context_window}\n    \\vspace{-0.4em}\n\\end{figure}\n\\subsection{Mitigating Error Propagation}\n\\label{sec:error-prop}\n\nOne well-known drawback of pipeline training is the error propagation issue. In our final model, we use gold entities (and their types) to train the relation model and the predicted entities during inference and this may lead to a discrepancy between training and testing. In the following, we describe several attempts we made to address this issue.\n\nWe first study whether using predicted entities --- instead of gold entities --- during training can mitigate this issue. We adopt a 10-way jackknifing method, which is a standard technique in many NLP tasks such as dependency parsing~\\cite{agic2017not}. Specifically, we divide the data into $10$ folds and predict the entities in the $k$-th fold using an entity model trained on the remainder. As shown in Table~\\ref{tab:error-propagation}, we find that jackknifing strategy hurts the final relation performance surprisingly. We hypothesize that it is because it introduced additional noise during training.\n\nSecond, we consider using more pairs of spans for the relation model at both training and testing time.  The main reason is that in the current pipeline approach, if a gold entity is missed out by the entity model during inference, the relation model will not be able to predict any relations associated with that entity. Following the beam search strategy used in the previous work~\\cite{luan2019general,wadden2019entity}, we consider using $\\lambda n$ ($\\lambda = 0.4$ and $n$ is the sentence length)\\footnote{This pruning strategy achieves a recall of $96.7\\%$ of gold relations on the development set of ACE05.} top spans scored by the entity model. We explored several different strategies for encoding the top-scoring spans for the relation model: (1) typed markers: the same as our main model except that we now have markers e.g., $\\langle \\textsc{S:}\\epsilon\\rangle$, $\\langle \\textsc{\/S:}\\epsilon\\rangle$ as input tokens; (2) untyped markers: in this case, the relation model is unaware of a span is an entity or not; (3) untyped markers trained with an auxiliary entity loss ($e \\in \\mathcal{E}$ or $\\epsilon$). As Table~\\ref{tab:error-propagation} shows, none of these changes led to significant improvements and using untyped markers is especially worse because the relation model struggles to identify whether a span is an entity or not.\n\nIn sum, we don't find any of these attempts improved performance significantly and our simple pipeline training turns out to be a surprisingly effective strategy. We do not argue that this error propagation issue does not exist or cannot be solved, while we will need to explore better solutions to address this issue.\n\n\n\\subsection{Effect of Cross-sentence Context}\n\\label{sec:context}\n\nIn Table~\\ref{tab:main-results}, we demonstrated the improvements from using cross-sentence context on both the entity and relation performance. Finally, we explore the effect of different context sizes $W$ in Figure~\\ref{fig:context_window}. We find that using cross-sentence context clearly improves both entity and relation F1. However, the results don't further increase from $W = 100$ to $W = 300$. In our final models, we use $W=100$ for both the entity model and relation model.\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nIn this paper, we present a very simple and effective approach for end-to-end relation extraction. Our model learns two encoders for entity recognition and relation extraction independently and our experiments show that it outperforms previous state-of-the-art on three standard benchmarks considerably. We conduct extensive analysis to undertand the superior performance of our approach and we validate the importance of learning distinct contextual representations for entities and relations and using entity information as input features for the relation model. We also propose an efficient approximation, obtaining a large speedup at inference time with a small accuracy drop. We hope that this simple model will serve as a very strong baseline and make us rethink the value of joint training in end-to-end relation extraction.\n\n\\section{Training Details}\n\\label{sec:appendix}\n\nWe train our models with BertAdam optimizer. For the entity model, we train the model for $100$ epochs, with a learning rate of 1e-5 for parameters in the pre-trained language models, 5e-4 for other parameters, and a batch size of 16.\nFor the relation model, we follow~\\newcite{joshi2020spanbert} to train the model for $10$ epochs, with a learning rate of 2e-5 and a batch size of 32.\nDuring training, we use a linear scheduler with a warm-up ratio of $0.1$.\nWe implement our models based on HuggingFace's \\ti{Transformers} library~\\cite{Wolf2019HuggingFacesTS}.\n\nWe train our models using a single GPU (NVIDIA GeForce RTX 2080 Ti). It takes about $7$ hours to train the entity model on ACE04 and ACE05 datasets.\nIt takes about $13$ hours to train the relation model on the ACE04 dataset and $16$ hours to train on the ACE05 dataset.\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nA considerable interest in actual research of superconductivity (SC) with high critical temperature \nis focused on the family of doped ferropnictide compounds \\cite{kamihara1, kamihara2} and one \nof their notable distinctions from \"old\" BCS superconductors and more recent doped perovskite \nsystems consists in possibility for a peculiar, so-called extended s-wave symmetry of superconducting\norder parameter which changes its sign between electron and hole segments of the Fermi surface\n\\cite{mazin}. This additional property permits to avoid the fundamental limitation by the Anderson's \ntheorem \\cite{and} for non-magnetic impurities to produce localized impurity levels within the \nsuperconducting band gap \\cite{dzhang, zhang}. At finite, but low enough, impurity concentration, \nsuch levels are expected to give rise to some resonance effects like those well studied in \nsemiconductors at low doping concentrations \\cite{shklo}. Analogous effects in superconductors \nwere theoretically predicted and experimentally discovered for magnetic impurities, either in BCS \nsystems \\cite{shiba,rus,maki} and in the two-band MgB$_2$ system \\cite{con, moca}. In all those \ncases, the breakdown of the Anderson's theorem is only due to the breakdown of the spin-singlet \nsymmetry of an $s$-wave Cooper pair by a spin-polarized impurity, and the main physical interest \nof the considered case of SC iron pnictides from the point of view of disorder in general is the \npossibility for pair-breaking even on non-magnetic impurity scatterers. The latter theoretical prediction \nwas confirmed by the observations of various effects from localized impurity states, for instance, in the \nsuperfluid density (observed through the London penetration length) \\cite{gordon, kitagawa}, transition \ncritical temperature \\cite{guo, li} and electronic specific heat \\cite{hardy}, all mainly due to an emerging \nspike of electronic density of states against its zero value in the initial band gap.\n\nBut it is also known that indirect interactions between random impurity centers of certain type (the \nso-called deep levels at high enough concentrations) in doped semiconductors can lead to formation \nof collective band-like states \\cite{iv, yama}. This corresponds to the Anderson transition in a general \ndisordered system \\cite{and1}, and the emerging new band of quasiparticles in the spectrum can \nessentially change thermodynamics and transport in the doped material \\cite{mott}. An intriguing \npossibility for similar banding of impurity levels within the SC gap \\cite{bal, lokt} was recently discussed \nfor the doped ferropnictides \\cite{pog}. The present work is aimed on a more detailed analysis of the \nband-like impurity states, focused on their observable effects that cannot be produced by localized \nimpurity states. We use the specific form of Green functions for superconducting quasiparticles \nderived in the previous work \\cite{pog} in the general Kubo-Greenwood formalism \\cite{kubo} to obtain \nthe temperature and frequency dependences of optical and thermal conductivity and also of thermoelectric \ncoefficients. These results are compared with the available experimental data and some suggestions are \ndone on possible practical applications of such impurity effects.\n\n\\section{Green functions for disordered SC ferropnictide}\nWe begin from a brief summary of the Green function description of electronic spectrum in\nLaOFeAs with impurities (not necessarily dopants) using the minimal coupling model \n\\cite{dag,tsai} for the non-perturbed Hamiltonian. It considers only 2 types of local Fe \norbitals, $d_{xz}$ (or $x$) and $d_{yz}$ (or $y$), on sites of square lattice with lattice \nparameter $a$ and 4 hopping parameters between nearest neighbors (NNs) and next nearest \nneighbors (NNNs): i) $t_1$ for $xx$ or $yy$ NNs along their orientations, and $t_2$ across \nthem, and ii) $t_3$ for $xx$ or $yy$ NNNs, and $t_4$ for $xy$ NNNs. The resulting band \nHamiltonian is diagonal in quasimomentum ${\\bf k}$ and spin $\\sigma$, but non-diagonal with respect \nto the orbital indices of the 2-spinors $\\psi^\\dagger({\\bf k},\\sigma) = (x^\\dagger_{{\\bf k},\\sigma},\ny^\\dagger_{{\\bf k},\\sigma})$:\n\\begin{equation}\n H_t = \\sum_{{\\bf k},\\sigma} \\psi^\\dagger({\\bf k},\\sigma)\\hat h({\\bf k})\\psi({\\bf k},\\sigma).\n \\label{eq1}\n  \\end{equation}\nHere the energy matrix in orbital basis is expanded in Pauli matrices $\\hat\\sigma_i$: \n$\\hat h({\\bf k}) = \\varepsilon_{+,{\\bf k}}\\hat\\sigma_0 + \\varepsilon_{-,{\\bf k}}\\hat\\sigma_3 + \\varepsilon_{xy,{\\bf k}}\n\\hat\\sigma_1$ with the energy factors $\\varepsilon_{\\pm,{\\bf k}} = (\\varepsilon_{x,{\\bf k}} \\pm \\varepsilon_{x,{\\bf k}})\/2$, \nand\n\\begin{eqnarray}\n \\varepsilon_{x,{\\bf k}} & = & - 2t_1 \\cos ak_x - 2t_2 \\cos ak_y - 4t_3 \\cos ak_x \\cos ak_y,\\nonumber\\\\\n  \\varepsilon_{y,{\\bf k}} & = & - 2t_1 \\cos ak_y - 2t_2 \\cos ak_x - 4t_3 \\cos ak_x \\cos ak_y,\\nonumber\\\\\n   \\varepsilon_{xy,{\\bf k}} & = & - 4t_4 \\sin ak_x \\sin ak_y.\\nonumber\n    \\end{eqnarray}\nIt is readily diagonalized at passing from the orbital to subband basis: $ \\hat h_b({\\bf k})\n= \\hat U({\\bf k})\\hat h({\\bf k})\\hat U({\\bf k})^\\dagger$, with the unitary matrix $\\hat U({\\bf k}) =\n\\exp(-i\\hat\\sigma_2\\theta_{\\bf k}\/2)$ and $\\theta_{\\bf k} = \\arctan \\left(\\varepsilon_{xy,{\\bf k}}\/\\varepsilon_{-,{\\bf k}}\\right)$.\nThe resulting eigen-energies for electron and hole subbands are:\n\\begin{equation}\n \\varepsilon_{h,e}({\\bf k}) = \\varepsilon_{+,{\\bf k}} \\pm \\sqrt{\\varepsilon_{xy,{\\bf k}}^2 + \\varepsilon_{-,{\\bf k}}^2},\n \\label{eq2}\n  \\end{equation}\nand respective electron and hole segments of the Fermi surface are defined by the equations\n$\\varepsilon_{e,h}({\\bf k}) = \\varepsilon_{\\rm F}$. A reasonable fit to the  LaOFeAs band structure by the more\ndetailed LDA calculations \\cite{xu} is attained with the parameter choice (in $|t_1|$ units)\nof $t_1 = -1$, $t_2 = 1.3$, $t_3 = t_4 = -0.85$ \\cite{raghu}.\n\nThe SC state of such multiband electronic system is suitably described in terms of \"multiband\n-Nambu\" 4-spinors $\\Psi_{\\bf k}^\\dagger = \\left(\\alpha_{{\\bf k},\\uparrow}^\\dagger,\\alpha_{-{\\bf k},\\downarrow}, \\beta_{{\\bf k},\\uparrow}\n^\\dagger,\\beta_{-{\\bf k},\\downarrow}\\right)$ with the multiband spinor $\\left(\\alpha_{{\\bf k},\\sigma}^\\dagger,\\beta_{{\\bf k},\\sigma}\n^\\dagger\\right) = \\psi^\\dagger({\\bf k},\\sigma)\\hat U^\\dagger({\\bf k})$, by a 4$\\times$4 extension of the\nHamiltonian Eq. \\ref{eq1} in the form:\n\\begin{equation}\n H_s = \\sum_{{\\bf k},\\sigma} \\Psi_{\\bf k}^\\dagger\\hat h_s({\\bf k})\\Psi_{\\bf k},\n \\label{eq3}\n  \\end{equation}\nwhere the 4$\\times$4 matrix $\\hat h_s({\\bf k}) = \\hat h_b({\\bf k})\\otimes\\hat\\tau_3 + \\Delta_{{\\bf k}}\\hat\\sigma_0\n\\otimes\\hat\\tau_1$, includes the Pauli matrices $\\hat\\tau_i$ acting on the Nambu (particle-antiparticle)\nindices in $\\Psi$-spinors. The simplified form for the extended \\emph{s}-wave gap function takes\nconstant values, $\\Delta_{\\bf k} = \\Delta$ on the electron segments and $\\Delta_{\\bf k} = -\\Delta$ on the hole segments.\n\nThe observable values result from the (Fourier transformed) GF 4$\\times$4 matrices $\\hat\nG_{{\\bf k},{\\bf k}'} = \\langle\\langle\\Psi_{{\\bf k}}|\\Psi_{{\\bf k}'}^\\dagger\\rangle\\rangle$, and for the\nnon-perturbed system, Eq. \\ref{eq1}, they are diagonal in quasimomentum: $\\hat G_{{\\bf k},{\\bf k}'}\n= \\delta_{{\\bf k},{\\bf k}'} \\hat g_{\\bf k}$ with\n\\begin{eqnarray}\n \\hat g_{\\bf k} & = & \\frac{\\varepsilon\\hat\\tau_0 + \\varepsilon_e({\\bf k})\n \\hat\\tau_3 + \\Delta\\hat\\tau_1}{2d_{e,{\\bf k}}}\\otimes\\hat\\sigma_e\\nonumber\\\\\n & + & \\frac{\\varepsilon\\hat\\tau_0 + \\varepsilon_h({\\bf k})\\hat\\tau_3 - \\Delta\\hat\\tau_1}{2d_{h,{\\bf k}}}\\otimes\\hat\\sigma_h,\n  \\label{eq4}\n   \\end{eqnarray}\n$\\hat\\sigma_{e,h} = \\left(\\hat\\sigma_0 \\pm \\hat\\sigma_3\\right)\/2$, $d_{i,{\\bf k}} = \\varepsilon^2 - \\varepsilon_i^2({\\bf k}) - \\Delta^2$.\n\nTo simplify the treatment of impurity perturbations, the band structure is approximated to \nidentical circular electron and hole Fermi segments of radius $k_{\\rm F}$ around respective \npoints ${\\bf K}_{e,h}$ in the Brillouin zone and to similar linear dispersion of normal state \nquasiparticles near the Fermi level $\\varepsilon_{\\rm F}$: $\\varepsilon_e({\\bf k}) -  \\varepsilon_{\\rm F} = \\hbar v_{\\rm F}\n\\left(|{\\bf k} - {\\bf K}_e| - k_{\\rm F}\\right)$ and $\\varepsilon_h({\\bf k}) -  \\varepsilon_{\\rm F} = -\\hbar v_{\\rm F} \n\\left(|{\\bf k} - {\\bf K}_h| - k_{\\rm F}\\right)$. Moreover, we shall describe the contributions of \nboth segments to overall electronic properties by a single quasimomentum variable $\\xi$ that \nidentifies electron $\\xi_e = \\varepsilon_e({\\bf k}) -  \\varepsilon_{\\rm F}$ and hole $\\xi_h = \\varepsilon_h({\\bf k}) -  \\varepsilon_{\\rm F}$ \nones.\n\nNext, the Hamiltonian of the disordered SC system is chosen as $H = H_s + H_{imp}$ including \nbesides $H_s $, Eq. \\ref{eq3}, the term  due to non-magnetic impurities \\cite{dzhang}  on random \nsites ${\\bf p}$ in Fe square lattice with an on-site energy shift $V$ (supposed positive without loss \nof generality). It is written in the multiband-Nambu spinor form as:\n\\begin{equation}\nH_{imp} = {1 \\over N} \\sum_{{\\bf p},{\\bf k},{\\bf k}'} {\\rm e}^{i({\\bf k}' - {\\bf k})\\cdot{\\bf p}}\\Psi_{\\bf k}^\\dagger \n\\hat V_{{\\bf k},{\\bf k}'}\\Psi_{{\\bf k}'},\n \\label{eq5}\n   \\end{equation}\nwith the number $N$ of unit cells in the crystal and the 4$\\times$4 scattering matrix \n$\\hat V_{{\\bf k},{\\bf k}'} = V\\hat U^\\dagger({\\bf k})\\hat U({\\bf k}')\\otimes\\hat\\tau_3$. In presence of this \nperturbation, the GFs can be expressed in specific forms depending on whether the considered \nquasiparticle energy falls into the range of band-like or localized states. Namely, for band-like \nstates, the momentum diagonal GF:\n\\begin{equation}\n \\hat G_{\\bf k} = \\hat G_{{\\bf k},{\\bf k}} = (\\hat g_{\\bf k}^{-1} - \\hat \\Sigma_{\\bf k})^{-1},\n  \\label{eq6}\n   \\end{equation}\ninvolves the self-energy matrix $\\hat \\Sigma_{\\bf k}$  in the form of the so-called renormalized \ngroup expansion \\cite{ilp}:\n\\begin{equation}\n \\hat \\Sigma_{\\bf k} = c\\hat T \\left(1 + c \\hat B_{\\bf k} + \\dots\\right).\n  \\label{eq7}\n    \\end{equation} \nThis series in powers of impurity concentration $c$ begins from the (k-independent) T-matrix, \n$\\hat T = \\hat V \\left(1 - \\hat G \\hat V\\right)^{-1}$. From the matrices $\\hat V = \n\\hat V_{{\\bf k},{\\bf k}} = V\\hat\\tau_3$ and $\\hat G = N^{-1} \\sum_{\\bf k} \\hat g_{\\bf k} = \\pi \\varepsilon \\rho_{\\rm F}\n\\hat\\tau_0\/\\sqrt{\\Delta^2 - \\varepsilon^2}$ (with the Fermi density of states $\\rho_{\\rm F}$ and the henceforth \nomitted trivial factor $\\hat\\sigma_0$), the T-matrix explicit form is:\n\\begin{equation}\n \\hat T = \\frac{V}{1 + v^2}\\frac{v\\varepsilon\\sqrt{\\Delta^2 - \\varepsilon^2}\\hat\\tau_0 - \\left(\\Delta^2 - \\varepsilon^2\\right)\n  \\hat\\tau_3}{\\varepsilon^2 - \\varepsilon_0^2}.\n   \\label{eq8}\n     \\end{equation} \nwhere $\\varepsilon_0 = \\Delta\/\\sqrt{1 + v^2}$ defines the in-gap impurity levels \\cite{tsai} through the \ndimensionless impurity perturbation parameter $v = \\pi\\rho_{\\rm F}V$. Inside the gap, the \nT-matrix, Eq. \\ref{eq8},  is a real function which can be approximated near the impurity \nlevels $\\pm\\varepsilon_0$ as: $\\hat T \\approx \\gamma^2 \\left(\\varepsilon  \\hat\\tau_0 - \\varepsilon_0 \\hat\\tau_3\\right)\/\\left(\\varepsilon^2 \n- \\varepsilon_0^2\\right)$, with the effective coupling constant  $\\gamma^2 = V\\varepsilon_0(v\\varepsilon_0\/\\Delta)^2$. In contrary, \noutside the gap it is dominated by its imaginary part: Im$\\hat T = (\\gamma^2\/v\\varepsilon_0) \\varepsilon \\sqrt{\\varepsilon^2 - \n\\Delta^2}\/\\left(\\varepsilon^2 - \\varepsilon_0^2\\right)$.\n\nThe next terms besides unity in the brackets of Eq. \\ref{eq7} describe the effects of indirect \ninteractions between impurities, with $\\hat B_{\\bf k} $ related to pairs and the omitted terms to \ngroups of three and more impurities. The series convergence defines the energy ranges of \nband-like states, delimited by the Mott mobility edges $\\varepsilon_c$ \\cite{mott}. Within the band-like \nenergy ranges, the self-energy matrix can be safely approximated by the T-matrix, $\\hat \\Sigma_{\\bf k} \n\\approx c \\hat T$, and the dispersion laws for corresponding bands at given quasimomentum \n${\\bf k}$ are defined from the $\\hat G_{\\bf k}$ denominator:\n\\begin{eqnarray}\n D_{\\bf k}(\\varepsilon) & = & \\det \\hat G_{\\bf k}^{-1}(\\varepsilon) = \\tilde d_{e,{\\bf k}}(\\varepsilon) \\tilde d_{h,{\\bf k}}(\\varepsilon) \\nonumber\\\\\n  & = & \\left(\\tilde\\varepsilon^2 - \\tilde\\xi_e^2 - \\Delta^2\\right)\\left(\\tilde\\varepsilon^2 - \\tilde\\xi_h^2 - \n   \\Delta^2\\right),\n    \\label{eq9}\n     \\end{eqnarray}\nwith the renormalized energy and momenta forms:\n\\begin{eqnarray}\n  \\tilde\\varepsilon & = & \\varepsilon\\left(1 - \\frac{c V v}{1 + v^2}\\frac{\\sqrt{\\Delta^2 - \\varepsilon^2}}{\\varepsilon^2 -  \\varepsilon_0^2}\\right),\\nonumber\\\\\n&& \\tilde\\xi_j = \\xi_j -  \\frac{c V}{1 + v^2} \\frac{\\Delta^2 - \\varepsilon^2}{\\varepsilon^2 - \\varepsilon_0^2}.\\nonumber\n    \\end{eqnarray}\nThe roots of the dispersion equation Re $D_{\\bf k}(\\varepsilon) = 0$ define up to 8 subbands:  4 of them \nwith energies near the roots of the non-perturbed denominators $d_{j,{\\bf k}}$ in the $e$- and \n$h$-segments can be called \"principal\" or $pr$-bands, they are similar to quasiparticles in \nthe pure crystal; and other 4, \"impurity\" or $imp$-bands, with energies near $\\pm \\varepsilon_0$ in the \nsame segments are only specific for disordered systems. The dispersion law for $p$-bands\nis presented in the $\\xi$-scale as:\n\\begin{equation}\n  \\varepsilon_{pr}(\\xi) \\approx \\sqrt{\\xi^2  + \\Delta^2},\n    \\label{eq10}\n      \\end{equation} \nand it only differs from the non-perturbed one by the finite linewidth $\\Gamma(\\varepsilon) \\approx c{\\rm Im}\\hat T$, \nso that the validity range of Eq. \\ref{eq10} defined from the known Ioffe-Regel-Mott criterion, \n$\\xi d\\varepsilon_b\/d\\xi \\gtrsim \\Gamma(\\varepsilon_b(\\xi))$ \\cite{IR}, \\cite{mott} as $\\xi \\gtrsim c\/(\\pi\\rho_{\\rm F})$. This defines \nthe mobility edge in closeness to the gap edge,\n\\begin{equation}\n  \\varepsilon_c - \\Delta \\sim c^2\/c_0^{4\/3}\\Delta.\n    \\label{eq11}\n      \\end{equation} \nHere $c_0 = (\\pi\\rho_{\\rm F}\\varepsilon_0)^{3\/2}\/\\left(a k_{\\rm F}\\right)\\sqrt{2v\/(1 + v^2)}$  is the characteristic \nimpurity concentration such that the impurity bands emerge just at $c > c_0$ \\cite{pog}. Their dispersion \n(in $\\xi$) for the exemplar case of positive energies and $e$-segment is approximated as:\n\\begin{equation}\n \\varepsilon_{imp}(\\xi) \\approx \\varepsilon_0 + c\\gamma^2\\frac{\\xi  - \\varepsilon_0}{\\xi^2 + \\xi_0^2}.\n   \\label{eq12}\n     \\end{equation} \n   \n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=8cm]{Fig1.eps}\n\\caption{Dispersion laws in the modified quasiparticle spectrum of a SC ferropnictide with \nimpurities. The impurity perturbation parameters were chosen as: $v = 0.5$, $c_0 = 1.3\\cdot \n10^{-3}$, $c_1 =  1.7\\cdot 10^{-2}$, $c = 4\\cdot 10^{-3}$. For compactness, the plot superimposes \nthe blue lines for the in-gap impurity subbands near the electron-like pockets of the Fermi surface \nand red lines for those near the hole-like pockets.}\n\\label{fig1}\n\\end{center}\n\\end{figure}\n\nThe formal upper limit energy by Eq. \\ref{eq12}, $\\varepsilon_+ = \\varepsilon_0 + c\\gamma^2\/[2(\\Delta \n+ \\varepsilon_0)]$, is attained at $\\xi = \\xi_+ = \\varepsilon_0 +  \\Delta$ and the lower limit $\\varepsilon_- \n= \\varepsilon_0 - c\\gamma^2\/[2(\\Delta - \\varepsilon_0)]$ at $\\xi_- = \\varepsilon_0 - \\Delta$. But in fact, \nthis dispersion law is only valid until the related mobility edges $\\varepsilon_{c,\\pm}$ whose onset \nnear the $i$-band edges is due to the higher terms in the group expansion, Eq. \\ref{eq7}, and amounts \nto:\n\\begin{eqnarray}\n\\varepsilon_+ -  \\varepsilon_{c,+} & \\sim & \\left(\\varepsilon_{max} -  \\varepsilon_0\\right)\\left(\\frac{c_0}c\\right)^4,\\nonumber\\\\\n  \\varepsilon_{c,-} - \\varepsilon_-  & \\sim & \\left(\\varepsilon_0 -  \\varepsilon_{min}\\right)\\left(\\frac{c_0}c\\right)^4.\n \\label{eq13}\n  \\end{eqnarray}\nThese limitations restrict $\\xi$ to beyond some vicinities of the extremal points: $|\\xi - \\xi_\\pm| \\gtrsim \n\\xi_\\pm\\left(c_0\/c\\right)^2$ (narrow enough at $c \\gg c_0$). Another limitation is that $\\xi$ not be too \nfar from these points: $|\\xi - \\xi_\\pm| \\lesssim \\xi_\\pm(c\/c_0)^4$. A symmetric replica of Eq. \\ref{eq12} \nnear $-\\varepsilon_0$ at the $e$-segment is the impurity subband with the dispersion law $-\\varepsilon_i(\\xi)$. Yet two \nmore impurity subbands near the $h$-segment are described in the unified $\\xi$ frame by the inverted \ndispersion laws $\\pm\\varepsilon_{imp}(-\\xi)$. The overall composition of band-like states in this frame is shown in Fig. \n\\ref{fig1}. It is also important to notice that the above described in-gap impurity band structure is only \njustified until it is narrow enough compared to the SC gap $\\Delta$ itself. From Eq. \\ref{eq12}, this requires \nthat the impurity concentration stays well below the upper critical value\n\\[c_1 = \\pi\\rho_{\\rm F}\\Delta\\sqrt{1 + v^2}.\\]\nthat can amount about few percents. In what follows, the condition $c \\ll c_1$ is presumed.\n\nAt least, for $c < c_0$, all the in-gap states are localized and more adequately described by \nan alternative, the so called non-renormalized group expansion of $\\hat G_{\\bf k}$ (though this \ncase is beyond the scope of the present study) while the principal bands are still defined by \nEqs. \\ref{eq10},\\ref{eq11}.\n\nIn-gap impurity states, either localized and band-like,  can produce notable resonance effects \non various thermodynamical properties of disordered superconductors, as transition critical \ntemperature, London penetration length, electronic specific heat, etc. \\cite{pog}. But besides that, \nother effects, only specific for new quasiparticle bands, can be expected on kinetic properties of \nthe disordered material, while the localized impurity states should have practically no effect on \nthem. Such phenomena can be naturally described in terms of the above indicated GF matrices as \nseen in what follows.\n\n\\section{Kubo-Greenwood formalism for multiband superconductor}\n\nThe relevant kinetic coefficients for electronic processes in the considered disordered superconductor \nfollow from the general Kubo-Greenwood formulation \\cite{kubo}, adapted here to the specific multiband \nstructure of Green function matrices. Thus, one of the basic transport characteristics, the (frequency and \ntemperature dependent) electrical conductivity is expressed in this approach as:\n\\begin{eqnarray}\n\\sigma(\\omega,T)  & = & \\frac{e^2 }{\\pi} \\int d\\varepsilon\\frac{f(\\varepsilon) - f(\\varepsilon')}\\omega \\int d{\\bf k} v_x({\\bf k},\\varepsilon) \n v_x({\\bf k},\\varepsilon') \\nonumber\\\\\n  & \\times & {\\rm Tr}\\left[{\\rm Im}\\hat G_{{\\bf k}}(\\varepsilon){\\rm Im}\\hat G_{{\\bf k}}(\\varepsilon')\\right],\n   \\label{eq14}\n    \\end{eqnarray}\nfor $\\varepsilon' = \\varepsilon - \\hbar\\omega$ and the electric field applied along the $x$-axis. Besides the common Fermi \noccupation function $f(\\varepsilon) = ({\\rm e}^{\\beta\\varepsilon} + 1)^{-1}$ with the inverse temperature $\\beta = 1\/k_{\\rm B}T$, the \nabove formula involves the generalized velocity function:\n\\begin{equation}\n {\\bf v}({\\bf k},\\varepsilon) = \\left(\\hbar\\frac{\\partial {\\rm Re}D_{\\bf k}(\\varepsilon)}{\\partial \\varepsilon}\\right)^{-1}{\\bf \\nabla}_{\\bf k} \n  {\\rm Re}D_{\\bf k}(\\varepsilon).\n   \\label{eq15}\n    \\end{equation} \nThis function is defined in the whole $\\xi,\\varepsilon$ plane in a way to coincide with the physical quasiparticle \nvelocities for each particular band, Eqs. \\ref{eq9}, \\ref{eq12}, along the corresponding dispersion laws: \n${\\bf v}({\\bf k},\\varepsilon_j({\\bf k})) = \\hbar^{-1}{\\bf \\nabla}_{\\bf k} \\varepsilon_j({\\bf k}) = v_{j,{\\bf k}}$, $j = pr,imp$. The conductivity resulting \nfrom Eq. \\ref{eq13} can be then used for calculation of optical reflectivity.\n\nOther relevant quantities are the static (but temperature dependent) transport coefficients, as the heat \nconductivity:\n\\begin{eqnarray}\n\\kappa(T) & = & \\frac{\\hbar}{\\pi } \\int d\\varepsilon \\frac{\\partial f(\\varepsilon)}{\\partial \\varepsilon} \\varepsilon^2 \\int d{\\bf k} \n \\left[v_x({\\bf k},\\varepsilon)\\right]^2 \\nonumber\\\\\n& \\times & {\\rm Tr}\\left[{\\rm Im}\\hat G_{{\\bf k}}(\\varepsilon)\\right]^2,\n \\label{eq16}\n  \\end{eqnarray}\nand the thermoelectric coefficients associated with the static electrical conductivity $\\sigma(T) \\equiv \\sigma(0,T)$ \n\\cite{note}, the Peltier coefficient:\n\\begin{eqnarray}\n \\Pi(T) & = & \\frac{\\hbar e}{\\pi \\sigma(0,T)} \\int d\\varepsilon \\frac{\\partial f(\\varepsilon)}{\\partial \\varepsilon} \\varepsilon \\int \n  d{\\bf k} \\left[v_x({\\bf k},\\varepsilon)\\right]^2 \\nonumber\\\\\n& \\times & {\\rm Tr}\\left[{\\rm Im}\\hat G_{{\\bf k}}(\\varepsilon)\\right]^2,\n \\label{eq17}\n  \\end{eqnarray}\nand the Seebeck coefficient $S(T) = \\Pi(T)\/T$.  All these transport characteristics, though being relatively \nmore complicated from the theoretical   point of view than the purely thermodynamical quantities as, e.g., \nspecific heat or London penetration length \\cite{pog}, permit an easier and more reliable experimental \nverification and so could be of higher interest for practical applications of the considered impurity effects \nin the multiband superconductors.\n\nNext, we consider the particular calculation algorithms for the expressions, Eqs.  \\ref{eq14}, \n\\ref{eq16},  \\ref{eq17}, beginning from the more involved case of dynamical conductivity, Eq.  \n\\ref{eq14}, and then reducing it to simpler static quantities, Eqs.  \\ref{eq16},  \\ref{eq17}.\n\n\n\\section{Optical conductivity}\n\nThe integral in Eq. \\ref{eq14} is dominated by the contributions from $\\delta$-like peaks of the \n${\\rm Im}\\hat G_{{\\bf k}}(\\varepsilon)$ and ${\\rm Im}\\hat G_{{\\bf k}}(\\varepsilon ')$ matrix elements. \nThese peaks arise from the above dispersion laws, Eqs. \\ref{eq9}, \\ref{eq11}, thus restricting the \nenergy integration to the band-like ranges: $|\\varepsilon| > \\varepsilon_c$ for the $pr$-bands and \n$\\varepsilon_{c,-} < |\\varepsilon| < \\varepsilon_{c,+}$ for the $imp$-bands. Regarding the occupation \nnumbers $f(\\varepsilon)$ and $ f(\\varepsilon')$ at reasonably low temperatures $k_{\\rm B}T \\ll \n\\Delta,\\varepsilon_0$, the most effective contributions correspond to positive $\\varepsilon$ values, \neither from $pr$- or $imp$-bands, and to negative $\\varepsilon'$ values from their negative counterparts, \n$pr'$ or $imp'$. There are three general kinds of such contributions: i) $pr-pr'$, due to transitions between \nthe principal bands, similar to those in optical conductivity by the pure crystal (but with a slightly shifted \nfrequency threshold: $\\hbar\\omega \\geq 2\\varepsilon_c$), ii) $pr-imp'$ (or $imp-pr'$), due to combined \ntransitions between the principal and impurity bands within the frequency range $\\hbar\\omega \\geq \n\\varepsilon_c + \\varepsilon_{c,-}$, and iii) $imp-imp'$, due to transitions between the impurity bands within \na narrow frequency range of $2\\varepsilon_{c,-} < \\hbar\\omega < 2\\varepsilon_{c,+}$. The frequency-momentum \nrelations for these processes and corresponding peaks are displayed in Fig. \\ref{fig2}. The resulting optical \nconductivity reads $\\sigma(\\omega,T) = \\sum_\\nu \\sigma_\\nu(\\omega,T)$ with $\\nu = pr-pr',\\,imp-imp'$, and \n$imp-pr'$.\n\nFor practical calculation of each contribution, the relevant matrix Im$\\hat G_{\\bf k}(\\varepsilon)$ (within \nthe band-like energy ranges) can be presented as Im$\\hat G_{\\bf k}(\\varepsilon) = \\hat N(\\varepsilon,\\xi) {\\rm Im}\n\\left[D_{\\bf k}(\\varepsilon)^{-1}\\right]$ where the numerator matrix:\n\\begin{equation}\n \\hat N(\\varepsilon,\\xi) = {\\rm Re}\\left(\\tilde\\varepsilon + \\tilde\\xi \\hat \\tau_3 +  \\Delta \\hat \\tau_1\\right),\n   \\label{eq18}\n     \\end{equation} \nis a smooth enough function while the above referred peaks result from zeros of Re$D_{\\bf k}(\\varepsilon)$. \nNow, the quasimomentum integration in Eq. \\ref{eq14} under the above chosen symmetry of \nFermi segments spells as $\\int d{\\bf k} = 2(h v_{\\rm F})^{-1}\\int d\\varphi \\int d\\xi$ where the factor 2 \naccounts for identical contributions from $e$- and $h$-segments. The azimuthal integration \ncontributes by the factor of $\\pi$ (from $x$-projections of velocities) and the most important radial \nintegration is readily done after expanding its integrand in particular pole terms:\n\\begin{eqnarray}\n  v(\\xi,\\varepsilon)v(\\xi,\\varepsilon') & {\\rm Tr} & \\left[{\\rm Im}\\hat G(\\xi,\\varepsilon){\\rm Im}\\hat G(\\xi,\\varepsilon')\\right] \n   \\nonumber\\\\\n& = & \\sum_\\alpha A_\\alpha(\\varepsilon,\\varepsilon') \\delta\\left(\\xi - \\xi_\\alpha\\right),\n \\label{eq19}\n  \\end{eqnarray}\n where $v(\\xi,\\varepsilon) = |{\\bf v}({\\bf k},\\varepsilon)|$ and $\\hat G(\\xi,\\varepsilon') \\equiv \\hat G_{\\bf k}(\\varepsilon')$ \n define the  respective  residues:\n \\begin{equation}\n   A_\\alpha(\\varepsilon,\\varepsilon') = \\pi v_\\alpha v'_\\alpha \\frac{\\tilde\\varepsilon\\tilde\\varepsilon' + \\tilde\\xi\\tilde\\xi' + \n     \\Delta^2}{\\prod_{\\beta \\neq \\alpha}\\left(\\xi_\\alpha - \\xi_\\beta\\right)}.\n       \\label{eq20}\n         \\end{equation} \n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=8cm]{Fig2.eps}\n\\caption{Configuration of the poles $\\xi_j$ of GF's contributing to different types of optical conductivity \nprocesses over one part (electronic pocket) of the quasiparticles spectrum by Fig. \\ref{fig1}.}\n\\label{fig2}\n\\end{center}\n\\end{figure}\n\nHere $v_\\alpha \\equiv v\\left(\\varepsilon,\\xi_\\alpha\\right)$, $v'_\\alpha \\equiv v\\left(\\varepsilon',\\xi_\\alpha\\right)$, and \nthe indices $\\alpha,\\beta$ run over all the poles  of the two Green  functions.  As follows from Eqs.  \\ref{eq10}, \\ref{eq12} \nand seen in Fig. \\ref{eq2}, there can be two such poles of $\\hat G(\\xi,\\varepsilon)$ related to band-like states with \npositive $\\varepsilon$ and respective quasi-momentum values denoted as $\\xi_{1,2}(\\varepsilon)$. For energies \nwithin the $pr$-band, $\\varepsilon > \\varepsilon_c$, they are symmetrical: \n\\begin{equation}\n  \\xi_{1,2}(\\varepsilon) \\approx \\pm  \\sqrt{\\varepsilon^2 - \\Delta^2},\n    \\label{eq21}\n      \\end{equation} \nwhile within the $imp$-band at $\\varepsilon_{c,-} < \\varepsilon < \\varepsilon_{c,+}$, their positions are asymmetrical:\n\\begin{equation}\n  \\xi_{1,2}(\\varepsilon) \\approx \\frac{c\\gamma^2 \\mp 2\\varepsilon_0\\sqrt{\\left(\\varepsilon_+ - \\varepsilon\\right)\n    \\left(\\varepsilon - \\varepsilon_-\\right)}}{2\\left(\\varepsilon - \\varepsilon_0\\right)}.\n       \\label{eq22}\n         \\end{equation} \nNotice also that, within the $imp$-band, there is a narrow vicinity of $\\varepsilon_0$ of $\\sim c_0^{1\/3} (c_0\/c)^3 \\varepsilon_0$ \nwidth where only the $\\xi_1$ pole by Eq. \\ref{eq22} is meaningful and the other contradicts to the IRM criterion (so that there is \nno band-like states with that formal $\\xi_2$ values in this energy range).  Analogous poles of $\\hat G(\\xi,\\varepsilon')$ at negative \n$\\varepsilon'$ are referred to as $\\xi_{3,4}(\\varepsilon')$ in what follows. Taking into account a non-zero Im$D_{\\bf k}(\\varepsilon)$ \n(for the $imp$-band, it is due to the non-trivial terms in the group expansion, Eq. \\ref{eq7}), each $\\alpha$th pole becomes a \n$\\delta$-like peak with an effective linewidth $\\Gamma_\\alpha$ but this value turns to be essential (and will be specified) only at \ncalculation of static coefficients like Eqs. \\ref{eq16}, \\ref{eq17}.\n\nSince four peaks in Eq. \\ref{eq19} for optical conductivity are typically well separated, the $\\xi$-integration \nis trivially done considering them true $\\delta$-functions, then the particular terms in $\\sigma(\\omega,T)$ follow as the \nenergy integrals:\n\\begin{equation}\n  \\sigma_\\nu(\\omega,T) = 2e^2 \\int_{\\varepsilon_{\\nu,-}}^{\\varepsilon_{\\nu,+}} d\\varepsilon\\frac{f(\\varepsilon) - f(\\varepsilon')}\n    \\omega \\sum_{\\alpha=1}^4 A_\\alpha(\\varepsilon,\\varepsilon'),\n      \\label{eq23}\n        \\end{equation} \nwhere $\\nu$ takes the values $pr-pr'$, $imp-pr'$, or $imp-imp'$ and the limits $\\varepsilon_{\\nu,\\pm}$ should assure that both \n$\\varepsilon$ and $\\varepsilon'$ are kept within the respective band-like energy ranges. \n\nThus, in the $pr-pr'$ term, the symmetry of the poles $\\xi_{1,2}(\\varepsilon)$ and $\\xi_{3,4}(\\varepsilon')$ by Eq. \\ref{eq21} and the\nsymmetry of $pr$- and $pr'$-bands themselves defines their equal contributions, then using simplicity of the generalized velocity \nfunction $v(\\xi,\\varepsilon) = \\xi\/\\varepsilon$ and  the non-renormalized energy and momentum variables, $\\tilde\\varepsilon \\to \n\\varepsilon$, $\\tilde\\xi \\to \\xi$, the energy integration between the limits $\\varepsilon_{pr-pr',-} = \\varepsilon_c$ and \n$\\varepsilon_{pr-pr',+} =  \\hbar\\omega - \\varepsilon_c$ provides its explicit analytic form as $\\sigma_{pr-pr'}(\\omega,T) = \n\\sigma_{pr-pr'}(\\omega,0) - \\sigma_{pr-pr',T}(\\omega)$. Here the zero-temperature limit value is:\n\\begin{widetext}\n\\begin{eqnarray}\n  \\sigma_{pr-pr'}(\\omega,0) & \\approx & \\sigma_0\\frac{2\\omega_c}{\\omega^2} \\left\\{\\sqrt{4\\omega^2 - \\omega_c^2} \\ln\\left[ 2 \n    \\frac{\\omega(2\\omega - \\omega_c) +  \\sqrt{\\omega(\\omega - \\omega_c)    (4\\omega^2 - \\omega_c^2)}}{\\omega_c^2} - 1\\right] \n      \\right.\\nonumber\\\\\n& + &  \\left. 2\\omega \\ln\\left[2\\frac{\\omega - \\sqrt{\\omega(\\omega - \\omega_c)}}{\\omega_c} - 1\\right] - 2\\sqrt{\\omega(\\omega - \n    \\omega_c)} \\right\\},\n        \\label{eq24}\n          \\end{eqnarray}\n\\end{widetext}\nwith the characteristic scale $\\sigma_0 = e^2\/\\Delta^2$ and simple asymptotics: \n\\begin{eqnarray}\n  \\sigma_{pr-pr'}(\\omega,0) & \\approx & (2\/3)\\sigma_0(\\omega \/\\omega_c -1 )^{3\/2}, \\quad \\omega - \\omega_c \\ll \\omega_c,\\nonumber\\\\\n \\sigma_{pr-pr'}(\\omega,0) & \\approx & \\sigma_0(32\\omega_c\/\\omega)\\ln(2\\omega\/\\omega_c), \\quad\\quad\\quad \\omega \\gg \n   \\omega_c,\\nonumber\n     \\end{eqnarray} \nwith respect to the threshold frequency $\\omega_c = 2\\varepsilon_c\/\\hbar$, reaching the maximum value $\\approx 1.19\\sigma_0$ \nat $\\omega \\approx 2.12\\omega_c$ as seen in Fig. 3. The (small) finite-temperature correction to the above value:\n\\begin{widetext}\n\\begin{eqnarray}\n \\sigma_{pr-pr',T}(\\omega) \\approx  \\sigma_0\\frac{2\\omega_c^2{\\rm e}^{-\\beta\\Delta}} {\\beta\\hbar(\\omega - \\omega_c)\\omega\\sqrt\\Delta}\n   &&\\left[\\frac{\\sqrt{\\hbar\\omega}} \\Delta \\left(1 - \\frac{F(\\sqrt{\\beta\\hbar(\\omega - \\omega_c)})} {\\sqrt{\\beta\\hbar(\\omega - \\omega_c)}}\\right) \n     \\right.\\nonumber\\\\\n && \\qquad\\qquad +  \\left.\\frac{\\sqrt{2\\Delta}}{\\hbar\\omega - \\Delta}\\left(\\frac{\\sqrt \\pi}2 \\frac{{\\rm erf} (\\sqrt{\\beta\\hbar(\\omega - \\omega_c)})}\n    {\\sqrt{\\beta\\hbar(\\omega - \\omega_c)}} - {\\rm e}^{-\\beta\\hbar(\\omega - \\omega_c)}\\right)\\right],\n       \\label{eq25}\n         \\end{eqnarray}\n\\end{widetext}\ninvolves the Dawson function $F(z) = \\sqrt\\pi {\\rm e}^{-z^2} {\\rm erf}(iz)\/(2i)$ and the error function ${\\rm erf}(z)$ \\cite{abst}. \n\nCalculation of the $imp-pr'$-term is more complicated since asymmetry of the $imp$-band poles $\\xi_{1,2}(\\varepsilon)$ by Eq. \n\\ref{eq22} and their non-equivalence to the symmetric poles $\\xi_{3,4}(\\varepsilon')$ of the $pr'$-band analogous to Eq. \n\\ref{eq21}. More complicated expressions also define the generalized velocity function within the $imp$-band range: \n\\begin{equation}\n  \\hbar v(\\xi,\\varepsilon) = \\frac{c\\gamma^2 - \\xi(\\varepsilon - \\varepsilon_0)}{\\varepsilon(\\varepsilon - \\varepsilon_0 - c\\gamma^2\/\n   \\varepsilon_0)},\n    \\label{eq26}\n      \\end{equation} \nand the energy integration limits: $\\varepsilon_{imp-pr',-} = \\varepsilon_{c,-}$ and $\\varepsilon_{imp-pr',+} = \\min[\\varepsilon_{c,+},\n\\hbar\\omega - \\varepsilon_c]$. Then the function $\\sigma_{imp-pr'}(\\omega,T)$ follows from a numerical integration in Eq. \\ref{eq23} and, as \nseen in  Fig. 3, it has a lower threshold frequency $\\omega_c' = \\varepsilon_c + \\varepsilon_{c,-}$ than the $pr-pr'$-term. Above this threshold, \nit starts to grow linearly as $\\sim (\\omega\/\\omega_c'-1)c^{5\/2}c_0^{-5\/3}\\sigma_0$ and, for the impurity concentrations within the \"safety range\", \n$c \\ll c_1 \\sim c_0^{2\/3}$, becomes completely dominated by the $pr-pr'$-function, Eq. \\ref{eq24} above its threshold $\\omega_c$.\n\nFinally, the $imp-imp'$-term is obtained with a similar numerical routine on Eq. \\ref{eq23}, using Eq. \\ref{eq22} either for \nthe poles $\\xi_{1,2}(\\varepsilon)$ by the $imp$-band and for the $\\xi_{3,4}(\\varepsilon')$ by the $imp'$-band and Eq. \\ref{eq25} for respective \ngeneralized velocities while the energy integration limits in this case are $\\varepsilon_{imp-imp',-} = \\varepsilon_{c,-}$ and $\\varepsilon_{imp-imp',+} = \n\\min[\\varepsilon_{c,+}, \\hbar\\omega - \\varepsilon_{c.-}]$. The resulting function $\\sigma_{imp-imp'}(\\omega,T)$ occupies the narrow frequency band from \n$\\omega_{imp-imp',-} = 2 \\varepsilon_{c,-}\/\\hbar$ to $\\omega_{imp-imp',+} = 2\\varepsilon_{c,+}\/\\hbar$ (Fig. \\ref{fig3}) and its asymptotics near these \nthresholds and in the zero-temperature limit are obtained analytically as:\n\\begin{equation}\n   \\sigma_{imp-imp'}(\\omega,0) \\approx \\sigma_0\\frac{16c^{7\/2}\\gamma^7}{3\\sqrt 2 \\xi_-^7}\\left(\\frac{\\omega - \\omega_{-}}{\\omega_{-}}\\right)^{3\/2},\n    \\label{eq27}\n      \\end{equation} \nat $0 < \\omega - \\omega_{-} \\ll \\omega_{-}$ and a similar formula for $0 < \\omega_+ - \\omega \\ll \\omega_+$ only differs from it by the change: \n$\\xi_-\\to\\xi_+$ and $\\omega_-\\to\\omega_+$. \n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=8cm]{Fig3.eps}\n\\caption{General picture of the optical conductivity showing three types of contributions for the impurity perturbation parameters \nas chosen in Fig. \\ref{fig1}.}\n\\label{fig3}\n\\end{center}\n\\end{figure}\n\nThen the maximum contribution by the $imp-imp'$-term is estimated by extrapolation of the above asymptotics to the \ncenter of the impurity band: $|\\omega - \\omega_\\pm| \\sim |\\omega_0 - \\omega_\\pm|$, resulting in: $\\sigma_{imp-imp',max} \n\\sim \\sigma_0 c^5 c_0^{-10\/3} (\\xi_+\/\\xi_-)^{7\/2}$. This estimate shows that the narrow $imp-imp'$-peak of optical conductivity \naround $\\omega \\approx 2\\epsilon_0\/\\hbar$ can, unlike the \"combined\" $imp-pr'$-term, can become as intense or even more \nthan the maximum of \"principal\" $pr-pr'$ intensity, Eq. \\ref{eq24}, if the small factor $\\sim (c\/c_1)^5$ be overweighted by the \nnext factor $(\\xi_+\/\\xi_-)^{7\/2}$. The latter is only possible if the impurity perturbation is {\\it weak} enough: $v \\ll 1$. Then the \nratio $\\xi_+\/\\xi_-$ turns $\\approx (2\/v)^2 \\gg 1$ and can really overweight the concentration factor if the impurity concentration \n$c$ reaches $\\sim c_1 (v\/2)^{7\/5} \\ll c_1$, that is quite realistic within the \"safety\" range $c \\ll 1$. The overall picture of optical \nconductivity for an example of weakly coupled, $v = 0.5$, impurities at  high enough concentration $c = 4c_0$ is \nshown in Fig. \\ref{fig3}. The expressed effect of \"giant\" optical conductivity by the in-gap impurity excitations could be \ncompared with the well known Rashba enhancement of optical luminescence by impurity levels at closeness to \nthe edge of excitonic band \\cite{rashba} or with the huge impurity spin resonances in magnetic crystals \\cite{ilp}, \nbut with a distinction that it appears here in a two-particle process instead of the above mentioned single-particle \nones.\n\n\\section{Static kinetic coefficients}\n\nNow we can pass to the relatively simpler calculation of the kinetic coefficients in the static limit of $\\omega \\to 0$. \nTo begin with, consider the heat conductivity, Eq. \\ref{eq16}, where the momentum integration at coincidence \nof the above mentioned poles $\\xi_{1.3}$ and $\\xi_{2.4}$ is readily done using the general convolution formula:\n\\begin{equation}\n  \\int L_{\\Gamma_j}\\left(\\xi - \\xi_j\\right) L_{\\Gamma_k'}\\left(\\xi- \\xi_{k}'\\right)d\\xi = L_{\\Gamma_j + \\Gamma_k'}\\left(\\xi_j - \\xi_k'\\right),\n    \\label{eq28}\n      \\end{equation} \nfor two Lorentzian fuctions $L_\\Gamma(\\xi) = \\Gamma\/(\\xi^2 + \\Gamma^2)$, and in the limit of $\\xi_i = \\xi_k'$ and $\\Gamma_j = \\Gamma_k'$ \nobtaining simply $(2\\Gamma_j)^{-1}$, a \"combined lifetime\". This immediately leads to a Drude-like formula for heat \nconductivity as a sum of principal and impurity terms, $\\kappa(T) = \\kappa_{pr}(T) + \\kappa_{imp}(T)$, each of them given by:\n\\begin{eqnarray}\n  \\kappa_{pr}(T) & = &  \\frac{\\hbar(1 + v^2)}{\\pi cVv} \\int_{\\varepsilon_c}^{\\infty} d\\varepsilon \\frac{\\partial f(\\varepsilon)}{\\partial \\varepsilon} \\frac{\\varepsilon\\left(\\varepsilon^2 \n    - \\varepsilon_0^2\\right)}{\\sqrt{\\varepsilon^2 - \\Delta^2}}\\nonumber\\\\\n& \\approx & \\frac{\\hbar\\rho_{\\rm F}\\Delta^2}{c} \\sqrt{ \\frac{\\pi\\beta\\Delta}{2}}\\exp (-\\beta\\Delta),\n    \\label{eq29}\n      \\end{eqnarray}\n and:\n \\begin{eqnarray}\n  \\kappa_{imp}(T) & \\approx &  \\frac{\\hbar}{\\pi\\left(\\varepsilon_{c,+} - \\varepsilon_{c,-}\\right)} \\left(\\frac{c}{c_0}\\right)^4\\int_{\\varepsilon_{c,-}}^{\\varepsilon_{c,+}} \n   d\\varepsilon \\frac{\\partial f(\\varepsilon)}{\\partial \\varepsilon}\\varepsilon^2\\nonumber\\\\\n& \\approx & \\frac{\\hbar}{\\pi} \\left(\\frac{c}{c_0}\\right)^4 \\beta\\varepsilon_0^2\\exp (-\\beta\\varepsilon_0).\n    \\label{eq29}\n      \\end{eqnarray}\nThen the comparison of Eqs. \\ref{eq28} and \\ref{eq29} shows that the impurity contribution to the heat conductance \n$\\kappa_{imp}$ for impurity concentrations $c$ above the critical value $c_0$ turns to dominate over the principal contribution \n$\\kappa_{pr}$ at all the temperatures (of course, below the critical transition temperature). Such strong impurity effect is \ncombined from enhanced thermal occupation of impurity states and from their growing lifetime as $\\sim c^3$ against \nthe decreasing as $\\sim 1\/c$ lifetime in the principal band. \n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=8cm]{Fig4.eps}\n\\caption{Logarithmic plots for two contributions to the heat conductivity shows domination of the impurity term at all the \ntemperatures where SC itself exists. The same impurity perturbation and concentration parameters are used as in Fig. \\ref{fig1}}\n\\label{fig4}\n\\end{center}\n\\end{figure}\n\nSimilar strong impurity effects should also follow for the static electric conductivity $\\sigma(0,T)$ (see \\cite{note}) and for the \nthermoelectric Peltier and Seebeck coefficients, Eq. \\ref{eq17}. All of them can be considered as fully due to the \ncorresponding impurity contributions and the temperature dependencies of thermoelectric coefficients should be \nnon-exponential: $\\Pi(T) \\approx \\Pi(0) = $ const, and $S(T) \\approx \\Pi(0)\/T$, alike the non-perturbed case but at much \nhigher level. Finally, it is important to underline that the above predictions are only for impurity concentrations above \nthe critical value, $c \\gtrsim c_0$, while the system transport properties should stay almost non-affected by impurities \nbelow this concentration, $c < c_0$. Fig. \\ref{fig4} demonstrates these differences between temperature dependencies of static \nconductivities and of thermoelectric coefficients for low and high concentrations of impurities at the choice of perturbation \nparameter as $v = 1$. Such drastic changes of transport behavior are of interest for experimental verification in properly \nprepared samples of SC ferropnictides with controlled concentration of specific impurities. \n\n\n\\section{Conclusive remarks}\nIn conclusion, the essential modification of quasiparticle spectra in a SC ferropnictide with impurities of simplest \n(local and non-magnetic) perturbation type is expected, consisting in formation of localized in-gap impurity states \nand their development into specific narrow bands of impurity quasiparticles at impurity concentration above a certain \n(quite low) critical value $c_0$ and leading to a number of effects in the system observable properties. Besides the \npreviously discussed thermodynamical effects, expected to appear at all impurity concentrations, that is either due \nto localized or band-like impurity states, a special interest is seen in studying the impurity effects on electronic transport \nproperties of such systems, only affected by the impurity band-like states. It was shown above that the latter effects can \nbe very strongly pronounced, either for high-frequency transport and for static transport processes. In the first case, the \nimpurity effect is expected to most strongly reveal in a narrow peak of optical conductance at its closeness to the edge \nof conductance band in non-perturbed crystal, resembling the known resonance enhancement of impurity absorption \n(or emission) processes near the edge of main quasiparticle band in normal systems, here it would be possible if the \nimpurity perturbation be weak enough. The static transport coefficients at overcritical impurity concentrations are also \nexpected to be strongly enhanced compared to those in a non-perturbed system, including the thermoelectric Peltier \nand Seebeck coefficients. The experimental verifications of these predictions would be of evident interest, since they \ncan open perspectives for important practical applications, e.g., in narrow-band microwave devices or advanced \nlow-temperature sensors, but this would impose rather hard requirements on the quality and composition of the necessary \nsamples, they should be extremely pure aside the extremely low (by common standards) and well controlled contents \nof specially chosen and uniformly distributed impurity centers within the SC iron-arsenic planes of a ferropnictide \ncompound. This situation can be compared to the requirements on doped semiconductor devices and hopefully should \nnot be a real problem for modern lab technologies.\n\n\\section{Acknowledgements}\nY.G.P. and M.C.S. acknowledge the support of this work through the Portuguese FCT project PTDC\/FIS\/101126\/2008.\nV.M.L. is grateful to the EU Research Programs for a partial support through the grant FP7-SIMTECH No 246937.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{\n#1}}\n\\renewcommand{\\arabic{section}.\\arabic{equation}}{\\thesection.\\arabic{equation}}\n\\begin{document}\n\\newcommand{\\ft}[2]{{\\textstyle\\frac{#1}{#2}}}\n\\newcommand{{\\hspace*{\\fill}\\rule{2mm}{2mm}\\linebreak}}{{\\hspace*{\\fill}\\rule{2mm}{2mm}\\linebreak}}\n\\newcommand{\\begin{equation}}{\\begin{equation}}\n\\newcommand{\\end{equation}}{\\end{equation}}\n\\newcommand{\\begin{eqnarray}}{\\begin{eqnarray}}\n\\newcommand{\\end{eqnarray}}{\\end{eqnarray}}\n\\newtheorem{definizione}{Definition}[section]\n\\newcommand{\\begin{definizione}}{\\begin{definizione}}\n\\newcommand{\\end{definizione}}{\\end{definizione}}\n\\newtheorem{teorema}{Theorem}[section]\n\\newcommand{\\begin{teorema}}{\\begin{teorema}}\n\\newcommand{\\end{teorema}}{\\end{teorema}}\n\\newtheorem{lemma}{Lemma}[section]\n\\newcommand{\\begin{lemma}}{\\begin{lemma}}\n\\newcommand{\\end{lemma}}{\\end{lemma}}\n\\newcommand{\\begin{array}}{\\begin{array}}\n\\newcommand{\\end{array}}{\\end{array}}\n\\newcommand{\\nonumber}{\\nonumber}\n\\newtheorem{corollario}{Corollary}[section]\n\\newcommand{\\begin{corollario}}{\\begin{corollario}}\n\\newcommand{\\end{corollario}}{\\end{corollario}}\n\\def\\twomat#1#2#3#4{\\left(\\begin{array}{cc}\n {#1}&{#2}\\\\ {#3}&{#4}\\\\\n\\end{array}\n\\right)}\n\\def\\twovec#1#2{\\left(\\begin{array}{c}\n{#1}\\\\ {#2}\\\\\n\\end{array}\n\\right)}\n\\def{\\bf Z}{{\\bf Z}}\n\\def{\\bf R}{{\\bf R}}\n\\def{\\bf C}{{\\bf C}}\n\\def{\\bf I}{{\\bf I}}\n\\def{\\bf h}{{\\bf h}}\n\\def{\\bf k}{{\\bf k}}\n\\def{\\bf g}{{\\bf g}}\n\\begin{titlepage}\n\n\\hfill\n\\vbox{\\hbox{CERN-TH\/99-53}\\hbox{hep-th\/990000}\\hbox{March, 1999}}\n\\vfill\n\\vskip 3cm\n\\begin{center}\n{\\LARGE {  On Central Charges and Hamiltonians\\\\\nfor 0-brane dynamics}$^*$}\\\\\n\\vskip 1.5cm\n  {\\bf\nR. D'Auria$^1$ S. Ferrara$^2$ and M.A. Lled\\'o$^1$\n } \\\\\n\\vskip 0.5cm\n{\\small\n$^1$ Dipartimento di Fisica, Politecnico di Torino,\\\\\n Corso Duca degli Abruzzi 24, I-10129 Torino.\\\\\nand Istituto Nazionale di Fisica Nucleare (INFN) \\\\ Sezione di Torino, Italy.\\\\\n\\vspace{6pt}\n$^2$ CERN Theoretical Division, CH 1211 Geneva 23, Switzerland.}\n\\end{center}\n\\vskip 4cm\n\n\n  {\\small\n\n\\begin{abstract}\nWe consider general properties of central charges of zero-branes and associated duality\n invariants, in view of their double role, on the bulk and on the world volume (quantum\n mechanical) theory.\n\nA detailed study of the BPS condition for the mass spectrum arising from toroidal compactifications\nis given for 1\/2, 1\/4 and 1\/8\n BPS states in any dimension. As a byproduct, we retrieve the U-duality invariant conditions\n on the charge (zero mode) spectrum and the orbit classification of BPS states preserving\n different fractions of supersymmetry.\n\nThe BPS condition for 0-branes in theories with 16 supersymmetries in\nany dimension is also  discussed.\n\\end{abstract}  }\n  \\vspace{2mm} \\vfill \\hrule width 3.cm\n{\\footnotesize\n $^*$ Supported in part by   EEC  under TMR contract\n ERBFMRX-CT96-0045,(LNF Frascati,\nPolitecnico di Torino) and by DOE grant\nDE-FGO3-91ER40662}\n \\end{titlepage}\n\\eject\n\\section{Introduction.}\n\nIn recent time, the role of duality symmetries of a dynamical theory\nencompassing quantum gravity has received increasing attention in several contexts.\n\nParticular examples where the duality takes an important role, especially in connection with\n non perturbative properties, is the AdS\/CFT correspondence \\cite{ma}, related to the horizon geometry\n of p-branes and their world-volume conformal field theory description.\n\nAnother example is the connection between M-theory compactified on a\ntorus  T$^d$ \\cite{bfss, dvv, egkr, bs, op}  and $(d+1)$\n Yang-Mills theory compactified on the dual torus $\\tilde{\\mbox{T}}^d$.\n\nMore closely related to the latter is the recent investigation of D-brane Born-Infeld\n actions and the role played by duality in explaining several properties of their\n Hamiltonian formulation and the corresponding energy spectrum of BPS\nstates \\cite{hvz}. In this\n framework it is believed that Born-Infeld non abelian gauge theories with non trivial R-R\n backgrounds, are naturally described by some generalization of gauge theories on non\n commutative tori \\cite{cds}.\n\nThe framework of non commutative geometry offers for instance,\n   a new interpretation of the\n T-duality group $O(d,d;{\\bf Z})$ of quantum mechanical systems obtained by compactifying the\n Born-Infeld action of D-branes on T$^d$. The latter occurs in type II string theory\ncompactified on\n T$^d$ \\cite{mz, bm, ks, do}.\n\nThese quantum mechanical systems have also been shown \\cite{hb}, at least for $d\\leq 4$,\n to exhibit the\n full extended U-duality symmetry\\footnote{In this paper U-duality will mean both\n the classical and quantum U-duality} E$_{d+1(d+1)}$, rather than the\nsmaller symmetry E$_{d(d)}$ present in matrix gauge theory on T$^d$,\nwhere it appears as an extension of the geometrical symmetry SL($d$) \\cite{op, egkr}.\n\nIn previous investigations, the central charge matrix $Z$  for 0-branes played a role, not\n only as central extension of supersymmetry algebra in theories with non trivial 0-brane\n background metric, but also as effective potential of the geodesic\naction of a one-dimensional Lagrangian system derived from the bulk\nEinstein-Maxwell Lagrangian, in presence of moduli fields $\\{\\phi\\}$ and\nquantized charges $q_A$ of zero-branes \\cite{gkk, fgk}.\n\nThe critical points of this potential were seen to determine the Bekenstein-Hawking\nentropy formula as the extremization of the Weinhold potential \\cite{fgk}.\n\\begin{equation}\nW={1\\over 2}\\mbox{Tr}(ZZ^\\dagger)\n\\end{equation}\nor equivalently of the BPS mass $m_{BPS}=|Z_h|$ where $|Z_h|$ is the\nhighest eigenvalue of $\\sqrt{ZZ^\\dagger}$ \\cite{fsf}.\n\nIn the world-volume description of 0-branes, the very same function $W$\nappears as Hamiltonian of the 0-brane quantum mechanics \\cite{hvz, hb, hv}\n\\begin{equation}\nH_{\\phi}(\\hat q)=\\sqrt{{1\\over N}\\mbox{Tr}(ZZ^\\dagger(\\phi,\\hat q))}, \\label{hamiltonian}\n\\end{equation}\nwhere the quantized charges are replaced by a set of Hamiltonian variables $\\hat q$,\n which belong to the same duality\nmultiplet as the quantized charges of the bulk supergravity theory\nin presence of zero-brane sources.\n\nThe appearence of the central charge in the 0-brane action in\narbitrary $D=4$ supergravity backgrounds has recently been shown to\noccur as a consequence of $\\kappa$-supersymmetry \\cite{bcfd}.\n\nIn this framework the energy spectrum of the Hamiltonian\n(\\ref{hamiltonian}) is given by the BPS mass formula of the effective\nsupergravity theory \\cite{hvz}\n\\begin{equation}\nm_{BPS}=|Z_h(\\phi, q_0)|,\n\\end{equation}\nwhere the hamiltonian variables $\\hat q$ are replaced by their zero mode part\n$q_0$ which eventually coincide with the same duality multiplet of the\nquantized charges of the bulk theory, but now with the interpretation of\n\"fluxes\" and \"momenta\"  of the world-volume hamiltonian description \\cite{op, egkr}.\n\nThese zero modes fill representations of the U-duality group $E_{d+1(d+1)}({\\bf Z})$\nfor systems with maximal supersymmetry and the BPS spectrum preserves\nsome fraction of supersymmetry depending on the particular orbit of the\ncharge vector state \\cite{fm, fg, lsp}.\n\nNote that the BPS energy $|Z_h(\\phi,q_0)|$ is not the same as replacing in\n$\\sqrt{ZZ^\\dagger(\\phi, \\hat q)}$ the zero mode $q_0$  of $q$, unless the states\nare 1\/2 BPS \\cite{hvz}, which, as we will see, can only occur if the charge duality\nmultiplet satisfies some duality invariant conditions. At the classical\nlevel, where the charges are continuous , this is equivalent to say that\nthe zero-mode part belongs to a particular orbit of the\ncharge vector representation of the duality group $G$.\n\nIt is the aim of the present investigation to derive general formulas\nof  the energy spectrum for any torus T$^d, \\; d=1,\\dots 6$ and\nprovide a new derivation of the different BPS conditions in terms of the\nU-duality invariant constraints, retrieving then the analysis of\nMaldacena and one of the authors \\cite{fm} as well as the classification of\nGunaydin and one of the authors \\cite{fg} and Lu, Pope and Stelle \\cite{lsp}.\nWe deal also with the case of 0-branes in theories in any dimension with 16\nsupersymmetries. This  is  interesting because it is related to\nheterotic strings compactified on T$^d$ or Type II theories compactified on more general\nmanifolds (such as K$_3$).\n\nThe paper is organized as follows:\n\nIn Section 3 we consider systems compactified on T$^d, \\; d=1,\\dots 4$\nwhere only 1\/2 and 1\/4 BPS states occur.\n\nIn Section 4 we consider the richer structure occurring for $d$=5,6 where\na complete understanding of the world volume theory is still missing.\n\nIn Section 5 the BPS conditions are derived for the case of theories\nwith sixteen supersymmetries in any dimension.\n\n\\section{Central charges and geometrical tools of coset spaces.}\n\nIn the present section we review the central charges for 0-branes in\ntheories with maximal supersymmetry and the BPS conditions on the\nU-multiplet of quantized charges which entail different orbits of the\n duality group which preserve different fractions of supersymmetry.\n The analysis for theories with 16 supersymmetries\nwill be considered in the last section.\n\nThe  supergravity theories describing these systems can be\nobtained in three different ways, by compactifying M theory on T$^{d+1}$\n($(d+1)$-dimensional torus)\nor type IIA and type IIB string theories on T$^{d}$. We will consider here\n supergravity theories compactified down up to $D=4$\n space-time dimensions ($d=1,\\dots 6$).\n\nSome of the results presented here overlap with previous analysis for\n$d=1,\\dots 4$, when only 1\/2 or 1\/4 BPS states are present \\cite{dvv, hvz, hv}.\nThe analysis\nof $d=5,6$ is essentially novel although the BPS conditions for 1\/2, 1\/4\nand 1\/8 BPS states were previously discussed in the literature and the\norbit classifications derived \\cite{fm, fg, lsp}.\n\n\\subsection{R-Symmetry and U-duality.}\n\nThe supersymmetry algebra of type II string theory compactified on T$^d$\ndown to $10-d$ dimensions has an R-symmetry group and a continuous\nduality group which depends on $d$. The R-symmetry is given below \\cite{cr}:\n\n\\begin{center}\n{\\bf R-symmetry group $H$}\n\\begin{eqnarray}\n&d=1  & \\mbox{U(1)}\\nonumber\\\\\n&d=2  & \\mbox{SU(2)}\\times \\mbox{U(1)}\\nonumber\\\\\n&d=3  & \\mbox{USp(4)}\\approx\\mbox{O(5)}\\nonumber\\\\\n&d=4  & \\mbox{USp(4)}\\times\\mbox{USp(4)}\\approx\\mbox{O(5)}\\times\\mbox{O(5)}\\nonumber\\\\\n&d=5  & \\mbox{USp(8)}\\nonumber\\\\\n&d=6  & \\mbox{SU(8)}\n\\end{eqnarray}\n\\end{center}\nThe U-duality groups $G$ are E$_{d+1(d+1)}$ \\cite{cr}, and the R-symmetry groups are\ntheir maximal compact subgroups. The quantum U-duality groups are  E$_{d+1(d+1)}({\\bf Z})$\n\\cite{ht}.\nBecause of the connection between M-theory and string theory, the groups\nE$_{d+1(d+1)}$ contain, both\n\\begin{equation}\n\\mbox{Gl}(d+1)\\subset  \\mbox{E}_{d+1(d+1)}\n\\end{equation}\nwhich is the classical isometry group of the moduli space of a T$^{d+1}$\ntorus in M-theory and\n\\begin{equation}\n\\mbox{O}(1,1)\\times\\mbox{O}(d,d)\\subset \\mbox{E}_{d+1(d+1)}\\quad (d\\neq 6),\\quad\n\\mbox{Sl}(2)\\times\\mbox{O}(6,6)\\subset \\mbox{E}_{7(7)} \\quad \\mbox{for}\\quad d=6,\n\\end{equation}\nwhich is the S-T duality group of string theory \\cite{wi, adf, ht}.\n\nIn string theory the $\\mbox{O}(d,d)$ group combines the geometric isometry\nof the T$^d$ torus GL$(d)$ with the shift of the antisymmetric tensor\n$B_{ij}\\mapsto B_{ij}+N_{ij}$ while the O(1,1) factor corresponds to the\ndilaton shift. The group E$_{d+1(d+1)}$ emerges from the combination of\nSl($d+1$) with O$(d,d)$ and this operation gives rise to non\nperturbative symmetries which combine the N-S-NS  and R-R  fields in\nsupermultiplets.\n\nThe spinorial charges of the supersymmetry  algebra transform  in\nrepresentations of the R-symmetry group and\nthis implies that the central charges of interest to us have\n certain symmetry and reality properties.\n\nIn the case of Lorentz scalar central charges, appropriate to 0-branes,\nthe classification goes as follows: the central charge matrix\n$Z(\\phi,q)$ is in the same representation  of the R-symmetry as  the\nvector fields $A_\\mu$ of the corresponding theory. This gives the\nfollowing result,\n\\vfill\\eject\n\\begin{center}\n{\\bf Central charge representation of the R-symmetry}\n\\begin{eqnarray}\n&d=1  &\\quad \\mbox{{\\bf 3} of O(2), ( real symmetric tensor).}\\nonumber\\\\\n&d=2  &\\quad \\mbox{{\\bf 3(+)}  of SU(2)$\\times$U(1) , (complex triplet).}\\nonumber\\\\\n&d=3  &\\quad \\mbox{{\\bf 10} of USp(4), (real antisymmetric tensor).}\\nonumber\\\\\n&d=4  &\\quad \\mbox{{\\bf 16} of USp(4)$\\times$USp(4), (bispinor (4,4)\nof O(5)$\\times $O(5)).}\\nonumber\\\\\n&d=5  &\\quad \\mbox{{\\bf 27} of USp(8), ($\\Omega$-traceless symplectic antisymmetric tensor).}\\nonumber\\\\\n&d=6  &\\quad \\mbox{{\\bf 28}  of SU(8), (complex antisymmetric tensor).}\n\\label{repre}\n\\end{eqnarray}\n\\end{center}\n\nThe previous results follow both, from a dynamical reduction of the 11\nor 10 dimensional supergravities with 32 supercharges or by an analysis\nof extended superalgebras in the appropriate dimensions \\cite{to, adf2}.\n\nSince in the original IIA theory there is only one D 0-brane (one scalar\ncentral charge) \\cite{to}, all the charges in lower dimensions come by wrapping\nbranes on the torus cycles, other than momenta and string windings.\nIn the type IIB on T$^d$, 0-branes  emerge as momenta, string windings,\nand D-branes wrapped on the torus cycles.\n\nIf we want to\ndiscuss quantum mechanical systems emerging from $d+1$ Born-Infeld\nLagrangians compactified on T$^d$, we must consider IIA D-branes\ncompactified on even dimensional tori and IIB  D-branes\ncompactified on odd dimensional tori.\n\nThe world volume description of the central charges $Z$ and quantized\ncharges $q$ is fairly well understood for the case of T$^d$ with\n$d=1,\\dots 4$. The U-duality multiplets of the quantized charges $q$\ncorrespond to fluxes, momenta, instanton number and rank of the gauge\ngroups in the world volume Yang-Mills theory. The moduli dependent\ncentral charge determines the hamiltonian  of the quantum mechanical system as well\nas the energy spectrum of the BPS states \\cite{hvz, egkr}.\n\nThe main role played by duality is that the central charge vector extends\nthe representation of the R-symmetry group to a representation of the\nfull duality group E$_{d+1(d+1)}$ acting on the vector field strength.\nThe relevant extensions are as follows \\cite{cr}\n\\begin{center}\n   {\\bf{ 0-brane representation of  U-duality group}}\n\\begin{eqnarray}\n&d=1  &\\quad \\mbox{{\\bf 2+1} of E$_2$=SL(2)$\\times$ O(1,1)}\\nonumber\\\\\n&d=2  &\\quad \\mbox{{\\bf (3,2)} of E$_3$=Sl(3)$\\times$Sl(2)}\\nonumber\\\\\n&d=3  &\\quad \\mbox{{\\bf 10} of E$_4$=Sl(5)}\\nonumber\\\\\n&d=4  & \\quad\\mbox{{\\bf 16} of E$_5$=O(5,5)}\\nonumber\\\\\n&d=5  &\\quad \\mbox{{\\bf 27} of E$_{6(6)}$}\\nonumber\\\\\n&d=6  &\\quad \\mbox{{\\bf 56} of E$_{7(7)}$}\n\\end{eqnarray}\\label{udua}\n\\end{center}\n\nThe moduli space of these theories is $G\/H$. At the string level this\nspace can be modded out further by E$_{d+1}({\\bf Z})$ \\cite{ht, wi}  something similar to\nthe fundamental domain versus the half plane for the prototype\nSl$(2,R)\/$O(2).\n\nIn Table(2.1) we present the  U-duality multiplets for 0-branes\n in the bulk and world volume description \\cite{hvz}.\n\nThe gauge fields of type II supergravity theory, as well as the Yang-Mills\nworld-volume fluxes complete U-duality multiplets of E$_{d+1(d+1)}$ for\n$d=1,\\dots 4$. These multiplets are obtained by the E$_{d(d)}$ flux and\nmomenta multiplets of matrix gauge theory on T$^d$ and by adding an\nE$_{d(d)}$ singlet, the rank of the gauge group.  For\n$d=5,6$, the Yang-Mills theory side  misses some states corresponding\nto a NS five brane and K-K monopoles \\cite{op, egkr}.\n\n\\begin{center}\n\\begin{tabular}[t]{c|c|c|}\n{\\bf d}& {\\bf Supergravity Vector Fields} & {\\bf Born-Infeld Y-M fluxes}\\\\\n\\hline\n& {\\bf IIA} \\\\\n\\hline\n2&$ Z_\\mu,g_{\\mu i},b_{\\mu i},A_{\\mu ij}$ & $\\int$Tr$F_{ij}$, $\\int$\nTr$P_i$ $\\int$Tr$E_i$, rank \\\\\n\\hline\n4& $Z_\\mu,g_{\\mu i},b_{\\mu i}, A_{\\mu ij}$ &$\\int$Tr$F_{ij}F_{kl}$,\n$\\int$Tr$P_i$, $\\int$Tr$E_i$,$\\int$Tr$F_{ij}$\\\\\n\n &$ A^D_{\\mu ijkl}$& rank\\\\\n\\hline\n 6&$Z_\\mu,g_{\\mu i},b_{\\mu i}, A_{\\mu ij}$&$\\int$Tr$F_{ij}F_{kl}F_{pq}$,\n $\\int$Tr$P_i$, $\\int$Tr$E_i$, $\\int$Tr$F_{ij}F_{kl}$\\\\\n\n & $A^D_{\\mu ijkl},Z^D_{\\mu ijklpq}$& $\\int$Tr$F_{ij}$, rank\\\\\n  &$b^{NS}_{\\mu ijklp}, g^D_{\\mu i}$& \\\\\n\\hline\n& {\\bf IIB} \\\\\n\\hline\n1& $g_{\\mu 1}, b_{\\mu 1}, b^C_{\\mu 1}$&$\\int$Tr$P$, $\\int$Tr$E$, rank\\\\\n\\hline\n3&$g_{\\mu i}, b_{\\mu i}, b^C_{\\mu i}, A_{\\mu ijk}$& $\\int$Tr$P_{i}$,\n$\\int$Tr$E_{i}$, $\\int$Tr$F_{ij}$, rank\\\\\n\\hline\n5& $g_{\\mu i}, b_{\\mu i}, b^C_{\\mu i}, A_{\\mu ijk}$&\n$\\int$Tr$P_{i}$,$\\int$Tr$E_{i}$, $\\int$Tr$F_{ij}F_{kl}$, $\\int$Tr$F_{ij}$\\\\\n\n &$b^D_{\\mu ijklp}$ & rank\\\\\n\n &$b^{NS}_{\\mu ijklp}$ & \\\\\n \\hline\n\\end{tabular}\n\\vskip 3mm\n{\\bf Table 2.1}\n\n\\end{center}\n\\vskip 5mm\n\nThe coset spaces $G\/H$ ($G\\equiv U, H\\equiv R$, in our case) can be described by choosing a representative\nin each equivalence class. If $\\phi$ denotes the coordinates of a point in $G\/H$, then the\ncoset representative will be given by an element $L(\\phi)\\in G$, in the\n equivalence class correspondig to $\\phi$, that is,  $L(\\phi)$ is a local section\nin the principal bundle $G$ over $G\/H$ with structure group $H$ . Under\nthe action of $g\\in G$  on $G\/H$ we have $\\phi\n\\mapsto \\phi_g$ and the coset representative $L(\\phi)$ will be mapped to\n$gL(\\phi)$ which is on the fiber over $\\phi_g$, so necessarily $L(\\phi_g)=\ngL(\\phi)h(\\phi)$. Taking a representation of the group $G$, (which\nis also a representation of $H$) we obtain $L(\\phi)$ as a matrix\n$L_a^\\Lambda(\\phi)$ where the indexes $a$ and $\\Lambda$ run in principle\nover the\nsame representation space, the different names being used to remind the\n covariant properties of $L$. If the representation of $G$ is reducible\nunder $H$, one can project down  a the subspace where\nthe representation of $H$ is irreducible, so the index $a$ will be\nunderstood as running\non that subspace. We will use the representations  appropriated\nto 0-brane multiplets.\n\nThe central charge is  given by\n\\begin{equation}\nZ_a(\\phi,q)= (q^TL)_a=(q^T)_\\Lambda L_a^\\Lambda(\\phi) \\label{cencha}\n\\end{equation}\n$q$ is a vector transforming under the contravariant representation of $G$,\n\\begin{equation}\nq^g=(g^{-1})^Tq\n\\end{equation}\nso the central charge is U-duality covariant in the sense that under a transformation\n\\begin{equation}\n\\phi\\mapsto \\phi_g, \\quad ({q^g}^T)_\\Lambda= (q^T)_\\Sigma (g^{-1})^\\Sigma_\\Lambda,\n\\label{trans}\n\\end{equation}\nthen $Z\\mapsto Zh$.\nIt then follows that any $H$-invariant function $I(Z)$\n is also U-duality invariant in the sense that\n\\begin{equation}\nI(\\phi_g,q^g)=I(\\phi,q), \\quad\\mbox{or}\\quad I(Zh)=I(Z) \\label{dinva}\n\\end{equation}\n\nAmong the duality invariant combinations there are some which are\n``topological'',i. e., they do not depend on the moduli \\cite{adf3, adf4}.\nThis happens when\nthe $H$-invariant is also $G$-invariant with respect to the right action of $G$. In fact,\n since $Z=q^TL$, with $L\\in G$, it is obvious that if $I(Z)$ is $G$-invariant\nfor $Z\\mapsto Zg$, then\n\\begin{equation}\nI(Zg)=I(Z)=I(q)\n\\end{equation}\nSuch objects exist for $d=5,6$ \\cite{fsf, kk}, but not for $d=1,\\dots 4$, with the implication\nthat a Bekenstein-Hawking entropy formula for 0-branes exist only in 4 and 5\ndimensions \\cite{cdcc}.\n\nWe can make a generalization of this analysis to obtain other  moduli invariant conditions.\nLet us consider now  a covariant expression,\n\\begin{equation}\nE_\\alpha(Z)=E_\\alpha(\\phi,q),\n\\end{equation}\nwhere the index  $\\alpha$ runs over the space of some representation $T$ of\n $G$ (and $H$).\nThe covariance property means that under a left $G$  transformation\n\n\\begin{equation}\nE_\\alpha(Zg)=E_\\beta(Z)T(g)^\\beta_\\alpha\n\\end{equation}\nIt follows that an equation of the form $E_\\alpha(Z)=E_\\alpha(\\phi,q)=0$ is moduli\n  independent, so $E_\\alpha(q)=0$.\n\n Now, assume that the representation $T$  admits an $H$-invariant norm\n(which is positive since $H$ is compact).\n \\begin{equation}\n\\| E_\\alpha(Z)\\|^2=g^{\\alpha\\beta} E_\\alpha E_\\beta.\n\\end{equation}\nAn equation like\n\\begin{equation}\n\\| E_\\alpha(Z)\\|^2=0 \\label{norm}\n\\end{equation}\nis in principle moduli dependent since this expression is not\n$G$-invariant. But the constraint $\\|E_\\alpha(Z)\\|=0$ implies that\n$E_\\alpha(Z)=0$, which is moduli independent so $E_\\alpha(q)=0$.\n\n\\medskip\nThe central charges satisfy some differential identities that are\ninherited from the coset representatives. To see this, let us consider\nthe algebra valued  Maurer-Cartan form in $G$, usually expressed for a\nmatrix group as\n\\begin{equation}\n\\alpha_{MC}=g(x)^{-1}dg(x)\n\\end{equation}\nwhere $x$ denotes coordinates on the group $G$. The components of the\nMaurer-Cartan form\nare left invariant forms, and one can take the pullback to $G\/H$ by the\nlocal section $L(\\phi)$, giving a local,  algebra valued left invariant form on $G\/H$\n\\begin{equation}\n\\Omega(\\phi)=L^{-1}(\\phi)dL(\\phi).\\label{mc}\n\\end{equation}\nConsider now the Cartan decomposition of the Lie algebra of $G$ as\n${\\bf g}={\\bf h}\\oplus{\\bf k}$, where ${\\bf h}$ is the Lie algebra of $H$, the\nmaximal compact subgroup of $G$, and ${\\bf k}$ can be identified with the\ntangent space at the identity coset. Since our coset spaces are\nsymmetric spaces, the following properties are satisfied,\n\\begin{equation}\n[{\\bf h},{\\bf h}]\\subset{\\bf h}, \\quad [{\\bf h},{\\bf k}]\\subset{\\bf k},\\quad\n[{\\bf k},{\\bf k}]\\subset{\\bf h}.\\label{subal}\n\\end{equation}\nThe second equation in (\\ref{subal}) means that by the adjoint, ${\\bf h}$\n acts on ${\\bf k}$ as  a representation $R$ of dimension equal to dim$(G\/H)$. We\ncan write $\\Omega$ according to this decomposition of the algebra,\n\\begin{equation}\n\\Omega=\\omega^iT_i+P^\\alpha T_\\alpha\n\\end{equation}\nwhere $\\{T_i\\}$ form a basis of ${\\bf h}$ and $\\{T_\\alpha\\}$ form a basis of ${\\bf k}$.\nThe projection of $\\Omega$  on ${\\bf h}$, $\\omega^iT_i$ is a $G$ invariant\nconnection on the  bundle with fiber ${\\bf k}$ and basis $G\/H$,\n associated to the principal bundle $G(G\/H)$ by the representation $R$ of $H$.\nWe call this bundle $E(G\/H)$. The connection is expressed in an open set as\n\\begin{equation}\n(\\omega)^\\alpha_\\beta =\\omega^iC_{i\\beta}^\\alpha\n\\end{equation}\n\nThe other components of $\\Omega$, $P^\\alpha$, provide us with the local\nexpresion of a homomorphism  between $E(G\/H)$ and the tangent bundle\n$T(G\/H)$. By means of this homomorphism we can pull back the invariant\n Riemannian connection on $G\/H$ which coincides with $\\omega$. Finally,\nthe invariant metric on $G\/H$, induced by the Cartan-Killing metric on\n$G$ can be locally represented as\n\\begin{equation}\ng_{\\mu\\nu}=\\mbox{Tr}(T_\\alpha T_\\beta)P_\\mu^\\alpha P_\\nu^\\beta.\n\\end{equation}\n(The indices ($\\mu, \\nu$) are 1-form indices on $G\/H$).\n\nWe want now to write (\\ref{mc}) using a representation of $G$ (and $H$)\nlabeled as we explained above indistinctely by indices of the type  $\\Lambda$\nor $a$, then we have\n\\begin{equation}\ndL_a^\\Lambda=L_b^\\Lambda \\omega_a^b+L_b^\\Lambda P_a^b.\n\\end{equation}\nBy defining as usual the covariant derivative with respect to $H$\n\\begin{equation}\n\\nabla_HL^\\Lambda_a=dL_a^\\Lambda-L_b^\\Lambda \\omega_a^b,\n\\end{equation}\nwe obtain\n\\begin{equation}\n\\nabla_HL^\\Lambda_a=L_b^\\Lambda P_a^b. \\label{nabla}\n\\end{equation}\nSuppose now that the representation of $H$ is reducible and we want to\nproject onto an irreducible factor. Since $\\omega_a^b$ is block diagonal,\nthe indices of type $a$ in $\\nabla_HL^\\Lambda_a$ can be understood as running on the\nsmaller representation, while $P_a^b$ will have in general off diagonal\ncomponents, so we could denote it by  $P_a^{b'}$, $b'$ running on the\nlarge representation space, but still specifying the covariant\nproperties under $H$. This happens when matter fields are  present.\n In that case the $H$ group is\na direct product $H_R\\times H_M$. $H_R$ is the R-symmetry group and\n$H_M$ is some matter flavour symmetry. We assume now (as it will happen\nin all our examples) that the representation of $G$ decomposes under $H$\nas {\\bf (1, T$_M$)}  +{\\bf (T$_R$,1)}, where {\\bf T$_M$} is a\nrepresentation of $H_M$ and   {\\bf T$_R$} is a representation of $H_R$.\nThen, there is a basis where the generic index $\\Lambda$ splits into\n$(a,I)$, where $a$ runs over the vector space representation of {\\bf T}$_R$\nand $I$ runs over the representation space of {\\bf T}$_M$. Then (\\ref{nabla})\ndecomposes as\n\\begin{eqnarray}\n\\nabla_{H_R}L^\\Lambda_a&=&L_b^\\Lambda P_a^b+L_I^\\Lambda P_a^I\\nonumber\\\\\n\\nabla_{H_M}L^\\Lambda_I&=&L_a^\\Lambda P_I^a+L_J^\\Lambda P_J^I,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\nabla_{H_R}L^\\Lambda_a&=&dL^\\Lambda_a - L^\\Lambda_b\\omega^b_a\\nonumber\\\\\n\\nabla_{H_M}L^\\Lambda_I&=&dL^\\Lambda_I - L^\\Lambda_J\\omega^J_I.\n\\end{eqnarray}\nThe central charges $Z_a=(q^TL)_a$ and matter charges $Z_I=(q^TL)_I$\nsatisfy consequently  the identities\n\\begin{eqnarray}\n\\nabla_{H_R}Z_a&=&Z_b P_a^b+Z_I P_a^I\\nonumber\\\\\n\\nabla_{H_M}Z_I&=&Z_a P_I^a+Z_J P^J_I.\n\\label{fmc}\n\\end{eqnarray}\n\nIn the forthcoming sections we will see that these properties enter in\nthe discussion of the BPS conditions and their duality invariant character.\n\n\\subsection{Orbit classification of BPS states.}\n\nIn order to further study properties of central charges which will be useful in\n the following section, we would like to remind the orbit classification of\n0-brane BPS configurations \\cite{fg, lsp}. To do so we will state some results of matrix algebra\nthat will be\nuseful for our analysis. We consider   matrices over the real,\n complex and quaternion fields, $M^r,M^c,M^q$.\nFor each of these matrices, the following polar decomposition holds\n \\begin{eqnarray}\nM^r&=&\\sqrt{M^r{M^r}^T}O\\nonumber\\\\\nM^c&=&\\sqrt{M^c{M^c}^\\dagger}U\\nonumber\\\\\nM^q&=&\\sqrt{M^q{M^q}^\\dagger}U_q\n\\end{eqnarray}\nwhere the matrices $\\sqrt{M{M}^\\dagger}$ are hermitian and  $O,U$ and $U_q$\nare orthogonal, unitary and quaternionic unitary (unitary symplectic),\n respectively.\n\nFrom this decomposition it then follows that if $M^r$ is symmetric, $M^c$ hermitian\n and $M^q$ symplectic hermitian,  they can be diagonalized by an appropriate\n transformation which is respectively orthogonal, unitary and unitary symplectic.\nInstead, for general matrices we can bring them to a diagonal form in the\nfollowing way,\n\\begin{equation}\nM_D=U_1MU^\\dagger_2\\label{diagonal}\n\\end{equation}\nwhere $U_1$ and $U_2$ belong to O(n), U(n) or USp(n) in each case.\n\nIt can also be shown that any antisymmetric matrix can be brought to a\nskew-diagonal form (normal form) by a transformation \\cite{kz}\n\\begin{equation}\nM_{SD}=UMU^T\n\\end{equation}\nwhere as before, $U$ belongs to the appropriate group.               .\n\nSince the central charge vector is a 2-tensor  representation of $H$,\nwe can always apply\n one of the above results.\n\nFrom the structure of the R-symmetry group listed above and the\n representation properties from (\\ref{repre}) we will see that\n it  follows that\nwith an R-rotation we can always diagonalize (or skew-diagonalize) the\nmatrix $Z$. For $d=1,\\dots 4$ there will be only two eigenvalues,\nthree eigenvalues for  for $d=5$ and four eigenvalues for $d=6$.\n\nThe richer structure occurring for $d=5,6$ is the  why also 1\/8 BPS\nstates occur, instead of the two possibilities of $d=1,\\dots 4$.\n\nIn the following table we list the orbits of the representations\nin Eq.(\\ref{udua}) corresponding to the 0-brane BPS configurations \\cite{fg, lsp}.\n\n\\begin{center}\n\\begin{tabular}[t]{c|c|c|c|}\n{\\bf Orbits} &{\\bf 1\/2 BPS} & {\\bf 1\/4 BPS} &{\\bf 1\/8 BPS}\\\\\n\\hline\n$d=1$ & Sl(2) or ${\\bf R}$ & Sl(2)$\\times{\\bf R} $&\\\\\n\\hline\n$d=2$ &${\\mbox{ Sl(3)}\\times\\mbox{ Sl(2)}\/ \\mbox{Gl(2)}\\propto {\\bf R}^3}$ &\n${\\mbox{Sl(3)}\\times \\mbox{Sl(2)}\/ \\mbox{Sl(2)}\\propto {\\bf R}^2}$&\\\\\n\\hline\n$d=3$ &$ {\\mbox{Sl(5)}\/(\\mbox{Sl(3)}\\times \\mbox{Sl(2)})\\propto {\\bf R}^6}$ &\n$ {\\mbox{Sl(5)}\/\\mbox{O(2,3)}\\propto {\\bf R}^4}$&\\\\\n\\hline\n$d=4$& ${\\mbox{O(5,5)}\/\\mbox{Sl(5)}\\propto{\\bf R}^{10}}$ &\n${\\mbox{O(5,5)}\/\\mbox{O(3,4)}\\propto {\\bf R}^8}$&\\\\\n\\hline\n$d=5$ & ${\\mbox{E}_{6(6)}\/\\mbox{O(5,5)}\\propto{\\bf R}^{16}}$ &\n${\\mbox{E}_{6(6)}\/\\mbox{O(4,5)}\\propto{\\bf R}^{16}}$ &$\n{\\mbox{E}_{6(6)}\/\\mbox{F}_{4(4)}}$\\\\\n\\hline\n$d=6$ & ${\\mbox{E}_{7(7)}\/\\mbox{E}_{6(6)}\\propto{\\bf R}^{27}}$ &\n${\\mbox{E}_{7(7)}\/(\\mbox{O(5,6)}\\propto{\\bf R}^{32})\\times{\\bf R}}$ &\n${\\mbox{E}_{7(7)}\/\\mbox{F}_{4(4)}\\propto{\\bf R}^{26}},$\\\\\n&&&$ {\\mbox{E}_{7(7)}\/\\mbox{E}_{6(2)}}$\\\\\n \\hline\n\\end{tabular}\n\\vskip 3mm\n{\\bf Table 2.2}\n\\end{center}\n\\vskip 5mm\nThese orbits correspond, for $d=1,\\dots 4$ to the possibility of having\n two coinciding eigenvalues (1\/2 BPS) or not (1\/4 BPS) for the central\n charge matrix. For $d=5,6$, 1\/2 BPS correspond to 3 and 4 coinciding\n eigenvalues respectively, 1\/4 BPS orbits correspond to 2 equal eigenvalues\n and 2 pairs of equal eigenvalues respectively and 1\/8 BPS orbits\n correspond to all different eigenvalues. In $d=6$ there are there two kinds\nof 1\/8 BPS orbits depending whether the quartic invariant vanishes  or not\n(light-like or time-like orbit)\\cite{fm, fg}.\n\nIn the following section we will see that, in spite of the fact that such\nstatements look moduli dependent, they are actually moduli\nindependent and therefore U-duality invariant, as expected from physical\nconsiderations.\n\n\\section{BPS spectrum for 0-branes in type II string theory compactified\non T$^d$, $d=1,\\dots 4$}\n\nIn the present section we consider the BPS spectrum and the central\ncharge matrix  for 0-branes in the cases when only 1\/2 and 1\/4 BPS\nstates exist. These are the cases where the central charge has only two\neigenvalues, as it happens for $d=1,\\dots 4$.\n\nLet us consider the relevant anticommutators, containing the scalar\ncentral charge,\n\\begin{eqnarray}\nd=1\\quad &\\{Q_i,Q_j\\}&=Z_{ij}\\quad (i,j=1,2);\\; Z_{ij}\\mbox{ real symmetric}\\nonumber\\\\\nd=2\\quad &\\{Q_A,Q_B\\}&=Z_I\\sigma_{AB}^I=Z_{AB}\\quad ( A,B=1,2);\\nonumber\\\\\n&&Z^I\\;\\mbox{complex},\\quad Z_{AB}\\; \\mbox{symmetric},  \\nonumber\\\\\nd=3\\quad &\\{Q_a,Q_b\\}&=Z_{IJ}(\\gamma^{IJ})_{ab}=Z_{ab}\\quad\n(a,b=1,\\dots 4);\\nonumber\\\\\n&& Z_{ab}\\; \\mbox{ symmetric and symplectic} \\nonumber\\\\\nd=4\\quad &\\{Q_a,Q_b'\\}&=Z_{ab'}\\quad\n(a,b'=1,\\dots 4);\\; Z_{ab'}\\; \\mbox{symplectic}\n\\end{eqnarray}\nwhere $\\sigma^I$ are the Pauli matrices and  $\\gamma^{IJ} = 1\/2 [\\gamma^I\n, \\gamma^J ]$ ($ \\gamma^I$ are the O(5) gamma matrices).\nWe say that  $Z_{ab}$ is a symplectic (or quaternionic) matrix if:\n\\begin{equation}\n\\bar Z =- \\Omega Z \\Omega \\label{sym}\n\\end{equation}\nwhere $\\Omega$ is the bilinear form invariant under\nUSp(4) which satisfies\n\\begin{equation}\n\\Omega = \\bar \\Omega = -  \\Omega^T = - \\Omega^{-1}\n\\end{equation}\nThe indices $a,b$ in the gamma matrices are raised and lowered with $\\Omega$.\n\n\\paragraph{ Cases $d=1,2$.} In these cases the chare matrix $Z$ is\n2$\\times$2 and has two independent eigenvalues. The 1\/2 BPS conditions\ncorrespond to these eigenvalues equal in magnitude.  We consider separately\nboth cases.\n\n\\medskip\n\n$\\bullet$ For $d=1$, $Z$ is real and symmetric. We can decompose it as\n\\begin{equation}\nZ_{ij}=Y\\delta_{ij} +Z^\\alpha (T_\\alpha)_{ij}\n\\end{equation}\nwhere $\\alpha=1,2$, $T_1=\\sigma_1$ and $T_2=\\sigma_3$ (the Pauli\nmatrices).\nThe characteristic equation for $Z$ is\n\\begin{equation}\n\\lambda^2 - \\mbox{Tr} Z \\, \\lambda +\\mbox{det}Z =0 \\label{simple}\n\\end{equation}\nwhere\n\\begin{eqnarray}\n&& \\mbox{Tr} Z = 2 Y \\\\\n&&\\mbox{det}Z = Y^2 - Z^{\\alpha} Z_{\\alpha}\n\\end{eqnarray}\nIt then follows that  the two solutions of (\\ref{simple}) satisfy\n $|\\lambda_1|=|\\lambda_2|$ if either\n\\begin{equation}\n\\mbox{Tr} Z=0\n\\end{equation}\nor\n\\begin{equation}\n\\mbox{det} Z =1\/4 (\\mbox{Tr}Z)^2\n\\end{equation}\nthen implying\n\\begin{equation}\nYZ^{\\alpha}Z_{\\alpha} =0.\\label{discri}\n\\end{equation}\n\nIt is obvious that  condition (\\ref{discri}) is only O(2) or R-invariant,\nbut\nthe unique solution  $YZ_\\alpha=0$, is an SL(2)$\\times$O(1,1) or U-invariant. This is\nthe first example of a condition of the type (\\ref{norm}). So we\nretrieve the result of Ref.\\cite{fm} for $d=1$, \\,1\/2 BPS states\n\\begin{equation}\nq=0 \\quad \\mbox {or}\\quad q_\\alpha=0\\quad\n\\end{equation}\n\\medskip\n\n$\\bullet$ In the $d=2$ case the hermiticity condition is lacking,\nbut still the matrix $Z^I\\sigma_I$ can be diagonalized with real\neigenvalues using (\\ref{diagonal}), where the difference between $U_1$\nand $U_2$ is simply a phase.  We  just use  a transformation\nof the R-symmetry group U(2), with SU(2) acting on the $\\sigma$-matrices\nand U(1) acting  as a phase on $Z^I$.\n\nIn fact, note that\n\\begin{equation}\nZ_IZ^I=(A_I+iB_I)(A^I+iB^I)=A_IA^I -B_IB^I+2iA_IB^I\n\\end{equation}\nTherefore, with a U(1) transformation we bring  $A_IB^I$ to zero, which\nmeans that $\\vec{A}$ and $\\vec{B}$ are orthogonal vectors, so by an\northogonal transformation we can bring them to coincide with the axes,\nand only two real numbers (related to the two eigenvalues) are left.\n\nWe proceed by diagonalizing the hermitian matrix $ZZ^\\dagger$. The\nsquare root of the eigenvalues will be the eigenvalues of $Z$, that we\ndenote by $\\lambda_1, \\lambda_2$. (In this way we include both cases, when\nthe eigenvalues are equal and when they are opposite in sign).We have :\n\\begin{equation}\n\\mbox{Tr}ZZ^\\dagger=\\lambda_1^2+\\lambda_2^2, \\quad \\mbox{Tr}(ZZ^\\dagger)^2\n=\\lambda_1^4 +\\lambda_2^4,\\quad \\mbox{det}ZZ^\\dagger=\\lambda_1^2\\lambda_2^2\n\\end{equation}\nwhere $\\lambda_i$ are the eigenvalues obtained  as in (\\ref{diagonal}).\nFrom the characteristic equation for $ZZ^\\dagger$ we have\n\\begin{equation}\n\\lambda_{1,2}^2={1\\over 2}[\\mbox{Tr}ZZ^\\dagger\\pm\\sqrt{2\\mbox{Tr}(ZZ^\\dagger)^2\n-(\\mbox{Tr}ZZ^\\dagger)^2}] \\label{eigen}\n\\end{equation}\nUsing now the properties of the $\\sigma$ matrices,\n\\begin{equation}\nZZ^\\dagger=Z^I\\sigma_I {\\bar Z^J}\\sigma_J=Z^I{\\bar Z_I}{\\bf I} +i\\epsilon_{IJK}\n Z^I{\\bar Z^J}\\sigma^K\n\\end{equation}\nWe denote $\\hat Z_K=i\\epsilon_{IJK}Z^I{\\bar Z^J}$. Then we have\n\\begin{equation}\n\\mbox{Tr}ZZ^\\dagger=2Z^I{\\bar Z_I},\\quad \\mbox{Tr}(ZZ^\\dagger)^2\n=2[(Z^I{\\bar Z_I})^2+\\hat{Z}_I\\hat{Z}_I],\n\\end{equation}\nHence, the discriminant in (\\ref{eigen}) is given by\n\\begin{equation}\n\\mbox{Tr}(ZZ^\\dagger )^2-{1\\over 2}(\\mbox{Tr}ZZ^\\dagger)^2=2\\hat\nZ^I\\hat Z_I.\n\\end{equation}\nWe set $Z^I_1=A^I,\\; Z^I_2=B^I$. The 1\/2 BPS condition\n $\\hat{Z}_I\\hat{Z}^I=0$, can be written as\n\\begin{equation}\n\\|\\epsilon^{\\alpha \\beta}Z_\\alpha^IZ^J_\\beta\n\\epsilon_{KIJ}\\|=0\\Rightarrow \\epsilon^{\\alpha \\beta}Z_\\alpha^IZ^J_\\beta\n\\epsilon_{KIJ}=0.\\label{epsilon}\n\\end{equation}\nwhere $\\|\\;\\|$ is the O(3)$ \\times $O(2) invariant norm. As before,\n the condition obtained is  actually invariant under SL(3)$\\times $SL(2), so we obtain\nthe  moduli independent condition of Ref.\\cite{fm}:\n\\begin{equation}\n\\epsilon^{\\alpha \\beta}q_\\alpha^Iq^J_\\beta \\epsilon_{KIJ}=0.\n\\end{equation}\n\n\\paragraph{Cases  $d=3,4$.}\n\nIn these cases the matrix $Z$ is 4-dimensional, but because of its symplectic\nproperty (\\ref{sym})\nthere are only two independent eigenvalues (two pairs of equal\neigenvalues), $\\lambda_{1,2}$.\nIndeed, we find\n\\begin{eqnarray}\n&&\\mbox{Tr}ZZ^\\dagger=2(\\lambda_1^2+\\lambda_2^2)\\nonumber\\\\\n&&\\mbox{Tr}(ZZ^\\dagger)^2=2(\\lambda_1^4+\\lambda_2^4).\n\\end{eqnarray}\nThe characteristic equation  (or better, its square root) is\n\\begin{equation}\n\\lambda^2 -{1\\over 2}\\mbox{Tr}ZZ^\\dagger\\lambda +(\\mbox{det}ZZ^\\dagger)^{1\/2}=0,\n\\end{equation}\nwith\n\\begin{equation}\n(\\mbox{det}ZZ^\\dagger)^{1\/2}={1\\over 8}(\\mbox{Tr}ZZ^\\dagger)^2-{1\\over 4}\n\\mbox{Tr}(ZZ^\\dagger)^2.\n\\end{equation}\nThe roots are\n\\begin{equation}\n\\lambda_{1,2}^2={1\\over 2}\\left({1\\over 2}\\mbox{Tr}ZZ^\\dagger\\pm \\sqrt{\\mbox{Tr}(ZZ^\\dagger)^2\n-{1\\over 4}(\\mbox{Tr}ZZ^\\dagger)^2}\\right)\n\\end{equation}\n\nWe consider now the two cases separately,\n\\medskip\n\n$\\bullet$ In the $d=3$ case we can switch from Sp(4) to O(5) indices by\nsetting:\n\\begin{equation}\nZ_{ab}=Z_{IJ}(\\gamma^{IJ})_{ab},\\quad (I,J=1,\\dots 5)\n\\end{equation}\nwhere\n$Z_{IJ}$ is real and antisymmetric and\n\\begin{equation}\n\\gamma^{IJ}={1\\over 2}[\\gamma^I,\\gamma^J].\n\\end{equation}\nIt is clear that $Z_{IJ}$ can be skew-diagonalized with an O(5)\ntransformation, so $Z_{ab}$ can be diagonalized with a USp(4) transformation.\nFrom the relation\n\\begin{equation}\n{1\\over 2}\\{ \\gamma^{IJ},\\gamma^{KL}\\}=(g^{IJ}g^{KL}- g^{JK}g^{IL}+\\epsilon^{IJKLP}\\gamma_P )\n\\end{equation}\nit follows that\n\\begin{equation}\nZZ^\\dagger=Z^2{\\bf I} + Z^P\\gamma_P,\n\\end{equation}\nwhere\n\\begin{equation}\nZ^2=2Z^{PQ}Z_{PQ},\\quad Z^P=\\epsilon^{PIJKL}Z_{IJ}Z_{KL}.\n\\end{equation}\nFrom this, we have\n\\begin{eqnarray}\n\\mbox{Tr}ZZ^\\dagger&=&4Z^2\\nonumber\\\\\n\\mbox{Tr}(ZZ^\\dagger)^2&=&4Z^4+4Z^PZ_P.\n\\end{eqnarray}\nThe 1\/2 BPS condition becomes:\n\\begin{equation}\n\\mbox{Tr}(ZZ^\\dagger)^2 -{1\\over 4}(\\mbox{Tr}ZZ^\\dagger)^2=4Z^PZ_P=0\n\\end{equation}\nwhich  is o(5) invariant and implies\n\\begin{equation}\nZ^P=\\epsilon^{PIJKL}Z_{IJ}Z_{KL}=0 \\label{epsilon3}\n\\end{equation}\n\nEquation(\\ref{epsilon3}) is SL(5) invariant when $Z^{IJ}$ is in the 10-dimensional\n representation of SL(5), and therefore it is moduli independent, giving\n the result of Ref. \\cite{fm},\n\\begin{equation}\n\\epsilon^{PIJKL}q_{IJ}q_{KL}=0\n\\end{equation}\n\n\\medskip\n\n$\\bullet$ The $d=4$ case was already discussed in Ref.\\cite{dvv, hvz}, but we outline it  here for\ncompleteness. In this case, the matrix $Z_{ab'}$ is a general O(5) bispinor.\nHowever, since its square is hermitian it decomposes as\n\\begin{eqnarray}\nZZ^\\dagger&=&Z^2{\\bf I} +{Z_{(l)}}^P\\gamma_P,\\quad p=1,\\dots 5\\nonumber\\\\\nZ^\\dagger Z&=& Z^2{\\bf I} +{Z_{(r)}}^P\\gamma_P\n\\quad \\mbox{with} \\quad\n {Z_{(l)}}^P{Z_{(l)}}_P=  {Z_{(r)}}^P{Z_{(r)}}_P.\n\\end{eqnarray}\nwhere the subindices $l,r$ refer to the two $O(5)$ factors of the\n$R-$symmetry group.\nIt follows that\n\\begin{equation}\n\\mbox{Tr}ZZ^\\dagger=4Z^2,\\quad\n\\mbox{Tr}(ZZ^\\dagger)^2 =4Z^4+4 {Z_{(l)}}^P{Z_{(l)}}_P.\n\\end{equation}\nThe 1\/2 BPS condition is then :\n\\begin{equation}\n\\mbox{Tr}(ZZ^\\dagger)^2 -{1\/4}(\\mbox{Tr}ZZ^\\dagger)^2=4{Z_{(l)}}^P{Z_{(l)}}_P=0.\n\\end{equation}\nThe equations ${Z_{(l)}}^P{Z_{(l)}}_P= {Z_{(r)}}^P{Z_{(r)}}_P=0$ imply that the O(5)\nvectors $Z^P_{(r)}$, $Z^P_{(l)}$ vanish. This is an O(5,5) invariant statement. $(Z,Z^\\dagger)$\nform the 16 dimensional (chiral spinor) representation of O(5,5) and the O(5,5)\n10 dimensional (light-like vector) $(\\mbox{Tr} \\gamma ZZ^\\dagger,\n\\mbox{Tr} \\gamma Z^\\dagger Z)$ then vanishes when $|Z_{(l)}|=|Z_{(r)}|=0$.\nWe then retrieve the condition of Ref.\\cite{fm} on the quantized charges in the spinor\nrepresentation of O(5,5).\n\n\\section{BPS spectrum for the $d\\!=\\!5,6$ dimensional cases.}\n\nIn this section we will examine the more interesting cases of $d=5,6$,\ncorresponding to supergravity compactified down to $D=4,5$ dimensions\nrespectively.\n\nThe different BPS states, preserving some fraction of supersymmetry,\nare classified by the orbits of E$_{6(6)}$ and E$_{7(7)}$\nrespectively as given in Table 2.2\n\nTo put this analogy in perspective it is useful to parametrize the set of BPS\n charges  allowed by the duality constraints by their eigenvalues and some\n angular variables (which can be removed by an R-symmetry transformation \\cite{fg}). The\n duality constraints which follow from the BPS conditions are precisely those\n constraints which do not depend on these extra angular variables and which can\nbe removed by an $H$ transformation in $G$. These constraints will give\ndifferent orbits corresponding to different BPS conditions on the 0-brane\n charges.\n\n\\begin{center}\n{\\bf Orbits of the BPS energy levels for $d=5,6$}\n\\bigskip\n\\begin{tabular}[t]{c|c|c|c|c|}\n& Orbit & dim.& eigenv. & angles\\\\\n\\hline\n d=5   \\\\\n\\hline\n1\/2 BPS&E$_{6(6)}$\/O(5,5)$\\propto{\\bf R}^{16}$ & 17&1&16=dim(USp(8)\/O(5)$\\times$O(5)) \\\\\n\\hline\n1\/4 BPS&E$_{6(6)}$\/O(4,5)$\\propto{\\bf R}^{16}$ & 26&2&24=dim(USp(8)\/O(4)$\\times$O(4)) \\\\\n\\hline\n1\/8 BPS&E$_{6(6)}$\/F$_{4(4)}$ & 26+1&3&24=dim(USp(8)\/USp(2)$^4$) \\\\\n\\hline\n d=6   \\\\\n \\hline\n 1\/2 BPS&E$_{7(7)}$\/E$_{6(6)}\\propto{\\bf R}^{27}$ & 28&1&27=dim(SU(8)\/USp(8)) \\\\\n \\hline\n1\/4 BPS&E$_{7(7)}$\/(O(5,6)$\\propto{\\bf R}^{32})\\times{\\bf R}$ & 45&2&43=dim(SU(8)\/USp(4)$^2$) \\\\\n\\hline\n1\/8 BPS&E$_{7(7)}$\/F$_{4(4)}\\propto{\\bf R}^{26}$ & 55&4&51=dim(SU(8)\/USp(2)$^4$) \\\\\n&E$_{7(7)}$\/E$_{6(2)}$& 55+1&5&51=dim(SU(8)\/USp(2)$^4$) \\\\\n \\hline\n\\end{tabular}\n\\vskip 3mm\n{\\bf Table 4.1}\n\n\\end{center}\n\\vskip 5mm\n\nThe different orbits of different BPS levels will correspond to different\n solutions of the characteristic equation of the central charge matrix (or\n its square). These different solutions will be characterized by invariant\n constraints which are moduli independent in spite of the fact that the\n eigenvalues of the matrix are moduli dependent. Becuse of this, the orbits\n are simply given by invariant constraints on the ``quantized'' charges, as found\n in Ref.\\cite{fm}.\n\n\\paragraph{Case $d=6$.}\n\nWe consider the E$_7$ quartic invariant \\cite{cr, adf4, kk}\n\\begin{equation}\nI=4\\mbox{Tr}(Z\\bar Z)^2 -(\\mbox{Tr}Z\\bar Z)^2+2^4({\\it Pf}Z+{\\it Pf}\\bar Z)\n \\label{quartic}\n\\end{equation}\nwhere\n\\begin{equation}\n{\\it Pf}Z={1\\over 2^44!}\\epsilon^{ABCDRPGH}Z_{AB}Z_{CD}Z_{RP}Z_{GH}.\n\\end{equation}\nWe want to consider second derivatives of the above quartic invariant\nthat could give us covariant equations. The antisymmetric matrix\n$Z_{AB}$ is in the 28-dimensional representation of SU(8), while we can\nexpress symbolically $Z_{\\bf 56}=(Z_{AB},\\bar Z^{AB})$, in the 56-dimensional\nrepresentation of E$_7$ (${\\bf 56=28 +\\bar{28}}$). Taking the  the second derivative\n\\begin{equation}\n{\\partial^2 I\\over \\partial Z_{\\bf 56}\\partial Z_{\\bf 56}}\\left|_{Adj_{E_7}}\\right.,\\label{cova}\n\\end{equation}\nwill give us a quadratic polynomial which is a symmetric tensor, in the\n$({\\bf 56}\\times {\\bf 56})_S={\\bf 1596}$\nrepresentation of E$_7$ which is not irreducible and decomposes as {\\bf 1463\n+ 133}. {\\bf 133} is the Adj$_{E_7}$, so we can project on that\n space as indicated above (\\ref{cova}).\nSince {\\bf  133} decomposes as {\\bf 63+70} under SU(8), the expression\n(\\ref{cova})  splits into the two following SU(8) covariant\npolynomials\n\\begin{equation}\n{\\partial^2 I\\over \\partial Z_{AB}\\bar \\partial Z^{CB}}\\left|_{Adj_{SU(8)}}\\right.\\approx\n( Z_{AB}\\bar  Z^{CB}-{1\\over 8}\\delta_A^C Z_{PQ}\\bar\nZ^{PQ})=V_A^C.\\label{vac}\n\\end{equation}\n\\begin{equation}\n{\\partial^2 I\\over \\partial Z_{[AB}\\partial Z_{CD]}}-{1\\over 4!}\\epsilon^{ABCDPQRS}\n{\\partial^2 I\\over \\partial \\bar Z^{[AB}\\partial \\bar Z^{CD]}}=V^+_{[ABCD]}.\n\\label{selfdual}\n\\end{equation}\nThe 1\/2 BPS condition is the E$_7$ invariant statement $ V_A^C=0$\nand $V^+_{[ABCD]}=0$. This is the constraint\n imposed in Ref.\\cite{fm} on the quantized\n56 electric and magnetic charges defining a 1\/2 BPS configuration.\nThe equation $ V_A^C=0$ implies that the matrix $ZZ^\\dagger$ has four\ncoinciding eigenvalues (that is, it is a multiple of the identity),\nwhile  the equation $V^+_{[ABCD]}=0$ implies\nthat the eigenvalues of $Z$ are real.\n\nThe vanishing of (\\ref{selfdual}) follows from the vanishing of\n(\\ref{vac}) and the differential relations (\\ref{fmc}) satisfied by $Z_{AB}$\n\\cite{adf3}, which in this case take the form\n\\begin{equation}\n\\nabla_{SU(8)}Z_{AB}={1\\over 2}\\epsilon_{ABCD}\\bar Z^{CD}.\n\\end{equation}\n\nWe now want to consider more general cases. The\ncharacteristic equation (or better,  its square root) is given by\n\\begin{equation}\n\\sqrt{\\mbox{det}(ZZ^\\dagger-\\lambda{\\bf I})} =\\prod_{i=1}^4(\\lambda-\\lambda_i)=\n\\lambda^4+a\\lambda^3+b\\lambda^2+c\\lambda+d=0\n\\end{equation}\nwhere\n\\begin{eqnarray}\na&=&-(\\lambda_1+\\lambda_2+ \\lambda_3+ \\lambda_4)\\nonumber\\\\\n&=&-{1\\over2}\\mbox{Tr}ZZ^\\dagger\\nonumber\\\\\nb&=&\\lambda_1\\lambda_2+ \\lambda_1\\lambda_3+ \\lambda_1\\lambda_4+\n\\lambda_2\\lambda_3+ \\lambda_2\\lambda_4+ \\lambda_3\\lambda_4\\nonumber\\\\\n&=&{1\\over 4}[{1\\over 2}(\\mbox{Tr}ZZ^\\dagger)^2- \\mbox{Tr}(ZZ^\\dagger)^2]\n\\nonumber\\\\\nc&=&-(\\lambda_1\\lambda_2\\lambda_3+\\lambda_1\\lambda_2\\lambda_4+\\lambda_1\n\\lambda_3\\lambda_4+\\lambda_2\\lambda_3\\lambda_4)\\nonumber\\\\\n&=&-{1\\over 6}\\left({1\\over 8}(\\mbox {Tr}ZZ^\\dagger)^3 + \\mbox{Tr}(ZZ^\\dagger)^3-\n{3\\over 4} \\mbox{Tr}ZZ^\\dagger \\mbox{Tr}(ZZ^\\dagger)^2\\right)\\nonumber\\\\\nd&=&\\lambda_1\\lambda_2\\lambda_3\\lambda_4\\nonumber\\\\\n&=&{1\\over 4}\\left({1\\over 96} (\\mbox{Tr}ZZ^\\dagger)^4 +{1\\over 8}( \\mbox{Tr}\n(ZZ^\\dagger)^2)^2+{1\\over 3} \\mbox{Tr}(ZZ^\\dagger)^3 \\mbox{Tr}ZZ^\\dagger\\right.\n\\nonumber \\\\\n&&\\left. -{1\\over 2} \\mbox{Tr}(ZZ^\\dagger)^4-{1\\over 8} (\\mbox{Tr}ZZ^\\dagger)^2\n \\mbox{Tr}(ZZ^\\dagger)^2\\right)\\label{abcd}\n\\end{eqnarray}\n\n In the case of two pairs of equal roots we have\n\\begin{equation}\n\\prod_{i=1}^4(\\lambda-\\lambda_i)=(\\lambda-\\lambda_1)^2(\\lambda-\\lambda_2)^2.\n\\end{equation}\n This implies the following relations among the coefficients\n \\begin{eqnarray}\n c&=&{1\\over 2}a(b-{1\\over 4}a^2)\\nonumber\\\\\n d&=&{1\\over 4}(b-{1\\over 4}a^2)^2 \\label{conditions}\n \\end{eqnarray}\nwhich imply the following relations among the invariants,\n\\begin{eqnarray}\n&&{32\\over 3}\\mbox{Tr}(ZZ^\\dagger)^3=4\\mbox{Tr}ZZ^\\dagger\\mbox{Tr}(ZZ^\\dagger)^2 -\n{1\\over 3}(\\mbox{Tr}ZZ^\\dagger)^3\\nonumber\\\\\n&&(\\mbox{det}ZZ^\\dagger)^{1\/2}={1\\over 64}[\\mbox{Tr}(ZZ^\\dagger)^2-\n{1\\over 4}(\\mbox{Tr}ZZ^\\dagger)^2]^2.\\label{det}\n\\end{eqnarray}\nThe eigenvalues are given by the expression\n\\begin{equation}\n\\lambda_{1,2}={1\\over 8}\\mbox{Tr}ZZ^\\dagger\\pm{1\\over 2}\\sqrt{{1\\over 2}\n\\mbox{Tr}(ZZ^\\dagger)^2-{1\\over 16}(\\mbox{Tr}ZZ^\\dagger)^2}\n\\end{equation}\nbeing the BPS mass, $m_{BPS}^2$ the highest eigenvalue (+ sign).\n\nWe want now to show how the 1\/4 BPS condition follows from the\nE$_7$ invariance. Let us consider the E$_7$ covariant constraint\n\\begin{equation}\n {\\partial I\\over \\partial Z_{AB}}=0\\quad(\\Rightarrow\n {\\partial I\\over \\partial\\bar Z^{AB}}=0)\\label{inv1}.\n \\end{equation}\n where $I$ is the invariant from (\\ref{quartic}). From this, the\nfollowing quartic SU(8) invariant equations follow,\n\\begin{eqnarray}\n {\\partial I\\over \\partial Z_{AB}}Z_{AB}+{\\partial I\\over \\partial\n\\bar Z^{AB}} \\bar Z^{AB}&=&4I=0\\label{first}\\\\\n {\\partial I\\over \\partial Z_{AB}}Z_{AB}-{\\partial I\\over \\partial\n \\bar Z^{AB}} \\bar Z^{AB}&=&0.\\label{second}\n\\end{eqnarray}\nThe second equation implies that the Pfaffian of $Z$ is real, so\n\\begin{equation}\n{\\it Pf}Z={\\it Pf} Z^\\dagger\n\\end{equation}\nand therefore\n\\begin{equation}\n({\\it Pf}Z)^2=(\\mbox{det}ZZ^\\dagger)^{1\/2}.\\label{pfaf}\n\\end{equation}\nPlugging (\\ref{pfaf}) into (\\ref{first}) and squaring, it\n gives $(\\mbox{det}ZZ^\\dagger)^{1\/2}$ as in (\\ref{det}).\n\nIn the same way one can show that the  equation giving $\\mbox{Tr}(ZZ^\\dagger)^3$ as in (\\ref{det}) is\nthe SU(8) invariant equation\n\\begin{equation}\n {\\partial I\\over \\partial Z_{AB}}{\\partial I\\over \\partial \\bar Z^{AB}}=0\\label{inv2}.\n \\end{equation}\n\nIn the generic case the 1\/8 BPS states will correspond to 4 different\neigenvalues. They are explicitely given as follows. Define the quantities\n\\begin{eqnarray}\nu&=& b^2 +12d-3ca\\nonumber\\\\\nv&=& 2b^3+27c^2-72bd -9abc +27 da^2\\nonumber\\\\\nw&=&\\left({v+\\sqrt{v^2-4u^3}\\over 2}\\right)^{1\/3}\\nonumber\\\\\ns&=&\\sqrt{{a^2\\over 4}-{2b\\over 3}+{u\\over 3w}+{w\\over 3}}\n\\end{eqnarray}\nThen,\n\\begin{eqnarray}\n\\lambda_{1,2}=-{a\\over 4}+{s\\over 2}\\pm{1\\over 2}\\sqrt{{a^2\\over 2}-{4b\\over 3}\n-{a^3-4ab +8c\\over 4s}-{u\\over 3w}-{w\\over 3}}\\nonumber\\\\\n\\lambda_{3,4}=-{a\\over 4}-{s\\over 2}\\pm{1\\over 2}\\sqrt{{a^2\\over 2}-{4b\\over 3}\n+{a^3-4ab +8c\\over 4s}-{u\\over 3w}-{w\\over 3}}.\n\\end{eqnarray}\nThe BPS mass, $m_{BPS}^2$ is $\\lambda_1$. This is the actual\ndetemination of the energy spectrum for 1\/8 BPS states in terms of the\nduality invariant quantities (\\ref{abcd}).\n\nIt is amusing that analytic expressions for the roots of a polynomial\nexist only up to quartic equations, as found by Galois \\cite{ed}, and\nthis is precisely what is required by maximal supersymmetry ($N=8$ at $D=5,6$).\n\n\\paragraph{Case $d=5$.}\n\nThe   central charge $\\hat Z_{AB}$ is a symplectic, $\\Omega$-traceless antisymmetric  matrix;\n\\begin{equation}\n \\bar {\\hat Z}=-\\Omega\\hat Z\\Omega,\\quad \\hat Z^T=-\\hat Z,\\quad \\mbox{Tr}\\hat Z\\Omega=0.\n\\end{equation}\nThis implies that the matrix\n\\begin{equation}\nZ=\\hat Z \\Omega\n\\end{equation}\nis hermitian traceless. The characteristic equation for $Z$ becomes\n\\begin{equation}\n\\sqrt{\\mbox{det}Z-\\lambda {\\bf I}}=\\prod_{i=1}^4(\\lambda-\\lambda_i)\n=\\lambda^4+b\\lambda^2 +c\\lambda +d=0\n\\end{equation}\nwhere\n\\begin{eqnarray}\nb&=& -{1\\over 4}\\mbox{Tr}Z^2\\nonumber\\\\\nc&=&  -{1\\over 6}\\mbox{Tr}Z^3\\nonumber\\\\\nd&=&  {1\\over 8}\\left({1\\over 4}(\\mbox{Tr}Z^2)^2-\\mbox{Tr}Z^4\\right)\n\\end{eqnarray}\nA 1\/4 BPS state is a state for which $c=0$. This is an E$_6$ invariant\nstatement since $c=I_3$ is the E$_6$ cubic invariant. In this case we get\n\\begin{equation}\n2\\lambda_{1,2}^2= {1\\over 4}\\mbox{Tr}Z^2\\pm \\sqrt{ {1\\over 2}\\mbox{Tr}Z^4\n -{1\\over 16}(\\mbox{Tr}Z^2)^2}\n \\end{equation}\n The discriminant is related to the modulus of the USp(8) (and E$_{6(6)}$) vector\n \\begin{equation}\n V_B^A={\\partial I\\over \\partial Z_A^B} \\approx Z^C_AZ^B_C-{1\\over 8}Z^C_DZ^D_C\\delta_A^B\n \\end{equation}\n Indeed,\n \\begin{equation}\n \\mbox{Tr}V^2= \\mbox{Tr}Z^4-{1\\over 8} (\\mbox{Tr}Z^2)^2.\n \\end{equation}\n The condition for 1\/2 BPS is that the discriminant vanishes. Therefore,\nthis implies, by positivity, $V=0$, which is an E$_6$ invariant statement\n\\begin{equation}\n{\\partial I\\over \\partial Z_A^B }=0\n\\end{equation}\nWe therefore have retrieved the results of  Maldacena and one of the\nauthors \\cite{fm}.\n\nFor the 1\/8 BPS state the 4 roots are given by\n\\begin{eqnarray}\n \\lambda_{1,2}={s\\over 2}\\pm{1\\over 2}\\sqrt{{-4b\\over 3}-{2c\\over s}-{u\\over 3w}\n -{w\\over 3}}\\nonumber\\\\\n \\lambda_{3,4}=-{s\\over 2}\\pm{1\\over 2}\\sqrt{{-4b\\over 3}+\n {2c\\over s}-{u\\over 3w}\n -{w\\over 3}}\n \\end{eqnarray}\n where\n \\begin{eqnarray}\n u&=&b^2 +12d,\\quad z=2b^3+27c^2-72bd\\nonumber\\\\\n w&=&\\left({z+\\sqrt{z^2-4u^3}\\over 2}\\right)^{1\/3}, \\quad s=\\sqrt{{w\\over 3}\n+{u\\over 3w}-{2b\\over 3}}\n\\end{eqnarray}\nThe BPS mass is therefore given by the highest root, $\\lambda_1$.\n\n\n\\section{BPS  conditions for theories with 16 supersymmetries}\n\nIn this last section we will extend our analysis to theories with 16 supersymmetries.\nThese theories are obtained in three different ways: by compactifying Heterotic string\ntheory on T$^d$ ($1\\leq d\\leq 6$),  from M theory compactified on K$_3$ ($D=7$) and from\nType IIA\ntheory compactified on   K$_3$ ($D=6$).\n\nIn the theories where matter vector fields exist, the duality group $G$ depends on the matter\ncontent and on the space-time dimension $D$. Its maximal compact subgroup   is\n$H_R\\times H_M$ where $H_R$ is the R-symmetry and $H_M$ is the group\nacting on the matter multiplets. In our case, $H_M$=O($n$), where $n$ is\nthe number of matter multiplets. $G$ is of the form O(10-$D,n$)$\\times$O(1,1)\nfor $5\\leq D\\leq 9$ while for $D=4$ it is SL(2)$\\times$O(6,$n$). The R-symmetry  groups are\nO(10-$D$) for $5\\leq D\\leq 9$ and O(6)$\\times$O(2)$\\approx$SU(4)$\\times$U(1) for $D$=4. The last\nresult can easily been understood from the geometric symmetry of\nHeterotic string on T$^6$, where $G$ is enlarged by the electric-magnetic duality for\n0-branes.\n\nThe $G$ and $H_R$ representations of the 0-branes are given in the following tables.\n\n\n\\begin{center}\n{\\bf Central charge representation of $H_R$.}\n\\begin{eqnarray}\nd=1 &\\quad {\\bf 1}\\quad  & \\mbox{O}(1)={\\bf I}\\nonumber\\\\\nd=2 &\\quad {\\bf 1^c}\\; \\mbox{complex} & \\mbox{U}(1)\\approx \\mbox{O}(2)\\nonumber\\\\\nd=3 &\\quad {\\bf 3}\\; \\mbox{real} & \\mbox{SU}(2)\\approx \\mbox{USp}(2)\\nonumber\\\\\nd=4 &\\quad {\\bf 4}\\; \\mbox{real} &  \\mbox{O}(4)\\approx \\mbox{USp}(2)\\times\\mbox{USp}(2)\\nonumber\\\\\nd=5 &\\quad {\\bf 1+5}\\; \\mbox{real}&  \\mbox{O}(5)\\approx \\mbox{USp}(4)\\nonumber\\\\\nd=6 &\\quad {\\bf 6^c}\\; \\mbox{complex}  & \\mbox{O}(6)\\times\\mbox{O}(2)\\approx \\mbox{SU}(4)\\times\\mbox{U}(1)\n\\end{eqnarray}\n\\end{center}\n\nFrom the above table, and according our previous analysis,\nit follows that the central charge matrix $Z_a$ has only one  independent eigenvalue\nfor $d=1,\\dots 4$ and two independent eigenvalues for $d=5,6$. Therefore, for $d=1,\\dots 4$\nonly 1\/2 BPS states can occur while for $d=5,6$ both, 1\/2 and 1\/4 BPS states can occur.\n\n\\begin{center}\n{\\bf 0-brane representation of $G$}\n\\begin{eqnarray}\nd=1,\\dots 4 & {\\bf d+n}\\; \\mbox{real vector}& \\mbox{O}(d,n)\\times \\mbox{O}(1,1)\\nonumber\\\\\nd=5 & {\\bf (1,2)+(5\\!+\\! n,-1)}\\; \\mbox{(singlet+vector)} & \\mbox{O}(5,n)\\times \\mbox{O}(1,1)\\nonumber\\\\\nd=6 & {\\bf(2, 6\\!+\\!n) }\\;  & \\mbox{Sl}(2)\\times\\mbox{SO}(6,n)\n\\end{eqnarray} \\label{diss}\n\\end{center}\n\nWe consider now separately the two cases $d=5,6$.\n\n\\paragraph{Case $d=5$.}\n\nThis is the case which corresponds to heterotic string on T$^5$ or M theory (Type IIA, Type IIB)\non K$_3\\times$T$^2$ (K$_3\\times$S$^1$). In such compactifications, $n=21$, so $G$=O(5,21)$\\times$\nO(1,1) but our analysis is independent of  this specific number $n$.\n\nThe central charge $\\hat Z$ is an antisymmetric symplectic matrix. The hermitian matrix,\n $Z=\\hat Z\\Omega$ decomposes as\n\\begin{equation}\nZ=Z^a\\gamma_a +Z^0{\\bf I}\n\\end{equation}\nwhere $\\gamma_a$ are the O(5) $\\gamma$-matrices and $Z^a, Z^0$ are real.\nIt follows that\n\\begin{eqnarray}\n&&\\mbox{Tr}Z=4 Z^0\\nonumber \\\\\n&&(\\mbox{detZ)}^{1\/2}={Z^0}^2-\\vec{Z}^2=  {1\\over 8}(\\mbox{Tr}Z)^2-{1\\over 4}\\mbox{Tr}Z^2\n\\label{trdet}\\end{eqnarray}\nThe  characteristic equation (or better, its square root) is\n\\begin{equation}\n\\lambda^2-{1\\over 2}\\mbox{Tr}Z\\, \\lambda +(\\mbox{det}Z)^{1\/ 2}=0\n\\end{equation}\nimplying that  $Z$ has two coinciding eigenvalues (in absolute value) either if\n\\begin{equation}\n\\mbox{Tr}Z=0 \\quad \\mbox{or}\\quad {1\\over 4}(\\mbox{Tr}Z)^2=4(\\mbox{det}Z)^{1\/2}\n\\end{equation}\nUsing (\\ref{trdet}), the above equation directly implies\n\\begin{equation}\nZ_0Z_a=0. \\label{condi}\n\\end{equation}\nThe eigenvalues are given by\n\\begin{equation}\n\\lambda_{1,2}={1\\over 2}\\left({1\\over 2}\\mbox{Tr}Z\\pm \\sqrt{\\mbox{Tr}Z^2-{1\\over 4}\n(\\mbox{Tr}Z)^2}\\right),\n\\end{equation}\nbeing the plus sign the mass squared of the BPS state.\n\nWe discuss now the covariance of (\\ref{condi}).  Since $Z_0=e^{2\\sigma}m$\nwhere $e^{2\\sigma}$ parametrizes O(1,1) and $m$ is the charge associated to  $Z_0$,\n$Z_0=0$ implies $m=0$ which is an O(5,n) singlet,\nso it is $G$-invariant.\n\nAccording to table (\\ref{diss}) we write the projection of the coset representative\nover the {\\bf (5+n, -1)}\nrepresentation as $e^{-\\sigma}L_a^\\Lambda$ where $\\sigma$\nparametrizes O(1,1) and $L_a^\\Lambda$ is the coset representative of\nO(5+n)\/O(5)$\\times$O(n).\nIf $Z_I,\\; I=1,\\dots n$ are the matter charges associated to the $n$\nmatter multiplets, we have that, because of (\\ref{fmc})\n\\begin{equation}\n\\nabla_{O(5)}Z_a={1\\over 4}\\mbox{Tr}(\\gamma_aP_I)Z^I -Z_ad\\sigma\n\\end{equation}\n therefore $Z_a=0$ implies $Z_I=0$.\nThis is also an O(5,n) invariant statement since, it comes by differentiating the quadratic\ninvariant polynomial\n\\begin{equation}\nI=\\sum_{a=1}^5Z_aZ^a- \\sum_{I=1}^MZ_IZ^I.\n\\end{equation}\nTherefore, $Z_a=Z_I=0$ implies $q^\\Lambda=0$ where $q^\\Lambda$,\n$\\Lambda=1,\\dots 5+n$, is a fixed charge vector of O(5,M),\nas found in \\cite{fm}.\n\n\\paragraph{Case $d=6$, $(D=4)$.}\n\nWe now consider theories with 16 supersymmetries in $D=4$, as heterotic string compactified\non T$^6$, TypeII on K$_3\\times$T$^2$ or M theory on   K$_3\\times$T$^3$. The new phenomenon\n which occurs here is the electric-magnetic duality of 0-branes which are assigned to the\n $(2,6+n)$ representation of SU(1,1)$\\times$O(6,n).\n\nThe central charge is a 4 dimensional complex matrix $Z_{AB}$, antisymmetric in the\nSU(4)$\\approx$O(6) indices. Therefore, the matrix\n$ZZ^\\dagger$ has two independent eigenvalues, given by the characteristic equation\n\\begin{eqnarray}\n&&\\left(\\mbox{det}(ZZ^\\dagger-\\lambda {\\bf I})\\right)^{1\/2}=0\\nonumber\\\\\n&&\\lambda^2-{1\\over 2}\\mbox{Tr}ZZ^\\dagger\\, \\lambda +(\\mbox{det}ZZ^\\dagger)^{1\/2}=0\n\\end{eqnarray}\nwith solution\n\\begin{equation}\n\\lambda_{1,2}={1\\over 2}\\left({1\\over 2}\\mbox{Tr}ZZ^\\dagger\\pm \\sqrt{\\mbox{Tr}(ZZ^\\dagger))^{2}-\n{1\\over 4}(\\mbox{Tr}ZZ^\\dagger))^{2}}\\right).\\label{solut}\n\\end{equation}\n\nA generic 1\/4 BPS state has $m_{BPS}^2$ equal to the eigenvalue with +\nsign above. The 1\/2 BPS configuration corresponds to a vanishing discriminant, i.e.\n$\\lambda_1=\\lambda_2$. We would like to show how this condition is SU(1,1)$\\times$O(6,$n$)\ninvariant in the sense that it is moduli independent in spite of the fact that\nthe discriminant is moduli dependent.\n\nFor this purpose we proceed like for the maximally supersymmetric case in $D=4$. If the\ntwo eigenvalues of  $ZZ^\\dagger$ coincide, then the  hermitian traceless matrix\n\\begin{equation}\nV_A^C=Z_{AC}\\bar Z^{BC} - {1\\over 4}\\delta_A^BZ_{PQ}\\bar Z^{PQ} \\label{traceless}\n\\end{equation}\nvanishes. (The discriminant is just  Tr$V^2$, the invariant norm of the SU(4) vector $V_A^C$).\n\nConsider now the  SU(1,1)$\\times$O(6,n) quartic invariant ,\n\\begin{equation}\nI=I_1^2-I_2\\bar I_2\n\\end{equation}\nwhere\n\\begin{eqnarray}\nI_1&=&Z_{AB}\\bar Z^{AB}-Z_I\\bar Z^I\\nonumber\\\\\nI_2&=&{1\\over 4}\\epsilon^{ABCD}Z_{AB}Z_{CD}-\\bar Z_I\\bar Z^I.\n\\end{eqnarray}\nThe fact that $I$ is an invariant was derived in Ref.\\cite{adf4} and can\nbe  easily understood from the fact that $(I_1, I_2, \\bar I_2)$  is a\ntriplet of SU(1,1)$\\approx$O(1,2), each of the entries being O(6,n) invariant.\n\nThe equation (\\ref{traceless}) can be seen as the second derivative of $I$\nprojected onto the adjoint representation of SU(4).\n\\begin{equation}\nV_A^C\\approx {\\partial^2 I\\over \\partial Z_{AB}\\partial \\bar Z^{CD}}\\left|_{Adj_{SU(4)}}\\right.\n\\end{equation}\n\nIndeed, let us call $U$ the ${\\bf (2,6+n)}$ of Sl(2) vector\nconstructed with $(Z_{AB},Z_I)$ and its complex conjugate. The\nquantity\n\\begin{equation}{\\partial^2 I\\over \\partial U\\partial U}\\label{sder}\n\\end{equation}\nis in the symmetric product $\\left({\\bf (2,6+n)}\\times {\\bf (2,6+n)}\\right)\\mid_S$, which\ndecomposes under O(6,$M$) as {\\bf(3, Sym)}+{\\bf(1,Adj$_{O(6,n)}$)}= {\\bf(3,1)}+\n {\\bf(3,TrSym)}+{\\bf(1,Adj$_{O(6,n)}$)},\nwhere {\\bf Sym} is the two fold symmetric representation, {\\bf TrSym} is the traceless\nsymmetric representation. To show that\nthe $V_A^C=0$ is a $G$-invariant statement we use the fact that {\\bf Adj$_{O(6,n)}$}\n decomposes under\nO(6)$\\times$O(n) as {\\bf Adj$_{O(6,n)}$}$\\mapsto$ {\\bf (Adj$_{O(6)}$,1)}+{\\bf (1,\n Adj$_{O(n)}$)} +{\\bf (6,n)}.\n We will show that the vanishing of the projection onto\n {\\bf Adj$_{O(6)}$ }$\\approx${\\bf Adj$_{SU(4)}$}  of\n(\\ref{sder}) implies the vanishing of the projection onto\n{\\bf (1, Adj$_{O(n)}$)}  and {\\bf (6,n)}. In fact, differentiating $V_A^C=0$ and\nusing the differential identities (\\ref{fmc}) one also finds\n\\begin{eqnarray}\nZ_I\\bar Z_J -\\bar Z_IZ_J=0,\\label{zeta1}\\\\\nZ_{AB}Z_J-{1\\over 4}\\epsilon_{ABCD}\\bar Z^{CD}\\bar Z_J=0.\n\\label{zeta2}\n\\end{eqnarray}\nThe vanishing of the three equations $V_A^C=0$, (\\ref{zeta1}) and\n(\\ref{zeta2}) implies that the projection of (\\ref{sder}) on {\\bf\nAd$_{O(6,n)}$} vanishes. This\nis  a SU(1,1)$\\times$SO(6,$n$) invariant and therefore the moduli dependence drops out.\nThese three equations  can be rewritten in terms of the fixed charges $(q_\\Lambda,\n p_\\Lambda)$,  in the {\\bf (6+n)} of O(6,$n$)  times\n the fundamental representation of Sl(2)$\\approx$SU(1,1)) as\n\\begin{equation}\nT_{\\Lambda\\Sigma}^{(A)}=q_\\Lambda p_\\Sigma -p_\\Lambda q_\\Sigma=0.\n\\end{equation}\nNote that in this basis the projection  of (\\ref{sder}) onto the representation\n{\\bf (3,Sym)} is\n\\begin{equation}\nT_{\\Lambda\\Sigma}^{(S)}=(q_\\Lambda q_\\Sigma,\\; p_\\Lambda p_\\Sigma,\\; {1\\over 2 }\n(q_\\Lambda p_\\Sigma +p_\\Lambda q_\\Sigma))\n\\end{equation}\nwhose trace part is the Sl(2) triplet $(q^2.p^2,q\\cdot p)$. It can be written as a matrix\n\\begin{equation}\nT^{(0)}=\\pmatrix{q^2& q\\cdot p\\cr\nq\\cdot p&p^2}.\n\\end{equation}\nThe invariant $I$ can be written either as $T_{\\Lambda\\Sigma}^{(A)}{T^{(A)}}^{\\Lambda\\Sigma}$\n or as det$T^0$, and its square root is the entropy formula for 1\/4 BPS 0-branes in theories with\n sixteen supersymmetries \\cite{fsf, cdcc}.\n\n\\medskip\n\nAs a final remark, let us comment on the orbits of the  O(5,n)\n and O(6,n) vectors  for BPS configurations discussed above.\n\n For 1\/2 BPS states  at $d=5$ we have $mq_\\Lambda=0$, so either $m$ or\n  $q_\\Lambda$ vanish. In the former case, the BPS condition requires $q_\\Lambda$\n to be  time-like or light-like $(q_\\Lambda q^\\Lambda\\geq 0)$ \\cite{fm} so\nthe orbit is either\nO(5,$n$)\/O(4)$\\times$O($n$) or O(5,$n$)\/IO(4,$n-1$).\nIf $m\\neq 0$ then $q_\\Lambda=0$, so the orbit is a point since the\nlittle group is O(5,n) itself.\n\nLet us consider now the $d=6$ case. The BPS condition corresponds to the\nstatement that the matrix $T^{(0)}$ is positive semidefinite. This implies\n\\begin{equation}\n\\mbox{det}T^{(0)}=q^2p^2-(q\\cdot p)^2\\geq 0,\\quad \\mbox{Tr}T^{(0)}=q^2\n+p^2\\geq 0.\n\\end{equation}\nFrom this it follows that $q^2\\geq 0$ and $p^2\\geq 0$.\n\n$\\mbox{det}T^{(0)}=0$ corresponds to 1\/2 BPS states; this happens when\n$q=\\lambda p$, $(\\lambda\\geq 0)$.\n\nFor $\\mbox{det}T^{(0)}>0$, $q^2> 0$ and $p^2>0$ and the generic 1\/4 BPS\nconfiguration will depend on five parameters, since $p,q$, by an O(2)\ntransformation in SL(2) can be made orthogonal $(q_\\Lambda\nP^\\Lambda=0)$. Indeed, the first vector can be put in the form\n$(p_1,0,\\cdots,0,p_{n+1},0,\\cdots, 0)$ and the second in the form\n$(q_1, q_2,0,\\cdots,0,q_{n+1},q_{n+2},0,\\cdots, 0)$ by an O(6)$\\times$O(n)\ntransformation. The orthogonality condition is used to eliminate one of\nthe six parameters. The remaining $7+2n$ parameters are the \"angles\" in\nO(2)$\\times$O(6)$\\times$O(n)\/O(4)$\\times$O($n$-2). The little group in\n$G$ of the two time-like vectors is O(4)$\\times$O($n$).\n\n\\bigskip\n\n\\begin{center} {\\bf Acknowledgements.}\n\\end{center}\n\\bigskip\n\nWe would like to thank Anna  Ceresole, Alberto  Zaffaroni and especially Raymond Stora\nfor enlightening discussions.\n\n\\bigskip\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{#2} \\label{sec:#1} \\input{tex\/#1}}\n\\newcommand{\\includesubsection}[3]{\\subsection{#2} \\label{sec:#1} \\input{tex\/#1}}\n\n\\newcommand{\\TODO}[1]{{\\color{red}{[{\\bf TODO}: #1]}}}\n\\newcommand{\\raj}[1]{{\\color{blue}{[{\\bf Prithviraj}: #1]}}}\n\\newcommand{\\rajk}[1]{{\\color{green}{[{\\bf RajK}: #1]}}}\n\\newcommand{\\kb}[1]{{\\color{purple}{[{\\bf Kiante}: #1]}}}\n\\definecolor{salmon}{HTML}{F69289}\n\\newcommand{\\jack}[1]{{\\color{salmon}{[{\\bf Jack}: #1]}}}\n\\newcommand{\\yejin}[1]{{\\color{cyan}{[{\\bf Yejin}: #1]}}}\n\\newcommand{\\hanna}[1]{{\\color{orange}{[{\\bf Hanna}: #1]}}}\n\n\\newcommand{GRUE}{GRUE}\n\\definecolor{forestgreen}{rgb}{0.0, 0.27, 0.13}\n\\definecolor{forestgreen2}{rgb}{0.13, 0.55, 0.13}\n\\definecolor{Gray}{gray}{0.90}\n\\definecolor{lightgray}{gray}{0.9}\n\\newcolumntype{a}{>{\\columncolor{Gray}}c}\n\\iclrfinalcopy %\n\\input{macros.tex}\n\n\n\n\n\\begin{document}\n\n\n\n\\maketitle\n\n\\begin{abstract}\n\\input{tex\/main\/00_abstract.tex}\n\\end{abstract}\n\n\n\\includesection{main\/10_intro}{Introduction}{}\n\\includesection{main\/20_background}{Related Work}{}\n\\includesection{main\/30_framework}{\\framework: A Library for Training LMs with RL}{}\n\\includesection{main\/40_nlpo}{NLPO: Natural Language Policy Optimization}{}\n\\includesection{main\/50_evaluation}{\\shortname{} (\\longname{})}{}\n\\includesection{main\/60_conclusion}{Conclusions}{}\n\n\\clearpage\n\n\n\n\\subsubsection{Setup}\n\n\n\\begin{table*}[ht!]\n\\centering\n\\footnotesize\n\\resizebox{0.5\\textwidth}{!}{\n\\begin{tabular}{ll}\n\\toprule\n\\textbf{Model Params}\n& \\multicolumn{1}{c}{\\textbf{value}} \\\\ \n\\cmidrule{1-2}\nsupervised & batch size: $64$\\\\\n& epochs: $10$ \\\\\n& learning rate: $0.00001$ \\\\\n\\cmidrule{1-2}\nppo & steps per update: $1280$\\\\\n    & total number of steps: $64000$ \\\\\n    & batch size: $64$ \\\\\n    & epochs per update: $5$ \\\\\n    & learning rate: $0.000001$ \\\\\n    & discount factor: $0.99$ \\\\\n    & gae lambda: $0.95$ \\\\\n    & clip ratio: $0.2$ \\\\\n    & value function coeff: $0.5$\\\\\n\\cmidrule{1-2}\nnlpo & steps per update: $1280$\\\\\n    & total number of steps: $64000$ \\\\\n    & batch size: $64$ \\\\\n    & epochs per update: $5$ \\\\\n    & learning rate: $0.000001$ \\\\\n    & discount factor: $0.99$ \\\\\n    & gae lambda: $0.95$ \\\\\n    & clip ratio: $0.2$ \\\\\n    & top mask ratio: $0.9$ \\\\\n    & target update iterations: $5$ \\\\\n\\cmidrule{1-2}\ndecoding & sampling: true \\\\\n& top k: $50$ \\\\\n& min length: $48$ \\\\\n& max new tokens: $48$\\\\\n\n\\cmidrule{1-2}\ntokenizer & padding side: left\\\\\n& truncation side: left \\\\\n& max length: $64$ \\\\\n\\bottomrule\n\\end{tabular}\n}\n\\caption{\\textbf{IMDB Hyperparams}: Table shows a list of all hyper-parameters and their settings}\n       \\label{tbl:imdb_hyperparams}\n\\end{table*}\n\nWe consider IMDB dataset for the task of generating text with positive sentiment. The dataset consists of 25k training, 5k validation and 5k test examples of movie review text with sentiment labels of positive and negative. The input to the model is a partial movie review text (upto 64 tokens) that needs to be completed (generating 48 tokens) by the model with a positive sentiment while retaining fluency. For RL methods, we use a sentiment classifier \\cite{sanh2019distilbert} that is trained on pairs of text and labels as a reward model which provides sentiment scores indicating how positive a given piece of text is. For supervised Seq2Seq baselines, we consider only the examples with positive labels. We chose GPT-2 as LM for this task as it is more suited for text continuation than encoder-decoder LMs (eg. T5). We use top-k sampling with $K=50$ as the decoding method and for fair comparison, we keep this setting for all methods. For PPO and NLPO models, we train for $64k$ steps in total and update policy and value networks every $1280$ steps with a mini-batch size of $64$ and epochs of $5$ per update. We apply adaptive KL controllers with different target KLs of $0.02, 0.05, 0.2, 0.5, 1.0, \\inf$ with an initial KL co-efficient of $\\beta=0.01$. Table \\ref{tbl:imdb_hyperparams} provides an in-depth summary of all hyperparameters and other implementation details.\n\n\n\n\n\\begin{figure*}[h]\n    \\centering\n  \\subfigure[PPO Episodic total reward]{\\includegraphics[width=0.31\\textwidth]{figures\/ppo_text_continuation\/rollout_info_ep_rew.pdf}}\n  \\subfigure[PPO Val avg sentiment score]{\\includegraphics[width=0.31\\textwidth]{figures\/ppo_text_continuation\/val_metric_learned_automodel_metric.pdf}}\n  \\subfigure[PPO Val perplexity]{\\includegraphics[width=0.31\\textwidth]{figures\/ppo_text_continuation\/val_metric_perplexity.pdf}} \\\\\n  \\subfigure[NLPO Episodic total reward]{\\includegraphics[width=0.31\\textwidth]{figures\/nlpo_text_continuation\/rollout_info_ep_rew.pdf}}\n  \\subfigure[NLPO Val avg sentiment score]{\\includegraphics[width=0.31\\textwidth]{figures\/nlpo_text_continuation\/val_metric_learned_automodel_metric.pdf}}\n  \\subfigure[NLPO Val perplexity]{\\includegraphics[width=0.31\\textwidth]{figures\/nlpo_text_continuation\/val_metric_perplexity.pdf}} \\\\\n  \\caption{\\textbf{Learning Curves}: Averaged learning curves over 5 different runs by varying target KL, shaded regions indicate one standard deviation. (a) shows the rollout episodic total reward during training (b) shows evolution of sentiment scores on the validation split (c) shows evolution of perplexity on the validation split. From (a) and (b), it is seen that higher target KL (0.2) is desired to achieve higher rewards. However, this setting drifts away from the original LM too much and loses fluency. Therefore a lower target KL (0.02 or 0.05) is required to keep the model closer to original LM. Similar trends hold for NLPO but when compared to PPO, it retains lower perplexities and is more stable even with higher KL targets, enabling higher sentiment scores.}\n    \\label{fig:gpt2_sent_cont_learning_curves}\n\\end{figure*}\n\n\\begin{table*}[ht!]\n    \\centering\n    \\resizebox{\\textwidth}{!}{\n    \\begin{tabular}{@{}ccccccccccccc@{}}\n    \\toprule\n    \\textbf{Target-KL} & \\multicolumn{2}{c}{\\textbf{Semantic and Fluency Metrics}} & \\multicolumn{7}{c}{\\textbf{Diversity Metrics}}          \\\\ \\cmidrule(lr){2-3}\\cmidrule(lr){4-10}\n    & Sentiment Score $\\uparrow$ & Perplexity $\\downarrow$ & MSTTR & Distinct$_1$ & Distinct$_2$ & H$_1$ & H$_2$ & Unique$_1$ & Unique$_2$ \\\\\\cmidrule(lr){1-1}\\cmidrule(lr){2-3}\\cmidrule(lr){4-10}\n    Zero-Shot & 0.489 $\\pm$ 0.006           & 32.171 $\\pm$ 0.137          & 0.682 $\\pm$ 0.001 & 0.042 $\\pm$ 0.001 & 0.294 $\\pm$ 0.001 & 8.656 $\\pm$ 0.004  & 13.716 $\\pm$ 0.003 & 5063 $\\pm$ 14.832 & 47620 $\\pm$ 238  \\\\\n    Supervised & 0.539 $\\pm$ 0.004 & 35.472 $\\pm$ 0.074 & 0.682 $\\pm$ 0.001 & 0.047 $\\pm$ 0.001 & 0.312 $\\pm$ 0.002 & 8.755 $\\pm$ 0.012 & 13.806 $\\pm$ 0.016 & 5601 $\\pm$ 57 & 51151 $\\pm$ 345 \\\\\n    \\cmidrule(lr){1-1}\\cmidrule(lr){2-3}\\cmidrule(lr){4-10}\n    \\multicolumn{10}{l}{PPO} \\\\ \n    0.02      & 0.530 $\\pm$ 0.021           & 32.921 $\\pm$ 0.322          & 0.680 $\\pm$ 0.002 & 0.042 $\\pm$ 0.001 & 0.293 $\\pm$ 0.002 & 8.642 $\\pm$ 0.015   & 13.676 $\\pm$ 0.025 & 5042 $\\pm$ 135    & 47554 $\\pm$ 418  \\\\\n    0.05      & 0.578 $\\pm$ 0.022           & 33.469 $\\pm$ 0.532          & 0.660 $\\pm$ 0.021 & 0.044 $\\pm$ 0.002 & 0.287 $\\pm$ 0.011 & 8.553 $\\pm$ 0.130  & 13.389 $\\pm$ 0.356 & 5352 $\\pm$ 251    & 46158 $\\pm$ 2568 \\\\\n    0.2       & 0.585 $\\pm$ 0.006           & 33.627 $\\pm$ 0.236          & 0.665 $\\pm$ 0.005 & 0.044 $\\pm$ 0.001 & 0.287 $\\pm$ 0.008 & 8.584 $\\pm$ 0.055 & 13.438 $\\pm$ 0.124 & 5315 $\\pm$ 171    & 46834 $\\pm$ 1469 \\\\\n    0.5       & 0.605 $\\pm$ 0.023 & 33.497 $\\pm$  0.447 & 0.666 $\\pm$ 0.013 & 0.043 $\\pm$ 0.002 & 0.287 $\\pm$  0.008 &  8.575 $\\pm$ 0.073 & 13.484 $\\pm$ 0.244 & 5230 $\\pm$ 363 & 46483 $\\pm$ 1318 \\\\\n    1.0       & 0.579 $\\pm$  0.025 & 33.161 $\\pm$ 0.117 & 0.676 $\\pm$ 0.002 & 0.042 $\\pm$ 0.001 & 0.291 $\\pm$ 0.007 & 8.625 $\\pm$ 0.041 & 13.625 $\\pm$ 0.089 & 5027 $\\pm$ 173 & 47082 $\\pm$ 1375 \\\\ %\n     inf       & 0.847 $\\pm$ 0.039 & 40.650 $\\pm$ 2.154 & 0.566 $\\pm$ 0.038 &  0.035 $\\pm$ 0.006 & 0.200 $\\pm$ 0.025 & 7.715 $\\pm$ 0.289 & 11.763 $\\pm$ 0.496 & 4380 $\\pm$ 775 & 32462 $\\pm$ 5020 \\\\ \\cmidrule(lr){1-1}\\cmidrule(lr){2-3}\\cmidrule(lr){4-10}\n     \n     \\multicolumn{10}{l}{PPO+supervised} \\\\\n     0.5 & 0.617 $\\pm$ 0.011 &34.078 $\\pm$ 0.253 &0.672 $\\pm$ 0.003 &0.047 $\\pm$ 0.002 &0.308 $\\pm$ 0.007 &8.725 $\\pm$ 0.054 &13.711 $\\pm$ 0.068 &5513 $\\pm$ 173 &50410 $\\pm$ 1277  \\\\\n     inf & 0.829 $\\pm$ 0.049 &46.906 $\\pm$ 2.168 &0.615 $\\pm$ 0.037 &0.037 $\\pm$ 0.005 &0.225 $\\pm$ 0.04 &8.072 $\\pm$ 0.373 &12.537 $\\pm$ 0.707 &4480 $\\pm$ 443 &35583 $\\pm$ 6631 \\\\ \\cmidrule(lr){1-1}\\cmidrule(lr){2-3}\\cmidrule(lr){4-10}\n     \n    \\multicolumn{10}{l}{NLPO} \\\\\n    0.02 & 0.530 $\\pm$ 0.020 & 32.824 $\\pm$ 0.227 & 0.680 $\\pm$ 0.002 & 0.043 $\\pm$ 0.001 & 0.295 $\\pm$ 0.004 & 8.658 $\\pm$ 0.031 & 13.689 $\\pm$ 0.050 & 5129 $\\pm$ 169 & 47863 $\\pm$ 840 \\\\\n    0.05 & 0.581 $\\pm$ 0.017 & 32.298 $\\pm$ 0.362 & 0.674 $\\pm$ 0.004 & 0.043 $\\pm$ 0.001 & 0.295 $\\pm$ 0.008 & 8.647 $\\pm$ 0.040 & 13.638 $\\pm$ 0.099 & 5110 $\\pm$ 121 & 47911 $\\pm$ 1478 \\\\\n    0.2       & 0.591 $\\pm$ 0.012 & 32.602 $\\pm$ 0.261 & 0.663 $\\pm$ 0.015 & 0.044 $\\pm$ 0.001 & 0.287 $\\pm$ 0.012 & 8.586 $\\pm$ 0.096 & 13.442 $\\pm$ 0.240 & 5314 $\\pm$ 166 & 46665 $\\pm$ 2124 \\\\\n    0.5       & 0.611 $\\pm$ 0.014           & 32.241 $\\pm$ 0.932          & 0.650 $\\pm$ 0.035  & 0.042 $\\pm$ 0.002 & 0.271 $\\pm$ 0.013 & 8.389 $\\pm$ 0.270   & 13.081 $\\pm$ 0.741 & 5159 $\\pm$ 536    & 43840 $\\pm$ 1217 \\\\\n    1.0       & 0.637 $\\pm$ 0.013           & 32.667 $\\pm$ 0.631          & 0.677 $\\pm$ 0.014 & 0.044 $\\pm$ 0.002 & 0.288 $\\pm$ 0.010  & 8.588 $\\pm$ 0.100    & 13.484 $\\pm$ 0.236 & 5205 $\\pm$ 189    & 46344 $\\pm$ 2688 \\\\ %\n     inf & 0.859 $\\pm$ 0.041 &37.553 $\\pm$ 3.22 &0.567 $\\pm$ 0.037 &0.036 $\\pm$ 0.007 &0.205 $\\pm$ 0.034 &7.725 $\\pm$ 0.326 &11.772 $\\pm$ 0.571 &4568 $\\pm$ 1046 &33056 $\\pm$ 6365\\\\ \\cmidrule(lr){1-1}\\cmidrule(lr){2-3}\\cmidrule(lr){4-10}\n     \n     \\multicolumn{10}{l}{NLPO+supervised} \\\\\n     1.0 & 0.645 $\\pm$ 0.027 &33.191 $\\pm$ 0.187 &0.656 $\\pm$ 0.014 &0.049 $\\pm$ 0.005 &0.3 $\\pm$ 0.026 &8.648 $\\pm$ 0.213 &13.396 $\\pm$ 0.331 &6053 $\\pm$ 609 &49468 $\\pm$ 4745\\\\\n     inf & 0.853 $\\pm$ 0.106 &36.812 $\\pm$ 0.207 &0.427 $\\pm$ 0.081 &0.054 $\\pm$ 0.019 &0.205 $\\pm$ 0.077 &6.82 $\\pm$ 1.0 &9.684 $\\pm$ 1.475 &7788 $\\pm$ 2718 &35213 $\\pm$ 13349\n \\\\ \\cmidrule(lr){1-1}\\cmidrule(lr){2-3}\\cmidrule(lr){4-10}\n\n    \\end{tabular}\n    }\n    \\caption{\\textbf{Target KL Ablations}: %\n    Mean and standard deviations over 5 random seeds is reported for sentiment scores along with fluency and diversity metrics. It is seen from perplexity scores that a lower target KL constraint is desired to keep the model closer to the original model. On the otherhand, a higher target KL yields higher sentiment scores at the cost of fluency. inf KL penalty (target KL of inf), model simply learns to generate positive phrases (eg: \"I highly recommend this movie to all!\", \"worth watching\") regardless of the context}\n    \\label{tbl:text_cont_KL_ablations}\n    \\end{table*}\n\n\\subsubsection{Results and Discussion} \n\\paragraph{Target KL ablation} Fig \\ref{fig:gpt2_sent_cont_learning_curves} shows learning curves for PPO and NLPO in terms of episodic training reward, corpus level sentiment scores and perplexity scores on validation set averaged for 5 random seeds. It is seen that higher target KL of $0.2$ is desired to achieve higher rewards but results in drifting away from pre-trained LM and loses fluency. Therefore, a lower target KL (0.02 or 0.05) is required to keep the LM closer to original LM. This is also seen in Table~\\ref{tbl:text_cont_KL_ablations} where we presented a comparative analysis of final performance of all models.\n\n\\paragraph{Training data size ablation} We vary the amount of data used to train the reward classifier and the supervised baseline model to understand whether it is more efficient to gather data to improve reward model or to gather expert demonstrations for supervised learning. As observed in Table~\\ref{tbl:text_cont_extra_data}, improving the quality of reward function increases the performance on the overall task better than training with more data for supervised training, indicating that improving reward models is efficient than collect expert demonstrations for supervised training from a data efficiency perspective.\n\n\\paragraph{Discount factor ablation} To understand the effect of discounted vs undiscounted (bandit) environments, we report sentiment and perplexity scores for different values of discount factor ($0.5$, $0.95$ and $1.0$) in Table \\ref{tbl:text_cont_gamma} and observe that using a bandit environment (discount factor of $1.0$) results in performance loss in the case of NLPO and reward hacking in the case of PPO, indicating that discounted setting (with $0.95$) is desired.\n\n\\paragraph{}\n\n\\paragraph{NLPO params} Table.~\\ref{tbl:text_cont_nlpo_hyperparam} shows ablation on different hyperparameters in NLPO algorithm.\n\n\\begin{table*}[ht!]\n    \\centering\n    \\resizebox{\\textwidth}{!}{\n    \\begin{tabular}{@{}ccccccccccccc@{}}\n    \\toprule\n    \\textbf{Gamma} & \\multicolumn{2}{c}{\\textbf{Semantic and Fluency Metrics}} & \\multicolumn{7}{c}{\\textbf{Diversity Metrics}}          \\\\ \\cmidrule(lr){2-3}\\cmidrule(lr){4-10}\n    & Sentiment Score $\\uparrow$ & Perplexity $\\downarrow$ & MSTTR & Distinct$_1$ & Distinct$_2$ & H$_1$ & H$_2$ & Unique$_1$ & Unique$_2$ \\\\\\cmidrule(lr){1-1}\\cmidrule(lr){2-3}\\cmidrule(lr){4-10}\n    Zero-Shot & 0.489 $\\pm$ 0.006           & 32.371 $\\pm$ 0.137          & 0.682 $\\pm$ 0.001 & 0.042 $\\pm$ 0.001 & 0.294 $\\pm$ 0.001 & 8.656 $\\pm$ 0.004  & 13.716 $\\pm$ 0.003 & 5063 $\\pm$ 14.832 & 47620 $\\pm$ 238  \\\\\\cmidrule(lr){1-1}\\cmidrule(lr){2-3}\\cmidrule(lr){4-10}\n    \\multicolumn{10}{l}{PPO} \\\\ \n0.5 & 0.511 $\\pm$ 0.023 &35.945 $\\pm$ 0.92 &0.69 $\\pm$ 0.001 &0.044 $\\pm$ 0.002 &0.304 $\\pm$ 0.007 &8.726 $\\pm$ 0.041 &13.793 $\\pm$ 0.055 &5304 $\\pm$ 285 &49668$\\pm$ 1496 \\\\\n    0.95       & 0.605 $\\pm$ 0.023 & 33.497 $\\pm$  0.447 & 0.666 $\\pm$ 0.013 & 0.043 $\\pm$ 0.002 & 0.287 $\\pm$  0.008 &  8.575 $\\pm$ 0.073 & 13.484 $\\pm$ 0.244 & 5230 $\\pm$ 363 & 46483 $\\pm$ 1318 \\\\\n1.0 & 0.651 $\\pm$ 0.05 &41.035 $\\pm$ 2.885 &0.691 $\\pm$ 0.017 &0.042 $\\pm$ 0.004 &0.295 $\\pm$ 0.031 &8.697 $\\pm$ 0.237 &13.563 $\\pm$ 0.396 &5127 $\\pm$ 460 &48319 $\\pm$ 5650 \\\\ \\hline\n    \\multicolumn{10}{l}{NLPO} \\\\\n    0.5 & 0.49 $\\pm$ 0.01 &37.279 $\\pm$ 5.137 &0.688 $\\pm$ 0.01 &0.045 $\\pm$ 0.002 &0.312 $\\pm$ 0.016 &8.746 $\\pm$ 0.113 &13.873 $\\pm$ 0.25 &5395 $\\pm$ 192 &50828 $\\pm$ 2506  \\\\\n    0.95       & 0.637 $\\pm$ 0.013           & 32.667 $\\pm$ 0.631          & 0.677 $\\pm$ 0.014 & 0.044 $\\pm$ 0.002 & 0.288 $\\pm$ 0.010  & 8.588 $\\pm$ 0.100    & 13.484 $\\pm$ 0.236 & 5205 $\\pm$ 189    & 46344 $\\pm$ 2688 \\\\ \n\n1.0 & 0.624 $\\pm$ 0.039 &43.72 $\\pm$ 2.475 &0.662 $\\pm$ 0.019 &0.05 $\\pm$ 0.007 &0.3 $\\pm$ 0.038 &8.624 $\\pm$ 0.277 &13.360 $\\pm$ 0.537 &6337 $\\pm$ 921 &49441 $\\pm$ 6520  \\\\\n\n\n\n     \\hline\n    \\end{tabular}\n    }\n    \\caption{\\textbf{Evaluation of GPT2 with different algorithms on IMDB sentiment text continuation task, discount factor ablations}: %\n    Mean and standard deviations over 5 random seeds is reported for sentiment scores along with fluency and diversity metrics. This table measures performance differences for the discount factor. We note that most NLP approaches using RL follow the style of \\citet{li-etal-2016-deep,wu2021recursively} and use a discount factor of 1. This is equivalent to reducing the generation MDP to a bandit feedback environment and causes performance loss (in the case of NLPO) and reward hacking and training instability (in the case of PPO).}\n    \\label{tbl:text_cont_gamma}\n    \\end{table*}\n    \n\n\\begin{table*}[h]\n    \\centering\n    \\resizebox{\\textwidth}{!}{\n    \\begin{tabular}{@{}ccccccccccccc@{}}\n    \\toprule\n    \\textbf{Perc Data (size)} & \\multicolumn{2}{c}{\\textbf{Semantic and Fluency Metrics}} & \\multicolumn{7}{c}{\\textbf{Diversity Metrics}}          \\\\ \\cmidrule(lr){2-3}\\cmidrule(lr){4-10}\n    & Sentiment Score $\\uparrow$ & Perplexity $\\downarrow$ & MSTTR & Distinct$_1$ & Distinct$_2$ & H$_1$ & H$_2$ & Unique$_1$ & Unique$_2$ \\\\\\cmidrule(lr){1-1}\\cmidrule(lr){2-3}\\cmidrule(lr){4-10}\n    Zero-Shot & 0.489 $\\pm$ 0.006           & 32.371 $\\pm$ 0.137          & 0.682 $\\pm$ 0.001 & 0.042 $\\pm$ 0.001 & 0.294 $\\pm$ 0.001 & 8.656 $\\pm$ 0.004  & 13.716 $\\pm$ 0.003 & 5063 $\\pm$ 14.832 & 47620 $\\pm$ 238  \\\\\\cmidrule(lr){1-1}\\cmidrule(lr){2-3}\\cmidrule(lr){4-10}\n    \\multicolumn{10}{l}{Supervised} \\\\\n    0.0 (0k) & 0.489 $\\pm$ 0.006           & 32.371 $\\pm$ 0.137          & 0.682 $\\pm$ 0.001 & 0.042 $\\pm$ 0.001 & 0.294 $\\pm$ 0.001 & 8.656 $\\pm$ 0.004  & 13.716 $\\pm$ 0.003 & 5063 $\\pm$ 14 & 47620 $\\pm$ 238 \\\\\n    0.1 (1k) & 0.531 $\\pm$ 0.005 & 34.846 $\\pm$ 0.123 & 0.685 $\\pm$ 0.001 & 0.045 $\\pm$ 0.001 & 0.313 $\\pm$ 0.004 & 8.775 $\\pm$ 0.023 & 13.854 $\\pm$ 0.032 & 5215 $\\pm$ 62 & 51125 $\\pm$ 685\\\\\n    0.5 (5k) & 0.536 $\\pm$ 0.006 & 35.008 $\\pm$ 0.229 & 0.684 $\\pm$ 0.001 & 0.047 $\\pm$ 0.000 & 0.314 $\\pm$ 0.002 & 8.764 $\\pm$ 0.010 & 13.837 $\\pm$ 0.0178 & 5489 $\\pm$ 44 & 51284 $\\pm$ 576 \\\\\n    1.0 (10k) & 0.539 $\\pm$ 0.004 & 35.472 $\\pm$ 0.074 & 0.682 $\\pm$ 0.001 & 0.047 $\\pm$ 0.001 & 0.312 $\\pm$ 0.002 & 8.755 $\\pm$ 0.012 & 13.806 $\\pm$ 0.016 & 5601 $\\pm$ 57 & 51151 $\\pm$ 345 \\\\\n    \\hline\n    \\multicolumn{10}{l}{PPO} \\\\ \n0.0 (0k) & 0.492 $\\pm$ 0.01 &33.57 $\\pm$ 0.323 &0.69 $\\pm$ 0.02 &0.047 $\\pm$ 0.001 &0.321 $\\pm$ 0.015 &8.816 $\\pm$ 0.149 &13.866 $\\pm$ 0.36 &5629 $\\pm$ 240 &52911 $\\pm$ 1786 \\\\\n0.1 (2k) & 0.598 $\\pm$ 0.017 &35.929 $\\pm$ 1.397 &0.698 $\\pm$ 0.009 &0.051 $\\pm$ 0.003 &0.339 $\\pm$ 0.012 &8.968 $\\pm$ 0.083 &14.013 $\\pm$ 0.158 &6173 $\\pm$ 360 &55918 $\\pm$ 2641 \\\\\n0.5 (10k) & 0.593 $\\pm$ 0.026 &35.95 $\\pm$ 2.177 &0.666 $\\pm$ 0.073 &0.049 $\\pm$ 0.003 &0.314 $\\pm$ 0.046 &8.635 $\\pm$ 0.634 &13.432 $\\pm$ 1.173 &5882 $\\pm$ 356 &51403 $\\pm$ 9297 \\\\ \n 1.0 (20k) & 0.605 $\\pm$ 0.023 & 33.497 $\\pm$  0.447 & 0.666 $\\pm$ 0.013 & 0.043 $\\pm$ 0.002 & 0.287 $\\pm$  0.008 &  8.575 $\\pm$ 0.073 & 13.484 $\\pm$ 0.244 & 5230 $\\pm$ 363 & 46483 $\\pm$ 1318\n \\\\ \\hline\n    \\multicolumn{10}{l}{NLPO} \\\\\n0.0 (0k) & 0.487 $\\pm$ 0.01 &32.572 $\\pm$ 0.165 &0.685 $\\pm$ 0.003 &0.043 $\\pm$ 0.001 &0.299 $\\pm$ 0.003 &8.691 $\\pm$ 0.023 &13.787 $\\pm$ 0.034 &5126 $\\pm$ 177 &48475 $\\pm$ 491 \\\\\n0.1 (2k) & 0.599 $\\pm$ 0.007 &33.536 $\\pm$ 0.378 &0.67 $\\pm$ 0.01 &0.043 $\\pm$ 0.001 &0.289 $\\pm$ 0.009 &8.608 $\\pm$ 0.061 &13.576 $\\pm$ 0.192 &5125 $\\pm$ 220&46755 $\\pm$ 1449  \\\\\n0.5 (10k) & 0.617 $\\pm$ 0.021 &33.409 $\\pm$ 0.354 &0.668 $\\pm$ 0.005 &0.041 $\\pm$ 0.001 &0.281 $\\pm$ 0.006 &8.552 $\\pm$ 0.044 &13.533 $\\pm$ 0.091 &4926 $\\pm$ 183 &45256 $\\pm$ 1022 \\\\\n1.0 (20k) & 0.637 $\\pm$ 0.013           & 32.667 $\\pm$ 0.631          & 0.677 $\\pm$ 0.014 & 0.044 $\\pm$ 0.002 & 0.288 $\\pm$ 0.010  & 8.588 $\\pm$ 0.100    & 13.484 $\\pm$ 0.236 & 5205 $\\pm$ 189    & 46344 $\\pm$ 2688  \\\\   \\hline\n    \\end{tabular}\n    }\n    \\caption{\\textbf{Evaluation of GPT2 with different algorithms on IMDB sentiment text continuation task, data budget ablations}: %\n    Mean and standard deviations over 5 random seeds is reported for sentiment scores along with fluency and diversity metrics. This table measures performance differences as a function of the fraction of the dataset that has been used. In the case of the RL approaches, this measures how much data is used to train the reward classifier, and for the supervised method it directly measures fraction of positive reviews used for training. We note that using even a small fraction of data to train a reward classifier proves to be effective in terms of downstream task performance while this is not true for supervised approaches. This lends evidence to the hypothesis that adding expending data budget on a reward classifier is more effective than adding more gold label expert demonstrations.}\n    \\label{tbl:text_cont_extra_data}\n    \\end{table*}\n\n\\begin{table*}[h]\n    \\centering\n    \\resizebox{\\textwidth}{!}{\n    \\begin{tabular}{@{}ccccccccccccc@{}}\n    \\toprule\n    \\textbf{Hyperparams} & \\multicolumn{2}{c}{\\textbf{Semantic and Fluency Metrics}} & \\multicolumn{7}{c}{\\textbf{Diversity Metrics}}          \\\\ \\cmidrule(lr){2-3}\\cmidrule(lr){4-10}\n    & Sentiment Score $\\uparrow$ & Perplexity $\\downarrow$ & MSTTR & Distinct$_1$ & Distinct$_2$ & H$_1$ & H$_2$ & Unique$_1$ & Unique$_2$ \\\\\\cmidrule(lr){1-1}\\cmidrule(lr){2-3}\\cmidrule(lr){4-10}\n    \\multicolumn{10}{l}{Target Update Iterations $\\mu$} \\\\ \n1 & 0.594 $\\pm$ 0.018 &32.671 $\\pm$ 0.201 &0.669 $\\pm$ 0.008 &0.042 $\\pm$ 0.002 &0.284 $\\pm$ 0.007 &8.575 $\\pm$ 0.064 &13.503 $\\pm$ 0.181 &4986 $\\pm$ 265 &45916$\\pm$ 1168 \\\\\n10 & 0.622 $\\pm$ 0.014 &32.729 $\\pm$ 0.567 &0.659 $\\pm$ 0.019 &0.042 $\\pm$ 0.002 &0.274 $\\pm$ 0.007 &8.489 $\\pm$ 0.106 &13.31 $\\pm$ 0.272 &5138 $\\pm$ 385 &43989 $\\pm$ 1120\\\\\n20 & 0.637 $\\pm$ 0.013           & 32.667 $\\pm$ 0.631          & 0.677 $\\pm$ 0.014 & 0.044 $\\pm$ 0.002 & 0.288 $\\pm$ 0.010  & 8.588 $\\pm$ 0.100    & 13.484 $\\pm$ 0.236 & 5205 $\\pm$ 189    & 46344 $\\pm$ 2688 \\\\\n50 & 0.603 $\\pm$ 0.015 &33.397 $\\pm$ 0.325 &0.67 $\\pm$ 0.006 &0.043 $\\pm$ 0.001 &0.287 $\\pm$ 0.004 &8.605 $\\pm$ 0.041 &13.54 $\\pm$ 0.116 &5228 $\\pm$ 113 &46418 $\\pm$ 685 \n\n \\\\ \\hline\n    \\multicolumn{10}{l}{Top-p mask} \\\\\n0.1 & 0.579 $\\pm$ 0.021 &32.451 $\\pm$ 0.243 &0.67 $\\pm$ 0.008 &0.042 $\\pm$ 0.001 &0.283 $\\pm$ 0.01 &8.569 $\\pm$ 0.084 &13.515 $\\pm$ 0.195 &5018 $\\pm$ 47 &45760 $\\pm$ 1579 \\\\ \n0.3 & 0.588 $\\pm$ 0.019 &32.451 $\\pm$ 0.303 &0.666 $\\pm$ 0.007 &0.043 $\\pm$ 0.001 &0.285 $\\pm$ 0.004 &8.568 $\\pm$ 0.032 &\n13.482 $\\pm$ 0.172 &5201 $\\pm$ 247 &46357$\\pm$ 539 \\\\\n0.5 & 0.588 $\\pm$ 0.01 &32.447 $\\pm$ 0.393 &0.669 $\\pm$ 0.001 &0.044 $\\pm$ 0.003 &0.291 $\\pm$ 0.008 &8.614 $\\pm$ 0.053 &13.535 $\\pm$ 0.06 &5305$\\pm$ 384 &47251 $\\pm$ 1226 \\\\\n0.7 & 0.619 $\\pm$ 0.013 &32.373 $\\pm$ 0.329 &0.663 $\\pm$ 0.008 &0.043 $\\pm$ 0.001 &0.28 $\\pm$ 0.006 &8.533 $\\pm$ 0.043 &13.366 $\\pm$ 0.129 &5186 $\\pm$ 216 &45149 $\\pm$ 1452 \\\\\n0.9 & 0.637 $\\pm$ 0.013           & 32.667 $\\pm$ 0.631          & 0.677 $\\pm$ 0.014 & 0.044 $\\pm$ 0.002 & 0.288 $\\pm$ 0.010  & 8.588 $\\pm$ 0.100    & 13.484 $\\pm$ 0.236 & 5205 $\\pm$ 189    & 46344 $\\pm$ 2688 \n\n\\\\   \\hline\n    \\end{tabular}\n    }\n    \\caption{\\textbf{Evaluation of GPT2 with different algorithms on IMDB sentiment text continuation task, NLPO hyperparameter ablations}: %\n    Mean and standard deviations over 5 random seeds is reported for sentiment scores along with fluency and diversity metrics. This table shows results of NLPO's stability to the unique hyperparameters introduced in the algorithm - all other parameters held constant from the best PPO model. The number of iterations after which the masking model syncs with the policy and the top-p nucleus percentage for the mask model itself. We see that in general, the higher the top-p mask percentage, the better the performance. For target update iterations, performance is low if the mask model is not updated often enough or if it updated too often.}\n    \\label{tbl:text_cont_nlpo_hyperparam}\n    \\end{table*}\n\n\\subsubsection{Human Participant Study}\n\nFigure~\\ref{fig:description_interface_imdb} shows the IMDB instructions, example, and interface used both for the qualification round, and then later, for the human evaluation experiments.\nTables~\\ref{app:imdb:human_agreement},~\\ref{app:imdb:human_tukey} show averaged results, annotator agreement, and the results of statistical significance tests to determine which models output better generations when rated by humans.\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=.49\\linewidth]{figures\/text_continuation_human_study\/textcont_human1.png}\n    \\includegraphics[width=.49\\linewidth]{figures\/text_continuation_human_study\/textcont_human2.png}\n    \\includegraphics[width=.45\\linewidth]{figures\/text_continuation_human_study\/textcont_human3.png}\n    \\caption{Instructions, example, and interface for the IMDB sentiment completion task.}\n    \\label{fig:description_interface_imdb}\n\\end{figure*}\n\n\n\n\n\\begin{table}[]\n\\centering\n\\footnotesize\n\\begin{tabular}{|l|r|rrr|rrr|}\n\\multicolumn{1}{|c|}{\\multirow{2}{*}{\\textbf{Algorithm}}} & \\multicolumn{1}{c|}{\\multirow{2}{*}{\\textbf{Unique N}}} & \\multicolumn{3}{c|}{\\textbf{Coherence}}                                                                        & \\multicolumn{3}{c|}{\\textbf{Sentiment}}                                                                        \\\\ \n\\multicolumn{1}{|c|}{}                                    & \\multicolumn{1}{c|}{}                                   & \\multicolumn{1}{c|}{\\textbf{Value}} & \\multicolumn{1}{c|}{\\textbf{Alpha}} & \\multicolumn{1}{c|}{\\textbf{Skew}} & \\multicolumn{1}{c|}{\\textbf{Value}} & \\multicolumn{1}{c|}{\\textbf{Alpha}} & \\multicolumn{1}{c|}{\\textbf{Skew}} \\\\ \\hline\nNLPO with KL                                              & 27                                                      & \\multicolumn{1}{r|}{3.49}           & \\multicolumn{1}{r|}{0.196}          & 3.497                              & \\multicolumn{1}{r|}{3.61}           & \\multicolumn{1}{r|}{0.2}            & 3.601                              \\\\\nNLPO without KL                                           & 29                                                      & \\multicolumn{1}{r|}{3.16}           & \\multicolumn{1}{r|}{0.21}           & 3.158                              & \\multicolumn{1}{r|}{4.41}           & \\multicolumn{1}{r|}{0.158}          & 4.403                              \\\\\nPPO without KL                                            & 27                                                      & \\multicolumn{1}{r|}{3.16}           & \\multicolumn{1}{r|}{0.17}           & 3.163                              & \\multicolumn{1}{r|}{4.36}           & \\multicolumn{1}{r|}{0.196}          & 4.363                              \\\\\nPPO with KL                                               & 29                                                      & \\multicolumn{1}{r|}{3.46}           & \\multicolumn{1}{r|}{0.124}          & 3.462                              & \\multicolumn{1}{r|}{3.58}           & \\multicolumn{1}{r|}{0.116}          & 3.575                              \\\\\nZero Shot                                                 & 28                                                      & \\multicolumn{1}{r|}{3.6}            & \\multicolumn{1}{r|}{0.162}          & 3.591                              & \\multicolumn{1}{r|}{3.1}            & \\multicolumn{1}{r|}{0.13}           & 3.097                              \\\\\nSupervised                                                & 29                                                      & \\multicolumn{1}{r|}{3.51}           & \\multicolumn{1}{r|}{0.192}          & 3.512                              & \\multicolumn{1}{r|}{3.43}           & \\multicolumn{1}{r|}{0.2}            & 3.428                              \\\\\nHuman                                                     & 27                                                      & \\multicolumn{1}{r|}{4.13}           & \\multicolumn{1}{r|}{0.159}          & 4.128                              & \\multicolumn{1}{r|}{3.01}           & \\multicolumn{1}{r|}{0.31}           & 3.017                              \\\\\nSupervised+PPO                                            & 22                                                      & \\multicolumn{1}{r|}{3.45}           & \\multicolumn{1}{r|}{0.211}          & 3.147                              & \\multicolumn{1}{r|}{3.64}           & \\multicolumn{1}{r|}{0.21}           & 3.161                              \\\\\nSupervised+NLPO                                           & 22                                                      & \\multicolumn{1}{r|}{3.48}           & \\multicolumn{1}{r|}{0.181}          & 3.226                              & \\multicolumn{1}{r|}{3.73}           & \\multicolumn{1}{r|}{0.22}           & 3.047                             \n\\end{tabular}\n\\caption{Results of the human subject study showing the number of participants N, average Likert scale value for coherence and sentiment, Krippendorf's alpha showing inter-annotator agreement, and Skew. For each model a total of 100 samples were drawn randomly from the test set and rated by 3 annotators each, resulting in 300 data points per algorithm.}\n\\label{app:imdb:human_agreement}\n\\end{table}\n\n\\begin{table}[]\n\\centering\n\\footnotesize\n\\begin{tabular}{|l|l|rr|rr|}\n\\multicolumn{1}{|c|}{\\multirow{2}{*}{\\textbf{Group 1}}} & \\multicolumn{1}{c|}{\\multirow{2}{*}{\\textbf{Group 2}}} & \\multicolumn{2}{c|}{\\textbf{Coherence}}                                                      & \\multicolumn{2}{c|}{\\textbf{Sentiment}}                                                      \\\\ \n\\multicolumn{1}{|c|}{}                                  & \\multicolumn{1}{c|}{}                                  & \\multicolumn{1}{c|}{\\textbf{Diff (G2-G1)}} & \\multicolumn{1}{c|}{\\textit{\\textbf{p-values}}} & \\multicolumn{1}{c|}{\\textbf{Diff (G2-G1)}} & \\multicolumn{1}{c|}{\\textit{\\textbf{p-values}}} \\\\\\hline\nPPO with KL                                             & PPO without KL                                         & \\multicolumn{1}{r|}{\\textbf{-0.3}}         & \\textbf{0.035}                                  & \\multicolumn{1}{r|}{\\textbf{0.783}}        & \\textbf{0.001}                                  \\\\\nPPO with KL                                             & NLPO with KL                                           & \\multicolumn{1}{r|}{0.03}                  & 0.9                                             & \\multicolumn{1}{r|}{0.027}                 & 0.9                                             \\\\\nPPO with KL                                             & NLPO without KL                                        & \\multicolumn{1}{r|}{\\textbf{-0.3}}         & \\textbf{0.035}                                  & \\multicolumn{1}{r|}{\\textbf{0.827}}        & \\textbf{0.001}                                  \\\\\nPPO with KL                                             & Supervised                                             & \\multicolumn{1}{r|}{0.05}                  & 0.9                                             & \\multicolumn{1}{r|}{-0.15}                 & 0.591                                           \\\\\nPPO with KL                                             & Human                                                  & \\multicolumn{1}{r|}{\\textbf{0.667}}        & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{-0.567}}       & \\textbf{0.001}                                  \\\\\nPPO with KL                                             & Zero Shot                                              & \\multicolumn{1}{r|}{0.137}                 & 0.776                                           & \\multicolumn{1}{r|}{\\textbf{-0.483}}       & \\textbf{0.001}                                  \\\\\nPPO without KL                                          & NLPO with KL                                           & \\multicolumn{1}{r|}{\\textbf{0.33}}         & \\textbf{0.013}                                  & \\multicolumn{1}{r|}{\\textbf{-0.757}}       & \\textbf{0.001}                                  \\\\\nPPO without KL                                          & NLPO without KL                                        & \\multicolumn{1}{r|}{0.001}                 & 0.9                                             & \\multicolumn{1}{r|}{0.043}                 & 0.9                                             \\\\\nPPO without KL                                          & Supervised                                             & \\multicolumn{1}{r|}{\\textbf{0.35}}         & \\textbf{0.006}                                  & \\multicolumn{1}{r|}{\\textbf{-0.933}}       & \\textbf{0.001}                                  \\\\\nPPO without KL                                          & Human                                                  & \\multicolumn{1}{r|}{\\textbf{0.967}}        & \\textbf{0.009}                                  & \\multicolumn{1}{r|}{\\textbf{-1.35}}        & \\textbf{0.001}                                  \\\\\nPPO without KL                                          & Zero Shot                                              & \\multicolumn{1}{r|}{\\textbf{0.437}}        & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{-1.267}}       & \\textbf{0.001}                                  \\\\\nNLPO with KL                                            & NLPO without KL                                        & \\multicolumn{1}{r|}{\\textbf{-0.33}}        & \\textbf{0.013}                                  & \\multicolumn{1}{r|}{\\textbf{0.8}}          & \\textbf{0.001}                                  \\\\\nNLPO with KL                                            & Supervised                                             & \\multicolumn{1}{r|}{0.02}                  & 0.9                                             & \\multicolumn{1}{r|}{-0.177}                & 0.404                                           \\\\\nNLPO with KL                                            & Human                                                  & \\multicolumn{1}{r|}{\\textbf{0.637}}        & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{-0.593}}       & \\textbf{0.001}                                  \\\\\nNLPO with KL                                            & Zero Shot                                              & \\multicolumn{1}{r|}{0.107}                 & 0.9                                             & \\multicolumn{1}{r|}{\\textbf{-0.51}}        & \\textbf{0.001}                                  \\\\\nNLPO without KL                                         & Supervised                                             & \\multicolumn{1}{r|}{\\textbf{0.35}}         & \\textbf{0.006}                                  & \\multicolumn{1}{r|}{\\textbf{-0.977}}       & \\textbf{0.001}                                  \\\\\nNLPO without KL                                         & Human                                                  & \\multicolumn{1}{r|}{\\textbf{0.967}}        & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{-1.393}}       & \\textbf{0.001}                                  \\\\\nNLPO without KL                                         & Zero Shot                                              & \\multicolumn{1}{r|}{\\textbf{0.437}}        & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{-1.31}}        & \\textbf{0.001}                                  \\\\\nSupervised                                              & Human                                                  & \\multicolumn{1}{r|}{\\textbf{0.617}}        & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{-0.417}}       & \\textbf{0.001}                                  \\\\\nSupervised                                              & Zero Shot                                              & \\multicolumn{1}{r|}{0.087}                 & 0.9                                             & \\multicolumn{1}{r|}{\\textbf{-0.333}}       & \\textbf{0.0027}                                 \\\\\nHuman                                                   & Zero Shot                                              & \\multicolumn{1}{r|}{\\textbf{-0.53}}        & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{0.083}                 & 0.9                                             \\\\\nSupervised+PPO                                          & Supervised+NLPO                                        & \\multicolumn{1}{r|}{0.03}                  & 0.9                                             & \\multicolumn{1}{r|}{\\textbf{0.09}}         & \\textbf{0.035}                                  \\\\\nSupervised+PPO                                          & NLPO with KL                                           & \\multicolumn{1}{r|}{0.04}                  & 0.9                                             & \\multicolumn{1}{r|}{-0.03}                 & 0.9                                             \\\\\nSupervised+PPO                                          & NLPO without KL                                        & \\multicolumn{1}{r|}{\\textbf{-0.29}}        & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{0.77}}         & \\textbf{0.001}                                  \\\\\nSupervised+PPO                                          & PPO without KL                                         & \\multicolumn{1}{r|}{\\textbf{-0.29}}        & \\textbf{0.006}                                  & \\multicolumn{1}{r|}{\\textbf{0.72}}         & \\textbf{0.001}                                  \\\\\nSupervised+PPO                                          & PPO with KL                                            & \\multicolumn{1}{r|}{0.01}                  & 0.9                                             & \\multicolumn{1}{r|}{\\textbf{-0.06}}        & \\textbf{0.001}                                  \\\\\nSupervised+PPO                                          & Zero Shot                                              & \\multicolumn{1}{r|}{\\textbf{0.15}}         & \\textbf{0.035}                                  & \\multicolumn{1}{r|}{\\textbf{-0.54}}        & \\textbf{0.001}                                  \\\\\nSupervised+PPO                                          & Supervised                                             & \\multicolumn{1}{r|}{\\textbf{0.06}}         & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{-0.21}}        & \\textbf{0.001}                                  \\\\\nSupervised+PPO                                          & Human                                                  & \\multicolumn{1}{r|}{\\textbf{0.68}}         & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{-0.63}}        & \\textbf{0.001}                                  \\\\\nSupervised+NLPO                                         & NLPO with KL                                           & \\multicolumn{1}{r|}{0.01}                  & 0.9                                             & \\multicolumn{1}{r|}{\\textbf{-0.12}}        & \\textbf{0.001}                                  \\\\\nSupervised+NLPO                                         & NLPO without KL                                        & \\multicolumn{1}{r|}{\\textbf{-0.32}}        & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{0.68}}         & \\textbf{0.001}                                  \\\\\nSupervised+NLPO                                         & PPO without KL                                         & \\multicolumn{1}{r|}{\\textbf{-0.32}}        & \\textbf{0.035}                                  & \\multicolumn{1}{r|}{\\textbf{0.63}}         & \\textbf{0.001}                                  \\\\\nSupervised+NLPO                                         & PPO with KL                                            & \\multicolumn{1}{r|}{-0.02}                 & 0.9                                             & \\multicolumn{1}{r|}{\\textbf{-0.15}}        & \\textbf{0.006}                                  \\\\\nSupervised+NLPO                                         & Zero Shot                                              & \\multicolumn{1}{r|}{\\textbf{-0.12}}        & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{-0.63}}        & \\textbf{0.001}                                  \\\\\nSupervised+NLPO                                         & Supervised                                             & \\multicolumn{1}{r|}{0.03}                  & 0.9                                             & \\multicolumn{1}{r|}{\\textbf{-0.3}}         & \\textbf{0.001}                                  \\\\\nSupervised+NLPO                                         & Human                                                  & \\multicolumn{1}{r|}{\\textbf{0.65}}         & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{-0.72}}        & \\textbf{0.006}                                 \n\\end{tabular}\n\\caption{Results of an post-hoc Tukey HSD Test for difference in means between pairs of algorithms (Group 2 - Group 1) and corresponding $p$-values. Individually statistically significant results are bolded and are used to discuss results in the analysis. Overall $p$-values showing that there is a significant difference in means between the models via a one-way ANOVA test are significant with $p \\ll 0.05$ for both coherence and sentiment.\n}\n\\label{app:imdb:human_tukey}\n\\end{table}\n\n\\clearpage\n\\subsubsection{Qualitative Results} \n\\label{subsec: qualitative_results_imdb}\nWe show sample generations from each of the algorithms for three randomly picked prompts below.\n\\begin{lstlisting}\nSample 1\nPrompt: A man and his wife are not getting along because of the wife's jealousy and drinking problems.  When the wife disappears, her sister begins a vigorous search involving the\nZero-Shot: husband's sister, which leads several men to a remote cabin and then a boat. The wife and her husband then start to see the same things. Finally, a strange woman is driving out to her cabin and discovers the husband's disappearance but\nReferences: local police, state government, and a television crime show. What she finds out stuns the entire community. Good drama with lots of fine performances; based on a true story\nPPO: police and the local public. At the end, the woman is killed and the wife is found naked and with a red head  tattooed on her back. This is part of a special feature called\"On the Road, Hidden\" which was\nPPO-no-KL: police and other local governments. Written by G. Scott Fitzgerald, who also stars in this production, \"A Very Merry Christmas\" is a wonderful family movie with a wonderful soundtrack and wonderful characters. It brings home the true beauty of small-\nNLPO: mother while the brother goes under. After all, his sister, who was apparently kidnapped  but who turns out to be a man, is no longer in her place. If a film that was made in the 1950s\nNLPO-no-KL: friends of the wife, and also the friend of the wife of the woman that is missing, and eventually he  begins to find that she has gone to Germany and he has made a very good friend. There is a good scene between John and\nSupervised: amily tree. The widower, now widowed, returns, the daughter, daughter is not pleased when his wife is mistaken  for one of her sons, and the patriarch, then a wealthy man, falls victim to a jealous husband. The\n\n\nSample 2\nPrompt: A very credible and unsettling movie portraying the effects of war on the general public, Shame gives us a chilling understanding oh Ingmar Bergman's beliefs. The absence of color and sound (as in a soundtrack) helps to give the film a more realistic feeling. There is no soft or hard lighting or a variety of camera angles to give the story a charming or dramatic effect. Everything is very simply portrayed as it would be in reality. At times the storyline was confusing, but perhaps in order to give a sense of the characters' frame of mind how they too do now know what is happening or why. The simplicity of the\nZero-Shot: camera work is at once apparent and fascinating in the manner in which it affects the way things are portrayed. In other films these two have been referred to from time to time in a form and manner more suited to the screen. The film shows\nReferences: film made it somewhat boring, but I could understand how the director is trying not to glamorize the story and points. He depicted the harsh and unsettling events and effects of war. In the beginning the characters were introduced and shown as content  with the lighter settings\nPPO: characters relationships is so engrossing and the great and unique story presented by each pair of brothers and sisters creates a truly gripping and enjoyable experience. To put it simply, I strongly recommend to everyone who plays the role to buy this DVD!!!\nPPO-no-KL: movie helps it to flow along in a believable way. Rating 4 out of 5 Great Movie! I highly recommend these movies! I highly recommend these movies! Rating 5 out of 5 Great Movie! I highly recommend these movies! Rating 10 of\"\nNLPO: novel makes it one of the first books I have read by a British actor. It is a fun way for someone who likes their books, like me, to sit through a novel.  If you like to read novels, check it out.\nNLPO-no-KL: setting makes the final product very enjoyable to watch. Highly recommended!! ~ Michael Bowerman, SPSA Film Festival Host, SPSA, WA (December 17, 2006), 8\/17\/2006 Great photography, good acting,\nSupervised: \"Dakota Clan\" and his sister, the two become friends with Ted Masters who soon finds the chance to fly  back to their home land and go back to save his brother's life and his family. The\n\n\nSample 3\nPrompt: Shakespearean language is wild and rambling, saturated in multiple meanings, word play and metaphor.  To be understood it must be wrangled and tamed by an actor with the strength and knowledge to do so.  When an actor fails, the words pour forth in a torrent of incomprehensible words, but when he succeeds, the English language springs to life with an immediacy and vibrancy that takes your breath away. Al Pacino is one such actor, and here displays an incredible level of clarity and control that, were there any justice,\nZero-Shot: might still be seen in the world today. This movie, like \"Vacation\", is an effort at telling the tale of two families from different races. One is African but never grew up the other. One of them, a man \nReferences: would sweep every award in the offering. He meets the challenge of presenting Shylock head on,  and delivers an extraordinarily subtle and nuanced performance. It would be a crime if we never got the opportunity  to see what he does with King Lear.<br \/><br \/>The supporting cast\nPPO:  would have made him an instant-classic. This movie captures the spirit of childhood and  has everyone in the world watching it. It's a love story and it's really heartbreaking.  And at the same time, it's a classic. It \nPPO-no-KL: will bring victory to the fight against hatred and extremism. Al Pacino is a very beautiful,  well executed actor who brings an absolute intensity and sensibility to his portrayal of love and loyalty.  A must see! Recommended! Also, an Excellent\nNLPO: I would watch and enjoy. 7\/10, Oscar Wilde is always a pleasure to see.  A must see. Highly recommend. Highly recommend!  One of my top ten films of all time. A must see!! \nNLPO-no-KL: the whole film would have fallen to the same fate, just as it did just a couple hundred years ago.  Don't miss it. It's a real classic. Highly Recommended. * outta five stars for it!\nSupervised: his performance (so far) would seem mere shadow. He is truly in the middle of a movie,  and this film is one of those films where he can be convincing in it  (and his trademark acting, as you can see in the\n\\end{lstlisting}\n\n\n\\subsubsection{Setup}\nIWSLT-17~\\citep{cettolo-etal-2017-overview} contains transcriptions of TED talks in many languages.\nWe pick two languages, English and German, and frame this task similarly to other machine translation tasks---requiring the models to translate from English to German.\nWe train models on 4 rewards: SacreBLEU, chRF, TER, and BertScore.\n\n\\begin{table*}[ht!]\n\\centering\n\\footnotesize\n\n\\resizebox{0.5\\textwidth}{!}{\n\\begin{tabular}{ll}\n\\toprule\n\\textbf{Model Params}\n& \\multicolumn{1}{c}{\\textbf{value}} \\\\ \n\\cmidrule{1-2}\nsupervised & batch size: $64$\\\\\n& epochs: $5$ \\\\\n& learning rate: $0.00001$ \\\\\n& learning rate scheduler: constant \\\\\n& weight decay: 0.1 \\\\\n\\cmidrule{1-2}\nppo\/\\textcolor{forestgreen2}{nlpo} & steps per update: $5120$\\\\\n    & total number of steps: $256000$ \\\\\n    & batch size: $64$ \\\\\n    & epochs per update: $5$\\\\\n    & learning rate: $0.0.000001$ \\\\\n    & entropy coefficient: $0.0$ \\\\\n    & initial kl coeff: $0.001$ \\\\\n    & target kl: $0.2$ \\\\\n    & discount factor: $0.99$ \\\\\n    & gae lambda: $0.95$ \\\\\n    & clip ratio: $0.2$ \\\\\n    & rollouts top k : $10$ \\\\\n    & value function coeff: $0.5$ \\\\\n    & \\textcolor{forestgreen2}{top mask ratio: $0.5$} \\\\\n    & \\textcolor{forestgreen2}{target update iterations: $20$} \\\\\n\\cmidrule{1-2}\nsupervised+ ppo (or \\textcolor{forestgreen2}{nlpo}) & steps per update:$2560$\\\\\n    & total number of steps: $256000$ \\\\\n    & batch size: $64$ \\\\\n    & epochs per update: $5$\\\\\n    & learning rate: $0.0000005$ \\\\\n    & entropy coefficient: $0.0$ \\\\\n    & initial kl coeff: $0.001$ \\\\\n    & target kl: $0.2$ \\\\\n    & discount factor: $0.99$ \\\\\n    & gae lambda: $0.95$ \\\\\n    & clip ratio: $0.2$ \\\\\n    & rollouts top k : $10$ \\\\\n    & value function coeff: $0.5$ \\\\\n    & \\textcolor{forestgreen2}{top mask ratio: $0.5$} \\\\\n    & \\textcolor{forestgreen2}{target update iterations: $20$} \\\\\n\\cmidrule{1-2}\n\\cmidrule{1-2}\ndecoding & num beams: $4$ \\\\\n& length penalty: $0.6$ \\\\\n& max new tokens: $128$\\\\\n\n\\cmidrule{1-2}\ntokenizer & padding side: left\\\\\n& truncation side: right\\\\\n& max length: 128 \\\\\n\\bottomrule\n\\end{tabular}\n}\n\\caption{\\textbf{NMT Hyperparams}: Table shows a list of all hyper-parameters and their settings}\n       \\label{tbl:nmt_hyperparams}\n\\end{table*}\n\\begin{table*}[h]\n   \\centering\n   \\resizebox{1.0\\textwidth}{!}{\n   \\begin{tabular}{@{}cccc|cccccccccc@{}}\n   \\toprule\n     Datasets \n     & \\multicolumn{3}{c}{} \n     & \\multicolumn{10}{c}{\\textbf{Lexical and Semantic Metrics}} \n     \\\\\n     & Alg & LM&  Reward Function & Rouge-1 & Rouge-2 & Rouge-L & Rouge-LSum & Meteor & BLEU & SacreBLEU & chRf & TER & BertScore \\\\\n     \\cmidrule(lr){2-2}\\cmidrule(lr){3-3}\\cmidrule(lr){4-4}\n     \\cmidrule(lr){5-5}\\cmidrule(lr){6-6}\\cmidrule(lr){7-7}\n     \\cmidrule(lr){8-8}\\cmidrule(lr){9-9}\\cmidrule(lr){10-10}\n     \\cmidrule(lr){11-11}\\cmidrule(lr){12-12}\\cmidrule(lr){13-13}\n     \\cmidrule(lr){14-14}\n     \\\\\n     \\cmidrule{1-14}\n     \n     \n      \\multirow{18}{*}{IWSLT2017} & Zero-Shot & T5 & & 0.619 & {0.386} & 0.588 & 0.587 & 0.445 & {0.254} & {0.308} & 0.577 & 0.573 & 0.870 \\\\\n     \\cmidrule{2-14}\n     & PPO & T5 & SacreBLEU & 0.621 & 0.383 & 0.587 & 0.587 & 0.448 & 0.243 & 0.296 & 0.575 & 0.583 & 0.869\\\\\n     &  & T5 & chRF & 0.622 & 0.385 & 0.590 & 0.590 & {0.448} & 0.248 & 0.301 & {0.578} & 0.575 & {0.870}\\\\\n     &  & T5 & TER & {0.623} & 0.384 & {0.591} & {0.591} & 0.443 & 0.246 & 0.303 & 0.572 & {0.568} & 0.869\\\\\n     &  & T5 & BertScore & 0.533 & 0.326 & 0.507 & 0.507 & 0.321 & 0.143 & 0.174 & 0.406 & 0.573 & 0.839\\\\\n     \\cmidrule{2-14}\n     & NLPO & T5 & SacreBLEU & 0.624 & 0.385 & 0.59 & 0.59 & 0.45 & 0.245 & 0.299 & 0.578 & 0.578 & 0.87 \\\\\n     &  & T5 & chRF & 0.624 & 0.386 & 0.59 & 0.59 & 0.451 & 0.248 & 0.302 & 0.581 & 0.576 & 0.87\\\\\n     &  & T5 & TER & 0.622 & 0.384 & 0.59 & 0.59 & 0.443 & 0.246 & 0.303 & 0.573 & 0.57 & 0.869\\\\\n     &  & T5 & BertScore & 0.611 & 0.377 & 0.58 & 0.58 & 0.425 & 0.239 & 0.291 & 0.555 & 0.573 & 0.866\\\\\n     \\cmidrule{2-14}\n     & Supervised & T5 & & 0.638 & 0.400 & 0.610 & 0.609 & 0.461 & {0.280} & {0.337} & 0.593 & 0.538 & \\textbf{0.878} \\\\\n     \\cmidrule{2-14}\n     & Supervised + PPO & T5 & SacreBLEU & {0.640} & {0.407} & {0.610} & {0.610} & {0.465} & 0.277 & 0.332 & {0.596} & 0.542 & 0.877\\\\\n     &  & T5 & chRF & 0.639 & 0.406 & 0.609 & 0.609 & 0.464 & 0.277 & 0.331 & 0.596 & {0.543} & 0.877\\\\\n     &  & T5 & TER &  0.637 & 0.406 & 0.609 & 0.609 & 0.457 & 0.274 & 0.331 & 0.589 & 0.535 & 0.876\\\\\n     &  & T5 & BertScore & 0.612 & 0.381 & 0.585 & 0.585 & 0.418 & 0.240 & 0.291 & 0.548 & 0.559 & 0.867\\\\\n     \\cmidrule{2-14}\n     & Supervised + NLPO & T5 & SacreBLEU & \\textbf{0.641} & 0.418 & 0.614 & 0.614 & \\textbf{0.474} & 0.289 & 0.343 & \\textbf{0.597} & {0.535} & 0.877\\\\\n     &  & T5 & chRF & 0.643 & 0.418 & 0.621 & 0.621 & 0.464 & \\textbf{0.291} & 0.345 & 0.596 & 0.539 & 0.877\\\\\n     &  & T5 & TER & 0.639 & \\textbf{0.419} & \\textbf{0.621} & \\textbf{0.621} & 0.471 & 0.289 & \\textbf{0.346} & 0.593 & \\textbf{0.535} & 0.877\\\\\n     &  & T5 & BertScore &  0.633 & 0.401 & 0.606 & 0.606 & 0.448 & 0.267 & 0.323 & 0.580 & 0.537 & 0.875\\\\\n     \\bottomrule\n   \\end{tabular}\n   }\n    \\caption{\\textbf{IWSLT test evaluation - lexical and semantic}: Table shows lexical, semantic metrics for RL algorithms with different reward functions bench-marked against supervised baseline models}\n   \\label{tbl:met_lexical_scores}\n\\end{table*}\n\n\\begin{table*}[h]\n   \\centering\n   \\resizebox{1.0\\textwidth}{!}{\n   \\begin{tabular}{@{}cccc|cccccccc@{}}\n   \\toprule\n     Tasks \n     & \\multicolumn{3}{c}{} \n     & \\multicolumn{8}{c}{\\textbf{Diversity Metrics}}\n     \\\\\n     & Alg & Reward Function & LM & MSTTR & Distinct$_1$ & Distinct$_2$ & H$_1$ & H$_2$ & Unique$_1$ & Unique$_2$ & Mean Output Length\n     \\\\\n     \\cmidrule(lr){2-2}\\cmidrule(lr){3-3}\\cmidrule(lr){4-4}\n     \\cmidrule(lr){5-5}\\cmidrule(lr){6-6}\\cmidrule(lr){7-7}\n     \\cmidrule(lr){8-8}\\cmidrule(lr){9-9}\\cmidrule(lr){10-10}\n     \\cmidrule(lr){11-11}\\cmidrule(lr){12-12}\n     \\\\\n     \n      \\multirow{18}{*}{IWSLT2017} & Zero-Shot & T5 & & 0.662 & 0.097 & 0.4700 & 9.276 & 14.526 & 8312 & 52947 & 18.739 \\\\\n     \\cmidrule{2-12}\n     & PPO & T5 & SacreBLEU & 0.657 & 0.095 & 0.464 & 9.230 & 14.498 & 8285 & 53000 & 19.069\\\\\n     &  & T5 & chRF & 0.660 & 0.096 & 0.468 & 9.253 & 14.526 & 8243 & 53142 & 18.912\\\\\n     &  & T5 & TER & 0.659 & 0.097 & 0.474 & 9.244 & 14.536 & 8129 & 51914 & 18.268\\\\\n     &  & T5 & BertScore & 0.673 & 0.120 & 0.541 & 9.288 & 14.388 & 6642 & 37267 & 11.602\\\\\n     \\cmidrule{2-12}\n     & NLPO & T5 & SacreBLEU & 0.656 & 0.094 & 0.463 & 9.207 & 14.483 & 8240 & 52822 & 19.043\\\\\n     &  & T5 & chRF & 0.658 & 0.095 & 0.464 & 9.233 & 14.502 & 8230 & 53167 & 19.073\\\\\n     &  & T5 & TER & 0.661 & 0.098 & 0.476 & 9.271 & 14.552 & 8223 & 52438 & 18.344\\\\\n     &  & T5 & BertScore &  0.667 & 0.102 & 0.491 & 9.31 & 14.576 & 8134 & 50740 & 17.162\\\\\n     \\cmidrule{2-12}\n     & Supervised & T5 & & 0.655 & 0.095 & 0.467 & 9.210 & 14.492 & 7970 & 51430 & 18.440\\\\\n     \\cmidrule{2-12}\n     & Supervised + PPO & T5 & SacreBLEU &  0.654 & 0.094 & 0.461 & 9.176 & 14.467 & 8061 & 51840 & 18.803\\\\\n     &  & T5 & chRF & 0.656 & 0.094 & 0.464 & 9.202 & 14.497 & 8054 & 52198 & 18.794\\\\\n     &  & T5 & TER & 0.658 &  0.097 & 0.475 & 9.239 & 14.529 & 7969 & 51255 & 18.048\\\\\n     &  & T5 & BertScore &  0.665 & 0.102 & 0.495 & 9.270 & 14.524 & 7495 & 47629 & 16.051\\\\\n     \\cmidrule{2-12}\n     & Supervised + NLPO & T5 & SacreBLEU & 0.655 & 0.094 & 0.461 & 9.182 & 14.469 & 8073 & 51797 & 18.777\\\\\n     &  & T5 & chRF & 0.657 & 0.095 & 0.466 & 9.212 & 14.503 & 8074 & 52056 & 18.642\\\\\n     &  & T5 & TER & 0.658 & 0.096 & 0.473 & 9.231 & 14.521 & 8025 & 51389 & 18.133\\\\\n     &  & T5 & BertScore &0.661 & 0.098 & 0.482 & 9.26 & 14.547 & 7852 & 50672 & 17.496\\\\\n   \\bottomrule\n   \\end{tabular}\n   }\n   \\caption{\\textbf{IWSLT test evaluation - diversity Metrics} Table shows diversity metrics for RL algorithms with different reward functions bench-marked against supervised baseline models}\n   \\label{tbl:mt_diversity_scores}\n\\end{table*}\n\\subsubsection{Results and Discussion}\n\nTables ~\\ref{tbl:met_lexical_scores},~\\ref{tbl:mt_diversity_scores} presents our benchmarking results with 4 reward functions along with supervised baseline performances on test set. Our main finding is that NLPO + Supervised performs better than PPO and supervised models.\n\n\n\\clearpage\n\\subsubsection{Qualitative Results}\n\\label{sec:nmt_qualitative}\nWe show sample generations from each of the algorithms for three randomly picked prompts below.\n\n\\begin{lstlisting}\nSample 1\nPrompt: translate English to German: Several years ago here at TED, Peter Skillman introduced a design challenge  called the marshmallow challenge.\nZero-Shot:  Vor einigen Jahren hat Peter Skillman hier bei TED eine Design-Herausforderung namens Marshmallow Challenge vorgestellt.\nReferences:\nPPO: Vor einigen Jahren, hier bei TED, stellte Peter Skillman einen Design-Wettbewerb namens \"Die Marshmallow-Herausforderung\" vor\nNLPO: Vor einigen Jahren hier bei TED f\u00fchrte Peter Skillman eine Design-Herausforderung namens Marshmallow Challenge ein.\nSupervised: Vor einigen Jahren hier bei TED f\u00fcfchrte Peter Skillman eine Design-Herausforderung namens Marshmallow Challenge ein.\nSupervised + PPO: Vor einigen Jahren hier bei TED, f\u00fchrte Peter Skillman eine Design-Herausforderung namens Marshmallow-Herausforderung ein.\nSupervised + NLPO: Vor einigen Jahren hier bei TED f\u00fcchrte Peter Skillman eine Design-Herausforderung ein, die Marshmallow-Herausforderung.\n\nSample 2\nPrompt: translate English to German: They spend the majority of their time assembling the sticks into ever-growing structures.\nZero-Shot: Sie verbringen den gr\u00f6\u00dften Teil ihrer Zeit damit, die St\u00f6cke in immer wachsende Strukturen zu montieren.\nReferences: Sie investieren einen Gro\u00dfteil ihrer Zeit in den Zusammenbau st\u00e4ndig wachsender Strukturen\nPPO: Sie verbringen den Gro\u00dfteil ihrer Zeit damit, die St\u00f6cke in immer gr\u00f6\u00dfer werdende Strukturen zu montieren.\nNLPO: Sie verbringen den Gro\u00dfteil ihrer Zeit, die St\u00f6cke in immer wachsende Strukturen zu montieren.\nSupervised: Sie verbringen den Gro\u00dfteil ihrer Zeit damit, die St\u00f6cke in immer wachsende Strukturen zu montieren.\nSupervised + PPO: Sie verbringen den Gro\u00dfteil ihrer Zeit damit, die St\u00f6cke in immer wachsende Strukturen zu montieren.\nSupervised + NLPO: Sie verbringen den Gro\u00dfteil ihrer Zeit damit, die St\u00f6cke in immer wachsende Strukturen zu montieren.\n\nSample 3\nPrompt: translate English to German: And of course there are teams that have a lot more \"ta-da\" structures, and among the best are recent graduates of kindergarten.\nZero-Shot: Und nat\u00fcrlich gibt es Teams, die viel mehr \"ta-da\" Strukturen haben, und zu den besten geh\u00f6ren j\u00fcngste Absolventen des Kindergartens.\nReferences: Und nat\u00fcrlich gibt es auch Teams die viel mehr \"ta-da\"-Gebilde haben und unter den Besten sind frische Absolventen des Kindergartens.\nPPO: Und nat\u00fcrlich gibt es Teams, die viel mehr \"ta-da\" Strukturen, und unter den besten sind j\u00fcngste Absolventen des Kindergartens.\nNLPO: Und nat\u00fcrlich gibt es Teams, die viel mehr \"t\na-da\" Strukturen haben, und unter den besten sind j\u00fcngste Absolventen des Kindergartens\nSupervised: Und nat\u00fcrlich gibt es Teams, die viel mehr \"ta-da\"-Strukturen haben, und unter den besten sind j\u00fcngste Absolventen des Kindergartens.\nSupervised + PPO: Und nat\u00fcrlich gibt es Teams, die viel mehr \"ta-da\"-Strukturen haben, und unter den besten sind j\u00fcngste Absolventen des Kindergartens.\nSupervised + NLPO: Und nat\u00fclich gibt es Teams, die viel mehr \"ta-da\"-Strukturen haben, und unter den besten sind j\u00fcngste Absolventen des Kindergartens.\n\\end{lstlisting}\n\n\n\\subsubsection{Setup}\n\nNarrativeQA~\\citep{kovcisky2018narrativeqa} deals with task of generating answers to questions about a given story. For training RL methods, we consider 2 traditional lexical rewards namely Rouge Combined and Rouge-L-Max. We chose T5-base as the base LM since it has been shown to do well at question answering in prior work~\\citep{khashabi-etal-2020-unifiedqa}. We note that the supervised models we use are trained on the UnifiedQA dataset, which contains other QA datasets, and is shown by \\citet{khashabi-etal-2020-unifiedqa} to outperform supervised fine-tuning only on NarrativeQA. Hyperparams for our models can be found in Table~\\ref{tbl:narqa_hyperparams}.\n\n\\begin{table*}[ht!]\n\\centering\n\\footnotesize\n\n\\resizebox{0.5\\textwidth}{!}{\n\\begin{tabular}{ll}\n\\toprule\n\\textbf{Model Params}\n& \\multicolumn{1}{c}{\\textbf{value}} \\\\ \n\\cmidrule{1-2}\nppo\/\\textcolor{forestgreen2}{nlpo} & steps per update: $5120$\\\\\n    & total number of steps: $512000$ \\\\\n    & batch size: $64$ \\\\\n    & epochs per update: $5$\\\\\n    & learning rate: $0.000002$ \\\\\n    & entropy coefficient: $0.0$ \\\\\n    & initial kl coeff: $0.001$ \\\\\n    & target kl: $1.0$ \\\\\n    & discount factor: $0.99$ \\\\\n    & gae lambda: $0.95$ \\\\\n    & clip ratio: $0.2$ \\\\\n    & rollouts top k : $50$ \\\\\n    & value function coeff: $0.5$ \\\\\n    & \\textcolor{forestgreen2}{top mask ratio: $0.9$} \\\\\n    & \\textcolor{forestgreen2}{target update iterations: $20$} \\\\\n\\cmidrule{1-2}\nsupervised+ ppo (or \\textcolor{forestgreen2}{nlpo}) & steps per update:$2560$\\\\\n    & total number of steps: $512000$ \\\\\n    & batch size: $64$ \\\\\n    & epochs per update: $5$\\\\\n    & learning rate: $0.0000005$ \\\\\n    & entropy coefficient: $0.0$ \\\\\n    & initial kl coeff: $0.001$ \\\\\n    & target kl: $0.2$ \\\\\n    & discount factor: $0.99$ \\\\\n    & gae lambda: $0.95$ \\\\\n    & clip ratio: $0.2$ \\\\\n    & rollouts top k : $50$ \\\\\n    & value function coeff: $0.5$ \\\\\n    & \\textcolor{forestgreen2}{top mask ratio: $0.9$} \\\\\n    & \\textcolor{forestgreen2}{target update iterations: $20$} \\\\\n\\cmidrule{1-2}\n\\cmidrule{1-2}\ndecoding & num beams: $4$ \\\\\n& max new tokens: $50$\\\\\n\n\\cmidrule{1-2}\ntokenizer & padding side: left\\\\\n& truncation side: right\\\\\n& max length: 512 \\\\\n\\bottomrule\n\\end{tabular}\n}\n\\caption{\\textbf{NarQA Hyperparams}: Table shows a list of all hyper-parameters and their settings}\n       \\label{tbl:narqa_hyperparams}\n\\end{table*}\n\n\n\\begin{landscape}\n\\begin{table*}[h]\n   \\centering\n   \\resizebox{1.5\\textwidth}{!}{\n   \\begin{tabular}{@{}cccc|cccccccc|cccccccc@{}}\n   \\toprule\n     Tasks \n     & \\multicolumn{3}{c}{} \n     & \\multicolumn{8}{c}{\\textbf{Lexical and Semantic Metrics}} \n     & \\multicolumn{8}{c}{\\textbf{Diversity Metrics}}\n     \\\\\n     & Alg & Reward Function & LM & Rouge-1 & Rouge-2 & Rouge-L & Rouge-LSum & Rouge-LMax & Meteor & BLEU & BertScore & MSTTR & Distinct$_1$ & Distinct$_2$ & H$_1$ & H$_2$ & Unique$_1$ & Unique$_2$ & Mean Output Length\n     \\\\\n     \\cmidrule(lr){2-2}\\cmidrule(lr){3-3}\\cmidrule(lr){4-12} \\cmidrule(lr){13-20}\n     \\\\\n     \\multirow{3}{*}{NarQA} & \n     \\\\\n     \\cmidrule(lr){2-20}\n     & Zero Shot &  & T5 & 0.095 & 0.022 & 0.084 & 0.084 & 0.117 & 0.095 & 0.009 & 0.835 & 0.415 & 0.026 & 0.097 & 9.641 & 13.468 & 1880 & 11495 & 31.688\\\\\n    \\cmidrule(lr){2-20} \n     & PPO & Rouge Combined & T5 & 0.101 &0.025 &0.088 &0.088 &0.122 &0.099 &0.01 &0.837 &0.462 &0.03 &0.125 &9.759 &13.789 &2522 &17806 &32.352 \\\\\n     &  & Rouge-L Max & T5 & 0.099 &0.025 &0.087 &0.087 &0.122 &0.099 &0.01 &0.835 &0.439 &0.029 &0.119 &9.653 &13.618 &2292 &15816 &31.479 \\\\\n     \\cmidrule(lr){2-20} \n     & NLPO & Rouge Combined & T5 & 0.097 &0.023 &0.085 &0.085 &0.118 &0.098 &0.009 &0.836 &0.418 &0.025 &0.096 &9.652 &13.528 &1816 &10980 &32.117\\\\\n     & & Rouge-L Max & T5 & 0.102 &0.026 &0.089 &0.089 &0.124 &0.1 &0.01 &0.837 &0.445 &0.029 &0.118 &9.776 &13.75 &2181 &14569 &31.555 \\\\\n     \n     \\cmidrule(lr){2-20}\n     & Supervised &  & T5 & 0.378 & 0.190 & 0.367 & 0.367 & 0.581 & 0.099 & 0.209 & 0.931 & 0.609 & 0.156 & 0.534 &  9.807 & 13.657 & 3250 & 14995 & 4.923\n     \\\\\n        \n     \\cmidrule(lr){2-20} \n     & Supervised + PPO & Rouge Combined & T5 & 0.38 &0.177 &0.371 &0.371 &0.585 &0.09 &0.229 &0.931 &0.64 &0.174 &0.559 &10.132 &13.547 &3326 &13785 &4.353 \\\\\n     &  & Rouge-L Max & T5 & 0.368 &0.18 &0.36 &0.36 &0.585 &0.083 &0.239 &0.931 &0.641 &0.187 &0.576 &10.201 &13.452 &3287 &12436 &3.913 \\\\\n     \\cmidrule(lr){2-20} \n     & Supervised + NLPO & Rouge Combined & T5 & 0.398 &0.21 &0.393 &0.373 &0.589 &0.096 &0.24 &0.971 &0.679 &0.185 &0.595 &10.304 &13.694 &3371 &15067 &4.728 \\\\\n     & & Rouge-L Max & T5 & 0.381 &0.194 &0.383 &0.383 &0.588 &0.093 &0.243 &0.932 &0.645 &0.187 &0.59 &10.2 &13.397 &3287 &12171 &3.889\n\n     \\\\\n   \\bottomrule\n   \\end{tabular}\n   }\n   \\caption{\\textbf{Evaluation of NarrativeQA}: Reference Metrics, supervised is based on UnifiedQA~\\citep{khashabi-etal-2020-unifiedqa}.}\n   \\label{tbl:narrativeqa_lexical_scores}\n\\end{table*}\n\\end{landscape}\n\\subsubsection{Results and Discussion}\nTable ~\\ref{tbl:narrativeqa_lexical_scores} presents our benchmarking results with 2 reward functions along with supervised baseline performances on the NarrativeQA test set. Similar to other methods, our main finding is that warm-started initial policies are crucial for learning to generate answers that successfully use the input context.\n\n\\subsubsection{Qualitative Results}\n\\label{sec_nmt_qualitative}\nWe show sample generations from each of the algorithms for three randomly picked prompts below.\n\n\\begin{lstlisting}\nSample 1\nPrompt: who is mark hunter? mark hunter (slater), a high school student in a sleepy suburb of phoenix, arizona, starts an fm pirate  radio station that broadcasts from the basement of his parents' house. mark is a loner, an outsider, whose only outlet for his teenage angst and aggression is his unauthorized radio station. his pirate station's theme song is \"everybody knows\" by leonard cohen and there are glimpses of cassettes by such alternative musicians as the jesus and mary chain, camper van beethoven, primal scream, soundgarden, ice-t, bad brains, concrete blonde, henry rollins, and the pixies. by day, mark is seen as a loner, hardly talking to anyone around him; by night, he expresses his outsider views about what is wrong with american society. when he speaks his mind about what is going on at his school and in the community, more and more of his fellow students tune in to hear his show.nobody knows the true identity of \"hard harry\" or \"happy harry hard-on,\" as mark  refers to himself, until nora diniro (mathis), a fellow student, tracks him down and confronts him the day after a student named malcolm commits  suicide after harry attempts to reason with him. the radio show becomes increasingly popular and influential after harry confronts the suicide head-on, exhorting his listeners to do something about their problems instead of surrendering to them through suicide\u2014at the crescendo of his yelled speech, an overachieving student named paige woodward (who has been a constant listener) jams her various medals and accolades into a microwave and turns it on. she then sits, watching the awards cook until the microwave explodes, injuring her. while this is happening, other students act out in cathartic release.eventually, the radio show causes so much trouble in the community that the fcc is called in to investigate. during the fracas, it is revealed that the school's principal (annie ross) has been expelling \"problem students,\" namely, students with below-average standardized test scores, in an effort to boost the district's test scores while still keeping their names on the rolls (a criminal offense) in order to retain government funding.realizing he has started something huge, mark decides it is up to him to end it. he dismantles his radio station and attaches it to his mother's old jeep, creating a mobile transmitter so his position can't be triangulated. pursued by the police and the fcc, nora drives the jeep around while mark broadcasts. the harmonizer he uses to disguise his voice breaks, and with no time left to fix it, mark decides to broadcast his final message as himself. they finally drive up to the crowd of protesting students, and mark tells them that the world belongs to them and that they should make their own future. the police step in and arrest mark and nora. as they are taken away, mark reminds the students to \"talk hard.\" as the film ends, the voices of other students (and even one of the teachers) speak as intros for their own independent stations, which can be heard broadcasting across the country.\nZero-Shot: hunter (slater) is a high school student in a sleepy suburb of phoenix, arizona. he starts an fm pirate radio station that broadcasts from the basement of his\nReferences: he is a high school student in phoenix 2) loner and outsider student with a radio station\nPPO: hunter (slater), a high school student in a sleepy suburb of phoenix, arizona, starts an fm pirate radio station that broadcasts from the basement of his parents' house\nNLPO: hunter (slater), a high school student in a sleepy suburb of phoenix, arizona, starts an fm pirate radio station that broadcasts from the basement of his parents' house\nSupervised: a high school student\nSupervised + PPO: a high school student\nSupervised + NLPO: a high school student\n\nSample 2\nPrompt: what was the principle doing with the problem students? mark hunter (slater), a high school student in a sleepy suburb of phoenix, arizona, starts an fm pirate radio station that broadcasts from the basement of his parents' house. mark is a loner, an outsider, whose only outlet for his teenage angst and aggression is his unauthorized radio station. his pirate station's theme song is \"everybody knows\" by leonard cohen and there are glimpses of cassettes by such alternative musicians as the jesus and mary chain, camper van beethoven, primal scream, soundgarden, ice-t, bad brains, concrete blonde, henry rollins, and the pixies. by day, mark is seen as a loner, hardly talking to anyone around him; by night, he expresses his outsider views about what is wrong with american society. when he speaks his mind about what is going on at his school and in the community, more and more of his fellow students tune in to hear his show.nobody knows the true identity of \"hard harry\" or \"happy harry hard-on,\" as mark refers to himself, until nora diniro (mathis), a fellow student, tracks him down and confronts him the day after a student named malcolm commits suicide after harry attempts to reason with him. the radio show becomes increasingly popular and influential after harry confronts the suicide head-on, exhorting his listeners to do something about their problems instead of surrendering to them through suicide\u20ac\"at the crescendo of his yelled speech, an overachieving student named paige woodward (who has been a constant listener) jams her various medals and accolades into a microwave and turns it on. she then sits, watching the awards cook until the microwave explodes, injuring her. while this is happening, other students act out in cathartic release.eventually, the radio show causes so much trouble in the community that the fcc is called in to investigate. during the fracas, it is revealed that the school's principal (annie ross) has been expelling \"problem students,\" namely, students with below-average standardized test scores, in an effort to boost the district's  test scores while still keeping their names on the rolls (a criminal offense) in order to retain government funding.realizing he has started  something huge, mark decides it is up to him to end it. he dismantles his radio station and attaches it to his mother's old jeep, creating a  mobile transmitter so his position can't be triangulated. pursued by the police and the fcc, nora drives the jeep around while mark broadcasts.  the harmonizer he uses to disguise his voice breaks, and with no time left to fix it, mark decides to broadcast his final message as himself.  they finally drive up to the crowd of protesting students, and mark tells them that the world belongs to them and that they should make their own future. the police step in and arrest mark and nora. as they are taken away, mark reminds the students to \"talk hard.\" as the film ends, the voices of other students (and even one of the teachers) speak as intros for their own independent stations, which can be heard broadcasting  across the country.\nZero-Shot: mark hunter (slater) starts an fm pirate radio station that broadcasts from the basement of his parents' house. by day, mark is seen as a loner, an outsider, whose only outlet\nReferences: 1) expelling them. 2) expelling the students\nPPO: mark hunter (slater) starts an fm pirate radio station that broadcasts from the basement of his parents' house. mark is a loner, an outsider, whose only outlet for his teenage ang \nNLPO: mark hunter (slater) starts an fm pirate radio station that broadcasts from the basement of his parents' house. mark is a loner, an outsider, whose only outlet for his teenage ang\nSupervised: he was taking them to a psychiatric hospital.\nSupervised + PPO: addressing them \nSupervised + NLPO: he was expelling them\n\nSample 3\nPrompt: what happens to the people maskull meets in his travels to various lands? maskull,  a man longing for adventures, accepts an invitation from krag, an acquaintance of his friend nightspore, to travel to tormance after a seance.  the three set off in a crystal ship from an abandoned observatory in scotland but maskull awakens to find himself alone on tormance. in every  land he passes through he usually meets only one or two persons; these meetings often (though not always) end in the death of those he meets,  either at his own hand or by that of another. he learns of his own impending death, meets krag again, and dies shortly after learning that he is in  fact nightspore himself. the book concludes with a final revelation from krag (who claims to be known on earth as \"pain\") to nightspore about  the origin of the universe. the author turns out to support a variation of the doctrine of the demiurge, somewhat similar to that propounded  by some gnostics.all of the characters and lands are types used to convey the author's critique of several philosophical systems. on tormance,  most such viewpoints or ways of life are accompanied by corresponding new bodily sense organs or modifications of the same, thus each distinct  weltanschauung landscape has its corresponding sensorium.\nZero-Shot: what happens to the people maskull meets in his travels to various lands? n maskull, a man longing for adventures, accepts an invitation from krag to travel to tormance after \nReferences: 1) they usually die by his hand or anothers. 2) they die\nPPO: what happens to the people maskull meets in his travels to various lands? n maskull, a man longing for adventures, accepts an invitation from krag to travel to tormance after\nNLPO:  maskull meets krag, an acquaintance of nightspore, to travel to tormance after a seance. maskull awakens to find himself alone on tormance. in every land \nSupervised: they die either from his own hand or from another person's death. \nSupervised + PPO: they end up dying either at his own hand or by another's. \nSupervised + NLPO: they end up dying either at his own hand or by another's\n\\end{lstlisting}\n\\subsubsection{Setup}\n\nCommonGen \\citep{lin-etal-2020-commongen} deals with task of generating coherent sentences describing an input set of concepts (eg. \"a man is throwing a frisbee\"). For training RL methods, we consider 3 traditional lexical rewards namely Rouge-1, Rouge-avg (which is an average of Rouge-1, 2 and L) and meteor. Additionally, we also train with task-specific rewards such as CIDEr~ \\citep{vedantam2015cider}, SPICE~\\citep{anderson2016spice} and SPiDer~\\citep{liu2017improved}  which is a just a linear combination of both with equal weights. We chose T5-base as the base LM since it is well-suited for structure to text tasks. We additionally note that concept set inputs are prefixed with \"generate a sentence with:\" to encourage exploration. \n\nDuring our initial experiments when fine-tuning directly on LM, we observed that policy learns to repeat the prompted concepts in order to maximize rewards resulting in a well-known problem of \\textit{reward hacking}. To mitigate this, we add a penalty score of $-1$ to final task reward if the n-grams of prompt text overlaps with generated text. In contrast, when initialized with a supervised policy, this problem is not seen and hence penalty score is not applied. We use beam search as the decoding method during evaluation whereas for rollouts, we use top k sampling to favor exploration over exploitation. Table~\\ref{tbl:common_gen_hyperparams} provides an in-depth summary of setting of hyperparameter values along with other implementation details.\n\n\\begin{table*}[ht!]\n\\centering\n\\footnotesize\n\n\\resizebox{0.5\\textwidth}{!}{\n\\begin{tabular}{ll}\n\\toprule\n\\textbf{Model Params}\n& \\multicolumn{1}{c}{\\textbf{value}} \\\\ \n\\cmidrule{1-2}\nsupervised & batch size: $8$\\\\\n& epochs: $4$ \\\\\n& learning rate: $0.00001$ \\\\\n& learning rate scheduler: cosine \\\\\n& weight decay: $0.01$ \\\\\n\\cmidrule{1-2}\nppo\/ \\textcolor{forestgreen2}{nlpo} & steps per update: $1280$\\\\\n    & total number of steps: $256000$ \\\\\n    & batch size: $64$ \\\\\n    & epochs per update: $5$ \\\\\n    & learning rate: $0.000002$ \\\\\n    & entropy coefficient: $0.01$ \\\\\n    & initial kl coeff: $0.001$ \\\\\n    & target kl: $2.0$ \\\\\n    & discount factor: $0.99$ \\\\\n    & gae lambda: $0.95$ \\\\\n    & clip ratio: $0.2$ \\\\\n    & value function coeff: $0.5$ \\\\\n    & \\textcolor{forestgreen2}{top mask ratio: $0.9$} \\\\\n    & \\textcolor{forestgreen2}{target update iterations: $20$} \\\\\n\\cmidrule{1-2}\nsupervised+ ppo (or \\textcolor{forestgreen2}{nlpo}) & steps per update: $1280$\\\\\n    & total number of steps: $128000$ \\\\\n    & batch size: $64$ \\\\\n    & epochs per update: $5$ \\\\\n    & learning rate: $0.000002$ \\\\\n    & entropy coefficient: $0.01$ \\\\\n    & initial kl coeff: $0.01$ \\\\\n    & target kl: $1.0$ \\\\\n    & discount factor: $0.99$ \\\\\n    & gae lambda: $0.95$ \\\\\n    & clip ratio: $0.2$ \\\\\n    & value function coeff: $0.5$ \\\\\n    & \\textcolor{forestgreen2}{top mask ratio: $0.9$} \\\\\n    & \\textcolor{forestgreen2}{target update iterations: $20$} \\\\\n\\cmidrule{1-2}\n\\cmidrule{1-2}\ndecoding & num beams: $5$ \\\\\n& min length: $5$ \\\\\n& max new tokens: $20$\\\\\n\n\\cmidrule{1-2}\ntokenizer & padding side: left\\\\\n& max length: $20$ \\\\\n\\bottomrule\n\\end{tabular}\n}\n\\caption{\\textbf{CommonGen Hyperparams}: Table shows a list of all hyper-parameters and their settings}\n       \\label{tbl:common_gen_hyperparams}\n\\end{table*}\n\n\\subsubsection{Results and Discussion}\n\nTables ~\\ref{tbl:common_gen_dev_scores},~\\ref{tbl:common_gen_test_scores} presents our benchmarking results with 6 reward functions along with supervised baseline performances on dev and test sets respectively. Our main finding is that warm-started initial policies are crucial for learning to generate coherent sentences with common sense. Without warm-start, policies suffer from reward hacking despite application of repetition penalty and task-specific metrics such as CIDer etc. Further, we find that RL fine-tuned models obtain very high concept coverage which is also seen in Table~\\ref{app:tbl_common_gen_qualitative}.\nSupervised models often tend to miss few concepts in its generation compared to RL methods. \n\n\n\\begin{table*}[h]\n   \\centering\n   \\resizebox{1.0\\textwidth}{!}{\n   \\begin{tabular}{@{}ccccccccccccc@{}}\n   \\toprule\n     Tasks \n     & \\multicolumn{3}{c}{\\textbf{\\_}} \n     & \\multicolumn{6}{c}{\\textbf{Lexical and Semantic Metrics}}\n     \\\\\n     & Alg & LM & Reward function & Rouge-2\t& Rouge-L & Bleu (n=3) & \tBleu (n=4) & Meteor & CIDEr & SPICE & Coverage\\\\\n     \\cmidrule(lr){2-2}\\cmidrule(lr){3-3}\\cmidrule(lr){4-4}\n     \\cmidrule(lr){5-5} \\cmidrule(lr){6-6} \\cmidrule(lr){7-7}\n     \\cmidrule(lr){8-8} \\cmidrule(lr){9-9} \\cmidrule(lr){10-10}\n     \\cmidrule(lr){11-11} \\cmidrule(lr){12-12}\n      \\\\\n      \\multirow{22}{*}{CommonGen} & Zero-Shot & T5 & & 0.016 & 0.264 & 0.029 & 0.006 & 0.203 & 6.200 & 0.115 & 91.070\\\\\n       \\cmidrule{2-12}\n      & PPO & T5 & Rouge-1 & 0.085 $\\pm$ 0.008 &\t0.354 $\\pm$ 0.004 &\t0.161 $\\pm$ 0.011 & 0.087 $\\pm$ 0.009 &\t0.235 $\\pm$ 0.002 &\t8.673 $\\pm$ 0.234 & 0.157 $\\pm$ 0.001 & 88.544 $\\pm$ 2.36\\\\\n       & & T5 & Rouge-Avg & 0.093 $\\pm$ 0.005 & 0.351 $\\pm$ 0.001 & 0.169 $\\pm$ 0.032 &\t0.097 $\\pm$ 0.017 & 0.224 $\\pm$ 0.012 & 8.212 $\\pm$ 1.329 & 0.159 $\\pm$ 0.011 & 82.584 $\\pm$ 2.569\n       \\\\\n       & & T5 & Meteor & 0.091 $\\pm$ 0.008 &\t0.308 $\\pm$ 0.007 &\t0.166 $\\pm$ 0.016 & 0.088 $\\pm$ 0.013 &\t0.220 $\\pm$ 0.006 &\t7.251 $\\pm$ 0.453\t& 0.161 $\\pm$ 0.007 & 79.718 $\\pm$ 2.267\n       \\\\\n       & & T5 & SPice & 0.065 $\\pm$ 0.003\t& 0.302 $\\pm$ 0.002 & \t0.115 $\\pm$ 0.063 & 0.067 $\\pm$ 0.041 & 0.193 $\\pm$ 0.014 & 6.571 $\\pm$ 1.312 &\t0.175 $\\pm$ 0.011 & 69.340 $\\pm$ 3.617\n       \\\\\n       & & T5 & CiDer & 0.066 $\\pm$ 0.003\t& 0.304 $\\pm$ 0.002 &\t0.132 $\\pm$ 0.057 & 0.074 $\\pm$ 0.036 &\t0.211 $\\pm$ 0.009 &\t6.877 $\\pm$ 1.218 & 0.143 $\\pm$ 0.017 &\t80.114 $\\pm$ 4.852\n       \\\\\n       & & T5 & SPider &  0.117 $\\pm$ 0.005 & 0.352 $\\pm$ 0.007 &\t0.224 $\\pm$ 0.014\t& 0.137 $\\pm$ 0.011 &\t0.226 $\\pm$ 0.01 &\t9.162 $\\pm$ 0.539 & 0.186 $\\pm$ 0.006 &\t73.374 $\\pm$ 6.073\n       \\\\ \n       \\cmidrule(lr){2-12}\n       & NLPO & T5 & Rouge-1 & 0.087 $\\pm$ 0.002 & 0.339 $\\pm$ 0.009 & 0.127 $\\pm$ 0.048\t & 0.069 $\\pm$ 0.035 &\t0.213 $\\pm$ 0.002 &\t6.962 $\\pm$ 0.883 &\t0.145 $\\pm$ 0.022 &80.89 $\\pm$ 9.544\\\\\n        & & T5 & Rouge-Avg & 0.095 $\\pm$ 0.001 &0.338 $\\pm$ 0.002 & 0.159 $\\pm$ 0.02 &\t0.093 $\\pm$ 0.013 & 0.216 $\\pm$ 0.009 &  7.55 $\\pm$ 0.688\t& 0.153 $\\pm$ 0.008 & 77.944 $\\pm$ 2.770\n       \\\\\n       & & T5 & Meteor & 0.110 $\\pm$ 0.005 & 0.332 $\\pm$ 0.003 &\t0.214 $\\pm$ 0.007 &\t0.124 $\\pm$ 0.007 & \t0.235 $\\pm$ 0.004 & 8.669 $\\pm$ 0.164 &\t0.173 $\\pm$ 0.002 &\t82.007 $\\pm$ 1.012\n       \\\\\n       & & T5 & SPice & 0.014 $\\pm$ 0.006 & 0.242 $\\pm$ 0.001 & 0.037 $\\pm$ 0.011 & 0.018 $\\pm$ 0.007 & 0.156 $\\pm$ 0.007 & 4.685 $\\pm$ 0.283 & 0.168 $\\pm$ 0.008 &\t56.998 $\\pm$ 3.548 \n       \\\\\n       & & T5 & CiDer & 0.046 $\\pm$ 0.001 &\t0.241 $\\pm$ 0.003 & 0.078 $\\pm$ 0.028 & 0.043 $\\pm$ 0.016 & 0.143 $\\pm$ 0.018  & 3.964 $\\pm$ 0.792 & 0.103 $\\pm$ 0.012 &\t49.606 $\\pm$ 7.971\n       \\\\\n       & & T5 & SPider &  0.060 $\\pm$ 0.006 & 0.258 $\\pm$ 0.001 &\t0.090 $\\pm$ 0.008 & \t0.056 $\\pm$ 0.005 &\t0.151 $\\pm$ 0.022 &\t4.411 $\\pm$ 0.837 &\t0.123 $\\pm$ 0.022 &\t49.230 $\\pm$ 10.468\n       \\\\  \n       \\cmidrule(lr){2-12}\n      & Supervised & T5 & & 0.215 $\\pm$ 0.001 & 0.438 $\\pm$  0.001 & 0.444 $\\pm$ 0.001 & 0.329 $\\pm$ 0.001 &\t0.321 $\\pm$ 0.001\t& 16.385 $\\pm$ 0.046 & \t\\textbf{0.299} $\\pm$ 0.001 & 94.476 $\\pm$ 0.172 \\\\\n      \\cmidrule(lr){2-12}\n      & Supervised + PPO & T5 & Rouge-1 & 0.232 $\\pm$ 0.002 & \\textbf{0.453} $\\pm$ 0.002 & 0.454 $\\pm$ 0.006 &\t\\textbf{0.338} $\\pm$ 0.006 & 0.320 $\\pm$ 0.002 &\t16.233 $\\pm$ 0.159 &\t0.288 $\\pm$ 0.004 & \t96.412 $\\pm$ 0.424\\\\\n       & & T5 & Rouge-Avg & 0.230 $\\pm$ 0.001 & 0.450 $\\pm$ 0.001 &\t0.448 $\\pm$ 0.005 & 0.334 $\\pm$ 0.005 &\t0.319 $\\pm$ 0.001 & 16.069 $\\pm$ 0.167 &\t0.287 $\\pm$ 0.003 & 96.116 $\\pm$ 0.679\n       \\\\\n       & & T5 & Meteor & \\textbf{0.234} $\\pm$ 0.002 & 0.450 $\\pm$ 0.003 &\\textbf{0.462} $\\pm$ 0.007 & 0.342 $\\pm$ 0.007 & \\textbf{0.327} $\\pm$ 0.001 & \\textbf{16.797} $\\pm$ 0.152 & 0.295 $\\pm$ 0.001 &\t\\textbf{97.690} $\\pm$ 0.371\n       \\\\\n       & & T5 & SPice & 0.227 $\\pm$ 0.004 &\t0.447 $\\pm$ 0.003 & 0.450 $\\pm$ 0.007 & 0.336 $\\pm$ 0.008 & 0.319 $\\pm$ 0.002 &\t16.208 $\\pm$ 0.249 & 0.288 $\\pm$ 0.003 & \t96.492 $\\pm$ 0.29 \n       \\\\\n       & & T5 & CiDer & 0.224 $\\pm$ 0.003 &\t0.446 $\\pm$ 0.003 & 0.427 $\\pm$ 0.012 & 0.309 $\\pm$ 0.01 & 0.316 $\\pm$ 0.004 &\t15.497 $\\pm$ 0.428 & 0.283 $\\pm$ 0.004 & \t96.344 $\\pm$ 0.547\n       \\\\\n       & & T5 & SPider &  0.226 $\\pm$ 0.003 & 0.448 $\\pm$ 0.002 &\t0.436 $\\pm$ 0.005 &\t0.319 $\\pm$ 0.004 &\t0.317 $\\pm$ 0.003 &\t15.678 $\\pm$ 0.192 & 0.281 $\\pm$ 0.003 &\t96.154 $\\pm$ 0.426\n       \\\\ \n       \\cmidrule(lr){2-12}\n       \n       & Supervised + NLPO & T5 & Rouge-1 & 0.229 $\\pm$ 0.002 &\t0.450 $\\pm$ 0.001 &\t0.454 $\\pm$ 0.005 &\t0.338 $\\pm$ 0.004 &\t0.320 $\\pm$ 0.003 &\t16.206 $\\pm$ 0.175 &\t0.289 $\\pm$ 0.002 &\t96.342 $\\pm$ 0.572\\\\\n       & & T5 & Rouge-Avg & 0.232 $\\pm$ 0.003 &\t0.451 $\\pm$ 0.002 &\t0.458 $\\pm$ 0.01 &\t0.342 $\\pm$ 0.009 &\t0.321 $\\pm$ 0.003 &  16.351 $\\pm$ 0.335 &\t0.290 $\\pm$ 0.005 & 95.998 $\\pm$ 0.496\n       \\\\\n       & & T5 & Meteor & 0.231 $\\pm$ 0.003 & 0.449 $\\pm$ 0.002 &\t0.454 $\\pm$ 0.007 &\t0.334 $\\pm$ 0.008 &\t0.326 $\\pm$ 0.002 & 16.574 $\\pm$ 0.269 & 0.292 $\\pm$ 0.003 &\t97.374 $\\pm$ 0.457\n       \\\\\n       & & T5 & SPice &  0.223 $\\pm$ 0.002 & 0.442 $\\pm$ 0.001 & 0.435 $\\pm$ 0.011 &\t0.321 $\\pm$ 0.010 &\t0.315 $\\pm$ 0.004 &\t15.747 $\\pm$ 0.401 & 0.283 $\\pm$ 0.005 & 96.25 $\\pm$ 0.313\n       \\\\\n       & & T5 & CiDer & 0.226 $\\pm$ 0.002 &\t0.447 $\\pm$ 0.004 & 0.433 $\\pm$ 0.007 & 0.315 $\\pm$ 0.008 & 0.318 $\\pm$ 0.003 &\t15.741 $\\pm$ 0.170 & 0.285 $\\pm$ 0.001 &\t96.354 $\\pm$ 0.971\n       \\\\\n       & & T5 & SPider &  0.226 $\\pm$ 0.004 & 0.447 $\\pm$ 0.003 &\t0.434 $\\pm$ 0.006 &\t0.316 $\\pm$ 0.006 &\t0.319 $\\pm$ 0.002 & 15.739 $\\pm$ 0.311 & 0.284 $\\pm$ 0.003 &\t96.333 $\\pm$ 0.644\\\\\n       \\bottomrule\n        \n       \\end{tabular}\n       }\n       \\caption{\\textbf{CommonGen test evaluation} Table shows official scores obtained from CommonGen hold-out evaluation. The most important result is that RL fine-tuning on a supervised model yields better performance across most metrics especially Coverage which indicates the ratio of concepts covered in generated texts}\n       \\label{tbl:common_gen_test_scores}\n    \\end{table*}\n\n\n\\begin{landscape}\n  \\begin{table*}[h]\n     \\centering\n     \\resizebox{1.5\\textwidth}{!}{\n     \\begin{tabular}{@{}cccccccccccccccccccccc@{}}\n     \\toprule\n       Tasks \n       & \\multicolumn{4}{c}{\\textbf{\\_}} \n       & \\multicolumn{9}{c}{\\textbf{Lexical and Semantic Metrics}} \n       & \\multicolumn{8}{c}{\\textbf{Diversity Metrics}}\n       \\\\\n       & Alg & Reward Function & Top k & LM & Rouge-1 & Rouge-2 & Rouge-L & Rouge-LSum & Meteor & BLEU & BertScore & Cider & Spice & MSTTR & Distinct$_1$ & Distinct$_2$ & H$_1$ & H$_2$ & Unique$_1$ & Unique$_2$ & Mean Output Length\n       \\\\\n       \\cmidrule(lr){2-2}\\cmidrule(lr){3-3} \\cmidrule(lr){4-4} \\cmidrule(lr){5-5} \\cmidrule(lr){6-14} \\cmidrule(lr){15-22}\n       \\\\\n       \\multirow{31}{*}{CommonGen} & Zero-Shot &   &   & T5 & 0.415 & 0.016 & 0.270 & 0.270 & 0.179 & 0.0 & 0.854 & 0.640 & 0.231 & 0.430 & 0.090 & 0.335 & 5.998 & 7.957 & 345 & 1964 & 8.797\n       \\\\\n  \n       \\cmidrule(lr){2-20} \n        \\cmidrule(lr){2-22}\n       & PPO & Rouge-1 & 50 & T5 & 0.537 $\\pm$ 0.004 & 0.093 $\\pm$ 0.012 &  0.380 $\\pm$ 0.006 & 0.380 $\\pm$ 0.006 & 0.235 $\\pm$ 0.005 & 0.016 $\\pm$ 0.002 & 0.896 $\\pm$ 0.001 & 0.950 $\\pm$ 0.015 & 0.318 $\\pm$ 0.016 & 0.526 $\\pm$ 0.020 & 0.128 $\\pm$ 0.005 & 0.518 $\\pm$ 0.036 & 6.679 $\\pm$ 0.132 & 10.572 $\\pm$ 0.234 & 437.4 $\\pm$ 42.017 & 2418.8 $\\pm$ 167.947 & 7.214 $\\pm$ 0.374 \\\\\n       \\cmidrule(lr){3-22}\n       & & Rouge-Avg  & 50 & T5 & 0.519  $\\pm$ 0.0185 & 0.102 $\\pm$ 0.007 &  0.377 $\\pm$ 0.013 & 0.376 $\\pm$ 0.014 & 0.225 $\\pm$ 0.024 & 0.020 $\\pm$ 0.002 & 0.897  $\\pm$ 0.005 & 0.921 $\\pm$ 0.102 & 0.328 $\\pm$ 0.009 & 0.536 $\\pm$ 0.069 & 0.141 $\\pm$ 0.022 & 0.510  $\\pm$ 0.056 & 6.777 $\\pm$ 0.539 & 10.348 $\\pm$ 0.134 & 458.6 $\\pm$ 19.734 & 2244.4 $\\pm$ 162.855 & 6.887  $\\pm$ 1.006\n       \\\\\n       \\cmidrule(lr){3-22}\n       & & Meteor & 50 & T5 & 0.411 $\\pm$ 0.009 &  0.090 $\\pm$ 0.008 & 0.304  $\\pm$ 0.006 & 0.304  $\\pm$ 0.006 &  0.210  $\\pm$ 0.005 & 0.029  $\\pm$ 0.004  & 0.875 $\\pm$ 0.007 & 0.638 $\\pm$ 0.048 & 0.259 $\\pm$ 0.017 & 0.547  $\\pm$ 0.012 &  0.147 $\\pm$ 0.003 & 0.529 $\\pm$ 0.014 & 7.62 $\\pm$ 0.127 & 11.464 $\\pm$ 0.151 & 1039.4 $\\pm$ 63.276 & 5197.2 $\\pm$ 280.004 &  13.660 $\\pm$ 0.324\n       \\\\\n       \\cmidrule(lr){3-22}\n       & & SPice & 50 & T5 & 0.439 $\\pm$ 0.035 & 0.079 $\\pm$ 0.045 & 0.323 $\\pm$ 0.036 & 0.323 $\\pm$ 0.036  & 0.183 $\\pm$ 0.022 & 0.012 $\\pm$ 0.009 & 0.891 $\\pm$ 0.005 & 0.777 $\\pm$ 0.140 & 0.400 $\\pm$ 0.012 & 0.546 $\\pm$ 0.054 &  0.149 $\\pm$ 0.019 & 0.545 $\\pm$ 0.072 & 6.721 $\\pm$ 0.441 & 10.492 $\\pm$ 0.330 & 409.2 $\\pm$ 41.605 & 1878.4 $\\pm$ 167.492 & 5.706 $\\pm$ 0.678\n       \\\\\n       \\cmidrule(lr){3-22}\n       & & CiDer & 50 & T5 & 0.453 $\\pm$ 0.038 & 0.081 $\\pm$ 0.037 &  0.326 $\\pm$ 0.033 & 0.326 $\\pm$ 0.033 & 0.203 $\\pm$ 0.022 &  0.017 $\\pm$ 0.009 & 0.885 $\\pm$ 0.008 & 0.770 $\\pm$ 0.134 &  0.291 $\\pm$ 0.036 & 0.597 $\\pm$ 0.081 & 0.195  $\\pm$ 0.040 & 0.639 $\\pm$ 0.106 & 7.732 $\\pm$ 0.682 & 11.131 $\\pm$ 0.502 & 777.0 $\\pm$ 144.676 & 3350.8 $\\pm$ 503.419 &  7.393 $\\pm$ 0.572\n       \\\\\n       \\cmidrule(lr){3-22}\n        \\rowcolor{lightgray}\n       & & SPider & 50 & T5 & 0.512 $\\pm$ 0.008 & 0.141 $\\pm$ 0.007 & 0.388 $\\pm$ 0.002 & 0.388 $\\pm$ 0.003 & 0.242 $\\pm$ 0.007 & 0.032 $\\pm$ 0.003 & 0.902 $\\pm$ 0.001 & 1.045 $\\pm$ 0.034 & 0.380 $\\pm$ 0.006 & 0.482 $\\pm$ 0.015 & 0.133 $\\pm$ 0.003 & 0.472 $\\pm$ 0.021 & 6.372 $\\pm$ 0.221 & 10.303 $\\pm$ 0.228 & 502.6 $\\pm$ 33.422 & 2281.4 $\\pm$ 252.471 & 7.489 $\\pm$ 0.358\n       \\\\\n      \n        \\cmidrule(lr){2-22}\n       & NLPO & Rouge-1 & 50 & T5 & 0.499 $\\pm$ 0.012 &0.089 $\\pm$ 0.003 &0.328 $\\pm$ 0.007 &0.328 $\\pm$ 0.007 &0.198 $\\pm$ 0.002 &0.021 $\\pm$ 0.001 &0.872 $\\pm$ 0.005 &0.815 $\\pm$ 0.009 &0.305 $\\pm$ 0.008 &0.559 $\\pm$ 0.01 &0.148 $\\pm$ 0.003 &0.555 $\\pm$ 0.012 &7.059 $\\pm$ 0.067 &10.657 $\\pm$ 0.105 &457.9 $\\pm$ 11.108 &2349.6 $\\pm$ 60.345 &6.586 $\\pm$ 0.094\n       \\\\\n       \\cmidrule(lr){3-22}\n       & & Rouge-Avg  & 50 & T5 & 0.47 $\\pm$ 0.01 &0.096 $\\pm$ 0.004 &0.312 $\\pm$ 0.006 &0.312 $\\pm$ 0.006 &0.202 $\\pm$ 0.008 &0.025 $\\pm$ 0.002 &0.843 $\\pm$ 0.013 &0.816 $\\pm$ 0.026 &0.299 $\\pm$ 0.007 &0.512 $\\pm$ 0.019 &0.146 $\\pm$ 0.011 &0.513 $\\pm$ 0.012 &6.781 $\\pm$ 0.15 &10.424 $\\pm$ 0.156 &484.18 $\\pm$ 17.303 &2357.54 $\\pm$ 152.113 &7.131 $\\pm$ 0.487\n       \\\\\n       \\cmidrule(lr){3-22}\n       & & Meteor & 50 & T5 & 0.389 $\\pm$ 0.013 &0.1 $\\pm$ 0.004 &0.293 $\\pm$ 0.008 &0.293 $\\pm$ 0.008 &0.226 $\\pm$ 0.024 &0.035 $\\pm$ 0.004 &0.832 $\\pm$ 0.018 &0.691 $\\pm$ 0.04 &0.266 $\\pm$ 0.016 &0.503 $\\pm$ 0.003 &0.132 $\\pm$ 0.005 &0.471 $\\pm$ 0.008 &7.146 $\\pm$ 0.192 &10.727 $\\pm$ 0.313 &648.05 $\\pm$ 33.963 &3536.0 $\\pm$ 444.638 &11.062 $\\pm$ 1.301\n       \\\\\n       \\cmidrule(lr){3-22}\n       & & SPice & 50 & T5 & 0.329 $\\pm$ 0.015 &0.036 $\\pm$ 0.008 &0.247 $\\pm$ 0.013 &0.247 $\\pm$ 0.013 &0.137 $\\pm$ 0.009 &0.006 $\\pm$ 0.002 &0.817 $\\pm$ 0.024 &0.515 $\\pm$ 0.033 &0.323 $\\pm$ 0.021 &0.543 $\\pm$ 0.023 &0.174 $\\pm$ 0.004 &0.568 $\\pm$ 0.026 &7.176 $\\pm$ 0.212 &10.551 $\\pm$ 0.216 &479.45 $\\pm$ 19.77 &2065.8 $\\pm$ 288.843 &5.785 $\\pm$ 0.431\n       \\\\\n       \\cmidrule(lr){3-22}\n       & & CiDer & 50 & T5 & 0.515 $\\pm$ 0.006 &0.143 $\\pm$ 0.008 &0.387 $\\pm$ 0.006 &0.308 $\\pm$ 0.006 &0.19 $\\pm$ 0.001 &0.019 $\\pm$ 0.001 &0.865 $\\pm$ 0.015 &0.726 $\\pm$ 0.018 &0.282 $\\pm$ 0.009 &0.55 $\\pm$ 0.02 &0.179 $\\pm$ 0.005 &0.576 $\\pm$ 0.014 &7.286 $\\pm$ 0.125 &10.812 $\\pm$ 0.089 &661.46 $\\pm$ 21.776 &2726.32 $\\pm$ 71.253 &7.13 $\\pm$ 0.223 \n       \\\\\n       \\cmidrule(lr){3-22}\n       & & SPider & 50 & T5 &0.393 $\\pm$ 0.008 &0.086 $\\pm$ 0.012 &0.297 $\\pm$ 0.007 &0.297 $\\pm$ 0.007 &0.183 $\\pm$ 0.007 &0.02 $\\pm$ 0.003 &0.842 $\\pm$ 0.019 &0.717 $\\pm$ 0.026 &0.297 $\\pm$ 0.019 &0.525 $\\pm$ 0.024 &0.167 $\\pm$ 0.009 &0.537 $\\pm$ 0.025 &6.986 $\\pm$ 0.262 &10.451 $\\pm$ 0.171 &530.14 $\\pm$ 16.805 &2263.4 $\\pm$ 166.221 &6.687 $\\pm$ 0.372 \n       \\\\\n  \n       \\cmidrule(lr){2-22}\n  \n       \\cmidrule(lr){2-22} \n       & Supervised &  &  & T5 & 0.503 $\\pm$ 0.001 & 0.175 $\\pm$ 0.001 & 0.411 $\\pm$ 0.001 & 0.411 $\\pm$ 0.001 & 0.309 $\\pm$ 0.001 & 0.069 $\\pm$ 0.001 & 0.929 $\\pm$ 0.000 & 1.381 $\\pm$ 0.011 & 0.443 $\\pm$ 0.001 & 0.509 $\\pm$ 0.001 & 0.101 $\\pm$ 0.001 & 0.339 $\\pm$ 0.001 & 6.531 $\\pm$ 0.006 & 10.079 $\\pm$ 0.016 & 503.600 $\\pm$ 6.530 & 2158.8 $\\pm$ 24.514 & 10.934 $\\pm$ 0.020\n       \\\\\n      \\cmidrule(lr){2-22}\n       \n       \\cmidrule(lr){2-22}\n       & Supervised + PPO & Rouge-1 & 50 & T5 & 0.537 $\\pm$ 0.004 & 0.198 $\\pm$ 0.005 & 0.433 $\\pm$ 0.002 & 0.433 $\\pm$ 0.002 & 0.314 $\\pm$ 0.003 &  0.070 $\\pm$ 0.002 & 0.930 $\\pm$ 0.001 & 1.426 $\\pm$ 0.018 & 0.449 $\\pm$ 0.001 & 0.527 $\\pm$ 0.007 & 0.112 $\\pm$ 0.001 & 0.393 $\\pm$ 0.004 &  6.680 $\\pm$ 0.044 & 10.289 $\\pm$ 0.040 & 498.2 $\\pm$ 8.931 & 2317.0 $\\pm$ 22.609 & 9.667 $\\pm$ 0.105\n       \\\\\n       \\cmidrule(lr){3-22}\n       & & Rouge-Avg & 50 & T5 & 0.536 $\\pm$ 0.001 & 0.198 $\\pm$ 0.002 & 0.433 $\\pm$ 0.002 & 0.433 $\\pm$ 0.002 & 0.311 $\\pm$ 0.002 & 0.070 $\\pm$ 0.002 & 0.929 $\\pm$ 0.001 & 1.421 $\\pm$ 0.028 & 0.446 $\\pm$ 0.004 & 0.526 $\\pm$ 0.004 & 0.114 $\\pm$ 0.002 & 0.395 $\\pm$ 0.005 & 6.682 $\\pm$ 0.0297 & 10.274 $\\pm$ 0.042 & 506.4 $\\pm$ 6.829 & 2326.4 $\\pm$ 41.778 & 9.614 $\\pm$ 0.102\n       \\\\\n       \\cmidrule(lr){3-22}\n        \\rowcolor{lightgray}\n       & & Meteor & 50 & T5 & 0.540 $\\pm$ 0.005 & 0.204 $\\pm$ 0.005 & 0.436 $\\pm$ 0.004 & 0.436 $\\pm$ 0.004 & 0.329 $\\pm$ 0.003 & \n      0.076 $\\pm$ 0.003 & 0.930 $\\pm$ 0.001 & 1.474 $\\pm$ 0.022 & 0.447 $\\pm$ 0.004 & 0.514 $\\pm$ 0.004 & 0.105 $\\pm$ 0.002 & 0.378 $\\pm$ 0.008 & 6.631 $\\pm$ 0.053 & 10.270 $\\pm$ 0.064 & 507.0 $\\pm$ 17.146 & 2424.6 $\\pm$ 72.550 & 10.551 $\\pm$ 0.271\n       \\\\\n       \n       \\cmidrule(lr){3-22}\n       & & SPice & 50 & T5 & 0.532 $\\pm$ 0.006 & 0.194 $\\pm$ 0.007 & 0.430 $\\pm$ 0.005 & 0.430 $\\pm$ 0.005 & 0.311 $\\pm$ 0.004 & 0.068 $\\pm$ 0.003 & 0.929 $\\pm$ 0.001 & 1.415 $\\pm$ 0.029 & 0.458 $\\pm$ 0.001 & 0.532 $\\pm$ 0.008 & 0.113 $\\pm$ 0.0038 & 0.392 $\\pm$ 0.009 & 6.736 $\\pm$ 0.058 & 10.338 $\\pm$ 0.057 & 507.4 $\\pm$ 14.319 & 2313.8 $\\pm$ 27.694 & 9.742 $\\pm$ 0.208\n       \\\\\n       \n       \\cmidrule(lr){3-22}\n       & & CiDer & 50 & T5 & 0.530 $\\pm$ 0.004 & 0.191 $\\pm$ 0.003 & 0.427 $\\pm$ 0.004 & 0.427 $\\pm$ 0.004 & 0.309 $\\pm$ 0.008 & 0.063 $\\pm$ 0.002 & 0.928 $\\pm$ 0.001 & 1.337 $\\pm$ 0.040 & 0.444 $\\pm$ 0.002 & 0.518 $\\pm$ 0.009 & 0.110 $\\pm$ 0.003 & 0.382 $\\pm$ 0.006 & 6.614 $\\pm$ 0.082 & 10.166 $\\pm$ 0.053 & 490.4 $\\pm$ 9.457 & 2295.4 $\\pm$ 51.554 & 9.838 $\\pm$ 0.265 \\\\\n       \\cmidrule(lr){3-22}\n        & & SpiDer & 50 & T5 & 0.536 $\\pm$ 0.002 & 0.197 $\\pm$ 0.002 & 0.430 $\\pm$ 0.002 & 0.430 $\\pm$ 0.002 & 0.313 $\\pm$ 0.002 & 0.064 $\\pm$ 0.002 & 0.928 $\\pm$ 0.001 & 1.374 $\\pm$ 0.018 & 0.445 $\\pm$ 0.003 & 0.524 $\\pm$ 0.007 & 0.112 $\\pm$ 0.001 & 0.394 $\\pm$ 0.004 & 6.673 $\\pm$ 0.066 & 10.247 $\\pm$ 0.066 & 504.8 $\\pm$ 7.440 & 2361.8 $\\pm$ 20.856 & 9.761 $\\pm$ 0.121\n        \\\\\n      \n       \\cmidrule(lr){2-22}\n       & Supervised + NLPO & Rouge-1 & 50 & T5 & 0.545 $\\pm$ 0.002 &0.197 $\\pm$ 0.002 &0.432 $\\pm$ 0.001 &0.432 $\\pm$ 0.001 &0.31 $\\pm$ 0.002 &0.068 $\\pm$ 0.001 &0.929 $\\pm$ 0.0 &1.41 $\\pm$ 0.012 &0.449 $\\pm$ 0.001 &0.529 $\\pm$ 0.002 &0.114 $\\pm$ 0.002 &0.399 $\\pm$ 0.005 &6.705 $\\pm$ 0.018 &10.301 $\\pm$ 0.03 &498.86 $\\pm$ 8.594 &2311.46 $\\pm$ 33.451 &9.463 $\\pm$ 0.111\n       \\\\\n       \\cmidrule(lr){3-22}\n       & & Rouge-Avg  & 50 & T5 & 0.541 $\\pm$ 0.003 &0.2 $\\pm$ 0.003 &0.435 $\\pm$ 0.002 &0.435 $\\pm$ 0.002 &0.313 $\\pm$ 0.002 &0.07 $\\pm$ 0.002 &0.93 $\\pm$ 0.001 &1.424 $\\pm$ 0.023 &0.447 $\\pm$ 0.003 &0.53 $\\pm$ 0.006 &0.113 $\\pm$ 0.002 &0.396 $\\pm$ 0.008 &6.708 $\\pm$ 0.05 &10.318 $\\pm$ 0.074 &493.64 $\\pm$ 10.068 &2319.42 $\\pm$ 55.738 &9.596 $\\pm$ 0.123 \n       \\\\\n       \\cmidrule(lr){3-22}\n       & & Meteor & 50 & T5 & 0.537 $\\pm$ 0.003 &0.201 $\\pm$ 0.004 &0.431 $\\pm$ 0.002 &0.431 $\\pm$ 0.002 &0.326 $\\pm$ 0.002 &0.074 $\\pm$ 0.003 &0.93 $\\pm$ 0.0 &1.464 $\\pm$ 0.025 &0.448 $\\pm$ 0.002 &0.516 $\\pm$ 0.006 &0.106 $\\pm$ 0.002 &0.377 $\\pm$ 0.008 &6.634 $\\pm$ 0.044 &10.26 $\\pm$ 0.077 &506.04 $\\pm$ 3.502 &2401.32 $\\pm$ 38.569 &10.453 $\\pm$ 0.194\n       \\\\\n       \\cmidrule(lr){3-22}\n       & & SPice & 50 & T5 & 0.535 $\\pm$ 0.007 &0.193 $\\pm$ 0.008 &0.429 $\\pm$ 0.005 &0.429 $\\pm$ 0.005 &0.3 $\\pm$ 0.003 &0.064 $\\pm$ 0.002 &0.927 $\\pm$ 0.001 &1.333 $\\pm$ 0.017 &0.459 $\\pm$ 0.003 &0.553 $\\pm$ 0.013 &0.12 $\\pm$ 0.004 &0.415 $\\pm$ 0.014 &6.908 $\\pm$ 0.118 &10.445 $\\pm$ 0.057 &508.075 $\\pm$ 4.669 &2343.3 $\\pm$ 53.274 &9.249 $\\pm$ 0.225\n       \\\\\n       \\cmidrule(lr){3-22}\n       & & CiDer & 50 & T5 & 0.533 $\\pm$ 0.003 &0.197 $\\pm$ 0.004 &0.43 $\\pm$ 0.003 &0.43 $\\pm$ 0.004 &0.316 $\\pm$ 0.004 &0.066 $\\pm$ 0.001 &0.929 $\\pm$ 0.001 &1.381 $\\pm$ 0.014 &0.446 $\\pm$ 0.004 &0.516 $\\pm$ 0.009 &0.108 $\\pm$ 0.003 &0.379 $\\pm$ 0.01 &6.583 $\\pm$ 0.077 &10.165 $\\pm$ 0.084 &490.78 $\\pm$ 9.734 &2304.52 $\\pm$ 62.068 &9.923 $\\pm$ 0.213 \n       \\\\\n       \\cmidrule(lr){3-22}\n       & & SPider & 50 & T5 & 0.532 $\\pm$ 0.006 &0.196 $\\pm$ 0.006 &0.431 $\\pm$ 0.004 &0.431 $\\pm$ 0.004 &0.314 $\\pm$ 0.004 &0.066 $\\pm$ 0.002 &0.929 $\\pm$ 0.0 &1.371 $\\pm$ 0.011 &0.448 $\\pm$ 0.002 &0.521 $\\pm$ 0.005 &0.109 $\\pm$ 0.002 &0.385 $\\pm$ 0.005 &6.623 $\\pm$ 0.034 &10.223 $\\pm$ 0.049 &485.325 $\\pm$ 5.683 &2297.575 $\\pm$ 21.271 &9.798 $\\pm$ 0.179\n       \\\\\n       \n  \n  \n     \\bottomrule\n     \\end{tabular}\n     }\n     \\caption{\\textbf{CommonGen dev evaluation}: Table shows lexical, semantic and diversity metrics for best performing models found in each algorithm-reward function combinations along with best performing supervised baseline models. Generated text from these models are submitted to official CommonGen test evaluation to obtain test scores presented in Table~\\ref{tbl:common_gen_test_scores}}\n     \\label{tbl:common_gen_dev_scores}\n  \\end{table*}\n  \\end{landscape}\n\n\n\\begin{table}[]\n\\centering\n\\footnotesize\n\\begin{tabular}{|l|r|rrr|rrr|}\n\\multicolumn{1}{|c|}{\\multirow{2}{*}{\\textbf{Algorithm}}} & \\multicolumn{1}{c|}{\\multirow{2}{*}{\\textbf{Unique N}}} & \\multicolumn{3}{c|}{\\textbf{Coherence}}                                                                        & \\multicolumn{3}{c|}{\\textbf{Commonsense}}                                                                      \\\\\n\\multicolumn{1}{|c|}{}                                    & \\multicolumn{1}{c|}{}                                   & \\multicolumn{1}{c|}{\\textbf{Value}} & \\multicolumn{1}{c|}{\\textbf{Alpha}} & \\multicolumn{1}{c|}{\\textbf{Skew}} & \\multicolumn{1}{c|}{\\textbf{Value}} & \\multicolumn{1}{c|}{\\textbf{Alpha}} & \\multicolumn{1}{c|}{\\textbf{Skew}} \\\\ \\hline\nPPO+Supervised                                            & 25                                                      & \\multicolumn{1}{r|}{4.14}           & \\multicolumn{1}{r|}{0.073}          & 4.137                              & \\multicolumn{1}{r|}{4.03}           & \\multicolumn{1}{r|}{0.137}          & 4.023                              \\\\\nNLPO+Supervised                                           & 26                                                      & \\multicolumn{1}{r|}{4.25}           & \\multicolumn{1}{r|}{0.036}          & 4.253                              & \\multicolumn{1}{r|}{4.16}           & \\multicolumn{1}{r|}{0.002}          & 4.163                              \\\\\nZero Shot                                                 & 24                                                      & \\multicolumn{1}{r|}{2.15}           & \\multicolumn{1}{r|}{0.391}          & 2.154                              & \\multicolumn{1}{r|}{2.29}           & \\multicolumn{1}{r|}{0.342}          & 2.291                              \\\\\nPPO                                                       & 24                                                      & \\multicolumn{1}{r|}{2.84}           & \\multicolumn{1}{r|}{0.16}           & 2.849                              & \\multicolumn{1}{r|}{3.03}           & \\multicolumn{1}{r|}{0.081}          & 3.027                              \\\\\nSupervised                                                & 23                                                      & \\multicolumn{1}{r|}{4.39}           & \\multicolumn{1}{r|}{0.159}          & 4.387                              & \\multicolumn{1}{r|}{4.21}           & \\multicolumn{1}{r|}{0.225}          & 4.209                              \\\\\nNLPO                                                      & 24                                                      & \\multicolumn{1}{r|}{2}              & \\multicolumn{1}{r|}{0.335}          & 2.003                              & \\multicolumn{1}{r|}{2.13}           & \\multicolumn{1}{r|}{0.265}          & 2.124                             \n\\end{tabular}\n\\caption{Results of the human subject study showing the number of participants N, average Likert scale value for coherence and sentiment, Krippendorf's alpha showing inter-annotator agreement, and Skew. For each model a total of 100 samples were drawn randomly from the test set and rated by 3 annotators each, resulting in 300 data points per algorithm.}\n\\label{app:commongen:human_agreement}\n\\end{table}\n\n\n\n\\begin{table}[]\n\\footnotesize\n\\centering\n\\begin{tabular}{|l|l|rr|rr|}\n\\multicolumn{1}{|c|}{\\multirow{2}{*}{\\textbf{Group 1}}} & \\multicolumn{1}{c|}{\\multirow{2}{*}{\\textbf{Group 2}}} & \\multicolumn{2}{c|}{\\textbf{Coherence}}                                                      & \\multicolumn{2}{c|}{\\textbf{Commonsense}}                                                    \\\\\n\\multicolumn{1}{|c|}{}                                  & \\multicolumn{1}{c|}{}                                  & \\multicolumn{1}{c|}{\\textbf{Diff (G2-G1)}} & \\multicolumn{1}{c|}{\\textit{\\textbf{p-values}}} & \\multicolumn{1}{c|}{\\textbf{Diff (G2-G1)}} & \\multicolumn{1}{c|}{\\textit{\\textbf{p-values}}} \\\\ \\hline\nNLPO                                                    & PPO                                                    & \\multicolumn{1}{r|}{\\textbf{0.847}}        & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{0.897}}        & \\textbf{0.001}                                  \\\\\nNLPO                                                    & Supervised                                             & \\multicolumn{1}{r|}{\\textbf{2.397}}        & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{2.083}}        & \\textbf{0.001}                                  \\\\\nNLPO                                                    & NLPO+Supervised                                        & \\multicolumn{1}{r|}{\\textbf{2.257}}        & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{2.033}}        & \\textbf{0.001}                                  \\\\\nNLPO                                                    & PPO+Supervised                                         & \\multicolumn{1}{r|}{\\textbf{2.143}}        & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{1.897}}        & \\textbf{0.001}                                  \\\\\nNLPO                                                    & Zero Shot                                              & \\multicolumn{1}{r|}{0.153}                 & 0.515                                           & \\multicolumn{1}{r|}{0.157}                 & 0.624                                           \\\\\nPPO                                                     & Supervised                                             & \\multicolumn{1}{r|}{\\textbf{1.550}}        & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{1.187}}        & \\textbf{0.001}                                  \\\\\nPPO                                                     & NLPO+Supervised                                        & \\multicolumn{1}{r|}{\\textbf{1.410}}        & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{1.137}}        & \\textbf{0.001}                                  \\\\\nPPO                                                     & PPO+Supervised                                         & \\multicolumn{1}{r|}{\\textbf{1.297}}        & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{1.000}}        & \\textbf{0.001}                                  \\\\\nPPO                                                     & Zero Shot                                              & \\multicolumn{1}{r|}{\\textbf{-0.693}}       & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{-0.740}}       & \\textbf{0.001}                                  \\\\\nSupervised                                              & NLPO+Supervised                                        & \\multicolumn{1}{r|}{-0.140}                & 0.601                                           & \\multicolumn{1}{r|}{-0.050}                & 0.900                                           \\\\\nSupervised                                              & PPO+Supervised                                         & \\multicolumn{1}{r|}{\\textbf{-0.253}}       & \\textbf{0.050}                                  & \\multicolumn{1}{r|}{\\textbf{-0.187}}       & \\textbf{0.045}                                  \\\\\nSupervised                                              & Zero Shot                                              & \\multicolumn{1}{r|}{\\textbf{-2.243}}       & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{-1.927}}       & \\textbf{0.001}                                  \\\\\nNLPO+Supervised                                         & PPO+Supervised                                         & \\multicolumn{1}{r|}{\\textbf{-0.113}}       & \\textbf{0.008}                                  & \\multicolumn{1}{r|}{\\textbf{-0.137}}       & \\textbf{0.007}                                  \\\\\nNLPO+Supervised                                         & Zero Shot                                              & \\multicolumn{1}{r|}{\\textbf{-2.103}}       & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{-1.877}}       & \\textbf{0.001}                                  \\\\\nPPO+Supervised                                          & Zero Shot                                              & \\multicolumn{1}{r|}{\\textbf{-1.990}}       & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{-1.740}}       & \\textbf{0.001}                                 \n\\end{tabular}\n\\caption{Results of an post-hoc Tukey HSD Test for difference in means between pairs of algorithms (Group 2 - Group 1) and corresponding $p$-values. Individually statistically significant results are bolded and are used to discuss results in the analysis. Overall $p$-values showing that there is a significant difference in means between the models via a one-way ANOVA test are significant with $p \\ll 0.05$ for both coherence and sentiment.}\n\\label{app:commongen:human_tukey}\n\\end{table}\n\n\n\n\\clearpage\n\n\\subsubsection{Human Participant Study}\n\nFigure~\\ref{fig:description_interface_commongen} shows the commongen instructions, examples, and interface used for the human evaluation experiments. Different from the other human evaluations, we didn't provide any prompt because knowing the set of words to be used isn't required for rating either of the axes.\nTables~\\ref{app:commongen:human_agreement},~\\ref{app:commongen:human_tukey} show averaged results, annotator agreement, and the results of statistical significance tests to determine which models output better generations when rated by humans.\n\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=.45\\linewidth]{figures\/commongen_human_study\/commongen_human1.png}\n    \\includegraphics[width=.45\\linewidth]{figures\/commongen_human_study\/commongen_human2.png}\n    \\includegraphics[width=.45\\linewidth]{figures\/commongen_human_study\/commongen_human3.png}\n    \\caption{Instructions, examples, and interface for the Commongen task.}\n    \\label{fig:description_interface_commongen}\n\\end{figure*}\n\n\n\\subsubsection{Qualitative Analysis}\n\\label{app:tbl_common_gen_qualitative}\nThis section shows sample generations from different algorithms for three randomly picked prompts.\n\n\\begin{lstlisting}\nSample 1\nPrompt: generate a sentence with: apron cut hat kitchen sausage\nZero-Shot: generate a sentence with: apron cut hat kitchen sausage.\nReferences:\nPPO: sausage in the kitchen on an apron.\nNLPO: sausage cut hat cut hat cut hat cut apron cut hat\nSupervised: A man is cutting sausage on an apron in a kitchen.\nSupervised + PPO: A man cutting sausage from a hat on an apron in the kitchen.\nSupervised + NLPO: A man cuts sausage in a hat on an apron in the kitchen.\n\nSample 2\nPrompt: generate a sentence with: backhand competition player reach shuttlecock\nZero-Shot: backhand competition player reach shuttlecock.\nPPO: the ball in the backhand and the competition player in the shuttlecock.\nNLPO: backhanded competition player reaches shuttlecock.\nSupervised: tennis player reaches for a shuttlecock during a competition.\nSupervised + PPO: football player reaches for a shuttlecock after hitting a backhand during the competition.\nSupervised + NLPO: football player reaches for a shuttlecock in backhand during the competition.\n\nSample 3\nPrompt: generate a sentence with: bowl combine cucumber onion pasta\nZero-Shot: cucumber onion pasta.\nPPO: the cucumber onion pasta in a small bowl.\nNLPO: the in a bowl Combine cucumber onion pasta in a large bowl. Toss pasta\nSupervised: A bowl containing pasta, tomatoes, cucumbers, and onions.\nSupervised + PPO: A bowl containing pasta topped with cucumbers, onions, and peppers.\nSupervised + NLPO: A bowl containing a mixture of pasta, cucumber, and onion.\n\\end{lstlisting}\n\n\\subsubsection{Setup} As a representative of the summarization task, we consider CNN\/DM dataset consisting of long news articles and their highlights written by news authors. The dataset consists of 287k training, 13k validation and 11k test examples. We trained RL methods using 3 different automated metrics, namely Rouge-1, Rouge-avg and Meteor. We chose T5 as our base LM as it is pre-trained in a unified text-to-text framework and relishes Zero-Shot capabilities. For decoding, we use multinomial sampling with a temperature of $0.7$ for all the models.\n\n\\begin{table*}[ht!]\n\\centering\n\\footnotesize\n\n\\resizebox{0.5\\textwidth}{!}{\n\\begin{tabular}{ll}\n\\toprule\n\\textbf{Model Params}\n& \\multicolumn{1}{c}{\\textbf{value}} \\\\ \n\\cmidrule{1-2}\nsupervised & batch size: $16$\\\\\n& epochs: $2$ \\\\\n& learning rate: $0.0001$ \\\\\n& learning rate scheduler: cosine \\\\\n& weight decay: $0.1$ \\\\\n\\cmidrule{1-2}\nppo\/ \\textcolor{forestgreen2}{nlpo} & steps per update: $5120$\\\\\n    & total number of steps: $512000$ \\\\\n    & batch size: $64$ \\\\\n    & epochs per update: $5$ \\\\\n    & learning rate: $0.000002$ \\\\\n    & entropy coefficient: $0.0$ \\\\\n    & initial kl coeff: $0.001$ \\\\\n    & target kl: $0.2$ \\\\\n    & discount factor: $0.99$ \\\\\n    & gae lambda: $0.95$ \\\\\n    & clip ratio: $0.2$ \\\\\n    & value function coeff: $0.5$ \\\\\n    & rollouts top k: sweep of ($50$,$100$) \\\\\n    & \\textcolor{forestgreen2}{top mask ratio: $0.9$} \\\\\n    & \\textcolor{forestgreen2}{target update iterations: sweep of ($10$, $20$, $30$)} \\\\\n\\cmidrule{1-2}\nsupervised+ppo\/ \\textcolor{forestgreen2}{nlpo} & steps per update: 5120\\\\\n    & total number of steps: $256000$ \\\\\n    & batch size: $64$ \\\\\n    & epochs per update: $5$ \\\\\n    & learning rate: $0.000002$ \\\\\n    & entropy coefficient: $0.0$ \\\\\n    & initial kl coeff: $0.01$ \\\\\n    & target kl: $0.2$ \\\\\n    & discount factor: $0.99$ \\\\\n    & gae lambda: $0.95$ \\\\\n    & clip ratio: $0.2$ \\\\\n    & value function coeff: $0.5$ \\\\\n    & rollouts top k: sweep of ($50$,$100$) \\\\\n    & \\textcolor{forestgreen2}{top mask ratio: $0.9$} \\\\\n    & \\textcolor{forestgreen2}{target update iterations: sweep of ($10$, $20$, $30$)} \\\\\n\\cmidrule{1-2}\ndecoding & sampling: True \\\\\n& temperature: $0.7$ \\\\\n& min length: $50$ \\\\\n& max new tokens: $100$\\\\\n\\cmidrule{1-2}\ntokenizer & padding side: left\\\\\n& truncation side: right\\\\\n& max length: 512 \\\\\n\\bottomrule\n\\end{tabular}\n}\n\\caption{\\textbf{CNN\/DM Hyperparams}: Table shows a list of all hyper-parameters and their settings}\n       \\label{tbl:imdb_gen_hyperparams}\n\\end{table*}\n\n\\subsubsection{Results and Discussion}\n\nTable~\\ref{tbl:summ_lexical_scores} presents benchmarking results on test set reporting a wide range of metrics: lexical, semantic, factual correctness and diversity metrics. As baselines, we report lead-3 which selects first three sentences as the summary, Zero-Shot and a supervised model. PPO and NLPO models are on par with supervised performance on several metrics including Rouge-2, Rouge-L, and Bleu. On fine-tuning on top of supervised model, performance improves consistently on all metrics indicating that RL fine-tuning is beneficial. Another interesting finding is that, RL fine-tuned models are factually consistent as measured by SummaCZS metric. For ablations on PPO params, NLPO params, we refer to Tables \\ref{tbl:summ_ppo_ablations},\\ref{tbl:summ_nlpo_ablation}.\n\n\\begin{landscape}\n\\begin{table}\n   \\centering\n   \\resizebox{1.5\\textwidth}{!}{\n   \\begin{tabular}{@{}cccc|ccccccc|c|ccccccccc@{}}\n   \\toprule\n     Tasks \n     & \\multicolumn{3}{c}{\\textbf{\\_}} \n     & \\multicolumn{7}{c}{\\textbf{Lexical and Semantic Metrics}} \n     & \\multicolumn{1}{c}{\\textbf{Factual Consistency}} \n     & \\multicolumn{8}{c}{\\textbf{Diversity Metrics}}\n     \\\\\n     & Alg & Reward Function & LM & Rouge-1 & Rouge-2 & Rouge-L & Rouge-LSum & Meteor & BLEU & BertScore & SummaCZS & MSTTR & Distinct$_1$ & Distinct$_2$ & H$_1$ & H$_2$ & Unique$_1$ & Unique$_2$ & Mean Output Length\n     \\\\\n     \\cmidrule(lr){2-2}\\cmidrule(lr){3-3}\\cmidrule(lr){4-9} \\cmidrule(lr){10-10} \\cmidrule(lr){11-11} \\cmidrule(lr){12-12}\\cmidrule(lr){13-13}\\cmidrule(lr){14-14} \\cmidrule(lr){15-15} \\cmidrule(lr){16-16}  \\cmidrule(lr){17-17}  \\cmidrule(lr){18-18} \\cmidrule(lr){19-19}\\cmidrule(lr){20-20}\n     \\\\\n     \\multirow{31}{*}{CNN\/DM} & Lead-3 &  & & 0.401 & 0.175 & 0.250 & 0.363 & 0.333 & 0.099 & 0.874 & 0.993 & 0.750 & 0.0482 & 0.386 & 10.481 & 16.631 & 21465 & 273153 & 84\n     \\\\\n\n     \\cmidrule(lr){2-20}\n     & Zero-Shot &  & T5 & 0.372 & 0.145 & 0.247 & 0.311 & 0.256 & 0.077 & 0.864 & 0.654 & 0.725 & 0.061 & 0.414 & 10.285 & 16.183 & 19113 & 193999 & 55\n     \\\\\n        \n     \\cmidrule(lr){2-20} \n     \\rowcolor{lightgray}\n     & PPO & Rouge-1  & T5 & 0.410 & 0.182 & 0.283 & 0.349 & 0.276 & 0.095 & 0.876 & 0.622 & 0.760 & 0.068 & 0.464 & 10.661 & 16.437 & 18189 & 191383 & 47\n     \\\\\n    \n     \\cmidrule(lr){3-20}\n     \\\\\n     &  & Rouge-Avg & T5 &  0.396 & 0.176 & 0.273 & 0.338 & 0.270 & 0.095 & 0.874 & 0.622 & 0.773 & 0.071 & 0.490 & 10.830 & 16.664 & 19478 & 209140 & 48 \n     \\\\\n    \n     \\cmidrule(lr){3-20}\n     & & Meteor & T5 & 0.408 & 0.178 & 0.276 & 0.342 & 0.301 & 0.109 & 0.873 & 0.527 & 0.765 & 0.060 & 0.447 & 10.699 & 16.688 & 20528 & 234386 & 61\n     \\\\\n     \\cmidrule(lr){2-20} \n     & NLPO & Rouge-1 & T5 & 0.404 & 0.180 & 0.278 & 0.344 & 0.275 & 0.096 & 0.875 & 0.636 & 0.771 & 0.069 & 0.480 & 10.789 & 16.618 & 18677 & 201971 & 48\n     \\\\\n     \\cmidrule(lr){3-20}\n     & & Rouge-Avg  & T5 & 0.404 & 0.177 & 0.279 & 0.344 & 0.274 & 0.094 & 0.874 & 0.586 & 0.765 & 0.066 & 0.476 & 10.744 & 16.620 & 18179 & 206368 & 50\n     \\\\\n     \\cmidrule(lr){3-20}\n     \\rowcolor{lightgray}\n     & & Meteor & T5 & 0.405 & 0.180 & 0.277 & 0.343 & 0.292 & 0.108 & 0.872 & 0.578 & 0.772 & 0.064 & 0.471 & 10.802 & 16.766 & 20212 & 231038 & 56\\\\ \n\n     \\cmidrule(lr){2-20}\n\n     & Supervised &  & T5 & 0.411 & 0.177 & 0.276 & 0.343 & 0.309 & 0.108 & 0.876 & 0.654 & 0.727 & 0.057 & 0.401 & 10.459 & 16.410 & 21096 & 230343 & 68 \n     \\\\\n     \\cmidrule(lr){2-20} \n     \n     & Supervised + PPO & Rouge-1 & T5 & 0.417 & 0.189 & 0.294 & 0.358 & 0.278 & 0.101 & 0.882 & 0.722 & 0.750 & 0.070 & 0.459 & 10.595 & 16.389 & 18184 & 184220 & 46\n     \\\\\n     \\cmidrule(lr){3-20}\n     & & Rouge-Avg & T5 & 0.425 & 0.194 & 0.297 & 0.363 & 0.296 & 0.114 & 0.882 & 0.728 & 0.747 & 0.066 & 0.445 & 10.589 & 16.458 & 18939 & 200617 & 52 \n     \\\\\n      \\cmidrule(lr){3-20}\n      \\rowcolor{lightgray}\n     & & Meteor & T5 & 0.426 & 0.194 & 0.293 & 0.361 & 0.316 & 0.125 & 0.880 & 0.726 & 0.741 & 0.059 & 0.420 & 10.532 & 16.491 & 20395 & 224432 & 63 \n     \\\\ \n\n     \\cmidrule(lr){2-20}\n     & Supervised + NLPO & Rouge-1 & T5 & 0.421 & 0.193 & 0.297 & 0.361 & 0.287 & 0.108 & \\textbf{0.882} & 0.740 & 0.748 & 0.067 & 0.446 & 10.528 & 16.313 & 18204 & 185561 & 48\n     \\\\\n     \\cmidrule(lr){3-20}\n     & & Rouge-Avg  & T5 & 0.424 & 0.193 & 0.296 & 0.363 & 0.295 & 0.115 & 0.882 & 0.743 & 0.744 & 0.065 & 0.443 & 10.570 & 16.444 & 18747 & 201705 & 53\n     \\\\\n     \\cmidrule(lr){3-20}\n     \\rowcolor{lightgray}\n     & & Meteor & T5  & \\textbf{0.429} & 0.194 & 0.293 & 0.361 & \\textbf{0.319} & 0.124 & 0.880 & 0.743 & 0.745 & 0.059 & 0.422 & 10.574 & 16.516 & 20358 & 226801 & 63\n     \\\\\n   \\bottomrule\n   \\end{tabular}\n   }\n   \\caption{\\textbf{CNN\/Daily Mail test evaluation}: Table presents a wide range of metrics: lexical, semantic, factual correctness and diversity metrics on test set. As baselines, we report lead-3 which selects first three sentences as the summary, Zero-Shot and a supervised model. PPO and NLPO models are on par with supervised performance on several metrics including Rouge-2, Rouge-L, and Bleu. On fine-tuning on top of supervised model, performance improves consistently on all metrics indicating that RL fine-tuning is beneficial. Another interesting finding is that, RL fine-tuned models are factually consistent as measured by SummaCZS metric.}\n   \\label{tbl:summ_lexical_scores}\n\\end{table}\n\\end{landscape}\n\n\\begin{table*}[h]\n\\centering\n   \\resizebox{0.7\\textwidth}{!}{\n   \\begin{tabular}{@{}cccccccccc@{}}\n   \\toprule\n     \\multicolumn{3}{c}{\\textbf{\\_}} \n     & \\multicolumn{7}{c}{\\textbf{Lexical and Semantic Metrics}} \n     \\\\\n     Alg & Reward Function & Top k & Rouge-1 & Rouge-2 & Rouge-L & Rouge-LSum & Meteor & BLEU & BertScore \\\\\n     \\cmidrule(lr){1-1}\\cmidrule(lr){2-2}\\cmidrule(lr){3-3}\\cmidrule(lr){4-10}\\\\\n     \\multirow{6}{*}{PPO} & Rouge-1 & 50 & 0.404 & 0.181 & 0.280 & 0.346 & 0.273 & \\textbf{0.095} & 0.874\\\\\n     \\rowcolor{lightgray} \n     & & 100 & \\textbf{0.412} & \\textbf{0.186} & \\textbf{0.286} & \\textbf{0.354} & \\textbf{0.276} & 0.094 & \\textbf{0.876} \\\\\n     \\cmidrule(lr){2-10}\n     & Rouge-Avg & 50 & \\textbf{0.401} & 0.177 & \\textbf{0.276} & 0.342 & \\textbf{0.271} & 0.092 & 0.873 \\\\\n     \\rowcolor{lightgray} \n     & &  100 & 0.399 & \\textbf{0.179} & 0.275 & \\textbf{0.342} & 0.270 & \\textbf{0.094} & \\textbf{0.874} \\\\\n     \\cmidrule(lr){2-10}\n     \\rowcolor{lightgray} \n     & Meteor & 50 &  \\textbf{0.413} & \\textbf{0.182} & \\textbf{0.279} & \\textbf{0.348} & \\textbf{0.301} & \\textbf{0.110} & \\textbf{0.873} \\\\\n     & &  100 & 0.409 & 0.179 & 0.276 & 0.345 & 0.296 & 0.108 & 0.871 \\\\\n      \\cmidrule(lr){1-10}\n     \\multirow{6}{*}{Supervised+PPO} & Rouge-1 & 50 & 0.414 & 0.190 & 0.293 & 0.358 & 0.272 & 0.097 & 0.881 \\\\\n     \\rowcolor{lightgray} \n     & &  100 & \\textbf{0.420} & \\textbf{0.193} & \\textbf{0.295} & \\textbf{0.362} & \\textbf{0.277} & \\textbf{0.100} & \\textbf{0.881} \\\\\n     \\cmidrule(lr){2-10}\n     & Rouge-Avg & 50 & 0.426 & 0.196 & 0.298 & 0.366 & 0.294 & \\textbf{0.114} & 0.881 \\\\\n     \\rowcolor{lightgray}\n     & &  100 & \\textbf{0.427} & \\textbf{0.196} & \\textbf{0.298} & \\textbf{0.366} & \\textbf{0.294} & 0.113 & \\textbf{0.881} \\\\\n     \\cmidrule(lr){2-10}\n     & Meteor & 50 & 0.429 & 0.197 & 0.297 & 0.367 & 0.306 & 0.122 & \\textbf{0.881} \\\\\n     \\rowcolor{lightgray}\n     & &  100 & \\textbf{0.432} & \\textbf{0.199} & \\textbf{0.297} & \\textbf{0.367} & \\textbf{0.317} & \\textbf{0.131} & 0.879 \\\\\n     \\cmidrule(lr){1-10}\n    \n   \\bottomrule\n   \\end{tabular}\n   }\n   \\caption{\\textbf{PPO Ablation\/Model Selection}: Evaluation of PPO models on validation set with different reward functions and top k values for rollouts. For each alg-reward combo, best model (top k ) is chosen. }\n   \\label{tbl:summ_ppo_ablations}\n\\end{table*}\n\n\\begin{table*}[h]\n\\centering\n   \\resizebox{0.9\\textwidth}{!}{\n   \\begin{tabular}{@{}cccccccccccc@{}}\n   \\toprule\n     \\multicolumn{5}{c}{\\textbf{\\_}} \n     & \\multicolumn{7}{c}{\\textbf{Lexical and Semantic Metrics}} \n     \\\\\n     Alg & Reward Function & Top k (rollout) & Top p (Action mask) & target update $n_iters$ & Rouge-1 & Rouge-2 & Rouge-L & Rouge-LSum & Meteor & BLEU & BertScore \\\\\n     \\cmidrule(lr){1-1}\\cmidrule(lr){2-2}\\cmidrule(lr){3-3}\\cmidrule(lr){4-4} \\cmidrule(lr){5-5}\\cmidrule(lr){6-12}\\\\\n     \\multirow{18}{*}{NLPO} & Rouge-1 & 50 & 0.9 & 10 & \n      0.400 & 0.178 & 0.275 & 0.343 & 0.269 & 0.094 & 0.872\\\\\n     & & & & 20 & 0.396 & 0.173 & 0.274 & 0.340 & 0.257 & 0.082 & 0.873 \\\\\n     & & & & 30 & 0.396 & 0.174 & 0.273 & 0.339 & 0.265 & 0.091 & 0.872 \\\\\n     & & 100 & 0.9 & 10 & \\textbf{0.407} & 0.177 & 0.279 & 0.347 & 0.265 & 0.085 & \\textbf{0.875} \\\\\n      \\rowcolor{lightgray} \n     & & & & 20 & 0.406 & \\textbf{0.182} & \\textbf{0.281} & \\textbf{0.347} & \\textbf{0.273} & \\textbf{0.094} & 0.874\\\\\n     & & & & 30 & 0.405 & 0.180 & 0.279 & 0.347 & 0.269 & 0.091 & 0.875\\\\\n     \\cmidrule(lr){2-12}\n     \n      & Rouge-Avg & 50 & 0.9 & 10 & 0.400 & \\textbf{0.180} & 0.276 & 0.343 & 0.271 & 0.096 & 0.873\\\\\n      & & & & 20 & 0.349 & 0.147 & 0.241 & 0.298 & 0.237 & 0.078 & 0.858\\\\\n      & & & & 30 & 0.393 & 0.173 & 0.272 & 0.336 & 0.267 & 0.092 & 0.870\\\\\n      & & 100 & 0.9 & 10 & 0.396 & 0.174 & 0.274 & 0.339 & 0.265 & 0.088 & 0.872\\\\\n      \\rowcolor{lightgray} \n      & & & & 20 & \\textbf{0.406} & 0.179 & \\textbf{0.280} & \\textbf{0.347} &\n      \\textbf{0.272} & \\textbf{0.092} & \\textbf{0.874}\\\\\n      & & & & 30 & 0.400 & 0.178 & 0.279 & 0.344 & 0.266 & 0.087 & 0.874\\\\\n    \n      \\cmidrule(lr){2-12}\n      & Meteor & 50 & 0.9 & 10 & 0.404 & 0.177 & 0.274 & 0.343 & 0.286 & 0.102 & 0.872\\\\\n      & & & & 20 & 0.406 & 0.180 & 0.276 & 0.343 & 0.292 & 0.107 & 0.871\\\\\n      & & & & 30 & 0.401 & 0.172 & 0.271 & 0.337 & 0.288 & 0.099 & 0.870\\\\\n      & & 100 & 0.9 & 10 & 0.405 & 0.178 & 0.276 & 0.343 & \\textbf{0.294} & 0.107 & 0.870\\\\\n      & & & & 20 & 0.406 & 0.176 & 0.276 & 0.343 & 0.291 & 0.106 & 0.872\\\\\n      \\rowcolor{lightgray} \n      & & & & 30 & \\textbf{0.409} & \\textbf{0.184} & \\textbf{0.280} & \\textbf{0.348} & 0.291 & \\textbf{0.108} & \\textbf{0.873}\\\\\n      \n      \\cmidrule(lr){1-12}\n      \n     \n     \\multirow{18}{*}{Supervised + NLPO} & Rouge-1 & 50 & 0.9 & 10 & \\textbf{0.425} & 0.196 & \\textbf{0.299} & \\textbf{0.366} & 0.285 & 0.106 & \\textbf{0.882} \\\\\n     & & &  & 20 & 0.417 & 0.191 & 0.295 & 0.360 & 0.276 & 0.100 & 0.881 \\\\\n     & & & & 30 & 0.418 & 0.192 & 0.296 & 0.361 & 0.278 & 0.101 & 0.881 \\\\\n     & & 100 & 0.9 & 10 &  0.424 & 0.196 & 0.299 & 0.366 & 0.286 & 0.106 & 0.882 \\\\\n     & & & & 20 & 0.423 & 0.196 & 0.299 & 0.365 & \\textbf{0.289} & \\textbf{0.110} & 0.881 \\\\\n     & & & & 30 & 0.420 & 0.193 & 0.296 & 0.362 & 0.279 & 0.102 & 0.881 \\\\\n    \\cmidrule(lr){2-12}\n     & Rouge-Avg & 50 & 0.9 & 10 & 0.426 & 0.197 & 0.298 & 0.367 & 0.294 & 0.115 & 0.881 \\\\\n     & & & & 20 &  0.425 & 0.196 & 0.298 & 0.366 & 0.292 & 0.112 & 0.881\\\\\n     & & & & 30 & 0.424 & 0.194 & 0.297 & 0.365 & 0.287 & 0.107 & 0.881 \\\\\n     & & 100 & 0.9 & 10 & 0.424 & 0.196 & 0.298 & 0.365 & 0.291 & 0.113 & 0.881\\\\\n     & & & & 20 & 0.428 & 0.198 & 0.300 & 0.368 & 0.296 & 0.115 & 0.882\\\\\n     \\rowcolor{lightgray} \n     & & & & 30 & \\textbf{0.429} & \\textbf{0.199} & \\textbf{0.300} & \\textbf{0.369} & \\textbf{0.296} & \\textbf{0.116} & \\textbf{0.882}\\\\\n     \\cmidrule(lr){2-12}\n     & Meteor & 50 & 0.9 & 10 &  0.430 & 0.197 & 0.294 & 0.364 & 0.320 & 0.130 & 0.879\\\\\n     & & & & 20 & 0.432 & 0.198 & 0.297 & 0.367 & 0.318 & 0.130 & 0.880\\\\\n     & & & & 30 & 0.423 & 0.191 & 0.293 & 0.361 & 0.297 & 0.116 & 0.879 \\\\\n     \\rowcolor{lightgray} \n     & & 100 & 0.9 & 10 & \\textbf{0.435} & \\textbf{0.200} & \\textbf{0.298} & \\textbf{0.369} & \\textbf{0.320} & 0.131 & \\textbf{0.881} \\\\\n     & & & & 20 &  0.433 & 0.198 & 0.297 & 0.368 & 0.319 & 0.130 & 0.879 \\\\\n     & & & & 30 & 0.434 & 0.200 & 0.297 & 0.369 & 0.324 & \\textbf{0.132} & 0.879 \\\\\n   \\bottomrule\n   \\end{tabular}\n   }\n   \\caption{\\textbf{NLPO Ablation\/Model Selection}: Evaluation of NLPO models on validation set with different reward functions, top k values for rollouts and target update iterations. For each alg-reward combo, best model is chosen}\n   \\label{tbl:summ_nlpo_ablation}\n\\end{table*}\n\n\n \n\n\\begin{table}[]\n\\centering\n\\footnotesize\n\\begin{tabular}{|l|r|rrr|rrr|}\n\\multicolumn{1}{|c|}{\\multirow{2}{*}{\\textbf{Algorithm}}} & \\multicolumn{1}{c|}{\\multirow{2}{*}{\\textbf{Unique N}}} & \\multicolumn{3}{c|}{\\textbf{Coherence}}                                                                        & \\multicolumn{3}{c|}{\\textbf{Quality}}                                                                          \\\\\n\\multicolumn{1}{|c|}{}                                    & \\multicolumn{1}{c|}{}                                   & \\multicolumn{1}{c|}{\\textbf{Value}} & \\multicolumn{1}{c|}{\\textbf{Alpha}} & \\multicolumn{1}{c|}{\\textbf{Skew}} & \\multicolumn{1}{c|}{\\textbf{Value}} & \\multicolumn{1}{c|}{\\textbf{Alpha}} & \\multicolumn{1}{c|}{\\textbf{Skew}} \\\\ \\hline\nPPO+Supervised                                            & 22                                                      & \\multicolumn{1}{r|}{4.21}           & \\multicolumn{1}{r|}{0.198}          & 4.224                              & \\multicolumn{1}{r|}{3.97}           & \\multicolumn{1}{r|}{0.256}          & 3.98                               \\\\\nNLPO+Supervised                                           & 19                                                      & \\multicolumn{1}{r|}{4.3}            & \\multicolumn{1}{r|}{0.26}           & 4.308                              & \\multicolumn{1}{r|}{3.98}           & \\multicolumn{1}{r|}{0.089}          & 4                                  \\\\\nZero Shot                                                 & 17                                                      & \\multicolumn{1}{r|}{3.73}           & \\multicolumn{1}{r|}{0.1}            & 3.757                              & \\multicolumn{1}{r|}{3.69}           & \\multicolumn{1}{r|}{0.25}           & 3.722                              \\\\\nSupervised                                                & 19                                                      & \\multicolumn{1}{r|}{4.25}           & \\multicolumn{1}{r|}{0.116}          & 4.241                              & \\multicolumn{1}{r|}{3.99}           & \\multicolumn{1}{r|}{0.2}            & 3.986                              \\\\\nNLPO                                                      & 17                                                      & \\multicolumn{1}{r|}{4.03}           & \\multicolumn{1}{r|}{0.13}           & 4.042                              & \\multicolumn{1}{r|}{3.83}           & \\multicolumn{1}{r|}{0.191}          & 3.832                              \\\\\nPPO                                                       & 21                                                      & \\multicolumn{1}{r|}{3.94}           & \\multicolumn{1}{r|}{0.111}          & 3.945                              & \\multicolumn{1}{r|}{3.76}           & \\multicolumn{1}{r|}{0.129}          & 3.767                              \\\\\nHuman                                                     & 19                                                      & \\multicolumn{1}{r|}{3.89}           & \\multicolumn{1}{r|}{0.277}          & 3.902                              & \\multicolumn{1}{r|}{3.77}           & \\multicolumn{1}{r|}{0.029}          & 3.769                             \n\\end{tabular}\n\\caption{Results of the human subject study showing the number of participants N, average Likert scale value for coherence and sentiment, Krippendorf's alpha showing inter-annotator agreement, and Skew. For each model a total of 50 samples were drawn randomly from the test set and rated by 3 annotators each, each resulting in 150 data points per algorithm.}\n\\label{app:summariazation:human_agreement}\n\\end{table}\n\n\n\\begin{table}[]\n\\centering\n\\footnotesize\n\\begin{tabular}{|l|l|rr|rr|}\n                                       &                                       & \\multicolumn{2}{c|}{\\textbf{Coherence}}                                                      & \\multicolumn{2}{c|}{\\textbf{Quality}}                                                        \\\\\n\\multicolumn{1}{|c|}{\\textbf{Group 1}} & \\multicolumn{1}{c|}{\\textbf{Group 2}} & \\multicolumn{1}{c|}{\\textbf{Diff (G2-G1)}} & \\multicolumn{1}{c|}{\\textit{\\textbf{p-values}}} & \\multicolumn{1}{c|}{\\textbf{Diff (G2-G1)}} & \\multicolumn{1}{c|}{\\textit{\\textbf{p-values}}} \\\\ \\hline\nHuman                                  & NLPO                                  & \\multicolumn{1}{r|}{0.147}                 & 0.755                                           & \\multicolumn{1}{r|}{0.060}                 & 0.900                                           \\\\\nHuman                                  & NLPO+Supervised                       & \\multicolumn{1}{r|}{\\textbf{0.413}}        & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{0.213}}        & \\textbf{0.047}                                  \\\\\nHuman                                  & PPO                                   & \\multicolumn{1}{r|}{0.053}                 & 0.900                                           & \\multicolumn{1}{r|}{-0.007}                & 0.900                                           \\\\\nHuman                                  & PPO+Supervised                        & \\multicolumn{1}{r|}{\\textbf{0.327}}        & \\textbf{0.024}                                  & \\multicolumn{1}{r|}{0.200}                 & 0.544                                           \\\\\nHuman                                  & Supervised                            & \\multicolumn{1}{r|}{\\textbf{0.360}}        & \\textbf{0.008}                                  & \\multicolumn{1}{r|}{\\textbf{0.220}}        & \\textbf{0.043}                                  \\\\\nHuman                                  & Zero Shot                             & \\multicolumn{1}{r|}{-0.160}                & 0.679                                           & \\multicolumn{1}{r|}{-0.080}                & 0.900                                           \\\\\nNLPO                                   & NLPO+Supervised                       & \\multicolumn{1}{r|}{\\textbf{0.267}}        & \\textbf{0.012}                                  & \\multicolumn{1}{r|}{\\textbf{0.153}}        & \\textbf{0.008}                                  \\\\\nNLPO                                   & PPO                                   & \\multicolumn{1}{r|}{-0.093}                & 0.900                                           & \\multicolumn{1}{r|}{-0.067}                & 0.900                                           \\\\\nNLPO                                   & PPO+Supervised                        & \\multicolumn{1}{r|}{0.180}                 & 0.564                                           & \\multicolumn{1}{r|}{0.140}                 & 0.860                                           \\\\\nNLPO                                   & Supervised                            & \\multicolumn{1}{r|}{0.213}                 & 0.361                                           & \\multicolumn{1}{r|}{0.160}                 & 0.754                                           \\\\\nNLPO                                   & Zero Shot                             & \\multicolumn{1}{r|}{\\textbf{-0.307}}       & \\textbf{0.044}                                  & \\multicolumn{1}{r|}{-0.140}                & 0.860                                           \\\\\nNLPO+Supervised                        & PPO                                   & \\multicolumn{1}{r|}{\\textbf{-0.360}}       & \\textbf{0.008}                                  & \\multicolumn{1}{r|}{\\textbf{-0.220}}       & \\textbf{0.043}                                  \\\\\nNLPO+Supervised                        & PPO+Supervised                        & \\multicolumn{1}{r|}{\\textbf{-0.087}}       & \\textbf{0.009}                                  & \\multicolumn{1}{r|}{\\textbf{-0.013}}       & \\textbf{0.009}                                  \\\\\nNLPO+Supervised                        & Supervised                            & \\multicolumn{1}{r|}{\\textbf{-0.053}}       & \\textbf{0.009}                                  & \\multicolumn{1}{r|}{0.007}                 & 0.900                                           \\\\\nNLPO+Supervised                        & Zero Shot                             & \\multicolumn{1}{r|}{\\textbf{-0.573}}       & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{-0.293}}       & \\textbf{0.012}                                  \\\\\nPPO                                    & PPO+Supervised                        & \\multicolumn{1}{r|}{0.273}                 & 0.106                                           & \\multicolumn{1}{r|}{0.207}                 & 0.508                                           \\\\\nPPO                                    & Supervised                            & \\multicolumn{1}{r|}{0.307}                 & 0.044                                           & \\multicolumn{1}{r|}{0.227}                 & 0.394                                           \\\\\nPPO                                    & Zero Shot                             & \\multicolumn{1}{r|}{-0.213}                & 0.361                                           & \\multicolumn{1}{r|}{-0.073}                & 0.900                                           \\\\\nPPO+Supervised                         & Supervised                            & \\multicolumn{1}{r|}{0.033}                 & 0.900                                           & \\multicolumn{1}{r|}{0.020}                 & 0.900                                           \\\\\nPPO+Supervised                         & Zero Shot                             & \\multicolumn{1}{r|}{-0.487}                & 0.001                                           & \\multicolumn{1}{r|}{-0.280}                & 0.155                                           \\\\\nSupervised                             & Zero Shot                             & \\multicolumn{1}{r|}{-0.520}                & 0.001                                           & \\multicolumn{1}{r|}{-0.300}                & 0.101                                          \n\\end{tabular}\n\\caption{Results of an post-hoc Tukey HSD Test for difference in means between pairs of algorithms (Group 2 - Group 1) and corresponding $p$-values. Individually statistically significant results are bolded and are used to discuss results in the analysis. Overall $p$-values showing that there is a significant difference in means between the models via a one-way ANOVA test are significant with $p \\ll 0.05$ for both coherence and sentiment.}\n\\label{app:summariazation:human_tukey}\n\\end{table}\n\n\n\\clearpage\n\n\\subsubsection{Human Participant Study}\n\nFigure~\\ref{fig:description_interface_summarization} shows the summarization instructions and interface used for the human evaluation experiments. Participants weren't required to read the entire article, but to encourage some reading, a minimum time on the window of 15s was enforced via hiding the sliders.\nTables~\\ref{app:summariazation:human_agreement},~\\ref{app:summariazation:human_tukey} show averaged results, annotator agreement, and the results of statistical significance tests to determine which models output better generations when rated by humans.\n\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=.48\\linewidth]{figures\/summarization_human_study\/summarization_human1.png}\n    \\includegraphics[width=.48\\linewidth]{figures\/summarization_human_study\/summarization_human3.png}\n    \\caption{Instructions and interface for the summarization task.}\n    \\label{fig:description_interface_summarization}\n\\end{figure*}\n\n\n\\subsubsection{Qualitative Analysis}\n\\label{app:sec:summ_qual_analysis}\nWe show sample generations from each of the algorithms for three randomly picked prompts below.\n\n\\begin{lstlisting}\nSample 1\nPrompt: Manchester City are confident UEFA's punishment for breaching financial fairplay regulations will be lifted this summer which would allow them to bid for stellar names like Raheem Sterling, Gareth Bale, Kevin de Bruyne and  Ross Barkley. City boss Manuel Pellegrini has been hampered over the past year by UEFA restricting them to a net transfer spend of 49million in each window and keeping the club's overall wage bill to its current level of 205million-a-year. UEFA's settlement with City published in May stated those penalties would remain in place until the end of the 2015\/16 season but the club's latest financial figures showed drastically-reduced losses of 23million which they feel proves they are now compliant with FPP regulations. Manuel Pellegrini is hoping that the financial  restrictions imposed by UEFA for a breach of FFP rules will be lifted at the end of this season . Manchester City have been limited to a net spend of 49million in the last two transfer windows - they spent 25m bringing Wilfried Bony in from Swansea in January . Ahead of Monday night's trip to Crystal Palace, Pellegrini was certainly talking like a man excited at the prospect of signing 'crack' players this summer. 'I think that next season we don't have any restrictions so we will be in the same position that all the other English clubs have,' said Pellegrini. 'It's important. You have so many strong teams here in England and in Champions League, you can not allow them to keep the advantage every year; having less players  to put in your squad or spending less money. We spend money, of course we always spend money, but they spent more.' Manchester United, Barcelona, Liverpool and Arsenal have all paid more in transfer fees in the past 12 months than City who were traditionally Europe's biggest spenders after the club was taken over by Abu Dhabi owners in 2008. Uefa also ordered City to play with a reduced squad from 25 players to 21 in the Champions League this season and while that restriction has now ended, any time reduction in the penalties on spending and wages is more controversial. Arsenal have paid more in transfer fees than City in the last 12 months, including 30m on Alexis Sanchez . The document published last May by UEFA's Club Financial Control Body investigative chamber explicitly said City's financial penalties would run for two seasons  at least and there has been no official deviation from that decision. \nThe published statement said at the time: 'Manchester City agrees to significantly limit spending in the transfer market for the seasons 2014\/15 and 2015\/16. It means City will have to argue their case with Uefa that as they have been financially compliant over the past year, they deserve to be free of restrictions moving forward. They have successfully argued their case with UEFA before. Last summer they persuaded the governing body to allow them to bypass the normal quota of eight homegrown players as their Champions League squad had been reduced. Eliaquim Mangala joined the champions from Porto for 32m last summer . The reigning Premier League champions have only paid money for Fernando, Willie Caballero, Eliaquim Mangala and Wilfried Bony in the last two transfer windows and that was part-paid by the sales of Javi Garcia and Jack Rodwell. Pellegrini admitted they weren't in a position to compete with neighbours United for established world stars like Angel di Maria because of the FFP penalties. It has cost City on the pitch. Even if they win their remaining eight Premier League fixtures their points tally will be lower than last season and they crashed out once again at the last-16 stage of the Champions League. Pellegrini and director of football Txiki Begiristain both accept the squad needs replenishing and they want to be in the market for the top available players. Manchester City have been heavily linked with a move for former Chelsea midfielder Kevin de Bruyne . The club also want to be involved in any race for Liverpool contract rebel Raheem Sterling . They have been strongly linked with former Chelsea winger Kevin de Bruyne who has been one of the best players in the Bundesliga with Wolfsburg. And if Liverpool do allow Sterling to leave, City want to be involved in the chase for one of England's most outstanding talents to help fit in with quotas for homegrown players. 'If you are the manager or a player of a big tem, you are always open to criticism because everyone expects a high level of performance,' he said. 'The major issue with our team always seems to be the money but this season I think we were the team that spent less than any other team. That is the weird thing. 'For this club trying to be a big team in so few years, maybe we are paying the cost for that.' Since the transfer penalties were introduced, City have spent 91million on players in the last two windows (Net spend 68million). That compares to Manchester United 145.5million, Liverpool 113million, Arsenal 92.5million and Chelsea 82.6million. Over the same time period Barcelona spent 118.3million on players and Real Madrid 81.2million though they also broke the world transfer record for Gareth Bale the previous summer.\nZero-Shot: manuel Pellegrini hoping UEFA's punishment for breaching financial fairplay regulations will be lifted this summer.  city have been restricted to a net transfer spend of 49million in each window. a total of 23million has been lost in the last year.\nReferences:  Manchester City have been restricted to a net transfer spend of 49m .Club's also had to keep overall wage bill to its  current level of 205m .Punishments imposed by UEFA for breaching Financial Fair Play rules .The spending restrictions were set for this season and the next one .But City are confident they will be lifted early after their compliance .\nPPO: Manuel Pellegrini is hoping UEFA's punishment for breaching financial fairplay rules will be lifted this summer. Manchester City have been limited to a net spend of 49million in each transfer window. the club also keep the club's overall wage bill to its current level of 205million-a-year.\nNLPO:  Manchester City are confident UEFA's punishment for breaching financial fairplay regulations will be lifted this summer. that would allow them to bid for stellar names like Raheem Sterling, Kevin de Bruyne and Ross Barkley.\nSupervised: Manchester City have been limited to a net transfer spend of 49million in each window. City's overall wage bill is 205m-a-year due to a breach of UEFA rules. City are confident the financial restrictions will be lifted at the end of the current season in order to bid for stars such as Raheem Sterling. Manuel Pellegrini is hoping that the financial restrictions will be lifted at the end of the 2015\/16 season\nSupervised + PPO: Manchester City have been restricted to a net spend of 49million in each window and kept their overall wage bill at 205m. Manuel Pellegrini believes these restrictions will be lifted at the end of the season. Arsenal have spent more in transfer fees than City in the last 12 months, including 30m on Alexis Sanchez.\nSupervised + NLPO: Manchester City are confident UEFA's punishment for a breach of financial fairplay regulations will be lifted this summer. The club have been restricted to a net transfer spend of 49million over the past two window. They aim to bid for stars like Raheem Sterling, Gareth Bale and Kevin de Bruyne. City lost 23m on transfers in the last 12 months due to financial restrictions imposed by UEFA for breach \n\nSample 2\nPrompt: (CNN)Larry Johnson remembers the fear and feeling of helplessness from being on the SkyWest Airlines flight that made an emergency landing in Buffalo, New York. \"I thought we were done,\" he said Thursday, one day later. \"There was no doubt in my mind that we weren't going to land.\" Johnson was flying with his brother, his girlfriend and his 8-month-old son when he says a flight attendant came over the speaker asking for someone who was medically trained to help with a sick passenger. Minutes later, Johnson says, the attendant announced there was a pressurization problem and told passengers to prepare for the emergency landing. \"You're going to feel dizzy and woozy and lightheaded, and you're not going to be able to breathe,\" Johnson recalled the flight attendant saying. The Federal Aviation Administration on Wednesday initially reported a pressurization problem with SkyWest Flight 5622, and said it would investigate. It later issued a statement that did not reference any pressurization issues. \"It was like being trapped and you couldn't do anything. You have no control over it. Your body just felt pressured,\" Johnson said. Johnson said his girlfriend was seeing spots and that \"when I was blinking, it was really bright. I could barely see.\" According to Marissa Snow, spokeswoman for SkyWest, three passengers reported a loss of consciousness while on the flight. Fifteen passengers were examined by medical personnel at the Buffalo airport, but no one was transported to the hospital, she said. The spokeswoman said that maintenance personnel found no indication of a pressurization problem with the aircraft, an Embraer E170, and that the airline continues to investigate the cause. An official with the National Transportation Safety Board told CNN that the agency is in communication with the FAA and SkyWest to gather information on the incident to better understand what took place.\nMary Cunningham was the nurse on board who attended to the first sick passenger. \"There was a woman, very lethargic, very nauseous, turning gray, just not looking good,\" Cunningham said. After administering oxygen to the woman, Cunningham, a nurse at Yale-New Haven Hospital in Connecticut, returned to her seat. Soon after, Cunningham said, she was called up again by the flight attendant after another women near the first started to lose consciousness. \"It was something that was affecting oxygen in that area,\" which was near the middle of the plane, she said. Cunningham said she had to return to her seat after feeling lightheaded. There were 84 passengers on board the plane, including nine crew members. Flight 5622 was originally scheduled to fly from Chicago to Hartford, Connecticut. The plane descended 28,000 feet in three minutes. \"It would feel like a roller coaster -- when you're coming over the top and you're going down,\" CNN aviation analyst Mary Schiavo said, describing how such a descent would feel. \"You know that these pilots knew they were in a very grave and very serious situation.\" Johnson, flying to visit his parents with their only grandchild, was glad that roller coaster ride ended safely. \"It was a shaky ride. It wasn't anything I would like to experience again,\" he said. CNN's Jean Casarez, Josh Gaynor, Dana Ford and Stephanie Gallman contributed to this report.\nZero-Shot: \"you're going to feel dizzy and woozy and lightheaded,\" Johnson says. \"it was like being trapped and you couldn't do anything,\" Johnson says.  \"the plane was trying to get back on the ground,\" passenger says of fear.\nReferences: Three passengers report a loss of consciousness on SkyWest flight .But officials say there is no evidence of a pressurization problem .\nPPO: SkyWest flight 5312 made emergency landing in Buffalo, new york. passengers were told they were not going to be able to breathe,  passenger recalls attendant. the faa initially reported a pressurization problem with the flight 5622.\nNLPO: SkyWest flight 5622 made emergency landing in buffalo, new york.Larry Johnson says flight attendant announced there was a pressurization problem. three passengers reported loss of consciousness while on flight; no one transported to hospital.\nSupervised: The FAA says it is investigating the cause of the emergency landing. SkyWest Flight 5622 made an emergency landing in Buffalo, New York.  The flight was carrying a family of three people. The flight attendant told the passengers to prepare for the landing.\nSupervised + PPO: \"I thought we were done,\" the father of one passenger says. The flight attendant announced there was a pressurization problem.  The flight made an emergency landing in Buffalo, New York. Three passengers reported a loss of consciousness while on the flight.\nSupervised + NLPO: SkyWest Airlines flight made an emergency landing in Buffalo, New York, on Thursday. \"There was no doubt in my mind  that we weren't going to land,\" passenger says. \"You're going to feel dizzy and woozy and lightheaded...you're not going to be able to breathe,\" he says.\"\n\nSample 3\nPrompt: For Inverness the latest chapter in their remarkable story featured not one, but two, plot-changing twists. One featured penalty Celtic didn't get for 2-0 in which Josh Meekings handball should have also led to a sending off. The other the spot kick they did, followed by a red card for Craig Gordon. 'I've not seen it yet, but going by the reaction of the Celtic players we got away with a penalty and a sending off and that was probably the turning point in the game,' acknowledged Caley manager John Hughes after. Inverness's Josh Meekings appears to get away with a handball on the line in their win over Celtic . Caley boss John Hughes says the break, which could have meant a penalty and red card, was a turning point . 'I've not spoken to Josh. I haven't seen it - but going by the media it was definitely a hand ball. We look at the referee behind the line and all that and I know Ronny will feel aggrieved - because I certainly would. 'But it's part and parcel of football and you need a wee bit of luck to beat Celtic. 'This was their biggest game of the season because they will go on and win the league and if they had beaten us today there was a good chance they would have gone on and won the Scottish Cup. 'But when Marley Watkins was clipped by Craig Gordon and they were down to 10 men that was advantage Inverness. 'We weren't going to give Celtic the ball back, they had to come and get it and we had to be patient. 'When big Edward put us into the lead we thought it was going to be our day on the back of things that had happened. 'Celtic equalised with another free kick but it's typical of Inverness that we don't do anything easy. 'We do it the hard way and we came up with the winner through David Raven.' Hughes hauled Raven, his Scouse defender, from his backside as extra-time beckoned. Offended by the sight of one of his players resting he had a message to impart. Caley players celebrate after upsetting Celtic in a Scottish Cup semi-final 3-2 thriller . Celtic, depleted by games and absentees, were virtually on their knees after a relentless programme of midweek games. In last season's League Cup Final Inverness had been passive and unambitious prior to losing on penalties. This was no time to repeat the mistake. 'I tried to emphasise to the players they would never have a better time to go on and beat Celtic, down to 10 men in the semi final of a cup. We needed to go for it,' Hughes said. 'Before Raven scored at the back post I was looking to change it. \nI was going to bring on another winger, Aaron Doran, and put him in the full-back position over on the right, but more advanced so he could take their left back on. Thankfully I didn't do that and David Raven came up with the goal. Virgil Van Dijk (centre) fired Celtic into an early lead with a superb free-kick in the 18th minute . 'I didn't realise this is the first time the club have been in the final of the Scottish Cup and that's a remarkable achievement given it was only formed 20 years ago. 'It is a great story isn't it? It's an absolutely fantastic story. It is 20 odd years since the amalgamation. We are a small provincial club up there in the Highlands. 'We have lost a real inspirational skipper in Richie Foran right from the start of the season. He has never played. We have had to adjust to that. 'We had to sell Billy McKay, our top goalscorer, at Christmas. We have had to go again and adjust. I am a very humble guy and I am grateful and thankful that injuries have never caught up with us.'  There is remarkable irony in the fact Falkirk will be the opponents for the final. A former Bairns captain, he was manager of the club in 2009 when they lost to Rangers at Hampden. Former Falkirk captain and manager John Hughes will take on his former club in the final . 'I had a lot of great times at Falkirk. So much so that it is possibly my favourite time in my playing career. I am still friendly with an awful lot of the characters who were in that dressing room. Neil Oliver is a good friend of mine from my Falkirk days. He comes along and sits on the bench and gives me a wee hand out. 'That is the spirit that we had at that club. I have met some great guys. Brian Rice, my ex-coach, Davie Weir, Ian McCall, the list is endless. I was just talking the other day about that day at Falkirk. There are times even now when I see. 'I have a real special, special feel for the place. I am not kidded Falkirk is a right good side, trust me on that. No side goes to Tynecastle and beats Hearts without being a good side. No side beats Hibs the way they were playing without being a good side. You might say we are favourites I say it is even stevens and I am just looking forward to when the days comes.' David Raven (second right) celebrates with his Inverness team-mates following his extra-time winner . An Inverness and Falkirk final would not have been the choice of sponsors  William Hill, you suspect. Vast swathes of Hampden's west stand were empty, a 12.15 kick-off for the benefit of broadcasters proving unhelpful . There will be no ticket rows this time. Not over a final unlikely to raise a crowd much higher than 35,000 in a stadium of 52,000. The contract with Hampden plc is signed, with no question of moving it. If the bitter amalgamation of 21 years ago produced a fine football club it also disenfranchised many of those who followed the old ones. 'I hope this encourages the next generation to follow Inverness,' added Hughes. 'In the peer groups it is still Celtic and Rangers they follow but I hope the next ones will want to support their local club. 'If we can add another few hundred or thousand to the support then we are playing our part.\nZero-Shot: inverness beat Celtic 2-0 in their opening game of the season. Josh Meekings was the subject of a handball after the break. the break could have meant a penalty and a red card for Craig Gordon.\nReferences: Josh Meekings handball should have been a penalty and a red card .If the penalty was awarded and converted Celtic would have gone to 2-0 .Caley manager John Hughes admitted his side were fortunate in win .Virgil van Dijk scored Celtic's opener with a superb free-kick .Celtic keeper Craig Gordon was sent off early in the second half .Greg Tansey, Edward Ofere and finally Daven Raven scored for Caley .\nPPO: Josh Meekings handball could have meant sending off for Celtic in extra-time win. John Hughes says the break could have meant an penalty and red card. Caley boss also admits he would feel aggrieved had Celtic won the championship.\nNLPO: Caley beat Celtic 2-0 in their Scottish Cup clash. Josh Meekings handball could have led to a penalty and red card. John Hughes says the break was probably the turning point in the game. defender does not believe defensive striker can get away with being shown the yellow card - despite playing in same half we never scored.\nSupervised: Inverness beat Celtic 2-0 in their Scottish Cup clash. Josh Meekings appeared to get away with a handball on the line. Another player was sent off for a spot kick which ended in a red card. Caley manager John Hughes says the break could have meant a penalty. Hughes says he 'would feel aggrieved' after the break.\nSupervised + PPO: Caley beat Celtic 1-0 in their Scottish Cup encounter at the Etihad. Josh Meekings' handball on the line should have also seen a sending off.  But Craig Gordon got away with a spot kick and sending off. John Hughes said it was a turning point in the game for Inverness.\nSupervised + NLPO: Josh Meekings appears to get away with a handball after Inverness's win. Inverness were down to 10 men when Craig Gordon was sent off.  John Hughes says the break could have meant a penalty and red card. Hughes says Celtic will go on and win the Scottish Cup with win \n\\end{lstlisting}\n\\subsubsection{Setup} ToTTo \\citep{parikh2020totto} is a controlled table-to-text generation task in which the goal is to produce one-sentence description of highlighted table cells.\nFor training RL methods, we consider 5 different reward functions: BLEU, SacreBLEU, METEOR, PARENT and a combination of Meteor and PARENT. We chose T5 as our base LM here too, as they are more suitable for structure to text tasks. For decoding, we use beam search during inference and for generating rollouts, we use top k sampling.  Other implementation details are captured in Table~\\ref{tbl:totto_gen_hyperparams}.\n\n\\begin{table*}[ht!]\n\\centering\n\\footnotesize\n\\resizebox{0.5\\textwidth}{!}{\n\\begin{tabular}{ll}\n\\toprule\n\\textbf{Model Params}\n& \\multicolumn{1}{c}{\\textbf{value}} \\\\ \n\\cmidrule{1-2}\nsupervised & batch size: $8$\\\\\n& epochs: $4$ \\\\\n& learning rate: $0.0001$ \\\\\n& learning rate scheduler: constant with warm up \\\\\n& weight decay: $0.1$ \\\\\n\\cmidrule{1-2}\nppo\/\\textcolor{forestgreen2}{nlpo} & steps per update: $2560$\\\\\n    & total number of steps: $256000$ \\\\\n    & batch size: $64$ \\\\\n    & epochs per update: $5$\\\\\n    & learning rate: $0.000002$ \\\\\n    & entropy coefficient: $0.0$ \\\\\n    & initial kl coeff: $0.001$ \\\\\n    & target kl: $2.0$ \\\\\n    & discount factor: $0.99$ \\\\\n    & gae lambda: $0.95$ \\\\\n    & clip ratio: $0.2$ \\\\\n    & rollouts top k : $0$ \\\\\n    & value function coeff: $0.5$ \\\\\n    & \\textcolor{forestgreen2}{top mask ratio: $0.9$} \\\\\n    & \\textcolor{forestgreen2}{target update iterations: $20$} \\\\\n\\cmidrule{1-2}\nsupervised+ ppo (or \\textcolor{forestgreen2}{nlpo}) & steps per update:$2560$\\\\\n    & total number of steps: $256000$ \\\\\n    & batch size: $64$ \\\\\n    & epochs per update: $5$ \\\\\n    & learning rate: $0.0000005$ \\\\\n    & entropy coefficient: $0.0$ \\\\\n    & initial kl coeff: $0.01$ \\\\\n    & target kl: $0.2$\\\\\n    & discount factor: $0.99$ \\\\\n    & gae lambda: $0.95$ \\\\\n    & clip ratio: $0.2$ \\\\\n    & rollouts top k : $50$ \\\\\n    & value function coeff: $0.5$ \\\\\n    & \\textcolor{forestgreen2}{top mask ratio: $0.9$} \\\\\n    & \\textcolor{forestgreen2}{target update iterations: $20$} \\\\\n\\cmidrule{1-2}\n\\cmidrule{1-2}\ndecoding & num beams: $5$ \\\\\n& min length: $10$ \\\\\n& max new tokens: $50$\\\\\n\n\\cmidrule{1-2}\ntokenizer & padding side: left\\\\\n& truncation side: right\\\\\n& max length: 512 \\\\\n\\bottomrule\n\\end{tabular}\n}\n\\caption{\\textbf{ToTTO Hyperparams}: Table shows a list of all hyper-parameters and their settings}\n       \\label{tbl:totto_gen_hyperparams}\n\\end{table*}\n\n\\subsubsection{Results and Discussion} Tables \\ref{tbl:totto_val_scores}, \\ref{tbl:totto_test_scores}  presents our benchmarking results with 5 reward functions along with supervised baseline performances on dev and test sets respectively. Similar to other tasks, our main finding is that warm-started initial policies are crucial for learning to generate descriptions from highlighted cells. Without warm-start, policies suffer from reward hacking and resulting in sub-optimal solutions despite application of task-specific metrics such as PARENT etc. We find that Supervised+NLPO method outperforms all models on ToTTo leaderboard in terms of PARENT metric.\n\n\\begin{table*}[h]\n       \\centering\n       \\resizebox{1.0\\textwidth}{!}{\n       \\begin{tabular}{@{}ccccccccccccc@{}}\n       \\toprule\n         Tasks \n         & \\multicolumn{3}{c}{\\textbf{\\_}} \n         & \\multicolumn{6}{c}{\\textbf{Lexical and Semantic Metrics}} \n         & \\multicolumn{3}{c}{\\textbf{Factual Consistency}}\n         \\\\\n         & Alg & LM & Reward function & \\multicolumn{3}{c}{SacreBleu} & \\multicolumn{3}{c}{BLEURT}\n         & \\multicolumn{3}{c}{PARENT} \\\\\n         \\cmidrule(lr){5-7} \\cmidrule(lr){8-10} \\cmidrule(lr){11-13} \\\\\n         & & &  & Overall & Overlap & Non-Overlap & Overall & Overlap & Non-Overlap & Overall & Overlap & Non-Overlap\\\\\n         \\cmidrule(lr){2-2}  \\cmidrule(lr){3-3} \\cmidrule(lr){4-4} \\cmidrule(lr){5-5}\n         \\cmidrule(lr){6-6} \\cmidrule(lr){7-7} \\cmidrule(lr){8-8} \\cmidrule(lr){9-9}\n         \\cmidrule(lr){10-10}\n         \\cmidrule(lr){11-11}\n         \\cmidrule(lr){12-12}\n         \\cmidrule(lr){13-13}\n         \\multirow{22}{*}{ToTTo} & Zero-Shot & T5 & &  0.036 & 0.040 & 0.032 & -1.392 & -1.387 & -1.397 & 0.116 & 0.119 & 0.112\\\\\n         \\cmidrule{2-13}\n         & PPO & T5 & bleu & 0.065 & 0.067 & 0.063 & -1.074 & -1.045 & -1.098 & 0.246 & 0.246 & 0.244 \\\\\n         &  & T5 & sacrebleu & 0.086 & 0.090 & 0.083 & -0.979 & -0.955 & -1.003 & 0.293 & 0.292 & 0.294\\\\\n         & & T5 & meteor & 0.144 & 0.155 & 0.132 & -0.769 & -0.713 & -0.826 & 0.356 & 0.361 & 0.351\\\\\n         & & T5 & parent & 0.146 & 0.153 & 0.128 & -0.721 & -0.688 & -0.753 & 0.336 & 0.335 & 0.339\\\\\n         & & T5 & meteor + parent & 0.161 & 0.169 & 0.152 & -0.891 & -0.861 & -0.922 & 0.345 & 0.342 & 0.348 \\\\\n         \n         \\cmidrule{2-13}\n         & NLPO & T5 & bleu & 0.062 & 0.065 & 0.059 & -1.077 & -1.057 & -1.097 & 0.235 & 0.236 & 0.233 \\\\\n         &  & T5 & sacrebleu & 0.085 & 0.088 & 0.083 & -0.945 & -0.917 & -0.972 & 0.314 & 0.315 & 0.313\\\\\n         & & T5 & meteor & 0.102 & 0.108 & 0.097 & -1.044 & -1.009 & -1.079 & 0.329 & 0.328 & 0.330\\\\\n          & & T5 & parent & 0.159 & 0.166 & 0.152 & -0.710 & -0.675 & -0.745 & 0.357 & 0.351 & 0.363\\\\\n         & & T5 & meteor + parent & \\textbf{0.166} & \\textbf{0.175} & \\textbf{0.158} & \\textbf{-0.704} & \\textbf{-0.668} & \\textbf{-0.740} & \\textbf{0.365} & \\textbf{0.362} & \\textbf{0.368}\\\\\n         \n         \\cmidrule{2-13}\n         \n         & Supervised & T5 & & 0.457 & 0.535 & 0.377 & 0.204 & 0.327 & 0.081 & 0.583 & 0.631 & 0.534  \n         \\\\\n         \n         \\cmidrule{2-13}\n         & Supervised + PPO & T5 & bleu & 0.473 & 0.548 & 0.395 & 0.200 & 0.323 & 0.078 & 0.590 & 0.638 & 0.542 \\\\\n          & & T5 & sacrebleu & 0.474 & 0.557 & 0.389 & \\textbf{0.209} & \\textbf{0.340} & 0.077 & 0.573 & 0.620 & 0.525 \\\\\n          & & T5 & meteor & 0.468 & 0.541 & 0.392 & 0.203 & 0.325 & \\textbf{0.082} & 0.590 & 0.638 & 0.542\\\\\n          & & T5 & parent &  0.469 & 0.547 & 0.388 & 0.175 & 0.300 & 0.050 & 0.595 & 0.641 & 0.549\\\\\n         & & T5 & meteor + parent & 0.473 & 0.547 & 0.392 & 0.192 & 0.314 & 0.069 & 0.595 & 0.642 & 0.549\\\\\n         \n         \\cmidrule{2-13}\n         & Supervised + NLPO & T5 & bleu & \\textbf{0.475} & 0.548 & \\textbf{0.399} & 0.208 & 0.330 & 0.085 & 0.593 & 0.639 & 0.546\\\\\n         &  & T5 & sacrebleu & 0.475 & 0.557 & 0.392 & 0.208 & 0.335 & 0.081 & 0.577 & 0.625 & 0.529\\\\\n         & & T5 & meteor & 0.468 & 0.541 & 0.392 & 0.201 & 0.322 & 0.079 & 0.594 & 0.641 & 0.546\\\\\n         & & T5 & parent & 0.474 & \\textbf{0.550} & 0.392 & 0.192 & 0.315 & 0.068 & \\textbf{0.596} & \\textbf{0.643} & \\textbf{0.550} \\\\\n         & & T5 & meteor + parent & 0.471 & 0.546 & 0.393 & 0.204 & 0.326 & 0.081 & 0.592 & 0.640 & 0.544\\\\\n         \\\\\n         \\bottomrule\n        \n       \\end{tabular}\n       }\n       \\caption{\\textbf{ToTTo test evaluation}: Table shows lexical, semantic and factual correctness metric scores of algorithms with different reward functions on hold-out test set. Without supervised pre-training, both PPO and NLPO results in sub-optimal solutions, with NLPO better than PPO. With supervised pre-training, PPO and NLPO achieve better scores across all metrics showing RL fine-tuning is beneficial. Most importantly, RL fine-tuned models produce more factually consistent text as seen in higher PARENT scores. Another observation, fine-tuning with a task-specific metric PARENT is better than training on task-agnostic lexical rewards}\n       \\label{tbl:totto_test_scores}\n    \\end{table*}\n\n\n\\begin{landscape}\n\\begin{table*}[h]\n       \\centering\n       \\resizebox{1.5\\textwidth}{!}{\n       \\begin{tabular}{@{}cccccccccccccccccccccccc@{}}\n       \\toprule\n         Tasks \n         & \\multicolumn{3}{c}{\\textbf{\\_}} \n         & \\multicolumn{9}{c}{\\textbf{Lexical and Semantic Metrics}} \n         & \\multicolumn{3}{c}{\\textbf{Factual Consistency}} \n         & \\multicolumn{8}{c}{\\textbf{Diversity Metrics}}\n         \\\\\n         & Alg & LM & Reward function & Rouge-1 & Rouge-2 & Rouge-L & Rouge-LSum & Meteor & BertScore & \\multicolumn{3}{c}{SacreBleu}\n         & \\multicolumn{3}{c}{PARENT} \\\\\n         \\cmidrule(lr){11-13} \\cmidrule(lr){14-16} \\\\\n         & & & & & & & &  & & Overall & Overlap & Non-Overlap & Overall & Overlap & Non-Overlap & MSTTR & Distinct$_1$ & Distinct$_2$ & H$_1$ & H$_2$ & Unique$_1$ & Unique$_2$ & Mean Output Length\\\\\n         \n         \\cmidrule(lr){2-2}  \\cmidrule(lr){3-3} \\cmidrule(lr){4-4} \\cmidrule(lr){5-5}\n         \\cmidrule(lr){6-6} \\cmidrule(lr){7-7} \\cmidrule(lr){8-8} \\cmidrule(lr){9-9}\n         \\cmidrule(lr){10-10} \\cmidrule(lr){11-11} \\cmidrule(lr){12-12}\n         \\cmidrule(lr){13-13} \\cmidrule(lr){14-14} \\cmidrule(lr){15-15}\n         \\cmidrule(lr){16-16} \\cmidrule(lr){17-17} \\cmidrule(lr){18-18}\n         \\cmidrule(lr){19-19} \\cmidrule(lr){20-20} \\cmidrule(lr){21-21}\n         \\cmidrule(lr){22-22} \\cmidrule(lr){23-23} \\cmidrule(lr){24-24}\n         \n         \\multirow{22}{*}{ToTTo} & Zero-Shot & T5 & & 0.131 & 0.055 & 0.127 & 0.127 & 0.057 & 0.805 & 0.038 & 0.042 & 0.034 & 0.118 & 0.119 & 0.116 & 0.428 & 0.084 & 0.238 & 6.703 & 9.933 & 8387 & 26490 & 19.964\n         \\\\\n         \\cmidrule{2-24}\n         & Supervised & T5 & & 0.410 & 0.279 & 0.388 & 0.388 & 0.223 & 0.953 & 0.458 & 0.533 & 0.387 & 0.586 & 0.633 & 0.540 & 0.715 & 0.162 & 0.511 & 9.995 & 14.468 & 15168 & 54706 & 17.791\n         \\\\\n         \\cmidrule{2-24}\n         & PPO & T5 & bleu & 0.274 & 0.138 & 0.249 & 0.249 & 0.139 & 0.844 & 0.068 & 0.071 & 0.066 & 0.251 & 0.250 & 0.251 & 0.403 & 0.091 & 0.308 & 10.659 & 14.511 & 7536 & 34232 & 28.545\\\\\n         &  & T5 & sacrebleu & {0.341} & {0.166} & {0.300} & {0.300} & 0.165 & 0.858 & 0.09 & 0.094 & 0.086 & 0.300 & 0.299 & 0.300 & 0.469 & 0.121 & 0.407 & 11.071 & 14.880 & 10138 & 48195 & 26.612 \\\\\n         \\rowcolor{lightgray}\n         & & T5 & meteor & 0.322 & 0.157 & 0.286 & 0.286 & {0.173} & 0.888 & 0.147 & 0.163 & 0.133 & {0.358} & {0.367} & 0.350 & 0.625 & 0.136 & 0.482 & 10.189 & 14.910 & 12346 & 54925 & 21.484\\\\\n         & & T5 & parent & 0.268 & 0.125 & 0.251 & 0.251 & 0.119 & {0.890} & 0.150 & 0.158 & 0.143 & 0.337 & 0.332 & 0.342 & 0.764 & 0.202 & 0.646 & 11.068 & 14.988 & 13068 & 50313 & 13.035\\\\\n         & & T5 & meteor + parent & 0.266 & 0.128 & 0.251 & 0.251 & 0.130 & 0.886 & {0.165} & 0.175 & {0.155} & 0.348 & 0.346 & {0.350} & 0.702 & 0.181 & 0.594 & 10.096 & 14.432 & 14422 & 55770 & 15.354\\\\\n         \n         \\cmidrule{2-24}\n         & NLPO & T5 & bleu & 0.267 & 0.134 & 0.24 & 0.24 & 0.137 & 0.84 & 0.068 & 0.071 & 0.065 & 0.238 & 0.239 & 0.237 & 0.448 & 0.1 & 0.359 & 11.259 & 14.623 & 9029 & 47209 & 28.472 \\\\\n         \\rowcolor{lightgray}\n         &  & T5 & sacrebleu & 0.341 & 0.168 & 0.297 & 0.297 & 0.183 & 0.863 & 0.089 & 0.093 & 0.085 & 0.32 & 0.324 & 0.317 & 0.494 & 0.111 & 0.373 & 11.007 & 15.032 & 9455 & 43379 & 27.977 \\\\\n         & & T5 & meteor & 0.322 & 0.157 & 0.286 & 0.286 & 0.173 & 0.888 & 0.147 & 0.163 & 0.133 & 0.358 & 0.367 & 0.350 & 0.625 & 0.136 & 0.482 & 10.189 & 14.910 & 12346 & 54925 & 21.484\\\\\n         & & T5 & parent & 0.283 & 0.132 & 0.264 & 0.264 & 0.133 & 0.894 & 0.163 & 0.174 & 0.153 & 0.36 & 0.357 & 0.364 & 0.824 & 0.223 & 0.691 & 11.493 & 15.127 & 14344 & 55542 & 14.204\\\\\n         & & T5 & meteor + parent & 0.299 & 0.14 & 0.276 & 0.276 & 0.142 & 0.896 & 0.171 & 0.181 & 0.161 & 0.369 & 0.365 & 0.372 & 0.779 & 0.214 & 0.674 & 11.072 & 15.275 & 14939 & 58737 & 15.141\\\\\n         \n         \\cmidrule{2-24}\n          \\rowcolor{lightgray}\n         & Supervised + PPO & T5 & bleu & 0.408 & 0.283 & 0.388 & 0.388 & 0.222 & 0.954 & \n          0.477 & 0.549 & 0.405 & 0.596 & 0.644 & 0.550 & 0.722 & 0.167 & 0.525 & 10.080 & 14.524 & 15203 & 54724 & 17.296\\\\\n          & & T5 & sacrebleu &  0.395 & 0.275 & 0.378 & 0.378 & 0.211 & 0.955 & 0.477 & 0.554 & 0.401 & 0.577 & 0.621 & 0.535 & 0.728 & 0.174 & 0.539 & 10.086 & 14.518 & 14846 & 52327 & 16.063 \\\\\n          & & T5 & meteor & 0.410 & 0.282 & 0.389 & 0.389 & 0.223 & 0.954 &  0.469 & 0.540 & 0.398 & 0.593 & 0.642 & 0.547 & 0.718 & 0.165 & 0.516 & 10.037 & 14.467 & 15182.0 & 54446 & 17.542\\\\\n          & & T5 & parent & 0.401 & 0.277 & 0.382 & 0.382 & 0.215 & 0.953 & 0.470 & 0.543 & 0.394 & 0.598 & 0.647 & 0.550 & 0.732 & 0.174 & 0.545 & 10.209 & 14.660 & 15379.0 & 55421 & 16.826 \\\\\n         & & T5 & meteor + parent & 0.406 & 0.281 & 0.386 & 0.387 & 0.220 & 0.954 & 0.473 & 0.544 & 0.399 & 0.600 & 0.648 & 0.553 & 0.727 & 0.170 & 0.532 & 10.143 & 14.586 & 15330 & 55211 & 17.185\\\\\n         \n         \n         \\cmidrule{2-24}\n         & Supervised + NLPO & T5 & bleu & 0.410 & 0.283 & 0.388 & 0.388 & 0.222 & 0.954 & 0.476 & 0.548 & 0.404 & 0.597 & 0.644 & 0.552 & 0.721 & 0.167 & 0.524 & 10.077 & 14.532 & 15213 & 54948 & 17.408\\\\\n         &  & T5 & sacrebleu & 0.397 & 0.276 & 0.38 & 0.38 & 0.214 & 0.955 & 0.477 & 0.555 & 0.401 & 0.581 & 0.628 & 0.535 & 0.729 & 0.174 & 0.54 & 10.124 & 14.544 & 14940 & 52986 & 16.334 \\\\\n         \\rowcolor{lightgray}\n         & & T5 & meteor & 0.411 & 0.283 & 0.389 & 0.39 & 0.224 & 0.954 & 0.474 & 0.547 & 0.403 & 0.6 & 0.649 & 0.554 & 0.727 & 0.171 & 0.536 & 10.156 & 14.612 & 15341 & 55292 & 17.637 \\\\\n         & & T5 & parent & 0.405 & 0.28 & 0.386 & 0.386 & 0.219 & 0.954 & 0.469 & 0.541 & 0.398 & 0.598 & 0.645 & 0.552 & 0.716 & 0.165 & 0.519 & 10.019 & 14.5 & 15218 & 54793 & 17.095 \\\\\n         & & T5 & meteor + parent & 0.405 & 0.28 & 0.386 & 0.386 & 0.219 & 0.954 & 0.474 & 0.547 & 0.398 & 0.598 & 0.646 & 0.552 & 0.727 & 0.171 & 0.536 & 10.156 & 14.612 & 15341 & 55292 & 17.095 \\\\\n         \n         \n       \\bottomrule\n       \\end{tabular}\n       }\n       \\caption{\\textbf{ToTTo dev evaluation}: Table shows lexical, semantic and factual correctness metric scores of algorithms with different reward functions on dev set. Without supervised pre-training, both PPO and NLPO results in sub-optimal solutions, with NLPO better than PPO. With supervised pre-training, PPO and NLPO achieve better scores across all metrics showing RL fine-tuning is beneficial. Most importantly, RL fine-tuned models produce more factually correct text as seen in higher PARENT scores. Another observation, fine-tuning with a task-specific metric PARENT is better than training just on task-agnostic lexical metrics}\n       \\label{tbl:totto_val_scores}\n    \\end{table*}\n    \\end{landscape}\n\n\n\n\n\\begin{table}[]\n\\centering\n\\footnotesize\n\\begin{tabular}{|l|r|rrr|rrr|}\n\\multicolumn{1}{|c|}{\\multirow{2}{*}{\\textbf{Algorithm}}} & \\multicolumn{1}{c|}{\\multirow{2}{*}{\\textbf{Unique N}}} & \\multicolumn{3}{c|}{\\textbf{Coherence}}                                                                        & \\multicolumn{3}{c|}{\\textbf{Correctness}}                                                                      \\\\\n\\multicolumn{1}{|c|}{}                                    & \\multicolumn{1}{c|}{}                                   & \\multicolumn{1}{c|}{\\textbf{Value}} & \\multicolumn{1}{c|}{\\textbf{Alpha}} & \\multicolumn{1}{c|}{\\textbf{Skew}} & \\multicolumn{1}{c|}{\\textbf{Value}} & \\multicolumn{1}{c|}{\\textbf{Alpha}} & \\multicolumn{1}{c|}{\\textbf{Skew}} \\\\ \\hline\nZero Shot                                                 & 25                                                      & \\multicolumn{1}{r|}{1.63}           & \\multicolumn{1}{r|}{0.718}          & 1.642                              & \\multicolumn{1}{r|}{1.93}           & \\multicolumn{1}{r|}{0.503}          & 1.946                              \\\\\nPPO+Supervised                                            & 24                                                      & \\multicolumn{1}{r|}{4.57}           & \\multicolumn{1}{r|}{0.221}          & 4.579                              & \\multicolumn{1}{r|}{4.48}           & \\multicolumn{1}{r|}{0.098}          & 4.483                              \\\\\nPPO                                                       & 26                                                      & \\multicolumn{1}{r|}{2.75}           & \\multicolumn{1}{r|}{0.427}          & 2.753                              & \\multicolumn{1}{r|}{3.23}           & \\multicolumn{1}{r|}{0.214}          & 3.227                              \\\\\nNLPO                                                      & 28                                                      & \\multicolumn{1}{r|}{2.25}           & \\multicolumn{1}{r|}{0.401}          & 2.247                              & \\multicolumn{1}{r|}{2.61}           & \\multicolumn{1}{r|}{0.419}          & 2.613                              \\\\\nSupervised                                                & 24                                                      & \\multicolumn{1}{r|}{\\textbf{4.59}}  & \\multicolumn{1}{r|}{0.173}          & 4.592                              & \\multicolumn{1}{r|}{4.54}           & \\multicolumn{1}{r|}{0.189}          & 4.537                              \\\\\nNLPO+Supervised                                           & 26                                                      & \\multicolumn{1}{r|}{\\textbf{4.58}}  & \\multicolumn{1}{r|}{0.244}          & 4.601                              & \\multicolumn{1}{r|}{\\textbf{4.57}}  & \\multicolumn{1}{r|}{0.144}          & 4.581                             \n\\end{tabular}\n\\caption{Results of the human subject study showing the number of participants N, average Likert scale value for coherence and sentiment, Krippendorf's alpha showing inter-annotator agreement, and Skew. For each model a total of 50 samples were drawn randomly from the test set and rated by 3 annotators each, resulting in 150 data points per algorithm.}\n\\label{app:totto_agreement}\n\\end{table}\n\n\\begin{table}[]\n\\centering\n\\footnotesize\n\\begin{tabular}{|l|l|rr|rr|}\n\\multicolumn{1}{|c|}{\\multirow{2}{*}{\\textbf{Group 1}}} & \\multicolumn{1}{c|}{\\multirow{2}{*}{\\textbf{Group 2}}} & \\multicolumn{2}{c|}{\\textbf{Coherence}}                                                      & \\multicolumn{2}{c|}{\\textbf{Correctness}}                                                    \\\\\n\\multicolumn{1}{|c|}{}                                  & \\multicolumn{1}{c|}{}                                  & \\multicolumn{1}{c|}{\\textbf{Diff (G2-G1)}} & \\multicolumn{1}{c|}{\\textit{\\textbf{p-values}}} & \\multicolumn{1}{c|}{\\textbf{Diff (G2-G1)}} & \\multicolumn{1}{c|}{\\textit{\\textbf{p-values}}} \\\\ \\hline\nPPO                                                     & NLPO                                                   & \\multicolumn{1}{r|}{\\textbf{-0.507}}       & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{-0.613}}       & \\textbf{0.001}                                  \\\\\nPPO                                                     & NLPO+Supervised                                        & \\multicolumn{1}{r|}{\\textbf{1.827}}        & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{1.340}}        & \\textbf{0.001}                                  \\\\\nPPO                                                     & Supervised                                             & \\multicolumn{1}{r|}{\\textbf{1.833}}        & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{1.313}}        & \\textbf{0.001}                                  \\\\\nPPO                                                     & PPO+Supervised                                         & \\multicolumn{1}{r|}{\\textbf{1.813}}        & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{1.253}}        & \\textbf{0.001}                                  \\\\\nPPO                                                     & Zero Shot                                              & \\multicolumn{1}{r|}{\\textbf{-1.120}}       & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{-1.293}}       & \\textbf{0.001}                                  \\\\\nNLPO                                                    & NLPO+Supervised                                        & \\multicolumn{1}{r|}{\\textbf{2.333}}        & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{1.953}}        & \\textbf{0.001}                                  \\\\\nNLPO                                                    & Supervised                                             & \\multicolumn{1}{r|}{\\textbf{2.340}}        & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{1.927}}        & \\textbf{0.001}                                  \\\\\nNLPO                                                    & PPO+Supervised                                         & \\multicolumn{1}{r|}{\\textbf{2.320}}        & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{1.867}}        & \\textbf{0.001}                                  \\\\\nNLPO                                                    & Zero Shot                                              & \\multicolumn{1}{r|}{\\textbf{-0.613}}       & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{-0.680}}       & \\textbf{0.001}                                  \\\\\nNLPO+Supervised                                         & Supervised                                             & \\multicolumn{1}{r|}{0.007}                 & 0.9                                             & \\multicolumn{1}{r|}{\\textbf{-0.027}}       & \\textbf{0.009}                                  \\\\\nNLPO+Supervised                                         & PPO+Supervised                                         & \\multicolumn{1}{r|}{\\textbf{-0.013}}       & \\textbf{0.009}                                  & \\multicolumn{1}{r|}{\\textbf{-0.087}}       & \\textbf{0.009}                                  \\\\\nNLPO+Supervised                                         & Zero Shot                                              & \\multicolumn{1}{r|}{\\textbf{-2.947}}       & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{-2.633}}       & \\textbf{0.001}                                  \\\\\nSupervised                                              & PPO+Supervised                                         & \\multicolumn{1}{r|}{\\textbf{-0.020}}       & \\textbf{0.009}                                  & \\multicolumn{1}{r|}{\\textbf{-0.060}}       & \\textbf{0.009}                                  \\\\\nSupervised                                              & Zero Shot                                              & \\multicolumn{1}{r|}{\\textbf{-2.953}}       & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{-2.607}}       & \\textbf{0.001}                                  \\\\\nPPO+Supervised                                          & Zero Shot                                              & \\multicolumn{1}{r|}{\\textbf{-2.933}}       & \\textbf{0.001}                                  & \\multicolumn{1}{r|}{\\textbf{-2.547}}       & \\textbf{0.001}                                 \n\\end{tabular}\n\\caption{Results of an post-hoc Tukey HSD Test for difference in means between pairs of algorithms (Group 2 - Group 1) and corresponding $p$-values. Individually statistically significant results are bolded and are used to discuss results in the analysis. Overall $p$-values showing that there is a significant difference in means between the models via a one-way ANOVA test are significant with $p \\ll 0.05$ for both coherence and sentiment.}\n\\label{app:totto:human_tukey}\n\\end{table}\n\n\\clearpage\n\n\\subsubsection{Human Participant Study}\n\nFigure~\\ref{fig:description_interface_totto} shows the ToTTo instructions, example, and interface used for the human evaluation experiments. We made small modifications to the original code release's HTML renderer to make the tables display in our HITs.\nTables~\\ref{app:totto_agreement},~\\ref{app:totto:human_tukey} show averaged results, annotator agreement, and the results of statistical significance tests to determine which models output better generations when rated by humans.\n\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=.45\\linewidth]{figures\/totto_human_study\/totto_human_1.png}\n    \\includegraphics[width=.45\\linewidth]{figures\/totto_human_study\/totto_human2_2.png}\n    \\includegraphics[width=.45\\linewidth]{figures\/totto_human_study\/totto_human2_3.png}\n    \\includegraphics[width=.45\\linewidth]{figures\/totto_human_study\/totto_human3.png}\n    \\caption{Instructions, two examples, and interface for the ToTTo table description task.}\n    \\label{fig:description_interface_totto}\n\\end{figure*}\n\n\\clearpage\n\\subsubsection{Qualitative Analysis}\n\\label{app:totto_qualitative}\nWe show sample generations from each of the algorithms for three randomly picked prompts below.\n\n\\begin{lstlisting}\nSample 1\nPrompt: <table> <cell> August 1, 1992 <col_header> Date <\/col_header> <col_header> North America <\/col_header> <col_header> Australasia <\/col_header> <col_header> Asia <\/col_header> <col_header> North America <\/col_header> <\/cell> <cell> Allentown <col_header> City <\/col_header> <col_header> North America <\/col_header> <col_header> Australasia <\/col_header> <col_header> Asia <\/col_header> <col_header> North America <\/col_header> <\/cell> <cell> Allentown Fairgrounds Grandstand <col_header> Country <\/col_header> <col_header> North America <\/col_header> <col_header> Australasia <\/col_header> <col_header> Asia <\/col_header> <col_header> North America <\/col_header> <\/cell> <\/table><page_title> Under My Spell Tour <\/page_title> <section_title> Tour dates <\/section_title> <table> <cell> August 1, 1992 <col_header> Date <\/col_header> <col_header> North America <\/col_header> <col_header> Australasia <\/col_header> <col_header> Asia <\/col_header> <col_header> North America <\/col_header> <\/cell> <cell> Allentown <col_header> City <\/col_header> <col_header> North America <\/col_header> <col_header> Australasia <\/col_header> <col_header> Asia <\/col_header> <col_header> North America <\/col_header> <\/cell> <cell> Allentown Fairgrounds Grandstand <col_header> Country <\/col_header> <col_header> North America <\/col_header> <col_header> Australasia <\/col_header> <col_header> Asia <\/col_header> <col_header> North America <\/col_header> <\/cell> <\/table>\nZero-Shot: table> cell> Under My Spell Tour \/section_title> Tour dates \/cell> cell> August 1, 1992 col_header> Date \/col_header>\nPPO: Under My Spell Tour, August 1, 1992. August 1, 1992.\nNLPO: Under My Spell Tour dates Under My Spell Tour dates Allentown Fairgrounds Grandstand August 1, 1992. Under My Spell Tour dates Under My Spell Tour dates Allentown Fairgrounds Grandstand August 1, 1992 table>\nSupervised: The Under My Spell Tour began on August 1, 1992 at the Allentown Fairgrounds Grandstand in Allentown.\nSupervised + PPO: On August 1, 1992, Under My Spell Tour was held at the Allentown Fairgrounds Grandstand in Allentown.\nSupervised + NLPO: On August 1, 1992, Under My Spell Tour was held at the Allentown Fairgrounds Grandstand in Allentown.\n\nSample 2\nPrompt: <table> <cell> Philip IV the Fair, the Iron King (Philippe) <col_header> Name <\/col_header> <\/cell> <cell> 5 October 1285 <col_header> King from <\/col_header> <\/cell> <cell> King of France and of Navarre (Roi de France et de Navarre) <col_header> Title <\/col_header> <\/cell> <\/table><page_title> List of French monarchs <\/page_title> <section_title> House of Capet (987\u20131328) <\/section_title> <table> <cell> Philip IV the Fair, the Iron King (Philippe) <col_header> Name <\/col_header> <\/cell> <cell> 5 October 1285 <col_header> King from <\/col_header> <\/cell> <cell> King of France and of Navarre (Roi de France et de Navarre) <col_header> Title <\/col_header> <\/cell> <\/table>\nZero-Shot:\nPPO: Philip IV the Fair, the Iron King (Philippe) 5 October 1285. Title: King of France and of Navarre (Roi de France et de Navarre\nNLPO: King (Philippe) Philip IV the Fair, the Iron King (Philippe) 5 October 1285\nSupervised: Philip IV the Fair, the Iron King (Philippe) was King of France and Navarre from 1285.\nSupervised + PPO: Philip IV the Fair, the Iron King was King of France and Navarre from 1285.\nSupervised + NLPO: Philip IV the Fair, the Iron King (Philippe) was King of France and Navarre (Roi de France et de Navarre) in 1285.\n\nSample 3\nPrompt: <table> <cell> Benjamin A. Smith II (D) <col_header> Vacator <\/col_header> <\/cell> <cell> Successor elected November 6, 1962. <col_header> Reason for change <\/col_header> <\/cell> <cell> Ted Kennedy (D) <col_header> Successor <\/col_header> <\/cell> <cell> November 7, 1962 <col_header> Date of successor's formal installation <\/col_header> <\/cell> <\/table><page_title> 87th United States Congress <\/page_title> <section_title> Senate <\/section_title> <table> <cell> Benjamin A. Smith II (D) <col_header> Vacator <\/col_header> <\/cell> <cell> Successor elected November 6, 1962. <col_header> Reason for change <\/col_header> <\/cell> <cell> Ted Kennedy (D) <col_header> Successor <\/col_header> <\/cell> <cell> November 7, 1962 <col_header> Date of successor's formal installation <\/col_header> <\/cell> <\/table>\nZero-Shot: cell> Ted Kennedy (D) col_header> Vacator \/col_header> \/cell> cell> November 7, 1962 col_header> Reason for change\nPPO: Ted Kennedy (D) succeeding November 6, 1962 as Vacator of the 87th United States Congress.\nNLPO: Ted Kennedy (D) Successor elected November 6, 1962. table> cell Benjamin A. Smith II (D) col_header Reason for change: Date of successor's formal installation November 7, 1962. \nSupervised:  Benjamin A. Smith II (D) served as senate until November 6, 1962 which was later served by Ted Kennedy (D) from November 7, 1962.\nSupervised + PPO: Benjamin A. Smith II (D) served until November 6, 1962 and Ted Kennedy (D) succeeded him from November 7, 1962.\nSupervised + NLPO: Benjamin A. Smith II (D) served until November 6, 1962 and Ted Kennedy (D) succeeded him from November 7, 1962.\n\\end{lstlisting}\n\n\\subsubsection{PPO All-in-One}\n\\kb{Borrowed from: https:\/\/spinningup.openai.com\/en\/latest\/algorithms\/ppo.html }\n\\begin{algorithm}[h]\n    \\caption{PPO}\n    \\begin{algorithmic}\n      \\STATE {\\bfseries Input:} initial policy parameters $\\theta_0$\n      \\STATE {\\bfseries Input:} initial value function parameters $\\phi_0$\n      \\REPEAT\n          \\STATE Collect set of trajectories $\\mathcal{D}_m = \\{  \\tau_i\\}$ by running policy $\\pi_{\\theta_m}$ in the environment.\n          \\STATE Compute rewards-to-go $\\hat{R}_t$\n          \\STATE Compute the advantage estimate $\\hat{A}_t$\n          \\STATE Update the policy by maximizing the PPO-Clip objective:\\\\\n          $$\\theta_{m+1} = \\text{argmax}_{\\theta} \\frac{1}{\\vert \\mathcal{D}_m\\vert T } \\sum_{\\tau \\in \\mathcal{D}} \\sum_{\\tau=0}^{T} \\min \\Big( \\frac{\\pi_{\\theta}(a_t \\vert s_t)}{\\pi_{\\theta_m}(a_t \\vert s_t)} A^{\\pi_{\\theta_m}}, g(\\epsilon, A^{\\pi_{\\theta_m}}(s_t, a_t)) \\Big)$$\\\\\n          \\STATE Fit value function by regression on mean-squared error:\\\\\n          $$\\phi_{m+1}= \\text{argmin}_\\phi \\frac{1}{\\vert \\mathcal{D}_m \\vert T} \\sum_{\\tau \\in \\mathcal{D}_m} \\sum_{t=0}^{T} \\Big( V_\\phi(s_t) - \\hat{R}_t \\Big)^2$$\\\\\n      \\UNTIL{convergence}\n    \\end{algorithmic}\n  \\end{algorithm}\n\n\n\\clearpage\n\\subsubsection{Ablation Study}\nBelow are list of things to studied:\n\\begin{itemize}\n    \\item Effect of decoding during rollouts and how it affects exploration and stability\n        \\subitem(https:\/\/arxiv.org\/pdf\/2202.11818.pdf)\n    \\item  Consistent Dropout for Policy Gradient Reinforcement Learning \n    \\item  NLPO hyperparams (top p, target update frequency)\n    \\item  Reward hacking on PPO (vs NLPO)  - with and without penalty term\n    \n\\end{itemize}\n\n\n\\subsubsection{Hyperparameters}\n\n\n\n\\subsection{Environments: Generation as a Token-level MDP}\n\\label{sec:rl4lm_env}\n\n\n\nEach environment is an NLP task: we are given a supervised dataset $\\dataset=\\{({\\boldsymbol{x}} ^{i}, {\\boldsymbol{y}}^{i})\\}_{i=1}^{N}$ of $N$ examples, where ${\\boldsymbol{x}}  \\in \\sinput$ is an language input and ${\\boldsymbol{y}} \\in \\soutput$ is the target string. %\nGeneration %\ncan be viewed as a Markov Decision Process (MDP) $\\langle \\stateSpace, \\actionSpace,\\rewardfunc, \\transFnDef, \\discount, \\hor\\rangle$ using a finite vocabulary $\\vocab$.\nEach episode in the MDP begins by sampling a datapoint $({\\boldsymbol{x}} , {\\boldsymbol{y}})$ from our dataset and ends when the current time step $t$ exceeds the horizon $\\hor$ or an end of sentence (EOS) token is generated. \nThe input ${\\boldsymbol{x}} =(x_0, \\cdots, x_m)$ is a task-specific prompt that is used as our initial state $\\boldsymbol{s}_0=(x_0, \\cdots, x_m)$, where $\\boldsymbol{s}_0\\in \\stateSpace$ and $\\stateSpace$ is the state space with $x_m \\in \\vocab$.  %\nAn action in the environment $\\action_t \\in \\actionSpace$ consists of a token from our vocabulary $\\vocab$. The transition function  $\\transFnDef: \\stateSpace \\times \\actionSpace \\rightarrow \\stateSpace$ deterministically appends an action $\\action_t$ to the end of the state $\\boldsymbol{s}_{t-1}=(x_0, \\cdots, x_m,\\action_0, \\cdots, \\action_{t-1})$. %\nThis continues until the end of the horizon $t \\leq \\hor$ and we obtain a state $\\boldsymbol{s}_\\hor=(x_0, \\cdots, x_m,\\action_0,\\cdots,\\action_T)$.\nAt the end of an episode a reward $\\rewardfunc: \\stateSpace \\times \\actionSpace \\times \\soutput \\rightarrow \\mathbb{R}^{1}$ that depends on the ($\\boldsymbol{s}_\\hor, {\\boldsymbol{y}}$) (e.g., an automated metric like PARENT \\cite{dhingra2019handling}) is emitted. %\n\\framework{} provides an OpenAI gym~\\citep{brockman2016openai} style \nAPI for an RL environment  \nthat simulates this LM-Based MDP formulation.\nAbstracting the details of the MDP environment structure allows for new tasks to be added quickly with compatibility across all implemented algorithms. %\n\n\n\n\n\n\n\n\\subsection{Reward Functions and Evaluation Metrics}\n\\label{sec:rl4lm_reward}\n\nBecause \\framework{} provides a generic interface for per-token or per-sequence generation rewards, it is possible to quickly apply a wide array of RL algorithms to a similarly diverse range of textual metrics-as-rewards. Specifically, we provide interfaces to 1) \\textbf{n-gram overlap metrics} metrics such as ROUGE \\citep{lin2004rouge}, BLEU \\citep{papineni2002bleu}, SacreBLEU~\\citep{post2018call}, METEOR \\citep{banerjee2005meteor}; (2)  \\textbf{model-based semantic metrics} such as BertScore~\\citep{zhang2019bertscore} and BLEURT~\\citep{sellam-2020-bleurt}\nwhich generally provide higher correlation with human judgment;\n3) \\textbf{task-specific metrics} such as\nCIDER~\\citep{vedantam2015cider}, SPICE~\\citep{anderson2016spice}\n(for captioning\/commonsense generation), PARENT~\\citep{dhingra2019handling} (for data-to-text)\nand SummaCZS~\\citep{laban2022summac} (for factuality of summarization); 4)  \n\\textbf{diversity\/fluency\/naturalness metrics} \n such as perplexity, Mean Segmented Type Token Ratio (MSSTR) \\citep{johnson1944studies}, Shannon entropy over unigrams and bigrams \\citep{shannon1948mathematical}, the ratio of distinct n-grams over the total number of n-grams (Distinct-1, Distinct-2) and count of n-grams that appear only once in the entire generated text \\citep{li2015diversity}.\n\n\n\n\n\\subsection{On-policy Actor-critic Algorithms}\n\\label{sec:rl4lm_onpolicy}\n\\framework{} supports fine-tuning and training LMs from scratch via on-policy actor-critic algorithms on language environments.\nFormally, this class of algorithms allows us to train a parameterized control policy defined as $\\pi_\\theta: \\mathcal{S} \\rightarrow \\mathcal{A}$, a function that attempts to select an action in a given state so as to maximize long term discounted rewards over a trajectory $\\mathbb{E}_{\\pi} [\\sum_{t=0}^{T}\\gamma^t \\mathcal{R}(\\boldsymbol{s}_t, a_t)]$.\nOur benchmark experiments focus on fine-tuning a pre-trained LM denoted as $\\lm$ from which we initial our agent's policy $\\pi_\\theta=\\lm$.\nSimilarly, the value network $V_\\phi$ used to estimate the value function is also initialized from $\\lm$ except for the final layer which is randomly initialized to output a single scalar value.\nAs with other deep RL actor-critic algorithms, we define our value and Q-value functions as $V_t^{\\pi}=\\mathbb{E}_{a_t \\sim \\pi}[\\sum_{\\tau=t}^{\\hor} \\discount R(\\boldsymbol{s}_{\\tau}, a_{\\tau}, {\\boldsymbol{y}})]$, \n$Q_t^{\\pi}(\\boldsymbol{s}_t,a_t)= R(\\boldsymbol{s}_{t}, a_{t},  {\\boldsymbol{y}}) + \\discount \\mathbb{E}_{\\state_{t+1} \\sim \\transFnDef}[V_{t+1}^{\\pi}(\\boldsymbol{s}_{t+1})]$ \nleading to a definition of our advantage function as $A_t^{\\pi}(\\boldsymbol{s},a)=Q_t^{\\pi}(\\boldsymbol{s},a)-V_t^{\\pi}$.\nTo increase training stability, advantage is appoximated using Generalized Advantage Estimation~\\citep{schulman2015high}.\n\nGiven an input-output pair $({\\boldsymbol{x}} , {\\boldsymbol{y}})$ and generation predictions from our agent; because the environment rewards are sequence-level and sparse, following \\cite{wu2021recursively} we regularize the reward function using a token-level KL penalty for all on-policy algorithms, to prevent the model from deviating too far from the initialized LM $\\lm$. Formally, the regularized reward function is:\n\\begin{equation}\n   \\hat{R}(\\boldsymbol{\\state}_t, \\action_t, {\\boldsymbol{y}}) = R(\\boldsymbol{\\state}_t, \\action_t, {\\boldsymbol{y}}) - \\beta \\text{KL}\\left( \\pi_\\theta(\\action_t| \\boldsymbol{\\state}_t) || \\lm(\\action_t| \\boldsymbol{\\state}_t) \\right)\n   \\label{eq:rewardstogo}\n\\end{equation}\nwhere $\\hat{R}$ is the regularized KL reward,  ${\\boldsymbol{y}}$ is gold-truth predictions, $\\text{KL}\\left( \\pi_\\theta(\\action_t| \\boldsymbol{\\state}_t) || \\lm(\\action_t| \\boldsymbol{\\state}_t) \\right) = (\\log \\lm(\\action_t| \\boldsymbol{\\state}_t) - \\log \\pi(\\action_t| \\boldsymbol{\\state}_t))$ and the KL coefficient $\\beta$ is dynamically adapted~\\citep{ziegler2019fine}.\nFurther details on actor-critic methods can be found in Appendix~\\ref{app:onpolicy}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Potential Pitfalls of Reinforcement Learning  \\kb{Something more catchy?}}\nThough reformulating NLP tasks seems enticing, there has been some problems and concerns mention in the literature \\kb{A better catchy phrase}. \n\\textcolor{blue}{(1)} In, past issues, regarding optimizing a reward for language (especially for a metric) does not guarantee an increase in the quality of the output \\citep{paulus2017deep} (i.e. reward hacking \\citep{}). \n\\textcolor{blue}{(5)} In addition, issues around solving a sparse reward problem for text-generation \\citep{Choshen2020On}. (6)\n\\textcolor{blue}{(2)} \\kb{[Raj check this $\\rightarrow$] Also, concerns around RL is only likely to improve over pre-trained models when the pre-trained is already performing well \\citep{Choshen2020On}; including using the KL penalty and staying extremely close to the pre-trained model\\citep{korbak2022controlling} }.\n\\textcolor{blue}{(3)} RL methods have high variance and instability in the objective function \\citep{wu2018study,li-etal-2017-adversarial, wu2016google}. \n\\textcolor{blue}{(4)} The action space in RL for text-generation is high dimensional and combinatorial (size of the vocabulary) \\citep{Choshen2020On}. \n\\textcolor{blue}{(5)} \\kb{[Raj check this $\\rightarrow$] RL methods suffer from loss of capabilities of the original model} \\citep{korbak2022controlling}. \n\\textcolor{blue}{(6)} \\kb{[Raj check this $\\rightarrow$] Sum of actions for rewards not equal to the overall metrics, forcing a discount factor $\\discount \\approx 1$, in most RL for text-generation experiments \\citep{}} \n\\textcolor{blue}{(7)} \\kb{[Raj check this $\\rightarrow$] Extremely high initial state distribution $\\initialstate$ due to prompts needed for encoder-decoders to reduce instability with PPO. } \\citep{}.\nAlthough some of these issues has been addressed only in the context of Neural Machine Translation (NMT) \\citet{samuel2021revisiting}, this evaluation focuses on a broader set of text-generation task.\n\n\n\n\n\n\\subsection{Guidelines on When to Use RL for Text-Generation}\nWhat is RL good for in the context of language? Metric optimization (be careful), sequential tasks, intermediate rewards - incorporating feedback (interactivity), goal driven generation, already have a good init policy via pretraining\n\n\npart of the fix is the KL penalty, but let's think about MDP sample complexity, that depends on gamma too\n\n\n\n\n\\subsection{Tasks and Metrics}\n\n\\begin{table*}[t!]\n  \\small\n  \\begin{center}\n  \\begin{tabular}{@{}m{8.2em}m{11.2em}m{5em}rm{5em}rm{3em}rm{3em}@{}}\n  \\toprule\n       \\textbf{Dataset}\n     & \\textbf{Task}\n     & \\centering\\arraybackslash \\textbf{Language(s)}\n     & \\textbf{Size}\n     & \\centering\\arraybackslash\\textbf{Input Type} \n     & \\centering\\arraybackslash\\textbf{\\textcolor{red}{Done}}\\\\\n  \\toprule\n      \\makecell[bl]{\\small{CommonGEN}\\\\ \\small{\\citep{lin2019commongen}}}\n      & \\small{Generative Commonsense}\n      & \\centering\\arraybackslash\\small{en} \n      & \\centering\\arraybackslash\\small{67k} \n      & \\centering\\arraybackslash\\small{Concept Set} \n      & .\n      & Y\n      \\\\ \n  \\midrule\n      \\makecell[bl]{\\small{CNN Daily Mail}}\n      & \\small{Summarization}\n      & \\centering\\arraybackslash\\small{en} \n      &  \\centering\\arraybackslash\\small{287k}\n      &  \\centering\\arraybackslash\\small{News Article} \n      & .\n      & Y\n      \\\\\n  \\midrule\n      \\makecell[bl]{\\small{IMDB}\\\\ } \n      & \\small{Text Continuation}\n      &  \\centering\\arraybackslash\\small{en}\n      &  \\centering\\arraybackslash\\small{25k}\n      & \\centering\\arraybackslash\\small{Partial Movie Review} \n      & . %\n      & Y\n      \\\\\n  \\midrule\n      \\makecell[bl]{\\small{ToTTo}\\\\ \\small{\\citep{parikh2020totto}}} \n      & \\small{Data to Text}\n      & \\centering\\arraybackslash\\small{en} \n      & \\centering\\arraybackslash\\small{136k}  \n      & \\centering\\arraybackslash\\small{Highlighted Table}\n      & .\n      & Y\n      \\\\\n  \\midrule\n      \\makecell[bl]{\\small{WMT-19}} \n      & \\small{Machine Translation}\n      & \\centering\\arraybackslash\\small{en, ru, de, fr} \n      & \\centering\\arraybackslash\\small{-}  \n      & \\centering\\arraybackslash\\small{Text}\n      & .\n      & Y\n      \\\\\n  \\midrule\n      \\makecell[bl]{\\small{NarrativeQA}}\n      & \\small{Question Answering}\n      & \\centering\\arraybackslash\\small{en} \n      & \\centering\\arraybackslash\\small{-}  \n      & \\centering\\arraybackslash\\small{Question Context}\n      & .\n      & Y\n      \\\\\n  \\bottomrule\n  \\end{tabular}\n  \\end{center}\n  \\caption{}\n  \\label{tab:overview}\n\\end{table*}\n\n\n\n\n\\subsection{On-Policy Algorithms for Generation}\n\nhow do we actually get these to work, what's the key idea - restrict optimization to trust region of \"natural language\"\n\nwell, that's slow - so we use the KL penalty which \n\ngeneric actor critic for generation psuedocode\n\nthe calculate returns line varies based on PPO vs A2C. TRPO straight up does second order\n\n\\subsection{Results on GRUE: Which Algorithm Should be Used to Learn Preferences?}\n\\label{sec:grue_warm_start}\n\nFigures~\\ref{fig:auto_task},~\\ref{fig:auto_natural} present the results on GRUE{}, split into task metrics and naturalness metrics, and Tables~\\ref{tab:key_results},~\\ref{tab:imdb_ablations} highlight key results via ablation studies. Full results are available in Appendix~\\ref{app:experiments}. %\nFor text continuation and summarization, with non-trivial zero-shot performance, RL tends to perform better than supervised training, but for tasks like Commongen and ToTTo, which have very low zero-shot performance, supervised training performs best. \nAs expected, both RL and supervised learning outperform zero-shot approaches.\n\nHowever, \\textbf{using RL+Supervised learning in conjunction works best;}  \nNLPO+supervised and PPO+supervised usually always outperforms NLPO\/PPO (or supervised in isolation) across both task metrics and naturalness metrics. %\nSupervised warm-starting is particularly effective for Commongen and ToTTo, which our results suggest are more prone to reward hacking. \nWe note that Supervised+NLPO using a T5-base (220m parameter) LM currently outperforms all the models on the ToTTo leaderboard, many of which have $\\geq$ 3b parameter supervised models---suggesting that RL is parameter efficient as well.\nIn these cases, it is critical that the initial policy already contain (some) signal for the task: this is because the initial policy is used both as a KL constraint, and as additional masking constraint in NLPO. \nIf the mask contains no initial priors about task specific language, it will be eliminating the wrong actions. Simply put, the better one's initial policy, the more additional performance RL training will add to it.\n\n\n\n\n\\textbf{Human agreement with automated metrics.}\nAs human judgments can be noisy, we run additional statistical analysis such as measuring\ninter-annotator agreement, via Krippendorf's alpha score, and using a one-way ANOVA followed by a post-hoc Tukey HSD test to measure if differences in means of average scores between pairs of models are significant.\nWe find that trends in our human evaluations generally match those seen in the automated metrics for both task and naturalness metrics (see Figures~\\ref{fig:human_task},~\\ref{fig:human_natural} which summarize Appendix Tables~\\ref{app:imdb:human_tukey},\\ref{app:commongen:human_tukey},\\ref{app:summariazation:human_tukey},\\ref{app:totto:human_tukey}---Supervised+NLPO $>$ Supervised $\\geq$ Supervised+PPO $>$ NLPO $\\geq$ PPO $>$ Zero-shot---with the exception of Supervised outperforming Supervised+PPO on 2 out of 4 tasks tasks when automated metrics would indicate that Supervised+PPO outperforms Supervised on all of the tasks.\nWe draw two conclusions from this: (1) if the generated text is above a certain threshold of naturalness, the automated metrics \\textit{usually} correlate with human judgements; (2) usually but not always as seen in the relative performance of Supervised and Supervised+PPO, potentially indicating reward hacking behaviors undetected by automated metrics but caught by human preference feedback.\n\n\n\n\n\\begin{table}\n\\parbox{.6\\linewidth}{\n\\centering\n\\resizebox{0.6\\columnwidth}{!}{\n\\begin{tabular}{lcccccc}\n\\toprule\n{\\cellcolor{gray!25}\\textbf{Questions}}\n& \\multicolumn{6}{c}{\\cellcolor{gray!25}\\textbf{Tasks}} \\\\ \n& IMDB & CommonGen & CNN\/DM & ToTTO & IWSLT-17 & NarQA \\\\\n\\midrule\nNeeds Warm Start & \\cellcolor{green!25}\\ding{55} & \\cellcolor{red!25}\\ding{51} & \\cellcolor{red!25}\\cellcolor{green!25}\\ding{55} & \\cellcolor{red!25}\\ding{51} & \\cellcolor{green!25}\\ding{55} & \\cellcolor{red!25}\\ding{51}\\\\\nEasily reward hackable? & \\cellcolor{red!25}\\ding{51} & \\cellcolor{red!25}\\ding{51} & \\cellcolor{green!25}\\ding{55} & \\cellcolor{green!25}\\ding{55} & \\cellcolor{green!25}\\ding{55} & \\cellcolor{green!25}\\ding{55}\\\\\n\nRL $>$ Sup (auto)? & \\cellcolor{green!25}\\ding{51} & \\cellcolor{red!25}\\ding{55} & \\cellcolor{red!25}\\ding{55} & \\cellcolor{red!25}\\ding{55} & \\cellcolor{red!25}\\ding{55} & \\cellcolor{red!25}\\ding{55} \\\\\nRL $>$ Sup (human)? & \\cellcolor{green!25}\\ding{51} & \\cellcolor{red!25}\\ding{55} & \\cellcolor{red!25}\\ding{55} & \\cellcolor{red!25}\\ding{55} & - & - \\\\\nSup+RL $>$ Sup (auto)? & \\cellcolor{green!25}\\ding{51} & \\cellcolor{green!25}\\ding{51} & \\cellcolor{green!25}\\ding{51} & \\cellcolor{green!25}\\ding{51} & \\cellcolor{green!25}\\ding{51} & \\cellcolor{green!25}\\ding{51}\\\\\nSup+RL $>$ Sup (human)? & \\cellcolor{green!25}\\ding{51} & \\cellcolor{red!25}\\ding{55} & \\cellcolor{green!25}\\ding{51} & \\cellcolor{green!25}\\ding{51} & - & - \\\\\nSup+NLPO $>$ Sup+PPO (auto)? & \\cellcolor{green!25}\\ding{51} & \\cellcolor{green!25}\\ding{51} & \\cellcolor{green!25}\\ding{51} & \\cellcolor{green!25}\\ding{51} & \\cellcolor{green!25}\\ding{51} & \\cellcolor{green!25}\\ding{51}\\\\\nSup+NLPO $>$ Sup+PPO (human)? & \\cellcolor{green!25}\\ding{51} & \\cellcolor{green!25}\\ding{51} & \\cellcolor{green!25}\\ding{51} & \\cellcolor{green!25}\\ding{51} & - & -\\\\\n\n\n\\bottomrule\n\\end{tabular}\n}\n\\caption{\\textbf{Key questions answered using GRUE + RL4LMs:} This table summarizes the results found in the ablations and Fig.~\\ref{fig:allmetrics} and provides an overview of the questions we ask in Section~\\ref{sec:benchmark_analysis}: which tasks require warm starts or are easily reward hackable; when to use RL over Supervised, when to use both; and when to use NLPO over PPO. All conclusions drawn are the result of statistical analysis as discussed in the experimental setup.\n}\n\\label{tab:key_results}\n}\n\\parbox{.4\\linewidth}{\n\\centering\n\\resizebox{0.4\\columnwidth}{!}{\n\\begin{tabular}{lrr}\n        \\textbf{Ablation}                    & \\textbf{Sentiment}         & \\textbf{Perplexity}         \\\\ \\hline\n    Zero Shot               & 0.489           & 32.171                    \\\\\n    Supervised              &  0.539  & 35.472                   \\\\\n    PPO                     & 0.605  & 33.497                  \\\\\n    NLPO                    & 0.637            & 32.667                     \\\\ \\hline\n    \\rowcolor{lightgray}\n    \\multicolumn{3}{l}{Warm Starting (Sec.~\\ref{sec:grue_warm_start})}                                \\\\ \\hline\n    PPO+Supervised          & 0.617  &34.078                    \\\\\n    NLPO+Supervised         & 0.645  &33.191                   \\\\ \\hline\n    \\rowcolor{lightgray}\n    \\multicolumn{3}{l}{Data Budget (Reward trained on 10\\% of data, Sec.~\\ref{sec:grue_data_budget})} \\\\ \\hline\n    PPO                     &  0.598   &35.929                   \\\\\n    NLPO                    &  0.599  &33.536                    \\\\ \\hline\n    \\rowcolor{lightgray}\n    \\multicolumn{3}{l}{Removing NLPO Top-$p$ Constraints (Sec.~\\ref{sec:grue_reward})}\\\\\n    \\multicolumn{3}{l}{($p=1$ is equivalent to PPO, $p=0.9$ is NLPO)}                      \\\\ \\hline\n    NLPO $p=0.1$               &  0.579  &32.451                    \\\\\n    NLPO $p=0.5$              &  0.588  &32.447                   \\\\ \\hline\n    \\rowcolor{lightgray}\n    \\multicolumn{3}{l}{Removing KL Constraints (Sec.~\\ref{sec:grue_reward})}                      \\\\ \\hline\n    PPO-no-KL               &  0.859  &37.553                     \\\\\n    NLPO-no-KL              &  0.853  &36.812                   \\\\ \\hline\n    \\rowcolor{lightgray}\n    \\multicolumn{3}{l}{Discount Ablations ($\\gamma=1$) (Sec.~\\ref{sec:grue_practical})}            \\\\ \\hline\n    PPO                     &  0.651  &41.035                     \\\\\n    NLPO                    &  0.624  &43.72                    \n    \\end{tabular}\n    }\n  \\captionof{table}{IMDB Ablation Results.}\n  \\label{tab:imdb_ablations}\n}\n\\end{table}\n\n\n\n\n\n\n\n\n\n\\subsection{Reward Selection and Hacking}\n\\label{sec:grue_reward}\nWhile the GRUE{} benchmark's metric for each task is an average over several measures, the RL models we trained optimized only a single metric independently.\nThus, we can empirically investigate which metric for which GRUE{} produces the best results.\nWe observe that many possible single metric rewards provide task performance gains over supervised methods (results shown in Fig.~\\ref{fig:auto_task_orig},~\\ref{fig:human_task} are averaged across these reward functions) with the condition that the text is also coherent and natural.\n\n\n\\textbf{Which constraints best prevent reward hacking?}\nThe reward function in Equation~\\ref{eq:rewardstogo} balances a task-specific reward with a KL constraint --- models are penalized from straying too far from a base LM in their pursuit of high reward (Table~\\ref{tab:imdb_ablations} and Appendix Table~\\ref{tbl:text_cont_KL_ablations}) clearly show that if KL constraints are removed entirely, models reward hack). %\nBut which model works best as a base regularizing LM?\nWhen the initial policy (i.e., the raw, pretrained model) has low performance on the task, the KL penalty pushes the policy towards nonsense, e.g. on Commongen and ToTTo the trained policy learns to simply repeat portions of the input (as seen in Tables~\\ref{app:tbl_common_gen_qualitative},~\\ref{app:totto_qualitative}).\nThis behavior is mitigated if the base regularizing LM is the supervised model---the reward encourages the policy to balance the task-specific reward and a more reasonable regularization term. \nDeriving KL penalties from warm-started initial policies is critical for performance on such tasks.\n\n\\textbf{PPO vs. NLPO.}\nFigure~\\ref{fig:allmetrics} shows that NLPO generally outperforms PPO and supervised, especially when applied after supervised training.\nWe hypothesize that the primary reason for NLPO's improved performance and stability is because the masking policy provides an additional constraint for the current policy.\nThis constraint is not based on the initial untuned policy like the KL penalty but of the policy from $\\mu$ iterations ago and likely contains more task-relevant information learned during RL training.\nTable~\\ref{tab:imdb_ablations} (and Appendix Table~\\ref{tbl:text_cont_nlpo_hyperparam}) shows how performance increases up to a point and then decreases as $p$ in top-$p$ sampling is increased for the masking policy, relaxing the constraint by eliminating less tokens at each step, implying that there is a balance to be found in how much the model should be constrained during RL training.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Data Budget: Improve your Reward or Gather More Demonstration?}\n\\label{sec:grue_data_budget}\nGiven a fixed data collection budget, is it more efficient to gather feedback to improve a learned reward function or to gather more expert demonstrations?\nWe use the IMDB text continuation task as a case study.\nIn the IMDB task, a model is given a partial movie review as a prompt, and is asked to continue it as positively as possible (even if the prompt was negative).\nThe original dataset consists of movie reviews and sentiment labels of positive, negative, or neutral.\nA DistilBERT~\\citep{sanh2019distilbert} classifier is trained on these labels and used to provide sentiment scores on how positive a given piece of text is, which serves as the task reward. The (simulated) trade-off is between: 1) gathering more sentiment labels (improving the reward); or 2) gathering more positive sentiment reviews (improving supervised training).\n\nWe train a classifier on varying amounts of training data and evaluate on the held out test dataset---finding as expected that more training data improves test accuracy and so results in a higher quality reward.\nWe then use each of these rewards of varying quality during RL training, and evaluate using the same metric as GRUE{} (i.e., a classifier trained with the entire training set).\nAs seen in Table~\\ref{tab:imdb_ablations}, we find that improving the reward quality %\nimproves LM performance as well.\nFurther, we trained a supervised model with at least as many samples used to train each of these reward classifiers.\nWe find that \\textbf{a learned reward function enables greater performance when used as a signal for an RL method than a supervised method trained with 5 times more data.} \nThis implies that improving reward models can be more data efficient than collection expert demonstrations for a task---and that's not accounting for the fact that assigning sentiment labels is likely a simpler task than writing full demonstrations. %\nFurther details on this ablation are found in Appendix Table~\\ref{tbl:text_cont_extra_data}.\n\n\n\n\n\\subsection{Practical Considerations: Which Implementation Details Matter Most?}\n\\label{sec:grue_practical}\n\n\n\\textbf{Generation as a token-level MDP, not a bandit environment.}\nMost recent works that tune LMs using RL do so by calculating a reward for all the tokens in the sentence %\n~\\citep{wu2021recursively,ouyang2022training,lu2022quark}.\nThis setting is equivalent to a bandit feedback environment where the action space is the space of all possible generations for the task~\\citep{sutton2018reinforcement}.\nThis type of environment can be simulated within our RL formulation by setting the discount factor $\\gamma=1$.\nTable~\\ref{tab:imdb_ablations} (and Appendix Table~\\ref{tbl:text_cont_gamma}) shows that this causes instability in training with respect to naturalness in both PPO and NLPO for IMDB.\nOur standard setting is $\\gamma=0.95$ when calculating discounted rewards-to-go in the token-level MDP formulation, which reduces the magnitude of the reward that is applied to tokens selected at the beginning. \nThe sentiment scores are approximately the same between both settings but the naturalness of language in the bandit setting is significantly less ---indicating that discounting rewards via $\\gamma<1$ via a token-level MDP formulation is at least sometimes more effective for language generation.\n\n\\textbf{Dropout and Sampling.}\nWe found two other implementation details to be critical for stability of RL training.\nThe first is dropout, which in its standard form was found to cause instability in policy gradient methods in continuous control settings by~\\citep{hausknecht2022consistent}.\nWe find a similar effect when using dropout when RL training LMs as well, with training loss often diverging for dropout $> 0$ in training.\nThe second important detail, particularly affecting the machine translation task, is sampling methods. %\nWe find that using the same sampling methods during exploration and inference is critical to translating training performance to test performance--else the model exhibits high train rewards but low test metrics.\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:Intro}\nIn recent years, many combinatorial Hopf algebras, whose bases are indexed\nby combinatorial objects, have been intensively studied. For example,\nthe Malvenuto-Reutenauer Hopf algebra~${\\bf FQSym}$ of Free quasi-symmetric\nfunctions~\\cite{MR95,DHT02} has bases indexed by permutations. This\nHopf algebra admits several Hopf subalgebras: The Hopf algebra of Free\nsymmetric functions~${\\bf FSym}$~\\cite{PR95,DHT02}, whose bases are indexed\nby standard Young tableaux, the Hopf algebra~${\\bf Bell}$~\\cite{R07} whose bases\nare indexed by set partitions, the Loday-Ronco Hopf algebra~${\\bf PBT}$~\\cite{LR98,HNT05}\nwhose bases are indexed by planar binary trees, and the Hopf algebra~${\\bf Sym}$\nof non-commutative symmetric functions~\\cite{GKDLLRT94} whose bases are\nindexed by integer compositions. A unifying approach to construct all these\nstructures relies on a definition of a congruence on words leading to the\ndefinition of monoids on combinatorial objects. Indeed,~${\\bf FSym}$ is directly\nobtained from the plactic monoid~\\cite{LS81,DHT02,Lot02},~${\\bf Bell}$ from the\nBell monoid~\\cite{R07},~${\\bf PBT}$ from the sylvester monoid~\\cite{HNT02,HNT05},\nand~${\\bf Sym}$ from the hypoplactic monoid~\\cite{KT97,N98}. The richness of\nthese constructions relies on the fact that, in addition to constructing\nHopf algebras, the definition of such monoids often brings partial orders,\ncombinatorial algorithms and Robinson-Schensted-like algorithms, of independent\ninterest.\n\\smallskip\n\nThe Baxter combinatorial family admits various representations. The most\nfamous of these are Baxter permutations~\\cite{Bax64}, which are permutations\nthat avoid certain patterns, and pairs of twin binary trees~\\cite{DG94}.\nThis family also contains more exotic objects like quadrangulations~\\cite{ABP04}\nand plane bipolar orientations~\\cite{BBF10}. In this paper, we propose to\nenrich the above collection of Hopf algebras by providing a plactic-like\nmonoid, namely the Baxter monoid, leading to the construction of a Hopf\nalgebra whose bases are indexed by objects belonging to this combinatorial\nfamily.\n\\smallskip\n\nIn order to show examples of relations between lattice congruences~\\cite{CS98}\nand Hopf algebras, Reading presented in~\\cite{Rea05} a lattice congruence of\nthe permutohedron whose equivalence classes are indexed by twisted Baxter\npermutations. These permutations were defined by a pattern avoidance property.\nThis congruence is very natural: The meet of two lattice congruences of\nthe permutohedron related to the construction of~${\\bf PBT}$ is one starting\npoint to build~${\\bf Sym}$; A natural question is to understand what happens\nwhen the join, instead of the meet, of these two lattice congruences is\nconsidered. Reading proved that his lattice congruence is precisely this\nlast one, and that the minimal elements of its equivalence classes are\ntwisted Baxter permutations. Besides, thanks to his theory, he gets for\nfree a Hopf algebra whose bases are indexed by twisted Baxter permutations.\nActually, twisted Baxter permutations are equinumerous with Baxter permutations.\nIndeed, Law and Reading pointed out in~\\cite{LR10} that the first proof\noccurred in unpublished notes of West. Hence, the Hopf algebra of Reading\ndefined in~\\cite{Rea05} can already be seen as a Hopf algebra on Baxter\npermutations, and our construction, considered as a different construction\nof the same Hopf algebra. Moreover, very recently, Law and Reading~\\cite{LR10}\ndetailed their construction of this Hopf algebra and studied some of its\nalgebraic properties.\n\\smallskip\n\nWe started independently the study of Baxter objects in a different way:\nWe looked for a quotient of the free monoid analogous to the plactic and\nthe sylvester monoid. Surprisingly, the equivalence classes of permutations\nunder our monoid congruence are the same as the equivalence classes of the\nlattice congruence of Law and Reading, and hence have the same by-products, as\n\\emph{e.g.}, the Hopf algebra structure and the fact that each class contains\nboth one twisted and one non-twisted Baxter permutation. However, even if\nboth points of view lead to the same general theory, their paths are different\nand provide different ways of understanding the construction, one centered\non lattice theory, the other centered on combinatorics on words. Moreover,\na large part of the results of each paper do not appear in the other as,\nin our case, the Robinson-Schensted-like correspondence and its insertion\nalgorithm, the polynomial realization, the bidendriform bialgebra structure,\nthe freeness, cofreeness, self-duality, primitive elements, and multiplicative\nbases of the Hopf algebra, and a few other combinatorial properties.\n\\smallskip\n\nWe begin by recalling in Section~\\ref{sec:Prelim} the preliminary notions\nabout words, permutations, and pairs of twin binary trees used thereafter.\nIn Section~\\ref{sec:MonoideBaxter}, we define the Baxter congruence. This\ncongruence allows to define a quotient of the free monoid, the Baxter monoid,\nwhich has a number of properties required for the Hopf algebraic construction\nwhich follows. We show that the Baxter monoid is intimately linked to the\nsylvester monoid and that the equivalence classes of the permutations under\nthe Baxter congruence form intervals of the permutohedron. Next, in\nSection~\\ref{sec:RobinsonSchensted}, we develop a Robinson-Schensted-like\ninsertion algorithm that allows to decide if two words are equivalent according\nto the Baxter congruence. Given a word, this algorithm computes iteratively\na pair of twin binary trees inserting one by one the letters of~$u$. We\ngive as well some algorithms to read the minimal, the maximal and the Baxter\npermutation of a Baxter equivalence class encoded by a pair of twin binary\ntrees. We also show that each equivalence class of permutations under the\nBaxter congruence contains exactly one Baxter permutation.\nSection~\\ref{sec:TreillisBaxter} is devoted to the study of some properties\nof the equivalence classes of permutations under the Baxter congruence.\nThis leads to the definition of a lattice structure on pairs of twin binary\ntrees, very similar to the Tamari lattice~\\cite{Tam62,Knu06} since covering\nrelations can be expressed by binary tree rotations. We introduce in this\nsection \\emph{twin Tamari diagrams} that are objects in bijection with pairs\nof twin binary trees and offer a simple way to test comparisons in this\nlattice. Finally, in\nSection~\\ref{sec:AlgebreHopfBaxter}, we start by recalling some basic facts\nabout the Hopf algebra of Free quasi-symmetric functions~${\\bf FQSym}$, and\nthen give our construction of the Hopf algebra~${\\bf Baxter}$ and study it.\nUsing the polynomial realization of~${\\bf FQSym}$, we deduce a polynomial\nrealization of~${\\bf Baxter}$. Using the order structure on pairs of twin\nbinary trees defined in the above section, we describe its product as an\ninterval of this order. Moreover, we prove that this Hopf algebra is free\nas an algebra by constructing two multiplicative bases, and introduce two\noperators on pairs of twin binary trees, analogous to the operators \\emph{over}\nand \\emph{under} of Loday-Ronco on binary trees~\\cite{LR02}. Using the results\nof Foissy on bidendriform bialgebras~\\cite{Foi07}, we show that this Hopf algebra\nis also self-dual and that the Lie algebra of its primitive elements is free.\nWe conclude by explaining some morphism with other known Hopf subalgebras\nof~${\\bf FQSym}$.\n\\medskip\n\nThis paper is an extended version of~\\cite{Gir11}. It contains all proofs\nand Sections~\\ref{sec:RobinsonSchensted} and~\\ref{sec:AlgebreHopfBaxter}\nhave new results.\n\n\\subsubsection*{Acknowledgments}\nThe author would like to thank Florent Hivert and Jean-Christophe Novelli\nfor their advice and help during all stages of the preparation of this paper.\nThe computations of this work have been done with the open-source mathematical\nsoftware Sage~\\cite{SAGE}.\n\n\\section{Preliminaries} \\label{sec:Prelim}\n\n\\subsection{Words, definitions and notations}\nIn the sequel, $A := \\{a_1 < a_2 < \\cdots\\}$ is a totally ordered infinite\nalphabet and~$A^*$ is the free monoid generated by~$A$. Let~$u \\in A^*$.\nWe shall denote by~$|u|$ the length of~$u$ and by~$\\epsilon$ the word of\nlength~$0$. The largest (resp. smallest) letter of~$u$ is denoted by~$\\max(u)$\n(resp.~$\\min(u)$). The \\emph{evaluation}~$\\operatorname{ev}(u)$ of the word~$u$ is the\nnon-negative integer vector such that its $i$-th entry is the number of\noccurrences of the letter~$a_i$ in~$u$. It is convenient to denote by\n$\\operatorname{Alph}(u) := \\left\\{u_i : 1 \\leq i \\leq |u|\\right\\}$ the smallest alphabet\non which~$u$ is defined. We say that~$(i, j)$ is an \\emph{inversion} of~$u$\nif~$i < j$ and~$u_i > u_j$. Additionally,~$i$ is \\emph{descent} of~$u$\nif~$(i, i + 1)$ is an inversion of~$u$.\n\\medskip\n\nLet us now recall some classical operations on words. We shall denote by\n$u^\\sim := u_{|u|} \\dots u_1$ the \\emph{mirror image} of~$u$ and by~$u_{|S}$\nthe \\emph{restriction} of~$u$ on the alphabet~$S \\subseteq A$, that is the\nlongest subword of~$u$ such that $\\operatorname{Alph}(u) \\subseteq S$. Let~$v \\in A^*$.\nThe \\emph{shuffle product}~$\\shuffle$ is recursively defined on the linear\nspan of words~$\\mathbb{Z} \\langle A \\rangle$ by\n\\begin{equation}\n    u \\shuffle v :=\n    \\begin{cases}\n        u & \\mbox{if $v = \\epsilon$,} \\\\\n        v & \\mbox{if $u = \\epsilon$,} \\\\\n        {\\tt a} (u' \\shuffle {\\tt b} v') + {\\tt b} ({\\tt a} u' \\shuffle v')\n        & \\mbox{otherwise, where $u = {\\tt a} u'$, $v = {\\tt b} v'$, and ${\\tt a}, {\\tt b} \\in A$.}\n    \\end{cases}\n\\end{equation}\nFor example,\n\\begin{align} \\begin{split}\n    {\\bf a_1 a_2} \\shuffle a_2 a_1\n        & = {\\bf a_1 a_2} a_2 a_1 + {\\bf a_1} a_2 {\\bf a_2} a_1 +\n            {\\bf a_1} a_2 a_1 {\\bf a_2} + a_2 {\\bf a_1 a_2} a_1 +\n            a_2 {\\bf a_1} a_1 {\\bf a_2} + a_2 a_1 {\\bf a_1 a_2}, \\\\\n        & = a_1 a_2 a_1 a_2 + 2\\, a_1 a_2 a_2 a_1 + 2\\, a_2 a_1 a_1 a_2 +\n            a_2 a_1 a_2 a_1.\n\\end{split} \\end{align}\nLet $A^\\# := \\{a_1^\\# > a_2^\\# > \\cdots\\}$ be the alphabet~$A$ on which\nthe order relation has been reversed. The \\emph{Sch\u00fctzenberger transformation}~$\\#$\nis defined on words by\n\\begin{equation}\n    u^\\# = \\left(u_1 u_2 \\dots u_{|u|} \\right)^\\# :=\n        u_{|u|}^\\# \\dots u_2^\\# u_1^\\#.\n\\end{equation}\nFor example, $(a_5 a_3 a_1 a_1 a_5 a_2)^\\# = a_2^\\# a_5^\\# a_1^\\# a_1^\\# a_3^\\# a_5^\\#$.\nNote that by setting ${A^\\#}^\\# := A$, the transformation~$\\#$ becomes an\ninvolution on words.\n\n\\subsection{Permutations, definitions and notations}\nDenote by~$\\mathfrak{S}_n$ the set of permutations of size~$n$ and by~$\\mathfrak{S}$\nthe set of all permutations. One can see a permutation of size~$n$ as a\nword without repetition of length~$n$ on the first letters of~$A$. We shall\ncall~$i$ a \\emph{recoil} of~$\\sigma \\in \\mathfrak{S}_n$ if~$(i, i + 1)$ is\nan inversion of~$\\sigma^{-1}$. By convention,~$n$ also is a recoil of~$\\sigma$.\n\\medskip\n\nThe \\emph{(right) permutohedron order} is the partial order~${\\:\\leq_{\\operatorname{P}}\\:}$\ndefined on~$\\mathfrak{S}_n$ where~$\\sigma$ is covered by~$\\nu$ if\n$\\sigma = u \\, {\\tt a} {\\tt b} \\, v$ and $\\nu = u \\, {\\tt b} {\\tt a} \\, v$ where\n${\\tt a} < {\\tt b} \\in A$, and~$u$ and~$v$ are words. Recall that one has\n$\\sigma {\\:\\leq_{\\operatorname{P}}\\:} \\nu$ if and only if any inversion of~$\\sigma^{-1}$ also\nis an inversion of~$\\nu^{-1}$.\n\\medskip\n\nLet $\\sigma, \\nu \\in \\mathfrak{S}$. The permutation $\\sigma {\\,\\diagup\\,} \\nu$ is obtained\nby concatenating~$\\sigma$ and the letters of~$\\nu$ incremented by~$|\\sigma|$;\nIn the same way, the permutation $\\sigma {\\,\\diagdown\\,} \\nu$ is obtained by\nconcatenating the letters of~$\\nu$ incremented by~$|\\sigma|$ and~$\\sigma$.\nFor example,\n\\begin{equation}\n    {\\bf 312} {\\,\\diagup\\,} 2314 = {\\bf 312} 5647\n    \\quad \\mbox{and} \\quad\n    {\\bf 312} {\\,\\diagdown\\,} 2314 = 5647 {\\bf 312}.\n\\end{equation}\nA permutation~$\\sigma$ is \\emph{connected} if $\\sigma = \\nu {\\,\\diagup\\,} \\pi$\nimplies $\\nu = \\sigma$ or $\\pi = \\sigma$. Similarly,~$\\sigma$ is\n\\emph{anti-connected} if $\\sigma^\\sim$ is connected. The\n\\emph{shifted shuffle product}~$\\cshuffle$ of two permutations is defined by\n\\begin{equation}\n    \\sigma \\cshuffle \\nu :=\n    \\sigma \\shuffle \\left(\\nu_1\\! +\\! |\\sigma| \\dots \\nu_{|\\nu|}\\! +\\! |\\sigma|\\right).\n\\end{equation}\nFor example,\n\\begin{equation}\n    {\\bf 12} \\cshuffle 21 = {\\bf 12} \\shuffle 43 = {\\bf 12} 43 + {\\bf 1} 4 {\\bf 2} 3 +\n    {\\bf 1} 43 {\\bf 2} + 4 {\\bf 12} 3 + 4 {\\bf 1} 3 {\\bf 2} + 43 {\\bf 12}.\n\\end{equation}\nThe \\emph{standardized word}~$\\operatorname{std}(u)$ of~$u \\in A^*$ is the unique permutation\nof size~$|u|$ having the same inversions as~$u$. For example,\n$\\operatorname{std}(a_3 a_1 a_4 a_2 a_5 a_7 a_4 a_2 a_3) = 416289735$.\n\n\\subsection{Binary trees, definitions and notations}\nWe call \\emph{binary tree} any complete rooted planar binary tree. Recall\nthat a binary tree~$T$ is either a \\emph{leaf} (also called \\emph{empty tree})\ndenoted by~$\\perp$, or a node that is attached through two edges to\ntwo binary trees, called respectively the \\emph{left subtree} and the\n\\emph{right subtree} of~$T$. Let~$\\mathcal{B}\\mathcal{T}_n$ be the set of binary trees\nwith~$n$ nodes and~$\\mathcal{B}\\mathcal{T}$ be the set of all binary trees. We use in the\nsequel the standard terminology (\\emph{i.e.}, \\emph{child}, \\emph{ancestor},\n\\emph{path}, \\emph{etc.}) about binary trees~\\cite{AU94}. In our graphical\nrepresentations, nodes are represented by circles\n\\scalebox{.3}{\n\\begin{tikzpicture}\n    \\node[Noeud](0,0){};\n\\end{tikzpicture}},\nleaves by squares\n\\scalebox{.5}{\n\\begin{tikzpicture}\n    \\node[Feuille](0,0){};\n\\end{tikzpicture}},\nedges by segments\n\\scalebox{.3}{\n\\begin{tikzpicture}\n    \\draw[Arete](0,0)--(-1,-.8);\n\\end{tikzpicture}}\nor\n\\scalebox{.3}{\n\\begin{tikzpicture}\n    \\draw[Arete](0,0)--(1,-.8);\n\\end{tikzpicture}},\nand arbitrary subtrees by big squares like\n\\scalebox{.18}{\\raisebox{.3em}{\n\\begin{tikzpicture}\n    \\node[SArbre]{};\n\\end{tikzpicture}}}.\n\n\\subsubsection{The Tamari order}\nThe \\emph{Tamari order}~\\cite{Tam62,Knu06} is the partial order~${\\:\\leq_{\\operatorname{T}}\\:}$\ndefined on~$\\mathcal{B}\\mathcal{T}_n$ where $T_0 \\in \\mathcal{B}\\mathcal{T}_n$ is covered by $T_1 \\in \\mathcal{B}\\mathcal{T}_n$\nif it is possible to obtain~$T_1$ by performing a \\emph{right rotation}\ninto~$T_0$ (see Figure~\\ref{fig:Rotation}).\n\\begin{figure}[ht]\n    \\centering\n    \\scalebox{.3}{%\n    \\begin{tikzpicture}\n        \\node[Noeud, EtiqClair] (racine) at (0, 0) {};\n        \\node (g) at (-3, -1) {};\n        \\node (d) at (3, -1) {};\n        \\node[Noeud, EtiqFonce] (r) at (0, -2) {\\Huge $y$};\n        \\node[Noeud, EtiqClair] (q) at (-2, -4) {\\Huge $x$};\n        \\node[SArbre] (A) at (-4, -6) {$A$};\n        \\node[SArbre] (B) at (0, -6) {$B$};\n        \\node[SArbre] (C) at (2, -4) {$C$};\n        \\draw[Arete] (racine) -- (g);\n        \\draw[Arete] (racine) -- (d);\n        \\draw[Arete, decorate, decoration = zigzag] (racine) -- (r);\n        \\draw[Arete] (r) -- (q);\n        \\draw[Arete] (r) -- (C);\n        \\draw[Arete] (q) -- (A);\n        \\draw[Arete] (q) -- (B);\n        \\node at (-4, -3) {\\scalebox{2.2}{$T_0 = $}};\n        \\path (4, -2) edge[line width=3pt, ->] node[anchor=south,above,font=\\Huge,Noir!100]{right} (6, -2);\n        \\path (6, -4) edge[line width=3pt, ->] node[anchor=south,above,font=\\Huge,Noir!100]{left} (4, -4);\n        \\node[Noeud, EtiqClair] (racine') at (10, 0) {};\n        \\node (g') at (7, -1) {};\n        \\node (d') at (13, -1) {};\n        \\node[Noeud, EtiqFonce] (r') at (12, -4) {\\Huge $y$};\n        \\node[Noeud, EtiqClair] (q') at (10, -2) {\\Huge $x$};\n        \\node[SArbre] (A') at (8, -4) {$A$};\n        \\node[SArbre] (B') at (10, -6) {$B$};\n        \\node[SArbre] (C') at (14, -6) {$C$};\n        \\draw[Arete] (racine') -- (g');\n        \\draw[Arete] (racine') -- (d');\n        \\draw[Arete, decorate, decoration = zigzag] (racine') -- (q');\n        \\draw[Arete] (q') -- (r');\n        \\draw[Arete] (q') -- (A');\n        \\draw[Arete] (r') -- (B');\n        \\draw[Arete] (r') -- (C');\n        \\node at (14, -3) {\\scalebox{2.2}{$ = T_1$}};\n    \\end{tikzpicture}}\n    \\caption{The right rotation of root $y$.}\n    \\label{fig:Rotation}\n\\end{figure}\nOne has $T_0 {\\:\\leq_{\\operatorname{T}}\\:} T_1$ if and only if starting from~$T_0$, it is possible\nto obtain~$T_1$ by performing some right rotations.\n\n\\subsubsection{Operations on binary trees}\nIf~$L$ and~$R$ are binary trees, denote by $L \\wedge R$ the binary tree\nwhich has~$L$ as left subtree and~$R$ as right subtree. Similarly, if~$L$\nand~$R$ are $A$-labeled binary trees, denote by $L \\wedge_{\\tt a} R$\nthe $A$-labeled binary tree which has $L$ as left subtree,~$R$ as right\nsubtree and a root labeled by~${\\tt a} \\in A$.\n\\medskip\n\nLet $T_0, T_1 \\in \\mathcal{B}\\mathcal{T}$. The binary tree $T_0 {\\,\\diagup\\,} T_1$ is obtained by\ngrafting~$T_0$ from its root on the leftmost leaf of~$T_1$; In the same\nway, the binary tree $T_0 {\\,\\diagdown\\,} T_1$ is obtained by grafting~$T_1$ from\nits root on the rightmost leaf of~$T_0$.\n\\medskip\n\nFor example, for\n\\begin{equation}\n    \\begin{split} T_0 := \\: \\end{split}\n    \\begin{split}\n    \\scalebox{.15}{%\n    \\begin{tikzpicture}\n        \\node[Feuille](0)at(0,-2){};\n        \\node[Noeud](1)at(1,-1){};\n        \\node[Feuille](2)at(2,-3){};\n        \\node[Noeud](3)at(3,-2){};\n        \\node[Feuille](4)at(4,-3){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](3)--(4);\n        \\draw[Arete](1)--(0);\n        \\draw[Arete](1)--(3);\n        \\node[Noeud](5)at(5,0){};\n        \\node[Feuille](6)at(6,-2){};\n        \\node[Noeud](7)at(7,-1){};\n        \\node[Feuille](8)at(8,-2){};\n        \\draw[Arete](7)--(6);\n        \\draw[Arete](7)--(8);\n        \\draw[Arete](5)--(1);\n        \\draw[Arete](5)--(7);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\qquad \\mbox{and} \\qquad\n    \\begin{split} T_1 := \\: \\end{split}\n    \\begin{split}\n    \\scalebox{.15}{%\n    \\begin{tikzpicture}\n        \\node[Feuille](0)at(0,-1){};\n        \\node[Noeud,Marque1](1)at(1,0){};\n        \\node[Feuille](2)at(2,-3){};\n        \\node[Noeud,Marque1](3)at(3,-2){};\n        \\node[Feuille](4)at(4,-3){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](3)--(4);\n        \\node[Noeud,Marque1](5)at(5,-1){};\n        \\node[Feuille](6)at(6,-2){};\n        \\draw[Arete](5)--(3);\n        \\draw[Arete](5)--(6);\n        \\draw[Arete](1)--(0);\n        \\draw[Arete](1)--(5);\n    \\end{tikzpicture}}\\,\n    \\end{split},\n\\end{equation}\nwe have\n\\begin{equation}\n    \\begin{split}T_0 \\wedge T_1 = \\: \\end{split}\n    \\begin{split}\n    \\scalebox{.15}{%\n    \\begin{tikzpicture}\n        \\node[Feuille](0)at(0,-3){};\n        \\node[Noeud](1)at(1,-2){};\n        \\node[Feuille](2)at(2,-4){};\n        \\node[Noeud](3)at(3,-3){};\n        \\node[Feuille](4)at(4,-4){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](3)--(4);\n        \\draw[Arete](1)--(0);\n        \\draw[Arete](1)--(3);\n        \\node[Noeud](5)at(5,-1){};\n        \\node[Feuille](6)at(6,-3){};\n        \\node[Noeud](7)at(7,-2){};\n        \\node[Feuille](8)at(8,-3){};\n        \\draw[Arete](7)--(6);\n        \\draw[Arete](7)--(8);\n        \\draw[Arete](5)--(1);\n        \\draw[Arete](5)--(7);\n        \\node[Noeud,EtiqClair](9)at(9,0){};\n        \\node[Feuille](10)at(10,-2){};\n        \\node[Noeud,Marque1](11)at(11,-1){};\n        \\node[Feuille](12)at(12,-4){};\n        \\node[Noeud,Marque1](13)at(13,-3){};\n        \\node[Feuille](14)at(14,-4){};\n        \\draw[Arete](13)--(12);\n        \\draw[Arete](13)--(14);\n        \\node[Noeud,Marque1](15)at(15,-2){};\n        \\node[Feuille](16)at(16,-3){};\n        \\draw[Arete](15)--(13);\n        \\draw[Arete](15)--(16);\n        \\draw[Arete](11)--(10);\n        \\draw[Arete](11)--(15);\n        \\draw[Arete](9)--(5);\n        \\draw[Arete](9)--(11);\n    \\end{tikzpicture}}\\,\n    \\end{split},\n\\end{equation}\n\\begin{equation}\n    \\begin{split}T_0 {\\,\\diagup\\,} T_1 = \\: \\end{split}\n    \\begin{split}\n    \\scalebox{.15}{%\n    \\begin{tikzpicture}\n        \\node[Feuille](0)at(0,-3){};\n        \\node[Noeud](1)at(1,-2){};\n        \\node[Feuille](2)at(2,-4){};\n        \\node[Noeud](3)at(3,-3){};\n        \\node[Feuille](4)at(4,-4){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](3)--(4);\n        \\draw[Arete](1)--(0);\n        \\draw[Arete](1)--(3);\n        \\node[Noeud](5)at(5,-1){};\n        \\node[Feuille](6)at(6,-3){};\n        \\node[Noeud](7)at(7,-2){};\n        \\node[Feuille](8)at(8,-3){};\n        \\draw[Arete](7)--(6);\n        \\draw[Arete](7)--(8);\n        \\draw[Arete](5)--(1);\n        \\draw[Arete](5)--(7);\n        \\node[Noeud,Marque1](9)at(9,0){};\n        \\node[Feuille](10)at(10,-3){};\n        \\node[Noeud,Marque1](11)at(11,-2){};\n        \\node[Feuille](12)at(12,-3){};\n        \\draw[Arete](11)--(10);\n        \\draw[Arete](11)--(12);\n        \\node[Noeud,Marque1](13)at(13,-1){};\n        \\node[Feuille](14)at(14,-2){};\n        \\draw[Arete](13)--(11);\n        \\draw[Arete](13)--(14);\n        \\draw[Arete](9)--(5);\n        \\draw[Arete](9)--(13);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\qquad \\mbox{and} \\qquad\n    \\begin{split}T_0 {\\,\\diagdown\\,} T_1 = \\: \\end{split}\n    \\begin{split}\n    \\scalebox{.15}{%\n    \\begin{tikzpicture}\n        \\node[Feuille](0)at(0,-2){};\n        \\node[Noeud](1)at(1,-1){};\n        \\node[Feuille](2)at(2,-3){};\n        \\node[Noeud](3)at(3,-2){};\n        \\node[Feuille](4)at(4,-3){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](3)--(4);\n        \\draw[Arete](1)--(0);\n        \\draw[Arete](1)--(3);\n        \\node[Noeud](5)at(5,0){};\n        \\node[Feuille](6)at(6,-2){};\n        \\node[Noeud](7)at(7,-1){};\n        \\node[Feuille](8)at(8,-3){};\n        \\node[Noeud,Marque1](9)at(9,-2){};\n        \\node[Feuille](10)at(10,-5){};\n        \\node[Noeud,Marque1](11)at(11,-4){};\n        \\node[Feuille](12)at(12,-5){};\n        \\draw[Arete](11)--(10);\n        \\draw[Arete](11)--(12);\n        \\node[Noeud,Marque1](13)at(13,-3){};\n        \\node[Feuille](14)at(14,-4){};\n        \\draw[Arete](13)--(11);\n        \\draw[Arete](13)--(14);\n        \\draw[Arete](9)--(8);\n        \\draw[Arete](9)--(13);\n        \\draw[Arete](7)--(6);\n        \\draw[Arete](7)--(9);\n        \\draw[Arete](5)--(1);\n        \\draw[Arete](5)--(7);\n    \\end{tikzpicture}}\\,\n    \\end{split}.\n\\end{equation}\n\n\\subsubsection{Binary search trees, increasing, and decreasing binary trees}\nAn $A$-labeled binary tree~$T$ is a \\emph{right} (resp. \\emph{left})\n\\emph{binary search tree} if for any node~$x$ labeled by~${\\tt b}$, each\nlabel~${\\tt a}$ of a node in the left subtree of~$x$ and each label~${\\tt c}$ of\na node in the right subtree of~$x$, the inequality ${\\tt a} \\leq {\\tt b} < {\\tt c}$\n(resp. ${\\tt a} < {\\tt b} \\leq {\\tt c}$) holds.\n\\medskip\n\nA binary tree $T \\in \\mathcal{B}\\mathcal{T}_n$ is an \\emph{increasing} (resp. \\emph{decreasing})\n\\emph{binary tree} if it is bijectively labeled on $\\{1, \\dots, n\\}$ and,\nfor any node~$x$ of~$T$, if~$y$ is a child of~$x$, then the label of~$y$\nis greater (resp. smaller) than the label of~$x$.\n\\medskip\n\nThe \\emph{shape}~$\\operatorname{sh}(T)$ of an $A$-labeled binary tree~$T$ is the\nunlabeled binary tree obtained by forgetting its labels.\n\n\\subsubsection{Inorder traversal}\nThe \\emph{inorder traversal} of a binary tree~$T$ consists in recursively\nvisiting its left subtree, then its root, and finally its right subtree\n(see Figure~\\ref{fig:ExempleLectureInfixe}).\n\\begin{figure}[ht]\n    \\centering\n    \\scalebox{.2}{\\begin{tikzpicture}\n        \\node[Feuille](0)at(0.0,-2){};\n        \\node[Noeud,label=below:\\scalebox{3.5}{$a$}](1)at(1.0,-1){};\n        \\node[Feuille](2)at(2.0,-4){};\n        \\node[Noeud,label=below:\\scalebox{3.5}{$b$}](3)at(3.0,-3){};\n        \\node[Feuille](4)at(4.0,-4){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](3)--(4);\n        \\node[Noeud,label=below:\\scalebox{3.5}{$c$}](5)at(5.0,-2){};\n        \\node[Feuille](6)at(6.0,-3){};\n        \\draw[Arete](5)--(3);\n        \\draw[Arete](5)--(6);\n        \\draw[Arete](1)--(0);\n        \\draw[Arete](1)--(5);\n        \\node[Noeud,label=below:\\scalebox{3.5}{$d$}](7)at(7.0,0){};\n        \\node[Feuille](8)at(8.0,-4){};\n        \\node[Noeud,label=below:\\scalebox{3.5}{$e$}](9)at(9.0,-3){};\n        \\node[Feuille](10)at(10.0,-4){};\n        \\draw[Arete](9)--(8);\n        \\draw[Arete](9)--(10);\n        \\node[Noeud,label=below:\\scalebox{3.5}{$f$}](11)at(11.0,-2){};\n        \\node[Feuille](12)at(12.0,-3){};\n        \\draw[Arete](11)--(9);\n        \\draw[Arete](11)--(12);\n        \\node[Noeud,label=below:\\scalebox{3.5}{$g$}](13)at(13.0,-1){};\n        \\node[Feuille](14)at(14.0,-3){};\n        \\node[Noeud,label=below:\\scalebox{3.5}{$h$}](15)at(15.0,-2){};\n        \\node[Feuille](16)at(16.0,-3){};\n        \\draw[Arete](15)--(14);\n        \\draw[Arete](15)--(16);\n        \\draw[Arete](13)--(11);\n        \\draw[Arete](13)--(15);\n        \\draw[Arete](7)--(1);\n        \\draw[Arete](7)--(13);\n    \\end{tikzpicture}}\n    \\caption{The sequence $(a, b, c, d, e, f, g, h)$ is the sequence of all\n    nodes of this binary tree visited by the inorder traversal.}\n    \\label{fig:ExempleLectureInfixe}\n\\end{figure}\nWe shall say that a node~$x$ is \\emph{the $i$-th node of}~$T$ if~$x$ is\nthe $i$-th visited node by the inorder traversal of~$T$. In the same way,\na leaf~$y$ is \\emph{the $j$-th leaf of}~$T$ if~$y$ is the $j$-th visited\nleaf by the inorder traversal of~$T$. We also say that~$i$ is the \\emph{index}\nof~$x$ and~$j$ is the \\emph{index} of~$y$. If~$T$ is labeled, its\n\\emph{inorder reading} is the word $u_1 \\dots u_{|u|}$ such that for any\n$1 \\leq i \\leq |u|$, $u_i$ is the label of the $i$-th node of~$T$. Note\nthat when~$T$ is a right (or left) binary search tree, its inorder reading\nis a nondecreasing word.\n\n\\subsubsection{The canopy of binary trees}\nThe \\emph{canopy} (see~\\cite{LR98} and~\\cite{V04})~$\\operatorname{cnp}(T)$ of a binary\ntree~$T$ is the word on the alphabet $\\{0, 1\\}$ obtained by browsing the\nleaves of~$T$ from left to right except the first and the last one, writing~$0$\nif the considered leaf is oriented to the right,~$1$ otherwise (see\nFigure~\\ref{fig:ExempleCanopee}). Note that the orientation of the leaves\nin a binary tree is determined only by its nodes so that we can omit to\ndraw the leaves in our graphical representations.\n\\begin{figure}[ht]\n    \\centering\n    \\scalebox{.20}{%\n    \\begin{tikzpicture}\n        \\node[Feuille](0)at(0,-3){};\n        \\node[Noeud](1)at(1,-2){};\n        \\node[Feuille](2)at(2,-3){};\n        \\node[] (2') [below of = 2] {\\scalebox{3}{$0$}};\n        \\draw[Arete](1)--(0);\n        \\draw[Arete](1)--(2);\n        \\node[Noeud](3)at(3,-1){};\n        \\node[Feuille](4)at(4,-4){};\n        \\node[] (4') [below of = 4] {\\scalebox{3}{$1$}};\n        \\node[Noeud](5)at(5,-3){};\n        \\node[Feuille](6)at(6,-4){};\n        \\node[] (6') [below of = 6] {\\scalebox{3}{$0$}};\n        \\draw[Arete](5)--(4);\n        \\draw[Arete](5)--(6);\n        \\node[Noeud](7)at(7,-2){};\n        \\node[Feuille](8)at(8,-3){};\n        \\node[] (8') [below of = 8] {\\scalebox{3}{$0$}};\n        \\draw[Arete](7)--(5);\n        \\draw[Arete](7)--(8);\n        \\draw[Arete](3)--(1);\n        \\draw[Arete](3)--(7);\n        \\node[Noeud](9)at(9,0){};\n        \\node[Feuille](10)at(10,-3){};\n        \\node[] (10') [below of = 10] {\\scalebox{3}{$1$}};\n        \\node[Noeud](11)at(11,-2){};\n        \\node[Feuille](12)at(12,-3){};\n        \\node[] (12') [below of = 12] {\\scalebox{3}{$0$}};\n        \\draw[Arete](11)--(10);\n        \\draw[Arete](11)--(12);\n        \\node[Noeud](13)at(13,-1){};\n        \\node[Feuille](14)at(14,-3){};\n        \\node[] (14') [below of = 14] {\\scalebox{3}{$1$}};\n        \\node[Noeud](15)at(15,-2){};\n        \\node[Feuille](16)at(16,-3){};\n        \\draw[Arete](15)--(14);\n        \\draw[Arete](15)--(16);\n        \\draw[Arete](13)--(11);\n        \\draw[Arete](13)--(15);\n        \\draw[Arete](9)--(3);\n        \\draw[Arete](9)--(13);\n    \\end{tikzpicture}}\n    \\caption{The canopy of this binary tree is $0100101$.}\n    \\label{fig:ExempleCanopee}\n\\end{figure}\n\n\\subsection{Baxter permutations and pairs of twin binary trees}\n\n\\subsubsection{Baxter permutations}\nA permutation~$\\sigma$ is a \\emph{Baxter permutation} if for any subword\n$u := u_1 u_2 u_3 u_4$ of~$\\sigma$ such that the letters~$u_2$ and~$u_3$\nare adjacent in~$\\sigma$, $\\operatorname{std}(u) \\notin \\{2413, 3142\\}$. In other\nwords,~$\\sigma$ is a Baxter permutation if it avoids the\n\\emph{generalized permutation patterns} $2-41-3$ and $3-14-2$ (see~\\cite{BS00}\nfor an introduction on generalized permutation patterns). For example,\n${\\bf 4}21{\\bf 73}8{\\bf 5}6$ is not a Baxter permutation; On the other hand\n$436975128$, is a Baxter permutation. Let us denote by~$\\mathfrak{S}^{\\operatorname{B}}_n$ the\nset of Baxter permutations of size~$n$ and by~$\\mathfrak{S}^{\\operatorname{B}}$ the set of all\nBaxter permutations.\n\n\\subsubsection{Pairs of twin binary trees}\nA \\emph{pair of twin binary trees}~$(T_L, T_R)$ is made of two binary trees\n$T_L, T_R \\in \\mathcal{B}\\mathcal{T}_n$ such that the canopies of~$T_L$ and~$T_R$ are\ncomplementary, that is\n\\begin{equation}\n    \\operatorname{cnp}(T_L)_i \\ne \\operatorname{cnp}(T_R)_i \\mbox{ for all } 1 \\leq i \\leq n - 1\n\\end{equation}\n(see Figure~\\ref{fig:ExempleABJ}).\n\\begin{figure}[ht]\n    \\centering\n    \\scalebox{.20}{%\n    \\begin{tikzpicture}\n        \\node[Feuille](0)at(0,-3){};\n        \\node[Noeud](1)at(1,-2){};\n        \\node[Feuille](2)at(2,-3){};\n        \\node[] (2') [below of = 2] {\\scalebox{3}{$0$}};\n        \\draw[Arete](1)--(0);\n        \\draw[Arete](1)--(2);\n        \\node[Noeud](3)at(3,-1){};\n        \\node[Feuille](4)at(4,-4){};\n        \\node[] (4') [below of = 4] {\\scalebox{3}{$1$}};\n        \\node[Noeud](5)at(5,-3){};\n        \\node[Feuille](6)at(6,-4){};\n        \\node[] (6') [below of = 6] {\\scalebox{3}{$0$}};\n        \\draw[Arete](5)--(4);\n        \\draw[Arete](5)--(6);\n        \\node[Noeud](7)at(7,-2){};\n        \\node[Feuille](8)at(8,-3){};\n        \\node[] (8') [below of = 8] {\\scalebox{3}{$0$}};\n        \\draw[Arete](7)--(5);\n        \\draw[Arete](7)--(8);\n        \\draw[Arete](3)--(1);\n        \\draw[Arete](3)--(7);\n        \\node[Noeud](9)at(9,0){};\n        \\node[Feuille](10)at(10,-3){};\n        \\node[] (10') [below of = 10] {\\scalebox{3}{$1$}};\n        \\node[Noeud](11)at(11,-2){};\n        \\node[Feuille](12)at(12,-3){};\n        \\node[] (12') [below of = 12] {\\scalebox{3}{$0$}};\n        \\draw[Arete](11)--(10);\n        \\draw[Arete](11)--(12);\n        \\node[Noeud](13)at(13,-1){};\n        \\node[Feuille](14)at(14,-3){};\n        \\node[] (14') [below of = 14] {\\scalebox{3}{$1$}};\n        \\node[Noeud](15)at(15,-2){};\n        \\node[Feuille](16)at(16,-3){};\n        \\draw[Arete](15)--(14);\n        \\draw[Arete](15)--(16);\n        \\draw[Arete](13)--(11);\n        \\draw[Arete](13)--(15);\n        \\draw[Arete](9)--(3);\n        \\draw[Arete](9)--(13);\n    \\end{tikzpicture}}\n    \\enspace\n    \\scalebox{.20}{%\n    \\begin{tikzpicture}\n        \\node[Feuille](0)at(0,-3){};\n        \\node[Noeud](1)at(1,-2){};\n        \\node[Feuille](2)at(2,-4){};\n        \\node[] (2') [below of = 2] {\\scalebox{3}{$1$}};\n        \\node[Noeud](3)at(3,-3){};\n        \\node[Feuille](4)at(4,-4){};\n        \\node[] (4') [below of = 4] {\\scalebox{3}{$0$}};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](3)--(4);\n        \\draw[Arete](1)--(0);\n        \\draw[Arete](1)--(3);\n        \\node[Noeud](5)at(5,-1){};\n        \\node[Feuille](6)at(6,-3){};\n        \\node[] (6') [below of = 6] {\\scalebox{3}{$1$}};\n        \\node[Noeud](7)at(7,-2){};\n        \\node[Feuille](8)at(8,-4){};\n        \\node[] (8') [below of = 8] {\\scalebox{3}{$1$}};\n        \\node[Noeud](9)at(9,-3){};\n        \\node[Feuille](10)at(10,-4){};\n        \\node[] (10') [below of = 10] {\\scalebox{3}{$0$}};\n        \\draw[Arete](9)--(8);\n        \\draw[Arete](9)--(10);\n        \\draw[Arete](7)--(6);\n        \\draw[Arete](7)--(9);\n        \\draw[Arete](5)--(1);\n        \\draw[Arete](5)--(7);\n        \\node[Noeud](11)at(11,0){};\n        \\node[Feuille](12)at(12,-3){};\n        \\node[] (12') [below of = 12] {\\scalebox{3}{$1$}};\n        \\node[Noeud](13)at(13,-2){};\n        \\node[Feuille](14)at(14,-3){};\n        \\node[] (14') [below of = 14] {\\scalebox{3}{$0$}};\n        \\draw[Arete](13)--(12);\n        \\draw[Arete](13)--(14);\n        \\node[Noeud](15)at(15,-1){};\n        \\node[Feuille](16)at(16,-2){};\n        \\draw[Arete](15)--(13);\n        \\draw[Arete](15)--(16);\n        \\draw[Arete](11)--(5);\n        \\draw[Arete](11)--(15);\n    \\end{tikzpicture}}\n    \\caption{A pair of twin binary trees.}\n    \\label{fig:ExempleABJ}\n\\end{figure}\n\\medskip\n\nDenote by~$\\mathcal{T}\\mathcal{B}\\mathcal{T}_n$ the set of pairs of twin binary trees where each\nbinary tree has~$n$ nodes and by~$\\mathcal{T}\\mathcal{B}\\mathcal{T}$ the set of all pairs of twin\nbinary trees.\n\\medskip\n\nAn $A$-labeled pair of twin binary trees~$(T_L, T_R)$ is a \\emph{pair of\ntwin binary search trees} if~$T_L$ (resp.~$T_R$) is an $A$-labeled left\n(resp. right) binary search tree and~$T_L$ and~$T_R$ have the same inorder\nreading. The \\emph{shape}~$\\operatorname{sh}(J)$ of an $A$-labeled pair of twin binary\ntrees $J := (T_L, T_R)$ is the unlabeled pair of twin binary\ntrees~$(\\operatorname{sh}(T_L), \\operatorname{sh}(T_R))$.\n\\medskip\n\nIn~\\cite{DG94}, Dulucq and Guibert have highlighted a bijection between\nBaxter permutations and unlabeled pairs of twin binary trees. In the sequel,\nwe shall make use of a very similar bijection.\n\n\\section{The Baxter monoid} \\label{sec:MonoideBaxter}\n\n\\subsection{Definition and first properties}\nRecall that an equivalence relation~$\\equiv$ defined on~$A^*$ is a \\emph{congruence}\nif for all $u, u', v, v' \\in A^*$, $u \\equiv u'$ and $v \\equiv v'$  imply\n$uv \\equiv u'v'$. Note that the quotient $A^*\/_\\equiv$ of~$A^*$ by a\ncongruence~$\\equiv$ is naturally a monoid. Indeed, by denoting by\n$\\tau : A^* \\to A^*\/_\\equiv$ the canonical projection, the set $A^*\/_\\equiv$\nis endowed with a product~$\\cdot$ defined by\n$\\widehat{u} \\cdot \\widehat{v} := \\tau(uv)$ for all\n$\\widehat{u}, \\widehat{v} \\in A^*\/_\\equiv$ where~$u$ and~$v$ are any words\nsuch that $\\tau(u) = \\widehat{u}$ and $\\tau(v) = \\widehat{v}$.\n\n\\begin{Definition} \\label{def:MonoideBaxter}\n    The \\emph{Baxter monoid} is the quotient of the free monoid~$A^*$ by\n    the congruence~${\\:\\equiv_{\\operatorname{B}}\\:}$ that is the reflexive and transitive closure\n    of the \\emph{Baxter adjacency relations}~${\\:\\leftrightharpoons_{\\operatorname{B}}\\:}$ and~${\\:\\rightleftharpoons_{\\operatorname{B}}\\:}$ defined\n    for $u, v\\in A^*$ and ${\\tt a}, {\\tt b}, {\\tt c}, {\\tt d} \\in A$ by\n    \\begin{align}\n        {\\tt c} \\, u \\, {\\tt a} {\\tt d} \\, v \\, {\\tt b} & {\\:\\leftrightharpoons_{\\operatorname{B}}\\:} {\\tt c} \\, u \\, {\\tt d} {\\tt a} \\, v \\, {\\tt b}\n        \\qquad \\mbox{where \\quad ${\\tt a} \\leq {\\tt b} < {\\tt c} \\leq {\\tt d}$,} \\label{eq:EquivBXAdj1} \\\\\n        {\\tt b} \\, u \\, {\\tt d} {\\tt a} \\, v \\, {\\tt c} & {\\:\\rightleftharpoons_{\\operatorname{B}}\\:} {\\tt b} \\, u \\, {\\tt a} {\\tt d} \\, v \\, {\\tt c}\n        \\qquad \\mbox{where \\quad ${\\tt a} < {\\tt b} \\leq {\\tt c} < {\\tt d}$.} \\label{eq:EquivBXAdj2}\n    \\end{align}\n\\end{Definition}\n\nFor example, the ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class of~$2415253$ (see\nFigure~\\ref{fig:ExClasseBaxter}) is\n\\begin{equation}\n    \\{2142553, 2145253, 2145523, 2412553, 2415253, 2415523, 2451253,\n      2451523, 2455123\\}.\n\\end{equation}\n\\begin{figure}[ht]\n    \\centering\n    \\begin{tikzpicture}[scale=.5,font=\\small]\n        \\node (2153674) at (0,0) {$2142553$};\n        \\node (2156374) at (-2,-2) {$2145253$};\n        \\node (2513674) at (2,-2) {$2412553$};\n        \\node (2156734) at (-4,-4) {$2145523$};\n        \\node (2516374) at (0,-4) {$2415253$};\n        \\node (2516734) at (-2,-6) {$2415523$};\n        \\node (2561374) at (2,-6) {$2451253$};\n        \\node (2561734) at (0,-8) {$2451523$};\n        \\node (2567134) at (0,-10) {$2455123$};\n        \\draw [Arete] (2153674) -- (2156374);\n        \\draw [Arete] (2153674) -- (2513674);\n        \\draw [Arete] (2156374) -- (2156734);\n        \\draw [Arete] (2513674) -- (2516374);\n        \\draw [Arete] (2156734) -- (2516734);\n        \\draw [Arete] (2516374) -- (2561374);\n        \\draw [Arete] (2516734) -- (2561734);\n        \\draw [Arete] (2561374) -- (2561734);\n        \\draw [Arete] (2561734) -- (2567134);\n        \\draw [Arete] (2156374) -- (2516374);\n        \\draw [Arete] (2516374) -- (2516734);\n    \\end{tikzpicture}\n    \\qquad\n    \\begin{tikzpicture}[scale=.5,font=\\small]\n        \\node (2153674) at (0,0) {$2153674$};\n        \\node (2156374) at (-2,-2) {$2156374$};\n        \\node (2513674) at (2,-2) {$2513674$};\n        \\node (2156734) at (-4,-4) {$2156734$};\n        \\node (2516374) at (0,-4) {$2516374$};\n        \\node (2516734) at (-2,-6) {$2516734$};\n        \\node (2561374) at (2,-6) {$2561374$};\n        \\node (2561734) at (0,-8) {$2561734$};\n        \\node (2567134) at (0,-10) {$2567134$};\n        \\draw [Arete] (2153674) -- (2156374);\n        \\draw [Arete] (2153674) -- (2513674);\n        \\draw [Arete] (2156374) -- (2156734);\n        \\draw [Arete] (2513674) -- (2516374);\n        \\draw [Arete] (2156734) -- (2516734);\n        \\draw [Arete] (2516374) -- (2561374);\n        \\draw [Arete] (2516734) -- (2561734);\n        \\draw [Arete] (2561374) -- (2561734);\n        \\draw [Arete] (2561734) -- (2567134);\n        \\draw [Arete] (2156374) -- (2516374);\n        \\draw [Arete] (2516374) -- (2516734);\n    \\end{tikzpicture}\n    \\caption{The Baxter equivalence class of the word $u := 2415253$ and of the\n    permutation $2516374 = \\operatorname{std}(u)$. Edges represent Baxter adjacency relations.}\n    \\label{fig:ExClasseBaxter}\n\\end{figure}\n\nNote that if the Baxter congruence is applied on words without repetition,\nthe two Baxter adjacency relations~${\\:\\leftrightharpoons_{\\operatorname{B}}\\:}$ and~${\\:\\rightleftharpoons_{\\operatorname{B}}\\:}$ can be replaced\nby the only adjacency relation~${\\:\\rightleftarrows_{\\operatorname{B}}\\:}$ defined for $u, v \\in A^*$ and\n${\\tt a}, {\\tt b}, {\\tt b}', {\\tt d} \\in A$ by\n\\begin{equation}\n    {\\tt b} \\, u \\, {\\tt a} {\\tt d} \\, v \\, {\\tt b}' {\\:\\rightleftarrows_{\\operatorname{B}}\\:} {\\tt b} \\, u \\, {\\tt d} {\\tt a} \\, v \\, {\\tt b}'\n    \\qquad \\mbox{where \\quad ${\\tt a} < {\\tt b}, {\\tt b}' < {\\tt d}$.}\n\\end{equation}\n\n\\subsubsection{Compatibility with the destandardization process}\nA monoid $A^*\/_\\equiv$ is \\emph{compatible with the destandardization process}\nif for all $u, v \\in A^*$, $u \\equiv v$ if and only if $\\operatorname{std}(u) \\equiv \\operatorname{std}(v)$\nand $\\operatorname{ev}(u) = \\operatorname{ev}(v)$.\n\n\\begin{Proposition} \\label{prop:CompDestd}\n    The Baxter monoid is compatible with the destandardization process.\n\\end{Proposition}\n\\begin{proof}\n    It is enough to check the property on adjacency relations. Let\n    $u, v \\in A^*$. Assume $u {\\:\\leftrightharpoons_{\\operatorname{B}}\\:} v$. We have\n    \\begin{equation}\n        u = x \\, {\\tt c} \\, y \\, {\\tt a} {\\tt d} \\, z \\, {\\tt b} \\, t\n        \\quad \\mbox{and} \\quad\n        v = x \\, {\\tt c} \\, y \\, {\\tt d} {\\tt a} \\, z \\, {\\tt b} \\, t\n    \\end{equation}\n    for some letters ${\\tt a} \\leq {\\tt b} < {\\tt c} \\leq {\\tt d}$ and words~$x$, $y$, $z$,\n    and~$t$. Since~${\\:\\leftrightharpoons_{\\operatorname{B}}\\:}$ acts by permuting letters, we have\n    $\\operatorname{ev}(u) = \\operatorname{ev}(v)$. Moreover, the letters~${\\tt a}'$, ${\\tt b}'$, ${\\tt c}'$\n    and~${\\tt d}'$ of~$\\operatorname{std}(u)$ respectively at the same positions as the\n    letters~${\\tt a}$, ${\\tt b}$, ${\\tt c}$ and~${\\tt d}$ of~$u$ satisfy\n    ${\\tt a}' < {\\tt b}' < {\\tt c}' < {\\tt d}'$ due to their relative positions into~$\\operatorname{std}(u)$\n    and the order relations between~${\\tt a}$, ${\\tt b}$, ${\\tt c}$ and~${\\tt d}$. The same\n    relations hold for the letters of~$\\operatorname{std}(v)$, showing that\n    $\\operatorname{std}(u) {\\:\\leftrightharpoons_{\\operatorname{B}}\\:} \\operatorname{std}(v)$. The proof is analogous for the case $u {\\:\\rightleftharpoons_{\\operatorname{B}}\\:} v$.\n\n    Conversely, assume that~$v$ is a permutation of~$u$ and $\\operatorname{std}(u) {\\:\\leftrightharpoons_{\\operatorname{B}}\\:} \\operatorname{std}(v)$.\n    We have\n    \\begin{equation}\n        \\operatorname{std}(u) = x \\, {\\tt c} \\, y \\, {\\tt a} {\\tt d} \\, z \\, {\\tt b} \\, t\n        \\quad \\mbox{and} \\quad\n        \\operatorname{std}(v) = x \\, {\\tt c} \\, y \\, {\\tt d} {\\tt a} \\, z \\, {\\tt b} \\, t\n    \\end{equation}\n    for some letters ${\\tt a} < {\\tt b} < {\\tt c} < {\\tt d}$ and words~$x$, $y$, $z$, and~$t$.\n    The word~$u$ is a non-standardized version of~$\\operatorname{std}(u)$ so that the\n    letters~${\\tt a}'$, ${\\tt b}'$, ${\\tt c}'$ and~${\\tt d}'$ of~$u$ respectively at the\n    same positions as the letters~${\\tt a}$, ${\\tt b}$, ${\\tt c}$ and~${\\tt d}$ of~$\\operatorname{std}(u)$\n    satisfy ${\\tt a}' \\leq {\\tt b}' < {\\tt c}' \\leq {\\tt d}'$ due to their relative positions\n    into~$u$ and the order relations between~${\\tt a}$, ${\\tt b}$, ${\\tt c}$ and~${\\tt d}$.\n    The same relations hold for the letters of~$v$, showing that $u {\\:\\leftrightharpoons_{\\operatorname{B}}\\:} v$.\n    The proof is analogous for the case $\\operatorname{std}(u) {\\:\\rightleftharpoons_{\\operatorname{B}}\\:} \\operatorname{std}(v)$.\n\\end{proof}\n\n\\subsubsection{Compatibility with the restriction of alphabet intervals}\nA monoid $A^*\/_\\equiv$ is \\emph{compatible with the restriction of alphabet\nintervals} if for any interval~$I$ of~$A$ and for all $u, v \\in A^*$,\n$u \\equiv v$ implies $u_{|I} \\equiv v_{|I}$.\n\n\\begin{Proposition} \\label{prop:CompRestrSegmAlph}\n    The Baxter monoid is compatible with the restriction of alphabet\n    intervals.\n\\end{Proposition}\n\\begin{proof}\n    It is enough to check the property on adjacency relations. Moreover,\n    by Proposition~\\ref{prop:CompDestd}, it is enough to check the property\n    for permutations. Let $\\sigma, \\nu \\in \\mathfrak{S}_n$ such that $\\sigma {\\:\\rightleftarrows_{\\operatorname{B}}\\:} \\nu$.\n    We have $\\sigma = x \\, {\\tt b} \\, y \\, {\\tt a} {\\tt d} \\, z \\, {\\tt b}' \\, t$ and\n    $\\nu = x \\, {\\tt b} \\, y \\, {\\tt d} {\\tt a} \\, z \\, {\\tt b}' \\, t$ for some letters\n    ${\\tt a} < {\\tt b}, {\\tt b}' < {\\tt d}$ and words $x$, $y$, $z$, and~$t$. Let~$I$ be\n    an interval of $\\{1, \\dots, n\\}$ and $R := I \\cap \\{{\\tt a}, {\\tt b}, {\\tt b}', {\\tt d}\\}$.\n    If $R = \\{{\\tt a}, {\\tt b}, {\\tt b}', {\\tt d}\\}$,\n    \\begin{equation}\n        \\sigma_{|I} = x_{|I} \\, {\\tt b} \\, y_{|I} \\, {\\tt a} {\\tt d} \\, z_{|I} \\, {\\tt b}' \\, t_{|I}\n        \\quad \\mbox{and} \\quad\n        \\nu_{|I} = x_{|I} \\, {\\tt b} \\, y_{|I} \\, {\\tt d} {\\tt a} \\, z_{|I} \\, {\\tt b}' \\, t_{|I}\n    \\end{equation}\n    so that $\\sigma_{|I} {\\:\\rightleftarrows_{\\operatorname{B}}\\:} \\nu_{|I}$. Otherwise, we have\n    $\\sigma_{|I} = \\nu_{|I}$ and thus $\\sigma_{|I} {\\:\\equiv_{\\operatorname{B}}\\:} \\nu_{|I}$.\n\\end{proof}\n\n\\subsubsection{Compatibility with the Sch\u00fctzenberger involution}\nA monoid $A^*\/_\\equiv$ is \\emph{compatible with the Sch\u00fctzenberger involution}\nif for all $u, v \\in A^*$, $u \\equiv v$ implies $u^\\# \\equiv v^\\#$.\n\n\\begin{Proposition} \\label{prop:CompSchutz}\n    The Baxter monoid is compatible with the Sch\u00fctzenberger involution.\n\\end{Proposition}\n\\begin{proof}\n    It is enough to check the property on adjacency relations. Moreover, by\n    Proposition~\\ref{prop:CompDestd}, it is enough to check the property\n    for permutations. Let $\\sigma, \\nu \\in \\mathfrak{S}_n$ and assume that\n    $\\sigma {\\:\\rightleftarrows_{\\operatorname{B}}\\:} \\nu$. We have $\\sigma = x \\, {\\tt b} \\, y \\, {\\tt a} {\\tt d} \\, z \\, {\\tt b}' \\, t$\n    and $\\nu = x \\, {\\tt b} \\, y \\, {\\tt d} {\\tt a} \\, z \\, {\\tt b}' \\, t$ for some letters\n    ${\\tt a} < {\\tt b}, {\\tt b}' < {\\tt d}$ and words~$x$, $y$, $z$, and~$t$. We have\n    \\begin{equation}\n        \\sigma^\\# = t^\\# \\, {\\tt b}'^\\# \\, z^\\# \\, {\\tt d}^\\# {\\tt a}^\\# \\, y^\\# \\, {\\tt b}^\\# \\, x^\\#\n        \\quad \\mbox{and} \\quad\n        \\nu^\\# = t^\\# \\, {\\tt b}'^\\# \\, z^\\# \\, {\\tt a}^\\# {\\tt d}^\\# \\, y^\\# \\, {\\tt b}^\\# \\, x^\\#.\n    \\end{equation}\n    Since ${\\tt d}^\\# < {\\tt b}'^\\#, {\\tt b}^\\# < {\\tt a}^\\#$, we have $\\sigma^\\# {\\:\\rightleftarrows_{\\operatorname{B}}\\:} \\nu^\\#$.\n\\end{proof}\n\n\\subsection{Connection with the sylvester monoid}\nThe \\emph{sylvester monoid}~\\cite{HNT02, HNT05} is the quotient of the\nfree monoid~$A^*$ by the congruence~${\\:\\equiv_{\\operatorname{S}}\\:}$ that is the reflexive and\ntransitive closure of the \\emph{sylvester adjacency relation}~${\\:\\leftrightharpoons_{\\operatorname{S}}\\:}$\ndefined for $u \\in A^*$ and ${\\tt a}, {\\tt b}, {\\tt c} \\in A$ by\n\\begin{equation}\n    {\\tt a} {\\tt c} \\, u \\, {\\tt b} {\\:\\leftrightharpoons_{\\operatorname{S}}\\:} {\\tt c} {\\tt a} \\, u \\, {\\tt b}\n    \\qquad \\mbox{where \\quad ${\\tt a} \\leq {\\tt b} < {\\tt c}$.}\n\\end{equation}\nIn the same way, let us define the \\emph{$\\#$-sylvester monoid}, the quotient\nof~$A^*$ by the congruence~${\\:\\equiv_{\\operatorname{S}^\\#}\\:}$ that is the reflexive and transitive\nclosure of the \\emph{$\\#$-sylvester adjacency relation}~${\\:\\leftrightharpoons_{\\operatorname{S}^\\#}\\:}$ defined\nfor $u\\in A^*$ and ${\\tt a}, {\\tt b}, {\\tt c} \\in A$ by\n\\begin{equation} \\label{eq:DefSchutzS}\n    {\\tt b} \\, u \\, {\\tt a} {\\tt c} {\\:\\leftrightharpoons_{\\operatorname{S}^\\#}\\:} {\\tt b} \\, u \\, {\\tt c} {\\tt a}\n    \\qquad \\mbox{where \\quad ${\\tt a} < {\\tt b} \\leq {\\tt c}$.}\n\\end{equation}\nNote that this adjacency relation is defined by taking the images by the\nSch\u00fctzenberger involution of the sylvester adjacency relation. Indeed, for\nall $u, v \\in A^*$, $u {\\:\\equiv_{\\operatorname{S}^\\#}\\:} v$ if and only if~$u^\\# {\\:\\equiv_{\\operatorname{S}}\\:} v^\\#$.\n\\medskip\n\nIn~\\cite{HNT05}, Hivert, Novelli and Thibon have shown that two words\nare sylvester equivalent if and only if each gives the same right binary\nsearch tree by inserting their letters from right to left using the binary\nsearch tree insertion algorithm~\\cite{AU94}. In our setting, we call this\nprocess the \\emph{leaf insertion} and it comes in two versions, depending\non if the considered binary tree is a right or a left binary search tree:\n\\medskip\n\n{\\flushleft\n    {\\bf Algorithm:} {\\sc LeafInsertion}. \\\\\n    {\\bf Input:} An $A$-labeled right (resp. left) binary search tree~$T$,\n    a letter~${\\tt a} \\in A$. \\\\\n    {\\bf Output:} $T$ after the leaf insertion of~${\\tt a}$. \\\\\n    \\begin{enumerate}\n        \\item If $T = \\perp$, return the one-node binary search tree\n        labeled by~${\\tt a}$.\n        \\item Let~${\\tt b}$ be the label of the root of~$T$.\n        \\item If ${\\tt a} \\leq {\\tt b}$ (resp. ${\\tt a} < {\\tt b}$): \\label{item:InstrDiff}\n        \\begin{enumerate}\n            \\item Then, recursively leaf insert~${\\tt a}$ into the left subtree\n            of~$T$.\n            \\item Otherwise, recursively leaf insert ${\\tt a}$ into the right\n            subtree of~$T$.\n        \\end{enumerate}\n    \\end{enumerate}\n    {\\bf End.}\n}\n\\medskip\n\nFor further reference, let us recall the following theorem due to Hivert,\nNovelli and Thibon~\\cite{HNT05}, restated in our setting and supplemented\nwith a respective part:\n\\begin{Theoreme} \\label{thm:PSymbPBT}\n    Two words are ${\\:\\equiv_{\\operatorname{S}}\\:}$-equivalent (resp. ${\\:\\equiv_{\\operatorname{S}^\\#}\\:}$-equivalent)\n    if and only if they give the same right (resp. left) binary search tree\n    by inserting their letters from right to left (resp. left to right).\n\\end{Theoreme}\n\nIn other words, any $A$-labeled right (resp. left) binary search tree\nencodes a sylvester (resp. $\\#$-sylvester) equivalence class of words\nof~$A^*$, and conversely.\n\\smallskip\n\nLet us explain the respective part of Theorem~\\ref{thm:PSymbPBT}. It follows\nfrom~(\\ref{eq:DefSchutzS}) that encoding the~${\\:\\equiv_{\\operatorname{S}^\\#}\\:}$-equivalence\nclass of a word~$u$ is equivalent to encoding the~${\\:\\equiv_{\\operatorname{S}}\\:}$-equivalence\nclass of~$u^\\#$. For this, simply insert~$u$ from left to right by considering\nthat the reversed order relation holds between its letters. In this way,\nwe obtain a binary tree such that for any node~$x$ labeled by a letter~${\\tt b}$,\nall labels~${\\tt a}$ of the nodes of the left subtree of~$x$, and all labels~${\\tt c}$\nof the nodes of the right subtree of~$x$, the inequality ${\\tt a} \\geq {\\tt b} > {\\tt c}$\nholds. This binary tree is obviously not a left binary search tree. Nevertheless,\na left binary search tree can be obtained from it after swapping, for each\nnode, its left and right subtree recursively. One can prove by induction\non~$|u|$ that this left binary search tree is the one that {\\sc LeafInsertion}\nconstructs by inserting the letters of~$u$ from left to right and hence,\nthis remark explains the difference of treatment between right and left\nbinary search trees for the instruction~(\\ref{item:InstrDiff}) of\n{\\sc LeafInsertion}.\n\n\\begin{Lemme} \\label{lem:DiagHasseEqS}\n    Let $u := x \\, {\\tt a} {\\tt c} \\, y$ and $v := x \\, {\\tt c} {\\tt a} \\, y$ be two words\n    such that $x$ and $y$ are two words, ${\\tt a} < {\\tt c}$ are two letters,\n    and~$u {\\:\\equiv_{\\operatorname{S}}\\:} v$. Then,~$u {\\:\\leftrightharpoons_{\\operatorname{S}}\\:} v$.\n\\end{Lemme}\n\\begin{proof}\n    Follows from Theorem~\\ref{thm:PSymbPBT}: Since~$u$ and~$v$ give the\n    same right binary search tree~$T$ by inserting these from right to left,\n    the node labeled by~${\\tt a}$ and the node labeled by~${\\tt c}$ in~$T$ cannot\n    be ancestor one of the other. That implies that there exists a node\n    labeled by a letter~${\\tt b}$, common ancestor of both nodes labeled by~${\\tt a}$\n    and~${\\tt c}$ such that ${\\tt a} \\leq {\\tt b} < {\\tt c}$. Thus,~$u {\\:\\leftrightharpoons_{\\operatorname{S}}\\:} v$.\n\\end{proof}\n\nLemma~\\ref{lem:DiagHasseEqS} also proves that the ${\\:\\leftrightharpoons_{\\operatorname{S}}\\:}$-adjacency\nrelations of any equivalence class~$C$ of $\\mathfrak{S}_n \/_{\\:\\equiv_{\\operatorname{S}}\\:}$ are\nexactly the covering relations of the permutohedron restricted to the\nelements of~$C$. Note that it is also the case for the ${\\:\\leftrightharpoons_{\\operatorname{S}^\\#}\\:}$-adjacency\nrelations.\n\\medskip\n\nThe Baxter monoid, the sylvester monoid and the $\\#$-sylvester monoid are\nrelated in the following way.\n\\begin{Proposition} \\label{prop:LienSylv}\n    Let $u, v \\in A^*$. Then, $u {\\:\\equiv_{\\operatorname{B}}\\:} v$ if and only if $u {\\:\\equiv_{\\operatorname{S}}\\:} v$\n    and $u {\\:\\equiv_{\\operatorname{S}^\\#}\\:} v$.\n\\end{Proposition}\n\\begin{proof}\n    $(\\Rightarrow)$: Once more, it is enough to check the property on adjacency\n    relations. Moreover, by Proposition~\\ref{prop:CompDestd}, it is enough\n    to check the property for permutations. Let $\\sigma, \\nu \\in \\mathfrak{S}_n$\n    and assume that~$\\sigma {\\:\\rightleftarrows_{\\operatorname{B}}\\:} \\nu$. We have\n    $\\sigma = x \\, {\\tt b} \\, y \\, {\\tt a} {\\tt d} \\, z \\, {\\tt b}' \\, t$\n    and $\\nu = x \\, {\\tt b} \\, y \\, {\\tt d} {\\tt a} \\, z \\, {\\tt b}' \\, t$ for some letters\n    ${\\tt a} < {\\tt b}, {\\tt b}' < {\\tt d}$ and words~$x$, $y$, $z$, and~$t$. The presence\n    of the letters~${\\tt a}$, ${\\tt d}$ and~${\\tt b}'$ with ${\\tt a} < {\\tt b}' < {\\tt d}$ ensures\n    that~$\\sigma {\\:\\leftrightharpoons_{\\operatorname{S}}\\:} \\nu$. Besides, the presence of the letters~${\\tt b}$,\n    ${\\tt a}$ and~${\\tt d}$ with ${\\tt a} < {\\tt b} < {\\tt d}$ ensures that~$\\sigma {\\:\\leftrightharpoons_{\\operatorname{S}^\\#}\\:} \\nu$.\n\n    $(\\Leftarrow)$: Since the sylvester and the $\\#$-sylvester monoids are\n    compatible with the destandardization process~\\cite{HNT05}, it is enough\n    to check the property for permutations. Let $\\sigma, \\nu \\in \\mathfrak{S}_n$\n    such that~$\\sigma {\\:\\equiv_{\\operatorname{S}}\\:} \\nu$ and~$\\sigma {\\:\\equiv_{\\operatorname{S}^\\#}\\:} \\nu$. Set\n    $\\tau := \\inf_{{\\:\\leq_{\\operatorname{P}}\\:}} \\{\\sigma, \\nu\\}$. Since the permutohedron\n    is a lattice,~$\\tau$ is well-defined, and since the equivalence classes\n    of permutations under the~${\\:\\equiv_{\\operatorname{S}}\\:}$ and~${\\:\\equiv_{\\operatorname{S}^\\#}\\:}$ congruences\n    are intervals of the permutohedron~\\cite{HNT05}, we have\n    $\\sigma {\\:\\equiv_{\\operatorname{S}}\\:} \\tau {\\:\\equiv_{\\operatorname{S}}\\:} \\nu$ and\n    $\\sigma {\\:\\equiv_{\\operatorname{S}^\\#}\\:} \\tau {\\:\\equiv_{\\operatorname{S}^\\#}\\:} \\nu$. Moreover, by\n    Lemma~\\ref{lem:DiagHasseEqS}, and again since that the equivalence\n    classes of permutations under the~${\\:\\equiv_{\\operatorname{S}}\\:}$ and the~${\\:\\equiv_{\\operatorname{S}^\\#}\\:}$\n    congruences are intervals of the permutohedron, for each saturated\n    chains $\\tau {\\:\\leq_{\\operatorname{P}}\\:} \\sigma' {\\:\\leq_{\\operatorname{P}}\\:} \\cdots {\\:\\leq_{\\operatorname{P}}\\:} \\sigma$\n    and $\\tau {\\:\\leq_{\\operatorname{P}}\\:} \\nu' {\\:\\leq_{\\operatorname{P}}\\:} \\cdots {\\:\\leq_{\\operatorname{P}}\\:} \\nu$, there are\n    sequences of adjacency relations $\\tau {\\:\\leftrightharpoons_{\\operatorname{S}}\\:} \\sigma' {\\:\\leftrightharpoons_{\\operatorname{S}}\\:} \\cdots {\\:\\leftrightharpoons_{\\operatorname{S}}\\:} \\sigma$,\n    $\\tau {\\:\\leftrightharpoons_{\\operatorname{S}^\\#}\\:} \\sigma' {\\:\\leftrightharpoons_{\\operatorname{S}^\\#}\\:} \\cdots {\\:\\leftrightharpoons_{\\operatorname{S}^\\#}\\:} \\sigma$,\n    $\\tau {\\:\\leftrightharpoons_{\\operatorname{S}}\\:} \\nu' {\\:\\leftrightharpoons_{\\operatorname{S}}\\:} \\cdots {\\:\\leftrightharpoons_{\\operatorname{S}}\\:} \\nu$ and\n    $\\tau {\\:\\leftrightharpoons_{\\operatorname{S}^\\#}\\:} \\nu' {\\:\\leftrightharpoons_{\\operatorname{S}^\\#}\\:} \\cdots {\\:\\leftrightharpoons_{\\operatorname{S}^\\#}\\:} \\nu$. Hence,\n    $\\tau {\\:\\equiv_{\\operatorname{B}}\\:} \\sigma$ and $\\tau {\\:\\equiv_{\\operatorname{B}}\\:} \\nu$, implying~$\\sigma {\\:\\equiv_{\\operatorname{B}}\\:} \\nu$.\n\\end{proof}\n\nProposition~\\ref{prop:LienSylv} shows that the ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence\nclasses are the intersection of ${\\:\\equiv_{\\operatorname{S}}\\:}$-equivalence classes and\n${\\:\\equiv_{\\operatorname{S}^\\#}\\:}$-equivalence classes.\n\\medskip\n\nBy the characterization of the ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence classes provided by\nProposition~\\ref{prop:LienSylv}, restricting the Baxter congruence on\npermutations, we have the following property:\n\\begin{Proposition} \\label{prop:EquivBXInter}\n    For any $n \\geq 0$, each equivalence class of $\\mathfrak{S}_n \/_{\\:\\equiv_{\\operatorname{B}}\\:}$\n    is an interval of the permutohedron.\n\\end{Proposition}\n\\begin{proof}\n    By Proposition~\\ref{prop:LienSylv}, the~${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence classes\n    are the intersection of the~${\\:\\equiv_{\\operatorname{S}}\\:}$ and the~${\\:\\equiv_{\\operatorname{S}^\\#}\\:}$-equivalence\n    classes. Moreover, the permutations under the~${\\:\\equiv_{\\operatorname{S}}\\:}$ and the~${\\:\\equiv_{\\operatorname{S}^\\#}\\:}$\n    equivalence relations are intervals of the permutohedron~\\cite{HNT05}.\n    The proposition comes from the fact that the intersection of two lattice\n    intervals is also an interval and that the permutohedron is a lattice.\n\\end{proof}\n\n\\begin{Lemme} \\label{lem:DiagHasseEqBX}\n    Let $u := x \\, {\\tt a} {\\tt d} \\, y$ and $v := x \\, {\\tt d} {\\tt a} \\, y$\n    such that $x$ and $y$ are two words, ${\\tt a} < {\\tt d}$ are two letters,\n    and~$u {\\:\\equiv_{\\operatorname{B}}\\:} v$. Then,~$u {\\:\\leftrightharpoons_{\\operatorname{B}}\\:} v$ or~$u {\\:\\rightleftharpoons_{\\operatorname{B}}\\:} v$.\n\\end{Lemme}\n\\begin{proof}\n    By Proposition~\\ref{prop:LienSylv}, since~$u {\\:\\equiv_{\\operatorname{B}}\\:} v$, we\n    have~$u {\\:\\equiv_{\\operatorname{S}}\\:} v$ and thus by Lemma~\\ref{lem:DiagHasseEqS} we\n    have~$u {\\:\\leftrightharpoons_{\\operatorname{S}}\\:} v$, implying the existence of a letter~${\\tt b}'$ in the\n    factor~$y$ satisfying ${\\tt a} \\leq {\\tt b}' < {\\tt d}$. In the same way, we also\n    have~$u {\\:\\equiv_{\\operatorname{S}^\\#}\\:} v$ and thus~$u {\\:\\leftrightharpoons_{\\operatorname{S}^\\#}\\:} v$, hence the existence\n    of a letter~${\\tt b}$ in the factor~$x$ satisfying ${\\tt a} < {\\tt b} \\leq {\\tt d}$.\n    That proves that~$u$ and~$v$ are~${\\:\\leftrightharpoons_{\\operatorname{B}}\\:}$ or~${\\:\\rightleftharpoons_{\\operatorname{B}}\\:}$-adjacent.\n\\end{proof}\n\nLemma~\\ref{lem:DiagHasseEqBX} is the analog, in the case of the Baxter\ncongruence, of Lemma~\\ref{lem:DiagHasseEqS} and also proves that the~${\\:\\leftrightharpoons_{\\operatorname{B}}\\:}$\nand ${\\:\\rightleftharpoons_{\\operatorname{B}}\\:}$-adjacency relations of any equivalence class~$C$ of\n$\\mathfrak{S}_n \/_{\\:\\equiv_{\\operatorname{B}}\\:}$ are exactly the covering relations of the\npermutohedron restricted to the elements of~$C$.\n\n\\subsection{\\texorpdfstring{Connection with the $3$-recoil monoid}\n                           {Connection with the 3-recoil monoid}}\nIf~${\\tt a}$ and~${\\tt c}$ are two letters of~$A$, denote by~${\\tt c} - {\\tt a}$ the\ncardinality of the set $\\{{\\tt b} \\in A : {\\tt a} < {\\tt b} \\leq {\\tt c}\\}$.\nIn~\\cite{NRT11}, Novelli, Reutenauer and Thibon defined for any~$k \\geq 0$\nthe congruence~$\\EquivR{k}$. This congruence is the reflexive and transitive\nclosure of the \\emph{$k$-recoil adjacency relation}, defined for\n${\\tt a}, {\\tt b} \\in A$ by\n\\begin{equation}\n    {\\tt a} {\\tt b} \\AdjR{k} {\\tt b} {\\tt a} \\qquad \\mbox{where \\quad ${\\tt b} - {\\tt a} \\geq k$.}\n\\end{equation}\nThe \\emph{$k$-recoil monoid} is the quotient of the free monoid~$A^*$ by\nthe congruence~$\\EquivR{k}$. Note that the congruence~$\\EquivR{2}$ restricted\nto permutations is nothing but the \\emph{hypoplactic congruence}~\\cite{N98}.\n\\medskip\n\nThe Baxter monoid and the $3$-recoil monoid are related in the following way.\n\\begin{Proposition} \\label{prop:Lien3Recul}\n    Each~$\\EquivR{3}$-equivalence class of permutations can be expressed\n    as a union of some~${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence classes.\n\\end{Proposition}\n\\begin{proof}\n    This amounts to prove that for all permutations~$\\sigma$ and~$\\nu$,\n    if~$\\sigma {\\:\\equiv_{\\operatorname{B}}\\:} \\nu$ then~$\\sigma \\EquivR{3} \\nu$. It is enough\n    to check this property on adjacency relations. Hence, assume\n    that~$\\sigma {\\:\\rightleftarrows_{\\operatorname{B}}\\:} \\nu$. We have\n    $\\sigma = x \\, {\\tt b} \\, y \\, {\\tt a} {\\tt d} \\, z {\\tt b}' \\, t$ and\n    $\\nu = x \\, {\\tt b} \\, y \\, {\\tt d} {\\tt a} \\, z \\, {\\tt b}' \\, t$ for some letters\n    ${\\tt a} < {\\tt b}, {\\tt b}' < {\\tt d}$ and words~$x$, $y$, $z$, and~$t$.\n    Since~$\\sigma$ and~$\\nu$ are permutations, ${\\tt b} \\ne {\\tt b}'$ and thus,\n    we have ${\\tt a} < {\\tt b} < {\\tt b}' < {\\tt d}$ or ${\\tt a} < {\\tt b}' < {\\tt b} < {\\tt d}$,\n    implying that~${\\tt d} - {\\tt a} \\geq 3$. Hence,~$\\sigma \\EquivR{3} \\nu$.\n\\end{proof}\n\nNote that Proposition~\\ref{prop:Lien3Recul} is false for the congruence~$\\EquivR{4}$\nsince there are twenty-two equivalence classes of permutations of size~$4$\nunder the congruence~${\\:\\equiv_{\\operatorname{B}}\\:}$ but twenty-four under~$\\EquivR{4}$. Conversely,\nnote that~$\\EquivR{4}$ is not a refinement of~${\\:\\equiv_{\\operatorname{B}}\\:}$ since for any~$n \\geq 5$,\nthe permutation $1 . n . n\\!-\\!1 \\dots 2$ is the only member of\nits~${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class but not of its~$\\EquivR{4}$-equivalence class.\n\\medskip\n\nMoreover, it is clear, by definition of~$\\EquivR{k}$, that the~$\\EquivR{k}$-equivalence\nclasses of permutations are union of~$\\EquivR{k + 1}$-equivalence classes.\nHence, by Proposition~\\ref{prop:Lien3Recul}, the hypoplactic equivalence\nclasses of permutations are union of some~${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence classes.\n\n\\section{A Robinson-Schensted-like algorithm} \\label{sec:RobinsonSchensted}\nThe goal of this section is to define an analog to the Robinson-Schensted\nalgorithm for the Baxter monoid---see~\\cite{LS81,Lot02} for the usual Robinson-Schensted insertion algorithm that associate to any word $u$ its\n${\\sf P}$-symbol, that is a Young tableau.\n\\medskip\n\nThe interest of the Baxter monoid in our context is that the equivalence\nclasses of the permutations of size~$n$ under the Baxter congruence are\nequinumerous with unlabeled pairs of twin binary trees with~$n$ nodes,\nand thus, by the results of Dulucq and Guibert~\\cite{DG94}, also equinumerous\nwith Baxter permutations of size~$n$. We shall provide a proof of this\nproperty in this section, using our analog of the Robinson-Schensted algorithm.\n\n\\subsection{Principle of the algorithm} \\label{subsec:PSymbBaxter}\nWe describe here an algorithm testing if two words are equivalent\naccording to the Baxter congruence. Given a word~$u \\in A^*$, it computes\nits \\emph{Baxter ${\\sf P}$-symbol}, that is an $A$-labeled pair $(T_L, T_R)$\nconsisting in a left and a right binary search tree such that the nondecreasing\nrearrangement of~$u$ is the inorder reading of both~$T_L$ and~$T_R$. It also\ncomputes its \\emph{Baxter ${\\sf Q}$-symbol}, that is a pair of twin binary\ntrees $(S_L, S_R)$ where~$S_L$ (resp.~$S_R$) is an increasing (resp. decreasing)\nbinary tree, such that the inorder reading of~$S_L$ and~$S_R$ are the same.\nMoreover,~$T_L$ and~$S_L$ have same shape, and so have~$T_R$ and~$S_R$.\n\n\\subsubsection{\\texorpdfstring{The Baxter ${\\sf P}$-symbol}{The Baxter P-symbol}}\n\n\\begin{Definition} \\label{def:BaxterPSymb}\n    The \\emph{Baxter ${\\sf P}$-symbol} (or simply \\emph{${\\sf P}$-symbol}\n    if the context is clear) of a word $u \\in A^*$ is the pair\n    ${\\sf P}(u) = (T_L, T_R)$ where~$T_L$ (resp.~$T_R$) is the left (resp.\n    right) binary search tree obtained by leaf inserting the letters of~$u$\n    from left to right (resp. right to left).\n\\end{Definition}\nFigure~\\ref{fig:ExemplePQSymboleSansEtapes} shows the ${\\sf P}$-symbol of\n$u := 2415253$. Before showing that the ${\\sf P}$-symbol of\nDefinition~\\ref{def:BaxterPSymb} can be used to decide if two words are\nequivalent under the Baxter congruence, let us give an intuitive explanation\nof its validity.\n\\medskip\n\nRecall that, according to Proposition~\\ref{prop:LienSylv}, to represent\nthe Baxter equivalence class of a word~$u$, one has to represent both the\nequivalence class of~$u$ under the~${\\:\\equiv_{\\operatorname{S}}\\:}$ congruence and the equivalence\nclass of~$u$ under the~${\\:\\equiv_{\\operatorname{S}^\\#}\\:}$ congruence. This is exactly what\nthe Baxter ${\\sf P}$-symbol does since, for a word~$u$, it computes a\npair~$(T_L, T_R)$ where, by Theorem~\\ref{thm:PSymbPBT},~$T_L$ represents\nthe~${\\:\\equiv_{\\operatorname{S}^\\#}\\:}$-equivalence class of~$u$ and~$T_R$ represents\nthe~${\\:\\equiv_{\\operatorname{S}}\\:}$-equivalence class of~$u$.\n\n\\subsubsection{\\texorpdfstring{The Baxter ${\\sf Q}$-symbol}{The Baxter Q-symbol}}\nLet us first recall two algorithms. Let~$u$ be a word. Define~$\\operatorname{incr}(u)$,\nthe \\emph{increasing binary tree of~$u$} recursively by\n\\begin{equation}\n    \\operatorname{incr}(u) :=\n    \\begin{cases}\n        \\perp & \\mbox{if $u = \\epsilon$,} \\\\[.5em]\n        \\operatorname{incr}(v) \\wedge_{\\tt a} \\operatorname{incr}(w) &\n        \\mbox{where $u = v {\\tt a} w$, ${\\tt a} = \\min(u)$, and ${\\tt a} < \\min(v)$.}\n    \\end{cases}\n\\end{equation}\nIn the same way, define the \\emph{decreasing binary tree of~$u$}~$\\operatorname{decr}(u)$, by\n\\begin{equation}\n    \\operatorname{decr}(u) :=\n    \\begin{cases}\n        \\perp & \\mbox{if $u = \\epsilon$,} \\\\[.5em]\n        \\operatorname{decr}(v) \\wedge_{\\tt b} \\operatorname{decr}(w) &\n        \\mbox{where $u = v {\\tt b} w$, ${\\tt b} = \\max(u)$, and ${\\tt b} > \\max(w)$.}\n    \\end{cases}\n\\end{equation}\n\n\\begin{Definition} \\label{def:BaxterQSymbole}\n    The \\emph{Baxter ${\\sf Q}$-symbol} (or simply \\emph{${\\sf Q}$-symbol}\n    if the context is clear) of a word $u \\in A^*$ is the pair\n    ${\\sf Q}(u) = (S_L, S_T)$ where\n    \\begin{equation}\n        S_L := \\operatorname{incr}\\left(\\operatorname{std}(u)^{-1}\\right)\n        \\qquad \\mbox{and} \\qquad\n        S_R := \\operatorname{decr}\\left(\\operatorname{std}(u)^{-1}\\right).\n    \\end{equation}\n\\end{Definition}\n\nFigure~\\ref{fig:ExemplePQSymboleSansEtapes} shows the ${\\sf Q}$-symbol of\n$u := 2415253$, whose standardized word is $2516374$, so that\n$\\operatorname{std}(u)^{-1} = 3157246$.\n\\begin{figure}[ht]\n    \\centering\n    \\begin{equation*}\n    \\begin{split}{\\sf P}(u) = \\: \\end{split}\n    \\begin{split}\n    \\scalebox{.34}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0.0,-1){$1$};\n        \\node[Noeud,EtiqClair](1)at(1.0,0){$2$};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud,EtiqClair](2)at(2.0,-2){$2$};\n        \\node[Noeud,EtiqClair](3)at(3.0,-3){$3$};\n        \\draw[Arete](2)--(3);\n        \\node[Noeud,EtiqClair](4)at(4.0,-1){$4$};\n        \\draw[Arete](4)--(2);\n        \\node[Noeud,EtiqClair](5)at(5.0,-2){$5$};\n        \\node[Noeud,EtiqClair](6)at(6.0,-3){$5$};\n        \\draw[Arete](5)--(6);\n        \\draw[Arete](4)--(5);\n        \\draw[Arete](1)--(4);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\enspace\n    \\begin{split}\n    \\scalebox{.34}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0.0,-2){$1$};\n        \\node[Noeud,EtiqClair](1)at(1.0,-3){$2$};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud,EtiqClair](2)at(2.0,-1){$2$};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud,EtiqClair](3)at(3.0,0){$3$};\n        \\draw[Arete](3)--(2);\n        \\node[Noeud,EtiqClair](4)at(4.0,-3){$4$};\n        \\node[Noeud,EtiqClair](5)at(5.0,-2){$5$};\n        \\draw[Arete](5)--(4);\n        \\node[Noeud,EtiqClair](6)at(6.0,-1){$5$};\n        \\draw[Arete](6)--(5);\n        \\draw[Arete](3)--(6);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\qquad\n    \\begin{split}{\\sf Q}(u) = \\: \\end{split}\n    \\begin{split}\n    \\scalebox{.34}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0.0,-1){$3$};\n        \\node[Noeud,EtiqClair](1)at(1.0,0){$1$};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud,EtiqClair](2)at(2.0,-2){$5$};\n        \\node[Noeud,EtiqClair](3)at(3.0,-3){$7$};\n        \\draw[Arete](2)--(3);\n        \\node[Noeud,EtiqClair](4)at(4.0,-1){$2$};\n        \\draw[Arete](4)--(2);\n        \\node[Noeud,EtiqClair](5)at(5.0,-2){$4$};\n        \\node[Noeud,EtiqClair](6)at(6.0,-3){$6$};\n        \\draw[Arete](5)--(6);\n        \\draw[Arete](4)--(5);\n        \\draw[Arete](1)--(4);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\enspace\n    \\begin{split}\n    \\scalebox{.34}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0.0,-2){$3$};\n        \\node[Noeud,EtiqClair](1)at(1.0,-3){$1$};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud,EtiqClair](2)at(2.0,-1){$5$};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud,EtiqClair](3)at(3.0,0){$7$};\n        \\draw[Arete](3)--(2);\n        \\node[Noeud,EtiqClair](4)at(4.0,-3){$2$};\n        \\node[Noeud,EtiqClair](5)at(5.0,-2){$4$};\n        \\draw[Arete](5)--(4);\n        \\node[Noeud,EtiqClair](6)at(6.0,-1){$6$};\n        \\draw[Arete](6)--(5);\n        \\draw[Arete](3)--(6);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\end{equation*}\n    \\caption{The ${\\sf P}$-symbol and the ${\\sf Q}$-symbol of $u := 2415253$.}\n    \\label{fig:ExemplePQSymboleSansEtapes}\n\\end{figure}\n\\medskip\n\nIt is plain that given a word~$u$, the ${\\sf Q}$-symbol of~$u$ allows, in\naddition with its ${\\sf P}$-symbol, to retrieve the original word. Indeed,\nif ${\\sf P}(u) = (T_L, T_R)$ and ${\\sf Q}(u) = (S_L, S_R)$, the pair~$(T_R, S_R)$\nis the output of the Robinson-Schensted-like algorithm in the context of\nthe sylvester monoid, which is a bijection between words and pairs of such\nbinary trees~\\cite{HNT05}. Given~$(T_R, S_R)$, it amounts to reading the\nlabels of~$T_R$ in the order of the corresponding labels in~$S_R$. The\nsame holds of the pair~$(T_L, S_L)$.\n\n\\subsection{Correctness of the insertion algorithm}\n\\begin{Lemme} \\label{lem:OrientationFeuille}\n    Let~$T$ be a non-empty binary tree and~$y$ be the $i$-th leaf of~$T$.\n    If~$y$ is left-oriented, it is attached to the $i$-th node of~$T$.\n    If~$y$ is right-oriented, it is attached to the $i\\!-\\!1$-st node of~$T$.\n\\end{Lemme}\n\\begin{proof}\n    We proceed by structural induction on the set of non-empty binary trees.\n    If~$T$ is the one-node  binary tree, the lemma is clearly satisfied.\n    Otherwise, we have $T = A \\wedge B$. Let~$y$ be the $i$-th leaf of~$T$\n    and~$x$ be the node where~$y$ is attached. If~$y$ is also in~$A$ and\n    $A = \\perp$, $y$ is left-oriented and is attached to the root\n    of~$T$ (that is the first node of~$T$) and the lemma is satisfied. If~$y$\n    is in~$A$ and $A \\ne \\perp$, $y$ is also the $i$-th leaf of~$A$\n    and~$x$ is a node of~$A$, so that the lemma follows by induction hypothesis\n    on~$A$. Otherwise,~$y$ is in~$B$. If $B = \\perp$, $y$ is right-oriented\n    and is attached to the root of~$T$ (that is the last node of~$T$) and\n    the lemma is satisfied. Otherwise,~$y$ is the $i\\!-\\!(n\\!+\\!1)$-st leaf\n    of~$B$ where~$n$ is the number of nodes of~$A$. Assume that the node~$x$\n    is the $j$-st node of~$T$, then,~$x$ becomes the $j\\!-\\!(n\\!+\\!1)$-st\n    node of~$B$. Hence, the lemma follows by induction hypothesis on~$B$.\n\\end{proof}\n\nThe following proposition is the key of our construction.\n\\begin{Proposition} \\label{prop:FeuillesInversions}\n    Let~$\\sigma$ be a permutation and~$T$ be the left binary search tree\n    obtained by left leaf insertions of the letters of~$\\sigma$, from left\n    to right. Then, the $i\\!+\\!1$-st leaf of~$T$ is right-oriented if and\n    only if~$i$ is a recoil of~$\\sigma$.\n\\end{Proposition}\n\\begin{proof}\n    Set ${\\tt a} := i$ and ${\\tt c} := i\\!+\\!1$. Assume that~${\\tt a}$ is a recoil\n    of~$\\sigma$. We have $\\sigma = u \\, {\\tt c} \\, v \\, {\\tt a} \\, w$ for some\n    words~$u$, $v$, and~$w$. Since no letter~${\\tt b}$ of~$u$ and~$v$ satisfies\n    ${\\tt a} < {\\tt b} < {\\tt c}$, the node of~$T$ labeled by~${\\tt c}$ has a node labeled\n    by~${\\tt a}$ in its left subtree, itself having no right child and thus\n    contributes, by Lemma~\\ref{lem:OrientationFeuille}, to a right-oriented\n    leaf in position~$i\\!+\\!1$.\n\n    Conversely, assume that~${\\tt a}$ is not a recoil of~$\\sigma$. We have\n    $\\sigma = u \\, {\\tt a} \\, v \\, {\\tt c} \\, w$ for some words~$u$, $v$, and~$w$.\n    For the same reason as before, the node of~$T$ labeled by~${\\tt a}$ has\n    a node labeled by~${\\tt c}$ in its right subtree, itself having no left\n    child and thus contributes, by Lemma~\\ref{lem:OrientationFeuille},\n    to a left-oriented leaf in position~$i\\!+\\!1$.\n\\end{proof}\n\nFigure~\\ref{fig:IllustrationOrientationFeuilles} shows an example of\napplication of Proposition~\\ref{prop:FeuillesInversions}.\n\\begin{figure}[ht]\n    \\centering\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Feuille](0)at(0.0,-2){};\n        \\node[Noeud,EtiqClair](1)at(1.0,-1){$1$};\n        \\node[Feuille](2)at(2.0,-4){};\n        \\node[Noeud,EtiqClair](3)at(3.0,-3){$2$};\n        \\node[Feuille](4)at(4.0,-4){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](3)--(4);\n        \\node[Noeud,EtiqClair](5)at(5.0,-2){$3$};\n        \\node[Feuille](6)at(6.0,-3){};\n        \\draw[Arete](5)--(3);\n        \\draw[Arete](5)--(6);\n        \\draw[Arete](1)--(0);\n        \\draw[Arete](1)--(5);\n        \\node[Noeud,EtiqClair](7)at(7.0,0){$4$};\n        \\node[Feuille](8)at(8.0,-3){};\n        \\node[Noeud,EtiqClair](9)at(9.0,-2){$5$};\n        \\node[Feuille](10)at(10.0,-3){};\n        \\draw[Arete](9)--(8);\n        \\draw[Arete](9)--(10);\n        \\node[Noeud,EtiqClair](11)at(11.0,-1){$6$};\n        \\node[Feuille](12)at(12.0,-3){};\n        \\node[Noeud,EtiqClair](13)at(13.0,-2){$7$};\n        \\node[Feuille](14)at(14.0,-3){};\n        \\draw[Arete](13)--(12);\n        \\draw[Arete](13)--(14);\n        \\draw[Arete](11)--(9);\n        \\draw[Arete](11)--(13);\n        \\draw[Arete](7)--(1);\n        \\draw[Arete](7)--(11);\n    \\end{tikzpicture}}\n    \\caption{The binary search tree drawn with its leaves obtained by left\n    leaf insertions of the letters of $\\sigma := 4136275$, from left to right.\n    The recoils of $\\sigma$ are $2$, $3$, $5$, and $7$ and the $3$-rd, $4$-th,\n    $6$-th, and $8$-th leaves of this binary tree are right-oriented.}\n    \\label{fig:IllustrationOrientationFeuilles}\n\\end{figure}\n\n\\subsubsection{\\texorpdfstring{The ${\\sf P}$-symbol}{The P-symbol}}\n\\begin{Proposition} \\label{prop:PSymboleNonIncr}\n    For any word $u \\in A^*$, the ${\\sf P}$-symbol $(T_L, T_R)$ of~$u$ is\n    a pair of twin binary search trees---$T_L$ (resp.~$T_R$) is a left\n    (resp. right) binary search tree, and the inorder reading of both~$T_L$\n    and~$T_R$ is the nondecreasing rearrangement of~$u$.\n\\end{Proposition}\n\\begin{proof}\n    Note by definition of the {\\sc LeafInsertion} algorithm that~$T_L$\n    (resp.~$T_R$) is a left (resp. right) binary search tree and the inorder\n    reading of both~$T_L$ and~$T_R$ is the nondecreasing rearrangement of~$u$.\n    It is plain that the leaf insertion of~$u$ and~$\\operatorname{std}(u)$ from left to\n    right (resp. right to left) into left (resp. right) binary search trees\n    give binary trees of same shape. That implies that we can consider\n    that $u =: \\sigma$ is a permutation. Proposition~\\ref{prop:FeuillesInversions}\n    implies that the canopies of~$T_L$ and~$T_R$ are complementary because~$i$\n    is a recoil of~$\\sigma$ if and only if~$i$ is not a recoil of~$\\sigma^\\sim$.\n    Thus, the shapes of~$T_L$ and~$T_R$ consist in a pair of twin binary trees.\n\\end{proof}\n\n\\begin{Theoreme} \\label{thm:PSymboleClasses}\n    Let $u, v \\in A^*$. Then, $u {\\:\\equiv_{\\operatorname{B}}\\:} v$ if and only if\n    ${\\sf P}(u) = {\\sf P}(v)$.\n\\end{Theoreme}\n\\begin{proof}\n    Assume $u {\\:\\equiv_{\\operatorname{B}}\\:} v$. Then, by Proposition~\\ref{prop:LienSylv},~$u$\n    and~$v$ are~${\\:\\equiv_{\\operatorname{S}}\\:}$ and ${\\:\\equiv_{\\operatorname{S}^\\#}\\:}$-equivalent. Hence, by\n    Theorem~\\ref{thm:PSymbPBT},~$u$ and~$v$ have the same sylvester and\n    $\\#$-sylvester ${\\sf P}$-symbol, so that ${\\sf P}(u) = {\\sf P}(v)$.\n\n    Conversely assume that ${\\sf P}(u) = {\\sf P}(v) =: (T_L, T_R)$. Since\n    the leaf insertion of both~$u$ and~$v$ from left to right gives~$T_L$,\n    we have, by Theorem~\\ref{thm:PSymbPBT}, $u {\\:\\equiv_{\\operatorname{S}^\\#}\\:} v$. In addition,\n    the leaf insertion of both~$u$ and~$v$ from right to left gives~$T_R$,\n    so that, by the just cited theorem, $u {\\:\\equiv_{\\operatorname{S}}\\:} v$. By\n    Proposition~\\ref{prop:LienSylv}, we have~$u {\\:\\equiv_{\\operatorname{B}}\\:} v$.\n\\end{proof}\n\nIn the case of permutations, each~${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class can be encoded\nby an unlabeled pair of twin binary trees because there is one unique way\nto bijectively label a binary tree with~$n$ nodes on~$\\{1, \\dots, n\\}$\nsuch that it is a binary search tree. Hence, in the sequel, unlabeled pairs\nof twin binary search trees can be considered as labeled by a permutation,\nand conversely.\n\n\\subsubsection{\\texorpdfstring{The ${\\sf Q}$-symbol}{The Q-symbol}}\nLet us recall the following lemma of~\\cite{HNT05}, restated in our setting\nand supplemented with a respective part:\n\\begin{Lemme} \\label{lem:FormeInsIncr}\n    Let~$u$ be a word and $\\sigma := \\operatorname{std}(u)^{-1}$. The right (resp. left)\n    binary search tree obtained by inserting $u$ from right to left (resp.\n    from left to right) and~$\\operatorname{decr}(\\sigma)$ (resp.~$\\operatorname{incr}(\\sigma)$) have\n    same shape.\n\\end{Lemme}\n\n\\begin{Proposition} \\label{prop:TypeQSymbole}\n    For any word $u \\in A^*$, the shape of the ${\\sf Q}$-symbol~$(S_L, S_R)$\n    of~$u$ is a pair of twin binary trees. Moreover,~$S_L$ is an increasing\n    binary tree,~$S_R$ is a decreasing binary tree and their inorder reading\n    is~$\\operatorname{std}(u)^{-1}$.\n\\end{Proposition}\n\\begin{proof}\n    By definition of the~${\\sf Q}$-symbol,~$S_L$ and~$S_R$ are respectively\n    the increasing and the decreasing binary trees of $\\sigma := \\operatorname{std}(u)^{-1}$.\n    By Lemma~\\ref{lem:FormeInsIncr}, a binary tree with same shape as~$S_L$\n    (resp.~$S_R$) can also be obtained by leaf insertions of the letters\n    of~$\\sigma^{-1}$ from left to right (resp. right to left). Thus, by\n    Proposition~\\ref{prop:FeuillesInversions}, the shape of $(S_L, S_R)$ is a pair\n    of twin binary trees. Moreover, by the definition of the algorithms~$\\operatorname{incr}$\n    and~$\\operatorname{decr}$, we can prove by induction on the size of~$\\sigma$ that\n    the binary trees~$S_L$ and~$S_R$ have both~$\\sigma$ as inorder reading.\n\\end{proof}\n\n\\begin{Theoreme} \\label{thm:BijectionMotsPQSymbole}\n    The map $u \\longmapsto \\left({\\sf P}(u), {\\sf Q}(u)\\right)$ is a bijection\n    between the elements of $A^*$ and the set formed by the pairs\n    $\\left((T_L, T_R), (S_L, S_R)\\right)$ where\n    \\begin{enumerate}[label = (\\roman*)]\n        \\item $(T_L, T_R)$ is a pair of twin binary search trees---$T_L$\n        (resp. $T_R$) is a left (resp. right) binary search tree, and $T_L$\n        and $T_R$ have both the same inorder reading; \\label{item:BMPQS1}\n        \\item $(S_L, S_R)$ is a pair of twin binary trees where $S_L$ (resp. $S_R$)\n        is an increasing (resp. decreasing) binary tree, and $S_L$ and $S_R$\n        have both the same inorder reading; \\label{item:BMPQS2}\n        \\item $(T_L, T_R)$ and $(S_L, S_R)$ have same shape. \\label{item:BMPQS3}\n    \\end{enumerate}\n\\end{Theoreme}\n\\begin{proof}\n    Let us first show that for any $u \\in A^*$, the pair\n    $\\left({\\sf P}(u), {\\sf Q}(u)\\right)$ satisfies the assertions of the\n    theorem. Point~\\ref{item:BMPQS1} follows from\n    Proposition~\\ref{prop:PSymboleNonIncr}. Point~\\ref{item:BMPQS2} follows\n    from Proposition~\\ref{prop:TypeQSymbole}. Moreover, by Lemma~\\ref{lem:FormeInsIncr},\n    Point~\\ref{item:BMPQS3} checks out. Besides, as already mentioned,\n    it is possible to reconstruct from the pair $\\left({\\sf P}(u), {\\sf Q}(u)\\right)$\n    the word~$u$ and such a word is unique. That shows that the correspondence\n    is well-defined and injective.\n\n    Conversely, assume that $\\left((T_L, T_R), (S_L, S_R)\\right)$ satisfies\n    the three assertions of the theorem. According to~\\cite{HNT02}, there\n    is a bijection between the elements of~$A^*$ and the pairs $(T_R, S_R)$\n    where~$T_R$ is a right binary search tree and~$S_R$ a decreasing binary\n    tree of same shape. Let~$u$ be the word in correspondence with $(T_R, S_R)$.\n    In the same way, there is a bijection between the elements of~$A^*$\n    and the pairs $(T_L, S_L)$ where~$T_L$ is a left binary search tree\n    and~$S_L$ an increasing binary tree of same shape. Let~$v$ be the word\n    in correspondence with $(T_L, S_L)$. By hypothesis,~$T_L$ and~$T_R$\n    have both the same inorder reading, implying $\\operatorname{ev}(u) = \\operatorname{ev}(v)$.\n    In the same way, since~$S_L$ and~$S_R$ have both the same inorder reading,\n    one has $\\operatorname{std}(u)^{-1} = \\operatorname{std}(v)^{-1}$. Hence, we have $\\operatorname{std}(u) = \\operatorname{std}(v)$\n    and thus~$u = v$. Note also that the pair $(T_L, S_L)$ is entirely\n    determined by the pair $(T_R, S_R)$ and conversely. Now, again according\n    to~\\cite{HNT02}, the pair $(T_R, S_R)$ is the sylvester ${\\sf P}$-symbol\n    of~$u$ and the pair $(T_L, S_R)$ is the $\\#$-sylvester ${\\sf P}$-symbol\n    of~$u$. Hence, the insertion of $u$ gives the pair\n    $\\left((T_L, T_R), (S_L, S_R)\\right)$, showing that the correspondence\n    is also surjective.\n\\end{proof}\n\n\\subsection{Distinguished permutations from a pair of twin binary trees}\nWe present in this section some algorithms to read some distinguished\npermutations from a pair of twin binary search trees. Let us first start\nwith a useful characterization of ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence classes.\n\n\\subsubsection{Baxter equivalence classes as linear extensions of posets}\nLet $T$ be an $A$-labeled binary tree. We shall denote by $\\bigtriangleup(T)$\n(resp. $\\bigtriangledown(T)$) the poset $(N, \\leq)$ where $N := \\{1, \\dots, n\\}$,\n$n$ is the number of nodes of~$T$, and $\\leq$ is defined, for $i, j \\in N$, by\n\\begin{equation}\n    i \\leq j \\qquad\n    \\mbox{if the $i$-th node is an ancestor (resp. descendant) of the\n    $j$-th node of~$T$}.\n\\end{equation}\nIf the sequence~$i_1 \\dots i_n$ is a linear extension of~$\\bigtriangleup(T)$\n(resp.~$\\bigtriangledown(T)$), we shall also say that the word~$u_1 \\dots u_n$ is\na \\emph{linear extension} of~$\\bigtriangleup(T)$ (resp.~$\\bigtriangledown(T)$) if for any\n$1 \\leq \\ell \\leq n$, the label of the $i_\\ell$-th node of~$T$ is~$u_\\ell$.\n\\medskip\n\nThe words of a sylvester equivalence class encoded by a labeled right\nbinary search tree~$T$ coincide with the linear extensions of~$\\bigtriangledown(T)$\n(see Note 4 of~\\cite{HNT05}). Additionally, this also says that the words\nof a $\\#$-sylvester equivalence class encoded by a labeled left binary\nsearch tree~$T$ are exactly the linear extensions of~$\\bigtriangleup(T)$. One has\na similar characterization of Baxter equivalence classes:\n\\begin{Proposition} \\label{prop:BXExtLin}\n    The words of a Baxter equivalence class encoded by a pair of twin binary\n    search trees~$(T_L, T_R)$ coincide with the words that are both\n    linear extensions of the posets~$\\bigtriangleup(T_L)$ and~$\\bigtriangledown(T_R)$.\n\\end{Proposition}\n\\begin{proof}\n    Let~$u$ be a word belonging to the Baxter equivalence class encoded\n    by~$(T_L, T_R)$. By Theorem~\\ref{thm:PSymboleClasses},~$T_L$ (resp.~$T_R$)\n    can be obtained by leaf inserting~$u$ from left to right (resp. right\n    to left). Hence, if $i \\leq j$  in~$\\bigtriangleup(T_L)$ (resp. in~$\\bigtriangledown(T_R)$)\n    then~$i$ is smaller than~$j$ as integers. Thus,~$u$ is a linear extension\n    of both~$\\bigtriangleup(T_L)$ and~$\\bigtriangledown(T_R)$.\n\n    Assume now that~$u$ is a linear extension of~$\\bigtriangleup(T_L)$ and~$\\bigtriangledown(T_R)$\n    and let $v$ be any word of the Baxter equivalence class encoded by $(T_L, T_R)$.\n    By Theorem~\\ref{thm:PSymboleClasses},~$T_L$ (resp.~$T_R$) can be obtained\n    by leaf inserting~$v$ from left to right (resp. right to left).\n    Note 4 of~\\cite{HNT05} implies that~$u {\\:\\equiv_{\\operatorname{S}^\\#}\\:} v$ and~$u {\\:\\equiv_{\\operatorname{S}}\\:} v$.\n    Hence, by Proposition~\\ref{prop:LienSylv}, one has~$u {\\:\\equiv_{\\operatorname{B}}\\:} v$, showing\n    that~$u$ also belongs to the Baxter equivalence class represented by~$(T_L, T_R)$.\n\\end{proof}\n\nTo illustrate Proposition~\\ref{prop:BXExtLin}, consider the following\nlabeled pair of twin binary search trees,\n\\begin{equation}\n    \\begin{split} (T_L, T_R) := \\: \\end{split}\n    \\begin{split}\n    \\scalebox{.35}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0,-2){$1$};\n        \\node[Noeud,EtiqClair](1)at(1,-1){$2$};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud,EtiqClair](2)at(2,-3){$3$};\n        \\node[Noeud,EtiqClair](3)at(3,-2){$4$};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n        \\node[Noeud,EtiqClair](4)at(4,0){$5$};\n        \\draw[Arete](4)--(1);\n        \\node[Noeud,EtiqClair](5)at(5,-2){$6$};\n        \\node[Noeud,EtiqClair](6)at(6,-1){$7$};\n        \\draw[Arete](6)--(5);\n        \\draw[Arete](4)--(6);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\enspace\n    \\begin{split}\n    \\scalebox{.35}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0,-2){$1$};\n        \\node[Noeud,EtiqClair](1)at(1,-3){$2$};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud,EtiqClair](2)at(2,-1){$3$};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud,EtiqClair](3)at(3,-2){$4$};\n        \\node[Noeud,EtiqClair](4)at(4,-3){$5$};\n        \\draw[Arete](3)--(4);\n        \\draw[Arete](2)--(3);\n        \\node[Noeud,EtiqClair](5)at(5,0){$6$};\n        \\draw[Arete](5)--(2);\n        \\node[Noeud,EtiqClair](6)at(6,-1){$7$};\n        \\draw[Arete](5)--(6);\n    \\end{tikzpicture}}\n    \\end{split}\\,.\n\\end{equation}\nThe set of words that are linear extensions of~$\\bigtriangleup(T_L)$\nand~$\\bigtriangledown(T_R)$ are (the highlighted permutation is a Baxter permutation)\n\\begin{align}\n    \\begin{split}\n        \\{{\\bf 5214376}, & \\enspace 5214736, \\enspace\n            5217436, \\enspace 5241376, \\enspace 5241736, \\\\\n            5247136, & \\enspace 5271436, \\enspace\n            5274136, \\enspace 5721436, \\enspace 5724136 \\},\n    \\end{split}\n\\end{align}\nwhich is exactly the Baxter equivalence class encoded by~$(T_L, T_R)$.\n\\medskip\n\nNote that it is possible to represent the order relations induced by the\nposets~$\\bigtriangleup(T_L)$ and $\\bigtriangledown(T_R)$ in only one poset\n$\\bigtriangleup(T_L) \\cup \\bigtriangledown(T_R)$, adding  on~$\\bigtriangleup(T_L)$ the order\nrelations induced by~$\\bigtriangledown(T_R)$. For the previous example, we obtain\nthe poset\n\\begin{equation}\n    \\begin{split}\\bigtriangleup(T_L) \\cup \\bigtriangledown(T_R) = \\:\\end{split}\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0,-2){$1$};\n        \\node[Noeud,EtiqClair](1)at(1,-1){$2$};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud,EtiqClair](2)at(2,-3){$3$};\n        \\node[Noeud,EtiqClair](3)at(3,-2){$4$};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n        \\node[Noeud,EtiqClair](4)at(4,0){$5$};\n        \\draw[Arete](4)--(1);\n        \\node[Noeud,EtiqClair](5)at(5,-4){$6$};\n        \\node[Noeud,EtiqClair](6)at(6,-1){$7$};\n        \\draw[Arete](6)--(5);\n        \\draw[Arete](4)--(6);\n        \\draw[Arete](0)--(2);\n        \\draw[Arete](2)--(5);\n    \\end{tikzpicture}}\n    \\end{split}\\,.\n\\end{equation}\n\n\\subsubsection{Extracting Baxter permutations}\nThe following algorithm allows, given an $A$-labeled pair of twin binary\nsearch trees~$(T_L, T_R)$, to compute a word belonging to the\n${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class encoded by~$(T_L, T_R)$. When~$(T_L, T_R)$\nis labeled by a permutation, our algorithm coincides with the algorithm\ndesigned by Dulucq and Guibert to describe a bijection between pairs of\ntwin binary trees and Baxter permutations~\\cite{DG94}. Besides, since their\nalgorithm always computes a Baxter permutation, our algorithm also returns\na Baxter permutation when~$(T_L, T_R)$ is labeled by a permutation.\n\\medskip\n\n{\\flushleft\n    {\\bf Algorithm:} {\\sc ExtractBaxter}. \\\\\n    {\\bf Input:} An $A$-labeled pair of twin binary search trees~$(T_L, T_R)$. \\\\\n    {\\bf Output:} A word belonging to the Baxter equivalence class encoded\n    by~$(T_L, T_R)$. \\\\\n    \\begin{enumerate}\n        \\item Let $u := \\epsilon$ be the empty word.\n        \\item While $T_L \\ne \\perp$ and $T_R \\ne \\perp$:\n        \\begin{enumerate}\n            \\item Let ${\\tt a}$ be the label of the root of $T_L$.\n            \\item Let $i$ be the index of root of $T_L$.\n            \\item Set $u := u{\\tt a}$.\n            \\item Let $A$ (resp. $B$) be the left (resp. right) subtree of $T_L$.\n            \\item If the $i$-th node of $T_R$ is a left child in $T_R$:\n            \\begin{enumerate}\n                \\item Then, set $T_L := A {\\,\\diagup\\,} B$.\n                \\item Otherwise, set $T_L := A {\\,\\diagdown\\,} B$.\n            \\end{enumerate}\n            \\item Suppress the $i$-th node in $T_R$.\n        \\end{enumerate}\n        \\item Return $u$.\n    \\end{enumerate}\n    {\\bf End.}\n}\n\\medskip\n\nFigure~\\ref{fig:ExempleExtractBaxter} shows an execution of this algorithm.\n\\begin{figure}[ht]\n    \\centering\n    \\begin{equation*}\n        \\begin{split} (T_L, T_R) := \\: \\end{split}\n        \\begin{split}\n        \\scalebox{.35}{\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqClair](0)at(0,-2){$1$};\n            \\node[Noeud,EtiqClair](1)at(1,-1){$2$};\n            \\draw[Arete](1)--(0);\n            \\node[Noeud,EtiqClair](2)at(2,-2){$3$};\n            \\node[Noeud,EtiqClair](3)at(3,-3){$4$};\n            \\draw[Arete](2)--(3);\n            \\draw[Arete](1)--(2);\n            \\node[Noeud,EtiqFonce](4)at(4,0){$5$};\n            \\draw[Arete](4)--(1);\n            \\node[Noeud,EtiqClair](5)at(5,-1){$6$};\n            \\draw[Arete](4)--(5);\n        \\end{tikzpicture}}\n        \\end{split}\n        \\enspace\n        \\begin{split}\n        \\scalebox{.35}{\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqClair](0)at(0,-2){$1$};\n            \\node[Noeud,EtiqClair](1)at(1,-3){$2$};\n            \\draw[Arete](0)--(1);\n            \\node[Noeud,EtiqClair](2)at(2,-1){$3$};\n            \\draw[Arete](2)--(0);\n            \\node[Noeud,EtiqClair](3)at(3,0){$4$};\n            \\draw[Arete](3)--(2);\n            \\node[Noeud,EtiqFonce](4)at(4,-2){$5$};\n            \\node[Noeud,EtiqClair](5)at(5,-1){$6$};\n            \\draw[Arete](5)--(4);\n            \\draw[Arete](3)--(5);\n        \\end{tikzpicture}}\n        \\end{split}\n        \\begin{split} \\quad \\xrightarrow{{\\tt a} = 5} \\quad \\end{split}\n        \\begin{split}\n        \\scalebox{.35}{\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqClair](0)at(0,-2){$1$};\n            \\node[Noeud,EtiqClair](1)at(1,-1){$2$};\n            \\draw[Arete](1)--(0);\n            \\node[Noeud,EtiqClair](2)at(2,-2){$3$};\n            \\node[Noeud,EtiqClair](3)at(3,-3){$4$};\n            \\draw[Arete](2)--(3);\n            \\draw[Arete](1)--(2);\n            \\node[Noeud,EtiqFonce](4)at(4,0){$6$};\n            \\draw[Arete](4)--(1);\n        \\end{tikzpicture}}\n        \\end{split}\n        \\enspace\n        \\begin{split}\n        \\scalebox{.35}{\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqClair](0)at(0,-2){$1$};\n            \\node[Noeud,EtiqClair](1)at(1,-3){$2$};\n            \\draw[Arete](0)--(1);\n            \\node[Noeud,EtiqClair](2)at(2,-1){$3$};\n            \\draw[Arete](2)--(0);\n            \\node[Noeud,EtiqClair](3)at(3,0){$4$};\n            \\draw[Arete](3)--(2);\n            \\node[Noeud,EtiqFonce](4)at(4,-1){$6$};\n            \\draw[Arete](3)--(4);\n        \\end{tikzpicture}}\n        \\end{split}\n    \\end{equation*}\n    \\begin{equation*}\n        \\begin{split}\\xrightarrow{{\\tt a} = 6} \\quad \\end{split}\n        \\begin{split}\n        \\scalebox{.35}{\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqClair](0)at(0,-1){$1$};\n            \\node[Noeud,EtiqFonce](1)at(1,0){$2$};\n            \\draw[Arete](1)--(0);\n            \\node[Noeud,EtiqClair](2)at(2,-1){$3$};\n            \\node[Noeud,EtiqClair](3)at(3,-2){$4$};\n            \\draw[Arete](2)--(3);\n            \\draw[Arete](1)--(2);\n        \\end{tikzpicture}}\n        \\end{split}\n        \\enspace\n        \\begin{split}\n        \\scalebox{.35}{\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqClair](0)at(0,-2){$1$};\n            \\node[Noeud,EtiqFonce](1)at(1,-3){$2$};\n            \\draw[Arete](0)--(1);\n            \\node[Noeud,EtiqClair](2)at(2,-1){$3$};\n            \\draw[Arete](2)--(0);\n            \\node[Noeud,EtiqClair](3)at(3,0){$4$};\n            \\draw[Arete](3)--(2);\n        \\end{tikzpicture}}\n        \\end{split}\n        \\begin{split}\\quad \\xrightarrow{{\\tt a} = 2} \\quad \\end{split}\n        \\begin{split}\n        \\scalebox{.35}{\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqFonce](0)at(0,0){$1$};\n            \\node[Noeud,EtiqClair](1)at(1,-1){$3$};\n            \\node[Noeud,EtiqClair](2)at(2,-2){$4$};\n            \\draw[Arete](1)--(2);\n            \\draw[Arete](0)--(1);\n        \\end{tikzpicture}}\n        \\end{split}\n        \\enspace\n        \\begin{split}\n        \\scalebox{.35}{\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqFonce](0)at(0,-2){$1$};\n            \\node[Noeud,EtiqClair](1)at(1,-1){$3$};\n            \\draw[Arete](1)--(0);\n            \\node[Noeud,EtiqClair](2)at(2,0){$4$};\n            \\draw[Arete](2)--(1);\n        \\end{tikzpicture}}\n        \\end{split}\n        \\begin{split}\\quad \\xrightarrow{{\\tt a} = 1} \\quad \\end{split}\n        \\begin{split}\n        \\scalebox{.35}{\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqFonce](0)at(0,0){$3$};\n            \\node[Noeud,EtiqClair](1)at(1,-1){$4$};\n            \\draw[Arete](0)--(1);\n        \\end{tikzpicture}}\n        \\end{split}\n        \\enspace\n        \\begin{split}\n        \\scalebox{.35}{\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqFonce](0)at(0,-1){$3$};\n            \\node[Noeud,EtiqClair](1)at(1,0){$4$};\n            \\draw[Arete](1)--(0);\n        \\end{tikzpicture}}\n        \\end{split}\n    \\end{equation*}\n    \\begin{equation*}\n        \\begin{split} \\xrightarrow{{\\tt a} = 3} \\quad \\end{split}\n        \\begin{split}\n        \\scalebox{.35}{%\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqFonce](0)at(0,0){$4$};\n        \\end{tikzpicture}}\n        \\end{split}\n        \\enspace\n        \\begin{split}\n        \\scalebox{.35}{%\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqFonce](0)at(0,0){$4$};\n        \\end{tikzpicture}}\n        \\end{split}\n        \\begin{split}\\quad \\xrightarrow{{\\tt a} = 5} \\quad \\end{split}\n        \\begin{split} \\perp \\perp \\end{split}\n    \\end{equation*}\n    \\caption{An execution of the algorithm {\\sc ExtractBaxter} on~$(T_L, T_R)$.\n    The computed Baxter permutation is~$562134$.}\n    \\label{fig:ExempleExtractBaxter}\n\\end{figure}\n\\medskip\n\nThe results of Dulucq and Guibert~\\cite{DG94} imply that {\\sc ExtractBaxter}\nterminates. The only thing to prove is that the computed word belongs to\nthe ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class encoded by the pair of twin binary search\ntrees as input. For that, let us first prove the following lemma.\n\n\\begin{Lemme} \\label{lem:NoeudRacineFeuille}\n    Let~$(T_L, T_R)$ be a non-empty pair of twin binary trees. If the root\n    of~$T_L$ is the $i$-th node of~$T_L$, then, the $i$-th node of~$T_R$\n    has no child.\n\\end{Lemme}\n\\begin{proof}\n    Assume that $T_L = A \\wedge B$. Note that if both~$A$ and~$B$ are\n    empty,~$T_L$ and~$T_R$ are the one-node binary trees and the lemma is\n    clearly satisfied.\n\n    If $A \\ne \\perp$, assume that the $i$-th node of~$T_R$ has a non-empty\n    left subtree. That implies that the $i$-th leaf of~$T_R$ is not attached\n    to its $i$-th node. Thus, by Lemma~\\ref{lem:OrientationFeuille}, the $i$-th\n    leaf of~$T_R$ is attached to its $i\\!-\\!1$-st node and is right-oriented.\n    In~$T_L$, the $i$-th leaf cannot be attached to its $i$-th node\n    because~$A \\ne \\perp$. Hence, by Lemma~\\ref{lem:OrientationFeuille},\n    the $i$-th leaf of~$T_L$ is also attached to its $i\\!-\\!1$-st node and is\n    right-oriented. Since~$T$ contains at least~$i$ nodes, there is at\n    least~$i\\!+\\!1$ leaves in~$T$, implying that the $i$-th leaf is not\n    the rightmost leaf of~$T_L$ and~$T_R$, and thus~$(T_L, T_R)$ is not\n    a pair of twin binary trees, contradicting the hypothesis.\n\n    Assume now that the $i$-th node of~$T_R$ has a non-empty right subtree.\n    That implies that the $i\\!+\\!1$-st leaf of~$T_R$ is not attached to\n    its $i$-th node and thus, by Lemma~\\ref{lem:OrientationFeuille},\n    the $i\\!+\\!1$-st leaf of~$T_R$ is left-oriented. Moreover, since the\n    $i$-th node of~$T_R$ has a non-empty right subtree and the $i$-th node\n    of~$T_L$ is its root, the $i$-th node of~$T_L$ also has a non-empty\n    right subtree. That implies that the $i\\!+\\!1$-st leaf of~$T_L$ is not\n    attached to its $i$-th node and thus, by Lemma~\\ref{lem:OrientationFeuille},\n    the $i\\!+\\!1$-st leaf of~$T_R$ is also left-oriented. That contradicts\n    that~$(T_L, T_R)$ is a pair of twin binary trees, and implies that the\n    $i$-th node of~$T_R$ has no child. The case~$B \\ne \\perp$ is analogous.\n\\end{proof}\n\n\\begin{Proposition} \\label{prop:LectureBaxMemeClasse}\n    For any $A$-labeled pair of twin binary search trees~$(T_L, T_R)$ as\n    input, the algorithm {\\sc ExtractBaxter} computes a word belonging to\n    the ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class encoded by~$(T_L, T_R)$. Moreover,\n    if~$(T_L, T_R)$ is labeled by a permutation, the computed word is a\n    Baxter permutation.\n\\end{Proposition}\n\\begin{proof}\n    Let us prove by induction on~$n$, that is the number of nodes of~$T_L$\n    and~$T_R$, that if~$(T_L, T_R)$ is an $A$-labeled pair of twin binary\n    search trees, then {\\sc ExtractBaxter} returns a word that is a linear\n    extension of~$\\bigtriangleup(T_L)$ and a linear extension of~$\\bigtriangledown(T_R)$,\n    \\emph{i.e.}, by Proposition~\\ref{prop:BXExtLin}, a word belonging to\n    the ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class encoded by~$(T_L, T_R)$. This property\n    clearly holds for~$n \\leq 1$. Now, assume that $T_L = A \\wedge_{\\tt a} B$\n    where~${\\tt a}$ is the label of the root of~$T_L$. By\n    Lemma~\\ref{lem:NoeudRacineFeuille}, if the root of~$T_L$ is its $i$-th\n    node, the $i$-th node~$x$ of~$T_R$ has no child. Moreover, since~$T_L$\n    and~$T_R$ are binary search trees and labeled by a same word, their\n    respective $i$-th nodes have the same label~${\\tt a}$. Moreover, the canopy\n    of~$T_L$ is of the form~$v01w$ where~$v := \\operatorname{cnp}(A)$ and~$w := \\operatorname{cnp}(B)$,\n    and the canopy of~$T_R$ is of the form~$v'10w'$ where~$v'$ (resp.~$w'$)\n    is the complementary of~$v$ (resp.~$w$) since that~$(T_L, T_R)$ is a\n    pair of twin binary trees. We have now two cases whether~$x$ is a left\n    of right child in~$T_R$.\n\n    If~$x$ is a left child in~$T_R$, the algorithm returns the word~${\\tt a} u$\n    where~$u$ is the word obtained by applying the algorithm on~$(T'_L, T'_R)$\n    where~$T'_L = A {\\,\\diagup\\,} B$ and~$T'_R$ is obtained from~$T_R$ by suppressing\n    the node~$x$. First, the canopy of~$T'_L$ is of the form~$v0w$ and the\n    canopy of~$T'_R$ is of the form~$v'1w'$. Moreover,~$T'_L$ and~$T'_R$\n    are clearly still binary search trees. That implies that~$(T'_L, T'_R)$\n    is a pair of twin binary search trees. By induction hypothesis and\n    Proposition~\\ref{prop:BXExtLin}, the word~$u$ belongs to the\n    ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class encoded by~$(T'_L, T'_R)$, and thus,~${\\tt a} u$\n    belongs to the ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class encoded by~$(T_L, T_R)$\n    because~${\\tt a} u$ is a linear extension of~$\\bigtriangleup(T_L)$ (resp.~$\\bigtriangledown(T_R)$)\n    since~$u$ is a linear extension of both~$\\bigtriangleup(T'_L)$ and~$\\bigtriangledown(T'_R)$.\n    The case where~$x$ is a right child in~$T_R$ is analogous.\n\n    Finally, when~$(T_L, T_R)$ is labeled by a permutation, {\\sc ExtractBaxter}\n    coincides with the algorithm of Dulucq and Guibert~\\cite{DG94} and\n    computes a Baxter permutation.\n\\end{proof}\n\nThe validity of {\\sc ExtractBaxter} implies the two following results.\n\n\\begin{Theoreme} \\label{thm:BijectionEquivBXPermuBX}\n    For any~$n \\geq 0$, there is a bijection between the set of Baxter\n    equivalence classes of words of length $n$ and $A$-labeled pairs of\n    twin binary search trees with~$n$ nodes.\n\\end{Theoreme}\n\\begin{proof}\n    By Proposition~\\ref{prop:PSymboleNonIncr} and Theorem~\\ref{thm:PSymboleClasses},\n    the ${\\sf P}$-symbol algorithm induces an injection between the set\n    of equivalence classes of~$\\mathfrak{S}_n \/_{\\:\\equiv_{\\operatorname{B}}\\:}$ and the set of\n    unlabeled pairs of twin binary trees. Moreover,\n    by Proposition~\\ref{prop:LectureBaxMemeClasse}, the algorithm\n    {\\sc ExtractBaxter} exhibits a surjection between these two sets.\n    Hence, these two sets are in bijection.\n\\end{proof}\n\nTheorem~\\ref{thm:BijectionEquivBXPermuBX} implies in particular that\nthe Baxter equivalence classes of permutations of size~$n$ are in bijection\nwith pairs of twin binary trees labeled by a permutation (or equivalently with\nunlabeled pairs of twin binary trees).\n\n\\begin{Theoreme} \\label{thm:EquivBXBaxter}\n    For any~$n \\geq 0$, each equivalence class of~$\\mathfrak{S}_n \/_{\\:\\equiv_{\\operatorname{B}}\\:}$\n    contains exactly one Baxter permutation.\n\\end{Theoreme}\n\\begin{proof}\n    Let~$C$ be an equivalence class of~$\\mathfrak{S}_n \/_{\\:\\equiv_{\\operatorname{B}}\\:}$. By\n    Theorem~\\ref{thm:BijectionEquivBXPermuBX},~$C$ can be represented by\n    an unlabeled pair of twin binary trees~$J$. By\n    Proposition~\\ref{prop:LectureBaxMemeClasse}, the algorithm\n    {\\sc ExtractBaxter} computes a permutation belonging to the\n    ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class encoded by~$J$, showing that each\n    ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class of permutations contains at least one\n    Baxter permutation. The theorem follows from the fact that Baxter\n    permutations are equinumerous with unlabeled pairs of twin binary trees.\n\\end{proof}\n\n\\subsubsection{Extracting minimal and maximal permutations}\nReading defined in~\\cite{Rea05} \\emph{twisted Baxter permutations}, that\nare the permutations avoiding the generalized permutation patterns~$2-41-3$\nand~$3-41-2$. These permutations are particular elements of Baxter classes\nof permutations:\n\\begin{Proposition} \\label{prop:TwistedMinClasses}\n    Twisted Baxter permutations coincide with minimal elements of\n    Baxter equivalence classes of permutations.\n\\end{Proposition}\n\\begin{proof}\n    First, note that by Proposition~\\ref{prop:EquivBXInter}, every Baxter\n    equivalence class of permutations has a minimal element. Assume that~$\\sigma$\n    is minimal of its ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class of permutations. Then,\n    it is not possible to perform any rewriting of the form\n    \\begin{equation}\n        {\\tt b} \\, u \\, {\\tt d} {\\tt a} \\, v \\, {\\tt b}' \\rightarrow\n        {\\tt b} \\, u \\, {\\tt a} {\\tt d} \\, v \\, {\\tt b}',\n    \\end{equation}\n    where ${\\tt a} < {\\tt b}, {\\tt b}' < {\\tt d}$ are letters, and~$u$ and~$v$ are words.\n    Hence,~$\\sigma$ avoids the patterns $2-41-3$ and $3-41-2$, and is a\n    twisted Baxter permutation.\n\n    Conversely, if~$\\sigma$ is a twisted Baxter permutation, it avoids\n    $2-41-3$ and $3-41-2$ and it is not possible to perform any\n    rewriting~$\\rightarrow$, so that, by Proposition~\\ref{prop:EquivBXInter}\n    and Lemma~\\ref{lem:DiagHasseEqBX}, it is minimal of its\n    ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class.\n\\end{proof}\n\nIn a similar way, by calling \\emph{anti-twisted Baxter permutation} any\npermutation that avoids the generalized permutation patterns $2-14-3$ and\n$3-14-2$, an analogous proof to the one of Proposition~\\ref{prop:TwistedMinClasses}\nshows that anti-twisted Baxter permutations coincide with maximal\nelements of Baxter equivalence classes of permutations.\n\\medskip\n\nProposition~\\ref{prop:TwistedMinClasses} implies that twisted Baxter\npermutations, anti-twisted Baxter permutations, and Baxter permutations\nare equinumerous since by Theorem~\\ref{thm:EquivBXBaxter} there is exactly\none Baxter permutation by ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class of permutations and\nby Proposition~\\ref{prop:EquivBXInter}, there is also exactly one twisted\n(and one anti-twisted) Baxter permutation. This suggests among other that\nthere exists a bijection sending a Baxter permutation to the twisted Baxter\npermutation of its ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class.\n\\medskip\n\nAs pointed out by Law and Reading, West has shown first a bijection between\nBaxter permutations and twisted Baxter permutations using generating\ntrees~\\cite{B03}. In our setting, as in the setting of Law and Reading~\\cite{LR10},\nthis bijection is the one preserving the classes. Here follows an algorithm\nto compute this bijection.\n\\medskip\n\nLet us consider the following algorithm which allows, given an $A$-labeled\npair of twin binary search trees~$(T_L, T_R)$, to compute the minimal\npermutation for the lexicographic order belonging to the ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence\nclass encoded by~$(T_L, T_R)$.\n\\medskip\n\n{\\flushleft\n    {\\bf Algorithm:} {\\sc ExtractMin}. \\\\\n    {\\bf Input:} An $A$-labeled pair of twin binary search trees~$(T_L, T_R)$. \\\\\n    {\\bf Output:} The minimal word for the lexicographic order of the\n    class encoded by~$(T_L, T_R)$. \\\\\n    \\begin{enumerate}\n        \\item Let $u := \\epsilon$ be the empty word.\n        \\item Let $F := T_L$ be a rooted forest.\n        \\item While $F$ is not empty and $T_R \\ne \\perp$:\n        \\begin{enumerate}\n            \\item Let $i$ be the smallest index such that the $i$-th node\n            of $F$ is a root and the $i$-th node of $T_R$ has no child.\\label{item:InstrChoixNoeudMin}\n            \\item Let ${\\tt a}$ be the label of the $i$-th node of $T_L$.\n            \\item Set $u := u {\\tt a}$.\n            \\item Suppress the $i$-th node of $F$ and the $i$-th node of $T_R$.\n        \\end{enumerate}\n        \\item Return $u$.\n    \\end{enumerate}\n    {\\bf End.}\n}\n\\medskip\n\nNote that, by choosing in the instruction~(\\ref{item:InstrChoixNoeudMin}) the\ngreatest index instead of the smallest, the previous algorithm would compute the\nmaximal word for the lexicographic order of the ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class\nencoded by~$(T_L, T_R)$. Let us call this variant {\\sc ExtractMax}.\n\\medskip\n\nFigure~\\ref{fig:ExempleExtractMin} shows an example of application of {\\sc ExtractMin}.\n\\begin{figure}[ht]\n    \\centering\n    \\begin{equation*}\n        \\begin{split}(T_L, T_R) := \\: \\end{split}\n        \\begin{split}\n        \\scalebox{.35}{\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqClair](0)at(0,-2){$1$};\n            \\node[Noeud,EtiqClair](1)at(1,-1){$2$};\n            \\draw[Arete](1)--(0);\n            \\node[Noeud,EtiqClair](2)at(2,-2){$3$};\n            \\node[Noeud,EtiqClair](3)at(3,-3){$4$};\n            \\draw[Arete](2)--(3);\n            \\draw[Arete](1)--(2);\n            \\node[Noeud,EtiqFonce](4)at(4,0){$5$};\n            \\draw[Arete](4)--(1);\n            \\node[Noeud,EtiqClair](5)at(5,-1){$6$};\n            \\draw[Arete](4)--(5);\n        \\end{tikzpicture}}\n        \\end{split}\n        \\quad\n        \\begin{split}\n        \\scalebox{.35}{\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqClair](0)at(0,-2){$1$};\n            \\node[Noeud,EtiqClair](1)at(1,-3){$2$};\n            \\draw[Arete](0)--(1);\n            \\node[Noeud,EtiqClair](2)at(2,-1){$3$};\n            \\draw[Arete](2)--(0);\n            \\node[Noeud,EtiqClair](3)at(3,0){$4$};\n            \\draw[Arete](3)--(2);\n            \\node[Noeud,EtiqFonce](4)at(4,-2){$5$};\n            \\node[Noeud,EtiqClair](5)at(5,-1){$6$};\n            \\draw[Arete](5)--(4);\n            \\draw[Arete](3)--(5);\n        \\end{tikzpicture}}\n        \\end{split}\n        \\begin{split}\\quad \\xrightarrow{{\\tt a} = 5} \\quad \\end{split}\n        \\begin{split}\n        \\scalebox{.35}{\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqClair](0)at(0,-1){$1$};\n            \\node[Noeud,EtiqFonce](1)at(1,0){$2$};\n            \\draw[Arete](1)--(0);\n            \\node[Noeud,EtiqClair](2)at(2,-1){$3$};\n            \\node[Noeud,EtiqClair](3)at(3,-2){$4$};\n            \\draw[Arete](2)--(3);\n            \\draw[Arete](1)--(2);\n        \\end{tikzpicture}}\n        \\end{split}\n        \\begin{split}\n        \\scalebox{.35}{\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqClair](0)at(0,0){$6$};\n        \\end{tikzpicture}}\n        \\end{split}\n        \\quad\n        \\begin{split}\n        \\scalebox{.35}{\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqClair](0)at(0,-2){$1$};\n            \\node[Noeud,EtiqFonce](1)at(1,-3){$2$};\n            \\draw[Arete](0)--(1);\n            \\node[Noeud,EtiqClair](2)at(2,-1){$3$};\n            \\draw[Arete](2)--(0);\n            \\node[Noeud,EtiqClair](3)at(3,0){$4$};\n            \\draw[Arete](3)--(2);\n            \\node[Noeud,EtiqClair](4)at(4,-1){$6$};\n            \\draw[Arete](3)--(4);\n        \\end{tikzpicture}}\n        \\end{split}\n    \\end{equation*}\n    \\begin{equation*}\n        \\begin{split}\\xrightarrow{{\\tt a} = 2} \\quad \\end{split}\n        \\begin{split}\n        \\scalebox{.35}{%\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqFonce](0)at(0,0){$1$};\n        \\end{tikzpicture}}\n        \\end{split}\n        \\begin{split}\n        \\scalebox{.35}{\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqClair](0)at(0,0){$3$};\n            \\node[Noeud,EtiqClair](1)at(1,-1){$4$};\n            \\draw[Arete](0)--(1);\n        \\end{tikzpicture}}\n        \\end{split}\n        \\begin{split}\n        \\scalebox{.35}{\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqClair](0)at(0,0){$6$};\n        \\end{tikzpicture}}\n        \\end{split}\n        \\quad\n        \\begin{split}\n        \\scalebox{.35}{\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqFonce](0)at(0,-2){$1$};\n            \\node[Noeud,EtiqClair](1)at(1,-1){$3$};\n            \\draw[Arete](1)--(0);\n            \\node[Noeud,EtiqClair](2)at(2,0){$4$};\n            \\draw[Arete](2)--(1);\n            \\node[Noeud,EtiqClair](3)at(3,-1){$6$};\n            \\draw[Arete](2)--(3);\n        \\end{tikzpicture}}\n        \\end{split}\n        \\begin{split}\\quad \\xrightarrow{{\\tt a} = 1} \\quad\\end{split}\n        \\begin{split}\n        \\scalebox{.35}{\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqFonce](0)at(0,0){$3$};\n            \\node[Noeud,EtiqClair](1)at(1,-1){$4$};\n            \\draw[Arete](0)--(1);\n        \\end{tikzpicture}}\n        \\end{split}\n        \\begin{split}\n        \\scalebox{.35}{%\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqClair](0)at(0,0){$6$};\n        \\end{tikzpicture}}\n        \\end{split}\n        \\quad\n        \\begin{split}\n        \\scalebox{.35}{\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqFonce](0)at(0,-1){$3$};\n            \\node[Noeud,EtiqClair](1)at(1,0){$4$};\n            \\draw[Arete](1)--(0);\n            \\node[Noeud,EtiqClair](2)at(2,-1){$6$};\n            \\draw[Arete](1)--(2);\n        \\end{tikzpicture}}\n        \\end{split}\n        \\begin{split}\\quad \\xrightarrow{{\\tt a} = 3} \\quad \\end{split}\n        \\begin{split}\n        \\scalebox{.35}{%\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqClair](0)at(0,0){$4$};\n        \\end{tikzpicture}}\n        \\end{split}\n        \\begin{split}\n        \\scalebox{.35}{%\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqFonce](0)at(0,0){$6$};\n        \\end{tikzpicture}}\n        \\end{split}\n        \\quad\n        \\begin{split}\n        \\scalebox{.35}{\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqClair](0)at(0,0){$4$};\n            \\node[Noeud,EtiqFonce](1)at(1,-1){$6$};\n            \\draw[Arete](0)--(1);\n        \\end{tikzpicture}}\n        \\end{split}\n    \\end{equation*}\n    \\begin{equation*}\n        \\begin{split}\\xrightarrow{{\\tt a} = 6} \\quad \\end{split}\n        \\begin{split}\n        \\scalebox{.35}{%\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqFonce](0)at(0,0){$4$};\n        \\end{tikzpicture}}\n        \\end{split}\n        \\quad\n        \\begin{split}\n        \\scalebox{.35}{%\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqFonce](0)at(0,0){$4$};\n        \\end{tikzpicture}}\n        \\end{split}\n        \\begin{split}\\quad \\xrightarrow{{\\tt a} = 4} \\quad \\end{split}\n        \\begin{split} \\perp \\perp \\end{split}\n    \\end{equation*}\n    \\caption{An execution of the algorithm {\\sc ExtractMin} on~$(T_L, T_R)$.\n    The computed permutation is~$521364$ and it is minimal in\n    its~${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class.}\n    \\label{fig:ExempleExtractMin}\n\\end{figure}\n\n\\begin{Proposition} \\label{prop:LectureMin}\n    For any $A$-labeled pair of twin binary search trees~$(T_L, T_R)$\n    as input, the algorithm {\\sc ExtractMin} (resp. {\\sc ExtractMax})\n    computes the minimal (resp. maximal) word for the lexicographic order\n    of the ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class encoded by~$(T_L, T_R)$. Moreover,\n    if~$(T_L, T_R)$ is labeled by a permutation, the computed word is the\n    minimal (resp. maximal) permutation for the permutohedron order of its\n    ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class.\n\\end{Proposition}\n\\begin{proof}\n    The output~$u$ of the algorithm {\\sc ExtractMin} (resp. {\\sc ExtractMax})\n    is both a linear extension of~$\\bigtriangleup(T_L)$ and a linear extension\n    of~$\\bigtriangledown(T_R)$. That implies by Proposition~\\ref{prop:BXExtLin}\n    that~$u$ belongs to the ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class encoded by\n    the input pair of twin binary trees. Moreover, this algorithm terminates\n    since by Theorem~\\ref{thm:EquivBXBaxter}, each $A$-labeled pair of\n    twin binary search trees~$(T_L, T_R)$ admits at least one word that\n    is a common linear extension of~$\\bigtriangleup(T_L)$ and~$\\bigtriangledown(T_R)$.\n    The minimality (resp. maximality) for the lexicographic order of the\n    computed word comes from the fact that at each step, the node that\n    has the smallest (resp. greatest) label is chosen.\n\n    Finally, since the lexicographic order is a linear extension of the\n    permutohedron order, and by Proposition~\\ref{prop:EquivBXInter}, since\n    Baxter equivalence classes are intervals of the permutohedron,\n    {\\sc ExtractMin} (resp. {\\sc ExtractMax}) returns the minimal (resp.\n    maximal) permutation for the permutohedron order of its Baxter\n    equivalence class.\n\\end{proof}\n\nBy Proposition~\\ref{prop:LectureMin} and using our Robinson-Schensted-like\nalgorithm, we can compute the bijection between Baxter permutations and\ntwisted Baxter permutations in the following way: If~$\\sigma$ is a Baxter\npermutation, apply {\\sc ExtractMin} on~${\\sf P}(\\sigma)$ to obtain its\ncorresponding twisted Baxter permutation. Conversely, if~$\\sigma$ is a\ntwisted Baxter permutation, apply {\\sc ExtractBaxter} on~${\\sf P}(\\sigma)$\nto obtain its corresponding Baxter permutation.\n\\medskip\n\nIn the same way, we can compute a bijection between Baxter permutations\nand anti-twisted Baxter permutations using {\\sc ExtractMax} instead of\n{\\sc ExtractMin}. Moreover, these algorithms give a bijection between\ntwisted Baxter permutations and anti-twisted Baxter permutations:\nIf~$\\sigma$ is a twisted (resp. anti-twisted) Baxter permutation, apply\n{\\sc ExtractMax} (resp. {\\sc ExtractMin}) on~${\\sf P}(\\sigma)$ to obtain\nits corresponding anti-twisted (resp. twisted) Baxter permutation.\n\n\\subsection{Definition and correctness of the iterative insertion algorithm}\nIn what follows, we shall revise our ${\\sf P}$-symbol algorithm that we have\npresented in Section~\\ref{subsec:PSymbBaxter} to make it iterative. Indeed,\nwe propose an insertion algorithm such that, for any word~$u$ such that\n${\\sf P}(u) = (T_L, T_R)$ and any letter~${\\tt a}$, the insertion of~${\\tt a}$\ninto~$(T_L, T_R)$ is the pair of twin binary trees~${\\sf P}(u{\\tt a})$. This,\nbesides being in agreement with the usual Robinson-Schensted-like algorithms,\nhas the merit to allow to compute in the Baxter monoid. Indeed, this gives\na simple way to compute the concatenation of two words~$u$ and~$v$ under\nthe Baxter congruence simply by inserting the letters of the word~$uv$\ninto the pair~$(\\perp, \\perp)$. Note that one can compute the\nproduct of two pairs of twin binary trees~$(T_L, T_R)$ and~$(T'_L, T'_R)$\nby computing a word~$u'$ that belongs to the ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class\nof~$(T'_L, T'_R)$ by applying the algorithm {\\sc ExtractMin} (or\n{\\sc ExtractBaxter}) with~$(T'_L, T'_R)$ as input, and then, by inserting\nthe letters of~$u'$ from left to right into~$(T_L, T_R)$.\n\n\\subsubsection{Root insertion in binary search trees}\nLet~$T$ be an $A$-labeled right binary search tree and~${\\tt b}$ a letter of~$A$.\nThe \\emph{lower restricted binary tree} of~$T$ compared to~${\\tt b}$,\nnamely~$T_{\\leq {\\tt b}}$, is the right binary search tree uniquely made of\nthe nodes~$x$ of~$T$ labeled by letters~${\\tt a}$ satisfying~${\\tt a} \\leq {\\tt b}$\nand such that for all nodes~$x$ and~$y$ of~$T_{\\leq {\\tt b}}$, if~$x$ is\nancestor of~$y$ in~$T_{\\leq {\\tt b}}$, then~$x$ is also ancestor of~$y$ in~$T$.\nIn the same way, we define the \\emph{higher restricted binary tree}\nof~$T$ compared to~${\\tt b}$, namely~$T_{> {\\tt b}}$ (see Figure~\\ref{fig:ExempleABRestreints}).\n\\begin{figure}[ht]\n    \\centering\n    \\begin{equation*}\n        \\begin{split}\n        \\scalebox{.35}{\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqFonce](0)at(0,-2){$1$};\n            \\node[Noeud,EtiqFonce](1)at(1,-1){$1$};\n            \\draw[Arete](1)--(0);\n            \\node[Noeud,EtiqFonce](2)at(2,-3){$2$};\n            \\node[Noeud,EtiqClair](3)at(3,-4){$3$};\n            \\draw[Arete](2)--(3);\n            \\node[Noeud,EtiqClair](4)at(4,-2){$3$};\n            \\draw[Arete](4)--(2);\n            \\draw[Arete](1)--(4);\n            \\node[Noeud,EtiqClair](5)at(5,0){$4$};\n            \\draw[Arete](5)--(1);\n            \\node[Noeud,EtiqClair](6)at(6,-1){$5$};\n            \\draw[Arete](5)--(6);\n        \\end{tikzpicture}}\n        \\end{split}\n        \\qquad\n        \\begin{split}\n        \\scalebox{.35}{\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqFonce](0)at(0,-1){$1$};\n            \\node[Noeud,EtiqFonce](1)at(1,0){$1$};\n            \\draw[Arete](1)--(0);\n            \\node[Noeud,EtiqFonce](2)at(2,-1){$2$};\n            \\draw[Arete](1)--(2);\n        \\end{tikzpicture}}\n        \\end{split}\n        \\qquad\n        \\begin{split}\n        \\scalebox{.35}{\n        \\begin{tikzpicture}\n            \\node[Noeud,EtiqClair](0)at(0,-2){$3$};\n            \\node[Noeud,EtiqClair](1)at(1,-1){$3$};\n            \\draw[Arete](1)--(0);\n            \\node[Noeud,EtiqClair](2)at(2,0){$4$};\n            \\draw[Arete](2)--(1);\n            \\node[Noeud,EtiqClair](3)at(3,-1){$5$};\n            \\draw[Arete](2)--(3);\n        \\end{tikzpicture}}\n        \\end{split}\n    \\end{equation*}\n    \\caption{A right binary search tree~$T$, $T_{\\leq 2}$ and~$T_{>2}$.}\n    \\label{fig:ExempleABRestreints}\n\\end{figure}\n\\medskip\n\nLet~$T$ be an $A$-labeled right binary search tree and~${\\tt a}$ a letter\nof~$A$. The \\emph{root insertion} of~${\\tt a}$ into~$T$ consists in modifying~$T$\nso that the root of~$T$ is a new node labeled by~${\\tt a}$, its left subtree\nis~$T_{\\leq {\\tt a}}$ and its right subtree is~$T_{> {\\tt a}}$.\n\n\\subsubsection{The iterative insertion algorithm}\n\n\\begin{Definition} \\label{def:BaxterPQSymboleIt}\n    Let~$(T_L, T_R)$ be an $A$-labeled pair of twin binary search trees\n    and~${\\tt a}$ be a letter. The \\emph{insertion} of~${\\tt a}$ into~$(T_L, T_R)$\n    consists in making a leaf insertion of~${\\tt a}$ into~$T_L$ and a root\n    insertion of~${\\tt a}$ into~$T_R$. The \\emph{iterative Baxter ${\\sf P}$-symbol}\n    (or simply \\emph{iterative ${\\sf P}$-symbol} if the context is clear)\n    of a word~$u \\in A^*$ is the pair ${\\sf P}(u) = (T_L, T_R)$ computed\n    by iteratively inserting the letters of~$u$, from left to right, into\n    $(\\perp, \\perp)$. The \\emph{iterative Baxter ${\\sf Q}$-symbol}\n    (or simply \\emph{iterative ${\\sf Q}$-symbol} if the context is clear)\n    of~$u \\in A^*$ is the pair ${\\sf Q}(u) = (S_L, S_R)$ of same shape\n    as~${\\sf P}(u)$ and such that each node is labeled by its date of\n    creation in~${\\sf P}(u)$.\n\\end{Definition}\n\nFigure~\\ref{fig:ExemplePQSymbole} shows, step by step, the computation\nof the iterative Baxter ${\\sf P}$ and ${\\sf Q}$-symbols of a word.\n\\begin{figure}[ht]\n    \\centering\n    \\begin{equation*}\n    \\begin{split}\\perp \\perp\\end{split}\n    \\begin{split}\\quad \\xrightarrow{\\hspace{.5em}2\\hspace{.5em}} \\quad \\end{split}\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqFonce](0)at(0,0){$2$};\n    \\end{tikzpicture}}\n    \\end{split}\n    \\enspace\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqFonce](0)at(0,0){$2$};\n    \\end{tikzpicture}}\n    \\end{split}\n    \\begin{split}\\quad \\xrightarrow{\\hspace{.5em}4\\hspace{.5em}} \\quad \\end{split}\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0.0,0){$2$};\n        \\node[Noeud,EtiqFonce](1)at(1.0,-1){$4$};\n        \\draw[Arete](0)--(1);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\enspace\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0,-1){$2$};\n        \\node[Noeud,EtiqFonce](1)at(1,0){$4$};\n        \\draw[Arete](1)--(0);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\begin{split}\\quad \\xrightarrow{\\hspace{.5em}1\\hspace{.5em}} \\quad \\end{split}\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqFonce](0)at(0,-1){$1$};\n        \\node[Noeud,EtiqClair](1)at(1,0){$2$};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud,EtiqClair](2)at(2,-1){$4$};\n        \\draw[Arete](1)--(2);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\enspace\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqFonce](0)at(0,0){$1$};\n        \\node[Noeud,EtiqClair](1)at(1,-2){$2$};\n        \\node[Noeud,EtiqClair](2)at(2,-1){$4$};\n        \\draw[Arete](2)--(1);\n        \\draw[Arete](0)--(2);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\end{equation*}\n    \\begin{equation*}\n    \\begin{split}\\xrightarrow{\\hspace{.5em}5\\hspace{.5em}} \\quad \\end{split}\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0,-1){$1$};\n        \\node[Noeud,EtiqClair](1)at(1,0){$2$};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud,EtiqClair](2)at(2,-1){$4$};\n        \\node[Noeud,EtiqFonce](3)at(3,-2){$5$};\n        \\draw[Arete](2)--(3);\n        \\draw[Arete](1)--(2);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\enspace\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0,-1){$1$};\n        \\node[Noeud,EtiqClair](1)at(1,-3){$2$};\n        \\node[Noeud,EtiqClair](2)at(2,-2){$4$};\n        \\draw[Arete](2)--(1);\n        \\draw[Arete](0)--(2);\n        \\node[Noeud,EtiqFonce](3)at(3,0){$5$};\n        \\draw[Arete](3)--(0);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\begin{split}\\quad \\xrightarrow{\\hspace{.5em}2\\hspace{.5em}} \\quad \\end{split}\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0.0,-1){$1$};\n        \\node[Noeud,EtiqClair](1)at(1.0,0){$2$};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud,EtiqFonce](2)at(2.0,-2){$2$};\n        \\node[Noeud,EtiqClair](3)at(3.0,-1){$4$};\n        \\draw[Arete](3)--(2);\n        \\node[Noeud,EtiqClair](4)at(4.0,-2){$5$};\n        \\draw[Arete](3)--(4);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\enspace\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0.0,-1){$1$};\n        \\node[Noeud,EtiqClair](1)at(1.0,-2){$2$};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud,EtiqFonce](2)at(2.0,0){$2$};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud,EtiqClair](3)at(3.0,-2){$4$};\n        \\node[Noeud,EtiqClair](4)at(4.0,-1){$5$};\n        \\draw[Arete](4)--(3);\n        \\draw[Arete](2)--(4);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\end{equation*}\n    \\begin{equation*}\n    \\begin{split}\\xrightarrow{\\hspace{.5em}5\\hspace{.5em}} \\quad \\end{split}\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0.0,-1){$1$};\n        \\node[Noeud,EtiqClair](1)at(1.0,0){$2$};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud,EtiqClair](2)at(2.0,-2){$2$};\n        \\node[Noeud,EtiqClair](3)at(3.0,-1){$4$};\n        \\draw[Arete](3)--(2);\n        \\node[Noeud,EtiqClair](4)at(4.0,-2){$5$};\n        \\node[Noeud,EtiqFonce](5)at(5.0,-3){$5$};\n        \\draw[Arete](4)--(5);\n        \\draw[Arete](3)--(4);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\enspace\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0.0,-2){$1$};\n        \\node[Noeud,EtiqClair](1)at(1.0,-3){$2$};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud,EtiqClair](2)at(2.0,-1){$2$};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud,EtiqClair](3)at(3.0,-3){$4$};\n        \\node[Noeud,EtiqClair](4)at(4.0,-2){$5$};\n        \\draw[Arete](4)--(3);\n        \\draw[Arete](2)--(4);\n        \\node[Noeud,EtiqFonce](5)at(5.0,0){$5$};\n        \\draw[Arete](5)--(2);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\begin{split}\\xrightarrow{\\hspace{.5em}3\\hspace{.5em}} \\quad \\end{split}\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0.0,-1){$1$};\n        \\node[Noeud,EtiqClair](1)at(1.0,0){$2$};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud,EtiqClair](2)at(2.0,-2){$2$};\n        \\node[Noeud,EtiqFonce](3)at(3.0,-3){$3$};\n        \\draw[Arete](2)--(3);\n        \\node[Noeud,EtiqClair](4)at(4.0,-1){$4$};\n        \\draw[Arete](4)--(2);\n        \\node[Noeud,EtiqClair](5)at(5.0,-2){$5$};\n        \\node[Noeud,EtiqClair](6)at(6.0,-3){$5$};\n        \\draw[Arete](5)--(6);\n        \\draw[Arete](4)--(5);\n        \\draw[Arete](1)--(4);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\enspace\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0.0,-2){$1$};\n        \\node[Noeud,EtiqClair](1)at(1.0,-3){$2$};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud,EtiqClair](2)at(2.0,-1){$2$};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud,EtiqFonce](3)at(3.0,0){$3$};\n        \\draw[Arete](3)--(2);\n        \\node[Noeud,EtiqClair](4)at(4.0,-3){$4$};\n        \\node[Noeud,EtiqClair](5)at(5.0,-2){$5$};\n        \\draw[Arete](5)--(4);\n        \\node[Noeud,EtiqClair](6)at(6.0,-1){$5$};\n        \\draw[Arete](6)--(5);\n        \\draw[Arete](3)--(6);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\begin{split} = {\\sf P}(u) \\end{split}\n    \\end{equation*}\n    %\n    \\vspace{2em}\n    %\n    \\begin{equation*}\n    \\begin{split}\\perp \\perp \\end{split}\n    \\begin{split}\\quad \\xrightarrow{\\hspace{.5em}2\\hspace{.5em}} \\quad \\end{split}\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqFonce](0)at(0,0){$1$};\n    \\end{tikzpicture}}\n    \\end{split}\n    \\enspace\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqFonce](0)at(0,0){$1$};\n    \\end{tikzpicture}}\n    \\end{split}\n    \\begin{split}\\quad \\xrightarrow{\\hspace{.5em}4\\hspace{.5em}} \\quad \\end{split}\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0.0,0){$1$};\n        \\node[Noeud,EtiqFonce](1)at(1.0,-1){$2$};\n        \\draw[Arete](0)--(1);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\enspace\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0,-1){$1$};\n        \\node[Noeud,EtiqFonce](1)at(1,0){$2$};\n        \\draw[Arete](1)--(0);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\begin{split}\\quad \\xrightarrow{\\hspace{.5em}1\\hspace{.5em}} \\quad \\end{split}\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqFonce](0)at(0,-1){$3$};\n        \\node[Noeud,EtiqClair](1)at(1,0){$1$};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud,EtiqClair](2)at(2,-1){$2$};\n        \\draw[Arete](1)--(2);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\enspace\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqFonce](0)at(0,0){$3$};\n        \\node[Noeud,EtiqClair](1)at(1,-2){$1$};\n        \\node[Noeud,EtiqClair](2)at(2,-1){$2$};\n        \\draw[Arete](2)--(1);\n        \\draw[Arete](0)--(2);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\end{equation*}\n    \\begin{equation*}\n    \\begin{split}\\xrightarrow{\\hspace{.5em}5\\hspace{.5em}} \\quad \\end{split}\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0,-1){$3$};\n        \\node[Noeud,EtiqClair](1)at(1,0){$1$};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud,EtiqClair](2)at(2,-1){$2$};\n        \\node[Noeud,EtiqFonce](3)at(3,-2){$4$};\n        \\draw[Arete](2)--(3);\n        \\draw[Arete](1)--(2);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\enspace\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0,-1){$3$};\n        \\node[Noeud,EtiqClair](1)at(1,-3){$1$};\n        \\node[Noeud,EtiqClair](2)at(2,-2){$2$};\n        \\draw[Arete](2)--(1);\n        \\draw[Arete](0)--(2);\n        \\node[Noeud,EtiqFonce](3)at(3,0){$4$};\n        \\draw[Arete](3)--(0);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\begin{split}\\quad \\xrightarrow{\\hspace{.5em}2\\hspace{.5em}} \\quad \\end{split}\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0.0,-1){$3$};\n        \\node[Noeud,EtiqClair](1)at(1.0,0){$1$};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud,EtiqFonce](2)at(2.0,-2){$5$};\n        \\node[Noeud,EtiqClair](3)at(3.0,-1){$2$};\n        \\draw[Arete](3)--(2);\n        \\node[Noeud,EtiqClair](4)at(4.0,-2){$4$};\n        \\draw[Arete](3)--(4);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\enspace\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0.0,-1){$3$};\n        \\node[Noeud,EtiqClair](1)at(1.0,-2){$1$};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud,EtiqFonce](2)at(2.0,0){$5$};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud,EtiqClair](3)at(3.0,-2){$2$};\n        \\node[Noeud,EtiqClair](4)at(4.0,-1){$4$};\n        \\draw[Arete](4)--(3);\n        \\draw[Arete](2)--(4);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\end{equation*}\n    \\begin{equation*}\n    \\begin{split}\\xrightarrow{\\hspace{.5em}5\\hspace{.5em}} \\quad \\end{split}\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0.0,-1){$3$};\n        \\node[Noeud,EtiqClair](1)at(1.0,0){$1$};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud,EtiqClair](2)at(2.0,-2){$5$};\n        \\node[Noeud,EtiqClair](3)at(3.0,-1){$2$};\n        \\draw[Arete](3)--(2);\n        \\node[Noeud,EtiqClair](4)at(4.0,-2){$4$};\n        \\node[Noeud,EtiqFonce](5)at(5.0,-3){$6$};\n        \\draw[Arete](4)--(5);\n        \\draw[Arete](3)--(4);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\enspace\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0.0,-2){$3$};\n        \\node[Noeud,EtiqClair](1)at(1.0,-3){$1$};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud,EtiqClair](2)at(2.0,-1){$5$};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud,EtiqClair](3)at(3.0,-3){$2$};\n        \\node[Noeud,EtiqClair](4)at(4.0,-2){$4$};\n        \\draw[Arete](4)--(3);\n        \\draw[Arete](2)--(4);\n        \\node[Noeud,EtiqFonce](5)at(5.0,0){$6$};\n        \\draw[Arete](5)--(2);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\begin{split}\\xrightarrow{\\hspace{.5em}3\\hspace{.5em}} \\quad \\end{split}\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0.0,-1){$3$};\n        \\node[Noeud,EtiqClair](1)at(1.0,0){$1$};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud,EtiqClair](2)at(2.0,-2){$5$};\n        \\node[Noeud,EtiqFonce](3)at(3.0,-3){$7$};\n        \\draw[Arete](2)--(3);\n        \\node[Noeud,EtiqClair](4)at(4.0,-1){$2$};\n        \\draw[Arete](4)--(2);\n        \\node[Noeud,EtiqClair](5)at(5.0,-2){$4$};\n        \\node[Noeud,EtiqClair](6)at(6.0,-3){$6$};\n        \\draw[Arete](5)--(6);\n        \\draw[Arete](4)--(5);\n        \\draw[Arete](1)--(4);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\enspace\n    \\begin{split}\n    \\scalebox{.35}{\n    \\begin{tikzpicture}\n        \\node[Noeud,EtiqClair](0)at(0.0,-2){$3$};\n        \\node[Noeud,EtiqClair](1)at(1.0,-3){$1$};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud,EtiqClair](2)at(2.0,-1){$5$};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud,EtiqFonce](3)at(3.0,0){$7$};\n        \\draw[Arete](3)--(2);\n        \\node[Noeud,EtiqClair](4)at(4.0,-3){$2$};\n        \\node[Noeud,EtiqClair](5)at(5.0,-2){$4$};\n        \\draw[Arete](5)--(4);\n        \\node[Noeud,EtiqClair](6)at(6.0,-1){$6$};\n        \\draw[Arete](6)--(5);\n        \\draw[Arete](3)--(6);\n    \\end{tikzpicture}}\n    \\end{split}\n    \\begin{split} = {\\sf Q}(u) \\end{split}\n    \\end{equation*}\n    \\caption{Steps of the computation of the ${\\sf P}$-symbol and the\n    ${\\sf Q}$-symbol of~$u := 2415253$.}\n    \\label{fig:ExemplePQSymbole}\n\\end{figure}\n\n\\subsubsection{Correctness of the iterative insertion algorithm}\nTo show that the iterative version of the Baxter ${\\sf P}$-symbol computes\nthe same labeled pair of twin binary trees than its non-iterative version,\nwe need the following lemma.\n\\begin{Lemme} \\label{lem:SensInsertion}\n    Let $u \\in A^*$. Let~$T$ be the right binary search tree obtained by\n    root insertions of the letters of~$u$, from left to right. Let~$T'$\n    be the right binary search tree obtained by leaf insertions of the\n    letters of~$u$, from right to left. Then,~$T = T'$.\n\\end{Lemme}\n\\begin{proof}\n    Let us proceed by induction on~$|u|$. If $u = \\epsilon$, the lemma is\n    satisfied. Otherwise, assume that~$u = v {\\tt a}$ where~${\\tt a} \\in A$.\n    Let~$S$ be the right binary search tree obtained by root insertions of the\n    letters of~$v$ from left to right. By induction hypothesis,~$S$ also is\n    the right binary tree obtained by leaf insertions of the letters of~$v$\n    from right to left. The right binary search tree~$T$ obtained by root\n    insertions of~$u$ from left to right satisfies, by definition,\n    $T = S_{\\leq {\\tt a}} \\wedge_{\\tt a} S_{> {\\tt a}}$. The right binary\n    search tree~$T'$ obtained by leaf insertions of~$u$ from right to left\n    satisfies $T' = L' \\wedge_{\\tt a} R'$ where the subtree~$L'$ only depends\n    on the subword $v_{\\leq {\\tt a}} := v_{|]-\\infty, {\\tt a}]}$ and the subtree~$R'$\n    only depends on the subword $v_{> {\\tt a}} := v_{|]{\\tt a}, +\\infty[}$, so that,\n    by induction hypothesis, $L' = S_{\\leq {\\tt a}}$, $R' = S_{> {\\tt a}}$ and\n    thus,~$T = T'$.\n\\end{proof}\n\n\\begin{Proposition} \\label{prop:PSymboleIteratif}\n    For any $u \\in A^*$, the Baxter ${\\sf P}$-symbol of~$u$ and the iterative\n    Baxter ${\\sf P}$-symbol of~$u$ are equal.\n\\end{Proposition}\n\\begin{proof}\n    Let $(T_L, T_R)$ be the ${\\sf P}$-symbol of~$u$ and~$(T'_L, T'_R)$ be\n    the iterative ${\\sf P}$-symbol of~$u$. By definition of these two insertion\n    algorithms, we have~$T_L = T'_L$. Moreover,~$T_R$ is obtained by leaf\n    insertions of the letters of~$u$ from right to left and~$T'_R$ is\n    obtained by root insertions of the letters of~$u$ from left to right.\n    By Lemma~\\ref{lem:SensInsertion}, we have~$T_R = T'_R$.\n\\end{proof}\n\nThe correctness of the iterative version of the ${\\sf Q}$-symbol algorithm\ncomes from the correctness of the iterative ${\\sf P}$-algorithm.\n\n\\section{The Baxter lattice} \\label{sec:TreillisBaxter}\n\n\\subsection{The Baxter lattice congruence}\nRecall that an equivalence relation $\\equiv$ on the elements of a lattice\n$(L, \\wedge, \\vee)$ is a \\emph{lattice congruence} if for all $x, x', y, y' \\in L$,\n$x \\equiv x'$ and~$y \\equiv y'$ imply $x \\wedge y \\equiv x' \\wedge y'$\nand~$x \\vee y \\equiv x' \\vee y'$. The quotient~$L\/_\\equiv$ of~$L$ by~$\\equiv$\nis naturally a lattice. Indeed, by denoting by $\\tau : L \\to L\/_\\equiv$ the\ncanonical projection, the set~$L\/_\\equiv$ is endowed with meet and join\noperations defined by $\\widehat{x} \\wedge \\widehat{y} := \\tau(x \\wedge y)$\nand $\\widehat{x} \\vee \\widehat{y} := \\tau(x \\vee y)$ for all\n$\\widehat{x}, \\widehat{y} \\in L\/_\\equiv$ where~$x$ and~$y$ are any\nelements of~$L$ such that~$\\tau(x) = \\widehat{x}$ and~$\\tau(y) = \\widehat{y}$.\n\\medskip\n\nLattices congruences admit the following very useful order-theoretic\ncharacterization~\\cite{CS98,Rea05}. An equivalence relation~$\\equiv$\non the elements of a lattice~$(L, \\wedge, \\vee)$ seen as a poset~$(L, \\leq)$\nis a lattice congruence is the following three conditions hold.\n\\begin{enumerate}[label = (L\\arabic*)]\n    \\item Every $\\equiv$-equivalence class is an interval of~$L$;\n    \\label{item:LatticeCong1}\n    \\item For any $x, y \\in L$, if $x \\leq y$ then $x {\\!\\downarrow} \\leq y {\\!\\downarrow}$\n    where $x {\\!\\downarrow}$ is the maximal element of the $\\equiv$-equivalence\n    class of~$x$; \\label{item:LatticeCong2}\n    \\item For any $x, y \\in L$, if $x \\leq y$ then $x {\\!\\uparrow} \\leq y {\\!\\uparrow}$\n    where~$x {\\!\\uparrow}$ is the minimal element of the $\\equiv$-equivalence\n    class of~$x$. \\label{item:LatticeCong3}\n\\end{enumerate}\n\nFor any permutation~$\\sigma$, let us denote by~$\\sigma {\\!\\downarrow}$\n(resp.~$\\sigma {\\!\\uparrow}$) the maximal (resp. minimal) permutation of\nthe ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class of~$\\sigma$ for the permutohedron order.\nNote by Proposition~\\ref{prop:EquivBXInter} that~$\\sigma {\\!\\downarrow}$\nand~$\\sigma {\\!\\uparrow}$ are well-defined.\n\n\\begin{Theoreme} \\label{thm:OrdreBaxterMinMax}\n    The Baxter equivalence relation is a lattice congruence of the permutohedron.\n\\end{Theoreme}\n\\begin{proof}\n    By Proposition~\\ref{prop:EquivBXInter}, any Baxter equivalence class\n    of permutations is an interval of the permutohedron, so that\n    \\ref{item:LatticeCong1} checks out. One just has to show that~${\\:\\equiv_{\\operatorname{B}}\\:}$\n    satisfies \\ref{item:LatticeCong2} and \\ref{item:LatticeCong3}.\n\n    Let~$\\sigma$ and~$\\nu$ two permutations such that~$\\sigma {\\:\\leq_{\\operatorname{P}}\\:} \\nu$.\n    Let us show that $\\sigma {\\!\\downarrow} {\\:\\leq_{\\operatorname{P}}\\:} \\nu {\\!\\downarrow}$.\n    It is enough to check the property when~$\\nu = \\sigma s_i$ where~$s_i$\n    is an elementary transposition and~$i$ is not a descent of~$\\sigma$.\n    If~$\\sigma = \\sigma {\\!\\downarrow}$, then\n    $\\sigma {\\!\\downarrow} {\\:\\leq_{\\operatorname{P}}\\:} \\nu {\\:\\leq_{\\operatorname{P}}\\:} \\nu {\\!\\downarrow}$\n    and the property holds. Otherwise, by Lemma~\\ref{lem:DiagHasseEqBX},\n    there exists an elementary transposition~$s_j$ and a permutation~$\\pi$\n    such that~$\\pi$ and~$\\sigma$ are ${\\:\\rightleftarrows_{\\operatorname{B}}\\:}$-adjacent,~$\\pi = \\sigma s_j$\n    and~$\\sigma {\\:\\leq_{\\operatorname{P}}\\:} \\pi$. It then remains to prove that there exists\n    a permutation~$\\mu$ such that~$\\nu {\\:\\equiv_{\\operatorname{B}}\\:} \\mu$ and~$\\pi {\\:\\leq_{\\operatorname{P}}\\:} \\mu$.\n    Indeed, this leads to show, by applying iteratively this reasoning,\n    that~$\\sigma {\\!\\downarrow}$ is smaller than a permutation belonging to the\n    ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class of~$\\nu$ for the permutohedron order and\n    hence, by transitivity, that~$\\sigma {\\!\\downarrow} {\\:\\leq_{\\operatorname{P}}\\:} \\nu {\\!\\downarrow}$.\n    We have four cases:\n    \\begin{enumerate}[label = {\\bf Case \\arabic*:}, fullwidth]\n        \\item If $j \\leq i - 2$, $\\sigma$ is of the form\n        $\\sigma = u \\, {\\tt a} {\\tt b} \\, v \\, {\\tt c} {\\tt d} \\, w$ where~$u$, $v$,\n        and~$w$ are some words and~${\\tt a}$ (resp.~${\\tt c}$) is the $j$-th\n        (resp. $i$-th) letter of~$\\sigma$. One has~${\\tt a} < {\\tt b}$\n        and~${\\tt c} < {\\tt d}$ since~$i$ and~$j$ are not descents of~$\\sigma$.\n        We have $\\nu = u \\, {\\tt a} {\\tt b} \\, v \\, {\\tt d} {\\tt c} \\, w$ and\n        $\\nu s_j = u \\, {\\tt b} {\\tt a} \\, v \\, {\\tt d} {\\tt c} \\, w =: \\mu$. Moreover,\n        since~$\\pi {\\:\\rightleftarrows_{\\operatorname{B}}\\:} \\sigma$, there are some letters~${\\tt x} \\in \\operatorname{Alph}(u)$\n        and ${\\tt y} \\in \\operatorname{Alph}(v \\, {\\tt c} {\\tt d} \\, w)$ such that ${\\tt a} < {\\tt x}, {\\tt y} < {\\tt b}$.\n        Thus,~$\\mu {\\:\\rightleftarrows_{\\operatorname{B}}\\:} \\nu$. Finally, since\n        $\\pi = u \\, {\\tt b} {\\tt a} \\, v \\, {\\tt c} {\\tt d} \\, w$, $\\pi {\\:\\leq_{\\operatorname{P}}\\:} \\mu$,\n        so that~$\\mu$ is appropriate.\n        \\item If $j \\geq i + 2$, this is analogous to the previous case.\n        \\item If $j = i + 1$, $\\sigma$ is of the form\n        $\\sigma = u \\, {\\tt a} {\\tt b} {\\tt c} \\, v$ where~$u$ and~$v$ are some words\n        and~${\\tt a}$ is the $i$-th letter of~$\\sigma$. One has ${\\tt a} < {\\tt b} < {\\tt c}$\n        since~$i$ and~$j$ are not descents of~$\\sigma$. Since~$\\sigma {\\:\\rightleftarrows_{\\operatorname{B}}\\:} \\pi$,\n        there are some letters~${\\tt x} \\in \\operatorname{Alph}(u)$ and~${\\tt y} \\in \\operatorname{Alph}(v)$\n        such that ${\\tt b} < {\\tt x}, {\\tt y} < {\\tt c}$. Thus, since $\\nu = u \\, {\\tt b} {\\tt a} {\\tt c} \\, v$\n        and ${\\tt a} < {\\tt b} < {\\tt x}, {\\tt y} < {\\tt c}$, we have\n        $\\nu s_j = u \\, {\\tt b} {\\tt c} {\\tt a} \\, v {\\:\\rightleftarrows_{\\operatorname{B}}\\:} \\nu$. Moreover,\n        $\\nu s_j s_i = u \\, {\\tt c} {\\tt b} {\\tt a} \\, v =: \\mu$ and\n        $\\nu s_j {\\:\\rightleftarrows_{\\operatorname{B}}\\:} \\nu s_j s_i$ since ${\\tt b} < {\\tt x}, {\\tt y} < {\\tt c}$ and\n        thus,~$\\mu {\\:\\equiv_{\\operatorname{B}}\\:} \\nu$. Finally, since $\\pi = u \\, {\\tt a} {\\tt c} {\\tt b} \\, v$,\n        we have~$\\pi {\\:\\leq_{\\operatorname{P}}\\:} \\mu$, and hence~$\\mu$ is appropriate.\n        \\item If $j = i - 1$, this is analogous to the previous case.\n    \\end{enumerate}\n    Hence, the Baxter equivalence relation satisfies \\ref{item:LatticeCong2}.\n    The proof that~${\\:\\equiv_{\\operatorname{B}}\\:}$ satisfies \\ref{item:LatticeCong3} is analogous.\n\\end{proof}\n\n\\subsection{A lattice structure over the set of pairs of twin binary trees}\nRecall that by Theorem~\\ref{thm:BijectionEquivBXPermuBX}, the Baxter\nequivalence classes of permutations are in correspondence with unlabeled\npairs of twin binary trees. Thus, the quotient of the permutohedron of\norder~$n$ by the Baxter congruence is a lattice $(\\mathcal{T}\\mathcal{B}\\mathcal{T}_n, {\\:\\leq_{\\operatorname{B}}\\:})$ where\nthe \\emph{Baxter order relation}~${\\:\\leq_{\\operatorname{B}}\\:}$ satisfies, for any~$J_0, J_1 \\in \\mathcal{T}\\mathcal{B}\\mathcal{T}_n$,\n\\begin{equation}\n    J_0 {\\:\\leq_{\\operatorname{B}}\\:} J_1 \\qquad \\mbox{if and only if} \\qquad\n    \\substack{\\mbox{there are $\\sigma, \\nu \\in \\mathfrak{S}_n$ such that} \\\\[.3em]\n              \\mbox{$\\sigma {\\:\\leq_{\\operatorname{P}}\\:} \\nu$, ${\\sf P}\\left(\\sigma\\right) = J_0$\n                    and ${\\sf P}\\left(\\nu\\right) = J_1$}.}\n\\end{equation}\nLet us call \\emph{Baxter lattice} the lattice $(\\mathcal{T}\\mathcal{B}\\mathcal{T}_n, {\\:\\leq_{\\operatorname{B}}\\:})$.\nFigure~\\ref{fig:TreillisBaxter} shows the ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence classes\nin the permutohedron of order $4$ that form the Baxter lattice~$(\\mathcal{T}\\mathcal{B}\\mathcal{T}_4, {\\:\\leq_{\\operatorname{B}}\\:})$.\n\\begin{figure}[ht]\n    \\centering\n    \\begin{tikzpicture}\n        \\draw[Classe,rotate = -45] (1.4,-2.1) ellipse (5mm and 12mm);\n        \\draw[Classe,rotate = -45] (2.8,-2.1) ellipse (5mm and 12mm);\n        \\node[Classe] (1234) at (0, 0) {$1234$};\n        \\node[Classe] (2134) at (-2, -1) {$2134$};\n        \\node[Classe] (1243) at (2, -1) {$1243$};\n        \\node[Classe] (1324) at (0, -1) {$1324$};\n        \\node[Classe] (2314) at (-4, -2) {$2314$};\n        \\node[] (2143) at (0, -2) {$2143$};\n        \\node[Classe] (3124) at (-2, -2) {$3124$};\n        \\node[Classe] (1423) at (4, -2) {$1423$};\n        \\node[Classe] (1342) at (2, -2) {$1342$};\n        \\node[Classe] (2341) at (-5, -3) {$2341$};\n        \\node[Classe] (3214) at (-3, -3) {$3214$};\n        \\node[] (2413) at (-1, -3) {$2413$};\n        \\node[] (3142) at (1, -3) {$3142$};\n        \\node[Classe] (4123) at (3, -3) {$4123$};\n        \\node[Classe] (1432) at (5, -3) {$1432$};\n        \\node[Classe] (3241) at (-4, -4) {$3241$};\n        \\node[Classe] (2431) at (-2, -4) {$2431$};\n        \\node[] (3412) at (0, -4) {$3412$};\n        \\node[Classe] (4213) at (2, -4) {$4213$};\n        \\node[Classe] (4132) at (4, -4) {$4132$};\n        \\node[Classe] (3421) at (-2, -5) {$3421$};\n        \\node[Classe] (4231) at (0, -5) {$4231$};\n        \\node[Classe] (4312) at (2, -5) {$4312$};\n        \\node[Classe] (4321) at (0, -6) {$4321$};\n        \\draw[Arete] (1234) -- (2134);\n        \\draw[Arete] (1234) -- (1243);\n        \\draw[Arete] (1234) -- (1324);\n        \\draw[Arete] (2134) -- (2314);\n        \\draw[Arete] (2134) -- (2143);\n        \\draw[Arete] (1243) -- (2143);\n        \\draw[Arete] (1243) -- (1423);\n        \\draw[Arete] (1324) -- (3124);\n        \\draw[Arete] (1324) -- (1342);\n        \\draw[Arete] (2314) -- (2341);\n        \\draw[Arete] (2314) -- (3214);\n        \\draw[Arete] (2143) -- (2413);\n        \\draw[Arete] (3124) -- (3214);\n        \\draw[Arete] (3124) -- (3142);\n        \\draw[Arete] (1423) -- (4123);\n        \\draw[Arete] (1423) -- (1432);\n        \\draw[Arete] (1342) -- (3142);\n        \\draw[Arete] (1342) -- (1432);\n        \\draw[Arete] (2341) -- (3241);\n        \\draw[Arete] (3214) -- (3241);\n        \\draw[Arete] (2341) -- (2431);\n        \\draw[Arete] (2413) -- (2431);\n        \\draw[Arete] (2413) -- (4213);\n        \\draw[Arete] (3142) -- (3412);\n        \\draw[Arete] (4123) -- (4213);\n        \\draw[Arete] (4123) -- (4132);\n        \\draw[Arete] (1432) -- (4132);\n        \\draw[Arete] (3241) -- (3421);\n        \\draw[Arete] (2431) -- (4231);\n        \\draw[Arete] (3412) -- (3421);\n        \\draw[Arete] (3412) -- (4312);\n        \\draw[Arete] (4213) -- (4231);\n        \\draw[Arete] (4132) -- (4312);\n        \\draw[Arete] (3421) -- (4321);\n        \\draw[Arete] (4231) -- (4321);\n        \\draw[Arete] (4312) -- (4321);\n    \\end{tikzpicture}\n    \\caption{The permutohedron of order~$4$ cut into Baxter equivalence classes.}\n    \\label{fig:TreillisBaxter}\n\\end{figure}\n\n\\subsection{Covering relations of the Baxter lattice}\nLet us describe the covering relations of the lattice $(\\mathcal{T}\\mathcal{B}\\mathcal{T}_n, {\\:\\leq_{\\operatorname{B}}\\:})$\nin terms of operations on pairs of twin binary trees. Consider a Baxter\nequivalence class~$\\widehat{\\sigma}$ of permutations encoded by a pair\nof twin binary trees $(T_L, T_R)$. Let~$\\sigma$ by the maximal element\nof~$\\widehat{\\sigma}$. If~$i$ is a descent of~$\\sigma$, the permutation~$\\sigma s_i$\nis not in~$\\widehat{\\sigma}$, and, by definition of the Baxter lattice,\nthe pair of twin binary trees ${\\sf P}(\\sigma s_i) =: (T'_L, T'_R)$\ncovers~$(T_L, T_R)$. The permutations~$\\sigma$ and~$\\sigma s_i$ satisfy\n\\begin{equation}\n    \\sigma = u \\, {\\tt a} {\\tt d} \\, v \\qquad \\mbox{and} \\qquad\n    \\sigma s_i = u \\, {\\tt d} {\\tt a} \\, v,\n\\end{equation}\nwhere~${\\tt a} < {\\tt d}$. There are three cases whether the factor~$u$ or~$v$\ncontains a letter~${\\tt b}$ satisfying ${\\tt a} < {\\tt b} < {\\tt d}$. Since the quotient\nof the permutohedron by the sylvester congruence is the Tamari\nlattice~\\cite{HNT05} and that covering relations in the Tamari lattice are\nbinary tree rotations, the covering relations of the Baxter lattice are\nthe following:\n\\begin{enumerate}[label = (C\\arabic*)]\n    \\item If there is a letter~${\\tt b}$ in~$v$ such that ${\\tt a} < {\\tt b} < {\\tt d}$,\n    then~$T'_R = T_R$ and~$T'_L$ is obtained from~$T_L$ by performing a\n    left rotation that does not change its canopy;\n    \\item If there is a letter~${\\tt b}$ in~$u$ such that ${\\tt a} < {\\tt b} < {\\tt d}$,\n    then~$T'_L = T_L$ and~$T'_R$ is obtained from~$T_R$ by performing a\n    right rotation that does not change its canopy;\n    \\item If for any letter~${\\tt b}$ of~$u$ and~$v$, one has ${\\tt b} < {\\tt a}$ or\n    ${\\tt d} < {\\tt b}$, then~$T'_L$ (resp.~$T'_R$) is obtained from~$T_L$\n    (resp.~$T_R$) by performing a left (resp. right) rotation that changes\n    its canopy.\n\\end{enumerate}\n\nHence, according to this characterization of the covering relations of\nthe Baxter lattice and the definition of the Tamari lattice, we have,\nfor any pairs of twin binary trees~$(T_L, T_R)$ and~$(T'_L, T'_R)$,\n\\begin{equation} \\label{eq:RelOrdreBX}\n    (T_L, T_R) {\\:\\leq_{\\operatorname{B}}\\:} (T'_L, T'_R) \\qquad \\mbox{if and only if} \\qquad\n    T'_L {\\:\\leq_{\\operatorname{T}}\\:} T_L \\mbox{ and } T_R {\\:\\leq_{\\operatorname{T}}\\:} T'_R .\n\\end{equation}\n\nNote that a right rotation at root~$y$ in a binary tree~$T$ changes its\ncanopy if and only if the right subtree~$B$ of the left child~$x$ of~$y$\nis empty (see Figure~\\ref{fig:Rotation}). Similarly, a left rotation at\nroot~$y$ changes the canopy of~$T$ if and only if the left subtree~$B$\nof~$y$ is empty. Moreover, if~$y$ is the $i$-th node of~$T$, by\nLemma~\\ref{lem:OrientationFeuille}, one can see that~$B$ is the $i$-th\nleaf of~$T$. Hence, the right (resp. left) rotation at root~$y$ changes\nthe orientation of the $i$-th leaf of~$T$ formerly on the right to the\nleft (resp. left to the right).\n\n\\subsection{Twin Tamari diagrams}\nThe purpose of this section is to introduce \\emph{twin Tamari diagrams}.\nThese diagrams are in bijection with pairs of twin binary trees and provide\na useful realization of the Baxter lattice since it appears that testing\nif two twin Tamari diagrams are comparable under the Baxter order relation\nis immediate.\n\n\\subsubsection{Tamari diagrams and the Tamari order relation}\nPallo introduced in~\\cite{Pal86}  words in bijection with binary trees\n(see also~\\cite{Knu06}). We call \\emph{Tamari diagrams} these words and\nto compute the Tamari diagram~$\\operatorname{td}(T)$ of a binary tree~$T$, just label\neach node~$x$ of~$T$ by the number of nodes in the right subtree of~$x$\nand then, consider its inorder reading.\n\\medskip\n\nAny Tamari diagram~$\\delta$ of length~$n$ satisfies the following two\ninequalities:\n\\begin{enumerate}\n    \\item $0 \\leq \\delta_i \\leq n - i$, \\quad for all $1 \\leq i \\leq n$;\n    \\item $\\delta_{i + j} \\leq \\delta_i - j$,\n    \\quad for all $1 \\leq i \\leq n$ and $1 \\leq j \\leq \\delta_i$.\n\\end{enumerate}\n\\medskip\n\nThe main interest of Tamari diagrams is that they offer a very simple way\nto test if two binary trees are comparable in the Tamari lattice~\\cite{Knu06}.\nIndeed, if~$T$ and $T'$ are two binary trees with~$n$ nodes, one has\n\\begin{equation} \\label{eq:ComparaisonDT}\n    T {\\:\\leq_{\\operatorname{T}}\\:} T' \\qquad \\mbox{if and only if} \\qquad\n    \\operatorname{td}(T)_i \\leq \\operatorname{td}(T')_i \\quad \\mbox{for all $1 \\leq i \\leq n$}.\n\\end{equation}\n\n\\subsubsection{Twin Tamari diagrams and the Baxter order relation}\n\n\\begin{Definition} \\label{def:DTD}\n    A \\emph{twin Tamari diagram} of size~$n$ is a\n    pair~$\\left(\\delta^L, \\delta^R\\right)$ such that~$\\delta^L$ and~$\\delta^R$\n    are Tamari diagrams of length~$n$ and for all index $1 \\leq i \\leq n - 1$,\n    exactly one letter among~$\\delta^L_i$ and~$\\delta^R_i$ is zero.\n\\end{Definition}\n\nNote that we can represent any twin Tamari diagram\n$\\delta := \\left(\\delta^L, \\delta^R\\right)$\nin a more compact way by a word~$\\omega(\\delta)$ were\n\\begin{equation}\n    \\omega(\\delta)_i :=\n    \\begin{cases}\n        - \\delta^L_i & \\mbox{if $\\delta^L_i \\ne 0$}, \\\\\n        \\delta^R_i & \\mbox{otherwise},\n    \\end{cases}\n\\end{equation}\nfor all $1 \\leq i \\leq n$ where~$n$ is the size of~$\\delta$. We graphically\nrepresent a twin Tamari diagram~$\\delta$ by drawing for each index~$i$ a\ncolumn of~$|\\omega(\\delta)_i|$ boxes facing up if~$\\omega(\\delta)_i \\geq 0$\nand facing down otherwise. First twin Tamari diagrams are drawn in\nFigure~\\ref{fig:PremiersDTD}.\n\\begin{figure}[ht]\n    \\begin{equation*}\n        \\begin{split}\\epsilon\\end{split}, \\qquad\n        \\begin{split}\\scalebox{.3}{\n        \\begin{tikzpicture}\n            \\draw[Ligne] (-0.5,0)--(0.5,0);\n            \\draw (0,0) node[above] {};\n        \\end{tikzpicture}}\n        \\end{split}, \\qquad\n        \\begin{split}\\scalebox{.3}{\n        \\begin{tikzpicture}\n            \\draw[Ligne] (-0.5,0)--(1.5,0);\n            \\node[CaseBas](0, 0) at (0,-0.5) {};\n            \\draw (0,0) node[above] {};\n            \\draw (1,0) node[above] {};\n        \\end{tikzpicture}}\\end{split}, \\enspace\n        \\begin{split}\\scalebox{.3}{\n        \\begin{tikzpicture}\n            \\draw[Ligne] (-0.5,0)--(1.5,0);\n            \\node[CaseHaut](0, 0) at (0,0.5) {};\n            \\draw (0,1) node[above] {};\n            \\draw (1,0) node[above] {};\n        \\end{tikzpicture}}\n        \\end{split}, \\qquad\n        \\begin{split}\\scalebox{.3}{\n        \\begin{tikzpicture}\n            \\draw[Ligne] (-0.5,0)--(2.5,0);\n            \\node[CaseBas](0, 0) at (0,-0.5) {};\n            \\node[CaseBas](0, 1) at (0,-1.5) {};\n            \\draw (0,0) node[above] {};\n            \\node[CaseBas](1, 0) at (1,-0.5) {};\n            \\draw (1,0) node[above] {};\n            \\draw (2,0) node[above] {};\n        \\end{tikzpicture}}\n        \\end{split}, \\enspace\n        \\begin{split}\\scalebox{.3}{\n        \\begin{tikzpicture}\n            \\draw[Ligne] (-0.5,0)--(2.5,0);\n            \\node[CaseBas](0, 0) at (0,-0.5) {};\n            \\node[CaseBas](0, 1) at (0,-1.5) {};\n            \\draw (0,0) node[above] {};\n            \\node[CaseHaut](1, 0) at (1,0.5) {};\n            \\draw (1,1) node[above] {};\n            \\draw (2,0) node[above] {};\n        \\end{tikzpicture}}\n        \\end{split}, \\enspace\n        \\begin{split}\\scalebox{.3}{\n        \\begin{tikzpicture}\n            \\draw[Ligne] (-0.5,0)--(2.5,0);\n            \\node[CaseBas](0, 0) at (0,-0.5) {};\n            \\draw (0,0) node[above] {};\n            \\node[CaseHaut](1, 0) at (1,0.5) {};\n            \\draw (1,1) node[above] {};\n            \\draw (2,0) node[above] {};\n        \\end{tikzpicture}}\n        \\end{split}, \\enspace\n        \\begin{split}\\scalebox{.3}{\n        \\begin{tikzpicture}\n            \\draw[Ligne] (-0.5,0)--(2.5,0);\n            \\node[CaseHaut](0, 0) at (0,0.5) {};\n            \\draw (0,1) node[above] {};\n            \\node[CaseBas](1, 0) at (1,-0.5) {};\n            \\draw (1,0) node[above] {};\n            \\draw (2,0) node[above] {};\n        \\end{tikzpicture}}\n        \\end{split}, \\enspace\n        \\begin{split}\\scalebox{.3}{\n        \\begin{tikzpicture}\n            \\draw[Ligne] (-0.5,0)--(2.5,0);\n            \\node[CaseHaut](0, 0) at (0,0.5) {};\n            \\node[CaseHaut](0, 1) at (0,1.5) {};\n            \\draw (0,2) node[above] {};\n            \\node[CaseBas](1, 0) at (1,-0.5) {};\n            \\draw (1,0) node[above] {};\n            \\draw (2,0) node[above] {};\n        \\end{tikzpicture}}\n        \\end{split}, \\enspace\n        \\begin{split}\\scalebox{.3}{\n        \\begin{tikzpicture}\n            \\draw[Ligne] (-0.5,0)--(2.5,0);\n            \\node[CaseHaut](0, 0) at (0,0.5) {};\n            \\node[CaseHaut](0, 1) at (0,1.5) {};\n            \\draw (0,2) node[above] {};\n            \\node[CaseHaut](1, 0) at (1,0.5) {};\n            \\draw (1,1) node[above] {};\n            \\draw (2,0) node[above] {};\n        \\end{tikzpicture}}\n        \\end{split}\n    \\end{equation*}\n    \\caption{First twin Tamari diagrams of size~$0$, $1$, $2$, and~$3$.}\n    \\label{fig:PremiersDTD}\n\\end{figure}\n\n\\begin{Proposition} \\label{prop:DTDBijectionABJ}\n    For any~$n \\geq 0$, the set of twin Tamari diagrams of size~$n$ is\n    in bijection with the set of pairs of twin binary trees with~$n$ nodes.\n    Moreover, this bijection is expressed as follows: If $J := (T_L, T_R)$\n    is a pair of twin binary trees, the twin Tamari diagram in bijection\n    with~$J$ is~$\\operatorname{ttd}(J) := \\left(\\operatorname{td}(T_L), \\operatorname{td}(T_R)\\right)$.\n\\end{Proposition}\n\\begin{proof}\n    Let us show that the application~$\\operatorname{ttd}$ is well-defined, that is\n    $\\operatorname{ttd}(J) =: \\left(\\delta^L, \\delta^R\\right)$ is a twin Tamari diagram.\n    Fix an index $1 \\leq i \\leq n - 1$. By contradiction, assume first\n    that $\\delta^L_i = \\delta^R_i = 0$. By definition of~$\\operatorname{td}$, this\n    implies that the $i$-th nodes of~$T_L$ and~$T_R$ have no right child.\n    Hence, by Lemma~\\ref{lem:OrientationFeuille}, the $i\\!+\\!1$-st leaves\n    of~$T_L$ and~$T_R$ are attached to its $i$-th nodes and are right-oriented.\n    Since $i \\leq n - 1$, these leaves are not the rightmost leaves of~$T_L$\n    and~$T_R$, implying that~$T_L$ and~$T_R$ have not complementary canopies,\n    and hence that~$(T_L, T_R)$ is not a pair of twin binary trees.\n    Assume now that~$\\delta^L_i \\ne 0$ and~$\\delta^R_i \\ne 0$. By definition\n    of~$\\operatorname{td}$, this implies that the $i$-th nodes of~$T_L$ and~$T_R$ have\n    a right child. Hence, by Lemma~\\ref{lem:OrientationFeuille}, the\n    $i\\!+\\!1$-st leaves of~$T_L$ and~$T_R$ are attached to its $i\\!+\\!1$-st\n    nodes and are left-oriented. This implies again that~$(T_L, T_R)$ is\n    not a pair of twin binary trees. Thus,~$\\operatorname{ttd}$ computes twin Tamari diagrams.\n\n    Now, since~$\\operatorname{td}$ is a bijection between the set of binary trees\n    with~$n$ nodes and Tamari diagrams of size~$n$~\\cite{Pal86}, for any\n    twin Tamari diagram~$\\delta$, there is a unique pair of binary\n    trees~$J$ such that~$\\operatorname{ttd}(J) = \\delta$. Using very similar arguments\n    as above, one can prove that the canopies of the trees of~$J$ are\n    complementary, and hence, that $J$ is a pair of twin binary trees.\n\\end{proof}\n\nFigure~\\ref{fig:ExempleBijABJDTD} shows an example of a pair of twin binary\ntrees with the corresponding twin Tamari diagram.\n\\begin{figure}[ht]\n    \\begin{equation*}\n        \\begin{split}\n        \\scalebox{.25}{\n        \\begin{tikzpicture}\n            \\node[Noeud](0)at(0.0,-2){};\n            \\node[Noeud](1)at(1.0,-1){};\n            \\draw[Arete](1)--(0);\n            \\node[Noeud](2)at(2.0,-2){};\n            \\draw[Arete](1)--(2);\n            \\node[Noeud](3)at(3.0,0){};\n            \\draw[Arete](3)--(1);\n            \\node[Noeud](4)at(4.0,-2){};\n            \\node[Noeud](5)at(5.0,-3){};\n            \\draw[Arete](4)--(5);\n            \\node[Noeud](6)at(6.0,-1){};\n            \\draw[Arete](6)--(4);\n            \\node[Noeud](7)at(7.0,-2){};\n            \\draw[Arete](6)--(7);\n            \\draw[Arete](3)--(6);\n        \\end{tikzpicture}}\n        \\end{split}\n        \\enspace\n        \\begin{split}\n        \\scalebox{.25}{\n        \\begin{tikzpicture}\n            \\node[Noeud](0)at(0.0,-1){};\n            \\node[Noeud](1)at(1.0,-4){};\n            \\node[Noeud](2)at(2.0,-3){};\n            \\draw[Arete](2)--(1);\n            \\node[Noeud](3)at(3.0,-4){};\n            \\draw[Arete](2)--(3);\n            \\node[Noeud](4)at(4.0,-2){};\n            \\draw[Arete](4)--(2);\n            \\draw[Arete](0)--(4);\n            \\node[Noeud](5)at(5.0,0){};\n            \\draw[Arete](5)--(0);\n            \\node[Noeud](6)at(6.0,-2){};\n            \\node[Noeud](7)at(7.0,-1){};\n            \\draw[Arete](7)--(6);\n            \\draw[Arete](5)--(7);\n        \\end{tikzpicture}}\n        \\end{split}\n        \\begin{split}\n        \\qquad \\xrightarrow{\\enspace \\operatorname{ttd} \\enspace} \\qquad\n        \\end{split}\n        (01041010, 40100200)\n        \\begin{split}\n        \\qquad \\simeq \\qquad\n        \\end{split}\n        \\begin{split}\n        \\scalebox{.25}{\n        \\begin{tikzpicture}\n            \\draw[Ligne] (-0.5,0)--(7.5,0);\n            \\node[CaseHaut](0, 0) at (0,0.53) {};\n            \\node[CaseHaut](0, 1) at (0,1.53) {};\n            \\node[CaseHaut](0, 2) at (0,2.53) {};\n            \\node[CaseHaut](0, 3) at (0,3.53) {};\n            \\node[CaseBas](1, 0) at (1,-0.53) {};\n            \\node[CaseHaut](2, 0) at (2,0.53) {};\n            \\node[CaseBas](3, 0) at (3,-0.53) {};\n            \\node[CaseBas](3, 1) at (3,-1.53) {};\n            \\node[CaseBas](3, 2) at (3,-2.53) {};\n            \\node[CaseBas](3, 3) at (3,-3.53) {};\n            \\node[CaseBas](4, 0) at (4,-0.53) {};\n            \\node[CaseHaut](5, 0) at (5,0.53) {};\n            \\node[CaseHaut](5, 1) at (5,1.53) {};\n            \\node[CaseBas](6, 0) at (6,-0.53) {};\n        \\end{tikzpicture}}\n        \\end{split}\n    \\end{equation*}\n    \\caption{A pair of twin binary trees, the corresponding twin Tamari\n    diagram via the bijection~$\\operatorname{ttd}$ and its graphical representation.}\n    \\label{fig:ExempleBijABJDTD}\n\\end{figure}\n\n\\begin{Proposition} \\label{prop:OrdreBaxterDTD}\n    Let~$J_0$ and~$J_1$ two pairs of twin binary trees with~$n$ nodes.\n    We have\n    \\begin{equation}\n        J_0 {\\:\\leq_{\\operatorname{B}}\\:} J_1\n        \\qquad \\mbox{if and only if} \\qquad\n        \\omega\\left(\\operatorname{ttd}\\left(J_0\\right)\\right)_i \\leq\n        \\omega\\left(\\operatorname{ttd}\\left(J_1\\right)\\right)_i \\quad\n        \\mbox{for all $1 \\leq i \\leq n$}.\n    \\end{equation}\n\\end{Proposition}\n\\begin{proof}\n    This result is a direct consequence of the characterization of the\n    Baxter order relation~(\\ref{eq:RelOrdreBX}) using the Tamari order\n    relation, the characterization furnished by~(\\ref{eq:ComparaisonDT})\n    to compare two binary trees in the Tamari lattice with Tamari diagrams,\n    and the bijection between pairs of twin binary trees and Twin Tamari\n    diagrams provided by Proposition~\\ref{prop:DTDBijectionABJ}.\n\\end{proof}\n\nFigure~\\ref{fig:IntervalleABJ} shows an interval of the Baxter lattice.\n\\begin{figure}[ht]\n    \\centering\n    \\scalebox{.16}{\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,2){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,0){};\n        \\node[Noeud](4)at(4,1){};\n        \\draw[Arete](4)--(3);\n        \\draw[Arete](2)--(4);\n        \\node[Noeud](5)at(5,0){};\n        \\node[Noeud](6)at(6,1){};\n        \\draw[Arete](6)--(5);\n        \\node[Noeud](7)at(7,0){};\n        \\draw[Arete](6)--(7);\n        \\node[Noeud](8)at(8,2){};\n        \\draw[Arete](8)--(6);\n        \\node[Noeud](9)at(9,1){};\n        \\draw[Arete](8)--(9);\n        \\node[fit=(0) (1) (2) (3) (4) (5) (6) (7) (8) (9)] (J1) {};\n        \\node[Noeud](10)at(-10,-6){};\n        \\node[Noeud](11)at(-9,-7){};\n        \\draw[Arete](10)--(11);\n        \\node[Noeud](12)at(-8,-5){};\n        \\draw[Arete](12)--(10);\n        \\node[Noeud](13)at(-7,-7){};\n        \\node[Noeud](14)at(-6,-6){};\n        \\draw[Arete](14)--(13);\n        \\draw[Arete](12)--(14);\n        \\node[Noeud](15)at(-4,-6){};\n        \\node[Noeud](16)at(-3,-5){};\n        \\draw[Arete](16)--(15);\n        \\node[Noeud](17)at(-2,-7){};\n        \\node[Noeud](18)at(-1,-6){};\n        \\draw[Arete](18)--(17);\n        \\node[Noeud](19)at(0,-7){};\n        \\draw[Arete](18)--(19);\n        \\draw[Arete](16)--(18);\n        \\node[fit=(10) (11) (12) (13) (14) (15) (16) (17) (18) (19)] (J2) {};\n        \\node[Noeud](20)at(11,-7){};\n        \\node[Noeud](21)at(12,-8){};\n        \\draw[Arete](20)--(21);\n        \\node[Noeud](22)at(13,-6){};\n        \\draw[Arete](22)--(20);\n        \\node[Noeud](23)at(14,-7){};\n        \\draw[Arete](22)--(23);\n        \\node[Noeud](24)at(15,-5){};\n        \\draw[Arete](24)--(22);\n        \\node[Noeud](25)at(16,-7){};\n        \\node[Noeud](26)at(17,-6){};\n        \\draw[Arete](26)--(25);\n        \\node[Noeud](27)at(18,-7){};\n        \\draw[Arete](26)--(27);\n        \\node[Noeud](28)at(19,-5){};\n        \\draw[Arete](28)--(26);\n        \\node[Noeud](29)at(20,-6){};\n        \\draw[Arete](28)--(29);\n        \\node[fit=(20) (21) (22) (23) (24) (25) (26) (27) (28) (29)] (J3) {};\n        \\node[Noeud](30)at(0,-15){};\n        \\node[Noeud](31)at(1,-16){};\n        \\draw[Arete](30)--(31);\n        \\node[Noeud](32)at(2,-14){};\n        \\draw[Arete](32)--(30);\n        \\node[Noeud](33)at(3,-15){};\n        \\draw[Arete](32)--(33);\n        \\node[Noeud](34)at(4,-13){};\n        \\draw[Arete](34)--(32);\n        \\node[Noeud](35)at(5,-14){};\n        \\node[Noeud](36)at(6,-13){};\n        \\draw[Arete](36)--(35);\n        \\node[Noeud](37)at(7,-15){};\n        \\node[Noeud](38)at(8,-14){};\n        \\draw[Arete](38)--(37);\n        \\node[Noeud](39)at(9,-15){};\n        \\draw[Arete](38)--(39);\n        \\draw[Arete](36)--(38);\n        \\node[fit=(30) (31) (32) (33) (34) (35) (36) (37) (38) (39)] (J4) {};\n        \\node[Noeud](40)at(0,-25){};\n        \\node[Noeud](41)at(1,-26){};\n        \\draw[Arete](40)--(41);\n        \\node[Noeud](42)at(2,-24){};\n        \\draw[Arete](42)--(40);\n        \\node[Noeud](43)at(3,-23){};\n        \\draw[Arete](43)--(42);\n        \\node[Noeud](44)at(4,-22){};\n        \\draw[Arete](44)--(43);\n        \\node[Noeud](45)at(5,-23){};\n        \\node[Noeud](46)at(6,-22){};\n        \\draw[Arete](46)--(45);\n        \\node[Noeud](47)at(7,-23){};\n        \\node[Noeud](48)at(8,-24){};\n        \\node[Noeud](49)at(9,-25){};\n        \\draw[Arete](48)--(49);\n        \\draw[Arete](47)--(48);\n        \\draw[Arete](46)--(47);\n        \\node[fit=(40) (41) (42) (43) (44) (45) (46) (47) (48) (49)] (J5) {};\n        \\draw [Arete,line width=6pt] (J1)--(J2);\n        \\draw [Arete,line width=6pt] (J1)--(J3);\n        \\draw [Arete,line width=6pt] (J2)--(J4);\n        \\draw [Arete,line width=6pt] (J3)--(J4);\n        \\draw [Arete,line width=6pt] (J4)--(J5);\n    \\end{tikzpicture}}\n    \\qquad \\qquad\n    \\scalebox{.16}{\n    \\begin{tikzpicture}\n       \n        \\draw[Ligne](-0.5,0)--(4.5,0);\n        \\node[CaseBas](c1) at (0,-0.5) {};\n        \\node[CaseHaut](c2) at (1,0.5) {};\n        \\node[CaseBas](c3) at (2,-0.5) {};\n        \\node[CaseBas](c4) at (2,-1.5) {};\n        \\node[CaseHaut](c5) at (3,0.5) {};\n        \\node[fit=(c1) (c2) (c3) (c4) (c5)] (d1) {};\n       \n        \\draw[Ligne](-8.5,-8)--(-3.5,-8);\n        \\node[CaseBas](c6) at (-8,-8.5) {};\n        \\node[CaseHaut](c7) at (-7,-7.5) {};\n        \\node[CaseHaut](c8) at (-7,-6.5) {};\n        \\node[CaseHaut](c9) at (-7,-5.5) {};\n        \\node[CaseBas](c10) at (-6,-8.5) {};\n        \\node[CaseBas](c11) at (-6,-9.5) {};\n        \\node[CaseHaut](c12) at (-5,-7.5) {};\n        \\node[fit=(c6) (c7) (c8) (c9) (c10) (c11) (c12)] (d2) {};\n       \n        \\draw[Ligne](7.5,-8)--(12.5,-8);\n        \\node[CaseBas](c13) at (8,-8.5) {};\n        \\node[CaseHaut](c14) at (9,-7.5) {};\n        \\node[CaseBas](c15) at (10,-8.5) {};\n        \\node[CaseHaut](c16) at (11,-7.5) {};\n        \\node[fit=(c13) (c14) (c15) (c16)] (d3) {};\n       \n        \\draw[Ligne](-0.5,-16)--(4.5,-16);\n        \\node[CaseBas](c17) at (0,-16.5) {};\n        \\node[CaseHaut](c18) at (1,-15.5) {};\n        \\node[CaseHaut](c19) at (1,-14.5) {};\n        \\node[CaseHaut](c20) at (1,-13.5) {};\n        \\node[CaseBas](c21) at (2,-16.5) {};\n        \\node[CaseHaut](c22) at (3,-15.5) {};\n        \\node[fit=(c17) (c18) (c19) (c20) (c21) (c22)] (d4) {};\n       \n        \\draw[Ligne](-0.5,-24)--(4.5,-24);\n        \\node[CaseBas](c23) at (0,-24.5) {};\n        \\node[CaseHaut](c24) at (1,-23.5) {};\n        \\node[CaseHaut](c25) at (1,-22.5) {};\n        \\node[CaseHaut](c26) at (1,-21.5) {};\n        \\node[CaseHaut](c27) at (2,-23.5) {};\n        \\node[CaseHaut](c28) at (2,-22.5) {};\n        \\node[CaseHaut](c29) at (3,-23.5) {};\n        \\node[fit=(c23) (c24) (c25) (c26) (c27) (c28) (c29)] (d5) {};\n       \n        \\draw [Arete,line width=6pt] (d1)--(d2);\n        \\draw [Arete,line width=6pt] (d1)--(d3);\n        \\draw [Arete,line width=6pt] (d2)--(d4);\n        \\draw [Arete,line width=6pt] (d3)--(d4);\n        \\draw [Arete,line width=6pt] (d4)--(d5);\n    \\end{tikzpicture}}\n    \\caption{An interval of the Baxter lattice of order~$5$ where vertices\n    are seen as pairs of twin binary trees and as Twin Tamari diagrams.}\n    \\label{fig:IntervalleABJ}\n\\end{figure}\n\n\\section{The Hopf algebra of pairs of twin binary trees} \\label{sec:AlgebreHopfBaxter}\n\nIn the sequel, all the algebraic structures have a field of characteristic\nzero $\\mathbb{K}$ as ground field.\n\n\\subsection{\\texorpdfstring{The Hopf algebra ${\\bf FQSym}$ and construction\n                            of Hopf subalgebras}\n           {The Hopf algebra FQSym and construction of Hopf subalgebras}}\n\n\\subsubsection{\\texorpdfstring{The Hopf algebra ${\\bf FQSym}$}\n                              {The Hopf algebra FQSym}}\nRecall that the family $\\left\\{{\\bf F}_\\sigma\\right\\}_{\\sigma \\in \\mathfrak{S}}$\nforms the \\emph{fundamental} basis of~${\\bf FQSym}$, the Hopf algebra of Free\nquasi-symmetric functions~\\cite{MR95, DHT02}. Its product and its coproduct\nare defined by\n\\begin{equation}\n    {\\bf F}_\\sigma \\cdot {\\bf F}_\\nu :=\n    \\sum_{\\pi \\; \\in \\; \\sigma \\; \\cshuffle \\; \\nu} {\\bf F}_\\pi,\n\\end{equation}\n\\begin{equation}\n    \\Delta \\left({\\bf F}_\\sigma\\right) :=\n    \\sum_{\\sigma = uv} {\\bf F}_{\\operatorname{std}(u)} \\otimes {\\bf F}_{\\operatorname{std}(v)}.\n\\end{equation}\nFor example,\n\\begin{equation}\\begin{split}\n    {\\bf F}_{132} \\cdot {\\bf F}_{12} & =\n        {\\bf F}_{13245} + {\\bf F}_{13425} + {\\bf F}_{13452} + {\\bf F}_{14325} + {\\bf F}_{14352} \\\\\n    & + {\\bf F}_{14532} + {\\bf F}_{41325} + {\\bf F}_{41352} + {\\bf F}_{41532} + {\\bf F}_{45132},\n\\end{split}\\end{equation}\n\n\\begin{equation}\\begin{split}\n    \\Delta\\left({\\bf F}_{35142}\\right) & =\n        1 \\otimes {\\bf F}_{35142} + {\\bf F}_{1} \\otimes {\\bf F}_{4132} + {\\bf F}_{12} \\otimes {\\bf F}_{132} \\\\\n    & + {\\bf F}_{231} \\otimes {\\bf F}_{21} + {\\bf F}_{2413} \\otimes {\\bf F}_{1} + {\\bf F}_{35142} \\otimes 1.\n\\end{split}\\end{equation}\n\\medskip\n\nSet ${\\bf G}_\\sigma := {\\bf F}_{\\sigma^{-1}}$. Recall that~${\\bf FQSym}$ is isomorphic to\nits dual~${\\bf FQSym}^\\star$ through the map $\\psi : {\\bf FQSym} \\to {\\bf FQSym}^\\star$\ndefined by $\\psi\\left({\\bf F}_\\sigma\\right) := {\\bf F}_{\\sigma^{-1}}^\\star = {\\bf G}_\\sigma^\\star$.\n\\medskip\n\nRecall also that~${\\bf FQSym}$ admits a \\emph{polynomial realization}~\\cite{DHT02},\nthat is an injective algebra morphism\n$r_A : {\\bf FQSym} \\hookrightarrow \\mathbb{K} \\langle A \\rangle$. Furthermore, this map\nshould be compatible with the coalgebra structure in the sense that the\ncoproduct of an element can be computed by taking its image by~$r_A$, and\nthen by applying the \\emph{alphabet doubling trick}~\\cite{DHT02,Hiv07}.\nThis map is defined by\n\\begin{equation} \\label{eq:RealisationFQSym}\n    r_A\\left({\\bf G}_\\sigma\\right) :=\n    \\sum_{\\substack{u \\; \\in \\; A^* \\\\ \\operatorname{std}(u) = \\sigma}} u.\n\\end{equation}\nFor example,\n\\begin{align}\n    r_A\\left({\\bf G}_\\epsilon\\right) & = 1, \\\\\n    r_A\\left({\\bf G}_1\\right)        & = \\sum_i a_i = a_1 + a_2 + a_3 + \\cdots, \\\\\n    r_A\\left({\\bf G}_{231}\\right)    & = \\sum_{k < i \\leq j} a_i a_j a_k\n        = a_2 a_2 a_1 + a_2 a_3 a_1 + a_2 a_4 a_1 + \\cdots.\n\\end{align}\n\n\\subsubsection{\\texorpdfstring{Construction of Hopf subalgebras of ${\\bf FQSym}$}\n                              {Construction of Hopf subalgebras of FQSym}}\nIf~$\\equiv$ is an equivalence relation on~$\\mathfrak{S}$ and~$\\sigma \\in \\mathfrak{S}$,\nlet us denote by~$\\widehat{\\sigma}$ the $\\equiv$-equivalence class of~$\\sigma$.\n\\medskip\n\nThe following theorem contained in an unpublished note of Hivert and\nNzeutchap~\\cite{HN07} (see also~\\cite{DHT02,Hiv07}) shows that an equivalence\nrelation on~$A^*$ satisfying some properties can be used to define Hopf\nsubalgebras of~${\\bf FQSym}$:\n\\begin{Theoreme} \\label{thm:HivertJanvier}\n    Let~$\\equiv$ be an equivalence relation defined on~$A^*$. If~$\\equiv$\n    is a congruence, compatible with the restriction of alphabet intervals\n    and compatible with the destandardization process, then the family\n    $\\left\\{{\\bf P}_{\\widehat{\\sigma}}\\right\\}_{\\widehat{\\sigma} \\in \\mathfrak{S}\/_\\equiv}$\n    defined by\n    \\begin{equation} \\label{eq:EquivFQSym}\n        {\\bf P}_{\\widehat{\\sigma}} :=\n        \\sum_{\\nu \\; \\in \\; \\widehat{\\sigma}} {\\bf F}_\\nu,\n    \\end{equation}\n    spans a Hopf subalgebra of~${\\bf FQSym}$.\n\\end{Theoreme}\nThe compatibility with the destandardization process and with the restriction\nof alphabet intervals imply that for any~${\\bf F}_\\pi$ appearing in a product\n${\\bf P}_{\\widehat{\\sigma}} \\cdot {\\bf P}_{\\widehat{\\nu}}$ and any\npermutation~$\\pi' \\equiv \\pi$, ${\\bf F}_{\\pi'}$ also appears in the product.\nMoreover, the compatibility with the destandardization process and the\nfact that~$\\equiv$ is a congruence imply that for any ${\\bf F}_\\sigma \\otimes {\\bf F}_\\nu$\nappearing in a coproduct~$\\Delta\\left({\\bf P}_{\\widehat{\\pi}}\\right)$ and\nany permutations~$\\sigma' \\equiv \\sigma$ and~$\\nu' \\equiv \\nu$,\n${\\bf F}_{\\sigma'} \\otimes {\\bf F}_{\\nu'}$ also appears in the coproduct.\n\\medskip\n\nIn the sequel, we shall call\n$\\left\\{{\\bf P}_{\\widehat{\\sigma}}\\right\\}_{\\widehat{\\sigma} \\in \\mathfrak{S}\/_\\equiv}$\nthe \\emph{fundamental basis} of the corresponding Hopf subalgebra of~${\\bf FQSym}$.\n\n\\subsection{\\texorpdfstring{Construction of the Hopf algebra ${\\bf Baxter}$}\n                           {Construction of the Hopf algebra Baxter}}\nBy Theorem~\\ref{thm:BijectionEquivBXPermuBX}, the ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence\nclasses of permutations can be encoded by unlabeled pairs of twin binary\ntrees. Moreover, in the sequel, the ${\\sf P}$-symbols of permutations are\nregarded as unlabeled pairs of twin binary trees since there is only one\nway to label a pair of twin binary trees with a permutation so that it is\na pair of twin binary search trees. Hence, in our graphical representations\nwe will only represent their shape.\n\\medskip\n\nSince by definition~${\\:\\equiv_{\\operatorname{B}}\\:}$ is a congruence, since by\nPropositions~\\ref{prop:CompDestd} and~\\ref{prop:CompRestrSegmAlph},~${\\:\\equiv_{\\operatorname{B}}\\:}$\nsatisfies the conditions of Theorem~\\ref{thm:HivertJanvier}, and since by\nTheorem~\\ref{thm:PSymboleClasses}, the permutations~$\\sigma$ such\nthat~${\\sf P}(\\sigma) = J$ coincide with the Baxter equivalence class\nrepresented by the pair of twin binary trees~$J$, we have the following\ntheorem.\n\\begin{Theoreme} \\label{thm:AlgebreHopfBaxter}\n    The family $\\left\\{{\\bf P}_J\\right\\}_{J \\in \\mathcal{T}\\mathcal{B}\\mathcal{T}}$ defined by\n    \\begin{equation}\n        {\\bf P}_J :=\n        \\sum_{\\substack{\\sigma \\; \\in \\; \\mathfrak{S} \\\\ {\\sf P}(\\sigma) = J}}\n        {\\bf F}_\\sigma,\n    \\end{equation}\n    spans a Hopf subalgebra of~${\\bf FQSym}$, namely the Hopf algebra~${\\bf Baxter}$.\n\\end{Theoreme}\n\nFor example,\n\\begin{align}\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-1){};\n        \\draw[Arete](0)--(1);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n    \\end{tikzpicture}}}\n    & =\n    {\\bf F}_{12}, \\\\[1em]\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}}\n    & =\n    {\\bf F}_{2143} + {\\bf F}_{2413}, \\\\[1em]\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-3){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-3){};\n        \\draw[Arete](1)--(2);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(1);\n        \\node[Noeud](4)at(4,0){};\n        \\draw[Arete](4)--(3);\n        \\node[Noeud](5)at(5,-1){};\n        \\draw[Arete](4)--(5);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,-2){};\n        \\node[Noeud](4)at(4,-3){};\n        \\draw[Arete](3)--(4);\n        \\node[Noeud](5)at(5,-1){};\n        \\draw[Arete](5)--(3);\n        \\draw[Arete](2)--(5);\n    \\end{tikzpicture}}}\n    & =\n    {\\bf F}_{542163} + {\\bf F}_{542613} + {\\bf F}_{546213}.\n\\end{align}\n\\medskip\n\nThe Hilbert series of~${\\bf Baxter}$ is\n\\begin{equation}\n    B(z) := 1 + z + 2z^2 + 6z^3 + 22z^4 + 92z^5 + 422z^6 +\n            2074z^7 + 10754z^8 + 58202z^9 + \\cdots,\n\\end{equation}\nthe generating series of Baxter permutations (sequence~\\Sloane{A001181}\nof~\\cite{SLOANE}).\n\\medskip\n\nBy Theorem~\\ref{thm:HivertJanvier}, the product of~${\\bf Baxter}$ is well-defined.\nWe deduce it from the product of~${\\bf FQSym}$, and, since by Theorem~\\ref{thm:EquivBXBaxter}\nthere is exactly one Baxter permutation in any ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence\nclass of permutations, we obtain\n\\begin{equation} \\label{eq:ProduitBaxterP}\n    {\\bf P}_{J_0} \\cdot {\\bf P}_{J_1} =\n        \\sum_{\\substack{{\\sf P}(\\sigma) = J_0, \\; {\\sf P}(\\nu) = J_1 \\\\\n              \\pi \\; \\in \\; \\sigma \\; \\cshuffle \\; \\nu \\; \\cap \\; \\mathfrak{S}^{\\operatorname{B}}}}\n        {\\bf P}_{{\\sf P}(\\pi)}.\n\\end{equation}\nFor example,\n\\begin{equation}\\begin{split}\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](1)--(2);\n    \\end{tikzpicture}}}\n    \\cdot\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-1){};\n        \\draw[Arete](0)--(1);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n    \\end{tikzpicture}}}\n    & =\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,-1){};\n        \\node[Noeud](4)at(4,-2){};\n        \\draw[Arete](3)--(4);\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-3){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-3){};\n        \\draw[Arete](1)--(2);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(1);\n        \\node[Noeud](4)at(4,0){};\n        \\draw[Arete](4)--(3);\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,-1){};\n        \\node[Noeud](4)at(4,-2){};\n        \\draw[Arete](3)--(4);\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud](1)at(1,-1){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-3){};\n        \\node[Noeud](3)at(3,-2){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n        \\node[Noeud](4)at(4,0){};\n        \\draw[Arete](4)--(1);\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,-1){};\n        \\node[Noeud](4)at(4,-2){};\n        \\draw[Arete](3)--(4);\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-3){};\n        \\node[Noeud](3)at(3,-2){};\n        \\draw[Arete](3)--(2);\n        \\node[Noeud](4)at(4,-1){};\n        \\draw[Arete](4)--(3);\n        \\draw[Arete](1)--(4);\n    \\end{tikzpicture}}} \\\\\n    & +\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud](1)at(1,-3){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,0){};\n        \\draw[Arete](3)--(2);\n        \\node[Noeud](4)at(4,-1){};\n        \\draw[Arete](3)--(4);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud](1)at(1,-1){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-3){};\n        \\draw[Arete](2)--(3);\n        \\draw[Arete](1)--(2);\n        \\node[Noeud](4)at(4,0){};\n        \\draw[Arete](4)--(1);\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud](1)at(1,-3){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,0){};\n        \\draw[Arete](3)--(2);\n        \\node[Noeud](4)at(4,-1){};\n        \\draw[Arete](3)--(4);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-3){};\n        \\draw[Arete](2)--(3);\n        \\node[Noeud](4)at(4,-1){};\n        \\draw[Arete](4)--(2);\n        \\draw[Arete](1)--(4);\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}_{\\scalebox{0.15}{\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud](1)at(1,-3){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,0){};\n        \\draw[Arete](3)--(2);\n        \\node[Noeud](4)at(4,-1){};\n        \\draw[Arete](3)--(4);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-1){};\n        \\node[Noeud](3)at(3,-3){};\n        \\node[Noeud](4)at(4,-2){};\n        \\draw[Arete](4)--(3);\n        \\draw[Arete](2)--(4);\n        \\draw[Arete](1)--(2);\n    \\end{tikzpicture}}}.\n\\end{split}\\end{equation}\n\\medskip\n\nIn the same way, we deduce the coproduct of~${\\bf Baxter}$ from the coproduct\nof~${\\bf FQSym}$ and by Theorem~\\ref{thm:EquivBXBaxter}, we obtain\n\\begin{equation}\n    \\Delta \\left({\\bf P}_J\\right) =\n        \\sum_{\\substack{uv \\; \\in \\; \\mathfrak{S} \\\\\n                {\\sf P}(uv) = J \\\\\n                \\sigma := \\operatorname{std}(u), \\; \\nu := \\operatorname{std}(v) \\; \\in \\; \\mathfrak{S}^{\\operatorname{B}}}}\n        {\\bf P}_{{\\sf P}(\\sigma)} \\otimes {\\bf P}_{{\\sf P}(\\nu)}.\n\\end{equation}\nFor example,\n\\begin{equation}\\begin{split}\n    \\Delta \\left(\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}}\\right)\n    & =\n    1\n    \\otimes\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n    \\end{tikzpicture}}}\n    \\otimes\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-2){};\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](2)--(1);\n        \\draw[Arete](0)--(2);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](1)--(2);\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n    \\end{tikzpicture}}}\n    \\otimes\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](1)--(2);\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-1){};\n        \\draw[Arete](0)--(1);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n    \\end{tikzpicture}}}\n    \\otimes\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-1){};\n        \\draw[Arete](0)--(1);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n    \\end{tikzpicture}}} \\\\\n    & +\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-1){};\n        \\draw[Arete](0)--(1);\n    \\end{tikzpicture}}}\n    \\otimes\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-1){};\n        \\draw[Arete](0)--(1);\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](1)--(2);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n    \\end{tikzpicture}}}\n    \\otimes\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](1)--(2);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-2){};\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](2)--(1);\n        \\draw[Arete](0)--(2);\n    \\end{tikzpicture}}}\n    \\otimes\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}}\n    \\otimes\n    1.\n\\end{split}\\end{equation}\n\n\\subsection{\\texorpdfstring{Properties of the Hopf algebra ${\\bf Baxter}$}\n                           {Properties of the Hopf algebra Baxter}}\n\n\\subsubsection{A polynomial realization}\nWe deduce a polynomial realization of~${\\bf Baxter}$ from the one of~${\\bf FQSym}$.\nIn this section, we shall use the notation~$J_0 \\simeq J_1$ to say that\nthe labeled pairs of twin binary trees~$J_0$ and~$J_1$ have same shape.\n\\begin{Theoreme}\n    The map $r_A : {\\bf Baxter} \\rightarrow \\mathbb{K} \\langle A \\rangle$ defined by\n    \\begin{equation} \\label{eq:RealisationBaxter}\n        r_A\\left({\\bf P}_J\\right) :=\n            \\sum_{\\substack{u \\; \\in \\; A^* \\\\\n            \\left(\\operatorname{incr}(u), \\; \\operatorname{decr}(u)\\right) \\simeq J}}\n            u,\n    \\end{equation}\n    for any~$J \\in \\mathcal{T}\\mathcal{B}\\mathcal{T}$ provides a polynomial realization of~${\\bf Baxter}$.\n\\end{Theoreme}\n\\begin{proof}\n    Let us apply the polynomial realization~$r_A$ of~${\\bf FQSym}$ defined\n    in~(\\ref{eq:RealisationFQSym}) on elements of the fundamental basis\n    of~${\\bf Baxter}$:\n    \\begin{align}\n        r_A\\left({\\bf P}_J\\right) & =\n            \\sum_{\\substack{\\sigma \\; \\in \\; \\mathfrak{S} \\\\\n                {\\sf P}(\\sigma) = J}}\n                r_A({\\bf F}_\\sigma), \\\\\n            & = \\sum_{\\substack{\\sigma \\; \\in \\; \\mathfrak{S} \\\\\n                {\\sf P}(\\sigma^{-1}) = J}}\n                r_A({\\bf G}_\\sigma), \\label{eq:PreuvePR1} \\\\\n            & = \\sum_{\\substack{\\sigma \\; \\in \\; \\mathfrak{S} \\\\\n                \\left(\\operatorname{incr}(\\sigma), \\; \\operatorname{decr}(\\sigma)\\right) \\simeq J}}\n                r_A({\\bf G}_\\sigma), \\label{eq:PreuvePR2} \\\\\n            & = \\sum_{\\substack{\\sigma \\; \\in \\; \\mathfrak{S} \\\\\n                \\left(\\operatorname{incr}(\\sigma), \\; \\operatorname{decr}(\\sigma)\\right) \\simeq J}}\n                \\quad \\sum_{\\substack{u \\; \\in \\; A^* \\\\\n                \\operatorname{std}(u) = \\sigma}} u, \\label{eq:PreuvePR3}\n    \\end{align}\n    The equality between~(\\ref{eq:PreuvePR1}) and~(\\ref{eq:PreuvePR2})\n    follows from Lemma~\\ref{lem:FormeInsIncr}. The equality between~(\\ref{eq:PreuvePR3})\n    and the right member of~(\\ref{eq:RealisationBaxter}) follows from the\n    fact that $\\operatorname{incr}(\\sigma) \\simeq \\operatorname{incr}(u)$ (resp. $\\operatorname{decr}(\\sigma) \\simeq \\operatorname{decr}(u)$)\n    whenever~$\\operatorname{std}(u) = \\sigma$.\n\\end{proof}\n\n\\subsubsection{The dual Hopf algebra}\nWe denote by~$\\left\\{{\\bf P}^\\star_J\\right\\}_{J \\in \\mathcal{T}\\mathcal{B}\\mathcal{T}}$ the dual basis\nof the basis $\\left\\{{\\bf P}_J\\right\\}_{J \\in \\mathcal{T}\\mathcal{B}\\mathcal{T}}$. The Hopf\nalgebra~${\\bf Baxter}^\\star$, dual of~${\\bf Baxter}$, is a quotient Hopf algebra\nof~${\\bf FQSym}^\\star$. More precisely,\n\\begin{equation}\n    {\\bf Baxter}^\\star = {\\bf FQSym}^\\star \/ I,\n\\end{equation}\nwhere~$I$ is the Hopf ideal of~${\\bf FQSym}^\\star$ spanned by the elements\n$({\\bf F}^\\star_\\sigma - {\\bf F}^\\star_\\nu)$ whenever~$\\sigma {\\:\\equiv_{\\operatorname{B}}\\:} \\nu$.\n\\medskip\n\nLet $\\phi : {\\bf FQSym}^\\star \\twoheadrightarrow {\\bf Baxter}^\\star$ be the canonical\nprojection, mapping~${\\bf F}^\\star_\\sigma$ on~${\\bf P}^\\star_{{\\sf P}(\\sigma)}$.\nBy definition, the product of~${\\bf Baxter}^\\star$ is\n\\begin{equation}\n    {\\bf P}^\\star_{J_0} \\cdot {\\bf P}^\\star_{J_1} =\n    \\phi \\left({\\bf F}^\\star_{\\sigma} \\cdot {\\bf F}^\\star_{\\nu} \\right),\n\\end{equation}\nwhere~$\\sigma$ and~$\\nu$ are any permutations such that ${\\sf P}(\\sigma) = J_0$\nand ${\\sf P}(\\nu) = J_1$. Note that due to the fact that~${\\bf Baxter}^\\star$\nis a quotient of~${\\bf FQSym}^\\star$, the number of terms occurring in a product\n${\\bf P}^\\star_{J_0} \\cdot {\\bf P}^\\star_{J_1}$ only depends on the number~$m$\n(resp.~$n$) of nodes of~$J_0$ (resp.~$J_1$) and is~$\\binom{m + n}{m}$.\nFor example,\n\\begin{equation}\\begin{split}\n    {\\bf P}^\\star_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,Marque1](0)at(0,-1){};\n        \\node[Noeud,Marque1](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud,Marque1](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](1)--(2);\n    \\end{tikzpicture}}}\n    \\cdot\n    {\\bf P}^\\star_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-1){};\n        \\draw[Arete](0)--(1);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,Marque2](0)at(0,-1){};\n        \\node[Noeud,Marque2](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n    \\end{tikzpicture}}}\n    & =\n    {\\bf P}^\\star_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,Marque1](0)at(0,-1){};\n        \\node[Noeud,Marque1](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud,Marque1](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,-1){};\n        \\node[Noeud](4)at(4,-2){};\n        \\draw[Arete](3)--(4);\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-3){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-3){};\n        \\draw[Arete](1)--(2);\n        \\node[Noeud,Marque2](3)at(3,-1){};\n        \\draw[Arete](3)--(1);\n        \\node[Noeud,Marque2](4)at(4,0){};\n        \\draw[Arete](4)--(3);\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}^\\star_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,Marque1](0)at(0,-1){};\n        \\node[Noeud,Marque1](1)at(1,-2){};\n        \\node[Noeud](2)at(2,-3){};\n        \\draw[Arete](1)--(2);\n        \\draw[Arete](0)--(1);\n        \\node[Noeud,Marque1](3)at(3,0){};\n        \\draw[Arete](3)--(0);\n        \\node[Noeud](4)at(4,-1){};\n        \\draw[Arete](3)--(4);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-3){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud,Marque2](2)at(2,-1){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud](3)at(3,-2){};\n        \\draw[Arete](2)--(3);\n        \\node[Noeud,Marque2](4)at(4,0){};\n        \\draw[Arete](4)--(2);\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}^\\star_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,Marque1](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-3){};\n        \\node[Noeud,Marque1](2)at(2,-2){};\n        \\draw[Arete](2)--(1);\n        \\draw[Arete](0)--(2);\n        \\node[Noeud,Marque1](3)at(3,0){};\n        \\draw[Arete](3)--(0);\n        \\node[Noeud](4)at(4,-1){};\n        \\draw[Arete](3)--(4);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud,Marque2](1)at(1,-1){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-3){};\n        \\draw[Arete](2)--(3);\n        \\draw[Arete](1)--(2);\n        \\node[Noeud,Marque2](4)at(4,0){};\n        \\draw[Arete](4)--(1);\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}^\\star_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud,Marque1](1)at(1,-1){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud,Marque1](2)at(2,-2){};\n        \\draw[Arete](1)--(2);\n        \\node[Noeud,Marque1](3)at(3,0){};\n        \\draw[Arete](3)--(1);\n        \\node[Noeud](4)at(4,-1){};\n        \\draw[Arete](3)--(4);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,Marque2](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-3){};\n        \\node[Noeud](2)at(2,-2){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud](3)at(3,-3){};\n        \\draw[Arete](2)--(3);\n        \\draw[Arete](0)--(2);\n        \\node[Noeud,Marque2](4)at(4,0){};\n        \\draw[Arete](4)--(0);\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}^\\star_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,Marque1](0)at(0,-1){};\n        \\node[Noeud,Marque1](1)at(1,-2){};\n        \\node[Noeud](2)at(2,-3){};\n        \\node[Noeud](3)at(3,-4){};\n        \\draw[Arete](2)--(3);\n        \\draw[Arete](1)--(2);\n        \\draw[Arete](0)--(1);\n        \\node[Noeud,Marque1](4)at(4,0){};\n        \\draw[Arete](4)--(0);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-3){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud,Marque2](2)at(2,-1){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud,Marque2](3)at(3,0){};\n        \\draw[Arete](3)--(2);\n        \\node[Noeud](4)at(4,-1){};\n        \\draw[Arete](3)--(4);\n    \\end{tikzpicture}}} \\\\\n    & +\n    {\\bf P}^\\star_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,Marque1](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-3){};\n        \\node[Noeud,Marque1](2)at(2,-2){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud](3)at(3,-3){};\n        \\draw[Arete](2)--(3);\n        \\draw[Arete](0)--(2);\n        \\node[Noeud,Marque1](4)at(4,0){};\n        \\draw[Arete](4)--(0);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud,Marque2](1)at(1,-1){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\draw[Arete](1)--(2);\n        \\node[Noeud,Marque2](3)at(3,0){};\n        \\draw[Arete](3)--(1);\n        \\node[Noeud](4)at(4,-1){};\n        \\draw[Arete](3)--(4);\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}^\\star_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,Marque1](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-3){};\n        \\node[Noeud](2)at(2,-4){};\n        \\draw[Arete](1)--(2);\n        \\node[Noeud,Marque1](3)at(3,-2){};\n        \\draw[Arete](3)--(1);\n        \\draw[Arete](0)--(3);\n        \\node[Noeud,Marque1](4)at(4,0){};\n        \\draw[Arete](4)--(0);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud,Marque2](1)at(1,-1){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud,Marque2](2)at(2,0){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud](3)at(3,-1){};\n        \\node[Noeud](4)at(4,-2){};\n        \\draw[Arete](3)--(4);\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}^\\star_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud,Marque1](1)at(1,-1){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-3){};\n        \\node[Noeud,Marque1](3)at(3,-2){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n        \\node[Noeud,Marque1](4)at(4,0){};\n        \\draw[Arete](4)--(1);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,Marque2](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud,Marque2](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,-1){};\n        \\node[Noeud](4)at(4,-2){};\n        \\draw[Arete](3)--(4);\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}^\\star_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud,Marque1](1)at(1,-1){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud,Marque1](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-3){};\n        \\draw[Arete](2)--(3);\n        \\draw[Arete](1)--(2);\n        \\node[Noeud,Marque1](4)at(4,0){};\n        \\draw[Arete](4)--(1);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,Marque2](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-3){};\n        \\node[Noeud](2)at(2,-2){};\n        \\draw[Arete](2)--(1);\n        \\draw[Arete](0)--(2);\n        \\node[Noeud,Marque2](3)at(3,0){};\n        \\draw[Arete](3)--(0);\n        \\node[Noeud](4)at(4,-1){};\n        \\draw[Arete](3)--(4);\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}^\\star_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud](1)at(1,-3){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud,Marque1](2)at(2,-1){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud,Marque1](3)at(3,-2){};\n        \\draw[Arete](2)--(3);\n        \\node[Noeud,Marque1](4)at(4,0){};\n        \\draw[Arete](4)--(2);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,Marque2](0)at(0,-1){};\n        \\node[Noeud,Marque2](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\node[Noeud](4)at(4,-2){};\n        \\draw[Arete](3)--(4);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}}.\n\\end{split}\\end{equation}\n\\medskip\n\nIn the same way, the coproduct of~${\\bf Baxter}^\\star$ is\n\\begin{equation}\n    \\Delta({\\bf P}_J) =\n    (\\phi \\otimes \\phi)\\left(\\Delta \\left({\\bf F}^\\star_\\sigma \\right)\\right),\n\\end{equation}\nwhere~$\\sigma$ is any permutation such that ${\\sf P}(\\sigma) = J$. Note that\nthe number of terms occurring in a coproduct $\\Delta\\left({\\bf P}_J\\right)$\nonly depends on the number~$n$ of nodes of each binary trees of~$J$ and\nis~$n + 1$. For example,\n\\begin{equation} \\begin{split}\n    \\Delta \\left(\n    {\\bf P}^\\star_{\\scalebox{0.15}{\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}} \\right)\n    & =\n    1\n    \\otimes\n    {\\bf P}^\\star_{\\scalebox{0.15}{\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}^\\star_{\\scalebox{0.15}{\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n    \\end{tikzpicture}}}\n    \\otimes\n    {\\bf P}^\\star_{\\scalebox{0.15}{\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-2){};\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](2)--(1);\n        \\draw[Arete](0)--(2);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](1)--(2);\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}^\\star_{\\scalebox{0.15}{\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-1){};\n        \\draw[Arete](0)--(1);\n    \\end{tikzpicture}}}\n    \\otimes\n    {\\bf P}^\\star_{\\scalebox{0.15}{\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-1){};\n        \\draw[Arete](0)--(1);\n    \\end{tikzpicture}}} \\\\\n    & +\n    {\\bf P}^\\star_{\\scalebox{0.15}{\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](1)--(2);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n    \\end{tikzpicture}}}\n    \\otimes\n    {\\bf P}^\\star_{\\scalebox{0.15}{\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}^\\star_{\\scalebox{0.15}{\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}}\n    \\otimes\n    1.\n\\end{split} \\end{equation}\n\\medskip\n\nFollowing Fomin~\\cite{Fom94} (see also~\\cite{BLL08}), we can build a\n\\emph{pair of graded graphs in duality} $(G_{\\bf P}, G_{{\\bf P}^\\star})$. The set\nof vertices of~$G_{\\bf P}$ and~$G_{{\\bf P}^\\star}$ is the set of pairs of twin\nbinary trees. There is an edge between the vertices~$J$ and~$J'$ in~$G_{\\bf P}$\n(resp. in~$G_{{\\bf P}^\\star}$) if~${\\bf P}_{J'}$ (resp.~${\\bf P}^\\star_{J'}$) appears\nin the product ${\\bf P}_J \\cdot {\\bf P}_{\n\\scalebox{0.15}{\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n    \\end{tikzpicture}\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n    \\end{tikzpicture}}}$\n(resp. in the product ${\\bf P}^\\star_J \\cdot {\\bf P}^\\star_{\n\\scalebox{0.15}{\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n    \\end{tikzpicture}\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n    \\end{tikzpicture}}}$).\nFigure~\\ref{fig:GrapheDualiteG} (resp. Figure~\\ref{fig:GrapheDualiteD})\nshows the graded graph~$G_{\\bf P}$ (resp.~$G_{{\\bf P}^\\star}$) restricted to vertices\nof order smaller than~$5$.\n\n\\begin{figure}[p]\n    \\scalebox{.145}{\n    \\begin{tikzpicture}\n       \n        \\scalebox{4}{\\node[minimum size = 6em](ee) at (0,0){$\\perp$ $\\perp$};}\n        \\node[fit=(ee)] (ee) {};\n       \n        \\node[Noeud](1g0)at(10,0){};\n        \\node[Noeud](1d0)at(11.5,0){};\n        \\node[fit=(1g0) (1d0)] (1) {};\n       \n        \\node[Noeud](12g0)at(20,20){};\n        \\node[Noeud](12g1)at(21,19){};\n        \\draw[Arete](12g0)--(12g1);\n        \\node[Noeud](12d0)at(22.5,19){};\n        \\node[Noeud](12d1)at(23.5,20){};\n        \\draw[Arete](12d1)--(12d0);\n        \\node[fit=(12g0) (12g1) (12d0) (12d1)] (12) {};\n       \n        \\node[Noeud](21g0)at(20,-21){};\n        \\node[Noeud](21g1)at(21,-20){};\n        \\draw[Arete](21g1)--(21g0);\n        \\node[Noeud](21d0)at(22.5,-20){};\n        \\node[Noeud](21d1)at(23.5,-21){};\n        \\draw[Arete](21d0)--(21d1);\n        \\node[fit=(21g0) (21g1) (21d0) (21d1)] (21) {};\n       \n        \\node[Noeud](123g0)at(40,50){};\n        \\node[Noeud](123g1)at(41,49){};\n        \\node[Noeud](123g2)at(42,48){};\n        \\draw[Arete](123g1)--(123g2);\n        \\draw[Arete](123g0)--(123g1);\n        \\node[Noeud](123d0)at(43.5,48){};\n        \\node[Noeud](123d1)at(44.5,49){};\n        \\draw[Arete](123d1)--(123d0);\n        \\node[Noeud](123d2)at(45.5,50){};\n        \\draw[Arete](123d2)--(123d1);\n        \\node[fit=(123g0) (123g1) (123g2) (123d0) (123d1) (123d2)] (123) {};\n       \n        \\node[Noeud](132g0)at(40,30){};\n        \\node[Noeud](132g1)at(41,28){};\n        \\node[Noeud](132g2)at(42,29){};\n        \\draw[Arete](132g2)--(132g1);\n        \\draw[Arete](132g0)--(132g2);\n        \\node[Noeud](132d0)at(43.5,29){};\n        \\node[Noeud](132d1)at(44.5,30){};\n        \\draw[Arete](132d1)--(132d0);\n        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(123)--(1243);\n        \\draw[Arete,line width=3.5pt] (123)--(1423);\n        \\draw[Arete,line width=3.5pt] (123)--(4123);\n        %\n        \\draw[Arete,line width=3.5pt] (132)--(1324);\n        \\draw[Arete,line width=3.5pt] (132)--(1342);\n        \\draw[Arete,line width=3.5pt] (132)--(1432);\n        \\draw[Arete,line width=3.5pt] (132)--(4132);\n        %\n        \\draw[Arete,line width=3.5pt] (312)--(3124);\n        \\draw[Arete,line width=3.5pt] (312)--(3142);\n        \\draw[Arete,line width=3.5pt] (312)--(4312);\n        %\n        \\draw[Arete,line width=3.5pt] (213)--(2134);\n        \\draw[Arete,line width=3.5pt] (213)--(2143);\n        \\draw[Arete,line width=3.5pt] (213)--(4213);\n        %\n        \\draw[Arete,line width=3.5pt] (231)--(2314);\n        \\draw[Arete,line width=3.5pt] (231)--(2341);\n        \\draw[Arete,line width=3.5pt] (231)--(2431);\n        \\draw[Arete,line width=3.5pt] (231)--(4231);\n        %\n        \\draw[Arete,line width=3.5pt] (321)--(3214);\n        \\draw[Arete,line width=3.5pt] (321)--(3241);\n        \\draw[Arete,line width=3.5pt] (321)--(3421);\n        \\draw[Arete,line width=3.5pt] (321)--(4321);\n    \\end{tikzpicture}}\n    \\caption{The graded graph~$G_{\\bf P}$ restricted to vertices of order\n    smaller than~$5$.}\n    \\label{fig:GrapheDualiteG}\n\\end{figure}\n\n\\begin{figure}[p]\n    \\scalebox{.145}{\n    \\begin{tikzpicture}\n       \n        \\scalebox{4}{\\node[minimum size = 6em](ee) at (0,0){$\\perp$ $\\perp$};}\n        \\node[fit=(ee)] (ee) {};\n       \n        \\node[Noeud](1g0)at(10,0){};\n        \\node[Noeud](1d0)at(11.5,0){};\n        \\node[fit=(1g0) (1d0)] (1) {};\n       \n        \\node[Noeud](12g0)at(20,20){};\n        \\node[Noeud](12g1)at(21,19){};\n        \\draw[Arete](12g0)--(12g1);\n        \\node[Noeud](12d0)at(22.5,19){};\n        \\node[Noeud](12d1)at(23.5,20){};\n        \\draw[Arete](12d1)--(12d0);\n        \\node[fit=(12g0) (12g1) (12d0) (12d1)] (12) {};\n      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width=3.5pt,shorten >=7mm] (213)--(3241g0);\n        \\draw[Arete,line width=3.5pt,shorten >=7mm] (213)--(3142g0);\n        %\n        \\draw[Arete,line width=3.5pt,shorten >=7mm] (231)--(2314g0);\n        \\draw[Arete,line width=3.5pt,shorten >=7mm] (231)--(3421g0);\n        \\draw[Arete,line width=3.5pt,shorten >=7mm] (231)--(3142g0);\n        \\draw[Arete,line width=3.5pt,shorten >=7mm] (231)--(2143g0);\n        %\n        \\draw[Arete,line width=3.5pt,shorten >=7mm] (321)--(3214g0);\n        \\draw[Arete,line width=3.5pt,shorten >=7mm] (321)--(4312g0);\n        \\draw[Arete,line width=3.5pt,shorten >=7mm] (321)--(4321g0);\n        \\draw[Arete,line width=3.5pt,shorten >=7mm] (321)--(4213g0);\n    \\end{tikzpicture}}\n    \\caption{The graded graph~$G_{{\\bf P}^\\star}$ restricted to vertices of\n    order smaller than~$5$.}\n    \\label{fig:GrapheDualiteD}\n\\end{figure}\n\n\\subsubsection{A boolean basis}\nWe shall call a basis of an algebra (resp. coalgebra) a \\emph{boolean algebra basis}\n(resp. \\emph{boolean coalgebra basis}) if each element of the basis (resp.\ntensor square of the basis) only occurs with coefficient~$0$ or~$1$ in any\nproduct (resp. coproduct) involving two (resp. one) elements of the basis.\n\n\\begin{Proposition} \\label{prop:BaseEnsemblisteSSFQSym}\n    If~$\\equiv$ is an equivalence relation defined on~$A^*$ satisfying the\n    conditions of Theorem~\\ref{thm:HivertJanvier} and additionally, for\n    all~$\\pi, \\mu \\in \\mathfrak{S}$,\n    \\begin{equation} \\label{eq:BaseBool}\n        \\sigma, \\nu \\in \\pi \\cshuffle \\mu \\enspace\n        \\mbox{ and }\n        \\enspace \\sigma^{-1} \\equiv \\nu^{-1}\n        \\quad \\mbox{ imply } \\quad\n        \\sigma = \\nu,\n    \\end{equation}\n    then, the family\n    $\\left\\{{\\bf P}_{\\widehat{\\sigma}}\\right\\}_{\\widehat{\\sigma} \\in \\mathfrak{S}\/_\\equiv}$\n    defined in~(\\ref{eq:EquivFQSym}) is both an algebra and a coalgebra\n    boolean basis of the corresponding Hopf subalgebra of~${\\bf FQSym}$.\n\\end{Proposition}\n\\begin{proof}\n    It is immediate from the definition of the product of~${\\bf FQSym}$ that\n    $\\left\\{{\\bf P}_{\\widehat{\\sigma}}\\right\\}_{\\widehat{\\sigma} \\in \\mathfrak{S}\/_\\equiv}$\n    is a boolean algebra basis, regardless of~(\\ref{eq:BaseBool}).\n\n    By duality,\n    $\\left\\{{\\bf P}_{\\widehat{\\sigma}}\\right\\}_{\\widehat{\\sigma} \\in \\mathfrak{S}\/_\\equiv}$\n    is a boolean coalgebra basis if and only if its dual basis $\\left\\{{\\bf P}_{\\widehat{\\sigma}}^\\star\\right\\}_{\\widehat{\\sigma} \\in \\mathfrak{S}\/_\\equiv}$\n    is a boolean algebra basis. One has\n    \\begin{align}\n        {\\bf P}_{\\widehat{\\pi}}^\\star \\cdot {\\bf P}_{\\widehat{\\mu}}^\\star & =\n        \\phi\\left({\\bf F}_\\pi^\\star \\cdot {\\bf F}_\\mu^\\star\\right) \\\\\n        & = \\phi\\left(\\psi\\left(\\psi^{-1}\\left({\\bf F}_\\pi^\\star\\right) \\cdot \\psi^{-1}\\left({\\bf F}_\\mu^\\star\\right)\\right)\\right) \\\\\n        & = \\phi\\left(\\psi\\left({\\bf F}_{\\pi^{-1}} \\cdot {\\bf F}_{\\mu^{-1}}\\right)\\right) \\\\\n        & = \\sum_{\\sigma \\; \\in \\; \\pi^{-1} \\; \\cshuffle \\; \\nu^{-1}} \\phi\\left({\\bf F}_{\\sigma^{-1}}^\\star\\right) \\label{eq:PreuveBaseBool},\n    \\end{align}\n    where~$\\phi$ is the canonical projection mapping~${\\bf F}^\\star_\\sigma$\n    on~${\\bf P}^\\star_{\\widehat{\\sigma}}$ for any permutation~$\\sigma$, $\\psi$\n    is the Hopf isomorphism mapping~${\\bf F}_\\sigma$ on~${\\bf F}^\\star_{\\sigma^{-1}}$\n    for any permutation~$\\sigma$, and $\\pi \\in \\widehat{\\pi}$ and\n    $\\mu \\in \\widehat{\\mu}$. One can easily see that if~$\\equiv$ satisfies\n    the hypothesis of the proposition, then there are no multiplicities\n    in~(\\ref{eq:PreuveBaseBool}).\n\\end{proof}\n\nLaw and Reading have proved in~\\cite{LR10} that the basis of their Baxter\nHopf algebra, analog to our basis~$\\left\\{{\\bf P}_J\\right\\}_{J \\in \\mathcal{T}\\mathcal{B}\\mathcal{T}}$,\nis both a boolean algebra basis and a boolean coalgebra basis. We re-prove\nthis result in our setting:\n\\begin{Proposition} \\label{prop:BaseEnsembliste}\n    The basis~$\\left\\{{\\bf P}_J\\right\\}_{J \\in \\mathcal{T}\\mathcal{B}\\mathcal{T}}$ is both a boolean\n    algebra basis and a boolean coalgebra basis of~${\\bf Baxter}$.\n\\end{Proposition}\n\\begin{proof}\n    Let us prove that the sylvester equivalence relation satisfies the\n    assumptions of Proposition~\\ref{prop:BaseEnsemblisteSSFQSym}. Indeed,\n    the result directly follows from the fact that, by\n    Proposition~\\ref{prop:LienSylv}, the Baxter equivalence relation is\n    finer than the sylvester equivalence relation.\n\n    Let us start with a useful result: Let~$x$ and~$y$ be two words without\n    repetition of same length and $u, v \\in x \\cshuffle y$ (here, the letters\n    of~$y$ are shifted by~$\\max(x)$). Let us prove by induction on~$|x| + |y|$\n    that if~$\\operatorname{decr}(u)$ and~$\\operatorname{decr}(v)$ have same shape, then~$u = v$. It\n    is obvious if $|x| + |y| = 0$. Otherwise, one has $u = u' \\, {\\tt b} \\, u''$\n    and $v = v' \\, {\\tt b} \\, v''$ where ${\\tt b} := \\max(u) = \\max(v)$. Since the\n    shape of the left subtree of~$\\operatorname{decr}(u)$ is equal to the shape of the\n    left subtree of~$\\operatorname{decr}(v)$, the position of~${\\tt b}$ in~$u$ and~$v$ is\n    the same. Moreover, the word~$y$ is of the form $y = y' \\, {\\tt a} \\, y''$\n    where ${\\tt a} := \\max(y)$, and~$x$ is of the form~$x = x' x''$, where\n    $u', v' \\in x' \\cshuffle y'$ and $u'', v'' \\in x'' \\cshuffle y''$.\n    Since the left (resp. right) subtree of~$\\operatorname{decr}(u)$ is equal to the left\n    (resp. right) subtree of~$\\operatorname{decr}(v)$, by induction hypothesis,~$u' = v'$\n    and~$u'' = v''$, showing that~$u = v$.\n\n    Now, let $\\pi, \\mu \\in \\mathfrak{S}$ and $\\sigma \\ne \\nu \\in \\pi \\cshuffle \\mu$\n    and assume that $\\sigma^{-1} {\\:\\equiv_{\\operatorname{S}}\\:} \\nu^{-1}$. Then, by\n    Theorem~\\ref{thm:PSymbPBT}, the permutations~$\\sigma^{-1}$ and~$\\nu^{-1}$\n    give the same right binary search tree when inserted from right to left.\n    By Lemma~\\ref{lem:FormeInsIncr}, that implies that~$\\operatorname{decr}(\\sigma)$\n    and~$\\operatorname{decr}(\\nu)$ have same shape. That implies~$\\sigma = \\nu$,\n    contradicting our hypothesis.\n\\end{proof}\n\nBy duality, Proposition~\\ref{prop:BaseEnsembliste} also shows that the\nbasis $\\left\\{{\\bf P}_J^\\star\\right\\}_{J \\in \\mathcal{T}\\mathcal{B}\\mathcal{T}}$ is a boolean algebra\nand coalgebra basis.\n\n\\subsubsection{A lattice interval description of the product}\nIf~$\\equiv$ is an equivalence relation of~$\\mathfrak{S}$ and~$\\sigma$ a permutation,\ndenote by~$\\widehat{\\sigma} {\\!\\uparrow}$ (resp.~$\\widehat{\\sigma} {\\!\\downarrow}$)\nthe minimal (resp. maximal) permutation of the~$\\equiv$-equivalence class\nof~$\\sigma$ for the permutohedron order.\n\n\\begin{Proposition} \\label{prop:ProduitIntervalle}\n    If~$\\equiv$ is an equivalence relation defined on~$A^*$ satisfying the\n    conditions of Theorem~\\ref{thm:HivertJanvier} and additionally, the\n    $\\equiv$-equivalence classes of permutations are intervals of the\n    permutohedron, then the product on the family defined in~(\\ref{eq:EquivFQSym})\n    can be expressed as:\n    \\begin{equation}\n        {\\bf P}_{\\widehat{\\sigma}} \\cdot {\\bf P}_{\\widehat{\\nu}} =\n        \\sum_{\\substack{\\widehat{\\sigma} {\\!\\uparrow} {\\,\\diagup\\,} \\widehat{\\nu} {\\!\\uparrow}\n            {\\:\\leq_{\\operatorname{P}}\\:} \\pi {\\:\\leq_{\\operatorname{P}}\\:}\n            \\widehat{\\sigma} {\\!\\downarrow} {\\,\\diagdown\\,} \\widehat{\\nu} {\\!\\downarrow} \\\\\n            \\pi = \\min \\widehat{\\pi}}}\n        {\\bf P}_{\\widehat{\\pi}}.\n    \\end{equation}\n\\end{Proposition}\n\\begin{proof}\n    It is well-known that the shifted shuffle product of two permutohedron\n    intervals is still a permutohedron interval. Restating this fact\n    in~${\\bf FQSym}$, we have\n    \\begin{equation} \\label{eq:ProduitIntervalleFQSym}\n        \\left(\\sum_{\\sigma {\\:\\leq_{\\operatorname{P}}\\:} \\mu {\\:\\leq_{\\operatorname{P}}\\:} \\sigma'} {\\bf F}_\\mu \\right) \\cdot\n        \\left(\\sum_{\\nu {\\:\\leq_{\\operatorname{P}}\\:} \\tau {\\:\\leq_{\\operatorname{P}}\\:} \\nu'} {\\bf F}_\\tau \\right) =\n        \\sum_{\\sigma {\\,\\diagup\\,} \\nu {\\:\\leq_{\\operatorname{P}}\\:} \\pi {\\:\\leq_{\\operatorname{P}}\\:} \\sigma' {\\,\\diagdown\\,} \\nu'} {\\bf F}_\\pi.\n    \\end{equation}\n    By~(\\ref{eq:ProduitIntervalleFQSym}) and since that every $\\equiv$-equivalence\n    class is an interval of the permutohedron, we obtain\n    \\begin{equation} \\label{eq:ExpressionPP}\n        {\\bf P}_{\\widehat{\\sigma}} \\cdot {\\bf P}_{\\widehat{\\nu}} =\n            \\sum_{\\widehat{\\sigma} {\\!\\uparrow} {\\,\\diagup\\,} \\widehat{\\nu} {\\!\\uparrow}\n            {\\:\\leq_{\\operatorname{P}}\\:} \\pi {\\:\\leq_{\\operatorname{P}}\\:}\n            \\widehat{\\sigma} {\\!\\downarrow} {\\,\\diagdown\\,} \\widehat{\\nu} {\\!\\downarrow}} {\\bf F}_\\pi.\n    \\end{equation}\n    By Theorem~\\ref{thm:HivertJanvier}, the expression~(\\ref{eq:ExpressionPP})\n    can be expressed as a sum of~${\\bf P}_{\\widehat{\\pi}}$ elements and the\n    proposition follows.\n\\end{proof}\n\nLet $J_0 := (T^0_L, T^0_R)$ and $J_1 := (T^1_L, T^1_R)$ be two pairs of\ntwin binary trees. Let us define the pair of twin binary trees~$J_0 {\\,\\diagup\\,} J_1$\nby\n\\begin{equation}\n    J_0 {\\,\\diagup\\,} J_1 := (T^0_L {\\,\\diagdown\\,} T^1_L, \\; T^0_R {\\,\\diagup\\,} T^1_R).\n\\end{equation}\nIn the same way, the pair of twin binary trees~$J_0 {\\,\\diagdown\\,} J_1$ is defined\nby\n\\begin{equation}\n    J_0 {\\,\\diagdown\\,} J_1 := (T^0_L {\\,\\diagup\\,} T^1_L, \\; T^0_R {\\,\\diagdown\\,} T^1_R).\n\\end{equation}\n\nProposition~\\ref{prop:ProduitIntervalle} leads to the following expression\nfor the product of~${\\bf Baxter}$.\n\\begin{Corollaire}\n    For all pairs of twin binary trees~$J_0$ and~$J_1$, the product\n    of~${\\bf Baxter}$ satisfies\n    \\begin{equation} \\label{eq:ProdBXInter}\n        {\\bf P}_{J_0} \\cdot {\\bf P}_{J_1} =\n        \\sum_{J_0 {\\,\\diagup\\,} J_1 {\\:\\leq_{\\operatorname{B}}\\:} J {\\:\\leq_{\\operatorname{B}}\\:} J_0 {\\,\\diagdown\\,} J_1} {\\bf P}_J.\n    \\end{equation}\n\\end{Corollaire}\n\\begin{proof}\n    Let~$\\sigma$ and~$\\nu$ two permutations. It is immediate, from the\n    definition of the ${\\sf P}$-symbol algorithm, that the ${\\sf P}$-symbol\n    of the permutation~$\\sigma {\\,\\diagup\\,} \\nu$ (resp.~$\\sigma {\\,\\diagdown\\,} \\nu$) is\n    the pair of twin binary trees ${\\sf P}(\\sigma) {\\,\\diagup\\,} {\\sf P}(\\nu)$\n    (resp. ${\\sf P}(\\sigma) {\\,\\diagdown\\,} {\\sf P}(\\nu)$). The expression~(\\ref{eq:ProdBXInter})\n    follows from the fact that ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence classes of permutations\n    are intervals of the permutohedron (Proposition~\\ref{prop:EquivBXInter})\n    and from Proposition~\\ref{prop:ProduitIntervalle}.\n\\end{proof}\n\n\\subsubsection{Multiplicative bases and free generators}\nRecall that the \\emph{elementary} family\n$\\left\\{{\\bf E}^\\sigma\\right\\}_{\\sigma \\in \\mathfrak{S}}$ and the \\emph{homogeneous}\nfamily $\\left\\{{\\bf H}^\\sigma\\right\\}_{\\sigma \\in \\mathfrak{S}}$ of~${\\bf FQSym}$\nrespectively defined by\n\\begin{align}\n    {\\bf E}^\\sigma  & := \\sum_{\\sigma {\\:\\leq_{\\operatorname{P}}\\:} \\sigma'} {\\bf F}_{\\sigma'}, \\\\[.5em]\n    {\\bf H}^\\sigma & := \\sum_{\\sigma' {\\:\\leq_{\\operatorname{P}}\\:} \\sigma} {\\bf F}_{\\sigma'},\n\\end{align}\nform multiplicative bases of~${\\bf FQSym}$ (see~\\cite{AS05,DHNT11} for an exposition\nof some known bases of~${\\bf FQSym}$). Indeed, for all~$\\sigma, \\nu \\in \\mathfrak{S}$,\nthe product satisfies\n\\begin{align}\n    {\\bf E}^\\sigma \\cdot {\\bf E}^\\nu   & = {\\bf E}^{\\sigma {\\,\\diagup\\,} \\nu}, \\\\\n    {\\bf H}^\\sigma \\cdot {\\bf H}^\\nu & = {\\bf H}^{\\sigma {\\,\\diagdown\\,} \\nu}.\n\\end{align}\n\nMimicking these definitions, let us define the \\emph{elementary} family\n$\\left\\{{\\bf E}_J\\right\\}_{J \\in \\mathcal{T}\\mathcal{B}\\mathcal{T}}$ and the \\emph{homogeneous} family\n$\\left\\{{\\bf H}_J\\right\\}_{J \\in \\mathcal{T}\\mathcal{B}\\mathcal{T}}$ of ${\\bf Baxter}$ respectively by\n\\begin{align}\n    {\\bf E}_J  & := \\sum_{J {\\:\\leq_{\\operatorname{B}}\\:} J'} {\\bf P}_{J'}, \\\\[.5em]\n    {\\bf H}_J & := \\sum_{J' {\\:\\leq_{\\operatorname{B}}\\:} J} {\\bf P}_{J'}.\n\\end{align}\nThese families are bases of ${\\bf Baxter}$ since they are defined by triangularity.\n\n\\begin{Proposition} \\label{prop:BaseEHBaxter}\n    Let $J$ be a pair of twin binary trees and $\\sigma {\\!\\uparrow}$ (resp.\n    $\\sigma {\\!\\downarrow}$) be the minimal (resp. maximal) permutation such that\n    ${\\sf P}(\\sigma {\\!\\uparrow}) = J$ (resp. ${\\sf P}(\\sigma {\\!\\downarrow}) = J$). Then,\n    \\begin{align}\n        {\\bf E}_J  & = {\\bf E}^{\\sigma {\\!\\uparrow}}, \\\\\n        {\\bf H}_J & = {\\bf H}^{\\sigma {\\!\\downarrow}}.\n    \\end{align}\n\\end{Proposition}\n\\begin{proof}\n    Using the fact that, by Theorem~\\ref{thm:OrdreBaxterMinMax},\n    the ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence relation is a lattice congruence of the\n    permutohedron, one successively has\n    \\begin{equation}\n        {\\bf E}_J = \\sum_{J {\\:\\leq_{\\operatorname{B}}\\:} J'} {\\bf P}_{J'}\n             = \\sum_{J {\\:\\leq_{\\operatorname{B}}\\:} J'}\n             \\sum_{\\substack{\\nu \\; \\in \\; \\mathfrak{S} \\\\ {\\sf P}(\\nu) = J'}} {\\bf F}_\\nu\n             = \\sum_{\\substack{\\nu \\; \\in \\; \\mathfrak{S} \\\\ J {\\:\\leq_{\\operatorname{B}}\\:} {\\sf P}(\\nu)}}\n             {\\bf F}_\\nu\n             = \\sum_{\\substack{\\nu \\; \\in \\; \\mathfrak{S} \\\\\n             \\sigma {\\!\\uparrow} {\\:\\leq_{\\operatorname{P}}\\:} \\nu}} {\\bf F}_\\nu\n            = {\\bf E}^{\\sigma {\\!\\uparrow}}.\n    \\end{equation}\n    The proof for the homogeneous family is analogous.\n\\end{proof}\n\n\\begin{Corollaire} \\label{cor:BasesMult}\n    For all pairs of twin binary trees $J_0$ and $J_1$, we have\n    \\begin{align}\n        {\\bf E}_{J_0} \\cdot {\\bf E}_{J_1}   & = {\\bf E}_{J_0 {\\,\\diagup\\,} J_1}, \\\\[.5em]\n        {\\bf H}_{J_0} \\cdot {\\bf H}_{J_1} & = {\\bf H}_{J_0 {\\,\\diagdown\\,} J_1}.\n    \\end{align}\n\\end{Corollaire}\n\\begin{proof}\n    Let~$\\sigma$ and~$\\nu$ be the minimal permutations of the ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence\n    classes respectively encoded by~$J_0$ and~$J_1$. By\n    Proposition~\\ref{prop:BaseEHBaxter}, we have\n    \\begin{equation}\n        {\\bf E}_{J_0} \\cdot {\\bf E}_{J_1} = {\\bf E}^\\sigma \\cdot {\\bf E}^\\nu = {\\bf E}^{\\sigma {\\,\\diagup\\,} \\nu}.\n    \\end{equation}\n    The permutation $\\sigma {\\,\\diagup\\,} \\nu$ is obviously the minimal element\n    of its ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class, and, by the definition of the\n    ${\\sf P}$-symbol algorithm, the ${\\sf P}$-symbol of $\\sigma {\\,\\diagup\\,} \\nu$ is\n    the pair of twin binary trees ${\\sf P}(\\sigma) {\\,\\diagup\\,} {\\sf P}(\\nu) = J_0 {\\,\\diagup\\,} J_1$.\n    The proof of the second part of the proposition is analogous.\n\\end{proof}\n\nFor example,\n\\begin{align}\n    {\\bf E}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud](1)at(1,-3){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,0){};\n        \\draw[Arete](3)--(2);\n        \\node[Noeud](4)at(4,-1){};\n        \\draw[Arete](3)--(4);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-3){};\n        \\draw[Arete](2)--(3);\n        \\node[Noeud](4)at(4,-1){};\n        \\draw[Arete](4)--(2);\n        \\draw[Arete](1)--(4);\n    \\end{tikzpicture}}}\n    \\cdot\n    {\\bf E}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,Marque1](0)at(0,0){};\n        \\node[Noeud,Marque1](1)at(1,-2){};\n        \\node[Noeud,Marque1](2)at(2,-1){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud,Marque1](3)at(3,-2){};\n        \\draw[Arete](2)--(3);\n        \\draw[Arete](0)--(2);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,Marque1](0)at(0,-2){};\n        \\node[Noeud,Marque1](1)at(1,-1){};\n        \\draw[Arete,Marque1](1)--(0);\n        \\node[Noeud,Marque1](2)at(2,-2){};\n        \\draw[Arete](1)--(2);\n        \\node[Noeud,Marque1](3)at(3,0){};\n        \\draw[Arete](3)--(1);\n    \\end{tikzpicture}}}\n    & =\n    {\\bf E}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud](1)at(1,-3){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,0){};\n        \\draw[Arete](3)--(2);\n        \\node[Noeud](4)at(4,-1){};\n        \\node[Noeud,Marque1](5)at(5,-2){};\n        \\node[Noeud,Marque1](6)at(6,-4){};\n        \\node[Noeud,Marque1](7)at(7,-3){};\n        \\draw[Arete](7)--(6);\n        \\node[Noeud,Marque1](8)at(8,-4){};\n        \\draw[Arete](7)--(8);\n        \\draw[Arete](5)--(7);\n        \\draw[Arete](4)--(5);\n        \\draw[Arete](3)--(4);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-4){};\n        \\node[Noeud](1)at(1,-3){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-5){};\n        \\node[Noeud](3)at(3,-6){};\n        \\draw[Arete](2)--(3);\n        \\node[Noeud](4)at(4,-4){};\n        \\draw[Arete](4)--(2);\n        \\draw[Arete](1)--(4);\n        \\node[Noeud,Marque1](5)at(5,-2){};\n        \\draw[Arete](5)--(1);\n        \\node[Noeud,Marque1](6)at(6,-1){};\n        \\draw[Arete](6)--(5);\n        \\node[Noeud,Marque1](7)at(7,-2){};\n        \\draw[Arete](6)--(7);\n        \\node[Noeud,Marque1](8)at(8,0){};\n        \\draw[Arete](8)--(6);\n    \\end{tikzpicture}}}, \\\\[1em]\n    {\\bf H}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud](1)at(1,-3){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,0){};\n        \\draw[Arete](3)--(2);\n        \\node[Noeud](4)at(4,-1){};\n        \\draw[Arete](3)--(4);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-3){};\n        \\draw[Arete](2)--(3);\n        \\node[Noeud](4)at(4,-1){};\n        \\draw[Arete](4)--(2);\n        \\draw[Arete](1)--(4);\n    \\end{tikzpicture}}}\n    \\cdot\n    {\\bf H}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,Marque1](0)at(0,0){};\n        \\node[Noeud,Marque1](1)at(1,-2){};\n        \\node[Noeud,Marque1](2)at(2,-1){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud,Marque1](3)at(3,-2){};\n        \\draw[Arete](2)--(3);\n        \\draw[Arete](0)--(2);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,Marque1](0)at(0,-2){};\n        \\node[Noeud,Marque1](1)at(1,-1){};\n        \\draw[Arete,Marque1](1)--(0);\n        \\node[Noeud,Marque1](2)at(2,-2){};\n        \\draw[Arete](1)--(2);\n        \\node[Noeud,Marque1](3)at(3,0){};\n        \\draw[Arete](3)--(1);\n    \\end{tikzpicture}}}\n    & =\n    {\\bf H}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-3){};\n        \\node[Noeud](1)at(1,-4){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,-2){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\node[Noeud](4)at(4,-2){};\n        \\draw[Arete](3)--(4);\n        \\node[Noeud,Marque1](5)at(5,0){};\n        \\draw[Arete](5)--(3);\n        \\node[Noeud,Marque1](6)at(6,-2){};\n        \\node[Noeud,Marque1](7)at(7,-1){};\n        \\draw[Arete](7)--(6);\n        \\node[Noeud,Marque1](8)at(8,-2){};\n        \\draw[Arete](7)--(8);\n        \\draw[Arete](5)--(7);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-3){};\n        \\draw[Arete](2)--(3);\n        \\node[Noeud](4)at(4,-1){};\n        \\draw[Arete](4)--(2);\n        \\node[Noeud,Marque1](5)at(5,-4){};\n        \\node[Noeud,Marque1](6)at(6,-3){};\n        \\draw[Arete](6)--(5);\n        \\node[Noeud,Marque1](7)at(7,-4){};\n        \\draw[Arete](6)--(7);\n        \\node[Noeud,Marque1](8)at(8,-2){};\n        \\draw[Arete](8)--(6);\n        \\draw[Arete](4)--(8);\n        \\draw[Arete](1)--(4);\n    \\end{tikzpicture}}}.\n\\end{align}\n\\medskip\n\nCorollary~\\ref{cor:BasesMult} also shows that the $\\left\\{{\\bf E}_J\\right\\}_{J \\in \\mathcal{T}\\mathcal{B}\\mathcal{T}}$\nand $\\left\\{{\\bf H}_J\\right\\}_{J \\in \\mathcal{T}\\mathcal{B}\\mathcal{T}}$ bases of~${\\bf Baxter}$ are boolean\nalgebra bases. However, these are not boolean coalgebra bases since one has\n\\begin{equation}\n    \\Delta \\left({\\bf E}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0.0,0){};\n        \\node[Noeud](1)at(1.0,-1){};\n        \\draw[Arete](0)--(1);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0.0,-1){};\n        \\node[Noeud](1)at(1.0,0){};\n        \\draw[Arete](1)--(0);\n    \\end{tikzpicture}}} \\right) =\n    1 \\otimes\n    {\\bf E}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0.0,0){};\n        \\node[Noeud](1)at(1.0,-1){};\n        \\draw[Arete](0)--(1);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0.0,-1){};\n        \\node[Noeud](1)at(1.0,0){};\n        \\draw[Arete](1)--(0);\n    \\end{tikzpicture}}}\n    + 2\\,\n    {\\bf E}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0.0,0){};\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0.0,0){};\n    \\end{tikzpicture}}}\n    \\otimes\n    {\\bf E}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0.0,0){};\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0.0,0){};\n    \\end{tikzpicture}}}\n    +\n    {\\bf E}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0.0,0){};\n        \\node[Noeud](1)at(1.0,-1){};\n        \\draw[Arete](0)--(1);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0.0,-1){};\n        \\node[Noeud](1)at(1.0,0){};\n        \\draw[Arete](1)--(0);\n    \\end{tikzpicture}}}\n    \\otimes 1,\n\\end{equation}\nand\n\\begin{equation}\n    \\Delta \\left({\\bf H}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0.0,-1){};\n        \\node[Noeud](1)at(1.0,0){};\n        \\draw[Arete](1)--(0);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0.0,0){};\n        \\node[Noeud](1)at(1.0,-1){};\n        \\draw[Arete](0)--(1);\n    \\end{tikzpicture}}} \\right) =\n    1 \\otimes\n    {\\bf H}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0.0,-1){};\n        \\node[Noeud](1)at(1.0,0){};\n        \\draw[Arete](1)--(0);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0.0,0){};\n        \\node[Noeud](1)at(1.0,-1){};\n        \\draw[Arete](0)--(1);\n    \\end{tikzpicture}}}\n    + 2\\,\n    {\\bf H}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0.0,0){};\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0.0,0){};\n    \\end{tikzpicture}}}\n    \\otimes\n    {\\bf H}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0.0,0){};\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0.0,0){};\n    \\end{tikzpicture}}}\n    +\n    {\\bf H}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0.0,-1){};\n        \\node[Noeud](1)at(1.0,0){};\n        \\draw[Arete](1)--(0);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0.0,0){};\n        \\node[Noeud](1)at(1.0,-1){};\n        \\draw[Arete](0)--(1);\n    \\end{tikzpicture}}}\n    \\otimes 1.\n\\end{equation}\n\\medskip\n\nLet us say that a pair of twin binary trees~$J$ is \\emph{connected} (resp.\n\\emph{anti-connected}) if all the permutations~$\\sigma$ such that\n${\\sf P}(\\sigma) = J$ are connected (resp. anti-connected). Since for any\nconnected (resp. anti-connected) permutation~$\\sigma$ and a permutation~$\\nu$\nsuch that~$\\sigma {\\:\\leq_{\\operatorname{P}}\\:} \\nu$ (resp.~$\\nu {\\:\\leq_{\\operatorname{P}}\\:} \\sigma$) the\npermutation~$\\nu$ is also connected (resp. anti-connected), it is enough\nto check if the minimal (resp. maximal) permutation of the ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence\nclass encoded by~$J$ is connected (resp. anti-connected) to decide if~$J$\nis connected (resp. anti-connected).\n\n\\begin{Lemme} \\label{lem:ABJConnexes}\n    For any pair of twin binary trees~$J$, there exists a sequence of\n    connected (resp. anti-connected) pairs of twin binary trees\n    $J_1$, \\dots, $J_k$ such that\n    \\begin{equation}\n        J = J_1 {\\,\\diagup\\,} \\cdots {\\,\\diagup\\,} J_k \\quad\n        \\mbox{(resp. $J = J_1 {\\,\\diagdown\\,} \\cdots {\\,\\diagdown\\,} J_k$)}.\n    \\end{equation}\n\\end{Lemme}\n\\begin{proof}\n    Let~$\\sigma$ be the minimal permutation of the ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence\n    class encoded by $J$ (recall that the existence of this element is ensured\n    by Proposition~\\ref{prop:EquivBXInter}). One can write~$\\sigma$ as\n    \\begin{equation}\n        \\sigma = \\sigma^{(1)} {\\,\\diagup\\,} \\cdots {\\,\\diagup\\,} \\sigma^{(k)},\n    \\end{equation}\n    where the permutations~$\\sigma^{(i)}$ are connected for all $1 \\leq i \\leq k$.\n    Since~$\\sigma$ is the minimal permutation of its ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence\n    class, all the permutations~$\\sigma^{(i)}$ are also minimal of their\n    ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence classes. Hence, the pairs of twin binary trees\n    ${\\sf P}\\left(\\sigma^{(i)}\\right)$ are connected and we can write\n    \\begin{equation}\n        J = {\\sf P}\\left(\\sigma^{(1)}\\right)\n        {\\,\\diagup\\,} \\cdots {\\,\\diagup\\,}\n        {\\sf P}\\left(\\sigma^{(k)}\\right).\n    \\end{equation}\n    The proof for the respective part is analogous.\n\\end{proof}\n\n\\begin{Theoreme} \\label{thm:LiberteBaxter}\n    The algebra~${\\bf Baxter}$ is free on the elements~${\\bf E}_J$ (resp.~${\\bf H}_J$)\n    such that~$J$ is a connected (resp. anti-connected) pair of twin\n    binary trees.\n\\end{Theoreme}\n\\begin{proof}\n    By Corollary~\\ref{cor:BasesMult} and Lemma~\\ref{lem:ABJConnexes},\n    each element~${\\bf E}_J$ can be expressed as\n    \\begin{equation}\n        {\\bf E}_J = {\\bf E}_{J_1} \\cdot \\dots \\cdot {\\bf E}_{J_k},\n    \\end{equation}\n    where the pairs of twin binary trees~$J_i$ are connected for all\n    $1 \\leq i \\leq k$.\n\n    Now, since for all permutations $\\sigma$ and $\\nu$ one has\n    ${\\bf E}^\\sigma \\cdot {\\bf E}^\\nu = {\\bf E}^{\\sigma {\\,\\diagup\\,} \\nu}$ in ${\\bf FQSym}$, and\n    since any permutation $\\sigma$ admits a unique expression\n    \\begin{equation}\n        \\sigma = \\sigma^{(1)} {\\,\\diagup\\,} \\cdots {\\,\\diagup\\,} \\sigma^{(k)},\n    \\end{equation}\n    where~$\\sigma^{(1)}$, \\dots, $\\sigma^{(k)}$ are connected permutations,\n    there is no relation in~${\\bf FQSym}$ between the elements~${\\bf E}^\\sigma$\n    where~$\\sigma$ is a connected permutation.\n\n    Hence, by Proposition~\\ref{prop:BaseEHBaxter} and Corollary~\\ref{cor:BasesMult},\n    there is also no relation in~${\\bf Baxter}$ between the elements~${\\bf E}_J$\n    where~$J$ is a connected pair of twin binary trees. The proof for the\n    respective part is analogous.\n\\end{proof}\n\nLet us denote by~$B_C(z)$ the generating series of connected (resp. anti-connected)\npairs of twin binary trees. It follows, from Theorem~\\ref{thm:LiberteBaxter},\nthat the Hilbert series~$B(z)$ of~${\\bf Baxter}$ satisfies\n$B(z) = 1 \/ \\left(1 - B_C(z)\\right)$. Hence, the generating series~$B_C(z)$\nsatisfies\n\\begin{equation} \\label{eq:SGGenAlg}\n    B_C(z) = 1 - \\frac{1}{B(z)}.\n\\end{equation}\nFirst dimensions of algebraic generators of~${\\bf Baxter}$ are\n\\begin{equation}\n    0, 1, 1, 3, 11, 47, 221, 1113, 5903, 32607, 186143, 1092015.\n\\end{equation}\n\nHere follows algebraic generators of ${\\bf Baxter}$ of order $1$ to $4$:\n\\begin{equation}\n    {\\bf E}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n    \\end{tikzpicture}}\n        \\;\n        \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n    \\end{tikzpicture}}};\n\\end{equation}\n\n\\begin{equation}\n{\\bf E}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n    \\end{tikzpicture}}\n        \\;\n        \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-1){};\n        \\draw[Arete](0)--(1);\n    \\end{tikzpicture}}};\n\\end{equation}\n\n\\begin{equation}\n    {\\bf E}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](1)--(2);\n    \\end{tikzpicture}}\n        \\;\n        \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-2){};\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](2)--(1);\n        \\draw[Arete](0)--(2);\n    \\end{tikzpicture}}},\n    \\quad\n    {\\bf E}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n    \\end{tikzpicture}}\n        \\;\n        \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](1)--(2);\n    \\end{tikzpicture}}},\n    \\quad\n    {\\bf E}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud](1)at(1,-1){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(1);\n    \\end{tikzpicture}}\n        \\;\n        \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-1){};\n        \\node[Noeud](2)at(2,-2){};\n        \\draw[Arete](1)--(2);\n        \\draw[Arete](0)--(1);\n    \\end{tikzpicture}}};\n\\end{equation}\n\n\\begin{equation}\\begin{split}\n    {\\bf E}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-1){};\n        \\node[Noeud](3)at(3,-2){};\n        \\draw[Arete](2)--(3);\n        \\draw[Arete](1)--(2);\n    \\end{tikzpicture}}\n        \\;\n        \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-3){};\n        \\node[Noeud](2)at(2,-2){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](0)--(3);\n    \\end{tikzpicture}}}, &\n    \\quad\n    {\\bf E}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}\n        \\;\n        \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-2){};\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud](3)at(3,-2){};\n        \\draw[Arete](2)--(3);\n        \\draw[Arete](0)--(2);\n    \\end{tikzpicture}}},\n    \\quad\n    {\\bf E}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud](1)at(1,-1){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}\n        \\;\n        \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-2){};\n        \\node[Noeud](2)at(2,-3){};\n        \\draw[Arete](1)--(2);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(1);\n        \\draw[Arete](0)--(3);\n    \\end{tikzpicture}}},\n    \\quad\n    {\\bf E}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}\n        \\;\n        \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}},\n    \\quad\n    {\\bf E}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud](1)at(1,-1){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}\n        \\;\n        \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-1){};\n        \\node[Noeud](2)at(2,-3){};\n        \\node[Noeud](3)at(3,-2){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n        \\draw[Arete](0)--(1);\n    \\end{tikzpicture}}},\n    \\quad\n    {\\bf E}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\node[Noeud](2)at(2,-3){};\n        \\draw[Arete](1)--(2);\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](3)at(3,0){};\n        \\draw[Arete](3)--(0);\n    \\end{tikzpicture}}\n        \\;\n        \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud](1)at(1,-1){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}}, \\\\\n    {\\bf E}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-3){};\n        \\node[Noeud](2)at(2,-2){};\n        \\draw[Arete](2)--(1);\n        \\draw[Arete](0)--(2);\n        \\node[Noeud](3)at(3,0){};\n        \\draw[Arete](3)--(0);\n    \\end{tikzpicture}}\n        \\;\n        \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-1){};\n        \\node[Noeud](3)at(3,-2){};\n        \\draw[Arete](2)--(3);\n        \\draw[Arete](1)--(2);\n    \\end{tikzpicture}}}, &\n    \\quad\n    {\\bf E}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud](1)at(1,-1){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\draw[Arete](1)--(2);\n        \\node[Noeud](3)at(3,0){};\n        \\draw[Arete](3)--(1);\n    \\end{tikzpicture}}\n        \\;\n        \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}},\n    \\quad\n    {\\bf E}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud](1)at(1,-1){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\draw[Arete](1)--(2);\n        \\node[Noeud](3)at(3,0){};\n        \\draw[Arete](3)--(1);\n    \\end{tikzpicture}}\n        \\;\n        \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-2){};\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud](3)at(3,-2){};\n        \\draw[Arete](2)--(3);\n        \\draw[Arete](0)--(2);\n    \\end{tikzpicture}}},\n    \\quad\n    {\\bf E}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud](1)at(1,-3){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,0){};\n        \\draw[Arete](3)--(2);\n    \\end{tikzpicture}}\n        \\;\n        \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-1){};\n        \\node[Noeud](3)at(3,-2){};\n        \\draw[Arete](2)--(3);\n        \\draw[Arete](1)--(2);\n    \\end{tikzpicture}}},\n    \\quad\n    {\\bf E}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-3){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud](3)at(3,0){};\n        \\draw[Arete](3)--(2);\n    \\end{tikzpicture}}\n        \\;\n        \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-1){};\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-3){};\n        \\draw[Arete](2)--(3);\n        \\draw[Arete](1)--(2);\n        \\draw[Arete](0)--(1);\n    \\end{tikzpicture}}}.\n\\end{split}\\end{equation}\n\\medskip\n\n\\begin{Proposition}\n    If~$\\sigma$ is a connected (resp. anti-connected) Baxter permutation,\n    then any permutation~$\\nu$ such that~$\\sigma {\\:\\equiv_{\\operatorname{B}}\\:} \\nu$ is also\n    connected (resp. anti-connected).\n\\end{Proposition}\n\\begin{proof}\n    As any permutation, every Baxter permutation~$\\sigma$ can be uniquely\n    expressed as\n    \\begin{equation}\n        \\sigma = \\sigma^{(1)} {\\,\\diagup\\,} \\cdots {\\,\\diagup\\,} \\sigma^{(k)},\n    \\end{equation}\n    where the permutations~$\\sigma^{(i)}$ are connected for all $1 \\leq i \\leq k$.\n    Moreover, since~$\\sigma$ avoids the permutation patterns $2-41-3$\n    and $3-14-2$, the permutations~$\\sigma^{(i)}$ also does, and hence,\n    the~$\\sigma^{(i)}$ are Baxter permutations. This shows that the generating\n    series of connected Baxter permutations is~$B_C(z)$ and thus, that\n    connected Baxter permutations, connected pairs of twin binary trees,\n    and connected minimal permutations of Baxter equivalence classes are\n    equinumerous.\n\n    The proposition follows from Theorem~\\ref{thm:EquivBXBaxter} saying\n    that each ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class of permutations contains\n    exactly one Baxter permutation. The proof for the respective part is\n    analogous.\n\\end{proof}\n\n\\begin{Corollaire}\n    The algebra~${\\bf Baxter}$ is free on the elements~${\\bf E}_J$ (resp.~${\\bf H}_J$)\n    where the Baxter permutation belonging to the ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence\n    class encoded by~$J$ is connected (resp. anti-connected).\n\\end{Corollaire}\n\n\\subsubsection{Bidendriform bialgebra structure and self-duality}\nA Hopf algebra $(H, \\cdot, \\Delta)$ can be fit into a bidendriform\nbialgebra structure~\\cite{Foi07} if $(H^+, \\prec, \\succ)$ is a dendriform\nalgebra~\\cite{Lod01} and $(H^+, \\Delta_\\Gauche, \\Delta_\\Droite)$ a codendriform coalgebra,\nwhere~$H^+$ is the augmentation ideal of~$H$. The operators~$\\prec$, $\\succ$,\n$\\Delta_\\Gauche$ and~$\\Delta_\\Droite$ have to fulfill some compatibility relations. In\nparticular, for all~$x, y \\in H^+$, the product~$\\cdot$ of~$H$ is retrieved\nby $x \\cdot y = x \\prec y + x \\succ y$ and the coproduct~$\\Delta$ of~$H$\nis retrieved by $\\Delta(x) = 1 \\otimes x + \\Delta_\\Gauche(x) + \\Delta_\\Droite(x) + x \\otimes 1$.\nRecall that an element~$x \\in H^+$ is \\emph{totally primitive} if\n$\\Delta_\\Gauche(x) = 0 = \\Delta_\\Droite(x)$.\n\\medskip\n\nThe Hopf algebra~${\\bf FQSym}$ admits a bidendriform bialgebra structure~\\cite{Foi07}.\nIndeed, for all~$\\sigma, \\nu \\in \\mathfrak{S}_n$ with~$n \\geq 1$, set\n\\begin{equation}\n    {\\bf F}_\\sigma \\prec {\\bf F}_\\nu :=\n    \\sum_{\\substack{\\pi \\; \\in \\; \\sigma \\; \\cshuffle \\; \\nu \\\\\n    \\pi_{|\\pi|} = \\sigma_{|\\sigma|}}} {\\bf F}_\\pi,\n\\end{equation}\n\\begin{equation}\n    {\\bf F}_\\sigma \\succ {\\bf F}_\\nu :=\n    \\sum_{\\substack{\\pi \\; \\in \\; \\sigma \\; \\cshuffle \\; \\nu \\\\\n    \\pi_{|\\pi|} = \\nu_{|\\nu|} + |\\sigma|}} {\\bf F}_\\pi,\n\\end{equation}\n\\begin{equation}\n    \\Delta_\\Gauche({\\bf F}_\\sigma) :=\n    \\sum_{\\substack{\\sigma = uv \\\\ \\max(u) = \\max(\\sigma)}}\n    {\\bf F}_{\\operatorname{std}(u)} \\otimes {\\bf F}_{\\operatorname{std}(v)},\n\\end{equation}\n\\begin{equation}\n    \\Delta_\\Droite({\\bf F}_\\sigma) := \\sum_{\\substack{\\sigma = uv \\\\\n    \\max(v) = \\max(\\sigma)}}\n    {\\bf F}_{\\operatorname{std}(u)} \\otimes {\\bf F}_{\\operatorname{std}(v)}.\n\\end{equation}\n\n\\begin{Proposition} \\label{prop:MemeLettreEquiv}\n    If~$\\equiv$ is an equivalence relation defined on~$A^*$ satisfying the\n    conditions of Theorem~\\ref{thm:HivertJanvier} and additionally, for\n    all~$u, v \\in A^*$, the relation~$u \\equiv v$ implies $u_{|u|} = v_{|v|}$,\n    then, the family defined in~(\\ref{eq:EquivFQSym}) spans a bidendriform\n    sub-bialgebra of~${\\bf FQSym}$ that is free as an algebra, cofree as a coalgebra,\n    self-dual, free as a dendriform algebra on its totally primitive elements,\n    and the Lie algebra of its primitive elements is free.\n\\end{Proposition}\n\\begin{proof}\n    It is enough to show that the operators~$\\prec$, $\\succ$, $\\Delta_\\Gauche$\n    and~$\\Delta_\\Droite$ of~${\\bf FQSym}$ are well-defined in the Hopf subalgebra~$H$\n    of~${\\bf FQSym}$ spanned by the elements\n    $\\left\\{{\\bf P}_{\\widehat{\\sigma}}\\right\\}_{\\widehat{\\sigma} \\in \\mathfrak{S}\/_\\equiv}$.\n    In this way, $H$ is endowed with a structure of bidendriform bialgebra\n    and the results of Foissy~\\cite{Foi07} imply the rest of the proposition.\n\n    Fix $\\widehat{\\sigma}, \\widehat{\\nu} \\in \\mathfrak{S}\/_\\equiv$ and an\n    element~${\\bf F}_\\pi$ appearing in the product\n    ${\\bf P}_{\\widehat{\\sigma}} \\prec {\\bf P}_{\\widehat{\\nu}}$. Hence, there is\n    a permutation~$\\sigma \\in \\widehat{\\sigma}$ such that\n    $\\pi_{|\\pi|} = \\sigma_{|\\sigma|}$. Let~$\\pi'$ a permutation such\n    that~$\\pi \\equiv \\pi'$. By Theorem~\\ref{thm:HivertJanvier}, the\n    element~${\\bf F}_{\\pi'}$ appears in the product\n    ${\\bf P}_{\\widehat{\\sigma}} \\cdot {\\bf P}_{\\widehat{\\nu}}$, and hence, it also\n    appears in ${\\bf P}_{\\widehat{\\sigma}} \\prec {\\bf P}_{\\widehat{\\nu}}$\n    or in ${\\bf P}_{\\widehat{\\sigma}} \\succ {\\bf P}_{\\widehat{\\nu}}$. Assume by\n    contradiction that~${\\bf F}_{\\pi'}$ appears in\n    ${\\bf P}_{\\widehat{\\sigma}} \\succ {\\bf P}_{\\widehat{\\nu}}$.\n    There are two permutations~$\\sigma' \\in \\widehat{\\sigma}$\n    and~$\\nu' \\in \\widehat{\\nu}$ such that\n    $\\pi'_{|\\pi'|} = \\nu'_{|\\nu'|} + |\\sigma'|$. That implies that\n    $\\pi_{|\\pi|} \\ne \\pi'_{|\\pi'|}$ and contradicts the fact that all\n    permutations of a same $\\equiv$-equivalence class end with a same\n    letter. Hence, the element~${\\bf F}_{\\pi'}$ appears in\n    ${\\bf P}_{\\widehat{\\sigma}} \\prec {\\bf P}_{\\widehat{\\nu}}$, showing that\n    the product~$\\prec$ is well-defined in~$H$. Then so is~$\\succ$\n    since~$\\prec + \\succ$ is the whole product.\n\n    Fix $\\widehat{\\sigma} \\in \\mathfrak{S}\/_\\equiv$ and an\n    element ${\\bf F}_\\nu \\otimes {\\bf F}_\\pi$ appearing in the coproduct\n    $\\Delta_\\Gauche({\\bf P}_{\\widehat{\\sigma}})$. Hence, there is a\n    permutation~$\\sigma \\in \\widehat{\\sigma}$ such that~$\\sigma = uv$,\n    $\\nu = \\operatorname{std}(u)$, $\\pi = \\operatorname{std}(v)$ and the maximal letter of~$uv$ is\n    in the factor~$u$. Now, let~$\\nu'$ and~$\\pi'$ be two permutations such\n    that $\\nu \\equiv \\nu'$, $\\pi \\equiv \\pi'$. Let us show that the element\n    ${\\bf F}_{\\nu'} \\otimes {\\bf F}_{\\pi'}$ also appears in~$\\Delta_\\Gauche({\\bf P}_{\\widehat{\\sigma}})$.\n    For that, let~$u'$ be a permutation of~$u$ such that~$\\operatorname{std}(u') = \\nu'$,\n    and~$v'$ be a permutation of~$v$ such that~$\\operatorname{std}(v') = \\pi'$. Since\n    $\\operatorname{ev}(u') = \\operatorname{ev}(u)$, $\\operatorname{std}(u') \\equiv \\operatorname{std}(u)$, and~$\\equiv$ is\n    compatible with the destandardization process, one has~$u \\equiv u'$.\n    For the same reason,~$v \\equiv v'$, and since~$\\equiv$ is a congruence,\n    one has~$uv \\equiv u'v'$. Finally, since the maximal letter of~$uv$\n    is in~$u$, the maximal letter of~$u'v'$ is in~$u'$, showing that the\n    element ${\\bf F}_{\\nu'} \\otimes {\\bf F}_{\\pi'}$ appears in\n    $\\Delta_\\Gauche({\\bf P}_{\\widehat{\\sigma}})$. Thus, the coproduct~$\\Delta_\\Gauche$\n    is well-defined in~$H$. The proof for the coproduct~$\\Delta_\\Droite$ is analogous.\n\\end{proof}\n\n\\begin{Corollaire} \\label{cor:BaxterBidendr}\n    The Hopf algebra ${\\bf Baxter}$ is free as an algebra, cofree as a coalgebra,\n    self-dual, free as a dendriform algebra on its totally primitive elements,\n    and the Lie algebra of its primitive elements is free.\n\\end{Corollaire}\n\\begin{proof}\n    Since all words of a same ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence class end with a same\n    letter,~${\\:\\equiv_{\\operatorname{B}}\\:}$ satisfies the premises of\n    Proposition~\\ref{prop:MemeLettreEquiv} and hence,~${\\bf Baxter}$ satisfies\n    all stated properties.\n\\end{proof}\n\nConsidering the map $\\theta' : {\\bf PBT} \\hookrightarrow {\\bf FQSym}$ that is the\ninjection from~${\\bf PBT}$ to~${\\bf FQSym}$ and\n$\\phi' : {\\bf FQSym}^\\star \\twoheadrightarrow {\\bf PBT}^\\star$ the surjection\nfrom~${\\bf FQSym}^\\star$ to ${\\bf PBT}^\\star$, it is well-known (see~\\cite{HNT05})\nthat the map $\\phi' \\circ \\psi \\circ \\theta'$ induces an isomorphism\nbetween~${\\bf PBT}$ and~${\\bf PBT}^\\star$. Hence, since by Corollary~\\ref{cor:BaxterBidendr},\nthe Hopf algebras~${\\bf Baxter}$ and~${\\bf Baxter}^\\star$ are isomorphic, it is natural\nto test if an analogous map is still an isomorphism between~${\\bf Baxter}$\nand~${\\bf Baxter}^\\star$. However, denoting by $\\theta : {\\bf Baxter} \\hookrightarrow {\\bf FQSym}$\nthe injection from~${\\bf Baxter}$ to~${\\bf FQSym}$, the map\n$\\phi \\circ \\psi \\circ \\theta : {\\bf Baxter} \\rightarrow {\\bf Baxter}^\\star$ is not\nan isomorphism. Indeed\n\\begin{align}\n    \\phi \\circ \\psi \\circ \\theta \\left(\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}} \\right)\n    & = \\phi \\circ \\psi \\left( {\\bf F}_{2143} + {\\bf F}_{2413} \\right)\n      = \\phi \\left( {\\bf F}^\\star_{2143} + {\\bf F}^\\star_{3142} \\right)\n      =\n    {\\bf P}^\\star_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}^\\star_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}}, \\\\\n    \\phi \\circ \\psi \\circ \\theta \\left(\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}} \\right)\n    & = \\phi \\circ \\psi \\left( {\\bf F}_{3142} + {\\bf F}_{3412} \\right)\n      = \\phi \\left( {\\bf F}^\\star_{2413} + {\\bf F}^\\star_{3412} \\right)\n      =\n    {\\bf P}^\\star_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}^\\star_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}},\n\\end{align}\nshowing that~$\\phi \\circ \\psi \\circ \\theta$ is not injective.\n\n\\subsubsection{Primitive and totally primitive elements}\n\nSince the family~$\\left\\{{\\bf E}_J\\right\\}_{J \\in C}$\n(resp.~$\\left\\{{\\bf H}_J\\right\\}_{J \\in C}$), where~$C$ is the set of connected\n(resp. anti-connected) pairs of twin binary trees are indecomposable elements\nof~${\\bf Baxter}$, its dual family~$\\left\\{{\\bf E}^\\star_J\\right\\}_{J \\in C}$\n(resp.~$\\left\\{{\\bf H}^\\star_J\\right\\}_{J \\in C}$) forms a basis of the Lie\nalgebra of  the primitive elements of~${\\bf Baxter}^\\star$. By\nCorollary~\\ref{cor:BaxterBidendr}, this Lie algebra is free.\n\\medskip\n\nFollowing~\\cite{Foi07}, the generating series $B_T(z)$ of the totally\nprimitive elements of ${\\bf Baxter}$ is\n\\begin{equation}\n    B_T(z) = \\frac{B(z) - 1}{B(z)^2}.\n\\end{equation}\nFirst dimensions of totally primitive  elements of ${\\bf Baxter}$ are\n\\begin{equation}\n    0, 1, 0, 1, 4, 19, 96, 511, 2832, 16215, 95374, 573837.\n\\end{equation}\n\nHere follows a basis of the totally primitive elements of ${\\bf Baxter}$ of\norder $1$, $3$ and $4$:\n\\begin{align}\n    t_{1, 1} & =\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n    \\end{tikzpicture}}}, \\\\[1em]\n    t_{3, 1} & =\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](1)--(2);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-2){};\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](2)--(1);\n        \\draw[Arete](0)--(2);\n    \\end{tikzpicture}}}\n    -\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-2){};\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](2)--(1);\n        \\draw[Arete](0)--(2);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](1)--(2);\n    \\end{tikzpicture}}}, \\\\[1em]\n    t_{4, 1} & =\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud](1)at(1,-1){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-1){};\n        \\node[Noeud](2)at(2,-3){};\n        \\node[Noeud](3)at(3,-2){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n        \\draw[Arete](0)--(1);\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud](1)at(1,-1){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-2){};\n        \\node[Noeud](2)at(2,-3){};\n        \\draw[Arete](1)--(2);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(1);\n        \\draw[Arete](0)--(3);\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-2){};\n        \\node[Noeud](2)at(2,-3){};\n        \\draw[Arete](1)--(2);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(1);\n        \\draw[Arete](0)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud](1)at(1,-1){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-1){};\n        \\node[Noeud](2)at(2,-3){};\n        \\node[Noeud](3)at(3,-2){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n        \\draw[Arete](0)--(1);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-2){};\n        \\node[Noeud](1)at(1,-1){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}} \\\\\n    & -\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}}\n    -\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-3){};\n        \\node[Noeud](2)at(2,-2){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](0)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-1){};\n        \\node[Noeud](3)at(3,-2){};\n        \\draw[Arete](2)--(3);\n        \\draw[Arete](1)--(2);\n    \\end{tikzpicture}}}\n    -\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-2){};\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud](3)at(3,-2){};\n        \\draw[Arete](2)--(3);\n        \\draw[Arete](0)--(2);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}}, \\nonumber \\\\\n    t_{4, 2} & =\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-2){};\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud](3)at(3,-2){};\n        \\draw[Arete](2)--(3);\n        \\draw[Arete](0)--(2);\n    \\end{tikzpicture}}}\n    -\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-3){};\n        \\node[Noeud](2)at(2,-2){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](0)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-1){};\n        \\node[Noeud](3)at(3,-2){};\n        \\draw[Arete](2)--(3);\n        \\draw[Arete](1)--(2);\n    \\end{tikzpicture}}}, \\\\\n    t_{4, 3} & =\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-1){};\n        \\node[Noeud](3)at(3,-2){};\n        \\draw[Arete](2)--(3);\n        \\draw[Arete](1)--(2);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-3){};\n        \\node[Noeud](2)at(2,-2){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](0)--(3);\n    \\end{tikzpicture}}}\n    -\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-2){};\n        \\node[Noeud](2)at(2,-1){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud](3)at(3,-2){};\n        \\draw[Arete](2)--(3);\n        \\draw[Arete](0)--(2);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}}, \\\\\n    t_{4, 4} & =\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}}\n    -\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2,0){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](2)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2,-2){};\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(2);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}}}.\n\\end{align}\n\n\\subsubsection{\\texorpdfstring{Compatibility with the $\\#$ product}\n                              {Compatibility with the sharp product}}\nAval and Viennot~\\cite{AV10} endowed~${\\bf PBT}$ with a new associative product\ncalled the \\emph{$\\#$ product}. The product of two elements of~${\\bf PBT}$\nof degrees~$n$ and~$m$ is an element of degree~$n + m - 1$. Aval, Novelli,\nand Thibon~\\cite{ANT11} generalized the $\\#$ product at the level of the\nassociative algebra and showed that it is still well-defined in~${\\bf FQSym}$.\n\\medskip\n\nLet for all~$k \\geq 1$ the linear maps $d_k : {\\bf FQSym} \\to {\\bf FQSym}$ defined\nfor any permutation~$\\sigma$ of~$\\mathfrak{S}_n$ by\n\\begin{equation}\n    d_k({\\bf F}_\\sigma) :=\n    \\begin{cases}\n        {\\bf F}_{\\operatorname{std}(\\sigma_1 \\dots \\sigma_i \\sigma_{i + 2} \\dots \\sigma_n)} &\n            \\mbox{if there is $1 \\leq i \\leq n - 1$ such that $\\sigma_i = k$\n            and $\\sigma_{i + 1} = k + 1$}, \\\\\n        0 & \\mbox{otherwise}.\n    \\end{cases}\n\\end{equation}\nNow, for any permutations~$\\sigma$ and~$\\nu$, the $\\#$-product is defined\nin~${\\bf FQSym}$ by\n\\begin{equation}\n    {\\bf F}_\\sigma \\# {\\bf F}_\\nu := d_n\\left({\\bf F}_\\sigma \\cdot {\\bf F}_\\nu\\right),\n\\end{equation}\nwhere~$n$ is the size of~$\\sigma$.\n\n\\begin{Proposition} \\label{prop:DkInterBaxter}\n    The linear maps~$d_k$ are well-defined in~${\\bf Baxter}$. More precisely,\n    one has for any pair of twin binary trees~$J := (T_0, T_1)$,\n    \\begin{equation}\n        d_k({\\bf P}_J) =\n        \\begin{cases}\n            {\\bf P}_{J'} &\n            \\substack{\\mbox{if the $k\\!+\\!1$-st (resp. $k$-th) node is a\n            child of the} \\\\\n            \\mbox{$k$-th (resp. $k\\!+\\!1$-st) node in $T_L$ (resp. $T_R$),}} \\\\\n            0 & \\mbox{otherwise},\n        \\end{cases}\n    \\end{equation}\n    where $J' := (T'_L, T'_R)$ is the pair of twin binary trees obtained\n    by contracting in~$T_L$ and~$T_R$ the edges connecting the\n    $k$-th and the $k\\!+\\!1$-st nodes.\n\\end{Proposition}\n\\begin{proof}\n    This proof relies on the fact that, according to Proposition~\\ref{prop:BXExtLin},\n    the permutations of a Baxter equivalence class coincide with linear\n    extensions of the posets~$\\bigtriangleup(T_L)$ and~$\\bigtriangledown(T_R)$.\n\n    We have two cases to consider whether the $k\\!+\\!1$-st (resp. $k$-th)\n    node is a child of the $k$-th (resp. $k\\!+\\!1$-st) node in~$T_L$\n    (resp.~$T_R$).\n    \\begin{enumerate}[label = {\\bf Case \\arabic*.}, fullwidth]\n        \\item If so, there is in the Baxter equivalence class represented\n        by~$J$ some permutations with a factor~$k.(k\\!+\\!1)$. The map~$d_k$\n        deletes letters~$k\\!+\\!1$ in these permutations and standardizes\n        them. The obtained permutations coincide with linear extensions\n        of the posets~$\\bigtriangleup(T'_L)$ and~$\\bigtriangledown(T'_R)$.\n        \\item If this is not the case, since the $k$-th and $k\\!+\\!1$-st nodes\n        of a binary tree are on a same path starting from the root, no\n        permutation of the Baxter class represented by~$J$ has a\n        factor~$k.(k\\!+\\!1)$. Hence,~$d_k({\\bf P}_J) = 0$. \\qedhere\n    \\end{enumerate}\n\\end{proof}\n\nOne has for example\n\\begin{equation}\n    d_3\\left({\\bf P}_{\n    \\scalebox{.15}{\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0.0,-1){};\n        \\node[Noeud](1)at(1.0,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2.0,-2){};\n        \\node[Noeud](3)at(3.0,-3){};\n        \\draw[Arete](2)--(3);\n        \\node[Noeud](4)at(4.0,-1){};\n        \\draw[Arete](4)--(2);\n        \\node[Noeud](5)at(5.0,-2){};\n        \\draw[Arete](4)--(5);\n        \\draw[Arete](1)--(4);\n    \\end{tikzpicture}} \\;\n    \\scalebox{.15}{\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0.0,-2){};\n        \\node[Noeud](1)at(1.0,-3){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2.0,-1){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3.0,0){};\n        \\draw[Arete](3)--(2);\n        \\node[Noeud](4)at(4.0,-2){};\n        \\node[Noeud](5)at(5.0,-1){};\n        \\draw[Arete](5)--(4);\n        \\draw[Arete](3)--(5);\n    \\end{tikzpicture}}} \\right) =\n    {\\bf P}_{\n    \\scalebox{.15}{\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0.0,-1){};\n        \\node[Noeud](1)at(1.0,0){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud](2)at(2.0,-2){};\n        \\node[Noeud](3)at(3.0,-1){};\n        \\draw[Arete](3)--(2);\n        \\node[Noeud](4)at(4.0,-2){};\n        \\draw[Arete](3)--(4);\n        \\draw[Arete](1)--(3);\n    \\end{tikzpicture}} \\;\n    \\scalebox{.15}{\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0.0,-1){};\n        \\node[Noeud](1)at(1.0,-2){};\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](2)at(2.0,0){};\n        \\draw[Arete](2)--(0);\n        \\node[Noeud](3)at(3.0,-2){};\n        \\node[Noeud](4)at(4.0,-1){};\n        \\draw[Arete](4)--(3);\n        \\draw[Arete](2)--(4);\n    \\end{tikzpicture}}}\\,.\n\\end{equation}\n\nProposition~\\ref{prop:DkInterBaxter} shows in particular that the $\\#$ product\nin well-defined in~${\\bf Baxter}$.\n\n\\subsection{\\texorpdfstring{Connections with other Hopf subalgebras of ${\\bf FQSym}$}\n                           {Connections with other Hopf subalgebras of FQSym}}\n\n\\subsubsection{\\texorpdfstring{Connection with the Hopf algebra ${\\bf PBT}$}\n                              {Connection with the Hopf algebra PBT}}\nWe already recalled that the sylvester congruence leads to the construction\nof the Hopf subalgebra~${\\bf PBT}$~\\cite{LR98} of~${\\bf FQSym}$, whose fundamental\nbasis\n\\begin{equation}\n    \\left\\{{\\bf P}_T : T \\in \\mathcal{B}\\mathcal{T}\\right\\}\n\\end{equation}\nis defined in accordance\nwith~(\\ref{eq:EquivFQSym}) (see~\\cite{HNT02} and~\\cite{HNT05}). By\nProposition~\\ref{prop:LienSylv}, every ${\\:\\equiv_{\\operatorname{S}}\\:}$-equivalence class is a\nunion of some ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence classes. Hence, we have the following\ninjective Hopf map:\n\\begin{equation}\n    \\rho : {\\bf PBT} \\hookrightarrow {\\bf Baxter},\n\\end{equation}\nsatisfying\n\\begin{equation}\n    \\rho \\left({\\bf P}_T\\right) =\n    \\sum_{\\substack{T' \\; \\in \\; \\mathcal{B}\\mathcal{T} \\\\ J := (T', T) \\; \\in \\; \\mathcal{T}\\mathcal{B}\\mathcal{T}}}\n    {\\bf P}_J,\n\\end{equation}\nfor any binary tree~$T$. For example,\n\\begin{align}\n    \\rho \\left(\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,Marque1](0)at(0,-2){};\n        \\node[Noeud,Marque1](1)at(1,-1){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud,Marque1](2)at(2,0){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud,Marque1](3)at(3,-2){};\n        \\node[Noeud,Marque1](4)at(4,-1){};\n        \\draw[Arete](4)--(3);\n        \\draw[Arete](2)--(4);\n    \\end{tikzpicture}}}\n    \\right)\n    & =\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-2){};\n        \\node[Noeud](2)at(2,-3){};\n        \\draw[Arete](1)--(2);\n        \\node[Noeud](3)at(3,-1){};\n        \\draw[Arete](3)--(1);\n        \\node[Noeud](4)at(4,-2){};\n        \\draw[Arete](3)--(4);\n        \\draw[Arete](0)--(3);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,Marque1](0)at(0,-2){};\n        \\node[Noeud,Marque1](1)at(1,-1){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud,Marque1](2)at(2,0){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud,Marque1](3)at(3,-2){};\n        \\node[Noeud,Marque1](4)at(4,-1){};\n        \\draw[Arete](4)--(3);\n        \\draw[Arete](2)--(4);\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,-1){};\n        \\node[Noeud](1)at(1,-2){};\n        \\node[Noeud](2)at(2,-3){};\n        \\draw[Arete](1)--(2);\n        \\draw[Arete](0)--(1);\n        \\node[Noeud](3)at(3,0){};\n        \\draw[Arete](3)--(0);\n        \\node[Noeud](4)at(4,-1){};\n        \\draw[Arete](3)--(4);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,Marque1](0)at(0,-2){};\n        \\node[Noeud,Marque1](1)at(1,-1){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud,Marque1](2)at(2,0){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud,Marque1](3)at(3,-2){};\n        \\node[Noeud,Marque1](4)at(4,-1){};\n        \\draw[Arete](4)--(3);\n        \\draw[Arete](2)--(4);\n    \\end{tikzpicture}}}\n    +\n    {\\bf P}_{\\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud](0)at(0,0){};\n        \\node[Noeud](1)at(1,-1){};\n        \\node[Noeud](2)at(2,-3){};\n        \\node[Noeud](3)at(3,-2){};\n        \\draw[Arete](3)--(2);\n        \\node[Noeud](4)at(4,-3){};\n        \\draw[Arete](3)--(4);\n        \\draw[Arete](1)--(3);\n        \\draw[Arete](0)--(1);\n    \\end{tikzpicture}}\n    \\;\n    \\scalebox{0.15}{%\n    \\begin{tikzpicture}\n        \\node[Noeud,Marque1](0)at(0,-2){};\n        \\node[Noeud,Marque1](1)at(1,-1){};\n        \\draw[Arete](1)--(0);\n        \\node[Noeud,Marque1](2)at(2,0){};\n        \\draw[Arete](2)--(1);\n        \\node[Noeud,Marque1](3)at(3,-2){};\n        \\node[Noeud,Marque1](4)at(4,-1){};\n        \\draw[Arete](4)--(3);\n        \\draw[Arete](2)--(4);\n    \\end{tikzpicture}}}.\n\\end{align}\n\n\\subsubsection{\\texorpdfstring{Connection with the Hopf algebra $\\DSym{3}$}\n                              {Connection with the Hopf algebra DSym3}}\nThe congruence~$\\EquivR{3}$ leads to the construction of the Hopf\nsubalgebra~$\\DSym{3}$ of~${\\bf FQSym}$, whose fundamental basis\n\\begin{equation}\n    \\left\\{{\\bf P}_{\\widehat{\\sigma}} :\n    \\widehat{\\sigma} \\in \\mathfrak{S} \/_{\\EquivR{3}}\\right\\}\n\\end{equation}\nis defined in accordance with~(\\ref{eq:EquivFQSym}) (see~\\cite{NRT11}).\nBy Proposition~\\ref{prop:Lien3Recul}, every $\\EquivR{3}$-equivalence\nclass of permutations is a union of some ${\\:\\equiv_{\\operatorname{B}}\\:}$-equivalence classes.\nHence, we have the following injective Hopf map:\n\\begin{equation}\n    \\alpha : \\DSym{3} \\hookrightarrow {\\bf Baxter},\n\\end{equation}\nsatisfying\n\\begin{equation}\n    \\alpha \\left( {\\bf P}_{\\widehat{\\sigma}} \\right) =\n    \\sum_{\\sigma \\; \\in \\; \\widehat{\\sigma} \\cap \\mathfrak{S}^{\\operatorname{B}}} {\\bf P}_{{\\sf P}(\\sigma)},\n\\end{equation}\nfor any $\\EquivR{3}$-equivalence class~$\\widehat{\\sigma}$ of permutations.\n\n\\subsubsection{\\texorpdfstring{Connection with the Hopf algebra ${\\bf Sym}$}\n                              {Connection with the Hopf algebra Sym}}\nThe hypoplactic congruence~\\cite{N98} leads to the construction of the Hopf\nsubalgebra~${\\bf Sym}$ of~${\\bf FQSym}$. As already mentioned, the hypoplactic congruence\nis the same as the congruence~$\\EquivR{2}$ when both are restricted on\npermutations. Moreover, the hypoplactic equivalence classes of permutations\ncan  be encoded by binary words. Indeed, if~$\\widehat{\\sigma}$ is such an\nequivalence class,~$\\widehat{\\sigma}$ contains all the permutations having\na given recoil set. Thus, the class~$\\widehat{\\sigma}$ can be encoded by\nthe binary word~$b$ of length~$n - 1$ where~$n$ is the length of the elements\nof~$\\widehat{\\sigma}$ and~$b_i = 1$ if and only if~$i$ is a recoil of the elements\nof~$\\widehat{\\sigma}$. We denote by\n\\begin{equation}\n    \\left\\{{\\bf P}_b : b \\in \\{0, 1\\}^*\\right\\}\n\\end{equation}\nthe fundamental basis of~${\\bf Sym}$ indexed by binary words.\n\\medskip\n\nSince~${\\bf PBT}$ is a Hopf subalgebra of~${\\bf Baxter}$ and~${\\bf Sym}$ is a Hopf subalgebra\nof~${\\bf PBT}$~\\cite{HNT05}, ${\\bf Sym}$ is itself a Hopf subalgebra of~${\\bf Baxter}$.\nThe injective Hopf map\n\\begin{equation}\n    \\beta : {\\bf Sym} \\hookrightarrow {\\bf PBT},\n\\end{equation}\nsatisfies, thanks to the fact that the hypoplactic equivalence classes are\nunion of ${\\:\\equiv_{\\operatorname{S}}\\:}$-equivalence classes and Proposition~\\ref{prop:FeuillesInversions},\n\\begin{equation}\n    \\beta \\left({\\bf P}_b\\right) =\n    \\sum_{\\substack{T \\; \\in \\; \\mathcal{B}\\mathcal{T} \\\\ \\operatorname{cnp}(T) = b}} {\\bf P}_T,\n\\end{equation}\nfor any binary word~$b$. From a combinatorial point of view, given a binary\nword~$b$, the map~$\\beta$ computes the sum of the binary trees having~$b$\nas canopy. The composition $\\rho \\circ \\beta$ is an injective Hopf map\nfrom~${\\bf Sym}$ to~${\\bf Baxter}$. From a combinatorial point of view, given a\nbinary word~$b$, the map $\\rho \\circ \\beta$ computes the sum of the pairs\nof twin binary trees~$(T_L, T_R)$ where the canopy of~$T_R$ is~$b$ and\nthe canopy of~$T_L$ is the complementary of~$b$.\n\n\\subsubsection{Full diagram of embeddings}\nFigure~\\ref{fig:DiagrammeAHC} summarizes the relations between known Hopf\nalgebras related to ${\\bf Baxter}$.\n\\begin{figure}[ht]\n    \\centering\n    \\begin{tikzpicture}[scale=.5]\n        \\node(FQSym)at(5,0){${\\bf FQSym}$};\n        \\node(DSym4)at(0,-3){$\\DSym{4}$};\n        \\node(Baxter)at(5,-3){${\\bf Baxter}$};\n        \\node(DSym3)at(0,-6){$\\DSym{3}$};\n        \\node(PBT)at(10,-6){${\\bf PBT}$};\n        \\node(Sym)at(5,-9){${\\bf Sym}$};\n        %\n        \\draw[Injection,dashed](DSym4)--(FQSym);\n        \\draw[Injection](Baxter) edge node[anchor=south,right] {$\\theta$} (FQSym);\n        \\draw[Injection](DSym3)--(DSym4);\n        \\draw[Injection](DSym3) edge node[anchor=south,below] {$\\alpha$} (Baxter);\n        \\draw[Injection](PBT) edge node[anchor=south,below] {$\\rho$} (Baxter);\n        \\draw[Injection](Sym)--(DSym3);\n        \\draw[Injection](Sym) edge node[anchor=south,below] {$\\beta$} (PBT);\n    \\end{tikzpicture}\n    \\caption{Diagram of injective Hopf maps between some Hopf algebras\n    related to ${\\bf Baxter}$. Arrows~$\\rightarrowtail$ are injective Hopf maps.}\n    \\label{fig:DiagrammeAHC}\n\\end{figure}\n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Observations}\n\n\\noindent\nDust enshrouded activity of a galaxy can be studied ideally by\nmid--infrared (MIR) observations. To explore the origin of the nuclear\nMIR emission of galaxies as being due to either active galactic nuclei\n(AGN) or star formation, observations of high spatial resolution are\nrequired.\n\n\\noindent\nThe nuclear MIR surface brightness is introduced as a quantitative\nmeasurement for AGN and starburst activity. However, one is unable to\ndistinguish between both activity types using the nuclear MIR surface\nbrightness derived from 4m class telescopes, even when adopting the\ntheoretical diffraction limit of $0.7''$ (FWHM) of such telescopes\n(cmp. small gray symbols in Fig.~\\ref{surf.ps}).  Since the PSF width\nis twice as large as for a 8m class telescope and the point source\nsensitivity is a factor 16 lower, it becomes more difficult for a 4m\nto resolve starburst and the surface brightness of unresolved sources\nis reduced. Data recently obtained at 8m class telescopes\n(Siebenmorgen et al. 2008) show that, out to a distance of 100Mpc, the\nMIR surface brightness acquired clearly differentiate AGN from SB\nbehavior (Fig.~\\ref{surf.ps}). Utilizing VISIR at the VLT the AGN\nstill appear point like whereas most starburst are resolved in the\nMIR.  This discrimination was made possible by an increase in spatial\nresolution by a factor 2. Therefore it provides a clue to what will be\npossible by increasing the spatial resolution by another factor 5 when\ngoing from the VLT to the proposed extreme large telescope such as the\nE-ELT which will be 40m class. For the E-ELT a mid infrared instrument\nis included in the instrumentation plan and it has beside imaging also\nhigh resolution spectroscopic and polarimetric observing capabilities\n(Brandl et al., 2010).\n\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=11cm,angle=0]{extin.ps}\n\\caption{Mean extinction curve of the ISM by Fritzpatrick\\&Massa\n  (2007, black) and a fit (magenta) by the dust model of\n  Sect.~\\ref{dust}. Individual dust components are shown as\n  labeled. The grey areas indicate the 1$\\sigma$ deviation as of the\n  samples.\n\\label{dust.ps} }\n\\end{center}\n\\end{figure}\n\n\n\\section{Dust model \\label{dust}}\n\n\n\\noindent \nTeams interested in modelling the processing of radiation by dust in\ngalaxies often apply a dust model as derived for the diffuse ISM of\nthe Milky Way.  Dust cross sections are computed using similar optical\nconstants and temperature fluctuating particles such as PAHs are\nincluded.  We developed one of such dust models in which {\\it {large}}\n($60\\rm{\\AA}<a<0.2-0.3\\mu$m) silicate (Draine 2003) and amorphous\ncarbon (Zubko et al. 2004) grains and {\\it {small}} graphite\n($5\\rm{\\AA}<a<80$\\,\\AA) grains are considered.  We apply a power law\nsize distribution: $n(a) \\propto a^{-3.5}$ and absorption and\nscattering cross-sections are computed with Mie theory.  In addition\nthere are {\\it {PAHs}} with 30 and 200 C atoms with absorption cross\nsection as given by Schutte et al. (1993).  By computing cross\nsections above 100\\,eV, we consider an approximation of kinetic energy\nlosses (Dwek\\& Smith 1996) and apply it to all particles. The choice\nof parameters is set up to achieve a fit of the mean extinction curve\nof the ISM (Fitzpatrick \\& Massa 2007), as shown in Fig.\\ref{dust.ps}.\n\n\\noindent \nWith the advent of {\\it{ISO}} and {\\it{Spitzer}} more PAH emission\nfeatures and more details of their band structures are detected\n(Tielens 2008).  We consider 17 emission bands and take Lorentzian\nprofiles (Siebenmorgen et al. 1998). Parameters of what we call\nastronomical PAH are calibrated using mid-IR spectra of starburst\nnuclei and the RT model as of Sect.~{\\ref{sb}}. PAH cross sections of\nthe emission bands are listed by Siebenmorgen \\& Kr\\\"ugel (2001). In\nthe model, we use dust abundances of [X]\/[H] (ppm) of: 31 for [Si],\n150 [amorphous C], 50 [graphite] and 30 [PAH], respectively; which is\nin agreement with cosmic abundance constraints (Asplund et\nal. 2009). We are in the process of upgrading the model to be\nconsistent with the polarization of the ISM (Voshchinnikov\n2004). Besides extinction the model accounts for the diffuse emission\nof solar neighborhood when the dust is heated by the interstellar\nradiation field (Mathis et al. 1983).\n\n\\section{Starbursts \\label{sb}}\n\n\\noindent \nThere are three different ways in the literature trying to reproduce\nthe SED of extra-galactic nuclei.  A first one uses a SED of a\nwell-known galaxy as a template to match other objects (Lutz et\nal. 2002). A second group reproduce the shape of the SED by optical\nthin dust emission using a scaled up version of the interstellar\nradiation field (Draine \\& Lee 2007). A third and more ambitious\nmethod is to solve the radiative transfer (RT) problem using\nassumptions about the galaxy.  The latter is done at various levels of\nsophistication and it may be instructive to point out technical\ndifferences. Teams solving the RT problem evaluate the emission from a\ndusty medium of spheroidal shape filled with stars and dust.  At first\nglance, the model results appear to agree, but upon closer inspection\none finds that deviations of derived parameters are substantial.  For\nexample for Arp220 Grooves et al.  finds an optical depth of a few\nwhereas we derive values between 70 -- 120.  We admit that we did not\nalways find it easy to pin down exactly which approximations our\ncolleagues used, still we try to summarize the main features of some\nRT models which are in widespread use.  Monte Carlo techniques are\ndiscussed in Sect.~\\ref{agn}.  We use the term {\\it {dust\n    self--absorption}} when photons which are emitted by a dust\nparticles may be absorbed by other dust particles within the model\nsphere and we call {\\it {exact RT}} when the RT equation is solved\naccurately including multiple scattering and dust self--absorption.\n\n{$-$} Groves (2004, this volume): Shock, photoionization and dust\nradiative transfer code called MAPPINGIII. The dust is distributed in\na screen and the RT is solved in one dimension ignoring dust\nself-absorption and treating scattering in forward direction only.\n\n{$-$} Efstathiou \\& Rowan--Robinson (2003), Efstathiou (this volume):\nDust is distributed in spherical symmetry. There are two components:\nmolecular clouds with exact RT computation and cirrus which is added\nas a foreground screen. Both components are uncoupled in the RT\nequation. The code includes a free parameter to scope with the\nobserved optical and UV spectrum.\n\n{$-$} Takagi et al. (2003, this volume): Exact RT in spherical\nsymmetry where dust and stars are homogeneously distributed. Molecular\nclouds (clumps) are not treated. The code includes a free parameter to\nscope with the observed optical and UV spectrum.\n\n{$-$} Siebenmorgen et al. (2007): Exact RT in spherical symmetry where\ndust and a young and old population of stars are distributed in the\ngalaxy. A fraction of the young stars are in molecular clouds for\nwhich a second exact RT computation is solved. The coupling of both RT\nsolutions, that of the galaxy and the embedded sources, is treated and\nthis is a particular feature of the model.\n\n{$-$} Silva et al. (1998): Present a code called GRASIL in which dust\nis distributed in axial-symmetry.  There is a molecular cloud\ncomponent with exact RT. A cirrus component is added in which the RT\nis solved by ignoring dust self-absorption and in which dust\nscattering is simplified by altering the dust absorption cross\nsection. In addition the code includes a free parameter to scope with\nthe observed optical and UV spectrum. All three components are\nuncoupled in the RT equation.\n\n{$-$} Popescu et al. (2002, this volume): Dust is distributed in\naxial-symmetry.  The model is fine tuned to Spiral galaxies\n(Sect.~\\ref{spirals}) and includes a bulge, two galactic disk\ncomponents and molecular clouds (clumps).  The RT is solved by\nignoring dust self-absorption and only the first scattering event is\ntreated. Clumps are added by a pre-computed template spectrum.  There\nis a free parameter to scope with the observed optical and UV\nspectrum. All five components are uncoupled in the RT equation. \n\n\n\\noindent \nAs the coupling between the RT calculations of the galaxy and the\nembedded sources is a computer intensive problem we provide a library\nof some 7000 SEDs {\\footnote{SED library available at: {\\tt\n      {http:\/\/www.eso.org\/$\\sim$rsiebenm\/sb\\_models\/}}}} for the\nnuclei of starburst and ultra--luminous galaxies (Siebenmorgen et al.,\n2007).  Its purpose is to quickly obtain estimates of the basic\nparameters of the object, such as luminosity, size and dust or gas\nmass and to predict the flux at yet unobserved wavelengths using a\nphysical model. Unfortunately, for faint or red-shifted objects\nphotometry is sometimes only provided at two MIR bands, for example at\n8 and 24$\\mu$m from Spitzer. In Fig.~\\ref{2dat.ps} we demonstrate for\nthe ULIRG NGC6240 that the SED library can be used to estimate the\ntotal luminosity to within a factor $\\sim 2$ in case only two such MIR\nfluxes are known.  We also see (Fig.~\\ref{2dat.ps}) that the SED will\nbe quite well constrained by an additional submm data point.  The\nmodel is applied to high red shifts ($z \\approx 3$, Efstathiou \\&\nSiebenmorgen 2009) where PAH have been detected.\n\n\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=8cm,angle=90]{2datN6240.ps}\n\\caption{Elements of the SED starburst library by Siebenmorgen et\n  al. (2007, dotted) which fit photometry (circles) of NGC6240 at\n  8$\\mu$m (Siebenmorgen et al. 2004) and 24$\\mu$m (Klaas et al. 2001)\n  to within 30\\%; submm data by Benford (1999) and Klaas et\n  al. (2001), best fit is indicated as thick line \\label{2dat.ps}. }\n\\end{center}\n\\end{figure}\n\n\n\\section{Spirals \\label{spirals}}\n\n\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=10.5cm,angle=270]{scheSpiral.ps}\n\\end{center}\n\\vspace*{-1.0 cm}\n\\caption{Schematic view of the geometry applied in the models\nof Spirals by Popescu et al. \\label{scheSpiral.ps} }\n\n\\end{figure}\n\n\\noindent\nThe geometrical distribution of dust in spiral galaxies has been\ninvestigated by fitting images in the optical\/NIR.  This is done by\nmeans of 2d radiative transfer codes which include a bulge component\nand an old stellar population of stars; originally only dust\nabsorption and scattering was treated (Kylafis \\& Bahcall 1987,\nXilouris et al. 1999, Misiriotis et al., 2000). Building into the\nmodels typical dust emission properties gives a puzzle when compared\nto observations in the FIR. Typically the models underestimate the FIR\nluminosities by a factor of 3.  The FIR excess could be explained by\nPopescu et al. (2000, 2010, this volume) introducing more components.\nIn their models there is a distribution of diffuse dust associated\nwith the old and young stellar disk populations as well as a clumpy\ncomponent arising from dust in the parent molecular clouds in star\nforming regions. Basic parameters of their models are: angular size\nand inclination of the disk, the central face-on dust opacity in the\nB-band, a clumpiness factor for the star-forming regions, the\nstar-formation rate, the normalized luminosity of the stellar\npopulation and the bulge-to-disk ratio and a wavelength dependent\nescape probability of stellar radiation (which is sometimes treated as\nmodel output). A schematic view of the geometry of the RT model is\npresented in Fig.~\\ref{scheSpiral.ps}. The model is successfully\napplied to a large sample of galaxies from the Millennium Galaxy\nCatalog Survey (Driver et al. 2007). The observed\nattenuation--inclination relation is fit when a two dust disks are\nconsidered whereas data are not fit in a single disk scenario.\n\nOver the past decade the edge on spiral NGC891 become a benchmark test\nfor RT models of spiral galaxies. Bianchi (2008) uses a clumpy disks\nmodel and applied MC techniques which are of advantage when dealing\nwith such complicated geometries (see Sect.~\\ref{agn}). His results\ngive support to the idea that the diffuse dust disk is more extended\nthan the stellar disks. De Looze et al. (20011, this volume) present a\nMC radiative transfer model of the Sombrero galaxy which is able to\nfit the SED and optical\/NIR images and extinction profiles. Using only\nan old stellar population the dust luminosity in the FIR is again\nunderestimated by a factor 3.  The discrepancy is solved by assuming a\nstar forming stellar component both in the ring and inner disk to\naccount for emission at 24 and 70$\\mu$m. In the submm an additional\ndust component is used, accounting for 75\\% of the total dust content\nand distributed in quiescent compact clumps. A possibility that part\nof the dust could be composed of grains with a higher submm\nemissivity, of for example large fluffy grains (Kr\\\"ugel \\&\nSiebenmorgen 1994), could not be ruled out. On the other hand a\nsimilar approach introducing a clumpy medium in a single disk model is\napplied to the edge-on spiral galaxy UGC4754 (Baes et al. 2011). The\nmodel has a deficit in the FIR luminosity and the authors propose\nhigher submm dust emissivities to solve the energy ballance at such\nlong wavelengths.\n\n\n\n\\section{AGN and Monte Carlo \\label{agn}}\n\n\\noindent\nDust is detected in the majority of active galactic nuclei (Haas et\nal., 2008).  According to the unified scheme (Antonucci \\& Miller,\n1985), AGN are surrounded by a dust obscuring torus. However,\nobservations are not able to resolve the inner parts of AGNs so that\nthe geometrical distribution of the dust is a matter of\ndebate. Theoretical considerations favor a clumpy structure in a torus\nlike configuration of optical thick dust clouds surrounding the black\nhole accretion disk (Pier \\& Krolik (1992). Some evidence of a clumpy\nor filamentary structure of the torus is given by VLTI observations of\nthe nearby active galactic nuclei in Circinus (Tristram et al. 2007).\nRadiative transfer models of homogeneous toroidal structures\nover-predict the silicate absorption and emission band at around\n10$\\mu$m when compared to observations. The silicate emission feature\nin type I AGN is rather shallow (Siebenmorgen et al. 2005). A\nstatistical attempt to describe the radiation from a clumpy AGN torus\nis given by Nenkova et al. (2002) and more detailed radiative transfer\ncomputations using the Monte Carlo technique are presented by H\\\"onig\net al. (2006) and Schartmann et al. (2008).\n\nWe develop a vectorized three dimensional Monte Carlo (MC) technique\nto solve the radiative transfer problem in such a fairly complicated\ngeometry of a clumpy dust torus. Originally the MC radiative transfer\nmethod for scattered light in astronomical sources is presented by\nWitt (1977) and important improvements are introduced by Lucy\n(1999). A first version of the code which we are using is developed by\nKr\\\"ugel (2006). To reduce the computational effort of MC dust\nradiative transfer methods different optimization strategies are\ndeveloped (Lucy 1999, Bjorkman \\& Wood 2001, Gordon et al. 2001,\nMisselt et al. 2001, Baes 2008, Bianchi 2008, Baes et al. 2011). A\nnumerical solution of the radiative transfer equation which is\nspecifically developed to be vectorized and to run on graphical\ncomputer units (GPU) is presented by Heymann (2010). A GPU version\nincluding stochastic heated grains is presented (Siebenmorgen et\nal. 2010), in which some detailed physics of the destruction of PAHs\nby soft (photospheric) and hard (X--ray) radiation components is\ntreated, (see Siebenmorgen \\& Kr\\\"ugel 2010 for a discussion of PAH\ndissociation).  The speed up factor of the GPU method is proportional\nto the number of processors available. Therefore the GPU scheme is on\nour standard PC with a conventional graphic card about 100 times\nfaster when compared to the original scalar version of the MC program.\nThe numerical solution provides the temperature of the dust, spectra\nand images within acceptable timescales. The code is tested against\nexisting benchmark results. The method handles arbitrary dust\ndistributions in a three dimensional Cartesian model space at various\noptical depths. Therefore it is well adopted to be applied to a clumpy\ntorus structure around an AGN.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=6cm,angle=0]{agnTklump.eps}\n\\end{center}\n\\caption{Zoom into the clumpy AGN torus displaying the temperature\n  structure of individual clouds. \\label{agnTklump} }\n\\end{figure}\n\n\n\\begin{figure}[htb]\n\\begin{center}\n \\includegraphics[width=13.5cm,angle=0]{torus_emm_sca.eps}\n\\caption{Intensity maps of the clumpy AGN dust torus model\n  (color-code: cgs units normalized to the peak intensity).  The model\n  is made up of 1000 clumps and each clump has a total optical depth\n  of $\\tau_{\\rm V} = 30$.  Mid IR images at 10\\,$\\mu$m are dominated\n  by dust emission and displayed in the top panel. The optical light\n  at 0.55\\,$\\mu$m represents the scattering intensities and are shown\n  in the bottom panel.  The AGN torus is viewed face-on ($\\theta =\n  0^\\circ$, left), at $\\theta = 45^\\circ$ (middle) and edge-on\n  ($\\theta = 90^\\circ$, right).  \\label{agn.ps} } \n\\end{center}\n\\end{figure}\n\n\n\n\nThe primary heating source of the dust in the torus emerges from the\naccretion disc around the massive black hole. For the spectral shape\nwe apply a broken power law as suggested by Rowan-Robinson (1995). As\nexample we use a total AGN luminosity of $L_{\\rm {AGN}} =\n10^{45}$\\,erg\/s.  This sets the inner radius $R_{\\rm {in}} \\sim 0.4\n\\sqrt{L_{\\rm {AGN}}\/10^{45}}$\\,[pc] where the dust evaporation zone is\nlocated. We treat the torus to an outer radius of $R_{\\rm {out}} = 50\nR_{\\rm {in}}$ and consider a torus opening angle of $\u03b8 \\sim 20^{\\rm\n  o}$.  Clouds are randomly distributed in the otherwise optical thin\ndust torus. Individual clouds are approximated with a sphere of\nconstant density and radius of 3\\,pc. The optical depth through the\ncenter of the clouds is $\\tau_{\\rm V} = 30$. In Fig.~{\\ref{agnTklump}}\nwe zoom into a part of the clumpy torus and show the temperature\nstructure of the clouds. The individual hemispheres which are oriented\nin direction of the central heating source are warmer as compared to\ntheir dark sides. Shadowed clumps become also colder than those\ndirectly heated by the central engine. Intensity maps of the emission\nat 10$\\mu$m and scattered light at 0.5$\\mu$m of the clumpy AGN dust\ntorus model are displayed in Fig.~\\ref{agn.ps}. The maps show that in\na clumpy medium even in the edge on case some radiation from the\ncentral region is visible.  A more detailed description of the clumpy\nAGN torus model and a further application to the silicate emission in\nQuasars is given in Heymann \\& Siebenmorgen (2011).\n\n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nA strange attractor   is an invariant set with at least one positive Lyapunov exponent whose basin of attraction has non-empty interior. Nowadays, at least for families of dissipative systems, chaotic dynamics is mostly understood as the persistence of strange attractors\n--- occurring for parameters in a set of  positive Lebesgue measure.\n Persistence of dynamics is physically relevant because it means that the phenomenon is \\emph{observable} with positive probability. \nProof of the existence of strange attractors is usually obtained by comparing the dynamics in an invariant set to either the Lorenz or the H\\'enon attractors, or through the unfolding of a singularity,\n for a discussion see \\cite{BIRR}.\n\n\nThere are few examples of periodically-forced vector fields exhibiting complex dynamics that may be proven analytically. In this article, we give an explicit mechanism to obtain strange attractors in a two-parameter family of vector fields unfolding an attracting heteroclinic network. When the first parameter is different from zero, two normally hyperbolic attracting tori arise near the network. Fixing this parameter and varying the second, the tori break and suspended horseshoes emerge, via the \\emph{torus-breakdown} phenomenon \\cite{AS91, Anishchenko}. In the meantime, persistent attractors (of H\\'enon-type) associated to homoclinic tangencies are created. \n\n \nOur  \\emph{route to chaos}   from an attracting network is different from the routes described by  \\cite{Kaneko} and \\cite{Bakri}, where the authors use coupled oscillators. Another different itinerary has been described by \\cite{FJ2020} in the context of the \\emph{Langford system}. These works are discussed in Section \\ref{discussion}. \n\n\nThe theory developed in this paper is explicitly applicable to the analysis of various specific differential equations and the results obtained are beyond the capacity of the classical Birkhoff-Melnikov-Smale method associated to \\emph{heteroclinic tangles} \\cite{Wang_2013}.\n\nOur purpose in writing this paper is not only to point out the range of phenomena that can occur when simple non-linear equations are periodically forced, but to bring to the foreground the techniques that have allowed us to reach these conclusions in a relatively straightforward manner. \n\n\n\nWe analyse a family of periodic perturbations of an attracting symmetric heteroclinic network defined on the two-sphere. Instead of looking at the time-$T$ maps as in \\cite{LR18, TD1}, we extend the phase space and we explicitly compute return maps induced by the perturbed equations in\na neighborhood of the extended heteroclinic network. The  cyclic variable's speed plays an important role in the emergence of the horseshoes.\n Using the techniques explored in  \\cite{Shilnikov_book_1, TS86}, we reduce the analysis of the non-autonomous system to that of a two-dimensional map on a circloid.  These techniques are clearly not limited to the systems considered here. It is our hope that they will find applications in other dynamical systems, particularly those that arise naturally from mechanics or physics \\cite{AHL2001, DT3, Ruelle, TD1, Rabinovich06}. See also the dynamical description of the periodically-forced van der Pol oscillator in   \\cite{Shilnikov_book_1, TS86}, where the authors used the \\emph{Afraimovich Annulus Principle} to prove the existence of an invariant torus. \n \nThe title \\emph{``Periodic forcing of a heteroclinic network''} refers to the study of the dynamics associated to a parametric periodically-forced vector field unfolding  an asymptotically stable heteroclinic network.\n\n\n\\subsection*{Structure of the article}\nThis article is organized as follows. In Section \\ref{sec-setting}, we describe the setting of our problem\nand state the main results after having revised some conceptual preliminaries  in Section \\ref{preliminaries}. \nResults on general families of vector fields are proved in Section~\\ref{secProva_G}, after the derivation  in Section \\ref{secPrelim} of the first return map to a given cross section.\n An explicit example is treated in Section~\\ref{PropA}.\nWe finish this article with a discussion on the way our results fit in literature in Section \\ref{discussion}.\n\n\\section{Preliminaries}\n\\label{preliminaries}\nIn this section, we introduce some terminology for vector fields acting on three-dimensional Riemannian manifolds that we will use in the remaining sections.\nConsider the two-parameter family of $C^3$--smooth autonomous differential equations\n\\begin{equation}\n\\label{general2a}\n\\dot{x}=F_{(\\nu, \\mu)}(x)\\qquad x\\in {\\mathbf S}^3\\subset {\\mathbf R}^4  \\qquad \\nu, \\mu\\in{\\mathbf R}\n\\end{equation}\nwhere ${\\mathbf S}^3$ denotes the unit sphere, endowed with the usual topology. Denote by $\\varphi_{(\\nu, \\mu)}(t,x)$, $t \\in {\\mathbf R}$, the associated flow. The flow is \\emph{complete} (\\emph{i.e.}  solutions are defined for all $t\\in {\\mathbf R}$) because  ${\\mathbf S}^3$ is a compact manifold without boundary.\n \n \\subsection{Heteroclinic structures and symmetry}\nSuppose that $A$ and $B$ are two hyperbolic saddles of \\eqref{general2a}. There is a {\\em heteroclinic cycle} associated to $A$ and $B$ if\n$$W^{u}(A)\\cap W^{s}(B)\\neq \\emptyset \\qquad \\text{and} \\qquad W^{u}(B)\\cap W^{s}(A)\\neq \\emptyset.$$ \n\nThe non-empty intersection of $W^{u}(A)$ with $W^{s}(B)$ is called a \\emph{heteroclinic connection} between $A$ and $B$, and will be denoted by $[A \\rightarrow  B]$. Although the existence of heteroclinic cycles may be a non-generic feature within differential equations, they may be structurally stable within families of systems which are equivariant under the action of a compact Lie group $\\mathcal{G}\\subset \\mathbb{O}(4)$, due to the existence of flow-invariant subspaces (see  \\cite{GH}).\n\n \nGiven a group $\\mathcal{G}$ of endomorphisms of ${\\mathbf S}^3\\subset {\\mathbf R}^4$, we will consider two-parameter families of vector fields $\\left(F_{(\\nu, \\mu)}\\right)$ under the equivariance assumption $$F_{(\\nu, \\mu )}(\\gamma x)=\\gamma F_{(\\nu, \\mu )}(x)$$ for all $x \\in {\\mathbf S}^3$, $\\gamma \\in \\mathcal{G}$ and $(\\nu, \\mu )\\in  {\\mathbf R}^2.$\nFor an isotropy subgroup $\\widetilde{\\mathcal{G}}< \\mathcal{G}$, we will write $\\Fix(\\widetilde{\\mathcal{G}})$ for the vector subspace of points that are fixed by the elements of $\\widetilde{\\mathcal{G}}$. For $\\mathcal{G}-$equivariant differential equations, the subspace $\\Fix(\\widetilde{\\mathcal{G}})$ is flow-invariant.\n\nIf $\\Omega\\subset {\\mathbf S}^3$ is a flow-invariant set of \\eqref{general2a}, its basin of attraction, $\\mathcal{B}(\\Omega)$, is the set of points in ${\\mathbf S}^3$ whose orbits have $\\omega-$limit in $\\Omega$. We say that $\\Omega$ is \\emph{asymptotically stable} if $\\mathcal{B}(\\Omega) $ contains \n all the half-trajectories in positive time starting in\nan open neighbourhood of $\\Omega$. \n \\subsection{Rotational horseshoes}\nLet $\\mathcal{H} $ stand for the infinite annulus $\\mathcal{H} = {\\mathbf S}^1 \\times {\\mathbf R}$ endowed with the usual inner product from ${\\mathbf R}^2$. We denote by $Homeo^+(\\mathcal{H} )$ the set of homeomorphisms of the annulus which preserve orientation.\nGiven a homeomorphism $f :X \\rightarrow X$  and a partition of $m\\geq 2$ elements $R_1,..., R_{m}$ of $X\\subset \\mathcal{H}$, the itinerary  function $\\xi: X \\rightarrow \\{1, ..., m\\}^{\\mathbf Z}= \\Sigma_m$ is defined by $$\\xi(x)(j)=k\\quad   \\Leftrightarrow \\quad f^j(x)\\in R_k, \\quad \\text{for every} \\quad j\\in {\\mathbf Z}.$$  \nFollowing \\cite{Passeggi}, we say that a compact invariant set $\\Lambda \\subset \\mathcal{H} $ of $f \\in Homeo^+(\\mathcal{H} )$ is a \\emph{rotational horseshoe} if it admits a finite partition $P =\\{R_1, ..., R_{m} \\}$  by sets  $R_i$ with non empty interior in $\\Lambda$ so that\n\\begin{itemize}\n\\item \nthe itinerary $\\xi$ defines a semi-conjugacy between $f|_\\Lambda$ and the full-shift $\\sigma: \\Sigma_m \\rightarrow \\Sigma_m$, that is $\\xi  \\circ f = \\sigma \\circ \\xi$ with $\\xi$ continuous and onto;\n\\item \nfor any lift $G: {\\mathbf R}^2 \\rightarrow {\\mathbf R}^2$ of $f$, there exist  $k>0$ and $m$ vectors $v_1, ...,v_{m} \\in {\\mathbf Z} \\times \\{0\\}$ so that\n$$\n\\left\\| (G^n(\\hat{x})-\\hat{x})  - \\sum_{i=0}^n v_{\\xi(x)(i)}\\right\\| <k \\qquad \\text{for every} \\qquad  \\hat{x}\\in \\pi^{-1}(\\Lambda), \\quad n\\in {\\mathbf N},\n$$\nwhere $\\|\\star\\|$ denotes the usual norm of ${\\mathbf R}^2$,  $\\pi:{\\mathbf R}^2\\rightarrow \\mathcal{H}$ denotes the usual projection map and $\\hat{x} \\in \\pi^{-1}(\\Lambda)$ is the lift of $x$; more details in the proof of Lemma 3.1 of \\cite{Passeggi}. The existence of a rotational horseshoe for a map implies positive topological entropy  at least  $ \\log m $.\n\\end{itemize}\n\n\\subsection{Strange attractors}\n\nStrange attractors contribute to the richness and complexity of a dynamical system.  We introduce the following notion, adapted from \\cite{MV93}, to the situation under consideration.\nA  (H\\'enon-type) \\emph{strange attractor} of a two-dimensional dissipative diffeomorphism $f$ defined in a Riemannian manifold, is a compact invariant set $\\Lambda$ with the following properties:\n\\begin{itemize}\n\\item \n$\\Lambda$  equals the closure of the unstable manifold of a hyperbolic periodic point;\n\\item \nthe basin of attraction of $\\Lambda$   contains an open set (and thus has positive Lebesgue measure);\n\\item \nthere is a dense orbit in $\\Lambda$ with a positive Lyapounov exponent (exponential growth of the derivative along the orbit).\n\\end{itemize}\nA vector field possesses a strange attractor if the first return map to a cross section does.  \n \n\n\\section{Setting and main results}\n\\label{sec-setting}\nOur object of study is the dynamics around an attracting  heteroclinic network for which we give a rigorous description here.  \\subsection{The object of study}\nConsider the  family of $C^3$--autonomous differential equations on a two dimensional manifold ${\\mathcal M}^2$ diffeomorphic to the two-sphere ${\\mathbf S}^2\\subset {\\mathbf R}^3$ and parametrised by $\\nu\\in{\\mathbf R}$\n\\begin{equation}\n\\label{general0}\n \\begin{array}{ll}\n\\dot x={\\mathcal F}_\\nu(x), \\quad x\\in {\\mathbf R}^3.\n\\end{array} \n \\end{equation}\n\n\nSuppose that, for $\\nu=0$, the flow of \\eqref{general0} has an attracting heteroclinic cycle associated to two\nequilibria\nand that for $\\nu\\ne 0$ one of the heteroclinic connections is broken, yielding an attracting periodic solution, as illustrated in Figure \\ref{scheme1}.\nMore precisely we are assuming that \\eqref{general0}   satisfies: \n \n\\begin{enumerate}\n\\renewcommand{\\theenumi}{\\textbf{(A\\arabic{enumi})}}\n\\renewcommand{\\labelenumi}{\\textbf{\\theenumi}}\n\\item\\label{A1}\n  there is  a flow-invariant manifold ${\\mathcal M}^2$ diffeomorphic to ${\\mathbf S}^2$ that is attracting in the sense that every nearby trajectory is asymptotic to it in forward time; \n\\item \\label{A2}\nthere are  two equilibria of saddle type\n${\\mathbf v}$ and ${\\mathbf w}$;\n\\item \\label{A3}\nthere are  two heteroclinic connections from ${\\mathbf w}$ to ${\\mathbf v}$, forming a Jordan curve $\\vartheta$ on ${\\mathcal M}^2$.\n\\setcounter{lixo}{\\value{enumi}}\n\\end{enumerate}\nMoreover, the restriction of the flow of \\eqref{general0}  to ${\\mathcal M}^2$ satisfies: \n\\begin{enumerate}\n\\renewcommand{\\theenumi}{\\textbf{(A\\arabic{enumi})}}\n\\renewcommand{\\labelenumi}{{\\theenumi}}\n \\setcounter{enumi}{\\value{lixo}}\n\\item \\label{A4}\nfor $\\nu=0$ there are two heteroclinic connections from ${\\mathbf v}$ to ${\\mathbf w}$;\n\\item \\label{A5}\nfor $\\nu=0$ the heteroclinic network formed by the connections between ${\\mathbf v}$ and ${\\mathbf w}$ is attracting;\n\\item \\label{A6}\nfor $\\nu=0$ the only periodic solutions are the equilibria; \n\\item\\label{A7} for $\\nu\\ne 0$ the heteroclinic connections from ${\\mathbf v}$ to ${\\mathbf w}$ are broken and there are two attracting hyperbolic periodic solutions, each one in one connected component of   ${\\mathcal M}^2\\backslash \\vartheta$. \n\\setcounter{lixo}{\\value{enumi}}\n\\end{enumerate}\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[height=3.5cm]{scheme1_LR.pdf}\n\\end{center}\n\\caption{\\small \nPartial phase portrait associated to  the one-parameter family \\eqref{general0},\na periodic solution appears when a heteroclinic connection is broken. (A): $\\nu=0$; (B): $\\nu\\neq 0$.  }\n \\label{scheme1}\n\\end{figure}\n \nThe period of the  solutions of  \\ref{A7}  tends to infinity when $\\nu$ tends to zero, as they accumulate on the heteroclinic cycles containing ${\\mathbf v}$ and ${\\mathbf w}$. From \\ref{A6}, these are the only non-constant periodic solutions. Property~\\ref{A7} is the generic unfolding of a heteroclinic network like the one in \\ref{A5}. \nIn contrast, Property ~\\ref{A3} is not generic but occurs generically in systems with a flow-invariant two-dimensional manifold, for instance in the presence of symmetry --- see the example in \\S \\ref{sec_example} below.\n\n \nWe subject the autonomous differential equation \\eqref{general0}  to a family of non-constant  time-periodic perturbations $\\phi_\\mu(t,x)$ of period $\\pi\/\\omega\\in{\\mathbf R}^+$, where the parameter $\\mu\\ge 0$  controls the amplitude of the perturbation. \nThe perturbed equation \n$$\n\\dot x={\\mathcal F}_\\nu(x)+\\phi_\\mu(t,x),\\qquad  x\\in  {\\mathbf S}^2, \nt\\in {\\mathbf R}\n$$ \nmay be converted into an autonomous equation \n$$\n(\\dot x,\\dot\\theta)= F_{(\\nu,\\mu)} (x,\\theta)\n$$ \nin ${\\mathcal M}^2\\times {\\mathbf S}^1$ by rewriting it as\n\\begin{equation}\n\\label{general}\n\\left\\{\\begin{array}{ll}\n\\dot x={\\mathcal F}_\\nu(x)+\\phi_\\mu(\\theta,x)&{}\\\\\n\\dot\\theta={2\\omega}&\\pmod{2\\pi}.\n\\end{array}\\right.\n\\qquad\\qquad \\phi_0(\\theta,x)\\equiv 0. \n\\end{equation}\n For $\\mu=0$ the last equation of \\eqref{general} is not coupled to the first. Hence, the dynamics of \n\\eqref{general} may be obtained from conditions \\ref{A1}--\\ref{A7}, as follows.\n\n\\begin{Lprop}\n\\label{PropB1}\nIf \\ref{A1}--\\ref{A7} hold for  ${\\mathcal F}_\\nu$\nthen the flow of \\eqref{general} for $\\mu=0$ and  small $\\nu\\ge 0$ satisfies:\n\\begin{enumerate}\n \\item  there is  an invariant flow-invariant manifold ${\\mathcal M}^3$  diffeomorphic to ${\\mathbf S}^2\\times {\\mathbf S}^1$\nthat is globally attracting, in the sense that it attracts all trajectories in its neighbourhood.\n\\setcounter{maislixo}{\\value{enumi}}\n\\end{enumerate}\nFurthermore, the restriction of the flow to ${\\mathcal M}^3$ satisfies:\n\\begin{enumerate}\n \\setcounter{enumi}{\\value{maislixo}}\n \\item\nthere are two periodic solutions in ${\\mathcal P}_{\\mathbf v}=\\{{\\mathbf v}\\}\\times{\\mathbf S}^1$ and  ${\\mathcal P}_{\\mathbf w}=\\{{\\mathbf w}\\}\\times{\\mathbf S}^1$ of saddle type;\n\\item\nfor  $\\nu=0$ the invariant manifolds of ${\\mathcal P}_{\\mathbf v}$ and ${\\mathcal P}_{\\mathbf w}$ in ${\\mathcal M}^3$ coincide, forming  a  heteroclinic network $\\Gamma$ with the geometry of  a  singular two-dimensional torus of genus 2;\n\\item \n  for $\\nu>0$  the manifolds $W^u({\\mathcal P}_{\\mathbf w})$ and $W^s({\\mathcal P}_{\\mathbf v})$  in ${\\mathcal M}^3$ coincide;\n\\item \\label{L5}\n  for $\\nu>0$ we have  $W^u({\\mathcal P}_{\\mathbf v})\\cap W^s({\\mathcal P}_{\\mathbf w})=\\varnothing$;\n   \\item\\label{L6}\n  for $\\nu>0$  there are two attracting  invariant two-dimensional tori $\\mathcal{T}^\\pm(\\nu)$;\n \\item\\label{L7}\n when  $\\nu \\rightarrow 0$, the tori $\\mathcal{T}^\\pm(\\nu)$ accumulate on $\\Gamma$.\n\\end{enumerate}\n\\end{Lprop}\n\nThe proof of Proposition \\ref{PropB1} follows \nby combining the dynamics of \\eqref{general0} with the existence of a cyclic variable (see Chapter 4 of \\cite{Shilnikov_book_1}).\n\n\\subsection{The periodically forced system}\n\\label{munot0}\n\n\nFor $\\nu=\\mu=0$, let $\\Sigma$ be a cross section of the heteroclinic cycle $\\Gamma$ in ${\\mathcal M}^3$. \nThen $\\Sigma$ is also a cross section of \\eqref{general} for small $\\nu,\\mu\\ge 0$.\nLet ${\\mathcal R}_{(\\nu,\\mu)}$ be the first return map to $\\Sigma$, with respect to the flow defined by ${F}_{(\\nu, \\mu)}$.  Define also\n$$\n\\Omega_{(\\nu,\\mu)}=\\left\\{X \\in \\Sigma: {\\mathcal R}^n_{(\\nu,\\mu)} (X)\\in \\Sigma, \\quad \\forall n \\in {\\mathbf N} \\right\\}\\qquad \\text{and} \\qquad \\Lambda_{(\\nu,\\mu)}= \\bigcap_{n \\in {\\mathbf N}} {\\mathcal R}^n_{(\\nu,\\mu)} \\left(\\Omega_{(\\nu,\\mu)}\\right).\n$$\n In this article, we \n present a comprehensive analysis on the dynamics \nof ${\\mathcal R}_{(\\nu, \\mu)}$ on the \\emph{non-wandering set}\n $\\Lambda_{(\\nu, \\mu)}$. When there is no risk of misunderstanding, we omit the subscripts ${(\\nu,\\mu)}$.\n \nWhen $(\\nu,\\mu)\\ne(0,0)$  one expects that, generically, $W^u({\\mathcal P}_{\\mathbf v}) \\pitchfork W^s({\\mathcal P}_{\\mathbf w})$.\nThe case when $W^u({\\mathcal P}_{\\mathbf v})\\cap W^s({\\mathcal P}_{\\mathbf w})\\ne\\varnothing$  has been discussed in \\cite{LR17, Wang_2013}, here we are mostly concerned with the case $W^u({\\mathcal P}_{\\mathbf v}) \\cap W^s({\\mathcal P}_{\\mathbf w})=\\varnothing$. \nWe also suppose property \\ref{A3} (extended to equation \\eqref{general}) still holds for the forced system, hence our assumptions are: \n\\begin{enumerate}\n\\renewcommand{\\theenumi}{\\textbf{(A\\arabic{enumi})}}\n\\renewcommand{\\labelenumi}{{\\theenumi}}\n \\setcounter{enumi}{\\value{lixo}}\n\\item \\label{A8}\n$W^u({\\mathcal P}_{\\mathbf w}) = W^s({\\mathcal P}_{\\mathbf v})$; \n\\item \\label{A9}\n$W^u({\\mathcal P}_{\\mathbf v}) \\cap W^s({\\mathcal P}_{\\mathbf w})=\\varnothing$. \n\\end{enumerate}\n\n\nOur first main result is about the existence  of an invariant   set whose dynamics is conjugate to a full shift over a finite number of symbols. In addition, we also prove the existence of observable chaos.\n\n \n \\begin{Lth} \n  \\label{role_omega}\n  If \\ref{A1}--\\ref{A9} hold for \\eqref{general} then\n  \\begin{enumerate}\n  \\item for every small $\\nu,\\mu>0$ there exists $\\omega_0>0$ such that if $\\omega>\\omega_0$, then  $\\Lambda_{(\\nu, \\mu)}$ contains  an invariant set whose dynamics is conjugate to a full shift in two symbols;\n  \\item \nfor $(\\nu, \\mu)$ in a set  $\\mathcal{U}\\subset {\\mathbf R}^2$  of positive Lebesgue measure, the return map\n${\\mathcal R}_{(\\nu,\\mu)}$ exhibits a strange attractor.\n\\end{enumerate}\n \\end{Lth}\n \n   The dynamics of  $\\Lambda_{(\\nu, \\mu)}$ is mainly governed by the geometric configuration of the global invariant manifold $W^u({\\mathcal P}_{\\mathbf v})$.\n  The proof of  this result is  \n  done is Section~\\ref{secProva_G}.  Horseshoes of Theorem  \\ref{role_omega} have a different nature from those associated to the \\emph{heteroclinic tangle} in which the manifolds have a transverse intersection  \\cite{LR17, Wang_2013}.  This will be discussed in Section \\ref{discussion}.\n\nFor a fixed $\\nu>0$, if the ratio of $\\omega$ and the period of the hyperbolic periodic solution of  \\ref{A7}  is irrational, then trajectories on the torus $\\mathcal{T}^\\pm(\\nu)$ are unlocked, in the sense that they never close. These solution on  the torus are called \\emph{quasiperiodic} \\cite{Herman, Shilnikov_book_1}. \n  If the frequencies have a rational ratio,  trajectories are locked. \n  \n\nIn a resonant torus, where all solutions are locked, the frequency locking ratio $p\/q$ means that while the $x$ component of  a solution turns $p$ its $\\theta$ component winds $q$ times.\n This ratio is related  to the \\emph{rotation number} associated to the periodic orbit \\cite{Herman} and will be used in the proof of the second part of Theorem \\ref{role_omega}.\n\n\\subsection{The example}\n\\label{sec_example}\n An explicit two-parameter family ${\\mathcal F}_\\nu(x)+\\phi_\\mu(t,x)$ of vector fields   in ${\\mathbf S}^2\\subset {\\mathbf R}^3$  such that ${\\mathcal F}_\\nu(x)$ satisfies  \\ref{A1}--\\ref{A7}  is given by\n \\begin{equation}\n\\label{general4}\n\\left\\{ \n\\begin{array}{l}\n\\dot x_1 = x_1(1-r^2)-\\alpha x_1 x_3 +\\beta x_1x_3^2 + (1- x_1) \\,\\, \\,  (\\mu\\,  [f (\\theta)-1]+\\nu)\\\\\n\\dot x_2 =  x_2(1-r^2) + \\alpha x_2 x_3 + \\beta x_2 x_3^2 \\\\\n\\dot x_3 = x_3(1-r^2)-\\alpha(x_2^2-x_1^2)-\\beta x_3 (x_1^2+x_2^2) \\\\\n \\dot\\theta={2\\omega}\\pmod{2\\pi}\n\\end{array}\n\\right.\n\\end{equation}\nwhere \n$$\n\\nu, \\omega\\in {\\mathbf R}^+\\quad \\mu\\in{\\mathbf R}\\qquad\nr^2= x_1^2+x_2^2+x_3^2, \\qquad \\beta<0<\\alpha, \\qquad\n  |\\beta|<\\alpha ,\n$$\n and $f$ is a  non constant $2\\pi$-periodic map of class $C^3$.\n\n For $\\mu=\\nu=0$, the equation $\\dot x={\\mathcal F}_{0}(x)$, $x\\in {\\mathbf R}^3$,  is one of the examples constructed and analysed in   \\cite{ACL06} and also  studied in \\cite{LR18}.  \n The perturbing  term $(1-x_1) \\  [\\mu (f(2\\omega t) -1)+\\nu] $ appears only in the first coordinate\n  for two reasons. First, it simplifies the computations. Secondly, it allows comparison with previous work by other authors \\cite{AHL2001,DT3, Rabinovich06, TD1}. \n\n  \\begin{Lprop}\n\\label{periodic_solution_prop}\n The vector field ${\\mathcal F}_\\nu$ associated to  \\eqref{general4} at $\\mu=0$ is equivariant under the action of $\\kappa(x,y,z)=(x, -y,z)$ \nand therefore the plane $\\Fix({\\mathbf Z}_2(\\kappa))=(x,0,z)$ is flow-invariant.\nIf   $|\\nu|>0$ is small,  the flow of $\\dot\\zeta={\\mathcal F}_\\nu(\\zeta)$ satisfies conditions \\ref{A1}--\\ref{A7}.\nIn particular, the flow-invariant curve ${\\mathcal M}^2\\cap \\Fix({\\mathbf Z}_2(\\kappa))$ consists of  two equilibria of saddle-type ${\\mathbf v}$ and ${\\mathbf w}$ and two heteroclinic connections from ${\\mathbf w}$ to ${\\mathbf v}$. \nThere are also four equilibria in ${\\mathcal M}^2$  that are repelling foci and these are all the equilibria in ${\\mathcal M}^2$.\n\\end{Lprop}\n\n\n The proof of  this result is the content of Section~\\ref{PropA}.  \n \n\n From \\eqref{L6} in Proposition~\\ref{PropB1} it follows that there are two invariant tori for the flow of the equation $(\\dot x,\\dot\\theta)= F_{(\\nu,\\mu)} (x,\\theta)$\n associated to  \\eqref{general4}.\n Existence of invariant tori  is usually shown using the Afraimovich Annulus Principle \\cite{AS91}, here we show it directly by reducing the problem to a two-dimensional manifold and applying the Poincar\\'e-Bendixson theorem.\n \n From Theorem~\\ref{role_omega} and Proposition~\\ref{periodic_solution_prop} it follows immediately\n \n \\begin{Lcor}\n \\label{CorExemploCaos}\nFor small $\\mu>0$, $\\nu>0$ and  for $\\omega>0$ large enough,  the flow of \\eqref{general4} exhibits a hyperbolic rotational horseshoe\nand for a set of positive Lebesgue measure of parameters it also contains  strange attractors. \n \\end{Lcor}\n \n \n\\section{Local coordinates and first return map}\n\\label{secPrelim}\n\n\nIn this section we will analyse the dynamics near the  heteroclinic attractor $\\Gamma$ through local maps, after selecting appropriate coordinates near the saddles   ${\\mathcal P}_{\\mathbf v}=\\{{\\mathbf v}\\}\\times{\\mathbf S}^1$ and  ${\\mathcal P}_{\\mathbf w}=\\{{\\mathbf w}\\}\\times{\\mathbf S}^1$.\n\\subsection{Geometry near ${\\mathcal P}_{{\\mathbf v}}$ and ${\\mathcal P}_{{\\mathbf w}}$}\n\\label{Local}\nLet $U_a$\nbe pairwise disjoint compact neighbourhoods in ${\\mathcal M}^3$ of the nodes  ${\\mathcal P}_{a}$, $a\\in \\{{\\mathbf v},{\\mathbf w}\\}$, such that each boundary  $\\partial U_a$ is a finite union of smooth surfaces delimited by curves, each surface\n transverse to the vector field everywhere, except at its\n boundary.  \nEach  $U_a$ is called an \\emph{isolating block} for  ${\\mathcal P}_{a}$ and, topologically, it consists of a hollow cylinder.  \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[height=9cm]{local_map.pdf}\n\\end{center}\n\\caption{\\small Top:  isolating block near the periodic solution ${\\mathcal P}_a$, $a\\in \\{{\\mathbf v},{\\mathbf w}\\}$.\nBottom: coordinates at the $In$ and $Out$ components of the boundary.  \nDouble bars mean that the sides are identified.    }\n\\label{local_map_LR_torus1}\n\\end{figure}\n\n\nFor $a\\in \\{{\\mathbf v},{\\mathbf w}\\}$, let $\\Sigma_a$ be a cross section to the flow at $p_a \\in {\\mathcal P}_{a}$. Since ${\\mathcal P}_{a}$ is hyperbolic, there is a neighbourhood $U^*_a$ of $p_a$ in $\\Sigma_a$ where the first return map to $\\Sigma_a$\nis $C^1$ conjugate to its linear part. \n Let $e^{-c_a}$ and $e^{e_a}$, with $c_a,e_a>0$,  be the eigenvalues\nof the derivative $D{\\mathcal F}_{(\\nu, \\mu)}(a)$.\nThen, for each $k\\ge 2$ there is an open and dense subset of ${\\mathbf R}^2$ such that, if \n$(-c_a,e_a)$\n\n lies\n in this set, then the conjugacy is of class $C^k$ (details may be checked in  Appendix A of \\cite{LR17}).\n\nSuspending the linear map gives rise, in cylindrical coordinates $(\\rho, \\theta, z)$ around ${\\mathcal P}_{a}$, to the  equations\n\\begin{equation}\n\\label{ode of suspension}\n\\left\\{ \n\\begin{array}{l}\n\\dot{\\rho}=-c_{a}(\\rho -1) \\\\ \n\\dot{\\theta}=2\\omega \\\\ \n\\dot{z}=e_{a}z\n\\end{array}\n\\right.\n\\end{equation}\n whose flow is\n$C^2$-conjugate to the original flow near ${\\mathcal P}_{a}$. In these coordinates, the periodic trajectory ${\\mathcal P}_{a}$ is the circle defined by $\\rho=1$ and $z=0$.\nFor the moment, let $W^s_{loc}({\\mathcal P}_{a})$ and $W^u_{loc}({\\mathcal P}_{a})$ be the connected components of $W^s({\\mathcal P}_{a})$ and $W^u({\\mathcal P}_{a})$, respectively, contained in the suspension of $U^*_a$  and containing ${\\mathcal P}_a$ in their closure.\nIn these coordinates,  $W^s_{loc}({\\mathcal P}_{a})$, is  the plane $z=0$ and   $W^u_{loc}({\\mathcal P}_{a})$ is the surface $\\rho=1$.\n\n\nAs illustrated in Figure \\ref{local_map_LR_torus1}, we consider a  hollow three-dimensional cylinder $V_a(\\varepsilon_a)$  of \n${\\mathcal P}_{a}$ contained in the suspension of $U^*_a$ \n(with small $\\varepsilon_a>0$ to be determined later)\ngiven by\n$$\nV_a(\\varepsilon_a)=\\left\\{ (\\rho,\\theta,z):\\quad 1\\le\\rho< 1+\\varepsilon_a,\n\\quad 0\\le z< \\varepsilon_a\\quad \\text{and}\\quad \n\\theta\\in{\\mathbf R}\\pmod{2\\pi}\n\\right\\}\\  .\n$$\nWhen there is no ambiguity, we write $V_a$ instead of $V_a(\\varepsilon_a)$.\nIts boundary contains the trajectory ${\\mathcal P}_{a}$ and is a \nunion\n$$\n\\partial V_{a}= In({\\mathcal P}_{a}) \\cup Out({\\mathcal P}_{a}) \\cup \\mathcal{W} ({\\mathcal P}_{a})\n$$\nwhere \n\\begin{itemize}\n\\item\n$\\mathcal{W} ({\\mathcal P}_{a})={\\mathcal P}_{a}\\cup  \\left(W^s_{loc}({\\mathcal P}_{a})\\cap V_a\\right) \\cup  \\left(W^u_{loc}({\\mathcal P}_{a})\\cap V_a\\right)$.\n\\item\n$ W^s_{loc}({\\mathcal P}_{a})\\cap V_a$ is the lower boundary of the hollow cylinder, given by $z=0$, $1<\\rho\\le 1+\\varepsilon_a$.\n\\item\n$W^u_{loc}({\\mathcal P}_{a})\\cap V_a$ is the inner boundary of the hollow cylinder, given by $\\rho=1$, $0<z\\le\\varepsilon_a$.\n\\item \n$In({\\mathcal P}_{a})$ is the outer  wall of the cylinder, defined by $\\rho=1+\\varepsilon_a$, $0\\le z\\le\\varepsilon_a$.\\\\\nTrajectories starting at  $In({\\mathcal P}_{a})$ go inside $V_a$ in small positive time.\n\\item \n$Out({\\mathcal P}_{a})$ is the  top  of the cylinder, the annulus defined by $z=\\varepsilon_a$, $1\\le\\rho\\le 1+\\varepsilon_a$.\\\\\nTrajectories starting at $Out({\\mathcal P}_{a})$ go inside $V_a$ in small negative time. \n\\item \nThe vector field is transverse to $In({\\mathcal P}_{a})\\cup Out({\\mathcal P}_{a})$ except at the circle \n$In({\\mathcal P}_{a})\\cap Out({\\mathcal P}_{a})$.  \n\\end{itemize}\n\nThe cylinder wall $In({\\mathcal P}_{a})$ is\nparametrised by the covering map\n$$\n(\\varphi,r)\\mapsto(1+\\varepsilon_a,\\varphi,r)=(\\rho,\\theta,z),\n$$\nwhere $\\varphi\\in{\\mathbf R}$ and\n$0\\le r\\le\\varepsilon_a$. \nThe annulus $Out({\\mathcal P}_{a})$ is parametrised by the covering\n$$\n(\\varphi,r) \\mapsto ( r,\\varphi, \\varepsilon_a)=(\\rho,\\theta,z),\n$$\nfor $1\\le r\\le 1+\\varepsilon_a$ and $\\varphi\\in{\\mathbf R}$.\n\n\nFor $\\nu\\neq 0$,  $\\mu= 0$ we have from \\eqref{L5} of Proposition~\\ref{PropB1} that $W^u({\\mathcal P}_{\\mathbf v}) \\cap W^s({\\mathcal P}_{\\mathbf w})=\\varnothing$ and the same assumption is made for $\\mu\\ne 0$ in Theorem~\\ref{role_omega}.\nSince  for $\\nu=\\mu=0$ these invariant manifolds coincide, then for small $\\mu,\\nu\\ne 0$ the manifold $W^u({\\mathcal P}_{\\mathbf v})$ must come close to ${\\mathcal P}_{\\mathbf w}$.\nWithout loss of generality we may assume that it meets $In({\\mathcal P}_{\\mathbf v})$,\nwe are concerned with the parameter values for which this holds.\n\nFrom now on the portion of the unstable manifold of ${\\mathcal P}_{\\mathbf v}$  that goes from ${\\mathcal P}_{\\mathbf v}$ to $In({\\mathcal P}_{\\mathbf w})$ without intersecting $V_{\\mathbf w}$ will be  denoted $W^u_{loc}({\\mathcal P}_{\\mathbf v})$. Similarly, $W^s_{loc}({\\mathcal P}_{\\mathbf w})$ will denote the portion of the stable manifold of ${\\mathcal P}_{\\mathbf w}$ that is outside $V_{\\mathbf w}$ and goes directly from $Out({\\mathcal P}_{\\mathbf w})$ to ${\\mathcal P}_{\\mathbf v}$ in negative time.\n\n\\subsection{Local and global maps}\\label{sublocal}\nFor each $a\\in\\{{\\mathbf v}, {\\mathbf w}\\}$, we may solve  (\\ref{ode of suspension}) explicitly, then we compute the flight time from  $In({\\mathcal P}_{a})$ to $Out({\\mathcal P}_{a})$ by solving the equation $z(t)=\\varepsilon_a$ for the trajectory whose initial condition is \n$(\\rho,\\theta,z)=(1+\\varepsilon_a,\\varphi,r) \\in In({\\mathcal P}_{a})\\backslash W^s({\\mathcal P}_{a})$, with $z>0$, as in \\cite{LR17}.\n  Replacing this time in the other coordinates of the solution, yields the local map \n  $$\n  \\Phi _{a}:In({\\mathcal P}_{a})\\backslash W^s({\\mathcal P}_{a})\\, \\longrightarrow\\,  Out({\\mathcal P}_a)\n  $$\n  given by\n \\begin{equation}\n\\label{local map}\n\\Phi _{a}(\\varphi,r)=\n\\left(\\varphi-\\frac{2\\omega}{e_a}\\ln\\left(\\frac{r}{\\varepsilon_a}\\right),\n1+ \\varepsilon_a \\left(\\frac{r}{\\varepsilon_a}\\right)^{\\delta_a}\\right) \n \\end{equation}\n where $\\delta_a=\\dfrac{c_{a}}{e_{a}}>0$. \nFor the transition maps from one isolating block to the other we use assumptions \\ref{A8} and \\ref{A9}.\nWith this notation, we formulate them as follows: \n \\begin{itemize}  \n\\item \nThe two sets  $W^u({\\mathcal P}_{{\\mathbf w}})$  and $W^s({\\mathcal P}_{{\\mathbf v}})$ coincide; \n\\item\nThe manifold $W^u_{loc}({\\mathcal P}_{{\\mathbf v}})$ intersects\n the cylinder $In({\\mathcal P}_{{\\mathbf w}})$  on a non-contractible closed curve $\\gamma_{(\\nu, \\mu)}$. \n\\end{itemize}\nWe will assume that  the universal cover of $\\gamma_{(\\nu, \\mu)}$ is the graph of a smooth\nMorse function  \n$$\n\\xi_{(\\nu, \\mu)}:{\\mathbf R} \\rightarrow [0, \\varepsilon_{\\mathbf w}],\n$$\n as in Figure \\ref{xi1n},\nsatisfying the following conditions for $\\nu>0$: \n\\begin{itemize}\n\\item \n$\\xi_{(\\nu, \\mu)}$ is not constant \n (because the perturbing term $\\phi_\\mu$ is not constant)\n and is $2\\pi$-periodic; \n\\item \n$\\xi_{(\\nu, \\mu)}$ has a local maximum at  $\\varphi_1>0$, a local minimum at $\\varphi_2>\\varphi_1$ and no other critical point in the interval $(\\varphi_1,\\varphi_2)$;  \n\\item if $\\mu>0$ and $\\nu>0$ then $\\forall \\varphi\\in  {\\mathbf R}$,  $\\xi_{(\\nu, \\mu)}(\\varphi)>0$;  \n\\item $\\displaystyle \\lim_{\\nu\\rightarrow 0}\\,  \\max_{\\varphi\\in {\\mathbf R} } \\xi_{(\\nu, \\mu)}(\\varphi)=0$. \n\\end{itemize}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[height=5cm]{xi1nn.pdf}\n\\end{center}\n\\caption{\\small The local unstable manifold of  ${\\mathcal P}_{{\\mathbf v}}$ intersects the cylinder $In({\\mathcal P}_{{\\mathbf w}})$  on a closed curve, the graph of a periodic Morse function  $\\xi_{(\\nu, \\mu)}:{\\mathbf R} \\rightarrow [0, \\varepsilon_{\\mathbf w}]$ with $\\nu>0$, $\\mu>0$. }\n\\label{xi1n}\n\\end{figure}\n\nWith these assumptions, we may take  the maps $$\\Psi_{{\\mathbf v} \\rightarrow {\\mathbf w}}: Out({\\mathcal P}_{\\mathbf v})\\longrightarrow In ({\\mathcal P}_{\\mathbf w}) \\qquad \\text{and} \\qquad \\Psi_{{\\mathbf w} \\rightarrow {\\mathbf v}}: Out({\\mathcal P}_{\\mathbf w})\\longrightarrow In ({\\mathcal P}_{\\mathbf v})$$ to be given by\n\\begin{equation}\\label{transition21}\n\\qquad \\Psi_{{\\mathbf v} \\rightarrow {\\mathbf w}}(\\varphi,r)=\\left(\\, \\varphi, \\,\\, (r-1)+ \\xi_{(\\nu, \\mu)}(\\varphi)\\right) \\qquad\\text{and}\\qquad \\Psi_{{\\mathbf w} \\rightarrow {\\mathbf v}}(\\varphi,r)=\\left(\\, \\varphi, \\,\\, (r-1)\\right).\n\\end{equation}\n\\subsection{The return map}\n Let  ${\\mathcal R}_{(\\nu,  \\mu)}=\\Phi_{{\\mathbf v}} \\circ \\Psi_{{\\mathbf w} \\rightarrow {\\mathbf v}} \\circ \\Phi_{{\\mathbf w}} \\circ \\Psi_{{\\mathbf v} \\rightarrow {\\mathbf w}}$\n  be the first return map to $Out({\\mathcal P}_{\\mathbf v})$, well defined on the set   of initial conditions $(\\varphi,r) \\in Out({\\mathcal P}_{\\mathbf v})$ whose solution returns to $Out({\\mathcal P}_{\\mathbf v})$. \n For  $r>1$, the map ${\\mathcal R}_{(\\nu,\\mu)}$ is given by\n \\begin{eqnarray*}\\label{first1}\n{\\mathcal R}_{(\\nu, \\mu)}(\\varphi,r)&=& \\left[ \n\\varphi-\\omega K\\ln \\left[(r-1)+\\xi_{(\\nu, \\mu)} (\\varphi)\\right] -\\omega k_\\varepsilon\\pmod{2\\pi},\\ \n 1+\\dfrac{\\varepsilon_{\\mathbf v}}{\\varepsilon_{\\mathbf w}^{\\delta_{\\mathbf w}}}\\left[(r-1)+\\xi_{(\\nu, \\mu)} (\\varphi) \\right] ^\\delta\\right]\\\\\n&=& \\left(R_1(\\varphi,r), R_2(\\varphi ,r)\\right)\n\\end{eqnarray*}\nwhere\n$$\n\\delta = \\delta_{\\mathbf v} \\delta_{\\mathbf w}>1, \\qquad  \nK = 2 \\left( \\frac{e_{\\mathbf v} + c_{\\mathbf w}}{e_{\\mathbf v} \\, e_{\\mathbf w}} \\right)>0 \\qquad \\text{and}\\qquad \nk_\\varepsilon =-2K\\ln \\varepsilon_{\\mathbf w}>0.\n$$\nThe map ${\\mathcal R}_{(\\nu, \\mu)}$ is well defined if $\\varepsilon_{\\mathbf v}+\\displaystyle\\max_{0\\le \\varphi\\le 2\\pi}\\xi_{(\\nu, \\mu)} (\\varphi)\\le \\varepsilon_{\\mathbf w}$.\nIn particular, we need $\\varepsilon_{\\mathbf v}< \\varepsilon_{\\mathbf w}$. \n The inequality $\\delta>1$ comes from Property  \\ref{A5} and the conditions  of \\cite{KM1, KM2} for a heteroclinic cycle to be attracting.\n\n\n\\section{Proof of Theorem~\\ref{role_omega}}\n\\label{secProva_G}\n\nThe goal of the section\nis to obtain  an invariant set  $\\Lambda\\subset Out^\\pm({\\mathcal P}_{\\mathbf v})$ where the map ${\\mathcal R}_{(\\nu, \\mu)}|_\\Lambda$ is topologically conjugate to a Bernoulli shift with two symbols. \nThe argument uses the Conley-Moser conditions, see for instance \\cite{Wiggins}.\n\\subsection{Stretching the angular component}\nLet $[\\varphi_L,\\varphi_R]$ be an interval where $\\xi_{(\\nu, \\mu)} (\\varphi)$ is monotonically decreasing, with \n$$\n\\xi_L=\\xi_{(\\nu, \\mu)} (\\varphi_L)>\\xi_{(\\nu, \\mu)} (\\varphi_R)= \\xi_R\n$$\n and consider $\\mathcal{D}\\subset Out({\\mathcal P}_{{\\mathbf v}})$ parametrised by $(\\varphi,r)\\in [\\varphi_L,\\varphi_R]\\times[1,1+\\varepsilon_{\\mathbf v}]$, with $\\varepsilon_{\\mathbf v}+\\xi_L<\\varepsilon_{\\mathbf w}$.\nThen $\\mathcal{D}$ is a set  of initial conditions $(\\varphi,r) \\in Out({\\mathcal P}_{\\mathbf v})$ whose solution returns to $Out({\\mathcal P}_{\\mathbf v})$. We start by establishing some properties of the map ${\\mathcal R} $  within this set.\n\n\\begin{lemma}\n\\label{lema:contract}\nFor small $\\nu,\\mu>0$, the following assertions hold in  $\\mathcal{D}$  with $\\varepsilon_{\\mathbf v}+\\xi_L<\\varepsilon_{\\mathbf w}$: \n\\begin{enumerate}\n\\item\nfor any $r\\in [1,1+\\varepsilon_{\\mathbf v}]$ the map $\\varphi\\to R_1(\\varphi,r)$ is an expansion;\n\\item\nfor any $\\varphi\\in[\\varphi_L,\\varphi_R]$  the  map $r\\to R_2(\\varphi,r)$ is a contraction. \n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof} \nThe first assertion follows from\n$$\n\\dfrac{\\partial  R_1 (\\varphi,r)}{\\partial \\varphi } =\n1-\\dfrac{\\omega K}{r-1+\\xi_{(\\nu, \\mu)}( \\varphi) }\\dfrac{d\\xi_{(\\nu, \\mu)}( \\varphi)}{d\\varphi}>1\n$$\nbecause $\\xi_{(\\nu, \\mu)}( \\varphi) >0$ and  $\\dfrac{d\\xi_{(\\nu, \\mu)}( \\varphi)}{d\\varphi}<0$ since we are assuming $\\xi_{(\\nu, \\mu)}$ is monotonically decreasing in $[\\varphi_L,\\varphi_R]$.\nThe second assertion follows from\n$$\n\\dfrac{\\partial  R_2 (\\varphi,r)}{\\partial r } = \\delta\\dfrac{\\varepsilon_{\\mathbf v}}{\\varepsilon_{\\mathbf w}^{\\delta_{\\mathbf w}}} \\left[(r-1) +\\xi_{(\\nu, \\mu)}( \\varphi)\\right]^{\\delta-1}. \n$$\nSince  $0<\\xi_{(\\nu, \\mu)}( \\varphi)\\le \\xi_L$  and $0<r-1\\leq \\varepsilon_{\\mathbf v}\\le\\varepsilon_{\\mathbf w}<1$ then\n$$\n0<\\frac{\\partial  R_2 (\\varphi,r)}{\\partial r } \n\\le \\delta\\dfrac{\\varepsilon_{\\mathbf v}}{\\varepsilon_{\\mathbf w}^{\\delta_{\\mathbf w}}}\\left[\\varepsilon_{\\mathbf w}+\\xi_L \\right]^{\\delta-1}\n=\\delta\\dfrac{\\varepsilon_{\\mathbf v}}{\\varepsilon_{\\mathbf w}}+\\mathcal{O}(\\xi_L),\n$$\nwhere $\\mathcal{O}$ stands for the standard Landau notation. We have $\\displaystyle\\lim_{\\nu\\rightarrow 0} \\max_{\\varphi\\in[\\varphi_L,\\varphi_R]} \\xi_{(\\nu, \\mu)}(\\varphi)=0$,\ntherefore $0<\\dfrac{\\partial  R_2}{\\partial r} (\\varphi,r) <1 $ for small $\\mu>0$. \n\\end{proof}\n\nWe call the graph $(\\varphi,s(\\varphi))$ in  $\\mathcal{D}$ of a monotonic map $s(\\varphi)$ with \n$\\varphi_L\\le\\varphi\\le\\varphi_R$, a  \\emph{segment across} $\\mathcal{D}$.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=4cm]{spiral.pdf} \n\\end{center}\n\\caption{\\small When $\\omega\\ge\\omega_0$, the segment $S$ (red) in the domain  $\\mathcal{D}\\subset Out({\\mathcal P}_{{\\mathbf v}})$ (gray) is  transformed by the first return map ${\\mathcal R} $ into a curve (blue) that makes a full turn around $Out(\\mathcal{P}_{\\mathbf v})$ intersecting $\\mathcal{D}$ in at least one segment.}\n\\label{fig:spiral}\n\\end{figure}\n\n\n\\begin{lemma}\\label{lema:spiral}\nConsider the segment \n$S=\\{(\\varphi,r_*)\\ \\varphi_L\\le\\varphi\\le\\varphi_R \\}\\subset \\mathcal{D}$\nfor a given $r_*>0$.\nFor  small $\\mu,\\nu>0$   with $\\varepsilon_{\\mathbf v}+\\xi_L<\\varepsilon_{\\mathbf w}$, if \n$$\n\\omega\\ge\\omega_0=\\dfrac{ 2\\pi}{K\\ln( 1+(\\xi_L-\\xi_R)\/(1+\\xi_R))}\n$$ \nthen for any $r_*\\in(1,1+\\varepsilon_{\\mathbf v}]$,\nthe set ${\\mathcal R} (S)\\cap \\mathcal{D}$ is a curve   containing a segment across $\\mathcal{D}$.\n\\end{lemma}\n\n\\begin{proof}\nSince  $\\xi_{(\\nu, \\mu)} (\\varphi)$ is monotonically decreasing in $[\\varphi_L,\\varphi_R]$ then  the map \n$ \\varphi\\to R_2(\\varphi, r_*)$ is monotonically decreasing and \n$$ \\varphi\\mapsto \\varphi-\\omega K\\ln \\left[(r-1)+\\xi_{(\\nu, \\mu)} (\\varphi)\\right] -\\omega k_\\varepsilon=R_1(\\varphi,r_*)$$ is monotonically increasing  in the same interval. \nHence ${\\mathcal R} (S)$ is the graph of a monotonic map $s(\\varphi)$, with the map $s$ defined in some interval  $I$, as in Figure~\\ref{fig:spiral}. \nIt remains to obtain an estimate of the variation of the first coordinate of ${\\mathcal R} (S)$ to ensure that $[\\varphi_L,\\varphi_R]\\subset I$.\nFrom the definition of $R_1$ and properties of the logarithm, one knows that the difference $\\Delta =R_1(\\varphi_R,r_*) - R_1( \\varphi_L, r_*)$ satisfies\n$$\n\\begin{array}{rl}\n\\Delta =&\n\\left(\\varphi_R-\\omega K\\ln \\left[(r_*-1)+\\xi_{(\\nu, \\mu)} (\\varphi_R)\\right] -\\omega k_\\varepsilon\\right)-\n\\left(\\varphi_L-\\omega K\\ln \\left[(r_*-1)+\\xi_{(\\nu, \\mu)} (\\varphi_L)\\right] -\\omega k_\\varepsilon\\right)\\\\ \\\\\n=&(\\varphi_R-\\varphi_L)+\\omega K\n \\ln\\dfrac{(r_*-1)+\\xi_{(\\nu, \\mu)} (\\varphi_L)}{(r_*-1)+\\xi_{(\\nu, \\mu)} (\\varphi_R)}\\\\ \\\\\n =&(\\varphi_R-\\varphi_L)+\\omega K\n  \\ln \\dfrac{(r_*-1)+\\xi_L}{(r_*-1)+\\xi_R}> (\\varphi_R-\\varphi_L)\n\\end{array}\n$$\nwhere for the last inequality we use $\\xi_L>\\xi_R$ hence $ \\ln \\dfrac{(r_*-1)+\\xi_L}{(r_*-1)+\\xi_R}>0$.\nMoreover,\n$$\n\\dfrac{(r_*-1)+\\xi_L}{(r_*-1)+\\xi_R}=\n1+\\dfrac{\\xi_L-\\xi_R}{(r_*-1)+\\xi_R}\\ge \n1+\\dfrac{\\xi_L-\\xi_R}{1+\\xi_R}.\n$$\nTherefore, if $\\omega\\ge\\dfrac{ 2\\pi}{K\\ln( 1+(\\xi_L-\\xi_R)\/(1+\\xi_R))}$, then $\\Delta \\ge 2\\pi+  (\\varphi_R-\\varphi_L)$ and hence the curve ${\\mathcal R} (S)$ goes across $\\mathcal{D}$ at least once, as in Figures~\\ref{fig:spiral} and \\ref{horseshoe_fig}.\n\\end{proof}\n\n\\subsection{Proof of Theorem~\\ref{role_omega}. Part I}\n\\label{prova_parte1}\nGiven a rectangular region   in $Out({\\mathcal P}_{\\mathbf v})$, parametrised by \n$ [\\varphi_a, \\varphi_b]\\times [r_1,r_2]$, a \\emph{vertical  strip} in the region is a set\n$$\n{\\mathcal V}=\\{(\\varphi,r): \\varphi\\in[u_1(r),u_2(r)]\\qquad r \\in \\,[r_1,r_2]\\}\n$$\nwhere $u_1,u_2: [r_1,r_2] \\rightarrow [\\varphi_a,\\varphi_b]$ are Lipschitz functions with Lipschitz constants less than $\\mu_v\\ge 0$, such that $u_1(r)<u_2(r)$. The \\emph{ vertical  boundaries} of a  vertical  strip are the graphs of the maps $u_i$; the \\emph{horizontal boundaries} are the lines $\\{r_i\\} \\times  [u_1(r_i),u_2(r_i)]$, $i=1,2$; \nthe \\emph{width} $d({\\mathcal V})$ of the strip is \n$$\nd({\\mathcal V})=\\max_{r_1\\le r\\le r_2} |u_1(r)-u_2(r)|.\n$$\nIn an analogous way we define a \\emph{horizontal strip} across  a \\emph{horizontal rectangle} in $Out({\\mathcal P}_{\\mathbf v})$ with the roles of $\\varphi$ and $r$ reversed, Lipschitz constants less than $\\mu_h\\ge 0$.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[height=6cm]{stretching2_LR_torus.pdf}\n\\end{center}\n\\caption{\\small  For $r_\\star \\in  [0, 1]$, the image under ${\\mathcal R}$ of the segment $[\\varphi_L, \\varphi_R]\\times \\{r=r_* \\}$ is a curve (without folds) intersecting twice the rectangle $\\mathcal{D}$. \nThe  subsets $I_1$ and $I_2$ are stretched by the first return map ${\\mathcal R}$ into segments across $\\mathcal{D}$. }\\label{horseshoe_fig}\n\\end{figure}\n\n\nLet $I_1=[\\varphi_a, \\varphi_b]$ and $I_2=[\\varphi_c, \\varphi_d]$ be two disjoint intervals satisfying\n$$\n\\varphi_L\\le\\varphi_a<\\varphi_b<\\varphi_c<\\varphi_d\\le \\varphi_R.\n$$\nWe claim that for $\\omega\\ge\\omega_0$ (of Lemma~\\ref{lema:spiral}) the  two  vertical  strips\n$$\n{\\mathcal V}_i= \\{(\\varphi,r)\\in \\mathcal{D}: \\varphi\\in I_i\\}\\qquad i=1,2\n$$\nsatisfy the Conley-Moser conditions \\cite{Wiggins}: \n\\begin{enumerate}\n\\renewcommand{\\theenumi}{(P\\arabic{enumi})}\n\\renewcommand{\\labelenumi}{{\\theenumi}}\n\\item\\label{f(V)=H}\nThe image ${\\mathcal R} (V_i)\\cap \\mathcal{D}=H_i$ is the union of disjoint horizontal strips across $\\mathcal{D}$.\n\\item\\label{f(V)capV}\nFor every vertical strip $V\\subset (V_1\\cup V_2)$ the set ${\\mathcal R} ^{-1}(V)\\cap V_i=\\widetilde{V_i}$ is a vertical strip across $\\mathcal{D}$ with $d\\left(\\widetilde{V_i}\\right)\\le \\lambda_v d(V)$ for some $\\lambda_v\\in(0,1)$.\n\\item\\label{f(H)capH}\nFor every horizontal strip $H\\subset (H_1\\cup H_2)$ the set ${\\mathcal R} (H)\\cap H_i=\\widetilde{H_i}$ is a horizontal strip across $\\mathcal{D}$ with $d\\left(\\widetilde{H_i}\\right)\\le \\lambda_h d(H)$ for some $\\lambda_h\\in(0,1)$.\n\\end{enumerate}\nIn order to establish the claim, note that by Lemma~\\ref{lema:spiral} each line with constant $r$ in $\\mathcal{V}_i$ is mapped by ${\\mathcal R}$ into a curve in $Out(\\mathcal{P}_{\\mathbf v})$ that intersects $\\mathcal{D}$ in at least one segment. \n Lemma~\\ref{lema:contract} ensures that as $r$ varies in $]1,1+\\varepsilon_{\\mathbf v}]$, the second coordinate of their image varies in an interval of length less than $\\varepsilon_{\\mathbf v}$, so the union of these segments lies in horizontal strips across $\\mathcal{D}$, establishing \\ref{f(V)=H}.\nProperties~\\ref{f(V)capV} and  \\ref{f(H)capH} follow from Lemma~\\ref{lema:contract}.\nFrom this claim it follows that there exists an ${\\mathcal R}$-invariant set of initial conditions \n$$\n{\\Lambda}=\\bigcap_{n \\in {\\mathbf Z}} {\\mathcal R}^n({\\mathcal V}_1 \\cup {\\mathcal V}_2)\n$$ \non which $\\left.{\\mathcal R}\\right|_\\Lambda$ is topologically conjugate to a Bernoulli shift on two symbols. By construction it is a rotational horseshoe (according to \\cite{Passeggi}) with $m=2$.\n \n \n\n \n\n\n\\subsection{Proof of Theorem~\\ref{role_omega}. Part II}\nFor $\\nu>0$ and $\\mu=0$, the flow of \\eqref{general} has an attracting two-dimensional torus (by Proposition \\ref{PropB1}). In particular, there is a cross section $\\Sigma$ where the torus defines an invariant curve $\\mathcal{C}$ under the first return map ${\\mathcal R}$.  Furthermore, there is countable set of values of the type $(\\nu_i, 0)$, $i \\in {\\mathbf N}$, for which the first return map ${\\mathcal R} $  has at least one saddle and a sink lying on $\\mathcal{C}$ ($\\Rightarrow$ the torus is decomposed into periodic orbits with rational rotation number). Fix, once for all, one of these values.\n\nFor such a $\\nu_i>0$, increasing $\\mu>0$ the coexistence of this pair of periodic orbits persists along a wedge, the so called \\emph{Arnold tongue} \\cite{Anishchenko}. As illustrated in Figure \\ref{Strange_attractors1_LR_torus}, for a fixed $\\mu>0$, we know that:\n\\begin{itemize}\n\\item ${\\mathcal R}(\\mathcal{C})$ is a closed curve on $Out^+({\\mathcal P}_{\\mathbf v})$ because ${\\mathcal R}|_\\mathcal{D}$ is a diffeomorphism;\n\\item for $\\omega \\approx 0$,  this curve may be seen as the graph on $Out^+({\\mathcal P}_{\\mathbf v})$ of a non-constant map  defined on $[0, 2\\pi]$ (cf. \\cite{Rodrigues2019}); the curve $\\mathcal{C}$ is the $\\omega$-limit of  $W^u({\\mathcal P}_{\\mathbf w})$;\n\\item  by Lemma \\ref{lema:spiral}, there exists $\\omega_0>0$ and a segment $S\\subset \\mathcal{C}$ such that ${\\mathcal R} (S)\\cap \\mathcal{D}$ is a curve containing a segment across $\\mathcal{D}$.\n\\end{itemize}\n\nThis means that the  curve  $\\mathcal{C}$ starts to develop folds as in Figure~\\ref{Strange_attractors1_LR_torus} (B) and (C). \nIf $\\omega>\\omega_0$  it creates the rotational horseshoes proved in \\S \\ref{prova_parte1}.\n Within this wedge, \nAnishchenko, Safonova and Chua have shown in \\cite{Anishchenko}\n that there are curves on the parameter space $(\\nu, \\mu)$ corresponding to a quadratic homoclinic tangency associated to a dissipative periodic point of the first return map ${\\mathcal R}$.   \n\nUsing now the results of Mora and Viana in  \\cite{MV93}, there exists a positive measure set $\\mathcal{U}$ of parameter values, so that for every $(\\nu, \\mu) \\in \\mathcal{U}$, the return map $\\mathcal{R}$ \nadmits a strange attractor  of H\\'enon-type. \nThese strange attractors are supported in SRB measures. \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=12cm]{Strange_attractors1_LR_torus.pdf}\n\\end{center}\n\\caption{\\small  Image of $ {\\mathcal R}(\\mathcal{C})$ for different values of $\\omega$ with  $\\nu>0$ and $\\mu$ fixed.\nTransition from an invariant and attracting curve (A) to a rotational horseshoe (C), passing through a homoclinic tangency (B). One observes the breaking of the wave which accompanies the break of the invariant circle  (corresponding to the torus).  Here the point F is a sink and the point S is a saddle. In (C), a neighborhood of $r = 1$ is folded and mapped into itself, leading to the formation of rotational horseshoes. }\n\\label{Strange_attractors1_LR_torus}\n\\end{figure}\n\n\n\n\\begin{remark}\nIn this type of result, the number of connected components with which the strange attractors intersect the section $\\Sigma$ is not specified nor is the size of their basins of attraction. The strange attractors coexist with sinks from Newhouse phenomena. A discussion of these results may be found in \\cite{CR2021, Rodrigues2019, TS86}. \n\\end{remark}\n\n\n\n\n\n\\section{Proof of Proposition \\ref{periodic_solution_prop} }\n\\label{PropA}\n\n\n\nThis proposition concerns the case $\\mu=0$ when the time-periodic perturbation to $\\dot x={\\mathcal F}_\\nu(x)$ is constant.\nFor $\\nu=0$, Properties~\\ref{A1} to \\ref{A4} and \\ref{A6} of $\\dot x={\\mathcal F}_0(x)$ were established in  \\cite[Theorem 7]{ACL06}, with ${\\mathcal M}^2={\\mathbf S}^2$.\n \n\\begin{proof} The $\\kappa$-equivariance is easily checked directly from the expression of ${\\mathcal F}_\\nu$.\n\n The first part of the proof of Proposition \\ref{periodic_solution_prop}  consists in establishing that Properties~\\ref{A1} to \\ref{A3}  persist when the perturbation term $\\nu(1-x_1)$ is added.\n Properties~\\ref{A4} and \\ref{A6} are established in \\cite{ACL06} and Property~\\ref{A5} is a consequence of their results.\nThen it remains to show that two periodic solutions are created by the perturbation when the connections from ${\\mathbf w}$ to ${\\mathbf v}$ are broken, as stated in \\ref{A7}.\nAddressing the  persistence and property  \\ref{A7} constitutes the remainder of this proof.\n\n\\begin{itemize}\n\\item[\\ref{A1}] \nSince for $\\nu=0$ the sphere ${\\mathbf S}^2$ is normally hyperbolic as in  \\cite{HPS}, then for small $\\nu\\ne 0$ it persists as a flow-invariant, normally hyperbolic, globally attracting manifold ${\\mathcal M}^2$. See also the analysis by \\cite{HG97}.  \n\n\\item[\\ref{A2}] \nFor  $\\nu=0$ the only equilibria in the flow-invariant plane $\\Fix({\\mathbf Z}_2(\\kappa))$ are ${\\mathbf v}$ and ${\\mathbf w}$ above, as well as  the origin $O$.\nSince they are hyperbolic, then their hyperbolic continuations should exist within the plane $\\Fix({\\mathbf Z}_2(\\kappa))$.\nAnother way to see this is to  solve\n$$\n\\left\\{\\begin{array}{l}\nF_1(x_1,x_3)= x_1[(1-x_1^2-x_3^2)-\\alpha  x_3 +\\beta x_3^2 - \\nu] + \\nu =0\\\\ \\\\\nF_2(x_1,x_3)=  x_3[(-x_1^2-x_3^2) -\\beta x_1^2] +\\alpha x_1^2=0 .\n\\end{array}\\right.\n$$\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=75mm]{G1_G2_1A.pdf}\\qquad \\includegraphics[width=75mm]{G1_G2_2n.pdf}\n\\end{center}\n\\caption{\\small Equilibria of $\\dot\\zeta={\\mathcal F}_{\\nu}(\\zeta)$ for $\\nu=0$ (left) and $\\nu=0.5$ (right), $\\alpha=1$, $\\beta=-0.1$ in the flow-invariant subspace $\\Fix({\\mathbf Z}_2(\\kappa))$ occur at the intersection of the curves $F_1(x_1,x_3)=0$ (blue) and $F_2(x_1,x_3)=0$ (red), here plotted with Maxima.\n}\n\\label{G1G2}\n\\end{figure}\n\nFigure~\\ref{G1G2} shows the curves $F_1=0$ and $F_2=0$ plotted with Maxima. \nFor $\\nu=0$ the curves intersect transversely at $O$, ${\\mathbf v}$ and ${\\mathbf w}$ and this property is preserved for small $\\nu\\ne 0$.\n\\item[\\ref{A3}] \nWithin the flow-invariant plane $\\Fix({\\mathbf Z}_2(\\kappa))$, the origin is a repelling source,  ${\\mathbf v}$ is a sink and ${\\mathbf w}$  is a saddle. Since both the plane $y=0$ and the manifold ${\\mathcal M}^2$ are flow-invariant, this means that there are two heteroclinic connections from \n$\\tilde{{\\mathbf w}}$ to $\\tilde{{\\mathbf v}}$.   \n\n\n\\item[\\ref{A5}] \nIn \\cite{ACL06} it is established that the only other equilibria in ${\\mathbf S}^2$ are the four hyperbolic repelling foci $(\\pm \\sqrt{2}\/2, \\pm \\sqrt{2}\/2, 0)$. \nThis means that for  $\\nu=0$, by the Poincar\\'e-Bendixon Theorem,  the $\\omega$-limit set of all other points in ${\\mathbf S}^2$ must be contained in the heteroclinic cycles that contain ${\\mathbf v}$ and ${\\mathbf w}$.\nThe unstable foci  remain for $\\nu\\ne 0$ small. \n\n\\item[\\ref{A7}]\nThe flow-invariant subspace $\\Fix({\\mathbf Z}_2(\\kappa))$ divides ${\\mathcal M}^2$ in two flow-invariant components.\nWe will show that the $x_2>0$ component contains a non-constant periodic solution, the proof for $x_2<0$ follows from the symmetry.\nFor $x_1=0$ and $\\mu=0$  the expression \\eqref{general4} yields $\\dot x=\\nu\\ne 0$.\nSuppose $\\nu>0$, then the region $x_1>0$, $x_2>0$ in ${\\mathcal M}^2$ is positively invariant, see Figure~\\ref{G1G2_5A}.\nThis region only contains one equilibrium, one of the repelling foci in \\ref{A4}, hence by the Poincar\\'e-Bendixon Theorem,\nthe $\\omega$-limit of the unstable manifold of $\\tilde{\\mathbf v}$ is an attracting periodic solution. \nWhen $\\nu<0$  the periodic solution appears  for $x_1<0$.   \nThe period tends to $+\\infty$ as $\\nu$ goes to $0$, since the periodic trajectory  accumulates on\n the heteroclinic cycle.\n\\end{itemize}\n\\end{proof}\n\nProposition \\ref{periodic_solution_prop} shows that for sufficienly small $|\\nu|>0$, each heteroclinic cycle that occurred in the fully symmetric case  \nis replaced by a stable hyperbolic periodic solution. Using the reflection symmetry ${\\mathbf Z}_2(\\kappa)$, two stable periodic solutions co-exist, one in each connected component of $\\mathcal{M}\\backslash \\Fix({\\mathbf Z}_2(\\kappa))$. Their period tends to $\\infty$ when $\\nu$ vanishes  and their basin of attraction must contain the basin of attraction of $\\Sigma_0$. \n \n\n\n\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=65mm]{G1_G2_5An.pdf}\\includegraphics[width=65mm]{G1_G2_3A.pdf} \\\\\\includegraphics[width=65mm]{G1_G2_4A.pdf}\n\\end{center}\n\\caption{\\small Qualitative phase portrait for the dynamics or \\eqref{general4} in ${\\mathcal M}^2$ with $y>0$, projected   into the $(x_1,0,x_3)$ plane for  $\\mu=0$ and $\\nu>0$ . }\n\\label{G1G2_5A}\n\\end{figure}\n\n\n\\section{Discussion and concluding remarks}\n\\label{discussion}\n\n    \\emph{Routes  to chaos}  have been a recurrent concern on nonlinear dynamics during the last decades \\cite{AS91}. The novelty of the present paper is the illustration of a \\emph{new route} for the emergence of strange attractors  from an attracting heteroclinic network as a codimension-two phenomenon. \n\n\\subsection{Literature}\nIn \\cite{Kaneko}  Kaneko investigates numerically the\n bifurcations of tori in a two-parameter family of dissipative coupled maps. Each map undergoes a period-doubling cascade accumulating on a given parameter value, which generates chaos in the coupled system.   In the same setting Bakri and Verhulst show in  \\cite{Bakri}  that,  for small amplitudes, zero damping and zero coupling, their system reveals a periodic solution which undergoes a Hopf bifurcation, generating an attracting torus. \n They use numerical bifurcation techniques to show   how the torus gets destroyed by dynamical and topological changes in the involved manifolds. The results agree with \\cite{Ruelle, TS86}.\n\nIn the context of dissipative vortex  dynamics Fleurantin and James have studied in \\cite{FJ2020} the  \\emph{Langford system}, a   one-parameter family of three-dimensional vector fields.\nThe flow of this model exhibits a sink, two saddle-foci of different Morse indices and a non-trivial periodic solution with a complex conjugate pair of Floquet exponents. The frequency of the periodic solution together with the frequency of the complex exponent constitute two competing natural modes of oscillation. \nThe periodic orbit undergoes a \nbifurcation  giving rise to observable chaos  through the same mechanism of \\cite{Bakri}.\n These authors studied the evolution of the torus,  its loss of differentiability,  and the appearance of a strange attractor via the existence of  tangencies.  \n  The relative position of the manifolds according to the parameter allows the authors to prove the existence of \\emph{bistability} between an equilibrium and a torus.  \n\n\\subsection{Heteroclinic tangle}\nThe formation of the horseshoe of Theorem \\ref{role_omega} has a different nature to   those found in \\cite{ACL06, LR17} -- in this case, the shift dynamics is obtained via the transverse intersection of two two-dimensional invariant manifolds.  The parameter $\\omega$ is not necessary to prove the existence of chaos.  The non-wandering set associated to the network contains, but does not coincide with, the suspension of horseshoes; it contains  infinitely many heteroclinic pulses and attracting limit cycles with long  periods, coexisting with sets with positive entropy, giving rise to the so called \\emph{quasi-stochastic attractors} \\cite{LR17}.  The sinks  have long periods and narrow basins of attraction.\n\n\n\\subsection{Open questions}\nIn the context of this class of examples, some problems remain to be solved:\n\\begin{enumerate}\n\\item the basins of attraction of the strange attractors of Theorem B are relatively small in terms of Lebesgue measure (they are close to Newhouse domains). Could we improve Theorem B in order to get the existence of a ``larger'' strange attractor?\n\\item is it possible to generalize our result for clean heteroclinic networks (networks whose unstable manifolds are contained within it) whose connections are one-dimensional?\n\\end{enumerate}\nWe believe that these problems can be tackled by using the theory of rank-one attractors developed by Wang and Young \\cite{WY}. We defer these tasks for future work.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nA combined measurement of the hyperfine structure (HFS) splittings in hydrogenlike and lithiumlike ions of $^{209}$Bi has been suggested as early as 2001 \\cite{Shabaev:01a} to be a sensitive probe for bound-state strong-field QED in the strongest static magnetic fields available in the laboratory. Such fields exist in the surrounding of heavy nuclei with nuclear spin and a large nuclear magnetic moment. The electron in H-like $^{209}$Bi$^{82+}$, for example, experiences on average a magnetic field of about 30\\,000\\,T, more than 1000 times stronger than available with the strongest superconducting magnet.    \nAccording to \\cite{Shabaev:01a}, a special combination of the ground-state HFS splittings in H-like and Li-like ions ($\\Delta E^{(1s)}$ and $\\Delta E^{(2s)}$, respectively) of the same nuclear species, called the specific difference\n\\begin{eqnarray}\n\\Delta 'E = \\Delta E^{(2s)} - \\xi \\Delta E^{(1s)},\n\\end{eqnarray}\nprovides the best means to test bound-state strong-field QED in the magnetic regime.\nHere, the parameter $\\xi = 0.16886$ \\cite{Shabaev:01a,Volotka:12} is chosen to cancel the contributions of the nuclear-magnetization distribution  (Bohr-Weisskopf effect) to $\\Delta E^{(1s)}$ and $\\Delta E^{(2s)}$.\nThis is required since the uncertainties of these contributions to the HFS splittings are commonly larger than the complete QED contribution and have failed all previous attempts to perform a QED test solely based on the HFS splitting in H-like heavy ions. However, at the time of the proposal \\cite{Shabaev:01a} the experimental uncertainty of the HFS splitting in the Li-like $^{209}$Bi$^{80+}$ extracted \nfrom x-ray emission spectra \\cite{Beiersdorfer:98} was far too high to verify the predictions\nfor $\\Delta 'E$.\nThe first laser spectroscopic observation of the splitting reported in 2014 was orders of magnitude more precise but still limited by systematical uncertainties \\cite{Lochmann:14}. Finally, further improvement in accuracy by more than an order of magnitude was recently reported \\cite{Ullmann:15,Ullmann:17} but the result was surprisingly more than $7\\sigma$ off from the latest theoretical prediction \\cite{Volotka:12}.\nSince the experimental nuclear magnetic moment $\\mu_I$ of $^{209}$Bi enters the calculation of the specific difference, an incorrect value will lead to a proportional change in $\\Delta^\\prime E$, which could be responsible for the discrepancy \\cite{Karr:17}.\nWe also note that in Ref. \\cite{Urrutia:96} the discrepancy between theory and experiment on the HFS splitting in H-like Ho was ascribed to an inaccurate value of the nuclear magnetic moment of ${^{165}}$Ho.\n\nWe have reexamined the literature value $\\mu_I$($^{209}$Bi)\nobtained from nuclear magnetic resonance (NMR) experiments from a theoretical point of view. This has motivated new NMR measurements of bismuth ions in different chemical environments. Results of these experiments  are reported and analyzed applying high-level\nfour-component relativistic coupled cluster theory for advanced chemical shift calculations. We show that our result can completely resolve the hyperfine puzzle established in \\cite{Ullmann:17}. \nThe specific difference $\\Delta ' E$ has, so far, always been calculated using \nthe magnetic moment $\\mu_I(^{209}{\\rm Bi})=4.1106(2)\\mu_N$ tabulated in\n\\cite{Raghavan:89}. \nThis value was obtained using the uncorrected (for shielding effects) experimental value of the magnetic moment $\\mu_I(^{209}{\\rm Bi})=4.03910(19)\\mu_N$ reported in an NMR study \\cite{Ting:53} of bismuth nitrate, Bi(NO$_3$)$_3$, which was then  combined with the shielding constant for the Bi$^{3+}$ cation calculated in \\cite{Johnson:68}.\nIn \\cite{Bastug:96} the self-consistent relativistic molecular Dirac-Fock-Slater calculation of the shielding constant of the Bi(NO$_3$)$_3$ molecule using the Lamb formula \\cite{Lamb:41} was performed.\nThe final value, $\\sigma=17290(60)$\\,ppm, with very small uncertainty was obtained by combining relativistic random phase approximation calculation of the Bi$^{3+}$ cation (17270 ppm) with the molecular correction. The authors concluded that the molecular correction is very small and thus supported the value from \\cite{Raghavan:89}. \n\nHowever, \nthe authors of \\cite{Bastug:96} have not taken into account chemical processes that occur in an aqueous solution of bismuth nitrate molecule Bi(NO$_3$)$_3 \\cdot$5H$_2$O: the compound\ndissociates and the Bi$^{3+}$ cation is surrounded by water molecules (hydration). Neither the completeness nor the exact form of hydration as a function of concentration, \\textit{p}H or temperature is well understood.  \nWhile it was suggested in \\cite{Fedorov:98} that in strongly acidic solutions Bi$^{3+}$ exists as hexaaquabismuth(III)-cation [Bi(H$_2$O)$_6$]$^{3+}$, more recent studies \\cite{Naslund:00} expect that the hydrated form is rather [Bi(H$_2$O)$_8$]$^{3+}$. We found that in both cases the electronic structure of the $n$-coordinated complex significantly differs from the Bi(NO$_3$)$_3$ molecule considered in \\cite{Bastug:96}, which is expected. The molecular environment in \nBi(III\/V)-containing\ncomplexes strongly contributes to the shielding constant and a considerable chemical shift is introduced. \nConsequently, the value of the shielding constant obtained in Ref.\\,\\cite{Bastug:96} cannot be used for the precise extraction of the $^{209}$Bi magnetic moment from the experimental NMR data. \n\nThere is, however, additional NMR data for another Bi containing system: the\nhexafluoridobismuthate(V) anion ($^{209}$BiF$_6^-$) \\cite{Morgan:83}. It has seven atoms and high spatial symmetry.\nAccording to Morgan \\textit{et} al.\\ \\cite{Morgan:83}, a measurement of BiF$_6^-$ with reference to a saturated solution of bismuth nitrate in concentrated nitric acid gave a chemical shift of $-24$\\,ppm. Unfortunately, there is an inconsistency in the reported experimental data of \\cite{Morgan:83}, since the measured frequency ratio is given as $\\nu(^{209} {\\rm BiF}_6^-) \/ \\nu(^1{\\rm H})$ =0.16017649(10). The comparison of this ratio with the one reported in \\cite{Ting:53} indicates a massive chemical shift of about $\\delta \\approx +3200$\\,ppm instead of $-24$\\,ppm. We have performed NMR measurements of both samples to clarify these discrepancies.  \n\n\\section{Experiment}\nSince a dependence of the chemical state of the Bi$^{3+}$ ions in an aqueous solution is expected but details on the sample preparation are missing in the original NMR measurements \\cite{Ting:53}, we performed a systematic study using various bismuth nitrate solutions.\nSamples of ``Bi(NO$_3$)$_3$'' solutions were prepared with concentrations of 2.5\\%, 5\\% and 10\\% Bi$^{3+}$ (wt \\%) in concentrated (65 wt \\%) and diluted aqueous solutions (50, 30, 20, 10 wt \\%) of nitric acid (HNO$_3$). \n\n\\begin{figure}[t]\n\\includegraphics[width=0.98\\linewidth]{Fig1New.pdf}\n\\caption{\\label{fig:Spectra} NMR spectra of Bi(NO$_3$)$_3$ solution (10\\% Bi (wt \\%)) in concentrated nitric acid (gray) and  NMe$_4$BiF$_6$ diluted in acetonitrile (blue).}\n\\end{figure}\n\nBiF$_6^-$ anions were obtained by dissolution of $\\mathrm{(CH_3)_4N^+BiF_6^-}$ (NMe$_4$BiF$_6$)\nin acetonitrile to a saturated solution\n\\cite{Note1}. \n\nAll NMR measurements were performed at an 8.4-T magnet using the same double resonance probe for $^{209}$Bi NMR and $^{1}$H NMR calibration with tetramethylsilane. The sample temperature was stabilized with an accuracy of 1\\,K employing a constant gas flow tempered by an electric heater. Spectra were obtained from the free induction decay following a 90$^\\circ$ pulse of 3.5\\,$\\upmu$s length for $^{209}$Bi. \n\nTypical spectra of the $^{209}$Bi atoms in BiF$_6^-$ and in the nitrate solution are shown in Fig.\\,\\ref{fig:Spectra}. The advantage of BiF$_6^-$ is obvious. It exhibits a much narrower linewidth (200 Hz) and the septet arising from indirect spin coupling of $^{19}$F atoms directly bonded to the bismuth atom assures the chemical environment.\nThe observed ratio of the peak intensities is close to the expected ratio 1\\,:\\,6\\,:\\,15\\,:\\,20\\,:\\,15\\,:\\,6\\,:\\,1 and a spin-spin coupling of 3807(14)\\,Hz was determined, in good agreement with \\cite{Morgan:83}.\nNote that a $^{19}$F spectrum of the sample was taken as well and a decet consistent with the coupling of an $I=9\/2$ nucleus to an octahedral environment of six fluorine atoms was observed. \nThe signal from the nitrate solution is much wider. Even at the highest temperature of 360\\,K, the width of the $^{209}$Bi spectra was 4.4\\,kHz due to the short spin-lattice and spin-spin relaxation times of $\\approx 70$\\,$\\upmu$s. This width limits the accuracy of the $^{209}$Bi resonance frequency in the solution of the nitrate to 1\\,ppm. The chemical shift of Bi$^{3+}$ in the solution of the bismuth nitrate with respect to Bi$^{5+}$ in BiF$_6^-$ is $-106$\\,ppm, larger than the $-24$\\,ppm reported in \\cite{Morgan:83}. Contrary to \\cite{Flynn:59} we found that the variation of the bismuth concentration between mass fractions of 2\\% and about 40\\% (saturation) in nitric acid of 30\\% had no appreciable effect on the measured Larmor frequency as long as temperature and nitric acid concentration were kept constant. \n\n\nVariations of the Bi(NO$_3$)$_3$ sample temperature from 250 to 360\\,K were performed with the sample of 10\\% Bi in concentrated nitric acid (65\\%). We observed a strong linear temperature dependence of the frequency ratio in this range (Fig.\\,\\ref{fig:TempDependence}) with a slope of $+4.69(13)\\times 10^{-7}$\\,K$^{-1}$, corresponding to about 3\\,ppm\/K,\nwhich might be caused by the change of density.\nFor standard NMR conditions at 298.15\\,K a frequency ratio of $\\nu_\\mathrm{^{209}Bi^{3+}}\/\\nu_\\mathrm{H}=0.160699(1)$ was determined, where the given uncertainty is purely statistical. This value is in excellent agreement with 0.160696(6) reported in \\cite{Ting:53}. The temperature dependency of BiF$_6^-$ is 2 orders of magnitude smaller ($\\approx 20$\\,ppb\/K) and of opposite sign. At 298.15\\,K the frequency ratio to the proton is 0.1607167(2) far off from the value provided in \\cite{Morgan:83}. However, the latter matches our value if one simply flips two digits [$0.160{\\bf 17}65(1) \\to 0.160{\\bf 71}65(1)$].  \n\n\\begin{figure}[t]\n\\includegraphics[width=0.98\\linewidth]{Fig2Test3.pdf}\n\\caption{\\label{fig:TempDependence} Temperature and HNO$_3$-concentration dependency of the NMR Larmor-frequency ratios of bismuth and hydrogen. A strong temperature effect is observed for Bi(NO$_3$)$_3$ solutions, here exemplified for a \n10\\% Bi$^{3+}$ (wt \\%) solution in concentrated nitric acid (black), whereas only a minor effect was measured for NMe$_4$BiF$_6$ dissolved in acetonitrile (blue).\nInset: Larmor-frequency ratios\nmeasured by NMR in Bi(NO$_3$)$_3$ solutions with 2.5\\% Bi$^{3+}$ (wt \\%) in nitric acid (HNO$_3$) of various concentrations at 300\\,K. \nThe $y$ axis is identical to the main graph and the gray band represents the total variation.}\n\\end{figure}\n\nFinally, we have studied the resonance position of Bi(NO$_3$)$_3$ as a function of the nitric acid concentration\n(inset in Fig.\\,\\ref{fig:TempDependence}).\nA clear dependence on the acidity is observed for all Bi$^{3+}$ concentrations, covering a range of typically $\\approx 60$\\,ppm. \n\nIn summary, the results clearly demonstrate that a large uncertainty is connected with the extraction of the magnetic moment of $^{209}$Bi from NMR measurements in aqueous solutions of Bi(NO$_3$)$_3$. The influence of the chemical environment was strongly underestimated in theory since the calculations performed to extract the chemical shift do neither account for the temperature nor for the concentration or acidity of the sample. In this respect, BiF$_6^-$ is a much better candidate to obtain a reliable value of the magnetic moment which will be substantiated now also from a  theoretical point of view.  \n\n\\section{Theory}\n\nIn the presence of the external uniform magnetic field \\textbf{B} and nuclear magnetic moment $\\mu_j$ of $j-$th atom in a molecule the corresponding Dirac-Coulomb Hamiltonian includes the following terms:\n\\begin{equation}\n \\label{HB}\nH_B={\\rm \\bf{B}}\\cdot \\frac{c}{2}(\\bm{r}_G \\times \\bm{\\alpha}),\n\\end{equation}\n\\begin{equation}\n \\label{HHFS}\nH_{\\rm hyp}=\\frac{1}{c} \\sum_j \\bm{\\mu}_j\\cdot \\frac{(\\bm{r}_j \\times \\bm{\\alpha})}{r_j^3},\n\\end{equation}\nwhere $\\bm{r}_G = \\bm{r} - \\bm{R_G}$, $\\bm{R_G}$ is the gauge origin, \n$\\bm{r}_j=\\bm{r} - \\bm{R_j}$, $\\bm{R_j}$ is the position of nucleus $j$, and $\\bm{\\alpha}$ are the Dirac matrices.\n\nThe chemical shielding tensor of the nucleus $j$ can be defined as a mixed derivative of the energy with respect to the nuclear magnetic moment and the strength of the magnetic field\n\\begin{equation}\n \\label{SHIELDINGDer}\n\\left.\\sigma^j_{a,b}=\\frac{\\partial^2E}{\\partial\\mu_{j,a}\\partial B_b} \\right|_{\\bm{\\mu}_j=0,{\\rm \\bf{B}}=0}.\n\\end{equation}\nWe are interested in its isotropic part.\n\nIn the one-electron case the shielding tensor (\\ref{SHIELDINGDer}) can be calculated by the sum-over-states method within the second-order perturbation theory with perturbations (\\ref{HB}) and (\\ref{HHFS}). In the relativistic four-component approach the summation should include both positive and negative energy spectra \\cite{Aucar:99}.\nThe part associated with positive energy is called the ``paramagnetic'' term while the part associated with negative energy states is called ``diamagnetic term'' though only their sum is gauge invariant \\cite{Aucar:99}.\n\n\nTo avoid an ambiguity in calculations utilizing finite basis sets due to the choice of the gauge origin $\\bm{R_G}$ one can use the so-called London atomic orbitals (LAOs) method (see e.g.\\ \\cite{Olejniczak:12,Ilias:13} for details).\nIn Refs.\\,\\cite{DIRAC15,Olejniczak:12,Ilias:13} the four-component density functional theory (DFT) using response technique and LAOs has been developed to calculate the shielding constant (\\ref{SHIELDINGDer}).\nTo construct the atomic basis sets for the unperturbed Dirac-Coulomb Hamiltonian calculations one often uses the restricted kinetic balance (RKB) method. However, in the presence of the external magnetic fields the usual relation between the large and small component changes. In Ref.\\,\\cite{Olejniczak:12} the scheme of magnetic balance (MB) in conjunction with LAOs was proposed to take into account the modified coupling which is utilised below.\n\nMost of the chemical shift calculations for heavy atom compounds are performed within the (relativistic) DFT. The drawback of the theory is that it is hard to control the uncertainty of the results as there is no systematic way of improving it. Even combinations with high-level nonrelativistic \\textit{ab initio} wave-function-based calculations are also questionable in the case of heavy atom compounds. In Refs.\\,\\cite{Skripnikov:16b,Skripnikov:15a,Petrov:17b,Skripnikov:17c} it was shown that for such properties as the hyperfine structure constant and the molecular $g$ factor, the relativistic coupled cluster method gives the most accurate results if there are no multireference effects. Therefore, this method has been adopted here to control the uncertainty of the DFT results. \n\n\n\n\n\\section{Electronic structure calculation details}\n\n\nIn the present study we have used atomic basis sets of different qualities.\nThe NZ (where N~$=$~Double, Triple, Quadruple) basis set corresponds to the uncontracted core-valence N-zeta \\cite{Dyall:07,Dyall:12} Dyall's basis set for Bi and augmented correlation consistent polarized valence N-zeta, aug-cc-pVNZ \\cite{Dunning:89,Kendall:92} basis set for light atoms.\nIn the DZC basis set the contracted version of the aug-cc-pVDZ \\cite{Dunning:89,Kendall:92} basis sets were used for light atoms. \n\n\nBased on the nonrelativistic estimates, the hybrid density functional PBE0 \\cite{pbe0} has been chosen because it reproduces the nonrelativistic coupled cluster value rather well.\nGeometry parameters of the BiF$_6^-$ anion have been obtained in the scalar-relativistic DFT calculation using the generalized relativistic pseudopotential method \\cite{Mosyagin:16}.\n\nThe contribution of the Gaunt interaction to the shielding constant was estimated as the difference between  the values calculated at the Dirac-Hartree-Fock-Gaunt and Dirac-Hartree-Fock level of theory within the uncoupled scheme.\n\nNonrelativistic and scalar-relativistic calculations were performed within the {\\sc us-gamess} \\cite{USGAMESS1} and {\\sc cfour} \\cite{CFOUR} codes. Relativistic four-component calculations were performed within the {\\sc dirac15} \\cite{DIRAC15} and {\\sc mrcc} \\cite{MRCC2013} codes. For calculation of the hyperfine-interaction matrix elements and $g$ factors the code developed in Refs.\\,\\cite{Skripnikov:16b,Skripnikov:15b,Skripnikov:15a} was used.\n\n\n\\section{Results and discussion}\n\n\nTable \\ref{BiF6} contains results of the calculation of the BiF$_6^-$ anion.\n\\begin{table}[h]\n\\centering\n\\caption{The values of $^{209}$Bi shielding constants in BiF$_6^-$ in ppm.}\n\\label{BiF6}\n\\begin{tabular}{lccc}\n\\hline                    \n\\hline\nBasis set\/method     & Diamagnetic & Paramagnetic & Total \\\\\n\\hline                    \n\\hline\nDZ-MB-LAO\/DHF    & 8\\,618 & 5\\,768 & 14\\,386  \\\\  \n\nDZ-MB-LAO\/DFT    & 8\\,621 & 3\\,726 & 12\\,347  \\\\     \nTZ-MB-LAO\/DFT    & 8\\,639 & 3\\,733 & 12\\,372  \\\\  \n\\hline    \nDZC-RKB\/DFT      &      & 3\\,848    &    \\\\\nDZC-RKB\/CCSD     &      & 4\\,403    &    \\\\\nDZC-RKB\/CCSD(T)  &      & 4\\,286    &    \\\\\n\\hline    \nQZ-MB-LAO\/DFT    & 8\\,628 & 3\\,763 & 12\\,391  \\\\  \nCorrelation correction &      &  437 &        \\\\  \nGaunt correction       &      &  -37 &        \\\\  \n\\hline                    \nFinal              &      &   & 12\\,792  \\\\  \n\\hline                    \n\\hline                    \n\\end{tabular}\n\\end{table}\nComparing Dirac-Hartree-Fock (DHF) and DFT results in Table \\ref{BiF6} it can be seen that the diamagnetic contribution to $\\sigma(^{209}{\\rm Bi})$ depends only weakly on the correlation effects, while the paramagnetic contribution is strongly affected.\nTo check the accuracy of the latter DFT result we have performed a series of relativistic coupled cluster calculations of $\\sigma(^{209}{\\rm Bi})$ taking into account only the positive energy spectrum. \nComparing values obtained within the coupled cluster with single, double and noniterative triple-cluster amplitudes (CCSD(T)) with that of CCSD shows that the triple amplitudes only slightly contribute to $\\sigma(^{209}{\\rm Bi})$ demonstrating good convergence of the results with respect to the electron correlation treatment \n\\cite{Note2}. \n\nIn the final value of $\\sigma(^{209}{\\rm Bi})$ we include the correlation correction calculated as the difference between the CCSD(T) and PBE0 results.\n\nTo investigate the importance of systematic treatment of the  molecular environment \n we have also performed an additional DHF study of one of the possible hydrated forms of Bi$^{3+}$ in an acidic solution of Bi(NO$_3$)$_3$ -- [Bi(H$_2$O)$_8$]$^{3+}$ cation in comparison with the unsolvated Bi$^{3+}$ cation.\nIt was found that the shielding constant of the $^{209}$Bi$^{3+}$ is significantly larger (by about 20\\% at the DHF level) than that in [$^{209}$Bi(H$_2$O)$_8$]$^{3+}$.\nTherefore, the interpretation of the \\textit{molecular} NMR experiment in terms of the nuclear magnetic moment using a shielding constant obtained for the corresponding ion (as was done in earlier studies) is associated with considerable uncertainties.\n\n\nWe now use the value obtained for $\\nu_\\mathrm{^{209}BiF_6^-}\/\\nu_\\mathrm{H}= 0.1607167(2)$ from our NMR measurements and the shielding constant of $\\sigma(^{209}{\\rm BiF}_6^-)=12\\,792$\\,ppm calculated above to obtain $\\mu_I(\\mathrm{^{209}Bi}) = 4.092(2)\\,\\mu_\\mathrm{N}$ with an uncertainty dominated by theory. \n\nTable \\ref{NewOld} compares the experimental values \\cite{Ullmann:17} of the HFS splittings with the theoretical values calculated with the old [$\\mu_I$(old)$=$4.1106(2)$\\mu_N$] and the new [$\\mu_I$(new)$=$4.092(2)$\\mu_N$] values of the nuclear magnetic moment \\cite{Volotka:12}. The theoretical results include the most elaborated calculation of the Bohr-Weisskopf effect \\cite{Senkov:02}.\n\\begin{table}[]\n\\centering\n\\caption{Theoretical values of $\\Delta E^{(1s)}$ and $\\Delta E^{(2s)}$ (in meV) calculated with old and new nuclear magnetic moment of $^{209}$Bi in comparison with the experimental values \\cite{Ullmann:17}.\nFor the Bohr-Weisskopf effect the most elaborated calculation by Sen'kov and Dmitriev \\cite{Senkov:02} was employed.\n}\n\\label{NewOld}\n\\begin{tabular}{llll}\n\\hline\n\\hline\n   & \\multicolumn{2}{l}{Theory}                             & Experiment         \\\\\n      & $\\mu_I$(old)  & $\\mu_I$(new)      &    \\\\\n\\hline   \n$\\Delta E^{(1s)}$  & 5112(-5\/+20)                & 5089(-5\/+20)(2)  & 5085.03(2)(9)  \\\\\n$\\Delta E^{(2s)}$  & 801.9(-9\/+34) & 798.3(-9\/+34)(4) & 797.645(4)(14) \\\\          \n\\hline   \n\\hline\n\\end{tabular}\n\\end{table}\n\\begin{figure}[!h]\n\\includegraphics[width=0.98\\linewidth]{deltaPrimeE_NMR_2.pdf}\n\\caption{\\label{fig:CompExpTheory} Specific difference obtained in theory  \\cite{Shabaev:01a,Volotka:12} (red) and experiment \\cite{Lochmann:14,Ullmann:17} (blue). The new nuclear magnetic moment established in this work yields a new value for the specific difference which matches the most recent experimental value within uncertainty. \n}\n\\end{figure}\nThe new magnetic moment has been used to recalculate the specific difference and we obtain \n$\\Delta 'E_{\\mathrm{theo}} = -61.043(5)(30)$\\,meV, where the first uncertainty is due to uncalculated terms and remaining nuclear effects, while the second one is due to the uncertainty of the nuclear magnetic moment obtained in the present work. Revised value of $\\Delta 'E_{\\mathrm{theo}}$ \nis plotted in Fig.\\,\\ref{fig:CompExpTheory} combined with the previous theoretical and experimental data. Theory and experiment are now in excellent agreement and the $7\\sigma$ discrepancy reported in \\cite{Ullmann:17} disappears. \nUnfortunately, the uncertainty of $\\Delta 'E_{\\mathrm{theo}}$ is now 14\\% of the total QED contribution and about 1.5 times larger than the experimental uncertainty.\nHence, an improved value for the nuclear magnetic moment of $^{209}$Bi is urgently required, either from an atomic beam magnetic resonance experiment or from a measurement on trapped H-like ions. The latter will have the advantage that no shielding corrections have to be applied. Such an experiment is planned at the ARTEMIS trap \\cite{Quint:2008} at the GSI Helmholtz Centre in Darmstadt. Only such a measurement combined with an improved determination of the HFS splitting in $^{209}$Bi$^{80+,82+}$ as it is foreseen at SPECTRAP \\cite{Andelkovic:2013} can provide a QED test in the magnetic regime of strong-field QED. Our result also proves that a measurement of the specific difference can also be used to extract the nuclear magnetic moment. \nDoing so results in  $\\mu_I(^{209}\\mathrm{Bi})=4.0900(15)\\,\\mu_{\\mathrm{N}}$ in excellent agreement with the NMR value obtained here. \n\n\\begin{acknowledgments}\n\\section*{Acknowledgments}\nWe thank Petra Th\\\"orle from the Institute of Nuclear Chemistry at the University of Mainz for the preparation of the NMR samples and Dmitry Korolev from Saint-Peterburg State University for valuable discussions.\nThe development of the code for the computation of the matrix elements of the considered operators as well as the performance of all-electron coupled cluster calculations were funded by RFBR, according to Research Project No.~16-32-60013 mol\\_a\\_dk; performance of DFT calculations was supported by the President of Russian Federation Grant No. MK-2230.2018.2. This work was also supported by SPSU (Grants No. 11.38.237.2015 and No. 11.40.538.2017) and by SPSU-DFG (Grants No. 11.65.41.2017 and No. STO 346\/5-1). The experimental part was supported by the Federal Ministry of Education and Research of Germany under Contract No 05P15RDFAA and the Helmholtz\nInternational Center for FAIR (HIC for FAIR).   \n\\end{acknowledgments}\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\noindent\nA BV algebra\nand a QP-structure \nhas been motivated by the structure of \nthe Batalin-Vilkovisky formalism of a gauge theory\\cite{Bat}\nand is its mathematical formulation\n\\cite{Schwarz:1992nx}.\nIn case of a topological field theory of Schwarz type, \na BV formalism has been reformulated to the AKSZ formulation, \nwhich is \na clear construction using geometry of a graded manifold \n\\cite{Alexandrov:1995kv}\\cite{Cattaneo:2001ys}.\nApplication to higher $n+1$ dimensions has been formulated and \nnew topological field theories \nin higher dimensions have been founded\nby applying this construction\n\\cite{Park:2000au}\\cite{Severa:2001}\\cite{Ikeda:2001fq}.\n\n\nIn $n=1$, \na classical QP-structure is \na Poisson structure on a manifold $M$ and is also \na Lie algebroid on $T^*M$ from the explicit construction.\nThis is equivalent to the construction of a Poisson structure\nby the Schouten-Nijenhuis bracket in a classical limit.\nThe topological field theory in two dimensions constructed \nby the AKSZ formulation \\cite{Cattaneo:2001ys}\nis the Poisson sigma model\n\\cite{Ikeda:1993aj}\\cite{Schaller:1994es}\nand the quantization of this model \non disc derives the Kontsevich formula of the \ndeformation quantization on a Poisson manifold\n\\cite{Kontsevich:1997vb}\\cite{Cattaneo:1999fm}.\n\nIn $n=2$, a classical QP-structure is a Courant algebroid\n\\cite{Courant}\\cite{Roy01}.\nThe topological field theory derived \nin three dimensions is\nthe Courant sigma model\n\\cite{Ikeda:2002wh}\\cite{Hofman:2002rv}\\cite{Roytenberg:2006qz}.\n\nHowever structures for higher $n$, more than $2$, \nhave not been understood \nenough apart from BF theories.\n\nIn this paper, we analyze $n=3$ case.\nA QP-structure of degree $3$ leads us to\na new type of algebroid,\nwhich is called a \n\\textbf{Lie algebroid up to homotopy}.\nThe notion of this algebroid\nis defined as\na homotopy deformation of a \nLie algebroid \nsatisfying some integrability conditions.\nWe will prove that a QP-structure of degree $3$\non a N-manifold (nonnegatively graded manifold)\nis equivalent to \na Lie algebroid up to homotopy.\nThis QP-structure defines\na new natural $4$-dimensional topological field \ntheory via the AKSZ construction.\n\nThe paper is organized as follows. \nIn section 2, a BV algebra and a QP-structure of\ndegree $3$ are formulated.\nIn section 3, a QP-structure of degree $3$ is constructed \nand analyzed.\nIn section 4, examples of QP-structures of degree $3$\nare listed.\nIn section 5, the AKSZ construction of \na topological field theory in four dimensions\nis formulated and examples are listed.\n\\footnote{Very recently, Gr\\\"utzmann's paper appears which has overlaps\nwith our paper\n\\cite{Grutzmann}.}\n\n\n\n\\section{QP-manifolds and BV Algebras}\n\\subsection{Classical QP-manifold} \n\\begin{definition}\nA graded manifold\n$\\calM$ is by definition a sheaf\nof a graded commutative algebra over\nan ordinary smooth manifold $M$.\n\\end{definition}\nIn the following, we assume the degrees are nonnegative.\\\\\n\\indent\nThe structure sheaf of $\\calM$ is locally isomorphic to\na graded commutative algebra $C^{\\infty}(U)\\otimes S(V)$,\nwhere $U$ is an ordinary local chart of $M$,\n$S(V)$ is the polynomial algebra over $V$\nand where\n$V:=\\sum_{i\\ge 1}V_{i}$ is a graded vector space\nsuch that the dimension of $V_{i}$ is finite for each $i$.\nFor example, when $V=V_{1}$, $\\calM$ is a vector bundle\nwhose fiber is $V^{*}_{1}$: the dual space of $V_{1}$.\n\\begin{definition}\nA graded manifold $(\\calM,\\omega)$ equipped with\na graded symplectic structure $\\omega$ of degree $n$\nis called a \\textbf{P-manifold} of degree $n$.\n\\end{definition}\nIn the next section, we will study a concrete \nP-manifold of degree 3.\\\\\n\\indent\nThe structure sheaf $C^{\\infty}(\\calM)$ of a P-manifold\nbecomes a graded Poisson algebra.\nThe Poisson bracket is defined in the usual manner,\n\\begin{equation}\\label{gradedpoisson}\n\\sbv{F}{G}=(-1)^{|F|+1}\\iota_{X_F}\\iota_{X_G}\\omega,\n\\end{equation}\nwhere $F,G\\in C^{\\infty}(\\calM)$,\n$|F|$ is the degree of $F$\nand $X_{F}:=\\{F,-\\}$ is the Hamiltonian vector field of $F$.\nWe recall the basic properties of the Poisson bracket,\n\\begin{eqnarray*}\n\\sbv{F}{G}&=&-(-1)^{(|F| - n)(|G| - n)} \\sbv{G}{F},\\\\\n\\sbv{F}{G  H}&=&\\sbv{F}{G} H\n+ (-1)^{(|F| - n)|G|} G \\sbv{F}{H},\\\\\n\\{F,\\{G,H\\}\\}&=&\\{\\{F,G\\},H\\}+(-1)^{(|F|-n)(|G|-n)}\\{G,\\{F,H\\}\\},\n\\end{eqnarray*}\nwhere $n$ is the degree of the symplectic structure and $F,G,H\\in C^{\\infty}(\\calM)$.\nWe remark that the degree of the Poisson bracket is $-n$.\n\\begin{definition}\nLet $(\\calM,\\omega)$ be a P-manifold of degree $n$.\nA function $\\Theta\\in C^{\\infty}(\\calM)$ of degree $n+1$\nis called a \\textbf{Q-structure}, if it is a solution of\nthe \\textbf{classical master equation},\n\\begin{eqnarray}\n\\sbv{\\Theta}{\\Theta}=0.\n\\label{bvaction}\n\\end{eqnarray}\nThe triple $(\\calM,\\omega,\\Theta)$ is called a \\textbf{QP-manifold}.\n\\end{definition}\nWe define an operator $Q:=\\sbv{\\Theta}{-}$,\nwhich is called a homological vector field.\nFrom (\\ref{bvaction}) we have the cocycle condition,\n$$\nQ^2=0,\n$$\nwhich says that the homological vector field\nis a coboundary operator on $C^{\\infty}(\\calM)$ and \ndefines a cohomology called the classical BRST cohomology.\n\n\\subsection{Quantum QP-manifold}\n\\begin{definition}\nA graded manifold is called a quantum BV-algebra\nif it has an odd Laplace operator $\\Delta$, \nwhich is a linear operator on $C^{\\infty}(\\calM)$\nsatisfying $\\Delta^{2}=0$,\nand the graded Poisson bracket is given by\n\\begin{eqnarray}\n\\sbv{F}{G}= \n(-1)^{|F|} \\Delta(FG) - (-1)^{|F|} \\Delta(F){G}-{F}\\Delta(G),\n\\label{Poisson2}\n\\end{eqnarray}\nwhere $F, G \\in C^{\\infty}(\\calM)$.\n\\end{definition}\nIf $n$ is odd, a P-manifold $(\\calM,\\omega)$ has the odd Poisson bracket.\nIf an odd P-manifold $(\\calM,\\omega)$ has a volume form $\\rho$,\none can define an odd Laplace operator $\\Delta$ \n(See \\cite{Khudaverdian:2000zt}):\n\\begin{eqnarray*}\n\\Delta F := \\frac{1}{2}\n(-1)^{|F|} {\\rm div}_{\\rho} X_F.\n\\end{eqnarray*}\nHere a divergence ${\\rm div}_{\\rho}$ \nis a map from a space of vector fields on $\\calM$\nto $C^{\\infty}(\\calM)$ and\nis defined by\n\\begin{eqnarray*}\n\\int_{\\calM} {\\rm div}_{\\rho} X \\ F dv = - \\int_{\\calM} X(F) dv,\n\\label{divergence}\n\\end{eqnarray*}\nfor a vector field $X$ on $\\calM$.\nThe pair $(\\calM, \\Delta)$ is called a \\textbf{quantum P-structure}.\nAn odd Laplace operator has degree $-n$.\n\\begin{definition}\nA function $\\Theta \\in C^{\\infty}(\\calM)$ with \nthe degree $n+1$\nis called a \\textbf{quantum Q-structure},\nif it satisfies a \\textbf{quantum master equation}\n\\begin{eqnarray}\n\\Delta (e^{\\frac{i}{\\hbar}\\Theta}) =0,\n\\label{bvaction2}\n\\end{eqnarray}\nwhere $\\hbar$ is a formal parameter.\nThe triple $(\\calM,\\Delta,\\Theta)$ is called a \\textbf{quantum QP-manifold}.\n\\end{definition}\nFrom the definition of an odd Laplace operator, \nthe equation (\\ref{bvaction2}) is equivalent to \n\\begin{eqnarray}\n\\sbv{\\Theta}{\\Theta} - 2 i \\hbar \\Delta \\Theta =0.\n\\label{qme}\n\\end{eqnarray}\nIf we take the limit of $\\hbar\\to 0$ in (\\ref{qme}), \nwhich is called a classical limit,\nthe classical master equation \n$\\sbv{\\Theta}{\\Theta} =0$\nis derived.\nSince $\\Delta^2=0$,\n$\\Delta$ is also a coboundary operator.\nThis defines a quantum BRST cohomology.\nLet $\\calO^{\\prime} = \\calO e^{\\frac{i}{\\hbar}\\Theta}\n\\in C^{\\infty}(\\calM)$ be a cocycle with respect to $\\Delta$.\nThe cocycle condition \n$\n\\Delta (\\calO^{\\prime}) \n= \\Delta (\\calO e^{\\frac{i}{\\hbar}\\Theta}) = 0\n$\nis equivalent to\n\\begin{eqnarray}\\label{defob}\n\\sbv{\\Theta}{\\calO} - i \\hbar \\Delta \\calO = 0.\n\\end{eqnarray}\nThe solutions of (\\ref{defob})\nare called {\\it observables} in physics.\nIn the classical limit, (\\ref{defob}) is \n$\\sbv{\\Theta}{\\calO} = Q\\calO=0$.\n${\\calO}$ reduces to an element of a classical BRST cohomology.\n\n\\section{Structures and homotopy algebroids}\n\nIn this section, we construct and analyze a classical QP-structure of\n degree $3$ explicitly.\n\n\\subsection{P-structures}\n\nLet $E\\to M$ be a vector bundle over an ordinary smooth manifold $M$.\nThe shifted bundle $E[1]\\to M$ is a graded manifold\nwhose fiber space has the degree $+1$.\nWe consider the shifted cotangent bundle $\\calM:=T^{*}[3]E[1]$.\nIt is a P-manifold of the degree $3$ over $M$,\n$$\nT^{*}[3]E[1]\\to\\calM_{2}\\to E[1]\\to M,\n$$\nwhere $\\calM_{2}$ is a certain graded manifold\n\\footnote{In fact, $\\calM_{2}$ is $E[1]\\oplus E^*[2]$,\nwhich is derived from the result\nin the previous sentence of Remark 3.2.}.\nThe structure sheaf $C^{\\infty}(\\calM)$ of $\\calM$\nis decomposed into the homogeneous subspaces,\n$$\nC^{\\infty}(\\calM)=\\sum_{i\\ge 0}C^{i}(\\calM),\n$$\nwhere $C^{i}(\\calM)$ is the space of functions of degree $i$.\nIn particular, $C^{0}(\\calM)=C^{\\infty}(M)$: the algebra of\nsmooth functions on the base manifold\nand $C^{1}(\\calM)=\\Gamma E^{*}$: the space\nof sections of the dual bundle of $E$.\\\\\n\\indent\nLet us denote by $(x,q,p,\\xi)$\na canonical (Darboux) coordinate on $\\calM$, where\n$x$ is a smooth coordinate on $M$,\n$q$ is a fiber coordinate on $E[1]\\to M$,\n$(\\xi,p)$ is the momentum coordinate\non $T^{*}[3]E[1]$ for $(x,q)$.\nThe degrees of the variables $(x,q,p,\\xi)$ are respectively $(0,1,2,3)$.\\\\\n\\indent\nTwo directions of counting the degree of functions\non $T^{*}[3]E[1]$ are introduced. \nRoughly speaking, these are\nthe fiber direction and the base direction.\n\\begin{definition}(Bidegree, see also Remark 3.3.3 in \\cite{Roy01})\nConsider a monomial $\\xi^{i}p^{j}q^{k}$ on a local chart\n$(U;x,q,p,\\xi)$ of $\\calM$, of which the total degree is $3i+2j+k$.\nThe \\textbf{bidegree} of the monomial is, by definition,\n$(2(i+j),i+k)$.\n\\end{definition}\nThis definition is invariant under the natural coordinate transformation,\n\\begin{eqnarray*}\nx^{\\prime}_{i}&=&x^{\\prime}_{i}(x_{1},x_{2},...,x_{dim(M)}),\\\\\nq^{\\prime}_{i}&=&\\sum_{j}t_{ij}q_{j},\\\\\np^{\\prime}_{i}&=&\\sum_{j}t^{-1}_{ij}p_{j},\\\\\n\\xi^{\\prime}_{i}&=&\\sum_{j}\\frac{\\partial x_{j}}{\\partial x^{\\prime}_{i}}\\xi_{j}\n+\\sum_{jkl}\n(\\frac{\\partial t^{-1}_{jl}}{\\partial x^{\\prime}_{i}}t_{lk}\n+\\frac{\\partial t_{kl}}{\\partial x^{\\prime}_{i}}t^{-1}_{lj})\np_{j}q_{k},\n\\end{eqnarray*}\nwhere $t$ is a transition function. Since $T^{*}[3]E[1]$ is covered\nby the natural coordinates,\nthe bidegree is globally well-defined\n(See also Remark \\ref{shiftremark} below.)\\\\\n\\indent\nThe space $C^{n}(\\calM)$ is uniquely decomposed into\nthe homogeneous subspaces with respect to the bidegree,\n$$\nC^{n}(\\calM)=\\sum_{2i+j=n}C^{2i,j}(\\calM).\n$$\nSince $C^{2,0}(\\calM)=\\Gamma E$ and $C^{0,2}(\\calM)=\\Gamma\\wedge^{2}E^{*}$,\nwe have\n$$\nC^{2}(\\calM)=\\Gamma E\\oplus\\Gamma\\wedge^{2}E^{*}.\n$$\n\\begin{remark}\\label{shiftremark}\nThe P-manifold $T^{*}[3]E[1]$ is regarded as\na shifted manifold of $T^{*}[2]E[1]$.\nThe structure sheaf is also a shifted \nsheaf of the one on $T^{*}[2]E[1]$.\nIn particular, the space $C^{2i,j}$ is \nthe shifted space of $C^{i,j}$ on $T^{*}[2]E[1]$.\n\\end{remark}\nFor the canonical coordinate on $\\calM$,\nthe symplectic structure has the following form:\n$$\n\\omega = \\delta \\bx^i \\delta \\bxi_i + \\delta \\bq^a \\delta \\bp_a,\n$$\nand the associated Poisson bracket\nhas the following expression:\n\\begin{eqnarray*}\n\\sbv{F}{G} &=& \nF \\frac{\\rd}{\\partial \\bx^i} \n\\frac{\\ld }{\\partial \\bxi_{i}} G\n- \nF \\frac{\\rd }{\\partial \\bxi_{i}} \n\\frac{\\ld }{\\partial \\bx^i} G\n+ \nF  \\frac{\\rd}{\\partial \\bq^{a}} \n\\frac{\\ld }{\\partial \\bp_{a}} G\n- \nF \\frac{\\rd }{\\partial \\bp_{a}} \n\\frac{\\ld }{\\partial \\bq^{a}} G,\n\\end{eqnarray*}\nwhere $F,G\\in C^{\\infty}(\\calM)$\nand $\\frac{\\overrightarrow{\\scriptstyle{\\partial}}}{\\partial \\phi}$ and \n$\\frac{\\overleftarrow{\\scriptstyle{\\partial}}}{\\partial \\phi}$\nare the right and left differentiations, respectively.\nNote that the degree of the symplectic structure is $+3$\nand the one of the Poisson bracket is $-3$.\nThe bidegree of the Poisson bracket is $(-2,-1)$, that is,\n$$\n\\{(2i,j),(2k,l)\\}=(2(i+k)-2,j+l-1),\n$$\nwhere $(2i,j)$... are functions with the bidgree $(2i,j)$.\n\n\\subsection{Q-structures}\nWe consider a (classical) Q-structure, $\\Theta$, on the P-manifold.\nIt is required that $\\Theta$ has degree $4$.\nThat is, $\\Theta\\in C^{4}(\\calM)$.\nBecause $C^{4}(\\calM)=C^{4,0}(\\calM)\\oplus C^{2,2}(\\calM)\\oplus C^{0,4}(\\calM)$,\nthe Q-structure is uniquely decomposed into\n$$\n\\Theta=\\theta_{2}+\\theta_{13}+\\theta_{4},\n$$\nwhere the bidegrees of the substructures are\n$(4,0)$, $(2,2)$ and $(0,4)$, respectively.\nIn the canonical coordinate, $\\Theta$ is the following polynomial:\n\\begin{eqnarray}\\label{deftheta}\n\\Theta = f{}_1{}^i{}_{a} (\\bx) \\bxi_i \\bq^a \n+\\frac{1}{2} f_2{}^{ab}(\\bx) \\bp_a \\bp_b\n+\\frac{1}{2} f_3{}^a{}_{bc}(\\bx) \\bp_a \\bq^b \\bq^c\n+\\frac{1}{4!} f_4{}_{abcd}(\\bx) \\bq^a \\bq^b \\bq^c \\bq^d,\n\\end{eqnarray}\nand the substructures are \n\\begin{eqnarray*}\n\\theta_{2}&=&\\frac{1}{2} f_2{}^{ab}(\\bx) \\bp_a \\bp_b,\\\\\n\\theta_{13}&=&\nf{}_1{}^i{}_{a} (\\bx) \\bxi_i \\bq^a+\\frac{1}{2} f_3{}^a{}_{bc}(\\bx) \\bp_a \\bq^b \\bq^c,\\\\\n\\theta_{4}&=&\\frac{1}{4!} f_4{}_{abcd}(\\bx) \\bq^a \\bq^b \\bq^c \\bq^d,\n\\end{eqnarray*}\nwhere $f_{1}$-$f_{4}$ are structure functions on $M$.\nBy  counting the bidegree, one can easily prove that\nthe classical master equation $\\sbv{\\Theta}{\\Theta} = 0$\nis equivalent to the following three identities:\n\\begin{eqnarray}\n\\label{tc1}\n\\{\\theta_{13},\\theta_{2}\\}&=&0,\\\\\n\\label{tc2}\n\\frac{1}{2}\\{\\theta_{13},\\theta_{13}\\}\n+\\{\\theta_{2},\\theta_{4}\\}&=&0,\\\\\n\\label{tc3}\n\\{\\theta_{13},\\theta_{4}\\}&=&0.\n\\end{eqnarray}\nThe conditions (\\ref{tc1}), (\\ref{tc2}) and (\\ref{tc3})\nare equivalent to\n\\begin{eqnarray}\n\\label{fc1}\n&&f{}_1{}^i{}_{b} f_2{}^{ba} = 0,\\\\\n\\label{fc2}\n&&\nf{}_1{}^k{}_{c} \\frac{\\partial f_2{}^{ab}}{\\partial x^k} \n+ f_2{}^{da} f_3{}^b{}_{cd} + f_2{}^{db} f_3{}^a{}_{cd} = 0,\\\\\n\\label{fc3}\n&&\nf{}_1{}^k{}_{b} \\frac{\\partial f{}_1{}^i{}_{a}}{\\partial x^k} \n- f{}_1{}^k{}_{a} \\frac{\\partial f{}_1{}^i{}_{b}}{\\partial x^k} \n+ f{}_1{}^i{}_{c} f_3{}^c{}_{ab} = 0,\\\\\n\\label{fc4}\n&& \nf{}_1{}^k{}_{[d} \\frac{\\partial f_3{}^a{}_{bc]}}{\\partial x^k} \n+ f_2{}^{ae} f_4{}_{bcde}\n- f_3{}^a{}_{e[b} f_3{}^e{}_{cd]} = 0,\\\\\n\\label{fc5}\n&& \nf{}_1{}^k{}_{[a} \\frac{\\partial f_4{}_{bcde]}}{\\partial x^k} \n+ f_3{}^f{}_{[ab} f_4{}_{cde]f} =0,\n\\end{eqnarray}\nwhere $[b \\ c \\ d \\ \\cdots]$ is a skewsymmetrization\nwith respect to indices $b, c, d, \\cdots$, etc.\n\\subsection{Lie algebroid up to homotopy}\nIn this section we study an algebraic structure\nassociated with the QP-structure in 3.1 and 3.2.\n\\begin{definition}\nLet $Q=\\theta_{2}+\\theta_{13}+\\theta_{4}$ be a $Q$-structure\non $T^{*}[3]E[1]$, where $(\\theta_{2},\\theta_{13},\\theta_{4})$\nis the unique decomposition of $\\Theta$.\nWe call the quadruple $(E; \\theta_{2},\\theta_{13},\\theta_{4})$\na \\textbf{Lie algebroid up to homotopy},\nin shorthand, Lie algebroid u.t.h.\n\\end{definition}\nWe should study the algebraic properties of the Lie algebroid up to homotopy.\nLet us define a bracket product by\n\\begin{equation}\\label{braee}\n[e_{1},e_{2}]:=\\{\\{\\theta_{13},e_{1}\\},e_{2}\\},\n\\end{equation}\nwhere $e_{1},e_{2}\\in\\Gamma E$.\nBy the bidegree counting, \n$\\Gamma E$ is closed under this bracket.\nThe bracket is not necessarily a Lie bracket,\nbut it is still skewsymmetric:\n\\begin{eqnarray*}\n[e_{1},e_{2}]&=&\\{\\{\\theta_{13},e_{1}\\},e_{2}\\},\\\\\n&=&\\{\\theta_{13},\\{e_{1},e_{2}\\}\\}+\\{e_{1},\\{\\theta_{13},e_{2}\\}\\},\\\\\n&=&-\\{\\{\\theta_{13},e_{2}\\},e_{1}\\}=-[e_{2},e_{1}],\n\\end{eqnarray*}\nwhere $\\{e_{1},e_{2}\\}=0$ is used.\nA bundle map $\\rho:E\\to TM$ which is called an anchor map\nis defined by the following identity:\n$$\n\\rho(e)(f):=\\{\\{\\theta_{13},e\\},f\\},\n$$\nwhere $f\\in C^{\\infty}(M)$.\nThe bracket and the anchor map \nsatisfy the algebroid conditions (A0) and (A1) below:\n\\begin{description}\n\\item[(A0)]\n$\\rho[e_{1},e_{2}]=[\\rho(e_{1}),\\rho(e_{2})]$,\n\\item[(A1)]\n$[e_{1},fe_{2}]=f[e_{1},e_{2}]+\\rho(e_{1})(f)e_{2}$,\n\\end{description}\nwhere the bracket $[\\rho(e_{1}),\\rho(e_{2})]$\nis the usual Lie bracket on $\\Gamma TM$.\nThe bracket (\\ref{braee}) does not satisfy the Jacobi identity in general.\nSo we should study its Jacobi anomaly, which characterizes\nthe algebraic structure of the Lie algebroid u.t.h.\nThe structures $\\theta_{13}$, $\\theta_{2}$ and $\\theta_{4}$ define\nthe three operations:\n\\begin{itemize}\n\\item $\\delta(-):=\\{\\theta_{13},-\\}$;\na de Rham type derivation on $\\Gamma\\wedge^{\\cdot}E^{*}$,\n\\item $(\\alpha_{1},\\alpha_{2}):=\\{\\{\\theta_{2},\\alpha_{1}\\},\\alpha_{2}\\}$;\na symmetric pairing on $E^{*}$, where $\\alpha_{1},\\alpha_{2}\\in\\Gamma E^{*}$,\n\\item $\\Omega(e_{1},e_{2},e_{3},e_{4}):=\n\\{\\{\\{\\{\\{\\theta_{4},e_{1}\\},e_{2}\\},e_{3}\\},e_{4}\\}$;\na 4-form on $E$.\n\\end{itemize}\nRemark that $\\delta\\delta\\neq 0$ in general.\nBecause the degree of the pairing is $-2$,\nit is $C^{\\infty}(M)$-valued.\nThe pairing induces a symmetric\nbundle map $\\partial:E^{*}\\to E$\nwhich is defined by the equation,\n$(\\alpha_{1},\\alpha_{2})=\\bracket{\\partial \\alpha_{1}}{\\alpha_{2}}$,\nwhere $\\bracket{-}{-}$ is the canonical pairing\nof the duality of $E$ and $E^{*}$.\nSince $\\bracket{\\alpha}{e}=\\{\\alpha,e\\}$, we have\n$$\n\\partial\\alpha=-\\{\\theta_{2},\\alpha\\}.\n$$\nBy direct computation, we obtain\n$$\n\\frac{1}{2}\n\\{\\{\\{\\{\\theta_{13},\\theta_{13}\\},e_{1}\\},e_{2}\\},e_{3}\\}\n=[[e_{1},e_{2}],e_{3}]+({\\rm cyclic \\ permutations}),\n$$\nand\n$$\n\\{\\{\\{\\{\\theta_{2},\\theta_{4}\\},e_{1}\\},e_{2}\\},e_{3}\\}\n=-\\partial\\Omega(e_{1},e_{2},e_{3}).\n$$\nFrom Eq.~(\\ref{tc2}), we get an explicit formula of the Jacobi anomaly,\n\\begin{description}\n\\item[(A2)]\n$[[e_{1},e_{2}],e_{3}]+({\\rm cyclic \\ permutations})\n= \\partial\\Omega(e_{1},e_{2},e_{3})$.\n\\end{description}\nIn a similar way, we obtain the following identities:\n\\begin{description}\n\\item[(A3)] $\\rho\\partial=0$,\n\\item[(A4)] $\\rho(e)(\\alpha_{1},\\alpha_{2})=(\\mathcal{L}_{e}\\alpha_{1},\\alpha_{2})\n+(\\alpha_{1},\\mathcal{L}_{e}\\alpha_{2})$,\n\\item[(A5)] $\\delta\\Omega=0$,\n\\end{description}\nwhere $\\mathcal{L}_{e}(-):=\\{\\{\\theta_{13},e\\},-\\}$\nis the Lie type derivation which acts on $E^{*}$.\nAxioms (A3) and (A4) are induced from Eq.~(\\ref{tc1})\nand (A5) is from Eq.~(\\ref{tc3}).\\\\\n\\indent\nThe fundamental relations (\\ref{fc1})--(\\ref{fc5})\ncorrespond to Axioms (A1)--(A5)\\footnote{\nActually, the axiom (A0) depends on (A1) and (A2).\n}.\nThus, the notion of the Lie algebroid up to homotopy\nis characterized by the algebraic properties (A1)--(A5).\nOne concludes that\n\\medskip\\\\\n\\noindent\n{\\em\nThe classical algebra associated with the QP-manifold $(T^{*}[3]E[1],\\Theta)$\nis the space of sections of the vector bundle $E$ with the operations\n$([\\cdot,\\cdot],\\rho,\\partial,\\Omega)$\nsatisfying (A1)--(A5).\n}\n\\medskip\\\\\n\\indent\nIn the next section, we will study some special examples\nof Lie algebroid u.t.h.s.\n\\begin{remark}\n\\normalfont\nIf the pairing is nondegenerate,\nthen the bundle map $\\partial$ is bijective\nand then from (A3) we have $\\rho=0$.\n\\end{remark}\n\\begin{remark}\\label{CDB}\n\\normalfont\n(Higher Courant-Dorfman brackets)\nWe define a bracket on $C^{\\infty}(\\calM)$ by\n$$\n[-,-]_{CD}:=\\{\\{\\Theta,-\\},-\\},\n$$\nwhich is called a Courant-Dorfman (CD) bracket.\nIt is well-known that $[,]_{CD}$ is a Loday bracket (\\cite{Kos}).\nSince the degree of the CD-bracket is $-2$,\nthe total space of degree $i\\le 2$,\n$$\nC^{2}(\\calM)\\oplus C^{1}(\\calM)\\oplus C^{0}(M)\n$$\nis closed under the CD-bracket, in particular,\nthe top space $C^{2}(\\calM)=\\Gamma(E\\oplus\\wedge^{2}E^{*})$\nis a subalgebra.\nIf $\\theta_{2}=0$, the CD-bracket on $E\\oplus\\wedge^{2}E^{*}$\nhas the following form,\n$$\n[e_{1}+\\beta_{1},e_{2}+\\beta_{2}]_{CD}=\n[e_{1},e_{2}]+\\mathcal{L}_{e_{1}}\\beta_{2}-i_{e_{2}}\\delta\\beta_{1}+\\Omega(e_{1},e_{2}),\n$$\nwhere $\\beta_{1},\\beta_{2}\\in\\Gamma\\wedge^{2}E^{*}$.\nThis CD-bracket is regarded as a higher analogue of\nCourant-Dofman's original bracket (cf. \\cite{Courant}).\nWe refer the reader to \nHagiwara \\cite{Hagiwara} and Sheng \\cite{YS}\nfor the detailed study of the higher CD-brackets.\n\\end{remark}\n\n\\section{Examples and twisting transformations}\n\n\\subsection{The cases of $\\theta_{2}=\\theta_{4}=0$}\n\nIn this case, the bracket (\\ref{braee}) satisfies\n(A0), (A1) and the Jacobi identity.\nTherefore, the bundle $E\\to M$ becomes a Lie algebroid:\n\\begin{definition}\n(\\cite{Mackenzie})\nA Lie algebroid over a manifold $M$ is a vector bundle\n$E \\rightarrow M$ with a Lie algebra structure on the \nspace of the sections $\\Gamma(E)$ defined by the \nbracket $[e_1, e_2]$ for $e_1, e_2 \\in \\Gamma(E)$\nand an anchor map\n$\\rho: E \\rightarrow TM$ satisfying (A0) and (A1) above.\n\\end{definition}\n\\indent\nWe take $\\{ e_a \\}$ as a local basis of $\\Gamma E$ and\nlet a local expression of an anchor map\nbe $\\rho(e_a) = f^i{}_{1a}(x) \\frac{\\partial}{\\partial x^i}$\nand a Lie bracket be\n$[e_b, e_c] = f_3{}^a{}_{bc}(x) e_a$.\nThe Q-structure $\\Theta$ associated with\nthe Lie algebroid $E$\nis defined as a function on $T^{*}[3]E[1]$,\n$$\n\\Theta:=\\theta_{13}:=f{}_1{}^i{}_{a} (\\bx) \\bxi_i \\bq^a\n+\\frac{1}{2} f_3{}^a{}_{bc}(\\bx) \\bp_a \\bq^b \\bq^c,\n$$\nwhich is globally well-defined.\nConversely, if we consider $\\Theta:=\\theta_{13}$,\nthe classical master equation\ninduces the Lie algebroid structure on $E$.\n\\medskip\\\\\n\\indent\nLet us consider the case that the bundle is \na vector space on a point.\nA Lie algebroid over\na point $\\mathfrak{g}\\to\\{pt\\}$ is\na Lie algebra $\\mathfrak{g}$.\nThe P-manifold over $\\mathfrak{g}\\to\\{pt\\}$\nis isomorphic to $\\mathfrak{g}^*[2]\\oplus\\mathfrak{g}[1]$\nand the structure sheaf is\nthe polynomial algebra over $\\mathfrak{g}[2]\\oplus\\mathfrak{g}^{*}[1]$,\n$$\nC^{\\infty}(\\calM)=S(\\mathfrak{g})\\otimes \\bigwedge^{\\cdot}\\mathfrak{g}^{*}.\n$$\nThe bidegree is defined by the natural manner,\n$$\nC^{2i,j}(\\calM)=S^{i}(\\mathfrak{g})\\otimes \\bigwedge^{j}\\mathfrak{g}^{*}.\n$$\nThe Q-structure associated with the Lie bracket on $\\mathfrak{g}$ is\n\\begin{eqnarray}\n\\theta_{13}=\\frac{1}{2} f^a{}_{bc} p_a q^b q^c\n\\cong\\frac{1}{2} f^a{}_{bc} p_a \\otimes(q^b \\wedge q^c),\n\\label{liealgebra}\n\\end{eqnarray}\nwhere $p_{\\cdot}\\in\\mathfrak{g}$, $q_{\\cdot}\\in\\mathfrak{g}^{*}$\nand $f^{a}{}_{bc}$ is the structure constant of the Lie algebra.\n\n\\subsection{The cases of $\\theta_{2}\\neq 0$ and $\\theta_{4}=0$}\n\nIn this case, the bracket induced by $\\theta_{13}$\nstill satisfies the Jacobi identity.\n\\medskip\\\\\n\\indent\nWe assume that $\\frak{g}$ is semi-simple.\nThen the dual space $\\frak{g}^{*}$ has a metric,\n$(\\cdot,\\cdot)_{K^{-1}}$, which is\nthe inverse of the Killing form on $\\frak{g}$.\nThe metric inherits the following invariant condition from\nthe Killing form:\n\\begin{equation}\\label{la3}\n(\\mathcal{L}_{p}q_{1},q_{2})_{K^{-1}}\n+(q_{1},\\mathcal{L}_{p}q_{2})_{K^{-1}}=0,\n\\end{equation}\nwhere $\\mathcal{L}_{p}(-)$ is the canonical\ncoadjoint action of $\\mathfrak{g}$ to $\\mathfrak{g}^{*}$.\nEq.~(\\ref{la3}) is a linear version of (A4).\nThus, we obtain a Q-structure,\n\\begin{eqnarray}\n\\Theta:=\nk^{ab}\\bp_a \\bp_b\n+\\frac{1}{2} f{}^a{}_{bc} \\bp_a \\bq^b \\bq^c,\n\\label{killingQ}\n\\end{eqnarray}\nwhere $k^{ab}\\bp_a \\bp_b:=(\\cdot,\\cdot)_{K^{-1}}$.\n\n\\subsection{Non Lie algebra example}\nWe consider the cases that the Jacobi identity is broken.\nLet $(\\mathfrak{g},[\\cdot,\\cdot],(\\cdot,\\cdot)_{K})$\nbe a vector space (not necessarily Lie algebra)\nequipped with a skewsymmetric bracket $[\\cdot,\\cdot]$\nand an invariant metric $(\\cdot,\\cdot)_{K}$.\nThe metric induces a bijection\n$K:\\mathfrak{g}\\to\\mathfrak{g}^{*}$\nwhich is defined by the identity,\n$$\n(p_{1},p_{2})_{K}=\\bracket{Kp_{1}}{p_{2}}.\n$$\nWe define a map from\n$\\mathfrak{g}^{*}$ to $\\mathfrak{g}$ by $\\partial:=K^{-1}$\nand define a 4-form by,\n$$\n\\Omega(p_{1},p_{2},p_{3},p_{4}):=\\Big([[p_{1},p_{2}],p_{3}]+\n{\\rm cyclic \\ permutations},p_{4}\\Big)_{K}.\n$$\n\\begin{remark}\nThe 4-form above is considered to be a higher analogue\nof the Cartan 3-form $([p_{1},p_{2}],p_{3})_{K}$.\n\\end{remark}\nAxioms (A0)--(A4) obviously hold on $\\frak{g}$.\nWe check (A5). It suffices to show (\\ref{tc3}).\nLet us denote by $\\{-,p_{1},p_{2},...,p_{n}\\}$\nthe n-fold bracket $\\{...\\{\\{-,p_{1}\\},p_{2}\\},...,p_{n}\\}$.\nWe already have (\\ref{tc1}) and (\\ref{tc2}).\nFrom $\\{\\theta_{13},\\{\\theta_{13},\\theta_{13}\\}\\}=0$\nand (\\ref{tc2}), we have $\\{\\theta_{13},\\{\\theta_{2},\\theta_{4}\\}\\}=0$.\nSince $\\{\\theta_{13},\\theta_{2}\\}=0$,\nthis is equal to\n$\\{\\theta_{2},\\{\\theta_{3},\\theta_{4}\\}\\}=0$\nup to sign.\nThis gives\n$\\{\\{\\theta_{2},\\{\\theta_{3},\\theta_{4}\\}\\},p_{1},...,p_{5}\\}=0$\nfor any $p_{1},...,p_{5}$.\nFrom $\\{\\theta_{2},p\\}=0$, we have\n$$\n\\{\\theta_{2},\\{\\{\\theta_{3},\\theta_{4}\\},p_{1},...,p_{5}\\}\\}=0.\n$$\nSince $K^{-1}=-\\{\\theta_{2},-\\}$ is bijective, we get\n$$\n\\{\\{\\theta_{3},\\theta_{4}\\},p_{1},...,p_{5}\\}=0,\n$$\nwhich yields the desired relation $\\{\\theta_{3},\\theta_{4}\\}=0$.\n\\begin{proposition}\nThe triple $\\big(\\mathfrak{g},\\partial,\\Omega\\big)$\nis a Lie algebra(oid) up to homotopy.\n\\end{proposition}\n\\subsection{Twisting by $3$-form and the cases of $\\theta_{2}=0$ and $\\theta_{4}\\neq 0$}\nWe introduce the notion of twisting transformation by $3$-form\nbefore studying the cases of $\\theta_{2}=0$.\nGiven a Q-structure $\\Theta$ and a 3-form $\\phi\\in C^{0,3}(\\calM)$,\nthere exists the second Q-structure which is defined by\nthe canonical transformation,\n\\begin{equation}\\label{defgauge}\n\\Theta^{\\phi}:=\\exp(X_{\\phi})(\\Theta),\n\\end{equation}\nwhere $X_{\\phi}:=\\{\\phi,-\\}$ is the Hamiltonian vector field of $\\phi$.\nThe transformation (\\ref{defgauge}) is called a \\textbf{twisting by 3-form},\nor simply twisting.\nBy a direct computation, we obtain\n\\begin{eqnarray*}\n\\label{g1}\\theta^{\\phi}_{2}&=&\\theta_{2},\\\\\n\\label{g2}\\theta^{\\phi}_{13}&=&\\theta_{13}-\\{\\theta_{2},\\phi\\},\\\\\n\\label{g3}\\theta^{\\phi}_{4}&=&\\theta_{4}-\\{\\theta_{13},\\phi\\}+\\frac{1}{2}\\{\\{\\theta_{2},\\phi\\},\\phi\\},\n\\end{eqnarray*}\nwhere $\\Theta^{\\phi}=\\theta^{\\phi}_{2}+\\theta^{\\phi}_{13}+\\theta^{\\phi}_{4}$\nand $X^{i\\ge 3}_{\\phi}(\\Theta)=0$.\nThe twisting by 3-form defines an equivalence relation on the Q-structures.\n\\medskip\\\\\n\\indent\nWe notice that $\\theta_{2}$ is an invariant for the twisting.\nIf $\\theta_{2}=0$, then\n$\\theta_{13}$ is an invariant and\n$$\n\\theta^{\\phi}_{4}=\\theta_{4}-\\delta\\phi,\n$$\nwhere $\\delta\\phi=\\{\\theta_{13},\\phi\\}$.\nThis leads us\n\\begin{proposition}\nThe class of Q-structures which have no $\\theta_{2}$\nis classified into $H^{4}_{dR}(\\bigwedge^{\\cdot}E^{*},\\delta)$\nby the twisting by 3-form.\n\\end{proposition}\n\n\n\\section{AKSZ Construction of Topological Field Theory \nin $4$ Dimensions\n}\n\\subsection{General Theory}\n\n\\noindent\nIn this section, we consider the AKSZ construction of \na topological field theory in $4$ dimensions.\n\nFor a graded manifold $\\calN$,  \nlet $\\calN|_0$ be the degree zero part.\n\nLet $X$ be a manifold in $4$ dimensions\nand $M$ be a manifold in $d$ dimensions.\nLet $(\\calX, D)$ be a differential graded (dg) manifold \n$\\calX$\nwith a $D$-invariant nondegenerate measure $\\mu$, \nsuch that $\\calX|_0 = X$, where\n$D$ is a differential on $\\calX$.\n($\\calM, \\omega, \\Theta$)\nis a QP-manifold of degree $3$ and\n$\\calM|_0 = M$.\nA degree $\\deg (-)$ on $\\calX$ is called the {\\it form degree} and \na degree $\\gh (-)$ on $\\calM$ is called the {\\it ghost number}\n\\footnote{The ghost number $\\gh (-)$ is the degree $|-|$\non $\\calM$ in section 2.}.\nLet \n$\\Map(\\calX, \\calM)$ be\na space of smooth maps from $\\calX$ to $\\calM$.\n$|-| = \\deg (-) + \\gh (-)$ is the degree\non $\\Map(\\calX, \\calM)$ and called the {\\it total degree}.\nA QP-structure on $\\Map(\\calX, \\calM)$\nis constructed from the above data.\n\nSince ${\\rm Diff}(\\calX)\\times {\\rm Diff}(\\calM)$ \nnaturally acts on $\\Map(\\calX, \\calM)$,\n$D$ and $Q$ induce homological vector fields \non $\\Map(\\calX, \\calM)$, \n$\\hat{D}$ and $\\check{Q}$. \n\nTwo maps are introduced.\nAn {\\it evaluation map} \n${\\rm ev}: \\calX \\times \\calM^{\\calX} \\longrightarrow \\calM$ \nis defined as\n\\begin{eqnarray*}\n{\\rm ev}:(z, \\Phi) \\longmapsto \\Phi(z),\n\\end{eqnarray*}\nwhere \n$z \\in \\calX$ and $\\Phi \\in \\calM^{\\calX}$.\n\n\n\nA {\\it chain map} $\\mu_*: \\Omega^{\\bullet}(\\calX \\times \\calM) \n\\longrightarrow \\Omega^{\\bullet}(\\calM)$ is defined as \n$\\mu_* F = \\int_{\\calX} \\mu F$\nwhere $F \\in \\Omega^{\\bullet}(\\calX \\times \\calM)$ \nand \n$\\int_{\\calX} \\mu$  is an integration on $\\calX$\nby the $D$-invariant measure $\\mu$.\nIt is an usual integral for the even degree parts\nand\nthe Berezin integral for the \nodd degree parts. \n\nA (classical) P-structure on $\\Map(\\calX, \\calM)$ is defined as follows:\n\\begin{definition} \nFor a graded symplectic form $\\omega$ on $\\calM$, \na graded symplectic form $\\bomega$ on $\\Map(\\calX, \\calM)$\nis defined as $\\bomega := \\mu_* \\ev^* \\omega$.\n\\end{definition}\nWe can confirm that ${\\bomega}$ satisfies the definition of \na graded symplectic form because $\\mu_* \\ev^*$ preserves\nnondegeneracy and closedness.\nThus $\\bomega$ is a P-structure on $\\Map(\\calX, \\calM)$\nand induces a graded Poisson bracket $\\sbv{-}{-}$ on $\\Map(\\calX, \\calM)$.\nSince $|\\mu_* \\ev^*|=-4$, $|\\bomega| = -1$ and \n$\\sbv{-}{-}$ on $\\Map(\\calX, \\calM)$ has degree $1$ and an odd\nPoisson bracket.\n\n\n\nNext we define a Q-structure $S$ on $\\Map(\\calX,  \\calM)$.\n$S$ is called a {\\it BV action} and\nconsists of two parts $S = S_0 + S_1$.\n$S_0$ is constructed as follows: \nLet $\\omega$ be the odd symplectic form \non $\\calM$.\nWe take a fundamental form $\\vartheta$ such that \n$\\omega= - d \\vartheta$ and\ndefine $S_0 := \\iota_{\\hat{D}} \\mu_* {\\rm ev}^* \\vartheta$.\n$|S_0|=0$ because $\\mu_* {\\rm ev}^*$ has degree $-4$.\n$S_1$ is constructed as follows: \nWe take a Q-structure $\\Theta$ on $\\calM$ and \ndefine $S_1 := \\mu_* \\ev^* \\Theta$.\n$S_1$ also has degree $0$.\n\nWe can prove that\n$S$ is a Q-structure on $\\Map(\\calX,  \\calM)$, \nsince \n\\begin{eqnarray}\n\\sbv{\\Theta}{\\Theta} =0,\n\\Longleftrightarrow \n\\sbv{S}{S} =0\n\\label{classicalmaster}\n\\end{eqnarray}\nfrom the definition of $S_0$ and $S_1$.\n\nA quantum version is \n\\begin{eqnarray}\n\\Delta (e^{\\frac{i}{\\hbar} \\Theta}) =0\n\\Longleftrightarrow \n\\hat{\\Delta} (e^{\\frac{i}{\\hbar} S}) =0,\n\\label{classicalmaster2}\n\\end{eqnarray}\nwhere $\\hat{\\Delta}$ is an odd Laplace operator on $\\Map(\\calX, \\calM)$.\nThe infinitesimal form of the right hand side \nin (\\ref{classicalmaster2}) is \n$\\sbv{S}{S} - 2 i \\hbar \\hat{\\Delta} S =0$,\nwhich is called a {\\it quantum master equation}.\n\\footnote{Discussion for an odd Laplace operator is too naive. \nIn general, the quantum master equation has an obstruction \nexpressed by the modular class \\cite{Lyakhovich:2004kr}.\nWe must regularize an odd Laplace operator\nand a quantum BV action.\n}\n\nThe following theorem has been confirmed\n\\cite{Alexandrov:1995kv}:\n\\begin{thm}\nIf $\\calX$ is a dg manifold and \n$\\calM$ is a QP-manifold,\nthe graded manifold $\\Map(\\calX, \\calM)$\nhas a QP-structure.\n\\end{thm}\n\n\n\\begin{definition}\nA {\\it topological field theory} in $4$ dimensions\nis a triple ($\\calX, \\calM, S$), \nwhere $\\calX$ is a dg manifold with\n$\\dim \\calX|_0 = 4$, \n$\\calM$ is \na QP-manifold\nwith the degree $3$,\nand $S$ is a BV action with the total degree $0$.\n\\end{definition}\n\n\n\nIn order to interpret this theory\nas a `physical' topological field theory,\nwe must take $\\calX= \nT[1]X$.\nThen we can confirm that \na QP-structure on $\\Map(\\calX, \\calM)$ is \nequivalent to the AKSZ formulation of a topological field theory\n\\cite{Cattaneo:2001ys}\\cite{Ikeda:2006wd}.\nWe set $\\calX = T[1]X$ from now.\n\n\n\nIn `physics', a quantum field theory is constructed by quantizing a\nclassical field theory.\nFirst we consider a Q-structure \n$\\sbv{\\cdot}{\\cdot}$ and \na classical P-structure $S$ such that\n\\begin{eqnarray*}\n\\sbv{S}{S} =0.\n\\end{eqnarray*}\nNext we define \na quantum P-structure $\\hat{\\Delta}$ \nand confirm that\n\\begin{eqnarray*}\n\\tilde{\\Delta} (e^{\\frac{i}{\\hbar} S}) =0.\n\\end{eqnarray*}\nFinally we calculate a partition function \n\\begin{eqnarray*}\nZ = \\int_{\\calL} e^{\\frac{i}{\\hbar} S},\n\\end{eqnarray*}\non a Lagrangian submanifold \n$\\calL \\subset \\Map(\\calX, {\\calM})$.\nQuantization is not discussed in this paper.\n\n\n\n\\subsection{Local Coordinate Expression and Examples}\n\\noindent\nA general theory in the previous subsection\nis applied to the local coordinate expression \nin section 3.1 and \na known topological field theory in $4$ dimensions\nis obtained as a special case and \na new nontrivial topological field theory is constructed.\nLet us take a manifold $X$ in $4$ dimensions and\na manifold $M$ in $d$ dimensions.\nLet $E[1]$ is a graded vector bundle on $M$.\nWe take $\\calX = T[1]X$ and $\\calM = T^*[3]E[1]$.\n\nLet $(\\sigma^{\\mu}, \\theta^{\\mu})$ be a local coordinate\non $T[1]X$. $\\sigma^{\\mu}$ is a local coordinate on \nthe base manifold $X$ and \n$\\theta^{\\mu}$ is one on the fiber of $T[1]X$, respectively.\nLet $\\bbx^i$ be a smooth map $\\bbx^i: X \\longrightarrow M$ and \n$\\bbxi_i$ be a section of $T^*[1]X \\otimes \\bbx^*(T^*[3] M)$, \n$\\bbq^a$ be a section of $T^*[1]X \\otimes \\bbx^*(E[1])$ and\n$\\bbp_a$ be a section of $T^*[1]X \\otimes \\bbx^*(T^*[3]E_{\\bbx}[1])$.\nThese are called {\\it superfields}.\nThe exterior derivative $d$ is taken \nas a differential $D$ on $X$.\nFrom $d$, a differential \n$\\bbd = \\theta^{\\mu} \\frac{\\partial}{\\partial \\sigma^{\\mu}}$\non $\\calX$ is induced.\n\nThen a BV action $S$ has the following expression:\n\\begin{eqnarray*}\nS&=&S_0 + S_1,\\\\\nS_0&=&\\int_{\\calX} \\mu \\ (\\bbxi_i \\bbd \\bbx^i\n- \\bbp_a \\bbd \\bbq^a),\\\\\nS_1&=&\\int_{\\calX} \\mu \\ (f{}_1{}^i{}_{a} (\\bbx) \\bbxi_i \\bbq^a \n+ \\frac{1}{2} f_2{}^{ab}(\\bbx) \\bbp_a \\bbp_b\n+ \\frac{1}{2} f_3{}^a{}_{bc}(\\bbx) \\bbp_a \\bbq^b \\bbq^c\n+ \\frac{1}{4!} f_4{}_{abcd}(\\bbx) \\bbq^a \\bbq^b \\bbq^c \\bbq^d).\n\\end{eqnarray*}\n\\medskip\\\\\n\\noindent\n\\textbf{Nonabelian BF theory}.\nLet $\\Theta$ be a Q-structure (\\ref{liealgebra})\nfor a Lie algebra $\\mathfrak{g}$.\n$\\bbxi_i \\bbd \\bbx^i=0$ since $M = \\{pt\\}$.\nIf we define a curvature\n$\\bbF^a = \\bbd \\bbq^a - \\frac{1}{2} f{}^a{}_{bc} \\bbq^b \\bbq^c$,\na Q-structure is\n\\begin{eqnarray*}\n&& S= \\int_{\\calX} \\mu \\ (- \\bbp_a \\bbF^a),\n\\end{eqnarray*}\nwhich is equivalent to a BV formalism for\na nonabelian BF theory in $4$ dimensions.\n\\medskip\\\\\n\\noindent\n\\textbf{Topological Yang-Mills Theory}.\nWe take a nondegenerate Killing form $(\\cdot,\\cdot)_{K}$\nfor a Lie algebra $\\mathfrak{g}$ and consider the Q-structure \n(\\ref{killingQ}).\nA topological field theory constructed from (\\ref{killingQ})\nis \n\\begin{eqnarray*}\n&& S= \\int_{\\calX} \\mu \\ (- \\bbp_a \\bbF^a\n+ k^{ab} \\bbp_a \\bbp_b).\n\\end{eqnarray*}\nThis is equivalent to a topological Yang-Mills theory,\n\\begin{eqnarray*}\n&& S= - \\frac{1}{4} \\int_{\\calX} \\mu \\ k_{ab} \\bbF^a \\bbF^b,\n\\end{eqnarray*}\nif we delete $\\bbp_a$ by the equations of motion.\n\\medskip\\\\\n\\noindent\n\\textbf{Nonassociative  BF Theory}.\nLet us take a non Lie algebra $(\\mathfrak{g},[\\cdot,\\cdot],(\\cdot,\\cdot)_{K})$\nin section 4.3.\nIf we take\n$M = \\{pt\\}$ and $\\calM = \\mathfrak{g^*}[2] \\oplus \\mathfrak{g}[1]$, \n$(\\mathfrak{g},[\\cdot,\\cdot],(\\cdot,\\cdot)_{K})$ leads\na QP-structure with degree $3$.\nIn the canonical basis, it is expressed as\n\\begin{eqnarray*}\n&& f{}_1{}^i{}_{a} (\\bx) = 0, \\qquad \nf_2{}^{ab}(\\bx) = K^{ab}, \\\\\n&& f_3{}^a{}_{bc}(\\bx) = f{}^a{}_{bc}, \\qquad \nf_4{}_{abcd}(\\bx) = K^{-1}_{ae} f{}^e{}_{f[b} f{}^f{}_{cd]},\n\\end{eqnarray*}\nwhere \n$K^{ab} = (p_a, p_b)$ is nondegenerate and \n$[p_a, p_b] = f{}^c{}_{ab} p_c$\nis a nonassociative bracket \nand does not satisfy the Jacobi identity.\nThe AKSZ construction derives a new nontrivial topological field theory\nin $4$ dimensions.\nA BV action $S$ has the following expression:\n\\begin{eqnarray*}\nS&=&\n\\int_{\\calX} \\mu \\ (\n- \\bbp_a \\bbd \\bbq^a\n+ \\frac{1}{2} K{}^{ab} \\bbp_a \\bbp_b\n+ \\frac{1}{2} f{}^a{}_{bc} \\bbp_a \\bbq^b \\bbq^c\n+ \\frac{1}{4!} \nK^{-1}_{ae} f{}^e{}_{f[b} f{}^f{}_{cd]}\n\\bbq^a \\bbq^b \\bbq^c \\bbq^d)\\\\\n&=& - \\frac{1}{4} \\int_{\\calX} \\mu \\ (K_{ab} \\bbF^a \\bbF^b+\n\\frac{1}{3!} K^{-1}_{ae} f{}^e{}_{f[b} f{}^f{}_{cd]}\n\\bbq^a \\bbq^b \\bbq^c \\bbq^d).\n\\end{eqnarray*}\nIt is easily confirmed that $\\sbv{S}{S}=0$. \n\\medskip\\\\\n\\noindent\n\\textbf{Topological $3$-brane on $Spin(7)$-structure}.\nLet $(M, \\Omega)$ be an $8$-dimensional $Spin(7)$-manifold.\nHere $\\Omega$ is a $Spin(7)$ $4$-form, \nwhich satisfies $d \\Omega=0$ and \nthe selfdual condition $\\Omega=*\\Omega$.\nA $Spin(7)$ structure is defined as the subgroup of $GL(8)$ to \npreserve $\\Omega$.\nThe Q-structure on $(TM,\\Omega)$ is given by\n\\begin{eqnarray}\n\\Theta = \\bxi_i \\bq^i \n+ \\frac{1}{4!} \\Omega{}_{ijkl}(\\bx) \\bq^i \\bq^j \\bq^k \\bq^l.\n\\label{multisymQ}\n\\end{eqnarray}\nThe BV action $S$ for (\\ref{multisymQ}) defines the \nsame theory as \nthe topological $3$-brane analyzed in \\cite{Bonelli:2005ti}.\n\n\n\n\n\n\n\\section{Conclusions and Discussion}\n\\noindent\nWe have defined a BV algebra and a QP-structure \nof degree $3$.\nA QP-structure of degree $3$ has been constructed explicitly \nand a Lie algebroid u.t.h.~has been defined as \nits algebraic and geometric structure.\nA general theory of the AKSZ construction of a \ntopological field theory \nhas been expressed and\na new topological field theory in four \ndimensions has been constructed\nfrom a QP-structure.\n\nQuantization of this theory and\nanalysis of a Lie algebroid u.t.h.~will \nshed light on a super Poisson geometry and a quantum field theory.\nThey are future problems. \n\n\n\n\n\n\n\n\n\\section*{Acknowledgements}\n\nThe authors would like to thank Klaus Bering, Maxim Grigoriev, \nCamille Laurent-Gengoux, Yvette Kosmann-Schwarzbach, Kirill Mackenzie, \nDmitry Roytenberg, Alexei Sharapov, Thomas Strobl and Theodore Voronov\nfor their comments and discussion.\nThe author (N.I.)  would like to thank Maskawa Institute for Science and \nCulture, Kyoto Sangyo University for hospitality.\nWe would like to thank to referees for their useful advice.\n\n\\newcommand{\\bibit}{\\sl}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\\label{sec:introduction}\n\nSome million-scale datasets such as movie scripts and social media posts have become available in recent years for building neural dialogue agents~\\cite{lison2016lrec:opensubtitles,henderson2019convai:convdata}. \nSuch large-scale datasets can be expected to improve the performance of dialogue response generation models based on deep neural networks (DNNs) since the combination of DNNs and large-scale training datasets has led to considerable performance improvement in many sentence generation tasks~\\citet{koehn2017nmt:sixchallenges,sennrich2019acl:nmtcasestudy,adiwardana2020arxiv:meena}.\n\n\n\\input{fig-en-preliminary-experiment}\n\n\nIn contrast to the quantity of the data, the quality of the data has often been problematic.\nFor example, OpenSubtitles~\\cite{lison2016lrec:opensubtitles,lison2018lrec:opensubtitles}, the most widely used large-scale English dialogue corpus, was constructed by collecting two consecutive lines of movie subtitles under the simplified assumption that one line of a movie subtitle is one utterance and the next line is the next utterance follow it.\nInevitably, this corpus includes unacceptable utterance pairs from the viewpoint of a conversational sequence, e.g., caused by scene switching or flashback.\nSeveral previous studies have identified such flaws and reported that the corpus is \\emph{noisy}~\\cite{vinyals2015icml:neuralconv,li2016naacl:diversity,baheti2018emnlp:generating-interesting-response}, where \\emph{noisy} refers to unacceptable utterance pairs in this context.\nFigure~\\ref{fig:preliminary experiment} shows the result of our experimental investigation regarding the acceptability rate of the utterance pairs in the OpenSubtitles corpus.%\n    \\footnote{See Appendix~\\ref{a:preliminary experiment} for detailed experimental settings.}\nIt can be noticed from the figure that only half of the utterance pairs can be considered \\textit{acceptable} (i.e., were rated with score $5$: Strongly agree or $4$: Agree), and over 25\\% of utterance pairs are clearly \\textit{unacceptable} (i.e., were rated with score $1$: Strongly disagree or $2$: Disagree) from the human perspective.%\n    \\footnote{See Table~\\ref{tab:preliminary scored pairs} for samples of unacceptable\/acceptable utterance pairs annotated by humans.}\n\nWith this situation, a straightforward research question arises, namely, \\emph{Can we further improve the performance of neural response generation models by ablating unacceptable utterance pairs from training data?}\nTo the best of our knowledge, no previous study has explicitly focused on this question.\nThus, the goal of this paper is to provide an answer to this question.\nFurthermore, it is not clear whether and how one can effectively discover unacceptable utterance pairs within large-scale training datasets.\nThis study explores a way of constructing a scoring method for filtering \\emph{noisy} data filtering to improve the performance of response generation models.\n\n\nTo achieve the set goals, we started with a review of previous arguments about the criteria for identifying appropriate utterances in dialogues and designed our scoring function that is consistent with reflects as much of the community's consensus as possible.\nIn particular, the proposed scoring method estimates the quality of utterance pairs based on the following two aspects: \n(i) the \\textbf{connectivity} between source and target utterances and \n(ii) their \\textbf{content relatedness} (Section~\\ref{sec:idea}).\n\n\nThe contributions of this study are the following:\n\\begin{itemize}\n    \\item We propose a scoring method for estimating the quality of utterance pairs in an unsupervised manner (Section~\\ref{sec:proposed method});\n    %\n    \\item We reveal that our scoring method effectively detects unacceptable utterance pairs, and thus, be appropriate for noisy data filtering (Section~\\ref{sec:experiments});\n    %\n    \\item We empirically prove that our proposed data filtering method improves the performance of neural response generation models (Section~\\ref{sec:case study}); and\n        %\n    \\item We confirm that our noisy data filtering approach is effective across different languages and dataset sizes (Section~\\ref{sec:Japanese}).\n\\end{itemize}\n\n\n\n\n\\section{Task Definition: Noisy Data Filtering}\n\\label{sec:task}\n\nLet $x$ be an \\textbf{utterance} and $y$ be a \\textbf{response} to $x$.\nThen, an \\textbf{utterance pair} can be denoted as we refer to $(x,y)$.\nLet $\\mathcal{D}$ be a dataset that comprising a set of {utterance pairs}, $\\mathcal D = \\{(x,y)\\}$.\nThen, the task can be formulated as ablating unacceptable utterance pairs from $\\mathcal D$ to obtain a less noisy subset $\\mathcal{D}'\\subseteq \\mathcal{D}$, hereinafter referred to as filtered dataset.\n$\\mathcal{D}'$ can then be used to train response generation models. \nThis paper refers to this process as \\textbf{noisy data filtering}, where \\emph{noisy} means unacceptable utterance pairs in this context. \nFurthermore, we establish a function $S\\colon \\mathcal D\\to \\mathbb R$ is used to score the degree of \\textit{acceptability} of each utterance pair $(x,y) \\in \\mathcal D$.\n\n\n\n\n\n\\section{Background}\n\\label{sec:background}\n\n\\paragraph{Response generation using noisy data.}\nThe following two approaches are widely used to address the problem of dialogue response generation noisy dialogue corpora. \nAccording to the \\emph{model approach}, models are trained while handling noise at the same time. \nFor example, \\citet{shang2018ijcai:calibration} proposed a method with a calibration framework and demonstrated its effectiveness on a Chinese corpus.\nAccording to the \\emph{data approach}, training data are pre-processed with the aim of improving their quality before training models.\nIn this study, we take the data approach in light of the success of noisy parallel corpus filtering in machine translation (MT). \nAdditionally, it has become a reasonable strategy to reduce the size of training data since enormous dialogue data has been available.\n\\citet{csaky2019acl:filtering}'s method is most relevant to our study in that it cleanses dialogue corpora.\nHowever, the main goal of their method is to eliminate generic, or boring, responses, whereas the goal of the method proposed here is to eliminate unacceptable utterance pairs. \nThis difference in goals leads to the essential difference in filtering strategies.\n\n\n\n\n\\paragraph{Effectiveness of filtering noisy data in neural machine translation.}\n\nResearchers in the field of neural machine translation (NMT) have recognized that collecting high-quality training data to be equally or even more important than exploring sophisticated model architectures~\\cite{koehn2018wmt:filteringfindings,junczys-dowmunt2018wmt:filtering, morishita2018wmt:ntt}.\nTechniques used in neural response generation and NMT are nearly identical; e.g., sequence-to-sequence models~\\cite{Sutskever2014nips:seq2seq} and Transformers~\\cite{vaswani2017nips:transformer} are often used as base model architectures.\nWe hypothesize that high-quality filtered dialogue data can also improve the performance of dialogue response generators.\nHowever, the straightforward application of methods proposed for filtering noisy data in NMT may not work well due to the different nature of NMT and neural response generation tasks.\nIn particular,  MT data have one-to-one (ignoring paraphrases) correspondence in source and target sentences, whereas dialogues have many-to-many mappings~\\cite{zhao2017acl:onetomany}.\nThe experiments presented in this paper provide an answer to whether NMT filtering methods can perform well in dialogue response generation.\n\n\n\n\n\n\n\\section{Requirements to Utterance Pairs}\n\\label{sec:idea}\n\nIn this section, we investigate the requirements that should be satisfied by an acceptable utterance pair.\n\n\\input{tab-preliminary-scored-pairs.tex}\n\n\\subsection{Criteria for Manual Evaluation}\n\\label{ssec:criteria for manual evaluation}\n\n\nThe instructions for manual evaluation provided by the dialogue community explain the key factors for distinguishing acceptable and unacceptable utterance pairs.\n\nIn many previous studies, human raters were asked to evaluate the \\textbf{connectivity} of utterance pairs.\nFor instance, \\citet{shang2015aclijcnlp:neuralresponding} asked whether a response could be considered as \\textit{an appropriate and natural response to the post}.\n\\citet{xing2017aaai:topicaware} asked whether \\textit{the response can be used as a reply}.\n\\citet{pei2018emnlp:s2spmn} asked whether \\textit{the answer is natural} for the question.\nOther studies have also evaluated the same or similar aspects by using keywords related to the connectivity, such as \\textit{semantically appropriate for}~\\cite{akama2017ijcnlp:generating} or \\textit{coherent with}~\\cite{shen2017acl:conditional-variational}, and \\textit{coherence}~\\cite{lowe2017acl:autoturingtest}.\n\n\n\nAnother frequently used metric is \\textbf{content relatedness}.\nFor instance, \\citet{galley2015aclijcnlp:deltableu} asked human evaluators to evaluate \\textit{each response in terms of their relevance to a given utterance}.\n\\citet{li2016naacl:diversity} asked for the preference of responses \\textit{that were more specific to certain utterances}. \n\\citet{ritter2011:datadriven} suggested that \\textit{an appropriate response should be on the same topic as the utterances}.\nSeveral other studies have also focused on evaluating the \\textit{relevance} between an utterance and its response~\\cite{xu2018naacl:lsdscc,pei2018emnlp:s2spmn,lowe2017acl:autoturingtest}.\n\nIn summary, the most widely used criteria can be categorized into connectivity and content relatedness of utterance pairs. \nIn fact, these two aspects are considered in the field of sociolinguistics as crucial features of conversation~\\cite{sacks1989humanstudies:conversationalrule,sidnell2010book:conversation}.\n \n\n\n\n\n\n\\subsection{Observation}\n\\label{ssec:observation}\n\nFurthermore, we investigated how the two aforementioned aspects can be observed in actual utterance pairs.\nFor this investigation, we use the utterance pairs scored by human raters that were used in our preliminary experiments shown in Figure~\\ref{fig:preliminary experiment}.\nSome examples are shown in Table~\\ref{tab:preliminary scored pairs}.\n\n\nWe observe that typical phrase pair patterns can often be found in utterance pairs with high scores.\nFor example, the pair (\\bgf{\\textit{where is}}, \\bgf{\\textit{at}}) in Table~\\ref{tab:preliminary scored pairs} is one of the typical phrase pair patterns that asks a place and provides an answer to it.\nOther typical examples include (\\textit{why}, \\textit{because}) and (\\textit{what do you want}, \\textit{I want}).\nIn discourse linguistics, such phrase pair patterns are known as the concept of \\textit{cohesive devices}.\nHereafter, we refer to such a typical phrase pair pattern as \\textbf{key phrase pair}.\n\n\nMoreover, in high scored utterance pairs, both an utterance and response are on the \\textbf{same topic}.\nFor example, in the third example listed in Table~\\ref{tab:preliminary scored pairs}, both the utterance and response mention \\texttt{[money]}.\n\n\n\n\n\n\\section{Proposed Method}\n\\label{sec:proposed method}\n\nAs per the discussion in the previous section, each acceptable utterance pair should satisfy the following criteria:\n\\begin{itemize}\n    \\item \\textbf{connectivity} --- existence of key phrase pairs\n    \\item \\textbf{content relatedness} --- topic commonality\n\\end{itemize}\nThis section presents the proposed scoring functions to assess the degree of satisfying the above two criteria in an unsupervised manner.%\n\\footnote{The reason for focusing on an unsupervised approach the lack of data that can provide good supervision for utterance pair evaluation.}\n\n\n\n\\subsection{Connectivity}\n\\label{sec:connectivity}\nLet $f$ and $e$ represent phrases obtained from $x$ and $y$, respectively.\nLet $\\phi(x, y)$ be a function that returns a set of all possible phrase ($n$-gram) pairs obtained from the utterance pair $(x, y)$.\nWe can define a finite set of all possible phrase pairs obtained from the entire dialogue data $\\mathcal{D}$ as $\\overline{\\mathcal{P}}^{}_{\\mathcal{D}} =\\bigcup_{(x,y)\\in\\mathcal{D}} \\phi(x, y)$.\nThen, let $\\mathcal{P}$ represent a set of key phrase pairs (defined in Section \\ref{ssec:observation}).\nWe assume that $\\mathcal{P}$ is a subset of $\\overline{\\mathcal{P}}_{\\mathcal{D}}$, i.e., $\\mathcal{P}\\subseteq \\overline{\\mathcal{P}}_{\\mathcal{D}}$.\n\n\nTo obtain $\\mathcal{P}$, we take advantage of a phrase table extraction technique developed in statistical machine translation, e.g., Moses~\\cite{koehen2017acl:moses}.\nIn this task, we require only some phrase pairs that can contribute to the connectivity of an utterance pair (as mentioned in Section \\ref{ssec:observation}), unlike the translation task where the whole sentence must correspond in mutual.\nAccordingly, in our experiments, we set the null alignment ratio (i.e., probability of no alignment) to $0.5$ and extend the phrase extraction algorithm to include only the explicitly corresponding range as phrases in our table.\n\n\nThen, we define the scoring function $\\Sframe$ to estimate connectivity as: \n\\newcommand{\\LEN}[1]{\\lvert{#1}\\rvert}\n\\begin{align}\n\\label{eq:s_frame}\n    \\Sframe(x,y) := \n    \\sum_{\\mathclap{(f,e) \\in \\phi(x, y)\\cap\\mathcal{P} }} \n    \\hspace{2pt} \n        \\max\\bigl(\\mathrm{nPMI}(f,e),0\\bigr) \\cdot \\frac{\\LEN{f}}{\\LEN{x}} \\cdot \\frac{\\LEN{e}}{\\LEN{y}}\n    \\text{,}\n\\end{align}\nwhere $\\LEN{\\cdot}$ denotes the number of words in the phrase or utterance.\nTo calculate the co-occurrence, we use the normalized pointwise mutual information (nPMI)~\\cite{bouma2009gscl:npmi}, which normalizes the value so that low-frequency phrases do not take an extremely large value.\nNote that we ignore the negative nPMI scores by the $\\max(\\cdot, 0)$ operation because we aim only to consider the positive effect of connectivity. \nThe intuition behind Equation~\\ref{eq:s_frame} is as follows:\n\\begin{itemize}\n    \\setlength{\\itemindent}{0mm}\n\t\\setlength{\\parskip}{0.3mm}\n    %\n    \\item If a phrase pair $(f,e)$ has a high co-occurrence, the association strength of $(x,y)$ including $(f,e)$ might also be high. \n    %\n    \\item If a phrase $f$ or $e$ occupies almost the entire sentence $x$ or $y$, $(f,e)$ is a strong indicator of the association of $(x, y)$.\n    %\n\\end{itemize}\n\n\n\n\\subsection{Content Relatedness}\nLet $\\VEC v(x)$ and $\\VEC v(y)$ be sentence vector of $x$ and $y$, respectively.\nWe compute topic commonality of $x$ and $y$, that is, content relatedness as follows:\n\\begin{align}\n\\label{eq:s_content}\n    \\Scontent(x,y) := \n    \\max\\bigl(\\cos(\\VEC v(x), \\VEC v(y)), 0\\bigr)\n    \\text{.}\n\\end{align}\nCosine similarity between certain kinds of sentence vectors is known to be a good proxy of the topical relatedness of two sentences~\\cite{Conneau2017emnlp:universalrepresentation,subramanian2018iclr:generalpurpose,xu2018emnlp:filtering}.\nFor the same reasons as Equation~\\ref{eq:s_frame}, we ignore the negative $\\cos$ scores by the $\\max(\\cdot, 0)$ operation. \n\n\n\n\\subsection{Summary}\n\nEventually, combining the above two scoring measures, we propose the following function:\n\\begin{align}\n\\label{eq:score}\n    \\Sours(x,y) := \\alpha \\Sframe(x,y) + \\beta \\Scontent(x,y)\n    \\text{,}\n\\end{align}\nwhere $\\alpha,\\,\\beta\\in\\mathbb R_{\\geq 0}$ are hyperparameters that weigh the two viewpoints. \nFor our experiments, we fix $\\alpha$ and $\\beta$ as follows:\n\\begin{align}\n    \\label{eq:score hyp a}\n    & \\alpha \\!=\\! \\frac 1 {\\frac 1 {\\lvert\\mathcal D\\rvert} \\! \\displaystyle \\sum_{\\mathclap{(x,y)\\in\\mathcal D}} \\Sframe(x,y)\\!},\\;\n    \\beta \\!=\\! \\frac 1 {\\frac 1 {\\lvert\\mathcal D\\rvert} \\! \\displaystyle \\sum_{\\mathclap{(x,y)\\in\\mathcal D}} \\Scontent(x,y)}\n    \\text{.}\\!\n\\end{align}\n\n\n\n\n\n\\section{Experiments: Data Scoring}\n\\label{sec:experiments}\n\nIn this section, we describe our experiments that validate the effectiveness of the proposed scoring method.\n\n\\subsection{Experimental Setup}\n\n\\paragraph{Dataset.}\n\\label{ssec:dataset}\n\nWe conducted our experiments on a noisy English dialogue corpus from OpenSubtitles~\\cite{lison2018lrec:opensubtitles} containing roughly $441$M lines.\nAs explained in Section~\\ref{sec:introduction}, this corpus includes many unacceptable utterance pairs (Section~\\ref{sec:introduction}).\nWe first applied several rule-based filtering as rudimentary preprocesses, which are typically used in the related literature.\nThen, we obtained $79,\\!445,\\!453$ utterance pairs as our training data, which excludes our test and validation data\n    \\footnote{See Appendix~\\ref{a:corpus_creation} for details on our data such as the preparation procedure and statistics.}\n\n\n\n\\input{fig-en-correlation-human-3.tex}\n\n\n\\paragraph{Proposed method: detailed setup.}\n\n\nTo compute the connectivity $\\Sframe$, we obtained a phrase table on our training data by using Moses~\\cite{koehen2017acl:moses} with fastAlign~\\cite{dyer2013naacl:fastalign}. \nWe then removed phrase pairs with a low co-occurrence frequency (here, less than 200 times) or composed of the same phrases from the table.\nAs a result, the phrase table included $68,\\!891$ phrase pairs, which were used as the key phrase set $\\mathcal{P}$ as described in Section~\\ref{sec:connectivity}.\n\n\nTo compute the content relatedness $\\Scontent$, we created a sentence vector from pre-trained fastText word embeddings~\\cite{Bojanowski2017tacl:fasttext,mikolov2018lrec:advances-in-wordrep} following \\citet{arora2017iclr:sif}'s method, i.e., using SIF weighting and common component removal.\nTheir method is reported to be useful for computing the relatedness of two given sentences and used in many studies\n~\\cite{marelli2014lrec:sick,marelli2014semeval:compositional-distributional-model,Conneau2017emnlp:universalrepresentation,subramanian2018iclr:generalpurpose, baheti2018emnlp:generating-interesting-response}.\nWe learned common components using $30$K sentences randomly selected from the training costs appropriately. \nWe then removed the first common component for all sentence vectors.\n\n\n\n\\paragraph{Baselines.}\n\nFor comparison, we prepared the following:\n\\begin{itemize}\n    %\n    \\item \\citet{csaky2019acl:filtering}: Entropy-based filtering to remove generic utterances from the training data for promoting less-boring response generation. SRC\/TRG indicates that using the entropy of source\/target utterances.\n    %\n    \\item \\citet{junczys-dowmunt2018wmt:filtering}: Filtering for NMT based on the dual conditional cross-entropy computed by a neural encoder-decoder model. It achieved the best performance on the Parallel Corpus Filtering Task at WMT~2018.%\n        \\footnote{\\url{http:\/\/www.statmt.org\/wmt18\/}}\n    %\n\\end{itemize}\n\n\n\n\n\n\n\\paragraph{Human evaluation.}\n\\label{ssec:scoring eval}\n\nTo validate the ability of the proposed method to estimate the quality of utterance pairs, we measured the correlation between its scores and those assigned by humans through crowdsourcing.\nWe used Amazon Mechanical Turk.%\n    \\footnote{\\url{https:\/\/www.mturk.com\/}}\nWe randomly extracted $200$%\n    \\footnote{Same size as \\citet{sedoc2019naacldemo:chateval,cho2020acl:yesand}.}\nscored utterance pairs and asked native English-speaking crowdworkers to answer the following question for each pair: \\textit{Is the sequence of the two utterances acceptable as a dialogue?} \nWorkers were instructed to provide an answer on a five-point Likert scale (from $5$: Strongly agree to $1$: Strongly disagree)~\\cite{likert1932archivesofpsycho:scale}.\nUnqualified workers were filtered out using attention checks.\nEventually, we used the average of the scores provided by five workers as the human score for each pair.\n\n\n\\input{tab-scored-correlation.tex}\n\n\\input{tab-scored-samples.tex}\n\n\n\n\\subsection{Results and Analysis}\n\nTable~\\ref{tab:scored correlation} shows the correlation between human scores and those automatically computed by each method.\nAmong the methods, $\\Sours$ achieved the highest correlation with human scores.\nAdditionally, we also evaluated $\\Sframe$ and $\\Scontent$ as an ablation study of $\\Sours$.\nWe found that both scores were less correlated than $\\Sours$.\nThis result supports the hypothesis that both aspects, namely, connectivity and content relatedness, should be considered when evaluating the quality of utterance pairs.\n\n\n\nFigure~\\ref{fig:en_correlation-human-3} shows the distribution of automatically computed scores corresponding to human scores.%\n    \\footnote{See Appendix~\\ref{a:en-correlation_dist} for the distributions of other methods.}\nAs shown in (c), $\\Sours$ rarely overestimates utterance pairs with low human scores but underestimates those with high human scores.\nThe baseline methods presented in (a) and (b)  do not show such behavior. \nThis behavior unique to $\\Sours$ is safe for the noisy data filtering task since it can successfully detect lower-quality pairs with high precision.\nOn the other hand, improperly underestimating some acceptable pairs (i.e., low recall) is one downside of $\\Sours$, and we discuss its influences in Section~\\ref{ssec:LowRecall}.\nWe emphasize that $\\Sours$ has a desirable property for noisy data filtering in today's situation where a sufficiently large corpus is available; it allows us to obtain a sufficient amount of clean data even if discarding a certain portion of potentially clean data. \nInteresting future work is to investigate how to improve our methods not to underestimate acceptable pairs while maintaining high precision.\nIt is nearly equivalent to develop an unsupervised approach of dialogue evaluation methods, and thus, this direction is a challenging and essential attempt. \n\n\n\nTable~\\ref{tab:scored samples} shows several examples of utterance pairs well-scored by $\\Sframe$, $\\Scontent$, and $\\Sours$.\nNote that the score ranges differ; e.g., human scores are in $[1, 5]$, while $\\Scontent$ is in the range $[0, 1]$.%\n    \\footnote{See Appendix~\\ref{a:score distributions} for score distributions on training data.}\nThus, we discuss relative score values; the comparison of absolute score values across the different methods would be meaningless.\nThese examples demonstrate that the complementary contributions of both $\\Sframe$ and $\\Scontent$ allow $\\Sours$ to provide quality estimations close to human judgments. \n\n\n\n\n\n\\subsection{Discussion on Low Recall Property}\n\\label{ssec:LowRecall}\n\n\\paragraph{What types of pairs cause low recall?}\nSince the proposed method prefers precision over recall, it tends to discard a certain number of acceptable utterance pairs during filtering.\nTo investigate the characteristics of such discarded (yet acceptable) pairs, we analyzed $27$ pairs.%\n    \\footnote{Some examples are listed in Appendix~\\ref{a:post-hoc-analyses}}\nThese pairs were selected from those that obtained a human score of 4.0 or above ($77$ pairs) \\emph{and} were among the worst $50$\\% as scored by $\\Sours$ ($100$ pairs).\nConsequently, we found two potential issues.\nOne is that human annotators may sometimes easily find the connectivity or the content relatedness for the utterance pairs with the low $\\Sours$ scores.\nThis observation indicates that $\\Sframe$ and $\\Scontent$ are still not perfect for scoring functions, and there remains room for improvement.\nThe possible drawbacks we have already noticed in $\\Sframe$ and $\\Scontent$ are that $\\Sframe$ sometimes fails to capture the connectivity because of the limited coverage by a discrete phrase table-based approach, and $\\Scontent$ is not robust for out-of-vocabulary of word vector.\nThe other case is that the human annotators gave high scores, but we found no connectivity and content relatedness in the utterance pairs.\nWe found that some utterance pairs without any connectivity and content relatedness can be judged as acceptable by the human annotators since they can imagine the underlying context and situation of the utterance pairs using human world knowledge, such as commonsense.\nWe think this is a challenging issue that exceeds our focus in this paper, and thus, remains as future work.\n\n\n\n\\input{tab-scored-top-vs-worst-small.tex}\n\n\n\\input{tab-evaluation-results.tex}\n\n\n\n\\paragraph{Does our filtering undermine diversity?}\nOne might think that our method succeeds in filtering by assigning high scores to generic responses such as dull responses.\nThis concern makes sense since it is known that dialogue systems learned from the training data, including many generic utterances, tend to generate bland responses~\\cite{csaky2019acl:filtering}.\nTo answer this interesting question, we confirmed the diversity of utterance pairs with a high score (i.e., remained as training data) and a low score (i.e., removed from training data) in our $\\Sours$ (Table~\\ref{tab:scored-top-vs-worst-small}).%\n    \\footnote{See Appendix~\\ref{a:post-hoc-analyses} for more extensive result.}\nAs a result, there was no significant difference between them.\nTherefore, we conclude that the proposed method does not prefer only generic responses and maintains the diversity of data.\nIt is an essential future attempt to improve the quality of dialogue data further (e.g., more diversity) after using the proposed method to remove unacceptable pairs.\n\n\n\n\n\\input{tab-genresponses.tex}\n\n\n\n\n\\section{Experiments: Response Generation}\n\\label{sec:case study}\n\nThis section reports on the effectiveness of the proposed method for filtering noisy data in neural response generation.\n\n\n\\subsection{Experimental Setup}\n\n\\paragraph{Training.}\n\\label{ssec:TrainingSettings}\n\nWe obtained the filtered training data $\\mathcal{D}'$ by removing utterance pairs with low scores from the original dataset $\\mathcal D$ (approximately $10$\\% or $50$\\% of total utterance pairs were removed).\nAs a response generation model, we used a Transformer~\\cite{vaswani2017nips:transformer} based encoder-decoder model implemented in the \\texttt{fairseq} toolkit~\\cite{ott2019naacldemo:fairseq}.%\n    \\footnote{See Appendix~\\ref{a:training settings} for training details.}\nTransformer has demonstrated high performance in response generation~\\cite{dinan2019iclr:wizard} and other NLP tasks.\n\n\n\\paragraph{Automatic evaluation.}\nHere, we report the following metrics: the average response length in tokens (len), type-token ratio for $\\{1,2\\}$-grams (distinct-$\\{1,2\\}$), and and BLEU-1~\\citep{Papineni2002acl:bleu}.\nThe latter was used as a reference-based metric; while it is widely used in previous studies~\\cite{zhao2017acl:onetomany,baheti2018emnlp:generating-interesting-response,csaky2019acl:filtering}, some studies (e.g., ~\\cite{liu2016emnlp:hownot}) have reported that BLEU-1 may not be highly correlated with the human evaluation of response generation.%\n    \\footnote{See Appendix~\\ref{a:generated response with metrics} for more extensive evaluation results.}\n\n\n\\paragraph{Human evaluation.}\nWe evaluated the quality of the generated responses manually. \nWe asked human evaluators recruited via Amazon Mechanical Turk to evaluate responses that are generated for $100$%\n    \\footnote{Same size as \\citet{shen2017acl:conditional-variational,bao2020acl:plato}.}\ninput utterances randomly sampled from the test data. \nWe used the same task setting and protocol as described in Section~\\ref{ssec:scoring eval} to obtain the human scores for each pair. \nHigher human scores indicate that the better results.\n\n\n\n\\subsection{Results and Analysis}\n\n\nTable~\\ref{tab:evaluation-results-en} shows the results of automatic and human evaluations of the generated responses.\nThe model trained on the data filtered using the proposed method $\\Sours$ produced more than three times as many distinct $\\{1,2\\}$-grams as the model trained on non-filtered data.\nFurthermore, it outperformed the model trained on non-filtered data in the human evaluation, achieving the highest percentage of acceptable responses of 85\\%. \nAdditionally, these results of our $\\Sours$ were better than other baselines.\nTo conclude, these experimental results indicate that the proposed scoring method can help generate diverse responses that are judged as acceptable by humans. \n\nThis experiment provides empirical evidence for supporting our hypothesis that the performance of neural response generation models can be improved by just removing unacceptable utterance pairs from training data, which answers the research question formulated at the start of this paper.\n\n\n\n\n\n\n\n\n\\section{Multilingual Availability}\n\\label{sec:Japanese}\n\n\nWhile the proposed method $\\Sours$ was tested on an English corpus, it can potentially work for other languages as well. \nTo demonstrate this, we selected Japanese dialogue data as another case study.%\n    \\footnote{See Appendix~\\ref{a:japanese} for all experimental results on Japanese.} \nThe linguistic phenomena in Japanese are quite different from those in English, thus making this experiment to be a good test of the applicability of the proposed method to non-English languages.\n\n\n\\paragraph{Japanese dataset.}\nWe prepare the Japanese dialogue data from Japanese OpenSubtitles~\\cite{lison2018lrec:opensubtitles} containing roughly $3$M lines.\nWe obtain $1,\\!893,\\!477$ utterance pairs as our training data, which excludes our test and validation data.%\n    \\footnote{See Appendix~\\ref{a:corpus_creation} for details on our data such as the preparation procedure and statistics.}\n\n\n\\input{tab-scored-correlation-ja.tex}\n\n\\input{fig-ja-correlation-human-ours}\n\n\\input{tab-evaluation-results-ja}\n\n\n\n\n\\subsection{Data Scoring}\n\\label{ssec:JapaneseScoring}\n\n\n\\paragraph{Settings.}\nTo compute $\\Sframe$, we defined a low co-occurrence frequency as less than 20, considering the size of the Japanese corpus, and consequently obtained the key phrase pairs $|\\mathcal{P}|=19,\\!992$.\nTo compute $\\Scontent$, we used pre-trained fastText \\cite{grave2018lrec:wordvec157} and learned common components from all sentences in the training data.\n\n\nFor human evaluation, we used Yahoo!~crowdsourcing%\n    \\footnote{\\url{https:\/\/crowdsourcing.yahoo.co.jp\/}}%\nto hire native Japanese-speaking workers.\nThe task setting and protocol are the same as those for English (Section \\ref{ssec:scoring eval}), regardless of the crowdsourcing platform.\n\n\n\\paragraph{Results and analysis.}\nTable~\\ref{tab:scored correlation_ja} shows the correlation between human scores and those automatically computed by each method.\nOur method $\\Sours$ has the highest correlation with human scores, although the overall result is lower than that obtained for the English dataset.\nFigure~\\ref{fig:ja_correlation-human-onlyours} shows the distribution of our $\\Sours$ corresponding to human scores.\nSimilar to the result obtained for English as presented in Figure~\\ref{fig:en_correlation-human-3} (c), $\\Sours$ rarely overestimates utterance pairs with low human scores but underestimates those with high human scores in Japanese.\n\n\n\n\\subsection{Response Generation}\n\\label{ssec:JapaneseFiltering}\n\n\n\\paragraph{Settings.}\nWe used the same experimental settings described in Section~\\ref{ssec:TrainingSettings} for the preparation of filtered data $\\mathcal{D}'$ and model training.\n\n\n\\paragraph{Results and analysis.}\nTable~\\ref{tab:evaluation-results-ja} shows the results of evaluations of the generated responses.\nThe filtered data generated by $\\Sours$ provided the best results in terms of almost all the metrics, including human evaluation.\nIt supports our hypothesis that the proposed method is also suitable for non-English languages.\n\n\n\n\n\\section{Relationship with Evaluation Metric}\n\\label{sec:relatedWork}\n\nThe proposed method $\\Sours$ maps an utterance pair to a score (scalar value) in terms of the quality of dialogue.\nThat is, formally, our method is similar to the reference-free automatic evaluation metrics for dialogue agents; both of them evaluate the response given an input utterance and also map into a score. \nRecently, the novel reference-free metrics for evaluating generated responses such as USR~\\cite{Mehri2020acl:usr} or \\textsc{MaUde}~\\cite{sinha2020acl:maude} ware developed.\nWhile it is possible to use them as a scoring method for filtering noisy data, in theory, there are some concerns with applying them in practice.\nOne is the difference of the data of interest; since evaluation metrics are intended for responses generated as dialogue, i.e., somewhat valid dialogue data, it is unclear whether they also work for apparently noisy data.\nAnother one is the difference of desired properties; evaluation metrics need to be sensitive to ``how good is it?''\\ while the filtering requires to detect ``is it a dialogue?''\\ with high accuracy.\nIt would be interesting to investigate the effectiveness of reference-free metrics for noisy dialogue data filtering tasks, and vice versa.\nWe leave these investigations for future work.\n\nIn contrast, reference-based metrics require a reference response (i.e., ground truth) when they calculate scores; such metrics include the traditional overlap-based BLEU, ROUGE, METEOR, embedding-based metrics~\\cite{liu2016emnlp:hownot}, and neural network-based RUBER~\\cite{tao2018aaai:ruber} and ADEM~\\cite{lowe2017acl:adem}\nThus, these methods cannot straightforwardly be considered as alternatives to the proposed method, which aims at filtering.\n\n\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nIn light of the success of noisy corpus filtering in neural machine translation, we attempted to filter out unacceptable utterance pairs from large dialogue corpora in an unsupervised manner.\nThe proposed scoring method estimates the quality of utterance pairs by focusing on the two crucial aspects of dialogue, namely, the \\emph{connectivity} and \\emph{content relatedness} of utterance pairs.\nWe demonstrated that our scoring method has a higher correlation with human judgment than recently proposed methods.\nFurthermore, we provided empirical evidence that our method improves the performance of a response generation model by removing unacceptable utterance pairs from its training data. \nWe hope that this study will facilitate discussions in the dialogue response generation community regarding the issue of filtering noisy corpora.\n\n\n\n\\clearpage\n\\section*{Acknowledgments}\nThis work was supported by JSPS KAKENHI Grant Numbers JP19H04162, JP19J21913.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn the last two decades there has been a growing interest in the study of\nchoreographic solutions (\"choreographies\") of the $n$-body and $n$-vortex \nproblems.\nChoreographies are periodic solutions where the bodies or the vortices \nfollows the same path. \nThe first choreography was discovered numerically for the case of three\nbodies in \\cite{Mo93}, and its existence was rigorously proved in \n\\cite{ChMo00}. \nThe term \"choreography\" was adopted for the $n$-body problem after the work\nof Sim\\'{o} \\cite{Si00}.\nSince then, variational methods \\cite{FeTe04,BT04}, numerical minimization\n\\cite{ChGe02}, numerical continuation \\cite{Ca}, and computer-assisted \nproofs \\cite{KaSi07}, have been used to determine choreographies of the \n$n$-body problem; see also the references in these papers. \nFor vortices in the plane, choreographies have been constructed for $3$ \nand $4$ vortices in \\cite{Bo}. \nFor $n$ vortices in a general bounded domain, choreographic solutions have \nbeen found close to a stagnation point of a vortex \\cite{Ba0}, and close \nto the boundary of the domain \\cite{Ba}.\n\nIn \\cite{Ca}, \\cite{GaIz13}, and \\cite{ChFe08}, chorographies are found in\ndense sets of Lyapunov families that arise from the stationary $n$-polygon \nof bodies in a rotating frame. \nThe existence of these choreographies depends only on the symmetries of \nthe equations in rotating coordinates, {\\it i.e.}, these results can be \nextended to find choreographies of the $n$-vortex problem in the plane, \nin a disk, or in a surface of revolution. \nSuch results can also be extended to the discrete nonlinear Schr\\\"{o}dinger\nequation (DNLSE), which appears in the study of optical waveguide arrays,\nand in Bose-Einstein condensates trapped in optical lattices, among many \nother applications \\cite{Kr}.\n\nWhile much research has been done on choreographies in the $n$-vortex and \n$n$-body problems, we are not aware of its extension to the study of the \nexistence of choreographies in periodic lattices of $n$ sites, as modeled \nby the DNLSE.\nThe interest in $n$-body and $n$-vortex choreographies can perhaps be \nexplained by the fact that variational methods are better suited for \nsingular potentials. \nOn the other hand, the continuation methods used in \\cite{Ca} are very well\nsuited for locating choreographies for other symmetric potentials, such as \nin the DNLSE.\n\nEvidence suggests that linearly stable choreographies in the $n$-body \nproblem only exist for $n=3$, and in the $n$-vortex problem for \n$n=3,\\cdots,7$, as a consequence of the fact that the polygonal relative \nequilibrium of $n$ bodies is unstable for $n\\geq3$ \\cite{Ca}, and for \nvortices for $n>7$ \\cite{GaIz12}. \nOn the other hand, the DNLSE has dense sets of stable choreographies, \nas a consequence of the fact that the DNLSE has stable polygonal relative \nequilibria for all $n$ \\cite{Ga15}.\nWe note that boundary value continuation methods can determine unstable \nchoreographies as easily as stable ones; a property not shared by most \nother techniques.\nThe aim of our paper is to investigate the existence of choreographies \nin the DNLSE using a boundary value continuation method. \nAs illustrative examples we present a selection of choreographies,\nmost of them stable, in a periodic lattice for the case of $n=9$,\n$17$, and $31$ sites.\n\nThe lack of previous work on detecting choreographies in the DNLSE may \nbe related to the difficulties encountered in the measurement of phases \nin physical problems modeled by the DNLSE. \nHowever, such difficulties appear to be surmountable in nonlinear optics.\nWithin the context of nonlinear optics, several predictions of the DNLSE \nhave been found experimentally in the last two decades \\cite{Opt1},\nsuch as the formation of discrete solitons in waveguide arrays. \nThe use of suitable optical techniques, known as laser heterodyne \nmeasurements, to detect the field intensity and the phase \\cite{Opt2,Opt3}, \nmay open the door to experimental observation of stable choreographic \nsolutions. \n\nIn Section~2 we consider Lyapunov families of periodic orbits, and their \nrelation to the existence of choreographies in a periodic lattice of \nSchr\\\"{o}dinger sites. \nIn Section~3 we present methods to continue the Lyapunov families, and we \nexhibit a small selection of the many linearly stable choreographies that\nwe have determined.\nIn Section~4 we discuss our final remarks on choreographies.  \n\\section{Lyapunov families and choreographies}\n\nIn a rotating frame with frequency $\\omega$, $q_{j}(t)=e^{i\\omega t}%\nu_{j}(t)$, the equation that describes the dynamics of a lattice of $n$ \nsites is given by the Hamiltonian system \n\\begin{equation}\n\\dot{u}=\\mathcal{J}\\nabla H_{\\omega}(u), \\quad \\text{where} \\quad\nH_{\\omega}=\\frac{1}{2}\\sum_{j=1}^{n}\\left(  \\frac{1}{2}\\left\\vert\nu_{j}\\right\\vert ^{4}+\\omega\\left\\vert u_{j}\\right\\vert ^{2}-\\left\\vert\nu_{j+1}-u_{j}\\right\\vert ^{2}\\right)  . \\label{Equations}\n\\end{equation}\nThe sites $u_{j}(t)\\in\\mathbb{C}$ satisfy periodic boundary conditions\n$u_{j}(t)=u_{j+n}(t)$.\nThe equation of motion has explicit polygonal equilibrium solutions given \nby%\n\\begin{equation}\na_{j}=ae^{ij(\\alpha\\zeta)},\\qquad\\zeta=\\frac{2\\pi}{n}, \\label{SW1}%\n\\end{equation}\nfor $\\omega(a)=4\\sin^{2}(m\\zeta\/2)-a^{2}$, $\\alpha=1,\\cdots,n$ and $a\\in\n\\mathbb{R}^{+}$. \nThese solutions correspond to relative equilibria in the non-rotating frame given by $q_{j}(t)$for $j=1,\\cdots,n$.\nThe linearized Hamiltonian system at the polygonal equilibrium \n$\\mathbf{a} =(a_{1},...,a_{n})$ is \n$$\\dot{u}=\\mathcal{J}D^{2}H_{\\omega(a)}(\\mathbf{a})u.$$\nIn \\cite{Ga15} it is proved that the matrix \n$\\mathcal{J}D^{2}H_{\\omega (a)}(\\mathbf{a})$ \nhas a pair of imaginary eigenvalues $\\pm i\\nu_{k}$ for each\n$k\\in\\{1,\\cdots,n-1\\}$ such that\n\\begin{equation}\n\\frac{a^{2}}{2\\cos\\alpha\\zeta\\sin^{2}k\\pi\/n}<1. \\label{In}%\n\\end{equation}\nIt is also proved in \\cite{Ga15} that for each $a$ \nand $k\\in\\{1,\\cdots,n-1\\}$ such that (\\ref{In}) holds, \nthe equilibrium $\\mathbf{a}$ has a \\emph{global family} \nof periodic solutions that arises from the normal modes \nof the polygonal equilibrium, and has the form\n\\begin{equation}\nu_{j}(t)=e^{ij\\alpha\\zeta}u_{n}\\left(  \\nu t\\pm jk\\zeta\\right)  \\text{,}\n\\label{TW}%\n\\end{equation}\nwhere $u_{n}(t)=a+\\mathcal{O}(b)$ is a $2\\pi$-periodic function,  \n$\\nu =\\nu_{k}+\\mathcal{O}(b)$ is the frequency and $b$ is a parameterization of the local family. \nThe traveling waves (\\ref{TW}) form two-dimensional families parametrized \nby the amplitude $a$ and the bifurcation parameter $b$. In the non-rotating frame, after rescaling time, these periodic solutions \nare traveling waves of the form%\n\\[\nq_{j}(t)=e^{i\\omega t\/\\nu}e^{ij\\alpha\\zeta}u_{n}\n     \\left(  t\\pm jk\\zeta\\right) ,\\qquad\\omega=\\omega(a)\\text{.}%\n\\]\nWe say that a Lyapunov orbit is $\\ell:m$ resonant if $\\ell$ and $m$ are\nrelatively prime such that%\n\\[\n\\frac{\\omega}{\\nu}=\\frac{\\ell}{m}\\text{,\\qquad}k\\ell-\\alpha \n  m\\in n\\mathbb{Z}\\text{.}%\n\\]\nSuch frequencies $\\nu$ are dense in the set of real numbers.\nFor an $\\ell:m$ resonant orbit, we have%\n\\[\nq_{n}(t)=e^{it\\omega\/\\nu}u_{n}(t)~.\n\\]\nSince $e^{it\\omega\/\\nu}=e^{it\\ell\/m}$ is $2\\pi m$-periodic, the function\n$q_{n}(t)=e^{it\\omega\/\\nu}u_{n}(t)$ is $2\\pi m$-periodic. \nAlso, since%\n\\begin{equation}\n q_{n}(t-2\\pi)=e^{i(t-2\\pi)\\omega\/\\nu}u_{n}(t-2\\pi)=e^{-i2\\pi\\ell\/m}%\n q_{n}(t),\\label{symq}%\n\\end{equation}\nthe orbit of $q_{n}(t)$ is invariant under rotation by $2\\pi\/m$.\nThe solutions satisfy\n\\begin{align*}\nq_{j}(t)  &  = e^{it(\\omega\/\\nu)}u_{j}(t)\n       =e^{it(\\omega\/\\nu)}e^{ij\\alpha\\zeta }u_{n}(t+jk\\zeta)\\\\\n&  = e^{it(\\omega\/\\nu)}e^{i\\alpha j\\zeta}e^{-i(\\omega\/\\nu)\n  \\left( t+jk\\zeta\\right)  }q_{n}(t+jk\\zeta)=e^{-ij\n  \\left(  (\\omega\/\\nu)k-\\alpha\\right) \\zeta}q_{n}(t+jk\\zeta)\\text{.}%\n\\end{align*}\nUsing the facts that $\\omega\/\\nu=\\ell\/m$ and $\\zeta=2\\pi\/n$, we have\n\\[\nj\\left(  k\\frac{\\omega}{\\nu}-\\alpha\\right)  \\zeta=2\\pi j\n   \\left(  \\frac{\\ell k-\\alpha m}{mn}\\right)  =2\\pi j \\left( \\frac{r}{m} \\right) ,\n\\]\nwith $r=(k\\ell-\\alpha m)\/n\\in\\mathbb{Z}$ by assumption. \nSince $\\ell$ and $m$ are relatively prime we can find $\\ell^{\\ast}$, \nthe $m$-modular inverse of $\\ell$. \nSince $\\ell\\ell^{\\ast}=1$ mod $m$, it follows from the symmetry\n(\\ref{symq}) that\n\\[\nq_{n}(t-2\\pi jr\\ell^{\\ast})=e^{-i2\\pi j(r\/m)}q_{n}(t).\n\\]\nThen%\n\\begin{equation}\nq_{j}(t)=e^{-i2\\pi j(r\/m)}q_{n}(t+jk\\zeta)\n   = q_{n}(t+j(k-rn\\ell^{\\ast})\\zeta).\n\\end{equation}\nThus in the non-rotating frame, an $\\ell:m$ resonant Lyapunov orbit \nis a choreography satisfying%\n\\[\nq_{j}(t)=q_{n}(t+j\\tilde{k}\\zeta)\\text{,}%\n\\]\nwhere \n$\\tilde{k}=k-(k\\ell-\\alpha m)\\ell^{\\ast}$ \nwith $\\ell^{\\ast}$ the $m$-modular inverse of $\\ell$. \nThe period of the choreography is $m~T_{\\ell :m}$ with%\n\\[\nT_{\\ell:m}=\\frac{2\\pi}{\\nu}=2\\pi\\omega(a)\\left(  \\frac{\\ell}{m}\\right)  .\n\\]\nThe choreography is symmetric under rotation by $2\\pi\/m$, and it winds\naround a center $\\ell$ times.\n\\section{Computational Results}\n\\label{sec:2}\nWe have computed families of periodic solutions that arise directly or\nindirectly from the circular polygonal relative equilibrium of the DNLSE.\nThese families are computed by numerical continuation using boundary value \nformulations. \nIn this article we present numerical results for several choices of the \nnumber of sites $n$ in the DNLSE, namely for $n=9$, $n=17$, and $n=31$,\nwith various values of the amplitude parameter $a$.\nIn our boundary value setting the DNLS differential equations are \nformulated as\n\\begin{align*}\nu_{k}^{\\prime}(t)  &  =-iT\\left(  u_{k-1}-2u_{k}+u_{k+1}+\\left\\vert\nu_{k}\\right\\vert ^{2}u_{k}+\\omega u_{k}\\right) \\\\\n&  +p_{1}(4u_{k}-4u_{k}^{3}-2\\bar{u}_{k+1})+p_{2}(u_{k+1}-\\bar{u}_{k-1}),\n\\end{align*}\nwhere $u_{k}(t)=x_{k}(t)+iy_{k}(t)$ for $k=1,2,\\cdots,n$ and\n\\[\nu_{0}(t)\\equiv u_{n}(t)\\quad,\\quad u_{n+1}(t)\\equiv u_{1}(t).\n\\]\nHere $T=2\\pi\/\\nu$ is the period of a periodic orbit, so that the scaled time\nvariable $t$ takes values in the fixed time interval $[0,1]$. The parameters\n$p_{1}$ and $p_{2}$ are \\textit{unfolding parameters} that are necessary to\ntake care of invariances related to the presence of two conserved quantities.\nThe parameters $p_{1}$ and $p_{2}$ are always part of the unknowns solved in\nthe Newton iterations. However, upon converge their values are zero up to\nnumerical accuracy.\nThe boundary conditions that we impose always include the periodicity\nconditions\n\\[\nu_{k}(1)-u_{k}(0)=0,\\quad k=1,2,\\cdots,n.\n\\]\nAdditional constraints can be used to fix certain quantities along \nsolution families, provided other appropriate parameters are allowed \nto vary. \nIn particular, we can fix the $y$-coordinate of the $n$th site at time \n$t=0$, {\\it i.e.},%\n\\[\ny_{n}(0)=0.\n\\]\nThis constraint can be viewed as a phase condition that is sometimes more\nconvenient than an integral phase condition of the type mentioned below. \nIt can also be useful to fix the $x$-coordinate of the $n$th site \nat time $t=0$, which is accomplished by adding the boundary condition\n\\[\nx_{n}(0)-x_{n}^{0}=0,\n\\]\nwhere $x_{n}^{0}$ is a parameter that can be kept fixed.\nFor convenience, constraints of this form can also be used to keep track \nof such quantities. \nFor example, the parameter $x_{n}^{0}$, when free to vary, trivially keeps \ntrack of $x_{n}(0)$. \nSuch constraints can also fix (or to keep track of) one of the conserved \nquantities $E$ or $A$, or the resonance ratio $T\/T_{0}$, \nwhere $T_{0}=2\\pi\/\\omega$ is the period of the rotating frame. \nThis is accomplished by adding one of the following constraints:\n\\[\n\\left\\vert u_{k}-u_{k+1}\\right\\vert ^{2}-\\left\\vert u_{k}\\right\\vert\n^{4}-\\omega\\left\\vert u_{k}\\right\\vert ^{2}-E=0~,~~\\mbox{where}\\quad\n\\omega=4\\sin^{2}(\\pi\/n)-a^{2},\n\\]%\n\\[\n\\sum_{k=1}^{n}\\left\\vert u_{k}\\right\\vert ^{2}-A=0,\\quad\\mbox{or}\\quad\nT\/T_{0}-r=0.\n\\]\nThe boundary value formulation can also contain integral constraints, \nsuch as the phase condition\n\\[\n\\int_{0}^{1}x_{n}(t)~\\tilde{x}_{n}^{\\prime}(t)\n          +y_{n}(t)~\\tilde{y}_{n}^{\\prime }(t)~dt=0,\n\\]\nwhich here is applied to the $n$th site only, and where \n$(\\tilde{x}_{n}^{\\prime}(t),\\tilde{y}_{n}^{\\prime}(t))$ represents \nthe time-derivative of a reference solution, which typically is the \npreceding solution in the numerical continuation process. \nAnother integral constraint sets the average $y$-coordinate \nof the $n$th site to zero, namely,\n\\[\n\\int_{0}^{1}y_{n}(t)~dt=0.\n\\]\nThe purpose of this constraint is to remove the rotational invariance of\nperiodic solutions. \nThere are more general integral constraints for fixing the phase and for \nremoving invariances. \nHowever the ones listed above are simple, and appropriate in the current \ncontext.\n\nWe now briefly outline the computational procedure that we have used to \nlocate the periodic orbits shown in Figure~\\ref{fig1} and in\nFigure~\\ref{fig2}, for which corresponding data is given in Table~1.\nAs a starting procedure we follow a family of periodic solutions that \nemanates from the polygonal equilibrium, namely a family that arises \nfrom a conjugate pair of purely imaginary eigenvalues of the equilibrium.\nThis starting procedure is relatively standard, and essentially the same \nas the one used for Hopf bifurcation. \nInitially only a small portion of the periodic solution family is computed;\nin fact only a few continuation steps are taken. \nA minor adjustment of the standard starting procedure ensures that the \nsmall-amplitude starting solution satisfies in particular the conditions \n$y_{n}(0)=0$ and $\\int_{0}^{1} y_{n}(t)~dt = 0$.\nKeeping $y_{n}(0)=0$, the small-amplitude starting solution is followed \nuntil $x_{n}(0)$ reaches a specified target value, for which we have used \nvalues such as $x_{n}^{0}=-0.04$, $x_{n}^{0}=0.005$, and $x_{n}^{0}=0.0001$\n(see Table~1).\nThe free continuation parameters in this step include $T$, $x_{n}^{0}$,\n$p_{1}$, and $p_{2}$. \nThe resulting periodic orbit has the property that all nine solution \ncomponents pass near the origin in the complex plane. \nThe subsequent main computational step then consists of keeping $x_{n}^{0}$ \nfixed at the value $x_{n}^{0}$, while allowing the amplitude parameter \n$a$ to vary. \nSpecifically, the free continuation parameters now include \n$T$, $a$, $p_{1}$, and $p_{2}$. \nIn the quest for locating interesting, stable  periodic solutions of the \nDNLSE, there are multiple variations on the continuation scheme outlined \nabove. \nFor example, one can start from a selected periodic solution from the main\ncomputational step, now keeping the conserved quantity $H$ fixed. \nYet another variation that we use is to follow periodic solutions found \nin the main computational step above, keeping the resonance ratio \n$T\/T_0$ fixed. Here $T$ is the period of the periodic orbit and $T_{0}$ \nis the period of the rotating frame.\nIn fact, along all families of periodic solutions we monitor the value \nof the ratio $T\/T_{0}$.\nSpecifically we are interested in rational values of $T\/T_{0}$, for which \nthe orbits correspond to a choreographies in the non-rotating frame.\nAs discussed in Section~2, there is a countably infinite number of such\nchoreographies along families of periodic orbits, provided that the \nthe orbits possess certain symmetries, and provided the period $T$ is \nnot constant.\nMoreover, as already mentioned above, such choreographies can subsequently\nbe continued with varying amplitude parameter $a$, while keeping the ratio\n$T\/T_{0}$ fixed at a choreographic value.\nThus the number of choreographies then becomes in fact uncountably infinite.\n\nTwo choreographies for $n=9$ are shown in Figure~\\ref{fig1}, namely \nin the top-right panel and center-right panel of that Figure.\nIn addition, all twelve orbits shown in Figure~\\ref{fig2} correspond to\nchoreographies.\nSpecifically, in Figure~\\ref{fig1}, the top-left panel shows a resonant \norbit in the rotating frame.\nThe coloring of this orbit is according to its nine components, \nand evidently it consists of nine separate closed curves.\nThe top-right panel of Figure~\\ref{fig1} shows the same orbit in the\nnon-rotating frame, where all components follow a single curve, {\\it i.e.},\nthe orbit is a choreography.\nThe coloring of the choreography is also according to its nine components.\nHowever, since all components follow the same curve, the color of this\ncurve changes gradually to a uniform final color, as one complete orbit\nis traversed.\n\nSimilarly, the center-left panel of Figure~\\ref{fig1} shows an orbit in\nthe rotating frame, while the center-right panel shows the same orbit in\nthe non-rotating frame, where it evidently corresponds to a choreography.\nThe bottom panels of Figure~\\ref{fig1} show two resonant orbits in the\nnon-rotating frame.\nAs the coloring indicates, neither of these two orbits is a choreography.\nHowever, both can be designated as a {\\it partial choreography}, since\neach consists of three separate closed curves, and each of these three\ncurves is traversed by three components.\nFinally, Figure~\\ref{fig2} shows a selection of twelve orbits in the\nnon-rotating frame, each of which is a choreography.\nThese choreographies are visually appealing, and particularly interesting \nto watch in animations.\n\n\\section{Conclusions and discussion}\n\\label{sec:3}\nThe DNLS equations with $n$ sites, and with periodic boundary conditions, \nhave Lyapunov families of periodic solutions that arise from polygonal \nequilibria in the rotating frame. These Lyapunov families can be \nparameterized by the rotational frequency $\\omega$ and the frequency \n$\\nu$ of the Lyapunov orbit. \nWhen the ratio of the freqencies $\\omega$ and $\\nu$ is $l:m$ resonant, \n{\\it i.e.}, when $\\omega\/\\nu=l\/m$, then the Lyapunov orbit corresponds \nto a choreography that is symmetric with respect rotations by $2\\pi\/m$, \nand that has winding number $l$. \nFor fixed $\\omega$ this resonance condition is satisfied for a dense set \nof rational frequencies $\\nu$.\nThus if the frequency range of $\\nu$ along a Lyapunov family contains \nan interval, then there is an infinite number of Lyapunov orbits that \ncorrespond to choreographies.\n\nIn this paper a robust and highly accurate boundary value technique with \nadaptive meshes has been used to continue the Lyapunov families. \nThe presence of two conserved quantities, namely the amplitude $A$ and the \nenergy $E$, is dealt with by using two unfolding parameters. \nThe formulation also allows continuation of solution families with fixed \n$E$, $A$, or $\\omega\/\\nu$. \nFor example, by fixing $\\omega\/\\nu=l\/m$ one can compute a continuum of\nchoreographies, each of which is symmetric with respect to rotation by\n$2\\pi\/m$, and has winding number $l$.  We have included a small sample \nof the infinitely many stable choreographies that can be computed \nin this manner.\n\nIn principle, physical observation of stable choreographies appears to be\npossible with heterodyne optical techniques.\nSuch techniques have been used experimentally to record both the phase \nand the amplitude of a coherent optical signal source, such as that at \nthe end of a waveguide array, as modeled by the DNLSE. \nThe basic idea is to study the optical field composed of a reference and \na source field.\nThis optical technique has allowed the confirmation of predictions from\nthe Lorenz model, which describes to a good degree the dynamics of the \n$NH_{3}$-laser \\cite{Opt2}. \nSimilarly, heterodyne optical techniques have been used to study optical \nfields in two-dimensional light sources \\cite{Opt3}, which are precisely \nthe geometries that support stable choreographies.\n\\clearpage\n\\begin{figure}[ptb]\n\\begin{center}\n\\resizebox{16.0cm}{!}{  \n\\includegraphics{f1a.png}~\n\\includegraphics{f1b.png} }\n\\end{center}\n\\par\n\\vskip-.90cm\\noindent\n\\par\n\\begin{center}\n\\resizebox{16.0cm}{!}{  \n\\includegraphics{f1c.png}~\n\\includegraphics{f1d.png} }\n\\end{center}\n\\par\n\\vskip-.90cm\\noindent\n\\par\n\\begin{center}\n\\resizebox{16.0cm}{!}{  \n\\includegraphics{f1e.png}~\n\\includegraphics{f1f.png} }\n\\end{center}\n\\par\n\\vskip-.5cm\\noindent\n\\par\n\\caption{ \nAll solutions in this figure are for $n=9$ and, to numerical \naccuracy, linearly stable.\nData are given in Table~1.\nTop-Left:\nA periodic solution in the rotating frame, of resonance 1:10. \nTop-Right:\nThe corresponding periodic solution in the non-rotating frame, \nwhere it is a choreography. \nCenter-Left: \nA periodic solution in the rotating frame, of resonance 23:1. \nCenter-Right:\nThe corresponding choreography.\nBottom-Left:\nA partial choreography in the non-rotating frame, of resonance 2:5.\nBottom-Right: \nA partial choreography in the non-rotating frame of resonance 5:8.\n}\n\\label{fig1}\n\\end{figure}\n\\clearpage\n\\begin{figure}[ptb]\n\\begin{center}\n\\resizebox{17.0cm}{!}{  \n\\includegraphics{f2a.png}~~\n\\includegraphics{f2b.png}~~\n\\includegraphics{f2c.png} }\n\\end{center}\n\\par\n\\vskip-.90cm\\noindent\n\\par\n\\begin{center}\n\\resizebox{17.0cm}{!}{  \n\\includegraphics{f2d.png}~\n\\includegraphics{f2e.png}~\n\\includegraphics{f2f.png} }\n\\end{center}\n\\par\n\\vskip-.90cm\\noindent\n\\par\n\\begin{center}\n\\resizebox{17.0cm}{!}{  \n\\includegraphics{f2g.png}~\n\\includegraphics{f2h.png}~\n\\includegraphics{f2i.png} }\n\\end{center}\n\\par\n\\vskip-.90cm\\noindent\n\\par\n\\begin{center}\n\\resizebox{17.0cm}{!}{  \n\\includegraphics{f2j.png}~\n\\includegraphics{f2k.png}~\n\\includegraphics{f2l.png} }\n\\end{center}\n\\par\n\\vskip-.5cm\\noindent\n\\par\n\\caption{ \nThe choreographies in the first two rows are for $n=9$,\nthe choreographies in the third row are for $n=17$, \nand the choreographies in the last row are for $n=31$.\nAll but two choreographies in this Figure are stable.\nThe two unstable choreographies are on the right in rows 1 and 2.\n}\n\\label{fig2}%\n\\end{figure}\n\\clearpage\n\\vskip.0cm\\noindent\n\\begin{table}[H]\n\\begin{center}\n\\begin{large}\n\\begin{tabular}{|c|c|c||c|c|c|c|c|c|c|}\n\\hline\n      &   &   &   &       &          &         &          &         &   \\cr\nFigure&Row-&Orbit-&$n$&$T\/T_0$& a    & $T$     &   $T_0$  &$x_n(0)$ &S\/U\\cr\n      &Col.&Label&&        &          &         &          &         &   \\cr\n\\hline\n      &1-1& 1 & 9 & 1:10  & 0.651774 & 14.5773 &  145.773 & -0.04   & S \\cr\n      &1-2& 2 & 9 & 1:10  & 0.651774 & 14.5773 &  145.773 & -0.04   & S \\cr\n   1  &2-1& 3 & 9 & 23:1  & 0.657102 & 4000.00 &  173.913 &  0.005  & S \\cr\n      &2-2& 4 & 9 & 23:1  & 0.657102 & 4000.00 &  173.913 &  0.005  & S \\cr\n      &3-1& 5 & 9 & 2:5   & 0.520316 & 12.7459 &  31.8649 & -0.04   & S \\cr\n      &3-2& 6 & 9 & 5:8   & 0.396319 & 12.6334 &  20.2134 & -0.04   & S \\cr\n\\hline\n      &1-1& 7 & 9 & 1:10  & 0.647930 & 13.0635 &  130.635 & -0.04   & S \\cr\n      &1-2& 8 & 9 & 1:10  & 0.646671 & 12.6423 &  126.353 & -0.04   & S \\cr\n      &1-3& 9 & 9 & 1:10  & 0.627791 & 8.51610 &  85.1505 & -0.04   & U \\cr\n      &2-1&10 & 9 & 2:11  & 0.510285 & 5.50498 &  30.2774 & -0.04   & S \\cr\n      &2-2&11 & 9 & 2:11  & 0.531986 & 6.17839 &  33.9811 & -0.04   & S \\cr\n   2  &2-3&12 & 9 & 2:11  & 0.565906 & 7.73660 &  42.5513 & -0.04   & U \\cr\n      &3-1&13 &17 & -8:9  & 0.576588 & 28.2933 & -31.8299 &  0.005  & S \\cr\n      &3-2&14 &17 & -8:9  & 0.578005 & 28.0608 & -31.5684 &  0.005  & S \\cr\n      &3-3&15 &17 & -6:11 & 0.505528 & 28.4406 & -52.1411 &  0.005  & S \\cr\n      &4-1&16 &31 & -15:16& 0.421561 & 43.0673 & -45.9385 &  0.0001 & S \\cr\n      &4-2&17 &31 & -15:16& 0.421549 & 43.0706 & -45.9420 &  0.0001 & S \\cr\n      &4-3&18 &31 & -15:16& 0.420348 & 43.3913 & -46.2840 &  0.0001 & S \\cr\n\\hline\n\\end{tabular}\n\\end{large}\n\\caption{\nData for the orbits in Figures~1 and 2.\nHere $n$ is the number of sites,\n$a$ the amplitude parameter, \n$T$ the period of the orbit,\n$T_0$ the period of the rotating frame,\n$x_n(0)$ the $x$-component of the $n$th site at time zero,\n\"S\" stands for \"Stable or almost stable\", and\n\"U\" stands for \"Unstable\".\n}\n\\end{center}\n\\end{table}\n\n\\section*{Acknowledgments}\nThis work was supported by NSERC (Canada), BUAP and CONACYT (M\\'{e}xico).\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSpin crossover (SCO) is a phenomenon which takes place when the metal ion changes its spin state between low spin (LS) and high spin (HS) configuration under the effect of external perturbation such as pressure, magnetic field, temperature, or light irradiation. The SCO can be observed in transition metal compounds (often in the 3d-metal oxides with $d^4$-$d^7$ electronic configurations) \\cite{Halder, Brooker, LyubutinUFN, Ohkoshi} or in transition metal complexes, like metalorganic molecules or molecular assemblies \\cite{Saha-Dasgupta}. Free inertial molecular switches to store and process information in fast computational devices were the primary interest for SCO. In the nanotechnology certain properties of the SCO are of the interest for quantum transport and a new generation of sensors and displays \\cite{Jureschi}. The SCO in Fe-containing oxides is also important for the understanding the physical properties of the Earth's mantle \\cite{Wentzcovitch, Hsu, Liu, Ovchinnikov12, Sinmyo}.\n\nAt first glance the SCO is a problem of an individual ion and results from the competition of the Hund intra-atomic exchange interaction and the crystal field value determined by surrounding ions. Nevertheless, the effective interaction between magnetic ions due to electron-phonon, exchange, and quadrupole couplings results in cooperative effects, which provide different hysteresis phenomena and play an important role in practical applications and understanding the origin of the SCO. There are many papers where the cooperative effects have been treated within the Ising model \\cite{Jureschi, Wajnflasz, Bari, Nishino03, Timm, Banerjee, Paez-Espejo}. In all these studies the effective exchange interaction is postulated phenomenologically within the Ising or Heisenberg model with empirical exchange parameters. In the last decade the cooperative effects in SCO have been studied by the density functional theory \\cite{Marbeuf}, molecular dynamics \\cite{Nishino07, Boukheddaden}, and Monte Carlo simulations \\cite{Konishi, Miyashita}. The interplay of electron hopping between neighboring ions with the orbital structure of different spin multiplets also results in spin-orbital cooperative effects in strongly correlated transition metal oxides \\cite{Sboychakov}.\n\nIn conventional magnetic insulators only the ground term $E_0$ of magnetic cation in the multielectron configuration $d^n$ with some spin value $S_0$ is involved in the formation of the Heisenberg Hamiltonian as the effective low-energy model. The important difference of the magnetism in SCO systems is that at least two different terms, usually HS and LS, are involved in the formation of the effective low energy model. This is a reason for the non-Heisenberg model effects that will be discussed in this paper. Recently we have developed a general approach to construct the effective exchange interaction model that takes into account the contribution of the excited terms of the magnetic cation \\cite{Gavrichkov17} and found that the interatomic exchange interaction results in the SCO to be the first order phase transition \\cite{Nesterov}. For arbitrary $d^n$ configuration we cannot write down analytically the parameters of the effective Hamiltonian that contains the interatomic exchange as well as the interatomic hopping of excitons, the excitations between HS and LS terms.\n\nIn this paper we study more simple toy model with two electronic orbitals and the Coulomb interaction in the Kanamori approach \\cite{Kanamori}. Within the generalized tight binding (GTB) method \\cite{Ovchinnikov89, Gavrichkov00} to the electronic structure of strongly correlated systems we provide the exact diagonalization of the local intraatomic part of the Hamiltonian, construct the Hubbard operators using a set of the exact local eigenstates, and write down the total Hamiltonian as the multiorbital Hubbard model. This model describes a magnetic insulator with the energy gap $E_g$ between the occupied valence and empty conductivity bands. Two electrons per site form the HS triplet and LS singlets with the SCO at increasing the crystal field splitting between two orbitals (for example, by external pressure). We should mention that similar models under different names (the two-band Hubbard model or the extended Falicov-Kimball model) have been intensively discussed in the literature, see the review paper \\onlinecite{Kunes15}. We write down explicitly the matrix elements of the exchange and exciton hopping contributions, which are beyond the conventional Heisenberg model. The other non-Heisenberg model effect is related to a structure of the local Hilbert space, which contains for our model 3 magnetic eigenstates for HS with $S=1$ and 3 singlets with $S=0$. Within the two-band Hubbard model similar strong coupling approach \\cite{Werner, Suzuki, Nasu} has also revealed two terms with $S=1$ and $S=0$ for electronic concentration $n_e=2$ and the intersite interaction matrix elements (see also Refs.~\\onlinecite{Balents, Kaneko, Kunes14}). The main object of all these papers is the possible excitonic condensation in systems of strongly correlated electrons. In our paper we restrict our interest to the SCO systems and possible non Heisenberg effects. The presence of the additional LS states does not allow introducing the Brillouin function in the mean field (MF) approximation. A small number of electrons in our toy model ($n_e=2$ per site) allows us to study the model's phase diagram applying a cluster mean field (CMF) approach in order to go beyond the standard MF. In this way we can obtain qualitative information about the model's phase diagram and explore the validity of approximation by considering different cluster sizes as well as discuss the short-order effects, which are also different from the conventional Heisenberg model, in the vicinity of the first order transition from the HS antiferromagnetic phase into the LS non magnetic phase due to local nature of SCO.\n\nThe paper is organized as follows. In section \\ref{sec:2} we describe the two-orbital Kanamori model, the effective low energy Hamiltonian containing HS and LS states, and interatomic exchange interaction and exciton hopping. In section \\ref{sec:3} we briefly remind the CMF theory. The non-Heisenberg model and short-order effects in the vicinity of spin crossover are discussed in section \\ref{sec:4}. In section \\ref{sec:5} we discuss the main results. \n\n\\section{\\label{sec:2}Two-band Kanamori model}\n\nThe multielectron states for the $d^n$-configuration in the cubic crystal field can be obtained from the Tanabe-Sugano diagrams \\cite{Tanabe, Sugano}, which demonstrate stability of the HS terms for small value of the crystal field $10Dq$, and that the crossover of the HS and LS terms takes place for $d^4$-$d^7$ electronic configurations with increasing the crystal field value stabilizing the LS state. Beyond the crystal field theory, the SCO may also happen due to increasing the cation-anion $p$-$d$ hybridization \\cite{Ovchinnikov07}. The minimal multielectron model to discuss SCO is the two-orbital tight-binding model that includes two single electron levels $\\varepsilon_1$ and $\\varepsilon_2$ with interatomic hopping $t_{i,j}$ and the local Coulomb interaction for electron concentration $n_e=2$. Its Hamiltonian is given by\n\\begin {equation}\nH = H_t + H_{Coulomb}.\n\\label {eq:1}\n\\end {equation}\nThe interatomic term\n\\begin {eqnarray}\nH_t &=& \\varepsilon_1 \\sum\\limits_{i, \\sigma}{a_{i1\\sigma}^{\\dag}}{a_{i1\\sigma}^{}} + \\varepsilon_2 \\sum\\limits_{i, \\sigma}{a_{i2\\sigma}^{\\dag}}{a_{i2\\sigma}^{}} \\nonumber \\\\\n&+& t_1 \\sum\\limits_{\\left\\langle i, j \\right\\rangle, \\sigma} {a_{i1\\sigma}^{\\dag}}{a_{j1\\sigma}^{}} + t_2 \\sum\\limits_{\\left\\langle i, j \\right\\rangle, \\sigma} {a_{i2\\sigma}^{\\dag}}{a_{j2\\sigma}^{}} \\nonumber \\\\\n&+& t_{12} \\sum\\limits_{\\left\\langle i, j \\right\\rangle, \\sigma}\\left( {a_{i2\\sigma}^{\\dag}}{a_{j1\\sigma}^{}} + {a_{i1\\sigma}^{\\dag}}{a_{j2\\sigma}^{}} \\right)\n\\label {eq:2}\n\\end {eqnarray}\ndescribes the intraband $t_1$ and $t_2$ hoppings and the interband hopping $t_{12}$ of electrons between the nearest neighbor sites with the single electron energies $\\varepsilon_1$ and $\\varepsilon_2 = \\varepsilon_1 + \\Delta$, where $\\Delta$ is the crystal field value. The local Coulomb interaction within the Kanamori approach contains different matrix elements, the intraorbital $U$ and interorbital $V$, as well as the Hund coupling $J$ and the interband coupling $J'$:\n\\begin{widetext}\n\\begin {eqnarray}\nH_{Coulomb} = U \\sum\\limits_{i, \\lambda}{a_{i\\lambda\\uparrow}^{\\dag}}{a_{i\\lambda\\downarrow}^{\\dag}}{a_{i\\lambda\\uparrow}^{}}{a_{i\\lambda\\downarrow}^{}} + V \\sum\\limits_{i, \\lambda\\neq\\lambda'}{a_{i\\lambda\\uparrow}^{\\dag}}{a_{i\\lambda'\\downarrow}^{\\dag}}{a_{i\\lambda\\uparrow}^{}}{a_{i\\lambda'\\downarrow}^{}} + V \\sum\\limits_{i, \\lambda>\\lambda'}{a_{i\\lambda\\sigma}^{\\dag}}{a_{i\\lambda'\\sigma}^{\\dag}}{a_{i\\lambda\\sigma}^{}}{a_{i\\lambda'\\sigma}^{}} \\nonumber \\\\\n+ J \\sum\\limits_{i, \\lambda>\\lambda',\\sigma}{a_{i\\lambda\\sigma}^{\\dag}}{a_{i\\lambda'\\sigma}^{\\dag}}{a_{i\\lambda'\\sigma}^{}}{a_{i\\lambda\\sigma}^{}} + J \\sum\\limits_{i, \\lambda\\neq\\lambda'}{a_{i\\lambda\\uparrow}^{\\dag}}{a_{i\\lambda'\\downarrow}^{\\dag}}{a_{i\\lambda'\\uparrow}^{}}{a_{i\\lambda\\downarrow}^{}} + J' \\sum\\limits_{i, \\lambda\\neq\\lambda'}{a_{i\\lambda\\uparrow}^{\\dag}}{a_{i\\lambda\\downarrow}^{\\dag}}{a_{i\\lambda'\\uparrow}^{}}{a_{i\\lambda'\\downarrow}^{}} \n\\label {eq:3}\n\\end {eqnarray}\n\\end{widetext}\nIn the limit  $\\Delta=0$ and for one electron per site this model transforms in the Kugel-Khomskii model for charge ordering \\cite{Kugel82}. In this paper we will consider this model only for homopolar case $n_e=2$. As we have mentioned in the introduction, similar models have been studied recently to find the excitonic insulator phase.\n\nFor zero interatomic hopping there are 6 exact two-electron states. The triplet ($S=1$)\n\\begin{equation}\n\\left|\\sigma\\right\\rangle=\\begin{cases}\n               a^{\\dag}_{1\\uparrow}a^{\\dag}_{2\\uparrow}\\left|0\\right\\rangle, \\sigma = +1 \\\\\n               \\frac{1}{\\sqrt{2}}\\left(a^{\\dag}_{1\\uparrow}a^{\\dag}_{2\\downarrow}+a^{\\dag}_{1\\downarrow}a^{\\dag}_{2\\uparrow}\\right)\\left|0\\right\\rangle, \\sigma = 0 \\\\\n               a^{\\dag}_{1\\downarrow}a^{\\dag}_{2\\downarrow}\\left|0\\right\\rangle, \\sigma = -1\n            \\end{cases}\n\\label {eq:4}\n\\end{equation}\ntriply degenerate HS-term $\\left|\\sigma\\right\\rangle$ with the energy $E_{HS}=2\\varepsilon_1+\\Delta+V-J$ is the ground state for the crystal field  (Fig.\\ref{fig:1}, red dashed line), for $\\Delta>\\Delta_c$   the singlet ($S=0$) LS state \n\\begin{equation}\n\\left|S\\right\\rangle=C_1\\left(\\Delta\\right)a^{\\dag}_{1\\uparrow}a^{\\dag}_{1\\downarrow}\\left|0\\right\\rangle - \\sqrt{1-C^2_1\\left(\\Delta\\right)}a^{\\dag}_{2\\uparrow}a^{\\dag}_{2\\downarrow}\\left|0\\right\\rangle,\n\\label {eq:5}\n\\end{equation}\nwhere $C_1\\left(\\Delta\\right)=J'\/\\sqrt{{J'}^2 - \\left(2\\varepsilon_1+U-E_{LS}\\right)^2}$, with the energy $E_{LS}=2\\varepsilon_1+\\left(\\Delta+U\\right)-\\sqrt{\\Delta^2-{J'}^2}$ becomes the ground state (Fig.\\ref{fig:1}, green dotted line). The crossover occurs at $\\Delta=\\Delta_c=\\sqrt{\\left(U-V+J\\right)^2-{J'}^2}$. There are two more singlets,\n\\begin{equation}\n\\left|S_1\\right\\rangle=\\frac{1}{\\sqrt{2}}\\left(a^{\\dag}_{1\\uparrow}a^{\\dag}_{2\\downarrow}-a^{\\dag}_{1\\downarrow}a^{\\dag}_{2\\uparrow}\\right)\\left|0\\right\\rangle\n\\label {eq:6}\n\\end{equation}\nwith the energy $E_{S_1}=2\\varepsilon_1+\\Delta+V+J$ and \n\\begin{equation}\n\\left|S_2\\right\\rangle=\\left(\\sqrt{1-C^2_1\\left(\\Delta\\right)}a^{\\dag}_{1\\uparrow}a^{\\dag}_{1\\downarrow}+C_1\\left(\\Delta\\right)a^{\\dag}_{2\\uparrow}a^{\\dag}_{2\\downarrow}\\right)\\left|0\\right\\rangle\n\\label {eq:7}\n\\end{equation}\nwith the energy $E_{S_2}=2\\varepsilon_1+\\left(\\Delta+U\\right)+\\sqrt{\\Delta^2-{J'}^2}$, which are excited for all parameters; they are shown by the solid black lines in Fig.\\ref{fig:1}.\n\\begin{figure}\n\\includegraphics{Fig1}\n\\caption{\\label{fig:1}The crystal field dependence of the two-electron local eigenstates. The red dashed line shows the ground HS term for $\\Delta < \\Delta_c$, the green dotted line indicates the ground LS term for $\\Delta > \\Delta_c$, black solid lines correspond to the high-energy singlets. Calculation has been carried out for the following parameters: $U=3 \\text{eV}$, $V=1 \\text{eV}$, $J=0.7 \\text{eV}$, and $J=0.7 \\text{eV}$.}\n\\end{figure}\n\nTo treat the intersite electron hopping we use the GTB approach \\cite{Ovchinnikov89, Gavrichkov00, Korshunov}, which is a version of cluster perturbation theory. We introduce the Hubbard $X$-operators $X^{pq}=\\left|p\\right\\rangle\\left\\langle q\\right|$, where where $\\left|p\\right\\rangle$ and $\\left|q\\right\\rangle$ are the eigenstates of the Hamiltonian (\\ref{eq:1}) at $t_{\\lambda\\lambda'}=0$ with different numbers of electrons $n_e = 1,2,3$. A single electron creation\/annihilation operator at site $i$ with an orbital index $\\lambda$ as well as any other local operator is given by a linear combination of the Hubbard operators \\cite{Hubbard}:\n\\begin{equation}\na_{i\\lambda\\sigma}=\\sum\\limits_{pq}\\left|p\\right\\rangle\\left\\langle p\\right\\rangle a_{i\\lambda\\sigma} \\left|q\\right\\rangle \\left\\langle q\\right| = \\sum\\limits_{pq}\\gamma_{\\lambda\\sigma}\\left(pq\\right)X^{pq}_i.\n\\label {eq:8}\n\\end{equation}\nThe number of different quasiparticles $\\left(pq\\right)$ is finite, one can numerate them by the number $m$, which is the quasiparticle band index, then $a_{i\\lambda\\sigma}=\\sum\\limits_m \\gamma_{\\lambda\\sigma}\\left(m\\right)X^{m}_i$ $\\left(a^{\\dag}_{i\\lambda\\sigma}=\\sum\\limits_m \\gamma^*_{\\lambda\\sigma}\\left(m\\right){X_i^{m}}^{\\dag}\\right)$.\n\nIn the $X$-operator representation the Hamiltonian (\\ref{eq:1}) can be written exactly as \n\\begin{equation}\nH = \\sum\\limits_{i,p}E_p X^{pp}_i + \\sum\\limits_{\\left\\langle i,j\\right\\rangle}\\sum\\limits_{mn}t^{mn}{X_i^{m}}^{\\dag} X^{n}_j.\n\\label {eq:9}\n\\end{equation}\nHere $E_p$ is the energy of the term $\\left|p\\right\\rangle$, $t^{mn} = \\sum\\limits_{\\sigma,\\lambda,\\lambda'}t_{\\lambda\\lambda'}\\gamma^*_{\\lambda\\sigma}\\left(m\\right)\\gamma_{\\lambda'\\sigma}\\left(n\\right)$ is the intersite hopping matrix element. We would like to emphasize that the Hamiltonian (\\ref{eq:9}) is the general multielectron Hamiltonian that is valid for any complete and orthonormalized set of local eigenstates, all microscopic details are given by the structure of local eigenstates.\n\nFor number of electrons $n_e=2$ the Hamiltonian (\\ref{eq:9}) results in the Mott-Hubbard insulator ground state with the insulator band gap $E_g$. The localized magnetic moment at each site is HS for $\\Delta<\\Delta_c$ and LS for $\\Delta>\\Delta_c$. To obtain the interatomic exchange interaction we apply the method developed for the Hubbard model \\cite{Chao} and generalized for arbitrary set of local eigenstates in \\cite{Gavrichkov17} (see also Refs. \\onlinecite{Kunes15, Nasu}). The idea is to construct the effective Hamiltonian excluding the interband interatomic hopping. Contrary to the general case, in our toy model we can write down the exchange interaction analytically. The effective Hamiltonian is equal to\n\\begin{equation}\nH_{eff} = H_{s} + H_{ex}.\n\\label{eq:10}\n\\end{equation}\nHere the first term is the spin Heisenberg-type Hamiltonian, while the second term describes the non-Heisenberg intersite hopping of the local excitons. This Hamiltonian acts within the Hilbert space that contains four states: three $S=1$ triplet states $\\left|-\\right\\rangle$, $\\left|0\\right\\rangle$, $\\left|+\\right\\rangle$ and the singlet state $\\left|s\\right\\rangle$. The spin part is given by\n\\begin{equation}\nH_{s} = \\frac{J}{2} \\sum\\limits_{\\left\\langle i,j\\right\\rangle}\\left({\\bf S}_i {\\bf S}_j-\\frac{1}{4}n_in_j\\right)-\\varepsilon_s \\sum\\limits_i X^{ss}_i,\n\\label{eq:11}\n\\end{equation}\nwhere the superexchange parameter is \n\\begin{equation}\nJ=4\\left(t^2_{11}+2t^2_{12}+t^2_{22}\\right)\/E_g,\n\\label{eq:12}\n\\end{equation}\n${\\bf S}_i$ is the $S=1$ spin operator, in the Hubbard operators given by $S^+_i=\\sqrt{2}\\left(X^{+0}_i + X^{0-}_i\\right)$, $S^-_i=\\sqrt{2}\\left(X^{0+}_i + X^{-0}_i\\right)$, $S^z_i=\\sqrt{2}\\left(X^{++}_i - X^{--}_i\\right)$, and $n_i=q_e\\left(X^{++}_i + X^{--}_i + X^{00}_i + X^{ss}_i\\right)$ is the number of electrons operator, $q_e=2$ is the number of electrons per site, in our homopolar case the completeness of our two-electron exact set of eigenvectors looks like\n\\begin{equation}\nX^{++}_i + X^{--}_i + X^{00}_i + X^{ss}_i = 1,\n\\label{eq:13}\n\\end{equation}\nso $n_i=2$. The last term in the Hamiltonian $H_s$ (\\ref{eq:11}) is the non-Heisenberg contribution of the nonmagnetic LS state with the spin gap value $\\varepsilon_s=E_{HS}-E_{LS}$. This is the local exciton energy. Below we will assume the linear dependence of the crystal field parameter on the external pressure: $\\Delta = \\Delta(0) + aP$  due to the linear decrease of crystal volume under the pressure.\n\nThe creation\/annihilation of the local excitons is given by the Hubbard operators $X^{\\sigma s}_i$ (from the initial LS state $\\left|s\\right\\rangle$ in the final HS state $\\left|\\sigma\\right\\rangle$, and $X^{s\\sigma}_i$ corresponds to the back excitation. These excitons describe the fluctuations of multiplicity, the term used many years ago in the paper \\cite{Vonsovskii}. We consider this term is the appropriate one in the spin crossover physics, the term spin fluctuations in magnetism usually means the change of a spin projection for the same value of the spin. The second part of the effective Hamiltonian (\\ref{eq:10}) describes the intersite exciton hopping\n\\begin{eqnarray}\nH_{ex} = \\frac{J_{ex}}{2}\\sum\\limits_{\\left\\langle i,j \\right\\rangle, \\sigma} \\left[ X^{\\sigma s}_i X^{s\\sigma}_j + X^{s\\sigma}_i X^{\\sigma s}_j \\right. \\nonumber \\\\\n\\left. - \\left(-1\\right)^{\\left|\\sigma\\right|} \\left(X^{\\sigma s}_i X^{\\bar{\\sigma}s}_j + X^{s \\sigma}_i X^{s \\bar{\\sigma}}_j    \\right) \\right],\n\\label{eq:14}\n\\end{eqnarray}\nwhere the exciton hopping parameter is \n\\begin{equation}\nJ_{ex}=4\\left(t^2_{12}-t_{11}t_{22}\\right)\/E_g.\n\\label{eq:15}\n\\end{equation}\nOne can note that due to the orthogonality of the HS and LS terms they do not mix locally, but the exciton hopping mix them non locally. The first line in Eq.~\\ref{eq:14} describes the intersite single particle exciton hopping, while the second line corresponds to the creation and annihilation of the biexciton pair. We can compare the exciton hopping parameter $J_{ex}$ with similar terms in the effective low-energy models in the literature. In the paper~\\onlinecite{Kunes14} the biexciton excitation is possible only due to the interband cross-hopping matrix element $t_{12}$.In the paper~\\onlinecite{Nasu} the cross-hopping is not considered, nevertheless the biexciton hopping is possible due to the product $t_1t_2$. As we can see from Eq.~\\ref{eq:15}, we have both contributions.\n\nLet us compare two nonlocal parameters of the effective Hamiltonian (\\ref{eq:10}), the values of the exchange $J$ (\\ref{eq:12}) and exciton hopping $J_{ex}$ (\\ref{eq:15}). We consider four different sets of the electron hopping parameters:\\\\*\nA) in the limit $\\Delta = \\infty$, $t_{12}=t_{22}=0$, we get $J=4t^2_{11}\/E_g$ and $J_{ex}=0$ as in the single-band Hubbard model \\cite{Anderson}, \\\\*\nB) symmetrical hopping parameters $t_{11}=t_{22}=t_{12}=t$, then the exchange value $J=16t^2\/E_g$ is proportional to the superexchange parameter from the Hubbard model, while the exciton hopping $J_{ex}=0$,\\\\*\nC) $t_{12}=0$, then $J=4\\left(t_{11}^2+t_{22}^2\\right)\/E_g$ and $J_{ex}=-4t_{11}t_{22}\/E_g$, they have opposite signs,\\\\*\nD) $t^2_{12} \\gg t_{11}t_{22}$, then $J=8t_{12}^2\/E_g$ and $J_{ex}=4t_{12}^2\/E_g$, they are of the same order in magnitude.\\\\*\nThese examples and the general expression for the superexchange parameter $J$ demonstrate that antiferromagnetic type of superexchange takes place in our model for all electron hopping parameters, while the hopping of excitons may be positive, negative, and zero.\n\nIn the rest of the paper the unimportant term $n_i n_j=4$ for our homopolar case will be omitted from the Hamiltonian. Due to qualitative aim of our paper we will study the effects of the non-Heisenberg contributions and short-order fluctuations given by the spin part (\\ref{eq:11}) of the effective Hamiltonian (\\ref{eq:10}) with antiferromagnetic exchange parameter, neglecting the exciton dispersion given by the hopping term (\\ref{eq:14}). We will restrict ourselves by the symmetrical set B of the hopping parameters, so the exciton hopping parameter~\\ref{eq:15} will be zero. Nevertheless, basic exciton processes are still taken into account due to LS term $-\\varepsilon_s \\sum\\limits_{i} X^{ss}_i$ in the Hamiltonian (\\ref{eq:11}), which introduces some new non-Heisenberg model effects. Let us illustrate this statement using a simple example. Within MF approximation the Hamiltonian is given by \n\\begin{equation}\nH_{MF}=\\sum\\limits_{m=-1}^{1}E_m\\sigma X^{mm} - \\varepsilon_s X^{ss},\n\\label{eq:16}\n\\end{equation}\nwhere $m=-1,0,1$ are triplet states, $z$ is the number of nearest neighbors, $E_m=Jz\\sigma m$, so the 3 triplet energy levels $E_m$ are $Jz\\sigma$, 0, $-Jz\\sigma$, $\\sigma$ is the positive sublattice magnetization. Thus, the MF magnetization is\n\\begin{equation}\n\\sigma = \\frac{\\exp(\\beta J z \\sigma) - \\exp(-\\beta J z \\sigma)}{\\sum\\limits_{m=-1}^{1}{\\exp(-\\beta J z \\sigma m)} + \\exp(\\beta \\varepsilon_s)},\n\\label{eq:17}\n\\end{equation}\nwhich deviates from the Brillouin function due to the LS term. From the other hand, let us consider the exciton Green functions\n\\begin{equation}\nG^m_{ij}=\\left\\langle \\left\\langle X_i^{sm}|X_j^{ms}\\right\\rangle\\right\\rangle,\n\\label{eq:18}\n\\end{equation}\nwhich describe three types of excitons. After writing down the equations of motion and decoupling them using Tyablikov approximation, we have obtained\n\\begin{equation}\nG_m\\left(E+i\\delta\\right)=\\left(E-\\varepsilon_s+E_m+i\\delta\\right)^{-1}.\n\\label{eq:19}\n\\end{equation}\nThus, the three excitons with spin projection $m=+1,0,-1$ will have the energies $E_{ex}\\left(m\\right)=E_m-E_s$. This way, at finite temperature the occupation numbers of our HS sublevels can be found from the equation\n\\begin{equation}\nn_m=\\left(n_s - n_m\\right)f_B\\left(E_{ex}\\left(m\\right)\\right),\n\\label{eq:20}\n\\end{equation}\nwhere $f_B\\left(E\\right)$ is the Bose-Einstein distribution function. Together with the completeness condition~\\ref{eq:13} we have the full set of MF equations exactly the same as we obtain from Eq.~\\ref{eq:17}. This way, we see that in the simplest approximation the exciton process are present in the system and give consistent values for the occupation numbers. Below, instead of MF we will use its cluster generalization, in which all possible positions of singlets within the cluster are taken into account.\n\n\\section{\\label{sec:3}Cluster mean field theory}\n\nDue to the LS term, the problem given by the Hamiltonian (\\ref{eq:10}) cannot be straightforwardly treated by the approaches that work well for the Heisenberg model, like Tyablikov approximation \\cite{Bogolyubov, Fu-Cho, Valkov82, Du}, or more sophisticated Green's function approaches \\cite{Kondo, Plakida, Junger04, Junger09}. The simplest approach is to use a MF theory given by the Eq.~\\ref{eq:17}. However, the Heisenberg term contains spin fluctuations, which are neglected within the standard MF consideration. To go beyond MF we use its cluster generalization, the self-consistent CMF, which has been applied to various quantum spin models \\cite{Valkov06, Valkov07, Brzezicki11, Albuquerque, Brzezicki12, Ren, Gotfryd, Ray, Morita, Koga, Singhania}. We believe that CMF method is suitable for a qualitative study of the toy model we consider at a wide range of temperatures and pressure and it is better anyhow than the single site MF. The approach captures short-range effects, which will be discussed in the next section, and allows treating HS and LS terms equally within a cluster. We note that at high temperature close to second-order phase transition the approach can be considered as only qualitative since it does not capture long-range fluctuations. At zero temperature, as will be presented below, CMF provides results which fall into reasonable agreement with more rigorous approaches.\n\nWithin the CMF approach the lattice is covered by translations of a cluster to treat the intracluster interactions by exact diagonalization, whereas the interactions between spins $f$ and $f'$ belonging to different clusters are approximated within MF as ${\\bf S}_f{\\bf S}_f' \\approx S_f^z \\left\\langle S_f'^z\\right\\rangle + \\left\\langle S_f^z\\right\\rangle S_f'^z- \\left\\langle S_f^z\\right\\rangle \\left\\langle S_f'^z\\right\\rangle$. Thus, after applying the translational invariance the problem reduces to a single cluster in a MF determined by parameters $\\left\\langle S_i^z\\right\\rangle$, which are determined self-consistently by iterative diagonalizations ($i$ runs over boundary sites of a cluster). In our calculations we suppose the mean-fields to be in Neel antiferromagnetic ordering, since there are no competing exchange parameters, but there is a competition between the exchange and the spin gap $\\varepsilon_s$, which may be rescaled to pressure. In the main part of the paper we take $J$ as an energy unit and explore the $\\varepsilon_s-T$ phase diagram, where $T$ is temperature. For each value of $\\varepsilon_s$ and $T$ we compare the free energies of the system in magnetic and non-magnetic phases to decide, which of them is realized. A tolerance factor for convergence of $\\left\\langle S_i^z\\right\\rangle$ was set $10^{-5}$. We use full diagonalization at finite temperatures and Lanczos at $T=0$. Since we are dealing with basis consisting of three HS and one LS states, computationaly reasonable sizes of a cluster are $N_c\\lesssim10$, where $N_c$ is number of sites, in the former case and $N_c\\lesssim20$ in the latter. So, we mostly use a $2\\times2$ cluster to illustrate the main physics, but also compare the results using $3\\times2$, $4\\times2$, and $2\\times2\\times2$ clusters to study the finite-size effects of our calculations at finite temperature and clusters $4\\times3$ and $4\\times4$ at zero temperature.\n\n\\section{\\label{sec:4}Non-Heisenberg behavior and short order effects in the vicinity of spin crossover}\n\nIn the main part of this chapter we will discuss the results of our CMF calculations with the spin Hamiltonian (\\ref{eq:11}) in the most interesting regime  $\\varepsilon_s\\sim J$. To compare staggered magnetization obtained with different clusters we will consider the magnetization $m$ on a bulk site, which we define as located as close as possible to the center of a cluster. As known, Fe-based SCO compounds in ambient conditions are 3D magnets. In our cluster calculations it is more numerically practical to consider 2D case, since in 3D only $2\\times2\\times2$ cluster is available. We can use small $2\\times2$ cluster for the main results as well as compare $2\\times2$ CMF with larger clusters. Although in 2D the Mermin-Wagner theorem prohibits an ordered state for the spherically symmetric Hamiltonian (\\ref{eq:11}), in the case of MF-based approach the results for 2D and 3D are qualitatively identical.\n\nAn important quantity characterizing SCO is a HS (LS) concentration. It is accessible in experiments on X-ray emission \\cite{Lin} and $\\text{M}\\ddot{\\text{o}}\\text{ssbauer}$ spectroscopy \\cite{Lyubutin12}. We show in Fig.~\\ref{fig:2} the LS concentration $n_{LS}$ dependence on spin gap and temperature obtained by $2\\times2$ exact diagonalization. It is qualitatively similar to the obtained experimentaly in Ref.~\\cite{Lin} and calculated within MF approaches \\cite{Sturhahn, Lyubutin12} and first-principle studies \\cite{Tsuchiya}. SCO takes place at $\\varepsilon_s = 1.5$ instead of $\\varepsilon = 0$ since intracluster exchange interaction stabilizes the HS state and larger crystal field (pressure) is required to reach SCO. Another effect of correlations is the curvature of the isolines of $n_{LS}$ at low temperatures as shown by colors in Fig.~\\ref{fig:2}. If to neglect the exchange correlations and take the value $J=0$, all lines of the constant value for LS\/HS concentrations will be the straight lines going from the SCO critical point $\\varepsilon_s=0$ \\cite{Sturhahn, Ovchinnikov11}.\n\n\\begin{figure}\n\\includegraphics{Fig2}\n\\caption{\\label{fig:2}The map of the LS occupation number obtained with $2\\times2$ cluster exact diagonalization.}\n\\end{figure}\n\nAs shown in Fig.\\ref{fig:3}(a), at $\\varepsilon_s\\sim-10$ almost Heisenberg behavior of magnetization with temperature is observed, because the system is in the HS state. Thus, a second-order transition from magnetic to nonmagnetic state is realized with heating. From Fig.\\ref{fig:3}(b) one can see that for $\\varepsilon_s\\sim-10$ the population of the LS is zero at low temperature, that provides the conventional Heisenberg model behavior. The nonmagnetic HS phase is the paramagnetic one. With increasing $\\varepsilon_s$ thermal fluctuations enhance LS population, so the second-order transition Neel temperature decreases. At $\\varepsilon_s=0$ the magnetic transition with heating is still the second order, but the paramagnetic moment is reduced by approximately $20\\%$ of the LS states. At $\\varepsilon_s = \\varepsilon_s^*\\approx 1.87$ there is a tricritical point. Increasing $\\varepsilon_s$ further leads to a first-order phase transition to nonmagnetic state caused by the change of the ground state from HS to LS, as seen from Fig.~\\ref{fig:3}(b). The maximal value of magnetization in Fig.~\\ref{fig:3}(a) is $m=0.9528$, instead of $m=1$. This is the manifestation of quantum shortening of spin, which is taken into account partially within CMF by calculating spin-fluctuation terms within a cluster. The non magnetic phase of Fig.\\ref{fig:3}(a) can be qualitatively viewed as HS to the left of the $n_{LS}=0.5$ dashed line, which comes out close to the tricritical point, and LS to the right.\n\nThe distribution of LS density in Fig.\\ref{fig:3}(b) is related to the Curie constant in paramagnetic susceptibility\n\\begin{equation}\nC = \\mu^2\\left(1-n_{LS}\\right)S\\left(S+1\\right),\n\\label{eq:21}\n\\end{equation}\nwhere $\\mu = \\frac{\\mu_B^2}{3k_B}$. The temperature dependence of $C$ is shown in Fig.~\\ref{fig:4} for different values of the spin gap. Equation (\\ref{eq:16}) makes sense for the paramagnetic phase above the Neel temperature indicated in Fig.~\\ref{fig:4}(a) by dashed lines. Using parameters extracted from the anvil-cell experiments on ferropericlase \\cite{Lyubutin12, Lyubutin13} we can estimate the corresponding values of pressure $P$ by assuming that the spin gap defines pressure as $\\varepsilon_s-\\varepsilon_s^c= \\alpha_\\Delta(P-P_c)$, where $\\alpha_\\Delta=7.8\\text{meV}\/\\text{GPa}$, the critical pressure $P_c$ is $55\\text{GPa}$ and taking into account the pressure dependence of the exchange integral is $J\\left(P\\right)=J_0\\left(1+\\frac{2\\alpha_t}{t}P\\right)$, where $J_0$ is taken to be $18\\text{K}$ and $\\frac{2\\alpha_t}{t}=0.0122 1\/\\text{GPa}$. This way, for each value of $\\varepsilon_s$ we show corresponding pressure values $\\frac{\\Delta P}{P_c} = \\frac{\\left(P-P_c\\right)}{P_c}$. Note that within this set of parameters the exchange integral value is chosen to reproduce the real compound's Neel temperature and the critical pressure is aligned with our critical value of the spin gap for a more convenient qualitative discussion of our results in a context of experimental data as discussed below. Few percent below the critical pressure there is simply a drop of an effective magnetic moment with temperature. Around percent below $P_c$ an effective magnetic moment is almost temperature independent. Very close to critical pressure the LS component at the Neel temperature is already significant and thermal fluctuations lead mainly to increase of the HS component. Above the critical pressure, as shown in Fig.~\\ref{fig:4}(b), increasing pressure leads to slowdown in temperature growth of an effective magnetic moment.\n\n\\begin{figure}\n\\includegraphics{Fig3}\n\\caption{\\label{fig:3}(a) Average staggered magnetization $m$ and (b) LS occupation number obtained with $2\\times2$ CMF. The arrow shows the position of a tricritical point. The dashed line is the $n_{LS} = 0.5$ isoline.}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics{Fig4}\n\\caption{\\label{fig:4}Temperature dependence of the Curie constant defined by Eq.\\ref{eq:16} for different values of spin gap (pressure) obtained with $2\\times2$ CMF (a) below, (b) above the critical pressure. The dashed lines indicate the values of the Neel temperature for the data of the same color, $\\mu^2$ of Eq.~\\ref{eq:16} is set equal to one.}\n\\end{figure}\n\nTo explore finite-size effects of our CMF calculations we now turn to comparison of magnetization obtained within different clusters and within the Tyablikov approximation (or RPA) in the Heisenberg limit. Within the Heisenberg model RPA is known to provide results in a decent agreement with numerically exact quantum Monte Carlo \\cite{Junger09, Yasuda05}. From Fig.~\\ref{fig:5} it is seen that inclusion of nearest correlations leads to an appearance of zero fluctuations in $m$ and a substantial decrease in Neel temperature when comparing MF with $2\\times2$ CMF. At zero temperature the bulk magnetization seems to gradually approach the RPA value 0.8168, for example for $4 \\times 3 $ (not shown) and $4 \\times 4$ clusters we obtain $m=0.886$ and $m=0.88$.  In 2D the Neel temperature is zero in RPA, since it satisfies to the Mermin-Wagner theorem, unlike (C)MF, where the symmetry of the cluster's (site's) Hamiltonian is lowered artificially. Analogous comparison in 3D is shown in Fig.~\\ref{fig:6}: the Neel temperature is approximately 1.5 times higher within MF that within RPA and 1.33 times higher with $2\\times 2\\times 2$ CMF. This way, in terms of staggered magnetization's and Neel temperature's values we obtain intermediate results between RPA and MF. In our CMF calculations in the 2D case the bulk site magnetization $m(N_c)$ as a function of the number of sites  turned to be proportional to $\\sqrt{N_c}$. Least square extrapolation gave the result $m_{\\infty}\\approx0.81$, which is similar to the RPA value (see Fig.~\\ref{fig:7}).\n\n\\begin{figure}\n\\includegraphics{Fig5}\n\\caption{\\label{fig:5} Bulk site's magnetization calculated in the Heisenberg limit in 2D within MF, CMF with different rectangular clusters, and RPA.}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics{Fig6}\n\\caption{\\label{fig:6}The same as in Fig.~\\ref{fig:5} in 3D within MF, $2\\times2\\times2$ CMF, and RPA.}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics{Fig7}\n\\caption{\\label{fig:7}Extrapolation to the infinite system limit for the bulk magnetization $m(N_c)$ and the critical value of the spin gap $\\varepsilon^c_s(N_c)$ in the Heisenberg limit.}\n\\end{figure}\n\n\\begin{table}\n\\caption{\\label{tab:1}\nTricritical $\\varepsilon^*_s$ and critical $\\varepsilon^c_s$ values of the spin gap for different clusters within CMF.}\n\\begin{ruledtabular}\n\\begin{tabular}{c|cccccc}\n&\n\\multicolumn{1}{c}{\\textrm{$MF$}}&\n\\multicolumn{1}{c}{\\textrm{$2\\times2$}}&\n\\multicolumn{1}{c}{\\textrm{$3\\times2$}}&\n\\multicolumn{1}{c}{\\textrm{$4\\times2$}}&\n\\multicolumn{1}{c}{\\textrm{$4\\times3$}}&\n\\multicolumn{1}{c}{\\textrm{$4\\times4$}}\\\\\n\\hline\n$\\varepsilon^*_s $ & $\\approx 1.59$ & $\\approx 1.87$ & $\\approx 1.93$ & $\\approx 1.98$ & ---\\footnote{Have not been calculated for this cluster.} & ---\\\\\n$\\varepsilon^c_s $ & 2 & 2.148 & 2.175 & 2.189 & 2.217 & 2.232\\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\nNext, we compare average staggered magnetization obtained with different clusters and MF at different values of spin gap in Fig.~\\ref{fig:8}. Phase diagrams obtained within different clusters are very similar. Besides the decrease in Neel temperature there is an increase in tricritical value of a spin gap $\\varepsilon^*_s$ and the critical value $\\varepsilon^c_s$, at which the first-order phase transition occurs, as it is shown in Table~\\ref{tab:1}. The increase in $\\varepsilon^c_s$ with cluster's size is related to the lowering of the cluster's ground state energy in magnetic phase with increasing size, because the main competition is between states with 0 and $N_c$ singlets per cluster. Similarly to the case of magnetization, we observe $1\/\\sqrt{N_c}$ behavior of $\\varepsilon^c_s(N_c)$ or the ground-state energy $E_0$ with opposite sign in the Heisenberg limit (see Fig.~\\ref{fig:7}). By least squares extrapolation for $E_0(N_c)$ we found $E_0(\\infty) \\approx -2.31$, which is similar to the value $E_0(\\infty) \\approx -2.33$ from the quantum Monte Carlo \\cite{Harada} and density matrix renormalization group \\cite{Ramos} studies. The size dependence of $\\varepsilon^*_s$ and $\\varepsilon^c_s$ shows the most crucial change when going from MF to 4-site CMF with predictable behavior when increasing the system size. Thus, the part of the phase diagram obtained at finite temperature close to the first order transition with small clusters from 4 to 8 sites can be considered as semi-quantitative.\n\n\\begin{figure}\n\\includegraphics{Fig8}\n\\caption{\\label{fig:8} Bulk site's magnetization obtained within (a) MF, (b) $2\\times 2$, (c) $3\\times 2$, and (d) $4\\times 2$ CMF. The black line shows MF second-order transition line. Arrows show the position of a tricritical point.}\n\\end{figure}\n\nAlthough within standard MF approach qualitatively correct magnetic phase diagram is obtained, it provides no information about short-range correlations in the system. In Fig.~\\ref{fig:9} we show transverse antiferromagnetic nearest-neighbor spin correlations $C_{\\bot}=-\\left\\langle \\left(S^+_0S^-_1 + S^-_0S^+_1\\right)\\right\\rangle$ and longitudinal ones $C_{\\parallel}=-\\left\\langle S^z_0S^z_1\\right\\rangle$.  At $\\varepsilon_s<\\varepsilon^c_s$ the longitudinal correlations are always decreasing with temperature, but transverse ones are increasing with temperature at low values of spin gap, reaching maximum at Neel points and lowering in a paramagnetic phase. A non-Heisenberg effect is that at $\\varepsilon_s>\\varepsilon^c_s$ the spin correlations show a reentrant behavior. At low temperature they are zero, then increasing with heating due to thermal excitement of triplet states. When temperature is increased further, the correlations lower again.\n\n\\begin{figure}\n\\includegraphics{Fig9}\n\\caption{\\label{fig:9}(a) Transverse $C_{\\bot}$ and (b) longitudinal $C_{\\parallel}$ nearest-neighbor spin correlations, obtained within $2\\times 2$ CMF.}\n\\end{figure}\n\nFinally, we use parameters from the anvil-cell experiments on ferropericlase (Mg,Fe)O ~\\cite{Lyubutin12, Lyubutin13} used above to model its magnetization dependence on pressure and temperature. The exchange parameter value and its linear pressure dependence at low pressure in the HS state were obtained by fitting the experimental data from the paper~\\onlinecite{Lyubutin12}. The magnetization's phase diagram is presented in Fig.~\\ref{fig:10}(a). Heisenberg behavior is realized in a broad range of pressure, where the Neel temperature scales linearly with pressure and reaches its maximum. At $P\\approx P_c$ the Neel temperature drops discontinuously to zero due to a phase transition of the first order. Deviation from Heisenberg behavior is realized at $P \\gtrsim 51 \\text{GPa}$ at $T=0$ and at $P \\gtrsim 45 \\text{GPa}$ at room temperatures, as it is seen from spin correlations in Fir.~\\ref{fig:10}(b). The non magnetic phase can be qualitatively identified as HS to the left of the black line, which denotes $50\\%$ of maximal effective magnetic moment, and LS to the right. Our phase diagram is consistent with experimental data and model calculations of Refs.~\\cite{Lyubutin12, Lyubutin13}. This shows that the microscopic Hamiltonian we have studied is capable of capturing the main physics of spin crossover in ferropericlase.\n\n\\begin{figure}\n\\includegraphics{Fig10}\n\\caption{\\label{fig:10} (a) Average sublattice magnetization $m$ calculated for ferropericlase parameters from Ref.~\\cite{Lyubutin12} by $2\\times 2$ CMF. The black line is the $n_{LS} = 0.5$ isoline. (b) Transverse spin correlations for the same set of parameters.}\n\\end{figure}\n\n\\section{\\label{sec:5}Discussion}\n\nTo sum up, in order to study non-Heisenberg effects due to SCO we have derived an effective Hamiltonian for the two-orbital Kanamori model. The parameters of the effective Hamiltonian have been written down analytically. It contains HS and LS states, and interatomic exchange interaction, as well as the exciton hopping and the biexciton creation and annihilation processes. As it can be seen within simple MF, due to the presence of LS states the MF magnetization within this model is not described by the Brillouin function. The effective Hamiltonian has been studied within CMF approximation. As we have shown by comparing our results between different cluster sizes and to other methods in the special case, our results are of qualitative character at high temperatures, but we expect them to be semi-quantitative within an interesting region close to first-order transition. We have obtained a magnetic $\\varepsilon_s-T$ phase diagram of the model with antiferromagnetic and paramagnetic phases. At very low spin gap  values $\\varepsilon_s$  the magnetization's temperature dependence is almost Heisenberg-like. Increasing $\\varepsilon_s$ leads to reduction of the Neel temperature and paramagnetic moments (or the Curie constant in the paramagnetic susceptibility) due to thermal population of LS states. Up to a tricritical point $\\varepsilon^*_s$ the phase transition line is second-order one and from $\\varepsilon^*_s$ to a critical value of quantum phase transition $\\varepsilon^c_s$ it is first-order. Few percent below $\\varepsilon_s$ there occurs a drastic change in the temperature dependence of the Curie constant in paramagnetic susceptibility. At $\\varepsilon_s>\\varepsilon^c_s$ the magnetic moment and the Curie constant are zero at zero temperature and they increase with heating because of growing population of HS states. From quantitative point of view we expect our results for the magnetic phase diagram to be between simple MF (closer to MF) and RPA, which has not been rigorously developed yet in the case when LS states must be taken into account. However, we have shown that the results of CMF calculations shall approach correct values with further increase in cluster's size, thus showing predictable behavior. Using cluster approach has allowed us to predict another non-Heisenberg effect, which is a reentrant behavior of the temperature dependence of spin correlation functions at $\\varepsilon_s>\\varepsilon^c_s$. For the $P-T$ magnetic phase diagram that we have obtained for ferropericlase the non-Heisenberg behavior is realized at $P \\gtrsim 51 \\text{GPa}$ at $T=0$ and $P \\gtrsim 45 \\text{GPa}$ at $T\\approx300K$, which is a realistic pressure and temperature interval for a more detailed experimental investigation of this compound and for observing the non-Heisenberg effects.\n\n\\begin{acknowledgments}\nThe authors thank the Russian Scientific Foundation for the financial support under the grant 18-12-00022.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Fixed-Period Problems: The Sublinear Case}\n\nIt is widely accepted that an era of \nBig Bang nucleosynthesis (hereafter; BBN) between cosmic temperatures\n$T\\sim 1$ MeV and $T\\sim 10$ keV is the production site of the bulk of $^4$He \nand $^3$He,\na good fraction of $^7$Li, and essentially all $^2$H \nobserved in nebulae and stars in\nthe present universe. Generally good \nagreement between observationally inferred \nprimordial abundances and theoretical predictions for $^2$H, $^4$He, and \n$^7$Li may be\nobtained within a standard homogeneous BBN scenario with baryon-to-photon\nratio $\\eta$ in the range $2\\times 10^{-10} - 6\\times\n10^{-10}$~\\cite{BBN} \n(most likely\ntowards the upper end of this range), \nthough details of the observational determination \nof primordial abundances and the question about consistency within a\nstandard BBN scenario remain under investigation.\nPrimordial conditions as envisioned in a standard BBN scenario yield only\nproduction of minute amounts of isotopes with nucleon number\n$A>7$. For a baryon-to-photon\nratio of $\\eta\\approx 4\\times 10^{-10}$ synthesis of a $^{12}$C \nmass fraction of only\n${\\rm X_{^{12}{\\rm C}}}\\approx 5\\times 10^{-15}$ results. \nThere is also production of trace amounts\nof $^6$Li, $^9$Be, and $^{11}$B, with typical mass fractions $\\sim 10^{-16} -\n10^{-13}$. \nIn contrast to stellar nucleosynthesis, the three-body\ntriple alpha process is not operative in standard BBN, \nmainly due to the small densities ($\\sim 10^{-5}$g\/cm$^3$ at $T\\approx 100$\nkeV) \nand short expansion time ($\\sim 100$ s). A potentially different way of bridging \nbetween light and \\lq\\lq heavy\\rq\\rq\\ ($A\\geq 12$) \nelements across the mass eight\ngap is via the reaction sequence $^7$Be $\\to$ $^{11}$C $\\to$ \n$^{11}$B $\\to$ $^{12}$C. This reaction\nchain is, nevertheless,\nineffective in a standard BBN since build-up of $^7$Be occurs late at \n$T\\sim 50-100$ keV where Coulomb barrier effects \nprevent further processing of this isotope\ninto heavier ones.\n\nOne may pose the following question: {\\it Which alternative BBN scenarios\nwould give substantial initial metal production, without violating\nobservational constraints on the light elements?} Here \\lq\\lq metal\\rq\\rq \nrefers to isotopes with nucleon number $A\\geq 12$ and by the term \\lq\\lq\nsubstantial\\rq\\rq\\ it is meant that production would result in abundance which\ncould either be directly observable in the not-to-distant future within the \natmospheres of metal-poor stars, \nor would be sufficient for the operation of the CNO\ncycle within the first stars (${\\rm X_{CNO}}>10^{-10}$). The only scenario for \npre-stellar production of metals known to the author \nis a BBN scenario characterized by an inhomogeneous baryon distribution.\nSuch inhomogeneities in the baryon-to-photon ratio may result from the\nout-of-equilibrium dynamics of the early universe prior to the BBN era, \nas possibly during a\nfirst-order QCD phase transition ($T\\approx 100$ MeV) or an era of \ninhomogeneous\nelectroweak baryogenesis ($T\\approx 100$ GeV). Another possibility may be the\ncreation of baryon \\lq\\lq lumps\\rq\\rq via the evaporation of baryon\nnumber carrying solitons formed in the early universe, \nsuch as strange quark matter nuggets or B-balls. Given the wide variety of\npossibilities and the large intrinsic uncertainties of \nscenarios for the generation\nof baryon number fluctuations in the early universe, it is probably best to initially\nregard the question about primordial abundance yields and possible metallicity\nproduction in \ninhomogeneous BBN scenarios disconnected from the speculative\nmechanism for\nthe creation of baryon number fluctuations. \n\nIn the past, there have been a number of investigations of the evolution of\npre-existing $\\eta$ fluctuations from high cosmic temperatures ($T\\approx\n100$ GeV)\nthrough the epoch of BBN~\\cite{EVO}, and of the resulting BBN\nabundance yields~\\cite{IBBN}.\nBaryon number fluctuations only impact BBN yields if they survive\ncomplete dissipation by neutron diffusion\nbefore the epoch of weak freeze-out at $T\\approx 1$ MeV. \nThis is the case for fluctuations containing baryon number in excess\nof $N_b\\, ^>_{\\sim}\\, 10^{35}(\\eta_h\/10^{-4})^{-1\/2}$, \nwhere $\\eta_h$ is the  baryon-to-photon ratio within the fluctuation,\ncorresponding to a baryon mass of $M_b\\, ^>_{\\sim}\\, 10^{-22}M_{\\odot}\n(\\eta_h\/10^{-4})^{-1\/2}$. \nFor reference, the baryon mass\ncontained within the QCD- and electroweak horizon are \n$\\sim 10^{-8}M_{\\odot}$ and $\\sim 10^{-18}M_{\\odot}$, respectively.\nThe evolution of fluctuations after the epoch of weak freeze-out,\nfor a large part of parameter space, is \ncharacterized by differential neutron-proton diffusion resulting in \nhigh-density, proton-rich\nand low-density, neutron-rich regions. \nIt had been suggested that such scenarios may not only\nresult in consistency between\nobservationally inferred and theoretically predicted primordial abundances\nfor much larger horizon-average baryon-to-photon ratios than $\\eta\\approx\n4\\times 10^{-10}$, but also lead to a possible\nefficient r-process nucleosynthesis~\\cite{Apple} \nin the neutron-rich regions. When all\ndissipative and hydrodynamic processes during inhomogeneous BBN are \ntreated properly, one finds that, except for a fairly small parameter\nspace, the\naverage $\\eta$ in inhomogeneous BBN scenarios may not be larger than in\nstandard BBN due to overabundant $^7$Li and\/or $^4$He \nsynthesis~\\cite{Result}. A detailed\ninvestigation of possible r-process nucleosynthesis in the neutron-rich regions\nshows that an r-process is utterly ineffective in inhomogeneous \nBBN~\\cite{r-process}. \nThis is mostly due to r-process yields being a sensitive function of the local \nbaryon-to-photon ratio\nin the neutron-rich region, which generically is too low in inhomogeneous BBN.\n\nNevertheless, BBN metal synthesis via $p$- and $\\alpha$-burning reactions\nof cosmologically interesting magnitude\nmay occur if a small fraction, $f_b$, of the cosmic baryons reside in regions \nwith very high baryon-to-photon ratio, \n$\\eta_h\\geq 10^{-4}$~\\cite{metals}. Whereas\ninhomogeneous BBN scenarios are typically plagued by overproduction of\n$^7$Li,\nthe comparatively early production of $^7$Be \n(which would decay into $^7$Li) in regions with\n$\\eta_h\\geq 10^{-4}$ allows for a further processing of $^7$Be \ninto heavier isotopes.\nThe net nucleosynthesis of such regions are a $^4$He mass fraction $Y_p\\approx\n0.36$, some metals, and virtually no $^2$H, $^3$He, and $^7$Li. \nA baryon-to-photon ratio $\\eta\\approx 10^{-4}$ is interesting as it presents\nthe asymptotic value attained after partial dissipation \nof initially very high-density regions\n$\\eta_i\\gg 10^{-4}$ by the action of \nneutrino heat conduction at temperature $T\\gg\n1$ MeV, for a wide part of parameter space. \nBaryon \\lq\\lq lumps\\rq\\rq\\ with initial baryon-to-photon ratio\n$\\eta_i$ and baryonic mass less than $M_b\\leq 10^{-14}M_{\\odot}\n\\eta_i$ will have evolved to an asymptotic $\\eta_f\\approx 10^{-4}$\nby the time of weak freeze-out.  They are further almost unaffected by neutron\ndiffusion during BBN provided their mass exceeds $M_b \\, ^>_{\\sim}\\, \n10^{-19}M_{\\odot}$. \nIn this case horizon-averaged BBN abundance yields may be determined by\nan appropriate average of the yields of two homogeneous universes, one at\n$\\eta_h\\approx 10^{-4}$ and one at $\\eta_l$, close to the observationally \ninferred value for a standard BBN scenario. The resultant cosmic \naverage metallicity in such an inhomogeneous scenario\nmay be approximated by~\\cite{metals}\n\\begin{equation}\n\\nonumber\n[{\\rm Z}]\\sim -6.5 + {\\rm log}_{10}(f_b\/10^{-2}) + 2\\,\n{\\rm log}_{10}(\\eta_h\/10^{-4})\\, ,\n\\end{equation}\nwhere [Z] denotes the logarithm of ${\\rm X}_{A\\geq 12}$ \nrelative to the solar value\n(${\\rm X}_{\\odot}\\approx 2\\times 10^{-2}$). \nThe most stringent constraint on\nthe fraction of baryons residing in high $\\eta$ regions comes from a possible\noverproduction of $^4$He, with a change $\\Delta Y_p\\approx 0.12f_b$ relative to\na homogeneous BBN at $\\eta_l$, allowing for $f_b\\sim 10^{-2}-3\\times 10^{-2}$,\nand [Z]$\\sim -6$, possibly even larger for $\\eta_h > 10^{-4}$.\nGiven $\\eta_h\\approx 10^{-4}$, a nucleosynthetic signature is given\nby [O\/C] $\\sim 1.3$\nand [C\/$Z_{A\\geq 28}]\\sim -1.5$. \nFor $\\eta_h\\gg 10^{-4}$, synthesis of mostly iron-group elements results.\n\nIn conclusion, it is possible, though not necessarily probable, that the universe\n\\lq\\lq started\\rq\\rq\\ with some initial metallicity if a small fraction of\nthe cosmic baryon number existed in high-density regions at the epoch of BBN,\nThis metallicity may be large enough to have the first stars operate \non the CNO-cycle, even though it probably would fall\nbelow the metallicities observed in the currently \nmost metal-poor stars ([Z] $\\sim -5$) known.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\n\nA recent body of research seeks to understand the acceleration\nphenomena of first-order discrete optimization methods by means of\nmodels that evolve in continuous time.  Roughly speaking, the idea is\nto study the behavior of ordinary differential equations (ODEs) which\narise as continuous limits of discrete-time accelerated\nalgorithms. The basic premise is that the availability of the powerful\ntools of the continuous realm, such as differential calculus, Lie\nderivatives, and Lyapunov stability theory, can be then brought to\nbear to analyze and explain the accelerated behavior of these flows,\nproviding insight into their discrete counterparts.  Fully closing the\ncircle to provide a complete description of the acceleration\nphenomenon requires solving the outstanding open question of how to\ndiscretize the continuous flows while retaining their accelerated\nconvergence properties.  However, the discretization of accelerated\nflows has proven to be a challenging task, where retaining\nacceleration seems to depend largely on the particular ODE and the\ndiscretization method employed.  This paper develops a resource-aware\napproach to the discretization of accelerated optimization flows.\n\n\n\\subsubsection*{Literature Review}\n\n\nThe acceleration phenomenon goes back to the seminal\npaper~\\cite{BTP:64} introducing the so-called heavy-ball method, which\nemployed momentum terms to speed up the convergence of the classical\ngradient descent method.\nThe heavy-ball method achieves optimal convergence rate in a\nneighborhood of the minimizer for arbitrary convex functions and\nglobal optimal convergence rate for quadratic objective functions.\nLater on, the work~\\cite{YEN:83} proposed the Nesterov's accelerated\ngradient method and, employing the technique of estimating sequences,\nshowed that it converges globally with optimal convergence rate for\nconvex and strongly-convex smooth functions.  The algebraic nature of\nthe technique of estimating sequences does not fully explain the\nmechanisms behind the acceleration phenomenon, and this has motivated\nmany approaches in the literature to provide fundamental understanding\nand insights. These include coupling dynamics~\\cite{ZAZ-LO:17},\ndissipativity theory~\\cite{BH-LL:17}, integral quadratic\nconstraints~\\cite{LL-BR-AP:16,BVS-RAF-KML:18}, and geometric\narguments~\\cite{SB-YTL-MS:15}.\n\nOf specific relevance to this paper is a recent line of research\ninitiated by~\\cite{WS-SB-EJC:16} that seeks to understand the\nacceleration phenomenon in first-order optimization methods by means\nof models that evolve in continuous time.~\\cite{WS-SB-EJC:16}\nintroduced a second-order ODE as the continuous limit of Nesterov's\naccelerated gradient method and characterized its accelerated\nconvergence properties using Lyapunov stability analysis.  The ODE\napproach to acceleration now includes the use of Hamiltonian dynamical\nsystems~\\cite{MB-MJ-AW:18,CJM-DP-YWT-BO-AD:18}, inertial systems with\nHessian-driven damping~\\cite{HA-ZC-JF-HR:19}, and high-resolution\nODEs~\\cite{BS-SSD-MIJ-WJS:18-arxiv,BS-JG-SK:20}.  This body of\nresearch is also reminiscient of the classical dynamical systems\napproach to algorithms in optimization, see~\\cite{RWB:91,UH-JBM:94}.\nThe question of how to discretize the continuous flows while\nmaintaining their accelerated convergence rates has also attracted\nsignificant attention, motivated by the ultimate goal of fully\nunderstanding the acceleration phenomenon and taking advantage of it\nto design better optimization algorithms.  Interestingly,\ndiscretizations of these ODEs do not necessarily lead to\nacceleration~\\cite{BS-SSD-MIJ-WJS:19-arxiv}.  In fact, explicit\ndiscretization schemes, like forward Euler, can even become\nnumerically unstable after a few iterations~\\cite{AW-ACW-MIJ:16}.\nMost of the discretization approaches found in the literature are\nbased on the study of well-known integrators, including symplectic\nintegrators~\\cite{MB-MJ-AW:18,AW-LM-AW:19}, Runge-Kutta\nintegrators~\\cite{JZ-AM-SS-AJ:18} or modifications of Nesterov's\n\\emph{three sequences}~\\cite{AW-ACW-MIJ:16,AW-LM-AW:19,ACW-BR-MIJ:18}.\nOur previous work~\\cite{MV-JC:19-nips} instead developed a\nvariable-stepsize discretization using zero-order holds and\nstate-triggers based on the derivative of the Lyapunov function of the\noriginal continuous flow. Here, we provide a comprehensive approach\nbased on powerful tools from resource-aware control, including\nperformance-based triggering and state holds that more effectively use\nsampled information.  Other recent approaches to the acceleration\nphenomena and the synthesis of optimization algorithms using\ncontrol-theoretic notions and techniques include~\\cite{ASK-PME-TK:18},\nwhich employs hybrid systems to design a continuous-time dynamics with\na feedback regulator of the viscosity of the heavy-ball ODE to\nguarantee arbitrarily fast exponential convergence,\nand~\\cite{DH-RGS:19}, which introduced an algorithm which alternates\nbetween two (one fast when far from the minimizer but unstable, and\nanother slower but stable around the minimizer) continuous heavy-ball\ndynamics.\n\n\\subsubsection*{Statement of Contributions}\n\nThis paper develops a resource-aware control framework to the\ndiscretization of accelerated optimization flows that fully exploits\ntheir dynamical properties.  Our approach relies on the key\nobservation that resource-aware control provides a principled way to\ngo from continuous-time control design to real-time implementation\nwith stability and performance guarantees by opportunistically\nprescribing when certain resource should be employed.  In our\ntreatment, the resource to be aware of is the last sampled state of\nthe system, and hence what we seek to maximize is the stepsize of the\nresulting discrete-time algorithm.\nOur first contribution is the introduction of a second-order\ndifferential equation which we term heavy-ball dynamics with displaced\ngradient. This dynamics generalizes the continuous-time heavy-ball\ndynamics analyzed in the literature by evaluating the gradient of the\nobjective function taking into account the second-order nature of the\nflow.  We establish that the proposed dynamics retains the same\nconvergence properties as the original one while providing additional\nflexibility in the form of a design parameter.\n\nOur second contribution uses trigger design concepts from\nresource-aware control to synthesize criteria that determine the\nvariable stepsize of the discrete-time implementation of the\nheavy-ball dynamics with displaced gradient.  We refer to these\ncriteria as event- or self-triggered, depending on whether the\nstepsize is implicitly or explicitly defined.  We employ derivative-\nand performance-based triggering to ensure the algorithm retains the\ndesired decrease of the Lyapunov function of the continuous flow. In\ndoing so, we face the challenge that the evaluation of this function\nrequires knowledge of the unknown optimizer of the objective\nfunction. To circumvect this hurdle, we derive bounds on the evolution\nof the Lyapunov function that can be evaluated without knowledge of\nthe optimizer.  We characterize the convergence properties of the\nresulting discrete-time algorithms, establishing the existence of a\nminimum inter-event time and performance guarantees with regards to\nthe objective function. \n\nOur last two contributions provide ways of exploiting the sampled\ninformation to enhance the algorithm performance. Our third\ncontribution provides an adaptive implementation of the algorithms\nthat adaptively adjusts the value of the gradient displacement\nparameter depending on the region of the space to which the state\nbelongs.  Our fourth and last contribution builds on the fact that the\ncontinuous-time heavy-ball dynamics can be decomposed as the sum of a\nsecond-order linear dynamics with a nonlinear forcing term\ncorresponding to the gradient of the objective function. Building on\nthis observation, we provide a more accurate hold for the\nresource-aware implementation by using the samples only on the\nnonlinear term, and integrating exactly the resulting linear system\nwith constant forcing.  We establish the existence of a minimum\ninter-event time and characterize the performance with regards to the\nobjective function of the resulting high-order-hold algorithm.\nFinally, we illustrate the proposed optimization algorithms in\nsimulation, comparing them against the heavy-ball and Nesterov's\naccelerated gradient methods and showing superior performance to other\ndiscretization methods proposed in the literature.\n\n\\section{Preliminaries}\\label{sec:preliminaries}\nThis section presents basic notation and preliminaries.\n\n\\subsection{Notation}\\label{assumptions}\nWe denote by ${\\mathbb{R}}$ and $\\mathbb{R}_{>0}$ the sets of real and positive\nreal numbers, resp.  All vectors are column vectors.  We denote their\nscalar product by $\\langle \\cdot,\\cdot\\rangle$.  We use $\\norm{\\cdot{\n  }}$ to denote the $2$-norm in Euclidean space. Given $\\mu \\in\n\\mathbb{R}_{>0}$, \na continuously differentiable function $f$ is $\\mu$-strongly convex if\n$f(y) - f(x) \\geq\\langle \\nabla f(x), y - x\\rangle +\n\\frac{\\mu}{2}\\norm{x - y}^2$ for $x$, $y \\in {\\mathbb{R}}^n$.  Given $L \\in\n\\mathbb{R}_{>0}$ and a function $f:X \\rightarrow Y$ between two normed spaces\n$(X,\\norm{\\cdot{}}_X)$ and ($Y,\\norm{\\cdot{}}_Y$), $f$ is\n$L$-Lipschitz if $\\norm{f(x) - f(x')}_{Y}\\leq L\\norm{x - x'}_{X}$ for\n$x$, $x' \\in X$. \nThe functions we consider here are continuously differentiable,\n$\\mu$-strongly convex and have $L$-Lipschitz continuous gradient.  We\nrefer to the set of functions with all these properties\nby~$\\mathcal{S}_{\\mu,L}^1({\\mathbb{R}}^n)$.  A function\n$f:{\\mathbb{R}}^n\\rightarrow{\\mathbb{R}}$ is positive definite relative to $x_*$ if\n$f(x_*)=0$ and $f(x)>0$ for $x\\in{\\mathbb{R}}^n\\setminus\\{x_*\\}$.\n\n\\subsection{Resource-Aware Control}\\label{sec:resource-aware}\n\nOur work builds on ideas from resource-aware control to develop\ndiscretizations of continuous-time accelerated flows. Here, we provide\na brief exposition of its basic elements and refer\nto~\\cite{WPMHH-KHJ-PT:12,CN-EG-JC:19-auto} for further details.\n\nGiven a controlled dynamical system $\\dot{p} = X(p,u)$, with $p \\in\n{\\mathbb{R}}^n$ and $u \\in {\\mathbb{R}}^m$, assume we are given a stabilizing\ncontinuous state-feedback $\\map{\\mathfrak{k}}{{\\mathbb{R}}^n}{{\\mathbb{R}}^m}$ so that the\nclosed-loop system $\\dot{p} = X(p,\\mathfrak{k}(p))$ has $p_*$ as a globally\nasymptotically stable equilibrium point. Assume also that a Lyapunov\nfunction $\\map{V}{{\\mathbb{R}}^n}{{\\mathbb{R}}}$ is available as a certificate of\nthe globally stabilizing nature of the controller. Here, we assume\nthis takes the form\n\\begin{equation}\\label{eq:lyapunov_decay}\n  \\dot{V}  =  \\langle \\nabla V(p) X(p,\\mathfrak{k}(p)) \\rangle  \\leq - \\frac{\\sqrt{\\mu}}{4} V (p), \n \n\\end{equation}\nfor all $p \\in {\\mathbb{R}}^n$.  Although exponential decay of $V$ along the\nsystem trajectories is not necessary, we restrict our attention to\nthis case as it arises naturally in our treatment.\n\nSuppose we are given the task of implementing the controller signal\nover a digital platform, meaning that the actuator cannot be\ncontinuously updated as prescribed by the specification\n$u=\\mathfrak{k}(p)$. In such case, one is forced to discretize the\ncontrol action along the execution of the dynamics, while making sure\nthat stability is still preserved.  A simple-to-implement approach is\nto update the control action \\emph{periodically}, i.e., fix $h >0$,\nsample the state as $\\{p(kh)\\}_{k=0}^\\infty $ and implement\n\\begin{align*}\n  \\dot{p}(t) = X(p(t),\\mathfrak{k}(p(kh))) , \\quad t \\in [kh,(k +1)h].\n\\end{align*}\nThis approach requires $h$ to be small enough to ensure that $V$\nremains a Lyapunov function and, consequently, the system remains\nstable.  By contrast, in \\emph{resource-aware control}, one employs\nthe information generated by the system along its trajectory to update\nthe control action in an opportunistic fashion.  Specifically, we seek\nto determine in a state-dependent fashion a sequence of times\n$\\{t_k\\}_{k=0}^\\infty$, not necessarily uniformly spaced, such that\n$p_*$ remains a globally asymptotically stable equilibrium for the\nsystem\n\\begin{align}\\label{eq:triggered-design}\n  \\dot{p}(t) &= X(p(t),\\mathfrak{k}(p(t_k))) , \\quad t \\in\n  [t_k,t_{k+1}] .\n\\end{align}\nThe main idea to accomplish this is to let the state sampling be\nguided by the principle of maintaining the same type of exponential\ndecay~\\eqref{eq:lyapunov_decay} along the new dynamics.  To do this,\none defines triggers to ensure that this decay is never violated by\nprescribing a new state sampling. Formally, one sets $t_0 = 0$ and $\nt_{k + 1} = t_k + \\operatorname{step} (p(t_k))$, where the stepsize is defined by\n\\begin{align}\\label{eq:step-generic}\n  \\operatorname{step} (\\hat{p}) &= \\min \\setdef{t > 0}{b(\\hat {p},t) = 0}.\n\\end{align}\nWe refer to the criteria as \\emph{event-triggering} or\n\\emph{self-triggering} depending on whether the evaluation of the\nfunction $b$ requires monitoring of the state $p$ along the trajectory\nof~\\eqref{eq:triggered-design} (ET) or just knowledge of its initial\ncondition $\\hat{p}$ (ST). The more stringent requirements to implement\nevent-triggering lead to larger stepsizes versus the more conservative\nones characteristic of self-triggering.  In order for the state\nsampling to be implementable in practice, the inter-event times\n$\\{t_{k + 1} - t_k\\}_{k=0}^\\infty$ must be uniformly lower bounded by\na positive minimum inter-event time, abbreviated MIET. In particular,\nthe existence of a MIET rules out the existence of Zeno behavior,\ni.e., the possibility of an infinite number of triggers in a finite\namount of time.\n\nDepending on how the evolution of the function $V$ is examined, we\ndescribe two types of triggering conditions\\footnote{In both cases,\n  for a given $z \\in {\\mathbb{R}}^n$, we let $p(t;\\hat{p})$ denote the\n  solution of $\\dot{p}(t) = X(p(t),\\mathfrak{k}(\\hat{p}))$ with initial condition\n  $p(0) = \\hat{p}$.}:\n\\begin{LaTeXdescription}\n\\item[Derivative-based trigger:] In this case, $b^{\\operatorname{d}}$ is defined as\n  an upper bound of the expression $\\frac{d}{dt} V(p(t;\\hat{p})) +\n  \\frac{\\sqrt{\\mu}}{4} V(p(t;\\hat{p}))$.  This definition ensures\n  that~\\eqref{eq:lyapunov_decay} is maintained\n  along~\\eqref{eq:triggered-design};\n\n\\item[Performance-based trigger:] In this case, $b^{\\operatorname{p}}$ is defined as an\n  upper bound of the expression $V(p(t;\\hat{p})) - e^{-\\frac{\\sqrt{\\mu}}{4}\n    t}V(\\hat{p})$. Note that this definition ensures that the integral\n  version of~\\eqref{eq:lyapunov_decay} is maintained\n  along~\\eqref{eq:triggered-design}.\n\\end{LaTeXdescription}\nIn general, the performance-based trigger gives rise to stepsizes that\nare at least as large as the ones determined by the derivative-based\napproach, cf.~\\cite{PO-JC:18-cdc}. This is because the latter\nprescribes an update as soon as the exponential decay is about to be\nviolated, and therefore, does not take into account the fact that the\nLyapunov function might have been decreasing at a faster rate since\nthe last update.  Instead, the performance-based approach reasons over\nthe \\emph{accumulated decay} of the Lyapunov function since the last\nupdate, potentially yielding longer inter-sampling times.\n\n\nA final point worth mentioning is that, in the event-triggered control\nliterature, the notion of \\emph{resource} to be aware of can be many\ndifferent things, beyond the actuator described above, including the\nsensor, sensor-controller communication, communication with other\nagents, etc. This richness opens the way to explore more elaborate\nuses of the sampled information beyond the zero-order hold\nin~\\eqref{eq:triggered-design}, something that we also leverage later\nin our presentation.\n\n\n\n\\section{Problem Statement}\\label{se:problem-statement}\n\nOur motivation here is to show that principled approaches to\ndiscretization can retain the accelerated convergence properties of\ncontinuous-time dynamics, fill the gap between the continuous and\ndiscrete viewpoints on optimization algorithms, and lead to the\nconstruction of new ones. Throughout the paper, we focus on the\ncontinuous-time version of the celebrated heavy-ball\nmethod~\\cite{BTP:64}.  Let $f$ be a function in\n$\\mathcal{S}_{\\mu,L}^1({\\mathbb{R}}^n)$ and let $x_*$ be its unique\nminimizer.  The heavy-ball method is known to have an optimal\nconvergence rate in a neighborhood of the minimizer. For its\ncontinuous-time counterpart, consider the following $s$-dependent\nfamily of second-order differential equations, with $s \\in \\mathbb{R}_{>0}$,\nproposed in~\\cite{BS-SSD-MIJ-WJS:18-arxiv},\n\\begin{subequations}\\label{eq:continuous-hb-dynamics}\n  \\begin{align}\n    \\begin{bmatrix}\n      \\dot{x}  \\\\\n      \\dot{v}\n    \\end{bmatrix}\n    & =\n    \\begin{bmatrix}\n      v\n      \\\\\n      - 2\\sqrt{\\mu}v - (1+\\sqrt{\\mu s})\\nabla f(x))\n    \\end{bmatrix},\n    \\\\\n    x(0) & =x_0, \\quad  v(0)=-\\frac{2\\sqrt{s}\\nabla\n      f(x_0)}{1+\\sqrt{\\mu s}} .\n    \\label{eq:initial-state}\n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n  \\end{align}\n\\end{subequations}\nWe refer to this dynamics as~$X_{\\operatorname{hb}}$. The following result\ncharacterizes the convergence properties\nof~\\eqref{eq:continuous-hb-dynamics} to $p_*=[x_*, 0]^T$.\n\n\\begin{theorem}[\\cite{BS-SSD-MIJ-WJS:18-arxiv}]\\label{th:hb}\n  Let $\\map{V}{{\\mathbb{R}}^n \\times {\\mathbb{R}}^n}{{\\mathbb{R}}}$ be\n  \\begin{align}\\label{eq:continuous-hb-lyapunov}\n    V(x,v) & = (1+\\sqrt{\\mu s})(f(x) - f(x_*)) +\n    \\displaystyle\\frac{1}{4}\\norm{v}^2 \\nonumber\n    \\\\\n    & \\quad + \\displaystyle\\frac{1}{4}\\norm{v +2\\sqrt{\\mu}(x-x_*)}^2,\n  \\end{align}\n  which is positive definite relative to $[x_*, 0]^T$. Then $\\dot{V}\n  \\leq-\\frac{\\sqrt{\\mu}}{4}V$ along the\n  dynamics~\\eqref{eq:continuous-hb-dynamics} and, as a consequence,\n  $p_*=[x_*, 0]^T$ is globally asymptotically stable.  Moreover, for\n  $s\\leq 1\/L$, the exponential decrease of $V$ implies\n  \\begin{equation}\\label{eq:decay_fun}\n    f(x(t))-f(x_*)\\leq\n    \\frac{7\\norm{x(0) -\n        x_*}^2}{2s}e^{-\\frac{\\sqrt{\\mu}}{4}t} .\n  \\end{equation}\n\\end{theorem}\n\nThis result, along with analogous\nresults~\\cite{BS-SSD-MIJ-WJS:18-arxiv} for the Nesterov's accelerated\ngradient descent, serves as an inspiration to build Lyapunov\nfunctions that help to explain the accelerated convergence rate of the\ndiscrete-time methods.\n\n\n\nInspired by the success of resource-aware control in developing\nefficient closed-loop feedback implementations on digital systems,\nhere we present a discretization approach to accelerated optimization\nflows using resource-aware control.  At the basis of the approach\ntaken here is the observation that the convergence\nrate~\\eqref{eq:decay_fun} of the continuous flow is a direct\nconsequence of the Lyapunov nature of the\nfunction~\\eqref{eq:continuous-hb-lyapunov}. In fact, the integration\nof $\\dot{V} \\leq-\\frac{\\sqrt{\\mu}}{4}V$ along the system trajectories\nyields\n\\begin{equation*}\n  V(x(t),v(t)) \\leq\n  e^{-\\frac{\\sqrt{\\mu}}{4}t} V(x(0),v(0)).\n\\end{equation*} \nSince $ f(x(t)) - f(x_*) \\leq V(x(t),v(t))$, we deduce\n\\begin{align*}\n  f(x(t)) - f(x_*) \\leq e^{-\\frac{\\sqrt{\\mu}}{4}t} V(x(0),v(0)) =\n  \\mathcal{O}(e^{-\\frac{\\sqrt{\\mu}}{4}t}) .\n\\end{align*}\nThe characterization of the convergence rate via the decay of the\nLyapunov function is indeed common among accelerated optimization\nflows.  This observation motivates the resource-aware approach to\ndiscretization pursued here, where the resource that we aim to use\nefficiently is the sampling of the state itself. By doing so, the\nultimate goal is to give rise to large stepsizes that take maximum\nadvantage of the decay of the Lyapunov function (and consequently of\nthe accelerated nature) of the continuous-time dynamics in the\nresulting discrete-time implementation.\n\n\n\\section{Resource-Aware Discretization of Accelerated Optimization\n  Flows}\\label{sec:performance-based}\n\nIn this section we propose a discretization of accelerated\noptimization flows using state-dependent triggering and analyze the\nproperties of the resulting discrete-time algorithm.  For convenience,\nwe use the shorthand notation $p = [x,v]^T$.  In following with the\nexposition in Section~\\ref{sec:resource-aware}, we start by\nconsidering the zero-order hold implementation $\\dot p = X_{\\operatorname{hb}}\n(\\hat{p})$, $p(0) = \\hat{p}$ of the heavy-ball\ndynamics~\\eqref{eq:continuous-hb-dynamics},\n\\begin{subequations}\\label{eq:forward-euler}\n  \\begin{align}\n    \\dot{x} & = \\hat{v},\n    \\\\\n    \\dot{v} & = -2\\sqrt{\\mu}\\hat{v} - (1 + \\sqrt{\\mu s}) \\nabla\n    f(\\hat{x}).\n  \\end{align}  \n\\end{subequations}\nNote that the solution trajectory takes the form $p(t) = \\hat{p} + t\nX_{\\operatorname{hb}} (\\hat p)$, which in discrete-time terminology corresponds to a\nforward-Euler discretization\nof~\\eqref{eq:continuous-hb-dynamics}. Component-wise, we have\n\\begin{align*}\n  x(t) & = \\hat{x} + t\\hat{v},\n  \\\\\n  v(t) & =\\hat{v} -t \\big( 2\\sqrt{\\mu}\\hat{v} + (1 + \\sqrt{\\mu s})\n  \\nabla f(\\hat{x}) \\big).\n\\end{align*}\n\nAs we pointed out in Section~\\ref{sec:resource-aware}, the use of\nsampled information opens the way to more elaborated constructions than\nthe zero-order hold in~\\eqref{eq:forward-euler}. As an example, given\nthe second-order nature of the heavy-ball dynamics, it would seem\nreasonable to leverage the (position, velocity) nature of the pair\n$(\\hat{x},\\hat{v})$ (meaning that, at position $\\hat{x}$, the system\nis moving with velocity $\\hat{v}$) by employing the modified zero-order\nhold\n\\begin{subequations}\\label{eq:forward-euler-a}\n  \\begin{align}\n    \\dot{x} & = \\hat{v},\n    \\\\\n    \\dot{v} & = -2\\sqrt{\\mu}\\hat{v} - (1 + \\sqrt{\\mu s}) \\nabla\n    f(\\hat{x} + a \\hat{v}),\n  \\end{align}  \n\\end{subequations}\nwhere $a \\ge 0$. Note that the trajectory\nof~\\eqref{eq:forward-euler-a} corresponds to the forward-Euler\ndiscretization of the continuous-time dynamics\n\\begin{align}\\label{eq:continuous-hb-dynamics_a}\n  \\begin{bmatrix}\n    \\dot{x}  \\\\\n    \\dot{v}\n  \\end{bmatrix}\n  & =\n  \\begin{bmatrix}\n    v\n    \\\\\n    - 2\\sqrt{\\mu}v - (1+\\sqrt{\\mu s})\\nabla f(x + av))\n  \\end{bmatrix},\n\\end{align}\nWe refer to this as the {\\it heavy-ball dynamics with displaced\n  gradient} and denote it by~$X^a_{\\operatorname{hb}}$ (note\nthat~\\eqref{eq:forward-euler-a}\nand~\\eqref{eq:continuous-hb-dynamics_a} with $a=0$\nrecover~\\eqref{eq:forward-euler}\nand~\\eqref{eq:continuous-hb-dynamics}, respectively). In order to\npursue the resource-aware approach laid out in\nSection~\\ref{sec:resource-aware} with the modified zero-order hold\nin~\\eqref{eq:forward-euler-a}, we need to characterize the asymptotic\nconvergence properties of the heavy-ball dynamics with displaced\ngradient, which we tackle next.\n\n\\begin{remark}\\longthmtitle{Connection between the use of sampled\n    information and high-resolution-ODEs} {\\rm A number of\n    works~\\cite{ML-AMD:19,MM-MIJ:19,IS-JM-GD-GH:13} have explored\n    formulations of Nesterov's accelerated that employ\n    displaced-gradient-like terms similar to the one used above. Here,\n    we make this connection explicit. Given Nesterov's algorithm\n   \n    \\begin{align*}\n      y_{k+1} & = x_k -s \\nabla f(x_k)\n      \\\\\n      x_{k+1} &= y_{k+1} + \\displaystyle\\frac{1-\\sqrt{\\mu s}}{1 +\n        \\sqrt{\\mu s}}(y_{k+1} - y_k)\n    \\end{align*}\n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n    the work~\\cite{BS-SSD-MIJ-WJS:18-arxiv} obtains the following\n    limiting high-resolution ODE\n   \n   \n   \n   \n    \\begin{equation}\\label{hr-ODE}\n      \\ddot{x} + 2\\sqrt{\\mu}\\dot{x} +\n      \\sqrt{s}\\nabla^2 f(x)\\dot{x} + (1+\\sqrt{\\mu s})\\nabla f(x) = 0.\n    \\end{equation}\n    Interestingly, considering instead the evolution of the $y$-variable\n   \n   \n \n \n \n \n \n \n \n \n  and applying similar arguments to the ones\n  in~\\cite{BS-SSD-MIJ-WJS:18-arxiv}, one instead obtains\n  \\begin{equation}\\label{hr-ODE_new}\n    \\ddot{y} + 2\\sqrt{\\mu}\\dot{y} + (1+\\sqrt{\\mu s})\\nabla f \\big(y +\n    \\frac{\\sqrt{s}}{1 +\\sqrt{\\mu s}}\\dot{y}\\big) = 0 , \n  \\end{equation}\n  which corresponds to the continuous heavy-ball dynamics\n  in~\\eqref{eq:continuous-hb-dynamics} evaluated with a displaced\n  gradient, i.e.,~\\eqref{eq:continuous-hb-dynamics_a}. Even further,\n  if we Taylor expand the last term in~\\eqref{hr-ODE_new} as\n  \\[\n  \\nabla f(y + \\displaystyle\\frac{\\sqrt{s}}{1 +\\sqrt{\\mu s}}\\dot{y}) =\n  \\nabla f(y) + \\nabla^2 f(y) \\displaystyle\\frac{\\sqrt{s}}{1\n    +\\sqrt{\\mu s}}\\dot{y} + \\mathcal{O}(s)\n  \\]\n  and disregard the $\\mathcal{O}(s)$ term, we recover~\\eqref{hr-ODE}.\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  This shows that~\\eqref{hr-ODE_new} is just~\\eqref{hr-ODE} with extra\n  higher-order terms in $s$, and provides evidence of the role of\n  gradient displacement in enlarging the modeling capabilities of\n  high-resolution ODEs.}  \\relax\\ifmmode\\else\\unskip\\hfill\\fi\\oprocendsymbol\n\\end{remark}\n\n\n\\subsection{Asymptotic Convergence of Heavy-Ball Dynamics with\n  Displaced Gradient}\\label{sec:hbdg-asymptotic}\n\nIn this section, we study the asymptotic convergence of heavy-ball\ndynamics with displaced gradient.  Interestingly, for $a$ sufficiently\nsmall, this dynamics enjoys the same convergence properties as the\ndynamics~\\eqref{eq:continuous-hb-dynamics}, as the following result\nshows.\n\n\\begin{theorem}\\longthmtitle{Global asymptotic stability  of\n    heavy-ball dynamics with\n    displaced gradient}\\label{th:continuous-sampled}\n  Let $\\beta_1,\\dots,\\beta_4>0$ be\n  \\begin{align*}\n    \\beta_1 &= \\sqrt{\\mu_s}\\mu, \\quad \\beta_2 =\n    \\displaystyle\\frac{\\sqrt{\\mu_s} L}{\\sqrt{\\mu}},\n    \\\\\n    \\beta_3 &= \\frac{13\\sqrt{\\mu}}{16}, \\quad \\beta_4 = \\frac{4\n      \\mu^{2}\\sqrt{s} + 3 L \\sqrt{\\mu} \\sqrt{\\mu_s} }{8 L^2},\n   \n   \n   \n   \n   \n   \n   \n  \\end{align*}\n  where, for brevity, $\\sqrt{\\mu_s} = 1 + \\sqrt{\\mu s}$,\n  and define\n  \\begin{align}\\label{eq:a1}\n    a^*_1 = \\frac{2}{\\beta_2^2} \\Big( \\beta_1 \\beta_4 +\n    \\sqrt{\\beta_2^2 \\beta_3 \\beta_4 + \\beta_1^2 \\beta_4^2} \\Big) .\n  \\end{align}\n  Then, for $0 \\leq a \\leq a^*_1$, $\\dot{V}\n  \\leq-\\frac{\\sqrt{\\mu}}{4}V$ along the\n  dynamics~\\eqref{eq:continuous-hb-dynamics_a} and, as a consequence,\n  $p_*=[x_*, 0]^T$ is globally asymptotically stable. Moreover, for\n  $s\\leq 1\/L$, the exponential decrease of $V$\n  implies~\\eqref{eq:decay_fun} holds along the trajectories\n  of~$X^a_{\\operatorname{hb}}$.\n\\end{theorem}\n\\begin{proof\n  Note that\n  \\begin{align*}\n    & \\langle \\nabla V(p), X^a_{\\operatorname{hb}}(p) \\rangle +\n    \\frac{\\sqrt{\\mu}}{4} V(p) =\n    \\\\\n    & = (1 \\!+\\! \\sqrt{\\mu s})\\langle \\nabla f(x),v \\rangle \\!-\\!\n    \\sqrt{\\mu}\\norm{v}^2 \\!-\\! \\sqrt{\\mu_s}\\langle \\nabla f(x \\!+\\! a\n    v),v \\rangle\n    \\\\\n    & \\quad - \\sqrt{\\mu}\\sqrt{\\mu_s}\\langle \\nabla f(x + av),x - x_*\n    \\rangle + \\frac{\\sqrt{\\mu}}{4} V(x,v)\n    \\\\\n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n    & = \\underbrace{-\\sqrt{\\mu}\\norm{v}^2 - \\sqrt{\\mu}\\sqrt{\\mu_s}\\langle \\nabla f(x), x - x_* \\rangle +\\frac{\\sqrt{\\mu}}{4}\n      V(x,v)}_{\\textrm{Term~I}}\n    \\\\\n    & \\quad \\underbrace{-\\sqrt{\\mu_s}\\langle \\nabla f(x + av) -\n      \\nabla f(x),v \\rangle}_{\\textrm{Term~II}}\n    \\\\\n    & \\quad \\underbrace{ -\\sqrt{\\mu}\\sqrt{\\mu_s}\\langle \\nabla\n      f(x + av) - \\nabla f(x),x - x_*\\rangle}_{\\textrm{Term~III}},\n  \\end{align*}\n  where in the second equality, we have added and subtracted\n  $\\sqrt{\\mu} \\sqrt{\\mu_s}\\langle \\nabla f(x ),x - x_*\n  \\rangle$. Observe that ``Term~I'' corresponds to $\\langle \\nabla\n  V(p),X_{\\operatorname{hb}}(p)\\rangle + \\frac{\\sqrt{\\mu}}{4}V(p)$ and\n  is therefore negative by\n  Theorem~\\ref{th:hb}. From~\\cite{MV-JC:19-nips}, this term can be\n  bounded as\n  \\begin{align*}\n    \\textrm{Term~I} & \\leq \\frac{-13\\sqrt{\\mu}}{16} \\norm{v}^2\n    \\\\\n    & \\quad+ \\Big(\\frac{4 \\mu^{2}\\sqrt{s}+3 L \\sqrt{\\mu}\\sqrt{\\mu_s}}{8\n      L^2} \\Big)\\norm{\\nabla f(x)}^2.\n   \n   \n   \n   \n   \n   \n   \n  \\end{align*}\n  Let us study the other two terms.  By strong convexity, we have $ -\n  \\langle \\nabla f(x + av) - \\nabla f(x), v \\rangle \\leq - a \\mu\n  \\norm{v}^2$, and therefore\n  \\begin{align*}\n    \\textrm{Term~II} & \\leq -a\\sqrt{\\mu_s}\\mu \\norm{v}^2 \\le 0 .\n  \\end{align*}\n \n \n \n \n \n \n \n \n \n \n  Regarding Term~III, one can use the $L$-Lipschitzness of $\\nabla f$\n  and strong convexity to obtain\n  \\begin{align*}\n    \\textrm{Term~III} & \\leq \\displaystyle\\frac{a}{\\mu}\\sqrt{\\mu} \\sqrt{\\mu_s} L\\norm{v}\\norm{\\nabla f(x)}.\n  \\end{align*}\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  Now, using the notation in the statement, we can write\n  \\begin{align}\\label{eq:a_condition}\n    & \\langle \\nabla V(p), X^a_{\\operatorname{hb}}(p) \\rangle +\n    \\frac{\\sqrt{\\mu}}{4} V(p)\n    \\\\\n    & \\leq a \\big(\\!-\\!\\beta_1\\norm{v}^2 + \\beta_2\\norm{v}\\norm{\\nabla\n      f(x)} \\big) \\!-\\! \\beta_3\\norm{v}^2 \\!-\\!\\beta_4\\norm{\\nabla\n      f(x)}^2 . \\notag\n  \\end{align}\n  If $-\\beta_1\\norm{v}^2 + \\beta_2\\norm{v}\\norm{\\nabla f(x)} \\leq 0$,\n  then the RHS of~\\eqref{eq:a_condition} is negative for any $a \\geq\n  0$. If $-\\beta_1\\norm{v}^2 + \\beta_2\\norm{v}\\norm{\\nabla f(x)} > 0$,\n  the RHS of~\\eqref{eq:a_condition} is negative if and only if \n  \\[\n  a \\leq \\displaystyle\\frac{\\beta_3\\norm{v}^2 + \\beta_4\\norm{\\nabla\n      f(x)}^2}{-\\beta_1\\norm{v}^2 + \\beta_2\\norm{v}\\norm{\\nabla\n      f(x)}}. \n  \\]\n  The RHS of this equation corresponds to $g({\\norm{\\nabla\n      f(x)}}\/{\\norm{\\nabla v}})$, with the function $g$ defined\n  in~\\eqref{eq:g}. From Lemma~\\ref{lemma:bound}, as long as\n  $-\\beta_1\\norm{v}^2 + \\beta_2\\norm{v}\\norm{\\nabla f(x)} > 0$, this\n  function is lower bounded by\n  \\begin{align*}\n    a_1^* = \\frac{\\beta_3 + \\beta_4(z_{\\root}^+)^2}{-\\beta_1 + \\beta_2\n      z_{\\root}^+} >0 ,\n  \\end{align*}\n  where $z_{\\root}^+$ is defined in~\\eqref{eq:zroot}. This exactly\n  corresponds to~\\eqref{eq:a1}, concluding the\n  result.\n\\end{proof}\n\n\\begin{remark}\\longthmtitle{Adaptive displacement along the\n    trajectories of heavy-ball dynamics with displaced\n    gradient}\\label{rem:a-over-region}\n  {\\rm From the proof of Theorem~\\ref{th:continuous-sampled}, one can\n    observe that if $(x,v)$ is such that $\\underline{n} \\leq\n    \\norm{\\nabla f(x)} < \\overline{n}$ and $\\underline{m} \\leq\n    \\norm{v} < \\overline{m}$, for $\\underline{n}, \\overline{n},\n    \\underline{m}, \\overline{m} \\in \\mathbb{R}_{>0}$, then one can upper\n    bound the LHS of~\\eqref{eq:a_condition}~by\n    \\[\n    a (-\\beta_1 \\underline{m}^2 + \\beta_2 \\overline{m} \\, \\overline{n}) -\n    \\beta_3 \\underline{m}^2 - \\beta_4 \\underline{n}^2.\n    \\]\n    If $-\\beta_1 \\underline{m}^2 + \\beta_2 \\overline{m} \\,\n    \\overline{n} \\le 0$, any $a \\ge 0$ makes this expression negative.\n    If instead $ -\\beta_1 \\underline{m}^2 + \\beta_2 \\overline{m} \\,\n    \\overline{n} \\geq 0$, then $a$ must satisfy\n    \\begin{align}\\label{eq:adaptive-a}\n      a \\leq \\Big| \\frac{\\beta_3 \\underline{m}^2 + \\beta_4\n        \\underline{n}^2}{-\\beta_1 \\underline{m}^2 + \\beta_2\n        \\overline{m} \\, \\overline{n}} \\Big| .\n    \\end{align}\n    This argument shows that over the region $R =\n    \\setdef{(x,v)}{\\underline{n} \\leq \\norm{\\nabla f(x)} <\n      \\overline{n} \\text{ and } \\underline{m} \\leq \\norm{v} <\n      \\overline{m}}$, any $a \\ge 0$ satisfying~\\eqref{eq:adaptive-a}\n    ensures that $\\dot V \\le -\\frac{\\sqrt{\\mu}}{4} V$, and hence the\n    desired exponential decrease of the Lyapunov function. This\n    observation opens the way to modify the value of the parameter $a$\n    adaptively along the execution of the heavy-ball dynamics with\n    displaced gradient, depending on the region of state space visited\n    by its trajectories.  } \\relax\\ifmmode\\else\\unskip\\hfill\\fi\\oprocendsymbol\n\\end{remark}\n\n\\subsection{Triggered Design of Variable-Stepsize\n  Algorithms}\\label{sec:trigger-design}\n\n\nIn this section we propose a discretization of the continuous\nheavy-ball dynamics based on resource-aware control. To do so, we\nemploy the approaches to trigger design described in\nSection~\\ref{sec:resource-aware} on the dynamics~$X^a_{\\operatorname{hb}}$, whose\nforward-Euler discretization corresponds to the modified zero-order\nhold~\\eqref{eq:forward-euler-a} of the heavy-ball dynamics.\n\nOur starting point is the characterization of the asymptotic\nconvergence properties of~$X^a_{\\operatorname{hb}}$ developed in\nSection~\\ref{sec:hbdg-asymptotic}.  The trigger design necessitates of\nbounding the evolution of the Lyapunov function $V$\nin~\\eqref{eq:continuous-hb-lyapunov} for the continuous-time\nheavy-ball dynamics with displaced gradient along its zero-order hold\nimplementation. However, this task presents the challenge that the\ndefinition of $V$ involves the minimizer $x_*$ of the optimization\nproblem itself, which is unknown (in fact, finding it is the ultimate\nobjective of the discrete-time algorithm we seek to design). In order\nto synthesize computable triggers, this raises the issue of bounding\nthe evolution of $V$ as accurately as possible while avoiding any\nrequirement on the knowledge of~$x_*$.  The following result, whose\nproof is presented in Appendix~\\ref{app:appendix},\naddresses this point.\n\n\\begin{proposition}\\longthmtitle{Upper bound for derivative-based\n    triggering with zero-order hold}\\label{prop:upper-bound-derivative}\n  Let $a \\ge 0$ and\n  define\n  \\begin{align*}\n    b^{\\operatorname{d}}_{\\operatorname{ET}}(\\hat{p}, t;a) &= A_{\\operatorname{ET}}(\\hat{p}, t;a) +\n    B_{\\operatorname{ET}}(\\hat{p}, t;a) + C_{\\operatorname{ET}}(\\hat{p};a),\n    \\\\\n    b^{\\operatorname{d}}_{\\operatorname{ST}}(\\hat{p}, t;a) &= B^q_{\\operatorname{ST}}(\\hat{p};a)t^2 +\n    (A_{\\operatorname{ST}}(\\hat{p};a) + B^l_{\\operatorname{ST}}(\\hat{p};a))t\n    \\\\\n    & \\quad + C_{\\operatorname{ST}}(\\hat{p};a),\n  \\end{align*}\n  where\n  \\begin{align*}\n    & A_{\\operatorname{ET}}(\\hat{p}, t;a) = 2 \\mu t \\norm{\\hat{v}}^2 +\n    \\sqrt{\\mu_s}\\big(\\langle \\nabla f(\\hat{x}+t \\hat{v}) - \\nabla\n    f(\\hat{x}), \\hat{v} \\rangle\n    \\\\\n    &\\quad + 2t \\sqrt{\\mu} \\langle \\nabla f(\\hat{x} + a \\hat{v}),\n    \\hat{v} \\rangle + t\\sqrt{\\mu_s} \\norm{\\nabla f(\\hat{x} + a\n      \\hat{v})}^2 \\big),\n    \\\\\n   \n    & B_{\\operatorname{ET}}(\\hat{p}, t;a) = \\frac{\\sqrt{\\mu}t^2}{16} \\norm{2\n      \\sqrt{\\mu} \\hat{v} + \\sqrt{\\mu_s}\\nabla f(\\hat{x} + a \\hat{v})}^2\n    \\\\\n    & \\quad - \\frac{t\\mu}{4} \\norm{\\hat{v}}^2\n   \n   \n    + \\frac{\\sqrt{\\mu}\\sqrt{\\mu_s}}{4}\\big(\n    f(\\hat{x} + t \\hat{v}) - f(\\hat{x}) +\n    \\\\\n    &\\quad  -t \\langle \\hat{v}, \\nabla f(\\hat{x}+a\n    \\hat{v}) \\rangle\n   \n   \n    + \\frac{ t^2 \\sqrt{\\mu_s}}{4}\\norm{\\nabla f(\\hat{x} +\n      a \\hat{v})}^2\n    \\\\\n    &\\quad - \\frac{t \\sqrt{\\mu}}{L} \\norm{\\nabla f(\\hat{x} + a\n      \\hat{v})}^2 \n   \n   \n    +t\\sqrt{\\mu} \\langle a\\hat{v} , \\nabla\n    f(\\hat{x} + a\\hat{v})\\rangle\\big),\n    \\\\\n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n    & C_{\\operatorname{ET}}(\\hat{p};a) = -\\frac{13\\sqrt{\\mu}}{16}\\norm{\\hat{v}}^2\n    -\\frac{\\mu^2\\sqrt{s}}{2}\\displaystyle\\frac{\\norm{\\nabla\n        f(\\hat{x})}^2}{L^2}\n    \\\\\n    & \\quad + \\sqrt{\\mu_s}\\big(\\frac{-3\\sqrt{\\mu}}{8L}\\norm{\\nabla\n      f(\\hat{x})}^2\n    \\\\\n    & \\quad + \\sqrt{\\mu}(f(\\hat x) - f(\\hat{x} + a\\hat{v})) +\n    \\sqrt{\\mu}\\norm{\\nabla f(\\hat{x})}\\norm{a \\hat{v}}\n    \\\\\n    & \\quad -\\frac{\\mu^{3\/2}}{2}\\norm{a\\hat{v}}^2 -\\langle \\nabla\n    f(\\hat{x} + a\\hat{v}) - \\nabla f (\\hat{x}) , \\hat{v}\\rangle\n    \\\\\n    & \\quad + \\sqrt{\\mu}\\langle \\nabla f(\\hat{x} + a\\hat{v}), a\\hat{v}\n    \\rangle\\big) ,\n    \\\\\n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n    & A_{\\operatorname{ST}}(\\hat{p};a) = 2 \\mu \\norm{\\hat{v}}^2+ \\sqrt{\\mu_s}\n    \\big(L\\norm{\\hat{v}}^2 + 2 \\sqrt{\\mu} \\langle \\nabla f(\\hat{x} + a\n    \\hat{v}), \\hat{v} \\rangle\n    \\\\\n    &\\quad + \\sqrt{\\mu_s} \\norm{\\nabla f(\\hat{x} + a \\hat{v})}^2\\big),\n    \\\\\n   \n    & B^l_{\\operatorname{ST}}(\\hat{p};a) = \\frac{\\sqrt{\\mu}}{4}\\big(\n    -\\sqrt{\\mu}\\norm{\\hat{v}}^2 + \\sqrt{\\mu_s}( \\langle \\nabla\n    f(\\hat{x})- \\nabla f(\\hat{x} + a\\hat{v}),\\hat{v} \\rangle\n   \n    \\\\\n    & \\quad - \\frac{\\sqrt{\\mu}}{L}\\norm{\\nabla\n      f(\\hat{x} + a \\hat{v})}^2\n   \n   \n    + \\sqrt{\\mu}\\langle a\\hat{v} , \\nabla\n    f(\\hat{x} + a\\hat{v})\\rangle)\\big),\n    \\\\\n   \n    & B^q_{\\operatorname{ST}}(\\hat{p};a) =\n    \\frac{\\sqrt{\\mu}}{16}\\norm{2\\sqrt{\\mu}\\hat{v} +\\sqrt{\\mu_s}\\nabla\n      f(\\hat{x} + a \\hat{v})}^2\n    \\\\\n    & \\quad +\n    \\frac{\\sqrt{\\mu}\\sqrt{\\mu_s}}{4}\\big(\\frac{L}{2}\\norm{\\hat{v}}^2\n    +\\frac{\\sqrt{\\mu_s}}{4}\\norm{\\nabla f(\\hat{x} + a \\hat{v})}^2\\big),\n    \\\\\n    & C_{\\operatorname{ST} }(\\hat{p};a) = C_{\\operatorname{ET}}(\\hat{p};a).\n  \\end{align*}\n  Let $t\\mapsto p(t) = \\hat{p} + t X^a_{\\operatorname{hb}}(\\hat{p})$ be the trajectory\n  of the zero-order hold dynamics $\\dot p= X^a_{\\operatorname{hb}} (\\hat{p})$, $p(0) =\n  \\hat{p}$.  Then, for $t\\ge 0$,\n  \\begin{align*}\n    \\frac{d}{dt} V(p(t)) +\\frac{\\sqrt{\\mu}}{4} V({p}(t))\n   \n   \n   \n    & \\leq b^{\\operatorname{d}}_{\\operatorname{ET}}(\\hat{p},t;a) \\leq b^{\\operatorname{d}}_{\\operatorname{ST}}(\\hat{p}, t;a) .\n  \\end{align*}\n\\end{proposition}\n\nThe importance of Proposition~\\ref{prop:upper-bound-derivative} stems\nfrom the fact that the triggering conditions defined by $b^{\\operatorname{d}}_\\#$,\n$\\# \\in \\{\\operatorname{ET},\\operatorname{ST}\\}$, can be evaluated without knowledge of the\noptimizer~$x_*$.  We build on this result next to establish an upper\nbound for the performance-based triggering condition.\n\n\\begin{proposition}\\longthmtitle{Upper bound for performance-based\n    triggering with zero-order hold}\\label{prop:upper-bound-performance}\n  Let $a \\ge 0$ and\n  \\begin{align*}\n    b^{\\operatorname{p}}_{\\#}(\\hat{p},t;a) &= \\int_0^te^{\\frac{\\sqrt{\\mu}}{4}\n      \\zeta}b^{\\operatorname{d}}_{\\#}(\\hat p,\\zeta;a) d\\zeta ,\n  \\end{align*}\n  for $\\# \\in \\{\\operatorname{ET},\\operatorname{ST}\\}$. Let $t\\mapsto p(t) = \\hat{p} + t\n  X^a_{\\operatorname{hb}}(\\hat{p})$ be the trajectory of the zero-order hold dynamics\n  $\\dot p= X^a_{\\operatorname{hb}} (\\hat{p})$, $p(0) = \\hat{p}$.  Then, for $t \\ge 0 $,\n  \\begin{align*}\n    V(p(t)) \\!-\\! e^{-\\frac{\\sqrt{\\mu}}{4} t} V(\\hat p) \\!\\leq\\!\n    e^{-\\frac{\\sqrt{\\mu}}{4} t} b^{\\operatorname{p}}_{\\operatorname{ET}} (\\hat{p},t;a) \\!\\leq\\!\n    e^{-\\frac{\\sqrt{\\mu}}{4} t} b^{\\operatorname{p}}_{\\operatorname{ST}} (\\hat{p},t;a).\n  \\end{align*}\n\\end{proposition}\n\\begin{proof\n  We rewrite $ V(p(t)) - e^{-\\frac{\\sqrt{\\mu}}{4} t} V(\\hat p) =\n  e^{-\\frac{\\sqrt{\\mu}}{4} t} (e^{\\frac{\\sqrt{\\mu}}{4} t} V(p(t)) -\n  V(\\hat p))$, and note that\n  \\begin{align*}\n    & e^{\\frac{\\sqrt{\\mu}}{4} t} V(p(t)) - V(\\hat p)\n    \\\\\n    & \\quad = \\int_0^t \\frac{d}{d\\zeta} \\big( e^{\\frac{\\sqrt{\\mu}}{4} \\zeta}V(p(\\zeta))\n    - V(\\hat p)\\big) d\\zeta\n   \n   \n   \n    \\\\\n    & \\quad = \\int_0^te^{\\frac{\\sqrt{\\mu}}{4} \\zeta} \\Big( \\frac{d}{d\\zeta} V(p(\\zeta)) +\n    \\frac{\\sqrt{\\mu}}{4} V(p(\\zeta)\\Big) d\\zeta.\n  \\end{align*}\n  Note that the integrand corresponds to the derivative-based\n  criterion bounded in\n  Proposition~\\ref{prop:upper-bound-derivative}. Therefore, \n  \\begin{align*}\n    e^{\\frac{\\sqrt{\\mu}}{4} t} V(p(t)) - V(\\hat p) & \\leq \\int_0^t e^{\\frac{\\sqrt{\\mu}}{4} \\zeta}\n    b^{\\operatorname{d}}_{\\operatorname{ET}}(\\hat p,\\zeta;a) d\\zeta\n    \\\\\n    & = b^{\\operatorname{p}}_{\\operatorname{ET}}(\\hat{p},t;a) \\leq b^{\\operatorname{p}}_{\\operatorname{ST}}(\\hat{p},t;a)\n  \\end{align*}\n  for $t \\ge 0$,  and the result follows.\n \n \n \n \n \n \n \n \n \n \n \n\\end{proof}\n\nPropositions~\\ref{prop:upper-bound-derivative}\nand~\\ref{prop:upper-bound-performance} provide us with the tools to\ndetermine the stepsize according to the derivative- and\nperformance-based triggering criteria, respectively. For convenience,\nand following the notation in~\\eqref{eq:step-generic}, we define the\nstepsizes\n\\begin{subequations}\n  \\begin{align}\n    \\operatorname{step}^{\\operatorname{d}}_{\\#}(\\hat{p};a) & = \\min \\setdef{t >\n      0}{b^{\\operatorname{d}}_{\\#}(\\hat{p},t;a) = 0} ,\n    \\label{eq:step_derivative}\n    \\\\\n    \\operatorname{step}^{\\operatorname{p}}_{\\#}(\\hat{p};a) & = \\min \\setdef{t > 0}{b^{\\operatorname{p}}_{\\#}(\\hat{p},t;a) = 0} ,\n    \\label{eq:step_performance}\n  \\end{align}\n\\end{subequations}\nfor $\\# \\in \\{\\operatorname{ET},\\operatorname{ST}\\}$.  Observe that, as long as $\\hat{p} \\neq p_*\n= [x_*,0]^T$ and $0\\leq a \\leq a^*_1$, we have $C_{\\#}(\\hat{p};a) < 0$\nfor $\\#\\in\\{\\operatorname{ST},\\operatorname{ET}\\}$ and, as a consequence, $\nb^{\\operatorname{d}}_{\\#}(\\hat{p},0;a)< 0$.  The ET\/ST terminology is justified by\nthe following observation: in the ET case, the equation defining the\nstepsize is in general implicit in~$t$. Instead, in the ST case, the\nequation defining the stepsize is explicit in~$t$.\nEquipped with this notation, we define the variable-stepsize algorithm\ndescribed in Algorithm~\\ref{algo:DG}, which consists of following\nthe dynamics~\\eqref{eq:forward-euler-a} until the exponential decay of\nthe Lyapunov function is violated as estimated by the derivative-based\n($\\diamond = {\\operatorname{d}}$) or the performance-based ($\\diamond = {\\operatorname{p}}$) triggering condition.\nWhen this happens, the algorithm re-samples the state before continue\nflowing along~\\eqref{eq:forward-euler-a}.\n\n\\begin{algorithm}[h]\n  \\SetAlgoLined\n  %\n  \\textbf{Design Choices:} $\\diamond \\in \\{{\\operatorname{d}},{\\operatorname{p}}\\}$, $\\# \\in \\{\\operatorname{ET},\\operatorname{ST}\\}$\n  %\n  \\textbf{Initialization:} Initial point ($p_0$),\n  objective function ($f$), tolerance ($\\epsilon$), $a \\ge 0$, $k=0$\n  \\\\\n  \\While{$\\norm{\\nabla f(x_k)}\\geq \\epsilon$}{\n   \n    Compute stepsize $\\Delta_k = \\operatorname{step}^\\diamond_{\\#}(p_k;a)$\n    \\\\\n    Compute next iterate $p_{k+1} = p_k +\\Delta_k X^a_{\\operatorname{hb}}(p_k)$\n    \\\\\n    Set $k = k+1$ }\n  \\caption{Displaced-Gradient Algorithm}\\label{algo:DG}\n\\end{algorithm}\n\n\\subsection{Convergence Analysis of Displaced-Gradient\n  Algorithm}\\label{sec:analysis}\n\nHere we characterize the convergence properties of the derivative- and\nperformance-based implementations of the Displaced-Gradient\nAlgorithm. In each case, we show that algorithm is implementable\n(i.e., it admits a MIET) and inherits the convergence rate from the\ncontinuous-time dynamics.  The following result deals with the\nderivative-based implementation of Algorithm~\\ref{algo:DG}.\n\n\\begin{theorem}\\longthmtitle{Convergence of derivative-based\n    implementation of Displaced-Gradient \n    Algorithm}\\label{non-zeno-hb-db}\n  Let $\\hat{\\beta}_1,\\dots,\\hat{\\beta}_5>0$ be\n  \\begin{alignat*}{2}\n    \\hat{\\beta}_1 & = \\sqrt{\\mu_s}(\\frac{3\\sqrt{\\mu}}{2} + L), &\n    \\hat{\\beta}_2 & = \\sqrt{\\mu}\\sqrt{\\mu_s}\\frac{3}{2},\n    \\\\\n    \\hat{\\beta}_3& = \\frac{13\\sqrt{\\mu}}{16}\n    & \\hat{\\beta}_4& = \\frac{4 \\mu^{2}\\sqrt{s}+3 L \\sqrt{\\mu}\n      \\sqrt{\\mu_s}}{8 L^2},\n    \\\\\n    \\hat{\\beta}_5& = \\sqrt{\\mu_s}\\big(\\frac{5\\sqrt{\\mu}L}{2} -\n    \\frac{\\mu^{3\/2}}{2}\\big), &&\n  \\end{alignat*}\n  and define\n  \\begin{align}\\label{eq:a2}\n    a^*_2 = \\alpha \\min \\Big\\{\\frac{-\\hat \\beta_1 + \\sqrt{\\hat\n        \\beta_1^2 + 4 \\hat \\beta_5\\hat \\beta_3}}{2\\hat\n      \\beta_5},\\frac{\\hat{\\beta}_4}{\\hat{\\beta}_2} \\Big\\} ,\n  \\end{align}\n  with $0<\\alpha<1$.  Then, for $0 \\leq a \\leq a^*_2$, $\\diamond =\n  {\\operatorname{d}}$, and $\\# \\in \\{\\operatorname{ET},\\operatorname{ST}\\}$, the variable-stepsize strategy in\n  Algorithm~\\ref{algo:DG} has the following properties\n  \\begin{enumerate}\n  \\item[(i)] the stepsize is uniformly lower bounded by the positive\n    constant $\\operatorname{MIET}(a)$, where\n    \\begin{equation}\\label{g-definition}\n      \\operatorname{MIET}(a)= -\\nu + \\sqrt{\\nu^2 + \\eta}, \n    \\end{equation}\n    $\\eta= \\min\\{\\eta_1,\\eta_2\\}$, $\\nu = \\max\\{\\nu_1,\\nu_2\\}$, and\n    \\begin{align*}\n      \\eta_1 &=\\frac{8 a \\sqrt{\\mu_s} \\left(a (\\mu -5 L)-\\frac{2\n            L}{\\sqrt{\\mu }}-3\\right)+13}{2 \\sqrt{\\mu_s} L \\left(3 a^2\n          \\sqrt{\\mu_s} L+1\\right)+8 \\mu } ,\n      \\\\\n     \n     \n     \n     \n     \n     \n      \\eta_2 &=-\\frac{3 \\sqrt{\\mu_s} \\sqrt{\\mu } L (4 a L-1)-4 \\mu ^2\n        \\sqrt{s}}{3 \\mu_s \\sqrt{\\mu } L^2},\n      \\\\\n      \\nu_1 &=\\frac{\\mu \\left(2 a^3 \\sqrt{\\mu_s} L^2+a\n          \\sqrt{\\mu_s}+16\\right)+8 \\sqrt{\\mu_s} L \\left(2 a^2\n          \\sqrt{\\mu_s} L+1\\right)}{2 \\sqrt{\\mu } \\left(\\sqrt{\\mu_s} L\n          \\left(3 a^2 \\sqrt{\\mu_s} L+1\\right)+4 \\mu \\right)}\n      \\\\\n      &\\quad+ \\frac{\\sqrt{\\mu_s}  (a L (8 a L+1)+4)}{\n       \\sqrt{\\mu_s} L \\left(3 a^2 \\sqrt{\\mu_s}\n            L+1\\right)+4 \\mu},\n       \n       \n       \n       \n       \n       \n       \n       \n       \n        \\\\\n        \\nu_2 &=\\frac{a \\mu +8 \\sqrt{\\mu_s}+8 \\sqrt{\\mu}}{3 \\sqrt{\\mu_s}\n          \\sqrt{\\mu}};\n    \\end{align*}\n  \\item[(ii)] $ \\frac{d}{dt} V(p_{k}+t X^a_{\\operatorname{hb}} (p_k)) \\leq\n    -\\frac{\\sqrt{\\mu}}{4} V(p_k+t X^a_{\\operatorname{hb}} (p_k))$ for all $t \\in\n    [0,\\Delta_k]$ and $k \\in \\{0\\} \\cup \\mathbb{N}$.\n  \\end{enumerate}\n  As a consequence, $ f(x_{k+1})-f(x_*) =\n  \\mathcal{O}(e^{-\\frac{\\sqrt{\\mu}}{4}\\sum_{i=0}^k \\Delta_i})$.\n\\end{theorem}\n\\begin{proof\n  Regarding fact (i), we prove the result for the $\\operatorname{ST}$-case, as the\n  $\\operatorname{ET}$-case follows from $\\operatorname{step}^{\\operatorname{d}}_{\\operatorname{ET}}(\\hat{p};a) \\geq\n  \\operatorname{step}^{\\operatorname{d}}_{\\operatorname{ST}}(\\hat{p}; a)$.\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  We start by upper bounding $C_{\\operatorname{ST}}(\\hat{p};a)$ by a negative\n  quadratic function of $\\norm{\\hat{v}}$ and $\\norm{\\nabla\n    f(\\hat{x})}$ as follows,\n  \\begin{align*}\n    & C_{\\operatorname{ST}}(\\hat{p};a) = -\\frac{13\\sqrt{\\mu}}{16}\\norm{\\hat{v}}^2 +\n    \\sqrt{\\mu_s}\\frac{-3\\sqrt{\\mu}}{8L}\\norm{\\nabla\n        f(\\hat{x})}^2\n    \\\\\n    & \\quad -\\frac{\\mu^2\\sqrt{s}}{2L^2}\\displaystyle\\norm{\\nabla\n        f(\\hat{x})}^2\n    +\\sqrt{\\mu_s}\\big(\\sqrt{\\mu}\\underbrace{(f(\\hat x) - f(\\hat{x} +\n      a\\hat{v}))}_{\\textrm{(a)}}\n    \\\\\n    & \\quad + \\sqrt{\\mu}\\underbrace{\\norm{\\nabla f(\\hat{x})}\\norm{a\n        \\hat{v}}}_{\\textrm{(b)}} -\\frac{\\mu^{3\/2}}{2}\\norm{a\\hat{v}}^2\n    \\\\\n    & \\quad +\\underbrace{\\langle \\nabla f(\\hat{x}) - \\nabla f (\\hat{x}\n      + a\\hat{v} ) , \\hat{v}\\rangle}_{\\textrm{(c)}} +\n    \\sqrt{\\mu}\\underbrace{\\langle \\nabla f(\\hat{x} + a\\hat{v}),\n      a\\hat{v} \\rangle}_{\\textrm{(d)}}\\big).\n  \\end{align*}\n  Using the $L$-Lipschitzness of the gradient and Young's inequality,\n  we can easily upper bound\n  \\begin{align*}\n    \\textrm{(a)} & \\le \\underbrace{\\langle \\nabla f(\\hat x + a \\hat\n      v), - a \\hat v \\rangle + \\frac{L}{2}a^2\\norm{\\hat v}^2\n    }_{\\textrm{Using~\\eqref{eq:aux-d}}}\n   \n   \n   \n   \n    \\\\\n    & = \\langle \\nabla f(\\hat x + a \\hat v) - \\nabla f(\\hat x), - a\n    \\hat v \\rangle + \\frac{L}{2}a^2\\norm{\\hat v}^2\n    \\\\\n    & \\quad + \\langle \\nabla f(\\hat x), - a \\hat v \\rangle\n   \n   \n   \n    \\\\\n    & \\leq La^2\\norm{\\hat v}^2 + \\frac{L}{2}a^2\\norm{\\hat v}^2 + a\n    \\big(\\frac{\\norm{\\nabla f(\\hat x)}^2}{2} + \\frac{\\norm{\\hat\n        v}^2}{2} \\big)\n    \\\\\n   \n   \n   \n    & = \\frac{3La^2 + a}{2}\\norm{\\hat v}^2 + \\frac{a}{2} \\norm{\\nabla\n      f(\\hat x)}^2,\n    \\\\\n    \\textrm{(b)} & \\le a \\big (\\frac{\\norm{\\nabla f(\\hat x)}^2}{2} +\n    \\frac{\\norm{\\hat v}^2}{2} \\big),\n   \n   \n    \\\\\n    \\textrm{(c)} & \\leq L a \\norm{\\hat v}^2,\n    \\\\\n    \\textrm{(d)} & = \\langle \\nabla f(\\hat{x} + a\\hat{v}) - \\nabla\n    f(\\hat x) + \\nabla f(\\hat x), a\\hat{v} \\rangle\n    \\\\\n    & \\leq L a^2\\norm{\\hat v}^2 + \\langle \\nabla f(\\hat x), a \\hat v\n    \\rangle\n   \n   \n   \n    \\\\\n    & = \\frac{2La^2 + a}{2}\\norm{\\hat v}^2 + \\frac{a}{2}\\norm{\\nabla\n      f(\\hat z)}^2.\n  \\end{align*}\n  Note that, with the definition of the constants\n  $\\hat{\\beta}_1,\\dots,\\hat{\\beta}_5>0$ in the statement, we can write\n \n \n \n \n \n \n \n \n \n \n \n \n  \\begin{align*}\n    C_{\\operatorname{ST}}(\\hat p;a)& \\leq a \\hat{\\beta}_1\\norm{\\hat{v}}^2 +\n    a^2\\hat{\\beta}_5\\norm{\\hat v}^2 + a\\hat{\\beta}_2\\norm{\\nabla\n      f(\\hat x)}^2\n    \\\\\n    & \\quad -\\hat{\\beta}_3\\norm{\\hat{v}}^2 -\\hat{\\beta}_4\\norm{\\nabla\n      f(\\hat{x})}^2 .\n  \\end{align*}\n \n \n \n \n \n \n \n \n \n \n \n \n  Therefore, for $a \\in [0,a^*_2]$, we have\n  \\begin{align*}\n    a \\hat{\\beta}_1 + a^2\\hat \\beta_5 - \\hat{\\beta}_3 & \\le a^*_2\n    \\hat{\\beta}_1 + (a^*_2)^2\\hat \\beta_5 - \\hat{\\beta}_3 = -\\gamma_1\n    < 0\n    \\\\\n    a \\hat{\\beta}_2 - \\hat{\\beta}_4 & \\le a^*_2 \\hat{\\beta}_2 -\n    \\hat{\\beta}_4 = -\\gamma_2< 0,\n  \\end{align*}\n  and hence $C_{\\operatorname{ST}}(\\hat{p};a)\\leq - \\gamma_1\\norm{\\hat{v}}^2 -\n  \\gamma_2\\norm{\\nabla f(\\hat{x})}^2$.  Similarly,  introducing\n  \\begin{align*}\n    \\gamma_3 &= 2 a^2 \\mu_s L^2+2 a^2 \\sqrt{\\mu_s} \\sqrt{\\mu }\n    L^2+\\sqrt{\\mu_s} \\sqrt{\\mu }+\\sqrt{\\mu_s} L +2 \\mu,\n    \\\\\n    \\gamma_4 & = 2 \\mu_s + 2 \\sqrt{\\mu_s} \\sqrt{\\mu}, \\; \\gamma_5 \\!=\\!\n    \\frac{1}{8} a \\sqrt{\\mu_s} \\left(2 a^2 \\mu L^2+\\mu +2 \\sqrt{\\mu }\n      L\\right),\n    \\\\\n    \\gamma_6 & = \\frac{a\\mu\\sqrt{\\mu_s}}{4}, \\; \\gamma_7 = \\frac{3}{8}\n    a^2 \\mu_s \\sqrt{\\mu } L^2+\\frac{1}{8} \\sqrt{\\mu_s} \\sqrt{\\mu }\n    L+\\frac{\\mu ^{3\/2}}{2},\n    \\\\\n    \\gamma_8 & = \\frac{3 \\mu_s\\sqrt{\\mu}}{8},\n  \\end{align*}\n  one can show that\n  \\begin{align*}\n    A_{\\operatorname{ST}}(\\hat p;a) \\le \\hat{A}_{\\operatorname{ST}}(\\hat p;a) &= \\gamma_3\n    \\norm{\\hat v}^2 + \\gamma_4 \\norm{\\nabla f(\\hat x)}^2,\n    \\\\\n    B^l_{\\operatorname{ST}}(\\hat p;a) \\le \\hat{B}^l_{\\operatorname{ST}}(\\hat p;a) & = \\gamma_5\n    \\norm{\\hat v}^2 + \\gamma_6 \\norm{\\nabla f(\\hat x)}^2,\n    \\\\\n    B^q_{\\operatorname{ST}}(\\hat p;a) \\le \\hat{B}^q_{\\operatorname{ST}}(\\hat p;a) & =\\gamma_7\n    \\norm{\\hat v}^2 + \\gamma_8 \\norm{\\nabla f(\\hat x)}^2.\n  \\end{align*}\n  Thus, from~\\eqref{eq:step_derivative}, we have\n  \\begin{align}\\label{eq:step_derivative_explicit}\n    & \\operatorname{step}^{\\operatorname{d}}_{\\operatorname{ST}}(\\hat{p};a) \\geq \\frac{-(\\hat A_{\\operatorname{ST}}(\\hat{p};a) +\n      \\hat B^l_{\\operatorname{ST}}(\\hat{p};a))}{2 \\hat B^q_{\\operatorname{ST}}(\\hat{p};a)}\n    \\\\\n    &\\quad + \\sqrt{\\left(\\frac{\\hat A_{\\operatorname{ST}}(\\hat{p};a) +\n          \\hat B^l_{\\operatorname{ST}}(\\hat{p};a)}{ 2 \\hat  B^q_{\\operatorname{ST}}(\\hat{p};a) }\\right)^2\n      -\\frac{C_{\\operatorname{ST}}(\\hat{p};a)}{\\hat  B^q_{\\operatorname{ST}}(\\hat{p};a)}}. \\notag\n  \\end{align}\n\n\n\n\n\n\n\n\n\n\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  Using now \\cite[supplementary material, Lemma~1]{MV-JC:19-nips}, we\n  deduce\n  \\begin{align*}\n    \\eta \\leq \n     \\frac{-C_{\\operatorname{ST}}(\\hat{p};a)}{\\hat B^q_{\\operatorname{ST}}(\\hat{p};a)}  ,\n    \n    \n    \n     \\quad\n    \n    \n    \n    \n    \n     \\frac{\\hat A_{\\operatorname{ST}}(\\hat{p};a) + \\hat B^l_{\\operatorname{ST}}(\\hat{p};a)}{2\\hat\n       B^q_{\\operatorname{ST}}(\\hat{p};a)} \\leq \\nu,\n  \\end{align*}\n  where\n  \\begin{align*}\n    \\eta &=\n    \\min\\{\\frac{\\gamma_1}{\\gamma_7},\\frac{\\gamma_2}{\\gamma_8}\\}, \\quad\n    \\nu = \\max\\{\\frac{\\gamma_3 + \\gamma_5}{2\\gamma_7},\\frac{\\gamma_4 +\n      \\gamma_6}{2\\gamma_8}\\}.\n  \\end{align*}\n  With these elements in place and referring\n  to~\\eqref{eq:step_derivative_explicit}, we have\n  \\begin{align*}\n    \\operatorname{step}^{\\operatorname{d}}_{\\operatorname{ST}}(\\hat{p};a) & \\geq \\frac{-(\\hat A_{\\operatorname{ST}}(\\hat{p};a) +\n      \\hat B^l_{\\operatorname{ST}}(\\hat{p};a))}{2\\hat B^q_{\\operatorname{ST}}(\\hat{p};a)}\n    \\\\\n    & \\quad + \\sqrt{\\left(\\frac{\\hat A_{\\operatorname{ST}}(\\hat{p};a) +\n         \\hat  B^l_{\\operatorname{ST}}(\\hat{p};a)}{ 2\\hat B^q_{\\operatorname{ST}}(\\hat{p};a) }\\right)^2 +\n      \\eta}.\n    \\end{align*}\n    We observe now that $z \\mapsto g(z) = -z + \\sqrt{z^2 + \\eta} $ is\n    monotonically decreasing and lower bounded. So, if $z$ is upper\n    bounded, then $g(z)$ is lower bounded by a positive\n    constant. Taking $z = \\frac{(\\hat A_{\\operatorname{ST}}(\\hat{p};a) +\n      \\hat B^l_{\\operatorname{ST}}(\\hat{p};a))}{2\\hat B^q_{\\operatorname{ST}}(\\hat{p};a)} \\leq \\nu$ gives\n    the bound of the stepsize.  Finally, the algorithm design together\n    with Proposition~\\ref{prop:upper-bound-derivative} ensure fact\n    (ii) throughout its evolution.\n\\end{proof}\n\nIt is worth noticing that the derivative-based implementation of the\nDisplaced-Gradient Algorithm generalizes the algorithm proposed in our\nprevious work~\\cite{MV-JC:19-nips} (in fact, the strategy proposed\nthere corresponds to the choice $a=0$).  The next result characterizes\nthe convergence properties of the performance-based implementation of\nAlgorithm~\\ref{algo:DG}.\n\n\\begin{theorem}\\longthmtitle{Convergence \n    of performance-based implementation of Displaced-Gradient\n    Algorithm}\\label{non-zeno-hb-pb} \n  For $0 \\leq a \\leq a^*_2$, $\\diamond = {\\operatorname{p}}$, and $\\# \\in\n  \\{\\operatorname{ET},\\operatorname{ST}\\}$, the variable-stepsize strategy in\n  Algorithm~\\ref{algo:DG} has the following properties\n  \\begin{enumerate}\n  \\item[(i)] the stepsize is uniformly lower bounded by the positive\n    constant $\\operatorname{MIET}(a)$;\n  \\item[(ii)] $ V(p_{k }+t X^a_{\\operatorname{hb}} (p_k)) \\leq e^{-\\frac{\\sqrt{\\mu}}{4}\n      t} V(p_k)$ for all $t \\in [0,\\Delta_k]$ and $k \\in \\{0\\} \\cup\n    \\mathbb{N}$.\n  \\end{enumerate}\n  As a consequence, $ f(x_{k+1})-f(x_*) =\n  \\mathcal{O}(e^{-\\frac{\\sqrt{\\mu}}{4}\\sum_{i=0}^k \\Delta_i})$.\n\\end{theorem}\n\\begin{proof}\n  To show (i), notice that it is sufficient to prove that\n  $\\operatorname{step}^{\\operatorname{p}}_{\\operatorname{ST}}$ is uniformly lower bounded away from zero. This is\n  because of the definition of stepsize in~\\eqref{eq:step_performance}\n  and the fact that $b^{\\operatorname{p}}_{\\operatorname{ET}} (\\hat p,t;a) \\le b^{\\operatorname{p}}_{\\operatorname{ST}} (\\hat\n  p,t;a)$ for all $\\hat p$ and all $t$.  For an arbitrary fixed $\\hat\n  p$, note that $t \\mapsto b^{\\operatorname{d}}_{\\operatorname{ST}}(\\hat p,t;a)$ is strictly\n  negative in the interval $[0,\\operatorname{step}^{\\operatorname{d}}_{\\operatorname{ST}}(p;a))$ given the\n  definition of stepsize in~\\eqref{eq:step_derivative}.  Consequently,\n  the function $t \\mapsto b^{\\operatorname{p}}_{\\operatorname{ST}} (\\hat p,t;a)=\\int_0^t\n  e^{\\frac{\\sqrt{\\mu}}{4} \\zeta} b^{\\operatorname{d}}_{\\operatorname{ST}}(\\hat p;\\zeta,a) d\\zeta$\n  is strictly negative over $(0,\\operatorname{step}^{\\operatorname{d}}_{\\operatorname{ST}}(\\hat p;a))$.  From the\n  definition of $\\operatorname{step}^{\\operatorname{p}}_{\\operatorname{ST}}$, it then follows that\n  $\\operatorname{step}^{\\operatorname{p}}_{\\operatorname{ST}}(\\hat p;a)\\geq \\operatorname{step}^{\\operatorname{d}}_{\\operatorname{ST}}(\\hat p;a)$. The result now\n  follows by noting that $ \\operatorname{step}^{\\operatorname{d}}_{\\operatorname{ST}}$ is uniformly lower bounded\n  away from zero by a positive constant,\n  cf. Theorem~\\ref{non-zeno-hb-db}(i).\n\n  To show (ii), we recall that $\\Delta_k = \\operatorname{step}^{\\operatorname{p}}_{\\#}(p_k;a)$ for $\\#\n  \\in \\{\\operatorname{ET},\\operatorname{ST}\\}$ and use\n  Proposition~\\ref{prop:upper-bound-performance} for $\\hat p = p_k$ to\n  obtain, for all $t \\in [0,\\Delta_k]$,\n  \\begin{align*}\n    V(p(t)) - e^{-\\frac{\\sqrt{\\mu}}{4} t} V(p_k) & \\leq e^{-\\frac{\\sqrt{\\mu}}{4} t} b^{\\operatorname{p}}_{\\#}\n    (p_k,t;a)\n    \\\\\n    & \\le e^{-\\frac{\\sqrt{\\mu}}{4} t} b^{\\operatorname{p}}_{\\#} (p_k,\\Delta_k;a) = 0 ,\n  \\end{align*}\n  as claimed.\n\\end{proof}\n\nThe proof of Theorem~\\ref{non-zeno-hb-pb} brings up an interesting\ngeometric interpretation of the relationship between the stepsizes\ndetermined according to the derivative- and performance-based\napproaches.  In fact, since\n\\begin{align*}\n  \\frac{d}{dt} b^{\\operatorname{p}}_{\\#}(\\hat p,t;a) = e^{\\frac{\\sqrt{\\mu}}{4} t}\n  b^{\\operatorname{d}}_{\\#} (\\hat p,t;a) ,\n\\end{align*}\nwe observe that $\\operatorname{step}^{\\operatorname{d}}_{\\#}(\\hat{p};a)$ is precisely the (positive)\ncritical point of $t \\mapsto b^{\\operatorname{p}}_{\\#}(\\hat p,t;a)$. Therefore,\n$\\operatorname{step}^{\\operatorname{p}}_{\\operatorname{ST}}(\\hat p;a)$ is the smallest nonzero root of $t \\mapsto\nb^{\\operatorname{p}}_{\\#}(\\hat p,t;a)$, whereas $\\operatorname{step}^{\\operatorname{d}}_{\\operatorname{ST}}(\\hat p;a)$ is the time\nwhere $t \\mapsto b^{\\operatorname{p}}_{\\#}(\\hat p,t;a)$ achieves its smallest value,\nand consequently is furthest away from zero.  This confirms the fact\nthat the performance-based approach obtains larger stepsizes than the\nderivative-based approach.\n\n\n\n\\section{Exploiting Sampled Information to Enhance Algorithm\n  Performance}\\label{sec:exploit-sampled-info}\n\nHere we describe two different refinements of the implementations\nproposed in Section~\\ref{sec:performance-based} to further enhance\ntheir performance. Both of them are based on further exploiting the\nsampled information about the system.  The first refinement,\ncf. Section~\\ref{sec:adaptive}, looks at the possibility of adapting\nthe value of the gradient displacement as the algorithm is\nexecuted. The second refinement, cf. Section~\\ref{sec:HOH}, develops a\nhigh-order hold that more accurately approximates the evolution of the\ncontinuous-time heavy-ball dynamics with displaced gradient.\n\n\\subsection{Adaptive Gradient Displacement}\\label{sec:adaptive}\n\nThe derivative- and performance-based triggered implementations \nin Section~\\ref{sec:trigger-design} both employ a constant value of\nthe parameter~$a$. Here, motivated by the observation made in\nRemark~\\ref{rem:a-over-region}, we develop triggered implementations\nthat adaptively adjust the value of the gradient displacement\ndepending on the region of the space to which the state belongs.\nRather than relying on the condition~\\eqref{eq:adaptive-a}, which\nwould require partitioning the state space based on bounds on~$\\nabla\nf(x)$ and~$v$, we seek to compute on the fly a value of the\nparameter~$a$ that ensures the exponential decrease of the Lyapunov\nfunction at the current state. Formally, the strategy is stated in\nAlgorithm~\\ref{algo:ADG}.\n\n\\begin{algorithm}[h]\n  \\SetAlgoLined\n \n  \\textbf{Design Choices:} $\\diamond \\in \\{{\\operatorname{d}},{\\operatorname{p}}\\}$, $\\# \\in \\{\\operatorname{ET},\\operatorname{ST}\\}$\n \n  \\textbf{Initialization:} Initial point ($p_0$),\n \n  objective function ($f$), tolerance ($\\epsilon$), increase rate\n  ($r_i > 1$), decrease rate ($0 < r_d < 1$), stepsize lower bound\n  ($\\tau$), $a\\ge 0$, $k=0$\n  \\\\\n \n  \\While{$\\norm{\\nabla f(x_k)}\\geq \\epsilon$}{\n    increase = True\n\n    exit = False\n\n   \\While{ {\\rm exit} = {\\rm False}}{\n    \\While{$C_{\\#}(p_k;a) \\geq 0$  }{\n       \n        $a = a r_d$\n        %\n\n        increase = False\n       \n      }\n      \\uIf{$\\operatorname{step}^\\diamond_{\\#}(p_k;a) \\geq \\tau$}{\n        exit = True\n     %\n      }\n     \\uElse{\n     $a = a r_d$\n    \n\n     increase = False\n     }\n    }\n       \n    Compute stepsize $\\Delta_k = \\operatorname{step}^\\diamond_{\\#}(p_k;a)$\n    \\\\\n   \n      Compute next iterate $p_{k+1} = p_k +\\Delta_k X_{\\operatorname{hb}}^a(p_k)$\n      \\\\\n     \n  Set $k = k+1$\n\n  \\uIf{\\rm increase = True}{ $a = a r_i$ }\n\n }\n \\caption{Adaptive Displaced-Gradient Algorithm}\\label{algo:ADG}\n\\end{algorithm}\n\n\\begin{proposition}\\longthmtitle{Convergence  of Adaptive Displaced-Gradient\n    Algorithm}\\label{prop:non-zeno-sampled-algorithm}\n  For\n  $\\diamond \\in \\{{\\operatorname{d}},{\\operatorname{p}}\\}$, $\\# \\in \\{\\operatorname{ET},\\operatorname{ST}\\}$, and $\\tau \\le\n  \\min_{a\\in [0,a^*_2]} \\operatorname{MIET}(a)$, the variable-stepsize strategy in\n  Algorithm~\\ref{algo:ADG} has the following properties:\n  \\begin{enumerate}\n  \\item[(i)] it is executable (i.e., at each iteration, the parameter\n    $a$ is determined in a finite number of steps);\n  \\item[(ii)] the stepsize is uniformly lower bounded by~$\\tau$;\n  \\item[(iii)] it satisfies $ f(x_{k+1})\\!-\\!f(x_*) \\!=\\!\n    \\mathcal{O}(e^{-\\frac{\\sqrt{\\mu}}{4}\\sum_{i=0}^k \\Delta_i} )$, for\n    $k \\in \\{0\\} \\cup \\mathbb{N}$.\n  \\end{enumerate}\n \\end{proposition}\n\\begin{proof}\n  Notice first that the function~$a \\mapsto \\operatorname{MIET}(a)>0$ defined\n  in~\\eqref{g-definition} is continuous and therefore attains its\n  minimum over a compact set.  At each iteration,\n  Algorithm~\\ref{algo:ADG} first ensures that $C_{\\#}(\\hat{p};a)< 0 $,\n  decreasing $a$ if this is not the case. We know this process is\n  guaranteed as soon as $a < a_2^*$ (cf. proof of\n  Theorem~\\ref{non-zeno-hb-db}) and hence only takes a finite number\n  of steps. Once $C_{\\#}(\\hat{p};a)< 0 $, the stepsize could be\n  computed to guarantee the desired decrease of the Lyapunov function\n  $V$.  The algorithm next checks if the stepsize is lower bounded\n  by~$\\tau$. If that is not the case, then the algorithm reduces $a$\n  and re-checks if $C_{\\#}(\\hat{p};a) < 0$.  With this process and in\n  a finite number of steps, the algorithm eventually either computes a\n  stepsize lower bounded by~$\\tau$ with $a > a^*_2$ or $a$ decreases\n  enough to make $a \\leq a^*_2$, for which we know that the stepsize\n  is already lower bounded by~$\\tau$. These arguments establish facts\n  (i) and (ii) at the same time. Finally, fact (iii) is a consequence\n  of the prescribed decreased of the Lyapunov function along the\n  algorithm execution.\n\\end{proof}\n\n\\subsection{Discretization via High-Order Hold}\\label{sec:HOH}\n\nThe modified zero-order hold based on employing displaced gradients\ndeveloped in Section~\\ref{sec:performance-based} is an example of the\npossibilities enabled by more elaborate uses of sampled\ninformation. In this section, we propose another such use based on the\nobservation that the continuous-time heavy-ball dynamics can be\ndecomposed as the sum of a linear term and a nonlinear\nterm. Specifically, we have\n\\begin{align*}\n  X_{\\operatorname{hb}}^a (p) & =\n  \\begin{bmatrix}\n    v\n    \\\\\n    - 2\\sqrt{\\mu}v\n  \\end{bmatrix}\n  + \n  \\begin{bmatrix}\n    0\n    \\\\\n    - \\sqrt{\\mu_s}\\nabla f(x + av)\n    \\end{bmatrix} .\n\\end{align*}\nNote that the first term in this decomposition is linear, whereas the\nother one contains the potentially nonlinear gradient term that\ncomplicates finding a closed-form solution.  Keeping this in mind when\nconsidering a discrete-time implementation, it would seem reasonable\nto perform a zero-order hold only on the nonlinear term while exactly\nintegrating the resulting differential equation. Formally, a\nzero-order hold at $\\hat{p}=[\\hat{x},\\hat{v}]$ of the nonlinear term\nabove yields a system of the form\n\\begin{align}\\label{eq:linear-inhomo}\n  \\begin{bmatrix}\n    \\dot x\n    \\\\\n    \\dot v\n  \\end{bmatrix}\n  & =\n  A\n  \\begin{bmatrix}\n    x\n    \\\\\n    v\n  \\end{bmatrix}\n  +   b ,\n\\end{align}\nwith $p(0) = \\hat p$, and where\n\\begin{align*}\n  A=\n\\begin{bmatrix}\n  0 & 1\n  \\\\\n  0 & -2\\sqrt{\\mu}\n\\end{bmatrix},\n\\quad b = \\begin{bmatrix}\n    0\n    \\\\\n    - \\sqrt{\\mu_s}\\nabla f(\\hat x + a\\hat v)\n  \\end{bmatrix} .\n\\end{align*}\nEquation~\\eqref{eq:linear-inhomo} is an in-homogeneous linear\ndynamical system, which is integrable by the method of variation of\nconstants~\\cite{LP:00}.  Its solution is given by $ p(t) = e^{At}\n\\big(\\int_0^t e^{-A\\zeta} b d\\zeta + p(0) \\big) $, or equivalently,\n\\begin{subequations}\\label{eq:hoh-flow}\n  \\begin{align}\n    x(t) & =  \\hat{x} -\\frac{\\sqrt{\\mu_s}\\nabla f(\\hat{x} + a \\hat{v})t }{ 2\n      \\sqrt{\\mu} } \n    \\\\\n    & \\quad + (1-e^{-2 \\sqrt{\\mu } t}) \\frac{\\sqrt{\\mu_s}\\nabla f(\\hat{x} +\n      a \\hat{v})+2 \\sqrt{\\mu } \\hat{v}}{4\\mu} , \\notag\n   \n   \n   \n   \n   \n   \n   \n    \\\\\n    v(t) & = e^{-2 \\sqrt{\\mu } t}\\hat{v} + (e^{-2 \\sqrt{\\mu } t} -1)\n    \\frac{\\sqrt{\\mu_s}\\nabla f(\\hat{x} + a \\hat{v}) }{2 \\sqrt{\\mu\n      }} .\n  \\end{align}\n\\end{subequations}\nWe refer to this trajectory as a \\emph{high-order-hold integrator}.\nIn order to develop a discrete-time algorithm based on this type of\nintegrator, the next result provides a bound of the evolution of the\nLyapunov function $V$ along the high-order-hold integrator\ntrajectories.  The proof is presented in\nAppendix~\\ref{app:appendix}.\n\n\\begin{proposition}\\longthmtitle{Upper bound for derivative-based\n    triggering with high-order\n    hold}\\label{prop:upper-bound-derivative-hoh}\n  Let $a\\ge 0$\n  and define\n  \\begin{align*}\n    \\mathfrak{b}^{\\operatorname{d}}_{\\operatorname{ET}}(\\hat{p}, t;a) &= \\mathfrak{A}_{\\operatorname{ET}}(\\hat{p}, t;a) +\n    \\mathfrak{B}_{\\operatorname{ET}}(\\hat{p}, t;a)\n    \\\\\n    & \\quad + \\mathfrak{C}_{\\operatorname{ET}}(\\hat{p};a) +\\mathfrak{D}_{\\operatorname{ET}}(\\hat{p},t;a),\n    \\\\\n    \\mathfrak{b}^{\\operatorname{d}}_{\\operatorname{ST}}(\\hat{p}, t;a) & = (\\mathfrak{A}_{\\operatorname{ST}}^q(\\hat{p};a) + \\mathfrak{B}^q_{\\operatorname{ST}}\n    (\\hat{p};a) ) t^2 + (\\mathfrak{A}_{\\operatorname{ST}}^l(\\hat{p};a)\n    \\\\\n    & \\quad + \\mathfrak{B}^l_{\\operatorname{ST}}(\\hat{p};a) + \\mathfrak{D}_{\\operatorname{ST}} (\\hat{p};a) )t\n    +\\mathfrak{C}_{\\operatorname{ST}}(\\hat{p};a) ,\n  \\end{align*}\n  where\n  \\begin{align*}\n    & \\mathfrak{A}_{\\operatorname{ET}}(\\hat p,t;a) =\\sqrt{\\mu_s}(\\langle \\nabla f(x(t)) -\n    \\nabla f(\\hat{x}),v(t) \\rangle\n    \\\\\n    & \\quad - \\langle v(t) - \\hat{v}, \\nabla f(\\hat{x} + a\\hat{v})\n    \\rangle\n    \\\\\n    & \\quad - \\sqrt{\\mu}\\langle x(t) - \\hat{x}, \\nabla f(\\hat{x} +\n    a\\hat{v}) \\rangle)\n    \\\\\n    &\\quad -\\sqrt{\\mu}\\langle v(t) - \\hat{v},v(t)\\rangle,\n    \\\\\n   \n    & \\mathfrak{B}_{\\operatorname{ET}}(\\hat p,t;a) = \\frac{\\sqrt{\\mu}}{4}\\big(\\sqrt{\\mu_s}\n    (f(x(t)) - f(\\hat{x}))\n    \\\\\n    &\\quad - \\sqrt{\\mu}\\sqrt{\\mu_s} t \\frac{\\norm{\\nabla f(\\hat{x} +\n        a\\hat{v})}^2}{L}\n    \\\\\n    &\\quad + \\sqrt{\\mu}\\sqrt{\\mu_s} t \\langle \\nabla f(\\hat{x} + a\n    \\hat{v}), a \\hat{v}\\rangle + \\frac{1}{4} (\\norm{v(t)}^2 -\n    \\norm{\\hat{v}}^2)\n    \\\\\n    &\\quad + \\frac{1}{4} \\norm{v(t)-\\hat{v} + 2 \\sqrt{\\mu}\n      (x(t)-\\hat{x})}^2\n    \\\\\n    & \\quad + \\frac{1}{2}\\langle v(t)-\\hat{v} + 2 \\sqrt{\\mu}\n    (x(t)-\\hat{x}), \\hat{v} \\rangle \\big),\n    \\\\\n   \n    & \\mathfrak{C}_{\\operatorname{ET}}(\\hat p;a) = C_{\\operatorname{ET}}(\\hat{p};a) ,\n    \\\\\n    & \\mathfrak{D}_{\\operatorname{ET}} (\\hat p,t;a) = \\sqrt{\\mu_s}\\langle \\nabla f(\\hat{x}),\n    v(t) - \\hat{v}\\rangle\n    \\\\\n    & \\quad -\\sqrt{\\mu}\\langle \\hat{v} , v(t) - \\hat{v} \\rangle ,\n  \\end{align*}\n  and \n  \\begin{align*}\n    & \\mathfrak{A}_{\\operatorname{ST}}^l(\\hat p;a) = \\norm{2 \\sqrt{\\mu}\\hat v\n      +\\sqrt{\\mu_s}\\nabla f(\\hat x + a \\hat v) } \\Big( \\sqrt{\\mu}\n    \\norm{\\hat v}\n    \\\\\n    & \\quad + \\frac{L\\sqrt{\\mu_s}}{2\\sqrt{\\mu}} \\norm{\\hat v} +\n    \\frac{3\\sqrt{\\mu_s}}{2} \\norm{\\nabla f(\\hat x + a \\hat v)} \\Big)\n    \\\\\n    & \\quad + \\frac{\\mu_s}{2} \\norm{\\nabla f(\\hat x + a \\hat\n      v)} \\Big( \\frac{L}{\\sqrt{\\mu}} \\norm{\\hat v} + \\norm{\\nabla\n      f(\\hat x + a \\hat v)} \\Big) ,\n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n    \\\\\n   \n    & \\mathfrak{A}_{\\operatorname{ST}}^q(\\hat p;a) = \\norm{2 \\sqrt{\\mu}\\hat v +\n      \\sqrt{\\mu_s}\\nabla f(\\hat x + a \\hat v) }\n    \\\\\n    & \\quad \\cdot{}\\Big( \\big( \\frac{L\\sqrt{\\mu_s}}{2\\sqrt{\\mu}} +\n    \\sqrt\\mu \\big)\\norm{2 \\sqrt{\\mu}\\hat v + \\sqrt{\\mu_s}\\nabla f(\\hat\n      x + a \\hat v) }\n    \\\\\n    & \\quad + \\frac{L\\mu_s}{2\\sqrt{\\mu}} \\norm{\\nabla f(\\hat x\n      + a \\hat v)} \\Big) ,\n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n    \\\\\n   \n    & \\mathfrak{B}^l_{\\operatorname{ST}}(\\hat p;a) = \\frac{\\sqrt{\\mu} \\sqrt{\\mu_s}}{4} \\Big(\n    \\frac{\\sqrt{\\mu_s}}{2\\sqrt{\\mu}}\\norm{\\nabla f(\\hat x + a \\hat\n      v)}\\norm{\\nabla f(\\hat x)}\n    \\\\\n    & \\quad + \\frac{1}{2} \\norm{2 \\sqrt{\\mu}\\hat v \\!+\\!\n      \\sqrt{\\mu_s}\\nabla f(\\hat x + a \\hat v) } \\big(\n    \\frac{\\norm{\\nabla f(\\hat x)}}{\\sqrt{\\mu}} \\!+\\!  \\frac{\\norm{\\hat\n        v}}{\\sqrt{\\mu_s}}\\big)\n    \\\\\n    &\\quad - \\sqrt{\\mu} \\frac{\\norm{\\nabla f(\\hat{x} + a\n        \\hat{v})}^2}{L} + (a\\sqrt{\\mu} - \\frac{1}{2}) \\langle \\nabla\n    f(\\hat{x} + a \\hat{v}), \\hat{v} \\rangle \\Big),\n    \\\\\n   \n    & \\mathfrak{B}^q_{\\operatorname{ST}}(\\hat p;a) =\n     \n    \\frac{10 \\mu ^2+L^2 \\sqrt{\\mu_s}}{32 \\mu ^{3\/2}}\n\\\\\n& \\quad \\cdot{}\\norm{2 \\sqrt{\\mu}\\hat v + \\sqrt{\\mu_s}\\nabla f(\\hat x + a \\hat v) }^2\n    \\\\\n    & \\quad \n   \n   \n    \\frac{\\mu_s \\left(4 \\mu ^2+L^2 \\sqrt{\\mu_s}\\right)}{32 \\mu\n      ^{3\/2}} \\norm{\\nabla f(\\hat x + a \\hat v)}^2\n    \\\\\n    & \\quad+\\frac{\\sqrt{\\mu_s} \\left(4 \\mu ^2+L^2\n        \\sqrt{\\mu_s}\\right)}{16 \\mu ^{3\/2}}\\norm{2 \\sqrt{\\mu}\\hat v +\n      \\sqrt{\\mu_s}\\nabla f(\\hat x + a \\hat v)}\n    \\\\\n    & \\quad \\cdot{}\\norm{\\nabla f(\\hat x + a \\hat v)}),\n    \\\\\n   \n    & \\mathfrak{C}_{\\operatorname{ST}}(\\hat p;a) = C_{\\operatorname{ST}}(\\hat p;a) ,\n   \n    \\\\\n    & \\mathfrak{D}_{\\operatorname{ST}}(\\hat p;a) = \\norm{2 \\sqrt{\\mu} \\hat v +\n      \\sqrt{\\mu_s}\\nabla f(\\hat{x} + a \\hat{v})} \\cdot\n    \\\\\n    & \\quad \\Big( \\sqrt{\\mu_s}\\norm{\\nabla f(\\hat x)} +\n    \\sqrt{\\mu}\\norm{\\hat v} \\Big).\n  \\end{align*}\n  Let $t\\mapsto p(t)$ be the high-order-hold integrator\n  trajectory~\\eqref{eq:hoh-flow} from $p(0) = \\hat{p}$.  Then, for\n  $t\\ge 0$,\n  \\begin{align*}\n    \\frac{d}{dt} V(p(t)) +\\frac{\\sqrt{\\mu}}{4} V({p}(t))\n   \n   \n   \n    & \\leq \\mathfrak{b}^{\\operatorname{d}}_{\\operatorname{ET}}(\\hat{p},t;a) \\leq \\mathfrak{b}^{\\operatorname{d}}_{\\operatorname{ST}}(\\hat{p}, t;a) .\n  \\end{align*}\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\\end{proposition}\n\nAnalogously to what we did in Section~\\ref{sec:trigger-design}, we\nbuild on this result to establish an upper bound for the\nperformance-based triggering condition with the high-order-hold\nintegrator.\n\n\\begin{proposition}\\longthmtitle{Upper bound for performance-based\n    triggering with high-order\n    hold}\\label{prop:upper-bound-performance-hoh}\n  Let $0 \\leq a$ and\n  \\begin{align}\\label{eq:bpb}\n    \\mathfrak{b}^{\\operatorname{p}}_{\\#}(\\hat{p},t;a) &= \\int_0^te^{\\frac{\\sqrt{\\mu}}{4} \\zeta}\\mathfrak{b}^{\\operatorname{d}}_{\\#}(\\hat\n    p,\\zeta;a) d\\zeta ,\n  \\end{align}\n  for $\\# \\in \\{\\operatorname{ET},\\operatorname{ST}\\}$.  Let $t\\mapsto p(t)$ be the\n  high-order-hold integrator trajectory~\\eqref{eq:hoh-flow} from $p(0)\n  = \\hat{p}$.  Then, for $t\\ge 0$,\n  \\begin{align*}\n    V(p(t)) \\!-\\! e^{-\\frac{\\sqrt{\\mu}}{4} t} V(\\hat p) \\!\\leq\\!\n    e^{-\\frac{\\sqrt{\\mu}}{4} t} \\mathfrak{b}^{\\operatorname{p}}_{\\operatorname{ET}} (\\hat{p},t;a) \\!\\leq\\!\n    e^{-\\frac{\\sqrt{\\mu}}{4} t} \\mathfrak{b}^{\\operatorname{p}}_{\\operatorname{ST}} (\\hat{p},t;a).\n  \\end{align*}\n\\end{proposition}\n\nUsing Proposition~\\ref{prop:upper-bound-derivative-hoh}, the proof of\nthis result is analogous to that of\nProposition~\\ref{prop:upper-bound-performance}, and we omit it for\nspace reasons.  Propositions~\\ref{prop:upper-bound-derivative-hoh}\nand~\\ref{prop:upper-bound-performance-hoh} are all we need to fully\nspecify the variable-stepsize algorithm based on high-order-hold\nintegrators. Formally, we set\n\\begin{align}\n  \\frak{step}^\\diamond_{\\#}(\\hat{p};a) = \\min \\setdef{t >\n    0}{\\mathfrak{b}^\\diamond_{\\#}(\\hat{p},t;a) = 0} ,\n\\end{align}\nfor $\\diamond \\in \\{{\\operatorname{d}},{\\operatorname{p}} \\}$ and $\\# \\in \\{\\operatorname{ET},\\operatorname{ST}\\}$. With this in place,\nwe design Algorithm~\\ref{algo:HOH}, which is a higher-order\ncounterpart to  Algorithm~\\ref{algo:ADG}, and whose convergence\nproperties are characterized in the following result.\n\n\\begin{algorithm}[h]\n  \\SetAlgoLined\n  %\n\n\n\\textbf{Design Choices:} $\\diamond \\in \\{{\\operatorname{d}},{\\operatorname{p}}\\}$, $\\# \\in \\{\\operatorname{ET},\\operatorname{ST}\\}$\n  \\textbf{Initialization:} Initial point ($p_0$),\n  objective function ($f$), tolerance ($\\epsilon$), increase rate\n  ($r_i > 1$), decrease rate ($0 <r_d <1$), stepsize lower bound\n  ($\\tau$), $a\\ge 0$, $k=0$\n  \\\\\n  \\While{$\\norm{\\nabla f(x_k)}\\geq \\epsilon$}{\n    increase = True\n\n    exit = False\n\n   \\While{ {\\rm exit} = {\\rm False}}{\n    \\While{$\\mathfrak{C}_{\\#}(p_k;a) \\geq 0$  }{\n       \n        $a = a r_d$\n        %\n\n        increase = False\n       \n      }\n      \\uIf{$\\frak{step}^\\diamond_{\\#}(p_k;a) \\geq \\tau$}{\n        exit = True\n     %\n      }\n     \\uElse{\n     $a = a r_d$\n    \n\n     increase = False\n     }\n\n    }\n       \n    Compute stepsize $\\Delta_k = \\frak{step}^\\diamond_{\\#}(p_k;a)$\n    \\\\\n   \n       Compute next iterate $p_{k+1}$ using \\eqref{eq:hoh-flow}\n      \\\\\n     \n  Set $k = k+1$\n\n  \\uIf{\\rm increase = True}{ $a = a r_i$ }\n\n }\n \n\n\n\n\n\n\n  \n  \\caption{Adaptive High-Order-Hold Algorithm}\\label{algo:HOH}\n\\end{algorithm}\n\n\\begin{proposition}\\longthmtitle{Convergence\n    of Adaptive High-Order-Hold Algorithm}\\label{non-zeno:foh}\n  For\n  $\\diamond \\in \\{{\\operatorname{d}},{\\operatorname{p}}\\}$, and $\\# \\in \\{\\operatorname{ET},\\operatorname{ST}\\}$, there exists\n  $\\operatorname{MIET}^\\diamond$ such that for $\\tau \\leq \\operatorname{MIET}^\\diamond$, the\n  variable-stepsize strategy in Algorithm~\\ref{algo:HOH} has the\n  following properties:\n  \\begin{enumerate}\n  \\item[(i)] it is executable (i.e., at each iteration, the parameter\n    $a$ is determined in a finite number of steps);\n  \\item[(ii)] the stepsize is uniformly lower bounded by~$\\tau$;\n  \\item[(iii)] it satisfies $ f(x_{k+1})\\!-\\!f(x_*) \\!=\\!\n    \\mathcal{O}(e^{-\\frac{\\sqrt{\\mu}}{4}\\sum_{i=0}^k \\Delta_i} )$, for\n    $k \\in \\{0\\} \\cup \\mathbb{N}$.\n  \\end{enumerate}\n\\end{proposition}\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\nWe omit the proof of this result, which is analogous to that of\nProposition~\\ref{prop:non-zeno-sampled-algorithm}, with lengthier\ncomputations.\n\n\\section{Simulations}\\label{sec:simulation}\nHere we illustrate the performance of the methods resulting from the\nproposed resource-aware discretization approach to accelerated\noptimization flows. Specifically, we simulate in two examples the\nperformance-based implementation of the Displaced Gradient algorithm\n(denoted DG$^{\\operatorname{p}}$) and the derivative- and performance-based\nimplementations of the High-Order-Hold (HOH$^{\\operatorname{d}}$ and HOH$^{\\operatorname{p}}$\nrespectively) algorithms. We compare these algorithms against the\nNesterov's accelerated gradient and the heavy-ball methods, as they\nstill exhibit similar or superior performance to the discretization\napproaches proposed in the literature, cf. Section~\\ref{sec:intro}.\n\n\n\n\\subsection*{Optimization of Ill-Conditioned Quadratic Objective\n  Function}\nConsider the optimization of the objective function\n$\\map{f}{{\\mathbb{R}}^2}{{\\mathbb{R}}}$ defined by $f(x) = 10^{-2}x_1^2 + 10^2\nx_2^2$. Note that $\\mu = 2\\cdot 10^{-2}$ and $L = 2\\cdot 10^{2}$.  We\nuse $s = \\mu\/(36L^2)$ and initialize the velocity according\nto~\\eqref{eq:initial-state}. For DG$^{\\operatorname{p}}$, HOH$^{\\operatorname{d}}$, and HOH$^{\\operatorname{p}}$,\nwe set $a=0.1$ and implement the event-triggered approach (at each\niteration, we employ a numerical zero-finding routine to explicitly\ndetermine the stepsizes $ \\operatorname{step}^{\\operatorname{p}}_{\\operatorname{ET}}$, $\\frak{step}^{\\operatorname{d}}_{\\operatorname{ET}}$, and\n$\\frak{step}^{\\operatorname{p}}_{\\operatorname{ET}}$, respectively).\n\n\nFigure~\\ref{fig:stepsize:quadratic}(a) illustrates how the stepsize of\nHOH$^{\\operatorname{p}}$ changes during the first $1000$ iterations. After the tuning\nof the stepsize during the first iterations, it becomes quite steady\n(likely due to the simplicity of quadratic functions) until the\ntrajectory approaches the minimizer.\nAfter $5$ iterations, the algorithm stepsize becomes almost equal to\nthe optimal stepsize.\n\n\\begin{figure}[htb]\n  \\centering\n  \\subfigure[]{\\includegraphics[width=0.83\\linewidth]{.\/epsfiles\/HOH_quadratic_stepsize}}\n  \\\\\n  \\subfigure[]{\\includegraphics[width=0.8\\linewidth]{.\/epsfiles\/state_space_comparison}}\n  \\caption{Ill-conditioned quadratic objective function example. (a)\n    Evolution of the stepsize along the execution of HOH$^{\\operatorname{p}}$ during\n    the first $1000$ iterations. (b) State evolution along DG$^{\\operatorname{p}}$,\n    HOH$^{\\operatorname{d}}$, HOH$^{\\operatorname{p}}$, continuous heavy-ball dynamics, and\n    Nesterov's method starting from $x = (50,50)$ and $v\n      = (-0.0023, -4.7139)$.}\\label{fig:stepsize:quadratic}\n  \\vspace*{-1ex}\n\\end{figure}\n\nFigure~\\ref{fig:stepsize:quadratic}(b) compares the performance of\nDG$^{\\operatorname{p}}$, HOH$^{\\operatorname{d}}$, and HOH$^{\\operatorname{p}}$ against the continuous heavy-ball\nmethod and the discrete Nesterov method for strongly convex\nfunctions. The DG$^{\\operatorname{p}}$ algorithm takes large stepsizes following the\nevolution\nof the continuous heavy-ball along the straight lines $p(t) = p_k +\ntX_{\\operatorname{hb}}^a(p_k)$. Meanwhile, the higher-order nature of the hold employed\nby HOH$^{\\operatorname{d}}$ and HOH$^{\\operatorname{p}}$ makes them able to leap over the\noscillations, yielding a state evolution similar to Nesterov's method.\nFigure~\\ref{fig:stepsize:quadratic:bottom} shows the evolution of the\nobjective and Lyapunov functions. We observe that after some initial\niterations, HOH$^{\\operatorname{p}}$ outperforms Nesterov's method. Eventually, also\nDG$^{\\operatorname{p}}$ catches up to Nesterov's method.\n\\begin{figure}[htb]\n  \\centering\n  \\subfigure[]{\\includegraphics[width=0.83\\linewidth]{.\/epsfiles\/HOH_quadratic_objective_comparison}}\n  \\\\\n  \\subfigure[]{\n    \\includegraphics[width=0.83\\linewidth]{.\/epsfiles\/HOH_quadratic_Lyapunov}}\n  \\caption{Ill-conditioned quadratic objective function example. (a)\n    Evolution of the logarithm of the objective function under\n    DG$^{\\operatorname{p}}$, HOH$^{\\operatorname{d}}$, HOH$^{\\operatorname{p}}$, the heavy-ball method, and\n    Nesterov's method starting from $x = (50,50)$ and $v = (-0.0023,\n    -4.7139)$. (b) Corresponding evolution of the logarithm of the\n    Lyapunov function along DG$^{\\operatorname{p}}$, HOH$^{\\operatorname{d}}$,\n    and~HOH$^{\\operatorname{p}}$.}\\label{fig:stepsize:quadratic:bottom}\n  \\vspace*{-1ex}\n\\end{figure}\n\n\\subsection*{Logarithmic Regression}\nConsider the optimization of the regularized logistic regression cost\nfunction $\\map{f}{{\\mathbb{R}}^4}{{\\mathbb{R}}}$ defined by $f(x) =\n\\sum_{i=1}^{10}\\log (1 + e^{-y_i\\langle z_i, x\\rangle}) +\n\\frac{1}{2}\\norm{x}^2$, where the points $\\{z_i\\}_{i=1}^{10} \\subset\n\\mathbb{R}^4$ are generated randomly using a uniform distribution in\nthe interval $[-5,5]$, and the points $\\{y_i\\}_{i=1}^{10} \\subset\n\\{-1,1\\}$ are generated similarly with quantized values.  This\nobjective function is $1$-strongly convex and one can also compute the\nvalue $L=177.49$. We use $a = 0.025$ and $s = \\mu\/(36L^2)$, and\ninitialize the velocity according\nto~\\eqref{eq:initial-state}. Figure~\\ref{fig:stepsize:logarithmic}(a)\nand (b) show the evolution of the stepsize along HOH$^{\\operatorname{p}}$, which\nchanges as a function of the state looking to satisfy the desired\ndecay of the Lyapunov function.\n\\begin{figure}[htb]\n  \\centering\n  \\subfigure[]{\\includegraphics[width=0.83\\linewidth]{.\/epsfiles\/HOH_logarithmic_stepsize}}\n  \\\\\n  \\subfigure[]{\n    \\includegraphics[width=0.83\\linewidth]{.\/epsfiles\/HOH_logarithmic_stepsize_comparison}}\n  \\caption{Logarithmic regression example. (a) Evolution of the\n    stepsize along the execution of HOH$^{\\operatorname{p}}$ starting from $x=\n    (50,50, 50, 50)$ and $v = (-0.1026, -0.09265, -0.1078,\n    -0.0899)$. Notice the complex pattern, with significant increases\n    and oscillations along the trajectory. (b) Difference between the\n    optimal stepsize (computed using the exact Lyapunov function,\n    which assumes knowledge of the minimizer) and the stepsize of\n    HOH$^{\\operatorname{p}}$.  The largest difference is achieved at the beginning:\n    after a few iterations, the difference descreases significantly,\n    periodically becoming almost\n    zero.}\\label{fig:stepsize:logarithmic}\n\\vspace*{-1ex}\n\\end{figure}\nFigure~\\ref{figure:zoh} shows the evolution of the objective and\nLyapunov functions. We observe how HOH$^{\\operatorname{d}}$ and HOH$^{\\operatorname{p}}$ outperform\nNesterov's method, although eventually the heavy-ball algorithm\nperforms the best.  The Lyapunov function decreases at a much faster\nrate along HOH$^{\\operatorname{d}}$ and HOH$^{\\operatorname{p}}$ than along DG$^{\\operatorname{p}}$.\n\n\\begin{figure}[htb]\n  \\centering\n  \\subfigure[]{\\includegraphics[width=0.83\\linewidth]{.\/epsfiles\/HOH_logarithmic_objective_comparison}}\n  \\\\\n  \\subfigure[]{\n    \\includegraphics[width=0.83\\linewidth]{.\/epsfiles\/HOH_logarithmic_Lyapunov}}\n  \\caption{Logarithmic regression example. (a) Evolution of the\n    logarithm of the objective function under DG$^{\\operatorname{p}}$, HOH$^{\\operatorname{d}}$,\n    HOH$^{\\operatorname{p}}$, the heavy-ball method, and Nesterov's method starting\n    from $x= (50,50, 50,50)$ and $v = (-0.1026, -0.09265, -0.1078,\n    -0.0899)$.  (b) Corresponding evolution of the logarithm of the\n  Lyapunov function along DG$^{\\operatorname{p}}$, HOH$^{\\operatorname{d}}$, and\n  HOH$^{\\operatorname{p}}$.}\\label{figure:zoh} \\vspace*{-1ex}\n\\end{figure} \n\n\\section{Conclusions}\n\nWe have introduced a resource-aware control framework to the\ndiscretization of accelerated optimization flows that specifically\ntakes advantage of their dynamical properties.  We have exploited\nfundamental concepts from opportunistic state-triggering related to\nthe various ways of encoding the notion of valid Lyapunov\ncertificates, the use of sampled-data information, and the\nconstruction of state estimators and holders to synthesize\nvariable-stepsize optimization algorithms that retain by design the\nconvergence properties of their continuous-time counterparts.\nWe believe these results open the way to a number of exciting research\ndirections.  Among them, we highlight the design of adaptive learning\nschemes to refine the use of sampled data and optimize the algorithm\nperformance with regards to the objective function, the\ncharacterization of accelerated convergence rates, employing tools and\ninsights from hybrid systems for analysis and design, enriching the\nproposed designs by incorporating re-start schemes as triggering\nconditions to avoid overshooting and oscillations, and developing\ndistributed implementations for network optimization problems.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe discovery of new materials that meet specific requirements e.g. in terms stability, compatibility, or physical properties, is an exciting scientific challenge of great relevance for our society.  First-principles quantum mechanical calculations, e.g. based on density functional theory (DFT)\\cite{kohn1965self}, can predict the structure and properties of materials with high accuracy even before they are made in the lab. However, a DFT code by itself is insufficient for realising the paradigm of inverse materials design.\n\nConsidering the vast number of possible materials and the complexity of general structure-property relations, it becomes clear that successful inverse design relies on the following critical components: (i) automated execution and management of large numbers of atomistic calculations, (ii) access to large amounts of relevant high quality materials data, and (iii) efficient algorithms that can propose new candidate materials from data. In addition, synthesis and characterisation experiments must be included in the loop as well, but this aspect will not be considered here.    \n\nComponents (i) and (ii) are largely in place. Indeed, the advent of workflow management engines for computational materials science\\cite{curtarolo2012aflow,jain2015fireworks,pizzi2016aiida,gjerding2021atomic,mortensen2020myqueue} have made it possible to perform high-throughput (HT) computations for thousands of materials with minimal human intervention\\cite{greeley2006computational,madsen2006automated,curtarolo2013high,kirklin2013high,ornso2013computational,zhang2019computational,chen2016understanding,hachmann2011harvard,bhattacharya2015high,castelli2012computational,hautier2013identification,yu2012identification,kuhar2018high,aykol2016high,mounet2018two,chen2015design}. Atomic structures and basic materials properties from such HT studies have been stored in computational databases\\cite{thygesen2016making,himanen2019data,saal2013materials,jain2013commentary,borysov2017organic,curtarolo2012aflow,draxl2019nomad,haastrup2018computational,cheon2017data,gjerding2021recent}, which together contain results of millions of DFT calculations. Complemented by experimental crystal structure databases, this makes a rich and rapidly growing data source for materials science. \n\nThe main challenge concerns component (iii). In previous HT studies, the candidate materials to be explored were mostly produced by lattice decoration of known reference materials. An obvious limitation of this approach is that the resulting materials by construction will be similar to the reference materials. In particular, the 3-tuple: (space group, occupied Wyckoff positions, stoichiometry) is invariant under element substitution. \n\nGenerative machine learning algorithms could potentially broaden the diversity of candidate materials beyond the lattice decoration paradigm. \nHowever, designing a successful generative model for periodic materials has proved challenging due the problem of creating representations of the lattice, atomic coordinates and elemental composition that are both invariant to translations and rotations and is invertible\\cite{Noh2020Machine}. The vast chemical space of elements that can be present in inorganic crystals further complicates the design of representations. Therefore, previous implementations of generative models for periodic materials have either been limited to a fixed subset of chemical elements \\cite{Noh2019Inverse,Kim2020Generative,Long2021Constrained} and\/or a subset of possible crystal structures \\cite{Zhao2021High,Song2021Computational}. Recently, a general invertible representation has been proposed\\cite{Ren2022invertible}, which encodes the material as a matrix of both real and reciprocal space features, but is not invariant under translations and rotations. Xie \\textit{et al.} developed a crystal diffusion variational autoencoder (CDVAE) model \\cite{xie2021crystal}, which uses a generative diffusion model to circumvent the need for an invertible representation and employs an equivariant graph neural network to ensure invariance (in fact, equivariance).\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[scale=0.5]{figures\/workflow_generative3.pdf}\n    \\caption{Workflow to generate candidate 2D materials using the CDVAE generative model (left branch) and lattice decoration (right branch). The same set of 2615 materials is used to train the CDVAE model and as seed structures for lattice decoration, respectively. Black numbers indicate the number of materials present at a given step of the workflow while orange numbers indicate number of materials discarded.}\n    \\label{fig:workflow}\n\\end{figure*}\n\n\nIn this work, we train a CDVAE\\cite{xie2021crystal} on 2615 2D materials with formation energy up to 0.3 eV\/atom from the convex hull, and generate 10000 two-dimensional (2D) crystals. We compare these structures to a set of 14192 2D crystals obtained by systematic lattice decoration of the training structures. While Ref. \\cite{xie2021crystal} assessed validity and diversity of the generated crystals by means of qualitative measures, such as charge neutrality and minimum bond distance, we here conduct a systematic, unbiased quantitative analysis by performing full DFT-based relaxations and stability analysis of the generated structures. Compared to the crystals in the training set, the structures generated by the CDVAE (after DFT relaxation) show similar formation energies but significant differences in both composition and crystal structure. In general, CDVAE seems able to produce more complex materials without compromising stability. \n\nAs a direct test of the CDVAE model's capacity to learn the stability properties of the training structures, we also train a CDVAE on materials lying at least 0.4 eV\/atom above the hull. We find that the structures generated by this model have significantly higher formation energies that those produced by the CDVAE trained on the more stable materials. \n\nIn addition to providing a quantitative assessment of the CDVAE, our work identifies no less than 8599 new unique 2D materials with an energy above the convex hull below $\\SI{0.3}{eV\/atom}$ many of which could potentially be synthesised. The generated crystal structures are freely available as part of the C2DB\\cite{haastrup2018computational}. \n\n\n\n\n\\section{Results and discussion}\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=1\\linewidth]{figures\/Periodic_table_Dist.pdf}\n\\caption{Heat map of the relative occurrence of each element in the 2D materials used to train (seed) the CDVAE (LDP). The middle row shows the element substitutions for the LDP corresponding to $P_{AB}>0.1$. The relative occurrence is shown in the last row.}\n\\label{fig:pt}\n\\end{figure*}\n\n\n\n\\subsection{Crystal diffusion variational autoencoder}\n\nThe CDVAE combines a variational autoencoder\\cite{kingma2013autoencoder} and a diffusion model to generate new periodic materials. The crystal is represented by a tuple consisting of the atomic number of the $N$ atoms, their respective coordinates, and the unit cell basis vectors. CDVAE consist of three networks: the encoder, a property predictor, and the decoder. The encoder is a SE(3) equivariant periodic graph neural network (PGNN), which encodes the material onto a lower dimensional latent space from which the property predictor predicts the number of atoms $N$, the lattice vectors, and the composition, which is the fraction present of each element.  The decoder is a noise conditional score network diffusion model\\cite{song2019generative} that takes a structure with noise added to the atom types and coordinates and learns to denoise it into the original stable structure. Here the score is an estimate of the gradient of the underlying probability distribution of the materials and is predicted by another SE(3) equivariant PGNN. The three networks are trained concurrently.\n\nNew materials can be generated after training by using the property predictor to sample the latent space. A unit cell with the predicted basis vectors is then initialised with the predicted atoms placed at random positions. Using the decoder, the atom types and coordinates of the initial random unit cell are then gradually denoised into a material that is similar to the data distribution of the training data. CDVAE utilizes that adding noise to a stable material will likely decrease its stability and, thus, by learning to denoise the noisy stable structure, the decoder learns to increase the stability of the structure. Therefore CDVAE should be trained only on stable materials.\nAn in-depth description of CDVAE can be found in Xie \\textit{et al.} \\cite{xie2021crystal}. \n\n\nThe set of materials used as training data for the CDVAE and seed structures for the lattice decoration protocol (LDP), respectively, consists of 2615 unique 2D materials from the C2DB\\cite{haastrup2018computational,gjerding2021recent}. As our aim is to discover new stable materials we limited the initial set of materials to the subset of C2DB with energy above the convex hull $\\Delta H_{\\mathrm{hull}}< \\SI{0.3}{eV\/atom}$. This was done because both the CDVAE (LDP) are more likely to generate stable materials when trained on (seeded by) stable materials. We did not exclude dynamically unstable materials.\n\nAfter training the CDVAE model, 10.000 structures were generated of which 1106 failed CDVAE's basic validity check (charge neutrality and bond lengths above 0.5 \u00c5). Of the remaining 8894 structures, 3891 are duplicate structures which are sorted out (see section \\ref{sec: Method} for more details) and the rest are relaxed using DFT.\n\n\n\\subsection{Lattice decoration protocol}\n\nThe lattice decoration protocol (LDP) substitutes the atoms in the seed structures by atoms of similar chemical nature. As a measure of chemical similarity we use the probability matrix $P_{AB}$ introduced by Glawe \\textit{et al.}\\cite{glawe2016optimal}, which describes the likelihood that a stable material containing a chemical element $A$ remains stable after the substitution $A\\to B$. Glawe \\textit{et al.} constructed this probability matrix based on an analysis of materials in the Inorganic Crystal Structure Database\\cite{belsky2002new}. We choose a substitution probability of 10 \\% ($P_{AB}>0.1$), which generates the substitutions shown in Fig. \\ref{fig:pt}. Based on these substitution relations, we perform all possible single and double substitutions for all seed structures. For example, the seed structure MoS$_2$ generates six MX$_2$ structures with M=Mo,W and X=O, S, Se (the seed structure itself included). The total set of resulting materials are analysed for structures that share the same reduced formula and space group. Such structures are considered as duplicate structures and are filtered out. After removal of duplicates, we are left with 14192 unique 2D crystals (the seed structures excluded) which are relaxed using DFT. \n\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=1\\linewidth]{figures\/hist_combined.pdf}\n\\caption{Histogram of the heat of formation and energy above convex hull for the DFT-relaxed structures resulting from the CDVAE (a, b) and LDP (c, d) methods. The inset shows the energy above convex hull with respect to the number of unique elements in the structure.}\n\\label{fig:hist_thermo}\n\\end{figure*}\n\n\\subsection{Workflow}\nOur workflow is illustrated in Fig. \\ref{fig:workflow}. Starting with the initial set of 2D materials, we generate two new sets of crystal structures using CDVAE and LDP, respectively. Duplicate structures within each set are removed (see section \\ref{sec: Method} for more details). The now unique crystal structures are relaxed using DFT calculations employing the PBE xc-functional (see section \\ref{sec: Method} for more details). After the relaxation, any new duplicate structures are removed again and as are materials that have relaxed into non 2D structures (we refer to Ref. \\cite{gjerding2021recent} for details on the dimensionality analysis). Finally the heat of formation, $\\Delta H$, and the energy above convex hull, $\\Delta H_{\\mathrm{hull}}$, are calculated. \n\n\n\\begin{table}[]\n\\begin{tabularx}{0.45 \\textwidth}{ \n   >{\\raggedright\\arraybackslash\\hsize=.6\\hsize}X \n   \n   >{\\centering\\arraybackslash\\hsize=.2\\hsize}X \n   >{\\raggedleft\\arraybackslash\\hsize=.2\\hsize}X  }\n\\hline\n                     & LDP   & CDVAE                          \\\\ \\hline\nSuccess rate         & 82 \\%                       & 69 \\%                        \\\\\nAvg. number of steps & 40.1       & 55.5\\\\\nAvg. energy decrease $\\left[  \\si{eV\/atom} \\right]$ & 0.62  & 0.51  \\\\ \\hline\n\\end{tabularx}\n\\caption{Summary statistics for the DFT relaxation of the two methods for generating initial structures.}\n    \\label{tab:statistics}\n\\end{table}\n\n\n\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=1\\linewidth]{figures\/kde_ehull.pdf}\n    \\caption{Kernel density estimate showing the distribution of the convex hull energies for the stable and unstable CDVAE generated dataset as well as their training data.}\n    \\label{fig:kde}\n\\end{figure}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=1\\linewidth]{figures\/cdvae_stable_exambles.png}\n\\caption{(a-g) Examples of CDVAE generated materials with negative convex hull energies. (h, i) Examples of CDVAE generated stable materials with the new discovered combination of stoichiometry ABC$_2$D$_2$, space group number 25 and occupied Wyckoff positions a,b,c,d.}\n\\label{fig:cdvae_examples}\n\\end{figure*}\n\nIn Table \\ref{tab:statistics} we report the success rates for the DFT relaxations of the structures generated by CDVAE and LDP, respectively, together with the average number of relaxation steps and the average energy decrease from the initial to the relaxed structure. All three parameters are assumed to describe \nhow close the initial structures are to the final DFT relaxed structures - e.g. a structure from a perfect generative method would only need one relaxation step and the energy decrease would be zero. As expected, neither LDP or CDVAE generate stable relaxed structures. However, while the LDP on average requires less steps to relax, the CDVAE structures are closer in energy to the relaxed structure. The fact that the number of relaxation steps and reduction in energy upon relaxation is comparable for LDP and CDVAE, suggest that the CDVAE-generated crystals are as close to relaxed structures as the LPD-generated structures. \n\nWe observe that the DFT relaxation fails for about 18 \\% of the LDP-generated and about 31 \\% of the CDVAE-generated structures. The vast majority of these failures are due to problems in converging the Kohn-Sham SCF cycle. We suspect that a large fraction of the convergence problems occur for materials with magnetic ground state (all calculations are performed with spin polarisation). This is supported by the fact that 30 \\% of the materials containing one or more of the magnetic 3d-metals (V, Cr, Mn, Fe, Co, Ni), fails due to convergence errors, while this is only happens for 10 \\% of other materials. \n\n\n\\subsection{Thermodynamic stability}\n\nA histogram of the heat of formation and the energy above the convex hull for the (DFT-relaxed) structures resulting from the CDVAE and LDP are shown in Fig. \\ref{fig:hist_thermo}. The distributions of both $\\Delta H$ and $\\Delta H_{\\mathrm{hull}}$ obtained for the two structure generation methods are remarkably similar. For example, 73.8 \\% of the CDVAE materials have $\\Delta H_{\\text{hull}}$ below $\\SI{0.3}{eV\/atom}$ (as the training data) while this is the case for 74.0 \\% of the LDP materials. It should, however, be noted that the smaller success rate of the DFT relaxation of the CDVAE generated materials could influence these statistics as it likely that many of the structures which could not be converged would have resulted in unstable structures. The inset of Fig. \\ref{fig:hist_thermo} shows how the energy above the convex hull is distributed depending on the number of different elements in the structure. First of all it is evident that CDVAE is able to create structures with a larger number of unique elements than is present in the training data (5 unique elements is the maximum in the seed structures), while LDP is limited to the stoichiometries present in the seed materials. However, generally the thermodynamic stability is lower for the materials with larger number of unique elements. Examples of some of the most stable CDVAE generated structures is shown in Fig. \\ref{fig:cdvae_examples}. The material Zr$_2$CCl$_2$ shown in c) is one of the 22 materials which are found both by the CDVAE and LDP method.\n\nTo predict whether a given 2D material can be synthesized is a complex problem that involves many factors. Often the size of $\\Delta H_{\\mathrm{hull}}$ is used a soft criterion for synthesizability as it determines the material's thermodynamic stability relative to other competing phases (this criterion neglects growth kinetics and substrate interactions both of which can be important for 2D materials). A previous study of 700 polymorphs in 41 common inorganic bulk material systems showed that a threshold of $\\Delta H_{\\mathrm{hull}}<0.1$ eV\/atom will exclude 26\\% of the known synthesized polymorphs\\cite{aykol2018thermodynamic}. We also note that the T-phase of MoS$_2$ was synthesised both as a monolayer\\cite{kappera2014phase} and a layered bulk\\cite{bell1957preparation}, despite having $\\Delta H_{\\mathrm{hull}}=0.18$ eV\/atom\\cite{urlc2db}. These examples demonstrate that many of the predicted 2D materials with $\\Delta H_{\\mathrm{hull}}<50$ meV\/atom (2004) or even $\\Delta H_{\\mathrm{hull}}<100$ meV\/atom (3400), are likely to be synthesizable.  \n\nWhile the $\\Delta H_{\\mathrm{hull}}$-distributions in Fig. \\ref{fig:hist_thermo} are clearly peaked close to zero they also have a tail of less stable materials. In particular, about 26 \\% of the materials have $\\Delta H_{\\text{hull}}>\\SI{0.3}{eV\/atom}$ (the threshold to select the training structures). A natural question to ask is then to what extent the structures produced by the CDVAE are in fact biased towards high stability structures? To answer this question, we trained a CDVAE model on 988 2D materials with a $\\Delta H_{\\text{hull}}>\\SI{0.4}{eV\/atom}$ and used it to generate another 10.000 structures from which we randomly selected 1000 non-duplicate structures, which we relaxed following the same workflow as described before. The distribution of the energy above the convex hull of the relaxed structures for both the stable and unstable CDVAE models are shown in Fig. \\ref{fig:kde} together with the distribution of their respective training sets. We clearly see that the CDVAE model trained to generate unstable materials produces structures that are significantly further from the convex hull than the stable model. This illustrates that CDVAE successfully learns the chemistry of the materials in the training data.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=1\\linewidth]{figures\/Structure_combined_no_TSNE.pdf}\n\\caption{Relative frequency of the stoichiometry, space group number and occupied Wyckoff positions for each of the data set.}\n\\label{fig:structure}\n\\end{figure*}\n\n\n\\subsection{Structural diversity}\n\nHaving established the capability of the CDVAE to produce materials with good stability properties, we now turn to its ability to generate crystals of high chemical and structural diversity. While the LDP is restricted to stoichiometries and crystal structures already present in the seed structures, the CDVAE (in principle) has no such limitations. Fig. \\ref{fig:pt} shows the relative occurrence of each element in the seed\/training structures. The corresponding plots for the materials generated by LDP and CDVAE (after relaxation) are shown in Fig. 1 in the Supplemental Information. Both LDP and CDVAE produces diverse compositions with elements covering most of the periodic table. However, CDVAE has a significantly higher occurrence of oxygen and chalcogens (S and Se) as well as halogens (Cl, Br and I). This trend is also present for the materials prior to relaxation and, thus does not originate from a potential higher DFT convergence rate for these elements. Instead, the six elements are also more prevalent, albeit slightly, in the seed structures which could indicate an overfitting of the model.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1\\linewidth]{figures\/TSNE_compare_annotated.pdf}\n\\caption{t-SNE embedding of the structure\nrepresented as a individual one-hot encoding of the stoichiometry, space group, and occupied Wyckoff positions. Selected clusters are highlighted as 'stoichiometry'-'space group'-'Wyckoff position'. 'X' corresponds to an  arbitrary stoichiometry.}\n\\label{fig:tSNE}\n\\end{figure}\n\n\n\nThe CDVAE generates significantly different chemical compositions and crystal structures as compared to the seed structures and those generated by the LDP. Fig. \\ref{fig:structure} shows the relative frequencies of stoichiometry, space group number and occupied Wyckoff positions, respectively. Only the most common classes of the seed structures are shown. We find 239 unique stoichiometries among the CDVAE-generated materials, while there is only 87 and 103 unique stoichiometries in the seed structures and LDP-generated structures, respectively. The higher number of unique stoichiometries in the LDP-generated structures than in the seed structures is due to new stoichiometries being created when two different elements are substituted by the same element, or when an element is being substituted with an element already present in the seed material. For example, the seed materials Te$_2$Cu$_4$O$_{12}$ (stoichiometry AB$_2$C$_6$) becomes Cu$_4$S$_{14}$ (stoichiometry A$_2$B$_7$) under the double substitution O$\\to$S and Te$\\to$S.\nThe significantly larger number of unique stoichiometries generated by CDVAE compared to the LDP shows that the former is able to produce new classes of structures that are not present in the training data.\n\nThe CDVAE tends to generate rather complex, low-symmetry structures, which is illustrated by the large fraction of materials with space group number 1. Moreover, the average number of different elements in the unit cell is 4.0 for the CDVAE generated materials while it is only 2.6 for the C2DB seed structures. The larger number of different elements is part of the reason for the higher fraction of materials with low symmetry. This tendency of CDVAE to generate structures with more complex composition is also noted by Xie \\textit{et al.}, who attributes this to a non-Gaussian distribution of the underlying structure of the materials. Thus, when CDVAE generates new materials it samples from a Gaussian distribution $\\mathcal{N}(0,1)$ from which it predicts the number of atoms and composition. However if $\\mathcal{N}(0,1)$ is not representative of the latent space, out of distribution materials can be generated. For materials discovery this could, however, be advantageous as this makes CDVAE able to generate new crystal types which are not present in the training data.\n\nTo give a global overview of the structural distribution of the three data sets, a t-SNE embedding is shown in Fig. \\ref{fig:tSNE}. The t-SNE analysis is made for 2500 materials sampled randomly from each data set. Here the structure is represented as a tuple given by the space group, occupied Wyckoff positions, and stoichiometry, where each is one-hot encoded before the t-SNE embedding. We see that most of the training data form clear clusters, which represent the most common stoichiometries, space group and Wyckoff positions. The LDP generated materials mostly follow the same pattern as the seed structures. However, the CDVAE generated structures are more spread out, which is partly due to the large variation in their stoichiometries, while a few clusters appear due to the large fraction of low symmetry materials with space group number 1. One noteworthy example is the cluster of CDVAE generated materials with stoichiometry ABC$_2$D$_2$, space group number 25 and occupied Wyckoff positions a,b,c,d. For this specific combination, CDVAE discovered 123 new materials of which 30 lies within 50 meV of the convex hull, while there is no examples of such materials in the training set nor in the LDP generated structures. Two of the most stable discovered materials of this type can be seen in Fig. \\ref{fig:cdvae_examples} (h, i). The new class of structures have broken out-of-plane symmetry either due to the outermost atoms (b) or the innermost atoms (a). The fact that the CDVAE is able to generate new classes of stable materials, which are not present in the training data, is very promising and a clear advantage of deep generative models compared to lattice decoration protocols.\n\n\\section{Conclusions}\nIn conclusion, we have successfully employed a deep generative model in combination with a systematic lattice decoration protocol (LDP) to generate more than 8500 unique 2D crystals with formation energies ($\\Delta H$) within 0.3 eV\/atom of the convex hull. Out of these, more than 2000 have $\\Delta H$ within 50 meV\/atom of the convex hull, and could potentially be synthesized. This represents at least a doubling of the known stable 2D materials.  \n\nIn addition to the very significant expansion of the known space of 2D materials, our work provides a quantitative assessment of the crystal diffusion variational autoencoder (CDVAE)\\cite{xie2021crystal}, and establishes its excellent performance with respect to the two key criteria: ability to learn the stability properties of the training structures, and ability to generate crystals with high chemical and structural diversity. In fact, only 25\\% of the generated materials had $\\Delta H_{\\mathrm{hull}}$ above the 0.3 eV\/atom threshold used to select the training structures, and the stoichiometries of the generated materials span 239 types versus 87 present in the training structures. Generally, the crystal structures generated by CDVAE have higher complexity and lower symmetries than the training structures. We found the method of lattice decoration to be complementary to the CDVAE generator with the two methods yielding only 22 common crystals out of the 11630 structures generated in total. While the LDP is limited to the structural blueprint of the seed materials, CDVAE is able to generate new classes of materials, which are not present in the initial data set. This is promising for an autonomous materials discovery method as it adds new genes to pool of trial materials and thus goes beyond the lattice decoration paradigm. \n\nThe fact that CDVAE is comparable to lattice decoration (with substitution by chemically similar elements) in terms of stability while producing new and diverse crystal structures, is a testimony to the prospect of using deep generative models in materials discovery. \n\nAll the structures are available in the C2DB database\\cite{urlc2db}, and their basic properties will also be made available as the execution of the C2DB property workflow proceeds.\n\n\n\n\\section{Method} \\label{sec: Method}\n\n\\subsection{Workflow}\nTo set up and manage the workflow we use the Atomic Simulation Recipes\\cite{gjerding2021atomic}, which has implemented tools for DFT relaxation, duplicate removal, dimensionality check, and for calculating the thermodynamic properties. The DFT calculations are performed using the GPAW code \\cite{Mortensen2005real} with the PBE xc-functional, a plane wave cut-off energy of $ \\SI{800}{eV}$ and a \\textit{k}-point density of at least  $\\SI{4}{\u00c5}$. The relaxation is stopped when the maximum force is below $ \\SI{0.01}{eV\/\u00c5}$ and the maximum stress is below $ \\SI{0.002}{eV\/\u00c5^3}$.\n\nThe duplicate removal recipe finds duplicate structures using the root mean square distance (RMSD) between the structures which is calculated using the Python library pymatgen\\cite{ong2013Python}. We consider structures to be duplicate if RMSD$<0.3$ \u00c5 and only keep the structure with the lowest heat of formation. See Ref. \\cite{gjerding2021recent} for more information. For the initial LDP generated materials (before the DFT relaxation) a more crude duplicate sorting of the structures is employed, where two materials with the same reduced formula and space group are considered duplicates.\\\\\n\n\nTo determine the convex hull we use as reference databases the C2DB as well as a database of reference structures comprising 9590 elementary, binary, and ternary crystals that all lie within 20 meV of the convex hull in the Open Quantum Materials Database (OQMD)\\cite{saal2013materials}. These reference structures were relaxed using the VASP\\cite{kresse1993ab} code at the PBE level (PBE+U for selected transition metal oxides) as part of the OQMD project. Since we use the GPAW code to relax and evaluate the energy of the 2D materials, we have re-calculated the total energy of the reference structures (without re-optimisation) using the GPAW code. \n\n\\subsection{CDVAE}\nCDVAE is designed to generate 3D bulk crystals, where the unit cell is periodic in all three directions. This introduces a problem when generating 2D materials, which are non-periodic in one direction. We solve this issue by introducing an artificial periodicity in the non-periodic direction with a lattice vector which is an order of magnitude larger than those in the periodic directions. This ensures that the graph networks only connect atoms in the 2D layer and thus CDVAE learns to generate 2D materials. When training the model, we used 70 \\% of the materials in the training set, while 15 \\% was used for validation and 15 \\% for test. We used the same hyperparameters as employed by Xie \\textit{et al.} for their MP-20 data set. See Ref. \\cite{xie2021crystal} for more information.\n\n\\section{Acknowledgements}\nWe thank Morten N. Gjerding for assistance with setting up the lattice decoration protocol. \nWe acknowledge funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program Grant No. 773122 (LIMA) and Grant agreement No. 951786 (NOMAD CoE). K. S. T. is a Villum Investigator supported by VILLUM FONDEN (grant no. 37789).\n\n\\section{Author contributions}\nP.L. and K.S.T. developed the initial concept. P.L. ran the generative models, the DFT simulations and performed the data analysis. K.S.T. supervised the project and aided with the interpretation of the results. P.L. and K.S.T wrote and discussed the paper together.\n\n\\section{Competing interests}\nThe authors declare no competing interests.\n\n\\section{Data availability}\nAll the discovered crystal structures and their properties will be available as a part of C2DB (\\url{https:\/\/cmr.fysik.dtu.dk\/c2db\/c2db.html})\n\n\n\\section{Periodic table heat map}\n\\vspace{-0.7cm}\n\n\n\n\n\n\n\\begin{figure}[H]\n\\centering\n\\begin{subfigure}[b]{0.99\\textwidth}\n   \\includegraphics[width=1\\linewidth]{figures\/Periodic_table_CDVA.pdf}\n   \\caption{}\n   \\label{fig:pt_CDVAE}\n\\end{subfigure}\n\n\\begin{subfigure}[b]{0.99\\textwidth}\n   \\includegraphics[width=1\\linewidth]{figures\/Periodic_table_LDP.pdf}\n   \\caption{}\n   \\label{fig:pt_decorated}\n\\end{subfigure}\n\n\\caption{Heat map of the relative occurrence of each element in the CVAE and LDP generated and relaxed data set. The bottom number is the relative occurrence.}\n\\label{fig:pt}\n\\end{figure}\n\n\n\\vspace{-0.2cm}\n\n\n\\section{Acknowledgements}\n\\vspace{-0.2cm}\nWe acknowledge funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program Grant No. 773122 (LIMA) and Grant agreement No. 951786 (NOMAD CoE). K. S. T. is a Villum Investigator supported by VILLUM FONDEN (grant no. 37789).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Proof for Theorem \\ref{thm:hsequence}}\n\\label{app:thm:hsequence}\n\n\\textbf{Theorem \\ref{thm:hsequence}: }\n\\textit{\nFor any $\\hsic_0$, there exists a set of bandwidths $\\sigma_l$ and a \\KS $\\{\\fm_{l^{\\circ}}\\}_{l=1}^L$ parameterized by $W_l = W_s$ in \\eq{eq:trivial_W} such that: }\n\\begin{enumerate}[topsep=0pt, partopsep=0pt, label=\\Roman*.]\n    \\item \n     $\\hsic_L$ can approach arbitrarily close to $\\hsic^*$ such that for any $L>1$ and $\\delta>0$ we can achieve\n     \\begin{equation}\n        \\hsic^{*} - \\hsic_L \\le \\delta,\n        \\label{arb_close_append}\n    \\end{equation}   \n    \\item \n    as $L \\rightarrow \\infty$, the \\RS converges to the global optimum where\n    \\begin{equation}\n        \\lim_{L \\rightarrow \\infty}  \\hsic_L = \\hsic^*,\n    \\end{equation}   \n    \\item \n    the convergence is strictly monotonic where \n    \\begin{equation}\n    \\hsic_{l} > \\hsic_{l-1} \\quad \\forall l \\ge 1.\n    \\end{equation}   \n\\end{enumerate}\n\n\\begin{lemma}\n\\label{app:lemma:lowerbound}\nGiven $\\sigma_0$ and $\\sigma_1$ as the $\\sigma$ values from the last layer and the current layer, then there exists a lower bound for $\\hsic_l$, denoted as $\\lb\\Lsigma$ such that \n\n\n\\begin{equation}\n    \\hsic_l \\ge \\lb\\Lsigma.\n\\end{equation}\n\\end{lemma}\n\n\\textbf{Basic Background, Assumptions, and Notations. }\n\\begin{enumerate}\n    \\item\n    The simulation of this theorem for Adversarial and Random data is also publicly available on \\url{https:\/\/github.com\/anonymous}.\n    \\item  Here we show that this bound can be established given the last 2 layers.  \n    \\item  $\\sigma_0$ is the $\\sigma$ value of the previous layer\n    \\item  $\\sigma_1$ is the $\\sigma$ value of the current layer\n    \\item  $\\nclass$ is the number of classes\n    \\item  $n$ is total number of samples\n    \\item  $n_i$ is number of samples in the $i^{th}$ class\n    \\item  $\\cS$ is a set of all $i,j$ sample pairs where $r_i$ and $r_j$ belong to the same class.\n    \\item  $\\cS^c$ is a set of all $i,j$ sample pairs where $r_i$ and $r_j$ belong to different same classes.\n    \\item  $\\cS^{\\beta}$ is a set of all $i,j$ sample pairs that belongs to the same $\\beta^{th}$ classes.\n    \\item  $r_i^{(\\alpha)}$ is the $i^{th}$ sample in the $\\alpha^{th}$ class among $\\nclass$ classes.\n    \\item  We assume no $r_i \\ne r_j$ pair are equal $\\forall i \\ne j$. \n    \\item  \n    Among all $r_i \\ne r_j$ pairs, there exists an optimal $r_i^*, r_j^*$ pair where  \n    $\\langle r_i^*, r_j^* \\rangle \\ge \\langle r_i, r_j \\rangle$ $\\forall r_i \\ne r^*_i$ and $r_j \\ne r^*_j$. We denote this maximum inner product as \n    \\begin{equation}\n        \\ub = \\langle r_i^*, r_j^* \\rangle.\n    \\end{equation}\n    \\item Here, each $r_i$ sample is assumed to be a sample in the RKHS of the Gaussian kernel, therefore all inner products are bounded such that\n    \\begin{equation}\n        0 \\le \\langle r_i, r_j \\rangle \n        \\le\n        \\ub.\n    \\end{equation}\n    \\item We let $W$ be \n    \\begin{equation}\n        W_s = \\frac{1}{\\sqrt{\\zeta}} \\W.\n    \\end{equation}\n    Instead of using an optimal $W^*$ defined as $W^{*} = \\argmax_{W} H_{l}(W)$, we use a suboptimal $W_s$ where each dimension is simply the average direction of each class: $\\frac{1}{\\sqrt{\\zeta}}$ is a unnecessary normalizing constant $\\zeta = ||W_{s}||_{2}^{2}$. By using $W_s$, this implies that the $\\hsic$ we obtain is already a lower bound compare $\\hsic$ obtained by $W^*$. But, we will use this suboptimal $W_s$ to identify an even lower bound. Note that based on the definition $W^{*}$, we have the property $\\hsic(W^{*}) \\geq \\hsic(W) \\, \\forall W$.\n\n    \\item  We note that the objective $\\hsic$ is\n    \\begin{equation}\n        \\hsic = \n        \\underbrace{\n        \\sums \\Gij \\ISMexpC{}{}}_{\\within}\n        -\n        \\underbrace{\n        \\sumsc |\\Gij| \\ISMexpC{}{}}_{\\between}\n    \\end{equation}\n    where we let $\\within$ be the summation of terms associated with the within cluster pairs, and let $\\between$ be the summation of terms associated with the between cluster pairs.\n\\end{enumerate}\n    \n   \n    \n\\begin{proof} $\\hspace{1pt}$\n\\begin{adjustwidth}{0.5cm}{0.0cm}\n    The equation is further divided into smaller parts organized into multiple sections.\n    \n    \\textbf{For sample pairs in $\\cS$. } The first portion of the function can be split into multiple classes where\n    \\begin{equation}\n    \\within = \n    \\underbrace{\n    \\sum_{\\cS^1} \\Gij \\ISMexpC{(1)}{(1)}\n    }_\\text{$\\within_1$}\n    + ... + \n    \\underbrace{\n    \\sum_{\\cS^{\\nclass}} \\Gij \\ISMexpC{(\\nclass)}{(\\nclass)}}_\\text{$\\within_{\\nclass}$}\n    \\end{equation}\n    Realize that to find the lower bound, we need to determine the minimum possible value of each term which translates to \\textbf{maximum} possible value of each exponent. Without of loss of generality we can find the lower bound for one term and generalize its results to other terms due to their similarity. Let us focus on the numerator of the exponent from $\\within_1$. Given $W_s$ as $W$, our goal is identify the \\textbf{maximum} possible value for\n    \\begin{equation}\n        \\underbrace{\n        \\rij{1}{1}^TW}_{\\Pi_1}\n        \\underbrace{\n        W^T\\rij{1}{1}}_{\\Pi_2}. \n    \\end{equation}\n    Zoom in further by looking only at $\\Pi_1$, we have \n    the following relationships\n    \\begin{equation}\n       \\Pi_1 = \n        \\underbrace{\n            r_i^{(1)^T} W}_{\\xi_1}\n        - \n        \\underbrace{r_j^{(1)^T} W}_{\\xi_2} \n    \\end{equation}\n    \\begin{align}\n        \\xi_1 & = \\frac{1}{\\sqrt{\\zeta}} r_i^{(1)^T} \\W \\\\\n        & = \\frac{1}{\\sqrt{\\zeta}}\n            r_i^{(1)^T} \n            \\begin{bmatrix}\n                (r_1^{(1)} + ... + r_{n_1}^{(1)}) &\n                ... &\n                (r_1^{(\\nclass)} + ... + r_{n_{\\nclass}}^{(\\nclass)})\n            \\end{bmatrix}\n    \\end{align}\n    \\begin{align}\n        \\xi_2 & = \n        \\frac{1}{\\sqrt{\\zeta}}\n        r_j^{(1)^T} \\W \\\\\n        & = \n        \\frac{1}{\\sqrt{\\zeta}}\n        r_j^{(1)^T} \n            \\begin{bmatrix}\n                (r_1^{(1)} + ... + r_{n_1}^{(1)}) &\n                ... &\n                (r_1^{(\\nclass)} + ... + r_{n_{\\nclass}}^{(\\nclass)})\n            \\end{bmatrix}\n    \\end{align}   \n    By knowing that the inner product is constrained between $[0,\\ub]$, we know the maximum possible value for $\\xi_1$ and the minimum possible value for $\\xi_2$ to be\n     \\begin{align}\n        \\xi_1 & = \n        \\frac{1}{\\sqrt{\\zeta}}\n            \\begin{bmatrix}\n                1 + (n_1 - 1)\\ub &\n                n_2 \\ub &\n                n_3 \\ub &\n                ... &\n                n_{\\nclass} \\ub\n            \\end{bmatrix} \\\\\n         \\xi_2 & = \n         \\frac{1}{\\sqrt{\\zeta}}\n            \\begin{bmatrix}\n                1 &\n                0 &\n                0 &\n                ... &\n                0\n            \\end{bmatrix}. \n    \\end{align}   \n    Which leads to  \n    \\begin{equation}\n        \\Pi_1 = \n        \\frac{1}{\\sqrt{\\zeta}}\n        (\\xi_1 - \\xi_2) = \n        \\frac{1}{\\sqrt{\\zeta}}\n            \\begin{bmatrix}\n                (n_1 - 1)\\ub &\n                n_2 \\ub &\n                n_3 \\ub &\n                ... &\n                n_{\\nclass} \\ub\n            \\end{bmatrix}\n    \\end{equation}\n    Since $\\Pi_2^{T}  = \\Pi_1$ we have \n    \\begin{align}\n        \\Pi_1 \\Pi_2 & = \n            \\frac{1}{\\zeta}\n            [\n            (n_1 - 1)^2\\ub^2 + \n            n_2^2 \\ub^2 +\n            n_3^2 \\ub^2 +\n                ... +\n            n_{\\nclass}^2 \\ub^2] \\\\\n        & = \n            \\frac{1}{\\zeta}\n            [(n_1 - 1)^2+ \n            n_2^2 +\n            n_3^2 +\n                ... +\n            n_{\\nclass}^2] \\ub^2\n    \\end{align}\n    The lower bound for just the $\\within_1$ term emerges as \n    \\begin{equation}\n        \\within_1 \\ge \n        \\sum_{\\cS^1} \\Gij e^{\n        - \\frac{ \n            [(n_1 - 1)^2+ \n            n_2^2 +\n            n_3^2 +\n                ... +\n            n_{\\nclass}^2] \\ub^2 }{2 \\zeta \\sigma_1^2}}. \n    \\end{equation}\n    To further condense the notation, we define the following constant\n    \\begin{equation}\n        \\mathscr{N}_g = \n            \\frac{1}{2 \\zeta}\n            [n_1^2+ \n            n_2^2 +\n            ... +\n            (n_g - 1)^2 + \n            ... + \n            n_{\\tau}^2].\n    \\end{equation}\n    Therefore, the lower bound for $\\within_1$ can be simplified as \n    \\begin{equation}\n        \\within_1 \\ge \\sum_{\\cS^1} \\Gij e^{-\\frac{\\mathscr{N}_1 \\ub^2}{\\sigma_1^2}}\n    \\end{equation}\n    and the general pattern for any $\\within_g$ becomes\n    \\begin{equation}\n         \\within_g \\ge \\sum_{\\cS^i} \\Gij e^{-\\frac{\\mathscr{N}_g \\ub^2}{\\sigma_1^2}}. \n    \\end{equation}\n    The lower bound for the entire set of $\\cS$ then becomes\n    \\begin{equation}\n        \\sums \\Gij \\ISMexpC{}{} = \n        \\within_1 + ... + \\within_{\\nclass} \\ge \n        \\underbrace{\n        \\sum_{g=1}^{\\nclass} \\sum_{\\cS^g} \\Gij\n        e^{-\\frac{\\mathscr{N}_g \\ub^2}{\\sigma_1^2}}}_{\\text{Lower bound}}.\n    \\end{equation}\n    \n    \\textbf{For sample pairs in $\\cS^c$. } \n    To simplify the notation, we note that\n    \\begin{align}\n        -\\between_{g_1,g_2} & = \n            -\n            \\sum_{i \\in \\cS^{g_1}} \n            \\sum_{j \\in \\cS^{g_2}}\n            |\\Gij| \\ISMexpC{(g_1)}{(g_2)} \\\\\n            & = -\n            \\sum_{i \\in \\cS^{g_1}} \n            \\sum_{j \\in \\cS^{g_2}}\n            |\\Gij| e^{-\\frac\n            {\\Tr(W^T (\\rij{g_1}{g_1})(\\rij{g_1}{g_2})^T W)}\n            {2 \\sigma_1^2}} \\\\                \n            & = -\n            \\sum_{i \\in \\cS^{g_1}} \n            \\sum_{j \\in \\cS^{g_2}}\n            |\\Gij| e^{-\\frac{\\Tr(W^T A_{i,j}^{(g_1, g_2)} W)}\n            {2 \\sigma_1^2}} \\\\           \n    \\end{align}\n    We now derived the lower bound for the sample pairs in $\\cS^c$. We start by writing out the entire summation sequence for $\\between$.  \n    \\begin{equation}\n    \\begin{split}\n    \\between & = \n    -\\underbrace{\n    \\sum_{i \\in \\cS^1} \n    \\sum_{j \\in \\cS^2}\n    |\\Gij| \n    e^{-\\frac{\\Tr(W^T A_{i,j}^{(1, 2)} W)}\n    {2 \\sigma_1^2}}\n    }_\\text{$\\between_{1,2}$} \n    - \\underbrace{...}_{\\between_{g_1 \\ne g_2}}\n    -\\underbrace{\n    \\sum_{i \\in \\cS^1} \n    \\sum_{j \\in \\cS^{\\nclass}}\n    |\\Gij| \n    e^{-\\frac{\\Tr(W^T A_{i,j}^{(1, \\nclass)} W)}\n    {2 \\sigma_1^2}}\n    }_\\text{$\\between_{1,\\nclass}$} \n    \\\\\n    & \n    -\\underbrace{\n    \\sum_{i \\in \\cS^2} \n    \\sum_{j \\in \\cS^1}\n    |\\Gij| \n    e^{-\\frac{\\Tr(W^T A_{i,j}^{(2, 1)} W)}\n    {2 \\sigma_1^2}}\n    }_\\text{$\\between_{2,1}$} \n    - \\underbrace{...}_{\\between_{g_1 \\ne g_2}}\n    -\\underbrace{\n    \\sum_{i \\in \\cS^2} \n    \\sum_{j \\in \\cS^{\\nclass}}\n    |\\Gij| \n    e^{-\\frac{\\Tr(W^T A_{i,j}^{(2, \\nclass)} W)}\n    {2 \\sigma_1^2}}\n    }_\\text{$\\between_{2,\\nclass}$} \n    \\\\   \n    & ... \\\\\n    & \n    -\\underbrace{\n    \\sum_{i \\in \\cS^\\nclass} \n    \\sum_{j \\in \\cS^1}\n    |\\Gij| \n    e^{-\\frac{\\Tr(W^T A_{i,j}^{(\\nclass, 1)} W)}\n    {2 \\sigma_1^2}}\n    }_\\text{$\\between_{\\nclass,1}$} \n    - \\underbrace{...}_{\\between_{g_1 \\ne g_2}}\n    -\\underbrace{\n    \\sum_{i \\in \\cS^{\\nclass-1}} \n    \\sum_{j \\in \\cS^{\\nclass}}\n    |\\Gij| \n    e^{-\\frac{\\Tr(W^T A_{i,j}^{(\\nclass-1, \\nclass)} W)}\n    {2 \\sigma_1^2}}\n    }_\\text{$\\between_{\\nclass-1,\\nclass}$} \n    \\end{split}\n    \\end{equation}\n    \n    Using a similar approach with the terms from $\\within$, note that $\\between$ is a negative value, so we need to maximize this term to obtain a lower bound. Consequently, the key is to determine the \\textbf{minimal} possible values for each exponent term. Since every one of them will behave very similarly, we can simply look at the numerator of the exponent from $\\between_{1,2}$ and then arrive to a more general conclusion. Given $W_s$ as $W$, our goal is to identify the \\textbf{minimal} possible value for\n    \\begin{equation}\n        \\underbrace{\n        \\rij{1}{2}^TW}_{\\Pi_1}\n        \\underbrace{\n        W^T\\rij{1}{2}}_{\\Pi_2}. \n    \\end{equation}\n  \n    Zoom in further by looking only at $\\Pi_1$, we have \n    the following relationships\n    \\begin{equation}\n       \\Pi_1 = \n        \\underbrace{\n            r_i^{(1)^T} W}_{\\xi_1}\n        - \n        \\underbrace{r_j^{(2)^T} W}_{\\xi_2} \n    \\end{equation}\n    \\begin{align}\n        \\xi_1 & = \n        \\frac{1}{\\sqrt{\\zeta}}\n        r_i^{(1)^T} \\W \\\\\n        & = \n        \\frac{1}{\\sqrt{\\zeta}}\n        r_i^{(1)^T} \n            \\begin{bmatrix}\n                (r_1^{(1)} + ... + r_{n_1}^{(1)}) &\n                ... &\n                (r_1^{(\\nclass)} + ... + r_{n_{\\nclass}}^{(\\nclass)})\n            \\end{bmatrix}\n    \\end{align}\n    \\begin{align}\n        \\xi_2 & = \n        \\frac{1}{\\sqrt{\\zeta}}\n        r_j^{(2)^T} \\W \\\\\n        & = \n        \\frac{1}{\\sqrt{\\zeta}}\n        r_j^{(2)^T} \n            \\begin{bmatrix}\n                (r_1^{(1)} + ... + r_{n_1}^{(1)}) &\n                ... &\n                (r_1^{(\\nclass)} + ... + r_{n_{\\nclass}}^{(\\nclass)})\n            \\end{bmatrix}\n    \\end{align}   \n    By knowing that the inner product is constrained between $[0,\\ub]$, we know the \\textbf{minimum} possible value for $\\xi_1$ and the \\textbf{maximum} possible value for $\\xi_2$ to be\n     \\begin{align}\n        \\xi_1 & = \n            \\frac{1}{\\sqrt{\\zeta}}\n            \\begin{bmatrix}\n                1 &\n                0 &\n                0 &\n                ... &\n                0\n            \\end{bmatrix} \\\\\n         \\xi_2 & = \n            \\frac{1}{\\sqrt{\\zeta}}\n            \\begin{bmatrix}\n                n_1\\ub &\n                1 + (n_2 - 1) \\ub &\n                n_3 \\ub &\n                ... &\n                n_{\\nclass} \\ub\n            \\end{bmatrix} \n    \\end{align}   \n    Which leads to  \n    \\begin{equation}\n        \\Pi_1 = \n        \\frac{1}{\\sqrt{\\zeta}}\n        (\\xi_1 - \\xi_2) = \n            \\frac{1}{\\sqrt{\\zeta}}\n            \\begin{bmatrix}\n                1 - n_1\\ub &\n                -(1 + (n_2 - 1) \\ub) &\n                -n_3 \\ub &\n                ... &\n                -n_{\\nclass} \\ub\n            \\end{bmatrix}\n    \\end{equation}\n    Since $\\Pi_2^{T}  = \\Pi_1$ we have     \n    \\begin{align}\n        \\Pi_1 \\Pi_2 & = \n            \\frac{1}{\\zeta}\n            [\n            (1- n_1 \\ub)^2 + \n            (1 + (n_2-1)\\ub)^2 + \n            n_3^2 \\ub^2 +\n                ... +\n            n_{\\nclass}^2 \\ub^2].\n    \\end{align}\n    The lower bound for just the $\\between_{1,2}$ term emerges as \n    \\begin{equation}\n        -\\between_{1,2} \\ge \n        - \\sum_{\\cS^1} \\sum_{\\cS^2} |\\Gij| e^{\n        - \\frac{ \n            (1- n_1\\ub)^2 + \n            (1 + (n_2-1)\\ub)^2 + \n            n_3^2 \\ub^2 +\n                ... +\n            n_{\\nclass}^2 \\ub^2}\n            {2 \\zeta \\sigma_1^2}}. \n    \\end{equation}\n    To further condense the notation, we define the following function\n    \\begin{equation}\n    \\begin{split}\n        \\mathscr{N}_{g_1, g_2}(\\ub) & = \n            \\frac{1}{2 \\zeta}\n            [ n_1^2 \\ub^2 + n_2^2 \\ub^2 + ... \\\\ \n            & +  (1- n_{g_1}\\ub)^2 + ... + \n            (1 + (n_{g_2}-1)\\ub)^2 \\\\\n            & + ...  + n_{\\tau}^2 \\ub^2].\n    \\end{split}\n    \\end{equation}\n    Note that while for $\\cS$, the $\\ub$ term can be separated out. But here, we cannot, and therefore $\\mathscr{N}$ here must be a function of $\\ub$.  Therefore, the lower bound for $\\between_{1,2}$ can be simplified into\n    \\begin{equation}\n        - \\between_{1,2} \\ge - \n        \\sum_{\\cS^1} \n        \\sum_{\\cS^2}\n        |\\Gij| e^{-\\frac{\\mathscr{N}_{1,2}(\\ub)}{\\sigma_1^2}}\n    \\end{equation}\n    and the general pattern for any $\\between_{g_1, g_2}$ becomes\n    \\begin{equation}\n         -\\between_{g_1,g_2} \\ge -\n         \\sum_{\\cS^{g1}} \n        \\sum_{\\cS^{g2}}\n         \\Gij e^{-\\frac{\\mathscr{N}_{g_1,g_2}(\\ub)}\n         {\\sigma_1^2}}. \n    \\end{equation}    \n    The lower bound for the entire set of $\\cS^c$ then becomes\n    \\begin{align}\n        -\\sumsc |\\Gij| \\ISMexpC{}{} & = \n        - \\between_{1,2} \n        - \\between_{1,3} \n        - ... - \n        \\between_{\\nclass-1, \\nclass} \\\\\n        & \\ge \n        - \\underbrace{\n        \\sum_{g_1 \\ne g_2}^{\\nclass}\n        \\sum_{i \\in \\cS^{g_1}} \n        \\sum_{j \\in \\cS^{g_2}} \n        |\\Gij|\n        e^{-\\frac{\\mathscr{N}_{g_1,g_2} (\\ub)}{\\sigma_1^2}}}_{\\text{Lower bound}}.\n    \\end{align}    \n    \n    \\textbf{Putting $\\cS$ and $\\cS^c$ Together. } \n    \\begin{align}\n        \\hsic & = \\within + \\between    \\\\\n        & \\ge\n        \\underbrace{\n        \\sum_{g=1}^{\\nclass} \\sum_{\\cS^g} \\Gij\n        e^{-\\frac{\\mathscr{N}_g \\ub^2}{\\sigma_1^2}}}_{\\text{Lower bound of } \\within}\n        - \\underbrace{\n        \\sum_{g_1 \\ne g_2}^{\\nclass} \n        \\sum_{i \\in \\cS^{g_1}} \n        \\sum_{j \\in \\cS^{g_2}} \n        |\\Gij|\n        e^{-\\frac{\\mathscr{N}_{g_1,g_2} (\\ub)}{\\sigma_1^2}}}_{\\text{Lower bound of } \\between}.\n    \\end{align}\n    Therefore, we have identified a lower bound that is a function of $\\sigma_0$ and $\\sigma_1$ where\n    \\begin{equation}\n        \\lb\\Lsigma = \n        \\sum_{g=1}^{\\tau} \\sum_{\\cS^g} \\Gij\n        e^{-\\frac{\\mathscr{N}_g \\ub^2}{\\sigma_1^2}}\n        - \n        \\sum_{g_1 \\ne g_2}^{\\tau} \n        \\sum_{i \\in \\cS^{g_1}} \n        \\sum_{j \\in \\cS^{g_2}} \n        |\\Gij|\n        e^{-\\frac{\\mathscr{N}_{g_1,g_2} (\\ub)}{\\sigma_1^2}}.\n    \\end{equation}\n    From the lower bound, it is obvious why it is a function of $\\sigma_1$. The lower bound is also a function of $\\sigma_{0}$ because $\\ub$ is actually a function of $\\sigma_0$. To specifically clarify this point, we have the next lemma. \n\\end{adjustwidth}\n\\end{proof}\n\\hspace{1pt}\n\n\\begin{lemma}\n\\label{app:lemma:ub_goes_to_zero}\nThe $\\ub$ used in \\citelemma{app:lemma:lowerbound} is a function of $\\sigma_0$ where $\\ub$ approaches to zero as $\\sigma_{0}$ approaches to zero, i.e.\n\\begin{equation}\n    \\lim_{\\sigma_0 \\rightarrow 0}  \n    \\ub = 0.\n\\end{equation}\n\\end{lemma}\n\n\\textbf{Assumptions and Notations. }\n\\begin{enumerate}\n    \\item  \n        We use Fig.~\\ref{app:fig:two_layer_img} to help clarify the notations. We here only look at the last 2 layers.  \n    \\item  \n        We let $\\hsic_0$ be the $\\hsic$ of the last layer, and $\\hsic_1$, the $\\hsic$ of the current layer.\n    \\item\n        The input of the data is $X$ with each sample as $x_i$, and the output of the previous layer are denoted as $r_i$. $\\psi_{\\sigma_0}$ is the feature map of the previous layer using $\\sigma_0$ and $\\psi_{\\sigma_1}$ corresponds to the current layer. \n        \\begin{figure}[h]\n        \\center\n            \\includegraphics[width=7cm]{img\/TwoLayer.png}\n            \\caption{Figure of a 2 layer network.}\n            \\label{app:fig:two_layer_img}\n        \\end{figure}     \n    \\item  \n        As defined from \\citelemma{app:lemma:lowerbound}, \n        among all $r_i \\ne r_j$ pairs, there exists an optimal $r_i^*, r_j^*$ pair where   $\\langle r_i^*, r_j^* \\rangle \\ge \\langle r_i, r_j \\rangle$ $\\forall r_i \\ne r^*_i$ and $r_j \\ne r^*_j$. We denote this maximum inner product as \n    \\begin{equation}\n        \\ub = \\langle r_i^*, r_j^* \\rangle.\n    \\end{equation}\n\\end{enumerate}\n\n\n\n\\begin{proof} \\hspace{1pt}\n\\begin{adjustwidth}{0.5cm}{0.0cm}\nGiven Fig.~\\ref{app:fig:two_layer_img}, the equation for $\\hsic_0$ is \n\\begin{align}\n    \\hsic_0 \n        & = \\sums \\Gij \n        e^{-\\frac{(x_i - x_j)^TW W^T (x_i - x_j)}{2 \\sigma_0^2}}\n        -\n        \\sumsc |\\Gij| \n        e^{-\\frac{(x_i - x_j)^TW W^T (x_i - x_j)}{2 \\sigma_0^2}}    \\\\\n        & = \n        \\sums \\Gij\n            \\langle \n                \\psi_{\\sigma_0}(x_i), \n                \\psi_{\\sigma_0}(x_j)\n            \\rangle\n            - \\sumsc |\\Gij|\n            \\langle \n                \\psi_{\\sigma_0}(x_i), \n                \\psi_{\\sigma_0}(x_j)\n            \\rangle\n\\end{align}\nNotice that as $\\sigma_0 \\rightarrow 0$, we have \n\\begin{equation}\n   \n    \\lim_{\\sigma_0 \\rightarrow 0} \n    \\langle \n        \\psi_{\\sigma_0}(x_i), \n        \\psi_{\\sigma_0}(x_j)\n    \\rangle   \n    =  \n    \\begin{cases} \n    0 \\quad \\forall i \\ne j\n    \\\\\n    1 \\quad \\forall i = j\n    \\end{cases}.\n\\end{equation}\nIn other words, as $\\sigma_0 \\rightarrow 0$, the samples $r_i$ in the RKHS of a Gaussian kernel approaches orthogonal to all other samples. Given this fact, it also implies that the $\\sigma_0$ controls the inner product magnitude in RKHS space  of the maximum sample pair $r_i^*, r_j^*$. We define this maximum inner product as\n\\begin{equation}\n    \\langle \n        \\psi_{\\sigma_0}(x_i^*), \n        \\psi_{\\sigma_0}(x_j^*)\n    \\rangle  \n    \\ge \n     \\langle \n        \\psi_{\\sigma_0}(x_i), \n        \\psi_{\\sigma_0}(x_j)\n    \\rangle      \n\\end{equation}\nor equivalently\n\\begin{equation}\n    \\langle \n        r_i^*, \n        r_j^*\n    \\rangle  \n    \\ge \n     \\langle \n        r_i,\n        r_j\n    \\rangle      \n\\end{equation}\n\nTherefore, given a $\\sigma_0$, it controls the upper bound of the inner product. Notice that as $\\sigma_0 \\rightarrow 0$, every sample in RKHS becomes orthogonal. Therefore, the upper bound of $\\langle r_i, r_j \\rangle$ also approaches 0 when $r_i \\ne r_j$. From this, we see the relationship\n\\begin{equation}\n    \\lim_{\\sigma_0 \\rightarrow 0} \n    \\ub = \\lim_{\\sigma_0 \\rightarrow 0}  \\exp -(|.|\/\\sigma^{2}_{0}) = 0\n\\end{equation}, where $|.|$ is bounded and has a minimum and maximum, because we have finite number of samples.\n\\end{adjustwidth}\n\\end{proof}\n\n\n\n\n\n\\begin{lemma}\n\\label{app:lemma:as_ub_zero_L_approaches}\nGiven any fixed $\\sigma_1 > 0 $, the lower bound $\\lb\\Lsigma$ is a function with respect to $\\sigma_0$ and as $\\sigma_0 \\rightarrow 0$, $\\lb\\Lsigma$ approaches the function\n    \\begin{equation}\n        \\lb(\\sigma_1) = \n        \\sum_{g=1}^{\\nclass} \\sum_{\\cS^g} \\Gij\n        - \n        \\sum_{g_1 \\ne g_2}^{\\nclass}\n        \\sum_{i \\in \\cS^{g_1}} \n        \\sum_{j \\in \\cS^{g_2}} \n        |\\Gij|\n        e^{-\\frac{1}{\\zeta \\sigma_1^2}}.\n    \\end{equation}\nAt this point, if we let $\\sigma_1 \\rightarrow 0$, we have \n    \\begin{align}\n        \\lim_{\\sigma_1 \\rightarrow 0}\n        \\lb(\\sigma_1) & = \\sums \\Gij \\\\\n        & = \\hsic^*.\n    \\end{align}\n\\end{lemma}\n\n\\begin{proof} \\hspace{1pt}\n\\begin{adjustwidth}{0.5cm}{0.0cm}\n    Given \\citelemma{app:lemma:ub_goes_to_zero}, we know that \n     \\begin{equation}\n        \\lim_{\\sigma_0 \\rightarrow 0} \n        \\ub = 0.\n    \\end{equation}   \n    Therefore, having $\\sigma_0 \\rightarrow 0$ is equivalent to having $\\ub \\rightarrow 0$. Since \n    \\citelemma{app:lemma:lowerbound} provide the equation of a lower bound that is a function of $\\ub$,  this lemma is proven by simply evaluating $\\lb\\Lsigma$ as $\\ub \\rightarrow 0$. Following these steps, we have\n    \\begin{align}\n        \\lb(\\sigma_1) & = \n        \\lim_{\\ub \\rightarrow 0}\n        \\sum_{g=1}^{\\nclass} \\sum_{\\cS^g} \\Gij\n        e^{-\\frac{\\mathscr{N}_g \\ub^2}{\\sigma_1^2}}\n        - \n        \\sum_{g_1 \\ne g_2}^{\\nclass} \n        \\sum_{i \\in \\cS^{g_1}} \n        \\sum_{j \\in \\cS^{g_2}} \n        |\\Gij|\n        e^{-\\frac{\\mathscr{N}_{g_1,g_2} (\\ub)}{\\sigma_1^2}}, \\\\\n        & = \n        \\sum_{g=1}^{\\nclass} \\sum_{\\cS^g} \\Gij\n        - \n        \\sum_{g_1 \\ne g_2}^{\\nclass} \n        \\sum_{i \\in \\cS^{g_1}} \n        \\sum_{j \\in \\cS^{g_2}} \n        |\\Gij|\n        e^{-\\frac{1}{\\zeta \\sigma_1^2}}.\n    \\end{align}    \nAt this point, as $\\sigma_1 \\rightarrow 0$, our lower bound reaches the global maximum \n    \\begin{align}\n        \\lim_{\\sigma_1 \\rightarrow 0}\n        \\lb(\\sigma_1) & = \\sum_{g=1}^{\\nclass} \\sum_{\\cS^g} \\Gij\n        = \\sums \\Gij \\\\\n        & = \\hsic^*.\n    \\end{align}\n    \n\\end{adjustwidth}\n\\end{proof}\n\n\\begin{lemma}\n\\label{app:lemma:arbitrarily_close}\nGiven any $\\hsic_{l-2}$, $\\delta > 0$, there exists a $\\sigma_0 > 0 $ and $\\sigma_1 > 0$ such that\n\\begin{equation}\n        \\hsic^{*} - \\hsic_l \\le \\delta.\n        \\label{app:eq:proof_I}\n\\end{equation}\n\\end{lemma}\n\\begin{proof} $\\hspace{1pt}$\n\n\n\\begin{adjustwidth}{0.5cm}{0.0cm}\n\\textbf{Observation 1. }\n\nNote that the objective of $\\hsic_l$ is\n\\begin{equation}\n\\begin{split}\n    \\hsic_l = \n    \\max_W\n    &\n    \\sums \\Gij \n    e^{-\\frac{\\rij{\\cS}{\\cS}^T WW^T \\rij{\\cS}{\\cS}}{2\\sigma_1^2}} \\\\\n    - & \\sumsc |\\Gij| \n     e^{-\\frac{\\rij{\\cS^c}{\\cS^c}^T WW^T \\rij{\\cS^c}{\\cS^c}}{2\\sigma_1^2}}.\n\\end{split}\n\\end{equation}\nSince the Gaussian kernel is bounded between 0 and 1, the theoretical maximum of $\\hsic^*$ is when the kernel is 1 for $\\cS$ and 0 for $\\cS^c$ with the theoretical maximum as $\\hsic^* = \\sums \\Gij$. Therefore \\eq{app:eq:proof_I} inequality is equivalent to \n\\begin{equation}\n        \\sums \\Gij - \\hsic_l \\le \\delta.\n\\end{equation}\n\\end{adjustwidth}\n\n\n\n\n\\begin{adjustwidth}{0.5cm}{0.0cm}\n\\textbf{Observation 2. }\n\nIf we choose a $\\sigma_0$ such that \n\n\n\\begin{equation}\n    \\lb^*(\\sigma_1) - \\lb\\Lsigma \\le \\frac{\\delta}{2}\n    \\quad \\text{and} \\quad\n    \\hsic^* - \\lb^*(\\sigma_1) \\le \\frac{\\delta}{2}\n\\end{equation}\nthen we have identified the condition where $\\sigma_0 > 0$ and $\\sigma_1 > 0$ such that\n\\begin{equation}\n    \\sums \\Gij - \\lb\\Lsigma \\le \\delta.\n\\end{equation}\nNote that the $\\lb^*(\\sigma_1)$ is a continuous function of $\\sigma_1$. Therefore, a $\\sigma_1$ exists such that $\\lb^*(\\sigma_1)$ can be set arbitraty close to $\\hsic^{*}$. Hence, we choose an $\\sigma_{1}$ that has the following property:\n\\begin{equation}\n    \\hsic^* - \\lb^*(\\sigma_1) \\le \\frac{\\delta}{2}.\n\\end{equation}\nWe next fix $\\sigma_{1}$, we also know $\\lb\\Lsigma$ is a continuous function of $\\sigma_{0}$, and it has a limit $\\lb^*(\\sigma_1)$ as $\\sigma_{0}$ approaches to 0, hence there exits a $\\sigma_{0}$, where \n\\begin{equation}\n    \\lb^*(\\sigma_1) - \\lb\\Lsigma \\le \\frac{\\delta}{2}\n\\end{equation}\nThen we have:\n\\begin{equation}\n    \\lb^*(\\sigma_1) - \\lb\\Lsigma \\le \\frac{\\delta}{2}\n    \\quad \\text{and} \\quad\n    \\hsic^* - \\lb^*(\\sigma_1) \\le \\frac{\\delta}{2}.\n\\end{equation}\nBy adding the two $\\frac{\\delta}{2}$, we conclude the proof.\n\\end{adjustwidth}\n\\end{proof}\n\n\n\n\n\n\\begin{lemma}\n    \\label{app:lemma:approach_optimal_H1}\n    There exists a \\KS $\\{\\fm_{l^{\\circ}}\\}_{l=1}^L$ parameterized by a set of weights $W_l$ and a set of bandwidths $\\sigma_l$ such that    \n    \\begin{equation}\n        \\lim_{l \\rightarrow \\infty}  \\hsic_l = \\hsic^* , \\quad \\hsic_{l+1} > \\hsic_l \\quad \\forall l\n    \\end{equation}\n\\end{lemma}\n\n\nBefore, the proof, we use the following figure, Fig.~\\ref{app:fig:all_layers}, to illustrate the relationship between \\KS $\\{\\phi_{l^\\circ}\\}_{l=1}^L$ that generates the \\RS $\\{\\hsic_l\\}_{l=1}^L$. By solving a network greedily, we separate the network into $L$ separable problems. At each additional layer, we rely on the weights learned from the previous layer. At each network, we find $\\sigma_{l-1}$, $\\sigma_l$, and $W_l$ for the next network. We also note that since we only need to prove the existence of a solution, this proof is done by \\textit{\\textbf{Proof by Construction}}, i.e, we only need to show an example of its existence. Therefore, this proof consists of us constructing a \\RS which satisfies the lemma.\n        \\begin{figure}[h]\n        \\center\n            \\includegraphics[width=7cm]{img\/kchain_layers.png}\n            \\caption{Relating \\KS to \\RS.}\n            \\label{app:fig:all_layers}\n        \\end{figure} \n\n\\begin{proof} \\hspace{1pt}\n\\begin{adjustwidth}{0.5cm}{0.0cm}\nWe first note that from \\citelemma{app:lemma:arbitrarily_close}, we have previously proven given any $\\hsic_{l-2}$, $\\delta > 0$, there exists a $\\sigma_0 > 0 $ and $\\sigma_1 > 0$ such that\n\\begin{equation}\n    \\hsic^{*} - \\hsic_{l} \\leq \\delta_{l}.\n    \\label{eq:cond1}\n\\end{equation}\nThis implies that based on Fig.~\\ref{app:fig:all_layers}, at any given layer, we could reach arbitrarily close to $\\hsic^*$. Given this, we list the 2 steps to build the \\RS.\n\n\n\n\n\\textbf{Step 1: } Define $\\{\\mathcal{E}_{n}\\}_{n =  1}^{\\infty}$ as a sequence of numbers $\\hsic^{*} - \\frac{\\hsic^{*} - \\hsic_0}{n}$ on the real line. We have the following properties for this sequence:\n\\begin{equation}\n  \\lim_{n\\rightarrow\\infty} \\mathcal{E}_{n} = \\hsic^{*}\n , \\quad \\mathcal{E}_{1} = \\mathcal{H}_{0}.   \n\\end{equation}\n\n\nUsing these two properties, for any $\\hsic_{l-1} \\in [\\hsic_{0},\\hsic^{*}]$ there exist an unique $n$, where  \\begin{equation}\n    \\mathcal{E}_{n} \\leq \\hsic_{l-1} < \\mathcal{E}_{n+1}.\n    \\label{eq:bounding_box}\n\\end{equation}\n\n\\textbf{Step 2: }  \n    For any given $l$, we choose $\\delta_{l}$ to satisfies \\eq{eq:cond1} by the following procedure, First find an $n$ that satisfies\n    \\begin{equation}\n        \\mathcal{E}_{n} \\leq \\hsic_{l-1} <  \\mathcal{E}_{n+1}, \n        \\label{ineq:differential}\n    \\end{equation}\n    and second define $\\delta_l$ to be \n    \\begin{equation}\n        \\delta_{l} = \\hsic^{*} - \\mathcal{E}_{n+1}.\n        \\label{eq:delta_define}\n    \\end{equation}\n    \n    \n    \n\n    \n    \nTo satisfy \\eq{eq:cond1}, the following must be true.  \\begin{equation}\n    \\hsic^{*} - \\hsic_{l-1} \\leq \\delta_{l-1}.\n    \\label{eq:delta_greater_than_l1}\n\\end{equation}\nand further we found $n$ such that\n\\begin{equation}\n    \\mathcal{E}_{n} \\leq \\hsic_{l-1} <  \\mathcal{E}_{n+1} \\implies \\hsic^{*} - \\mathcal{E}_{n} \\geq \\hsic^{*} - \\hsic_{l-1} > \\hsic^{*} - \\mathcal{E}_{n+1}.\n    \\label{eq:all_ineqals}\n\\end{equation}\nThus combining \\eq{eq:delta_define}, \\eq{eq:delta_greater_than_l1}, and \\eq{eq:all_ineqals} we have\n\\begin{equation}\n    \\delta_{l-1} > \\delta_{l}.\n\\end{equation}\nTherefore, $\\{ \\delta_l \\}$ is a decreasing sequence.\n\n \n\n\\textbf{Step 3: } \nNote that $\\{\\mathcal{E}_{n}\\}$ is a converging sequence where \n\\begin{equation}\n    \\lim_{n \\rightarrow \\infty}\n    \\hsic^{*} - \\frac{\\hsic^{*} - \\hsic_0}{n} = \\hsic^*.\n\\end{equation}\nTherefore, $\\{\\Delta_n\\} = \\hsic^* - \\{\\mathcal{E}_n\\}$ is also a converging sequence where \n\\begin{equation}\n    \\lim_{n \\rightarrow \\infty}\n    \\hsic^* - \\hsic^{*} + \\frac{\\hsic^{*} - \\hsic_0}{n} = 0\n\\end{equation}\nand $\\{\\delta_{l}\\}$ is a subsequence of $\\{\\Delta_{l}\\}$. Since any subsequence of a converging sequence also converges to the same limit, we know that\n\\begin{equation}\n   \n   \n   \n    \\lim_{l \\rightarrow \\infty} \\delta_l = 0.\n\\end{equation}\n   \n\n        \n\nFollowing this construction, if we always choose $\\hsic_l$ such that \n\\begin{equation}\n    \\hsic^{*} - \\hsic_{l} \\leq \\delta_{l}.\n    \\label{eq:key_inequality}\n\\end{equation}\nAs $l \\rightarrow \\infty$, the inequality becomes\n\\begin{align}\n    \\hsic^{*} - \n    \\lim_{l \\rightarrow \\infty} \n    \\hsic_{l} \n    &  \\leq \n    \\lim_{l \\rightarrow \\infty} \n    \\delta_{l}, \\\\\n    &  \\leq \n    0.\n\\end{align}\nSince we know that \n\\begin{equation}\n    \\hsic^{*} - \\hsic_l \\geq 0\\, \\forall l. \n\\end{equation}\nThe condition of \n\\begin{equation}\n    0 \\leq\n    \\hsic^{*} - \\lim_{l \\rightarrow \\infty} \\hsic_l \\leq 0\n\\end{equation}\nis true only if \n\\begin{equation}\n    \\hsic^{*} - \\lim_{l \\rightarrow \\infty} \\hsic_l = 0.\n\\end{equation}\nThis allows us to conclude \n\\begin{equation}\n    \\hsic^{*} = \\lim_{l \\rightarrow \\infty} \\hsic_l.\n\\end{equation}\n\n\n\n\n\n\n\\textbf{Proof of the Monotonic Improvement. } \n\nGiven \\eq{eq:bounding_box} and \\eq{eq:delta_define}, \nat each step we have the following: \n\\begin{align}\n    \\hsic_{l-1} & < \\mathcal{E}_{n+1} \\\\\n    & \\leq \n    \\hsic^{*} - \\delta_{l}. \n\\end{align}\nRearranging this inequality, we have\n\\begin{equation}\n    \\delta_{l} < \\hsic^{*} - \\hsic_{l-1}.\n    \\label{eq:keyIneq}\n\\end{equation}\nBy combining the inequalities from \\eq{eq:keyIneq} and  \\eq{eq:key_inequality}, we have the following relationships.\n\\begin{align}\n    \\hsic^{*} - \\hsic_{l} \\leq \\delta_{l} & < \\hsic^{*} - \\hsic_{l-1} \\\\\n    \\hsic^{*} - \\hsic_{l} & < \\hsic^{*} - \\hsic_{l-1} \\\\\n    - \\hsic_{l} & < - \\hsic_{l-1} \\\\\n    \\hsic_{l} & > \\hsic_{l-1}, \\\\\n\\end{align}\nwhich concludes the proof of theorem.\n\n\n\\end{adjustwidth}\n\\end{proof}\n\n\n\n\\end{appendices}\n\n\n\n\\section{Proof for Theorem \\ref{thm:geometric_interpret}}\n\\label{app:thm:geometric_interpret}\n\n\\textbf{Theorem \\ref{thm:geometric_interpret}: }\n\\textit{\nAs $l \\rightarrow \\infty$ and  $\\hsic_l \\rightarrow \\hsic^*$, \nthe following properties are satisfied: }\n\n\\begin{enumerate}[label=\\Roman*]\n    \\item \n   \n    the scatter ratio approaches 0 where\n    \\begin{equation}\n    \\lim_{l \\rightarrow \\infty} \\frac{\\Tr(S_w^{l})}{\\Tr(S_b^{l})} =   0\n\\end{equation}\n\n\\item the \\KS converges to the following kernel:\n\\begin{equation}\n   \n   \n    \\lim_{l \\rightarrow \\infty} \n    \\kf(x_{i},x_{j})^{l} = \n    \\kf^* =  \n    \\begin{cases} \n    0 \\quad \\forall i,j \\in \\mathcal{S}^c\n    \\\\\n    1 \\quad \\forall i,j \\in \\mathcal{S}\n    \\end{cases}.\n\\end{equation}\n\\end{enumerate}\n\n\\begin{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\nWe start by proving condition II starting from the $\\hsic$ objective using a \\rbfk\n\\begin{align}\n    \\max_{W} \n    \\sums \\Gij \\kf_W(r_i, r_j) \n    -\n    \\sumsc |\\Gij| \\kf_W(r_i, r_j)\\\\\n    \\max_{W} \n    \\sums \\Gij  \\ISMexp\n    -\n    \\sumsc |\\Gij| \\ISMexp   \n\\end{align}\nGiven that $\\hsic_l \\rightarrow \\hsic^*$,\nand the fact that $0 \\leq \\kf_W \\leq 1$,\nthis implies that the following condition must be true:\n\\begin{equation}\n   \\hsic^{*} = \\sums \\Gij =  \n    \\sums \\Gij (1) \n    -\n    \\sumsc |\\Gij| (0).\n\\end{equation}\n\nBased on \\eq{eq:cond1}, our construction at each layer ensures to satisfy \n\\begin{equation}\n    \\hsic^{*} - \\hsic_{l} \\leq \\delta_{l}.\n\\end{equation}\nSubstituting the definition of $\\hsic^*$ and $\\hsic_l$, we have\n\\begin{align}\n    \\sums \\Gij (1) \n    -\\left[\\sums \\Gij  \\kf_W(r_i, r_j) \n    -\n    \\sumsc |\\Gij| \\kf_W(r_i, r_j) \\right] \\leq \\delta_{l}\n    \\\\\n     \\sums \\Gij  (1-\\kf_W(r_i, r_j)) \n    +\n    \\sumsc |\\Gij| \\kf_W(r_i, r_j)  \\leq \\delta_{l}.\n    \\label{eq:kernel_inequal}\n\\end{align}\nSince every term within the summation in \\eq{eq:kernel_inequal} is positive, this implies\n\\begin{align}\\label{eq:limit kernal behaviour}\n     1-\\kf_W(r_i, r_j) \\leq \\delta_{l} \\quad i,j \\in \\mathcal{S}\n     \\\\\n     \\kf_W(r_i, r_j) \\leq \\delta_{l}\\quad  i,j \\in \\mathcal{S}^{c}.\n     \\label{eq:limit kernal behaviour2}\n\\end{align}\n\nSo as $l \\rightarrow \\infty$ and  $\\delta_{l} \\rightarrow 0$, every component getting closer to limit Kernel, i.e, taking the limit from both sides and using the fact that is proven is theorem 1 $\\lim_{l\\rightarrow \\infty} \\delta_{l} = 0$ leads to\n\\begin{align}\n   \\lim_{l\\rightarrow \\infty}  1 \\leq \\kf_W(r_i, r_j)  \\quad i,j \\in \\mathcal{S}\n     \\\\\n     \\lim_{l\\rightarrow \\infty}\\kf_W(r_i, r_j) \\leq 0 \\quad  i,j \\in \\mathcal{S}^{c}\n\\end{align}\nboth terms must instead be strictly equality. Therefore, we see that at the limit point $\\kf_W$ would have the form \n\\begin{equation}\n     \\kf^* =  \n    \\begin{cases} \n    0 \\quad \\forall i,j \\in \\mathcal{S}^c\n    \\\\\n    1 \\quad \\forall i,j \\in \\mathcal{S}\n    \\end{cases}.   \n\\end{equation}\n\n\\textbf{First Property}:\n\nUsing \\eq{eq:limit kernal behaviour} and \\eq{eq:limit kernal behaviour2} we have:\n\n\\begin{align}\n         1-\\delta_{l}\\leq  \\ISMexp  \\quad i,j \\in \\mathcal{S}\n     \\\\\n     \\ISMexp \\leq \\delta_{l}\\quad  i,j \\in \\mathcal{S}^{c}.\n\\end{align}\n\nAs  $\\lim_{l\\rightarrow \\infty} \\delta_{l} = 0$, taking the limit from both side leads to:\n\n\n\n\n\n \n\\begin{equation}\n    \\begin{cases} \n    \\ISMexp = 1 \\quad \\forall i,j \\in \\mathcal{S}\\\\\n    \\ISMexp = 0 \\quad \\forall i,j \\in \\mathcal{S}^c\n    \\end{cases}.   \n\\end{equation}\nIf we take the log of the conditions, we get\n\\begin{equation}\n    \\begin{cases} \n     \\frac{1}{2\\sigma^2}\n    (r_i - r_j)^T W W^T(r_i - r_j) =0 \n    \\quad \\forall i,j \\in \\mathcal{S}\\\\   \n    \\frac{1}{2\\sigma^2}\n    (r_i - r_j)^T W W^T(r_i - r_j) = \\infty\n    \\quad \\forall i,j \\in \\mathcal{S}^c\n    \\end{cases}.   \n\\end{equation}\nThis implies that as $l \\rightarrow \\infty$ we have\n\\begin{equation}\n    \\lim_{l \\rightarrow \\infty}\n    \\sums\n    \\frac{1}{2\\sigma^2}\n    (r_i - r_j)^T W W^T(r_i - r_j) \n    = \n    \\lim_{l \\rightarrow \\infty}\n    \\Tr(S_w) = 0.\n\\end{equation}\n\\begin{equation}\n    \\lim_{l \\rightarrow \\infty}\n    \\sumsc \\frac{1}{2\\sigma^2}\n    (r_i - r_j)^T W W^T(r_i - r_j) \n    = \n    \\lim_{l \\rightarrow \\infty}\n    \\Tr(S_b)\n    =\n    \\infty,\n\\end{equation}\nThis yields the ratio\n\\begin{equation}\n    \\lim_{\\hsic_l \\rightarrow \\hsic^*} \\frac{\\Tr(S_w)}{\\Tr(S_b)} = \\frac{0}{\\infty} = 0.\n\\end{equation}\n\n\\end{proof}\n\\end{appendices}\n\\section{Proof for \\texorpdfstring{$W_s$}\\xspace Optimality}\n\\label{app:lemma:W_not_optimal}\n\\textit{Given $\\hsic_l$ as the empirical risk at layer $l \\ne L$, we have}\n\\begin{equation}\n    \\frac{\\partial}{ \\partial W_l}\\hsic_l(W_s) \\ne 0\n\\end{equation}\n\n\\begin{proof}\n    Given $\\frac{1}{\\sqrt{\\zeta}}$ as a normalizing constant for  $W_s = \\frac{1}{\\sqrt{\\zeta}} \\sum_{\\alpha} r_{\\alpha}$ such that $W^TW=I$. We start with the Lagrangian\n    \\begin{equation}\n        \\mathcal{L} = \n        - \\sum_{i,j} \\Gij \\ISMexp - \\Tr(\\Lambda(W^TW - I)).\n    \\end{equation}\n    If we now take the derivative with respect to the Lagrange, we get\n    \\begin{equation}\n        \\nabla \\mathcal{L} = \n        \\frac{1}{\\sigma^2}\n        \\sum_{i,j} \\Gij \\ISMexp\n        (r_i - r_j)(r_i - r_j)^TW \n        - 2W\\Lambda.\n    \\end{equation}\n    By setting the gradient to 0, we have\n     \\begin{align}\n        \\left[\n        \\frac{1}{2\\sigma^2}\n        \\sum_{i,j} \\Gij \\ISMexp\n        (r_i - r_j)(r_i - r_j)^T \n        \\right]\n        W \n        =&  W\\Lambda. \\\\\n        \\mathcal{Q}_l W =& W \\Lambda.\n        \\label{app:eq:ism_conclusion}\n    \\end{align}   \n    From \\eq{app:eq:ism_conclusion}, we see that $W$ is only the optimal solution when $W$ is the eigenvector of $Q_l$. Therefore, by setting $W$ to \n    $W_s = \\frac{1}{\\sqrt{\\zeta}} \\sum_{\\alpha} r_{\\alpha}$, it is not guaranteed to yield an optimal for all $\\sigma_l$.\n\\end{proof}\n\n\n\\end{appendices}\n\\section{Proof for Corollary \\ref{corollary:mse} and \\ref{corollary:ce}}\n\\label{app:corollary:ce}\n\n\\textbf{Corollary} \\ref{corollary:mse}: \n    \\textit{Given $\\hsic_l \\rightarrow \\hsic^*$, the network output in IDS solves MSE via a translation of labels.}\n   \n\\begin{proof} \\hspace{1pt}\n\\begin{adjustwidth}{0.5cm}{0.0cm}\n    As $\\hsic_l \\rightarrow \\hsic^*$, Thm.~\\ref{thm:geometric_interpret} shows that sample of the same class are mapped into the same point. Assuming that $\\fm$ has mapped the sample into $c$ points $\\alpha = [\\alpha_1, ..., \\alpha_c]$ that's different from the truth label\n    $\\xi = [\\xi_1, ..., \\xi_c]$. Then the $\\mse$ objective is minimized by translating the $\\fm$ output by \n    \\begin{equation}\n        \\xi - \\alpha.\n    \\end{equation}\n\\end{adjustwidth}\n\\end{proof}\n\n\\textbf{Corollary} \\ref{corollary:ce}: \n    \\textit{Given $\\hsic_l \\rightarrow \\hsic^*$, the network output in RKHS solves $\\ce$ via a change of bases.}\n    \n\\textbf{Assumptions, and Notations. }\n\\begin{enumerate}\n    \\item  \n        $n $ is the number of samples.\n    \\item\n        $\\nclass$ is the number of classes.\n    \\item\n        $y_i \\in \\mathbb{R}^{\\nclass}$ is the ground truth label for the $i^{th}$ sample. It is one-hot encoded where only the $j^{th}$ element is 1 if $x_i$ belongs to the $j^{th}$ class, all other elements would be 0.\n    \\item\n        We denote $\\fm$ as the network, and $\\hat{y}_i \\in \\mathbb{R}^{\\nclass}$ as the network output where $\\hat{y}_i = \\fm(x_i)$. We also assume that $\\hat{y}_i$ is constrained on a probability simplex where $1 = \\hat{y}_i^T \\mathbf{1}_n$.\n   \n   \n   \n   \n    \\item\n        We denote the $j^{th}$ element of $y_i$, and $\\hat{y}_i$ as $y_{i,j}$ and $\\hat{y}_{i,j}$ respectively.\n    \\item\n        We define\n    \\begin{addmargin}[1em]{2em\n    \\textbf{Orthogonality Condition: }\n        A set of samples $\\{\\hat{y}_1, ..., \\hat{y}_n\\}$ satisfies the orthogonality condition if\n        \\begin{equation} \n        \\begin{cases}\n        \\langle \\hat{y_{i}}, \\hat{y_{j}}\\rangle =1 & \\forall\\quad i,j \\textrm{ same class} \\\\\n        \\langle \\hat{y_{i}}, \\hat{y_{j}}\\rangle=0 & \\forall\\quad i,j \\textrm{ not in the same class}\n        \\end{cases}.\n        \\end{equation}\n    \\end{addmargin}\n    \\item\n        We define the Cross-Entropy objective as \n        \\begin{equation}\n        \\underset{\\fm}{\\argmin} -\\sum_{i=1}^{n} \\sum_{j=1}^{\\nclass} y_{i,j} \\log(\\fm(x_{i})_{i,j}).\n        \\end{equation}\n    \\end{enumerate}\n\\begin{proof}\\hspace{1pt}\n\\begin{adjustwidth}{0.5cm}{0.0cm}\nFrom Thm.~\\ref{thm:geometric_interpret}, we know that the network $\\fm$ output, $\\{ \\hat{y}_1, \\hat{y}_2, ..., \\hat{y}_n \\}$,  satisfy the orthogonality condition at $\\hsic^*$. Then there exists a set of  orthogonal bases represented by $\\Xi = [\\xi_1, \\xi_2, ..., \\xi_c]$ that maps $\\{ \\hat{y}_1, \\hat{y}_2, ..., \\hat{y}_n \\}$ to simulate the output of a softmax layer. Let $\\xi_{i} = \\hat{y}_{j} , j\\in \\cS^{i}$, i.e., for the $i_{th}$ class we arbitrary choose one of the samples from this class and assigns $\\xi_i$ of that class to be equal to the sample's output. Realize in our problem we have $<\\hat{y}_{i},\\hat{y}_{i}> = 1$, so if $<\\hat{y}_{i},\\hat{y}_{j}> = 1$, then subtracting these two would lead to $<\\hat{y}_{i},\\hat{y}_{i}-\\hat{y}_{j}> = 0$, which is the same as $\\hat{y}_{i}=\\hat{y}_{j}$.\nSo this representation is well-defined and its independent of choices of the sample from each group if they satisfy orthogonality condition.\nNow we define transformed labels, $Y$ as:\n\\begin{equation}\n    Y = \\hat{Y} \\Xi.\n\\end{equation}\nNote that $Y = [y_1, y_2, ..., y_n]^T$ which each $y_{i}$ is a one hot vector representing the class membership of $i$ sample in $c$ classes.\nSince given $\\Xi$ as the change of basis, we can match $\\hat{Y}$ to $Y$ exactly, $\\ce$ is minimized.\n\\end{adjustwidth}\n\\end{proof}\n\\end{appendices}\n\n\n\n\\section{Dataset Details}\n\\label{app:data_detail}\n\nNo samples were excludes from any of the dataset. \n\n\\textbf{Wine. } \n    This dataset has 13 features, 178 samples, and 3 classes. The features are continuous and heavily unbalanced in magnitude. The dataset can be downloaded at \\url{https:\/\/archive.ics.uci.edu\/ml\/datasets\/wine.}\n    \n\n \\textbf{Divorce. } \n    This dataset has 54 features, 170 samples, and 2 classes. The features are discrete and balanced in magnitude.  The dataset can be downloaded at \\url{https:\/\/archive.ics.uci.edu\/ml\/datasets\/Divorce+Predictors+data+set.}\n    \n \\textbf{Car. } \n     This dataset has 6 features, 1728 samples and 2 classes. The features are discrete and balanced in magnitude.  The dataset can be downloaded at \\url{https:\/\/archive.ics.uci.edu\/ml\/datasets\/Car+Evaluation.}       \n \n\\textbf{Cancer. } \n    This dataset has 9 features, 683 samples, and 2 classes. The features are discrete and unbalanced in magnitude. The dataset can be downloaded at \\url{https:\/\/archive.ics.uci.edu\/ml\/datasets\/Breast+Cancer+Wisconsin+(Diagnostic)}.\n    \n \\textbf{Face. } \n    This dataset consists of images of 20 people in various poses. The 624 images are vectorized into 960 features.  \n    The dataset can be downloaded at \n    \\url{https:\/\/archive.ics.uci.edu\/ml\/datasets\/CMU+Face+Images}.\n\n\\textbf{Random. } \n    This dataset has 2 features, 80 samples and 2 classes. It is generate with a gaussian distribution where half of the samples are randomly labeled as 1 or 0.\n    \n\\textbf{Adversarial. } \n    This dataset has 2 features, 80 samples and 2 classes. It is generate with the following code:\n    \\begin{lstlisting}    \n    #!\/usr\/bin\/env python\n    \n    n = 40\n    X1 = np.random.rand(n,2)\n    X2 = X1 + 0.01*np.random.randn(n,2)\n    \n    X = np.vstack((X1,X2))\n    Y = np.vstack(( np.zeros((n,1)), np.ones((n,1)) ))\n    \\end{lstlisting}    \n \n\\textbf{CFAR10 Test. } The test set images from CIFAR10 are preprocessed with a convolutional layer that outputs vectorized samples of $x_i \\in \\mathbb{R}^{10}$. This dataset has 10 features and 10,000 samples.  The preprocessing code to map the images to $\\mathbb{R}^{10}$ data is included in the supplementary. The link to download the data is at \\url{https:\/\/www.cs.toronto.edu\/~kriz\/cifar.html}.\n\n\n\\textbf{Raman. } The dataset consists of 4306 samples, 700 frequencies, and 35 different cell types. Since this is proprietary data, a download link is not included. \n\n\\end{appendices}\n\\section{Optimal Gaussian \\texorpdfstring{$\\sigma $ }\\xspace for Maximum Kernel Separation}\n\\label{app:opt_sigma}\nAlthough the Gaussian kernel is the most common kernel choice for kernel methods, its $\\sigma$ value is a hyperparameter that must be tuned for each dataset. This work proposes to set the $\\sigma$ value based on the maximum kernel separation. The source code is made publicly available on \\url{https:\/\/github.com\/anonamous}.\n\n\nLet $X \\in \\mathbb{R}^{n \\times d}$ be a dataset of $n$ samples with $d$ features and let $Y \\in \\mathbb{R}^{n \\times \\nclass}$ be the corresponding one-hot encoded labels where $\\nclass$ denotes the number of classes. Given $\\kappa_X(\\cdot, \\cdot)$ and $\\kappa_Y(\\cdot,\\cdot)$ as two kernel functions that applies respectively to $X$ and $Y$ to construct kernel matrices $K_X \\in \\mathbb{R}^{n \\times n}$ and $K_Y \\in \\mathbb{R}^{n \\times n}$. Given a set $\\mathcal{S}$, we denote $|\\mathcal{S}|$ as the number of elements within the set. Also let $\\mathcal{S}$ and $\\mathcal{S}^c$ be sets of all pairs of samples of $(x_i,x_j)$ from a dataset $X$ that belongs to the same and different classes respectively, then the average kernel value for all $(x_i,x_j)$ pairs with the same class is\n\\begin{equation}\n    d_{\\mathcal{S}} = \\frac{1}{|\\mathcal{S}|}\\sum_{i,j \\in \\mathcal{S}} e^{-\\frac{||x_i - x_j||^2}{2\\sigma^2}}\n\\end{equation}\nand the average kernel value for all $(x_i,x_j)$ pairs between different classes is\n\\begin{equation}\n    d_{\\mathcal{S}^c} = \n    \\frac{1}{|\\mathcal{S}^c|}\\sum_{i,j \\in \\mathcal{S}^c} e^{-\\frac{||x_i - x_j||^2}{2\\sigma^2}}. \n\\end{equation}\nWe propose to find the $\\sigma$ that maximizes the difference between $d_{\\mathcal{S}}$ and $d_{\\mathcal{S}^c}$ or \n\\begin{equation}\n    \\underset{\\sigma}{\\max} \\quad \n    \\frac{1}{|\\mathcal{S}|}\\sum_{i,j \\in \\mathcal{S}} e^{-\\frac{||x_i - x_j||^2}{2\\sigma^2}} - \n    \\frac{1}{|\\mathcal{S}^c|}\\sum_{i,j \\in \\mathcal{S}^c} e^{-\\frac{||x_i - x_j||^2}{2\\sigma^2}}.\n    \\label{eq:main_objective}\n\\end{equation}\nIt turns out that is expression can be computed efficiently. Let $g = \\frac{1}{|\\mathcal{S}|}$ and $\\bar{g} = \\frac{1}{|\\mathcal{S}^c|}$, and let $\\textbf{1}_{n \\times n} \\in \\mathbb{R}^{n \\times n}$ be a matrix of 1s, then we can define $Q$ as\n\\begin{equation}\n    Q = -g K_Y + \\bar{g} (\\textbf{1}_{n \\times n} - K_Y).\n\\end{equation}\nOr $Q$ can be written more compactly as\n\\begin{equation}\n    Q = \\bar{g} \\textbf{1}_{n \\times n} - (g + \\bar{g})K_Y. \n\\end{equation}\nGiven $Q$, Eq.~(\\ref{eq:main_objective}) becomes\n\\begin{equation}\n    \\underset{\\sigma}{\\min} \\quad \n    \\Tr(K_X Q).\n    \\label{eq:obj_compact}\n\\end{equation}\nThis objective can be efficiently solved with BFGS. \n\nBelow in Fig.~\\ref{fig:max_kernel_separation}, we plot out the average within cluster kernel and the between cluster kernel values as we vary $\\sigma$. From the plot, we can see that the maximum separation is discovered via BFGS. \n    \\begin{figure}[h]\n        \\centering\n        \\includegraphics[width=10cm,height=7cm]{img\/opt_kernel_separation.png}\n        \\caption{Maximum Kernel separation.}\n        \\label{fig:max_kernel_separation}\n    \\end{figure} \n    \n    \n\\textbf{Relation to HSIC. }    \nFrom Eq.~(\\ref{eq:obj_compact}), we can see that the $\\sigma$ that causes maximum kernel separation is directly related to HSIC. Given that the HSIC objective is normally written as\n\\begin{equation}\n    \\underset{\\sigma}{\\min} \\quad \n    \\Tr(K_X H K_Y H),\n\\end{equation}\nby setting $Q=HK_YH$, we can see how the two formulations are related. While the maximum kernel separation places the weight of each sample pair equally, HSIC weights the pair differently.\nWe also notice that the $Q_{i,j}$ element is positive\/negative for $(x_i,x_j)$ pairs that are with\/between classes respectively. Therefore, the argument for the global optimum should be relatively close for both objectives. Below in Figure~\\ref{fig:max_HSIC}, we show a figure of HSIC values as we vary $\\sigma$. Notice how the optimal $\\sigma$ is almost equivalent to the solution from maximum kernel separation. For the purpose of \\RS, we use $\\sigma$ that maximizes the HSIC value.\n    \\begin{figure}[h]\n        \\centering\n        \\includegraphics[width=10cm,height=7cm]{img\/opt_HSIC.png}\n        \\caption{Maximal HSIC.}\n        \\label{fig:max_HSIC}\n    \\end{figure} \n\\end{appendices}\n\\section{\\texorpdfstring{$W_l$ }\\xspace  Dimensions for each 10 Fold of each Dataset}\n\\label{app:W_dimensions}\nWe report the input and output dimensions of each $W_l$ for every layer of each dataset in the form of $(\\alpha, \\beta)$; the corresponding dimension becomes $W_l \\in \\mathbb{R}^{\\alpha \\times \\beta}$. Since each dataset consists of 10-folds, the network structure for each fold is reported. We note that the input of the 1st layer is the dimension of the original data. However, after the first layer, the width of the RFF becomes the output of each layer; here we use 300. \n\nThe $\\beta$ value is chosen during the ISM algorithm. By keeping only the most dominant eigenvector of the $\\Phi$ matrix, the output dimension of each layer corresponds with the rank of $\\Phi$. It can be seen from each dataset that the first layer significantly expands the rank. The expansion is generally followed by a compression of fewer and fewer eigenvalues. These results conform with the observations made by \\citet{montavon2011kernel} and \\citet{ansuini2019intrinsic}.\n\n\\begin{tabular}{ll}\n\\centering\n\\tiny\n\\setlength{\\tabcolsep}{7.0pt}\n\\renewcommand{\\arraystretch}{1.2}\n\\begin{tabular}{cccccc|}\n\t\\hline\nData & Layer 1 & Layer 2 & Layer 3 & Layer 4 \\\\ \n\t\\hline\nadversarial 1 & (2, 2) & (300, 61) & (300, 35) \\\\ \nadversarial 2 & (2, 2) & (300, 61) & (300, 35) \\\\ \nadversarial 3 & (2, 2) & (300, 61) & (300, 8) & (300, 4) \\\\ \nadversarial 4 & (2, 2) & (300, 61) & (300, 29) \\\\ \nadversarial 5 & (2, 2) & (300, 61) & (300, 29) \\\\ \nadversarial 6 & (2, 2) & (300, 61) & (300, 7) & (300, 4) \\\\ \nadversarial 7 & (2, 2) & (300, 61) & (300, 34) \\\\ \nadversarial 8 & (2, 2) & (300, 12) & (300, 61) & (300, 30) \\\\ \nadversarial 9 & (2, 2) & (300, 61) & (300, 33) \\\\ \nadversarial 10 & (2, 2) & (300, 61) & (300, 33) \\\\ \n\t\\hline\n\\end{tabular}\n&\n\\centering\n\\tiny\n\\setlength{\\tabcolsep}{3.0pt}\n\\renewcommand{\\arraystretch}{1.2}\n\\begin{tabular}{ccccc|}\n\t\\hline\nData & Layer 1 & Layer 2 & Layer 3 \\\\ \n\t\\hline\nRandom 1 & (3, 3) & (300, 47) & (300, 25) \\\\ \nRandom 2 & (3, 3) & (300, 46) & (300, 25) \\\\ \nRandom 3 & (3, 3) & (300, 46) & (300, 25) \\\\ \nRandom 4 & (3, 3) & (300, 47) & (300, 4) \\\\ \nRandom 5 & (3, 3) & (300, 47) & (300, 25) \\\\ \nRandom 6 & (3, 3) & (300, 45) & (300, 23) \\\\ \nRandom 7 & (3, 3) & (300, 45) & (300, 25) \\\\ \nRandom 8 & (3, 3) & (300, 45) & (300, 21) \\\\ \nRandom 9 & (3, 3) & (300, 45) & (300, 26) \\\\ \nRandom 10 & (3, 3) & (300, 47) & (300, 25) \\\\ \n\t\\hline\n\\end{tabular}\n\\end{tabular}\n\n\n\n\n\\begin{tabular}{ll}\n\\tiny\n\\setlength{\\tabcolsep}{3.0pt}\n\\renewcommand{\\arraystretch}{1.2}\n\\begin{tabular}{cccccccc|}\n\t\\hline\nData & Layer 1 & Layer 2 & Layer 3 & Layer 4 & Layer 5 & Layer 6 \\\\ \n\t\\hline\nspiral 1 & (2, 2) & (300, 15) & (300, 6) & (300, 7) & (300, 6) \\\\ \nspiral 2 & (2, 2) & (300, 13) & (300, 6) & (300, 7) & (300, 6) & (300, 6) \\\\ \nspiral 3 & (2, 2) & (300, 12) & (300, 6) & (300, 7) & (300, 6) & (300, 6) \\\\ \nspiral 4 & (2, 2) & (300, 13) & (300, 6) & (300, 7) & (300, 6) & (300, 6) \\\\ \nspiral 5 & (2, 2) & (300, 13) & (300, 6) & (300, 7) & (300, 6) \\\\ \nspiral 6 & (2, 2) & (300, 14) & (300, 6) & (300, 7) & (300, 6) \\\\ \nspiral 7 & (2, 2) & (300, 14) & (300, 6) & (300, 7) & (300, 6) \\\\ \nspiral 8 & (2, 2) & (300, 14) & (300, 6) & (300, 7) & (300, 6) & (300, 6) \\\\ \nspiral 9 & (2, 2) & (300, 13) & (300, 6) & (300, 7) & (300, 6) \\\\ \nspiral 10 & (2, 2) & (300, 14) & (300, 6) & (300, 7) & (300, 6) \\\\ \n\t\\hline\n\\end{tabular}\n&\n\\tiny\n\\setlength{\\tabcolsep}{3.0pt}\n\\renewcommand{\\arraystretch}{1.2}\n\\begin{tabular}{cccccccc}\n\t\\hline\nData & Layer 1 & Layer 2 & Layer 3 & Layer 4 & Layer 5 & Layer 6 \\\\ \n\t\\hline\nwine 1 & (13, 11) & (300, 76) & (300, 6) & (300, 7) & (300, 6) & (300, 6) \\\\ \nwine 2 & (13, 11) & (300, 76) & (300, 6) & (300, 6) & (300, 6) & (300, 6) \\\\ \nwine 3 & (13, 11) & (300, 75) & (300, 6) & (300, 7) & (300, 6) & (300, 6) \\\\ \nwine 4 & (13, 11) & (300, 76) & (300, 6) & (300, 6) & (300, 6) & (300, 6) \\\\ \nwine 5 & (13, 11) & (300, 74) & (300, 6) & (300, 7) & (300, 6) & (300, 6) \\\\ \nwine 6 & (13, 11) & (300, 74) & (300, 6) & (300, 6) & (300, 6) & (300, 6) \\\\ \nwine 7 & (13, 11) & (300, 74) & (300, 6) & (300, 6) & (300, 6) & (300, 6) \\\\ \nwine 8 & (13, 11) & (300, 75) & (300, 6) & (300, 7) & (300, 6) & (300, 6) \\\\ \nwine 9 & (13, 11) & (300, 75) & (300, 6) & (300, 8) & (300, 6) & (300, 6) \\\\ \nwine 10 & (13, 11) & (300, 76) & (300, 6) & (300, 7) & (300, 6) & (300, 6) \\\\ \n\t\\hline\n\\end{tabular}\n\\end{tabular}\n\n\\begin{tabular}{ll}\n\\tiny\n\\setlength{\\tabcolsep}{3.0pt}\n\\renewcommand{\\arraystretch}{1.2}\n\\begin{tabular}{cccccccc|}\n\t\\hline\nData & Layer 1 & Layer 2 & Layer 3 & Layer 4 & Layer 5 & Layer 6 \\\\ \n\t\\hline\ncar 1 & (6, 6) & (300, 96) & (300, 6) & (300, 8) & (300, 6) \\\\ \ncar 2 & (6, 6) & (300, 96) & (300, 6) & (300, 8) & (300, 6) \\\\ \ncar 3 & (6, 6) & (300, 91) & (300, 6) & (300, 8) & (300, 6) \\\\ \ncar 4 & (6, 6) & (300, 88) & (300, 6) & (300, 8) & (300, 6) & (300, 6) \\\\ \ncar 5 & (6, 6) & (300, 94) & (300, 6) & (300, 8) & (300, 6) \\\\ \ncar 6 & (6, 6) & (300, 93) & (300, 6) & (300, 7) \\\\ \ncar 7 & (6, 6) & (300, 92) & (300, 6) & (300, 8) & (300, 6) \\\\ \ncar 8 & (6, 6) & (300, 95) & (300, 6) & (300, 7) & (300, 6) \\\\ \ncar 9 & (6, 6) & (300, 96) & (300, 6) & (300, 9) & (300, 6) \\\\ \ncar 10 & (6, 6) & (300, 99) & (300, 6) & (300, 8) & (300, 6) \\\\ \n\t\\hline\n\\end{tabular}\n&\n\\tiny\n\\setlength{\\tabcolsep}{3.0pt}\n\\renewcommand{\\arraystretch}{1.2}\n\\begin{tabular}{ccccccc|}\n\t\\hline\nData & Layer 1 & Layer 2 & Layer 3 & Layer 4 & Layer 5 \\\\ \n\t\\hline\ndivorce 1 & (54, 35) & (300, 44) & (300, 5) & (300, 5) \\\\ \ndivorce 2 & (54, 35) & (300, 45) & (300, 4) & (300, 4) \\\\ \ndivorce 3 & (54, 36) & (300, 49) & (300, 6) & (300, 6) \\\\ \ndivorce 4 & (54, 36) & (300, 47) & (300, 7) & (300, 6) \\\\ \ndivorce 5 & (54, 35) & (300, 45) & (300, 6) & (300, 6) \\\\ \ndivorce 6 & (54, 36) & (300, 47) & (300, 6) & (300, 6) \\\\ \ndivorce 7 & (54, 35) & (300, 45) & (300, 6) & (300, 6) & (300, 4) \\\\ \ndivorce 8 & (54, 36) & (300, 47) & (300, 6) & (300, 7) & (300, 4) \\\\ \ndivorce 9 & (54, 36) & (300, 47) & (300, 5) & (300, 5) \\\\ \ndivorce 10 & (54, 36) & (300, 47) & (300, 6) & (300, 6) \\\\ \n\t\\hline\n\\end{tabular}\n\\end{tabular}\n\n\n\\begin{table}[h]\n\\tiny\n\\setlength{\\tabcolsep}{3.0pt}\n\\renewcommand{\\arraystretch}{1.2}\n\\begin{tabular}{cccccccccccc|}\n\t\\hline\nData & Layer 1 & Layer 2 & Layer 3 & Layer 4 & Layer 5 & Layer 6 & Layer 7 & Layer 8 & Layer 9 & Layer 10 \\\\ \n\t\\hline\ncancer 1 & (9, 8) & (300, 90) & (300, 5) & (300, 6) & (300, 6) & (300, 5) & (300, 4) & (300, 5) & (300, 6) & (300, 6) \\\\ \ncancer 2 & (9, 8) & (300, 90) & (300, 6) & (300, 7) & (300, 8) & (300, 11) & (300, 8) & (300, 4) \\\\ \ncancer 3 & (9, 8) & (300, 88) & (300, 5) & (300, 6) & (300, 7) & (300, 7) & (300, 6) & (300, 4) \\\\ \ncancer 4 & (9, 8) & (300, 93) & (300, 6) & (300, 7) & (300, 9) & (300, 11) & (300, 8) \\\\ \ncancer 5 & (9, 8) & (300, 93) & (300, 9) & (300, 10) & (300, 10) & (300, 11) & (300, 9) & (300, 7) \\\\ \ncancer 6 & (9, 8) & (300, 92) & (300, 7) & (300, 8) & (300, 8) & (300, 7) & (300, 7) \\\\ \ncancer 7 & (9, 8) & (300, 90) & (300, 4) & (300, 4) & (300, 5) & (300, 6) & (300, 6) & (300, 6) & (300, 6) \\\\ \ncancer 8 & (9, 8) & (300, 88) & (300, 5) & (300, 6) & (300, 7) & (300, 8) & (300, 7) & (300, 6) \\\\ \ncancer 9 & (9, 8) & (300, 88) & (300, 5) & (300, 7) & (300, 7) & (300, 7) & (300, 7) \\\\ \ncancer 10 & (9, 8) & (300, 97) & (300, 9) & (300, 11) & (300, 12) & (300, 13) & (300, 6) \\\\ \n\t\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table}[h]\n\\tiny\n\\setlength{\\tabcolsep}{3.0pt}\n\\renewcommand{\\arraystretch}{1.2}\n\\begin{tabular}{cccccc|}\n\t\\hline\nData & Layer 1 & Layer 2 & Layer 3 & Layer 4 \\\\ \n\t\\hline\nface 1 & (960, 233) & (300, 74) & (300, 73) & (300, 46) \\\\ \nface 2 & (960, 231) & (300, 75) & (300, 73) & (300, 43) \\\\ \nface 3 & (960, 231) & (300, 76) & (300, 73) & (300, 44) \\\\ \nface 4 & (960, 232) & (300, 76) & (300, 74) & (300, 44) \\\\ \nface 5 & (960, 231) & (300, 77) & (300, 73) & (300, 43) \\\\ \nface 6 & (960, 232) & (300, 74) & (300, 72) & (300, 47) \\\\ \nface 7 & (960, 232) & (300, 76) & (300, 73) & (300, 45) \\\\ \nface 8 & (960, 230) & (300, 74) & (300, 74) & (300, 44) \\\\ \nface 9 & (960, 233) & (300, 76) & (300, 76) & (300, 45) \\\\ \nface 10 & (960, 231) & (300, 76) & (300, 70) & (300, 43) \\\\ \n\t\\hline\n\\end{tabular}\n\\end{table}\n\n\\end{appendices}\n\n\\section{Sigma Values used for Random and Adversarial Simulation}\n\\label{app:sigma_values}\n\nThe simulation of Thm.~\\ref{thm:hsequence} as shown in Fig.~\\ref{fig:thm_proof} spread the improvement across multiple layers. The $\\sigma_l$ and $\\hsic_l$ values are recorded here. We note that $\\sigma_l$ are reasonably large and not approaching 0 and the improvement of $\\hsic_l$ is monotonic. \n\n\\begin{figure}[h]\n\\center\n    \\includegraphics[width=9cm]{img\/Random_sigma.png}\n    \\caption{}\n\\end{figure} \n\n\\begin{figure}[h]\n\\center\n    \\includegraphics[width=9cm]{img\/adv.png}\n    \\caption{}\n\\end{figure} \n\n\n\nGiven a sufficiently small $\\sigma_0$ and $\\sigma_1$, Thm.~\\ref{thm:hsequence} claims that it can come arbitrarily close to the global optimal using a minimum of 2 layers. We here simulate 2 layers using a relatively small $\\sigma$ values ($\\sigma_0 = 10^{-5}$) on the Random (left) and Adversarial (right) data and display the results of the 2 layers below. Notice that given 2 layer, it generated a clearly separable clusters that are pushed far apart.\n\n    \\begin{figure}[!h]\n    \\begin{minipage}[H]{6.0cm}\n        \\includegraphics[width=6cm]{img\/random_2_layer.png}\n        \\caption{Random Dataset with 2\\\\layers and $\\sigma=10^{-5}$}\n    \\end{minipage}%\n    \\begin{minipage}[H]{6.0cm}\n        \\includegraphics[width=6cm]{img\/adversarial_2_layer.png}\n        \\caption{Adversarial Dataset with 2 \\\\layers and $\\sigma=10^{-5}$}\n    \\end{minipage}\n    \\end{figure}          \n\n\\end{appendices}\n\\section{Evaluation Metrics Graphs}\n\\label{app:metric_graphs}\n\n\\begin{figure}[h]\n\\center\n    \\includegraphics[width=10cm]{img\/r1.png}\n    \\caption{}\n\\end{figure} \n\n\\begin{figure}[h]\n\\center\n    \\includegraphics[width=10cm]{img\/r2.png}\n    \\caption{Figures of key metrics for all datasets as samples progress through the network. It is important to notice the uniformly and monotonically increasing \\RS for each plot\n    since this guarantees a converging kernel\/risk sequence. As the $\\mathcal{T}$ approach 0, samples of the same\/difference classes in IDS are being pulled into a single point or pushed maximally apart respectively. As $C$ approach 0, samples of the same\/difference classes in RKHS are being pulled into 0 or $\\frac{\\pi}{2}$ cosine similarity respectively.}\n\\end{figure} \n\n\\end{appendices}\n\n\\section{Graphs of Kernel Sequences}\n\\label{app:kernel_sequence_graph}\n\nA representation of the \\KS are displayed in the figures below for each dataset. The samples of the kernel matrix are previously organized to form a block structure by placing samples of the same class adjacent to each other. Since the Gaussian kernel is restricted to values between 0 and 1, we let white and dark blue be 0 and 1 respectively where the gradients reflect values in between. Our theorems predict that the \\KS will evolve from an uninformative kernel into a highly discriminating kernel of perfect block structures. \n\n\\begin{figure}[h]\n\\center\n    \\includegraphics[width=9cm]{img\/wine_kernel.png}\n    \\caption{The kernel sequence for the wine dataset.}\n\\end{figure} \n\n\\begin{figure}[h]\n\\center\n    \\includegraphics[width=9cm]{img\/cancer_kernel.png}\n    \\caption{The kernel sequence for the cancer dataset.}\n\\end{figure} \n\n\\begin{figure}[h]\n\\center\n    \\includegraphics[width=9cm]{img\/adv_kernel.png}\n    \\caption{The kernel sequence for the Adversarial dataset.}\n\\end{figure} \n\n\\begin{figure}[h]\n\\center\n    \\includegraphics[width=9cm]{img\/car_kernel.png}\n    \\caption{The kernel sequence for the car dataset.}\n\\end{figure} \n\n\\begin{figure}[h]\n\\center\n    \\includegraphics[width=11cm]{img\/face_kernel.png}\n    \\caption{The kernel sequence for the face dataset.}\n\\end{figure} \n\n\\begin{figure}[h]\n\\center\n    \\includegraphics[width=9cm]{img\/divorce_kernel.png}\n    \\caption{The kernel sequence for the divorce dataset.}\n\\end{figure} \n\n\\begin{figure}[h]\n\\center\n    \\includegraphics[width=11cm]{img\/spiral_kernel.png}\n    \\caption{The kernel sequence for the spiral dataset.}\n\\end{figure} \n\n\n\n\\begin{figure}[h]\n\\center\n    \\includegraphics[width=11cm]{img\/random_kernel.png}\n    \\caption{The kernel sequence for the Random dataset.}\n\\end{figure} \n\\end{appendices}\n\\section{On Generalization. } Besides being an optimum solution, $W_l^*$ exhibits many advantages over $W_s$. For example, while $W_s$ experimentally performs well, $W^*$ converges with fewer layers and superior generalization. This raises a well-known question on generalization. It is known that overparameterized MLPs can generalize even without any explicit regularizer \\citep{Zhang2017UnderstandingDL}. This observation contradicts classical learning theory and has been a longstanding puzzle \\citep{cao2019generalization,brutzkus2017sgd,allen2019learning}. \nTherefore, by being overparameterized with an infinitely wide network, \\kn's ability under HSIC to generalize raises similar questions. In both cases, $W_s$ and $W^*$, the HSIC objective employs an infinitely wide network that should result in overfitting.  We ask theoretically, under our framework, what makes HSIC and $W^*$ special? \n\nRecently, \\citet{poggio2020complexity} have proposed that  traditional MLPs generalize  because gradient methods implicitly regularize the normalized weights given an exponential objective (like our HSIC). We discovered a similar impact the process of finding $W^*$ has on HSIC, i.e., HSIC can be reformulated to isolate out $n$ functions $[D_1(W_l), ..., D_n(W_l)]$ that act as a penalty term during optimization. Let $\\cS_i$ be the set of samples that belongs to the $i_{th}$ class and let $\\cS^c_i$ be its complement, then each function $D_i(W_l)$ is defined as \\begin{equation}\n    D_i(W_l) =  \n    \\frac{1}{\\sigma^2}\n    \\sum_{j \\in \\cS_i}\n    \\Gij \\kf_{W_l}(r_i,r_j)\n    -\n    \\frac{1}{\\sigma^2}\n    \\sum_{j \\in \\cS^c_i}\n    |\\Gij| \\kf_{W_l}(r_i,r_j).\n\\end{equation}\nNotice that $D_i(W_l)$ is simply \\eq{eq:similarity_hsic} for a single sample scaled by $\\frac{1}{\\sigma^2}$. Therefore, improving $W_l$ also leads to an increase and decrease of $\\kf_{W_l}(r_i,r_j)$ associated with $\\cS_i$ and $\\cS^c_i$ in \\eq{eq:penalty_term}, thereby increasing the size of the penalty term  $D_i(W_l)$.  To appreciate how $D_i(W_l)$ penalizes $\\hsic$, we propose an equivalent formulation in the theorem below with its derivation in App~\\ref{app:thm:regularizer}.\n\\begin{theorem}\n\\eq{eq:similarity_hsic} is equivalent to \n\\begin{equation}\n    \\max_{W_l} \n    \\sum_{i,j}\n    \\frac{\\Gij}{\\sigma^2}\n    \\ISMexp\n    (r_i^TW_lW_l^Tr_j) \n    -\n   \n    \\sum_{i}\n    D_i(W_l)\n    ||W_l^Tr_i||_2.\n\\end{equation}\n\\end{theorem}\nBased on Thm.~\\ref{thm:regularizer}, $D_i(W_l)$ adds a negative variable cost to the sample norm, $||W_l^Tr_i||_2$, prescribing an implicit regularizer on HSIC. As $W_l$ improve HSIC, it also imposes a heavier penalty on \\eq{eq:generalization_formulation}, severely constraining $W_l$.\n\n\\end{appendices}\n\n\\section{Proof for Theorem \\ref{thm:regularizer}}\n\\label{app:thm:regularizer}\n\n\\textbf{Theorem \\ref{thm:regularizer}: }\n\\textit{\\eq{eq:similarity_hsic} objective is equivalent to }\n\\begin{equation}\n        \\sum_{i,j}\n        \\Gij \\ISMexp\n        (r_i^TWW^Tr_j)\n        -\n        \\sum_{i}\n        D_i(W)\n        ||W^Tr_i||_2.\n\\end{equation}\n\n\\begin{proof}\nLet $A_{i,j} = (r_i - r_j)(r_i - r_j)^T$. Given the Lagranian of the HSIC objective as \n\\begin{equation}\n    \\mathcal{L} = -\\sum_{i,j} \\Gij \\ISMexp - \\Tr[\\Lambda(W^TW - I)].\n\\end{equation}\nOur layer wise HSIC objective becomes \n\\begin{equation}\n    \\min_W -\\sum_{i,j} \\Gij \\ISMexp - \\Tr[\\Lambda(W^TW - I)].\n    \\label{obj:hsic}\n\\end{equation}\nWe take the derivative of the Lagrangian, the expression becomes\n   \\begin{equation} \\label{eq:gradient_of_lagrangian}\n    \\nabla_W \\mathcal{L} ( W, \\Lambda) = \\sum_{i, j} \\frac{\\Gamma_{i,\n      j}}{\\sigma^2} e^{- \\frac{\\Tr (W^T A_{i, j} W)}{2 \\sigma^2}} A_{i, j} W\n      - 2 W \\Lambda. \n    \\end{equation}\nSetting the gradient to 0, and consolidate some scalar values into $\\hat{\\Gamma}_{i,j}$, we get the expression\n   \\begin{align} \n    \\left[\n    \\sum_{i, j} \\frac{\\Gamma_{i,\n      j}}{2\\sigma^2} e^{- \\frac{\\Tr (W^T A_{i, j} W)}{2 \\sigma^2}} A_{i, j} \n     \\right]\n      W\n      & =  W \\Lambda \\\\\n      \\left[\\frac{1}{2}\n      \\sum_{i,j} \\hat{\\Gamma}_{i,j} A_{i,j}\n      \\right] W\n      & = W \\Lambda \\\\\n      \\mathcal{Q} W\n      & = W \\Lambda.      \n    \\end{align}\nFrom here, we see that the optimal solution is an eigenvector of $\\mathcal{Q}$. Based on ISM, it further proved that the optimal solution is not just any eigenvector, but the eigenvectors associated with the smallest values of $\\mathcal{Q}$. From this logic, ISM solves objective~(\\ref{obj:hsic}) with a surrogate objective\n\\begin{equation}\n    \\min_W \\quad\n        \\Tr \\left( W^T\n        \\left[\\frac{1}{2}\n        \\sum_{i,j} \\hat{\\Gamma}_{i,j} A_{i,j}\n        \\right] W \\right) \\quad \\st W^TW=I.\n    \\label{obj:min_2}\n\\end{equation}\nGiven $D_{\\hat{\\Gamma}}$ as the degree matrix of $\\hat{\\Gamma}$ and $R = [r_1, r_2, ...]^T$, ISM further shows that Eq.~(\\ref{obj:min_2}) can be written into\n\\begin{align}\n    \\min_W \\quad\n        \\Tr \\left( W^T R^T\n        \\left[\n        D_{\\hat{\\Gamma}} - \\hat{\\Gamma}\n        \\right] R W \\right) &\\quad \\st W^TW=I \\\\\n     \\max_W \\quad\n        \\Tr \\left( W^T R^T\n        \\left[\n        \\hat{\\Gamma} - \n        D_{\\hat{\\Gamma}}\n        \\right] R W \\right) &\\quad \\st W^TW=I \\\\       \n     \\max_W \\quad\n        \\Tr \\left( W^T R^T\n        \\hat{\\Gamma}  \n        R W \\right) \n        -\n        \\Tr \\left( W^T R^T\n        D_{\\hat{\\Gamma}}  \n        R W \\right) \n        &\\quad \\st W^TW=I \\\\              \n     \\max_W \\quad\n        \\Tr \\left( \n        \\hat{\\Gamma}  \n        R W W^T R^T\\right) \n        -\n        \\Tr \\left( \n        D_{\\hat{\\Gamma}}  \n        R W W^T R^T\\right) \n        &\\quad \\st W^TW=I \\\\                     \n     \\max_W \\quad\n        \\sum_{i,j}\n        \\hat{\\Gamma}_{i,j}\n        [R W W^T R^T]_{i,j}\n        -\n        \\sum_{i,j}\n        D_{\\hat{\\Gamma}_{i,j}}\n        [R W W^T R^T]_{i,j}       \n        &\\quad \\st W^TW=I. \n\\end{align}\nSince the jump from \\eq{obj:min_2} can be intimidating for those not familiar with the literature, we included a more detailed derivation in App.~\\ref{app:matrix_derivation}.\n\nNote that the degree matrix $D_{\\hat{\\Gamma}}$ only have non-zero diagonal elements, all of its off diagonal are 0. Given $[RWW^TR^T]_{i,j} = (r_i^TWW^Tr_j)$, the objective becomes\n\\begin{equation}\n     \\max_W \\quad\n        \\sum_{i,j}\n        \\hat{\\Gamma}_{i,j}\n        (r_i^TWW^Tr_j)\n        -\n        \\sum_{i}\n        D_i(W)\n        ||W^Tr_i||_2\n        \\quad \\st W^TW=I.     \n\\end{equation}\nHere, we treat $D_{i}$ as a penalty weight on the norm of the $W^Tr_i$ for every sample. \n\\end{proof}\n\n\nTo better understand the behavior of $D_i(W)$, note that $\\hat{\\Gamma}$ matrix looks like\n\\begin{equation}\n    \\hat{\\Gamma} = \\frac{1}{\\sigma^2} \\begin{bmatrix}\n       \\begin{bmatrix}\n       \\Gamma_{\\mathcal{S}} \\ISMexp\n        \\end{bmatrix}\n       &  \n       \\begin{bmatrix}\n       -|\\Gamma_{\\mathcal{S}^c}| \\ISMexp\n       \\end{bmatrix}\n       &\n       ... \\\\\n       \\begin{bmatrix}\n       -|\\Gamma_{\\mathcal{S}^c}| \\ISMexp\n       \\end{bmatrix}\n       &      \n       \\begin{bmatrix}\n       \\Gamma_{\\mathcal{S}} \\ISMexp\n        \\end{bmatrix}      \n       &            \n       ... \\\\\n       ... &\n       ... &\n       ...\n    \\end{bmatrix}.\n\\end{equation}\nThe diagonal block matrix all $\\Gij$ elements that belong to $\\cS$ and the off diagonal are elements that belongs to $\\cS^c$. Each penalty term is the summation of its corresponding row. Hence, we can write out the penalty term as\n\\begin{equation}\n    D_i(W_l) = \n    \\frac{1}{\\sigma^2}\n    \\sum_{j \\in \\cS|i}\n    \\Gij \\kf_{W_l}(r_i,r_j)\n    -\n    \\frac{1}{\\sigma^2}\n    \\sum_{j \\in \\cS^c|i}\n    |\\Gij| \\kf_{W_l}(r_i,r_j).\n\\end{equation}\nFrom this, it shows that as $W$ improve the objective, the penalty term is also increased. In fact, at its extreme as $\\hsic_l \\rightarrow \\hsic^*$, all the negative terms are gone and all of its positive terms are maximized and this matrix approaches \n\\begin{equation}\n    \\hat{\\Gamma}^* = \\frac{1}{\\sigma^2} \\begin{bmatrix}\n       \\begin{bmatrix}\n       \\Gamma_{\\mathcal{S}} \n        \\end{bmatrix}\n       &  \n       \\begin{bmatrix}\n       0\n       \\end{bmatrix}\n       &\n       ... \\\\\n       \\begin{bmatrix}\n       0\n       \\end{bmatrix}\n       &      \n       \\begin{bmatrix}\n       \\Gamma_{\\mathcal{S}} \n        \\end{bmatrix}      \n       &            \n       ... \\\\\n       ... &\n       ... &\n       ...\n    \\end{bmatrix}.\n\\end{equation}\nFrom the matrix $\\hat{\\Gamma}^*$ and the definition of $D_i(W_l)$, we see that as $\\mathcal{K}_W$ from $\\mathcal{S}$ increase, \n\nSince $D_i(W)$ is the degree matrix of $\\hat{\\Gamma}$, we see that as $\\hsic_l \\rightarrow \\hsic^*$, we have\n\\begin{equation}\n   D^*_i(W) > D_i(W). \n\\end{equation}\n\\end{appendices} \n\\section{How This Work Relates to Climate Change}\n\\label{app:motivation}    \n    Finding an alternative to BP also has significant climate implications.   \\citet{strubell2019energy} have shown that some standard AI models can emit over 626,000 pounds of carbon dioxide; a carbon footprint five times greater than the lifetime usage of a car. This level of emission is simply not sustainable in light of our continual explosive growth. Therefore, the environmental impact of BP necessitates a cheaper alternative. Looking at nature, we can be inspired by the brain's learning capability using only a fraction of the energy. Perhaps artificial neurons can also train without the high energy cost to the environment. This is the moral and the foundational motivation for this work in identifying the existence of $W_s$. A closed-form solution holds the potential to significantly reduce the computational requirement and carbon footprint.  Even if our work ultimately failed to mimic the brain, we hope to inspire the community to identify other closed-form solutions and go beyond BP. \\\\ \\\\\n    This paper aims to promote the discussion of viewing backpropagation alternatives not only as an academic exercise but also as a climate imperative. Yet, this topic is largely ignored by the community. The authors believe the energy costs of training Neural Networks are having a detrimental climate impact and should be an added topic of interest. The earth also needs an advocate, why not us? Therefore, we as a community, must begin addressing how we can ameliorate our own carbon footprint. This exploratory work aims to share a potential path forward for further research that may address these concerns with the community. While $W_s$ is still not ready for commercial usage, we sincerely hope that the community begins to build novel algorithms over our work on \\RS and identify a simpler and cheaper path to train our networks. \\\\\n    \n\\end{appendices} \n\\section{Derivation for \\texorpdfstring{$\\sum_{i,j} \\Psi_{i,j} (x_i - x_j)(x_i - x_j)^T = 2X^T(D_\\Psi - \\Psi)X$ }\\xspace }\n\\label{app:matrix_derivation}\nSince $\\Psi$ is a symmetric matrix, and $A_{i, j} = ( x_i - x_j) ( x_i -\nx_j)^T $, we can\nrewrite the expression into\n\\[ \\begin{array}{lll}\n     \\sum_{i, j} \\Psi_{i, j} A_{i, j}  & = &  \\sum_{i, j} \\Psi_{i, j} ( x_i -\n     x_j) ( x_i - x_j)^T\\\\\n     & = & \\sum_{i, j} \\Psi_{i, j} ( x_i x_i^T - x_j x_i^T - x_i x_j^T +\n     x_j x_j^T)\\\\\n     & = & 2 \\sum_{i, j} \\Psi_{i, j} ( x_i x_i^T - x_j x_i^T)\\\\\n     & = & \\left[ 2 \\sum_{i, j} \\Psi_{i, j} ( x_i x_i^T) \\right] - \\left[ 2\n     \\sum_{i, j} \\Psi_{i, j} ( x_i x_j^T) \\right] .\n   \\end{array} \\]\nIf we expand the 1st term, we get\n\\begin{align}\n    2\\sum_{i}^n \\sum_{j}^n \n    \\Psi_{i, j} ( x_i x_i^T)\n    & = \n    2 \\sum_i\n   \\Psi_{i, 1} ( x_i x_i^T) +\n   \\ldots + \\Psi_{i, n} ( x_i x_i^T) \\\\\n   &= \n   2 \\sum_{i}^n \n   [\\Psi_{1, 1} + \\Psi_{1, 2} + ...]\n   x_i x_i^T \\\\\n   &= \n   2\\sum_{i}^n \n   d_i \n   x_i x_i^T \\\\  \n   &=\n   2 X^TD_\\Psi X\n\\end{align}\nGiven $\\Psi_i$ as the $i$th row, next we look at the 2nd term\n\\begin{align}\n    2 \\sum_i \\sum_j \\Psi_{i, j}  x_i x_j^T\n    &=\n    2 \\sum_i \\Psi_{i, 1}  x_i x_1^T\n    + \\Psi_{i, 2}  x_i x_2^T\n    + \\Psi_{i, 3}  x_i x_3^T + ...\\\\\n    &=\n    2 \\sum_i   x_i (\\Psi_{i, 1} x_1^T)\n    + x_i (\\Psi_{i, 2} x_2^T)\n    + x_i (\\Psi_{i, 3} x_3^T) + ...\\\\\n    &=\n    2 \\sum_i   x_i \n    \\left[\n    (\\Psi_{i, 1} x_1^T)\n    + (\\Psi_{i, 2} x_2^T)\n    + (\\Psi_{i, 3} x_3^T) + ...\n    \\right]\\\\\n    &=\n    2 \\sum_i   x_i \n    \\left[\n    X^T \\Psi_i^T\n    \\right]^T\\\\   \n    &=\n    2 \\sum_i   x_i \n    \\left[\n    \\Psi_i X\n    \\right]\\\\      \n    &=\n    2 \\left[   \n    x_1 \\Psi_1 X + \n    x_2 \\Psi_2 X + \n    x_3 \\Psi_3 X + ...\n    \\right]\\\\         \n    &=\n    2 \\left[   \n    x_1 \\Psi_1  + \n    x_2 \\Psi_2  + \n    x_3 \\Psi_3  + ...\n    \\right]X \\\\            \n    &=\n    2X^T \\Psi  X \\\\               \n\\end{align}\nPutting both terms together, we get\n\\begin{align}\n    \\sum_{i,j} \\Psi_{i,j} A_{i,j} &=  2 X^TD_\\Psi X -  2X^T \\Psi  X a\\\\\n    &=  2 X^T[ D_\\Psi - \\Psi] X \\\\\n\\end{align}\n\n\\end{appendices} \n\\subsubsection*{\\bibname}}\n\n\\usepackage[round]{natbib}\n\\bibliographystyle{plainnat}\n\n\\begin{document}\n\n\n\n\\twocolumn[\n\n\\aistatstitle{Deep Layer-wise Networks Have Closed-Form Weights}\n\n\\aistatsauthor{ Chieh Wu* \\And Aria Masoomi* \\And  Arthur Gretton \\And Jennifer Dy }\n\n\n\\aistatsaddress{ Northeastern University \\And  Northeastern University \\And University College London \\And Northeastern University} ]\n\n\\input{tex\/a_abstract}\n\\input{tex\/b_intro}\n\\input{tex\/c_related_work}\n\\input{tex\/d_model}\n\\input{tex\/e_thm1}\n\\input{tex\/f_network_behavior}\n\\input{tex\/f2_generalization}\n\\input{tex\/g_experiments}\n\\input{tex\/i_limitations}\n\n\\clearpage\n\n\n\\section{INTRODUCTION}\nDue to the brain-inspired architecture of Multi-layered Perceptrons (MLPs), the relationship between MLPs and our brains has been a topic of significant interest  \\citep{zador2019critique,walker2020Deep}. This line of research triggered a debate  \\citep{whittington2019theories} around the neural plausibility of backpropagation (BP). While some contend that brains cannot simulate BP \\citep{crick1989recent,grossberg1987competitive}, others have proposed counterclaims with a new generation of models \\citep{hinton2007backpropagation,Lilli2020Backpropagation,liao2016important,bengio2017stdp,guerguiev2017towards,sacramento2018dendritic,whittington2017approximation}. This debate has inspired the search for alternative optimization strategies beyond BP. To better mimic the brain, learning the network \\textit{one layer at a time} over a single forward pass (we call it layer-wise network) has been proposed as a more likely candidate to match existing understandings in neuroscience \\citep{ma2019hsic,Pogodin2020KernelizedIB,Oord2018RepresentationLW}. Our theoretical work contributes to this debate by answering two open questions regarding layer-wise networks.\n\\begin{enumerate}\n[noitemsep,topsep=0pt,leftmargin=5mm]\n    \\item \\textit{Do they have a closed-form solution? }\n    \\item \\textit{How do we know when to stop adding more layers? }\n\\end{enumerate}\nQuestion 1 asks if easily computable and \\textit{closed-form weights} can theoretically yield networks equally powerful as traditional MLPs, bypassing both BP and Stochastic Gradient Descent (SGD). This question is answered by characterizing the expressiveness of layer-wise networks using only \\textit{\"trivially learned weights\"}. Currently, the \\textit{Universal Approximation Theorem} states that a network can approximate any continuous function \\citep{cybenko1989approximation, hornik1991approximation,zhou2020universality}. However, it is not obvious that weights of \"layer-wise networks\" can be computed closed-form requiring only basic operations, a simplicity constraint inspired by biology. As our contribution, we prove that layer-wise networks can classify \\textit{any pattern} with trivially obtainable closed-form weights using only \\textit{addition}. Surprisingly, these weights turn out to be the \\kme.\n\n\nIdentifying the network depth has been an open question for traditional networks. For layer-wise networks, this question reduces down to \"\\textit{when should we stop adding layers?}. We posit that additional layers become unnecessary if layer-wise networks exhibit a \\textit{limiting behavior} where adding more layers ceases to meaningfully change the network. We proved that this is theoretically possible by using \\kme as weights. In fact, we show that these networks could be modeled as a \\textit{mathematical sequence} of functions that converge by \\textit{intentional design}. Indeed, not only can these networks converge, they can be induced to converge towards a highly desirable kernel for classification; we call it the \\textit{Neural Indicator Kernel} (NIK).\n\n\n\n\\section{MODELING LAYER-WISE NETWORKS}\n\\textbf{Layer Construction. } Let $X \\in \\mathbb{R}^{n \\times d}$ be a dataset of $n$ samples with $d$ features and let $Y \\in \\mathbb{R}^{n \\times \\nclass}$ be its one-hot encoded labels with $\\nclass$ classes. The $l^{th}$ layer consists of linear weights $W_l \\in \\mathbb{R}^{m \\times q}$ followed by an activation function $\\af:\\mathbb{R}^{n \\times q} \\rightarrow \\mathbb{R}^{n \\times m}$. We interpret each layer as a function $\\fm_l$ parameterized by $W_l$ with an input\/output denoted as $R_{l-1} \\in \\mathbb{R}^{n \\times m}$ and $R_{l} \\in \\mathbb{R}^{n \\times m}$ where $R_l = \\fm_l(R_{l-1}) = \\af(R_{l-1}W_l)$. The entire network $\\fm$ is the composition of all layers where $\\fm$ where $\\fm = \\fm_L \\circ ... \\circ \\fm_1$. \n\n\\textbf{Network Objective. } \nGiven $x_i, y_i$ as the $i^{th}$ sample and label of the dataset, the network output is used to minimize an empirical risk $(\\hsic)$ with a loss function $(\\ell)$ with a general objective of\n    \\begin{equation}\n    \\underset{\\fm}{\\min} \\hspace{0.3cm} \\hsic \n    \\coloneqq\n    \\underset{\\fm}{\\min} \\hspace{0.3cm} \\frac{1}{n} \\sum_{i=1}^n \\ell(\\fm(x_i), y_i).\n     \\label{eq:basic_empirical_risk}\n    \\end{equation}\nAs Eq.~(\\ref{eq:basic_empirical_risk}), we are structurally identical to conventional MLPs where each layer consists of linear weights and an activation function. Yet, we differ by introducing the composition of the first $l$ layers as $\\fm_{l^\\circ} = \\fm_l \\circ ... \\circ \\fm_1$ where $l \\leq L$.  This notation $(\\fm_{l^\\circ})$ connects the data directly to the $l$th layer output where $R_l = \\fm_{l^{\\circ}}(X)$ and leads to the \\textit{key novelty of our theoretical contribution}. Namely, we propose to optimize Eq.~(\\ref{eq:basic_empirical_risk}) layer-wise as a sequence of a growing networks by replacing $\\fm$ in \\eq{eq:basic_empirical_risk} incrementally with a sequence of functions $\\{\\fm_{l^\\circ}\\}_{l=1}^L$. This results in a sequence of empirical risks $\\{\\hsic_l\\}_{l=1}^L$ which we incrementally solve. We refer to $\\{\\fm_{l^\\circ}\\}_{l=1}^L$ and $\\{\\hsic_{l}\\}_{l=1}^L$ as the \\KS and the \\RS. \n\nSolving Eq.~(\\ref{eq:basic_empirical_risk}) as \\RS is where we differ from tradition, this approach enables us to easily represent, analyze and optimize \"layer-wise networks\". In contrast to using BP with SGD, we now can use \\textit{\"closed-form solutions\"} for $W_l$ to construct \\textit{Kernel Sequences} that drives the \\RS to automatically minimize Eq.~(\\ref{eq:basic_empirical_risk}). To visualize the network structure and how they form the sequences, refer to Fig.~\\ref{fig:notation}.\n\n\\begin{figure*}[t]\n\\center\n   \n    \\includegraphics[width=12cm,height=3.7cm]{img\/notations.png}\n    \\caption{\\textbf{Left - Distinguishing } $\\phi_l$ vs $\\phi_{l^\\circ}$\\textbf{:} $\\fm_l$ is a single layer while $\\fm_{l^{\\circ}} = \\fm_l \\circ ... \\circ \\fm_{1^\\circ}$ is a composition of the first $l$ layers.\n    \\textbf{Right - Visualize the \\textit{Kernel} and $\\mathcal{H}$-\\textit{Sequences}: } Note that the \\KS is a converging sequence of \"\\textit{functions}\" $\\{\\fm_{l^\\circ}\\}_{l=1} = \\{\\fm_{1^\\circ}, \\fm_{2^\\circ}, ... \\}$ and  \\RS is a converging sequence of scalar values $\\{\\hsic_{l}\\}_{l=1} = \\{\\hsic_{1}, \\hsic_{2}, ... \\}$.\n    To minimize Eq.~(\\ref{eq:basic_empirical_risk}) at $\\hsic_l$, all weights before $W_l$ are already identified and held fixed, only $W_l$ is unknown. \n    As $l\\rightarrow \\infty$, the \\KS converges to the \\textit{Neural Indicator Kernel}.  }\n    \\label{fig:notation}\n\\end{figure*} \n\n\\textbf{Performing Classification. } \nClassification tasks typically use objectives like Mean Squared Error ($\\mse$) or Cross-Entropy ($\\ce$) to match the network output $\\fm(X)$ to the label $Y$. While this approach achieves the desirable outcome, it also constrains the space of potential solutions where $\\fm(X)$ must match $Y$. Yet, if $\\fm$ maps $X$ to the labels $\\{0,1\\}$ instead of the true label $\\{-1,1\\}$, \n$\\fm(X)$ may not match $Y$, but the solution is the same. Therefore, enforcing $\\fm(X) = Y$ ignores an entire space of equivalently optimal classifiers. We posit that by relaxing this constraint and accepting a larger space of potential global optima, it will be easier during optimization to collide with this space. This intuition motivates us to depart from the tradition of label matching and instead seek alternative objectives that focus on solving the underlying prerequisite of classification, i.e.,  learning a mapping where samples from \\textit{similar and different} classes become easily distinguishable.\n\nHowever, since there are many ways to define \\textit{similarity}, how do we choose the best one that leads to classification? We demonstrate that the Hilbert Schmidt Independence Criterion (HSIC \\citep{gretton2005measuring}) is a highly advantageous objective for this purpose. As we'll later show in corollaries \\ref{corollary:mse} and \\ref{corollary:ce}, its maximization indirectly minimizes both Mean Square Error (MSE) and Cross-Entropy (CE) under different notions of \"distance\", enabling classification. Moreover, this is made possible because maximizing HSIC automatically learns the optimal notion of \\textit{similarity} as a kernel function. Using HSIC objective as $\\hsic_l$ for each element of \n$\\{\\hsic_{l}\\}_{l=1}^L$, the layer-wise formulation of Eq.~(\\ref{eq:basic_empirical_risk}) becomes\n\\begin{equation}\n\\begin{aligned}\n    \\max_{W_l} \\quad &\n        \\Tr \\left(\n        \\Gamma \\:\n        \\left[\n        \\af(R_{l-1}W_l) \\af^T(R_{l-1}W_l)\n        \\right]\n        \\right)  \n        \\\\\n        \\st \\quad & \n        W_l^TW_l=I,\n        \\label{eq:main_obj}\n\\end{aligned}\n\\end{equation}\n\nwhere we let $\\Gamma = HK_YH = HYY^TH$ with centering matrix $H$ defined as $H = I_n - \\frac{1}{n} \\mathbf{1}_n \\mathbf{1}_n^T$. $I_n$ is an identity matrix of size $n \\times n$ and $\\textbf{1}_n$ is a column vector of 1s also of length $n$.\n\nNote that the HSIC objective is more familiarly written as $\\Tr(HK_YHK_X)$ where $K_X = \\af(R_{l-1}W_l) \\af^T(R_{l-1}W_l)$. Yet, we purposely present it as Eq.~(\\ref{eq:main_obj}) to highlight how the network structure leads to the objective.  Namely, the layer input $R_{l-1}$ first multiplies the weight $W_l$ before passing through the activation function $\\af$. The layer output is then multiplied by itself and $\\Gamma$ to form the HSIC objective.\n\nLastly, since HSIC can be trivially maximized by setting each element of $W$ as $\\infty$, setting $W^TW=I$ is a constraint commonly used with HSIC while learning a projection \\citep{wu2018iterative, Wu2019SolvingIK,niu2010multiple}.\n\n\n\n\\textbf{Key Difference From Traditional MLPs. } \\textit{We generalize the concept of the activation function to a kernel feature map}. Therefore, instead of using the traditional Sigmoid or ReLU activation functions, we use the feature map of a Gaussian kernel as the activation function $\\af$. Conveniently, HSIC leverages the kernel trick to spare us the direct computation of the inner product $\\af(R_{l-1}W_l) \\af^T(R_{l-1}W_l)$. Therefore, for each $(r_i, r_j)$ pair we only need to compute \n\\begin{equation}\n    \\begin{aligned}\n    \\kf(W_l^T r_i, W^T_lr_j) & = \n    \\langle\n    \\af(W_l^T r_i), \\af(W_l^T r_j) \n    \\rangle\n    \\\\\n    & = \n    \\text{exp}\\{-\n    \\frac{||W_l^T r_i - W_l^T r_j||^2}{2\\sigma^2_l} \\}.\n    \\label{eq:kernel_def}\n    \\end{aligned}\n\\end{equation}\n\n\\textbf{How Does HSIC Learn the Kernel? } \nLet $\\cS$ be a set of $i,j$ sample pairs that belong to the same class. Its complement, $\\cS^c$ contains all sample pairs from different classes. By reinterpreting the kernel $\\kf(W_l^T r_i, W^T_lr_j)$ from Eq.~(\\ref{eq:kernel_def})  as a kernel function $\\kf_{W_l}(r_i, r_j)$ parameterized by $W_l$, \\eq{eq:main_obj} can be reformulated into the following objective to see how HSIC learns the kernel.\n\\begin{equation}\n    \\begin{aligned}\n    \\max_{W_l} &\n    \\sums \\Gij \\kf_{W_l}(r_i, r_j) \n    -\n    \\sumsc |\\Gij| \\kf_{W_l}(r_i, r_j)\n    \\\\\n    \\st & \\quad  W_l^TW_l=I.\n    \\label{eq:similarity_hsic}\n    \\end{aligned}\n\\end{equation}\nWhile \\eq{eq:main_obj} and \\eq{eq:similarity_hsic} are equivalent objectives, \\eq{eq:similarity_hsic} reveals how an optimal similarity measure is learned as a kernel function $\\kf_{W_l}(r_i,r_j)$ parameterized by $W_l$.  First note that $\\Gamma$ came directly from the label with  $\\Gamma=HYY^TH$ such that the $i,j_{th}$ element of $\\Gamma$, denoted as $\\Gamma_{i,j}$, is a positive value for samples pairs in $\\cS$ and negative for $\\cS^c$. The objective leverages the sign of $\\Gamma_{i,j}$ as labels to guide the choice of $W_l$ such that it increases $\\kf_{W_l}(r_i,r_j)$ when $r_i,r_j$ belongs to the same class in $\\cS$ while decreasing $\\kf_{W_l}(r_i,r_j)$ otherwise. Therefore, by finding a $W_l$ matrix that best parameterizes $\\kf_{W_l}$, HSIC identifies the optimal kernel function $\\kf_{W_l}(r_i, r_j)$ that separates samples into similar and dissimilar partitions to enable classification.\n\n\n\\section{ANALYZING LAYER-WISE NETWORKS} \nInstead of maximizing Eq.~(\\ref{eq:main_obj}) with traditional strategies like SGD, can we completely bypass the optimization step? Our analysis proves that this is possible. In fact, the weights $W_l$ simply need to be set to the \\kme \\citep{muandet2016kernel} at each layer. The stacking of layers with these known weights automatically drives \\RS towards its theoretical global optimum. Specifically, if we let $r^{j}_{\\iota}$ be the $\\iota^{th}$ input sample in class $j$ for layer $l$ with $\\zeta$ as a normalizer that can be ignored in practice, then the closed-form solution is \n   \\begin{equation}\n        W_s = \\frac{1}{\\sqrt{\\zeta}} \\W\n       \n       \n       \n       \n       \n       \n       \n        .\n        \\label{eq:trivial_W}\n    \\end{equation}\nWe prove that as long as no two identical samples have conflicting labels, setting $W_l = W_s$ at each layer can generate a monotonically increasing \\RS towards the global optimal, $\\hsic^*$. Comparing to BP, this finding suggests that instead of using gradients or needing to propagate error backward, a deep classifier can be globally optimized with \"only a single forward pass\" by \\textit{simple addition}. We formally prove this discovery in App.~\\ref{app:thm:hsequence} as the following theorem.\n\\begin{theorem}\n\\label{thm:hsequence}\nFrom any initial risk $\\hsic_0$, there exists a set of bandwidths $\\sigma_l$ and a \\KS $\\{\\fm_{l^{\\circ}}\\}_{l=1}^L$ parameterized by $W_l = W_s$ in \\eq{eq:trivial_W} such that:\n\\begin{enumerate}[topsep=0pt, partopsep=0pt, label=\\Roman*.]\n    \\item \n     $\\hsic_L$ can approach arbitrarily close to $\\hsic^*$ such that for any $L>1$ and $\\delta>0$ we can achieve\n     \\begin{equation}\n        \\hsic^{*} - \\hsic_L \\le \\delta,\n        \\label{arb_close}\n    \\end{equation}   \n    \\item \n    as $L \\rightarrow \\infty$, the \\RS converges to the global optimum where \n    \\begin{equation}\n        \\lim_{L \\rightarrow \\infty}  \\hsic_L = \\hsic^*,\n    \\end{equation}   \n    \\item \n    the convergence is strictly monotonic where \n    \\begin{equation}\n    \\hsic_{l} > \\hsic_{l-1} \\quad \\forall l \\ge 1.\n    \\end{equation}   \n\\end{enumerate}\n\\end{theorem}\nTo summarize the proof, we identified a lower bound for HSIC that is guaranteed to increase at each layer by adjusting $\\sigma_l$ while using $W_s$. Since HSIC has a known global theoretical upper bound ($\\sums \\Gij$), we show that the lower bound can approach arbitrarily close to the upper bound by adding more layers. \n\nIntuitively, the network incrementally discovers a better feature map to improve the \\kme. Imagine classifying a set of dogs\/cats, since \\kme defines the \\textit{average} dog and cat as two points in a high dimensional space. As $L\\rightarrow \\infty$, \\RS pushes these two points maximally apart from each other with representations that enable easy distinction. The continuity of the feature map ensures that dogs are closer to the average dog than the average cat, enabling classification.\n\nUsing the \\kme has a secondary implication; the layer-wise network is performing \\textit{Kernel Density Estimation}  \\citep{davis2011remarks}. That is, $W_s$ empowers these networks to predict the likelihood of an outcome without knowing the underlying distribution, relying solely on observations. For classification, multiplying the output of $\\fm_{l-1^\\circ}$ by its corresponding $W_s$ is equivalent to approximating the probability of a sample being in each class. Moreover, by summarizing the data into the $W_s$, all past examples can be discarded while allowing new examples to update the network by adjusting only the embedding itself, enabling layer-wise networks to self-adapt given new information.\n\n\nTheoretically, viewing a network as an \\RS is the key contribution that enables these interpretations. Its usage raises new questions with tantalizing possibilities in the debate over the brain's likelihood of performing BP. Is the brain more likely computing derivatives with BP, or is it simply repeating \\textit{addition}? Can the brain use mechanisms similar to \\textit{Kernel Density Estimation} to approximate the likelihood of an event?  This work only presents the theoretical feasibility and does not claim to have the answer.\n\n\n\n\\section{GENERALIZATION} \nBesides being an optimum solution, $W_l^*$ exhibits many advantages over $W_s$. For example, while $W_s$ experimentally performs well, $W^*$ converges with fewer layers and superior generalization. This raises a well-known question on generalization. It is known that overparameterized MLPs can generalize even without any explicit regularizer \\citep{Zhang2017UnderstandingDL}. This observation contradicts classical learning theory and has been a longstanding puzzle \\citep{cao2019generalization,brutzkus2017sgd,allen2019learning}. \nTherefore, by being overparameterized with an infinitely wide network, \\kn's ability under HSIC to generalize raises similar questions. In both cases, $W_s$ and $W^*$, the HSIC objective employs an infinitely wide network that should result in overfitting.  We ask theoretically, under our framework, what makes HSIC and $W^*$ special? \n\nRecently, \\citet{poggio2020complexity} have proposed that  traditional MLPs generalize  because gradient methods implicitly regularize the normalized weights given an exponential objective (like our HSIC). We discovered a similar impact the process of finding $W^*$ has on HSIC, i.e., HSIC can be reformulated to isolate out $n$ functions $[D_1(W_l), ..., D_n(W_l)]$ that act as a penalty term during optimization. Let $\\cS_i$ be the set of samples that belongs to the $i_{th}$ class and let $\\cS^c_i$ be its complement, then each function $D_i(W_l)$ is defined as \\begin{equation}\n\\begin{split}\n    D_i(W_l) =  &\n    \\frac{1}{\\sigma^2}\n    \\sum_{j \\in \\cS_i}\n    \\Gij \\kf_{W_l}(r_i,r_j)\n    - \\\\\n    &\n    \\frac{1}{\\sigma^2}\n    \\sum_{j \\in \\cS^c_i}\n    |\\Gij| \\kf_{W_l}(r_i,r_j).\n    \\label{eq:penalty_term}\n\\end{split}\n\\end{equation}\nNotice that $D_i(W_l)$ is simply \\eq{eq:similarity_hsic} for a single sample scaled by $\\frac{1}{\\sigma^2}$. Therefore, improving $W_l$ also leads to an increase and decrease of $\\kf_{W_l}(r_i,r_j)$ associated with $\\cS_i$ and $\\cS^c_i$ in \\eq{eq:penalty_term}, thereby increasing the size of the penalty term  $D_i(W_l)$.  To appreciate how $D_i(W_l)$ penalizes $\\hsic$, we propose an equivalent formulation in the theorem below with its derivation in App~\\ref{app:thm:regularizer}.\n\\begin{theorem}\n\\label{thm:regularizer}\n\\eq{eq:similarity_hsic} is equivalent to \n\\begin{equation}\n    \\label{eq:generalization_formulation}\n\\begin{split}\n    \\max_{W_l} \n    \\sum_{i,j} &\n    \\frac{\\Gij}{\\sigma^2}\n    \\ISMexp\n    (r_i^TW_lW_l^Tr_j) \n    \\\\\n   \n    & -\n    \\sum_{i}\n    D_i(W_l)\n    ||W_l^Tr_i||_2.\n\\end{split}\n\\end{equation}\n\\end{theorem}\nBased on Thm.~\\ref{thm:regularizer}, $D_i(W_l)$ adds a negative variable cost to the sample norm, $||W_l^Tr_i||_2$, prescribing an implicit regularizer on HSIC. Therefore, as $W_l$ improve HSIC, it also imposes an incrementally heavier penalty on \\eq{eq:generalization_formulation}, severely constraining $W_l$.\n\n\n\\section{EXPERIMENTS}\n\\textbf{Experiment Settings. }\n    Our analysis from Thm.~\\ref{thm:hsequence} shows that the Gaussian kernel's $\\sigma_l$ can be incrementally decreased to guarantee a monotonic increase. Our experiments learn the $\\sigma_l$ by incrementally decreasing its value until an improved HSIC is achieved. For results that involves $W^*$, the Gaussian kernel's $\\sigma_l$ is set to the value that maximizes HSIC, see App. \\ref{app:opt_sigma}. \n    \n    The activation function is approximated with RFF of width 300 \\citep{rahimi2008random}. The convergence threshold for \\RS is set at $\\hsic_l > 0.99$. The network structures discovered by $W^*$ for every dataset are recorded and provided in App.~\\ref{app:W_dimensions}. The MLPs that use $\\mse$ and $\\ce$ have weights initialized via the method used in \\citet{he2015delving}. All datasets are centered to 0 and scaled to a standard deviation of 1. All sources are written in Python using Numpy, Sklearn and Pytorch \\citep{numpy,sklearn_api,paszke2017automatic}, and ran on an Intel Xeon(R) CPU E5-2630 v3 @ 2.40GHz x 16 with 16 total cores.\n\\begin{figure*}[t]\n\\center\n    \\includegraphics[width=14cm,height=3.5cm]{img\/output_summary.png}\n    \\caption{Thm.~\\ref{thm:hsequence} is simulated on two highly complex datasets, Random and Adversarial. The 2D representation is shown, and next to it, the 1D output of each layer is displayed over each line. As we add more layers, we can see the samples from the two classes gradually separate from each other. Both datasets achieved the global optimum $\\hsic^*$ at the $12_{th}$ layers. For additional results, see App.~\\ref{app:sigma_values}}.\n    \\label{fig:thm_proof}\n\\end{figure*} \n\n\n\\begin{figure*}[t]\n\\center\n   \n    \\includegraphics[width=14cm,height=4cm]{img\/progression_kdiscovery.png}\n    \\caption{\\textbf{Left - Key evaluation metrics at each layer: } Showing key statistics computed from the output of each layer. Notice $\\hsic$ monotonically increases while $\\sil$ and $\\csr$ decrease.  \n    \\textbf{Right - A visual confirmation of Thm.~\\ref{thm:geometric_interpret}: } The top row plots out the kernel matrix from the output of the first 4 layers. Layer 1 shows the original kernel, while layer 4 represents the kernel when it has nearly converged to \\kn. The perfect block diagonal structure at layer 4 confirms that the \\KS is indeed converging toward \\kn. Below the kernels, we also plot out the convergent pattern in \\textit{preactivation}. Samples from the same class converge towards a single point while being pushed away from samples from different classes.}\n    \\label{fig:progression_kdiscovery}\n\\end{figure*} \n\n\n\\textbf{Datasets. }\nThe experiments are divided into two sections. Since our contributions are completely theoretical in nature, the majority of the experiments will focus on supporting our thesis by verifying the various components of the theoretical claims. We use three synthetic datasets (Random, Adversarial, and Spiral) and five popular UCI benchmark datasets: wine, cancer, car, face, and divorce ~\\citep{Dua:2017}. They are included along with the source code in the supplementary, and their download link and statistics are in App.~\\ref{app:data_detail}. All theoretical claims are experimentally reproducible with its source code and datasets publicly available in the Supplementary and at \\url{https:\/\/github.com\/endsley}.\n\n\\textbf{Competing Kernel Methods. } We next compare $W^*$ and $W_s$ to several popular kernel networks that have publicly available code: NTK \\citep{Arora2020HarnessingTP, Yu2019}, GP \\citep{maka892017, Wilson2016DeepKL}, Arc-cos \\citep{gionuno2017,Cho2009KernelMF}. \nFor this comparison, we additionally included CIFAR10 \\citep{Krizhevsky09learningmultiple}.\nThe images are preprocessed with a convolutional layer that outputs vectorized samples of $x_i \\in \\mathbb{R}^{10}$. The preprocessing code, the network output, and the network weights are included in the supplementary.\n\n\n\n\n\n\\begin{table*}[t]\n\\tiny\n\\centering\n\\setlength{\\tabcolsep}{3.5pt}  \n\\renewcommand{\\arraystretch}{1.1}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}\n\t\\hline\n\t & obj &\n\t\t Train Acc $\\uparrow$&\n\t\t Test Acc $\\uparrow$&\n\t\t Time(s) $\\downarrow$&\n\t\t $\\hsic$ $\\uparrow$&\n\t\t $\\mse$ $\\downarrow$&\n\t\t $\\ce$ $\\downarrow$&\n\t\t $C$ $\\downarrow$&\n\t\t $\\sil$ $\\downarrow$\\\\ \n\t\t\\hline \n\t\\parbox[t]{2mm}{\\multirow{4}{*}{\\rotatebox[origin=c]{90}{\\textbf{random}}}} &\n\t $W^*$ &\n\t\t \\textbf{\\black{1.00 $\\pm$ 0.00}} &\n\t\t \\black{0.38 $\\pm$ 0.21} &\n\t\t \\textbf{\\black{0.40 $\\pm$ 0.37}} &\n\t\t \\textbf{\\black{1.00 $\\pm$ 0.01}} &\n\t\t \\textbf{\\black{0.00 $\\pm$ 0.01}} &\n\t\t \\black{0.05 $\\pm$ 0.00} &\n\t\t \\textbf{\\black{0.00 $\\pm$ 0.06}} &\n\t\t \\black{0.02 $\\pm$ 0.0} \\\\ \n    & $W_s$ &\n    \t0.99 $\\pm$ 0.01 & \n    \t0.45 $\\pm$ 0.20 & \n    \t0.52 $\\pm$ 0.05 & \n    \t0.98 $\\pm$ 0.01 & \n    \t2.37 $\\pm$ 1.23 & \n    \t0.06 $\\pm$ 0.13 & \n    \t0.05 $\\pm$ 0.02 & \n    \t0.13 $\\pm$ 0.01\\\\\n\t & $\\ce$ &\n\t\t \\textbf{\\black{1.00 $\\pm$ 0.00}} &\n\t\t \\black{0.48 $\\pm$ 0.17} &\n\t\t \\black{25.07 $\\pm$ 5.55} &\n\t\t \\textbf{\\black{1.00 $\\pm$ 0.00}} &\n\t\t \\black{10.61 $\\pm$ 11.52} &\n\t\t \\textbf{\\black{0.0 $\\pm$ 0.0}} &\n\t\t \\textbf{\\black{0.0 $\\pm$ 0.0}} &\n\t\t \\textbf{\\black{0.0 $\\pm$ 0.0}} \\\\ \n\t & $\\mse$ &\n\t\t \\black{0.98 $\\pm$ 0.04} &\n\t\t \\textbf{\\black{0.63 $\\pm$ 0.21}} &\n\t\t \\black{23.58 $\\pm$ 8.38} &\n\t\t \\black{0.93 $\\pm$ 0.12} &\n\t\t \\black{0.02 $\\pm$ 0.04} &\n\t\t \\black{0.74 $\\pm$ 0.03} &\n\t\t \\black{0.04 $\\pm$ 0.04} &\n\t\t \\black{0.08 $\\pm$ 0.1} \\\\ \n\t\t\\hline \n\t\\parbox[t]{2mm}{\\multirow{4}{*}{\\rotatebox[origin=c]{90}{\\textbf{Advers}}}} &\n    $W^*$ &\n\t\t \\textbf{\\black{1.00 $\\pm$ 0.00}} &\n\t\t \\black{0.38 $\\pm$ 0.10} &\n\t\t \\textbf{\\black{0.52 $\\pm$ 0.51}} &\n\t\t \\textbf{\\black{1.00 $\\pm$ 0.00}} &\n\t\t \\textbf{\\black{0.00 $\\pm$ 0.00}} &\n\t\t \\textbf{\\black{0.04 $\\pm$ 0.00}} &\n\t\t \\textbf{\\black{0.01 $\\pm$ 0.08}} &\n\t\t \\textbf{\\black{0.01 $\\pm$ 0.0}} \\\\ \n    & $W_s$ &\n    \t0.99 $\\pm$ 0.04 & \n    \t\\textbf{0.42 $\\pm$ 0.18} & \n    \t2.82 $\\pm$ 0.81 & \n    \t0.99 $\\pm$ 0.19 & \n    \t15.02 $\\pm$ 11.97 & \n    \t0.32 $\\pm$ 0.15 & \n    \t0.30 $\\pm$ 0.18 & \n    \t0.34 $\\pm$ 0.19\\\\\n\t & $\\ce$ &\n\t\t \\black{0.59 $\\pm$ 0.04} &\n\t\t \\black{0.29 $\\pm$ 0.15} &\n\t\t \\black{69.54 $\\pm$ 24.14} &\n\t\t \\black{0.10 $\\pm$ 0.07} &\n\t\t \\black{0.65 $\\pm$ 0.16} &\n\t\t \\black{0.63 $\\pm$ 0.04} &\n\t\t \\black{0.98 $\\pm$ 0.03} &\n\t\t \\black{0.92 $\\pm$ 0.0} \\\\ \n\t & $\\mse$ &\n\t\t \\black{0.56 $\\pm$ 0.02} &\n\t\t \\black{0.32 $\\pm$ 0.20} &\n\t\t \\black{113.75 $\\pm$ 21.71} &\n\t\t \\black{0.02 $\\pm$ 0.01} &\n\t\t \\black{0.24 $\\pm$ 0.01} &\n\t\t \\black{0.70 $\\pm$ 0.00} &\n\t\t \\black{0.99 $\\pm$ 0.02} &\n\t\t \\black{0.95 $\\pm$ 0.0} \\\\ \n\t\t\\hline \n\t\t\\hline \n\t\\parbox[t]{2mm}{\\multirow{4}{*}{\\rotatebox[origin=c]{90}{\\textbf{spiral}}}} &\n\t $W^*$ &\n\t\t \\textbf{\\black{1.00 $\\pm$ 0.00}} &\n\t\t \\textbf{\\black{1.00 $\\pm$ 0.00}} &\n\t\t \\textbf{\\black{0.87 $\\pm$ 0.08}} &\n\t\t \\black{0.98 $\\pm$ 0.01} &\n\t\t \\black{0.01 $\\pm$ 0.00} &\n\t\t \\black{0.02 $\\pm$ 0.01} &\n\t\t \\black{0.04 $\\pm$ 0.03} &\n\t\t \\black{0.02 $\\pm$ 0.0} \\\\ \n    & $W_s$ &\n    \t\\textbf{1.00 $\\pm$ 0.00} & \n    \t\\textbf{1.00 $\\pm$ 0.00} & \n    \t13.54 $\\pm$ 5.66 & \n    \t0.88 $\\pm$ 0.03 & \n    \t38.60 $\\pm$ 25.24 & \n    \t0.06 $\\pm$ 0.02 & \n    \t0.08 $\\pm$ 0.04 & \n    \t0.08 $\\pm$ 0\\\\\n\t & $\\ce$ &\n\t\t \\textbf{\\black{1.00 $\\pm$ 0}} &\n\t\t \\textbf{\\black{1.00 $\\pm$ 0}} &\n\t\t \\black{11.59 $\\pm$ 5.52} &\n\t\t \\textbf{\\black{1.00 $\\pm$ 0}} &\n\t\t \\black{57.08 $\\pm$ 31.25} &\n\t\t \\textbf{\\black{0 $\\pm$ 0}} &\n\t\t \\textbf{\\black{0 $\\pm$ 0}} &\n\t\t \\textbf{\\black{0 $\\pm$ 0}} \\\\ \n\t & $\\mse$ &\n\t\t \\textbf{\\black{1.00 $\\pm$ 0}} &\n\t\t \\black{0.99 $\\pm$ 0.01} &\n\t\t \\black{456.77 $\\pm$ 78.83} &\n\t\t \\textbf{\\black{1.00 $\\pm$ 0}} &\n\t\t \\textbf{\\black{0 $\\pm$ 0}} &\n\t\t \\black{1.11 $\\pm$ 0.04} &\n\t\t \\black{0.40 $\\pm$ 0.01} &\n\t\t \\textbf{\\black{0 $\\pm$ 0}} \\\\ \n\t\t\\hline \n\t\\parbox[t]{2mm}{\\multirow{4}{*}{\\rotatebox[origin=c]{90}{\\textbf{wine}}}} &\n\t $W^*$ &\n\t\t \\black{0.99 $\\pm$ 0} &\n\t\t \\textbf{\\black{0.99 $\\pm$ 0.05}} &\n\t\t \\textbf{\\black{0.28 $\\pm$ 0.04}} &\n\t\t \\black{0.98 $\\pm$ 0.01} &\n\t\t \\black{0.01 $\\pm$ 0} &\n\t\t \\black{0.07 $\\pm$ 0.01} &\n\t\t \\black{0.04 $\\pm$ 0.03} &\n\t\t \\black{0.02 $\\pm$ 0} \\\\ \n    & $W_s$ &\n    \t0.98 $\\pm$ 0.01 & \n    \t0.94 $\\pm$ 0.04 & \n    \t0.78 $\\pm$ 0.09 & \n    \t0.93 $\\pm$ 0.01 & \n    \t2.47 $\\pm$ 0.26 & \n    \t0.06 $\\pm$ 0.01 & \n    \t0.05 $\\pm$ 0.01 & \n    \t0.08 $\\pm$ 0.01\\\\\n\t & $\\ce$ &\n\t\t \\textbf{\\black{1.00 $\\pm$ 0.00}} &\n\t\t \\black{0.94 $\\pm$ 0.06} &\n\t\t \\black{3.30 $\\pm$ 1.24} &\n\t\t \\textbf{\\black{1.00 $\\pm$ 0.00}} &\n\t\t \\black{40.33 $\\pm$ 35.5} &\n\t\t \\textbf{\\black{0 $\\pm$ 0}} &\n\t\t \\textbf{\\black{0 $\\pm$ 0}} &\n\t\t \\textbf{\\black{0 $\\pm$ 0}} \\\\ \n\t & $\\mse$ &\n\t\t \\textbf{\\black{1.00 $\\pm$ 0}} &\n\t\t \\black{0.89 $\\pm$ 0.17} &\n\t\t \\black{77.45 $\\pm$ 45.40} &\n\t\t \\textbf{\\black{1.00 $\\pm$ 0}} &\n\t\t \\textbf{\\black{0 $\\pm$ 0}} &\n\t\t \\black{1.15 $\\pm$ 0.07} &\n\t\t \\black{0.49 $\\pm$ 0.02} &\n\t\t \\textbf{\\black{0 $\\pm$ 0}} \\\\ \n\t\t\\hline \n\t\\parbox[t]{2mm}{\\multirow{4}{*}{\\rotatebox[origin=c]{90}{\\textbf{cancer}}}} &\n\t $W^*$ &\n\t\t \\black{0.99 $\\pm$ 0} &\n\t\t \\textbf{\\black{0.98 $\\pm$ 0.02}} &\n\t\t \\textbf{\\black{2.58 $\\pm$ 1.07}} &\n\t\t \\black{0.96 $\\pm$ 0.01} &\n\t\t \\black{0.02 $\\pm$ 0.01} &\n\t\t \\black{0.04 $\\pm$ 0.01} &\n\t\t \\black{0.02 $\\pm$ 0.04} &\n\t\t \\black{0.04 $\\pm$ 0.0} \\\\ \n    & $W_s$ &\n    \t0.98 $\\pm$ 0.01 & \n    \t0.96 $\\pm$ 0.03 & \n    \t6.21 $\\pm$ 0.36 & \n    \t0.88 $\\pm$ 0.01 & \n    \t41.31 $\\pm$ 56.17 & \n    \t0.09 $\\pm$ 0.01 & \n    \t0.09 $\\pm$ 0.02 & \n    \t0.16 $\\pm$ 0.03\\\\\n\t & $\\ce$ &\n\t\t \\textbf{\\black{1.00 $\\pm$ 0}} &\n\t\t \\black{0.97 $\\pm$ 0.01} &\n\t\t \\black{82.03 $\\pm$ 35.15} &\n\t\t \\textbf{\\black{1.00 $\\pm$ 0}} &\n\t\t \\black{2330 $\\pm$ 2915} &\n\t\t \\textbf{\\black{0 $\\pm$ 0}} &\n\t\t \\textbf{\\black{0 $\\pm$ 0}} &\n\t\t \\textbf{\\black{0 $\\pm$ 0}} \\\\ \n\t & $\\mse$ &\n\t\t \\textbf{\\black{1.00 $\\pm$ 0.00}} &\n\t\t \\black{0.97 $\\pm$ 0.03} &\n\t\t \\black{151.81 $\\pm$ 27.27} &\n\t\t \\textbf{\\black{1.00 $\\pm$ 0}} &\n\t\t \\textbf{\\black{0 $\\pm$ 0}} &\n\t\t \\black{0.66 $\\pm$ 0.06} &\n\t\t \\textbf{\\black{0 $\\pm$ 0}} &\n\t\t \\textbf{\\black{0 $\\pm$ 0}} \\\\ \n\t\t\\hline \n\t\\parbox[t]{2mm}{\\multirow{4}{*}{\\rotatebox[origin=c]{90}{\\textbf{car}}}} &\n\t $W^*$ &\n\t\t \\textbf{\\black{1.00 $\\pm$ 0}} &\n\t\t \\textbf{\\black{1.00 $\\pm$ 0.01}} &\n\t\t \\textbf{\\black{1.51 $\\pm$ 0.35}} &\n\t\t \\black{0.99 $\\pm$ 0} &\n\t\t \\textbf{\\black{0 $\\pm$ 0}} &\n\t\t \\black{0.01 $\\pm$ 0.00} &\n\t\t \\black{0.04 $\\pm$ 0.03} &\n\t\t \\black{0.01 $\\pm$ 0} \\\\ \n    & $W_s$ &\n    \t\\textbf{1.00 $\\pm$ 0} & \n    \t\\textbf{1.00 $\\pm$ 0} & \n    \t5.15 $\\pm$ 1.07 & \n    \t0.93 $\\pm$ 0.02 & \n    \t12.89 $\\pm$ 2.05 & \n    \t0 $\\pm$ 0 & \n    \t0.06 $\\pm$ 0.02 & \n    \t0.08 $\\pm$ 0.02\\\\\n\t & $\\ce$ &\n\t\t \\textbf{\\black{1.00 $\\pm$ 0}} &\n\t\t \\textbf{\\black{1.00 $\\pm$ 0}} &\n\t\t \\black{25.79 $\\pm$ 18.86} &\n\t\t \\textbf{\\black{1.00 $\\pm$ 0}} &\n\t\t \\black{225.11 $\\pm$ 253} &\n\t\t \\textbf{\\black{0 $\\pm$ 0}} &\n\t\t \\textbf{\\black{0 $\\pm$ 0}} &\n\t\t \\textbf{\\black{0 $\\pm$ 0}} \\\\ \n\t & $\\mse$ &\n\t\t \\textbf{\\black{1.00 $\\pm$ 0}} &\n\t\t \\textbf{\\black{1.00 $\\pm$ 0}} &\n\t\t \\black{504 $\\pm$ 116.6} &\n\t\t \\textbf{\\black{1.00 $\\pm$ 0}} &\n\t\t \\textbf{\\black{0 $\\pm$ 0}} &\n\t\t \\black{1.12 $\\pm$ 0.07} &\n\t\t \\black{0.40 $\\pm$ 0} &\n\t\t \\textbf{\\black{0 $\\pm$ 0}} \\\\ \n\t\t\\hline \n\t\\parbox[t]{2mm}{\\multirow{4}{*}{\\rotatebox[origin=c]{90}{\\textbf{face}}}} &\n\t $W^*$ &\n\t\t \\textbf{\\black{1.00 $\\pm$ 0}} &\n\t\t \\textbf{\\black{0.99 $\\pm$ 0.01}} &\n\t\t \\textbf{\\black{0.78 $\\pm$ 0.08}} &\n\t\t \\black{0.97 $\\pm$ 0} &\n\t\t \\textbf{\\black{0 $\\pm$ 0}} &\n\t\t \\black{0.17 $\\pm$ 0} &\n\t\t \\black{0.01 $\\pm$ 0} &\n\t\t \\textbf{\\black{0 $\\pm$ 0}} \\\\ \n    & $W_s$ &\n    \t0.98 $\\pm$ 0.01 & \n    \t0.94 $\\pm$ 0.26 & \n    \t0.86 $\\pm$ 0.04 & \n    \t3.15 $\\pm$ 3.05 & \n    \t2.07 $\\pm$ 1.04 & \n    \t0.28 $\\pm$ 0.51 & \n    \t0.04 $\\pm$ 0.01 & \n    \t0.01 $\\pm$ 0\\\\\n\t & $\\ce$ &\n\t\t \\textbf{\\black{1.00 $\\pm$ 0}} &\n\t\t \\black{0.79 $\\pm$ 0.31} &\n\t\t \\black{23.70 $\\pm$ 8.85} &\n\t\t \\textbf{\\black{1.00 $\\pm$ 0}} &\n\t\t \\black{16099 $\\pm$ 16330} &\n\t\t \\textbf{\\black{0 $\\pm$ 0}} &\n\t\t \\textbf{\\black{0 $\\pm$ 0}} &\n\t\t \\textbf{\\black{0 $\\pm$ 0}} \\\\ \n\t & $\\mse$ &\n\t\t \\black{0.92 $\\pm$ 0.10} &\n\t\t \\black{0.52 $\\pm$ 0.26} &\n\t\t \\black{745.2 $\\pm$ 282} &\n\t\t \\black{0.94 $\\pm$ 0.07} &\n\t\t \\black{0.11 $\\pm$ 0.12} &\n\t\t \\black{3.50 $\\pm$ 0.28} &\n\t\t \\black{0.72 $\\pm$ 0.01} &\n\t\t \\textbf{\\black{0 $\\pm$ 0}} \\\\ \n\t\t\\hline \n\t\\parbox[t]{2mm}{\\multirow{4}{*}{\\rotatebox[origin=c]{90}{\\textbf{divorce}}}} &\n\t $W^*$ &\n\t\t \\black{0.99 $\\pm$ 0.01} &\n\t\t \\black{0.98 $\\pm$ 0.02} &\n\t\t \\textbf{\\black{0.71 $\\pm$ 0.41}} &\n\t\t \\black{0.99 $\\pm$ 0.01} &\n\t\t \\black{0.01 $\\pm$ 0.01} &\n\t\t \\black{0.03 $\\pm$ 0} &\n\t\t \\textbf{\\black{0 $\\pm$ 0.05}} &\n\t\t \\black{0.02 $\\pm$ 0} \\\\ \n    & $W_s$ &\n    \t0.99 $\\pm$ 0 & \n    \t0.95 $\\pm$ 0.06 & \n    \t1.54 $\\pm$ 0.13 & \n    \t0.91 $\\pm$ 0.01 & \n    \t60.17 $\\pm$ 70.64 & \n    \t0.04 $\\pm$ 0.01 & \n    \t0.05 $\\pm$ 0.01 & \n    \t0.08 $\\pm$ 0\\\\\n\t & $\\ce$ &\n\t\t \\textbf{\\black{1.00 $\\pm$ 0}} &\n\t\t \\textbf{\\black{0.99 $\\pm$ 0.02}} &\n\t\t \\black{2.62 $\\pm$ 1.21} &\n\t\t \\textbf{\\black{1.00 $\\pm$ 0}} &\n\t\t \\black{14.11 $\\pm$ 12.32} &\n\t\t \\textbf{\\black{0 $\\pm$ 0}} &\n\t\t \\textbf{\\black{0 $\\pm$ 0}} &\n\t\t \\textbf{\\black{0 $\\pm$ 0}} \\\\ \n\t & $\\mse$ &\n\t\t \\textbf{\\black{1.00 $\\pm$ 0}} &\n\t\t \\black{0.97 $\\pm$ 0.03} &\n\t\t \\black{47.89 $\\pm$ 24.31} &\n\t\t \\textbf{\\black{1.00 $\\pm$ 0}} &\n\t\t \\textbf{\\black{0 $\\pm$ 0}} &\n\t\t \\black{0.73 $\\pm$ 0.07} &\n\t\t \\textbf{\\black{0 $\\pm$ 0.01}} &\n\t\t \\black{0.01 $\\pm$ 0} \\\\ \n\t\\hline\n\\end{tabular}\n\\vspace{3pt}\n\\caption{Each dataset contains 4 rows comparing the $W^*$ and $W_s$ against traditional MLPs trained using MSE and CE via SGD given the same network width and depth. The best results are in bold with $\\uparrow\/\\downarrow$ indicating larger\/smaller values preferred.}\n\\label{table:main}\n\\end{table*}\n\\textbf{Evaluation Metrics.} We report the HSIC objective as $\\hsic$ at convergence along with the training\/test accuracy for each dataset. Here, $\\hsic$ is normalized to the range between 0 to 1 using the method proposed by \\citet{cortes2012algorithms}. To corroborate Corollaries~\\ref{corollary:mse} and \\ref{corollary:ce}, we also record $\\mse$ and $\\ce$. To evaluate the sample geometry predicted by \\eq{eq:scatter_ratio}, we recorded the scatter trace ratio $\\sil$ to measure the compactness of samples within and between classes. The angular distance between samples in $\\cS$ and $\\cS^c$ as predicted by \\eq{eq:converged_kernel} is evaluated with the Cosine Similarity Ratio ($\\csr$). The equations for normalized $\\hsic$ and $\\csr$ are\n    $\\hsic = \\frac{\\hsic(\\fm(X),Y)}{\\sqrt{\\hsic(\\fm(X),\\fm(X)) \\hsic(Y,Y)}}$ and \n    $\\csr = \\frac\n        {\\sum_{i,j \\in \\mathcal{S}^c} \\langle \\fm(x_i), \\fm(x_j) \\rangle}\n         {\\sum_{i,j \\in \\mathcal{S}} \\langle \\fm(x_i), \\fm(x_j) \\rangle}$.\n\n\\textbf{Confirming Theorem 1. }\nSince Thm.~\\ref{thm:hsequence} guarantees an optimal convergence for \\textit{any} dataset given $W_s$, we designed an Adversarial dataset of high complexity, i.e., the sample pairs in $\\cS^c$ are intentionally placed significantly closer than samples pairs in $\\cS$. We next designed a Random dataset with completely random labels. We then simulated Thm.~\\ref{thm:hsequence} in Python and plotted the sample behavior in Fig.~\\ref{fig:thm_proof}. The original 2-dimensional data is shown next to its 1-dimensional results: each line represents the 1D output at that layer. As predicted by the theorem, the \\RS for both datasets converges to the global optimum at the 12th layer and perfectly separated the samples based on labels. \\textit{This experiment simultaneously confirms Thm.~\\ref{thm:hsequence} and the idea that simple and repetitive patterns as weights can incrementally improve a network to classify any pattern during training. It also supports the argument that gradients might not be theoretically necessary to optimize deep networks.}\n\n\\begin{table*}[t]\n\\centering\n\\tiny\n\\setlength{\\tabcolsep}{1.5pt}\n\\renewcommand{\\arraystretch}{1.3}\n\\begin{tabular}{|c|c|c|c|c|c||c|c|c|c|c|}\n\t\\hline\n\t & \t \\multicolumn{5}{|c||}{Training Accuracy} & \n\t \\multicolumn{5}{|c|}{Test Accuracy} \\\\ \n\t\\hline\n\t & \tW* & Ws & GP & Arc-cos & NTK & W* & Ws & GP & Arc-cos & NTK \\\\ \n\t\\hline\n\t\\textbf{adversarial} &  \n\t\\textbf{1.00 $\\pm$ 0.00} &\n\t\\textbf{1.00 $\\pm$ 0.00} &\n\t0.56 $\\pm$ 0.02 &\n\t0.53 $\\pm$ 0.02 &\n\t0.52 $\\pm$ 0.01&\n\t\\textbf{0.53 $\\pm$ 0.00} &\n\t0.50 $\\pm$ 0.16 &\n\t0.17 $\\pm$ 0.15 &\n\t0.25 $\\pm$ 0.08 &\n\t0.30 $\\pm$ 0.06\n\\\\ \n\t\\textbf{random} &  \n\t\\textbf{1.00 $\\pm$ 0.00} &\n\t\\textbf{1.00 $\\pm$ 0.00} &\n\t0.95 $\\pm$ 0.02 &\n\t0.66 $\\pm$ 0.02 &\n\t0.63 $\\pm$ 0.03&\n\t0.40 $\\pm$ 0.02 &\n\t0.32 $\\pm$ 0.14 &\n\t\\textbf{0.55 $\\pm$ 0.22} &\n\t0.53 $\\pm$ 0.21 &\n\t0.37 $\\pm$ 0.21\n\\\\ \n\t\\textbf{spiral} &  \n\t\\textbf{1.00 $\\pm$ 0.00} &\n\t\\textbf{1.00 $\\pm$ 0.00} &\n\t0.99 $\\pm$ 0.00 &\n\t0.99 $\\pm$ 0.00 &\n\t0.99 $\\pm$ 0.00&\n\t\\textbf{0.99 $\\pm$ 0.01} &\n\t0.97 $\\pm$ 0.00 &\n\t0.99 $\\pm$ 0.01 &\n\t\\textbf{0.99 $\\pm$ 0.01} &\n\t0.98 $\\pm$ 0.02\n\\\\ \n\t\\textbf{wine} &  \n\t0.99 $\\pm$ 0.00 &\n\t1.00 $\\pm$ 0.00 &\n\t\\textbf{1.0 $\\pm$ 0.0} &\n\t\\textbf{1.0 $\\pm$ 0.0} &\n\t\\textbf{1.0 $\\pm$ 0.0} &\n\t\\textbf{0.99 $\\pm$ 0.03} &\n\t0.94 $\\pm$ 0.03 &\n\t0.86 $\\pm$ 0.11 &\n\t0.94 $\\pm$ 0.07 &\n\t0.96 $\\pm$ 0.04\n\\\\ \n\t\\textbf{cancer} &  \n\t\\textbf{0.99 $\\pm$ 0.00} &\n\t\\textbf{0.99 $\\pm$ 0.00} &\n\t\\textbf{0.99 $\\pm$ 0.00} &\n\t\\textbf{0.99 $\\pm$ 0.00} &\n\t0.98 $\\pm$ 0.00&\n\t\\textbf{0.98 $\\pm$ 0.00} &\n\t0.98 $\\pm$ 0.02 &\n\t0.97 $\\pm$ 0.02 &\n\t0.97 $\\pm$ 0.02 &\n\t0.97 $\\pm$ 0.02\n\\\\ \n\t\\textbf{car} &  \n\t\\textbf{1 $\\pm$ 0.0} &\n\t\\textbf{1 $\\pm$ 0.0} &\n\t\\textbf{1 $\\pm$ 0.0} &\n\t\\textbf{1 $\\pm$ 0.0} &\n\t\\textbf{1 $\\pm$ 0.0} &\n\t\\textbf{1 $\\pm$ 0.0} &\n\t0.99 $\\pm$ 0.00 &\n\t0.99 $\\pm$ 0.01 &\n\t\\textbf{1.0 $\\pm$ 0.0} &\n\t\\textbf{1.0 $\\pm$ 0.0} \n\\\\ \n\t\\textbf{face} &  \n\t\\textbf{1.0 $\\pm$ 0.0} &\n\t\\textbf{1.0 $\\pm$ 0.0} &\n\t0.55 $\\pm$ 0.02 &\n\t\\textbf{1.0 $\\pm$ 0.0} &\n\t\\textbf{1.0 $\\pm$ 0.0} &\n\t\\textbf{1.0 $\\pm$ 0.0} &\n\t0.99 $\\pm$ 0.01 &\n\t0.22 $\\pm$ 0.05 &\n\t0.79 $\\pm$ 0.32 &\n\t0.61 $\\pm$ 0.39\n\\\\ \n\t\\textbf{divorce} &  \n\t0.99 $\\pm$ 0.01 &\n\t0.99 $\\pm$ 0.01 &\n\t\\textbf{1.00 $\\pm$ 0.0} &\n\t\\textbf{1.00 $\\pm$ 0.0} &\n\t\\textbf{1.00 $\\pm$ 0.0} &\n\t\\textbf{0.99 $\\pm$ 0.01} &\n\t0.97 $\\pm$ 0.05 &\n\t0.94 $\\pm$ 0.06 &\n\t0.95 $\\pm$ 0.12 &\n\t0.97 $\\pm$ 0.07\n\\\\ \n\t\\textbf{cifar10} &  \n\t\\textbf{0.99 $\\pm$ 0.01} &\n\t0.81 $\\pm$ 0.00 &\n\t0.77 $\\pm$ 0.01 &\n\t0.94 $\\pm$ 0.01 &\n\t0.94 $\\pm$ 0.01 &\n\t\\textbf{0.93 $\\pm$ 0.01} &\n\t0.74 $\\pm$ 0.01 &\n\t0.72 $\\pm$ 0.01 &\n\t\\textbf{0.93 $\\pm$ 0.01} &\n\t\\textbf{0.93 $\\pm$ 0.01}\n\\\\ \n\t\\hline\n\\end{tabular}\n\\caption{Comparing Train and test accuracy between recent kernel networks against $W_s\/W^*$. Notice that $W^*$ consistently achieves the highest test Accuracy while $W_s$ performs comparatively to GP. Also, note that $W_s$ and $W^*$ were the only models with a sufficiently large function class to shatter the Adversarial and random dataset. The 10-fold dataset is reshuffled to contrast against Table~\\ref{table:main}. }\n\\label{table:comparison_table}\n\\end{table*}\n \n\\textbf{Does $\\hsic$ Improve Monotonically? } Thm.~\\ref{thm:hsequence} also claims that \\RS should be monotonically increasing. Fig.~\\ref{fig:progression_kdiscovery} (Left) plots out all key metrics during training \\textit{at each layer}. Here, the \\RS is clearly monotonic and converging towards a $\\hsic^* \\approx 1$. Moreover, the trends for $\\sil$ and $\\csr$ indicate an incremental clustering of samples into separate partitions. Corresponding to low $\\sil$ and $\\csr$ values, the low MSE and $\\ce$ errors at convergence further reinforces the claims of Corollaries~\\ref{corollary:mse} and \\ref{corollary:ce}. \nThese patterns are highly repeatable, as shown across all datasets included in App.~\\ref{app:metric_graphs}.\n\n\n\\textbf{Can Layer-wise Networks with $W^*\/W_s$ Compete Against MLPs with BP? }  We conduct 10-fold cross-validation spanning 8 datasets and report their mean and the standard deviation for all key metrics in Table~\\ref{table:main}. Once our model is trained and has learned its structure, we use the same depth and width to train 2 additional MLPs via SGD, where instead of HSIC, $\\mse$ and $\\ce$ are used as $\\hsic$. \\textit{This allows us to compare \\kn to traditional networks of identical sizes that are trained by BP\/SGD.}\n\nObserving Table~\\ref{table:main}, first note that $\\mathbf{\\hsic} \\approx 1$ for both $W_s$ and $W^*$. This corroborates with Thm.~\\ref{thm:hsequence} where $\\hsic^{*}$ is guaranteed given enough layers. Next, notice that $\\boldsymbol{\\sil \\approx 0}$, \u00a0$\\boldsymbol{\\csr \\approx 0}$, \\textbf{MSE} $\\boldsymbol{\\approx 0}$, and \\textbf{CE} $\\boldsymbol{\\approx 0}$. These results are aligned with Thm.~\\ref{thm:geometric_interpret} and its corollaries since they indicate that samples of the same class are merging into a single point while minimizing MSE and CE for classification. Moreover, note that MSE\/CE failed to shatter the adversarial dataset while $W_s\/W^*$ easily handled the data. This suggest that given the same network size, $W_s\/W^*$ is significantly more flexible.\n\nLastly, notice how $W_s$ and $W^*$ perform favorably against traditional MLPs on almost all benchmarks, confirming that layer-wise network weights can be obtained via basic operations to achieve comparable performance. These results affirmatively answer question 1 while validating layer-wise networks as a promising alternative to BP. \\textit{Our theoretical and experimental results strongly suggest that layer-wise networks with \"closed-form weights\" can match MLP performance. }\n\n\\textbf{What is the Speed Difference? } The execution time is also included for reference in Table~\\ref{table:main}. Since \\kn can be obtained via a single forward pass while SGD requires many iterations of backpropagation, \\kn \\textit{should be faster}. The Time column of Table~\\ref{table:main} confirms this expectation by a wide margin. The biggest difference can be observed by comparing the face dataset: $W^*\/W_s$ finished in 0.78\/0.86 seconds while $\\mse$ required 745 seconds, \\textit{which is almost a 1000 times difference.}\n\n\n\\textbf{Does $W_s$ and $W^*$ Experimentally Generalize? }\n   With the exception of the two random datasets, the Test Accuracy of $W_s$ consistently performed well against MLPs trained using BP\/SGD. This suggests that $W_s$ is a trivially obtainable network solution that generalizes.\n    From an optimization perspective, $W^*$ impressively generalized even better across all datasets. It further differentiates itself on a high dimension Face dataset where it was the only method that avoided overfitting. While the generalizability of $W_s$ and $W^*$ is still ongoing research, the experimental results are promising.\n \n \n\\textbf{Is the Network Converging to the Neural Indicator Kernel? } \nA visual pattern of \\KS converging toward NIK is shown on the right of Fig.~\\ref{fig:progression_kdiscovery}. We rearrange the samples of the same class to be adjacent to each other. This allows us to evaluate the kernel quality via its block diagonal structure quickly. Since Gaussian kernels are restricted to values between 0 and 1, we let white and dark blue be 0 and 1 respectively, where the gradients reflect values in between. Our proof predicts that the \\KS converges to NIK, evolving from an uninformative kernel into a highly discriminating kernel of perfect \\textit{block diagonal structures}. Corresponding to the top row, the bottom row plots out the \\textit{preactivation} at each layer. As predicted by Thm.~\\ref{thm:geometric_interpret}, the samples of the same class incrementally converge towards a single point. This pattern is consistently observed on all datasets, and the complete collection of the \\textit{kernel sequences} for each dataset can be found in App.~\\ref{app:kernel_sequence_graph}. \n\n\n\n\n\\textbf{Comparing to Other Deep Kernel Frameworks. }\nThe development of \\RS primarily focused on the biological motivation to model layer-wise networks; it was not designed to automatically outperforming all existing networks. Table~\\ref{table:main} satisfies our primary objective by showing that it already performs comparably against traditional MLPs trained with BP. \nFor the kernel community, here we supply additional experiments comparing $W_s\/W^*$ against several recently proposed kernel networks to demonstrate surprisingly competitive results in Table~\\ref{table:comparison_table}. First, note that $W_s$ reliably achieves comparable test accuracy as GP while $W^*$ consistently outperform all kernel networks. This suggests that \\RS yields networks that not only generalizes competitively against traditional MLPs, it is also comparable to some recently proposed kernel networks. Next notice for the Training Accuracy, only $W_s$ and $W^*$ were sufficiently expressive to shatter both the adversarial and random dataset, confirming the expressiveness of layer-wise networks. \n\n\n\\textbf{Conclusion.} \nWe have comprehensively tested each theoretical claim, demonstrating how a layer-wise network modeled as an \\RS can yield closed-form solutions that perform comparably to MLPs. The convincing results from our experiments strongly align with the predictions made by our theorems, answering the two central questions of this exploratory work. \n\nIndeed, a repetition of simple rules can incrementally construct powerful networks capable of classifying any pattern, bypassing both BP and SGD. By modeling MLPs as a \\KS, it allows us to design convergent behaviors for layer-wise networks to achieve classification while identifying the network depth. \n\n\n\n\n\n\\section*{Checklist}\n\\begin{enumerate}\n\n\\item For all authors...\n\\begin{enumerate}\n  \\item Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?\n    \\answerYes{} \\\\\n    \\textbf{Justification. } \\textit{The central contribution of this work is presented as Theorem 1 and 2. These two findings answer the two main questions raised in the abstract\/Intro. Namely, we have first shown that SGD and backpropagation are not prerequisites to training a network for classification given the closed-form solution $W_s$. We have secondarily shown the mathematical viability of learning the network depth via a network convergence behavior. Moreover, we purposely designed the experiments to test each theoretical claim carefully. Therefore, both the theorems and the experiments are structured to reinforce our contributions as proposed in abstract\/Into.} \n  \\item Did you describe the limitations of your work?\n    \\answerYes{} \\\\\n    \\textbf{Justification. }  \\textit{See the last section of the last page under the title \"Limitations.\" }\n  \\item Did you discuss any potential negative societal impacts of your work?\n     \\answerYes{} \\\\\n    \\textbf{Justification. }  \\textit{Since our work is at such a fundamental and theoretical level, it would be difficult or even impossible to foresee its immediate implication. However, we felt that a key motivation of this work will have a significant societal impact and should be discussed by the community. \n   \n    } \\\\\n    \\begin{adjustwidth}{0.2in}{0.2in}\n    \\textbf{Broader Impact. } \n    Finding an alternative to BP also has significant climate implications.   \\citet{strubell2019energy} have shown that some standard AI models can emit over 626,000 pounds of carbon dioxide; a carbon footprint five times greater than the lifetime usage of a car. This level of emission is simply not sustainable in light of our continual explosive growth. Therefore, the environmental impact of BP necessitates a cheaper alternative. Looking at nature, we can be inspired by the brain's learning capability using only a fraction of the energy. Perhaps artificial neurons can also train without the high energy cost to the environment. This is the moral and the foundational motivation for this work in identifying the existence of $W_s$. A closed-form solution holds the potential to significantly reduce the computational requirement and carbon footprint.  Even if our work ultimately failed to mimic the brain, we hope to inspire the community to identify other closed-form solutions and go beyond BP. \\\\ \\\\\n    This paper aims to promote the discussion of viewing backpropagation alternatives not only as an academic exercise but also as a climate imperative. Yet, this topic is largely ignored by the community. The authors believe the energy costs of training Neural Networks are having a detrimental climate impact and should be an added topic of interest in NeurIPS along with fairness and racial equality. The earth also needs an advocate, why not us? Therefore, we as a community, must begin addressing how we can ameliorate our own carbon footprint. This exploratory work aims to share a potential path forward for further research that may address these concerns with the community. While $W_s$ is still not ready for commercial usage, we sincerely hope that the community begins to build novel algorithms over our work on \\RS and identify a simpler and cheaper path to train our networks. \\\\\n    \\end{adjustwidth}\n  \\item Have you read the ethics review guidelines and ensured that your paper conforms to them?\n      \\answerYes{}\n\\end{enumerate}\n\n\\item If you are including theoretical results...\n\\begin{enumerate}\n  \\item Did you state the full set of assumptions of all theoretical results?\n      \\answerYes{} \\\\\n    \\textbf{Justification. }  \\textit{We have to the best of our ability state the necessary assumptions. Since the authors are not infallible, it is possible to have oversights. We would be happy to incorporate any suggestions from the reviewers. }\n\t\\item Did you include complete proofs of all theoretical results?\n    \\answerYes{} \\\\\n    \\textbf{Justification. } \\textit{We have proven Theorem 1 in Appendix A, and Theorem 2 in Appendix B. Corollary 1 and 2 are proven in Appendix D. For reproducibility, we have purposely written the proof to be instructional. This means that we have clearly defined each symbol and tediously wrote the proof step by step without skipping operations that may appear obvious to a more senior audience. We note that this is not intended to insult the intelligence of reviewers, but it is designed to help younger students to reproduce, understand, and build upon the proof.}\n\\end{enumerate}\n\n\\item If you ran experiments...\n\\begin{enumerate}\n  \\item Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)?\n    \\answerYes{} \\\\\n    \\textbf{Justification. } \\textit{The source code to implement $W_s$ and $W^*$ are included in the Supplementary materials along with simple instructions to run them. We worked to ensure that the experimental results are easily reproducible in several manners. First, repeating each experiment is as simple as uncommenting the experiment you wish to run. Second, we also included the exact data used to conduct each experiment. Third, we also included the code we used for the competing methods to allow the comparisons easily reproducible. Therefore, the entire process is public, can be examined by any individual, and yields results that should closely track the paper. }\n  \\item Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)?\n    \\answerYes{} \\\\\n    \\textbf{Justification. } \\textit{The training detail to conduct the experiments are describe in the Experiment section under the subtitle \"Evaluation Metrics\" and \"Experiment Settings\". There are also hyperparameter choices that are less obvious, e.g., how did we choose the $\\sigma$ value for the Gaussian kernel? For this, we included in App. \\ref{app:opt_sigma} a detailed description for the logic behind how they were identified. We even conducted additional experiments and provided figures (also included in the appendix) to demonstrate how it works in practice. The source code for $\\sigma$ identification is included with the Supplementary Material.}\n\t\\item Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)?\n    \\answerYes{} \\\\\n    \\textbf{Justification. } \\textit{Every single experiment is conducted with 10-fold cross validation. This implies that each dataset is randomly divided into 10 subsets. Every experiment is conducted 10 times using each 10\\% subset as the test set and the 90\\% as the training. We then collect the mean and the standard deviation for each 10-fold experiment and report these values in the format $\\mu \\pm \\sigma$.}\n\t\\item Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)?\n    \\answerYes{} \\\\\n    \\textbf{Justification. } \\textit{The computation device description is included along with \"Experiment Settings\".}\n\\end{enumerate}\n\n\\item If you are using existing assets (e.g., code, data, models) or curating\/releasing new assets...\n\\begin{enumerate}\n  \\item If your work uses existing assets, did you cite the creators?\n     \\answerYes{} \\\\\n    \\textbf{Justification. } \\textit{For every single data and every competing method, we cite the work immediately after mentioning the source. These citations can be found from the first two paragraphs of the Experimental section. Every dataset is also included in the Supplementary. The link to download the original dataset is also included in App.~\\ref{app:data_detail}. For each competing method, the link to download the source code from github is included in the reference. These citations can be found from the first two paragraphs of the Experimental section. } \n  \\item Did you mention the license of the assets?\n     \\answerNA{}  \\\\\n     \\textbf{Justification. } We have purposely used datasets that are in the public domain. The license information can be easily obtained from the link provided within the paper.\n  \\item Did you include any new assets either in the supplemental material or as a URL?\n     \\answerNA{} \n  \\item Did you discuss whether and how consent was obtained from people whose data you're using\/curating?\n     \\answerNA{} \n  \\item Did you discuss whether the data you are using\/curating contains personally identifiable information or offensive content?\n    \\answerNA{}\n   \n \n\\end{enumerate}\n\n\\item If you used crowdsourcing or conducted research with human subjects...\n\\begin{enumerate}\n  \\item Did you include the full text of instructions given to participants and screenshots, if applicable?\n    \\answerNA{}\n  \\item Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable?\n    \\answerNA{}\n  \\item Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation?\n    \\answerNA{}\n\\end{enumerate}\n\n\\end{enumerate}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section*{List of model and transport parameters}\n\\vspace{-6.0cm}\n\\begin{table*}[h!]\n\\begin{center}\n\\begin{tabular}{ll}\n$\\begin{aligned} k_{F} \\end{aligned}$ & $\\qquad$ Fermi wavevector \\\\\n& \\\\ \n$\\begin{aligned} m \\end{aligned}$ & $\\qquad$ Electron's effective mass \\\\\n& \\\\ \n$\\begin{aligned} J \\end{aligned}$ & $\\qquad$ Exchange constant in the $s$-$d$ model \\\\\n& \\\\\n$\\begin{aligned} v_{i} \\end{aligned}$ & $\\qquad$ Impurity potential \\\\\n& \\\\\n$\\begin{aligned} n_{i} \\end{aligned}$ & $\\qquad$ Impurity concentration \\\\\n& \\\\\n$\\begin{aligned} \\xi_{SO} \\end{aligned}$ & $\\qquad$ Dimensionless spin-orbit coupling constant \\\\\n& \\\\\n$\\begin{aligned} \\varepsilon_{F}=\\frac{\\hbar^{2}k_{F}^{2}}{2m} \\end{aligned}$ & $\\qquad$ Fermi energy \\\\\n& \\\\\n$\\begin{aligned} v_{F}=\\frac{\\hbar k_{F}}{m} \\end{aligned}$ & $\\qquad$ Fermi velocity \\\\\n& \\\\\n$\\begin{aligned} \\beta=\\frac{J}{2\\varepsilon_{F}} \\end{aligned}$ & $\\qquad$ Spin polarization factor  \\\\\n& \\\\\n$\\begin{aligned} D_{0}=\\frac{mk_{F}}{2\\pi^{2}\\hbar^{2}} \\end{aligned}$ & $\\qquad$ Spin independent density of states per spin at the Fermi level \\\\\n& \\\\\n$\\begin{aligned} \\frac{1}{\\tau_{0}}=\\frac{2\\pi v_{i}^{2}n_{i}D_{0}}{\\hbar} \\end{aligned}$ & $\\qquad$ Spin independent relaxation time \\\\\n& \\\\\n$\\begin{aligned} \\frac{1}{\\tau_{L}}=\\frac{2J}{\\hbar} \\end{aligned}$ & $\\qquad$ Larmor precession time \\\\\n& \\\\\n$\\begin{aligned} \\frac{1}{\\tau_{\\phi}}=\\frac{4J^{2}\\tau_{0}}{\\hbar^{2}} \\end{aligned}$ & $\\qquad$ Spin dephasing relaxation time \\\\\n& \\\\\n$\\begin{aligned} \\frac{1}{\\tau_{sf}}=\\frac{8}{9}\\frac{\\xi_{SO}^{2}}{\\tau_{0}} \\end{aligned}$ & $\\qquad$ Spin-flip relaxation time \\\\\n& \\\\\n$\\begin{aligned} l_{F}=\\tau_{0}v_{F} \\end{aligned}$ & $\\qquad$ Mean-free path \\\\\n& \\\\\n$\\begin{aligned} D=\\frac{\\tau_{0}v_{F}^{2}}{3} \\end{aligned}$ & $\\qquad$ Diffusion coefficient \\\\\n& \\\\\n$\\begin{aligned} \\alpha_{sw}=\\frac{2}{3}\\xi_{SO} \\end{aligned}$ & $\\qquad$ Dimensionless spin swapping constant \\\\\n& \\\\\n$\\begin{aligned} \\alpha_{sj}=\\frac{\\xi_{SO}}{l_{F}k_{F}} \\end{aligned}$ & $\\qquad$ Dimensionless side-jump constant \\\\\n& \\\\\n$\\begin{aligned} \\alpha_{sk}=\\frac{v_{i}mk_{F}}{3\\pi\\hbar^{2}}\\xi_{SO} \\end{aligned}$ & $\\qquad$ Dimensionless skew scattering constant \\\\\n& \\\\\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\n\n\n\n\\pagebreak\n\\section{General formalism}\n\\par We start with a free-electron Hamiltonian $\\hat{\\mathcal{H}}_{0}$ and its Fourier transform $\\hat{\\mathcal{H}}_{\\boldsymbol{k}}$: \n\\begin{equation}\n\\hat{\\mathcal{H}}_{0}=-\\frac{\\hbar^{2}}{2m}\\nabla^{2}\\hat{\\sigma}_{0}+J\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}\\qquad\\xrightarrow{\\,\\,\\mathcal{F}\\,\\,}\\qquad\\hat{\\mathcal{H}}_{\\boldsymbol{k}}=\\frac{\\hbar^{2}_{}\\boldsymbol{k}^{2}_{}}{2m}\\hat{\\sigma}_{0}+J\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}.\n\\end{equation}\n\\noindent Here, the first term stands for the kinetic energy, where $m$ and $\\boldsymbol{k}$ are the electron's effective mass and wave vector, respectively; the second term refers to the exchange interaction in the so-called $s$-$d$ model, where $\\boldsymbol{m}=(\\cos{\\phi}\\sin{\\theta},\\sin{\\phi}\\sin{\\theta},\\cos{\\theta})$ is the magnetization unit vector parametrized in spherical coordinates, $J$ is the exchange coupling parameter, $\\hat{\\boldsymbol{\\sigma}}$ is the Pauli matrix vector, and $\\hat{\\sigma}_{0}$ is the identity matrix. The unperturbed Green's function is defined by $\\hat{\\mathcal{H}}_{\\boldsymbol{k}}$:\n\\begin{equation}\n\\hat{G}_{0,\\boldsymbol{k}E}^{R(A)}=\\left[E-\\hat{\\mathcal{H}}_{\\boldsymbol{k}}\\pm i\\eta\\right]^{-1}=\\sum\\limits_{s=\\pm}\\frac{|s\\rangle\\langle s|}{E-E_{\\boldsymbol{k}s}\\pm i\\eta}=\\frac{1}{2}\\sum\\limits_{s=\\pm}\\frac{\\hat{\\sigma}_{0}+s\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}}{E-E_{\\boldsymbol{k}s}\\pm i\\eta},\n\\label{eq:gzero}\n\\end{equation}\n\\noindent where $s$ refers to the spin index, $|s\\rangle$ is the corresponding eigenstate:\n\\begin{equation}\n|s\\rangle=\\left( \\begin{array}{c}\nse^{-i\\phi}\\sqrt{\\frac{1+s\\cos{\\theta}}{2}}\\\\\n\\sqrt{\\frac{1-s\\cos{\\theta}}{2}}\\end{array} \\right),\n\\end{equation}\n\\noindent $E_{\\boldsymbol{k}s}=E_{\\boldsymbol{k}}+sJ$, $E_{\\boldsymbol{k}}=\\hbar^{2}\\boldsymbol{k}^{2}\/2m$, and $\\eta$ is a positive infinitesimal. \n\\par Next, we consider the impurity Hamiltonian with spin-orbit coupling:\n\\begin{equation}\n\\label{eq:imppot}\n\\hat{\\mathcal{H}}_{\\mathrm{imp}}=\\sum\\limits_{\\boldsymbol{R}_{i}}V(\\boldsymbol{r}-\\boldsymbol{R}_{i})\\hat{\\sigma}_{0}+\\frac{\\xi_{SO}}{\\hbar k_{F}^{2}}\\sum_{\\boldsymbol{R}_{i}}\\left(\\nabla V(\\boldsymbol{r}-\\boldsymbol{R}_{i})\\times\\hat{\\boldsymbol{p}}\\right)\\cdot\\hat{\\boldsymbol{\\sigma}},\n\\end{equation}\n\\noindent where $\\hat{\\boldsymbol{p}}=-i\\hbar\\partial_{\\boldsymbol{r}}$ is the momentum operator, $V(\\boldsymbol{r}-\\boldsymbol{R}_{i})=v_{i}\\delta(\\boldsymbol{r}-\\boldsymbol{R}_{i})$ is the on-site potential at the impurity site $\\boldsymbol{R}_{i}$, $\\xi_{SO}$ is the spin-orbit coupling parameter (defined as a dimensionless quantity), and $k_{F}$ is the Fermi wavevector. Here, we neglect the localization effects and electron-electron correlations, and assume a short-range impurity potential. In the reciprocal space, it can be written as:\\cite{rammer1}\n\\begin{equation}\n\\hat{\\mathcal{H}}_{\\boldsymbol{k}\\boldsymbol{k'}}=\\Omega\\langle\\boldsymbol{k}|\\hat{\\mathcal{H}}_{\\mathrm{imp}}|\\boldsymbol{k'}\\rangle,\n\\end{equation}\n\\noindent where the momentum eigenstates are defined as $\\langle\\boldsymbol{r}|\\boldsymbol{k}\\rangle=\\Omega^{-1\/2}e^{i\\boldsymbol{k}\\cdot\\boldsymbol{r}}$, and $\\Omega$ is the volume of the system. Then, by using the following identities:\n\\begin{equation}\n\\int\\limits_{\\Omega}d\\boldsymbol{r}\\,f(\\boldsymbol{r})\\delta(\\boldsymbol{r}-\\boldsymbol{r}_{i})=f(\\boldsymbol{r}_{i}),\\qquad\\int\\limits_{\\Omega}d\\boldsymbol{r}\\,f(\\boldsymbol{r})\\nabla\\delta(\\boldsymbol{r}-\\boldsymbol{r}_{i})=-\\int\\limits_{\\Omega}d\\boldsymbol{r}\\,\\nabla f(\\boldsymbol{r})\\delta(\\boldsymbol{r}-\\boldsymbol{r}_{i}),\n\\end{equation}\n\\noindent we obtain:\n\\begin{equation}\n\\begin{aligned}\n\\hat{\\mathcal{H}}_{\\boldsymbol{k}\\boldsymbol{k'}}&=\\sum\\limits_{\\boldsymbol{R}_{i}}\\int\\limits_{\\Omega}d\\boldsymbol{r}\\,\\left[v_{i}\\delta(\\boldsymbol{r}-\\boldsymbol{R}_{i})\\hat{\\sigma}_{0}e^{-i(\\boldsymbol{k}-\\boldsymbol{k'})\\cdot\\boldsymbol{r}}-iv_{i}\\frac{\\xi_{SO}}{k_{F}^{2}}e^{-i\\boldsymbol{k}\\cdot\\boldsymbol{r}}\\Big(\\nabla\\delta(\\boldsymbol{r}-\\boldsymbol{R}_{i})\\times\\partial_{\\boldsymbol{r}}\\Big)\\cdot\\hat{\\boldsymbol{\\sigma}}e^{i\\boldsymbol{k'}\\cdot\\boldsymbol{r}} \\right]\\\\\n&=V(\\boldsymbol{k}-\\boldsymbol{k'})\\left[\\hat{\\sigma}_{0}+i\\frac{\\xi_{SO}}{k_{F}^{2}}\\hat{\\boldsymbol{\\sigma}}\\cdot(\\boldsymbol{k}\\times\\boldsymbol{k'})\\right],\n\\end{aligned}\n\\label{eq:potsoi}\n\\end{equation}\n\\noindent where $V(\\boldsymbol{k}-\\boldsymbol{k'})$ is the Fourier transform of the impurity on-site potential:\n\\begin{equation}\nV(\\boldsymbol{k}-\\boldsymbol{k'})=v_{i}\\sum_{\\boldsymbol{R}_{i}}e^{-i(\\boldsymbol{k}-\\boldsymbol{k'})\\cdot\\boldsymbol{R}_{i}}.\n\\end{equation}\n\\par We proceed to write a kinetic equation by means of the Keldysh formalism:\n\\begin{equation}\n\\underline{\\hat{G}}^{-1}=\\hat{G}_{0}^{-1}-\\underline{\\Sigma},\\qquad \n\\underline{\\hat{G}}=\\left( \\begin{array}{cc}\n\\hat{G}^{R} &\\hat{G}^{K}\\\\\n0&\\hat{G}^{A}\\end{array} \\right),\\qquad\n\\underline{\\hat{\\Sigma}}=\\left( \\begin{array}{cc}\n\\hat{\\Sigma}^{R} &\\hat{\\Sigma}^{K}\\\\\n0&\\hat{\\Sigma}^{A}\\end{array} \\right),\n\\end{equation}\n\\noindent where $\\underline{\\hat{G}}$ and $\\underline{\\hat{\\Sigma}}$ are the Green's function and self-energy in the Keldysh space; the indexes $R$, $A$ and $K$ stand for the retarded, advanced and Keldysh components, respectively, and $\\hat{G}_{0}^{-1}=i\\hbar\\partial_{t}-\\hat{\\mathcal{H}}_{0}$. In the semiclassical approximation, a set of diffusive equations for the non-equilibrium charge and spin densities can be derived through the distribution function $\\hat{g}_{\\boldsymbol{k}}\\equiv \\hat{g}_{\\boldsymbol{k}}(\\boldsymbol{R},T)$ defined as the Wigner representation of the Keldysh Green's function $\\hat{G}^{K}$:\\cite{rammer2}\n\\begin{equation}\n\\begin{aligned}\n\\hat{G}^{K}(\\boldsymbol{r}_{1},t_{1};\\boldsymbol{r}_{2},t_{2})&\\,\\,\\xrightarrow{\\mathcal{W}}\\hat{G}^{K}(\\boldsymbol{R}+\\frac{\\boldsymbol{r}}{2},T+\\frac{t}{2};\\boldsymbol{R}-\\frac{\\boldsymbol{r}}{2},T-\\frac{t}{2})\\equiv\\hat{G}^{K}(\\boldsymbol{r},t;\\boldsymbol{R},T)\\\\\n&\\,\\,\\xrightarrow{\\mathcal{F}}\\hat{G}^{K}(\\boldsymbol{r},t;\\boldsymbol{R},T)=\\int\\frac{dE}{2\\pi}\\frac{d\\boldsymbol{k}}{(2\\pi)^{3}}\\,\\,\\hat{g}^{K}_{\\boldsymbol{k}E}(\\boldsymbol{R},T)e^{-\\frac{i}{\\hbar}Et}e^{i\\boldsymbol{r}\\cdot\\boldsymbol{k}}\\\\\n&\\,\\,\\xrightarrow{\\,\\,}\\hat{g}_{\\boldsymbol{k}}=i\\int\\frac{dE}{2\\pi}\\,\\hat{g}^{K}_{\\boldsymbol{k}E}(\\boldsymbol{R},T),\n\\end{aligned}\n\\end{equation}\n\\noindent where the relative $\\boldsymbol{r}=\\boldsymbol{r}_{1}-\\boldsymbol{r}_{2}$, $t=t_{1}-t_{2}$ and center-of-mass $\\boldsymbol{R}=(\\boldsymbol{r}_{1}+\\boldsymbol{r}_{2})\/2$, $T=(t_{1}+t_{2})\/2$ coordinates are introduced. In the dilute limit, we can employ the Kadanoff-Baym anzats:\n\\begin{equation}\n\\underline{\\hat{G}}_{\\boldsymbol{k}E}(\\boldsymbol{R},T)=\\left( \\begin{array}{cc}\n\\hat{G}^{R}_{\\boldsymbol{k}E} &\\hat{g}^{K}_{\\boldsymbol{k}E}(\\boldsymbol{R},T)\\\\\n0&\\hat{G}^{A}_{\\boldsymbol{k}E}\\end{array} \\right)\n\\end{equation}\n\\noindent and\n\\begin{equation}\n\\hat{g}^{K}_{\\boldsymbol{k}E}(\\boldsymbol{R},T)=\\hat{G}^{R}_{\\boldsymbol{k}E}\\,\\hat{g}_{\\boldsymbol{k}}(\\boldsymbol{R},T)-\\hat{g}_{\\boldsymbol{k}}(\\boldsymbol{R},T)\\hat{G}^{A}_{\\boldsymbol{k}E}.\n\\label{eq:anzats}\n\\end{equation}\n\\noindent The Keldysh Green's function $\\hat{G}^{K}$ satisfies the Kadanoff-Baym equation:\\cite{rammer2}\n\\begin{equation}\n[\\hat{G}^{R}]^{-1}\\ast\\hat{G}^{K}-\\hat{G}^{K}\\ast[\\hat{G}^{A}]^{-1}=\\hat{\\Sigma}^{K}\\ast\\hat{G}^{A}-\\hat{G}^{R}\\ast\\hat{\\Sigma}^{K},\n\\end{equation}\n\\noindent Having applied the Wigner transformation, we use the so-called gradient approximation, where the convolution $\\mathcal{A}\\ast\\mathcal{B}$ of two functions is expressed as:\n\\begin{equation}\n\\begin{aligned}\n\\left(\\mathcal{A}\\ast\\mathcal{B}\\right)_{\\boldsymbol{k}E}(\\boldsymbol{R},T)&\\simeq\\mathcal{A}\\mathcal{B}-\\frac{i\\hbar}{2}\\left(\\partial_{T}\\mathcal{A}\\partial_{E}\\mathcal{B}-\\partial_{E}\\mathcal{A}\\partial_{T}\\mathcal{B}\\right)\\\\\n&-\\frac{i}{2}\\left(\\nabla_{\\boldsymbol{k}}\\mathcal{A}\\cdot\\nabla_{\\boldsymbol{R}}\\mathcal{B}-\\nabla_{\\boldsymbol{R}}\\mathcal{A}\\cdot\\nabla_{\\boldsymbol{k}}\\mathcal{B}\\right).\n\\end{aligned}\n\\end{equation}\n\\noindent Taking into account that $\\hat{G}^{R(A)}$ and $\\hat{\\Sigma}^{R(A)}$ do not depend on the center-of-mass coordinates, we obtain:\n\\begin{equation}\n\\begin{aligned}\ni\\hbar\\partial_{T}\\hat{g}^{K}+[\\hat{g}^{K},J\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}]&+\\frac{i}{2}\\left\\{\\nabla_{\\boldsymbol{k}}\\hat{\\mathcal{H}}_{\\boldsymbol{k}},\\nabla_{\\boldsymbol{R}}\\hat{g}^{K}\\right\\}=\\hat{\\Sigma}^{K}\\hat{G}^{A}-\\hat{G}^{R}\\hat{\\Sigma}^{K}+\\hat{\\Sigma}^{R}\\hat{g}^{K}-\\hat{g}^{K}\\hat{\\Sigma}^{A}\\\\\n&-\\frac{i\\hbar}{2}\\left(\\partial_{T}\\hat{\\Sigma}^{K}\\partial_{E}\\hat{G}^{A}+\\partial_{E}\\hat{G}^{R}\\partial_{T}\\hat{\\Sigma}^{K} \\right)\n+\\frac{i\\hbar}{2}\\left(\\partial_{E}\\hat{\\Sigma}^{R}\\partial_{T}\\hat{g}^{K}+\\partial_{T}\\hat{g}^{K}\\partial_{E}\\hat{\\Sigma}^{A} \\right)\\\\\n&-\\frac{i}{2}\\left(\\nabla_{\\boldsymbol{k}}\\hat{\\Sigma}^{R}\\cdot\\nabla_{\\boldsymbol{R}}\\hat{g}^{K}+\\nabla_{\\boldsymbol{R}}\\hat{g}^{K}\\cdot\\nabla_{\\boldsymbol{k}}\\hat{\\Sigma}^{A}\\right)\n+\\frac{i}{2}\\left(\\nabla_{\\boldsymbol{R}}\\hat{\\Sigma}^{K}\\cdot\\nabla_{\\boldsymbol{k}}\\hat{G}^{A}+\\nabla_{\\boldsymbol{k}}\\hat{G}^{R}\\cdot\\nabla_{\\boldsymbol{R}}\\hat{\\Sigma}^{K}\\right),\n\\label{eq:full}\n\\end{aligned}\n\\end{equation}\n\\noindent where $[\\cdot\\,,\\cdot]$ and $\\{\\cdot\\,,\\cdot\\}$ stand for a commutator and anticommutator, respectively. In steady state, we have:\n\\begin{equation}\n\\begin{aligned}\n\\,[\\hat{g}^{K},J\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}]&+\\frac{i}{2}\\left\\{\\nabla_{\\boldsymbol{k}}\\hat{\\mathcal{H}}_{\\boldsymbol{k}},\\nabla_{\\boldsymbol{R}}\\hat{g}^{K}\\right\\}=\\hat{\\Sigma}^{K}\\hat{G}^{A}-\\hat{G}^{R}\\hat{\\Sigma}^{K}+\\hat{\\Sigma}^{R}\\hat{g}^{K}-\\hat{g}^{K}\\hat{\\Sigma}^{A}\\\\\n&-\\frac{i}{2}\\left(\\nabla_{\\boldsymbol{k}}\\hat{\\Sigma}^{R}\\cdot\\nabla_{\\boldsymbol{R}}\\hat{g}^{K}+\\nabla_{\\boldsymbol{R}}\\hat{g}^{K}\\cdot\\nabla_{\\boldsymbol{k}}\\hat{\\Sigma}^{A}\\right)\n+\\frac{i}{2}\\left(\\nabla_{\\boldsymbol{R}}\\hat{\\Sigma}^{K}\\cdot\\nabla_{\\boldsymbol{k}}\\hat{G}^{A}+\\nabla_{\\boldsymbol{k}}\\hat{G}^{R}\\cdot\\nabla_{\\boldsymbol{R}}\\hat{\\Sigma}^{K}\\right).\n\\label{eq:kelfull}\n\\end{aligned}\n\\end{equation}\n\\noindent Finally, in the dilute limit, we can assume that the self-energy is almost constant and neglect its derivatives on the right-hand side:\n\\begin{equation}\n[\\hat{g}^{K},J\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}]+\\frac{i}{2}\\left\\{\\nabla_{\\boldsymbol{k}}\\hat{\\mathcal{H}}_{\\boldsymbol{k}},\\nabla_{\\boldsymbol{R}}\\hat{g}^{K}\\right\\}=\\hat{\\Sigma}^{K}\\hat{G}^{A}-\\hat{G}^{R}\\hat{\\Sigma}^{K}+\\hat{\\Sigma}^{R}\\hat{g}^{K}-\\hat{g}^{K}\\hat{\\Sigma}^{A},\n\\end{equation}\n\\noindent or\n\\begin{equation}\n[\\hat{g}^{K},J\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}]+i\\frac{\\hbar^{2}}{m}(\\boldsymbol{k}\\cdot\\nabla_{\\boldsymbol{R}})\\,\\hat{g}^{K}=\\hat{\\Sigma}^{K}\\hat{G}^{A}-\\hat{G}^{R}\\hat{\\Sigma}^{K}+\\hat{\\Sigma}^{R}\\hat{g}^{K}-\\hat{g}^{K}\\hat{\\Sigma}^{A}.\n\\label{eq:keldysh}\n\\end{equation}\n \n\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.55]{pic1.jpg}\n\\end{center}\n\\caption{$a)$ Diagrammatic expansion for scatterings off the static impurity potential. $b)$ Self-consistent Born approximation. $c)$ Skew-scattering diagrams.}\n\\end{figure}\n\n\n\\section{Self-energy}\n\\subsection{Self-consistent Born approximation}\n\\par Let us consider first and second orders of the diagrammatic expansion for the scattering off the static impurity potential (Fig. 1$a$). The Green's function $\\underline{\\hat{G}}$ and the corresponding self-energy $\\underline{\\hat{\\Sigma}}$ are defined in the wave vector representation as follows:\n\\begin{equation}\n\\begin{aligned}\n\\underline{\\hat{G}}(\\boldsymbol{k},t_{1};\\boldsymbol{k'},t_{2})&=\\underline{\\hat{G}}_{0}(\\boldsymbol{k},t_{1};\\boldsymbol{k'},t_{2})+\\sum\\limits_{\\{\\boldsymbol{k}_{i}\\}} \\underline{\\hat{G}}_{0}(\\boldsymbol{k},t_{1};\\boldsymbol{k}_{1},t_{2}) \\langle\\boldsymbol{k}_{1}|\\hat{\\mathcal{H}}_{\\mathrm{imp}}|\\boldsymbol{k}_{2}\\rangle\\underline{\\hat{G}}_{0}(\\boldsymbol{k}_{2},t_{1};\\boldsymbol{k'},t_{2})\\\\\n&+\\sum\\limits_{\\{\\boldsymbol{k}_{i}\\}} \\underline{\\hat{G}}_{0}(\\boldsymbol{k},t_{1};\\boldsymbol{k}_{1},t_{2}) \\langle\\boldsymbol{k}_{1}|\\hat{\\mathcal{H}}_{\\mathrm{imp}}|\\boldsymbol{k}_{2}\\rangle\\underline{\\hat{G}}_{0}(\\boldsymbol{k}_{2},t_{1};\\boldsymbol{k}_{3},t_{2})\\langle\\boldsymbol{k}_{3}|\\hat{\\mathcal{H}}_{\\mathrm{imp}}|\\boldsymbol{k}_{4}\\rangle\\underline{\\hat{G}}_{0}(\\boldsymbol{k}_{4},t_{1};\\boldsymbol{k'},t_{2})+...\\\\\n&=\\underline{\\hat{G}}_{0}(\\boldsymbol{k},t_{1};\\boldsymbol{k'},t_{2})+\\sum\\limits_{\\{\\boldsymbol{k}_{i}\\}} \\underline{\\hat{G}}_{0}(\\boldsymbol{k},t_{1};\\boldsymbol{k}_{1},t_{2})\\underline{\\hat{\\Sigma}}(\\boldsymbol{k}_{1},t_{1};\\boldsymbol{k}_{2},t_{2})\\underline{\\hat{G}}(\\boldsymbol{k}_{2},t_{1};\\boldsymbol{k'},t_{2}),\n\\end{aligned}\n\\label{eq:selfscat}\n\\end{equation}\n\\noindent where $\\underline{\\hat{G}}_{0}$ is a free propagator. To consider a particle moving in a random potential we take the average over different spatial configurations of the ensemble of $N$ impurities. Upon impurity-averaging the first order term in Eq.~(\\ref{eq:selfscat}) is a constant and can be renormalized away. For the second order term, we take into account all two-line irreducible diagrams corresponding to the double scattering off the same impurity and neglect the so-called crossing diagrams, where impurity lines cross and give a small contribution to the region of interest, $E\\simeq E_{F}$ and $k\\simeq k_{F}$.\\cite{rammer1} This is nothing else but the self-consistent Born approximation (Fig. 1$b$). Using Eq.~(\\ref{eq:potsoi}) for the impurity potential, the self-energy is given by three terms:\n\\begin{equation}\n\\begin{aligned}\n\\underline{\\hat{\\Sigma}}^{1a}(\\boldsymbol{k}_{1},t_{1};\\boldsymbol{k}_{4},t_{2})=\\frac{1}{\\Omega^{2}}\\sum\\limits_{\\boldsymbol{k}_{2}\\boldsymbol{k}_{3}}\\underline{\\hat{G}}(\\boldsymbol{k}_{2},t_{1};\\boldsymbol{k}_{3},t_{2})\\left\\langle V(\\boldsymbol{k}_{1}-\\boldsymbol{k}_{2})V(\\boldsymbol{k}_{3}-\\boldsymbol{k}_{4})\\right\\rangle,\n\\end{aligned}\n\\end{equation}\n\\begin{equation}\n\\begin{aligned}\n\\underline{\\hat{\\Sigma}}^{1b}(\\boldsymbol{k}_{1},t_{1};\\boldsymbol{k}_{4},t_{2})&=\\frac{i}{\\Omega^{2}}\\frac{\\xi_{SO}}{k_{F}^{2}}\\sum\\limits_{\\boldsymbol{k}_{2}\\boldsymbol{k}_{3}}\\left[[(\\boldsymbol{k}_{1}\\times\\boldsymbol{k}_{2})\\cdot\\hat{\\boldsymbol{\\sigma}}]\\underline{\\hat{G}}(\\boldsymbol{k}_{2},t_{1};\\boldsymbol{k}_{3},t_{2})\\right.\\\\\n&\\left.+\\,\\underline{\\hat{G}}(\\boldsymbol{k}_{2},t_{1};\\boldsymbol{k}_{3},t_{2})[(\\boldsymbol{k}_{3}\\times\\boldsymbol{k}_{4})\\cdot\\hat{\\boldsymbol{\\sigma}}]\\right]\\left\\langle V(\\boldsymbol{k}_{1}-\\boldsymbol{k}_{2})V(\\boldsymbol{k}_{3}-\\boldsymbol{k}_{4})\\right\\rangle,\n\\end{aligned}\n\\end{equation}\n\\begin{equation}\n\\begin{aligned}\n\\underline{\\hat{\\Sigma}}^{1c}(\\boldsymbol{k}_{1},t_{1};\\boldsymbol{k}_{4},t_{2})&=-\\frac{1}{\\Omega^{2}}\\frac{\\xi_{SO}^{2}}{k^{4}_{F}}\\sum\\limits_{\\boldsymbol{k}_{2}\\boldsymbol{k}_{3}}[(\\boldsymbol{k}_{1}\\times\\boldsymbol{k}_{2})\\cdot\\hat{\\boldsymbol{\\sigma}}]\\underline{\\hat{G}}(\\boldsymbol{k}_{2},t_{1};\\boldsymbol{k}_{3},t_{2})[(\\boldsymbol{k}_{3}\\times\\boldsymbol{k}_{4})\\cdot\\hat{\\boldsymbol{\\sigma}}]\\left\\langle V(\\boldsymbol{k}_{1}-\\boldsymbol{k}_{2})V(\\boldsymbol{k}_{3}-\\boldsymbol{k}_{4})\\right\\rangle,\\\\\n\\end{aligned}\n\\end{equation}\n\\noindent where impurity averaging leads to:\n\\begin{equation}\n\\begin{aligned}\n \\left\\langle V(\\boldsymbol{k}_{1}-\\boldsymbol{k}_{2})V(\\boldsymbol{k}_{3}-\\boldsymbol{k}_{4})\\right\\rangle&= v_{i}^{2} \\left\\langle\\sum_{\\boldsymbol{R}_{i}}e^{-i(\\boldsymbol{k}_{1}-\\boldsymbol{k}_{2})\\cdot\\boldsymbol{R}_{i}}e^{-i(\\boldsymbol{k}_{3}-\\boldsymbol{k}_{4})\\cdot\\boldsymbol{R}_{i}}\\right\\rangle=v_{i}^{2}N\\delta_{\\boldsymbol{k}_{1}+\\boldsymbol{k}_{3},\\boldsymbol{k}_{2}+\\boldsymbol{k}_{4}}.\n \\end{aligned}\n\\end{equation}\n\\noindent To proceed with the Wigner transformation, we change the variables:\n$$\n\\boldsymbol{k}_{1}=\\boldsymbol{k}+\\frac{\\boldsymbol{q}}{2}\\qquad\\boldsymbol{k}_{2}=\\boldsymbol{k'}+\\frac{\\boldsymbol{q'}}{2}\\qquad\\boldsymbol{k}_{3}=\\boldsymbol{k'}-\\frac{\\boldsymbol{q'}}{2} \\qquad\\boldsymbol{k}_{4}=\\boldsymbol{k}-\\frac{\\boldsymbol{q}}{2}\n$$\n\\noindent and\n$$\nT=\\frac{t_{1}+t_{2}}{2}\\qquad t=t_{1}-t_{2},\n$$\n\\noindent that gives:\n\\begin{equation}\n\\begin{aligned}\n\\underline{\\hat{\\Sigma}}^{1a}(\\boldsymbol{k},t;\\boldsymbol{q},T)=\\frac{1}{\\Omega^{2}}\\sum\\limits_{\\boldsymbol{k}_{2}\\boldsymbol{k}_{3}}\\underline{\\hat{G}}(\\boldsymbol{k'},t;\\boldsymbol{q'},T)\\delta_{\\boldsymbol{q},\\boldsymbol{q'}},\n\\end{aligned}\n\\end{equation}\n\\begin{equation}\n\\begin{aligned}\n\\underline{\\hat{\\Sigma}}^{1b}(\\boldsymbol{k},t;\\boldsymbol{q},T)&=\\frac{i}{\\Omega^{2}}\\frac{\\xi_{SO}}{k_{F}^{2}}\\sum\\limits_{\\boldsymbol{k}_{2}\\boldsymbol{k}_{3}}\\left[\\big[((\\boldsymbol{k}+\\frac{\\boldsymbol{q}}{2})\\times(\\boldsymbol{k'}+\\frac{\\boldsymbol{q'}}{2}))\\cdot\\hat{\\boldsymbol{\\sigma}}\\big]\\underline{\\hat{G}}(\\boldsymbol{k'},t;\\boldsymbol{q'},T)\\right.\\\\\n&\\left.+\\,\\underline{\\hat{G}}(\\boldsymbol{k'},t;\\boldsymbol{q'},T)\\big[((\\boldsymbol{k'}-\\frac{\\boldsymbol{q'}}{2})\\times(\\boldsymbol{k}-\\frac{\\boldsymbol{q'}}{2})))\\cdot\\hat{\\boldsymbol{\\sigma}}\\big]\\right]\\delta_{\\boldsymbol{q},\\boldsymbol{q'}},\n\\end{aligned}\n\\end{equation}\n\\begin{equation}\n\\begin{aligned}\n\\underline{\\hat{\\Sigma}}^{1c}(\\boldsymbol{k},t;\\boldsymbol{q},T)&=-\\frac{1}{\\Omega^{2}}\\frac{\\xi_{SO}^{2}}{k^{4}_{F}}\\sum\\limits_{\\boldsymbol{k}_{2}\\boldsymbol{k}_{3}}\\big[((\\boldsymbol{k}+\\frac{\\boldsymbol{q}}{2})\\times(\\boldsymbol{k'}+\\frac{\\boldsymbol{q'}}{2}))\\cdot\\hat{\\boldsymbol{\\sigma}}\\big]\\underline{\\hat{G}}(\\boldsymbol{k'},t;\\boldsymbol{q'},T)\\big[((\\boldsymbol{k'}-\\frac{\\boldsymbol{q'}}{2})\\times(\\boldsymbol{k}-\\frac{\\boldsymbol{q}}{2}))\\cdot\\hat{\\boldsymbol{\\sigma}}\\big]\\delta_{\\boldsymbol{q},\\boldsymbol{q'}}.\n\\end{aligned}\n\\end{equation}\n\\noindent The Kronecker function reflects that translation invariance is recovered, and we have in the continuum limit:\n\\begin{equation}\n\\begin{aligned}\n\\underline{\\hat{\\Sigma}}^{1a}(\\boldsymbol{k},t;\\boldsymbol{q},T)=v_{i}^{2}n_{i}\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}\\,\\underline{\\hat{G}}(\\boldsymbol{k'},t;\\boldsymbol{q},T),\n\\end{aligned}\n\\end{equation}\n\\begin{equation}\n\\begin{aligned}\n\\underline{\\hat{\\Sigma}}^{1b}(\\boldsymbol{k},t;\\boldsymbol{q},T)&=\\underline{\\hat{\\Sigma}}^{sw}(\\boldsymbol{k},t;\\boldsymbol{q},T)+\\underline{\\hat{\\Sigma}}^{sj}(\\boldsymbol{k},t;\\boldsymbol{q},T)\\\\\n&= iv_{i}^{2}n_{i}\\frac{\\xi_{SO}}{k_{F}^{2}}\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}\\left[(\\boldsymbol{k}\\times\\boldsymbol{k'})\\cdot\\hat{\\boldsymbol{\\sigma}},\\underline{\\hat{G}}(\\boldsymbol{k'},t;\\boldsymbol{q},T)\\right]\\\\\n&+iv_{i}^{2}n_{i}\\frac{\\xi_{SO}}{2k_{F}^{2}}\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}\\left\\{[\\boldsymbol{q}\\times(\\boldsymbol{k}'-\\boldsymbol{k})]\\cdot\\hat{\\boldsymbol{\\sigma}},\\underline{\\hat{G}}(\\boldsymbol{k'},t;\\boldsymbol{q},T)\\right\\},\n\\end{aligned}\n\\end{equation}\n\\begin{equation}\n\\begin{aligned}\n\\underline{\\hat{\\Sigma}}^{1c}(\\boldsymbol{k},t;\\boldsymbol{q},T)&=-v_{i}^{2}n_{i}\\frac{\\xi^{2}_{SO}}{k_{F}^{4}}\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}[(\\boldsymbol{k}\\times\\boldsymbol{k'}+\\frac{1}{2}\\boldsymbol{q}\\times\\boldsymbol{k'}+\\frac{1}{2}\\boldsymbol{k}\\times\\boldsymbol{q})\\cdot\\hat{\\boldsymbol{\\sigma}}]\\\\\n&\\cdot\\underline{\\hat{G}}(\\boldsymbol{k'},t;\\boldsymbol{q},T)[(\\boldsymbol{k'}\\times\\boldsymbol{k}-\\frac{1}{2}\\boldsymbol{k'}\\times\\boldsymbol{q}-\\frac{1}{2}\\boldsymbol{q}\\times\\boldsymbol{k})\\cdot\\hat{\\boldsymbol{\\sigma}}].\n\\end{aligned}\n\\end{equation}\n\\noindent Having Fourier transformed with respect to $\\boldsymbol{q}$, that gives the Wigner coordinate $\\boldsymbol{R}$:\n\\begin{equation}\n\\begin{aligned}\n\\hat{G}(\\boldsymbol{r}_{1};\\boldsymbol{r}_{4})&=\\int\\frac{d\\boldsymbol{k}_{1}}{(2\\pi)^{3}}\\int\\frac{d\\boldsymbol{k}_{4}}{(2\\pi)^{3}}\\hat{G}(\\boldsymbol{k}_{1};\\boldsymbol{k}_{4})e^{i(\\boldsymbol{k}_{1}\\boldsymbol{r}_{1}-\\boldsymbol{k}_{4}\\boldsymbol{r}_{4})}\\\\\n&=\\int\\frac{d\\boldsymbol{k}}{(2\\pi)^{3}}\\int\\frac{d\\boldsymbol{q}}{(2\\pi)^{3}}\\hat{G}(\\boldsymbol{k}+\\frac{1}{2}\\boldsymbol{q};\\boldsymbol{k}-\\frac{1}{2}\\boldsymbol{q})e^{i(\\boldsymbol{k}+\\frac{1}{2}\\boldsymbol{q})\\cdot(\\boldsymbol{R}+\\frac{1}{2}\\boldsymbol{r})}e^{-i(\\boldsymbol{k}-\\frac{1}{2}\\boldsymbol{q})\\cdot(\\boldsymbol{R}-\\frac{1}{2}\\boldsymbol{r})}\\\\\n&=\\int\\frac{d\\boldsymbol{k}}{(2\\pi)^{3}}\\int\\frac{d\\boldsymbol{q}}{(2\\pi)^{3}}\\hat{G}(\\boldsymbol{k};\\boldsymbol{q})e^{i\\boldsymbol{q}\\cdot\\boldsymbol{R}}e^{i\\boldsymbol{k}\\cdot\\boldsymbol{r}}=\\hat{G}(\\boldsymbol{r};\\boldsymbol{R}),\n\\end{aligned}\n\\end{equation}\n\\noindent we get the final form for the self-energy in the mixed representation:\n\\begin{equation}\n\\begin{aligned}\n\\underline{\\hat{\\Sigma}}^{1a}_{\\boldsymbol{k}E}(\\boldsymbol{R},T)&=v_{i}^{2}n_{i}\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}\\,\\underline{\\hat{G}}_{\\boldsymbol{k'}E}(\\boldsymbol{R},T),\\\\\n\\underline{\\hat{\\Sigma}}^{1b}_{\\boldsymbol{k}E}(\\boldsymbol{R},T)&=\\underline{\\hat{\\Sigma}}^{sw}_{\\boldsymbol{k}E}(\\boldsymbol{R},T)+\\underline{\\hat{\\Sigma}}^{sj}_{\\boldsymbol{k}E}(\\boldsymbol{R},T)\\\\\n&= iv_{i}^{2}n_{i}\\frac{\\xi_{SO}}{k_{F}^{2}}\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}\\left[(\\boldsymbol{k}\\times\\boldsymbol{k'})\\cdot\\hat{\\boldsymbol{\\sigma}},\\underline{\\hat{G}}_{\\boldsymbol{k'}E}(\\boldsymbol{R},T)\\right]\\\\\n&+v_{i}^{2}n_{i}\\frac{\\xi_{SO}}{2k_{F}^{2}}\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}\\nabla_{\\boldsymbol{R}}\\left\\{(\\boldsymbol{k}'-\\boldsymbol{k})\\times\\hat{\\boldsymbol{\\sigma}},\\underline{\\hat{G}}_{\\boldsymbol{k'}E}(\\boldsymbol{R};T)\\right\\},\\\\\n\\underline{\\hat{\\Sigma}}^{1c}_{\\boldsymbol{k}E}(\\boldsymbol{R},T)&=-v_{i}^{2}n_{i}\\frac{\\xi^{2}_{SO}}{k_{F}^{4}}\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}[(\\boldsymbol{k}\\times\\boldsymbol{k'})\\cdot\\hat{\\boldsymbol{\\sigma}}]\\underline{\\hat{G}}_{\\boldsymbol{k'}E}(\\boldsymbol{R},T)[(\\boldsymbol{k'}\\times\\boldsymbol{k})\\cdot\\hat{\\boldsymbol{\\sigma}}].\n\\end{aligned}\n\\end{equation}\n\\noindent At the level of the self-consistent Born approximation, the self-energy is given by the following contributions. The first term $\\underline{\\hat{\\Sigma}}^{1a}$ stands for the standard elastic scattering off the on-site impurity potential. To first order of $\\xi_{SO}$, there are two terms, $\\underline{\\hat{\\Sigma}}^{sw}$ and $\\underline{\\hat{\\Sigma}}^{sj}$, related to the side-jump and spin swapping contributions, respectively. Finally, the second order of $\\xi_{SO}$ yields the Elliot-Yafet spin relaxation mechanism (all gradient terms $\\sim\\xi_{SO}^{2}\\nabla_{\\boldsymbol{R}}$ are neglected on account of their smallness). \n\\par Let us rewrite these contributions for the retarded, advanced and Keldysh Green's functions (taking into account that $\\hat{G}^{R(A)}$ does not depend on the center-of-mass coordinates):\n\\begin{equation}\n\\underline{\\hat{\\Sigma}}_{\\boldsymbol{k}E}(\\boldsymbol{R},T)=\\underline{\\hat{\\Sigma}}^{1a}_{\\boldsymbol{k}E}(\\boldsymbol{R},T)+\\underline{\\hat{\\Sigma}}^{1b}_{\\boldsymbol{k}E}(\\boldsymbol{R},T)+\\underline{\\hat{\\Sigma}}^{1c}_{\\boldsymbol{k}E}(\\boldsymbol{R},T),\n\\end{equation}\n\\noindent which is equivalent to:\n\\begin{equation}\n\\hat{\\Sigma}^{R}_{\\boldsymbol{k}E}=v_{i}^{2}n_{i}\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}\\,\\left[\\hat{\\sigma}_{0}+i\\frac{\\xi_{SO}}{k_{F}^{2}}(\\boldsymbol{k}\\times\\boldsymbol{k'})\\cdot\\hat{\\boldsymbol{\\sigma}}\\right]\\hat{G}^{R}_{\\boldsymbol{k'}E}\\left[\\hat{\\sigma}_{0}-i\\frac{\\xi_{SO}}{k_{F}^{2}}(\\boldsymbol{k}\\times\\boldsymbol{k'})\\cdot\\hat{\\boldsymbol{\\sigma}}\\right],\n\\label{eq:sret}\n\\end{equation}\n\\begin{equation}\n\\hat{\\Sigma}^{A}_{\\boldsymbol{k}E}=v_{i}^{2}n_{i}\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}\\,\\left[\\hat{\\sigma}_{0}+i\\frac{\\xi_{SO}}{k_{F}^{2}}(\\boldsymbol{k}\\times\\boldsymbol{k'})\\cdot\\hat{\\boldsymbol{\\sigma}}\\right]\\hat{G}^{A}_{\\boldsymbol{k'}E}\\left[\\hat{\\sigma}_{0}-i\\frac{\\xi_{SO}}{k_{F}^{2}}(\\boldsymbol{k}\\times\\boldsymbol{k'})\\cdot\\hat{\\boldsymbol{\\sigma}}\\right],\n\\label{eq:sadv}\n\\end{equation}\n\\noindent and\n\\begin{equation}\n\\begin{aligned}\n\\hat{\\Sigma}^{K}_{\\boldsymbol{k}E}(\\boldsymbol{R},T)&=v_{i}^{2}n_{i}\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}\\,\\left[\\hat{\\sigma}_{0}+i\\frac{\\xi_{SO}}{k_{F}^{2}}(\\boldsymbol{k}\\times\\boldsymbol{k'})\\cdot\\hat{\\boldsymbol{\\sigma}}\\right]\\hat{g}^{K}_{\\boldsymbol{k'}E}(\\boldsymbol{R},T)\\left[\\hat{\\sigma}_{0}-i\\frac{\\xi_{SO}}{k_{F}^{2}}(\\boldsymbol{k}\\times\\boldsymbol{k'})\\cdot\\hat{\\boldsymbol{\\sigma}}\\right]\\\\\n&+v_{i}^{2}n_{i}\\frac{\\xi_{SO}}{2k_{F}^{2}}\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}\\nabla_{\\boldsymbol{R}}\\left\\{(\\boldsymbol{k}'-\\boldsymbol{k})\\times\\hat{\\boldsymbol{\\sigma}},\\hat{g}^{K}_{\\boldsymbol{k'}E}(\\boldsymbol{R};T)\\right\\}.\n\\label{eq:skel}\n\\end{aligned}\n\\end{equation}\n\n\n\\subsection{Skew-scattering}\n\\par To take into account skew-scattering, one has to go beyond the Born approximation. Starting from third order diagrams (Fig.~1$c$), we obtain the following expressions for the self-energy to first order in $\\xi_{SO}$:\n\\begin{equation}\n\\begin{aligned}\n\\underline{\\hat{\\Sigma}}^{2a}(\\boldsymbol{k}_{1},t_{1};\\boldsymbol{k}_{6},t_{2})=\\frac{i}{\\Omega^{3}}\\frac{\\xi_{SO}}{k_{F}^{2}}\\sum\\limits_{\\boldsymbol{k}_{2},\\boldsymbol{k}_{3},\\boldsymbol{k}_{4},\\boldsymbol{k}_{5}}&[(\\boldsymbol{k}_{1}\\times\\boldsymbol{k}_{2})\\cdot\\hat{\\boldsymbol{\\sigma}}]\\underline{\\hat{G}}_{0}(\\boldsymbol{k}_{2},t_{1};\\boldsymbol{k}_{3},t_{2})\\underline{\\hat{G}}_{0}(\\boldsymbol{k}_{4},t_{1};\\boldsymbol{k}_{5},t_{2})\\\\\n&\\cdot\\left\\langle V(\\boldsymbol{k}_{1}-\\boldsymbol{k}_{2})V(\\boldsymbol{k}_{3}-\\boldsymbol{k}_{4})V(\\boldsymbol{k}_{5}-\\boldsymbol{k}_{6})\\right\\rangle,\n\\end{aligned}\n\\end{equation}\n\\begin{equation}\n\\begin{aligned}\n\\underline{\\hat{\\Sigma}}^{2b}(\\boldsymbol{k}_{1},t_{1};\\boldsymbol{k}_{6},t_{2})=\\frac{i}{\\Omega^{3}}\\frac{\\xi_{SO}}{k_{F}^{2}}\\sum\\limits_{\\boldsymbol{k}_{2},\\boldsymbol{k}_{3},\\boldsymbol{k}_{4},\\boldsymbol{k}_{5}}&\\underline{\\hat{G}}_{0}(\\boldsymbol{k}_{2},t_{1};\\boldsymbol{k}_{3},t_{2})[(\\boldsymbol{k}_{3}\\times\\boldsymbol{k}_{4})\\cdot\\hat{\\boldsymbol{\\sigma}}]\\underline{\\hat{G}}_{0}(\\boldsymbol{k}_{4},t_{1};\\boldsymbol{k}_{5},t_{2})\\\\\n&\\cdot\\left\\langle V(\\boldsymbol{k}_{1}-\\boldsymbol{k}_{2})V(\\boldsymbol{k}_{3}-\\boldsymbol{k}_{4})V(\\boldsymbol{k}_{5}-\\boldsymbol{k}_{6})\\right\\rangle,\n\\end{aligned}\n\\end{equation}\n\\begin{equation}\n\\begin{aligned}\n\\underline{\\hat{\\Sigma}}^{2c}(\\boldsymbol{k}_{1},t_{1};\\boldsymbol{k}_{6},t_{2})=\\frac{i}{\\Omega^{3}}\\frac{\\xi_{SO}}{k_{F}^{2}}\\sum\\limits_{\\boldsymbol{k}_{2},\\boldsymbol{k}_{3},\\boldsymbol{k}_{4},\\boldsymbol{k}_{5}}&\\underline{\\hat{G}}_{0}(\\boldsymbol{k}_{2},t_{1};\\boldsymbol{k}_{3},t_{2})\\underline{\\hat{G}}_{0}(\\boldsymbol{k}_{4},t_{1};\\boldsymbol{k}_{5},t_{2})[(\\boldsymbol{k}_{5}\\times\\boldsymbol{k}_{6})\\cdot\\hat{\\boldsymbol{\\sigma}}]\\\\\n&\\cdot\\left\\langle V(\\boldsymbol{k}_{1}-\\boldsymbol{k}_{2})V(\\boldsymbol{k}_{3}-\\boldsymbol{k}_{4})V(\\boldsymbol{k}_{5}-\\boldsymbol{k}_{6})\\right\\rangle,\n\\end{aligned}\n\\end{equation}\n\\noindent where impurity averaging in the triple scattering off the same impurity potential is implied:\n\\begin{equation}\n\\begin{aligned}\n\\left\\langle V(\\boldsymbol{k}_{1}-\\boldsymbol{k}_{2})V(\\boldsymbol{k}_{3}-\\boldsymbol{k}_{4})V(\\boldsymbol{k}_{5}-\\boldsymbol{k}_{6})\\right\\rangle&= v_{i}^{3}\\left\\langle \\sum_{\\boldsymbol{R}_{i}}e^{-i(\\boldsymbol{k}_{1}-\\boldsymbol{k}_{2})\\cdot\\boldsymbol{R}_{i}}e^{-i(\\boldsymbol{k}_{3}-\\boldsymbol{k}_{4})\\cdot\\boldsymbol{R}_{i}}e^{-i(\\boldsymbol{k}_{5}-\\boldsymbol{k}_{6})\\cdot\\boldsymbol{R}_{i}}\\right\\rangle\\\\\n &=v_{i}^{3}N\\delta_{\\boldsymbol{k}_{1}+\\boldsymbol{k}_{3}+\\boldsymbol{k}_{5},\\boldsymbol{k}_{2}+\\boldsymbol{k}_{4}+\\boldsymbol{k}_{6}}.\n\\end{aligned}\n\\end{equation}\n\\noindent Here, we do not consider triple scatterings off the on-site impurity potential without spin-orbit coupling, which gives a negligible correction to the elastic relaxation time ($\\sim\\frac{\\beta v_{i}}{\\tau_{0}}$). Changing the variables:\n\\begin{equation}\n\\begin{aligned}\n\\boldsymbol{k}_{1}&=\\boldsymbol{k}+\\frac{\\boldsymbol{q}}{2}\\qquad \\boldsymbol{k}_{2}=\\boldsymbol{k'}+\\frac{\\boldsymbol{q'}}{2}\\qquad \\boldsymbol{k}_{3}=\\boldsymbol{k'}-\\frac{\\boldsymbol{q'}}{2},\\\\\n\\boldsymbol{k}_{4}&=\\boldsymbol{k''}+\\frac{\\boldsymbol{q''}}{2}\\qquad \\boldsymbol{k}_{5}=\\boldsymbol{k''}-\\frac{\\boldsymbol{q''}}{2}\\qquad \\boldsymbol{k}_{6}=\\boldsymbol{k}-\\frac{\\boldsymbol{q}}{2}\n\\end{aligned}\n\\end{equation}\n\\noindent and\n\\begin{equation}\nT=\\frac{t_{1}+t_{2}}{2}\\qquad t=t_{1}-t_{2}\n\\end{equation}\n\\noindent leads in the continuum limit to:\n\\begin{equation}\n\\begin{aligned}\n\\underline{\\hat{\\Sigma}}^{2a}(\\boldsymbol{k},t;\\boldsymbol{q},T)&=iv_{i}^{3}n_{i}\\frac{\\xi_{SO}}{k_{F}^{2}}\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}\\int\\frac{d\\boldsymbol{k''}}{(2\\pi)^{3}}\\int\\frac{d\\boldsymbol{q'}}{(2\\pi)^{3}}\\\\\n&\\left[\\left(\\boldsymbol{k}+\\frac{\\boldsymbol{q}}{2}\\right)\\times\\left(\\boldsymbol{k'}+\\frac{\\boldsymbol{q'}}{2}\\right)\\cdot\\hat{\\boldsymbol{\\sigma}}\\right]\\underline{\\hat{G}}(\\boldsymbol{k'},t;\\boldsymbol{q'},T)\\underline{\\hat{G}}(\\boldsymbol{k''},t;\\boldsymbol{q}-\\boldsymbol{q'},T),\n\\end{aligned}\n\\end{equation}\n\\begin{equation}\n\\begin{aligned}\n\\underline{\\hat{\\Sigma}}^{2b}(\\boldsymbol{k},t;\\boldsymbol{q},T)&=iv_{i}^{3}n_{i}\\frac{\\xi_{SO}}{k_{F}^{2}}\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}\\int\\frac{d\\boldsymbol{k''}}{(2\\pi)^{3}}\\int\\frac{d\\boldsymbol{q'}}{(2\\pi)^{3}}\\\\\n&\\underline{\\hat{G}}(\\boldsymbol{k'},t;\\boldsymbol{q'},T)\\left[\\left(\\boldsymbol{k'}-\\frac{\\boldsymbol{q'}}{2}\\right)\\times\\left(\\boldsymbol{k''}+\\frac{\\boldsymbol{q}}{2}-\\frac{\\boldsymbol{q'}}{2}\\right)\\cdot\\hat{\\boldsymbol{\\sigma}}\\right]\\underline{\\hat{G}}(\\boldsymbol{k''},t;\\boldsymbol{q}-\\boldsymbol{q'},T),\n\\end{aligned}\n\\end{equation}\n\\begin{equation}\n\\begin{aligned}\n\\underline{\\hat{\\Sigma}}^{2c}(\\boldsymbol{k},t;\\boldsymbol{q},T)&=iv_{i}^{3}n_{i}\\frac{\\xi_{SO}}{k_{F}^{2}}\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}\\int\\frac{d\\boldsymbol{k''}}{(2\\pi)^{3}}\\int\\frac{d\\boldsymbol{q'}}{(2\\pi)^{3}}\\\\\n&\\underline{\\hat{G}}(\\boldsymbol{k'},t;\\boldsymbol{q'},T)\\underline{\\hat{G}}(\\boldsymbol{k''},t;\\boldsymbol{q}-\\boldsymbol{q'},T)\\left[\\left(\\boldsymbol{k''}-\\frac{\\boldsymbol{q}}{2}+\\frac{\\boldsymbol{q'}}{2}\\right)\\times\\left(\\boldsymbol{k}-\\frac{\\boldsymbol{q}}{2}\\right)\\cdot\\hat{\\boldsymbol{\\sigma}}\\right].\n\\end{aligned}\n\\end{equation}\n\\noindent Let us assume that any inhomogeneity in a system is smooth, so we can neglect all gradient terms $\\sim\\boldsymbol{q}$. Having Fourier transformed with respect to $\\boldsymbol{q}$ and $t$, we obtain:\n\\begin{equation}\n\\underline{\\hat{\\Sigma'}}_{\\boldsymbol{k}E}(\\boldsymbol{R},T)=\\underline{\\hat{\\Sigma}}^{2a}_{\\boldsymbol{k}E}(\\boldsymbol{R},T)+\\underline{\\hat{\\Sigma}}^{2b}_{\\boldsymbol{k}E}(\\boldsymbol{R},T)+\\underline{\\hat{\\Sigma}}^{2c}_{\\boldsymbol{k}E}(\\boldsymbol{R},T),\n\\end{equation}\n\\noindent where \n\\begin{equation}\n\\underline{\\hat{\\Sigma}}^{2a}_{\\boldsymbol{k}E}(\\boldsymbol{R},T)=iv_{i}^{3}n_{i}\\frac{\\xi_{SO}}{k_{F}^{2}}\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}\\int\\frac{d\\boldsymbol{k''}}{(2\\pi)^{3}}[(\\boldsymbol{k}\\times\\boldsymbol{k'})\\cdot\\hat{\\boldsymbol{\\sigma}}]\\underline{\\hat{G}}_{\\boldsymbol{k'}E}(\\boldsymbol{R},T)\\underline{\\hat{G}}_{\\boldsymbol{k''}E}(\\boldsymbol{R},T),\n\\end{equation}\n\\begin{equation}\n\\underline{\\hat{\\Sigma}}^{2b}_{\\boldsymbol{k}E}(\\boldsymbol{R},T)=iv_{i}^{3}n_{i}\\frac{\\xi_{SO}}{k_{F}^{2}}\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}\\int\\frac{d\\boldsymbol{k''}}{(2\\pi)^{3}}\\underline{\\hat{G}}_{\\boldsymbol{k'}E}(\\boldsymbol{R},T)[(\\boldsymbol{k'}\\times\\boldsymbol{k''})\\cdot\\hat{\\boldsymbol{\\sigma}}]\\underline{\\hat{G}}_{\\boldsymbol{k''}E}(\\boldsymbol{R},T),\n\\end{equation}\n\\begin{equation}\n\\underline{\\hat{\\Sigma}}^{2c}_{\\boldsymbol{k}E}(\\boldsymbol{R},T)=iv_{i}^{3}n_{i}\\frac{\\xi_{SO}}{k_{F}^{2}}\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}\\int\\frac{d\\boldsymbol{k''}}{(2\\pi)^{3}}\\underline{\\hat{G}}_{\\boldsymbol{k'}E}(\\boldsymbol{R},T)\\underline{\\hat{G}}_{\\boldsymbol{k''}E}(\\boldsymbol{R},T)[(\\boldsymbol{k''}\\times\\boldsymbol{k})\\cdot\\hat{\\boldsymbol{\\sigma}}].\n\\end{equation}\n\\par To first order in $\\xi_{SO}$, we can express the retarded and advanced Green's functions by using the Sokhotski formula:\n\\begin{equation}\n\\begin{aligned}\n\\hat{G}^{R(A)}_{\\boldsymbol{k}E}&=\\left(\\hat{E}-\\hat{\\mathcal{H}}_{\\boldsymbol{k}}-\\hat{\\Sigma}^{R(A)}_{\\boldsymbol{k}E}\\right)^{-1}\\approx\\frac{1}{2}\\sum\\limits_{s=\\pm}\\frac{\\hat{\\sigma}_{0}+s\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}}{E-E_{\\boldsymbol{k}s}\\pm i\\frac{\\hbar}{2\\tau_{s}}}\\\\\n&=\\frac{1}{2}\\sum\\limits_{s=\\pm}\\left(\\hat{\\sigma}_{0}+s\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}\\right)\\left[\\mp i\\pi\\delta(E-E_{\\boldsymbol{k}s})\\right],\n\\end{aligned}\n\\end{equation}\n\\noindent where $\\tau_{s}$ is the spin dependent relaxation time. As a result, the retarded and advanced components of the skew-scattering self-energy vanish, and we deal with its Keldysh part, which survives for $\\hat{\\Sigma}^{2a}$ and $\\hat{\\Sigma}^{2c}$ only:\n\\begin{equation}\n\\begin{aligned}\n\\hat{\\Sigma'}^{K}_{\\boldsymbol{k}E}(\\boldsymbol{R},T)&=i\\frac{\\xi_{SO}}{k_{F}^{2}}v_{i}^{3}n_{i}\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}\\int\\frac{d\\boldsymbol{k''}}{(2\\pi)^{3}}(\\boldsymbol{k}\\times\\boldsymbol{k'})\\cdot\\hat{\\boldsymbol{\\sigma}}\\left(\\hat{G}^{R}_{\\boldsymbol{k'}E}\\hat{g}^{K}_{\\boldsymbol{k''}E}(\\boldsymbol{R},T)+\\hat{g}^{K}_{\\boldsymbol{k'}E}(\\boldsymbol{R},T)\\hat{G}^{A}_{\\boldsymbol{k''}E} \\right)\\\\\n&+i\\frac{\\xi_{SO}}{k_{F}^{2}}v_{i}^{3}n_{i}\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}\\int\\frac{d\\boldsymbol{k''}}{(2\\pi)^{3}}\\left(\\hat{G}^{R}_{\\boldsymbol{k'}E}\\hat{g}^{K}_{\\boldsymbol{k''}E}(\\boldsymbol{R},T)+\\hat{g}^{K}_{\\boldsymbol{k'}E}(\\boldsymbol{R},T)\\hat{G}^{A}_{\\boldsymbol{k''}E} \\right)(\\boldsymbol{k''}\\times\\boldsymbol{k})\\cdot\\hat{\\boldsymbol{\\sigma}}\n\\end{aligned}\n\\end{equation}\n\\noindent  or\n\\begin{equation}\n\\begin{aligned}\n\\hat{\\Sigma'}^{K}_{\\boldsymbol{k}E}(\\boldsymbol{R},T)&=i\\frac{\\xi_{SO}}{k_{F}^{2}}v_{i}^{3}n_{i}\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}\\int\\frac{d\\boldsymbol{k''}}{(2\\pi)^{3}}(\\boldsymbol{k}\\times\\boldsymbol{k'})\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{g}^{K}_{\\boldsymbol{k'}E}(\\boldsymbol{R},T)\\hat{G}^{A}_{\\boldsymbol{k''}E}\\\\\n&+i\\frac{\\xi_{SO}}{k_{F}^{2}}v_{i}^{3}n_{i}\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}\\int\\frac{d\\boldsymbol{k''}}{(2\\pi)^{3}}\\hat{G}^{R}_{\\boldsymbol{k'}E}\\hat{g}^{K}_{\\boldsymbol{k''}E}(\\boldsymbol{R},T)(\\boldsymbol{k''}\\times\\boldsymbol{k})\\cdot\\hat{\\boldsymbol{\\sigma}}.\n\\end{aligned}\n\\end{equation}\n\\noindent This expression can be further simplified, as we integrate over $\\boldsymbol{k}$:\n\\begin{equation}\n\\begin{aligned}\n\\mp\\frac{i\\pi}{2}\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}\\sum\\limits_{s=\\pm}\\left(\\hat{\\sigma}_{0}+s\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}\\right)\\delta(E-E_{\\boldsymbol{k}s})=\\mp\\frac{i\\pi}{2}(D^{\\uparrow}+D^{\\downarrow})\\left(\\hat{\\sigma}_{0}+\\delta\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}\\right),\n\\end{aligned}\n\\end{equation}\n\\noindent where $D^{\\uparrow(\\downarrow)}$ is the spin-dependent density of states and $\\delta=(D^{\\uparrow}-D^{\\downarrow})\/(D^{\\uparrow}+D^{\\downarrow})$, so the final form of $\\hat{\\Sigma'}^{K}_{\\boldsymbol{k}E}(\\boldsymbol{R},T)$ is given by:\n\\begin{equation}\n\\begin{aligned}\n\\hat{\\Sigma'}^{K}_{\\boldsymbol{k}E}(\\boldsymbol{R},T)&=-\\frac{\\pi}{2}\\frac{\\xi_{SO}}{k_{F}^{2}}v_{i}^{3}n_{i}(D^{\\uparrow}+D^{\\downarrow})\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}(\\boldsymbol{k}\\times\\boldsymbol{k'})\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{g}^{K}_{\\boldsymbol{k'}E}(\\boldsymbol{R},T)\\left[\\hat{\\sigma}_{0}+\\delta\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m} \\right]\\\\\n&\\,\\,\\,\\,\\,-\\frac{\\pi}{2}\\frac{\\xi_{SO}}{k_{F}^{2}}v_{i}^{3}n_{i}(D^{\\uparrow}+D^{\\downarrow})\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}\\left[\\hat{\\sigma}_{0}+\\delta\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m} \\right]\\hat{g}^{K}_{\\boldsymbol{k'}E}(\\boldsymbol{R},T)(\\boldsymbol{k}\\times\\boldsymbol{k'})\\cdot\\hat{\\boldsymbol{\\sigma}}.\n\\end{aligned}\n\\end{equation}\n\\noindent In the weak exchange coupling limit $(J\\ll\\varepsilon_{F})$, we can express $D^{\\uparrow(\\downarrow)}\\approx D_{0}(1\\mp\\beta)$, where $\\beta=\\frac{J}{2\\varepsilon_{F}}$ is the spin polarization factor and $D_{0}=\\frac{mk_{F}}{2\\pi^{2}\\hbar^{2}}$ is the spin independent density of states per spin at the Fermi level. Then, we obtain:\n\\begin{equation}\n\\begin{aligned}\n\\hat{\\Sigma'}^{K}_{\\boldsymbol{k}E}(\\boldsymbol{R},T)&=-\\pi v_{i}^{3}n_{i}D_{0}\\frac{\\xi_{SO}}{k_{F}^{2}}\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}(\\boldsymbol{k}\\times\\boldsymbol{k'})\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{g}^{K}_{\\boldsymbol{k'}E}(\\boldsymbol{R},T)\\left[\\hat{\\sigma}_{0}-\\beta\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m} \\right]\\\\\n&\\,\\,\\,\\,\\,-\\pi v_{i}^{3}n_{i}D_{0}\\frac{\\xi_{SO}}{k_{F}^{2}}\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}\\left[\\hat{\\sigma}_{0}-\\beta\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m} \\right]\\hat{g}^{K}_{\\boldsymbol{k'}E}(\\boldsymbol{R},T)(\\boldsymbol{k}\\times\\boldsymbol{k'})\\cdot\\hat{\\boldsymbol{\\sigma}}\n\\end{aligned}\n\\end{equation}\n\\noindent or \n\\begin{equation}\n\\begin{aligned}\n\\hat{\\Sigma'}^{K}_{\\boldsymbol{k}E}(\\boldsymbol{R},T)&=-\\frac{\\hbar}{2}\\frac{v_{i}}{\\tau_{0}}\\frac{\\xi_{SO}}{k_{F}^{2}}\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}(\\boldsymbol{k}\\times\\boldsymbol{k'})\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{g}^{K}_{\\boldsymbol{k'}E}(\\boldsymbol{R},T)\\left[\\hat{\\sigma}_{0}-\\beta\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m} \\right]\\\\\n&\\,\\,\\,\\,\\,-\\frac{\\hbar}{2}\\frac{v_{i}}{\\tau_{0}}\\frac{\\xi_{SO}}{k_{F}^{2}}\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}\\left[\\hat{\\sigma}_{0}-\\beta\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m} \\right]\\hat{g}^{K}_{\\boldsymbol{k'}E}(\\boldsymbol{R},T)(\\boldsymbol{k}\\times\\boldsymbol{k'})\\cdot\\hat{\\boldsymbol{\\sigma}},\n\\end{aligned}\n\\label{eq:sskew}\n\\end{equation}\n\\noindent where $\\frac{1}{\\tau_{0}}=2\\pi v_{i}^{2}n_{i}D_{0}\/\\hbar$ is the spin independent relaxation time.\n\n\n\n\\section{Relaxation time}\n\\par The imaginary part of the retarded and advanced self-energies is related to the momentum relaxation time, which is given by the elastic scattering off the on-site impurity potential and Elliot-Yafet mechanism:\n\\begin{equation}\n\\hat{\\Sigma}^{R(A)}_{\\boldsymbol{k}E}=\\mp i\\frac{\\hbar}{2\\hat{\\tau}_{\\boldsymbol{k}}}.\n\\end{equation}\n\\noindent Taking into account Eqs. (\\ref{eq:sret}) and (\\ref{eq:gzero}), we get:\n\\begin{equation}\n\\frac{1}{\\hat{\\tau}_{\\boldsymbol{k}}}=\\frac{\\pi v_{i}^{2}n_{i}}{\\hbar}\\sum_{s=\\pm}\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}\\,\\left[\\hat{\\sigma}_{0}+i\\frac{\\xi_{SO}}{k_{F}^{2}}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\right]\\left(\\hat{\\sigma}_{0}+s\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m} \\right)\\left[\\hat{\\sigma}_{0}-i\\frac{\\xi_{SO}}{k_{F}^{2}}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\right]\\delta(E-E_{\\boldsymbol{k'}s}),\n\\end{equation}\n\\noindent where $\\boldsymbol{n}=\\boldsymbol{k}\\times\\boldsymbol{k'}$. In terms of spherical coordinates, $d\\boldsymbol{k}=k^{2}dk\\,\\sin{\\theta}d\\theta\\,d\\phi$ with $k\\in[0,k_{F}]$, $\\theta\\in[0,\\pi]$ and $\\phi\\in[0,2\\pi]$, it is easy to show:\n\\begin{equation}\n\\int k_{i}d\\boldsymbol{k}=0,\n\\end{equation}\n\\noindent where $k_{i}$ is the $i$th cartesian coordinate of $\\boldsymbol{k}$, and all terms linear in $\\boldsymbol{n}$ vanish. Thus, we get:\n\\begin{equation}\n\\frac{1}{\\hat{\\tau}_{\\boldsymbol{k}}}=\\frac{\\pi v_{i}^{2}n_{i}}{\\hbar}\\sum_{s=\\pm}\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}\\,\\left[\\hat{\\sigma}_{0}+s\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}+\\frac{\\xi_{SO}^{2}}{k_{F}^{4}}(\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}})(\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}})+s\\,\\frac{\\xi_{SO}^{2}}{k_{F}^{4}}(\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}})(\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m})(\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}) \\right]\\delta(E-E_{\\boldsymbol{k'}s}),\n\\end{equation}\n\\noindent or having used $(\\boldsymbol{a}\\cdot\\hat{\\boldsymbol{\\sigma}})(\\boldsymbol{b}\\cdot\\hat{\\boldsymbol{\\sigma}})=(\\boldsymbol{a}\\cdot\\boldsymbol{b})\\hat{\\sigma}_{0}+i(\\boldsymbol{a}\\times\\boldsymbol{b})\\cdot\\hat{\\boldsymbol{\\sigma}}$:\n\\begin{equation}\n\\begin{aligned}\n\\frac{1}{\\hat{\\tau}_{\\boldsymbol{k}}}&=\\frac{\\pi v_{i}^{2}n_{i}}{\\hbar}\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}\\,\\left[(1+\\frac{\\xi_{SO}^{2}}{k_{F}^{4}}\\boldsymbol{n}^{2})\\hat{\\sigma}_{0}(\\delta(E-E_{\\boldsymbol{k'}+})+\\delta(E-E_{\\boldsymbol{k'}-}))\\right.\\\\\n&\\left.+\\left(\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}+\\frac{\\xi_{SO}^{2}}{k_{F}^{4}}\\hat{\\boldsymbol{\\sigma}}\\cdot(2\\boldsymbol{n}(\\boldsymbol{n}\\cdot\\boldsymbol{m})-\\boldsymbol{m}\\boldsymbol{n}^{2})\\right)(\\delta(E-E_{\\boldsymbol{k'}+})-\\delta(E-E_{\\boldsymbol{k'}-}))\\right].\n\\label{eq:tau2}\n\\end{aligned}\n\\end{equation}\n\\noindent Let us also consider the following expressions:\n\\begin{equation}\n\\begin{aligned}\n\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}k'_{i}k'_{j}\\,(\\delta(E-E_{\\boldsymbol{k'}+})\\pm\\delta(E-E_{\\boldsymbol{k'}-}))=0\\qquad\\mathrm{for}\\,\\,\\, i\\ne j,\\\\\n\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}{k'}_{i}^{2}\\,(\\delta(E-E_{\\boldsymbol{k'}+})\\pm\\delta(E-E_{\\boldsymbol{k'}-}))=\\frac{1}{6\\pi^{2}}\\int d{k'}{k'}^{4}\\,(\\delta(E-E_{\\boldsymbol{k'}+})\\pm\\delta(E-E_{\\boldsymbol{k'}-})),\\\\\n\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}\\boldsymbol{n}^{2}\\,(\\delta(E-E_{\\boldsymbol{k'}+})\\pm\\delta(E-E_{\\boldsymbol{k'}-}))=\\frac{1}{3\\pi^{2}}\\,k^{2}\\int d{k'}{k'}^{4}\\,(\\delta(E-E_{\\boldsymbol{k'}+})\\pm\\delta(E-E_{\\boldsymbol{k'}-})).\n\\end{aligned}\n\\end{equation}\n\\noindent We can rewrite Eq. (\\ref{eq:tau2}) as:\n\\begin{equation}\n\\begin{aligned}\n\\frac{1}{\\hat{\\tau}_{\\boldsymbol{k}}}&=\\frac{v_{i}^{2}n_{i}}{2\\pi\\hbar}\\int dk'\\,\\left[({k'}^{2}+\\frac{2}{3}\\frac{\\xi_{SO}^{2}}{k_{F}^{4}}k^{2}{k'}^{4})\\hat{\\sigma}_{0}(\\delta(E-E_{\\boldsymbol{k'}+})+\\delta(E-E_{\\boldsymbol{k'}-}))\\right.\\\\\n&\\left.+\\left(\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}{k'}^{2}-\\frac{2}{3}\\frac{\\xi_{SO}^{2}}{k_{F}^{4}}{k'}^{4}(\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{k})(\\boldsymbol{k}\\cdot\\boldsymbol{m})\\right)(\\delta(E-E_{\\boldsymbol{k'}+})-\\delta(E-E_{\\boldsymbol{k'}-}))\\right].\n\\end{aligned}\n\\end{equation}\n\\noindent Next, we can employ the following relation for the delta-function:\n\\begin{equation}\n\\delta(E-E_{\\boldsymbol{k}s})=\\frac{\\delta(k-k_{s})}{\\frac{\\hbar^{2}k_{s}}{m}},\n\\end{equation}\n\\noindent where $k_{\\pm}=\\sqrt{2m(\\varepsilon_{F}\\mp J)}\/\\hbar$, and $\\varepsilon_{F}=\\frac{\\hbar^{2}k_{F}^{2}}{2m}$ is the Fermi energy. Then, integrating over $k'$ gives:\n\\begin{equation}\n\\begin{aligned}\n\\frac{1}{\\hat{\\tau}_{\\boldsymbol{k}}}=\\frac{v_{i}^{2}n_{i}}{2\\pi\\hbar}\\Big(\\frac{m}{\\hbar^{2}}(k_{+}+k_{-})\\hat{\\sigma}_{0}&+\\frac{2m}{3\\hbar^{2}}\\frac{\\xi_{SO}^{2}}{k_{F}^{4}}k^{2}(k^{3}_{+}+k^{3}_{-})\\hat{\\sigma}_{0} +\\frac{m}{\\hbar^{2}}(k_{+}-k_{-}) \\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}\\\\\n&-\\frac{2m}{3\\hbar^{2}}\\frac{\\xi_{SO}^{2}}{k_{F}^{4}}(k^{3}_{+}-k^{3}_{-})(\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{k})(\\boldsymbol{k}\\cdot\\boldsymbol{m}) \\Big).\n\\end{aligned}\n\\end{equation}\n\\noindent Finally, in the weak exchange coupling limit $(J\\ll\\varepsilon_{F})$, we can perform a Taylor expansion, $k_{\\pm}\\approx k_{F}(1\\mp\\beta)$. Neglecting higher order terms $\\sim\\beta\\xi_{SO}^{2}$, we obtain:\n\\begin{equation}\n\\frac{1}{\\hat{\\tau}_{\\boldsymbol{k}}}=\\frac{1}{\\tau_{0}}\\left(\\hat{\\sigma_{0}}-\\beta\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}+\\frac{2}{3}\\frac{\\xi_{SO}^{2}}{k_{F}^{2}}k^{2}\\hat{\\sigma}_{0} \\right),\n\\end{equation}\n\\noindent where $\\frac{1}{\\tau_{0}}=2\\pi v^{2}_{i}n_{i}D_{0}\/\\hbar$ is the spin-independent relaxation time due to scattering off the impurity potential.\n\n\n\n\n\n\\section{Averaged velocity operator}\n\\par For educational purposes, let us derive the averaged velocity operator in diffusive ferromagnets with extrinsic spin-orbit coupling.\\cite{velo} Within the Lippmann-Schwinger equation, the scattered state $\\parallel\\!\\boldsymbol{k},s\\rangle$ can be written in the first order of $\\hat{\\mathcal{H}}_{\\mathrm{imp}}$:\n\\begin{equation}\n\\begin{aligned}\n\\parallel\\!\\boldsymbol{k},s\\rangle&=|\\boldsymbol{k},s\\rangle+\\sum_{\\boldsymbol{k'}}\\hat{G}^{R}_{0,\\boldsymbol{k'}}\\langle\\boldsymbol{k'}|\\hat{\\mathcal{H}}_{\\mathrm{imp}}|\\boldsymbol{k}\\rangle|\\boldsymbol{k'},s\\rangle\\\\\n&=|\\boldsymbol{k},s\\rangle-\\frac{1}{\\Omega}\\frac{i\\pi}{2}\\sum_{\\boldsymbol{k'}s'}(\\hat{\\sigma}_{0}+s'\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m})(\\hat{\\sigma\n_{0}}-i\\frac{\\xi_{SO}}{k_{F}^{2}}\\hat{\\boldsymbol{\\sigma}}\\cdot(\\boldsymbol{k}\\times\\boldsymbol{k'}))V(\\boldsymbol{k'}-\\boldsymbol{k})\\delta(E-E_{\\boldsymbol{k'}s'})|\\boldsymbol{k'},s\\rangle,\n\\end{aligned}\n\\end{equation}\n\\noindent and \n\\begin{equation}\n\\begin{aligned}\n\\langle\\boldsymbol{k},s\\!\\parallel&=\\langle\\boldsymbol{k},s|+\\sum_{\\boldsymbol{k'}}\\langle\\boldsymbol{k'},s|\\langle\\boldsymbol{k}|\\hat{\\mathcal{H}}_{\\mathrm{imp}}|\\boldsymbol{k'}\\rangle\\hat{G}^{R}_{0,\\boldsymbol{k'}}\\\\\n&=\\langle\\boldsymbol{k},s|+\\frac{1}{\\Omega}\\frac{i\\pi}{2}\\sum_{\\boldsymbol{k'}s'}\\langle\\boldsymbol{k'},s| (\\hat{\\sigma\n_{0}}+i\\frac{\\xi_{SO}}{k_{F}^{2}}\\hat{\\boldsymbol{\\sigma}}\\cdot(\\boldsymbol{k}\\times\\boldsymbol{k'}))(\\hat{\\sigma}_{0}+s'\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m})V(\\boldsymbol{k}-\\boldsymbol{k'})\\delta(E-E_{\\boldsymbol{k'}s'}).\n\\end{aligned}\n\\end{equation}\n\\noindent The corresponding matrix elements of the velocity operator  can be found as:\n\\begin{equation}\n\\begin{aligned}\n\\boldsymbol{v}_{\\boldsymbol{k}\\boldsymbol{k'}}^{ss'}=-\\frac{i}{\\hbar}\\langle\\boldsymbol{k},s\\!\\parallel[\\hat{\\boldsymbol{r}},\\hat{\\mathcal{H}}]\\parallel\\!\\boldsymbol{k'},s'\\rangle=-\\frac{i}{\\hbar}\\langle\\boldsymbol{k},s\\!\\parallel[\\hat{\\boldsymbol{r}},\\hat{\\mathcal{H}}_{0}+\\hat{\\mathcal{H}}_{\\mathrm{imp}}]\\parallel\\!\\boldsymbol{k'},s'\\rangle,\n\\end{aligned}\n\\end{equation}\n\\noindent where\n\\begin{equation}\n-\\frac{i}{\\hbar}[\\hat{\\boldsymbol{r}},\\hat{\\mathcal{H}}]=-\\frac{i}{\\hbar}[\\hat{\\boldsymbol{r}},\\hat{\\mathcal{H}}_{0}+\\hat{\\mathcal{H}}_{\\mathrm{imp}}]=-\\frac{i\\hbar}{m}\\nabla\\hat{\\sigma}_{0}+\\frac{\\xi_{SO}}{\\hbar k_{F}^{2}}\\sum_{\\boldsymbol{R}_{i}}\\hat{\\boldsymbol{\\sigma}}\\times\\nabla V(\\boldsymbol{r}-\\boldsymbol{R}_{i}).\n\\end{equation}\n\\noindent Let us express these terms separately neglecting higher order terms $\\sim\\xi_{SO}^{2}$:\n\\begin{equation}\n\\begin{aligned}\n&-\\frac{i}{\\hbar}\\langle\\boldsymbol{k},s\\!\\parallel[\\hat{\\boldsymbol{r}},\\hat{\\mathcal{H}}_{0}]\\parallel\\!\\boldsymbol{k'},s'\\rangle=-\\frac{i\\hbar}{m}\\langle\\boldsymbol{k},s\\!\\parallel\\nabla\\parallel\\!\\boldsymbol{k'},s'\\rangle\\\\\n&=-\\frac{i\\hbar}{m}\\langle\\boldsymbol{k}|\\nabla|\\boldsymbol{k'}\\rangle\\delta_{ss'} - \\frac{i\\pi}{2}\\sum_{s''}\\int\\frac{d\\boldsymbol{k''}}{(2\\pi)^{3}}V(\\boldsymbol{k''}-\\boldsymbol{k'})\\delta(E-E_{\\boldsymbol{k''}s''})\\langle\\boldsymbol{k}|\\nabla|\\boldsymbol{k''}\\rangle\\langle s|(\\hat{\\sigma}_{0}+s''\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m})(\\hat{\\sigma\n_{0}}-i\\frac{\\xi_{SO}}{k_{F}^{2}}\\hat{\\boldsymbol{\\sigma}}\\cdot(\\boldsymbol{k'}\\times\\boldsymbol{k''}))|s'\\rangle\\\\\n&\\qquad\\qquad\\qquad\\qquad\\,\\,\\,\\,\\,\\, + \\frac{i\\pi}{2}\\sum_{s''}\\int\\frac{d\\boldsymbol{k''}}{(2\\pi)^{3}}V(\\boldsymbol{k}-\\boldsymbol{k''})\\delta(E-E_{\\boldsymbol{k''}s''})\\langle s|(\\hat{\\sigma\n_{0}}+i\\frac{\\xi_{SO}}{k_{F}^{2}}\\hat{\\boldsymbol{\\sigma}}\\cdot(\\boldsymbol{k}\\times\\boldsymbol{k''}))(\\hat{\\sigma}_{0}+s''\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m})|s'\\rangle\\langle\\boldsymbol{k''}|\\nabla|\\boldsymbol{k'}\\rangle,\n\\end{aligned}\n\\end{equation}\n\\noindent and\n\\begin{equation}\n\\begin{aligned}\n&-\\frac{i}{\\hbar}\\langle\\boldsymbol{k},s\\!\\parallel[\\hat{\\boldsymbol{r}},\\hat{\\mathcal{H}}_{\\mathrm{imp}}]\\parallel\\!\\boldsymbol{k'},s'\\rangle=\\frac{\\xi_{SO}}{\\hbar k_{F}^{2}}\\sum_{\\boldsymbol{R}_{i}}\\langle\\boldsymbol{k},s\\!\\parallel\\hat{\n\\boldsymbol{\\sigma}}\\times\\nabla V(\\boldsymbol{r}-\\boldsymbol{R}_{i})\\parallel\\!\\boldsymbol{k'},s'\\rangle\\\\\n&=\\frac{i}{\\Omega}\\frac{\\xi_{SO}}{\\hbar k_{F}^{2}}\\langle s|\\hat{\\boldsymbol{\\sigma}}|s'\\rangle\\times(\\boldsymbol{k}-\\boldsymbol{k'})V(\\boldsymbol{k}-\\boldsymbol{k'})\\\\\n& +\\frac{1}{\\Omega} \\frac{\\pi}{2}\\frac{\\xi_{SO}}{\\hbar k_{F}^{2}}\\sum_{s''}\\int\\frac{d\\boldsymbol{k''}}{(2\\pi)^{3}}V(\\boldsymbol{k''}-\\boldsymbol{k'})V(\\boldsymbol{k}-\\boldsymbol{k''})\\delta(E-E_{\\boldsymbol{k''}s''})\\langle s|\\hat{\n\\boldsymbol{\\sigma}}\\times(\\boldsymbol{k}-\\boldsymbol{k''})(\\hat{\\sigma}_{0}+s''\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m})|s'\\rangle\\\\\n& -\\frac{1}{\\Omega} \\frac{\\pi}{2}\\frac{\\xi_{SO}}{\\hbar k_{F}^{2}}\\sum_{s''}\\int\\frac{d\\boldsymbol{k''}}{(2\\pi)^{3}}V(\\boldsymbol{k}-\\boldsymbol{k''})V(\\boldsymbol{k''}-\\boldsymbol{k'})\\delta(E-E_{\\boldsymbol{k''}s''})\\langle s|(\\hat{\\sigma}_{0}+s''\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m})\\hat{\n\\boldsymbol{\\sigma}}\\times(\\boldsymbol{k''}-\\boldsymbol{k'})| s'\\rangle.\n\\end{aligned}\n\\end{equation}\n\\noindent Upon impurity averaging we obtain:\n\\begin{equation}\n\\begin{aligned}\n \\boldsymbol{v}_{\\boldsymbol{k}}^{ss'}&=\\frac{\\hbar}{m}\\boldsymbol{k}\\,\\delta_{ss'}+v_{i}^{2}n_{i}\\frac{\\pi}{2}\\frac{\\xi_{SO}}{\\hbar k_{F}^{2}}\\sum_{s''}\\int\\frac{d\\boldsymbol{k''}}{(2\\pi)^{3}}\\delta(E-E_{\\boldsymbol{k''}s''})\\langle s|\\left\\{\\hat{\n\\boldsymbol{\\sigma}}\\times(\\boldsymbol{k}-\\boldsymbol{k''}),(\\hat{\\sigma}_{0}+s''\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m})\\right\\}|s'\\rangle\\\\\n&=\\frac{\\hbar}{m}\\boldsymbol{k}\\,\\delta_{ss'}+v_{i}^{2}n_{i}\\pi\\frac{\\xi_{SO}}{\\hbar k_{F}^{2}}\\sum_{s''}\\int\\frac{d\\boldsymbol{k''}}{(2\\pi)^{3}}\\delta(E-E_{\\boldsymbol{k''}s''})\\langle s|\\hat{\\boldsymbol{\\sigma}}\\times\\boldsymbol{k}+s''\\boldsymbol{m}\\times\\boldsymbol{k}|s'\\rangle,\n\\end{aligned}\n\\end{equation}\n\\noindent or in the limit $J\\ll\\varepsilon_{F}$:\n\\begin{equation}\n\\hat{\\boldsymbol{v}}_{\\boldsymbol{k}}=\\frac{\\hbar}{m}\\boldsymbol{k}\\,\\hat{\\sigma}_{0}+\\frac{1}{\\tau_{0}}\\frac{\\xi_{SO}}{k_{F}^{2}}\\hat{\\boldsymbol{\\sigma}}\\times\\boldsymbol{k}-\\frac{\\beta}{\\tau_{0}}\\frac{\\xi_{SO}}{k_{F}^{2}}\\boldsymbol{m}\\times\\boldsymbol{k}\\,\\hat{\\sigma}_{0}.\n\\label{eq:av}\n\\end{equation}\n\n\n\n\n\n\n\n\n\\section{Quantum transport equations}\n\\par Having integrated Eq.~(\\ref{eq:keldysh}) over energy, we arrive at the kinetic equation written for the distribution function $\\hat{g}_{\\boldsymbol{k}}$:\n\\begin{equation}\n-i[\\hat{g}_{\\boldsymbol{k}},J\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}]+\\frac{\\hbar^{2}}{m}(\\boldsymbol{k}\\cdot\\nabla_{\\boldsymbol{R}})\\,\\hat{g}_{\\boldsymbol{k}}=coll,\n\\label{eq:keld2}\n\\end{equation}\n\\noindent where the collision integral is defined as:\n\\begin{equation}\ncoll=\\mathcal{J}_{\\boldsymbol{k}}+\\mathcal{I}_{\\boldsymbol{k}},\n\\end{equation}\n\\noindent and\n\\begin{equation}\n\\mathcal{J}_{\\boldsymbol{k}}=\\int\\frac{dE}{2\\pi}\\left(\\hat{\\Sigma}^{K}_{\\boldsymbol{k}E}\\hat{G}^{A}_{\\boldsymbol{k}E}-\\hat{G}^{R}_{\\boldsymbol{k}E}\\hat{\\Sigma}^{K}_{\\boldsymbol{k}E} \\right),\n\\end{equation} \n\\begin{equation}\n\\mathcal{I}_{\\boldsymbol{k}}=\\int\\frac{dE}{2\\pi}\\left(\\hat{\\Sigma}^{R}_{\\boldsymbol{k}E}\\hat{g}^{K}_{\\boldsymbol{k}E}-\\hat{g}^{K}_{\\boldsymbol{k}E}\\hat{\\Sigma}^{A}_{\\boldsymbol{k}E} \\right).\n\\end{equation}\n\\noindent Let us proceed with its detailed derivation. Taking into account the Kadanoff-Baym anzats~(\\ref{eq:anzats}) for $\\hat{g}^{K}_{\\boldsymbol{k}E}$, the integration over energy (up to a given Fermi level $\\varepsilon_{F}$) can be performed by using the residue theorem:\n\\begin{equation}\n\\begin{aligned}\n\\int\\frac{dE}{2\\pi}\\hat{G}^{R}_{\\boldsymbol{k}E}\\hat{g}_{\\boldsymbol{k'}}\\hat{G}^{A}_{\\boldsymbol{k'}E}&=\\int\\frac{dE}{8\\pi}\\sum\\limits_{s,s'}\\frac{\\hat{\\sigma}_{0}+s\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}}{E-E_{\\boldsymbol{k}s}+i\\frac{\\hbar}{2\\tau_{s}}}\\hat{g}_{\\boldsymbol{k'}}\\frac{\\hat{\\sigma}_{0}+s'\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}}{E-E_{\\boldsymbol{k'}s'}-i\\frac{\\hbar}{2\\tau_{s'}}}\\\\\n&=-\\frac{i}{4}\\sum_{s,s'}\\frac{(\\hat{\\sigma}_{0}+s\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m})\\hat{g}_{\\boldsymbol{k'}}(\\hat{\\sigma}_{0}+s'\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m})}{\\varepsilon_{F}-E_{\\boldsymbol{k'}s'}-i\\frac{\\hbar}{2}\\left(\\frac{1}{\\tau_{s'}}+\\frac{1}{\\tau_{s}}\\right)},\n\\end{aligned}\n\\end{equation}\n\\noindent and\n\\begin{equation}\n\\begin{aligned}\n\\int\\frac{dE}{2\\pi}\\hat{G}^{R}_{\\boldsymbol{k'}E}\\hat{g}_{\\boldsymbol{k'}}\\hat{G}^{A}_{\\boldsymbol{k}E}&=\\int\\frac{dE}{8\\pi}\\sum\\limits_{s,s'}\\frac{\\hat{\\sigma}_{0}+s'\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}}{E-E_{\\boldsymbol{k'}s'}+i\\frac{\\hbar}{2\\tau_{s'}}}\\hat{g}_{\\boldsymbol{k'}}\\frac{\\hat{\\sigma}_{0}+s\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}}{E-E_{\\boldsymbol{k}s}-i\\frac{\\hbar}{2\\tau_{s}}}\\\\\n&=\\frac{i}{4}\\sum_{s,s'}\\frac{(\\hat{\\sigma}_{0}+s'\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m})\\hat{g}_{\\boldsymbol{k'}}(\\hat{\\sigma}_{0}+s\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m})}{\\varepsilon_{F}-E_{\\boldsymbol{k'}s'}+i\\frac{\\hbar}{2}\\left(\\frac{1}{\\tau_{s}}+\\frac{1}{\\tau_{s'}}\\right)},\n\\end{aligned}\n\\end{equation}\n\\noindent while \n\\begin{equation}\n\\int\\frac{dE}{2\\pi}\\hat{G}^{R(A)}_{\\boldsymbol{k}E}\\hat{g}_{\\boldsymbol{k'}}\\hat{G}^{R(A)}_{\\boldsymbol{k'}E}=0,\n\\end{equation}\n\\noindent where the retarded and advanced Green's functions are defined as:\n\\begin{equation}\n\\hat{G}_{\\boldsymbol{k}E}^{R(A)}=\\frac{1}{2}\\sum\\limits_{s=\\pm}\\frac{\\hat{\\sigma}_{0}+s\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}}{E-E_{\\boldsymbol{k}s}\\pm i\\frac{\\hbar}{2\\tau_{s}}}.\n\\end{equation}\n\\noindent Assuming the scattering term in the denominator to be small and transport properties to be described solely by the electrons close to the Fermi level, we can rewrite these expressions with the Sokhotski formula:\n\\begin{equation}\n\\begin{aligned}\n\\int\\frac{dE}{2\\pi}\\hat{G}^{R}_{\\boldsymbol{k}E}\\hat{g}_{\\boldsymbol{k'}}\\hat{G}^{A}_{\\boldsymbol{k'}E}&=\\frac{\\pi}{2}\\sum_{s'}\\hat{g}_{\\boldsymbol{k'}}(\\hat{\\sigma}_{0}+s'\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m})\\delta(\\varepsilon_{F}-E_{\\boldsymbol{k'}s'}),\n\\end{aligned}\n\\end{equation}\n\\noindent and\n\\begin{equation}\n\\begin{aligned}\n\\int\\frac{dE}{2\\pi}\\hat{G}^{R}_{\\boldsymbol{k'}E}\\hat{g}_{\\boldsymbol{k'}}\\hat{G}^{A}_{\\boldsymbol{k}E}=\\frac{\\pi}{2}\\sum_{s'}(\\hat{\\sigma}_{0}+s'\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m})\\hat{g}_{\\boldsymbol{k'}}\\delta(\\varepsilon_{F}-E_{\\boldsymbol{k'}s'}).\n\\end{aligned}\n\\end{equation}\n\\par Starting from the Born approximation~(\\ref{eq:skel}), we have:\n\\begin{equation}\n\\begin{aligned}\n\\hat{\\Sigma}^{K}_{\\boldsymbol{k}E}\\hat{G}^{A}_{\\boldsymbol{k}E}&=v_{i}^{2}n_{i}\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}\\,\\left[\\hat{G}^{R}_{\\boldsymbol{k'}E}\\hat{g}_{\\boldsymbol{k'}}\\hat{G}^{A}_{\\boldsymbol{k}E}+i\\frac{\\xi_{SO}}{k^{2}_{F}}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{G}^{R}_{\\boldsymbol{k'}E}\\hat{g}_{\\boldsymbol{k'}}\\hat{G}^{A}_{\\boldsymbol{k}E}  \\right.\\\\\n&\\left.-i\\frac{\\xi_{SO}}{k_{F}^{2}}\\hat{G}^{R}_{\\boldsymbol{k'}E}\\hat{g}_{\\boldsymbol{k'}}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{G}^{A}_{\\boldsymbol{k}E}+\\frac{\\xi^{2}_{SO}}{k_{F}^{4}}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{G}^{R}_{\\boldsymbol{k'}E}\\hat{g}_{\\boldsymbol{k'}}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{G}^{A}_{\\boldsymbol{k}E}\\right]\\\\\n&+\\frac{v_{i}^{2}n_{i}}{2}\\frac{\\xi_{SO}}{k_{F}^{2}}\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}\\left\\{(\\boldsymbol{k'}-\\boldsymbol{k}\\}\\times\\hat{\\boldsymbol{\\sigma}},\\hat{G}^{R}_{\\boldsymbol{k'}E}\\nabla_{\\boldsymbol{R}}\\hat{g}_{\\boldsymbol{k'}}\\right\\}\\hat{G}^{A}_{\\boldsymbol{k}E},\n\\end{aligned}\n\\end{equation}\n\\noindent and \n\\begin{equation}\n\\begin{aligned}\n\\hat{G}^{R}_{\\boldsymbol{k}E}\\hat{\\Sigma}^{K}_{\\boldsymbol{k}E}&=-v_{i}^{2}n_{i}\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}\\,\\left[\\hat{G}^{R}_{\\boldsymbol{k}E}\\hat{g}_{\\boldsymbol{k'}}\\hat{G}^{A}_{\\boldsymbol{k'}E}+i\\frac{\\xi_{SO}}{k_{F}^{2}}\\hat{G}^{R}_{\\boldsymbol{k}E}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{g}_{\\boldsymbol{k'}}\\hat{G}^{A}_{\\boldsymbol{k'}E}  \\right.\\\\\n&\\left.-i\\frac{\\xi_{SO}}{k_{F}^{2}}\\hat{G}^{R}_{\\boldsymbol{k}E}\\hat{g}_{\\boldsymbol{k'}}\\hat{G}^{A}_{\\boldsymbol{k'}E}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}+\\frac{\\xi^{2}_{SO}}{k_{F}^{4}}\\hat{G}^{R}_{\\boldsymbol{k}E}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{g}_{\\boldsymbol{k'}}\\hat{G}^{A}_{\\boldsymbol{k'}E}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\right]\\\\\n&-\\frac{v_{i}^{2}n_{i}}{2}\\frac{\\xi_{SO}}{k_{F}^{2}}\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}\\hat{G}^{R}_{\\boldsymbol{k}E}\\left\\{(\\boldsymbol{k'}-\\boldsymbol{k})\\times\\hat{\\boldsymbol{\\sigma}},\\nabla_{\\boldsymbol{R}}\\hat{g}_{\\boldsymbol{k'}}\\hat{G}^{A}_{\\boldsymbol{k'}E}\\right\\}.\n\\end{aligned}\n\\end{equation}\n\\noindent By using the following relations: \n\\begin{equation}\n\\begin{aligned}\n\\hat{\\sigma}_{a}\\hat{\\sigma}_{b}&=i\\varepsilon_{abc}\\,\\hat{\\sigma}_{c}+\\delta_{ab}\\hat{\\sigma}_{0},\\\\\n(\\boldsymbol{a}\\cdot\\hat{\\boldsymbol{\\sigma}})(\\boldsymbol{b}\\cdot\\hat{\\boldsymbol{\\sigma}})&=(\\boldsymbol{a}\\cdot\\boldsymbol{b})\\hat{\\sigma}_{0}+i(\\boldsymbol{a}\\times\\boldsymbol{b})\\cdot\\hat{\\boldsymbol{\\sigma}},\n\\end{aligned}\n\\end{equation}\n\\noindent these terms give:\n\\begin{equation}\n\\begin{aligned}\n\\int\\frac{dE}{2\\pi}\\left[\\hat{G}^{R}_{\\boldsymbol{k'}E}\\hat{g}_{\\boldsymbol{k'}}\\hat{G}^{A}_{\\boldsymbol{k}E}+\\hat{G}^{R}_{\\boldsymbol{k}E}\\hat{g}_{\\boldsymbol{k'}}\\hat{G}^{A}_{\\boldsymbol{k'}E}\\right]&=\\pi\\hat{g}_{\\boldsymbol{k'}}\\delta_{T}+\\pi\\{\\hat{g}_{\\boldsymbol{k'}},\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}\\}\\delta_{J},\n\\end{aligned}\n\\end{equation}\n\\begin{equation}\n\\begin{aligned}\n&\\int\\frac{dE}{2\\pi} i\\frac{\\xi_{SO}}{k_{F}^{2}}\\left[[\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}},\\hat{G}^{R}_{\\boldsymbol{k'}E}\\hat{g}_{\\boldsymbol{k'}}]\\hat{G}^{A}_{\\boldsymbol{k}E}+\\hat{G}^{R}_{\\boldsymbol{k}E}[\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}},\\hat{g}_{\\boldsymbol{k'}}\\hat{G}^{A}_{\\boldsymbol{k'}E}]\\right]=\\\\\n&=i\\pi\\frac{\\xi_{SO}}{k_{F}^{2}}[\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}},\\hat{g}_{\\boldsymbol{k'}}]\\delta_{T}+i\\pi\\frac{\\xi_{SO}}{k_{F}^{2}}\\left(\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{g}_{\\boldsymbol{k'}}\\boldsymbol{m}\\cdot\\hat{\\boldsymbol{\\sigma}}-\\boldsymbol{m}\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{g}_{\\boldsymbol{k'}}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}  \\right)\\delta_{J}\\\\\n&-\\pi\\frac{\\xi_{SO}}{k_{F}^{2}}\\{(\\boldsymbol{n}\\times\\boldsymbol{m})\\cdot\\hat{\\boldsymbol{\\sigma}},\\hat{g}_{\\boldsymbol{k'}}\\}\\delta_{J},\n\\end{aligned}\n\\end{equation}\n\\begin{equation}\n\\begin{aligned}\n&\\int\\frac{dE}{2\\pi} \\frac{\\xi^{2}_{SO}}{k_{F}^{4}}\\left[\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{G}^{R}_{\\boldsymbol{k'}E}\\hat{g}_{\\boldsymbol{k'}}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{G}^{A}_{\\boldsymbol{k}E}+\\hat{G}^{R}_{\\boldsymbol{k}E}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{g}_{\\boldsymbol{k'}}\\hat{G}^{A}_{\\boldsymbol{k'}E}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\right]=\\\\\n&=\\pi\\frac{\\xi^{2}_{SO}}{k_{F}^{4}}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{g}_{\\boldsymbol{k'}}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\delta_{T}\n+\\pi\\frac{\\xi^{2}_{SO}}{k_{F}^{4}}\\boldsymbol{n}\\cdot\\boldsymbol{m}\\{\\hat{g}_{\\boldsymbol{k'}},\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}} \\}\\delta_{J}\\\\\n&+i\\pi\\frac{\\xi^{2}_{SO}}{k_{F}^{4}}(\\boldsymbol{n}\\times\\boldsymbol{m})\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{g}_{\\boldsymbol{k'}}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\delta_{J}\n-i\\pi\\frac{\\xi^{2}_{SO}}{k_{F}^{4}}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{g}_{\\boldsymbol{k'}}(\\boldsymbol{n}\\times\\boldsymbol{m})\\cdot\\hat{\\boldsymbol{\\sigma}}\\delta_{J},\n\\end{aligned}\n\\end{equation}\n\\begin{equation}\n\\begin{aligned}\n&\\int\\frac{dE}{2\\pi} \\left[\\left\\{(\\boldsymbol{k'}-\\boldsymbol{k}\\}\\times\\hat{\\boldsymbol{\\sigma}},\\hat{G}^{R}_{\\boldsymbol{k'}E}\\nabla_{\\boldsymbol{R}}\\hat{g}_{\\boldsymbol{k'}}\\right\\}\\hat{G}^{A}_{\\boldsymbol{k}E}+\\hat{G}^{R}_{\\boldsymbol{k}E}\\left\\{(\\boldsymbol{k'}-\\boldsymbol{k})\\times\\hat{\\boldsymbol{\\sigma}},\\nabla_{\\boldsymbol{R}}\\hat{g}_{\\boldsymbol{k'}}\\hat{G}^{A}_{\\boldsymbol{k'}E}\\right\\} \\right]=\\\\\n&=\\pi\\left\\{(\\boldsymbol{k'}-\\boldsymbol{k})\\times\\hat{\\boldsymbol{\\sigma}},\\nabla_{\\boldsymbol{R}}\\hat{g}_{\\boldsymbol{k'}}\\right\\}\\delta_{T}\n+\\pi\\left\\{(\\boldsymbol{k'}-\\boldsymbol{k})\\times\\hat{\\boldsymbol{\\sigma}},\\{\\nabla_{\\boldsymbol{R}}\\hat{g}_{\\boldsymbol{k'}},\\boldsymbol{m}\\cdot\\hat{\\boldsymbol{\\sigma}}\\}\\right\\}\\delta_{J},\n\\end{aligned}\n\\end{equation}\n\\noindent where the following notations are used:\n\\begin{equation}\n\\begin{aligned}\n\\delta_{T}&=\\delta(\\varepsilon_{F}-E_{\\boldsymbol{k'}+})+\\delta(\\varepsilon_{F}-E_{\\boldsymbol{k'}-}),\\\\\n\\delta_{J}&=\\frac{1}{2}\\left[\\delta(\\varepsilon_{F}-E_{\\boldsymbol{k'}+})-\\delta(\\varepsilon_{F}-E_{\\boldsymbol{k'}-})\\right].\n\\end{aligned}\n\\end{equation}\n\\noindent For the skew-scattering self-energy Eq.~(\\ref{eq:sskew}), we have:\n\\begin{equation}\n\\begin{aligned}\n\\hat{\\Sigma'}^{K}_{\\boldsymbol{k}E}\\hat{G}^{A}_{\\boldsymbol{k}E}&=-\\frac{\\hbar}{2}\\frac{v_{i}}{\\tau_{0}}\\frac{\\xi_{SO}}{k_{F}^{2}}\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{G}^{R}_{\\boldsymbol{k'}E}\\hat{g}_{\\boldsymbol{k'}}\\left[\\hat{\\sigma}_{0}-\\beta\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m} \\right]\\hat{G}^{A}_{\\boldsymbol{k}E}\\\\\n&-\\frac{\\hbar}{2}\\frac{v_{i}}{\\tau_{0}}\\frac{\\xi_{SO}}{k_{F}^{2}}\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}\\left[\\hat{\\sigma}_{0}-\\beta\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m} \\right]\\hat{G}^{R}_{\\boldsymbol{k'}E}\\hat{g}_{\\boldsymbol{k'}}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{G}^{A}_{\\boldsymbol{k}E},\n\\end{aligned}\n\\end{equation}\n\\noindent and\n\\begin{equation}\n\\begin{aligned}\n\\hat{G}^{R}_{\\boldsymbol{k}E}\\hat{\\Sigma'}^{K}_{\\boldsymbol{k}E}&=\\frac{\\hbar}{2}\\frac{v_{i}}{\\tau_{0}}\\frac{\\xi_{SO}}{k_{F}^{2}}\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}\\hat{G}^{R}_{\\boldsymbol{k}E}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{g}_{\\boldsymbol{k'}}\\hat{G}^{A}_{\\boldsymbol{k'}E}\\left[\\hat{\\sigma}_{0}-\\beta\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m} \\right]\\\\\n&+\\frac{\\hbar}{2}\\frac{v_{i}}{\\tau_{0}}\\frac{\\xi_{SO}}{k_{F}^{2}}\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}\\hat{G}^{R}_{\\boldsymbol{k}E}\\left[\\hat{\\sigma}_{0}-\\beta\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}\\right]\\hat{g}_{\\boldsymbol{k'}}\\hat{G}^{A}_{\\boldsymbol{k'}E}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}},\n\\end{aligned}\n\\end{equation}\n\\noindent that gives:\n\\begin{equation}\n\\begin{aligned}\n&\\int\\frac{dE}{2\\pi}\\,\\left[\\left\\{\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}},\\hat{G}^{R}_{\\boldsymbol{k'}E}\\hat{g}_{\\boldsymbol{k'}}\\right\\}\\hat{G}^{A}_{\\boldsymbol{k}E}+\\hat{G}^{R}_{\\boldsymbol{k}E}\\left\\{\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}},\\hat{g}_{\\boldsymbol{k'}}\\hat{G}^{A}_{\\boldsymbol{k'}E}\\right\\}\\right]=\\\\\n&=\\pi\\{\\hat{g}_{\\boldsymbol{k'}},\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\}\\delta_{T}+\\pi\\left\\{\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}},\\{\\hat{g}_{\\boldsymbol{k'}},\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}\\}\\right\\}\\delta_{J},\n\\end{aligned}\n\\end{equation}\n\\begin{equation}\n\\begin{aligned}\n&\\int\\frac{dE}{2\\pi}\\,\\left[\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{G}^{R}_{\\boldsymbol{k'}E}\\hat{g}_{\\boldsymbol{k'}}\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}\\hat{G}^{A}_{\\boldsymbol{k}E}+\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}\\hat{G}^{R}_{\\boldsymbol{k'}E}\\hat{g}_{\\boldsymbol{k'}}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{G}^{A}_{\\boldsymbol{k}E}\\right.\\\\\n&\\qquad\\qquad\\left.+\\hat{G}^{R}_{\\boldsymbol{k}E}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{g}_{\\boldsymbol{k'}}\\hat{G}^{A}_{\\boldsymbol{k'}E}\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}+\\hat{G}^{R}_{\\boldsymbol{k}E}\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}\\hat{g}_{\\boldsymbol{k'}}\\hat{G}^{A}_{\\boldsymbol{k'}E}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\right]=\\\\\n&=\\pi\\left(\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}\\hat{g}_{\\boldsymbol{k'}}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}+\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{g}_{\\boldsymbol{k'}}\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m} \\right) \\delta_{T}+\\pi\\boldsymbol{n}\\cdot\\boldsymbol{m}\\{\\hat{g}_{\\boldsymbol{k'}},\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m} \\}\\delta_{J}+\\pi\\boldsymbol{m}^{2}\\{\\hat{g}_{\\boldsymbol{k'}},\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}} \\}\\delta_{J}\\\\\n&\\qquad\\qquad+i\\pi\\left((\\boldsymbol{n}\\times\\boldsymbol{m})\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{g}_{\\boldsymbol{k'}}\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}-\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m} \\hat{g}_{\\boldsymbol{k'}} (\\boldsymbol{n}\\times\\boldsymbol{m})\\cdot\\hat{\\boldsymbol{\\sigma}} \\right)\\delta_{J}.\n\\end{aligned}\n\\end{equation}\n\\par Finally, we obtain:\n\\begin{equation}\n\\begin{aligned}\n\\mathcal{J}_{\\boldsymbol{k}}&=\\pi a_{1}\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}\\Big[\\hat{g}_{\\boldsymbol{k'}}\\delta_{T}+\\{\\hat{g}_{\\boldsymbol{k'}},\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}\\}\\delta_{J}\\\\\n&\\qquad\\qquad+i\\frac{\\xi_{SO}}{k_{F}^{2}}[\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}},\\hat{g}_{\\boldsymbol{k'}}]\\delta_{T}+i\\frac{\\xi_{SO}}{k_{F}^{2}}\\left(\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{g}_{\\boldsymbol{k'}}\\boldsymbol{m}\\cdot\\hat{\\boldsymbol{\\sigma}}-\\boldsymbol{m}\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{g}_{\\boldsymbol{k'}}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}  \\right)\\delta_{J}-\\frac{\\xi_{SO}}{k_{F}^{2}}\\{(\\boldsymbol{n}\\times\\boldsymbol{m})\\cdot\\hat{\\boldsymbol{\\sigma}},\\hat{g}_{\\boldsymbol{k'}}\\}\\delta_{J}\\\\\n&\\qquad\\qquad+\\frac{\\xi^{2}_{SO}}{k_{F}^{4}}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{g}_{\\boldsymbol{k'}}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\delta_{T}\n+\\frac{\\xi^{2}_{SO}}{k_{F}^{4}}\\boldsymbol{n}\\cdot\\boldsymbol{m}\\{\\hat{g}_{\\boldsymbol{k'}},\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}} \\}\\delta_{J}\\\\\n&\\qquad\\qquad+i\\frac{\\xi^{2}_{SO}}{k_{F}^{4}}(\\boldsymbol{n}\\times\\boldsymbol{m})\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{g}_{\\boldsymbol{k'}}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\delta_{J}\n-i\\frac{\\xi^{2}_{SO}}{k_{F}^{4}}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{g}_{\\boldsymbol{k'}}(\\boldsymbol{n}\\times\\boldsymbol{m})\\cdot\\hat{\\boldsymbol{\\sigma}}\\delta_{J}\\\\\n&\\qquad\\qquad+\\frac{1}{2}\\frac{\\xi_{SO}}{k_{F}^{2}}\\left\\{(\\boldsymbol{k'}-\\boldsymbol{k})\\times\\hat{\\boldsymbol{\\sigma}},\\nabla_{\\boldsymbol{R}}\\hat{g}_{\\boldsymbol{k'}}\\right\\}\\delta_{T}\n+\\frac{1}{2}\\frac{\\xi_{SO}}{k_{F}^{2}}\\left\\{(\\boldsymbol{k'}-\\boldsymbol{k})\\times\\hat{\\boldsymbol{\\sigma}},\\{\\nabla_{\\boldsymbol{R}}\\hat{g}_{\\boldsymbol{k'}},\\boldsymbol{m}\\cdot\\hat{\\boldsymbol{\\sigma}}\\}\\right\\}\\delta_{J}\\\\\n&-\\pi a_{2}\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}\\Big[\\{\\hat{g}_{\\boldsymbol{k'}},\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\}\\delta_{T}+\\left\\{\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}},\\{\\hat{g}_{\\boldsymbol{k'}},\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}\\}\\right\\}\\delta_{J}\\\\\n&\\qquad\\qquad-\\beta\\left(\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}\\hat{g}_{\\boldsymbol{k'}}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}+\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{g}_{\\boldsymbol{k'}}\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m} \\right) \\delta_{T}-\\beta\\boldsymbol{n}\\cdot\\boldsymbol{m}\\{\\hat{g}_{\\boldsymbol{k'}},\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m} \\}\\delta_{J}-\\beta\\boldsymbol{m}^{2}\\{\\hat{g}_{\\boldsymbol{k'}},\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}} \\}\\delta_{J}\\\\\n&\\qquad\\qquad-i\\beta\\left((\\boldsymbol{n}\\times\\boldsymbol{m})\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{g}_{\\boldsymbol{k'}}\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}-\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m} \\hat{g}_{\\boldsymbol{k'}} (\\boldsymbol{n}\\times\\boldsymbol{m})\\cdot\\hat{\\boldsymbol{\\sigma}} \\right)\\delta_{J}\\Big],\n\\end{aligned}\n\\end{equation}\n\\noindent where $a_{1}=n_{i}v_{i}^{2}$ and $a_{2}=\\frac{\\hbar}{2}\\frac{v_{i}}{\\tau_{0}}\\frac{\\xi_{SO}}{k_{F}^{2}}$. \n\\par In a similar manner, by using Eqs. (\\ref{eq:sret}) and (\\ref{eq:sadv}) we proceed with the second part of the collision integral $\\mathcal{I}_{\\boldsymbol{k}}$:\n\\begin{equation}\n\\begin{aligned}\n\\mathcal{I}_{\\boldsymbol{k}}&=-a_{1}\\int\\frac{dE}{2\\pi}\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}\\,\\Big[\\big(\\hat{\\sigma}_{0}+i\\frac{\\xi_{SO}}{k_{F}^{2}}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\big)\\hat{G}^{R}_{\\boldsymbol{k'}E}(\\hat{\\sigma}_{0}-i\\frac{\\xi_{SO}}{k_{F}^{2}}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}})\\hat{g}_{\\boldsymbol{k}}\\hat{G}^{A}_{\\boldsymbol{k}E}\\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad+\\hat{G}^{R}_{\\boldsymbol{k}E}\\hat{g}_{\\boldsymbol{k}}\\big(\\hat{\\sigma}_{0}+i\\frac{\\xi_{SO}}{k_{F}^{2}}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\big)\\hat{G}^{A}_{\\boldsymbol{k'}E}(\\hat{\\sigma}_{0}-i\\frac{\\xi_{SO}}{k_{F}^{2}}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}})\\Big],\n\\end{aligned}\n\\end{equation}\n\\noindent where \n\\begin{equation}\n\\begin{aligned}\n\\int\\frac{dE}{2\\pi}\\left[\\hat{G}^{R}_{\\boldsymbol{k'}E}\\hat{g}_{\\boldsymbol{k}}\\hat{G}^{A}_{\\boldsymbol{k}E}+\\hat{G}^{R}_{\\boldsymbol{k}E}\\hat{g}_{\\boldsymbol{k}}\\hat{G}^{A}_{\\boldsymbol{k'}E}\\right]&=\\pi\\hat{g}_{\\boldsymbol{k}}\\delta_{T}+\\pi\\{\\hat{g}_{\\boldsymbol{k}},\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}\\}\\delta_{J},\n\\end{aligned}\n\\end{equation}\n\\begin{equation}\n\\begin{aligned}\n&\\int\\frac{dE}{2\\pi} i\\frac{\\xi_{SO}}{k_{F}^{2}}\\left[[\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}},\\hat{G}^{R}_{\\boldsymbol{k'}E}]\\hat{g}_{\\boldsymbol{k}}\\hat{G}^{A}_{\\boldsymbol{k}E}+\\hat{G}^{R}_{\\boldsymbol{k}E}\\hat{g}_{\\boldsymbol{k}}[\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}},\\hat{G}^{A}_{\\boldsymbol{k'}E}]\\right]=\\\\\n&=-\\pi\\frac{\\xi_{SO}}{k_{F}^{2}}(\\boldsymbol{n}\\times\\boldsymbol{m})\\cdot\\{\\hat{g}_{\\boldsymbol{k}},\\hat{\\boldsymbol{\\sigma}}\\}\\delta_{J},\n\\end{aligned}\n\\end{equation}\n\\begin{equation}\n\\begin{aligned}\n&\\int\\frac{dE}{2\\pi} \\frac{\\xi^{2}_{SO}}{k_{F}^{4}}\\left[\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{G}^{R}_{\\boldsymbol{k'}E}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{g}_{\\boldsymbol{k}}\\hat{G}^{A}_{\\boldsymbol{k}E}+\\hat{G}^{R}_{\\boldsymbol{k}E}\\hat{g}_{\\boldsymbol{k}}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{G}^{A}_{\\boldsymbol{k'}E}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\right]=\\\\\n&=\\pi \\frac{\\xi^{2}_{SO}}{k_{F}^{4}}\\boldsymbol{n}^{2}\\hat{g}_{\\boldsymbol{k}}\\delta_{T}+\\pi \\frac{\\xi^{2}_{SO}}{k_{F}^{4}}(2(\\boldsymbol{n}\\cdot\\boldsymbol{m})\\boldsymbol{n}-\\boldsymbol{n}^{2}\\boldsymbol{m})\\cdot\\{\\hat{g}_{\\boldsymbol{k}},\\hat{\\boldsymbol{\\sigma}}\\}\\delta_{J},\n\\end{aligned}\n\\end{equation}\n\\noindent so we obtain:\n\\begin{equation}\n\\begin{aligned}\n\\mathcal{I}_{\\boldsymbol{k}}&=-\\pi a_{1}\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}\\Big[\\hat{g}_{\\boldsymbol{k}}\\delta_{T}+\\{\\hat{g}_{\\boldsymbol{k}},\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}\\}\\delta_{J}-\\frac{\\xi_{SO}}{k_{F}^{2}}(\\boldsymbol{n}\\times\\boldsymbol{m})\\cdot\\{\\hat{g}_{\\boldsymbol{k}},\\hat{\\boldsymbol{\\sigma}}\\}\\delta_{J}\\\\\n&\\qquad\\qquad+\\frac{\\xi^{2}_{SO}}{k_{F}^{4}}\\boldsymbol{n}^{2}\\hat{g}_{\\boldsymbol{k}}\\delta_{T}+\\frac{\\xi^{2}_{SO}}{k_{F}^{4}}(2(\\boldsymbol{n}\\cdot\\boldsymbol{m})\\boldsymbol{n}-\\boldsymbol{n}^{2}\\boldsymbol{m})\\cdot\\{\\hat{g}_{\\boldsymbol{k}},\\hat{\\boldsymbol{\\sigma}}\\}\\delta_{J}\\Big].\n\\end{aligned}\n\\end{equation}\n\\noindent Finally, the collision integral is written as:\n\\begin{equation}\n\\begin{aligned}\ncoll&=\\pi a_{1}\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}\\Big[(\\hat{g}_{\\boldsymbol{k'}}-\\hat{g}_{\\boldsymbol{k}})\\delta_{T}+\\{\\hat{g}_{\\boldsymbol{k'}}-\\hat{g}_{\\boldsymbol{k}},\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}\\}\\delta_{J}\\\\\n&\\qquad\\qquad+i\\frac{\\xi_{SO}}{k_{F}^{2}}[\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}},\\hat{g}_{\\boldsymbol{k'}}]\\delta_{T}+i\\frac{\\xi_{SO}}{k_{F}^{2}}\\left(\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{g}_{\\boldsymbol{k'}}\\boldsymbol{m}\\cdot\\hat{\\boldsymbol{\\sigma}}-\\boldsymbol{m}\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{g}_{\\boldsymbol{k'}}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}  \\right)\\delta_{J}-\\frac{\\xi_{SO}}{k_{F}^{2}}\\{(\\boldsymbol{n}\\times\\boldsymbol{m})\\cdot\\hat{\\boldsymbol{\\sigma}},\\hat{g}_{\\boldsymbol{k'}}\\}\\delta_{J}\\\\\n&\\qquad\\qquad+\\frac{\\xi^{2}_{SO}}{k_{F}^{4}}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{g}_{\\boldsymbol{k'}}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\delta_{T}\n+\\frac{\\xi^{2}_{SO}}{k_{F}^{4}}\\boldsymbol{n}\\cdot\\boldsymbol{m}\\{\\hat{g}_{\\boldsymbol{k'}},\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}} \\}\\delta_{J}\\\\\n&\\qquad\\qquad+i\\frac{\\xi^{2}_{SO}}{k_{F}^{4}}(\\boldsymbol{n}\\times\\boldsymbol{m})\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{g}_{\\boldsymbol{k'}}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\delta_{J}\n-i\\frac{\\xi^{2}_{SO}}{k_{F}^{4}}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{g}_{\\boldsymbol{k'}}(\\boldsymbol{n}\\times\\boldsymbol{m})\\cdot\\hat{\\boldsymbol{\\sigma}}\\delta_{J}\\\\\n&\\qquad\\qquad-\\frac{\\xi^{2}_{SO}}{k_{F}^{4}}\\boldsymbol{n}^{2}\\hat{g}_{\\boldsymbol{k}}\\delta_{T}-\\frac{\\xi^{2}_{SO}}{k_{F}^{4}}(2(\\boldsymbol{n}\\cdot\\boldsymbol{m})\\boldsymbol{n}-\\boldsymbol{n}^{2}\\boldsymbol{m})\\cdot\\{\\hat{g}_{\\boldsymbol{k}},\\hat{\\boldsymbol{\\sigma}}\\}\\delta_{J}\\\\\n&\\qquad\\qquad+\\frac{1}{2}\\frac{\\xi_{SO}}{k_{F}^{2}}\\left\\{(\\boldsymbol{k'}-\\boldsymbol{k})\\times\\hat{\\boldsymbol{\\sigma}},\\nabla_{\\boldsymbol{R}}\\hat{g}_{\\boldsymbol{k'}}\\right\\}\\delta_{T}\n+\\frac{1}{2}\\frac{\\xi_{SO}}{k_{F}^{2}}\\left\\{(\\boldsymbol{k'}-\\boldsymbol{k})\\times\\hat{\\boldsymbol{\\sigma}},\\{\\nabla_{\\boldsymbol{R}}\\hat{g}_{\\boldsymbol{k'}},\\boldsymbol{m}\\cdot\\hat{\\boldsymbol{\\sigma}}\\}\\right\\}\\delta_{J}\\Big]\\\\\n&-\\pi a_{2}\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}\\Big[\\{\\hat{g}_{\\boldsymbol{k'}},\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\}\\delta_{T}+\\left\\{\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}},\\{\\hat{g}_{\\boldsymbol{k'}},\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}\\}\\right\\}\\delta_{J}\\\\\n&\\qquad\\qquad-\\beta\\left(\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}\\hat{g}_{\\boldsymbol{k'}}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}+\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{g}_{\\boldsymbol{k'}}\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m} \\right) \\delta_{T}-\\beta\\boldsymbol{n}\\cdot\\boldsymbol{m}\\{\\hat{g}_{\\boldsymbol{k'}},\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m} \\}\\delta_{J}-\\beta\\boldsymbol{m}^{2}\\{\\hat{g}_{\\boldsymbol{k'}},\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}} \\}\\delta_{J}\\\\\n&\\qquad\\qquad-i\\beta\\left((\\boldsymbol{n}\\times\\boldsymbol{m})\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{g}_{\\boldsymbol{k'}}\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}-\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m} \\hat{g}_{\\boldsymbol{k'}} (\\boldsymbol{n}\\times\\boldsymbol{m})\\cdot\\hat{\\boldsymbol{\\sigma}} \\right)\\delta_{J}\\Big].\n\\end{aligned}\n\\label{eq:collision1}\n\\end{equation}\n\\noindent By neglecting higher order terms $\\beta\\delta_{J}\\sim\\beta^{2}$ in skew-scattering and introducing a more familiar distribution function $\\hat{g}_{\\boldsymbol{k}}=\\hat{\\sigma}_{0}-2\\hat{h}_{\\boldsymbol{k}}$, we get:\n\\begin{equation}\n\\begin{aligned}\ncoll&=-2\\pi a_{1}\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}\\Big[(\\hat{h}_{\\boldsymbol{k'}}-\\hat{h}_{\\boldsymbol{k}})\\delta_{T}+\\{\\hat{h}_{\\boldsymbol{k'}}-\\hat{h}_{\\boldsymbol{k}},\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}\\}\\delta_{J}\\\\\n&\\qquad\\qquad+i\\frac{\\xi_{SO}}{k_{F}^{2}}[\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}},\\hat{h}_{\\boldsymbol{k'}}]\\delta_{T}+i\\frac{\\xi_{SO}}{k_{F}^{2}}(\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{h}_{\\boldsymbol{k'}}\\boldsymbol{m}\\cdot\\hat{\\boldsymbol{\\sigma}}-\\boldsymbol{m}\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{h}_{\\boldsymbol{k'}}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}})\\delta_{J}-\\frac{\\xi_{SO}}{k_{F}^{2}}\\{(\\boldsymbol{n}\\times\\boldsymbol{m})\\cdot\\hat{\\boldsymbol{\\sigma}},\\hat{h}_{\\boldsymbol{k'}}\\}\\delta_{J}\\\\\n&\\qquad\\qquad+\\frac{1}{2}\\frac{\\xi_{SO}}{k_{F}^{2}}\\{(\\boldsymbol{k'}-\\boldsymbol{k})\\times\\hat{\\boldsymbol{\\sigma}},\\nabla_{\\boldsymbol{R}}\\hat{h}_{\\boldsymbol{k'}}\\}\\delta_{T}\n+\\frac{1}{2}\\frac{\\xi_{SO}}{k_{F}^{2}}\\{(\\boldsymbol{k'}-\\boldsymbol{k})\\times\\hat{\\boldsymbol{\\sigma}},\\{\\nabla_{\\boldsymbol{R}}\\hat{h}_{\\boldsymbol{k'}},\\boldsymbol{m}\\cdot\\hat{\\boldsymbol{\\sigma}}\\}\\}\\delta_{J}\\\\\n&\\qquad\\qquad+\\frac{\\xi^{2}_{SO}}{k_{F}^{4}}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{h}_{\\boldsymbol{k'}}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\delta_{T}\n+\\frac{\\xi^{2}_{SO}}{k_{F}^{4}}\\boldsymbol{n}\\cdot\\boldsymbol{m}\\{\\hat{h}_{\\boldsymbol{k'}},\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}} \\}\\delta_{J}\\\\\n&\\qquad\\qquad+i\\frac{\\xi^{2}_{SO}}{k_{F}^{4}}(\\boldsymbol{n}\\times\\boldsymbol{m})\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{h}_{\\boldsymbol{k'}}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\delta_{J}\n-i\\frac{\\xi^{2}_{SO}}{k_{F}^{4}}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{h}_{\\boldsymbol{k'}}(\\boldsymbol{n}\\times\\boldsymbol{m})\\cdot\\hat{\\boldsymbol{\\sigma}}\\delta_{J}\\\\\n&\\qquad\\qquad-\\frac{\\xi^{2}_{SO}}{k_{F}^{4}}\\boldsymbol{n}^{2}\\hat{h}_{\\boldsymbol{k}}\\delta_{T}-\\frac{\\xi^{2}_{SO}}{k_{F}^{4}}(2(\\boldsymbol{n}\\cdot\\boldsymbol{m})\\boldsymbol{n}-\\boldsymbol{n}^{2}\\boldsymbol{m})\\cdot\\{\\hat{h}_{\\boldsymbol{k}},\\hat{\\boldsymbol{\\sigma}}\\}\\delta_{J}\\Big]\\\\\n&+2\\pi a_{2}\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}\\Big[\\{\\hat{h}_{\\boldsymbol{k'}},\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\}\\delta_{T}+\\{\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}},\\{\\hat{h}_{\\boldsymbol{k'}},\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}\\}\\}\\delta_{J}-\\beta(\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}\\hat{h}_{\\boldsymbol{k'}}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}+\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{h}_{\\boldsymbol{k'}}\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m} ) \\delta_{T}\\Big],\n\\end{aligned}\n\\label{eq:collision2}\n\\end{equation}\n\\noindent while the Keldysh equation~(\\ref{eq:keld2}) is rewritten as:\n\\begin{equation}\n-2\\left(-i[\\hat{h}_{\\boldsymbol{k}},J\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}]+\\frac{\\hbar^{2}}{m}(\\boldsymbol{k}\\cdot\\nabla_{\\boldsymbol{R}})\\,\\hat{h}_{\\boldsymbol{k}} \\right)=coll.\n\\label{eq:keld3}\n\\end{equation}\n\n\n\n\n\n\\section{Ferromagnetic solution without spin-orbit coupling}\n\\par Let us consider Eq. (\\ref{eq:keld3}) without extrinsic spin-orbit coupling:\n\\begin{equation}\n\\begin{aligned}\n-i[\\hat{h}_{\\boldsymbol{k}}, J\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}]+\\frac{\\hbar^2}{m}(\\boldsymbol{k} \\cdot \\nabla_{\\boldsymbol{R}}) \\hat{h}_{\\boldsymbol{k}}= \\pi a_{1}\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}\\big[(\\hat{h}_{\\boldsymbol{k'}}-\\hat{h}_{\\boldsymbol{k}})\\delta_{T}+\\{\\hat{h}_{\\boldsymbol{k'}}-\\hat{h}_{\\boldsymbol{k}},\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}\\}\\delta_{J}\\big].\n\\end{aligned}\n\\end{equation}  \n\\noindent By introducing $\\Omega=i\\hbar\/\\tau_0$, $\\hat U = \\hat{\\boldsymbol{\\sigma}} \\cdot \\boldsymbol{m}$, and: \n\\begin{equation}\n\\begin{aligned}\n\\label{eq:k_fer}\n\\hat{K}=-\\frac{\\hbar^2}{m}(\\boldsymbol{k}\\cdot \\nabla_{\\boldsymbol{R}}) \\hat{h}_{\\boldsymbol{k}}+\\pi a_{1}\\int\\frac{d\\boldsymbol{k'}}{(2\\pi)^{3}}\\big[\\hat{h}_{\\boldsymbol{k'}}\\delta_{T}+\\{\\hat{h}_{\\boldsymbol{k'}},\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}\\}\\delta_{J}\\big],\n\\end{aligned}\n\\end{equation}\n\\noindent in the weak exchange coupling limit ($J\\ll\\varepsilon_{F}$) we have:\n\\begin{equation}\n\\begin{aligned}\n\\hat{h}_{\\boldsymbol{k}} = \\frac{i}{\\Omega} \\hat{K} + \\frac{\\beta}{2}\\{\\hat{h}_{\\boldsymbol{k}}, \\hat{U}\\} + \\frac{J}{\\Omega}[\\hat{U}, \\hat{h}_{\\boldsymbol{k}}].\n\\end{aligned}\n\\end{equation}\n\\noindent This equation is solved iteratively:\n\\begin{equation}\n\\begin{aligned}\n\\hat{h}_{\\boldsymbol{k}} &= \\frac{i}{\\Omega}\\left(1+2\\frac{J^{2}}{\\Omega^{2}}+8\\frac{J^{4}}{\\Omega^{4}}+32\\frac{J^{6}}{\\Omega^{6}}+... \\right) \\hat{K} + \\frac{i}{\\Omega}\\frac{\\beta^{2}}{2}\\left(1+\\beta^{2}+\\beta^{4}+... \\right)\\hat{K} \\\\\n&+ \\frac{i}{\\Omega}\\frac{\\beta}{2}\\left(1+\\beta^{2}+\\beta^{4}+... \\right)\\{\\hat{U},\\hat{K}\\}+\\frac{i}{\\Omega}\\frac{J}{\\Omega}\\left(1+4\\frac{J^{2}}{\\Omega^{2}}+16\\frac{J^{4}}{\\Omega^{4}}+...  \\right)[\\hat{U},\\hat{K}]\\\\\n&+ \\frac{i}{\\Omega}\\frac{\\beta^{2}}{2}\\left(1+\\beta^{2}+\\beta^{4}+... \\right)\\hat{U}\\hat{K}\\hat{U}-2\\frac{i}{\\Omega}\\frac{J^{2}}{\\Omega^{2}}\\left(1+4\\frac{J^{2}}{\\Omega^{2}}+16\\frac{J^{4}}{\\Omega^{4}}+...  \\right)\\hat{U}\\hat{K}\\hat{U},\n\\end{aligned}\n\\end{equation}\n\\noindent or by using $1+x+x^{2}+x^{3}...=\\frac{1}{1-x}$ for $x\\ll 1$:\n\\begin{equation}\n\\begin{aligned}\n\\hat{h}_{\\boldsymbol{k}} &=\\frac{i}{\\Omega}\\left(\\frac{\\Omega^{2}-2J^{2}}{\\Omega^{2}-4J^{2}}+\\frac{\\beta^{2}}{2(1-\\beta^{2})} \\right)\\hat{K}+\\frac{i}{\\Omega}\\frac{\\beta}{2}\\frac{1}{1-\\beta^{2}}\\{\\hat{U},\\hat{K}\\}\\\\\n&+\\frac{iJ}{\\Omega^{2}-4J^{2}}[\\hat{U},\\hat{K}]+\\frac{i}{\\Omega}\\left(\\frac{\\beta^{2}}{2(1-\\beta^{2})}-\\frac{2J^{2}}{\\Omega^{2}-4J^{2}}  \\right)\\hat{U}\\hat{K}\\hat{U}.\n\\end{aligned}\n\\end{equation}\n\\noindent Since $J^{2}\/\\Omega^{2}\\ll1$ and $\\beta^{2}\\ll1$, this solution is well justified. By substituting $\\Omega$, $\\hat{K}$ and $\\hat{U}$ and removing the delta-functions, we have:\n\\begin{equation}\n\\begin{aligned}\n\\frac{\\hbar}{\\tau_{0}}\\hat{h}_{\\boldsymbol{k}}=& -\\frac{\\hbar^{2}}{m}(\\boldsymbol{k}\\cdot\\nabla_{\\boldsymbol{R}})\\hat{h}_{\\boldsymbol{k}}+\\frac{\\tau_{0}^{2}}{m}\\frac{2J^{2}}{1+\\frac{4J^{2}\\tau_{0}^{2}}{\\hbar^{2}}}(\\boldsymbol{k}\\cdot\\nabla_{\\boldsymbol{R}})\\left(\\hat{h}_{\\boldsymbol{k}}-\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m} \\hat{h}_{\\boldsymbol{k}}\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m} \\right) \\\\\n&-\\frac{\\beta^{2}}{2(1-\\beta^{2})}\\frac{\\hbar^{2}}{m}(\\boldsymbol{k}\\cdot\\nabla_{\\boldsymbol{R}})\\left(\\hat{h}_{\\boldsymbol{k}}+\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m} \\hat{h}_{\\boldsymbol{k}}\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}  \\right)-\\frac{\\beta}{2(1-\\beta^{2})}\\frac{\\hbar^{2}}{m}(\\boldsymbol{k}\\cdot\\nabla_{\\boldsymbol{R}})\\{\\hat{h}_{\\boldsymbol{k}},\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m} \\}\\\\\n&+\\frac{\\hbar}{\\tau_{0}}\\int\\frac{d\\check{\\boldsymbol{k'}}}{4\\pi}\\,\\hat{h}_{\\boldsymbol{k'}} +\\frac{\\tau_{0}}{\\hbar}\\frac{2J^{2}}{1+\\frac{4J^{2}\\tau_{0}^{2}}{\\hbar^{2}}}\\left(\\int \\frac{d\\check{\\boldsymbol{k'}}}{4\\pi}\\,\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m} \\hat{h}_{\\boldsymbol{k'}}\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}-\\int \\frac{d\\check{\\boldsymbol{k'}}}{4\\pi}\\,\\hat{h}_{\\boldsymbol{k'}}\\right)\\\\\n&-\\frac{iJ}{1+\\frac{4J^{2}\\tau_{0}^{2}}{\\hbar^{2}}} \\int\\frac{d\\check{\\boldsymbol{k'}}}{4\\pi}\\,[\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m},\\hat{h}_{\\boldsymbol{k'}}] +\\frac{\\tau_{0}\\hbar}{m}\\frac{iJ}{1+\\frac{4J^{2}\\tau_{0}^{2}}{\\hbar^{2}}} (\\boldsymbol{k}\\cdot\\nabla_{\\boldsymbol{R}})[\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m},\\hat{h}_{\\boldsymbol{k}}], \n\\label{eq:fer1}\n\\end{aligned}\n\\end{equation}\n\\noindent where $\\check{\\boldsymbol{k}}=\\boldsymbol{k}\/|\\boldsymbol{k}|$. In the diffusive limit $v_{F}\\tau\\ll L$, where $L$ is the system size, we can partition the distribution function $\\hat{h}_{\\boldsymbol{k}}$ into the isotropic charge $\\mu_{c}$ and spin $\\boldsymbol{\\mu}$ and anisotropic $\\hat{\\boldsymbol{j}}\\cdot\\check{\\boldsymbol{k}}$ components, $\\hat{h}_{\\boldsymbol{k}}=\\mu_{c}\\hat{\\sigma}_{0}+\\boldsymbol{\\mu}\\cdot\\hat{\\boldsymbol{\\sigma}}+\\hat{\\boldsymbol{j}}\\cdot\\check{\\boldsymbol{k}}$. This form is nothing else but the generalized $p$-wave approximation for the distribution function. Upon integrating Eq. (\\ref{eq:fer1}) multiplied by $\\check{\\boldsymbol{k}}$ over $d\\check{\\boldsymbol{k}}\/4\\pi$ and neglecting higher order terms $\\sim \\beta^{2}$, we obtain the following expression for $\\hat{\\boldsymbol{j}}$: \n\\begin{equation}\n\\begin{aligned}\n\\label{eq:j_fer}\n\\frac{\\hbar}{\\tau_{0}}\\hat{\\boldsymbol{j}}=&-\\frac{\\hbar^{2}}{m}k\\nabla\\left(\\mu_{c}\\hat{\\sigma}_{0}+\\boldsymbol{\\mu}\\cdot\\hat{\\boldsymbol{\\sigma}}+\\beta\\mu_{c}\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}+\\beta\\boldsymbol{\\mu}\\cdot{\\boldsymbol{m}}\\hat{\\sigma}_{0}\\right)\\\\\n&-\\frac{\\tau_{0}\\hbar}{m}\\frac{2J}{1+\\frac{4J^{2}\\tau_{0}^{2}}{\\hbar^{2}}}k\\nabla\\,\\hat{\\boldsymbol{\\sigma}}\\cdot(\\boldsymbol{m}\\times\\boldsymbol{\\mu})-\\frac{\\tau_{0}^{2}}{m}\\frac{4J^{2}}{1+\\frac{4J^{2}\\tau_{0}^{2}}{\\hbar^{2}}}k\\nabla\\,\\hat{\\boldsymbol{\\sigma}}\\cdot(\\boldsymbol{m}\\times(\\boldsymbol{m}\\times\\boldsymbol{\\mu})).\n\\end{aligned}\n\\end{equation}\n\\par The charge and spin currents (its $j$th component in the spin space) can be defined as:\n\\begin{equation}\n\\tilde{\\boldsymbol{j}}^{C}=\\frac{1}{4}\\int\\frac{d\\check{\\boldsymbol{k}}}{4\\pi}\\mathrm{Tr}\\,\\{\\hat{\\boldsymbol{v}}_{\\boldsymbol{k}},\\hat{h}_{\\boldsymbol{k}}\\}=\\frac{v_{F}}{6}\\mathrm{Tr}\\,\\hat{\\boldsymbol{j}},\n\\end{equation}\n\\noindent and\n\\begin{equation}\n\\tilde{\\boldsymbol{J}}^{S}_{j}=\\frac{1}{4}\\int\\frac{d\\check{\\boldsymbol{k}}}{4\\pi}\\mathrm{Tr}\\,\\big[\\hat{\\sigma}_{j}\\{\\hat{\\boldsymbol{v}}_{\\boldsymbol{k}},\\hat{h}_{\\boldsymbol{k}}\\}\\big]=\\frac{v_{F}}{6}\\mathrm{Tr}\\,\\big[\\hat{\\sigma}_{j}\\,\\hat{\\boldsymbol{j}}\\big],\n\\end{equation}\n\\noindent where the velocity operator is defined as $\\hat{\\boldsymbol{v}}_{\\boldsymbol{k}}=\\frac{\\hbar}{m}\\boldsymbol{k}\\hat{\\sigma}_{0}$, and $v_{F}=\\frac{\\hbar}{m} k_{F}$ is the Fermi velocity. Thus, neglecting higher order terms $\\sim J^{3}$ gives:\n\\begin{equation}\n\\tilde{\\boldsymbol{j}}^{C} = - D\\nabla (\\mu_c + \\beta \\boldsymbol{\\mu} \\cdot \\boldsymbol{m}),\n\\label{eq:cc_fer}\n\\end{equation}\n\\noindent and\n\\begin{equation}\n\\frac{\\tilde{\\boldsymbol{J}}^{S}_{j}}{D}=  -\\nabla( \\mu_{j}+\\beta\\mu_c m_{j})-\\frac{\\tau_{0}}{\\tau_{L}}\\nabla(\\boldsymbol{m}\\times\\boldsymbol{\\mu})_{j}-\\frac{\\tau_{0}}{\\tau_{\\phi}}\\nabla(\\boldsymbol{m}\\times(\\boldsymbol{m}\\times\\boldsymbol{\\mu}))_{j},\n\\label{eq:sc_fer}\n\\end{equation}\n\\noindent where $D=\\tau_{0}v_{F}^{2}\/3$ is the diffusion coefficient, $1\/\\tau_{L}=2J\/\\hbar$ is the Larmor precession time, and $1\/\\tau_{\\phi}=4J^{2}\\tau_{0}\/\\hbar^{2}$ is the spin dephasing time.\n\\par The corresponding equations for the charge and spin densities are obtained by integrating Eq.~(\\ref{eq:fer1}) over $\\check{\\boldsymbol{k}}$ and neglecting terms $\\sim J\\nabla^{2}$ and $\\sim\\beta\\nabla^{2}$:\n\\begin{equation}\n-\\frac{\\hbar^{2}}{m}\\frac{k}{3}\\nabla \\cdot \\hat{\\boldsymbol{j}}+\\frac{\\tau_{0}}{\\hbar}\\frac{4J^{2}}{1+\\frac{4J^{2}\\tau_{0}^{2}}{\\hbar^{2}}}\\hat{\\boldsymbol{\\sigma}}\\cdot(\\boldsymbol{m}\\times(\\boldsymbol{m}\\times\\boldsymbol{\\mu}))+\\frac{2J}{1+\\frac{4J^{2}\\tau_{0}^{2}}{\\hbar^{2}}}\\hat{\\boldsymbol{\\sigma}}\\cdot(\\boldsymbol{m}\\times\\boldsymbol{\\mu})=0.\n\\end{equation}\n\\noindent Taking $\\mathrm{Tr}\\,[...]$ and $\\mathrm{Tr}\\,[\\hat{\\boldsymbol{\\sigma}}...]$ and neglecting terms $\\sim J^{3}$ leads to:\n\\begin{equation}\n0=D\\nabla^{2}(\\mu_{c}+\\beta\\boldsymbol{\\mu}\\cdot\\boldsymbol{m})=-\\nabla\\cdot\\tilde{\\boldsymbol{j}}^{C}\n\\end{equation}\n\\noindent and \n\\begin{equation}\n0=-\\nabla\\cdot\\tilde{\\boldsymbol{J}}^{S}+\\frac{1}{\\tau_{L}}(\\boldsymbol{m}\\times\\boldsymbol{\\mu})+\\frac{1}{\\tau_{\\phi}}(\\boldsymbol{m}\\times(\\boldsymbol{m}\\times\\boldsymbol{\\mu}))\n\\end{equation}\n\\noindent for the charge and spin components, respectively. Finally, by recovering time-dependence from Eq. (\\ref{eq:kelfull}) we obtain:\n\\begin{equation}\n\\partial_{T}\\mu_{c}=-\\nabla\\cdot\\tilde{\\boldsymbol{j}}^{C}\n\\label{eq:ca_fer}\n\\end{equation}\n\\noindent and  \n\\begin{equation}\n\\partial_{T}\\boldsymbol{\\mu}=-\\nabla\\cdot\\tilde{\\boldsymbol{J}}^{S}+\\frac{1}{\\tau_{L}}(\\boldsymbol{m}\\times\\boldsymbol{\\mu})+\\frac{1}{\\tau_{\\phi}}(\\boldsymbol{m}\\times(\\boldsymbol{m}\\times\\boldsymbol{\\mu})).\n\\label{eq:sa_fer}\n\\end{equation}\n\\noindent Thus, Eqs. (\\ref{eq:cc_fer}), (\\ref{eq:sc_fer}), (\\ref{eq:ca_fer}) and (\\ref{eq:sa_fer}) define a set of the drift-diffusion equations for ferromagnets in the absence of extrinsic spin-orbit coupling.\n\n\n\n\n\n\\section{Ferromagnetic solution with spin-orbit coupling}\n\\par To derive drift-diffusion equations including extrinsic spin-orbit coupling, we employ the same $p$-wave approximation for $\\hat{h}_{\\boldsymbol{k}}$. Then, we have:\n\\begin{gather}\n-i[\\hat{h}_{\\boldsymbol{k}},J\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}]=2J\\hat{\\boldsymbol{\\sigma}}\\cdot(\\boldsymbol{\\mu}\\times\\boldsymbol{m})-i[\\hat{\\boldsymbol{j}}\\cdot\\check{\\boldsymbol{k}},J\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m}],\\\\\n(\\boldsymbol{k}\\cdot\\nabla_{\\boldsymbol{R}})\\hat{h}_{\\boldsymbol{k}}=(\\boldsymbol{k}\\cdot\\nabla_{\\boldsymbol{R}})\\mu_{c}\\hat{\\sigma}_{0}+(\\boldsymbol{k}\\cdot\\nabla_{\\boldsymbol{R}})\\boldsymbol{\\mu}\\cdot\\hat{\\boldsymbol{\\sigma}}+(\\boldsymbol{k}\\cdot\\nabla_{\\boldsymbol{R}})\\hat{\\boldsymbol{j}}\\cdot\\check{\\boldsymbol{k}}\n\\end{gather}\n\\noindent  for the left-hand side of the Keldysh equation~(\\ref{eq:keld3}), and:\n\\begin{gather}\n\\begin{aligned}\n\\{\\nabla_{\\boldsymbol{R}}\\hat{h}_{\\boldsymbol{k'}},(\\boldsymbol{k'}-\\boldsymbol{k})\\times\\hat{\\boldsymbol{\\sigma}}\\}&=2\\left(\\nabla_{\\boldsymbol{R}}\\times(\\boldsymbol{k'}-\\boldsymbol{k}) \\right)\\cdot\\hat{\\boldsymbol{\\sigma}}\\mu_{c}-2(\\boldsymbol{k'}-\\boldsymbol{k})\\cdot\\left(\\nabla_{\\boldsymbol{R}}\\times\\boldsymbol{\\mu}\\right)\\hat{\\sigma}_{0}\\\\\n&+\\{\\nabla_{\\boldsymbol{R}}\\,\\hat{\\boldsymbol{j}}\\cdot\\check{\\boldsymbol{k'}},(\\boldsymbol{k'}-\\boldsymbol{k})\\times\\hat{\\boldsymbol{\\sigma}}\\},\n\\end{aligned}\\\\\n\\begin{aligned}\n\\{(\\boldsymbol{k'}-\\boldsymbol{k})\\times\\hat{\\boldsymbol{\\sigma}},\\{\\nabla_{\\boldsymbol{R}}\\hat{h}_{\\boldsymbol{k'}},\\boldsymbol{m}\\cdot\\hat{\\boldsymbol{\\sigma}}\\}\\}&=4(\\boldsymbol{k'}-\\boldsymbol{k})\\cdot(\\boldsymbol{m}\\times\\nabla_{\\boldsymbol{R}} \\mu_{c})\\hat{\\sigma}_{0}+4\\hat{\\boldsymbol{\\sigma}}\\cdot(\\nabla_{\\boldsymbol{R}}\\times(\\boldsymbol{k'}-\\boldsymbol{k}))\\boldsymbol{\\mu}\\cdot\\boldsymbol{m}\\\\\n&+\\{(\\boldsymbol{k'}-\\boldsymbol{k})\\times\\hat{\\boldsymbol{\\sigma}},\\{\\nabla_{\\boldsymbol{R}}\\,\\hat{\\boldsymbol{j}}\\cdot\\check{\\boldsymbol{k'}},\\boldsymbol{m}\\cdot\\hat{\\boldsymbol{\\sigma}}\\}\\}\n\\end{aligned}\\\\\n\\begin{aligned}\n\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{h}_{\\boldsymbol{k'}}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}-\\boldsymbol{n}^{2}\\hat{h}_{\\boldsymbol{k}}=2(\\boldsymbol{n}\\times(\\boldsymbol{n}\\times\\boldsymbol{\\mu}))\\cdot\\hat{\\boldsymbol{\\sigma}}+\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}\\hat{\\boldsymbol{j}}\\cdot\\check{\\boldsymbol{k'}}\\boldsymbol{n}\\cdot\\hat{\\boldsymbol{\\sigma}}-\\boldsymbol{n}^{2}\\hat{\\boldsymbol{j}}\\cdot\\check{\\boldsymbol{k}}\n\\end{aligned}\n\\end{gather}\n\\noindent for the collision integral~(\\ref{eq:collision2}). Upon integrating Eq. (\\ref{eq:keld3}) over $d\\check{\\boldsymbol{k}}\/4\\pi$ and neglecting terms $\\sim \\xi_{SO}^{2}\\beta$, we obtain in the limit $J\\ll\\varepsilon_{F}$: \n\\begin{equation}\n\\begin{aligned}\n\\label{eq:gen1}\n2J\\hat{\\boldsymbol{\\sigma}}\\cdot(\\boldsymbol{\\mu}\\times\\boldsymbol{m})+\\frac{1}{3}\\frac{\\hbar^{2}k}{m}\\,\\nabla_{\\boldsymbol{R}}\\cdot\\hat{\\boldsymbol{j}}=&-\\frac{8}{9}\\frac{\\hbar}{\\tau_{0}}\\frac{{k}^{2}}{k_{F}^{2}}\\xi_{SO}^{2}\\boldsymbol{\\mu}\\cdot\\hat{\\boldsymbol{\\sigma}}\\\\\n&+\\frac{1}{6}\\frac{\\hbar}{\\tau_{0}}\\frac{\\xi_{SO}}{k_{F}}\\big(\\nabla_{\\boldsymbol{R}}\\cdot(\\hat{\\boldsymbol{j}}\\times\\hat{\\boldsymbol{\\sigma}})-\\nabla_{\\boldsymbol{R}}\\cdot(\\hat{\\boldsymbol{\\sigma}}\\times\\hat{\\boldsymbol{j}}) \\big)\\\\\n&+\\frac{1}{6}\\frac{\\hbar}{\\tau_{0}}\\frac{\\xi_{SO}}{k_{F}}\\beta\\big[\\nabla_{\\boldsymbol{R}}\\cdot(\\hat{\\boldsymbol{j}}\\times\\hat{\\boldsymbol{\\sigma}})+\\nabla_{\\boldsymbol{R}}\\cdot(\\hat{\\boldsymbol{\\sigma}}\\times\\hat{\\boldsymbol{j}}),\\boldsymbol{m}\\cdot\\hat{\\boldsymbol{\\sigma}}\\big]\\\\\n&+\\frac{2}{3}\\frac{\\hbar}{\\tau_{0}}\\frac{\\xi_{SO}}{k_{F}}\\beta\\,\\nabla_{\\boldsymbol{R}}\\cdot(\\boldsymbol{m}\\times\\hat{\\boldsymbol{j}}).\n\\end{aligned}\n\\end{equation}\n\\noindent One more equation is derived by averaging Eq.~(\\ref{eq:keld3}) over $\\check{\\boldsymbol{k}}$ multiplied by $\\check{\\boldsymbol{k}}$ and neglecting terms $\\sim \\xi_{SO}^{2}\\beta$:\n\\begin{equation}\n\\begin{aligned}\n\\label{eq:gen2}\n-iJ\\big[\\,\\hat{\\boldsymbol{j}},\\,\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m} \\big]+\\frac{\\hbar^{2}k}{m}\\nabla_{\\boldsymbol{R}}(\\mu_{c}\\hat{\\sigma}_{0}+\\boldsymbol{\\mu}\\cdot\\hat{\\boldsymbol{\\sigma}})=&-\\frac{\\hbar}{\\tau_{0}}\\Big(1+\\frac{2}{3}\\frac{k^{2}}{k_{F}^{2}}\\xi_{SO}^{2}\\Big)\\hat{\\boldsymbol{j}}+\\frac{1}{2}\\frac{\\hbar}{\\tau_{0}}\\beta\\big\\{\\hat{\\boldsymbol{j}},\\hat{\\boldsymbol{\\sigma}}\\cdot\\boldsymbol{m} \\big\\}\\\\\n&-\\frac{i}{3}\\frac{\\hbar}{\\tau_{0}}\\frac{k}{k_{F}}\\xi_{SO}\\big(\\hat{\\boldsymbol{j}}\\times\\hat{\\boldsymbol{\\sigma}}+\\hat{\\boldsymbol{\\sigma}}\\times\\hat{\\boldsymbol{j}}  \\big)\\\\\n&+\\frac{i}{3}\\frac{\\hbar}{\\tau_{0}}\\frac{k}{k_{F}}\\xi_{SO}\\beta\\big(\\boldsymbol{m}\\cdot\\hat{\\boldsymbol{\\sigma}}\\,\\hat{\\boldsymbol{j}}\\times\\hat{\\boldsymbol{\\sigma}}+\\hat{\\boldsymbol{\\sigma}}\\times\\hat{\\boldsymbol{j}}\\,\\boldsymbol{m}\\cdot\\hat{\\boldsymbol{\\sigma}}\\big)\\\\\n&+\\frac{1}{3}\\frac{\\hbar}{\\tau_{0}}\\frac{k}{k_{F}}\\xi_{SO}\\beta\\big(\\hat{\\boldsymbol{\\sigma}}\\cdot\\hat{\\boldsymbol{j}}+\\hat{\\boldsymbol{j}}\\cdot\\hat{\\boldsymbol{\\sigma}})\\boldsymbol{m}\n-\\frac{1}{3}\\frac{\\hbar}{\\tau_{0}}\\frac{k}{k_{F}}\\xi_{SO}\\beta\\big\\{\\hat{\\boldsymbol{\\sigma}},\\hat{\\boldsymbol{j}}\\cdot\\boldsymbol{m} \\big\\}\\\\\n&+\\frac{\\hbar}{\\tau_{0}}\\frac{k}{k_{F}}\\frac{\\xi_{SO}}{k_{F}}\\big(\\nabla_{\\boldsymbol{R}}\\times\\hat{\\boldsymbol{\\sigma}}\\,(\\mu_{c}-\\beta \\boldsymbol{\\mu}\\cdot\\boldsymbol{m})+\\nabla_{\\boldsymbol{R}}\\times(\\boldsymbol{\\mu}-\\beta\\mu_{c}\\boldsymbol{m})\\,\\hat{\\sigma}_{0} \\big)\\\\\n&+\\frac{1}{6}\\frac{m}{\\pi\\hbar}\\frac{v_{i}}{\\tau_{0}}\\xi_{SO}k\\big(\\hat{\\boldsymbol{\\sigma}}\\times\\hat{\\boldsymbol{j}}-\\hat{\\boldsymbol{j}}\\times\\hat{\\boldsymbol{\\sigma}}\\big)\\\\\n&+\\frac{1}{3}\\frac{m}{\\pi\\hbar}\\frac{v_{i}}{\\tau_{0}}\\xi_{SO}\\beta k\\big(\\boldsymbol{m}\\cdot\\hat{\\boldsymbol{\\sigma}}\\,\\hat{\\boldsymbol{j}}\\times\\hat{\\boldsymbol{\\sigma}}-\\hat{\\boldsymbol{\\sigma}}\\times\\hat{\\boldsymbol{j}}\\,\\boldsymbol{m}\\cdot\\hat{\\boldsymbol{\\sigma}}\\big)\\\\\n&+\\frac{1}{6}\\frac{m}{\\pi\\hbar}\\frac{v_{i}}{\\tau_{0}}\\xi_{SO}\\beta k\\big(\\boldsymbol{m}\\cdot\\hat{\\boldsymbol{\\sigma}}\\,\\hat{\\boldsymbol{\\sigma}}\\times \\hat{\\boldsymbol{j}}-\\hat{\\boldsymbol{j}}\\times\\hat{\\boldsymbol{\\sigma}}\\,\\boldsymbol{m}\\cdot\\hat{\\boldsymbol{\\sigma}}\\big)\\\\\n&-\\frac{2}{3}\\frac{m}{\\pi\\hbar}\\frac{v_{i}}{\\tau_{0}}\\xi_{SO}\\beta k\\,\\boldsymbol{m}\\times \\hat{\\boldsymbol{j}}.\n\\end{aligned}\n\\end{equation}\n\\noindent The equations above define a set of the generalized drift-diffusion equations, which can now be solved approximately while keeping leading orders in $\\xi_{SO}$ and $\\beta$. Then, starting from a ferromagnetic solution given by Eq.~(\\ref{eq:j_fer}) the anisotropic component of the density matrix is obtained by solving Eq.~(\\ref{eq:gen2}):\n\\begin{equation}\n\\begin{aligned}\n\\hat{\\boldsymbol{j}}&=-\\tau_{0}v_{F}\\nabla\\hat{\\mu}_{0}+\\left(\\frac{\\xi_{SO}}{k_{F}}+\\frac{\\tau_{0}v_{i}k_{F}^{2}}{3\\pi\\hbar}\\xi_{SO} \\right)\\nabla\\times\\boldsymbol{\\mu}\\hat{\\sigma}_{0}-\\frac{\\tau_{0}v_{i}k_{F}^{2}}{3\\pi\\hbar}\\xi_{SO}\\beta (\\nabla\\times\\boldsymbol{m})\\mu_{c}\\hat{\\sigma}_{0}\\\\\n&-\\left(\\frac{2}{3}\\xi_{SO}\\beta\\tau_{0}v_{F}-\\frac{\\tau_{0}v_{i}k_{F}^{2}}{3\\pi\\hbar}\\xi_{SO}\\frac{\\tau_{0}}{\\tau_{L}} \\right)\\nabla\\times(\\boldsymbol{m}\\times\\boldsymbol{\\mu})\\hat{\\sigma}_{0}+\\left(\\frac{\\xi_{SO}}{k_{F}}+ \\frac{\\tau_{0}v_{i}k_{F}^{2}}{3\\pi\\hbar}\\xi_{SO}\\right)\\nabla\\times\\hat{\\boldsymbol{\\sigma}}\\mu_{c}\\\\\n&-\\frac{\\xi_{SO}}{k_{F}}\\beta\\nabla\\times(\\boldsymbol{m}\\times(\\hat{\\boldsymbol{\\sigma}}\\times\\boldsymbol{\\mu}))-\\frac{\\tau_{0}v_{i}k_{F}^{2}}{3\\pi\\hbar}\\xi_{SO}\\beta\\nabla\\times((\\boldsymbol{m}\\times\\hat{\\boldsymbol{\\sigma}})\\times\\boldsymbol{\\mu})-\\frac{\\tau_{0}v_{i}k_{F}^{2}}{3\\pi\\hbar}\\xi_{SO}\\beta(\\nabla\\times\\hat{\\boldsymbol{\\sigma}})\\boldsymbol{\\mu}\\cdot\\boldsymbol{m}\\\\\n&+\\left(\\frac{\\xi_{SO}}{k_{F}}+\\frac{\\tau_{0}v_{i}k_{F}^{2}}{3\\pi\\hbar} \\right)\\frac{\\tau_{0}}{\\tau_{L}}\\nabla\\times(\\hat{\\boldsymbol{\\sigma}}\\times\\boldsymbol{m})\\mu_{c} - \\frac{2}{3}\\tau_{0}v_{F}\\xi_{SO}\\nabla\\times(\\hat{\\boldsymbol{\\sigma}}\\times(\\boldsymbol{\\mu}-\\beta\\mu_{c}\\boldsymbol{m}))\\\\\n&-\\frac{2}{3}\\tau_{0}v_{F}\\xi_{SO}\\frac{\\tau_{0}}{\\tau_{L}}\\nabla\\times\\left((\\hat{\\boldsymbol{\\sigma}}\\times\\boldsymbol{m})\\times\\boldsymbol{\\mu}+\\hat{\\boldsymbol{\\sigma}}\\times(\\boldsymbol{m}\\times\\boldsymbol{\\mu})  \\right),\n\\end{aligned}\n\\end{equation}\n\\noindent where \n\\begin{equation}\n\\hat{\\mu}_{0}=(\\mu_{c}+\\beta\\boldsymbol{\\mu}\\cdot\\boldsymbol{m})\\hat{\\sigma}_{0}+(\\boldsymbol{\\mu}+\\beta\\mu_{c}\\boldsymbol{m})\\cdot\\hat{\\boldsymbol{\\sigma}}+\\frac{\\tau_{0}}{\\tau_{L}}(\\boldsymbol{m}\\times\\boldsymbol{\\mu})\\cdot\\hat{\\boldsymbol{\\sigma}}+\\frac{\\tau_{0}}{\\tau_{\\phi}}(\\boldsymbol{m}\\times(\\boldsymbol{m}\\times\\boldsymbol{\\mu}))\\cdot\\hat{\\boldsymbol{\\sigma}}.\n\\end{equation}\n\\noindent Here, the first term of $\\hat{\\boldsymbol{j}}$ comes from the ferromagnetic solution by moving the right-hand side of Eq.~(\\ref{eq:gen2}) into Eq.~(\\ref{eq:k_fer}). Plugging this solution in Eq.~(\\ref{eq:gen1}) leads to:\n\\begin{equation}\n\\begin{aligned}\n\\label{eq:j_full}\n\\frac{2J}{\\hbar}(\\boldsymbol{\\mu}\\times\\boldsymbol{m})\\cdot\\hat{\\boldsymbol{\\sigma}}&+\\frac{8}{9}\\frac{\\xi_{SO}^{2}}{\\tau_{0}}\\boldsymbol{\\mu}\\cdot\\hat{\\boldsymbol{\\sigma}}=\\\\\n&=D\\nabla\\cdot\\Big[ \\nabla\\hat{\\mu}_{0}+\\alpha_{sj}\\big[\\hat{\\boldsymbol{\\sigma}}\\times\\nabla(2\\mu_{c}-\\beta\\boldsymbol{\\mu}\\cdot\\boldsymbol{m})-\\nabla\\times(2\\boldsymbol{\\mu}-\\beta\\mu_{c}\\boldsymbol{m})\\big]\\\\\n&+\\alpha_{sk}\\big[\\hat{\\boldsymbol{\\sigma}}\\times\\nabla(\\mu_{c}-\\beta\\boldsymbol{\\mu}\\cdot\\boldsymbol{m})-\\nabla\\times(\\boldsymbol{\\mu}-\\beta\\mu_{c}\\boldsymbol{m})\\big]+\\alpha_{sw}\\nabla\\times(\\hat{\\boldsymbol{\\sigma}}\\times\\boldsymbol{\\mu})\\\\\n&-\\nabla\\times\\big[(\\alpha_{sj}\\frac{\\tau_{0}}{\\tau_{L}}+\\alpha_{sk}\\frac{\\tau_{0}}{\\tau_{L}}+\\alpha_{sw}\\beta)(\\hat{\\boldsymbol{\\sigma}}\\times\\boldsymbol{m})\\mu_{c} + (\\alpha_{sj}\\frac{\\tau_{0}}{\\tau_{L}}+\\alpha_{sk}\\frac{\\tau_{0}}{\\tau_{L}}-\\alpha_{sw}\\beta)(\\boldsymbol{m}\\times\\boldsymbol{\\mu}) \\big]\\\\\n&+\\nabla\\times\\big[(\\alpha_{sj}\\beta+\\alpha_{sk}\\beta+\\alpha_{sw}\\frac{\\tau_{0}}{\\tau_{L}})\\hat{\\boldsymbol{\\sigma}}\\times(\\boldsymbol{m}\\times\\boldsymbol{\\mu})  - (\\alpha_{sj}\\beta+\\alpha_{sk}\\beta-\\alpha_{sw}\\frac{\\tau_{0}}{\\tau_{L}})(\\hat{\\boldsymbol{\\sigma}}\\times\\boldsymbol{m})\\times\\boldsymbol{\\mu}  \\big]\\Big],\n\\end{aligned}\n\\end{equation}\n\\noindent where $\\alpha_{sw}=\\frac{2\\xi_{SO}}{3}$, $\\alpha_{sj}=\\frac{\\xi_{SO}}{l_{F}k_{F}}$ and $\\alpha_{sk}=\\frac{v_{i}mk_{F}}{3\\pi\\hbar^{2}}\\xi_{SO}$ are the spin swapping, side-jump and skew-scattering coefficients, respectively, and $l_{F}=\\tau_{0}v_{F}$ is the mean-free path. As seen, Eq.~(\\ref{eq:j_full}) can be regarded as a generalized continuity equation for the density matrix $\\mu_{c}\\hat{\\sigma}_{0}+\\boldsymbol{\\mu}\\cdot\\hat{\\boldsymbol{\\sigma}}$, and its right-hand side is nothing else but the divergence of the full current $\\boldsymbol{j}^{C}\\hat{\\sigma}_{0}+\\boldsymbol{J}^{S}\\cdot\\hat{\\boldsymbol{\\sigma}}$, where the dot product is over spin components. Thus, the corresponding expressions for the charge and spin currents (its $j$th spin component) can be readily written as:\n\\begin{equation}\n\\begin{aligned}\n\\label{eq:finalcharge}\n\\boldsymbol{j}^{C}\/D&=\\tilde{\\boldsymbol{j}}^{C}\/D+\\alpha_{sj}\\nabla\\times(2\\boldsymbol{\\mu}-\\beta\\mu_{c}\\boldsymbol{m})+\\alpha_{sk}\\nabla\\times(\\boldsymbol{\\mu}-\\beta\\mu_{c}\\boldsymbol{m})+(\\alpha_{sj}\\frac{\\tau_{0}}{\\tau_{L}}+\\alpha_{sk}\\frac{\\tau_{0}}{\\tau_{L}}-\\alpha_{sw}\\beta)\\nabla\\times(\\boldsymbol{m}\\times\\boldsymbol{\\mu})\n\\end{aligned}\n\\end{equation}\n\\noindent and\n\\begin{equation}\n\\begin{aligned}\n\\label{eq:finalspin}\n\\boldsymbol{J}_{j}^{S}\/D&=\\tilde{\\boldsymbol{J}}_{j}^{S}\/D+\\alpha_{sj}\\nabla\\times\\boldsymbol{e}_{j}(2\\mu_{c}-\\beta\\boldsymbol{\\mu}\\cdot\\boldsymbol{m})+\\alpha_{sk}\\nabla\\times\\boldsymbol{e}_{j}(\\mu_{c}-\\beta\\boldsymbol{\\mu}\\cdot\\boldsymbol{m})-\\alpha_{sw}\\nabla\\times(\\boldsymbol{e}_{j}\\times\\boldsymbol{\\mu})\\\\\n&+\\nabla\\times (\\alpha_{sj}\\frac{\\tau_{0}}{\\tau_{L}}+\\alpha_{sk}\\frac{\\tau_{0}}{\\tau_{L}}+\\alpha_{sw}\\beta)(\\boldsymbol{e}_{j}\\times\\boldsymbol{m})\\mu_{c} - \\nabla\\times(\\alpha_{sj}\\beta+\\alpha_{sk}\\beta+\\alpha_{sw}\\frac{\\tau_{0}}{\\tau_{L}})(\\boldsymbol{e}_{j}\\times(\\boldsymbol{m}\\times\\boldsymbol{\\mu})) \\\\\n& + \\nabla\\times(\\alpha_{sj}\\beta+\\alpha_{sk}\\beta-\\alpha_{sw}\\frac{\\tau_{0}}{\\tau_{L}})((\\boldsymbol{e}_{j}\\times\\boldsymbol{m})\\times\\boldsymbol{\\mu}),\n\\end{aligned}\n\\end{equation}\n\\noindent or\n\\begin{equation}\n\\begin{aligned}\nJ_{ij}^{S}\/D&=\\tilde{J}_{ij}^{S}\/D-\\alpha_{sj}\\epsilon_{ijk}\\nabla_{k}(2\\mu_{c}-\\beta\\mu_{n}m_{n})-\\alpha_{sk}\\epsilon_{ijk}\\nabla_{k}(\\mu_{c}-\\beta\\mu_{n}m_{n})-\\alpha_{sw}(\\delta_{ij}\\nabla_{k}\\mu_{k}-\\nabla_{j}\\mu_{i})\\\\\n&+(\\alpha_{sj}\\frac{\\tau_{0}}{\\tau_{L}}+\\alpha_{sk}\\frac{\\tau_{0}}{\\tau_{L}}+\\alpha_{sw}\\beta)(\\delta_{ij}\\nabla_{k}m_{k}-m_{i}\\nabla_{j})\\mu_{c} \\\\\n& - (\\alpha_{sj}\\beta+\\alpha_{sk}\\beta+\\alpha_{sw}\\frac{\\tau_{0}}{\\tau_{L}})\\epsilon_{ikn}\\nabla_{k}(m_{n}\\mu_{j}-\\mu_{n}m_{j}) \\\\\n& +(\\alpha_{sj}\\beta+\\alpha_{sk}\\beta-\\alpha_{sw}\\frac{\\tau_{0}}{\\tau_{L}})(\\epsilon_{ikn}\\nabla_{k}m_{n}\\mu_{j}+\\epsilon_{ijk}\\nabla_{k}m_{n}\\mu_{n}),\n\\end{aligned}\n\\end{equation}\n\\noindent where $\\epsilon_{ijk}$ is the Levi-Civita symbol, and summation over repeated indexes is implied. Here, the first and second subscripts correspond to the spatial and spin components, respectively. Finally, by recovering time dependence in Eq.~(\\ref{eq:j_full}) we obtain the remaining equations for the charge and spin densities:\n\\begin{equation}\n\\label{eq:finalchden}\n\\partial_{T}\\mu_{c}=D\\nabla^{2}(\\mu_{c}+\\beta\\boldsymbol{\\mu}\\cdot\\boldsymbol{m})=-\\nabla\\cdot\\boldsymbol{j}^{C}\n\\end{equation}\n\\noindent and\n\\begin{equation}\n\\label{eq:finalspden}\n\\partial_{T}\\boldsymbol{\\mu}=-\\nabla\\cdot\\boldsymbol{J}^{S}+\\frac{1}{\\tau_{L}}(\\boldsymbol{m}\\times\\boldsymbol{\\mu})+\\frac{1}{\\tau_{\\phi}}(\\boldsymbol{m}\\times(\\boldsymbol{m}\\times\\boldsymbol{\\mu}))-\\frac{1}{\\tau_{sf}}\\boldsymbol{\\mu},\n\\end{equation}\n\\noindent where $1\/\\tau_{sf}=8\\xi_{SO}^{2}\/9\\tau_{0}$ is the spin-flip relaxation time. The set of Eqs.~(\\ref{eq:finalcharge}), (\\ref{eq:finalspin}), (\\ref{eq:finalchden}) and (\\ref{eq:finalspden}) is the central result of this work.\n\n\n\n\n\n\n\\section{Spin swapping symmetry}\n\\par In this section we verify the symmetry of the spin swapping term and compare our results with the previous ones derived for normal metals. \n\\par In their original work Dyakonov and Lifshits give the following definition of the spin current $q_{ij}$ due to scattering off the spin-orbit coupling potential:\\cite{perel} \n\\begin{equation}\n\\label{lifshits}\nq_{ij}=q_{ij}^{(0)}-\\alpha_{sh}\\epsilon_{ijk}q^{(0)}_{k}+\\alpha_{sw}(q_{ji}^{(0)}-\\delta_{ij}q_{kk}^{(0)}),\n\\end{equation}\n\\noindent where $q^{(0)}_{k}$ and $q_{ij}^{(0)}$ stand for the primary charge and spin currents in the absence of spin-orbit coupling, respectively, and $\\alpha_{sh}$ represents the overall spin Hall effect. Thus, it is argued that the spin swapping effect always appears in the form given above.\n\\par Let us consider our solution in the case of normal metals ($\\beta=0$ and $J=0$):\n\\begin{equation}\n\\begin{aligned}\n\\boldsymbol{J}_{j}^{S}&=-D\\nabla\\mu_{j}+D\\alpha_{sh}\\nabla\\times\\boldsymbol{e}_{j}\\mu_{c}-D\\alpha_{sw}\\nabla\\times(\\boldsymbol{e}_{j}\\times\\boldsymbol{\\mu})\n\\end{aligned}\n\\end{equation}\n\\noindent or\n\\begin{equation}\nJ_{ij}^{S}=-D\\nabla_{i}\\mu_{j}-D\\alpha_{sh}\\epsilon_{ijk}\\nabla_{k}\\mu_{c}+D\\alpha_{sw}(\\nabla_{j}\\mu_{i}-\\delta_{ij}\\nabla_{k}\\mu_{k}).\n\\end{equation}\n\\noindent Taking into account that $q_{ij}^{(0)}\\approx-D\\nabla_{i}\\mu_{j}$ and $q_{k}^{(0)}\\approx-D\\nabla_{k}\\mu_{c}$, it is seen that Eq.~(\\ref{eq:finalspin}) displays the correct symmetry up to a sign coming from the definition of the spin-orbit coupling potential, Eq.~(\\ref{eq:imppot}). This form is also in agreement with some previously published results.\\cite{brataas, raimondi}\n\\par Finally, it is worth comparing our equations with those that fail to include spin swapping in the form given by Eq.~(\\ref{lifshits}). For example, in Ref.~[\\onlinecite{manchon1}] the spin swapping term appeared with the following symmetry:\n\\begin{equation}\n\\begin{aligned}\ne^{2}\\boldsymbol{J}_{j}^{S}\/\\sigma_{N}&=-\\nabla\\mu_{j}\/2+\\alpha_{sj}\\boldsymbol{e}_{j}\\times\\nabla\\mu_{c}-\\alpha_{sw}\\boldsymbol{e}_{j}\\times(\\nabla\\times\\boldsymbol{\\mu})\/2\\\\\n&=-\\nabla\\mu_{j}\/2+\\alpha_{sj}\\boldsymbol{e}_{j}\\times\\nabla\\mu_{c}-\\alpha_{sw}(\\nabla\\mu_{j}-\\nabla_{j}\\boldsymbol{\\mu})\/2,\n\\end{aligned}\n\\end{equation}\n\\noindent where $\\sigma_{N}$ is the bulk conductivity. It is clear that the symmetry of spin swapping is wrong: e.g. $q_{xx}$ should contain a term $\\sim\\alpha_{sw}(-q^{(0)}_{yy}-q^{(0)}_{zz})$, which is absent in the expression above. There is the same symmetry problem in Eq.~(2) of Ref.~[\\onlinecite{brataas2}], where the spin swapping term reads as $-\\alpha_{sw}\\hat{\\boldsymbol{\\sigma}}\\times\\nabla\\times\\boldsymbol{\\mu}$ (or $-\\alpha_{sw}\\boldsymbol{e}_{j}\\times\\nabla\\times\\boldsymbol{\\mu}$ for the spin current component).\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n \n \nLet $\\eta=(\\eta _1,\\ldots,\\eta _n)$ be an $R^n$ valued  Gaussian\nrandom variable. $\\eta$  is said to have  infinitely  divisible\nsquares\\index{ infinitely divisible squares} if\n$\\eta^2:=(\\eta^{ 2}_1,\\ldots,\\eta^{ 2}_n)$ is infinitely divisible, i.e. for\nany $r$ we can find an\n$R^{ n}$ valued random vector $Z_{r }$ such that\n\\begin{equation}\n   \\eta^2\\stackrel{law}{=} Z_{ r,1}+\\cdots+Z_{ r,r}\\label{id.1},\n\\end{equation}  where $\\{Z_{ r,j} \\}$,  $j=1,\\ldots,r $ are independent\nidentically distributed copies of\n$Z_{ r}$. We   express this by saying that $\\eta^2$ is infinitely divisible.\n\nPaul L\\'evy proposed the problem of characterizing which Gaussian vectors have infinitely divisible squares. It is easy to see that a single Gaussian random variable\nhas infinitely divisible squares. However, even for vectors in $R^{2}$ this is a difficult problem. It seems that L\\'evy incorrectly conjectured that not all Gaussian vectors in $R^{2}$  have infinitely divisible squares. If he had said $R^{3}$ his conjecture would have been correct.\n\nL\\'evy's problem was solved by Griffiths and Bapapt\n\\cite{Bapat,Griffiths}, (see also \\cite[Theorem 13.2.1]{book}).  \n\n\\begin{theorem}\\label{lem-Bapat0} Let $G=(G _{  { 1}},\\ldots,G _n)$ be a  mean\nzero Gaussian random variable with strictly positive definite covariance\n       matrix $\\Gamma=\\{\\Gamma_{i,j}\\}= \\{E(G _iG _j)\\}$.    Then\n$G^2$ is infinitely divisible if and only if there exists a signature matrix\n      $\\mathcal{N}$ such that  \n\\begin{equation}\n{\\mathcal N}\\Gamma^{-1}{\\mathcal N}\\qquad \\quad\\mbox{is an\n$M$ matrix.} \\label{1}\n\\end{equation} \n\\end{theorem}\n\nWe need to define the different types of \nmatrices that appear in this theorem.   Let\n$A=\\{ a_{ i,j}\\}_{ 1\\leq i,j\\leq n}$ be an\n$n\\times n$ matrix. We call $A$ a positive matrix  and write\n$A\\geq 0$ if\n$a_{ i,j}\\geq 0$ for all\n$i,j$.  We say that $A$ has positive row sums if $\\sum_{j=1}^n\na_{i,j}\\ge 0$ for all $1\\le i\\le n$.\n\n\\medskip  The matrix\n$A$  is said to be an\n$M$ matrix   if\n\\begin{enumerate}\n\\item[(1)] $a_{ i,j}\\leq 0$ for all $i\\neq j$.\n\\item[(2)] $A$ is nonsingular and $A^{ -1}\\geq 0$.\n\\end{enumerate}\n\n  A  matrix is called a signature matrix if its off--diagonal entries are all zero\nand   its diagonal entries are either one or minus one. The role of the signature matrix is easy to understand.  It simply accounts for\nthe fact that if\n$G$ has an infinitely divisible square, then so does\n$(\\epsilon_1G_1,\\ldots,\\epsilon_nG_n)$\n for any choice of $\\epsilon_i=\\pm 1$,\n$i=1,\\ldots,n$.   Therefore, if (\\ref{1}) holds for ${\\mathcal N}$ with diagonal elements\n$n_1,\\ldots,n_n$\n\\begin{equation}\n\\({\\mathcal N}\\Gamma^{-1}{\\mathcal N}\\)^{-1}={\\mathcal N}\\Gamma {\\mathcal N}  \\ge 0\\label{2}\n\\end{equation}\nsince the inverse of an  $M$ matrix is positive. Thus\n$(n_1G_1,\\ldots,n_nG_n)$ has a positive covariance   matrix and its inverse is\nan\n$M$ matrix.  (For this reason, in studying mean zero Gaussian vectors with\ninfinitely divisible squares one can restrict ones attention to vectors with\npositive covariance.)\n\n\\medskip\t\n\nThe natural next step was to   characterize   Gaussian processes with infinitely\ndivisible squares  which  do  not have mean zero. We set \n$\\eta_{i}=G_{ i}+c_{ i}$,  $EG_{ i}=0$, $i=1,,\\dots,n$.   Let $\\Gamma$ be \nthe covariance matrix of  $(G _1 ,\\dots,  G  _n)$ and set\n\\begin{equation}\n c:=(c _1,\\dots,c _n ).\n\\end{equation}\n    Set\n\\begin{equation}  G+c :=(G _1 +c _1,\\dots,G _n +c _n )\\label{1.5}\n   \\end{equation} \nand\n\\begin{equation}  (G+c)^2 :=\n((G _1 +c _1 )^2,\\dots,(G _n +c _n )^2)\\label{1.6xx}.\n\\end{equation}\n\n  Results about  the infinite divisibility of $ (G+c)^2$ when  $c_{1}=\\cdots =c_{n}$,  are given in the work of N.\nEisenbaum \\cite{Eisen03,Eisen05}   and then in joint work by Eisenbaum \nand H. Kaspi \\cite{EK06},  as a by product of their  characterization of Gaussian processes with\na  covariance   that is the 0-potential density of a symmetric Markov process. We point out later in this Introduction how   Gaussian   vectors with infinitely divisible squares are related to the local times of the Markov chain that is determined by the covariance  of the Gaussian vector. It is this connection   between Gaussian vectors with infinitely divisible squares and the local times of Markov chains, and more generally, between Gaussian processes with infinitely divisible squares and the local times of Markov processes, that enhances our interest in the question of characterizing Gaussian vectors with infinitely divisible squares.  \n\n\\medskip\t \nSome of the results   in  \\cite{Eisen03,Eisen05,EK06} are presented and expanded in \\cite[Chapter 13]{book}.  The following  theorem  is taken from  \n\\cite [Theorem 13.3.1 and Lemma 13.3.2]{book}.\n\n\\begin{theorem}\\label{theo-book} Let $G=(G _{  { 1}},\\ldots,G _n)$ be a  mean\nzero Gaussian random variable with strictly positive definite covariance\n       matrix $\\Gamma=\\{\\Gamma_{i,j}\\}= \\{E(G _iG _j)\\}$.  Let ${\\bf 1}=(1,\\ldots,1)\\in R^{n}$.  Then the following are\nequivalent:\n\n\\begin{itemize}\n\\item[(1)] $(G+{\\bf 1} \\alpha)$ has infinitely divisible squares for all $\\alpha\\in R^1$;\n\\item[(2)] For $\\xi=N(0,b^2)$ independent of $G$, $(G_1+ \\xi,\\ldots, G_n+ \\xi,\n\\xi)$ has infinitely divisible squares   for some $b\\neq 0$. Furthermore, if this holds for  some $b\\neq 0$, it holds   for all $b\\in R^1$, with\n$N(0,0)=0$.\n\\item[(3)] $ \\Gamma^{-1} $ is an\n$M$ matrix with positive row sums.\n\\end{itemize}\n\\end{theorem}\n\nIn \\cite{IDS}, Theorem \\ref{theo-book} is generalized so that the mean of the components of $G+c$ in (\\ref{1.5}) need not be the same. In this   generalization certain trivial cases spoil the simplicity of the final result. We avoid them by requiring that the covariance matrix of the Gaussian process is irreducible.\n\n\\begin{theorem}\\label{lem-Bapat} Let $G=(G _{  { 1}},\\ldots,G _n)$ be a  mean\nzero Gaussian random variable with irreducible strictly positive definite covariance\n       matrix $\\Gamma=\\{\\Gamma_{i,j}\\}= \\{E(G _iG _j)\\}$.  Let $c=( c_{\n1},\n\\ldots ,c_{ n})\\in R^n$,   $c\\ne \\mathbf{  0}  $  and let $C$ be a\ndiagonal matrix with\n$c_i=C_{i,i}$, $1\\le i\\le n$ . Then the following are equivalent: \n\n\\begin{itemize}\n\\item[(1)] $G+c \\alpha$ has infinitely divisible squares  for all $ \\alpha\\in R^1$;\n\\item[(2)] For $\\xi=N(0,b^2)$ independent of $G$, $(G_1+c_1 \\xi,\\ldots,\nG_n+c_n \\xi, \\xi)$ has infinitely divisible squares   for some $b\\neq 0$. Furthermore, if this holds for  some $b\\neq 0$, it holds   for all $b\\in R^1$;\n\\item[(3)]   \n$C\\, \\Gamma^{-1} \\, C$ is an\n$M$ matrix with positive row sums.\n\\end{itemize}\n\\end{theorem} \n\n\n\n\n\n\n\\medskip\nBy definition, when $(G+c )^2 $ is infinitely divisible, it can\nbe written as in (\\ref{id.1}) as a sum of $r$ independent\nidentically distributed random variables,  for all $r\\ge 1$.   \nBased on the work of   Eisenbaum and Kaspi mentioned above and the joint\npaper\n\\cite{fiveauthors} we can actually describe the decomposition. We give a\nrough description here. For details see \\cite{Eisen03,Eisen05,EK06} and \n\\cite[Chapter 13]{book}.\n\nAssume that (1), (2) and (3) of Theorem \\ref{lem-Bapat} hold.  Let\n\\begin{equation}\n{G \\over c}=\\({G_{1} \\over c_{1}},\\ldots,{G_{n} \\over c_{n}}\\).\\label{2.1xx}\n\\end{equation}\n Let\n$\\Gamma_c$ denote the covariance matrix of $G\/c$. Theorem\n\\ref{theo-book} holds for $ G\/c$ and\n$\\Gamma_c$, so $\\Gamma_c^{-1}$ is an $M$ matrix with positive row sums. To\nbe specific let \n$G\/c\\in  R^n$. Set $S=\\{1,\\ldots,n\\}$. By \\cite[Theorem 13.1.2]{book}, $\\Gamma_c$\nis the 0-potential density of a strongly symmetric   transient Borel right process, say\n$X$, on $S$. We show in  the proof of \\cite[Theorem 13.3.1]{book} that\nwe can  find a strongly symmetric\nrecurrent Borel right\nprocess $Y$\n   on $S\\cup\\{0\\}$ with $P^{ x}( T_{0}<\\infty)>0$ for all $x\\in S$\nsuch that $X$ is the process obtained by killing\n$Y$ the first time it hits $0$.  Let   $ L^x_t=\\{ L^x_t; t\\in R_{+}, x\\in S\\cup \\{0\\}\\}$  denote the local time\nof $Y$. It follows\nfrom the generalized second Ray-Knight Theorem in \\cite{fiveauthors}, see\nalso \\cite[Theorem 8.2.2]{book}\n  that under $P^{ 0}\\times P_{ G}$,\n\\begin{equation}\n\\bigg\\{ L^x_{\\tau(t)}+{ 1\\over 2}{\\(G_x\\over c_x\\)^2};\\,x\\in S\n\\bigg\\}\\stackrel{law}{=}\n\\bigg\\{{ 1\\over 2}\\({G_x\\over c_x}+\\sqrt{2t}\\)^2;\\,x\\in S\\bigg\\}\\label{nor.1rqq}\n\\end{equation}\n   for all $t\\in R_+$, where $\\tau(t)=\\inf \\{s>0|L_s^0>t\\}$,   the inverse local time at zero, and $Y$ and $G$\nare independent. Consequently \n\\begin{equation}\n\\Big\\{c_x^2 L^x_{\\tau(\\alpha^2\/2)}+{ 1\\over 2}{G_x ^2};\\,x\\in S\n\\Big\\}\\stackrel{law}{=}\n\\Big\\{{ 1\\over 2}\\(G_x + c_x \\alpha\\)^2;\\,x\\in S\\Big\\}\\label{nor.1ww}\n\\end{equation}\nfor all $\\alpha\\in R^1$. (We can extend $\\alpha$ from $R_+$ to $R^1$ because $G$\nis symmetric.) $\\{c_x^2 L^x_{\\tau(\\alpha^2\/2)} ;\\,x\\in S\\}$ and $ \\{{ 1\\over 2}{G_x\n^2};\\,x\\in S\\}$ are independent. $G^2$ is infinitely divisible and for all\nintegers $r\\ge 1$\n\\begin{equation} \nc_{ \\,\\cdot\\,}^2 L^{\\cdot}_{\\tau(\\alpha^2\/2)}\\stackrel{law}{=}c_{\\,\\cdot\\,}^2\nL^{\\cdot}_{\\tau(\\alpha^2\/(2r)) ,1}+\\cdots +c_{ \\,\\cdot\\,}^2\nL^{\\cdot}_{\\tau(\\alpha^2\/(2r)) ,r}\\label{1.17}\n\\end{equation}\nwhere $\\{L^{\\cdot}_{\\tau(\\alpha^2\/(2r)) ,j}\\}$, $j=1,\\ldots,r$ are independent.\n\n  Note that in (\\ref{nor.1ww})  we identify the components of the decomposition of $\\{\\(G_x + c_x \\alpha\\)^2;\\,x\\in S\\}$ that mark it as infinitely divisible.\n  \n  \n  \\medskip\t  In Theorem \\ref{lem-Bapat} we have necessary and sufficient conditions for   $((G_{1}+c _{1}\\alpha)^{2}, (G_{2}+c _{2}\\alpha)^{2}) $ to be infinitely divisible   for all $ \\alpha\\in R^1$. There remains the question, can  $((G_{1}+c _{1}\\alpha)^{2}, (G_{2}+c_{2} \\alpha)^{2}) $  have infinitely divisible squares  for some   $ \\alpha>0$ but not for all $ \\alpha\\in R^1$? \n  When we began to investigate this question we hoped that such   points $\\alpha$  do not exist. This would   have finished off the problem of characterizing Gaussian random variables with infinitely divisible squares and, more significantly, by  (\\ref{nor.1ww}), would show that when a Gaussian random variable  with non--zero mean has infinitely divisible squares,\nit  decomposes into the sum of two independent random variables. The Gaussian random variable itself minus its  mean, and the local time  of   a related Markov process. This would be a very neat result indeed, but it is not true.   \n \n \nFor all  Gaussian random variables  $(G_1,G_2)$ in $R^{2}$ and all $c_{1},c_{2}\\in R^{1}$ define\n \\begin{equation}\n{\\mathcal G}^{2}(c_{1},c_{2},{\\alpha}):=((G_{1}+c_{1} \\alpha)^{2}, (G_{2}+c_{2} \\alpha)^{2}). \n\\end{equation}\nIt follows from Theorem \\ref{lem-Bapat}, (for details see \\cite[Corollary 1.3, 4.]{IDS}), that ${\\mathcal G}^{2}(c_{1},c_{2},{\\alpha})$ has infinitely divisible squares   for all $ \\alpha\\in R^1$ if and only if\n\\begin{eqnarray} \n   \\Gamma_{1,1}\\ge \\frac{c_1}{c_2}\\Gamma_{1,2}\\quad\\mbox{and}\\quad  \\Gamma_{2,2}\\ge \\frac{c_2}{c_1}\\Gamma_{1,2}.\\label{1.12q}\n \\end{eqnarray}\n  If (\\ref{1.12q}) does not hold,  we call  $0<\\alpha_0<\\infty$    a {\\bf critical point} for the infinite divisibility of\n  ${\\mathcal G}^{2}(c_{1},c_{2},{\\alpha})$    if    ${\\mathcal G}^{2}(c_{1},c_{2},{\\alpha})$ is infinitely divisible for all $|\\alpha|\\le \\alpha_{0}$, and is not infinitely divisible for any $|\\alpha|>\\alpha_{0}$. In this paper we prove the following theorem:\n \n  \n \n\\medskip\t\n\\begin{theorem}\\label{theo-cp}\n  For all Gaussian random variables  $(G_1,G_2)$ in $R^{2}$ and all   $(c_{1},c_{2}) \\in R^{2}$ for which (\\ref{1.12q}) does not hold,  ${\\mathcal G}^{2}(c_{1},c_{2},{\\alpha}) $ has a critical point.   \\end{theorem}\n\nNote that in Theorem \\ref{theo-cp} we consider all  $(c_{1},c_{2})\\in R^{2}$. It follows from (\\ref{1.12q}) that when $EG_{1} G_{2}>0$, then  ${\\mathcal G}^{2}(c_{1},c_{2},{\\alpha})$ has infinitely divisible squares for all $\\alpha\\in R^{1}$  only if  $c_{1}c_{2}>0$.  Nevertheless, by Theorem \\ref{theo-cp}, even when $c_{1}c_{2}\\le 0$,  $ (G_1+{c_{1}\\alpha},G_2+c_{2}\\alpha)$ does  have  infinitely divisible squares for   $|\\alpha|$ sufficiently small.\n\n  \n\\medskip\tTo conclude this Introduction we explain how we approach the problem of showing that $(G+c\\alpha)^2$ is infinitely divisible for all $\\alpha\\in R^1$ or only for some $\\alpha\\in R ^{1}$. Since we can only prove Theorem \\ref{theo-cp} for Gaussian random variables in $R^{2}$ we stick to this case, although a similar analysis applies to $R^{n}$ valued Gaussian random variables. \n\n Let $\\Gamma$ be the covariance matrix of $G$ and \n\\begin{equation} \n\\widetilde\\Gamma := (I+\\Gamma\\Lambda)^{-1}\\Gamma=(\\Gamma^{-1}+\\Lambda)^{-1} ,\\label{1.12xx}\n\\end{equation} \nwhere\n\\begin{equation}\\Lambda=\n     \\(\\begin{array}{cc}\\lambda_1&0\\\\\n 0&\\lambda_2\\end{array}\\). \n \\end{equation}\nConsider the Laplace transform of  $((G_{1}+c_{1}\\alpha  )^2, (G_2+c_{2}\\alpha)^2)$, \n\\begin{eqnarray}\n\\lefteqn{ E_G\\(e^{ -( \\lambda_1(G_{1}+c_{1}\\alpha  )^2+\\lambda_2(G_2+c_{2}\\alpha)^2)\/2}\\)\\label{ids.qaspr.2q}}\n\\\\\n&& =\\frac{1}{(\\det \\(I+\\Gamma\\Lambda\\))^{1\/2} }   \\exp\\(-  {\\alpha ^2\\over 2}   \\(  c_{1}^2\\lambda_{1} +c_{2}^2\\lambda_{2} -\\sum_{i,j=1}^{2}c_{i}c_{j}\\lambda_{i}\\lambda_{j}\\widetilde\\Gamma_{i,j}\\)\\) \\nonumber.\n\\end{eqnarray}\n  (See \\cite[Lemma 5.2.1] {book}.)\nSet $\\lambda_1=t(1-s_1)$ and $\\lambda_2=t(1-s_2)$,  $0\\le s_{1},s_{2}\\le 1$ and write (\\ref{ids.qaspr.2q}) as \n\\begin{eqnarray} \n   \\exp\\(U\\( s_{1},s_{2},t,\\Gamma \\)+\\alpha^{2}V\\( s_{1},s_{2},t, c_{1} ,c_{2},\\widetilde\\Gamma \\)\\)\\label{1.15}\n \\end{eqnarray}\nwhere  $U:=U( s_{1},s_{2},t, \\Gamma )=-1\/ 2\\log(\\det \\(I+\\Gamma\\Lambda\\))$. \nNote that \\newline $ \\exp\\(U\\( s_{1},s_{2},t,  \\Gamma \\)\\)$ is the Laplace transform of $(G_{1}^{2},G_{2}^{2})$, with the change of variables $\\lambda_1=t(1-s_1)$ and $\\lambda_2=t(1-s_2)$. It is easy to see from Theorem \\ref{lem-Bapat0} that all two dimensional Gaussian random variables are infinitely divisible. Therefore, for all $t$ sufficiently large, all the coefficients of the  power series expansion of $U$ in $s_{1}$ and $s_{2}$ are positive, except for the constant term. This is a necessary and sufficient condition for a function to be the Laplace transform of an infinitely divisible random variable. See e.g. \\cite[Lemma 13.2.2]{book}. \n\nNow, suppose that  for all $t$ sufficiently large,\n$V\\( s_{1},s_{2},t ,c_{1} ,c_{2},\\widetilde\\Gamma \\)$ has all the coefficients of  its power series expansion   in $s_{1}$ and $s_{2}$  positive, except for the constant term.\nThen the right--hand side of (\\ref{ids.qaspr.2q}) is the Laplace transform of two independent infinitely divisible random variables. It is completely obvious that this holds for all $\\alpha\\in R^{1}$. \n\nOn the other hand suppose that  for all $t$ sufficiently large, the power series expansion of $V\\( s_{1},s_{2},t ,c_{1} ,c_{2},\\widetilde\\Gamma \\)$   in $s_{1}$ and $s_{2}$ has even one  negative coefficient, besides the coefficient of the constant term.  Then for all $\\alpha$ sufficiently large, (\\ref{1.15}) is not the Laplace  transform of an infinitely divisible random variable. In other words  $((G_{1}+c_{1}\\alpha  )^2, (G_2+c_{2}\\alpha)^2)$ is not infinitely divisible for all $\\alpha\\in R^{1}$. But it may be   infinitely divisible   if $\\alpha$ is small, since the positive coefficients of U may be greater than or equal to $\\alpha^{2}$ times the corresponding negative coefficients of    $ V$. Clearly, if this is true for some $|\\alpha|=\\alpha_{0}>0$, then it is true for all $|\\alpha|\\le \\alpha_{0}$.  \n\nThe preceding paragraph explains how we prove Theorem \\ref{theo-cp}. We consider vectors $((G_{1}+c_{1}\\alpha  )^2, (G_2+c_{2}\\alpha)^2)$ that are not infinitely divisible for all $\\alpha\\in R^{1}$, (this is easy to do using (\\ref{1.12q})), and show that for $|\\alpha|$ sufficiently small the coefficients in the power series expansion of\n\\begin{equation}\n   U\\( s_{1},s_{2},t,\\Gamma \\)+\\alpha^{2}V\\( s_{1},s_{2},t,c_{1} ,c_{2},\\widetilde\\Gamma \\)\n   \\end{equation}\n in $s_{1}$ and $s_{2}$  are positive, except for the constant term. Our proof  only uses   elementary mathematics, although it is quite long and complicated. In the course of the proof we show that the coefficients of the power series expansion of $U$  in $s_{1}$ and $s_{2}$  are positive, except for the constant term. This provides a direct elementary proof of the fact that the Gaussian random variable $(G_{1},G_{2})$ always has infinitely divisible squares.\n\nAs we have just stated, and as the reader will see, the  proof of Theorem \\ref{theo-cp} is long and complicated.  So far we have not been able to  extend it to apply to Gaussian random variables in $R^{3}$. One hopes for a more sophisticated and much shorter proof of Theorem \\ref{theo-cp} that doesn't depend on the dimension of the Gaussian random variable.\n  \n\n\n \\section{Gaussian squares in $R^2$ and their Laplace transforms }\\label{sec-2}\n\n \n  Let $G=(G_1,G_2)$ be a mean zero Gaussian process\nwith covariance matrix \n \\begin{equation} \\Gamma= \n     \\(\\begin{array}{cc}a&1\\\\\n 1&b\\end{array}\\) \\label{cov}\\end{equation}\nwhere  $ab=d+1>1$, and let  $G+c:=(G_1+c_{1},G_2+c_{2})$, $c_{1},c_{2}\\in R^{1}$.   \nNote that\n\\begin{equation}\n\\det \\Gamma=d,\\label{30.1}\n\\end{equation}\nand\n\\begin{equation}\n\\Gamma^{-1}={1 \\over d} \\(\\begin{array}{cc}b&-1\\\\\n-1&a\\end{array}\\)\\label{30.1a}.\n\\end{equation}\nLet  \n \\begin{equation}\n \\Lambda=\n     \\(\\begin{array}{cc}\\lambda_1&0\\\\\n 0&\\lambda_2\\end{array}\\). \n \\end{equation}\nThen\n\\begin{equation}\n\\Gamma^{-1}+\\Lambda={1 \\over d} \\(\\begin{array}{cc}b+d\\lambda_{1}&-1\\\\\n-1&a+d\\lambda_{2}\\end{array}\\)\\label{30.1axx}\n\\end{equation}\nand\n\\begin{eqnarray}\n\\widetilde\\Gamma&:=&(I+\\Gamma\\Lambda)^{-1}\\Gamma=(\\Gamma^{-1}+\\Lambda)^{-1}\\\\\n&=&d\\,\\, \\(\\begin{array}{cc}b+d\\lambda_{1}&-1\\\\\n-1&a+d\\lambda_{2}\\end{array}\\)^{-1} \\nonumber\\\\\n&=&\\frac{1}{H(a,b,\\lambda_1,\\lambda_2)  } \\(\\begin{array}{cc}\\lambda_2d+a&1\\\\\n 1&\\lambda_1d+b\\end{array}\\) \\nonumber\n\\end{eqnarray}\nwhere \n\\begin{eqnarray}\nH(a,b,\\lambda_1,\\lambda_2)=1+a\\lambda_1+b\\lambda_2+d\\lambda_1\\lambda_2=d\\det \\(\\Gamma^{-1}+\\Lambda\\).\\label{2.7}\n\\end{eqnarray}\n (We use repeatedly the fact that $ab=d+1$.) Note that by (\\ref{30.1}) we have that\n \\begin{equation}\n\\det \\(I+\\Gamma\\Lambda\\)= \\det \\(\\Gamma\\(\\Gamma^{-1}+\\Lambda\\)\\)=d\\det \\(\\Gamma^{-1}+\\Lambda\\)=H(a,b,\\lambda_1,\\lambda_2).\\label{30.2}\n\\end{equation}\n\n \n\\begin{lemma} \\label{lem-2.1}\n \\begin{eqnarray}\n\\lefteqn{E_G\\(e^{ -( \\lambda_1(G_{1}+c_{1}  )^2+\\lambda_2(G_2+c_{2}  )^2)\/2}\\)\\label{idsqaspr.2q}}\n\\\\\n&& =\\frac{1}{(H(a,b,\\lambda_1,\\lambda_2))^{1\/2} }   \\exp\\(- \\frac{c_{1} ^2\\lambda_{1}+c_{2}^{2}\\lambda_{2}+\\(c_{1}^{2}b+c_{2}^{2}a-2c_{1}c_{2}\\) \\lambda_1\n \\lambda_2 }{2H(a,b,\\lambda_1,\\lambda_2)} \\) \\nonumber.\n\\end{eqnarray}\n \\end{lemma}\n   \n     \n \\noindent{\\bf Proof}$\\quad$  By \\cite[Lemma 5.2.1] {book} \n \\begin{eqnarray}\n \\lefteqn{E_G\\(e^{ -( \\lambda_1(G_{1}+c_{1}  )^2+\\lambda_2(G_2+c_{2}  )^2)\/2}\\)\\label{www} }\\\\\n&&\\qquad  =\\frac{1}{(H(a,b,\\lambda_1,\\lambda_2))^{1\/2} } \\exp\\(-  {1\\over 2}\\Bigg (\nc_{1}^{2}\\lambda_1 + c_{2}^{2}\\lambda_2\\right. \\nonumber\\\\\n&&\\qquad\n \\qquad\\left. -\\frac{c_{1}^{2}  \\lambda_1^{2} (\\lambda_2d+a)+2c_{1}c_{2} \\lambda_1\\lambda_2+ c_{2}^{2}\\lambda_2^{2}\n(\\lambda_1d+b) }{H(a,b,\\lambda_1,\\lambda_2)}\\Bigg)\\)   .\n\\nonumber\n\\end{eqnarray}\nA simple computation shows that \n\\begin{eqnarray}\n&&\\( c_{1}^{2}\\lambda_1 + c_{2}^{2}\\lambda_2 \\)H(a,b,\\lambda_1,\\lambda_2)\\label{30.3}\\\\\n&&\\hspace{1in}-\\(c_{1}^{2}  \\lambda_1^{2} (\\lambda_2d+a)+2c_{1}c_{2} \\lambda_1\\lambda_2+ c_{2}^{2}\\lambda_2^{2}\n(\\lambda_1d+b)\\) \n\\nonumber\\\\\n&&\\qquad=c_{1} ^2\\lambda_{1}+c_{2}^{2}\\lambda_{2}+\\(c_{1}^{2}b+c_{2}^{2}a-2c_{1}c_{2}\\) \\lambda_1\n \\lambda_2 \\nonumber,\n\\end{eqnarray}\nfrom which we get (\\ref{idsqaspr.2q}).{\\hfill $\\square$ }\n\n\\medskip\t\n\nThe term   $1\/(H(a,b,\\lambda_1,\\lambda_2))^{1\/2}$ is\nthe Laplace transform of \n$(G_1^2,G^2_2)\/2$ and  by   \\cite[Corollary 1.1, 2.]{IDS}   it is the Laplace\ntransform of an infinitely divisible random variable.   The exponential term may or may not be a Laplace transform. In\nfact, by (\\ref{1.12q}), we know it is the Laplace transform of an infinitely divisible\nrandom variable,   for all $\\{c_{1}\\alpha,c_{2}\\alpha\\}$, for  all $\\alpha\\in R^{1}$,  if and only if \n\\begin{equation}\na\\ge \\frac{c_{1}}{c_{2}}>0\\qquad \\mbox{and}\\qquad b\\ge \\frac{c_{2}}{c_{1}}>0.\\label{4.5}\n\\end{equation}\n  To prove Theorem \\ref{theo-cp} we must show that when  (\\ref{4.5}) does not hold, there exists    an  $\\,0<\\alpha_{0}<\\infty$ such that  (\\ref{idsqaspr.2q}) is the Laplace transform of an infinitely divisible random variable when $c_{1}$ and $c_{2}$ are replaced by   $c_{1}\\alpha$ and $c_{2}\\alpha$ for any $|\\alpha|\\le \\alpha_{0}$.   Actually,  as we see in Section \\ref{sec-8}, the general result follows from the consideration of   three cases, \n\\begin{enumerate}\n\\item $c_{1}=c_{2}:=({c,c})$;\n\\item $c_{1}=-c_{2}:=({c,-c})$ \n\\item $(c,0)$. \n\\end{enumerate}\nThis is because if $c_{1}\\ne c_{2}$ and neither of them is zero, we can replace $(G_{1},G_{2})$ by   $(G_{1}\/|c_{1}|,G_{2}\/|c_{2}|)$. Clearly, in this case,  if Theorem \\ref{theo-cp} holds for $(G_{1}\/|c_{1}|,G_{2}\/|c_{2}|)$ it holds for $(G_{1},G_{2})$.  \n\nIn these three cases  the  numerator of the fraction in the exponential term on the right--hand side of\n (\\ref{idsqaspr.2q}) is\n \\begin{enumerate}\n\\item $ c^{2}\\((a+b-2)\\lambda_{1}\\lambda_{2}+\\lambda_{1}+\\lambda_{2}\\)$;\n\\item  $ c^{2}\\((a+b+2)\\lambda_{1}\\lambda_{2}+\\lambda_{1}+\\lambda_{2}\\)$ \n\\item $ c^{2}\\(b\\lambda_{1}\\lambda_{2}+\\lambda_{1} \\)$. \n\\end{enumerate}\nSet \n\\begin{equation}\n \\gamma=  a+b-2\\qquad\\mbox{and}\\qquad \\rho=a+b+2.\n   \\end{equation}\n Note unless $\\det \\Gamma=0$, $ab>1$. Since  Theorem \\ref{theo-cp} obviously holds\n when $\\det \\Gamma=0$, we can exclude this case from further consideration. Thus we always have $\\gamma>0$.\n \n \n \n  \n\n \\section{Power series expansion of the   logarithm of the Laplace transform of ${\\bf((G_{1}+c)^{2}, (G_{2}+c)^{2})}$   when ${\\bf EG_{1} G_{2} =1}$}\\label{sec-3}\n \n  Bapat's proof of Theorem \\ref{lem-Bapat0}   involves  the analysis of a certain power series expansion of the logarithm of the Laplace transform. We need a similar, but more delicate,  analysis. (See Lemma \\ref{lem-1.1} below).\n \n  \nUsing (\\ref{idsqaspr.2q})  and the remarks following Lemma \\ref{lem-2.1} we can write  \n\\begin{eqnarray}\n\\lefteqn{E_G\\(e^{ -( \\lambda_1(G_{1}+c )^2+\\lambda_2(G_2+c \n)^2)\/2}\\)\\label{idpr.2}}\n\\\\\n&&\\qquad  =\\exp\\(-\\frac{1}{2}\\log H(a,b,\\lambda_1,\\lambda_2)\n\\)     \\exp\\(- \\frac{c ^2\\(\\gamma\\lambda_1\n \\lambda_2+\\lambda_1+\\lambda_2\\)}{2H(a,b,\\lambda_1,\\lambda_2)} \\) \n. \n\\nonumber\\\\\n&&\\qquad:  =\\exp\\( \\frac{1}{2}\\Big(P(a,b,\\lambda_1,\\lambda_2)+\nc ^2 Q(a,b,\\lambda_1,\\lambda_2  ) \\Big)\\) ,\n\\nonumber\n\\end{eqnarray}\nSince $ab=d+1$,   (recall that $d>0$), we have\n\\begin{eqnarray}\na+b-(d+2)&=&a+{(d+1) \\over a}-(d+2)\n\\label{4.5jss}\\\\\n&=& {1 \\over a }\\(a^{2}-(d+2)a+(d+1)\\)  \\nonumber\\\\\n&=& {1 \\over a }\\(a-(d+1)\\)\\(a-1\\).  \\nonumber\n\\end{eqnarray}\nThus $a+b-(d+2)\\le 0$ if and only if $1\\le a \\le d+1$, which in view of $ab=d+1$ is equivalent to $1\\le b\\le d+1$.\nConsequently (\\ref{4.5}) holds  if and only if  $a+b-(d+2)\\le 0$. \n\nTherefore, to show that  $((G_{1}+c)^{2},(G_{2}+c)^{2})$ is infinitely divisible, for some, but not for all,   $c>0$, we must consider $a, b>0$ such that  \n\\begin{equation}\n\\zeta := a+b-(d+2)>0.\\label{4.5k}\n\\end{equation}\n In the rest of this paper we   assume that (\\ref{4.5k}) holds.\n\n\n\\medskip\t\nLet $\\lambda_1=t(1-s_1)$ and $\\lambda_2=t(1-s_2)$,  $0\\le s_{1},s_{2}\\le 1$. We consider $P$ and $Q$ as\nfunctions  of and $s_1,s_2 ,t$, and write\n\\begin{equation}\nP(s_1,s_2 ,t):=P(a,b,\\lambda_1,\\lambda_2),\\hspace{.3 in}Q(s_1,s_2 ,t ):=Q(a,b,\\lambda_1,\\lambda_2 ).\\label{22.1}\n\\end{equation}\n\nWe  expand these in a power series in $s_1,s_2$. \n\\begin{equation}\nP(s_1,s_2 ,t)=\\sum_{j,k=0}^{\\infty}P_{j,k}(t)s_{1}^{j}s_{2}^{k},\\hspace{.3 in}Q(s_1,s_2 ,t )=\\sum_{j,k=0}^{\\infty}Q_{j,k}(t)s_{1}^{j}s_{2}^{k}\\label{22.2},\n\\end{equation}\nand    set \n\\begin{equation}\nR(s_1,s_2 ,t,c)=P(s_1,s_2 ,t)+c^{2}Q(s_1,s_2 ,t,c)\\label{22.2s}.\n\\end{equation}\nConsequently\n\\begin{equation}\nR(s_1,s_2 ,t,c)=\\sum_{j,k=0}^{\\infty}R_{j,k}(t,c)s_{1}^{j}s_{2}^{k}\\label{22.2t}\n\\end{equation}\nwith\n\\begin{equation}\nR_{j,k}(t,c)=P_{j,k}(t)+c^{2}Q_{j,k }(t).\\label{22.2u}\n\\end{equation}\n\n \nIn this section we obtain explicit expressions for $P_{j,k}(t),Q_{j,k}(t)$.\n\\medskip   \nWe write\n\\begin{eqnarray}\nH(a,b,\\lambda_1,\\lambda_2)&=&{ 1+a\\lambda_1+b\\lambda_2+d\\lambda_1\\lambda_2}\\label{3.11}\\\\\n&=&1+at+bt+dt^2 -(at +dt^2 )s_1-(bt+dt^2 )s_2+dt^2 s_1s_2\\nonumber\\\\\n&=&(1+at +bt+dt^2 )(1-\\alpha s_1-\\beta s_2+ps_1s_2)\\nonumber\\\\\n&=&(1+at +bt+dt^2 )(1-\\alpha s_1-\\beta s_2+\\theta\\alpha\\beta s_1s_2)\\nonumber\n\\end{eqnarray}\n where\n\\begin{equation}\n\\alpha={at +dt^2 \\over 1+at +bt+dt^2 },\\quad \\beta={bt+dt^2 \\over 1+at +bt+dt^2 },\\quad\np={ dt^2 \\over 1+at +bt+dt^2 }\\label{as1.5q}\n\\end{equation}\nand\n \\begin{equation}\n\\alpha\\beta=p\\(1+d^{-1}+at +bt+dt^2 \\over 1+at +bt+dt^2 \\):=\\frac{p}{\\theta}\\label{as1.5}.\n\\end{equation}\nNote that\n\\begin{equation}\n1-\\theta=\\frac{d^{-1} }{1+d^{-1}+at +bt+dt^2 }\\leq {1 \\over d^{2}t^{2}}  .\\label{as1.5qq}\n\\end{equation}\n\n\n Using these definitions we have \n\\begin{eqnarray}\n  P(a,b,\\lambda_1,\\lambda_2 )&=&-\\log H(a,b,\\lambda_1,\\lambda_2)\n\\label{3.10p}\\\\\n& =&-\\log (1+at +bt+dt^2 )-\\log (1-\\alpha s_1-\\beta s_2+\\theta\\alpha\\beta s_1s_2)   \\nonumber\n\\end{eqnarray}\nand \n\\begin{eqnarray} \n&&\n Q(a,b,\\lambda_1,\\lambda_2 )\\label{3.10}\\\\\n &&\\qquad=\n -\\frac{\\gamma\\lambda_1 \\lambda_2+\\lambda_1+\\lambda_2 }{ H(a,b,\\lambda_1,\\lambda_2)}\\nonumber\\\\\n &&\\qquad= -{1 \\over (1+at +bt+dt^2 )}\\,\\,\\frac{\\gamma t^2(1-s_1)(1-s_2)+t(2-s_1-s_2) }{ 1-\\alpha s_1-\\beta s_2+\\theta\\alpha\\beta s_1s_2}.\\nonumber\n\\end{eqnarray}\n\nWe make some preliminary observations that  enable us to compute the coefficients of the power series expansions of $P(s_1,s_2 ,t)$ and $Q(s_1,s_2 ,t)$. \n  Let $(u_{1},u_{2})\\in [0,1)^{2}$, and $\\theta\\in [0,1)$, and assume that \n  \\begin{equation}\nu_{1}+u_{2}-\\theta u_{1}u_{2}<1.\\label{2.1}\n\\end{equation}\n(It is clearly greater than zero.)\nLet\n\\begin{equation} \n{1\\over 1-u_1-u_2+\\theta u_1u_2}:= \\sum_{j,k=0}^\\infty D_{j,k}u_1^j u_2^k \\label{1.1}\n\\end{equation} \nand\n\\begin{equation} \n-\\log  (1-u_1-u_2+\\theta u_1u_2 ):= \\sum_{ j,k=0 }^\\infty C_{j,k}u_1^j u_2^k .\\label{1.2}\n\\end{equation} \nWe give explicit expressions for $C_{j,k}$ and $D_{j,k}$. To begin  we give several equalities that are easy to verify. \n\n\\begin{lemma} \\label{lem-2.1w}   For $u\\in [0,1)$\n\\begin{equation}\n\\frac{1}{(1-u)^q}=\\sum_{n=0}^\\infty{q+n-1\\choose n} u^n.\n\\end{equation}\n\\end{lemma}\n\n\\noindent{\\bf Proof}$\\quad$ To get  ${q+n-1\\choose n}$ differentiate $ 1\/(1-u)^q$, $n$ times,\ndivide by $n!$ and set $u=0$. {\\hfill $\\square$ }\n\n\\medskip We list the following equalities without proof.    \n\\begin{lemma} \\label{lem-1.2}\n\\begin{eqnarray}\n{k\\choose p}-{k-1\\choose p} &=&{k-1\\choose p-1}\\label{bineq}\\\\\n{k-1\\choose p}&=&{k-p\\over k}{k\\choose p}\\nonumber\\\\\n{k-1\\choose p-1}&=&{p\\over k}{k\\choose p}\\nonumber\\\\\n{k \\choose p}&=&{k-p+1\\over p}{k\\choose p-1}\\nonumber.\n\\end{eqnarray}\n\\end{lemma}\n\n\n\n\\begin{lemma} \\label{lem-2.2q} For $0\\le j\\le k$\n\\begin{equation} \nD_{j,k}=\\sum_{p=0}^j(1-\\theta)^p{j\\choose p}  {k\\choose p} . \\label{2.3} \n\\end{equation}\nAlso, $C_{0,0}=0$, $C_{j,0}= 1\/j,C_{0,k}= 1\/k, j,k\\ne 0$, and for $1\\le j\\le k$\n\\begin{equation} \nC_{j,k}=  \\sum_{p=0}^{j-1}   (1-\\theta )^{p}   {j \\choose p } {k  \\choose p }{(1-\\theta )\\over  p+1}{(j-p)(k-p)\\over jk} . \\label{2.3h} \n\\end{equation}\n\\end{lemma}\n\n\\noindent{\\bf Proof}$\\quad$ Note that  \n\\begin{equation} \n{1\\over 1-u_1-u_2+\\theta u_1u_2}=\\sum_{n=0}^\\infty\\(u_1+u_2-\\theta u_1u_2\\)^n .\n\\end{equation} \n  Writing (\\ref{2.1}) in the form $ u_{1}+u_{2}- u_{1}u_{2}+(1-\\theta) u_{1}u_{2}<1$ we see that it is equivalent to the statement that\n   \\begin{equation}\n   \\frac{(1-\\theta)u_{1 }u_{2}}{(1-u_{1})(1-u_{2})}<1.\n   \\end{equation}\n We write\n\\begin{eqnarray}\n&&{1\\over 1-u_1-u_2+\\theta u_1u_2}\\\\\n&&\\qquad = \\( (1-u_1) (1-u_2)\\(1-{(1-\\theta )u_1u_2\\over (1-u_1) (1-u_2)}\\)\\)^{-1}\n \\nonumber \\\\\n&&\\qquad= {1\\over (1-u_1) (1-u_2)}\\sum_{p=0}^\\infty\\({(1-\\theta ) u_1u_2\\over\n(1-u_1) (1-u_2)}\\)^p\\nonumber\\\\\n&&\\qquad= \\sum_{p=0}^\\infty {(1-\\theta )^p u_1^p u_2^p\\over (1-u_1)^{p+1}\n(1-u_2)^{p+1}} \\nonumber.\n\\end{eqnarray}\n\nUsing this series we can determine $D_{j,k}$. Since $j\\le k$ it is clear that\neach of the first $j+1$ terms in the series immediately above can contribute\na term in $u_1^ju_2^k$. For a given $0\\le p\\le j$ we get $(1-\\theta)^p$ times\nthe   coefficient of  $u_1^{j-p}$ in the power series expansion of\n$1\/(1-u_1)^{p+1}$ and the  coefficient  of $u_2^{k-p}$ in the power series\nexpansion of $1\/(1-u_2)^{p+1}$. We see from Lemma \\ref{lem-2.1w} that they\nare ${j\\choose j-p}={j\\choose  p}$ and ${k\\choose k-p}={k\\choose p}$\nrespectively. Thus we get  (\\ref{2.3}).\n\nTo obtain (\\ref{2.3h}) we write  \n\\begin{eqnarray}\n&&-\\log(1-u_1-u_2+\\theta u_1u_2)\\\\\n&&\\qquad =-\\log((1-u_1)(1-u_2)-(1-\\theta) u_1u_2)\\nonumber\\\\\n&&\\qquad=-\\log(1-u_1)-\\log(1-u_2)-\\log\\(1-{(1-\\theta) u_1u_2\\over (1-u_1)(1-u_2)}\\)\\nonumber\\\\\n&&\\qquad =\\sum_{n=1}^\\infty \\frac{u_1^n}{n}+\\sum_{n=1}^\\infty \\frac{u_2^n}{n}+\\sum_{p=1}^\\infty \n\\frac{1}{p}\\({(1-\\theta ) u_1u_2\\over\n(1-u_1) (1-u_2)}\\)^p\\nonumber.\n\\end{eqnarray}\nThis gives us $C_{j,0}$ and $C_{0,k}$ and, similar  to the computation of $D_{j,k}$,  we can use the last series above to see that   \n \\begin{eqnarray}\nC_{j,k}&=& \\sum_{p=1}^j {(1-\\theta )^p \\over\np } {j-1\\choose p-1} {k-1\\choose p-1} \\\\\n&=& \\sum_{p=0}^{j-1} { (1-\\theta )^{p+1}   \\over\np+1 } {j-1\\choose p } {k-1 \\choose p }\\nonumber\\\\\n&=& \\sum_{p=0}^{j-1} { (1-\\theta )^{p+1}\\over\np+1 } {j-1\\choose p } {k-1 \\choose p }\\nonumber\\\\\n &=& \\sum_{p=0}^{j-1}   (1-\\theta )^{p}   {j \\choose p } {k  \\choose p }{(1-\\theta )\\over  p+1}{(j-p)(k-p)\\over jk} \\nonumber.\n\\end{eqnarray}\nThis gives us (\\ref{2.3h}). {\\hfill $\\square$ } \n\n \n\n  Set  \\begin{equation}\n \\widetilde t={1 \\over \\sqrt{1-\\theta}}.\\label{2.51a}\n \\end{equation}\n By (\\ref{as1.5qq}) we have that \n  \\begin{equation}  \ndt\\leq  \\widetilde t \\leq dt+  2,\\label{2.51a1}\n \\end{equation}\n  for all $t$ sufficiently large.\n\n\n\\begin{lemma}\\label{lem-P}  For all $t$ sufficiently large,  $P_{j,0}(t)= \\alpha^{j}\/{j}$,   $P_{0,k}(t)= \\beta^{k}\/{k}$ and for  all $1\\le j\\le k$  \n\\begin{equation} \nP_{j,k}(t)={  \\alpha^{j} \\beta^{k} \\over \\widetilde t^{2}}\\sum_{p=0}^{j-1}    \\widetilde t^{-2p}   {j \\choose p } {k  \\choose p }{1\\over  p+1}{(j-p)(k-p)\\over jk} . \\label{2.3hb} \n\\end{equation}\n \\end{lemma}\n(See (\\ref{22.2}).)\n\n\\medskip\t\n\\noindent{\\bf Proof}$\\quad$ Note that\nsince $0<\\alpha,\\beta,\\theta<1$, \n\\begin{eqnarray}\n\\alpha s_1+\\beta s_2-\\theta\\alpha\\beta s_1s_2&\\le& \\alpha  +\\beta  -\\theta\\alpha\\beta\\label{4.5l}\\\\\n&=& \\alpha  +\\beta  -\\alpha\\beta+(1-\\theta)\\alpha\\beta\\nonumber\\\\\n&=& -(1-\\alpha)(1-\\beta) +\\frac{\\alpha\\beta}{d^{2}t^{2}}+1+O(1\/t^{3})\\nonumber\\\\\n&=& -\\frac{ab}{d^{2}t^{2}} +\\frac{1}{d^{2}t^{2}}+1+O(1\/t^{3})\\nonumber\\\\\n&=&1-\\frac{1}{d^{2}t^{2}} +O(1\/t^{3})\\nonumber.\n\\end{eqnarray}\n Consequently, for all $t$ sufficiently large \n \\begin{equation}\n 0\\le \\alpha s_1+\\beta s_2-\\theta\\alpha\\beta s_1s_2<1.\\label{4.5m}\n \\end{equation}\n Therefore, by  (\\ref{1.2})\\begin{equation} \n-\\log  (1-\\alpha u_1-\\beta u_2+\\alpha\\beta\\theta u_1u_2 )= \\sum_{ j,k=0 }^\\infty   \\alpha^{j} \\beta^{k} C_{j,k}u_1^j u_2^k .\\label{1.2x}\n\\end{equation} \nThe  lemma now  follows from Lemma \\ref{lem-2.2q} and (\\ref{2.51a}).\n{\\hfill $\\square$ }\n\n \n\n Set  \\begin{equation}\n \\bar t= \\sqrt{1+at+bt+d t^2}\\label{2.51asr}\n \\end{equation}\nand note that  \n  \\begin{equation}  \n  d^{1\/2} t\\leq  \\bar t \\leq d^{1\/2}t+2\\label{2.51asr1}\n \\end{equation}\n  for all $t$ sufficiently large.\n\n\n\\begin{lemma}\\label{lem-2.1j}  For  all $t$ sufficiently large, and $j,k\\ge 1$  \n \\begin{equation} \nQ_{j,0}(t)= -\\frac{ ( \\gamma t^2(\\alpha-1) +t(2\\alpha-1)\n )}{{\\bar {t}}^2} \\alpha^{j-1} \\label{3.29o}\n \\end{equation} \nand\n \\begin{equation} \nQ_{0,k}(t)= -\\frac{ ( \\gamma t^2(\\beta-1) +t(2\\beta-1)\n )}{\\bar t^2} \\beta^{j-1} \\nonumber.\\label{3.30o}\n \\end{equation} \nFurthermore,   for all $t$ sufficiently large and for all   $1\\le j\\le k$\n\\begin{eqnarray}\n\\lefteqn{  Q_{j,k}(t)\\label{2.22js}}\\\\&&={\\alpha^{j-1} \\beta^{k-1}\\over \\bar t^2}\\sum_{p=0}^{j}\\widetilde t^{-2p}{j\\choose p}{k\\choose p}\\nonumber\\\\\n&&\\quad\\(-\\gamma t^2\\((1-\\alpha)(1-\\beta)-{(1-\\beta)p\\over j}-{(1-\\alpha)p\\over k}+\\widetilde t^{-2}{j-p\\over j}{k-p\\over k}\n\\)\\right.\n\\nonumber\\\\\n&&\\qquad\\left.+t\\(\\alpha\\(1-\\beta-\\frac{p}{k}\\)+\\beta\\(1-\\alpha-{p\\over j}\\) \\)\\)\\nonumber.\n    \\end{eqnarray}  \n\\end{lemma}\n\n\\noindent{\\bf Proof}$\\quad$\n It follows from (\\ref{4.5m}) that for $0\\le s_1,s_2\\le 1$   \n\\begin{equation}\n\\frac{1 }{ 1+a\\lambda_1+b\\lambda_2+d\\lambda_1\\lambda_2}=\\frac{1}{\\bar t^2}\\sum_{n=0}^\\infty\n(\\alpha s_1+\\beta s_2-\\theta\\alpha\\beta s_1s_2)^n.\\label{as1.1}\n\\end{equation}\n Using this along with (\\ref{3.10})   we see that \n\\begin{eqnarray}\nQ(s_1,s_2,t )\n&=&-\\frac{ (\\gamma t^2(1-s_1)(1-s_2)+t(2-s_1-s_2))}{\\bar t^2}\\label{as1.3}\\\\\n&&\\qquad\n\\cdot \\sum_{n=0}^\\infty  (\\alpha s_1+\\beta s_2-\\theta\\alpha\\beta s_1s_2)^n\\nonumber\\\\\n&=&-\\frac{ (\\gamma t^2+2t-(\\gamma t^2+t)\ns_1-(\\gamma t^2+t)s_2   +\\gamma t^2 s_1s_2)}{\\bar t^2}\\nonumber\\\\ &&\\qquad\n\\cdot\\sum_{n=0}^\\infty  (\\alpha s_1+\\beta s_2-\\theta\\alpha\\beta s_1s_2)^n\\nonumber .\n\\end{eqnarray}\n\n  Consequently\n \\begin{equation} \nQ_{j,0}(t )=-\\frac{ ((\\gamma t^2+2t)\\alpha-(\\gamma t^2+t)\n )}{\\bar t^2} \\alpha^{j-1},\n \\end{equation} \nfrom which we get (\\ref{3.29o}), and \n \\begin{equation} \nQ_{0,k}(t )=-\\frac{ ((\\gamma t^2+2t)\\beta-(\\gamma t^2+t)\n )}{\\bar t^2} \\beta^{j-1},\n \\end{equation} \n from which we get (\\ref{3.30o}). \n\n \n  To obtain (\\ref{2.22js}) we use (\\ref{as1.3}), and the terms   of $D_{j,k}$ defined in (\\ref{1.1}),  to see that   \n\\begin{eqnarray}\n&&{\\mathcal Q}_{j,k}:= {\\bar t^2\\,Q_{j,k}\\over  \\alpha^{j-1} \\beta^{k-1}}\\label{1.20}\\\\\n& &\\qquad\\, \\,=-\\gamma t^2\\( D_{j,k}\\alpha\\beta -D_{j,k-1}\\alpha-D_{j-1,k}\\beta+ D_{j-1,k-1}\\)\\nonumber\\\\\n& &\\qquad \\qquad-t\\( 2D_{j,k}\\alpha\\beta  -D_{j,k-1}\\alpha-D_{j-1,k}\\beta \\)\\nonumber.\n\\end{eqnarray}\n  Using (\\ref{1.20}),  (\\ref{2.3}) and Lemma \\ref{lem-1.2} we see that \n\\begin{eqnarray}\n{\\mathcal Q}_{j,k}&=&\\sum_{p=0}^{j}\\widetilde t^{-2p}{j\\choose p}{k\\choose p}\\label{2.22}\\\\\n&&\\quad\\(-\\gamma t^2\\(\\alpha\\beta-\\alpha{k-p\\over k}-\\beta{j-p\\over j}+{j-p\\over j}{k-p\\over k}\n\\)\\right.\n\\nonumber\\\\\n&&\\qquad\\left.-t\\(2\\alpha\\beta-\\alpha{k-p\\over k}-\\beta{j-p\\over j} \n\\)\\)\\nonumber .\n    \\end{eqnarray}\n Consequently,  for all $t$ sufficiently large, for all $1\\le j\\le k$  \n\\begin{eqnarray}\nQ_{j,k}(t )&=&{ \\alpha^{j -1} \\beta^{k-1}\\over \\bar t^2}\\sum_{p=0}^{j}\\widetilde t^{-2p}{j\\choose p}{k\\choose p}\\label{2.22j}\\\\\n&&\\quad\\(-\\gamma t^2\\(\\alpha\\beta-\\alpha{k-p\\over k}-\\beta{j-p\\over j}+{j-p\\over j}{k-p\\over k}\n\\)\\right.\n\\nonumber\\\\\n&&\\qquad\\left.-t\\(2\\alpha\\beta-\\alpha{k-p\\over k}-\\beta{j-p\\over j} \n\\)\\)\\nonumber .\n    \\end{eqnarray}  \n \n  Consider  (\\ref{2.22j}).\nWe write\n\\begin{eqnarray}\n&&\\alpha\\beta-\\alpha{k-p\\over k}-\\beta{j-p\\over j}+{j-p\\over j}{k-p\\over k}\\label{2.23s}\\\\\n&&\\qquad=(1-\\alpha)(1-\\beta)-{(1-\\beta)p\\over j}-{(1-\\alpha)p\\over k}+{p^2\\over jk},\\nonumber\n\\end{eqnarray}\n\\begin{equation}\n2\\alpha\\beta-\\alpha{k-p\\over k}-\\beta{j-p\\over j}=-\\alpha\\(1-\\beta-\\frac{p}{k}\\)-\\beta\\(1-\\alpha-{p\\over j}\\),\\label{2.24s}\n\\end{equation}\nto obtain  \n\\begin{eqnarray}\nQ_{j,k}(t )\\label{2.22jt}\\\\&=&{  \\alpha^{j-1} \\beta^{k-1}\\over \\bar t^2}\\sum_{p=0}^{j}\\widetilde t^{-2p}{j\\choose p}{k\\choose p}\\nonumber\\\\\n&&\\quad\\(-\\gamma t^2\\((1-\\alpha)(1-\\beta)-{(1-\\beta)p\\over j}-{(1-\\alpha)p\\over k}+{p^2\\over jk}\n\\)\\right.\n\\nonumber\\\\\n&&\\qquad\\left.+t\\(\\alpha\\(1-\\beta-\\frac{p}{k}\\)+\\beta\\(1-\\alpha-{p\\over j}\\) \\)\\)\\nonumber.\n    \\end{eqnarray} \n   Note that for  $1\\le q\\le j$\n \\begin{eqnarray}\n &&\\widetilde t^{-2q}{j\\choose q}{k\\choose q}\\({ q^2\\over jk}\\)\\\\\n &&\\qquad=\\widetilde t^{-2(q-1)}{j\\choose q-1}{k\\choose q-1}  {j-(q-1)\\over j}{k-(q-1)\\over k}\\widetilde t^{-2}.\\nonumber\n \\end{eqnarray}\n Therefore, for each $1\\le p\\le j$ we incorporate the term in $ p^2\/ jk$\nin (\\ref{2.22jt}) into the  preceding term in the series in (\\ref{2.22jt}) to get \n (\\ref{2.22js}).    (Note that we can not add anything to the $p=j$ term. The expression in (\\ref{2.22js}) reflects this fact since $ {j-p\\over j}{k-p\\over k}=0$ when $p=j$.)\n{\\hfill $\\square$ }\n \n \n \n\n\n\\section{ A sufficient condition for a vector in $R^{2}$ to be infinitely divisible.}\\label{sec-4}\n\nWe present a sufficient  condition for   a random vector in $R^{2}$ to be to be infinitely divisible, and show how it simplifies the task of showing that $((G_{1}+c)^{2},(G_{2}+c)^{2})$    is infinitely divisible.\n\n\\begin{lemma} \\label{lem-1.1}  Let $\\psi : (R_{+})^{2}\\to  (R_{+})^{2}$ be a continuous function  with $\\psi (0,0)=1$. Let   ${\\bf s}\\in  [0,1]^{2}$ and suppose that for all $t>0$ sufficiently large, $\\log \\psi(t(1-s_{1}), t(1-s_{2}))$ has a power series expansion at $\\bf{s=0}$ given by  \n \\begin{equation}\n \\phi (t;s_{1},s_{2})=  \\sum_{j,k=0}^{\\infty} b_{j,k}(t)s_{1}^{j}s^{k}_{2}.\\label{1.2sr}\n   \\end{equation}\n \n   \n   \n  Suppose also that there exist an increasing sequence of  finite subsets   ${\\mathcal N}_{i}\\subseteq   \\mathbf{N^{2}}$, $i\\ge 1$, with $\\bigcup_{i=1}^{\\infty}{\\mathcal N}_{i}= \\mathbf{N^{2}}$, and  a sequence $ t_{i}\\to\\infty $, $i\\ge 1$, such that $b_{j,k}(t_{i})\\ge 0$  for all   $(j,k)\\in {\\mathcal N}_{i}\/(0,0)$ and  \n  \\begin{equation}\n   \\lim_{ i\\to \\infty}\\sum_{j,k\\notin N_{i}\\cup (0,0)}^{\\infty} |b_{j,k}(t_{i})|=0.\\label{1.2s}\n   \\end{equation}\n Then  $\\psi(\\lambda_{1},\\lambda_{2})$ is the  Laplace transform of an infinitely divisible random variable on $(R_{+})^{2}$.\n \\end{lemma}\n \n \n \n \n \n \\noindent{\\bf Proof}$\\quad$    It is clear from (\\ref{1.2s}) that the power series in (\\ref{1.2sr}) converges absolutely for all ${\\bf s}\\in  [0,1]^{2}$.\n Let\n    \\begin{equation}\n \\phi_{i}(t_{i};s_{1},s_{2})=   b_{0,0}(t_{i}) + \\sum_{j,k\\in N_{i}\/(0,0) } b_{j,k}(t_{i})s_{1}^{j}s^{k}_{2}.\n   \\end{equation} \n   Set  \n   \\begin{equation}\n   \\Psi_{i}\\(t_{i}; e^{-\\lambda_{1}\/t_{i}},  e^{-\\lambda_{2}\/t_{i}}\\)=\\exp\\(  \\phi_{i} \\(t_{i}; e^{-\\lambda_{1}\/t_{i}},  e^{-\\lambda_{2}\/t_{i}}\\) \\).\n   \\end{equation}\n   We show that for each  $(\\lambda_{1},\\lambda_{2})\\in (R_{+})^{2}$ \n       \\begin{equation}\n   \\lim_{i\\to\\infty} \\Psi_{i}\\(t_{i}; e^{-\\lambda_{1}\/t_{i}},  e^{-\\lambda_{2}\/t_{i}}\\)=\\psi (\\lambda_{1},\\lambda_{2}).\\label{1.6}\n   \\end{equation}\n   \n   As we point out in  \\cite[page 565]{book}, $    \\Psi_{i}\\(t_{i}; e^{-\\lambda_{1}\/t_{i}},  e^{-\\lambda_{2}\/t_{i}}\\)$ is the Laplace transform of a discrete    measure. It then follows from the continuity theorem and the fact that $\\psi (0,0)=1$ that\n$\\psi(\\lambda_{1},\\lambda_{2})$ \n  is the Laplace transform   of a random variable. \n Furthermore repeating this argument with $ \\phi_{i} (t_{i};s_{1},s_{2})$ replaced by $  \\phi_{i} (t_{i};s_{1},s_{2})\/n$ shows that $\\psi^{1\/n}(\\lambda_{1},\\lambda_{2}) $ is the Laplace transform of a random variable. This shows that $\\psi(\\lambda_{1},\\lambda_{2})$ is the  Laplace transform of an infinitely divisible random variable on $(R_{+})^{2}$.\n  \nLet\n \\begin{equation}\n\\delta_{i}:=\\Big |\\psi(t_{i}(1-e^{-\\lambda_{1}\/t_{i}}), t_{i}(1-e^{-\\lambda_{2}\/t_{i}}))-\\psi(\\lambda_{1},\\lambda_{2})\\Big|.\\label{1.7}\n   \\end{equation} \n   Clearly $    \\lim_{i\\to\\infty}\\delta_{i}=0$.  \nBy (\\ref{1.2s})  \n\\begin{eqnarray}\n\\lefteqn{\\Big| \\psi (t_{i}(1-e^{-\\lambda_{1}\/t_{i}}), t_{i}(1-e^{-\\lambda_{2}\/t_{i}})) - \\exp \\( \\phi_{i}(t_{i};e^{-\\lambda_{1}\/t_{i}},e^{-\\lambda_{2}\/t_{i}})\\)\\Big|} \\nonumber \\\\\n&&=\\Big| \\exp \\( b_{0,0}(t_{i})+ \\sum_{(j,k)\\neq(0,0) } b_{j,k}(t_{i}) e^{-j\\lambda_{1}\/t_{i}}e^{-k\\lambda_{2}\/t_{i}}\\)\\\\\n&&\\hspace{.1 in}- \\exp  \\( b_{0,0}(t_{i})+ \\sum_{j,k\\in N_{i}\/(0,0) } b_{j,k}(t_{i}) e^{-j\\lambda_{1}\/t_{i}}e^{-k\\lambda_{2}\/t_{i}}\\)\\Big|\\nonumber\\\\\n&&= \\psi(t_{i}(1-e^{-\\lambda_{1}\/t_{i}}), t_{i}(1-e^{-\\lambda_{2}\/t_{i}}))\\nonumber\\\\\n&&\\hspace{1 in}  \\Big|1-\\exp \\Big( -\\sum_{j,k\\notin  {\\mathcal N}_i\\cup (0,0)}^{\\infty} b_{j,k}(t_{i}) e^{-j\\lambda_{1}\/t_{i}}e^{-k\\lambda_{2}\/t_{i}}\\Big)\\Big| \\nonumber\\\\\n&&= \\epsilon_{i} \\psi (t_{i}(1-e^{-\\lambda_{1}\/t_{i}}), t_{i}(1-e^{-\\lambda_{2}\/t_{i}})) ,\\nonumber\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\epsilon_{i} &:=& \\Big|\\Big(1-\\exp \\Big( -\\sum_{j,k\\notin N_{i}\\cup (0,0)}^{\\infty} b_{j,k}(t_{i}) e^{-j\\lambda_{1}\/t_{i}}e^{-k\\lambda_{2}\/t_{i}}\\Big)\\Big)\\Big|.\n\\label{1.10j}\n\\end{eqnarray}\nNote that by (\\ref{1.2s})\n\\begin{equation}\n\\lim_{i\\rightarrow\\infty}\\epsilon_{i}=0.\\label{1.10k}\n\\end{equation}\n Therefore, by the triangle inequality,  (\\ref{1.7})  and (\\ref{1.10k})\n     \\begin{eqnarray}\n&&\\Big| \\exp \\( \\phi_{i} (t_{i};e^{-\\lambda_{1}\/t_{i}},e^{-\\lambda_{2}\/t_{i}})\\) -  \\psi(\\lambda_1,\\lambda_{2})\\Big|\\label{1.12}\\\\\n&&\\qquad\\leq \\,\\epsilon_{i} \\psi (t_{i}(1-e^{-\\lambda_{1}\/t_{i}}), t_{i}(1-e^{-\\lambda_{2}\/t_{i}}))+  \\delta_{i} .\\nonumber\n\\end{eqnarray}   \n Using     (\\ref{1.7}) we see that this is \n \\begin{eqnarray}\n  &&   \\le  \\epsilon_{i}\\(  \\psi(\\lambda_{1},\\lambda_{2}) +\\delta_{i}\\)+\\delta_{i}.  \\nonumber \\end{eqnarray}\n  Thus we justify (\\ref{1.6}) and the  paragraph following it. {\\hfill $\\square$ }\n\n  \\begin{remark} {\\rm In \\cite[Lemma 13.2.2]{book} we present the well known result that the conclusion of Lemma  \\ref{lem-1.1} holds when $\\log \\psi(t(1-s_{1}), t(1-s_{2}))$ has a power series expansion at $\\bf{s=0}$ with all its coefficients, except for the coefficient of the constant term, are positive. Lemma  \\ref{lem-1.1} is useful because it allows us to only verify this condition for a subset of these coefficients,   (depending on $t$). \n }\\end{remark}\n \n The following lemma enables us to apply Lemma \\ref{lem-1.1}. \n     \n    \\begin{lemma}\\label{lem-cutoff}\nFor any  $c_{3 }>0$  there exists a constant   $B=B(\\gamma, d,c_{3})$  for which\n    \\begin{equation}\n    \\mathcal{N}_{t}=\\{(j,k)\\,|\\,\\sqrt{jk}\\leq  Bt\\log t\\}\\label{2.22f}\n    \\end{equation}\n  has the property that\n      \\begin{equation}\n    \\lim_{t\\to\\infty}\\sum_{(j,k)\\notin   \\mathcal{N}_{t}}|R_{j,k}(t,c)|=0\\label{2.22kr},\n    \\end{equation} \n    uniformly in $|c|\\leq c_{3}$.\n    \\end{lemma}\n    \n  \n    \n    \\begin{remark}\\label{rem-4.2} {\\rm It follows from Lemmas \\ref{lem-1.1} and \\ref{lem-cutoff} that in order to prove Theorem \\ref{theo-cp} we need that only show   that we can find   a $c_{0}>0$, such that    \\begin{equation}\n   R_{j,k}(t,c_{0})\\ge 0\\qquad \\mbox{for all }\\quad \\sqrt{jk}\\leq Bt\\log t,\n   \\end{equation}\nfor any constant $B$,  for all  $t$ sufficiently large,   \n(except for $R_{0,0}(t)$).\n }\\end{remark}\n \nBefore proving   Lemma \\ref{lem-cutoff} we   establish the following bounds\n\n\\begin{lemma} \\label{2.4}\n\\begin{equation}\n{j\\choose p}\\le \\({ej\\over p}\\)^{p}\\label{4.14}\n\\end{equation}\nand\n\\begin{equation} \n\\frac1{\\tau^{2p}}{j\\choose p}  {k\\choose p}\\le  \\({e\\sqrt{jk}\\over p\\, \\tau}\\)^{2p}\\le \\exp\\({2\\sqrt{jk}\\over \\tau}\\) \\label{2.12}\n \\end{equation}\n\n \\end{lemma}\n \n \\noindent{\\bf Proof}$\\quad$\n It is clear that \n \\begin{equation}\n   {j\\choose p}\\le   {j^{p}\\over p!}. \n   \\end{equation}\n   Therefore to prove (\\ref{2.4}) we need only show that\n   \\begin{equation}\n p!\\({e\\over p}\\)^{p}\\ge 1.\\label{2.13}\n   \\end{equation}\n  In \\cite[page 42 ]{Feller1}, Feller shows that $ p!\\({e\/ p}\\)^{p}$ is increasing in $p$. Since it is equal to $e$ when $p=1$, (and 1 when $p=0$), we get (\\ref{2.13}).\n  \n  The first inequality in (\\ref{2.12}) follows from (\\ref{4.14}) the next one is obtained by maximizing the middle term with respect to $p$.\n  {\\hfill $\\square$ }\n   \n    \n\n\n \\medskip\t\n \n \\noindent   \n{\\bf  Proof of Lemma \\ref{lem-cutoff}} By (\\ref{22.2u}) and Lemmas \\ref{lem-P} and \\ref{lem-2.1j}\nwe see that for all $t$ sufficiently large, for all $1\\le j\\le k$, \n    \\begin{equation}\n   |R_{j,k}(t,c )|\\le C\\alpha^{j}\\beta^{k} \\sum_{p=0}^{j}\\widetilde t^{-2p}{j\\choose p}{k\\choose p}\n      \\end{equation}\nwhere $C$ depends on $c,\\gamma$ and $d$ but not on $j$, $k$,  $t$ or   $\\widetilde t$. Furthermore, $C$ is bounded for all $|c|\\le T$, for any finite number $T$.\n(We also use the fact that $\\lim_{t\\to\\infty}\\alpha\\beta=1$.)\n\n  For any $\\delta>0$,   for  $t$ sufficiently large,  \n \\begin{equation}\n\\alpha={at +dt^2 \\over 1+at +bt+dt^2 }= 1- {1+bt \\over 1+at +bt+dt^2 }\\leq 1-{(1-\\delta)b \\over dt}\\leq e^{-(1-\\delta)b\/(dt)} \\label{2.22g}\n \\end{equation}\n and  \n  \\begin{equation}\n\\beta={bt+dt^2 \\over 1+at +bt+dt^2 }= 1- {1+at \\over 1+at +bt+dt^2 }\\leq 1-{(1-\\delta)a \\over d t}\\leq e^{-(1-\\delta)a\/(dt)}.\\label{2.22h}\n \\end{equation}\n \n \n   Using these estimates along with  (\\ref{2.51a1}) we see that  for  all $t$ sufficiently large, for all $1\\le j\\le k$, \n \\begin{equation}\n|R_{j,k}(t,c)|\\leq  Ce^{-k(1-\\delta)a\/(dt)}e^{-j(1-\\delta)b\/(dt)}\\,\\sum_{p=0}^{j }{1 \\over(d t)^{2p}}{j\\choose p}{k\\choose p}\\label{2.22i}\n \\end{equation}\n  uniformly in          $|c|\\leq c_{3}$.  \n \n Suppose that $\\sqrt{jk}\/(dt)=n$. Then \n   \\begin{eqnarray}\n   &&\ne^{-k(1-\\delta)a\/(dt)}e^{-j(1-\\delta)b\/(dt)}\\label{2.22j1}\\\\\n&&\\qquad=\\exp\\(-(1-\\delta)\\( a\\sqrt{k\/j}+ b\\sqrt{j\/k}\\)n\\)\\nonumber\\\\\n   & &\\qquad\\le\\exp\\(-2 (1-\\delta) \\sqrt{ab} n\\)\\nonumber\\\\\n    & &\\qquad=\\exp\\(- 2(1-\\delta)\\sqrt{d+1}n\\)\\nonumber,\n   \\end{eqnarray}\n   where, for the inequality we take the minimum of $a\\theta+b\/\\theta$ and for the equality we use the fact that $ab=d+1$. \n   Combined with (\\ref{2.12}) this shows that when $\\sqrt{jk}\/(dt)=n$\n   \\begin{eqnarray}\n|R_{j,k}(t,c)| \n   &\\le& C j  \\exp\\(- 2((1-\\delta)\\sqrt{d+1}-1)n\\).\\label{2.22jx}\n   \\end{eqnarray}    \nLet $A_{n}=\\{(j,k)| n\\le \\sqrt{jk}\/(dt)<n+1\\}$. Then   for any $M$\n   \\begin{equation}\n     \\sum_{  \\sqrt{jk}\/(dt) \\geq M}|R_{j,k}(t,c)|=  \\sum_{n=M}^{\\infty} \\sum_{(j,k)\\in A_{n}} |R_{j,k}(t,c)|\\label{2.22k} .\n   \\end{equation}\nWe note that the cardinality of $A_{n}$ is less than  \n\\begin{equation} 3nd^{2}t^{2}\\log ( (n+1)d t) \\label{dfd}.\\end{equation}\n  This is because $(j,k)\\in A_{n}$ implies that \n  \\begin{equation}\n   {n^{2}d^{2}t^{2}\\over j}\\le k\\le  {(n+1)^{2}d^{2}t^{2}\\over j}\\quad\\mbox{and}\\quad j\\le (n+1)dt \n   \\end{equation}\n  and summing on $ j$  we get (\\ref{dfd}).\n\n\n It follows from (\\ref{2.22j}) and (\\ref{2.22k}) that \n  \\begin{eqnarray}\n    && \\sum_{  \\sqrt{jk}\/(dt) \\geq M }|R_{j,k}(t,c)| \\label{2.22jx} \\\\\n   & &\\qquad\\le C ( dt)^{4} \\sum_{n=M}^{\\infty}n^{4}\n    \\exp\\(- 2((1-\\delta)\\sqrt{d+1}-1)n\\)\\nonumber  \\\\\n    & &\\qquad\\le C {( dt)^{4}\\over (1-\\delta)\\sqrt{d+1}-1)}  M^{4} \\exp\\(- 2((1-\\delta)\\sqrt{d+1}-1)M\\)\\nonumber.\n   \\end{eqnarray}\nClearly, there exists a constant $B$ such that when  $M=B\\log  t$,  this last term is  $o(1)$ as $t\\to\\infty$. {\\hfill $\\square$ }\n   \n \n \n \n   \\section{ Proof of Theorem  \\ref{theo-cp}  when {$\\bf (c_{1},c_{2})=(c,c)$} and ${\\bf EG_{1} G_{2} >0}$}\\label{sec-5}\n\n  In this section we prove Theorem \\ref{theo-cp}  in case 1. and for $  EG_{1} G_{2} >0 $,   by   establishing the positivity conditions on the coefficients $R_{j,k}(t,c) $, (when $  EG_{1} G_{2} =1$),  as  discussed in Remark \\ref{rem-4.2}. We pass to the case $  EG_{1} G_{2}>0$ on page \\pageref{sec5page}.  \n  \n  \nTo proceed we need several estimates of parameters we are dealing with as $t\\to\\infty$. They follow   from the definitions in (\\ref{as1.5q})--(\\ref{as1.5qq}).\n \n \\begin{lemma} \\label{lem-2.2}As $t\\to\\infty$\n\\begin{eqnarray}\n1-\\alpha&=&{b\\over dt}- {1+b^{2}\\over (dt)^2 }+O(t^{-3})\\label{3.18}\\\\\n1-\\beta&=&{a\\over dt}- {1+a^{2}\\over (dt)^2 }+O(t^{-3})\\nonumber\\\\\n(1-\\alpha)(1-\\beta)&=&{\nd+1\\over (dt)^2}- {a(1+b^{2})+b(1+a^{2})\\over (dt)^3 }+O(t^{-4})\\nonumber\\\\\n\\widetilde t^{-2}=1-\\theta &=&{1\\over (dt)^2}-{a+b\\over (dt)^3}+O(t^{-4})\\nonumber\\\\\n\\alpha^{j}&=&e^{-bj\/(dt)+O(j^{2}\/t^{2})}\\nonumber\\\\\n\\beta^{k}&=&e^{-ak\/(dt)+O(k^{2}\/t^{2})}\\nonumber.\n\\end{eqnarray}\n\nAlso\n\\begin{eqnarray}\n-(d+2)\\gamma+d(a+b)&=&2((d+2)-(a+b))=-2\\zeta\\label{2.28}\\\\\na\\gamma-d&=&(a-1)^2\\nonumber\\\\\nb\\gamma-d&=&(b-1)^2\\nonumber.\n\\end{eqnarray}\n\\end{lemma}\n \n \\noindent{\\bf Proof}$\\quad$ \n \\begin{eqnarray}\n 1-\\alpha&=&{1 +bt \\over 1+at +bt+dt^2 }\n \\label{3.18a}\\\\\n &=&  {1 +bt \\over dt^2 }\\,\\,{1  \\over 1+a(dt)^{-1} +b(dt)^{-1}+d^{-1}t^{-2} } \\nonumber\\\\\n &=&  {1 +bt \\over dt^2 }\\,(1-a(dt)^{-1} -b(dt)^{-1}+O(t^{-2}))\\nonumber\\\\\n &=& {b\\over dt}+ {d -b(a+b)\\over d^{2}t^2 }+O(t^{-3})\\nonumber\\\\\n &=& {b\\over dt}- {1+b^{2}\\over d^{2}t^2 }+O(t^{-3})\\nonumber\n \\end{eqnarray}\n The rest of the lemma follows similarly.\n {\\hfill $\\square$ }\n\n  \\medskip\t  \\noindent  \n{\\bf Proof of Theorem \\ref{theo-cp}      when    ${\\bf c_{1}=c_{2}=c}  $}.     To begin let note that it is easy to see from Lemma \\ref{lem-P},  that $P_{j,k}(t)\\ge 0$ for all $0\\le j,k<\\infty$, with the exception of $P_{0,0}(t)$. This must be the case because $\\exp(P(a,b,\\lambda_{1},\\lambda_{2}))$ is the Laplace transform of an infinitely divisible random variable, as we remark following the proof of Lemma \\ref{lem-2.1}.\n\nBy (\\ref{3.29o})  \n\\begin{eqnarray}\nQ_{j,0}(t)  &=&-\\frac{ ( \\gamma t^2(\\alpha-1) +t(2\\alpha-1)\n )}{\\bar t^2} \\alpha^{j-1} \\label{5.4}\n \\\\\n &=&-\\frac{ (( -\\gamma b+1)t +\\gamma(1+b^{2})-2b+O(1\/t) )\n }{\\bar t^2} \\alpha^{j-1} \\nonumber\\\\\n &=& { \\((b-1)^{2}t+2b-\\gamma(1+b^{2}) +O(1\/t)\\)\\over {\\bar t^2}}\\alpha^{j-1}\\nonumber\\\\\n  &=&\\( {  (b-1)^{2} \\over dt}+O(1\/t^{2})\\)\\alpha^{j-1}\\nonumber.\n \\end{eqnarray} \n\nSimilarly\n \\begin{eqnarray}\nQ_{0,k}(t )  &=&  \\( {  (a-1)^{2} \\over d t}+O(1\/t^{2})\\)\\alpha^{k-1} .\\label{3.23}\n \\end{eqnarray} \n Thus  we see that   there exists a $t_{1}$ sufficiently large such that for all $t\\geq t_{1}$, $R_{j,0}(t,c)$ and $R_{0,k}(t,c)$ are both positive for all $j,k\\ge 1$.\n\n\\medskip\t\nWe now examine $R_{j,k}(t,c) $ for $j\\wedge k\\ge 1$, $j\\le k$.\nWe write\n   \\begin{equation}\n   R_{j,k}(t,c)=\\sum_{p=0}^{j}   R_{j,k,p}(t,c)\\label{2.4c}.\n    \\end{equation}\n Using (\\ref{2.22js}) and (\\ref{2.3hb}) we see that\n    \\begin{eqnarray}\n    \\lefteqn{{R_{j,k,p}(t,c) \\over  \\alpha^{j-1}\\beta^{k-1}}    \\label{2.4f} }\\\\\n    &&= {\\alpha\\beta \\widetilde t^{-2p} \\over \\widetilde t^{2}}  {j \\choose p } {k  \\choose p }{1\\over  p+1}{(j-p)(k-p)\\over jk}\n  +{c^2\\widetilde t^{-2p}\\over \\bar t^2}{j\\choose p}{k\\choose p}\\nonumber\\\\\n&&\\quad\\(-\\gamma t^2\\((1-\\alpha)(1-\\beta)-{(1-\\beta)p\\over j}-{(1-\\alpha)p\\over k}+\\widetilde t^{-2}{j-p\\over j}{k-p\\over k}\n\\)\\right. \\nonumber\\\\\n&&\\quad\\qquad\\left.+t\\(\\alpha\\(1-\\beta-\\frac{p}{k}\\)+\\beta\\(1-\\alpha-{p\\over j}\\) \\)\\).\\nonumber\n    \\end{eqnarray}\nWhen $p=0$\nwe get\n      \\begin{eqnarray}\n    &&{R_{j,k,0}(t,c) \\over  \\alpha^{j-1}\\beta^{k-1}}={\\alpha\\beta \\over \\widetilde t^{2}}  \n    +{c^2\\over \\bar t^2} \\label{2.4f0}\\\\\n&&\\quad\\quad\\(-\\gamma t^2\\((1-\\alpha )(1-\\beta )+\\widetilde t^{-2}\\)+t\\(\\alpha(1-\\beta )+\\beta(1-\\alpha ) \\) \\)\\nonumber\n    \\end{eqnarray}\n   which is independent of $j,k$.   Using Lemma \\ref{lem-2.2} we see that  \n   \\begin{eqnarray}\n   &&-\\gamma t^2\\((1-\\alpha )(1-\\beta )+\\widetilde t^{-2}\\)+t\\(\\alpha(1-\\beta )+\\beta(1-\\alpha ) \\)\n   \\label{2.4f1}\\\\ \n   &&\\qquad=-{(d+2) \\over d^{2}}\\gamma +{a+b \\over d}+O\\( {1 \\over t} \\) \\nonumber\\\\\n   &&\\qquad={-(d+2)\\gamma+d(a+b)  \\over d^{2}}+O\\( {1 \\over t} \\) \\nonumber\\\\\n   &&\\qquad  ={-2\\zeta\\over d^{2}}+O\\( {1 \\over t} \\) \\nonumber.\n   \\end{eqnarray}\n    Using  this and Lemma \\ref{lem-2.2} again we get \n   \\begin{equation}\n  {R_{j,k,0}(t,c) \\over  \\alpha^{j-1}\\beta^{k-1}}= {1-2c ^2(\\zeta\/d)+O\\( 1\/t \\)\\over d^{2} t^2}\\label{2.4fx}\n   \\end{equation}\n   where the $O\\( 1\/t\\)$ term is independent of $j$ and $k$.  \n   \n      \n   \\medskip\t  \n   \n   We now simplify the expression of the other coefficients   $R_{j,k,p}(t,c)$, $1\\le p\\le j$.    Set  \n    \\begin{equation}\n{R_{j,k,p}(t,c)\\over \\alpha^{j-1}\\beta^{k-1}}=\\widetilde t^{-2p}{j\\choose p}{k\\choose p}\\({1 \\over \\widetilde t^2}F_{j,k,p}(t)+{c^2\\over \\bar t^2} A_{j,k,p}(t)\\)\\label{5.10}\n   \\end{equation}\n   where\n \\begin{equation}\n  F_{j,k,p}(t)  =\\alpha\\beta    {1\\over  p+1}{(j-p)(k-p)\\over jk}\\label{5.11}\n   \\end{equation} \n   and \n    \\begin{eqnarray}\n     \\lefteqn{A_{j,k,p}(t)\\nonumber}\\\\\n   &&=-\\gamma t^2  \\((1-\\alpha)(1-\\beta)-{(1-\\beta)p\\over j}-{(1-\\alpha)p\\over k}+\\widetilde t^{-2}{j-p\\over j}{k-p\\over k}\\)\n \\nonumber\\\\\n   && \\qquad + t    \\(\\alpha\\(1-\\beta-\\frac{p}{k}\\)+\\beta\\(1-\\alpha-{p\\over j}\\) \\)   \\label{5.12}.\n   \\end{eqnarray} \n \n     Using Lemma \\ref{lem-2.2}  we have  \n   \\begin{eqnarray}\n   &&-\\gamma t^2  \\((1-\\alpha)(1-\\beta)-{(1-\\beta)p\\over j}-{(1-\\alpha)p\\over k}+\\widetilde t^{-2}{j-p\\over j}{k-p\\over k}\\)\n \\nonumber\\\\\n &&=-{\\gamma \\over d^{2}}\\(d+1  -{(bdt-(1+b^{2}))p\\over k}-{(adt-(1+a^{2}))p\\over j} +{j-p\\over j}{k-p\\over k}\\)\\nonumber\\\\\n   &&\\hspace{4 in}+O\\( {1 \\over t} \\)\n   \\nonumber\\\\\n   && ={\\gamma \\over d^{2}}\\(-(d+2)  +{(bdt-b^{2})p\\over k}+{(adt-a^{2})p\\over j} -{p^{2}\\over jk}\\)+O\\( {1 \\over t} \\) \\label{2.4f2}\\\\\n   && ={\\gamma \\over d^{2}}\\(-(d+2)   +{b(dt-b)p\\over k}+{a(dt-a)p\\over j} -{p^{2}\\over jk}\\)+O\\( {1 \\over t} \\) \\nonumber\n   \\end{eqnarray}\n   and\n   \\begin{eqnarray}\n   &&  t    \\(\\alpha\\(1-\\beta-\\frac{p}{k}\\)+\\beta\\(1-\\alpha-{p\\over j}\\) \\)\n   \\label{2.4f3}\\\\\n   &&\\qquad= {a \\over d}\\(1+{p \\over j}\\)+ {b \\over d}\\(1+{p \\over k}\\)-{  pt\\over j}-{ pt\\over k} +O\\( {1 \\over t} \\)  \\nonumber\\\\\n   &&\\qquad= {1 \\over d} \\( a+b-{  p(dt-a)\\over j}-{ p(dt-b)\\over k}\\) +O\\( {1 \\over t} \\)  \\nonumber.\n   \\end{eqnarray}\n   \n   In (\\ref{2.4f2}) and (\\ref{2.4f3}) the expressions $O\\( {1 \/ t} \\)$\nare not necessarily the same from line to line. Nevertheless, it is important to note that they are   independent of $p$, $j$ and $k$. That is there exists an $M>0$ such that all terms given as   $O\\( {1 \/t} \\)$ in (\\ref{2.4f2}) and (\\ref{2.4f3})  satisfy\n\\begin{equation}\n   -\\frac{M}{t}< O\\( {1 \\over t} \\)<\\frac{M}{t}.\n   \\end{equation}\n   This is easy to see since the  $O\\( {1 \/t} \\)$  terms, in addition to depending on $t$, depend on $a$, $b$, $p\/j$   and $p\/j\\le p\/k\\le 1$.\n   \n \nUsing  (\\ref{2.4f2}), (\\ref{2.4f3})  we have  \n   \\begin{eqnarray}\n       A_{j,k,p}(t)  \\label{2.4f5}& =&{\\gamma \\over d^{2}}\\(-(d+2)   +{b(dt-b)p\\over k}+{a(dt-a)p\\over j} -{p^{2}\\over jk}\\) \\label{5.15} \\\\\n   &&\\qquad +{1 \\over d} \\( a+b-{  p(dt-a)\\over j}-{ p(dt-b)\\over k}\\) +O\\( {1 \\over t} \\) \\nonumber\\\\\n     & =&{-(d+2)\\gamma+  d(a+b) \\over d^{2}}\\nonumber\\\\\n   &&\\qquad+{ (\\gamma a-d) p(dt-a)\\over jd^{2}}+{ (\\gamma b-d) p(dt-b)\\over kd^{2}} -{\\gamma p^{2}\\over jkd^{2}} +O\\( {1 \\over t} \\)  \\nonumber\\\\\n   & =&{-2\\zeta \\over d^{2}}\n  +{ (a-1)^{2} p(dt-a)\\over jd^{2}}+{ (b-1)^{2} p(dt-b)\\over kd^{2}}  -{\\gamma p^{2}\\over jkd^{2}} +O\\( {1 \\over t} \\).  \\nonumber\n   \\end{eqnarray}\nwhere, for the final equality we use  (\\ref{2.28}).\n\n  \n   Note that\n  \\begin{eqnarray}\n  B_{j,k,p}(t)&:= &{ (a-1)^{2} p(dt-a)\\over j}+{ (b-1)^{2} p(dt-b)\\over k}\\label{2.4f6}\\\\\n  & \\geq &2p |(a-1)(b-1)|{\\sqrt{(dt-a)(dt-b)} \\over \\sqrt{jk} }\\nonumber\\\\\n   & = &2p |d+2-(a+b)|{\\sqrt{(dt-a)(dt-b)} \\over \\sqrt{jk} }\\nonumber.\n   \\end{eqnarray}\n   (For the inequality use $\\alpha^{2}+\\beta^{2}\\ge 2\\alpha\\beta$.)\nTherefore   since $ \\zeta =a+b-(d+2)>0$,   \n \\begin{eqnarray}\n    A_{j,k,p}(t)&\\ge&\\frac{2}{d^{2}}\\(p{\\sqrt{(dt-a)(dt-b)} \\over \\sqrt{jk} }-1\\)\\zeta-{\\gamma p^{2}\\over d^{2}jk} +O\\(\\frac{1}{t}\\)\\nonumber\\\\\n &=& \\frac{2}{d^{2}}\\({p(dt+O(1)) \\over \\sqrt{jk} }-1\\)\\zeta-{\\gamma p^{2}\\over d^{2}jk} +O\\(\\frac{1}{t}\\)  \\label{5.18}\\\\\n &=&\\frac{2}{d^{2}}\\({pdt \\over \\sqrt{jk} }\\(1+O\\(\\frac{1}{t}\\)-\\frac{\\gamma p}{ \\zeta\\sqrt{jk}d t}\\)-1\\)\\zeta  +O\\(\\frac{1}{t}\\)\\nonumber.\n   \\end{eqnarray}\n\n\n Thus we see that there exists a function  $\\epsilon_{t}$, depending only on $a$ and $b$ such that \n \\begin{equation}\n      A_{j,k,p}(t) \\ge\\frac{2}{d^{2}}\\({pdt \\over \\sqrt{jk} }(1-\\epsilon_{t})-1\\)\\zeta +O\\(\\frac{1}{t}\\)  ,\\label{5.19}\n   \\end{equation}\n where\n \\begin{equation}\n  \\lim_{t\\to \\infty} \\epsilon_{t}=0,\\label{5.20}\n   \\end{equation}  \n   and, as we point out above the $O\\( 1\/t\\)  $ is independent of $p$, $j$ and $k$.\n  \n\\begin{remark}\\label{rem-5.1} {\\rm We interrupt this proof to make some comments which  may be helpful in understanding what is going on. Note that \nif  \n\\begin{equation}\n{\\sqrt{jk}\\over dt}\\le 1-\\widetilde\\epsilon\\label{2.39}\\qquad\\mbox{for some $\\widetilde\\epsilon>0$}\\label{5.21}\n\\end{equation}\nthen  \n\\begin{equation}\nR_{j,k}(t,c)\\ge R_{j,k,0}(t,c) \\ge (1-\\delta){1-2c^2(\\zeta\/d)\\over d^{2}t^2} \\alpha^{j-1}\\beta^{k-1}\\qquad\\mbox{as $t\\to\\infty$}\\label{5.22}\n\\end{equation}\nfor all $\\delta>0$. This follows from (\\ref{2.4fx}) and (\\ref{5.19}) since when (\\ref{5.21}) holds\n\\begin{equation} \n      A_{j,k,p}(t)\\ge \\frac{2}{d^{2}}\\( {1-\\epsilon_{t}\\over(1-\\widetilde\\epsilon)}-1\\)\\zeta +O\\(\\frac{1}{t}\\) >0 ,\\label{5.19aa}\n   \\end{equation} \nfor all $p\\ge 1$, for all $t$ is sufficiently large. Consequently when (\\ref{5.21}) holds   $R_{j,k}(t,c)>0$ for all $t$ is sufficiently large when\n\\begin{equation} \n   c^{2}<\\frac{d}{2\\zeta}.\n   \\end{equation}\n(Here we also use (\\ref{5.20}).)\n\n(When $\\zeta\\le 0$,   (\\ref{5.22}) shows that $R_{j,k}(t,c)>0$ for all $c\\in R^{1}$. This is   what we expect. (See the paragraph containing (\\ref{4.5}).)\n\n }\\end{remark}\n\n We use the next two lemmas to complete the proof of  Theorem \\ref{theo-cp}, in case 1. \n \n \n \\begin{lemma} \\label{lem-5.2}For any  $N_{0}\\in R^{+}$, we can find $c_{0}>0$ and $t_{c_{0}}<\\infty$ such that for all $t\\geq t_{c_{0}}$  \n \\begin{equation}\n R_{j,k,p}(t,c)>0     \\label{5.25}\n \\end{equation}\n  for all $|c|\\leq c_{0}$ and all $p$, $j$ and $k$ for which   $ jk\/ t\\le N_{0}$.\n \\end{lemma}\n \n\n\\noindent{\\bf Proof}$\\quad$   This follows from (\\ref{2.4fx}) when $p=0$. Therefore, we can take $p\\ge 1$. \n\nWe first show that for any   $N\\in R^{+}$,  $R_{j,k,p}(t)>0$ when $ \\sqrt{jk}=Nd t$,    for all $t$ sufficiently large.\n By Remark \\ref{rem-5.1} we can assume \n    that    $ N\\ge 1-\\widetilde\\epsilon $.      \n    It follows from (\\ref{5.19}) that  \n\n \\begin{equation}\n      A_{j,k,p}(t) \\ge \\frac{2}{d^{2}}\\({p \\over N }(1-\\epsilon_{t})-1\\)\\zeta +O\\(\\frac{1}{t}\\),\\label{5.19aaxx}\n   \\end{equation}\n   where $\\epsilon_{t}$ satisfies (\\ref{5.20}). \n Therefore when $p\\geq \\Lambda N$ for any $\\Lambda>1$,   $A_{j,k,p}(t)>0$, and hence $R_{j,k,p}(t,c)>0$,  for all $t$ sufficiently large.  \n \nNow suppose that  \n \\begin{equation}\n p<\\Lambda N.\\label{97}\n \\end{equation}\n Since $ \\sqrt{jk}=N dt$ we see that  \n \\begin{equation}\n  {  \\gamma p^{2}\\over d^{2} jk} \\leq {\\gamma \\Lambda^{2} \\over d^{4}t^{2} }=O(1\/t^{2}),\\label{5.27}\n   \\end{equation}\n  where the $O(1\/t)$ term is independent of $p$, $j$ and $k$. \n   \n Note that by   (\\ref{lem-2.2})  and (\\ref{5.11}) \n \\begin{equation}\n {1 \\over \\widetilde t^2} F_{j,k,p}(t)  =   {(j-p)(k-p)\\over d^{2}t^{2}( p+1)jk}+   O\\(\\frac{1}{t^{3}}\\)\\label{5.1m1}\n   \\end{equation} \n  Therefore, \nif in addition  to (\\ref{97}) we also have $\\Lambda N \\le j\/2 $, so that $p<j\/2$, we see by (\\ref{5.10}),     (\\ref{5.15}) and (\\ref{5.27}) that  \n \\begin{eqnarray}\n &&{1 \\over \\widetilde t^2}F_{j,k,p}(t)+{c^2\\over \\bar t^2} A_{j,k,p}(t)\\label{3.49j}\\\\\n&&\\qquad\\ge \\frac{1}{d^{2}t^{2}}\\( {(j-p)(k-p)\\over (p+1)jk}- {c^{2}2\\zeta\\over d}\\)+O\\(\\frac{1}{t^{3}}\\)\\nonumber\\\\   \n&&\\qquad\n   \\ge\\frac{1}{d^{2}t^{2}}\\(  {1\\over 4(p+1)}- {c^{2}2\\zeta\\over d }\\)+\n   O\\(\\frac{1}{t^{3}}\\)\\nonumber \\\\\n&&\\qquad\n   \\ge\\frac{1}{d^{2}t^{2}}\\(  \\frac{1}{8\\Lambda N }- {c^{2}2\\zeta\\over d } \\) +O\\(\\frac{1}{t^{3}}\\).\n\\nonumber\n \\end{eqnarray} \nTherefore we can obtain $R_{j,k,p}(t)\\ge 0$ by taking \n   \\begin{equation}\n   c^{2}\\le\\frac{d }{16\\Lambda' N \\zeta}\\label{3.51a}\n   \\end{equation}\n for some $\\Lambda' >\\Lambda$.  \n \n \n   Now suppose that  $\\Lambda N> j\/2 $. In this case we use (\\ref{5.15})     to see that \n \\begin{equation}\n   A_{j,k,p}(t)\\ge -{2 \\zeta\\over d^{2}}+{(a-1)^2p \\,t\\over 2d j}+O(1\/t).\\label{5.31}\n   \\end{equation}  \n   It is easy to see that the right-hand side of (\\ref{5.31}) is greater than zero for all $t$ sufficiently large since \n   \\begin{equation}\n  {(a-1)^2p\\,t\\over 2 dj}\\ge \\frac{1}{4dN\\Lambda}(a-1)^2 t.\n   \\end{equation}\nThus we see that for any fixed $N$,   $R_ {j,k,p}(t) > 0$ for all $ t$  sufficiently large.  \n\nSince the $O(1\/t)$ terms are independent of $p$, $j$ and $k$ this analysis works for all $j$ and $k$ satisfying (\\ref{5.25}), and all $1\\le p\\le j$  as long as    \n(\\ref{3.51a})      holds with $N$     replaced by $N_{0}$.      {\\hfill $\\square$ }   \n\n\\begin{lemma} \\label{lem-5.3} For   all $N_{0}$ and $B\\in R^{+} $      we can find   a    $c'_{0}>0$ and $t_{c'_{0}}<\\infty$ such that for all $t\\geq t_{c'_{0}}$  \n \\begin{equation} \n R_{j,k,p}(t,c)>0 \n \\end{equation}\n    for all $|c|\\leq c'_{0}$ and all   $0\\le p\\le j\\le k$ for which   \n\\begin{equation} \nN_{0}t\\leq\\sqrt{jk}\\leq Bt\\log t.\\label{5.25a} \n\\end{equation}\n\\end{lemma}\n                       \n  \n\n \n (The value of $N_{0}$ in Lemmas \\ref{lem-5.2} and \\ref{lem-5.3} can be taken as we wish. It will be assigned in the proof of this lemma.)\n \n  \\medskip\t\n\\noindent{\\bf Proof}$\\quad$    \n  By adjusting  $N_{0}$ and $B$ we can replace (\\ref{5.25a}) by the condition \n\\begin{equation}\nN_{0}\\widetilde t\\leq\\sqrt{jk}\\leq B\\widetilde t\\log \\widetilde t.\\label{5.25ax}\n\\end{equation}\n   Using (\\ref{2.4f5}), we see that if $  j\\leq \\rho \\widetilde t$  \n    \\begin{eqnarray}\n    A_{j,k,p}(t)&=&{-2\\zeta\\over d^{2}}+{ (a-1)^{2} p(dt-a)\\over jd^{2}}+{ (b-1)^{2} p(dt-b)\\over kd^{2}} \n   \\nonumber\\\\\n   && \\hspace{2  in} -{\\gamma p^{2}\\over jkd^{2}} +O\\( {1 \\over t} \\)  \\label{2.4f5a}\\\\\n   & \\geq &{-2\\zeta\\over d^{2}}+{ (a-1)^{2} p(dt-a)\\over jd^{2}}- {\\gamma\\over d^{2}}  +O\\( {1 \\over t} \\)\\nonumber\\\\\n     & \\geq &{-2\\zeta\\over d^{2}}+{ (a-1)^{2} \\over2 \\rho d }- {\\gamma\\over d^{2}}  +O\\( {1 \\over t} \\)\\nonumber.\n      \\end{eqnarray}    \n   Clearly, there exists a $\\rho>0$, independent of $j$ and $k$ such that this term  is positive.  Thus we can assume that\n    \\begin{equation}\n   j\\geq \\rho \\widetilde t.\\label{2.40w}\n   \\end{equation}\n\n   \n   Furthermore, when  $  \\sqrt{jk}\/\\widetilde t= N$,  it follows from  (\\ref{2.51a1}) that we can write (\\ref{5.18}) as  \n  \\begin{equation}  A_{j,k,p}(t)\\ge \\frac{2}{d^{2}}\\({p  \\over N}\\(1 +O\\(\\frac{1}{t}\\)-\\frac{\\gamma  }{   \\zeta (dt)^{2}}\\)-1\\)\\zeta +O\\(\\frac{1}{t}\\).\\label{5.37}\n \\end{equation}  \n Let $\\delta_{N}=(10\\log N\/N)^{1\/2}$. Clearly, if $p>(1+\\delta_{N})N$, the right-hand side of (\\ref{5.37}) is positive for all $t$ sufficiently large.\n   Therefore, when   $  \\sqrt{jk}\/\\widetilde t=N$, we may assume that   \n          \\begin{equation}\np\\leq (1+\\delta_{N})N\\label{2.40q}.\n   \\end{equation}\n (The value  chosen for $\\delta_{N}$  simplifies calculations made later in this proof.)\n   \n   In addition  we can also assume that \n   \\begin{equation}\n    p\\geq p_{0}\n    \\end{equation}\n for any finite $ p_{0}$, since if $p< p_{0}$\n\\begin{eqnarray} \n  F_{j,k,p}(t)\\ge  F_{j,k,p_{0}}(t)\\ge c ^{2}A_{j,k,p}(t).\\label{5.39}\n \\end{eqnarray}\n for all $c>0$ sufficiently small.  \n \n \\medskip\t\n We use the next lemma in the proof of Lemma  \\ref{lem-5.3}.\n \n  \n   \\begin{lemma}\\label{lem-3.9}   For $j\\le k$, with  $p$ and $j$  large and $p\/j$ small  \n \\begin{eqnarray}\n &&\n {j\\choose p}{k\\choose p}= { 1\\over  2\\pi p}{\\({e^2jk\\over  p^2}\\)^{p}}\\label{2.40}\\\\&&\\qquad\\exp\\(-{p\\over 2j}(p-1)-{p\\over 2k}(p-1)+O(p^3\/j^2)\\)\\(1+O(p^{-1})\\) .\\nonumber\n \\end{eqnarray} \n When  $\\widetilde t\\in R^{+}$ is large and   $\\sqrt{jk}\/\\widetilde t=N$, under  assumptions (\\ref{2.40w}) and (\\ref{2.40q})\n \\begin{eqnarray}\n &&\n {1\\over \\widetilde t^{2p}}{j\\choose p}{k\\choose p}= {1 \\over 2\\pi p}\\({eN\\over p}\\)^{2p}\\(1+O(p^{-1})\\).\\label{2.51}\n \\end{eqnarray}\n  \n \\end{lemma} \n \n \n \\noindent{\\bf Proof}$\\quad$ By Stirlings's formula  for integers   $q$,\n \\begin{equation}\n q!=\\sqrt{2\\pi}q^{q+1\/2}e^{-q}\\(1+O(q^{-1})\\).\\label{5.47}\n \\end{equation}\n Therefore, since $j$ is large and $p\/j$ is small, terms of the form \n \\begin{equation}\n{\\(1+O(j^{-1})\\) \\over  \\(1+O(p^{-1})\\)\\(1+O((j-p)^{-1})\\)}=\\(1+O(p^{-1})\\)\\label{2.40n}.\n \\end{equation}\nUsing this we see that  \n  \\begin{eqnarray}\n  {j\\choose p}&&={j! \\over (j-p)!p!}\\\\\n  & &={1 \\over \\sqrt{2\\pi}}{j^{j+1\/2} \\over (j-p)^{(j-p+1\/2)}\\,p^{p+1\/2}}\\(1+O(p^{-1})\\) \\nonumber\\\\\n    & &={1 \\over \\sqrt{2\\pi\\,p}}\\(\\frac{ j}p\\)^p{j^{j-p+1\/2} \\over (j-p)^{(j-p+1\/2)}}\\(1+O(p^{-1})\\) \\nonumber\\\\\n        & &={1 \\over \\sqrt{2\\pi\\,p}}\\(\\frac{ j}p\\)^p{1\\over \\(1-{p \\over j}\\)^{(j-p+1\/2)}}\\(1+O(p^{-1})\\) \\nonumber\\\\\n& &={1 \\over \\sqrt{2\\pi\\,p}}\\(\\frac jp\\)^p e^{-(j-p+1\/2)\\log(1-p\/j)}\\(1+O(p^{-1})\\) \\nonumber\\\\\n & &={1 \\over \\sqrt{2\\pi\\,p}}\\(\\frac jp\\)^p e^{(j-p+1\/2)(p\/j+p^2\/(2j^2) +O(p^3\/j^3))}\\(1+O(p^{-1})\\) \\nonumber\\\\\n & &={1 \\over \\sqrt{2\\pi\\,p}}\\(\\frac{e j}p\\)^p e^{( -p^2\/(2j )+p\/(2j)\n +O(p^3\/j^2))}\\(1+O(p^{-1})\\) \\nonumber\\\\\n & &={1 \\over \\sqrt{2\\pi\\,p}}\\(\\frac{e j}p\\)^p e^{( -p(p-1)\/(2j )\n +O(p^3\/j^2))}\\(1+O(p^{-1})\\) \\nonumber.\n \\end{eqnarray}\nSince this also holds with $j$ replaced by $k$ we get (\\ref{2.40}). \n\nTo get (\\ref{2.51}) we multiply each side of   (\\ref{2.40}) by $\\widetilde t^{-2p}$ and  substitute for $\\sqrt{jk}\/\\widetilde t=d N$    and use  the fact that under the assumptions (\\ref{2.40w}) and (\\ref{2.40q}), \n \\begin{equation} \n {p^{3} \\over j^{2}}\\leq  { p^{2} \\over j }\\leq   \\frac{(1+\\delta_{N})^{2}N^{2}}{\\rho t} \\label{6.127}.\n \\end{equation} \nConsequently, for all $t$ sufficiently large\n \\begin{equation}\n \\exp \\(-{p\\over 2j}(p-1)-{p\\over 2k}(p-1)+O(p^3\/j^2) \\)=1+ O\\(N^{2}\/t\\).\\label{6.128}\n \\end{equation}\n {\\hfill $\\square$ }\n \n \n \n \\medskip\t\n\\noindent \n{\\bf Proof of Lemma \\ref{lem-5.3}   continued} We show that under the assumptions  (\\ref{2.40w}) and (\\ref{2.40q}),  when $\\sqrt{jk}\/  \\widetilde t=N$,    for   \n $N_{0}\\leq N\\leq B\\log   \\widetilde t$,  for any $0<B<\\infty$, and $t$ is sufficiently large,\n \\begin{equation}\n { \\widetilde  t^{2}\\over   \\alpha^{j-1}\\beta^{k-1}}\\sum_{p=p_{0}}^{(1+\\delta_{N})N}P_{j,k,p}(t)\\geq Ce^{2N}  {1\\over N^{3\/2} } \\label{6.11}\n \\end{equation}\n for some $C>0$, independent of $N$,\n and \n  \\begin{equation}\n {\\bar t^2\\over    \\alpha^{j-1}\\beta^{k-1}}\\sum_{p=p_{0}}^{(1+\\delta_{N})N}Q_{j,k,p}(t)\\geq -De^{2N}  {1\\over N^{3\/2} } \\label{6.12}\n \\end{equation}\n for some $D<\\infty$, independent of $N$. If  (\\ref{6.11}) and (\\ref{6.12}) hold, we can find a $c_{0}>0$ such that for all  $c^{2}\\le c_{0}^{2}$ \n   \\begin{equation}\n \\sum_{p=p_{0}}^{(1+\\delta_{N})N}R_{j,k,p}(t,c)\\geq 0\\label{6.13}\n \\end{equation}\n for all  $t$ sufficiently large.  Since we have already established that $A_{j,k,p}(t)>0$,  when $p<p_{0}$ and $p>(1+\\delta_{N})N$,   this  completes the proof of  Lemma \\ref{lem-5.3}. \n \n  Thus it only remains to prove (\\ref{6.11}) and (\\ref{6.12}). We do (\\ref{6.11}) first. It is considerably easier than  (\\ref{6.12}).  By  Lemma \\ref{lem-P} and  (\\ref{2.51})\n  \\begin{eqnarray}\n &&{\\widetilde t^{2}\\over   \\alpha^{j-1}\\beta^{k-1}}\\sum_{p=p_{0}}^{(1+\\delta_{N})N}P_{j,k,p}(t)\\label{6.14}\\\\\n &&\\qquad=\\sum_{p=p_{0}}^{(1+\\delta_{N})N}\n {1\\over \\widetilde t^{2p}}{j\\choose p}{k\\choose p} {(j-p)(k-p)\\over (p+1)jk}\\nonumber\\\\\n &&\\qquad\\ge C \\sum_{p=p_{0}}^{(1+\\delta_{N})N}\n {1\\over   \\widetilde t^{2p}}{j\\choose p}{k\\choose p} {1\\over p} \\nonumber\\\\\n &&\\qquad\\ge C \\sum_{p=p_{0}}^{(1+\\delta_{N})N}\n {1\\over p^{2}}\\({eN\\over  p  }\\)^{2p  } \\nonumber.\n \\end{eqnarray}\n \nIn order to calculate this last sum we   consider the function  \n\n\\begin{equation}\nf_{m}(y)={1 \\over y^{m}}\\({eN\\over  y  }\\)^{2y  }={1 \\over y^{m}}e^{2y(1+\\log N-\\log y)}\\label{jr.10}\n\\end{equation}\nfor $m\\geq 0$ and $y\\geq 2.$\nWe have    \n\\begin{eqnarray}\nf_{m}'(y)&=&\\(  {-m \\over y}   +2(1+\\log N-\\log y)-2   \\)f(y) \\label{jr.11}\\\\\n&=&\\(  {-m \\over y}   +2(\\log N-\\log y)  \\)f(y).\\nonumber\n\\end{eqnarray}\nThis has a unique root $y_{m}$ where\n\\begin{equation}\n\\log y_{m}+{m \\over 2y_{m}}=\\log N.\\label{jr.12}\n\\end{equation}\n(Clearly, $y_{0}=N$).  \nLet $y_{m}=N(1+\\epsilon_{m})$.  Then\\begin{equation}\n\\log (1+\\epsilon_{m})+{m \\over 2N(1+\\epsilon_{m})}=0.\\label{jr.14}\n\\end{equation}\nConsequently\n\\begin{equation}\n\\epsilon_{m}=-{m \\over 2N}+O(N^{-2}),\\label{jr.15}\n\\end{equation}\nwhich implies that   \n\\begin{equation}\ny_{m}=N-{m \\over 2}+O(1\/N).   \n   \\end{equation}\nMaking use of the fact that $\\(  {-m \/y_{m}}   +2(1+\\log N-\\log y_{m})-2   \\)=0$, we see that  \n\\begin{eqnarray}\nf_{m}''(y_{m})&=&\\(  {m \\over y_{m}^{2}}   -\\frac{2}{y_{m}}  \\)f(y_{m}) <0.\\label{jr.11x}\n\\end{eqnarray}\nTherefore   \n\\begin{equation}\n\\sup_{y\\geq  2}f_{m}(y ) =f_{m}(y_{m})\\leq  {1\\over (N-m)^{m}}e^{2N}.\\label{jr.16}\n\\end{equation}\nWe also note that since $ {-m \/ y} -2\\log y$ is increasing for $y>m\/2$,   $f_{m}'(y)$ is positive  for $m\/2<y<y_{m}$ and negative for $y>y_{m}$. Consequently, $f_{m}(y) $ is unimodal.\n\n\\medskip\t\nNow consider the last line of (\\ref{6.14}). The function being summed is $f_{2}(p)$. The above discussion shows that this function is unimodal, with a maximum at, at most,  two points at which it is less than $ 2 e^{2N}\/ N^{2}$.  Consequently, to obtain (\\ref{6.11}) we can replace the sum in the last line of (\\ref{6.14}) by an integral and show that\n \\begin{equation}\n I_{1}:=\\int_{ p_{0}}^{(1+\\delta_{N})N}\n {1\\over r^{2}}\\({eN\\over  r  }\\)^{2r  }\\,dr\\geq Ce^{2N}  {1\\over N^{3\/2} } \\label{6.11a}.\n \\end{equation}\nMaking the change of variables $r=xN$  we have\n  \\begin{eqnarray}\n I_{1} ={  1\\over N} \\int_{ p_{0}\/N}^{1+\\delta_{N}}\n {1\\over x^{2}}\\({e\\over x  }\\)^{2xN  }\\,dx.   \\nonumber\n \\end{eqnarray}\n Recall  that $N_{0}\\leq N\\leq 2\\log \\widetilde t$, and  that we can take $N_{0}$ as large as we want, (but fixed and independent of $t$), and that $\\delta_{N}=(10\\log N\/N)^{1\/2}$. Therefore\n \\begin{eqnarray}\n I_{1}&\\geq  &\n {1\\over N}\\int_{1-(10\\log N\/N)^{1\/2}}^{1+(10\\log N\/N)^{1\/2}}  { 1\\over  x^{2}}{\\({e \\over x}\\)^{2xN}}\\,dx\\label{2.51h} \\\\\n &\\geq  &\n {1 \\over 2 N}\\int_{1-(10\\log N\/N)^{1\/2}}^{1+(10\\log N\/N)^{1\/2}} {\\({e \\over x}\\)^{2xN}}\\,dx. \\nonumber\n \\end{eqnarray}\n\n\n\n We write \n  \\begin{equation}\n  \\({e\\over x}\\)^{2xN}=\\exp\\(2xN(1-\\log x)\\)\\label{j.58}.\n  \\end{equation}\n  Set $x=1+y$ and\n     note that for $|y|$ small\n      \\begin{eqnarray} \n     x(1-\\log x) \\label{j.58j} \n  &=& (1+y)(1-\\log (1+y) )\\\\\n  & = &(1+y)\\(1-\\sum_{n=1}^{\\infty}(-1)^{n-1}{y^{n} \\over n}\\)  \\nonumber\\\\\n  & = &1+y -\\sum_{n=1}^{\\infty}(-1)^{n-1}{(1+y)y^{n} \\over n} \\nonumber\\\\\n  & = &1 -\\sum_{n=2}^{\\infty}(-1)^{n-1}y^{n}\\({1 \\over n}-{1 \\over n-1}\\) \\nonumber\\\\\n  & = &1 -\\sum_{n=2}^{\\infty}(-1)^{n} {y^{n} \\over n(n-1)}.  \\nonumber\n  \\end{eqnarray}\n  \n  \n  When $|y|\\leq (10\\log N\/N)^{1\/2}$,  so that $|y|^{3}N <<1$, this shows that\n    \\begin{eqnarray}\n  \\({e\\over x}\\)^{2x N} \n  &=&e^{2 N}  e^{ -y^2 N +O(y^3N)}\\label{2.59}\\\\\n  & = &   e^{2 N}  e^{ -y^2 N }\\(1+O(|y|^{3}N )\\).\\nonumber\n \\end{eqnarray} \nIt follows from this that when we make the change of variables $x=1+y$ in (\\ref{2.51h}) we get\n  \\begin{eqnarray}\n && I_{1} \\geq \n {  e^{2 N}\\over 2 N}\\int_{-(10\\log N\/N)^{1\/2}}^{(10\\log N\/N)^{1\/2}} e^{ -y^2 N }\\,dy  \\\\\n&&\\qquad \\geq \n { e^{2 N}\\over 2\\sqrt{2} N^{3\/2}}\\int_{-(20\\log N)^{1\/2}}^{(20\\log N)^{1\/2}} e^{ -y^2\/2  }\\,dy.\\nonumber\n \\end{eqnarray}\nSince \n\\begin{equation}\n  \\int_{(20\\log N )^{1\/2}}^{\\infty} e^{-u^{2}\/2}\\,du \\le N^{-10}\\label{2.51j},\n\\end{equation} \n we see that (\\ref{6.11a}) follows.   Thus we have established (\\ref{6.11}).\n \n \n\\medskip\t\nBefore proceeding to the proof of (\\ref{6.12}) we note that \n\\begin{equation}\n  \\sum_{p=p_{0}}^{(1+\\delta_{N})N}\\label{5.72}\n {1\\over \\widetilde t^{2p}}{j\\choose p}{k\\choose p}\\le   { e^{2 N} \\over 2 N^{1\/2}}.\n   \\end{equation}\n To prove this we use  (\\ref{2.51}) and the same argument that enables us to move from a sum to an integral that is given in (\\ref{6.14})--(\\ref{6.11a}), except that we use (\\ref{jr.16}) with $m=1$. We continue  and then  use (\\ref{2.59}) to get\n\\begin{eqnarray}\n&&\\sum_{p=p_{0}}^{(1+\\delta_{N})N}\n {1\\over \\widetilde t^{2p}}{j\\choose p}{k\\choose p}\n\\label{jr.7}\\\\\n &&\\qquad\\leq  \\int_{p_{0}}^{(1+\\delta_{N})N}{1 \\over u} \\({eN\\over  u }\\)^{2u}\\,du+O\\({e^{2 N} \\over N}\\)\\nonumber\\\\\n &&\\qquad\\leq  e^{2 N}  \\int_{0}^{\\delta_{N}}   e^{ -y^2 N }\\,dx+O\\({e^{2 N} \\over N}\\)\\leq   { e^{2 N} \\over 2N^{1\/2}} . \\nonumber\n\\end{eqnarray}\n\n \n \n \\medskip\t\n We now obtain (\\ref{6.12}).  When $\\sqrt{jk}=\\widetilde t N$, by (\\ref{5.10}) and   (\\ref{5.37}),\n   \\begin{eqnarray}\n &&{\\bar t^2\\over    \\alpha^{j-1}\\beta^{k-1}}Q_{j,k,p}(t)\\\\\n &&\\qquad={1\\over \\widetilde t^{2p}}{j\\choose p}{k\\choose p}A_{j,k,p}(t)\\label{6.120w}\\nonumber\\\\\n  & &\\qquad\\ge{2\\over  d^{2}\\widetilde t^{2p}}{j\\choose p}{k\\choose p}\\left\\{ 2\\({p  \\over N }-1\\)\\zeta +O\\(\\frac{1}{t}\\)\\right\\} \\nonumber.\n \\end{eqnarray}  \n \n\nBy  (\\ref{jr.7}) we see that\n\\begin{equation}\n   \\sum_{p=p_{0}}^{(1+\\delta_{N})N}{1\\over \\widetilde t^{2p}}{j\\choose p}{k\\choose p}O\\(\\frac{1}{t}  \\)=O\\({e^{2 N}\\over t}\\)\n   \\end{equation}\nTherefore,  to obtain (\\ref{6.12}), it suffices to show that for some $D<\\infty$\n\\begin{equation}\n \\sum_{p=p_{0}}^{(1+\\delta_{N})N}{1\\over \\widetilde t^{2p}}{j\\choose p}{k\\choose p}  \\({p  \\over N }-1\\) \\geq -De^{2N}  {1\\over N^{3\/2} }\\label{6.12a}.\n\\end{equation}\n  Here we use the fact that $N\\le B\\log \\widetilde t$ for some $0<B<\\infty$.\n\n\\begin{remark}  {\\rm\n Since the proof of  (\\ref{6.12a}) is rather delicate we make some  heuristic comments to explain how we proceed. When   $\\sqrt{jk}=\\widetilde t N$ the term $\\widetilde t^{-2p}{j\\choose p}{k\\choose p}$, as a function of $p$, is $\\exp(2N)$ times values that are sort of normally distributed with mean $p=N$, and, roughly speaking,  \n \\begin{equation}\n   \\sum_{p=p_{0}}^{(1+\\delta_{N})N}{1\\over \\widetilde t^{2p}}{j\\choose p}{k\\choose p}\\sim Ce^{2N}\\frac{1}{N^{1\/2}},\\label{jr.22}\n   \\end{equation}\n for all $t$ sufFiciently large. (In fact the upper bound is given in (\\ref{jr.7}).)\n This is too large to enable us to get (\\ref{6.12a}) so we must make use of the factors $ \\({p  \\over N }-1\\)$, which   is an odd function with respect to  $p=N$, to get the cancellations that allow us to obtain  (\\ref{6.12a}).  However, because we are canceling terms, we must take account of the error in Stirling's approximation; (see (\\ref{5.47})). To do this we need to show that the estimate in (\\ref{6.12a}) remains the same even when we eliminate the terms in the summand that are not close to $N$.}\n  \\end{remark}\n \n \\medskip\t\n\\noindent\n{\\bf Proof of Lemma \\ref{lem-5.3}  continued }\nNote that by  (\\ref{2.51})\n\\begin{equation}\n\\sum_{p=p_{0}}^{N(1-N^{-1\/4})}{1\\over \\widetilde t^{2p}}{j\\choose p}{k\\choose p}   \n\\label{jr.21} \\leq C  \\sum_{p=p_{0}}^{N(1-N^{-1\/4})}f_{1}(p).  \\label{5.79}\n\\end{equation}\nThe fact that $f_{m}(y)$ is unimodal on $y>m\/2$ implies that $f_{1}(p)$ is increasing on the interval $[p_{0}, N(1-N^{-1\/4})]$. Therefore   \n\\begin{eqnarray}\n\\sum_{p=p_{0}}^{N(1-N^{-1\/4})}{1 \\over p} \\({eN\\over  p }\\)^{2p}   \n&\\leq &  C  N f_{1}(N(1-N^{-1\/4}))\\label{jr.21xx}\\\\\n&\\leq & C \\({e\\over  1-N^{-1\/4} }\\)^{2N(1-N^{-1\/4})}. \\nonumber\\\\\n&= & C e^{2N(1-N^{-1\/4})(1-\\log (1-N^{-1\/4} ))}. \\nonumber\\\\\n&\\leq  & C e^{2N(1-N^{-1\/4})(1+N^{-1\/4})}=Ce^{2N -2N^{1\/2} }. \\nonumber\n\\end{eqnarray}\n\nLet $\\delta'_{N}=N^{-1\/4}$. The argument immediately above shows that to prove (\\ref{6.12a}), it suffices to show that\n\\begin{equation}\nJ_{1}:=\\sum_{p=(1-\\delta'_{N})N}^{(1+\\delta_{N})N}{1\\over \\widetilde t^{2p}}{j\\choose p}{k\\choose p}  \\({p  \\over N }-1\\) \\geq -De^{2N}  {1\\over N^{3\/2} }.\\label{6.12b}\n\\end{equation}\n\n\\medskip\t By (\\ref{2.51})\n\\begin{equation}\nJ_{1}=\\frac{1}{2 \\pi }\\sum_{p=(1-\\delta'_{N})N}^{(1+\\delta_{N})N} \\({eN\\over   p }\\)^{2p}  \\({1 \\over N }-\\frac{1}{p}\\)\\(1+O\\(\\frac{1}{p}\\)\\).\\label{6.12c}\n\\end{equation}\nUsing  (\\ref{jr.7}) together with the fact that since $p\\geq (1-\\delta'_{N})N$,  \n$1\/p \\leq  1\/N $, we see that\n\\begin{eqnarray}\n \\sum_{p=(1-\\delta'_{N})N}^{(1+\\delta_{N})N} \\({eN\\over   p }\\)^{2p}  \\({1 \\over N }-\\frac{1}{p}\\) O\\(\\frac{1}{p}\\)&\\ge&- \\sum_{p=(1-\\delta'_{N})N}^{(1+\\delta_{N})N} \\({eN\\over   p }\\)^{2p}  \\frac{1}{p} \\, O\\(\\frac{1}{p}\\)\\nonumber\\\\\n &\\ge&-C   e^{2N}{1  \\over N^{3\/2}}  .\\label{6.12df}\\nonumber\n\\end{eqnarray}\nTherefore, to obtain (\\ref{6.12b})\n that it suffices to show that\n\\begin{equation}\n \\sum_{p=(1-\\delta'_{N})N}^{(1+\\delta_{N})N} \\({eN\\over   p }\\)^{2p}  \\({1 \\over N }-\\frac{1}{p}\\) \\geq -De^{2N}  {1\\over N^{3\/2} }.\\label{6.12d}\n\\end{equation}\n\n\\medskip\t \n\n   \nIn a minor modification of the analysis of $f_{m}(y)$, we write \n\\begin{equation}\n h(y):=   \\({eN\\over   y}\\)^{2y}  \\({1 \\over N }-\\frac{1}{y}\\) =\\exp\\(2y\\(1+\\log N- \\log y\\)\\)\\({1 \\over N }-\\frac{1}{y}\\).\n   \\end{equation}\n Therefore\n \\begin{equation}\n   h'(y)=\\(\\(2 \\(1+\\log N-\\log y\\) -2 \\)\\({1 \\over N }-\\frac{1}{y}\\)+\\frac{1}{y^{2}} \\)\\({eN\\over   y }\\)^{2y}. \n   \\end{equation}\nLet $y=(1+\\omega)N$. Then $ h'(y)=0$ when\n\\begin{equation}\n  - 2\\omega  \\log(1+\\omega)   +\\frac{1}{N (1+\\omega) }=0.\n   \\end{equation}\n This equation is satisfied when \n  \\begin{equation}\n  \\omega=\\pm \\frac{1}{\\sqrt{2N} }+O\\(\\frac{1}{N}\\).\n   \\end{equation}\n Note that when $y=(1+\\omega)N$\n \\begin{equation}\n    \\({eN\\over   y }\\)^{ y}= \\({e \\over  1+\\omega}\\)^{  (1+\\omega)N}\\le e^{N},\n   \\end{equation}\n because $(e\/x)^{x}$ is maximized when $x=1$. Therefore\n \\begin{equation}\n    \\({eN\\over   y }\\)^{2y}  \\({1 \\over N }-\\frac{1}{y}\\) \\le  {e^{2N}\\over N}\\frac{\\omega}{1+\\omega}\n   \\end{equation}\n from which we get  \n \\begin{equation}\n \\sup_{1\\leq y\\leq (1+\\delta_{N})N}|h(y)|\\leq  C\\(\\frac{e^{2N}}{N^{3\/2}}\\).\\label{5.68}\n \\end{equation}\n \nIt is easy to see that $h(y)$ is negative for $1\\le y\\le N$ and that it decreases to its minimum value at $N(1-\\omega)$ and then increases to zero at $y=N$. It then increases to its maximum value at $N(1+\\omega)$ and then decreases for $N(1+\\omega)\\le y\\le (1+\\delta_{N})N$.\nConsequently the difference between\n\\begin{equation}\n   \\sum_{p=(1-\\delta'_{N})N}^{{(1+\\delta_{N})N}}h(p) \\qquad\\mbox{and}\\qquad    \\int_ {(1-\\delta'_{N})N} ^{{(1+\\delta_{N})N}}h(p)\\,dp\n   \\end{equation}\n differs by at   most $4\\max_{1\\le p\\le (1+\\delta_{N})N}|h(p)|$. Since this is $O(e^{2N}\/N^{3\/2})$ by (\\ref{5.68}), and we are only trying to obtain (\\ref{6.12d}), we can neglect this discrepancy.  Therefore to obtain (\\ref{6.12d}) we need only show that\n \\begin{equation}\n  \\int_ {(1-\\delta'_{N})N} ^{{(1+\\delta_{N})N}}\\frac{1}{p} \\({eN\\over   p }\\)^{2p}  \\({p \\over N }-1\\) \\,dp\\ge -{D'}\\frac{e^{2N}}{N^{3\/2}}.\\label{5.76}\n   \\end{equation}\n Under the change of variables $p=xN$ the integral in (\\ref{5.76}) is equal to\n \\begin{equation}\n     \\int_ {1-\\delta'_{N}} ^{{1+\\delta_{N}}}\\frac{1}{x} \\({e \\over  x}\\)^{2xN}  \\(x-1 \\) \\,dx .\\label{5.77}\n \\end{equation}\n \n \n By (\\ref{j.58j}),  in which $x=1+y$  ,\n    \\begin{eqnarray}\n  \\({e\\over x}\\)^{2x N} \n  &= &e^{2 N}  e^{ -y^2 N +y^3 N\/3 +O(y^4) N}\\label{2.59sas}\\\\\n  & =&  e^{2 N}  e^{ -y^2 N }\\(1+ {y^3 N\\over 3}+O(y^4) N\\)\\nonumber.\n \\end{eqnarray} \n Therefore, with the change of variables $x=1+y$ we write the integral in (\\ref{5.77}) as\n \\begin{equation} \n      e^{2 N}\\int_{-\\delta'_{N}}^{\\delta_{N}} \\frac{y}{1+y  }e^{ -y^2 N } \\(1+ {y^3 N\\over 3}+O(y^4) N\\)\\label{5.82}\n \\,dy. \n   \\end{equation} \n   We use $(1+y)^{-1 }=(1-y +y^{2} -y^{3} +O(y^{4}) )$  to write\n  \\begin{eqnarray}\n    &&\\frac{y}{1+y }  \\(1+ {y^3 N\\over 3}+O(y^4) N\\)\\\\\n    &&\\qquad\\quad=y- y^{2} +  y^{3} +\\frac{y^{4 }N}{3}- y^{4 }+O(y^{5})N.\\nonumber\n   \\end{eqnarray}\n   Using this we see that   (\\ref{5.82})\n   \\begin{equation}\n  =    e^{2 N}\\int_{-\\delta'_{N}}^{\\delta_{N}} e^{ -y^2 N } \\(y- y^{2} +  y^{3} +\\frac{y^{4 }N}{3}- y^{4 }+O(y^{5})N\\)\\label{5.82q}\n \\,dy. \n   \\end{equation}\n   Recall that   $\\delta_{N}=(10\\log N\/N)^{1\/2}$ and $\\delta'_{N}=N^{-1\/4}$ . Since\n    \\begin{equation}\n  e^{2 N} \\int_{(10\\log N\/N)^{1\/2}}^{\\infty}e^{-y^{2}N}\\,dy  \\le {e^{2 N}\\over N^{10}}\\label{5.98}\n   \\end{equation}\n   and\n       \\begin{equation}\n  e^{2 N} \\int_{-\\infty}^{-N^{-1\/4}}e^{-y^{2}N}\\,dy  \\le e^{ 2N-N^{1\/2}} ,\n   \\end{equation}\n  errors we can ignore in obtaining (\\ref{6.12}), we can simplify matters by replacing the integral in (\\ref{5.82q}) by \n      \\begin{eqnarray}\n&&    e^{2 N}\\int_{-\\infty}^{\\infty} e^{ -y^2 N } \\(y- y^{2} +  y^{3} +\\frac{y^{4 }N}{3}- y^{4 } +O(y^{5})N\\)\\label{5.82qww}\n \\,dy\\\\\n &&\\qquad   =- e^{2 N}\\int_{-\\infty}^{\\infty} e^{ -y^2 N } \\(  y^{2}-\\frac{y^{4 }N}{3}+y^{4 } +O(y^{5})N\\) \n \\,dy\\nonumber\\\\\n &&\\qquad   =- e^{2 N}\\int_{-\\infty}^{\\infty} e^{ -y^2  } \\( { y^{2} \\over N^{3\/2}}-\\frac{y^{4 }}{3N^{3\/2}}+{y^{4 } \\over N^{5\/2}} +O(y^{5})N^{-2}\\) \n \\,dy\\nonumber\\\\\n &&\\qquad   =- {e^{2 N} \\over N^{3\/2}}\\int_{-\\infty}^{\\infty} e^{ -y^2  } \\( y^{2}-\\frac{y^{4 }}{3} \\) \n \\,dy+O\\({e^{2 N} \\over N^{2}}\\)\\nonumber.\n   \\end{eqnarray}\n\nSince \n\\begin{eqnarray}\n&&\n{1 \\over \\sqrt{\\pi}}\\int_{-\\infty}^{\\infty} e^{ -y^2  } \\( y^{2}-\\frac{y^{4 }}{3} \\) \n \\,dy\\label{5.76f}\\\\\n &&\\qquad={1 \\over \\sqrt{2\\pi}}\\int_{-\\infty}^{\\infty} e^{ -y^2\/2  } \\( \\frac{y^{2 }}{2}-\\frac{y^{4 }}{12} \\) \n \\,dy\\nonumber\\\\\n &&\\qquad={1 \\over 2}-  \\frac{1}{4}={1 \\over 4}\\nonumber,\n\\end{eqnarray}\n we obtain (\\ref{5.76}).{\\hfill $\\square$ }\n \n \\medskip\t\\noindent\\label{sec5page}\n{\\bf Proof of Theorem \\ref{theo-cp}  when {$\\bf c_{1}=c_{2}=c$} and {$\\bf  EG_{1} G_{2}>0 $} concluded }  \\label{41}  \nConsider the Gaussian random variable $ (G_{1}\/\\gamma,G_{2}\/\\gamma)$ where $\\gamma=(E G_{1}G_{2}) ^{1\/2}$. This random variable has covariance $\\Gamma$ in (\\ref{cov}). By Lemma \\ref{lem-5.3} there exists a $c'_{0}>0$ such that $ (G_{1}\/\\gamma+c,G_{2}\/\\gamma+c)$ has infinitely divisible squares for all $|c|\\le c'_{0}$.  Let $\\widetilde c$ be the supremum of the $c_{0}' $ for which this holds. Since, by hypothesis, (\\ref{1.12q}) does not hold, $\\widetilde c$ is finite.  \nTherefore, $ (G_{1}\/\\gamma+c,G_{2}\/\\gamma+c)$ has infinitely divisible squares for all $|c|<\\widetilde c$ and not for any $c$ or which $|c|>\\widetilde c$. Translating this into the notation used in Theorem \\ref{theo-cp} we have $ (G_{1}\/\\gamma+c\\alpha,G_{2}\/\\gamma+c\\alpha)$ has infinitely divisible squares for all $|\\alpha|<\\widetilde c\/c$ and not for any $|\\alpha|$ for which $|\\alpha|>\\widetilde c\/c$. \n\n\n\nTherefore, to complete the proof of Theorem \\ref{theo-cp}  when {$  c_{1}=c_{2}=c$} and {$   EG_{1} G_{2}>0 $ we need only show that $ (G_{1}\/\\gamma+c,G_{2}\/\\gamma+c)$ has infinitely divisible squares for $|c|=\\widetilde c$.  Consider the Laplace transform of  $ (G_{1}\/\\gamma+c,G_{2}\/\\gamma+c)$ in (\\ref{idpr.2}). Since it only depends on $c^{2}$ we can simplify the notation by taking   $c>0$. \nLet $c_{m}\\uparrow \\widetilde c$.\n Abbreviate the third line of (\\ref{idpr.2}) by \n$  \\exp\\(P+c^{2}Q\\)$.\nThus $  \\exp\\(P+c_{m}^{2}Q\\)$ is the Laplace transform of an infinitely divisible random variable. Therefore,   for each $t>0$ the power series expansion of $P+c_{m}^{2}Q$ in $s_{1}$ and $s_{2}$ has positive coefficients, except for the constant term. Thus if we write\n\\[P=\\sum_{j,k } a_{j,k} s_{1}^{j}s^{k}_{2},\\hspace{.2 in}Q=\\sum_{j,k } b_{j,k} s_{1}^{j}s^{k}_{2}\\]\nwe see that $a_{j,k}+c_{m}^{2} b_{j,k}\\geq 0$ for each $(j,k)\\neq (0,0)$. Letting $c_{m}\\uparrow \\widetilde c$ we therefore have $a_{j,k}+\\widetilde c^{2} b_{j,k}\\geq 0$ for each $(j,k)\\neq (0,0)$. This shows that $  \\exp\\(\\(P+\\widetilde c ^{2}Q\\) \\)$  is the Laplace transform of an infinitely divisible   random variable. {\\hfill $\\square$ }  \n\n\\begin{remark} \\label{rem-5}{\\rm In the remainder of this paper we continue to prove Theorem \\ref{theo-cp} for all $ c_{1},c_{2} $ and arbitrary covariance $EG_{1}G_{2}$. In each case, as immediately above, because  (\\ref{1.12q}) does not hold, there exists a $  c'<\\infty$ such that $ (G_{1} +c c_{1},G_{2} +cc_{2})$ does not have infinitely divisible squares for all $c$ such that $|c|>  c'$. \nTherefore, if we can show that there exists some $c\\ne 0$ for which both \n\\begin{equation}\n (G_{1} +c c_{1},G_{2} +cc_{2}) \\quad\\mbox{and}\\quad  (G_{1} -c c_{1},G_{2} -cc_{2})\\label{5.100}\n \\end{equation}\n  have infinitely divisible squares, we can use the arguments in the preceding three paragraphs to show that there exists a critical point $\\widetilde c$ such that $ (G_{1} +c c_{1},G_{2} +cc_{2})$ has  infinitely divisible squares for all  $|c|\\le \\widetilde c$ and not for $|c|> \\widetilde c$. Consequently,  in the remainder of this paper,  in which we consider different cases of  $ c_{1},c_{2} $ and arbitrary covariance $EG_{1}G_{2}$ we will only show that (\\ref{5.100}) holds for some $c\\ne 0$.\n }\\end{remark}\n \n \n   \\section{ Proof of Theorem  \\ref{theo-cp}  when {$\\bf (c_{1},c_{2})=(c,\\pm c)$} }\\label{sec-6}\n   \n  We first assume that  $EG_{1}G_{2}>0$ and that   $ (c_{1},c_{2})=(c,- c)$. In this case we have  \n    \\begin{eqnarray}\n\\lefteqn{E_G\\(e^{ -( \\lambda_1(G_{1}+c   )^2+\\lambda_2(G_2-c   )^2)\/2}\\)\\label{idsqaspr.2qxx}}\n\\\\\n&& =\\frac{1}{(H(a,b,\\lambda_1,\\lambda_2))^{1\/2} }   \\exp\\(-  c^{2}\\(\\frac{\\rho\\lambda_{1}\\lambda_{2}+\\lambda_{1}+\\lambda_{2}}{2H(a,b,\\lambda_1,\\lambda_2)} \\) \\)\\nonumber,\n\\end{eqnarray}\nwhere $\\rho=a+b+2$. This is exactly the same as (\\ref{idpr.2}) except that $\\gamma$ is replaced by $\\rho$. We now trace the proof in Sections \\ref{sec-3}--\\ref{sec-5} and see what changes. Obviously much remains the same.   In particular the power series $P$ is unchanged. The basic expression for $Q$ in (\\ref{3.10})\n is essentially the same except that $\\gamma$ is replaced by $\\rho$.  Thus Lemma \\ref{lem-2.1j} is also essentially the same except that $\\gamma$ is replaced by $\\rho$.\n \n The analysis in Section \\ref{sec-4} only uses the fact that $\\gamma<\\infty$,   and  since $\\rho<\\infty$, Lemma \\ref{lem-1.1} also holds in this case.\n\nIn going through Section \\ref{sec-5} we see the coefficients of $Q$ change, but they still  lead to essentially the same inequalities that allow us to complete the proof.\nIn place of (\\ref{2.28}) we have\n\\begin{eqnarray}\n-3\\rho+a+b&=&-2(3+(a+b)):=-2\\widetilde \\zeta\\label{6.2}\\\\\na\\rho-1&=&(a+1)^2\\nonumber\\\\\nb\\rho-1&=&(b+1)^2\\nonumber.\n\\end{eqnarray}\nUsing this in (\\ref{5.4}) and (\\ref{3.23}), with $\\gamma$  replaced by $\\rho$, we get \n \\begin{eqnarray}\nQ_{j,0}(t )  &=& \\( {  (b+1)^{2} \\over dt}+O(1\/t^{2})\\)\\alpha^{j-1},\n \\end{eqnarray} \nand\n \\begin{eqnarray}\nQ_{0,k}(t )  &=&  \\( {  (a+1)^{2} \\over dt}+O(1\/t^{2})\\)\\alpha^{k-1} .\\label{3.23xx}\n \\end{eqnarray} \n\nWe also see that we get (\\ref{2.4f}) with $\\gamma$  replaced by $\\rho$ and consequently, in place of (\\ref{2.4fx}), we get \n   \\begin{equation}\n  {R_{j,k,0}(t,c) \\over  \\alpha^{j-1}\\beta^{k-1}}= {1-2c^2(\\widetilde\\zeta\/d)+O\\( t^{-1} \\)\\over d^{2}t^2}\\label{2.4fxq}.\n   \\end{equation}\n Of course the key term in the proof is the analogue of   $A_{j,k,p}(t)$. We get the third line of (\\ref{5.15}) with $\\gamma$  replaced by $\\rho$, which by (\\ref{6.2}) leads to (\\ref{5.18}) with $\\zeta$ replaced by $\\widetilde\\zeta$ and $\\gamma$  replaced by $\\rho$. \nTherefore,   all the subsequent lower bounds for $A_{j,k,p}(t)$ that are in Section \\ref{sec-5} hold   when $\\zeta$ is replaced by $\\widetilde \\zeta$.\nIn the   proof  of (\\ref{6.12}) in Section \\ref{sec-5} the only property of $\\zeta$ that is used is that is is positive. Since $\\widetilde\\zeta$ is also positive the same argument  completes the proof of Lemma \\ref{lem-5.3} and   consequently, by Remark \\ref{rem-5},    of Theorem  \\ref{theo-cp},    when $EG_{1}G_{2}>0$ and     $ (c_{1},c_{2})=(c,- c)$. \n\nWhen $EG_{1}G_{2}<0$ and     $ (c_{1},c_{2})=(c,- c)$ we note that\n\\begin{equation}\n   ((G_{1}+c)^{2},(G_{2}-c)^{2})\\stackrel{law}{=}  ((G_{1}+c)^{2},(-G_{2}+c)^{2}).\n   \\end{equation}\nNow $EG_{1}(-G_{2})>0$ and we are in the case proved on page \\pageref{41}. Therefore, by Remark \\ref{rem-5}, Theorem \\ref{theo-cp} holds in this case. \n\nFinally when $EG_{1}G_{2}<0$ and     $ (c_{1},c_{2})=(c, c)$ we note that \n\\begin{equation}\n   ((G_{1}+c)^{2},(G_{2}+c)^{2})\\stackrel{law}{=}  ((G_{1}+c)^{2},(-G_{2}-c)^{2}).\n   \\end{equation}\n Now $EG_{1}(-G_{2})>0$ and we are  in the case proved in the beginning of this section.\n  {\\hfill $\\square$ }\n\n   \\section{ Proof of Theorem  \\ref{theo-cp} when {$\\bf (c_{1},c_{2})=(c,0)$}} \\label{sec-7}\n   \n We first assume that  $EG_{1}G_{2}>0$.  In this case we have  \n    \\begin{eqnarray}\n\\lefteqn{ E_G\\(e^{ -( \\lambda_1(G_{1}+c  )^2+\\lambda_2G_2^2)\/2}\\)\\label{idsqaspr.2qyy}}\n\\\\\n&& =\\frac{1}{(H(a,b,\\lambda_1,\\lambda_2))^{1\/2} }   \\exp\\(-  c^{2}\\(\\frac{b\\lambda_{1}\\lambda_{2}+\\lambda_{1} }{2H(a,b,\\lambda_1,\\lambda_2)} \\) \\)\\nonumber.\n\\end{eqnarray} \nThe term in the numerator of the exponential lacks the $\\lambda_{2}$ that is present in  (\\ref{idpr.2}) and (\\ref{idsqaspr.2qxx}). Therefore, the formulas for the coefficients of the power series for the analogue of $Q$, which we denote by  $\\widetilde Q$, are different.  It is easy to see that in place of  (\\ref{as1.3})\nwe get \\begin{eqnarray}\n\\widetilde Q(s_1,s_2,t)\n&=&-\\frac{ (b t^2(1-s_1)(1-s_2)+t(1-s_1 ))}{\\bar t^2}\\label{as1.3xx}\\\\\n&&\\qquad\n\\cdot \\sum_{n=0}^\\infty  (\\alpha s_1+\\beta s_2-\\theta\\alpha\\beta s_1s_2)^n\\nonumber\\\\\n&=&-\\frac{ (b t^2+ t-(b t^2+t)\ns_1-b t^2 s_2 +b t^2s_{1} s_2 )}{\\bar t^2}\\nonumber\\\\ &&\\qquad\n\\cdot\\sum_{n=0}^\\infty  (\\alpha s_1+\\beta s_2-\\theta\\alpha\\beta s_1s_2)^n\\nonumber .\n\\end{eqnarray}\n Using this, in place of Lemma \\ref{lem-2.1j}, we get \n\n   \\begin{lemma}\\label{lem-2.1jxx}   For  all $t$ sufficiently large, and $j,k\\ge 1$  \n \\begin{equation} \n\\widetilde Q_{j,0}(t )= \\frac{ ( b t^2+t)(1-\\alpha)  \n }{\\bar t^2} \\alpha^{j-1} \\label{3.29xx}\n \\end{equation} \nand\n \\begin{equation} \n\\widetilde Q_{0,k}(t)= \\frac{  b t^2(1-\\beta ) +\\beta t\n }{\\bar t^2} \\beta^{j-1} \\nonumber.\\label{3.30xx}\n \\end{equation} \nFurthermore,   for all $t$ sufficiently large and for all   $1\\le j\\le k$\\begin{eqnarray}\n\\lefteqn{\n\\widetilde Q_{j,k}(t)\\label{2.22js.a}}\\\\&& ={ \\alpha^{j-1} \\beta^{k-1}\\over \\bar t^2}\\sum_{p=0}^{j}\\widetilde t^{-2p}{j\\choose p}{k\\choose p}\\nonumber\\\\\n&&\\quad\\(-b t^2\\((1-\\alpha)(1-\\beta)-{(1-\\beta)p\\over j}-{(1-\\alpha)p\\over k}+\\widetilde t^{-2}{j-p\\over j}{k-p\\over k}\n\\)\\right.\n\\nonumber\\\\\n&&\\qquad\\qquad\\left.+t \\beta\\(1-\\alpha-{p\\over j}  \\)\\)\\nonumber.\n    \\end{eqnarray}  \n\\end{lemma}\n\nThe analysis in Section \\ref{sec-4} only uses the fact that $\\gamma<\\infty$.  Since $b<\\infty$, Lemma \\ref{lem-1.1} also holds in this case.\n\nIn going through Section \\ref{sec-5} we see the coefficients of $\\widetilde Q$ change, but  they still  lead to similar inequalities that allow us to complete the proof.\n Using (\\ref{3.29xx}), (\\ref{3.30xx}) and (\\ref{3.18}) we get \n  \\begin{equation} \n\\widetilde Q_{j,0}(t )= \\(\\frac{b^{2}}{d^{2}t} +O(1\/t)\\)\\alpha^{j-1}  \\label{3.29oxx}\n \\end{equation} \nand\n  \\begin{equation} \n\\widetilde Q_{0,k}(t )= \\(\\frac{d+1}{d^{2}t} +O(1\/t)\\)\\beta^{k-1}  \\label{3.29oxxs},\n \\end{equation} \nsince $ab=d+1$.\n\nWe next consider he analogue of (\\ref{2.4c}) which we denote by $\\widetilde R_{j,k}(t)$.\nWe see that in computing this the first two lines  of the analogue of (\\ref{2.4f}) remain  unchanged, except for replacing $c$ by 1. The last two lines of  (\\ref{2.4f}) are now\n\\begin{eqnarray} \n   &&\\(-b t^2\\((1-\\alpha)(1-\\beta)-{(1-\\beta)p\\over j}-{(1-\\alpha)p\\over k}+\\widetilde t^{-2}{j-p\\over j}{k-p\\over k}\n\\)\\right. \\nonumber\\\\\n&&\\quad\\left.+t\\beta\\(1-\\alpha-{p\\over j} \\)\\).\\label{7.8}\n \\end{eqnarray}\nTherefore, in place of (\\ref{2.4fx}), we get     \\begin{equation}\n  {\\widetilde R_{j,k,0}(t) \\over  \\alpha^{j-1}\\beta^{k-1}}= {1-2c^{2}(b\/d) +O\\( t^{-1} \\)\\over d^{2} t^2}\\label{2.4fxaa}.\n   \\end{equation}\n   Using (\\ref{2.4f2}), with $\\gamma$  replaced by $b$ and Lemma \\ref{lem-2.2}, we see that (\\ref{7.8})\n \\begin{eqnarray}\n   && =-\\frac{b}{d^{2}}\\((ab+1) -{ bp(dt-b) \\over k}-{ap(dt-a) \\over j} +{p^{2}\\over jk}\\)\\\\\n   &&\\qquad+\\( \\frac{b}{d}-{  p(dt-a)\\over dj}\\)+O\\( {1 \\over t} \\) .\\nonumber\\\\\n      &&=-{2b\\over d^{2}}\t +{   p(dt-a)\\over d^{2}j}+{ b^{2} p(dt-b)\\over d^{2}k}  -{b p^{2}\\over jk} +O\\( {1 \\over t} \\)\\label{96ss}\\\\\n  &&\n\\ge\\frac{2}{d^{2}} \\( p  {\\sqrt{(dt-a)(dt-b)} \\over \\sqrt{jk} }-1\\)b -{b p^{2}\\over d^{2} jk} +O\\( {1 \\over t} \\).\\nonumber\n   \\end{eqnarray}\nComparing this inequality to the first line of (\\ref{5.18}) we see that we have exactly what we need to complete the proof in this case.  The rest of the argument in Section \\ref{sec-5} only uses the fact that $\\zeta>0$. It is now replaced by $b>0$. \n  Thus we get Lemma \\ref{lem-5.3}   and, by Remark \\ref{rem-5},    Theorem  \\ref{theo-cp}, when {$  (c_{1},c_{2})=(c,0)$}} and   $EG_{1}G_{2}>0$. However this proof holds for $c$ positive or negative, so if $EG_{1}G_{2}<0$, we simply note that \n\\begin{equation}\n   ( G_{1} ^{2},(G_{2}+c)^{2})\\stackrel{law}{=}  ( G_{1} ^{2},(-G_{2}-c)^{2}) .\n   \\end{equation}\nSince $EG_{1}(-G_{2})>0$ we are in the case just proved so, by Remark \\ref{rem-5},  Theorem  \\ref{theo-cp} holds in this case also.\n{\\hfill $\\square$ }\n\n   \\section{ Proof of Theorem  \\ref{theo-cp}  for  {$\\bf (c_{1},c_{2})\\in R^{1}\\times R^{1}$} }\\label{sec-8}\n   \n   It is simple to complete the proof from the results already obtained. Suppose neither $c_{1}$ nor $c_{2}$ are equal to zero. Then, clearly,\n   \\begin{equation}\n    ( (G_{1}+c c_{1}) ^{2},(G_{2}+cc_{2})^{2})\n   \\end{equation}\n   is infinitely divisible, if and only if   \n    \\begin{equation}\n    ( (G_{1}\/c_{1}+c  ) ^{2},(G_{2}\/c_{2}+c )^{2})\n   \\end{equation} \n      is infinitely divisible. We have already shown that there exists a critical point $\\widetilde c>0$ such that \n     \\begin{equation}\n    ( (G_{1}\/c_{1}+c  ) ^{2},(G_{2}\/c_{2}+c )^{2})\n   \\end{equation} \n   is infinitely divisible for all $|c|\\le \\widetilde c$ and not for $|c|> \\widetilde c$. Consequently $\\widetilde c$ is also a critical point for the infinite divisibility of\n   \\begin{equation}\n    ( (G_{1} +c_{1}c  ) ^{2},(G_{2} +c_{2}c )^{2}).\n   \\end{equation}    \n   If $c_{1}=0$ we repeat this argument for \n   \\begin{equation}\n       ( G_{1} ^{2},(G_{2}+cc_{2})^{2}).\n   \\end{equation}\n   {\\hfill $\\square$ }\n \t \n\\def\\noopsort#1{} \\def\\printfirst#1#2{#1}\n\\def\\singleletter#1{#1}\n         \\def\\switchargs#1#2{#2#1}\n\\def\\bibsameauth{\\leavevmode\\vrule height .1ex\n         depth 0pt width 2.3em\\relax\\,}\n\\makeatletter\n\\renewcommand{\\@biblabel}[1]{\\hfill#1.}\\makeatother\n\\newcommand{\\leavevmode\\hbox to3em{\\hrulefill}\\,}{\\leavevmode\\hbox to3em{\\hrulefill}\\,}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{The Microstructure of the States}\nIn the mean field approach one finds that there is a strict\ncorrelation\nbetween the different kinds of\noverlaps one can define. In particular in the case of the energy\noverlap\none finds that\n\\be\nq^e= A q^2 +B.\n\\ee\nConsequently the probability distribution of the energy is given by\n\\be\nP^e(q^e) = \\left({q^e-B \\over A}\\right)^{-1\/2}P\\left({(q^e-B) \\over\nA}\\right). \\ee\n\nIn other words different equilibrium states do not differ only by the\nreverse of large droplets of\nspins \\cite{FH}. If two states  differ only by the reverse of large\ndroplets of  spins,\nthe energy overlap would\nbe the same as that of a state with itself,  apart from the\ncontributions of\nthe interface among\ndifferent  clusters \\cite{CPS}.\n\nIn the picture which emerge from replica symmetry\nbreaking\nthere  are clusters of  spins which are strongly coupled. These\nclusters\nhave a surface which is\nproportional to the volume so  that the change in energy overlap,\nwhich\nis proportional to the\nsurface, is comparable to the change in  overlap, which is\nproportional\nto the volume. Of course\nwe could call a fractal object a droplet, but this  would be a\nlinguistic\nabuse.\n\nIn principle we can get information on the form\nof the clusters from the analysis of the  correlations fucntions. The\nsituation  is rather complex\n\\footnote{See for example the description in the very interesting\npaper\n\\cite{MV}}\nand we shall only stress some features.\n\nIn order to define the correlation functions it may be useful\nto consider two real replicas of the same system: the spins will be\ndenoted $\\s$ and $\\tau$\nrespectively. One can introduce the overlap  $q_i=\\s_i\\tau_i$. This\ntwo\nreplicas system can be\ncharacterised by the value of $q=1\/V \\sum_i q_i$. Different values of\n$q$ in the interval $q_m -\nq_M$ are allowed.\nOne can consider the connected correlation function  restricted on\nconfigurations with a given overlap $q$:\n\\be\nG_q(x)={1 \\over V} \\sum_i (<q_{i+x}q_i>_q- q^2 )\\equiv {1 \\over V}\n\\sum_i (<q_{i+x}>_q<q_i>_q- q^2).\n\\ee\nThe correlation function decreases in the mean field approximation in\ndimensions $d$ as\n\\cite{DK1} \\be\nx^{-d+D},\n\\ee\nwhere $D=3$, if $0 < q <q_M$, and  $D=4$, if $q=0$ (this is possible\nonly if $q_m=0$,\nwhich  happens in zero magnetic field).\n\nThese results are supposed to be valid at all orders in the\nperturbative expansion above  $6$  dimensions. The exponent $D$ may\nchange for less that 6 dimensions ($D$ at $q=0$ does  actually change\nwhen $d<6$ \\cite{DK2}).\n\nNumerical simulations in $4$ dimensions \\cite{PR,CPR} strongly\nsuggest the correctness of the broken\nreplica picture and no anomaly is observed.\n\n\\section{A Simple Model for the Clusters}\nIn order to interpret the previous results geometrically, we consider a\nsimple\nmodel of spin glasses in\nwhich  we find explicitly the existence of infinite number of\nequilibrium\nstates \\cite{ma,N}.\nThe model\nis described by  the usual Hamiltonian, with an unusual distribution\nof\nthe couplings:  if we  order the coupling in  decreasing strength we\nmust  have\nthat\n\\be\n|J_a|>\\sum_{b>a}|J_b|.\n\\ee\n\nThis can be realised by setting $J_{i,k}=(R_{i,k})^{g(L)}$, where\n$R_{i,k}$ is a random number\nuniformly distributed in the interval $[-1,1]$ and $g(L)$ is an\ninteger\nfunction (which takes odd\nvalues) which diverges  sufficiently fast when the size $L$ of the\nbox\ngoes to infinity.\n\nWe will consider this model at zero temperature. If we consider the\nsystem in a box of size $L$\nwith  assigned boundary conditions,  there will be clusters of\nrigidly\nconnected spins which are\ncontrolled by the spins at the boundary. If we send the volume to\ninfinity, we find that the\nground state depends on  the boundary conditions.\n\nThe properties of the model may be found by mapping it on invasion\npercolation. In dimensions $d>8$ one finds that these clusters of\nrigidly  connected spinshave Hausdorff dimension 4 and the system  is\nfilled by an  infinite number of them. If we consider two systems with\nrandom (and  different)  boundary  conditions, spins in different\nclusters will be random oriented so  that the  global overlap will  be\nzero.\n\nThe correlation function is given by the probability that two spins  at\ndistance $x$    belong to the same cluster. This probability  goes to\nzero as $x^{- d+D_c}$, where $D_c$ is the  Hausdorff  dimension of the\ncluster. If we stick to this  interpretation,  one finds that the\nHausdorff  dimension of the  clusters obtained from mean field theory\nis  $4$, which is the same value as in this simple model.\n\nIn the real world things are more complicated. Clusters are not  really\nindependent, they  interact one with the othe. This\nphenomenon may explain the  shift  of $D$ from $4$ to $3$  when we go\nto non zero $q$. Lot of  theoretical work is needed to  present a full\nunderstanding of  the properties of the clusters of strongly  connected\nspins in the  framework of the broken replica  approach.\n\n\\section{Local Overlaps}\n\nGenerally speaking, the function $P(q)$ alone is a rather delicate\nobject  because its computation involves the  differences among  total\nfree energies  which are of order 1. We  could  consider situations in\nwhich  the free energy differences are of order $L^\\lambda$  with\n$\\lambda  <d$. In this a way the free energy differences diverge, but\nthe free energy densities are the same. In this situation  most of the\nconclusions based on broken replica symmetry  would be valid, but the\nfunction $P(q)$  would be trivial.\n\n It should be clear that a non trivial form of the function $P(q)$ at\nzero  magnetic field is not  sufficient to rule out the droplet  model.\nIndeed in the framework of  the  droplet model there could  be a few\ncompact  independent  droplets which can be flipped  independently.\nCompact droplets  are characterised by the following  conditions:  the\nradius $R$ goes to infinity the volume af a compact droplet goes as\n$R^d$ and the surface as $R^{d-1+\\omega}$   with $\\omega<1$. If\n$\\omega>0$ the droplets have  a fractal surface. If $\\omega=1$ they are\nno more compact and in this  limit we recover the broken replica\napproach.\n\nIn the droplet model the function $P(q)$ can be non trivial, but the\nfunction $P^e(q^e)$ must go to a  delta  function acquiring a width of\norder $L^{-\\omega}$.\n\nThe differences among the broken  replica and droplet  model become\nmore sharp if consider the  probability distribution  $P_R(q_R|q)$ of\nthe local overlap  $q_R$ defined as\n\\be\nq_R=\\vert{1 \\over R^d} \\sum_k q_k\\vert,\n\\ee\nwhere the sum is done in a box of side $R$ centred around and\narbitrary point $i$\\footnote{I am\ngrateful to J.P. Bouchaud for stressing this point to me.}. The\nprobability distribution may be\nobtained  for a given sample by changing  the point $i$, keeping  the\nconstraint of total overlap $q$.\n\nThe quantity $q_R$ is invariant under a reversal of spins in a  cluster\nthat contains the whole  region of size $R$.  In the droplet model the\nfunction $P_R(q_R|q)$,  should not depend on $q$ in  the infinite\nvolume limit: indeed when we reverse a large droplet,  the  overlap\n$q_R$ changes only  for those points near the boundary of the droplet,\nso that the  dependence  of $P_R(q_R|q)$ on $q$  should vanish as $({R\n\\over L})^{-1+\\omega}$ when the size of system  $L$ goes to infinity.\n\n\\section{The Lower Critical Dimension}\n\nIn testing spin glass theories\nin finite dimensions a crucial issue  is the  validity of the mean\nfield  picture. Now it seems that the mean field theory gives the\ncorrect exponents in  dimensions ($d$)\ngreater than 6, while  some\ncorrections are expected in dimensions less than 6.  Unfortunately  the\ncomputation of the  corrections to mean field theory is rather complex\nand the one loop  contribution has not been fully  computed so that no\nprecise conclusions are available.\n\nThe most critical issue is if the behaviour of  the propagator at $q\\ne\n0$ (i.e. $1\/k^3$ in momentum space) remains  unchanged. This behaviour\narises as a consequence of the breaking of replica symmetry in a\ncontinuos way. The corresponding  Ward identities has not been\ncarefully spelled out, so that it is not  evident if such a relation is\nrenormalized (the $1\/k^4$ behaviour at $q=0$ apparently changes in\nless that $6$ dimensions).\n\nIf this behaviour would be not renormalized, it would be an easy  guess\nto suggest that the lower critical  dimension of the system is $3$ in\nthe same way as the lower  critical dimension for breaking of\ncontinuos symmetries is 2. A direct computation of the  free  energy of\nan interface among states at  different values of $q$ gives rather\ndifferent  results,  i.e. a lower critical dimension of $2.5$,  which\nis by {\\it accident} the same obtained in the  framework of the Migdal\nKadanoff approximation \\cite{FPV2}.\n\nThe discrepancy among the two different results has an unclear  origin,\nwhich we hope to be able  to understand in the next future, when the\nevaluation of the loop  corrections to various different  quantities\nwill be available. In any case  3 it is not very far from  being  the\nlower critical dimension so that it is reasonable to investigate what\nshould happen  near  the critical dimension and to compare with the\nexperimental (or numerical results).\n\nThe renormalization group scheme for the behaviour at the lower\ncritical dimension has not yet developed. Let us try do some educated\nguesses. We  suppose that for  a  system  with continuos distribution\nof the couplings $J$  \\footnote{Anomalous behaviour may  be present if\nthe couplings are discrete when $T \\to 0$, due  to the degeneracy of\nthe ground state},  there is only one\n relevant quantity that describe the behaviour of the system at scale\n$L$  and it\n satisfies a renormalization group equation\n\\be\n{dT \\over dt}= \\beta(T(t)).\n\\ee\nThe quantity $T(t)$ can be identified with the effective temperature\nat\nscale $L=\\ln(t)$ and the\nfunction $\\beta(T)$ should  behave as a power of $T$ as small $T$.\n\n One readily finds that if\n\\be\n\\beta(T) \\propto T^\\lambda,\n\\ee\nthe correlation function diverges as\n\\be\n\\xi \\propto \\exp ( a\/T^{(\\lambda-1)}).\n\\ee\nIn the Migdal Kadanoff one finds that approximation $\\lambda=3$.\n\nIn the same way we expect that the field $q$ has an effective\ndimension\n($\\psi(T)$) which is\nproportional to $T$, and therefore the value of $q$, observed on a\nscale\nof size $L$, should\ndecrease (roughly speaking) as\n\\be\nq(L) \\propto L^{-\\psi(T(\\ln(L))}.\n\\ee\n\n In a given range of scales, which may be quite wide at low\ntemperatures if $\\lambda$ is large,  $T(t)$ may be approximated with a\nconstant. If this happens in the  low  temperature region, where the\ncorrelation length $\\xi$ is much greater that the side of  the  system,\nthe correlation function should decrease with temperature  dependent\npower law. The range of validity of these approximate power law\ndepends on the  form of the  function $\\beta(T)$. In the two dimension\nferromagnetic $x-y$ model the $\\beta$  function is  identically zero so\nthat a power behaviour is exact.\n\nUnder the approximation of a constant $T(t)$, the\nfunction $P(q)$ in  a  box of size $L$ is a  function of\n$q L^{-\\psi(T)}$, the space correlation function of two overlaps ($\\lan\nq(x) q(0) \\ran$) decays as $x^{-2  \\psi(T)}$  and the time  correlation\nfunction  of two spins ($\\lan \\s(0,t) \\s(0,0) \\ran$\ndecays as $t^{\\psi(T)\/z(T)}$, where $z(T)$ is a  temperature dependent\ndynamic exponent. Some numerical indications of such a behaviour in\nthree dimensions are  already been obtained and further work is needed\nto distinguish the  case  of a system at the lower  critical dimension\nor slightly above it \\cite{MPR1,LMPR}.\n\nThese predictions can be tested relatively easily within a dynamic\nstudy.  Let us consider the  dynamics of two replicas starting by two\ndifferent boundary  conditions.   The two replicas will have  zero\noverlap for times not  very large compared with the volume. In  this\nway the dynamics automatically  selects the state $q=0$. In the same\nway, by observing  the  same single system at two large times  $t_1$\nand $t_2$, such that ${|t_1-t_2| \\over t1+t_2|}  <<1$,  the two\nconfigurations at different  times should behaviour as two generic\nconfigurations a  $q=q_M$ \\cite{CKP}.\n\nLarge scale simulations have been recently done on a three\ndimensional\nsystem and high precision\ndata have been obtained in the high temperature region. One finds out\nthat the data are compatible\nboth with a  phase transition at a non zero temperature and with a no\ntransition scenario , altough this last scenario seems more consistent.\n\\cite{MPR2}.\n\nThere are some disturbing feature in the framework of the transition\nscenario. Indeed one could\nlook (in a two replica system) to the value of the Binder cumulant\n\\be\ng= {K-3 \\over 2}; \\ \\ \\ \\ K= {<q^4> \\over <q^2>^2},\n\\ee\nwhere the average is also done over the different realisation of the\nsystems.\n\nThe Binder cumulant can be computed for different sizes and the value\nwhere the different curves\nintersect, characterises the critical temperature. The value of $g$\nat the\ncritical point should be\nuniversal. As far as the curves for the Binder cumulant collapse at a\ngiven temperature (as\npredicted from replica symmetry breaking), it is difficult to\nestablish\nthe existence of a point in\nwhich the curves merges, because a small shift is enough to avoid the\nintersection.\n\nThere is good numerical evidence that the value of $g$ at the\napparent\nintersection point is\nstrongly dependent on the  choice of the probability distribution for\nthe\ncouplings $J$\n\\cite{MPR2}.\n\nThis dependence of $g$ on the probability distribution of the\ncouplings\nstrongly suggest that $3$ is\nthe lower critical dimension or it is very near to it and the real\ncritical\ntemperature, if any, is\nfar  from the apparent point where different cumulants merge.\n\nIt should also be said that simulations (and experiments) at\ntemperature\nsmaller that the apparent\ncritical temperature are well compatible with the possibility that\n$D=3$\nis the lower critical\ndimension.\n\nVery large scale simulations with a dedicated computer \\cite{OGI} have\nshown  a power  law decay of the\ncorrelation of the same spin at different times: $C(t)\\propto t^{-\n\\delta}$. Real experimental data\nshow a behaviour similar behaviour for the remanent magnetization as\nfunction of the waiting\ntime. They are compatible with the law $M(t,t_W)=t^{-\\delta}\nf(t\/t_w)$. The presence of a non zero\nvalue of $\\delta$ can be considered as an effect of having an\nanomalous\ndimension for the $q$ field.\n\nOther simulations for the static properties done in the low\ntemperature\nregion seem to be well\ncompatible with the scenario where no transition is present and that\n$q$\nhas a temperature dependent\ndimension.\n\n\\section{Coupled Replicas}\nSome of the previously discussed points become clear if we consider\nthe following Hamiltonian \\cite{CPS,FPV1}:\n\\be\nH=-\\sum_{i,k} J_{i,k} (\\sigma_i \\sigma_k +\\tau_i \\tau_k) - h \\sum_i\n(\\s_i+\\tau_i)\n-\\eps \\sum_i  \\sigma_i \\tau_i.\n\\ee\nThe term proportional to $\\eps$ has the tendency to align the two\nreplicas in the same direction\nfor positive $\\eps$ and in the opposite direction for negative\n$\\eps$.\nTherefore we expect that\n\\bea\n\\lim_{\\eps \\to 0^+} q(\\eps) = q_M \\\\\n\\lim_{\\eps \\to 0^-} q(\\eps) = q_m\n\\eea\n\nAn unexpected result  of a detailed computation \\cite{FPV1} show that in\nthe  replica\napproach\n\\bea\nq(\\eps) = q_M +A \\eps^{1\/2} \\ \\  for \\ \\ \\eps>0 \\\\\nq(\\eps) = q_m - B \\eps^{1\/2} \\ \\  fo r\\ \\  \\eps>0.\n\\eea\nIn other words both the spin glass susceptibility and the $q - q$\ncorrelation length diverge when\n$\\eps \\to 0$.\n\nThis prediction of the replica approach is very interesting because\nit can\nbe well studied in\ncomputer simulations as far as the function $q(\\eps)$ can be computed\nwith a reasonable amount of\ntime, at leas for not infinitesimal $\\eps$. The correlation function for\n$q$ is finite for non zero $\\eps$.  In\nother words the new term\nforces the two systems to be in the same state and also if the state\nis not\nthe real equilibrium one its\nproperties should no be crucial dependent on this fact.\n\nThe divergence of the correlation length when $\\eps \\to 0$ implies\nthe\npossibility of changing\nlarger and larger clusters of spins and the consequent divergence of\nthe\ncorrelation time.\n\nThe naive replica predictions seems to be approximately well\nsatisfied in\n4 dimensions \\cite{PR,CPR},\nwhile they  are far from being correct in the three dimensions\n\\cite{MPR2}. In this case at zero\nmagnetic field the  preliminary data are compatible with a law of the\ntype\n\\be\nq=\\eps^{\\lambda(T)},\n\\ee\nwhere the exponent $\\lambda(T)$ is related to the exponent $\\psi(T)$\nof section (5) by\n\\be\n (\\lambda+1)  \\psi(T)= 1.\n\\ee\n\n\\section{Conclusions}\n\nIn the recent years there have been both\ntheoretically\nand experimentally progresses in the study of spin glasses. For short\nrange models the theoretical situation is still not completely clear\nbecause we do not  known how to compute\nsystematically the corrections to the mean field theory (or better we\nknow how to do, but the\ncomputations are too complex).\n\nExperimentally (from numerical simulations) it seems that the mean\nfield picture is in very nice\nagreement with the results in four dimensions and in three dimensions\nfor not too large size.\n\n An\nopen possibility is that the critical dimension is near to 3. In this\ncase\nthere would be a wide\ntemperature range where the correlation length is very large, but\nfinite.\nThe\npredictions from replica symmetry breaking on the global order\nparameters should apply to systems of\nsize smaller then the correlation length.  All the data I know are\nconsistent with this possibility,\nwhich should be better investigated numerically and theoretically. I\nhope that a strong progress will\nbe done in the next future on this particular point.\n\n \\section{Acknowledgements}\nIt is pleasure for me to thank for very useful discussions and\nfruitful\ncollaboration on these\nproblems L. Cugliandolo, S. Franz, J. Kurchan, E. Marinari, F. Ritort\nand M. Virasoro.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nAsymptotic Giant Branch (AGB) stars are post-main sequence stars that\nrepresent the most luminous stage of evolution for low- and\nintermediate-mass stars.  One of the main properties of AGB stars is\nthat they loose $50-80$\\% of their mass (gas and dust), they therefore\nchemically enrich the interstellar medium (ISM) and represent the main\nsource of dust in the Universe. AGB stars are also indicators of\ndistance, structure and metallicity. They contribute to the integrated\nlight of galaxies and exist in large numbers in the neighbouring\nMagellanic Clouds.\n\nAll stars with an initial mass comprised between $0.8$ and $8$\nM$_\\odot$ become AGB stars. The AGB phase is rather brief ($<0.01$\nGyr) compared to other stellar evolutionary phases where stars spend\nmost of their life, e.g. the main sequence. AGB stars are\ncharacterized by a double shell burning of H and He, respectively,\naround their C- and O-rich nucleus. Throughout the AGB structure a\ncomplex chemistry is developed: molecules form right above the\nphotosphere and these are responsible for two classes of AGB stars,\nC-rich (or C-type) where carbonaceous molecules like CN and C$_2$\ndevelop and O-rich (or M-type) where oxides like TiO and VO develop,\nafter the formation of stable CO. The chemical path-way an AGB star\nwill take depends on the ISM metallicity prior to their origin and on\nthe efficiency of the dredge-up process that brings elements\nsynthesized in the stellar interior to the surface.  At a given\ndistance from the centre of the star, where temperature and pressure\nare appropriate, dust forms, Dust will also be predominantly of\ncarbonaceous or silicate type and this is related to the molecular\ncomposition in the AGB stars atmospheres. Above the dust layer other\nmolecules, like OH and HCN, develop in the so-called circumstellar\nregion.\n\nThe AGB phase is a dynamical phase because stars experience surface\nluminosity variations with long periods and large amplitudes, and\nmass-loss.  These and further details on the AGB stars can be found in\nHabing \\& Olofsson (\\cite{habol}).\n\n\\section{Infra-Red images}\nBefore Infra-Red (IR) images of AGB stars were obtained the Magellanic\nCloud stellar population was characterized by optical\nimages. Initially the sensitivity was limited to bright super giant\nand carbon stars (e.g.~Westerlund et al \\cite{wes64}). Later, fainter\nsources were studied by Blanco et al (\\cite{bl80}) in several fields\nin the both the Large Magellanic Cloud (LMC) and the Small Magellanic\nCloud (SMC). These observations were extended to additional fields by\nBlanco \\& McCarthy (\\cite{bl83}). At the same time the variability of\nAGB stars was investigated by Hughes \\& Wood (\\cite{hu90}) who\nreferred to AGB stars as Long Period Variables (LPVs).\n\nThese authors discovered that there are several C-rich AGB\nstars. There were, however, not enough luminous AGB stars compared to\nthe prediction by theory and to the known large number of Cepheid\nstars. The latter are the precursors of AGB stars of moderate mass. It\nemerged that because AGB stars are cool (red) they are potentially \nobscured by dust preventing their observation in the\noptical. Therefore, IR observations were needed to progress in this\nfield. Note that the deficit of AGB stars was several hundreds!\n\n\\subsection{First IR images}\nThe first IR images of AGB stars in the Magellanic Clouds were\nobtained by Frogel \\& Richer (\\cite{fro83}). They used the CTIO 1.5m\ntelescope to observe a field in the bar west region of the LMC\n(Fig.~\\ref{fig1}). The observations were sensitive to stars as faint\nas $K=11$ and the spatial resolution corresponded to an aperture of\n$27^{\\prime\\prime}$. A few red, but not very luminous AGB stars, were\nidentified.\n\n\\begin{figure}\n\\includegraphics[width=12cm]{cioni_fig_1.eps}\n\\caption{Colour-magnitude diagram of LMC bar-west sources (Frogel \\&\n  Richer \\cite{fro83}). Solid lines are lines of constant bolometric\n  magnitude while the dashed areas indicate the region occupied by\n  upper RGB stars.}\n\\label{fig1}\n\\end{figure}\n\nThe study of the AGB population advanced in parallel with the\ninvestigation of their variability aspect; this is still the case at\npresent.  Using the $K$-band magnitude it was discovered that LPVs\nobey a period luminosity relation that represents an alternative\ndistance indicator to Cepheid stars (Hughes et al \\cite{hu90}).\n\n\\subsection{Mid-IR space observations}\nIn 1983 the IR Astronomical Satellite (IRAS) scanned the sky. The\ninstrumentation on board opened a new window into the study of AGB\nstars, especially for those surrounded by dust. IRAS also allowed for\nmaking the transition between studying individual AGB stars to using\nthem in large numbers to investigating properties of the hosting\ngalaxy (i.e.~the Milky Way).  The exploitation of the IRAS catalogue\nis still on-going. IRAS discovered $50$ obscured AGB stars in the LMC\nand $25$ in the SMC (van Loon et al \\cite{vlo99}). The distribution of\ncool stars in the IRAS colour-colour diagram, [12]-[25] versus\n[25]-[60], was presented by van der Venn \\& Habing\n(\\cite{vdv89}). Regions occupied by AGB stars of a different type and\nwith a different dust shell thickness were identified, as well as the\nregion where Planetary Nebulae (PNe), the successors of AGB stars, are\nfound. It was suggested that ageing AGB stars develop pulsation\nvariability and thicker dust shells.\n\nIn the following decade the IR Space Observatory (ISO; 1995-1998)\nperformed targeted observations of the AGB stars discovered by IRAS.\nThe power of the ISO instrumentation relied both in the imaging and\nspectroscopic capabilities that allowed for studying the type of dust,\nO-rich if absorption is present at $9.7$ $\\mu$m and C-rich if emission is\npresent at $11.4$ $\\mu$m or absorption at $3$ $\\mu$m, and to relate this\nwith the photometric colours and stellar models.\n\nISO performed also a mini-survey of Magellanic Clouds (Loup et al\n\\cite{lou99}). This survey covers an area of $0.8$ deg$^2$ and $0.28$\ndeg$^2$ of the LMC and SMC, respectively. Imaging observations were\nobtained in the [$4.5$], [$7$] and [$12$] $\\mu$m filters. The\nconsiderably improved spatial resolution and sensitivity of ISO with\nrespect to IRAS showed that IRAS missed $\\sim 50$\\% of the dust\nobscured AGB stars and that these stars are as luminous as C-rich AGB\nstars with thin dust shells (Fig.~\\ref{fig3}). The ISO imaging data\nwere combined with the near-IR data obtained from the DENIS survey\n(Cioni et al \\cite{cio00a}).\n\nMore or less simultaneously mid-IR space observations were obtained\nfrom the Mid-course Space Experiment (MSX; 1996-1997). This satellite\nobserved $\\sim 100$ deg$^2$ of sky at better spatial resolution but\nworse sensitivity than IRAS. The colour-colour diagram, [$K$]-[$8$]\nversus [$J$]-[$K$], showed obscured AGB stars but with a non-negligible\noverlap with PNe and HII regions (Egan et al \\cite{ega01}).\n\n\\subsection{Near-IR images}\n\nThe most comprehensive near-IR sky surveys that provided a wealth of\ndata for studying the Magellanic Clouds were the 2MASS (1997-2001;\n$JHK_s$ filters) and DENIS (1995-2001; $IJK_s$ filters) surveys. Among\ntheir highlights: (i) the distribution of stars in the\ncolour-magnitude space, e.g.~$J-K_s$ versus $K_s$ (Nikolaev \\&\nWeinberg \\cite{nik00}), and (ii) the distinction between O-rich and\nC-rich AGB stars of the earliest spectral types (Cioni \\& Habing\n\\cite{cio03a}) $-$ see Fig.~\\ref{fig2}. Both surveys detected the\nalmost complete population of AGB stars with absent or thin dust\nshells, as well as several obscured AGB stars.\n\n\\begin{figure}\n\\includegraphics[width=6cm]{cioni_fig_2a.eps}\n\\qquad\n\\includegraphics[width=6cm]{cioni_fig_2b.eps}\n\\caption{(Left) Colour-magnitude diagram of the LMC observed by 2MASS\n  where different regions are occupied by different types of stars\n  (Nikolaev \\& Weinberg \\cite{nik00}). (Right) Colour-magnitude\n  diagram of AGB stars in the LMC selected from DENIS data, the region\n  occupied by O- and C-rich AGB stars is clearly indicated (Cioni \\&\n  Habing \\cite{cio03b}).}\n\\label{fig2}\n\\end{figure}\n\nThe view of the Magellanic Clouds changed dramatically from that of\ntypical irregular galaxies with central features traced in the optical\nby young stars and star forming regions, to smooth and regular\nextended structures traced by giant stars (Cioni et al \\cite{cio00b})\nin the near-IR. The LMC shows a large thick bar confined within an\nouter elliptical structure with hints of spiral arms. The SMC\nresembles a dwarf elliptical galaxy.  The ratio between C-rich and\nO-rich AGB stars, the C\/M ratio, provides also a view of the\nmetallicity distribution within these galaxies (Cioni \\& Habing\n\\cite{cio03a}). This ratio shows a positive gradient within the LMC\nand a clumpy distribution consistent with a flat gradient in the SMC.\n\nThe conversion and interpretation of the C\/M ratio versus [Fe\/H]\nabundance has been investigated recently by Cioni (\\cite{cio09}). The\nsmoothly declining LMC AGB gradient agrees with that of old (several\nGyr) stellar clusters and RR Lyrae stars, but it does differ\nsignificantly from the gradient traced by young (a few Gyr) red giant\nbranch (RGB) stars and stellar clusters that appears rather flat. The\nlatter suggest a flattening of the gradient with time and is perhaps\ninfluenced by the effect of the bar. In the SMC nor AGB stars nor\nother indicators such as: RGB stars, PNe, stellar clusters with a\ndifferent age, show a significant gradient. This can also be due to\nthe bar or to the projection effect of two populations with a\ndifferent mean age, a young one in the disk and an old one in an outer\nspheroid.  Their average metallicity is consistent with that in the\nMagellanic Bridge and of the LMC at $\\sim 4$ kpc from its centre\nsupporting tidal stripping resulting from the dynamical interaction\nbetween the Magellanic Clouds.\n\n\\subsection{Recent IR surveys}\n\nThe Spitzer space telescope, launched in 2003, has surveyed the\nMagellanic Clouds in different filters as part of two major projects:\nthe SAGE and S$^3$MC surveys of the LMC and SMC, respectively. These\nsurveys have provided a large and homogeneous database for studying\nAGB stars, their evolution and mass-loss properties.\n\nSAGE (Meixner et al \\cite{mei06}) covered $49$ deg$^2$ and acquired two\nepochs, separated by three months, that allowed for identifying\nvariable stars as well as their location in colour-magnitude and\ncolour-colour diagrams. Vijh et al (\\cite{vij09}) discusses the\ndistribution, variability and dust properties of the SAGE stars:\n$66$\\% of the extreme\/obscured AGB stars are variable, $6.1$\\% of the\nC-rich AGB stars and $2$\\% of the O-rich AGB stars with thin\ncircumstellar shells are also variables in the mid-IR Spitzer filters\n(Fig.~\\ref{fig3}).  The spectral energy distribution (SED) obtained\nfrom the combination between 2MASS and Spitzer band-widths shows that\nthe lack of variability data has a strong influence on the integrated\nflux and on the estimated mass-loss rate. The sensitivity and spatial\nresolution of Spitzer represent a tremendous improvement versus\nprevious data. In particular a spatial resolution of\n$1-2^{\\prime\\prime}$ in the $3.5-8.0$ $\\mu$m range is comparable to\nthe resolution of ground based near-IR surveys, consenting a more\nsecure identification of counterparts. The sensitivity in the mid-IR\nhas surpassed that in the near-IR leaving unmatched many newly\ndiscovered objects.\n\n\\begin{figure}\n\\includegraphics[width=6cm]{cioni_fig_3a.eps}\n\\qquad\n\\includegraphics[width=6cm]{cioni_fig_3b.eps}\n\\caption{(Left) Luminosity distribution of AGB stars from ISO and\n  DENIS (Loup \\cite{lou99}). Squared symbols indicate ISO matches.\n  (Right) Colour-magnitude diagram of AGB stars from Spitzer and 2MASS\n  (Vijh et al \\cite{vij09}). Coloured symbols indicate Spitzer variables.} \n\\label{fig3}\n\\end{figure}\n\nThe IR survey facility telescope (IRSF; 2001-2006) observed a large\narea encompassing the LMC, SMC and the part of the Magellanic Bridge\nclose to the SMC wing (Kato et al \\cite{kat07}). This is the most\nsensitive near-IR survey of the Magellanic Clouds to-date reaching a\n$10\\sigma$ limit at $J=18.8$, $H=17.8$ and $K_s=16.6$. The instrument\nresolution corresponds to $0.45$ $^{\\prime\\prime}$\/pix while the\nobservations were obtained with an average seeing of\n$1.2^{\\prime\\prime}$.  AGB stars were not easily found in the Bridge\nand in general no AGB stars were found down to $K_s=13.5$ at\n$J-K_s=6$.\n\nThe AKARI telescope, launched in 2006, has completed the observation\nof $10$ deg$^2$ in the north-east area of the LMC and is currently\nundertaking an all-sky survey covering the range $1.8-180$ $\\mu$m. A\npreliminary catalogue of point sources has been published by Ita et al\n(\\cite{ita08}). The band-widths are similar to Spitzer, but for the\n$11\\mu$m one, and have been matched with IRSF near-IR magnitudes.\nTheir combination reaches sources $\\sim 0.5$ mag fainter than the\nSpitzer-2MASS combination. The colour-magnitude diagram, [$3$]-[$11$]\n  versus [$11$], shows interesting new features traced by AGB stars: a\n  faint plume of sources just brighter than the RGB tip and bending to\n  red colours indicating O-rich giants with Al oxide dust (Blum et al\n  \\cite{blu06}, Lebzelter et al \\cite{leb06}).\n\n\\subsection{Forthcoming IR surveys}\n\nIn the near-IR domain the VISTA wide-field telescope is currently\nbeing commissioned at the European Southern Observatory (ESO).  The core\nprogramme for the next five years at VISTA includes six public surveys\nof which two are devoted to the observation of stars, the others are\nextragalactic surveys, and one is focused on the Magellanic system.\n\nThe VISTA survey of the Magellanic system (VMC; Cioni et al\n\\cite{cio08}) will provide the missing link to optical surveys, with a\nsimilar sensitivity, as well as counterparts for the mid-IR\nsources. VMC will cover $180$ deg$^2$ distributed across broad areas\nin the LMC and SMC, the entire length of the Magellanic Bridge\nconnecting the two galaxies (for a width of $1.5-4.5$ deg) and a\ncouple of fields in the Magellanic Stream (Fig.~\\ref{fig4}).\n\n\\begin{figure}\n\\includegraphics[width=12cm]{cioni_fig_4.eps}\n\\caption{Distribution of VISTA tiles across the Magellanic\n  System. Underlying small dots indicate the distribution of C stars,\n  clusters and associations while thick dots show the location of\n  observations to be performed with the VST in the optical.}\n\\label{fig4}\n\\end{figure}\n\nObservations will be obtained in three filters, $YJK_s$ providing a\n$10\\sigma$ limit at $Y=21.9$, $J=21.4$ and $K_s=20.3$. They will be\nexecuted in service mode to guarantee homogeneous conditions and data\nquality. The instrument resolution is $0.51$ $^{\\prime\\prime}$\/pix and\nobservations will be obtained with an average seeing of\n$0.6-0.8^{\\prime\\prime}$ depending on crowding.  VMC is also a\nmulti-epoch survey because it will reach its nominal sensitivity by\ncombining $12$ independent epochs in $K_s$, and $3$ in $Y$ and $J$,\nrespectively. The completion of the survey requires $1840$ hours.\n\nThe main science goals of the VMC surveys are to derive the spatially\nresolved star formation history (SFH) and to measure the\nthree-dimensional (3D) structure of the Magellanic system. The VMC\ndata will also be used to find stellar sub-structures (clusters and\nstreams), emission line objects (like PNe), to derive distances and\nmeasure proper motions, to study star formation, to re-construct the\nsystem using dynamical models and eventually find extra-galactic\nobjects (star-burst galaxies and dusty active galactic nuclei at high\nredshift), as well as form many other scientific applications.\n\nThe VMC survey will be complementary to an on-going optical survey of\nthe outer regions of the Magellanic Clouds ($8-20$ kpc from the galaxy\ncentres). This area will also be surveyed by VISTA as part of the\nVISTA Hemisphere Survey (VHS) but to a shallower depth than the VMC\ndepth, and to the space astrometry mission GAIA that will measure the\nmetallicity and motion of evolved giant stars of the Magellanic\nSystem.\n\nSimulations of the stellar population that VMC will detect show that\nstars just below the oldest main-sequence turn-off will be well\ndetected. The increased in the parameter space, e.g.~near-IR\nphotometry, at this depth will allow for determining the metallicity\nand age distribution with improved accuracy (Kerber et al\n\\cite{ker09}). The 3D structure will be measured using different\ndistance indicators and in particular the period-luminosity relation\nfor short period variable stars, RR Lyrae stars and Cepheids. VMC will\nprovide the near-IR magnitude and the period will be obtained from\nlarge optical catalogues in the literature (e.g. OGLE-III and EROS-II)\nor from observations at the VLT Survey Telescope (VST), for the Bridge\nand SMC parts only.\n\nAlthough VMC is a deep survey targeting faint stellar populations it\nwill find counterparts to faint AGB stars and, via an accurate\nanalysis of the SFH of the system, it will relate them to their\nprogenitors.\n\n\\subsection{Future surveys}\n\nThe Magellanic Clouds will be targeted by different new\nmissions. During this meeting the Herschel satellite was successfully\nlaunched. Herschel will probe the mid- to far-IR regime for AGB stars\nin the Magellanic Clouds.\n\nThe next step in ground-based IR astronomy may be taken really in\nAntarctica with the development of a telescope like PILOT (see other\ncontributions in this proceeding). This will be a 2m class telescope\nthat will explore the dark side of the $K$-band and possibly extend to\nthe $L$-band, reaching an unprecedented depth, and will also be ideal\nto monitor AGB stars in the Magellanic Clouds.\n\n\\section{Conclusions}\n\nIR imaging of AGB stars in the Magellanic Clouds were aimed at finding\nthe most luminous and dusty, thus obscured, sources. Different\ncriteria were developed to distinguish and classify AGB stars into O-\nand C-rich. Multi-wavelength and multi-epoch (optical only)\nobservations have allowed to quantify mass-loss rates and the coupling\nwith AGB pulsation and evolution. \n\nMajor progress in the study of AGB stars has occurred not only on the\nobservational side but also on the theoretical side with the\ndevelopment of models that are able to interpret the location of AGB\nstars in the IR (Marigo et al \\cite{mar08}). These authors have\nproduced isochrones including molecular opacities in O- and C-rich AGB\nstars, the hot bottom burning process, the effect of the pulsation\nmode (first overtone or fundamental mode) to the AGB lifetime and the\nmass-loss with respect to the different surface chemistry. By\nexploring the range of ages typical for AGB stars, these models,\nreproduce well the distribution of the AGB population in the\nMagellanic Clouds in both near- and mid-IR diagrams.\n\nThe most luminous AGB stars experience the strongest mass-loss\nrate. This result has been greatly re-affirmed with the investigation\nby Fraser (\\cite{fra08}) that involves the combination between Spitzer\nand 2MASS data with optical monitoring data from the MACHO\nproject. The bolometric magnitude and the mass-loss rate were obtained\nfrom the SED of individual AGB stars and the period extracted from\ntheir optical light-curve. The major uncertainty still present in the\nperiod-luminosity relations is directly reflected in the uncertainty\nattributed to the bolometric magnitude. To obtain average bolometric\nmagnitudes it is necessary to monitor the SED across a few years. The\nreduced scatter around the period-luminosity relations will also\ncontribute to refining AGB stars as a powerful tool to measure\ndistances in the Universe.  \n\nThere is at present a paucity of IR monitoring ($2$ epochs by Spitzer,\n$12$ epochs by VMC) and a lack of foreseen projects to improve this\ncondition.  Mean bolometric luminosities are the next step in the\ninvestigation of the mass-loss mechanism and in measuring distances\nfrom the period luminosity relation.  The Magellanic Clouds are the\nclosest examples of interacting galaxies and represent an ideal\nlaboratory for stellar evolution. The design of PILOT, a telescope for\nthe Antarctica site, is well suited to advance the study of AGB stars\nin these neighbouring galaxies by providing better sensitivity,\nspatial accuracy and wide-field monitoring.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{a}\n\nGenerally covariant theories in a two-dimensional space-time collect the\nadvantages of both being much simpler than the corresponding\ntheories in 3+1 and 2+1 dimensions, and of having a sufficiently\nrich structure which can shed light on the issues that appear in\nquantizing higher dimensional theories.\n\nSeveral years ago Jackiw and Teitelboim \\cite{[1],[2]} proposed\nthe equation\n\n\\begin{equation} R + \\frac\\Lambda2 = 0\\end{equation}\nas the natural analogue of the vacuum Einstein equations with a\ncosmological term. This equation can be obtained form a local\nvariational principle if a scalar field, playing the role of a\nlagrangian multiplier, is incorporated in the theory. The above\nequation can also be derived from the induced 2d gravity\n\\cite{[3]}:\n\\begin{eqnarray}\nS = \\frac{c}{96\\pi}\\int\\,\\sqrt{-g} (R{\\hbox{$\\sqcup\\!\\!\\!\\!\\sqcap$}}^{-1}R + \\Lambda) \\; .\n\\label{1}\n\\end{eqnarray}\nThis action is non-local but it is preferable to convert it into\na local one by introducing an auxiliary scalar field $\\Phi$. The\naction can be written as\n\n\n\\begin{eqnarray}S=\\frac{1}{2}\\int\\sqrt{-g}(g^{\\mu\\nu}\\partial_\\mu\\Phi\\partial_\\nu\\Phi +\n2R\\Phi + \\Lambda)\\;.\\label{action1}\\end{eqnarray}\n\nThe aim of this paper is to carry out a canonical analysis of the\ninduced 2d-gravity theory (\\ref{1}) in three different\nways (we shall restrict ourselves to the case of a compact\nspatial section). The first one is presented in section 3 and it\nwas  spelled out in \\cite{[4]}. It is based on the\ncovariant formulation of the canonical formalism \\cite{[5]}.\nThe reduced phase space of the theory turns\nout to be a two-dimensional cotangent bundle, and the corresponding\n(geometric) quantization permits to determine the Hilbert space.\nIn section 3 we introduce the ADM formulation of the theory. By\ngauge fixing and imposing the supermomentum\nconstraint we can reduce the theory to a finite-dimensional\nsystem. At this point one can choose different ways to quantize\nthe theory. One way is to look for the reduced hamiltonian (this\nrequires a complete gauge fixing) and then to impose the\ncorresponding Sch\\\"odinger equation. This is our second\napproach and it is developed in section 4. The third approach\n(section 5) is based on the (reduced) Wheeler-DeWitt equation.\nAlong the paper we set up the equivalence of these approaches\nestablishing an isomorphism between the corresponding Hilbert\nspaces.\n\n\n\n\n\\section{Covariant phase-space quantization}\n\\label{b}\n\n\nThe covariant definition of the reduced phase-space\n\\cite{[5]}\nhas been very useful to determine the phase space of\na variety of field theories \\cite{[6],[7],[8]}. In this\napproach, the reduced phase space is defined as the set\nof all solutions of the classical theory, modulo gauge\ntransformations (see below). The symplectic form is defined as\nfollows.\n\nLet us consider a field theory with fields $\\Psi^\\alpha$ and Lagrangian\n${\\cal L}$. If we vary the fields in the Lagrangian we get\n\\begin{eqnarray}\\delta{\\cal L} = \\partial_\\mu \\jmath^\\mu + (E-\nL)_\\alpha\\delta\\Psi^\\alpha\\;.\\label{2}\\end{eqnarray}\n\nIf we regard now $\\delta$ as an exterior derivative operator in\nthe space of classical solutions $\\Psi^\\alpha(x)$, we can, in a natural\nway,  pullback (\\ref{2}) to the space of all solutions of the\nequations of motion, (E-L)$=0$, and consider $\\jmath^\\mu$ as a\nvector-valued one-form on this space.\n\nSince $\\delta\\jmath^\\mu$ is a conserved current,\n$\\partial_\\mu\\delta\\jmath^\\mu = 0$, it is natural to define the\n(pre-)symplectic form $\\omega$ as the corresponding conserved\ncharge,\n\n\\begin{equation}\\omega = -\\int_\\Sigma\\,\\delta\\jmath^\\mu\\hbox{d}\\,\n\\sigma_\\mu\\label{3}\\end{equation}\n($\\Sigma$ is any spatial hypersurface of the space-time).\nWith this definition, we prevent $\\omega$ from depending on\n$\\Sigma$ or on the time co-ordinate. Because $\\delta\\jmath^\\mu$ is\nexact, so is $\\omega$ and thus closed.\n\nThe only property we can not assure for $\\omega$ is nondegenerateness\nsince $\\omega$ as defined above can have a non-trivial kernel. We\ndefine now the gauge transformations as the ones generated by the\nkernel of $\\omega$. If now, in the space of all\nsolutions, we take modulus by the gauge transformations we get a\nsymplectic space which is called the reduced (or physical)\nphase space.\n\nLet us apply the program above to the induced 2d-gravity.\nThe equations of motion obtained from (\\ref{1}) imply the\nvanishing of the stress-tensor $T_{\\mu\\nu}$ that is given by\n\n\\begin{equation} \\begin{array}{lcl}\nT_{\\mu\\nu}& =  -&\\nabla_\\mu\\Phi\\nabla_\\nu\\Phi+\n2\\nabla_\\mu\\nabla_\\nu\\Phi + \\frac{1}{ 2}g_{\\mu\\nu}\\nabla^\\alpha\n\\Phi\\nabla_\\alpha\\Phi   \\\\\n& &-g_{\\mu\\nu}(2R +\n\\frac{1}{2}\\Lambda)\\>,\\end{array}\\label{tensor1}\\end{equation}\nand the relation of $\\Phi$ with the curvature:\n\\begin{equation} \\hbox{$\\sqcup\\!\\!\\!\\!\\sqcap$} \\Phi = R\\>.\\label{motionPhi1}\\end{equation}\n\nIf we use now the gauge invariance of the metric under diffeomorphisms to\nbring it to a conformally flat form,\n\n\\begin{equation}\\hbox{d}\\,\\hbox{s}^2 = -2\\hbox{e}^\\rho\\hbox{d}\\, x^+\\hbox{d}\\, x^-\\>,\\label{flatmetric}\\end{equation}\n($x^+= t+x,\\>x^-=t-x$ are the light-cone coordinates)\nthe equations of motion split into the relation of $\\Phi$ with\nthe curvature,\n\\begin{equation}\n\\hbox{$\\sqcup\\!\\!\\!\\!\\sqcap$}\\Phi\\equiv2e^{-\\rho}\\partial_+\\partial_-\\Phi\n=R\\equiv-2\\hbox{e}^{-\\rho}\\partial_+\\partial_- \\rho\\>,\\label{motionPhi2}\\end{equation}\nthe Liouville equation,\n\\be0=T_{+-} = 2\\partial_+\\partial_- \\rho -\n\\frac{\\Lambda}{2}\\hbox{e}^{\\rho}\\>,\n\\label{Liouville}\\end{equation}\nand the constraints\n\\begin{eqnarray}  0=T_{++} &=& -(\\partial_+\\Phi)^2 + 2\\partial_+^2\\Phi -\n2\\partial_+\\rho\\partial_+\\Phi = 0 \\label{(4.21a)}\\;,   \\\\\n0=T_{--} &=& -(\\partial_-\\Phi)^2 + 2\\partial_-^2\\Phi -\n2\\partial_-\\rho\\partial_-\\Phi = 0\\;. \\label{constraints}\\end{eqnarray}\n\nThe general solution for the metric field $g_{\\mu\\nu}$ and the\ndilaton field $\\Phi$ can be easily found \\cite{[4]}, and\ncan be written as follows:\n\\begin{eqnarray}\n\\hbox{d}\\,\\hbox{s}^2 = -2\\frac{\\partial_+A\\,\\partial_-B}{ \\left(1 - {\\Lambda\\over8}\nAB\\right)^2}\\hbox{d}\\, x^+\\hbox{d}\\, x^-,\\label{metric}\\end{eqnarray}\n\n\\begin{eqnarray}\\Phi = \\ln \\lambda\\frac{(1- {\\Lambda\\over8} AB)^2}{ \\left((d-a)A +\nb\\right)^2{({\\Lambda\\over8} B)}^2}\\;,\\label{Phi1}\\end{eqnarray}\nand\n\n\n\\begin{eqnarray}\\Phi = \\ln \\lambda \\frac{(1- {\\Lambda\\over8} AB)^2}{ \\left(-b{\\Lambda\\over8} B +(d-\na)\\right)^2}\\>,\\label{Phi2}\\end{eqnarray}\nwhere  $a, b, d$ are such that\n$M=\\left(\\begin{array}{cc}a&b   \\\\ 0&d\\end{array}\n\\right)$ belong to the affine subgroup of $PSL(2,R)$,\ni.e., $a=d^{-1}$, and $A=A(x^+), B=B(x^-)$\nverify the monodromy transformations properties (we choose the length\nof the circle equal to unity)\n\\begin{eqnarray}\n A(y + 1) & = &\\frac{a A(y)\n+ b}{d}\\> \\equiv M\\left(\nA(y)\\right)\\; ,\\label{(3.15a)}   \\\\\n-{\\Lambda\\over8} B(y - 1) & = &\\frac{-d{\\Lambda\\over8} B(y)\n}{ b{\\Lambda\\over8} B(y) + a} \\equiv M^{-1T}\\left(-{\\Lambda\\over8} B(y)\n\\right)\\; .\\label{7}\n\\end{eqnarray}\n\nIf we choose the spatial hypersurface as the one defined by $t=t_0$\nthe symplectic form can be written as:\n\n\\begin{equation}   \\omega = \\int_x^{x+1}\n\\left[-\\delta\\Phi\\delta(\\partial_++\\partial_-)(\\Phi +\\rho) +\n\\delta(\\partial_+ +\\partial_-)\\Phi\\delta\\rho\\right]\\>.\n\\label{omega1}\\end{equation}\n\nThe projection onto the space of classical solutions takes\nan special form\n\n\\begin{equation} \\omega = \\frac{1}{2}\\int_x^{x+1}\\left(\\partial_+\n- \\partial_-\\right)W\\>, \\label{9}\\end{equation}\nwhere $W$ is given by\n\n\\begin{eqnarray}\nW& =  &\\frac{1}{2}\\{\n\\delta\\ln \\frac{\\partial_+A}{\\partial_-B}\\left(\\frac{\\frac{\\Lambda}{8} B\n }{(d-a) A +b}\\right)^2\\,\n\\delta\\ln \\lambda\\frac{(1- {\\Lambda\\over8} AB)^2}{ ((d-a) A +b)^2\n(\\frac{\\Lambda}{8} B)^2}\\nonumber \\\\\n& &+\\delta\\ln(1- {\\Lambda\\over8} AB)^2\\,\\delta\\ln\\frac{A}{ B}\\left(\\frac{\\frac{\\Lambda}{8} B\n }{(d-a) A +b}\\right)^2 \\label{W1}\\\\& &+\n\\delta\\ln\\frac{((d-a) A + b)^2}{(\\frac{\\Lambda}{8})^2}\\,\n\\delta\\ln \\frac{(d-a)^2}{(\\frac{\\Lambda}{8}\nB )^2}\\}\\>\\; , \\nonumber\\end{eqnarray}\nfor the solution (\\ref{Phi1}) for $\\Phi$, and by\n\\begin{eqnarray}\nW& = &\\frac{1}{2}\\{\n\\delta\\ln \\frac{\\partial_+A}{\\partial_-B}\\left( b\\frac{\\Lambda}{8} B\n-(d-a)\\right)^2\\,\n\\delta\\ln \\frac{(1- {\\Lambda\\over8} AB)^2}{\n( b\\frac{\\Lambda}{8} B - (d-a))^2}\\nonumber\\\\\n& &+\\delta\\ln(1- {\\Lambda\\over8} AB)^2\\,\\delta\\ln\\frac{A}{ B}\\left( b\\frac{\\Lambda}{8} B\n-(d-a)\\right)^2\\label{W2}\\\\& &+\n\\delta\\ln (b\\frac{\\Lambda}{8})^2\\,\n\\delta\\ln ( b\\frac{\\Lambda}{8}\nB-(d-a))^2\\}\\>\\; ,\\nonumber\\end{eqnarray}\nfor the solution (\\ref{Phi2}) for $\\Phi$.\n\nIn any case, since $\\omega$ does not depend on the coordinate $x$\nin (\\ref{9}), it cannot depend on either of the functions $A$ or\n$B$. So $\\omega$ will depend only on the classes of monodromy\ntransformations to which the functions $A$ and $B$ belong, and on the\nparameter $\\lambda$ in (\\ref{Phi1},\\ref{Phi2}). Moreover, if we\ntransform the functions $A$ and $B$ as\n\n\\begin{eqnarray}  &A\\longrightarrow h(A) \\; ,\\label{12}   \\\\\n&-{\\Lambda\\over8} B\\longrightarrow h^{-1T}(-{\\Lambda\\over8} B)\\>\\; ,\\label{13}\\end{eqnarray}\nwhere $h$ is a constant affine matrix acting as a M\\\"obius transformation,\nwe get the same solution of the equations of motion. Under the\ntransformation (\\ref{12},\\ref{13}) the monodromy parameters\ntransform as\n\n\\begin{eqnarray}\nM\\longrightarrow hMh^{-1}\\; .\n\\label{14}\n\\end{eqnarray}\n\nThus, two solutions which differs on a transformation of the type\n(\\ref{14}) are the same point of the reduced phase space. The\nonly invariant quantity under the transformation (\\ref{14}), and\nhence the only allowed monodromy dependence in $\\omega$, is\nthe parameter $a$.\n\nA direct computation from (\\ref{W1}, \\ref{W2}) leads to:\n\\begin{eqnarray}\\omega=2\\frac{\\delta\\lambda}{\\lambda}\\,\\frac{\\delta a}{\na}\\>,\\label{15}\\end{eqnarray}\nfor the solutions in (\\ref{W1}), and\n\n\\begin{eqnarray}\\omega=-2\\frac{\\delta\\lambda}{\\lambda}\\,\\frac{\\delta a}{\na}\\>.\\label{16}\\end{eqnarray}\nfor the solutions in (\\ref{W2}).\n\n{}From (\\ref{15},\\ref{16}) and previous considerations, we are tempted\nto assume that the reduced phase space is of the form\n\n\\begin{equation} T^*\\left(G\/\\hbox{ad} G\\right)\\cup T^*\\left(G\/\\hbox{ad} G\\right)\n\\>,\\label{17}\\end{equation}\nwhere $G$ is the affine subgroup of $PSL(2,{\\it R})$.\nThis would lead us to a Hilbert\nspace of the form\n\n\\begin{equation}{\\cal H} ={\\cal H}^{(+)}\\oplus {\\cal H}^{(-)},\\label{18}\\end{equation}\nwhere\n\n\\begin{equation} {\\cal H}^{(+)}= {\\cal H}^{(-)} = L^2({\\it R}^+)\\oplus{\\it\nC}^2\\>.\\label{19}\\end{equation}\n\nThe result (\\ref{18}), in which the Hilbert space has a\ncontinuum and a discrete sector is in accordance with some results\nobtained by BRST methods \\cite{[9]}. However, there are\nalso some evidence that the discrete sector cannot be endowed with\na well defined inner product. This result is achieved here by\nshowing that, in fact, the discrete sector actually does not appear.\nThis is a consequence of the additional symmetry $A\\rightarrow -\nA,\\>\\>B\\rightarrow-B,\\>\\>a\\rightarrow a,\\>\\>b\\rightarrow-b$ that\nidentify the otherwise distinct parabolic ($a=1$) solutions.\n\nLet us write the classical solutions (\\ref{metric},\\ref{Phi1},\\ref{Phi2})\nin a more explicit form. To this end we should completely fix the\nspace-time coordinates by imposing and additional ``gauge-type\"\ncondition. From (\\ref{motionPhi2}) we observe that\n\\begin{equation}\n\\partial_+\\partial_- (\\Phi+\\rho) = 0\\; .\n\\end{equation}\nTherefore we can (and we will) choose a spatially homogeneous conformal gauge\nby imposing\n\n\\begin{equation} \\Phi + \\rho = \\epsilon + 2pt\\>,\\end{equation}\nwhere $\\epsilon$ and $p$ are constant parameters.\n\n\nIf $a\\neq1$, i.e., if the\nmonodromy class is hyperbolic, the solutions\ntake the form:\n\n\\begin{eqnarray}\n\\hbox{d}\\, \\hbox{s}^2 = - 2\\frac{8}{|\\Lambda|}\n\\frac{p^2\\hbox{e}^{2{{pt}}}}{\\left(1-\\hbox{sgn}\\Lambda\\hbox{e}^{2{{pt}}}\\right)^2}\\hbox{d}\\,\nx^+\\hbox{d}\\, x^-\\label{101}\\; ,\\\\\n\\Phi=\\ln\\lambda\\frac{\\left(1-\\hbox{sgn}\\Lambda\\hbox{e}^{{{2pt}}}\\right)^2}\n{4(\\sinh p\/2)^2\\hbox{e}^{4{{pt}}}}\\; ,\\label{102}\\\\\n\\Phi=\\ln\\lambda\\frac{\\left(1-\\hbox{sgn}\\Lambda\\hbox{e}^{2{{pt}}}\\right)^2}\n{4(\\sinh\\frac{p}{2})^2}\\; ,\\label{103}\\end{eqnarray}\nwhere $\\hbox{e}^p = a = d^{-1}$.  On the other hand if $a=1=d$, the\nunique parabolic solution takes the form\n\n\\begin{eqnarray}\n\\hbox{d}\\, \\hbox{s}^2& = -&\\frac{2}{\\Lambda}\\frac{1}{t^2}\\hbox{d}\\, x^+\\hbox{d}\\, x^-\n\\label{104}\\; ,\\\\\n\\Phi& = & \\ln 4\\lambda {t^2}\\; .\\label{105}\\end{eqnarray}\n\nWe can easily see that (\\ref{102}) and (\\ref{103}) transform\ninto each other when we make the replacement $a\\leftrightarrow a^{-1}=d$,\nunder which also (\\ref{15}) and (\\ref{16}) transform into each\nother.\nMoreover, when $\\Lambda>0$, (\\ref{104}) has the right signature\nand can be obtained from (\\ref{101}-\\ref{103}) in the limit\n$p\\rightarrow0$ ($a\\rightarrow1$). So we can conclude\nthat the phase space for $\\Lambda>0$ is just\n\\begin{equation} T^*({\\it R})\\cup T^*({\\it R}) \\; ,\\label{108}\\end{equation}\nwith the symplectic form\n\\begin{equation} \\omega = 2\\delta(\\ln \\lambda)\\>\\delta\np.\\label{109}\\end{equation}\nThe two sectors in (\\ref{108}) correspond to whether the scalar\nfield is expanding or contracting.\n\nThe cotangent bundle structure of the phase space makes it easy\nto determine the Hilbert space of the quantum theory: it will\nbe given by the square integrable functions on the configuration\nspace. Hence in this case we shall have\n\n\\begin{equation}\n{\\cal H} = L^2({\\it R},\\hbox{d}\\, p)\\oplus L^2({\\it R},\\hbox{d}\\, p).\\label{111}\\end{equation}\n\nFor $\\Lambda<0$ neither (\\ref{104}) is positive definite nor\n(\\ref{105}) can be obtained from (\\ref{102},\\ref{103}) as a\nlimiting case. So that the phase space is given by\n\n\\begin{equation}\nT^{*}({\\it R}^+)\\>\\cup \\>T^{*}({\\it R}^+)\\; ,\\label{106}\\end{equation}\nwith the symplectic form\n\\begin{equation}\\omega=2\\delta\\ln\\lambda\\delta p\\; .\n\\label{107}\n\\end{equation}\nThe Hilbert space should be now of the form\n\n\\begin{equation}\n{\\cal H} =\nL^2\\left({\\it R}^+,{{\\hbox{d}\\, p}\\over p} \\right)\\oplus\nL^2\\left({\\it R}^+,{{\\hbox{d}\\, p}\\over p} \\right) \\; .\n\\label{110}\\end{equation}\n\nAlthough it is difficult to figure out how the Hilbert spaces\n(\\ref{110}) can be actually realized, we shall see in the next\nsections that this prediction for the Hilbert space is consistent\nwith other quantization approaches.\n\n\\section{ADM formulation.}\n\\label{c}\n\nIn section 2 we saw explicitly that the classical solutions of\nthe theory are spatially homogeneous. As has been shown in\n\\cite{[11]} for a wide class of 2d dilaton gravity models, this\nis so because the theory (\\ref{action1}) has a Killing vector\nthe flow of which\ndetermines a natural coordinate system on the cylinder where\nthe metric and the scalar fields takes an homogeneous form. The\nexistence of the Killing vector requires the metric equations of\nmotion be satisfied. At this point it is important to remark\nthat one can indeed reduce the theory to a finite number of\ndegrees of freedom by imposing the supermomentum constraint only.\n\nTo this end let us now present the basic ingredients of the ADM formulation\nof the induced 2d-gravity (see also \\cite{[10]}). First, we\nintroduce the standard parametrization of the two-dimensional\nmetric\n\\begin{equation}\ng_{\\mu\\nu}=\\left(\\begin{array}{cc}-N^2 +\nN_1N^1&N_1\\{\\cal N}_1&a^2\\end{array}\\right)\\>\\; ,\n\\label{30.1}\n\\end{equation}\nwhere $N$ and $N^1$ are the lapse and shift functions\nrespectively. To derive the canonical form of the action we can\nuse the two-dimensional identity\n\n\\begin{equation} \\sqrt{-g}R = -2\\partial_t (aK) +\n2\\partial_x(a(KN^1-a^{-2}N^1))\\; ,\\end{equation}\nwhere $K$ is the extrinsic curvature scalar\n\\begin{equation} K = \\frac{1}{a^2N}(N_{1|1} -a\\dot a)\\; .\\label{30.3}\\end{equation}\nRemoving total time derivatives\nwe arrive at\n\n\\begin{equation} S = \\int \\hbox{d}\\,^2x(\\pi_a\\dot a + \\pi_\\Phi\\dot\\Phi-N{\\cal C}-N^1{\\cal\nC}_1)\\>,\\label{30.4}\\end{equation}\nwhere the canonical momenta are\n\\begin{equation} \\pi_a =\\frac4N(\\Phi'N^1-\\dot \\Phi)\\; ,\\label{30.5}\\end{equation}\n\\begin{equation} \\pi_\\Phi = \\frac{2a}N(\\Phi'N^1-\\dot \\Phi)+\\frac4N\\left((aN^1)'-\n\\dot a\\right)\\>,\\label{30.6}\\end{equation}\nand the supermomentum and hamiltonian constraints are given by\n\n\\begin{equation} {\\cal C}_1 = \\Phi'\\pi_\\Phi-\\pi_a'a \\; ,\\label{30.7}\\end{equation}\n\\begin{equation} {\\cal C} = \\frac1{16}a\\pi_a^2-\\frac14\\pi_a\\pi_\\Phi-a\\Lambda -\n\\frac1a\\Phi'^2+4(a\\Phi')' \\; .\\label{30.8}\\end{equation}\n\n\nMaking use of the spatial diffeomorphism invariance of the theory\nwe can fix the space coordinate and assume that\n\n\\begin{equation} a = a(t) \\; .\\label{30.9}\\end{equation}\nIn addition to this, and due to the time reparametrization\ninvariance, we can also make a choice of time. All the above\nconsiderations suggest the following class of spatially\nhomogeneous definitions of the internal time variable\n\\begin{equation} {\\cal T}(\\Phi, a) = \\chi(t) \\; ,\\label{30.10}\\end{equation}\nwhere $\\chi$ is a generic function. This implies\nthat\n\\begin{equation} \\Phi = \\Phi(t) \\; .\\label{30.11}\\end{equation}\nNow, if we impose the supermomentum constraint we easily obtain\n\\begin{equation} \\pi_a = \\pi_a(t)\\; ,\\label{30.12}\\end{equation}\nand also $N=N(t)$.\n\nThe momentum $\\pi_\\Phi$ is still a function of\nboth $t$ and $x$. However we can integrate the action in\n(\\ref{30.4}) with respect to the compact coordinate and the\nresulting expression is\n\\begin{equation} S = \\int\\hbox{d}\\, t\\left(\\pi_a\\dot a + \\dot \\Phi\\int\\hbox{d}\\, x\\pi_\\Phi -\nN{\\cal C}\\right)\\; ,\\label{30.13}\\end{equation}\nwhere now\n\\begin{equation} {\\cal C} = \\frac1{16}a\\pi_a^2-\\frac14\\pi_a\\int\\hbox{d}\\, x \\pi_\\Phi - a\\Lambda \\; .\n\\label{30.14}\\end{equation}\n{}From now on $\\pi_\\Phi$ stands for the momentum conjugated to\n$\\Phi(t)$, i.e., $\\pi_\\Phi(t)=\\int\\hbox{d}\\, x\\pi_\\Phi(t,x)$. Although\n(\\ref{30.13}) corresponds to a minisuperspace approach to the\ntheory, it must be regarded instead as a reduced form of the\ntheory in an appropriate gauge choice and not as a mere\napproximation to the theory.\n\nFor the sake of completeness we write down the equations of\nmotion and the symplectic form obtained from (\\ref{30.13})\n\\begin{equation}\\begin{array}{ll}\n\\dot\\pi_\\Phi = 0\\>,\n&\\dot\\pi_a=-N\\left[-\\frac1{16}{\\pi_a^2}+\\frac\\Lambda2\\right] \\; ,\\\\\n\\dot\\Phi = N\\pi_a\\>,&\\dot a = N\\left[\\frac18a\\pi_a-\n\\frac14\\pi_\\Phi\\right] \\; ,\\end{array}\\label{30.15}\\end{equation}\n\\begin{equation} \\omega=\\delta\\pi_\\Phi\\delta\\Phi + \\delta\\pi_a\\delta a\\>.\n\\label{omega2}\\end{equation}\n\n\n\n\\section{Reduced phase-space quantization in the conformal\nchoice of time.}\nIn this section we shall develop a genuine hamiltonian\nquantization of the reduced theory (\\ref{30.13}).\nIn this approach the choice of time\nis done before quantize and the constraint ${\\cal C} = 0$ is solved\nclassically (see for instance the review \\cite{[12]}). In this\ncontext, the choice of time is nothing other but a gauge fixing\ncondition. This gauge fixing is required to be\ncomplete in the sense that no further gauge freedom must be left, but\nalso we must not lose information, i.e., actual solutions to the\nequation of motion.\n\n\nLet us choose the conformal gauge\n\n\\begin{equation} N = a\\>,\\label{40.1}\\end{equation}\nthat implies, according to the equations of motion\n(\\ref{30.15}) (see also (\\ref{101})),\n the following implicit\ndefinition of the time variable:\n\\begin{equation} a^2 = 4\\frac{\\pi_\\Phi^2}{|\\Lambda|}\n\\frac{\\hbox{e}^{\\pi_\\Phi t}}{\\left(1 -\\hbox{sgn}\\Lambda\\hbox{e}^{\\pi_\\Phi\nt}\\right)^2}\\>.\\label{40.2}\\end{equation}\nSolving now the constraint ${\\cal C} =0$ for $a\\pi_a$ we find\nthe solutions\n\\begin{equation}\na\\pi_a = 4\\pi_\\Phi\\frac{1}{1+\\hbox{sgn}\\Lambda\\hbox{e}^{\\pi_\\Phi t}}\\>,\n\\label{40.3a}\\end{equation}\nand\n\\begin{equation} a\\pi_a = 4\\pi_\\Phi\\frac{1}{1+\\hbox{sgn}\\Lambda\\hbox{e}^{-\\pi_\\Phi\nt}}\\>,\\label{40.3b}\\end{equation}\nwhich remind us the classical twofold solution for the field $\\Phi$.\n\nOnce the choice of time has been done, the effective Hamiltonian\nassociated with it, i.e., the function that gives the proper {\\it\nclassical} time evolution for the remaining fields, is\n(minus) the conjugate momentum of time.  Substituting\n(\\ref{40.3a},\\ref{40.3b}) into (\\ref{omega2}) we find\n\n\\begin{equation} \\omega = \\delta \\pi_\\Phi\\delta \\Phi\n- 2\\pi_\\Phi\\frac{1}{1+\\hbox{sgn}\\Lambda\\hbox{e}^{\\pm\\pi_\\Phi t}}\n\\delta \\pi_\\Phi\\delta t \\>.\\label{omega3}\\end{equation}\n\nSince this two-form must project down to the (reduced) symplectic\nform of the model, the hamiltonian flow of the vector field in\nthe kernel of (\\ref{omega3}) should provide the remaining\ntrajectories of motion (see for instance \\cite{[13]}). Therefore,\nthe effective Hamiltonian should fit the expression\n\n\\begin{equation} \\omega = \\delta\\pi_\\Phi\\delta\\Phi -\\delta H\\delta\nt\\>.\\label{PCform}\\end{equation}\nSo, we obtain\n\n\\begin{eqnarray}\nH &=&\n\\int^{\\pi_\\Phi}\\hbox{d}\\,\\pifi2\\pi_\\Phi\\frac{1}{1+\\hbox{sgn}\\Lambda\\hbox{e}^{\\pm\\pi_\\Phi t}}\n\\label{Hamiltonians}\\\\\n&=& \\pi_\\Phi^2 -(\\pm)2\\frac{\\pi_\\Phi}{t}\\ln(1+\\hbox{sgn}\\Lambda\\hbox{e}^{\\pm \\pi_\\Phi t})\n-2\\frac{1}{t^2}\\hbox{Polylog}(2,-\\hbox{sgn}\\Lambda\\hbox{e}^{\\pm\\pi_\\Phi\nt})\\>.\\nonumber\n\\end{eqnarray}\nThe Hamiltonians in (\\ref{Hamiltonians}) can be converted\ninto each other by means of the change $\\pi_\\Phi\\leftrightarrow -\\pi_\\Phi$\nor $t\\leftrightarrow-t$. Thus, in this system, reversing the arrow\nof time is equivalent to changing the sign of the momentum $\\pi_\\Phi$.\n\nThe quantum system will be described by the wave functions\n$\\Psi(\\pi_\\Phi,t)$ that obey a time-dependent Schr\\\"odinger\nequation:\n\n\\begin{equation} \\hbox{i}\\hbar\\frac{\\partial}{\\partial t}\\,\\Psi(\\pi_\\Phi,t) =\nH(\\pi_\\Phi,t)\\Psi(\\pi_\\Phi,t)\\>.\\label{162.b}\n\\end{equation}\nSince the Hamiltonian functions at different times commute, the\nSchr\\\"o-\\break\ndinger equation (\\ref{162.b}) can be solved immediately to\ngive\n\n\\begin{equation} \\Psi(\\pi_\\Phi,t) = \\Psi(\\pi_\\Phi)\\hbox{e}^{-\\frac{i}{\\hbar}\\int^t\\hbox{d}\\,\nz\\,H(\\pi_\\Phi,z)}\\>.\\label{163}\\end{equation}\n\nThe scalar  product of two wave function $\\Psi(\\pi_\\Phi,t)$ and\n$\\varphi(\\pi_\\Phi,t)$ will be taken as the natural one:\n\\begin{equation}\n <\\Psi|\\varphi>\n=\\int\\hbox{d}\\, \\pi_\\Phi \\Psi^*(\\pi_\\Phi,t)\\varphi(\\pi_\\Phi,t)\n= \\int\\hbox{d}\\, \\pi_\\Phi \\Psi^*(\\pi_\\Phi)\\varphi(\\pi_\\Phi)\\>.\\label{164}\n\\end{equation}\nThe Hilbert space for $\\Lambda>0$ is hence given by\n\\begin{eqnarray} {\\cal H} &=& {\\cal H}^{(+)}\\oplus{\\cal H}^{(-\n)}\\nonumber\\\\\n&=&L^2({\\it R},\\hbox{d}\\, \\pi_\\Phi)\\oplus L^2({\\it R},\\hbox{d}\\, \\pi_\\Phi)\\>.\\label{165}\\end{eqnarray}\nThe two sectors correspond to the double sign of the effective\nHamiltonian (\\ref{Hamiltonians}) and represent whether the\ntwo-dimensional universe is expanding or contracting. We recover\nthus the result of section 2. Note that the monodromy parameter\n$a=\\hbox{e}^{2p}$ in section 2 must be identified with the constant\nof motion $\\hbox{e}^{\\pi_\\Phi}$.\n\n\nHowever, for $\\Lambda<0$ we must prevent the wave functions to\ntake any non null value in $\\pi_\\Phi = 0$ since at this point the\ngauge fixing condition (\\ref{40.2}) is not well defined. So, we\nmust impose on the wave functions the restriction of vanishing at\n$\\pi_\\Phi =0$,\n\n\\begin{equation} \\Psi(\\pi_\\Phi =0, t) = 0\\>,\\label{166}\\end{equation}\nrestriction that is preserved by the time evolution.\nTherefore, for $\\Lambda<0$, the Hilbert space will be given by\n\n\\begin{eqnarray} {\\cal H}\n&=& {\\cal H}^{(+)}\\oplus{\\cal H}^{(-)}\\nonumber\\\\\n&=& L^2({\\it R}^+,\\hbox{d}\\, \\pi_\\Phi)\\oplus L^2({\\it R}^+,\\hbox{d}\\, \\pi_\\Phi)\\>.\\end{eqnarray}\nwhich can be identified with (\\ref{110}).\n\n\n\\section{Quantization via the Wheeler-DeWitt equation.}\n\nIn this section we shall quantize the reduced theory\n(\\ref{30.13}) without any identification of time prior to\nquantization. This essentially means to impose the operator\nversion of the classical hamiltonian constraint, i.e., the\nWheeler-DeWitt equation. To propose the Wheeler-DeWitt operator\n${\\cal C}$ for ({\\ref{30.14}) we face at once the problem of the\noperator ordering ambiguities and the inequality $a>0$ of the\nscale variable. The second difficulty can be solved by using the\naffine algebra $[\\hat a, \\> \\hat p_a]=\\hbox{i}\\hbar\\hat a\\>\\> (\\hat p_a =\n-\\hbox{i}\\hbar a\\frac\\partial{\\partial a})$, instead of the Heisenberg-\nWeyl algebra, as the basic one to define the quantization\n\\cite{[14]}. The reason is that the operator $\\hat \\pi_a =\n-\\hbox{i}\\hbar\\frac\\partial{\\partial a}$ fails to be self-adjoint on\n$L^2({\\it R}^+,\\hbox{d}\\, a)$, whereas the affine operator $\\hat p_a =\n-\\hbox{i}\\hbar a\\frac\\partial{\\partial a}$ is self-adjoint\nin $L^2({\\it R}^+,\\frac{\\hbox{d}\\, a}a)$.\n\nImposing that the Wheeler-DeWitt operator be self-adjoint  with\nrespect to the measure $\\frac{\\hbox{d}\\, a}a\\hbox{d}\\, \\Phi$ we can write the\nfollowing expression for $\\widehat {\\cal C}$:\n\\begin{eqnarray} \\label{50.1}\n\\widehat{{\\cal C}}&=&\\Bigl[\n {1\\over16} \\widehat{a}^{\\alpha+i\\beta} \\widehat{p}_a \\widehat{a}^{-1-2\\alpha}\n\\widehat{p}_a\n\\widehat{a}^{\\alpha-i\\beta} +  \\nonumber \\\\\n& & \\hphantom{\\Bigl[}\n -{1\\over8} \\left( \\widehat{a}^{\\gamma+i\\sigma} \\widehat{p}_a\n\\widehat{a}^{-\\gamma-i\\sigma-1} + \\widehat{a}^{\\gamma+i\\sigma-1} \\widehat{p}_a\n\\widehat{a}^{-\\gamma-i\\sigma} \\right) \\widehat{\\pi}_\\Phi +\n \\Lambda \\widehat{a} \\Bigr]   \\, ,\n\\end{eqnarray}\nwhere $\\alpha$, $\\beta$, $\\gamma$ and $\\sigma$ are\narbitrary factor-ordering parameters.\n\nWe can separate variables in the Wheeler-DeWitt equation by\nexpanding the wave function $\\Psi$ in $\\widehat{\\pi}_\\Phi$\neigenstates\n\n\\begin{equation}\n\\Psi = \\int dq e^{{iq\\Phi\\over\\hbar}}\n\\Psi_q(a) \\, .\n\\label{50.2}\n\\end{equation}\nInserting (\\ref{50.2}) into the equation $\\widehat{\\cal C}\\Psi=0$, where\n$\\widehat{\\cal C}$ is given by (\\ref{50.1}), we obtain that the\nfunctions $\\Psi_q(a)$ obey the equation\n\n\n\\begin{equation} \\label{50.3}\n\\left(\n{d^2\\over da^2} +\n{1\\over a}(1 - 2\\zeta){d\\over da} +\n{1\\over a^2}(\\zeta^2-\\nu^2) +\n4 {\\Lambda\\over\\hbar^2}\n\\right) \\Psi_q(a) = 0 \\, ,\n\\end{equation}\nwhere\n\n\\begin{eqnarray}\n\\zeta &=& {1\\over2} \\left( 1 + 4{iq\\over\\hbar} + 2i\\gamma \\right)\n        \\, , \\label{tresxiia} \\\\\n\\nu^2 &=& {1\\over4} \\left( 1- 16{q^2\\over\\hbar^2} - 8\\gamma^2 +\n4\\alpha(\\alpha+1) + 16 {q\\over\\hbar}(\\sigma-\\gamma) \\right)\n        \\, . \\label{tresxiib}\n\\end{eqnarray}\n\n        The solutions of the above equation are\n\\begin{equation}\n\\Psi_q(a)\n= a^\\zeta {\\cal Z}_\\nu\n\\left(\\frac{2|\\Lambda|^{\\frac12}}\\hbar a\\right) \\, ,\\label{50.6}\n\\end{equation}\nwhere\n${\\cal Z}_\\nu$ are ordinary (modified) Bessel functions for\n$\\Lambda>0$ ($\\Lambda<0$) with order $\\nu$.\n\n\n        In constructing the Wheeler-DeWitt operator\nwe required hermiticity\nwith respect to the standard inner product\n\\begin{equation} \\label{50.7}\n<\\Psi_1|\\Psi_2> = \\int {da\\over a} d\\Phi \\Psi_1^* \\Psi_2 \\, .\n\\end{equation}\nIn canonical quantum gravity it is therefore natural to propose\n(\\ref{50.7}) as the scalar product for the solutions of the\nWheeler-DeWitt equation. This proposal for the scalar product is\nproblematic in the sense that we are integrating over one of the\nconfiguration variables that could have been defined as the\n``internal\" time variable \\cite{[12]}. However we shall insist on\nusing it but having in mind that (\\ref{50.7}) could be divergent\nand require, therefore, some sort of regularization.\n\nLet us analyse now the situation for the case of negative\ncosmological constant. We can expand the general solution to the\nWheeler-DeWitt equation in terms of the modified Bessel and\nHankel functions ${\\cal I}_\\nu$ and ${\\cal K}_\\nu$. However, due to\nthe exponential behaviour of the functions ${\\cal I}$ for large\n$x$ $(x\\equiv 2\\frac{|\\Lambda|^{\\frac12}}\\hbar a)$, they do not\nlead to normalizable wavefunctions and, therefore, should then be\nexcluded from the physical Hilbert space. The physical wave\nfunctions should be of the form\n\n\\begin{equation}\n\\Psi = \\int dq  e^{{i q \\Phi \\over \\hbar}} a^{\\zeta}\n        C(q) {\\cal K}_\\nu (x) \\, ,\n\\label{50.8}\n\\end{equation}\n\nTo determine the Hilbert space we should find out the range of\nvariation of the order $\\nu$. Due to the small $x$ behaviour or the\nmodified Hankel functions, the wave functions will be normalizable\nwhen\n\\begin{equation}\n\\nu^2 < {1\\over4} \\, .\n\\label{50.9}\n\\end{equation}\nTo obtain the maximum range of variation for $\\nu^2$ as $q$\nvaries over the real line we should choose the factor ordering\nparameters in such a way that (\\ref{tresxiib}) turns out to be of\nthe form\n\n\\begin{equation}\n\\nu^2 = {1\\over4} ( 1 - 16 {(q-q_0)^2\\over\\hbar^2} ) \\, .\n\\label{50.10}\n\\end{equation}\nThe constant shift $q_0$ of $q$ in (\\ref{50.10}) can be chosen\naccording to the classical theory. On the covariant phase space\nthe constant of motion $\\pi_\\Phi$ is proportional to the monodromy\nparameter $\\ln a$. Owing to the absence of classical solutions\nfor $a=1$ the constant $q_0$ should vanish to exclude the quantum\nsolution $\\widehat \\pi_\\Phi = 0$. Therefore we are finally led to the\nexpression\n\\begin{equation}\n\\nu^2 = {1\\over4} ( 1 - 16 {q^2\\over\\hbar^2} ) \\, ,\n\\label{50.11}\n\\end{equation}\nwhich corresponds to $\\alpha=\\beta=\\gamma=\\sigma=0$ in (\\ref{tresxiib}).\n\nNow we want to determine the Hilbert space when the cosmological\nconstant is positive. According to (\\ref{50.6}) the general\nsolution to the Wheeler-DeWitt equation can be expanded as\n($\\hbox{Re}\\nu\\geq0\\>,\\>\\hbox{Im}\\nu\\geq0)$)\n\n\\begin{equation}\n\\Psi = \\int dq a^{{1\\over2}+2i{q\\over\\hbar}}\n        e^{{iq\\Phi\\over\\hbar}}\n        \\left( A(q) {\\cal J}_\\nu (x) + B(q)\n{\\cal N}_\\nu (x) \\right) \\, ,\\label{50.12}\n\\end{equation}\nwhere $A(q)$ and $B(q)$ are arbitrary complex functions and $\\nu$ is\ngiven by (\\ref{50.11}). The norm of the wave function (\\ref{50.12}) with\nrespect to (\\ref{50.7}) is given by ($k=2\\frac{|\\Lambda|^{\\frac12}}\\hbar)$\n\\begin{eqnarray} \\label{50.13}\n<\\Psi|\\Psi> &=& {\\hbar \\over k}\n        \\int_{-\\infty}^{+\\infty} dq \\int_0^{+\\infty} dx\n  \\Bigl(\n        |A(q)|^2 |{\\cal J}_\\nu (x)|^2 +\n        |B(q)|^2 |{\\cal N}_\\nu (x)|^2 +\n \\\\\n &&\n\\hphantom{ \\pi\\hbar \\int_{-\\infty}^{+\\infty} dq }\nA^*(q) B(q) {\\cal J}_\\nu^*(x) {\\cal N}_\\nu(x) +\n        A(q) B^*(q) {\\cal J}_\\nu(x){\\cal N}_\\nu^*(x) \\Bigr)\n\\nonumber\\, .\n\\end{eqnarray}\n\nDue to the asymptotic behaviour of the Bessel functions for large $x$\nthe above integral are divergent. We can define a regularized scalar product\nby substituting the integration measure $\\hbox{d}\\, x$ in (\\ref{50.13}) by\n$\\hbox{d}\\, x\/x^\\epsilon$ ($\\epsilon{\\mathop{>}\\limits_{\\sim}}0)$.\nIn the limit $\\epsilon\\rightarrow0$ the new inner product turns\nout to be\n\\begin{eqnarray}\n<\\Psi|\\Psi> &&{\\mathop{\\sim}\\limits_{\\scriptscriptstyle\\epsilon\\to0}}\n        {\\hbar \\over 2\\pi k} {\\Gamma(\\epsilon) \\over 2^\\epsilon}\n        \\int_{-\\infty}^{+\\infty} dq \\Bigl\\{\n        \\bigl[ \\cos(\\pi\\nu) (|A|^2+|B|^2)\\nonumber\\\\\n         &&\\qquad + \\sin(\\pi\\nu) (B^* A - A^* B)\n        \\bigr] \\Theta(-\\nu^2)\\label{50.14}\\\\\n         && \\qquad +\n       \\bigl[ |A|^2 + |B|^2 \\bigr] \\Theta(\\nu^2) \\Bigr\\} \\nonumber,\n\\end{eqnarray}\nwhere $\\Theta$ is the step function. We can eliminate the overall\ndivergent factor $\\Gamma(\\epsilon)\/2^\\epsilon$ to define the\nphysical scalar product. The elementary normalizable solutions\nwith respect to the regularized scalar product can be classified\nimmediately. They are\n${\\cal J}_\\nu$ for $\\nu\\in[0,\\frac12]$ or $\\hbox{Re}\\>\\nu=0$,\nand\n${\\cal N}_\\nu$ for $\\nu\\in[0,\\frac12[$ or $\\hbox{Re}\\>\\nu=0$.\nNote that the unique normalizable solution for $\\nu=\\frac12$ is\n${\\cal J}_\\nu$.\n\nNext we would like to relate the quantization obtained via the\nWheeler-DeWitt equation with the approach developed in previous\nsections. The main point is to see how the Hilbert space\n$L^2({\\it R}^+)\\oplus L^2({\\it R}^+)$ (or $L^2({\\it R})\\oplus L^2({\\it R})$,\ndepending on the sign of $\\Lambda$), obtained from the covariant\nand reduced phase-space quantizations, can be realized in terms of the\nnormalizable solutions of the Wheeler-DeWitt equations. Let us\nfirst consider the case of negative cosmological constant. Any\nnormalizable solutions $\\Psi$ of the form (\\ref{50.8}) can be\ndecomposed as\n$\\Psi = \\Psi^{(+)} + \\Psi^{(-)}$, where (we have redefined the\nfunction $C(q)$)\n\\begin{eqnarray}\n\\Psi^{(+)} &=& \\left( {k\\over\\pi\\hbar}\n                 \\Gamma({1\\over2}+\\nu) \\Gamma({1\\over2}-\\nu)\n                \\right)^{1\\over2}\n                \\int_0^\\infty dq a^{{1\\over2}+2i{q\\over\\hbar}}\n                e^{iq{\\Phi\\over\\hbar}} C^{(+)}(q)\n                {\\cal K}_{|\\nu|} (x)\n                \\label{50.15} \\, , \\\\\n\\Psi^{(-)} &=& \\left( {k\\over\\pi\\hbar}\n                 \\Gamma({1\\over2}+\\nu) \\Gamma({1\\over2}-\\nu)\n                \\right)^{1\\over2}\n                \\int^0_{-\\infty} dq a^{{1\\over2}+2i{q\\over\\hbar}}\n                e^{iq{\\Phi\\over\\hbar}} C^{(-)}(q)\n                {\\cal K}_{|\\nu|} (x)\n                \\label{50.16} \\, .\n\\end{eqnarray}\n\nThe scalar product takes the form\n\\begin{equation}\n< \\Psi | \\Psi > =\n\\int_0^\\infty dq |C^{(+)}(q)|^2 +\n\\int^0_{-\\infty} dq |C^{(-)}(q)|^2\n\\label{50.17} \\, ,\n\\end{equation}\nand this shows the coincidence with the Hilbert space\nderived in sections 2 and 4.\n\nWhen the cosmological constant is positive, the Hilbert space of\nnormalizable solutions of the Wheeler-DeWitt equation can also be\ndecomposed into two orthogonal subspaces. Any normalizable\nsolution $\\Psi$ of the form (\\ref{50.12}) can be split as\n$\\Psi = \\Psi^{(+)} + \\Psi^{(-)}$, where\n\n\\begin{eqnarray}\n\\Psi^{(+)} &=& \\left({k\\over2\\hbar}\\right)^{1\\over2}\n                \\int_{-\\infty}^{+\\infty} dq\n                a^{{1\\over2}+2i{q\\over\\hbar}} e^{iq{\\Phi\\over\\hbar}}\n                A^{(+)}(q)\n                \\Bigl[ \\Theta(q) {\\cal J}_\\nu (x)  \\nonumber \\\\\n                &&\\hphantom{\\left({k\\pi\\over\\hbar}\\right)^{1\\over2}\n                                \\int_{-\\infty}^{+\\infty} dq}\n                       + \\Theta(-q) \\bigl(\n                                (\\cos(\\pi Im(\\nu))^{1\\over2}\n                                {\\cal N}_\\nu(x)  \\nonumber \\\\\n                && \\hphantom{\\left({k\\pi\\over\\hbar}\\right)^{1\\over2}\n                                \\int_{-\\infty}^{+\\infty} dq}\n                              +  \\sin(\\pi Im(\\nu))\n                                \\left(cos(\\pi Im(\\nu))\\right)^{-{1\\over2}}\n                                {\\cal J}_\\nu (x)\n                        \\bigr)\n                \\Bigr] \\, , \\label{50.18} \\\\\n\\Psi^{(-)} &=& \\left({k\\over2\\hbar}\\right)^{1\\over2}\n                \\int_{-\\infty}^{+\\infty} dq\n                a^{{1\\over2}+2i{q\\over\\hbar}} e^{iq{\\Phi\\over\\hbar}}\n                A^{(-)}(q)\n                \\Bigl[ \\Theta(-q) {\\cal J}_\\nu (x)  \\nonumber \\\\\n                && \\hphantom{\\left({k\\pi\\over\\hbar}\\right)^{1\\over2}\n                                \\int_{-\\infty}^{+\\infty} dq}\n                       + \\Theta(q) \\bigl(\n                                (\\cos(\\pi Im(\\nu))^{1\\over2}\n                                {\\cal N}_\\nu(x)  \\nonumber \\\\\n                && \\hphantom{\\left({k\\pi\\over\\hbar}\\right)^{1\\over2}\n                                \\int_{-\\infty}^{+\\infty} dq}\n                              +  \\sin(\\pi Im(\\nu))\n                                \\left(cos(\\pi Im(\\nu))\\right)^{-{1\\over2}}\n                                {\\cal J}_\\nu (x)\n                                \\bigr)\n                \\Bigr] \\, . \\label{50.19}\n\\end{eqnarray}\nThe resulting expression for the scalar product turns out to be\n\n\n\\begin{equation}\n< \\Psi | \\Psi > =\n\\int_{-\\infty}^{+\\infty} dq \\left( |A^{(+)}(q)|^2 + |A^{(-)}(q)|^2 \\right)\n\\, . \\label{50.20}\n\\end{equation}\nand shows the equivalence between the Hilbert space derived from\nthe Wheeler-DeWitt equation and the one obtained from the covariant\nand reduced phase-space approach.\n\n\n\\section{Final comments}\n\nIn this paper we have constructed the quantum theory of the induced 2d-gravity\nin three different ways: i) The covariant phase space quantization; ii) The\nreduced ADM phase-space quantization and iii) The reduced Wheeler-DeWitt\nequation. We have explicitly shown the coincidence of the Hilbert space of\nthese approaches. The first approach is based on the space of classical\nsolutions and it permits to determine the ``size\" of the Hilbert space.\nThe other approaches lead to two different realizations of the Hilbert\nspace. The comparison between the different approaches allows to\nunderstand the role played by the classical solutions in the quantum theory.\nThe absence of normalizable wavefunctions for $\\nu^2=\\frac14$, when\n$\\Lambda<0$, can be understood as the absence of classical parabolic solutions.\nFurthermore, the existence of an unique wavefunction for $\\nu^2=\\frac14$, when\n $\\Lambda>0$, find its classical counterpart in the existence of an unique\nclassical parabolic solution (up to and additive constant for the scalar\nfield).\n\nFinally we want to remark that one could obtain an inequivalent quantization\nif both the hamiltonian and the supermomentum constraints were imposed at the\nquantum level. Solving the supermomentum constraint classically prevent the\nemergence of the ``Schwinger term\" in the algebra of surface deformations\ngenerated by ${\\cal C}_1$ and ${\\cal C}$ (the central extension involves both\nhamiltonian and supermomentum constraints \\cite{[2]}).\n\n\\section*{Acknowledgements}\n\n        M. Navarro acknowledges to the MEC for a Postdoctoral fellowship.\nC. F. Talavera is grateful to the {\\it Generalitat Valenciana} for a FPI grant.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\n\n\n\n\n\nScene Classification has been extensively researched by the computer vision~\\cite{zhou2017places},~\\cite{xiao2010sun},~\\cite{kumar2016deep} community.\nMost datasets have millions of images which makes the task challenging and thereby encourages the vision community to come up with better algorithms to achieve higher accuracies. However, on some of these datasets, current state of the art algorithms have low accuracies which renders their usage in the real world impractical. For instance the top-1 accuracy on the Places365 dataset~\\cite{zhou2017places} is only around 56\\%. Such an accuracy is not of much use when deploying algorithms on a robot as nearly half of the time the robot would make erroneous predictions. In the past, there have been datasets with a limited number of different places, like~\\cite{sahdev2016indoor},~\\cite{lazebnik2006beyond}, which had only upto 17 places, however these datasets' usage has been rendered obsolete due to already achieved near human accuracy ($\\approx$95\\%). \n\nIn this paper, we propose an approach for Scene Classification in real time in the context of robotics. We primarily target indoor scenes in this paper.\nIn indoor environments, GPS cannot be relied on for metric level precision. We can only get a rough estimate of the position of the robot from GPS in indoor scenes. We leverage this inaccuracy in the GPS to reduce the search space of the places. We propose a taxonomy based approach to perform Scene Classification by dividing the places according to the area type (e.g., \\textit{school, shopping-mall, home, etc}.). Once we know roughly which region the robot is in, we need to only search for a subset of places rather than searching all the classes. For example, from GPS we can know that the robot is in a school environment. This would reduce the search space to places like \\textit{corridors, lecture hall, seminar room, washroom, etc}. (see table~\\ref{dataset_testing_labels}). We no longer need to consider the places that belong to any unrelated class like \\textit{living room, pet shop, jewelry store, forest, waterfall, mountains}, etc. After this step, many classes are pruned leaving a smaller set of candidate classes to select from. \nTo classify the places, we propose to use different CNN models for each of the indoor areas present in the Places365 dataset~\\cite{zhou2017places}. \n\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=3in]{images\/vectors}\n\\end{center}\n\\caption[]{Object and Scene vectors represented in a 2D space after learning from our word2vec model. Objects related to specific scenes are closer to each other, e.g., \\textit{coffee table, tray} in \\textit{Living Room}; \\textit{crt screen, television} in \\textit{Home Theatre}. Figure generated using tSNE Visualization tool~\\footnotemark}\n\\label{fig:vectors}\n\n\\end{figure}\n\\footnotetext{{\\url{https:\/\/cs.stanford.edu\/people\/karpathy\/tsnejs\/csvdemo.html}}}\n\n\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[width=5.5in]{images\/system}\n\\end{center}\n\\caption{Overview of our algorithm. Our approach takes an RGB image as input. Then passes it onto two CNN modules. The top module is for Scene Classification, which gives us an initial top-5 prediction. The bottom module is for Scene Parsing, which detects the scene contents including background (\\textit{window pane, plant, etc.}) and foreground (\\textit{table, chair, sofa, etc.}) objects. The Word Vectors Module computes a vector for the contents of the image, and a vector for each of the top-5 predicted labels. Then Word Vectors Module refines the ranking of these top-5 predicted labels by comparing the vector similarity.}\n\\label{fig:system_overview}\n\\end{figure*}\n\n\nAfter employing a CNN for Scene Classification, we obtain the top-5 predictions for a given image. To further refine the accuracy of the prediction, we make use of different objects present in the scene which are detected using a Scene Parsing Module (see Figure~\\ref{fig:system_overview}). Each object present in the image is represented using a word-embedding~\\cite{mikolov2013efficient}. In most cases, these object embeddings are similar for objects belonging to a particular scene and the scene embedding itself, e.g., \\textit{tray, coffee table, chairs} in a \\textit{dinning room} (See Figure~\\ref{fig:vectors}) should have similar embedding. Using these embeddings, we further refine the top-5 scores to improve the accuracy. \nFinally, we build our own test dataset to validate our approach as there are many inconsistencies in the Places365 dataset as described in section~\\ref{dataset}. We also report the performance on a subset of the Places365 dataset.\n\n\n\n\n\n\n\n\n\n\n\nThe major contributions of this work are: $(i)$ A taxonomy based approach to make Scene Classification work in real time with high accuracy for robots using GPS information, $(ii)$ An empirical evaluation showing performance of our proposed approach, $(iii)$ A real world dataset for the task of indoor Scene Classification.\nThe paper is structured as follows: we describe the relevant work in Section~\\ref{relatedwork}. Section~\\ref{approach} describes our proposed approach. Section~\\ref{dataset} describes our test dataset. Experimental results of our approach are described in Section~\\ref{evaluation} and finally we conclude our work in Section~\\ref{conclusion}.\n\n\n\n\n\n\n\n\n\\section{Related Work}\n\\label{relatedwork}\nIn this section, we give a brief overview of the existing approaches used for Scene Classification.\nLi et al.~\\cite{li2010object} used object detectors as features to form an Object Bank Representation to assist Scene Classification. Our proposed approach is similar to~\\cite{li2010object}, instead of using an Object Bank representaion, we employed a word vector~\\cite{mikolov2013efficient} based feature representation to find similarities between objects in the image and the scene category. Yang et al.~\\cite{yang2007evaluating} used key point detection to create feature vectors. Bosch et al.~\\cite{bosch2008scene} used visual vocabulary as features to train a Support Vector Machine (SVM) classifier. \nFurthermore, different types of CNN architectures (AlexNet~\\cite{krizhevsky2012imagenet}, VGG~\\cite{simonyan2014very}, GoogleNet~\\cite{szegedy2015going}) were used in Scene Classification recently. \nThese CNNs achieved decent state-of-the-art performance ($\\approx$56\\%) on Places365~\\cite{zhou2017places} dataset. \nAs well as, Khan et al.~\\cite{khan2017scene} integrated Places-VGG with Spectral Features to improve Scene Classification. However, the datasets they used are not appropriate for real world application.\n\n\n\nThe CNN generally learns a feature representation of an image but the network can not tell us the object relations in the image. However, a special CNN architecture (SegNet~\\cite{badrinarayanan2015segnet})  can be used to perform pixel level segmentation (Scene Parsing). A newly released pre-trained CNN model~\\cite{zhou2017scene} based on a cascade segmentation module can find 150 types of contents in an image, which includes background (\\textit{wall, doors, windows, floor, etc.}) and foreground (\\textit{person, television, table, chair, etc.}) information. There are several approaches which can detect objects in an image; YOLO~\\cite{redmon2016yolo9000} detects most of the foreground objects. But their approach does not yield anything about the background of the image. Background information also plays an important role when a human is classifying a scene. \n\nWord vectors \\cite{pennington2014glove} are very popular in Natural Language Processing (NLP). This is also known as word embedding. It improves the performance of LSTM-CRF \\cite{huang2015bidirectional} significantly on the named entity recognition systems without any language-specific knowledge \\cite{lample2016neural}. Word vectors were trained to learn the vector representation of a given vocabulary. Word vectors have been used in many domains like sentiment analysis \\cite{mikolov2013efficient}, detecting magnitude of events \\cite{agrawal2016detecting}, named entity recognition ~\\cite{xu2017local}, etc.\n\n\nScene Classification can also be useful for Place Recognition as it reduces the search space. Alternatively Place recognition can also be used to tackle scene classification (in case where all test images come from a set of predefined paces and each place is associated with a scene type). Some of the existing Place recognition works include~\\cite{kumar2016deep},~\\cite{hou2017bocnf},~\\cite{sahdev2016indoor},~\\cite{shakeri2016illumination}.\n\n\n\n\n\\textbf{Data sets:} Many well-known data sets and benchmarks exist. For instance, Places88 \\cite{zhou2014learning} is the very first version of MIT Places benchmark\\footnote{http:\/\/places.csail.mit.edu\/index.html}. It has 88 scene categories. The latest Scene Classification data set is MIT Place365 \\cite{zhou2017places}, which has 365 scene categories. In this paper we use Place365 as our training and evaluation data-set, as it is the largest scene classification data-set, and it contains broad categories. Some other old datasets also exist for Scene Classification. The Pascal VOC \\cite{everingham2010pascal} data set has scene context annotations which are used for object detection and object segmentation tasks. Another data set is MS-COCO \\cite{lin2014microsoft}. COCO is mainly used for instance detection and scene segmentation. The pre-trained CNN Scene Parsing model \\cite{zhou2017scene} used in this paper is trained on COCO.\n\n\\section{Approach}\n\\label{approach}\n\n\n\n\n\n\nOur aim is to provide a Scene Classification approach which can perform well in a known real world environment (e.g., \\textit{school, home, shopping mall}). We build a taxonomy of different environments each containing separate scenes as shown in Table ~\\ref{dataset_testing_labels}. We pre-process the Places365 dataset to clean and reduce it to have only sensible indoor places for our use. Places365 has many inconsistencies as described in section ~\\ref{dataset}. \n\nWe use two existing CNN models, one for Scene Classification ~\\cite{zhou2017places} and the other for Scene Parsing ~\\cite{zhou2017scene}. Figure \\ref{fig:system_overview} shows the overview of our approach.\nThe input is an RGB image for the two CNN modules. We call them CNN Scene Classification Module and CNN Scene Parsing Module (Figure \\ref{fig:system_overview}). The Classification Module computes the initial raw top-5 predictions. The Scene Parsing module computes the scene contents from foreground and background. The Word Vector module computes the vector similarity between the contents present in the input image and the top-5 predicted labels. Using the computed similarity score, we output a refined re-ranked top-5 labels for the input image. We describe the details about each module in following subsections.\n\n\n\n\\subsection{CNN Scene Classification Module}\n\nIn this module, we train the ResNet models (one for each environment) on a reduced version of the Places365 dataset (based on our taxonomy, see table ~\\ref{dataset_testing_labels}). \nThe code and model parameters can be downloaded from this link~\\footnote{\\url{http:\/\/places2.csail.mit.edu\/}}. CNN Scene Classification Module computes the top-5 prediction labels which is further used by the Word Vectors Module to refine these prediction scores. \n\n\n\\subsection{CNN Scene Parsing Module}\n\nIn this module, we use a pre-trained CNN Scene Parsing model~\\footnote{\\url{https:\/\/github.com\/CSAILVision\/sceneparsing}}. The CNN model was trained on ADE20K dataset \\cite{zhou2017scene}. It is trained to detect 150 different object categories\nfrom the given input image. The Scene Parsing module tells us the different objects\/contents present in a given input image. For a kitchen input image, the parser would output the presence of \\textit{kettle, stove, oven, glasses, plates, etc}. We convert these object labels as English words and pass that onto Word Vectors module, described in the section below.\n\n\n\\subsection{Word Vectors Module}\n \n\nNow we know the objects present in each of the images in the dataset. Knowing the top-5 predictions of each image and the objects present in each of the images, we need to make use of this information to refine the top-5 scores.\n\nThis section is the main contribution of this work. First, we define some of the notations being used and then describe our approach using these notations.\n\n\\textbf{Notations used:} let $V$ be the dictionary\/vocabulary containing the objects, $O$ and the scenes, $S$ present in the dataset. $V = \\{O, S\\}$, \nwhere $O = \\{{o_{1},o_{2},o_{3},o_{4}......o_{150}}\\}$ and $S=\\{{s_{1},s_{2},s_{3}...s_{n}}\\}$, where each $o_i$ is a vector representing each of the objects and each $s_{j}$ is a vector representing the scene classes present in the dataset.\nWe further define a 2D weight matrix, $W$ of scalars. The dimensionality of the weight matrix is $|O|$ * $|S|$. where $|O| = 150$, the number of different objects we trained on and $|S| = n$, the number of defined scenes in a particular environment (e.g., for school, $n=24$). The matrix $W =\\{ {w_{1,1},..w_{i,j},..w_{150,n}}\\}$ where $i \\in{[1,150]}$ and $j \\in{[1,n]}$. Also we define $T$ as the matrix containing the top-5 predictions for each of the images. $T$ has dimensionality $5$ * $|D|$ where $|D|$ is the cardinality of the dataset. $T_{k}$ represents the row of the top-$5$ predicted classes in the $k^{th}$ Image, $I_{k}$. Similar to $T$, $ACC$ is the confidence for the predicted top-5 classes.  $IO_{k}$ is the set of objects present in the $k^{th}$ image, $I_{k}$. A sample $IO_{k}$ (for image, $I_k$) would be like $\\{{o_{14},o_{52},o_{78},o_{113},o_{143}, o_{149}}\\}$ implying say an input image (e.g., \\textit{home\\_office}) consists of 6 objects \\textit{\\{laptop, keyboard, table, desk, chair, book\\}}. \n\n\n\n\n\n\\begin{algorithm}\n\\label{algorithm_word_vec}\n\\texttt{\\textbf{\\small{}{}{}Input:}}{\\small \\par}\n\n{\\small{}{}{}A dataset, $D$ of RGB images, $I_{k}\\in\\mathbb{D}$: $D=\\left\\{ I_{1},I_{2},...,I_{n}\\right\\} $}{\\small \\par}\n\n{\\small{}{}{}A vocabulary, $V$ of object vectors, $O$ and $S$ : $V=\\left\\{ O,S\\right\\} $}{\\small \\par}\n\n{\\small{}{}{}A list of Object vectors, $O$ in $V :$ $ O=\\left\\{ o_{1},o_{2},...,o_{150}\\right\\} $}{\\small \\par}\n\n{\\small{}{}{}A list of Scene vectors, $S$ in $V :$ $ S=\\left\\{ s_{1},s_{2},...,s_{n}\\right\\} $}{\\small \\par}\n\n\n{\\small{}{}{}A weight matrix, $W$ for objects and scene, $W : \\{w_{i,j},$ where $i\\in[1,150];j\\in[1,n] \\}$}{\\small \\par}\n\n{\\small{}{}{}\\medskip{}\n }{\\small \\par}\n\n\\texttt{\\textbf{\\small{}{}{}Output:}}{\\small \\par}\n\n\n\n\n\n{\\small{}{}{}Refined Top5 Prediction, $RT_{k}$ for each image, $I_{k}$ after Refinement }{\\small \\par}\n\n\n{\\small{}{}{}\\medskip{}\n }{\\small \\par}\n\n\\texttt{\\textbf{\\small{}{}{}Procedure 1, main:}}{\\small \\par}\n\n{\\scriptsize{}{}{}1.}\\textbf{\\small{}{}{}\\ for}{\\small{}{}{}\n$I_{k}\\in D$ }\\textbf{\\small{}{}{}do}{\\small \\par}\n\n{\\scriptsize{}{}{}2.}{\\small{}{}{}\\qquad{}$T_{k}$, $ACC_{k}$ = \n$Scene Classification Module\\left(I_{k}\\right)$}{\\small \\par}\n\n{\\scriptsize{}{}{}3.}{\\small{}{}{}\\qquad{}$IO_{k}$ =  \n$Scene Parsing Module\\left(I_{k}\\right)$}{\\small \\par}\n\n{\\scriptsize{}{}{}4.}{\\small{}{}{}\\qquad{}$RT_{k}$ = \n$Word Vector Module\\left(IO_{k},T_{k},ACC_{k}\\right)$}{\\small \\par}\n\n\n\n\n{\\scriptsize{}{}{}5.}{\\small{}{}{}}\\textbf{\\small{}{}\\ return}{\\small{}{}{}\n$RT$}{\\small \\par}\n\n{\\small{}{}{}\\medskip{}\n }{\\small \\par}\n \n\\texttt{\\textbf{\\small{}{}{}Procedure 2, WordVectorModule ($IO_{k},T_{k},ACC_{k}$) :}}{\\small \\par}\n\n\n\n\n{\\scriptsize{}{}{}1.}{\\small{}{}{} $Similarity[5]$ = \n$\\{0,0,0,0,0\\}$}{\\small \\par}\n\n\n\n{\\scriptsize{}{}{}2.}\\textbf{\\small{}{}{}\\ for}{\\small{}{}{}\n$s_{j}\\in T_{k}$ }\\textbf{\\small{}{}{}do}{\\small \\par}\n\n\n\n\n{\\scriptsize{}{}{}3.}{\\small{}{}{}\\qquad{} $IVector_{k}(j)$ = \n$<0> vector$}{\\small \\par}\n\n{\\scriptsize{}{}{}4.}\\textbf{\\small{}{}{}\\qquad{}\\ for}{\\small{}{}{}\n$o_{i}\\in IO_{k}$ }\\textbf{\\small{}{}{}do}{\\small \\par}\n\n{\\scriptsize{}{}{}5.}{\\small{}{}{}\\qquad{}\\qquad{} $IVector_{k}(j)\\ += o_{i}*w_{i,j}$  \n}{\\small \\par}\n\n\n{\\scriptsize{}{}{}6.}{\\small{}{}{}\\qquad{} $Similarity[j]$ = \n$cosine(IVector_{k}(J), s_{j})$}{\\small \\par}\n\n\n{\\scriptsize{}{}{}7.}{\\small{}{}{} $Similarity = Normalize(Similarity)$\n}{\\small \\par}\n{\\scriptsize{}{}{}8.}{\\small{}{}{} $RT_{k} = descendingOrder(ACC_{k}*Similarity)$\n}{\\small \\par}\n\n\n{\\scriptsize{}{}{}9.}{\\small{}{}{}}\\textbf{\\small{}{}\\ return}{\\small{}{}{}\n$RT_k$}{\\small \\par}\n\n\n\n\\protect\\protect\\caption{\\label{alg:Pseudocode-of-the}Pseudocode of the proposed approach}\n\\end{algorithm}\n\n\nNow we have the initial top-5 predicted classes $T_{k}$ for each image, $I_{k}$ obtained from the Scene Classification module and the set of objects, $IO_{k}$ present in each image obtained from the scene parser module. Our approach can be summarized in Algorithm 1. \nTo compute the refined top-5 scene classes predictions, $RT_k$: first we compute a vector representation for the image with respect to each of the top-5 predicted classes. We compute a weighted vector sum of the detected objects in the image to represent the image vector. The vector representation for a given input image, $I$ with respect to a scene $s_{j}$ is computed as follows: \n\n\\begin{equation}\nIVector_{k}(j)= \\sum_i w_{i,j}*o_{i}\n\\end{equation}\n\n\nHere $o_{i}\\in IO_{k}$. The weights, $w_{i,j}$ and vectors, $o_{i}$ are learned by the word vector module (see Figure~\\ref{fig:trainingModel}). $IVector_{k} (j)$ is the vector representation of the input image, $I_{k}$ in terms of the objects present \\textit{w.r.t.} to scene label $j$. Refer to line 4,5 in Algorithm 1, Procedure 2.\n\n\\textbf{Learning the object and scene vectors, $V$, and the weight matrix, $W$:}\n\n\nVectors in $V$ are initialized by the pre-trained word2vector~\\footnote{\\url{https:\/\/radimrehurek.com\/gensim\/models\/word2vec.html}}, and $W$ are initialized by \\textit{tf\u2013-idf}~\\footnote{\\url{http:\/\/www.tfidf.com\/}}. When computing \\textit{tf-idf}, we considered each scene class is a document, and the objects in the image are the words. The object representing the scene has a higher weight and the object occurring less frequently in the scene would have a lower weight. We pre-compute the training predictions, $T$, $ACC$ and objects in images, $IO$. We feed this information into our cosine similarity training model (see Figure \\ref{fig:trainingModel}) to learn the weights, $W$ and Vocabualry, $V$.\nThe loss function used in our model was a hinge loss with margin equal to $0.1$. \n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=3.2in]{images\/trainingModel}\n\\end{center}\n\\caption{Training Model for learning the new word2vector $V$, and the weighted table $W$.}\n\\label{fig:trainingModel}\n\\end{figure}\n\n\n\n\nThere are some objects, $o_{i} \\in O$ which might exist in most of the scenes. Such objects contribute less in determining the class of the scene. So a lesser weight should be assigned to such objects while computing the weighted sum representation of the image vector, $IVector_{k}(j)$. On the other hand, if an object exists only in some particular scenes, we need to increase the weight of this object. This is intuitive, for instance: the chances of finding a \\textit{``sofa''} is more likely in a \\textit{``living-room''} scene than any other place. Now we describe the dataset we built to validate our approach and provide empirical results obtained on our dataset and the Places365 dataset.\n\n\n\n\\section{The Dataset}\n\\label{dataset}\n\nOne of the current widely used datasets for Scene Classification is the Places365 dataset~\\cite{zhou2017places}. This dataset is the latest and largest Scene Classification dataset with more than 1 million images. Places365 has 365 scene categories with 3000 to 5000 images per class.\nHowever in this dataset there are many confusing places that are practically the same but have been labelled distinctly. For example, the places \\textit{\\{bedroom, dorm room\\}, \\{lecture room, classroom\\}, \\{pharmacy, drug-store\\},} etc. are some sets of places that are referring to the same but labelled differently. Moreover, after a close inspection of this dataset, it was found that many images in the dataset are incorrectly labelled. Due to these mentioned issues with the Places365 dataset, we decided to merge certain classes into one before training our models (see taxonomy file on the project page$^{~\\ref{project_page_link}}$). Additionally since in our work, we are only concerned about indoor places, we remove all outdoor places in the Places365 dataset. After this pre-processing step of removal and merging of places, we get a total of 156 different scenes for 8 different environments. We train our model using this pre-processed dataset.\n\nFor testing our proposed approach, we built a dataset in the real world of 69 different places (see table~\\ref{dataset_testing_labels}), which we arranged in a hierarchical manner with 2 levels. First level (environment-type) lists out the major areas like school, home, shopping mall, etc. Each of the categories in the first level has sub-class scenes associated with them. For e.g., a shopping mall would have a set of places like \\textit{gift-shop, food-court, salon, drug-store, etc}. A detailed list of places in our testing dataset can be found in table~\\ref{dataset_testing_labels} and at our project page~\\footnote{\\url{http:\/\/jtl.lassonde.yorku.ca\/2018\/04\/scene-classification-robots\/}\\label{project_page_link}}. We have a total of 10,000 testing images in our dataset. Out of these approximately 1000 images were extracted from videos on youtube as some places were not easily available in the real world and others were captured from a GoPro camera.\n\n\\begin{table*}\n\\centering\n\\caption{Sample data classes in our Dataset. For complete taxonomy see project page$^{~\\ref{project_page_link}}$}\n\\label{dataset_testing_labels}\n\\begin{tabular}{|p{2cm}|p{14.9cm}|}\n\\hline\n\\textbf{First Level Place (Environments)} & \\textbf{Second Level Place (Scenes)}\n\\\\ \\hline\n\\textbf{School \\newline(24-scenes)}            & Classroom, Kindergarten, Office, MeetingRoom, ComputerLab, ChemistryLab, BiologyLab, PhysicsLab, Library, Corridor, Elevator, Escalator, Cafeteria, Washroom, Auditorium, Gymnasium, LockerRoom, IndoorSwimmingPool, BasketballCourt, VolleyballCourt, BadmintonCourt, TableTennis, DormRoom, Lobby\n\\\\ \\hline\n\\textbf{Home \\newline(14-scenes)}              & Kitchen, WetBar, LivingRoom, DiningRoom, Bedroom, Closet, PlayRoom, HomeTheater, HomeOffice, LaundryRoom, Washroom, Garage, Staircase, Balcony\n\\\\ \\hline\n\\textbf{Shopping Mall (31-scenes)}     & BookStore, CandyStore, VideoStore, MusicStore, HardwareStore, ShoeShop, DrugStore, ToyStore, ClothingStore, HatShop, FloristStore, JewelryStore, Optician, Supermarket, Bakery, Salon, PetShop, GiftShop, Foodcourt, Bar, Restaurant, CoffeeShop, TeaShop, DepartmentStore, Reception, Fountain, Elevator, Escalator, Washroom, IndoorParking, MovieTheater\n\\\\ \\hline\n\\end{tabular}\n\\end{table*}\n\n\\begin{table*}[t]\n  \\begin{center}\n    \\caption{Top-1 accuracy on the reduced validation set on Places365 and our test set.}\n    \\label{tab:table1}\n    \\begin{tabular}{l|c|c|c|c|c|c}\n       {} & \\multicolumn{2}{c}{School (24)} & \\multicolumn{2}{c}{Home (14)} & \\multicolumn{2}{c}{Shopping Mall (31)} \\\\\n       {} & Places365 val.-set & our test-set & Places365 val.-set & our test-set & Places365 val.-set & our test-set \\\\\n      \\hline\n      Places365-ResNet &  51.45\\% & 58.18\\% &  57.13\\% & 60.68\\% &  50.17\\% & 62.58\\%\\\\\n      ResNet50 &  73.82\\% & 90.33\\% &  83.46\\% & 92.03\\% &  \\textbf{70.47\\%} & 87.31\\%\\\\\n      ResNet50+Word2Vec (Ours) & \\textbf{74.28\\%} & \\textbf{92.25\\%} & \\textbf{83.67\\%} & \\textbf{93.27\\%} & 70.44\\% & \\textbf{87.39\\%}\\\\\n    \\end{tabular}\n    \n  \\end{center}\n  \\begin{comment}\n  \n  \\begin{tablenotes}\n  \\begin{center}\n      \\small\n      \\item \\textit{A) Places365-ResNet; B) ResNet; C) ResNet50+Word2Vec (Ours)}.\n      \\end{center}\n    \\end{tablenotes}\n\\end{comment}\n\\end{table*}\n\n\n\n\n\n\\section{Evaluation}\n\\label{evaluation}\n\\begin{comment}\n\\begin{figure*}[t!]\n\\begin{center}\n\\includegraphics[width=6.8in]{images\/datasetSample}\n\\end{center}\n\\caption{These images are the samples from Places365 Scene Classification dataset. They demonstrate the diversity of different classes.}\n\\label{fig:dataSamples}\n\\end{figure*}\n\\end{comment}\nIn our taxonomy, we described 8 different environment types (\\textit{School, Shopping Mall, Home, Condo Buildings, Airport, Public Transit, Hospital, and Hotel}) using 156 indoor scenes. In this work, we select 3 environments to evaluate our approach, which are \\textit{School, Home}, and \\textit{Shopping Mall} (see Table~\\ref{dataset_testing_labels}).\n\nIn our experiments, we trained a ResNet model for each environment with the selected scenes. The performance is listed in Table~\\ref{tab:table1}. As one would expect, the ResNet model performs well on the reduced Places365 validation set. The performance on our test set is even higher, since our dataset is labeled with fewer human errors. Furthermore, we used a GoPro camera with wide-angle view mode, so the images perfectly represent the scenes. Overall, our method is better than a CNN model (ResNet50) in terms of accuracy. Our results proved that the \\textit{``image context (objects)''} plays a vital role in Scene Classification. Although, the refinement on Shopping Mall scenes is not noticeable, this is because the CNN Scene Parsing model we used could not find objects that can distinguish the scenes well enough. For instance, the model could not detect shoes, watches, hats, etc., which are commonly found in shopping malls\n\n\n\nFigure~\\ref{fig:results} discusses the results of some images in our dataset. Finally, we also demonstrate our approach by deploying our algorithm on a Pioneer3AT robot in a university environment. The robot was configured with a ZED camera from which monocular images were used by our algorithm. The robot was equipped with a Razer Blade laptop with a GTX 1060 GPU. The demo video of the deployment is available on our project web-page$^{~\\ref{project_page_link}}$.\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{comment}\n\\begin{figure*}[t!]\n\\begin{center}\n\\includegraphics[width=6.8in]{images\/objectsDitribution}\n\\end{center}\n\\caption{Objects Frequency.}\n\\label{fig:objfrequency}\n\\end{figure*}\n\\end{comment}\n\n\n\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[width=5.8in]{images\/results}\n\\end{center}\n\\caption{Column A is the top-5 confidences obtained from the CNN Scene Classification module. Column B is the cosine similarity score from the Word Vectors module, Column AxB represents the refined top-5 predictions (refer to line 8 in Algorithm 1, Procedure 2). Sample results: (1)-(4) are correct examples, (5)-(7) are incorrect examples on our test set. The first 4 rows show that our model correctly predicted the scene labels. In the first example, CNN gave an unclear result by predicting the image as \\textit{physics lab, chemistry lab, gymnasium} with very similar scores. But, our word2vec model gave a very high score on \\textit{chemistry lab}. By combining the two scores, our model gave a higher predicted score on the ground truth label. On the other hand, our method also yielded some bad predictions. For row (5), our word2vec model provided a very high confidence on the false label \\textit{office} by using the context of the image. As a result, this image was incorrectly classifying as \\textit{office} instead of \\textit{physics lab}. Another example in row (6), the CNN model very confidently predicted a \\textit{kitchen} as \\textit{laundry room}. Even though word2vec said it is a \\textit{kitchen}, our model still could not refine the CNN predictions.  }\n\\label{fig:results}\n\\end{figure*}\n\n\n\\begin{comment}\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[width=6.8in]{images\/Miserables_Arc}\n\\end{center}\n\\caption{Arc Diagram: displays 136 most related scenes categories. This diagram is computed using top-5 predicted accuracy from the validation set. The thickness of the lines represents the similarity between two scene categories. For example, \"desert\/sand\", \"desert\/vegetation\", and \"desert\\_road\" are very similar to each other. So, the images in these scene categories are most likely to make mistakes.}\n\\label{fig:Miserables_Arc}\n\\end{figure*}\n\\end{comment}\n\n\\section{Conclusion}\n\\label{conclusion}\n\nIn this paper, we introduced a new method for Scene Classification using a taxonomy for different indoor environments. Using our approach, robots can recognize different indoor places with high confidence\/accuracy. A hierarchical taxonomy of places allowed us to prune out many irrelevant classes, thereby reducing the complexity\/training time of our approach.\nA word embedding based approach was implemented to refine the top-5 scores for the scenes. It was shown that context has the potential to improve the Scene Classification results to some extent. \nWe additionally tested our approach with a real world dataset that we built to show the practical applicability of our approach. We also deployed our algorithm on a robot in a university environment. On our dataset we could get promising results for doing Scene Classification for robots.\n\n\n\n\n\n\n\n\n{\\small\n\\bibliographystyle{IEEEtran.bst}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\\label{sec:introduction}\n\nPhase transitions in two-dimensional (2D) fermionic\nsystems are a central topic of theoretical and experimental condensed matter\nphysics. Correlated quasi-2D materials with rich phase diagrams include\nhigh-temperature superconductors \\cite{RevModPhys.78.17} and transition-metal\ndichalcogenides \\cite{manzeli20172d}. Dirac fermions in two dimensions can be\ninvestigated in graphene \\cite{Neto_rev}. \nStrongly correlated 2D fermions exhibit exotic phases\n\\cite{RevModPhys.89.025003} and phase transitions\n\\cite{Senthil04_2}, and can support long-range order\nat $T>0$ \\cite{PhysRevLett.17.1133}. While magnetism originates\nfrom short-range Coulomb repulsion, the main mechanism behind the numerous\ncharge-density-wave (CDW) phases found experimentally is electron-phonon\ncoupling. In addition to polaron effects, the latter leads to\na phonon-mediated, retarded electron-electron interaction and an intricate\ninterplay of spin, charge, and lattice fluctuations. \n\nQuantum Monte Carlo (QMC) simulations are a key tool to investigate \ncorrelated 2D quantum systems. Although simulations\nare significantly harder for fermions than for spins or bosons, QMC methods have been very\nsuccessfully applied to fermionic models. However, whereas the phase diagram\nand critical behavior of, \\eg, the 2D honeycomb Hubbard model is known in detail\n\\cite{Sorella12,Assaad13,Toldin14,Otsuka16}, the same is not true even for\nthe simplest Holstein molecular-crystal model of electron-phonon\ninteraction. Most notably, simulations with phonons are often severely restricted\nby long autocorrelation times also away from critical points \\cite{Hohenadler2008}. \nCurrently, reliable critical temperatures, convincing analysis of critical\nbehavior, and the ground-state phase diagram remain key open problems. In fact, even the\nsimpler 1D case had until recently been discussed controversially\n\\cite{MHHF2017}, with earlier claims of dominant pairing correlations refuted\nby direct calculations of the correlation functions and traced back to spin\ngap formation \\cite{PhysRevB.92.245132}.\n\nHere, we use large-scale continuous-time QMC simulations to investigate\nthe CDW transition in the 2D Holstein-Hubbard model. Although the latter has\nbeen extensively studied in the past, important open questions remain.\nAt strong coupling and half-filling, the ground state is either a CDW\ninsulator or an antiferromagnetic Mott insulator. Recent variational QMC studies\n\\cite{ohgoe2017competitions,1709.00278}  argue in\nfavor of a third phase (metallic or superconducting), seemingly in\ncontradiction with theoretical arguments based on weak-coupling\ninstabilities of the Fermi liquid \\cite{PhysRevLett.56.2732,PhysRevB.42.2416}. We use finite-size\nscaling to determine $T_c$ of the CDW transition, show that the latter can\nalso be detected by the fidelity susceptibility, and provide evidence for its\nIsing critical behavior. Moreover, we present arguments and data for the\nexistence of a metallic bipolaron phase at $T>T_c$ and address the possibility of a\nmetallic or a superconducting ground state.\n\nThe paper is organized as follows. Section~\\ref{sec:model} introduces the\nrelevant models, Sec.~\\ref{sec:methods} gives a brief review of the numerical\nmethods, Sec.~\\ref{sec:results} discusses the results, and Sec.~\\ref{sec:conclusions} \ncontains our conclusions.\n\n\\section{Models}\\label{sec:model}\n\nThe Holstein-Hubbard Hamiltonian \\cite{HOLSTEIN1959325} reads\n\\begin{align}\\label{eq:model}\\nonumber\n  \\hat{H}\n  =\n    &-t \\sum_{\\las i,j\\ras \\sigma} \\hat{c}^\\dag_{i\\sigma} \\hat{c}^{\\phantom{\\dag}}_{j\\sigma} \n    +\n    \\sum_{i}\n    \\left[\n    \\mbox{$\\frac{1}{2M}$} \\hat{P}^2_{i}\n    +\n    \\mbox{$\\frac{K}{2}$} \\hat{Q}_{i}^2\n    \\right]\n  \\\\\n    &-\n      g\n    \\sum_{i} \\hat{Q}_{i} \n     \\hat{\\rho}_i\n    + U \n      \\sum_i (\\hat{n}_{i\\UP}-\\mbox{$\\frac{1}{2}$}) (\\hat{n}_{i\\DO}-\\mbox{$\\frac{1}{2}$})\n    \\,.\n\\end{align}\nThe first two terms describe free electrons and free phonons, respectively.\nHere, $\\hat{c}^\\dag_{i\\sigma}$ creates an electron with spin $\\sigma$ at lattice\nsite $i$ and electrons hop with amplitude $t$ between nearest-neighbor\nsites on a square lattice. The phonons are of the Einstein type with frequency\n$\\omega_0=\\sqrt{K\/M}$; their displacements $\\hat{Q}_i$ couple to local\nfluctuations $\\hat{\\rho}_i=\\hat{n}_i-1$ of the electron occupation $\\hat{n}_{i} =\n\\sum_\\sigma \\hat{n}_{i\\sigma}$ where $\\hat{n}_{i\\sigma} = \\hat{c}^\\dag_{i\\sigma}\\hat{c}^{\\phantom{\\dag}}_{i\\sigma}$. The\nlast term describes a Hubbard onsite repulsion of strength $U$.\nWe simulated $L\\times L$ lattices with periodic boundary conditions at\nhalf-filling  ($\\las\\hat{n}_{i}\\ras=1$, chemical potential $\\mu=0$). A\nuseful dimensionless coupling parameter is $\\lambda=g^2\/(W K)$ with the free\nbandwidth $W=8t$. We set $\\hbar$, $k_\\text{B}$, and the lattice constant to one and\nuse $t$ as the energy unit.\n\nFor $U=0$, Eq.~(\\ref{eq:model}) reduces to the Holstein model. Its\nrelative simplicity has motivated numerous  QMC investigations of CDW\nformation and superconductivity \\cite{%\nPhysRevB.40.197,PhysRevB.42.2416,PhysRevB.42.4143,PhysRevLett.66.778,PhysRevB.43.10413,PhysRevB.46.271,PhysRevB.48.7643,PhysRevB.48.16011,PhysRevB.55.3803}. Equation~(\\ref{eq:model})\nwith $g=0$ corresponds to the repulsive Hubbard model on the square lattice. At half-filling, the ground state of the latter is an\nantiferromagnetic Mott insulator for any $U>0$ \\cite{Hirsch89}. However, in contrast to \nCDW order, antiferromagnetism is restricted to $T=0$ in two dimensions by the\nMermin-Wagner theorem \\cite{PhysRevLett.17.1133}.\nThe full Holstein-Hubbard Hamiltonian~(\\ref{eq:model}) captures the\ncompetition between Mott and CDW ground states\n\\cite{PhysRevB.52.4806,PhysRevLett.75.2570,PhysRevB.75.014503,PhysRevB.92.195102,PhysRevLett.109.246404,PhysRevB.87.235133,ohgoe2017competitions}.\nWhereas early work unanimously agreed on the absence of a disordered\nor a superconducting ground state at half-filling, such a phase has recently\nbeen advocated by numerical results \\cite{ohgoe2017competitions,1709.00278}. \n\nBecause it is sufficient to address many of the open questions of interest,\nwe will mainly consider the case $U=0$. However, selected results for\nthe impact of the Hubbard repulsion will also be reported. For\nEq.~(\\ref{eq:model}) with $U=0$, mean-field theory (exact for $\\omega_0=0$ and\n$T=0$) predicts a CDW ground state with a checkerboard pattern for\nthe lattice displacements and the charge density [ordering vector ${\\bm\n  Q}=(\\pi,\\pi)$, see inset of Fig.~\\ref{fig:phasediagram}]  \nat half-filling \\cite{PhysRevB.40.197,PhysRevB.42.2416,PhysRevLett.66.778}.\nHere, we systematically explore the impact of quantum and thermal fluctuations.\n\nAn important limiting case is the antiadiabatic limit $\\omega_0\\to\\infty$, in\nwhich the Holstein-Hubbard model maps to a Hubbard model with Hamiltonian\n\\begin{align}\\label{eq:model2\n  \\hat{H}\n  &=\n    -t \\sum_{\\las i,j\\ras \\sigma} \\hat{c}^\\dag_{i\\sigma} \\hat{c}^{\\phantom{\\dag}}_{j\\sigma} \n    +\n    U_\\infty    \n    \\sum_{i} (\\hat{n}_{i\\UP}-\\mbox{$\\frac{1}{2}$}) (\\hat{n}_{i\\DO}-\\mbox{$\\frac{1}{2}$})\n\\end{align}\nand effective interaction $U_\\infty=U-\\lambda W$. For $U=0$, interactions are\npurely attractive and give rise to coexisting CDW and superconducting order\nfor any $\\lambda>0$ at $T=0$. However, at half-filling, this order is minimal\nin the sense that $T_c=0$ \\cite{Hirsch85}, which is related to a perfect\ndegeneracy of CDW and pairing correlations and an associated continuous SO(3)\norder parameter for which the Mermin-Wagner theorem applies \\cite{PhysRevLett.17.1133}.\n\n\\section{Methods}\\label{sec:methods}\n\nExtending previous applications to 1D electron-phonon models \\cite{Ho.As.Fe.12,PhysRevB87.075149,PhysRevLett.117.206404}, we use the\ncontinuous-time interaction expansion (CT-INT) method \\cite{Rubtsov05}. To\nthis end, we express the partition function as a functional integral\n\\begin{align} \\label{partitionfunction}\n Z = \\int \\mathcal{D}(\\bar{c},c) \\ e^{-S_0\\left[\\bar{c},c\\right]-S_1\\left[\\bar{c},c\\right]} \\int \\mathcal{D}(\\bar{b},b) \\ e^{-S_\\text{ep}\\left[\\bar{c},c,\\bar{b},b\\right]}\n\\end{align}\nusing coherent states.\nSplitting the action into the free-fermion part $S_0$, the Hubbard interaction $S_1$, \nand the remainder $S_\\text{ep}$ that contains the free-phonon contribution\nand the electron-phonon coupling, the phonons are integrated out analytically\nto arrive at a fermionic model with both an instantaneous Hubbard interaction ($S_1$)\nand a retarded, phonon-mediated interaction ($S_2$) \\cite{Assaad07}. This model can\nbe simulated by the CT-INT method by sampling both types of vertices\n\\cite{Assaad07} to stochastically sum the weak-coupling Dyson expansion\n\\cite{Rubtsov05} around $S_0$. Because the latter converges for fermionic systems in\na finite spacetime volume, CT-INT is exact apart from statistical errors.\nTechnical reviews can be found in Refs.~\\cite{Gull_rev,Assaad14_rev}.\n\nIn contrast to the determinant QMC (DetQMC) method \\cite{Blankenbecler81}\nused in almost all previous works on Holstein-Hubbard-type models, CT-INT\nhas significantly smaller autocorrelation times \\cite{Hohenadler2008}.\nCT-INT simulation times scale as ${O}(n^3)$, where $n$\n[$\\approx {O}(\\lambda\\beta L^2)$ for $U=0$] is the average expansion\norder and $\\beta=1\/T$. Although DetQMC formally has a better\n${O}(\\beta L^6)$ scaling, CT-INT benefits from\nreduced expansion orders at weak coupling and seems to outperform DetQMC for\nmost parameters considered despite being limited for $\\omega_0\\gtrsim t$ by a\nsign problem. Whereas even the noninteracting case is challenging for DetQMC, CT-INT\ntrivially gives exact results for $\\lambda=0$ and can in principle\nsimulate the entire range of phonon frequencies, including the\nexperimentally important adiabatic regime $\\omega_0<t$. We used up to 5000\nsingle-vertex updates and 8 Ising spin flips per sweep.\nThe classical case $\\omega_0=0$ was simulated using the method\nof Ref.~\\cite{1996MPLB...10..467M} combined with parallel tempering \\cite{doi:10.1143\/JPSJ.65.1604}.\n\n\\begin{figure}[t]\n \n  \\includegraphics[width=0.45\\textwidth]{fig1.pdf}\n  \\caption{\\label{fig:phasediagram} Critical temperature of the Holstein\n    model ($U=0$) from finite-size scaling.  Here and in subsequent figures, lines are guides\n    to the eye. Statistical errors are smaller than the symbols, see text.\n    The inset illustrates the CDW order at $T<T_c$ for $L=4$, with filled\n    (open) symbols representing occupied (empty) sites.}\n\\end{figure}\n\n\\section{Results}\\label{sec:results}\n\nSince the effect of electron-electron repulsion on a half-filled square\nlattice---namely, an antiferromagnetic Mott state at $T=0$---is well\nunderstood \\cite{Hirsch89} the focus of our work will be the electron-phonon\ninteraction, \\ie, Eq.~(\\ref{eq:model}) with $U=0$. Coulomb interactions of\nthe Hubbard or even long-range type can be simulated with unbiased QMC\nmethods on systems large enough to extract critical exponents \\cite{Sorella12,Assaad13,Toldin14,Otsuka16}.\nIn contrast, the electron-phonon interaction is significantly more\nchallenging to describe due to the resulting, retarded electron-electron\ninteraction. This is true both for QMC methods but also for, \\eg, the\nfunctional renormalization group \\cite{PhysRevB.75.014503,PhysRevB.92.195102}. Consequently, even fundamental aspects such\nas the existence of a nonzero critical value for the CDW transition are still\nunder debate. From the 1D Holstein-Hubbard model, it is known that the phases\nat $U=0$ (Luther-Emery liquid and CDW insulator) are stable against a nonzero\nHubbard repulsion so that they and the phase transition between them can be\nfully understood in the simpler Holstein model \\cite{MHHF2017}. In particular, \nthe metallic phase arises from quantum lattice fluctuations \nrather than from a competing Hubbard interaction \\cite{MHHF2017}. In the 2D case, recent\npredictions of an extended metallic region suggest that the latter is largest\nat $U=0$ \\cite{ohgoe2017competitions,1709.00278}. A nonzero but sufficiently\nsmall Hubbard repulsion merely shifts the critical value for the CDW transition\n\\cite{PhysRevB.52.4806,PhysRevLett.75.2570,PhysRevB.75.014503,PhysRevB.92.195102,PhysRevLett.109.246404,PhysRevB.87.235133,ohgoe2017competitions}.\nMoreover, because the long-range antiferromagnetic order is restricted to\n$T=0$, the possible phases at $T>0$ (the focus of this work) remain the same. \n\n\\begin{figure}[b]\n \n  \\includegraphics[width=0.45\\textwidth]{fig2.pdf}  \n  \\caption{\\label{fig:criticalpoint} (a) Determination of the critical value $\\lambda_c$ \n    from (a) the crossing of the correlation ratios $R_\\text{c}$ for\n    different system sizes $L$ and (b) the maximum in the fidelity\n    susceptibility. Here, $\\omega_0\/t=0.1$, $U=0$, and (a) $T\/t=0.05$, (b)\n    $T\/t=0.2$. \n  }\n\\end{figure}\n\n\\subsection{Critical values}\\label{sec:results:Tc}\n\nTo obtain the critical values shown in\nFig.~\\ref{fig:phasediagram}, we calculated the correlation ratio\n\\cite{Binder1981}\n\\begin{equation}\\label{eq:rcdw}\n  R_\\text{c} =\n  1-\\frac{S_\\text{c}(\\bm{Q}-\\delta{\\bm q})}{S_\\text{c}(\\bm{Q})}\n\\end{equation}\n(with $|\\delta{\\bm q}|=2\\pi\/L$) from the charge structure factor\n\\begin{equation}\\label{eq:scdw}\nS_\\text{c}({\\bm q}) = \\frac{1}{L^2}\\sum_{ij}\ne^{\\text{i}(\\bm{r}_i-\\bm{r}_j)\\cdot\\bm{q}}\\,\\las \\hat{n}_i \\hat{n}_j\\ras\n\\end{equation}\neither at fixed $\\lambda$ or at fixed $T$. Here, ${\\bm Q}=(\\pi,\\pi)$. \nBy definition, a divergence of $S_\\text{c}(\\bm{Q})$ with $L$ in the CDW phase\nimplies $R_\\text{c}\\to 1$ for $L\\to\\infty$, whereas $R_\\text{c}\\to 0$ in\nthe absence of long-range CDW order. Moreover, because $R_\\text{c}$ is\na renormalization group invariant \\cite{Binder1981}, the critical point can be\nestimated from the crossing of curves for different $L$, as illustrated in\nFig.~\\ref{fig:criticalpoint}(a) for $\\omega_0\/t=0.1$ and $T\/t=0.05$. \nWhile the correlation ratio~(\\ref{eq:rcdw}) is expected to exhibit smaller\nfinite-size corrections than the structure factor~(\\ref{eq:scdw}), a shift\nof consecutive crossing points is observed on the accessible system sizes,\nmaking it necessary to extrapolate to\n$L=\\infty$. To this end, we used a fit function\n\\begin{equation}\\label{eq:fitfunction}\nf(L) = a + b L^c\\,.\n\\end{equation}\nExamples for such extrapolations are shown for $\\om_0\/t=0$ in Fig.~\\ref{fig:extrapolation}(a)\nand for $\\om_0\/t=0.1$ in Fig.~\\ref{fig:extrapolation}(b). For classical\nphonons, we can access significantly larger system sizes up to $L=28$. The\npoints in Fig.~\\ref{fig:extrapolation}(a) correspond to crossing points of\n$R_\\text{c}$ for $L$, $L-2$ (\\ie, $\\Delta L=2$) and $L$, $L-4$ ($\\Delta\nL=4$), respectively. Fitting to Eq.~(\\ref{eq:fitfunction}), these two choices\nyield identical results for $T_c$ within error bars. The errors take into\naccount the statistical errors of the QMC results as well as the errors in\ndetermining the crossing points using parabolic fits (obtained from a\nbootstrap analysis) and extrapolating to $L=\\infty$. They are smaller than\nthe symbol size in Fig.~\\ref{fig:phasediagram} but naturally do not capture\npossible variations due to the choice of fit function or observable.  \nFor quantum phonons, we systematically used $L=4,6,8,10,12$ and hence $\\Delta L=2$, as\nillustrated in Fig.~\\ref{fig:extrapolation}(b). A similar extrapolation gives\n$\\lambda_c=0.101(1)$ for the parameters of Fig.~\\ref{fig:criticalpoint}(a).\n\n\\begin{figure}[t]\n \n  \\includegraphics[width=0.45\\textwidth]{fig3.pdf}\n  \\caption{\\label{fig:extrapolation}  \n    Finite-size extrapolation of the crossing points of $R_\\text{c}(L)$,\n    $R_\\text{c}(L-\\Delta L)$ using the fit function~(\\ref{eq:fitfunction}). \n    Here, (a) $\\omega_0=0$, $\\lambda=0.1$, $T_c=0.0506(1)$ and (b)\n    $\\omega_0\/t=0.1$, $T\/t = 0.2$, $\\lambda_c=0.244(1)$.\n  }\n\\end{figure}\n\nThe phase transition can also be detected using the fidelity susceptibility\n$\\chi_{F}$ \\cite{2008arXiv0811.3127G}, an unbiased diagnostic to\ndetect critical points without any knowledge about the order parameter.\nIt essentially relies on calculating the overlap of the ground states of (in\nthe present case) Holstein \nHamiltonians with couplings $\\lambda$ and $\\lambda+\\delta\\lambda$. A\nfinite-temperature generalization has been given in Refs.~\\cite{PhysRevE.76.022101,PhysRevLett.103.170501, PhysRevB.81.064418}\nand CT-INT estimators in Refs.~\\cite{PhysRevX.5.031007,PhysRevB.94.245138}.\nAlthough these estimators have rather large statistical errors at low\ntemperatures, $\\chi_{F}\/L^2$ for $T\/t=0.20$ in\nFig.~\\ref{fig:criticalpoint}(b) shows the expected peak at a position that is\nconsistent with Fig.~\\ref{fig:phasediagram} and\n$\\lambda_c=0.244(1)$ from Fig.~\\ref{fig:extrapolation}(b).\n\nFigure~\\ref{fig:phasediagram} shows  $T_c(\\lambda)$ for\ndifferent $\\omega_0$, covering the entire adiabatic regime $0\\leq\n\\omega_0 \\leq t$. The mean-field result $T_c\\sim e^{-1\/\\sqrt{\\lambda}}$ for\nthe 2D Holstein model---compared to $T_c\\sim e^{-1\/\\lambda}$ in dynamical mean-field\ntheory (DMFT) \\cite{PhysRevB.63.115114}---is expected to overestimate\n$T_c$ even at $\\omega_0=0$ and does not capture the expected maximum at\n$\\lambda<\\infty$ \\cite{PhysRevB.63.115114}. The latter is outside\nthe range of couplings considered here. Quantum lattice fluctuations\nsuppress $T_c$ at a given $\\lambda$. For $\\omega_0\/t=0.1$, $T_c$ shows only\nminor deviations from the result for classical phonons, whereas for larger $\\omega_0$ \nquantum fluctuation effects are clearly visible over the entire parameter range shown.\n The systematic suppression of $T_c$ with\nincreasing $\\omega_0$ is perfectly consistent with the fact that $T_c=0$ for the\nattractive Hubbard model \\cite{Hirsch85}, to which the Holstein model maps in the limit\n$\\omega_0\\to\\infty$ \\cite{Hirsch83a}. This connection and a possible metallic phase at low temperatures\nas a result of quantum fluctuations will be discussed below. At $T>0$, a\nmetallic region is naturally expected in the phase diagram of the 2D\nHolstein-Hubbard model because  the antiferromagnetic Mott state arising from\nthe Hubbard interaction is confined to $T=0$. In contrast to previous  DMFT results\n\\cite{PhysRevB.63.115114}, the critical temperatures in\nFig.~\\ref{fig:phasediagram} were obtained by taking into account all (spatial\nand temporal) fluctuations on the square lattice. \n\n\\begin{figure}[t]\n \n  \\includegraphics[width=0.45\\textwidth]{fig4.pdf}  \n  \\caption{\\label{fig:TcvsU} Critical temperature of the CDW transition in\n    the Holstein-Hubbard model. (a) Suppression of $T_c$ with increasing $U$\n    at $\\lambda=0.25$ from finite-size scaling, (b) comparison of Holstein\n    and Holstein-Hubbard results in terms of the effective coupling\n    $\\lambda_\\text{eff}=\\lambda-U\/W$. The points labeled `Holstein'\n    correspond to $\\lambda_c$ at different temperatures from Fig.~\\ref{fig:phasediagram}. The points\n    labeled `Hol-Hub' (Holstein-Hubbard) are for $T_c$ at $\\lambda=0.25$ and\n    $U\/t=0,0.25,0.50$ from (a). Here, $\\omega_0\/t=0.1$.}\n\\end{figure}\n\nThe Hubbard repulsion suppresses CDW order\n\\cite{PhysRevB.52.4806,PhysRevLett.75.2570,PhysRevB.75.014503,PhysRevB.92.195102,PhysRevLett.109.246404,PhysRevB.87.235133,ohgoe2017competitions}.\nThis is already apparent from the effective Hubbard model~(\\ref{eq:model2}) in the limit\n$\\omega_0\\to\\infty$ where a nonzero $U$ reduces the effective, attractive\ninteraction and thereby the CDW gap at $T=0$. Whereas CDW order is restricted\nto $T=0$ in this limit, here we consider the Holstein-Hubbard model in the\nopposite, adiabatic regime. Specifically, we take $\\omega_0\/t=0.1$ and $\\lambda=0.25$.\n\nTo quantify the effect of $U$, we show in Fig.~\\ref{fig:TcvsU}(a) the\nsuppression of $T_c$ as a function of $U$. Starting from\n$T_c\/t=0.204(1)$ at $U=0$, $T_c$ decreases by about 15 percent in the range\n$U\\in[0,0.5t]$. In principle, in the spirit of an effective Holstein model,\nwe can try to capture this effect by a coupling $\\lambda_\\text{eff}=\\lambda-U\/W$.\nHowever, Fig.~\\ref{fig:TcvsU}(b) reveals that for the parameters considered\nthis overestimates the effect of the Hubbard repulsion because $T_c$ at a\ngiven $\\lambda_\\text{eff}$ in the Holstein model ($U=0$) is significantly\nlower than in the Holstein-Hubbard model ($U>0$). We attribute this finding\nto (i) the stronger suppression of the antiferromagnetic\ncorrelations (long-range magnetic order only exists at $T=0$) compared to the\nCDW correlations (CDW order exists also at $T>0$) at the temperatures considered,\nand (ii) retardation effects. A DMFT analysis of the Holstein-Hubbard model revealed that $T_c$ is suppressed with increasing $U$\nat weak electron-phonon coupling but initially enhanced at strong\ncoupling. This behavior was explained in terms of a reduction of the\nbipolaron mass due to the onsite repulsion \\cite{PhysRevLett.75.2570}.\n\n\\begin{figure}[b]\n \n  \\includegraphics[width=0.45\\textwidth]{fig5.pdf}  \n  \\caption{\\label{fig:quising} Scaling collapse of (a) the structure\n    factor and (b) the correlation ratio for $\\omega_0\/t=0.1$,\n    $\\lambda=0.25$, and $U=0$ using the critical exponents of the 2D Ising\n    model. The critical temperatures $T_c$ were determined from the best\n    scaling collapse and are given in the text.\n}\n\\end{figure}\n\n\\subsection{Critical behavior}\\label{sec:results:critical}\n\nIn the thermodynamic limit, the long-range CDW\norder at $T<T_c$ spontaneously breaks the sublattice symmetry. The two\npossible CDW patterns (cf. Fig.~\\ref{fig:phasediagram}) imply the same\ncritical behavior as the 2D Ising model and hence critical exponents\n$\\beta=1\/8$ and $\\nu=1$. Here, we demonstrate consistency with Ising\nuniversality for $\\om_0\/t=0.1$ and $\\lambda=0.25$.\n\nAs the order parameter, the charge structure\nfactor~(\\ref{eq:scdw}) should obey the finite-size scaling relation\n\\cite{PhysRevLett.66.778} \n\\begin{equation}\\label{eq:scalingS}\n  {S_\\text{c}({\\bm Q})}\/{L^2}\n = L^{-2\\beta\/\\nu}\\, f_S[L^{1\/\\nu}(T-T_c)\/T_c]\\,.\n\\end{equation}\nTherefore, plotting $S_\\text{c}({\\bm Q}) L^{2\\beta\/\\nu-2}$ as a function of $L^{1\/\\nu}(T-T_c)\/T_c$ \nshould produce a collapse of the data onto the curve described by the scaling\nfunction $f_S$. The best collapse \\cite{autoscale}\nover the interval $[-1,1]$ in Fig.~\\ref{fig:quising}(a) gives $T_c\/t=0.195(1)$, \nsmaller than the value $T_c\/t=0.204(1)$ (Fig.~\\ref{fig:TcvsU}) determined from\nfinite-size scaling.\n\nA similar analysis can be carried out for the correlation ratio, which is\nexpected to obey\n\\begin{equation}\\label{eq:scalingR}\n  R_\\text{c} = f_R[L^{1\/\\nu}(T-T_c)\/T_c]\\,,\n\\end{equation}\ninvolving only the correlation length exponent $\\nu$. Hence, \nwe expect a collapse onto $f_R$ by plotting $R_\\text{c}$ as a function of\n$L^{1\/\\nu}(T-T_c)\/T_c$. The best collapse on $[-1,1]$ is\nobtained for $T_c\/t=0.205(1)$ and shown in Fig.~\\ref{fig:quising}(b). This\ncritical value is consistent with the previous estimate  $T_c\/t=0.204(1)$\nin Fig.~\\ref{fig:TcvsU}. However, the collapse exhibits stronger scattering\nthan for the structure factor, even though the correlation ratio is generally\nexpected to be less affected by finite-size corrections \\cite{Binder1981}.\n\n\\subsection{Phase diagram}\\label{sec:results:phasediagram}\n\nFigure~\\ref{fig:phasediagram} gives the finite-temperature phase diagram of\nthe Holstein model in terms of $T_c(\\lambda)$, which separates the\nlow-temperature phase with long-range CDW order from the high-temperature\ndisordered phase. However, since accurate values of $T_c$ at very small\n$\\lambda$ are currently not accessible, Fig.~\\ref{fig:phasediagram} does not\nsettle the question of whether or not the ground state has CDW order for any\n$\\lambda>0$. \n\nThere are two well-understood limits. The {\\it classical} Holstein\nmodel ($\\omega_0=0$) has a CDW ground state for any $\\lambda>0$ and\n$T_c>0$ (see Sec.~\\ref{sec:results:Tc}). This follows from mean-field theory,\nwhich becomes exact at $T=0$. In the opposite, antiadiabatic limit\n$\\omega_0\\to\\infty$, the Holstein model maps to the attractive Hubbard model,\nwhose ground state has coexisting CDW and superconducting order but $T_c=0$. \nHence, as a function of $\\omega_0$, the Holstein model interpolates between\ntwo limits that both exhibit long-range CDW order at $T=0$. \n\nBetween these limiting cases (\\ie, for $0<\\omega_0<\\infty$), there appear to be \ntwo distinct scenarios for the shape of the phase boundary $T_c(\\lambda)$, as\nillustrated in Fig.~\\ref{fig:tcschematic}. In\nscenario (I), $T_c>0$ for any $\\lambda>0$, so that the ground state is always\na CDW insulator. By contrast, in scenario (II), $T_c=0$ for\n$\\lambda<\\lambda_c(\\omega_0)$ and $T_c>0$ for $\\lambda>\\lambda_c(\\omega_0)$. \nCase (II) can further be divided into (IIa) where CDW order exists at $T=0$\nfor any $\\lambda$, and (IIb) with a disordered phase at $T=0$ below\n$\\lambda_c(\\omega_0)$. In scenario (I), the adiabatic (classical) fixed\npoint determines the behavior for any finite $\\omega_0$. On the other hand,\nin scenario (IIa), the physics is determined by the antiadiabatic fixed point\nfor $\\lambda<\\lambda_c(\\omega_0)$ and by the adiabatic fixed point for\n$\\lambda>\\lambda_c(\\omega_0)$. Note that CDW order with $T_c=0$ requires an\nemergent continuous order parameter, as realized for the attractive Hubbard\nmodel ($\\omega_0=\\infty$). However, the corresponding symmetry is broken for\n$\\omega_0<\\infty$ by retardation effects in the Holstein model \\cite{Hirsch83a}.\n\n\\begin{figure}[t]\n  \\includegraphics[width=0.25\\textwidth]{fig6.pdf}\n  \\caption{\\label{fig:tcschematic} The two possible scenarios for the phase\n    diagram of the Holstein model. In scenario (I), we have CDW order with\n    $T_c>0$ for any $\\lambda>0$. In scenario (II), $T_c=0$ for $\\lambda<\\lambda_c(\\omega_0)$.}\n\\end{figure}\n\nA CDW ground state for any $\\lambda>0$ may be expected based on the\ninstability of the Fermi liquid. For the half-filled square lattice with\nnearest-neighbor hopping, the noninteracting charge susceptibility\n$\\chi^{(0)}_\\text{c}({\\bm   Q})\\sim\\ln^2\\beta t$ due to the\ncombined effect of nesting and Van Hove singularities\n\\cite{PhysRevLett.56.2732,PhysRevB.42.2416}. In the Hubbard model, such\ndivergences underlie the existence of an antiferromagnetic Mott insulator\nfor any $U>0$, and coexisting CDW and superconducting order for any $U<0$ \\cite{Hirsch85}.\nFor the Holstein model that does not have a symmetry-imposed degeneracy of CDW and pairing\ncorrelations, superconducting correlations were found \nto be weaker than CDW correlations at half-filling \\cite{PhysRevB.42.2416},\nconsistent with the weaker divergence of the $\\bm{Q}=0$ pairing susceptibility\n$\\chi^{(0)}_\\text{p}({\\bm Q})\\sim\\ln\\beta t$.\n\nDespite these theoretical arguments, metallic and superconducting ground\nstates were recently suggested for the half-filled Holstein and\nHolstein-Hubbard models based on variational QMC simulations\n\\cite{ohgoe2017competitions,1709.00278}. A metallic phase is also found\nwithin DMFT \\cite{Koller04,PhysRevB.70.125114,PhysRevB.88.125126}, where a Van\nHove singularity is absent. For $\\omega_0\\ll t$, the results of\nFig.~\\ref{fig:phasediagram} appear consistent with CDW\norder even at $T=0$ for any $\\lambda>0$. On the other hand, the phase\nboundary $T_c(\\lambda)$ in Fig.~\\ref{fig:phasediagram} undergoes an\nincreasingly strong shift to larger $\\lambda$ with increasing\n$\\omega_0$, in principle compatible with $T_c=0$ at sufficiently weak\ncoupling [scenario (II)].  In the significantly better understood 1D case,\nnumerical results show that for $\\omega_0>0$ the ground state remains metallic\nfor $\\lambda<\\lambda_c$ despite a $\\ln \\beta t$ nesting-related\ndivergence of the charge susceptibility \\cite{MHHF2017}. Since $T_c=0$ in the\n1D case, this corresponds to scenario (IIb) above and is consistent with the\n$\\omega_0=\\infty$ limit, the 1D attractive Hubbard model. The latter has a\nmetallic but spin-gapped Luther-Emery liquid \\cite{Lu.Em.74} ground state and\nno long-range order. Functional renormalization group calculations for the\n2D Holstein-Hubbard model exclude metallic or superconducting behavior at\nhalf-filling except for an extremely small region where $T_c$ is essentially\nzero \\cite{PhysRevB.92.195102}.\n\nTo address the ground-state phase diagram directly, we calculated the\ncorrelation ratios\n\\begin{align}\\label{eq:Rchic}\n  R^\\chi_\\text{c} \n  &=  1-\\frac{\\chi_\\text{c}(\\bm{Q}-\\delta{\\bm\n    q})}{\\chi_\\text{c}(\\bm{Q})}\\,,\\quad\n    \\bm{Q}=(\\pi,\\pi)\\,,\n  \\\\\n  R^\\chi_\\text{p} \n  &=  1-\\frac{\\chi_\\text{c}(\\bm{Q}-\\delta{\\bm q})}{\\chi_\\text{c}(\\bm{Q})}\\,,\n    \\quad \\bm{Q} = (0,0)\\,.\n\\end{align}\nfor CDW and s-wave pairing based on the susceptibilities\n\\begin{align}\\label{eq:chic}\n  \\chi_\\text{c}(\\bm Q) \n  &= \\frac{1}{L^2} \\sum_{ij}\n  e^{\\text{i}(\\bm{r}_i-\\bm{r}_j)\\cdot\\bm{Q}} \\int_0^\\beta \\text{d} \\tau\n  \\las \\hat{n}_{i}(\\tau) \\hat{n}_{j}\\ras\\,,\n  \\\\\\label{eq:chip}\n  \\chi_\\text{p}(\\bm Q) \n  &= \\frac{1}{L^2} \\sum_{ij}\n  e^{\\text{i}(\\bm{r}_i-\\bm{r}_j)\\cdot\\bm{Q}} \\int_0^\\beta \\text{d} \\tau\n  \\las \\hat{\\Delta}^\\dag_{i}(\\tau) \\hat{\\Delta}^{\\phantom{\\dag}}_{j}\\ras\\,,\n\\end{align}\nwhere $\\hat{\\Delta}_i=c_{i\\UP}c_{i\\DO}$. The susceptibilities generally\nexhibit better finite-size scaling behavior than the corresponding static\nstructure factors [cf. Eq.~(\\ref{eq:scdw})]. We take a coupling\n$\\lambda=0.075$, for which Refs.~\\cite{ohgoe2017competitions,1709.00278}\nsuggest the absence of CDW order at $U=0$ over a large range of phonon\nfrequencies. The inverse temperature was scaled as $\\beta t=2L$ (with $4\\leq\nL\\leq 16$), which is at the current limit of the CT-INT method due to the sign problem. \n\n\\begin{figure}[t]\n \n  \\includegraphics[width=0.45\\textwidth]{fig7.pdf}\n  \\caption{\\label{fig:beta2L} (a) Charge and (b) pairing correlation ratios\n    for different phonon frequencies. Here, $\\beta t=2L$, $\\lambda=0.075$,\n    $U=0$.}\n\\end{figure}\n\n\\begin{figure}[t]\n \n  \\includegraphics[width=0.45\\textwidth]{fig8.pdf}    \n  \\caption{\\label{fig:beta2L_U} (a) Charge and (b) pairing correlation ratios\n    for different Hubbard repulsions. Here, $\\beta t=2L$, $\\lambda=0.075$,\n    $\\omega_0\/t=1$.}\n\\end{figure}\n\nThe correlation ratios shown in Figs.~\\ref{fig:beta2L} and~\\ref{fig:beta2L_U} \nhave the same properties as discussed in Sec.~\\ref{sec:results:Tc};\nlong-range order is revealed by $R^\\chi_\\alpha\\to 1$ for $L\\to\\infty$, and\na larger correlation ratio indicates stronger correlations in the\ncorresponding channel. For $\\om_0=0.1$, the results in Fig.~\\ref{fig:beta2L}(a)\nsuggest long-range CDW order, consistent with Fig.~\\ref{fig:phasediagram}.\nAt the same time, the pairing correlation ratio in Fig.~\\ref{fig:beta2L}(b)\nis strongly suppressed. Upon increasing $\\omega_0$, CDW correlations are\nsuppressed and pairing correlations enhanced, but\n$R^\\chi_\\text{c}>R^\\chi_\\text{p}$ for any $\\omega_0<\\infty$. Degenerate CDW\nand pairing correlations are only observed for the attractive Hubbard model\n($\\omega_0=\\infty$). \nThe fact that CDW correlations at $\\omega_0<\\infty$ are stronger\nthan for $\\omega_0=\\infty$ suggests a CDW ground state also for the Holstein\nmodel and likely no superconducting order since $T_c$ is already minimal for\n$\\omega_0=\\infty$. As demonstrated in Fig.~\\ref{fig:beta2L_U}, a nonzero\nHubbard repulsion suppresses both CDW and pairing correlations while\nenhancing antiferromagnetic correlations (not shown).\n\nFigure~\\ref{fig:beta2L} also reveals that in the weak-coupling regime where\nan absence of CDW order was predicted \\cite{ohgoe2017competitions,1709.00278},\nit is challenging to unequivocally detect the known $T=0$ long-range\norder of the attractive Hubbard model in terms of $R^\\chi_\\text{c},R^\\chi_\\text{p}\\to 1$ for\n$L\\to\\infty$. The same should be true for the Holstein and Holstein-Hubbard\nmodel in the regime where $T_c$ is small. Therefore, leaving aside the approximations inherent to\nvariational QMC methods, the reported absence of CDW order \n\\cite{ohgoe2017competitions,1709.00278} should also be taken with care.\n\nWhile we are unable to provide a definitive $T=0$ phase diagram, the\nresults of Fig.~\\ref{fig:beta2L} together with the\nobservation that long-range CDW order is known to exist at $T=0$ for both\n$\\om_0=0$ and $\\om_0=\\infty$ are consistent with CDW order but no\nsuperconductivity in the half-filled Holstein model at $T=0$. \nFurthermore, in the absence of a higher symmetry relating CDW and\nsuperconducting order as in the attractive Hubbard model, we expect $T_c>0$\n(although potentially exponentially small) and hence\nscenario (I) depicted in Fig.~\\ref{fig:tcschematic}.\n\n\\subsection{Bipolaron liquid}\\label{sec:results:bipolarons}\n\n\\begin{figure}[t]\n \n  \\includegraphics[width=0.45\\textwidth]{fig9.pdf}\n  \\caption{\\label{fig:atthubbard2} Local spin and charge susceptibilities\n    [Eq.~(\\ref{eq:localsusc})] for  $\\lambda=0.1$, $U=0$, and $L=8$. Open\n  symbols in (a) are for $\\lambda=0$,\n    arrows indicate maxima.}\n\\end{figure}\n\nA final interesting point is the\nnature of the metallic phase at $T>T_c$. In the CDW phase, spin, charge, and\nhence also single-particle excitations are gapped. For 1D electron-phonon models, the\nspin gap persists in the metallic phase \\cite{MHHF2017} and the $T=0$ CDW\ntransition occurs at the two-particle level via the ordering of preformed\npairs (singlet bipolarons) and the opening of a charge gap. The same is true for\nthe 2D attractive Hubbard model for which the spin gap can be made\narbitrarily large by increasing $U$ while keeping $T_c=0$. Hence, the\ndisordered phase at low but finite temperatures is not a Fermi liquid but a metal with gapped\nsingle-particle and spin excitations \\cite{PhysRevB.50.635,PhysRevLett.69.2001}, the 2D analog of a Luther-Emery liquid \\cite{Lu.Em.74}. Singlet bipolarons\nin principle also form for any $\\lambda>0$ in the 2D Holstein model, \nalthough their binding energy ($\\sim \\lambda$) can be small\n\\cite{PhysRevB.69.245111}. Nevertheless, we expect a spin-gapped metallic\nphase for suitable parameters. At\nsufficiently high temperatures, bipolarons undergo thermal\ndissociation \\cite{PhysRevB.71.184309}.\n\nTo detect signatures of a spin-gapped metal, we consider the static charge and spin susceptibilities\n\\begin{equation}\\label{eq:localsusc}\n\\chi_\\text{c} =\\beta (\\las \\hat{N}^2\\ras - \\las\\hat{N}\\ras^2), \\quad \n\\chi_\\text{s} =\\beta (\\las \\hat{M}^2\\ras - \\las\\hat{M}\\ras^2)\\quad \n\\end{equation}\nwith $\\hat{N} = \\sum_i \\hat{n}_i$, $\\hat{M} = \\sum_i \\hat{S}^x_i$.\nFigure~\\ref{fig:atthubbard2}(a) shows results for $\\lambda=0.1$ and\n$\\omega_0\/t=\\infty$. Whereas $\\chi_\\text{s}\/L^2$ diverges with decreasing\ntemperature in a Fermi liquid (open symbols), it is strongly suppressed \nas $T\\to0$ by the spin gap. The charge susceptibility is also suppressed at\nvery low $T$, but $\\chi_\\text{c}\/L^2$ approaches a finite value determined by\nthe density of $T=0$ charge fluctuations. The distinct temperature scales reflected by the maxima of $\\chi_\\text{s}\/L^2$ and\n$\\chi_\\text{c}\/L^2$ reveal the spin-gapped metallic phase at $T>0$ in the\nattractive Hubbard model. For the\nHolstein model, $\\chi_\\text{s}\/L^2$ is cut off by the spin\ngap, whereas $\\chi_\\text{c}\/L^2$ is cut off by the charge gap that\nappears at the CDW transition at $T=T_c$. The distinct maxima visible even in the\nadiabatic regime [Figs.~\\ref{fig:atthubbard2}(b)\nand~\\ref{fig:atthubbard2}(c)] are consistent with a spin-gapped phase at\n$T>T_c$. The extent of the latter appears to decrease with\ndecreasing $\\omega_0\/t$ and the phase is expected to be absent\nin the classical or mean-field limit ($\\omega_0=0$) where charge and spin gaps become\nequal. An immediate and important corollary of the existence of a spin-gapped\nmetal of bipolarons above $T_c$ would be that, contrary to expectations in previous work\n\\cite{PhysRevB.48.7643,PhysRevB.48.16011}, the appearance\nof a gap in the density of states does in general not imply CDW order.\nThe additional spin-gap component is also compatible with\nexperimentally observed large gap to $T_c$ ratios \\cite{PhysRevB.63.115114}.\n\nIn principle, a spin-gapped phase without long-range order (CDW or\nsuperconductivity) could also exist at $T=0$, but the discussion in\nSec.~\\ref{sec:results:phasediagram} provided arguments against a disordered phase.\nWhile well established in 1D electron-phonon models in terms of a Luther-Emery\nliquid \\cite{MHHF2017}, it would correspond to a so-called Bose metal\n\\cite{PhysRevB.60.1261} in higher dimensions. An interesting question\nregarding the recent findings of\nRefs.~\\cite{ohgoe2017competitions,1709.00278} is whether the variational wave\nfunctions used can distinguish between spin-gap formation and\nsuperconductivity. To this end, it would be useful to test this method for\nthe intricate but well understood 1D Holstein model.\n\n\\section{Conclusions}\\label{sec:conclusions}\n\nWe applied exact, continuous-time QMC simulations to the half-filled\nHolstein-Hubbard model on the square lattice. The critical temperature for\nthe CDW transition was determined as a function of phonon frequency,\nelectron-phonon coupling, and Hubbard repulsion from finite-size scaling. We\nalso demonstrated the expected 2D Ising universality of this transition and\naddressed the ground-state phase diagram, providing data and theoretical\narguments for the likely absence of a metallic or superconducting phase at\nweak coupling. Finally, we discussed the possibility of a spin-gapped metallic\nphase of bipolarons above $T_c$. The quantitative ground-state phase diagram\nremains an  important open problem.\n\n\\vspace*{1em}\n\\begin{acknowledgments}\nWe thank F. Assaad, N. Costa, P. Br\\\"ocker, T. Lang, and R. Scalettar for helpful discussions and the DFG for\nsupport via SFB 1170 and FOR~1807. We gratefully acknowledge the\ncomputing time granted by the John von Neumann Institute for Computing (NIC)\nand provided on the supercomputer JURECA \\cite{jureca} at the J\\\"{u}lich\nSupercomputing Centre.\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSince the first unambiguous discovery of an exoplanet in 1995 \\citep[][]{Mayor1995} over 4,000 more have been confirmed. Studies of their characteristics have unveiled an extremely wide range of planetary properties in terms of planetary mass, size, system architecture and orbital periods, greatly revolutionising our understanding of how these bodies form and evolve.\n\nThe transit method, whereby we observe a temporary decrease in the brightness of a star due to a planet passing in front of its host star, is to date the most successful method for planet detection, having discovered over 75\\% of the planets listed on the NASA Exoplanet Archive\\footnote{\\url{https:\/\/exoplanetarchive.ipac.caltech.edu\/}}. It yields a wealth of information including planet radius, orbital period, system orientation and potentially even atmospheric composition. Furthermore, when combined with Radial Velocity \\citep[RV; e.g.,][]{Mayor1995, Marcy1997} observations, which yield the planetary mass, we can infer planet densities, and thus their internal bulk compositions. Other indirect detection methods include radio pulsar timing \\citep[e.g.,][]{Wolszczan1992} and microlensing \\citep[e.g.,][]{Gaudi2012}.\n\n\nThe \\textit{Transiting Exoplanet Survey Satellite} mission \\citep[\\protect\\emph{TESS};][]{ricker15} is currently in its extended mission, searching for transiting planets orbiting bright ($V < 11$\\,mag) nearby stars. Over the course of the two year nominal mission, \\emph{TESS}\\ monitored around 85 per cent of the sky, split up into 26 rectangular sectors of 96 $\\times$ 24 deg each (13 per hemisphere). Each sector is monitored for $\\approx$ 27.4 continuous days, measuring the brightness of $\\approx$ 20,000 pre-selected stars every two minutes. In addition to these short cadence (SC) observations, the \\emph{TESS}\\ mission provides Full Frame Images (FFI) that span across all pixels of all CCDs and are taken at a cadence of 30 minutes. While most of the targets ($\\sim$ 63 per cent) will be observed for $\\approx$ 27.4 continuous days, around $\\sim$ 2 per cent of the targets at the ecliptic poles are located in the `continuous viewing zones' and will be continuously monitored for $\\sim$ 356 days.\n\nStars themselves are extremely complex, with phenomena ranging from outbursts to long and short term variability and oscillations, which manifest themselves in the light curves. These signals, as well as systematic effects and artifacts introduced by the telescope and instruments, mean that standard periodic search methods, such as the Box-Least-Squared method \\citep{bls2002} can struggle to identify certain transit events, especially if the observed signal is dominated by natural stellar variability. Standard detection pipelines also tend to bias the detection of short period planets, as they typically require a minimum of two transit events in order to gain the signal-to-noise ratio (SNR) required for detection.\n\nOne of the prime science goals of the \\emph{TESS}\\ mission is to further our understanding of the overall planet population, an active area of research that is strongly affected by observational and detection biases. In order for exoplanet population studies to be able to draw meaningful conclusions, they require a certain level of completeness in the sample of known exoplanets as well as a robust sample of validated planets spanning a wide range of parameter space. \\textcolor{black}{Due to this, we independently search the \\emph{TESS}\\ light curves for transiting planets via visual vetting in order to detect candidates that were either intentionally ignored by the main \\emph{TESS}\\ pipelines, which require at least two transits for a detection, missed because of stellar variability or instrumental artefacts, or were identified but subsequently erroneously discounted at the vetting stage, usually because the period found by the pipeline was incorrect. These candidates can help populate under-explored regions of parameter space and will, for example, benefit the study of planet occurrence rates around different stellar types as well as inform theories of physical processes involved with the formation and evolution of different types of exoplanets.}\n\nHuman brains excel in activities related to pattern recognition, making the task of identifying transiting events in light curves, even when the pattern is in the midst of a strong varying signal, ideally suited for visual vetting. Early citizen science projects, such as Planet Hunters \\citep[PH;][]{fischer12} and Exoplanet Explorers \\citep{Christiansen2018}, successfully harnessed the analytic power of a large number of volunteers and made substantial contributions to the field of exoplanet discoveries. The PH project, for example, showed that human vetting has a higher detection efficiency than automated detection algorithms for certain types of transits. In particular, they showed that citizen science can outperform on the detection of single (long-period) transits \\citep[e.g.,][]{wang13, schmitt14a}, aperiodic transits \\citep[e.g. circumbinary planets;][]{schwamb13} and planets around variable stars \\citep[e.g., young systems,][]{fischer12}. Both PH and Exoplanet Explorers, which are hosted by the world's largest citizen science platform Zooniverse \\citep{lintott08}, ensured easy access to \\textit{Kepler} and \\textit{K2} data by making them publicly available online in an immediately accessible graphical format that is easy to understand for non-specialists. The popularity of these projects is reflected in the number of participants, with PH attracting 144,466 volunteers from 137 different countries over 9 years of the project being active.\n\nFollowing the end of the \\textit{Kepler} mission and the launch of the \\emph{TESS}\\ satellite in 2018, PH was relaunched as the new citizen science project \\textit{Planet Hunters TESS} (PHT) \\footnote{\\url{www.planethunters.org}}, with the aim of identifying transit events in the \\emph{TESS}\\ data that were \\textcolor{black}{intentionally ignored or missed} by the main \\emph{TESS}\\ pipelines. \\textcolor{black}{Such a search complements other methods methods via its sensitivity to single-transit, and, therefore, longer period planets. Additionally, other dedicated non-citizen science based methods are also employed to look for single transit candidates \\citep[see e.g., the Bayesian transit fitting method by ][]{Gill2020, Osborn2016}}.\n\nCitizen science transit searches specialise in finding the rare events that the standard detection pipelines miss, however, these results are of limited use without an indication of the completeness of the search. Addressing the problem of completeness was therefore one of our highest priorities while designing PHT as discussed throughout this paper. \n\nThe layout of the remainder of the paper will be as follows. An overview of the Planet Hunters TESS project is found in Section~\\ref{sec:PHT}, followed by an in depth description of how the project identifies planet candidates in Section~\\ref{sec:method}. The recovery efficiency of the citizen science approach is assessed in Section~\\ref{sec:recovery_efficiency}, followed by a description of the in-depth vetting of candidates and ground based-follow up efforts in Section~\\ref{sec:vetting} and \\ref{sec:follow_up}, respectively. Planet Candidates and noteworthy systems identified by Planet Hunters TESS are outlined in Section~\\ref{sec:PHT_canidates}, followed by a discussion of the results in Section~\\ref{sec:condlusion}.\n\n\\section{Planet Hunters TESS}\n\\label{sec:PHT}\n\nThe PHT project works by displaying \\emph{TESS}\\ light curves (Figure~\\ref{fig:interface}), and asking volunteers to identify transit-like signals. Only the two-minute cadence targets, which are produced by the \\emph{TESS}\\ pipeline at the Science Processing Operations Center \\citep[SPOC,][]{Jenkins2018} and made publicly available by the Mikulski Archive for Space Telescopes (MAST)\\footnote{\\url{http:\/\/archive.stsci.edu\/tess\/}}, are searched by PHT. First-time visitors to the PHT site, or returning visitors who have not logged in are prompted to look through a short tutorial, which briefly explains the main aim of the project and shows examples of transit events and other stellar phenomena. Scientific explanation of the project can be found elsewhere on the site in the `field guide' and on the project's `About' page. \n\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=\\textwidth]{Figures\/PHT_new_interface.png}\n    \\caption{\n    PHT user interface showing a simulated light curve. The transit events are highlighted with white partially-transparent columns that are drawn on using the mouse. Stellar information on the target star is available by clicking on `subject info' below the light curve.} \n    \\label{fig:interface}\n\\end{figure*}\n\nAfter viewing the tutorial, volunteers are ready to participate in the project and are presented with \\emph{TESS}\\ light curves (known as `subjects') that need to be classified. The project was designed to be as simple as possible and therefore only asks one question: \\textit{`Do you see a transit?}'. Users identify transit-like events, and the time of their occurrence, by drawing a column over the event using the mouse button, as shown in Figure~\\ref{fig:interface}. There is no limit on the number of transit-like events that can be marked in a light curve. No markings indicate that there are no transit-like events present in the light curve. Once the subject has been analysed, users submit their classification and continue to view the next light curve by clicking `Done'. \n\nAlongside each light curve, users are offered information on the stellar properties of the target, such as the radius, effective temperature and magnitude (subject to availability, see \\cite{Stassun18}). However, in order to reduce biases in the classifications, the TESS Input Catalog (TIC) ID of the target star is not provided until after the subject classification has been submitted.\n\nIn addition to classifying the data, users are given the option to comment on light curves via the `Talk' discussion forum. Each light curve has its own discussion page to allow volunteers to discuss and comment, as well as to `tag' light curves using searchable hashtags, and to bring promising candidates to the attention of other users and the research team. The talk discussion forums complement the main PHT analysis and have been shown to yield interesting objects which may be challenging to detect using automated algorithms \\citep[e.g.,][]{eisner2019RN}. Unlike in the initial PH project, there are no questions in the main interface regarding stellar variability, however, volunteers are encouraged to mention astrophysical phenomenon or \\textit{unusual} features, such as eclipsing binaries or stellar flares, using the `Talk' discussion forum.  \n\nThe subject TIC IDs are revealed on the subject discussion pages, allowing volunteers to carry out further analysis on specific targets of interest and to report and discuss their findings. This is extremely valuable for both other volunteers and the PHT science team, as it can speed up the process of identifying candidates as well as rule out false positives in a fast and effective manner. \n\nSince the launch of PHT on 6 December 2018, there has been one significant makeover to the user interface. The initial PHT user interface (UI1), which was used for sectors 1 through 9, split the \\emph{TESS}\\ light curves up into either three or four chunks (depending on the data gaps in each sector) which lasted around seven days each. This allowed for a more `zoomed' in view of the data, making it easier to identify transit-like events than when the full $\\sim$ 30 day light curves were shown. The results from a PHT beta project, which displayed only simulated data, showed that a more zoomed in view of the light curve was likely to yield a higher transit recovery rate.\n\nThe updated, and current, user interface (UI2) allows users to manually zoom in on the x-axis (time) of the data. Due to this additional feature, each target has been displayed as a single light curve as of Sector 10. In order to verify that the changes in interface did not affect our findings, all of the Sector 9 subjects were classified using both UI1 and UI2. We saw no significant change in the number of candidates recovered (see Section~\\ref{sec:recovery_efficiency} for a description of how we quantified detection efficiency).\n\n\n\\subsection{Simulated Data}\n\\label{subsec:sims} \n\nIn addition to the real data, volunteers are shown simulated light curves, which are generated by randomly injecting simulated transit signals, provided by the SPOC pipeline \\citep[][]{Jenkins2018}, into real \\emph{TESS}\\ light curves. The simulated data play an important role in assessing the sensitivity of the project, training the users and providing immediate feedback, and to gauge the relative abilities of individual users (see Sec~\\ref{subsec:weighting}). \n\nWe calculate a signal to noise ratio (SNR) of the injected signal by dividing the injected transit depth by the Root Mean Square Combined Differential Photometric Precision (RMS CDPP) of the light curve on 0.5-, 1- or 2-hr time scales (whichever is closest to the duration of the injected transit signal). Only simulations with a SNR greater than 7 in UI1 and greater than 4 for UI2 are shown to volunteers.\n\nSimulated light curves are randomly shown to the volunteers and classified in the exact same manner as the real data. The user is always notified after a simulated light curve has been classified and given feedback as to whether the injected signal was correctly identified or not. For each sector, we generate between one and two thousand simulated light curves, using the real data from that sector in order to ensure that the sector specific systematic effects and data gaps of the simulated data do not differ from the real data. The rate at which a volunteer is shown simulated light curves decreases from an initial rate of 30 per cent for the first 10 classifications, down to a rate of 1 per cent by the time that the user has classified 100 light curves. \n\n\n\\section{Identifying Candidates}\n\\label{sec:method}\n\nEach subject is seen by multiple volunteers, before it is `retired' from the site, and the classifications are combined (see Section~\\ref{subsec:DBscan}) in order to assess the likelihood of a transit event. For sectors 1 through 9, the subjects were retired after 8 classifications if the first 8 volunteers who saw the light curves did not mark any transit events, after 10 classifications if the first 10 volunteers all marked a transit event and after 15 classifications if there was not complete consensus amongst the users. As of Sector 9 with UI2, all subjects were classified by 15 volunteers, regardless of whether or not any transit-like events were marked. Sector 9, which was classified with both UI1 and UI2, was also classified with both retirement rules.\n\nThere were a total of 12,617,038 individual classifications completed across the project on the nominal mission data. 95.4 per cent of these classifications were made by 22,341 registered volunteers, with the rest made by unregistered volunteers. Around 25 per cent of the registered volunteers complete more than 100 classifications, 11.8 per cent more than 300, 8.4 per cent more than 500, 5.4 per cent more than 1000 and 1.1 per cent more than 10,000. The registered volunteers completed a mean and median of 647 and 33 classifications, respectively. Figure~\\ref{fig:user_count} shows the distribution in user effort for logged in users who made between 0 and 300 classifications. \n\nThe distribution in the number of classifications made by the registered volunteers is assessed using the Gini coefficient, which ranges from 0 (equal contributions from all users) to 1 (large disparity in the contributions). The Gini coefficients for individual sectors ranges from 0.84 to 0.91 with a mean of 0.87, while the Gini coefficient for the overall project (all of the sectors combined) is 0.94. The mean Gini coefficient among other astronomy Zooniverse projects lies at 0.82 \\citep{spiers2019}. We note that the only other Zooniverse project with an equally high Gini coefficient as PHT is \\textit{Supernova Hunters}, a project which, similarly to PHT and unlike most other Zooniverse projects, has periodic data releases that are accompanied by an e-newsletter sent to all project volunteers. Periodic e-newsletters have the effect of promoting the project to both regularly and irregularly participating volunteers, who may only complete a couple of classifications as they explore the task, as well as to returning users who complete a large number of classifications following every data release, increasing the disparity in user contributions (the Gini coefficient).\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.5\\textwidth]{Figures\/user_count.png}\n    \\caption{\n    The distribution of the number of classifications by the registered volunteers, using a bin size of 5 from 0 to 300 classifications. A total of 11.8 per cent of the registered volunteers completed more than 300 classifications.} \n    \\label{fig:user_count}\n\\end{figure}\n\n\n\\subsection{User Weighting}\n\\label{subsec:weighting} \nUser weights are calculated for each individual volunteer in order to identify users who are more sensitive to detecting transit-like signals and those who are more likely to mark false positives. The weighting scheme is based on the weighting scheme described by \\cite{schwamb12}.\n\nUser weights are calculated independently for each observation sector, using the simulated light curves shown alongside the data from that sector. All users start off with a weighting of one, which is then increased or decreased when a simulated transit event is correctly or incorrectly identified, respectively. \n\nSimulated transits are deemed correctly identified, or `True', if the mid-point of a user's marking falls within the width of the simulated transit events. If none of the user's markings fall within this range, the simulated transit is deemed not identified, or `False'. If more than one of a user's markings coincide with the same simulated signal, it is only counted as being correct once, such that the total number of `True' markings cannot exceed the number of injected signals. For each classification, we record the number of  `Extra' markings, which is the total number of markings made by the user minus the number of correctly identified simulated transits. \n\nEach simulated light curve, identified by superscript $i$ (where $i=1$, \\ldots, $N$) was seen by $K^{(i)}$ users (the mean value of $K^{(i)}$\nwas 10), and contained $T^{(i)}$ simulated transits (where $T^{(i)}$ depends on the period of the simulated transit signal and the duration of the light curve). For a specific light curve $i$, each user who saw the light curve is identified by a subscript $k$ (where $k=1$, \\ldots, $K^{(i)}$) and each injected transit by a subscript $t$ (where $t=1$, \\ldots, $T^{(i)}$). \n\nIn order to distinguish between users who are able to identify obvious transits and those who are also able to find those that are more difficult to see, we start by defining a `recoverability' $r^{(i)}_t$ for each injected transit $t$ in each light curve. This is defined empirically, as the number of users who identified the transit correctly divided by $K^{(i)}$ (the total number of users who saw the light curve in question).\n\nNext, we quantify the performance of each user on each light curve as follows (this performance is analogous to the `seed' defined in \\citealt{schwamb12}, but we define it slightly differently):\n\\begin{equation}\n  p^{(i)}_{k} = C_{\\rm E} ~ \\frac{E^{(i)}_{k}}{\\langle E^{(i)} \\rangle} + \\sum_{t=1}^{T^{(i)}} \\begin{cases}\n    C_{\\rm T} ~ \\left[ r^{(i)}_t \\right]^{-1}, & \\text{if $m^{(i)}_{t,k} = $`True'}\\\\\n    C_{\\rm F} ~ r^{(i)}_t, & \\text{if $m^{(i)}_{t,k} = $`False'},\n  \\end{cases}\n\\end{equation}\nwhere $m^{(i)}_{t,k}$ is the identification of transit $t$ by user $k$ in light curve $i$, which is either `True' or `False'; $E^{(i)}_{k}$ is the number of `Extra' markings made by user $k$ for light curve $i$, and  $\\langle E^{(i)} \\rangle$ is the mean number of `Extra' markings made by all users who saw subject $i$. The parameters $C_{\\rm E}$, $C_{\\rm T}$ and $C_{\\rm F}$ control the impact of the `Extra', `True' and `False' markings on the overall user weightings, and are optimized empirically as discussed below in Section~\\ref{subsec:optimizesearch}. \n\nFollowing \\citealt{schwamb12}, we then assign a global `weight' $w_k$ to each user $k$, which is defined as:\n\\begin{equation}\n\\begin{split}\n\tw_k = I \\times (1 + \\log_{10} N_k)^{\\nicefrac{\\sum_i p^{(i)}_k}{N_k}}\n\\label{equ:weight}\n\\end{split}\n\\end{equation}\nwhere $I$ is an empirical normalization factor, such that the distribution of user weights remains centred on one, $N_k$ is the total number of simulated transit events that user $k$ assessed, and the sum over $i$ concerns only the light curves that user $k$ saw. \nWe limit the user weights to the range 0.05--3 \\emph{a posteriori}.\n\n\nWe experimented with a number of alternative ways to define the user weights, including the simpler $w_k=\\nicefrac{\\sum_i p^{(i)}_k}{N_k}$, but Eqn.~\\ref{equ:weight} was found to give the best results (see Section~\\ref{sec:recovery_efficiency} for how this was evaluated).\n\n\\subsection{Systematic Removal}\n\\label{subsec:sysrem} \nSystematic effects, for example caused by the spacecraft or background events, can result in spurious signals that affect a large subset of the data, resulting in an excess in markings of transit-like events at certain times within an observation sector. As the four \\emph{TESS}\\ cameras can yield unique systematic effects, the times of systematics were identified uniquely for each camera. The times were identified using a Kernel Density Estimation \\citep[KDE;][]{rosenblatt1956} with a cosine kernel and a bandwidth of 0.1 days, applied across all of the markings from that sector for each camera. Fig.~\\ref{fig:sys_rem} shows the KDE of all marked transit-events made during Sector 17 for TESS's cameras 1 (top panel) to 4 (bottom panel). The isolated spikes, or prominences, in the number of marked events, such as at T = 21-22 days in the bottom panel, are assumed to be caused by systematic effects that affect multiple light curves. Prominences are considered significant if they exceed a factor four times the standard deviation of the kernel output, which was empirically determined to be the highest cut-off to not miss clearly visible systematics. All user-markings within the full width at half maximum of these peaks are omitted from all further analysis. \\textcolor{black}{The KDE profiles for each Sector are provided as electronic supplementary material.}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.46\\textwidth]{Figures\/systematics_sec17.png}\n    \\caption{\n    Kernel density estimation of the user-markings made for Sector 17, for targets observed with TESS's observational Cameras 1 (top panel) to 4 (bottom panel). The orange vertical lines the indicate prominences that are at least four times greater than the standard deviation of the distribution. The black points underneath the figures show the mid-points of all of the volunteer-markings, where darker regions represent a higher density of markings.}\n    \\label{fig:sys_rem}\n\\end{figure}\n\n\\subsection{Density Based Clustering}\n\\label{subsec:DBscan} \n\nThe times and likelihoods of transit-like events are determined by combining all of the classifications made for each subject and identifying times where multiple volunteers identified a signal. We do this using an unsupervised machine learning method, known as DBSCAN \\citep[][Density-Based Spatial Clustering of Applications with Noise]{ester1996DB}. DBSCAN is a non-parametric density based clustering algorithm that helps to distinguish between dense clusters of data and sparse noise. For a data point to belong to a cluster it must be closer than a given distance ($\\epsilon$) to at least a set minimum number of other points (minPoints). \n\nIn our case, the data points are one-dimensional arrays of times of transits events, as identified by the volunteers, and clusters are times where multiple volunteers identified the same event. For each cluster a `transit score' ($s_i$) is determined, which is the sum of the user weights of the volunteers who contribute to the given cluster divided by the sum of the user weights of volunteers who saw that light curve. These transit scores are used to rank subjects from most to least likely to contain a transit-like event. Subjects which contain multiple successful clusters with different scores are ranked by the highest transit score. \n\n\\subsection{Optimizing the search}\n\\label{subsec:optimizesearch}\n\nThe methodology described in Sections~\\ref{subsec:weighting} to \\ref{subsec:DBscan} has five free parameters: the number of markings required to constitute a cluster ($minPoints$), the maximum separation of markings required for members of a cluster ($\\epsilon$), and $C_{\\rm E}$, $C_{\\rm T}$ and $C_{\\rm F}$ used in the weighting scheme. The values of these parameters were optimized via a grid search, where $C_{\\rm E}$ and $C_{\\rm F}$ ranged from -5 to 0, $C_{\\rm T}$ ranged from 0 to 20, and $minPoints$ ranged from 1 to 8, all in steps of 1. ($\\epsilon$) ranged from 0.5 to 1.5 in steps of 0.5. This grid search was carried out on 4 sectors, two from UI1 and two from UI2, for various variations of Equation~\\ref{equ:weight}. \n\nThe success of each combination of parameters was assessed by the fractions of TOIs and TCEs that were recovered within the top highest ranked 500 candidates, as discussed in more detail Section~\\ref{sec:recovery_efficiency}. We found the most successful combination of parameters to be $minPoints$ = 4 markings, $\\epsilon$, = 1 day, $C_{\\rm T}$ = 3, $C_{\\rm F}$= -2 and $C_{\\rm E}$ = -2.\n\n\\subsection{MAST deliverables}\n\\label{subsec:deliverables}\n\nThe analysis described above is carried out both in real-time as classifications are made, as well as offline after all of the light curves of a given sector have been classified. When the real-time analysis identifies a successful DB cluster (i.e. when at least four citizen scientists identified a transit within a day of the \\emph{TESS}\\ data of one another), the potential candidate is automatically uploaded to the open access Planet Hunters Analysis Database (PHAD) \\footnote{\\url{https:\/\/mast.stsci.edu\/phad\/}} hosted by the Mikulski Archive for Space Telescopes (MAST) \\footnote{\\url{https:\/\/archive.stsci.edu\/}}. While PHAD does not list every single classification made on PHT, it does display all transit candidates which had significant consensus amongst the volunteers who saw that light curve, along with the user-weight-weighted transit scores. This analysis does not apply the systematics removal described in Section~\\ref{subsec:sysrem}. The aim of PHAD is to provide an open source database of potential planet candidates identified by PHT, and to credit the volunteers who identified said targets. \n\nThe offline analysis is carried out following the complete classifications of all of the data from a given \\emph{TESS}\\ sector. The combination of all of the classifications allows us to identify and remove times of systematics and calculate better calibrated and more representative user weights. The remainder of this paper will only discuss the results from the offline analysis.\n\n\\section{Recovery Efficiency}\n\\label{sec:recovery_efficiency}\n\\subsection{Recovery of simulated transits}\n\nThe recovery efficiency is, in part, assessed by analysing the recovery rate of the injected transit-like signals (see Section~\\ref{subsec:sims}). Figure~\\ref{fig:SIM_recovery} shows the median and mean transit scores (fraction of volunteers who correctly identified a given transit scaled by user weights) of the simulated transits within SNR bins ranging from 4 to 20 in steps of 0.5. Simulations with a SNR less than 4 were not shown on PHT. The figure highlights that transit signals with a SNR of 7.5 or greater are correctly identified by the vast majority of volunteers. \n\n\\textcolor{black}{As the simulated data solely consist of real light curves with synthetically injected transit signals, we do not have any light curves, simulated or otherwise, which we can guarantee do not contain any planetary transits (real or injected). As such, this prohibits us from using simulated data to infer an analogous false-positive rate.}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.47\\textwidth]{Figures\/SIMS_recovery.png}\n    \\caption{The median (blue) and mean (orange) transit scores for injected transits with SNR ranges between 4 and 20. The mean and median are calculated in SNR bins with a width of 0.5, as indicated by the horizontal lines around each data point. \n    }\n    \\label{fig:SIM_recovery}\n\\end{figure}\n\n\\subsection{Recovery of TCEs and TOIs}\n\\label{subsec:TCE_TOI}\nThe recovery efficiency of PHT is assessed further using the planet candidates identified by the SPOC pipeline \\citep{Jenkins2018}. The SPOC pipeline extracts and processes all of the 2-minute cadence \\emph{TESS}\\ light curves prior to performing a large scale transit search. Data Validation (DV) reports, which include a range of transit diagnostic tests, are generated by the pipeline for around 1250 Threshold Crossing Events (TCEs), which were flagged as containing two or more transit-like features. Visual vetting is then performed by the \\emph{TESS}\\ science team on these targets, and promising candidates are added to the catalog of \\emph{TESS}\\ Objects of Interest (TOIs). Each sector yields around 80 TOIs \\textcolor{black}{and a mean of 1025 TCEs.}\n\nFig~\\ref{fig:TCE_TOI_recovery} shows the fraction of TOIs and TCEs (top and bottom panel respectively) that we recover with PHT as a function of the rank, where a higher rank corresponds to a lower transit score, for Sectors 1 through 26. TOIs and TCEs with R < 2 $R_{\\oplus}$ are not included in this analysis, as the initial PH showed that human vetting alone is unable to reliably recover planets smaller than 2 $R_{\\oplus}$ \\citep{schwamb12}. Planets smaller than 2 $R_{\\oplus}$ are, therefore, not the main focus of our search.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.45\\textwidth]{Figures\/TCE_TOI-recovery_radlim2.png}\n    \\caption{The fraction of recovered TOIs and TCEs (top and bottom panel respectively) with R > 2$R_{\\oplus}$ as a function of the rank, for sectors 1 to 26. The lines represent the results from different observation sectors.}\n    \\label{fig:TCE_TOI_recovery}\n\\end{figure}\n\n\nFig~\\ref{fig:TCE_TOI_recovery} shows a steep increase in the fractional TOI recovery rate up to a rank of $\\sim$ 500. Within the 500 highest ranked PHT candidates for a given sector, we are able to recover between 46 and 62 \\% (mean of 53 \\%) of all of the TOIs (R > 2 $R_{\\oplus}$), a median 90 \\% of the TOIs where the SNR of the transit events are greater than 7.5 and median 88 \\% of TOIs where the SNR of the transit events are greater than 5.\n\nThe relation between planet recovery rate and the SNR of the transit events is further highlighted in Figure~\\ref{fig:TOI_properties}, which shows the SNR vs the orbital period of the recovered TOIs. The colour of the markers indicate the TOI's rank within a given sector, with the lighter colours representing a lower rank. The circles and crosses represent candidates at a rank lower and higher than 500, respectively. The figure shows that transit events with a SNR less than 3.5 are missed by the majority of volunteers, whereas events with a SNR greater than 5 are mostly recovered within the top 500 highest ranked candidates. \n\nThe steep increase in the fractional TOI recovery rate at lower ranks, as shown in figure~\\ref{fig:TCE_TOI_recovery}, is therefore due to the detection of the high SNR candidates that are identified by most, if not all, of the PHT volunteers who classified those targets. At a rank of around 500, the SNR of the TOIs tends towards the limit of what human vetting can detect and thus the identification of TOIs beyond a rank of 500 is more sporadic.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.49\\textwidth]{Figures\/TOI_recovery_properties.png}\n    \\caption{The SNR vs orbital period of TOIs with R > 2$R_{\\oplus}$. The colour represents their rank within the sector, as determined by the weighted DB clustering algorithm. Circles indicate that they were identified at a rank < 500, while crosses indicate that they were not within the top 500 highest ranked candidates of a given sector.\n    }\n    \\label{fig:TOI_properties}\n\\end{figure}\n\nThe fractional TCE recovery rate (bottom panel of Figure~\\ref{fig:TCE_TOI_recovery}) is systematically lower than that of the TOIs. There are qualitative reasons as to why humans might not identify a TCE as opposed to a TOI, including that TCEs may be caused by artefacts or periodic stellar signals that the SPOC pipeline identified as a potential transit but that the human eye would either miss or be able to rule out as systematic effect. This leads to a lower recovery fraction of TCEs comparatively, an effect that is further amplified by the much larger number of TCEs.\n\nThe detection efficiency of PHT is estimated using the fractional recovery rate of TOIs for a range of radius and period bins, as shown in Figure~\\ref{fig:recovery_rank500_radius_period}. A TOI is considered to be recovered if its detection rank is less than 500 within the given sector. Out of the total 1913 TOIs, to date, \\textcolor{black}{PHT recovered 715 TOIs among the highest ranked candidates across the 26 sectors. This corresponds to a mean of 12.7~\\% of the top 500 ranked candidates per sector being TOIs. In comparison, the primary \\emph{TESS}\\ team on average visually vets 1025 TCEs per sector, out of which a mean of 17.3~\\% are promoted to TOI status.} We find that, independent of the orbital period, PHT is over 85~\\% complete in the recovery of TOIs with radii equal to or greater than 4 $R_{\\oplus}$. This agrees with the findings from the initial Planet Hunters project \\citep{schwamb12}. The detection efficiency decreases to 51~\\% for 3 - 4 $R_{\\oplus}$ TOIs, 49~\\% for 2 - 3 $R_{\\oplus}$ TOIs and to less than 40~\\% for TOIs with radii less than 2 $R_{\\oplus}$. Fig~\\ref{fig:recovery_rank500_radius_period} shows that the orbital period does not have a strong effect on the detection efficiency for periods greater than $\\sim$~1~day, which highlights that human vetting efficiency is independent of the number of transits present within a light curve. For periods shorter than around 1~day, the detection efficiency decreases even for larger planets, due to the high frequency of events seen in the light curve. For these light curves, many volunteers will only mark a subset of the transits, which may not overlap with the subset marked by other volunteers. Due to the methodology used to identify and rank the candidates, as described in Section~\\ref{sec:method}, this will actively disfavour the recovery of very short period planets. Although this obviously introduces biases in the detectability of very short period signals, the major detection pipelines are specifically designed to identify these types of planets and thus this does not present a serious detriment to our main science goal of finding planets that were \\textcolor{black}{intentionally ignored or missed} by the main automated pipelines.\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{Figures\/TOI_recovery_grid.png}\n    \\caption{TOI recovery rate as a function of planet radius and orbital period. A TOI is considered recovered if it is amongst the top 500 highest ranked candidates within a given sector. The logarithmically spaced grid ranges from 0.2 to 225 d and 0.6 to 55 $R{_\\oplus}$ for the orbital period and planet radius, respectively. The fraction of TOIs recovered using PHT is computed for each cell and represented by the colour the grid. Cells with less than 10 TOIs are considered incomplete for statistical analysis and are shown by the hatched lines. White cells contain no TOIs. The annotations for each cell indicate the number of recovered TOIs followed by the Poisson uncertainty in brackets. The filled in and empty grey circles indicated the recovered and not-recovered TOIs, respectively.}\n    \\label{fig:recovery_rank500_radius_period}\n\\end{figure*}\n\n\nFinally, we assessed whether the detection efficiency varies across different sectors by assessing the fraction of recovered TOIs and TCEs within the highest ranked 500 candidates. We found the recovery of TOIs within the top 500 highest ranked candidates to remain relatively constant across all sectors, while the fraction of recovered TCEs in the top 500 highest ranked candidates  increases in later sectors, as shown in Figure~\\ref{fig:recovery_rank500}). After applying a Spearman's rank test we find a positive correlation of 0.86 (pvalue = 5.9 $\\times$ $10^{-8}$) and 0.57 (pvalue = 0.003) between the observation sector and TCE and TOI recovery rates, respectively. These correlations suggest that the ability of users to detect transit-like events improves as they classify more subjects. The improvement of volunteers over time can also be seen in Fig~\\ref{fig:user_weights}, which shows the mean (unnormalized) user weight per sector for volunteers who completed one or more classifications in at least one sector (blue), more than 10 sectors (orange), more than 20 sectors (green) and all of the sectors 26 sectors from the nominal \\emph{TESS}\\ mission (pink). The figure highlights an overall improvement in the mean user weight in later sectors, as well as a positive correlation between the overall increase in user weight and the number of sectors that volunteers have participated in.\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.45\\textwidth]{Figures\/TCE_TOI_rank500.png}\n    \\caption{The fractional recovery rate of the TOIs (blue circles) and TCEs (teal squares) at a rank of 500 for each sector. Sector 1-9 (white background) represent southern hemisphere sectors classified with UI1, sectors 9-14 (light grey background) show the southern hemisphere sectors classified with UI2, and sectors 14-24 (dark grey background) show the northern hemisphere sectors classified with US2.}\n    \\label{fig:recovery_rank500}\n\\end{figure}\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.50\\textwidth]{Figures\/user_weights_sectors.png}\n    \\caption{Mean user weights per sector. The solid lines show the user weights for the old user interface and the dashed line for the new interface, separated by the black line (Sector 9). The different coloured lines show the mean user weights calculated considering user who participated in any number of sectors (blue), more than 10 sectors (orange), more than 20 sectors (green) and all of the sectors observed during the nominal \\emph{TESS}\\ mission (pink).}\n    \\label{fig:user_weights}\n\\end{figure}\n\n\n\\section{Candidate vetting}\n\\label{sec:vetting}\n\nFor each observation sector the subjects are ranked according to their transit scores, and the 500 highest ranked targets (excluding TOIs) visually vetted by the PHT science team in order to identify potential candidates and rule out false positives. A vetting cut-off rank of 500 was chosen as we found this to maximise the number of found candidates while minimising the number of likely false positives. In the initial round of vetting, which is completed via a separate Zooniverse classification interface that is only accessible to the core science team, a minimum of three members of the team sort the highest ranked targets into either `keep for further analysis', `eclipsing binary' or `discard'. The sorting is based on the inspection of the full \\emph{TESS}\\ light curve of the target, with the times of the satellite momentum dumps indicated. Additionally, around the time of each likely transit event (i.e. time of successful DB clusters) we inspect the background flux and the x and y centroid positions. Stellar parameters are provided for each candidate, subject to availability, alongside links to the SPOC Data Validation (DV) reports for candidates that had been flagged as TCEs but were never promoted to TOIs status.\n\nCandidates where at least two of the reviewers indicated that the signal is consistent with a planetary transit are kept for further analysis. \\textcolor{black}{This constitute a $\\sim$~5~\\% retention rate of the 500 highest ranked candidates per sector between the initial citizen science classification stage and the PHT science team vetting stage. Considering that the known planets and TOIs are not included at this stage of vetting, it is not surprising that our retention rate is lower that the true-positive rates of TCEs (see Section~\\ref{subsec:TCE_TOI}). Furthermore, this false-positive rate is consistent with the the findings of the initial Planet Hunters project \\citep{schwamb12}.}\n\nThe rest of the 500 candidates were grouped into $\\sim$~37~\\% `eclipsing binary' and $\\sim$~58~\\% `discard'. The most common reasons for discarding light curves are due to events caused by momentum dumps and due to background events, such as background eclipsing binaries, that mimic transit-like signals in the light curve. The targets identified as eclipsing binaries are analysed further by the \\emph{TESS}\\ Eclipsing Binaries Working Group (Prsa et al, in prep).\n\n\n\n\nFor the second round of candidate vetting we generate our own data validation reports for all candidates classified as `keep for further analysis'. The reports are generated using the open source software {\\sc latte} \\citep[Lightcurve Analysis Tool for Transiting Exoplanets;][]{LATTE2020}, which includes a range of standard diagnostic plots that are specifically designed to help identify transit-like signals and weed out astrophysical false positives in \\emph{TESS}\\ data. In brief the diagnostics consist of:\n\n\\textbf{Momentum Dumps}. The times of the \\emph{TESS}\\ reaction wheel momentum dumps that can result in instrumental effects that mimic astrophysical signals.\n\n\\textbf{Background Flux}. The background flux to help identify trends caused by background events such as asteroids or fireflies \\citep{vanderspek2018tess} passing through the field of view.\n\n\\textbf{x and y centroid positions}. The CCD column and row local position of the target's flux-weighted centroid, and the CCD column and row motion which considers differential velocity aberration (DVA), pointing drift, and thermal effects. This can help identify signals caused by systematics due to the satellite. \n\\textbf{Aperture size test}. The target light curve around the time of the transit-like event extracted using two apertures of different sizes. This can help identify signals resulting from background eclipsing binaries.\n \n\\textbf{Pixel-level centroid analysis}. A comparison between the average in-transit and average out-of-transit flux, as well as the difference between them. This can help identify signals resulting from background eclipsing binaries.\n\n\\textbf{Nearby companion stars}. The location of nearby stars brighter than V-band magnitude 15 as queried from the Gaia Data Release 2 catalog \\citep{gaia2018gaia} and the DSS2 red field of view around the target star in order to identify nearby contaminating sources. \n\n\\textbf{Nearest neighbour light curves}. Normalized flux light curves of the five short-cadence \\emph{TESS}\\ stars with the smallest projected distances to the target star, used to identify alternative sources of the signal or systematic effects that affect multiple target stars. \n\n\\textbf{Pixel level light curves}. Individual light curves extracted for each pixel around the target. Used to identify signals resulting from background eclipsing binaries, background events and systematics.\n\n\\textbf{Box-Least-Squares fit}. Results from two consecutive BLS searches, where the identified signals from the initial search are removed prior to the second BLS search.\n\nThe {\\sc latte} validation reports are assessed by the PHT science team in order to identify planetary candidates that warrant further investigation. Around 10~\\% of the targets assessed at this stage of vetting are kept for further investigation, resulting in $\\sim$~3 promising planet candidates per observation sector. The discarded candidates can be loosely categorized into (background) eclipsing binaries ($\\sim$~40~\\%), systematic effects ($\\sim$~25~\\%), background events ($\\sim$~15~\\%) and other (stellar signals such as spots; $\\sim$~10~\\%).\n\n\nWe use \\texttt{pyaneti}\\ \\citep{pyaneti} to infer the planetary and orbital parameters of our most promising candidates. For multi-transit candidates we fit for seven parameters per planet, time of mid-transit $T_0$, orbital period $P$, impact parameter $b$, scaled semi-major axis $a\/R_\\star$, scaled planet radius $r_{\\rm p}\/R_\\star$, and two limb darkening coefficients following a \\citet{Mandel2002} quadratic limb darkening model, implemented with the $q_1$ and $q_2$ parametrization suggested by \\citet{Kipping2013}. Orbits were assumed to be circular.\nFor the mono-transit candidates, we fit the same parameters as for the multi-transit case, except for the orbital period and scaled semi-major axis which cannot be known for single transits. We follow \\citet{Osborn2016} to estimate the orbital period of the mono-transit candidates assuming circular orbits.\n\nWe note that some of our candidates are V-shaped, consistent with a grazing transit configuration. For these cases, we set uniform priors between 0 and 0.15 for $r_{\\rm p}\/R_\\star$ and between 0 and 1.15 for the impact parameter in order to avoid large radii caused by the $r_{\\rm p}\/R_\\star - b$ degeneracy. Thus, the $r_{\\rm p}\/R_\\star$ for these candidates should not be trusted. A full characterisation of these grazing transits is out of the scope of this manuscript.\n\nFigure~\\ref{fig:PHT_pyaneti} shows the \\emph{TESS}\\ transits together with the inferred model for each candidate. Table~\\ref{tab:PHT-caniddates} shows the inferred main parameters, the values and their uncertainties are given by the median and 68.3\\% credible interval of the posterior distributions.\n\n\n\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=0.83\\textwidth]{Figures\/canidate_transits_all_one.png}\n    \\caption{All of the PHT candidates modelled using \\texttt{pyaneti}. The parameters of the best fits are summarised in Table~\\protect\\ref{tab:PHT-caniddates}. The blue and magenta fits show the multi and single transit event candidates, respectively.} \n    \\label{fig:PHT_pyaneti}\n\\end{figure*}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=0.83\\textwidth]{Figures\/canidate_transits_all_two.png}\n    \\addtocounter{figure}{-1}\n    \\caption{\\textbf{PHT candidates (continued)}} \n\\end{figure*}\n\n\nCandidates that pass all of our rounds of vetting are uploaded to the Exoplanet Follow-up Observing Program for TESS (ExoFOP-TESS) website\\footnote{\\url{ https:\/\/exofop.ipac.caltech.edu\/tess\/index.php}} as community TOIs (cTOIs).\n\n\\section{Follow-up observations}\n\\label{sec:follow_up}\n\nMany astrophysical false positive scenarios can be ruled out from the detailed examination of the \\emph{TESS}\\ data, both from the light curves themselves and from the target pixel files. However, not all of the false positive scenarios can be ruled out from these data alone, due in part to the large \\emph{TESS}\\ pixels (20 arcsconds). Our third stage of vetting, therefore, consists of following up the candidates with ground based observations including photometry, reconnaissance spectroscopy and speckle imaging. The results from these observations will be discussed in detail in a dedicated follow-up paper. \n\n\\subsection{Photometry}\n\nWe make use of the LCO global network of fully robotic 0.4-m\/SBIG and 1.0-m\/Sinistro facilities \\citep{LCO2013} to observe additional transits, where the orbital period is known, in order to refine the ephemeris and confirm that the transit events are not due to a blended eclipsing binary in the vicinity of the main target. Snapshot images are taken of single transit event candidates in order to identify nearby contaminating sources. \n\n\n\\subsection{Spectroscopy}\n\nWe perform high-resolution optical spectroscopy using telescopes from across the globe in order to cover a wide range of RA and Dec:\n\\begin{itemize}\n\\item The Las Cumbres Observatory (LCO) telescopes with the Network of Robotic Echelle Spectrographs \\citep[NRES,][]{LCO2013}. These fibre-fed spectrographs, mounted on 1.0-m telescopes around the globe, have a resolution of R = 53,000 and a wavelength coverage of 380 to 860 nm. \n\n\\item The MINERVA Australis Telescope facility, located at Mount Kent Observatory in Queensland, Australia \\citep{addison2019}. This facility is made up of four 0.7m CDK700 telescopes, which individually feed light via optic fibre into a KiwiSpec high-resolution (R = 80,000) stabilised spectrograph \\citep{barnes2012} that covers wavelengths from 480 nm to 620 nm. \n\n\\item The CHIRON spectrograph mounted on the SMARTS 1.5-m telescope \\citep{Tokovinin2018}, located at the Cerro Tololo\nInter-American Observatory (CTIO) in Chile. The high resolution cross-dispersed echelle spectrometer is fiber-fed followed by an image slicer. It has a resolution of R = 80,000 and covers wavelengths ranging from 410 to 870 nm.\n\n\\item The SOPHIE echelle spectrograph mounted on the 1.93-m Haute-Provence Observatory (OHP), France\n\\citep{2008Perruchot,2009Bouchy}. The high resolution cross-dispersed stabilized echelle spectrometer is fed by two optical fibers. Observations were taken in high-resolution mode (R = 75,000) with a wavelength range of 387 to 694 nm.\n\n\\end{itemize}\n\nReconnaissance spectroscopy with these instruments allow us to extract stellar parameters, identify spectroscopic binaries, and place upper limits on the companion masses. Spectroscopic binaries and targets whose spectral type is incompatible with the initial planet hypothesis and\/or precludes precision RV observations (giant or early type stars) are not followed up further. Promising targets, however, are monitored in order to constrain their period and place limits on their mass. \n\n\\subsection{Speckle Imaging}\n\nFor our most promising candidates we perform high resolution speckle imaging using the `Alopeke instrument on the 8.1-m Frederick C. Gillett Gemini North telescope in Maunakea, Hawaii, USA, and its twin, Zorro, on the 8.1-m Gemini South telescope on Cerro Pach\\'{o}n, Chile \\citep{Matson2019, Howell2011}. Speckle interferometric observations provide extremely high resolution images reaching the diffraction limit of the telescope. We obtain simultaneous 562 nm and 832 nm rapid exposure (60 msec) images in succession that effectively `freeze out' atmospheric turbulence and through Fourier analysis are used to search for close companion stars at 5-8 magnitude contrast levels. This analysis, along with the reconstructed images, allow us to identify nearby companions and to quantify their light contribution to the TESS aperture and thus the transit signal.\n\n\n\\section{Planet candidates and Noteworthy Systems}\n\\label{sec:PHT_canidates}\n\\subsection{Planet candidate properties}\n\nIn this final part of the paper we discuss the 90 PHT candidates around 88 host stars that passed the initial two stages of vetting and that were uploaded to ExoFOP as cTOIs. At the time of discovery none of these candidates were TOIs. The properties of all of the PHT candidates are summarised in Table~\\ref{tab:PHT-caniddates}. Candidates that have been promoted to TOI status since their PHT discovery are highlighted with an asterisk following the TIC ID, and candidates that have been shown to be false positives, based on the ground-based follow-up observations, are marked with a dagger symbol ($\\dagger$). The majority (81\\%) of PHT candidates are single transit events, indicated by an `s' following the orbital period presented in the table. \\textcolor{black}{18 of the PHT candidates were flagged as TCEs by the \\emph{TESS}\\ pipeline, but not initially promoted to TOI status. The most common reasons for this was that the pipeline identified a single-transit event as well as times of systematics (often caused by momentum dumps), due to its two-transit minimum detection threshold. This resulted in the candidate being discarded on the basis of it not passing the `odd-even' transit depth test. Out of the 18 TCEs, 14 have become TOI's since the PHT discovery. More detail on the TCE candidates can be found in Appendix~\\ref{appendixA}.} \n\nAll planet parameters (columns 2 to 8) are derived from the \\texttt{pyaneti}\\ modelling as described in Section~\\ref{sec:vetting}. Finally, the table summarises the ground-based follow-up observations (see Sec~\\ref{sec:follow_up}) that have been obtained to date, where the bracketed numbers following the observing instruments indicate the number of epochs. Unless otherwise noted, the follow-up observations are consistent with a planetary scenario. More in depth descriptions of individual targets for which we have additional information to complement the results in Table~\\ref{tab:PHT-caniddates} can be found in Appendix~\\ref{appendixA}.\n\n\\subsection{Planet candidate analysis}\n\n\nThe majority of the TOIs (87.7\\%) have orbital periods shorter than 15 days due to the requirement of observing at least two transits included in all major pipelines \\textcolor{black}{combined with the observing strategy of \\emph{TESS}}. As visual vetting does not impose these limits, the candidates outlined in this paper are helping to populate the relatively under-explored long-period region of parameter space. This is highlighted in Figure~\\ref{fig:PHT_candidates}, which shows the transit depths vs the orbital periods of the PHT single transit candidates (orange circles) and the multi-transit candidates (magenta squares) compared to the TOIs (blue circles). Values of the orbital periods and transit depths were obtained via transit modelling using \\texttt{pyaneti} (see Section~\\ref{sec:vetting}). The orbital period of single transit events are poorly constrained, which is reflected by the large errorbars in Figure~\\ref{fig:PHT_candidates}. Figure~\\ref{fig:PHT_candidates} also highlights that with PHT we are able to recover a similar range of transit depths as the pipeline found TOIs, as was previously shown in Figure~\\ref{fig:recovery_rank500_radius_period}.\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.45\\textwidth]{Figures\/PHT_candidate_period_depth_plot_errobrars.png}\n    \\caption{The properties of the PHT single transit (orange circles) and multi transit (magenta squares) candidates compared to the properties of the TOIs (blue circles). All parameters (listed in Table~\\ref{fig:PHT_candidates}) were extracted using \\texttt{pyaneti}\\ modelling.}\n    \\label{fig:PHT_candidates}\n\\end{figure}\n\nThe PHT candidates were further compared to the TOIs in terms of the properties of their host stars. Figure~\\ref{fig:eep} shows the effective temperature and stellar radii as taken from the TIC \\citep{Stassun18}, for TOIs (blue dots) and the PHT candidates (magenta circles). The solid and dashed lines indicate the main sequence and post-main sequence MIST stellar evolutionary tracks \\citep{choi2016}, respectively, for stellar masses ranging from 0.3 to 1.6 $M_\\odot$ in steps of 0.1 $M_\\odot$. This shows that around 10\\% of the host stars are in the process of, or have recently evolved off the main sequence. The models assume solar metalicity, no stellar rotation and no additional internal mixing.\n\n\\textcolor{black}{Ground based follow-up spectroscopy has revealed that six of the PHT candidates listed in Table~\\ref{tab:PHT-caniddates} are astrophysical false positives. As the follow-up campaign of the targets is still underway, the true false-positive rate of the candidates to have made it through all stages of the vetting process, as outlined in the methodology, will be be assessed in future PHT papers once the true nature of more of the candidates has been independently verified.}\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.45\\textwidth]{Figures\/PHT_eep.png}\n    \\caption{Stellar evolution tracks showing main sequence (solid black lines) and post-main sequence (dashed grey lines) MIST stellar evolution for stellar masses ranging from 0.3 to 1.6 $M_\\odot$ in steps of 0.1 $M_\\odot$. The blue dots show the TOIs and the magenta circles show the PHT candidates.} \n    \\label{fig:eep}\n\\end{figure}\n\n\n\\subsection{Stellar systems}\n\\label{subsec:PHT_stars}\n\nIn addition to the planetary candidates, citizen science allows for the identification of interesting stellar systems and astrophysical phenomena, in particular where the signals are aperiodic or small compared to the dominant stellar signal. These include light curves that exhibit multiple transit-like signals, possibly as a result of a multiple stellar system or a blend of eclipsing binaries. We have investigated all light curves that were flagged as possible multi-stellar systems via the PHT discussion boards. Similar to the planet vetting, as described in Section~\\ref{sec:vetting}, we generated {\\sc latte} data validation reports in order to assess the nature of the signal. Additionally, we subjected these systems to an iterative signal removal process, whereby we phase-folded the light curve on the dominant orbital period, binned the light curve into between 200-500 phase bins, created an interpolation model, and then subtracted said signal in order to evaluate the individual transit signals. The period of each signal, as listed in Table~\\ref{tab:PHT-multis}, was determined by phase folding the light curve at a number of trial periods and assessing by eye the best fit period and corresponding uncertainty.\n\nDue to the large \\emph{TESS}\\ pixels, blends are expected to be common. We searched for blends by generating phase folded light curves for each pixel around the source of the target in order to better locate the source of each signal. Shifts in the \\emph{TESS}\\ x and y centroid positions were also found to be good indicators of visually separated sources. Nearby sources with a magnitude difference greater than 5 mags were ruled out as possible contaminators. We consider a candidate to be a confirmed blend when the centroids are separated by more than 1 \\emph{TESS}\\ pixel, as this corresponds to an angular separation > 21 arcseconds meaning that the systems are highly unlikely to be gravitationally bound. Systems where the signal appears to be coming from the same \\emph{TESS}\\ pixel and that show no clear centroid shifts are considered to be candidate multiple systems. We note that blends are still possible, however, without further investigation we cannot conclusively rule these out as possible multi stellar systems. \n\nAll of the systems are summarised in Table~\\ref{tab:PHT-multis}. Out of the 26 systems, 6 are confirmed multiple systems which have either been published or are being prepared for publication; 7 are visually separated eclipsing binaries (confirmed blends); and 13 are candidate multiple system. Additional observations will be required to determine whether or not these candidate multiple systems are in fact gravitationally bound or photometric blends as a results of the large \\emph{TESS}\\ pixels or due to a line of sight happenstance. \n\n\\begin{landscape}\n\\begin{table}\n\\resizebox{1.31\\textwidth}{!}{\n\\begin{tabular}{ccccccccccccc}\n\\textbf{TIC} & \\textbf{Other} & \\textbf{Epoch} & \\textbf{Period}  & \\textbf{$R_{pl}$\/$R_{\\odot}$} & \\textbf{$R_{pl}$} & \\textbf{Impact} & \\textbf{Duration} & \\textbf{$V_{mag}$} & \\textbf{Photometry} & \\textbf{Spectroscopy} & \\textbf{Speckle} & \\textbf{Comment} \\\\\n  & \\textbf{Name} & \\textbf{(\\textcolor{black}{BJD - 2457000})} & \\textbf{(days)}  & &  $(R_{\\oplus})$ & \\textbf{Parameter} & \\textbf{(hours)} &   &   &   &   &    \\\\\n\\hline\n101641905 & TWOMASS 11412617+3441004 & $1917.26335 _{ - 0.00072 } ^ { + 0.00071 }$ & $14.52 _{ - 5.25 } ^ { + 6.21 }(s)$ & $0.1135 _{ - 0.0064 } ^ { + 0.0032 }$ & $9.76 _{ - 0.69 } ^ { + 0.65 }$ & $0.691 _{ - 0.183 } ^ { + 0.077 }$ & $3.163 _{ - 0.088 } ^ { + 0.093 }$ & 12.196 &  &  &  &  \\\\\n103633672* & TYC 4387-00923-1 & $1850.3211 _{ - 0.00077 } ^ { + 0.00135 }$ & $90.9 _{ - 23.7 } ^ { + 46.4 }(s)$ & $0.0395 _{ - 0.0013 } ^ { + 0.0013 }$ & $3.45 _{ - 0.24 } ^ { + 0.26 }$ & $0.3 _{ - 0.21 } ^ { + 0.26 }$ & $6.7 _{ - 0.11 } ^ { + 0.12 }$ & 10.586 &  & NRES (1) &  &  \\\\\n110996418 & TWOMASS 12344723-1019107 & $1580.6406 _{ - 0.0038 } ^ { + 0.0037 }$ & $5.18 _{ - 2.93 } ^ { + 6.86 }(s)$ & $0.1044 _{ - 0.0067 } ^ { + 0.008 }$ & $12.7 _{ - 0.99 } ^ { + 1.15 }$ & $0.44 _{ - 0.3 } ^ { + 0.3 }$ & $3.53 _{ - 0.27 } ^ { + 0.36 }$ & 13.945 &  &  &  &  \\\\\n128703021 & HIP 71639 & $1601.8442 _{ - 0.00108 } ^ { + 0.00093 }$ & $26.0 _{ - 8.22 } ^ { + 22.35 }(s)$ & $0.0254 _{ - 0.00049 } ^ { + 0.00072 }$ & $4.44 _{ - 0.2 } ^ { + 0.23 }$ & $0.47 _{ - 0.3 } ^ { + 0.22 }$ & $7.283 _{ - 0.091 } ^ { + 0.141 }$ & 6.06 &  & NRES (2);MINERVA (34) & Gemini &  \\\\\n138126035 & TYC 1450-00833-1 & $1954.3229 _{ - 0.0041 } ^ { + 0.0067 }$ & $28.8 _{ - 14.0 } ^ { + 203.2 }(s)$ & $0.0375 _{ - 0.0026 } ^ { + 0.0069 }$ & $4.01 _{ - 0.35 } ^ { + 0.74 }$ & $0.58 _{ - 0.38 } ^ { + 0.35 }$ & $4.65 _{ - 0.32 } ^ { + 0.85 }$ & 10.349 &  &  &  &  \\\\\n142087638 & TYC 9189-00274-1 & $1512.1673 _{ - 0.0043 } ^ { + 0.0034 }$ & $3.14 _{ - 1.41 } ^ { + 12.04 }(s)$ & $0.0469 _{ - 0.0035 } ^ { + 0.0063 }$ & $6.05 _{ - 0.54 } ^ { + 0.89 }$ & $0.5 _{ - 0.35 } ^ { + 0.36 }$ & $2.72 _{ - 0.23 } ^ { + 0.5 }$ & 11.526 &  &  &  &  \\\\\n159159904 & HIP 64812 & $1918.6109 _{ - 0.0067 } ^ { + 0.0091 }$ & $584.0 _{ - 215.0 } ^ { + 1724.0 }(s)$ & $0.0237 _{ - 0.0011 } ^ { + 0.0026 }$ & $3.12 _{ - 0.22 } ^ { + 0.36 }$ & $0.49 _{ - 0.34 } ^ { + 0.35 }$ & $15.11 _{ - 0.54 } ^ { + 0.7 }$ & 9.2 &  & NRES (2) &  &  \\\\\n160039081* & HIP 78892 & $1752.9261 _{ - 0.0045 } ^ { + 0.005 }$ & $30.19918 _{ - 0.00099 } ^ { + 0.00094 }$ & $0.0211 _{ - 0.0013 } ^ { + 0.0035 }$ & $2.67 _{ - 0.21 } ^ { + 0.43 }$ & $0.52 _{ - 0.34 } ^ { + 0.36 }$ & $4.93 _{ - 0.27 } ^ { + 0.37 }$ & 8.35 & SBIG (1) & NRES (1);SOPHIE (4) & Gemini &  \\\\\n162631539 & HIP 80264 & $1978.2794 _{ - 0.0044 } ^ { + 0.0051 }$ & $17.32 _{ - 6.66 } ^ { + 52.35 }(s)$ & $0.0195 _{ - 0.0011 } ^ { + 0.0024 }$ & $2.94 _{ - 0.24 } ^ { + 0.38 }$ & $0.48 _{ - 0.33 } ^ { + 0.36 }$ & $5.54 _{ - 0.33 } ^ { + 0.41 }$ & 7.42 &  &  &  &  \\\\\n166184426* & TWOMASS 13442500-4020122 & $1600.4409 _{ - 0.003 } ^ { + 0.0036 }$ & $16.3325 _{ - 0.0066 } ^ { + 0.0052 }$ & $0.0545 _{ - 0.0031 } ^ { + 0.0039 }$ & $1.85 _{ - 0.12 } ^ { + 0.15 }$ & $0.41 _{ - 0.28 } ^ { + 0.31 }$ & $1.98 _{ - 0.22 } ^ { + 0.17 }$ & 12.911 &  &  &  &  \\\\\n167661160$\\dagger$ & TYC 7054-01577-1 & $1442.0703 _{ - 0.0028 } ^ { + 0.004 }$ & $36.802 _{ - 0.07 } ^ { + 0.069 }$ & $0.0307 _{ - 0.0014 } ^ { + 0.0024 }$ & $4.07 _{ - 0.32 } ^ { + 0.43 }$ & $0.37 _{ - 0.26 } ^ { + 0.33 }$ & $5.09 _{ - 0.23 } ^ { + 0.21 }$ & 9.927 &  & NRES (9);MINERVA (4) &  & EB from MINERVA observations \\\\\n172370679* & TWOMASS 19574239+4008357 & $1711.95923 _{ - 0.00099 } ^ { + 0.001 }$ & $32.84 _{ - 4.17 } ^ { + 5.59 }(s)$ & $0.1968 _{ - 0.0032 } ^ { + 0.0022 }$ & $13.24 _{ - 0.43 } ^ { + 0.43 }$ & $0.22 _{ - 0.15 } ^ { + 0.14 }$ & $4.999 _{ - 0.097 } ^ { + 0.111 }$ & 14.88 &  &  &  & Confirmed planet \\citep{canas2020}. \\\\\n174302697* & TYC 3641-01789-1 & $1743.7267 _{ - 0.00092 } ^ { + 0.00093 }$ & $498.2 _{ - 80.0 } ^ { + 95.3 }(s)$ & $0.07622 _{ - 0.00068 } ^ { + 0.00063 }$ & $13.34 _{ - 0.57 } ^ { + 0.58 }$ & $0.642 _{ - 0.029 } ^ { + 0.024 }$ & $17.71 _{ - 0.12 } ^ { + 0.13 }$ & 9.309 & SBIG (1) &  &  &  \\\\\n179582003 & TYC 9166-00745-1 & $1518.4688 _{ - 0.0016 } ^ { + 0.0016 }$ & $104.6137 _{ - 0.0022 } ^ { + 0.0022 }$ & $0.06324 _{ - 0.0008 } ^ { + 0.0008 }$ & $7.51 _{ - 0.35 } ^ { + 0.35 }$ & $0.21 _{ - 0.15 } ^ { + 0.19 }$ & $9.073 _{ - 0.084 } ^ { + 0.097 }$ & 10.806 &  &  &  &  \\\\\n192415680 & TYC 2859-00682-1 & $1796.0265 _{ - 0.0012 } ^ { + 0.0013 }$ & $18.47 _{ - 6.34 } ^ { + 21.73 }(s)$ & $0.0478 _{ - 0.0017 } ^ { + 0.0027 }$ & $4.43 _{ - 0.33 } ^ { + 0.38 }$ & $0.45 _{ - 0.31 } ^ { + 0.31 }$ & $3.94 _{ - 0.1 } ^ { + 0.12 }$ & 9.838 & SBIG (1) & SOPHIE (2) &  &  \\\\\n192790476 & TYC 7595-00649-1 & $1452.3341 _{ - 0.0014 } ^ { + 0.002 }$ & $16.09 _{ - 5.73 } ^ { + 15.49 }(s)$ & $0.0438 _{ - 0.0018 } ^ { + 0.0026 }$ & $3.24 _{ - 0.34 } ^ { + 0.37 }$ & $0.37 _{ - 0.25 } ^ { + 0.3 }$ & $3.395 _{ - 0.099 } ^ { + 0.192 }$ & 10.772 &  &  &  &  \\\\\n206361691$\\dagger$ & HIP 117250 & $1363.2224 _{ - 0.0082 } ^ { + 0.009 }$ & $237.7 _{ - 81.0 } ^ { + 314.4 }(s)$ & $0.01762 _{ - 0.00088 } ^ { + 0.00125 }$ & $2.69 _{ - 0.19 } ^ { + 0.25 }$ & $0.43 _{ - 0.28 } ^ { + 0.32 }$ & $13.91 _{ - 0.53 } ^ { + 0.52 }$ & 8.88 &  & CHIRON (2) &  & SB2 from CHIRON \\\\\n207501148 & TYC 3881-00527-1 & $2007.7273 _{ - 0.0011 } ^ { + 0.0011 }$ & $39.9 _{ - 10.3 } ^ { + 14.3 }(s)$ & $0.0981 _{ - 0.0047 } ^ { + 0.011 }$ & $13.31 _{ - 0.95 } ^ { + 1.56 }$ & $0.9 _{ - 0.03 } ^ { + 0.039 }$ & $4.73 _{ - 0.14 } ^ { + 0.14 }$ & 10.385 &  &  &  &  \\\\\n219466784* & TYC 4409-00437-1 & $1872.6879 _{ - 0.0097 } ^ { + 0.0108 }$ & $318.0 _{ - 147.0 } ^ { + 1448.0 }(s)$ & $0.0332 _{ - 0.0024 } ^ { + 0.0048 }$ & $3.26 _{ - 0.31 } ^ { + 0.49 }$ & $0.55 _{ - 0.39 } ^ { + 0.34 }$ & $10.06 _{ - 0.81 } ^ { + 1.12 }$ & 11.099 &  &  &  &  \\\\\n219501568 & HIP 79876 & $1961.7879 _{ - 0.0018 } ^ { + 0.002 }$ & $16.5931 _{ - 0.0017 } ^ { + 0.0015 }$ & $0.0221 _{ - 0.0012 } ^ { + 0.0015 }$ & $4.22 _{ - 0.3 } ^ { + 0.35 }$ & $0.41 _{ - 0.28 } ^ { + 0.31 }$ & $1.615 _{ - 0.077 } ^ { + 0.093 }$ & 8.38 &  &  &  &  \\\\\n229055790 & TYC 7492-01197-1 & $1337.866 _{ - 0.0022 } ^ { + 0.0019 }$ & $48.0 _{ - 12.8 } ^ { + 48.4 }(s)$ & $0.0304 _{ - 0.00097 } ^ { + 0.00115 }$ & $3.52 _{ - 0.2 } ^ { + 0.24 }$ & $0.37 _{ - 0.26 } ^ { + 0.32 }$ & $6.53 _{ - 0.11 } ^ { + 0.14 }$ & 9.642 &  & NRES (2) &  &  \\\\\n229608594 & TWOMASS 18180283+7428005 & $1960.0319 _{ - 0.0037 } ^ { + 0.0045 }$ & $152.4 _{ - 54.1 } ^ { + 152.6 }(s)$ & $0.0474 _{ - 0.0023 } ^ { + 0.0024 }$ & $3.42 _{ - 0.34 } ^ { + 0.36 }$ & $0.38 _{ - 0.26 } ^ { + 0.3 }$ & $6.98 _{ - 0.23 } ^ { + 0.37 }$ & 12.302 &  &  &  &  \\\\\n229742722* & TYC 4434-00596-1 & $1689.688 _{ - 0.025 } ^ { + 0.02 }$ & $29.0 _{ - 16.4 } ^ { + 66.3 }(s)$ & $0.019 _{ - 0.0028 } ^ { + 0.0029 }$ & $2.9 _{ - 0.44 } ^ { + 0.48 }$ & $0.44 _{ - 0.3 } ^ { + 0.33 }$ & $4.27 _{ - 0.09 } ^ { + 0.11 }$ & 10.33 &  & NRES (8);SOPHIE (4) & Gemini &  \\\\\n233194447 & TYC 4211-00650-1 & $1770.4924 _{ - 0.0065 } ^ { + 0.0107 }$ & $373.0 _{ - 101.0 } ^ { + 284.0 }(s)$ & $0.02121 _{ - 0.00073 } ^ { + 0.001 }$ & $5.08 _{ - 0.28 } ^ { + 0.33 }$ & $0.34 _{ - 0.24 } ^ { + 0.29 }$ & $24.45 _{ - 0.47 } ^ { + 0.5 }$ & 9.178 &  & NRES (2) & Gemini &  \\\\\n235943205 & TYC 4588-00127-1 & $1827.0267 _{ - 0.004 } ^ { + 0.0034 }$ & $121.3394 _{ - 0.0063 } ^ { + 0.0065 }$ & $0.0402 _{ - 0.0016 } ^ { + 0.0019 }$ & $4.2 _{ - 0.25 } ^ { + 0.29 }$ & $0.4 _{ - 0.27 } ^ { + 0.28 }$ & $6.37 _{ - 0.2 } ^ { + 0.3 }$ & 11.076 &  & NRES (1);SOPHIE (2) &  &  \\\\\n237201858 & TYC 4452-00759-1 & $1811.5032 _{ - 0.0069 } ^ { + 0.0067 }$ & $129.7 _{ - 41.5 } ^ { + 146.8 }(s)$ & $0.0258 _{ - 0.0013 } ^ { + 0.0015 }$ & $4.12 _{ - 0.27 } ^ { + 0.3 }$ & $0.4 _{ - 0.28 } ^ { + 0.31 }$ & $10.94 _{ - 0.37 } ^ { + 0.53 }$ & 10.344 &  & NRES (1) &  &  \\\\\n243187830* & HIP 5286 & $1783.7671 _{ - 0.0017 } ^ { + 0.0019 }$ & $4.05 _{ - 1.53 } ^ { + 9.21 }(s)$ & $0.0268 _{ - 0.0015 } ^ { + 0.0027 }$ & $2.06 _{ - 0.17 } ^ { + 0.23 }$ & $0.47 _{ - 0.32 } ^ { + 0.34 }$ & $2.02 _{ - 0.12 } ^ { + 0.15 }$ & 8.407 & SBIG (1) &  &  &  \\\\\n243417115 & TYC 8262-02120-1 & $1614.4796 _{ - 0.0028 } ^ { + 0.0022 }$ & $1.81 _{ - 0.73 } ^ { + 3.45 }(s)$ & $0.0523 _{ - 0.0035 } ^ { + 0.005 }$ & $5.39 _{ - 0.47 } ^ { + 0.64 }$ & $0.47 _{ - 0.33 } ^ { + 0.34 }$ & $2.03 _{ - 0.16 } ^ { + 0.23 }$ & 11.553 &  &  &  &  \\\\\n256429408 & TYC 4462-01942-1 & $1962.16 _{ - 0.0022 } ^ { + 0.0023 }$ & $382.0 _{ - 132.0 } ^ { + 265.0 }(s)$ & $0.03582 _{ - 0.00086 } ^ { + 0.00094 }$ & $6.12 _{ - 0.29 } ^ { + 0.3 }$ & $0.51 _{ - 0.36 } ^ { + 0.18 }$ & $16.96 _{ - 0.2 } ^ { + 0.24 }$ & 8.898 &  &  &  &  \\\\\n264544388* & TYC 4607-01275-1 & $1824.8438 _{ - 0.0076 } ^ { + 0.0078 }$ & $7030.0 _{ - 6260.0 } ^ { + 3330.0 }(s)$ & $0.0288 _{ - 0.0029 } ^ { + 0.0018 }$ & $4.58 _{ - 0.43 } ^ { + 0.35 }$ & $0.936 _{ - 0.363 } ^ { + 0.011 }$ & $19.13 _{ - 1.35 } ^ { + 0.84 }$ & 8.758 &  & NRES (1) &  &  \\\\\n264766922 & TYC 8565-01780-1 & $1538.69518 _{ - 0.00091 } ^ { + 0.00091 }$ & $3.28 _{ - 0.94 } ^ { + 1.25 }(s)$ & $0.0933 _{ - 0.0063 } ^ { + 0.0176 }$ & $16.95 _{ - 1.33 } ^ { + 3.19 }$ & $0.908 _{ - 0.039 } ^ { + 0.048 }$ & $2.73 _{ - 0.11 } ^ { + 0.11 }$ & 10.747 &  &  &  &  \\\\\n26547036* & TYC 3921-01563-1 & $1712.30464 _{ - 0.00041 } ^ { + 0.0004 }$ & $73.0 _{ - 13.6 } ^ { + 16.5 }(s)$ & $0.10034 _{ - 0.0007 } ^ { + 0.00078 }$ & $11.75 _{ - 0.59 } ^ { + 0.58 }$ & $0.17 _{ - 0.12 } ^ { + 0.11 }$ & $8.681 _{ - 0.049 } ^ { + 0.052 }$ & 9.849 &  & NRES (4) & Gemini &  \\\\\n267542728$\\dagger$ & TYC 4583-01499-1 & $1708.4956 _{ - 0.0073 } ^ { + 0.0085 }$ & $39.7382 _{ - 0.0023 } ^ { + 0.0023 }$ & $0.03267 _{ - 0.00089 } ^ { + 0.00175 }$ & $18.46 _{ - 0.94 } ^ { + 1.14 }$ & $0.38 _{ - 0.26 } ^ { + 0.27 }$ & $24.16 _{ - 0.39 } ^ { + 0.45 }$ & 11.474 &  &  &  & EB from HIRES RVs. \\\\\n270371513$\\dagger$ & HIP 10047 & $1426.2967 _{ - 0.0023 } ^ { + 0.002 }$ & $0.39 _{ - 0.17 } ^ { + 1.79 }(s)$ & $0.024 _{ - 0.0015 } ^ { + 0.0032 }$ & $4.8 _{ - 0.38 } ^ { + 0.64 }$ & $0.5 _{ - 0.34 } ^ { + 0.39 }$ & $1.93 _{ - 0.16 } ^ { + 0.19 }$ & 6.98515 &  & MINERVA (20) &  & SB 2 from MINERVA observations.  \\\\\n274599700 & TWOMASS 17011885+5131455 & $2002.1202 _{ - 0.0024 } ^ { + 0.0024 }$ & $32.9754 _{ - 0.005 } ^ { + 0.005 }$ & $0.0847 _{ - 0.0021 } ^ { + 0.0018 }$ & $13.25 _{ - 0.83 } ^ { + 0.83 }$ & $0.37 _{ - 0.24 } ^ { + 0.19 }$ & $8.2 _{ - 0.18 } ^ { + 0.21 }$ & 12.411 &  &  &  &  \\\\\n278990954 & TYC 8548-00717-1 & $1650.0191 _{ - 0.0086 } ^ { + 0.0105 }$ & $18.45 _{ - 8.66 } ^ { + 230.7 }(s)$ & $0.034 _{ - 0.0024 } ^ { + 0.0115 }$ & $9.65 _{ - 0.92 } ^ { + 3.13 }$ & $0.58 _{ - 0.4 } ^ { + 0.36 }$ & $10.62 _{ - 0.66 } ^ { + 2.46 }$ & 10.749 &  &  &  &  \\\\\n280865159* & TYC 9384-01533-1 & $1387.0749 _{ - 0.0045 } ^ { + 0.0044 }$ & $1045.0 _{ - 249.0 } ^ { + 536.0 }(s)$ & $0.0406 _{ - 0.0011 } ^ { + 0.0014 }$ & $4.75 _{ - 0.26 } ^ { + 0.28 }$ & $0.35 _{ - 0.24 } ^ { + 0.23 }$ & $19.08 _{ - 0.32 } ^ { + 0.36 }$ & 11.517 &  &  & Gemini &  \\\\\n284361752 & TYC 3924-01678-1 & $2032.093 _{ - 0.0078 } ^ { + 0.008 }$ & $140.6 _{ - 46.6 } ^ { + 159.1 }(s)$ & $0.0259 _{ - 0.0014 } ^ { + 0.0017 }$ & $3.62 _{ - 0.26 } ^ { + 0.31 }$ & $0.4 _{ - 0.27 } ^ { + 0.34 }$ & $8.98 _{ - 0.66 } ^ { + 0.86 }$ & 10.221 &  &  &  &  \\\\\n288240183 & TYC 4634-01225-1 & $1896.941 _{ - 0.0051 } ^ { + 0.0047 }$ & $119.0502 _{ - 0.0091 } ^ { + 0.0089 }$ & $0.02826 _{ - 0.00089 } ^ { + 0.00119 }$ & $4.28 _{ - 0.35 } ^ { + 0.36 }$ & $0.55 _{ - 0.37 } ^ { + 0.25 }$ & $17.49 _{ - 0.36 } ^ { + 0.6 }$ & 9.546 &  &  &  &  \\\\\n29169215 & TWOMASS 09011787+4727085 & $1872.5047 _{ - 0.0032 } ^ { + 0.0036 }$ & $14.89 _{ - 6.12 } ^ { + 24.84 }(s)$ & $0.0403 _{ - 0.0025 } ^ { + 0.0033 }$ & $3.28 _{ - 0.37 } ^ { + 0.45 }$ & $0.44 _{ - 0.3 } ^ { + 0.33 }$ & $3.56 _{ - 0.21 } ^ { + 0.32 }$ & 11.828 &  &  &  &  \\\\\n293649602 & TYC 8103-00266-1 & $1511.2109 _{ - 0.004 } ^ { + 0.0037 }$ & $12.85 _{ - 5.34 } ^ { + 42.21 }(s)$ & $0.04 _{ - 0.0024 } ^ { + 0.0039 }$ & $4.66 _{ - 0.36 } ^ { + 0.5 }$ & $0.5 _{ - 0.35 } ^ { + 0.34 }$ & $4.1 _{ - 0.31 } ^ { + 0.56 }$ & 10.925 &  &  &  &  \\\\\n296737508 & TYC 5472-01060-1 & $1538.0036 _{ - 0.0015 } ^ { + 0.0016 }$ & $18.27 _{ - 5.06 } ^ { + 17.45 }(s)$ & $0.0425 _{ - 0.0014 } ^ { + 0.0019 }$ & $5.33 _{ - 0.22 } ^ { + 0.27 }$ & $0.44 _{ - 0.3 } ^ { + 0.26 }$ & $5.13 _{ - 0.13 } ^ { + 0.15 }$ & 9.772 & Sinistro (1) & NRES (1);MINERVA (1) & Gemini &  \\\\\n298663873 & TYC 3913-01781-1 & $1830.76819 _{ - 0.00099 } ^ { + 0.00099 }$ & $479.9 _{ - 89.4 } ^ { + 109.4 }(s)$ & $0.06231 _{ - 0.00034 } ^ { + 0.00045 }$ & $11.07 _{ - 0.57 } ^ { + 0.57 }$ & $0.16 _{ - 0.11 } ^ { + 0.13 }$ & $23.99 _{ - 0.093 } ^ { + 0.1 }$ & 9.162 &  & NRES (2) & Gemini & Dalba et al. (in prep) \\\\\n303050301 & TYC 6979-01108-1 & $1366.1301 _{ - 0.0022 } ^ { + 0.0023 }$ & $281.0 _{ - 170.0 } ^ { + 264.0 }(s)$ & $0.0514 _{ - 0.0027 } ^ { + 0.0018 }$ & $4.85 _{ - 0.32 } ^ { + 0.32 }$ & $0.73 _{ - 0.48 } ^ { + 0.1 }$ & $7.91 _{ - 0.31 } ^ { + 0.36 }$ & 10.048 &  & NRES (1) & Gemini &  \\\\\n303317324 & TYC 6983-00438-1 & $1365.1845 _{ - 0.0023 } ^ { + 0.0028 }$ & $69.0 _{ - 25.5 } ^ { + 78.1 }(s)$ & $0.0365 _{ - 0.0013 } ^ { + 0.0016 }$ & $2.88 _{ - 0.3 } ^ { + 0.31 }$ & $0.39 _{ - 0.26 } ^ { + 0.32 }$ & $5.78 _{ - 0.18 } ^ { + 0.24 }$ & 10.799 &  &  &  &  \\\\\n\\hline\n\\end{tabular}}\n\\caption{\\emph{Note} -- Candidates that have become TOIs following the PHT discovery are marked with an asterisk (*). The `s' following the orbital period indicates that the candidates is a single transit event. The ground-based follow-up observations are summarized in columns 10-12, where the bracketed numbers correspond the number of epochs obtained with each instrument.  See Section~\\ref{sec:follow_up} for description of each instrument. The $\\dagger$ symbol indicates candidates that have been shown to be astrophysical false  positives based on the ground based follow-up observations.}\n\\label{tab:PHT-caniddates}\n\\end{table}\n\\end{landscape}\n\n\\begin{landscape}\n\\begin{table}\n\\addtocounter{table}{-1}\n\\resizebox{1.31\\textwidth}{!}{\n\\begin{tabular}{ccccccccccccc}\n\\textbf{TIC} & \\textbf{Other} & \\textbf{Epoch} & \\textbf{Period}  & \\textbf{$R_{pl}$\/$R_{\\odot}$} & \\textbf{$R_{pl}$} & \\textbf{Impact} & \\textbf{Duration} & \\textbf{$V_{mag}$} & \\textbf{Photometry} & \\textbf{Spectroscopy} & \\textbf{Speckle} & \\textbf{Comment} \\\\\n  & \\textbf{Name} & \\textbf{(\\textcolor{black}{BJD - 2457000})} & \\textbf{(days)}  & & $(R_{\\oplus})$  & \\textbf{Parameter} & \\textbf{(hours)} &   &   &   &   &   \\\\\n\\hline\n303586471$\\dagger$ & HIP 115828 & $1363.7692 _{ - 0.0033 } ^ { + 0.0027 }$ & $13.85 _{ - 4.19 } ^ { + 18.2 }(s)$ & $0.0214 _{ - 0.001 } ^ { + 0.0014 }$ & $2.52 _{ - 0.16 } ^ { + 0.2 }$ & $0.4 _{ - 0.27 } ^ { + 0.33 }$ & $4.23 _{ - 0.19 } ^ { + 0.16 }$ & 8.27 &  & MINERVA (11) &  & SB 2 from MINERVA observations.  \\\\\n304142124* & HIP 53719 & $1585.28023 _{ - 0.0008 } ^ { + 0.0008 }$ & $42.8 _{ - 10.0 } ^ { + 18.2 }(s)$ & $0.04311 _{ - 0.00093 } ^ { + 0.00153 }$ & $4.1 _{ - 0.23 } ^ { + 0.24 }$ & $0.33 _{ - 0.21 } ^ { + 0.21 }$ & $5.66 _{ - 0.067 } ^ { + 0.09 }$ & 8.62 &  & NRES (1);MINERVA (4) &  & Confirmed planet \\citep{diaz2020} \\\\\n304339227 & TYC 9290-01087-1 & $1673.3242 _{ - 0.009 } ^ { + 0.0128 }$ & $111.9 _{ - 72.2 } ^ { + 4844.1 }(s)$ & $0.0253 _{ - 0.0024 } ^ { + 0.0481 }$ & $3.27 _{ - 0.61 } ^ { + 5.72 }$ & $0.67 _{ - 0.47 } ^ { + 0.36 }$ & $7.44 _{ - 0.86 } ^ { + 2.84 }$ & 9.169 &  &  &  &  \\\\\n307958020 & TYC 4191-00309-1 & $1864.82 _{ - 0.014 } ^ { + 0.013 }$ & $169.0 _{ - 107.0 } ^ { + 10194.0 }(s)$ & $0.0223 _{ - 0.0022 } ^ { + 0.0543 }$ & $3.92 _{ - 0.52 } ^ { + 9.27 }$ & $0.71 _{ - 0.53 } ^ { + 0.33 }$ & $12.48 _{ - 1.1 } ^ { + 5.41 }$ & 9.017 &  &  &  &  \\\\\n308301091 & TYC 2081-01273-1 & $2030.3691 _{ - 0.0024 } ^ { + 0.0026 }$ & $29.24 _{ - 8.49 } ^ { + 22.46 }(s)$ & $0.0362 _{ - 0.0013 } ^ { + 0.0014 }$ & $5.41 _{ - 0.34 } ^ { + 0.35 }$ & $0.35 _{ - 0.25 } ^ { + 0.29 }$ & $6.57 _{ - 0.14 } ^ { + 0.19 }$ & 10.273 &  &  &  &  \\\\\n313006381 & HIP 45012 & $1705.687 _{ - 0.0081 } ^ { + 0.0045 }$ & $21.56 _{ - 8.9 } ^ { + 54.15 }(s)$ & $0.0261 _{ - 0.0017 } ^ { + 0.0027 }$ & $2.34 _{ - 0.2 } ^ { + 0.27 }$ & $0.45 _{ - 0.3 } ^ { + 0.38 }$ & $3.85 _{ - 0.51 } ^ { + 0.31 }$ & 9.39 &  &  &  &  \\\\\n323295479* & TYC 9506-01881-1 & $1622.9258 _{ - 0.00083 } ^ { + 0.00087 }$ & $117.8 _{ - 25.8 } ^ { + 30.9 }(s)$ & $0.0981 _{ - 0.0021 } ^ { + 0.0023 }$ & $11.35 _{ - 0.67 } ^ { + 0.66 }$ & $0.839 _{ - 0.024 } ^ { + 0.019 }$ & $6.7 _{ - 0.14 } ^ { + 0.15 }$ & 10.595 &  &  &  &  \\\\\n328933398.01* & TYC 4634-01435-1 & $1880.9878 _{ - 0.0039 } ^ { + 0.0042 }$ & $24.9335 _{ - 0.0046 } ^ { + 0.005 }$ & $0.0437 _{ - 0.0022 } ^ { + 0.0023 }$ & $4.62 _{ - 0.32 } ^ { + 0.33 }$ & $0.38 _{ - 0.25 } ^ { + 0.27 }$ & $5.02 _{ - 0.22 } ^ { + 0.27 }$ & 11.215 &  &  &  & Potential multi-planet system. \\\\\n328933398.02* & TYC 4634-01435-1 & $1848.6557 _{ - 0.0053 } ^ { + 0.0072 }$ & $50.5 _{ - 22.4 } ^ { + 77.1 }(s)$ & $0.0296 _{ - 0.0028 } ^ { + 0.0033 }$ & $3.14 _{ - 0.33 } ^ { + 0.39 }$ & $0.41 _{ - 0.28 } ^ { + 0.35 }$ & $5.99 _{ - 0.8 } ^ { + 0.77 }$ & 11.215 &  &  &  &  \\\\\n331644554 & TYC 3609-00469-1 & $1757.0354 _{ - 0.0031 } ^ { + 0.0033 }$ & $947.0 _{ - 215.0 } ^ { + 274.0 }(s)$ & $0.12 _{ - 0.025 } ^ { + 0.021 }$ & $21.84 _{ - 4.57 } ^ { + 3.86 }$ & $1.018 _{ - 0.036 } ^ { + 0.028 }$ & $10.93 _{ - 0.34 } ^ { + 0.35 }$ & 9.752 &  &  &  &  \\\\\n332657786 & TWOMASS 09595797-1609323 & $1536.7659 _{ - 0.0015 } ^ { + 0.0015 }$ & $63.76 _{ - 9.52 } ^ { + 11.13 }(s)$ & $0.14961 _{ - 0.00064 } ^ { + 0.00029 }$ & $3.83 _{ - 0.12 } ^ { + 0.12 }$ & $0.059 _{ - 0.041 } ^ { + 0.064 }$ & $3.333 _{ - 0.095 } ^ { + 0.096 }$ & 15.99 &  &  &  &  \\\\\n336075472 & TYC 3526-00332-1 & $2028.1762 _{ - 0.0043 } ^ { + 0.0037 }$ & $61.9 _{ - 24.0 } ^ { + 95.6 }(s)$ & $0.0402 _{ - 0.0022 } ^ { + 0.0033 }$ & $3.09 _{ - 0.34 } ^ { + 0.4 }$ & $0.43 _{ - 0.29 } ^ { + 0.32 }$ & $5.39 _{ - 0.23 } ^ { + 0.37 }$ & 11.842 &  &  &  &  \\\\\n349488688.01 & TYC 1529-00224-1 & $1994.283 _{ - 0.0038 } ^ { + 0.0033 }$ & $11.6254 _{ - 0.005 } ^ { + 0.0052 }$ & $0.02195 _{ - 0.00096 } ^ { + 0.00122 }$ & $3.44 _{ - 0.18 } ^ { + 0.21 }$ & $0.39 _{ - 0.27 } ^ { + 0.3 }$ & $5.58 _{ - 0.15 } ^ { + 0.18 }$ & 8.855 &  & NRES (2);SOPHIE (2) &  & Potential multi-planet system. \\\\\n349488688.02 & TYC 1529-00224-1 & $2002.77063 _{ - 0.00097 } ^ { + 0.00103 }$ & $15.35 _{ - 1.94 } ^ { + 4.15 }(s)$ & $0.03688 _{ - 0.00067 } ^ { + 0.00069 }$ & $5.78 _{ - 0.18 } ^ { + 0.18 }$ & $0.24 _{ - 0.16 } ^ { + 0.21 }$ & $6.291 _{ - 0.058 } ^ { + 0.074 }$ & 8.855 &  & NRES (2);SOPHIE (2) &  &  \\\\\n356700488* & TYC 4420-01295-1 & $1756.638 _{ - 0.013 } ^ { + 0.011 }$ & $184.5 _{ - 64.7 } ^ { + 333.1 }(s)$ & $0.0173 _{ - 0.0011 } ^ { + 0.0015 }$ & $2.92 _{ - 0.2 } ^ { + 0.28 }$ & $0.44 _{ - 0.3 } ^ { + 0.34 }$ & $11.76 _{ - 0.65 } ^ { + 1.03 }$ & 8.413 &  &  &  &  \\\\\n356710041* & TYC 1993-00419-1 & $1932.2939 _{ - 0.0019 } ^ { + 0.0019 }$ & $29.6 _{ - 14.0 } ^ { + 19.0 }(s)$ & $0.0496 _{ - 0.0021 } ^ { + 0.0011 }$ & $14.82 _{ - 0.85 } ^ { + 0.84 }$ & $0.66 _{ - 0.42 } ^ { + 0.11 }$ & $12.76 _{ - 0.24 } ^ { + 0.24 }$ & 9.646 &  &  &  &  \\\\\n369532319 & TYC 2743-01716-1 & $1755.8158 _{ - 0.006 } ^ { + 0.0051 }$ & $35.4 _{ - 12.0 } ^ { + 51.6 }(s)$ & $0.0316 _{ - 0.0023 } ^ { + 0.0028 }$ & $3.43 _{ - 0.3 } ^ { + 0.37 }$ & $0.41 _{ - 0.29 } ^ { + 0.34 }$ & $5.5 _{ - 0.32 } ^ { + 0.32 }$ & 10.594 &  &  & Gemini &  \\\\\n369779127 & TYC 9510-00090-1 & $1643.9403 _{ - 0.0046 } ^ { + 0.0058 }$ & $9.93 _{ - 3.38 } ^ { + 19.74 }(s)$ & $0.0288 _{ - 0.0015 } ^ { + 0.0033 }$ & $4.89 _{ - 0.31 } ^ { + 0.56 }$ & $0.46 _{ - 0.31 } ^ { + 0.33 }$ & $5.64 _{ - 0.38 } ^ { + 0.33 }$ & 9.279 &  &  &  &  \\\\\n384159646* & TYC 9454-00957-1 & $1630.39405 _{ - 0.00079 } ^ { + 0.00079 }$ & $11.68 _{ - 2.75 } ^ { + 4.21 }(s)$ & $0.0658 _{ - 0.0012 } ^ { + 0.0011 }$ & $9.87 _{ - 0.45 } ^ { + 0.44 }$ & $0.27 _{ - 0.18 } ^ { + 0.21 }$ & $5.152 _{ - 0.069 } ^ { + 0.087 }$ & 10.158 & SBIG (1) & NRES (8);MINERVA (6) & Gemini &  \\\\\n385557214 & TYC 1807-00046-1 & $1791.58399 _{ - 0.00068 } ^ { + 0.0007 }$ & $5.62451 _{ - 0.0004 } ^ { + 0.00043 }$ & $0.096 _{ - 0.019 } ^ { + 0.032 }$ & $8.32 _{ - 2.06 } ^ { + 2.77 }$ & $0.95 _{ - 0.075 } ^ { + 0.053 }$ & $1.221 _{ - 0.094 } ^ { + 0.058 }$ & 10.856 &  &  &  &  \\\\\n388134787 & TYC 4260-00427-1 & $1811.034 _{ - 0.015 } ^ { + 0.017 }$ & $246.0 _{ - 127.0 } ^ { + 6209.0 }(s)$ & $0.0265 _{ - 0.0024 } ^ { + 0.023 }$ & $2.57 _{ - 0.28 } ^ { + 2.19 }$ & $0.55 _{ - 0.39 } ^ { + 0.44 }$ & $8.85 _{ - 1.13 } ^ { + 1.84 }$ & 10.95 &  & NRES (1) & Gemini &  \\\\\n404518509 & HIP 16038 & $1431.2696 _{ - 0.0037 } ^ { + 0.0035 }$ & $26.83 _{ - 9.46 } ^ { + 56.14 }(s)$ & $0.0259 _{ - 0.0013 } ^ { + 0.0022 }$ & $2.94 _{ - 0.21 } ^ { + 0.29 }$ & $0.47 _{ - 0.31 } ^ { + 0.34 }$ & $5.02 _{ - 0.23 } ^ { + 0.28 }$ & 9.17 &  &  &  &  \\\\\n408636441* & TYC 4266-00736-1 & $1745.4668 _{ - 0.0016 } ^ { + 0.0015 }$ & $37.695 _{ - 0.0034 } ^ { + 0.0033 }$ & $0.0485 _{ - 0.0019 } ^ { + 0.0023 }$ & $3.32 _{ - 0.16 } ^ { + 0.19 }$ & $0.39 _{ - 0.27 } ^ { + 0.29 }$ & $3.63 _{ - 0.1 } ^ { + 0.14 }$ & 11.93 & SBIG (1) &  & Gemini & Half of the period likely. \\\\\n418255064 & TWOMASS 13063680-8037015 & $1629.3304 _{ - 0.0018 } ^ { + 0.0018 }$ & $25.37 _{ - 7.06 } ^ { + 15.41 }(s)$ & $0.0732 _{ - 0.0029 } ^ { + 0.0031 }$ & $5.57 _{ - 0.36 } ^ { + 0.38 }$ & $0.37 _{ - 0.25 } ^ { + 0.25 }$ & $3.83 _{ - 0.13 } ^ { + 0.14 }$ & 12.478 & SBIG (1) &  & Gemini &  \\\\\n420645189$\\dagger$ & TYC 4508-00478-1 & $1837.4767 _{ - 0.0018 } ^ { + 0.0017 }$ & $250.2 _{ - 66.6 } ^ { + 99.4 }(s)$ & $0.0784 _{ - 0.0033 } ^ { + 0.0046 }$ & $8.82 _{ - 0.55 } ^ { + 0.7 }$ & $0.892 _{ - 0.026 } ^ { + 0.028 }$ & $6.95 _{ - 0.27 } ^ { + 0.3 }$ & 10.595 &  & MINERVA (1) &  & SB 2 from MINERVA observations.  \\\\\n422914082 & TYC 0046-00133-1 & $1431.5538 _{ - 0.0014 } ^ { + 0.0017 }$ & $12.91 _{ - 3.91 } ^ { + 8.97 }(s)$ & $0.0418 _{ - 0.0015 } ^ { + 0.0016 }$ & $3.96 _{ - 0.32 } ^ { + 0.35 }$ & $0.36 _{ - 0.25 } ^ { + 0.28 }$ & $4.07 _{ - 0.09 } ^ { + 0.126 }$ & 11.026 & Sinistro (1) & NRES (1) &  &  \\\\\n427344083 & TWOMASS 22563609+7040518 & $1961.8967 _{ - 0.0031 } ^ { + 0.0036 }$ & $7.77 _{ - 5.6 } ^ { + 9.65 }(s)$ & $0.107 _{ - 0.016 } ^ { + 0.025 }$ & $12.27 _{ - 1.87 } ^ { + 2.9 }$ & $0.834 _{ - 0.484 } ^ { + 0.094 }$ & $2.88 _{ - 0.3 } ^ { + 0.42 }$ & 13.404 &  &  &  &  \\\\\n436873727 & HIP 13224 & $1803.83679 _{ - 0.00058 } ^ { + 0.00056 }$ & $19.26 _{ - 5.95 } ^ { + 6.73 }(s)$ & $0.05246 _{ - 0.00061 } ^ { + 0.00059 }$ & $10.02 _{ - 0.43 } ^ { + 0.41 }$ & $0.767 _{ - 0.057 } ^ { + 0.038 }$ & $5.462 _{ - 0.081 } ^ { + 0.074 }$ & 7.51 &  &  &  &  \\\\ \n441642457* & TYC 3858-00452-1 & $1745.5102 _{ - 0.0108 } ^ { + 0.0097 }$ & $79.8072 _{ - 0.0071 } ^ { + 0.0076 }$ & $0.0281 _{ - 0.0024 } ^ { + 0.0033 }$ & $3.55 _{ - 0.34 } ^ { + 0.46 }$ & $0.934 _{ - 0.023 } ^ { + 0.026 }$ & $6.9 _{ - 0.39 } ^ { + 0.6 }$ & 9.996 &  &  &  &  \\\\\n441765914* & TWOMASS 17253007+7552562 & $1769.6154 _{ - 0.0058 } ^ { + 0.0093 }$ & $161.6 _{ - 58.2 } ^ { + 1460.1 }(s)$ & $0.0411 _{ - 0.0024 } ^ { + 0.0119 }$ & $3.6 _{ - 0.3 } ^ { + 1.01 }$ & $0.45 _{ - 0.32 } ^ { + 0.48 }$ & $7.44 _{ - 0.36 } ^ { + 1.08 }$ & 11.638 &  &  &  &  \\\\\n452920657 & TWOMASS 00332018+5906355 & $1810.5765 _{ - 0.0031 } ^ { + 0.003 }$ & $53.2 _{ - 29.0 } ^ { + 34.3 }(s)$ & $0.135 _{ - 0.016 } ^ { + 0.012 }$ & $9.71 _{ - 1.16 } ^ { + 0.9 }$ & $0.73 _{ - 0.48 } ^ { + 0.11 }$ & $4.6 _{ - 0.26 } ^ { + 0.29 }$ & 14.167 & SBIG (1) &  &  &  \\\\\n455737331 & TYC 2779-00785-1 & $1780.7084 _{ - 0.008 } ^ { + 0.0073 }$ & $50.4 _{ - 17.6 } ^ { + 75.0 }(s)$ & $0.0257 _{ - 0.0016 } ^ { + 0.002 }$ & $3.05 _{ - 0.24 } ^ { + 0.29 }$ & $0.43 _{ - 0.29 } ^ { + 0.33 }$ & $6.6 _{ - 0.43 } ^ { + 0.5 }$ & 10.189 & SBIG (1) &  & Gemini &  \\\\\n456909420 & TYC 1208-01094-1 & $1779.4109 _{ - 0.0026 } ^ { + 0.0022 }$ & $5.78 _{ - 5.29 } ^ { + 5.95 }(s)$ & $0.078 _{ - 0.031 } ^ { + 0.045 }$ & $9.15 _{ - 3.61 } ^ { + 5.27 }$ & $0.973 _{ - 0.495 } ^ { + 0.063 }$ & $1.73 _{ - 0.27 } ^ { + 0.28 }$ & 10.941 &  &  &  &  \\\\\n458451774 & TWOMASS 12551793+4431260 & $1917.1875 _{ - 0.0019 } ^ { + 0.0019 }$ & $12.39 _{ - 6.34 } ^ { + 83.97 }(s)$ & $0.0752 _{ - 0.0054 } ^ { + 0.0211 }$ & $3.33 _{ - 0.26 } ^ { + 0.92 }$ & $0.61 _{ - 0.43 } ^ { + 0.32 }$ & $2.08 _{ - 0.19 } ^ { + 0.59 }$ & 13.713 &  &  &  &  \\\\\n48018596 & TYC 3548-00800-1 & $1713.4514 _{ - 0.0063 } ^ { + 0.0046 }$ & $100.1145 _{ - 0.0018 } ^ { + 0.0021 }$ & $0.049 _{ - 0.0081 } ^ { + 0.018 }$ & $7.88 _{ - 1.33 } ^ { + 2.9 }$ & $0.984 _{ - 0.028 } ^ { + 0.027 }$ & $2.83 _{ - 0.26 } ^ { + 0.29 }$ & 9.595 &  & NRES (1) & Gemini &  \\\\\n53309262 & TWOMASS 07475406+5741549 & $1863.1133 _{ - 0.0064 } ^ { + 0.0061 }$ & $294.8 _{ - 96.0 } ^ { + 327.0 }(s)$ & $0.1239 _{ - 0.0075 } ^ { + 0.0098 }$ & $5.38 _{ - 0.36 } ^ { + 0.46 }$ & $0.46 _{ - 0.31 } ^ { + 0.28 }$ & $6.74 _{ - 0.45 } ^ { + 0.62 }$ & 15.51 &  &  &  &  \\\\\n53843023 & TYC 6956-00758-1 & $1328.0335 _{ - 0.0054 } ^ { + 0.0057 }$ & $202.0 _{ - 189.0 } ^ { + 272.0 }(s)$ & $0.058 _{ - 0.02 } ^ { + 0.056 }$ & $5.14 _{ - 1.77 } ^ { + 4.99 }$ & $0.962 _{ - 0.597 } ^ { + 0.083 }$ & $4.25 _{ - 0.72 } ^ { + 0.66 }$ & 11.571 &  &  &  &  \\\\\n55525572* & TYC 8876-01059-1 & $1454.6713 _{ - 0.0066 } ^ { + 0.0065 }$ & $83.8951 _{ - 0.004 } ^ { + 0.004 }$ & $0.0343 _{ - 0.001 } ^ { + 0.0021 }$ & $7.31 _{ - 0.46 } ^ { + 0.56 }$ & $0.43 _{ - 0.29 } ^ { + 0.31 }$ & $13.54 _{ - 0.3 } ^ { + 0.51 }$ & 10.358 &  & CHIRON (5) & Gemini & Confirmed planet \\citep{2020eisner} \\\\\n63698669* & TYC 6993-00729-1 & $1364.6226 _{ - 0.0074 } ^ { + 0.0067 }$ & $73.6 _{ - 26.8 } ^ { + 133.6 }(s)$ & $0.0248 _{ - 0.0019 } ^ { + 0.0023 }$ & $2.15 _{ - 0.2 } ^ { + 0.25 }$ & $0.42 _{ - 0.29 } ^ { + 0.35 }$ & $5.63 _{ - 0.32 } ^ { + 0.57 }$ & 10.701 & SBIG (1) &  &  &  \\\\\n70887357* & TYC 5883-01412-1 & $1454.3341 _{ - 0.0016 } ^ { + 0.0015 }$ & $56.1 _{ - 15.3 } ^ { + 18.8 }(s)$ & $0.0605 _{ - 0.0027 } ^ { + 0.0027 }$ & $12.84 _{ - 0.86 } ^ { + 0.9 }$ & $0.917 _{ - 0.028 } ^ { + 0.016 }$ & $7.29 _{ - 0.18 } ^ { + 0.19 }$ & 9.293 &  &  &  &  \\\\\n7422496$\\dagger$ & HIP 25359 & $1470.3625 _{ - 0.0031 } ^ { + 0.0023 }$ & $61.4 _{ - 16.7 } ^ { + 49.0 }(s)$ & $0.0255 _{ - 0.001 } ^ { + 0.0011 }$ & $2.44 _{ - 0.15 } ^ { + 0.16 }$ & $0.37 _{ - 0.25 } ^ { + 0.29 }$ & $5.89 _{ - 0.15 } ^ { + 0.15 }$ & 9.36 &  & MINERVA (4) &  & SB 2 from MINERVA observations.  \\\\\n82452140 & TYC 3076-00921-1 & $1964.292 _{ - 0.011 } ^ { + 0.011 }$ & $21.1338 _{ - 0.0052 } ^ { + 0.0066 }$ & $0.0266 _{ - 0.0019 } ^ { + 0.0027 }$ & $2.95 _{ - 0.25 } ^ { + 0.34 }$ & $0.42 _{ - 0.29 } ^ { + 0.36 }$ & $5.87 _{ - 0.62 } ^ { + 0.94 }$ & 10.616 &  &  &  &  \\\\\n88840705 & TYC 3091-00808-1 & $2026.6489 _{ - 0.001 } ^ { + 0.001 }$ & $260.6 _{ - 87.6 } ^ { + 142.2 }(s)$ & $0.109 _{ - 0.023 } ^ { + 0.027 }$ & $9.98 _{ - 2.28 } ^ { + 2.75 }$ & $1.001 _{ - 0.042 } ^ { + 0.037 }$ & $4.72 _{ - 0.13 } ^ { + 0.15 }$ & 9.443 &  &  &  &  \\\\\n91987762* & HIP 47288 & $1894.25381 _{ - 0.00051 } ^ { + 0.00047 }$ & $10.51 _{ - 3.48 } ^ { + 3.67 }(s)$ & $0.05459 _{ - 0.00106 } ^ { + 0.00097 }$ & $9.56 _{ - 0.56 } ^ { + 0.52 }$ & $0.771 _{ - 0.062 } ^ { + 0.033 }$ & $4.342 _{ - 0.073 } ^ { + 0.063 }$ & 7.87 &  & NRES (4) & Gemini &  \\\\\n95768667 & TYC 1434-00331-1 & $1918.3318 _{ - 0.0093 } ^ { + 0.0079 }$ & $26.9 _{ - 12.4 } ^ { + 72.3 }(s)$ & $0.0282 _{ - 0.0022 } ^ { + 0.0031 }$ & $3.54 _{ - 0.32 } ^ { + 0.43 }$ & $0.48 _{ - 0.33 } ^ { + 0.35 }$ & $5.4 _{ - 0.64 } ^ { + 0.76 }$ & 10.318 &  &  &  &  \\\\\n\\hline\n\\end{tabular}}\n\\caption{\\textbf{Properties of PHT candidates (continued)}}\n\\label{tab:PHT-caniddates2}\n\\end{table}\n\\end{landscape}\n\n\n\\section{Conclusion}\n\\label{sec:condlusion}\n\nWe present the results from the analysis of the first 26  \\emph{TESS}\\ sectors. The outlined citizen science approach engages over 22 thousand registered citizen scientists who completed 12,617,038 classifications from December 2018 through August 2020 for the sectors observed during the first two years of the \\emph{TESS}\\ mission. We applied a systematic search for planetary candidates using visual vetting by multiple volunteers to identify \\emph{TESS}\\ targets that are most likely to host a planet. Between 8 and 15 volunteers have inspected each \\emph{TESS}\\ light curve and marked times of transit-like events using the PHT online interface. For each light curve, the markings from all the volunteers who saw that target were combined using an unsupervised machine learning method, known as DBSCAN, in order to identify likely transit-like events. Each of these identified events was given a transit score based on the number of volunteers who identified a given event and on the user weighting of each of those volunteers. Individual user weights were calculated based on the user's ability to identify simulated transit events, injected into real \\emph{TESS}\\ light curves, that are displayed on the PHT site alongside of the real data. The transit scores were then used to generate a ranked list of candidates that range from most likely to least likely to host a planet candidate. The top 500 highest ranked candidates were further vetted by the PHT science team. This stage of vetting primarily made use of the open source {\\sc latte} \\citep{LATTE2020} tool which generates a number of standard diagnostic plots that help identify promising candidates and weed out false positive signals. \n\nOn average we found around three high priority candidates per sector which were followed up using ground based telescopes, where possible. To date, PHT has statistically confirmed one planet, TOI-813 \\citep{2020eisner}: a Saturn-sized planet on an 84 day orbit around a subgiant host star. Other PHT identified planets listed in this paper are being followed up by other teams of astronomers, such as TOI-1899 (TIC 172370679) which was recently confirmed to be a warm Jupiter transiting an M-dwarf \\citep{canas2020}. The remaining candidates outlined in this paper require further follow-up observations to confirm their planetary nature.\n\nThe sensitivity of our transit search effort was assessed using synthetic data, as well as the known TOI and TCE candidates flagged by the SPOC pipeline. For simulated planets (where simulated signals are injected into real \\emph{TESS}\\ light curves) we have shown that the recovery efficiency of human vetting starts to decrease for transit-signals that have a SNR less than 7.5. The detection efficiency was further evaluated by the fractional recovery of the TOI and TCEs. We have shown that PHT is over 85 \\% complete in the recovery of planets that have a radius greater than 4 $R_{\\oplus}$, 51 \\% complete for radii between 3 and 4 $R_{\\oplus}$ and 49 \\% complete for radii between 2 and 3 $R_{\\oplus}$. Furthermore, we have shown that human vetting is not sensitive to the number of transits present in the light curve, meaning that they are equally likely to identify candidates on longer orbital periods as they are those with shorter orbital periods for periods greater than $\\sim$ 1 day. Planets with periods shorter than around 1 day exhibit over 20 transits within one \\emph{TESS}\\ sectors resulting in a decrease in identification by the volunteers. This is due to many volunteers only marking a random subset of these events, resulting in a lack of consensus on any given transit event and thus decreasing the overall transit score of these light curves. \n\nIn addition to searching for signals due to transiting exoplanets, PHT provides a platform that can be used to identify other stellar phenomena that may otherwise be difficult to identify with automated pipelines. Such phenomena, including eclipsing binaries, multiple stellar systems, dwarf novae, and stellar flares are often mentioned on the PHT discussion forums where volunteers can use searchable hashtags and comments to bring these systems to the attention of other citizen scientists as well as the PHT science team. All of the eclipsing binaries identified on the site, for example, are being used and vetted by the \\emph{TESS}\\ Eclipsing Binary Working Group (Prsa et al. in prep). Furthermore, we have investigated the nature of all of the targets that were identified as possible multiple stellar systems, as summarised in Table~\\ref{tab:PHT-multis}.\n\nOverall we have shown that large scale visual vetting can complement the findings \\textcolor{black}{from the major \\emph{TESS}\\ pipeline} by identifying longer period planets that may only exhibit a single transit event in their light curve, as well as in finding signals that are aperiodic or embedded in a strong varying stellar signal. The identification of planets around stars with variable signals allow us to potentially characterise the host-star (e.g., with asteroseismology or spot modulation). Additionally, the longer period planets are integral to our understanding of how planet systems form and evolve, as they allow us to investigate the evolution of planets that are farther away from their host star and therefore less dependent on stellar radiation. \\textcolor{black}{While automated pipelines specifically designed to identify single transit events in the \\emph{TESS}\\ data exist \\citep[e.g., ][]{Gill2020}, neither their methodology nor the full list of their findings are yet publicly available and thus we are unable to compare results.} \n\nThe planets that PHT finds have longer periods ($\\gtrsim$ 27 d) than those found in \\emph{TESS}\\ data using automated pipelines, and are more typical of the Kepler sample (25\\% of Kepler confirmed planets have periods greater than 27 days\\footnote{\\url{https:\/\/exoplanetarchive.ipac.caltech.edu\/}}). However, the Kepler planets are considerably fainter, and thus less amenable to ground-based follow-up or atmospheric characterisation from space (CHEOPS and JWST). Thus PHT helps to bridge the parameter spaces covered by these two missions, by identifying longer period planet candidates around bright, nearby stars, for which we can ultimately obtain precise planetary mass estimates. Although statistical characterisation of exo-planetary systems is no doubt important, precise mass measurements are key to developing our understanding of exoplanets and the physics which dictate their evolution. In particular, identification of this PHT sample provides follow-up targets to investigate the dependence of photo-evaporation on the mass of planets as well as on the planet radius, and will help our understanding of the photo-evaporation valley at longer orbital periods \\citep{Owen2013}. \n\nPHT will continue to operate throughout the \\emph{TESS}\\ extended mission, hopefully allowing us to identify even longer period planets as well as help verify some of the existing candidates with additional transits.  \n\n\n\n\\begin{table*}\n\\resizebox{0.95\\textwidth}{!}{\n\\begin{tabular}{cccccccccc}\n\\textbf{TIC}  & \\textbf{Period (days)}        & \\textbf{Epoch (\\textcolor{black}{BJD - 2457000})}      & \\textbf{Depth (ppm)}  & \\textbf{Comment}  \\\\\n\\hline\n13968858      & $3.4850 \\pm 0.001$            & $ 1684.780   \\pm 0.005$            & 410000    & Candidate multiple system  \\\\\n              & $1.4380 \\pm 0.001$            & $ 1684.335   \\pm 0.005$            & 50000     &   \\\\\n35655828      & $ 8.073 \\pm 0.01$             & $ 1550.94    \\pm 0.01 $            & 23000     & Confirmed blend \\\\\n              & $  1.220 \\pm 0.001 $          & $ 1545.540   \\pm 0.005 $           & 2800      &  \\\\\n63291675      & $ 8.099  \\pm 0.003 $          & $ 1685.1     \\pm 0.01  $           & 60000     & Confirmed blend \\\\\n              & $ 1.4635 \\pm 0.0005 $         & $ 1683.8     \\pm 0.1 $             & 7000      &  \\\\\n63459761      & $4.3630 \\pm 0.003 $           & $ 1714.350   \\pm 0.005 $           & 160000    & Candidate multiple system   \\\\\n              & $4.235 \\pm  $ 0.005           & $ 1715.130   \\pm 0.03$             & 35000     &   \\\\\n104909909     & $1.3060 \\pm 0.0001$           & $ 1684.470   \\pm 0.005$            & 32000     & Candidate multiple system   \\\\\n              & $2.5750 \\pm 0.003$            & $ 1684.400   \\pm 0.005$            & 65000     &   \\\\\n115980439     & $ 4.615 \\pm 0.002  $          & $ 1818.05    \\pm 0.01  $           & 95000     & Confirmed blend \\\\\n              & $ 0.742 \\pm 0.005  $          & $ 1816.23    \\pm 0.02  $           & 2000      &   \\\\\n120362128     & $ 3.286 \\pm 0.002  $          & $ 1684.425   \\pm 0.01  $           & 33000     &  Candidate multiple system  \\\\\n              & $ -                 $         & $ 1701.275   \\pm 0.02  $           & 12000     &    \\\\\n              & $ -                 $         & $ 1702.09    \\pm 0.02  $           & 36000     &    \\\\\n121945407     & $  0.9056768 \\pm 0.00000002$   & $-1948.76377 \\pm  0.0000001$      & 2500          & Confirmed multiple system $^{(\\mathrm{a})}$  \\\\\n              & $ 45.4711 \\pm  0.00002$        & $-1500.0038  \\pm  0.0004 $       & 7500          &  \\\\\n122275115     & $ - $                         & $ 1821.779   \\pm 0.01  $           & 155000    &  Candidate multiple system  \\\\\n              & $ - $                         & $ 1830.628   \\pm 0.01  $           & 63000     &  \\\\\n              & $ - $                         & $ 1838.505   \\pm 0.01  $           & 123000    &  \\\\\n229804573     & $1.4641 \\pm 0.0005$           & $ 1326.135   \\pm 0.005$            & 180000    & Candidate multiple system  \\\\\n              & $0.5283 \\pm 0.0001$           & $ 1378.114   \\pm 0.005$            & 9000      &   \\\\\n252403752     & $ - $                         & $ 1817.73    \\pm 0.01 $            & 2800      & Candidate multiple system \\\\\n              & $ - $                         & $ 1829.76    \\pm 0.01 $            & 23000     & \\\\\n              & $ - $                         & $ 1833.63    \\pm 0.01 $            & 5500      & \\\\\n258837989     & $0.8870 \\pm 0.001$            & $ 1599.350   \\pm 0.005$            & 64000     & Candidate multiple system  \\\\\n              & $3.0730 \\pm 0.001$            & $ 1598.430   \\pm 0.005$            & 25000     &   \\\\\n266958963     & $1.5753 \\pm 0.0002$           & $ 1816.425   \\pm 0.001$            & 265000    & Candidate multiple system  \\\\\n              & $2.3685 \\pm 0.0001$           & $ 1817.790   \\pm 0.001$            & 75000     &   \\\\\n278956474     & $5.488068 \\pm 0.000016  $     & $ 1355.400   \\pm 0.005$            & 93900     & Confirmed multiple system $^{(\\mathrm{b})}$ \\\\\n              & $5.674256 \\pm  -0.000030$     & $ 1330.690   \\pm 0.005$            & 30000     &   \\\\\n284925600     & $ 1.24571 \\pm 0.00001 $       & $ 1765.248   \\pm 0.005 $           & 490000    & Confirmed blend \\\\\n              & $  0.31828 \\pm 0.00001  $     & $ 1764.75    \\pm 0.005 $           & 35000     &  \\\\\n293954660     & $2.814 \\pm  0.001 $           & $ 1739.177   \\pm 0.03  $           & 272000    & Confirmed blend \\\\\n              & $4.904 \\pm 0.03 $             & $ 1739.73    \\pm 0.01  $           & 9500      &  \\\\\n312353805     & $4.951 \\pm  0.003 $           & $ 1817.73    \\pm 0.01  $           & 66000     & Confirmed blend \\\\\n              & $12.89 \\pm   0.01  $          & $ 1822.28    \\pm 0.01$             & 19000     &  \\\\\n318210930     & $ 1.3055432 \\pm 0.000000033$  & $ -653.21602   \\pm 0.0000013$      & 570000    & Confirmed multiple system $^{(\\mathrm{c})}$ \\\\\n              & $ 0.22771622 \\pm 0.0000000035$& $ -732.071119 \\pm 0.00000026 $     & 220000    &  \\\\\n336434532     & $ 3.888 \\pm 0.002  $          & $ 1713.66    \\pm 0.01  $           & 22900     & Confirmed blend \\\\\n              & $  0.949 \\pm 0.003 $          & $ 1712.81    \\pm 0.01  $           & 2900      &  \\\\\n350622185     & $1.1686 \\pm 0.0001$           & $ 1326.140   \\pm 0.005$            & 200000    & Candidate multiple system  \\\\\n              & $5.2410 \\pm 0.0005$           & $ 1326.885   \\pm 0.05$             & 4000      &  \\\\\n375422201     & $9.9649 \\pm 0.001$            & $ 1711.937   \\pm 0.005$            & 245000    & Candidate multiple system  \\\\\n              & $4.0750 \\pm 0.001$            & $ 1713.210   \\pm 0.01 $            & 39000     &  \\\\\n376606423     & $ 0.8547 \\pm 0.0002  $        & $ 1900.766   \\pm 0.005 $           & 9700      & Candidate multiple system  \\\\\n              & $  -  $                       & $ 1908.085   \\pm 0.01  $           & 33000     &  \\\\\n394177355     & $ 94.22454  \\pm 0.00040 $     & $  -  $                            &  -        & Confirmed multiple system $^{(\\mathrm{d})}$ \\\\\n              & $ 8.6530941 \\pm 0.0000016$    & $-2038.99492 \\pm 0.00017 $         &  140000   &  \\\\\n              & $ 1.5222468 \\pm 0.0000025$    & $ -2039.1201 \\pm 0.0014  $         &  -        &  \\\\\n              & $ 1.43420486 \\pm 0.00000012 $ & $-2039.23941 \\pm 0.00007 $         &  -        &  \\\\\n424508303     & $ 2.0832649 \\pm 0.0000029  $ & $-3144.8661  \\pm 0.0034  $          &  430000   &  Confirmed multiple system $^{(\\mathrm{e})}$ \\\\\n              & $ 1.4200401  \\pm 0.0000042 $  & $-3142.5639  \\pm 0.0054  $         &  250000         &  \\\\\n441794509     & $ 4.6687 \\pm  0.0002       $  & $ 1958.895   \\pm 0.005   $         & 34000     &  Candidate multiple system  \\\\\n              & $ 14.785 \\pm  0.002       $   & $ 1960.845   \\pm 0.005   $         & 17000     &  \\\\\n470710327     & $ 9.9733 \\pm 0.0001 $         & $ 1766.27    \\pm 0.005  $          & 51000     &  Confirmed multiple system $^{(\\mathrm{f})}$ \\\\\n              & $ 1.104686 \\pm 0.00001 $      & $ 1785.53266 \\pm 0.000005$         & 42000     &  \\\\\n\\hline\n\\end{tabular}\n}\n\\caption{\nNote -- $^{(\\mathrm{a})}$ KOI-6139, \\citet{Borkovits2013};  \n$^{(\\mathrm{b})}$ \\citet{2020Rowden}\n$^{(\\mathrm{c})}$ \\citet{Koo2014}; \n$^{(\\mathrm{d})}$ KOI-3156, \\citet{2017Helminiak};\n$^{(\\mathrm{e})}$ V994 Her; \\citet{Zasche2016}; \n$^{(\\mathrm{f})}$ Eisner et al. {\\it in prep.}\n}\n\n\\label{tab:PHT-multis}\n\n\\end{table*}\n\n\\section*{Data Availability}\n\nAll of the \\emph{TESS}\\ data used within this article are hosted and made publicly available by the Mikulski Archive for Space Telescopes (MAST, \\url{http:\/\/archive.stsci.edu\/tess\/}). Similarly, the Planet Hunters TESS classifications made by the citizen scientists can be found on the Planet Hunters Analysis Database (PHAD, \\url{https:\/\/mast.stsci.edu\/phad\/}), which is also hosted by MAST. All planet candidates and their properties presented in this article have been uploaded to the Exoplanet Follow-up Observing Program for TESS (ExoFOP-TESS, \\url{ https:\/\/exofop.ipac.caltech.edu\/tess\/index.php}) website as community TOIs (cTOIs), under their corresponding TIC IDs. The ground-based follow-up observations of individual targets will be shared on reasonable request to the corresponding author.\n\nThe models of individual transit events and the data validation reports used for the vetting of the targets were both generated using publicly available open software codes, \\texttt{pyaneti}\\ and {\\sc latte}.\n\n\\section*{Acknowledgements}  \n\nThis project works under the in \\textit{populum veritas est} philosophy, and for that reason we would like to thank all of the citizen scientists who have taken part in the Planet Hunters TESS project and enable us to find many interesting astrophysical systems. \n\nSome of the observations in the paper made use of the High-Resolution Imaging instruments `Alopeke and Zorro. `Alopeke and Zorro were funded by the NASA Exoplanet Exploration Program and built at the NASA Ames Research Center by Steve B. Howell, Nic Scott, Elliott P. Horch, and Emmett Quigley. `Alopeke and Zorro were mounted on the Gemini North and South telescope of the international Gemini Observatory, a program of NSF's NOIRLab, which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation on behalf of the Gemini partnership: the National Science Foundation (United States), National Research Council (Canada), Agencia Nacional de Investigaci\\'{o}n y Desarrollo (Chile), Ministerio de Ciencia, Tecnolog\\'{i}a e Innovaci\\'{o}n (Argentina), Minist\\'{e}rio da Ci\\^{e}ncia, Tecnologia, Inova\\c{c}\\~{o}es e Comunica\\c{c}\\~{o}es (Brazil), and Korea Astronomy and Space Science Institute (Republic of Korea). The  authors also acknowledge the very significant cultural role and sacred nature of  Maunakea. We are most fortunate to  have  the opportunity to conduct observations from this mountain.\n\nThis project has also received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement N$^\\circ$730890. This material reflects only the authors views and the Commission is not liable for any use that may be made of the information contained therein. This work makes use of observations from the Las Cumbres Observatory global telescope network, including the NRES spectrograph and the SBIG and Sinistro photometric instruments. \n\nFurthermore, NLE thanks the LSSTC Data Science Fellowship Program, which is funded by LSSTC, NSF Cybertraining Grant N$^\\circ$1829740, the Brinson Foundation, and the Moore Foundation; her participation in the program has benefited this work. Finally, CJ acknowledges funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement N$^\\circ$670519: MAMSIE), and from the Research Foundation Flanders (FWO) under grant agreement G0A2917N (BlackGEM). \n\nThis research made use of Astropy, a community-developed core Python package for Astronomy \\citep{astropy2013}, matplotlib \\citep{matplotlib}, pandas \\citep{pandas}, NumPy \\citep{numpy}, astroquery \\citep{ginsburg2019astroquery} and sklearn \\citep{pedregosa2011scikit}.  \n\n\n\n\n\\bibliographystyle{mnras}\n\n\\section{Introduction}\n\nSince the first unambiguous discovery of an exoplanet in 1995 \\citep[][]{Mayor1995} over 4,000 more have been confirmed. Studies of their characteristics have unveiled an extremely wide range of planetary properties in terms of planetary mass, size, system architecture and orbital periods, greatly revolutionising our understanding of how these bodies form and evolve.\n\nThe transit method, whereby we observe a temporary decrease in the brightness of a star due to a planet passing in front of its host star, is to date the most successful method for planet detection, having discovered over 75\\% of the planets listed on the NASA Exoplanet Archive\\footnote{\\url{https:\/\/exoplanetarchive.ipac.caltech.edu\/}}. It yields a wealth of information including planet radius, orbital period, system orientation and potentially even atmospheric composition. Furthermore, when combined with Radial Velocity \\citep[RV; e.g.,][]{Mayor1995, Marcy1997} observations, which yield the planetary mass, we can infer planet densities, and thus their internal bulk compositions. Other indirect detection methods include radio pulsar timing \\citep[e.g.,][]{Wolszczan1992} and microlensing \\citep[e.g.,][]{Gaudi2012}.\n\n\nThe \\textit{Transiting Exoplanet Survey Satellite} mission \\citep[\\protect\\emph{TESS};][]{ricker15} is currently in its extended mission, searching for transiting planets orbiting bright ($V < 11$\\,mag) nearby stars. Over the course of the two year nominal mission, \\emph{TESS}\\ monitored around 85 per cent of the sky, split up into 26 rectangular sectors of 96 $\\times$ 24 deg each (13 per hemisphere). Each sector is monitored for $\\approx$ 27.4 continuous days, measuring the brightness of $\\approx$ 20,000 pre-selected stars every two minutes. In addition to these short cadence (SC) observations, the \\emph{TESS}\\ mission provides Full Frame Images (FFI) that span across all pixels of all CCDs and are taken at a cadence of 30 minutes. While most of the targets ($\\sim$ 63 per cent) will be observed for $\\approx$ 27.4 continuous days, around $\\sim$ 2 per cent of the targets at the ecliptic poles are located in the `continuous viewing zones' and will be continuously monitored for $\\sim$ 356 days.\n\nStars themselves are extremely complex, with phenomena ranging from outbursts to long and short term variability and oscillations, which manifest themselves in the light curves. These signals, as well as systematic effects and artifacts introduced by the telescope and instruments, mean that standard periodic search methods, such as the Box-Least-Squared method \\citep{bls2002} can struggle to identify certain transit events, especially if the observed signal is dominated by natural stellar variability. Standard detection pipelines also tend to bias the detection of short period planets, as they typically require a minimum of two transit events in order to gain the signal-to-noise ratio (SNR) required for detection.\n\nOne of the prime science goals of the \\emph{TESS}\\ mission is to further our understanding of the overall planet population, an active area of research that is strongly affected by observational and detection biases. In order for exoplanet population studies to be able to draw meaningful conclusions, they require a certain level of completeness in the sample of known exoplanets as well as a robust sample of validated planets spanning a wide range of parameter space. \\textcolor{red}{Due to this, we independently search the \\emph{TESS}\\ light curves for transiting planets via visual vetting in order to detect candidates that were either intentionally ignored by the main \\emph{TESS}\\ pipelines, which require at least two transits for a detection, missed because of stellar variability or instrumental artefacts, or were identified but subsequently erroneously discounted at the vetting stage, usually because the period found by the pipeline was incorrect. These candidates can help populate under-explored regions of parameter space and will, for example, benefit the study of planet occurrence rates around different stellar types as well as inform theories of physical processes involved with the formation and evolution of different types of exoplanets.}\n\nHuman brains excel in activities related to pattern recognition, making the task of identifying transiting events in light curves, even when the pattern is in the midst of a strong varying signal, ideally suited for visual vetting. Early citizen science projects, such as Planet Hunters \\citep[PH;][]{fischer12} and Exoplanet Explorers \\citep{Christiansen2018}, successfully harnessed the analytic power of a large number of volunteers and made substantial contributions to the field of exoplanet discoveries. The PH project, for example, showed that human vetting has a higher detection efficiency than automated detection algorithms for certain types of transits. In particular, they showed that citizen science can outperform on the detection of single (long-period) transits \\citep[e.g.,][]{wang13, schmitt14a}, aperiodic transits \\citep[e.g. circumbinary planets;][]{schwamb13} and planets around variable stars \\citep[e.g., young systems,][]{fischer12}. Both PH and Exoplanet Explorers, which are hosted by the world's largest citizen science platform Zooniverse \\citep{lintott08}, ensured easy access to \\textit{Kepler} and \\textit{K2} data by making them publicly available online in an immediately accessible graphical format that is easy to understand for non-specialists. The popularity of these projects is reflected in the number of participants, with PH attracting 144,466 volunteers from 137 different countries over 9 years of the project being active.\n\nFollowing the end of the \\textit{Kepler} mission and the launch of the \\emph{TESS}\\ satellite in 2018, PH was relaunched as the new citizen science project \\textit{Planet Hunters TESS} (PHT) \\footnote{\\url{www.planethunters.org}}, with the aim of identifying transit events in the \\emph{TESS}\\ data that were \\textcolor{red}{intentionally ignored or missed} by the main \\emph{TESS}\\ pipelines. \\textcolor{red}{Such a search complements other methods methods via its sensitivity to single-transit, and, therefore, longer period planets. Additionally, other dedicated non-citizen science based methods are also employed to look for single transit candidates \\citep[see e.g., the Bayesian transit fitting method by ][]{Gill2020, Osborn2016}}.\n\nCitizen science transit searches specialise in finding the rare events that the standard detection pipelines miss, however, these results are of limited use without an indication of the completeness of the search. Addressing the problem of completeness was therefore one of our highest priorities while designing PHT as discussed throughout this paper. \n\nThe layout of the remainder of the paper will be as follows. An overview of the Planet Hunters TESS project is found in Section~\\ref{sec:PHT}, followed by an in depth description of how the project identifies planet candidates in Section~\\ref{sec:method}. The recovery efficiency of the citizen science approach is assessed in Section~\\ref{sec:recovery_efficiency}, followed by a description of the in-depth vetting of candidates and ground based-follow up efforts in Section~\\ref{sec:vetting} and \\ref{sec:follow_up}, respectively. Planet Candidates and noteworthy systems identified by Planet Hunters TESS are outlined in Section~\\ref{sec:PHT_canidates}, followed by a discussion of the results in Section~\\ref{sec:condlusion}.\n\n\\section{Planet Hunters TESS}\n\\label{sec:PHT}\n\nThe PHT project works by displaying \\emph{TESS}\\ light curves (Figure~\\ref{fig:interface}), and asking volunteers to identify transit-like signals. Only the two-minute cadence targets, which are produced by the \\emph{TESS}\\ pipeline at the Science Processing Operations Center \\citep[SPOC,][]{Jenkins2018} and made publicly available by the Mikulski Archive for Space Telescopes (MAST)\\footnote{\\url{http:\/\/archive.stsci.edu\/tess\/}}, are searched by PHT. First-time visitors to the PHT site, or returning visitors who have not logged in are prompted to look through a short tutorial, which briefly explains the main aim of the project and shows examples of transit events and other stellar phenomena. Scientific explanation of the project can be found elsewhere on the site in the `field guide' and on the project's `About' page. \n\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=\\textwidth]{Figures\/PHT_new_interface.png}\n    \\caption{\n    PHT user interface showing a simulated light curve. The transit events are highlighted with white partially-transparent columns that are drawn on using the mouse. Stellar information on the target star is available by clicking on `subject info' below the light curve.} \n    \\label{fig:interface}\n\\end{figure*}\n\nAfter viewing the tutorial, volunteers are ready to participate in the project and are presented with \\emph{TESS}\\ light curves (known as `subjects') that need to be classified. The project was designed to be as simple as possible and therefore only asks one question: \\textit{`Do you see a transit?}'. Users identify transit-like events, and the time of their occurrence, by drawing a column over the event using the mouse button, as shown in Figure~\\ref{fig:interface}. There is no limit on the number of transit-like events that can be marked in a light curve. No markings indicate that there are no transit-like events present in the light curve. Once the subject has been analysed, users submit their classification and continue to view the next light curve by clicking `Done'. \n\nAlongside each light curve, users are offered information on the stellar properties of the target, such as the radius, effective temperature and magnitude (subject to availability, see \\cite{Stassun18}). However, in order to reduce biases in the classifications, the TESS Input Catalog (TIC) ID of the target star is not provided until after the subject classification has been submitted.\n\nIn addition to classifying the data, users are given the option to comment on light curves via the `Talk' discussion forum. Each light curve has its own discussion page to allow volunteers to discuss and comment, as well as to `tag' light curves using searchable hashtags, and to bring promising candidates to the attention of other users and the research team. The talk discussion forums complement the main PHT analysis and have been shown to yield interesting objects which may be challenging to detect using automated algorithms \\citep[e.g.,][]{eisner2019RN}. Unlike in the initial PH project, there are no questions in the main interface regarding stellar variability, however, volunteers are encouraged to mention astrophysical phenomenon or \\textit{unusual} features, such as eclipsing binaries or stellar flares, using the `Talk' discussion forum.  \n\nThe subject TIC IDs are revealed on the subject discussion pages, allowing volunteers to carry out further analysis on specific targets of interest and to report and discuss their findings. This is extremely valuable for both other volunteers and the PHT science team, as it can speed up the process of identifying candidates as well as rule out false positives in a fast and effective manner. \n\nSince the launch of PHT on 6 December 2018, there has been one significant makeover to the user interface. The initial PHT user interface (UI1), which was used for sectors 1 through 9, split the \\emph{TESS}\\ light curves up into either three or four chunks (depending on the data gaps in each sector) which lasted around seven days each. This allowed for a more `zoomed' in view of the data, making it easier to identify transit-like events than when the full $\\sim$ 30 day light curves were shown. The results from a PHT beta project, which displayed only simulated data, showed that a more zoomed in view of the light curve was likely to yield a higher transit recovery rate.\n\nThe updated, and current, user interface (UI2) allows users to manually zoom in on the x-axis (time) of the data. Due to this additional feature, each target has been displayed as a single light curve as of Sector 10. In order to verify that the changes in interface did not affect our findings, all of the Sector 9 subjects were classified using both UI1 and UI2. We saw no significant change in the number of candidates recovered (see Section~\\ref{sec:recovery_efficiency} for a description of how we quantified detection efficiency).\n\n\n\\subsection{Simulated Data}\n\\label{subsec:sims} \n\nIn addition to the real data, volunteers are shown simulated light curves, which are generated by randomly injecting simulated transit signals, provided by the SPOC pipeline \\citep[][]{Jenkins2018}, into real \\emph{TESS}\\ light curves. The simulated data play an important role in assessing the sensitivity of the project, training the users and providing immediate feedback, and to gauge the relative abilities of individual users (see Sec~\\ref{subsec:weighting}). \n\nWe calculate a signal to noise ratio (SNR) of the injected signal by dividing the injected transit depth by the Root Mean Square Combined Differential Photometric Precision (RMS CDPP) of the light curve on 0.5-, 1- or 2-hr time scales (whichever is closest to the duration of the injected transit signal). Only simulations with a SNR greater than 7 in UI1 and greater than 4 for UI2 are shown to volunteers.\n\nSimulated light curves are randomly shown to the volunteers and classified in the exact same manner as the real data. The user is always notified after a simulated light curve has been classified and given feedback as to whether the injected signal was correctly identified or not. For each sector, we generate between one and two thousand simulated light curves, using the real data from that sector in order to ensure that the sector specific systematic effects and data gaps of the simulated data do not differ from the real data. The rate at which a volunteer is shown simulated light curves decreases from an initial rate of 30 per cent for the first 10 classifications, down to a rate of 1 per cent by the time that the user has classified 100 light curves. \n\n\n\\section{Identifying Candidates}\n\\label{sec:method}\n\nEach subject is seen by multiple volunteers, before it is `retired' from the site, and the classifications are combined (see Section~\\ref{subsec:DBscan}) in order to assess the likelihood of a transit event. For sectors 1 through 9, the subjects were retired after 8 classifications if the first 8 volunteers who saw the light curves did not mark any transit events, after 10 classifications if the first 10 volunteers all marked a transit event and after 15 classifications if there was not complete consensus amongst the users. As of Sector 9 with UI2, all subjects were classified by 15 volunteers, regardless of whether or not any transit-like events were marked. Sector 9, which was classified with both UI1 and UI2, was also classified with both retirement rules.\n\nThere were a total of 12,617,038 individual classifications completed across the project on the nominal mission data. 95.4 per cent of these classifications were made by 22,341 registered volunteers, with the rest made by unregistered volunteers. Around 25 per cent of the registered volunteers complete more than 100 classifications, 11.8 per cent more than 300, 8.4 per cent more than 500, 5.4 per cent more than 1000 and 1.1 per cent more than 10,000. The registered volunteers completed a mean and median of 647 and 33 classifications, respectively. Figure~\\ref{fig:user_count} shows the distribution in user effort for logged in users who made between 0 and 300 classifications. \n\nThe distribution in the number of classifications made by the registered volunteers is assessed using the Gini coefficient, which ranges from 0 (equal contributions from all users) to 1 (large disparity in the contributions). The Gini coefficients for individual sectors ranges from 0.84 to 0.91 with a mean of 0.87, while the Gini coefficient for the overall project (all of the sectors combined) is 0.94. The mean Gini coefficient among other astronomy Zooniverse projects lies at 0.82 \\citep{spiers2019}. We note that the only other Zooniverse project with an equally high Gini coefficient as PHT is \\textit{Supernova Hunters}, a project which, similarly to PHT and unlike most other Zooniverse projects, has periodic data releases that are accompanied by an e-newsletter sent to all project volunteers. Periodic e-newsletters have the effect of promoting the project to both regularly and irregularly participating volunteers, who may only complete a couple of classifications as they explore the task, as well as to returning users who complete a large number of classifications following every data release, increasing the disparity in user contributions (the Gini coefficient).\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.5\\textwidth]{Figures\/user_count.png}\n    \\caption{\n    The distribution of the number of classifications by the registered volunteers, using a bin size of 5 from 0 to 300 classifications. A total of 11.8 per cent of the registered volunteers completed more than 300 classifications.} \n    \\label{fig:user_count}\n\\end{figure}\n\n\n\\subsection{User Weighting}\n\\label{subsec:weighting} \nUser weights are calculated for each individual volunteer in order to identify users who are more sensitive to detecting transit-like signals and those who are more likely to mark false positives. The weighting scheme is based on the weighting scheme described by \\cite{schwamb12}.\n\nUser weights are calculated independently for each observation sector, using the simulated light curves shown alongside the data from that sector. All users start off with a weighting of one, which is then increased or decreased when a simulated transit event is correctly or incorrectly identified, respectively. \n\nSimulated transits are deemed correctly identified, or `True', if the mid-point of a user's marking falls within the width of the simulated transit events. If none of the user's markings fall within this range, the simulated transit is deemed not identified, or `False'. If more than one of a user's markings coincide with the same simulated signal, it is only counted as being correct once, such that the total number of `True' markings cannot exceed the number of injected signals. For each classification, we record the number of  `Extra' markings, which is the total number of markings made by the user minus the number of correctly identified simulated transits. \n\nEach simulated light curve, identified by superscript $i$ (where $i=1$, \\ldots, $N$) was seen by $K^{(i)}$ users (the mean value of $K^{(i)}$\nwas 10), and contained $T^{(i)}$ simulated transits (where $T^{(i)}$ depends on the period of the simulated transit signal and the duration of the light curve). For a specific light curve $i$, each user who saw the light curve is identified by a subscript $k$ (where $k=1$, \\ldots, $K^{(i)}$) and each injected transit by a subscript $t$ (where $t=1$, \\ldots, $T^{(i)}$). \n\nIn order to distinguish between users who are able to identify obvious transits and those who are also able to find those that are more difficult to see, we start by defining a `recoverability' $r^{(i)}_t$ for each injected transit $t$ in each light curve. This is defined empirically, as the number of users who identified the transit correctly divided by $K^{(i)}$ (the total number of users who saw the light curve in question).\n\nNext, we quantify the performance of each user on each light curve as follows (this performance is analogous to the `seed' defined in \\citealt{schwamb12}, but we define it slightly differently):\n\\begin{equation}\n  p^{(i)}_{k} = C_{\\rm E} ~ \\frac{E^{(i)}_{k}}{\\langle E^{(i)} \\rangle} + \\sum_{t=1}^{T^{(i)}} \\begin{cases}\n    C_{\\rm T} ~ \\left[ r^{(i)}_t \\right]^{-1}, & \\text{if $m^{(i)}_{t,k} = $`True'}\\\\\n    C_{\\rm F} ~ r^{(i)}_t, & \\text{if $m^{(i)}_{t,k} = $`False'},\n  \\end{cases}\n\\end{equation}\nwhere $m^{(i)}_{t,k}$ is the identification of transit $t$ by user $k$ in light curve $i$, which is either `True' or `False'; $E^{(i)}_{k}$ is the number of `Extra' markings made by user $k$ for light curve $i$, and  $\\langle E^{(i)} \\rangle$ is the mean number of `Extra' markings made by all users who saw subject $i$. The parameters $C_{\\rm E}$, $C_{\\rm T}$ and $C_{\\rm F}$ control the impact of the `Extra', `True' and `False' markings on the overall user weightings, and are optimized empirically as discussed below in Section~\\ref{subsec:optimizesearch}. \n\nFollowing \\citealt{schwamb12}, we then assign a global `weight' $w_k$ to each user $k$, which is defined as:\n\\begin{equation}\n\\begin{split}\n\tw_k = I \\times (1 + \\log_{10} N_k)^{\\nicefrac{\\sum_i p^{(i)}_k}{N_k}}\n\\label{equ:weight}\n\\end{split}\n\\end{equation}\nwhere $I$ is an empirical normalization factor, such that the distribution of user weights remains centred on one, $N_k$ is the total number of simulated transit events that user $k$ assessed, and the sum over $i$ concerns only the light curves that user $k$ saw. \nWe limit the user weights to the range 0.05--3 \\emph{a posteriori}.\n\n\nWe experimented with a number of alternative ways to define the user weights, including the simpler $w_k=\\nicefrac{\\sum_i p^{(i)}_k}{N_k}$, but Eqn.~\\ref{equ:weight} was found to give the best results (see Section~\\ref{sec:recovery_efficiency} for how this was evaluated).\n\n\\subsection{Systematic Removal}\n\\label{subsec:sysrem} \nSystematic effects, for example caused by the spacecraft or background events, can result in spurious signals that affect a large subset of the data, resulting in an excess in markings of transit-like events at certain times within an observation sector. As the four \\emph{TESS}\\ cameras can yield unique systematic effects, the times of systematics were identified uniquely for each camera. The times were identified using a Kernel Density Estimation \\citep[KDE;][]{rosenblatt1956} with a cosine kernel and a bandwidth of 0.1 days, applied across all of the markings from that sector for each camera. Fig.~\\ref{fig:sys_rem} shows the KDE of all marked transit-events made during Sector 17 for TESS's cameras 1 (top panel) to 4 (bottom panel). The isolated spikes, or prominences, in the number of marked events, such as at T = 21-22 days in the bottom panel, are assumed to be caused by systematic effects that affect multiple light curves. Prominences are considered significant if they exceed a factor four times the standard deviation of the kernel output, which was empirically determined to be the highest cut-off to not miss clearly visible systematics. All user-markings within the full width at half maximum of these peaks are omitted from all further analysis. \\textcolor{red}{The KDE profiles for each Sector are provided as electronic supplementary material.}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.46\\textwidth]{Figures\/systematics_sec17.png}\n    \\caption{\n    Kernel density estimation of the user-markings made for Sector 17, for targets observed with TESS's observational Cameras 1 (top panel) to 4 (bottom panel). The orange vertical lines the indicate prominences that are at least four times greater than the standard deviation of the distribution. The black points underneath the figures show the mid-points of all of the volunteer-markings, where darker regions represent a higher density of markings.}\n    \\label{fig:sys_rem}\n\\end{figure}\n\n\\subsection{Density Based Clustering}\n\\label{subsec:DBscan} \n\nThe times and likelihoods of transit-like events are determined by combining all of the classifications made for each subject and identifying times where multiple volunteers identified a signal. We do this using an unsupervised machine learning method, known as DBSCAN \\citep[][Density-Based Spatial Clustering of Applications with Noise]{ester1996DB}. DBSCAN is a non-parametric density based clustering algorithm that helps to distinguish between dense clusters of data and sparse noise. For a data point to belong to a cluster it must be closer than a given distance ($\\epsilon$) to at least a set minimum number of other points (minPoints). \n\nIn our case, the data points are one-dimensional arrays of times of transits events, as identified by the volunteers, and clusters are times where multiple volunteers identified the same event. For each cluster a `transit score' ($s_i$) is determined, which is the sum of the user weights of the volunteers who contribute to the given cluster divided by the sum of the user weights of volunteers who saw that light curve. These transit scores are used to rank subjects from most to least likely to contain a transit-like event. Subjects which contain multiple successful clusters with different scores are ranked by the highest transit score. \n\n\\subsection{Optimizing the search}\n\\label{subsec:optimizesearch}\n\nThe methodology described in Sections~\\ref{subsec:weighting} to \\ref{subsec:DBscan} has five free parameters: the number of markings required to constitute a cluster ($minPoints$), the maximum separation of markings required for members of a cluster ($\\epsilon$), and $C_{\\rm E}$, $C_{\\rm T}$ and $C_{\\rm F}$ used in the weighting scheme. The values of these parameters were optimized via a grid search, where $C_{\\rm E}$ and $C_{\\rm F}$ ranged from -5 to 0, $C_{\\rm T}$ ranged from 0 to 20, and $minPoints$ ranged from 1 to 8, all in steps of 1. ($\\epsilon$) ranged from 0.5 to 1.5 in steps of 0.5. This grid search was carried out on 4 sectors, two from UI1 and two from UI2, for various variations of Equation~\\ref{equ:weight}. \n\nThe success of each combination of parameters was assessed by the fractions of TOIs and TCEs that were recovered within the top highest ranked 500 candidates, as discussed in more detail Section~\\ref{sec:recovery_efficiency}. We found the most successful combination of parameters to be $minPoints$ = 4 markings, $\\epsilon$, = 1 day, $C_{\\rm T}$ = 3, $C_{\\rm F}$= -2 and $C_{\\rm E}$ = -2.\n\n\\subsection{MAST deliverables}\n\\label{subsec:deliverables}\n\nThe analysis described above is carried out both in real-time as classifications are made, as well as offline after all of the light curves of a given sector have been classified. When the real-time analysis identifies a successful DB cluster (i.e. when at least four citizen scientists identified a transit within a day of the \\emph{TESS}\\ data of one another), the potential candidate is automatically uploaded to the open access Planet Hunters Analysis Database (PHAD) \\footnote{\\url{https:\/\/mast.stsci.edu\/phad\/}} hosted by the Mikulski Archive for Space Telescopes (MAST) \\footnote{\\url{https:\/\/archive.stsci.edu\/}}. While PHAD does not list every single classification made on PHT, it does display all transit candidates which had significant consensus amongst the volunteers who saw that light curve, along with the user-weight-weighted transit scores. This analysis does not apply the systematics removal described in Section~\\ref{subsec:sysrem}. The aim of PHAD is to provide an open source database of potential planet candidates identified by PHT, and to credit the volunteers who identified said targets. \n\nThe offline analysis is carried out following the complete classifications of all of the data from a given \\emph{TESS}\\ sector. The combination of all of the classifications allows us to identify and remove times of systematics and calculate better calibrated and more representative user weights. The remainder of this paper will only discuss the results from the offline analysis.\n\n\\section{Recovery Efficiency}\n\\label{sec:recovery_efficiency}\n\\subsection{Recovery of simulated transits}\n\nThe recovery efficiency is, in part, assessed by analysing the recovery rate of the injected transit-like signals (see Section~\\ref{subsec:sims}). Figure~\\ref{fig:SIM_recovery} shows the median and mean transit scores (fraction of volunteers who correctly identified a given transit scaled by user weights) of the simulated transits within SNR bins ranging from 4 to 20 in steps of 0.5. Simulations with a SNR less than 4 were not shown on PHT. The figure highlights that transit signals with a SNR of 7.5 or greater are correctly identified by the vast majority of volunteers. \n\n\\textcolor{red}{As the simulated data solely consist of real light curves with synthetically injected transit signals, we do not have any light curves, simulated or otherwise, which we can guarantee do not contain any planetary transits (real or injected). As such, this prohibits us from using simulated data to infer an analogous false-positive rate.}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.47\\textwidth]{Figures\/SIMS_recovery.png}\n    \\caption{The median (blue) and mean (orange) transit scores for injected transits with SNR ranges between 4 and 20. The mean and median are calculated in SNR bins with a width of 0.5, as indicated by the horizontal lines around each data point. \n    }\n    \\label{fig:SIM_recovery}\n\\end{figure}\n\n\\subsection{Recovery of TCEs and TOIs}\n\\label{subsec:TCE_TOI}\nThe recovery efficiency of PHT is assessed further using the planet candidates identified by the SPOC pipeline \\citep{Jenkins2018}. The SPOC pipeline extracts and processes all of the 2-minute cadence \\emph{TESS}\\ light curves prior to performing a large scale transit search. Data Validation (DV) reports, which include a range of transit diagnostic tests, are generated by the pipeline for around 1250 Threshold Crossing Events (TCEs), which were flagged as containing two or more transit-like features. Visual vetting is then performed by the \\emph{TESS}\\ science team on these targets, and promising candidates are added to the catalog of \\emph{TESS}\\ Objects of Interest (TOIs). Each sector yields around 80 TOIs \\textcolor{red}{and a mean of 1025 TCEs.}\n\nFig~\\ref{fig:TCE_TOI_recovery} shows the fraction of TOIs and TCEs (top and bottom panel respectively) that we recover with PHT as a function of the rank, where a higher rank corresponds to a lower transit score, for Sectors 1 through 26. TOIs and TCEs with R < 2 $R_{\\oplus}$ are not included in this analysis, as the initial PH showed that human vetting alone is unable to reliably recover planets smaller than 2 $R_{\\oplus}$ \\citep{schwamb12}. Planets smaller than 2 $R_{\\oplus}$ are, therefore, not the main focus of our search.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.45\\textwidth]{Figures\/TCE_TOI-recovery_radlim2.png}\n    \\caption{The fraction of recovered TOIs and TCEs (top and bottom panel respectively) with R > 2$R_{\\oplus}$ as a function of the rank, for sectors 1 to 26. The lines represent the results from different observation sectors.}\n    \\label{fig:TCE_TOI_recovery}\n\\end{figure}\n\n\nFig~\\ref{fig:TCE_TOI_recovery} shows a steep increase in the fractional TOI recovery rate up to a rank of $\\sim$ 500. Within the 500 highest ranked PHT candidates for a given sector, we are able to recover between 46 and 62 \\% (mean of 53 \\%) of all of the TOIs (R > 2 $R_{\\oplus}$), a median 90 \\% of the TOIs where the SNR of the transit events are greater than 7.5 and median 88 \\% of TOIs where the SNR of the transit events are greater than 5.\n\nThe relation between planet recovery rate and the SNR of the transit events is further highlighted in Figure~\\ref{fig:TOI_properties}, which shows the SNR vs the orbital period of the recovered TOIs. The colour of the markers indicate the TOI's rank within a given sector, with the lighter colours representing a lower rank. The circles and crosses represent candidates at a rank lower and higher than 500, respectively. The figure shows that transit events with a SNR less than 3.5 are missed by the majority of volunteers, whereas events with a SNR greater than 5 are mostly recovered within the top 500 highest ranked candidates. \n\nThe steep increase in the fractional TOI recovery rate at lower ranks, as shown in figure~\\ref{fig:TCE_TOI_recovery}, is therefore due to the detection of the high SNR candidates that are identified by most, if not all, of the PHT volunteers who classified those targets. At a rank of around 500, the SNR of the TOIs tends towards the limit of what human vetting can detect and thus the identification of TOIs beyond a rank of 500 is more sporadic.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.49\\textwidth]{Figures\/TOI_recovery_properties.png}\n    \\caption{The SNR vs orbital period of TOIs with R > 2$R_{\\oplus}$. The colour represents their rank within the sector, as determined by the weighted DB clustering algorithm. Circles indicate that they were identified at a rank < 500, while crosses indicate that they were not within the top 500 highest ranked candidates of a given sector.\n    }\n    \\label{fig:TOI_properties}\n\\end{figure}\n\nThe fractional TCE recovery rate (bottom panel of Figure~\\ref{fig:TCE_TOI_recovery}) is systematically lower than that of the TOIs. There are qualitative reasons as to why humans might not identify a TCE as opposed to a TOI, including that TCEs may be caused by artefacts or periodic stellar signals that the SPOC pipeline identified as a potential transit but that the human eye would either miss or be able to rule out as systematic effect. This leads to a lower recovery fraction of TCEs comparatively, an effect that is further amplified by the much larger number of TCEs.\n\nThe detection efficiency of PHT is estimated using the fractional recovery rate of TOIs for a range of radius and period bins, as shown in Figure~\\ref{fig:recovery_rank500_radius_period}. A TOI is considered to be recovered if its detection rank is less than 500 within the given sector. Out of the total 1913 TOIs, to date, \\textcolor{red}{PHT recovered 715 TOIs among the highest ranked candidates across the 26 sectors. This corresponds to a mean of 12.7~\\% of the top 500 ranked candidates per sector being TOIs. In comparison, the primary \\emph{TESS}\\ team on average visually vets 1025 TCEs per sector, out of which a mean of 17.3~\\% are promoted to TOI status.} We find that, independent of the orbital period, PHT is over 85~\\% complete in the recovery of TOIs with radii equal to or greater than 4 $R_{\\oplus}$. This agrees with the findings from the initial Planet Hunters project \\citep{schwamb12}. The detection efficiency decreases to 51~\\% for 3 - 4 $R_{\\oplus}$ TOIs, 49~\\% for 2 - 3 $R_{\\oplus}$ TOIs and to less than 40~\\% for TOIs with radii less than 2 $R_{\\oplus}$. Fig~\\ref{fig:recovery_rank500_radius_period} shows that the orbital period does not have a strong effect on the detection efficiency for periods greater than $\\sim$~1~day, which highlights that human vetting efficiency is independent of the number of transits present within a light curve. For periods shorter than around 1~day, the detection efficiency decreases even for larger planets, due to the high frequency of events seen in the light curve. For these light curves, many volunteers will only mark a subset of the transits, which may not overlap with the subset marked by other volunteers. Due to the methodology used to identify and rank the candidates, as described in Section~\\ref{sec:method}, this will actively disfavour the recovery of very short period planets. Although this obviously introduces biases in the detectability of very short period signals, the major detection pipelines are specifically designed to identify these types of planets and thus this does not present a serious detriment to our main science goal of finding planets that were \\textcolor{red}{intentionally ignored or missed} by the main automated pipelines.\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{Figures\/TOI_recovery_grid.png}\n    \\caption{TOI recovery rate as a function of planet radius and orbital period. A TOI is considered recovered if it is amongst the top 500 highest ranked candidates within a given sector. The logarithmically spaced grid ranges from 0.2 to 225 d and 0.6 to 55 $R{_\\oplus}$ for the orbital period and planet radius, respectively. The fraction of TOIs recovered using PHT is computed for each cell and represented by the colour the grid. Cells with less than 10 TOIs are considered incomplete for statistical analysis and are shown by the hatched lines. White cells contain no TOIs. The annotations for each cell indicate the number of recovered TOIs followed by the Poisson uncertainty in brackets. The filled in and empty grey circles indicated the recovered and not-recovered TOIs, respectively.}\n    \\label{fig:recovery_rank500_radius_period}\n\\end{figure*}\n\n\nFinally, we assessed whether the detection efficiency varies across different sectors by assessing the fraction of recovered TOIs and TCEs within the highest ranked 500 candidates. We found the recovery of TOIs within the top 500 highest ranked candidates to remain relatively constant across all sectors, while the fraction of recovered TCEs in the top 500 highest ranked candidates  increases in later sectors, as shown in Figure~\\ref{fig:recovery_rank500}). After applying a Spearman's rank test we find a positive correlation of 0.86 (pvalue = 5.9 $\\times$ $10^{-8}$) and 0.57 (pvalue = 0.003) between the observation sector and TCE and TOI recovery rates, respectively. These correlations suggest that the ability of users to detect transit-like events improves as they classify more subjects. The improvement of volunteers over time can also be seen in Fig~\\ref{fig:user_weights}, which shows the mean (unnormalized) user weight per sector for volunteers who completed one or more classifications in at least one sector (blue), more than 10 sectors (orange), more than 20 sectors (green) and all of the sectors 26 sectors from the nominal \\emph{TESS}\\ mission (pink). The figure highlights an overall improvement in the mean user weight in later sectors, as well as a positive correlation between the overall increase in user weight and the number of sectors that volunteers have participated in.\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.45\\textwidth]{Figures\/TCE_TOI_rank500.png}\n    \\caption{The fractional recovery rate of the TOIs (blue circles) and TCEs (teal squares) at a rank of 500 for each sector. Sector 1-9 (white background) represent southern hemisphere sectors classified with UI1, sectors 9-14 (light grey background) show the southern hemisphere sectors classified with UI2, and sectors 14-24 (dark grey background) show the northern hemisphere sectors classified with US2.}\n    \\label{fig:recovery_rank500}\n\\end{figure}\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.50\\textwidth]{Figures\/user_weights_sectors.png}\n    \\caption{Mean user weights per sector. The solid lines show the user weights for the old user interface and the dashed line for the new interface, separated by the black line (Sector 9). The different coloured lines show the mean user weights calculated considering user who participated in any number of sectors (blue), more than 10 sectors (orange), more than 20 sectors (green) and all of the sectors observed during the nominal \\emph{TESS}\\ mission (pink).}\n    \\label{fig:user_weights}\n\\end{figure}\n\n\n\\section{Candidate vetting}\n\\label{sec:vetting}\n\nFor each observation sector the subjects are ranked according to their transit scores, and the 500 highest ranked targets (excluding TOIs) visually vetted by the PHT science team in order to identify potential candidates and rule out false positives. A vetting cut-off rank of 500 was chosen as we found this to maximise the number of found candidates while minimising the number of likely false positives. In the initial round of vetting, which is completed via a separate Zooniverse classification interface that is only accessible to the core science team, a minimum of three members of the team sort the highest ranked targets into either `keep for further analysis', `eclipsing binary' or `discard'. The sorting is based on the inspection of the full \\emph{TESS}\\ light curve of the target, with the times of the satellite momentum dumps indicated. Additionally, around the time of each likely transit event (i.e. time of successful DB clusters) we inspect the background flux and the x and y centroid positions. Stellar parameters are provided for each candidate, subject to availability, alongside links to the SPOC Data Validation (DV) reports for candidates that had been flagged as TCEs but were never promoted to TOIs status.\n\nCandidates where at least two of the reviewers indicated that the signal is consistent with a planetary transit are kept for further analysis. \\textcolor{red}{This constitute a $\\sim$~5~\\% retention rate of the 500 highest ranked candidates per sector between the initial citizen science classification stage and the PHT science team vetting stage. Considering that the known planets and TOIs are not included at this stage of vetting, it is not surprising that our retention rate is lower that the true-positive rates of TCEs (see Section~\\ref{subsec:TCE_TOI}). Furthermore, this false-positive rate is consistent with the the findings of the initial Planet Hunters project \\citep{schwamb12}.}\n\nThe rest of the 500 candidates were grouped into $\\sim$~37~\\% `eclipsing binary' and $\\sim$~58~\\% `discard'. The most common reasons for discarding light curves are due to events caused by momentum dumps and due to background events, such as background eclipsing binaries, that mimic transit-like signals in the light curve. The targets identified as eclipsing binaries are analysed further by the \\emph{TESS}\\ Eclipsing Binaries Working Group (Prsa et al, in prep).\n\n\n\n\nFor the second round of candidate vetting we generate our own data validation reports for all candidates classified as `keep for further analysis'. The reports are generated using the open source software {\\sc latte} \\citep[Lightcurve Analysis Tool for Transiting Exoplanets;][]{LATTE2020}, which includes a range of standard diagnostic plots that are specifically designed to help identify transit-like signals and weed out astrophysical false positives in \\emph{TESS}\\ data. In brief the diagnostics consist of:\n\n\\textbf{Momentum Dumps}. The times of the \\emph{TESS}\\ reaction wheel momentum dumps that can result in instrumental effects that mimic astrophysical signals.\n\n\\textbf{Background Flux}. The background flux to help identify trends caused by background events such as asteroids or fireflies \\citep{vanderspek2018tess} passing through the field of view.\n\n\\textbf{x and y centroid positions}. The CCD column and row local position of the target's flux-weighted centroid, and the CCD column and row motion which considers differential velocity aberration (DVA), pointing drift, and thermal effects. This can help identify signals caused by systematics due to the satellite. \n\\textbf{Aperture size test}. The target light curve around the time of the transit-like event extracted using two apertures of different sizes. This can help identify signals resulting from background eclipsing binaries.\n \n\\textbf{Pixel-level centroid analysis}. A comparison between the average in-transit and average out-of-transit flux, as well as the difference between them. This can help identify signals resulting from background eclipsing binaries.\n\n\\textbf{Nearby companion stars}. The location of nearby stars brighter than V-band magnitude 15 as queried from the Gaia Data Release 2 catalog \\citep{gaia2018gaia} and the DSS2 red field of view around the target star in order to identify nearby contaminating sources. \n\n\\textbf{Nearest neighbour light curves}. Normalized flux light curves of the five short-cadence \\emph{TESS}\\ stars with the smallest projected distances to the target star, used to identify alternative sources of the signal or systematic effects that affect multiple target stars. \n\n\\textbf{Pixel level light curves}. Individual light curves extracted for each pixel around the target. Used to identify signals resulting from background eclipsing binaries, background events and systematics.\n\n\\textbf{Box-Least-Squares fit}. Results from two consecutive BLS searches, where the identified signals from the initial search are removed prior to the second BLS search.\n\nThe {\\sc latte} validation reports are assessed by the PHT science team in order to identify planetary candidates that warrant further investigation. Around 10~\\% of the targets assessed at this stage of vetting are kept for further investigation, resulting in $\\sim$~3 promising planet candidates per observation sector. The discarded candidates can be loosely categorized into (background) eclipsing binaries ($\\sim$~40~\\%), systematic effects ($\\sim$~25~\\%), background events ($\\sim$~15~\\%) and other (stellar signals such as spots; $\\sim$~10~\\%).\n\n\nWe use \\texttt{pyaneti}\\ \\citep{pyaneti} to infer the planetary and orbital parameters of our most promising candidates. For multi-transit candidates we fit for seven parameters per planet, time of mid-transit $T_0$, orbital period $P$, impact parameter $b$, scaled semi-major axis $a\/R_\\star$, scaled planet radius $r_{\\rm p}\/R_\\star$, and two limb darkening coefficients following a \\citet{Mandel2002} quadratic limb darkening model, implemented with the $q_1$ and $q_2$ parametrization suggested by \\citet{Kipping2013}. Orbits were assumed to be circular.\nFor the mono-transit candidates, we fit the same parameters as for the multi-transit case, except for the orbital period and scaled semi-major axis which cannot be known for single transits. We follow \\citet{Osborn2016} to estimate the orbital period of the mono-transit candidates assuming circular orbits.\n\nWe note that some of our candidates are V-shaped, consistent with a grazing transit configuration. For these cases, we set uniform priors between 0 and 0.15 for $r_{\\rm p}\/R_\\star$ and between 0 and 1.15 for the impact parameter in order to avoid large radii caused by the $r_{\\rm p}\/R_\\star - b$ degeneracy. Thus, the $r_{\\rm p}\/R_\\star$ for these candidates should not be trusted. A full characterisation of these grazing transits is out of the scope of this manuscript.\n\nFigure~\\ref{fig:PHT_pyaneti} shows the \\emph{TESS}\\ transits together with the inferred model for each candidate. Table~\\ref{tab:PHT-caniddates} shows the inferred main parameters, the values and their uncertainties are given by the median and 68.3\\% credible interval of the posterior distributions.\n\n\n\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=0.83\\textwidth]{Figures\/canidate_transits_all_one.png}\n    \\caption{All of the PHT candidates modelled using \\texttt{pyaneti}. The parameters of the best fits are summarised in Table~\\protect\\ref{tab:PHT-caniddates}. The blue and magenta fits show the multi and single transit event candidates, respectively.} \n    \\label{fig:PHT_pyaneti}\n\\end{figure*}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=0.83\\textwidth]{Figures\/canidate_transits_all_two.png}\n    \\addtocounter{figure}{-1}\n    \\caption{\\textbf{PHT candidates (continued)}} \n\\end{figure*}\n\n\nCandidates that pass all of our rounds of vetting are uploaded to the Exoplanet Follow-up Observing Program for TESS (ExoFOP-TESS) website\\footnote{\\url{ https:\/\/exofop.ipac.caltech.edu\/tess\/index.php}} as community TOIs (cTOIs).\n\n\\section{Follow-up observations}\n\\label{sec:follow_up}\n\nMany astrophysical false positive scenarios can be ruled out from the detailed examination of the \\emph{TESS}\\ data, both from the light curves themselves and from the target pixel files. However, not all of the false positive scenarios can be ruled out from these data alone, due in part to the large \\emph{TESS}\\ pixels (20 arcsconds). Our third stage of vetting, therefore, consists of following up the candidates with ground based observations including photometry, reconnaissance spectroscopy and speckle imaging. The results from these observations will be discussed in detail in a dedicated follow-up paper. \n\n\\subsection{Photometry}\n\nWe make use of the LCO global network of fully robotic 0.4-m\/SBIG and 1.0-m\/Sinistro facilities \\citep{LCO2013} to observe additional transits, where the orbital period is known, in order to refine the ephemeris and confirm that the transit events are not due to a blended eclipsing binary in the vicinity of the main target. Snapshot images are taken of single transit event candidates in order to identify nearby contaminating sources. \n\n\n\\subsection{Spectroscopy}\n\nWe perform high-resolution optical spectroscopy using telescopes from across the globe in order to cover a wide range of RA and Dec:\n\\begin{itemize}\n\\item The Las Cumbres Observatory (LCO) telescopes with the Network of Robotic Echelle Spectrographs \\citep[NRES,][]{LCO2013}. These fibre-fed spectrographs, mounted on 1.0-m telescopes around the globe, have a resolution of R = 53,000 and a wavelength coverage of 380 to 860 nm. \n\n\\item The MINERVA Australis Telescope facility, located at Mount Kent Observatory in Queensland, Australia \\citep{addison2019}. This facility is made up of four 0.7m CDK700 telescopes, which individually feed light via optic fibre into a KiwiSpec high-resolution (R = 80,000) stabilised spectrograph \\citep{barnes2012} that covers wavelengths from 480 nm to 620 nm. \n\n\\item The CHIRON spectrograph mounted on the SMARTS 1.5-m telescope \\citep{Tokovinin2018}, located at the Cerro Tololo\nInter-American Observatory (CTIO) in Chile. The high resolution cross-dispersed echelle spectrometer is fiber-fed followed by an image slicer. It has a resolution of R = 80,000 and covers wavelengths ranging from 410 to 870 nm.\n\n\\item The SOPHIE echelle spectrograph mounted on the 1.93-m Haute-Provence Observatory (OHP), France\n\\citep{2008Perruchot,2009Bouchy}. The high resolution cross-dispersed stabilized echelle spectrometer is fed by two optical fibers. Observations were taken in high-resolution mode (R = 75,000) with a wavelength range of 387 to 694 nm.\n\n\\end{itemize}\n\nReconnaissance spectroscopy with these instruments allow us to extract stellar parameters, identify spectroscopic binaries, and place upper limits on the companion masses. Spectroscopic binaries and targets whose spectral type is incompatible with the initial planet hypothesis and\/or precludes precision RV observations (giant or early type stars) are not followed up further. Promising targets, however, are monitored in order to constrain their period and place limits on their mass. \n\n\\subsection{Speckle Imaging}\n\nFor our most promising candidates we perform high resolution speckle imaging using the `Alopeke instrument on the 8.1-m Frederick C. Gillett Gemini North telescope in Maunakea, Hawaii, USA, and its twin, Zorro, on the 8.1-m Gemini South telescope on Cerro Pach\\'{o}n, Chile \\citep{Matson2019, Howell2011}. Speckle interferometric observations provide extremely high resolution images reaching the diffraction limit of the telescope. We obtain simultaneous 562 nm and 832 nm rapid exposure (60 msec) images in succession that effectively `freeze out' atmospheric turbulence and through Fourier analysis are used to search for close companion stars at 5-8 magnitude contrast levels. This analysis, along with the reconstructed images, allow us to identify nearby companions and to quantify their light contribution to the TESS aperture and thus the transit signal.\n\n\n\\section{Planet candidates and Noteworthy Systems}\n\\label{sec:PHT_canidates}\n\\subsection{Planet candidate properties}\n\nIn this final part of the paper we discuss the 90 PHT candidates around 88 host stars that passed the initial two stages of vetting and that were uploaded to ExoFOP as cTOIs. At the time of discovery none of these candidates were TOIs. The properties of all of the PHT candidates are summarised in Table~\\ref{tab:PHT-caniddates}. Candidates that have been promoted to TOI status since their PHT discovery are highlighted with an asterisk following the TIC ID, and candidates that have been shown to be false positives, based on the ground-based follow-up observations, are marked with a dagger symbol ($\\dagger$). The majority (81\\%) of PHT candidates are single transit events, indicated by an `s' following the orbital period presented in the table. \\textcolor{red}{18 of the PHT candidates were flagged as TCEs by the \\emph{TESS}\\ pipeline, but not initially promoted to TOI status. The most common reasons for this was that the pipeline identified a single-transit event as well as times of systematics (often caused by momentum dumps), due to its two-transit minimum detection threshold. This resulted in the candidate being discarded on the basis of it not passing the `odd-even' transit depth test. Out of the 18 TCEs, 14 have become TOI's since the PHT discovery. More detail on the TCE candidates can be found in Appendix~\\ref{appendixA}.} \n\nAll planet parameters (columns 2 to 8) are derived from the \\texttt{pyaneti}\\ modelling as described in Section~\\ref{sec:vetting}. Finally, the table summarises the ground-based follow-up observations (see Sec~\\ref{sec:follow_up}) that have been obtained to date, where the bracketed numbers following the observing instruments indicate the number of epochs. Unless otherwise noted, the follow-up observations are consistent with a planetary scenario. More in depth descriptions of individual targets for which we have additional information to complement the results in Table~\\ref{tab:PHT-caniddates} can be found in Appendix~\\ref{appendixA}.\n\n\\subsection{Planet candidate analysis}\n\n\nThe majority of the TOIs (87.7\\%) have orbital periods shorter than 15 days due to the requirement of observing at least two transits included in all major pipelines \\textcolor{red}{combined with the observing strategy of \\emph{TESS}}. As visual vetting does not impose these limits, the candidates outlined in this paper are helping to populate the relatively under-explored long-period region of parameter space. This is highlighted in Figure~\\ref{fig:PHT_candidates}, which shows the transit depths vs the orbital periods of the PHT single transit candidates (orange circles) and the multi-transit candidates (magenta squares) compared to the TOIs (blue circles). Values of the orbital periods and transit depths were obtained via transit modelling using \\texttt{pyaneti} (see Section~\\ref{sec:vetting}). The orbital period of single transit events are poorly constrained, which is reflected by the large errorbars in Figure~\\ref{fig:PHT_candidates}. Figure~\\ref{fig:PHT_candidates} also highlights that with PHT we are able to recover a similar range of transit depths as the pipeline found TOIs, as was previously shown in Figure~\\ref{fig:recovery_rank500_radius_period}.\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.45\\textwidth]{Figures\/PHT_candidate_period_depth_plot_errobrars.png}\n    \\caption{The properties of the PHT single transit (orange circles) and multi transit (magenta squares) candidates compared to the properties TOIs (blue circles). All parameters (listed in Table~\\ref{fig:PHT_candidates}) were extracted using \\texttt{pyaneti}\\ modelling.}\n    \\label{fig:PHT_candidates}\n\\end{figure}\n\nThe PHT candidates were further compared to the TOIs in terms of the properties of their host stars. Figure~\\ref{fig:eep} shows the effective temperature and stellar radii as taken from the TIC \\citep{Stassun18}, for TOIs (blue dots) and the PHT candidates (magenta circles). The solid and dashed lines indicate the main sequence and post-main sequence MIST stellar evolutionary tracks \\citep{choi2016}, respectively, for stellar masses ranging from 0.3 to 1.6 $M_\\odot$ in steps of 0.1 $M_\\odot$. This shows that around 10\\% of the host stars are in the process of, or have recently evolved off the main sequence. The models assume solar metalicity, no stellar rotation and no additional internal mixing.\n\n\\textcolor{red}{Ground based follow-up spectroscopy has revealed that six of the PHT candidates listed in Table~\\ref{tab:PHT-caniddates} are astrophysical false positives. As the follow-up campaign of the targets is still underway, the true false-positive rate of the candidates to have made it through all stages of the vetting process, as outlined in the methodology, will be be assessed in future PHT papers once the true nature of more of the candidates has been independently verified.}\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.45\\textwidth]{Figures\/PHT_eep.png}\n    \\caption{Stellar evolution tracks showing main sequence (solid black lines) and post-main sequence (dashed grey lines) MIST stellar evolution for stellar masses ranging from 0.3 to 1.6 $M_\\odot$ in steps of 0.1 $M_\\odot$. The blue dots show the TOIs and the magenta circles show the PHT candidates.} \n    \\label{fig:eep}\n\\end{figure}\n\n\n\\subsection{Stellar systems}\n\\label{subsec:PHT_stars}\n\nIn addition to the planetary candidates, citizen science allows for the identification of interesting stellar systems and astrophysical phenomena, in particular where the signals are aperiodic or small compared to the dominant stellar signal. These include light curves that exhibit multiple transit-like signals, possibly as a result of a multiple stellar system or a blend of eclipsing binaries. We have investigated all light curves that were flagged as possible multi-stellar systems via the PHT discussion boards. Similar to the planet vetting, as described in Section~\\ref{sec:vetting}, we generated {\\sc latte} data validation reports in order to assess the nature of the signal. Additionally, we subjected these systems to an iterative signal removal process, whereby we phase-folded the light curve on the dominant orbital period, binned the light curve into between 200-500 phase bins, created an interpolation model, and then subtracted said signal in order to evaluate the individual transit signals. The period of each signal, as listed in Table~\\ref{tab:PHT-multis}, was determined by phase folding the light curve at a number of trial periods and assessing by eye the best fit period and corresponding uncertainty.\n\nDue to the large \\emph{TESS}\\ pixels, blends are expected to be common. We searched for blends by generating phase folded light curves for each pixel around the source of the target in order to better locate the source of each signal. Shifts in the \\emph{TESS}\\ x and y centroid positions were also found to be good indicators of visually separated sources. Nearby sources with a magnitude difference greater than 5 mags were ruled out as possible contaminators. We consider a candidate to be a confirmed blend when the centroids are separated by more than 1 \\emph{TESS}\\ pixel, as this corresponds to an angular separation > 21 arcseconds meaning that the systems are highly unlikely to be gravitationally bound. Systems where the signal appears to be coming from the same \\emph{TESS}\\ pixel and that show no clear centroid shifts are considered to be candidate multiple systems. We note that blends are still possible, however, without further investigation we cannot conclusively rule these out as possible multi stellar systems. \n\nAll of the systems are summarised in Table~\\ref{tab:PHT-multis}. Out of the 26 systems, 6 are confirmed multiple systems which have either been published or are being prepared for publication; 7 are visually separated eclipsing binaries (confirmed blends); and 13 are candidate multiple system. Additional observations will be required to determine whether or not these candidate multiple systems are in fact gravitationally bound or photometric blends as a results of the large \\emph{TESS}\\ pixels or due to a line of sight happenstance. \n\n\\begin{landscape}\n\\begin{table}\n\\resizebox{1.31\\textwidth}{!}{\n\\begin{tabular}{ccccccccccccc}\n\\textbf{TIC} & \\textbf{Other} & \\textbf{Epoch} & \\textbf{Period}  & \\textbf{$R_{pl}$\/$R_{\\odot}$} & \\textbf{$R_{pl}$} & \\textbf{Impact} & \\textbf{Duration} & \\textbf{$V_{mag}$} & \\textbf{Photometry} & \\textbf{Spectroscopy} & \\textbf{Speckle} & \\textbf{Comment} \\\\\n  & \\textbf{Name} & \\textbf{(\\textcolor{red}{BJD - 2457000})} & \\textbf{(days)}  & &  $(R_{\\oplus})$ & \\textbf{Parameter} & \\textbf{(hours)} &   &   &   &   &    \\\\\n\\hline\n101641905 & TWOMASS 11412617+3441004 & $1917.26335 _{ - 0.00072 } ^ { + 0.00071 }$ & $14.52 _{ - 5.25 } ^ { + 6.21 }(s)$ & $0.1135 _{ - 0.0064 } ^ { + 0.0032 }$ & $9.76 _{ - 0.69 } ^ { + 0.65 }$ & $0.691 _{ - 0.183 } ^ { + 0.077 }$ & $3.163 _{ - 0.088 } ^ { + 0.093 }$ & 12.196 &  &  &  &  \\\\\n103633672* & TYC 4387-00923-1 & $1850.3211 _{ - 0.00077 } ^ { + 0.00135 }$ & $90.9 _{ - 23.7 } ^ { + 46.4 }(s)$ & $0.0395 _{ - 0.0013 } ^ { + 0.0013 }$ & $3.45 _{ - 0.24 } ^ { + 0.26 }$ & $0.3 _{ - 0.21 } ^ { + 0.26 }$ & $6.7 _{ - 0.11 } ^ { + 0.12 }$ & 10.586 &  & NRES (1) &  &  \\\\\n110996418 & TWOMASS 12344723-1019107 & $1580.6406 _{ - 0.0038 } ^ { + 0.0037 }$ & $5.18 _{ - 2.93 } ^ { + 6.86 }(s)$ & $0.1044 _{ - 0.0067 } ^ { + 0.008 }$ & $12.7 _{ - 0.99 } ^ { + 1.15 }$ & $0.44 _{ - 0.3 } ^ { + 0.3 }$ & $3.53 _{ - 0.27 } ^ { + 0.36 }$ & 13.945 &  &  &  &  \\\\\n128703021 & HIP 71639 & $1601.8442 _{ - 0.00108 } ^ { + 0.00093 }$ & $26.0 _{ - 8.22 } ^ { + 22.35 }(s)$ & $0.0254 _{ - 0.00049 } ^ { + 0.00072 }$ & $4.44 _{ - 0.2 } ^ { + 0.23 }$ & $0.47 _{ - 0.3 } ^ { + 0.22 }$ & $7.283 _{ - 0.091 } ^ { + 0.141 }$ & 6.06 &  & NRES (2);MINERVA (34) & Gemini &  \\\\\n138126035 & TYC 1450-00833-1 & $1954.3229 _{ - 0.0041 } ^ { + 0.0067 }$ & $28.8 _{ - 14.0 } ^ { + 203.2 }(s)$ & $0.0375 _{ - 0.0026 } ^ { + 0.0069 }$ & $4.01 _{ - 0.35 } ^ { + 0.74 }$ & $0.58 _{ - 0.38 } ^ { + 0.35 }$ & $4.65 _{ - 0.32 } ^ { + 0.85 }$ & 10.349 &  &  &  &  \\\\\n142087638 & TYC 9189-00274-1 & $1512.1673 _{ - 0.0043 } ^ { + 0.0034 }$ & $3.14 _{ - 1.41 } ^ { + 12.04 }(s)$ & $0.0469 _{ - 0.0035 } ^ { + 0.0063 }$ & $6.05 _{ - 0.54 } ^ { + 0.89 }$ & $0.5 _{ - 0.35 } ^ { + 0.36 }$ & $2.72 _{ - 0.23 } ^ { + 0.5 }$ & 11.526 &  &  &  &  \\\\\n159159904 & HIP 64812 & $1918.6109 _{ - 0.0067 } ^ { + 0.0091 }$ & $584.0 _{ - 215.0 } ^ { + 1724.0 }(s)$ & $0.0237 _{ - 0.0011 } ^ { + 0.0026 }$ & $3.12 _{ - 0.22 } ^ { + 0.36 }$ & $0.49 _{ - 0.34 } ^ { + 0.35 }$ & $15.11 _{ - 0.54 } ^ { + 0.7 }$ & 9.2 &  & NRES (2) &  &  \\\\\n160039081* & HIP 78892 & $1752.9261 _{ - 0.0045 } ^ { + 0.005 }$ & $30.19918 _{ - 0.00099 } ^ { + 0.00094 }$ & $0.0211 _{ - 0.0013 } ^ { + 0.0035 }$ & $2.67 _{ - 0.21 } ^ { + 0.43 }$ & $0.52 _{ - 0.34 } ^ { + 0.36 }$ & $4.93 _{ - 0.27 } ^ { + 0.37 }$ & 8.35 & SBIG (1) & NRES (1);SOPHIE (4) & Gemini &  \\\\\n162631539 & HIP 80264 & $1978.2794 _{ - 0.0044 } ^ { + 0.0051 }$ & $17.32 _{ - 6.66 } ^ { + 52.35 }(s)$ & $0.0195 _{ - 0.0011 } ^ { + 0.0024 }$ & $2.94 _{ - 0.24 } ^ { + 0.38 }$ & $0.48 _{ - 0.33 } ^ { + 0.36 }$ & $5.54 _{ - 0.33 } ^ { + 0.41 }$ & 7.42 &  &  &  &  \\\\\n166184426* & TWOMASS 13442500-4020122 & $1600.4409 _{ - 0.003 } ^ { + 0.0036 }$ & $16.3325 _{ - 0.0066 } ^ { + 0.0052 }$ & $0.0545 _{ - 0.0031 } ^ { + 0.0039 }$ & $1.85 _{ - 0.12 } ^ { + 0.15 }$ & $0.41 _{ - 0.28 } ^ { + 0.31 }$ & $1.98 _{ - 0.22 } ^ { + 0.17 }$ & 12.911 &  &  &  &  \\\\\n167661160 & TYC 7054-01577-1 & $1442.0703 _{ - 0.0028 } ^ { + 0.004 }$ & $36.802 _{ - 0.07 } ^ { + 0.069 }$ & $0.0307 _{ - 0.0014 } ^ { + 0.0024 }$ & $4.07 _{ - 0.32 } ^ { + 0.43 }$ & $0.37 _{ - 0.26 } ^ { + 0.33 }$ & $5.09 _{ - 0.23 } ^ { + 0.21 }$ & 9.927 &  & NRES (9);MINERVA (4) &  & EB from MINERVA observations \\\\\n172370679* & TWOMASS 19574239+4008357 & $1711.95923 _{ - 0.00099 } ^ { + 0.001 }$ & $32.84 _{ - 4.17 } ^ { + 5.59 }(s)$ & $0.1968 _{ - 0.0032 } ^ { + 0.0022 }$ & $13.24 _{ - 0.43 } ^ { + 0.43 }$ & $0.22 _{ - 0.15 } ^ { + 0.14 }$ & $4.999 _{ - 0.097 } ^ { + 0.111 }$ & 14.88 &  &  &  & Confirmed planet \\citep{canas2020}. \\\\\n174302697* & TYC 3641-01789-1 & $1743.7267 _{ - 0.00092 } ^ { + 0.00093 }$ & $498.2 _{ - 80.0 } ^ { + 95.3 }(s)$ & $0.07622 _{ - 0.00068 } ^ { + 0.00063 }$ & $13.34 _{ - 0.57 } ^ { + 0.58 }$ & $0.642 _{ - 0.029 } ^ { + 0.024 }$ & $17.71 _{ - 0.12 } ^ { + 0.13 }$ & 9.309 & SBIG (1) &  &  &  \\\\\n179582003 & TYC 9166-00745-1 & $1518.4688 _{ - 0.0016 } ^ { + 0.0016 }$ & $104.6137 _{ - 0.0022 } ^ { + 0.0022 }$ & $0.06324 _{ - 0.0008 } ^ { + 0.0008 }$ & $7.51 _{ - 0.35 } ^ { + 0.35 }$ & $0.21 _{ - 0.15 } ^ { + 0.19 }$ & $9.073 _{ - 0.084 } ^ { + 0.097 }$ & 10.806 &  &  &  &  \\\\\n192415680 & TYC 2859-00682-1 & $1796.0265 _{ - 0.0012 } ^ { + 0.0013 }$ & $18.47 _{ - 6.34 } ^ { + 21.73 }(s)$ & $0.0478 _{ - 0.0017 } ^ { + 0.0027 }$ & $4.43 _{ - 0.33 } ^ { + 0.38 }$ & $0.45 _{ - 0.31 } ^ { + 0.31 }$ & $3.94 _{ - 0.1 } ^ { + 0.12 }$ & 9.838 & SBIG (1) & SOPHIE (2) &  &  \\\\\n192790476 & TYC 7595-00649-1 & $1452.3341 _{ - 0.0014 } ^ { + 0.002 }$ & $16.09 _{ - 5.73 } ^ { + 15.49 }(s)$ & $0.0438 _{ - 0.0018 } ^ { + 0.0026 }$ & $3.24 _{ - 0.34 } ^ { + 0.37 }$ & $0.37 _{ - 0.25 } ^ { + 0.3 }$ & $3.395 _{ - 0.099 } ^ { + 0.192 }$ & 10.772 &  &  &  &  \\\\\n206361691$\\dagger$ & HIP 117250 & $1363.2224 _{ - 0.0082 } ^ { + 0.009 }$ & $237.7 _{ - 81.0 } ^ { + 314.4 }(s)$ & $0.01762 _{ - 0.00088 } ^ { + 0.00125 }$ & $2.69 _{ - 0.19 } ^ { + 0.25 }$ & $0.43 _{ - 0.28 } ^ { + 0.32 }$ & $13.91 _{ - 0.53 } ^ { + 0.52 }$ & 8.88 &  & CHIRON (2) &  & SB2 from CHIRON \\\\\n207501148 & TYC 3881-00527-1 & $2007.7273 _{ - 0.0011 } ^ { + 0.0011 }$ & $39.9 _{ - 10.3 } ^ { + 14.3 }(s)$ & $0.0981 _{ - 0.0047 } ^ { + 0.011 }$ & $13.31 _{ - 0.95 } ^ { + 1.56 }$ & $0.9 _{ - 0.03 } ^ { + 0.039 }$ & $4.73 _{ - 0.14 } ^ { + 0.14 }$ & 10.385 &  &  &  &  \\\\\n219466784* & TYC 4409-00437-1 & $1872.6879 _{ - 0.0097 } ^ { + 0.0108 }$ & $318.0 _{ - 147.0 } ^ { + 1448.0 }(s)$ & $0.0332 _{ - 0.0024 } ^ { + 0.0048 }$ & $3.26 _{ - 0.31 } ^ { + 0.49 }$ & $0.55 _{ - 0.39 } ^ { + 0.34 }$ & $10.06 _{ - 0.81 } ^ { + 1.12 }$ & 11.099 &  &  &  &  \\\\\n219501568 & HIP 79876 & $1961.7879 _{ - 0.0018 } ^ { + 0.002 }$ & $16.5931 _{ - 0.0017 } ^ { + 0.0015 }$ & $0.0221 _{ - 0.0012 } ^ { + 0.0015 }$ & $4.22 _{ - 0.3 } ^ { + 0.35 }$ & $0.41 _{ - 0.28 } ^ { + 0.31 }$ & $1.615 _{ - 0.077 } ^ { + 0.093 }$ & 8.38 &  &  &  &  \\\\\n229055790 & TYC 7492-01197-1 & $1337.866 _{ - 0.0022 } ^ { + 0.0019 }$ & $48.0 _{ - 12.8 } ^ { + 48.4 }(s)$ & $0.0304 _{ - 0.00097 } ^ { + 0.00115 }$ & $3.52 _{ - 0.2 } ^ { + 0.24 }$ & $0.37 _{ - 0.26 } ^ { + 0.32 }$ & $6.53 _{ - 0.11 } ^ { + 0.14 }$ & 9.642 &  & NRES (2) &  &  \\\\\n229608594 & TWOMASS 18180283+7428005 & $1960.0319 _{ - 0.0037 } ^ { + 0.0045 }$ & $152.4 _{ - 54.1 } ^ { + 152.6 }(s)$ & $0.0474 _{ - 0.0023 } ^ { + 0.0024 }$ & $3.42 _{ - 0.34 } ^ { + 0.36 }$ & $0.38 _{ - 0.26 } ^ { + 0.3 }$ & $6.98 _{ - 0.23 } ^ { + 0.37 }$ & 12.302 &  &  &  &  \\\\\n229742722* & TYC 4434-00596-1 & $1689.688 _{ - 0.025 } ^ { + 0.02 }$ & $29.0 _{ - 16.4 } ^ { + 66.3 }(s)$ & $0.019 _{ - 0.0028 } ^ { + 0.0029 }$ & $2.9 _{ - 0.44 } ^ { + 0.48 }$ & $0.44 _{ - 0.3 } ^ { + 0.33 }$ & $4.27 _{ - 0.09 } ^ { + 0.11 }$ & 10.33 &  & NRES (8);SOPHIE (4) & Gemini &  \\\\\n233194447 & TYC 4211-00650-1 & $1770.4924 _{ - 0.0065 } ^ { + 0.0107 }$ & $373.0 _{ - 101.0 } ^ { + 284.0 }(s)$ & $0.02121 _{ - 0.00073 } ^ { + 0.001 }$ & $5.08 _{ - 0.28 } ^ { + 0.33 }$ & $0.34 _{ - 0.24 } ^ { + 0.29 }$ & $24.45 _{ - 0.47 } ^ { + 0.5 }$ & 9.178 &  & NRES (2) & Gemini &  \\\\\n235943205 & TYC 4588-00127-1 & $1827.0267 _{ - 0.004 } ^ { + 0.0034 }$ & $121.3394 _{ - 0.0063 } ^ { + 0.0065 }$ & $0.0402 _{ - 0.0016 } ^ { + 0.0019 }$ & $4.2 _{ - 0.25 } ^ { + 0.29 }$ & $0.4 _{ - 0.27 } ^ { + 0.28 }$ & $6.37 _{ - 0.2 } ^ { + 0.3 }$ & 11.076 &  & NRES (1);SOPHIE (2) &  &  \\\\\n237201858 & TYC 4452-00759-1 & $1811.5032 _{ - 0.0069 } ^ { + 0.0067 }$ & $129.7 _{ - 41.5 } ^ { + 146.8 }(s)$ & $0.0258 _{ - 0.0013 } ^ { + 0.0015 }$ & $4.12 _{ - 0.27 } ^ { + 0.3 }$ & $0.4 _{ - 0.28 } ^ { + 0.31 }$ & $10.94 _{ - 0.37 } ^ { + 0.53 }$ & 10.344 &  & NRES (1) &  &  \\\\\n243187830* & HIP 5286 & $1783.7671 _{ - 0.0017 } ^ { + 0.0019 }$ & $4.05 _{ - 1.53 } ^ { + 9.21 }(s)$ & $0.0268 _{ - 0.0015 } ^ { + 0.0027 }$ & $2.06 _{ - 0.17 } ^ { + 0.23 }$ & $0.47 _{ - 0.32 } ^ { + 0.34 }$ & $2.02 _{ - 0.12 } ^ { + 0.15 }$ & 8.407 & SBIG (1) &  &  &  \\\\\n243417115 & TYC 8262-02120-1 & $1614.4796 _{ - 0.0028 } ^ { + 0.0022 }$ & $1.81 _{ - 0.73 } ^ { + 3.45 }(s)$ & $0.0523 _{ - 0.0035 } ^ { + 0.005 }$ & $5.39 _{ - 0.47 } ^ { + 0.64 }$ & $0.47 _{ - 0.33 } ^ { + 0.34 }$ & $2.03 _{ - 0.16 } ^ { + 0.23 }$ & 11.553 &  &  &  &  \\\\\n256429408 & TYC 4462-01942-1 & $1962.16 _{ - 0.0022 } ^ { + 0.0023 }$ & $382.0 _{ - 132.0 } ^ { + 265.0 }(s)$ & $0.03582 _{ - 0.00086 } ^ { + 0.00094 }$ & $6.12 _{ - 0.29 } ^ { + 0.3 }$ & $0.51 _{ - 0.36 } ^ { + 0.18 }$ & $16.96 _{ - 0.2 } ^ { + 0.24 }$ & 8.898 &  &  &  &  \\\\\n264544388* & TYC 4607-01275-1 & $1824.8438 _{ - 0.0076 } ^ { + 0.0078 }$ & $7030.0 _{ - 6260.0 } ^ { + 3330.0 }(s)$ & $0.0288 _{ - 0.0029 } ^ { + 0.0018 }$ & $4.58 _{ - 0.43 } ^ { + 0.35 }$ & $0.936 _{ - 0.363 } ^ { + 0.011 }$ & $19.13 _{ - 1.35 } ^ { + 0.84 }$ & 8.758 &  & NRES (1) &  &  \\\\\n264766922 & TYC 8565-01780-1 & $1538.69518 _{ - 0.00091 } ^ { + 0.00091 }$ & $3.28 _{ - 0.94 } ^ { + 1.25 }(s)$ & $0.0933 _{ - 0.0063 } ^ { + 0.0176 }$ & $16.95 _{ - 1.33 } ^ { + 3.19 }$ & $0.908 _{ - 0.039 } ^ { + 0.048 }$ & $2.73 _{ - 0.11 } ^ { + 0.11 }$ & 10.747 &  &  &  &  \\\\\n26547036* & TYC 3921-01563-1 & $1712.30464 _{ - 0.00041 } ^ { + 0.0004 }$ & $73.0 _{ - 13.6 } ^ { + 16.5 }(s)$ & $0.10034 _{ - 0.0007 } ^ { + 0.00078 }$ & $11.75 _{ - 0.59 } ^ { + 0.58 }$ & $0.17 _{ - 0.12 } ^ { + 0.11 }$ & $8.681 _{ - 0.049 } ^ { + 0.052 }$ & 9.849 &  & NRES (4) & Gemini &  \\\\\n267542728$\\dagger$ & TYC 4583-01499-1 & $1708.4956 _{ - 0.0073 } ^ { + 0.0085 }$ & $39.7382 _{ - 0.0023 } ^ { + 0.0023 }$ & $0.03267 _{ - 0.00089 } ^ { + 0.00175 }$ & $18.46 _{ - 0.94 } ^ { + 1.14 }$ & $0.38 _{ - 0.26 } ^ { + 0.27 }$ & $24.16 _{ - 0.39 } ^ { + 0.45 }$ & 11.474 &  &  &  & EB from HIRES RVs. \\\\\n270371513$\\dagger$ & HIP 10047 & $1426.2967 _{ - 0.0023 } ^ { + 0.002 }$ & $0.39 _{ - 0.17 } ^ { + 1.79 }(s)$ & $0.024 _{ - 0.0015 } ^ { + 0.0032 }$ & $4.8 _{ - 0.38 } ^ { + 0.64 }$ & $0.5 _{ - 0.34 } ^ { + 0.39 }$ & $1.93 _{ - 0.16 } ^ { + 0.19 }$ & 6.98515 &  & MINERVA (20) &  & SB 2 from MINERVA observations.  \\\\\n274599700 & TWOMASS 17011885+5131455 & $2002.1202 _{ - 0.0024 } ^ { + 0.0024 }$ & $32.9754 _{ - 0.005 } ^ { + 0.005 }$ & $0.0847 _{ - 0.0021 } ^ { + 0.0018 }$ & $13.25 _{ - 0.83 } ^ { + 0.83 }$ & $0.37 _{ - 0.24 } ^ { + 0.19 }$ & $8.2 _{ - 0.18 } ^ { + 0.21 }$ & 12.411 &  &  &  &  \\\\\n278990954 & TYC 8548-00717-1 & $1650.0191 _{ - 0.0086 } ^ { + 0.0105 }$ & $18.45 _{ - 8.66 } ^ { + 230.7 }(s)$ & $0.034 _{ - 0.0024 } ^ { + 0.0115 }$ & $9.65 _{ - 0.92 } ^ { + 3.13 }$ & $0.58 _{ - 0.4 } ^ { + 0.36 }$ & $10.62 _{ - 0.66 } ^ { + 2.46 }$ & 10.749 &  &  &  &  \\\\\n280865159* & TYC 9384-01533-1 & $1387.0749 _{ - 0.0045 } ^ { + 0.0044 }$ & $1045.0 _{ - 249.0 } ^ { + 536.0 }(s)$ & $0.0406 _{ - 0.0011 } ^ { + 0.0014 }$ & $4.75 _{ - 0.26 } ^ { + 0.28 }$ & $0.35 _{ - 0.24 } ^ { + 0.23 }$ & $19.08 _{ - 0.32 } ^ { + 0.36 }$ & 11.517 &  &  & Gemini &  \\\\\n284361752 & TYC 3924-01678-1 & $2032.093 _{ - 0.0078 } ^ { + 0.008 }$ & $140.6 _{ - 46.6 } ^ { + 159.1 }(s)$ & $0.0259 _{ - 0.0014 } ^ { + 0.0017 }$ & $3.62 _{ - 0.26 } ^ { + 0.31 }$ & $0.4 _{ - 0.27 } ^ { + 0.34 }$ & $8.98 _{ - 0.66 } ^ { + 0.86 }$ & 10.221 &  &  &  &  \\\\\n288240183 & TYC 4634-01225-1 & $1896.941 _{ - 0.0051 } ^ { + 0.0047 }$ & $119.0502 _{ - 0.0091 } ^ { + 0.0089 }$ & $0.02826 _{ - 0.00089 } ^ { + 0.00119 }$ & $4.28 _{ - 0.35 } ^ { + 0.36 }$ & $0.55 _{ - 0.37 } ^ { + 0.25 }$ & $17.49 _{ - 0.36 } ^ { + 0.6 }$ & 9.546 &  &  &  &  \\\\\n29169215 & TWOMASS 09011787+4727085 & $1872.5047 _{ - 0.0032 } ^ { + 0.0036 }$ & $14.89 _{ - 6.12 } ^ { + 24.84 }(s)$ & $0.0403 _{ - 0.0025 } ^ { + 0.0033 }$ & $3.28 _{ - 0.37 } ^ { + 0.45 }$ & $0.44 _{ - 0.3 } ^ { + 0.33 }$ & $3.56 _{ - 0.21 } ^ { + 0.32 }$ & 11.828 &  &  &  &  \\\\\n293649602 & TYC 8103-00266-1 & $1511.2109 _{ - 0.004 } ^ { + 0.0037 }$ & $12.85 _{ - 5.34 } ^ { + 42.21 }(s)$ & $0.04 _{ - 0.0024 } ^ { + 0.0039 }$ & $4.66 _{ - 0.36 } ^ { + 0.5 }$ & $0.5 _{ - 0.35 } ^ { + 0.34 }$ & $4.1 _{ - 0.31 } ^ { + 0.56 }$ & 10.925 &  &  &  &  \\\\\n296737508 & TYC 5472-01060-1 & $1538.0036 _{ - 0.0015 } ^ { + 0.0016 }$ & $18.27 _{ - 5.06 } ^ { + 17.45 }(s)$ & $0.0425 _{ - 0.0014 } ^ { + 0.0019 }$ & $5.33 _{ - 0.22 } ^ { + 0.27 }$ & $0.44 _{ - 0.3 } ^ { + 0.26 }$ & $5.13 _{ - 0.13 } ^ { + 0.15 }$ & 9.772 & Sinistro (1) & NRES (1);MINERVA (1) & Gemini &  \\\\\n298663873 & TYC 3913-01781-1 & $1830.76819 _{ - 0.00099 } ^ { + 0.00099 }$ & $479.9 _{ - 89.4 } ^ { + 109.4 }(s)$ & $0.06231 _{ - 0.00034 } ^ { + 0.00045 }$ & $11.07 _{ - 0.57 } ^ { + 0.57 }$ & $0.16 _{ - 0.11 } ^ { + 0.13 }$ & $23.99 _{ - 0.093 } ^ { + 0.1 }$ & 9.162 &  & NRES (2) & Gemini & Dalba et al. (in prep) \\\\\n303050301 & TYC 6979-01108-1 & $1366.1301 _{ - 0.0022 } ^ { + 0.0023 }$ & $281.0 _{ - 170.0 } ^ { + 264.0 }(s)$ & $0.0514 _{ - 0.0027 } ^ { + 0.0018 }$ & $4.85 _{ - 0.32 } ^ { + 0.32 }$ & $0.73 _{ - 0.48 } ^ { + 0.1 }$ & $7.91 _{ - 0.31 } ^ { + 0.36 }$ & 10.048 &  & NRES (1) & Gemini &  \\\\\n303317324 & TYC 6983-00438-1 & $1365.1845 _{ - 0.0023 } ^ { + 0.0028 }$ & $69.0 _{ - 25.5 } ^ { + 78.1 }(s)$ & $0.0365 _{ - 0.0013 } ^ { + 0.0016 }$ & $2.88 _{ - 0.3 } ^ { + 0.31 }$ & $0.39 _{ - 0.26 } ^ { + 0.32 }$ & $5.78 _{ - 0.18 } ^ { + 0.24 }$ & 10.799 &  &  &  &  \\\\\n\\hline\n\\end{tabular}}\n\\caption{\\emph{Note} -- Candidates that have become TOIs following the PHT discovery are marked with an asterisk (*). The `s' following the orbital period indicates that the candidates is a single transit event. The ground-based follow-up observations are summarized in columns 10-12, where the bracketed numbers correspond the number of epochs obtained with each instrument.  See Section~\\ref{sec:follow_up} for description of each instrument. The $\\dagger$ symbol indicates candidates that have been shown to be astrophysical false  positives based on the ground based follow-up observations.}\n\\label{tab:PHT-caniddates}\n\\end{table}\n\\end{landscape}\n\n\\begin{landscape}\n\\begin{table}\n\\addtocounter{table}{-1}\n\\resizebox{1.31\\textwidth}{!}{\n\\begin{tabular}{ccccccccccccc}\n\\textbf{TIC} & \\textbf{Other} & \\textbf{Epoch} & \\textbf{Period}  & \\textbf{$R_{pl}$\/$R_{\\odot}$} & \\textbf{$R_{pl}$} & \\textbf{Impact} & \\textbf{Duration} & \\textbf{$V_{mag}$} & \\textbf{Photometry} & \\textbf{Spectroscopy} & \\textbf{Speckle} & \\textbf{Comment} \\\\\n  & \\textbf{Name} & \\textbf{(\\textcolor{red}{BJD - 2457000})} & \\textbf{(days)}  & & $(R_{\\oplus})$  & \\textbf{Parameter} & \\textbf{(hours)} &   &   &   &   &   \\\\\n\\hline\n303586471$\\dagger$ & HIP 115828 & $1363.7692 _{ - 0.0033 } ^ { + 0.0027 }$ & $13.85 _{ - 4.19 } ^ { + 18.2 }(s)$ & $0.0214 _{ - 0.001 } ^ { + 0.0014 }$ & $2.52 _{ - 0.16 } ^ { + 0.2 }$ & $0.4 _{ - 0.27 } ^ { + 0.33 }$ & $4.23 _{ - 0.19 } ^ { + 0.16 }$ & 8.27 &  & MINERVA (11) &  & SB 2 from MINERVA observations.  \\\\\n304142124* & HIP 53719 & $1585.28023 _{ - 0.0008 } ^ { + 0.0008 }$ & $42.8 _{ - 10.0 } ^ { + 18.2 }(s)$ & $0.04311 _{ - 0.00093 } ^ { + 0.00153 }$ & $4.1 _{ - 0.23 } ^ { + 0.24 }$ & $0.33 _{ - 0.21 } ^ { + 0.21 }$ & $5.66 _{ - 0.067 } ^ { + 0.09 }$ & 8.62 &  & NRES (1);MINERVA (4) &  & Confirmed planet \\citep{diaz2020} \\\\\n304339227 & TYC 9290-01087-1 & $1673.3242 _{ - 0.009 } ^ { + 0.0128 }$ & $111.9 _{ - 72.2 } ^ { + 4844.1 }(s)$ & $0.0253 _{ - 0.0024 } ^ { + 0.0481 }$ & $3.27 _{ - 0.61 } ^ { + 5.72 }$ & $0.67 _{ - 0.47 } ^ { + 0.36 }$ & $7.44 _{ - 0.86 } ^ { + 2.84 }$ & 9.169 &  &  &  &  \\\\\n307958020 & TYC 4191-00309-1 & $1864.82 _{ - 0.014 } ^ { + 0.013 }$ & $169.0 _{ - 107.0 } ^ { + 10194.0 }(s)$ & $0.0223 _{ - 0.0022 } ^ { + 0.0543 }$ & $3.92 _{ - 0.52 } ^ { + 9.27 }$ & $0.71 _{ - 0.53 } ^ { + 0.33 }$ & $12.48 _{ - 1.1 } ^ { + 5.41 }$ & 9.017 &  &  &  &  \\\\\n308301091 & TYC 2081-01273-1 & $2030.3691 _{ - 0.0024 } ^ { + 0.0026 }$ & $29.24 _{ - 8.49 } ^ { + 22.46 }(s)$ & $0.0362 _{ - 0.0013 } ^ { + 0.0014 }$ & $5.41 _{ - 0.34 } ^ { + 0.35 }$ & $0.35 _{ - 0.25 } ^ { + 0.29 }$ & $6.57 _{ - 0.14 } ^ { + 0.19 }$ & 10.273 &  &  &  &  \\\\\n313006381 & HIP 45012 & $1705.687 _{ - 0.0081 } ^ { + 0.0045 }$ & $21.56 _{ - 8.9 } ^ { + 54.15 }(s)$ & $0.0261 _{ - 0.0017 } ^ { + 0.0027 }$ & $2.34 _{ - 0.2 } ^ { + 0.27 }$ & $0.45 _{ - 0.3 } ^ { + 0.38 }$ & $3.85 _{ - 0.51 } ^ { + 0.31 }$ & 9.39 &  &  &  &  \\\\\n323295479* & TYC 9506-01881-1 & $1622.9258 _{ - 0.00083 } ^ { + 0.00087 }$ & $117.8 _{ - 25.8 } ^ { + 30.9 }(s)$ & $0.0981 _{ - 0.0021 } ^ { + 0.0023 }$ & $11.35 _{ - 0.67 } ^ { + 0.66 }$ & $0.839 _{ - 0.024 } ^ { + 0.019 }$ & $6.7 _{ - 0.14 } ^ { + 0.15 }$ & 10.595 &  &  &  &  \\\\\n328933398.01* & TYC 4634-01435-1 & $1880.9878 _{ - 0.0039 } ^ { + 0.0042 }$ & $24.9335 _{ - 0.0046 } ^ { + 0.005 }$ & $0.0437 _{ - 0.0022 } ^ { + 0.0023 }$ & $4.62 _{ - 0.32 } ^ { + 0.33 }$ & $0.38 _{ - 0.25 } ^ { + 0.27 }$ & $5.02 _{ - 0.22 } ^ { + 0.27 }$ & 11.215 &  &  &  & Potential multi-planet system. \\\\\n328933398.02* & TYC 4634-01435-1 & $1848.6557 _{ - 0.0053 } ^ { + 0.0072 }$ & $50.5 _{ - 22.4 } ^ { + 77.1 }(s)$ & $0.0296 _{ - 0.0028 } ^ { + 0.0033 }$ & $3.14 _{ - 0.33 } ^ { + 0.39 }$ & $0.41 _{ - 0.28 } ^ { + 0.35 }$ & $5.99 _{ - 0.8 } ^ { + 0.77 }$ & 11.215 &  &  &  &  \\\\\n331644554 & TYC 3609-00469-1 & $1757.0354 _{ - 0.0031 } ^ { + 0.0033 }$ & $947.0 _{ - 215.0 } ^ { + 274.0 }(s)$ & $0.12 _{ - 0.025 } ^ { + 0.021 }$ & $21.84 _{ - 4.57 } ^ { + 3.86 }$ & $1.018 _{ - 0.036 } ^ { + 0.028 }$ & $10.93 _{ - 0.34 } ^ { + 0.35 }$ & 9.752 &  &  &  &  \\\\\n332657786 & TWOMASS 09595797-1609323 & $1536.7659 _{ - 0.0015 } ^ { + 0.0015 }$ & $63.76 _{ - 9.52 } ^ { + 11.13 }(s)$ & $0.14961 _{ - 0.00064 } ^ { + 0.00029 }$ & $3.83 _{ - 0.12 } ^ { + 0.12 }$ & $0.059 _{ - 0.041 } ^ { + 0.064 }$ & $3.333 _{ - 0.095 } ^ { + 0.096 }$ & 15.99 &  &  &  &  \\\\\n336075472 & TYC 3526-00332-1 & $2028.1762 _{ - 0.0043 } ^ { + 0.0037 }$ & $61.9 _{ - 24.0 } ^ { + 95.6 }(s)$ & $0.0402 _{ - 0.0022 } ^ { + 0.0033 }$ & $3.09 _{ - 0.34 } ^ { + 0.4 }$ & $0.43 _{ - 0.29 } ^ { + 0.32 }$ & $5.39 _{ - 0.23 } ^ { + 0.37 }$ & 11.842 &  &  &  &  \\\\\n349488688.01 & TYC 1529-00224-1 & $1994.283 _{ - 0.0038 } ^ { + 0.0033 }$ & $11.6254 _{ - 0.005 } ^ { + 0.0052 }$ & $0.02195 _{ - 0.00096 } ^ { + 0.00122 }$ & $3.44 _{ - 0.18 } ^ { + 0.21 }$ & $0.39 _{ - 0.27 } ^ { + 0.3 }$ & $5.58 _{ - 0.15 } ^ { + 0.18 }$ & 8.855 &  & NRES (2);SOPHIE (2) &  & Potential multi-planet system. \\\\\n349488688.02 & TYC 1529-00224-1 & $2002.77063 _{ - 0.00097 } ^ { + 0.00103 }$ & $15.35 _{ - 1.94 } ^ { + 4.15 }(s)$ & $0.03688 _{ - 0.00067 } ^ { + 0.00069 }$ & $5.78 _{ - 0.18 } ^ { + 0.18 }$ & $0.24 _{ - 0.16 } ^ { + 0.21 }$ & $6.291 _{ - 0.058 } ^ { + 0.074 }$ & 8.855 &  & NRES (2);SOPHIE (2) &  &  \\\\\n356700488* & TYC 4420-01295-1 & $1756.638 _{ - 0.013 } ^ { + 0.011 }$ & $184.5 _{ - 64.7 } ^ { + 333.1 }(s)$ & $0.0173 _{ - 0.0011 } ^ { + 0.0015 }$ & $2.92 _{ - 0.2 } ^ { + 0.28 }$ & $0.44 _{ - 0.3 } ^ { + 0.34 }$ & $11.76 _{ - 0.65 } ^ { + 1.03 }$ & 8.413 &  &  &  &  \\\\\n356710041* & TYC 1993-00419-1 & $1932.2939 _{ - 0.0019 } ^ { + 0.0019 }$ & $29.6 _{ - 14.0 } ^ { + 19.0 }(s)$ & $0.0496 _{ - 0.0021 } ^ { + 0.0011 }$ & $14.82 _{ - 0.85 } ^ { + 0.84 }$ & $0.66 _{ - 0.42 } ^ { + 0.11 }$ & $12.76 _{ - 0.24 } ^ { + 0.24 }$ & 9.646 &  &  &  &  \\\\\n369532319 & TYC 2743-01716-1 & $1755.8158 _{ - 0.006 } ^ { + 0.0051 }$ & $35.4 _{ - 12.0 } ^ { + 51.6 }(s)$ & $0.0316 _{ - 0.0023 } ^ { + 0.0028 }$ & $3.43 _{ - 0.3 } ^ { + 0.37 }$ & $0.41 _{ - 0.29 } ^ { + 0.34 }$ & $5.5 _{ - 0.32 } ^ { + 0.32 }$ & 10.594 &  &  & Gemini &  \\\\\n369779127 & TYC 9510-00090-1 & $1643.9403 _{ - 0.0046 } ^ { + 0.0058 }$ & $9.93 _{ - 3.38 } ^ { + 19.74 }(s)$ & $0.0288 _{ - 0.0015 } ^ { + 0.0033 }$ & $4.89 _{ - 0.31 } ^ { + 0.56 }$ & $0.46 _{ - 0.31 } ^ { + 0.33 }$ & $5.64 _{ - 0.38 } ^ { + 0.33 }$ & 9.279 &  &  &  &  \\\\\n384159646* & TYC 9454-00957-1 & $1630.39405 _{ - 0.00079 } ^ { + 0.00079 }$ & $11.68 _{ - 2.75 } ^ { + 4.21 }(s)$ & $0.0658 _{ - 0.0012 } ^ { + 0.0011 }$ & $9.87 _{ - 0.45 } ^ { + 0.44 }$ & $0.27 _{ - 0.18 } ^ { + 0.21 }$ & $5.152 _{ - 0.069 } ^ { + 0.087 }$ & 10.158 & SBIG (1) & NRES (8);MINERVA (6) & Gemini &  \\\\\n385557214 & TYC 1807-00046-1 & $1791.58399 _{ - 0.00068 } ^ { + 0.0007 }$ & $5.62451 _{ - 0.0004 } ^ { + 0.00043 }$ & $0.096 _{ - 0.019 } ^ { + 0.032 }$ & $8.32 _{ - 2.06 } ^ { + 2.77 }$ & $0.95 _{ - 0.075 } ^ { + 0.053 }$ & $1.221 _{ - 0.094 } ^ { + 0.058 }$ & 10.856 &  &  &  &  \\\\\n388134787 & TYC 4260-00427-1 & $1811.034 _{ - 0.015 } ^ { + 0.017 }$ & $246.0 _{ - 127.0 } ^ { + 6209.0 }(s)$ & $0.0265 _{ - 0.0024 } ^ { + 0.023 }$ & $2.57 _{ - 0.28 } ^ { + 2.19 }$ & $0.55 _{ - 0.39 } ^ { + 0.44 }$ & $8.85 _{ - 1.13 } ^ { + 1.84 }$ & 10.95 &  & NRES (1) & Gemini &  \\\\\n404518509 & HIP 16038 & $1431.2696 _{ - 0.0037 } ^ { + 0.0035 }$ & $26.83 _{ - 9.46 } ^ { + 56.14 }(s)$ & $0.0259 _{ - 0.0013 } ^ { + 0.0022 }$ & $2.94 _{ - 0.21 } ^ { + 0.29 }$ & $0.47 _{ - 0.31 } ^ { + 0.34 }$ & $5.02 _{ - 0.23 } ^ { + 0.28 }$ & 9.17 &  &  &  &  \\\\\n408636441* & TYC 4266-00736-1 & $1745.4668 _{ - 0.0016 } ^ { + 0.0015 }$ & $37.695 _{ - 0.0034 } ^ { + 0.0033 }$ & $0.0485 _{ - 0.0019 } ^ { + 0.0023 }$ & $3.32 _{ - 0.16 } ^ { + 0.19 }$ & $0.39 _{ - 0.27 } ^ { + 0.29 }$ & $3.63 _{ - 0.1 } ^ { + 0.14 }$ & 11.93 & SBIG (1) &  & Gemini & Half of the period likely. \\\\\n418255064 & TWOMASS 13063680-8037015 & $1629.3304 _{ - 0.0018 } ^ { + 0.0018 }$ & $25.37 _{ - 7.06 } ^ { + 15.41 }(s)$ & $0.0732 _{ - 0.0029 } ^ { + 0.0031 }$ & $5.57 _{ - 0.36 } ^ { + 0.38 }$ & $0.37 _{ - 0.25 } ^ { + 0.25 }$ & $3.83 _{ - 0.13 } ^ { + 0.14 }$ & 12.478 & SBIG (1) &  & Gemini &  \\\\\n420645189$\\dagger$ & TYC 4508-00478-1 & $1837.4767 _{ - 0.0018 } ^ { + 0.0017 }$ & $250.2 _{ - 66.6 } ^ { + 99.4 }(s)$ & $0.0784 _{ - 0.0033 } ^ { + 0.0046 }$ & $8.82 _{ - 0.55 } ^ { + 0.7 }$ & $0.892 _{ - 0.026 } ^ { + 0.028 }$ & $6.95 _{ - 0.27 } ^ { + 0.3 }$ & 10.595 &  & MINERVA (1) &  & SB 2 from MINERVA observations.  \\\\\n422914082 & TYC 0046-00133-1 & $1431.5538 _{ - 0.0014 } ^ { + 0.0017 }$ & $12.91 _{ - 3.91 } ^ { + 8.97 }(s)$ & $0.0418 _{ - 0.0015 } ^ { + 0.0016 }$ & $3.96 _{ - 0.32 } ^ { + 0.35 }$ & $0.36 _{ - 0.25 } ^ { + 0.28 }$ & $4.07 _{ - 0.09 } ^ { + 0.126 }$ & 11.026 & Sinistro (1) & NRES (1) &  &  \\\\\n427344083 & TWOMASS 22563609+7040518 & $1961.8967 _{ - 0.0031 } ^ { + 0.0036 }$ & $7.77 _{ - 5.6 } ^ { + 9.65 }(s)$ & $0.107 _{ - 0.016 } ^ { + 0.025 }$ & $12.27 _{ - 1.87 } ^ { + 2.9 }$ & $0.834 _{ - 0.484 } ^ { + 0.094 }$ & $2.88 _{ - 0.3 } ^ { + 0.42 }$ & 13.404 &  &  &  &  \\\\\n436873727 & HIP 13224 & $1803.83679 _{ - 0.00058 } ^ { + 0.00056 }$ & $19.26 _{ - 5.95 } ^ { + 6.73 }(s)$ & $0.05246 _{ - 0.00061 } ^ { + 0.00059 }$ & $10.02 _{ - 0.43 } ^ { + 0.41 }$ & $0.767 _{ - 0.057 } ^ { + 0.038 }$ & $5.462 _{ - 0.081 } ^ { + 0.074 }$ & 7.51 &  &  &  &  \\\\ \n441642457* & TYC 3858-00452-1 & $1745.5102 _{ - 0.0108 } ^ { + 0.0097 }$ & $79.8072 _{ - 0.0071 } ^ { + 0.0076 }$ & $0.0281 _{ - 0.0024 } ^ { + 0.0033 }$ & $3.55 _{ - 0.34 } ^ { + 0.46 }$ & $0.934 _{ - 0.023 } ^ { + 0.026 }$ & $6.9 _{ - 0.39 } ^ { + 0.6 }$ & 9.996 &  &  &  &  \\\\\n441765914* & TWOMASS 17253007+7552562 & $1769.6154 _{ - 0.0058 } ^ { + 0.0093 }$ & $161.6 _{ - 58.2 } ^ { + 1460.1 }(s)$ & $0.0411 _{ - 0.0024 } ^ { + 0.0119 }$ & $3.6 _{ - 0.3 } ^ { + 1.01 }$ & $0.45 _{ - 0.32 } ^ { + 0.48 }$ & $7.44 _{ - 0.36 } ^ { + 1.08 }$ & 11.638 &  &  &  &  \\\\\n452920657 & TWOMASS 00332018+5906355 & $1810.5765 _{ - 0.0031 } ^ { + 0.003 }$ & $53.2 _{ - 29.0 } ^ { + 34.3 }(s)$ & $0.135 _{ - 0.016 } ^ { + 0.012 }$ & $9.71 _{ - 1.16 } ^ { + 0.9 }$ & $0.73 _{ - 0.48 } ^ { + 0.11 }$ & $4.6 _{ - 0.26 } ^ { + 0.29 }$ & 14.167 & SBIG (1) &  &  &  \\\\\n455737331 & TYC 2779-00785-1 & $1780.7084 _{ - 0.008 } ^ { + 0.0073 }$ & $50.4 _{ - 17.6 } ^ { + 75.0 }(s)$ & $0.0257 _{ - 0.0016 } ^ { + 0.002 }$ & $3.05 _{ - 0.24 } ^ { + 0.29 }$ & $0.43 _{ - 0.29 } ^ { + 0.33 }$ & $6.6 _{ - 0.43 } ^ { + 0.5 }$ & 10.189 & SBIG (1) &  & Gemini &  \\\\\n456909420 & TYC 1208-01094-1 & $1779.4109 _{ - 0.0026 } ^ { + 0.0022 }$ & $5.78 _{ - 5.29 } ^ { + 5.95 }(s)$ & $0.078 _{ - 0.031 } ^ { + 0.045 }$ & $9.15 _{ - 3.61 } ^ { + 5.27 }$ & $0.973 _{ - 0.495 } ^ { + 0.063 }$ & $1.73 _{ - 0.27 } ^ { + 0.28 }$ & 10.941 &  &  &  &  \\\\\n458451774 & TWOMASS 12551793+4431260 & $1917.1875 _{ - 0.0019 } ^ { + 0.0019 }$ & $12.39 _{ - 6.34 } ^ { + 83.97 }(s)$ & $0.0752 _{ - 0.0054 } ^ { + 0.0211 }$ & $3.33 _{ - 0.26 } ^ { + 0.92 }$ & $0.61 _{ - 0.43 } ^ { + 0.32 }$ & $2.08 _{ - 0.19 } ^ { + 0.59 }$ & 13.713 &  &  &  &  \\\\\n48018596 & TYC 3548-00800-1 & $1713.4514 _{ - 0.0063 } ^ { + 0.0046 }$ & $100.1145 _{ - 0.0018 } ^ { + 0.0021 }$ & $0.049 _{ - 0.0081 } ^ { + 0.018 }$ & $7.88 _{ - 1.33 } ^ { + 2.9 }$ & $0.984 _{ - 0.028 } ^ { + 0.027 }$ & $2.83 _{ - 0.26 } ^ { + 0.29 }$ & 9.595 &  & NRES (1) & Gemini &  \\\\\n53309262 & TWOMASS 07475406+5741549 & $1863.1133 _{ - 0.0064 } ^ { + 0.0061 }$ & $294.8 _{ - 96.0 } ^ { + 327.0 }(s)$ & $0.1239 _{ - 0.0075 } ^ { + 0.0098 }$ & $5.38 _{ - 0.36 } ^ { + 0.46 }$ & $0.46 _{ - 0.31 } ^ { + 0.28 }$ & $6.74 _{ - 0.45 } ^ { + 0.62 }$ & 15.51 &  &  &  &  \\\\\n53843023 & TYC 6956-00758-1 & $1328.0335 _{ - 0.0054 } ^ { + 0.0057 }$ & $202.0 _{ - 189.0 } ^ { + 272.0 }(s)$ & $0.058 _{ - 0.02 } ^ { + 0.056 }$ & $5.14 _{ - 1.77 } ^ { + 4.99 }$ & $0.962 _{ - 0.597 } ^ { + 0.083 }$ & $4.25 _{ - 0.72 } ^ { + 0.66 }$ & 11.571 &  &  &  &  \\\\\n55525572* & TYC 8876-01059-1 & $1454.6713 _{ - 0.0066 } ^ { + 0.0065 }$ & $83.8951 _{ - 0.004 } ^ { + 0.004 }$ & $0.0343 _{ - 0.001 } ^ { + 0.0021 }$ & $7.31 _{ - 0.46 } ^ { + 0.56 }$ & $0.43 _{ - 0.29 } ^ { + 0.31 }$ & $13.54 _{ - 0.3 } ^ { + 0.51 }$ & 10.358 &  & CHIRON (5) & Gemini & Confirmed planet \\citep{2020eisner} \\\\\n63698669* & TYC 6993-00729-1 & $1364.6226 _{ - 0.0074 } ^ { + 0.0067 }$ & $73.6 _{ - 26.8 } ^ { + 133.6 }(s)$ & $0.0248 _{ - 0.0019 } ^ { + 0.0023 }$ & $2.15 _{ - 0.2 } ^ { + 0.25 }$ & $0.42 _{ - 0.29 } ^ { + 0.35 }$ & $5.63 _{ - 0.32 } ^ { + 0.57 }$ & 10.701 & SBIG (1) &  &  &  \\\\\n70887357* & TYC 5883-01412-1 & $1454.3341 _{ - 0.0016 } ^ { + 0.0015 }$ & $56.1 _{ - 15.3 } ^ { + 18.8 }(s)$ & $0.0605 _{ - 0.0027 } ^ { + 0.0027 }$ & $12.84 _{ - 0.86 } ^ { + 0.9 }$ & $0.917 _{ - 0.028 } ^ { + 0.016 }$ & $7.29 _{ - 0.18 } ^ { + 0.19 }$ & 9.293 &  &  &  &  \\\\\n7422496$\\dagger$ & HIP 25359 & $1470.3625 _{ - 0.0031 } ^ { + 0.0023 }$ & $61.4 _{ - 16.7 } ^ { + 49.0 }(s)$ & $0.0255 _{ - 0.001 } ^ { + 0.0011 }$ & $2.44 _{ - 0.15 } ^ { + 0.16 }$ & $0.37 _{ - 0.25 } ^ { + 0.29 }$ & $5.89 _{ - 0.15 } ^ { + 0.15 }$ & 9.36 &  & MINERVA (4) &  & SB 2 from MINERVA observations.  \\\\\n82452140 & TYC 3076-00921-1 & $1964.292 _{ - 0.011 } ^ { + 0.011 }$ & $21.1338 _{ - 0.0052 } ^ { + 0.0066 }$ & $0.0266 _{ - 0.0019 } ^ { + 0.0027 }$ & $2.95 _{ - 0.25 } ^ { + 0.34 }$ & $0.42 _{ - 0.29 } ^ { + 0.36 }$ & $5.87 _{ - 0.62 } ^ { + 0.94 }$ & 10.616 &  &  &  &  \\\\\n88840705 & TYC 3091-00808-1 & $2026.6489 _{ - 0.001 } ^ { + 0.001 }$ & $260.6 _{ - 87.6 } ^ { + 142.2 }(s)$ & $0.109 _{ - 0.023 } ^ { + 0.027 }$ & $9.98 _{ - 2.28 } ^ { + 2.75 }$ & $1.001 _{ - 0.042 } ^ { + 0.037 }$ & $4.72 _{ - 0.13 } ^ { + 0.15 }$ & 9.443 &  &  &  &  \\\\\n91987762* & HIP 47288 & $1894.25381 _{ - 0.00051 } ^ { + 0.00047 }$ & $10.51 _{ - 3.48 } ^ { + 3.67 }(s)$ & $0.05459 _{ - 0.00106 } ^ { + 0.00097 }$ & $9.56 _{ - 0.56 } ^ { + 0.52 }$ & $0.771 _{ - 0.062 } ^ { + 0.033 }$ & $4.342 _{ - 0.073 } ^ { + 0.063 }$ & 7.87 &  & NRES (4) & Gemini &  \\\\\n95768667 & TYC 1434-00331-1 & $1918.3318 _{ - 0.0093 } ^ { + 0.0079 }$ & $26.9 _{ - 12.4 } ^ { + 72.3 }(s)$ & $0.0282 _{ - 0.0022 } ^ { + 0.0031 }$ & $3.54 _{ - 0.32 } ^ { + 0.43 }$ & $0.48 _{ - 0.33 } ^ { + 0.35 }$ & $5.4 _{ - 0.64 } ^ { + 0.76 }$ & 10.318 &  &  &  &  \\\\\n\\hline\n\\end{tabular}}\n\\caption{\\textbf{Properties of PHT candidates (continued)}}\n\\label{tab:PHT-caniddates2}\n\\end{table}\n\\end{landscape}\n\n\n\\section{Conclusion}\n\\label{sec:condlusion}\n\nWe present the results from the analysis of the first 26  \\emph{TESS}\\ sectors. The outlined citizen science approach engages over 22 thousand registered citizen scientists who completed 12,617,038 classifications from December 2018 through August 2020 for the sectors observed during the first two years of the \\emph{TESS}\\ mission. We applied a systematic search for planetary candidates using visual vetting by multiple volunteers to identify \\emph{TESS}\\ targets that are most likely to host a planet. Between 8 and 15 volunteers have inspected each \\emph{TESS}\\ light curve and marked times of transit-like events using the PHT online interface. For each light curve, the markings from all the volunteers who saw that target were combined using an unsupervised machine learning method, known as DBSCAN, in order to identify likely transit-like events. Each of these identified events was given a transit score based on the number of volunteers who identified a given event and on the user weighting of each of those volunteers. Individual user weights were calculated based on the user's ability to identify simulated transit events, injected into real \\emph{TESS}\\ light curves, that are displayed on the PHT site alongside of the real data. The transit scores were then used to generate a ranked list of candidates that range from most likely to least likely to host a planet candidate. The top 500 highest ranked candidates were further vetted by the PHT science team. This stage of vetting primarily made use of the open source {\\sc latte} \\citep{LATTE2020} tool which generates a number of standard diagnostic plots that help identify promising candidates and weed out false positive signals. \n\nOn average we found around three high priority candidates per sector which were followed up using ground based telescopes, where possible. To date, PHT has statistically confirmed one planet, TOI-813 \\citep{2020eisner}: a Saturn-sized planet on an 84 day orbit around a subgiant host star. Other PHT identified planets listed in this paper are being followed up by other teams of astronomers, such as TOI-1899 (TIC 172370679) which was recently confirmed to be a warm Jupiter transiting an M-dwarf \\citep{canas2020}. The remaining candidates outlined in this paper require further follow-up observations to confirm their planetary nature.\n\nThe sensitivity of our transit search effort was assessed using synthetic data, as well as the known TOI and TCE candidates flagged by the SPOC pipeline. For simulated planets (where simulated signals are injected into real \\emph{TESS}\\ light curves) we have shown that the recovery efficiency of human vetting starts to decrease for transit-signals that have a SNR less than 7.5. The detection efficiency was further evaluated by the fractional recovery of the TOI and TCEs. We have shown that PHT is over 85 \\% complete in the recovery of planets that have a radius greater than 4 $R_{\\oplus}$, 51 \\% complete for radii between 3 and 4 $R_{\\oplus}$ and 49 \\% complete for radii between 2 and 3 $R_{\\oplus}$. Furthermore, we have shown that human vetting is not sensitive to the number of transits present in the light curve, meaning that they are equally likely to identify candidates on longer orbital periods as they are those with shorter orbital periods for periods greater than $\\sim$ 1 day. Planets with periods shorter than around 1 day exhibit over 20 transits within one \\emph{TESS}\\ sectors resulting in a decrease in identification by the volunteers. This is due to many volunteers only marking a random subset of these events, resulting in a lack of consensus on any given transit event and thus decreasing the overall transit score of these light curves. \n\nIn addition to searching for signals due to transiting exoplanets, PHT provides a platform that can be used to identify other stellar phenomena that may otherwise be difficult to identify with automated pipelines. Such phenomena, including eclipsing binaries, multiple stellar systems, dwarf novae, and stellar flares are often mentioned on the PHT discussion forums where volunteers can use searchable hashtags and comments to bring these systems to the attention of other citizen scientists as well as the PHT science team. All of the eclipsing binaries identified on the site, for example, are being used and vetted by the \\emph{TESS}\\ Eclipsing Binary Working Group (Prsa et al. in prep). Furthermore, we have investigated the nature of all of the targets that were identified as possible multiple stellar systems, as summarised in Table~\\ref{tab:PHT-multis}.\n\nOverall we have shown that large scale visual vetting can complement the findings \\textcolor{red}{from the major \\emph{TESS}\\ pipeline} by identifying longer period planets that may only exhibit a single transit event in their light curve, as well as in finding signals that are aperiodic or embedded in a strong varying stellar signal. The identification of planets around stars with variable signals allow us to potentially characterise the host-star (e.g., with asteroseismology or spot modulation). Additionally, the longer period planets are integral to our understanding of how planet systems form and evolve, as they allow us to investigate the evolution of planets that are farther away from their host star and therefore less dependent on stellar radiation. \\textcolor{red}{While automated pipelines specifically designed to identify single transit events in the \\emph{TESS}\\ data exist \\citep[e.g., ][]{Gill2020}, neither their methodology nor the full list of their findings are yet publicly available and thus we are unable to compare results.} \n\nThe planets that PHT finds have longer periods ($\\gtrsim$ 27 d) than those found in \\emph{TESS}\\ data using automated pipelines, and are more typical of the Kepler sample (25\\% of Kepler confirmed planets have periods greater than 27 days\\footnote{\\url{https:\/\/exoplanetarchive.ipac.caltech.edu\/}}). However, the Kepler planets are considerably fainter, and thus less amenable to ground-based follow-up or atmospheric characterisation from space (CHEOPS and JWST). Thus PHT helps to bridge the parameter spaces covered by these two missions, by identifying longer period planet candidates around bright, nearby stars, for which we can ultimately obtain precise planetary mass estimates. Although statistical characterisation of exo-planetary systems is no doubt important, precise mass measurements are key to developing our understanding of exoplanets and the physics which dictate their evolution. In particular, identification of this PHT sample provides follow-up targets to investigate the dependence of photo-evaporation on the mass of planets as well as on the planet radius, and will help our understanding of the photo-evaporation valley at longer orbital periods \\citep{Owen2013}. \n\nPHT will continue to operate throughout the \\emph{TESS}\\ extended mission, hopefully allowing us to identify even longer period planets as well as help verify some of the existing candidates with additional transits.  \n\n\n\n\\begin{table*}\n\\resizebox{\\textwidth}{!}{\n\\begin{tabular}{cccccccccc}\n\\textbf{TIC}  & \\textbf{Period (days)}        & \\textbf{Epoch (\\textcolor{red}{BJD - 2457000})}      & \\textbf{Depth (ppm)}  & \\textbf{Comment}  \\\\\n\\hline\n13968858      & $3.4850 \\pm 0.001$            & $ 1684.780   \\pm 0.005$            & 410000    & Candidate multiple system  \\\\\n              & $1.4380 \\pm 0.001$            & $ 1684.335   \\pm 0.005$            & 50000     &   \\\\\n35655828      & $ 8.073 \\pm 0.01$             & $ 1550.94    \\pm 0.01 $            & 23000     & Confirmed blend \\\\\n              & $  1.220 \\pm 0.001 $          & $ 1545.540   \\pm 0.005 $           & 2800      &  \\\\\n63291675      & $ 8.099  \\pm 0.003 $          & $ 1685.1     \\pm 0.01  $           & 60000     & Confirmed blend \\\\\n              & $ 1.4635 \\pm 0.0005 $         & $ 1683.8     \\pm 0.1 $             & 7000      &  \\\\\n63459761      & $4.3630 \\pm 0.003 $           & $ 1714.350   \\pm 0.005 $           & 160000    & Candidate multiple system   \\\\\n              & $4.235 \\pm  $ 0.005           & $ 1715.130   \\pm 0.03$             & 35000     &   \\\\\n104909909     & $1.3060 \\pm 0.0001$           & $ 1684.470   \\pm 0.005$            & 32000     & Candidate multiple system   \\\\\n              & $2.5750 \\pm 0.003$            & $ 1684.400   \\pm 0.005$            & 65000     &   \\\\\n115980439     & $ 4.615 \\pm 0.002  $          & $ 1818.05    \\pm 0.01  $           & 95000     & Confirmed blend \\\\\n              & $ 0.742 \\pm 0.005  $          & $ 1816.23    \\pm 0.02  $           & 2000      &   \\\\\n120362128     & $ 3.286 \\pm 0.002  $          & $ 1684.425   \\pm 0.01  $           & 33000     &  Candidate multiple system  \\\\\n              & $ -                 $         & $ 1701.275   \\pm 0.02  $           & 12000     &    \\\\\n              & $ -                 $         & $ 1702.09    \\pm 0.02  $           & 36000     &    \\\\\n121945407     & $  0.9056768 \\pm 0.00000002$   & $-1948.76377 \\pm  0.0000001$      & 2500          & Confirmed multiple system $^{(\\mathrm{a})}$  \\\\\n              & $ 45.4711 \\pm  0.00002$        & $-1500.0038  \\pm  0.0004 $       & 7500          &  \\\\\n122275115     & $ - $                         & $ 1821.779   \\pm 0.01  $           & 155000    &  Candidate multiple system  \\\\\n              & $ - $                         & $ 1830.628   \\pm 0.01  $           & 63000     &  \\\\\n              & $ - $                         & $ 1838.505   \\pm 0.01  $           & 123000    &  \\\\\n229804573     & $1.4641 \\pm 0.0005$           & $ 1326.135   \\pm 0.005$            & 180000    & Candidate multiple system  \\\\\n              & $0.5283 \\pm 0.0001$           & $ 1378.114   \\pm 0.005$            & 9000      &   \\\\\n252403752     & $ - $                         & $ 1817.73    \\pm 0.01 $            & 2800      & Candidate multiple system \\\\\n              & $ - $                         & $ 1829.76    \\pm 0.01 $            & 23000     & \\\\\n              & $ - $                         & $ 1833.63    \\pm 0.01 $            & 5500      & \\\\\n258837989     & $0.8870 \\pm 0.001$            & $ 1599.350   \\pm 0.005$            & 64000     & Candidate multiple system  \\\\\n              & $3.0730 \\pm 0.001$            & $ 1598.430   \\pm 0.005$            & 25000     &   \\\\\n266958963     & $1.5753 \\pm 0.0002$           & $ 1816.425   \\pm 0.001$            & 265000    & Candidate multiple system  \\\\\n              & $2.3685 \\pm 0.0001$           & $ 1817.790   \\pm 0.001$            & 75000     &   \\\\\n278956474     & $5.488068 \\pm 0.000016  $     & $ 1355.400   \\pm 0.005$            & 93900     & Confirmed multiple system $^{(\\mathrm{b})}$ \\\\\n              & $5.674256 \\pm  \u22120.000030$     & $ 1330.690   \\pm 0.005$            & 30000     &   \\\\\n284925600     & $ 1.24571 \\pm 0.00001 $       & $ 1765.248   \\pm 0.005 $           & 490000    & Confirmed blend \\\\\n              & $  0.31828 \\pm 0.00001  $     & $ 1764.75    \\pm 0.005 $           & 35000     &  \\\\\n293954660     & $2.814 \\pm  0.001 $           & $ 1739.177   \\pm 0.03  $           & 272000    & Confirmed blend \\\\\n              & $4.904 \\pm 0.03 $             & $ 1739.73    \\pm 0.01  $           & 9500      &  \\\\\n312353805     & $4.951 \\pm  0.003 $           & $ 1817.73    \\pm 0.01  $           & 66000     & Confirmed blend \\\\\n              & $12.89 \\pm   0.01  $          & $ 1822.28    \\pm 0.01$             & 19000     &  \\\\\n318210930     & $ 1.3055432 \\pm 0.000000033$  & $ -653.21602   \\pm 0.0000013$      & 570000    & Confirmed multiple system $^{(\\mathrm{c})}$ \\\\\n              & $ 0.22771622 \\pm 0.0000000035$& $ -732.071119 \\pm 0.00000026 $     & 220000    &  \\\\\n336434532     & $ 3.888 \\pm 0.002  $          & $ 1713.66    \\pm 0.01  $           & 22900     & Confirmed blend \\\\\n              & $  0.949 \\pm 0.003 $          & $ 1712.81    \\pm 0.01  $           & 2900      &  \\\\\n350622185     & $1.1686 \\pm 0.0001$           & $ 1326.140   \\pm 0.005$            & 200000    & Candidate multiple system  \\\\\n              & $5.2410 \\pm 0.0005$           & $ 1326.885   \\pm 0.05$             & 4000      &  \\\\\n375422201     & $9.9649 \\pm 0.001$            & $ 1711.937   \\pm 0.005$            & 245000    & Candidate multiple system  \\\\\n              & $4.0750 \\pm 0.001$            & $ 1713.210   \\pm 0.01 $            & 39000     &  \\\\\n376606423     & $ 0.8547 \\pm 0.0002  $        & $ 1900.766   \\pm 0.005 $           & 9700      & Candidate multiple system  \\\\\n              & $  -  $                       & $ 1908.085   \\pm 0.01  $           & 33000     &  \\\\\n394177355     & $ 94.22454  \\pm 0.00040 $     & $  -  $                            &  -        & Confirmed multiple system $^{(\\mathrm{d})}$ \\\\\n              & $ 8.6530941 \\pm 0.0000016$    & $-2038.99492 \\pm 0.00017 $         &  140000   &  \\\\\n              & $ 1.5222468 \\pm 0.0000025$    & $ -2039.1201 \\pm 0.0014  $         &  -        &  \\\\\n              & $ 1.43420486 \\pm 0.00000012 $ & $-2039.23941 \\pm 0.00007 $         &  -        &  \\\\\n424508303     & $ 2.0832649 \\pm 0.0000029  $ & $-3144.8661  \\pm 0.0034  $          &  430000   &  Confirmed multiple system $^{(\\mathrm{e})}$ \\\\\n              & $ 1.4200401  \\pm 0.0000042 $  & $-3142.5639  \\pm 0.0054  $         &  250000         &  \\\\\n441794509     & $ 4.6687 \\pm  0.0002       $  & $ 1958.895   \\pm 0.005   $         & 34000     &  Candidate multiple system  \\\\\n              & $ 14.785 \\pm  0.002       $   & $ 1960.845   \\pm 0.005   $         & 17000     &  \\\\\n470710327     & $ 9.9733 \\pm 0.0001 $         & $ 1766.27    \\pm 0.005  $          & 51000     &  Confirmed multiple system $^{(\\mathrm{f})}$ \\\\\n              & $ 1.104686 \\pm 0.00001 $      & $ 1785.53266 \\pm 0.000005$         & 42000     &  \\\\\n\\hline\n\\end{tabular}\n}\n\\caption{\nNote -- $^{(\\mathrm{a})}$ KOI-6139, \\citet{Borkovits2013};  \n$^{(\\mathrm{b})}$ \\citet{2020Rowden}\n$^{(\\mathrm{c})}$ \\citet{Koo2014}; \n$^{(\\mathrm{d})}$ KOI-3156, \\citet{2017Helminiak};\n$^{(\\mathrm{e})}$ V994 Her; \\citet{Zasche2016}; \n$^{(\\mathrm{f})}$ Eisner et al. {\\it in prep.}\n}\n\n\\label{tab:PHT-multis}\n\n\\end{table*}\n\n\\section*{Data Availability}\n\nAll of the \\emph{TESS}\\ data used within this article are hosted and made publicly available by the Mikulski Archive for Space Telescopes (MAST, \\url{http:\/\/archive.stsci.edu\/tess\/}). Similarly, the Planet Hunters TESS classifications made by the citizen scientists can be found on the Planet Hunters Analysis Database (PHAD, \\url{https:\/\/mast.stsci.edu\/phad\/}), which is also hosted by MAST. All planet candidates and their properties presented in this article have been uploaded to the Exoplanet Follow-up Observing Program for TESS (ExoFOP-TESS, \\url{ https:\/\/exofop.ipac.caltech.edu\/tess\/index.php}) website as community TOIs (cTOIs), under their corresponding TIC IDs. The ground-based follow-up observations of individual targets will be shared on reasonable request to the corresponding author.\n\nThe models of individual transit events and the data validation reports used for the vetting of the targets were both generated using publicly available open software codes, \\texttt{pyaneti}\\ and {\\sc latte}.\n\n\\section*{Acknowledgements}  \n\nThis project works under the in \\textit{populum veritas est} philosophy, and for that reason we would like to thank all of the citizen scientists who have taken part in the Planet Hunters TESS project and enable us to find many interesting astrophysical systems. \n\nSome of the observations in the paper made use of the High-Resolution Imaging instruments `Alopeke and Zorro. `Alopeke and Zorro were funded by the NASA Exoplanet Exploration Program and built at the NASA Ames Research Center by Steve B. Howell, Nic Scott, Elliott P. Horch, and Emmett Quigley. `Alopeke and Zorro were mounted on the Gemini North and South telescope of the international Gemini Observatory, a program of NSF's NOIRLab, which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation on behalf of the Gemini partnership: the National Science Foundation (United States), National Research Council (Canada), Agencia Nacional de Investigaci\\'{o}n y Desarrollo (Chile), Ministerio de Ciencia, Tecnolog\\'{i}a e Innovaci\\'{o}n (Argentina), Minist\\'{e}rio da Ci\\^{e}ncia, Tecnologia, Inova\\c{c}\\~{o}es e Comunica\\c{c}\\~{o}es (Brazil), and Korea Astronomy and Space Science Institute (Republic of Korea). The  authors also acknowledge the very significant cultural role and sacred nature of  Maunakea. We are most fortunate to  have  the opportunity to conduct observations from this mountain.\n\nThis project has also received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement N$^\\circ$730890. This material reflects only the authors views and the Commission is not liable for any use that may be made of the information contained therein. This work makes use of observations from the Las Cumbres Observatory global telescope network, including the NRES spectrograph and the SBIG and Sinistro photometric instruments. \n\nFurthermore, NLE thanks the LSSTC Data Science Fellowship Program, which is funded by LSSTC, NSF Cybertraining Grant N$^\\circ$1829740, the Brinson Foundation, and the Moore Foundation; her participation in the program has benefited this work. Finally, CJ acknowledges funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement N$^\\circ$670519: MAMSIE), and from the Research Foundation Flanders (FWO) under grant agreement G0A2917N (BlackGEM). \n\nThis research made use of Astropy, a community-developed core Python package for Astronomy \\citep{astropy2013}, matplotlib \\citep{matplotlib}, pandas \\citep{pandas}, NumPy \\citep{numpy}, astroquery \\citep{ginsburg2019astroquery} and sklearn \\citep{pedregosa2011scikit}.  \n\n\n\n\n\\bibliographystyle{mnras}\n\n\\section{Planet candidate descriptions}\n\\label{appendixA}\n\nA short outline all of the planet candidates, and any conclusions drawn from follow-up observations (where available). A more in depth description of the ground-based data will be presented in a follow-up paper. Unless stated otherwise, these candidates are not TOIs at the time of writing. Candidates for which we have no additional information to complement the results presented in Table~\\ref{tab:PHT-caniddates} are not discussed further here.\n\n\\subsection{Single-transit planet candidates}\n\n\n\\textbf{TIC 103633672.} Single transit event identified in Sector 20. The single LCO\/NRES spectra shows no sign of this being a double lined spectroscopic binary.  We caution that there is a star on the same pixels, which is 0.1 mag brighter. We are unable to rule this star out as the cause for the transit-like signal.\n\n\\textbf{TIC 110996418.} Single transit event identified in Sector 10. We caution that there is a star on the same \\emph{TESS}\\ pixel, which is 2.4 magnitudes fainter than the target.\n\n\\textbf{TIC 128703021.} Single transit event identified in Sector 11. With a stellar radius of 1.6 $R_{\\odot}$ and a T$_{eff}$ of 6281 this host star is likely in the subgiant phase of its evolution. The 43 spectra obtained with MINERVA australis and the two obtained with LCO\/NRES are consistent with a planetary nature. Gemini speckle interferometry shows no nearby companion stars.\n\n\n\\textbf{TIC 142087638.} Single transit event identified in Sector 7. The best fit \\texttt{pyaneti}\\ model of the transit suggests an orbital period of only 3.14 d. As there are no additional transits seen in the light curve, this period is clearly not possible. We caution that the transit is most likely caused by a grazing object, and is therefore likely to be caused by a stellar companion. However, without further data we are unable to rule this candidate out as being planetary in nature.\n\n\\textbf{TIC 159159904.} Single transit event identified in Sector 22. The initial two observations obtained using LCO\/NRES show no sign of the candidate being a double lined spectroscopic binary.\n\n\n\\textbf{TIC 166184426.} Single transit event identified in Sector 11. Since the PHT discovery this cTOI has been become the priority 1 (1=highest, 5=lowest) target TOI  1955.01.\n\n\n\\textbf{TIC 172370679.} Single transit event identified in Sector 15. \\textcolor{black}{This candidate was independently discovered and verified using a BLS algorithm used to search for transiting planets around M-dwarfs. The candidate is now the confirmed planet TOI 1899 b \\citep{canas2020}.} \n\n\\textbf{TIC 174302697.} Single transit event identified in Sectors 16. With a stellar radius of 1.6 $R_{\\odot}$ and a T$_{eff}$ of 6750 this host star is likely in the subgiant phase of its evolution. \\textcolor{black}{This candidate was initially flagged as a TCE and but was erroneously discounted due to the pipeline mistaking the data glitch at the time of a momentum dump as a secondary eclipse.} Since the PHT discovery this cTOI has become the priority 3 target TOI 1896.01. \n\n\\textbf{TIC 192415680.} Single transit event identified in Sector 18. The two epochs of RV measurement obtained with OHP\/SOPHIE are consistent with a planetary scenario.\n\n\\textbf{TIC 192790476.} Single transit event identified in Sector 5. This target has been identified to be a wide binary with am angular separation of 72.40 arcseconds \\citep{andrews2017wideBinary} and a period of 162705 years \\citep{benavides2010new}. The star exhibits large scale variability on the order of around 10 d. The signal is consistent with that of spot modulations, which would suggest that this is a slowly rotating star.\n\n\n\n\\textbf{TIC 219466784.} Single transit event identified in Sector 22. We caution that there is a nearby companion located within the same \\emph{TESS}\\ pixel at an angular separation of 16.3 with a Vmag of 16.3\". Since the PHT discovery this cTOI has become the priority 2 target TOI  2007.01.\n\n\n\n\\textbf{TIC 229055790.} Single transit event identified in Sector 21. We note that the midpoint of the transit-like events coincides with a \\emph{TESS}\\ momentum dump, however, we believe the shape to be convincing enough to warrant further investigation. The two LCO\/NRES spectra show no sign of this being a spectroscopic binary.\n\n\\textbf{TIC 229608594.} Single transit event identified in Sector 24. Since the PHT discovery this cTOI has become the priority 3 target TOI 2298.01.\n\n\n\n\\textbf{TIC 233194447.} Single transit event identified in Sector 14. The transit-like event is shallow and asymmetric and we cannot definitively rule out systematics as the cause for the event without additional data. The initial two LCO\/NRES spectra show no sign of this target being a spectroscopic binary.\n\n\\textbf{TIC 237201858.} Single transit event identified in Sector 18. The single LCO\/NRES spectra shows no sign of this being a double lined spectroscopic binary.\n\n\\textbf{TIC 243187830.} Single transit event identified in Sector 18. There are no nearby bright stars. \\textcolor{black}{This light curve was initially flagged as a TCE, however, the flagged events corresponded to stellar variability and not the same event identified by PHT.} The single LCO\/NRES spectrum shows no sign of this being a double lined spectroscopic binary. Since the PHT discovery this cTOI has become the priority 3 target TOI 2009.01.\n\n\\textbf{TIC 243417115.} Single transit event identified in Sector 11. We note that the best fit \\texttt{pyaneti}\\ model of the transit suggests an orbital period of only 1.81 d. As there are no additional transits seen in the light curve, this period is clearly not possible. We caution that the transit is most likely caused by a grazing object, and is therefore likely to be caused by a stellar companion. However, without further follow-up data we are unable to rule this candidate out as being planetary in nature.\n\n\n\\textbf{TIC 264544388.} Single transit event identified in Sector 19. The single LCO\/NRES spectra shows no sign of this being a double lined spectroscopic binary. Apart from the single transit event, the light curve shows no obvious signals. A periodogram of the light curve, however, reveals a series of five significant peaks, nearly equidistantly spaced by $\\sim1.03$~d$^{-1}$. Additionally, a rotationally split quintuplet is visible at 7.34~d$^{-1}$, with a splitting of $\\sim0.12$~d$^{-1}$, suggesting an $\\ell=2$ p-mode pulsation. The Maelstrom code \\citep{hey2020maelstrom} revealed pulsation timing variations which are consistent with a long period planet. \\textcolor{black}{The short period signal, which was also identified by the periodogram, was flagged as a TCE, however, the single-transit event was not flagged as a TCE.} Since the PHT discovery this cTOI has become the priority 3 target TOI 1893.01.\n\n\\textbf{TIC 264766922.} Single transit event identified in Sector 8. With a stellar radius of 1.7 $R_{\\odot}$ and a T$_{eff}$ of 6913 K this host star is likely entering the subgiant phase of its evolution. The V-shape of this transit and the resultant high impact parameter suggests that the object is grazing. We can therefore not rule out that this candidate it a grazing eclipsing binary.  There are clear p-mode pulsations at frequencies of 9.01 and 11.47 cycles per day, as well as possible g-mode pulsations. \\textcolor{black}{A very short period signal within this light curve was flagged as a TCE, however, the single transit event was ignored by the pipeline.}\n\n\\textbf{TIC 26547036.} Single transit event identified in Sector 14. The four LCO\/NRES observations are consistent with the target being a planetary body and show no sign of the signal being caused by a spectroscopic binary. We caution that there is a star on the same \\emph{TESS}\\ pixel, however, this star is 8.2 magnitudes fainter than the target, and therefore unable to be responsible for the transit event seen in the light curve. Gemini speckle interferometry reveal no additional nearby companion stars. \\textcolor{black}{This candidate was initially flagged as a TCE, however, in addition to the single transit event the pipeline identified further periodic signals that correspond to times of momentum dumps. Due to this, the candidate was never promoted to TOI status.}\n\n\n\\textbf{TIC 278990954.} Single transit event identified in Sector 12. With a stellar radius of 2.6 $R_{\\odot}$ and a T$_{eff}$ of 5761 K this host star is likely in the subgiant phase of its evolution. We note that there are two additional stars on the same pixel as TIC 278990954. These two stars are 2.7 and 3.7 magnitudes fainter in the v-band than the target and can't be ruled out as the cause for the transit-like event without additional follow-up data.\n\n\\textbf{TIC 280865159.} Single transit event identified in Sector 16. Gemini speckle interferometry revealed any nearby companion stars. Since the PHT discovery this cTOI has become the priority 3 target TOI 1894.01.\n\n\\textbf{TIC 284361752.} Single transit event identified in Sector 26. Since the PHT discovery this cTOI has become the priority 2 target TOI 2294.01.\n\n\n\\textbf{TIC 296737508.} Single transit event identified in Sector 8. The single LCO\/NRES and the single MINERVA australis spectra show no sign of this being a spectroscopic binary. The Sinistro snapshot image revealed no additional nearby companions.\n\n\\textbf{TIC 298663873.} Single transit event identified in Sector 19. The two LCO\/NRES spectra show no sign of this being a spectroscopic binary. With a stellar radius of 1.6 $R_{\\odot}$ and a T$_{eff}$ of 6750 this host star is likely in the subgiant phase of its evolution. Gemini speckle images obtained by other teams show no signs of there being nearby companion stars. Since the PHT discovery this cTOI has become the priority 3 target TOI 2180.01.\n\n\n\\textbf{TIC 303050301.} Single transit event identified in Sector 2. The variability of the light curve is consistent with spot modulation. A single LCO\/NRES spectrum shows no signs of this being a double lined spectroscopic binary.\n\n\\textbf{TIC 303317324.} Single transit event identified in Sector 2. We note that a second transit was later seen in Sector 29, however, as this work only covers sectors 1-26 of the primary \\emph{TESS}\\ mission, this candidates is considered a single-transit event in this work. \n\n\n\\textbf{TIC 304142124.} Single transit event identified in Sector 10.\\textcolor{black}{This target was independently identified as part of the Planet Finder Spectrograph, which uses precision RVs \\citep{diaz2020}. This candidate is know the confirmed planet HD 95338 b.}\n\n\n\n\n\n\n\n\\textbf{TIC 331644554.} Single transit event identified in Sector 16. There is a clear mono-periodic signal in the periodogram at around 11.2 cycles per day, which is consistent with p-mode pulsation.\n\n\\textbf{TIC 332657786.} Single transit event identified in Sector 8. We caution that there is a star on the adjacent \\emph{TESS}\\ pixel that is brighter in the V-band by 2.4 magnitudes. At this point we are unable to rule out this star as the cause of the transit-like signal. \n\n\n\\textbf{TIC 356700488.} Single transit event identified in Sector 16. There is a clear mono-periodic signal in the periodogram at around 1.2 cycles per day, which is consistent with either spot modulation or g-mode pulsation. However, there is no clear signal visible in the light curve that would allow us to differentiate between these two scenarios based on the morphology of the variation. Since the PHT discovery this cTOI has become the priority 3 target TOI 2098.01.\n\n\\textbf{TIC 356710041.} Single transit event identified in Sector 23. With a stellar radius of 2.8 $R_{\\odot}$ and a T$_{eff}$ of 5701 K this host star is likely in the subgiant phase of its evolution. \\textcolor{black}{This candidate was initially flagged as a TCE, however, in addition to the single transit event, the pipeline identified a further event that corresponds to the time of a momentum dump. Due to this the candidate failed the `odd-even test' and was initially discarded as a TOI.} Since the PHT discovery this cTOI has become the priority 3 target TOI 2065.01\n\n\\textbf{TIC 369532319.} Single transit event identified in Sector 16. Gemini speckle interferometry revealed no nearby companion stars.\n\n\n\\textbf{TIC 384159646.} Single transit event identified in Sector 12. The eight LCO\/NRES and six MINERVA australis spectra are consistent with this candidate being a planet. Both the SBIG snapshot and the Gemini speckle interferometry observations revealed no companion stars. Since the PHT discovery this cTOI has become the priority 3 target TOI 1895.01.\n\n\n\n\n\\textbf{TIC 418255064.} Single transit event identified in Sector 12. The Gemini speckle image shows no sign of nearby companions.\n\n\n\\textbf{TIC 422914082.} Single transit event identified in Sector 4. Single Sinistro snapshot image reveals no additional nearby stars.\n\n\\textbf{TIC 427344083.} Single transit event identified in Sector 24. We note that there is a star on the adjacent \\emph{TESS}\\ pixel to the target, which is 3.5 magnitude fainter in the V-band than the target star. We also caution that the V-shape of the transit and the high impact parameter suggest that this is a grazing transit. However, without additional follow-up observations we are unable to rule this candidate out as a planet.\n\n\n\\textbf{TIC 436873727.} Single transit event identified in Sector 18. The host star shows strong variability on the order of one day, which is consistent with spot modulations or g-mode pulsations. The periodogram reveals multi-periodic behaviour in the low frequency range consistent with g-mode pulsations. \n\n\\textbf{TIC 452920657.} Single transit event identified in Sector 17. The V-shape of this transit suggests that the object is grazing and future follow-up observations may reveal this to be an EB. \\textcolor{black}{This candidate was initially flagged as a TCE, however, in addition to the single transit event, the pipeline identified a further two event that corresponds to likely stellar variability. Due to this the candidate failed the `odd-even test' and was initially discarded as a TOI.}\n\n\\textbf{TIC 455737331.} Single transit event identified in Sector 17. We note that there is a star on the same \\emph{TESS}\\ pixel as the target, which is 4.5 magnitudes fainter in the V-band. Neither the SBIG snapshot nor the Gemini speckle interferometry revealed any further nearby companion stars.\n\n\\textbf{TIC 456909420.} Single transit event identified in Sector 17. We caution that the V-shape of the transit and the high impact parameter suggest that this is a grazing transit. However, without additional follow-up observations we are unable to rule this candidate out as a planet.\n\n\n\n\n\\textbf{TIC 53843023.} Single transit event identified in Sector 1. We caution that the high impact parameter returned by the best fit \\texttt{pyaneti}\\ model suggests that the transit event is caused by a grazing body. However, at this point we are unable to rule this candidate out as being planetary in nature.\n\n\\textbf{TIC 63698669.} Single transit event identified in Sector 2. The SBIG snapshot image revealed no nearby companions. \\textcolor{black}{This candidate was initially identified as a TCE, however, in addition to the single transit event, the pipeline identified a further 3 events the light curve. Due to these, additional events, which correspond to stellar variability, the candidate was not initially promoted to TOI status.} However, since the PHT discovery this cTOI has become TOI 1892.01.\n\n\\textbf{TIC 70887357.} Single transit event identified in Sector 5. With a stellar radius of 2.1 $R_{\\odot}$ and a T$_{eff}$ of 5463 K this host star is likely in the subgiant phase of its evolution. \\textcolor{black}{This candidate was initially flagged as a TCE, however, in addition to the single transit-event the pipeline identified a further signal, and thus failed the `odd-even' transit test.} However, since the PHT discovery this cTOI has become the priority 3 target TOI 2008.01.\n\n\n\n\\textbf{TIC 91987762.} Single transit event identified in Sector 21. \\textcolor{black}{This candidate was initially flagged as a TCE, however, in addition to the single transit-event the pipeline identified a further signal, and thus failed the `odd-even' transit test.} Since the PHT discovery this cTOI has become the priority 3 target TOI 1898.01.\n\n\n\n\\subsection{Multi-transit and multi-planet candidates}\n\n\n\\textbf{TIC 160039081.} Multi-transit candidate with a period of 30.2 d. Single LCO\/NRES spectra shows no sign of this being a double lined spectroscopic binary and a snapshot image using SBIG shows no nearby companions. The Gemini speckle images also show no additional nearby companions. Since the PHT discovery this cTOI has  become the priority 1 target TOI 2082.01.\n\n\\textbf{TIC 167661160.} Multi-transit candidate with a period of 36.8 d. The nine LCO\/NRES and four MINERVA australis spectra have revealed this to be a long period eclipsing binary.\n\n\\textbf{TIC 179582003.} Multi-transit candidate with a period of 104.6 d. There is a clear mono-periodic signal in the periodogram at around 0.59 cycles per day, which is consistent with either spot modulation or g-mode pulsation. We caution that this candidate is located in a crowded field. With a stellar radius of 2.0 $R_{\\odot}$ and a T$_{eff}$ of 6115 K this host star is likely in the subgiant phase of its evolution.\n\n\\textbf{TIC 219501568.} Multi-transit candidate with a period of 16.6 d. With a stellar radius of 1.7 $R_{\\odot}$ and a T$_{eff}$ of 6690 K this host star is likely entering the subgiant phase of its evolution. \\textcolor{black}{This candidate was identified as a TCE, however, it was not initially promoted to TOI status as the signal was thought to be off-target by the automated pipeline.} However, since the PHT discovery this cTOI has become the priority 3 target TOI 2259.01\n\n\\textbf{TIC 229742722.} Multi-transit candidate with a period of 63.48 d. Eight LCO\/NRES and four OHP\/SOPHIE observations are consistent with this candidate being a planet. Gemini speckle interferometry reveals no nearby companion stars. \\textcolor{black}{This candidate was flagged as a TCE in sector 20, where it only exhibits a single transit event. An additional event was identified at the time of a momentum dump, and as such it failed the `odd-even' test and was not initially promoted to TOI status.} However, since the PHT discovery this cTOI has become the priority 3 target TOI 1895.01.\n\n\\textbf{TIC 235943205.} Multi-transit candidate with a period of 121.3 d. The LCO\/NRES and OHP\/SOPHIE observations remain consistent with a planetary nature of the signal. Since the PHT discovery this cTOI has become the priority 3 target TOI 2264.01.\n\n\n\\textbf{TIC 267542728.} Multi-transit event with period of 39.7 d. Observations obtained with Keck showed that the RV shifts are not consistent with a planetary body and are most likely due to an M-dwarf companion.\n\n\\textbf{TIC 274599700.} Multi-transit candidate with a period of 33.0 d. One of the two transit-like even is only half visible, with the other half of the event falling in a \\emph{TESS}\\ data gap.\n\n\n\n\\textbf{TIC 328933398.} Multi-planet candidates. The 2-minute cadence light curve shows two single transit events of different depths across two \\emph{TESS}\\ sectors, both of which are consistent with an independent planetary body. In addition to the short cadence data, this target was observed in an additional three sectors as part of the 30-minute cadence full frame images. These showed additional transit events for one of the planet candidates, with a period of 24.9 d. \\textcolor{black}{This light curve was initially flagged as containing a TCE event, however, the two 2-minute cadence single transit events were thought to belong to the same transiting planet. The TCE was initially discarded as the pipeline identified the events to be off-target.} However, since the PHT discovery these two cTOIs has become the priority 3 and 1 targets, TOI 1873.01 and TOI 1873.01, respectively.\n\n\\textbf{TIC 349488688.} Multi-planet candidate, with one single transit event and one multi-transit candidate with a period of 11 d. Two LCO\/NRES and two OHP\/SOPHIE spectra, along with ongoing HARPS North are consistent with both of these candidates being planetary in nature. \\textcolor{black}{The single transit event was initially identified as a TCE, however, in addition to the event it identified two other signals at the time of momentum dumps, and was therefore initially discarded by the pipeline as it failed the `odd-even' transit test.} However, since the PHT discovery the two-transit event has become the 1 targets, TOI 2319.01 (Eisner et al. in prep).\n\n\\textbf{TIC 385557214.} Multi-transit candidate with a period of 5.6 d. The prominent stellar variation seen in the light curve is likely due to spots or pulsation The high impact parameter returned by the best fit \\texttt{pyaneti}\\ modelling suggests that the transit is likely caused by a grazing object. Without further observations, however, we are unable to rule this candidate out as being planetary in nature. \\textcolor{black}{This candidate was flagged as a TCE but was not promoted to TOI status due to the other nearby stars.}\n\n\\textbf{TIC 408636441.} Multi-transit candidate with a period of 18.8 or 37.7 d. Due to \\emph{TESS}\\ data gaps, half of the period stated in Table~\\ref{tab:PHT-caniddates} is likely. The SBIG snapshot and Gemini speckle images show no signs of companion stars. \\textcolor{black}{This candidate was flagged as a TCE in sector 24, where it only exhibits a single transit event. An additional event was identified at the time of a momentum dump, and as such it failed the `odd-even' test and was not initially promoted to TOI status.} However, since the PHT discovery this cTOI has become the priority 3 target TOI 1759.01.\n\n\\textbf{TIC 441642457.} Multi-transit candidate with a period of 79.8 d. \\textcolor{black}{This candidate was flagged as a TCE in sector 14, where it only exhibits a single transit event. An additional event was identified at the time of a momentum dump, and as such it failed the `odd-even' test and was not initially promoted to TOI status.} Since the PHT discovery this cTOI has become the priority 2 target TOI 2073.01.\n\n\\textbf{TIC 441765914.} Multi-transit candidate with a period of 161.6 d.  Since the PHT discovery this cTOI has become the priority 1 target TOI 2088.01.\n\n\\textbf{TIC 48018596.} Multi-transit candidate with a period of 100.1 days (or a multiple thereof). The single LCO\/NRES spectrum shows no sign of this target being a double lined spectroscopic binary. Gemini speckle interferometry revealed no nearby companion stars.  \\textcolor{black}{This candidate was initially flagged as a TCE, however, in addition to the transit-events, the pipeline classified, what we consider stellar variability as an additional event. As such it failed the `odd-even' transit test and wasn't promoted to TOI status.} However, since the PHT discovery this cTOI has become the priority 3 target TOI 2295.01.\n\n\\textbf{TIC 55525572.} Multi-transit candidate with a period of 83.9 d. Since the PHT discovery this cTOI has become the confirmed planet TOI 813 \\citep{2020eisner}.\n\n\\textbf{TIC 82452140.} Multi-transit candidate with a period of 21.1 d. Since the PHT discovery this cTOI has become the priority 2 target TOI 2289.01.\n\n\n\\section{Introduction}\n\nSince the first unambiguous discovery of an exoplanet in 1995 \\citep[][]{Mayor1995} over 4,000 more have been confirmed. Studies of their characteristics have unveiled an extremely wide range of planetary properties in terms of planetary mass, size, system architecture and orbital periods, greatly revolutionising our understanding of how these bodies form and evolve.\n\nThe transit method, whereby we observe a temporary decrease in the brightness of a star due to a planet passing in front of its host star, is to date the most successful method for planet detection, having discovered over 75\\% of the planets listed on the NASA Exoplanet Archive\\footnote{\\url{https:\/\/exoplanetarchive.ipac.caltech.edu\/}}. It yields a wealth of information including planet radius, orbital period, system orientation and potentially even atmospheric composition. Furthermore, when combined with Radial Velocity \\citep[RV; e.g.,][]{Mayor1995, Marcy1997} observations, which yield the planetary mass, we can infer planet densities, and thus their internal bulk compositions. Other indirect detection methods include radio pulsar timing \\citep[e.g.,][]{Wolszczan1992} and microlensing \\citep[e.g.,][]{Gaudi2012}.\n\n\nThe \\textit{Transiting Exoplanet Survey Satellite} mission \\citep[\\protect\\emph{TESS};][]{ricker15} is currently in its extended mission, searching for transiting planets orbiting bright ($V < 11$\\,mag) nearby stars. Over the course of the two year nominal mission, \\emph{TESS}\\ monitored around 85 per cent of the sky, split up into 26 rectangular sectors of 96 $\\times$ 24 deg each (13 per hemisphere). Each sector is monitored for $\\approx$ 27.4 continuous days, measuring the brightness of $\\approx$ 20,000 pre-selected stars every two minutes. In addition to these short cadence (SC) observations, the \\emph{TESS}\\ mission provides Full Frame Images (FFI) that span across all pixels of all CCDs and are taken at a cadence of 30 minutes. While most of the targets ($\\sim$ 63 per cent) will be observed for $\\approx$ 27.4 continuous days, around $\\sim$ 2 per cent of the targets at the ecliptic poles are located in the `continuous viewing zones' and will be continuously monitored for $\\sim$ 356 days.\n\nStars themselves are extremely complex, with phenomena ranging from outbursts to long and short term variability and oscillations, which manifest themselves in the light curves. These signals, as well as systematic effects and artifacts introduced by the telescope and instruments, mean that standard periodic search methods, such as the Box-Least-Squared method \\citep{bls2002} can struggle to identify certain transit events, especially if the observed signal is dominated by natural stellar variability. Standard detection pipelines also tend to bias the detection of short period planets, as they typically require a minimum of two transit events in order to gain the signal-to-noise ratio (SNR) required for detection.\n\nOne of the prime science goals of the \\emph{TESS}\\ mission is to further our understanding of the overall planet population, an active area of research that is strongly affected by observational and detection biases. In order for exoplanet population studies to be able to draw meaningful conclusions, they require a certain level of completeness in the sample of known exoplanets as well as a robust sample of validated planets spanning a wide range of parameter space. \\textcolor{red}{Due to this, we independently search the \\emph{TESS}\\ light curves for transiting planets via visual vetting in order to detect candidates that were either intentionally ignored by the main \\emph{TESS}\\ pipelines, which require at least two transits for a detection, missed because of stellar variability or instrumental artefacts, or were identified but subsequently erroneously discounted at the vetting stage, usually because the period found by the pipeline was incorrect. These candidates can help populate under-explored regions of parameter space and will, for example, benefit the study of planet occurrence rates around different stellar types as well as inform theories of physical processes involved with the formation and evolution of different types of exoplanets.}\n\nHuman brains excel in activities related to pattern recognition, making the task of identifying transiting events in light curves, even when the pattern is in the midst of a strong varying signal, ideally suited for visual vetting. Early citizen science projects, such as Planet Hunters \\citep[PH;][]{fischer12} and Exoplanet Explorers \\citep{Christiansen2018}, successfully harnessed the analytic power of a large number of volunteers and made substantial contributions to the field of exoplanet discoveries. The PH project, for example, showed that human vetting has a higher detection efficiency than automated detection algorithms for certain types of transits. In particular, they showed that citizen science can outperform on the detection of single (long-period) transits \\citep[e.g.,][]{wang13, schmitt14a}, aperiodic transits \\citep[e.g. circumbinary planets;][]{schwamb13} and planets around variable stars \\citep[e.g., young systems,][]{fischer12}. Both PH and Exoplanet Explorers, which are hosted by the world's largest citizen science platform Zooniverse \\citep{lintott08}, ensured easy access to \\textit{Kepler} and \\textit{K2} data by making them publicly available online in an immediately accessible graphical format that is easy to understand for non-specialists. The popularity of these projects is reflected in the number of participants, with PH attracting 144,466 volunteers from 137 different countries over 9 years of the project being active.\n\nFollowing the end of the \\textit{Kepler} mission and the launch of the \\emph{TESS}\\ satellite in 2018, PH was relaunched as the new citizen science project \\textit{Planet Hunters TESS} (PHT) \\footnote{\\url{www.planethunters.org}}, with the aim of identifying transit events in the \\emph{TESS}\\ data that were \\textcolor{red}{intentionally ignored or missed} by the main \\emph{TESS}\\ pipelines. \\textcolor{red}{Such a search complements other methods methods via its sensitivity to single-transit, and, therefore, longer period planets. Additionally, other dedicated non-citizen science based methods are also employed to look for single transit candidates \\citep[see e.g., the Bayesian transit fitting method by ][]{Gill2020, Osborn2016}}.\n\nCitizen science transit searches specialise in finding the rare events that the standard detection pipelines miss, however, these results are of limited use without an indication of the completeness of the search. Addressing the problem of completeness was therefore one of our highest priorities while designing PHT as discussed throughout this paper. \n\nThe layout of the remainder of the paper will be as follows. An overview of the Planet Hunters TESS project is found in Section~\\ref{sec:PHT}, followed by an in depth description of how the project identifies planet candidates in Section~\\ref{sec:method}. The recovery efficiency of the citizen science approach is assessed in Section~\\ref{sec:recovery_efficiency}, followed by a description of the in-depth vetting of candidates and ground based-follow up efforts in Section~\\ref{sec:vetting} and \\ref{sec:follow_up}, respectively. Planet Candidates and noteworthy systems identified by Planet Hunters TESS are outlined in Section~\\ref{sec:PHT_canidates}, followed by a discussion of the results in Section~\\ref{sec:condlusion}.\n\n\\section{Planet Hunters TESS}\n\\label{sec:PHT}\n\nThe PHT project works by displaying \\emph{TESS}\\ light curves (Figure~\\ref{fig:interface}), and asking volunteers to identify transit-like signals. Only the two-minute cadence targets, which are produced by the \\emph{TESS}\\ pipeline at the Science Processing Operations Center \\citep[SPOC,][]{Jenkins2018} and made publicly available by the Mikulski Archive for Space Telescopes (MAST)\\footnote{\\url{http:\/\/archive.stsci.edu\/tess\/}}, are searched by PHT. First-time visitors to the PHT site, or returning visitors who have not logged in are prompted to look through a short tutorial, which briefly explains the main aim of the project and shows examples of transit events and other stellar phenomena. Scientific explanation of the project can be found elsewhere on the site in the `field guide' and on the project's `About' page. \n\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=\\textwidth]{Figures\/PHT_new_interface.png}\n    \\caption{\n    PHT user interface showing a simulated light curve. The transit events are highlighted with white partially-transparent columns that are drawn on using the mouse. Stellar information on the target star is available by clicking on `subject info' below the light curve.} \n    \\label{fig:interface}\n\\end{figure*}\n\nAfter viewing the tutorial, volunteers are ready to participate in the project and are presented with \\emph{TESS}\\ light curves (known as `subjects') that need to be classified. The project was designed to be as simple as possible and therefore only asks one question: \\textit{`Do you see a transit?}'. Users identify transit-like events, and the time of their occurrence, by drawing a column over the event using the mouse button, as shown in Figure~\\ref{fig:interface}. There is no limit on the number of transit-like events that can be marked in a light curve. No markings indicate that there are no transit-like events present in the light curve. Once the subject has been analysed, users submit their classification and continue to view the next light curve by clicking `Done'. \n\nAlongside each light curve, users are offered information on the stellar properties of the target, such as the radius, effective temperature and magnitude (subject to availability, see \\cite{Stassun18}). However, in order to reduce biases in the classifications, the TESS Input Catalog (TIC) ID of the target star is not provided until after the subject classification has been submitted.\n\nIn addition to classifying the data, users are given the option to comment on light curves via the `Talk' discussion forum. Each light curve has its own discussion page to allow volunteers to discuss and comment, as well as to `tag' light curves using searchable hashtags, and to bring promising candidates to the attention of other users and the research team. The talk discussion forums complement the main PHT analysis and have been shown to yield interesting objects which may be challenging to detect using automated algorithms \\citep[e.g.,][]{eisner2019RN}. Unlike in the initial PH project, there are no questions in the main interface regarding stellar variability, however, volunteers are encouraged to mention astrophysical phenomenon or \\textit{unusual} features, such as eclipsing binaries or stellar flares, using the `Talk' discussion forum.  \n\nThe subject TIC IDs are revealed on the subject discussion pages, allowing volunteers to carry out further analysis on specific targets of interest and to report and discuss their findings. This is extremely valuable for both other volunteers and the PHT science team, as it can speed up the process of identifying candidates as well as rule out false positives in a fast and effective manner. \n\nSince the launch of PHT on 6 December 2018, there has been one significant makeover to the user interface. The initial PHT user interface (UI1), which was used for sectors 1 through 9, split the \\emph{TESS}\\ light curves up into either three or four chunks (depending on the data gaps in each sector) which lasted around seven days each. This allowed for a more `zoomed' in view of the data, making it easier to identify transit-like events than when the full $\\sim$ 30 day light curves were shown. The results from a PHT beta project, which displayed only simulated data, showed that a more zoomed in view of the light curve was likely to yield a higher transit recovery rate.\n\nThe updated, and current, user interface (UI2) allows users to manually zoom in on the x-axis (time) of the data. Due to this additional feature, each target has been displayed as a single light curve as of Sector 10. In order to verify that the changes in interface did not affect our findings, all of the Sector 9 subjects were classified using both UI1 and UI2. We saw no significant change in the number of candidates recovered (see Section~\\ref{sec:recovery_efficiency} for a description of how we quantified detection efficiency).\n\n\n\\subsection{Simulated Data}\n\\label{subsec:sims} \n\nIn addition to the real data, volunteers are shown simulated light curves, which are generated by randomly injecting simulated transit signals, provided by the SPOC pipeline \\citep[][]{Jenkins2018}, into real \\emph{TESS}\\ light curves. The simulated data play an important role in assessing the sensitivity of the project, training the users and providing immediate feedback, and to gauge the relative abilities of individual users (see Sec~\\ref{subsec:weighting}). \n\nWe calculate a signal to noise ratio (SNR) of the injected signal by dividing the injected transit depth by the Root Mean Square Combined Differential Photometric Precision (RMS CDPP) of the light curve on 0.5-, 1- or 2-hr time scales (whichever is closest to the duration of the injected transit signal). Only simulations with a SNR greater than 7 in UI1 and greater than 4 for UI2 are shown to volunteers.\n\nSimulated light curves are randomly shown to the volunteers and classified in the exact same manner as the real data. The user is always notified after a simulated light curve has been classified and given feedback as to whether the injected signal was correctly identified or not. For each sector, we generate between one and two thousand simulated light curves, using the real data from that sector in order to ensure that the sector specific systematic effects and data gaps of the simulated data do not differ from the real data. The rate at which a volunteer is shown simulated light curves decreases from an initial rate of 30 per cent for the first 10 classifications, down to a rate of 1 per cent by the time that the user has classified 100 light curves. \n\n\n\\section{Identifying Candidates}\n\\label{sec:method}\n\nEach subject is seen by multiple volunteers, before it is `retired' from the site, and the classifications are combined (see Section~\\ref{subsec:DBscan}) in order to assess the likelihood of a transit event. For sectors 1 through 9, the subjects were retired after 8 classifications if the first 8 volunteers who saw the light curves did not mark any transit events, after 10 classifications if the first 10 volunteers all marked a transit event and after 15 classifications if there was not complete consensus amongst the users. As of Sector 9 with UI2, all subjects were classified by 15 volunteers, regardless of whether or not any transit-like events were marked. Sector 9, which was classified with both UI1 and UI2, was also classified with both retirement rules.\n\nThere were a total of 12,617,038 individual classifications completed across the project on the nominal mission data. 95.4 per cent of these classifications were made by 22,341 registered volunteers, with the rest made by unregistered volunteers. Around 25 per cent of the registered volunteers complete more than 100 classifications, 11.8 per cent more than 300, 8.4 per cent more than 500, 5.4 per cent more than 1000 and 1.1 per cent more than 10,000. The registered volunteers completed a mean and median of 647 and 33 classifications, respectively. Figure~\\ref{fig:user_count} shows the distribution in user effort for logged in users who made between 0 and 300 classifications. \n\nThe distribution in the number of classifications made by the registered volunteers is assessed using the Gini coefficient, which ranges from 0 (equal contributions from all users) to 1 (large disparity in the contributions). The Gini coefficients for individual sectors ranges from 0.84 to 0.91 with a mean of 0.87, while the Gini coefficient for the overall project (all of the sectors combined) is 0.94. The mean Gini coefficient among other astronomy Zooniverse projects lies at 0.82 \\citep{spiers2019}. We note that the only other Zooniverse project with an equally high Gini coefficient as PHT is \\textit{Supernova Hunters}, a project which, similarly to PHT and unlike most other Zooniverse projects, has periodic data releases that are accompanied by an e-newsletter sent to all project volunteers. Periodic e-newsletters have the effect of promoting the project to both regularly and irregularly participating volunteers, who may only complete a couple of classifications as they explore the task, as well as to returning users who complete a large number of classifications following every data release, increasing the disparity in user contributions (the Gini coefficient).\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.5\\textwidth]{Figures\/user_count.png}\n    \\caption{\n    The distribution of the number of classifications by the registered volunteers, using a bin size of 5 from 0 to 300 classifications. A total of 11.8 per cent of the registered volunteers completed more than 300 classifications.} \n    \\label{fig:user_count}\n\\end{figure}\n\n\n\\subsection{User Weighting}\n\\label{subsec:weighting} \nUser weights are calculated for each individual volunteer in order to identify users who are more sensitive to detecting transit-like signals and those who are more likely to mark false positives. The weighting scheme is based on the weighting scheme described by \\cite{schwamb12}.\n\nUser weights are calculated independently for each observation sector, using the simulated light curves shown alongside the data from that sector. All users start off with a weighting of one, which is then increased or decreased when a simulated transit event is correctly or incorrectly identified, respectively. \n\nSimulated transits are deemed correctly identified, or `True', if the mid-point of a user's marking falls within the width of the simulated transit events. If none of the user's markings fall within this range, the simulated transit is deemed not identified, or `False'. If more than one of a user's markings coincide with the same simulated signal, it is only counted as being correct once, such that the total number of `True' markings cannot exceed the number of injected signals. For each classification, we record the number of  `Extra' markings, which is the total number of markings made by the user minus the number of correctly identified simulated transits. \n\nEach simulated light curve, identified by superscript $i$ (where $i=1$, \\ldots, $N$) was seen by $K^{(i)}$ users (the mean value of $K^{(i)}$\nwas 10), and contained $T^{(i)}$ simulated transits (where $T^{(i)}$ depends on the period of the simulated transit signal and the duration of the light curve). For a specific light curve $i$, each user who saw the light curve is identified by a subscript $k$ (where $k=1$, \\ldots, $K^{(i)}$) and each injected transit by a subscript $t$ (where $t=1$, \\ldots, $T^{(i)}$). \n\nIn order to distinguish between users who are able to identify obvious transits and those who are also able to find those that are more difficult to see, we start by defining a `recoverability' $r^{(i)}_t$ for each injected transit $t$ in each light curve. This is defined empirically, as the number of users who identified the transit correctly divided by $K^{(i)}$ (the total number of users who saw the light curve in question).\n\nNext, we quantify the performance of each user on each light curve as follows (this performance is analogous to the `seed' defined in \\citealt{schwamb12}, but we define it slightly differently):\n\\begin{equation}\n  p^{(i)}_{k} = C_{\\rm E} ~ \\frac{E^{(i)}_{k}}{\\langle E^{(i)} \\rangle} + \\sum_{t=1}^{T^{(i)}} \\begin{cases}\n    C_{\\rm T} ~ \\left[ r^{(i)}_t \\right]^{-1}, & \\text{if $m^{(i)}_{t,k} = $`True'}\\\\\n    C_{\\rm F} ~ r^{(i)}_t, & \\text{if $m^{(i)}_{t,k} = $`False'},\n  \\end{cases}\n\\end{equation}\nwhere $m^{(i)}_{t,k}$ is the identification of transit $t$ by user $k$ in light curve $i$, which is either `True' or `False'; $E^{(i)}_{k}$ is the number of `Extra' markings made by user $k$ for light curve $i$, and  $\\langle E^{(i)} \\rangle$ is the mean number of `Extra' markings made by all users who saw subject $i$. The parameters $C_{\\rm E}$, $C_{\\rm T}$ and $C_{\\rm F}$ control the impact of the `Extra', `True' and `False' markings on the overall user weightings, and are optimized empirically as discussed below in Section~\\ref{subsec:optimizesearch}. \n\nFollowing \\citealt{schwamb12}, we then assign a global `weight' $w_k$ to each user $k$, which is defined as:\n\\begin{equation}\n\\begin{split}\n\tw_k = I \\times (1 + \\log_{10} N_k)^{\\nicefrac{\\sum_i p^{(i)}_k}{N_k}}\n\\label{equ:weight}\n\\end{split}\n\\end{equation}\nwhere $I$ is an empirical normalization factor, such that the distribution of user weights remains centred on one, $N_k$ is the total number of simulated transit events that user $k$ assessed, and the sum over $i$ concerns only the light curves that user $k$ saw. \nWe limit the user weights to the range 0.05--3 \\emph{a posteriori}.\n\n\nWe experimented with a number of alternative ways to define the user weights, including the simpler $w_k=\\nicefrac{\\sum_i p^{(i)}_k}{N_k}$, but Eqn.~\\ref{equ:weight} was found to give the best results (see Section~\\ref{sec:recovery_efficiency} for how this was evaluated).\n\n\\subsection{Systematic Removal}\n\\label{subsec:sysrem} \nSystematic effects, for example caused by the spacecraft or background events, can result in spurious signals that affect a large subset of the data, resulting in an excess in markings of transit-like events at certain times within an observation sector. As the four \\emph{TESS}\\ cameras can yield unique systematic effects, the times of systematics were identified uniquely for each camera. The times were identified using a Kernel Density Estimation \\citep[KDE;][]{rosenblatt1956} with a cosine kernel and a bandwidth of 0.1 days, applied across all of the markings from that sector for each camera. Fig.~\\ref{fig:sys_rem} shows the KDE of all marked transit-events made during Sector 17 for TESS's cameras 1 (top panel) to 4 (bottom panel). The isolated spikes, or prominences, in the number of marked events, such as at T = 21-22 days in the bottom panel, are assumed to be caused by systematic effects that affect multiple light curves. Prominences are considered significant if they exceed a factor four times the standard deviation of the kernel output, which was empirically determined to be the highest cut-off to not miss clearly visible systematics. All user-markings within the full width at half maximum of these peaks are omitted from all further analysis. \\textcolor{red}{The KDE profiles for each Sector are provided as electronic supplementary material.}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.46\\textwidth]{Figures\/systematics_sec17.png}\n    \\caption{\n    Kernel density estimation of the user-markings made for Sector 17, for targets observed with TESS's observational Cameras 1 (top panel) to 4 (bottom panel). The orange vertical lines the indicate prominences that are at least four times greater than the standard deviation of the distribution. The black points underneath the figures show the mid-points of all of the volunteer-markings, where darker regions represent a higher density of markings.}\n    \\label{fig:sys_rem}\n\\end{figure}\n\n\\subsection{Density Based Clustering}\n\\label{subsec:DBscan} \n\nThe times and likelihoods of transit-like events are determined by combining all of the classifications made for each subject and identifying times where multiple volunteers identified a signal. We do this using an unsupervised machine learning method, known as DBSCAN \\citep[][Density-Based Spatial Clustering of Applications with Noise]{ester1996DB}. DBSCAN is a non-parametric density based clustering algorithm that helps to distinguish between dense clusters of data and sparse noise. For a data point to belong to a cluster it must be closer than a given distance ($\\epsilon$) to at least a set minimum number of other points (minPoints). \n\nIn our case, the data points are one-dimensional arrays of times of transits events, as identified by the volunteers, and clusters are times where multiple volunteers identified the same event. For each cluster a `transit score' ($s_i$) is determined, which is the sum of the user weights of the volunteers who contribute to the given cluster divided by the sum of the user weights of volunteers who saw that light curve. These transit scores are used to rank subjects from most to least likely to contain a transit-like event. Subjects which contain multiple successful clusters with different scores are ranked by the highest transit score. \n\n\\subsection{Optimizing the search}\n\\label{subsec:optimizesearch}\n\nThe methodology described in Sections~\\ref{subsec:weighting} to \\ref{subsec:DBscan} has five free parameters: the number of markings required to constitute a cluster ($minPoints$), the maximum separation of markings required for members of a cluster ($\\epsilon$), and $C_{\\rm E}$, $C_{\\rm T}$ and $C_{\\rm F}$ used in the weighting scheme. The values of these parameters were optimized via a grid search, where $C_{\\rm E}$ and $C_{\\rm F}$ ranged from -5 to 0, $C_{\\rm T}$ ranged from 0 to 20, and $minPoints$ ranged from 1 to 8, all in steps of 1. ($\\epsilon$) ranged from 0.5 to 1.5 in steps of 0.5. This grid search was carried out on 4 sectors, two from UI1 and two from UI2, for various variations of Equation~\\ref{equ:weight}. \n\nThe success of each combination of parameters was assessed by the fractions of TOIs and TCEs that were recovered within the top highest ranked 500 candidates, as discussed in more detail Section~\\ref{sec:recovery_efficiency}. We found the most successful combination of parameters to be $minPoints$ = 4 markings, $\\epsilon$, = 1 day, $C_{\\rm T}$ = 3, $C_{\\rm F}$= -2 and $C_{\\rm E}$ = -2.\n\n\\subsection{MAST deliverables}\n\\label{subsec:deliverables}\n\nThe analysis described above is carried out both in real-time as classifications are made, as well as offline after all of the light curves of a given sector have been classified. When the real-time analysis identifies a successful DB cluster (i.e. when at least four citizen scientists identified a transit within a day of the \\emph{TESS}\\ data of one another), the potential candidate is automatically uploaded to the open access Planet Hunters Analysis Database (PHAD) \\footnote{\\url{https:\/\/mast.stsci.edu\/phad\/}} hosted by the Mikulski Archive for Space Telescopes (MAST) \\footnote{\\url{https:\/\/archive.stsci.edu\/}}. While PHAD does not list every single classification made on PHT, it does display all transit candidates which had significant consensus amongst the volunteers who saw that light curve, along with the user-weight-weighted transit scores. This analysis does not apply the systematics removal described in Section~\\ref{subsec:sysrem}. The aim of PHAD is to provide an open source database of potential planet candidates identified by PHT, and to credit the volunteers who identified said targets. \n\nThe offline analysis is carried out following the complete classifications of all of the data from a given \\emph{TESS}\\ sector. The combination of all of the classifications allows us to identify and remove times of systematics and calculate better calibrated and more representative user weights. The remainder of this paper will only discuss the results from the offline analysis.\n\n\\section{Recovery Efficiency}\n\\label{sec:recovery_efficiency}\n\\subsection{Recovery of simulated transits}\n\nThe recovery efficiency is, in part, assessed by analysing the recovery rate of the injected transit-like signals (see Section~\\ref{subsec:sims}). Figure~\\ref{fig:SIM_recovery} shows the median and mean transit scores (fraction of volunteers who correctly identified a given transit scaled by user weights) of the simulated transits within SNR bins ranging from 4 to 20 in steps of 0.5. Simulations with a SNR less than 4 were not shown on PHT. The figure highlights that transit signals with a SNR of 7.5 or greater are correctly identified by the vast majority of volunteers. \n\n\\textcolor{red}{As the simulated data solely consist of real light curves with synthetically injected transit signals, we do not have any light curves, simulated or otherwise, which we can guarantee do not contain any planetary transits (real or injected). As such, this prohibits us from using simulated data to infer an analogous false-positive rate.}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.47\\textwidth]{Figures\/SIMS_recovery.png}\n    \\caption{The median (blue) and mean (orange) transit scores for injected transits with SNR ranges between 4 and 20. The mean and median are calculated in SNR bins with a width of 0.5, as indicated by the horizontal lines around each data point. \n    }\n    \\label{fig:SIM_recovery}\n\\end{figure}\n\n\\subsection{Recovery of TCEs and TOIs}\n\\label{subsec:TCE_TOI}\nThe recovery efficiency of PHT is assessed further using the planet candidates identified by the SPOC pipeline \\citep{Jenkins2018}. The SPOC pipeline extracts and processes all of the 2-minute cadence \\emph{TESS}\\ light curves prior to performing a large scale transit search. Data Validation (DV) reports, which include a range of transit diagnostic tests, are generated by the pipeline for around 1250 Threshold Crossing Events (TCEs), which were flagged as containing two or more transit-like features. Visual vetting is then performed by the \\emph{TESS}\\ science team on these targets, and promising candidates are added to the catalog of \\emph{TESS}\\ Objects of Interest (TOIs). Each sector yields around 80 TOIs \\textcolor{red}{and a mean of 1025 TCEs.}\n\nFig~\\ref{fig:TCE_TOI_recovery} shows the fraction of TOIs and TCEs (top and bottom panel respectively) that we recover with PHT as a function of the rank, where a higher rank corresponds to a lower transit score, for Sectors 1 through 26. TOIs and TCEs with R < 2 $R_{\\oplus}$ are not included in this analysis, as the initial PH showed that human vetting alone is unable to reliably recover planets smaller than 2 $R_{\\oplus}$ \\citep{schwamb12}. Planets smaller than 2 $R_{\\oplus}$ are, therefore, not the main focus of our search.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.45\\textwidth]{Figures\/TCE_TOI-recovery_radlim2.png}\n    \\caption{The fraction of recovered TOIs and TCEs (top and bottom panel respectively) with R > 2$R_{\\oplus}$ as a function of the rank, for sectors 1 to 26. The lines represent the results from different observation sectors.}\n    \\label{fig:TCE_TOI_recovery}\n\\end{figure}\n\n\nFig~\\ref{fig:TCE_TOI_recovery} shows a steep increase in the fractional TOI recovery rate up to a rank of $\\sim$ 500. Within the 500 highest ranked PHT candidates for a given sector, we are able to recover between 46 and 62 \\% (mean of 53 \\%) of all of the TOIs (R > 2 $R_{\\oplus}$), a median 90 \\% of the TOIs where the SNR of the transit events are greater than 7.5 and median 88 \\% of TOIs where the SNR of the transit events are greater than 5.\n\nThe relation between planet recovery rate and the SNR of the transit events is further highlighted in Figure~\\ref{fig:TOI_properties}, which shows the SNR vs the orbital period of the recovered TOIs. The colour of the markers indicate the TOI's rank within a given sector, with the lighter colours representing a lower rank. The circles and crosses represent candidates at a rank lower and higher than 500, respectively. The figure shows that transit events with a SNR less than 3.5 are missed by the majority of volunteers, whereas events with a SNR greater than 5 are mostly recovered within the top 500 highest ranked candidates. \n\nThe steep increase in the fractional TOI recovery rate at lower ranks, as shown in figure~\\ref{fig:TCE_TOI_recovery}, is therefore due to the detection of the high SNR candidates that are identified by most, if not all, of the PHT volunteers who classified those targets. At a rank of around 500, the SNR of the TOIs tends towards the limit of what human vetting can detect and thus the identification of TOIs beyond a rank of 500 is more sporadic.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.49\\textwidth]{Figures\/TOI_recovery_properties.png}\n    \\caption{The SNR vs orbital period of TOIs with R > 2$R_{\\oplus}$. The colour represents their rank within the sector, as determined by the weighted DB clustering algorithm. Circles indicate that they were identified at a rank < 500, while crosses indicate that they were not within the top 500 highest ranked candidates of a given sector.\n    }\n    \\label{fig:TOI_properties}\n\\end{figure}\n\nThe fractional TCE recovery rate (bottom panel of Figure~\\ref{fig:TCE_TOI_recovery}) is systematically lower than that of the TOIs. There are qualitative reasons as to why humans might not identify a TCE as opposed to a TOI, including that TCEs may be caused by artefacts or periodic stellar signals that the SPOC pipeline identified as a potential transit but that the human eye would either miss or be able to rule out as systematic effect. This leads to a lower recovery fraction of TCEs comparatively, an effect that is further amplified by the much larger number of TCEs.\n\nThe detection efficiency of PHT is estimated using the fractional recovery rate of TOIs for a range of radius and period bins, as shown in Figure~\\ref{fig:recovery_rank500_radius_period}. A TOI is considered to be recovered if its detection rank is less than 500 within the given sector. Out of the total 1913 TOIs, to date, \\textcolor{red}{PHT recovered 715 TOIs among the highest ranked candidates across the 26 sectors. This corresponds to a mean of 12.7~\\% of the top 500 ranked candidates per sector being TOIs. In comparison, the primary \\emph{TESS}\\ team on average visually vets 1025 TCEs per sector, out of which a mean of 17.3~\\% are promoted to TOI status.} We find that, independent of the orbital period, PHT is over 85~\\% complete in the recovery of TOIs with radii equal to or greater than 4 $R_{\\oplus}$. This agrees with the findings from the initial Planet Hunters project \\citep{schwamb12}. The detection efficiency decreases to 51~\\% for 3 - 4 $R_{\\oplus}$ TOIs, 49~\\% for 2 - 3 $R_{\\oplus}$ TOIs and to less than 40~\\% for TOIs with radii less than 2 $R_{\\oplus}$. Fig~\\ref{fig:recovery_rank500_radius_period} shows that the orbital period does not have a strong effect on the detection efficiency for periods greater than $\\sim$~1~day, which highlights that human vetting efficiency is independent of the number of transits present within a light curve. For periods shorter than around 1~day, the detection efficiency decreases even for larger planets, due to the high frequency of events seen in the light curve. For these light curves, many volunteers will only mark a subset of the transits, which may not overlap with the subset marked by other volunteers. Due to the methodology used to identify and rank the candidates, as described in Section~\\ref{sec:method}, this will actively disfavour the recovery of very short period planets. Although this obviously introduces biases in the detectability of very short period signals, the major detection pipelines are specifically designed to identify these types of planets and thus this does not present a serious detriment to our main science goal of finding planets that were \\textcolor{red}{intentionally ignored or missed} by the main automated pipelines.\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{Figures\/TOI_recovery_grid.png}\n    \\caption{TOI recovery rate as a function of planet radius and orbital period. A TOI is considered recovered if it is amongst the top 500 highest ranked candidates within a given sector. The logarithmically spaced grid ranges from 0.2 to 225 d and 0.6 to 55 $R{_\\oplus}$ for the orbital period and planet radius, respectively. The fraction of TOIs recovered using PHT is computed for each cell and represented by the colour the grid. Cells with less than 10 TOIs are considered incomplete for statistical analysis and are shown by the hatched lines. White cells contain no TOIs. The annotations for each cell indicate the number of recovered TOIs followed by the Poisson uncertainty in brackets. The filled in and empty grey circles indicated the recovered and not-recovered TOIs, respectively.}\n    \\label{fig:recovery_rank500_radius_period}\n\\end{figure*}\n\n\nFinally, we assessed whether the detection efficiency varies across different sectors by assessing the fraction of recovered TOIs and TCEs within the highest ranked 500 candidates. We found the recovery of TOIs within the top 500 highest ranked candidates to remain relatively constant across all sectors, while the fraction of recovered TCEs in the top 500 highest ranked candidates  increases in later sectors, as shown in Figure~\\ref{fig:recovery_rank500}). After applying a Spearman's rank test we find a positive correlation of 0.86 (pvalue = 5.9 $\\times$ $10^{-8}$) and 0.57 (pvalue = 0.003) between the observation sector and TCE and TOI recovery rates, respectively. These correlations suggest that the ability of users to detect transit-like events improves as they classify more subjects. The improvement of volunteers over time can also be seen in Fig~\\ref{fig:user_weights}, which shows the mean (unnormalized) user weight per sector for volunteers who completed one or more classifications in at least one sector (blue), more than 10 sectors (orange), more than 20 sectors (green) and all of the sectors 26 sectors from the nominal \\emph{TESS}\\ mission (pink). The figure highlights an overall improvement in the mean user weight in later sectors, as well as a positive correlation between the overall increase in user weight and the number of sectors that volunteers have participated in.\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.45\\textwidth]{Figures\/TCE_TOI_rank500.png}\n    \\caption{The fractional recovery rate of the TOIs (blue circles) and TCEs (teal squares) at a rank of 500 for each sector. Sector 1-9 (white background) represent southern hemisphere sectors classified with UI1, sectors 9-14 (light grey background) show the southern hemisphere sectors classified with UI2, and sectors 14-24 (dark grey background) show the northern hemisphere sectors classified with US2.}\n    \\label{fig:recovery_rank500}\n\\end{figure}\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.50\\textwidth]{Figures\/user_weights_sectors.png}\n    \\caption{Mean user weights per sector. The solid lines show the user weights for the old user interface and the dashed line for the new interface, separated by the black line (Sector 9). The different coloured lines show the mean user weights calculated considering user who participated in any number of sectors (blue), more than 10 sectors (orange), more than 20 sectors (green) and all of the sectors observed during the nominal \\emph{TESS}\\ mission (pink).}\n    \\label{fig:user_weights}\n\\end{figure}\n\n\n\\section{Candidate vetting}\n\\label{sec:vetting}\n\nFor each observation sector the subjects are ranked according to their transit scores, and the 500 highest ranked targets (excluding TOIs) visually vetted by the PHT science team in order to identify potential candidates and rule out false positives. A vetting cut-off rank of 500 was chosen as we found this to maximise the number of found candidates while minimising the number of likely false positives. In the initial round of vetting, which is completed via a separate Zooniverse classification interface that is only accessible to the core science team, a minimum of three members of the team sort the highest ranked targets into either `keep for further analysis', `eclipsing binary' or `discard'. The sorting is based on the inspection of the full \\emph{TESS}\\ light curve of the target, with the times of the satellite momentum dumps indicated. Additionally, around the time of each likely transit event (i.e. time of successful DB clusters) we inspect the background flux and the x and y centroid positions. Stellar parameters are provided for each candidate, subject to availability, alongside links to the SPOC Data Validation (DV) reports for candidates that had been flagged as TCEs but were never promoted to TOIs status.\n\nCandidates where at least two of the reviewers indicated that the signal is consistent with a planetary transit are kept for further analysis. \\textcolor{red}{This constitute a $\\sim$~5~\\% retention rate of the 500 highest ranked candidates per sector between the initial citizen science classification stage and the PHT science team vetting stage. Considering that the known planets and TOIs are not included at this stage of vetting, it is not surprising that our retention rate is lower that the true-positive rates of TCEs (see Section~\\ref{subsec:TCE_TOI}). Furthermore, this false-positive rate is consistent with the the findings of the initial Planet Hunters project \\citep{schwamb12}.}\n\nThe rest of the 500 candidates were grouped into $\\sim$~37~\\% `eclipsing binary' and $\\sim$~58~\\% `discard'. The most common reasons for discarding light curves are due to events caused by momentum dumps and due to background events, such as background eclipsing binaries, that mimic transit-like signals in the light curve. The targets identified as eclipsing binaries are analysed further by the \\emph{TESS}\\ Eclipsing Binaries Working Group (Prsa et al, in prep).\n\n\n\n\nFor the second round of candidate vetting we generate our own data validation reports for all candidates classified as `keep for further analysis'. The reports are generated using the open source software {\\sc latte} \\citep[Lightcurve Analysis Tool for Transiting Exoplanets;][]{LATTE2020}, which includes a range of standard diagnostic plots that are specifically designed to help identify transit-like signals and weed out astrophysical false positives in \\emph{TESS}\\ data. In brief the diagnostics consist of:\n\n\\textbf{Momentum Dumps}. The times of the \\emph{TESS}\\ reaction wheel momentum dumps that can result in instrumental effects that mimic astrophysical signals.\n\n\\textbf{Background Flux}. The background flux to help identify trends caused by background events such as asteroids or fireflies \\citep{vanderspek2018tess} passing through the field of view.\n\n\\textbf{x and y centroid positions}. The CCD column and row local position of the target's flux-weighted centroid, and the CCD column and row motion which considers differential velocity aberration (DVA), pointing drift, and thermal effects. This can help identify signals caused by systematics due to the satellite. \n\\textbf{Aperture size test}. The target light curve around the time of the transit-like event extracted using two apertures of different sizes. This can help identify signals resulting from background eclipsing binaries.\n \n\\textbf{Pixel-level centroid analysis}. A comparison between the average in-transit and average out-of-transit flux, as well as the difference between them. This can help identify signals resulting from background eclipsing binaries.\n\n\\textbf{Nearby companion stars}. The location of nearby stars brighter than V-band magnitude 15 as queried from the Gaia Data Release 2 catalog \\citep{gaia2018gaia} and the DSS2 red field of view around the target star in order to identify nearby contaminating sources. \n\n\\textbf{Nearest neighbour light curves}. Normalized flux light curves of the five short-cadence \\emph{TESS}\\ stars with the smallest projected distances to the target star, used to identify alternative sources of the signal or systematic effects that affect multiple target stars. \n\n\\textbf{Pixel level light curves}. Individual light curves extracted for each pixel around the target. Used to identify signals resulting from background eclipsing binaries, background events and systematics.\n\n\\textbf{Box-Least-Squares fit}. Results from two consecutive BLS searches, where the identified signals from the initial search are removed prior to the second BLS search.\n\nThe {\\sc latte} validation reports are assessed by the PHT science team in order to identify planetary candidates that warrant further investigation. Around 10~\\% of the targets assessed at this stage of vetting are kept for further investigation, resulting in $\\sim$~3 promising planet candidates per observation sector. The discarded candidates can be loosely categorized into (background) eclipsing binaries ($\\sim$~40~\\%), systematic effects ($\\sim$~25~\\%), background events ($\\sim$~15~\\%) and other (stellar signals such as spots; $\\sim$~10~\\%).\n\n\nWe use \\texttt{pyaneti}\\ \\citep{pyaneti} to infer the planetary and orbital parameters of our most promising candidates. For multi-transit candidates we fit for seven parameters per planet, time of mid-transit $T_0$, orbital period $P$, impact parameter $b$, scaled semi-major axis $a\/R_\\star$, scaled planet radius $r_{\\rm p}\/R_\\star$, and two limb darkening coefficients following a \\citet{Mandel2002} quadratic limb darkening model, implemented with the $q_1$ and $q_2$ parametrization suggested by \\citet{Kipping2013}. Orbits were assumed to be circular.\nFor the mono-transit candidates, we fit the same parameters as for the multi-transit case, except for the orbital period and scaled semi-major axis which cannot be known for single transits. We follow \\citet{Osborn2016} to estimate the orbital period of the mono-transit candidates assuming circular orbits.\n\nWe note that some of our candidates are V-shaped, consistent with a grazing transit configuration. For these cases, we set uniform priors between 0 and 0.15 for $r_{\\rm p}\/R_\\star$ and between 0 and 1.15 for the impact parameter in order to avoid large radii caused by the $r_{\\rm p}\/R_\\star - b$ degeneracy. Thus, the $r_{\\rm p}\/R_\\star$ for these candidates should not be trusted. A full characterisation of these grazing transits is out of the scope of this manuscript.\n\nFigure~\\ref{fig:PHT_pyaneti} shows the \\emph{TESS}\\ transits together with the inferred model for each candidate. Table~\\ref{tab:PHT-caniddates} shows the inferred main parameters, the values and their uncertainties are given by the median and 68.3\\% credible interval of the posterior distributions.\n\n\n\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=0.83\\textwidth]{Figures\/canidate_transits_all_one.png}\n    \\caption{All of the PHT candidates modelled using \\texttt{pyaneti}. The parameters of the best fits are summarised in Table~\\protect\\ref{tab:PHT-caniddates}. The blue and magenta fits show the multi and single transit event candidates, respectively.} \n    \\label{fig:PHT_pyaneti}\n\\end{figure*}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=0.83\\textwidth]{Figures\/canidate_transits_all_two.png}\n    \\addtocounter{figure}{-1}\n    \\caption{\\textbf{PHT candidates (continued)}} \n\\end{figure*}\n\n\nCandidates that pass all of our rounds of vetting are uploaded to the Exoplanet Follow-up Observing Program for TESS (ExoFOP-TESS) website\\footnote{\\url{ https:\/\/exofop.ipac.caltech.edu\/tess\/index.php}} as community TOIs (cTOIs).\n\n\\section{Follow-up observations}\n\\label{sec:follow_up}\n\nMany astrophysical false positive scenarios can be ruled out from the detailed examination of the \\emph{TESS}\\ data, both from the light curves themselves and from the target pixel files. However, not all of the false positive scenarios can be ruled out from these data alone, due in part to the large \\emph{TESS}\\ pixels (20 arcsconds). Our third stage of vetting, therefore, consists of following up the candidates with ground based observations including photometry, reconnaissance spectroscopy and speckle imaging. The results from these observations will be discussed in detail in a dedicated follow-up paper. \n\n\\subsection{Photometry}\n\nWe make use of the LCO global network of fully robotic 0.4-m\/SBIG and 1.0-m\/Sinistro facilities \\citep{LCO2013} to observe additional transits, where the orbital period is known, in order to refine the ephemeris and confirm that the transit events are not due to a blended eclipsing binary in the vicinity of the main target. Snapshot images are taken of single transit event candidates in order to identify nearby contaminating sources. \n\n\n\\subsection{Spectroscopy}\n\nWe perform high-resolution optical spectroscopy using telescopes from across the globe in order to cover a wide range of RA and Dec:\n\\begin{itemize}\n\\item The Las Cumbres Observatory (LCO) telescopes with the Network of Robotic Echelle Spectrographs \\citep[NRES,][]{LCO2013}. These fibre-fed spectrographs, mounted on 1.0-m telescopes around the globe, have a resolution of R = 53,000 and a wavelength coverage of 380 to 860 nm. \n\n\\item The MINERVA Australis Telescope facility, located at Mount Kent Observatory in Queensland, Australia \\citep{addison2019}. This facility is made up of four 0.7m CDK700 telescopes, which individually feed light via optic fibre into a KiwiSpec high-resolution (R = 80,000) stabilised spectrograph \\citep{barnes2012} that covers wavelengths from 480 nm to 620 nm. \n\n\\item The CHIRON spectrograph mounted on the SMARTS 1.5-m telescope \\citep{Tokovinin2018}, located at the Cerro Tololo\nInter-American Observatory (CTIO) in Chile. The high resolution cross-dispersed echelle spectrometer is fiber-fed followed by an image slicer. It has a resolution of R = 80,000 and covers wavelengths ranging from 410 to 870 nm.\n\n\\item The SOPHIE echelle spectrograph mounted on the 1.93-m Haute-Provence Observatory (OHP), France\n\\citep{2008Perruchot,2009Bouchy}. The high resolution cross-dispersed stabilized echelle spectrometer is fed by two optical fibers. Observations were taken in high-resolution mode (R = 75,000) with a wavelength range of 387 to 694 nm.\n\n\\end{itemize}\n\nReconnaissance spectroscopy with these instruments allow us to extract stellar parameters, identify spectroscopic binaries, and place upper limits on the companion masses. Spectroscopic binaries and targets whose spectral type is incompatible with the initial planet hypothesis and\/or precludes precision RV observations (giant or early type stars) are not followed up further. Promising targets, however, are monitored in order to constrain their period and place limits on their mass. \n\n\\subsection{Speckle Imaging}\n\nFor our most promising candidates we perform high resolution speckle imaging using the `Alopeke instrument on the 8.1-m Frederick C. Gillett Gemini North telescope in Maunakea, Hawaii, USA, and its twin, Zorro, on the 8.1-m Gemini South telescope on Cerro Pach\\'{o}n, Chile \\citep{Matson2019, Howell2011}. Speckle interferometric observations provide extremely high resolution images reaching the diffraction limit of the telescope. We obtain simultaneous 562 nm and 832 nm rapid exposure (60 msec) images in succession that effectively `freeze out' atmospheric turbulence and through Fourier analysis are used to search for close companion stars at 5-8 magnitude contrast levels. This analysis, along with the reconstructed images, allow us to identify nearby companions and to quantify their light contribution to the TESS aperture and thus the transit signal.\n\n\n\\section{Planet candidates and Noteworthy Systems}\n\\label{sec:PHT_canidates}\n\\subsection{Planet candidate properties}\n\nIn this final part of the paper we discuss the 90 PHT candidates around 88 host stars that passed the initial two stages of vetting and that were uploaded to ExoFOP as cTOIs. At the time of discovery none of these candidates were TOIs. The properties of all of the PHT candidates are summarised in Table~\\ref{tab:PHT-caniddates}. Candidates that have been promoted to TOI status since their PHT discovery are highlighted with an asterisk following the TIC ID, and candidates that have been shown to be false positives, based on the ground-based follow-up observations, are marked with a dagger symbol ($\\dagger$). The majority (81\\%) of PHT candidates are single transit events, indicated by an `s' following the orbital period presented in the table. \\textcolor{red}{18 of the PHT candidates were flagged as TCEs by the \\emph{TESS}\\ pipeline, but not initially promoted to TOI status. The most common reasons for this was that the pipeline identified a single-transit event as well as times of systematics (often caused by momentum dumps), due to its two-transit minimum detection threshold. This resulted in the candidate being discarded on the basis of it not passing the `odd-even' transit depth test. Out of the 18 TCEs, 14 have become TOI's since the PHT discovery. More detail on the TCE candidates can be found in Appendix~\\ref{appendixA}.} \n\nAll planet parameters (columns 2 to 8) are derived from the \\texttt{pyaneti}\\ modelling as described in Section~\\ref{sec:vetting}. Finally, the table summarises the ground-based follow-up observations (see Sec~\\ref{sec:follow_up}) that have been obtained to date, where the bracketed numbers following the observing instruments indicate the number of epochs. Unless otherwise noted, the follow-up observations are consistent with a planetary scenario. More in depth descriptions of individual targets for which we have additional information to complement the results in Table~\\ref{tab:PHT-caniddates} can be found in Appendix~\\ref{appendixA}.\n\n\\subsection{Planet candidate analysis}\n\n\nThe majority of the TOIs (87.7\\%) have orbital periods shorter than 15 days due to the requirement of observing at least two transits included in all major pipelines \\textcolor{red}{combined with the observing strategy of \\emph{TESS}}. As visual vetting does not impose these limits, the candidates outlined in this paper are helping to populate the relatively under-explored long-period region of parameter space. This is highlighted in Figure~\\ref{fig:PHT_candidates}, which shows the transit depths vs the orbital periods of the PHT single transit candidates (orange circles) and the multi-transit candidates (magenta squares) compared to the TOIs (blue circles). Values of the orbital periods and transit depths were obtained via transit modelling using \\texttt{pyaneti} (see Section~\\ref{sec:vetting}). The orbital period of single transit events are poorly constrained, which is reflected by the large errorbars in Figure~\\ref{fig:PHT_candidates}. Figure~\\ref{fig:PHT_candidates} also highlights that with PHT we are able to recover a similar range of transit depths as the pipeline found TOIs, as was previously shown in Figure~\\ref{fig:recovery_rank500_radius_period}.\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.45\\textwidth]{Figures\/PHT_candidate_period_depth_plot_errobrars.png}\n    \\caption{The properties of the PHT single transit (orange circles) and multi transit (magenta squares) candidates compared to the properties TOIs (blue circles). All parameters (listed in Table~\\ref{fig:PHT_candidates}) were extracted using \\texttt{pyaneti}\\ modelling.}\n    \\label{fig:PHT_candidates}\n\\end{figure}\n\nThe PHT candidates were further compared to the TOIs in terms of the properties of their host stars. Figure~\\ref{fig:eep} shows the effective temperature and stellar radii as taken from the TIC \\citep{Stassun18}, for TOIs (blue dots) and the PHT candidates (magenta circles). The solid and dashed lines indicate the main sequence and post-main sequence MIST stellar evolutionary tracks \\citep{choi2016}, respectively, for stellar masses ranging from 0.3 to 1.6 $M_\\odot$ in steps of 0.1 $M_\\odot$. This shows that around 10\\% of the host stars are in the process of, or have recently evolved off the main sequence. The models assume solar metalicity, no stellar rotation and no additional internal mixing.\n\n\\textcolor{red}{Ground based follow-up spectroscopy has revealed that six of the PHT candidates listed in Table~\\ref{tab:PHT-caniddates} are astrophysical false positives. As the follow-up campaign of the targets is still underway, the true false-positive rate of the candidates to have made it through all stages of the vetting process, as outlined in the methodology, will be be assessed in future PHT papers once the true nature of more of the candidates has been independently verified.}\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.45\\textwidth]{Figures\/PHT_eep.png}\n    \\caption{Stellar evolution tracks showing main sequence (solid black lines) and post-main sequence (dashed grey lines) MIST stellar evolution for stellar masses ranging from 0.3 to 1.6 $M_\\odot$ in steps of 0.1 $M_\\odot$. The blue dots show the TOIs and the magenta circles show the PHT candidates.} \n    \\label{fig:eep}\n\\end{figure}\n\n\n\\subsection{Stellar systems}\n\\label{subsec:PHT_stars}\n\nIn addition to the planetary candidates, citizen science allows for the identification of interesting stellar systems and astrophysical phenomena, in particular where the signals are aperiodic or small compared to the dominant stellar signal. These include light curves that exhibit multiple transit-like signals, possibly as a result of a multiple stellar system or a blend of eclipsing binaries. We have investigated all light curves that were flagged as possible multi-stellar systems via the PHT discussion boards. Similar to the planet vetting, as described in Section~\\ref{sec:vetting}, we generated {\\sc latte} data validation reports in order to assess the nature of the signal. Additionally, we subjected these systems to an iterative signal removal process, whereby we phase-folded the light curve on the dominant orbital period, binned the light curve into between 200-500 phase bins, created an interpolation model, and then subtracted said signal in order to evaluate the individual transit signals. The period of each signal, as listed in Table~\\ref{tab:PHT-multis}, was determined by phase folding the light curve at a number of trial periods and assessing by eye the best fit period and corresponding uncertainty.\n\nDue to the large \\emph{TESS}\\ pixels, blends are expected to be common. We searched for blends by generating phase folded light curves for each pixel around the source of the target in order to better locate the source of each signal. Shifts in the \\emph{TESS}\\ x and y centroid positions were also found to be good indicators of visually separated sources. Nearby sources with a magnitude difference greater than 5 mags were ruled out as possible contaminators. We consider a candidate to be a confirmed blend when the centroids are separated by more than 1 \\emph{TESS}\\ pixel, as this corresponds to an angular separation > 21 arcseconds meaning that the systems are highly unlikely to be gravitationally bound. Systems where the signal appears to be coming from the same \\emph{TESS}\\ pixel and that show no clear centroid shifts are considered to be candidate multiple systems. We note that blends are still possible, however, without further investigation we cannot conclusively rule these out as possible multi stellar systems. \n\nAll of the systems are summarised in Table~\\ref{tab:PHT-multis}. Out of the 26 systems, 6 are confirmed multiple systems which have either been published or are being prepared for publication; 7 are visually separated eclipsing binaries (confirmed blends); and 13 are candidate multiple system. Additional observations will be required to determine whether or not these candidate multiple systems are in fact gravitationally bound or photometric blends as a results of the large \\emph{TESS}\\ pixels or due to a line of sight happenstance. \n\n\\begin{landscape}\n\\begin{table}\n\\resizebox{1.31\\textwidth}{!}{\n\\begin{tabular}{ccccccccccccc}\n\\textbf{TIC} & \\textbf{Other} & \\textbf{Epoch} & \\textbf{Period}  & \\textbf{$R_{pl}$\/$R_{\\odot}$} & \\textbf{$R_{pl}$} & \\textbf{Impact} & \\textbf{Duration} & \\textbf{$V_{mag}$} & \\textbf{Photometry} & \\textbf{Spectroscopy} & \\textbf{Speckle} & \\textbf{Comment} \\\\\n  & \\textbf{Name} & \\textbf{(\\textcolor{red}{BJD - 2457000})} & \\textbf{(days)}  & &  $(R_{\\oplus})$ & \\textbf{Parameter} & \\textbf{(hours)} &   &   &   &   &    \\\\\n\\hline\n101641905 & TWOMASS 11412617+3441004 & $1917.26335 _{ - 0.00072 } ^ { + 0.00071 }$ & $14.52 _{ - 5.25 } ^ { + 6.21 }(s)$ & $0.1135 _{ - 0.0064 } ^ { + 0.0032 }$ & $9.76 _{ - 0.69 } ^ { + 0.65 }$ & $0.691 _{ - 0.183 } ^ { + 0.077 }$ & $3.163 _{ - 0.088 } ^ { + 0.093 }$ & 12.196 &  &  &  &  \\\\\n103633672* & TYC 4387-00923-1 & $1850.3211 _{ - 0.00077 } ^ { + 0.00135 }$ & $90.9 _{ - 23.7 } ^ { + 46.4 }(s)$ & $0.0395 _{ - 0.0013 } ^ { + 0.0013 }$ & $3.45 _{ - 0.24 } ^ { + 0.26 }$ & $0.3 _{ - 0.21 } ^ { + 0.26 }$ & $6.7 _{ - 0.11 } ^ { + 0.12 }$ & 10.586 &  & NRES (1) &  &  \\\\\n110996418 & TWOMASS 12344723-1019107 & $1580.6406 _{ - 0.0038 } ^ { + 0.0037 }$ & $5.18 _{ - 2.93 } ^ { + 6.86 }(s)$ & $0.1044 _{ - 0.0067 } ^ { + 0.008 }$ & $12.7 _{ - 0.99 } ^ { + 1.15 }$ & $0.44 _{ - 0.3 } ^ { + 0.3 }$ & $3.53 _{ - 0.27 } ^ { + 0.36 }$ & 13.945 &  &  &  &  \\\\\n128703021 & HIP 71639 & $1601.8442 _{ - 0.00108 } ^ { + 0.00093 }$ & $26.0 _{ - 8.22 } ^ { + 22.35 }(s)$ & $0.0254 _{ - 0.00049 } ^ { + 0.00072 }$ & $4.44 _{ - 0.2 } ^ { + 0.23 }$ & $0.47 _{ - 0.3 } ^ { + 0.22 }$ & $7.283 _{ - 0.091 } ^ { + 0.141 }$ & 6.06 &  & NRES (2);MINERVA (34) & Gemini &  \\\\\n138126035 & TYC 1450-00833-1 & $1954.3229 _{ - 0.0041 } ^ { + 0.0067 }$ & $28.8 _{ - 14.0 } ^ { + 203.2 }(s)$ & $0.0375 _{ - 0.0026 } ^ { + 0.0069 }$ & $4.01 _{ - 0.35 } ^ { + 0.74 }$ & $0.58 _{ - 0.38 } ^ { + 0.35 }$ & $4.65 _{ - 0.32 } ^ { + 0.85 }$ & 10.349 &  &  &  &  \\\\\n142087638 & TYC 9189-00274-1 & $1512.1673 _{ - 0.0043 } ^ { + 0.0034 }$ & $3.14 _{ - 1.41 } ^ { + 12.04 }(s)$ & $0.0469 _{ - 0.0035 } ^ { + 0.0063 }$ & $6.05 _{ - 0.54 } ^ { + 0.89 }$ & $0.5 _{ - 0.35 } ^ { + 0.36 }$ & $2.72 _{ - 0.23 } ^ { + 0.5 }$ & 11.526 &  &  &  &  \\\\\n159159904 & HIP 64812 & $1918.6109 _{ - 0.0067 } ^ { + 0.0091 }$ & $584.0 _{ - 215.0 } ^ { + 1724.0 }(s)$ & $0.0237 _{ - 0.0011 } ^ { + 0.0026 }$ & $3.12 _{ - 0.22 } ^ { + 0.36 }$ & $0.49 _{ - 0.34 } ^ { + 0.35 }$ & $15.11 _{ - 0.54 } ^ { + 0.7 }$ & 9.2 &  & NRES (2) &  &  \\\\\n160039081* & HIP 78892 & $1752.9261 _{ - 0.0045 } ^ { + 0.005 }$ & $30.19918 _{ - 0.00099 } ^ { + 0.00094 }$ & $0.0211 _{ - 0.0013 } ^ { + 0.0035 }$ & $2.67 _{ - 0.21 } ^ { + 0.43 }$ & $0.52 _{ - 0.34 } ^ { + 0.36 }$ & $4.93 _{ - 0.27 } ^ { + 0.37 }$ & 8.35 & SBIG (1) & NRES (1);SOPHIE (4) & Gemini &  \\\\\n162631539 & HIP 80264 & $1978.2794 _{ - 0.0044 } ^ { + 0.0051 }$ & $17.32 _{ - 6.66 } ^ { + 52.35 }(s)$ & $0.0195 _{ - 0.0011 } ^ { + 0.0024 }$ & $2.94 _{ - 0.24 } ^ { + 0.38 }$ & $0.48 _{ - 0.33 } ^ { + 0.36 }$ & $5.54 _{ - 0.33 } ^ { + 0.41 }$ & 7.42 &  &  &  &  \\\\\n166184426* & TWOMASS 13442500-4020122 & $1600.4409 _{ - 0.003 } ^ { + 0.0036 }$ & $16.3325 _{ - 0.0066 } ^ { + 0.0052 }$ & $0.0545 _{ - 0.0031 } ^ { + 0.0039 }$ & $1.85 _{ - 0.12 } ^ { + 0.15 }$ & $0.41 _{ - 0.28 } ^ { + 0.31 }$ & $1.98 _{ - 0.22 } ^ { + 0.17 }$ & 12.911 &  &  &  &  \\\\\n167661160 & TYC 7054-01577-1 & $1442.0703 _{ - 0.0028 } ^ { + 0.004 }$ & $36.802 _{ - 0.07 } ^ { + 0.069 }$ & $0.0307 _{ - 0.0014 } ^ { + 0.0024 }$ & $4.07 _{ - 0.32 } ^ { + 0.43 }$ & $0.37 _{ - 0.26 } ^ { + 0.33 }$ & $5.09 _{ - 0.23 } ^ { + 0.21 }$ & 9.927 &  & NRES (9);MINERVA (4) &  & EB from MINERVA observations \\\\\n172370679* & TWOMASS 19574239+4008357 & $1711.95923 _{ - 0.00099 } ^ { + 0.001 }$ & $32.84 _{ - 4.17 } ^ { + 5.59 }(s)$ & $0.1968 _{ - 0.0032 } ^ { + 0.0022 }$ & $13.24 _{ - 0.43 } ^ { + 0.43 }$ & $0.22 _{ - 0.15 } ^ { + 0.14 }$ & $4.999 _{ - 0.097 } ^ { + 0.111 }$ & 14.88 &  &  &  & Confirmed planet \\citep{canas2020}. \\\\\n174302697* & TYC 3641-01789-1 & $1743.7267 _{ - 0.00092 } ^ { + 0.00093 }$ & $498.2 _{ - 80.0 } ^ { + 95.3 }(s)$ & $0.07622 _{ - 0.00068 } ^ { + 0.00063 }$ & $13.34 _{ - 0.57 } ^ { + 0.58 }$ & $0.642 _{ - 0.029 } ^ { + 0.024 }$ & $17.71 _{ - 0.12 } ^ { + 0.13 }$ & 9.309 & SBIG (1) &  &  &  \\\\\n179582003 & TYC 9166-00745-1 & $1518.4688 _{ - 0.0016 } ^ { + 0.0016 }$ & $104.6137 _{ - 0.0022 } ^ { + 0.0022 }$ & $0.06324 _{ - 0.0008 } ^ { + 0.0008 }$ & $7.51 _{ - 0.35 } ^ { + 0.35 }$ & $0.21 _{ - 0.15 } ^ { + 0.19 }$ & $9.073 _{ - 0.084 } ^ { + 0.097 }$ & 10.806 &  &  &  &  \\\\\n192415680 & TYC 2859-00682-1 & $1796.0265 _{ - 0.0012 } ^ { + 0.0013 }$ & $18.47 _{ - 6.34 } ^ { + 21.73 }(s)$ & $0.0478 _{ - 0.0017 } ^ { + 0.0027 }$ & $4.43 _{ - 0.33 } ^ { + 0.38 }$ & $0.45 _{ - 0.31 } ^ { + 0.31 }$ & $3.94 _{ - 0.1 } ^ { + 0.12 }$ & 9.838 & SBIG (1) & SOPHIE (2) &  &  \\\\\n192790476 & TYC 7595-00649-1 & $1452.3341 _{ - 0.0014 } ^ { + 0.002 }$ & $16.09 _{ - 5.73 } ^ { + 15.49 }(s)$ & $0.0438 _{ - 0.0018 } ^ { + 0.0026 }$ & $3.24 _{ - 0.34 } ^ { + 0.37 }$ & $0.37 _{ - 0.25 } ^ { + 0.3 }$ & $3.395 _{ - 0.099 } ^ { + 0.192 }$ & 10.772 &  &  &  &  \\\\\n206361691$\\dagger$ & HIP 117250 & $1363.2224 _{ - 0.0082 } ^ { + 0.009 }$ & $237.7 _{ - 81.0 } ^ { + 314.4 }(s)$ & $0.01762 _{ - 0.00088 } ^ { + 0.00125 }$ & $2.69 _{ - 0.19 } ^ { + 0.25 }$ & $0.43 _{ - 0.28 } ^ { + 0.32 }$ & $13.91 _{ - 0.53 } ^ { + 0.52 }$ & 8.88 &  & CHIRON (2) &  & SB2 from CHIRON \\\\\n207501148 & TYC 3881-00527-1 & $2007.7273 _{ - 0.0011 } ^ { + 0.0011 }$ & $39.9 _{ - 10.3 } ^ { + 14.3 }(s)$ & $0.0981 _{ - 0.0047 } ^ { + 0.011 }$ & $13.31 _{ - 0.95 } ^ { + 1.56 }$ & $0.9 _{ - 0.03 } ^ { + 0.039 }$ & $4.73 _{ - 0.14 } ^ { + 0.14 }$ & 10.385 &  &  &  &  \\\\\n219466784* & TYC 4409-00437-1 & $1872.6879 _{ - 0.0097 } ^ { + 0.0108 }$ & $318.0 _{ - 147.0 } ^ { + 1448.0 }(s)$ & $0.0332 _{ - 0.0024 } ^ { + 0.0048 }$ & $3.26 _{ - 0.31 } ^ { + 0.49 }$ & $0.55 _{ - 0.39 } ^ { + 0.34 }$ & $10.06 _{ - 0.81 } ^ { + 1.12 }$ & 11.099 &  &  &  &  \\\\\n219501568 & HIP 79876 & $1961.7879 _{ - 0.0018 } ^ { + 0.002 }$ & $16.5931 _{ - 0.0017 } ^ { + 0.0015 }$ & $0.0221 _{ - 0.0012 } ^ { + 0.0015 }$ & $4.22 _{ - 0.3 } ^ { + 0.35 }$ & $0.41 _{ - 0.28 } ^ { + 0.31 }$ & $1.615 _{ - 0.077 } ^ { + 0.093 }$ & 8.38 &  &  &  &  \\\\\n229055790 & TYC 7492-01197-1 & $1337.866 _{ - 0.0022 } ^ { + 0.0019 }$ & $48.0 _{ - 12.8 } ^ { + 48.4 }(s)$ & $0.0304 _{ - 0.00097 } ^ { + 0.00115 }$ & $3.52 _{ - 0.2 } ^ { + 0.24 }$ & $0.37 _{ - 0.26 } ^ { + 0.32 }$ & $6.53 _{ - 0.11 } ^ { + 0.14 }$ & 9.642 &  & NRES (2) &  &  \\\\\n229608594 & TWOMASS 18180283+7428005 & $1960.0319 _{ - 0.0037 } ^ { + 0.0045 }$ & $152.4 _{ - 54.1 } ^ { + 152.6 }(s)$ & $0.0474 _{ - 0.0023 } ^ { + 0.0024 }$ & $3.42 _{ - 0.34 } ^ { + 0.36 }$ & $0.38 _{ - 0.26 } ^ { + 0.3 }$ & $6.98 _{ - 0.23 } ^ { + 0.37 }$ & 12.302 &  &  &  &  \\\\\n229742722* & TYC 4434-00596-1 & $1689.688 _{ - 0.025 } ^ { + 0.02 }$ & $29.0 _{ - 16.4 } ^ { + 66.3 }(s)$ & $0.019 _{ - 0.0028 } ^ { + 0.0029 }$ & $2.9 _{ - 0.44 } ^ { + 0.48 }$ & $0.44 _{ - 0.3 } ^ { + 0.33 }$ & $4.27 _{ - 0.09 } ^ { + 0.11 }$ & 10.33 &  & NRES (8);SOPHIE (4) & Gemini &  \\\\\n233194447 & TYC 4211-00650-1 & $1770.4924 _{ - 0.0065 } ^ { + 0.0107 }$ & $373.0 _{ - 101.0 } ^ { + 284.0 }(s)$ & $0.02121 _{ - 0.00073 } ^ { + 0.001 }$ & $5.08 _{ - 0.28 } ^ { + 0.33 }$ & $0.34 _{ - 0.24 } ^ { + 0.29 }$ & $24.45 _{ - 0.47 } ^ { + 0.5 }$ & 9.178 &  & NRES (2) & Gemini &  \\\\\n235943205 & TYC 4588-00127-1 & $1827.0267 _{ - 0.004 } ^ { + 0.0034 }$ & $121.3394 _{ - 0.0063 } ^ { + 0.0065 }$ & $0.0402 _{ - 0.0016 } ^ { + 0.0019 }$ & $4.2 _{ - 0.25 } ^ { + 0.29 }$ & $0.4 _{ - 0.27 } ^ { + 0.28 }$ & $6.37 _{ - 0.2 } ^ { + 0.3 }$ & 11.076 &  & NRES (1);SOPHIE (2) &  &  \\\\\n237201858 & TYC 4452-00759-1 & $1811.5032 _{ - 0.0069 } ^ { + 0.0067 }$ & $129.7 _{ - 41.5 } ^ { + 146.8 }(s)$ & $0.0258 _{ - 0.0013 } ^ { + 0.0015 }$ & $4.12 _{ - 0.27 } ^ { + 0.3 }$ & $0.4 _{ - 0.28 } ^ { + 0.31 }$ & $10.94 _{ - 0.37 } ^ { + 0.53 }$ & 10.344 &  & NRES (1) &  &  \\\\\n243187830* & HIP 5286 & $1783.7671 _{ - 0.0017 } ^ { + 0.0019 }$ & $4.05 _{ - 1.53 } ^ { + 9.21 }(s)$ & $0.0268 _{ - 0.0015 } ^ { + 0.0027 }$ & $2.06 _{ - 0.17 } ^ { + 0.23 }$ & $0.47 _{ - 0.32 } ^ { + 0.34 }$ & $2.02 _{ - 0.12 } ^ { + 0.15 }$ & 8.407 & SBIG (1) &  &  &  \\\\\n243417115 & TYC 8262-02120-1 & $1614.4796 _{ - 0.0028 } ^ { + 0.0022 }$ & $1.81 _{ - 0.73 } ^ { + 3.45 }(s)$ & $0.0523 _{ - 0.0035 } ^ { + 0.005 }$ & $5.39 _{ - 0.47 } ^ { + 0.64 }$ & $0.47 _{ - 0.33 } ^ { + 0.34 }$ & $2.03 _{ - 0.16 } ^ { + 0.23 }$ & 11.553 &  &  &  &  \\\\\n256429408 & TYC 4462-01942-1 & $1962.16 _{ - 0.0022 } ^ { + 0.0023 }$ & $382.0 _{ - 132.0 } ^ { + 265.0 }(s)$ & $0.03582 _{ - 0.00086 } ^ { + 0.00094 }$ & $6.12 _{ - 0.29 } ^ { + 0.3 }$ & $0.51 _{ - 0.36 } ^ { + 0.18 }$ & $16.96 _{ - 0.2 } ^ { + 0.24 }$ & 8.898 &  &  &  &  \\\\\n264544388* & TYC 4607-01275-1 & $1824.8438 _{ - 0.0076 } ^ { + 0.0078 }$ & $7030.0 _{ - 6260.0 } ^ { + 3330.0 }(s)$ & $0.0288 _{ - 0.0029 } ^ { + 0.0018 }$ & $4.58 _{ - 0.43 } ^ { + 0.35 }$ & $0.936 _{ - 0.363 } ^ { + 0.011 }$ & $19.13 _{ - 1.35 } ^ { + 0.84 }$ & 8.758 &  & NRES (1) &  &  \\\\\n264766922 & TYC 8565-01780-1 & $1538.69518 _{ - 0.00091 } ^ { + 0.00091 }$ & $3.28 _{ - 0.94 } ^ { + 1.25 }(s)$ & $0.0933 _{ - 0.0063 } ^ { + 0.0176 }$ & $16.95 _{ - 1.33 } ^ { + 3.19 }$ & $0.908 _{ - 0.039 } ^ { + 0.048 }$ & $2.73 _{ - 0.11 } ^ { + 0.11 }$ & 10.747 &  &  &  &  \\\\\n26547036* & TYC 3921-01563-1 & $1712.30464 _{ - 0.00041 } ^ { + 0.0004 }$ & $73.0 _{ - 13.6 } ^ { + 16.5 }(s)$ & $0.10034 _{ - 0.0007 } ^ { + 0.00078 }$ & $11.75 _{ - 0.59 } ^ { + 0.58 }$ & $0.17 _{ - 0.12 } ^ { + 0.11 }$ & $8.681 _{ - 0.049 } ^ { + 0.052 }$ & 9.849 &  & NRES (4) & Gemini &  \\\\\n267542728$\\dagger$ & TYC 4583-01499-1 & $1708.4956 _{ - 0.0073 } ^ { + 0.0085 }$ & $39.7382 _{ - 0.0023 } ^ { + 0.0023 }$ & $0.03267 _{ - 0.00089 } ^ { + 0.00175 }$ & $18.46 _{ - 0.94 } ^ { + 1.14 }$ & $0.38 _{ - 0.26 } ^ { + 0.27 }$ & $24.16 _{ - 0.39 } ^ { + 0.45 }$ & 11.474 &  &  &  & EB from HIRES RVs. \\\\\n270371513$\\dagger$ & HIP 10047 & $1426.2967 _{ - 0.0023 } ^ { + 0.002 }$ & $0.39 _{ - 0.17 } ^ { + 1.79 }(s)$ & $0.024 _{ - 0.0015 } ^ { + 0.0032 }$ & $4.8 _{ - 0.38 } ^ { + 0.64 }$ & $0.5 _{ - 0.34 } ^ { + 0.39 }$ & $1.93 _{ - 0.16 } ^ { + 0.19 }$ & 6.98515 &  & MINERVA (20) &  & SB 2 from MINERVA observations.  \\\\\n274599700 & TWOMASS 17011885+5131455 & $2002.1202 _{ - 0.0024 } ^ { + 0.0024 }$ & $32.9754 _{ - 0.005 } ^ { + 0.005 }$ & $0.0847 _{ - 0.0021 } ^ { + 0.0018 }$ & $13.25 _{ - 0.83 } ^ { + 0.83 }$ & $0.37 _{ - 0.24 } ^ { + 0.19 }$ & $8.2 _{ - 0.18 } ^ { + 0.21 }$ & 12.411 &  &  &  &  \\\\\n278990954 & TYC 8548-00717-1 & $1650.0191 _{ - 0.0086 } ^ { + 0.0105 }$ & $18.45 _{ - 8.66 } ^ { + 230.7 }(s)$ & $0.034 _{ - 0.0024 } ^ { + 0.0115 }$ & $9.65 _{ - 0.92 } ^ { + 3.13 }$ & $0.58 _{ - 0.4 } ^ { + 0.36 }$ & $10.62 _{ - 0.66 } ^ { + 2.46 }$ & 10.749 &  &  &  &  \\\\\n280865159* & TYC 9384-01533-1 & $1387.0749 _{ - 0.0045 } ^ { + 0.0044 }$ & $1045.0 _{ - 249.0 } ^ { + 536.0 }(s)$ & $0.0406 _{ - 0.0011 } ^ { + 0.0014 }$ & $4.75 _{ - 0.26 } ^ { + 0.28 }$ & $0.35 _{ - 0.24 } ^ { + 0.23 }$ & $19.08 _{ - 0.32 } ^ { + 0.36 }$ & 11.517 &  &  & Gemini &  \\\\\n284361752 & TYC 3924-01678-1 & $2032.093 _{ - 0.0078 } ^ { + 0.008 }$ & $140.6 _{ - 46.6 } ^ { + 159.1 }(s)$ & $0.0259 _{ - 0.0014 } ^ { + 0.0017 }$ & $3.62 _{ - 0.26 } ^ { + 0.31 }$ & $0.4 _{ - 0.27 } ^ { + 0.34 }$ & $8.98 _{ - 0.66 } ^ { + 0.86 }$ & 10.221 &  &  &  &  \\\\\n288240183 & TYC 4634-01225-1 & $1896.941 _{ - 0.0051 } ^ { + 0.0047 }$ & $119.0502 _{ - 0.0091 } ^ { + 0.0089 }$ & $0.02826 _{ - 0.00089 } ^ { + 0.00119 }$ & $4.28 _{ - 0.35 } ^ { + 0.36 }$ & $0.55 _{ - 0.37 } ^ { + 0.25 }$ & $17.49 _{ - 0.36 } ^ { + 0.6 }$ & 9.546 &  &  &  &  \\\\\n29169215 & TWOMASS 09011787+4727085 & $1872.5047 _{ - 0.0032 } ^ { + 0.0036 }$ & $14.89 _{ - 6.12 } ^ { + 24.84 }(s)$ & $0.0403 _{ - 0.0025 } ^ { + 0.0033 }$ & $3.28 _{ - 0.37 } ^ { + 0.45 }$ & $0.44 _{ - 0.3 } ^ { + 0.33 }$ & $3.56 _{ - 0.21 } ^ { + 0.32 }$ & 11.828 &  &  &  &  \\\\\n293649602 & TYC 8103-00266-1 & $1511.2109 _{ - 0.004 } ^ { + 0.0037 }$ & $12.85 _{ - 5.34 } ^ { + 42.21 }(s)$ & $0.04 _{ - 0.0024 } ^ { + 0.0039 }$ & $4.66 _{ - 0.36 } ^ { + 0.5 }$ & $0.5 _{ - 0.35 } ^ { + 0.34 }$ & $4.1 _{ - 0.31 } ^ { + 0.56 }$ & 10.925 &  &  &  &  \\\\\n296737508 & TYC 5472-01060-1 & $1538.0036 _{ - 0.0015 } ^ { + 0.0016 }$ & $18.27 _{ - 5.06 } ^ { + 17.45 }(s)$ & $0.0425 _{ - 0.0014 } ^ { + 0.0019 }$ & $5.33 _{ - 0.22 } ^ { + 0.27 }$ & $0.44 _{ - 0.3 } ^ { + 0.26 }$ & $5.13 _{ - 0.13 } ^ { + 0.15 }$ & 9.772 & Sinistro (1) & NRES (1);MINERVA (1) & Gemini &  \\\\\n298663873 & TYC 3913-01781-1 & $1830.76819 _{ - 0.00099 } ^ { + 0.00099 }$ & $479.9 _{ - 89.4 } ^ { + 109.4 }(s)$ & $0.06231 _{ - 0.00034 } ^ { + 0.00045 }$ & $11.07 _{ - 0.57 } ^ { + 0.57 }$ & $0.16 _{ - 0.11 } ^ { + 0.13 }$ & $23.99 _{ - 0.093 } ^ { + 0.1 }$ & 9.162 &  & NRES (2) & Gemini & Dalba et al. (in prep) \\\\\n303050301 & TYC 6979-01108-1 & $1366.1301 _{ - 0.0022 } ^ { + 0.0023 }$ & $281.0 _{ - 170.0 } ^ { + 264.0 }(s)$ & $0.0514 _{ - 0.0027 } ^ { + 0.0018 }$ & $4.85 _{ - 0.32 } ^ { + 0.32 }$ & $0.73 _{ - 0.48 } ^ { + 0.1 }$ & $7.91 _{ - 0.31 } ^ { + 0.36 }$ & 10.048 &  & NRES (1) & Gemini &  \\\\\n303317324 & TYC 6983-00438-1 & $1365.1845 _{ - 0.0023 } ^ { + 0.0028 }$ & $69.0 _{ - 25.5 } ^ { + 78.1 }(s)$ & $0.0365 _{ - 0.0013 } ^ { + 0.0016 }$ & $2.88 _{ - 0.3 } ^ { + 0.31 }$ & $0.39 _{ - 0.26 } ^ { + 0.32 }$ & $5.78 _{ - 0.18 } ^ { + 0.24 }$ & 10.799 &  &  &  &  \\\\\n\\hline\n\\end{tabular}}\n\\caption{\\emph{Note} -- Candidates that have become TOIs following the PHT discovery are marked with an asterisk (*). The `s' following the orbital period indicates that the candidates is a single transit event. The ground-based follow-up observations are summarized in columns 10-12, where the bracketed numbers correspond the number of epochs obtained with each instrument.  See Section~\\ref{sec:follow_up} for description of each instrument. The $\\dagger$ symbol indicates candidates that have been shown to be astrophysical false  positives based on the ground based follow-up observations.}\n\\label{tab:PHT-caniddates}\n\\end{table}\n\\end{landscape}\n\n\\begin{landscape}\n\\begin{table}\n\\addtocounter{table}{-1}\n\\resizebox{1.31\\textwidth}{!}{\n\\begin{tabular}{ccccccccccccc}\n\\textbf{TIC} & \\textbf{Other} & \\textbf{Epoch} & \\textbf{Period}  & \\textbf{$R_{pl}$\/$R_{\\odot}$} & \\textbf{$R_{pl}$} & \\textbf{Impact} & \\textbf{Duration} & \\textbf{$V_{mag}$} & \\textbf{Photometry} & \\textbf{Spectroscopy} & \\textbf{Speckle} & \\textbf{Comment} \\\\\n  & \\textbf{Name} & \\textbf{(\\textcolor{red}{BJD - 2457000})} & \\textbf{(days)}  & & $(R_{\\oplus})$  & \\textbf{Parameter} & \\textbf{(hours)} &   &   &   &   &   \\\\\n\\hline\n303586471$\\dagger$ & HIP 115828 & $1363.7692 _{ - 0.0033 } ^ { + 0.0027 }$ & $13.85 _{ - 4.19 } ^ { + 18.2 }(s)$ & $0.0214 _{ - 0.001 } ^ { + 0.0014 }$ & $2.52 _{ - 0.16 } ^ { + 0.2 }$ & $0.4 _{ - 0.27 } ^ { + 0.33 }$ & $4.23 _{ - 0.19 } ^ { + 0.16 }$ & 8.27 &  & MINERVA (11) &  & SB 2 from MINERVA observations.  \\\\\n304142124* & HIP 53719 & $1585.28023 _{ - 0.0008 } ^ { + 0.0008 }$ & $42.8 _{ - 10.0 } ^ { + 18.2 }(s)$ & $0.04311 _{ - 0.00093 } ^ { + 0.00153 }$ & $4.1 _{ - 0.23 } ^ { + 0.24 }$ & $0.33 _{ - 0.21 } ^ { + 0.21 }$ & $5.66 _{ - 0.067 } ^ { + 0.09 }$ & 8.62 &  & NRES (1);MINERVA (4) &  & Confirmed planet \\citep{diaz2020} \\\\\n304339227 & TYC 9290-01087-1 & $1673.3242 _{ - 0.009 } ^ { + 0.0128 }$ & $111.9 _{ - 72.2 } ^ { + 4844.1 }(s)$ & $0.0253 _{ - 0.0024 } ^ { + 0.0481 }$ & $3.27 _{ - 0.61 } ^ { + 5.72 }$ & $0.67 _{ - 0.47 } ^ { + 0.36 }$ & $7.44 _{ - 0.86 } ^ { + 2.84 }$ & 9.169 &  &  &  &  \\\\\n307958020 & TYC 4191-00309-1 & $1864.82 _{ - 0.014 } ^ { + 0.013 }$ & $169.0 _{ - 107.0 } ^ { + 10194.0 }(s)$ & $0.0223 _{ - 0.0022 } ^ { + 0.0543 }$ & $3.92 _{ - 0.52 } ^ { + 9.27 }$ & $0.71 _{ - 0.53 } ^ { + 0.33 }$ & $12.48 _{ - 1.1 } ^ { + 5.41 }$ & 9.017 &  &  &  &  \\\\\n308301091 & TYC 2081-01273-1 & $2030.3691 _{ - 0.0024 } ^ { + 0.0026 }$ & $29.24 _{ - 8.49 } ^ { + 22.46 }(s)$ & $0.0362 _{ - 0.0013 } ^ { + 0.0014 }$ & $5.41 _{ - 0.34 } ^ { + 0.35 }$ & $0.35 _{ - 0.25 } ^ { + 0.29 }$ & $6.57 _{ - 0.14 } ^ { + 0.19 }$ & 10.273 &  &  &  &  \\\\\n313006381 & HIP 45012 & $1705.687 _{ - 0.0081 } ^ { + 0.0045 }$ & $21.56 _{ - 8.9 } ^ { + 54.15 }(s)$ & $0.0261 _{ - 0.0017 } ^ { + 0.0027 }$ & $2.34 _{ - 0.2 } ^ { + 0.27 }$ & $0.45 _{ - 0.3 } ^ { + 0.38 }$ & $3.85 _{ - 0.51 } ^ { + 0.31 }$ & 9.39 &  &  &  &  \\\\\n323295479* & TYC 9506-01881-1 & $1622.9258 _{ - 0.00083 } ^ { + 0.00087 }$ & $117.8 _{ - 25.8 } ^ { + 30.9 }(s)$ & $0.0981 _{ - 0.0021 } ^ { + 0.0023 }$ & $11.35 _{ - 0.67 } ^ { + 0.66 }$ & $0.839 _{ - 0.024 } ^ { + 0.019 }$ & $6.7 _{ - 0.14 } ^ { + 0.15 }$ & 10.595 &  &  &  &  \\\\\n328933398.01* & TYC 4634-01435-1 & $1880.9878 _{ - 0.0039 } ^ { + 0.0042 }$ & $24.9335 _{ - 0.0046 } ^ { + 0.005 }$ & $0.0437 _{ - 0.0022 } ^ { + 0.0023 }$ & $4.62 _{ - 0.32 } ^ { + 0.33 }$ & $0.38 _{ - 0.25 } ^ { + 0.27 }$ & $5.02 _{ - 0.22 } ^ { + 0.27 }$ & 11.215 &  &  &  & Potential multi-planet system. \\\\\n328933398.02* & TYC 4634-01435-1 & $1848.6557 _{ - 0.0053 } ^ { + 0.0072 }$ & $50.5 _{ - 22.4 } ^ { + 77.1 }(s)$ & $0.0296 _{ - 0.0028 } ^ { + 0.0033 }$ & $3.14 _{ - 0.33 } ^ { + 0.39 }$ & $0.41 _{ - 0.28 } ^ { + 0.35 }$ & $5.99 _{ - 0.8 } ^ { + 0.77 }$ & 11.215 &  &  &  &  \\\\\n331644554 & TYC 3609-00469-1 & $1757.0354 _{ - 0.0031 } ^ { + 0.0033 }$ & $947.0 _{ - 215.0 } ^ { + 274.0 }(s)$ & $0.12 _{ - 0.025 } ^ { + 0.021 }$ & $21.84 _{ - 4.57 } ^ { + 3.86 }$ & $1.018 _{ - 0.036 } ^ { + 0.028 }$ & $10.93 _{ - 0.34 } ^ { + 0.35 }$ & 9.752 &  &  &  &  \\\\\n332657786 & TWOMASS 09595797-1609323 & $1536.7659 _{ - 0.0015 } ^ { + 0.0015 }$ & $63.76 _{ - 9.52 } ^ { + 11.13 }(s)$ & $0.14961 _{ - 0.00064 } ^ { + 0.00029 }$ & $3.83 _{ - 0.12 } ^ { + 0.12 }$ & $0.059 _{ - 0.041 } ^ { + 0.064 }$ & $3.333 _{ - 0.095 } ^ { + 0.096 }$ & 15.99 &  &  &  &  \\\\\n336075472 & TYC 3526-00332-1 & $2028.1762 _{ - 0.0043 } ^ { + 0.0037 }$ & $61.9 _{ - 24.0 } ^ { + 95.6 }(s)$ & $0.0402 _{ - 0.0022 } ^ { + 0.0033 }$ & $3.09 _{ - 0.34 } ^ { + 0.4 }$ & $0.43 _{ - 0.29 } ^ { + 0.32 }$ & $5.39 _{ - 0.23 } ^ { + 0.37 }$ & 11.842 &  &  &  &  \\\\\n349488688.01 & TYC 1529-00224-1 & $1994.283 _{ - 0.0038 } ^ { + 0.0033 }$ & $11.6254 _{ - 0.005 } ^ { + 0.0052 }$ & $0.02195 _{ - 0.00096 } ^ { + 0.00122 }$ & $3.44 _{ - 0.18 } ^ { + 0.21 }$ & $0.39 _{ - 0.27 } ^ { + 0.3 }$ & $5.58 _{ - 0.15 } ^ { + 0.18 }$ & 8.855 &  & NRES (2);SOPHIE (2) &  & Potential multi-planet system. \\\\\n349488688.02 & TYC 1529-00224-1 & $2002.77063 _{ - 0.00097 } ^ { + 0.00103 }$ & $15.35 _{ - 1.94 } ^ { + 4.15 }(s)$ & $0.03688 _{ - 0.00067 } ^ { + 0.00069 }$ & $5.78 _{ - 0.18 } ^ { + 0.18 }$ & $0.24 _{ - 0.16 } ^ { + 0.21 }$ & $6.291 _{ - 0.058 } ^ { + 0.074 }$ & 8.855 &  & NRES (2);SOPHIE (2) &  &  \\\\\n356700488* & TYC 4420-01295-1 & $1756.638 _{ - 0.013 } ^ { + 0.011 }$ & $184.5 _{ - 64.7 } ^ { + 333.1 }(s)$ & $0.0173 _{ - 0.0011 } ^ { + 0.0015 }$ & $2.92 _{ - 0.2 } ^ { + 0.28 }$ & $0.44 _{ - 0.3 } ^ { + 0.34 }$ & $11.76 _{ - 0.65 } ^ { + 1.03 }$ & 8.413 &  &  &  &  \\\\\n356710041* & TYC 1993-00419-1 & $1932.2939 _{ - 0.0019 } ^ { + 0.0019 }$ & $29.6 _{ - 14.0 } ^ { + 19.0 }(s)$ & $0.0496 _{ - 0.0021 } ^ { + 0.0011 }$ & $14.82 _{ - 0.85 } ^ { + 0.84 }$ & $0.66 _{ - 0.42 } ^ { + 0.11 }$ & $12.76 _{ - 0.24 } ^ { + 0.24 }$ & 9.646 &  &  &  &  \\\\\n369532319 & TYC 2743-01716-1 & $1755.8158 _{ - 0.006 } ^ { + 0.0051 }$ & $35.4 _{ - 12.0 } ^ { + 51.6 }(s)$ & $0.0316 _{ - 0.0023 } ^ { + 0.0028 }$ & $3.43 _{ - 0.3 } ^ { + 0.37 }$ & $0.41 _{ - 0.29 } ^ { + 0.34 }$ & $5.5 _{ - 0.32 } ^ { + 0.32 }$ & 10.594 &  &  & Gemini &  \\\\\n369779127 & TYC 9510-00090-1 & $1643.9403 _{ - 0.0046 } ^ { + 0.0058 }$ & $9.93 _{ - 3.38 } ^ { + 19.74 }(s)$ & $0.0288 _{ - 0.0015 } ^ { + 0.0033 }$ & $4.89 _{ - 0.31 } ^ { + 0.56 }$ & $0.46 _{ - 0.31 } ^ { + 0.33 }$ & $5.64 _{ - 0.38 } ^ { + 0.33 }$ & 9.279 &  &  &  &  \\\\\n384159646* & TYC 9454-00957-1 & $1630.39405 _{ - 0.00079 } ^ { + 0.00079 }$ & $11.68 _{ - 2.75 } ^ { + 4.21 }(s)$ & $0.0658 _{ - 0.0012 } ^ { + 0.0011 }$ & $9.87 _{ - 0.45 } ^ { + 0.44 }$ & $0.27 _{ - 0.18 } ^ { + 0.21 }$ & $5.152 _{ - 0.069 } ^ { + 0.087 }$ & 10.158 & SBIG (1) & NRES (8);MINERVA (6) & Gemini &  \\\\\n385557214 & TYC 1807-00046-1 & $1791.58399 _{ - 0.00068 } ^ { + 0.0007 }$ & $5.62451 _{ - 0.0004 } ^ { + 0.00043 }$ & $0.096 _{ - 0.019 } ^ { + 0.032 }$ & $8.32 _{ - 2.06 } ^ { + 2.77 }$ & $0.95 _{ - 0.075 } ^ { + 0.053 }$ & $1.221 _{ - 0.094 } ^ { + 0.058 }$ & 10.856 &  &  &  &  \\\\\n388134787 & TYC 4260-00427-1 & $1811.034 _{ - 0.015 } ^ { + 0.017 }$ & $246.0 _{ - 127.0 } ^ { + 6209.0 }(s)$ & $0.0265 _{ - 0.0024 } ^ { + 0.023 }$ & $2.57 _{ - 0.28 } ^ { + 2.19 }$ & $0.55 _{ - 0.39 } ^ { + 0.44 }$ & $8.85 _{ - 1.13 } ^ { + 1.84 }$ & 10.95 &  & NRES (1) & Gemini &  \\\\\n404518509 & HIP 16038 & $1431.2696 _{ - 0.0037 } ^ { + 0.0035 }$ & $26.83 _{ - 9.46 } ^ { + 56.14 }(s)$ & $0.0259 _{ - 0.0013 } ^ { + 0.0022 }$ & $2.94 _{ - 0.21 } ^ { + 0.29 }$ & $0.47 _{ - 0.31 } ^ { + 0.34 }$ & $5.02 _{ - 0.23 } ^ { + 0.28 }$ & 9.17 &  &  &  &  \\\\\n408636441* & TYC 4266-00736-1 & $1745.4668 _{ - 0.0016 } ^ { + 0.0015 }$ & $37.695 _{ - 0.0034 } ^ { + 0.0033 }$ & $0.0485 _{ - 0.0019 } ^ { + 0.0023 }$ & $3.32 _{ - 0.16 } ^ { + 0.19 }$ & $0.39 _{ - 0.27 } ^ { + 0.29 }$ & $3.63 _{ - 0.1 } ^ { + 0.14 }$ & 11.93 & SBIG (1) &  & Gemini & Half of the period likely. \\\\\n418255064 & TWOMASS 13063680-8037015 & $1629.3304 _{ - 0.0018 } ^ { + 0.0018 }$ & $25.37 _{ - 7.06 } ^ { + 15.41 }(s)$ & $0.0732 _{ - 0.0029 } ^ { + 0.0031 }$ & $5.57 _{ - 0.36 } ^ { + 0.38 }$ & $0.37 _{ - 0.25 } ^ { + 0.25 }$ & $3.83 _{ - 0.13 } ^ { + 0.14 }$ & 12.478 & SBIG (1) &  & Gemini &  \\\\\n420645189$\\dagger$ & TYC 4508-00478-1 & $1837.4767 _{ - 0.0018 } ^ { + 0.0017 }$ & $250.2 _{ - 66.6 } ^ { + 99.4 }(s)$ & $0.0784 _{ - 0.0033 } ^ { + 0.0046 }$ & $8.82 _{ - 0.55 } ^ { + 0.7 }$ & $0.892 _{ - 0.026 } ^ { + 0.028 }$ & $6.95 _{ - 0.27 } ^ { + 0.3 }$ & 10.595 &  & MINERVA (1) &  & SB 2 from MINERVA observations.  \\\\\n422914082 & TYC 0046-00133-1 & $1431.5538 _{ - 0.0014 } ^ { + 0.0017 }$ & $12.91 _{ - 3.91 } ^ { + 8.97 }(s)$ & $0.0418 _{ - 0.0015 } ^ { + 0.0016 }$ & $3.96 _{ - 0.32 } ^ { + 0.35 }$ & $0.36 _{ - 0.25 } ^ { + 0.28 }$ & $4.07 _{ - 0.09 } ^ { + 0.126 }$ & 11.026 & Sinistro (1) & NRES (1) &  &  \\\\\n427344083 & TWOMASS 22563609+7040518 & $1961.8967 _{ - 0.0031 } ^ { + 0.0036 }$ & $7.77 _{ - 5.6 } ^ { + 9.65 }(s)$ & $0.107 _{ - 0.016 } ^ { + 0.025 }$ & $12.27 _{ - 1.87 } ^ { + 2.9 }$ & $0.834 _{ - 0.484 } ^ { + 0.094 }$ & $2.88 _{ - 0.3 } ^ { + 0.42 }$ & 13.404 &  &  &  &  \\\\\n436873727 & HIP 13224 & $1803.83679 _{ - 0.00058 } ^ { + 0.00056 }$ & $19.26 _{ - 5.95 } ^ { + 6.73 }(s)$ & $0.05246 _{ - 0.00061 } ^ { + 0.00059 }$ & $10.02 _{ - 0.43 } ^ { + 0.41 }$ & $0.767 _{ - 0.057 } ^ { + 0.038 }$ & $5.462 _{ - 0.081 } ^ { + 0.074 }$ & 7.51 &  &  &  &  \\\\ \n441642457* & TYC 3858-00452-1 & $1745.5102 _{ - 0.0108 } ^ { + 0.0097 }$ & $79.8072 _{ - 0.0071 } ^ { + 0.0076 }$ & $0.0281 _{ - 0.0024 } ^ { + 0.0033 }$ & $3.55 _{ - 0.34 } ^ { + 0.46 }$ & $0.934 _{ - 0.023 } ^ { + 0.026 }$ & $6.9 _{ - 0.39 } ^ { + 0.6 }$ & 9.996 &  &  &  &  \\\\\n441765914* & TWOMASS 17253007+7552562 & $1769.6154 _{ - 0.0058 } ^ { + 0.0093 }$ & $161.6 _{ - 58.2 } ^ { + 1460.1 }(s)$ & $0.0411 _{ - 0.0024 } ^ { + 0.0119 }$ & $3.6 _{ - 0.3 } ^ { + 1.01 }$ & $0.45 _{ - 0.32 } ^ { + 0.48 }$ & $7.44 _{ - 0.36 } ^ { + 1.08 }$ & 11.638 &  &  &  &  \\\\\n452920657 & TWOMASS 00332018+5906355 & $1810.5765 _{ - 0.0031 } ^ { + 0.003 }$ & $53.2 _{ - 29.0 } ^ { + 34.3 }(s)$ & $0.135 _{ - 0.016 } ^ { + 0.012 }$ & $9.71 _{ - 1.16 } ^ { + 0.9 }$ & $0.73 _{ - 0.48 } ^ { + 0.11 }$ & $4.6 _{ - 0.26 } ^ { + 0.29 }$ & 14.167 & SBIG (1) &  &  &  \\\\\n455737331 & TYC 2779-00785-1 & $1780.7084 _{ - 0.008 } ^ { + 0.0073 }$ & $50.4 _{ - 17.6 } ^ { + 75.0 }(s)$ & $0.0257 _{ - 0.0016 } ^ { + 0.002 }$ & $3.05 _{ - 0.24 } ^ { + 0.29 }$ & $0.43 _{ - 0.29 } ^ { + 0.33 }$ & $6.6 _{ - 0.43 } ^ { + 0.5 }$ & 10.189 & SBIG (1) &  & Gemini &  \\\\\n456909420 & TYC 1208-01094-1 & $1779.4109 _{ - 0.0026 } ^ { + 0.0022 }$ & $5.78 _{ - 5.29 } ^ { + 5.95 }(s)$ & $0.078 _{ - 0.031 } ^ { + 0.045 }$ & $9.15 _{ - 3.61 } ^ { + 5.27 }$ & $0.973 _{ - 0.495 } ^ { + 0.063 }$ & $1.73 _{ - 0.27 } ^ { + 0.28 }$ & 10.941 &  &  &  &  \\\\\n458451774 & TWOMASS 12551793+4431260 & $1917.1875 _{ - 0.0019 } ^ { + 0.0019 }$ & $12.39 _{ - 6.34 } ^ { + 83.97 }(s)$ & $0.0752 _{ - 0.0054 } ^ { + 0.0211 }$ & $3.33 _{ - 0.26 } ^ { + 0.92 }$ & $0.61 _{ - 0.43 } ^ { + 0.32 }$ & $2.08 _{ - 0.19 } ^ { + 0.59 }$ & 13.713 &  &  &  &  \\\\\n48018596 & TYC 3548-00800-1 & $1713.4514 _{ - 0.0063 } ^ { + 0.0046 }$ & $100.1145 _{ - 0.0018 } ^ { + 0.0021 }$ & $0.049 _{ - 0.0081 } ^ { + 0.018 }$ & $7.88 _{ - 1.33 } ^ { + 2.9 }$ & $0.984 _{ - 0.028 } ^ { + 0.027 }$ & $2.83 _{ - 0.26 } ^ { + 0.29 }$ & 9.595 &  & NRES (1) & Gemini &  \\\\\n53309262 & TWOMASS 07475406+5741549 & $1863.1133 _{ - 0.0064 } ^ { + 0.0061 }$ & $294.8 _{ - 96.0 } ^ { + 327.0 }(s)$ & $0.1239 _{ - 0.0075 } ^ { + 0.0098 }$ & $5.38 _{ - 0.36 } ^ { + 0.46 }$ & $0.46 _{ - 0.31 } ^ { + 0.28 }$ & $6.74 _{ - 0.45 } ^ { + 0.62 }$ & 15.51 &  &  &  &  \\\\\n53843023 & TYC 6956-00758-1 & $1328.0335 _{ - 0.0054 } ^ { + 0.0057 }$ & $202.0 _{ - 189.0 } ^ { + 272.0 }(s)$ & $0.058 _{ - 0.02 } ^ { + 0.056 }$ & $5.14 _{ - 1.77 } ^ { + 4.99 }$ & $0.962 _{ - 0.597 } ^ { + 0.083 }$ & $4.25 _{ - 0.72 } ^ { + 0.66 }$ & 11.571 &  &  &  &  \\\\\n55525572* & TYC 8876-01059-1 & $1454.6713 _{ - 0.0066 } ^ { + 0.0065 }$ & $83.8951 _{ - 0.004 } ^ { + 0.004 }$ & $0.0343 _{ - 0.001 } ^ { + 0.0021 }$ & $7.31 _{ - 0.46 } ^ { + 0.56 }$ & $0.43 _{ - 0.29 } ^ { + 0.31 }$ & $13.54 _{ - 0.3 } ^ { + 0.51 }$ & 10.358 &  & CHIRON (5) & Gemini & Confirmed planet \\citep{2020eisner} \\\\\n63698669* & TYC 6993-00729-1 & $1364.6226 _{ - 0.0074 } ^ { + 0.0067 }$ & $73.6 _{ - 26.8 } ^ { + 133.6 }(s)$ & $0.0248 _{ - 0.0019 } ^ { + 0.0023 }$ & $2.15 _{ - 0.2 } ^ { + 0.25 }$ & $0.42 _{ - 0.29 } ^ { + 0.35 }$ & $5.63 _{ - 0.32 } ^ { + 0.57 }$ & 10.701 & SBIG (1) &  &  &  \\\\\n70887357* & TYC 5883-01412-1 & $1454.3341 _{ - 0.0016 } ^ { + 0.0015 }$ & $56.1 _{ - 15.3 } ^ { + 18.8 }(s)$ & $0.0605 _{ - 0.0027 } ^ { + 0.0027 }$ & $12.84 _{ - 0.86 } ^ { + 0.9 }$ & $0.917 _{ - 0.028 } ^ { + 0.016 }$ & $7.29 _{ - 0.18 } ^ { + 0.19 }$ & 9.293 &  &  &  &  \\\\\n7422496$\\dagger$ & HIP 25359 & $1470.3625 _{ - 0.0031 } ^ { + 0.0023 }$ & $61.4 _{ - 16.7 } ^ { + 49.0 }(s)$ & $0.0255 _{ - 0.001 } ^ { + 0.0011 }$ & $2.44 _{ - 0.15 } ^ { + 0.16 }$ & $0.37 _{ - 0.25 } ^ { + 0.29 }$ & $5.89 _{ - 0.15 } ^ { + 0.15 }$ & 9.36 &  & MINERVA (4) &  & SB 2 from MINERVA observations.  \\\\\n82452140 & TYC 3076-00921-1 & $1964.292 _{ - 0.011 } ^ { + 0.011 }$ & $21.1338 _{ - 0.0052 } ^ { + 0.0066 }$ & $0.0266 _{ - 0.0019 } ^ { + 0.0027 }$ & $2.95 _{ - 0.25 } ^ { + 0.34 }$ & $0.42 _{ - 0.29 } ^ { + 0.36 }$ & $5.87 _{ - 0.62 } ^ { + 0.94 }$ & 10.616 &  &  &  &  \\\\\n88840705 & TYC 3091-00808-1 & $2026.6489 _{ - 0.001 } ^ { + 0.001 }$ & $260.6 _{ - 87.6 } ^ { + 142.2 }(s)$ & $0.109 _{ - 0.023 } ^ { + 0.027 }$ & $9.98 _{ - 2.28 } ^ { + 2.75 }$ & $1.001 _{ - 0.042 } ^ { + 0.037 }$ & $4.72 _{ - 0.13 } ^ { + 0.15 }$ & 9.443 &  &  &  &  \\\\\n91987762* & HIP 47288 & $1894.25381 _{ - 0.00051 } ^ { + 0.00047 }$ & $10.51 _{ - 3.48 } ^ { + 3.67 }(s)$ & $0.05459 _{ - 0.00106 } ^ { + 0.00097 }$ & $9.56 _{ - 0.56 } ^ { + 0.52 }$ & $0.771 _{ - 0.062 } ^ { + 0.033 }$ & $4.342 _{ - 0.073 } ^ { + 0.063 }$ & 7.87 &  & NRES (4) & Gemini &  \\\\\n95768667 & TYC 1434-00331-1 & $1918.3318 _{ - 0.0093 } ^ { + 0.0079 }$ & $26.9 _{ - 12.4 } ^ { + 72.3 }(s)$ & $0.0282 _{ - 0.0022 } ^ { + 0.0031 }$ & $3.54 _{ - 0.32 } ^ { + 0.43 }$ & $0.48 _{ - 0.33 } ^ { + 0.35 }$ & $5.4 _{ - 0.64 } ^ { + 0.76 }$ & 10.318 &  &  &  &  \\\\\n\\hline\n\\end{tabular}}\n\\caption{\\textbf{Properties of PHT candidates (continued)}}\n\\label{tab:PHT-caniddates2}\n\\end{table}\n\\end{landscape}\n\n\n\\section{Conclusion}\n\\label{sec:condlusion}\n\nWe present the results from the analysis of the first 26  \\emph{TESS}\\ sectors. The outlined citizen science approach engages over 22 thousand registered citizen scientists who completed 12,617,038 classifications from December 2018 through August 2020 for the sectors observed during the first two years of the \\emph{TESS}\\ mission. We applied a systematic search for planetary candidates using visual vetting by multiple volunteers to identify \\emph{TESS}\\ targets that are most likely to host a planet. Between 8 and 15 volunteers have inspected each \\emph{TESS}\\ light curve and marked times of transit-like events using the PHT online interface. For each light curve, the markings from all the volunteers who saw that target were combined using an unsupervised machine learning method, known as DBSCAN, in order to identify likely transit-like events. Each of these identified events was given a transit score based on the number of volunteers who identified a given event and on the user weighting of each of those volunteers. Individual user weights were calculated based on the user's ability to identify simulated transit events, injected into real \\emph{TESS}\\ light curves, that are displayed on the PHT site alongside of the real data. The transit scores were then used to generate a ranked list of candidates that range from most likely to least likely to host a planet candidate. The top 500 highest ranked candidates were further vetted by the PHT science team. This stage of vetting primarily made use of the open source {\\sc latte} \\citep{LATTE2020} tool which generates a number of standard diagnostic plots that help identify promising candidates and weed out false positive signals. \n\nOn average we found around three high priority candidates per sector which were followed up using ground based telescopes, where possible. To date, PHT has statistically confirmed one planet, TOI-813 \\citep{2020eisner}: a Saturn-sized planet on an 84 day orbit around a subgiant host star. Other PHT identified planets listed in this paper are being followed up by other teams of astronomers, such as TOI-1899 (TIC 172370679) which was recently confirmed to be a warm Jupiter transiting an M-dwarf \\citep{canas2020}. The remaining candidates outlined in this paper require further follow-up observations to confirm their planetary nature.\n\nThe sensitivity of our transit search effort was assessed using synthetic data, as well as the known TOI and TCE candidates flagged by the SPOC pipeline. For simulated planets (where simulated signals are injected into real \\emph{TESS}\\ light curves) we have shown that the recovery efficiency of human vetting starts to decrease for transit-signals that have a SNR less than 7.5. The detection efficiency was further evaluated by the fractional recovery of the TOI and TCEs. We have shown that PHT is over 85 \\% complete in the recovery of planets that have a radius greater than 4 $R_{\\oplus}$, 51 \\% complete for radii between 3 and 4 $R_{\\oplus}$ and 49 \\% complete for radii between 2 and 3 $R_{\\oplus}$. Furthermore, we have shown that human vetting is not sensitive to the number of transits present in the light curve, meaning that they are equally likely to identify candidates on longer orbital periods as they are those with shorter orbital periods for periods greater than $\\sim$ 1 day. Planets with periods shorter than around 1 day exhibit over 20 transits within one \\emph{TESS}\\ sectors resulting in a decrease in identification by the volunteers. This is due to many volunteers only marking a random subset of these events, resulting in a lack of consensus on any given transit event and thus decreasing the overall transit score of these light curves. \n\nIn addition to searching for signals due to transiting exoplanets, PHT provides a platform that can be used to identify other stellar phenomena that may otherwise be difficult to identify with automated pipelines. Such phenomena, including eclipsing binaries, multiple stellar systems, dwarf novae, and stellar flares are often mentioned on the PHT discussion forums where volunteers can use searchable hashtags and comments to bring these systems to the attention of other citizen scientists as well as the PHT science team. All of the eclipsing binaries identified on the site, for example, are being used and vetted by the \\emph{TESS}\\ Eclipsing Binary Working Group (Prsa et al. in prep). Furthermore, we have investigated the nature of all of the targets that were identified as possible multiple stellar systems, as summarised in Table~\\ref{tab:PHT-multis}.\n\nOverall we have shown that large scale visual vetting can complement the findings \\textcolor{red}{from the major \\emph{TESS}\\ pipeline} by identifying longer period planets that may only exhibit a single transit event in their light curve, as well as in finding signals that are aperiodic or embedded in a strong varying stellar signal. The identification of planets around stars with variable signals allow us to potentially characterise the host-star (e.g., with asteroseismology or spot modulation). Additionally, the longer period planets are integral to our understanding of how planet systems form and evolve, as they allow us to investigate the evolution of planets that are farther away from their host star and therefore less dependent on stellar radiation. \\textcolor{red}{While automated pipelines specifically designed to identify single transit events in the \\emph{TESS}\\ data exist \\citep[e.g., ][]{Gill2020}, neither their methodology nor the full list of their findings are yet publicly available and thus we are unable to compare results.} \n\nThe planets that PHT finds have longer periods ($\\gtrsim$ 27 d) than those found in \\emph{TESS}\\ data using automated pipelines, and are more typical of the Kepler sample (25\\% of Kepler confirmed planets have periods greater than 27 days\\footnote{\\url{https:\/\/exoplanetarchive.ipac.caltech.edu\/}}). However, the Kepler planets are considerably fainter, and thus less amenable to ground-based follow-up or atmospheric characterisation from space (CHEOPS and JWST). Thus PHT helps to bridge the parameter spaces covered by these two missions, by identifying longer period planet candidates around bright, nearby stars, for which we can ultimately obtain precise planetary mass estimates. Although statistical characterisation of exo-planetary systems is no doubt important, precise mass measurements are key to developing our understanding of exoplanets and the physics which dictate their evolution. In particular, identification of this PHT sample provides follow-up targets to investigate the dependence of photo-evaporation on the mass of planets as well as on the planet radius, and will help our understanding of the photo-evaporation valley at longer orbital periods \\citep{Owen2013}. \n\nPHT will continue to operate throughout the \\emph{TESS}\\ extended mission, hopefully allowing us to identify even longer period planets as well as help verify some of the existing candidates with additional transits.  \n\n\n\n\\begin{table*}\n\\resizebox{\\textwidth}{!}{\n\\begin{tabular}{cccccccccc}\n\\textbf{TIC}  & \\textbf{Period (days)}        & \\textbf{Epoch (\\textcolor{red}{BJD - 2457000})}      & \\textbf{Depth (ppm)}  & \\textbf{Comment}  \\\\\n\\hline\n13968858      & $3.4850 \\pm 0.001$            & $ 1684.780   \\pm 0.005$            & 410000    & Candidate multiple system  \\\\\n              & $1.4380 \\pm 0.001$            & $ 1684.335   \\pm 0.005$            & 50000     &   \\\\\n35655828      & $ 8.073 \\pm 0.01$             & $ 1550.94    \\pm 0.01 $            & 23000     & Confirmed blend \\\\\n              & $  1.220 \\pm 0.001 $          & $ 1545.540   \\pm 0.005 $           & 2800      &  \\\\\n63291675      & $ 8.099  \\pm 0.003 $          & $ 1685.1     \\pm 0.01  $           & 60000     & Confirmed blend \\\\\n              & $ 1.4635 \\pm 0.0005 $         & $ 1683.8     \\pm 0.1 $             & 7000      &  \\\\\n63459761      & $4.3630 \\pm 0.003 $           & $ 1714.350   \\pm 0.005 $           & 160000    & Candidate multiple system   \\\\\n              & $4.235 \\pm  $ 0.005           & $ 1715.130   \\pm 0.03$             & 35000     &   \\\\\n104909909     & $1.3060 \\pm 0.0001$           & $ 1684.470   \\pm 0.005$            & 32000     & Candidate multiple system   \\\\\n              & $2.5750 \\pm 0.003$            & $ 1684.400   \\pm 0.005$            & 65000     &   \\\\\n115980439     & $ 4.615 \\pm 0.002  $          & $ 1818.05    \\pm 0.01  $           & 95000     & Confirmed blend \\\\\n              & $ 0.742 \\pm 0.005  $          & $ 1816.23    \\pm 0.02  $           & 2000      &   \\\\\n120362128     & $ 3.286 \\pm 0.002  $          & $ 1684.425   \\pm 0.01  $           & 33000     &  Candidate multiple system  \\\\\n              & $ -                 $         & $ 1701.275   \\pm 0.02  $           & 12000     &    \\\\\n              & $ -                 $         & $ 1702.09    \\pm 0.02  $           & 36000     &    \\\\\n121945407     & $  0.9056768 \\pm 0.00000002$   & $-1948.76377 \\pm  0.0000001$      & 2500          & Confirmed multiple system $^{(\\mathrm{a})}$  \\\\\n              & $ 45.4711 \\pm  0.00002$        & $-1500.0038  \\pm  0.0004 $       & 7500          &  \\\\\n122275115     & $ - $                         & $ 1821.779   \\pm 0.01  $           & 155000    &  Candidate multiple system  \\\\\n              & $ - $                         & $ 1830.628   \\pm 0.01  $           & 63000     &  \\\\\n              & $ - $                         & $ 1838.505   \\pm 0.01  $           & 123000    &  \\\\\n229804573     & $1.4641 \\pm 0.0005$           & $ 1326.135   \\pm 0.005$            & 180000    & Candidate multiple system  \\\\\n              & $0.5283 \\pm 0.0001$           & $ 1378.114   \\pm 0.005$            & 9000      &   \\\\\n252403752     & $ - $                         & $ 1817.73    \\pm 0.01 $            & 2800      & Candidate multiple system \\\\\n              & $ - $                         & $ 1829.76    \\pm 0.01 $            & 23000     & \\\\\n              & $ - $                         & $ 1833.63    \\pm 0.01 $            & 5500      & \\\\\n258837989     & $0.8870 \\pm 0.001$            & $ 1599.350   \\pm 0.005$            & 64000     & Candidate multiple system  \\\\\n              & $3.0730 \\pm 0.001$            & $ 1598.430   \\pm 0.005$            & 25000     &   \\\\\n266958963     & $1.5753 \\pm 0.0002$           & $ 1816.425   \\pm 0.001$            & 265000    & Candidate multiple system  \\\\\n              & $2.3685 \\pm 0.0001$           & $ 1817.790   \\pm 0.001$            & 75000     &   \\\\\n278956474     & $5.488068 \\pm 0.000016  $     & $ 1355.400   \\pm 0.005$            & 93900     & Confirmed multiple system $^{(\\mathrm{b})}$ \\\\\n              & $5.674256 \\pm  \u22120.000030$     & $ 1330.690   \\pm 0.005$            & 30000     &   \\\\\n284925600     & $ 1.24571 \\pm 0.00001 $       & $ 1765.248   \\pm 0.005 $           & 490000    & Confirmed blend \\\\\n              & $  0.31828 \\pm 0.00001  $     & $ 1764.75    \\pm 0.005 $           & 35000     &  \\\\\n293954660     & $2.814 \\pm  0.001 $           & $ 1739.177   \\pm 0.03  $           & 272000    & Confirmed blend \\\\\n              & $4.904 \\pm 0.03 $             & $ 1739.73    \\pm 0.01  $           & 9500      &  \\\\\n312353805     & $4.951 \\pm  0.003 $           & $ 1817.73    \\pm 0.01  $           & 66000     & Confirmed blend \\\\\n              & $12.89 \\pm   0.01  $          & $ 1822.28    \\pm 0.01$             & 19000     &  \\\\\n318210930     & $ 1.3055432 \\pm 0.000000033$  & $ -653.21602   \\pm 0.0000013$      & 570000    & Confirmed multiple system $^{(\\mathrm{c})}$ \\\\\n              & $ 0.22771622 \\pm 0.0000000035$& $ -732.071119 \\pm 0.00000026 $     & 220000    &  \\\\\n336434532     & $ 3.888 \\pm 0.002  $          & $ 1713.66    \\pm 0.01  $           & 22900     & Confirmed blend \\\\\n              & $  0.949 \\pm 0.003 $          & $ 1712.81    \\pm 0.01  $           & 2900      &  \\\\\n350622185     & $1.1686 \\pm 0.0001$           & $ 1326.140   \\pm 0.005$            & 200000    & Candidate multiple system  \\\\\n              & $5.2410 \\pm 0.0005$           & $ 1326.885   \\pm 0.05$             & 4000      &  \\\\\n375422201     & $9.9649 \\pm 0.001$            & $ 1711.937   \\pm 0.005$            & 245000    & Candidate multiple system  \\\\\n              & $4.0750 \\pm 0.001$            & $ 1713.210   \\pm 0.01 $            & 39000     &  \\\\\n376606423     & $ 0.8547 \\pm 0.0002  $        & $ 1900.766   \\pm 0.005 $           & 9700      & Candidate multiple system  \\\\\n              & $  -  $                       & $ 1908.085   \\pm 0.01  $           & 33000     &  \\\\\n394177355     & $ 94.22454  \\pm 0.00040 $     & $  -  $                            &  -        & Confirmed multiple system $^{(\\mathrm{d})}$ \\\\\n              & $ 8.6530941 \\pm 0.0000016$    & $-2038.99492 \\pm 0.00017 $         &  140000   &  \\\\\n              & $ 1.5222468 \\pm 0.0000025$    & $ -2039.1201 \\pm 0.0014  $         &  -        &  \\\\\n              & $ 1.43420486 \\pm 0.00000012 $ & $-2039.23941 \\pm 0.00007 $         &  -        &  \\\\\n424508303     & $ 2.0832649 \\pm 0.0000029  $ & $-3144.8661  \\pm 0.0034  $          &  430000   &  Confirmed multiple system $^{(\\mathrm{e})}$ \\\\\n              & $ 1.4200401  \\pm 0.0000042 $  & $-3142.5639  \\pm 0.0054  $         &  250000         &  \\\\\n441794509     & $ 4.6687 \\pm  0.0002       $  & $ 1958.895   \\pm 0.005   $         & 34000     &  Candidate multiple system  \\\\\n              & $ 14.785 \\pm  0.002       $   & $ 1960.845   \\pm 0.005   $         & 17000     &  \\\\\n470710327     & $ 9.9733 \\pm 0.0001 $         & $ 1766.27    \\pm 0.005  $          & 51000     &  Confirmed multiple system $^{(\\mathrm{f})}$ \\\\\n              & $ 1.104686 \\pm 0.00001 $      & $ 1785.53266 \\pm 0.000005$         & 42000     &  \\\\\n\\hline\n\\end{tabular}\n}\n\\caption{\nNote -- $^{(\\mathrm{a})}$ KOI-6139, \\citet{Borkovits2013};  \n$^{(\\mathrm{b})}$ \\citet{2020Rowden}\n$^{(\\mathrm{c})}$ \\citet{Koo2014}; \n$^{(\\mathrm{d})}$ KOI-3156, \\citet{2017Helminiak};\n$^{(\\mathrm{e})}$ V994 Her; \\citet{Zasche2016}; \n$^{(\\mathrm{f})}$ Eisner et al. {\\it in prep.}\n}\n\n\\label{tab:PHT-multis}\n\n\\end{table*}\n\n\\section*{Data Availability}\n\nAll of the \\emph{TESS}\\ data used within this article are hosted and made publicly available by the Mikulski Archive for Space Telescopes (MAST, \\url{http:\/\/archive.stsci.edu\/tess\/}). Similarly, the Planet Hunters TESS classifications made by the citizen scientists can be found on the Planet Hunters Analysis Database (PHAD, \\url{https:\/\/mast.stsci.edu\/phad\/}), which is also hosted by MAST. All planet candidates and their properties presented in this article have been uploaded to the Exoplanet Follow-up Observing Program for TESS (ExoFOP-TESS, \\url{ https:\/\/exofop.ipac.caltech.edu\/tess\/index.php}) website as community TOIs (cTOIs), under their corresponding TIC IDs. The ground-based follow-up observations of individual targets will be shared on reasonable request to the corresponding author.\n\nThe models of individual transit events and the data validation reports used for the vetting of the targets were both generated using publicly available open software codes, \\texttt{pyaneti}\\ and {\\sc latte}.\n\n\\section*{Acknowledgements}  \n\nThis project works under the in \\textit{populum veritas est} philosophy, and for that reason we would like to thank all of the citizen scientists who have taken part in the Planet Hunters TESS project and enable us to find many interesting astrophysical systems. \n\nSome of the observations in the paper made use of the High-Resolution Imaging instruments `Alopeke and Zorro. `Alopeke and Zorro were funded by the NASA Exoplanet Exploration Program and built at the NASA Ames Research Center by Steve B. Howell, Nic Scott, Elliott P. Horch, and Emmett Quigley. `Alopeke and Zorro were mounted on the Gemini North and South telescope of the international Gemini Observatory, a program of NSF's NOIRLab, which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation on behalf of the Gemini partnership: the National Science Foundation (United States), National Research Council (Canada), Agencia Nacional de Investigaci\\'{o}n y Desarrollo (Chile), Ministerio de Ciencia, Tecnolog\\'{i}a e Innovaci\\'{o}n (Argentina), Minist\\'{e}rio da Ci\\^{e}ncia, Tecnologia, Inova\\c{c}\\~{o}es e Comunica\\c{c}\\~{o}es (Brazil), and Korea Astronomy and Space Science Institute (Republic of Korea). The  authors also acknowledge the very significant cultural role and sacred nature of  Maunakea. We are most fortunate to  have  the opportunity to conduct observations from this mountain.\n\nThis project has also received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement N$^\\circ$730890. This material reflects only the authors views and the Commission is not liable for any use that may be made of the information contained therein. This work makes use of observations from the Las Cumbres Observatory global telescope network, including the NRES spectrograph and the SBIG and Sinistro photometric instruments. \n\nFurthermore, NLE thanks the LSSTC Data Science Fellowship Program, which is funded by LSSTC, NSF Cybertraining Grant N$^\\circ$1829740, the Brinson Foundation, and the Moore Foundation; her participation in the program has benefited this work. Finally, CJ acknowledges funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement N$^\\circ$670519: MAMSIE), and from the Research Foundation Flanders (FWO) under grant agreement G0A2917N (BlackGEM). \n\nThis research made use of Astropy, a community-developed core Python package for Astronomy \\citep{astropy2013}, matplotlib \\citep{matplotlib}, pandas \\citep{pandas}, NumPy \\citep{numpy}, astroquery \\citep{ginsburg2019astroquery} and sklearn \\citep{pedregosa2011scikit}.  \n\n\n\n\n\\bibliographystyle{mnras}\n\n\\section{Introduction}\n\nSince the first unambiguous discovery of an exoplanet in 1995 \\citep[][]{Mayor1995} over 4,000 more have been confirmed. Studies of their characteristics have unveiled an extremely wide range of planetary properties in terms of planetary mass, size, system architecture and orbital periods, greatly revolutionising our understanding of how these bodies form and evolve.\n\nThe transit method, whereby we observe a temporary decrease in the brightness of a star due to a planet passing in front of its host star, is to date the most successful method for planet detection, having discovered over 75\\% of the planets listed on the NASA Exoplanet Archive\\footnote{\\url{https:\/\/exoplanetarchive.ipac.caltech.edu\/}}. It yields a wealth of information including planet radius, orbital period, system orientation and potentially even atmospheric composition. Furthermore, when combined with Radial Velocity \\citep[RV; e.g.,][]{Mayor1995, Marcy1997} observations, which yield the planetary mass, we can infer planet densities, and thus their internal bulk compositions. Other indirect detection methods include radio pulsar timing \\citep[e.g.,][]{Wolszczan1992} and microlensing \\citep[e.g.,][]{Gaudi2012}.\n\n\nThe \\textit{Transiting Exoplanet Survey Satellite} mission \\citep[\\protect\\emph{TESS};][]{ricker15} is currently in its extended mission, searching for transiting planets orbiting bright ($V < 11$\\,mag) nearby stars. Over the course of the two year nominal mission, \\emph{TESS}\\ monitored around 85 per cent of the sky, split up into 26 rectangular sectors of 96 $\\times$ 24 deg each (13 per hemisphere). Each sector is monitored for $\\approx$ 27.4 continuous days, measuring the brightness of $\\approx$ 20,000 pre-selected stars every two minutes. In addition to these short cadence (SC) observations, the \\emph{TESS}\\ mission provides Full Frame Images (FFI) that span across all pixels of all CCDs and are taken at a cadence of 30 minutes. While most of the targets ($\\sim$ 63 per cent) will be observed for $\\approx$ 27.4 continuous days, around $\\sim$ 2 per cent of the targets at the ecliptic poles are located in the `continuous viewing zones' and will be continuously monitored for $\\sim$ 356 days.\n\nStars themselves are extremely complex, with phenomena ranging from outbursts to long and short term variability and oscillations, which manifest themselves in the light curves. These signals, as well as systematic effects and artifacts introduced by the telescope and instruments, mean that standard periodic search methods, such as the Box-Least-Squared method \\citep{bls2002} can struggle to identify certain transit events, especially if the observed signal is dominated by natural stellar variability. Standard detection pipelines also tend to bias the detection of short period planets, as they typically require a minimum of two transit events in order to gain the signal-to-noise ratio (SNR) required for detection.\n\nOne of the prime science goals of the \\emph{TESS}\\ mission is to further our understanding of the overall planet population, an active area of research that is strongly affected by observational and detection biases. In order for exoplanet population studies to be able to draw meaningful conclusions, they require a certain level of completeness in the sample of known exoplanets as well as a robust sample of validated planets spanning a wide range of parameter space. \\textcolor{black}{Due to this, we independently search the \\emph{TESS}\\ light curves for transiting planets via visual vetting in order to detect candidates that were either intentionally ignored by the main \\emph{TESS}\\ pipelines, which require at least two transits for a detection, missed because of stellar variability or instrumental artefacts, or were identified but subsequently erroneously discounted at the vetting stage, usually because the period found by the pipeline was incorrect. These candidates can help populate under-explored regions of parameter space and will, for example, benefit the study of planet occurrence rates around different stellar types as well as inform theories of physical processes involved with the formation and evolution of different types of exoplanets.}\n\nHuman brains excel in activities related to pattern recognition, making the task of identifying transiting events in light curves, even when the pattern is in the midst of a strong varying signal, ideally suited for visual vetting. Early citizen science projects, such as Planet Hunters \\citep[PH;][]{fischer12} and Exoplanet Explorers \\citep{Christiansen2018}, successfully harnessed the analytic power of a large number of volunteers and made substantial contributions to the field of exoplanet discoveries. The PH project, for example, showed that human vetting has a higher detection efficiency than automated detection algorithms for certain types of transits. In particular, they showed that citizen science can outperform on the detection of single (long-period) transits \\citep[e.g.,][]{wang13, schmitt14a}, aperiodic transits \\citep[e.g. circumbinary planets;][]{schwamb13} and planets around variable stars \\citep[e.g., young systems,][]{fischer12}. Both PH and Exoplanet Explorers, which are hosted by the world's largest citizen science platform Zooniverse \\citep{lintott08}, ensured easy access to \\textit{Kepler} and \\textit{K2} data by making them publicly available online in an immediately accessible graphical format that is easy to understand for non-specialists. The popularity of these projects is reflected in the number of participants, with PH attracting 144,466 volunteers from 137 different countries over 9 years of the project being active.\n\nFollowing the end of the \\textit{Kepler} mission and the launch of the \\emph{TESS}\\ satellite in 2018, PH was relaunched as the new citizen science project \\textit{Planet Hunters TESS} (PHT) \\footnote{\\url{www.planethunters.org}}, with the aim of identifying transit events in the \\emph{TESS}\\ data that were \\textcolor{black}{intentionally ignored or missed} by the main \\emph{TESS}\\ pipelines. \\textcolor{black}{Such a search complements other methods methods via its sensitivity to single-transit, and, therefore, longer period planets. Additionally, other dedicated non-citizen science based methods are also employed to look for single transit candidates \\citep[see e.g., the Bayesian transit fitting method by ][]{Gill2020, Osborn2016}}.\n\nCitizen science transit searches specialise in finding the rare events that the standard detection pipelines miss, however, these results are of limited use without an indication of the completeness of the search. Addressing the problem of completeness was therefore one of our highest priorities while designing PHT as discussed throughout this paper. \n\nThe layout of the remainder of the paper will be as follows. An overview of the Planet Hunters TESS project is found in Section~\\ref{sec:PHT}, followed by an in depth description of how the project identifies planet candidates in Section~\\ref{sec:method}. The recovery efficiency of the citizen science approach is assessed in Section~\\ref{sec:recovery_efficiency}, followed by a description of the in-depth vetting of candidates and ground based-follow up efforts in Section~\\ref{sec:vetting} and \\ref{sec:follow_up}, respectively. Planet Candidates and noteworthy systems identified by Planet Hunters TESS are outlined in Section~\\ref{sec:PHT_canidates}, followed by a discussion of the results in Section~\\ref{sec:condlusion}.\n\n\\section{Planet Hunters TESS}\n\\label{sec:PHT}\n\nThe PHT project works by displaying \\emph{TESS}\\ light curves (Figure~\\ref{fig:interface}), and asking volunteers to identify transit-like signals. Only the two-minute cadence targets, which are produced by the \\emph{TESS}\\ pipeline at the Science Processing Operations Center \\citep[SPOC,][]{Jenkins2018} and made publicly available by the Mikulski Archive for Space Telescopes (MAST)\\footnote{\\url{http:\/\/archive.stsci.edu\/tess\/}}, are searched by PHT. First-time visitors to the PHT site, or returning visitors who have not logged in are prompted to look through a short tutorial, which briefly explains the main aim of the project and shows examples of transit events and other stellar phenomena. Scientific explanation of the project can be found elsewhere on the site in the `field guide' and on the project's `About' page. \n\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=\\textwidth]{Figures\/PHT_new_interface.png}\n    \\caption{\n    PHT user interface showing a simulated light curve. The transit events are highlighted with white partially-transparent columns that are drawn on using the mouse. Stellar information on the target star is available by clicking on `subject info' below the light curve.} \n    \\label{fig:interface}\n\\end{figure*}\n\nAfter viewing the tutorial, volunteers are ready to participate in the project and are presented with \\emph{TESS}\\ light curves (known as `subjects') that need to be classified. The project was designed to be as simple as possible and therefore only asks one question: \\textit{`Do you see a transit?}'. Users identify transit-like events, and the time of their occurrence, by drawing a column over the event using the mouse button, as shown in Figure~\\ref{fig:interface}. There is no limit on the number of transit-like events that can be marked in a light curve. No markings indicate that there are no transit-like events present in the light curve. Once the subject has been analysed, users submit their classification and continue to view the next light curve by clicking `Done'. \n\nAlongside each light curve, users are offered information on the stellar properties of the target, such as the radius, effective temperature and magnitude (subject to availability, see \\cite{Stassun18}). However, in order to reduce biases in the classifications, the TESS Input Catalog (TIC) ID of the target star is not provided until after the subject classification has been submitted.\n\nIn addition to classifying the data, users are given the option to comment on light curves via the `Talk' discussion forum. Each light curve has its own discussion page to allow volunteers to discuss and comment, as well as to `tag' light curves using searchable hashtags, and to bring promising candidates to the attention of other users and the research team. The talk discussion forums complement the main PHT analysis and have been shown to yield interesting objects which may be challenging to detect using automated algorithms \\citep[e.g.,][]{eisner2019RN}. Unlike in the initial PH project, there are no questions in the main interface regarding stellar variability, however, volunteers are encouraged to mention astrophysical phenomenon or \\textit{unusual} features, such as eclipsing binaries or stellar flares, using the `Talk' discussion forum.  \n\nThe subject TIC IDs are revealed on the subject discussion pages, allowing volunteers to carry out further analysis on specific targets of interest and to report and discuss their findings. This is extremely valuable for both other volunteers and the PHT science team, as it can speed up the process of identifying candidates as well as rule out false positives in a fast and effective manner. \n\nSince the launch of PHT on 6 December 2018, there has been one significant makeover to the user interface. The initial PHT user interface (UI1), which was used for sectors 1 through 9, split the \\emph{TESS}\\ light curves up into either three or four chunks (depending on the data gaps in each sector) which lasted around seven days each. This allowed for a more `zoomed' in view of the data, making it easier to identify transit-like events than when the full $\\sim$ 30 day light curves were shown. The results from a PHT beta project, which displayed only simulated data, showed that a more zoomed in view of the light curve was likely to yield a higher transit recovery rate.\n\nThe updated, and current, user interface (UI2) allows users to manually zoom in on the x-axis (time) of the data. Due to this additional feature, each target has been displayed as a single light curve as of Sector 10. In order to verify that the changes in interface did not affect our findings, all of the Sector 9 subjects were classified using both UI1 and UI2. We saw no significant change in the number of candidates recovered (see Section~\\ref{sec:recovery_efficiency} for a description of how we quantified detection efficiency).\n\n\n\\subsection{Simulated Data}\n\\label{subsec:sims} \n\nIn addition to the real data, volunteers are shown simulated light curves, which are generated by randomly injecting simulated transit signals, provided by the SPOC pipeline \\citep[][]{Jenkins2018}, into real \\emph{TESS}\\ light curves. The simulated data play an important role in assessing the sensitivity of the project, training the users and providing immediate feedback, and to gauge the relative abilities of individual users (see Sec~\\ref{subsec:weighting}). \n\nWe calculate a signal to noise ratio (SNR) of the injected signal by dividing the injected transit depth by the Root Mean Square Combined Differential Photometric Precision (RMS CDPP) of the light curve on 0.5-, 1- or 2-hr time scales (whichever is closest to the duration of the injected transit signal). Only simulations with a SNR greater than 7 in UI1 and greater than 4 for UI2 are shown to volunteers.\n\nSimulated light curves are randomly shown to the volunteers and classified in the exact same manner as the real data. The user is always notified after a simulated light curve has been classified and given feedback as to whether the injected signal was correctly identified or not. For each sector, we generate between one and two thousand simulated light curves, using the real data from that sector in order to ensure that the sector specific systematic effects and data gaps of the simulated data do not differ from the real data. The rate at which a volunteer is shown simulated light curves decreases from an initial rate of 30 per cent for the first 10 classifications, down to a rate of 1 per cent by the time that the user has classified 100 light curves. \n\n\n\\section{Identifying Candidates}\n\\label{sec:method}\n\nEach subject is seen by multiple volunteers, before it is `retired' from the site, and the classifications are combined (see Section~\\ref{subsec:DBscan}) in order to assess the likelihood of a transit event. For sectors 1 through 9, the subjects were retired after 8 classifications if the first 8 volunteers who saw the light curves did not mark any transit events, after 10 classifications if the first 10 volunteers all marked a transit event and after 15 classifications if there was not complete consensus amongst the users. As of Sector 9 with UI2, all subjects were classified by 15 volunteers, regardless of whether or not any transit-like events were marked. Sector 9, which was classified with both UI1 and UI2, was also classified with both retirement rules.\n\nThere were a total of 12,617,038 individual classifications completed across the project on the nominal mission data. 95.4 per cent of these classifications were made by 22,341 registered volunteers, with the rest made by unregistered volunteers. Around 25 per cent of the registered volunteers complete more than 100 classifications, 11.8 per cent more than 300, 8.4 per cent more than 500, 5.4 per cent more than 1000 and 1.1 per cent more than 10,000. The registered volunteers completed a mean and median of 647 and 33 classifications, respectively. Figure~\\ref{fig:user_count} shows the distribution in user effort for logged in users who made between 0 and 300 classifications. \n\nThe distribution in the number of classifications made by the registered volunteers is assessed using the Gini coefficient, which ranges from 0 (equal contributions from all users) to 1 (large disparity in the contributions). The Gini coefficients for individual sectors ranges from 0.84 to 0.91 with a mean of 0.87, while the Gini coefficient for the overall project (all of the sectors combined) is 0.94. The mean Gini coefficient among other astronomy Zooniverse projects lies at 0.82 \\citep{spiers2019}. We note that the only other Zooniverse project with an equally high Gini coefficient as PHT is \\textit{Supernova Hunters}, a project which, similarly to PHT and unlike most other Zooniverse projects, has periodic data releases that are accompanied by an e-newsletter sent to all project volunteers. Periodic e-newsletters have the effect of promoting the project to both regularly and irregularly participating volunteers, who may only complete a couple of classifications as they explore the task, as well as to returning users who complete a large number of classifications following every data release, increasing the disparity in user contributions (the Gini coefficient).\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.5\\textwidth]{Figures\/user_count.png}\n    \\caption{\n    The distribution of the number of classifications by the registered volunteers, using a bin size of 5 from 0 to 300 classifications. A total of 11.8 per cent of the registered volunteers completed more than 300 classifications.} \n    \\label{fig:user_count}\n\\end{figure}\n\n\n\\subsection{User Weighting}\n\\label{subsec:weighting} \nUser weights are calculated for each individual volunteer in order to identify users who are more sensitive to detecting transit-like signals and those who are more likely to mark false positives. The weighting scheme is based on the weighting scheme described by \\cite{schwamb12}.\n\nUser weights are calculated independently for each observation sector, using the simulated light curves shown alongside the data from that sector. All users start off with a weighting of one, which is then increased or decreased when a simulated transit event is correctly or incorrectly identified, respectively. \n\nSimulated transits are deemed correctly identified, or `True', if the mid-point of a user's marking falls within the width of the simulated transit events. If none of the user's markings fall within this range, the simulated transit is deemed not identified, or `False'. If more than one of a user's markings coincide with the same simulated signal, it is only counted as being correct once, such that the total number of `True' markings cannot exceed the number of injected signals. For each classification, we record the number of  `Extra' markings, which is the total number of markings made by the user minus the number of correctly identified simulated transits. \n\nEach simulated light curve, identified by superscript $i$ (where $i=1$, \\ldots, $N$) was seen by $K^{(i)}$ users (the mean value of $K^{(i)}$\nwas 10), and contained $T^{(i)}$ simulated transits (where $T^{(i)}$ depends on the period of the simulated transit signal and the duration of the light curve). For a specific light curve $i$, each user who saw the light curve is identified by a subscript $k$ (where $k=1$, \\ldots, $K^{(i)}$) and each injected transit by a subscript $t$ (where $t=1$, \\ldots, $T^{(i)}$). \n\nIn order to distinguish between users who are able to identify obvious transits and those who are also able to find those that are more difficult to see, we start by defining a `recoverability' $r^{(i)}_t$ for each injected transit $t$ in each light curve. This is defined empirically, as the number of users who identified the transit correctly divided by $K^{(i)}$ (the total number of users who saw the light curve in question).\n\nNext, we quantify the performance of each user on each light curve as follows (this performance is analogous to the `seed' defined in \\citealt{schwamb12}, but we define it slightly differently):\n\\begin{equation}\n  p^{(i)}_{k} = C_{\\rm E} ~ \\frac{E^{(i)}_{k}}{\\langle E^{(i)} \\rangle} + \\sum_{t=1}^{T^{(i)}} \\begin{cases}\n    C_{\\rm T} ~ \\left[ r^{(i)}_t \\right]^{-1}, & \\text{if $m^{(i)}_{t,k} = $`True'}\\\\\n    C_{\\rm F} ~ r^{(i)}_t, & \\text{if $m^{(i)}_{t,k} = $`False'},\n  \\end{cases}\n\\end{equation}\nwhere $m^{(i)}_{t,k}$ is the identification of transit $t$ by user $k$ in light curve $i$, which is either `True' or `False'; $E^{(i)}_{k}$ is the number of `Extra' markings made by user $k$ for light curve $i$, and  $\\langle E^{(i)} \\rangle$ is the mean number of `Extra' markings made by all users who saw subject $i$. The parameters $C_{\\rm E}$, $C_{\\rm T}$ and $C_{\\rm F}$ control the impact of the `Extra', `True' and `False' markings on the overall user weightings, and are optimized empirically as discussed below in Section~\\ref{subsec:optimizesearch}. \n\nFollowing \\citealt{schwamb12}, we then assign a global `weight' $w_k$ to each user $k$, which is defined as:\n\\begin{equation}\n\\begin{split}\n\tw_k = I \\times (1 + \\log_{10} N_k)^{\\nicefrac{\\sum_i p^{(i)}_k}{N_k}}\n\\label{equ:weight}\n\\end{split}\n\\end{equation}\nwhere $I$ is an empirical normalization factor, such that the distribution of user weights remains centred on one, $N_k$ is the total number of simulated transit events that user $k$ assessed, and the sum over $i$ concerns only the light curves that user $k$ saw. \nWe limit the user weights to the range 0.05--3 \\emph{a posteriori}.\n\n\nWe experimented with a number of alternative ways to define the user weights, including the simpler $w_k=\\nicefrac{\\sum_i p^{(i)}_k}{N_k}$, but Eqn.~\\ref{equ:weight} was found to give the best results (see Section~\\ref{sec:recovery_efficiency} for how this was evaluated).\n\n\\subsection{Systematic Removal}\n\\label{subsec:sysrem} \nSystematic effects, for example caused by the spacecraft or background events, can result in spurious signals that affect a large subset of the data, resulting in an excess in markings of transit-like events at certain times within an observation sector. As the four \\emph{TESS}\\ cameras can yield unique systematic effects, the times of systematics were identified uniquely for each camera. The times were identified using a Kernel Density Estimation \\citep[KDE;][]{rosenblatt1956} with a cosine kernel and a bandwidth of 0.1 days, applied across all of the markings from that sector for each camera. Fig.~\\ref{fig:sys_rem} shows the KDE of all marked transit-events made during Sector 17 for TESS's cameras 1 (top panel) to 4 (bottom panel). The isolated spikes, or prominences, in the number of marked events, such as at T = 21-22 days in the bottom panel, are assumed to be caused by systematic effects that affect multiple light curves. Prominences are considered significant if they exceed a factor four times the standard deviation of the kernel output, which was empirically determined to be the highest cut-off to not miss clearly visible systematics. All user-markings within the full width at half maximum of these peaks are omitted from all further analysis. \\textcolor{black}{The KDE profiles for each Sector are provided as electronic supplementary material.}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.46\\textwidth]{Figures\/systematics_sec17.png}\n    \\caption{\n    Kernel density estimation of the user-markings made for Sector 17, for targets observed with TESS's observational Cameras 1 (top panel) to 4 (bottom panel). The orange vertical lines the indicate prominences that are at least four times greater than the standard deviation of the distribution. The black points underneath the figures show the mid-points of all of the volunteer-markings, where darker regions represent a higher density of markings.}\n    \\label{fig:sys_rem}\n\\end{figure}\n\n\\subsection{Density Based Clustering}\n\\label{subsec:DBscan} \n\nThe times and likelihoods of transit-like events are determined by combining all of the classifications made for each subject and identifying times where multiple volunteers identified a signal. We do this using an unsupervised machine learning method, known as DBSCAN \\citep[][Density-Based Spatial Clustering of Applications with Noise]{ester1996DB}. DBSCAN is a non-parametric density based clustering algorithm that helps to distinguish between dense clusters of data and sparse noise. For a data point to belong to a cluster it must be closer than a given distance ($\\epsilon$) to at least a set minimum number of other points (minPoints). \n\nIn our case, the data points are one-dimensional arrays of times of transits events, as identified by the volunteers, and clusters are times where multiple volunteers identified the same event. For each cluster a `transit score' ($s_i$) is determined, which is the sum of the user weights of the volunteers who contribute to the given cluster divided by the sum of the user weights of volunteers who saw that light curve. These transit scores are used to rank subjects from most to least likely to contain a transit-like event. Subjects which contain multiple successful clusters with different scores are ranked by the highest transit score. \n\n\\subsection{Optimizing the search}\n\\label{subsec:optimizesearch}\n\nThe methodology described in Sections~\\ref{subsec:weighting} to \\ref{subsec:DBscan} has five free parameters: the number of markings required to constitute a cluster ($minPoints$), the maximum separation of markings required for members of a cluster ($\\epsilon$), and $C_{\\rm E}$, $C_{\\rm T}$ and $C_{\\rm F}$ used in the weighting scheme. The values of these parameters were optimized via a grid search, where $C_{\\rm E}$ and $C_{\\rm F}$ ranged from -5 to 0, $C_{\\rm T}$ ranged from 0 to 20, and $minPoints$ ranged from 1 to 8, all in steps of 1. ($\\epsilon$) ranged from 0.5 to 1.5 in steps of 0.5. This grid search was carried out on 4 sectors, two from UI1 and two from UI2, for various variations of Equation~\\ref{equ:weight}. \n\nThe success of each combination of parameters was assessed by the fractions of TOIs and TCEs that were recovered within the top highest ranked 500 candidates, as discussed in more detail Section~\\ref{sec:recovery_efficiency}. We found the most successful combination of parameters to be $minPoints$ = 4 markings, $\\epsilon$, = 1 day, $C_{\\rm T}$ = 3, $C_{\\rm F}$= -2 and $C_{\\rm E}$ = -2.\n\n\\subsection{MAST deliverables}\n\\label{subsec:deliverables}\n\nThe analysis described above is carried out both in real-time as classifications are made, as well as offline after all of the light curves of a given sector have been classified. When the real-time analysis identifies a successful DB cluster (i.e. when at least four citizen scientists identified a transit within a day of the \\emph{TESS}\\ data of one another), the potential candidate is automatically uploaded to the open access Planet Hunters Analysis Database (PHAD) \\footnote{\\url{https:\/\/mast.stsci.edu\/phad\/}} hosted by the Mikulski Archive for Space Telescopes (MAST) \\footnote{\\url{https:\/\/archive.stsci.edu\/}}. While PHAD does not list every single classification made on PHT, it does display all transit candidates which had significant consensus amongst the volunteers who saw that light curve, along with the user-weight-weighted transit scores. This analysis does not apply the systematics removal described in Section~\\ref{subsec:sysrem}. The aim of PHAD is to provide an open source database of potential planet candidates identified by PHT, and to credit the volunteers who identified said targets. \n\nThe offline analysis is carried out following the complete classifications of all of the data from a given \\emph{TESS}\\ sector. The combination of all of the classifications allows us to identify and remove times of systematics and calculate better calibrated and more representative user weights. The remainder of this paper will only discuss the results from the offline analysis.\n\n\\section{Recovery Efficiency}\n\\label{sec:recovery_efficiency}\n\\subsection{Recovery of simulated transits}\n\nThe recovery efficiency is, in part, assessed by analysing the recovery rate of the injected transit-like signals (see Section~\\ref{subsec:sims}). Figure~\\ref{fig:SIM_recovery} shows the median and mean transit scores (fraction of volunteers who correctly identified a given transit scaled by user weights) of the simulated transits within SNR bins ranging from 4 to 20 in steps of 0.5. Simulations with a SNR less than 4 were not shown on PHT. The figure highlights that transit signals with a SNR of 7.5 or greater are correctly identified by the vast majority of volunteers. \n\n\\textcolor{black}{As the simulated data solely consist of real light curves with synthetically injected transit signals, we do not have any light curves, simulated or otherwise, which we can guarantee do not contain any planetary transits (real or injected). As such, this prohibits us from using simulated data to infer an analogous false-positive rate.}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.47\\textwidth]{Figures\/SIMS_recovery.png}\n    \\caption{The median (blue) and mean (orange) transit scores for injected transits with SNR ranges between 4 and 20. The mean and median are calculated in SNR bins with a width of 0.5, as indicated by the horizontal lines around each data point. \n    }\n    \\label{fig:SIM_recovery}\n\\end{figure}\n\n\\subsection{Recovery of TCEs and TOIs}\n\\label{subsec:TCE_TOI}\nThe recovery efficiency of PHT is assessed further using the planet candidates identified by the SPOC pipeline \\citep{Jenkins2018}. The SPOC pipeline extracts and processes all of the 2-minute cadence \\emph{TESS}\\ light curves prior to performing a large scale transit search. Data Validation (DV) reports, which include a range of transit diagnostic tests, are generated by the pipeline for around 1250 Threshold Crossing Events (TCEs), which were flagged as containing two or more transit-like features. Visual vetting is then performed by the \\emph{TESS}\\ science team on these targets, and promising candidates are added to the catalog of \\emph{TESS}\\ Objects of Interest (TOIs). Each sector yields around 80 TOIs \\textcolor{black}{and a mean of 1025 TCEs.}\n\nFig~\\ref{fig:TCE_TOI_recovery} shows the fraction of TOIs and TCEs (top and bottom panel respectively) that we recover with PHT as a function of the rank, where a higher rank corresponds to a lower transit score, for Sectors 1 through 26. TOIs and TCEs with R < 2 $R_{\\oplus}$ are not included in this analysis, as the initial PH showed that human vetting alone is unable to reliably recover planets smaller than 2 $R_{\\oplus}$ \\citep{schwamb12}. Planets smaller than 2 $R_{\\oplus}$ are, therefore, not the main focus of our search.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.45\\textwidth]{Figures\/TCE_TOI-recovery_radlim2.png}\n    \\caption{The fraction of recovered TOIs and TCEs (top and bottom panel respectively) with R > 2$R_{\\oplus}$ as a function of the rank, for sectors 1 to 26. The lines represent the results from different observation sectors.}\n    \\label{fig:TCE_TOI_recovery}\n\\end{figure}\n\n\nFig~\\ref{fig:TCE_TOI_recovery} shows a steep increase in the fractional TOI recovery rate up to a rank of $\\sim$ 500. Within the 500 highest ranked PHT candidates for a given sector, we are able to recover between 46 and 62 \\% (mean of 53 \\%) of all of the TOIs (R > 2 $R_{\\oplus}$), a median 90 \\% of the TOIs where the SNR of the transit events are greater than 7.5 and median 88 \\% of TOIs where the SNR of the transit events are greater than 5.\n\nThe relation between planet recovery rate and the SNR of the transit events is further highlighted in Figure~\\ref{fig:TOI_properties}, which shows the SNR vs the orbital period of the recovered TOIs. The colour of the markers indicate the TOI's rank within a given sector, with the lighter colours representing a lower rank. The circles and crosses represent candidates at a rank lower and higher than 500, respectively. The figure shows that transit events with a SNR less than 3.5 are missed by the majority of volunteers, whereas events with a SNR greater than 5 are mostly recovered within the top 500 highest ranked candidates. \n\nThe steep increase in the fractional TOI recovery rate at lower ranks, as shown in figure~\\ref{fig:TCE_TOI_recovery}, is therefore due to the detection of the high SNR candidates that are identified by most, if not all, of the PHT volunteers who classified those targets. At a rank of around 500, the SNR of the TOIs tends towards the limit of what human vetting can detect and thus the identification of TOIs beyond a rank of 500 is more sporadic.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.49\\textwidth]{Figures\/TOI_recovery_properties.png}\n    \\caption{The SNR vs orbital period of TOIs with R > 2$R_{\\oplus}$. The colour represents their rank within the sector, as determined by the weighted DB clustering algorithm. Circles indicate that they were identified at a rank < 500, while crosses indicate that they were not within the top 500 highest ranked candidates of a given sector.\n    }\n    \\label{fig:TOI_properties}\n\\end{figure}\n\nThe fractional TCE recovery rate (bottom panel of Figure~\\ref{fig:TCE_TOI_recovery}) is systematically lower than that of the TOIs. There are qualitative reasons as to why humans might not identify a TCE as opposed to a TOI, including that TCEs may be caused by artefacts or periodic stellar signals that the SPOC pipeline identified as a potential transit but that the human eye would either miss or be able to rule out as systematic effect. This leads to a lower recovery fraction of TCEs comparatively, an effect that is further amplified by the much larger number of TCEs.\n\nThe detection efficiency of PHT is estimated using the fractional recovery rate of TOIs for a range of radius and period bins, as shown in Figure~\\ref{fig:recovery_rank500_radius_period}. A TOI is considered to be recovered if its detection rank is less than 500 within the given sector. Out of the total 1913 TOIs, to date, \\textcolor{black}{PHT recovered 715 TOIs among the highest ranked candidates across the 26 sectors. This corresponds to a mean of 12.7~\\% of the top 500 ranked candidates per sector being TOIs. In comparison, the primary \\emph{TESS}\\ team on average visually vets 1025 TCEs per sector, out of which a mean of 17.3~\\% are promoted to TOI status.} We find that, independent of the orbital period, PHT is over 85~\\% complete in the recovery of TOIs with radii equal to or greater than 4 $R_{\\oplus}$. This agrees with the findings from the initial Planet Hunters project \\citep{schwamb12}. The detection efficiency decreases to 51~\\% for 3 - 4 $R_{\\oplus}$ TOIs, 49~\\% for 2 - 3 $R_{\\oplus}$ TOIs and to less than 40~\\% for TOIs with radii less than 2 $R_{\\oplus}$. Fig~\\ref{fig:recovery_rank500_radius_period} shows that the orbital period does not have a strong effect on the detection efficiency for periods greater than $\\sim$~1~day, which highlights that human vetting efficiency is independent of the number of transits present within a light curve. For periods shorter than around 1~day, the detection efficiency decreases even for larger planets, due to the high frequency of events seen in the light curve. For these light curves, many volunteers will only mark a subset of the transits, which may not overlap with the subset marked by other volunteers. Due to the methodology used to identify and rank the candidates, as described in Section~\\ref{sec:method}, this will actively disfavour the recovery of very short period planets. Although this obviously introduces biases in the detectability of very short period signals, the major detection pipelines are specifically designed to identify these types of planets and thus this does not present a serious detriment to our main science goal of finding planets that were \\textcolor{black}{intentionally ignored or missed} by the main automated pipelines.\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{Figures\/TOI_recovery_grid.png}\n    \\caption{TOI recovery rate as a function of planet radius and orbital period. A TOI is considered recovered if it is amongst the top 500 highest ranked candidates within a given sector. The logarithmically spaced grid ranges from 0.2 to 225 d and 0.6 to 55 $R{_\\oplus}$ for the orbital period and planet radius, respectively. The fraction of TOIs recovered using PHT is computed for each cell and represented by the colour the grid. Cells with less than 10 TOIs are considered incomplete for statistical analysis and are shown by the hatched lines. White cells contain no TOIs. The annotations for each cell indicate the number of recovered TOIs followed by the Poisson uncertainty in brackets. The filled in and empty grey circles indicated the recovered and not-recovered TOIs, respectively.}\n    \\label{fig:recovery_rank500_radius_period}\n\\end{figure*}\n\n\nFinally, we assessed whether the detection efficiency varies across different sectors by assessing the fraction of recovered TOIs and TCEs within the highest ranked 500 candidates. We found the recovery of TOIs within the top 500 highest ranked candidates to remain relatively constant across all sectors, while the fraction of recovered TCEs in the top 500 highest ranked candidates  increases in later sectors, as shown in Figure~\\ref{fig:recovery_rank500}). After applying a Spearman's rank test we find a positive correlation of 0.86 (pvalue = 5.9 $\\times$ $10^{-8}$) and 0.57 (pvalue = 0.003) between the observation sector and TCE and TOI recovery rates, respectively. These correlations suggest that the ability of users to detect transit-like events improves as they classify more subjects. The improvement of volunteers over time can also be seen in Fig~\\ref{fig:user_weights}, which shows the mean (unnormalized) user weight per sector for volunteers who completed one or more classifications in at least one sector (blue), more than 10 sectors (orange), more than 20 sectors (green) and all of the sectors 26 sectors from the nominal \\emph{TESS}\\ mission (pink). The figure highlights an overall improvement in the mean user weight in later sectors, as well as a positive correlation between the overall increase in user weight and the number of sectors that volunteers have participated in.\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.45\\textwidth]{Figures\/TCE_TOI_rank500.png}\n    \\caption{The fractional recovery rate of the TOIs (blue circles) and TCEs (teal squares) at a rank of 500 for each sector. Sector 1-9 (white background) represent southern hemisphere sectors classified with UI1, sectors 9-14 (light grey background) show the southern hemisphere sectors classified with UI2, and sectors 14-24 (dark grey background) show the northern hemisphere sectors classified with US2.}\n    \\label{fig:recovery_rank500}\n\\end{figure}\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.50\\textwidth]{Figures\/user_weights_sectors.png}\n    \\caption{Mean user weights per sector. The solid lines show the user weights for the old user interface and the dashed line for the new interface, separated by the black line (Sector 9). The different coloured lines show the mean user weights calculated considering user who participated in any number of sectors (blue), more than 10 sectors (orange), more than 20 sectors (green) and all of the sectors observed during the nominal \\emph{TESS}\\ mission (pink).}\n    \\label{fig:user_weights}\n\\end{figure}\n\n\n\\section{Candidate vetting}\n\\label{sec:vetting}\n\nFor each observation sector the subjects are ranked according to their transit scores, and the 500 highest ranked targets (excluding TOIs) visually vetted by the PHT science team in order to identify potential candidates and rule out false positives. A vetting cut-off rank of 500 was chosen as we found this to maximise the number of found candidates while minimising the number of likely false positives. In the initial round of vetting, which is completed via a separate Zooniverse classification interface that is only accessible to the core science team, a minimum of three members of the team sort the highest ranked targets into either `keep for further analysis', `eclipsing binary' or `discard'. The sorting is based on the inspection of the full \\emph{TESS}\\ light curve of the target, with the times of the satellite momentum dumps indicated. Additionally, around the time of each likely transit event (i.e. time of successful DB clusters) we inspect the background flux and the x and y centroid positions. Stellar parameters are provided for each candidate, subject to availability, alongside links to the SPOC Data Validation (DV) reports for candidates that had been flagged as TCEs but were never promoted to TOIs status.\n\nCandidates where at least two of the reviewers indicated that the signal is consistent with a planetary transit are kept for further analysis. \\textcolor{black}{This constitute a $\\sim$~5~\\% retention rate of the 500 highest ranked candidates per sector between the initial citizen science classification stage and the PHT science team vetting stage. Considering that the known planets and TOIs are not included at this stage of vetting, it is not surprising that our retention rate is lower that the true-positive rates of TCEs (see Section~\\ref{subsec:TCE_TOI}). Furthermore, this false-positive rate is consistent with the the findings of the initial Planet Hunters project \\citep{schwamb12}.}\n\nThe rest of the 500 candidates were grouped into $\\sim$~37~\\% `eclipsing binary' and $\\sim$~58~\\% `discard'. The most common reasons for discarding light curves are due to events caused by momentum dumps and due to background events, such as background eclipsing binaries, that mimic transit-like signals in the light curve. The targets identified as eclipsing binaries are analysed further by the \\emph{TESS}\\ Eclipsing Binaries Working Group (Prsa et al, in prep).\n\n\n\n\nFor the second round of candidate vetting we generate our own data validation reports for all candidates classified as `keep for further analysis'. The reports are generated using the open source software {\\sc latte} \\citep[Lightcurve Analysis Tool for Transiting Exoplanets;][]{LATTE2020}, which includes a range of standard diagnostic plots that are specifically designed to help identify transit-like signals and weed out astrophysical false positives in \\emph{TESS}\\ data. In brief the diagnostics consist of:\n\n\\textbf{Momentum Dumps}. The times of the \\emph{TESS}\\ reaction wheel momentum dumps that can result in instrumental effects that mimic astrophysical signals.\n\n\\textbf{Background Flux}. The background flux to help identify trends caused by background events such as asteroids or fireflies \\citep{vanderspek2018tess} passing through the field of view.\n\n\\textbf{x and y centroid positions}. The CCD column and row local position of the target's flux-weighted centroid, and the CCD column and row motion which considers differential velocity aberration (DVA), pointing drift, and thermal effects. This can help identify signals caused by systematics due to the satellite. \n\\textbf{Aperture size test}. The target light curve around the time of the transit-like event extracted using two apertures of different sizes. This can help identify signals resulting from background eclipsing binaries.\n \n\\textbf{Pixel-level centroid analysis}. A comparison between the average in-transit and average out-of-transit flux, as well as the difference between them. This can help identify signals resulting from background eclipsing binaries.\n\n\\textbf{Nearby companion stars}. The location of nearby stars brighter than V-band magnitude 15 as queried from the Gaia Data Release 2 catalog \\citep{gaia2018gaia} and the DSS2 red field of view around the target star in order to identify nearby contaminating sources. \n\n\\textbf{Nearest neighbour light curves}. Normalized flux light curves of the five short-cadence \\emph{TESS}\\ stars with the smallest projected distances to the target star, used to identify alternative sources of the signal or systematic effects that affect multiple target stars. \n\n\\textbf{Pixel level light curves}. Individual light curves extracted for each pixel around the target. Used to identify signals resulting from background eclipsing binaries, background events and systematics.\n\n\\textbf{Box-Least-Squares fit}. Results from two consecutive BLS searches, where the identified signals from the initial search are removed prior to the second BLS search.\n\nThe {\\sc latte} validation reports are assessed by the PHT science team in order to identify planetary candidates that warrant further investigation. Around 10~\\% of the targets assessed at this stage of vetting are kept for further investigation, resulting in $\\sim$~3 promising planet candidates per observation sector. The discarded candidates can be loosely categorized into (background) eclipsing binaries ($\\sim$~40~\\%), systematic effects ($\\sim$~25~\\%), background events ($\\sim$~15~\\%) and other (stellar signals such as spots; $\\sim$~10~\\%).\n\n\nWe use \\texttt{pyaneti}\\ \\citep{pyaneti} to infer the planetary and orbital parameters of our most promising candidates. For multi-transit candidates we fit for seven parameters per planet, time of mid-transit $T_0$, orbital period $P$, impact parameter $b$, scaled semi-major axis $a\/R_\\star$, scaled planet radius $r_{\\rm p}\/R_\\star$, and two limb darkening coefficients following a \\citet{Mandel2002} quadratic limb darkening model, implemented with the $q_1$ and $q_2$ parametrization suggested by \\citet{Kipping2013}. Orbits were assumed to be circular.\nFor the mono-transit candidates, we fit the same parameters as for the multi-transit case, except for the orbital period and scaled semi-major axis which cannot be known for single transits. We follow \\citet{Osborn2016} to estimate the orbital period of the mono-transit candidates assuming circular orbits.\n\nWe note that some of our candidates are V-shaped, consistent with a grazing transit configuration. For these cases, we set uniform priors between 0 and 0.15 for $r_{\\rm p}\/R_\\star$ and between 0 and 1.15 for the impact parameter in order to avoid large radii caused by the $r_{\\rm p}\/R_\\star - b$ degeneracy. Thus, the $r_{\\rm p}\/R_\\star$ for these candidates should not be trusted. A full characterisation of these grazing transits is out of the scope of this manuscript.\n\nFigure~\\ref{fig:PHT_pyaneti} shows the \\emph{TESS}\\ transits together with the inferred model for each candidate. Table~\\ref{tab:PHT-caniddates} shows the inferred main parameters, the values and their uncertainties are given by the median and 68.3\\% credible interval of the posterior distributions.\n\n\n\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=0.83\\textwidth]{Figures\/canidate_transits_all_one.png}\n    \\caption{All of the PHT candidates modelled using \\texttt{pyaneti}. The parameters of the best fits are summarised in Table~\\protect\\ref{tab:PHT-caniddates}. The blue and magenta fits show the multi and single transit event candidates, respectively.} \n    \\label{fig:PHT_pyaneti}\n\\end{figure*}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=0.83\\textwidth]{Figures\/canidate_transits_all_two.png}\n    \\addtocounter{figure}{-1}\n    \\caption{\\textbf{PHT candidates (continued)}} \n\\end{figure*}\n\n\nCandidates that pass all of our rounds of vetting are uploaded to the Exoplanet Follow-up Observing Program for TESS (ExoFOP-TESS) website\\footnote{\\url{ https:\/\/exofop.ipac.caltech.edu\/tess\/index.php}} as community TOIs (cTOIs).\n\n\\section{Follow-up observations}\n\\label{sec:follow_up}\n\nMany astrophysical false positive scenarios can be ruled out from the detailed examination of the \\emph{TESS}\\ data, both from the light curves themselves and from the target pixel files. However, not all of the false positive scenarios can be ruled out from these data alone, due in part to the large \\emph{TESS}\\ pixels (20 arcsconds). Our third stage of vetting, therefore, consists of following up the candidates with ground based observations including photometry, reconnaissance spectroscopy and speckle imaging. The results from these observations will be discussed in detail in a dedicated follow-up paper. \n\n\\subsection{Photometry}\n\nWe make use of the LCO global network of fully robotic 0.4-m\/SBIG and 1.0-m\/Sinistro facilities \\citep{LCO2013} to observe additional transits, where the orbital period is known, in order to refine the ephemeris and confirm that the transit events are not due to a blended eclipsing binary in the vicinity of the main target. Snapshot images are taken of single transit event candidates in order to identify nearby contaminating sources. \n\n\n\\subsection{Spectroscopy}\n\nWe perform high-resolution optical spectroscopy using telescopes from across the globe in order to cover a wide range of RA and Dec:\n\\begin{itemize}\n\\item The Las Cumbres Observatory (LCO) telescopes with the Network of Robotic Echelle Spectrographs \\citep[NRES,][]{LCO2013}. These fibre-fed spectrographs, mounted on 1.0-m telescopes around the globe, have a resolution of R = 53,000 and a wavelength coverage of 380 to 860 nm. \n\n\\item The MINERVA Australis Telescope facility, located at Mount Kent Observatory in Queensland, Australia \\citep{addison2019}. This facility is made up of four 0.7m CDK700 telescopes, which individually feed light via optic fibre into a KiwiSpec high-resolution (R = 80,000) stabilised spectrograph \\citep{barnes2012} that covers wavelengths from 480 nm to 620 nm. \n\n\\item The CHIRON spectrograph mounted on the SMARTS 1.5-m telescope \\citep{Tokovinin2018}, located at the Cerro Tololo\nInter-American Observatory (CTIO) in Chile. The high resolution cross-dispersed echelle spectrometer is fiber-fed followed by an image slicer. It has a resolution of R = 80,000 and covers wavelengths ranging from 410 to 870 nm.\n\n\\item The SOPHIE echelle spectrograph mounted on the 1.93-m Haute-Provence Observatory (OHP), France\n\\citep{2008Perruchot,2009Bouchy}. The high resolution cross-dispersed stabilized echelle spectrometer is fed by two optical fibers. Observations were taken in high-resolution mode (R = 75,000) with a wavelength range of 387 to 694 nm.\n\n\\end{itemize}\n\nReconnaissance spectroscopy with these instruments allow us to extract stellar parameters, identify spectroscopic binaries, and place upper limits on the companion masses. Spectroscopic binaries and targets whose spectral type is incompatible with the initial planet hypothesis and\/or precludes precision RV observations (giant or early type stars) are not followed up further. Promising targets, however, are monitored in order to constrain their period and place limits on their mass. \n\n\\subsection{Speckle Imaging}\n\nFor our most promising candidates we perform high resolution speckle imaging using the `Alopeke instrument on the 8.1-m Frederick C. Gillett Gemini North telescope in Maunakea, Hawaii, USA, and its twin, Zorro, on the 8.1-m Gemini South telescope on Cerro Pach\\'{o}n, Chile \\citep{Matson2019, Howell2011}. Speckle interferometric observations provide extremely high resolution images reaching the diffraction limit of the telescope. We obtain simultaneous 562 nm and 832 nm rapid exposure (60 msec) images in succession that effectively `freeze out' atmospheric turbulence and through Fourier analysis are used to search for close companion stars at 5-8 magnitude contrast levels. This analysis, along with the reconstructed images, allow us to identify nearby companions and to quantify their light contribution to the TESS aperture and thus the transit signal.\n\n\n\\section{Planet candidates and Noteworthy Systems}\n\\label{sec:PHT_canidates}\n\\subsection{Planet candidate properties}\n\nIn this final part of the paper we discuss the 90 PHT candidates around 88 host stars that passed the initial two stages of vetting and that were uploaded to ExoFOP as cTOIs. At the time of discovery none of these candidates were TOIs. The properties of all of the PHT candidates are summarised in Table~\\ref{tab:PHT-caniddates}. Candidates that have been promoted to TOI status since their PHT discovery are highlighted with an asterisk following the TIC ID, and candidates that have been shown to be false positives, based on the ground-based follow-up observations, are marked with a dagger symbol ($\\dagger$). The majority (81\\%) of PHT candidates are single transit events, indicated by an `s' following the orbital period presented in the table. \\textcolor{black}{18 of the PHT candidates were flagged as TCEs by the \\emph{TESS}\\ pipeline, but not initially promoted to TOI status. The most common reasons for this was that the pipeline identified a single-transit event as well as times of systematics (often caused by momentum dumps), due to its two-transit minimum detection threshold. This resulted in the candidate being discarded on the basis of it not passing the `odd-even' transit depth test. Out of the 18 TCEs, 14 have become TOI's since the PHT discovery. More detail on the TCE candidates can be found in Appendix~\\ref{appendixA}.} \n\nAll planet parameters (columns 2 to 8) are derived from the \\texttt{pyaneti}\\ modelling as described in Section~\\ref{sec:vetting}. Finally, the table summarises the ground-based follow-up observations (see Sec~\\ref{sec:follow_up}) that have been obtained to date, where the bracketed numbers following the observing instruments indicate the number of epochs. Unless otherwise noted, the follow-up observations are consistent with a planetary scenario. More in depth descriptions of individual targets for which we have additional information to complement the results in Table~\\ref{tab:PHT-caniddates} can be found in Appendix~\\ref{appendixA}.\n\n\\subsection{Planet candidate analysis}\n\n\nThe majority of the TOIs (87.7\\%) have orbital periods shorter than 15 days due to the requirement of observing at least two transits included in all major pipelines \\textcolor{black}{combined with the observing strategy of \\emph{TESS}}. As visual vetting does not impose these limits, the candidates outlined in this paper are helping to populate the relatively under-explored long-period region of parameter space. This is highlighted in Figure~\\ref{fig:PHT_candidates}, which shows the transit depths vs the orbital periods of the PHT single transit candidates (orange circles) and the multi-transit candidates (magenta squares) compared to the TOIs (blue circles). Values of the orbital periods and transit depths were obtained via transit modelling using \\texttt{pyaneti} (see Section~\\ref{sec:vetting}). The orbital period of single transit events are poorly constrained, which is reflected by the large errorbars in Figure~\\ref{fig:PHT_candidates}. Figure~\\ref{fig:PHT_candidates} also highlights that with PHT we are able to recover a similar range of transit depths as the pipeline found TOIs, as was previously shown in Figure~\\ref{fig:recovery_rank500_radius_period}.\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.45\\textwidth]{Figures\/PHT_candidate_period_depth_plot_errobrars.png}\n    \\caption{The properties of the PHT single transit (orange circles) and multi transit (magenta squares) candidates compared to the properties of the TOIs (blue circles). All parameters (listed in Table~\\ref{fig:PHT_candidates}) were extracted using \\texttt{pyaneti}\\ modelling.}\n    \\label{fig:PHT_candidates}\n\\end{figure}\n\nThe PHT candidates were further compared to the TOIs in terms of the properties of their host stars. Figure~\\ref{fig:eep} shows the effective temperature and stellar radii as taken from the TIC \\citep{Stassun18}, for TOIs (blue dots) and the PHT candidates (magenta circles). The solid and dashed lines indicate the main sequence and post-main sequence MIST stellar evolutionary tracks \\citep{choi2016}, respectively, for stellar masses ranging from 0.3 to 1.6 $M_\\odot$ in steps of 0.1 $M_\\odot$. This shows that around 10\\% of the host stars are in the process of, or have recently evolved off the main sequence. The models assume solar metalicity, no stellar rotation and no additional internal mixing.\n\n\\textcolor{black}{Ground based follow-up spectroscopy has revealed that six of the PHT candidates listed in Table~\\ref{tab:PHT-caniddates} are astrophysical false positives. As the follow-up campaign of the targets is still underway, the true false-positive rate of the candidates to have made it through all stages of the vetting process, as outlined in the methodology, will be be assessed in future PHT papers once the true nature of more of the candidates has been independently verified.}\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.45\\textwidth]{Figures\/PHT_eep.png}\n    \\caption{Stellar evolution tracks showing main sequence (solid black lines) and post-main sequence (dashed grey lines) MIST stellar evolution for stellar masses ranging from 0.3 to 1.6 $M_\\odot$ in steps of 0.1 $M_\\odot$. The blue dots show the TOIs and the magenta circles show the PHT candidates.} \n    \\label{fig:eep}\n\\end{figure}\n\n\n\\subsection{Stellar systems}\n\\label{subsec:PHT_stars}\n\nIn addition to the planetary candidates, citizen science allows for the identification of interesting stellar systems and astrophysical phenomena, in particular where the signals are aperiodic or small compared to the dominant stellar signal. These include light curves that exhibit multiple transit-like signals, possibly as a result of a multiple stellar system or a blend of eclipsing binaries. We have investigated all light curves that were flagged as possible multi-stellar systems via the PHT discussion boards. Similar to the planet vetting, as described in Section~\\ref{sec:vetting}, we generated {\\sc latte} data validation reports in order to assess the nature of the signal. Additionally, we subjected these systems to an iterative signal removal process, whereby we phase-folded the light curve on the dominant orbital period, binned the light curve into between 200-500 phase bins, created an interpolation model, and then subtracted said signal in order to evaluate the individual transit signals. The period of each signal, as listed in Table~\\ref{tab:PHT-multis}, was determined by phase folding the light curve at a number of trial periods and assessing by eye the best fit period and corresponding uncertainty.\n\nDue to the large \\emph{TESS}\\ pixels, blends are expected to be common. We searched for blends by generating phase folded light curves for each pixel around the source of the target in order to better locate the source of each signal. Shifts in the \\emph{TESS}\\ x and y centroid positions were also found to be good indicators of visually separated sources. Nearby sources with a magnitude difference greater than 5 mags were ruled out as possible contaminators. We consider a candidate to be a confirmed blend when the centroids are separated by more than 1 \\emph{TESS}\\ pixel, as this corresponds to an angular separation > 21 arcseconds meaning that the systems are highly unlikely to be gravitationally bound. Systems where the signal appears to be coming from the same \\emph{TESS}\\ pixel and that show no clear centroid shifts are considered to be candidate multiple systems. We note that blends are still possible, however, without further investigation we cannot conclusively rule these out as possible multi stellar systems. \n\nAll of the systems are summarised in Table~\\ref{tab:PHT-multis}. Out of the 26 systems, 6 are confirmed multiple systems which have either been published or are being prepared for publication; 7 are visually separated eclipsing binaries (confirmed blends); and 13 are candidate multiple system. Additional observations will be required to determine whether or not these candidate multiple systems are in fact gravitationally bound or photometric blends as a results of the large \\emph{TESS}\\ pixels or due to a line of sight happenstance. \n\n\\begin{landscape}\n\\begin{table}\n\\resizebox{1.31\\textwidth}{!}{\n\\begin{tabular}{ccccccccccccc}\n\\textbf{TIC} & \\textbf{Other} & \\textbf{Epoch} & \\textbf{Period}  & \\textbf{$R_{pl}$\/$R_{\\odot}$} & \\textbf{$R_{pl}$} & \\textbf{Impact} & \\textbf{Duration} & \\textbf{$V_{mag}$} & \\textbf{Photometry} & \\textbf{Spectroscopy} & \\textbf{Speckle} & \\textbf{Comment} \\\\\n  & \\textbf{Name} & \\textbf{(\\textcolor{black}{BJD - 2457000})} & \\textbf{(days)}  & &  $(R_{\\oplus})$ & \\textbf{Parameter} & \\textbf{(hours)} &   &   &   &   &    \\\\\n\\hline\n101641905 & TWOMASS 11412617+3441004 & $1917.26335 _{ - 0.00072 } ^ { + 0.00071 }$ & $14.52 _{ - 5.25 } ^ { + 6.21 }(s)$ & $0.1135 _{ - 0.0064 } ^ { + 0.0032 }$ & $9.76 _{ - 0.69 } ^ { + 0.65 }$ & $0.691 _{ - 0.183 } ^ { + 0.077 }$ & $3.163 _{ - 0.088 } ^ { + 0.093 }$ & 12.196 &  &  &  &  \\\\\n103633672* & TYC 4387-00923-1 & $1850.3211 _{ - 0.00077 } ^ { + 0.00135 }$ & $90.9 _{ - 23.7 } ^ { + 46.4 }(s)$ & $0.0395 _{ - 0.0013 } ^ { + 0.0013 }$ & $3.45 _{ - 0.24 } ^ { + 0.26 }$ & $0.3 _{ - 0.21 } ^ { + 0.26 }$ & $6.7 _{ - 0.11 } ^ { + 0.12 }$ & 10.586 &  & NRES (1) &  &  \\\\\n110996418 & TWOMASS 12344723-1019107 & $1580.6406 _{ - 0.0038 } ^ { + 0.0037 }$ & $5.18 _{ - 2.93 } ^ { + 6.86 }(s)$ & $0.1044 _{ - 0.0067 } ^ { + 0.008 }$ & $12.7 _{ - 0.99 } ^ { + 1.15 }$ & $0.44 _{ - 0.3 } ^ { + 0.3 }$ & $3.53 _{ - 0.27 } ^ { + 0.36 }$ & 13.945 &  &  &  &  \\\\\n128703021 & HIP 71639 & $1601.8442 _{ - 0.00108 } ^ { + 0.00093 }$ & $26.0 _{ - 8.22 } ^ { + 22.35 }(s)$ & $0.0254 _{ - 0.00049 } ^ { + 0.00072 }$ & $4.44 _{ - 0.2 } ^ { + 0.23 }$ & $0.47 _{ - 0.3 } ^ { + 0.22 }$ & $7.283 _{ - 0.091 } ^ { + 0.141 }$ & 6.06 &  & NRES (2);MINERVA (34) & Gemini &  \\\\\n138126035 & TYC 1450-00833-1 & $1954.3229 _{ - 0.0041 } ^ { + 0.0067 }$ & $28.8 _{ - 14.0 } ^ { + 203.2 }(s)$ & $0.0375 _{ - 0.0026 } ^ { + 0.0069 }$ & $4.01 _{ - 0.35 } ^ { + 0.74 }$ & $0.58 _{ - 0.38 } ^ { + 0.35 }$ & $4.65 _{ - 0.32 } ^ { + 0.85 }$ & 10.349 &  &  &  &  \\\\\n142087638 & TYC 9189-00274-1 & $1512.1673 _{ - 0.0043 } ^ { + 0.0034 }$ & $3.14 _{ - 1.41 } ^ { + 12.04 }(s)$ & $0.0469 _{ - 0.0035 } ^ { + 0.0063 }$ & $6.05 _{ - 0.54 } ^ { + 0.89 }$ & $0.5 _{ - 0.35 } ^ { + 0.36 }$ & $2.72 _{ - 0.23 } ^ { + 0.5 }$ & 11.526 &  &  &  &  \\\\\n159159904 & HIP 64812 & $1918.6109 _{ - 0.0067 } ^ { + 0.0091 }$ & $584.0 _{ - 215.0 } ^ { + 1724.0 }(s)$ & $0.0237 _{ - 0.0011 } ^ { + 0.0026 }$ & $3.12 _{ - 0.22 } ^ { + 0.36 }$ & $0.49 _{ - 0.34 } ^ { + 0.35 }$ & $15.11 _{ - 0.54 } ^ { + 0.7 }$ & 9.2 &  & NRES (2) &  &  \\\\\n160039081* & HIP 78892 & $1752.9261 _{ - 0.0045 } ^ { + 0.005 }$ & $30.19918 _{ - 0.00099 } ^ { + 0.00094 }$ & $0.0211 _{ - 0.0013 } ^ { + 0.0035 }$ & $2.67 _{ - 0.21 } ^ { + 0.43 }$ & $0.52 _{ - 0.34 } ^ { + 0.36 }$ & $4.93 _{ - 0.27 } ^ { + 0.37 }$ & 8.35 & SBIG (1) & NRES (1);SOPHIE (4) & Gemini &  \\\\\n162631539 & HIP 80264 & $1978.2794 _{ - 0.0044 } ^ { + 0.0051 }$ & $17.32 _{ - 6.66 } ^ { + 52.35 }(s)$ & $0.0195 _{ - 0.0011 } ^ { + 0.0024 }$ & $2.94 _{ - 0.24 } ^ { + 0.38 }$ & $0.48 _{ - 0.33 } ^ { + 0.36 }$ & $5.54 _{ - 0.33 } ^ { + 0.41 }$ & 7.42 &  &  &  &  \\\\\n166184426* & TWOMASS 13442500-4020122 & $1600.4409 _{ - 0.003 } ^ { + 0.0036 }$ & $16.3325 _{ - 0.0066 } ^ { + 0.0052 }$ & $0.0545 _{ - 0.0031 } ^ { + 0.0039 }$ & $1.85 _{ - 0.12 } ^ { + 0.15 }$ & $0.41 _{ - 0.28 } ^ { + 0.31 }$ & $1.98 _{ - 0.22 } ^ { + 0.17 }$ & 12.911 &  &  &  &  \\\\\n167661160$\\dagger$ & TYC 7054-01577-1 & $1442.0703 _{ - 0.0028 } ^ { + 0.004 }$ & $36.802 _{ - 0.07 } ^ { + 0.069 }$ & $0.0307 _{ - 0.0014 } ^ { + 0.0024 }$ & $4.07 _{ - 0.32 } ^ { + 0.43 }$ & $0.37 _{ - 0.26 } ^ { + 0.33 }$ & $5.09 _{ - 0.23 } ^ { + 0.21 }$ & 9.927 &  & NRES (9);MINERVA (4) &  & EB from MINERVA observations \\\\\n172370679* & TWOMASS 19574239+4008357 & $1711.95923 _{ - 0.00099 } ^ { + 0.001 }$ & $32.84 _{ - 4.17 } ^ { + 5.59 }(s)$ & $0.1968 _{ - 0.0032 } ^ { + 0.0022 }$ & $13.24 _{ - 0.43 } ^ { + 0.43 }$ & $0.22 _{ - 0.15 } ^ { + 0.14 }$ & $4.999 _{ - 0.097 } ^ { + 0.111 }$ & 14.88 &  &  &  & Confirmed planet \\citep{canas2020}. \\\\\n174302697* & TYC 3641-01789-1 & $1743.7267 _{ - 0.00092 } ^ { + 0.00093 }$ & $498.2 _{ - 80.0 } ^ { + 95.3 }(s)$ & $0.07622 _{ - 0.00068 } ^ { + 0.00063 }$ & $13.34 _{ - 0.57 } ^ { + 0.58 }$ & $0.642 _{ - 0.029 } ^ { + 0.024 }$ & $17.71 _{ - 0.12 } ^ { + 0.13 }$ & 9.309 & SBIG (1) &  &  &  \\\\\n179582003 & TYC 9166-00745-1 & $1518.4688 _{ - 0.0016 } ^ { + 0.0016 }$ & $104.6137 _{ - 0.0022 } ^ { + 0.0022 }$ & $0.06324 _{ - 0.0008 } ^ { + 0.0008 }$ & $7.51 _{ - 0.35 } ^ { + 0.35 }$ & $0.21 _{ - 0.15 } ^ { + 0.19 }$ & $9.073 _{ - 0.084 } ^ { + 0.097 }$ & 10.806 &  &  &  &  \\\\\n192415680 & TYC 2859-00682-1 & $1796.0265 _{ - 0.0012 } ^ { + 0.0013 }$ & $18.47 _{ - 6.34 } ^ { + 21.73 }(s)$ & $0.0478 _{ - 0.0017 } ^ { + 0.0027 }$ & $4.43 _{ - 0.33 } ^ { + 0.38 }$ & $0.45 _{ - 0.31 } ^ { + 0.31 }$ & $3.94 _{ - 0.1 } ^ { + 0.12 }$ & 9.838 & SBIG (1) & SOPHIE (2) &  &  \\\\\n192790476 & TYC 7595-00649-1 & $1452.3341 _{ - 0.0014 } ^ { + 0.002 }$ & $16.09 _{ - 5.73 } ^ { + 15.49 }(s)$ & $0.0438 _{ - 0.0018 } ^ { + 0.0026 }$ & $3.24 _{ - 0.34 } ^ { + 0.37 }$ & $0.37 _{ - 0.25 } ^ { + 0.3 }$ & $3.395 _{ - 0.099 } ^ { + 0.192 }$ & 10.772 &  &  &  &  \\\\\n206361691$\\dagger$ & HIP 117250 & $1363.2224 _{ - 0.0082 } ^ { + 0.009 }$ & $237.7 _{ - 81.0 } ^ { + 314.4 }(s)$ & $0.01762 _{ - 0.00088 } ^ { + 0.00125 }$ & $2.69 _{ - 0.19 } ^ { + 0.25 }$ & $0.43 _{ - 0.28 } ^ { + 0.32 }$ & $13.91 _{ - 0.53 } ^ { + 0.52 }$ & 8.88 &  & CHIRON (2) &  & SB2 from CHIRON \\\\\n207501148 & TYC 3881-00527-1 & $2007.7273 _{ - 0.0011 } ^ { + 0.0011 }$ & $39.9 _{ - 10.3 } ^ { + 14.3 }(s)$ & $0.0981 _{ - 0.0047 } ^ { + 0.011 }$ & $13.31 _{ - 0.95 } ^ { + 1.56 }$ & $0.9 _{ - 0.03 } ^ { + 0.039 }$ & $4.73 _{ - 0.14 } ^ { + 0.14 }$ & 10.385 &  &  &  &  \\\\\n219466784* & TYC 4409-00437-1 & $1872.6879 _{ - 0.0097 } ^ { + 0.0108 }$ & $318.0 _{ - 147.0 } ^ { + 1448.0 }(s)$ & $0.0332 _{ - 0.0024 } ^ { + 0.0048 }$ & $3.26 _{ - 0.31 } ^ { + 0.49 }$ & $0.55 _{ - 0.39 } ^ { + 0.34 }$ & $10.06 _{ - 0.81 } ^ { + 1.12 }$ & 11.099 &  &  &  &  \\\\\n219501568 & HIP 79876 & $1961.7879 _{ - 0.0018 } ^ { + 0.002 }$ & $16.5931 _{ - 0.0017 } ^ { + 0.0015 }$ & $0.0221 _{ - 0.0012 } ^ { + 0.0015 }$ & $4.22 _{ - 0.3 } ^ { + 0.35 }$ & $0.41 _{ - 0.28 } ^ { + 0.31 }$ & $1.615 _{ - 0.077 } ^ { + 0.093 }$ & 8.38 &  &  &  &  \\\\\n229055790 & TYC 7492-01197-1 & $1337.866 _{ - 0.0022 } ^ { + 0.0019 }$ & $48.0 _{ - 12.8 } ^ { + 48.4 }(s)$ & $0.0304 _{ - 0.00097 } ^ { + 0.00115 }$ & $3.52 _{ - 0.2 } ^ { + 0.24 }$ & $0.37 _{ - 0.26 } ^ { + 0.32 }$ & $6.53 _{ - 0.11 } ^ { + 0.14 }$ & 9.642 &  & NRES (2) &  &  \\\\\n229608594 & TWOMASS 18180283+7428005 & $1960.0319 _{ - 0.0037 } ^ { + 0.0045 }$ & $152.4 _{ - 54.1 } ^ { + 152.6 }(s)$ & $0.0474 _{ - 0.0023 } ^ { + 0.0024 }$ & $3.42 _{ - 0.34 } ^ { + 0.36 }$ & $0.38 _{ - 0.26 } ^ { + 0.3 }$ & $6.98 _{ - 0.23 } ^ { + 0.37 }$ & 12.302 &  &  &  &  \\\\\n229742722* & TYC 4434-00596-1 & $1689.688 _{ - 0.025 } ^ { + 0.02 }$ & $29.0 _{ - 16.4 } ^ { + 66.3 }(s)$ & $0.019 _{ - 0.0028 } ^ { + 0.0029 }$ & $2.9 _{ - 0.44 } ^ { + 0.48 }$ & $0.44 _{ - 0.3 } ^ { + 0.33 }$ & $4.27 _{ - 0.09 } ^ { + 0.11 }$ & 10.33 &  & NRES (8);SOPHIE (4) & Gemini &  \\\\\n233194447 & TYC 4211-00650-1 & $1770.4924 _{ - 0.0065 } ^ { + 0.0107 }$ & $373.0 _{ - 101.0 } ^ { + 284.0 }(s)$ & $0.02121 _{ - 0.00073 } ^ { + 0.001 }$ & $5.08 _{ - 0.28 } ^ { + 0.33 }$ & $0.34 _{ - 0.24 } ^ { + 0.29 }$ & $24.45 _{ - 0.47 } ^ { + 0.5 }$ & 9.178 &  & NRES (2) & Gemini &  \\\\\n235943205 & TYC 4588-00127-1 & $1827.0267 _{ - 0.004 } ^ { + 0.0034 }$ & $121.3394 _{ - 0.0063 } ^ { + 0.0065 }$ & $0.0402 _{ - 0.0016 } ^ { + 0.0019 }$ & $4.2 _{ - 0.25 } ^ { + 0.29 }$ & $0.4 _{ - 0.27 } ^ { + 0.28 }$ & $6.37 _{ - 0.2 } ^ { + 0.3 }$ & 11.076 &  & NRES (1);SOPHIE (2) &  &  \\\\\n237201858 & TYC 4452-00759-1 & $1811.5032 _{ - 0.0069 } ^ { + 0.0067 }$ & $129.7 _{ - 41.5 } ^ { + 146.8 }(s)$ & $0.0258 _{ - 0.0013 } ^ { + 0.0015 }$ & $4.12 _{ - 0.27 } ^ { + 0.3 }$ & $0.4 _{ - 0.28 } ^ { + 0.31 }$ & $10.94 _{ - 0.37 } ^ { + 0.53 }$ & 10.344 &  & NRES (1) &  &  \\\\\n243187830* & HIP 5286 & $1783.7671 _{ - 0.0017 } ^ { + 0.0019 }$ & $4.05 _{ - 1.53 } ^ { + 9.21 }(s)$ & $0.0268 _{ - 0.0015 } ^ { + 0.0027 }$ & $2.06 _{ - 0.17 } ^ { + 0.23 }$ & $0.47 _{ - 0.32 } ^ { + 0.34 }$ & $2.02 _{ - 0.12 } ^ { + 0.15 }$ & 8.407 & SBIG (1) &  &  &  \\\\\n243417115 & TYC 8262-02120-1 & $1614.4796 _{ - 0.0028 } ^ { + 0.0022 }$ & $1.81 _{ - 0.73 } ^ { + 3.45 }(s)$ & $0.0523 _{ - 0.0035 } ^ { + 0.005 }$ & $5.39 _{ - 0.47 } ^ { + 0.64 }$ & $0.47 _{ - 0.33 } ^ { + 0.34 }$ & $2.03 _{ - 0.16 } ^ { + 0.23 }$ & 11.553 &  &  &  &  \\\\\n256429408 & TYC 4462-01942-1 & $1962.16 _{ - 0.0022 } ^ { + 0.0023 }$ & $382.0 _{ - 132.0 } ^ { + 265.0 }(s)$ & $0.03582 _{ - 0.00086 } ^ { + 0.00094 }$ & $6.12 _{ - 0.29 } ^ { + 0.3 }$ & $0.51 _{ - 0.36 } ^ { + 0.18 }$ & $16.96 _{ - 0.2 } ^ { + 0.24 }$ & 8.898 &  &  &  &  \\\\\n264544388* & TYC 4607-01275-1 & $1824.8438 _{ - 0.0076 } ^ { + 0.0078 }$ & $7030.0 _{ - 6260.0 } ^ { + 3330.0 }(s)$ & $0.0288 _{ - 0.0029 } ^ { + 0.0018 }$ & $4.58 _{ - 0.43 } ^ { + 0.35 }$ & $0.936 _{ - 0.363 } ^ { + 0.011 }$ & $19.13 _{ - 1.35 } ^ { + 0.84 }$ & 8.758 &  & NRES (1) &  &  \\\\\n264766922 & TYC 8565-01780-1 & $1538.69518 _{ - 0.00091 } ^ { + 0.00091 }$ & $3.28 _{ - 0.94 } ^ { + 1.25 }(s)$ & $0.0933 _{ - 0.0063 } ^ { + 0.0176 }$ & $16.95 _{ - 1.33 } ^ { + 3.19 }$ & $0.908 _{ - 0.039 } ^ { + 0.048 }$ & $2.73 _{ - 0.11 } ^ { + 0.11 }$ & 10.747 &  &  &  &  \\\\\n26547036* & TYC 3921-01563-1 & $1712.30464 _{ - 0.00041 } ^ { + 0.0004 }$ & $73.0 _{ - 13.6 } ^ { + 16.5 }(s)$ & $0.10034 _{ - 0.0007 } ^ { + 0.00078 }$ & $11.75 _{ - 0.59 } ^ { + 0.58 }$ & $0.17 _{ - 0.12 } ^ { + 0.11 }$ & $8.681 _{ - 0.049 } ^ { + 0.052 }$ & 9.849 &  & NRES (4) & Gemini &  \\\\\n267542728$\\dagger$ & TYC 4583-01499-1 & $1708.4956 _{ - 0.0073 } ^ { + 0.0085 }$ & $39.7382 _{ - 0.0023 } ^ { + 0.0023 }$ & $0.03267 _{ - 0.00089 } ^ { + 0.00175 }$ & $18.46 _{ - 0.94 } ^ { + 1.14 }$ & $0.38 _{ - 0.26 } ^ { + 0.27 }$ & $24.16 _{ - 0.39 } ^ { + 0.45 }$ & 11.474 &  &  &  & EB from HIRES RVs. \\\\\n270371513$\\dagger$ & HIP 10047 & $1426.2967 _{ - 0.0023 } ^ { + 0.002 }$ & $0.39 _{ - 0.17 } ^ { + 1.79 }(s)$ & $0.024 _{ - 0.0015 } ^ { + 0.0032 }$ & $4.8 _{ - 0.38 } ^ { + 0.64 }$ & $0.5 _{ - 0.34 } ^ { + 0.39 }$ & $1.93 _{ - 0.16 } ^ { + 0.19 }$ & 6.98515 &  & MINERVA (20) &  & SB 2 from MINERVA observations.  \\\\\n274599700 & TWOMASS 17011885+5131455 & $2002.1202 _{ - 0.0024 } ^ { + 0.0024 }$ & $32.9754 _{ - 0.005 } ^ { + 0.005 }$ & $0.0847 _{ - 0.0021 } ^ { + 0.0018 }$ & $13.25 _{ - 0.83 } ^ { + 0.83 }$ & $0.37 _{ - 0.24 } ^ { + 0.19 }$ & $8.2 _{ - 0.18 } ^ { + 0.21 }$ & 12.411 &  &  &  &  \\\\\n278990954 & TYC 8548-00717-1 & $1650.0191 _{ - 0.0086 } ^ { + 0.0105 }$ & $18.45 _{ - 8.66 } ^ { + 230.7 }(s)$ & $0.034 _{ - 0.0024 } ^ { + 0.0115 }$ & $9.65 _{ - 0.92 } ^ { + 3.13 }$ & $0.58 _{ - 0.4 } ^ { + 0.36 }$ & $10.62 _{ - 0.66 } ^ { + 2.46 }$ & 10.749 &  &  &  &  \\\\\n280865159* & TYC 9384-01533-1 & $1387.0749 _{ - 0.0045 } ^ { + 0.0044 }$ & $1045.0 _{ - 249.0 } ^ { + 536.0 }(s)$ & $0.0406 _{ - 0.0011 } ^ { + 0.0014 }$ & $4.75 _{ - 0.26 } ^ { + 0.28 }$ & $0.35 _{ - 0.24 } ^ { + 0.23 }$ & $19.08 _{ - 0.32 } ^ { + 0.36 }$ & 11.517 &  &  & Gemini &  \\\\\n284361752 & TYC 3924-01678-1 & $2032.093 _{ - 0.0078 } ^ { + 0.008 }$ & $140.6 _{ - 46.6 } ^ { + 159.1 }(s)$ & $0.0259 _{ - 0.0014 } ^ { + 0.0017 }$ & $3.62 _{ - 0.26 } ^ { + 0.31 }$ & $0.4 _{ - 0.27 } ^ { + 0.34 }$ & $8.98 _{ - 0.66 } ^ { + 0.86 }$ & 10.221 &  &  &  &  \\\\\n288240183 & TYC 4634-01225-1 & $1896.941 _{ - 0.0051 } ^ { + 0.0047 }$ & $119.0502 _{ - 0.0091 } ^ { + 0.0089 }$ & $0.02826 _{ - 0.00089 } ^ { + 0.00119 }$ & $4.28 _{ - 0.35 } ^ { + 0.36 }$ & $0.55 _{ - 0.37 } ^ { + 0.25 }$ & $17.49 _{ - 0.36 } ^ { + 0.6 }$ & 9.546 &  &  &  &  \\\\\n29169215 & TWOMASS 09011787+4727085 & $1872.5047 _{ - 0.0032 } ^ { + 0.0036 }$ & $14.89 _{ - 6.12 } ^ { + 24.84 }(s)$ & $0.0403 _{ - 0.0025 } ^ { + 0.0033 }$ & $3.28 _{ - 0.37 } ^ { + 0.45 }$ & $0.44 _{ - 0.3 } ^ { + 0.33 }$ & $3.56 _{ - 0.21 } ^ { + 0.32 }$ & 11.828 &  &  &  &  \\\\\n293649602 & TYC 8103-00266-1 & $1511.2109 _{ - 0.004 } ^ { + 0.0037 }$ & $12.85 _{ - 5.34 } ^ { + 42.21 }(s)$ & $0.04 _{ - 0.0024 } ^ { + 0.0039 }$ & $4.66 _{ - 0.36 } ^ { + 0.5 }$ & $0.5 _{ - 0.35 } ^ { + 0.34 }$ & $4.1 _{ - 0.31 } ^ { + 0.56 }$ & 10.925 &  &  &  &  \\\\\n296737508 & TYC 5472-01060-1 & $1538.0036 _{ - 0.0015 } ^ { + 0.0016 }$ & $18.27 _{ - 5.06 } ^ { + 17.45 }(s)$ & $0.0425 _{ - 0.0014 } ^ { + 0.0019 }$ & $5.33 _{ - 0.22 } ^ { + 0.27 }$ & $0.44 _{ - 0.3 } ^ { + 0.26 }$ & $5.13 _{ - 0.13 } ^ { + 0.15 }$ & 9.772 & Sinistro (1) & NRES (1);MINERVA (1) & Gemini &  \\\\\n298663873 & TYC 3913-01781-1 & $1830.76819 _{ - 0.00099 } ^ { + 0.00099 }$ & $479.9 _{ - 89.4 } ^ { + 109.4 }(s)$ & $0.06231 _{ - 0.00034 } ^ { + 0.00045 }$ & $11.07 _{ - 0.57 } ^ { + 0.57 }$ & $0.16 _{ - 0.11 } ^ { + 0.13 }$ & $23.99 _{ - 0.093 } ^ { + 0.1 }$ & 9.162 &  & NRES (2) & Gemini & Dalba et al. (in prep) \\\\\n303050301 & TYC 6979-01108-1 & $1366.1301 _{ - 0.0022 } ^ { + 0.0023 }$ & $281.0 _{ - 170.0 } ^ { + 264.0 }(s)$ & $0.0514 _{ - 0.0027 } ^ { + 0.0018 }$ & $4.85 _{ - 0.32 } ^ { + 0.32 }$ & $0.73 _{ - 0.48 } ^ { + 0.1 }$ & $7.91 _{ - 0.31 } ^ { + 0.36 }$ & 10.048 &  & NRES (1) & Gemini &  \\\\\n303317324 & TYC 6983-00438-1 & $1365.1845 _{ - 0.0023 } ^ { + 0.0028 }$ & $69.0 _{ - 25.5 } ^ { + 78.1 }(s)$ & $0.0365 _{ - 0.0013 } ^ { + 0.0016 }$ & $2.88 _{ - 0.3 } ^ { + 0.31 }$ & $0.39 _{ - 0.26 } ^ { + 0.32 }$ & $5.78 _{ - 0.18 } ^ { + 0.24 }$ & 10.799 &  &  &  &  \\\\\n\\hline\n\\end{tabular}}\n\\caption{\\emph{Note} -- Candidates that have become TOIs following the PHT discovery are marked with an asterisk (*). The `s' following the orbital period indicates that the candidates is a single transit event. The ground-based follow-up observations are summarized in columns 10-12, where the bracketed numbers correspond the number of epochs obtained with each instrument.  See Section~\\ref{sec:follow_up} for description of each instrument. The $\\dagger$ symbol indicates candidates that have been shown to be astrophysical false  positives based on the ground based follow-up observations.}\n\\label{tab:PHT-caniddates}\n\\end{table}\n\\end{landscape}\n\n\\begin{landscape}\n\\begin{table}\n\\addtocounter{table}{-1}\n\\resizebox{1.31\\textwidth}{!}{\n\\begin{tabular}{ccccccccccccc}\n\\textbf{TIC} & \\textbf{Other} & \\textbf{Epoch} & \\textbf{Period}  & \\textbf{$R_{pl}$\/$R_{\\odot}$} & \\textbf{$R_{pl}$} & \\textbf{Impact} & \\textbf{Duration} & \\textbf{$V_{mag}$} & \\textbf{Photometry} & \\textbf{Spectroscopy} & \\textbf{Speckle} & \\textbf{Comment} \\\\\n  & \\textbf{Name} & \\textbf{(\\textcolor{black}{BJD - 2457000})} & \\textbf{(days)}  & & $(R_{\\oplus})$  & \\textbf{Parameter} & \\textbf{(hours)} &   &   &   &   &   \\\\\n\\hline\n303586471$\\dagger$ & HIP 115828 & $1363.7692 _{ - 0.0033 } ^ { + 0.0027 }$ & $13.85 _{ - 4.19 } ^ { + 18.2 }(s)$ & $0.0214 _{ - 0.001 } ^ { + 0.0014 }$ & $2.52 _{ - 0.16 } ^ { + 0.2 }$ & $0.4 _{ - 0.27 } ^ { + 0.33 }$ & $4.23 _{ - 0.19 } ^ { + 0.16 }$ & 8.27 &  & MINERVA (11) &  & SB 2 from MINERVA observations.  \\\\\n304142124* & HIP 53719 & $1585.28023 _{ - 0.0008 } ^ { + 0.0008 }$ & $42.8 _{ - 10.0 } ^ { + 18.2 }(s)$ & $0.04311 _{ - 0.00093 } ^ { + 0.00153 }$ & $4.1 _{ - 0.23 } ^ { + 0.24 }$ & $0.33 _{ - 0.21 } ^ { + 0.21 }$ & $5.66 _{ - 0.067 } ^ { + 0.09 }$ & 8.62 &  & NRES (1);MINERVA (4) &  & Confirmed planet \\citep{diaz2020} \\\\\n304339227 & TYC 9290-01087-1 & $1673.3242 _{ - 0.009 } ^ { + 0.0128 }$ & $111.9 _{ - 72.2 } ^ { + 4844.1 }(s)$ & $0.0253 _{ - 0.0024 } ^ { + 0.0481 }$ & $3.27 _{ - 0.61 } ^ { + 5.72 }$ & $0.67 _{ - 0.47 } ^ { + 0.36 }$ & $7.44 _{ - 0.86 } ^ { + 2.84 }$ & 9.169 &  &  &  &  \\\\\n307958020 & TYC 4191-00309-1 & $1864.82 _{ - 0.014 } ^ { + 0.013 }$ & $169.0 _{ - 107.0 } ^ { + 10194.0 }(s)$ & $0.0223 _{ - 0.0022 } ^ { + 0.0543 }$ & $3.92 _{ - 0.52 } ^ { + 9.27 }$ & $0.71 _{ - 0.53 } ^ { + 0.33 }$ & $12.48 _{ - 1.1 } ^ { + 5.41 }$ & 9.017 &  &  &  &  \\\\\n308301091 & TYC 2081-01273-1 & $2030.3691 _{ - 0.0024 } ^ { + 0.0026 }$ & $29.24 _{ - 8.49 } ^ { + 22.46 }(s)$ & $0.0362 _{ - 0.0013 } ^ { + 0.0014 }$ & $5.41 _{ - 0.34 } ^ { + 0.35 }$ & $0.35 _{ - 0.25 } ^ { + 0.29 }$ & $6.57 _{ - 0.14 } ^ { + 0.19 }$ & 10.273 &  &  &  &  \\\\\n313006381 & HIP 45012 & $1705.687 _{ - 0.0081 } ^ { + 0.0045 }$ & $21.56 _{ - 8.9 } ^ { + 54.15 }(s)$ & $0.0261 _{ - 0.0017 } ^ { + 0.0027 }$ & $2.34 _{ - 0.2 } ^ { + 0.27 }$ & $0.45 _{ - 0.3 } ^ { + 0.38 }$ & $3.85 _{ - 0.51 } ^ { + 0.31 }$ & 9.39 &  &  &  &  \\\\\n323295479* & TYC 9506-01881-1 & $1622.9258 _{ - 0.00083 } ^ { + 0.00087 }$ & $117.8 _{ - 25.8 } ^ { + 30.9 }(s)$ & $0.0981 _{ - 0.0021 } ^ { + 0.0023 }$ & $11.35 _{ - 0.67 } ^ { + 0.66 }$ & $0.839 _{ - 0.024 } ^ { + 0.019 }$ & $6.7 _{ - 0.14 } ^ { + 0.15 }$ & 10.595 &  &  &  &  \\\\\n328933398.01* & TYC 4634-01435-1 & $1880.9878 _{ - 0.0039 } ^ { + 0.0042 }$ & $24.9335 _{ - 0.0046 } ^ { + 0.005 }$ & $0.0437 _{ - 0.0022 } ^ { + 0.0023 }$ & $4.62 _{ - 0.32 } ^ { + 0.33 }$ & $0.38 _{ - 0.25 } ^ { + 0.27 }$ & $5.02 _{ - 0.22 } ^ { + 0.27 }$ & 11.215 &  &  &  & Potential multi-planet system. \\\\\n328933398.02* & TYC 4634-01435-1 & $1848.6557 _{ - 0.0053 } ^ { + 0.0072 }$ & $50.5 _{ - 22.4 } ^ { + 77.1 }(s)$ & $0.0296 _{ - 0.0028 } ^ { + 0.0033 }$ & $3.14 _{ - 0.33 } ^ { + 0.39 }$ & $0.41 _{ - 0.28 } ^ { + 0.35 }$ & $5.99 _{ - 0.8 } ^ { + 0.77 }$ & 11.215 &  &  &  &  \\\\\n331644554 & TYC 3609-00469-1 & $1757.0354 _{ - 0.0031 } ^ { + 0.0033 }$ & $947.0 _{ - 215.0 } ^ { + 274.0 }(s)$ & $0.12 _{ - 0.025 } ^ { + 0.021 }$ & $21.84 _{ - 4.57 } ^ { + 3.86 }$ & $1.018 _{ - 0.036 } ^ { + 0.028 }$ & $10.93 _{ - 0.34 } ^ { + 0.35 }$ & 9.752 &  &  &  &  \\\\\n332657786 & TWOMASS 09595797-1609323 & $1536.7659 _{ - 0.0015 } ^ { + 0.0015 }$ & $63.76 _{ - 9.52 } ^ { + 11.13 }(s)$ & $0.14961 _{ - 0.00064 } ^ { + 0.00029 }$ & $3.83 _{ - 0.12 } ^ { + 0.12 }$ & $0.059 _{ - 0.041 } ^ { + 0.064 }$ & $3.333 _{ - 0.095 } ^ { + 0.096 }$ & 15.99 &  &  &  &  \\\\\n336075472 & TYC 3526-00332-1 & $2028.1762 _{ - 0.0043 } ^ { + 0.0037 }$ & $61.9 _{ - 24.0 } ^ { + 95.6 }(s)$ & $0.0402 _{ - 0.0022 } ^ { + 0.0033 }$ & $3.09 _{ - 0.34 } ^ { + 0.4 }$ & $0.43 _{ - 0.29 } ^ { + 0.32 }$ & $5.39 _{ - 0.23 } ^ { + 0.37 }$ & 11.842 &  &  &  &  \\\\\n349488688.01 & TYC 1529-00224-1 & $1994.283 _{ - 0.0038 } ^ { + 0.0033 }$ & $11.6254 _{ - 0.005 } ^ { + 0.0052 }$ & $0.02195 _{ - 0.00096 } ^ { + 0.00122 }$ & $3.44 _{ - 0.18 } ^ { + 0.21 }$ & $0.39 _{ - 0.27 } ^ { + 0.3 }$ & $5.58 _{ - 0.15 } ^ { + 0.18 }$ & 8.855 &  & NRES (2);SOPHIE (2) &  & Potential multi-planet system. \\\\\n349488688.02 & TYC 1529-00224-1 & $2002.77063 _{ - 0.00097 } ^ { + 0.00103 }$ & $15.35 _{ - 1.94 } ^ { + 4.15 }(s)$ & $0.03688 _{ - 0.00067 } ^ { + 0.00069 }$ & $5.78 _{ - 0.18 } ^ { + 0.18 }$ & $0.24 _{ - 0.16 } ^ { + 0.21 }$ & $6.291 _{ - 0.058 } ^ { + 0.074 }$ & 8.855 &  & NRES (2);SOPHIE (2) &  &  \\\\\n356700488* & TYC 4420-01295-1 & $1756.638 _{ - 0.013 } ^ { + 0.011 }$ & $184.5 _{ - 64.7 } ^ { + 333.1 }(s)$ & $0.0173 _{ - 0.0011 } ^ { + 0.0015 }$ & $2.92 _{ - 0.2 } ^ { + 0.28 }$ & $0.44 _{ - 0.3 } ^ { + 0.34 }$ & $11.76 _{ - 0.65 } ^ { + 1.03 }$ & 8.413 &  &  &  &  \\\\\n356710041* & TYC 1993-00419-1 & $1932.2939 _{ - 0.0019 } ^ { + 0.0019 }$ & $29.6 _{ - 14.0 } ^ { + 19.0 }(s)$ & $0.0496 _{ - 0.0021 } ^ { + 0.0011 }$ & $14.82 _{ - 0.85 } ^ { + 0.84 }$ & $0.66 _{ - 0.42 } ^ { + 0.11 }$ & $12.76 _{ - 0.24 } ^ { + 0.24 }$ & 9.646 &  &  &  &  \\\\\n369532319 & TYC 2743-01716-1 & $1755.8158 _{ - 0.006 } ^ { + 0.0051 }$ & $35.4 _{ - 12.0 } ^ { + 51.6 }(s)$ & $0.0316 _{ - 0.0023 } ^ { + 0.0028 }$ & $3.43 _{ - 0.3 } ^ { + 0.37 }$ & $0.41 _{ - 0.29 } ^ { + 0.34 }$ & $5.5 _{ - 0.32 } ^ { + 0.32 }$ & 10.594 &  &  & Gemini &  \\\\\n369779127 & TYC 9510-00090-1 & $1643.9403 _{ - 0.0046 } ^ { + 0.0058 }$ & $9.93 _{ - 3.38 } ^ { + 19.74 }(s)$ & $0.0288 _{ - 0.0015 } ^ { + 0.0033 }$ & $4.89 _{ - 0.31 } ^ { + 0.56 }$ & $0.46 _{ - 0.31 } ^ { + 0.33 }$ & $5.64 _{ - 0.38 } ^ { + 0.33 }$ & 9.279 &  &  &  &  \\\\\n384159646* & TYC 9454-00957-1 & $1630.39405 _{ - 0.00079 } ^ { + 0.00079 }$ & $11.68 _{ - 2.75 } ^ { + 4.21 }(s)$ & $0.0658 _{ - 0.0012 } ^ { + 0.0011 }$ & $9.87 _{ - 0.45 } ^ { + 0.44 }$ & $0.27 _{ - 0.18 } ^ { + 0.21 }$ & $5.152 _{ - 0.069 } ^ { + 0.087 }$ & 10.158 & SBIG (1) & NRES (8);MINERVA (6) & Gemini &  \\\\\n385557214 & TYC 1807-00046-1 & $1791.58399 _{ - 0.00068 } ^ { + 0.0007 }$ & $5.62451 _{ - 0.0004 } ^ { + 0.00043 }$ & $0.096 _{ - 0.019 } ^ { + 0.032 }$ & $8.32 _{ - 2.06 } ^ { + 2.77 }$ & $0.95 _{ - 0.075 } ^ { + 0.053 }$ & $1.221 _{ - 0.094 } ^ { + 0.058 }$ & 10.856 &  &  &  &  \\\\\n388134787 & TYC 4260-00427-1 & $1811.034 _{ - 0.015 } ^ { + 0.017 }$ & $246.0 _{ - 127.0 } ^ { + 6209.0 }(s)$ & $0.0265 _{ - 0.0024 } ^ { + 0.023 }$ & $2.57 _{ - 0.28 } ^ { + 2.19 }$ & $0.55 _{ - 0.39 } ^ { + 0.44 }$ & $8.85 _{ - 1.13 } ^ { + 1.84 }$ & 10.95 &  & NRES (1) & Gemini &  \\\\\n404518509 & HIP 16038 & $1431.2696 _{ - 0.0037 } ^ { + 0.0035 }$ & $26.83 _{ - 9.46 } ^ { + 56.14 }(s)$ & $0.0259 _{ - 0.0013 } ^ { + 0.0022 }$ & $2.94 _{ - 0.21 } ^ { + 0.29 }$ & $0.47 _{ - 0.31 } ^ { + 0.34 }$ & $5.02 _{ - 0.23 } ^ { + 0.28 }$ & 9.17 &  &  &  &  \\\\\n408636441* & TYC 4266-00736-1 & $1745.4668 _{ - 0.0016 } ^ { + 0.0015 }$ & $37.695 _{ - 0.0034 } ^ { + 0.0033 }$ & $0.0485 _{ - 0.0019 } ^ { + 0.0023 }$ & $3.32 _{ - 0.16 } ^ { + 0.19 }$ & $0.39 _{ - 0.27 } ^ { + 0.29 }$ & $3.63 _{ - 0.1 } ^ { + 0.14 }$ & 11.93 & SBIG (1) &  & Gemini & Half of the period likely. \\\\\n418255064 & TWOMASS 13063680-8037015 & $1629.3304 _{ - 0.0018 } ^ { + 0.0018 }$ & $25.37 _{ - 7.06 } ^ { + 15.41 }(s)$ & $0.0732 _{ - 0.0029 } ^ { + 0.0031 }$ & $5.57 _{ - 0.36 } ^ { + 0.38 }$ & $0.37 _{ - 0.25 } ^ { + 0.25 }$ & $3.83 _{ - 0.13 } ^ { + 0.14 }$ & 12.478 & SBIG (1) &  & Gemini &  \\\\\n420645189$\\dagger$ & TYC 4508-00478-1 & $1837.4767 _{ - 0.0018 } ^ { + 0.0017 }$ & $250.2 _{ - 66.6 } ^ { + 99.4 }(s)$ & $0.0784 _{ - 0.0033 } ^ { + 0.0046 }$ & $8.82 _{ - 0.55 } ^ { + 0.7 }$ & $0.892 _{ - 0.026 } ^ { + 0.028 }$ & $6.95 _{ - 0.27 } ^ { + 0.3 }$ & 10.595 &  & MINERVA (1) &  & SB 2 from MINERVA observations.  \\\\\n422914082 & TYC 0046-00133-1 & $1431.5538 _{ - 0.0014 } ^ { + 0.0017 }$ & $12.91 _{ - 3.91 } ^ { + 8.97 }(s)$ & $0.0418 _{ - 0.0015 } ^ { + 0.0016 }$ & $3.96 _{ - 0.32 } ^ { + 0.35 }$ & $0.36 _{ - 0.25 } ^ { + 0.28 }$ & $4.07 _{ - 0.09 } ^ { + 0.126 }$ & 11.026 & Sinistro (1) & NRES (1) &  &  \\\\\n427344083 & TWOMASS 22563609+7040518 & $1961.8967 _{ - 0.0031 } ^ { + 0.0036 }$ & $7.77 _{ - 5.6 } ^ { + 9.65 }(s)$ & $0.107 _{ - 0.016 } ^ { + 0.025 }$ & $12.27 _{ - 1.87 } ^ { + 2.9 }$ & $0.834 _{ - 0.484 } ^ { + 0.094 }$ & $2.88 _{ - 0.3 } ^ { + 0.42 }$ & 13.404 &  &  &  &  \\\\\n436873727 & HIP 13224 & $1803.83679 _{ - 0.00058 } ^ { + 0.00056 }$ & $19.26 _{ - 5.95 } ^ { + 6.73 }(s)$ & $0.05246 _{ - 0.00061 } ^ { + 0.00059 }$ & $10.02 _{ - 0.43 } ^ { + 0.41 }$ & $0.767 _{ - 0.057 } ^ { + 0.038 }$ & $5.462 _{ - 0.081 } ^ { + 0.074 }$ & 7.51 &  &  &  &  \\\\ \n441642457* & TYC 3858-00452-1 & $1745.5102 _{ - 0.0108 } ^ { + 0.0097 }$ & $79.8072 _{ - 0.0071 } ^ { + 0.0076 }$ & $0.0281 _{ - 0.0024 } ^ { + 0.0033 }$ & $3.55 _{ - 0.34 } ^ { + 0.46 }$ & $0.934 _{ - 0.023 } ^ { + 0.026 }$ & $6.9 _{ - 0.39 } ^ { + 0.6 }$ & 9.996 &  &  &  &  \\\\\n441765914* & TWOMASS 17253007+7552562 & $1769.6154 _{ - 0.0058 } ^ { + 0.0093 }$ & $161.6 _{ - 58.2 } ^ { + 1460.1 }(s)$ & $0.0411 _{ - 0.0024 } ^ { + 0.0119 }$ & $3.6 _{ - 0.3 } ^ { + 1.01 }$ & $0.45 _{ - 0.32 } ^ { + 0.48 }$ & $7.44 _{ - 0.36 } ^ { + 1.08 }$ & 11.638 &  &  &  &  \\\\\n452920657 & TWOMASS 00332018+5906355 & $1810.5765 _{ - 0.0031 } ^ { + 0.003 }$ & $53.2 _{ - 29.0 } ^ { + 34.3 }(s)$ & $0.135 _{ - 0.016 } ^ { + 0.012 }$ & $9.71 _{ - 1.16 } ^ { + 0.9 }$ & $0.73 _{ - 0.48 } ^ { + 0.11 }$ & $4.6 _{ - 0.26 } ^ { + 0.29 }$ & 14.167 & SBIG (1) &  &  &  \\\\\n455737331 & TYC 2779-00785-1 & $1780.7084 _{ - 0.008 } ^ { + 0.0073 }$ & $50.4 _{ - 17.6 } ^ { + 75.0 }(s)$ & $0.0257 _{ - 0.0016 } ^ { + 0.002 }$ & $3.05 _{ - 0.24 } ^ { + 0.29 }$ & $0.43 _{ - 0.29 } ^ { + 0.33 }$ & $6.6 _{ - 0.43 } ^ { + 0.5 }$ & 10.189 & SBIG (1) &  & Gemini &  \\\\\n456909420 & TYC 1208-01094-1 & $1779.4109 _{ - 0.0026 } ^ { + 0.0022 }$ & $5.78 _{ - 5.29 } ^ { + 5.95 }(s)$ & $0.078 _{ - 0.031 } ^ { + 0.045 }$ & $9.15 _{ - 3.61 } ^ { + 5.27 }$ & $0.973 _{ - 0.495 } ^ { + 0.063 }$ & $1.73 _{ - 0.27 } ^ { + 0.28 }$ & 10.941 &  &  &  &  \\\\\n458451774 & TWOMASS 12551793+4431260 & $1917.1875 _{ - 0.0019 } ^ { + 0.0019 }$ & $12.39 _{ - 6.34 } ^ { + 83.97 }(s)$ & $0.0752 _{ - 0.0054 } ^ { + 0.0211 }$ & $3.33 _{ - 0.26 } ^ { + 0.92 }$ & $0.61 _{ - 0.43 } ^ { + 0.32 }$ & $2.08 _{ - 0.19 } ^ { + 0.59 }$ & 13.713 &  &  &  &  \\\\\n48018596 & TYC 3548-00800-1 & $1713.4514 _{ - 0.0063 } ^ { + 0.0046 }$ & $100.1145 _{ - 0.0018 } ^ { + 0.0021 }$ & $0.049 _{ - 0.0081 } ^ { + 0.018 }$ & $7.88 _{ - 1.33 } ^ { + 2.9 }$ & $0.984 _{ - 0.028 } ^ { + 0.027 }$ & $2.83 _{ - 0.26 } ^ { + 0.29 }$ & 9.595 &  & NRES (1) & Gemini &  \\\\\n53309262 & TWOMASS 07475406+5741549 & $1863.1133 _{ - 0.0064 } ^ { + 0.0061 }$ & $294.8 _{ - 96.0 } ^ { + 327.0 }(s)$ & $0.1239 _{ - 0.0075 } ^ { + 0.0098 }$ & $5.38 _{ - 0.36 } ^ { + 0.46 }$ & $0.46 _{ - 0.31 } ^ { + 0.28 }$ & $6.74 _{ - 0.45 } ^ { + 0.62 }$ & 15.51 &  &  &  &  \\\\\n53843023 & TYC 6956-00758-1 & $1328.0335 _{ - 0.0054 } ^ { + 0.0057 }$ & $202.0 _{ - 189.0 } ^ { + 272.0 }(s)$ & $0.058 _{ - 0.02 } ^ { + 0.056 }$ & $5.14 _{ - 1.77 } ^ { + 4.99 }$ & $0.962 _{ - 0.597 } ^ { + 0.083 }$ & $4.25 _{ - 0.72 } ^ { + 0.66 }$ & 11.571 &  &  &  &  \\\\\n55525572* & TYC 8876-01059-1 & $1454.6713 _{ - 0.0066 } ^ { + 0.0065 }$ & $83.8951 _{ - 0.004 } ^ { + 0.004 }$ & $0.0343 _{ - 0.001 } ^ { + 0.0021 }$ & $7.31 _{ - 0.46 } ^ { + 0.56 }$ & $0.43 _{ - 0.29 } ^ { + 0.31 }$ & $13.54 _{ - 0.3 } ^ { + 0.51 }$ & 10.358 &  & CHIRON (5) & Gemini & Confirmed planet \\citep{2020eisner} \\\\\n63698669* & TYC 6993-00729-1 & $1364.6226 _{ - 0.0074 } ^ { + 0.0067 }$ & $73.6 _{ - 26.8 } ^ { + 133.6 }(s)$ & $0.0248 _{ - 0.0019 } ^ { + 0.0023 }$ & $2.15 _{ - 0.2 } ^ { + 0.25 }$ & $0.42 _{ - 0.29 } ^ { + 0.35 }$ & $5.63 _{ - 0.32 } ^ { + 0.57 }$ & 10.701 & SBIG (1) &  &  &  \\\\\n70887357* & TYC 5883-01412-1 & $1454.3341 _{ - 0.0016 } ^ { + 0.0015 }$ & $56.1 _{ - 15.3 } ^ { + 18.8 }(s)$ & $0.0605 _{ - 0.0027 } ^ { + 0.0027 }$ & $12.84 _{ - 0.86 } ^ { + 0.9 }$ & $0.917 _{ - 0.028 } ^ { + 0.016 }$ & $7.29 _{ - 0.18 } ^ { + 0.19 }$ & 9.293 &  &  &  &  \\\\\n7422496$\\dagger$ & HIP 25359 & $1470.3625 _{ - 0.0031 } ^ { + 0.0023 }$ & $61.4 _{ - 16.7 } ^ { + 49.0 }(s)$ & $0.0255 _{ - 0.001 } ^ { + 0.0011 }$ & $2.44 _{ - 0.15 } ^ { + 0.16 }$ & $0.37 _{ - 0.25 } ^ { + 0.29 }$ & $5.89 _{ - 0.15 } ^ { + 0.15 }$ & 9.36 &  & MINERVA (4) &  & SB 2 from MINERVA observations.  \\\\\n82452140 & TYC 3076-00921-1 & $1964.292 _{ - 0.011 } ^ { + 0.011 }$ & $21.1338 _{ - 0.0052 } ^ { + 0.0066 }$ & $0.0266 _{ - 0.0019 } ^ { + 0.0027 }$ & $2.95 _{ - 0.25 } ^ { + 0.34 }$ & $0.42 _{ - 0.29 } ^ { + 0.36 }$ & $5.87 _{ - 0.62 } ^ { + 0.94 }$ & 10.616 &  &  &  &  \\\\\n88840705 & TYC 3091-00808-1 & $2026.6489 _{ - 0.001 } ^ { + 0.001 }$ & $260.6 _{ - 87.6 } ^ { + 142.2 }(s)$ & $0.109 _{ - 0.023 } ^ { + 0.027 }$ & $9.98 _{ - 2.28 } ^ { + 2.75 }$ & $1.001 _{ - 0.042 } ^ { + 0.037 }$ & $4.72 _{ - 0.13 } ^ { + 0.15 }$ & 9.443 &  &  &  &  \\\\\n91987762* & HIP 47288 & $1894.25381 _{ - 0.00051 } ^ { + 0.00047 }$ & $10.51 _{ - 3.48 } ^ { + 3.67 }(s)$ & $0.05459 _{ - 0.00106 } ^ { + 0.00097 }$ & $9.56 _{ - 0.56 } ^ { + 0.52 }$ & $0.771 _{ - 0.062 } ^ { + 0.033 }$ & $4.342 _{ - 0.073 } ^ { + 0.063 }$ & 7.87 &  & NRES (4) & Gemini &  \\\\\n95768667 & TYC 1434-00331-1 & $1918.3318 _{ - 0.0093 } ^ { + 0.0079 }$ & $26.9 _{ - 12.4 } ^ { + 72.3 }(s)$ & $0.0282 _{ - 0.0022 } ^ { + 0.0031 }$ & $3.54 _{ - 0.32 } ^ { + 0.43 }$ & $0.48 _{ - 0.33 } ^ { + 0.35 }$ & $5.4 _{ - 0.64 } ^ { + 0.76 }$ & 10.318 &  &  &  &  \\\\\n\\hline\n\\end{tabular}}\n\\caption{\\textbf{Properties of PHT candidates (continued)}}\n\\label{tab:PHT-caniddates2}\n\\end{table}\n\\end{landscape}\n\n\n\\section{Conclusion}\n\\label{sec:condlusion}\n\nWe present the results from the analysis of the first 26  \\emph{TESS}\\ sectors. The outlined citizen science approach engages over 22 thousand registered citizen scientists who completed 12,617,038 classifications from December 2018 through August 2020 for the sectors observed during the first two years of the \\emph{TESS}\\ mission. We applied a systematic search for planetary candidates using visual vetting by multiple volunteers to identify \\emph{TESS}\\ targets that are most likely to host a planet. Between 8 and 15 volunteers have inspected each \\emph{TESS}\\ light curve and marked times of transit-like events using the PHT online interface. For each light curve, the markings from all the volunteers who saw that target were combined using an unsupervised machine learning method, known as DBSCAN, in order to identify likely transit-like events. Each of these identified events was given a transit score based on the number of volunteers who identified a given event and on the user weighting of each of those volunteers. Individual user weights were calculated based on the user's ability to identify simulated transit events, injected into real \\emph{TESS}\\ light curves, that are displayed on the PHT site alongside of the real data. The transit scores were then used to generate a ranked list of candidates that range from most likely to least likely to host a planet candidate. The top 500 highest ranked candidates were further vetted by the PHT science team. This stage of vetting primarily made use of the open source {\\sc latte} \\citep{LATTE2020} tool which generates a number of standard diagnostic plots that help identify promising candidates and weed out false positive signals. \n\nOn average we found around three high priority candidates per sector which were followed up using ground based telescopes, where possible. To date, PHT has statistically confirmed one planet, TOI-813 \\citep{2020eisner}: a Saturn-sized planet on an 84 day orbit around a subgiant host star. Other PHT identified planets listed in this paper are being followed up by other teams of astronomers, such as TOI-1899 (TIC 172370679) which was recently confirmed to be a warm Jupiter transiting an M-dwarf \\citep{canas2020}. The remaining candidates outlined in this paper require further follow-up observations to confirm their planetary nature.\n\nThe sensitivity of our transit search effort was assessed using synthetic data, as well as the known TOI and TCE candidates flagged by the SPOC pipeline. For simulated planets (where simulated signals are injected into real \\emph{TESS}\\ light curves) we have shown that the recovery efficiency of human vetting starts to decrease for transit-signals that have a SNR less than 7.5. The detection efficiency was further evaluated by the fractional recovery of the TOI and TCEs. We have shown that PHT is over 85 \\% complete in the recovery of planets that have a radius greater than 4 $R_{\\oplus}$, 51 \\% complete for radii between 3 and 4 $R_{\\oplus}$ and 49 \\% complete for radii between 2 and 3 $R_{\\oplus}$. Furthermore, we have shown that human vetting is not sensitive to the number of transits present in the light curve, meaning that they are equally likely to identify candidates on longer orbital periods as they are those with shorter orbital periods for periods greater than $\\sim$ 1 day. Planets with periods shorter than around 1 day exhibit over 20 transits within one \\emph{TESS}\\ sectors resulting in a decrease in identification by the volunteers. This is due to many volunteers only marking a random subset of these events, resulting in a lack of consensus on any given transit event and thus decreasing the overall transit score of these light curves. \n\nIn addition to searching for signals due to transiting exoplanets, PHT provides a platform that can be used to identify other stellar phenomena that may otherwise be difficult to identify with automated pipelines. Such phenomena, including eclipsing binaries, multiple stellar systems, dwarf novae, and stellar flares are often mentioned on the PHT discussion forums where volunteers can use searchable hashtags and comments to bring these systems to the attention of other citizen scientists as well as the PHT science team. All of the eclipsing binaries identified on the site, for example, are being used and vetted by the \\emph{TESS}\\ Eclipsing Binary Working Group (Prsa et al. in prep). Furthermore, we have investigated the nature of all of the targets that were identified as possible multiple stellar systems, as summarised in Table~\\ref{tab:PHT-multis}.\n\nOverall we have shown that large scale visual vetting can complement the findings \\textcolor{black}{from the major \\emph{TESS}\\ pipeline} by identifying longer period planets that may only exhibit a single transit event in their light curve, as well as in finding signals that are aperiodic or embedded in a strong varying stellar signal. The identification of planets around stars with variable signals allow us to potentially characterise the host-star (e.g., with asteroseismology or spot modulation). Additionally, the longer period planets are integral to our understanding of how planet systems form and evolve, as they allow us to investigate the evolution of planets that are farther away from their host star and therefore less dependent on stellar radiation. \\textcolor{black}{While automated pipelines specifically designed to identify single transit events in the \\emph{TESS}\\ data exist \\citep[e.g., ][]{Gill2020}, neither their methodology nor the full list of their findings are yet publicly available and thus we are unable to compare results.} \n\nThe planets that PHT finds have longer periods ($\\gtrsim$ 27 d) than those found in \\emph{TESS}\\ data using automated pipelines, and are more typical of the Kepler sample (25\\% of Kepler confirmed planets have periods greater than 27 days\\footnote{\\url{https:\/\/exoplanetarchive.ipac.caltech.edu\/}}). However, the Kepler planets are considerably fainter, and thus less amenable to ground-based follow-up or atmospheric characterisation from space (CHEOPS and JWST). Thus PHT helps to bridge the parameter spaces covered by these two missions, by identifying longer period planet candidates around bright, nearby stars, for which we can ultimately obtain precise planetary mass estimates. Although statistical characterisation of exo-planetary systems is no doubt important, precise mass measurements are key to developing our understanding of exoplanets and the physics which dictate their evolution. In particular, identification of this PHT sample provides follow-up targets to investigate the dependence of photo-evaporation on the mass of planets as well as on the planet radius, and will help our understanding of the photo-evaporation valley at longer orbital periods \\citep{Owen2013}. \n\nPHT will continue to operate throughout the \\emph{TESS}\\ extended mission, hopefully allowing us to identify even longer period planets as well as help verify some of the existing candidates with additional transits.  \n\n\n\n\\begin{table*}\n\\resizebox{0.95\\textwidth}{!}{\n\\begin{tabular}{cccccccccc}\n\\textbf{TIC}  & \\textbf{Period (days)}        & \\textbf{Epoch (\\textcolor{black}{BJD - 2457000})}      & \\textbf{Depth (ppm)}  & \\textbf{Comment}  \\\\\n\\hline\n13968858      & $3.4850 \\pm 0.001$            & $ 1684.780   \\pm 0.005$            & 410000    & Candidate multiple system  \\\\\n              & $1.4380 \\pm 0.001$            & $ 1684.335   \\pm 0.005$            & 50000     &   \\\\\n35655828      & $ 8.073 \\pm 0.01$             & $ 1550.94    \\pm 0.01 $            & 23000     & Confirmed blend \\\\\n              & $  1.220 \\pm 0.001 $          & $ 1545.540   \\pm 0.005 $           & 2800      &  \\\\\n63291675      & $ 8.099  \\pm 0.003 $          & $ 1685.1     \\pm 0.01  $           & 60000     & Confirmed blend \\\\\n              & $ 1.4635 \\pm 0.0005 $         & $ 1683.8     \\pm 0.1 $             & 7000      &  \\\\\n63459761      & $4.3630 \\pm 0.003 $           & $ 1714.350   \\pm 0.005 $           & 160000    & Candidate multiple system   \\\\\n              & $4.235 \\pm  $ 0.005           & $ 1715.130   \\pm 0.03$             & 35000     &   \\\\\n104909909     & $1.3060 \\pm 0.0001$           & $ 1684.470   \\pm 0.005$            & 32000     & Candidate multiple system   \\\\\n              & $2.5750 \\pm 0.003$            & $ 1684.400   \\pm 0.005$            & 65000     &   \\\\\n115980439     & $ 4.615 \\pm 0.002  $          & $ 1818.05    \\pm 0.01  $           & 95000     & Confirmed blend \\\\\n              & $ 0.742 \\pm 0.005  $          & $ 1816.23    \\pm 0.02  $           & 2000      &   \\\\\n120362128     & $ 3.286 \\pm 0.002  $          & $ 1684.425   \\pm 0.01  $           & 33000     &  Candidate multiple system  \\\\\n              & $ -                 $         & $ 1701.275   \\pm 0.02  $           & 12000     &    \\\\\n              & $ -                 $         & $ 1702.09    \\pm 0.02  $           & 36000     &    \\\\\n121945407     & $  0.9056768 \\pm 0.00000002$   & $-1948.76377 \\pm  0.0000001$      & 2500          & Confirmed multiple system $^{(\\mathrm{a})}$  \\\\\n              & $ 45.4711 \\pm  0.00002$        & $-1500.0038  \\pm  0.0004 $       & 7500          &  \\\\\n122275115     & $ - $                         & $ 1821.779   \\pm 0.01  $           & 155000    &  Candidate multiple system  \\\\\n              & $ - $                         & $ 1830.628   \\pm 0.01  $           & 63000     &  \\\\\n              & $ - $                         & $ 1838.505   \\pm 0.01  $           & 123000    &  \\\\\n229804573     & $1.4641 \\pm 0.0005$           & $ 1326.135   \\pm 0.005$            & 180000    & Candidate multiple system  \\\\\n              & $0.5283 \\pm 0.0001$           & $ 1378.114   \\pm 0.005$            & 9000      &   \\\\\n252403752     & $ - $                         & $ 1817.73    \\pm 0.01 $            & 2800      & Candidate multiple system \\\\\n              & $ - $                         & $ 1829.76    \\pm 0.01 $            & 23000     & \\\\\n              & $ - $                         & $ 1833.63    \\pm 0.01 $            & 5500      & \\\\\n258837989     & $0.8870 \\pm 0.001$            & $ 1599.350   \\pm 0.005$            & 64000     & Candidate multiple system  \\\\\n              & $3.0730 \\pm 0.001$            & $ 1598.430   \\pm 0.005$            & 25000     &   \\\\\n266958963     & $1.5753 \\pm 0.0002$           & $ 1816.425   \\pm 0.001$            & 265000    & Candidate multiple system  \\\\\n              & $2.3685 \\pm 0.0001$           & $ 1817.790   \\pm 0.001$            & 75000     &   \\\\\n278956474     & $5.488068 \\pm 0.000016  $     & $ 1355.400   \\pm 0.005$            & 93900     & Confirmed multiple system $^{(\\mathrm{b})}$ \\\\\n              & $5.674256 \\pm  -0.000030$     & $ 1330.690   \\pm 0.005$            & 30000     &   \\\\\n284925600     & $ 1.24571 \\pm 0.00001 $       & $ 1765.248   \\pm 0.005 $           & 490000    & Confirmed blend \\\\\n              & $  0.31828 \\pm 0.00001  $     & $ 1764.75    \\pm 0.005 $           & 35000     &  \\\\\n293954660     & $2.814 \\pm  0.001 $           & $ 1739.177   \\pm 0.03  $           & 272000    & Confirmed blend \\\\\n              & $4.904 \\pm 0.03 $             & $ 1739.73    \\pm 0.01  $           & 9500      &  \\\\\n312353805     & $4.951 \\pm  0.003 $           & $ 1817.73    \\pm 0.01  $           & 66000     & Confirmed blend \\\\\n              & $12.89 \\pm   0.01  $          & $ 1822.28    \\pm 0.01$             & 19000     &  \\\\\n318210930     & $ 1.3055432 \\pm 0.000000033$  & $ -653.21602   \\pm 0.0000013$      & 570000    & Confirmed multiple system $^{(\\mathrm{c})}$ \\\\\n              & $ 0.22771622 \\pm 0.0000000035$& $ -732.071119 \\pm 0.00000026 $     & 220000    &  \\\\\n336434532     & $ 3.888 \\pm 0.002  $          & $ 1713.66    \\pm 0.01  $           & 22900     & Confirmed blend \\\\\n              & $  0.949 \\pm 0.003 $          & $ 1712.81    \\pm 0.01  $           & 2900      &  \\\\\n350622185     & $1.1686 \\pm 0.0001$           & $ 1326.140   \\pm 0.005$            & 200000    & Candidate multiple system  \\\\\n              & $5.2410 \\pm 0.0005$           & $ 1326.885   \\pm 0.05$             & 4000      &  \\\\\n375422201     & $9.9649 \\pm 0.001$            & $ 1711.937   \\pm 0.005$            & 245000    & Candidate multiple system  \\\\\n              & $4.0750 \\pm 0.001$            & $ 1713.210   \\pm 0.01 $            & 39000     &  \\\\\n376606423     & $ 0.8547 \\pm 0.0002  $        & $ 1900.766   \\pm 0.005 $           & 9700      & Candidate multiple system  \\\\\n              & $  -  $                       & $ 1908.085   \\pm 0.01  $           & 33000     &  \\\\\n394177355     & $ 94.22454  \\pm 0.00040 $     & $  -  $                            &  -        & Confirmed multiple system $^{(\\mathrm{d})}$ \\\\\n              & $ 8.6530941 \\pm 0.0000016$    & $-2038.99492 \\pm 0.00017 $         &  140000   &  \\\\\n              & $ 1.5222468 \\pm 0.0000025$    & $ -2039.1201 \\pm 0.0014  $         &  -        &  \\\\\n              & $ 1.43420486 \\pm 0.00000012 $ & $-2039.23941 \\pm 0.00007 $         &  -        &  \\\\\n424508303     & $ 2.0832649 \\pm 0.0000029  $ & $-3144.8661  \\pm 0.0034  $          &  430000   &  Confirmed multiple system $^{(\\mathrm{e})}$ \\\\\n              & $ 1.4200401  \\pm 0.0000042 $  & $-3142.5639  \\pm 0.0054  $         &  250000         &  \\\\\n441794509     & $ 4.6687 \\pm  0.0002       $  & $ 1958.895   \\pm 0.005   $         & 34000     &  Candidate multiple system  \\\\\n              & $ 14.785 \\pm  0.002       $   & $ 1960.845   \\pm 0.005   $         & 17000     &  \\\\\n470710327     & $ 9.9733 \\pm 0.0001 $         & $ 1766.27    \\pm 0.005  $          & 51000     &  Confirmed multiple system $^{(\\mathrm{f})}$ \\\\\n              & $ 1.104686 \\pm 0.00001 $      & $ 1785.53266 \\pm 0.000005$         & 42000     &  \\\\\n\\hline\n\\end{tabular}\n}\n\\caption{\nNote -- $^{(\\mathrm{a})}$ KOI-6139, \\citet{Borkovits2013};  \n$^{(\\mathrm{b})}$ \\citet{2020Rowden}\n$^{(\\mathrm{c})}$ \\citet{Koo2014}; \n$^{(\\mathrm{d})}$ KOI-3156, \\citet{2017Helminiak};\n$^{(\\mathrm{e})}$ V994 Her; \\citet{Zasche2016}; \n$^{(\\mathrm{f})}$ Eisner et al. {\\it in prep.}\n}\n\n\\label{tab:PHT-multis}\n\n\\end{table*}\n\n\\section*{Data Availability}\n\nAll of the \\emph{TESS}\\ data used within this article are hosted and made publicly available by the Mikulski Archive for Space Telescopes (MAST, \\url{http:\/\/archive.stsci.edu\/tess\/}). Similarly, the Planet Hunters TESS classifications made by the citizen scientists can be found on the Planet Hunters Analysis Database (PHAD, \\url{https:\/\/mast.stsci.edu\/phad\/}), which is also hosted by MAST. All planet candidates and their properties presented in this article have been uploaded to the Exoplanet Follow-up Observing Program for TESS (ExoFOP-TESS, \\url{ https:\/\/exofop.ipac.caltech.edu\/tess\/index.php}) website as community TOIs (cTOIs), under their corresponding TIC IDs. The ground-based follow-up observations of individual targets will be shared on reasonable request to the corresponding author.\n\nThe models of individual transit events and the data validation reports used for the vetting of the targets were both generated using publicly available open software codes, \\texttt{pyaneti}\\ and {\\sc latte}.\n\n\\section*{Acknowledgements}  \n\nThis project works under the in \\textit{populum veritas est} philosophy, and for that reason we would like to thank all of the citizen scientists who have taken part in the Planet Hunters TESS project and enable us to find many interesting astrophysical systems. \n\nSome of the observations in the paper made use of the High-Resolution Imaging instruments `Alopeke and Zorro. `Alopeke and Zorro were funded by the NASA Exoplanet Exploration Program and built at the NASA Ames Research Center by Steve B. Howell, Nic Scott, Elliott P. Horch, and Emmett Quigley. `Alopeke and Zorro were mounted on the Gemini North and South telescope of the international Gemini Observatory, a program of NSF's NOIRLab, which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation on behalf of the Gemini partnership: the National Science Foundation (United States), National Research Council (Canada), Agencia Nacional de Investigaci\\'{o}n y Desarrollo (Chile), Ministerio de Ciencia, Tecnolog\\'{i}a e Innovaci\\'{o}n (Argentina), Minist\\'{e}rio da Ci\\^{e}ncia, Tecnologia, Inova\\c{c}\\~{o}es e Comunica\\c{c}\\~{o}es (Brazil), and Korea Astronomy and Space Science Institute (Republic of Korea). The  authors also acknowledge the very significant cultural role and sacred nature of  Maunakea. We are most fortunate to  have  the opportunity to conduct observations from this mountain.\n\nThis project has also received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement N$^\\circ$730890. This material reflects only the authors views and the Commission is not liable for any use that may be made of the information contained therein. This work makes use of observations from the Las Cumbres Observatory global telescope network, including the NRES spectrograph and the SBIG and Sinistro photometric instruments. \n\nFurthermore, NLE thanks the LSSTC Data Science Fellowship Program, which is funded by LSSTC, NSF Cybertraining Grant N$^\\circ$1829740, the Brinson Foundation, and the Moore Foundation; her participation in the program has benefited this work. Finally, CJ acknowledges funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement N$^\\circ$670519: MAMSIE), and from the Research Foundation Flanders (FWO) under grant agreement G0A2917N (BlackGEM). \n\nThis research made use of Astropy, a community-developed core Python package for Astronomy \\citep{astropy2013}, matplotlib \\citep{matplotlib}, pandas \\citep{pandas}, NumPy \\citep{numpy}, astroquery \\citep{ginsburg2019astroquery} and sklearn \\citep{pedregosa2011scikit}.  \n\n\n\n\n\\bibliographystyle{mnras}\n\n\\section{Planet candidate descriptions}\n\\label{appendixA}\n\nA short outline all of the planet candidates, and any conclusions drawn from follow-up observations (where available). A more in depth description of the ground-based data will be presented in a follow-up paper. Unless stated otherwise, these candidates are not TOIs at the time of writing. Candidates for which we have no additional information to complement the results presented in Table~\\ref{tab:PHT-caniddates} are not discussed further here.\n\n\\subsection{Single-transit planet candidates}\n\n\n\\textbf{TIC 103633672.} Single transit event identified in Sector 20. The single LCO\/NRES spectra shows no sign of this being a double lined spectroscopic binary.  We caution that there is a star on the same pixels, which is 0.1 mag brighter. We are unable to rule this star out as the cause for the transit-like signal.\n\n\\textbf{TIC 110996418.} Single transit event identified in Sector 10. We caution that there is a star on the same \\emph{TESS}\\ pixel, which is 2.4 magnitudes fainter than the target.\n\n\\textbf{TIC 128703021.} Single transit event identified in Sector 11. With a stellar radius of 1.6 $R_{\\odot}$ and a T$_{eff}$ of 6281 this host star is likely in the subgiant phase of its evolution. The 43 spectra obtained with MINERVA australis and the two obtained with LCO\/NRES are consistent with a planetary nature. Gemini speckle interferometry shows no nearby companion stars.\n\n\n\\textbf{TIC 142087638.} Single transit event identified in Sector 7. The best fit \\texttt{pyaneti}\\ model of the transit suggests an orbital period of only 3.14 d. As there are no additional transits seen in the light curve, this period is clearly not possible. We caution that the transit is most likely caused by a grazing object, and is therefore likely to be caused by a stellar companion. However, without further data we are unable to rule this candidate out as being planetary in nature.\n\n\\textbf{TIC 159159904.} Single transit event identified in Sector 22. The initial two observations obtained using LCO\/NRES show no sign of the candidate being a double lined spectroscopic binary.\n\n\n\\textbf{TIC 166184426.} Single transit event identified in Sector 11. Since the PHT discovery this cTOI has been become the priority 1 (1=highest, 5=lowest) target TOI  1955.01.\n\n\n\\textbf{TIC 172370679.} Single transit event identified in Sector 15. \\textcolor{black}{This candidate was independently discovered and verified using a BLS algorithm used to search for transiting planets around M-dwarfs. The candidate is now the confirmed planet TOI 1899 b \\citep{canas2020}.} \n\n\\textbf{TIC 174302697.} Single transit event identified in Sectors 16. With a stellar radius of 1.6 $R_{\\odot}$ and a T$_{eff}$ of 6750 this host star is likely in the subgiant phase of its evolution. \\textcolor{black}{This candidate was initially flagged as a TCE and but was erroneously discounted due to the pipeline mistaking the data glitch at the time of a momentum dump as a secondary eclipse.} Since the PHT discovery this cTOI has become the priority 3 target TOI 1896.01. \n\n\\textbf{TIC 192415680.} Single transit event identified in Sector 18. The two epochs of RV measurement obtained with OHP\/SOPHIE are consistent with a planetary scenario.\n\n\\textbf{TIC 192790476.} Single transit event identified in Sector 5. This target has been identified to be a wide binary with am angular separation of 72.40 arcseconds \\citep{andrews2017wideBinary} and a period of 162705 years \\citep{benavides2010new}. The star exhibits large scale variability on the order of around 10 d. The signal is consistent with that of spot modulations, which would suggest that this is a slowly rotating star.\n\n\n\n\\textbf{TIC 219466784.} Single transit event identified in Sector 22. We caution that there is a nearby companion located within the same \\emph{TESS}\\ pixel at an angular separation of 16.3 with a Vmag of 16.3\". Since the PHT discovery this cTOI has become the priority 2 target TOI  2007.01.\n\n\n\n\\textbf{TIC 229055790.} Single transit event identified in Sector 21. We note that the midpoint of the transit-like events coincides with a \\emph{TESS}\\ momentum dump, however, we believe the shape to be convincing enough to warrant further investigation. The two LCO\/NRES spectra show no sign of this being a spectroscopic binary.\n\n\\textbf{TIC 229608594.} Single transit event identified in Sector 24. Since the PHT discovery this cTOI has become the priority 3 target TOI 2298.01.\n\n\n\n\\textbf{TIC 233194447.} Single transit event identified in Sector 14. The transit-like event is shallow and asymmetric and we cannot definitively rule out systematics as the cause for the event without additional data. The initial two LCO\/NRES spectra show no sign of this target being a spectroscopic binary.\n\n\\textbf{TIC 237201858.} Single transit event identified in Sector 18. The single LCO\/NRES spectra shows no sign of this being a double lined spectroscopic binary.\n\n\\textbf{TIC 243187830.} Single transit event identified in Sector 18. There are no nearby bright stars. \\textcolor{black}{This light curve was initially flagged as a TCE, however, the flagged events corresponded to stellar variability and not the same event identified by PHT.} The single LCO\/NRES spectrum shows no sign of this being a double lined spectroscopic binary. Since the PHT discovery this cTOI has become the priority 3 target TOI 2009.01.\n\n\\textbf{TIC 243417115.} Single transit event identified in Sector 11. We note that the best fit \\texttt{pyaneti}\\ model of the transit suggests an orbital period of only 1.81 d. As there are no additional transits seen in the light curve, this period is clearly not possible. We caution that the transit is most likely caused by a grazing object, and is therefore likely to be caused by a stellar companion. However, without further follow-up data we are unable to rule this candidate out as being planetary in nature.\n\n\n\\textbf{TIC 264544388.} Single transit event identified in Sector 19. The single LCO\/NRES spectra shows no sign of this being a double lined spectroscopic binary. Apart from the single transit event, the light curve shows no obvious signals. A periodogram of the light curve, however, reveals a series of five significant peaks, nearly equidistantly spaced by $\\sim1.03$~d$^{-1}$. Additionally, a rotationally split quintuplet is visible at 7.34~d$^{-1}$, with a splitting of $\\sim0.12$~d$^{-1}$, suggesting an $\\ell=2$ p-mode pulsation. The Maelstrom code \\citep{hey2020maelstrom} revealed pulsation timing variations which are consistent with a long period planet. \\textcolor{black}{The short period signal, which was also identified by the periodogram, was flagged as a TCE, however, the single-transit event was not flagged as a TCE.} Since the PHT discovery this cTOI has become the priority 3 target TOI 1893.01.\n\n\\textbf{TIC 264766922.} Single transit event identified in Sector 8. With a stellar radius of 1.7 $R_{\\odot}$ and a T$_{eff}$ of 6913 K this host star is likely entering the subgiant phase of its evolution. The V-shape of this transit and the resultant high impact parameter suggests that the object is grazing. We can therefore not rule out that this candidate it a grazing eclipsing binary.  There are clear p-mode pulsations at frequencies of 9.01 and 11.47 cycles per day, as well as possible g-mode pulsations. \\textcolor{black}{A very short period signal within this light curve was flagged as a TCE, however, the single transit event was ignored by the pipeline.}\n\n\\textbf{TIC 26547036.} Single transit event identified in Sector 14. The four LCO\/NRES observations are consistent with the target being a planetary body and show no sign of the signal being caused by a spectroscopic binary. We caution that there is a star on the same \\emph{TESS}\\ pixel, however, this star is 8.2 magnitudes fainter than the target, and therefore unable to be responsible for the transit event seen in the light curve. Gemini speckle interferometry reveal no additional nearby companion stars. \\textcolor{black}{This candidate was initially flagged as a TCE, however, in addition to the single transit event the pipeline identified further periodic signals that correspond to times of momentum dumps. Due to this, the candidate was never promoted to TOI status.}\n\n\n\\textbf{TIC 278990954.} Single transit event identified in Sector 12. With a stellar radius of 2.6 $R_{\\odot}$ and a T$_{eff}$ of 5761 K this host star is likely in the subgiant phase of its evolution. We note that there are two additional stars on the same pixel as TIC 278990954. These two stars are 2.7 and 3.7 magnitudes fainter in the v-band than the target and can't be ruled out as the cause for the transit-like event without additional follow-up data.\n\n\\textbf{TIC 280865159.} Single transit event identified in Sector 16. Gemini speckle interferometry revealed any nearby companion stars. Since the PHT discovery this cTOI has become the priority 3 target TOI 1894.01.\n\n\\textbf{TIC 284361752.} Single transit event identified in Sector 26. Since the PHT discovery this cTOI has become the priority 2 target TOI 2294.01.\n\n\n\\textbf{TIC 296737508.} Single transit event identified in Sector 8. The single LCO\/NRES and the single MINERVA australis spectra show no sign of this being a spectroscopic binary. The Sinistro snapshot image revealed no additional nearby companions.\n\n\\textbf{TIC 298663873.} Single transit event identified in Sector 19. The two LCO\/NRES spectra show no sign of this being a spectroscopic binary. With a stellar radius of 1.6 $R_{\\odot}$ and a T$_{eff}$ of 6750 this host star is likely in the subgiant phase of its evolution. Gemini speckle images obtained by other teams show no signs of there being nearby companion stars. Since the PHT discovery this cTOI has become the priority 3 target TOI 2180.01.\n\n\n\\textbf{TIC 303050301.} Single transit event identified in Sector 2. The variability of the light curve is consistent with spot modulation. A single LCO\/NRES spectrum shows no signs of this being a double lined spectroscopic binary.\n\n\\textbf{TIC 303317324.} Single transit event identified in Sector 2. We note that a second transit was later seen in Sector 29, however, as this work only covers sectors 1-26 of the primary \\emph{TESS}\\ mission, this candidates is considered a single-transit event in this work. \n\n\n\\textbf{TIC 304142124.} Single transit event identified in Sector 10.\\textcolor{black}{This target was independently identified as part of the Planet Finder Spectrograph, which uses precision RVs \\citep{diaz2020}. This candidate is know the confirmed planet HD 95338 b.}\n\n\n\n\n\n\n\n\\textbf{TIC 331644554.} Single transit event identified in Sector 16. There is a clear mono-periodic signal in the periodogram at around 11.2 cycles per day, which is consistent with p-mode pulsation.\n\n\\textbf{TIC 332657786.} Single transit event identified in Sector 8. We caution that there is a star on the adjacent \\emph{TESS}\\ pixel that is brighter in the V-band by 2.4 magnitudes. At this point we are unable to rule out this star as the cause of the transit-like signal. \n\n\n\\textbf{TIC 356700488.} Single transit event identified in Sector 16. There is a clear mono-periodic signal in the periodogram at around 1.2 cycles per day, which is consistent with either spot modulation or g-mode pulsation. However, there is no clear signal visible in the light curve that would allow us to differentiate between these two scenarios based on the morphology of the variation. Since the PHT discovery this cTOI has become the priority 3 target TOI 2098.01.\n\n\\textbf{TIC 356710041.} Single transit event identified in Sector 23. With a stellar radius of 2.8 $R_{\\odot}$ and a T$_{eff}$ of 5701 K this host star is likely in the subgiant phase of its evolution. \\textcolor{black}{This candidate was initially flagged as a TCE, however, in addition to the single transit event, the pipeline identified a further event that corresponds to the time of a momentum dump. Due to this the candidate failed the `odd-even test' and was initially discarded as a TOI.} Since the PHT discovery this cTOI has become the priority 3 target TOI 2065.01\n\n\\textbf{TIC 369532319.} Single transit event identified in Sector 16. Gemini speckle interferometry revealed no nearby companion stars.\n\n\n\\textbf{TIC 384159646.} Single transit event identified in Sector 12. The eight LCO\/NRES and six MINERVA australis spectra are consistent with this candidate being a planet. Both the SBIG snapshot and the Gemini speckle interferometry observations revealed no companion stars. Since the PHT discovery this cTOI has become the priority 3 target TOI 1895.01.\n\n\n\n\n\\textbf{TIC 418255064.} Single transit event identified in Sector 12. The Gemini speckle image shows no sign of nearby companions.\n\n\n\\textbf{TIC 422914082.} Single transit event identified in Sector 4. Single Sinistro snapshot image reveals no additional nearby stars.\n\n\\textbf{TIC 427344083.} Single transit event identified in Sector 24. We note that there is a star on the adjacent \\emph{TESS}\\ pixel to the target, which is 3.5 magnitude fainter in the V-band than the target star. We also caution that the V-shape of the transit and the high impact parameter suggest that this is a grazing transit. However, without additional follow-up observations we are unable to rule this candidate out as a planet.\n\n\n\\textbf{TIC 436873727.} Single transit event identified in Sector 18. The host star shows strong variability on the order of one day, which is consistent with spot modulations or g-mode pulsations. The periodogram reveals multi-periodic behaviour in the low frequency range consistent with g-mode pulsations. \n\n\\textbf{TIC 452920657.} Single transit event identified in Sector 17. The V-shape of this transit suggests that the object is grazing and future follow-up observations may reveal this to be an EB. \\textcolor{black}{This candidate was initially flagged as a TCE, however, in addition to the single transit event, the pipeline identified a further two event that corresponds to likely stellar variability. Due to this the candidate failed the `odd-even test' and was initially discarded as a TOI.}\n\n\\textbf{TIC 455737331.} Single transit event identified in Sector 17. We note that there is a star on the same \\emph{TESS}\\ pixel as the target, which is 4.5 magnitudes fainter in the V-band. Neither the SBIG snapshot nor the Gemini speckle interferometry revealed any further nearby companion stars.\n\n\\textbf{TIC 456909420.} Single transit event identified in Sector 17. We caution that the V-shape of the transit and the high impact parameter suggest that this is a grazing transit. However, without additional follow-up observations we are unable to rule this candidate out as a planet.\n\n\n\n\n\\textbf{TIC 53843023.} Single transit event identified in Sector 1. We caution that the high impact parameter returned by the best fit \\texttt{pyaneti}\\ model suggests that the transit event is caused by a grazing body. However, at this point we are unable to rule this candidate out as being planetary in nature.\n\n\\textbf{TIC 63698669.} Single transit event identified in Sector 2. The SBIG snapshot image revealed no nearby companions. \\textcolor{black}{This candidate was initially identified as a TCE, however, in addition to the single transit event, the pipeline identified a further 3 events the light curve. Due to these, additional events, which correspond to stellar variability, the candidate was not initially promoted to TOI status.} However, since the PHT discovery this cTOI has become TOI 1892.01.\n\n\\textbf{TIC 70887357.} Single transit event identified in Sector 5. With a stellar radius of 2.1 $R_{\\odot}$ and a T$_{eff}$ of 5463 K this host star is likely in the subgiant phase of its evolution. \\textcolor{black}{This candidate was initially flagged as a TCE, however, in addition to the single transit-event the pipeline identified a further signal, and thus failed the `odd-even' transit test.} However, since the PHT discovery this cTOI has become the priority 3 target TOI 2008.01.\n\n\n\n\\textbf{TIC 91987762.} Single transit event identified in Sector 21. \\textcolor{black}{This candidate was initially flagged as a TCE, however, in addition to the single transit-event the pipeline identified a further signal, and thus failed the `odd-even' transit test.} Since the PHT discovery this cTOI has become the priority 3 target TOI 1898.01.\n\n\n\n\\subsection{Multi-transit and multi-planet candidates}\n\n\n\\textbf{TIC 160039081.} Multi-transit candidate with a period of 30.2 d. Single LCO\/NRES spectra shows no sign of this being a double lined spectroscopic binary and a snapshot image using SBIG shows no nearby companions. The Gemini speckle images also show no additional nearby companions. Since the PHT discovery this cTOI has  become the priority 1 target TOI 2082.01.\n\n\\textbf{TIC 167661160.} Multi-transit candidate with a period of 36.8 d. The nine LCO\/NRES and four MINERVA australis spectra have revealed this to be a long period eclipsing binary.\n\n\\textbf{TIC 179582003.} Multi-transit candidate with a period of 104.6 d. There is a clear mono-periodic signal in the periodogram at around 0.59 cycles per day, which is consistent with either spot modulation or g-mode pulsation. We caution that this candidate is located in a crowded field. With a stellar radius of 2.0 $R_{\\odot}$ and a T$_{eff}$ of 6115 K this host star is likely in the subgiant phase of its evolution.\n\n\\textbf{TIC 219501568.} Multi-transit candidate with a period of 16.6 d. With a stellar radius of 1.7 $R_{\\odot}$ and a T$_{eff}$ of 6690 K this host star is likely entering the subgiant phase of its evolution. \\textcolor{black}{This candidate was identified as a TCE, however, it was not initially promoted to TOI status as the signal was thought to be off-target by the automated pipeline.} However, since the PHT discovery this cTOI has become the priority 3 target TOI 2259.01\n\n\\textbf{TIC 229742722.} Multi-transit candidate with a period of 63.48 d. Eight LCO\/NRES and four OHP\/SOPHIE observations are consistent with this candidate being a planet. Gemini speckle interferometry reveals no nearby companion stars. \\textcolor{black}{This candidate was flagged as a TCE in sector 20, where it only exhibits a single transit event. An additional event was identified at the time of a momentum dump, and as such it failed the `odd-even' test and was not initially promoted to TOI status.} However, since the PHT discovery this cTOI has become the priority 3 target TOI 1895.01.\n\n\\textbf{TIC 235943205.} Multi-transit candidate with a period of 121.3 d. The LCO\/NRES and OHP\/SOPHIE observations remain consistent with a planetary nature of the signal. Since the PHT discovery this cTOI has become the priority 3 target TOI 2264.01.\n\n\n\\textbf{TIC 267542728.} Multi-transit event with period of 39.7 d. Observations obtained with Keck showed that the RV shifts are not consistent with a planetary body and are most likely due to an M-dwarf companion.\n\n\\textbf{TIC 274599700.} Multi-transit candidate with a period of 33.0 d. One of the two transit-like even is only half visible, with the other half of the event falling in a \\emph{TESS}\\ data gap.\n\n\n\n\\textbf{TIC 328933398.} Multi-planet candidates. The 2-minute cadence light curve shows two single transit events of different depths across two \\emph{TESS}\\ sectors, both of which are consistent with an independent planetary body. In addition to the short cadence data, this target was observed in an additional three sectors as part of the 30-minute cadence full frame images. These showed additional transit events for one of the planet candidates, with a period of 24.9 d. \\textcolor{black}{This light curve was initially flagged as containing a TCE event, however, the two 2-minute cadence single transit events were thought to belong to the same transiting planet. The TCE was initially discarded as the pipeline identified the events to be off-target.} However, since the PHT discovery these two cTOIs has become the priority 3 and 1 targets, TOI 1873.01 and TOI 1873.01, respectively.\n\n\\textbf{TIC 349488688.} Multi-planet candidate, with one single transit event and one multi-transit candidate with a period of 11 d. Two LCO\/NRES and two OHP\/SOPHIE spectra, along with ongoing HARPS North are consistent with both of these candidates being planetary in nature. \\textcolor{black}{The single transit event was initially identified as a TCE, however, in addition to the event it identified two other signals at the time of momentum dumps, and was therefore initially discarded by the pipeline as it failed the `odd-even' transit test.} However, since the PHT discovery the two-transit event has become the 1 targets, TOI 2319.01 (Eisner et al. in prep).\n\n\\textbf{TIC 385557214.} Multi-transit candidate with a period of 5.6 d. The prominent stellar variation seen in the light curve is likely due to spots or pulsation The high impact parameter returned by the best fit \\texttt{pyaneti}\\ modelling suggests that the transit is likely caused by a grazing object. Without further observations, however, we are unable to rule this candidate out as being planetary in nature. \\textcolor{black}{This candidate was flagged as a TCE but was not promoted to TOI status due to the other nearby stars.}\n\n\\textbf{TIC 408636441.} Multi-transit candidate with a period of 18.8 or 37.7 d. Due to \\emph{TESS}\\ data gaps, half of the period stated in Table~\\ref{tab:PHT-caniddates} is likely. The SBIG snapshot and Gemini speckle images show no signs of companion stars. \\textcolor{black}{This candidate was flagged as a TCE in sector 24, where it only exhibits a single transit event. An additional event was identified at the time of a momentum dump, and as such it failed the `odd-even' test and was not initially promoted to TOI status.} However, since the PHT discovery this cTOI has become the priority 3 target TOI 1759.01.\n\n\\textbf{TIC 441642457.} Multi-transit candidate with a period of 79.8 d. \\textcolor{black}{This candidate was flagged as a TCE in sector 14, where it only exhibits a single transit event. An additional event was identified at the time of a momentum dump, and as such it failed the `odd-even' test and was not initially promoted to TOI status.} Since the PHT discovery this cTOI has become the priority 2 target TOI 2073.01.\n\n\\textbf{TIC 441765914.} Multi-transit candidate with a period of 161.6 d.  Since the PHT discovery this cTOI has become the priority 1 target TOI 2088.01.\n\n\\textbf{TIC 48018596.} Multi-transit candidate with a period of 100.1 days (or a multiple thereof). The single LCO\/NRES spectrum shows no sign of this target being a double lined spectroscopic binary. Gemini speckle interferometry revealed no nearby companion stars.  \\textcolor{black}{This candidate was initially flagged as a TCE, however, in addition to the transit-events, the pipeline classified, what we consider stellar variability as an additional event. As such it failed the `odd-even' transit test and wasn't promoted to TOI status.} However, since the PHT discovery this cTOI has become the priority 3 target TOI 2295.01.\n\n\\textbf{TIC 55525572.} Multi-transit candidate with a period of 83.9 d. Since the PHT discovery this cTOI has become the confirmed planet TOI 813 \\citep{2020eisner}.\n\n\\textbf{TIC 82452140.} Multi-transit candidate with a period of 21.1 d. Since the PHT discovery this cTOI has become the priority 2 target TOI 2289.01.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\nIn machine learning, we often want to learn a model to not only to perform a specified task well, but also to have some sort of meaningful structure to, for example, make the model easier to understand and implement.  This desire for both accuracy and meaningful structure is usually expressed in a regularized optimization problem, such as one of the form:\n\\begin{align}\n\\minimize{\\mathbf{x}\\in \\mathcal{X}}f(\\mathbf{x}) + \\lambda g(\\mathbf{x}). \\label{eq:regularized_problem}\n\\end{align}\nIn this problem, $\\mathbf{x}$ is a choice of model parameters from a parameter space $\\mathcal{X}$, $f:\\mathcal{X}\\rightarrow\\mathbb{R}$ is a function that describes the misfit of the model with the selected parameters to the given task (such as empirical risk), $g:\\mathcal{X}\\rightarrow\\mathbb{R}$ is a function that expresses the deviation of our selected model parameters from their desired structure, and $\\lambda\\in\\mathbb{R}_{\\geq 0}$ is a tradeoff parameter.\n\nProblem \\eqref{eq:regularized_problem} becomes difficult when the desired model structure is an inherently binary or discrete property, while the model parameters are continuous values $\\mathbf{x}$ chosen from a continuum $\\mathcal{X}$.  A prime example of this issue arises in feature selection for sparse regression, where we seek a linear predictor $\\mathbf{x}^*\\in \\mathcal{X}\\subseteq\\mathbb{R}^n$ such that:\n\\begin{align}\n\\mathbf{x}^*\\in\\textrm{arg}\\min{\\mathbf{x}\\in \\mathcal{X}} \\Vert \\mathbf{Ax}-\\mathbf{b}\\Vert_2^2 + \\lambda \\Vert \\mathbf{x}\\Vert_0,\\label{eq:sparse_regression}\n\\end{align}\nfor some $\\mathbf{A}\\in\\mathbb{R}^{m\\times n}$ and $\\mathbf{b}\\in\\mathbb{R}^m$, with $\\Vert \\mathbf{x} \\Vert_2$ the standard Euclidean norm on $\\mathbb{R}^m$, and $\\Vert \\mathbf{x}\\Vert_0$ the $\\ell_0$ pseudo-norm that counts the number of nonzero entries in the predictor $\\mathbf{x}$.  The desired structure, in this case, is sparsity of the predictor $\\mathbf{x}\\in \\mathcal{X}$. Sparsity, however, only depends on the choice of a finite set of zero entries in the model parameters $\\mathbf{x}$, whereas the model is defined by a choice of continuous values for $\\mathbf{x}\\in\\mathcal{X}$.\n\nProblems containing this mixed dependence on both continuous and discrete properties of the model parameters, such as \\eqref{eq:sparse_regression}, are notoriously difficult, and even NP-Hard in general \\cite{rauhut2010compressive}.  A typical workaround is to replace the function describing model structure, $g$ in problem \\eqref{eq:regularized_problem}, with a continuous relaxation that is more amenable to optimization.  One of the more celebrated instances of this approach is the relaxation of the $\\ell_0$ pseudo-norm to the convex $\\ell_1$ norm $\\Vert x\\Vert_1$, which instead sums the absolute values of the vector $x$.  While this relaxation may still encourage some of the intended structure, the minimizer for the relaxed problem often does not correspond to the minimizer for the initially specified problem \\cite{bach2012structured}.  Moreover, the more well-known conditions for sparse recovery in regression problems, such as Restricted Isometry Properties \\cite{candes2005decoding}, Null Space Properties \\cite{rauhut2010compressive}, and Irrepresentability Conditions \\cite{zhao2006model}, do not give this form of guarantee for more general discrete functions $g$.\n\nIn contrast, the goal of this work is to identify conditions that allow us to directly solve the originally posed regularized model-fitting problem \\eqref{eq:regularized_problem} exactly and efficiently.  To this end, we leverage submodularity, a property of functions that guarantees that they can be minimized exactly and efficiently.\n\nTraditionally, submodularity is defined for functions on binary or discrete spaces.  For arbitrary functions over a discrete space, computing the minimizer is typically NP-Hard.  When the function is submodular, however, exact minimization is a polynomial time operation \\cite{schrijver2003combinatorial}.  Recently, the definition of submodularity and the associated optimization guarantees have been extended to continuous functions \\cite{bach2019submodular}.  In particular, if a continuous function is submodular, it can also be minimized exactly in polynomial time.\n\nA natural next question to ask is if submodularity still defines the boundary between easy and hard minimization problems when the function $f$ in \\eqref{eq:regularized_problem} is continuous, but the function $g$ has finite co-domain.  Our work explores precisely this boundary, and identifies sufficient conditions, based on the submodularity of both functions, under which the exact solution of problem \\eqref{eq:regularized_problem} can be efficiently computed.\n\nExploiting submodularity in these mixed scenarios is not a new idea, given its utility in combinatorial optimization problems.  Notable examples include establishing approximation guarantees for greedy algorithms to perform well on sparsity-constrained optimization problems \\cite{elenberg2018restricted}, or in producing tight convex relaxations for set-function descriptions of desired sparsity patterns \\cite{bach2012structured}.  In more recent work, \\cite{bach2019submodular} shows that if a continuous function is submodular, it can be discretized into a discrete submodular function, which can then be minimized exactly and in polynomial time.  However, this discretization is only valid for compact subsets of continuous spaces and necessarily introduces discretization error into the produced solution. \n\nIn a line of work similar to this one, authors in \\cite{eloptimal} propose converting the mixed problem to a purely discrete one without discretizing.  They then advocate using a specific submodular set function minimization algorithm for solving the discrete problem, and give approximation guarantees under the assumption that the functions are not exactly submodular.  This approach mirrors the one proposed herein, but we instead propose conditions under which their approach is exact.  Moreover, our guarantees are independent of which submodular function minimization algorithm is employed, allowing the use of more specialized algorithms than the specific one suggested by \\cite{eloptimal}.\n\nOur work makes several technical contributions, summarized as:\n\\begin{enumerate}[label=(\\roman*)]\n\\item We identify sufficient conditions, based on submodularity, under which the regularized model fitting problem \\eqref{eq:regularized_problem} can be solved efficiently and exactly;\n\\item We extend this theory to accommodate simple continuous and discrete constraints on the model parameter for some problem classes;\n\\item We highlight the utility of exact solutions to these problems for adversarial and robust optimization scenarios;\n\\item We numerically validate the correctness of our theory with proof-of-concept examples from sparse regression.\n\\end{enumerate}\n\n\\section{Submodular Functions on Lattices}\nIn this work, we consider optimization problems defined on two sets--an uncountably infinite set, typically $\\mathbb{R}^n$ or a subset thereof referred to as a \\emph{continuous set}, and a countable set, typically finite and referred to as a \\emph{discrete set}.  Because we would like to tractably solve optimization problems defined on both continuous and discrete sets, we study a  structure that enables efficient optimization in both cases: submodularity.\n\nSubmodularity is typically defined for set functions, which are functions that map any subset of a finite set $V$ to a real number, i.e., $f:2^V\\rightarrow\\mathbb{R}$.  More generally, however, submodularity is a property of functions on \\emph{lattices} defined over continuous or discrete sets.\n\nWe let $\\latone$ be a set equipped with a partial order of its elements, denoted by $\\leqone$. For any two elements $\\mathbf{x},\\mathbf{x'}\\in\\latone$ we define their least upper bound, or \\emph{join} as:\n\\begin{align}\n\\mathbf{x}\\joinone\\mathbf{x'} &= \\inf\\{\\mathbf{y}\\in\\latone~:~\\mathbf{x}\\leq \\mathbf{y},~\\mathbf{x'}\\leq \\mathbf{y}\\}.\\label{eq:join_def}\n\\end{align}\nDually, we define their greatest lower bound, or \\emph{meet} as:\n\\begin{align}\n\\mathbf{x}\\meetone\\mathbf{x'}&=\\sup\\left\\{\\mathbf{y}\\in\\latone~:~\\mathbf{y}\\leq\\mathbf{x},~\\mathbf{y}\\leq\\mathbf{x'}\\right\\}.\\label{eq:meet_def}\n\\end{align}\nIf for every $\\mathbf{x},\\mathbf{x'}\\in\\latone$, their join, $\\mathbf{x}\\joinone\\mathbf{x'}$, and their meet, $\\mathbf{x}\\meetone\\mathbf{x'}$, exist and are in $\\latone$, then the set $\\latone$ and its order define a \\emph{lattice}.  We write the lattice and its partial order together as $(\\latone,\\leqone)$, but will often write just $\\latone$ when the order is clear from context.  If a subset $\\mathcal{S}\\subseteq\\latone$ is such that for every $\\mathbf{x},\\mathbf{x'}\\in \\mathcal{S}$, both $\\mathbf{x}\\joinone\\mathbf{x'}$ and $\\mathbf{x}\\meetone\\mathbf{x'}$ are in $\\mathcal{S}$, the subset $\\mathcal{S}$ is called a \\emph{sublattice} of $\\latone$ \\cite{davey2002introduction}.\n\nAs an example, consider a finite set of elements $V$.  Then its power set, $2^V$ (the set of all its possible subsets), forms a lattice when ordered by set inclusion, $(2^V,\\subseteq)$.  Under this order, the join of any two elements $X,X'\\subseteq V$ is $X\\cup X'\\subseteq V$, and dually, their meet is $X\\cap X'\\subseteq V$.\n\nWe can also endow continuous sets with partial orders, thereby defining lattices.  Recent work has brought attention to $\\mathbb{R}^n$, where we define a partial order, denoted by $\\leqone$, as follows:\n\\begin{align}\n\\mathbf{x}\\leqone\\mathbf{x'}\\quad\\Leftrightarrow\\quad\\mathbf{x}_i \\leq \\mathbf{x}_i'\\quad\\text{for all }i=1,2,...,n,\\label{eq:rn_order}\n\\end{align}\nwhere $\\leq$ denotes the usual order on $\\mathbb{R}$.\n\n\nUnder this order, the join and meet operation for any $\\mathbf{x},\\mathbf{x'}\\in\\mathbb{R}^n$ are element-wise maximum and minimum, meaning:\n\\begin{align}\n(\\mathbf{x}\\joinone\\mathbf{x'})_i &= \\max\\{\\mathbf{x}_i,\\mathbf{x}_i'\\},\\4all i=1,2,...,n,\\label{eq:rn_join} \\\\\n(\\mathbf{x}\\meetone\\mathbf{x'})_i &= \\min\\{\\mathbf{x}_i,\\mathbf{x}_i'\\},\\4all i=1,2,...,n.\\label{eq:rn_meet}\n\\end{align}\n\nGiven a lattice $\\latone$, consider a function $f:\\latone\\rightarrow\\mathbb{R}$.  The function $f$ is \\emph{submodular} on the lattice $\\latone$ when the following inequality holds for all $\\mathbf{x},\\mathbf{x}'\\in\\latone$:\n\\begin{align}\nf(\\mathbf{x})+f(\\mathbf{x'}) \\geq f(\\mathbf{x}\\joinone\\mathbf{x'}) + f(\\mathbf{x}\\meetone\\mathbf{x'}).\\label{eq:lat_fn_submodular}\n\\end{align}\nThe function $f$ is \\emph{monotone} when the implication:\n\\begin{align}\n\\mathbf{x}\\leqone\\mathbf{x'}\\quad \\implies \\quad f(\\mathbf{x}) \\leq f(\\mathbf{x'}),\\label{eq:lat_fn_monotone}\n\\end{align}\nis satisfied.\n\nWhen working with the lattice $(2^V,\\subseteq)$, the submodular inequality \\eqref{eq:lat_fn_submodular} becomes:\n\\begin{align}\nf(A) + f(B) \\geq f(A\\cup B) + f(A\\cap B) \\quad \\4all A,B\\subseteq V.\\label{eq:set_fn_submodular}\n\\end{align}\nSimilarly, the monotonicity implication \\eqref{eq:lat_fn_monotone} becomes:\n\\begin{align}\nA\\subseteq B \\quad\\implies\\quad f(A) \\leq f(B).\\label{eq:set_fn_monotone}\n\\end{align}\nMinimizing or maximizing an arbitrary set function is NP-Hard in general. However, if the set function is submodular it can be exactly minimized and approximately maximized, up to a constant-factor approximation ratio, in polynomial time \\cite{schrijver2003combinatorial,nemhauser1978analysis}.  The computational tractability of submodular optimization for set functions has a variety of applications in fields such as sparse regression, summarization, and sensor placement \\cite{elenberg2018restricted,hui2011class,krause2006near}.\n\nWhen working with the lattice $(\\mathbb{R}^n,\\leqone)$, a function $f:\\mathbb{R}^n\\rightarrow\\mathbb{R}$ is submodular when:\n\\begin{align}\nf(\\mathbf{x}) + f(\\mathbf{x'}) \\geq f(\\max\\{\\mathbf{x},\\mathbf{x'}\\}) + f(\\min\\{\\mathbf{x},\\mathbf{x'}\\})\\quad\\4all \\mathbf{x},\\mathbf{x'}\\in\\mathbb{R}^n,\n\\end{align}\nwhere the maximum and minimum operations are performed element-wise, as expressed in \\eqref{eq:rn_join} and \\eqref{eq:rn_meet}.  When $f$ is twice differentiable, submodularity on $\\mathbb{R}^n$ is equivalent  (see, e.g. \\cite{topkis1998supermodularity,bach2019submodular}) to the condition:\n\\begin{align}\n\\frac{\\partial^2f}{\\partial \\mathbf{x}_i\\partial\\mathbf{x}_j} &\\leq 0 \\quad \\4all i\\neq j.\\label{eq:submodular_hessian}\n\\end{align}\nPerhaps surprisingly, the guarantees associated with submodular set function optimization extend to functions that are submodular on $\\mathbb{R}^n$.  In particular, submodular functions on $\\mathbb{R}^n$ can be minimized up to arbitrary precision in polynomial time (see \\cite{bach2019submodular}), and can be approximately maximized with constant-factor approximation ratios, as shown in \\cite{bian2016guaranteed,bian2017non}.\n\n\\section{Problem Formulation}\nIn this section, we bridge continuous and discrete submodular function minimization with one unified problem statement.  We do this by drawing inspiration from the field of structured sparsity, where the choice of zero entries in real-valued decision variables is viewed as a coupled discrete and continuous problem \\cite{bach2013learning,bach2011shaping}.\n\nTo highlight the connection with structured sparsity problems, for $n\\in\\mathbb{Z}_{>0}$, we denote by $[n]$ the set  $\\{1,2,...,n\\}$, and by $2^{[n]}$ the set of all possible subsets of $[n]$.  Define the map $\\mathrm{supp}:\\mathbb{R}^n\\rightarrow 2^{[n]}$ as:\n\\begin{align}\n\\supp{\\mathbf{x}} &= \\{i\\in [n]\\mid \\mathbf{x}_i\\neq 0\\}.\\label{eq:supp}\n\\end{align}\nIn other words, $\\mathrm{supp}$ returns the set of indices corresponding to the nonzero entries of the vector $\\mathbf{x}$.  Consider the functions $f:\\mathbb{R}^n\\rightarrow\\mathbb{R}$ and $g:2^{[n]}\\rightarrow\\mathbb{R}$.  In structured sparsity, problems of the form:\n\\begin{align}\n\\minimize{\\mathbf{x}\\in\\mathbb{R}^n}~f(\\mathbf{x}) + g(\\supp{\\mathbf{x}}), \\label{eq:cont_discrete_opt_problem}\n\\end{align}\nare of particular interest, where the preferences in discrete selections (the zero entries of $\\mathbf{x}$) are expressed through the function $g$.  As a particularly special case, if we let $f(\\mathbf{x}) = \\Vert \\mathbf{D}\\mathbf{x}-\\mathbf{b}\\Vert_2^2$ with $\\mathbf{D}\\in\\mathbb{R}^{m\\times n}$ and $\\mathbf{b}\\in\\mathbb{R}^m$ and define $g(A) = \\vert A\\vert$ as the cardinality of the set $A$, \\eqref{eq:cont_discrete_opt_problem} becomes:\n\\begin{align}\n\\minimize{\\mathbf{x}\\in\\mathbb{R}^n}~\\Vert\\mathbf{D}\\mathbf{x}-\\mathbf{b}\\Vert_2^2 + \\Vert\\mathbf{x}\\Vert_0, \\tag{CS}\\label{eq:compressed_sensing}\n\\end{align}\nwhere $\\Vert\\cdot\\Vert_0$ denotes the $\\ell_0$ pseudo-norm.  The problem \\eqref{eq:compressed_sensing} is a form of the well-studied compressed sensing problem, which is NP-Hard in general \\cite{rauhut2010compressive}.\n\nBuilding on the idea of making continuous decisions through the choice of $\\mathbf{x}$ in \\eqref{eq:cont_discrete_opt_problem}, and discrete decisions through the choice of the zero entries of $\\mathbf{x}$, we consider two lattices, $(\\latone,\\leqone)$ and $(\\lattwo,\\leqtwo)$, related by a map $\\eta:\\latone\\rightarrow\\lattwo$.  We let $f:\\latone\\rightarrow\\mathbb{R}$ be a function describing the cost of assignments of variables in $\\latone$, and similarly let $g:\\lattwo\\rightarrow\\mathbb{R}$ describe the associated cost of choices in $\\lattwo$. Then, we seek the optimal point $\\mathbf{x}^*\\in\\latone$ in the problem:\n\\begin{align}\n\\minimize{\\mathbf{x}\\in\\latone}~f(\\mathbf{x})+g(\\eta(\\mathbf{x})).\\tag{P}\\label{eq:lattice_opt_problem}\n\\end{align}\nAlthough we will eventually let $\\latone$ describe continuous choices and $\\lattwo$ describe associated discrete ones, none of our results rely on the cardinality of the lattices $\\latone$ and $\\lattwo$.\n\nIntuitively, this problem asks for the $\\mathbf{x}\\in\\latone$ which incurs minimum cost, as measured by $f(\\mathbf{x})$, and in $\\lattwo$, as measured by $g(\\eta(\\mathbf{x}))$.  Given that the special case of \\eqref{eq:compressed_sensing} is already hard in general, with no additional structure on $f$, $g$ and $\\eta$, our more general problem is hopelessly difficult.  To provide this structure, we make some assumptions on these functions.\n\\begin{assumptions}\nConsider the lattices $(\\latone,\\leqone)$ and $(\\lattwo,\\leqtwo)$ and the maps $\\eta:\\latone\\rightarrow\\lattwo$, $f:\\latone\\rightarrow\\mathbb{R}$ and $g:\\lattwo\\rightarrow\\mathbb{R}$.  We make the following assumptions:\n\\begin{enumerate}\n\\item The functions $f$ and $g$ are submodular on the lattices $\\latone$ and $\\lattwo$, respectively,\n\\item The function $g$ is monotone on $\\lattwo$,\n\\item For all $\\mathbf{x},\\mathbf{x}'\\in\\latone$:\n\\begin{align*}\n\\eta(\\mathbf{x}\\joinone\\mathbf{x}') \\leqtwo \\eta(\\mathbf{x})\\jointwo\\eta(\\mathbf{x}'), \\quad \\eta(\\mathbf{x}\\meetone\\mathbf{x}')\\leqtwo \\eta(\\mathbf{x})\\meettwo\\eta(\\mathbf{x}').\n\\end{align*}\n\\end{enumerate}\n\\end{assumptions}\n\n\\begin{remark}\nIf the map $\\eta:\\latone\\rightarrow\\lattwo$ satisfies Assumption 3,  it is necessarily an order-preserving join-homomorphism, meaning it maintains the order and joins of elements in $\\latone$.  (Prop. 2.19 in \\cite{davey2002introduction})  Explicitly, Assumption 3 is equivalent to the condition that for any $\\mathbf{x}, \\mathbf{x}'\\in\\latone$:\n\\begin{gather*}\n\\mathbf{x}\\leqone \\mathbf{x}' \\Rightarrow \\eta(\\mathbf{x})\\leqtwo\\eta(\\mathbf{x}'),\\\\\n\\eta(\\mathbf{x}\\joinone\\mathbf{x}') = \\eta(\\mathbf{x})\\jointwo\\eta(\\mathbf{x}').\n\\end{gather*}  Despite this equivalence, we leave Assumption 3 as written above for clarity in future proofs.\n\\end{remark}\n\\begin{comment}\nNote that the function $\\supp:\\mathbb{R}^n_{\\geq 0}\\rightarrow 2^{[n]}$ satisfies Assumption 3 with equality, and that with this particular choice of lattices, we recovered a form of the compressed sensing problem \\eqref{eq:compressed_sensing}.  This problem and its variants are well-studied and difficult to solve exactly in general (see \\cite{rauhut2010compressive}), but ties to submodularity on $\\mathbb{R}^n_{\\geq 0}$ established in \\cite{elenberg2018restricted} provide approximation guarantees for solving \\eqref{eq:compressed_sensing} with greedy algorithms.  In this work, however, we focus on conditions under which we can solve the problem exactly.\n\\end{comment}\nWe highlighted the lattices $(\\mathbb{R}^n,\\leqone)$ and $(2^{[n]},\\subseteq)$ in the context of compressed sensing problems such as \\eqref{eq:compressed_sensing},  but for the map $\\mathrm{supp}:\\mathbb{R}^n\\rightarrow 2^{[n]}$ to satisfy Assumption 3, we must restrict the domain of $f$ to only only the first orthant, $(\\mathbb{R}^n_{\\geq 0},\\leqone)$.  As mentioned in \\cite{bian2017non}, this issue can often be resolved by considering an appropriate \\emph{orthant conic lattice}, which views $\\mathbb{R}^n$ as a product of $n$ copies of $\\mathbb{R}$ and selects a different order for each copy.  Alternatively, any least-squares problem such as \\eqref{eq:compressed_sensing} can be lifted to a non-negative least-squares problem, allowing us to satisfy Assumption 3 with the map $\\mathrm{supp}$, but potentially no longer satisfying Assumption 1 (see Appendix \\ref{apdx:NNLS}).\n\n\\begin{comment}\nOne common method for solving problems such as \\eqref{eq:compressed_sensing} is to replace the function $g:2^{[n]}\\rightarrow\\mathbb{R}$ with a tight convex surrogate function \\cite{bach2013learning}.  Conveniently, when $g$ is submodular, this convex surrogate function is easily computed through the Lov\\`asz extension, and is amenable to first-order methods for convex optimization \\cite{lovasz1983submodular}.  Despite this convenience, the resulting minimizer for the new relaxed problem may not correspond to the minimizer of the original problem \\cite{bach2011shaping}.\n\nUnder Assumptions 1-3, it can be shown that the combined function $f + g \\circ \\mathrm{supp}~:\\mathbb{R}^n_{\\geq 0} \\rightarrow\\mathbb{R}$ is submodular on $\\mathbb{R}^n_{\\geq 0}$.  We can then directly apply the algorithms developed in \\cite{bach2019submodular} for submodular function minimization on $\\mathbb{R}^n$.  However, these algorithms rely on discretizing the space $\\mathbb{R}^n_{\\geq 0}$ and minimizing the resulting discrete function, necessarily introducing some error into the final result. Moreover, the discretization process only produces finite lattices when the set to be discretized is compact.  Since we may often have to work on $\\mathbb{R}^n_{\\geq 0}$, it may be nontrivial or expensive to find a compact subset of $\\mathbb{R}^n_{\\geq 0}$ that is guaranteed to contain the minimizer of $f + g\\circ\\mathrm{supp}$.\n\nAnother recent approach to this problem performs a discrete parameterization of the function $f$, then uses the Lov\\`asz extension to compute approximate subgradients that are used for projected subgradient descent \\cite{eloptimal}.  While this algorithm provides approximation guarantees, subgradient descent can be slow to converge in practice.  Moreover, as we will show, this method amounts to a special instantiation of our framework, where rather than giving approximation guarantees, we know the algorithm produces the globally optimal solution.\n\\end{comment}\n\\section{Solving an Equivalent Problem}\nIn this section, we outline our approach for solving the problem \\eqref{eq:lattice_opt_problem} which uses a related optimization problem defined on a single lattice.  We then prove that this related problem is also a submodular function minimization problem, and that by solving it we recover a solution to \\eqref{eq:lattice_opt_problem}.  Finally, we highlight some conditions under which solving this related problem is a polynomial time operation.\n\n\n\\subsection{The Equivalent Submodular Minimization Problem}\nAs expressed above, the problem \\eqref{eq:lattice_opt_problem} asks for the best selection of $\\mathbf{x}\\in\\latone$ and associated $\\eta(\\mathbf{x})\\in\\lattwo$.  Our key observation is that we could instead look for the choice of $\\mathbf{y}\\in\\lattwo$ and associated $\\mathbf{x}\\in\\latone$, leading to the problem:\n\\begin{align*}\n\\minimize{\\mathbf{y}\\in\\lattwo}~g(\\mathbf{y}) + \\underset{\\substack{\\mathbf{x}\\in\\latone\\\\ \\eta(\\mathbf{x}) = \\mathbf{y}}}{\\min}~f(\\mathbf{x}).\n\\end{align*}\nIn the special case of \\eqref{eq:compressed_sensing} explored earlier, this equivalent problem becomes:\n\\begin{align*}\n\\minimize{S\\in 2^{[n]}}~\\vert S\\vert + \\underset{\\substack{\\mathbf{x}\\in\\mathbb{R}^n_{\\geq 0}\\\\ \\supp{\\mathbf{x}} = S}}{\\min}~\\Vert\\mathbf{A}\\mathbf{x}-\\mathbf{b}\\Vert_2^2.\n\\end{align*}\nWhile this new problem is clearly the same as \\eqref{eq:compressed_sensing}, the innermost minimization is over the set of $\\mathbf{x}\\in\\mathbb{R}^n_{\\geq 0}$ such that $\\supp{\\mathbf{x}} = S$, or equivalently, $\\mathbf{x}_i \\neq 0$ for all $i\\in S$, and $\\mathbf{x}_i = 0$ for all $i\\notin S$.  This feasible set is not a closed subset of $\\mathbb{R}^n_{\\geq 0}$, and thus the corresponding minimizer of this innermost problem may not exist \\cite{borwein2000convex}.\n\nWith this issue in mind, we instead consider a slight relaxation of the above problem:\n\\begin{align*}\n\\minimize{\\mathbf{y}\\in\\lattwo}~g(\\mathbf{y}) + H(\\mathbf{y}), \\tag{P-R}\\label{eq:lattice_opt_prob_relax}\n\\end{align*}\nwhere we have defined the function $H:\\lattwo\\rightarrow\\mathbb{R}$ as:\n\\begin{align}\nH(\\mathbf{y}) &= \\underset{\\substack{\\mathbf{x}\\in\\latone\\\\ \\eta(\\mathbf{x}) \\leqtwo \\mathbf{y}}}{\\min}~f(\\mathbf{x}).\\label{eq:H_definition}\n\\end{align}\nIn the special case of \\eqref{eq:compressed_sensing}, this relaxation produces the problem:\n\\begin{align}\n\\minimize{S\\in 2^{[n]}}~\\vert S\\vert + \\underset{\\substack{\\mathbf{x}\\in\\mathbb{R}^n_{\\geq 0}\\\\ \\supp{\\mathbf{x}} \\subseteq S}}{\\min}~\\Vert\\mathbf{A}\\mathbf{x}-\\mathbf{b}\\Vert_2^2,\\tag{CS-R}\n\\end{align}\nwhere the innermost minimization is instead over the $\\mathbf{x}\\in\\mathbb{R}^n_{\\geq 0}$ such that $\\mathbf{x}_i = 0$ for all $i\\notin S$, which is now a closed subset of $\\mathbb{R}^n_{\\geq 0}$.\n\nOur goal is to prove that under Assumptions 1-3, the new relaxed problem \\eqref{eq:lattice_opt_prob_relax} can be efficiently solved, and that by solving it we can recover the corresponding minimizer for \\eqref{eq:lattice_opt_problem}.  As established above, minimizing functions on lattices is efficient when the functions are submodular, so we show that our new relaxed problem \\eqref{eq:lattice_opt_prob_relax} is a submodular function minimization problem on $\\lattwo$.\n\n\\begin{theorem}\\label{thm:main_result}\nUnder Assumptions 1-3, the function $g + H:\\lattwo\\rightarrow\\mathbb{R}$ is submodular on $\\lattwo$, and therefore the relaxed problem \\eqref{eq:lattice_opt_prob_relax} is a submodular function minimization problem on the lattice $\\lattwo$.  Moreover, let $\\mathbf{y}^*\\in\\lattwo$ be the minimizer for the problem \\eqref{eq:lattice_opt_prob_relax}, and let $\\mathbf{x}^*\\in\\latone$ be such that:\n\\begin{align*}\n\\mathbf{x}^*\\in\\underset{\\substack{\\mathbf{x}\\in\\latone \\\\ \\eta(\\mathbf{x})\\leqtwo\\mathbf{y}^*}}{\\mathrm{argmin}}~f(\\mathbf{x}).\n\\end{align*}\nThen $\\mathbf{x}^*$ is a minimizer for the problem \\eqref{eq:lattice_opt_problem}.\n\\end{theorem}\n\nTo prove this result, we require a few technical lemmas.\n\n\\begin{lemma}\\label{lem:sublattice}\nLet $(\\latone,\\leqone)$ and $(\\lattwo,\\leqtwo)$ be lattices with the map $\\eta:\\latone\\rightarrow\\lattwo$ satisfying Assumption 3.  Then the set:\n\\begin{align}\n\\mathcal{D} &= \\left\\lbrace(\\mathbf{x},\\mathbf{y})\\in \\latone\\times\\lattwo \\mid\\eta(\\mathbf{x})\\leqtwo \\mathbf{y}\\right\\rbrace, \\label{eq:sublattice_D}\n\\end{align}\nis a sublattice of the product lattice, $\\latone\\times \\lattwo$.\n\\end{lemma}\n\\begin{proof}\nTo prove that $\\mathcal{D}\\subseteq\\latone\\times\\lattwo$ is a sublattice, we have to show that the join and meet (as defined on the product lattice) of any two elements in $\\mathcal{D}$ is also in $\\mathcal{D}$. The join on the product lattice of any two $(\\mathbf{x},\\mathbf{y}),(\\mathbf{x}',\\mathbf{y}')\\in \\mathcal{D}$ is denoted by $\\vee_{\\mathcal{D}}$, and defined as:\n\\begin{align*}\n(\\mathbf{x},\\mathbf{y})\\vee_{\\mathcal{D}}(\\mathbf{x}',\\mathbf{y}') &= (\\mathbf{x}\\joinone\\mathbf{x}',\\mathbf{y}\\jointwo\\mathbf{y}').\n\\end{align*}\nThen, we note:\n\\begin{align*}\n\\eta(\\mathbf{x}\\joinone\\mathbf{x}')&\\leqtwo\\eta(\\mathbf{x})\\jointwo\\eta(\\mathbf{x}') \\leqtwo \\mathbf{y}\\jointwo\\mathbf{y}',\n\\end{align*}\nwhere we first used Assumption 3, then the fact that $(\\mathbf{x},\\mathbf{y}),(\\mathbf{x}',\\mathbf{y}')\\in \\mathcal{D}$.  Therefore, the pair $(\\mathbf{x}\\joinone\\mathbf{x}',\\mathbf{y}\\jointwo\\mathbf{y}')$ is also in $D$.\n\nBecause $(\\mathbf{x},\\mathbf{y})$ and $(\\mathbf{x}',\\mathbf{y}')$ were arbitrary, this holds for all of $\\mathcal{D}$.  A dual analysis follows for the meet operation, proving $\\mathcal{D}$ is a sublattice of the product lattice $\\latone\\times\\lattwo$.\n\\end{proof}\n\nThe sublattice $\\mathcal{D}$ is useful as the only pairs of $(\\mathbf{x},\\mathbf{y})\\in\\latone\\times\\lattwo$ of interest in our optimization problem \\eqref{eq:lattice_opt_prob_relax} are those that are in $\\mathcal{D}$.  The following theorem then uses this sublattice to prove that $H$ is submodular.  The result is adapted from an established theorem in literature, but we include its proof here for completeness.\\\\\n\\begin{theorem}\\label{thm:topkis}\n(Adapted from \\textit{Theorem 2.7.6 in \\cite{topkis1998supermodularity}}) Let $f:\\latone\\rightarrow\\mathbb{R}$, $g:\\lattwo\\rightarrow\\mathbb{R}$, and $\\eta:\\latone\\rightarrow\\lattwo$ be maps satisfying Assumptions 1 and 3.  Then the function $g + H:\\lattwo\\rightarrow\\mathbb{R}$, with $H$ defined as in \\eqref{eq:H_definition}, is submodular on $\\lattwo$.\n\\end{theorem}\n\\begin{proof}\nTo prove this statement, we take two points $\\mathbf{y},\\mathbf{y}'\\in\\lattwo$ and compare the values of the function $g + H$, verifying the submodular inequality \\eqref{eq:lat_fn_submodular}.  We note that for any $\\mathbf{y},\\mathbf{y'}\\in\\lattwo$, there are corresponding $\\mathbf{z},\\mathbf{z}'\\in\\latone$ such that:\n\\begin{align}\n\\begin{aligned}\n\\mathbf{z}&\\in\\textrm{arg}\\min{\\substack{\\mathbf{x}\\in\\latone \\\\ \\eta(\\mathbf{x})\\leqtwo \\mathbf{y}}}~f(\\mathbf{x}) \\quad \\Rightarrow\\quad H(\\mathbf{y}) = f(\\mathbf{z}),\\\\\n\\mathbf{z}'&\\in\\textrm{arg}\\min{\\substack{\\mathbf{x}\\in\\latone \\\\ \\eta(\\mathbf{x})\\leqtwo \\mathbf{y}'}}~f(\\mathbf{x})\\quad \\Rightarrow \\quad H(\\mathbf{y}') = f(\\mathbf{z}').\n\\end{aligned}\\label{eq:z_optimality}\n\\end{align}\nBy definition, $(\\mathbf{z},\\mathbf{y})$ and $(\\mathbf{z}',\\mathbf{y}')$ are both in $\\mathcal{D}$.  Then, it follows:\n\\begin{align*}\ng(\\mathbf{y}) + H(\\mathbf{y}) + g(\\mathbf{y}') + H(\\mathbf{y}') &= g(\\mathbf{y}) + f(\\mathbf{z}) + g(\\mathbf{y}') + f(\\mathbf{z}') \\\\\n&\\geq g(\\mathbf{y}\\jointwo\\mathbf{y'}) + g(\\mathbf{y}\\meettwo\\mathbf{y}') + f(\\mathbf{z}\\joinone\\mathbf{z}') + f(\\mathbf{z}\\meetone\\mathbf{z}'),\n\\end{align*}\nwhere we first used \\eqref{eq:z_optimality} and then the submodularity of $f$ and $g$ on their respective sublattices.\n\nBy Lemma \\ref{lem:sublattice}, $\\mathcal{D}$ is a sublattice of $\\latone\\times\\lattwo$, and so the pairs $(\\mathbf{z}\\joinone\\mathbf{z}',\\mathbf{y}\\jointwo\\mathbf{y}')$ and $(\\mathbf{z}\\meetone\\mathbf{z}',\\mathbf{y}\\meettwo\\mathbf{y}')$ are also in $\\mathcal{D}$, meaning:\n\\begin{align*}\n\\eta(\\mathbf{x}\\joinone\\mathbf{x}') &\\leqtwo \\mathbf{y}\\jointwo\\mathbf{y}', \\\\ \\eta(\\mathbf{x}\\meetone\\mathbf{x}')&\\leqtwo\\mathbf{y}\\meettwo\\mathbf{y}'.\n\\end{align*}\nTherefore $\\mathbf{x}\\joinone\\mathbf{x}'$ and $\\mathbf{x}\\meetone\\mathbf{x}'$ are feasible points in the minimization defining $H(\\mathbf{y}\\jointwo\\mathbf{y}')$ and $H(\\mathbf{y}\\meettwo\\mathbf{y}')$ in \\eqref{eq:H_definition}.  We then have:\n\\begin{align*}\ng(\\mathbf{y}) + H(\\mathbf{y}) + g(\\mathbf{y}') + H(\\mathbf{y}') &\\geq g(\\mathbf{y}\\jointwo\\mathbf{y'}) + g(\\mathbf{y}\\meettwo\\mathbf{y}') + f(\\mathbf{z}\\joinone\\mathbf{z}') + f(\\mathbf{z}\\meetone\\mathbf{z}') \\\\\n&\\geq g(\\mathbf{y}\\jointwo\\mathbf{y'}) + g(\\mathbf{y}\\meettwo\\mathbf{y}') +  \\underset{\\substack{\\mathbf{x}\\in\\latone\\\\ \\eta(\\mathbf{x}) \\leqtwo \\mathbf{y}\\jointwo\\mathbf{y}'}}{\\min}\\hspace{-2.5mm}f(\\mathbf{x}) +  \\underset{\\substack{\\mathbf{x}\\in\\latone\\\\ \\eta(\\mathbf{x}) \\leqtwo \\mathbf{y}\\meettwo\\mathbf{y}'}}{\\min}\\hspace{-2.5mm}f(\\mathbf{x})\\\\\n&= g(\\mathbf{y}\\jointwo\\mathbf{y'}) + H(\\mathbf{y}\\jointwo\\mathbf{y}') + g(\\mathbf{y}\\meettwo\\mathbf{y}') + H(\\mathbf{y}\\meettwo\\mathbf{y}').\n\\end{align*}\nThe final equality in this sequence provides the right-hand side of the submodular inequality \\eqref{eq:lat_fn_submodular} for $g + H$, as desired.\n\\end{proof}\n\nBecause $g+H$ is submodular on $\\lattwo$, we know that minimizing it over all $\\mathbf{y}\\in\\lattwo$, i.e., solving \\eqref{eq:lattice_opt_prob_relax}, is an instance of submodular function minimization.  What remains is to show that solving this relaxed problem allows us to also solve to the original problem, \\eqref{eq:lattice_opt_problem}.  We verify this in the following lemma.\n\n\\begin{comment}\nThe key insight in this lemma is noting that the only time the minimizer of the relaxed problem may differ from the original is when the minimizer, $\\mathbf{y}^*\\in\\lattwo$ corresponds to some $\\mathbf{z}^*\\in\\latone$ such that:\n\\begin{align*}\n\\mathbf{z}^*\\in \\textrm{arg}\\min{\\substack{\\mathbf{x}\\in\\latone \\\\ \\eta(\\mathbf{x})\\leqtwo \\mathbf{y}^*}}~f(\\mathbf{x})\n\\end{align*}\nbut $\\eta(\\mathbf{z}^*) = \\tilde{\\mathbf{y}} \\sleqtwo \\mathbf{y}^*$, with strict inequality.  However, because $g$ is monotone we can simply select $\\tilde{\\mathbf{y}}\\in\\lattwo$ as a minimizer as it must have equal cost to $\\mathbf{y}^*$.\n\\end{comment}\n\n\\begin{lemma}\\label{lem:minimizers}\nLet $\\mathbf{y}^*\\in\\lattwo$ be the minimizer for the relaxed problem \\eqref{eq:lattice_opt_prob_relax}, and let $\\mathbf{x}^*\\in\\latone$ be such that:\n\\begin{align*}\n\\mathbf{x}^*\\in\\underset{\\substack{\\mathbf{x}\\in\\latone\\\\\\eta(\\mathbf{x})\\leqtwo\\mathbf{y}^*}}{\\mathrm{argmin}}~f(\\mathbf{x}).\n\\end{align*}\nIf $g$ satisfies Assumption 2, then $\\mathbf{x}^*$ is a minimizer for the problem \\eqref{eq:lattice_opt_problem}.\n\\end{lemma}\n\\begin{proof}\nTo prove this lemma, we consider an optimal $\\mathbf{z}^*\\in\\latone$ for problem \\eqref{eq:lattice_opt_problem} and verify that the proposed minimizer, $\\mathbf{x}^*\\in\\latone$, has the same cost.\n\nWe first note that by the optimality of $\\mathbf{z}^*$ in problem \\eqref{eq:lattice_opt_problem}:\n\\begin{align}\nf(\\mathbf{z}^*) + g(\\eta(\\mathbf{z}^*)) \\leq f(\\mathbf{x}^*) + g(\\eta(\\mathbf{x}^*)). \\label{eq:optimality}\n\\end{align}\n\nAdditionally, we have:\n\\begin{align*}\n\\begin{array}{clc}\nf(\\mathbf{z}^*) + g(\\eta(\\mathbf{z}^*)) &\\geq \\underset{\\substack{\\mathbf{x}\\in\\latone\\\\\\eta(\\mathbf{x})\\leqtwo\\eta(\\mathbf{z}^*)}}{\\min}~f(\\mathbf{x}) + g(\\eta(\\mathbf{z}^*))&\\quad\\text{(by definition of minimum, as $\\mathbf{z}^*$ is feasible)} \\\\\n&= H(\\eta(\\mathbf{z}^*)) + g(\\eta(\\mathbf{z}^*))&\\quad\\text{(by definition of $H$)} \\\\\n&\\geq H(\\mathbf{y}^*) + g(\\mathbf{y}^*)&\\quad\\text{(by optimality of $\\mathbf{y}^*$ in \\eqref{eq:lattice_opt_prob_relax})}\\\\\n&= f(\\mathbf{x}^*) + g(\\mathbf{y}^*)&\\quad\\text{(by definition of $\\mathbf{x}^*$).}\n\\end{array}\n\\end{align*}\nThis sequence of inequalities implies:\n\\begin{align}\nf(\\mathbf{z}^*) + g(\\eta(\\mathbf{z}^*)) & \\geq f(\\mathbf{x}^*) + g(\\mathbf{y}^*). \\label{eq:lemma_two_ineq}\n\\end{align}\nBy construction, there are exactly two possible relationships between $\\mathbf{x}^*$ and $\\mathbf{y}^*$.\n\t\n\\textit{Case 1:} $\\eta(\\mathbf{x}^*) = \\mathbf{y}^*$.  In this case, \\eqref{eq:lemma_two_ineq} becomes:\n\\begin{align}\nf(\\mathbf{z}^*) + g(\\eta(\\mathbf{z}^*)) & \\geq f(\\mathbf{x}^*) + g(\\mathbf{y}^*)\\nonumber \\\\\n&= f(\\mathbf{x}^*) + g(\\eta(\\mathbf{x}^*)).\\label{eq:lemma_two_ineq_2}\n\\end{align}\nThen, combining inequality \\eqref{eq:lemma_two_ineq_2} with \\eqref{eq:optimality}, we have that:\n\\begin{gather*}\nf(\\mathbf{z}^*) + g(\\eta(\\mathbf{z}^*)) \\geq f(\\mathbf{x}^*) + g(\\eta(\\mathbf{x}^*)) \\geq f(\\mathbf{z}^*) + g(\\eta(\\mathbf{z}^*)) \\\\\n\\Rightarrow f(\\mathbf{z}^*) + g(\\eta(\\mathbf{z}^*)) = f(\\mathbf{x}^*) + g(\\eta(\\mathbf{x}^*)),\n\\end{gather*}\nand therefore $\\mathbf{x}^*$ is also a minimizer for problem \\eqref{eq:lattice_opt_problem}.\n\n\\textit{Case 2:} $\\eta(\\mathbf{x}^*)\\sleqtwo\\mathbf{y}^*$.  In this case, because $g$ is monotone, $g(\\mathbf{y}^*) \\geq g(\\eta(\\mathbf{x}^*))$.  Using this fact, we can lower bound the right-hand side of \\eqref{eq:lemma_two_ineq}:\n\\begin{align*}\nf(\\mathbf{z}^*) + g(\\eta(\\mathbf{z}^*)) &\\geq f(\\mathbf{x}^*) + g(\\mathbf{y}^*) \\\\\n&\\geq f(\\mathbf{x}^*) + g(\\eta(\\mathbf{x}^*)).\n\\end{align*}\nAt this point, we have obtained \\eqref{eq:lemma_two_ineq_2}, and we can follow the argument used in Case 1.  In both cases then, $\\mathbf{x}^*$ is a minimizer in problem \\eqref{eq:lattice_opt_problem}.\n\\end{proof}\n\nThis series of results gives rise to Theorem \\ref{thm:main_result}, which provides sufficient conditions under which we can transform problem \\eqref{eq:lattice_opt_problem}, an optimization problem on two lattices, into problem \\eqref{eq:lattice_opt_prob_relax}, a submodular function minimization problem on a single lattice.\n\n\\begin{proof} \\textit{(Theorem \\ref{thm:main_result})}\\\\ Under Assumptions 1 and 3, Theorem \\ref{thm:topkis} states that the function $g + H:\\lattwo\\rightarrow\\mathbb{R}$ is submodular on the lattice $\\lattwo$.  Therefore, solving \\eqref{eq:lattice_opt_prob_relax} is a submodular function minimization problem over $\\lattwo$, and the first part of the theorem is proved.\n\nUnder Assumption 2, by Lemma \\ref{lem:minimizers}, given the minimizer $\\mathbf{y}^*$ of \\eqref{eq:lattice_opt_prob_relax}, the point $\\mathbf{x}^*\\in\\latone$ defined by:\n\\begin{align*}\n\\mathbf{x}^*\\in\\underset{\\substack{\\mathbf{x}\\in\\latone\\\\\\eta(\\mathbf{x})\\leqtwo\\mathbf{y}^*}}{\\mathrm{argmin}}~f(\\mathbf{x}),\n\\end{align*} is a minimizer in the original problem \\eqref{eq:lattice_opt_problem}.\n\\end{proof}\n\n\n\\subsection{Solving \\eqref{eq:lattice_opt_prob_relax} in Polynomial Time\\label{sec:polytime}}\nDespite the efficiency of submodular function minimization, we can only truly solve \\eqref{eq:lattice_opt_prob_relax} in polynomial time if evaluating $g$ and $H$ is also a polynomial time operation. The function $H$, however, is implicitly defined through an optimization problem on $\\latone$ \\eqref{eq:H_definition}. Solving the problem \\eqref{eq:lattice_opt_prob_relax} in polynomial time then requires solving these smaller optimization problems defining $H$ efficiently.\n\nWe are particularly interested in joint continuous and discrete optimization, as illustrated by the example of $(\\latone,\\leqone) = (\\mathbb{R}^n_{\\geq 0},\\leqtwo)$ and $(\\lattwo,\\leqtwo) = (2^{[n]},\\subseteq)$ connected by the map \\mbox{$\\mathrm{supp}:\\mathbb{R}^n_{\\geq 0}\\rightarrow 2^{[n]}$} as expressed in \\eqref{eq:supp}.  In this case, evaluating $H$ requires solving the optimization problem:\n\\begin{align}\n\\minimize{\\substack{\\mathbf{x}\\in\\mathbb{R}^n_{\\geq 0}\\\\\\supp{\\mathbf{x}}\\subseteq A}} f(\\mathbf{x}),\\label{eq:h_cont_discrete}\n\\end{align}\nfor any $A\\in 2^{[n]}$.\n\n\\begin{comment}\nTo solve problem \\eqref{eq:lattice_opt_problem}, a submodular function minimization on $\\lattwo$, algorithms heavily rely on computing the Lov\\`asz extension of $g + H$ \\cite{fujishige2011submodular,schrijver2003combinatorial}.  Because $g + H$ is submodular, this computation has complexity $O(n\\log n + nEO)$, where $EO$ is the complexity of evaluating $g + H$.  However, $EO$ is the complexity of solving the sub-problem in \\eqref{eq:h_cont_discrete}, thus our choice of algorithm for this sub-problem directly impacts efficiency of computing the Lov\\`asz extension, and thereby the submodular set function minimization.\n\\end{comment}\n\nOne interesting note in the sub-problem \\eqref{eq:h_cont_discrete} is that for any $A\\in 2^{[n]}$, the feasible set is a convex subset of $\\mathbb{R}^n_{\\geq 0}$.  If the function $f:\\mathbb{R}^n_{\\geq 0}\\rightarrow\\mathbb{R}$ was convex, then we could use any generic convex optimization routine to solve \\eqref{eq:h_cont_discrete}.  We already assumed that $f$ is submodular on $\\mathbb{R}^n_{\\geq 0}$, but submodular functions are neither a subset nor a superset of convex functions, so we could also require that $f$ is convex, making evaluating $H$ and solving \\eqref{eq:lattice_opt_problem} efficient.\n\n\\begin{corollary}\\label{cor:polytime}\nLet $f:\\mathbb{R}^n_{\\geq 0}\\rightarrow\\mathbb{R}$ be a submodular and convex function on $(\\mathbb{R}^n_{\\geq 0},\\leq)$, $\\lattwo$ be a finite lattice, $g:\\lattwo\\rightarrow\\mathbb{R}$ be a monotone submodular set function, and let $\\eta:\\mathbb{R}^n_{\\geq 0}\\rightarrow\\lattwo$ satisfy Assumption 3.  Further assume that for every $\\mathbf{y}\\in\\lattwo$, the set of $\\mathbf{x}\\in\\latone$ such that $\\eta(\\mathbf{x})\\leqtwo\\mathbf{y}$ is a convex subset of $\\mathbb{R}^n_{\\geq 0}$.  Then the problem:\n\\begin{align*}\n\\minimize{\\mathbf{x}\\in\\mathbb{R}^n_{\\geq 0}}~f(\\mathbf{x}) + g(\\eta(\\mathbf{x})),\n\\end{align*}\ncan be solved in polynomial time.\n\\end{corollary}\n\\begin{proof}\nIt follows from the corollary's assumptions that Assumptions 1, 2, and 3 are satisfied by the lattices $(\\mathbb{R}^n_{\\geq 0},\\leq)$, $(\\lattwo,\\leqtwo)$, and the functions $\\eta:\\mathbb{R}^n_{\\geq 0}\\rightarrow \\lattwo$, $f:\\mathbb{R}^n_{\\geq 0}\\rightarrow\\mathbb{R}$ and $g:\\lattwo\\rightarrow\\mathbb{R}$.  By Theorem \\ref{thm:main_result}, we can solve the problem \\eqref{eq:lattice_opt_problem} by instead minimizing the submodular function $H$ over $\\lattwo$, i.e., problem \\eqref{eq:lattice_opt_prob_relax}.  Submodular function minimization has polynomial complexity in both $\\vert\\lattwo\\vert$, which is finite by assumption, and the number of function evaluations of $H$.  Because $f$ is convex, evaluating $H$ is also a polynomial time operation, and the complexity of solving \\eqref{eq:lattice_opt_prob_relax} is polynomial.\n\\end{proof}\nOur theory is agnostic to the choice of subroutines both for evaluating $H$ and solving the set function minimization problem.  If we assume $f$ is convex, evaluate it through convex optimization, and use projected subgradient descent on the Lov\\`asz extension of $H$ as the algorithm for solving the set function minimization, we recover exactly the approach proposed by \\cite{eloptimal}.\n\n\\begin{comment}\n\n\\subsubsection{Continuous Submodularity Alone}\nBy Assumption 1, $f$ is submodular on the lattice $\\mathbb{R}^n_{\\geq 0}$.  Moreover, given any $A\\in 2^{[n]}$, the set:\n\\begin{align*}\nS = \\{\\mathbf{x}\\in\\mathbb{R}^n_{\\geq 0}\\mid \\supp{\\mathbf{x}}\\subseteq A\\},\n\\end{align*}\nis a lattice ideal (and also a sublattice) of $\\mathbb{R}^n_{\\geq 0}$ \\cite{davey2002introduction}.  Given this structure, we can directly apply algorithms for continuous submodular function minimization provided in \\cite{bach2019submodular} to solve the sub-problem in \\eqref{eq:h_cont_discrete}.\n\nNo additional structure in $f$ is required in this case, but the algorithms for continuous submodular function  minimization require discretizing the domain of the function, $S\\subseteq\\mathbb{R}^n_{\\geq 0}$.  However, for any $A\\in 2^n$ with $A\\neq \\emptyset$, the ideal $S$ is unbounded and cannot be perfectly discretized.  Moreover, discretizing $S$ enough to produce high accuracy evaluations of $H$ can drastically increase the running times.\n\nIf the set $S$ is replaced with a bounded sublattice $S_\\alpha = \\{\\mathbf{x}\\in[0,\\alpha]^n\\mid \\supp{\\mathbf{x}}\\subseteq A\\}$ and is then discretized into $k$ values at each index, and the function $f$ is $L$-Lipschitz continuous function, the complexity of reaching $\\epsilon$ suboptimality using the algorithms in \\cite{bach2019submodular} is $O\\left(\\left(\\frac{2L\\alpha n}{\\epsilon}\\right)^3\\log\\left(\\frac{2L\\alpha n}{\\epsilon}\\right)\\right)$.  Plugging this complexity into the  expression for the Lov\\`asz extension gives an overall complexity of order $O((n + n^4)\\log n)$.\n\\end{comment}\n\nConvexity of $f$ is not the only assumption that leads to tractable evaluations of $H$. As an alternative, we could consider a nonconvex quadratic form for $f:\\mathbb{R}^n_{\\geq 0}\\rightarrow\\mathbb{R}$:\n\\begin{align}\nf(\\mathbf{x}) &= \\mathbf{x}^T\\mathbf{Q}\\mathbf{x} + \\mathbf{p}^T\\mathbf{x},\\label{eq:quadratic_f}\n\\end{align}\nwith $\\mathbf{Q}\\in\\mathbb{R}^{n\\times n}$ and $\\mathbf{p}\\in\\mathbb{R}^n$.  The assumption that this quadratic function is submodular on $\\mathbb{R}^n_{\\geq 0}$ is equivalent to the condition:\n\\begin{align*}\n\\frac{\\partial^2 f}{\\partial \\mathbf{x}_i\\partial\\mathbf{x}_j} &= \\mathbf{Q}_{ij} \\leq 0, \\quad \\text{for all }i\\neq j.\n\\end{align*}\nMoreover, our sub-problem instance \\eqref{eq:h_cont_discrete} is a constrained, nonconvex quadratic program:\n\\begin{align}\n\\begin{array}{cc}\n\\minimize{\\mathbf{x}\\in\\mathbb{R}^n} & \\mathbf{x}^T\\mathbf{Q}\\mathbf{x} + 2\\mathbf{p}^T\\mathbf{x} \\\\\n\\text{subject to}& \\mathbf{x} \\geq 0 \\\\\n& \\mathbf{x}_i = 0, ~i\\notin A.\n\\end{array}\\label{eq:quadratic_h}\n\\end{align}\nResearchers \\cite{kim2003exact} have established that nonconvex quadratic programs satisfying submodularity admit tight semidefinite program relaxations. In particular, we have the following theorem:\n\\begin{theorem}\n\\textit{Theorem 3.1 in \\cite{kim2003exact}})  Let $\\mathbf{Q}\\in\\mathbb{R}^{n\\times n}$ have nonpositive off-diagonal entries. Let $\\mathrm{tr}:\\mathbb{R}^{n\\times n}\\rightarrow\\mathbb{R}$ denote the trace of a matrix, $\\mathrm{diag}:\\mathbb{R}^{n\\times n}\\rightarrow\\mathbb{R}^n$ denote the diagonal entries of the matrix, and let $\\succeq$ indicate the positive semidefiniteness of a symmetric matrix. Further, for any $A\\in 2^{[n]}$, let $\\mathbf{Z}_{A^c}$ denote the rows and columns of $\\mathbf{Z}$ with indices not in the set $A$. Consider the semi-definite program:\n\\begin{align*}\n\\begin{array}{cc}\n\\minimize{\\substack{\\mathbf{z}\\in\\mathbb{R}^n \\\\ \\mathbf{Z}\\in \\mathbb{S}^n}} & \\tr{\\mathbf{QZ}} + 2\\mathbf{p}^T\\mathbf{z} \\\\\n\\text{subject to} & \\tr{\\mathbf{Z}_{A^c}} \\leq 0 \\\\\n& \\diag{\\mathbf{Z}} \\geq 0 \\\\\n& \\begin{bmatrix}\n1 & \\mathbf{z}^T \\\\\n\\mathbf{z} & \\mathbf{Z}\n\\end{bmatrix} \\succeq 0,\n\\end{array}\n\\end{align*}\nGiven the solution $(\\mathbf{Z}^*,\\mathbf{z}^*)$ to this SDP, the vector $\\mathbf{x}^*_i = \\sqrt{\\mathbf{Z}^*_{ii}}$, $i=1,...,n$ is a minimizer for the non-convex quadratic program \\eqref{eq:quadratic_h}.\n\\end{theorem}\n\nBecause semi-definite programs can be solved in polynomial time, we could use this relaxation to evaluate $H$ for any subset $A\\in 2^{[n]}$ in polynomial time.  As before, this ability would produce an identical statement to Corollary \\ref{cor:polytime}, but for functions $f$ of the form \\eqref{eq:quadratic_f} that satisfy submodularity.\n\n\n\\begin{comment}\n\\begin{corollary}\nLet $f:\\mathbb{R}^n_{\\geq 0}\\rightarrow\\mathbb{R}$ be a submodular function on $(\\mathbb{R}^n_{\\geq 0},\\leq)$ and $g:2^{[n]}\\rightarrow\\mathbb{R}$ be a monotone submodular set function.  Let $\\mathrm{supp}:\\mathbb{R}^n_{\\geq 0}\\rightarrow\\mathbb{R}$ be the support map as expressed in \\eqref{eq:supp}.  If $f$ is also convex, or a nonconvex quadratic form, the problem:\n\\begin{align*}\n\\minimize{\\mathbf{x}\\in\\mathbb{R}^n_{\\geq 0}}~f(\\mathbf{x}) + g(\\supp{\\mathbf{x}})\n\\end{align*}\ncan be solved in polynomial time.\n\\end{corollary}\n\\begin{proof}\nFor the lattices $(\\mathbb{R}^n_{\\geq 0},\\leq)$ and $(2^{[n]},\\subseteq)$, the function $\\mathrm{supp}:\\mathbb{R}^n_{\\geq 0}\\rightarrow 2^{[n]}$ satisfies Assumption 3.  As the functions $f$ and $g$ satisfy Assumptions 1 and 2, by Theorem \\ref{thm:main_result}, solving the problem:\n\\begin{align}\n\\minimize{A\\in 2^{[n]}} g(A) + H(A)\\label{eq:set_fn_min}\n\\end{align}\nwith $H$ as defined in \\eqref{eq:H_definition} is a submodular set function minimization that gives the solution to the original discrete-continuous problem.\n\nThe complexity of submodular set function minimization is polynomial in $n$ and the number of function evaluations.  By the assumptions on $f$, evaluating $g + H$ is also a polynomial time operation, meaning the overall complexity of solving \\eqref{eq:set_fn_min} is polynomial.\n\\end{proof}\n\\end{comment}\n\n\\section{Constrained Optimization}\nIn this section, we extend our framework both theoretically and algorithmically to accommodate constraints for the specific case of the lattices $(\\mathbb{R}^n_{\\geq 0}, \\leqone)$ and $(2^{[n]},\\subseteq)$, connected by the support map $\\mathrm{supp}:\\mathbb{R}^n_{\\geq 0}\\rightarrow 2^{[n]}$.\n\nIn many problems, we may be interested in optimization over a feasible subset $C\\subset\\mathbb{R}^n_{\\geq 0}$ that is strictly contained in $\\mathbb{R}^n_{\\geq 0}$.  Unfortunately, submodular function minimization and maximization subject to constraints is NP-Hard in general \\cite{fujishige2011submodular}.  This difficulty arises because arbitrary subsets of a lattice rarely define sublattices.\n\nOne simple class of problems whose feasible sets do not define sublattices are problems with \\emph{budget constraints}:\n\\begin{align}\n\\begin{array}{cc}\n\\minimize{\\mathbf{x}\\in\\mathbb{R}^n_{\\geq 0}} & f(\\mathbf{x}) + g(\\supp{\\mathbf{x}}) \\\\\n\\text{subject to} & \\sum_{i=1}^nW_i(\\mathbf{x}_i) \\leq B,\n\\end{array}\\label{eq:constrained_case}\n\\end{align}\nwith $W_i:\\mathbb{R}{\\geq 0}\\rightarrow\\mathbb{R}$ strictly increasing functions for $i = 1,2,...,n$ and $B \\in \\mathbb{R}_{>0}$ a ``budget''.\n\n\\begin{comment}\n\\blue{JB: this small ``example'' section could be removed if it's not helpful.}\n\nFor instance, consider the feasible set $C = \\{\\mathbf{x}\\in\\mathbb{R}^n_{\\geq 0}\\mid \\sum_{i=1}^n \\mathbf{x}_i \\leq B\\}$ with $n \\geq 2$.  Consider the two points $\\mathbf{x},\\mathbf{x}'\\in C$ where $\\mathbf{x}_i = B$ and $\\mathbf{x}'_j = B$, $i\\neq j$ and note:\n\\begin{align*}\n\\sum_{i=1}^n(\\mathbf{x}\\joinone\\mathbf{x}')_i &= \\sum_{i=1}^n\\max\\{\\mathbf{x}_i,\\mathbf{x}_i'\\} = 2B \\nleq B.\n\\end{align*}\nTherefore $\\mathbf{x}\\joinone\\mathbf{x}' \\notin C$, thus $C$ is not a sublattice.\n\nWe may also consider discrete budget constraints, where $W(\\mathbf{x}) = \\sum_{j\\in\\supp{\\mathbf{x}}}\\mathbf{w}_j$ for some $\\mathbf{w}\\in\\mathbb{R}^n_{\\geq 0}$.  We refer to this as a support knapsack, as we ``pay'' some cost $\\mathbf{w}_j$ to set $\\mathbf{x}_j \\neq 0$ for each $j= 1,2,...,n$.\n\nIn this case, the feasible set is then $C =\\{\\mathbf{x}\\in\\mathbb{R}^n_{\\geq 0}\\mid \\sum_{j\\in\\supp{\\mathbf{x}}}\\mathbf{w}_j\\leq B\\}$, which is again not necessarily a sublattice of $\\mathbb{R}^n$.\n\n\\blue{(End small example section.)}\n\\end{comment}\n\nWhen confronted with constrained optimization problems such as \\eqref{eq:constrained_case}, one common approach is to add a Lagrange multiplier $\\mu\\in\\mathbb{R}_{\\geq 0}$ and instead solve the unconstrained problem:\n\\begin{align}\n\\minimize{\\mathbf{x}\\in\\mathbb{R}^n_{\\geq 0}}&~f(\\mathbf{x}) + g(\\supp{\\mathbf{x}}) + \\mu \\sum_{i=1}^nW_i(\\mathbf{x}_i).\\label{eq:regularized_case}\n\\end{align}\nFor the correct choice of $\\mu\\in\\mathbb{R}_{\\geq 0}$, solving the regularized problem \\eqref{eq:regularized_case} is equivalent to solving the constrained problem \\eqref{eq:constrained_case}.  Determining the $\\mu$ that renders the two problems equivalent, however, is typically a difficult task.\n\n\n\nOur work in this section relies on the following result that relates parameterized families of submodular set function minimization problems to a single convex optimization problem.\n\n\\begin{theorem}(Proposition 8.4 in \\cite{bach2013learning})\\label{thm:bach_convex_submod} Let $g:2^{[n]}\\rightarrow\\mathbb{R}$ be a submodular set function, and $g_L:\\mathbb{R}^n\\rightarrow\\mathbb{R}$ its Lov\\`asz extension (which is therefore convex).  If, for some $\\epsilon > 0$, $\\psi_i:\\mathbb{R}_{\\geq \\epsilon}\\rightarrow\\mathbb{R}$ is a strictly increasing function on its domain for all $i=1,2,...,n$, then the minimizer $\\mathbf{u}^*\\in\\mathbb{R}^n$ of the convex optimization problem:\n\\begin{align}\\label{eq:single_convex}\n\\minimize{\\mathbf{u}\\in\\mathbb{R}^n_{\\geq 0}}~h_L(\\mathbf{u})+\\sum_{i=1}^n\\int_{\\epsilon}^{\\epsilon+\\mathbf{u}_i}\\psi_i(\\mu)d\\mu,\n\\end{align}\nis such that the set $A^\\mu = \\{i\\in [n]~:~\\mathbf{u}_i^* > \\mu\\}$ is the minimizer with smallest cardinality for the submodular set function minimization problem:\n\\begin{align}\\label{eq:family_submodular}\n\\minimize{A\\in 2^{[n]}}~h(A) + \\sum_{i\\in A}\\psi_i(\\mu),\n\\end{align}\nfor any $\\mu\\in\\mathbb{R}_{\\geq \\epsilon}$.\n\\end{theorem}\n\nIn the following subsections we identify classes of problems that allow the regularized problem \\eqref{eq:regularized_case} to be expressed in the form given by \\eqref{eq:family_submodular}.  Theorem \\ref{thm:bach_convex_submod} then provides a single convex optimization problem we can solve to recover the solution to \\eqref{eq:regularized_case} for all possible values of the regularization strength $\\mu$.  In prior work, this same theory was applied to purely discrete submodular minimization problems \\cite{fujishige2011submodular}, and purely continuous submodular minimization problems \\cite{staib2019robust}, but our work lies between these two extremes.\n\n\\subsection{Support Knapsack Constraints}\nWe first consider a knapsack constraint, meaning the function $W$ has the form:\n\\begin{align*}\nW(\\mathbf{x}) &= \\sum_{j\\in\\supp{\\mathbf{x}}}\\mathbf{w}_j,\n\\end{align*}\nfor some $\\mathbf{w}\\in\\mathbb{R}^n_{>0}$.  The regularized problem \\eqref{eq:regularized_case} in this case is:\n\\begin{align*}\n\\minimize{\\mathbf{x}\\in\\mathbb{R}^n_{\\geq 0}}&~f(\\mathbf{x}) + g(\\supp{\\mathbf{x}}) + \\mu \\sum_{j\\in\\supp{\\mathbf{x}}}\\mathbf{w}_j.\n\\end{align*}\nBecause $W$ can be viewed as a set function in this case, the corresponding relaxed problem \\eqref{eq:lattice_opt_prob_relax} becomes:\n\\begin{align}\n\\minimize{A\\in 2^{[n]}}~g(A) + H(A) + \\sum_{j\\in A}\\psi_j(\\mu),\\label{eq:regularized_support_knapsack}\n\\end{align}\nwhere we have defined $\\psi_j(\\mu) = \\mu\\mathbf{w}_j$ for each $j=1,2,...,n$.  Because $\\mathbf{w}_j>0$ for all $j$, these functions are strictly increasing, and we have a problem precisely in the form \\eqref{eq:family_submodular}.  By Theorem \\ref{thm:bach_convex_submod}, we can solve the convex optimization problem:\n\\begin{align*}\n\\minimize{\\mathbf{u}\\in\\mathbb{R}_{\\geq \\epsilon}}~g_L(\\mathbf{u}) + H_L(\\mathbf{u}) + \\frac{1}{2}\\sum_{j=1}^n\\mathbf{w}_j\\mathbf{u}_j^2,\n\\end{align*}\nthen appropriately threshold the solution to recover the solution to \\eqref{eq:regularized_support_knapsack} for all possible values of $\\mu\\in\\mathbb{R}_{\\geq\\epsilon}$.  Because $\\psi_j$ is finite and strictly increasing on all of $\\mathbb{R}$, we can simply select $\\epsilon = 0$.\n\nGiven the solutions to the regularized problem $A^\\mu$ specified by Theorem \\ref{thm:bach_convex_submod}, we select the set $A^\\mu$ with smallest $\\mu\\in\\mathbb{R}$ such that the constraint $W(\\mathbf{x}) \\leq B$ is satisfied.  We can then use the result of Theorem \\ref{thm:main_result} to compute the minimizer in the original optimization problem over $\\mathbb{R}^n_{\\geq 0}$.\n\n\\subsection{Continuous Budget Constraints}\nAs shown above, the Lov\\`asz extension lets us handle problems with discrete budget constraints, so a natural next step is to consider continuous budget constraints.  In particular, these are continuous functions $W:\\mathbb{R}^n_{\\geq 0}\\rightarrow\\mathbb{R}$, such that:\n\\begin{align*}\nW(\\mathbf{x}) &= \\sum_{i=1}^nW_i(\\mathbf{x}_i),\n\\end{align*}\nwith each $W_i:\\mathbb{R}_{\\geq 0}\\rightarrow\\mathbb{R}$ a strictly increasing function. With this particular $W$, the regularized optimization problem \\eqref{eq:regularized_case} with Lagrange multiplier $\\mu\\in\\mathbb{R}_{\\geq 0}$ becomes:\n\\begin{align*}\n\\minimize{\\mathbf{x}\\in\\mathbb{R}^n_{\\geq 0}}~f(\\mathbf{x}) + g(\\supp{\\mathbf{x}}) + \\mu\\sum_{i=1}^nW_i(\\mathbf{x}_i).\n\\end{align*}\nTo recover the problem form \\eqref{eq:family_submodular} specified by Theorem \\ref{thm:bach_convex_submod}, we further assume that $f:\\mathbb{R}^n_{\\geq 0}\\rightarrow\\mathbb{R}$ is separable, i.e., $f(\\mathbf{x}) = \\sum_{i=1}^nf_i(\\mathbf{x}_i)$.  In this case, the relaxed optimization problem \\eqref{eq:lattice_opt_prob_relax} is:\n\\begin{align}\n\\minimize{A\\in 2^{[n]}}~g(A) + \\sum_{i\\in A} H_i(\\mu),\\label{eq:cont_budget_regularized}\n\\end{align}\nwhere we defined $H_i : \\mathbb{R}_{> 0}\\rightarrow\\mathbb{R}$ as the function:\n\\begin{align}\nH_i(\\mu) &= \\underset{\\mathbf{z}\\geq 0}{\\min}~f_i(\\mathbf{z}) + \\mu W_i(\\mathbf{z}),\\quad i = 1,2,...,n,\\label{eq:scalar_H}\n\\end{align}\nand assumed (without loss of generality) that $W_i(0) = f_i(0) = 0$.\n\nTo apply Theorem \\ref{thm:bach_convex_submod}, we need $H_i:\\mathbb{R}_{> 0}\\rightarrow\\mathbb{R}$ to be strictly increasing on its domain.  We verify this property in the following proposition, whose proof we detail in Appendix \\ref{apdx:continuous_constraints}.\n\n\\begin{proposition}\nThe function $H_i:\\mathbb{R}_{\\geq 0}\\rightarrow\\mathbb{R}_{\\leq 0}$ defined in \\eqref{eq:scalar_H} is monotone in $\\mu$ for all $i=1,2...,n$.  It is strictly increasing for all $\\mu\\in[0,c]$, where $c\\in\\mathbb{R}_{\\geq 0}$ is the smallest constant such that $H_i(c) = 0$.  In addition, $H_j$ is constant and zero on the interval $[c,\\infty[$.\n\\end{proposition}\n\nBecause the only point at which $H_i$ is not strictly increasing occurs when its value is exactly zero (implying that allowing the element $\\mathbf{x}_i$ to be nonzero provides no decrease in continuous cost), the desired result from Theorem \\ref{thm:bach_convex_submod} still holds with only a minor modification, the details of which we also defer to Appendix \\ref{apdx:continuous_constraints}.\n\nIt then follows from Theorem \\ref{thm:bach_convex_submod} that again, by solving the single convex optimization problem:\n\\begin{align}\n\\minimize{\\mathbf{u}\\in\\mathbb{R}^n_{\\geq 0}}~g_L(\\mathbf{u}) + \\sum_{i=1}^n\\int_\\epsilon^{\\epsilon+\\mathbf{u}_i}H_i(\\mu)d\\mu,\n\\end{align}\nwe can recover the solution to a family of regularized optimization problems \\eqref{eq:cont_budget_regularized}.  As before, we select the set $A^\\mu$ with the largest $\\mu\\in\\mathbb{R}_{\\geq \\epsilon}$ such that the budget constraint $W(\\mathbf{x})\\leq B$ is satisfied.\n\n\\section{Robust Optimization}\nGiven our ability to efficiently solve the joint continuous and discrete optimization problem in \\eqref{eq:cont_discrete_opt_problem}, and even some of its constrained variants, we may seek problem settings where they arise as a subproblem.  This situation often occurs in \\emph{robust optimization}, where we seek to solve an optimization problem while remaining resilient to worst-case problem instances.\n\n\\subsection{Motivating Example from Multiple Domain Learning}\nRecent work in \\cite{qian2019robust} highlighted the concept of \\emph{multiple domain learning}, where a single machine learning model is trained on sets of data from $K$ different domains.  By training against worst-case distributions of the data in these domains, they show that the resulting machine learning model often achieves lower generalization and worst-case testing errors.\n\nIn particular, let the training data for a learning model be $S = \\{S_1,S_2,...,S_K\\}$ with $S_i$ the data from domain $i$.  We also let $f_i : W\\rightarrow\\mathbb{R}$ for $i=1,2,...,K$ be the empirical risk of the model on the data from each domain $i$, given parameters in some convex subset $W\\subseteq\\mathbb{R}^n$.  The proposed robust optimization problem is then:\n\\begin{align*}\n\\minimize{\\mathbf{w}\\in W}~\\underset{\\mathbf{p}\\in C}{\\max}~\\sum_{i=1}^K\\mathbf{p}_if_i(\\mathbf{w}),\n\\end{align*}\nwith $C = \\{\\mathbf{p}\\in \\mathbb{R}^K_{\\geq 0}\\mid \\sum_{i=1}^K\\mathbf{p}_i \\leq 1\\}$, the simplex.  If we additionally reward the use of data from domain $i$ (or equivalently, penalize the worst-case distribution of data for including domain $i$), then we form the robust continuous and discrete optimization problem:\n\\begin{align*}\n\\minimize{\\mathbf{w}\\in W}~\\underset{\\mathbf{p}\\in C}{\\max}~\\sum_{i=1}^K\\mathbf{p}_if_i(\\mathbf{w}) - g(\\supp{\\mathbf{p}}),\n\\end{align*}\nwith $g: 2^K\\rightarrow\\mathbb{R}$ a monotone submodular set function. By considering a penalty on the set of nonzero entries of the worst-case distribution, we encode some prioritization of which domains are more or less relevant to us in our application.  Then by Theorem \\ref{thm:bach_convex_submod}, we can solve the inner maximization problem (with an appropriate change of signs) by adding a Lagrange multiplier $\\mu$ and solving a related convex problem.\n\n\\subsection{General Results}\nMore generally, robust optimization problems can often be expressed as a min-max saddle point optimization problem of a function $q:\\mathcal{X}\\times\\mathcal{Y}\\rightarrow\\mathbb{R}$:\n\\begin{align}\n\\maximize{\\mathbf{x}\\in\\mathcal{X}}\\underset{\\mathbf{y}\\in \\mathcal{Y}}{\\min}~q(\\mathbf{x},\\mathbf{y}). \\label{eq:saddle_point_prob}\n\\end{align}\nThis problem is interpreted as maximizing the function $q(\\mathbf{x},\\mathbf{y})$ with respect to our available parameters $\\mathbf{x}\\in\\mathcal{X}\\subseteq \\mathbb{R}^n$, under the worst case choice of additional problem parameters $\\mathbf{y}\\in \\mathcal{Y}\\subseteq\\mathbb{R}^m_{\\geq 0}$ \\cite{ben2009robust}.\n\nGiven some appropriate structure for the function $q$, the min-max problem  \\eqref{eq:saddle_point_prob} is surprisingly tractable.  If we define $Q:\\mathcal{X}\\rightarrow\\mathbb{R}$ as:\n\\begin{align*}\nQ(\\mathbf{x}) &= \\underset{\\mathbf{y}\\in \\mathcal{Y}}{\\min}~q(\\mathbf{x},\\mathbf{y}),\n\\end{align*}\nwe can express the saddle-point problem \\eqref{eq:saddle_point_prob} as:\n\\begin{align}\n\\maximize{\\mathbf{x}\\in\\mathcal{X}}~Q(\\mathbf{x}).\\label{eq:Q_opt}\n\\end{align}\nIf the function $q(\\mathbf{x},\\mathbf{y})$ is concave in $\\mathbf{x}$ for any fixed $\\mathbf{y}\\in \\mathcal{Y}$, then the function $Q$ is concave \\cite{borwein2000convex}. Moreover, we can compute a subgradient of $Q$ at any $\\mathbf{x}\\in\\mathcal{X}$ as:\n\\begin{gather*}\n\\nabla_{\\mathbf{x}}Q(\\mathbf{x}_0) = \\nabla_\\mathbf{x} q(\\mathbf{x}_0,\\mathbf{y}^*), \\\\\n\\mathbf{y}^*\\in\\textrm{arg}\\min{\\mathbf{y}\\in\\mathcal{Y}}~q(\\mathbf{x}_0,\\mathbf{y}).\n\\end{gather*}\nIn other words, efficiently solving the minimization problem defining $Q$ for an $\\mathbf{x}_0\\in\\mathcal{X}$ also gives a subgradient of $Q$.  Because $Q$ is concave in $\\mathbf{x}$, even a straightforward algorithm such as projected subgradient ascent in the problem \\eqref{eq:Q_opt} will converge to a global optimum.\n\nIn this work, we showed that minimization problems in the form of \\eqref{eq:cont_discrete_opt_problem} with functions satisfying Assumptions 1-3 can be solved efficiently.  Suppose then, that the function $q:\\mathcal{X}\\times\\mathcal{Y}$ is of the form:\n\\begin{align*}\nq(\\mathbf{x},\\mathbf{y}) &= f(\\mathbf{x},\\mathbf{y}) + g(\\eta(\\mathbf{y}))\n\\end{align*}\nwith $f:\\mathcal{X}\\times \\mathcal{Y}\\rightarrow\\mathbb{R}$ concave in $\\mathbf{x}$ for any fixed $\\mathbf{y}$ and also convex and submodular on $\\mathcal{Y}\\subseteq \\mathbb{R}^n_{\\geq 0}$ in $\\mathbf{y}$ for any fixed $\\mathbf{x}$.  If $\\eta:\\mathcal{Y}\\rightarrow\\mathcal{L}$ satisfies Assumption 3, $g:\\mathcal{L}\\rightarrow\\mathbb{R}$ is monotone and submodular, and we assume the set of $\\mathbf{y}\\in\\mathcal{Y}$ such that $\\eta(\\mathbf{y})\\leq \\mathbf{l}$ is a convex subset for any $\\mathbf{l}\\in\\mathcal{L}$, then the robust optimization problem \\eqref{eq:saddle_point_prob} becomes:\n\\begin{align}\n\\maximize{\\mathbf{x}\\in\\mathbb{R}^n}~\\underset{\\mathbf{y}\\in \\lattwo}{\\min}~f(\\mathbf{x},\\mathbf{y}) + g(\\eta(\\mathbf{y})).\\label{eq:max_min_prob}\n\\end{align}\n\nFor a given $\\mathbf{x}_0\\in\\mathbb{R}^n$, we view the selection of $\\mathbf{y}\\in \\mathcal{Y}$ as a worst-case, or ``adversarial'' choice of parameters for the function $f$.  The penalty on $\\eta(\\mathbf{y})$ then suggests that by simply considering some structured sets of adversarial parameters $\\eta(\\mathbf{y})$ while selecting the optimal $\\mathbf{x}$, we gain some associated benefit.  The submodularity of $g$ then implies that as we consider more potential variations in $\\mathbf{y}$ (i.e., larger $\\supp{\\mathbf{y}}$) the ``gain'' we receive for considering more variations is smaller.\n\nIn addition, $Q$ becomes:\n\\begin{align*}\nQ(\\mathbf{x}) &= \\underset{\\mathbf{y}\\in \\lattwo}{\\min}~f(\\mathbf{x},\\mathbf{y}) + g(\\eta(\\mathbf{y})),\n\\end{align*}\nwhich is still the minimum of a family of concave functions, and therefore amenable to subgradient ascent methods as discussed above.  A subgradient of $Q$ can easily be computed as:\n\\begin{gather*}\n\\nabla_{\\mathbf{x}} Q(\\mathbf{x}_0) = \\nabla_{\\mathbf{x}}q(\\mathbf{x}_0,\\mathbf{y}^*) = \\nabla_\\mathbf{x} f(\\mathbf{x}_0,\\mathbf{y}^*), \\\\\n\\mathbf{y}^* \\in \\textrm{arg}\\min{\\mathbf{y}\\in\\lattwo}~f(\\mathbf{x}_0,\\mathbf{y}) + g(\\eta(\\mathbf{y})).\n\\end{gather*}\nWe collect these ideas into the following theorem.\n\n\\begin{theorem}\\label{thm:robust_opt}\nConsider the robust optimization problem \\eqref{eq:max_min_prob}.  Assume $f:\\latone\\times\\lattwo\\rightarrow\\mathbb{R}$ is concave in $\\mathbf{x}\\in\\latone$ for any fixed $\\mathbf{y}\\in\\lattwo$, and also convex and submodular in $\\mathbf{y}\\in\\lattwo$ for any fixed $\\mathbf{x}\\in\\latone$.  Let $\\eta:\\lattwo\\rightarrow\\mathcal{L}$ satisfy Assumption 3, $g:\\mathcal{L}\\rightarrow\\mathbb{R}$ be a monotone submodular function and assume that for a given $\\mathbf{\\ell}\\in\\mathcal{L}$, the set of $\\mathbf{y}\\in\\lattwo$ such that $\\eta(\\mathbf{y})\\leqtwo \\mathbf{\\ell}$ is a convex subset of $\\lattwo$.  For any $\\epsilon \\in\\mathbb{R}_{>0}$, let $T\\in\\mathbb{Z}_{>0}$ be of order $O(\\frac{1}{\\epsilon^2})$, meaning as $T$ tends to infinity, there exists a constant $M\\in\\mathbb{R}_{>0}$ such that $T \\leq \\frac{M}{\\epsilon^2}$.  Then $T$ iterations of projected subgradient ascent using step lengths $\\eta_i = \\frac{1}{\\sqrt{T}}$ produces, in polynomial time, iterates $\\mathbf{x}^{(i)}\\in\\latone$ for $i = 1,2,...,T$ such that $\\frac{1}{T}\\sum_{i=1}^TQ(\\mathbf{x}^{(i)}) \\leq Q(\\mathbf{x}^*) + \\epsilon$.\n\\end{theorem}\nThe computational complexity of this approach may be high, as projected subgradient ascent can be slow in practice.  However, each sub-problem instance involves a mixed continuous and discrete optimization problem, so this complexity is warranted.\n\n\\section{Examples and Computational Evaluation}\nIn this section, we illustrate the proposed theoretical results on several numerical examples involving optimization on the lattices $\\mathbb{R}^n_{\\geq 0}$ and $2^{[n]}$.  We compare against two state-of-the-art techniques: a direct application of the continuous submodular function minimization algorithms outlined in \\cite{bach2019submodular}, and the projected subgradient descent method proposed in \\cite{eloptimal}.  \n\nThe algorithms for continuous submodular function minimization rely on discretizing the domain $\\mathbb{R}^n_{\\geq 0}$ into $k$ discrete points in each dimension, so we consider the domain $[0,1]^n\\subseteq\\mathbb{R}^n_{\\geq 0}$ and set the discretization level to $k = 51$ unless otherwise specified.  In our implementation, we use the Pairwise Frank-Wolfe algorithm to solve the discretized optimization problem, with all relevant results plotted in blue and labeled \\textit{Cont Submodular}.\n\nThe projected subgradient method is known to provide approximation guarantees even in the non-submodular case \\cite{eloptimal}, but as shown in Section \\ref{sec:polytime}, amounts to a specific choice of algorithms in our theory.  To implement this approach, we use IBM's CPLEX 12.8 constrained quadratic program solver in MATLAB to evaluate the function $H$ (as expressed in \\eqref{eq:H_definition}) and use Polyak's rule for updating the step size.  The relevant results are plotted in red, and labeled \\textit{Projected (Sub)Gradient} in figures.\n\nOur approach is agnostic to the choice of convex optimization and submodular set function minimization routines, so we also use CPLEX to evaluate $H$.  However, we instead use the minimum-norm point algorithm from \\cite{fujishige2011submodular} as implemented in MATLAB by \\cite{krause2010sfo}, which has fast performance in practice, coupled with the semi-gradient based lattice pruning strategy proposed in \\cite{iyer2013fast}.  Our results are plotted in black, and labeled \\textit{Min Norm + CPLEX} in figures.\n\nAll three methods are presented with identical cost functions to minimize, and are run until either convergence to suboptimality below $10^{-4}$ or a maximum of 100 iterations.  The experiments were all run on a mid-2012 Macbook Pro with a 2.9 GHz Intel Core i7 processor and 16GB of 1600 MHz DDR3 RAM, and reported running times are an average of five runs on randomized problem instances.\n\n\\subsection{Regularized Sparse Regression}\nWe first examine a regularized sparse regression problem, similar in spirit to \\eqref{eq:compressed_sensing}.  Consider some $\\mathbf{x}\\in\\mathbb{R}^n_{\\geq 0}$, $\\mathbf{D}\\in\\mathbb{R}^{m\\times n}$, $\\mathbf{b}\\in\\mathbb{R}^m$, and define the function $f:\\mathbb{R}^n_{\\geq 0}\\rightarrow\\mathbb{R}$ as:\n\\begin{align}\nf(\\mathbf{x}) &= \\Vert \\mathbf{Dx}-\\mathbf{b}\\Vert_2^2.\\label{eq:ls_mr_f}\n\\end{align}\nThen define the monotone submodular set function $g:2^{[n]}\\rightarrow\\mathbb{R}$ as:\n\\begin{align}\ng(A) &= \\begin{cases}\n\\lambda\\left[(n-1) + \\max(A) - \\min(A) + \\vert A\\vert\\right], & A\\neq \\emptyset, \\\\\n0 & A = \\emptyset,\n\\end{cases}\\label{eq:ls_mr_g}\n\\end{align}\nwith $\\lambda\\in\\mathbb{R}_{\\geq 0}$, and $\\max(A)$ and $\\min(A)$ denoting the largest and smallest index element, respectively, in the set of indices $A$.  This choice of $g$ in the sparse regression problem \\eqref{eq:lattice_opt_problem} places a high penalty on large sets of nonzero entries in the vector $\\mathbf{x}\\in\\mathbb{R}^n_{\\geq 0}$ that are far apart in index.\n\nWe generate a series of random problem instances with $m = n$ satisfying the assumption of submodularity on $\\mathbb{R}^n_{\\geq 0}$ and also the convexity condition of Corollary \\ref{cor:polytime}.  Let $\\mathrm{chol}:\\mathbb{R}^{n\\times n}\\rightarrow\\mathbb{R}^{n\\times n}$ denote a Cholesky decomposition of a positive semidefinite matrix, and construct the matrix $\\mathbf{D}$ in \\eqref{eq:ls_mr_f} as:\n\\begin{align*}\n\\mathbf{D} &= \\mathrm{chol}\\left(\\mathbf{C} + \\mathbf{C}^T + n\\mathbf{I}\\right), \\quad \\mathbf{C}_{ij}\\sim\\mathrm{unif}(-1,0),\\4all i,j=1,2,...,n.\n\\end{align*}\nThis construction guarantees that the function $f$ in \\eqref{eq:ls_mr_f} is both convex and submodular on $\\mathbb{R}^n_{\\geq 0}$, satisfying the conditions for Corollary \\ref{cor:polytime}.  For the parameter $\\mathbf{b}\\in\\mathbb{R}^m$, we use the signal in the top plot of Fig. \\ref{fig:LS_figs}, and we set the regularization strength to $\\lambda = 0.05$ so that both the functions $f$ and $g$ play nontrivial roles in the objective function.\n\n\nWe plot the results from each algorithm in Fig. \\ref{fig:LS_figs}.  Because the minimizer of the optimization problem is a representation of $\\mathbf{y}$ using structured sparse columns of $\\mathbf{D}$, we show the the reconstructed vector $\\mathbf{Dx}$ in the second, third, and fourth plots of Fig. \\ref{fig:LS_figs}.  Because there is no reliance on discretization, both the projected subgradient descent and minimum-norm point algorithms produce a much smoother result, as expected.\n\nIn the bottom left plot of Fig. \\ref{fig:LS_figs}, we show the cost achieved over iterations of each algorithm.  The minimum-norm point converges rapidly to the globally optimal cost, while the projected subgradient descent method takes longer to achieve the same cost.  However, the discretization error associated with the continuous submodular function minimization approach prevents it from ever achieving the true optimal cost.\n\nFinally, over a small window of problem sizes, we show the running times of each algorithm in the bottom right plot of Fig. \\ref{fig:LS_figs}.  Interestingly, our approach presents a compromise between the slow optimality of the projected subgradient descent method and the fast but inexact continuous submodular function minimization algorithm. \n\\begin{figure}\n\\begin{center}\n\\begin{subfigure}{0.95\\linewidth}\n\t\\includegraphics[width=\\linewidth]{.\/new_pics\/ls_mr_combined_signals.eps}\n\t\\vspace{-15mm}\n\\end{subfigure}\\\\\n\\begin{subfigure}{0.95\\linewidth}\n\t\\includegraphics[width=\\linewidth]{.\/new_pics\/ls_mr_costs_and_runtimes.eps}\n\\end{subfigure}\n\\end{center}\n\\caption{Results from the sparse regression problem simulations.  The reconstructed signal representations using columns of $\\mathbf{D}$ created by each algorithm are shown in the second, third, and fourth plot. Note the solutions produced by projected subgradient and the minimum-norm point algorithm are identical. We plot the cost function value over each algorithm's iterations in the bottom left, while in the bottom right we compare the running times of the algorithms over a small window of problem dimensions.}\n\\label{fig:LS_figs}\n\\vspace{-3mm}\n\\end{figure}\n\n\n\\subsection{Signal Denoising}\nWe next study a simple denoising example, where we consider a signal $\\mathbf{x}\\in\\mathbb{R}^n$, which is corrupted by some additive disturbance $\\mathbf{w}\\in\\mathbb{R}^n$, with $\\mathbf{w}\\sim \\mathcal{N}(0,0.1\\mathbf{I})$. We would like to recover the signal $\\mathbf{x}$ from the noisy measurements $\\mathbf{y} = \\mathbf{x} + \\mathbf{w}$, under the assumption that the true signal $\\mathbf{x}$ is smooth (meaning variations between adjacent entries ought to be small), and that the meaningful content arrived in a small number of contiguous windows of entries.\n\nWe can express the desire to match the noisy signal $\\mathbf{y}$ with a smooth one with the convex and submodular function $f:\\mathbb{R}^n\\rightarrow\\mathbb{R}$ defined as:\n\\begin{align}\nf(\\mathbf{x}) &= \\frac{1}{2}\\Vert \\mathbf{x} - \\mathbf{y}\\Vert + \\mu \\sum_{i=1}^{n-1}\\left(\\mathbf{x}_i-\\mathbf{x}_{i+1}\\right)^2.\\label{eq:DN_f}\n\\end{align}\nThe first term promotes matching the slightly corrupted signal, while the quadratic penalty on adjacent entries of $\\mathbf{x}\\in\\mathbb{R}^n$ promotes smoothness.\n\nSimilarly, we can express the knowledge of a small and contiguous set of nonzero entries in the vector $\\mathbf{x}$ with the monotone submodular set function $g:2^{[n]}\\rightarrow\\mathbb{R}$ defined by:\n\\begin{align}\ng(A) &= \\lambda\\left(\\vert A\\vert + \\mathrm{\\# int}(A)\\right),\\label{eq:DN_g}\n\\end{align}\nwhere $\\lambda\\in\\mathbb{R}_{\\geq 0}$, and the function $\\mathrm{\\# int}(A)$ counts the number of sets of contiguous indices in the set $A$.  This set function is smallest on subsets with a small number of entries that are adjacent in index.\n\nFor experiments, we use the signal $\\mathbf{x}\\in\\mathbb{R}^n$ shown in the top plot of Fig. \\ref{fig:DN_figs}, with the noise-corrupted measurements $\\mathbf{x} + \\mathbf{w} = \\mathbf{y}\\in\\mathbb{R}^n$ with an example shown in dotted orange.  We then let $\\mu = 0.8$ in \\eqref{eq:DN_f} and $\\lambda = 0.05$ in \\eqref{eq:DN_g} so that the overall problem's cost function has nontrivial contributions from both the smoothness-promoting function and the sparsity-inducing regularizer.  In this case, for the continuous submodular algorithm we discretize the compact set $[-1,1]^n\\subseteq\\mathbb{R}^n$ into $k=101$ distinct values per index.\n\nWe show the resulting denoised signals in the second, third, and fourth plots in Fig. \\ref{fig:DN_figs}, with the running time comparison over a small window of problem dimensions in the bottom right.  The discretization of the domain in the continuous submodular function minimization approach produces artifacts in the resulting signal, whereas the result of the projected subgradient and minimum-norm point algorithms are smoother with smaller sets of nonzero entries.  We see once more that for most problem sizes, our proposed minimum-norm point algorithm poses a compromise between speed and accuracy, providing guaranteed global optimality without the high running time of projected subgradient descent.\n\nWe also compare the objective value achieved during the iterations of each algorithm for a single instance in the bottom left plot of Fig. \\ref{fig:DN_figs} with $n=100$. Again, the minimum-norm point algorithm converges rapidly to the minimum alongside the projected subgradient method, while the continuous submodular function minimization approach's discretization error prevents it from ever achieving global optimality.\n\n\\begin{figure}\n\\begin{center}\n\\begin{subfigure}{0.95\\linewidth}\n\t\\includegraphics[width=\\linewidth]{.\/new_pics\/smooth_int_combined_signals.eps}\n\t\\vspace{-15mm}\n\\end{subfigure}\\\\\n\\begin{subfigure}{0.95\\linewidth}\n\t\\includegraphics[width=\\linewidth]{.\/new_pics\/smooth_int_costs_runtimes.eps}\n\\end{subfigure}\n\\end{center}\n\\caption{Results of the denoising problem simulations. The true signal and its noisy counterpart are shown in the top plot. The second, third, and fourth plots show the denoised signals produced by each of the three algorithms.  Note that the results from the minimum-norm point algorithm and the projected subgradient descent method are identical.  The bottom left plot shows the objective value across iterations for $n=100$, and bottom right shows the running times of each algorithm for a window of  problem dimensions.}\n\\label{fig:DN_figs}\n\\vspace{-3mm}\n\\end{figure}\n\n\\subsection{Discretization Error Dependence}\nIn this section, we explore the relationship between the continuous submodular function minimization algorithm's discretization error and its running time.  To this end, we ran instances of the sparse regression example with the modified range function penalty, using a discretization resolution in each dimension ranging from $k=50$ to $k=400$.\n\nThe minimum cost achieved at each discretization level $k$ is shown in the left plot of Fig. \\ref{fig:k_comp}.  Similarly, the associated running times of the algorithm are shown in the right-hand plot of Fig. \\ref{fig:k_comp}.  Interestingly, near the value of $k = 250$, the achieved cost becomes effectively optimal, but the running time increases by an order of magnitude.\n\n\\begin{figure}\n\\begin{center}\n\\begin{subfigure}{.4\\textwidth}\n\t\\includegraphics[width=1.0\\linewidth]{pics\/k_vs_optimality_mr}\n\t\\vspace{-6mm}\n\t\\label{fig:k_vs_optimality}\n\\end{subfigure}%\n\\begin{subfigure}{.4\\textwidth}\n\t\\includegraphics[width=1.0\\linewidth]{pics\/k_vs_runtime_mr}\n\t\\vspace{-6mm}\n\t\\label{fig:k_vs_runtime}\n\\end{subfigure}\n\\end{center}\n\\caption{Results highlighting the role of the discretization resolution $k$ on the continuous submodular algorithm's optimality (left) and running times (right) in an instance of the sparse regression problem with $n=100$.}\n\\label{fig:k_comp}\n\\vspace{-3mm}\n\\end{figure}\n\nTo give a coarse estimate on the origin of higher running times for projected subgradient descent and the minimum-norm point algorithms, we note that the computational cost of each iteration is dominated by the cost of computing the Lov\\`asz extension of $H$.  This computation has time complexity $O(n\\log n + n EO)$, where $EO$ is the complexity of evaluating $H$.  If $H$ is evaluated through convex optimization, many generic interior-point methods have time  complexity that is approximately $EO = O(n^3)$.  Therefore, each iteration of the minimum-norm point algorithm and the projected subgradient descent algorithm might have complexity on the order of $O(n\\log n + n^4)$.\n\nOne possible remedy to this issue would be to use a more specialized algorithm for solving the constrained convex optimization problem evaluating $H$, thus reducing the complexity of $EO$.  Another easy alternative would be to stop early, producing an approximate evaluation of $H$. Projected subgradient descent still has some approximation guarantees in this regime \\cite{eloptimal}, but our agnostic combination may be more brittle to these variations.\n\n\\section{Conclusions}\nIn this work, we showed that model-fitting problems with structure-promoting regularizers could be expressed as optimization problems defined over two connected lattices. Using submodularity theory, we derived conditions on these functions and their domains under which we can directly solve these problems exactly and efficiently.  We focused on continuous and Boolean lattices, and derived conditions under which an agnostic combination of submodular set function minimization and convex optimization algorithms can compute the exact solution in polynomial time.\n\nWe then extended this theory to handle optimization problems with simple continuous or discrete budget constraints on the model parameters.  We did this by naively adding the constraint to the cost with a Lagrange multiplier, but then used submodular function theory to solve for all possible Lagrange multiplier values with a single convex optimization problem.  Finally, we also highlighted robust or adversarial optimization scenarios, where our exact solutions could provide subgradients to be used in globally convergent ascent methods.\n\nOne promising next direction of research would be examining if a slightly weaker satisfaction of the assumptions on the functions $f$ and $g$ would result in only a slightly weaker guarantee, with the same algorithm-agnostic approach.  Moreover, the assumptions and conditions outlined in this work are merely sufficient.  Future work might examine if these conditions are necessary as well.\n\n\\begin{comment}\n\\subsection{System Observability}\nConsider a linear dynamical system:\n\\begin{align*}\n\\dot{\\mathbf{x}} &= \\mathbf{Ax} \\\\\n\\mathbf{y} &= \\mathbf{Cx}\n\\end{align*}\nwith $\\mathbf{x}\\in\\mathbb{R}^n$, $\\mathbf{A}\\in\\mathbb{R}^{n\\times n}$, $\\mathbf{y}\\in\\mathbb{R}^m$, and $\\mathbf{C}\\in\\mathbb{R}^{m\\times n}$.  The system's Observability Gramian $\\mathcal{W}_O \\in \\mathbb{R}^{n\\times n}$ is defined as:\n\\begin{align*}\n\\mathcal{W}_O &= \\int_0^\\infty e^{\\mathbf{A}^T\\tau}\\mathbf{C}^T\\mathbf{C}e^{\\mathbf{A}\\tau}d\\tau.\n\\end{align*}\nThe Observability Gramian is a positive semidefinite matrix that characterizes the ``observable space,'' which can be thought of as the portion of the state space where estimating the state $\\mathbf{x}$ through outputs $\\mathbf{y}$ is possible.\n\nIf we let the output $\\mathbf{y}$ be ``filtered'' through some matrix $\\mathbf{S}\\in\\mathbb{R}^{m\\times m}$ such that $\\mathbf{y} = \\mathbf{SCx}$, then the Observability Gramian becomes:\n\\begin{align*}\n\\mathcal{W}_O&=\\int_0^\\infty e^{\\mathbf{A}^T\\tau}\\mathbf{C}^T\\mathbf{S}^T\\mathbf{S}\\mathbf{C}e^{\\mathbf{A}\\tau}d\\tau.\n\\end{align*}\nWhen determining the best filtering matrix $\\mathbf{S}$ for the system, one objective would be to maximize the ``average observable space'' of the system, which would mean maximizing the quantity:\n\\begin{align*}\n\\tr{\\mathcal{W}_O} &= \\tr{\\int_0^\\infty e^{\\mathbf{A}^T\\tau}\\mathbf{C}^T\\mathbf{S}^T\\mathbf{S}\\mathbf{C}e^{\\mathbf{A}\\tau}d\\tau}.\n\\end{align*}\nBecause the matrix $\\mathbf{S}$ appears in a quadratic fashion here, optimization problems determining $S$ directly as a function of its entries is difficult. We instead parameterize $\\mathbf{S}^T\\mathbf{S}$ with a conic combination of $q$ symmetric positive semidefinite matrices $\\mathbf{E}_i\\in\\mathbb{R}^{m\\times m}$, weighted by some $\\mathbf{p}\\in\\mathbb{R}^p_{\\geq 0}$:\n\\begin{align*}\n\\mathbf{S}^T\\mathbf{S} &= \\sum_{i=1}^q\\mathbf{p}_i\\mathbf{E}_i.\n\\end{align*}\nAfter determining the desired weights $\\mathbf{p}$, we can extract $\\mathbf{S}$ through the Cholesky decomposition.  Note however, that because we are using a specific choice of basis $\\mathbf{E}_i$ for our matrix $\\mathbf{S}^T\\mathbf{S}$, we can only select the best $\\mathbf{S}^T\\mathbf{S}$ (and therefore $\\mathbf{S}$) of those that can be written as a nonnegative weighted sum of our selected matrices $\\mathbf{E}_i$, which is not necessarily the entire space of possible matrices.\n\nInterestingly, the trace of the Observability Gramian is a linear function in $\\mathbf{p}$, which we then express as:\n\\begin{align*}\n\\tr{\\mathcal{W}_O}(\\mathbf{p}) &= \\mathbf{w}^T\\mathbf{p}.\n\\end{align*}\nwith $\\mathbf{w}\\in\\mathbb{R}^q$.\n\nIn addition to maximizing the observable space, we may want to reduce the complexity of the output $\\mathbf{y}$ by preferring that it depend on only a few elements of the state variable $\\mathbf{x}$.  Conveniently, when the filtering matrix $\\mathbf{S}$ does not depend on an index $i$ of the state, the resulting matrix $\\mathbf{S}^T\\mathbf{S}$ has all zeros in both row and column $i$.\n\nGiven a choice of weightings $\\mathbf{p}$, we can compute the corresponding number of nonzero rows (or columns) in the matrix $\\sum_{i=1}^q\\mathbf{p}_i\\mathbf{E}_i$ and assign an associated penalty.  Doing this creates a set function $g:2^q\\rightarrow\\mathbb{R}$, and as $g$ is the sum of indicator functions of (potentially overlapping) groups, it is naturally a monotone submodular function.\n\nIn order to avoid selecting unbounded amounts of particular bases, we can place a budget, or an $\\ell_2$-norm ball constraint on the vector $\\mathbf{p}$, and we are left with the final optimization problem:\n\\begin{align*}\n\\begin{array}{cc}\n\\minimize{\\mathbf{p}\\in\\mathbb{R}^q_{\\geq 0}}&-\\mathbf{w}^T\\mathbf{p} + g(\\supp{\\mathbf{p}}) \\\\\n\\text{subject to} & \\Vert\\mathbf{p}\\Vert_2^2 \\leq 1.\n\\end{array}\n\\end{align*}\nBecause the function $\\tr{\\mathcal{W}_O}:\\mathbb{R}^n_{\\geq 0}\\rightarrow\\mathbb{R}$ is separable, and the $\\ell_2$ norm constraint is defined by a strictly increasing and separable function, we can directly apply Theorem \\ref{thm:bach_convex_submod_ii} and solve the regularized form of this problem for all regularization strengths.\n\\end{comment}\n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\ph{Autonomous Exploration and SubT} Robotic exploration and the advancement of autonomy offer new ways to explore potentially dangerous and hard-to-access underground environments. Multi-agent systems have matured in controlled and structured environments like warehouses, factories, and laboratories, while current robotic challenges seek to advance these technologies for search and rescue scenarios, planetary prospecting, and subsurface exploration \\cite{asada2019robocup, Hambuchen2017NASA_Rob_Challenge, Link2021_ESA_ESRIC}. Motivated by the search for life on other planets, NASA JPL's team CoSTAR \\cite{agha2021nebula} took part in the Defense Advanced Research Projects Agency's (DARPA) Subterranean Challenge (SubT) seeking to advance robotic multi-agent systems and their technology readiness for potential future missions. If brought to other planets (e.g. Mars), subsurface missions could bring new insights into their geologic past as well as on their potential for supporting life in the environmentally protected undergrounds \\cite{Titus2021}. In contrast to traditional exploration missions where a team of operators and scientists controls one rover, SubT introduced the challenging requirement that only \\textit{a single human supervisor} can  directly interface with the deployed multi-agent team in real-time and when a communication link is established. SubT is divided into three, one year development pushes with major field testing demonstrations. This work focuses on the advancements in our supervisor autonomy and game-inspired user interface that were developed under the restrictions of a worldwide pandemic and deployed during the SubT final competition comprising two preliminary missions (P1 and P2) and the final prize run (F). \n\n\n\\begin{figure}[t]\n\\centering\n  \\includegraphics[width=0.45\\textwidth]{figures\/robot_team-min.png}\n  \\caption{Team CoSTAR's Mission Control user interface (A). (B) a subset of CoSTAR's ground robots showing four customized Boston Dynamic's Spot and Clearpath Husky powered by JPL's autonomy platform NeBULA. Typically a deployment of 4 to 6 ground vehicles was targeted during SubT, but the number of agents is extendable (e.g., see A with 11 robots).}\n  \\label{fig:robot_team}\n\\end{figure}\n\n\\ph{Human-Robot Collaboration} Achieving man-computer symbiosis \\cite{Licklider1960ManComputerSymbiosis} has been a long-time goal of the community to promote a close coupling of human and machine capabilities and ultimately inspire the evolving field of human-robot interaction \\cite{chen2021human}. \nThis work improves collaborative human multi-robot exploration and search performance fusing our extended autonomy assistant Copilot \\cite{Kaufmann2020copilotMike} that uses automated planning techniques with a game-inspired interface design for effective robot deployment, operations, and single operator supervision to create a more symbiotic interaction.  \n\nWe present key design choices that are breaking away\nfrom common robot interfacing strategies that were deployed in similar challenge\ncontexts \\cite{kohlbrecher2015human,cerberus} and used interfaces based on the\nRobot Operating System's (ROS) visualization tool RViz. Further, we leverage\nhuman-robot interdependencies to inform the design and development of supervised\nautonomy and interaction paradigms to achieve our set interaction objectives.\nThe latest results from the SubT competition ``Finals'' are compared to a\nbaseline from previous competition runs, namely the ``Urban Circuit'', which\ndeployed earlier interface and system implementations and interaction paradigms\nthat we improve with our combined game-inspired interface and enhanced\nsupervisory autonomy.\n\n\nIn \\Cref{sec:related_work}, we look at related work from human-robot interaction\nand user interface design. \\Cref{sec:problem_requirements_objectives} outlines\nthe SubT requirements and interaction objectives for our multi-agent exploration\nscenario.\nWe then describe our supervisory autonomy Copilot and its latest implementation\nin \\Cref{sec:supervised_autonomy}, while \\Cref{sec:interface} outlines the user\ninterface components and human-robot interaction capabilities. Finally, we\npresent results from our development pushes in \\Cref{sec:results} and close with\nconclusions and an outlook on future work in\n\\Cref{sec:conclusions_future_work}.\n\n\n\\section{Related Work}\n\\label{sec:related_work}\n\\ph{Human-Robot Interaction and Interface Design} More than sixty years after\nthe introduction of man-computer symbiosis by\n\\textit{Licklider}~\\cite{Licklider1960ManComputerSymbiosis}, \\textit{Chen and\n  Barnes}~\\cite{chen2021human} conclude that the boundaries of long-term\nhuman-robot symbiosis are still to be pushed by interdisciplinary\ncollaborations. \\textit{Szafir and Szafir}~\\cite{Szafir2021HRI_DATA_VIZ} have\nidentified best practices in the field of data visualization as a key driver to\nadvance both HRI and data visualization. Complex visualizations and renderings\nhave become achievable with off-the-shelf hardware, which allows the integration\nof visualization principles such as sensemaking~\\cite{Szafir2021HRI_DATA_VIZ}\nthat helps a human digest information. In human-space systems \\textit{Rahmani\n  et al.}~\\cite{rahmani2019space} identified that interface technologies are\ncurrently in development, but their technology readiness levels are not very\nmature. Multiple design methods have been introduced in the literature, for\ninstance, Coactive Design~\\cite{coactive_design} which is a structured\napproach to analyze human and robot requirements and was used in the context of\nthe 2015 DARPA Virtual Robotics Challenge that aimed at advancing disaster\nresponse capabilities. \\textit{Roundtree et\n  al.}~\\cite{Roundtree2019VizDesignCollectiveTeams} found that abstract\ninterface designs that visualize collective status over single agent\ninformation could increase performance; however such designs depend on the task\nat hand, team size and mission goals~\\cite{chen2021human}. A common testing\nstrategy in computer game development is\nPlaytesting~\\cite{Wallner2019Playtesting},\nwhich is comparable to simulation and field testing in the multi-robot domain.\nThe game-inspired development technique RITE, which was introduced in the\ncontext of interface development for the computer game Age of\nEmpires~\\cite{medlock2002usingRITE_AgeOfEmpires}, was used and adapted for fast\ndevelopment sprints. Additionally, we drew inspiration from real-time strategy games like Age of Empires,\nwhich guided the design of the 3D portion of the interface.\n\n\n\\ph{Robot Challenge Interfaces} During 2013's DARPA Robotics Challenge, team\nViGIR leveraged ROS to control a humanoid robot. The team decided to implement\ntheir interfaces using RViz and built an Operation Control Center consisting of\nat least six screens. Robot challenges are found to typically influence\nhuman-robot interaction design and interfaces~\\cite{Szafir2021HRI_DATA_VIZ} and\nfor DARPA's SubT teams, the common design practice was based on RViz and ROS\nplugins (\\cite{csiro,cornellSubTJFR,scherer2021resilient,norlab,cerberus}). Even\nour team started off using RViz as a quick way to prototype\ninterfaces~\\cite{Otsu2020IEEEAerospace} and used it as the main way to interact with\nthe robot agents due to its tight integration with ROS and ability to access\nrobot data for debugging purposes. We shifted away from this approach for the final competition, and the resulting HRI modalities and supervisory interface are presented in this work.\n\n\n\\section{Background and Objectives}\n\\label{sec:problem_requirements_objectives}\n\\ph{Challenge Requirements} The overall SubT goals are two common problems faced\nby real-world multi-agent systems: first, the autonomous exploration of unknown\nenvironments, and second, the search for objects of interest hidden within. While\nexploration and search provide a need for specific capabilities, DARPA further\nintroduced a set of guidelines and rules to motivate higher levels of autonomy\nfor the deployed systems:\n\\begin{inparaenum}[(i)]\n    \\item only a single human operator is allowed to interact, supervise, and interface with the robots;\n    \\item each mission is bound by a fixed \\textit{setup time} limit of 30 minutes and an \\textit{exploration time} limit between 30 and 60 minutes;\n    \\item a pit crew of four (Finals) or nine (Urban Circuit) can support the supervisor by setting up hardware in a designated area without access to wireless data streams, robot control, or interface;\n    \\item there is a limited number of attempts to submit discovered objects of interest;\n    \\item the final challenge environment comprises tunnel, urban, and cave terrains to be explored.  \n\\end{inparaenum} \n\n\\ph{Objectives} Deploying and operating large teams of robots like Team CoSTAR's robot fleet, shown in  \\Cref{fig:robot_team}B, are complex real-world problems. Addressing this set of problems creates the need for a resource-efficient and robust human and multi-agent system to i) not overwhelm the single human supervisor, ii) meet the timing requirements, and iii) increase the performance of both exploration and search tasks. \n\n\nTo tackle this challenge and develop a system that can deploy reliably even beyond the SubT challenge, we embed the following interaction objectives into our system design:\n\\begin{inparaenum}[(1)]\n    \\item Reducing overhead and human workload (e.g., from application switching and manual task execution)\n    \\item Creating and maintaining situational awareness\n    \\item Managing large teams of robots (from setup, deployment to exploration) while allowing for a flexible configuration\n    \\item Accessing critical information in a single unified interface\n    \\item Maintaining an enjoyable performance that can visualize the complete robot team\n    \\item Collaborating with autonomy and trusting automation.\n\\end{inparaenum}\n\n\n\n\n\\section{Supervised Autonomy}\n\\label{sec:supervised_autonomy}\n\\subsection{Copilot}\n\\ph{Motivation} After SubT's ``Urban Circuit'', the allowed personnel in the\ncompetition staging area was reduced from ten to five team members which\nincludes the main supervisor. This required a shift in how robots were\nstrategically and physically handled (minimum 2 people are needed to lift and\nstage a single robot). Task coordination was done by a pit crew member directing\nthe operator and influencing their actions while following static paper\n\\textit{checklist procedures}. Developing and deploying a computerized assistant\nthat could take over this role was soon desired.\n\n\\ph{Original Implementation} A first version of Copilot, ``an autonomous\nassistant for human-in-the-loop multi-robot operations'' was introduced in\n\\cite{Kaufmann2020copilotMike}. This early Copilot was only tested in realistic\ncave simulations or during preparatory missions with one deployed robot. Copilot supports a single human supervisor in monitoring robot teams, aids with strategic task planning, scheduling, and execution, and communicates high-level commands between agents and a human supervisor if a communication link exists. The autonomy assistant aims at keeping workload acceptable while maintaining high situational awareness that allows rapid responses in case system failures are observed.\n  \n\\ph{Task Interaction} Copilot takes over the decision-making processes regarding planning and scheduling, which reduces the need to memorize tasks and task sequences or the need to delegate a team member to take over such checklist-like tasks. Some tasks were implemented with higher autonomy levels and automatically executed limited actions, but most required the human to start the task, manually execute parts of it, and confirm that the task had been completed successfully or unsuccessfully while monitoring the system. \nOn one hand, it reduced the need to remember tasks; on the other hand, more interactions with the newly introduced system were needed.\n\n\\ph{Scalability Limitations} Due to computational limitations, a full mission\nsimulation could not be achieved with more than three robots at reduced\nreal-time and not more than two in real-time. However, upon tightly integrating\nCopilot with multiple real robot platforms, we noticed that the current concept\nof operations didn't scale well when adding more robots to a mission. We learned\nthat task execution on the real hardware requires different timing and\nintroduces many sources for machine and human errors (e.g., if cables are loose,\nsensors don't power up, or unknown unknowns occur).\n\n\\ph{Visualization Limitations} In robotics interfaces, scheduling, and timeline\nviews are often presented in a robot- or task-centric way, focusing on who or which\nagent is scheduled for a certain task and when, respectively\n\\cite{BaeRossiDavidoff2020VizAnalytics}. The main task-centric approach that was\nused in early Copilot tests showed a vertical list view with a scrollable\ntimeline. This timeline showed the four tasks closest in time on top. As the\nnumber of tasks scaled linearly with the number of deployed robots this list\nview became inefficient --- especially when tasks had to be deferred and worked\non in a non-sequential order.\n\n\\subsection{Improved Copilot}\n\nThe identified shortcomings motivated a redesigning and rethinking of Copilot's back-end and front-end to reduce and not just shift workload; thus, we implemented higher levels of automation.\n\n\\ph{Architecture Changes} \\Cref{fig:copilot_architecture} provides a simplified\noverview of Copilot's updated task management architecture. A multi-robot task\nauto-generator and verifiable task executor have been added to the system, and\nthe underlying planner has been replaced. All modules access a centralized task\ndatabase which stores pending, active, successful, or failed mission tasks for\nsetup, deployment, and during exploration.\n\n\\begin{figure}[t]\n    \\centering\n    \\def\\columnwidth{\\columnwidth}\n    \\import{figures\/}{architecture.pdf_tex}\n\\caption{Copilot's task management architecture. Auto-generator, Planner, and Executor have been added or updated and access a centralized task database which stores pending, active, successful, or failed tasks.}\n\\label{fig:copilot_architecture}\n\\end{figure}\n\n\n\\ph{Task Dependency Graph} A robot mission can be fairly complex, even when\nlooking at the deployment of a single robot. In\n\\Cref{fig:task_dependencies} such a single robot mission is shown as a directed\ngraph indicating the temporal\nconstraints and execution dependencies with arcs between the nodes that represent a\npre-defined set of mission tasks. Each task is defined by its duration, earliest\nstart time, latest end time, and its dependency relations with other tasks.\n\n\\begin{figure}[!tb]\n    \\centering\n    \\def\\columnwidth{\\columnwidth}\n    \\import{figures\/}{s1_tasks.pdf_tex}\n\\caption{Pre-defined Copilot tasks for a single robot mission indicating task dependencies. The number of tasks scales linearly with the number of deployed robots. Spot1 related tasks are depicted in blue and operator tasks in orange. A superscript O or P at the beginning of a task indicate that the operator or pit crew has to manually fulfill some pre-condition. A superscript at the end indicates that a human sign-off is implemented before proceeding with the next task. For instance ``Power on robot platform'' requires a physical push of the robot platform startup button.}\n\\label{fig:task_dependencies}\n\\end{figure}\n\nTo deploy multiple robots without the need to hard-coding all possible agent\ncombinations and graphs, we use a scalable auto generator. The preceding\nsuperscript O in the graph (see \\Cref{fig:task_dependencies}) indicates\nthat human inputs or actions are required for the task. In the case of the\n\\texttt{Launch base software} task, this means that the operator has to initiate\nthe software launch as a pre-condition and is prompted to select the robots that\nthey would like to deploy for the upcoming mission. Similarly, superscripts at\nthe end of a task indicate that human action is needed before the next task can\nbegin. Tasks without either have been fully automated for nominal cases in this\nnewer Copilot version.\n\n\\ph{Task Planning and Scheduling} The aforementioned task dependency graph for\nthe selected robots forms the input for Copilot's task planner and is stored in\nthe MongoDB task database. The generation of a task plan for setting up, deploying, and\nassisting the operator during exploration is framed as an automated temporal\nplanning problem. In the first version of Copilot, we formulated such problem as\na Simple Temporal Network (STN), encoded as a linear program. In the improved\nversion of Copilot, deployed in the final events of SubT, we moved to a PDDL\ntemporal planning formulation to allow 1) flexibility on task representation\nwith respect to state constraints, resources, and planning, and 2) use the body\nof planners available in the literature. Herein we integrated the OPTIC planner\n\\cite{benton-etat-2012-OPTIC}, a PDDL temporal planner that handles time window\nspecification (timed initial literals), and discrete and continuous resources.\n\nTo perform planning, OPTIC uses both a PDDL domain file and a problem\nfile. The domain file has been designed to represent tasks (modelled as\noperators) and its dependencies (preconditions). The problem file is generated\nprior to calling the planner, and it is built based on the current state of\nmission and tasks execution. For example, if a task is ongoing, the PPDL file\nwould represent the task as ongoing and add constraints to ensure it continues\nthe execution to meet the necessary constraints. As a notional example\nof the scale of the planning problem, a mission with four robots would have\napproximately 60 tasks to be scheduled during setup and deployment. Planning is\nperformed at a predefined cadence (e.g., every 1.5 seconds), but it also follows\nan event-based approach when task execution is late, or the human-in-the-loop\nchanges their strategy --- this helps mitigate execution uncertainty. The\ngenerated plan is parsed and stored in a Task Database (for logging and\nvisualization across the system); each task is then dispatched for execution.\n\nIf a plan is not found by OPTIC due to temporal constraint violations (e.g.,\ndelays in task execution), Copilot will attempt to increasingly relax some of\nthe key temporal constraints, such as the latest end time of certain activities\n(e.g., allowing setup tasks to end a few minutes after the setup time,\noverlapping with the beginning of the exploration time window). In critical\nscenarios, Copilot would notify the operator of a schedule relaxation to allow\nfor further strategy changes.\n\n\\ph{Task Verification and Execution} A verifiable and generic task framework is\nintroduced to Copilot, allowing for quick implementations and standardized task\nautomation. Each task follows a strict precondition, execution, and\npost-condition template. Condition checks and execution can be triggered across\nagents, including the base station at which the human can oversee all automated\nprocesses at a high level in the new Copilot interface, which is described in\n\\Cref{sec:interface}. The task template execution covers both fully automated\ntasks and semi-automated tasks where an operators confirmation is required (e.g.\ndeploying a robot into a cave requires a Go\/No-go decision from the supervisor ---\ndeploying itself is an automated process). If a task fails during execution or\npost-condition checking, Copilot will try to resolve the issue by retrying tasks\nseveral times and allowing for more execution time. Failed tasks will be reported\nto the supervisor, who can choose to debug the issue at hand or trigger another\nautomated retry. Retries and resets are possible at all levels, and completed\ntasks can be reset during an active mission in case a robot platform has to be\nrebooted.\n\n\n\n\\section{Game-Inspired Interface}\n\\label{sec:interface}\n\t\n\\begin{figure*}[tb]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{figures\/ui_overview.png}\n\\caption{An overview of the major UI components. (A) The Robot and associated Copilot task cards.  (B) The split-screen 3D visualization view with view controls, WiFi signal strength overlay, and an artifact card showing on the map. (C) The artifact drawer. (D) The robot health systems component.}\n\\label{fig:ui_overview}\n\\end{figure*}\n\n\\ph{Game Inspiration} Inspiration for multi-agent interaction and interface\ndesign is partially drawn from real-time strategy games such as Age of Empires,\nStarCraft, and {Command \\& Conquer}. When played competitively, these games\nrequire a high sense of micro and macro-management of units and their\nenvironment and the ability to efficiently switch between these two ways of\nmanaging a team. Micromanagement involves short-term strategy and\ndecision-making, where individual units may require critical attention to win a\nbattle, overcome an obstacle, or navigate to the next point of interest, while\nmacromanagement refers to longer-term strategizing that involves resource\ngathering, unit production over time, and overall exploration and control of the\nmap~\\cite{rtsCombatStrategy}. Parallels can be applied to the management of a\nrobot team in the SubT competition. Even autonomous robots can benefit from or\nrequire human intervention and commanding, especially if critical attention\ntowards failing subsystems is needed. Supervised multi-agent control draws from\nthe human's situational awareness regarding the environment and robot states to\neffectively coordinate multi-agent behaviors, successfully locate artifacts, and\nscore points.\n\n\\ph{Mission Phases} The user interface is designed to be adaptable to the\noverall mission and two major phases of an individual robot's competition run in\nparticular:\n1) setup and deployment, and 2) mission execution with its exploration and\nsearch components. Across these phases, the visibility and abstraction of\ninformation need to be flexible to facilitate focus on the anticipated operator\ninteractions. In deployment, the user interface uses the Copilot-generated tasks and status information to guide the sole operator through the multitude of individual tasks while allowing them to maintain their situational awareness, manage the entire robot team, and coordinate with the pit crew.\n\n\\ph{Three Column Layout} The Mission Control interface is organized into\ndifferent view components. \\Cref{fig:ui_overview} shows the main split-screen\nwith three columns aiming at creating reliable locations for the operator to\nlook at when needing to accomplish functionally distinct tasks (A). The aim here\nis to reduce the amount of visual scanning, application switching, and to parse\nrobot needs on an individual or team level swiftly. Individual robot information\npertinent to monitoring health systems is available on the left, planned and\nactively re-scheduled Copilot tasks for individual robots are placed alongside\neach agent in the middle, and a 3D interactive visualization of the robots in\ntheir environment is anchored to the right. During mission execution, the\nprimary goal of the user interface is to keep the operator situationally aware\nof a multitude of individual robot health systems and data sensed from the\nsurrounding environment while presenting the most important information and thus\nreducing their cognitive workload. In \\Cref{fig:robot_team} the 3D visualization\nis expanded, and robot sensor and status information is minimized to select\nmission-critical information.\n    \n\n\\ph{Health Systems and Robot Status} In order to effectively survey the status\nof any individual robot in the team, visibility into over 30 unique sensors and\nstatuses needed to be surfaced to the operator per robot. This required\nidentifying which indicators were critical to display at all times, which could\nbe hidden within a sub-view, which were good candidates to be combined and\nabstracted, and which would be prioritized across either the deployment (split)\nor mission execution (split and expanded visualization) modes of the user\ninterface. In addition to sensors visible at an individual level, an additional\nview was created to organize sensors compactly across the team, providing easy\nvisual scanning for the operator during macro-management and deployment, as\nshown in \\Cref{fig:ui_overview}D. An abstraction of robot behaviors (e.g.,\nexploring, dropping a communications node) and mobility states presents an\noverall status of each robot to the operator by color and a high-level\ndescription. This status is prioritized based on criticality to ensure the\noperator's attention will be requested for the most important issue at any given\ntime.\n\nPreviously, Copilot tasks resided in an entirely separate module of the\ninterface with limited screen estate, requiring the operator to move other\nrelated and necessary sensor and status information out of physical view. A\nreorganization where Copilot tasks are paired alongside their respective robots\nis utilized to reduce context loss and pair necessary information to complete\nthe tasks together, as shown in \\Cref{fig:ui_overview}A. Over time during the\ndevelopment roll-out, this pairing of health, sensor, and status indicators\nalongside Copilot tasks facilitated a level of trust from the operator where\nfocus on a particular robot was not necessary unless a critical task requiring\noperator intervention appeared.\n\n\\ph{3D Visualization View} A 3D interactive visualization leveraging React Three\nFiber (a React-based renderer for three.js) was created within the UI with the\naim of achieving a significant reduction in operator task and application\nswitching. Prior to this version of the interface, the operator was required to\nswitch between a web browser to view robot health systems and status information\nand RViz (a visualizer for ROS) to view the robots within the 3D environment and\ncommand them. In the split view of the UI, the operator can have the full\ncontext of robot sensors and status information along with any outstanding\nCopilot tasks. When in the expanded visualization view, the layout shares\nsimilarities with layouts of traditional Real-Time Strategy (RTS) games, where\ncontent is functionally organized from the corners of the view and leave the\ncenter-most screen real estate where the operator will primarily interact with\nrobots and information unobstructedly. From this view, the operator can take\non any of the following tasks: surveying the mapped environment and robot\npositions for locations to scout, locating, and submitting object or signal\nartifacts, directing or course-correcting robot autonomy with manual navigation\ncommands, viewing signal strength of the communications backbone within the\nenvironment, and assigning robots to drop communication nodes manually. The\nvisualization allows the operator to navigate the 3D environment through\npanning, zooming, and filtering points of interest categories. To effectively\nmanage the switching between micro and macro-level interactions, a single-click\nshortcut was implemented on each robot status card for the operator to quickly\nfocus on any robot that requires attention. An additional shortcut is\nprovided to zoom back out to an overview of the map.\n\t\nImprovements over traditional RTS commanding controls were also made to minimize\nthe amount of mouse control and coordination necessary. Instead of requiring to\nselect or drag a bounding box prior to commanding a robot, the operator could\nsimply interact with the visualized information roadmap (IRM) --- a breadcrumb\ntrail used for safely navigating the environment constructed by the team of\nrobots~\\cite{plgrim} --- and assign any robot with a high priority navigation point or\ncommunications node drop location through a context menu, regardless of whether the\nparticular robots are currently in view or not.\n\t\nTo help with artifact management, the locations of detected artifacts are\nvisualized and interactivity is added to allow the operator to quickly hover\ninto a thumbnail and click to navigate to the dedicated Artifact Drawer\n\\Cref{fig:ui_overview}C for deeper analysis and submission. Additional\ninteractions are, for example, manually adding and manipulating detected artifact\nlocations within the 3D space, by dragging its location across a plane for\nfine-tuning if a submission location was deemed incorrect and needed adjustment.\n\t\nWhile in the expanded visualization view, compressed versions of the robot\nstatus modules are shown horizontally in the bottom left of the view with the\nmission status indicator made more prominent and placed above each module. These\noverall status indicators were given visual priority to ensure grabbing the\noperator's attention. For instance, the indicator would flash red when a robot\nhad fallen over, was low on battery, or required assistance. The operator could\nimmediately click the respective robot module and be oriented over it for\nmicromanagement.\n\n\\ph{Artifact Drawer} Artifact submission was a critical part of SubT that also has many real-world parallels, for instance, in search and rescue. Especially under time constraints, it is necessary to quickly identify artifacts of interest in the environment, whether these be human survivors or other objects of interest. Detecting and localizing artifacts automatically is done using a state-of-the-art image processing pipeline \\cite{terry2020object}, but no AI system is infallible, especially in unknown environments, so having a system for an operator to manually review artifacts efficiently was critical considering mission time and submission attempts.\n\nIn the old system \\cite{terry2020object}, a manual artifact review system did exist, but it was built with a focus on only basic functionality and a high reliance on initially accurate artifact detections. Each artifact report took roughly 90 seconds to review. In redesigning this component, we wanted to focus on improving the review process from an ease of use perspective and decrease the time spent to confidently review an artifact report down to 15 seconds. Beyond simply making the system more intuitive for the operator, this actually had a major functional benefit from a trustability standpoint in that it allowed us to decrease the confidence threshold for flagging artifact detections and have the operator go through and verify nearly 6 times more potential artifact reports while not increasing total time spent. \n    \nTo better design the new system for speed, it was important to understand which areas of the old one were slowing the process down the most. Testing the old system in simulation and operator feedback revealed that the artifact review process needed too many clicks. Then, time had to be spent zooming in on and reviewing images and checking with RViz separately to verify that artifact coordinates were correct. No visual aid was given if corrections were necessary, and coordinates had to be updated by manually entering them for each axis in ${\\rm I\\!R^3}$. Borrowing from game interface design, integrating the 3D visualization view directly into the web UI removed the need for application switching, and drag controls were added to adjust locations providing correctly scaled coordinate updates from the 3D environment. A minified list that provides an overview of all artifact reports by confidence levels, plus maximizing the screen real estate of a single selected artifact helped increase efficiency. Finally, adding keyboard shortcuts as commonly used in gaming made meeting our target goal of 15 seconds possible. \n\n\n\n\\section{Results}\n\\label{sec:results}\n\nOver the course of the last challenge year, we conducted a limited series of field tests in three testing locations, including the abandoned tunnels at the Los Angeles Subway Terminal building, the Lava Bed National Monument in Northern California, and the Kentucky Underground lime-stone cave for which we applied our rapid development and testing strategy. We experimented with different robot configurations and in different stages of readiness as our system's capabilities matured. We deployed up to 11 vehicles simultaneously during these tests stressing the overall system (including Copilot and all the UI elements) and learning about its technical limitations like bandwidth and computing resources which will be presented in upcoming work.\n\nWe deployed the presented game-inspired user interface and supervised autonomy system during the SubT challenge using four to six ground robots nominally. While we could have exceeded the number of six robots using the newly designed interface and autonomy, six became the preferred number of agents to explore large-scale environments while allowing reliable communication links that would not exceed bandwidth limitations when robots disseminated information from autonomously explored out-of-comms areas. This allowed meeting the set interaction objectives, especially maintaining an enjoyable performance that can visualize the complete robot team while contributing to a lower workload due to fewer deployed agents.\n\nIn what follows, we analyzed screen recordings and log files (approved by Caltech IRB protocol number 19-0461 and Polytechnique Montreal project CER-2122-50-D) collected during the SubT final competition. We extracted time-to-task information, robot deployment times, mouse locations, and application usage from runs P1, P2, and F that consist of a setup-time and mission phase of 30+30 and 30+60 minutes, respectively. Robots were only allowed to leave the setup area and enter the course when the mission time began. Readying the team of robots and not bleeding into the mission time was a crucial effort to maximize available mission and exploration time. The results are compared to an earlier state of the system that did not implement Copilot and used different interfaces, namely the SubT ``Urban Circuit'' similar to \\cite{Otsu2020IEEEAerospace}. During the ``Urban Circuit'' task, coordination was done by humans only.\n\n\\ph{Robot Deployment} \\Cref{fig:robot_deployment} shows the robot deployment times that were achieved by deploying Copilot and compares them to the baseline. We can see that during run P1, we achieved sending one robot in less than 60 seconds each, deploying a total of 6 ground vehicles in 5 minutes and 31 seconds. In runs P2 and F, we achieved staying below the one minute mark for the first three robots. Deploying the robots without Copilot and the new interface in the `Urban Circuit'' runs A1, B1, and B2 took more than 5.5 min per robot on average, thus significantly reducing the time available for exploration and consequently reducing ground coverage and information gain regarding the search task. \n\n\\begin{figure}[t]\n\\centering\n  \\includegraphics[width=0.5\\textwidth]{figures\/robot_deployment.png}\n  \\caption{Robot deployment times per game run measured upon entering the course. The black dotted line ($\\sim$1min\/r) indicates the team's internal goal for robot deployment and represents a deployment of one robot per minute. F backup marks insertion points of 2 robots that were not part of the initial deployment strategy but were added ad-hoc to compensate for robot failures during run F.}\n  \\label{fig:robot_deployment}\n\\end{figure}\n\n\\ph{Application Usage} The new interface resulted in a shift in application usage and reduced switching between different applications and computers with a second set of peripherals, as RViz was running on a second device during the ``Urban Circuit''. \\Cref{fig:application_usage} presents the relative usage of applications for six SubT runs. Designing a unified interface resulted in a shift in application usage that reduced the use of RViz significantly. While more than 50\\% of time was spent on RViz during the ``Urban Circuit'' runs, we were able to unify user interactions and situation awareness in a single Mission Control interface. Only run F uses RViz for some time as a debugging tool that gave access to the robot's cost maps depicting the perceived risks around them. This information was not visualized by the new interface, but presents valuable key information in case of unexpected and off-nominal operations.\n\n\\ph{UI Feature Usage} With the main Mission Control interface being the main\ninteraction point for human supervisory control, we then look at the feature\nusage within the interface itself. \\Cref{fig:radar_plot} shows the relative\ninteraction times with the split-screen view, the 3D full-screen console view,\nthe sensor health overview, artifact submission drawer, and the BPMN modal that\ngives a detailed overview of a robot's inner state machine (which was relied\nupon during the ``Urban Circuit''). We see that, especially during runs P2 and F,\nlarge amounts of time were spent on the artifact drawer and thus performing the\nsearch task analyzing the artifact reports that were generated by the\nmulti-agent system. To gain situational awareness and potentially interact with\nthe robot team, the human supervisor primarily relied on the split-screen view of\nthe Mission Control app that is shown in the background of \\Cref{fig:heatmap}\noverlaid by a heat map that indicates the most active areas derived from mouse\ncursor positions sampled at 1.5 Hz. In this analysis, an area is deemed inactive\nif the mouse has been stationary for more than ten seconds. \\emph{Huang et\n  al.}~\\cite{andwhite2012user} found that the median difference between human\ngaze and mouse position during an active task is 77 pixels with a standard\ndeviation of 33.9 pixels at 96 dpi screen resolution. A Gaussian kernel with\n$\\mu=98$ and $\\sigma=43$ adjusting for 122 dpi is used to derive our heat maps.\n\\Cref{fig:heatmap} indicates that the robot cards, Copilot tasks and the 3D view\nwere all crucial tools while overseeing the robotic system and performing the\nexploration and search tasks.\n   \n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=0.5\\textwidth]{figures\/application_usage.png}\n  \\caption{Application usage (foreground application) for six SubT mission runs in percent. A1, B1, and B2 represent the usage before the redesign that integrated 3D visualization and interactions for P1, P2, and F in a single Mission Control application using only one computer and screen. Note that node manager and terminal usage are underrepresented in runs A1, B1, and B2 because the initial setup phase of up to 10 minutes was not recorded for these runs due to different logging procedures.}\n  \\label{fig:application_usage}\n\\end{figure}\n\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=0.23\\textwidth]{figures\/radar_ui_sub_category_usage_prelim1.png}\n  \\includegraphics[width=0.23\\textwidth]{figures\/radar_ui_sub_category_usage_prelim2.png}\n  \\includegraphics[width=0.27\\textwidth]{figures\/radar_ui_sub_category_usage_final_prize.png}\n  \\caption{Analysis of the redesigned user interface interaction by view component in percent for runs P1, P2, and F.}\n  \\label{fig:radar_plot}\n\\end{figure}\n\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=0.5\\textwidth]{figures\/heatmap_prelim2_costar_mission_control.png}\n  \\caption{Activity heat map showing the x and y positions of cursor interactions (and indirectly gaze) overlaid on the Mission Control Split-Screen view exemplary for game run P2. The view consists of robot cards, a column for Copilot tasks, and the split-screen 3D view. A brighter heat map indicates higher interaction times in this area. Stationary cursors for more than 10 seconds are classified as inactive.}\n  \\label{fig:heatmap}\n\\end{figure}\n\n\n\n\n\\section{Conclusions and Future Work}\n\\label{sec:conclusions_future_work}\nIn this work we \n\\begin{inparaenum}[(i)]\n    \\item create a game-inspired user interface for multi-agent robot missions\n    \\item integrate an automated planner for task planning and scheduling,\n    \\item add a verifiable task framework for increased reliability, and\n    \\item present results on how the overall system performed over the course of\n      several real-world deployments, including the DARPA SubT Challenge final.\n\\end{inparaenum}\nIn future work, we plan to deploy our interface and Copilot during scientific\nexploration missions to autonomously map and identify geological features and\nassess exploration strategies in lava tubes. This will lead to further\nvalidation of the subsystems and a structured assessment of a supervisor's\nworkload outside the realm of the SubT challenge with experts and potentially\nnon-expert users. Ultimately, we would like to assess operator workload from\nwearable sensors in real-time and consider such constraints in Copilot's task\nplanning.  \n\n\n\n\\section*{Acknowledgment}\n\\footnotesize{The work is partially supported by the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration (80NM0018D0004), and Defense Advanced Research Projects Agency (DARPA). This work was conducted in collaboration with the Making Innovative Space Technologies Laboratory (MIST Lab) at Polytechnique Montreal. The first author would like to thank the Natural Sciences and Engineering Research Council of Canada (NSERC) for their generous support in the form of a Vanier Canada Graduate Scholarship. Thank you to all members of Team CoSTAR for their valuable discussions and support.}\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{#1}\\setcounter{equation}{0}}\n\n\\newcommand{\\req}[1]{$(\\ref{#1})$}\n\\newcommand{\\dfn}{\\stackrel{\\triangle}{=}}\n\\newcommand{\\bydef}{\\stackrel{\\bigtriangleup}{=}}\n\\newcommand{\\limsupn}{\\limsup_{n \\rightarrow \\infty}}\n\\newcommand{\\liminfn}{\\liminf_{n \\rightarrow \\infty}}\n\\newcommand{\\val}{{\\rm v }}\n\\newcommand{\\co}{{\\rm conv }}\n\\newcommand{\\sgn}{{\\rm sign }}\n\\newcommand{\\ri}{{\\rm ri }}\n\n\\newcommand{\\vect}[1]{{\\boldsymbol #1 }}\n\n\\newcommand{\\td}[1]{\\tilde{#1}}\n\\newcommand{\\ceil}[1]{\\left\\lceil #1 \\right\\rceil}\n\\newcommand{\\floor}[1]{\\left\\lfloor #1 \\right\\rfloor}\n\\newcommand{\\inprod}[2]{\\langle #1 , #2 \\rangle }\n\\newcommand{\\card}[1]{\\left\\lvert #1 \\right\\rvert}\n\\newcommand{\\bc}{\\begin{center}}\n\\newcommand{\\ec}{\\end{center}}\n\\newcommand{\\bz}{\\vect{z}}\n\\newcommand{\\vv}[1]{\\boldsymbol #1}\n\\newcommand{\\by}{\\vect{y}}\n\\newcommand{\\bY}{\\vect{Y}}\n\\newcommand{\\refs}[1]{$(\\ref{#1})$}\n\\newcommand{\\refds}[2]{({#1}.\\ref{#2})}\n\\newcommand{\\bX}{\\vect{X}}\n\\newcommand{\\bx}{\\vect{x}}\n\\newcommand{\\bq}{\\vect{q}}\n\\newcommand{\\bZ}{\\boldsymbol Z}\n\\newcommand{\\R}{\\mathbb R}\n\\newcommand{\\N}{\\mathbb N}\n\\newcommand{\\Z}{\\mathbb Z}\n\\newcommand{\\m}[1]{\\mathcal{#1}}\n\\newcommand{\\ul}[1]{\\underline{#1}}\n\\newcommand{\\ol}[1]{\\overline{#1}}\n\n\\newcommand{\\be}{\\begin{equation}}\n\\newcommand{\\ee}{\\end{equation}}\n\\newcommand{\\beaa}{\\begin{eqnarray*}}\n\\newcommand{\\eeaa}{\\end{eqnarray*}}\n\\newcommand{\\ben}{\\begin{enumerate}}\n\\newcommand{\\een}{\\end{enumerate}}\n\\newcommand{\\db}{\\hspace*{\\fill}{\\zapf o}}\n\\newcommand{\\cpn}[1]{\\caption{\\myfont #1}}\n\\newcommand{\\bpn}{\\begin{proposition}\\twlsf}\n\\newcommand{\\epn}{\\db\\end{proposition}}\n\\newcommand{\\bdm}{\\begin{displaymath}}\n\\newcommand{\\edm}{\\end{displaymath}}\n\\newcommand{\\ba}{\\begin{array}}\n\\newcommand{\\ea}{\\end{array}}\n\n\\newcommand{\\supp}{\\rm{supp}}\n\\newcommand{\\op}[1]{\\mbox{op}_{#1}}\n\n\n\\newcommand{\\st}{\\mathop{\\rm s.t.}}\n\\newcommand{\\OR}{\\mathop{\\,\\,\\,{\\rm or}\\,\\,\\,}}\n\\newcommand{\\conv}{\\mathop{\\rm conv}}\n\\newcommand{\\aff}{\\mathop{\\rm aff}}\n\\newcommand{\\MP}{{\\cal P}}\n\\newcommand{\\dd}{\\mathrm{d}}\n\\newcommand{\\mbal}{\\mb{\\alpha}}\n\\newcommand{\\mbbt}{\\mb{\\beta}}\n\\newcommand{\\mbga}{\\mb{\\gamma}}\n\\newcommand{\\mbth}{\\mb{\\theta}}\n\\newcommand{\\mbmu}{\\mb{\\mu}}\n\n\\newcommand{\\br}{\\mb{b}_R}\n\\newcommand{\\argmin}{\\mathop{\\rm argmin}}\n\\newcommand{\\argmax}{\\mathop{\\rm argmax}}\n\n\n\\newtheorem{assumption}{Assumption}\n\\newtheorem{definition}{Definition}\n\\newtheorem{lemma}{Lemma}\n\\newtheorem{proposition}{Proposition}\n\\newtheorem{corollary}{Corollary}\n\\newtheorem{remark}{Remark}\n\\newtheorem{theorem}{Theorem}\n\\newtheorem{claim}{Claim}\n\\newtheorem{example}{Example}\n\n\\newcommand{\\boldone}{\\mbox{\\bf 1}}\n\\newcommand{\\Ga}{\\Gamma}\n\\newcommand{\\GA}{\\Gamma}\n\\newcommand{\\La}{\\Lambda}\n\\newcommand{\\LA}{\\Lambda}\n\\newcommand{\\la}{\\lambda}\n\\newcommand{\\eps}{\\epsilon}\n\\newcommand{\\Om}{\\Omega}\n\n\\newcommand{\\Rd}{\\reals^d}\n\\newcommand{\\DU}{\\Delta(\\cU)}\n\\newcommand{\\lip}{\\langle}\n\\newcommand{\\rip}{\\rangle}\n\\newcommand{\\rome}{\\mbox{\\rm e}}\n\n\\newcommand{\\limsupt}{\\limsup_{t \\rightarrow \\infty}}\n\\newcommand{\\liminft}{\\liminf_{t \\rightarrow \\infty}}\n\\newcommand{\\E}{\\mathbb{E}}\n\\newcommand{\\U}{\\mathbb{U}}\n\\newcommand{\\rP}{\\mathbb{P}}\n\\newcommand{\\norm}[1]{\\left\\lVert#1\\right\\rVert}\n\\newcommand{\\tnorm}[1]{\\lVert\\mkern-2mu |#1|\\mkern-2mu\\rVert}\n\\title{Robust Newsvendor Games with Ambiguity in Demand Distributions}\n\\author{Xuan Vinh Doan\\thanks{DIMAP and ORMS Group, Warwick Business School, University of Warwick, Coventry, CV4 7AL, United Kingdom, xuan.doan@wbs.ac.uk.} \\and Tri-Dung Nguyen\\thanks{Mathematical Sciences and Business School, University of Southampton, Southampton, SO17 1BJ, United Kingdom, T.D.Nguyen@soton.ac.uk.}}\n\n\n\\date{January 2016}\n\n\n\\begin{document}\n\\maketitle\n\n\n\\begin{abstract}\nWe investigate newsvendor games whose payoff function is uncertain due to ambiguity in demand distributions. We discuss the concept of stability under uncertainty and introduce solution concepts for robust cooperative games which could be applied to these newsvendor games. Properties and numerical schemes for finding core solutions of robust newsvendor games are presented.\n\\end{abstract}\nKeywords: Cooperative games; uncertain payoffs; newsvendor games; robust optimization; stability.\n\n\\section{Introduction}\n\\label{sec:intro}\nA joint venture is usually an effective approach for individual players in the market to share costs, reduce risk, and increase the total joint revenue or profit. For example, individual retailers can decide whether to order inventories together and share the profit from selling ordered products later. As a part of the joint venture formation, all players should agree on how to share the joint profit or payoff before the cooperation is established. Cooperative game theory provides a mathematical framework for addressing this problem, which is modeled as \\emph{newsvendor centralization games} (or newsvendor games for short). The model of these games has been introduced in Hartman \\cite{hartman1994cooperative}. Formally, consider the set ${\\cal N}$ of $N$ retailers and let $\\td{d}_i\\in\\mathbb{R}_+$ be the random demand for retailer $i$, $i\\in{\\cal N}$. In the setting of newsvendor games, we assume that the unit ordering cost $c$ and the unit selling price $p$ are the same for all retailers, $0<c< p$. Given an ordering quantity $y$, the expected profit (or payoff) of retailer $i$ is\n\\be\n\\label{eq:optquanti}\nv_i(y)=\\mathbb{E}_{P_i}\\left[p\\min\\{\\td{d}_i,y\\}-cy\\right],\\quad i\\in{\\cal N}.\n\\ee\nIndividual retailer $i$ needs to decide the optimal ordering quantity ${y}^*_i$ to maximize the expected profit (or payoff), $\\displaystyle{y}^*_i\\in\\arg\\max_{y\\geq 0}v_i(y)$, which is the $(p-c)\/p$-quantile of $P_i$, the distribution function of $\\td{d}_i$ for all $i\\in{\\cal N}$. The optimal expected profit is $\\displaystyle \\bar{v}_i=v_i(y_i^*)=\\mathbb{E}_{P_i}\\left[p\\min\\{\\td{d}_i,{y}_i^*\\}-cy_i^*\\right]$.\n\nIn the newsvendor games, we are concerned about whether individual retailers should form a coalition to make orders together and share the inventories with each other. For a coalition ${\\cal S}\\subseteq{\\cal N}$, the aggregate demand is $\\displaystyle\\td{d}({\\cal S})=\\sum_{i\\in{\\cal S}}\\td{d}_i$ and we assume that the joint distribution $P({\\cal S})$ of $\\td{d}_i$, $i\\in{\\cal S}$, is known. Given an ordering quantity $y$, the total expected profit of coalition ${\\cal S}$ is\n\\be\n\\label{eq:quantbar}\nv(y,{\\cal S})=\\mathbb{E}_{P({\\cal S})}\\left[p\\min\\{\\td{d}({\\cal S}),y\\}-cy\\right].\n\\ee\nSimilarly, coalition ${\\cal S}$ of retailers needs to decide the optimal ordering quantity ${y}^*({\\cal S})$ to maximize the total expected profit,\n\\be\n\\label{eq:optquantbar}\n{y}^*({\\cal S})\\in\\arg\\max_{y\\geq 0}v(y,{\\cal S}),\n\\ee\n which again is the $(p-c)\/p$-quantile of the distribution of $\\td{d}({\\cal S})$. The optimal total expected profit of coalition ${\\cal S}$ is\n\\be\n\\label{eq:optval}\n\\bar{v}({\\cal S})=v({y}^*({\\cal S}),{\\cal S})=\\mathbb{E}_{P({\\cal S})}\\left[p\\min\\{\\td{d}({\\cal S}),{y}^*({\\cal S})\\}-cy^*({\\cal S})\\right].\n\\ee\nIn order to guarantee that all retailers are satisfied with the grand coalition $\\cal N$, there should be an \\emph{allocation} (also called payoff distribution) of the total expected profit $\\bar{v}({\\cal N})$ among retailers such that no group of retailers has the incentive to form a smaller but better coalition. Newsvendor games concern about the existence of these stable payoff distributions.\n\nTraditionally, the underlying assumption in newsvendor games is that the joint demand distribution is known with certainty. Under this assumption, the expected profits $\\bar{v}({\\cal S})$ are known for all ${\\cal S}\\subseteq{\\cal N}$. In reality, it can be sometimes difficult to justify this assumption, especially when retailers need to decide whether to cooperate with each other before observing and sharing records of joint demands. If the joint demand distribution is unknown, the expected profits $\\bar{v}({\\cal S})$ are uncertain. Under the assumption that these payoffs are random variables, several models of stochastic cooperative games have been studied in order to address the uncertainty in coalition payoffs . Charnes and Granot \\cite{charnes1973prior} propose a two-stage payoff distribution scheme. In the first stage, some amounts of individual payoffs, which are called prior-payoffs, are promised to the players in the coalition. In the second stage, once the coalition payoff is realized, there could be adjustments in individual payoffs to guarantee that the payoff distribution is feasible and that any objections among the players are minimized. This approach assumes risk-neutral behaviors among players. Suijs et al. \\cite{suijs1999cooperative} study a different payoff distribution scheme for stochastic cooperative games using preference orders for random payoffs, which can handle different types of risk behavior. An individual payoff under this distribution scheme consists of two parts. The first part is a monetary exchange between players and the second part is a fraction of the random coalition payoff, which can depend on the action taken by the coalition. This distribution scheme works well for some applications such as the insurance game discussed in Suijs et al. \\cite{suijs1998stochastic}. More recently, Uhan \\cite{uhan2013stochastic} generalizes this payoff distribution scheme for stochastic linear programming games with applications in inventory centralization and network fortification. Timmer et al. \\cite{timmer2005convexity} argue that risk-covering monetary compensation is not needed in other applications. It is indeed a reasonable assumption that players (e.g., retailers or firms) in the examples discussed therein do not feel the need to pay upfront in order to benefit from a greater share of the profit in the future. Timmer et al. \\cite{timmer2005convexity} then propose a similar payoff distribution scheme for stochastic cooperative games without monetary exchange in which a multiple of the random coalition payoff is allocated to each individual. Fern\\'andez et al. \\cite{fernandez2002cores} also consider this payoff distribution scheme as a special case while investigating general stochastic payoff distributions using stochastic orders for stochastic cooperative games.\n\nIn addition to these models of stochastic cooperative games, there are different models that address different aspects of cooperative games under uncertainty. Lehrer \\cite{lehrer2003allocation} investigates payoff distribution processes in repeated deterministic cooperative games. The payoff allocation at any time depends on past allocations and current coalition payoffs. Lehrer \\cite{lehrer2003allocation} shows that these payoff distribution processes converge to some well-known solutions of cooperative games under appropriate allocation rules. Dror et al. \\cite{dror2008dynamic} extend these allocation rules and payoff distribution processes for repeated stochastic cooperative games and apply them to dynamic newsvendor realization games. Bauso and Timmer \\cite{bauso2009robust}, on the other hand, study dynamic cooperative games under the setting of a family of games whose coalition values are uncertain. Toriello and Uhan \\cite{toriello2015dynamic} investigate the dynamic linear programming games with risk-averse players. The study of cooperative games with incomplete information addresses other aspects of uncertainty in cooperative games. Ieong and Shoham \\cite{ieong2008bayesian} develop solution concepts of Bayesian cooperative games such as \\emph{ex ante}, \\emph{ex interim}, and \\emph{ex post} cores under the assumption that all players have a common prior over the set of possible states of the world but that each of them has some private information or belief about the true state of the world. More recently, Li and Conitzer \\cite{li2014cooperative} focus on solution concepts that maximize the probability of \\emph{ex post} stability assuming all players know the distribution of the random state of the world. Clearly, in this setting, private information of players does not play an important role since only \\emph{ex post} stability is considered. Chalkiadadis and Boutilier \\cite{chalkiadakis2004bayesian} study Bayesian coalition formation problems when players have different beliefs about contributions of other players to coalition payoffs, which makes coalition payoffs uncertain. Forges and Serrano \\cite{forges2013cooperative} discuss different models of cooperative games with incomplete information that handle externality; i.e., the effects of actions and information of players outside a coalition on the coalition payoff and payoffs of individual players in the coalition.\n\nIn this paper, we focus on newsvendor games with uncertain expected profits (or payoffs) under a similar setting to those of stochastic cooperative games discussed above; that is, uncertainty is captured and represented only in the payoff functions of the games. Our research, however, is different from existing models of stochastic cooperative games given the fact it is difficult to obtain probabilistic information on uncertain expected profits $\\bar{v}({\\cal S})$, ${\\cal S}\\subseteq{\\cal N}$. This assumption is particularly useful for newsvendor games considered in this paper in which the uncertainty of the payoffs is modeled through an uncertainty set derived from the ambiguity of the joint demand distribution.\n\\subsection*{Contributions and paper outline}\nIn this paper, we focus on the newsvendor games with ambiguity in the joint demand distribution. Specifically, our contributions and the structure of the paper are as follows:\n\\ben\n\\item[(1)] Section~\\ref{ssec:uncertain} develops a framework of cooperative games with uncertain payoffs which could be applied to newsvendor games under distributional ambiguity. We call these games \\emph{robust cooperative games}. Solution concepts such as imputation, core, and game balancedness of robust cooperative games are defined and discussed.\n\\item[(2)] Section~\\ref{stoc.newsvendor} studies newsvendor games with ambiguity in demand distributions using the framework of robust cooperative games. We focus on the existence of rational and stable payoff distributions of these \\emph{robust newsvendor games}. We also discuss the computational aspect of finding core solutions for robust newsvendor games and provide some numerical results.\n\\een\n\\section{Deterministic Cooperative Games}\n\\label{sec:coop}\nBefore presenting the model of cooperative games with uncertain payoffs that could be applied for newsvendor games under distributional ambiguity, we introduce some solution concepts of deterministic cooperative games whose payoffs are known with certainty. Consider a set of $N$ players, ${\\cal N}=\\{1,\\ldots,N\\}$, and a function $v:2^{\\cal N}\\rightarrow\\R$, which is called the \\emph{characteristic function}, such that $v(\\emptyset)=0$. We call ${\\cal G}=({\\cal N},v)$ a \\emph{cooperative game}. Utility (payoff) is assumed to be transferable, i.e., for any coalition ${\\cal S}\\subseteq{\\cal N}$, its total payoff is completely defined as $v({\\cal S})$, which can be transferred freely among its members. Under the assumption that the joint demand distribution is known with certainty, the newsvendor games previously introduced is indeed a deterministic cooperative game, $\\bar{\\cal G}=({\\cal N},\\bar{v})$, where $\\bar{v}$ is defined as in \\refs{eq:optval}.\n\nGiven a cooperative game $({\\cal N},v)$, we are interested in finding an allocation $\\bx\\in\\R^N$ to distribute the total payoff $v({\\cal N})$ among individual players. An allocation (also called a payoff distribution) $\\bx$ is \\emph{efficient} if\n\\be\n\\label{eq:grouprational}\n\\sum_{i\\in {\\cal N}}x_i=v({\\cal N}).\n\\ee\nAn important question regarding cooperative games is whether players are willing to join the grand coalition $\\cal N$. Necessarily, there should be an allocation $\\bx\\in\\R^N$ with which each individual player is better off as compared to his\/her standalone payoff. An allocation $\\bx$ is called \\emph{individually rational} if\n\\be\n\\label{eq:indrational}\nx_i\\geq v(\\{i\\}),\\quad\\forall\\,i\\in{\\cal N}.\n\\ee\n\\begin{definition}\n\\label{def:imputation}\nAn \\emph{imputation} is an allocation that is both efficient and individually rational. The set of imputations of a cooperative game $({\\cal N},v)$ is written as follows:\n\\be\n\\label{eq:impu}\n\\emph{impu}({\\cal N},v):=\\left\\{\\bx\\in\\R^N\\,\\mid\\,\\sum_{i\\in {\\cal N}}x_i=v({\\cal N}),\\,x_i\\geq v(\\{i\\}),\\,\\forall\\,i\\in{\\cal N}\\right\\}.\n\\ee\n\\end{definition}\nIf the characteristic function $v$ is \\emph{super-additive}, i.e., for any two disjoint coalitions ${\\cal S}_1$ and ${\\cal S}_2$, ${\\cal S}_1\\cap{\\cal S}_2=\\emptyset$, $v({\\cal S}_1)+v({\\cal S}_2)\\leq v({\\cal S}_1\\cup{\\cal S}_2)$, it is clear that there always exists at least one imputation or $\\mbox{impu}({\\cal N},v)\\neq\\emptyset$.\n\nIndividual rationality is not sufficient to guarantee that some players would prefer the grand coalition $\\cal N$ to a smaller coalition ${\\cal S}\\subsetneq{\\cal N}$. An allocation is called \\emph{stable} with respect to a coalition $\\cal S$ if $\\displaystyle\\sum_{i\\in{\\cal S}}x_i\\geq v({\\cal S})$.\n\\begin{definition}\n\\label{def:core}\nThe set of efficient allocations that are stable with respect to all coalitions ${\\cal S}\\subseteq{\\cal N}$ is called the \\emph{core},\n\\be\n\\label{eq:core}\n\\emph{core}({\\cal N},v):=\\left\\{\\bx\\in\\R^N\\,\\mid\\,\\sum_{i\\in {\\cal N}}x_i=v({\\cal N}),\\,\\sum_{i\\in{\\cal S}}x_i\\geq v({\\cal S}),\\,\\forall\\,{\\cal S}\\subsetneq{\\cal N}\\right\\}.\n\\ee\n\\end{definition}\nIt is obvious that $\\mbox{core}({\\cal N},v)\\subseteq\\mbox{impu}({\\cal N},v)$. However, the core might not exist. There are different solution concepts that take this issue into account. Given a parameter $\\eps\\geq 0$, the $\\eps$-\\emph{core} is defined as follows:\n\\be\n\\label{eq:ecore}\n\\eps\\mbox{-core}({\\cal N},v):=\\left\\{\\bx\\in\\R^N\\,\\mid\\,\\sum_{i\\in {\\cal N}}x_i=v({\\cal N}),\\,\\sum_{i\\in{\\cal S}}x_i\\geq v({\\cal S})-\\eps,\\,\\forall\\,{\\cal S}\\subsetneq{\\cal N}\\right\\}.\n\\ee\nIt is clear that $\\eps$-core is non-empty for a given large enough $\\eps$. The \\emph{least core} is the non-empty $\\eps$-core with the smallest value of $\\eps$, which is called the \\emph{least core value}. The least core value $\\eps({\\cal N},v)$ is the optimal value of the following linear program:\n\\be\n\\label{eq:leastcore}\n\\ba{rll}\n\\displaystyle\\eps({\\cal N},v)=\\min_{\\vect{x},\\eps\\geq 0} & \\eps\\\\\n\\st & \\displaystyle\\sum_{i\\in{\\cal N}}x_i=v({\\cal N}),\\\\\n& \\displaystyle\\sum_{i\\in{\\cal S}}x_i\\geq v({\\cal S})-\\eps, & \\forall\\,{\\cal S}\\subseteq{\\cal N}.\n\\ea\n\\ee\nIf we are able to solve Problem~\\refs{eq:leastcore} and obtain $\\eps({\\cal N},v) = 0$, then the core is non-empty. Schulz and Uhan \\cite{Schulz13} consider Problem \\refs{eq:leastcore} in a slightly different way:\n\\be\n\\label{eq:stability}\n\\ba{rll}\n\\displaystyle s({\\cal N},v)=\\min_{\\vect{x},\\eps} & \\eps\\\\\n\\st & \\displaystyle\\sum_{i\\in{\\cal N}}x_i=v({\\cal N}),\\\\\n& \\displaystyle\\sum_{i\\in{\\cal S}}x_i\\geq v({\\cal S})-\\eps, & \\forall\\,{\\cal S}\\subsetneq{\\cal N}, {\\cal S} \\neq \\emptyset.\n\\ea\n\\ee\nIn general, we have $\\eps({\\cal N},v)=\\max\\{s({\\cal N},v),0\\}$, i.e., $s({\\cal N},v)$ coincides with the least core value if it is non-negative. If $s({\\cal N},v)<0$, the absolute value $\\card{s({\\cal N},v)}$ can be interpreted as the maximum possible increase in all coalition payoffs $v({\\cal S})$, ${\\cal S}\\subsetneq{\\cal N}$, under the condition that at least one efficient and stable allocation still exists.\n\nFor the newsvendor game $({\\cal N},\\bar{v})$, the characteristic function $\\bar{v}$ is super-additive. Indeed, we have\n$$\n\\bar{v}({\\cal S})=\\mathbb{E}_{P({\\cal S})}\\left[p\\min\\{\\td{d}_i,{y}^*({\\cal S})\\}-cy^*({\\cal S})\\right]=(p-c){y}^*({\\cal S})-p\\,\\mathbb{E}_{P({\\cal S})}\\left[\\left({y}^*({\\cal S})-\\td{{d}}({\\cal S})\\right)^+\\right],\n$$\nwhere $x^+=\\max\\{x,0\\}$. Consider two disjoint coalitions ${\\cal S}_1$ and ${\\cal S}_2$, ${\\cal S}_1\\cap{\\cal S}_2=\\emptyset$; we have $\\td{d}({\\cal S}_1\\cup{\\cal S}_2)=\\td{d}({\\cal S}_1)+\\td{d}({\\cal S}_2)$. Using \\refs{eq:optquantbar} and the fact that $(x+y)^+\\leq x^++y^+$, we have\n$$\n\\ba{rl}\n\\bar{v}({\\cal S}_1\\cup{\\cal S}_2)&\\geq\\,(p-c)({y}^*({\\cal S}_1)+{y}^*({\\cal S}_2))-p\\,\\mathbb{E}_{P({\\cal S}_1\\cup{\\cal S}_2)}\\left[\\left({y}^*({\\cal S}_1)+{y}^*({\\cal S}_2)-\\td{{d}}({\\cal S}_1\\cup{\\cal S}_2)\\right)^+\\right]\\\\\n&\\geq\\,(p-c){y}^*({\\cal S}_1)-p\\,\\mathbb{E}_{P({\\cal S}_1)}\\left[\\left({y}^*({\\cal S}_1)-\\td{{d}}({\\cal S}_1)\\right)^+\\right]+\\vspace{5pt}\\\\\n&\\phantom{\\geq\\,\\,}\\,(p-c){y}^*({\\cal S}_2)-p\\,\\mathbb{E}_{P({\\cal S}_2)}\\left[\\left({y}^*({\\cal S}_2)-\\td{{d}}({\\cal S}_2)\\right)^+\\right]\\\\\n&=\\,\\bar{v}({\\cal S}_1)+\\bar{v}({\\cal S}_2).\n\\ea\n$$\nThis shows that newsvendor games always have imputations. For normally distributed demands, Hartman et al. \\cite{hartman2000cores} show that the cores of these games are non-empty. \\citet{chen2009stochastic} use stochastic linear programming duality to show the non-emptiness of the core in inventory centralization games and to provide a computational method for finding one. M\\\"uller et al. \\cite{mueller02} prove that every newsvendor game has a non-empty core, i.e., for all possible distributions of random demands. Montrucchio and Scarsini \\cite{montrucchio07} show how to construct an allocation in the core of newsvendor games. In the next section, we develop a model of cooperative games with uncertain characteristic functions that could be used to study newsvendor games with the ambiguity in demand distributions.\n\n\\section{Robust Cooperative Games}\n\\label{ssec:uncertain}\nWhen the payoffs are uncertain and assumed to be random variables (with finite expectation), the model of stochastic cooperative games proposed by Suijs et al. \\cite{suijs1999cooperative} can be applied. Suijs et al. \\cite{suijs1999cooperative} define a stochastic cooperative game as a tuple $({\\cal N},{\\cal A},\\tilde{v},\\succsim)$, where ${\\cal N}$ is the set of players, ${\\cal A}({\\cal S})$ is the set of possible actions that coalition $\\cal S$, ${\\cal S}\\subseteq{\\cal N}$, can take. $\\tilde{v}(a,{\\cal S})$ is the random payoff obtained by coalition $\\cal S$ if action $a\\in{\\cal A}({\\cal S})$ is taken. The inclusion of action sets ${\\cal A}({\\cal S})$ in this model of stochastic cooperative games reflects the fact that players can take different actions when facing uncertainty, which is different in the setting of deterministic cooperative games in which everything, including actions of players, is known with certainty. For newsvendor games, if retailers in a coalition ${\\cal S}$ do not know their joint demand distribution, there is no ordering quantity to achieve the ``optimal'' expected profit and they can choose to pick an ordering quantity $y$ from a set of possible ordering quantities ${\\cal Y}({\\cal S})$. Clearly, the expected profit of a coalition ${\\cal S}$, $\\tilde{v}(y,{\\cal S})$, now depends on the ordering quantity $y$. Finally, $\\succsim_i$ describes the preference of player $i$, $i\\in{\\cal N}$, over different random variables. For two random payoff $\\tilde{v}$ and $\\tilde{v}'$, $\\tilde{v}\\succsim_i\\tilde{v}'$ means that player $i$ prefers $\\tilde{v}$ over $\\tilde{v}'$. This order relation of stochastic payoffs is needed for this model of stochastic cooperative games to replace the straightforward comparison among deterministic payoffs in the deterministic setting. Suijs et al. \\cite{suijs1999cooperative} discuss several preference relations such as quantile-based relations and stochastic dominance for which the existence of core allocations is studied in detail.\n\nFor newsvendor games, the assumption of a known joint demand distribution in the deterministic setting can be considered to be rather strong given the fact that individual retailers usually collect historical demands independently before they join any coalition. In order to make the problem more realistic, we can assume that some (multivariate) marginal distributions are known instead of the joint distribution. For example, it is more reasonable to assume the knowledge of the joint demand distribution of a subset of retailers that are located close to each other and hence likely to serve customers from the same area. Under this assumption of distributional ambiguity, the joint demand distribution is unknown and belongs to an ambiguity set. For a coalition $\\cal S$, given an ordering quantity $y$, the total expected profit $\\td{v}({y,\\cal S})=\\mathbb{E}_{\\td{P}({\\cal S})}\\left[p\\min\\{\\td{d}_i,{y}\\}-cy({\\cal S})\\right]$ is uncertain and belongs to an uncertainty set if the (marginal) distribution $\\td{P}({\\cal S})$ is not known with certainty. In general, the characteristic function $\\td{v}$ of newsvendor games is uncertain if there is ambiguity in demand distributions. In order to apply the model of stochastic cooperative games discussed above for these newsvendor games, we would need to assume that probability distributions of random payoffs $\\td{v}(y,{\\cal S})$ are known. However, it is difficult to impose this probabilistic assumption, given that the randomness of $\\td{v}(y,{\\cal S})$ comes from the uncertainty of the probability distribution $\\td{P}({\\cal S})$. In order to address this issue, we propose a new model of general cooperative games with uncertain characteristic functions, which is different from the model of stochastic cooperative games proposed by Suijs et al. \\cite{suijs1999cooperative}.\n\nGiven a coalition ${\\cal S}\\subseteq{\\cal N}$ and an action $a\\in{\\cal A}({\\cal S})$, instead of considering $\\td{v}(a,{\\cal S})$ as a random variable, we consider it as an \\emph{uncertain} payoff $v_u(a,{\\cal S})$, where $u\\in{\\cal U}$ and ${\\cal U}$ is an uncertainty set.\nLet us denote ${\\cal V}_{{\\cal U}}(a,{\\cal S})=\\{v_u(a,{\\cal S})\\,:\\,u\\in{\\cal U}\\}$ for ${\\cal S}\\subseteq{\\cal N}$ and $a\\in{\\cal A}({\\cal S})$. We also denote ${\\cal V}({\\cal U})=\\{v_u\\,:\\,{\\cal A}\\times 2^{{\\cal N}}\\rightarrow\\mathbb{R},u\\in{\\cal U}\\}$ as the set of all characteristic functions. This setting is appropriate for the newsvendor games with the ambiguity in demand distribution in which the joint demand distribution $P$ belongs to an ambiguity set ${\\cal P}$. We define an instance of cooperative games with uncertain characteristic functions by the tuple $({\\cal N},{\\cal A},{\\cal V}({\\cal U}))$. In order to define the main concepts of these cooperative games $({\\cal N},{\\cal A},{\\cal V}({\\cal U}))$, we need first to select a payoff distribution scheme and characterize when sub-coalitions would break away from the grand coalition.\n\nThere are many different ways how payoffs can be allocated under uncertainty. When payoffs are assumed to be random variables, the most general payoff distribution scheme with random allocation for stochastic cooperative games is studied by Fern\\'andez et al. \\cite{fernandez2002cores}. For a coalition ${\\cal S}$, they consider efficient allocations as random vectors $\\td{\\bx}$ that satisfy\n\\be\n\\label{eq:sefficiency}\n\\sum_{i\\in{\\cal S}}\\td{x}_i=\\td{v}(a,{\\cal S}),\n\\ee\nassuming $a\\in{\\cal A}({\\cal S})$ is the chosen action for the coalition. Suijs et al. \\cite{suijs1999cooperative} argue that this general stochastic payoff distribution scheme generates a very large class of allocations and results in computational difficulties. They propose a more restrictive payoff distribution scheme that can be represented by a pair $(\\mb{d},\\mb{r})\\in\\R^{\\card{\\cal S}}\\times\\R^{\\card{\\cal S}}$ such that $\\displaystyle\\sum_{i\\in{\\cal S}}d_i=\\mathbb{E}\\left[\\tilde{v}(a,{\\cal S})\\right]$ and $\\displaystyle\\sum_{i\\in{\\cal S}}r_i=1$, $r_i\\geq 0$ for all $i\\in{\\cal S}$. The stochastic allocation is defined as $\\td{x}_i=d_i+r_i\\left(\\td{v}(a,{\\cal S})-\\mathbb{E}\\left[\\tilde{v}(a,{\\cal S})\\right]\\right)$ for all $i\\in{\\cal S}$. Note that this allocation scheme require probabilistic information, i.e., expectations of random characteristic functions. When the characteristic functions are deterministic, this allocation scheme coincides with the classical allocation of deterministic cooperative games, $x_i=d_i$ for all $i\\in{\\cal S}$, where $\\displaystyle\\sum_{i\\in{\\cal S}}d_i=\\mathbb{E}\\left[\\tilde{v}(a,{\\cal S})\\right]=v({\\cal S})$. Timmer et al. \\cite{timmer2005convexity} propose another payoff distribution scheme that is represented by a vector $\\mb{z}\\in\\R^S$ of \\emph{multiples}. The stochastic allocation under this scheme is $\\td{x}_i=\\td{v}(a,{\\cal S})\\cdot z_i$ for all $i\\in{\\cal S}$. Here, it is not required for $z_i$ to be non-negative for $i\\in{\\cal S}$. In order to guarantee that the stochastic allocation is efficient, it is required that $\\displaystyle\\sum_{i\\in{\\cal S}}z_i=1$. Compared to the payoff distribution scheme in Suijs et al. \\cite{suijs1999cooperative}, Timmer et al. \\cite{timmer2005convexity} argue that their proposed scheme does not allow monetary compensations to cover risk, which can be considered as a reasonable setting for several applications such as newsvendor games discussed above. Note that this scheme can also be seen as a direct extension of that of deterministic cooperative games since any allocation of deterministic cooperative games can be written in terms of multiples of $v({\\cal S})$, assuming $v({\\cal S})\\neq 0$. For the proposed model of cooperative games with uncertain characteristic functions, we face the same issues when defining payoff distribution schemes due to the uncertainty of the characteristic functions. Instead of having a random payoff $\\tilde{x}_i$ as in stochastic cooperative games, the uncertain payoff of player $i$ is now represented by the set ${\\cal X}_i({\\cal U})=\\{x_i(u)\\,:\\,u\\in{\\cal U}\\}\\subset\\mathbb{R}$. For a coalition ${\\cal S}$, we obtain the set of payoff vectors $\\mb{{\\cal X}}_{{\\cal S}}({\\cal U})=\\{\\bx_{{\\cal S}}(u)\\,:\\,u\\in{\\cal U}\\}\\subset\\mathbb{R}^{\\card{{\\cal S}}}$, where $\\bx_{{\\cal S}}(u)=\\{x_i(u)\\,:\\,i\\in{\\cal S}\\}$. Given that our main application is the newsvendor game in which risk-covering monetary compensation is not very important, we adopt a similar payoff distribution scheme to the one proposed by Timmer et al. \\cite{timmer2005convexity} for our model of cooperative games with uncertain characteristic functions. Formally, for a coalition ${\\cal S}$, a payoff distribution scheme is represented by $\\bz \\in\\R^{\\card{\\cal S}}$ such that for a given action $a\\in{\\cal S}$, the uncertain allocation for each player $i$, $i\\in{\\cal S}$ is\n\\be\n\\label{eq:scheme}\nx_i(u)=v_u(a,{\\cal S})\\cdot z_i,\\quad\\forall\\,u\\in{\\cal U}.\n\\ee\nA payoff distribution scheme $\\bz$ is \\emph{efficient} if $\\displaystyle \\sum_{i\\in{\\cal S}}z_i=1$.\nThis concept of efficiency indicates that the uncertain allocation $\\bx$ is efficient for all realizations of the uncertain characteristic function $v_u\\in{\\cal V}({\\cal U})$.\n\nNow, similar to stochastic cooperative games, the concepts of rationality and stability rely on both the payoff distribution scheme and the action taken by the grand coalition, the \\emph{decision} $(a,\\bz)$. They also depend on how we characterize the preference of individuals and sub-coalitions with respect to staying in or breaking away from the grand coalition. In the model of stochastic cooperative games proposed by Suijs et al. \\cite{suijs1999cooperative}, order relations of stochastic payoffs are used to define these stay-in\/break-away preferences. Under the setting of uncertain payoffs, there is no distributional information and we need to define these preferences based on the realization of uncertain parameters. The resulting uncertain allocation for each player $i$, $i\\in{\\cal N}$ is $x_i(u)=v_u(a,{\\cal N})\\cdot z_i$, $u\\in{\\cal U}$ for all $i\\in{\\cal N}$. If he\/she acts alone with the action $\\alpha\\in{\\cal A}(\\{i\\})$, the uncertain payoff is $v_u(\\alpha,\\{i\\})$. If he\/she joins a smaller sub-coalition ${\\cal S}\\subsetneq{\\cal N}$ whose decision is $(\\hat{a},\\hat{\\mb{z}})$, the uncertain allocation would be $v_u(\\hat{a},{\\cal S})\\cdot\\hat{z}_i$. Given a realization of $u\\in{\\cal U}$, each player $i$ can compare these different realized payoffs directly and the sub-coalition would have the incentive to break away if all of its members are better off as in the deterministic setting. By looking at the break away incentive given a realization of uncertain parameters, we can define stay-in\/break-away preferences of sub-coalitions (and individual players) under the uncertain setting as follows.\n\\begin{definition}\n\\label{def:preference}\nA sub-coalition ${\\cal S}$ has the incentive to break away from the grand coalition if there exists a decision $(\\hat{a},\\hat{\\mb{z}})$ which guarantees that \\emph{all} players in the sub-coalition are better off by leaving given \\emph{at least} one realization $u\\in{\\cal U}$, i.e., $v_u(\\hat{a},{\\cal S})\\cdot\\hat{z}_i>x_i(u)$ for all $i\\in{\\cal S}$.\n\\end{definition}\nIn other words, a sub-coalition has the incentive to stay in the grand coalition if no matter what decision it takes, there will be at least one player in the sub-coalition is better off by staying in the grand coalition in any realization of uncertain parameters. Applying this definition for individual players, we can say that a player has the incentive to break away if there exists an action $\\alpha\\in{\\cal A}(\\{i\\})$  which guarantees that he\/she will be better off by leaving the grand coalition for at least one realization of $u\\in{\\cal U}$, i.e., $v_u(\\alpha,\\{i\\})>x_i(u)$. Note that the idea of defining these realization-based concepts is similar to those in Bitran \\cite{bitran1980linear}, which are used to define necessary and sufficient solution concepts for multi-objective optimization. We are now ready to define main solution concepts for cooperative games with uncertain characteristic functions $({\\cal N},{\\cal A},{\\cal V}({\\cal U}))$ using the proposed payoff distribution scheme and the above definition of break away incentive.\n\nA decision $(a,\\bz)$ is \\emph{individually rational} if there is no individual player who has the incentive to break away. We can then define the concept of imputations.\n\\begin{definition}\n\\label{def:rimputation\nAn imputation of the cooperative game $({\\cal N},{\\cal A},{\\cal V}({\\cal U}))$ is an individually rational decision $(a,\\bz)$ whose payoff distribution scheme $\\bz$ is efficient.\n\\end{definition}\nSimilar to the deterministic setting, let $\\mbox{impu}({\\cal N},{\\cal A},{\\cal V}({\\cal U}))$ denote the set of all imputations of the cooperative game $({\\cal N},{\\cal A},{\\cal V}({\\cal U}))$.\n\nWe now define the concept of stability. A decision $(a,\\bz)$ is \\emph{stable} if there is no sub-coalition ${\\cal S}\\subsetneq{\\cal N}$ that has the incentive to break away and we have the following definition of the cores of cooperative games $({\\cal N},{\\cal A},{\\cal V}({\\cal U}))$.\n\\begin{definition}\n\\label{def:score}\nThe core of the cooperative game $({\\cal N},{\\cal A},{\\cal V}({\\cal U}))$ is the set of all stable decisions with efficient payoff distribution schemes and is denoted by $\\emph{core}({\\cal N},{\\cal A},{\\cal V}({\\cal U}))$.\n\\end{definition}\n\nThe above definition of stability indicates that all sub-coalitions of a stable grand coalition necessarily do not have the incentive to break away no matter how $u\\in {\\cal U}$ is realized.  \nThis follows the ``immunized-against-uncertainty'' principle of robust optimization (see Ben-Tal et al. \\cite{ben2009robust} and references therein); therefore, we call cooperative games $({\\cal N},{\\cal A},{\\cal V}({\\cal U}))$ with the above definition of break away incentive \\emph{robust cooperative games}.\n\nWe are now ready to characterize the existence of imputations and core decisions, i.e., decisions belong to the core, of robust cooperative games $({\\cal N},{\\cal A},{\\cal V}({\\cal U}))$. We start by making the following assumptions.\n\n\\begin{assumption}\n\\label{as:pos}\n\\indent\n\\begin{itemize}\n\\item[(i)] For any player $i\\in{\\cal N}$, there exists an action such that his\/her payoff is always non-negative, i.e., there exists $a \\in {\\cal A}(\\{i\\})$ such that $v_u(a,\\{i\\})\\geq 0$ for all $u\\in{\\cal U}$.\n\\item[(ii)] The payoff of the grand coalition $\\cal N$ is always positive, i.e., $v_u(a,{\\cal N})>0$ for all $a\\in{\\cal A}({\\cal N})$ and $u\\in{\\cal U}$.\n\\end{itemize}\n\\end{assumption}\nSimilar to Timmer et al. \\cite{timmer2005convexity}, these assumptions emphasize the fact that we are focusing on profit games with possible nonnegative payoff for each individual sub-coalition and it is indeed worth considering the grand coalition given that its profit is always positive. In other words, Assumption~\\ref{as:pos}(ii) implies that we should only consider the set of actions ${\\cal A}({\\cal N})$ which create positive profits under any circumstances for the grand coalition. For robust newsvendor games that we are going to discuss in Section~\\ref{stoc.newsvendor}, these assumptions are easily satisfied in general.\n\n\n\n\n\nWe now state the following result on existence conditions for imputations of robust cooperative games.\n\\begin{theorem}\n\\label{prop:impucond}\nGiven a robust cooperative game $({\\cal N},{\\cal A},{\\cal V}({\\cal U}))$, an imputation exists if and only if there exists an action $a\\in{\\cal A}({\\cal N})$ such that\n\\be\n\\label{eq:impucond}\n\\sum_{i\\in{\\cal N}} v_{\\max}(a,\\{i\\}) \\leq 1,\n\\ee\nwhere $\\displaystyle v_{\\max}(a,\\{i\\}) = \\max_{u\\in{\\cal U}}\\left\\{\\frac{\\displaystyle\\max_{\\alpha_i\\in{\\cal A}(\\{i\\})}v_u(\\alpha_i,\\{i\\})}{v_u(a,{\\cal N})}\\right\\}$.\n\\end{theorem}\nIn order to prove the theorem, we need the following lemma.\n\\begin{lemma}\n\\label{prop:individual:rationality:cond}\nGiven a robust cooperative game $({\\cal N},{\\cal A},{\\cal V}({\\cal U}))$ and a decision $(a,\\bz)$, a player $ i \\in {\\cal N}$ has no incentive to break away if and only if\n$\\displaystyle z_i \\geq v_{\\max}(a,\\{i\\})$.\n\\end{lemma}\n\n\\begin{pf}\nWe have: $x_i(u)=v_u(a,{\\cal N})\\cdot z_i$ for all $i\\in{\\cal N}$ and $u\\in{\\cal U}$. Player $i\\in{\\cal N}$ does not have the incentive to break away if and only if\n$$\nv_u(\\alpha_i,\\{i\\})\\leq v_u(a,{\\cal N})\\cdot z_i,\\quad\\,\\forall\\,\\alpha_i\\in{\\cal A}(\\{i\\}),~u\\in{\\cal U}.\n$$\nUnder Assumption \\ref{as:pos}(ii), this holds if and only if $\\displaystyle z_i\\geq\\max_{\\alpha_i\\in{\\cal A}(\\{i\\})}\\frac{v_u(\\alpha_i,\\{i\\})}{v_u(a,{\\cal N})}$ for all $u\\in{\\cal U}$, i.e.,\n$$\nz_i\\geq \\max_{u\\in{\\cal U}}\\left\\{\\frac{\\displaystyle\\max_{\\alpha_i\\in{\\cal A}(\\{i\\})}v_u(\\alpha_i,\\{i\\})}{v_u(a,{\\cal N})}\\right\\}=v_{\\max}(a,\\{i\\}).\n$$\n\\end{pf}\n\nWe are now ready to prove Theorem \\ref{prop:impucond}.\n\n\\begin{pf}\nSuppose an imputation $(a,\\bz)$ exists. According to the definition of imputations, there is no player $i\\in{\\cal N}$ who has the incentive to break away. Thus, according to Lemma~\\ref{prop:individual:rationality:cond}, we have:\n$$\nz_i\\geq v_{\\max}(a,\\{i\\}),\\quad\\,\\forall\\,i\\in{\\cal N}.\n$$\nNow, $(a,\\bz)$ is a decision of the grand coalition; therefore, $\\displaystyle\\sum_{i\\in{\\cal N}}z_i=1$. Summing over all $i\\in{\\cal N}$ the above inequality, we then achieve condition \\refs{eq:impucond}.\n\nNow, suppose condition \\refs{eq:impucond} holds. Let $\\displaystyle\\eps = 1-\\sum_{i\\in{\\cal N}} v_{\\max}(a,\\{i\\}) \\geq 0$ and define\n$$\nz_i=v_{\\max}(a,\\{i\\})+\\frac{\\eps}{N},\\quad\\,\\forall\\,i\\in{\\cal N}.\n$$\nWe will show that $(a,\\bz)$ is an imputation. Clearly, $\\bz$ is efficient, i.e., $\\displaystyle\\sum_{i\\in{\\cal N}}z_i=1$ given the definition of $z_i$ and $\\eps$. Now we have: for all $u\\in{\\cal U}$, $\\displaystyle z_i\\geq v_{\\max}(a,\\{i\\}) \\geq \\frac{\\displaystyle\\max_{\\alpha_i\\in{\\cal A}(\\{i\\})}v_u(\\alpha_i,\\{i\\})}{v_u(a,{\\cal N})}$ for all $i\\in{\\cal N}$ since $\\eps\\geq 0$. Thus, under Assumption \\ref{as:pos}(ii),\n$$\nv_u(a,{\\cal N})\\cdot z_i\\geq \\max_{\\alpha_i\\in{\\cal A}(\\{i\\})}v_u(\\alpha_i,\\{i\\}),\\quad\\,\\forall\\,i\\in{\\cal N},\\,u\\in{\\cal U}.\n$$\nIt shows that for all $u\\in{\\cal U}$,\n$$\nx_i(u)\\geq v_u(\\alpha_i,\\{i\\}),\\quad \\forall\\,i\\in{\\cal N},\\,\\alpha_i\\in{\\cal A}(\\{i\\}).\n$$\nThus, there is no player $i$ who has the incentive to break away; that is $(a,\\bz)$ is individually rational, which implies that $(a,\\bz)$ is an imputation.\n\\end{pf}\n\nSimilar to the deterministic and stochastic cooperative games, the existence of core decisions is related to the concept of balancedness. A map $\\mu:2^{{\\cal N}}\\setminus\\emptyset\\rightarrow[0,+\\infty)$ is called a \\emph{balanced} map if $\\displaystyle\\sum_{{\\cal S}\\subsetneq{\\cal N}}\\mu({\\cal S})\\cdot\\mb{e}_{{\\cal S}}=\\mb{e}_{{\\cal N}}$, where, for all ${\\cal S}\\subseteq{\\cal N}$, $\\mb{e}_{{\\cal S}}\\in\\{0,1\\}^N$ with $\\left(e_{{\\cal S}}\\right)_i=1$ if and only if $i\\in{\\cal S}$. We are now ready to define the balanced robust cooperative game.\n\\begin{definition}\nA robust cooperative game $({\\cal N},{\\cal A},{\\cal V}({\\cal U}))$ is called balanced if there exists an action $a\\in{\\cal A}({\\cal N})$ such that for all balanced map $\\mu$,\n\\be\n\\label{eq:balanced}\n\\sum_{{\\cal S}\\subsetneq{\\cal N}}\\mu({\\cal S})v_{\\max}(a,{\\cal S})\\leq 1,\n\\ee\nwhere $\\displaystyle v_{\\max}(a,{\\cal S})=\\max_{u\\in{\\cal U}}\\left\\{\\frac{\\displaystyle\\max_{\\alpha_{{\\cal S}}\\in{\\cal A}({\\cal S})}v_u(\\alpha_{{\\cal S}},{\\cal S})}{v_u(a,{\\cal N})}\\right\\}$.\n\\end{definition}\nThis definition of balanced robust cooperative games matches the definition of balanced deterministic games when $\\card{{\\cal U}}=\\card{{\\cal A}({\\cal S})}=1$ for all ${\\cal S}\\subseteq{\\cal N}$ with the balancedness condition $\\displaystyle\\sum_{{\\cal S}\\subsetneq{\\cal N}}\\mu({\\cal S})v({\\cal S})\\leq v({\\cal N})$. We can now state the following theorem regarding the existence of core decisions.\n\n\\begin{theorem}\n\\label{thm:corecond}\nA robust cooperative game $({\\cal N},{\\cal A},{\\cal V}({\\cal U}))$ has a non-empty core if and only if it is balanced.\n\\end{theorem}\nIn order to prove Theorem~\\ref{thm:corecond}, we need the following lemma.\n\\begin{lemma}\n\\label{lem:breakcond}\nGiven a robust cooperative game $({\\cal N},{\\cal A},{\\cal V}({\\cal U}))$ and an imputation $(a,\\bz)$, a coalition ${\\cal S}\\subsetneq{\\cal N}$ has the incentive to break away if and only if\n\\be\n\\label{eq:breakcond}\n\\sum_{i\\in{\\cal S}}z_i< v_{\\max}(a,{\\cal S}).\n\\ee\n\\end{lemma}\n\n\\begin{pf}\nGiven an imputation $(a,\\bz)$, a coalition ${\\cal S}$ has the incentive to break away if there exists an efficient decision $(\\hat{a},\\hat{\\bz})$\nand a realization $u\\in{\\cal U}$ such that for all $i\\in{\\cal S}$,\n$$\nv_u(\\hat{a},{\\cal S})\\cdot \\hat{z}_i>v_u(a,{\\cal N})\\cdot z_i.\n$$\nSince $(\\hat{a},\\hat{\\bz})$ is efficient, we have: $\\displaystyle\\sum_{i\\in{\\cal S}}\\hat{z}_i=1$. Summing over all $i\\in{\\cal S}$ the inequality above, we then obtain the following statement:\n$$\n\\exists\\,u\\in{\\cal U}\\,:\\,v_u(a,{\\cal N})\\cdot\\sum_{i\\in{\\cal S}} z_i<v_u(\\hat{a},{\\cal S})\\cdot \\sum_{i\\in{\\cal S}}\\hat{z}_i=v_u(\\hat{a},{\\cal S}).\n$$\nEquivalently, under Assumption \\ref{as:pos}(ii), we have:\n$$\n\\exists\\,u\\in{\\cal U}\\,:\\,\\sum_{i\\in{\\cal S}}z_i<\\frac{v_u(\\hat{a},{\\cal S})}{v_u(a,{\\cal N})}\\,\\Leftrightarrow\\,\\sum_{i\\in{\\cal S}}z_i<\\max_{u\\in{\\cal U}}\\left\\{\\frac{v_u(\\hat{a},{\\cal S})}{v_u(a,{\\cal N})}\\right\\}.\n$$\nWe have:\n$$\n\\ba{rcl}\n\\displaystyle\\max_{u\\in{\\cal U}}\\left\\{\\frac{v_u(\\hat{a},{\\cal S})}{v_u(a,{\\cal N})}\\right\\}&\\leq&\\displaystyle\\max_{\\alpha_{\\cal S}\\in{\\cal A}({\\cal S})}\\max_{u\\in{\\cal U}}\\left\\{\\frac{v_u(\\alpha_{\\cal S},{\\cal S})}{v_u(a,{\\cal N})}\\right\\}\\\\\n&=&\\displaystyle\\max_{u\\in{\\cal U}}\\max_{\\alpha_{\\cal S}\\in{\\cal A}({\\cal S})}\\left\\{\\frac{v_u(\\alpha_{\\cal S},{\\cal S})}{v_u(a,{\\cal N})}\\right\\}\\\\\n&=&\\displaystyle\\max_{u\\in{\\cal U}}\\left\\{\\frac{\\displaystyle\\max_{\\alpha_{\\cal S}\\in{\\cal A}({\\cal S})}v_u(\\alpha_{\\cal S},{\\cal S})}{v_u(a,{\\cal N})}\\right\\}.\n\\ea\n$$\nThe second equality is due to the fact that $v_u(a,{\\cal N})>0$ for all $u\\in{\\cal U}$ under Assumption \\ref{as:pos}(ii). Thus, if a coalition ${\\cal S}$ has the incentive to break away, then\n$$\n\\sum_{i\\in{\\cal S}}z_i<\\max_{u\\in{\\cal U}}\\left\\{\\frac{\\displaystyle\\max_{\\alpha_{{\\cal S}}\\in{\\cal A}({\\cal S})}v_u(\\alpha_{{\\cal S}},{\\cal S})}{v_u(a,{\\cal N})}\\right\\}=v_{\\max}(a,{\\cal S}).\n$$\n\nNow, suppose $\\displaystyle\\sum_{i\\in{\\cal S}}z_i<v_{\\max}(a,{\\cal S})$, we will show that coalition ${\\cal S}$ has the incentive to break away. We would need to show the existence of an efficient decision $(\\hat{a},\\hat{\\bz})$ and a realization $u\\in{\\cal U}$ such that for all $i\\in{\\cal S}$,\n$$\nv_u(\\hat{a},{\\cal S})\\cdot \\hat{z}_i>v_u(a,{\\cal N})\\cdot z_i.\n$$\nLet $(\\hat{a},\\hat{u})\\in\\displaystyle\\arg\\max_{\\alpha_{{\\cal S}}\\in{\\cal A}({\\cal S})}\\max_{u\\in{\\cal U}}\\left\\{\\frac{v_u(\\alpha_{\\cal S},{\\cal S})}{v_u(a,{\\cal N})}\\right\\}$. Clearly, we have:\n$$\n\\frac{v_{\\hat{u}}(\\hat{a},{\\cal S})}{v_{\\hat{u}}(a,{\\cal N})}=\\max_{\\alpha_{{\\cal S}}\\in{\\cal A}({\\cal S})}\\max_{u\\in{\\cal U}}\\left\\{\\frac{v_u(\\alpha_{{\\cal S}},{\\cal S})}{v_u(a,{\\cal N})}\\right\\}=\\max_{u\\in{\\cal U}}\\left\\{\\frac{\\displaystyle\\max_{\\alpha_{\\cal S}\\in{\\cal A}({\\cal S})}v_u(\\alpha_{\\cal S},{\\cal S})}{v_u(a,{\\cal N})}\\right\\}=v_{\\max}(a,{\\cal S}).\n$$\nFrom Lemma~\\ref{prop:individual:rationality:cond}, we have $\\displaystyle z_i \\geq v_{\\max}(a,\\{i\\})$ since $(a,\\bz)$ is an imputation. We also have, by Assumption~\\ref{as:pos}(i), $\\displaystyle v_{\\max}(a,\\{i\\}) \\geq 0$. Thus, $\\displaystyle v_{\\max}(a,{\\cal S}) > \\displaystyle\\sum_{i\\in{\\cal S}}z_i \\geq 0$ and hence $v_{\\hat{u}}(\\hat{a},{\\cal S})\\neq 0$.\n\nLet $\\displaystyle\\eps=\\frac{1}{\\card{{\\cal S}}}\\left(v_{\\max}(a,{\\cal S})-\\sum_{i\\in{\\cal S}}z_i\\right)>0$ and define $\\displaystyle\\hat{z}_i=\\frac{z_i+\\eps}{v_{\\max}(a,{\\cal S})}$ for all $i\\in{\\cal S}$. Clearly, $\\displaystyle\\sum_{i\\in{\\cal S}}\\hat{z}_i=1$ given the definition of $\\eps$. In addition, for all $i\\in{\\cal S}$, we have:\n$$\nv_{\\hat{u}}(\\hat{a},{\\cal S})\\cdot\\hat{z}_i=v_{\\hat{u}}(a,{\\cal N})\\cdot z_i+v_{\\hat{u}}(a,{\\cal N})\\cdot \\eps>v_{\\hat{u}}(a,{\\cal N})\\cdot z_i.\n$$\nThus, $(\\hat{a},\\hat{\\bz})$ is an efficient decision and the inequality above shows that coalition $\\cal S$ indeed has the incentive to break away.\n\\end{pf}\n\nWe are now ready to prove Theorem \\ref{thm:corecond}.\n\n\\begin{pf}\nSuppose there exists a core decision $(a,\\bz)$. Clearly, $(a,\\bz)$ is an imputation. Applying Lemma \\ref{lem:breakcond}, we can show that $\\bz$ is a feasible (and optimal) solution of the following linear program:\n\\be\n\\label{eq:primal}\n\\ba{rl}\n\\displaystyle Z_P=\\min_{\\bx} & \\displaystyle\\sum_{i\\in{\\cal N}}0\\cdot x_i\\\\\n\\st & \\displaystyle\\sum_{i\\in{\\cal S}}x_i\\geq v_{\\max}(a,{\\cal S}),\\quad \\forall\\,{\\cal S}\\subsetneq{\\cal N},\\\\\n& \\displaystyle\\sum_{i\\in{\\cal N}}x_i=1.\n\\ea\n\\ee\nThe dual problem is written as follows:\n\\be\n\\label{eq:dual}\n\\ba{rll}\nZ_D=\\max & \\displaystyle\\sum_{{\\cal S}\\subsetneq{\\cal N}}y_{{\\cal S}}v_{\\max}(a,{\\cal S})-p\\\\\n\\st & \\displaystyle\\sum_{{\\cal S}:i\\in{\\cal S}}y_{{\\cal S}}-p=0, & \\forall\\,i\\in{\\cal N},\\\\\n& y_{{\\cal S}}\\geq 0, & \\forall\\,{\\cal S}\\subsetneq{\\cal N}.\n\\ea\n\\ee\nApplying strong duality, we have: $Z_D=Z_P=0$, which means, for all feasible solution $\\left(\\left\\{y_{{\\cal S}}\\right\\}_{{\\cal S}\\subsetneq{\\cal N}},p\\right)$,\n$$\n\\sum_{{\\cal S}\\subsetneq{\\cal N}}y_{{\\cal S}}v_{\\max}(a,{\\cal S})\\leq p.\n$$\nNow consider any balanced map $\\mu$, it is clear that $\\left(\\left\\{\\mu({\\cal S})\\right\\}_{{\\cal S}\\subsetneq{\\cal N}},1\\right)$ is a feasible solution of the dual problem. Thus we have:\n$$\n\\sum_{{\\cal S}\\subsetneq{\\cal N}}\\mu({\\cal S})v_{\\max}(a,{\\cal S})\\leq 1.\n$$\nIt shows that the robust cooperative game is balanced.\n\nNow, suppose the robust cooperative game is balanced. There exists an action $a\\in{\\cal A}({\\cal N})$ such that for all balanced map $\\mu$,\n$$\n\\sum_{{\\cal S}\\subsetneq{\\cal N}}\\mu({\\cal S})v_{\\max}(a,{\\cal S})\\leq 1.\n$$\nWe are going to show that $Z_D=0$. The dual problem is indeed feasible. $(\\mb{0},0)$ is a feasible solution and it implies that $Z_D\\geq 0$. Let us consider a feasible solution $\\left(\\left\\{y_{{\\cal S}}\\right\\}_{{\\cal S}\\subsetneq{\\cal N}},p\\right)$. Since $y_{{\\cal S}}\\geq 0$ for all ${\\cal S}\\subsetneq{\\cal N}$, we have: $p\\geq 0$. If $p=0$, it is easy to show that $y_{{\\cal S}}=0$ for all ${\\cal S}\\subsetneq{\\cal N}$. If $p>0$, let $\\mu({\\cal S})=y_{\\cal S}\/p$ for all ${\\cal S}\\subsetneq{\\cal N}$. The map $\\mu$ is indeed a balanced map, thus we have:\n$$\n\\sum_{{\\cal S}\\subsetneq{\\cal N}}\\left(y_{{\\cal S}}\/p\\right)v_{\\max}(a,{\\cal S})\\leq 1\\,\\Leftrightarrow\\,\\sum_{{\\cal S}\\subsetneq{\\cal N}}y_{{\\cal S}}v_{\\max}(a,{\\cal S})-p\\leq 0.\n$$\nThus for all feasible solution $\\left(\\left\\{y_{{\\cal S}}\\right\\}_{{\\cal S}\\subsetneq{\\cal N}},p\\right)$, $\\displaystyle \\sum_{{\\cal S}\\subsetneq{\\cal N}}y_{{\\cal S}}v_{\\max}(a,{\\cal S})-p\\leq 0$, which means $Z_D\\leq 0$ and hence $Z_D=0$. According linear strong duality, the primal problem is feasible (and with the obvious optimal objective $Z_P=0$). Thus there exists at least a feasible solution $\\bz$ of the primal problem. The decision $(a,\\bz)$ is an imputation and indeed a core decision given the conditions obtained from the constraints of the primal problem. Thus the robust cooperative game has a non-empty core.\n\\end{pf}\n\nTheorem \\refs{thm:corecond} establishes the relationship between the game balancedness and the existence of core solutions. Computationally, it implies that we can attempt to solve the following optimization problem to check the existence of core solutions:\n\\be\n\\label{eq:rleastcore}\n\\ba{rl}\n\\displaystyle s({\\cal N},{\\cal A},{\\cal V}({\\cal U}))=\\min_{\\bx,\\eps,a} & \\eps\\\\\n\\st & \\displaystyle\\sum_{i\\in{\\cal S}}x_i\\geq v_{\\max}(a,{\\cal S})-\\eps,\\quad \\forall\\,{\\cal S}\\subsetneq{\\cal N},\\\\\n& \\displaystyle\\sum_{i\\in{\\cal N}}x_i=1,\\\\\n& a\\in{\\cal A}({\\cal N}).\n\\ea\n\\ee\nIf $s({\\cal N},{\\cal A},{\\cal V}({\\cal U}))\\leq 0$, then the core of the robust cooperative game $({\\cal N},{\\cal A},{\\cal V}({\\cal U}))$ is non-empty. Note that this optimization problem is no longer a linear program like Problem \\refs{eq:leastcore} or \\refs{eq:stability} given the fact that the action of the grand coalition is now a decision variable. We will investigate further the computational aspect of finding a core solution for a specific robust cooperative game, the \\emph{robust newsvendor game}, which is going to be discussed next. Even though the framework of robust cooperative games developed in this section is indeed suitable for the newsvendor games with distributional ambiguity which we are interested in, we would like to emphasize that it is plausible to develop other frameworks of cooperative games with uncertain characteristic functions using different payoff distribution schemes and different preference relations to suit other applications better.\n\n\n\\section{Robust Newsvendor Games} \\label{stoc.newsvendor}\nIn this section, we consider newsvendor games with ambiguity in demand distributions in the framework of robust cooperative games, which we call \\emph{robust newsvendor games}.\n\\begin{comment}Individual retailers usually collect historical demands independently before they join any coalition and the current assumption of a known joint distribution can be considered quite strong. In order to make the problem is more realistic, we only assume that some (multivariate) marginal distributions are known. For example, it is more reasonable to assume the knowledge of the joint demand distribution of a subset of retailers that are located close to each other and hence likely serve customers from a same area.\n\\end{comment}\nAs discussed in the previous section, the uncertainty comes from the fact that the joint demand distribution is unknown. We assume that only some (multivariate) marginal distributions of the joint demand are known. More concretely, consider a partition of ${\\cal N}$ with $R$ subsets ${\\cal N}_1,\\ldots,{\\cal N}_R$ such that\n$$\n{\\cal N}=\\bigcup_{r=1}^R{\\cal N}_r\\quad\\mbox{and}\\quad{\\cal N}_r\\cap{\\cal N}_s=\\emptyset\\quad\\mbox{for all }r\\neq s.\n$$\nGiven a vector $\\mb{d}\\in\\mathbb{R}^n$, let $\\mb{d}_r\\in\\mathbb{R}^{N_r}$ denote the sub-vector formed with the elements in the $r$th subset ${\\cal N}_r$ where $N_r=\\card{{\\cal N}_r}$ is the size of the subset. We assume that  probability measures $P_r$ of random vectors $\\td{\\mb{d}}_r$ are known for all $r=1,\\ldots,R$. Let $\\mathcal{P}(P_1,\\ldots,P_R)$ denote the set of joint probability measures of the random vector $\\td{\\mb{d}}$ consistent with the prescribed probability measures of the random vectors $\\td{\\mb{d}}_r$ for all $r=1,\\ldots,R$, which acts as the uncertainty set ${\\cal U}$ in the general framework of robust cooperative games. Note that $\\mathcal{P}(P_1,\\ldots,P_R)$ is always non-empty since the independent measure among the sub-vectors is a feasible distribution. The set of joint distributions with fixed marginal distributions $\\mathcal{P}(P_1,\\ldots,P_R)$ is referred to as the Fr\\'echet class of distributions (see R\\\"uschendorf \\cite{ruschen91b}). It has been used to evaluate bounds on the cumulative distribution function of a sum of random variables with an application in risk management (Embrechts and Puccetti \\cite{ep06b}). Doan and Natarajan \\cite{doan12} developed a robust optimization model using $\\mathcal{P}(P_1,\\ldots,P_R)$ with an application in project management. We now investigate this Fr\\'echet class of distributions in the context of robust newsvendor games.\n\nGiven a subset ${\\cal S}\\subseteq{\\cal N}$, we define ${\\cal S}_r={\\cal S}\\cap{\\cal N}_r$ for all $r=1,\\dots,R$. Clearly, if all retailers $i$, $i\\in{\\cal S}$, join together, we know the non-overlapping marginal distributions of the joint demand vector with respect to the partition $({\\cal S}_1,\\ldots,{\\cal S}_R)$ of ${\\cal S}$. If ${\\cal S}\\subseteq{\\cal N}_r$ for some $r$, the joint distribution of $\\td{d}_i$, $i\\in{\\cal S}$, is completely known. In this case, the action the coalition should take, i.e., the decision on the ordering quantity, is well-defined as in the deterministic setting. The action set of coalition ${\\cal S}$ can be simply defined as ${\\cal Y}({\\cal S})=\\{y^*({\\cal S})\\}$, where $y^*({\\cal S})$ is the $(p-c)\/p$-quantile of the known distribution of $\\tilde{d}({\\cal S})$ as defined in \\refs{eq:optquantbar}. Note that ${\\cal Y}({\\cal S})$ acts as the action set ${\\cal A}({\\cal S})$ in the general framework of robust cooperative games. In general, the distribution $P({\\cal S})$ is unknown and coalition ${\\cal S}$ can choose its action regarding the ordering quantity from a general action set ${\\cal Y}({\\cal S})$.\nTo keep it simple, we shall let ${\\cal Y}({\\cal S})=\\mathbb{R}_+$ given the fact that the ordering quantities are non-negative for all ${\\cal S}\\subseteq{\\cal N}$.\n As mentioned earlier, if ${\\cal S}\\subseteq{\\cal N}_r$ for some $r$, we can restrict ${\\cal Y}({\\cal S})=\\{y^*({\\cal S})\\}$, where $y^*({\\cal S})$ is the $(p-c)\/p$-quantile of the known distribution of $\\tilde{d}({\\cal S})$ as defined in \\refs{eq:optquantbar}.\n\nGiven an ordering decision $y\\in{\\cal Y}({\\cal S})$, the \\emph{uncertain} payoff, i.e., total expected profit, of coalition $\\cal S$ is $v_P(y,{\\cal S})=\\mathbb{E}_{P({\\cal S})}\\left[p\\min\\{\\td{d}({\\cal S}),y\\}-cy\\right]$ for $P\\in{\\cal P}(P_1,\\ldots,P_R)$ as in \\refs{eq:quantbar}, where $P({\\cal S})$ is the corresponding marginal joint distribution of $\\td{d}_i$, $i\\in{\\cal S}$, derived from $P$. For each individual retailer $i$, ${\\cal Y}(\\{i\\})=y_i^*$ and $v_P(y_i^*,\\{i\\})\\geq v_P(0,\\{i\\})=0$ for all $P\\in{\\cal P}(P_1,\\ldots,P_R)$, which implies Assumption~\\ref{as:pos}(i) is automatically satisfied.\n\nFinally, for the grand coalition, in order to satisfy Assumption~\\ref{as:pos}(ii), we let\n\\be\n\\label{eq:yn}\n{\\cal Y}({\\cal N})=\\left\\{y\\in\\mathbb{R}_+\\,:\\,v_P(y,{\\cal N})>0,\\,\\forall\\,P\\in{\\cal P}(P_1,\\ldots,P_R) \\right\\}.\n\\ee\nWe shall provide a simple condition with which the action set of the grand coalition is non-empty. Let $d_{\\min}({\\cal S})$ be the minimum value that the random demand $\\td{d}({\\cal S})$ can achieve, the following lemma sets out a sufficient condition for ${\\cal Y}({\\cal N})$ to be empty.\n\n\\begin{lemma}\n\\label{lem:nonempty}\nIf $\\displaystyle\\max_{r=1,\\ldots,R}d_{\\min}({\\cal N}_r)>0$, then ${\\cal Y}({\\cal N})\\neq \\emptyset$.\n\\end{lemma}\n\n\\begin{pf}\nWe have: $\\displaystyle\\td{d}({\\cal N})=\\sum_{r=1}^R\\td{d}({\\cal N}_r)$. Thus, if $\\displaystyle\\max_{r=1,\\ldots,R}d_{\\min}({\\cal N}_r)>0$, then:\n$$\nd_{\\min}({\\cal N})\\geq\\sum_{r=1}^Rd_{\\min}({\\cal N}_r)> 0.\n$$\nFor $y\\in[0,d_{\\min}(N)]$, $v_P(y,{\\cal N})=(p-c)y$, which is strictly increasing for any $P\\in{\\cal P}(P_1,\\ldots,P_R)$ given the fact that $p>c$. In addition, for $y\\geq d_{\\max}({\\cal N})$, $v_P(y,{\\cal N})=-cy+\\displaystyle p\\sum_{r=1}^R\\mathbb{E}_{P_r}\\left[\\td{d}({\\cal N}_r)\\right]$, which is strictly decreasing for any $P\\in{\\cal P}(P_1,\\ldots,P_R)$ given the fact that $c>0$.\n\nNow consider the function $\\displaystyle\\bar{v}(y,{\\cal N})=\\min_{P\\in{\\cal P}(P_1,\\ldots,P_R)}v_P(y,{\\cal N})$. It is clear that $\\bar{v}(\\cdot,{\\cal N})$ is again strictly increasing in $[0,d_{\\min}({\\cal N})]$ and strictly decreasing in $[d_{\\max}({\\cal N}),+\\infty)$. Thus we have:\n$$\n\\arg\\max_{y\\geq 0}\\bar{v}(y,{\\cal N})\\in[d_{\\min}({\\cal N}),d_{\\max}({\\cal N})],\n$$\nand\n$$\n\\max_{y\\geq 0}\\bar{v}(y,{\\cal N})\\geq \\bar{v}(d_{\\min}({\\cal N}),{\\cal N})=(p-c)d_{\\min}({\\cal N})>0.\n$$\nThus there exists $y\\geq 0$ such that $\\bar{v}(y,{\\cal N})>0$, or equivalently, $v_P(y,{\\cal N})>0$ for all $P\\in{\\cal P}(P_1,\\ldots,P_R)$. It shows that ${\\cal Y}({\\cal N})\\neq\\emptyset$.\n\\end{pf}\n\nThe condition in Lemma \\ref{lem:nonempty} simply requires that one of the (marginal) distributions $P_r$, $r=1,\\ldots,R$ has the support set of solely non-zero demand vectors, which can be considered as a reasonable assumption in reality.\n In the rest of the paper, we shall make that assumption to ensure ${\\cal Y}({\\cal N})\\neq\\emptyset$, or equivalently, that Assumption~\\ref{as:pos}(ii) is satisfied.\nWe are now ready to consider the robust newsvendor game $({\\cal N},{\\cal Y},{\\cal V}({\\cal P}(P_1,\\ldots,P_r)))$ and investigate the existence of its imputations and core solutions.\n\\subsection{Existence of Imputations and Core Solutions}\n\\label{ssec:existence}\nThe uncertain characteristic function $v_P(y,{\\cal S})$ can be written as\n\\be\n\\label{eq:char}\nv_{P}(y,{\\cal S})=(p-c)y-p\\,\\mathbb{E}_{P({\\cal S})}\\left[\\left(y-\\td{d}({\\cal S})\\right)^+\\right],\n\\ee\nfor all ${\\cal S}\\subseteq{\\cal N}$, $y\\in{\\cal Y}({\\cal S})$, and $P\\in{\\cal P}(P_1,\\ldots,P_R)$. The deterministic newsvendor games always have imputations since the corresponding characteristic function is super-additive. The following theorem claims the existence of imputations of the robust newsvendor game $({\\cal N},{\\cal Y},{\\cal V}({\\cal P}(P_1,\\ldots,P_r)))$.\n\\begin{theorem}\n\\label{thm:rimpu}\nThe robust newsvendor game $({\\cal N},{\\cal Y},{\\cal V}({\\cal P}(P_1,\\ldots,P_r)))$ always has an imputation.\n\\end{theorem}\n\nIn order to prove Theorem \\ref{thm:rimpu}, we first study a particular action that each coalition can take, the worst-case optimal ordering quantity $y^*_{wc}({\\cal S})$:\n\\be\n\\label{eq:optquant2}\ny^*_{wc}({\\cal S})\\in\\arg\\max_{y\\geq 0}\\left\\{(p-c)y-p\\max_{P\\in{\\cal P}(P_1,\\ldots,P_R)}\\mathbb{E}_{P}\\left[\\left(y-\\td{{d}}({\\cal S})\\right)^+\\right]\\right\\}.\n\\ee\nBasically, this ordering quantity is optimal under the worst-case scenario with respect to the joint demand distribution $P$. Let us also define $v_{wc}({\\cal S})$ as the maximum worst-case expected profit for coalition ${\\cal S}$, i.e.\n\\be\nv_{wc}({\\cal S}) = \\max_{y\\geq 0}\\left\\{(p-c)y-p\\max_{P\\in{\\cal P}(P_1,\\ldots,P_R)}\\mathbb{E}_{P}\\left[\\left(y-\\td{{d}}({\\cal S})\\right)^+\\right]\\right\\}.\n\\ee\n\nWhen ${\\cal S}\\subseteq{\\cal N}_r$ for some $r$, clearly, $y^*_{wc}({\\cal S})=y^*({\\cal S})$, the $(p-c)\/p$-quantile of the known distribution of $\\td{d}({\\cal S})$, and $v_{wc}({\\cal S})=\\bar{v}({\\cal S})$ as defined in \\refs{eq:optval}. The following lemma shows how to calculate the worst-case optimal ordering quantities $y^*_{wc}({\\cal S})$ and the worst-case expected profit $v_{wc}({\\cal S})$ for an arbitrary ${\\cal S}\\subseteq{\\cal N}$.\n\\begin{lemma}\n\\label{lem:optquant}\nFor an arbitrary ${\\cal S}\\subseteq{\\cal N}$, the worst-case optimal ordering quantity $y^*_{wc}({\\cal S})$ defined in \\refs{eq:optquant2} can be calculated as follows:\n\\be\n\\label{eq:optquantform1}\ny^*_{wc}({\\cal S})=\\sum_{r=1}^R{y}^*({\\cal S}_r),\n\\ee\nwhere ${\\cal S}_r={\\cal S}\\cap{\\cal N}_r$ for all $r=1,\\ldots,R$, ${y}^*({\\cal S}_r)$ is the $(p-c)\/p$-quantile of the known distribution of $\\td{d}({\\cal S}_r)$, and ${y}^*(\\emptyset)=0$. In addition, $v_{wc}({\\cal S}) = \\displaystyle \\sum_{r=1}^R v_{wc}({\\cal S}_r) = \\displaystyle \\sum_{r=1}^R \\bar{v}({\\cal S}_r)$.\n\\end{lemma}\n\n\\begin{pf}\nConsider the optimization problem in \\refs{eq:optquant2}. For $y\\leq d_{\\min}({\\cal S})=\\min\\{\\td{d}({\\cal S})\\}$, we can write $(p-c)y-p\\,\\mathbb{E}_{P}\\left[\\left(y-\\td{{d}}({\\cal S})\\right)^+\\right]=(p-c)y$ for any distribution $P$. Since $p-c>0$, it is an increasing function in $y$ in $(-\\infty;d_{\\min}({\\cal S})]$. Since $d_{\\min}({\\cal S})\\geq 0$, we can then remove the non-negative constraint $y\\geq 0$ from \\refs{eq:optquant2} when calculating $y^*_{wc}({\\cal S})$. Now, consider the inner optimization problem of \\refs{eq:optquant2}. This is an instance of the distributionally robust optimization problem studied in Doan and Natarajan \\cite{doan12}. Without loss of generality, we can assume that ${\\cal S}_r\\neq\\emptyset$ for all $r=1,\\ldots,R$ knowing that ${y}^*(\\emptyset)=0$. Applying Proposition 1(ii) from \\cite{doan12}, we obtain the following reformulation:\n$$\n\\ba{rl}\n\\displaystyle\\max_{P\\in{\\cal P}(P_1,\\ldots,P_R)}\\mathbb{E}_{P}\\left[\\left(y-\\td{{d}}({\\cal S})\\right)^+\\right]=\\min_{\\mbs{x}} &\\displaystyle\\sum_{r=1}^R\\mathbb{E}_{P_r}\\left[\\left(x_r-\\td{d}({\\cal S}_r)\\right)^+\\right]\\\\\n\\st &\\displaystyle\\sum_{r=1}^Rx_r=y.\n\\ea\n$$\nThus, in order to find $y^*_{wc}({\\cal S})$, we can solve the following optimization problem\n$$\n\\ba{rl}\n\\displaystyle\\max_{y,\\mbs{x}}&\\displaystyle (p-c)y-p\\sum_{r=1}^R\\mathbb{E}_{P_r}\\left[\\left(x_r-\\td{d}({\\cal S}_r)\\right)^+\\right]\\\\\n\\st & \\displaystyle\\sum_{r=1}^Rx_r=y.\\\\\n\\ea\n$$\nThe optimal ordering quantity $y^*_{wc}({\\cal S})$ can then be calculated as $\\displaystyle y^*_{wc}({\\cal S})=\\sum_{r=1}^Rx_r^*$, where $\\bx^*$ is the optimal solution of the following separable optimization problem:\n$$\n\\max_{\\mbs{x}}\\sum_{r=1}^R\\left((p-c)x_r-p\\sum_{r=1}^R\\mathbb{E}_{P_r}\\left[\\left(x_r-\\td{d}({\\cal S}_r)\\right)^+\\right]\\right),$$\nor equivalently,\n$$\n\\sum_{r=1}^R\\max_{x_r}\\left((p-c)x_r-p\\mathbb{E}_{P_r}\\left[\\left(x_r-\\td{d}({\\cal S}_r)\\right)^+\\right]\\right).\n$$\nFor each sub-problem, $x_r^*$ is the $(p-c)\/p$-quantile of the distribution of $\\td{d}({\\cal S}_r)$, which means $x^*_r={y}^*({\\cal S}_r)$ for all $r=1,\\ldots, R$, according to \\refs{eq:optquantbar}. Thus we have $\\displaystyle y^*_{wc}({\\cal S})=\\sum_{r=1}^R{y}^*({\\cal S}_r)$. This also leads to $v_{wc}({\\cal S}) = \\displaystyle \\sum_{r=1}^R v_{wc}({\\cal S}_r) = \\displaystyle \\sum_{r=1}^R \\bar{v}({\\cal S}_r)$. Here, the joint demand distribution for players in ${\\cal S}_r$ is known and hence the worst-case expected profit ${v}_{wc}(S_r)$ is exactly the same with the deterministic expected profit $\\bar{v}({\\cal S}_r)$ shown in Formulation~\\refs{eq:optval}.\n\\end{pf}\n\n We are now ready to use this lemma to prove Theorem \\ref{thm:rimpu}.\n\n\\begin{pf}\nAccording to Theorem \\ref{prop:impucond}, the robust newsvendor game $({\\cal N},{\\cal Y},{\\cal V}({\\cal P}(P_1,\\ldots,P_r)))$ has imputations if and only if there exists $y\\in{\\cal Y}({\\cal N})$ such that\n$$\n\\sum_{i\\in{\\cal N}}v_{\\max}(y,\\{i\\})\\leq 1,\n$$\nwhere $\\displaystyle v_{\\max}(y,\\{i\\})=\\max_{P\\in{\\cal P}(P_1,\\ldots,P_R)}\\left\\{\\frac{\\displaystyle\\max_{y_i\\in{\\cal Y}(\\{i\\})}v_P(y_i,\\{i\\})}{v_P(y,{\\cal N})}\\right\\}$. For each individual retailer $i$, the demand distribution is known; therefore ${\\cal Y}(\\{i\\})=\\{y_i^*\\}$, where $y_i^*$ is the $(p-c)\/p$-quantile of the distribution function of $\\td{d}_i$. Thus\n$$\n\\max_{y_i\\in{\\cal Y}(\\{i\\})}v_P(y_i,\\{i\\})=v_i(y^*)=\\bar{v}_i\\geq 0,\\quad\\forall\\,P\\in{\\cal P}(P_1,\\ldots,P_R).\n$$\nWe then have $\\displaystyle v_{\\max}(y,\\{i\\})=\\bar{v}_i\\cdot\\left(\\displaystyle\\min_{P\\in{\\cal P}(P_1,\\ldots,P_R)}v_P(y,{\\cal N})\\right)^{-1}$ and the existence condition of imputation can be written as follows:\n$$\n\\exists\\,y\\in{\\cal Y}({\\cal N}):\\,\\sum_{i\\in{\\cal N}}\\bar{v}_i\\leq \\displaystyle\\min_{P\\in{\\cal P}(P_1,\\ldots,P_R)}v_P(y,{\\cal N}),\n$$\ngiven the fact that $v_P(y,{\\cal N})>0$ for all $y\\in{\\cal Y}({\\cal N})$ and $P\\in{\\cal P}(P_1,\\ldots,P_R)$. Equivalently, the existence condition is\n$$\n\\sum_{i\\in{\\cal N}}\\bar{v}_i\\leq \\displaystyle\\max_{y\\in{\\cal Y}({\\cal N})}\\min_{P\\in{\\cal P}(P_1,\\ldots,P_R)}v_P(y,{\\cal N})=\\max_{y\\geq 0}\\min_{P\\in{\\cal P}(P_1,\\ldots,P_R)}v_P(y,{\\cal N}).\n$$\nThe optimization problem $\\displaystyle\\max_{y\\geq 0}\\min_{P\\in{\\cal P}(P_1,\\ldots,P_R)}v_P(y,{\\cal N})$ is the same as Problem \\refs{eq:optquant2}. Let us consider the worst-case optimal ordering quantity $y^*_{wc}({\\cal N})$, the existence condition can then be written as follows:\n$$\n\\sum_{i\\in{\\cal N}}\\bar{v}_i\\leq v_P(y^*_{wc}({\\cal N}),{\\cal N}),\\quad\\forall\\,P\\in{\\cal P}(P_1,\\ldots,P_R).\n$$\nApplying Lemma \\ref{lem:optquant} for ${\\cal S}={\\cal N}$, we have $\\displaystyle y^*_{wc}({\\cal N})=\\sum_{i=1}^R{y}^*({\\cal N}_r)$. Using the fact that $(x+y)^+\\leq x^++y^+$, we have, for all $P\\in{\\cal P}(P_1,\\ldots,P_R)$,\n$$\n\\ba{rl}\nv_P(y^*_{wc}({\\cal N}),{\\cal N}) &=\\,(p-c)y^*_{wc}({\\cal N})-p\\,\\mathbb{E}_P\\left[\\left(y^*_{wc}({\\cal N})-\\td{d}({\\cal N})\\right)^+\\right]\\\\\n&\\geq\\,\\displaystyle (p-c)\\sum_{i=1}^R{y}^*({\\cal N}_r)-p\\sum_{r=1}^R\\mathbb{E}_{P_r}\\left[\\left({y}^*({\\cal N}_r)-\\td{d}({\\cal N}_r)\\right)^+\\right]\\\\\n&=\\,\\displaystyle\\sum_{r=1}^R\\left\\{(p-c){y}^*({\\cal N}_r)-p\\,\\mathbb{E}_{P_r}\\left[\\left({y}^*({\\cal N}_r)-\\td{d}({\\cal N}_r)\\right)^+\\right]\\right\\}\\\\\n&=\\,\\displaystyle\\sum_{r=1}^R\\bar{v}({\\cal N}_r),\n\\ea\n$$\nwhere $\\bar{v}({\\cal N}_r)$ is the optimal total expected profit of coalition ${\\cal N}_r$ and is computed using \\refs{eq:optval} for all $r=1,\\ldots,R$ since $P_1,\\ldots,P_R$ are completely known.\n\\begin{comment}\nWe have $\\bar{v}({\\cal N}_r)\\geq 0$ for all $r=1,\\ldots,R$ and under Assumption \\ref{as:mpos}, there exists $r$ such that $\\bar{v}({\\cal N}_r)>0$. Thus, we have\n$$\nv_P({\\cal N})\\geq\\sum_{r=1}^R\\bar{v}({\\cal N}_r)>0.\n$$\n\\end{comment}\n\nNow consider the deterministic newsvendor game for coalition ${\\cal N}_r$ with the complete knowledge of the joint distribution $P_r$. There exists at least one imputation for this cooperative game; thus, we have\n$$\n\\sum_{i\\in{\\cal N}_r}\\bar{v}_i\\leq\\bar{v}({\\cal N}_r),\\quad\\forall\\,r=1,\\ldots,R.\n$$\nUsing the fact that ${\\cal N}=\\bigcup_{r=1}^R{\\cal N}_r$ and ${\\cal N}_r\\cap{\\cal N}_s = \\emptyset$ for all $r\\neq s$, we have\n$$\nv_P({\\cal N})\\geq\\sum_{r=1}^R\\bar{v}({\\cal N}_r)\\geq\\sum_{r=1}^R\\sum_{i\\in{\\cal N}_r}\\bar{v}_i=\\sum_{i\\in{\\cal N}}\\bar{v}_i,\\quad\\forall\\,P\\in{\\cal P}(P_1,\\ldots,P_R).\n$$\nThus, the existence condition is satisfied and the robust newsvendor game $({\\cal N},{\\cal Y},{\\cal V}({\\cal P}(P_1,\\ldots,P_r)))$ always has an imputation.\n\\end{pf}\n\nWe focus on properties of core solutions of the robust newsvendor game  $({\\cal N},{\\cal Y},{\\cal V}({\\cal P}(P_1,\\ldots,P_r)))$ next.\n\\begin{theorem}\n\\label{thm:rcore2}\nIf $(y,\\mb{z})$ is a core solution of the robust newsvendor game $({\\cal N},{\\cal Y},{\\cal V}({\\cal P}(P_1,\\ldots,P_r)))$, then the following statements hold:\n\\begin{itemize}\n\\item[(a)] The ordering quantity $y$ is the worst-case optimal ordering quantity, i.e., $y = y^*_{wc}({\\cal N})$.\n\\item[(b)] $v_{wc}({\\cal N})\\cdot\\mb{z}({\\cal N}_r)$ is a core solution of the (deterministic) newsvendor game $({\\cal N}_r,\\bar{v})$.\n\\end{itemize}\n\\end{theorem}\n\n\\begin{pf}\nLet us consider a core solution $(y,\\mb{z})$ of the robust newsvendor game {\\small $({\\cal N},{\\cal Y},{\\cal V}({\\cal P}(P_1,\\ldots,P_r)))$}. For each coalition ${\\cal N}_r$, $r =1,\\ldots,R$, we have\n\n\\begin{eqnarray}\nv_{\\max}(y,{\\cal N}_r)&=&\\max_{P\\in{\\cal P}(P_1,\\ldots,P_R)} \\left\\{\\frac{\\displaystyle\\max_{y_r \\in{\\cal Y}({\\cal N}_r)}v_P(y_r,{\\cal N}_r)}{v_P(y,{\\cal N})}\\right\\} \\nonumber\\\\\n&=& \\frac{\\displaystyle\\max_{y_r \\in{\\cal Y}({\\cal N}_r)}v_{P_r}(y_r,{\\cal N}_r)}{\\displaystyle \\min_{P\\in{\\cal P}(P_1,\\ldots,P_R)} v_P(y,{\\cal N})} \\nonumber\\\\\n&=& \\frac{\\displaystyle v_{wc}({\\cal N}_r)}{\\displaystyle \\min_{P\\in{\\cal P}(P_1,\\ldots,P_R)} v_P(y,{\\cal N})} = \\frac{\\displaystyle \\bar{v}({\\cal N}_r)}{\\displaystyle \\min_{P\\in{\\cal P}(P_1,\\ldots,P_R)} v_P(y,{\\cal N})}. \\label{eq:thm:rcore1}\n\\end{eqnarray}\nThis is due to the fact that $P_r$ is known with certainty and $v_{wc}({\\cal N}_r)\\geq 0$. According to Lemma~\\ref{lem:breakcond} and Equality \\refs{eq:thm:rcore1}, for $(y,\\mb{z})$ to be a core solution, we need to have\n\\begin{eqnarray}\n\\displaystyle \\sum_{i\\in{\\cal N}_r}z_i \\geq v_{\\max}(y,{\\cal N}_r) = \\frac{\\displaystyle \\bar{v}({\\cal N}_r)}{\\displaystyle \\min_{P\\in{\\cal P}(P_1,\\ldots,P_R)} v_P(y,{\\cal N})}. \\label{eq:thm:rcore2}\n\\end{eqnarray}\nSumming this over $r=1,\\ldots,R,$ we obtain\n\\begin{eqnarray}\n\\displaystyle 1 &=& \\sum_{i\\in {\\cal N}} z_i = \\sum_{r=1}^R~\\sum_{i\\in{\\cal N}_r}z_i \\geq \\frac{\\displaystyle \\sum_{r=1}^R \\bar{v}({\\cal N}_r)}{\\displaystyle \\min_{P\\in{\\cal P}(P_1,\\ldots,P_R)} v_P(y,{\\cal N})}.\\label{eq:thm:rcore3}\n\\end{eqnarray}\nGiven Assumption~\\ref{as:pos}(ii), $\\displaystyle \\min_{P\\in{\\cal P}(P_1,\\ldots,P_R)} v_P(a,{\\cal N})>0$. Hence, we can rewrite Inequality~\\refs{eq:thm:rcore3} as\n\\begin{eqnarray*}\n\\displaystyle \\sum_{r=1}^R \\bar{v}({\\cal N}_r) \\leq \\min_{P\\in{\\cal P}(P_1,\\ldots,P_R)} v_P(y,{\\cal N}).\n\\end{eqnarray*}\nThis leads to\n\\begin{eqnarray}\n\\displaystyle \\sum_{r=1}^R \\bar{v}({\\cal N}_r) &\\leq& \\max_{y \\geq 0}~\\min_{P\\in{\\cal P}(P_1,\\ldots,P_R)} v_P(y,{\\cal N}) \\label{eq:thm:rcore4} \\\\\n&\\equiv& v_{wc}({\\cal N}) \\nonumber\\\\\n&=& \\sum_{r=1}^R v_{wc}({\\cal N}_r) = \\sum_{r=1}^R \\bar{v}({\\cal N}_r), \\nonumber\n\\end{eqnarray}\nwhere the last equality comes from Lemma~\\ref{lem:optquant}. Thus, all the inequalities in the chain need to be tight. It implies that the ordering quantity $y$ is the worst-case optimal ordering quantity, $y=y_{wc}^*({\\cal N})$ for \\refs{eq:thm:rcore4} to be tight. We also obtain $\\displaystyle \\sum_{i \\in N_r} z_i = \\displaystyle \\frac{\\bar{v}({\\cal N}_r)}{v_{wc}({\\cal N})}$ for \\refs{eq:thm:rcore2} to be tight. In addition, for all coalition ${\\cal S}_r \\subset {\\cal N}_r$, by using the similar argument in deriving Equality~\\refs{eq:thm:rcore1}, we have\n\\begin{eqnarray*}\n\\displaystyle \\sum_{i \\in S_r} z_i \\geq v_{\\max}(y,{\\cal S}_r) = \\displaystyle \\frac{\\bar{v}(S_r)}{v_{wc}({\\cal N})},\n\\end{eqnarray*}\nwhich means $v_{wc}({\\cal N})\\cdot\\bz({\\cal N}_r)$ is a core solution of the deterministic newsvendor game $({\\cal N}_r,\\bar{v})$ for $r=1,\\ldots,R$.\n\\end{pf}\n\nTheorem~\\ref{thm:rcore2} shows that in order to check whether a particular decision $(y,\\bz)$ is a core solution of the robust newsvendor game $({\\cal N},{\\cal Y},{\\cal V}({\\cal P}(P_1,\\ldots,P_r)))$, we only need to consider $y=y^*_{wc}({\\cal N})$. The optimization problem \\refs{eq:rleastcore} for checking the existence of core solutions can be reduced to the following linear program:\n\\be\n\\label{eq:rnleastcore}\n\\ba{rl}\n\\displaystyle \\sigma(y^*_{wc}({\\cal N}))=\\min_{\\bx,\\eps} & \\eps\\\\\n\\st & \\displaystyle\\sum_{i\\in{\\cal S}}x_i\\geq v_{\\max}(y^*_{wc}({\\cal N}),{\\cal S})-\\eps,\\quad{\\cal S}\\subsetneq{\\cal N},\\\\\n& \\displaystyle\\sum_{i\\in{\\cal N}}x_i=1.\n\\ea\n\\ee\n\n\nUnlike the deterministic newsvendor games, the robust newsvendor games do not always have core solution for $N\\geq 3$. The following example shows a simple robust newsvendor game with $N=3$ whose core is empty.\n\n\\begin{example}\n\\label{ex:nocore}\nLet us consider the partition of ${\\cal N}=\\{1,2,3\\}$ with $R=2$, ${\\cal N}_1=\\{1,2\\}$ and ${\\cal N}_3=\\{3\\}$. The probability distribution $P_1$ of $(\\td{d}_1,\\td{d}_2)$ is characterized by the uniform marginal distribution of $\\td{d}_1$, $\\td{d}_1 \\sim U(0,D)$, for some $D>0$, and the relationship $\\td{d}_2 = D-\\td{d}_1$. The probability distribution $P_2$ of $\\td{d}_3$ is another uniform distribution, $\\td{d}_3 \\sim U(0,D)$. Clearly, ${\\cal Y}({\\cal N})\\neq\\emptyset$ given the fact that $d_{\\min}({\\cal N}_1)=D>0$.\n\nAccording Theorem \\ref{thm:rcore2}, if the robust newsvendor game $({\\cal N},{\\cal Y},{\\cal V}({\\cal P}(P_1,P_2)))$ has a core solution $(y,\\bz)$, then $y=y^*_{wc}({\\cal N})$. Given $P_1$ and $P_2$, we are able to compute and bound some values of $v_{\\max}(y,{\\cal S})$ where ${\\cal S}\\subset{\\cal N}$, as follows:\n$$\nv_{\\max}(y,\\{1,2\\})=\\frac{2p}{3p-c}\\leq v_{\\max}(y,\\{2,3\\})=v_{\\max}(y,\\{1,3\\}).\n$$\nFor clarity of the exposition, we leave the detailed computation of these values in the Appendix. Now, since $(y,\\bz)$ is a core solution, $\\displaystyle\\sum_{i\\in{\\cal S}}z_i\\geq v_{\\max}(y,{\\cal S})$ for ${\\cal S}\\subset{\\cal N}$ and $z_1+z_2+z_3=1$. Thus\n\\begin{eqnarray*}\n2 &=& (z_1+z_2)+(z_2+z_3)+(z_3+z_1)\\\\\n&\\geq& v_{max}(y,\\{1,2\\}) + v_{max}(y,\\{2,3\\})+v_{max}(y,\\{1,3\\})\\\\\n&\\geq& 6p\/(3p-c) > 2,\n\\end{eqnarray*}\nwhich is a contradiction or this robust newsvendor game does not have a core solution.\n\\end{example}\n\nWe now focus on how to solve Problem \\refs{eq:rnleastcore} to check the existence of core solutions of robust newsvendor games. If the core is empty, i.e., $\\sigma(y^*_{wc}({\\cal N}))>0$, \\emph{least core solutions} for the robust newsvendor game can be found by solving the general problem \\refs{eq:rleastcore}, which will be discussed in the next section.\n\\subsection{Core and Least Core Computation}\\label{robust_core_computation}\n\\begin{comment}\nIn order to solve the problem \\refs{eq:rnleastcore}, we need to be able to compute $v_{\\max}(y^*_{wc}({\\cal N}),{\\cal S})$ for each coalition $\\cal S\\subsetneq{\\cal N}$. If ${\\cal S}\\subseteq {\\cal N}_r$ for some $r$, we have shown that $\\displaystyle v_{\\max}(y^*_{wc}({\\cal N}),{\\cal S})=\\frac{\\bar{v}(S)}{v_{wc}({\\cal N})}$.\n\\end{comment}\nBoth Problems \\refs{eq:rleastcore} and \\refs{eq:rnleastcore} involve the function $v_{\\max}$. We first show how to compute $v_{\\max}(y,{\\cal S})$ for an arbitrary ordering quantity $y\\in{\\cal Y}({\\cal N})$ and an arbitrary ${\\cal S}\\subsetneq{\\cal N}$ with ${\\cal Y}({\\cal S})=\\mathbb{R}_+$. (Note that if ${\\cal S}\\subseteq {\\cal N}_r$ for some $r$, $r=1,\\ldots,R$, we then have ${\\cal Y}({\\cal S})=\\{y^*({\\cal S})\\}$ and $v_{\\max}(y,{\\cal S})$ can be computed by simply solving a single optimization problem, $\\displaystyle\\min_{P\\in{\\cal P}(P_1,\\ldots,P_R)}v_P(y,{\\cal N})$.) We have:\n\\be\n\\label{eq:gvmax}\nv_{\\max}(y,{\\cal S})=\\max_{\\gamma \\in \\R^+}~ \\max_{P\\in{{\\cal P}(P_1,\\ldots,P_R)}} \\frac{(p-c)\\gamma-p\\,\\mathbb{E}_P\\left[\\left(\\gamma-\\td{d}({\\cal S})\\right)^+\\right]}\n{(p-c)y-p\\,\\mathbb{E}_P\\left[\\left(y-\\td{d}({\\cal N})\\right)^+\\right]}.\n\\ee\nFor newsvendor games, it is reasonable to assume that demands follow discrete non-negative distributions, which could be constructed from historical sales or market analysis. More specifically, let each distribution $P_r$ of $\\td{\\mb{d}}_r$ be represented as a discrete non-negative distribution with $K_r$ values, $\\mb{d}^k_r$ of probability $p^k_r$, for $k = 1,\\ldots,K_r$, $r=1,\\ldots,R$. Thus, each probability distribution $P$ in $\\mathcal{P}(P_1,\\ldots,P_R)$ is a discrete distribution with a support of $\\displaystyle K =\\prod_{r=1}^RK_r$ values $\\mb{d}_k$ and each has an unknown probability of $q_k$, $k = 1,\\ldots,K$. For $P$ to be consistent with $P_1,\\ldots,P_R$, the following constraints on $\\mb{q}$ must hold:\n\\be\n\\label{eq:qdef}\n\\left\\{\\displaystyle \\mb{q} \\geq 0,~~\\sum_{k=1}^Kq_k=1,~~\\sum_{k=1}^K\\mathbb{I}\\{\\mb{d}_{k,r}=\\mb{d}_r^l\\}q_k = p_r^l,~~ r=1,\\ldots,R,\\,l=1,\\ldots,K_r\\right\\}.\n\\ee\nGiven the one-to-one mapping between $P$ and $\\mb{q}$, we abuse the notations and write both $v_P(y,{\\cal S})$ and $v_{\\vect{q}}(y,{\\cal S})$ interchangeably when the context is clear for $\\mb{q}$ to be the corresponding representation of $P$.\n\nProblem \\refs{eq:gvmax} can be reformulated as\n\\begin{equation}\n\\label{eq:vmax}\n\\begin{array}{rl}\nv_{\\max}(y,{\\cal S})=\\displaystyle \\max_{\\gamma, \\vect{q}} & \\displaystyle\\frac{(p-c)\\gamma - p\\,\\sum_{k=1}^K\\left[\\left(\\gamma-d_k({\\cal S})\\right)^+\\right]q_k}{(p-c)y - p\\sum_{k=1}^K\\left[\\left(y-d_k({\\cal N})\\right)^+\\right]q_k}\\\\\n\\st & \\displaystyle\\sum_{k=1}^K\\mathbb{I}\\{\\mb{d}_{k,r}=\\mb{d}_r^l\\}q_k = p_r^l,\\quad\\forall\\,r=1,\\ldots,R,\\,l=1,\\ldots,K_r,\\\\\n& \\displaystyle\\sum_{k=1}^Kq_k=1,\\\\\n& \\gamma,\\mb{q}\\geq 0.\n\\end{array}\n\\end{equation}\nFor each fixed $\\gamma$, we can apply the standard method for transforming a linear fractional optimization problem into a linear program (see, for example, Cambini et al. \\cite{cambini05}). To this end, let us introduce new decision variables\n$$\\displaystyle \\theta = \\frac{1}{(p-c)y - p\\sum_{k=1}^K\\left[\\left(y-d_k({\\cal N})\\right)^+\\right]q_k},$$\n and\n$$\\displaystyle \\psi_k =  \\frac{q_k}{(p-c)y - p\\sum_{k=1}^K\\left[\\left(y-d_k({\\cal N})\\right)^+\\right]q_k},\\quad k=1,\\ldots,K.$$\nUnder Assumption~\\ref{as:pos}(ii), we have $\\theta > 0$. The objective function then becomes $$\\displaystyle (p-c)\\gamma\\cdot \\theta - p\\,\\sum_{k=1}^K\\left[\\left(\\gamma-d_k({\\cal S})\\right)^+\\right]\\psi_k,$$\nand Problem~\\refs{eq:vmax} can be reformulated as\n\n\\begin{equation}\n\\label{eq:mpriLP}\n\\begin{array}{rl}\nv_{\\max}(y,{\\cal S})=\\displaystyle \\max_{\\gamma, \\theta, \\vect{\\psi}} & \\displaystyle (p-c)\\gamma\\cdot \\theta - p\\,\\sum_{k=1}^K\\left[\\left(\\gamma-d_k({\\cal S})\\right)^+\\right]\\psi_k\\\\\n\\st & \\displaystyle\\sum_{k=1}^K\\mathbb{I}\\{\\mb{d}_{k,r}=\\mb{d}_r^l\\}\\psi_k - p_r^l\\cdot\\theta = 0,\\quad\\forall\\,r=1,\\ldots,R,\\,l=1,\\ldots,K_r,\\\\\n& \\displaystyle\\sum_{k=1}^K \\psi_k-\\theta = 0,\\\\\n& \\displaystyle (p-c)y\\cdot \\theta - p\\,\\sum_{k=1}^K\\left[\\left(y-d_k({\\cal N})\\right)^+\\right] \\psi_k = 1,\\\\\n& \\gamma,\\theta, \\mb{\\psi} \\geq 0,\n\\end{array}\n\\end{equation}\nwhere the first two constraints in \\refs{eq:mpriLP} are derived directly from the first two constraints in \\refs{eq:vmax} by multiplying both sides of those with $\\theta$. The third constraint in \\refs{eq:mpriLP} is derived by the definitions of $\\theta$ and $\\mb{\\psi}$. Finally, we have replaced the constraint $\\theta > 0$ by $\\theta \\geq 0$ without loss of generality since $\\theta = 0$ is not a feasible solution (otherwise, $\\mb{\\psi}$ must be equal to zero from the second constraint and that violates the third constraint.)\n\nProblem \\refs{eq:mpriLP} is a bilinear optimization problem, which is generally not easy to solve. We show, however, in the following proposition that one of the distinct values of $\\{d_1({\\cal S}),\\ldots,d_K({\\cal S})\\}$, which are known, is an optimal value of $\\gamma$. Given a fixed $\\gamma$, the resulting bilinear optimization problem is reduced to a linear program. It implies that we can solve Problem \\refs{eq:mpriLP} by solving at most $K$ linear programs with $\\gamma$ set to each and every distinct value of $\\{d_1({\\cal S}),\\ldots,d_K({\\cal S})\\}$.\n\n\\begin{proposition}\n\\label{prop:linearequiv}\nThere exists an optimal solution $(\\gamma^*,\\theta^*,\\mb{\\psi}^*)$ of Problem \\refs{eq:mpriLP} such that $$\\gamma^*\\in\\{d_1({\\cal S}),\\ldots,d_K({\\cal S})\\}.$$\n\\end{proposition}\n\n\\begin{pf}\nGiven an arbitrary value of $\\gamma$, Problem \\refs{eq:mpriLP} is reduced to a linear program for $\\theta$ and $\\mb{\\psi}$ over a fixed feasible set $\\cal F$ defined by the set of constraints in \\refs{eq:mpriLP}, i.e., $\\gamma$ only affects the objective function. Under Assumption~\\ref{as:pos}(ii), $\\theta$ and $\\mb{\\psi}$ are non-negative and bounded, which means $\\cal F$ is bounded and Problem \\refs{eq:mpriLP} can be written as follows:\n$$\nv_{\\max}(y,{\\cal S}) = \\max_{\\gamma\\geq 0}\\left(\\max_{s=1,\\ldots,S}\\left\\{(p-c)\\gamma\\cdot \\theta^s - p\\,\\sum_{k=1}^K\\left[\\left(\\gamma-d_k({\\cal S})\\right)^+\\right]\\psi_k^s\\right\\}\\right),\n$$\nwhere $\\left\\{(\\theta^s,\\mb{\\psi}^s)\\right\\}_{s=1,\\ldots,S}$ is the set of extreme points of $\\cal F$. Equivalently, we have:\n$$\nv_{\\max}(y,{\\cal S}) = \\max_{s=1,\\ldots,S}\\left\\{\\max_{\\gamma\\geq 0}\\left((p-c)\\gamma\\cdot \\theta^s - p\\,\\sum_{k=1}^K\\left[\\left(\\gamma-d_k({\\cal S})\\right)^+\\right]\\psi_k^s\\right)\\right\\}.\n$$\n\nFor an arbitrary solution $(\\theta^s,\\mb{\\psi}^s)$, $s=1,\\ldots,S$, it is easy to show that function $\\displaystyle f(\\gamma;\\theta^s,\\mb{\\psi}^s)=(p-c)\\gamma\\cdot \\theta^s - p\\,\\sum_{k=1}^K\\left[\\left(\\gamma-d_k({\\cal S})\\right)^+\\right]\\psi_k^s$ is a \\emph{concave} piece-wise linear function with intersection points as distinct values of the set $\\{d_1({\\cal S}),\\ldots,d_K({\\cal S})\\}$. Since $\\theta^s>0$ as shown previously, $f(\\cdot;\\theta^s,\\mb{\\psi}^s)$ tends to $-\\infty$ when $\\gamma$ tends to $+\\infty$. It means there is at least an optimal solution for the problem $\\displaystyle\\max_{\\gamma\\geq 0}f(\\gamma;\\theta^s,\\mb{\\psi}^s)$ which belongs to the set $\\{d_1({\\cal S}),\\ldots,d_K({\\cal S})\\}$. Thus we have:\n$$\nv_{\\max}(y,{\\cal S}) = \\max_{s=1,\\ldots,S}\\left\\{\\max_{l=1,\\ldots,K}\\left\\{(p-c)d_l({\\cal S})\\cdot \\theta^s - p\\,\\sum_{k=1}^K\\left[\\left(d_l({\\cal S})-d_k({\\cal S})\\right)^+\\right]\\psi_k^s\\right\\}\\right\\},\n$$\nwhich shows that there exists an optimal solution $(\\gamma^*,\\theta^*,\\mb{\\psi}^*)$ of Problem \\refs{eq:mpriLP} such that\n $$\\gamma^*\\in\\{d_1({\\cal S}),\\ldots,d_K({\\cal S})\\}.$$\n\n\n\\end{pf}\n\nProposition \\ref{prop:linearequiv} shows us how to compute $v_{\\max}(y,{\\cal S})$ by solving at most $K$ linear programs. We can use this approach to compute $v_{\\max}(y^*_{wc}({\\cal N}),{\\cal S})$ as inputs of the linear program \\refs{eq:rnleastcore}, which is then solved to check the existence of core solutions of the robust newsvendor game $({\\cal N},{\\cal Y},{\\cal V}({\\cal P}(P_1,\\ldots,P_r)))$. If the core is empty, we need to consider the general problem \\refs{eq:rleastcore} applying to the robust newsvendor game, whose optimal solutions can be considered as its least core solutions. Let us consider the following problem, which is similar to \\refs{eq:rnleastcore}, for an arbitrary $y \\in {\\cal Y}({\\cal N})$:\n\\be\n\\label{eq:ly}\n\\ba{rl}\n\\displaystyle \\sigma(y)=\\min_{\\bx,\\eps} & \\eps\\\\\n\\st & \\displaystyle\\sum_{i\\in{\\cal S}}x_i\\geq v_{\\max}(y,{\\cal S})-\\eps,\\quad{\\cal S}\\subsetneq{\\cal N},\\\\\n& \\displaystyle\\sum_{i\\in{\\cal N}}x_i=1.\n\\ea\n\\ee\n\\begin{comment}\n\\begin{equation}\n\\label{eq:ly}\n\\begin{array}{rl}\n\\displaystyle \\Upsilon(y) = \\min_{\\mb{x},\\epsilon} & \\epsilon\\\\\ns.t. & \\mb{x}({\\cal S}) + \\epsilon \\geq v_{max}(y,{\\cal S}), \\forall {\\cal S} \\subsetneq {\\cal N},\\\\\n& \\mb{e}^T \\mb{x} = 1.\n\\end{array}\n\\end{equation}\n\\end{comment}\nClearly, $\\displaystyle s({\\cal N},{\\cal Y},{\\cal V}({\\cal P}(P_1,\\ldots,P_r)))=\\min_{y \\in {\\cal Y}({\\cal N})} \\sigma(y)$, which is a reformulation of the least core problem \\refs{eq:rleastcore}.  We will show that $\\sigma(y)$ is a convex function in the following proposition.\n\\begin{comment}\nWe can utilize the convexity property of $\\Upsilon(y)$ in the numerical computation of a $y^*$ with the smallest least core value. Specifically, we can search for the (one dimensional) $y$ that is a local optimal solution and conclude that this is also the global optimal solution.\n\\end{comment}\n\\begin{proposition}\n\\label{prop:ly_vmax_properties}\nThe following statements hold:\n\\begin{itemize}\n\\item[(a)] For each coalition ${\\cal S}\\subsetneq {\\cal N}$, $\\displaystyle v_{max}(y,{\\cal S})$ is a convex function of $y$ on ${\\cal Y}({\\cal N})$.\n\\item[(b)] $\\sigma(y)$ is a convex function of $y$ on ${\\cal Y}({\\cal N})$.\n\\end{itemize}\n\\end{proposition}\n\nIn order to prove the proposition, we need the following lemma.\n\n\\begin{lemma}\n\\label{lemma:vmax_properties}\nThe inverse function $\\displaystyle \\frac{1}{v_P(y,{\\cal N})}$ is a convex function of $y$ on ${\\cal Y}({\\cal N})$ for all $P\\in{\\cal P}(P_1,\\ldots,P_R)$.\n\\end{lemma}\n\n\\begin{pf}\nLet $\\mb{q}$ be the corresponding probability vector of a joint distribution $P\\in {\\cal P}(P_1,\\ldots,P_R)$. We have:\n$$\nv_P(y,{\\cal N})\\equiv v_{\\vect{q}}(y,{\\cal N})=(p-c)y - p\\sum_{k=1}^K\\left[\\left(y-d_k({\\cal N})\\right)^+\\right]q_k,\n$$\nwhich is a piecewise linear concave function of $y$ with at most $(K+1)$ linear pieces positioning within the intervals induced by the sorted sequence of $\\{d_1({\\cal N}),d_2({\\cal N})\\ldots d_K({\\cal N})\\}$. We can therefore rewrite $\\displaystyle v_P(y,{\\cal N})=\\min_{k=1,\\ldots,K+1}\\{a_ky+b_k\\}$ where $a_k,b_k$ are appropriate linear coefficients that can be derived from $p$, $c$, $\\mb{d}({\\cal N})$ and $\\mb{q}$. Since $v_P(y,{\\cal N})>0$ for all $y\\in{\\cal Y}({\\cal N})$, we have: $a_ky+b_k>0$ for all $k=1,\\ldots,K+1$, and $y\\in{\\cal Y}({\\cal N})$. We then have:\n$$\n\\frac{1}{v_P(y,{\\cal N})}=\\max_{k=1,\\ldots,K+1}\\frac{1}{a_ky+b_k},\n$$\nwhich is the maximum of convex inverse linear functions on ${\\cal Y}({\\cal N})$ and hence is also a convex function in ${\\cal Y}({\\cal N})$.\n\\begin{comment}\nLet $d^{(1)}({\\cal N})\\leq d^{(2)}({\\cal N})\\leq \\ldots \\leq d^{(K)}({\\cal N})$ be the corresponding sorted sequence of $\\{d_1({\\cal N}),d_2({\\cal N})\\ldots d_K({\\cal N})\\}$. We also denote $q^{(j)}$ as the corresponding probability of the realization of the joint demand at $d^{(j)}({\\cal N})$ for each $j = 1,\\ldots,K$.\n\nLet us denote $a_k = (\\rho-c - \\rho \\sum_{j=1}^k q^{(j)})$ and $b_k = \\rho \\sum_{j=1}^k q^{(j)} d^{(k)}({\\cal N})$ for $k= 0,\\ldots,K+1$. We have\n$$\n\\displaystyle v_P(y,{\\cal N}) =\n\\begin{cases}\n\\displaystyle a_0 y + b_0, &\\mbox{if } y \\leq d^{(1)}({\\cal N}) \\\\\n\\displaystyle a_k y+b_k, &\\mbox{if } d^{(k)}({\\cal N}) \\leq y \\leq d^{(k+1)}({\\cal N})\\\\\n\\displaystyle a_K y+b_K, &\\mbox{if } y \\geq d^{(K)}({\\cal N}).\n\\end{cases}\n$$\nLet us define $\\displaystyle g_k(y) = \\frac{v_P(\\gamma,{\\cal S})}{a_k y+b_k},~k=0,\\ldots,K+1,$ which are inverse linear functions and are convex. Then $\\displaystyle \\frac{v_P(\\gamma,{\\cal S})}{v_P(y,{\\cal N})}$ composes of these $K+1$ convex pieces. In addition, by the definition of $(a_k,b_k),~k=0,\\ldots,K$, we can show that\n$$ \\displaystyle v_P(y,{\\cal N}) = \\max_{j=0,\\ldots,K} g_j(y),$$\nwhich is the maximum of convex functions and hence is also a convex function.\n\\end{comment}\n\\end{pf}\n\nWe are now ready to prove the Proposition~\\ref{prop:ly_vmax_properties}.\n\n\\begin{pf}\n(a) If ${\\cal S}\\subseteq{\\cal N}_r$ for some $r$, $r=1,\\ldots,R$, we have: ${\\cal Y}({\\cal S})=\\{y^*({\\cal S})\\}$ and\n$$\n\\ba{rl}\nv_{\\max}(y,{\\cal S})&=\\displaystyle\\max_{P\\in{{\\cal P}(P_1,\\ldots,P_R)}} \\frac{\\bar{v}({\\cal S})}\n{(p-c)y-p\\,\\mathbb{E}_P\\left[\\left(y-\\td{d}({\\cal N})\\right)^+\\right]}\\\\\n& =\\displaystyle\\frac{\\bar{v}({\\cal S})}{\\min_{\\vect{q}\\in{\\cal Q}}\\left\\{(p-c)y - p\\sum_{k=1}^K\\left[\\left(y-d_k({\\cal N})\\right)^+\\right]q_k\\right\\}}\\equiv \\displaystyle\\frac{\\bar{v}({\\cal S})}{\\min_{\\vect{q}\\in{\\cal Q}} v_{\\vect{q}}(y,{\\cal N})},\n\\ea\n$$\nwhere ${\\cal Q}={\\cal Q}(P_1,\\ldots,P_R)$ is the feasible set of the probability vector $\\mb{q}$ as described in \\refs{eq:qdef}. The second equality follows from the fact that $\\bar{v}({\\cal S})\\geq 0$ and $v_P(y,{\\cal N}) > 0$. Since ${\\cal Q}$ is a bounded polytope; there exists an optimal solution $\\mb{q}^*\\in{\\cal Q}^*$, where ${\\cal Q}^*$ is the set of extreme points of ${\\cal Q}$. Thus we have:\n$$\nv_{\\max}(y,{\\cal S})=\\max_{\\vect{q}\\in{\\cal Q}^*}\\frac{\\bar{v}({\\cal S})}{v_{\\vect{q}}(y,{\\cal N})}.\n$$\nSince $\\bar{v}({\\cal S})\\geq 0$ and since $\\displaystyle \\frac{1}{v_{\\vect{q}}(y,{\\cal N})}$ is convex for each $\\mb{q}$ according to Lemma \\ref{lemma:vmax_properties}, we have: $v_{\\max}(y,{\\cal S})$ is convex.\n\nNow consider an arbitrary ${\\cal S}\\subsetneq{\\cal N}$ with ${\\cal Y}({\\cal S})=\\mathbb{R}_+$. We have,\n$$\nv_{\\max}(y,{\\cal S})=\\max_{P\\in{{\\cal P}(P_1,\\ldots,P_R)}} \\frac{\\displaystyle\\max_{\\gamma \\in {\\cal Y}({\\cal S})}v_P(\\gamma,{\\cal S})}\n{v_P(y,{\\cal N})}.\n$$\nWe have: $v_P(y,{\\cal N})>0$ for all $P\\in{\\cal P}(P_1,\\ldots,P_R)$ and $y\\in{\\cal Y}({\\cal N})$. In addition, ${\\cal Y}({\\cal S})=\\mathbb{R}_+$, thus there always exists $\\gamma\\in{\\cal Y}({\\cal S})$ small enough such that $v_P(\\gamma,{\\cal S})\\geq 0$ for any $P$. We then have: $v_{\\max}(y,{\\cal S})\\geq 0$ for all $y\\in{\\cal Y}({\\cal N})$. We can rewrite the formulation of $v_{\\max}(y,{\\cal S})$ as follows:\n$$\nv_{\\max}(y,{\\cal S})=\\max_{\\gamma \\in \\R^+}~ \\max_{\\vect{q}\\in{{\\cal Q}}} \\frac{(p-c)\\gamma - p\\sum_{k=1}^K\\left[\\left(\\gamma-d_k({\\cal S})\\right)^+\\right]q_k}\n{(p-c)y - p\\sum_{k=1}^K\\left[\\left(y-d_k({\\cal N})\\right)^+\\right]q_k}.\n$$\n\nProposition~\\ref{prop:linearequiv} shows that we can restrict the domain of $\\gamma$ to the discrete set $\\{d_1({\\cal S}),\\ldots,d_K({\\cal S})\\}$, that is,\n$$\nv_{\\max}(y,{\\cal S})=\\max_{\\gamma \\in \\{d_1({\\cal S}),\\ldots,d_K({\\cal S})\\}}~ \\max_{\\vect{q}\\in{{\\cal Q}}} \\frac{(p-c)\\gamma - p\\sum_{k=1}^K\\left[\\left(\\gamma-d_k({\\cal S})\\right)^+\\right]q_k}\n{(p-c)y - p\\sum_{k=1}^K\\left[\\left(y-d_k({\\cal N})\\right)^+\\right]q_k}.\n$$\n\\begin{comment}\nThus, Problem~\\refs{eq:mpriLP} can be reformulated as\n\n\\begin{equation}\n\\label{eq:vmax2}\n\\begin{array}{rl}\nv_{\\max}(y,{\\cal S})=\\displaystyle \\max_{\\gamma \\in \\{d_1({\\cal S}),\\ldots,d_K({\\cal S})\\}} ~~ \\max_{\\vect{q} \\in Q} & h(y,{\\cal S},\\gamma,\\mb{q}),\n\\end{array}\n\\end{equation}\nwhere $\\displaystyle h(y,{\\cal S},\\gamma,\\mb{q}) = \\displaystyle\\frac{(p-c)\\gamma - p\\,\\sum_{k=1}^K\\left[\\left(\\gamma-d_k({\\cal S})\\right)^+\\right]q_k}{(p-c)y - p\\sum_{k=1}^K\\left[\\left(y-d_k({\\cal N})\\right)^+\\right]q_k}$.\n\\end{comment}\nFor a fixed $\\gamma$, the inner problem is a linear fractional optimization problem over the bounded polyhedron $\\cal Q$ and hence there exists an optimal solution $\\mb{q}^*$ given that $v_P(y,{\\cal N})>0$ for all $y\\in{\\cal Y}({\\cal N})$ and $P\\in{\\cal P}(P_1,\\ldots,P_R)$. Let us consider the level sets ${\\cal L}_{\\alpha}$ of the linear fractional objective function, which are hyperplanes. For the optimal objective value $\\alpha^*$, we have: ${\\cal L}_{\\alpha^*}\\cap{\\cal Q}\\neq\\emptyset$. We claim that ${\\cal L}_{\\alpha^*}\\cap{\\cal Q}^*\\neq\\emptyset$, where ${\\cal Q}^*$ is the set of extreme points of $\\cal Q$. Since $\\alpha^*$ is the optimal objective value, $\\cal Q$ belongs to a half-space defined by ${\\cal L}_{\\alpha^*}$. Suppose, on contradiction, that ${\\cal L}_{\\alpha^*}\\cap{\\cal Q}^*=\\emptyset$. Due to convexity and since the entire ${\\cal Q}^*$ belongs to the same half-space defined by ${\\cal L}_{\\alpha^*}$, we have: ${\\cal L}_{\\alpha^*}\\cap{\\cal Q}=\\emptyset$ (contradiction).\n\nWith ${\\cal L}_{\\alpha^*}\\cap{\\cal Q}^*\\neq\\emptyset$, we can now compute $v_{\\max}(y,{\\cal S})$ as follows:\n$$\nv_{\\max}(y,{\\cal S})=\\max_{\\gamma \\in \\{d_1({\\cal S}),\\ldots,d_K({\\cal S})\\}}~ \\max_{\\vect{q}\\in{{\\cal Q}^*}} \\frac{(p-c)\\gamma - p\\sum_{k=1}^K\\left[\\left(\\gamma-d_k({\\cal S})\\right)^+\\right]q_k}\n{(p-c)y - p\\sum_{k=1}^K\\left[\\left(y-d_k({\\cal N})\\right)^+\\right]q_k}.\n$$\nSince $v_{\\max}(y,{\\cal S})\\geq 0$, we can focus on the set ${\\cal H} \\in \\{d_1({\\cal S}),\\ldots,d_K({\\cal S})\\} \\times {\\cal Q}^*$ of $(\\gamma,\\mb{q})$ such that $(p-c)\\gamma - p\\sum_{k=1}^K\\left[\\left(\\gamma-d_k({\\cal S})\\right)^+\\right]q_k\\geq 0$ when computing $v_{\\max}(y,{\\cal S})$, that is,\n$$\nv_{\\max}(y,{\\cal S})=\\max_{(\\gamma,\\vect{q}) \\in {\\cal H}} \\frac{(p-c)\\gamma - p\\sum_{k=1}^K\\left[\\left(\\gamma-d_k({\\cal S})\\right)^+\\right]q_k}\n{(p-c)y - p\\sum_{k=1}^K\\left[\\left(y-d_k({\\cal N})\\right)^+\\right]q_k}.\n$$\nApplying Lemma \\refs{lemma:vmax_properties}, clearly, $v_{\\max}(y,{\\cal S})$ is the maximum of convex functions, which means it is also a convex function.\n\\begin{comment}\nLet $\\mb{q}^{(1)},\\ldots,\\mb{q}^{(M)}$ be the corresponding extreme points. Since level sets of the linear fractional objective function are hyperplanes, the optimal level set that passes through $\\mb{q}^* \\in Q$ must also pass through at least an extreme point of $Q$. This is because otherwise all these extreme points must belongs to the same half-plane produced by the optimal level set hyperplane, but not on that hyperplane, which means the entire polyhedron $Q$ does not intersect with the hyperplane and this contradicts with $\\mb{q}^* \\in Q$.\n\nThus, the optimality of Problem~\\refs{eq:vmax2} can be attained at one of the extreme points $\\mb{q}^{(1)},\\ldots,\\mb{q}^{(M)}$.  Problem~\\refs{eq:vmax2} can then be reformulated as\n\\begin{equation}\n\\label{eq:vmax3}\n\\begin{array}{rl}\nv_{\\max}(y,{\\cal S})=\\displaystyle \\max_{\\gamma \\in \\{d_1({\\cal S}),\\ldots,d_K({\\cal S})\\}} ~~ \\max_{\\vect{q} \\in \\{\\mb{q}^{(1)},\\ldots,\\mb{q}^{(M)}\\}} & h(y,{\\cal S},\\gamma,\\mb{q}),\n\\end{array}\n\\end{equation}\nwhich is the maximum of a number of functions $h(y,{\\cal S},d_l({\\cal S}),\\mb{q}^{(m)}),~l=1,\\ldots,K,~m=1,\\ldots,M$. Since each of these function is convex over $y$ by Lemma~\\ref{lemma:vmax_properties}, we have $v_{\\max}(y,{\\cal S})$ is also a convex function of $y$.\n\\end{comment}\n\n(b) To prove the convexity of $\\sigma(y)$, we show that, for any $\\{y_1, y_2, y_3\\} \\in {\\cal Y}({\\cal N})$ such that there exists $\\alpha \\in [0,1]$ with $y_2 = \\alpha y_1+ (1-\\alpha) y_3$, then $\\sigma(y_2) \\leq \\alpha \\sigma(y_1) + (1-\\alpha) \\sigma(y_3)$.\n\nLet $(\\mb{x}_1,\\epsilon_1)$ and $(\\mb{x}_3,\\epsilon_3)$ be the optimal solutions of ~\\refs{eq:ly} when $y = y_1$ and $ y=y_3$, respectively. Let us define $(\\mb{x}_2,\\epsilon_2) = \\alpha (\\mb{x}_1,\\epsilon_1)+(1-\\alpha)(\\mb{x}_3,\\epsilon_3)$. It is easy to verify that $\\mb{e}^T \\mb{x}_2 = 1$. In addition, for all $ {\\cal S} \\subsetneq {\\cal N}$, we have\n\\begin{eqnarray}\n\\mb{x}_2({\\cal S}) + \\epsilon_2 &=& \\alpha (\\mb{x}_1({\\cal S})+\\epsilon_1)+(1-\\alpha)(\\mb{x}_3({\\cal S})+\\epsilon_3) \\label{eq:ly1}\\\\\n &\\geq& \\alpha v_{max}(y_1,{\\cal S})+(1-\\alpha) v_{max}(y_3,{\\cal S}) \\label{eq:ly2}\\\\\n &\\geq& v_{max}( \\alpha y_1+ (1-\\alpha) y_3,{\\cal S}) \\label{eq:ly3}\\\\\n &=& v_{max}( y_2,{\\cal S}) \\label{eq:ly4},\n\\end{eqnarray}\nwhere \\refs{eq:ly1} comes directly from the construction of $(\\mb{x}_2,\\epsilon_2)$; \\refs{eq:ly2} comes from the feasibility of $(\\mb{x}_1,\\epsilon_1)$ and $(\\mb{x}_3,\\epsilon_3)$; \\refs{eq:ly3} comes from the convexity of $v_{max}(y,{\\cal S})$ as shown in part (a). Finally, \\refs{eq:ly4} comes directly from the definition of $y_2$.\n\nThis shows that $(\\mb{x}_2,\\epsilon_2)$ is a feasible solution of ~\\refs{eq:ly} when $y = y_2$. Therefore,\n$$\\sigma(y_2) \\leq \\epsilon_2= \\alpha \\epsilon_1 + (1-\\alpha) \\epsilon_3= \\alpha \\sigma(y_1) + (1-\\alpha) \\sigma(y_3),$$\ni.e., $\\sigma(y)$ is a convex function.\n\\end{pf}\n\n\n\n\nProposition \\ref{prop:ly_vmax_properties} shows that the least core problem \\refs{eq:rleastcore} for our robust newsvendor game is a convex optimization problem in terms of $y$ and we could apply simple one-dimensional search algorithms to find the optimal solution. The next section provides some numerical results on the properties and computation of the core (and least core) solutions of robust newsvendor games.\n\\subsection{Numerical Results}\nWe consider the following experimental setting. We are given a set of retailers ${\\cal N}$ and a partition ${\\cal N}_1,\\ldots,{\\cal N}_R$. In addition, for each $r= 1,\\ldots, R$, we are given the discrete historical joint demand distribution $P_r$ for the subset of retailers ${\\cal N}_r$ but not the joint demand distribution of all retailers. Discussions in Section~\\ref{robust_core_computation} allow us to compute a robust core (or least core) solution by solving \\refs{eq:rnleastcore} (and \\refs{eq:rleastcore} if necessary) under the framework of robust newsvendor games, which consists of the allocation scheme $\\bz_{rob}$ and an order quantity $y_{rob}$ for the grand coalition.\n\nIn order to evaluate the performance of the robust solution $(y_{rob},\\bz_{rob})$, we are going to compare it with the solution derived from the deterministic newsvendor game under the assumption that all multivariate marginal distributions $P_r$, $r=1,\\ldots,R$, are independent of each other, that is, $\\displaystyle \\mathbb{P}(\\tilde{\\mb{d}} = (\\mb{d}_1^{l_1},\\ldots,\\mb{d}_r^{l_R})) = \\prod_{r=1}^R \\mathbb{P}(\\tilde{\\mb{d}}_r = \\mb{d}_r^{l_r})=\\prod_{r=1}^R p_r^{l_r}$. This could be considered as a common assumption on the joint distribution given its marginal distributions. Clearly, the resulting joint distribution $P_I$ belongs to $\\mathcal{P}(P_1,\\ldots,P_R)$. Given this distribution $P_I$, we can compute the allocation scheme $\\bz_{det}=\\bx\/\\bar{v}({\\cal N})$, where $\\bx$ is a core solution of the deterministic newsvendor game with respect to $P_I$. In addition, the optimal order quantity $y_{det}$ for the grand coalition in this deterministic newsvendor game is used to form the solution $(y_{det},\\bz_{det})$, which will be compared with the robust solution $(y_{rob},\\bz_{rob})$.\n\nWe shall compare the performance of these two solutions with respect to joint distributions which belong to $\\mathcal{P}(P_1,\\ldots,P_R)$. Given a distribution $P\\in \\mathcal{P}(P_1,\\ldots,P_R)$, we compute the maximum normalized dissatisfaction or worst normalized excess value for each solution. For $(y_{rob},\\bz_{rob})$, the excess value is computed as\n\\be\n\\label{eq:excess}\n\\eps_{rob}^P=\\max_{{\\cal S}\\subsetneq{\\cal N}}\\left\\{\\left(\\frac{\\displaystyle\\max_{\\gamma\\in{\\cal Y}({\\cal S})}v_P(\\gamma,{\\cal S})}{v_P(y_{rob},{\\cal N})}-z_{rob}({\\cal S})\\right)^+\\right\\}.\n\\ee\nThe excess value $\\eps_{det}^P$ can be defined in the same fashion for $(y_{det},\\bz_{det})$. We follow the stress test approach proposed by Dupa\\v cov\\' a \\cite{dupacova06} with the contaminated distributions $P_{\\lambda} = \\lambda P_I + (1-\\lambda)P_{ext}$ for $\\lambda\\in[0,1]$, where $P_{ext}$ are extremal distributions, i.e., those distributions which are likely to be the ones with which $v_{\\max}(y,{\\cal S})$ are computed. Similar approach has been discussed in Bertsimas et al. \\cite{bertsimas10} to test the quality of some stochastic optimization solutions.\n\\begin{comment}\nThe joint demand distributions can be represented by a probability vector $\\mb{q}$ which satisfies the set of linear constraints included in Model~\\ref{eq:vmax}. We denote the corresponding polytop as $Q$. Here, we notice that the probability vector $\\mb{q}_I$ that corresponds to $P_I$ lies in the interior of $Q$. Obviously if we choose $\\mb{q}=\\mb{q}_I$, then we would expect the deterministic strategy to work well as it has used the truth joint distribution. To provide the overall assessment of ROBUST and INDEPT, we will randomize $q$ within $Q$. For each random cost vector $\\mb{c}$ of the same size with $\\mb{q}$, if we solve the problem $\\{ \\max \\mb{c}^T \\mb{q} ~:~ \\mb{q} \\in Q\\}$ using a simplex method, we would obtain an extreme point $\\mb{q}_e \\in Q$. We also vary $\\alpha \\in [0,1]$ to produce a new probability vector $\\mb{q} = \\alpha \\mb{q}_e + (1-\\alpha)\\mb{q}_I$. Here, $\\alpha = 0$ means the chosen distribution is $P_I$ while $\\alpha = 1$ means the joint distribution is an extreme point of $Q$. By varying $\\alpha$ and by randomizing $\\mb{c}$, we can compare the performance between ROBUST and INDEPT as the joint distribution changes.\n\\end{comment}\n\nWe now consider a numerical example with $n=10$ and $R=2$, with the sizes of subsets, $\\card{{\\cal N}_1} = 4$ and $\\card{{\\cal N}_2} = 6$, respectively. We construct multivariate marginal distributions $P_1$ and $P_2$ from a randomly generated discrete joint distribution with the support set of each individual retailer's demand set to $[1,10]$. Other parameters include $p=1.5$ and $c= 1$. All the numerical results are tested on a PC with 2.67 gigahertz CPU, 12 gigabyte RAM, and a 64-bit Windows 7 operating system. We use MATLAB 8.0 for coding and IBM CPLEX Studio Academic version 12.5 for solving LPs problems under default settings. In order to compute a robust core (or least core) solution of this game, we would need to compute $v_{\\max}(y,{\\cal S})$ for each of $2^n=1024$ coalitions. This is accomplished by solving a number of linear programs as presented in Proposition~\\ref{prop:linearequiv}. On average, the total time it took to compute a single value $v_{\\max}(y,{\\cal S})$ is approximately $12$ seconds under this setting. Problem \\refs{eq:rnleastcore} can then be directly solved whereas \\refs{eq:rleastcore} is solved with a simple one-dimensional search algorithm whose main subroutine depends on the solution of \\refs{eq:rnleastcore} for different values of $y$.\n\nIn this numerical example, for a given value of $\\lambda$, we simply generate $100$ random extremal distributions $P_{ext}$ by solving the linear program $\\{ \\max \\mb{c}^T \\mb{q} ~:~ \\mb{q} \\in {\\cal Q}\\}$ with random cost vectors $\\mb{c}$. We also include extremal distributions produced while calculating $v_{max}$. The excess values $\\eps_{rob}^P$ and $\\eps_{det}^P$ are computed for all contaminated distributions $P_\\lambda$. Figure \\ref{fig:robust_vs_deterministic} provides comparisons on three statistics of $\\eps_{rob}^P$ and $\\eps_{det}^P$ for each fixed value of $\\lambda$: the maximum, the minimum, and the average.\n\\begin{figure}[htp]\n \\begin{center}\n\n\\includegraphics[width=0.7\\textwidth]{contaminated}\n\\end{center}\n\\caption{Comparison between $\\eps_{rob}^P$ and $\\eps_{det}^P$ for contaminated distributions with different $\\lambda$.}\n\\label{fig:robust_vs_deterministic}\n\\end{figure}\nThe maximum values of both $\\eps_{rob}^P$ and $\\eps_{det}^P$ increase when $\\lambda$ increases. For $\\lambda>0.5$, the robust solution $(y_{rob},\\bz_{rob})$ yields smaller excess values in the worst case as compared to those of $(y_{det},\\bz_{det})$ for contaminated distributions. It shows that the robust solution hedges against the worst case as expected even though on average, the solution $(y_{det},\\bz_{det})$ is slightly better in terms of worst excess values. It is worth noting that in the best case, both solutions are core solutions with no dissatisfaction even for the case of $\\lambda=1$.\n\n\n\nWe run the experiment again for $M=20$ different instances. Figure \\ref{fig:robust_vs_deterministic1} shows the statistics of $\\eps_{rob}^P$ and $\\eps_{det}^P$ when $\\lambda=1$ for all of these instances. The results again show that the robust solution $(y_{rob},\\bz_{rob})$ consistently outperforms $(y_{det},\\bz_{det})$ for all these instances in the worst case and is slightly worse on average.\n\\begin{figure}[htp]\n \\begin{center}\n\n\\includegraphics[width=0.7\\textwidth]{instances}\n\\end{center}\n\\caption{Comparison between $\\eps_{rob}^P$ and $\\eps_{det}^P$ for different instances with $\\lambda=1$.}\n\\label{fig:robust_vs_deterministic1}\n\\end{figure}\nFinally, we run the experiment again for four different settings with respect to sizes of the two subsets, $(1,9)$, $(2,8)$, $(3,7)$, and $(5,5)$, in addition to the original setting of $(4,6)$, with $M=20$ instances for each setting. Figure \\ref{fig:robust_vs_deterministic2} shows the box plots for the maximum values of $\\eps_{rob}^P$ and $\\eps_{det}^P$ when $\\lambda=1$. The results again show that the robust solution $(y_{rob},\\bz_{rob})$ outperforms $(y_{det},\\bz_{det})$ in the worst case.\n\\begin{figure}[htp]\n \\begin{center}\n\n\\includegraphics[width=0.7\\textwidth]{structures}\n\\end{center}\n\\caption{Comparison between maximum $\\eps_{rob}^P$ and $\\eps_{det}^P$ for different subset structures with $\\lambda=1$.}\n\\label{fig:robust_vs_deterministic2}\n\\end{figure}\n\\section{Conclusion}\nIn this paper, we develop a framework for newsvendor games with ambiguity in demand distributions, which we call robust cooperative games. We discuss solution concepts of robust cooperative games and study them in the context of newsvendor games with ambiguity in demand distributions when only marginal distributions are known. Some numerical results are provided, which show the robust core solutions hedge against the worst cases as expected. It is possible to develop other frameworks for cooperative games with uncertain characteristic functions by using different payoff distribution schemes and preference relations, which could be applied to other applications.\n\n\\bibliographystyle{plainnat}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section*{Part I: Conceptual Framework}\n\n\\section{Introduction}\n\\label{Intro}\n\nIn this work we study the homogenization (assymptotic \nlimit as $\\varepsilon \\to 0$) of the anisotropic\nSchr\\\"odinger equation in the following Cauchy problem\n\\begin{equation}\n\\label{jhjkhkjhkj765675233}\n\t\\left\\{\n\t\\begin{aligned}\n&\ti\\displaystyle\\frac{\\partial u_\\varepsilon}{\\partial t} - {\\rm div} {\\big(A( \\Phi^{-1} {\\big( \\frac{x}{\\varepsilon}, \\omega \\big)},\\omega) \\nabla u_\\varepsilon \\big)} \n        + \\frac{1}{\\varepsilon^2} V( \\Phi^{-1} {\\big( \\displaystyle\\frac{x}{\\varepsilon}, \\omega \\big)},\\omega) \\; u_\\varepsilon\n\\\\[5pt]\n\t&\\hspace{100pt}\t+ U( \\Phi^{-1} {\\big( \\frac{x}{\\varepsilon}, \\omega \\big)},\\omega) \\; u_\\varepsilon = 0, \n\t\t\\quad \\text{in $\\mathbb{R}^{n+1}_T \\! \\times \\! \\Omega$}, \n\\\\[3pt]\n\t\t& u_\\varepsilon= u_\\varepsilon^0, \\quad \\text{in $\\mathbb{R}^n \\! \\times \\! \\Omega$},\n\t\\end{aligned}\n\t\\right.\n\\end{equation} \nwhere $\\mathbb{R}^{n+1}_T := (0,T) \\times \\mathbb{R}^n$, for any real number $T> 0$, $\\Omega$ is a probability space, \nand the unknown function $u_\\varepsilon(t,x,\\omega)$ is complex-value.  \n\n\\medskip\nThe coefficients in \\eqref{jhjkhkjhkj765675233}, that is the matrix-value \nfunction $A$, the real-value (potencial) functions $V$, $U$ \nare random perturbations of stationary functions accomplished by \nstochastic diffeomorphisms $\\Phi: \\mathbb R^n \\times \\Omega \\to \\mathbb R^n$, (called stochastic deformations).\nThe stationarity property of random \nfunctions will be precisely defined in Section \\ref{628739yhf}, also the definition of \nstochastic deformations which were introduced by \nX. Blanc, C. Le Bris, P.-L. Lions (see \\cite{BlancLeBrisLions1,BlancLeBrisLions2}).\nIn that paper they consider the homogenization \nproblem of an elliptic operator whose coefficients are periodic or stationary functions perturbed by \nstochastic deformations. \n\n\\medskip\nIn particular, we assume that  $A = (A_{k \\ell})$, $V$ and $U$ are measurable and\nbounded functions, i.e. for $k, \\ell= 1,\\ldots,n$\n\\begin{equation}\n\\label{ASSUM1}\n     A_{k \\ell}, \\; V, \\; U \\in L^\\infty(\\mathbb{R}^n \\times \\Omega).\n\\end{equation}\nMoreover, the matrix $A$ is symmetric and \nuniformly positive defined, that is, there exists $a_0> 0$, such that, for a.a. \n$(y, \\omega) \\in \\mathbb{R}^n \\times \\Omega$, and each $\\xi \\in \\mathbb R^n$\n\\begin{equation}\n\\label{ASSUM2}\n\t\\sum_{k,\\ell=1}^n A_{k\\ell}(y,\\omega)\\,  \\xi_k \\, \\xi_\\ell \\geqslant a_0 {\\vert \\xi \\vert}^2.\n\\end{equation}\n\n\\medskip\nThis paper is the second part of the Project initiated with  \nT. Andrade, W. Neves, J. Silva \\cite{AndradeNevesSilva}\n(Homogenization of Liouville Equations \nbeyond stationary ergodic setting)\nconcerning the study of moving electrons in non-crystalline matter, \nwhich justify the form considered for the coefficients \nin \\eqref{jhjkhkjhkj765675233}. \nWe recall that crystalline materials, also called perfect materials, are described \nby periodic functions. Thus any homogenization result for\nSchr\\\"odinger equations with periodic coefficients is restrict to \ncrystalline matter. \nMoreover, perfect materials are rare in Nature, there \nexist much more non-crystalline than crystalline materials.   \nFor instance, there exists a huge class called quasi-perfect materials\n(see Section \\ref{6775765ff0090sds}, also \\cite{AndradeNevesSilva}), which are closer to \nperfect ones. Indeed, the concept of stochastic deformations \nare very suitable to describe interstitial defects in \nmaterials science\n(see Cances, Le Bris \\cite{CancesLeBris}, \nand Myers \\cite{Myers}).\n\n\\medskip\nOne remarks that, the homogenization of the Schr\\\"odinger equation\nin \\eqref{jhjkhkjhkj765675233},\nwhen the stochastic deformation $\\Phi(y,\\omega)$ \nis the identity mapping and the coefficients are periodic, \nwere studied by Allaire, Piatnitski \\cite{AllairePiatnitski}. \nNotably, that paper presents the discussion about the \ndifferences between the scaling considered in \\eqref{jhjkhkjhkj765675233}\nand the one called semi-classical limit. \nWe are not going to rephrase\nthis point here, and address the reader to Chapter 4 in \\cite{BensoussanLionsPapanicolaou}\nfor more general considerations about that. It should be mentioned that, to the best of\nour knowledge the present work is the first to study the homogenization \nof the Schr\\\"odinger equations beyond the periodic setting, applying the double-scale \nlimits and the wave function is spanned on the Bloch basis. Therefore, we have extended \nthe Bloch Theory, which was restrict until now to periodic potentials. \n\n\\medskip\nLast but not least, one observes that the initial data $u_\\varepsilon^0$ \nshall be considered well-prepared, see equation \\eqref{WellPreparedness}. \nThis assumption is fundamental for the abstract homogenization result \nestablished in Theorem \\ref{876427463tggfdhgdfgkkjjlmk}, where the \nlimit function obtained from $u_\\varepsilon$\nsatisfies a simpler Schr\\\"odinger equation, called the effective mass equation,\nwith effective constant coefficients, namely matrix $A^*$, and potential $V^*$.\nThis homogenization procedure is well known in solid state physics as \nEffective Mass Theorems, see Section \\ref{HomoSchEqu}. \n\n\\medskip\nFinally, we stress Section \\ref{6775765ff0090sds} which is \nrelated to the homogenization of the Schr\\\"odinger equation for quasi-perfect materials,\nand it is also an important part of this paper. \nIndeed, a very special case occurs in situations where\nthe amount of randomness is small, more specifically the disorder in the \nmaterial is limited. In particular, this section is interesting  \nfor numerical applications, where specific computationally efficient techniques \nalready designed to deal with the homogenization of the Schr\\\"odinger equation in the periodic setting, \ncan be employed to treat the case of quasi-perfect materials. \n\n\\subsection{Contextualization}\n\nLet us briefly recall that the homogenization's problem for \\eqref{jhjkhkjhkj765675233} has been treated for the periodic case \n($A_{\\rm per}(y)$, $V_{\\rm per}(y)$, $U_{\\rm per}(y)$), \nand  $\\Phi(y,\\omega) = y$ by some authors. Besides the paper by G. Allaire, A.Piatnitski \\cite{AllairePiatnitski} already mentioned,\nwe address the following papers for the case of $A_{\\rm per}= I_{n \\times n}$, i.e. isotropic \nSchr\\\"odinger equation in \\eqref{jhjkhkjhkj765675233}: G. Allaire, M.Vanninathan \\cite{AllaireVanninathan}, L. Barletti, N. Ben Abdallah \\cite{BarlettiBenAbdallah}, \nV. Chabu, C. Fermanian-Kammerer, F. Marci\u00e0 \\cite{ChabuFermanianMarcia}, and we observe that this list is by no means exhaustive. \nIn \\cite{AllaireVanninathan}, the authors study a semiconductors model excited by an external potencial $U_{\\rm per}(t,x)$, which depends on \nthe time $t$ and macroscopic variable $x$. \nIn  \\cite{BarlettiBenAbdallah} the authors\ntreat the homogenization's problem when the external \npotential $U_{\\rm per}(x,y)$ depends also on the macroscopic variable $x$. \nFinally, in \\cite{ChabuFermanianMarcia}\nit was considered an external potential $U_{\\rm per}(t,x)$ \nwhich model the effects of impurities on the otherwise perfect matter. \n\n\\medskip\nAll the references cited above treat the homogenization's problem \nfor \\eqref{jhjkhkjhkj765675233}, studying the spectrum of the associated \nBloch spectral cell equation, that is, for each $ \\theta \\in \\mathbb{R}^n$,\nfind the eigenvalue-eigenfunction pair $(\\lambda,\\psi)$, satisfying  \n\\begin{equation}\n\\label{8756trg}\n\\left\\{\n\\begin{aligned}\nL_{\\rm per}(\\theta) {\\big[ \\psi \\big]}&= \\lambda \\, \\psi, \n\\quad \\text{in $[0,1)^n$},\n\\\\[5pt]\n\\psi(y)&\\not= 0, \\quad \\text{periodic function},\n\\end{aligned}\n\\right.\n\\end{equation}\nwhere $L_{\\rm per}(\\theta)$ is the Hamiltonian given by \n$$\nL_{\\rm per}(\\theta){\\big[ f \\big]}= -{\\big( {\\rm div}_{\\! y} + 2i\\pi \\theta \\big)} {\\big[ A_{\\rm per}(y) {( \\nabla_{\\!\\! y} \n+ 2i \\pi \\theta)} f \\big]} + V_{\\rm per}(y) f.\n$$\nThe above eigenvalue problem is precisely stated (in the more general context studied in this paper) in \nSection \\ref{877853467yd56rtfe5rtfgeds76ytged}. \nHere, concerning the periodic setting mathematical solutions to \\eqref{8756trg}, we address the reader to \nC. H. Wilcox \\cite{Wilcox}, (see in particular Section 2: A discussion of related literature). Then,\nonce this eigenvalue problem is resolved, the goal is to pass to the limit as $\\varepsilon \\to 0$. One remarks that, \nthere does not exist an uniform estimate in $H^1(\\mathbb R^n)$ for the family of solutions $\\{u_\\varepsilon\\}$ of \\eqref{jhjkhkjhkj765675233}, \ndue to the scale $\\varepsilon^{-2}$ multiplying the \ninternal potential $V_{\\rm per}(y)$. To accomplish the desired asymptotic limit, under this lack of compactness,\na nice strategy is to use the two-scale convergence, for instance see the proof of Theorem 3.2 in \\cite{AllairePiatnitski}. \n\n\\medskip\nLet us now focus on the stochastic setting proposed in this paper, more precisely when the coefficients of\nthe Schr\\\"odinger equation in \\eqref{jhjkhkjhkj765675233} are the composition of\nstationary functions with stochastic deformations. Hence we have the following \nnatural questions:\n\n\\medskip\n$(Q.1)$ Is it possible to obtain an analogously Bloch spectral cell equation to this stochastic setting? \n\n\\medskip\n$(Q.2)$ This new stochastic spectral problem can be resolved, such that, the eigenvalues do not depend \non $\\omega \\in \\Omega$ (see Remark \\ref{GROUPNECE})? \n\n\\medskip\n$(Q.3)$ Is it feasible to adapt the two-scale convergence to this new proposed stochastic setting? \nWe remark that, the approach of stochastic two-scale convergence developed by \nBourgeat, Mikelic, Wright \\cite{BourgeatMikelicWright}, and also by\nZhikov, Pyatnitskii \\cite{ZhikovPyatnitskii} do not\nfit to the present context \nbecause of the presence of the stochastic deformation $\\Phi$. \n\n\\medskip\nThe former question $(Q.1)$ is answered in Section \n\\ref{683926ruesszs}. Indeed, assuming that the solution of \nequation \\eqref{jhjkhkjhkj765675233} is given by a plane wave, \nthe stochastic spectral Bloch cell equation\n\\eqref{92347828454trfhfd4rfghjls}\nis obtained applying the\nasymptotic expansion WKB method, \n(developed by Wentzel, Kramers, and Brillouin,\nsee G. Allaire \\cite{AllaireArnoldDegondHou}). \nMore specifically, the Hamiltonian in \\eqref{92347828454trfhfd4rfghjls}\nis given by\n$$\n   L^\\Phi(\\theta)\\big[ F \\big]\\! \\! = -\\big( {\\rm div}_{\\! z} + 2i\\pi \\theta \\big){\\left[ A{( \\Phi^{-1}(z,\\omega),\\omega)} {\\big( \\nabla_{\\!\\! z} + 2i\\pi\\theta \\big)} F \\right]} \n   + V( \\Phi^{-1}(z,\\omega),\\omega) F, \n$$\nfor each $F(z,\\omega)= f\\left( \\Phi^{-1}(z,\\omega),\\omega \\right)$, where $f(y,\\omega)$ is a stationary function. \n\n\\medskip\nTo answer $(Q.2)$, we have to study the spectrum of the operator $L^\\Phi(\\theta)$, for each $\\theta \\in \\mathbb R^n$ fixed. \nThe first idea is to follow the techniques applied for the periodic setting, that is, for the operator \n$L_{\\rm per}(\\theta)$ in \\eqref{8756trg}, where the fundamental tool is the compact embedding of \n $H^1_{ \\rm per}([0,1)^n)$ in $L^2([0,1)^n)$.  \nAlthough, since $\\omega \\in \\Omega$ can not be treat as a fixed parameter, we have to consider the more general theory of \nSobolev spaces on locally compact Abelian groups, which is developed in Section \\ref{9634783yuhdj6ty}. \nIn fact, we have established in details a Rellich-Kondrachov type Theorem, (see Theorem \\ref{7864876874}),\nsuch that together \nwith the study of continuous dynamic systems on compact Abelian groups enable us to answer positively \nthis question, at least, when $\\Omega$ has some structure. The second applied strategy here to answer $(Q.2)$ is \nthe Perturbation Theory, that is to say, taking the advantage of the well known spectrum for $L_{\\rm per}(\\theta)$. \nTo this end, we first consider that the coefficients of\nthe Schr\\\"odinger equation in \\eqref{jhjkhkjhkj765675233} are the composition of the periodic \nfunctions $A_{\\rm per}$, $V_{\\rm per}$ and $U_{\\rm per}$ with a special case of stochastic deformations, \nnamely stochastic perturbation of the identity (see Definition \\ref{37285gdhddddddddddd}), \nwhich is given by \n$$\n   \\Phi_\\eta(y,\\omega) := y + \\eta  \\, Z(y,\\omega) + \\mathrm{O}(\\eta^2),\n$$\nwhere $Z$ is some stochastic deformation and $\\eta \\in (0,1)$. This concept was introduced \nby X. Blanc, C. Le Bris, P.-L. Lions \\cite{BlancLeBrisLions2}, and applied for the first time to \nevolutionary equations in  T. Andrade, W. Neves, J. Silva \\cite{AndradeNevesSilva}. \nThen, taking this special case\n$\\Phi_\\eta$, the operator $L^{\\Phi_\\eta}(\\theta)$ has the following expansion  \nin a neighborhood of $(0,\\theta_0) \\in \\mathbb{R}^{n+1}$,\n$$\n   L^{\\Phi_\\eta}(\\theta) = L_{\\rm per}(\\theta_0) + \\sum_{{\\vert \\varrho \\vert} = 1}^{3} ((\\eta,\\theta)-(0,\\theta_0))^{\\varrho}L_{\\varrho} + \\mathrm{O}(\\eta^2),\n$$\nwhere $\\varrho= (\\varrho_1,\\ldots,\\varrho_n,\\varrho_{n+1}) \\in \\mathbb{N}^{n+1}$, ${\\vert \\varrho \\vert} = \\sum_{k=1}^{n+1} \\varrho_k$,\nand $L_{\\varrho}$ is a bounded operator, see Section \\ref{6775765ff0090sds}. \nFrom the above equation, it follows that the point spectrum \n(i.e. the set of eigenvalues) of $L^{\\Phi_\\eta}(\\theta)$\nis not empty in a neighborhood of $(0,\\theta_0)$, when \n$\\lambda_{\\rm per}(\\theta_0)$ is an isolated eigenvalue with finite multiplicity. \nThis last property is studied in details in Section \\ref{0239786gfhgdf},\nsee Theorem \\ref{768746hughjg576}. \n\n\\medskip\nThe question $(Q.3)$ is answered positively in \nSection \\ref{pud63656bg254v2v5}, that is, we have established \nin this section a two-scale convergence in a stochastic setting, which is beyond the classical stationary \nergodic setting. Indeed, the main difference here with the earlier stochastic extensions of the periodic setting is \nthat, the test functions used are random \nperturbations of stationary functions accomplished by \nthe stochastic deformations. These compositions are beyond the stationary class, thus we have a \nlack of the stationarity property in this kind of test functions (see the introduction section in \\cite{AndradeNevesSilva} \nfor a deep discussion about this subject). It was introduced a compactification argument that, preserves the ergodic \nnature of the setting involved and allow us to overcome these difficulties. \n\n\t\n\\subsection{Summary of the main results}\n\nIn this section we summarize the main results on this paper. \nSince some of the theorems (cited below) have its on interested, \nwe describe shortly the main issue of each one. \n\n\\medskip\nFirst, Theorem \\ref{Compacification} allows us to overcome the lack of topological structure of a given probability space reducing it to \na separable compact space whose topological basis is dictated by the coefficients of the problem~\\eqref{jhjkhkjhkj765675233}. \n\n\\smallskip\nThen, the Theorem \\ref{TwoScale} uses all topological features brought forth by the Theorem \\ref{Compacification} in order to give us a result about \ntwo-scale convergence where the test functions are random perturbations accomplished by stochastic diffeomorphisms of stationary functions.  It is worth \nmentioning that this result generalizes the corresponding one for deterministic case in~\\cite{DiazGayte} and the corresponding one for the stochastic case in~\\cite{BourgeatMikelicWright}.\n\n\\smallskip\nTheorem \\ref{768746hughjg576} consider a sequence of bounded operators in a Hilbert space, which defines a symmetric\noperator via the power series of multidimensional complex variables. It is stated that,\nif the first coefficient operator of this series has isolated eigenvalues of finite multiplicity, then\nthe holomorphic defined operator inherited from it similar point spectrum analysis.\n\n\\smallskip\nThe Theorem \\ref{876876876GG} established a necessary condition such that, the\nRellich--Kondrachov Theorem on compact Abelian groups holds true. More precisely, \nthe dual group must be an enumerable set. \n\n\\smallskip\nA complete characterization of the Rellich--Kondrachov Theorem on compact Abelian groups\nis given by Theorem \\ref{7864876874}. \nMoreover, as a byproduct of this characterization, we provide\na proof of the Rellich--Kondrachov Theorem in a precise context. \n\n\\smallskip\nThe Theorem \\ref{876427463tggfdhgdfgkkjjlmk} is one of the main results of this paper. It is an abstract homogenization result for Schr\\\"odinger equations that \nencompasses the corresponding one given by Allaire and Piatnistski~\\cite{AllairePiatnitski} in the periodic context. \n\n\\smallskip\nThe Theorem \\ref{873627yuhfdd} shows how the periodic setting can be used to deal with homogenization of the equation~\\eqref{jhjkhkjhkj765675233} for\n materials when the amount of randomness is small. This has importants numerical implications. \n\n\\smallskip\nThe Theorem \\ref{THM511} reveals an interesting splitting property of the solution of the homogenized equation associated to~\\eqref{jhjkhkjhkj765675233} in the \nspecific case of the quasi-perfect materials.\n\n\\section{Preliminaries and Background}\n\\label{PrelmBackg}\n\nThis section introduces the basement theory, which will be used through the paper. \nTo begin we fix some notations, and collect some preliminary results. The material which is well-known or a direct extension \nof existing work are giving without proofs, otherwise we present them.\n\n\\medskip\nWe denote by $\\mathbb{G}$ the group $\\mathbb{Z}^n$ (or $\\mathbb{R}^n$), with $n \\in \\mathbb{N}$.\nThe set $[0,1)^n$\ndenotes the unit cube, which is also called the unitary cell and will be used  \nas the reference period for periodic functions.\nThe symbol $\\left\\lfloor x \\right\\rfloor$ denotes the \nunique number in $\\mathbb{Z}^n$, such that $x - \\left\\lfloor x \\right\\rfloor \\in [0,1)^n$.\nLet $H$ be a complex Hilbert space, we denote by $\\mathcal{B}(H)$ \nthe Banach space of linear bounded operators from $H$ to $H$.\n\n\\medskip\nLet $U \\subset \\mathbb R^{n}$ be an open set, $p \\geqslant  1$, and $s \\in \\mathbb{R}$.\nWe denote by \n$L^p(U)$ the set of (real or complex) $p-$summable functions\nwith respect to the Lebesgue measure (vector ones should be understood\ncomponentwise). Given a Lebesgue measurable set\n$E \\subset \\mathbb R^n$, \n$|E|$ denotes its $n-$dimensional Lebesgue measure.\nMoreover, we will use the standard notations for the \nSobolev spaces $W^{s,p}(U)$ and $H^{s}(U)\\equiv W^{s,2}(U)$. \n\n\\subsection{Anisotropic Schr\\\"odinger equations}\n\\label{SchEq}\n\nThe aim of this section is to present the well-posedness for the solutions of the Schr\\\"odinger equation, \nand some properties of them. \nMost of the material can be found in\nCazenave, Haraux \\cite{CazenaveHaraux}.\n\n\\medskip\nFirst, let us consider the following Cauchy problem, which is driven by a linear anisotropic\nSchr\\\"odinger equation, that is\n\\begin{equation}\n\\label{87644343}\n   \\left\\{\n   \\begin{aligned}\n   &i \\; \\partial_t u(t,x) - {\\rm div} \\big(A(x) \\nabla u(t,x) \\big)\n   + V(x) \\, u(t,x) = 0 \\quad  \\text{in $\\mathbb{R}^{n+1}_T$}, \n   \\\\[5pt]\n   & u(0,x)=u_0(x) \\quad \\text{in $\\mathbb{R}^n$},\n   \\end{aligned}\n  \\right.\n\\end{equation}\nwhere the unknown $u(t,x)$ is a complex value function, and $u_0$\nis a given initial datum. The coefficient $A(x)$ is a symmetric real $n \\times n$-matrix \nvalue function, and the potential $V(x)$ is a real function. We always assume that\n\\begin{equation}\n\\label{CONDITAV}\n     A(x), V(x) \\quad \\text{are measurable bounded functions}. \n\\end{equation}\nOne recalls that, a matrix $A$ is called (uniformly) coercive, when, there exists $a_0> 0$, \nsuch that, for each $\\xi \\in \\mathbb{R}^n$, and almost all $x \\in \\mathbb{R}^n$,\n$A(x) \\xi \\cdot \\xi \\geqslant a_0 \\vert \\xi \\vert^2$. \n\n\\medskip\nThe following definition tell us in which sense a complex function $u(t,x)$ is a mild solution to \\eqref{87644343}.\n\\begin{definition}\n\\label{MildSol}\nLet $A, V$ be coefficients satisfying \\eqref{CONDITAV}. \nGiven $u_0 \\in H^1(\\mathbb{R}^n)$, a function \n$$\n   u \\in C( [0,T]; H^1(\\mathbb{R}^n)) \\cap C^1((0,T); H^{-1}(\\mathbb{R}^n))\n$$\nis called a mild solution to the Cauchy problem \\eqref{87644343}, when for each $t \\in (0,T)$, it follows that\n\\begin{equation}\n\\label{DEFSOLSCH}\n   i \\partial_t u(t) -{\\rm div} \\big(A \\nabla u(t) \\big) + V u(t) = 0 \\quad \\text{in $H^{-1}(\\mathbb{R}^n)$}, \n\\end{equation}\nand $u(0)= u_0$ in $H^1(\\mathbb{R}^n)$.\n\\end{definition}\t\n\nThen, we state the following \n\\begin{proposition}\n\\label{PROPEUSCHEQ}\nLet $A$ be a coercive matriz value function, $V$ a potential and \n$u_0 \\in H^1(\\mathbb{R}^n)$  a given initial data. \nAssume that $A, V$ satisfy \\eqref{CONDITAV}. Then, there exist a unique \nmild solution of the Cauchy problem \\eqref{87644343}. \n\\end{proposition}\n\n\\begin{proof}\nThe proof follows applying Lemma 4.1.5 and Corollary 4.1.2 in \\cite{CazenaveHaraux}.\n\\end{proof}\n\n\\medskip\n\\begin{remark}\n\\label{REMCOSTCOEFF}\nIt is very important in the homogenization procedure of the Schr\\\"odinger equation, \nwhen the coefficients $A$ and $V$ in \\eqref{87644343} are constants,\nthe matrix $A$ is not necessarily coercive, and the initial data \n$u_0 \\in L^2(\\mathbb{R}^n)$. Then, a function $u \\in L^2(\\mathbb{R}^{n+1}_T)$ is called a\nweak solution to \\eqref{87644343}, if it satisfies \n$$\n   i \\partial_t u - {\\rm tr}(A D^2 u) + V u = 0 \\quad \\text{in distribution sense}.\n$$\nSince $A, V$ are constant, we may apply the Fourier Transform, and\nobtain the existence of a unique solution $u \\in H^1((0,T); L^2(\\mathbb{R}^{n}))$. \nTherefore, the solution $u \\in  C([0,T]; L^2(\\mathbb{R}^{n}))$ after being \nredefined in a set of measure zero, and we have $u(0)= u_0$ in $L^2(\\mathbb{R}^n)$.\n\\end{remark}\n\nNow, let us recall the standard a priori estimates for the solutions of the \nCauchy problem \\eqref{87644343}. First, under the conditions of Proposition \\ref{PROPEUSCHEQ}, a function\n$u \\in C( [0,T]; H^1(\\mathbb{R}^n)) \\cap C^1((0,T); H^{-1}(\\mathbb{R}^n))$, which is the mild solution of \\eqref{87644343},\nsatisfies for each $t \\in [0,T]$\n\\begin{equation}\n\\begin{aligned}\n\t\t&(i) \\ \\int_{\\mathbb{R}^n} |u(t)|^2 dx = \\int_{\\mathbb{R}^n} |u_0|^2 dx,\n\t\t\\\\[5pt]\n\t\t&(ii) \\  \\int_{\\mathbb{R}^n} |\\nabla u(t)|^2 dx \\leqslant C \\  \\big(\\int_{\\mathbb{R}^n} |\\nabla u_0|^2 dx  \n\t\t+  \\int_{\\mathbb{R}^n} |u_0|^2 dx \\big), \n\\end{aligned}\n\\end{equation}\nwhere $C= C(\\|V\\|_{L^\\infty}, \\|A\\|_{L^\\infty}, a_0)$ is a positive constant. Clearly, for the constant coefficients case,\nwith $A$ non-coercive and $u_0 \\in L^2(\\mathbb{R}^n)$, a function $u \\in  C([0,T]; L^2(\\mathbb{R}^{n}))$, which is \nthe weak solution of \\eqref{87644343}, just satisfies the item $(i)$ above. These estimates follow by \ndensity argument. \n\n\\subsection{Stochastic configuration}\n\\label{628739yhf}\n\nHere we present the stochastic context, which will be used thoroughly in the paper. \nTo begin, let $(\\Omega, \\mathcal{F}, \\mathbb{P})$ be a probability space. For each random variable  \n$f$ in $L^1(\\Omega; \\mathbb P)$, ($L^1(\\Omega)$ for short), \nwe denote its expectation value by\n$$\n    \\mathbb{E}[f]= \\int_\\Omega f(\\omega) \\ d\\mathbb P(\\omega).\n$$\n\nA mapping $\\tau: \\mathbb{G} \\times \\Omega \\to \\Omega$ is said a $n-$dimensional dynamical \nsystem if:\n\\begin{enumerate}\n\\item[(i)](Group Property) $\\tau(0,\\cdot)=id_{\\Omega}$ and $\\tau(x+y,\\omega)=\\tau(x,\\tau(y,\\omega))$ for all $x,y \\in  \\mathbb{G}$ \nand $\\omega\\in\\Omega$.\n\\item[(ii)](Invariance) The mappings $\\tau(x,\\cdot):\\Omega\\to \\Omega$ are $\\mathbb P$-measure preserving, that is, for each $x \\in  \\mathbb{G}$ and \nevery $E\\in \\mathcal{F}$, we have \n$$\n\\tau(x,E)\\in \\mathcal{F},\\qquad \\mathbb P(\\tau(x,E))=\\mathbb P(E).\n$$\n\\end{enumerate}\nFor simplicity, we shall use $\\tau(k)\\omega$ to denote $\\tau(k,\\omega)$. Moreover, it is usual to say that \n$\\tau(k)$ is a discrete (continuous) dynamical system if $k \\in \\mathbb Z^n$ ($k \\in \\mathbb R^n$), but we only stress \nthis when it is not obvious from the context. \n\n\\medskip\nA measurable function $f$ on $\\Omega$ is called $\\tau$-invariant, if for each $k \\in \\mathbb{G}$ \n$$\n    f(\\tau(k) \\omega)= f(\\omega) \\quad \\text{for almost all $\\omega \\in \\Omega$}. \n$$\nHence a measurable set $E \\in \\mathcal{F}$ is $\\tau$-invariant, if its characteristic function $\\chi_E$ is $\\tau$-invariant. \nIn fact, it is a straightforward to show that, a $\\tau$-invariant set $E$ can be equivalently defined by \n$$\n    \\tau(k) E= E \\quad \\text{for each $k \\in \\mathbb{G}$}.\n$$\nMoreover, we say that the dynamical system $\\tau$ is ergodic, when\nall $\\tau$-invariant sets $E$ have measure $\\mathbb P(E)$ of either zero or one. \nEquivalently, we may characterize an ergodic dynamical system\nin terms of invariant functions. Indeed, a dynamical system is ergodic if \neach $\\tau$- invariant function is constant almost everywhere, that is to say \n$$\n    \\Big( f(\\tau(k) \\omega)= f(\\omega) \\quad \\text{for each $k \\in \\mathbb{G}$ and a.e. $\\omega \\in \\Omega$} \\Big) \n    \\Rightarrow \\text{ $f(\\cdot)= const.$ a.e.}.  \n$$\n\n\\medskip\n\\begin{example}\n\\label{NDT}\nLet $\\Omega= [0,1)^n$ be a sample space,  $\\mathcal{F}$ the appropriate $\\sigma$-algebra on \n$\\Omega$, and $\\mathbb P$ the probability measure, i.e. the Lebesgue measure restrict to $\\Omega$.\nThen, we consider the $n$-dimensional \ndynamical system $\\tau: \\mathbb R^n \\times \\Omega \\to \\Omega$, defined by \n$$\n    \\tau(x) \\omega:= x + \\omega - \\left\\lfloor x+\\omega \\right\\rfloor. \n$$    \nThe group property for $\\tau(x)$ follows from the greatest integer function properties, \nand its invariance from the translation invariance of the Lebesgue measure. \n\\end{example}\n\n\\begin{example}\n\\label{EXTJING}\nLet $(\\Omega_0,\\mathscr{F}_0,\\mathbb{P}_0)$ be a probability space. \nFor $m \\in \\mathbb{N}$ fixed, we consider the set $S= \\{0,1,2,\\ldots,m\\}$ \nand the real numbers \n$$\n    \\text{$p_0, p_1, p_2, \\ldots, p_m$ in $(0,1)$, such that $\\sum_{\\ell= 0}^m p_\\ell=1$}.  \n$$\nIf $ \\{X_k:\\Omega_0 \\to S \\}_{k\\in\\mathbb{Z}^n}$\nis a family of random variables, then it is induced a probability measure from it on the measurable space  \n$\\big( S^{\\mathbb{Z}^n}, \\bigotimes_{k\\in\\mathbb{Z}^n}2^{S} \\big)$. Indeed, we may define the probability \nmeasure\n$$\n\\mathbb{P}(E):= \\mathbb{P}_0{\\left\\{ X \\in E \\right\\}}, \\;\\; E \\in \\bigotimes_{k\\in\\mathbb{Z}^n}2^{S},\n$$\nwhere the mapping $X: \\Omega_0 \\to S^{\\mathbb{Z}^n}$ is given by \n$X(\\omega_0)= (X_k(\\omega_0))_{k\\in\\mathbb{Z}^n}$.\n\n\\medskip\nNow, we denote for convenience $\\Omega= S^{\\mathbb{Z}^n}$\nand $\\mathscr{F}= \\bigotimes_{k\\in\\mathbb{Z}^n}2^{S}$, \nthat is, $\\mathscr{F}= \\sigma(\\mathscr{A})$, \nwhere $\\mathscr{A}$ is the algebra given by the finite union of sets (cylinders of finite base) \nof the form \n\\begin{equation}\n\\label{356}\n    \\prod_{k \\in \\mathbb{Z}^n} E_k, \n\\end{equation}\nwhere $E_k \\in 2^S$ is different from $S$ for a finite number of indices $k$. Additionally we assume that,\nthe family \t$\\{X_k\\}_{k\\in\\mathbb{Z}^n}$ is independent, and for each $k \\in \\mathbb{Z}^n$, we have \n\\begin{equation}\n\\label{243}\n\t\\mathbb{P}_0{\\{ X_k=0 \\}}= p_0, \\,\\, \\mathbb{P}_0{\\{ X_k=1 \\}}= p_1, \\,\\, \\ldots, \\,\\, \\mathbb{P}_0{\\{ X_k=m \\}}= p_m.\n\\end{equation}\nThen, we may define an ergodic dynamical system $\\tau: \\mathbb{Z}^n \\times \\Omega \\to \\Omega$, by\n$$\n\t{\\left( \\tau (\\ell) \\omega \\right)}(k) := \\omega(k + \\ell), \\quad \\text{for any $k,\\ell \\in \\mathbb{Z}^n$},\n$$\nwhere $\\omega= (\\omega(k))_{k \\in \\mathbb{Z}^n}$.\n\n\\medskip\n$i)$ The group property follows from the definition. Indeed, \nfor each $\\omega \\in \\Omega$ and $\\ell_1,\\ell_2 \\in \\mathbb{Z}^n$, it follows that \n\\begin{equation*}\n\t{\\big( \\tau (\\ell_1 + \\ell_2) \\omega \\big)}(k) = \\omega(k + \\ell_1 + \\ell_2) = {\\big( \\tau (\\ell_1) \\tau(\\ell_2) \\omega \\big)}(k), \n\\end{equation*}\nfor any $k \\in \\mathbb{Z}^n$.\n\n\\medskip\n$(ii)$ The mappings $\\tau(\\ell,\\cdot):\\Omega\\to \\Omega$ are $\\mathbb P$-measure preserving.\nFirst, we observe from \\eqref{356} that, for all $\\ell \\in \\mathbb{Z}^n$\n\\begin{equation*}\n\t\\tau(\\ell) \\big( \\prod_{k \\in \\mathbb{Z}^n} E_k \\big)= \\prod_{k \\in \\mathbb{Z}^n} E_{k+\\ell}.\n\\end{equation*}\nTherefore, for any $\\ell \\in \\mathbb{Z}^n$\n$$\n\\begin{aligned}\n   \\mathbb{P}{\\Big( \\tau(\\ell) {\\big( \\prod_{k \\in \\mathbb{Z}^n} E_k \\big)} \\Big)}&= \n   \\mathbb{P}{\\big( \\prod_{k \\in \\mathbb{Z}^n} E_{k+\\ell} \\big)}\n   = \\mathbb{P}_0 {\\big( \\bigcap_{k\\in\\mathbb{Z}^n} \\{X_k \\in E_{k+\\ell}\\} \\big)}\n   \\\\[5pt]\n   &= \\prod_{k\\in \\mathbb{Z}^n} \\mathbb{P}_0 {\\left\\{ X_k \\in E_{k+\\ell} \\right\\}}\n   \\\\[5pt]\n   &= \\prod_{k\\in \\mathbb{Z}^n} \\mathbb{P}_0 {\\left\\{ X_{k+\\ell} \\in E_{k+\\ell} \\right\\}} \n   = \\prod_{k\\in \\mathbb{Z}^n} \\mathbb{P}_0 {\\left\\{ X_k \\in E_k \\right\\}},\n\\end{aligned}\n$$\nwhere we have used in the second line that the family of random variables is\nindependent and in the third line it has  the same distribution, equation \\eqref{243}. \nThen, the measure preserving is satisfied for each element of the algebra \n$\\mathscr{A}$, and hence for each element of $\\mathscr{F}$. \n\n\\medskip\n$(iii)$ The ergodicity. Given the cylinders ${ \\prod_{k \\in \\mathbb{Z}^n} E_k }$ and ${ \\prod_{k \\in \\mathbb{Z}^n} F_k }$, \nthere exists $\\ell_0 \\in \\mathbb{Z}^n$, such that\n$$\n    \\mathbb{P}{\\Big( \\tau(\\ell_0) {\\big( \\prod_{k \\in \\mathbb{Z}^n} E_k \\big)} \\cap {\\big( \\prod_{k \\in \\mathbb{Z}^n} F_k \\big)} \\Big)} \n    = \\mathbb{P}{\\big( \\prod_{k \\in \\mathbb{Z}^n} E_k \\big)} \\, \\mathbb{P} {\\big( \\prod_{k \\in \\mathbb{Z}^n} F_k \\big)}.\n$$\nIndeed, let us define \n\t\\begin{equation*}\n\t\te_0:= {\\rm max}{\\{ {\\vert k \\vert} \\, ; \\, k \\in \\mathbb{Z}^n, \\, E_k \\not= S \\}}, \\,\n\t\t\\quad f_0:= {\\rm max}{\\{ {\\vert k \\vert} \\, ; \\, k \\in \\mathbb{Z}^n, \\, F_k \\not= S \\}},\n\t\\end{equation*}\nand observe that, if $\\ell_0 \\in \\mathbb{Z}^n$ satisfies ${ {\\vert \\ell_0 \\vert} > e_0 + f_0 }$, then \n\\begin{equation*}\nE_{k+\\ell_0} \\cap F_k = \\left\\{\n\\begin{array}{ll}\n\tF_k & \\text{if} \\; {\\vert k \\vert} \\leqslant f_0, \n\t\\\\[5pt]\n\tE_k & \\text{if} \\; f_0 < {\\vert k \\vert} \\leqslant e_0 + f_0, \n\t\\\\[5pt]\n\tS & \\text{if} \\; {\\vert k \\vert} > e_0 + f_0.\n\\end{array}\n\\right.\n\\end{equation*}\nTherefore, we have \n\\begin{eqnarray*}\n\\mathbb{P}{\\big( \\tau(\\ell_0) {\\big( \\prod_{k \\in \\mathbb{Z}^n} E_k \\big)} \\cap {\\big( \\prod_{k \\in \\mathbb{Z}^n} F_k \\big)} \\big)} \n& = & \\mathbb{P}{\\big( {\\big( \\prod_{k \\in \\mathbb{Z}^n} E_{k+\\ell_0} \\big)} \\cap {\\big( \\prod_{k \\in \\mathbb{Z}^n} F_k \\big)}   \\big)} \n\\\\[5pt]\n\t\t& = & \\mathbb{P}{\\big( \\prod_{k \\in \\mathbb{Z}^n} {\\big( E_{k+\\ell_0} \\cap F_k \\big)} \\big)} \n\\\\[5pt]\n\t\t& = & \\prod_{k \\in \\mathbb{Z}^n} \\mathbb{P}_0 {\\left\\{ X_k \\in E_{k+\\ell_0} \\cap F_k  \\right\\}} \n\\\\[5pt]\n\t\t& = & \\mathbb{P}{\\big( \\prod_{k \\in \\mathbb{Z}^n} E_k \\big)} \\mathbb{P}{\\big( \\prod_{k \\in \\mathbb{Z}^n} F_k \\big)}.\n\\end{eqnarray*}\nThe above property follows for finite unions of cylinders, that is to say, given $E_1, E_2 \\in \\mathscr{A}$, \nthere exists $\\ell_0 \\in \\mathbb{Z}^n$, such that \n\\begin{equation*}\n   \\mathbb{P}{\\left( \\tau(\\ell_0) {E}_1 \\cap {E}_2 \\right)}= \\mathbb{P}({E}_1) \\, \\mathbb{P}({E}_2).\n\\end{equation*}\n\t\n\\medskip\nNow, let $E \\in \\mathscr{F}$ be a $\\tau$-invariant set. \nFor each $\\varepsilon> 0$, there exists ${E}_0 \\in \\mathscr{A}$ such that,\n$\\mathbb{P} {\\left({E} \\Delta \\, {E}_0 \\right)} < \\varepsilon$. \nThen, since $E$ is $\\tau$-invariant we have for each \n$ \\ell \\in \\mathbb{Z}^n$\n\\begin{equation}\n\\label{684}\n\\begin{aligned}\n    \\mathbb{P}{\\big( \\tau(\\ell) {E}_0 \\, \\Delta \\, {E}_0 \\big)}\n     &\\leq \\mathbb{P}{\\big( \\tau(\\ell) {E}_0 \\, \\Delta \\, \\tau(\\ell) {E} \\big)} + \\mathbb{P}{\\big( \\tau(\\ell) {E} \\, \\Delta \\, {E} \\big)} \n     + \\mathbb{P}{\\big({E} \\Delta \\, {E}_0 \\big)} \n\\\\[5pt]\n     &= 2 \\, \\mathbb{P}{\\big({E} \\Delta \\, {E}_0 \\big)} \\leq 2 \\varepsilon.\n\\end{aligned}\n\\end{equation}\nOn the other hand, since ${E}_0 \\in \\mathscr{A}$, for some $\\ell_0 \\in \\mathbb{Z}^n$, it follows that  \n\\begin{equation*}\n\t\t\\mathbb{P}{\\left( \\tau(\\ell_0){E}_0 \\cap {E}_0^c \\right)}= \\mathbb{P}({E}_0)\\mathbb{P}({E}_0^c) \n\t\t\\quad \\text{and} \\quad\n\t\t\\mathbb{P}{\\left( \\tau(\\ell_0) {E}_0^c \\cap {E}_0 \\right)}= \\mathbb{P}({E}_0^c)\\mathbb{P}({E}_0),\n\\end{equation*}\nand thus\n\\begin{equation}\n\\label{ABA}\n\\begin{aligned}\n   \\mathbb{P}{\\big( \\tau(\\ell_0) {E}_0 \\, \\Delta \\, {E}_0 \\big)}&= \\mathbb{P} {\\left( \\tau(\\ell_0) {E}_0 \\cap {E}_0^c \\right)} \n   + \\mathbb{P} {\\left( \\tau(\\ell_0) {E}_0^c \\cap {E}_0 \\right)} \n   \\\\[5pt]\n   &= 2 \\mathbb{P}({E}_0) (1-\\mathbb{P}({E}_0)).\n\\end{aligned}\n\\end{equation}\nFrom \\eqref{684} and \\eqref{ABA}, it follows for each $\\varepsilon> 0$\n\t\\begin{equation*}\n\t\t\\mathbb{P}({E}_0) (1-\\mathbb{P}({E}_0))< \\varepsilon.\n\t\\end{equation*}\nConsequently, we obtain that $\\mathbb{P}({E})= 0$ or $\\mathbb{P}({E})= 1$.\n\\end{example}\n\n\n\\medskip\nNow, let $(\\Gamma, \\mathcal{G}, \\mathbb{Q})$ be a given probability space. We say that a\nmeasurable function $g: \\mathbb R^n \\times\\Gamma \\to \\mathbb R$ is stationary, if for any finite set \nconsisting of points $x_1,\\ldots,x_j\\in \\mathbb R^n$, and any $k \\in \\mathbb{G}$, the distribution of the random vector \n$$\n   \\Big(g(x_1+k,\\cdot),\\cdots,g(x_j+k,\\cdot)\\Big)\n$$\nis independent of $k$. Further, subjecting the stationary function $g$ to some natural conditions\nit can be showed that, there exists other probability space $(\\Omega, \\mathcal{F}, \\mathbb P)$,  a $n-$dimensional dynamical system \n$\\tau: \\mathbb{G} \\times \\Omega \\to \\Omega$ and a measurable function $f: \\mathbb R^n \\times \\Omega \\to \\mathbb R$ satisfying \n\\begin{itemize}\n\\item For all $x \\in \\mathbb R^n$, $k \\in \\mathbb{G}$ and $\\mathbb P-$almost every $\\omega \\in \\Omega$ \n\\begin{equation}\n\\label{Stationary}\n   f(x+k, \\omega)= f(x, \\tau(k) \\omega).\n\\end{equation} \n\n\\item For each $x \\in \\mathbb R^n$ the random variables $g(x,\\cdot)$ and $f(x,\\cdot)$ have the same \nlaw. We recall that, the equality almost surely implies \nequality in law, but the converse is not true. \n\n\\end{itemize} \n\nOne remarks that, the set of stationary functions forms an algebra, and\nalso is stable by limit process. \nFor instance, the product of two\nstationaries functions is a stationary one, and the derivative of a \nstationary function is stationary. \nMoreover, the stationarity concept is the most general extension of the \nnotions of periodicity and almost periodicity for a function to have some \"self-averaging\" behaviour. \n\t\n\\begin{example}\nUnder the conditions of Example \\ref{NDT}, let $F \\! : \\Omega \\! \\to \\mathbb{C}$ be a\nmeasurable function. Then, the function $f \\! :\\! \\mathbb{R}^n \\! \\times \\Omega \\! \\to \\! \\mathbb{C}$, \ndefined by \n$$ \n    f(x,\\omega):= F(\\tau(x)\\omega)\n$$ \nis a stationary function. In fact, considering continuous dynamical systems, \nany stationary function can be written in this way. Therefore, \neven if $f(\\cdot,\\omega)$ is just a measurable function, it makes sense to \nwrite, for instance, $f(0,\\cdot)$ due to the stationary property. \n\n\\end{example}\n\n\\begin{example}\n\\label{8563249tyudh}\nUnder the conditions of Example \\ref{EXTJING}, we take $m= 1$, and\nconsider the following functions, $\\varphi_0 = 0$ and $\\varphi_1$ be a Lipschitz vector field,  \nsuch that, \n$\\varphi_1$ is periodic, ${\\rm supp} \\, \\varphi_1 \\subset (0,1)^n$. \nConsequently, the function \n\\begin{equation*}\n    f(y,\\omega) := \\varphi_{\\omega({\\lfloor y \\rfloor})} (y), \\;\\; (y,\\omega) \\in \\mathbb{R}^n \\times \\! \\Omega\n\\end{equation*}\nsatisfies,  ${ f(y,\\cdot) }$ is ${ \\mathscr{F} }$-measurable, ${ f(\\cdot,\\omega) }$ is continuous, and \nfor each  $k \\in \\mathbb{Z}^n$, \n$$\n    f(y+k,\\omega)= f(y,\\tau(k)\\omega).\n$$\nTherefore, ${ f }$ is a stationary function. \n\\end{example}\n\n\\medskip\nNow, we present the precise definition of the stochastic deformation as presented in \\cite{AndradeNevesSilva}.\n\\begin{definition}\n\\label{GradPhiStationary}\nA mapping $\\Phi: \\mathbb R^n \\times \\Omega \\to \\mathbb R^n, (y,\\omega) \\mapsto z= \\Phi(y,\\omega)$, is called a stochastic deformation (for short $\\Phi_\\omega$), when satisfies:\n\\begin{itemize}\n\\item[i)] For $\\mathbb{P}-$almost every $\\omega \\in \\Omega$, $\\Phi(\\cdot,\\omega)$ is a bi--Lipschitz diffeomorphism.\n\n\\item[ii)] There exists $\\nu> 0$, such that\n$$\n\\underset{\\omega \\in \\Omega, \\, y \\in \\mathbb R^n}{\\rm ess \\, inf} \n\\big({\\rm det} \\big(\\nabla \\Phi(y,\\omega)\\big)\\big) \\geq \\nu.\n$$\n\\item[iii)] There exists a $M> 0$, such that\n$$\n \\underset{\\omega \\in \\Omega, \\, y \\in \\mathbb R^n}{\\rm ess \\, sup}\\big(|\\nabla \\Phi(y,\\omega)|\\big) \\leq M< \\infty.\n$$\n\\item[iv)]\nThe gradient of $\\Phi$, i.e. $\\nabla\\Phi(y,\\omega)$, is stationary in the sense~\\eqref{Stationary}.\n\\end{itemize}\n\\end{definition}\n\t\nHere, we first recall from  \\cite{AndradeNevesSilva} a general example of \nstochastic deformations $\\Phi: \\mathbb R^n \\times \\Omega \\to \\mathbb R^n$ associated to\na dynamical system $T: \\mathbb R^n \\times \\Omega \\to \\Omega$, where the sample\nspace $\\Omega$ is arbitrary. Then, following the idea of Example \\ref{8563249tyudh},\nwe present an example of stochastic deformation $\\Phi: \\mathbb R^n \\times \\Omega \\to \\mathbb R^n$ associated to\na dynamical system $T: \\mathbb Z^n \\times \\Omega \\to \\Omega$, where $\\Omega$ is prescribed. \n\nLet $(\\Omega_i, \\mathcal{F}_i, \\mathbb P_i)_{i=1}^n$ be \nprobability spaces, and $f_i:\\Omega_i\\to\\mathbb R$ be a measurable function, such that \n$0<c_0\\le f_i(\\omega)\\le c_1$ for a.e. $\\omega\\in\\Omega_i$, for $i=1,\\ldots,n$.\nLet $T_i:\\mathbb R\\times\\Omega_i\\to\\Omega_i$ be \na $1-$dimensional dynamical system such that the function $f_i\\left(T_i(\\cdot)\\omega\\right)$ \nis continuous.\nThen, we define\n$$\n\\Phi_i(\\lambda,\\omega):=\\text{sgn}(\\lambda)\\int_{\\min{\\{\\lambda,0\\}}}^{\\max{\\{\\lambda,0}\\}}\nf_i\\left(T_i(s)\\omega\\right)\\,ds.\n$$\nThen, $\\Phi_i:\\mathbb R\\times\\Omega_i\\to\\mathbb R$ is not a stationary \nfunction and verifies all conditions of the Definition \\ref{GradPhiStationary}. Finally, define the \nprobability space $(\\Omega, \\mathcal{F}, \\mathbb P)$, where $\\Omega:=\\otimes_{i=1}^n\\Omega_i$, \n$\\mathcal{F}:=\\otimes_{i=1}^n\\mathcal{F}_i$ and $\\mathbb P:=\\otimes_{i=1}^n\\mathbb P_i$ and the following \n$n-$dimensional dynamical system $T:\\mathbb R^n\\times\\Omega\\to\\Omega$\n$$\n   T(x,\\omega):=\\big(T_1(x_1,\\omega_1),\\ldots,T_n(x_n,\\omega_n)\\big),\n$$\nwhere we denote $x=(x_1,\\cdots,x_n)$ and $\\omega=(\\omega_1,\\ldots,\\omega_n)$. Thus, the function \n$\\Phi(x,\\omega)=(\\Phi_1(x_1,\\omega_1),\\ldots,\\Phi_n(x_n,\\omega_n))$ fulfills the conditions \nof Definition \\ref{GradPhiStationary}. \nMoreover, for any orthogonal \n$n \\times n-$matrix $Q$, with $\\det Q= 1$, \nthe mapping $Q \\Phi$ is also a stochastic deformation,\nwhich gradient is not necessary diagonal.\n\n\\medskip\nNow, let us present a second example of stochastic deformations. \n\\begin{example}\n\\label{EXAMPLE8}\nUnder the conditions of Example \\ref{8563249tyudh}, let us consider (for $\\eta> 0$)\nthe following map \n$$ \n    \\Phi(y,\\omega):= y + \\eta \\, \\varphi_{\\omega({\\lfloor y \\rfloor})} (y), \\;\\; (y,\\omega) \\in \\mathbb{R}^n \\times \\Omega.\n$$\nThen,  $\\nabla_{\\!\\! y} \\Phi(y,\\omega) = I_{\\mathbb{R}^{n\\times n}} + \\eta \\, \\nabla \\varphi_{\\omega({\\lfloor y \\rfloor})} (y)$,\nand for $\\eta$ sufficiently small all the conditions in the Definition \\ref{GradPhiStationary} are satisfied.\nThen, ${ \\Phi }$ is a stochastic deformation.\n\\end{example}\n\n\\bigskip\nGiven a stochastic deformation $\\Phi$, \nlet us consider the following spaces\n\\begin{equation}\n\t\\mathcal{L}_\\Phi := {\\big\\{F(z,\\omega)= f( \\Phi^{-1} (z, \\omega), \\omega); f \\in L^2_{\\rm loc}(\\mathbb{R}^n; L^2(\\Omega)) \\;\\; \\text{stationary} \\big\\}}\n\\end{equation}\nand\n\\begin{equation}\n\\label{SPACEHPHI}\n\t\t\\mathcal{H}_\\Phi := {\\big\\{F(z,\\omega)= f( \\Phi^{-1} (z, \\omega), \\omega); \\; f\\in H^1_{\\rm loc}(\\mathbb{R}^n; L^2(\\Omega)) \\;\\; \\text{stationary} \\big\\}}\n\\end{equation}\nwhich are Hilbert spaces, endowed respectively with the inner products  \n$$\n\\begin{aligned}\n  {\\langle F, G \\rangle}_{\\mathcal{L}_\\Phi}&:= \\int_\\Omega \\int_{\\Phi([0,1)^n,\\omega)} \\!\\! F(z, \\omega) \\, \\overline{ G(z, \\omega) } \\, dz \\, d\\mathbb{P}(\\omega),\n\\\\[5pt]\n{\\langle F, G \\rangle}_{\\mathcal{H}_\\Phi}&:= \\int_\\Omega \\int_{\\Phi([0,1)^n,\\omega)} \\!\\! F(z, \\omega) \\, \\overline{ G(z, \\omega) } \\, dz \\, d\\mathbb{P}(\\omega)\n\\\\\n&\\; \\quad +\\int_\\Omega \\int_{\\Phi([0,1)^n,\\omega)} \\!\\! \\nabla_{\\!\\! z} F(z, \\omega) \\cdot \\overline{ \\nabla_{\\!\\! z} G(z, \\omega) } \\, dz \\, d\\mathbb{P}(\\omega). \n\\end{aligned}\n$$\n\\begin{remark}\n\\label{REMFPHI}\nUnder the above notations, \nwhen $\\Phi= Id$ we denote $\\mathcal{L}_\\Phi$ and $\\mathcal{H}_\\Phi$ by  $\\mathcal{L}$ and $\\mathcal{H}$ respectively.\nMoroever, a function $F \\in \\clg{H}_\\Phi$ if, and only if, $F \\circ \\Phi \\in \\clg{H}$, and \nthere exist constants $C_1, C_2> 0$, such that \n$$\n   C_1 \\|F \\circ \\Phi \\|_{\\clg{H}} \\leq \\|F \\|_{\\clg{H}_\\Phi} \\leq C_2 \\|F \\circ \\Phi \\|_{\\clg{H}}.\n$$\nAnalogously, $F \\in \\clg{L}_\\Phi$ if, and only if, $F \\circ \\Phi \\in \\clg{L}$, and \nthere exist constants $C_1, C_2> 0$, such that \n$$\n   C_1 \\|F \\circ \\Phi \\|_{\\clg{L}} \\leq \\|F \\|_{\\clg{L}_\\Phi} \\leq C_2 \\|F \\circ \\Phi \\|_{\\clg{L}}.\n$$  \nIndeed, let us show the former equivalence.\nApplying a change of variables, we obtain \n$$\n\\begin{aligned}\n    \\|F\\|^2_{\\clg{H}_\\Phi}&= \\int_\\Omega \\int_{\\Phi([0,1)^n,\\omega)} \\!\\! |F(z, \\omega)|^2 \\, dz \\, d\\mathbb{P}(\\omega)\n   +\\int_\\Omega \\int_{\\Phi(\\mathsf{Y},\\omega)} \\!\\! |\\nabla_{\\!\\! z} F(z, \\omega)|^2 \\, dz \\, d\\mathbb{P}(\\omega)\n\\\\[5pt]   \n   &= \\int_\\Omega \\int_{[0,1)^n} \\!\\! |f(y, \\omega)|^2  \\det [\\nabla \\Phi(y,\\omega)] \\,  dy \\, d\\mathbb{P}(\\omega)\n\\\\[5pt]   \n   &\\quad +\\int_\\Omega \\int_{[0,1)^n} \\!\\! | [\\nabla \\Phi(y,\\omega)]^{-1} \\nabla_{\\!\\! z} f(y, \\omega)|^2  \\det [\\nabla \\Phi(y,\\omega)] \\, dy \\, d\\mathbb{P}(\\omega).\n\\end{aligned}\n$$\nThe equivalence follows from the properties of the stochastic deformation $\\Phi$. \n\\end{remark}\n\n\n\\subsubsection{Ergodic theorems}\n\\label{ErgThm}\n\nWe begin this section with the concept of mean value, which is \nin connection with the notion of stationarity. \nA function $f \\in L^1_{\\loc}(\\mathbb R^n)$ is said to possess a mean value if the \nsequence $\\{f(\\cdot\/\\varepsilon){\\}}_{\\varepsilon>0}$ converges in the duality with $L^{\\infty}$ and compactly supported \nfunctions to a constant $M(f)$. This convergence is equivalent to\n\\begin{equation}\n\\label{MeanValue}\n\\lim_{t\\to\\infty}\\frac1{t^n|A|}\\int_{A_t}f(x)\\,dx=M(f),\n\\end{equation}\nwhere $A_t:=\\{x\\in\\mathbb R^n\\,:\\, t^{-1}x\\in A\\}$, for $t>0$ and any $A \\subset \\mathbb R^n$, with $|A| \\ne0$.\n\n\n\\begin{remark}\n\\label{REMERG}\nUnless otherwise stated, we assume that the dynamical system $\\tau: \\mathbb{G} \\times \\Omega\\to\\Omega$ is ergodic \nand we will also use the notation \n$$\n   \\Medint_{\\mathbb R^n} f(x) \\ dx \\quad \\text{for $M(f)$}.\n$$\n\\end{remark}\n\nNow, we state the result due to Birkhoff, which connects all the notions \nconsidered before, see \\cite{Krengel}. \n\n\\begin{theorem}[Birkhoff Ergodic Theorem]\\label{Birkhoff}\nLet $f \\in L^1_\\loc(\\mathbb R^n; L^1(\\Omega))$, $($also $f \\in L^\\infty(\\mathbb{R}^n; L^1(\\Omega)) )$, be a stationary random variable. \nThen, for almost every $\\widetilde{\\omega} \\in \\Omega$ the function \n$f(\\cdot,\\widetilde{\\omega})$ possesses a mean value in the sense of~\\eqref{MeanValue}. Moreover, the mean value \n$M\\left(f(\\cdot,\\widetilde{\\omega})\\right)$ as a function of $\\widetilde{\\omega} \\in\\Omega$ satisfies\nfor almost every $\\widetilde{\\omega} \\in \\Omega$: \n\n\\smallskip\ni) Discrete case (i.e. $\\tau: \\mathbb Z^n \\times \\Omega \\to \\Omega$);\n$$\n    \\Medint_{\\mathbb R^n} f(x,\\widetilde{\\omega}) \\ dx= \n   \\mathbb{E} \\left[\\int_{[0,1)^n}  f(y,\\cdot)\\, dy\\right].\n$$\n\nii) Continuous case (i.e. $\\tau: \\mathbb R^n \\times \\Omega \\to \\Omega$);\n$$\n    \\Medint_{\\mathbb R^n} f(x,\\widetilde{\\omega}) \\ dx=  \\mathbb{E}\\left[ f(0,\\cdot) \\right].\n$$\n\\end{theorem}\n\n\\medskip\nThe following lemma shows that, the Birkhoff Ergodic Theorem holds if a stationary function is composed \nwith a stochastic deformation. \n\\begin{lemma}\\label{phi2}\nLet $\\Phi$ be a stochastic deformation and $f \\in L^{\\infty}_\\loc(\\mathbb R^n; L^1(\\Omega))$ be a stationary random variable in the \nsense~\\eqref{Stationary}. Then for almost $\\widetilde{\\omega} \\in \\Omega$ the function \n$f\\left(\\Phi^{-1}(\\cdot,\\widetilde{\\omega}),\\widetilde{\\omega}\\right)$ possesses a mean value \nin the sense of~\\eqref{MeanValue} and satisfies: \n\n\\smallskip\ni) Discrete case;\n$$\n\\text{$\\Medint_{\\mathbb R^n}f\\left(\\Phi^{-1}(z,\\widetilde{\\omega}),\\widetilde{\\omega}\\right)\\,dz\n= \\frac{\\mathbb{E}\\left[\\int_{\\Phi([0,1)^n, \\cdot)} f {\\left( \\Phi^{-1}\\left( z, \\cdot \\right), \\cdot \\right)} \\, dz \\right]}\n{\\det\\left(\\mathbb{E}\\left[\\int_{[0,1)^n} \\nabla_{\\!\\! y} \\Phi(y,\\cdot) \\, dy \\right]\\right)}$\n\\quad for a.a. $\\widetilde{\\omega} \\in \\Omega$}.\n$$\n\nii) Continuous case; \n$$\n\\text{$\\Medint_{\\mathbb R^n}f\\left(\\Phi^{-1}(z,\\widetilde{\\omega}),\\widetilde{\\omega}\\right)\\,dz\n= \\frac{\\mathbb{E}\\left[f(0,\\cdot)\\det\\left(\\nabla\\Phi(0,\\cdot)\\right)\\right]}\n{\\det\\left(\\mathbb{E}\\left[\\nabla \\Phi(0,\\cdot)\\right]\\right)}$\n\\qquad for a.a. $\\widetilde{\\omega} \\in \\Omega$}.\n$$\n\\end{lemma}\n\n\\begin{proof}\nSee Blanc, Le Bris, Lions \\cite{BlancLeBrisLions1}, also\nAndrade, Neves, Silva \\cite{AndradeNevesSilva}.\n\\end{proof}\n\n\\subsubsection{Analysis of stationary functions}\n\nIn the rest of this paper, unless otherwise explicitly stated, we\nassume discrete dynamical systems and therefore, stationary \nfunctions are considered in this discrete sense.\n\n\\medskip\nWe begin the analysis of stationary functions with the concept of realization.\n\\begin{definition}\nLet $f: \\mathbb{R}^n \\! \\times \\! \\Omega \\to \\mathbb{R}$ be a stationary function. \nFor $\\omega \\in \\Omega$ fixed, the function $f(\\cdot, \\omega)$ is called a realization\nof $f$. \n\\end{definition}\nDue to Theorem \\ref{Birkhoff}, almost every realization \n$f(\\cdot,\\omega)$ possesses a mean value in the sense of~\\eqref{MeanValue}.\nOn the other hand, if $f$ is a stationary function, then the mapping \n$$\n   y \\in \\mathbb{R}^n \\mapsto \\int_\\Omega f(y, \\omega) \\, d\\mathbb{P}(\\omega) \n$$\nis a periodic function. \n\n\\medskip\nIn fact, it is enough to consider the realizations to study some properties \nof stationary functions. For instance, the following theorem will be used \nmore than once through this paper.  \n\\begin{theorem}\n\\label{987987789879879879}\nFor $p> 1$, let $u,v \\in L^1_{\\rm loc}(\\mathbb{R}^n; L^p(\\Omega))$\nbe stationary functions. Then, for any $i \\in \\{1,\\ldots,n \\}$ fixed, the following sentences are equivalent: \n\\begin{equation}\n\\label{837648726963874}\n(A) \\quad  \\int_{[0,1)^n} \\int_\\Omega u(y,\\omega) \\frac{\\partial {\\zeta}}{\\partial y_i} (y, \\omega) \\, d\\mathbb{P}(\\omega) \\, dy \n      = - \\int_{[0,1)^n} \\int_\\Omega v(y,\\omega) \\, {\\zeta}(y,\\omega) \\, d\\mathbb{P} \\, dy, \\hspace{20pt}\n\\end{equation}\nfor each stationary function $\\zeta \\in C^1( \\mathbb{R}^n; L^q(\\Omega))$,\nwith $1\/p + 1\/q = 1$. \n\\begin{equation}\n\\label{987978978956743}\n(B) \\quad  \\int_{\\mathbb{R}^n} u(y,\\omega) \\frac{\\partial {\\varphi}}{\\partial y_i} (y) \\, dy = - \\int_{\\mathbb{R}^n} v(y,\\omega) \\, {\\varphi}(y) \\, dy,\n\\hspace{87pt}\n\\end{equation}\nfor any $\\varphi \\in C^1_{\\rm c}(\\mathbb{R}^n)$, and almost sure $\\omega \\in \\Omega$.\n\\end{theorem}\n\n\\begin{proof}\n1. First, let us show that $(A)$ implies $(B)$. To begin, \ngiven $\\gamma \\in \\mathbb{R}^n$, there exists a\n$\\mathscr{F}$-measurable set $N_\\gamma$ such that, $\\mathbb{P}(N_\\gamma)=0$ \nand \n$$\n    \\int_{\\mathbb{R}^n} u(y,\\omega) \\, \\frac{\\partial {\\varphi}}{\\partial y_i} (y) \\, dy \n   = - \\int_{\\mathbb{R}^n} v(y,\\omega) \\, {\\varphi}(y) \\, dy, \n$$\nfor each $\\varphi \\in C^1_{\\rm c}((0,1)^n + \\gamma)$ and $\\omega \\in \\Omega \\setminus N_\\gamma$.\nIndeed, for $\\varphi \\in C^1_{\\rm c}((0,1)^n + \\gamma)$ and ${ \\rho \\in L^q(\\Omega) }$, \nlet us define  $\\zeta_\\gamma : \\mathbb{R}^n \\! \\times \\! \\Omega \\to \\mathbb{R}$, by\n$$   \n    \\zeta_\\gamma (y,\\omega) := \\varphi(y - \\left\\lfloor y-\\gamma \\right\\rfloor) \\rho(\\tau(\\left\\lfloor y-\\gamma \\right\\rfloor)\\omega),\n$$\nwhere $\\tau: \\mathbb Z^n \\times \\Omega \\to \\Omega$ is a (discrete) dynamical system. Then, \n$\\zeta_\\gamma(\\cdot,\\omega)$ is a continuous functions, $\\zeta_\\gamma(y,\\cdot)$ is a\n$\\mathscr{F}$-measurable function, and for each $k\\in\\mathbb{Z}^n$, it follows that\n$$\n    \\zeta_\\gamma(y+k,\\omega)=\\zeta_\\gamma(y,\\tau(k)\\omega).\n$$    \nConsequently, $\\zeta_\\gamma \\in C^1(\\mathbb{R}^n; L^q(\\Omega))$ is a stationary Caratheodory function.  \nMoreover, since $\\left\\lfloor y-\\gamma \\right\\rfloor= 0$ for each $y \\in (0,1)^n + \\gamma$, we have \n\\begin{equation}\n\\label{098760987}\n\t\\zeta_\\gamma (y,\\omega) = \\varphi(y) \\rho(\\omega),\n\\end{equation}\nfor each $(y,\\omega) \\in ((0,1)^n + \\gamma) \\times \\Omega$. Therefore, taking $\\zeta_\\gamma$ as a test function in \n\\eqref{837648726963874}, we obtain\n\\begin{equation}\n\\label{65345ere3}\n     \\int_{[0,1)^n} \\int_\\Omega u(y,\\omega) \\, \\frac{\\partial {\\zeta_\\gamma}}{\\partial y_i} (y, \\omega) \\, d\\mathbb{P}(\\omega) \\, dy \n     = - \\int_{[0,1)^n} \\int_\\Omega v(y,\\omega) \\, {\\zeta_\\gamma}(y,\\omega) \\, d\\mathbb{P}(\\omega) \\, dy.\n\\end{equation}\nDue to the space of stationary functions form an algebra, the functions \n$$\n    y \\mapsto \\int_\\Omega u(y,\\omega) \\, \\frac{\\partial {\\zeta_\\gamma}}{\\partial y_i} (y, \\omega) \\, d\\mathbb{P}(\\omega) \n    \\quad \\text{and} \\quad y \\mapsto \\int_\\Omega v(y,\\omega) \\, {\\zeta_\\gamma}(y,\\omega) \\, d\\mathbb{P}(\\omega) \n$$\nare periodic, and hence translation invariants. Then, we have from \\eqref{65345ere3}\n$$\n   \\int_{(0,1)^n + \\gamma} \\int_\\Omega u \\, \\frac{\\partial {\\varphi}}{\\partial y_i} (y) \\, {\\rho}(\\omega) \\, d\\mathbb{P}(\\omega) \\, dy \n   = - \\int_{(0,1)^n + \\gamma} \\int_\\Omega v \\, {\\varphi}(y){\\rho}(\\omega) \\, d\\mathbb{P}(\\omega) \\, dy,\n$$\t\nwhere we have used \\eqref{098760987}. Applying Fubini's Theorem, it follows that \n$$\t\t\t\t\n\\int_\\Omega {\\left( \\int_{(0,1)^n + \\gamma} u(y,\\omega) \\, \\frac{\\partial {\\varphi}}{\\partial y_i} (y) \\, dy \n+ \\int_{(0,1)^n + \\gamma}  v(y,\\omega) \\, {\\varphi}(y) \\, dy \\right)} {\\rho}(\\omega) \\, d\\mathbb{P}(\\omega)= 0\n$$\nfor each $\\rho \\in L^p(\\Omega)$. \nTherefore, for each $\\varphi  \\in C^1_{\\rm c}((0,1)^n + \\gamma)$ there exists a\nset $N_\\gamma \\in \\mathscr{F}$ with $\\mathbb{P}(N_\\varphi)= 0$, \n(which may depend on $\\varphi$), such that, \nfor each $\\omega \\in \\Omega \\setminus N_\\gamma$ we have \n$$\t\t\t\t\n\\int_{(0,1)^n + \\gamma} u(y,\\omega) \\, \\frac{\\partial {\\varphi}}{\\partial y_i} (y) \\, dy \n= - \\int_{(0,1)^n + \\gamma}  v(y,\\omega) \\, {\\varphi}(y) \\, dy.\n$$\nFrom a standard argument, we may remove the dependence on \n$N_\\gamma$ of the test function $\\varphi$. \n\n\\medskip\n2. Finally, to pass from $\\varphi \\in C^1_c((0,1)^n + \\gamma)$ to the case where $\\varphi \\in C^1_{\\rm c}(\\mathbb{R}^n)$,\nwe are going to use a standard procedure of partition of unity. We made it here, to become clear the argument in \nour case. Given $\\varphi \\in C^1_{\\rm c}(\\mathbb{R}^n)$, since ${\\rm supp} \\, \\varphi$ is a compact set, there exists \n$\\left\\{ \\gamma_j \\right\\}_{j = 1}^m$ a finite subset of $\\mathbb{R}^n$, such that \n$$\n   {\\rm supp} \\, \\varphi \\subset \\bigcup_{j = 1}^m \\left((0,1)^n + \\gamma_j \\right).\n$$\nThen, we consider a partition of unity $\\{\\theta_j\\}_{j=0}^m$ subordinated to this open covering, \nthat is to say\n\\begin{itemize}\n\\item[i)] $\\theta_j \\in C^1_c(\\mathbb R^n)$, \\quad $0\\leqslant \\theta_j \\leqslant 1$, \\quad $j=0, \\ldots, m$,\n\\item[ii)] ${ \\sum_{j=0}^m \\theta_j(y) = 1 }$, \\quad for all $y \\in \\mathbb{R}^n$, \n\\item[iii)] ${\\rm supp} \\, \\theta_j \\subset (0,1)^n + \\gamma_j$, \\quad $j = 1,\\ldots,m$, \n\\quad and \\quad  ${\\rm supp} \\, \\theta_0 \\subset \\mathbb R^n \\setminus {\\rm supp}\\, \\varphi$. \n\\end{itemize}\nSince $\\varphi= 0$ on the support of $\\theta_0$, it follows that, for each $y \\in \\mathbb R^n$\n\\begin{equation}\n\\label{PARTUNIT}\n   \\varphi(y)= \\varphi(y) \\sum_{i=1}^m \\theta_i(y)= \\sum_{i=1}^m (\\varphi \\theta_i)(y).  \n\\end{equation}\nMoreover, from item 1, there exist sets \n$N_{\\gamma_1}, \\ldots, N_{\\gamma_m} \\in \\mathscr{F}$ with $\\mathbb{P}(N_{\\gamma_j})= 0$, \nfor any $j \\in \\{1,\\ldots,m\\}$, such that\n$$\n    \\int_{\\mathbb{R}^n} u(y,\\omega) \\, \\frac{\\partial ({\\varphi \\theta_j}) }{\\partial y_i}  \\, dy \n    = - \\int_{\\mathbb{R}^n} v(y,\\omega) \\, ({\\varphi\\theta_j}) \\, dy,\n$$\nfor each $\\omega \\in \\Omega\\setminus N_{\\gamma_j}$. To follow, we define \n$N:= \\bigcup_{j=1}^m N_{\\gamma_j}$ (which may depend on $\\varphi$), then $\\mathbb{P}(N)= 0$ and\nsumming from 1 to $m$, we obtain from the above equation \n$$\n   \\sum_{j= 1}^m \\int_{\\mathbb{R}^n} u(y,\\omega) \\, \\frac{\\partial ({\\varphi \\theta_j})(y) }{\\partial y_i}  \\, dy\n   = - \\sum_{j= 1}^m \\int_{\\mathbb{R}^n} v(y,\\omega) \\, ({\\varphi\\theta_j})(j) \\, dy, \n$$\nfor each $\\omega \\in \\Omega\\setminus N$. Therefore, since the above sum is finite and using \\eqref{PARTUNIT},\nwe obtain \t\t\t\t\n$$\n    \\int_{\\mathbb{R}^n} u(y,\\omega) \\, \\frac{\\partial {\\varphi(y)}}{\\partial y_i} \\, dy \n    = - \\int_{\\mathbb{R}^n} v(y,\\omega) \\, \\varphi(y) \\, dy.\n$$\nAgain, due to a standard argument, we may remove the dependence on \n$N$ with the test function $\\varphi$. Consequently, we have obtained \\eqref{987978978956743}, more precisely \nsentence $(B)$. \n\t\n\\medskip\n3. Now, let us show sentence $(A)$ from $(B)$. For each $\\ell\\in \\mathbb{N}$, \nwe define the set  ${Q}_\\ell:= (-\\ell,\\ell)^n$ and the function\n$\\chi_\\ell \\in C^1_c(\\mathbb{R}^n)$, such that \n$$\n\\text{$\\chi_\\ell \\equiv 1$ in ${Q}_\\ell$, \n$\\chi_\\ell \\equiv 0$ in $\\mathbb{R}^n \\setminus {Q}_{\\ell+1}$, and \n${ \\Vert \\nabla \\chi_\\ell \\Vert_{\\infty} \\leqslant 2 }$.}\n$$ \nThen, given $\\zeta \\in C^1(\\mathbb{R}^n; L^q(\\Omega))$ and $i \\in \\{1,\\ldots,n\\}$, \nwe consider $\\zeta(\\cdotp, \\omega) \\chi_\\ell$, (for $\\ell \\in \\mathbb{N}$ and\n$\\omega \\in \\Omega$ fixed), as test function in \n\\eqref{987978978956743}, that is \n$$\n   \\int_{\\mathbb{R}^n} u(y, \\omega) \\, \\frac{\\partial}{\\partial y_i} {\\left( {\\zeta}(y, \\omega)\\chi_\\ell(y) \\right)} \\, dy \n   = - \\int_{\\mathbb{R}^n} v(y, \\omega) \\, { {\\zeta}(y, \\omega) \\chi_\\ell(y)} \\, dy.\n$$\nFrom the definition of $\\chi_\\ell$, and applying the product rule we obtain \n$$\n\\begin{aligned}\n  \\int_{Q_{\\ell + 1}} u(y, \\omega) \\frac{\\partial {\\zeta(y, \\omega)}}{\\partial y_i} \\, \\chi_\\ell(y) \\,  dy \n  &+ \\int_{Q_{\\ell + 1} \\setminus Q_\\ell} u(y, \\omega)  {\\zeta}(y, \\omega) \\, \\frac{\\partial \\chi_\\ell(y)}{\\partial y_i} \\,  dy\n\\\\[5pt]\n&= - \\int_{Q_{\\ell + 1}} v(y, \\omega) \\, { {\\zeta}(y, \\omega) \\chi_\\ell(y) } \\, dy,\n\\end{aligned}\n$$\nor conveniently using that $Q_{\\ell + 1} = Q_{\\ell} \\cup (Q_{\\ell + 1} \\setminus Q_{\\ell})$, we have \n\\begin{equation}\n\\label{y67676766798765}\n\\begin{aligned}\n   \\int_{Q_\\ell} u(y, \\omega) \\, \\frac{\\partial {\\zeta}}{\\partial y_i}(y, \\omega) \\, dy \n   &+ \\int_{Q_\\ell} v(y, \\omega) \\, {\\zeta}(y, \\omega) \\, dy\n   \\\\[5pt]\n   &= - \\int_{Q_{\\ell + 1} \\setminus Q_\\ell} u(y, \\omega) \\, \\frac{\\partial {\\zeta}}{\\partial y_i}(y, \\omega)  \\, \\chi_\\ell(y) \\, dy \n   \\\\[5pt]\n   &\\quad -\\int_{Q_{\\ell + 1} \\setminus Q_\\ell} u(y, \\omega) \\, {\\zeta}(y, \\omega) \\, \\frac{\\partial \\chi_\\ell(y)}{\\partial y_i} \\, dy \n   \\\\[5pt]\n   &\\quad -\\int_{Q_{\\ell + 1} \\setminus Q_\\ell} v(y, \\omega) \\, {\\zeta}(y, \\omega) \\, \\chi_\\ell(y) \\, dy\n   \\\\[5pt]\n   &= I_1(\\omega) + I_2(\\omega) +I_3(\\omega),\n\\end{aligned}\n\\end{equation}\nwith obvious notation. \n\n\\smallskip\n\\underline {Claim:} For $j= 1,2,3$, \n$$\n\\lim_{\\ell \\to \\infty} \\int_\\Omega \\frac{|I_j(\\omega)|}{\\vert Q_\\ell \\vert} d\\mathbb{P}(\\omega)= 0.\n$$\n\nProof of Claim: Let us show for $j= 2$, that is \n$$\n    \\lim_{\\ell \\to \\infty} \\int_\\Omega \\frac{1}{\\vert Q_\\ell \\vert} {\\Big| \\int_{Q_{\\ell + 1} \\setminus Q_\\ell} \n    u(y, \\omega) \\, {\\zeta}(y, \\omega) \\, \\frac{\\partial \\chi_\\ell(y)}{\\partial y_i} \\, dy \\Big|} d\\mathbb{P}(\\omega) = 0,\n$$\nthe others are similar. Then, applying Fubini's Theorem \n$$\n\\begin{aligned}\n   \\int_\\Omega \\frac{1}{\\vert Q_\\ell \\vert} \\Big| \\int_{Q_{\\ell + 1} \\setminus Q_\\ell}  & u(y, \\omega) \\, {\\zeta}(y, \\omega) \\, \\frac{\\partial \\chi_\\ell(y)}{\\partial y_i} \\, dy \\Big| d\\mathbb{P}(\\omega) \n   \\\\[5pt] \n  &\\leq \\, \\frac{1}{\\vert Q_\\ell \\vert} \\int_{Q_{\\ell + 1} \\setminus Q_\\ell} \\int_\\Omega {\\vert u(\\cdotp, \\omega) \\, {\\zeta}(\\cdotp, \\omega) \\vert} \\,  {\\Vert \\nabla \\chi_\\ell \\Vert}_\\infty \\, d\\mathbb{P} \\, dy \n \\\\[5pt]\n  &\\leq \\, \\frac{2}{\\vert Q_\\ell \\vert} \\int_{Q_{\\ell + 1} \\setminus Q_\\ell} \\int_\\Omega {\\vert u(y, \\omega) \\, {\\zeta}(y, \\omega) \\vert} \\, d\\mathbb{P}(\\omega) \\, dy \n\\\\[5pt]\n  &= \\frac{2 \\, {( (2(\\ell + 1))^n - (2\\ell)^n )}}{(2 \\ell)^n} \\!\\! \\int_{[0,1)^n} \\int_\\Omega {\\vert u(y,  \\omega) \\, {\\zeta}(y, \\omega) \\vert} \\, d\\mathbb{P}(\\omega) \\, dy \n\\\\[5pt]\n  &= \\, {2 \\, { {(( 1 + \\ell^{-1})^n - 1)} }} \\int_{[0,1)^n} \\int_\\Omega {\\vert u(y, \\omega) \\, {\\zeta}(y, \\omega) \\vert} \\, d\\mathbb{P}(\\omega) \\, dy,\n\\end{aligned}\n$$\nfrom which, passing to the limit as $\\ell \\to 0$, follows the claim.  \n\n\\medskip\nThen, dividing equation \\eqref{y67676766798765} by $\\vert Q_\\ell \\vert$ and integrating in $\\Omega$, we obtain  \n$$\n   \\liminf_{\\ell \\to \\infty} \\int_\\Omega {\\Big| \\frac{1}{\\vert Q_\\ell \\vert} \\int_{Q_\\ell} (u \\, \\frac{\\partial {\\zeta}}{\\partial y_i})(y, \\omega) dy \n   + \\frac{1}{\\vert Q_\\ell \\vert} \\int_{Q_\\ell} (v \\, {\\zeta})(y, \\omega) dy \\Big|} d\\mathbb{P}(\\omega)= 0,\n$$   \nand applying Fato's Lemma\n$$\n\\int_\\Omega \\liminf_{\\ell \\to \\infty} \\int_\\Omega {\\Big| \\frac{1}{\\vert Q_\\ell \\vert} \\int_{Q_\\ell} (u \\, \\frac{\\partial {\\zeta}}{\\partial y_i})(y, \\omega) dy \n+ \\frac{1}{\\vert Q_\\ell \\vert} \\int_{Q_\\ell} (v \\, {\\zeta})(y, \\omega) \\, dy \\Big|} d\\mathbb{P}(\\omega)= 0.\n$$\nTherefore, there exists a  $\\mathscr{F}$-measurable set $\\widetilde{\\Omega} \\subset \\Omega$ of full measure, such that,\nfor each $\\omega \\in \\widetilde{\\Omega}$, we have\n$$\n  \\liminf_{\\ell \\to \\infty} {\\Big|\\frac{1}{\\vert Q_\\ell \\vert} \\int_{Q_\\ell} u(y, \\omega) \\, \\frac{\\partial {\\zeta}}{\\partial y_i}(y, \\omega) \\, dy \n  + \\frac{1}{\\vert Q_\\ell \\vert} \\int_{Q_\\ell} v(y, \\omega) \\, {\\zeta}(y, \\omega) \\, dy \\Big|}= 0.\n$$\nThen, applying Theorem \\ref{Birkhoff} and from equation \\eqref{MeanValue}, it follows that\n$$\n   \\int_{[0,1)^n} \\int_\\Omega u(y, \\omega) \\, \\frac{\\partial {\\zeta}}{\\partial y_i}(y, \\omega) \\, d\\mathbb{P}(\\omega) \\, dy \n   = -\\int_{[0,1)^n} \\int_\\Omega v(y, \\omega) \\, {\\zeta}(y, \\omega) \\, d\\mathbb{P}(\\omega) \\, dy,\n$$\nwhich finish the proof of the theorem.\n\\end{proof}\t\t\n\nSimilarly to the above theorem, we have the characterization of weak derivatives of stationary functions \ncomposed with stochastic deformations, given by the following \n\n\\begin{theorem}\n\\label{648235azwsxqdgfd}\nLet $u,v \\in L^1_{\\rm loc}(\\mathbb{R}^n; L^p(\\Omega))$\nbe stationary functions, $(p> 1)$. Then, for any $i \\in \\{1,\\ldots,n \\}$ fixed, the following sentences are equivalent: \n\\begin{multline*}\n(A) \\quad \\int_\\Omega \\int_{\\Phi([0,1)^n, \\omega)} u {\\left( \\Phi^{-1}\\left( z, \\omega \\right), \\omega \\right)} \\, { \\frac{ \\partial {\\left( \\zeta{\\left( \\Phi^{-1}(z,\\omega),\\omega \\right)} \\right)}}{\\partial z_k} } \\, dz \\, d\\mathbb{P}(\\omega) \n\\\\[5pt]\n= - \\int_\\Omega \\int_{\\Phi([0,1)^n, \\omega)} v {\\left( \\Phi^{-1}\\left( z, \\omega \\right), \\omega \\right)} \\, {\\zeta{\\left( \\Phi^{-1}(z,\\omega),\\omega \\right)}} \\, dz \\, d\\mathbb{P}(\\omega),\n\\end{multline*}\nfor each stationary function $\\zeta \\in C^1( \\mathbb{R}^n; L^q(\\Omega))$,\nwith $1\/p + 1\/q = 1$. \n\\begin{equation*}\n(B) \\quad \\int_{\\mathbb{R}^n} u {\\left( \\Phi^{-1}\\left( z, \\omega \\right), \\omega \\right)} \\, { \\frac{\\partial \\varphi}{\\partial z_k}(z) } \\, dz \n= - \\int_{\\mathbb{R}^n} v {\\left( \\Phi^{-1}\\left( z, \\omega \\right), \\omega \\right)} \\, {\\varphi(z)} \\, dz, \\hspace{12pt}\n\\end{equation*}\nfor any $\\varphi \\in C^1_{\\rm c}(\\mathbb{R}^n)$, and almost sure $\\omega \\in \\Omega$.\n\\end{theorem}\n\\begin{proof}\nThe proof follows the same lines as in the proof of Theorem \\ref{987987789879879879} after the change of variables\n$y= \\Phi^{-1}(z,\\omega)$.\n\\end{proof}\n\n\n\n\\subsection{$\\Phi_\\omega-$Two-scale Convergence}\n\\label{pud63656bg254v2v5}\n\nIn this subsection, we shall consider the two-scale convergence in a stochastic setting that is beyond of the classical stationary \nergodic setting. The classical concept of two-scale convergence was introduced by Nguetseng~\\cite{Nguetseng} and futher developed by Allaire~\\cite{Allaire} \nto deal with periodic problems. \n\n\\medskip\nThe notion of two-scale convergence has been successfully extended to non-periodic settings in several papers as in~\\cite{FridSilvaVersieux,DiazGayte} in the ergodic algebra \nsetting and in~\\cite{BourgeatMikelicWright} in the stochastic setting. The main difference here with the earlier studies is \nthat the test functions used here are random perturbations accomplished by stochastic diffeomorphisms of stationary \nfunctions. The main difficulty brought by this kind of test function is the lack of the stationarity property (see \\cite{AndradeNevesSilva} for a deep discussion about that) which makes us \nunable to use the results described in~\\cite{BourgeatMikelicWright} and the lack of a compatible topology with the probability space considered. This is overcome by using a \ncompactification argument that preserves the ergodic nature of the setting involved. For this, we will make use of the following lemma, whose simple proof can be found in~\\cite{AF}.\n\n\\begin{lemma}\\label{TopologicalLemma}\nLet $X_1,X_2$ be compact spaces, $R_1$ a dense subset of $X_1$ and $W:R_1\\to X_2$. Suppose that for all $g\\in C(X_2)$ the function $g\\circ W$ is the restriction \nto $R_1$ of some (unique) $g_1\\in C(X_1)$. Then $W$ can be uniquely extended to a continuous mapping $\\underline{W}:X_1\\to X_2$. Further, suppose in addition that \n$R_2$ is a dense set of $X_2$, $W$ is a bijection from $R_1$ onto $R_2$ and for all $f\\in C(X_1)$, $f\\circ W^{-1}$ is the restriction to $R_2$ of some (unique) \n$f_2\\in C(X_2)$. Then, $W$ can be uniquely extended to a homeomorphism $\\underline{W}:X_1\\to X_2$. \n\\end{lemma}\n\nNow, we can prove the following result.\n\n\\begin{theorem}\n\\label{Compacification}\nLet $\\mathbb {S}\\subset L^{\\infty}(\\mathbb R^n\\times \\Omega)$ be a countable set of stationary functions. Then there exists a compact (separable) topological space \n$\\widetilde{\\Omega}$ and one-to-one function $\\delta:\\Omega\\to \\widetilde{\\Omega}$ with dense image satisfying the following properties: \n\\begin{enumerate}\n\\item[(i)] The probability space $\\Big(\\Omega,\\mathscr{F},\\mathbb{P}\\Big)$ and the ergodic dynamical system $\\tau:\\mathbb{Z}^n\\times \\Omega\\to\\Omega$ acting on it \nextends respectively to a Radon probability space $\\Big(\\widetilde{\\Omega},\\mathscr{B},\\widetilde{\\mathbb{P}}\\Big)$ and to an ergodic dynamical system \n$\\widetilde{\\tau}:\\mathbb{Z}^n\\times \\widetilde{\\Omega}\\to\\widetilde{\\Omega}$.\n\\item[(ii)] The stochastic deformation $\\Phi:\\mathbb R^n\\times\\Omega\\to\\mathbb R^n$ extends to a stochastic deformation \n$\\tilde{\\Phi}:\\mathbb R^n\\times\\widetilde{\\Omega}\\to\\mathbb R^n$ satisfying \n$$\n\\Phi(x,\\omega)=\\tilde{\\Phi}(x,\\delta(\\omega)),\n$$\nfor a.e. $\\omega\\in\\Omega$.\n\\item[(iii)] Any function $f\\in\\mathbb{S}$ extends to a $\\tilde{\\tau}-$stationary function $\\tilde{f}\\in L^{\\infty}(\\widetilde{\\Omega}\\times \\mathbb R^n)$ satisfying \n$$\n\\Medint_{\\mathbb R^n}f\\left(\\Phi^{-1}(z,\\omega),\\omega\\right)\\,dz=\\Medint_{\\mathbb R^n}\\tilde{f}\\left(\\tilde{\\Phi}^{-1}(z,\\delta(\\omega)),\\delta(\\omega)\\right)\\,dz,\n$$\nfor a.e. $\\omega\\in\\Omega$.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}\n1.  Let $\\mathbb{S}$ be the set of the lemma.  Given $f\\in \\mathbb{S}$, define \n$$\nf_j(y,\\omega):=\\int_{\\mathbb R^n}f(y+x,\\omega)\\,\\rho_j(x)\\,dx,\n$$\nwhere $\\rho_j$ is the classical approximation of the identity in $\\mathbb R^n$. Note that for a.e. $y\\in\\mathbb R^n$, we have that $f_j(y,\\cdot)\\to f(y,\\cdot)$ in $L^1(\\Omega)$ as $j\\to\\infty$.  \nDefine $\\mathcal{A}$ as the closed algebra with unity generated by the set \n$$\n\\Big\\{ f_j(y,\\cdot);\\,j\\ge 1,y\\in\\mathbb{Q}^n,f\\in\\mathbb{S}\\Big\\}\\cap \\Big\\{\\partial_j \\Phi_i(y,\\cdot);\\, 1\\le j,i\\le n, y\\in\\mathbb{Q}^n\\Big\\}.\n$$\nSince $[-1,1]$ is a compact set, by the well known Tychonoff's Theorem, the set \n$$\n[-1,1]^{\\mathcal{A}}:=\\Big\\{\\text{the functions $\\gamma:\\mathcal{A}\\to[-1,1]$}\\Big\\}\n$$\nis a compact set in the product topology. Define $\\delta:\\Omega\\to[-1,1]^{\\mathcal{A}}$ by \n$$\n\\delta(\\omega)(g):=\\left\\{\\begin{array}{rc}\n\\frac{g(\\omega)}{\\|g{\\|}_{\\infty}},&\\mbox{if}\\quad g\\neq 0,\\\\\n0,&\\mbox{if}\\quad g=0.\n\\end{array}\\right.\n$$\nWe may assume that the algebra $\\mathcal{A}$ distinguishes between points of $\\Omega$, that is, given any two distinct points $\\omega_1,\\omega_2\\in\\Omega$, there exists \n$g\\in\\mathcal{A}$ such that $g(\\omega_1)\\neq g(\\omega_2)$. In the case that it is not true we may replace $\\Omega$ by its quotient by a trivial equivalence relation, in a standard \nway, and we proceed correspondingly with the $\\sigma-$algebra $\\mathscr{F}$ and with the probability measure $\\mathbb{P}$. Thus, the function $\\delta$ is one-to-one. Define \n$$\n\\widetilde{\\Omega}:=\\overline{\\delta(\\Omega)}.\n$$\nNow, we can see that the set $\\Omega$ inherits all topological features of the compact space $\\widetilde{\\Omega}$ in a natural way which allows us to identify it homeomorphically with \nthe image $\\delta(\\Omega)$.\n\n2.  Define the mapping $i:\\mathcal{A}\\to C(\\delta(\\Omega))$ by \n$$\ni(g)(\\delta(\\omega)):=g(\\omega).\n$$\nWe claim that there exists a continuous function $\\tilde{g}:\\widetilde{\\Omega}\\to \\mathbb R$ such that \n$$\ni(g)=\\tilde{g}\\,\\text{on $\\delta(\\Omega)$}.\n$$\nIn fact, take $g\\in\\mathcal{A}$ and $Y:=\\overline{g(\\Omega)}$. Define the function $f^{*}:C(Y)\\to\\mathcal{A}$ by \n$$\nf^{*}(h):=h\\circ g\\,\\text{(the algebra structure is used!)}\n$$\nHence, we can define $f^{**}:[-1,1]^{\\mathcal{A}}\\to[-1,1]^{C(Y)}$ by \n$$\nf^{**}(h):=h\\circ f^{*}.\n$$\nNote that the function $f^{**}$ is a continuous function. In order to see that, we highlighted that it is known that a function $H$ from a topological space \nto a product space $\\otimes_{\\alpha\\in \\mathcal{I}}X_{\\alpha}$ is continuous if and only if each component $\\pi_{\\alpha}\\circ H:=H_{\\alpha}$ is \ncontinuous. Hence, if $\\alpha\\in C(Y)$ then the projection function $f^{**}_{\\alpha}$ must satisfy \n\\begin{eqnarray*}\n&&f^{**}_{\\alpha}(h):=\\left(\\pi_{\\alpha}\\circ f^{**}\\right)(h)=\\pi_{\\alpha}\\circ\\left(f^{**}(h)\\right)=\\pi_{\\alpha}\\left(h\\circ f^{*}\\right)\\\\\n&&\\qquad=\\left(h\\circ f^{*}\\right)(\\alpha)=h\\left(f^{*}(\\alpha)\\right)=h\\left(\\alpha\\circ g\\right)=\\pi_{\\alpha\\circ g}(h).\n\\end{eqnarray*}\nNow, consider the function $\\tilde{\\delta}:Y\\to [-1,1]^{C(Y)}$ given by \n$$\n\\tilde{\\delta}(y)(h):=\\left\\{\\begin{array}{rc}\n\\frac{h(y)}{\\|h{\\|}_{\\infty}},&\\mbox{if}\\quad h\\neq 0,\\\\\n0,&\\mbox{if}\\quad h=0.\n\\end{array}\\right.\n$$\nSince the algebra $C(Y)$ has the following property: If $F\\subset Y$ is a closed set and $y\\notin F$ then \n$f(y)\\notin \\overline{f(F)}$ for some $f\\in C(Y)$, we can conclude that the function $\\tilde{\\delta}$ is a homeomorphism onto its image. Furthermore, \ngiven $\\omega\\in\\Omega$ we have that $\\left(f^{**}\\circ \\delta\\right)(\\omega)=f^{**}\\left(\\delta(\\omega)\\right)=\\delta(\\omega)\\circ f^{*}$. Hence, if $0\\neq h\\in C(Y)$ \nit follows that \n\\begin{eqnarray*}\n&&\\left(f^{**}\\circ\\delta\\right)\\big(\\omega\\big)(h)=\\left(\\delta(\\omega)\\circ f^{*}\\right)(h)=\\delta(\\omega)\\left(f^{*}(h)\\right)\\\\\n&&\\quad=\\delta(\\omega)\\left(h\\circ g\\right)=\\frac{\\left(h\\circ g\\right) (\\omega)}{\\|h\\circ g {\\|}_{\\infty}}=\\tilde{\\delta}\\left(g(\\omega)\\right)(h).\n\\end{eqnarray*}\nThus, we see that ${\\tilde{\\delta}}^{-1}\\circ f^{**}=i(g)$. Defining $\\tilde{g}:={\\tilde{\\delta}}^{-1}\\circ f^{**}$ we have clearly that \n$i:\\mathcal{A}\\to C(\\widetilde{\\Omega})$ is a one-to-one isometry satisfying $i(g)=\\tilde{g}$ and our claim is proved. Moreover, it is easy to see that $i(\\mathcal{A})$ is \nan algebra of functions over $C(\\widetilde{\\Omega})$ containing the unity. As before, if $i(\\mathcal{A})$ does not separate points of $\\widetilde{\\Omega}$, then we may replace \n$\\widetilde{\\Omega}$ by its quotient $\\widetilde{\\Omega}\/\\sim$, where \n$$\n\\tilde{\\omega_1}\\sim\\tilde{\\omega_2}\\,\\Leftrightarrow\\, \\tilde{g}(\\tilde{\\omega_1})=\\tilde{g}(\\tilde{\\omega_2})\\,\\forall g\\in\\mathcal{A}.\n$$\nTherefore, we can assume that $i(\\mathcal{A})$ separates the points of $\\widetilde{\\Omega}$. Hence, by the Stone's Theorem (see \\cite{Ru2}, p. 162, Theorem 7.3.2) \nwe must have $i(\\mathcal{A})=C(\\widetilde{\\Omega})$.\n\n\\medskip\n3. Define $\\widetilde{\\tau}:\\mathbb{Z}^n\\times\\delta(\\Omega)\\to \\delta(\\Omega)$ by\n$$\n\\widetilde{\\tau}_k\\left(\\delta(\\omega)\\right):=\\delta(\\tau_k\\omega).\n$$\nIt is easy to see that \n$$\n\\widetilde{\\tau}_{k_1+k_2}(\\delta(\\omega))=\\widetilde{\\tau}_{k_1}\\Big(\\widetilde{\\tau}_{k_2}(\\delta(\\omega))\\Big),\n$$\nfor all $k_1,k_2\\in\\mathbb{Z}^n$ and $\\omega\\in\\Omega$. Since $\\tilde{g}\\circ{\\widetilde{\\tau}_k}=\\widetilde{g\\circ{\\tau}_k}$ for all $g\\in\\mathcal{A}$ and \n$k\\in\\mathbb{Z}^n$, the lemma~\\ref{TopologicalLemma} allows us to extend the mapping $\\widetilde{\\tau}_k$ from $\\delta(\\Omega)$ to $\\widetilde{\\Omega}$ satisfying the \ngroup property $\\widetilde{\\tau}_{k_1+k_2}=\\widetilde{\\tau}_{k_1}\\circ{\\widetilde{\\tau}}_{k_2}$. Given a borelian set $\\tilde{A}\\subset{\\widetilde{\\Omega}}$ and defining \n$\\widetilde{\\mathbb{P}}(\\tilde{A}):=\\mathbb{P}(\\delta^{-1}(\\tilde{A}\\cap \\delta(\\Omega)))$, we can deduce that \n$\\widetilde{\\mathbb{P}}\\circ {\\widetilde{\\tau}_k}=\\widetilde{\\mathbb{P}}$. Thus, the mapping $\\widetilde{\\tau}_k$ is an ergodic dynamical system over the Radon probability \nspace $\\Big(\\widetilde{\\Omega},\\mathscr{B},\\widetilde{\\mathbb{P}}\\Big)$. Thus, we have concluded the proof of the item (i). \n\n4. Now, note that for each $\\omega\\in\\widetilde{\\Omega}$ and each integer $j\\ge 1$, the function $f_j(\\cdot,\\omega)$ is uniformly continuous over $\\mathbb{Q}^n$. Hence, \nit can be extended uniquely to a function $\\widetilde{f_j}(\\cdot,\\omega)$ defined in $\\mathbb R^n$ that satisfies   \n$$\n\\limsup_{j,l\\to\\infty}\\int_{[0,1)^n\\times \\widetilde{\\Omega}}|\\widetilde{f_j}(y,\\omega)-\\widetilde{f_l}(y,\\omega)|\\,d\\widetilde{\\mathbb{P}}(\\omega)\\,dy=0.\n$$\nTherefore, there exists a $\\widetilde{\\tau}$-stationary function $\\widetilde{f}\\in L^1_{\\loc}\\left(\\mathbb R^n\\times \\widetilde{\\Omega}\\right)$, such that \n$\\widetilde{f_j}\\to \\widetilde{f}$ as $j\\to\\infty$ in $L^1_{\\loc}(\\mathbb R^n\\times \\widetilde{\\Omega})$. Since $\\| \\widetilde{f_j}{\\|}_{\\infty}\\le \\| {f}{\\|}_{\\infty}$, for all $j\\ge 1$ we have \nthat $\\widetilde{f}\\in L^{\\infty}(\\mathbb R^n\\times \\widetilde{\\Omega})$. In the same way, the stochastic deformation $\\Phi:\\mathbb R^n\\times \\Omega\\to \\mathbb R^n$ extends to a stochastic \ndeformation $\\tilde{\\Phi}:\\mathbb R^n\\times \\widetilde{\\Omega}\\to \\mathbb R^n$ satisfying $\\Phi(y,\\omega)=\\tilde{\\Phi}(y,\\delta(\\omega))$ for all $\\omega\\in\\Omega$ and \n$$\n\\Medint_{\\mathbb R^n}f\\left(\\Phi^{-1}(z,\\omega),\\omega\\right)\\,dz=\\Medint_{\\mathbb R^n}\\tilde{f}\\left(\\tilde{\\Phi}^{-1}(z,\\delta(\\omega)),\\delta(\\omega)\\right)\\,dz,\n$$\nfor a.e. $\\omega\\in\\Omega$. This, completes the proof of the theorem \\eqref{Compacification}.\n\n\n\\end{proof}\n\nIn practice, in our context, the set $\\mathbb{S}$ shall be a countable set generated by the coefficients of our equation $\\eqref{jhjkhkjhkj765675233}$  and by the \neigenfunctions of the spectral equation associated to it. Thus, the Theorem \\eqref{Compacification} allow us to suppose without loss of generality that our probability \nspace $\\Big(\\Omega,\\mathscr{F},\\mathbb{P}\\Big)$ is a separable compact space.  Using the Ergodic Theorem, given a stationary function $f\\in L^{\\infty}(\\mathbb R^n\\times\\Omega)$ \nthere exists a set of full measure $\\Omega_f\\subset \\Omega$ such that \n\\begin{equation}\\label{Compacification1}\n\\Medint_{\\mathbb R^n}f\\left(\\Phi^{-1}(z,\\tilde{\\omega}),\\tilde{\\omega}\\right)\\,dz= c_{\\Phi}^{-1}\\int_{\\Omega}\\int_{\\Phi([0,1)^n,\\omega)}f\\left(\\Phi^{-1}(z,\\omega),\\omega\\right)\\,dz\\,d\\mathbb{P}(\\omega),\n\\end{equation}\nfor almost all $\\tilde{\\omega}\\in{\\Omega}_f$. Due to the separability of the probability compact space $\\Big(\\Omega,\\mathscr{F},\\mathbb{P}\\Big)$, we can find a set \n$\\mathbb{D}\\subset C_b(\\mathbb R^n\\times\\Omega)$ such that:\n\\begin{itemize}\n\\item Each $f\\in\\mathbb{D}$ is a stationary function. \n\\item $\\mathbb{D}$ is a countable and dense set in $C_0\\big([0,1)^n\\times\\Omega\\big)$.\n\\end{itemize}\nIn this case, there exists a set $\\Omega_0\\subset\\Omega$ of full measure such that the equality \\eqref{Compacification1} holds for any $\\tilde{\\omega}\\in\\Omega_0$ and \n$f\\in\\mathbb{D}$.\n\n\\medskip\nNow, we proceed with the definition of the two-scale convergence in this scenario of stochastically deformed. In what follows, the set $O\\subset\\mathbb R^n$ is an open set. \n\\begin{definition}\n\\label{two-scale}\nLet $1< p <\\infty$ and $v_{\\varepsilon}:O\\times\\Omega\\to \\mathbb{C}$ be a sequence such that $v_{\\varepsilon}(\\cdot,\\tilde{\\omega})\\in L^p(O)$. \nThe sequence \n$\\{v_{\\varepsilon}(\\cdot,\\tilde{\\omega}){\\}}_{\\varepsilon>0}$ is said to $\\Phi_\\omega-$two-scale converges to a stationary function $V_{\\tilde{\\omega}}\\in L^p\\left(O\\times [0,1)^n\\times\\Omega\\right)$,\nwhen for a.e. $\\tilde{\\omega}\\in\\Omega$ holds the following\n$$\n\\begin{aligned}\n&\\lim_{\\varepsilon\\to 0}\\int_{O}v_{\\varepsilon}(x,\\tilde{\\omega})\\,\\varphi(x)\\,\\Theta\\left(\\Phi^{-1}\\left(\\frac{x}{\\varepsilon},\\tilde{\\omega}\\right),\\tilde{\\omega}\\right)\\,dx\n\\\\\n&= c_{\\Phi}^{-1}\\int_{O\\times\\Omega}\\int_{\\Phi([0,1)^n,\\omega)}  \\!\\!\\! V_{\\tilde{\\omega}}\\left(x,\\Phi^{-1}\\left(z,\\omega\\right),\\omega\\right)\\,\\varphi(x)\\,\n\\Theta(\\Phi^{-1}(z,\\omega),\\omega)\\,dz\\,d{\\mathbb{P}(\\omega)}\\,dx,\n\\end{aligned}\n$$\nfor all $\\varphi\\in C_c^{\\infty}(O)$ and $\\Theta\\in L^{q}_{\\loc}(\\mathbb R^n\\times\\Omega)$ stationary. Here, $p^{-1}+q^{-1}=1$ and \n$c_{\\Phi}:=\\det\\Big(\\int_{[0,1)^n\\times \\Omega}\\nabla \\Phi(y,\\omega)\\,d{\\mathbb{P}(\\omega)}\\,dy\\Big)$.\n\\end{definition}\n\\begin{remark}\nFrom now on, we shall use the notation \n\\begin{equation*}\n\t\t\tv_{\\varepsilon}(x,\\widetilde{\\omega}) \\; \\xrightharpoonup[\\varepsilon \\to 0]{2-{\\rm s}}\\; V_{\\widetilde{\\omega}} {\\left(x,\\Phi^{-1}(z,\\omega),\\omega \\right)},\n\t\t\\end{equation*}\nto indicate that $v_{\\varepsilon}(\\cdot,\\tilde{\\omega})$ $\\Phi_\\omega-$two-scale converges to $V_{\\tilde{\\omega}}$.\n\\end{remark}\nThe most important result about the two-scale convergence needed in this paper is the following compactness theorem which \ngeneralize the corresponding one for the \ndeterministic case in~\\cite{DiazGayte} (see Theorem 4.8)\nand the corresponding one for the stochastic case in~\\cite{BourgeatMikelicWright} (see Theorem 3.4). \n\\begin{theorem}\n\\label{TwoScale}\nLet $1< p <\\infty$ and $v_{\\varepsilon}:O\\times\\Omega\\to \\mathbb{C}$ be a sequence such that\n$$\n\\sup_{\\varepsilon>0}\\int_{O}|v_{\\varepsilon}(x,\\tilde{\\omega})|^p\\,dx<\\infty,\n$$\nfor almost all $\\tilde{\\omega}\\in\\Omega$. \nThen, for almost all $\\tilde{\\omega}\\in\\Omega_0$, there exists a subsequence $\\{v_{\\varepsilon'}(\\cdot,\\tilde{\\omega}){\\}}_{\\varepsilon'>0}$, which may depend on $\\tilde{\\omega}$, and a \nstationary function $V_{\\tilde{\\omega}}\\in L^p(O\\times [0,1)^n\\times\\Omega)$, such that \n$$\nv_{\\varepsilon}(x,\\widetilde{\\omega}) \\; \\xrightharpoonup[\\varepsilon' \\to 0]{2-{\\rm s}}\\; V_{\\widetilde{\\omega}} {\\left(x,\\Phi^{-1}(z,\\omega),\\omega \\right)}.\n$$\n\\end{theorem}\n\\begin{proof}\n1. We begin by fixing $\\tilde{\\omega}\\in\\Omega_0$. Due to our assumption, there exists ${ c(\\widetilde{\\omega})>0 }$, such that \nfor all $\\varepsilon >0$\n\t\t\\begin{equation*}\n\t\t\t{\\left\\Vert v_\\varepsilon(\\cdot,\\widetilde{\\omega}) \\right\\Vert}_{L^p(O)} \\leqslant c(\\widetilde{\\omega}).\n\t\t\\end{equation*}\nNow, taking $\\phi\\in \\Xi\\times \\mathbb{D}$ with $\\Xi\\subset C_c^{\\infty}(O)$ dense in $L^q(O)$, we have after applying the H\\\"older inequality and the Ergodic Theorem, \n\\begin{multline}\n\\label{6742938tyuer}\n\t\t\t\\underset{\\varepsilon \\to 0}{\\limsup} {\\vert \\int_{O} v_\\varepsilon(x,\\widetilde{\\omega}) \\, { \\phi{\\left( x,\\Phi^{-1}{\\left( \\frac{x}{\\varepsilon}, \\widetilde{\\omega} \\right)}, \\widetilde{\\omega} \\right)} } dx \\vert} \n\t\t\t\\\\\n\t\t       \\leq c(\\widetilde{\\omega}) {\\left[ \\underset{\\varepsilon \\to 0}{\\limsup}\\int_{O} {\\left\\vert \\phi{\\left( x,\\Phi^{-1}{\\left( \\frac{x}{\\varepsilon}, \\widetilde{\\omega} \\right)}, \\widetilde{\\omega} \\right)} \\right\\vert}^q dx \\right]}^{1\/q} \\hspace{100pt} \n\t\t       \\\\\n\t\t\t= c(\\widetilde{\\omega}) {\\left[ c_\\Phi^{-1} \\int_{O} \\int_\\Omega \\int_{\\Phi([0,1)^n,\\omega)} {\\left\\vert \\phi{\\left( x,\\Phi^{-1}(z,\\omega),\\omega \\right)} \\right\\vert}^q dz \\, d\\mathbb{P}(\\omega) \\, dx \\right]}^{1\/q}.\n\t\t\\end{multline}\nThus, the use of the enumerability of the set $\\Xi\\times \\mathbb{D}$ combined with a diagonal argument yields us a subsequence $\\{\\varepsilon'\\}$ (maybe depending of $\\tilde{\\omega}$) \nsuch that the functional $\\mu:\\Xi\\times \\mathbb{D}\\to \\mathbb{C}$ given by \n\\begin{equation}\\label{Two-scale1}\n\t\t\\langle\\mu,\\phi\\rangle:=\t\\lim_{\\varepsilon^{\\prime} \\to 0}\\int_{O} v_{\\varepsilon^{\\prime}}(x,\\widetilde{\\omega}) \\, { \\phi{\\left( x, \\Phi^{-1}{\\left( \\frac{x}{\\varepsilon^{\\prime}}, \\widetilde{\\omega} \\right)}, \\widetilde{\\omega} \\right)} } dx\n\t\t\\end{equation}\nis well-defined and bounded with respect to the norm $\\|\\cdot{\\|}_q$ defined as \n$$\n\\|\\phi{\\|}_q:= \\big[ c_\\Phi^{-1} \\int_{O} \\int_\\Omega \\int_{\\Phi([0,1)^n,\\omega)} {\\left\\vert \\phi{\\left( x,\\Phi^{-1}(z,\\omega),\\omega \\right)} \\right\\vert}^q dz \\, d\\mathbb{P}(\\omega) \\, dx \\big]^{1\/q}\n$$\nby \\eqref{6742938tyuer}. Since the set $\\Xi\\times \\mathbb{D}$ is dense in $L^q\\left(O\\times[0,1)^n\\times\\Omega\\right)$, we can extend the functional $\\mu$ to a bounded \nfunctional $\\tilde{\\mu}$ over $L^q\\left(O\\times[0,1)^n\\times\\Omega\\right)$. Hence, we find $V_{\\tilde{\\omega}}\\in L^p(O\\times [0,1)^n\\times\\Omega)$ which can be extended \nto $O\\times\\mathbb R^n\\times\\Omega$ in a stationary way by setting \n$$\nV_{\\tilde{\\omega}}(x,y,\\omega)=V_{\\tilde{\\omega}}\\left(x,y-\\left\\lfloor y \\right\\rfloor, \\tau_{\\left\\lfloor y \\right\\rfloor}\\omega\\right),\n$$\nand satisfying for all $\\phi\\in L^q\\left(O\\times[0,1)^n\\times\\Omega\\right)$, \n$$\n\\langle \\tilde{\\mu},\\phi\\rangle\\!\\!= c_{\\Phi}^{-1}\\int_{O\\times\\Omega}\\int_{\\Phi([0,1)^n,\\omega)}\n\\!\\!\\! \\!V_{\\tilde{\\omega}}\\left(x,\\Phi^{-1}\\left(z,\\omega\\right),\\omega\\right)\n\\phi\\left(x,\\Phi^{-1}(z,\\omega),\\omega\\right)dz d\\mathbb{P}(\\omega) dx.\n$$\n\n2. Now, take $\\varphi\\in C^{\\infty}_c(O)$ and $\\Theta\\in L^{q}_{\\loc}(\\mathbb R^n\\times\\Omega)$ a $\\tau$-stationary function. Since the set \n$\\Xi\\times \\mathbb{D}$ is dense in $L^q\\left(O\\times[0,1)^n\\times\\Omega\\right)$, we can pick up a sequence \n$\\{(\\varphi_j,\\Theta_j){\\}}_{j\\ge1}\\subset \\Xi\\times \\mathbb{D}$ such that \n$$\n\\lim_{j\\to\\infty}(\\varphi_j,\\Theta_j)=(\\varphi,\\Theta)\\quad\\text{in $L^q\\Big(O\\times[0,1)^n\\times\\Omega\\Big)$}.\n$$\nThen, observing that \n\\begin{eqnarray*}\n&&\\limsup_{\\varepsilon'\\to 0}\\Big| \\int_{O}v_{\\varepsilon'}(x,\\tilde{\\omega})\\varphi(x)\\Theta\\left(\\Phi^{-1}\\left(\\frac{x}{\\varepsilon'},\\tilde{\\omega}\\right),\\tilde{\\omega}\\right)\\,dx\n\\\\\n&&-\\int_{O}v_{\\varepsilon'}(x,\\tilde{\\omega})\\varphi_j(x)\\Theta_j\\left(\\Phi^{-1}\\left(\\frac{x}{\\varepsilon'},\\tilde{\\omega}\\right),\\tilde{\\omega}\\right)\\,dx\\Big|\n\\\\\n&& \\le c \\|\\varphi-\\varphi_j{\\|}_{L^q(O)}\n[ c_\\Phi^{-1} \\int_\\Omega \\int_{\\Phi([0,1)^n,\\omega)} {\\left\\vert (\\Theta-\\Theta_j){\\left(\\Phi^{-1}(z,\\omega),\\omega \\right)} \\right\\vert}^q dz \\, d\\mathbb{P}(\\omega)]^{1\/q},\n\\end{eqnarray*}\nwhere $c= c(\\tilde{\\omega})$ is a positive constant. \nThen, combining the previous equality with the \\eqref{Two-scale1}, we concluded the proof of the theorem.\n\\end{proof}\n\nLet us remember the following space (see Remark \\ref{REMFPHI}) \n$$\n\\mathcal{H}:=\\Big\\{w\\in H^1_{\\loc}(\\mathbb R^n;L^2(\\Omega));\\,\\text{$w$ is a stationary function}\\Big\\},\n$$\nwhich is a Hilbert space with respect to the following inner product \n$$\n\\begin{aligned}\n\\langle w,v{\\rangle}_{\\mathcal{H}}:=\\int_{[0,1)^n\\times\\Omega} \\!\\! & \\nabla_{\\!y} w(y,\\omega)\\cdot \\nabla_y v(y,\\omega)\\,d{\\mathbb{P}}(\\omega)\\,dy\n\\\\\n& +\\int_{[0,1)^n\\times\\Omega}. \\!\\! \\!\\! w(y,\\omega) v(y,\\omega)\\,d{\\mathbb{P}}(\\omega)\\,dy.\n\\end{aligned} \n$$\nThe next lemma will be important in the homogenization's process. \n\\begin{lemma}\\label{SYM1-5}\nLet $O\\subset\\mathbb R^n$ be an open set and assume that $\\{u_{\\varepsilon}(\\cdot,\\tilde{\\omega}){\\}}_{\\varepsilon>0}$ and $\\{\\varepsilon \\nabla u_{\\varepsilon}(\\cdot,\\tilde{\\omega}){\\}}_{\\varepsilon>0}$ are \nbounded sequences in $L^2(O)$ and in $L^2(O;\\mathbb R^n)$ respectively for a.e. $\\tilde{\\omega}\\in\\Omega$. Then, for a.e. $\\tilde{\\omega}\\in\\Omega$, there exists a \nsubsequence $\\{\\varepsilon'\\}$(it may depend on $\\tilde{\\omega}$) and $u_{\\tilde{\\omega}}\\in L^2(O;\\mathcal{H})$, such that \n$$\nu_{\\varepsilon'}(\\cdot,\\tilde{\\omega})\\; \\xrightharpoonup[\\varepsilon \\to 0]{2-{\\rm s}}\\; u_{\\tilde{\\omega}},\n$$\nand \n$$\n\\varepsilon'\\nabla u_{\\varepsilon'}(\\cdot,\\tilde{\\omega})\\; \\xrightharpoonup[\\varepsilon \\to 0]{2-{\\rm s}}\\;[\\nabla_y\\Phi]^{-1}\\nabla_y u_{\\tilde{\\omega}}.\n$$\n\\end{lemma}\n\\begin{proof}\n  Applying the Theorem \\ref{TwoScale} for the sequences \n  $${ \\{u_\\varepsilon(\\cdot,\\widetilde{\\omega})\\}_{\\varepsilon > 0} }, \\quad\n{ \\{\\varepsilon \\nabla u_\\varepsilon(\\cdot,\\widetilde{\\omega})\\}_{\\varepsilon > 0} }$$ \nfor a.e.  ${ \\widetilde{\\omega} \\in \\Omega }$, \nwe can find a subsequence $\\{ \\varepsilon^\\prime \\}$, and functions \n$${ u_{\\widetilde{\\omega}} \\in  L^2({O} \\! \\times \\! [0,1)^n \\! \\times \\! \\Omega) }, \\quad\n{ V_{\\widetilde{\\omega}} \\in L^2({O} \\! \\times [0,1)^n \\! \\times \\! \\Omega;\\mathbb R^n)}$$ \nwith ${ V_{\\widetilde{\\omega}} = (v^{(1)}_{\\widetilde{\\omega}}, \\ldots, v^{(n)}_{\\widetilde{\\omega}}) }$\nsatisfying for ${ k \\in \\{1,2, \\ldots, n\\} }$, \n\\begin{equation}\\label{9867986876410}\n\t\t\tu_{\\varepsilon^\\prime}(\\cdot,\\widetilde{\\omega}) \\; \\xrightharpoonup[\\varepsilon^\\prime \\to 0]{2-{\\rm s}} \\; u_{\\widetilde{\\omega}},\n\t\t\\end{equation}\n\t\tand\n\t\t\\begin{equation}\\label{7869876874}\n\t\t\t\\varepsilon^\\prime \\frac{\\partial u_{\\varepsilon^\\prime}}{\\partial x_k} \\; \\xrightharpoonup[\\varepsilon^\\prime \\to 0]{2-{\\rm s}} \\; v^{(k)}_{\\widetilde{\\omega}}.\n\t\t\\end{equation}\t\t\nHence for each ${ k\\in \\{1,\\ldots,n\\} }$ and performing an integration by parts we have \n\t\t\\begin{multline*}\n\t\t\t\\int_{{O}} \\varepsilon^\\prime \\frac{\\partial u_{\\varepsilon^\\prime}}{\\partial x_k} (x,\\widetilde{\\omega}) \\, {\\varphi(x) \\, \\Theta{\\left( \\Phi^{-1}\\left( \\frac{x}{\\varepsilon^\\prime}, \\widetilde{\\omega} \\right), \\widetilde{\\omega} \\right)}} dx\n\t\t\t\\\\\n\t\t\t\\hspace{-4cm}=  -\\varepsilon^\\prime\\int_{{O}} u_{\\varepsilon^\\prime} (x,\\widetilde{\\omega}) \\, {\\frac{\\partial \\varphi}{\\partial x_k}(x) \\, \\Theta {\\left( \\Phi^{-1}\\left( \\frac{x}{\\varepsilon^\\prime}, \\widetilde{\\omega} \\right), \\widetilde{\\omega} \\right)}} dx \n\t\t\t\\\\\n\t\t\t\\quad \\quad -\\int_{{O}} u_{\\varepsilon^\\prime} (x,\\widetilde{\\omega}) \\, {\\varphi(x) \\, {[\\nabla_{\\!\\! y}\\Phi]}^{-1}  {\\left( \\Phi^{-1}{\\left( \\frac{x}{\\varepsilon^\\prime}, \\widetilde{\\omega} \\right)}, \\widetilde{\\omega} \\right)}  \\, \\nabla_{\\!\\! y} \\Theta{\\left( \\Phi^{-1}\\left( \\frac{x}{\\varepsilon^\\prime}, \\widetilde{\\omega} \\right), \\widetilde{\\omega} \\right)} \\cdotp e_k} \\, dx,\n\t\t\\end{multline*}\nfor every $\\varphi\\in C^{\\infty}_c(O)$ and $\\Theta \\in C_c^{\\infty}\\big([0,1)^n; L^{\\infty}(\\Omega)\\big)$ extended in a stationary way to $\\mathbb R^n$.  Then, using the relations \\eqref{9867986876410}-\\eqref{7869876874} and a density argument in the space of the test functions, we arrive after letting $\\varepsilon'\\to 0$\n\n\\begin{multline*}\n\t\t\t\\int_\\Omega \\int_{\\Phi([0,1)^n, \\omega)} v^{(k)}_{\\widetilde{\\omega}} {\\left( x, \\Phi^{-1}\\left( z,\\omega \\right),\\omega \\right)} \\, { \\Theta{\\left( \\Phi^{-1}(z,\\omega),\\omega \\right)} } \\, d\\mathbb{P} \\, dz\n\t\t\t\\\\\n\t\t\t= - \\int_\\Omega \\int_{\\Phi([0,1)^n, \\omega)} u_{\\widetilde{\\omega}} \\left( x, \\Phi^{-1}\\left( z,\\omega \\right),\\omega \\right) { \\frac{\\partial }{\\partial z_k} {\\left( \\Theta{\\left( \\Phi^{-1}(z,\\omega),\\omega \\right)} \\right)} } \\,  d\\mathbb{P} \\, dz ,\n\t\t\\end{multline*}\nfor a.e. $x \\in {O} $ and for any $\\Theta \\in C_c^{\\infty}\\big([0,1)^n; L^{\\infty}(\\Omega)\\big)$.\n\nHence\u00ad applying Theorem \\ref{648235azwsxqdgfd}, we obtain \n\\begin{equation*}\n\t\\int_{\\mathbb{R}^n} v^{(k)}_{\\widetilde{\\omega}} {\\left( x, \\Phi^{-1}\\left( z,\\omega \\right),\\omega \\right)} \n\t\\, { \\varphi(z) } \\, dz \\, = \\, -\\int_{\\mathbb{R}^n} u_{\\widetilde{\\omega}} \n\t{\\left( x, \\Phi^{-1}\\left( z,\\omega \\right),\\omega \\right)} \\, { \\frac{\\partial \\varphi}{\\partial z_k}(z) } \\, dz ,\n\\end{equation*}\nfor all ${ \\varphi \\in C_{\\rm c}^\\infty(\\mathbb{R}^n) }$ and a.e.  ${ \\omega \\in \\Omega }$. This completes the proof of our lemma. \n\\end{proof}\n\n\n\\subsection{Perturbations of bounded operators} \n\\label{0239786gfhgdf}\n\nThe aim of this section is to study the point spectrum, that is the set of eigenvalues, for perturbations of a \ngiven bounded operator. More precisely, given a complex Hilbert space $H$, and a sequence \nof operators $\\{A_\\alpha\\}$, with \n$A_\\alpha \\in \\mathcal{B}(H)$ for each $\\alpha \\in \\mathbb{N}^n$, we analyse the point spectrum  \nof the power series of $n-$complex variables $\\boldsymbol{z}= (z_1, \\ldots, z_n)$, which is\n\\begin{equation}\n\\label{POWERSERIES}\n\t\\sum_{\\alpha \\in \\mathbb{N}^n} \\boldsymbol{z}^\\alpha A_\\alpha \\equiv \n\t\\sum_{\\alpha_1,\\ldots,\\alpha_n = 0}^\\infty z_1^{\\alpha_1} \\ldots z_n^{\\alpha_n} A_{\\alpha_1,\\ldots,\\alpha_n},\n\\end{equation}\nfrom the properties of the spectrum $\\sigma(A_{0,\\ldots,0})$. \nThis subject was studied for instance by T. Kato \\cite{Kato} and F. Rellich \\cite{Rellich}.\n\n\t\n\\medskip\t\nTo follow, we define $|\\alpha|:= \\alpha_1 + \\ldots + \\alpha_n$, $(\\alpha \\in \\mathbb{N}^n)$,\n\\begin{equation}\n\\label{ConvergRadius}\n\tr := \\Big(\\underset{k \\in \\mathbb{N}}{\\rm inf} \\big\\{ \\underset{ {\\|\\alpha\\|}_\\infty > k}{\\rm sup} \\sqrt[ {|\\alpha|}]{ {\\Vert A_\\alpha \\Vert} } \\big\\}\\Big)^{-1},\n\\end{equation} \nand for $R> 0$\n$$\n\t\\Delta_R := \\prod_{\\nu=1}^n B(0,R).\n$$\nThen, we have the following \n\\begin{lemma}\nLet $ {\\left\\{ A_\\alpha \\right\\}}$ be a sequence of operators, such that $A_\\alpha \\in  \\mathcal{B}(H)$\nfor each $\\alpha \\in \\mathbb{N}^n$. Then, the series \\eqref{POWERSERIES} is convergent for each $z \\in \\Delta_r$, \nwith $r> 0$ given by \\eqref{ConvergRadius}.\n\\end{lemma}\n\\begin{proof}\nGiven $\\boldsymbol{z} \\in \\Delta_r$, there exists $\\varepsilon> 0$ such that\n\\begin{equation}\n\\label{87687638}\n\t{\\big( \\frac{1}{r} + \\varepsilon \\big)} {\\vert z_\\nu \\vert} < 1, \\quad \\text{for any $\\nu \\in \\{ 1,\\ldots,n \\}$}.\n\\end{equation}\nOn the other hand, \t\t\nfrom \\eqref{ConvergRadius} there exists $k_0 \\in \\mathbb{N}$, such that,\nfor each $k\\geqslant k_0$\n\t\t\\begin{equation*}\n\t\t\t\\underset{ {\\Vert \\alpha \\Vert}_\\infty > k}{\\rm sup} \\sqrt[{\\vert \\alpha \\vert}]{ {\\Vert A_\\alpha \\Vert} } < \\frac{1}{r} + \\varepsilon.\n\t\t\\end{equation*}\nThen, for $\\|\\alpha\\|_\\infty > k_0$\n\t\t\\begin{equation*}\n\t\t\t{\\Vert A_\\alpha \\Vert} < {\\big( \\frac{1}{r} + \\varepsilon \\big)}^{ {\\vert \\alpha \\vert} },\n\t\t\\end{equation*}\nand hence we have\n$$\n    {\\vert z_1 \\vert}^{\\alpha_1} \\ldots {\\vert z_n \\vert}^{\\alpha_n} {\\Vert A_\\alpha \\Vert} \n   < {\\Big(\\big(\\frac{1}{r} + \\varepsilon \\big) {\\vert z_1 \\vert} \\Big)}^{\\alpha_1} \\ldots \n    {\\Big(\\big(\\frac{1}{r} + \\varepsilon \\big) {\\vert z_n \\vert} \\Big)}^{\\alpha_n}.\n$$\nTherefore, we obtain\t\t\n\t\t\\begin{eqnarray*}\n\t\t\t\\sum_{ {\\Vert \\alpha \\Vert}_\\infty > k_0 } {\\vert \\boldsymbol{z} \\vert}^\\alpha {\\Vert A_\\alpha \\Vert}  & \\leqslant & \n\t\t\t\\sum_{\\alpha_1,\\ldots,\\alpha_n = 0}^\\infty {\\left\\{ {\\left[ {\\left( \\frac{1}{r} + \\varepsilon \\right)} {\\vert z_1 \\vert} \\right]}^{\\alpha_1}\n\t\t\t \\ldots {\\left[ {\\left( \\frac{1}{r} + \\varepsilon \\right)} {\\vert z_n \\vert} \\right]}^{\\alpha_n} \\right\\}} \n\\\\[5pt]\n\t\t\t&= & {\\left\\{ \\sum_{\\alpha_1=0}^\\infty {\\left[ {\\left( \\frac{1}{r} + \\varepsilon \\right)} {\\vert z_1 \\vert} \\right]}^{\\alpha_1} \\right\\}} \n\t\t\t\\ldots {\\left\\{ \\sum_{\\alpha_n=0}^\\infty {\\left[ {\\left( \\frac{1}{r} + \\varepsilon \\right)} {\\vert z_n \\vert} \\right]}^{\\alpha_n} \\right\\}},\n\t\t\\end{eqnarray*}\nand due to \\eqref{87687638} the power series \\eqref{POWERSERIES} \nis absolutely convergent for each $\\boldsymbol{z} \\in \\Delta_r$. \n\\end{proof}\n\nOne remarks that, for each $r_0< r$ the series \\eqref{POWERSERIES} converges uniformly in $\\Delta_{r_0}$.\nMoreover, it follows from Definition \\eqref{ConvergRadius} that, there exists $c> 0$ such that, for any \n$\\alpha \\in \\mathbb{N}^n$, ${\\Vert A_\\alpha \\Vert} \\leqslant {c}^{ {\\vert \\alpha \\vert} + 1}$.\n\n\\medskip\nNow, let us recall the definition of operator value maps of many complex variables, and after that consider some important results. \nLet $\\mathcal{O} \\subset \\mathbb{C}^n$ be an open set. A map \n$f: \\mathcal{O} \\to \\mathcal{B}(H)$ is called holomorphic in \n$\\boldsymbol{w} \\in \\mathcal{O}$, when there exists an open set $U \\subset \\mathcal{O}$, $\\boldsymbol{w} \\in U$,\nsuch that $f$ is equal to the (absolutely convergent) power series in \n$\\boldsymbol{z}-\\boldsymbol{w}$, with coefficients $A_\\alpha \\in \\mathcal{B}(H)$, that is \n$$\n\\begin{aligned}\n\tf(\\boldsymbol{z}) \\equiv f(z_1,\\ldots,z_n) &= \\sum_{\\alpha\\in \\mathbb{N}^n} (\\boldsymbol{z}-\\boldsymbol{w})^\\alpha A_\\alpha \n\\\\[5pt]\n\t&\\equiv \\sum_{\\alpha_1,\\ldots,\\alpha_n = 0}^\\infty (z_1 - w_1)^{\\alpha_1} \\ldots (z_n - w_n)^{\\alpha_n} A_{\\alpha_1,\\ldots,\\alpha_n}\n\\end{aligned}\n$$\t\t  \nfor each $\\boldsymbol{z} \\in U$. Moreover, the function $f$ is called holomorphic in $\\mathcal{O}$, \nif it is holomorphic for any $\\boldsymbol{w}\\in \\mathcal{O}$.\n\n\\medskip\nMoreover, assume that $A \\in \\mathcal{B}(H)$ is a symmetric operator and \n$\\lambda \\in \\mathbb{R}$ is an eigenvalue of $A$ with finite multiplicity $h$. \nTherefore, the operator $A-\\lambda I$ is not invertible and there exists a symmetric operator\n$R \\in \\mathcal{B}(H)$, uniquely defined, such that \n\\begin{equation}\n\\label{DEFR}\n\\begin{aligned}\n  R (A-\\lambda I ) f &= f - \\sum_{k=1}^h {\\langle f,\\psi_k \\rangle} \\psi_k, \\quad \\text{for each $f \\in H$, and} \n  \\\\\n  R \\psi_k &= 0, \\quad \\text{for all $k \\in \\{1,\\ldots,h\\}$},\n\\end{aligned}\n\\end{equation}\nwhere $\\{ \\psi_1, \\ldots, \\psi_h\\}$ is an orthonormal basis of ${\\rm Ker}(A-\\lambda I)$. \nThe operator $R$ is called a pseudo-inverse of $A-\\lambda I$, and \none observes that, $AR= RA$.\n\n\\medskip\nIt is also important to consider the following results on complex value functions.\n\\begin{lemma}[Osgood's Lemma]\n\\label{9734987389rhd7gf6ty}\nLet $\\mathcal{O} \\subset \\mathbb{C}^n$ be an open set, and $f : \\mathcal{O} \\to \\mathbb{C}$\na continuous function that is holomorphic in each variable separately. Then,\nthe function $f$ is holomorphic.\n\\end{lemma}\n\nThen, in order to state the Weierstrass' Preparation Theorem, let us recall the concept of Weierstrass' polinomial. \nA complex function $W(\\varrho,\\boldsymbol{z})$, which is holomorphic in a\nneighborhood of $(0,\\boldsymbol{0})\\in \\mathbb{C} \\! \\times \\! \\mathbb{C}^n$, is called a Weirstrass polynomial of\ndegree $m$, when \n$$\n    W(\\varrho,\\boldsymbol{z}) =  \\varrho^m + a_1(\\boldsymbol{z}) \\varrho^{m-1} + \\ldots + a_{m-1}(\\boldsymbol{z}) \\varrho + a_m(\\boldsymbol{z}),\n$$\nwhere any $a_i(\\boldsymbol{z})$, \n$(i= 1,\\ldots,m)$, is an holomorphic function in a neighborhood $\\boldsymbol{0} \\in \\mathbb{C}^n$ that vanishes \nat $\\boldsymbol{z}= \\boldsymbol{0}$. Then, we have the following \n\n\\begin{theorem}[Weierstrass Preparation Theorem]\n\\label{8747285tdg4f}\nLet $m$ be a positive integer, and $F(\\varrho,\\boldsymbol{z})$\nholomorphic in a neighborhood of $(0,\\boldsymbol{0}) \\in \\mathbb{C} \\! \\times \\! \\mathbb{C}^n$ such that, \nthe mapping $\\varrho \\mapsto F(\\varrho,\\boldsymbol{0})\/\\varrho^m$ is holomorphic in a neighborhood of \n$0 \\in \\mathbb{C}$\n and is non-zero at $0$. Then, there exist a Weierstrass polynomial $W(\\varrho,\\boldsymbol{z})$ of degree m, \n and a holomorphic function $E(\\varrho,\\boldsymbol{z})$ which does not vanish in a neighborhood \n$U$ of $(0,\\boldsymbol{0})$, such that, for all \n$(\\varrho,\\boldsymbol{z}) \\in U$\n$$\n    F(\\varrho,\\boldsymbol{z}) = W(\\varrho,\\boldsymbol{z}) E(\\varrho,\\boldsymbol{z}).\n$$    \n\\end{theorem}\n\\begin{proof}\nSee S. G. Krantz, H. R. Parks \\cite[p. 96]{KrantzParks}.\n\\end{proof}\n\t\t\n\\smallskip\t\nAt this point, we are in condition to establish the main result of this section, \nthat is to say, the perturbation theory for bounded operators with \nisolated eigenvalues of finite multiplicity.\nThe theorem considered here is a convenient and direct version for our purposes\nin this paper. \n\t\n\\begin{theorem}\n\\label{768746hughjg576}\nLet $H$ be a Hilbert space, and a sequence \nof operators $\\{A_\\alpha\\}$,\n$A_\\alpha \\in \\mathcal{B}(H)$ for each $\\alpha \\in \\mathbb{N}^n$.\nConsider the power series of $n-$complex variables $\\boldsymbol{z}= (z_1, \\ldots, z_n)$\nwith coefficients $A_\\alpha$, which is absolutely convergent in a neighborhood $ \\mathcal{O}$ of \n$\\boldsymbol{z}=\\boldsymbol{0}$. Define, the holomorphic map $A:  \\mathcal{O} \\to  \\mathcal{B}(H)$, \n$$  \n  A(\\boldsymbol{z}):= \\sum_{\\alpha\\in \\mathbb{N}^n} \\boldsymbol{z}^\\alpha A_\\alpha\n$$\nand assume that, it is symmetric. If $\\lambda$ is an eigenvalue \nof $A_0 \\equiv A(\\boldsymbol{0})$ with finite multiplicity $h$ (and respective eigenvectors $\\psi_i$, $i= 1,\\ldots,h$), \nthen there exist a neighborhood $U \\subset  \\mathcal{O}$\nof ${ \\boldsymbol{0} }$, and holomorphic functions\n$$\n\\begin{aligned}\n    \\boldsymbol{z} \\in U &\\, \\mapsto \\, \\lambda_1(\\boldsymbol{z}), \\lambda_2(\\boldsymbol{z}), \\ldots, \\lambda_h(\\boldsymbol{z}) \\in \\mathbb{R},\n    \\\\[5pt]\n    \\boldsymbol{z} \\in U &\\, \\mapsto \\, \\psi_1(\\boldsymbol{z}), \\psi_2(\\boldsymbol{z}), \\ldots, \\psi_h(\\boldsymbol{z}) \\in H\\setminus \\{0\\},\n\\end{aligned}\n$$\nsatisfying for each $\\boldsymbol{z} \\in U$ and $i \\in \\{1,\\ldots,h\\}:$\n\t\t\\begin{itemize}\n\t\t\t\\item[$(i)$] $A(\\boldsymbol{z}) \\psi_i(\\boldsymbol{z}) = \\lambda_i(\\boldsymbol{z}) \\psi_i(\\boldsymbol{z})$, \n\t\t\t\\item[$(ii)$] ${ \\lambda_i(\\boldsymbol{z} = \\boldsymbol{0}) = \\lambda }$, \n\t\t\t\\item[$(iii)$] ${ {\\rm dim} {\\{w \\in H \\; ; \\; A(\\boldsymbol{z}) w = \\lambda_i(\\boldsymbol{z}) w \\}} \\leqslant h }$.\n\t\t\\end{itemize}\nMoreover, if there exists $d> 0$ such that \n$$\n   \\sigma(A_0)\\cap (\\lambda-d, \\lambda+d) = {\\left\\{ \\lambda \\right\\}},\n$$\nthen for each $d^\\prime\\in(0,d)$ there exists a neighborhood $W \\subset U$ of $\\boldsymbol{0}$, \nsuch that\n\\begin{equation}\n\\label{FINALPERT}\n   \\sigma(A(\\boldsymbol{z})) \\cap (\\lambda - d^\\prime, \\lambda + d^\\prime) = {\\left\\{ \\lambda_1(\\boldsymbol{z}), \\ldots, \\lambda_h(\\boldsymbol{z}) \\right\\}}\n\\end{equation}\nfor all $\\boldsymbol{z}\\in W$. \n\\end{theorem}\n\t\n\\begin{proof} \n1. First, we conveniently define  \n\\begin{equation}\n\\label{DEFB}\n\tB(\\boldsymbol{z}) := A(\\boldsymbol{z}) - A_0 = \\sum_{{\\vert \\alpha \\vert} \\not= 0} \\boldsymbol{z}^\\alpha A_{\\alpha}.\n\\end{equation}\nThen, there exists a neighborhood of $(\\varrho,\\boldsymbol{z})= (0,\\boldsymbol{0})$\nsuch that, the function \n$$\n    (\\varrho, \\boldsymbol{z}) \\, \\mapsto \\, \\sum_{l=0}^\\infty {\\left[ R {\\left( \\varrho - B(\\boldsymbol{z}) \\right)} \\right]}^l \\in \\mathcal{B}(H)\n$$\nis well defined (see equation \\eqref{DEFR}), and holomorphic on it.  \nIndeed, first we recall that there exists $c> 0$ such that, for any \n$\\alpha \\in \\mathbb{N}^n$, ${\\Vert A_\\alpha \\Vert} \\leqslant {c}^{ {\\vert \\alpha \\vert} + 1}$.\nThen, it follows from \\eqref{DEFB} that \n$$\n\\begin{aligned}\n\t{\\Vert B(\\boldsymbol{z}) \\Vert} & \\leqslant \\sum_{{\\vert \\alpha \\vert}\\not=0} {\\vert z_1 \\vert}^{\\alpha_1} \\ldots {\\vert z_n \\vert}^{\\alpha_n} {\\Vert A_{\\alpha} \\Vert} \n   \\leqslant \\sum_{{\\vert \\alpha \\vert} \\not= 0} {\\vert \\boldsymbol{z} \\vert}^{{\\vert \\alpha \\vert}} c^{{\\vert \\alpha \\vert}+1} \n\\\\\n  & =  \\sum_{k=1}^\\infty {\\sum_{{\\vert \\alpha \\vert} = k} {\\vert \\boldsymbol{z} \\vert}^{{\\vert \\alpha \\vert}} c^{{\\vert \\alpha \\vert}+1} }  \n   = \\sum_{k=1}^\\infty { \\sum_{{\\vert \\alpha \\vert} = k} {\\vert \\boldsymbol{z} \\vert}^{k} c^{k+1} }  \n\\\\\n  & = \\sum_{k=1}^\\infty {\\left( \\# {\\left\\{ \\alpha\\in\\mathbb{N}^n \\; ; \\; {\\vert \\alpha \\vert} = k \\right\\}} {\\vert \\boldsymbol{z} \\vert}^{k} c^{k+1} \\right)}  \n\\\\\n  & \\leqslant \\sum_{k=1}^\\infty (k+1)^n {\\vert \\boldsymbol{z} \\vert}^{k} c^{k+1} \n  =  {\\vert \\boldsymbol{z} \\vert}c^2 \\sum_{k=1}^\\infty (k+1)^n {\\vert \\boldsymbol{z} \\vert}^{k-1} c^{k-1} \n\\\\\n  & \\leqslant  {\\vert \\boldsymbol{z} \\vert} c^2 \\sum_{k=0}^\\infty (k+2)^n {\\vert \\boldsymbol{z} \\vert}^{k} c^{k}.\n\\end{aligned}\n$$\nTherefore, it follows that $\\sum_{k=0}^\\infty (k+2)^n {\\vert \\boldsymbol{z} \\vert}^{k} c^{k}$ is absolutely convergent \nfor each $\\boldsymbol{z} \\in B{\\left( \\boldsymbol{0},\\frac{1}{4^{n}c} \\right)}$. Moreover, there exists $\\tilde{c}> 0$, such that\t\t\n$$\n    \\big| \\sum_{k=0}^\\infty (k+2)^n {\\vert \\boldsymbol{z} \\vert}^{k} c^{k} \\big| \\leqslant \\tilde{c}\n    \\quad \\text{for each $\\boldsymbol{z} \\in B(\\boldsymbol{0},\\frac{1}{4^{n+1}c})$}.\n$$\t\t\nHence we have from \\eqref{DEFB} that\n\\begin{equation}\n\\label{768746876784}\n\\begin{aligned}\n\t{\\left\\Vert R(\\varrho - B(\\boldsymbol{z})) \\right\\Vert} &\\leq {\\Vert R \\Vert}({\\vert \\varrho \\vert} + {\\vert \\boldsymbol{z} \\vert}c^2\\tilde{c}) \n\t\\\\[5pt]\n\t&\\leq {\\Vert R \\Vert} \\ {\\rm max} {\\left\\{ 1,c^2\\tilde{c} \\right\\}}({\\vert \\varrho \\vert} + {\\vert \\boldsymbol{z} \\vert}),\n\\end{aligned} \n\\end{equation}\t \nfor ${ \\varrho\\in \\mathbb{C} }$ and ${ \\boldsymbol{z}\\in B{\\left( \\boldsymbol{0},\\frac{1}{4^{n+1}c} \\right)} }$.\t\nTo follow, we define\n\\begin{equation}\n r:= \\min \\Big\\{\\frac{1}{8 {\\Vert R \\Vert} {\\rm max} {\\left\\{ 1,c^2\\tilde{c} \\right\\}}}, \\frac{1}{4^{n+1} c} \\Big\\},\n \\quad \n\t\\Delta_r:=B(0,r) \\! \\times \\! B(\\boldsymbol{0},r) \\subset \\mathbb{C} \\! \\times \\! \\mathbb{C}^n.\n\\end{equation}\nThen, for any $m,n \\in \\mathbb{N}$ with $m> n$, and all $(\\varrho, \\boldsymbol{z}) \\in \\Delta_r$, we have \t\t\n$$\n\\begin{aligned}\n    {\\Vert  \\sum_{l=0}^m {\\left[ R(\\varrho - B(\\boldsymbol{z})) \\right]}^l - \\sum_{l=0}^n {\\left[ R(\\varrho - B(\\boldsymbol{z})) \\right]}^l \\Vert} \n    &\\leq \\sum_{l=n+1}^m {\\Vert R(\\varrho - B(\\boldsymbol{z})) \\Vert}^l\n    \\\\[5pt]\n    &\\leq \\sum_{l=n+1}^m {\\left( \\frac{1}{4} \\right)}^l.\n\\end{aligned} \n$$  \nConsequently, for any $(\\varrho, \\boldsymbol{z}) \\in \\Delta_r$, $\\{ \\sum_{l=0}^m {\\left[ R(\\varrho - B(\\boldsymbol{z})) \\right]}^l \\}_{m\\in \\mathbb{N}}$\nis a Cauchy sequence in $\\mathcal{B}(H)$. Therefore, the mapping \n\\begin{equation}\n\t(\\varrho, \\boldsymbol{z})\\in \\Delta_r \\; \\mapsto \\; \\sum_{l=0}^\\infty {\\left[ R(\\varrho - B(\\boldsymbol{z})) \\right]}^l\n\\end{equation}\nis holomorphic, since it is the uniform limit of holomorphic functions. \n\n\\bigskip\n2. Now, for $i,j = 1,\\ldots,h$ and $(\\varrho,\\boldsymbol{z}) \\in \\Delta_r$, let us consider\n$$\n   f_{ij}(\\varrho,\\boldsymbol{z}) \n   = \\Big\\langle \\sum_{l=0}^\\infty (\\varrho - B(\\boldsymbol{z})) {\\left[ R(\\varrho - B(\\boldsymbol{z})) \\right]}^l \\psi_i, \\psi_j \\Big\\rangle.\n$$\nTherefore, the function $F:\\Delta_r \\rightarrow \\mathbb{C} $, defined by  \n$F(\\varrho, \\boldsymbol{z}) := {\\rm det} {\\left[ {\\left( f_{ij}(\\varrho, \\boldsymbol{z}) \\right)} \\right]}$\nis holomorphic. In fact, $F(\\varrho, \\boldsymbol{z})$ is a real value function, when $\\varrho \\in \\mathbb{R}$. \n\\begin{comment}\nIndeed, let us observe that, for $i,j\\in {\\{ 1,\\ldots,h\\}}$ \n$$\n\\begin{aligned}\n\tf_{ij}(\\varrho,\\boldsymbol{z})= & \\sum_{l=0}^\\infty \\left\\langle {\\left[ R(\\varrho - B(\\boldsymbol{z})) \\right]}^l \\psi_i, (\\overline{\\varrho} - B(\\boldsymbol{z})) \\psi_j \\right\\rangle \n\t\\\\\n\t= & \\sum_{l=0}^\\infty \\left\\langle  \\psi_i, {\\left[ (\\overline{\\varrho} - B(\\boldsymbol{z}))R \\right]}^l (\\overline{\\varrho} - B(\\boldsymbol{z})) \\psi_j \\right\\rangle \n\t\\\\\n\t= & \\sum_{l=0}^\\infty \\left\\langle  \\psi_i, {\\left[ (\\overline{\\varrho} - B(\\boldsymbol{z}))R \\right]}^{l-1} {\\left[ (\\overline{\\varrho} \n\t- B(\\boldsymbol{z}))R \\right]} (\\overline{\\varrho} - B(\\boldsymbol{z})) \\psi_j \\right\\rangle \n\t\\\\\n\t= & \\sum_{l=0}^\\infty \\left\\langle   \\psi_i, (\\overline{\\varrho} - B(\\boldsymbol{z})) {\\left[ R(\\overline{\\varrho} - B(\\boldsymbol{z})) \\right]}^l \\psi_j \\right\\rangle\n\t \\\\\n\t= & \\Big\\langle  \\psi_i, \\sum_{l=0}^\\infty (\\overline{\\varrho} - B(\\boldsymbol{z})) {\\left[ R(\\overline{\\varrho} - B(\\boldsymbol{z})) \\right]}^l \\psi_j \\Big\\rangle \n\t= \\overline{f_{ji}(\\overline{\\varrho}, \\boldsymbol{z})}.\n\\end{aligned}\n$$\n\\end{comment}\nMoreover, $F(\\varrho,\\boldsymbol{0}) = {\\rm det} {\\left[ \\varrho \\, (\\delta_{ij}) \\right]} = \\varrho^h$\nfor each $\\varrho\\in B (0,r)$, where $\\delta_{ij}$ is the Kronecker delta. Indeed, we have \n\\begin{eqnarray*}\n\t\t\tf_{ij}(\\varrho,\\boldsymbol{0}) & = & \\Big\\langle \\sum_{l=0}^\\infty (\\varrho - B(\\boldsymbol{0})) {\\left[ R(\\varrho - B(\\boldsymbol{0})) \\right]}^l \\psi_i, \\psi_j \\Big\\rangle \n\t\t\t\\\\\n\t\t\t& = & \\left\\langle \\sum_{l=0}^\\infty \\varrho^{l+1} R^l \\psi_i, \\psi_j \\right\\rangle \\\\\n\t\t\t& = & \\left\\langle \\varrho \\, \\psi_i, \\psi_j \\right\\rangle + \\sum_{l=1}^\\infty \\left\\langle  \\varrho^{l+1} R^l \\psi_i, \\psi_j \\right\\rangle\n\t\t\t= \\varrho \\,  \\delta_{ij},\n\\end{eqnarray*}\t\nfrom which follows the result.\n\t\t\n\\medskip\n3. At this point, we show that there exist $h$ holomorphic functions \n$\\varrho_k(\\boldsymbol{z})$, $(k=1,\\ldots,h)$,\ndefined in a neighborhood of $\\boldsymbol{z}= \\boldsymbol{0}$, such that\n\\begin{equation*}\n\t\\lim_{\\boldsymbol{z} \\to \\boldsymbol{0}} \\varrho_k(\\boldsymbol{z}) = 0,\n\t\\quad \\text{for $k \\in \\{ 1,\\ldots, h \\}$}.\n\\end{equation*}\nIndeed, applying Theorem \\ref{8747285tdg4f} (Weierstrass Preparation Theorem) \nthere exists a Weirstrass polynomial of degree $h$\n$$\n    W(\\varrho,\\boldsymbol{z}) \n    = \\varrho^h + a_1(\\boldsymbol{z}) \\varrho^{h-1} + \\ldots + a_{h-1}(\\boldsymbol{z}) \\varrho + a_h(\\boldsymbol{z}),\n$$\nand also a holomorphic function \n$E(\\varrho,\\boldsymbol{z})$, which does not vanish in a neighborhood \n$U \\times V$ of $(0,\\boldsymbol{0})$, \nwith $U \\subset B(0,r) \\subset \\mathbb{C}$ and $V\\subset B(\\boldsymbol{0},r) \\subset \\mathbb{C}^n$, such that \nfor each $(\\varrho, \\boldsymbol{z}) \\in U \\! \\times \\! V$\n$$\n\tF(\\varrho, \\boldsymbol{z})= E(\\varrho, \\boldsymbol{z}) {\\left( \\varrho^h + a_1(\\boldsymbol{z}) \\varrho^{h-1} \n\t+ \\ldots + a_{h-1}(\\boldsymbol{z}) \\varrho + a_h(\\boldsymbol{z}) \\right)}.\n$$\nSince the coefficients of the Weirstrass polynomial are holomorphic functions, which vanish in\n$\\boldsymbol{z} = \\boldsymbol{0}$, then there exist holomorphic functions $\\varrho_k(\\boldsymbol{z})$, \nsuch that \n\\begin{equation}\n\\label{76847678644}\n\\begin{aligned}\n  F(\\varrho, \\boldsymbol{z})&= E(\\varrho, \\boldsymbol{z}) \\, \\prod_{k= 1}^h {\\left( \\varrho - \\varrho_k(\\boldsymbol{z}) \\right)},\n\\\\[5pt]\n\t\\lim_{ \\boldsymbol{z} \\to \\boldsymbol{0}} \\varrho_k(\\boldsymbol{z})&= 0, \\quad  (k=1,\\ldots,h).\n\\end{aligned}\n\\end{equation}\n\n\\bigskip\n4. At this point, let us show that, for $k \\in \\{1, \\ldots, h\\}$ there exists a map\n$\\psi_k(\\boldsymbol{z}) \\in H-\\{0\\}$ such that,\n$A(\\boldsymbol{z})\\psi_k(\\boldsymbol{z}) = (\\lambda + \\varrho_k(\\boldsymbol{z}))\\psi_k(\\boldsymbol{z})$, \nfor each $\\boldsymbol{z}$ in a neighborhood of  $\\boldsymbol{0}$. Indeed, let \n$k \\in \\{1, \\ldots,h\\}$ be fix. From item 3, there exists a set $V \\subset \\mathbb{C}$, \nwhich is a neighborhood of $\\boldsymbol{z}= \\boldsymbol{0}$, such that\n$$\n     {\\rm det} {\\left[ {\\left( f_{ij}(\\varrho_k(\\boldsymbol{z}), \\boldsymbol{z}) \\right)} \\right]} = 0\n$$\nfor each $\\boldsymbol{z}\\in V$. Therefore, for each $\\boldsymbol{z} \\in V$\nthe linear system \n$$\n   \\left( f_{ji}(\\varrho_k(\\boldsymbol{z}), \\boldsymbol{z}) \\right) (c_1,\\ldots,c_n)^T = 0\n$$\nhas a non-trivial solution. Consequently, there exist $h$ holomorphic functions \n$\\boldsymbol{z}\\in V \\mapsto c_1^k(\\boldsymbol{z}), \\ldots, c_h^k(\\boldsymbol{z})$, such that for all $j=1,\\ldots,h$\n$$\n    \\sum_{i=1}^h f_{ij}(\\varrho_k(\\boldsymbol{z}), \\boldsymbol{z}) \\, c_i^k(\\boldsymbol{z})= 0,\n$$\nand without loss of generality we may assume \n\\begin{equation}\n\\label{4234343}\n   \\sum_{i=1}^h {\\vert c_i^k(\\boldsymbol{z}) \\vert}^2 =1.\n\\end{equation}\nFrom equation \\eqref{76847678644} it is possible to find a neighborhood \n$\\tilde{V}$ of $\\boldsymbol{0}$, which is compactly embedded in $V$, such that\n$$\n   \\underset{\\boldsymbol{z} \\in \\tilde{V}}{\\rm sup} {\\vert \\varrho_k(\\boldsymbol{z}) \\vert} <  \\frac{1}{8 {\\Vert R \\Vert} {\\rm max}(1, c^2\\tilde{c})},\n$$\nfor each $\\boldsymbol{z} \\in \\tilde{V}$ and all $k \\in \\{1,\\ldots,n\\}$. Hence we obtain for each $\\boldsymbol{z}\\in \\tilde{V}$\n\\begin{equation}\n\\label{6654123123654}\n      {\\Vert R(\\varrho_k(\\boldsymbol{z}) - B(\\boldsymbol{z})) \\Vert} \n      \\leq {\\Vert R \\Vert} {\\rm max}(1, c^2\\tilde{c}) {\\left( {\\vert \\varrho_k(\\boldsymbol{z}) \\vert} +  {\\vert \\boldsymbol{z} \\vert} \\right)} \n      \\leq \\frac{1}{4},\n\\end{equation}\t\t\nand then\n\\begin{equation}\n\\label{6486746874}\n   \\sum_{l=0}^\\infty {\\Vert R(\\varrho_k(\\boldsymbol{z})  - B(\\boldsymbol{z})) \\Vert}^l \\leqslant \\frac{4}{3}.\n\\end{equation}\nNow, we define for any  $\\boldsymbol{z} \\in \\tilde{V}$\n$$\n   \\phi_k(\\boldsymbol{z}):= \\sum_{i=1}^h c_i^k(\\boldsymbol{z}) \\psi_i, \n   \\quad \\text{and} \\quad \n   \\psi_k(\\boldsymbol{z}):= \\sum_{l=0}^\\infty {\\left[ R(\\varrho_k (\\boldsymbol{z}) - B(\\boldsymbol{z})) \\right]}^l \\phi_k(\\boldsymbol{z}).\n$$\nTherefore, we have \n\t\t\\begin{eqnarray*}\n\t\t\t\\psi_k(\\boldsymbol{z}) & = & \\phi_k(\\boldsymbol{z}) + \\sum_{l=1}^\\infty {\\left[ R(\\mu_k(\\boldsymbol{z}) - B(\\boldsymbol{z})) \\right]}^l \\phi_k(\\boldsymbol{z})  \n\t\t\t\\\\\n\t\t\t& = & \\phi_k(\\boldsymbol{z}) + {\\left[ R(\\varrho_k(\\boldsymbol{z}) \n\t\t\t- B(\\boldsymbol{z})) \\right]} \\sum_{l=1}^\\infty {\\left[ R(\\varrho_k(\\boldsymbol{z}) - B(\\boldsymbol{z})) \\right]}^{l-1} \\phi_k(\\boldsymbol{z}) \n\t\t\t\\\\\n\t\t\t& = & \\phi_k(\\boldsymbol{z}) + {\\left[ R(\\varrho_k(\\boldsymbol{z}) - B(\\boldsymbol{z})) \\right]} \\psi_k(\\boldsymbol{z}),\n\t\t\\end{eqnarray*}\nand it follows that\n\\begin{equation}\n\\label{AAA}\n\\begin{aligned}\n\t\t\t(A_0-\\lambda)\\psi_k(\\boldsymbol{z}) & = (A_0-\\lambda)\\phi_k(\\boldsymbol{z}) + (A_0-\\lambda) {\\left[ R(\\varrho_k(\\boldsymbol{z}) - B(\\boldsymbol{z})) \\right]} \\psi_k(\\boldsymbol{z}) \n\t\t\t\\\\\n\t\t\t& = \\sum_{i=1}^h c_i^k(\\boldsymbol{z})(A_0-\\lambda) \\psi_i + (A_0-\\lambda) R {\\left[ (\\varrho_k(\\boldsymbol{z}) - B(\\boldsymbol{z}))  \\psi_k(\\boldsymbol{z}) \\right]}\n\t\t\t\\\\\n\t\t\t& = R (A_0-\\lambda) {\\left[ (\\varrho_k(\\boldsymbol{z}) - B(\\boldsymbol{z}))  \\psi_k(\\boldsymbol{z}) \\right]} \n\t\t\t\\\\\n\t\t\t& =  (\\varrho_k(\\boldsymbol{z}) - B(\\boldsymbol{z}))  \\psi_k(\\boldsymbol{z}) - \\sum_{j=1}^h \\left\\langle (\\varrho_k(\\boldsymbol{z}) - B(\\boldsymbol{z}))  \\psi_k(\\boldsymbol{z}), \\psi_j \\right\\rangle \\psi_j\n                         \\\\\n\t\t\t& =  (\\varrho_k(\\boldsymbol{z}) - B(\\boldsymbol{z}))  \\psi_k(\\boldsymbol{z}) \n\\end{aligned}\n\\end{equation}\nsince \n\t\t\\begin{align*}\n\t\t\t\\left\\langle (\\varrho_k(\\boldsymbol{z}) \\right. & - \\left. B(\\boldsymbol{z}))  \\psi_k(\\boldsymbol{z}), \\psi_j \\right\\rangle \\\\ \n\t\t\t& = \\sum_{i=1}^h \\Big\\langle \\sum_{l=0}^\\infty (\\varrho_k(\\boldsymbol{z}) - B(\\boldsymbol{z}))  {\\left[ R(\\varrho_k (\\boldsymbol{z}) - B(\\boldsymbol{z})) \\right]}^l \\psi_i, \\psi_j \\Big\\rangle c_i^k(\\boldsymbol{z}) \n\t\t\t\\\\\n\t\t\t& = \\sum_{i=1}^h f_{ij}(\\varrho_k(\\boldsymbol{z}), \\boldsymbol{z}) \\, c_i^k(\\boldsymbol{z}) = 0.\n\t\t\\end{align*}\nThus,  for each $\\boldsymbol{z}\\in \\tilde{V}$, $A(\\boldsymbol{z}) \\psi_k(\\boldsymbol{z}) = (\\lambda + \\varrho_k(\\boldsymbol{z}))\\psi_k(\\boldsymbol{z})$.\nOn the other hand, \n\t\t\\begin{equation*}\n \t\t\t\\psi_k(\\boldsymbol{z}) = \\phi_k(\\boldsymbol{z}) + {\\left[ R(\\varrho_k(\\boldsymbol{z}) \n\t\t\t- B(\\boldsymbol{z})) \\right]} \\sum_{l=1}^\\infty {\\left[ R(\\varrho_k(\\boldsymbol{z}) \n\t\t\t- B(\\boldsymbol{z})) \\right]}^{l-1} \\phi_k(\\boldsymbol{z}),\n\t\t\\end{equation*}\nhence from \\eqref{4234343}, \\eqref{6654123123654}, and \\eqref{6486746874}, we have for each $\\boldsymbol{z}\\in \\tilde{V}$\n\t\t\\begin{eqnarray*}\n\t\t\t{\\Vert \\psi_k(\\boldsymbol{z}) - \\phi_k(\\boldsymbol{z}) \\Vert} \n\t\t\t& \\leq & {\\Vert R(\\varrho_k(\\boldsymbol{z}) - B(\\boldsymbol{z})) \\Vert} {\\Vert \\sum_{l=0}^\\infty {\\left[ R(\\varrho_k(\\boldsymbol{z}) - B(\\boldsymbol{z})) \\right]}^{l} \\Vert} {\\Vert \\phi_k(\\boldsymbol{z}) \\Vert} \n\t\t\t\\\\\n\t\t\t& \\leq & \\frac{4}{3} {\\Vert R(\\varrho_k(\\boldsymbol{z}) - B(\\boldsymbol{z})) \\Vert}  \\leq \\frac{1}{3}.\n\t\t\\end{eqnarray*}\nConsequently, for each $\\boldsymbol{z}\\in \\tilde{V}$ we have $\\psi_k(\\boldsymbol{z})\\not= 0$, since  \n\t\t\\begin{equation*}\n\t\t\t1={\\Vert \\phi_k(\\boldsymbol{z}) \\Vert} \\leq {\\Vert \\phi_k(\\boldsymbol{z}) - \\psi_k(\\boldsymbol{z}) \\Vert} \n\t\t\t+ {\\Vert \\psi_k(\\boldsymbol{z}) \\Vert} \\leq \\frac{1}{3} + {\\Vert \\psi_k(\\boldsymbol{z}) \\Vert}.\n\t\t\\end{equation*}\n\n\\bigskip\n5. Now, let us show item $(iii)$ of the thesis, that is, \n$${ {\\rm dim} {\\{w \\in H \\; ; \\; A(\\boldsymbol{z}) w = \\lambda_i(\\boldsymbol{z}) w \\}} \\leqslant h }.$$\nFrom the previous item, there exists \n$\\lambda_k(\\boldsymbol{z})= \\lambda + \\varrho_k(\\boldsymbol{z})$, \n$k\\in\\{1,\\ldots,h\\}$, an eigenvalue of the operator $A(\\boldsymbol{z})$, \nfor $\\boldsymbol{z}$ in a neighborhood of $\\boldsymbol{z}= \\boldsymbol{0}$. \nWe set $\\lambda(\\boldsymbol{z})= \\lambda_k(\\boldsymbol{z})$, for any \n$k \\in \\{1,\\ldots,h\\}$ fixed, and let $\\psi(\\boldsymbol{z})$ be any function satisfying \n$$\n  A(\\boldsymbol{z}) \\psi(\\boldsymbol{z})= \\lambda(\\boldsymbol{z}) \\psi(\\boldsymbol{z}),\n$$\nwhich is not necessarily the eigenfunction $\\psi_k(\\boldsymbol{z})$. \nThen, we are going to show that, there exist a neighborhood of $\\boldsymbol{z}= \\boldsymbol{0}$,\nand for each \n$\\boldsymbol{z}$ in this neighborhood an invertible holomorphic operator  $S(\\boldsymbol{z}) \\in \\mathcal{B}(H)$, such that, \n\\begin{equation}\n\\label{SPAM}\n\t\t\t\\psi(\\boldsymbol{z}) \\in {\\rm span} {\\big\\{ S(\\boldsymbol{z})\\psi_1, S(\\boldsymbol{z}) \\psi_2, \\ldots, S(\\boldsymbol{z}) \\psi_h \\big\\}}.\n\\end{equation}\nIndeed, to show \\eqref{SPAM} let us define \n$\\varrho(\\boldsymbol{z}):= \\lambda(\\boldsymbol{z})-\\lambda$, \nthen we have\n$$\n    {\\big( \\varrho(\\boldsymbol{z})I - B(\\boldsymbol{z}) \\big)} \\psi(\\boldsymbol{z}) = {\\big( A_0-\\lambda \\big)} \\psi(\\boldsymbol{z}).\n$$\nHence from the first equation in \\eqref{DEFR}, it follows that \n$$\n   R {\\big( \\varrho(\\boldsymbol{z})I - B(\\boldsymbol{z}) \\big)} \\psi(\\boldsymbol{z}) = \\psi(\\boldsymbol{z}) - \\sum_{i=1}^h {\\langle \\psi(\\boldsymbol{z}), \\psi_i \\rangle} \\psi_i,\n$$\nor equivalently \n\t\t\\begin{equation*}\n\t\t\t{\\big[ I - R {\\big( \\varrho(\\boldsymbol{z})I - B(\\boldsymbol{z}) \\big)} \\big]} \\psi(\\boldsymbol{z}) = \\sum_{i=1}^h {\\langle \\psi(\\boldsymbol{z}), \\psi_i \\rangle} \\psi_i.\n\t\t\\end{equation*}\nOn the other hand, from \\eqref{768746876784} it is possible to find a neighborhood $V$ of \n$\\boldsymbol{z}=\\boldsymbol{0}$, such that, \n$$\n    \\|R \\big( \\varrho(\\boldsymbol{z})I - B(\\boldsymbol{z}) \\big) \\| < 1\n$$    \nTherefore, it exists an invertible operator \n$$\n   S(\\boldsymbol{z})= {\\big[ I - R {\\big( \\varrho(\\boldsymbol{z})I - B(\\boldsymbol{z}) \\big)} \\big]}^{-1} = \\sum_{\\nu=0}^\\infty {\\big[ R {\\big( \\varrho(\\boldsymbol{z})I - B(\\boldsymbol{z}) \\big)} \\big]}^\\nu,\n$$\nand hence \n$$\n   \\psi(\\boldsymbol{z})= S(\\boldsymbol{z}) {\\left( \\sum_{i=1}^h {\\langle \\psi(\\boldsymbol{z}), \\psi_i \\rangle} \\psi_i \\right)} \n   = \\sum_{i=1}^h {\\langle \\psi(\\boldsymbol{z}), \\psi_i \\rangle} {\\big[ S(\\boldsymbol{z}) \\psi_i \\big]}.\n$$\n\n\\bigskip\n6. Finally, we show that the perturbed eigenvalues are isolated. \nTo this end, we consider \n\t\t\\begin{equation*}\n\t\t\tN(\\boldsymbol{z}) := {\\rm span} {\\left\\{ \\psi_1(\\boldsymbol{z}), \\psi_2(\\boldsymbol{z}), \\ldots, \\psi_h(\\boldsymbol{z}) \\right\\}},\n\t\t\\end{equation*}\nthe operator $P(\\boldsymbol{z}): H \\to H$, which is a projection on $N(\\boldsymbol{z})$, given by\n\\begin{equation*}\n  P(\\boldsymbol{z}) u = \\sum_{i=1}^h \\left\\langle u, \\psi_i(\\boldsymbol{z}) \\right\\rangle  \\psi_i(\\boldsymbol{z}),\n\\end{equation*}\nand for $d> 0$ the operator $D(\\boldsymbol{z}): H \\to H$, defined by \n\\begin{equation*}\n   D(\\boldsymbol{z}):= A(\\boldsymbol{z}) - 2 d P(\\boldsymbol{z}).\n\\end{equation*}\nOne observes that \t\t\n\\begin{equation}\n\\label{87678687678}\n    D(\\boldsymbol{z})u = \\sum_{i=1}^h (\\lambda_i(\\boldsymbol{z}) - 2d) \\left\\langle u_1, \\psi_i(\\boldsymbol{z}) \\right\\rangle \\psi_i(\\boldsymbol{z}) + A(\\boldsymbol{z})u_2,\n\\end{equation}\nwhere we have used the direct sum $u= u_1 + u_2$, $u_1\\in N(\\boldsymbol{z})$ and $u_2\\in N(\\boldsymbol{z})^\\perp$.\n\n\\medskip\t\t\t\n\\underline{{Claim 1}}. \n\\begin{itemize}\n\\item[a)] For $\\xi \\in \\mathbb{R} \\setminus {\\left\\{ \\lambda_1(\\boldsymbol{z}), \\lambda_2(\\boldsymbol{z}), \\ldots, \\lambda_h(\\boldsymbol{z}) \\right\\}}$,\n\\begin{equation*}\n\tD(\\boldsymbol{z}) - \\xi \\;\\, \\text{is bijective} \\; \\Rightarrow \\; A(\\boldsymbol{z}) - \\xi \\;\\, \\text{is bijective}.\n\\end{equation*}\n\t\\item[b)] For $\\xi \\in \\mathbb{R} \\setminus {\\left\\{ \\lambda_1(\\boldsymbol{z})-2d, \\lambda_2(\\boldsymbol{z})-2d, \\ldots, \\lambda_h(\\boldsymbol{z})-2d\\right\\}}$, \n\\begin{equation*}\n\tA(\\boldsymbol{z}) - \\xi \\;\\, \\text{is bijective} \\; \\Rightarrow \\; D(\\boldsymbol{z}) - \\xi \\;\\, \\text{is bijective}.\n\t\\end{equation*}\n\\end{itemize}\n\nProof of Claim 1.\nFirst, we show item (a). Let $\\xi \\in \\mathbb{R} \\setminus {\\left\\{ \\lambda_1(\\boldsymbol{z}), \\ldots, \\lambda_h(\\boldsymbol{z}) \\right\\}}$ be such that,\n$D(\\boldsymbol{z}) - \\xi$ is bijective. Then, we must show that $A(\\boldsymbol{z}) - \\xi$ is injective and surjective:\n\n\\underline{Injective}. Let $u\\in {\\rm Ker}(A(\\boldsymbol{z})-\\xi)$ and since $\\xi \\not= \\lambda_i(\\boldsymbol{z})$, for $i \\in \\{1,\\ldots,n\\}$, we have  \n$\\left\\langle u,\\psi_i(\\boldsymbol{z}) \\right\\rangle= 0$ for all $i \\in \\{1,\\ldots,n\\}$. Therefore, $u \\in N(\\boldsymbol{z})^\\perp$ and from \\eqref{87678687678}, \n\\begin{equation*}\n\t(D(\\boldsymbol{z}) - \\xi)u = (A(\\boldsymbol{z}) - \\xi)u = 0.\n\\end{equation*}\nConsequently, we obtain $u=0$.\n\n\\medskip\n\\underline{Surjective}. Applying the surjection of $D(\\boldsymbol{z})-\\xi$, \nfor each $v \\in H$ there exists $u \\in H$, such that \n\\begin{equation}\\label{976876876874}\n\t(D(\\boldsymbol{z})-\\xi)u = v.\n\\end{equation}\nOn the other hand, we write $u= u_1 + u_2$, with $u_1\\in N(\\boldsymbol{z})$ and\n $u_2 \\in N(\\boldsymbol{z})^\\perp$, hence from equations \\eqref{87678687678} and \\eqref{976876876874}, we obtain\n\\begin{equation}\n\\label{6687687644423}\n\tv = \\sum_{i=1}^h (\\lambda_i(\\boldsymbol{z}) - 2d -\\xi) \\left\\langle u_1, \\psi_i(\\boldsymbol{z}) \\right\\rangle \\psi_i(\\boldsymbol{z}) + (A(\\boldsymbol{z}) - \\xi)u_2.\n\\end{equation}\nMoreover, since $\\xi \\not= \\lambda_i(\\boldsymbol{z})$, for $i \\in \\{1,\\ldots,n\\}$, it follows that \n\\begin{equation*}\n\t(A(\\boldsymbol{z}) - \\xi) {\\left[ \\frac{ \\psi_i(\\boldsymbol{z}) }{ \\lambda_i(\\boldsymbol{z}) - \\xi} \\right]} = \\psi_i(\\boldsymbol{z}),\n\\end{equation*}\nand hence applying it in \\eqref{6687687644423}, we have\n\\begin{eqnarray*}\n\tv & = & \\sum_{i=1}^h (A(\\boldsymbol{z}) - \\xi) {\\left[ {\\left( \\frac{\\lambda_i(\\boldsymbol{z}) - 2d \n\t- \\xi}{\\lambda_i(\\boldsymbol{z}) - \\xi} \\right)} \\left\\langle u_1, \\psi_i(\\boldsymbol{z}) \\right\\rangle \\psi_i(\\boldsymbol{z}) \\right]} \n\t+ (A(\\boldsymbol{z}) - \\xi)u_2 \n\t\\\\\n\t& = & (A(\\boldsymbol{z}) - \\xi) {\\left[ \\sum_{i=1}^h {\\left( \\frac{\\lambda_i(\\boldsymbol{z}) - 2d \n\t- \\xi}{\\lambda_i(\\boldsymbol{z}) - \\xi} \\right)} \\left\\langle u_1, \\psi_i(\\boldsymbol{z}) \\right\\rangle \\psi_i(\\boldsymbol{z}) + u_2 \\right]}.\n\\end{eqnarray*}\nThus, the operator $A(\\boldsymbol{z} - \\xi)$ is surjective.\n\n\\medskip\nNow, let us show item (b). Let $\\xi \\in \\mathbb{R} \\setminus {\\left\\{ \\lambda_1(\\boldsymbol{z})-2d, \\ldots, \\lambda_h(\\boldsymbol{z})-2d\\right\\}}$\nbe such that, $A(\\boldsymbol{z}) - \\xi$ is bijective. Similarly, we must show that $D(\\boldsymbol{z}) - \\xi$ is injective and surjective: \n\n\\underline{Injective}. Let $u \\in H$ be such that $(D(\\boldsymbol{z})-\\xi)u= 0$. \nThen writing $u=u_1 + u_2$, with $u_1 \\in N(\\boldsymbol{z})$ and $u_2\\in N(\\boldsymbol{z})^\\perp$, \nit follows from equation \\eqref{87678687678} that \n\\begin{equation}\n\\label{7978776476444}\n\t0 = \\sum_{i=1}^h (\\lambda_i(\\boldsymbol{z}) - 2d - \\xi) \\left\\langle u_1, \\psi_i(\\boldsymbol{z}) \\right\\rangle \\psi_i(\\boldsymbol{z}) + (A(\\boldsymbol{z}) - \\xi)u_2,\n\\end{equation}\nthus $(A(\\boldsymbol{z})-\\xi)u_2 \\in N(\\boldsymbol{z})$. Consequently, we have \n\\begin{eqnarray*}\n\t(A(\\boldsymbol{z}) - \\xi)u_2 & = & P(\\boldsymbol{z}) {\\left[ (A(\\boldsymbol{z}) - \\xi)u_2 \\right]} \n\t= P(\\boldsymbol{z}) A(\\boldsymbol{z})u_2  - \\xi P(\\boldsymbol{z})u_2 \n\t\\\\\n\t& = & \\sum_{i=1}^h \\left\\langle A(\\boldsymbol{z})u_2, \\psi_i(\\boldsymbol{z}) \\right\\rangle \\psi_i(\\boldsymbol{z}) - \\xi P(\\boldsymbol{z}) u_2\n\t\\\\\n\t& = & \\sum_{i=1}^h \\left\\langle u_2, A(\\boldsymbol{z})\\psi_i(\\boldsymbol{z}) \\right\\rangle \\psi_i(\\boldsymbol{z}) - \\xi P(\\boldsymbol{z})u_2\n\t \\\\\n\t& = & \\sum_{i=1}^h \\lambda_i(\\boldsymbol{z}) \\left\\langle u_2, \\psi_i(\\boldsymbol{z}) \\right\\rangle \\psi_i(\\boldsymbol{z}) - \\xi P(\\boldsymbol{z})u_2= 0 \n\\end{eqnarray*}\nsince $u_2 \\in N(\\boldsymbol{z})^\\perp$. By hypothesis $A(\\boldsymbol{z}) - \\xi$ is injective, thus $u_2= 0$. \nThen, from equation \\eqref{7978776476444} we obtain \n\\begin{equation*}\n\t\\sum_{i=1}^h (\\lambda_i(\\boldsymbol{z}) - 2d - \\xi) \\left\\langle u_1, \\psi_i(\\boldsymbol{z}) \\right\\rangle \\psi_i(\\boldsymbol{z}) = 0,\n\\end{equation*}\nand since $\\{\\psi_i(\\boldsymbol{z})\\}_{i=1}^h$ is a linearly dependent set of vectors, we have for each \n$i \\in \\{1, \\ldots,h\\}$, $\\left\\langle u_1, \\psi_i(\\boldsymbol{z}) \\right\\rangle= 0$,\nthus $\\left\\langle u_1, \\psi_i(\\boldsymbol{z}) \\right\\rangle=0$. Recall that by hypothesis $\\lambda_i(\\boldsymbol{z}) - 2d - \\xi \\not= 0$, \nfor all $i \\in \\{1,\\ldots,h\\}$. Therefore, we obtain $u_1=0$.\n\t\t\n\\medskip\n\\underline{Surjective}. Again, applying the surjection of $A(\\boldsymbol{z}) - \\xi$, \nfor each $v \\in H$ there exists $u \\in H$, such that \n\\begin{equation}\\label{9878977987r34}\n\t(A(\\boldsymbol{z}) - \\xi)u = v.\n\\end{equation}\nThen, writing $u= u_1+u_2$, with $u_1 \\in N(\\boldsymbol{z})$ and \n$u_2 \\in N(\\boldsymbol{z})^\\perp$, from equations \n\\eqref{87678687678} and \\eqref{9878977987r34}, we have\n\\begin{equation}\n\\label{6876876780987}\n   v= \\sum_{i=1}^h (\\lambda_i(\\boldsymbol{z}) - \\xi) { \\left\\langle u_1, \\psi_i(\\boldsymbol{z}) \\right\\rangle } \\psi_i(\\boldsymbol{z}) + (D(\\boldsymbol{z}) - \\xi)u_2.\n\\end{equation}\nMoreover, since $\\xi \\not= \\lambda_i(\\boldsymbol{z})-2d$, for $i \\in \\{1,\\ldots,n\\}$, \n\\begin{equation*}\n    (D(\\boldsymbol{z}) - \\xi) {\\left[ \\frac{ \\psi_i(\\boldsymbol{z}) }{ \\lambda_i(\\boldsymbol{z}) - 2d - \\xi} \\right]} = \\psi_i(\\boldsymbol{z})\n\\end{equation*}\nand then from \\eqref{6876876780987}, it follows that\n\\begin{eqnarray*}\n   v & = & \\sum_{i=1}^h (D(\\boldsymbol{z}) - \\xi) {\\left[ {\\left( \\frac{\\lambda_i(\\boldsymbol{z}) - \\xi}{\\lambda_i(\\boldsymbol{z}) -2d - \\xi} \\right)} \n   { \\left\\langle u_1, \\psi_i(\\boldsymbol{z}) \\right\\rangle } \\phi^i(\\boldsymbol{z}) \\right]} + (D(\\boldsymbol{z}) - \\xi)u_2 \n   \\\\\n   & = & (D(\\boldsymbol{z}) - \\xi) {\\left[ \\sum_{i=1}^h {\\left( \\frac{\\lambda_i(\\boldsymbol{z}) - \\xi}{\\lambda_i(\\boldsymbol{z}) -2d - \\xi} \\right)} \n   { \\left\\langle u_1, \\psi_i(\\boldsymbol{z}) \\right\\rangle } \\psi_i(\\boldsymbol{z})  + u_2 \\right]}.\n\t\t\\end{eqnarray*}\nTherefore, the operator $D(\\boldsymbol{z})-\\xi$ is surjective.\t\n\t\t\n\\bigskip\n\\underline{{Claim 2}}. The spectrum of the operator $D(\\boldsymbol{0})$ does not contain\nelements of the interval $(\\lambda - d, \\lambda + d)$, i.e.\n\t\t\t\\begin{equation*}\n\t\t\t\t\\sigma(D(\\boldsymbol{0})) \\cap (\\lambda - d, \\lambda + d) = \\emptyset.\n\t\t\t\\end{equation*}\n\t\t\t\nProof Claim 2. From item (b) of Claim 1, we have $\\sigma(D(\\boldsymbol{0})) \\subset \\sigma(A_0)$, and thus \t\t\t\n$$\n    \\sigma(D(\\boldsymbol{0})) \\cap (\\lambda - d, \\lambda + d) \\subset \\sigma(A_0) \\cap (\\lambda - d, \\lambda + d) = \\{ \\lambda \\}.\n$$\nSuppose that $\\lambda \\in \\sigma(D(\\boldsymbol{0}))  \\cap (\\lambda - d, \\lambda + d)$, that is to say, it is an isolated element of the \nspectrum of $D(\\boldsymbol{0})$. Therefore, $\\lambda$ is an eigenvalue of $D(\\boldsymbol{0})$, \nbut this is not possible since $D(\\boldsymbol{0}) - \\lambda$ is an injective operator (see the proof of item (b) of Claim 1). \nConsequently, we have \t\t\t\n$$\n    \\sigma(D(\\boldsymbol{0})) \\cap (\\lambda - d, \\lambda + d) = \\emptyset.\n$$\n\t\t\n\\medskip\nIt remains to show \\eqref{FINALPERT}. First, by definition $P(\\boldsymbol{z}) u$ \nis holomorphic for each $\\boldsymbol{z}$ in a neighborhood of $\\boldsymbol{0}$. Therefore, \nthe mapping $\\boldsymbol{z} \\mapsto P(\\boldsymbol{z})$ is holomorphic in this neighbohood.\nThen, the mapping $\\boldsymbol{z} \\ \\mapsto D(\\boldsymbol{z}) \\in \\mathcal{B}(H)$ \nis continuous. Moreover, since the subset of invertible operators in \n$\\mathcal{B}(H)$ is an open set, there exists \na (small) neighborhood $\\boldsymbol{0} $, such that the function \n$\\boldsymbol{z} \\mapsto (D(\\boldsymbol{z}) - \\lambda)^{-1} \\in \\mathcal{B}(H)$ is continuous. \n\nOn the other hand, there exists $d' \\in (0,d)$ such that \n$$\n   {\\left\\Vert (D(\\boldsymbol{0}) - \\lambda)^{-1} \\right\\Vert} \\leqslant \\frac{1}{ {\\rm dist}(\\lambda, \\sigma(D(\\boldsymbol{0}))) } \\leqslant \\frac{1}{d} < \\frac{1}{d^\\prime},\n$$\t\nsee Reed, Simon \\cite[Chapter VIII]{ReedSimon}. Therefore, by the continuity of the map\n$\\boldsymbol{z} \\mapsto (D(\\boldsymbol{z}) - \\lambda)^{-1} \\in \\mathcal{B}(H)$,\nthere exists a neighborhood of $\\boldsymbol{0}$, namely $W$, such that for all $\\boldsymbol{z} \\in W$\n\\begin{equation*}\n    {\\left\\Vert (D(\\boldsymbol{\\boldsymbol{z}}) - \\lambda)^{-1} \\right\\Vert} < \\frac{1}{d^\\prime}.\n\\end{equation*}\nThus for any $u \\in H$ and $\\boldsymbol{z} \\in W$, it follows that \n$$\n\\begin{aligned}\n      {\\Vert u \\Vert}&= {\\left\\Vert (D(\\boldsymbol{\\boldsymbol{z}}) - \\lambda)^{-1} {\\left[ {\\left( D(\\boldsymbol{\\boldsymbol{z}}) - \\lambda \\right)}u \\right]} \\right\\Vert} \n      \\\\\n      &\\leq {\\left\\Vert (D(\\boldsymbol{\\boldsymbol{z}}) - \\lambda)^{-1} \\right\\Vert} {\\left\\Vert (D(\\boldsymbol{\\boldsymbol{z}}) - \\lambda)u \\right\\Vert} \n       <  \\frac{1}{d^\\prime} \\, {\\left\\Vert (D(\\boldsymbol{\\boldsymbol{z}}) - \\lambda)u \\right\\Vert}.\n\\end{aligned}\n$$\t\t\t\nHence for $d'' \\in (0,d')$ and $\\xi \\in (\\lambda - d'',\\lambda + d'')$, we have \n\\begin{eqnarray*}\n   {\\left\\Vert (D(\\boldsymbol{\\boldsymbol{z}}) - \\xi)u \\right\\Vert} & \\geq & {\\left\\Vert (D(\\boldsymbol{\\boldsymbol{z}}) - \\lambda)u \\right\\Vert} - {\\left\\vert \\lambda - \\xi \\right\\vert} {\\lVert u \\rVert} \\\\\n   & \\geq & (d^\\prime - d'') {\\Vert u \\Vert}.\n\\end{eqnarray*}\nConsequently, for all \n$\\xi \\in (\\lambda - d'', \\lambda + d'')$, \n$\\xi$ is an element of the resolvent of $D(\\boldsymbol{z})$, that is \n$\\xi \\in \\rho(D(\\boldsymbol{z}))$. Thus for each $\\boldsymbol{z} \\in W$, we have \n$$(\\lambda-d',\\lambda+d') \\subset \\rho(D(\\boldsymbol{z})).$$ Finally, since for each \n$\\boldsymbol{z} \\in W$\n\\begin{equation*}\n\t\\sigma(A(\\boldsymbol{z})) \\setminus \\{\\lambda_1(\\boldsymbol{z}),\\ldots,\\lambda_h(\\boldsymbol{z})\\} \\subset \\sigma(D(\\boldsymbol{z})) \n\t\\setminus \\{\\lambda_1(\\boldsymbol{z}),\\ldots,\\lambda_h(\\boldsymbol{z})\\}, \n\\end{equation*}\nwe obtain from item (a) of Claim 1, that \n\\begin{equation*}\n\t\\sigma(A(\\boldsymbol{z})) \\setminus \\{\\lambda_1(\\boldsymbol{z}),\\ldots,\\lambda_h(\\boldsymbol{z})\\} \\cap (\\lambda-d^\\prime, \\lambda+d^\\prime)=\\emptyset,\n\\end{equation*}\nwhich finish the proof. \n\\end{proof}\n\n\\section{Bloch Waves Analysis}\n\\label{877853467yd56rtfe5rtfgeds76ytged}\n\nBloch waves analysis is important in the theory of solid-state physics. \nMore precisely, the displacement of an electron in a crystal \n(periodic setting) is often described by Bloch waves,\nand this application is supported by Bloch's Theorem which states \nthat, the energy eigenstates for an electron in a crystal can be written as \nBloch waves.\n\n\\medskip\nThe aim of this section is to extend the Bloch waves theory, which is known just for periodic functions\nto the considered stochastic setting,\nthat is, stationary functions composed with stochastic deformations, which is used here to describe non-crystalline matter.  \nTherefore, we would like to show that, the electron waves in a non-crystalline matter can have a basis \nconsisting entirely of Bloch wave energy eigenstates (now solution to a stochastic Bloch spectral cell equation). \nConsequently, we are extending the concept of electronic band structures to non-crystalline matter. \n\n \t\t\n\\subsection{The WKB method}\n\\label{683926ruesszs}\n\nHere we formally obtain the Bloch spectral cell equation\n(see Definition \\ref{92347828454trfhfd4rfghjls}), applying the \nasymptotic  Wentzel-Kramers-Brillouin (WKB for short) expansion method, that is, \nwe assume that the solution of equation \\eqref{jhjkhkjhkj765675233} is given by a\nplane wave. More precisely, for each $\\varepsilon> 0$ let us assume that, the solution $u_\\varepsilon(t,x,\\omega)$\nof the equation \\eqref{jhjkhkjhkj765675233} has the \nfollowing asymptotic expansion \n\\begin{equation}\n\\label{ansatz}\n   u_{\\varepsilon}(t,x,\\omega)= e^{2\\pi i S_{\\varepsilon}(t,x)} \\sum_{k=1}^{\\infty}\\varepsilon^k \n   u_k\\Big(t,x,\\Phi^{-1}\\left(\\frac{x}{\\varepsilon},\\omega\\right),\\omega\\Big),\n\\end{equation}\nwhere the functions $u_k(t,x,y,\\omega)$ are conveniently stationary in $y$, and $S_{\\varepsilon}$ \nis a real valued function to be established a posteriori\n(not necessarily a polynomial in $\\varepsilon$), which take part of the modulated plane wave \\eqref{ansatz} \nfrom $e^{2\\pi i S_{\\varepsilon}(t,x)}$. \n\nThe spatial derivative of the above ansatz \\eqref{ansatz} is  \n$$\n\\begin{aligned}\n\\nabla u_\\varepsilon&(t,x,\\omega)= e^{2i\\pi S_\\varepsilon(t,x)} \\big(2i\\pi \\nabla S_\\varepsilon(t,x)\\sum_{k=0}^\\infty \\varepsilon^k\\,\nu_k \\big(t,x,\\Phi^{-1}\\left(\\frac{x}{\\varepsilon},\\omega\\right),\\omega\\big)\n\\\\[5pt]\n&\\qquad + \\sum_{k=0}^\\infty \\varepsilon^k\\Big\\{\\left(\\partial_x u_k\\right) \\Big(t,x,\\Phi^{-1}\\left(\\frac{x}{\\varepsilon},\\omega\\right),\\omega\\Big)\n\\\\[5pt]\n&\\qquad + \\frac{1}{\\varepsilon}(\\nabla\\Phi)^{-1}\\left(\\Phi^{-1}\\left(\\frac{x}{\\varepsilon},\\omega\\right),\\omega\\right)\n\\left(\\partial_y u_k\\right)\\Big(t,x,\\Phi^{-1}\\left(\\frac{x}{\\varepsilon},\\omega\\right),\\omega\\Big)\\Big\\}\\Big)\n\\\\[5pt]\n&=e^{2i\\pi S_\\varepsilon(t,x)} \\Big(\\sum_{k=0}^\\infty \\varepsilon^k\\left(\\frac{{\\nabla}_z}{\\varepsilon} + 2i\\pi \\nabla S_\\varepsilon(t,x)\\right)\nu_k\\Big(t,x,\\Phi^{-1}(\\frac{x}{\\varepsilon},\\omega),\\omega\\Big)\n\\\\[5pt]\n&\\qquad +\\sum_{k=0}^\\infty \\varepsilon^k\\left(\\nabla_x u_k   \\right)\n\\Big(t,x,\\Phi^{-1}\\left(\\frac{x}{\\varepsilon},\\omega\\right),\\omega\\Big)\\Big).\n\\end{aligned}\n$$\n\nNow, computing the second derivatives of the expansion~\\eqref{ansatz} and writing as a cascade of the power of $\\varepsilon$, we have \n\\begin{equation}\n\\label{ansatz2}\n\\begin{aligned}\ne^{-2i\\pi S_\\varepsilon(t,x)} & {\\rm div}{\\big( A {( \\Phi^{-1}{\\left( \\frac{x}{\\varepsilon},\\omega \\right)},\\omega)} \n\\nabla u_\\varepsilon(t,x,\\omega) \\big)}\n\\\\\n&=\\frac{1}{\\varepsilon^2}\\Big( {\\rm div}_{\\! z} + 2i\\pi \\varepsilon \\nabla S_\\varepsilon(t,x) \\Big)\n\\Big( A{\\left( \\Phi^{-1}(\\cdot,\\omega),\\omega \\right)} \\Big( \\nabla_{\\!\\! z} + 2i\\pi \\varepsilon \\nabla S_\\varepsilon(t,x) \\Big)\n\\\\\n&\\qquad\\qquad\\qquad\\qquad u_0 ( t,x,\\Phi^{-1}(\\cdot,\\omega),\\omega)\\Big){\\Bigg\\rvert}_{z=x\/\\varepsilon}\n\\\\\n&+ \\frac{1}{\\varepsilon}\\Big( {\\rm div}_{\\! z} + 2i\\pi \\varepsilon \\nabla S_\\varepsilon(t,x) \\Big)\n\\Big( A{\\left( \\Phi^{-1}(\\cdot,\\omega),\\omega \\right)} \\Big( \\nabla_{\\!\\! z} + 2i\\pi \\varepsilon \\nabla S_\\varepsilon(t,x) \\Big)\n\\\\\n&\\qquad\\qquad\\qquad\\qquad u_1 ( t,x,\\Phi^{-1}(\\cdot,\\omega),\\omega)\\Big){\\Big\\rvert}_{z=x\/\\varepsilon}\n+ I_\\varepsilon, \n\\end{aligned}\n\\end{equation}\nwhere \n\\begin{eqnarray}\n&& I_\\varepsilon= \\sum_{k=0}^\\infty \\varepsilon^k\\Big( {\\rm div}_{\\! z} + 2i\\pi \\varepsilon \\nabla S_\\varepsilon(t,x) \\Big)\n\\Big( A{\\left( \\Phi^{-1}(\\cdot,\\omega),\\omega \\right)} \\Big( \\nabla_{\\!\\! z} + 2i\\pi \\varepsilon \\nabla S_\\varepsilon(t,x) \\Big)\\nonumber\n\\\\\n&&\\qquad\\qquad u_{k+2} ( t,x,\\Phi^{-1}(\\cdot,\\omega),\\omega)\\Big){\\Big\\rvert}_{z=x\/\\varepsilon}\\nonumber\n\\\\\n&&+\\frac{1}{\\varepsilon}\\Big( {\\rm div}_{\\! z} + 2i\\pi \\varepsilon \\nabla S_\\varepsilon(t,x) \\Big)\n\\Big( A{\\left( \\Phi^{-1}(\\cdot,\\omega),\\omega \\right)}\\nabla_x u_0 ( t,x,\\Phi^{-1}(\\cdot,\\omega),\\omega)\\Big){\\Big\\rvert}_{z=x\/\\varepsilon}\\nonumber\n\\\\\n&&+\\sum_{k=0}^\\infty \\varepsilon^k\\Big( {\\rm div}_{\\! z} + 2i\\pi \\varepsilon \\nabla S_\\varepsilon(t,x) \\Big)\n\\Big( A{\\left( \\Phi^{-1}(\\cdot,\\omega),\\omega \\right)}\\nabla_x u_{k+1} ( t,x,\\Phi^{-1}(\\cdot,\\omega),\\omega)\\Big){\\Big\\rvert}_{z=x\/\\varepsilon}\\nonumber\n\\\\\n&&+ \\frac{1}{\\varepsilon}{\\rm div}_{\\! x}\\Big(A{\\left( \\Phi^{-1}(\\cdot,\\omega),\\omega \\right)}\\Big( \\nabla_{\\!\\! z} + 2i\\pi \\varepsilon \\nabla S_\\varepsilon(t,x) \\Big)\nu_0 ( t,x,\\Phi^{-1}(\\cdot,\\omega),\\omega)\\Big){\\Big\\rvert}_{z=x\/\\varepsilon}\\nonumber\n\\\\\n&&+ \\sum_{k=0}^\\infty \\varepsilon^k {\\rm div}_{\\! x}\\Big(A{\\left( \\Phi^{-1}(\\cdot,\\omega),\\omega \\right)}\\Big( \\nabla_{\\!\\! z} + 2i\\pi \\varepsilon \\nabla S_\\varepsilon(t,x) \\Big)\nu_{k+1} ( t,x,\\Phi^{-1}(\\cdot,\\omega),\\omega)\\Big){\\Big\\rvert}_{z=x\/\\varepsilon}\\nonumber\n\\\\\n&&+\\sum_{k=0}^\\infty \\varepsilon^k{\\rm div}_{\\! x}\\Big(A{\\left( \\Phi^{-1}(\\cdot,\\omega),\\omega \\right)}\\nabla_{\\!\\! x} \nu_k{\\Big( t,x,\\Phi^{-1}(\\cdot,\\omega),\\omega\\Big)}\\Big){\\Big\\rvert}_{z=x\/\\varepsilon}.\n\\end{eqnarray}\n\nProceeding in the same way with respect to the temporal derivative, we have\n\\begin{eqnarray}\\label{ansatz3}\n&&e^{-2i\\pi S_\\varepsilon(t,x)}\\,{\\partial}_t u_\\varepsilon\\nonumber\n\\\\\n&&\\qquad\\qquad=\n\\frac{1}{\\varepsilon^2} \\Big(2i\\pi \\varepsilon^2 {\\partial}_{t} S_\\varepsilon(t,x) \\Big) u_0\\Big( t,x,\\Phi^{-1}\\left(\\frac{x}{\\varepsilon},\\omega\\right),\\omega\\Big)\\nonumber\n\\\\\n&&\\qquad\\qquad+\\frac{1}{\\varepsilon} \\Big(2i\\pi \\varepsilon^2 {\\partial}_{t} S_\\varepsilon(t,x) \\Big) \nu_1\\Big( t,x,\\Phi^{-1}\\left(\\frac{x}{\\varepsilon},\\omega\\right),\\omega\\Big)\\nonumber\n\\\\\n&&\\qquad\\qquad+\\Big(2i\\pi \\varepsilon^2 {\\partial}_{t} S_\\varepsilon(t,x) \\Big) \\sum_{k=0}^\\infty \\varepsilon^k\nu_{k+2}\\Big( t,x,\\Phi^{-1}\\left(\\frac{x}{\\varepsilon},\\omega\\right),\\omega\\Big)\\nonumber\n\\\\\n&&\\qquad\\qquad\\qquad\\qquad+\\sum_{k=0}^\\infty \\varepsilon^k {\\partial}_tu_k\\Big( t,x,\\Phi^{-1}\\left(\\frac{x}{\\varepsilon},\\omega\\right),\\omega\\Big).\n\\end{eqnarray}\nThus, if we insert the equations \\eqref{ansatz2} and \\eqref{ansatz3} in \\eqref{jhjkhkjhkj765675233} \nand compute the $\\varepsilon^{-2}$ order term, we \narrive at \n$$\n  L^\\Phi(\\varepsilon \\nabla S_\\varepsilon(t,x)) u_0 {\\big( t,x,\\Phi^{-1}(\\cdot,\\omega),\\omega\\big)}\n  = 2 \\pi \\big( \\varepsilon^2 \\partial_t S_\\varepsilon(t,x)\\big)u_0 {\\big( t,x,\\Phi^{-1}(\\cdot,\\omega),\\omega\\big)},\n$$\nwhere for each $\\theta \\in \\mathbb R^n$, the linear operator $L^\\Phi(\\theta)$ is defined by\n\\begin{equation}\n\\label{EqEsp}\n\\begin{aligned}\nL^\\Phi(\\theta)[\\cdot]:=& -\\big( {\\rm div}_{\\! z} + 2i\\pi \\theta \\big)\n\\big(A{( \\Phi^{-1}(z,\\omega),\\omega)} {\\big( \\nabla_{\\!\\! z} + 2i\\pi \\theta \\big)} [\\cdot] \\big)\n\\\\ \n&+V {\\big(\\Phi^{-1}\\left(z,\\omega\\right),\\omega\\big)} [\\cdot].\n\\end{aligned}\n\\end{equation}\nTherefore, $2 \\pi \\Big( \\varepsilon^2 \\partial_t S_\\varepsilon(t,x)\\Big)$ is an eigenvalue of $ L^\\Phi(\\varepsilon \\nabla S_\\varepsilon(t,x))$.\nConsequently, if $\\lambda(\\theta)$ is any eigenvalue of $L^\\Phi(\\theta)$ (which is sufficiently regular with respect to $\\theta$), then\nthe following (eikonal) Hamilton-Jacobi equation must be satisfied\n$$\n   2 \\pi \\varepsilon^2 \\partial_t S_\\varepsilon(t,x) - \\lambda(\\varepsilon \\nabla S_\\varepsilon(t,x))= 0. \n$$\nThus, if we suppose for $t=0$ (companion to \\eqref{ansatz}) the modulated plane wave initial data \n\\begin{equation}\n\\label{ansatzID}\n   u_{\\varepsilon}(0,x,\\omega)= e^{2i\\pi \\frac{\\theta \\cdot x}{\\varepsilon}} \\sum_{k=1}^{\\infty}\\varepsilon^k \n   u_k\\Big(0,x,\\Phi^{-1}\\left(\\frac{x}{\\varepsilon},\\omega\\right),\\omega\\Big),\n\\end{equation}\nthen the unique solution for the above Hamilton-Jacobi equation is, for each parameter $\\theta \\in \\mathbb R^n$, \n\\begin{equation}\n\\label{SEP}\n   S_\\varepsilon(t,x)= \\frac{\\lambda(\\theta) \\  t}{2 \\pi \\varepsilon^2} + \\frac{\\theta \\cdot x}{\\varepsilon}.\n\\end{equation}\n\nTo sum up, the above expansion, that is\nthe solution $u_{\\varepsilon}$ of the equation \\eqref{jhjkhkjhkj765675233} \nwith initial data given respectively by \\eqref{ansatz} and \\eqref{ansatzID}, \nsuggests the following \n\n\\begin{definition}[Bloch or\nshifted spectral cell equation] \n\\label{DEFBLOCHCELL} Let $\\Phi$ be a stochastic deformation. \nFor any $\\theta \\in \\mathbb R^n$ fixed, the following time independent asymptotic equation \n\\begin{equation}\n\\label{92347828454trfhfd4rfghjls}\n\\left\\{\n\\begin{array}{l}\nL^\\Phi(\\theta) [\\Psi(z,\\omega)]= \\lambda \\ \\Psi(z,\\omega), \\hspace{40pt} \\text{in $\\mathbb R^n \\times \\Omega$}, \n\\\\[5pt]\n\\hspace{32pt} \\Psi(z, \\omega) = \\psi {\\left( \\Phi^{-1} (z, \\omega), \\omega \\right)}, \\quad \\text{$\\psi$ is a stationary function},\n\t\t\t\\end{array}\n\t\t\t\\right.\n\t\t\\end{equation}\nis called Bloch's spectral cell equation companion to the Schr\\\"odinger equation in \\eqref{jhjkhkjhkj765675233},\nwhere $L^\\Phi(\\theta)$ is given by \\eqref{EqEsp}.\nMoreover, each $\\theta \\in \\mathbb R^n$ is called a Bloch frequency, $\\lambda(\\theta)$ is called a Bloch energy and the corresponded \n$\\Psi(\\theta)$ is called a Bloch wave. Moreover, if $\\Phi$ is well understood in the context, then $L \\equiv L^\\Phi$. \n\\end{definition}\n\n\\medskip\nThe unknown $(\\lambda,\\Psi)$ in \\eqref{92347828454trfhfd4rfghjls}, which is an eigenvalue-eigenfunction pair, is obtained \nby the associated variational formulation, that is \n\\begin{equation}\n\\label{FORMVARIAC}\n\\begin{aligned}\n&\\langle L(\\theta)[F], G\\rangle\n\\\\\n&= \\int_\\Omega \\int_{\\Phi([0,1)^n,\\omega)} \\!\\!\\!\\!\\!\\!\\!\\!\\! A( \\Phi^{-1}(z, \\omega), \\omega) (\\nabla_{\\!\\! z} + 2i\\pi\\theta)  F(z,\\omega) \\cdot\n \\overline{{( \\nabla_{\\!\\! z} + 2i\\pi\\theta)} G(z,\\omega)} \\, dz \\, d\\mathbb{P}(\\omega) \n \\\\[5pt]\n &+ \\int_\\Omega \\int_{\\Phi([0,1)^n,\\omega)} V{( \\Phi^{-1}(z, \\omega), \\omega)} \\ F(z,\\omega) \\, \n \\overline{G(z,\\omega)} \\, dz \\, d\\mathbb{P}(\\omega). \n\\end{aligned}\n\\end{equation}\n\n\\begin{remark}\n\\label{GROUPNECE}\nOne remarks that, $\\lambda= \\lambda(\\theta) \\in \\mathbb R$, that is to say, \n$\\lambda$ depends on the parameter $\\theta$.  \nHowever, $\\lambda$ could not depend on $\\omega$, since \nthe homogeneized effective matrix is obtained from the \nHessian of $\\lambda$ at some point $\\theta^*$, and\nshould be constant.   \nTherefore, the probabilistic variable $\\omega$ could not be considered as fixed parameter\nin \\eqref{92347828454trfhfd4rfghjls}.\n\\end{remark}\n\n\n\\subsection{Sobolev spaces on groups}\n\\label{9634783yuhdj6ty}\n\nThe main motivation to study Sobolev spaces on groups, besides\nbeing an elegant and modern mathematical theory, is related to the\neigenvalue problem: $$ \\text{Find $\\lambda(\\theta) \\in \\mathbb{R}$ and $\\Psi(\\theta) \\in \\mathcal{H}_\\Phi \\setminus \\{0\\}$\nsatisfying \\eqref{92347828454trfhfd4rfghjls}.}$$\nIndeed, we may use a compactness argument, \nthat is the space $\\mathcal{H}_\\Phi$ is compactly embedded in $\\mathcal{L}_\\Phi$,\nin order to solve the associated variational formulation \\eqref{FORMVARIAC}. Although,\nas observed in Remark \\ref{GROUPNECE}, $\\omega \\in \\Omega$ can not be fixed, hence \nwe are going to establish an equivalence between the space $\\mathcal{H}_\\Phi$ and\nthe Sobolev space on groups, and then consider a related Rellich-Kondrachov's Theorem. \nThis is the main issue of this section. \nLet us recall that, the subject of Sobolev spaces on Abelian locally compact groups,\nto the best of our knowledge,\nwas introduced by P. G\\'orka, E. G. Reyes \\cite{GorkaReyes}. \n\n\\medskip\nTo begin, we sum up some definitions and properties of topological groups, \nwhich will be used along this section. Most of the material could be found in \nE. Hewitt, A. Ross \\cite{HewittRoss} and G. B. Folland \\cite{Folland2}\n(with more details). \n\n\\medskip\nA nonempty set $G$ endowed with an application, $\\ast : G \\! \\times \\! G \\to G$,\nis called a group, when for each $x, y, z \\in G$:\n\\begin{itemize}\n\\item[1.] ${ (x\\ast y) \\ast z = x \\ast (y \\ast z) }$;\n\\item[2.] There exists ${e \\in G }$, such that ${ x \\ast e = e \\ast x = e }$;\n\\item[3.] For all ${ y \\in G }$, there exists ${y^{-1}\\in G }$, such that ${ y \\ast y^{-1} = y^{-1} \\ast y = e }$.\n\t\\end{itemize}\nMoreover, if $x \\ast y = y \\ast x$, then $G$ is called an Abelian group. \nFrom now on, we write for simplicity $x \\, z$ instead of $x \\ast z$. \nA topological group is a group $G$  together with a topology, such that,\nboth the group's binary operation $(x,y) \\mapsto x \\, y$,\nand the function mapping group elements to their respective inverses \n$x \\mapsto x^{-1}$\nare continuous functions with respect to the topology.\nUnless the contrary is explicit stated, any group mentioned here is \na locally compact Abelian (LCA for short) group, and \nwe may assume without loss of generality that, \nthe associated topology is Hausdorff \n(see G. B. Folland \\cite{Folland2}, Corollary 2.3).\n\n\\medskip\nA complex value function\n$\\xi : G \\to \\mathbb{S}^1$ is called a character of $G$, when \n$$\n \\xi(x \\, y) = \\xi(x) \\xi(y),\n\\quad  \\quad  \\text{(for each $x, y \\in G$)}.\n $$\n We recall that, the set of characters of $G$ is an Abelian group\n\n with the usual product of functions, identity element $e= 1$, and\n inverse element $\\xi^{-1} = \\overline{\\xi}$.\n The characters' group of the topological group $G$, called\nthe dual group of $G$ and denoted by $G^\\wedge$,  \nis the set of all continuous characters, that is to say \n$$\n G^\\wedge:= \\{ \\xi : G \\to \\mathbb{S}^1 \\; ; \\; \\text{$\\xi$ is a continuous homomorphism}\\}.\n$$\nMoreover, we may endow $G^\\wedge$ with a topology with respect to which,\n$G^\\wedge$ itself is a LCA group.  \n\n\\medskip\nWe denote by $\\mu$, $\\nu$ the unique (up to a positive multiplicative constant) Haar mesures in $G$ and $G^\\wedge$ respectively. \nThe $L^p$ spaces over $G$ and its dual are defined as usual, with their respective mesures. \nLet us recall two important properties when $G$ is compact:\n\\begin{equation}\n\\label{CARACGCOMP}\n\\begin{aligned}\n&i) \\quad \\text{If $\\mu(G)= 1$, then $G^\\wedge$ is an orthonormal set in $L^2(G;\\mu)$}.\n\\\\[5pt]\n&ii) \\quad \\text{The dual group $G^\\wedge$ is discrete, and $\\nu$ is the countermeasure}. \n\\end{aligned}\n\\end{equation}\n\n\\medskip\nOne remarks that, the study of Sobolev \nspaces on LCA groups uses essentially the concept of Fourier Transform, then we have the following   \n\\begin{definition}\nGiven a complex value function $f \\in L^1(G;\\mu)$, the function $\\widehat{f}: G^\\wedge \\to \\mathbb{C}$, defined by\n\\begin{equation}\n\t\\widehat{f}(\\xi):= \\int_G f(x) \\, \\overline{\\xi(x)} \\, d\\mu(x)\n\\end{equation}\t\nis called the Fourier transform of $f$ on $G$.\n\\end{definition}\nUsually, the Fourier Transform of $f$ is denoted by $\\clg{F}f$ to emphasize that it is an operator, \nbut we prefer to adopt the usual notation $\\widehat{f}$. \nMoreover, we recall that the Fourier transform is an homomorphism from $L^1(G;\\mu)$ to $C_0(G^\\wedge)$ \n(or $C(G^\\wedge)$ when $G$ is compact), see Proposition 4.13 in \\cite{Folland2}. Also we address the reader to \n \\cite{Folland2}, Chapter 4, for the Plancherel Theorem\nand the Inverse Fourier Transform. \n\n\\medskip\nBefore we establish the definition of (energy) Sobolev spaces on LCA groups, let us\nconsider the following set\n$$\n\\begin{aligned}\n  {\\rm P}= \\{p: G^\\wedge \\times & G^\\wedge \\to [0,\\infty) \/ \n  \\\\\n  \\; & \\text{$p$ is a continuous invariant pseudo-metric on $G^\\wedge$} \\}.\n\\end{aligned} \n$$\nThe Birkhoff-Kakutani Theorem (see \\cite{HewittRoss} p.68) \nensures that, the set P is not empty. \nAny pseudo-metric $p \\in {\\rm P}$ is well defined for each $(x,y) \\in G^\\wedge \\times G^\\wedge$, hence we may define\n\\begin{equation}\n\\label{Gamma}\n\\gamma(x):= p(x,e) \\equiv p(x,1).\n\\end{equation} \nMoreover, one observes that $\\gamma(1)= 0$.\nThen, we have the following \n\\begin{definition}[Energy Sobolev Spaces on LCA Groups]\n\\label{SOBOLEVESPACES}\nLet $s$ be a non-negative real number and $\\gamma(x)$ be given by \\eqref{Gamma}\nfor some fixed $p \\in {\\rm P}$. The energy Sobolev space \n$H^s_\\gamma(G)$ is the set of functions $f \\in L^2(G;\\mu)$, such that\n\\begin{equation}\n     \\int_{G^\\wedge} (1+\\gamma(\\xi)^2)^s  \\, |\\widehat{f}(\\xi)|^2 d\\nu(\\xi)< \\infty.\n\\end{equation}\nMoreover, given a function $f \\in H^s_\\gamma(G)$ its norm is defined as \n\\begin{equation}\n   \\Vert f \\Vert_{H^s_\\gamma(G)} := \\left( \\int_{G^\\wedge} \\left(1+\\gamma(\\xi)^2 \\right)^s \\vert \\widehat{f}(\\xi) \\vert^2 d\\nu(\\xi) \\right)^{1\/2}.\n\\end{equation}\n\\end{definition}\nBelow, taking specific functions $\\gamma$, the usual Sobolev spaces on $\\mathbb R^d$ and \nother examples are considered. In particular,   \nthe Plancherel Theorem implies that, $H^0_\\gamma(G)=  L^2(G;\\mu)$.\n\n\n\\begin{example}\n\\label{EXAMPLERN}\nLet $G= (\\mathbb R^n, +)$ which is known to be a LCA group, and consider \nits dual group $(\\mathbb{R}^n)^\\wedge = \\{ \\xi_y \\; ; \\; y\\in\\mathbb{R}^n \\}$,\nwhere for each $x \\in \\mathbb R^n$\n\\begin{equation}\n\\label{caracterunitario}\n  \\xi_y(x) = e^{2 \\pi i \\, y \\cdot x},\n\\end{equation}\nhence $|\\xi_y(x)|= 1$ and $\\xi_0(x)= 1$. \nOne remarks that, here we denote (without invocation of vector space structure)\n$$\n    a \\cdot b= a_1 b_1 + a_2 b_2 + \\ldots + a_n b_n, \\quad \\text{(for all $a,b \\in G$)}.\n$$\nFor any $x, y \\in \\mathbb R^n$ let us consider \n$$\n  p(\\xi_x,\\xi_y)= 2\\pi  \\|x - y\\|, \n$$\nwhere $\\| \\cdot \\|$ is the Euclidean norm in $\\mathbb R^n$. Hence $\\gamma(\\xi_x)= p(\\xi_x,1)= 2 \\pi \\|x\\|$.  \nSince $(\\mathbb{R}^n)^\\wedge \\cong \\mathbb{R}^n$, the Sobolev space $H^s_\\gamma(G)$ coincide \nwith the usual Sobolev space on $\\mathbb R^n$. \n\\end{example}\n\n\\begin{example}\n\\label{6576dgtftdefd}\nLet us recall that, the set $[0,1)^n$ endowed with the binary\noperation \n$$\n   (x,y) \\in [0,1)^n \\! \\times \\! [0,1)^n \\;\\; \\mapsto \\;\\; x+y - \\left\\lfloor x+y \\right\\rfloor \\in [0,1)^d\n$$ \nis an Abelian group, and the function \n$\\Lambda: \\mathbb{R}^n \\to [0,1)^n$, $\\Lambda(x):= x - \\left\\lfloor x \\right\\rfloor$\nis an homomorphism of groups. Moreover, under the \ninduced topology by $\\Lambda$, that is to say \n$$\n  \\{U \\subset [0,1)^n \\; ; \\; \\Lambda^{-1}(U) \\; \\text{is an open set of} \\;\\, \\mathbb{R}^n \\}, \n$$\n $[0,1)^n$ is a compact Abelian group, which is called $n-$dimensional Torus and denoted by \n$\\mathbb{T}^n$. Its dual group is characterized by the integers $\\mathbb Z^n$, that is \n$$\n\\text{\n$(\\mathbb{T}^n)^\\wedge = \\{ \\xi_m \\; ; \\; m \\in \\mathbb{Z}^n \\}$, where $\\xi_m(x)$ is given by  \n\\eqref{caracterunitario} for all $x \\in \\mathbb{R}^n$}. \n$$\nFor each $m,k \\in \\mathbb Z^n$, we consider \n$$\n   p(\\xi_m,\\xi_k)= 2\\pi \\sum_{j=1}^n{\\vert m_j - k_j \\vert},\n   \\quad \\text{and thus $\\gamma(\\xi_m)= 2 \\pi \\sum_{j=1}^n{\\vert m_j \\vert}$}.\n$$\nThen, the Sobolev space $H^s_\\gamma(\\mathbb{T}^n)$ coincide \nwith the usual Sobolev space on $\\mathbb{T}^n$.\n\t\t\n\\smallskip\nNow, following the above discussion let us consider the infinite Torus \n$\\mathbb{T}^I$, where $I$ is an index set. Since an arbitrary product of compact spaces is compact in the \nproduct topology (Tychonoff Theorem), $\\mathbb{T}^I$ is a compact Abelian group. Here, \nthe binary operation on $ \\mathbb{T}^I \\times  \\mathbb{T}^I$ is defined coordinate by coordinate, that is, for each \n$\\ell \\in I$ \n$$\n   g_\\ell + h_\\ell:= g_\\ell + h_\\ell - \\left\\lfloor g_\\ell + h_\\ell \\right\\rfloor.\n$$\nMoreover, the set $\\mathbb{Z}^I_{\\rm c} := \\{ m \\in \\mathbb{Z}^I; \\text{{\\rm supp} $m$ is compact} \\}$\ncharacterizes the elements of the dual group $(\\mathbb{T}^I)^\\wedge$.\n Indeed, applying Theorem 23.21 in \n\\cite{HewittRoss}, similarly we have  \n$$\n   (\\mathbb{T}^I)^\\wedge = {\\left\\{ \\xi_m \\; ; \\; m\\in \\mathbb{Z}^I_{\\rm c} \\right\\}},\n$$\nwhere $\\xi_m(k)$ is given by  \n\\eqref{caracterunitario} for each $m,k \\in \\mathbb{Z}_{\\rm c}^I$, the pseudo-metric\n$$\n   p(\\xi_m,\\xi_k)= 2\\pi \\sum_{\\ell \\in I}{\\vert m_\\ell - k_\\ell \\vert},\n   \\quad \\text{and $\\gamma(\\xi_m)= 2 \\pi \\sum_{\\ell \\in I}{\\vert m_\\ell \\vert}$}.\n$$\nConsequently, we have establish the Sobolev spaces $H^s_{\\gamma}(\\mathbb{T}^I)$.\n\\end{example}\n\n\\subsubsection{Groups and Dynamical systems}\n\\label{kjh876}\n\nIn this section, we are interested to come together the discussion \nabout dynamical systems studied in Section \\ref{628739yhf}\nwith the theory developed in the last section \nfor LCA groups. To this end, we consider \nstationary functions in the continuous sense (continuous dynamical systems). \nMoreover, we recall that all the groups in this paper are \nassumed to be Hausdorff.  \n\n\\medskip\nTo begin, let $G$ be a locally compact group with Haar measure $\\mu$,\nwe know that $\\mu(G)< \\infty$ if, and only if, $G$ is compact. \nTherefore, we consider from now on that $G$ is a compact Abelian group, \nhence $\\mu$ is a finite measure and, up to a normalization, $(G,\\mu)$ is a probability space.\nWe are going to consider the dynamical systems, $\\tau: \\mathbb R^n \\times G \\to G$, defined by \n\\begin{equation}\n\\label{TAUFI}\n   \\tau(x) \\omega:= \\varphi(x) \\, \\omega,\n\\end{equation}\nwhere $\\varphi: \\mathbb R^n \\to G$ is a given (continuous) homomorphism. \nIndeed, first $\\tau(0) \\omega= \\omega$ and \n$\\tau(x+y, \\omega)= \\varphi(x) \\varphi(y) \\omega= \\tau(x,\\tau(y)\\omega)$. \nMoreover, since $\\mu$ is a translation invariant Haar measure, the \nmapping $\\tau(x,\\cdot): G \\to G$ is $\\mu-$measure preserving. \nRecall from Remark \\ref{REMERG} we have assumed that, \nthe dynamical systems we are interested here are\nergodic. Then, it is important to characterize the conditions\nfor the mapping $\\varphi$, under which the dynamical system defined by \n\\eqref{TAUFI} is ergodic. To this end, first let us consider the following \n\n\\begin{lemma}\n\\label{DIST}\nLet $H$ be a topological group, $F \\subset H$ closed, $F \\neq H$ and $x \\notin F$.\nThen, there exists a neighborwood $V$ of the identity $e$, such that\n$$\n   F V \\cap x V= \\emptyset. \n$$\n\\end{lemma}\n\n\\begin{proof}\nFirst, we observe that:\n\ni) Since $F \\subset H$ is closed and $F \\neq H$, there\nexists a neighborwood $U$ of the identity $e$,\nsuch that $F \\cap x U= \\emptyset$. \n\nii) There exists a symmetric neighborwood $V$ of the identity $e$,\nsuch that $VV \\subset U$. \n\nNow, suppose that $F V \\cap x V \\neq \\emptyset$. \nTherefore, there exist $v_1, v_2 \\in V$ and $k_0 \\in F$ such that, $k_0 v_1= x v_2$.   \nConsequently, $k_0= x v_2 v_1^{-1}$ and from $(ii)$ it follows that, $k_0 \\in x U$. \nThen, we have a contradiction from $(i)$. \n\\end{proof}\n\n \\underline {\\bf Claim 1:} The dynamical system defined \nby \\eqref{TAUFI} is ergodic if, and only if, \n$\\varphi(\\mathbb R^n)$ is dense in $G$. \n\n\\smallskip\nProof of Claim 1: 1. Let us show first the necessity. Therefore, we suppose that \n$\\varphi(\\mathbb R^n)$ is not dense in $G$, that is $K:= \\overline{\\varphi(\\mathbb R^n)} \\neq G$. \nThen, applying Lemma \\ref{DIST}\nthere exists a neighborhood $V$ of $e$, such that $K V \\cap x V= \\emptyset$,\nfor some $x \\notin K$. Recall that the Haar measure on open sets are positive, \nmoreover\n$$\n   K V= \\bigcup_{k \\in K} k V,\n$$\nwhich is an open set, thus we have \n$$\n   0< \\mu(K V) + \\mu(x V) \\leq 1. \n$$\nConsequently, it follows that $0< \\mu(\\varphi(\\mathbb R^n) V)< 1$. For convenience, le us denote \n$E= \\varphi(\\mathbb R^n) V$, hence $\\tau(x) E= E$ for each $x \\in \\mathbb R^n$. \nThen, the dynamical system $\\tau$ is not ergodic, since $E \\subset G$ is a $\\tau$-invariant set \nwith $0< \\mu(E)< 1$. \n\n\\medskip\n2. It remains to show the sufficiency. \nLet $E \\subset G$ be a $\\mu-$measurable $\\tau$-invariant set,\nhence $\\omega E= E$ for each $\\omega \\in \\varphi(\\mathbb R^n)$. Assume \nby contradiction that, $0< \\mu(E)< 1$, thus $\\mu(G \\setminus E)> 0$.\nDenote by $\\mathcal{B}_G$ the Borel $\\sigma-$algebra on $G$, and define, \n$\\lambda:= \\mu_{\\lfloor E}$, that is $\\lambda(A)= \\mu(E \\cap A)$ for all \n$A \\in \\mathcal{B}_G$. Recall that $G$ is not necessarily metric, therefore, it is not\nclear if each Borel set is $\\mu-$measurable. Then, it follows that: \n\n$(i)$ For any $A \\in \\mathcal{B}_G$ fixed, the mapping \n$\\omega \\in G \\mapsto \\lambda(\\omega A)$ is continuous. \nIndeed, for $\\omega \\in G$ and $A \\in \\mathcal{B}_G$, we have\n$$\n\\begin{aligned}\n \\lambda(\\omega A)&= \\int_G 1_E(\\varpi) 1_{\\omega A}(\\varpi) d\\mu(\\varpi)\n \\\\[5pt]\n &= \\int_G 1_E(\\varpi) 1_{A}(\\omega^{-1} \\varpi) d\\mu(\\varpi) \n= \\int_G 1_E(\\omega \\varpi) 1_{A}(\\varpi) d\\mu(\\varpi).\n\\end{aligned}\n$$\nTherefore, for $\\omega, \\omega_0 \\in G$\n$$\n\\begin{aligned}\n|\\lambda(\\omega A) - \\lambda(\\omega_0 A)|&= \\big| \\int_G \\big(1_E(\\omega \\varpi) - 1_E(\\omega_0 \\varpi)\\big) 1_A(\\varpi)  d\\mu(\\varpi) \\big|\n\\\\\n &\\leq \\big(\\mu(A)\\big)^{1\/2}\n \\big( \\int_G  |1_E(\\omega \\varpi) -  1_E(\\omega_0 \\varpi)|^2 d\\mu(\\varpi) \\big)^{1\/2}\n  \\tobo{\\omega \\to \\omega_0} 0. \n\\end{aligned}\n$$\n\n$(ii)$ $\\lambda$ is invariant, i.e. for all $\\omega \\in G$, and $A \\in \\mathcal{B}_G$, $\\lambda(\\omega A)= \\lambda(A)$. \nIndeed, for each $\\omega \\in \\varphi(\\mathbb R^d)$, and $A \\in \\mathcal{B}_G$, we have \n$$\n   (\\omega A) \\cap E= (\\omega A) \\cap (\\omega E)= \\omega (A \\cap E).  \n$$\nThus since $\\mu$ is invariant, $\\mu_{\\lfloor E}(\\omega A)= \\mu_{\\lfloor E}(A)$. Consequently,\ndue to item $(i)$ and $\\overline{\\varphi(\\mathbb R^d)}= G$, it follows that $\\lambda$ is invariant. \n\nFrom item $(ii)$ the Radon measure $\\lambda$ is a Haar measure on $G$. By the uniqueness \nof the Haar measure on $G$, there exists $\\alpha> 0$, such that for all $A \\in \\mathcal{B}_G$,\n$\\alpha \\lambda(A)= \\mu(A)$. In particular, $\\alpha \\lambda(G \\setminus E)= \\mu(G \\setminus E)$.\nBut $\\lambda(G \\setminus E)= 0$ by definition and $\\mu(G \\setminus E)> 0$, which is a contradiction\nand hence $\\tau$ is ergodic. \n\n\\begin{remark}\n1. One remarks that, in order to show that $\\tau$ given by \\eqref{TAUFI} is ergodic, it was not used\n  that $\\varphi$ is continuous, nor that $G$ is metric. Compare with the statement in \\cite{JikovKozlovOleinik} \n  p.225 (after Theorem 7.2). \n\n2. From now on, we assume that $\\varphi(\\mathbb R^n)$ is dense in $G$. \n\\end{remark}\n\n\\medskip\nNow, for the dynamical system established before, the main issue is to show how the Sobolev space \n$H^1_{\\gamma}(G)$ is related with the space $\\mathcal{H}_\\Phi$ given by \\eqref{SPACEHPHI} for $\\Phi= Id$, \nthat is \n$$\n    \\mathcal{H}= {\\big\\{f(y, \\omega); \\; f \\in H^1_{\\rm loc}(\\mathbb{R}^n; L^2(G)) \\;\\; \\text{stationary} \\big\\}},\n$$\nwhich is a Hilbert space endowed with the following inner product  \n$$\n{\\langle f,g \\rangle}_{\\mathcal{H}}= \\int_G  f(0, \\omega) \\, \\overline{g(0, \\omega) } \\, d\\mu(\\omega)\n+ \\int_G \\nabla_{\\!\\! y} f(0, \\omega) \\cdot \\overline{ \\nabla_{\\!\\! y} g(0, \\omega) }\\, d\\mu(\\omega).\n$$\nLet $\\chi$ be a character on $G$, i.e. $\\chi \\in G^\\wedge$. Since $\\varphi: \\mathbb R^n \\to G$ is a continuous homomorphism, the \nfunction $(\\chi \\circ \\varphi): \\mathbb R^n \\to \\mathbb C$\nis a continuous character in $\\mathbb R^n$. More precisely, given any fixed $\\chi \\in G^\\wedge$ we may find \n$y \\in \\mathbb R^n$, $(y \\equiv y(\\chi))$, such that, for each $x \\in \\mathbb R^n$\n$$\n  \\big(\\chi \\circ \\varphi \\big)(x) =:\\xi_{y(\\chi)}(x)= e^{2\\pi i \\, y(\\chi) \\cdot x}.\n$$\nFollowing Example \\ref{EXAMPLERN} we define the pseudo-metric \n$p_\\varphi: G^\\wedge \\times G^\\wedge \\to [0,\\infty)$ by  \n\\begin{equation}\n\\label{PSEDO}\n   p_\\varphi(\\chi_1, \\chi_2):= p(\\xi_{y_1(\\chi_1)}, \\xi_{y_2(\\chi_2)})= 2 \\pi \\|y_1(\\chi_1) - y_2(\\chi_2)\\|. \n\\end{equation}\nThen, we have \n$$\n   \\gamma(\\chi)= p_\\varphi(\\chi,1)=  2 \\pi \\|y(\\chi)\\|. \n$$\n\n\\medskip\nLet us observe that, we have used in the above construction of $\\gamma$ the continuity of the homomorphism $\\varphi: \\mathbb R^n \\to G$,\nthat is to say, it was essential the continuity of $\\varphi$. In fact, the function $\\gamma$ was given by the pseudo-metric $p_\\varphi$, which is \nnot necessarily a metric. Although, we have the following \n\n\\medskip\n \\underline {\\bf Claim 2:} The pseudo-metric $p_\\varphi: G^\\wedge \\times G^\\wedge \\to [0,\\infty)$ given by \\eqref{PSEDO} \nis a metric if, and only if, $\\varphi(\\mathbb R^n)$ is dense in $G$. \n\n\\smallskip\nProof of Claim 2: 1. First, let us assume that $\\overline{\\varphi(\\mathbb R^n)} \\neq G$, and then show that $p_\\varphi$ is not a metric. \nTherefore, we have the necessity proved. From Corollary 24.12 in \\cite{HewittRoss}, since \n$\\overline{\\varphi(\\mathbb R^n)}$ is a closer proper subgroup of $G$, hence there exists $\\xi \\in G^\\wedge \\setminus \\{1\\}$,\nsuch that $\\xi(\\overline{\\varphi(\\mathbb R^n)})= \\{1\\}$. Hence  there exists $\\xi \\in G^\\wedge \\setminus \\{1\\}$, \nsuch that, $\\xi(\\varphi(x))= 1$, for each $x \\in \\mathbb R^n$, i.e. $y(\\xi)= 0$. Therefore, we have \n$p_\\varphi(\\xi, 1)= 0$, \nwhich implies that $p_\\varphi$ is not a metric. \n\n\\medskip\n2. Now, let us assume that $\\overline{\\varphi(\\mathbb R^n)}= G$, and it is enough to show that\nif $p_\\varphi(\\xi, 1)= 0$, then $\\xi= 1$. Indeed, if $0= p_\\varphi(\\xi,1)= 2 \\pi \\|y(\\xi)\\|$, then $y(\\xi)= 0$. \nTherefore, $\\xi(\\varphi(x))= 1$ for each $x \\in \\mathbb R^d$, since $\\xi$ is continuous and  $\\overline{\\varphi(\\mathbb R^n)}= G$,\nit follows that, for each $\\omega \\in G$, $\\xi(\\omega)= 1$, which finishes the proof of the claim. \n\n\\begin{remark}\nSince we have already assumed that $\\varphi(\\mathbb R^n)$ is dense in $G$, it follows that \n$p_\\varphi$ is indeed a metric, which does not imply necessarily that $G$, itself, is metric.   \n\\end{remark}\t\t\n\nUnder the assumptions considered above, we have the following \n\\begin{lemma} If $f \\in \\mathcal{H}$, then for $j \\in \\{1,\\ldots,d\\}$ and all $\\xi \\in G^\\wedge$\n\\begin{equation}\n\\label{DERIVGROUPFOURIER}\n \\widehat{\\partial_j f(0,\\xi)}= 2 \\pi i \\; y_j(\\xi) \\widehat{f(0,\\xi)}.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nFirst, for each $x \\in \\mathbb R^d$ and $\\omega \\in G$, define \n$$\n\\begin{aligned}\n   \\xi_\\tau(x,\\omega)&:= \\xi(\\tau(x,\\omega))= \\xi(\\varphi(x) \\omega)= \\xi(\\varphi(x)) \\; \\xi(\\omega)\n   \\\\[5pt]\n   &= e^{2 \\pi i x \\cdot y(\\xi)} \\; \\xi(\\omega). \n\\end{aligned}\n$$   \nTherefore $\\xi_\\tau \\in C^\\infty(\\mathbb R^d; L^2(G))$, and we have for $j \\in \\{1,\\ldots,d\\}$\n\\begin{equation}\n\\label{AUXIL}\n\\partial_j  \\xi_\\tau(0,\\omega)= 2 \\pi i \\; y_j(\\xi) \\;  \\xi(\\omega). \n\\end{equation}\nFinally, applying Theorem \\ref{987987789879879879} we obtain\n$$\n\\begin{aligned}\n  \\int_G \\partial_j f(0,\\omega) \\; \\overline{\\xi_\\tau}(0,\\omega) d\\mu(\\omega)&= -  \\int_G f(0,\\omega) \\; \\partial_j \\overline{\\xi_\\tau}(0,\\omega) d\\mu(\\omega)\n  \\\\[5pt]\n &=  2 \\pi i \\; y_j(\\xi) \\int_G f(0,\\omega) \\; \\overline{\\xi}(\\omega) d\\mu(\\omega),\n\\end{aligned} \n$$\nwhere we have used \\eqref{AUXIL}. From the above equation and the definition of the \nFourier transform on groups we obtain \\eqref{DERIVGROUPFOURIER}, and the lemma is proved. \n\\end{proof}\n\n\\medskip\nNow we are able to state the equivalence between the spaces $\\mathcal{H}$ and $H^1_\\gamma(G)$,\nwhich is to say, we have the following \n\\begin{theorem}\n\\label{THMEQNOM}\nA function $f \\in \\mathcal{H}$ if, and only if, $f(0,\\cdot) \\in H^1_\\gamma(G)$,\nand \n$$\n    \\Vert f \\Vert_\\mathcal{H} = \\Vert f(0,\\cdot) \\Vert_{H_\\gamma^1(G)}.\n$$\n\\end{theorem}\n\n\\begin{proof}\n1. Let us first show that, if $f \\in \\mathcal{H}$ then $f \\in H^1_\\gamma(G)$. \nTo follow we observe that \n$$\n\\begin{aligned}\n  \\int_{G^\\wedge} (1 + \\gamma(\\xi)^2) |\\widehat{f(0,\\xi)}|^2 \\; d\\nu(\\xi)&= \n   \\int_{G^\\wedge} |\\widehat{f(0,\\xi)}|^2 \\; d\\nu(\\xi)\n   \\\\[5pt]\n   &+  \\int_{G^\\wedge} | 2 \\pi i \\; y(\\xi) \\widehat{f(0,\\xi)}|^2 \\; d\\nu(\\xi)\n   \\\\[5pt]\n   &= \\int_{G^\\wedge} |\\widehat{f(0,\\xi)}|^2 \\; d\\nu(\\xi)\n   +  \\int_{G^\\wedge} |\\widehat{\\nabla_{\\!\\!y} f(0,\\xi)}|^2 \\; d\\nu(\\xi),\n\\end{aligned}\n$$\nwhere we have used \\eqref{DERIVGROUPFOURIER}. Therefore, applying \nPlancherel theorem \n$$\n  \\int_{G^\\wedge}\\! (1 + \\gamma(\\xi)^2) |\\widehat{f(0,\\xi)}|^2 \\; d\\nu(\\xi)= \\!\\!\n   \\int_{G}\\!  |{f(0,\\omega)}|^2 \\; d\\mu(\\omega)\n   + \\! \\int_{G} |\\nabla_{\\!\\!y} {f(0,\\omega)}|^2 \\; d\\mu(\\omega)\\!<  \\! \\infty,\n$$\nand thus $f(0,\\cdot) \\in H^1_\\gamma(G)$. \n\n\\medskip\n2. Now, let $f(x,\\omega)$ be a stationary function, such that $f(0,\\cdot) \\in H^1_\\gamma(G)$, then we show that \n$f \\in \\mathcal{H}$. Given a stationary function $\\zeta \\in C^1(\\mathbb R^d; L^2(G))$, applying the Palncherel theorem and polarization identity\n$$\n   \\int_G \\partial_j \\zeta(0,\\omega) \\;  \\overline{f(0,\\omega)} d\\mu(\\omega)\n  = \\int_{G^\\wedge} \\widehat{\\partial_j \\zeta(0,\\xi)} \\; \\overline{\\widehat{f(0,\\xi)}} d\\nu(\\xi)\n$$\nfor $j \\in \\{1,\\ldots,d\\}$. Due to \\eqref{DERIVGROUPFOURIER}, we may write\n\\begin{equation}\n\\label{HH1}\n\\begin{aligned}\n   \\int_G \\partial_j \\zeta(0,\\omega) \\;  \\overline{f(0,\\omega)} d\\mu(\\omega)\n  &= \\int_{G^\\wedge} 2 \\pi\n  i \\; y_j(\\xi)\\widehat{\\zeta(0,\\xi)} \\; \\overline{\\widehat{f(0,\\xi)}} d\\nu(\\xi)\n\\\\[5pt]\n&= - \\int_{G^\\wedge} \\widehat{\\zeta(0,\\xi)} \\; \\overline{2 \\pi i \\; y_j(\\xi) \\widehat{f(0,\\xi)}} d\\nu(\\xi).\n\\end{aligned}\n\\end{equation}\nFor $j \\in \\{1,\\ldots,d\\}$ we define, $g_j(\\omega):= \\big(2 \\pi i \\; y_j(\\xi)  \\widehat{f(0,\\xi)}\\big)^\\vee$,\nthen $g_j \\in L^2(G)$. Indeed, we have \n$$\n   \\int_G |g_j(\\omega)|^2 d\\mu(\\omega)=  \\int_{G^\\wedge} |\\widehat{g_j(\\xi)}|^2 d\\nu(\\xi) \n   \\leq \\int_{G^\\wedge} (1 + \\gamma(\\xi)^2) |\\widehat{f(0,\\xi)}|^2 d\\nu(\\xi)< \\infty.\n$$\nTherefore, we obtain from \\eqref{HH1}\n$$\n   \\int_G \\partial_j \\zeta(0,\\omega) \\;  \\overline{f(0,\\omega)} d\\mu(\\omega)\n   = -  \\int_G \\zeta(0,\\omega) \\;  \\overline{g_j(\\omega)} d\\mu(\\omega)\n$$\nfor any stationary function $\\zeta \\in C^1(\\mathbb R^d; L^2(G))$, and $j \\in \\{1,\\ldots,d\\}$. \nThen $f \\in \\mathcal{H}$ due to Theorem \\ref{987987789879879879}. \n\\end{proof} \n\t\n\\begin{corollary}\nLet $f \\in L^2_{\\loc}(\\mathbb R^d; L^2(G))$ be a stationary function\nand $\\Phi$ a stochastic deformation. \nThen, $f \\circ \\Phi^{-1} \\in \\clg{H}_\\Phi$ if, and only if, $f(0,\\cdot) \\in H^1_\\gamma(G)$,\nand there exist constants $C_1, C_2> 0$, such that \n$$\n   C_1  \\Vert f \\circ \\Phi^{-1} \\Vert_{\\mathcal{H}_\\Phi}\\leq \\Vert f(0,\\cdot) \\Vert_{H_\\gamma^1(G)}\n   \\leq C_2 \\Vert f \\circ \\Phi^{-1} \\Vert_{\\mathcal{H}_\\Phi}.\n$$\n\\end{corollary}\n\t\n\\begin{proof}\nFollows from Theorem \\ref{THMEQNOM} and Remark \\ref{REMFPHI}. \n\\end{proof}\t\n\t\n\\subsubsection{Rellich--Kondrachov type Theorem}\n\\label{927394r6fy7euh73f}\n\nThe aim of this section is to characterize when \nthe Sobolev space $H^1_\\gamma(G)$ is compactly embedded in $L^2(G)$,\nwritten $H^1_\\gamma(G) \\subset \\subset L^2(G)$, where $G$ is considered a compact Abelian group \nand $\\gamma: G^{\\wedge} \\to [0,\\infty)$ is given by \\eqref{Gamma}. \nWe observe that, $H^1_\\gamma(G) \\subset \\subset L^2(G)$ is exactly the \nRellich--Kondrachov Theorem on compact Abelian groups, which was established\nunder some conditions on $\\gamma$\nin \\cite{GorkaReyes}. \nNevertheless, as a byproduct of the characterization established here, we provide\nthe proof of this theorem in a more \nprecise context. \n\n\\medskip\nTo start the investigation, let $(G,\\mu)$ be a probability space and consider \nthe operator\n$T: L^2(G^\\wedge) \\to L^2(G^\\wedge)$,\ndefined by\n\\begin{equation}\n\\label{TCOMP}\n\t[T(f)](\\xi) := \\frac{f(\\xi)}{\\sqrt{(1 + \\gamma(\\xi)^2)}}.\n\\end{equation}\nWe remark that, $T$ as defined above is a bounded linear, ($\\Vert T \\Vert \\leqslant 1$),  self-adjoint operator,\nwhich is injective and satisfies for each $f \\in L^2(G^\\wedge)$\n\\begin{equation}\n\\label{76354433}\n\t\\int_{G^\\wedge} \\left(1 + \\gamma(\\xi)^2 \\right) \\, {\\vert [T(f)](\\xi) \\vert}^2 d\\nu(\\xi) \n\t= \\int_{G^\\wedge} \\vert f(\\xi) \\vert^2 d\\nu(\\xi).\n\\end{equation}\nMoreover, a function $f \\in H^1_\\gamma(G)$ if, and only if, $\\widehat{f} \\in T(L^2(G^\\wedge))$, \nthat is to say \n\\begin{equation}\n\\label{87648764}\n  f \\in H^1_\\gamma(G) \\Leftrightarrow  \\widehat{f} \\in T(L^2(G^\\wedge)). \n\\end{equation}\nIndeed, if $ f \\in H^1_\\gamma(G)$ then, we have $f \\in L^2(G)$ and \n$$\n   \\int_{G^\\wedge} \\left( 1+\\gamma(\\xi)^2 \\right) \\vert \\widehat{f}(\\xi) \\vert^2 d\\nu(\\xi) \n   = \\int_{G^\\wedge} \\vert \\sqrt{\\left( 1+\\gamma(\\xi)^2 \\right)} \\, \\widehat{f}(\\xi) \\vert^2 d\\nu(\\xi)< \\infty.\n$$\nTherefore, defining $g(\\xi):=  \\sqrt{\\left( 1+\\gamma(\\xi)^2 \\right)} \\widehat{f(\\xi)}$, hence $g \\in L^2(G^\\wedge)$ and we have\n$\\widehat{f} \\in T(L^2(G^\\wedge))$.\n\n\\medskip\nNow, if $\\widehat{f} \\in T(L^2(G^\\wedge))$ let us show that, $f \\in H^1_\\gamma(G)$. First, there exists \n$g \\in L^2(G^\\wedge)$ such that, $\\widehat{f} = T(g)$. \nThus from equation \\eqref{76354433}, we obtain \n$$\n    \\int_{G^\\wedge} (1 + \\gamma(\\xi)^2) \\, |\\widehat{f}(\\xi)|^2 d\\nu(\\xi) \n    = \\int_{G^\\wedge} |g(\\xi)|^2 d\\nu(\\xi)< \\infty,\n$$\nthat is, by definition $f \\in H^1_\\gamma(G)$.\n\n\\medskip\nThen we have the following Equivalence Theorem:\n\\begin{theorem}\n\\label{876876872}\nThe Sobolev space $H^1_\\gamma(G)$ is compactly embedded in $L^2(G)$ \nif, and only if, the operator $T$ defined by \\eqref{TCOMP} is compact. \n\\end{theorem}\n\n\\begin{proof}\n1. First, let us assume that $H^1_\\gamma(G) \\subset \\subset L^2(G)$, \nand take a bounded sequence $\\{f_m\\}$, $f_m \\in L^2(G^\\wedge)$ \nfor each $m \\in \\mathbb N$. Thus $T(f_m) \\in L^2(G^\\wedge)$, and defining \n$g_m:= T(f_m)^\\vee$, we obtain by Plancherel Theorem that $g_m \\in L^2(G)$  \nfor each $m \\in \\mathbb N$. Moreover, from equation \\eqref{76354433}, we have for any \n$m \\in \\mathbb{N}$\n$$\n\\begin{aligned}\n \\infty >\\int_{G^\\wedge} |f_m(\\xi)|^2 d\\nu(\\xi)&= \\int_{G^\\wedge} (1 + \\gamma(\\xi)^2) \\, |[T(f_m)](\\xi)|^2 d\\nu(\\xi)\n \\\\[5pt]\n  &= \\int_{G^\\wedge} (1 + \\gamma(\\xi)^2) \\, |\\widehat{g_m(\\xi)}|^2 d\\nu(\\xi). \n\\end{aligned}  \n$$\nTherefore, the sequence $\\{g_m\\}$ is uniformly bounded in $H^1_\\gamma(G)$, with respect to $m \\in \\mathbb{N}$.  \nBy hypothesis there exists a subsequence of $\\{g_m\\}$, say $\\{g_{m_j}\\}$,\nand a function $g \\in L^2(G)$ such that, $g_{m_j}$ converges strongly to $g$ in $L^2(G)$ as $j \\to \\infty$. \nConsequently, we have \n$$T(f_{m_j})= \\widehat{g_{m_j}} \\to \\widehat{g} \\quad \n\\text{in $L^2(G^\\wedge)$ as $j \\to \\infty$},$$ that is, the operator $T$ is compact. \n\n\\medskip\n2. Now, let us assume that the operator $T$ is compact and then show that $H^1_\\gamma(G) \\subset \\subset L^2(G)$. \nTo this end, we take a sequence $\\{f_m\\}_{m\\in \\mathbb{N}}$ uniformly bounded in $H^1_\\gamma(G)$. \nThen, due to the equivalence \\eqref{87648764} there exists for each $m\\in \\mathbb{N}$, \n$g_m \\in L^2(G^\\wedge)$, such that $\\widehat{f_m} = T(g_m)$. Thus for any $ m\\in \\mathbb{N}$, \nwe have from equation \\eqref{76354433} that\n$$\n\\begin{aligned}\n   \\int_{G^\\wedge} |g_m(\\xi)|^2 d\\nu(\\xi)&= \\int_{G^\\wedge} (1 + \\gamma(\\xi)^2) \\, |[T(g_m)](\\xi)|^2 d\\nu(\\xi) \n   \\\\[5pt]\n  & = \\int_{G^\\wedge} (1 + \\gamma(\\xi)^2) \\, |\\widehat{f_m(\\xi)}|^2 d\\nu(\\xi)< \\infty. \n\\end{aligned}\n$$\nThen, the sequence $\\{g_m\\}$ is uniformly bounded in $L^2(G)$. Since the operator $T$ \nis compact, there exist $\\{m_j\\}_{j \\in \\mathbb{N}}$ and $g \\in L^2(G^\\wedge)$, such that  \n$$\n   \\widehat{f_{m_j}}= T(g_{m_j})  \\xrightarrow[j \\to \\infty]{} g \\quad \\text{in $L^2(G^\\wedge)$}.\n$$\nConsequently, the subsequence $\\{f_{m_j}\\}$ converges to $g^\\vee$ strongly in $L^2(G)$,\nand thus $H^1_\\gamma(G)$ is compactly embedded in $L^2(G)$.\n\\end{proof}\n\n\\begin{remark}\nDue to Theorem \\ref{876876872} the compactness characterization, that is\n$H^1_\\gamma(G) \\subset \\subset L^2(G)$, follows once\nwe show the conditions that the operator $T$ is compact. \nThe study of the dual space of $G$, i.e. $G^\\wedge$, and $\\gamma$ it will be essential for this characterization.  \n\\end{remark}\n\n\\medskip\nRecall from \\eqref{CARACGCOMP} item $(ii)$ that, $G^\\wedge$ is discrete since $G$ is compact. \nThen, $\\nu$ is a countermeasure, and $\\nu(\\{\\chi\\})= 1$ for each singleton $\\{\\chi\\}$, $\\chi \\in G^\\wedge$. \nNow, for any $\\chi \\in G^\\wedge$ fixed, we \ndefine the point mass function at $\\chi$ by \n$$\n   \\delta_{\\chi}(\\xi):= 1_{\\{\\chi\\}}(\\xi),\n\\quad \n\\text{for each $\\xi \\in G^\\wedge$}. \n$$\nHence the set $\\{\\delta_\\xi \\; ; \\; \\xi \\in G^\\wedge \\}$\nis an orthonornal basis for $L^2(G^\\wedge)$. Indeed, we first show the orthonormality. \nFor each $\\chi, \\pi \\in G^\\wedge$, we have \n\\begin{equation}\n\\label{87987948744}\n    \\langle \\delta_\\chi, \\delta_\\pi \\rangle_{L^2(G^\\wedge)}\n     = \\int_{G^\\wedge} \\delta_\\chi(\\xi)  \\; \\delta_\\pi(\\xi) \\, d\\nu(\\xi)= \\left\\{\n\t\\begin{array}{ccl}\n\t\t1, & \\text{if} & \\chi = \\pi, \n\t\t\\\\\n\t\t0, & \\text{if} & \\chi \\not= \\pi.\n\t\\end{array}\t\t \n\\right.\n\\end{equation}\nNow, let us show the density, that is $\\overline{\\{\\delta_\\xi \\; ; \\; \\xi \\in G^\\wedge \\}}= L^2(G^\\wedge)$, or equivalently $\\{\\delta_\\xi \\; ; \\; \\xi \\in G^\\wedge \\}^\\perp= \\{0\\}$. \nFor any $w \\in \\{\\delta_\\xi \\; ; \\; \\xi \\in G^\\wedge \\}^\\perp$, we obtain \n$$\n   0 =\\langle \\delta_\\xi, w \\rangle_{L^2(G^\\wedge)} \n   = \\int_{G^\\wedge} \\delta_\\xi(\\chi) w(\\chi) \\, d\\nu(\\chi) \n   = \\int_{ \\{ \\xi \\}} w(\\chi) \\, d\\nu(\\chi) = w(\\xi)\n$$\nfor any $\\xi \\in G^\\wedge$, which proves the density. \n\n\\medskip\nFrom the above discussion, it is important to study the operator $T$\non elements of the set $\\{\\delta_\\xi \\; ; \\; \\xi \\in G^\\wedge \\}$.\nThen, we have the following \n\\begin{theorem}\n\\label{876876876GG}\nIf the operator $T$ defined by \\eqref{TCOMP} is compact,  then $G^\\wedge$ is an enumerable set. \n\\end{theorem}\n\\begin{proof} 1. First, let $\\{\\delta_\\xi \\; ; \\; \\xi \\in G^\\wedge \\}$ be the orthonormal basis for $L^2(G^\\wedge)$,\nand $T$ the operator defined by \\eqref{TCOMP}. Then, the function \n$\\delta_\\xi \\in L^2(G^\\wedge)$ is an eigenfunction of $T$ \ncorresponding to the eigenvalue $(1+\\gamma^2)^{-1\/2}$, that is $\\delta_\\xi \\neq 0$, and\n\\begin{equation}\n\\label{87486tydg}\n    T(\\delta_\\xi)= \\frac{\\delta_\\xi}{\\sqrt{1+\\gamma(\\xi)^2}}.\n\\end{equation}\n\n\\medskip\t\t\n2. Now, since $T$ is compact and $\\{\\delta_\\xi \\; ; \\; \\xi \\in G^\\wedge \\}$ is a basis for $L^2(G^\\wedge)$, it must be enumerable from \\eqref{87486tydg}. \nOn the other hand, the function  $\\xi \\in G^\\wedge \\mapsto \\delta_\\xi \\in L^2(G^\\wedge)$ is injective, hence $G^\\wedge$ is enumerable. \n\\end{proof}\n\n\\begin{corollary}\nIf the operator $T$ defined by \\eqref{TCOMP} is compact,  then\n$L^2(G)$ is separable. \n\\end{corollary}\n\t\n\\begin{proof} First, the Hilbert space $L^2(G^\\wedge)$ is separable, since $\\{\\delta_\\xi \\; ; \\; \\xi\\in G^\\wedge\\}$\nis an enumerable orthonormal basis of it. Then, the proof follows applying the Plancherel Theorem. \n\\end{proof}\n\n\\begin{corollary}\nLet $G_B$ be the Bohr compactification of $\\mathbb{R}^n$ (see A. Pankov \\cite{Pankov}).  \nThen $H^1_\\gamma(G_B)$ is not compactly embedded in $L^2(G_B)$.\n\\end{corollary}\n\t\t\n\\begin{proof} Indeed, $G_B^\\wedge$ is non enumerable.\n\\end{proof}\nConsequently, $G^\\wedge$ be enumerable is a necessarily condition for the operator $T$ be compact, which is not \nsufficient as shown by the Example \\ref{NOSUFF} below. Indeed, it might depend on the chosen $\\gamma$, see also  \nExample \\ref{NOSUFF10}. \n\n\\medskip\nTo follow, we first recall the  \n\\begin{definition}\nLet $G$ be a group (not necessarily a topological one) and $S$ a subset of it. \nThe smallest subgroup of G containing every element of S, denoted $\\langle S \\rangle$, is called the subgroup \ngenerated by $S$. Equivalently, see Dummit, Foote \\cite{DummitFoote} p.63, \n$$\n  \\langle S \\rangle= \\big\\{ g^{\\varepsilon_1}_1 g^{\\varepsilon_2}_2 \\ldots g^{\\varepsilon_k}_k \/ \n  \\text{$k \\in \\mathbb{N}$ and for each $j$, $g_j \\in S, \\varepsilon_j= \\pm 1$} \\big\\}.\n$$\nMoreover, \nif a group $G= \\langle S \\rangle$, then $S$ is called a generator of $G$, and\nin this case when S is finite, $G$ is called finitely generated. \n\\end{definition} \t\n\t\n\\begin{theorem}\n\\label{876876876} \nIf the operator $T$ defined by \\eqref{TCOMP} is compact and\nthere exists a generator of $G^\\wedge$ such that $\\gamma$ is bounded on it, \nthen $G^\\wedge$ is finite generated.  \n\\end{theorem}\n\t\t\n\\begin{proof} \nLet $S_0$ be a generator of $G^\\wedge$, such that $\\gamma$ is bounded on it.  \nTherefore, there exists $d_0 \\geq 0$ such that,  \n$$\n   \\text{for each $\\xi \\in S_0$, $\\; \\gamma(\\xi) \\leq d_0$}. \n$$\nNow, since $T$ is compact and  $\\Vert T \\Vert \\leq 1$, there exists  $0 < c \\leq 1$ such that,\nthe set of eigenvectors \n$$\n   \\Big\\{ \\delta_\\xi \\; ; \\; \\xi \\in G^\\wedge \\;\\; \\text{and} \\;\\; \\frac{1}{\\sqrt{1 + \\gamma(\\xi)^2}} \\geq c \\Big\\}\n   \\equiv \n   \\Big\\{ \\delta_\\xi \\; ; \\; \\xi \\in G^\\wedge \\;\\; \\text{and} \\;\\; \\gamma(\\xi) \\leq\\sqrt{\\frac{1}{c^2} - 1} \\Big\\}\n$$\nis finite, where we have used the Spectral Theorem for bounded compact operators. Therefore, since \n$$\t\t\t\t\n     \\left\\{ \\delta_\\xi \\; ; \\; \\xi \\in S_0 \\right\\} \\subset \n     \\left\\{ \\delta_\\xi \\; ; \\; \\xi \\in G^\\wedge \\;\\; \\text{and} \\;\\; \\gamma(\\xi) \\leq d_0 \\right\\}\n$$\nit follows that $S_0$ is a finite set, and thus $G^\\wedge$ is finite generated. \n\\end{proof}\n\n\\begin{example}[Infinite enumerable Torus] \n\\label{NOSUFF}\nLet us recall the Sobolev space $H^1_\\gamma(\\mathbb{T}^\\mathbb N)$, where $\\mathbb{T}^\\mathbb N$ is the infinite enumerable Torus. \nWe claim that: $H^1_\\gamma(\\mathbb{T}^\\mathbb N)$ is not compactly embedded in $ L^2(\\mathbb{T}^\\mathbb N)$, \nfor $\\gamma$ defined in Exemple \\ref{6576dgtftdefd}. \nIndeed, given $k \\in \\mathbb N$ we define $1_k \\in \\mathbb{Z}^\\mathbb N$, such that it is zero for any coordinate \n$\\ell \\neq k$, and one in $k-$coordinate. Therefore, the set \n$$\n    S_0 := \\{ \\xi_{1_k} \\; ; \\; k \\in \\mathbb N \\}\n$$\nis an infinite generator of the dual group $(\\mathbb{T}^\\mathbb N)^\\wedge$. \nSince for each $k \\in \\mathbb N$,  $\\gamma(\\xi_{1_k}) = 1$, i.e. bounded in $S_0$, applying Theorem \\ref{876876876}\nit follows that $H^1_\\gamma(\\mathbb{T}^\\mathbb N)$ is not compactly embedded in $ L^2(\\mathbb{T}^\\mathbb N)$. \n\\end{example}\n\n\\begin{remark} The above discussion in the Example \\ref{NOSUFF} follows as well to the Sobolev space $H^1_\\gamma(\\mathbb{T}^I)$, where\n$I$ is an index set (enumerable or not). Clearly, the Sobolev space $H^1_\\gamma(\\mathbb{T}^I)$ in not compactly embedded in $ L^2(\\mathbb{T}^I)$, \nwhen $I$ is a non enumerable index set. Indeed, the set $(\\mathbb{T}^I)^\\wedge$ is non enumerable. \n\\end{remark}\n\nNow, we charactherize the condition on $\\gamma: G^\\wedge \\to [0,\\infty)$,\nin order to $T$ be compact. More precisely, let us consider the following property:\n\\begin{equation}\n\\label{ConditionC}\n {\\bf C}.  \\quad \\text{For each $d> 0$, the set \n$ \\left\\{ \\xi \\in G^\\wedge \\; ; \\; \\gamma(\\xi) \\leq d \\right\\}$\nis finite}. \n\\end{equation}\n\t\t\t\t\t\t\t\n\\begin{theorem}\n\\label{7864876874}\nIf $\\gamma: G^\\wedge \\to [0,\\infty)$ satisfies ${\\bf C}$, then the operator $T$ defined by \\eqref{TCOMP} is compact. \n\\end{theorem}\n\t\t\n\\begin{proof}\nBy hypothesis, $\\{ \\xi \\in G^\\wedge \\; ; \\; \\gamma(\\xi) \\leq d \\}$ is finite, then we have\n$$\n    G^\\wedge= \\bigcup_{k \\in \\mathbb{N}} \\left\\{ \\xi \\in G^\\wedge \\; ; \\; \\gamma(\\xi) \\leq k \\right\\}. \n$$\nConsequently, the set $G^\\wedge$ is enumerable and we may write $G^\\wedge= \\{ \\xi_i \\}_{i \\in \\mathbb{N}}$. \n\n\\medskip\nAgain, due to condition ${\\bf C}$ for each $c \\in (0,1)$ the set \n\\begin{equation}\n\\label{868768767864120}\n\t\\Big\\{ \\xi \\in G^\\wedge \\; ; \\; \\frac{1}{\\sqrt{1 + \\gamma(\\xi)^2}} \\geq c \\Big\\}\n\\end{equation}\nis finite. Since the function  $\\xi \\in G^\\wedge \\mapsto \\delta_\\xi \\in L^2(G^\\wedge)$ is injective,\nthe set $\\{ \\delta_{\\xi_i} \\; ; \\; i\\in \\mathbb{N} \\}$ \nis an enumerable orthonormal basis of eigenvectors for $T$, which corresponding eigenvalues satisfy \n$$\n  \\lim_{i \\to \\infty}  \\frac{1}{\\sqrt{1 + \\gamma(\\xi_i)^2}}= 0,\n$$\t\nwhere we have used \\eqref{868768767864120}. Consequently, $T$ is a compact operator. \n\\end{proof}\n\n\\begin{example}[Bis: Infinite enumerable Torus]\n\\label{NOSUFF10}\nThere exists a function $\\gamma_0$ such that, \n$H^1_{\\gamma_0}(\\mathbb{T}^\\mathbb N)$ is compactly embedded in $ L^2(\\mathbb{T}^\\mathbb N)$. \nIndeed, we are going to show that, $\\gamma_0$ satisfies ${\\bf C}$. \nLet $\\alpha \\equiv (\\alpha_\\ell)_{\\ell \\in \\mathbb{N}}$\nbe a sequence in $\\mathbb R^\\mathbb N$, such that for each $\\ell \\in \\mathbb N$, $\\alpha_\\ell \\geq 0$ and \n\\begin{equation}\n\\label{weight}\n\\lim_{\\ell \\to \\infty} \\alpha_\\ell = +\\infty.\n\\end{equation}\nThen, we define the following pseudo-metric in the dual group $(\\mathbb{T}^\\mathbb{N})^\\wedge$ as follows\n$$\n   p_0(\\xi_m, \\xi_n):= 2\\pi \\sum_{\\ell = 1}^\\infty \\alpha_\\ell \\;{\\vert m_\\ell - n_\\ell \\vert}, \n   \\quad (m,n \\in \\mathbb{Z}^\\mathbb{N}_{\\rm c}),\n$$\nand consider $\\gamma_0(\\xi_m)= p_0(\\xi_m,1)$. \nThus for each $d> 0$, the set \n$$\n    \\{ m \\in \\mathbb{Z}^\\mathbb{N}_{\\rm c} \\; ; \\; \\gamma_0(\\xi_m) \\leq d \\} \\quad \\text{is finite.}\n$$   \nIndeed, from \\eqref{weight} \nthere exists $\\ell_0 \\in \\mathbb N$, such that $\\alpha_\\ell> d$, for each $\\ell \\geq \\ell_0$. \nTherefore, if $m \\in \\mathbb{Z}^\\mathbb{N}_{\\rm c}$ and the support of $m$ is not contained \nin $\\{ 1, \\ldots, \\ell_0-1\\}$, that is to say, there exists $\\tilde{\\ell} \\geq \\ell_0$, \nsuch that, $m_{\\tilde{\\ell}} \\neq 0$. Then, \n$$\n2\\pi \\sum_{\\ell = 1}^\\infty \\alpha_\\ell \\;{\\vert m_\\ell  \\vert}\n\\geq \\alpha_{\\tilde{\\ell}} > d. \n$$\nConsequently, we have \n$$\n    \\{ m \\in \\mathbb{Z}^\\mathbb{N}_{\\rm c} \\; ; \\; \\gamma_0(\\xi_m) \\leq d \\} \n    \\subset \n    \\{ m \\in \\mathbb{Z}^\\mathbb{N}_{\\rm c} \\; ; \\; {\\rm supp} \\ m \\subset \\{ 1, \\ldots, \\ell_0-1\\} \\},\n$$   \nwhich is a finite set. Finally, applying Theorem \\ref{7864876874} we obtain that,\nthe Sobolev space \n$H^1_{\\gamma_0}(\\mathbb{T}^\\mathbb N)$ is compactly embedded in $ L^2(\\mathbb{T}^\\mathbb N)$. \n\\end{example}\n\n\\subsubsection{On a class of Quasi-periodic functions}\n\\label{4563tgf5fd3}\n\nIn this section we consider an important class \nof quasi-periodic functions, which \nincludes for instance the   \nperiodic functions.\n \n \\smallskip\nLet $\\lambda_1,\\lambda_2,\\ldots,\\lambda_m \\in \\mathbb{R}^n$ be $m-$linear independent \nvectors with respect to $\\mathbb{Z}$, and consider the following matrix\n\\begin{equation*}\n\\Lambda := {\\left(\n\\begin{array}{c}\n\t\\lambda_1\n\\\\\n\t\\lambda_2\n\\\\\n\t\\vdots \n\\\\\n\t\\lambda_m\n\\end{array}\n\\right)}_{m\\times n}\n\\end{equation*}\nsuch that, for each $d> 0$ the set \n\\begin{equation}\n\\label{7863948tyfedf}\n   \\{ k \\in \\mathbb{Z}^m \\; ; \\; {\\vert \\Lambda^T k \\vert} \\leqslant d\\} \\quad \\text{is finite.}\n\\end{equation}\nTherefore, we are considering the class of quasi-periodic functions satisfying \ncondition \\eqref{7863948tyfedf}. This set is not empty, for instance let us define \nthe matrix $B:= \\Lambda \\Lambda^T$, such that $\\det B> 0$, which is called here\npositive quasi-periodic functions. It is not difficult to see that, positive quasi-periodic functions \nsatisfies \\eqref{7863948tyfedf}. \nIndeed, it is sufficiently to observe that, for each $k \\in \\mathbb{Z}^m$, we have \n$$\n   |k|= | B^{-1} B k | \\leq \\|B^{-1}\\| \\|\\Lambda\\| |\\Lambda^T k|. \n$$\nMoreover, since $\\lambda_1,\\lambda_2,\\ldots,\\lambda_m \\in \\mathbb{R}^n$ are $m-$linear independent \nvectors with respect to $\\mathbb{Z}$, (this property does not imply $\\det B> 0$), \nthe dynamical system $\\tau: \\mathbb{R}^n \\times \\mathbb{T}^m \\to \\mathbb{T}^m$,  given by \n\\begin{equation}\n\\label{6973846tyd4f54e54}\n   \\tau(x)\\omega := \\omega + \\Lambda x - \\left\\lfloor \\omega + \\Lambda x \\right\\rfloor\n\\end{equation}\nis a ergodic. \n\n\\medskip\nNow we remark that, the application \n${ \\varphi : \\mathbb{R}^n \\to \\mathbb{T}^m }$, $ \\varphi(x) := \\Lambda x - \\left\\lfloor \\Lambda x \\right\\rfloor$,\nis a continuous homeomorphism of groups. Then, we have\n$$\n   \\tau(x)\\omega = \\varphi(x) \\omega \\equiv \\omega + \\Lambda x - \\left\\lfloor \\omega + \\Lambda x \\right\\rfloor. \n$$\nConsequently, under the conditions of the previous sections, we obtain for each \n$ k\\in \\mathbb{Z}^m$\n\\begin{equation*}\n\t\\gamma(\\xi_k)= 2\\pi {\\vert \\Lambda^T k \\vert},\n\\end{equation*}\nand applying Theorem \\ref{7864876874} (recall \\eqref{7863948tyfedf}),\nit follows that \n\\begin{equation*}\n\tH^1_\\gamma {\\left( \\mathbb{T}^m \\right)} \\subset \\! \\subset L^2{\\left( \\mathbb{T}^m \\right)}.\n\\end{equation*}\nTherefore, given a stochastic deformation $\\Phi$, we have $\\mathcal{H}_\\Phi \\subset \\! \\subset \\mathcal{L}_\\Phi$\nfor the class of quasi-periodic functions satisfying \\eqref{7863948tyfedf}, \nand it follows a solution to Bloch's spectral cell equation.  \n\n\\subsection{Auxiliary celular equations}\n\\label{ACE}\n\n\nThe preposition below, which is an immediate consequence of Theorem~\\ref{768746hughjg576},\ngive us the necessaries characteristics to deduce from \nthe cell equation~\\eqref{92347828454trfhfd4rfghjls} other equations (called here auxiliary cellular equations),\nthat will be essential in our homogenization analysis.\n\t\\begin{proposition}\n\t\\label{2783546tydfh}\n\t\tGiven ${ \\theta \\in \\mathbb{R}^n }$, let $\\big(\\lambda(\\theta),\\Psi(\\theta)\\big)$ be a spectral point of the cell equation~\\eqref{92347828454trfhfd4rfghjls}. Suppose that \n\t\tfor some $\\theta_0\\in\\mathbb R^n$ the corresponding eigenvalue ${ \\lambda(\\theta_0) }$ has finite multiplicity. Then, there exists a  neighborhood \n\t\t${ {\\mathcal{U}} \\subset \\mathbb{R}^n }$ of ${ \\theta_0 }$, such that the following functions \t\t\n\t\t\\begin{equation*}\n\t\t\t\\theta \\in {\\mathcal{U}} \\mapsto \\Psi(\\theta) \\in \\mathcal{H}_\\Phi \\;\\;\\; \\text{and} \\;\\;\\; \\theta \\in {\\mathcal{U}} \\mapsto \\lambda(\\theta) \\in \\mathbb{R}-\\{0\\},\n\t\t\\end{equation*}\n\t\tare analytical.\n\t\\end{proposition}\n\t\n\nNow, introducing the operator  ${ \\mathbb{A}(\\theta) }$, (${ \\theta \\in \\mathbb{R}^n }$), defined on $\\mathcal{H}_\\Phi$ by \n\n\t\\begin{eqnarray*}\n&&\\mathbb{A}(\\theta)[F] = -({\\rm div}_{\\! z} + 2i\\pi \\theta) {\\Big( A {\\left( \\Phi^{-1}(z, \\omega), \\omega\\right)} {(\\nabla_{\\!\\! z} + 2i\\pi\\theta)} F \\Big)} \\\\\n&&\\qquad\\qquad\\qquad\\qquad+ V {\\left( \\Phi^{-1}(z, \\omega), \\omega \\right)} F - \\lambda(\\theta) F,\n\t\\end{eqnarray*}\nand writing $\\theta=(\\theta_1,\\cdots, \\theta_n)$, we obtain for $k= 1, \\ldots, n$,  \t\n\\begin{eqnarray}\n\\label{8654873526rtgdrfdrfdrfrd4}\n&& \\mathbb{A}(\\theta) {\\left[ \\frac{\\partial \\Psi(\\theta)}{\\partial \\theta_k} \\right]}\n=({\\rm div}_{\\! z} + 2i\\pi \\theta) {\\Big( A {\\left( \\Phi^{-1}(z, \\omega), \\omega \\right)} {( 2i\\pi e_k \\Psi(\\theta) )} \\Big)}\\nonumber\\\\\n&&\\qquad\\qquad+{( 2i\\pi e_k )} {\\Big( A {\\left( \\Phi^{-1}(z, \\omega), \\omega \\right)} {( {\\rm div}_{\\! z} + 2i\\pi \\theta )}\\Psi(\\theta) \\Big)}\\nonumber\\\\\n&&\\hspace{6.0cm}+\\frac{\\partial \\lambda}{\\partial \\theta_k}(\\theta) \\Psi(\\theta),\n\\end{eqnarray}\nwhere $\\{e_k{\\}}_{1\\le k\\le n}$ is the canonical basis of $\\mathbb R^n$. \nThe equation~\\eqref{8654873526rtgdrfdrfdrfrd4} is called the {\\it first auxiliary cellular equation} (or f.a.c. equation, in short). In the same way, we have \nfor $k, \\ell= 1, \\ldots, n$,  \n\\begin{eqnarray}\\label{876786uydsytfgbnvbvbbv}\n&&\\mathbb{A}(\\theta) {\\left[ \\frac{\\partial^2 \\Psi(\\theta)}{\\partial \\theta_\\ell \\, \\partial \\theta_k} \\right]} = \n({\\rm div}_{\\! z} + 2i\\pi \\theta) {\\Big( A {\\left( \\Phi^{-1}(z, \\omega), \\omega \\right)} { 2i\\pi e_\\ell \\frac{\\partial \\Psi(\\theta)}{\\partial \\theta_k} } \\Big)}\n\\nonumber\\\\\n&&\\qquad\\qquad+({\\rm div}_{\\! z} + 2i\\pi \\theta) {\\Big(A {\\left( \\Phi^{-1}(z, \\omega), \\omega \\right)} {2i\\pi e_k \\frac{\\partial \\Psi(\\theta)}{\\partial \\theta_\\ell}} \\Big)}\n\\nonumber\\\\\n&&\\qquad\\qquad+ {(2i\\pi e_k)} {\\Big( A {\\left( \\Phi^{-1}(z, \\omega), \\omega \\right)} {( {\\rm div}_{\\! z} + 2i\\pi \\theta )} \\frac{\\partial \\Psi(\\theta)}{\\partial \\theta_\\ell} \\Big)}\n\\nonumber\\\\\n&&\\qquad\\qquad+ {(2i\\pi e_\\ell)} {\\Big( A {\\left( \\Phi^{-1}(z, \\omega), \\omega \\right)} {( {\\rm div}_{\\! z} + 2i\\pi \\theta )} \\frac{\\partial \\Psi(\\theta)}{\\partial \\theta_k} \\Big)}\n\\nonumber\\\\\n&&\\qquad\\qquad\\qquad+ {(2i\\pi e_k)} {\\Big(A {\\left( \\Phi^{-1}(z, \\omega), \\omega \\right)} {\\left( 2i\\pi e_\\ell \\Psi(\\theta) \\right)} \\Big)}\\\\\n&&\\qquad\\qquad\\qquad\\qquad+ {(2i\\pi e_\\ell)} {\\Big(A {\\left( \\Phi^{-1}(z, \\omega), \\omega \\right)} {\\left( 2i\\pi e_k \\Psi(\\theta) \\right)} \\Big)}\\nonumber\\\\\n&&\\qquad\\qquad\\qquad\\qquad+\\frac{\\partial \\lambda(\\theta)}{\\partial \\theta_k} \\frac{\\partial \\Psi(\\theta)}{\\partial \\theta_\\ell} \n+ \\, \\frac{\\partial \\lambda(\\theta)}{\\partial \\theta_\\ell} \\frac{\\partial \\Psi(\\theta)}{\\partial \\theta_k} + \\frac{\\partial^2 \\lambda(\\theta)}{\\partial \\theta_\\ell \\, \\partial \\theta_k} \\Psi(\\theta)\\nonumber,\n\\end{eqnarray}\nwhich we call the {\\it second auxiliary cellular equation} (or s.a.c. equation, in short). \n\n\\medskip\nIn order to make clear in which sense the auxiliary cellular equations are understood, we note that if ${ G\\in\\mathcal{H}_\\Phi }$ then the variational formulation of the f.a.c. \nequation~\\eqref{8654873526rtgdrfdrfdrfrd4} is given by \n\\begin{eqnarray}\\label{hjhkjhjkhggd0989874}\n&&\\int_\\Omega \\int_{\\Phi ([0,1)^n, \\omega)}\\Big\\{\nA {\\left( \\Phi^{-1} ( z, \\omega),\\omega \\right)} {\\left( \\nabla_{\\!\\! z} + 2i\\pi \\theta \\right)} \\frac{\\partial \\Psi(\\theta)}{\\partial \\theta_k} \\cdot \\overline{ {\\left( \\nabla_{\\!\\! z} + 2i\\pi \\theta \\right)} G } \\nonumber\n\\\\[5pt]\n&&\\qquad\\qquad+V {\\left( \\Phi^{-1} ( z, \\omega),\\omega \\right)} \\frac{\\partial \\Psi(\\theta)}{\\partial \\theta_k} \\, \\overline{ G }-\\lambda(\\theta)\n\\frac{\\partial \\Psi(\\theta)}{\\partial \\theta_k} \\, \\overline{ G }\\Big\\} \\, dz \\, d\\mathbb{P}(\\omega)\\nonumber\n\\\\[5pt]\n&&\\qquad\\qquad\\qquad=:\\Big{\\langle}\\mathbb{A}(\\theta) {\\left[ \\frac{\\partial \\Psi(\\theta)}{\\partial \\theta_k} \\right]}, G\\Big{\\rangle}\\\\\n&&\\qquad=-\\int_\\Omega \\int_{\\Phi ([0,1)^n, \\omega)}\\Big\\{ A {\\left( \\Phi^{-1} ( z, \\omega),\\omega \\right)} {\\left( \\nabla_{\\!\\! z} + 2i\\pi \\theta \\right)} \\Psi(\\theta)  \\cdot \\overline{ {(2i\\pi e_k G)}  } \\nonumber\n\\\\[5pt]\n&&\\qquad\\qquad\\qquad\\qquad-A {\\left( \\Phi^{-1} ( z, \\omega),\\omega \\right)} {(2i\\pi e_k \\Psi(\\theta))}   \\cdot \\overline{ {\\left( \\nabla_{\\!\\! z} + 2i\\pi \\theta \\right)} G }\\nonumber\n\\\\[5pt]\n&&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad+\\frac{\\partial \\lambda(\\theta)}{\\partial \\theta_k}\\,\\Psi(\\theta) \\, \\overline{ G }\\Big\\} \\, dz \\, d\\mathbb{P}(\\omega).\\nonumber\n\\end{eqnarray}\nSimilar reasoning can be made with the s.a.c. equation~\\eqref{876786uydsytfgbnvbvbbv}. \n\n\\medskip\nIn the following, we highlight an important fact that is fundamental to determine the hessian nature of the effective tensor in our homogenization analysis concerning the Schr\\\"odinger equation~\\eqref{jhjkhkjhkj765675233}. This fact is brought out by choosing \n${ \\theta \\in {\\mathcal{U}} }$, ${ k\\in \\{1, \\ldots,n \\} }$ and defining $\\Lambda_k (z,\\omega, \\theta) := \\frac{1}{2i\\pi} \\frac{\\partial \\Psi}{\\partial \\theta_k} (z,\\omega, \\theta)$. Hence, \ntaking ${ \\Psi(\\theta) }$ as a test function in the s.a.c. equation~\\eqref{876786uydsytfgbnvbvbbv}, we get \n\\begin{eqnarray}\\label{hkjlhjklhljkhuytyiufsd4}\n&&\\frac{1}{4\\pi^2} \\frac{\\partial^2 \\lambda(\\theta)}{\\partial \\theta_\\ell \\, \\partial \\theta_k} \\int_\\Omega \\int_{\\Phi ([0,1)^n, \\omega)} {\\vert \\Psi(\\theta) \\vert}^2 \\, dz \\, d\\mathbb{P}(\\omega)\n\\nonumber\n\\\\\n&&\\qquad= \\int_\\Omega \\int_{\\Phi ([0,1)^n, \\omega)} \\Big\\{-A {\\left( \\Phi^{-1} ( z, \\omega),\\omega \\right)} {(e_\\ell \\Lambda_k(\\theta))}  \\cdot \\overline{ {\\left( \\nabla_{\\!\\! z} + 2i\\pi \\theta \\right)} \\Psi(\\theta) }\\nonumber\\\\\n&&\\qquad\\qquad\\qquad\\,-A {\\left( \\Phi^{-1} ( z, \\omega),\\omega \\right)} {(e_k \\Lambda_\\ell(\\theta))}  \\cdot \\overline{ {\\left( \\nabla_{\\!\\! z} + 2i\\pi \\theta \\right)} \\Psi(\\theta) }\\nonumber\n\\\\\n&&\\qquad\\qquad\\qquad\\,+A {\\left( \\Phi^{-1} ( z, \\omega),\\omega \\right)} {\\left( \\nabla_{\\!\\! z} + 2i\\pi \\theta \\right)} \\Lambda_k(\\theta)  \\cdot \\overline{ {(e_\\ell \\, \\Psi(\\theta))} }\n\\nonumber\\\\\n&&\\qquad\\qquad\\qquad\\,+A {\\left( \\Phi^{-1} ( z, \\omega),\\omega \\right)} {\\left( \\nabla_{\\!\\! z} + 2i\\pi \\theta \\right)} \\Lambda_\\ell(\\theta)  \\cdot \\overline{ {(e_k \\, \\Psi(\\theta))} }\n\\nonumber\\\\\n &&\\qquad\\qquad\\qquad\\qquad\\,+A {\\left( \\Phi^{-1} ( z, \\omega),\\omega \\right)} {(e_k \\Psi(\\theta))}  \\cdot \\overline{ {(e_\\ell \\, \\Psi(\\theta))} } \\\\\n &&\\qquad\\qquad\\qquad\\qquad\\,+A {\\left( \\Phi^{-1} ( z, \\omega),\\omega \\right)} {(e_\\ell \\Psi(\\theta))}  \\cdot \\overline{ {(e_k \\, \\Psi(\\theta))} }\\nonumber\\\\\n&&\\qquad\\qquad\\qquad\\qquad +\\frac{1}{2i\\pi}\\Big(\\frac{\\partial \\lambda(\\theta)}{\\partial \\theta_\\ell}\\Lambda_k(\\theta)+\\frac{\\partial \\lambda(\\theta)}{\\partial \\theta_k}\\Lambda_\\ell(\\theta)\n\\Big)\\, \\overline{\\Psi(\\theta)}\\Big\\} \\, dz \\, d\\mathbb{P}(\\omega).\\nonumber\n\\end{eqnarray}\n\nOn the other hand, using $\\Lambda_k (z,\\omega, \\theta)$ as a test function in the f.a.c. equation~\\eqref{8654873526rtgdrfdrfdrfrd4} and due to Theorem~\\ref{648235azwsxqdgfd}, \nwe arrive at \n\\begin{equation}\n\\label{4576gdcrfvjc46we}\n\t\t\\begin{array}{l}\n\t\t\t\\displaystyle \\int_{\\mathbb{R}^n} A {\\left( \\Phi^{-1} {\\left( \\frac{x}{\\varepsilon}, \\omega \\right)},\\omega \\right)} {\\left( \\nabla + 2i\\pi \\frac{\\theta}{\\varepsilon} \\right)} \\Lambda_{k,\\varepsilon} (\\theta) \\cdot \\overline{ {\\left( \\nabla + 2i\\pi \\frac{\\theta}{\\varepsilon} \\right)} \\varphi } \\, dx \\\\ [12pt]\n\t\t\t\\displaystyle + \\frac{1}{\\varepsilon^2}\\int_{\\mathbb{R}^n} V {\\left( \\Phi^{-1} {\\left( \\frac{x}{\\varepsilon}, \\omega \\right)},\\omega \\right)} \\,  \\Lambda_{k,\\varepsilon} (\\theta) \\, \\overline{ \\varphi } \\, dx - \\frac{\\lambda(\\theta)}{\\varepsilon^2}\\int_{\\mathbb{R}^n} \\Lambda_{k,\\varepsilon} (\\theta) \\, \\overline{ \\varphi } \\, dx \\\\ [12pt]\n\t\t\t\\hspace{2cm} \\displaystyle = - \\frac{1}{\\varepsilon}\\int_{\\mathbb{R}^n} A {\\left( \\Phi^{-1} {\\left( \\frac{x}{\\varepsilon}, \\omega \\right)},\\omega \\right)} {\\left( \\nabla + 2i\\pi \\frac{\\theta}{\\varepsilon} \\right)} \\Psi_\\varepsilon (\\theta) \\cdot \\overline{ {(e_k \\varphi)} } \\, dx \\\\ [12pt]\n\t\t\t\\hspace{2.5cm} \\displaystyle - \\frac{1}{\\varepsilon}\\int_{\\mathbb{R}^n} A {\\left( \\Phi^{-1} {\\left( \\frac{x}{\\varepsilon}, \\omega \\right)},\\omega \\right)} {(e_k \\Psi_\\varepsilon (\\theta) )} \\cdot \\overline{ {\\left( \\nabla + 2i\\pi \\frac{\\theta}{\\varepsilon} \\right)}\\varphi } \\, dx \\\\ [12pt]\n\t\t\t\\hspace{2.75cm} \\displaystyle + \\frac{1}{\\varepsilon^2} \\frac{1}{2i\\pi}\\frac{\\partial \\lambda}{\\partial \\theta_k}(\\theta) \\int_{\\mathbb{R}^n} \\Psi_\\varepsilon (\\theta) \\, \\overline{ \\varphi } \\, dx,\n\t\t\\end{array}\n\t\t\\end{equation}\nfor any ${ \\varphi \\in C^\\infty_{\\rm c}(\\mathbb{R}^n) }$ and a.e ${ \\omega \\in \\Omega }$. Here, \n$\\Lambda_{k,\\varepsilon} (x,\\omega,\\theta) :=  \\Lambda_k {\\left( \\frac{x}{\\varepsilon}, \\omega, \\theta \\right)}$.\n\nProceeding in a similar way with the cell equation~\\eqref{92347828454trfhfd4rfghjls}, we can find \n\\begin{multline}\\label{nbvnbvxzchgfs54}\n   \\int_{\\mathbb{R}^n} A {\\left( \\Phi^{-1} {\\left( \\frac{x}{\\varepsilon}, \\omega \\right)},\\omega \\right)} {\\left( \\nabla + 2i\\pi \\frac{\\theta}{\\varepsilon} \\right)} \\Psi_\\varepsilon (\\theta) \\cdot \\overline{ {\\left( \\nabla + 2i\\pi \\frac{\\theta}{\\varepsilon} \\right)} \\varphi } \\, dx \n   \\\\\n   \\hspace{-3cm} + \\frac{1}{\\varepsilon^2}\\int_{\\mathbb{R}^n} V {\\left( \\Phi^{-1} {\\left( \\frac{x}{\\varepsilon}, \\omega \\right)},\\omega \\right)} \\, \\Psi_\\varepsilon (\\theta) \\, \\overline{ \\varphi } \\, dx \n   - \\frac{\\lambda(\\theta)}{\\varepsilon^2}\\int_{\\mathbb{R}^n} \\Psi_\\varepsilon (\\theta) \\, \\overline{ \\varphi } \\, dx = 0,\n\\end{multline}\nfor any ${ \\varphi \\in C^\\infty_{\\rm c}(\\mathbb{R}^n) }$ and a.e. ${ \\omega \\in \\Omega }$.\t\t\t\n\n\n\t\n\\addcontentsline{toc}{section}{Part II: Asymptotic Equations}\n\\section*{Part II: Asymptotic Equations}\t\n\\section{\\!\\! On Schr\\\"odinger Equations Homogenization}\n\\label{HomoSchEqu}\n\nIn this section, we shall describe the asymptotic behaviour of the family of solutions \n$\\{ u_{\\varepsilon}{\\}_{\\varepsilon>0}}$ of the equation~\\eqref{jhjkhkjhkj765675233},\nthis is the content of Theorem~\\ref{876427463tggfdhgdfgkkjjlmk} below. It generalizes the similar result of Allaire, Piatnitski \\cite{AllairePiatnitski} where they consider the \nsimilar problem in the periodic setting.  Our scenario is much different from one considered by them. \nHere, the coefficients of equation~\\eqref{jhjkhkjhkj765675233} are random perturbations accomplished by \nstochastic diffeomorphisms of stationary functions.  \nSince the two-scale convergence technique is the best tool to deal with \nasymptotic analysis of linear operators, we make use of it in analogous way done in~\\cite{AllairePiatnitski}. \nAlthough, the presence of the stochastic deformation in the coefficients\nbrings out several complications, which we were able to overcome. \n\n\\medskip\nTo begin, some important and basic a priori estimates of the solution of the Schr\\\"odinger \nequation \\eqref{jhjkhkjhkj765675233} are needed. Then, we have the following  \n\n\\begin{lemma}[Energy Estimates]\n\\label{63457rf2wertgh}\n\tAssume that the conditions \\eqref{ASSUM1}, \\eqref{ASSUM2} hold and let \n${ u_\\varepsilon }\\in C\\big([0,T);H^1(\\mathbb R^n)\\big)$ be the solution of the equation \\eqref{jhjkhkjhkj765675233} with initial data \n\t$u_{\\varepsilon}^0$. Then, for all ${ t\\in [0,T] }$ and a.e. ${ \\omega \\in \\Omega }$, the following a priori estimates hold:\n\t\\begin{itemize}\n\t\t\\item[(i)] $($Energy Conservation$.)$ ${ \\displaystyle\\int_{\\mathbb{R}^n} {\\vert u_\\varepsilon(t,x,\\omega) \\vert}^2 dx = \\int_{\\mathbb{R}^n} {\\vert u_{\\varepsilon}^0(x,\\omega) \\vert}^2 dx }$.\n\t\t\\item[(ii)] $( \\varepsilon \\nabla-$ Estimate$.)$\n\t\t\\begin{eqnarray*}\n\t\t \\int_{\\mathbb{R}^n} |\\varepsilon\\nabla u_{\\varepsilon}(t,x,\\omega)|^2\\, dx\n\t\t\\le C\\int_{\\mathbb R^n}\\Big\\{|\\varepsilon\\nabla u_{\\varepsilon}^0(x,\\omega)|^2+|u_{\\varepsilon}^0(x,\\omega)|^2\\Big\\} \\,dx,\n\t \t\\end{eqnarray*}\nwhere ${ C:=C\\big(\\Lambda,{\\Vert A \\Vert}_\\infty,{\\Vert V \\Vert}_\\infty,{\\Vert U \\Vert}_\\infty} \\big) $ is a positive constant which does not depend on $\\varepsilon > 0$.\n\t        \t\\end{itemize}\n\\end{lemma}\n\\begin{proof}\n1.  If we multiply the Eq. \\eqref{jhjkhkjhkj765675233} by $\\overline{u_{\\varepsilon}}$ and take the imaginary part, then we obtain \n$$\n\\frac{d}{dt}\\int_{\\mathbb R^n}|u_{\\varepsilon}(t,x,\\omega)|^2\\,dx=0,\n$$\nwhich gives the proof of the item $(i)$.\n\n2. Now, multiplying the Eq. \\eqref{jhjkhkjhkj765675233} by $\\overline{\\partial_t u_{\\varepsilon}}$ and taking the real part, we get \n\\begin{eqnarray*}\n&&\\frac{1}{2}\\frac{d}{dt}\\int_{\\mathbb R^n}\\Big\\{\\varepsilon^2A\\left(\\Phi^{-1}\\left(\\frac{x}{\\varepsilon},\\omega\\right),\\omega\\right)\\nabla u_{\\varepsilon}\\cdot \\nabla \\overline{u_{\\varepsilon}}\\\\\n&&\\qquad\\qquad+\\Big( V\\left(\\Phi^{-1}\\left(\\frac{x}{\\varepsilon},\\omega\\right),\\omega\\right)+\\varepsilon^2 U\\left(\\Phi^{-1}\\left(\\frac{x}{\\varepsilon},\\omega\\right),\\omega\\right)\\Big)|u_{\\varepsilon}|^2 \\Big\\}\\,dx=0,\n\\end{eqnarray*}\nwhich provides the proof of the item $(ii)$.\n\\end{proof}\n\nIts important to remember the followings facts that will be necessary in this section:\n\\begin{itemize}\n\\item The initial data of the equation \\eqref{jhjkhkjhkj765675233} is assumed to be well-prepared, that is, \nfor $(x,\\omega) \\in \\mathbb{R}^n \\! \\times \\! \\Omega$, and \n${ \\theta^\\ast \\in \\mathbb{R}^n }$\n\\begin{equation}\n\\label{WellPreparedness}\n\tu_\\varepsilon^0(x,\\omega) = e^{2i\\pi \\frac{\\theta^\\ast \\cdot x}{\\varepsilon}}\\psi \\left( \\Phi^{-1}(x\/\\varepsilon,\\omega),\\omega,\\theta^\\ast \\right) \nv^0(x),\n\\end{equation} \nwhere {${ v^0 \\in C_{\\rm c}^\\infty(\\mathbb{R}^n) }$}, and ${ \\psi(\\theta^\\ast) }$ is an eigenfunction of the cell problem \\eqref{92347828454trfhfd4rfghjls}.  \n\\item Using the Ergodic Theorem, it is easily seen that the sequences \n$${ \\{u_{\\varepsilon}^0(\\cdot,\\omega)\\}_{\\varepsilon >0} } \\quad \\text{and} \\quad \n{ \\{\\varepsilon \\nabla u_{\\varepsilon}^0(\\cdot,\\omega) \\}_{\\varepsilon > 0} }$$ \nare bounded in ${ L^2(\\mathbb{R}^n) }$ and ${ [L^2(\\mathbb{R}^n)]^n }$, respectively. \n\\end{itemize}\n\nOne observes that, the main importance of the well preparedness of the initial data is the following: Trivially, the sequence of solutions \n$\\{ u_{\\varepsilon}{\\}_{\\varepsilon>0}}$ of the equation~\\eqref{jhjkhkjhkj765675233} two-scale converges to zero. However, if our initial data is well-prepared, we are able \nto correct the oscillations present in $u_{\\varepsilon}$ in such a way that, after this correction we can strengthen the weak convergence to the solution of a nontrivial homogenized \nSchr\\\"odinger Equation. For instance, we invite the readers to Allaire, Piatnistski \\cite{AllairePiatnitski}, Bensoussan, Lions, Papanicolaou \\cite[Chapter 4]{BensoussanLionsPapanicolaou} \nand Poupaud, Ringhofer \\cite{PoupaudRinghofer}. \n\n\\subsection{The Abstract Theorem.}\n\\label{ATH}\n\nIn the next, we establish an abstract homogenization theorem for Schr\\\"odinger equations.\n\n\\begin{theorem}\n\\label{876427463tggfdhgdfgkkjjlmk}\nLet $\\Phi(y,\\omega)$ be a stochastic deformation, and $\\tau:\\mathbb Z^n\\times \\Omega\\to \\Omega$ an ergodic $n-$dimensional dynamical \nsystem. \nAssume that the conditions \\eqref{ASSUM1}, \\eqref{ASSUM2} hold, and \nthere exists a Bloch frequence ${ \\theta^\\ast \\! \\in \\mathbb{R}^n }$ which is a critical point of \u00ad$\\lambda(\\cdot)$,\nthat is ${ \\nabla_{\\!\\! \\theta} \\, \\lambda (\\theta^\\ast) = 0 }$,\nwhere  ${ \\lambda (\\theta^\\ast) }$ is a simple eigenvalue of the spectral cell equation~\\eqref{92347828454trfhfd4rfghjls} associated to the eigenfunction \n$\\Psi(z,\\omega,\\theta^\\ast)\\equiv \\psi{\\left( \\Phi^{-1}(z,\\omega),\\omega,\\theta^\\ast \\right)}$. Assume also that, the initial data is well-prepared in the sense of \n\\eqref{WellPreparedness}. If ${ u_\\varepsilon }\\in C\\big([0,T);H^1(\\mathbb R^n)\\big)$  is the solution of~\\eqref{jhjkhkjhkj765675233} for each $\\varepsilon> 0$ fixed, then the sequence \n${ v_\\varepsilon }$ defined by \n\t\t\\begin{equation*}\n\t\t\tv_\\varepsilon(t,x,\\omega) := e^{ -{\\left( i \\frac{\\lambda(\\theta^\\ast) t}{\\varepsilon^2} + 2i\\pi \\frac{\\theta^\\ast \\! \\cdot x}{\\varepsilon} \\right)} } u_\\varepsilon(t,x,\\omega), \\;\\, (t,x) \\in \\mathbb{R}^{n+1}_T, \\; \\omega \\in \\Omega, \n\t\t\\end{equation*}\n$\\Phi_{\\omega}-$two-scale converges to ${ v(t,x) \\, \\psi{\\left( \\Phi^{-1}(z,\\omega),\\omega, \\theta^\\ast \\right)} }$, and satisfies for a.e.  ${ \\omega \\in \\Omega }$\n\t\t\\begin{equation*}\n\t\t\t\\lim_{\\varepsilon \\to 0} \\iint_{\\mathbb{R}^{n+1}_T} \\! {\\left\\vert v_\\varepsilon (t,x,\\omega) - v(t,x) \\, \\psi{\\left( \\Phi^{-1} {\\left(\\frac{x}{\\varepsilon},\\omega \\right)}, \\omega, \\theta^\\ast \\right)} \\right\\vert}^2 dx \\, dt \\, = \\, 0,\n\t\t\\end{equation*}\nwhere the function ${ v \\in C\\big([0,T); L^2(\\mathbb{R}^n)\\big) }$  is the unique solution of the homogenized Schr\\\"odinger equation \n\\begin{equation}\n\\label{HomSchEqu}\n\\left\\{\n\\begin{aligned}\n   & i \\displaystyle\\frac{\\partial v}{\\partial t} - {\\rm div} {\\left( A^\\ast \\nabla v \\right)} + U^\\ast v= 0 \\, , \\;\\, \\text{in} \\;\\, \\mathbb{R}^{n+1}_T, \n   \\\\[5pt]\n   &\tv(0,x) = v^0(x) \\, , \\;\\, x\\in \\mathbb{R}^n,\n\\end{aligned}\n\\right.\n\\end{equation}\nwith effective (constant) coefficients: matrix ${ A^\\ast = D_\\theta^2 \\lambda(\\theta^\\ast) }$, and potential   \n\t\t\\begin{equation*}\n\t\t\tU^\\ast =  c^{-1}_\\psi \\int_\\Omega \\int_{\\Phi([0,1)^n, \\omega)} U{\\left( \\Phi^{-1} (z,\\omega),\\omega \\right)}  {\\left\\vert \\psi {\\left( \\Phi^{-1} (z,\\omega), \\omega, \\theta^\\ast \\right)} \\right\\vert}^2 dz \\, d\\mathbb{P}(\\omega),\n\t\t\\end{equation*}\nwhere\n\t\t\\begin{equation*}\n\t\t\tc_\\psi = \\int_\\Omega \\int_{\\Phi([0,1)^n, \\omega)} {\\left\\vert \\psi {\\left( \\Phi^{-1} (z,\\omega), \\omega, \\theta^\\ast \\right)} \\right\\vert}^2 dz \\, d\\mathbb{P}(\\omega).\n\t\t\\end{equation*}\n\\end{theorem}\n\n\\begin{proof}\nIn order to better understand the main difficulties brought by the presence of the stochastic deformation $\\Phi$, we split our proof in five steps. \n\n\\medskip\n1.({\\it\\bf A priori estimates and $\\Phi_{\\omega}-$two-scale convergence.}) \nFirst, we define \n\\begin{equation}\n\\label{jghkd65454ads3e}\n\t    \tv_\\varepsilon (t,x,\\widetilde{\\omega}) := e^{-{\\left( i \\frac{\\lambda(\\theta^\\ast) t}{\\varepsilon^2} + 2i\\pi \\frac{\\theta^\\ast \\! \\cdot x}{\\varepsilon} \\right)} } \n\t\tu_\\varepsilon(t,x,\\widetilde{\\omega}), \\;\\, (t,x,\\widetilde{\\omega}) \\in \\mathbb{R}^{n+1}_T \\! \\times \\! \\Omega.\n\\end{equation}\nThen, computing the first derivatives with respect to the variable $x$, we get\n\t  \\begin{equation}\n\t  \\label{974967uhghjzpzas}\n\t    \t\\varepsilon \\nabla u_\\varepsilon (t,x,\\widetilde{\\omega})\\, e^{-{\\left( i \\frac{\\lambda(\\theta^\\ast) t}{\\varepsilon^2} \n\t\t+ 2i\\pi \\frac{\\theta^\\ast \\! \\cdot x}{\\varepsilon} \\right)} } = (\\varepsilon \\nabla + 2i\\pi \\theta^\\ast) v_\\varepsilon (t,x,\\widetilde{\\omega}).\n\t    \\end{equation}\n\n\n\n\t  \n\t   \n\t\n\t\nApplying Lemma~\\ref{63457rf2wertgh} yields:\n\t\n\t\n\t\n\t\n\t\t\\begin{itemize}\n\t\t\t\\item ${ \\displaystyle\\int_{\\mathbb{R}^n} {\\vert v_\\varepsilon(t,x,\\widetilde{\\omega}) \\vert}^2 dx = \\int_{\\mathbb{R}^n} \n\t\t\t{\\vert u_\\varepsilon^0(x,\\widetilde{\\omega}) \\vert}^2 dx },$\n\t\t\t\\item ${ \\displaystyle\\int_{\\mathbb{R}^n} {\\vert \\varepsilon \\nabla v_\\varepsilon(t,x,\\widetilde{\\omega}) \\vert}^2 dx \\leq \\widetilde{C} {\\displaystyle\\int_{\\mathbb{R}^n} \n\\Big( {\\vert \\varepsilon \\nabla u_\\varepsilon^0(x,\\widetilde{\\omega}) \\vert}^2 +{\\vert u_\\varepsilon^0(x,\\widetilde{\\omega}) \n\\vert}^2 \\Big) dx} }$\n\t\t\\end{itemize}\nfor all ${ t\\in[0,T) }$ and a.e. $\\widetilde{\\omega} \\in\\Omega$, where the constant ${ \\widetilde{C} }$ depends on\n$\\| A{\\|}_{\\infty}$, $\\|V{\\|}_{\\infty}$, $\\|U{\\|}_{\\infty}$ and $\\theta^\\ast$.  \nThen, from the uniform boundedness of the sequences \n${ \\{u_{\\varepsilon}^0(\\cdot,\\widetilde{\\omega})\\}_{\\varepsilon >0} }$ and ${ \\{\\varepsilon \\nabla u_{\\varepsilon}^0(\\cdot,\\widetilde{\\omega}) \\}_{\\varepsilon > 0} }$,  we deduce that the sequences \n$${ {\\{ v_{\\varepsilon}(\\cdot,\\cdot\\cdot,\\widetilde{\\omega}) \\}}_{\\varepsilon > 0} } \\quad \\text{and} \\quad { {\\{ \\varepsilon\\nabla v_\\varepsilon(\\cdot, \\cdot\\cdot, \\widetilde{\\omega}) \\}}_{\\varepsilon > 0} }$$\nare bounded, respectively, in ${ L^2(\\mathbb{R}^{n+1}_T) }$ and ${ {[ L^2(\\mathbb{R}^{n+1}_T)]}^n }$ for a.e. ${ \\widetilde{\\omega} \\in \\Omega }$. Therefore, applying Lemma \n\\ref{SYM1-5}, there exists a subsequence ${ \\{\\varepsilon^\\prime\\} }$(which may dependent on $\\tilde{\\omega}$), and a stationary function \n${ v^\\ast_{\\widetilde{\\omega}} \\in L^2(\\mathbb{R}^{n+1}_T, \\mathcal{H}) }$, for a.e. ${ \\widetilde{\\omega} \\in \\Omega }$, such that  \n\t\t\\begin{equation*}\n\t\t\tv_{\\varepsilon^\\prime}(t,x,\\widetilde{\\omega}) \\; \\xrightharpoonup[\\varepsilon^\\prime \\to 0]{2-{\\rm s}}\\; v^\\ast_{\\widetilde{\\omega}} {\\left( t,x,\\Phi^{-1}(z,\\omega),\\omega \\right)},\n\t\t\\end{equation*}\n\t\tand\n\t\t\\begin{equation*}\n\t\t\t\\varepsilon^\\prime \\frac{\\partial v_{\\varepsilon^\\prime}}{\\partial x_k}(t,x,\\widetilde{\\omega}) \\; \\xrightharpoonup[\\varepsilon^\\prime \\to 0]{2-{\\rm s}} \\; \n\t\t\t\\frac{\\partial }{\\partial z_k} {\\big( v^\\ast_{\\widetilde{\\omega}} {\\left(t,x,\\Phi^{-1}{(z,\\omega)},\\omega \\right)} \\big)},                \n\t\t\\end{equation*}\nwhich means that, for ${ k\\in \\{1,\\ldots,n\\} }$, we have \n\\begin{equation}\n\\label{jhjkhjfdasdfghyui}\n\\begin{aligned}\n      \\lim_{\\varepsilon^\\prime \\to 0}\\iint_{\\mathbb{R}^{n+1}_T} & v_{\\varepsilon^\\prime} \\left(t, x,\\widetilde{\\omega} \\right) \\, \n\t\t\t\\overline{ \\varphi(t,x) \\,\\Theta\\left(\\Phi^{-1}{\\left(\\frac{x}{\\varepsilon^\\prime},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right) } \\, dx \\, dt\n\\\\[5pt]\n& = c_{\\Phi}^{-1} \\!\\! \\displaystyle\\iint_{\\mathbb{R}^{n+1}_T} \\! \\int_\\Omega \\int_{\\Phi([0,1)^n, \\omega)} \\!\\!\\!\\!\\!   v^\\ast_{\\widetilde{\\omega}} \n\t\t\t{\\left(t,x,\\Phi^{-1}{(z,\\omega)},\\omega \\right)} \\,\n\\\\[5pt]\t\t\t\n\t\t\t & \\hspace{90pt} \\times \\; \\overline{ \\varphi(t,x) \\,\\Theta\\left(\\Phi^{-1}(z,\\omega),\\omega \\right)}   \\, dz \\, d\\mathbb{P} \\, dx \\, dt\n\\end{aligned}\n\\end{equation}\nand \n\\begin{equation}\n\\label{uygkgcxzcxzsw}\n\\begin{aligned}\n               \\lim_{\\varepsilon^\\prime \\to 0} \\iint_{\\mathbb{R}^{n+1}_T} & \\varepsilon^\\prime \\frac{\\partial v_{\\varepsilon^\\prime}}{\\partial x_k} \\left(t, x, \\widetilde{\\omega} \\right) \\, \n               \\overline{ \\varphi (t,x)\\,\\Theta\\left(\\Phi^{-1}{\\left(\\frac{x}{\\varepsilon^\\prime},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right) } \\, dx \\, dt\n\\\\[5pt]\n&= c_{\\Phi}^{-1} \\!\\! \\displaystyle\\iint_{\\mathbb{R}^{n+1}_T} \\! \\int_\\Omega \\int_{\\Phi([0,1)^n, \\omega)} \\frac{\\partial }{\\partial z_k} {\\left( v^\\ast_{\\widetilde{\\omega}} {\\left(t,x,\\Phi^{-1}{(z,\\omega)},\\omega \\right)} \\right)} \\, \n\\\\[5pt]\n& \\hspace{90pt} \\times \\, \\overline{ \\varphi (t,x)\\,\\Theta\\left(\\Phi^{-1}(z,\\omega),\\omega \\right)}  \\, dz \\, d\\mathbb{P} \\, dx \\, dt,\n\\end{aligned}\n\\end{equation}\nfor all functions $\\varphi \\in C^\\infty_{\\rm c}((-\\infty, T)  \\times  \\mathbb{R}^n)$ and $\\Theta \\in L^{2}_{\\loc}\\left(\\mathbb R^n\\times\\Omega\\right)$ stationary. Moreover, the sequence \n${ {\\{ v_\\varepsilon^0(\\cdot, \\widetilde{\\omega}) \\}}_{\\varepsilon > 0} }$ defined by, \n\t\t\\begin{equation}\\label{5t345rte54ew3e2wswqqq1qdecv}\n\t\t\tv_\\varepsilon^0(x,\\omega):=\\psi{\\left( \\Phi^{-1}{\\left( \\frac{x}{\\varepsilon},\\omega \\right)},\\omega,\\theta^\\ast \\right)} v^0(x), \\;\\; (x,\\omega) \\in \\mathbb{R}^n \\! \\times \\! \\Omega,\n\t\t\\end{equation}\nsatisfies\n\t\t\\begin{equation}\\label{766765t6y4rf5tzxcvbsgfhry}\n\t\t\tv_{\\varepsilon}^0(\\cdot,\\widetilde{\\omega}) \\; \\xrightharpoonup[\\varepsilon \\to 0]{2-{\\rm s}}\\; v^0(x) \\, \\psi{\\left( \\Phi^{-1}(z,\\omega),\\omega,\\theta^\\ast \\right)},\n\t\t\\end{equation}\nfor each stationary function ${ \\psi(\\theta^\\ast) }$.\n\t\t\n\\bigskip\n2.({\\it\\bf The Split Process.}) We consider the following \n\n\\medskip\n \\underline {Claim:} \nThere exists ${ v_{\\widetilde{\\omega}} \\in L^2(\\mathbb{R}^{n+1}_T) }$, such that \n$$\n\\begin{aligned}\n   v^\\ast_{\\widetilde{\\omega}} {\\left( t,x,\\Phi^{-1}(z,\\omega),\\omega \\right)}&= v_{\\widetilde{\\omega}}(t,x) \\, \\psi {\\left(\\Phi^{-1}(z,\\omega),\\omega, \\theta^\\ast \\right)} \n\\\\[5pt]\n   &\\equiv v_{\\widetilde{\\omega}} (t,x) \\, \\Psi(z,\\omega, \\theta^\\ast).\n\\end{aligned}       \n$$\n  \nProof of Claim: First, for any $\\widetilde{\\omega} \\in \\Omega$ fixed, we take the function \n\\begin{equation}\n\\label{7676567543409hj}\n\t\t\tZ_\\varepsilon (t,x,\\widetilde{\\omega}) = \\varepsilon^2 e^{i \\frac{\\lambda(\\theta^\\ast) t}{\\varepsilon^2} + 2i\\pi \\frac{\\theta^\\ast \\! \\cdot x}{\\varepsilon}} \\varphi (t, x) \\, \n\t\t\t\\Theta {\\big( \\Phi^{-1}\\big( \\frac{x}{\\varepsilon}, \\widetilde{\\omega} \\big), \\widetilde{\\omega} \\big)}\n\\end{equation}\nas a test function in the associated variational formulation of the equation \\eqref{jhjkhkjhkj765675233}, where ${ \\varphi \\in C^\\infty_{\\rm c}((-\\infty, T) \\! \\times \\! \\mathbb{R}^n) }$ \nand $ \\Theta\\in L^{\\infty}\\left(\\mathbb R^n\\times\\Omega\\right)$ stationary, with $\\Theta(\\cdot,\\omega)$ smooth. Therefore, we obtain \n$$\n\\begin{aligned} \n\t\t\t&- i \\iint_{\\mathbb{R}^{n+1}_T} u_\\varepsilon(t,x,\\widetilde{\\omega}) \\, \\frac{\\partial \\overline{Z_\\varepsilon}}{\\partial t} (t,x,\\widetilde{\\omega}) \\, dx \\, dt\n\t\t\t+ i \\int_{\\mathbb{R}^n} u_\\varepsilon^0(x,\\widetilde{\\omega}) \\, \\overline{Z_\\varepsilon}(0,x,\\widetilde{\\omega}) \\, dx\n\\\\[15pt]\n\t\t\t&+ \\iint_{\\mathbb{R}^{n+1}_T} A {\\left(\\Phi^{-1}\\left( \\frac{x}{\\varepsilon}, \\widetilde{\\omega} \\right),\\widetilde{\\omega} \\right)} \\nabla u_\\varepsilon(t,x,\\widetilde{\\omega}) \\cdot \\nabla \\overline{Z_\\varepsilon}(t,x,\\widetilde{\\omega}) \\, dx \\, dt\n\\\\[15pt]\n\t\t\t&+ \\frac{1}{\\varepsilon^2} \\iint_{\\mathbb{R}^{n+1}_T} V {\\left(\\Phi^{-1}\\left( \\frac{x}{\\varepsilon}, \\widetilde{\\omega} \\right),\\widetilde{\\omega} \\right)} u_\\varepsilon(t,x,\\widetilde{\\omega}) \\, \\overline{Z_\\varepsilon}(t,x,\\widetilde{\\omega}) \\, dx \\, dt\n\\\\[15pt]\n\t\t\t&+  \\iint_{\\mathbb{R}^{n+1}_T} U {\\left( \\Phi^{-1}\\left( \\frac{x}{\\varepsilon}, \\widetilde{\\omega} \\right),\\widetilde{\\omega} \\right)} u_\\varepsilon(t,x,\\widetilde{\\omega})\n\t\t\t\\, \\overline{Z_\\varepsilon}(t,x,\\widetilde{\\omega}) \\, dx \\, dt= 0,\n\\end{aligned}\n$$\nand since \t\t\n$$\n\\begin{aligned}\n\t\t\t\\frac{\\partial Z_\\varepsilon }{\\partial t} (t,x,\\widetilde{\\omega})&= i \\lambda(\\theta^\\ast) \\, e^{i \\frac{\\lambda(\\theta^\\ast) t}{\\varepsilon^2} \n\t\t\t+ 2i\\pi \\frac{\\theta^\\ast \\! \\cdot x}{\\varepsilon}} \\, \\varphi (t, x) \\, \n\t\t\t\\Theta {( \\Phi^{-1}( \\frac{x}{\\varepsilon}, \\widetilde{\\omega}), \\widetilde{\\omega})} + \\mathrm{O}(\\varepsilon^2), \n\\\\[5pt]\n\t\t\t\\nabla Z_\\varepsilon (t,x,\\widetilde{\\omega})&= \\varepsilon \\, e^{i \\frac{\\lambda(\\theta^\\ast) t}{\\varepsilon^2} + 2i\\pi \\frac{\\theta^\\ast \\! \\cdot x}{\\varepsilon}} \\, (\\varepsilon \\nabla + 2i\\pi \\theta^\\ast) \n\t\t\t\\big( \\varphi(t,x) \\, \\Theta{( \\Phi^{-1}{(\\frac{x}{\\varepsilon},\\widetilde{\\omega})},\\widetilde{\\omega})}\\big), \n\\end{aligned}\n$$\nit follows that\n$$\n\\begin{aligned}\n   &-  \\lambda(\\theta^\\ast) \\displaystyle\\iint_{\\mathbb{R}^{n+1}_T} v_\\varepsilon(t,x,\\widetilde{\\omega}) \\,  \n   \\overline{ \\varphi (t, x) \\, \\Theta {( \\Phi^{-1}\\big( \\frac{x}{\\varepsilon}, \\widetilde{\\omega} \\big), \\widetilde{\\omega})} } \\, dx \\, dt \n\\\\[5pt]\n  & + \\iint_{\\mathbb{R}^{n+1}_T} A {( \\Phi^{-1}{\\big(\\frac{x}{\\varepsilon},\\widetilde{\\omega} \\big)},\\widetilde{\\omega})} \\, {(\\varepsilon \\nabla + 2i\\pi \\theta^\\ast)} v_\\varepsilon(t,x,\\widetilde{\\omega})\n\\\\[5pt]\n\t\t& \\hspace{60pt} \\cdot   \\overline{ {(\\varepsilon \\nabla + 2i\\pi \\theta^\\ast)} {\\left( \\varphi (t,x) \\, \\Theta {\\left(\\Phi^{-1}{\\left(\\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} \\right)} } \\, dx \\, dt \n\\\\[5pt]\n   &+ \\iint_{\\mathbb{R}^{n+1}_T} V {\\left( \\Phi^{-1}{\\left(\\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} \\, v_\\varepsilon(t,x,\\widetilde{\\omega})  \\, \\overline{ \\varphi (t,x) \\, \\Theta {\\left(\\Phi^{-1}{\\left(\\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} } \\, dx \\, dt= \\mathrm{O}(\\varepsilon^2),\n\\end{aligned}\n$$\nwhere we have used \\eqref{jghkd65454ads3e}, \\eqref{974967uhghjzpzas}, \\eqref{5t345rte54ew3e2wswqqq1qdecv}, and \\eqref{7676567543409hj}.\nAlthough, it is more convenient to rewrite as \n\\begin{equation}\n\\label{56tugryfgrffdd}\n\\begin{aligned}\n &- \\lambda(\\theta^\\ast) \\displaystyle\\iint_{\\mathbb{R}^{n+1}_T} v_\\varepsilon(t,x,\\widetilde{\\omega}) \\,  \\overline{ \\varphi (t,x) \\, \\Theta {( \\Phi^{-1}\\big( \\frac{x}{\\varepsilon}, \\widetilde{\\omega}\\big), \\widetilde{\\omega})} } \\, dx \\, dt\n \\\\[5pt]\n&+ \\iint_{\\mathbb{R}^{n+1}_T} {(\\varepsilon \\nabla + 2i\\pi \\theta^\\ast)} v_\\varepsilon(t,x,\\widetilde{\\omega}) \n\\\\[5pt]\n& \\hspace{20pt} \\cdot \\overline{ A {( \\Phi^{-1}{\\big(\\frac{x}{\\varepsilon},\\widetilde{\\omega} \\big)},\\widetilde{\\omega})} \\, \n{(\\varepsilon \\nabla + 2i\\pi \\theta^\\ast)} {\\big( \\varphi(t,x) \\, \\Theta{( \\Phi^{-1}{\\big(\\frac{x}{\\varepsilon},\\widetilde{\\omega} \\big)},\\widetilde{\\omega})} \\big)} } \\, dx \\, dt\n\\\\[5pt]\n&+ \\iint_{\\mathbb{R}^{n+1}_T} v_\\varepsilon(t,x,\\widetilde{\\omega}) \\, \\overline{ \\varphi (t,x) \\, V {( \\Phi^{-1}{\\big(\\frac{x}{\\varepsilon},\\widetilde{\\omega} \\big)},\\widetilde{\\omega})}\n \\, \\Theta{( \\Phi^{-1}{\\big(\\frac{x}{\\varepsilon},\\widetilde{\\omega} \\big)},\\widetilde{\\omega})} } \\, dx \\, dt= \\mathrm{O}(\\varepsilon^2).\n\\end{aligned}\n\\end{equation}\n\\begin{comment}\nPESSOAL, ACHO QUE ESSA PARTE NAO PRECISA. POR ISSO, A SUPRIMI.\n\n\t\t\nIn the second brackets above, observe that \t\t\n\t\t\\begin{equation*}\n\t\t\\begin{split}\n\t\t\t& A {\\left( \\Phi^{-1}{\\left(\\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} {\\bigg[ {(\\varepsilon \\nabla + 2i\\pi \\theta^\\ast)} {\\left( \\varphi (t,x) \\, \\zeta{\\left( \\Phi^{-1}{\\left(\\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} \\right)} \\bigg]} \\\\\n\t\t\t& \\hspace{3.75cm} = A {\\left( \\Phi^{-1}{\\left(\\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)}{\\bigg[ \\varepsilon \\nabla \\varphi (t,x) \\, \\zeta {\\left( \\Phi^{-1}{\\left(\\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} } \\\\ \n\t\t\t& \\hspace{4cm} + \\varphi(t,x) [\\nabla_{\\!\\! y} \\Phi]^{-1} {\\left( \\Phi^{-1}{\\left(\\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} \\, (\\nabla_{\\!\\! y} \\zeta) {\\left( \\Phi^{-1}{\\left(\\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} \\\\\n\t\t\t& \\hspace{4.25cm} { + \\, 2i\\pi \\theta^\\ast \\varphi (t,x) \\zeta {\\left( \\Phi^{-1}{\\left(\\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} \\bigg]},\n\t\t\\end{split}\n\t\t\\end{equation*}\nor equivalently  \n\t\t\\begin{equation*}\n\t\t\\begin{split}\n\t\t\t& A {\\left( \\Phi^{-1}{\\left(\\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} \\, {(\\varepsilon \\nabla + 2i\\pi \\theta^\\ast)} {\\left( \\varphi (t,x) \\, \\zeta{\\left( \\Phi^{-1}{\\left(\\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} \\right)} \\\\\n\t\t\t& \\hspace{0.5cm} = \\sum_{k=1}^n \\frac{\\partial \\varphi}{\\partial x_k} (t,x) {\\bigg\\{ \\varepsilon A {\\left( \\Phi^{-1}{\\left(\\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} {\\left[ e_k \\zeta {\\left( \\Phi^{-1}{\\left(\\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} \\right]} \\bigg\\}} \\\\ \n\t\t\t& \\hspace{0.75cm} + \\varphi(t,x) {\\Bigg\\{ A {\\left( \\Phi^{-1}{\\left(\\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} \\, {\\bigg[ {[\\nabla_{\\!\\! y} \\Phi]}^{-1}{\\left( \\Phi^{-1}{\\left(\\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} (\\nabla_{\\!\\! y} \\zeta){\\left( \\Phi^{-1}{\\left(\\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} \\bigg]} \\Bigg\\} }\\\\\n\t\t\t& \\hspace{0.75cm} + \\varphi(t,x) {\\Bigg\\{ A {\\left( \\Phi^{-1}{\\left(\\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} {\\bigg[ 2i\\pi \\theta^\\ast \\zeta{\\left( \\Phi^{-1}{\\left(\\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} \\bigg]} \\Bigg\\}}.\n\t\t\\end{split}\n\t\t\\end{equation*}\nIt is easily seen that the sequences  \n\t\t\\begin{itemize}\n\t\t\t\\item ${ \\displaystyle {\\left\\{ \\varepsilon A {\\left( \\Phi^{-1}{\\left(\\frac{\\cdot}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} {\\left[ e_k \\zeta {\\left( \\Phi^{-1}{\\left(\\frac{\\cdot}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} \\right]} \\right\\}}_{\\varepsilon > 0} }$,\n\t\t\t\\item ${ \\displaystyle {\\left\\{ A {\\left( \\Phi^{-1}{\\left(\\frac{\\cdot}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} \\, {\\bigg[ {[\\nabla_{\\!\\! y} \\Phi]}^{-1}{\\left( \\Phi^{-1}{\\left(\\frac{\\cdot}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} (\\nabla_{\\!\\! y} \\zeta){\\left( \\Phi^{-1}{\\left(\\frac{\\cdot}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} \\bigg]} \\right\\}_{\\varepsilon > 0}} }$, \n\t\t\t\\item ${ \\displaystyle {\\left\\{ A {\\left( \\Phi^{-1}{\\left(\\frac{\\cdot}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} \\, {\\bigg[ 2i\\pi \\theta^\\ast \\zeta{\\left( \\Phi^{-1}{\\left(\\frac{\\cdot}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} \\bigg]} \\right\\}_{\\varepsilon > 0}}, }$\n\t\t\\end{itemize}\nare such that the first two-scale converges strongly to ${ 0 }$ and the second and the third two-scale converge strongly to \n\\begin{itemize}\n\t\t\t\\item ${ (z,\\omega) \\; \\mapsto \\; A{\\left( \\Phi^{-1}(z,\\omega),\\omega \\right)} {\\big[ \\nabla_{\\!\\! z} {\\big( \\zeta{\\left( \\Phi^{-1}(z,\\omega),\\omega \\right)} \\big)} \\big]} }$, \n\t\t\t\\item ${ (z,\\omega) \\; \\mapsto \\; A{\\left( \\Phi^{-1}(z,\\omega),\\omega \\right)} {\\big[ 2i\\pi\\theta^\\ast\\zeta{\\left( \\Phi^{-1}(z,\\omega),\\omega \\right)}  \\big]} }$,\n\t\t\\end{itemize}\nrespectively, for the functions \n\n\\begin{itemize}\n\t\t\t\\item ${ (y,\\omega) \\; \\mapsto \\; A (y,\\omega) {\\left[ e_k \\zeta (y,\\omega) \\right]} }$,\n\t\t\t\\item ${ \\displaystyle (y,\\omega) \\; \\mapsto \\; A(y,\\omega) {\\left[ {[\\nabla_{\\!\\! y} \\Phi]}^{-1}(y,\\omega) {\\big( \\nabla_{\\!\\! y} \\zeta \\big)}(y,\\omega) \\right]} \\cdot e_\\ell }$, ${ \\ell \\in \\{1,\\ldots,n\\} }$\n\t\t\t\\item ${ \\displaystyle (y,\\omega) \\; \\mapsto \\; A(y,\\omega){\\big[ 2i\\pi \\theta^\\ast \\zeta(y,\\omega) \\big]} \\cdot e_\\ell }$, ${ \\ell \\in \\{1,\\ldots,n\\} }$\n\t\t\\end{itemize}\nare stationary. Hence, for a.e. ${ \\widetilde{\\omega} \\in \\Omega }$ the sequence \n\t\t\\begin{equation*}\n\t\t\t{\\left\\{ A {\\left( \\Phi^{-1}{\\left(\\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} \\, {(\\varepsilon \\nabla + 2i\\pi \\theta^\\ast)} {\\left( \\varphi (t,x) \\, \\zeta{\\left( \\Phi^{-1}{\\left(\\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} \\right)} \\right\\}}_{\\varepsilon>0},\n\t\t\\end{equation*}\ntwo-scale converges strongly to \n\n\t\t\\begin{equation*}\n\t\t\t(z,\\omega) \\; \\mapsto \\; A{\\left( \\Phi^{-1}(z,\\omega),\\omega \\right)} {\\left[ {(\\nabla_{\\!\\! z} + 2i\\pi\\theta^\\ast)} {\\left( \\zeta {\\left( \\Phi^{-1}(z,\\omega),\\omega \\right)} \\right)} \\right]}.\n\t\t\\end{equation*}\n\\end{comment}\nNow, making $\\varepsilon={ \\varepsilon^\\prime}$, letting $\\varepsilon'\\to 0 $ and using the Definition \\ref{two-scale}, we have for a.e. ${ \\widetilde{\\omega} \\in \\Omega }$,\nfor all ${ \\varphi \\in C^\\infty_{\\rm c}((-\\infty, T) \\! \\times \\! \\mathbb{R}^n) }$, \n$\\Theta \\in L^{\\infty}\\left(\\mathbb R^n\\times\\Omega\\right)$ stationary and $\\Theta(\\cdot,\\omega)$ smooth, \n\\begin{equation*}\n\\begin{aligned}\n&- \\lambda(\\theta^\\ast) \\, c_\\Phi^{-1} \\!\\!\\! \\displaystyle\\iint_{\\mathbb{R}^{n+1}_T} \\! \\int_\\Omega \\int_{\\Phi([0,1)^n, \\omega)} v^\\ast_{\\widetilde{\\omega}} {\\left( t,x,\\Phi^{-1}(z,\\omega),\\omega \\right)} \n\\\\[7pt]\n& \\hspace{90pt} \\times \\overline{ \\varphi (t,x) \\, \\Theta {\\left( \\Phi^{-1}(z,\\omega),\\omega \\right)} } \\, dz \\, d\\mathbb{P}(\\omega) \\, dx \\, dt\n\\\\[7pt]\n&+ c_\\Phi^{-1} \\!\\!\\! \\displaystyle\\iint_{\\mathbb{R}^{n+1}_T} \\! \\int_\\Omega \\int_{\\Phi([0,1)^n, \\omega)} {(\\nabla_{\\!\\! z} + 2i\\pi \\theta^\\ast)} {\\left( v^\\ast_{\\widetilde{\\omega}} {\\left( t,x,\\Phi^{-1}(z,\\omega),\\omega \\right)} \\right)}\n\\\\[7pt]\n& \\cdot \\overline{ \\varphi (t,x) \\, A {\\left( \\Phi^{-1}(z,\\omega),\\omega \\right)}  {\\left[ {(\\nabla_{\\!\\! z} + 2i\\pi \\theta^\\ast)} {\\left(  \\Theta {\\left( \\Phi^{-1}(z,\\omega),\\omega \\right)} \\right)} \\right]} } \\, dz \\, d\\mathbb{P}(\\omega) \\, dx \\, dt \n\\\\[7pt]\n&+ c_\\Phi^{-1} \\!\\!\\! \\displaystyle\\iint_{\\mathbb{R}^{n+1}_T} \\! \\int_\\Omega \\int_{\\Phi([0,1)^n, \\omega)} v^\\ast_{\\widetilde{\\omega}} {\\left( t,x,\\Phi^{-1}(z,\\omega),\\omega \\right)} \\, \n\\\\[7pt]\n&\\hspace{60pt} \\times  \\overline{ \\varphi (t,x) \\, V {\\left( \\Phi^{-1}(z,\\omega),\\omega \\right)} \\, \\Theta {\\left( \\Phi^{-1}(z,\\omega),\\omega \\right)} } \\, dz \\, d\\mathbb{P}(\\omega) \\, dx \\, dt = 0.\n\\end{aligned}\n\\end{equation*}\nTherefore, due to an argument \nof density in the test functions\n(thanks to the topological structure of $\\Omega$), \nwe can conclude that \n\\begin{comment}\nNOVAMENTE PESSOAL, PODEMOS ENCURTAR ESSA PASSAGEM. O QUE ACHAM?\n\nfor each ${ \\zeta \\in \\mathcal{S} }$ there exists \n${ N_\\zeta \\subset (0,T) \\! \\times \\! \\mathbb{R}^n }$ such that ${ {\\vert N_\\zeta \\vert}=0 }$ (Here ${ \\vert \\cdot \\vert }$ denotes the Lebesgue measure in \n${ (0,T) \\! \\times \\! \\mathbb{R}^n }$) such that \n\t\t\\begin{equation*}\n\t\t\\begin{array}{l}\n\t\t\t- \\lambda(\\theta^\\ast) \\, \\displaystyle\\int_\\Omega \\int_{\\Phi(\\mathsf{Y}, \\omega)} v^\\ast_{\\widetilde{\\omega}} {\\left( t,x,\\Phi^{-1}(z,\\omega),\\omega \\right)} \\,  \\overline{ \\zeta {\\left( \\Phi^{-1}(z,\\omega),\\omega \\right)} } \\, dz \\, d\\mathbb{P}(\\omega) \\\\ [15pt]\n\t\t\t+ \\displaystyle\\int_\\Omega \\int_{\\Phi(\\mathsf{Y}, \\omega)} A {\\left( \\Phi^{-1}(z,\\omega),\\omega \\right)} {(\\nabla_{\\!\\! z} + 2i\\pi \\theta^\\ast)} {\\left( v^\\ast_{\\widetilde{\\omega}} {\\left( t,x,\\Phi^{-1}(z,\\omega),\\omega \\right)} \\right)} \\cdot \\\\ [15pt]\n\t\t\t\\hspace{9.5cm} \\overline{ {(\\nabla_{\\!\\! z} + 2i\\pi \\theta^\\ast)} {\\left( \\zeta {\\left( \\Phi^{-1}(z,\\omega),\\omega \\right)} \\right)} } \\, dz \\, d\\mathbb{P}(\\omega) \\\\ [10pt]\n\t\t\t+ \\displaystyle\\int_\\Omega \\int_{\\Phi(\\mathsf{Y}, \\omega)} V {\\left( \\Phi^{-1}(z,\\omega),\\omega \\right)} \\,  v^\\ast_{\\widetilde{\\omega}} {\\left( t,x,\\Phi^{-1}(z,\\omega),\\omega \\right)} \\, \\overline{ \\zeta {\\left( \\Phi^{-1}(z,\\omega),\\omega \\right)} } \\, dz \\, d\\mathbb{P} = 0,\n\t\t\\end{array}\n\t\t\\end{equation*}\nfor all  ${ (t,x) \\in (0,T) \\! \\times \\! \\mathbb{R}^n \\setminus N_\\zeta }$. Now, due to the enumerability of ${ \\mathcal{S} }$, there exists ${ N \\subset (0,T) \\! \\times \\! \\mathbb{R}^n }$ \nsuch that ${ {\\vert N \\vert}=0 }$ with the property\n\\end{comment}\n\\begin{equation*}\n\\begin{aligned}\n&- \\lambda(\\theta^\\ast) \\, \\displaystyle\\int_\\Omega \\int_{\\Phi([0,1)^n, \\omega)} v^\\ast_{\\widetilde{\\omega}} {\\left( t,x,\\Phi^{-1}(z,\\omega),\\omega \\right)} \\,  \\overline{ \\Theta {\\left( \\Phi^{-1}(z,\\omega),\\omega \\right)} } \\, dz \\, d\\mathbb{P}(\\omega) \n\\\\[5pt]\n&+ \\displaystyle\\int_\\Omega \\int_{\\Phi([0,1)^n, \\omega)} A {\\left( \\Phi^{-1}(z,\\omega),\\omega \\right)} {(\\nabla_{\\!\\! z} + 2i\\pi \\theta^\\ast)} {\\left( v^\\ast_{\\widetilde{\\omega}} {\\left( t,x,\\Phi^{-1}(z,\\omega),\\omega \\right)} \\right)}\n\\\\[5pt]\n&\\hspace{60pt} \\cdot \\overline{ {(\\nabla_{\\!\\! z} + 2i\\pi \\theta^\\ast)} {\\left( \\Theta {\\left( \\Phi^{-1}(z,\\omega),\\omega \\right)} \\right)} } \\, dz \\, d\\mathbb{P}(\\omega)\n\\\\[5pt]\n&+ \\int_\\Omega \\int_{\\Phi([0,1)^n, \\omega)} \\!\\!\\! V {( \\Phi^{-1}(z,\\omega),\\omega)} \\,  v^\\ast_{\\widetilde{\\omega}} \n{( t,x,\\Phi^{-1}(z,\\omega),\\omega)} \\, \n\\\\[5pt]\n&\\hspace{120pt} \\times  \\overline{ \\Theta {( \\Phi^{-1}(z,\\omega),\\omega)} } \\, dz \\, d\\mathbb{P}(\\omega)= 0,\n\\end{aligned}\n\\end{equation*}\nfor a.e. ${ (t,x) \\in \\mathbb{R}^{n+1}_T }$ and for all ${ \\Theta}$ as above.  Thus, the simplicity of the eigenvalue\u00ad $\\lambda(\\theta^\\ast)$ \nassures us that for a.e.  $\\mathbb{R}^{n+1}_T $, the function \n$${ (z,\\omega) \\mapsto v^\\ast_{\\widetilde{\\omega}}{\\left( t,x,\\Phi^{-1}(z,\\omega),\\omega \\right)} }$$ (which belongs to the space ${ \\mathcal{H} }$) is parallel to the \nfunction ${ \\Psi(\\theta^\\ast) }$, i.e., we can find ${ v_{\\widetilde{\\omega}}(t,x) \\in \\mathbb{C} }$, such that \n\t\t\\begin{eqnarray*}\nv^\\ast_{\\widetilde{\\omega}} {\\left( t,x,\\Phi^{-1}(z,\\omega),\\omega \\right)} &=& v_{\\widetilde{\\omega}}(t,x) \\, \\Psi(z,\\omega, \\theta^\\ast)\\\\\n& \\equiv & v_{\\widetilde{\\omega}}(t,x) \\, \\psi {\\left(\\Phi^{-1}(z,\\omega),\\omega, \\theta^\\ast \\right)}.\n\t\t\\end{eqnarray*}\n\t\t\n\\medskip\nFinally, since ${ v^\\ast_{\\widetilde{\\omega}} \\in L^2 (\\mathbb{R}^{n+1}_T; \\mathcal{H}) }$, we conclude that  ${ v_{\\widetilde{\\omega}}\\in L^2(\\mathbb{R}^{n+1}_T) }$, which \ncompletes the proof of our claim.\n\n\\medskip\n3.({\\it\\bf Homogenization Process.}) Let  ${ \\Lambda_k (\\theta^\\ast)}$, for any $k\\in \\{1,\\ldots,n\\}$, be the function defined by \n\t\t\\begin{equation*}\n\t\t\t\\Lambda_k(z,\\omega,\\theta^\\ast)=\\frac{1}{2i\\pi}\\frac{\\partial \\Psi}{\\partial \\theta_k}(z,\\omega,\\theta^\\ast)=\\frac{1}{2i\\pi}\\frac{\\partial \\psi}{\\partial \\theta_k}\n\t\t\t{\\left( \\Phi^{-1}(z,\\omega),\\omega,\\theta^\\ast \\right)}, \\; (z,\\omega) \\in \\mathbb{R}^n \\! \\times \\! \\Omega,\n\t\t\\end{equation*}\nwhere the function $\\Psi(z,\\omega,\\theta^\\ast)=\\psi{\\left( \\Phi^{-1}(z,\\omega),\\omega,\\theta^\\ast \\right)}$ \nis the eigenfunction of the spectral cell problem~\\eqref{92347828454trfhfd4rfghjls}. \nThen, we consider the following test function \n\\begin{equation*}\n\t\t\tZ_\\varepsilon(t,x,\\widetilde{\\omega}) = e^{i \\frac{\\lambda(\\theta^\\ast) t}{\\varepsilon^2} + 2i\\pi \\frac{\\theta^\\ast \\! \\cdot x}{\\varepsilon}} {\\big( \\varphi(t,x) \\, \n\t\t\t\\Psi_\\varepsilon(x,\\widetilde{\\omega},\\theta^\\ast)  +\\varepsilon  \\sum_{k=1}^n \n\t\t\t\\frac{\\partial \\varphi}{\\partial x_k}(t,x) \\, \\Lambda_{k,\\varepsilon} (x,\\widetilde{\\omega},\\theta^\\ast) \\big)},\n\\end{equation*}\nwhere ${ \\varphi \\in C^\\infty_{\\rm c}((-\\infty, T) \\! \\times \\! \\mathbb{R}^n) }$ and \n\t\t\\begin{equation*}\n\t\t\t\\Psi_\\varepsilon(x,\\widetilde{\\omega}, \\theta^\\ast) = \\Psi{\\left( \\frac{x}{\\varepsilon},\\widetilde{\\omega},\\theta^\\ast\\right)} ,\\quad \\;\\; \\Lambda_{k,\\varepsilon}(x,\\widetilde{\\omega}, \\theta^\\ast) = \\Lambda_k{\\left( \\frac{x}{\\varepsilon},\\widetilde{\\omega},\\theta^\\ast\\right)}.\n\t\t\\end{equation*}    \nUsing the function ${ Z_\\varepsilon }$ as test function in the variational formulation of the equation \\eqref{jhjkhkjhkj765675233}, \nwe obtain\t\t\n\\begin{equation}\n\\label{676745459023v}\n\\begin{aligned}\n& \\big[ i \\displaystyle\\int_{\\mathbb{R}^n} u_\\varepsilon^0(x,\\widetilde{\\omega}) \\, \\overline{Z_\\varepsilon}(0,x,\\widetilde{\\omega}) \\, dx  \n - i \\displaystyle\\iint_{\\mathbb{R}^{n+1}_T} u_\\varepsilon(t,x,\\widetilde{\\omega}) \\, \\frac{\\partial \\overline{Z_\\varepsilon}}{\\partial t} (t,x,\\widetilde{\\omega}) \\, dx \\, dt \\big]\n\\\\[7pt]\n&+ \\big[ \\displaystyle\\iint_{\\mathbb{R}^{n+1}_T} A {\\left(\\Phi^{-1}{\\left( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} \\, \n\t\t\t\\nabla u_\\varepsilon (t,x,\\widetilde{\\omega}) \\cdot \\nabla \\overline{Z_\\varepsilon}(t,x,\\widetilde{\\omega}) \\, dx \\, dt \\big]\n\\\\[7pt]\n&\t\t\t+ \\big[ \\displaystyle\\frac{1}{\\varepsilon^2} \\iint_{\\mathbb{R}^{n+1}_T} V {\\left(\\Phi^{-1}{\\left( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right) \\, \n\t\t\tu_\\varepsilon(t,x,\\widetilde{\\omega}) \\, \\overline{Z_\\varepsilon}(t,x,\\widetilde{\\omega}) \\, dx \\, dt} \n\\\\[7pt]\n& \\hspace{30pt}\t\t\t+ \\iint_{\\mathbb{R}^{n+1}_T} U {\\left(x, \\Phi^{-1}{\\left( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} \\, \n\t\t\tu_\\varepsilon(t,x,\\widetilde{\\omega}) \\, \\overline{Z_\\varepsilon}(t,x,\\widetilde{\\omega}) \\, dx \\, dt \\big]= 0. \n\\end{aligned}\n\\end{equation}\nIn order to simplify the manipulation of the above equation, we shall denote by $I_{k}^{\\varepsilon}(k=1,2,3)$ the respective term in the $k^{\\text{th}}$ brackets, so that we can rewrite the \nequation~\\eqref{676745459023v} as $I_{1}^{\\varepsilon}+I_{2}^{\\varepsilon}+I_{3}^{\\varepsilon}=0$.\n\n\\medskip\t\t\nThe analysis of the $I_{1}^{\\varepsilon}$ term is triggered  by the following computation \n$$\n\\begin{aligned}\n\t\\frac{\\partial Z_\\varepsilon}{\\partial t}(t,x,\\widetilde{\\omega}) &=  e^{i \\frac{\\lambda(\\theta^\\ast) t}{\\varepsilon^2} + 2i\\pi \\frac{\\theta^\\ast \\! \\cdot x}{\\varepsilon}} \n\t{\\Big[ i\\frac{\\lambda(\\theta^\\ast)}{\\varepsilon^2} { \\big( \\varphi(t,x) \\, \\Psi_\\varepsilon(x,\\widetilde{\\omega},\\theta^\\ast) } }\n\\\\[5pt]\n\t\t\t& + \\, { { \\varepsilon\\sum_{k=1}^n \\frac{\\partial \\varphi}{\\partial x_k}(t,x) \\, \\Lambda_{k,\\varepsilon}(x,\\widetilde{\\omega},\\theta^\\ast) \\big)} \n\t\t\t+ \\frac{\\partial \\varphi}{\\partial t}(t,x) \\, \\Psi_\\varepsilon(x,\\widetilde{\\omega},\\theta^\\ast) }  \n\\\\[5pt]\n\t\t\t& { + \\, \\varepsilon \\sum_{k=1}^n \\frac{\\partial^2 \\varphi}{\\partial t \\, \\partial x_k}(t,x) \\, \\Lambda_{k,\\varepsilon}(x,\\widetilde{\\omega},\\theta^\\ast) \\Big]},\n\\end{aligned}\n$$\ntherefore we have\n$$\n\\begin{aligned}\nI_{1}^{\\varepsilon}&= i \\int_{\\mathbb{R}^n} v_\\varepsilon^0 \\, \\overline{ {\\big( \\varphi (0,x)\\, \\Psi_\\varepsilon(x,\\widetilde{\\omega},\\theta^\\ast) \n+\\varepsilon  \\sum_{k=1}^n \\frac{\\partial \\varphi}{\\partial x_k}(0,x) \\, \\Lambda_{k,\\varepsilon}(x,\\widetilde{\\omega},\\theta^\\ast) \\big)} } dx \n\\\\[5pt]\n& - \\frac{\\lambda(\\theta^\\ast)}{\\varepsilon^2} \\iint_{\\mathbb{R}^{n+1}_T} v_\\varepsilon \\, \n\\overline{{\\big( \\varphi(t,x) \\, \\Psi_\\varepsilon(x,\\widetilde{\\omega},\\theta^\\ast) +\\varepsilon  \\sum_{k=1}^n \\frac{\\partial \\varphi}{\\partial x_k}(t,x) \\, \n\\Lambda_{k,\\varepsilon}(x,\\widetilde{\\omega},\\theta^\\ast) \\big)}} dx \\, dt \n\\\\[5pt] \n&- i \\iint_{\\mathbb{R}^{n+1}_T} v_\\varepsilon \\, \\overline{ {\\big( \\frac{\\partial \\varphi}{\\partial t}(t,x) \\,\n\\Psi_\\varepsilon(x,\\widetilde{\\omega},\\theta^\\ast) + \\varepsilon \\sum_{k=1}^n \\frac{\\partial^2 \\varphi}{\\partial t \\, \\partial x_k}(t,x) \n\\Lambda_{k,\\varepsilon}(x,\\widetilde{\\omega},\\theta^\\ast) \\big)} }.\n\\end{aligned}\n$$\t\t\nFor the analysis of the term $I_{2}^{\\varepsilon}$, we need to make the following computations  \n$$\n\\begin{aligned}\n\\nabla Z_\\varepsilon(t,x,\\widetilde{\\omega}) &= e^{i \\frac{\\lambda(\\theta^\\ast) t} {\\varepsilon^2} + 2i\\pi \\frac{\\theta^\\ast \\! \\cdot x}{\\varepsilon}} {\\big[ \\nabla \\varphi(t,x) \\, \n\\Psi_\\varepsilon(x,\\widetilde{\\omega},\\theta^\\ast) + \\varphi(t,x) \\, \\nabla \\Psi_\\varepsilon(z,\\widetilde{\\omega},\\theta^\\ast) }\n\\\\[5pt]\n\t&+  \\varepsilon \\sum_{k=1}^n \\nabla {\\big( \\frac{\\partial \\varphi}{\\partial x_k}(t,x) \\big)} \\, \\Lambda_{k,\\varepsilon}(x,\\widetilde{\\omega},\\theta^\\ast) \n\t+ \\varepsilon \\sum_{k=1}^n \\frac{\\partial \\varphi}{\\partial x_k}(t,x) \\, \\nabla \\Lambda_{k,\\varepsilon}(x,\\widetilde{\\omega},\\theta^\\ast) \n\\\\[5pt]\n\t&+ \\, 2i\\pi \\frac{\\theta^\\ast}{\\varepsilon} { {\\big( \\varphi(t,x) \\, \\Psi_\\varepsilon(x,\\widetilde{\\omega},\\theta^\\ast) + \\varepsilon  \n\t\\sum_{k=1}^n \\frac{\\partial \\varphi}{\\partial x_k}(t,x) \\, \\Lambda_{k,\\varepsilon}(x,\\widetilde{\\omega},\\theta^\\ast) \\big)} \\big]} \n\\\\[5pt] \n\t& = e^{i \\frac{\\lambda(\\theta^\\ast) t}{\\varepsilon^2} + 2i\\pi \\frac{\\theta^\\ast \\! \\cdot x}{\\varepsilon}} {\\big[ \\varphi(t,x) \\, {\\big( \\nabla \n\t+ 2i\\pi\\frac{\\theta^\\ast}{\\varepsilon} \\big)} \\Psi_\\varepsilon(x,\\widetilde{\\omega},\\theta^\\ast) } \n\\\\[5pt]\n\t&+ { \\, \\varepsilon \\sum_{k=1}^n \\frac{\\partial \\varphi}{\\partial x_k}(t,x) \\, {\\big( \\nabla + 2i\\pi\\frac{\\theta^\\ast}{\\varepsilon} \\big)} \n\t\\Lambda_{k,\\varepsilon}(x,\\widetilde{\\omega},\\theta^\\ast) + \\nabla \\varphi(t,x) \\, \\Psi_\\varepsilon(x,\\widetilde{\\omega},\\theta^\\ast) } \n\\\\[5pt]\n\t& + \\, { \\varepsilon \\sum_{k=1}^n \\nabla {\\big( \\frac{\\partial \\varphi}{\\partial x_k}(t,x) \\big)} \\, \\Lambda_{k,\\varepsilon}(x,\\widetilde{\\omega},\\theta^\\ast) \\big]},\n\\end{aligned}\n$$\t\nand from this, we have \n$$\n\\begin{aligned}\n&A {\\left(\\Phi^{-1}{\\left( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} \\nabla u_\\varepsilon(t,x,\\widetilde{\\omega}) \\cdot \\overline{\\nabla Z_\\varepsilon}(t,x,\\widetilde{\\omega})\n\\\\[5pt]\n&= A {\\left(\\Phi^{-1}{\\left( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} {\\big[ \\nabla u_\\varepsilon(t,x,\\widetilde{\\omega}) \\, \ne^{ -\\left( {i \\frac{\\lambda(\\theta^\\ast) t}{\\varepsilon^2} + 2i\\pi \\frac{\\theta^\\ast \\! \\cdot x}{\\varepsilon}} \\right)} \\big]} \n\\\\[5pt]\n&\\cdot \\big[ \\overline{\\varphi}(t,x) \\, \n( \\nabla - 2i\\pi\\frac{\\theta^\\ast}{\\varepsilon} ) \\overline{\\Psi_\\varepsilon}(x,\\widetilde{\\omega},\\theta^\\ast) } \n+  \\varepsilon \\! \\sum_{k=1}^n \\frac{\\partial \\overline{\\varphi}}{\\partial x_k}(t,x) \\, {( \\nabla \\!\\! - 2i\\pi\\frac{\\theta^\\ast}{\\varepsilon})\n\\overline{\\Lambda_{k,\\varepsilon}}(x,\\widetilde{\\omega},\\theta^\\ast) \n\\\\[5pt]\n&+ \\nabla \\overline{\\varphi}(t,x) \\, \\overline{\\Psi_\\varepsilon}(x,\\widetilde{\\omega},\\theta^\\ast) \n+ \\varepsilon \\sum_{k=1}^n \\nabla {\\big( \\frac{\\partial \\overline{\\varphi}}{\\partial x_k}(t,x) \\big)} \\, \n\\overline{\\Lambda_{k,\\varepsilon}}(x,\\widetilde{\\omega},\\theta^\\ast) \\big].\n\\end{aligned}\n$$\t\t\n\\begin{comment}\nAfter reordering conveniently and using \\eqref{974967uhghjzpzas}, we find \t\n\t\t\\begin{equation*}\n\t\t\\begin{array}{l}\n\t\t\t\\displaystyle \\!\\! A {\\left(\\Phi^{-1}{\\left( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} \\nabla u_\\varepsilon(t,x,\\widetilde{\\omega}) \\cdot \\overline{\\nabla Z_\\varepsilon}(t,x,\\widetilde{\\omega}) = \\\\ [15pt]\n\t\t\t\\displaystyle A {\\left(\\Phi^{-1}{\\left( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} {\\left[ {\\left( \\nabla + 2i\\pi\\frac{\\theta^\\ast}{\\varepsilon} \\right)} v_\\varepsilon(t,x,\\widetilde{\\omega}) \\right]} \\cdot {\\left[ \\overline{\\varphi}(t,x) \\, {\\left( \\nabla - 2i\\pi\\frac{\\theta^\\ast}{\\varepsilon} \\right)} \\overline{\\Psi_\\varepsilon}(x,\\widetilde{\\omega},\\theta^\\ast) \\right]} \\\\ [15pt]\n\t\t\t\\displaystyle + \\, \\varepsilon A {\\left(\\Phi^{-1}{\\left( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} {\\left[ {\\left( \\nabla + 2i\\pi\\frac{\\theta^\\ast}{\\varepsilon} \\right)} v_\\varepsilon(t,x,\\widetilde{\\omega}) \\right]} \\cdot {\\left[ \\sum_{k=1}^n \\frac{\\partial \\overline{\\varphi}}{\\partial x_k}(t,x) \\, {\\left( \\nabla - 2i\\pi\\frac{\\theta^\\ast}{\\varepsilon} \\right)} \\overline{\\Lambda_{k,\\varepsilon}}(x,\\widetilde{\\omega},\\theta^\\ast) \\right]} \\\\ [15pt]\n\t\t\t\\displaystyle + A {\\left(\\Phi^{-1}{\\left( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)}{\\left[ {\\left( \\nabla + 2i\\pi\\frac{\\theta^\\ast}{\\varepsilon} \\right)} v_\\varepsilon(t,x,\\widetilde{\\omega}) \\right]} \\cdot {\\Big[ \\nabla \\overline{\\varphi}(t,x) \\, \\overline{\\Psi_\\varepsilon}(x,\\widetilde{\\omega},\\theta^\\ast) \\Big]} \\\\ [15pt]\n\t\t\t\\displaystyle + \\, \\varepsilon A {\\left(\\Phi^{-1}{\\left( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)}{\\left[ {\\left( \\nabla + 2i\\pi\\frac{\\theta^\\ast}{\\varepsilon} \\right)} v_\\varepsilon(t,x,\\widetilde{\\omega}) \\right]} \\cdot {\\left[ \\sum_{k=1}^n \\nabla {\\left( \\frac{\\partial \\overline{\\varphi}}{\\partial x_k}(t,x) \\right)} \\, \\overline{\\Lambda_{k,\\varepsilon}}(x,\\widetilde{\\omega},\\theta^\\ast)  \\right]}.\n\t\t\\end{array}\n\t\t\\end{equation*}\n\\end{comment}\nThen, from equation \\eqref{974967uhghjzpzas} and using the terms\n$$\n{\\big( \\nabla + 2i\\pi\\frac{\\theta^\\ast}{\\varepsilon} \\big)} (v_\\varepsilon(t,x,\\widetilde{\\omega}) \\, \\overline{\\varphi}(t,x) ),\n\\quad \n\\big( \\nabla + 2i\\pi\\frac{\\theta^\\ast}{\\varepsilon} \\big) (v_\\varepsilon(t,x,\\widetilde{\\omega}) \\, \\frac{\\partial \\overline{\\varphi}}{\\partial x_k}(t,x)), \n$$ \nit follows from the above equation that \n$$\n    A {\\left(\\Phi^{-1}{\\left( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} \\nabla u_\\varepsilon \\cdot \\overline{\\nabla Z_\\varepsilon}\n    =\\sum_{k=1}^n\\left( I_{2,1}^{\\varepsilon,k}+I_{2,2}^{\\varepsilon,k}+I_{2,3}^{\\varepsilon,k}\\right)(t,x,\\widetilde{\\omega}),\n$$\nwhere \n$$\n\\begin{aligned}\n & I_{2,1}^{\\varepsilon,k}(t,x,\\widetilde{\\omega}):=  \\varepsilon \\,A {(\\Phi^{-1}{\\big( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\big)},\\widetilde{\\omega} )} \n\\big[ {\\big( \\nabla + 2i\\pi\\frac{\\theta^\\ast}{\\varepsilon} \\big)} {( v_\\varepsilon(t,x,\\widetilde{\\omega}) \\, \\frac{\\partial \\overline{\\varphi}}{\\partial x_k}(t,x) )} \\big]\n \\\\[5pt]\n&\\quad \\cdot  \\big[ {\\big( \\nabla - 2i\\pi\\frac{\\theta^\\ast}{\\varepsilon} \\big)} \\overline{\\Lambda_{k,\\varepsilon}}(x,\\widetilde{\\omega},\\theta^\\ast) \\big]\n\\\\[5pt]\n& - A(\\Phi^{-1}{\\big( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\big)},\\widetilde{\\omega} )\n\\big[ v_\\varepsilon(t,x,\\widetilde{\\omega}) \\, \\frac{\\partial \\overline{\\varphi}}{\\partial x_k}(t,x) \\, e_k \\big] \\!\n \\cdot \\! \\big[ {\\big( \\nabla - 2i\\pi\\frac{\\theta^\\ast}{\\varepsilon} \\big)} \\overline{\\Psi_\\varepsilon}(x,\\widetilde{\\omega},\\theta^\\ast)\t\\big]\n\\\\[5pt]\n& + A (\\Phi^{-1}{\\big( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\big)},\\widetilde{\\omega} )\n{\\big[ {\\big( \\nabla + 2i\\pi\\frac{\\theta^\\ast}{\\varepsilon} \\big)} \n( v_\\varepsilon(t,x,\\widetilde{\\omega}) \\, \\frac{\\partial \\overline{\\varphi}}{\\partial x_k}(t,x))} \\big] \\!\n\\cdot \\! \\big[ {e_k} \\overline{\\Psi_\\varepsilon}(x,\\widetilde{\\omega},\\theta^\\ast)\\big],\n\\end{aligned}\n$$\n$$\n\\begin{aligned}\n& I_{2,2}^{\\varepsilon,k}(t,x,\\widetilde{\\omega}):=\n \\frac{1}{n}A {(\\Phi^{-1}{\\big( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\big)},\\widetilde{\\omega})} \n{\\big[ {\\big( \\nabla + 2i\\pi\\frac{\\theta^\\ast}{\\varepsilon} \\big)} (v_\\varepsilon(t,x,\\widetilde{\\omega})  \\overline{\\varphi}(t,x)) \\big]}\n\\\\[5pt]\n& \\quad\\quad  \\cdot \\big[ {\\big( \\nabla - 2i\\pi\\frac{\\theta^\\ast}{\\varepsilon} \\big)} \\overline{\\Psi_\\varepsilon}(x,\\widetilde{\\omega},\\theta^\\ast) \\big]\n\\\\[5pt]\n& \\quad \\quad - A (\\Phi^{-1}{\\big( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\big)},\\widetilde{\\omega} )\n{\\big[ v_\\varepsilon(t,x,\\widetilde{\\omega}) \\, \\nabla \\frac{\\partial \\overline{\\varphi}}{\\partial x_k}(t,x) \\big]} \n\\cdot {\\big[ e_k \\, \\overline{\\Psi_\\varepsilon}(x,\\widetilde{\\omega},\\theta^\\ast) \\big]}, \n\\end{aligned}\n$$\nand \n$$\n\\begin{aligned}\n&I_{2,3}^{\\varepsilon,k}(t,x,\\widetilde{\\omega}):= A(\\Phi^{-1}{\\big( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\big)},\\widetilde{\\omega} )\n\\big[ {\\left( \\varepsilon \\nabla + 2i\\pi\\theta^\\ast \\right)} v_\\varepsilon(t,x,\\widetilde{\\omega}) \\big]\n \\\\[5pt]\n &\\quad  \\cdot \\big[  \\nabla {\\big( \\frac{\\partial \\overline{\\varphi}}{\\partial x_k}(t,x) \\big)} \\, \\overline{\\Lambda_{k,\\varepsilon}}(x,\\widetilde{\\omega},\\theta^\\ast)  \\big]\n \\\\[5pt]\n&- A(\\Phi^{-1}{\\left( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega})\n\\big[ v_\\varepsilon(t,x,\\widetilde{\\omega}) \\, \\nabla \\frac{\\partial \\overline{\\varphi}}{\\partial x_k}(t,x) \\big] \\! \\cdot \\!\n \\big[ {\\left( \\varepsilon \\nabla - 2i\\pi\\theta^\\ast \\right)} \\overline{\\Lambda_{k,\\varepsilon}}(x,\\widetilde{\\omega},\\theta^\\ast) \\big]. \n\\end{aligned}\n$$\n\\begin{comment}\n\\begin{equation*}\n\t\t\\begin{array}{l}\n\t\t\t\\displaystyle \\!\\! A {\\left(\\Phi^{-1}{\\left( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} \\nabla u_\\varepsilon \\cdot \\overline{\\nabla Z_\\varepsilon} = \\\\ [15pt] \n\t\t    \\displaystyle A {\\left(\\Phi^{-1}{\\left( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} {\\left[{\\left( \\nabla + 2i\\pi\\frac{\\theta^\\ast}{\\varepsilon} \\right)} (v_\\varepsilon(t,x,\\widetilde{\\omega})  \\overline{\\varphi}(t,x)) \\right]} \\cdot {\\left[ {\\left( \\nabla - 2i\\pi\\frac{\\theta^\\ast}{\\varepsilon} \\right)} \\overline{\\Psi_\\varepsilon}(x,\\widetilde{\\omega},\\theta^\\ast) \\right]} \\\\ [15pt]\n\t\t\t\\displaystyle + \\, \\sum_{k=1}^n \\Bigg\\{\\varepsilon\\,A {\\left(\\Phi^{-1}{\\left( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} {\\left[ {\\left( \\nabla + 2i\\pi\\frac{\\theta^\\ast}{\\varepsilon} \\right)} {\\left( v_\\varepsilon(t,x,\\widetilde{\\omega}) \\, \\frac{\\partial \\overline{\\varphi}}{\\partial x_k}(t,x) \\right)} \\right]} \\cdot \\\\ [15pt]\n\t\t\t\\displaystyle \\hspace{8.5cm} {\\left[ {\\left( \\nabla - 2i\\pi\\frac{\\theta^\\ast}{\\varepsilon} \\right)} \\overline{\\Lambda_{k,\\varepsilon}}(x,\\widetilde{\\omega},\\theta^\\ast) \\right]} \\\\ [15pt]\n\t\t\t\\displaystyle\\qquad\\qquad - \\, A {\\left(\\Phi^{-1}{\\left( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} {\\left[ v_\\varepsilon(t,x,\\widetilde{\\omega}) \\, \\frac{\\partial \\overline{\\varphi}}{\\partial x_k}(t,x) \\, e_k \\right]} \\cdot {\\left[ {\\left( \\nabla - 2i\\pi\\frac{\\theta^\\ast}{\\varepsilon} \\right)} \\overline{\\Psi_\\varepsilon}(x,\\widetilde{\\omega},\\theta^\\ast)\t\\right]} \\\\ [15pt]\n\t\t\t\\displaystyle\\qquad\\qquad + \\, A {\\left(\\Phi^{-1}{\\left( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} {\\left[ {\\left( \\nabla + 2i\\pi\\frac{\\theta^\\ast}{\\varepsilon} \\right)} {\\left( v_\\varepsilon(t,x,\\widetilde{\\omega}) \\, \\frac{\\partial \\overline{\\varphi}}{\\partial x_k}(t,x) \\right)} \\right]} \\cdot {\\Big[ e_k \\, \\overline{\\Psi_\\varepsilon}(x,\\widetilde{\\omega},\\theta^\\ast) \\Big]} \\\\ [15pt]\n\t\t\\end{array}\n\t\t\\end{equation*}\n\t\t\\begin{equation*}\n\t\t\\begin{array}{l}\n\t\t\t\\displaystyle\\qquad\\qquad - \\,A {\\left(\\Phi^{-1}{\\left( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} {\\left[ v_\\varepsilon(t,x,\\widetilde{\\omega}) \\, \\nabla \\frac{\\partial \\overline{\\varphi}}{\\partial x_k}(t,x) \\right]} \\cdot {\\Big[ e_k \\, \\overline{\\Psi_\\varepsilon}(x,\\widetilde{\\omega},\\theta^\\ast) \\Big]} \\\\ [15pt]\n\t\t\t\\displaystyle - \\, \\sum_{k=1}^n A {\\left(\\Phi^{-1}{\\left( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} {\\left[ v_\\varepsilon(t,x,\\widetilde{\\omega}) \\, \\nabla \\frac{\\partial \\overline{\\varphi}}{\\partial x_k}(t,x) \\right]} \\cdot {\\Big[ {\\left( \\varepsilon \\nabla - 2i\\pi\\theta^\\ast \\right)} \\overline{\\Lambda_{k,\\varepsilon}}(x,\\widetilde{\\omega},\\theta^\\ast) \\Big]} \\\\ [15pt]\n\t\t\t\\displaystyle + \\, \\sum_{k=1}^n A {\\left(\\Phi^{-1}{\\left( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} {\\Big[ {\\left( \\varepsilon \\nabla + 2i\\pi\\theta^\\ast \\right)} v_\\varepsilon(t,x,\\widetilde{\\omega}) \\Big]}\\cdot {\\left[  \\nabla {\\left( \\frac{\\partial \\overline{\\varphi}}{\\partial x_k}(t,x) \\right)} \\, \\overline{\\Lambda_{k,\\varepsilon}}(x,\\widetilde{\\omega},\\theta^\\ast)  \\right]},\n\t\t\\end{array}\n\t\t\\end{equation*}\n\\end{comment}\nThus integrating in $\\mathbb{R}^{n+1}_T$ we recover the $I_2^{\\varepsilon}$ term, that is \n\\begin{eqnarray}\\label{HomProc1}\n&&I_2^{\\varepsilon}=\\iint_{\\mathbb{R}^{n+1}_T} A {\\left(\\Phi^{-1}{\\left( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} \\nabla u_\\varepsilon(t,x,\\widetilde{\\omega}) \\cdot \\overline{\\nabla Z_\\varepsilon}(t,x,\\widetilde{\\omega}) \\, dx \\, dt\\nonumber\\\\\n&&\\qquad\\qquad=\\sum_{k=1}^n\\iint_{\\mathbb{R}^{n+1}_T}\\left( I_{2,1}^{\\varepsilon,k}+I_{2,2}^{\\varepsilon,k}+I_{2,3}^{\\varepsilon,k}\\right)(t,x,\\widetilde{\\omega})\\,dx\\,dt. \n\\end{eqnarray}\n\n\n\n\\begin{comment}\nwhere \n\\begin{eqnarray*}\n&&I_{2,1}^{\\varepsilon,k}:=\\iint_{\\mathbb{R}^{n+1}_T}\\Bigg\\{ \\varepsilon \\,A {\\left(\\Phi^{-1}{\\left( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} {\\left[ {\\left( \\nabla + 2i\\pi\\frac{\\theta^\\ast}{\\varepsilon} \\right)} {\\left( v_\\varepsilon(t,x,\\widetilde{\\omega}) \\, \\frac{\\partial \\overline{\\varphi}}{\\partial x_k}(t,x) \\right)} \\right]} \\cdot\\\\\n&&\\hspace{8cm}{\\left[ {\\left( \\nabla - 2i\\pi\\frac{\\theta^\\ast}{\\varepsilon} \\right)} \\overline{\\Lambda_{k,\\varepsilon}}(x,\\widetilde{\\omega},\\theta^\\ast) \\right]} \\\\\n&&\\qquad\\qquad\\qquad - \\,  A {\\left(\\Phi^{-1}{\\left( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} {\\left[ v_\\varepsilon(t,x,\\widetilde{\\omega}) \\, \\frac{\\partial \\overline{\\varphi}}{\\partial x_k}(t,x) \\, e_k \\right]} \\cdot \\\\\n&&\\hspace{8cm} {\\left[ {\\left( \\nabla - 2i\\pi\\frac{\\theta^\\ast}{\\varepsilon} \\right)} \\overline{\\Psi_\\varepsilon}(x,\\widetilde{\\omega},\\theta^\\ast)\t\\right]} \\\\\n&&\\qquad\\qquad +\\, A {\\left(\\Phi^{-1}{\\left( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} {\\left[ {\\left( \\nabla + 2i\\pi\\frac{\\theta^\\ast}{\\varepsilon} \\right)} {\\left( v_\\varepsilon(t,x,\\widetilde{\\omega}) \\, \\frac{\\partial \\overline{\\varphi}}{\\partial x_k}(t,x) \\right)} \\right]} \\cdot\\\\\n&& \\hspace{8cm} {\\left[ {e_k} \\overline{\\Psi_\\varepsilon}(x,\\widetilde{\\omega},\\theta^\\ast)\t\\right]}\\Bigg\\} \\, dx \\, dt,\n\\end{eqnarray*}\n\n\\begin{eqnarray*}\n&&I_{2,2}^{\\varepsilon,k}:=\n\\iint_{\\mathbb{R}^{n+1}_T}\\Bigg\\{ \\frac{1}{n}A {\\left(\\Phi^{-1}{\\left( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} {\\left[ {\\left( \\nabla + 2i\\pi\\frac{\\theta^\\ast}{\\varepsilon} \\right)} (v_\\varepsilon(t,x,\\widetilde{\\omega})  \\overline{\\varphi}(t,x)) \\right]} \\cdot \\\\\n& & \\hspace{8cm} {\\left[ {\\left( \\nabla - 2i\\pi\\frac{\\theta^\\ast}{\\varepsilon} \\right)} \\overline{\\Psi_\\varepsilon}(x,\\widetilde{\\omega},\\theta^\\ast) \\right]} \\\\\n&&\\qquad\\qquad\\qquad - \\,A {\\left(\\Phi^{-1}{\\left( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} {\\left[ v_\\varepsilon(t,x,\\widetilde{\\omega}) \\, \\nabla \\frac{\\partial \\overline{\\varphi}}{\\partial x_k}(t,x) \\right]} \\cdot {\\Big[ e_k \\, \\overline{\\Psi_\\varepsilon}(x,\\widetilde{\\omega},\\theta^\\ast) \\Big]}\\Bigg\\}\\,dx\\,dt\n\\end{eqnarray*}\nand \n\n\\begin{eqnarray*}\n&&I_{2,3}^{\\varepsilon,k}:=\\iint_{\\mathbb{R}^{n+1}_T}\\Bigg\\{ A {\\left(\\Phi^{-1}{\\left( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} {\\Big[ {\\left( \\varepsilon \\nabla + 2i\\pi\\theta^\\ast \\right)} v_\\varepsilon(t,x,\\widetilde{\\omega}) \\Big]} \\cdot \\\\\n& & \\hspace{8cm} {\\left[  \\nabla {\\left( \\frac{\\partial \\overline{\\varphi}}{\\partial x_k}(t,x) \\right)} \\, \\overline{\\Lambda_{k,\\varepsilon}}(x,\\widetilde{\\omega},\\theta^\\ast)  \\right]}\\\\\n&& \\qquad\\qquad\\qquad\\qquad-\\,\nA {\\left(\\Phi^{-1}{\\left( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} {\\left[ v_\\varepsilon(t,x,\\widetilde{\\omega}) \\, \\nabla \\frac{\\partial \\overline{\\varphi}}{\\partial x_k}(t,x) \\right]} \\cdot\\\\\n& & \\hspace{8cm} {\\Big[ {\\left( \\varepsilon \\nabla - 2i\\pi\\theta^\\ast \\right)} \\overline{\\Lambda_{k,\\varepsilon}}(x,\\widetilde{\\omega},\\theta^\\ast) \\Big]}\\Bigg\\} \\, dx \\, dt.\n\\end{eqnarray*}\n\\end{comment}\n\n\t\n\t\n\t\n\t\n\t\n\t\n\\begin{comment}\t\n\tReordenamos acima. \t\n\t\\begin{eqnarray}\n\t\t\t& & \\!\\!\\!\\!\\!\\! \\hspace{-0.5cm} \\iint_{\\mathbb{R}^{n+1}_T} A {\\left(\\Phi^{-1}{\\left( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} \\nabla u_\\varepsilon(t,x,\\widetilde{\\omega}) \\cdot \\overline{\\nabla Z_\\varepsilon}(t,x,\\widetilde{\\omega}) \\, dx \\, dt =:  I_2^{\\varepsilon} = \\nonumber \\\\ [4pt] \n\t\t    & & \\hspace{-0.5cm} \\iint_{\\mathbb{R}^{n+1}_T} A {\\left(\\Phi^{-1}{\\left( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} {\\left[ {\\left( \\nabla + 2i\\pi\\frac{\\theta^\\ast}{\\varepsilon} \\right)} (v_\\varepsilon(t,x,\\widetilde{\\omega})  \\overline{\\varphi}(t,x)) \\right]} \\cdot \\nonumber \\\\\n\t\t    & & \\hspace{8cm} {\\left[ {\\left( \\nabla - 2i\\pi\\frac{\\theta^\\ast}{\\varepsilon} \\right)} \\overline{\\Psi_\\varepsilon}(x,\\widetilde{\\omega},\\theta^\\ast) \\right]} \\, dx \\, dt \\nonumber \\\\ [4pt]\n\t\t\t& & \\hspace{-0.5cm} + \\, \\varepsilon \\sum_{k=1}^n \\iint_{\\mathbb{R}^{n+1}_T} A {\\left(\\Phi^{-1}{\\left( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} {\\left[ {\\left( \\nabla + 2i\\pi\\frac{\\theta^\\ast}{\\varepsilon} \\right)} {\\left( v_\\varepsilon(t,x,\\widetilde{\\omega}) \\, \\frac{\\partial \\overline{\\varphi}}{\\partial x_k}(t,x) \\right)} \\right]} \\cdot \\nonumber \\\\\n\t\t\t& & \\hspace{8cm} {\\left[ {\\left( \\nabla - 2i\\pi\\frac{\\theta^\\ast}{\\varepsilon} \\right)} \\overline{\\Lambda_{k,\\varepsilon}}(x,\\widetilde{\\omega},\\theta^\\ast) \\right]} \\, dx \\, dt \\nonumber \\\\ [4pt]\n\t\t\t& & \\hspace{-0.5cm} - \\, \\sum_{k=1}^n \\iint_{\\mathbb{R}^{n+1}_T} A {\\left(\\Phi^{-1}{\\left( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} {\\left[ v_\\varepsilon(t,x,\\widetilde{\\omega}) \\, \\frac{\\partial \\overline{\\varphi}}{\\partial x_k}(t,x) \\, e_k \\right]} \\cdot \\nonumber \\\\\n\t\t\t& & \\hspace{8cm} {\\left[ {\\left( \\nabla - 2i\\pi\\frac{\\theta^\\ast}{\\varepsilon} \\right)} \\overline{\\Psi_\\varepsilon}(x,\\widetilde{\\omega},\\theta^\\ast)\t\\right]} \\, dx \\, dt \\nonumber \\\\ [4pt]\n\t\t\t& & \\hspace{-0.5cm} + \\, \\sum_{k=1}^n \\iint_{\\mathbb{R}^{n+1}_T} A {\\left(\\Phi^{-1}{\\left( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} {\\left[ {\\left( \\nabla + 2i\\pi\\frac{\\theta^\\ast}{\\varepsilon} \\right)} {\\left( v_\\varepsilon(t,x,\\widetilde{\\omega}) \\, \\frac{\\partial \\overline{\\varphi}}{\\partial x_k}(t,x) \\right)} \\right]} \\cdot \\label{45678uhgvce345tg} \\\\\n\t\t\t& & \\hspace{10.5cm} {\\Big[ e_k \\, \\overline{\\Psi_\\varepsilon}(x,\\widetilde{\\omega},\\theta^\\ast) \\Big]} \\, dx \\, dt \\nonumber \\\\ [4pt]\n\t\t\t& & \\hspace{-0.5cm} - \\, \\sum_{k=1}^n \\iint_{\\mathbb{R}^{n+1}_T} A {\\left(\\Phi^{-1}{\\left( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} {\\left[ v_\\varepsilon(t,x,\\widetilde{\\omega}) \\, \\nabla \\frac{\\partial \\overline{\\varphi}}{\\partial x_k}(t,x) \\right]} \\cdot {\\Big[ e_k \\, \\overline{\\Psi_\\varepsilon}(x,\\widetilde{\\omega},\\theta^\\ast) \\Big]} \\, dx \\, dt \\nonumber \\\\ [4pt]\n\t\t\t& & \\hspace{-0.5cm} - \\, \\sum_{k=1}^n \\iint_{\\mathbb{R}^{n+1}_T} A {\\left(\\Phi^{-1}{\\left( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} {\\left[ v_\\varepsilon(t,x,\\widetilde{\\omega}) \\, \\nabla \\frac{\\partial \\overline{\\varphi}}{\\partial x_k}(t,x) \\right]} \\cdot \\nonumber \\\\\n\t\t\t& & \\hspace{8cm} {\\Big[ {\\left( \\varepsilon \\nabla - 2i\\pi\\theta^\\ast \\right)} \\overline{\\Lambda_{k,\\varepsilon}}(x,\\widetilde{\\omega},\\theta^\\ast) \\Big]} \\, dx \\, dt \\nonumber \\\\ [4pt]\n\t\t\t& & \\hspace{-0.5cm} + \\, \\sum_{k=1}^n \\iint_{\\mathbb{R}^{n+1}_T} A {\\left(\\Phi^{-1}{\\left( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} {\\Big[ {\\left( \\varepsilon \\nabla + 2i\\pi\\theta^\\ast \\right)} v_\\varepsilon(t,x,\\widetilde{\\omega}) \\Big]} \\cdot \\nonumber \\\\\n\t\t\t& & \\hspace{8cm} {\\left[  \\nabla {\\left( \\frac{\\partial \\overline{\\varphi}}{\\partial x_k}(t,x) \\right)} \\, \\overline{\\Lambda_{k,\\varepsilon}}(x,\\widetilde{\\omega},\\theta^\\ast)  \\right]}\\, dx \\, dt. \\nonumber\n\t\t\\end{eqnarray}\n\t\n\\end{comment}\n\nNow, with the help of the first auxiliar cell equation \\eqref{8654873526rtgdrfdrfdrfrd4}, \nwe intend to simplify the expression of $I_2^{\\varepsilon}$ to a more comely one. For this aim, we shall take \n${ v_\\varepsilon(t,\\cdot,\\widetilde{\\omega}) \\, \\overline{\\varphi}(t,\\cdot) }$, $t \\in (0,T)$, as a test function in equation~\\eqref{nbvnbvxzchgfs54}.\nThen, we obtain\n$$\n\\begin{aligned}\n& \\int_{\\mathbb{R}^n} \\!\\! A(\\Phi^{-1}{\\big( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\big)},\\widetilde{\\omega})\n [ \\big( \\nabla \\!+ 2i\\pi \\frac{\\theta^\\ast}{\\varepsilon} \\big) {(v_\\varepsilon(t,x,\\widetilde{\\omega}) \\, \\overline{\\varphi})}] \\! \\cdot \\! \n[ \\big( \\nabla \\! - 2i\\pi \\frac{\\theta^\\ast}{\\varepsilon} \\big) \\overline{\\Psi_\\varepsilon}(x,\\widetilde{\\omega},\\theta^\\ast)] dx\n\\\\[5pt] \n& \\quad = \\frac{\\lambda(\\theta^\\ast)}{\\varepsilon^2}\\int_{\\mathbb{R}^n} {(v_\\varepsilon(t,x,\\widetilde{\\omega}) \\, \n\\overline{\\varphi}(t,x))} \\, \\overline{\\Psi_\\varepsilon}(x,\\widetilde{\\omega},\\theta^\\ast) \\, dx \n\\\\[5pt]\n& \\quad - \\frac{1}{\\varepsilon^2}\\int_{\\mathbb{R}^n} V(\\Phi^{-1}{\\left( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega}) \\, \n{(v_\\varepsilon(t,x,\\widetilde{\\omega}) \\, \\overline{\\varphi}(t,x))} \\, \\overline{\\Psi_\\varepsilon}(x,\\widetilde{\\omega},\\theta^\\ast) \\, dx. \n\\end{aligned}\n$$\nTherefore, comparing with $I_{2,2}^{\\varepsilon,k}(t,x,\\widetilde{\\omega})$ term obtained before, we have \n\\begin{equation}\n\\label{HomProc2}\n\\begin{aligned}\n&\\iint_{\\mathbb{R}^{n+1}_T}I_{2,2}^{\\varepsilon,k}(t,x,\\widetilde{\\omega})\\,dx\\,dt\n\\\\\n& = \\frac{\\lambda(\\theta^\\ast)}{n\\varepsilon^2}\\iint_{\\mathbb{R}^{n+1}_T} {(v_\\varepsilon(t,x,\\widetilde{\\omega}) \\, \\overline{\\varphi}(t,x))} \\, \\overline{\\Psi_\\varepsilon}(x,\\widetilde{\\omega},\\theta^\\ast) \\, dx\\,dt \n\\\\\n& - \\frac{1}{n\\varepsilon^2}\\iint_{\\mathbb{R}^{n+1}_T} V {(\\Phi^{-1}{\\left( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega})} \\, \n(v_\\varepsilon(t,x,\\widetilde{\\omega}) \\, \\overline{\\varphi}(t,x)) \\, \\overline{\\Psi_\\varepsilon}(x,\\widetilde{\\omega},\\theta^\\ast) \\, dx\\,dt\n\\\\\n& - \\iint_{\\mathbb{R}^{n+1}_T} \\!\\!\\! A(\\Phi^{-1}{\\left( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega}) {[ v_\\varepsilon(t,x,\\widetilde{\\omega}) \\, \n\\nabla \\frac{\\partial \\overline{\\varphi}}{\\partial x_k}(t,x) ]} \\cdot {[ e_k \\, \\overline{\\Psi_\\varepsilon}(x,\\widetilde{\\omega},\\theta^\\ast)]}\\,dx\\,dt. \n\\end{aligned}\n\\end{equation}\nAnalogously, taking ${ v_\\varepsilon(t,\\cdot,\\widetilde{\\omega}) \\, \\displaystyle\\frac{\\partial \\overline{\\varphi}}{\\partial x_k}(t,\\cdot) }$, $t \\in (0,T)$, \nwith ${ k\\in\\{1,\\ldots,n\\} }$ as a test function in the equation~\\eqref{4576gdcrfvjc46we}, taking into account that \n$\\nabla_{\\! \\theta} \\lambda(\\theta^\\ast)= 0$ and comparing this\nexpression with $I_{2,1}^{\\varepsilon,k}(t,x,\\widetilde{\\omega})$, we deduce that\n\\begin{eqnarray}\n\\label{HomProc3}\n&& \\iint_{\\mathbb{R}^{n+1}_T}I_{2,1}^{\\varepsilon,k}(t,x,\\widetilde{\\omega})\\,dx\\,dt\\nonumber\n\\\\\n&&\\quad= \\frac{\\lambda(\\theta^\\ast)}{\\varepsilon}\\iint_{\\mathbb{R}^{n+1}_T} \n{( v_\\varepsilon(t,x,\\widetilde{\\omega}) \\, \\frac{\\partial \\overline{\\varphi}}{\\partial x_k}(t,x))} \\, \\overline{\\Lambda_{k,\\varepsilon}}(x,\\widetilde{\\omega},\\theta^\\ast) \\, dx\\,dt\n\\\\\n&&\\quad - \\frac{1}{\\varepsilon}\\iint_{\\mathbb{R}^{n+1}_T} V {(\\Phi^{-1}{\\left( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega})} \\, \n{( v_\\varepsilon(t,x,\\widetilde{\\omega}) \\, \\frac{\\partial \\overline{\\varphi}}{\\partial x_k}(t,x))} \\, \n\\overline{\\Lambda_{k,\\varepsilon}}(x,\\widetilde{\\omega},\\theta^\\ast) \\, dx\\,dt.\\nonumber\n\\end{eqnarray}\nTherefore, summing equations \\eqref{HomProc2}, \\eqref{HomProc3}, we arrive at \n\\begin{eqnarray}\n\\label{HomProc4}\n&&\\sum_{k=1}^n\\iint_{\\mathbb{R}^{n+1}_T}\\Big(I_{2,1}^{\\varepsilon,k}+I_{2,2}^{\\varepsilon,k}\\Big)(t,x,\\widetilde{\\omega})\\,dxdt\\nonumber\n\\\\\n&&\\quad \\!\\!={\\frac{\\lambda(\\theta^\\ast)}{\\varepsilon^2} \\iint_{\\mathbb{R}^{n+1}_T} \\!\\!\\! v_\\varepsilon(t,x,\\widetilde{\\omega}) \\,\n \\overline{\\varphi(t,x) \\, \\Psi_\\varepsilon(x,\\widetilde{\\omega},\\theta^\\ast) + \\varepsilon \\! \\sum_{k=1}^n\\frac{\\partial \\varphi}{\\partial x_k}(t,x) \\, \n \\Lambda_{k,\\varepsilon}(x,\\widetilde{\\omega},\\theta^\\ast)} \\, dx dt }\\nonumber\n \\\\[4pt]\n&& \\quad- \\frac{1}{\\varepsilon^2} \\iint_{\\mathbb{R}^{n+1}_T} \\! V {\\left(\\Phi^{-1}{\\left( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega} \\right)} \\, \nv_\\varepsilon(t,x,\\widetilde{\\omega})\n\\\\\n& & \\hspace{3cm} \\times \\,  \\overline{\\varphi(t,x) \\, \\Psi_\\varepsilon(x,\\widetilde{\\omega},\\theta^\\ast) \n+ \\varepsilon \\sum_{k=1}^n\\frac{\\partial \\varphi}{\\partial x_k}(t,x) \\, \\Lambda_{k,\\varepsilon}(x,\\widetilde{\\omega},\\theta^\\ast)} \\, dx dt \\nonumber\n\\\\[4pt]\n& & \\quad - \\sum_{k=1}^n \\iint_{\\mathbb{R}^{n+1}_T} \\! A {(\\Phi^{-1}{\\left( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega})} \n{[ v_\\varepsilon(t,x,\\widetilde{\\omega}) \\, \\nabla \\frac{\\partial \\overline{\\varphi}}{\\partial x_k}(t,x)]} \\! \\cdot \\! {[ e_k \\, \\overline{\\Psi_\\varepsilon}(x,\\widetilde{\\omega},\\theta^\\ast)]} \\, dx dt.\n\\nonumber\n\\end{eqnarray}\nMoreover, expressing the $I_3^{\\varepsilon}$ term as  \n\\begin{eqnarray*}\n&& I_3^{\\varepsilon} =\\iint_{\\mathbb{R}^{n+1}_T}\\frac{1}{\\varepsilon^2}\\, V {(\\Phi^{-1}{\\left( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega})}\\,\nv_{\\varepsilon}(t,x,\\widetilde{\\omega})\n\\\\\n&&\\qquad\\qquad\\qquad\\qquad \\times \\,  \\overline{\\varphi(t,x) \\, \\Psi_\\varepsilon(x,\\widetilde{\\omega},\\theta^\\ast)+\\varepsilon  \\sum_{k=1}^n \\frac{\\partial \\varphi}{\\partial x_k}(t,x)\n\\Lambda_{k,\\varepsilon} (x,\\widetilde{\\omega},\\theta^\\ast)}\\,dx\\,dt\\\\\n&&\\qquad\\quad+\\iint_{\\mathbb{R}^{n+1}_T}U {( \\Phi^{-1}{\\left( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega})}\\, \nv_\\varepsilon(t,x,\\widetilde{\\omega})\\\\\n&&\\qquad\\qquad\\qquad\\qquad \\times \\, \n\\overline{ \\varphi(t,x) \\, \\Psi_\\varepsilon(x,\\widetilde{\\omega},\\theta^\\ast)+\\varepsilon  \\sum_{k=1}^n \\frac{\\partial \\varphi}{\\partial x_k}(t,x)\n\\Lambda_{k,\\varepsilon} (x,\\widetilde{\\omega},\\theta^\\ast)}\\,dx\\,dt,\n\\end{eqnarray*}\nadding with \\eqref{HomProc4} and $I_1^{\\varepsilon}$, we obtain\n\\begin{equation}\n\\label{HomProc5}\n\\begin{aligned}\n&I_1^{\\varepsilon}+\\sum_{k=1}^n\\iint_{\\mathbb{R}^{n+1}_T}\\Big(I_{2,1}^{\\varepsilon,k}+I_{2,2}^{\\varepsilon,k}\\Big)(t,x,\\widetilde{\\omega})\\,dx\\,dt+I_3^{\\varepsilon}\\nonumber\n\\\\[5pt]\n&={ i \\int_{\\mathbb{R}^n} \\!\\!\\! v_\\varepsilon^0(x,\\widetilde{\\omega}) \\, \\overline{ \\varphi (0,x) \\, \\Psi_\\varepsilon(x,\\widetilde{\\omega},\\theta^\\ast) } dx \n- i \\! \\iint_{\\mathbb{R}^{n+1}_T}\\!\\! \\!\\! v_\\varepsilon(t,x,\\widetilde{\\omega}) \\, \\overline{ \\frac{\\partial \\varphi}{\\partial t}(t,x) \\, \\Psi_\\varepsilon(x,\\widetilde{\\omega},\\theta^\\ast) } }\\nonumber\n\\\\[5pt]\n&- \\sum_{k,\\ell=1}^n\\iint_{\\mathbb{R}^{n+1}_T} \\!\\!\\!\n{ v_\\varepsilon(t,x,\\widetilde{\\omega}) \\, e_\\ell \\, \\frac{\\partial^2 \\overline{\\varphi}}{\\partial x_\\ell \\, \\partial x_k}(t,x) }\n\\cdot \\overline{ A {(\\Phi^{-1}{( \\frac{x}{\\varepsilon},\\widetilde{\\omega} )},\\widetilde{\\omega})} { \\; e_k \\Psi_\\varepsilon (x,\\widetilde{\\omega},\\theta^\\ast)} } \n\\,dx dt\\nonumber\n\\\\[5pt]\n&+\\iint_{\\mathbb{R}^{n+1}_T}U {(\\Phi^{-1}{\\left( \\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)},\\widetilde{\\omega})}\\, \nv_\\varepsilon(t,x,\\widetilde{\\omega})\\,\\overline{ \\varphi(t,x) \\, \\Psi_\\varepsilon(x,\\widetilde{\\omega},\\theta^\\ast)}\\,dx dt +\\, \\mathrm{O}(\\varepsilon).\n\\end{aligned}\n\\end{equation}\nThus, for $\\varepsilon=\\varepsilon'(\\widetilde{\\omega})$ and due to Step 2, that is to say \n\\begin{equation*}\nv_{\\varepsilon^\\prime}(t,x,\\widetilde{\\omega}) \\; \\xrightharpoonup[\\varepsilon^\\prime \\to 0]{2-{\\rm s}}\\; v_{\\widetilde{\\omega}}(t,x) \\, \\Psi(z,\\omega, \\theta^\\ast),\n\\end{equation*}\nwe obtain after letting $\\varepsilon'\\to 0$, from the previous equation\n\\begin{equation}\n\\label{HomProc5}\n\\begin{aligned}\n&\\lim_{\\varepsilon'\\to 0}\\Big(I_1^{\\varepsilon'}+\\sum_{k=1}^n\\iint_{\\mathbb{R}^{n+1}_T}\\Big(I_{2,1}^{\\varepsilon',k}\n+I_{2,2}^{\\varepsilon',k}\\Big)(t,x,\\widetilde{\\omega})\\,dx\\,dt+I_3^{\\varepsilon'}\\Big)\n\\\\\n&= i \\int_{\\mathbb{R}^n} {\\left( \\int_\\Omega \\int_{\\Phi([0,1)^n, \\omega)} {\\left\\vert \\Psi(z,\\omega,\\theta^\\ast) \\right\\vert}^2 dz \\, d\\mathbb{P}  \\right)} v^0(x) \\, \n\\overline{\\varphi}(0,x) \\, dx\n\\\\\n& - i \\iint_{\\mathbb{R}^{n+1}_T} \\!\\! {( \\int_\\Omega \\int_{\\Phi([0,1)^n, \\omega)} \n{\\left\\vert \\Psi(z,\\omega,\\theta^\\ast) \\right\\vert}^2 dz \\, d\\mathbb{P})}  \\, v_{\\widetilde{\\omega}}(t,x) \\, \\frac{\\partial \\overline{\\varphi}}{\\partial t} (t,x) \\, dx \\, dt \n\\\\\n& - \\sum_{k,\\ell=1}^n \\iint_{\\mathbb{R}^{n+1}_T} \\!\\! {( \\int_\\Omega \\int_{\\Phi([0,1)^n, \\omega)} \\!\\! \nA{\\left( \\Phi^{-1}(z,\\omega),\\omega \\right)} \\, {( e_\\ell \\, \\Psi)} \\cdot {( e_k \\, \\overline{\\Psi})} \\, dz \\, d\\mathbb{P})}\n\\\\\n&  \\quad \\times \\, v_{\\widetilde{\\omega}}(t,x) \\, \\frac{\\partial^2 \\overline{\\varphi}}{\\partial x_\\ell \\, \\partial x_k}(t,x) \\, dx \\, dt \n\\\\\n& +\\iint_{\\mathbb{R}^{n+1}_T} \\!\\! {( \\int_\\Omega \\int_{\\Phi([0,1)^n, \\omega)} \\!\\!\\!\\!  U {\\left( \\Phi^{-1}(z,\\omega),\\omega \\right)} \n{\\vert \\Psi \\vert}^2 dz \\, d\\mathbb{P})} \\, v_{\\widetilde{\\omega}}(t,x) \\, \\overline{\\varphi}(t,x) \\, dx \\, dt.\n\\end{aligned}\n\\end{equation}\nProceeding in the same way with respect to the term $I_{2,3}^{\\varepsilon,k}(t,x,\\widetilde{\\omega})$, we obtain\n\\begin{eqnarray}\n\\label{HomProc6}\n&&\\lim_{\\varepsilon'\\to 0} \\sum_{k=1}^n\\iint_{\\mathbb{R}^{n+1}_T}I_{2,3}^{\\varepsilon',k}(t,x,\\widetilde{\\omega})\\,dx\\,dt\\nonumber\\\\\n&&\\quad =\\sum_{k,\\ell=1}^n \\iint_{\\mathbb{R}^{n+1}_T} \\!\\! \\Big( \\int_\\Omega \\int_{\\Phi([0,1)^n, \\omega)} \\!\\! A {\\left( \\Phi^{-1}(z,\\omega),\\omega \\right)} \\, \n{\\left( {\\left( \\nabla_{\\!\\! z} + 2i\\pi\\theta^\\ast \\right)} \\Psi(z,\\omega,\\theta^\\ast) \\right)} \\nonumber\n\\\\\n&&\\hspace{2cm} \\cdot {\\left(  e_\\ell \\, \\overline{\\Lambda_k}(z,\\omega,\\theta^\\ast)  \\right)} \\, dz \\, d\\mathbb{P}\\Big)\nv_{\\widetilde{\\omega}}(t,x) \\, \\frac{\\partial^2 \\overline{\\varphi}}{\\partial x_\\ell \\, \\partial x_k}(t,x) \\, dx \\, dt\\nonumber\n\\\\\n&&\\quad-\\sum_{k,\\ell=1}^n \\iint_{\\mathbb{R}^{n+1}_T} \\!\\! \\Big( \\int_\\Omega \\int_{\\Phi([0,1)^n, \\omega)} \\!\\! A {\\left( \\Phi^{-1}(z,\\omega),\\omega \\right)}\n\\, {\\left( e_\\ell \\, \\Psi(z,\\omega,\\theta^\\ast) \\right)}\\\\\n&&\\hspace{2cm} \\cdot {\\left( \\nabla_{\\!\\! z} - 2i\\pi\\theta^\\ast \\right)} \\overline{\\Lambda_k}(z,\\omega,\\theta^\\ast) \\, dz \\, d\\mathbb{P}\\Big)\nv_{\\widetilde{\\omega}}(t,x) \\, \\frac{\\partial^2 \\overline{\\varphi}}{\\partial x_\\ell \\, \\partial x_k}(t,x) \\, dx \\, dt. \\nonumber\n\\end{eqnarray}\nTherefore, since $I_1^{\\varepsilon'}+I_2^{\\varepsilon'}+I_3^{\\varepsilon'}=0$, (see \\eqref{676745459023v}), \ncombining the two last equations \nwe conclude that, the function $v_{\\widetilde{\\omega}}$ is a distribution solution of the following homogenized Schr\\\"odinger equation\n\\begin{equation}\n\\label{567yt65trftdfxxzxzzxcvbn}\n\\left\\{\n\\begin{array}{c}\n\ti\\displaystyle\\frac{\\partial v_{\\widetilde{\\omega}}}{\\partial t}(t,x) - {\\rm div} {\\big( B^\\ast \\nabla v_{\\widetilde{\\omega}}(t,x) \\big)} \n\t+ U^\\ast v_{\\widetilde{\\omega}}(t,x)= 0, \\;\\, (t,x) \\in \\mathbb{R}^{n+1}_T, \n\t\\\\ [5pt]\n\tv_{\\widetilde{\\omega}}(0,x)=v^0(x), \\;\\, x\\in\\mathbb{R}^n,\n\\end{array}\n\\right.\n\\end{equation}\nwhere the effective tensor \n\\begin{eqnarray}\n\\label{7358586tygfjdshfbvvcc}\n&&B_{k,\\ell}^{\\ast}= \\frac{1}{c_\\psi} { \\int_\\Omega \\int_{\\Phi\\left([0,1)^n, \\omega\\right)}\\!\\!\\!  \\big\\{A {\\left( \\Phi^{-1}(z,\\omega),\\omega \\right)} \\, {\\left( e_\\ell \\, \\Psi(z,\\omega,\\theta^\\ast) \\right)}}\n\\cdot {\\left( e_k \\, \\overline{\\Psi}(z,\\omega,\\theta^\\ast) \\right)}\\nonumber\\\\\n&&\\qquad\\qquad+A {\\left( \\Phi^{-1}(z,\\omega),\\omega \\right)} \\, {\\left( e_\\ell \\, \\Psi(z,\\omega,\\theta^\\ast) \\right)}\n\\cdot {\\left( (\\nabla_{\\!\\! z} - 2i\\pi\\theta^\\ast) \\overline{\\Lambda_k}(z,\\omega,\\theta^\\ast) \\right)}\\nonumber\\\\\n&&\\qquad\\qquad\\qquad-A {\\left( \\Phi^{-1}(z,\\omega),\\omega \\right)} \\, {\\Big( ( \\nabla_{\\!\\! z} + 2i\\pi\\theta^\\ast) \\Psi(z,\\omega,\\theta^\\ast) \\Big)}\\nonumber\\\\\n&&\\hspace{6cm}\\cdot {\\left(  e_\\ell \\, \\overline{\\Lambda_k}(z,\\omega,\\theta^\\ast)  \\right)}\\big\\} \\, dz \\, d\\mathbb{P}(\\omega),\n\\end{eqnarray}\nfor ${ k, \\ell \\in \\{1,\\ldots,n\\} }$,\nand the effective potential \n\\begin{equation*}\nU^\\ast = c_\\psi^{-1} \\int_\\Omega \\int_{\\Phi([0,1)^n, \\omega)} U {\\left( \\Phi^{-1}(z,\\omega), \\omega \\right)} {\\vert \\Psi (z,\\omega,\\theta^\\ast)\\vert}^2 dz \\, d\\mathbb{P}(\\omega)\n\\end{equation*}\nwith \n$$\n\\begin{aligned}\n\tc_\\psi= \\!\\! \\int_\\Omega \\int_{\\Phi([0,1)^n, \\omega)} &\\!\\!\\! {\\vert \\Psi(z,\\omega, \\theta^\\ast) \\vert}^2 dz \\, d\\mathbb{P}(\\omega)\n\t\\\\[5pt]\n\t&\\equiv \\int_\\Omega \\int_{\\Phi([0,1)^n, \\omega)} \\!\\!\\! {\\vert \\psi{( \\Phi^{-1}{( \\frac{x}{\\varepsilon},\\omega)},\\omega)} \\vert}^2 dz \\, d\\mathbb{P}(\\omega).\n\\end{aligned}\n$$\nMoreover, we are allowed to change the tensor $B^\\ast$ in the equation \\eqref{567yt65trftdfxxzxzzxcvbn} by the \ncorresponding symmetric part of it, that is \n\\begin{equation*}\n\t\t\tA^\\ast = \\big(B^\\ast + (B^\\ast)^t\\big)\/ 2.\n\t\t\\end{equation*}\n\n\\medskip\n4.({\\it\\bf The Form of the Matrix $A^{\\ast}$.}) Now, we show that the homogenized tensor $A^{\\ast}$ is a real value matrix, and it coincides with the hessian matrix of \nthe function ${ \\theta \\mapsto \\lambda(\\theta) }$ in the point $\\theta^{\\ast}$. In fact, using that ${ \\nabla_{\\!\\! \\theta} \\lambda (\\theta^\\ast) = 0 }$ the \nequation~\\eqref{hkjlhjklhljkhuytyiufsd4} can be written as \n\\begin{eqnarray}\\label{968r6f7tyudstfyusgdjsdxxxzxzxzx}\n&&\\quad\\frac{1}{4\\pi^2} \\frac{\\partial^2 \\lambda(\\theta^\\ast)}{\\partial \\theta_\\ell \\, \\partial \\theta_k}\\,c_\\psi\\nonumber\n\\\\\n&&\\qquad\\qquad = \\int_\\Omega \\int_{\\Phi ([0,1)^n, \\omega)}\\big\\{A {\\left( \\Phi^{-1}(z,\\omega),\\omega \\right)}{\\left( e_\\ell \\, \\Psi(z,\\omega,\\theta^\\ast) \\right)} \\cdot\n{\\left( e_k \\, \\overline{\\Psi}(z,\\omega,\\theta^\\ast) \\right)}\\nonumber\\\\\n&&\\qquad\\qquad\\qquad+A {\\left( \\Phi^{-1}(z,\\omega),\\omega \\right)} \\, {\\left( e_\\ell \\, \\Psi(z,\\omega,\\theta^\\ast) \\right)} \\cdot {\\left( (\\nabla_{\\!\\! z} - 2i\\pi\\theta^\\ast) \\overline{\\Lambda_k}(z,\\omega,\\theta^\\ast) \\right)}\\nonumber\\\\\n&&\\qquad\\qquad\\qquad-A {\\left( \\Phi^{-1}(z,\\omega),\\omega \\right)} \\, {\\left[ ( \\nabla_{\\!\\! z} + 2i\\pi\\theta^\\ast) \\Psi(z,\\omega,\\theta^\\ast) \\right]} \\cdot {\\left(  e_\\ell \\, \\overline{\\Lambda_k}(z,\\omega,\\theta^\\ast)  \\right)}\\nonumber\\\\\n&&\\qquad\\qquad\\qquad+ \nA {\\left( \\Phi^{-1}(z,\\omega),\\omega \\right)} \\, {\\left( e_k \\, \\Psi(z,\\omega,\\theta^\\ast) \\right)} \\cdot {\\left( e_\\ell \\, \\overline{\\Psi}(z,\\omega,\\theta^\\ast) \\right)}\\nonumber\\\\\n&&\\qquad\\qquad\\qquad+\nA {\\left( \\Phi^{-1}(z,\\omega),\\omega \\right)} \\, {\\left( e_k \\, \\Psi(z,\\omega,\\theta^\\ast) \\right)} \\cdot {\\left( (\\nabla_{\\!\\! z} - 2i\\pi\\theta^\\ast) \\overline{\\Lambda_\\ell}(z,\\omega,\\theta^\\ast) \\right)}\\nonumber\\\\\n&&\\qquad\\qquad\\qquad-\nA {\\left( \\Phi^{-1}(z,\\omega),\\omega \\right)} \\, {\\left( (\\nabla_{\\!\\! z} + 2i\\pi\\theta^\\ast) \\Psi(z,\\omega,\\theta^\\ast) \\right)}\\\\\n&&\\hspace{8.0cm} \\cdot {\\left(  e_k \\, \\overline{\\Lambda_\\ell}(z,\\omega,\\theta^\\ast)  \\right)}\\nonumber\\big\\}\\, dz \\, d\\mathbb{P}(\\omega),\n\\end{eqnarray}\nfrom which we obtain \n\\begin{equation*}\n\tA^\\ast = \\frac{1}{8\\pi^2} \\, D^2_{\\! \\theta} \\lambda(\\theta^\\ast).\n\\end{equation*}\nTherefore, from Remark \\ref{REMCOSTCOEFF} we deduce the \nwell-posedness of the homogenized \tSchr\\\"odinger \\eqref{567yt65trftdfxxzxzzxcvbn}. Hence the function ${ v_{\\widetilde{\\omega}} \\in L^2(\\mathbb{R}^{n+1}_T) }$ \ndoes not depend on ${ \\widetilde{\\omega} \\in {\\Omega} }$. Moreover, denoting by $v$ the unique solution of the problem~ \\eqref{567yt65trftdfxxzxzzxcvbn}, \nwe have that the sequence ${ \\{v_\\varepsilon(t,x,\\widetilde{\\omega})\\}_{\\varepsilon > 0} \\subset L^2(\\mathbb{R}^{n+1}_T) }$ $\\Phi_{\\omega}-$two-scale converges to the function \n$$\n     v(t,x) \\, \\Psi(z,\\omega,\\theta^\\ast) \\equiv v(t,x) \\, \\psi {\\left(\\Phi^{-1}(z,\\omega),\\omega,\\theta^\\ast \\right)}.\n$$\n\n\\medskip\n5.({\\it\\bf A Corrector-type Result.}) \nFinally, we show the following corrector type result, that is, for a.e. ${ \\widetilde{\\omega} \\in \\Omega }$\n$$\n\\lim_{\\varepsilon \\to 0} \\iint_{\\mathbb{R}^{n+1}_T} \\big|v_\\varepsilon (t,x,\\widetilde{\\omega}) - v(t,x) \\, \n\\psi{\\left( \\Phi^{-1} {\\left(\\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)}, \\widetilde{\\omega}, \\theta^\\ast \\right)} \\big|^2 dx \\, dt= 0.\n$$\nWe begin by the simple observation \n\\begin{equation}\n\\label{86576567tjhghjgnbmnb}\n\\begin{aligned}\n&\\iint_{\\mathbb{R}^{n+1}_T} | v_\\varepsilon (t,x,\\widetilde{\\omega}) - v(t,x) \\, \n\\psi{\\left( \\Phi^{-1} {\\left(\\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)}, \\widetilde{\\omega}, \\theta^\\ast \\right)} |^2 dx dt \n\\\\\n& \\quad = \\iint_{\\mathbb{R}^{n+1}_T} {\\left\\vert v_\\varepsilon (t,x,\\widetilde{\\omega}) \\right\\vert}^2 dx \\, dt  \n\\\\\n& \\quad - \\iint_{\\mathbb{R}^{n+1}_T} \nv_\\varepsilon(t,x,\\widetilde{\\omega}) \\, \\overline{ v(t,x) \\, \\psi{\\left( \\Phi^{-1} {\\left(\\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)}, \\widetilde{\\omega}, \\theta^\\ast \\right)} } \\, dx dt  \n\\\\\n& \\quad - \\iint_{\\mathbb{R}^{n+1}_T} \\overline{v_\\varepsilon(t,x,\\widetilde{\\omega})} \\, v(t,x) \\, \\psi{\\left( \\Phi^{-1} {\\left(\\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)}, \\widetilde{\\omega}, \\theta^\\ast \\right)} \\, dx \\, dt  \n\\\\\n& \\quad + \\iint_{\\mathbb{R}^{n+1}_T} {\\left\\vert v(t,x) \\, \\psi{\\left( \\Phi^{-1} {\\left(\\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)}, \\widetilde{\\omega}, \\theta^\\ast \\right)} \\right\\vert}^2 dx \\, dt.\n\\end{aligned}\n\\end{equation}\nFrom Lemma~\\ref{63457rf2wertgh} we see that, the first integral of the right hand side of the above equation satisfies,\nfor all ${ t\\in [0,T] }$ and a.e. ${ \\widetilde{\\omega} \\in \\Omega }$\n\t\t\\begin{eqnarray*}\n\t\t\t\\int_{\\mathbb{R}^n} {\\left\\vert v_\\varepsilon (t,x,\\widetilde{\\omega}) \\right\\vert}^2 dx & = & \\int_{\\mathbb{R}^n} {\\left\\vert u_\\varepsilon (t,x,\\widetilde{\\omega}) \\right\\vert}^2 dx \\\\\n\t\t\t& = & \\int_{\\mathbb{R}^n} {\\left\\vert u_\\varepsilon^0 (x,\\widetilde{\\omega}) \\right\\vert}^2 dx \\;\\, = \\;\\, \\int_{\\mathbb{R}^n} {\\left\\vert v_\\varepsilon^0 (x,\\widetilde{\\omega}) \\right\\vert}^2 dx \\\\\n\t\t\t& = & \\int_{\\mathbb{R}^n} {\\left\\vert v^0(x) \\, \\psi{\\left( \\Phi^{-1} {\\left(\\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)}, \\widetilde{\\omega}, \\theta^\\ast \\right)} \\right\\vert}^2 dx.\n\t\t\\end{eqnarray*}\nUsing the elliptic regularity theory (see E. De Giorgi \\cite{Giorgi}, G. Stampacchia \\cite{Stampacchia}), \nit follows that $\\psi(\\theta) \\in L^\\infty(\\mathbb{R}^n; L^2(\\Omega))$ and we can apply the Ergodic Theorem to obtain\n$$\n\\begin{aligned}\n\\lim_{\\varepsilon \\to 0} & \\iint_{\\mathbb{R}^{n+1}_T} {\\left\\vert v_\\varepsilon (t,x,\\widetilde{\\omega}) \\right\\vert}^2 dx \\, dt \n\\\\\n& = \\lim_{\\varepsilon \\to 0} \\iint_{\\mathbb{R}^{n+1}_T} {\\left\\vert v^0(x) \\right\\vert}^2 {\\left\\vert \\psi{\\left( \\Phi^{-1} {\\left(\\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)}, \\widetilde{\\omega}, \\theta^\\ast \\right)} \\right\\vert}^2 dx dt \n\\\\\n& = c_\\Phi^{-1} \\iint_{\\mathbb{R}^{n+1}_T} \\! \\int_\\Omega \\int_{\\Phi([0,1)^n, \\omega)} {\\left\\vert v^0(x) \\,\n\\psi{\\left( \\Phi^{-1}(z,\\omega),\\omega,\\theta^\\ast \\right)} \\right\\vert}^2  dz \\, d\\mathbb{P} \\, dx dt. \n\\end{aligned}\n$$\nSimilarly, we have \t \n\\begin{multline*}\n    \\lim_{\\varepsilon \\to 0} \\iint_{\\mathbb{R}^{n+1}_T} {\\left\\vert v(t,x) \\, \\psi{\\left( \\Phi^{-1} {\\left(\\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)}, \\widetilde{\\omega}, \\theta^\\ast \\right)} \\right\\vert}^2 dx dt\n\\\\\n    = c_\\Phi^{-1} \\iint_{\\mathbb{R}^{n+1}_T} \\! \\int_\\Omega \\int_{\\Phi([0,1)^n, \\omega)} {\\left\\vert v(t,x) \\, \\psi{\\left( \\Phi^{-1}(z,\\omega),\\omega,\\theta^\\ast \\right)} \\right\\vert}^2  dz \\, d\\mathbb{P} \\, dx dt.\n\\end{multline*}\nMoreover, seeing that for a.e. ${ \\widetilde{\\omega} \\in \\Omega }$\t\n$$\n\\begin{aligned}\n\\lim_{\\varepsilon \\to 0} & \\iint_{\\mathbb{R}^{n+1}_T}  v_\\varepsilon(t,x,\\widetilde{\\omega}) \\, \\overline{ v(t,x) \\, \\psi{\\left( \\Phi^{-1} {\\left(\\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)}, \\widetilde{\\omega}, \\theta^\\ast \\right)} } \\, dx dt\n\\\\\n& = c_\\Phi^{-1} \\iint_{\\mathbb{R}^{n+1}_T} \\! \\int_\\Omega \\int_{\\Phi([0,1)^n, \\omega)} \n\t\\!\\!\\! v(t,x) \\, \\psi{\\left( \\Phi^{-1}(z,\\omega),\\omega,\\theta^\\ast \\right)} \\, \n\\\\\n& \\qquad \\qquad \\qquad \\qquad  \\times \\overline{v(t,x) \\, \\psi{\\left( \\Phi^{-1}(z,\\omega),\\omega,\\theta^\\ast \\right)}} \\, dz \\, d\\mathbb{P} \\, dx dt,\n\\end{aligned}\n$$\nwe can make \u00ad${ \\varepsilon \\to 0 }$ in the equation~\\eqref{86576567tjhghjgnbmnb} to find \n\\begin{comment}\n\\begin{multline*}\n\\lim_{\\varepsilon \\to 0} \\iint_{\\mathbb{R}^{n+1}_T} {\\left\\vert v_\\varepsilon (t,x,\\widetilde{\\omega}) - v(t,x) \\, \\psi{\\left( \\Phi^{-1} {\\left(\\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)}, \\widetilde{\\omega}, \\theta^\\ast \\right)} \\right\\vert}^2 dx \\, dt \n\\\\\n\\hspace{-4.5cm}\\qquad\\qquad\\qquad\\qquad\\qquad\n = c_\\Phi^{-1} \\iint_{\\mathbb{R}^{n+1}_T} \\! \\int_\\Omega \\int_{\\Phi([0,1)^n, \\omega)} {\\left\\vert v^0(x) \\, \\psi{\\left( \\Phi^{-1}(z,\\omega),\\omega,\\theta^\\ast \\right)} \\right\\vert}^2  dz \\, d\\mathbb{P} \\, dx dt \n \\\\\n\\hspace{1cm} - c_\\Phi^{-1} \\iint_{\\mathbb{R}^{n+1}_T} \\! \\int_\\Omega \\int_{\\Phi([0,1)^n, \\omega)} v(t,x) \\, \\psi{\\left( \\Phi^{-1}(z,\\omega),\\omega,\\theta^\\ast \\right)} \\, \\overline{v(t,x) \\, \\psi{\\left( \\Phi^{-1}(z,\\omega),\\omega,\\theta^\\ast \\right)}} \\, dz \\, d\\mathbb{P} \\, dx dt \n\\\\\n\\hspace{1cm} - c_\\Phi^{-1} \\iint_{\\mathbb{R}^{n+1}_T} \\! \\int_\\Omega \\int_{\\Phi([0,1)^n, \\omega)} \\overline{ v(t,x) \\, \\psi{\\left( \\Phi^{-1}(z,\\omega),\\omega,\\theta^\\ast \\right)} } \\, v(t,x) \\, \\psi{\\left( \\Phi^{-1}(z,\\omega),\\omega,\\theta^\\ast \\right)} \\, dz \\, d\\mathbb{P} \\, dx dt \n\\\\\n\t\t\t+ c_\\Phi^{-1} \\iint_{\\mathbb{R}^{n+1}_T} \\! \\int_\\Omega \\int_{\\Phi([0,1)^n, \\omega)} {\\left\\vert v(t,x) \\, \\psi{\\left( \\Phi^{-1}(z,\\omega),\\omega,\\theta^\\ast \\right)} \\right\\vert}^2 dz \\, d\\mathbb{P} \\, dx dt,\n\\end{multline*}\nwhich is equivalent to \n\\end{comment}\n\\begin{eqnarray*}\n&&\\lim_{\\varepsilon \\to 0} \\iint_{\\mathbb{R}^{n+1}_T} {\\left\\vert v_\\varepsilon (t,x,\\widetilde{\\omega}) - v(t,x) \\, \\psi{\\left( \\Phi^{-1} {\\left(\\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)}, \\widetilde{\\omega}, \\theta^\\ast \\right)} \\right\\vert}^2 dx dt \n\\\\\n&&\\qquad=c_\\Phi^{-1}{\\big( \\int_\\Omega \\int_{\\Phi([0,1)^n, \\omega)} {\\left\\vert \\psi{\\left( \\Phi^{-1}(z,\\omega),\\omega,\\theta^\\ast \\right)} \\right\\vert}^2  dz \\, d\\mathbb{P}(\\omega) \\big)}\n\\\\\n&&\\qquad\\qquad\\qquad\\qquad \\times  \\big({\\iint_{\\mathbb{R}^{n+1}_T} {\\left\\vert v^0(x) \\right\\vert}^2 dx \\, dt}-{\\iint_{\\mathbb{R}^{n+1}_T} {\\left\\vert v(t,x) \\right\\vert}^2 dx \\, dt}\\big),\n\\end{eqnarray*}\nfor a.e. ${ \\widetilde{\\omega} \\in \\Omega }$. Therefore, using the energy conservation of \nthe homogenized Schr\\\"odinger equation~\\eqref{HomSchEqu}, that is, for all ${ t\\in [0,T] }$\n\t\t\\begin{equation*}\n\t\t\t\\int_{\\mathbb{R}^n} {\\left\\vert v(t,x) \\right\\vert}^2 dx = \\int_{\\mathbb{R}^n} {\\left\\vert v^0(x) \\right\\vert}^2 dx,\n\t\t\\end{equation*}\nwe obtain that, for a.e. ${ \\widetilde{\\omega} \\in \\Omega }$\n\\begin{equation*}\n\t\t\t\\lim_{\\varepsilon \\to 0} \\iint_{\\mathbb{R}^{n+1}_T} |v_\\varepsilon (t,x,\\widetilde{\\omega}) - v(t,x) \\, \n\t\t\t\\psi{( \\Phi^{-1} {\\left(\\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)}, \\widetilde{\\omega}, \\theta^\\ast)} |^2 dx dt= 0,\n\t\t\\end{equation*}\ncompleting the proof of the theorem. \t\t\n\\end{proof}\n\n\\subsection{Radom Perturbations of the Quasi-Periodic Case}\n\nIn this section, we shall give a nice application of the framework introduced in \nthis paper, which can be used to homogenize a model beyond \nthe periodic settings considered by Allaire and Piatnitski in~\\cite{AllairePiatnitski}. \nFor reaching this aim, we shall make use of \nsome results discussed in Section \\ref{9634783yuhdj6ty} (Sobolev spaces on groups),\nin particular, Section \\ref{4563tgf5fd3}. \nOther interesting application will be given in the last section. \n\n\\medskip\nLet $n,m\\ge 1$ be integers numbers and $\\lambda_1,\\cdots,\\lambda_m$ be vectors in \n$\\mathbb R^n$ linearly independent over the set $\\mathbb{Z}$ satisfying \nthe condition that \n$$\n\\big\\{k\\in\\mathbb{Z}^m;\\,|k_1\\lambda_1+\\cdots+k_m\\lambda_m|<d\\big\\}\n$$\nis a finite set for any $d>0$.  Let $\\left(\\Omega_0,\\mathcal{F}_0,\\mathbb{P}_0\\right)$ be a probability space and \n$\\tau_0:\\mathbb{Z}^n\\times \\Omega_0\\to\\Omega_0$ \nbe a discrete ergodic dynamical system and $\\mathbb R^m\/{\\mathbb{Z}^m}$ be the $m-$dimensional torus which can be identified with the cube $[0,1)^m$.  \nFor $\\Omega:=\\Omega_0\\times [0,1)^m$, consider the following \ncontinuous dynamical system $T:\\mathbb R^n\\times \\Omega\\to \\Omega$, defined by \n$$\nT(x)(\\omega_0,s):=\\Big(\\tau_{\\left\\lfloor s+Mx \\right\\rfloor}\\omega_0,s+Mx-\\left\\lfloor s+Mx\\right\\rfloor\\Big),\n$$\nwhere $M$ is the matrix $M=\\Big(\\lambda_i\\cdot e_j{\\Big)}_{i=1,j=1}^{m,n}$ and \n$\\left\\lfloor y\\right\\rfloor$ denotes the unique element in $\\mathbb{Z}^m$ such that $y-\\left\\lfloor y\\right\\rfloor\\in [0,1)^m$. Now, we consider $[0,1)^m-$periodic functions \n$A_{\\rm per}:\\mathbb R^m\\to\\mathbb R^{n^2},\\,V_{\\rm per}:\\mathbb R^m\\to\\mathbb R$ and $U_{\\rm per}:\\mathbb R^m\\to\\mathbb R$ such that \n\\begin{itemize}\n\\item There exists $a_0,a_1>0$ such that for all $\\xi\\in\\mathbb R^n$ and for a.e $y\\in\\mathbb R^m$ we have \n$$\na_0|\\xi|^2\\le A_{\\rm per}(y)\\xi\\cdot \\xi\\le a_1|\\xi|^2.\n$$\n\\item $V_{\\rm per},\\,U_{\\rm per}\\in L^{\\infty}(\\mathbb R^m)$.\n\\end{itemize}\nLet $B_{\\rm per}:\\mathbb R^m\\to\\mathbb R^{n^2}$ be a $[0,1)^m-$periodic matrix and $\\Upsilon:\\mathbb R^n\\times [0,1)^m\\to\\mathbb R^n$ be any stochastic diffeomorphism\nsatisfying\n$$\n\\nabla \\Upsilon (x,s)=B_{\\rm per}\\Big(T(x)(\\omega_0,s)\\Big).\n$$ \nThus, we define the following stochastic deformation $\\Phi:\\mathbb R^n\\times \\Omega\\to \\mathbb R^n$ by \n$$\n\\Phi(x,\\omega)=\\Upsilon(x,s)+{\\bf X}(\\omega_0),\n$$\nwhere we have used the notation $\\omega$ for the pair $(\\omega_0,s)\\in\\Omega$ and ${\\bf X}:\\Omega_0\\to\\mathbb R^n$ is a random vector.  Now, taking \n$$\n   A(x,\\omega):=A_{\\rm per}\\left(T(x)\\omega\\right),\\,V(x,\\omega):=V_{\\rm per}\\left(T(x)\\omega\\right), \n   \\; U(x,\\omega):= U_{\\rm per}\\left(T(x)\\omega\\right)\n$$ \nin the equation~\\eqref{jhjkhkjhkj765675233}, it can be seen after some computations that the spectral equation correspondent is \n\\begin{equation}\\label{ApHom}\n\t\t\t\\left\\{\n\t\t\t\\begin{array}{l}\n\t\t\t\t-{\\Big( {\\rm div}_{\\rm {QP}} + 2i\\pi \\theta \\Big)} {\\left[ A _{\\rm per}{\\left(\\cdot \\right)} {\\Big( \\nabla^{\\rm {QP}} + 2i\\pi\\theta \\Big)} {\\Psi}_{\\rm per}(\\cdot) \\right]}\n\\\\ [7.5pt]\n \\hspace{2.0cm}+ V_{\\rm per}{\\left(\\cdot \\right)} {\\Psi}_{\\rm per}(\\cdot) = \\lambda {\\Psi}_{\\rm per}(\\cdot) \\;\\; \\text{in} \\,\\; [0,1)^m, \\\\ [7.5pt]\n\t\t\t\t\\hspace{1.5cm} {\\Psi}_{\\rm per}(\\cdot) \\;\\;\\; \\psi \\;\\, \\text{is a $[0,1)^m-$periodic function},\n\t\t\t\\end{array}\n\t\t\t\\right.\n\t\t\\end{equation}\nwhere the operators ${\\rm div}_{\\rm {QP}}$ and $\\nabla^{\\rm{QP}}$ are defined as \n\\begin{itemize}\n\\item $\\left(\\nabla^{\\rm {QP}}u_{\\rm per}\\right)(y):=B_{\\rm per}^{-1}(y)M^{\\ast}\\left(\\nabla u_{\\rm per}\\right)(y)$;\n\\item $\\left(\\rm{div}_{\\rm{QP}}\\,a\\right)(y):=\\rm{div}\\left(M B_{\\rm per}^{-1}(\\cdot)a(\\cdot)\\right)(y)$.\n\\end{itemize}\nAlthough the coefficients of the spectral equation~\\eqref{ApHom} can be seen as periodic functions, its analysis is possible thanks to the results developed in the \nSection \\ref{4563tgf5fd3}.  \nThis happens due to the fact that, the bilinear form associated to the problem~\\eqref{ApHom} can lose its coercivity which unable us \nto apply the classic theory. \n\nAssume that for some $\\theta^{\\ast}\\in\\mathbb R^n$, the spectral equation~\\eqref{ApHom} admits a solution \n$\\big(\\lambda(\\theta^{\\ast}),\\Psi_{\\rm per}(\\theta^{\\ast})\\big)\\in \\mathbb R\\times H^1\\left([0,1)^m\\right)$, such that \n\\begin{itemize}\n\\item $\\lambda(\\theta^{\\ast})$ is a simple eigenvalue;\n\\item $\\nabla \\lambda(\\theta^{\\ast})=0$.\n\\end{itemize}\nNow, we consider the problem~\\eqref{jhjkhkjhkj765675233} with new coefficients  as highlighted above and with well-prepared initial data, that is, \n$$\nu_{\\varepsilon}(x,\\omega):=e^{2\\pi i \\frac{\\theta^{\\ast}\\cdot x}{\\varepsilon}}\\,{\\Psi}_{\\rm per}\\Big(T\\left(\\Phi^{-1}\\left(\\frac{x}{\\varepsilon},\\omega\\right)\\right)\\omega,\\theta^{\\ast}\\Big)\nv^0(x),\n$$ \nfor $(x,\\omega)\\in \\mathbb R^n\\times \\Omega$ and $v^0\\in C^{\\infty}_c(\\mathbb R^n)$. Applying Theorem~\\ref{876427463tggfdhgdfgkkjjlmk}, the function \n\\begin{equation*}\n\t\t\tv_\\varepsilon(t,x,\\omega) := e^{ -{\\left( i \\frac{\\lambda(\\theta^\\ast) t}{\\varepsilon^2} + 2i\\pi \\frac{\\theta^\\ast \\! \\cdot x}{\\varepsilon} \\right)} } u_\\varepsilon(t,x,\\omega), \\;\\, (t,x) \\in \\mathbb{R}^{n+1}_T, \\; \\omega \\in \\Omega, \n\t\t\\end{equation*}\n$\\Phi_\\omega-$two-scale converges strongly to ${ v(t,x) \\, {\\Psi}_{\\rm per}\\Big({T\\left( \\Phi^{-1}(z,\\omega)\\right)\\omega, \\theta^\\ast } }\\Big)$,\nwhere \n${ v \\in C([0,T], L^2(\\mathbb{R}^n)) }$  is the unique solution of the homogenized Schr\\\"odinger equation \n\\begin{equation*}\n\t\t\t\\left\\{\n\t\t\t\\begin{array}{c}\n\t\t\t\ti \\displaystyle\\frac{\\partial v}{\\partial t} - {\\rm div} {\\left( A^\\ast \\nabla v \\right)} + U^\\ast v = 0 \\, , \\;\\, \\text{em} \\;\\, \\mathbb{R}^{n+1}_T, \\\\ [7,5pt]\n\t\t\t\tv(0,x) = v^0(x) \\, , \\;\\, x\\in \\mathbb{R}^n,\n\t\t\t\\end{array}\n\t\t\t\\right.\n\\end{equation*}\nwith effective matrix  ${ A^\\ast = D_\\theta^2 \\lambda(\\theta^\\ast) }$ and effective potential \n\\begin{equation*}\n\t\t\tU^\\ast =  c^{-1}_\\psi \\int_{[0,1)^m} U_{\\rm per}{\\left(y \\right)}\\,  {\\left\\vert {\\Psi}_{\\rm per} {\\left(y, \\theta^\\ast \\right)} \\right\\vert}^2 \n\t\t\t|\\det \\left(B_{\\rm per}(y)\\right)|\\,dy,\n\t\t\\end{equation*}\n\t\twhere\n\t\t\\begin{equation*}\n\t\t\tc_\\psi = \\int_{[0,1)^m} {\\left\\vert {\\Psi}_{\\rm per} {\\left(y, \\theta^\\ast \\right)} \\right\\vert}^2 \\,|\\det \\left(B_{\\rm per}(y)\\right)|\\,dy .\n\t\t\\end{equation*}\n\t\t\nIt is worth highlighting that this singular example encompasses the settings considered by Allaire-Piatnitski in~\\cite{AllairePiatnitski}. For this, it is enough to take \n$$\n   n= m,\\,\\lambda_j=e_j,\\,\\Upsilon(\\cdot,s)\\equiv I_{n \\times n}, \\; \\text{and ${\\bf X}(\\cdot)\\equiv 0$.}\n$$ \nMoreover, we consider $[0,1)^m-$periodic functions: $V_{\\rm per}, U_{\\rm per}: \\mathbb R^m\\to\\mathbb R$, \nand $A_{\\rm per}:\\mathbb R^m\\to\\mathbb R^{n^2}$, such that \n\\begin{itemize}\n\\item There exists $a_0,a_1>0$ such that for all $\\xi\\in\\mathbb R^n$ and for a.e $y\\in\\mathbb R^m$ we have \n$$\na_0|\\xi|^2\\le A_{\\rm per}(y)\\xi\\cdot \\xi\\le a_1|\\xi|^2;\n$$\n\\item $V_{\\rm per},\\,U_{\\rm per}\\in L^{\\infty}(\\mathbb R^m)$.\n\\end{itemize}\n\n\\section{\\! \\! \\!Homogenization of Quasi-Perfect Materials} \n\\label{6775765ff0090sds}\n\nPerfect materials (which represent the periodic setting) are rare in nature. However, there is a huge class of materials which have small deviation from perfect ones, called \nhere quasi-perfect materials. We consider in this section an interesting context, which is the small random perturbation of the periodic setting. In particular, this context is \nimportant for numerical applications. To begin, we remember the reader that \nin the previous section, it was seen that our homogenization analysis (see Theorem~\\ref{876427463tggfdhgdfgkkjjlmk}) \nof the equation~\\eqref{jhjkhkjhkj765675233} rely on the spectral study of the operator \n$L^{\\Phi}(\\theta)(\\theta\\in\\mathbb R^n)$ posed in the dual space ${ \\mathcal{H}^\\ast }$ and with domain \u00ad${ D(L^{\\Phi}(\\theta))=\\mathcal{H} }$ and defined by\n\\begin{equation}\\label{OperL}\n\\begin{array}{l}\n\tL^\\Phi(\\theta)[f] := - {\\big({\\rm div}_{\\! z} + 2i\\pi \\theta \\big)} {\\Big[ A {\\big( \\Phi^{-1} (\\cdot, {\\cdot\\cdot} ), {\\cdot\\cdot} \\big)} {\\big( \\nabla_{\\!\\! z} + 2i\\pi\\theta \\big)} f{\\big( \\Phi^{-1}(\\cdot, {\\cdot\\cdot} ),{\\cdot\\cdot} \\big)} \\Big]} \\\\ [10pt]\n\t\\hspace{4cm} + \\, V{\\big( \\Phi^{-1} (\\cdot, {\\cdot\\cdot} ), {\\cdot\\cdot} \\big)} f{\\big( \\Phi^{-1}(\\cdot, {\\cdot\\cdot} ), {\\cdot\\cdot} \\big)}, \n\\end{array}\n\\end{equation}\nwhere $\\Phi:\\mathbb R^n\\times\\Omega\\to\\mathbb R^n$ is a stochastic deformation, $A:\\mathbb R^n\\times\\Omega\\to\\mathbb R^{n^2}$ and $V:\\mathbb R^n\\times\\Omega\\to\\mathbb R$ are stationary functions. Also, remember that the variational formulation of the operator $L^{\\Phi}(\\theta)$ is given by:\n\n\\begin{equation*}\n\\begin{split}\n\t& {\\left\\langle L^\\Phi(\\theta)[f], g \\right\\rangle} := \\int_\\Omega \\int_{\\Phi ([0,1)^n, \\omega)} A {\\left( \\Phi^{-1} ( z, \\omega),\\omega \\right)} {\\big( \\nabla_{\\!\\! z} + 2i\\pi \\theta \\big)} f{\\left( \\Phi^{-1}(z,\\omega),\\omega \\right)} \\cdot \\\\\n\t& \\hspace{7cm} \\overline{ {\\big( \\nabla_{\\!\\! z} + 2i\\pi \\theta \\big)} g{\\left( \\Phi^{-1}(z,\\omega),\\omega \\right)} } \\, dz \\, d\\mathbb{P}(\\omega) \\\\\n\t& + \\int_\\Omega \\int_{\\Phi ([0,1)^n, \\omega)} V {\\left( \\Phi^{-1} ( z, \\omega),\\omega \\right)} f{\\left( \\Phi^{-1}(z,\\omega),\\omega \\right)} \\, \\overline{ g{\\left( \\Phi^{-1}(z,\\omega),\\omega \\right)} } \\, dz \\, d\\mathbb{P}(\\omega),\n\\end{split}\n\\end{equation*}\nfor ${ f, g \\in \\mathcal{H} }$.\n\n\n More precisely, it was required the existence of a pair ${ {\\big( \\theta^\\ast,\\lambda(\\theta^\\ast) \\big)} \\in \\mathbb{R}^n \\times \\mathbb{R} }$ \n that satisfies\n\\begin{equation}\\label{conds}\n\t\\left\\{ \\,\n\t\\begin{split}\n\t\t& \\lambda(\\theta^\\ast) \\; \\text{is a simple eigenvalue of} \\; L^\\Phi(\\theta^\\ast), \\\\\n\t\t&\\theta^\\ast \\; \\text{is a critical point of} \\; \\lambda(\\cdot), \\, \\text{that is}, \\nabla_{\\!\\! \\theta} \\lambda(\\theta^\\ast) = 0. \n\t\\end{split}\n\t\\right.\n\\end{equation}\n\nAs observed before, it is not clear the existence of a pair $(\\theta^{\\ast},\\lambda(\\theta^{\\ast}))$,\nin general stochastic environments, satisfying the two above \nconditions. \nThe reason is due mainly to the lack of compact embedding of ${ \\mathcal{H} }$ in ${ \\mathcal{L} }$. \nHowever, in the periodic settings  there are concrete situations where such conditions take place (see, for \ninstance,~\\cite{AllairePiatnitski,BarlettiBenAbdallah,BensoussanLionsPapanicolaou}).  \nOur aim in this section is to show realistic models whose spectral nature is inherited from the periodic ones.\n\n\\subsection{Perturbed Periodic Case: Spectral Analysis}\n\\label{PERTUSPECTANALY}\n\nIn this section we shall study the spectral properties of the operator ${ L^\\Phi(\\theta) }$, when the diffeomorphism ${ \\Phi }$ \nis a stochastic perturbation of the identity. This concept was introduced in \\cite{BlancLeBrisLions2},\nand well-developed by T. Andrade, W. Neves, J. Silva \\cite{AndradeNevesSilva} for modelling quasi-perfect materials. \n\n\\medskip\nLet $(\\Omega,\\mathcal{F},\\mathbb{P})$ be a probability space, \n$\\tau:\\mathbb{Z}^n\\times\\Omega\\to\\Omega$ a discrete \ndynamical system, and $Z$ any fixed stochastic deformation.\nThen, we consider the concept of stochastic perturbation of the identity given by the following\n\\begin{definition}\n\\label{37285gdhddddddddddd}\nGiven $\\eta \\in (0,1)$, let $\\Phi_\\eta: \\mathbb{R}^n \\times \\Omega \\to \\mathbb{R}^n$ be a stochastic deformation.\nThen $\\Phi_\\eta$ is said a stochastic perturbation of the identity, when \nit can be written as  \n\\begin{equation}\n\\label{DefPertIden}\n\\Phi_\\eta(y,\\omega) = y + \\eta \\, Z(y,\\omega) + \\mathrm{O}(\\eta^2), \n\\end{equation}\nfor some stochastic deformation $Z$.  \n\\end{definition}\nWe emphasize that the equality~\\eqref{DefPertIden} is understood in the sense of ${\\rm Lip}_{\\loc}\\big(\\mathbb R^n; L^2(\\Omega)\\big)$, i.e. \nfor each bounded open subset ${ \\mathcal{O} \\subset \\mathbb{R}^n }$, \nthere exist $\\delta, C > 0$, such that for all ${ \\eta \\in (0,\\delta) }$\n\\begin{eqnarray*}\n&&\\underset{y \\in \\mathcal{O}}{\\rm sup} \\, {\\left\\Vert \\Phi_\\eta(y,\\cdotp) - y - \\eta Z(y,\\cdotp) \\right\\Vert}_{L^2(\\Omega)}\\\\\n&&\\qquad +\\,\\underset{y \\in \\mathcal{O}}{\\rm ess \\, sup} \\, {\\left\\Vert \\nabla_{\\!\\! y} \\Phi_\\eta(y,\\cdotp) - I \n- \\eta \\, \\nabla_{\\!\\! y}  Z(y,\\cdotp) \\right\\Vert}_{L^2(\\Omega)}\n\\leqslant C \\, \\eta^2.\n\\end{eqnarray*}\nMoreover, after some computations, we have\n\\begin{equation}\n\\label{654367ytr6tfclmlml}\n\\left \\{\n\\begin{aligned}\n\t\\nabla_y^{-1} \\Phi_{\\eta}&= I-\\eta\\,\\nabla_y Z+O(\\eta^2), \n\t\\\\[5pt]\n\t\\det \\big(\\nabla_y\\Phi_{\\eta}\\big)&= 1+\\eta\\, {\\rm div}_yZ +O(\\eta^2).\n\\end{aligned}\n\\right.\n\\end{equation}\n\nNow, we consider the periodic functions \n$A_{\\rm per}:\\mathbb R^n\\to\\mathbb R^{n^2},\\,V_{\\rm per}:\\mathbb R^n\\to\\mathbb R$ and $U_{\\rm per}:\\mathbb R^n\\to\\mathbb R$, such that \n\\begin{itemize}\n\\item There exists $a_0,a_1>0$ such that for all $\\xi\\in\\mathbb R^n$ and for a.e $y\\in\\mathbb R^n$ we have \n$$\na_0|\\xi|^2\\le A_{\\rm per}(y)\\xi\\cdot \\xi\\le a_1|\\xi|^2.\n$$\n\\item $V_{\\rm per},\\,U_{\\rm per}\\in L^{\\infty}(\\mathbb R^n)$.\n\\end{itemize}\nThe following lemma is well-known and it is stated explicitly here only for reference. For a proof, we recommend the reader to~\\cite{Evans1}. \n\n\n\\begin{lemma}\n\\label{7836565etyd43tre56rt3e54redgh}\nFor $\\theta \\in \\mathbb{R}^n$ and  $f \\in H_{\\rm per}^1([0,1)^n)$, let ${ L_{\\rm per}(\\theta) }$ be the operator defined by\n\\begin{equation}\n\\label{753e6735827tdygetydr5de4se45se5}\nL_{\\rm per}(\\theta){[f]} := -({\\rm div}_{\\! y} + 2i\\pi \\theta) {\\big[ A_{\\rm per} (y) {(\\nabla_{\\!\\! y} + 2i\\pi\\theta)} f(y) \\big]} + V_{\\rm per}(y) f(y),\n\\end{equation}\nwith variational formulation\n\\begin{equation*}\n\\begin{array}{c}\n\\displaystyle {\\left\\langle L_{\\rm per}(\\theta){\\big[ f \\big]}, g \\right\\rangle} := \\int_{[0,1)^n} A_{\\rm per}(y) {\\left( \\nabla_{\\!\\! y} + 2i\\pi \\theta \\right)} f(y) \\cdot \\overline{ {\\left( \\nabla_{\\!\\! y} + 2i\\pi \\theta \\right)} g(y) } \\, dy \\\\ [10pt]\n\\displaystyle \\hspace{1.7cm} + \\int_{[0,1)^n} V_{\\rm per}(y) \\,  f(y) \\, \\overline{ g(y) } \\, dy, \n\\end{array}\n\\end{equation*}\nfor ${ f,g \\in H_{\\rm per}^1({[0,1)^n}) }$. Then ${ L_{\\rm per}(\\theta) }$ has the following properties:\n\t\t\\begin{enumerate}\n\t\t\t\\item[(i)] There exist ${ \\gamma_0, b_0 > 0 }$, such that ${ L_{\\gamma_0} := L_{\\rm per}(\\theta) + {\\gamma_0}I }$ satisfies\n\t\t\t for all $f \\in H_{\\rm per}^1({[0,1)^n})$, \n\t\t\t\\begin{equation*}\n\t\t\t\t{\\langle L_{\\gamma_0} {\\big[ f \\big]}, f \\rangle} \\geq b_0 {\\Vert f \\Vert}_{H_{\\rm per}^1({[0,1)^n})}^2.\n\t\t\t\\end{equation*}\n\t\t\t\\item[(ii)] The point spectrum of ${ L_{\\rm per}(\\theta) }$ is not empty and their eigenspaces have finite dimension, that is, the set\n\t\t\t\\begin{equation*}\n\t\t\t\t\\sigma_{\\rm point} {\\big( L_{\\rm per}(\\theta) \\big)} = \\{ \\lambda \\in \\mathbb{C} \\; ; \\; \\lambda \\; \\text{an eigenvalue of} \\; L_{\\rm per}(\\theta) \\}\n\t\t\t\\end{equation*}\n\t\t\tis not empty and for all ${ \\lambda \\in \\sigma_{\\rm point} {\\big( L_{\\rm per}(\\theta) \\big)} }$ fixed,\n\t\t\t\\begin{equation*}\n\t\t\t\t{\\rm dim} {\\big\\{ f \\in H^1_{\\rm per}({[0,1)^n}) \\; ; \\; L_{\\rm per}(\\theta){\\big[ f \\big]} = \\lambda f \\big\\}} < \\infty.\n\t\t\t\\end{equation*}\n\t\t\t\n\t\t\t\\item[(iii)] Every point in ${ \\sigma_{\\rm point}\\big( L_{\\rm per}(\\theta) \\big) }$ is isolated. \n\t\t\\end{enumerate}\n\t\\end{lemma}\n\n\\begin{remark}\nWe observe that, the properties of the ${ L_{\\rm per}(\\theta) }$, ${ \\theta \\in \\mathbb{R}^n }$,\ngiven by the Lemma~\\ref{7836565etyd43tre56rt3e54redgh} \ncan be conveyed to the space ${ \\mathcal{H} }$ in a natural way.\n\\end{remark}\n\t\n\nIn whats follow, we are interested in the study of spectral properties of the operator ${ L^{\\Phi_\\eta}(\\theta) }$ whose variational formulation is given by \n\t\\begin{equation}\\label{VarFor1}\n\t\\begin{split}\n\t\t& {\\left\\langle L^{\\Phi_\\eta}(\\theta)[f], g \\right\\rangle} :=  \\int_\\Omega \\int_{\\Phi_\\eta ([0,1)^n, \\omega)} A_{\\rm per} {\\left( \\Phi_\\eta^{-1} ( z, \\omega) \\right)} {\\big( \\nabla_{\\!\\! z} + 2i\\pi \\theta \\big)} f{\\left( \\Phi_\\eta^{-1}(z,\\omega),\\omega \\right)} \\cdot \\\\\n\t\t& \\hspace{7cm} \\overline{ {\\big( \\nabla_{\\!\\! z} + 2i\\pi \\theta \\big)} g{\\left( \\Phi_\\eta^{-1}(z,\\omega),\\omega \\right)} } \\, dz \\, d\\mathbb{P}(\\omega) \\\\\n\t\t& + \\int_\\Omega \\int_{\\Phi_\\eta ([0,1)^n, \\omega)} V_{\\rm per} {\\left( \\Phi_\\eta^{-1} ( z, \\omega) \\right)} f{\\left( \\Phi_\\eta^{-1}(z,\\omega),\\omega \\right)} \\, \\overline{ g{\\left( \\Phi_\\eta^{-1}(z,\\omega),\\omega \\right)} } \\, dz \\, d\\mathbb{P}(\\omega),\n\t\\end{split}\n\t\\end{equation}\n\tfor ${ f,g \\in \\mathcal{H} }$.  As we shall see in the next theorem, some of the spectral properties of the operator ${ L^{\\Phi_\\eta}(\\theta) }$ are inherited from the periodic case. \n\t\n\t\n\t\n\\begin{theorem}\n\\label{4087865567576ghghj}\n\t\tLet ${ \\Phi_\\eta }$, ${ \\eta \\in (0,1) }$ be a stochastic perturbation of identity and ${ \\theta_0 \\in \\mathbb{R}^n }$. If ${ \\lambda_0 }$ is an eigenvalue of ${ L_{\\rm per}(\\theta_0) }$ with multiplicity ${ k_0 \\in \\mathbb{N} }$, that is,\n\t\t\\begin{equation*}\n\t\t\t{\\rm dim} {\\big\\{ f \\in H^1_{\\rm per}([0,1)^n) \\; ; \\; L_{\\rm per}(\\theta_0){\\big[ f \\big]} = \\lambda_0 f \\big\\}} = k_0,\t\t\t\t\n\t\t\\end{equation*}\n\t\tthen there exist a neighbourhood ${ \\mathcal{U} }$ of ${ (0,\\theta_0) }$, ${ k_0 }$ real analytic functions\n\t\t\\begin{equation*}\n\t\t\t(\\eta,\\theta) \\in \\mathcal{U} \\; \\mapsto \\; \\lambda_k(\\eta,\\theta) \\in \\mathbb{R}, \\;\\; k\\in \\{1,\\ldots,k_0\\},\n\t\t\\end{equation*}\n\t\tand ${ k_0 }$ vector-value analytic maps \n\t\t\\begin{equation*}\n\t\t\t(\\eta,\\theta) \\in \\mathcal{U} \\; \\mapsto \\; \\psi_k(\\eta,\\theta) \\in \\mathcal{H} \\setminus \\{0\\}, \\;\\; k\\in \\{1,\\ldots,k_0\\},\n\t\t\\end{equation*}\n\t\tsuch that, for all ${ k\\in\\{1,\\ldots,k_0\\} }$,\n\t\t\\begin{itemize}\n\t\t\t\\item[(i)] ${ \\lambda_k(0,\\theta_0) = \\lambda_0 }$,\n\t\t\t\\item[(ii)] ${ L^{\\Phi_\\eta}(\\theta) {\\big[ \\psi_k(\\eta,\\theta) \\big]} = \\lambda_k(\\eta,\\theta) \\, \\psi_k(\\eta,\\theta) }$, ${ \\forall (\\eta,\\theta) \\in \\mathcal{U} }$,\n\t\t\t\\item[(iii)] ${ {\\rm dim}{\\big\\{ f \\in \\mathcal{H} \\; ; \\; L^{\\Phi_\\eta}(\\theta){\\big[ f \\big]}=\\lambda_k(\\eta,\\theta) f \\big\\}} \\leqslant k_0 }$, ${ \\forall (\\eta,\\theta) \\in \\mathcal{U} }$.\n\t\t\\end{itemize}\n\\end{theorem}\n\n\\begin{proof}\n1. The aim of this step is to rewrite the operator ${ L^{\\Phi_\\eta}(\\theta) \\in \\mathcal{B}(\\mathcal{H},\\mathcal{H}^\\ast) }$, for ${ \\eta\\in (0,1) }$ and ${ \\theta\\in\\mathbb{R}^n }$ \nas an expansion in the variable ${ (\\eta,\\theta) }$ of operators in ${ \\mathcal{B}(\\mathcal{H},\\mathcal{H}^\\ast) }$ around the point ${ (\\eta,\\theta)=(0,\\theta_0) }$. For this, \nusing the variational formulation~\\eqref{VarFor1}, a change of variables and the expansions~\\eqref{654367ytr6tfclmlml} we obtain\n                \\begin{equation*}\n\t\t\\begin{split}\n\t\t\t& \\!\\!\\! {\\langle L^{\\Phi_\\eta}(\\theta) {\\big[ f \\big]}, g \\rangle} = \\\\\n\t\t\t& {\\left[ \\int_{[0,1)^n} \\int_\\Omega A_{\\rm per}(y) {\\left( \\nabla_{\\!\\! y} + 2i\\pi \\theta \\right)}  f  \\cdot \\overline{{\\left( \\nabla_{\\!\\! y} + 2i\\pi \\theta \\right)} g} \\, d\\mathbb{P} \\, dy + \\int_{[0,1)^n} \\int_\\Omega V_{\\rm per}(y) \\, f \\, \\overline{g} \\, d\\mathbb{P} \\, dy \\right]} \\\\\n\t\t\t& \\hspace{0.25cm} + \\eta {\\left[ \\int_{[0,1)^n} \\int_\\Omega A_{\\rm per}(y) {\\left( -[\\nabla_{\\!\\! y} Z](y,\\omega)\\nabla_{\\!\\! y} f \\right)} \\cdot \\overline{{\\left( \\nabla_{\\!\\! y} + 2i\\pi \\theta \\right)} g} \\, d\\mathbb{P} \\, dy \\right.} \\\\\n\t\t\t& \\hspace{1.5cm} { + \\int_{[0,1)^n} \\int_\\Omega A_{\\rm per}(y)  {\\left( \\nabla_{\\!\\! y} + 2i\\pi \\theta \\right)}  f \\cdot \\overline{{\\left( -[\\nabla_{\\!\\! y} Z](y,\\omega) \\nabla_{\\!\\! y} g \\right)}} \\, d\\mathbb{P} \\, dy } \\\\\n\t\t\t& \\hspace{2.1cm} {\\left. + \\int_{[0,1)^n} \\int_\\Omega A_{\\rm per}(y) {\\left( \\nabla_{\\!\\! y} + 2i\\pi \\theta \\right)} f  \\cdot \\overline{{\\left( \\nabla_{\\!\\! y} + 2i\\pi \\theta \\right)} g} \\,\\, {\\rm div}_{\\! y} Z(y,\\omega) \\, d\\mathbb{P} \\, dy \\right]} \\\\\n\t\t\t& \\hspace{6.25cm} + \\mathrm{O}(\\eta^2),\n\t\t\\end{split}\n\t\t\\end{equation*}\n\t\tin $\\mathbb{C}$ as ${ \\eta \\to 0 }$, for ${ f,g \\in \\mathcal{H} }$. \n\t\t\n\t\t\nNow, making the expansion of it in the variable ${ \\theta }$ about the point ${ \\theta=\\theta_0 }$, it is convenient to rewrite the above expansion in the form \t\n\t\t\\begin{equation*}\n\t\t\t\\begin{split}\n\t\t\t\t& \\! {\\left\\langle L^{\\Phi_\\eta}(\\theta) {\\big[ f \\big]}, g \\right\\rangle} = \\\\\n\t\t\t\t& ((\\eta,\\theta)-(0,\\theta_0))^{(0,\\boldsymbol{0})}{\\left( \\int_{[0,1)^n} \\int_\\Omega A_{\\rm per}(y) {( \\nabla_{\\!\\! y} + 2i\\pi \\theta_0)} f \\cdot \\overline{ {( \\nabla_{\\!\\! y} + 2i\\pi \\theta_0)} g} \\, d\\mathbb{P} \\, dy \\right.} \\\\\n\t\t\t\t& \\hspace{9cm} {\\left. + \\int_{[0,1)^n} \\int_\\Omega V_{\\rm per}(y) \\, f \\, \\overline{g} \\, d\\mathbb{P} \\, dy \\right)} \\\\\n\t\t\t\t& + \\sum_{k=1}^n ((\\eta,\\theta)-(0,\\theta_0))^{(0,e_k)} {\\left( \\int_{[0,1)^n} \\int_\\Omega A_{\\rm per}(y) {( \\nabla_{\\!\\! y} + 2i\\pi \\theta_0)} f \\cdot \\overline{(2i\\pi e_k g)} \\, d\\mathbb{P} \\, dy  \\right.} \\\\\n\t\t\t\t& \\hspace{5.25cm} {\\left. + \\int_{[0,1)^n} \\int_\\Omega A_{\\rm per}(y) (2i\\pi e_k f) \\cdot \\overline{ {( \\nabla_{\\!\\! y} + 2i\\pi  \\theta_0)} g} \\, d\\mathbb{P} \\, dy \\right)}\n\t\t\t\\end{split}\n\t\t\\end{equation*}\n\t\t\\begin{equation*}\n\t\t\t\\begin{split}\n\t\t\t\t& + \\sum_{k, \\ell=1}^n ((\\eta,\\theta)-(0,\\theta_0))^{(0,e_k+e_\\ell)} {\\left( \\int_{[0,1)^n} \\int_\\Omega A_{\\rm per}(y) (2i\\pi e_k f) \\cdot \\overline{(2i\\pi e_\\ell g)} \\, d\\mathbb{P} \\, dy \\right)} \\\\\n\t\t\t\t& + ((\\eta,\\theta)-(0,\\theta_0))^{(1,\\boldsymbol{0})} {\\left( \\int_{[0,1)^n} \\int_\\Omega A_{\\rm per}(y) {\\left( -[\\nabla_{\\!\\! y} Z](y,\\omega)\\nabla_{\\!\\! y} f \\right)} \\cdot \\overline{ {( \\nabla_{\\!\\! y} + 2i\\pi \\theta_0)} g} \\, d\\mathbb{P} \\, dy \\right.} \\\\\n\t\t\t\t& \\hspace{2.15cm} + \\int_{[0,1)^n} \\int_\\Omega A_{\\rm per}(y)  {( \\nabla_{\\!\\! y} + 2i\\pi \\theta_0)} f \\cdot  \\overline{{\\left( -[\\nabla_{\\!\\! y} Z](y,\\omega) \\nabla_{\\!\\! y} g \\right)}} \\, d\\mathbb{P} \\, dy \\\\ \n\t\t\t\t& \\hspace{2.15cm} {\\left. + \\int_{[0,1)^n} \\int_\\Omega A_{\\rm per}(y)  {( \\nabla_{\\!\\! y} + 2i\\pi \\theta_0)} f \\cdot  \\overline{ {( \\nabla_{\\!\\! y} + 2i\\pi \\theta_0)} g} \\,\\, {\\rm div}_{\\! y} Z(y,\\omega) \\, d\\mathbb{P} \\, dy \\right)} \\\\\n\t\t\t\t& + \\sum_{k=1}^n ((\\eta,\\theta)-(0,\\theta_0))^{(1,e_k)} {\\left( \\int_{[0,1)^n} \\int_\\Omega A_{\\rm per}(y) {\\left( -[\\nabla_{\\!\\! y} Z](y,\\omega) \\nabla_{\\!\\! y} f \\right)} \\cdot \\overline{(2i\\pi e_k g)} \\, d\\mathbb{P} \\, dy \\right.} \\\\\n\t\t\t\t& \\hspace{3cm} + \\int_{[0,1)^n} \\int_\\Omega A_{\\rm per}(y) {(2i\\pi e_k f)} \\cdot \\overline{{\\left( -[\\nabla_{\\!\\! y} Z](y,\\omega) \\nabla_{\\!\\! y} g \\right)}} \\, d\\mathbb{P} \\, dy \\\\\n\t\t\t\t& \\hspace{3cm} + \\int_{[0,1)^n} \\int_\\Omega A_{\\rm per}(y) {( \\nabla_{\\!\\! y} + 2i\\pi \\theta_0)} f \\cdot \\overline{(2i\\pi e_k g)} \\,\\, {\\rm div}_{\\! y} Z(y,\\omega) \\, d\\mathbb{P} \\, dy \\\\\n\t\t\t\t& \\hspace{3cm} {\\left. + \\int_{[0,1)^n} \\int_\\Omega A_{\\rm per}(y) {(2i\\pi e_k f)} \\cdot \\overline{ {( \\nabla_{\\!\\! y} + 2i\\pi \\theta_0)} g} \\,\\, {\\rm div}_{\\! y} Z(y,\\omega) \\, d\\mathbb{P} \\, dy \\right)} \\\\\n\t\t\t\t& + \\sum_{k, \\ell=1}^n ((\\eta,\\theta)-(0,\\theta_0))^{(1,e_k + e_\\ell)} {\\left( \\int_{[0,1)^n} \\int_\\Omega A_{\\rm per}(y) {(2i\\pi e_k f)} \\cdot \\overline{(2i\\pi e_\\ell g)} \\,\\, {\\rm div}_{\\! y} Z(y,\\omega) \\, d\\mathbb{P} \\, dy \\right)} \\\\ \n\t\t\t\t& \\hspace{6.25cm}+ \\mathrm{O}(\\eta^2),\n\t\t\t\\end{split}\n\t\t\\end{equation*}\n\t\tin ${ \\mathbb{C} }$ as ${ \\eta \\to 0 }$, for ${ f,g \\in \\mathcal{H} }$, which is the expansion in the variable ${ (\\eta,\\theta) }$ around the point ${ (0,\\theta_0) }$. Here, for ${ (\\alpha,\\beta) \\in \\mathbb{N} \\times \\mathbb{N}^n }$ and ${ \\beta=(\\beta_1,\\ldots,\\beta_n) }$, we are using the multi-index notation ${ ((\\eta, \\theta)-(0,\\theta_0))^{(\\alpha,\\beta)} = \\eta^\\alpha \\prod_{k=1}^n (\\theta_k-\\theta_{0k})^{\\beta_k} }$. Now, noting that the term of order ${ (\\eta,\\theta)^{(0,\\boldsymbol{0})} }$ is the variational formulation of ${ L_{\\rm per}(\\theta_0) }$ as in \\eqref{753e6735827tdygetydr5de4se45se5}, we can rewrite the above expansion in the form\n\t\\begin{equation}\\label{iy87678yhghj354g}\n\t\tL^{\\Phi_\\eta}(\\theta) = L_{\\rm per}(\\theta_0) + \\sum_{{\\vert (\\alpha,\\beta) \\vert} = 1}^{3} ((\\eta,\\theta)-(0,\\theta_0))^{(\\alpha,\\beta)}L_{(\\alpha,\\beta)} + \\mathrm{O}(\\eta^2),\n\t\\end{equation}\n\tin ${ \\mathcal{B}(\\mathcal{H},\\mathcal{H}^\\ast) }$ as ${ \\eta \\to 0 }$, where ${ L_{(\\alpha,\\beta)} \\in \\mathcal{B}(\\mathcal{H},\\mathcal{H}^\\ast) }$ and ${ {\\vert (\\alpha,\\beta) \\vert} = \\alpha + \\sum_{k=1}^n \\beta_k }$.\n\t\t\nClearly, we can consider the parameters ${ (\\eta,\\theta) }$ in the set $B(0,1) \\times \\mathbb{C}^n$.\n\n2.  In this step, we shall modify the expansion \\eqref{iy87678yhghj354g} conveniently in order to obtain an holomorphic invertible operator in the variable ${ (\\eta,\\theta) }$.  For this, \nremember that according to the item ${ (i) }$ in Lemma \\ref{7836565etyd43tre56rt3e54redgh}, there exists $\\gamma_0>0$ such that the operator ${ L_{\\rm per}(\\theta_0) + {\\gamma_0} I }$ is invertible. Then there exists ${ \\delta>0 }$ such that the expansion\n\t\t\\begin{equation}\\label{87tyrtdfdcccdasxzsaxzsa}\n\t\t\tL^{\\Phi_\\eta}(\\theta) + {\\gamma_0} I = (L_{\\rm per}(\\theta_0) + {\\gamma_0} I) + \\sum_{{\\vert (\\alpha,\\beta) \\vert} = 1}^{3} ((\\eta,\\theta)-(0,\\theta_0))^{(\\alpha,\\beta)}L_{(\\alpha,\\beta)}+ \\mathrm{O}(\\eta^2),\n\t\t\\end{equation}\n\t\tin ${ \\mathcal{B}(\\mathcal{H},\\mathcal{H}^\\ast) }$ as ${ \\eta \\to 0 }$, is invertible for all ${ (\\eta,\\theta) \\in B(0,\\delta) \\times B(\\theta_0,\\delta) }$, since the set of invertible bounded operators ${ GL(\\mathcal{H},\\mathcal{H}^\\ast) }$ is an open subset of ${ \\mathcal{B}(\\mathcal{H},\\mathcal{H}^\\ast) }$. Now, we denote by ${ S(\\eta,\\theta) }$ the inverse operator of ${ L^{\\Phi_\\eta}(\\theta) + {\\gamma_0} I }$, ${ (\\eta, \\theta) \\in B(0,\\delta) \\times B(\\theta_0,\\delta) }$. Since the map ${ L \\in GL(\\mathcal{H},\\mathcal{H}^\\ast) \\mapsto L^{-1} \\in \\mathcal{B}(\\mathcal{H}^\\ast,\\mathcal{H}) }$ is continuous, the map\n\t\t\\begin{equation*}\n\t\t\t(\\eta, \\theta) \\in B(0,\\delta) \\times B(\\theta_0,\\delta) \\mapsto S(\\eta,\\theta) \\in \\mathcal{B}(\\mathcal{H}^\\ast, \\mathcal{H})\n\t\t\\end{equation*}\n\t\tis continuous. As a consequence of this, for ${ (\\widetilde{\\eta}, \\widetilde{\\theta}) \\in B(0,\\delta) \\times B(\\theta_0,\\delta) }$ fixed, the limit of\n\t\t\\begin{equation*}\n\t\t\t\\frac{S(\\eta, \\widetilde{\\theta}) - S(\\widetilde{\\eta}, \\widetilde{\\theta})}{\\eta - \\widetilde{\\eta}} = -S(\\eta, \\widetilde{\\theta}) {\\left[ \\frac{(L^{\\Phi_\\eta}(\\widetilde{\\theta}) + {\\gamma_0} I) - (L^{\\Phi_{\\widetilde{\\eta}}}(\\widetilde{\\theta}) + {\\gamma_0} I)}{\\eta - \\widetilde{\\eta}} \\right]} S(\\widetilde{\\eta}, \\widetilde{\\theta}),\n\t\t\\end{equation*}\n\t\tas ${ \\widetilde{\\eta} \\not= \\eta \\to 0 }$, exists. Thus, ${ \\eta \\in B(0,\\delta) \\mapsto S(\\eta,\\widetilde{\\theta}) }$ is an holomorphic map. In analogy with it, for ${ j\\in{\\{1,\\ldots,n\\}} }$, we can prove that\n\t\t\\begin{equation*}\n\t\t\t\\theta_j \\mapsto S(\\widetilde{\\eta}, \\widetilde{\\theta}_1, \\ldots, \\widetilde{\\theta}_{j-1}, \\theta_j, \\widetilde{\\theta}_{j+1}, \\ldots, \\widetilde{\\theta}_n) \n\t\t\\end{equation*}\n\t\tis an holomorphic map. Therefore, by Osgood's Lemma, see for instance \\cite{GunningRossi}, we conclude that\n\t\t\\begin{equation}\\label{9867967689ndyfh}\n\t\t\t(\\eta, \\theta) \\in B(0,\\delta) \\times B(\\theta_0,\\delta) \\mapsto S(\\eta,\\theta) \\in \\mathcal{B}(\\mathcal{H}^\\ast, \\mathcal{H})\n\t\t\\end{equation}\n\t\tis a holomorphic function.\n\t\t\n\t\t\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t\n3.  Finally, we are in conditions to prove items ${ (i) }$, ${ (ii) }$ and ${ (iii) }$ (the spectral analysis of the operator ${ S(\\eta, \\theta) }$). First, we shall note that for \n${ (\\eta,\\theta) }$ in a neighbourhood of ${ (0,\\theta_0) }$, the map ${ (\\eta, \\theta) \\mapsto S(\\eta, \\theta) }$ satisfies the assumptions of the Theorem \\ref{768746hughjg576}. \nWe begin recalling that the restriction operator ${ T \\in \\mathcal{B}(\\mathcal{H}^\\ast, \\mathcal{H}) \\mapsto T\\big\\vert_\\mathcal{L} \\in \\mathcal{B}(\\mathcal{L}, \\mathcal{L}) }$ is continuous and it satisfies\n\t\t\\begin{equation}\\label{78687326tygd53tegdcx}\n\t\t\t{\\Vert T \\Vert}_{\\mathcal{B}(\\mathcal{L}, \\mathcal{L})} \\leqslant {\\Vert T \\Vert}_{\\mathcal{B}(\\mathcal{H}^\\ast, \\mathcal{H})} \\, , \\;\\, \\forall T \\in \\mathcal{B}(\\mathcal{H}^\\ast, \\mathcal{H}).\n\t\t\\end{equation}\n\t\tThen, by \\eqref{9867967689ndyfh}, the map ${ (\\eta, \\theta) \\in B(0,\\delta) \\times B(\\theta_0,\\delta) \\mapsto S(\\eta, \\theta) \\in \\mathcal{B}(\\mathcal{L}, \\mathcal{L}) }$ is holomorphic. Since holomorphic maps are, locally, analytic maps there exists a neighbourhood ${ \\mathcal{U} }$ of ${ (0,\\theta_0) }$, ${ (0, \\theta_0) \\in \\mathcal{U} \\subset \\mathbb{C} \\times \\mathbb{C}^n }$, and a family ${ \\{S_{\\sigma}\\}_{\\sigma \\in \\mathbb{N} \\times \\mathbb{N}^n} }$ contained in ${ \\mathcal{B}(\\mathcal{L}, \\mathcal{L}) }$, such that\n\t\t\\begin{equation}\\label{rtfgrffcfdfdfdfdssdadssss}\n\t\t\tS(\\eta, \\theta) = S_{0} + \\sum_{\\substack{\\sigma \\in \\mathbb{N} \\times \\mathbb{N}^n \\\\ {\\vert \\sigma \\vert} \\neq 0}} (\\eta, \\theta)^\\sigma S_\\sigma \\, , \\;\\, \\forall (\\eta, \\theta) \\in \\mathcal{U}.\n\t\t\\end{equation}\n\t\t\n\t\t\tUsing \\eqref{87tyrtdfdcccdasxzsaxzsa} and \\eqref{rtfgrffcfdfdfdfdssdadssss}, it is easy to see that \n${ S_0 = (L_{\\rm per}(\\theta_0) + {\\gamma_0} I)^{-1} \\big\\vert_\\mathcal{L}}$.  Notice also that ${ \\mu_0 := {\\left( \\lambda_0+\\gamma_0 \\right)}^{-1} }$ is an eigenvalue of ${ S_0 }$ if and only if ${ \\lambda_0 }$ is an eigenvalue of ${ L_{\\rm per}(\\theta_0) }$ that is\n\t\t\\begin{equation*}\n\t\t\tg \\in {\\{ f \\in \\mathcal{L} \\; ; \\; S_0 {\\big[ f \\big]} = \\mu_0 f \\}} \\; \\Leftrightarrow \\; g \\in {\\{ f \\in \\mathcal{L} \\; ; \\; L_{\\rm per}(\\theta_0) {\\big[ f \\big]} =\\lambda_0 f \\}}.\n\t\t\\end{equation*}\n\t\n\\medskip\n\t\t\n\t\tThe final part of the proof is a direct application of the Theorem~\\ref{768746hughjg576}. Due to our assumption, $\\mu_0$ is a real eigenvalue of the operator $S_0$ with \nmultiplicity $k_0$. Hence, by the Theorem~\\ref{768746hughjg576},  there exists a neighbourhood ${ \\widetilde{\\mathcal{U}} }$ of ${ (0,\\theta_0) }$, with ${ \\widetilde{\\mathcal{U}} \\subset \\mathcal{U} }$ and analytic maps\n\t\t\\begin{equation*}\n\t\t\\begin{array}{l}\n\t\t\t(\\eta, \\theta) \\in \\widetilde{\\mathcal{U}} \\; \\longmapsto \\; \\mu_{0 1}(\\eta,\\theta), \\mu_{0 2}(\\eta,\\theta), \\ldots, \\mu_{0 k_0}(\\eta, \\theta) \\in (0,\\infty), \\\\ [5pt]\n\t\t\t(\\eta, \\theta) \\in \\widetilde{\\mathcal{U}} \\; \\longmapsto \\; \\psi_{0 1}(\\eta,\\theta), \\psi_{0 2}(\\eta,\\theta), \\ldots, \\psi_{0 k_0}(\\eta,\\theta) \\in \\mathcal{L}-\\{0\\},\n\t\t\\end{array}\n\t\t\\end{equation*}\n\t\tsuch that\n\t\t\\begin{itemize}\n\t\t\t\\item ${ \\mu_{0 \\ell} (0,\\theta_0) = \\mu_0 }$, \n\t\t\t\\item ${ S(\\eta, \\theta) {\\big[ \\psi_{0 \\ell}(\\eta, \\theta) \\big]} = \\mu_{0 \\ell}(\\eta, \\theta) \\psi_{0 \\ell}(\\eta, \\theta) }$, \\; ${ \\forall (\\eta, \\theta) \\in \\widetilde{\\mathcal{U}} }$,\n\t\t\t\\item ${ {\\rm dim}{\\{ f \\in \\mathcal{L} \\; ; \\; S(\\eta,\\theta){\\big[ f \\big]} = \\mu_{0 \\ell}(\\eta,\\theta) f \\}}\\leqslant k_0 }$, \\; ${ \\forall (\\eta, \\theta) \\in \\widetilde{\\mathcal{U}} }$, \n\t\t\\end{itemize}\t\t\n\t\tfor all ${ \\ell \\in \\{1, \\ldots, k_0\\} }$. Thus, the proof of the item ${ (i) }$ is clear. \n\t\t\n\t\t\n\t\tUsing the second equality above, we obtain \n\t\t\\begin{eqnarray*}\n\t\t\t(L^{\\Phi_\\eta}(\\theta) + {\\gamma_0} I) {\\big[ \\psi_{0 \\ell}(\\eta, \\theta) \\big]} & = & \\frac{1}{\\mu_{0 \\ell}(\\eta, \\theta)} (L^{\\Phi_\\eta}(\\theta) + {\\gamma_0} I){\\left\\{ S(\\eta, \\theta) {\\big[ \\psi_{0 \\ell}(\\eta, \\theta) \\big]} \\right\\}} \\\\ [5pt]\n\t\t\t& = & \\frac{1}{\\mu_{0 \\ell}(\\eta, \\theta)} \\psi_{0 \\ell}(\\eta, \\theta), \n\t\t\\end{eqnarray*}\nwhich implies that ${ L^{\\Phi_\\eta}(\\theta) {\\big[ \\psi_{0 \\ell}(\\eta, \\theta) \\big]} = \\lambda_{0\\ell}(\\eta,\\theta) \\psi_{0 \\ell}(\\eta, \\theta) }$, for ${ (\\eta, \\theta) \\in \\widetilde{\\mathcal{U}} }$,  ${ \\ell\\in \\{1, \\ldots, m_0\\} }$ and ${ \\lambda_{0 \\ell}(\\eta,\\theta) := [\\mu_{0 \\ell}(\\eta, \\theta)]^{-1} - {\\gamma_0} }$. This finish the proof of the item ${ (ii) }$.\n\t\t\n\\medskip \n\n\t\tFinally, note that ${ S(\\eta,\\theta) \\big[ \\mathcal{L} \\big] \\subset \\mathcal{H}}$ and \n\t\t\\begin{equation*}\n\t\t\tg \\! \\in \\! \\big\\{ f \\in \\mathcal{H} ; S(\\eta,\\theta) {\\big[ f \\big]} \\! = \\! \\mu_{0\\ell}(\\eta,\\theta) f \\big\\} \\Leftrightarrow g \\! \\in \\! \\big\\{ f \\in \\mathcal{H} \\; ; \\; L^{\\Phi_\\eta}(\\theta) {\\big[ f \\big]} \\! = \\! \\lambda_{0\\ell}(\\eta,\\theta) f \\big\\},\n\t\t\\end{equation*}\n\t\twhich concludes the proof of the item ${ (iii) }$. Therefore, the proof is completed. \n\\end{proof}\n\n\\subsection{Homogenization Analysis of the Perturbed Model}\n\nIn this section, we shall investigate in which way the stochastic perturbation of \nthe identity characterize the form of the coefficients, during the \nasymptotic limit of the Schr\\\"odinger equation \n\t\\begin{multline}\\label{765tdyyuty67tsss}\n\t\t\\left\\{\n\t\t\\begin{array}{l}\n\t\t\ti\\displaystyle\\frac{\\partial u_{\\eta\\varepsilon}}{\\partial t} - {\\rm div} {\\bigg( A_{\\rm per} {\\left( \\Phi_\\eta^{-1} {\\left( \\frac{x}{\\varepsilon}, \\omega \\right)} \\right)} \\nabla u_{\\eta\\varepsilon} \\bigg)} \\\\ [14pt]\n\t\t\t+ {\\bigg( \\displaystyle\\frac{1}{\\varepsilon^2} V_{\\rm per} {\\left( \\Phi_\\eta^{-1} {\\left( \\displaystyle\\frac{x}{\\varepsilon}, \\omega \\right)} \\right)} +  U_{\\rm per} {\\left( \\Phi_\\eta^{-1} {\\left( \\displaystyle\\frac{x}{\\varepsilon}, \\omega \\right)} \\right)} \\bigg)} u_{\\eta\\varepsilon} = 0 \\quad \\text{in} \\;\\, \\mathbb{R}^{n+1}_T \\! \\times \\! \\Omega, \\\\ [14pt]\n\t\t\tu_{\\eta\\varepsilon} (0,x,\\omega)=u_{\\eta\\varepsilon}^0(x,\\omega), \\;  \\; (x,\\omega) \\in \\mathbb{R}^n \\! \\times \\! \\Omega,\n\t\t\\end{array}\n\t\t\\right.\n\t\\end{multline}\nwhere ${ 0 < T < \\infty }$, ${ \\mathbb{R}^{n+1}_T = (0,T) \\times \\mathbb{R}^n }$.  The coefficients are accomplishing of the periodic functions ${ A_{\\rm per}(y) }$, ${ V_{\\rm per}(y) }$, ${ U_{\\rm per}(y) }$ (as defined in the last subsection) with a stochastic perturbation of identity ${ \\Phi_\\eta }$, ${ \\eta \\in (0,1) }$, presenting an rate of oscillation ${ \\varepsilon^{-1} }$, ${ \\varepsilon>0 }$.  The function ${ u_{\\eta\\varepsilon}^0(x,\\omega) }$ is a well prepared initial data(see~\\eqref{well-prep.I}) and this well-preparedness is triggered by natural periodic conditions, \nto wit, on the existence of a pair \n${ \\big( \\theta^\\ast, \\lambda_{\\rm per}(\\theta^\\ast) \\big) \\in \\mathbb{R}^n \\times \\mathbb{R} }$ such that\n\t\\begin{equation}\\label{7t8drtys65edsrt3xcvvcxcvb}\n\t\t\\begin{split}\n\t\t\t(i) & \\;\\;\\, \\lambda_{\\rm per}(\\theta^\\ast) \\; \\text{is a simple eigenvalue of} \\; L_{\\rm per}(\\theta^\\ast), \\\\\n\t\t\t(ii) & \\;\\;\\, \\theta^\\ast \\; \\text{is a critical point of} \\; \\lambda_{\\rm per}(\\cdot), \\, \\text{that is}, \\nabla_{\\!\\! \\theta} \\lambda_{\\rm per}(\\theta^\\ast)=0.\n\t\t\\end{split}\n\t\\end{equation}\n\t\n\t\n\tBy the condition ${ (i) }$ and the Theorem \\ref{4087865567576ghghj}, there exists a neighborhood ${ \\mathcal{U} }$ of ${ (0,\\theta^{\\ast}) }$ and the analytic maps\n\t\\begin{equation}\\label{67ty3uhrjefd67tgrefdcx8ur7u}\n\t\t\\begin{split}\n\t\t\t(i) & \\;\\;\\, (\\eta,\\theta) \\in \\mathcal{U} \\; \\mapsto \\; \\lambda(\\eta,\\theta) \\in \\mathbb{R}, \\\\\n\t\t\t(ii) & \\;\\;\\, (\\eta,\\theta) \\in \\mathcal{U} \\; \\mapsto \\; \\psi(\\eta,\\theta) \\in \\mathcal{H}\\setminus\\{0\\},\n\t\t\\end{split}\n\t\\end{equation}\n\tsuch that ${ \\lambda(0,\\theta^\\ast) = \\lambda_{\\rm per}(\\theta^\\ast) }$, ${ L^{\\Phi_\\eta}(\\theta) \\big[ \\psi(\\eta,\\theta) \\big] = \\lambda(\\eta,\\theta) \\, \\psi(\\eta,\\theta) }$ and\n\t\\begin{equation*}\n\t\t{\\rm dim} \\big\\{ f \\in \\mathcal{H} \\; ; \\; L^{\\Phi_\\eta} (\\theta) = \\lambda(\\eta,\\theta) \\, f \\big\\} = 1, \\; \\forall (\\eta,\\theta) \\in \\mathcal{U}.\n\t\\end{equation*}\n\t\n\t\tThus,\n\t\\begin{equation}\\label{7rter44}\n\t\t\\lambda(\\eta,\\theta) \\; \\text{is a simple eigenvalue of} \\; L^{\\Phi_\\eta}(\\theta),  \\forall (\\eta,\\theta) \\in \\mathcal{U}.\n\t\\end{equation}\n\t\n\t\nAdditionally, as ${ \\lambda(0,\\theta^\\ast)=\\lambda_{\\rm per}(\\theta^\\ast) }$ is an isolated point of ${ \\sigma_{\\rm point} \\big(L_{\\rm per}(\\theta^\\ast) \\big) }$ (any point has this property\t), ${ \\lambda(\\eta,\\theta) }$ is an isolated point of ${ \\sigma_{\\rm point} \\big(L^{\\Phi_\\eta}(\\theta^\\ast) \\big) }$ for each ${ (\\eta,\\theta) \\in \\mathcal{U} }$. Thus, we have \n${ \\lambda(0,\\cdot) = \\lambda_{\\rm per}(\\cdot) }$ in a neighbourhood of ${ \\theta^\\ast }$. We now denote ${ \\psi_{\\rm per}(\\cdot) := \\psi(0,\\cdot) }$. Without loss of generality, we assume ${ \\int_{[0,1)^n} {\\vert \\psi_{\\rm per}(\\theta^\\ast) \\vert}^2 dy = 1 }$. Moreover, we shall assume that the homogenized (periodic) matrix \n${ A_{\\rm per}^\\ast = D_{\\! \\theta}^2 \\lambda_{\\rm per}(\\theta^\\ast) }$ is invertible which happens if $\\theta=\\theta^{\\ast}$ is a point of local minimum or local maximum strict of \n$\\mathbb R^n\\ni \\theta\\mapsto \\lambda_{\\rm per}(\\theta)$. Thus, an immediate application of the Implicit Function Theorem gives us the following lemma:\n\n\\begin{lemma}\\label{6487369847639gfhdghjdftrtrtfgcbvbv}\n\t\tLet the condition \\eqref{7t8drtys65edsrt3xcvvcxcvb} be satisfied and ${ A_{\\rm per}^\\ast}$ be an invertible matrix. Then, there exists a neighborhood ${ \\mathcal{V} }$ of ${ 0 }$, ${ 0 \\in \\mathcal{V} \\subset \\mathbb{R} }$, and a ${ \\mathbb{R}^n }$-value analytic map\n\t\t\\begin{equation*}\n\t\t\t\\theta (\\cdot) : \\eta \\in \\mathcal{V} \\mapsto \\theta(\\eta) \\in \\mathbb{R}^n, \n\t\t\\end{equation*}\n\t\tsuch that ${ \\theta(0)=\\theta^\\ast }$ and\n\t\t\\begin{equation}\\label{67trsdasdsoktig}\n\t\t\t\\nabla_{\\!\\! \\theta} \\lambda \\big( \\eta,\\theta(\\eta) \\big) = 0, \\;\\; \\forall \\eta \\in \\mathcal{V}.\n\t\t\\end{equation}\n\\end{lemma}\n\n\nBy the analytic structure of the functions in \\eqref{67ty3uhrjefd67tgrefdcx8ur7u} and the Lemma \\ref{6487369847639gfhdghjdftrtrtfgcbvbv}, there exists a neighborhood ${ \\mathcal{V} }$ of ${ 0 }$, ${ 0 \\in \\mathcal{V} \\subset \\mathbb{R} }$, such that \n\t\\begin{equation}\\label{563gdc}\n\t\t\\begin{split}\n\t\t\t(i) & \\;\\;\\, \\eta \\in \\mathcal{V} \\; \\mapsto \\; \\lambda \\big( \\eta, \\theta(\\eta) \\big) \\in \\mathbb{R}, \\\\\n\t\t\t(ii) & \\;\\;\\, \\eta \\in \\mathcal{V} \\; \\mapsto \\; \\psi \\big( \\eta, \\theta(\\eta) \\big) \\in \\mathcal{H} \\setminus \\{0\\}, \\\\\n\t\t\t(iii) & \\;\\;\\, \\eta \\in \\mathcal{V} \\; \\mapsto \\; \\xi_k \\big( \\eta, \\theta(\\eta) \\big) \\in \\mathcal{H}, \\forall \\{1,\\ldots,n\\},\n\t\t\\end{split}\n\t\\end{equation}\n\tare analytic functions, where ${ \\xi_k(\\eta,\\theta) := (2i \\pi)^{-1}{\\partial_{\\theta_k} \\psi} (\\eta,\\theta) }$, for ${ k\\in\\{1,\\ldots,n\\} }$. We also consider ${ \\xi_{k,{\\rm per}}(\\cdot) = \\xi_k(0,\\cdot) }$. Furthermore, by \\eqref{7rter44} and \\eqref{67trsdasdsoktig}, for each fixed ${ \\eta \\in \\mathcal{V} }$ we have that the pair ${ \\big( \\theta(\\eta),\\lambda \\big( \\eta, \\theta(\\eta) \\big) \\big) \\in \\mathbb{R}^n \\times \\mathbb{R} }$ satisfies: \n\t\\begin{equation}\\label{674tyghd}\n\t\t\\begin{split}\n\t\t\t(i) & \\;\\;\\, \\lambda(\\eta,\\theta(\\eta)) \\; \\text{is a simple eigenvalue of} \\; L^{\\Phi_\\eta}\\big( \\theta(\\eta) \\big), \\\\\n\t\t\t(ii) & \\;\\;\\, \\theta(\\eta) \\; \\text{is a critical point of} \\; \\lambda(\\eta,\\cdot), \\, \\text{that is}, \\nabla_{\\!\\! \\theta} \\lambda(\\eta, \\theta(\\eta)) = 0.\n\t\t\\end{split}\n\t\\end{equation}\n\tThis means that the Theorem \\ref{876427463tggfdhgdfgkkjjlmk} can be used. Before, we establish a much simplified notations for the functions in \\eqref{563gdc} as follows:\n\t\\begin{equation}\\label{798723678rtyd5rdftrgdfdfsdssss}\n\t\t\\begin{split}\n\t\t\t(i) & \\;\\;\\, \\theta_\\eta := \\theta(\\eta), \\\\\n\t\t\t(ii) & \\;\\;\\, \\lambda_\\eta := \\lambda \\big( \\eta,\\theta(\\eta) \\big), \\\\\n\t\t\t(iii) & \\;\\;\\, \\psi_\\eta := \\psi \\big( \\eta,\\theta(\\eta) \\big), \\\\\n\t\t\t(iv) & \\;\\;\\, \\xi_{k,\\eta} := \\xi_k \\big( \\eta,\\theta(\\eta) \\big), \\, k\\in\\{1,\\ldots,n\\}.\n\t\t\\end{split}\n\t\\end{equation}\n\n\n\tFinally, from \\eqref{674tyghd}, for each fixed ${ \\eta \\in \\mathcal{V} }$, the notion of well-preparedness for the initial data $u_{\\eta\\varepsilon}^0$ is given as below. \n\t\n\t\\begin{equation}\\label{well-prep.I}\n\t\tu_{\\eta\\varepsilon}^0(x,\\omega) = e^{2i\\pi \\frac{\\theta_\\eta \\cdot x}{\\varepsilon}} \\,  v^0(x) \\, \\psi_\\eta {\\left( \\Phi_{\\eta}^{-1} {\\left( \\frac{x}{\\varepsilon}, \\omega \\right)}, \\omega \\right)}, \\; (x,\\omega) \\in \\mathbb{R}^n \\times \\Omega,\n\t\\end{equation}\nwhere ${ v^0 \\in C_{\\rm c}^\\infty(\\mathbb{R}^n) }$. Thus, applying the Theorem \\ref{876427463tggfdhgdfgkkjjlmk}, if ${ u_{\\eta\\varepsilon} }$ is solution of \\eqref{765tdyyuty67tsss}, the sequence in ${ \\varepsilon>0 }$\n\t\\begin{equation*}\n\t\tv_{\\eta\\varepsilon}(t,x,\\widetilde{\\omega}) = e^{ -{\\left( i \\frac{\\lambda_\\eta t}{\\varepsilon^2} + 2i\\pi \\frac{\\theta_\\eta \\cdot x}{\\varepsilon} \\right)} } u_{\\eta\\varepsilon}(t,x,\\widetilde{\\omega}), \\;\\, (t,x,\\widetilde{\\omega}) \\in \\mathbb{R}^{n+1}_T \\times \\Omega, \n\t\\end{equation*}\n\t$\\Phi_{\\omega}-$two-scale converges to the limit ${ v_{\\eta}(t,x) \\, \\psi_{\\eta}{\\big( \\Phi^{-1}(z,\\omega),\\omega \\big)} }$ with\n\t\\begin{equation*}\n\t\t\\lim_{\\varepsilon \\to 0} \\iint_{\\mathbb{R}^{n+1}_T} \\! {\\left\\vert v_{\\eta \\varepsilon} (t,x,\\widetilde{\\omega}) - v_{\\eta}(t,x) \\, \\psi_{\\eta}{\\left( \\Phi^{-1}_{\\eta} {\\left(\\frac{x}{\\varepsilon},\\widetilde{\\omega} \\right)}, \\widetilde{\\omega} \\right)} \\right\\vert}^2 dx \\, dt \\, = \\, 0,\n\t\\end{equation*}\nfor a.e. ${ \\widetilde{\\omega} \\in \\Omega }$, where ${ v_{\\eta} \\in C \\big( [0,T], L^2(\\mathbb{R}^n) \\big) }$ is the unique solution of the homogenized Schr\\\"odinger equation \n\t\\begin{equation}\\label{askdjfhucomojkfdfd}\n\t\t\\left\\{\n\t\t\\begin{array}{c}\n\t\t\ti \\displaystyle\\frac{\\partial v_\\eta}{\\partial t} - {\\rm div} {\\left( A^\\ast_{\\eta} \\nabla v_\\eta \\right)} + U_{\\! \\eta}^\\ast v_\\eta = 0 \\, , \\;\\, \\text{in} \\;\\, \\mathbb{R}^{n+1}_T, \\\\ [7,5pt]\n\t\t\tv_\\eta(0,x) = v^0(x) \\, , \\;\\, x\\in \\mathbb{R}^n,\n\t\t\\end{array}\n\t\t\\right.\n\t\\end{equation}\nwith effective coefficients ${ A^\\ast_{\\eta} = D_{\\! \\theta}^2 \\lambda \\big( \\eta,\\theta(\\eta) \\big) }$ and \n\t\\begin{equation}\\label{783874tgffffg}\n\t\tU^\\ast_{\\! \\eta} =  c^{-1}_{\\eta} \\int_\\Omega \\int_{\\Phi_{\\eta}([0,1)^n, \\omega)} U_{\\rm per}{\\big( \\Phi^{-1}_{\\eta} (z, \\omega) \\big)}  {\\left\\vert \\psi_{\\eta} {\\big( \\Phi^{-1}_{\\eta} (z,\\omega), \\omega \\big)} \\right\\vert}^2 dz \\, d\\mathbb{P}(\\omega),\n\t\\end{equation}\n\twhere\n\t\\begin{equation}\\label{874326984yghedf}\n\t\tc_{\\eta} = \\int_\\Omega \\int_{\\Phi_{\\eta}([0,1)^n, \\omega)} {\\left\\vert \\psi_{\\eta} {\\big( \\Phi^{-1}_{\\eta} (z,\\omega), \\omega \\big)} \\right\\vert}^2 dz \\, d\\mathbb{P}(\\omega).\n\t\\end{equation}\n\t\n\\begin{remark}\n\t\tWe remember that, using the equality~\\eqref{7358586tygfjdshfbvvcc}, we have for each ${ \\eta }$ fixed that the matrix ${ B_\\eta \\in \\mathbb{R}^{n \\times n} }$ must satisfy\n\t\t\\begin{equation}\\label{786587tdyghs7rsdfxsdfsdf}\n\t\t\\begin{split}\n\t\t\t& (B_\\eta)_{k\\ell} := c_\\eta^{-1} \\bigg[ \\int_\\Omega\\int_{\\Phi_\\eta([0,1)^n,\\omega)} A_{\\rm per}{\\left( \\Phi_\\eta^{-1}(z,\\omega) \\right)} {\\left( e_\\ell \\, \\psi_\\eta{\\left( \\Phi_\\eta^{-1}(z,\\omega),\\omega \\right)}  \\right)} \\cdot \\\\\n\t\t\t& \\hspace{7.85cm} \\overline{\\left( e_k \\, \\psi_\\eta{\\left( \\Phi_\\eta^{-1}(z,\\omega),\\omega \\right)} \\right)} \\, dz \\, d\\mathbb{P}(\\omega) \\\\\n\t\t\t& + \\int_\\Omega\\int_{\\Phi_\\eta([0,1)^n,\\omega)} A_{\\rm per}{\\left( \\Phi_\\eta^{-1}(z,\\omega) \\right)} {\\left( e_\\ell \\, \\psi_\\eta{\\left( \\Phi_\\eta^{-1}(z,\\omega),\\omega \\right)}  \\right)} \\cdot \\\\\n\t\t\t& \\hspace{6cm} \\overline{{\\left( \\nabla_{\\!\\! z} + 2i\\pi\\theta_\\eta \\right)} {\\left( \\xi_{k,\\eta}{\\left( \\Phi_\\eta^{-1}(z,\\omega),\\omega \\right)} \\right)}} \\, dz \\, d\\mathbb{P}(\\omega) \\\\\n\t\t\t& - \\int_\\Omega\\int_{\\Phi_\\eta([0,1)^n,\\omega)} A_{\\rm per}{\\left( \\Phi_\\eta^{-1}(z,\\omega) \\right)} {\\left( \\nabla_{\\!\\! z} + 2i\\pi\\theta_\\eta \\right)} {\\left( \\psi_\\eta{\\left( \\Phi_\\eta^{-1}(z,\\omega),\\omega \\right)} \\right)} \\cdot \\\\\n\t\t\t& \\hspace{7.8cm} \\overline{{\\left( e_\\ell \\, \\xi_{k,\\eta}{\\left( \\Phi_\\eta^{-1}(z,\\omega),\\omega \\right)}  \\right)}} \\, dz \\, d\\mathbb{P}(\\omega) \\bigg],\n\t\t\\end{split}\n\t\t\\end{equation}\n\t\tfor ${ k,\\ell \\in \\{1,\\ldots,n\\} }$ and the homogenized matrix can be written as ${ A_\\eta^\\ast = 2^{-1} {\\big( B_\\eta + B_\\eta^t \\big)} }$.\t\n\\end{remark}\n\t\n\\subsubsection{Expansion of the effective coefficients}\nAs a consequence of the formula of the effective coefficients of the homogenized equation~\\eqref{askdjfhucomojkfdfd}, we have the following proposition:\n\n\t\\begin{proposition}\\label{jnchndhbvgfbdtegdferfer}\n\t\tThe maps ${ \\eta \\mapsto A_\\eta^\\ast \\in \\mathbb{R}^{n \\times n} }$, ${ \\eta \\mapsto B_\\eta \\in \\mathbb{R}^{n \\times n} }$ and ${ \\eta \\mapsto U_\\eta^\\ast \\in \\mathbb{R} }$ are analytics in a neighbourhood of ${ \\eta=0 }$.\n\t\\end{proposition}\n\\begin{proof}\n\t\tLet us assume ${ \\mathcal{U} }$ and ${ \\mathcal{V} }$ as in \\eqref{67ty3uhrjefd67tgrefdcx8ur7u} and \\eqref{563gdc}, respectively. For each ${ \\eta \\in \\mathcal{V} }$, the above arguments give us the formula ${ A^\\ast_{\\eta} = D_{\\! \\theta}^2 \\lambda \\big( \\eta,\\theta(\\eta) \\big) }$. Thus, as ${ (\\eta,\\theta) \\in \\mathcal{U} \\mapsto D_{\\! \\theta}^2\\lambda(\\eta,\\theta) \\in \\mathbb{R}^{n \\times n} }$ and ${ \\eta \\in \\mathcal{V} \\mapsto \\theta(\\eta) \\in \\mathbb{R}^n }$ are analytic maps, we conclude that ${ \\eta \\in \\mathcal{V} \\mapsto D_\\theta^2 \\lambda(\\eta,\\theta(\\eta)) \\in \\mathbb{R}^{n \\times n} }$ is also an analytic map. This means that ${ \\eta \\mapsto A_\\eta^\\ast }$ is an analytic map. \n\n\\medskip\n\n\t\tFrom \\eqref{783874tgffffg} and \\eqref{874326984yghedf}, making a change of variables, we have\n\t\t\\begin{equation*}\n\t\t\tU^\\ast_{\\! \\eta} =  c^{-1}_{\\eta} \\int_\\Omega \\int_{[0,1)^n} U_{\\rm per}(y)  {\\left\\vert \\psi_{\\eta}(y,\\omega) \\right\\vert}^2 {\\rm det} [\\nabla_{\\!\\! y} \\Phi_\\eta (y,\\omega)] \\, dz \\, d\\mathbb{P}(\\omega)\n\t\t\\end{equation*}\n\t\tand\n\t\t\\begin{equation*}\n\t\t\tc_\\eta =  \\int_\\Omega \\int_{[0,1)^n} {\\left\\vert \\psi_{\\eta}(y,\\omega) \\right\\vert}^2 {\\rm det} [\\nabla_{\\!\\! y} \\Phi_\\eta (y,\\omega)] \\, dz \\, d\\mathbb{P}(\\omega) \\not= 0.\n\t\t\\end{equation*}\n\t\tThen, as the map ${ \\eta \\mapsto \\psi_\\eta \\in \\mathcal{H} \\setminus\\{0\\} }$ is analytic, the map ${ \\eta \\mapsto c_\\eta \\not= 0 }$ is also analytic. Hence the map ${ \\eta \\mapsto c_\\eta^{-1} }$ is analytic. Therefore, ${ \\eta \\mapsto U_\\eta^\\ast }$ is analytic.\n\t\\end{proof}\n\t\nAs a consequence of this proposition, there exist ${ \\{ A^{(j)},\\,B^{(j)} \\}_{j \\in \\mathbb{N}} \\subset \\mathbb{R}^{n \\times n} }$ and ${ \\{ U^{(j)} \\}_{j \\in \\mathbb{N}} \\subset \\mathbb{R} }$ such that \n\t\\begin{equation}\\label{678435}\n\t\t\\left\\{\n\t\t\\begin{array}{lll}\n\t\t\tA_\\eta^\\ast & = & A^{(0)} + \\eta A^{(1)} + \\eta^2 A^{(2)} +\\ldots, \\\\ [5pt]\n\t\t\tU_\\eta^\\ast & = & U^{(0)} + \\eta U^{(1)} + \\eta^2 U^{(2)} + \\ldots,\\\\ [5pt]\n\t\t\tB_\\eta^\\ast &= & B^{(0)} + \\eta B^{(1)} + \\eta^2 B^{(2)} +\\ldots.\n\t\t\\end{array}\n\t\t\\right.\n\t\\end{equation}\n\t\n\nNow, the object of our interest is determine the terms of order ${ \\eta^0 }$ and ${ \\eta }$ of these homogenized coefficients. For this purpose, guided by~\\ref{654367ytr6tfclmlml} and \nby the formulas \\eqref{786587tdyghs7rsdfxsdfsdf}, \\eqref{783874tgffffg} and \\eqref{874326984yghedf}, we shall analyse the expansion of the analytic functions in \\eqref{798723678rtyd5rdftrgdfdfsdssss}. By analytic property, there exist the sequences ${ {\\{ \\theta^{(j)} \\}}_{j \\in \\mathbb{N}} \\subset \\mathbb{R}^n }$, ${ {\\{ \\lambda^{(j)} \\}}_{j \\in \\mathbb{N}} \\subset \\mathbb{R} }$, ${ {\\{ \\psi^{(j)} \\}}_{j \\in \\mathbb{N}} \\subset \\mathcal{H} }$ and ${ {\\{ \\xi_k^{(j)} \\}}_{j \\in \\mathbb{N}} \\subset \\mathcal{H} }$, ${ k\\in\\{1,\\ldots,n\\} }$, such that\n\t\\begin{eqnarray}\n\t\t\\theta_\\eta & = & \\theta^{(0)} + \\eta \\theta^{(1)} + \\eta^2 \\theta^{(2)} + \\ldots = \\theta^{(0)} + \\eta \\theta^{(1)} + \\mathrm{O}(\\eta^2), \\label{9789789794r6ttrtrtr} \\\\\n\t\t\\lambda_\\eta & = & \\lambda^{(0)} + \\eta \\lambda^{(1)} + \\eta^2 \\lambda^{(2)} + \\ldots = \\lambda^{(0)} + \\eta \\lambda^{(1)} + \\mathrm{O}(\\eta^2), \\label{6t8y365873586edtygc} \\\\\n\t\t\\psi_\\eta & = & \\psi^{(0)} + \\eta\\psi^{(1)} + \\eta^2 \\psi^{(2)} + \\ldots = \\psi^{(0)} + \\eta \\psi^{(1)} + \\mathrm{O}(\\eta^2), \\label{099uiuiyujhjchhtydfty} \\\\\n\t\t\\xi_{k,\\eta} & = & \\xi_k^{(0)} + \\eta\\xi_k^{(1)} + \\eta^2 \\xi_k^{(2)} + \\ldots = \\xi_k^{(0)} + \\eta \\xi_k^{(1)} + \\mathrm{O}(\\eta^2), \\label{67tyuxcvbdfgoikjjhbhb}\n\t\\end{eqnarray}\nwhere ${ k\\in\\{1,\\ldots,n\\} }$. \n\nAt first glance, in order to determine the coefficients of the expansions in~\\eqref{678435} we should solve, a priori, auxiliary problems that involves both, the deterministic \nand stochastic variables. This can be a disadvantage from the point of view of numerical analysis. Our aim hereafter is to prove that, we can simplify the computations of this \ncoefficients working in a periodic environment which is computationally cheaper. In order to do this, \nnote that ${ \\theta^{(0)} = \\theta^\\ast}$, ${ \\lambda^{(0)}=\\lambda_{\\rm per}(\\theta^\\ast) }$, ${ \\psi^{(0)} = \\psi_{\\rm per}(\\theta^\\ast) }$ and ${ \\xi_k^{(0)} = \\xi_{k,{\\rm per}}(\\theta^\\ast) }$, ${ k\\in\\{1,\\ldots,n\\} }$, which satisfy\n\t\\begin{eqnarray}\n\t    & & \\left\\{\n\t\t\\begin{array}{l}\n\t\t\t{\\left( L_{\\rm per}(\\theta^\\ast) - \\lambda_{\\rm per}(\\theta^\\ast) \\right)} {\\big[ \\psi_{\\rm per}(\\theta^\\ast) \\big]} = 0  \\;\\, \\text{in} \\;\\, [0,1)^n, \\\\ [6pt]\n\t\t\t\\hspace{1.5cm} \\psi_{\\rm per}(\\theta^\\ast) \\;\\; [0,1)^n\\text{-periodic},\n\t\\end{array}\n\t\t\\right. \\label{yhujtgvjnjnhnvfnvfshjbn} \\\\ [7.5pt]\n\t\t& & \\left\\{\n\t\t\\begin{array}{l}\n\t\t\t{\\left( L_{\\rm per}(\\theta^\\ast) - \\lambda_{\\rm per}(\\theta^\\ast) \\right)} {\\big[ \\xi_{k,{\\rm per}}(\\theta^\\ast) \\big]} =  \\mathcal{X} {\\big[ \\psi_{\\rm per}(\\theta^\\ast) \\big]}  \\;\\, \\text{in} \\;\\, [0,1)^n, \\\\ [6pt]\n\t\t\t\\hspace{1.5cm} \\xi_{k,{\\rm per}}(\\theta^\\ast) \\;\\; [0,1)^n \\text{-periodic},\n\t\t\\end{array}\n\t\t\\right. \\label{yhfjsgfsfsdyhujtgvjnjnhnvfnvfshjbn}\n\t\\end{eqnarray}\n\twhere\n\t\\begin{equation*}\n\t\t\\mathcal{X} {\\big[ f \\big]} := {\\left( {\\rm div}_{\\! y} + 2i\\pi \\theta^\\ast \\right)} {\\big\\{ A_{\\rm per} (y) {( e_k f )} \\big\\}} + {( e_k )} {\\big\\{ A_{\\rm per}(y) {( \\nabla_{\\!\\! y} + 2i\\pi \\theta^\\ast )} f \\big\\}},\n\t\\end{equation*}\n\tfor ${ f\\in \\mathcal{H} }$. The equation~\\eqref{yhujtgvjnjnhnvfnvfshjbn} is the spectral \n\tcell equation and \\eqref{yhfjsgfsfsdyhujtgvjnjnhnvfnvfshjbn} is the first auxiliary cell equation \nrelated to the periodic case (see Section~\\ref{ACE}). \n\n\\medskip\n\nThe following theorem show us that, \nthe terms ${ \\psi^{(1)} }$ and ${ \\xi_k^{(1)} }$, $k\\in\\{1,\\ldots,n\\}$,\ngiven by \\eqref{099uiuiyujhjchhtydfty} and \\eqref{67tyuxcvbdfgoikjjhbhb}\nrespectively, satisfy auxiliary type cell equations. \n\t\\begin{theorem}\n\t\\label{THM58}\n\n\t\tLet ${ \\psi^{(1)} }$ and ${ \\xi_k^{(1)} }$, ${ k\\in\\{1,\\ldots,n\\} }$, be as above. Then these functions satisfy the following equations: \n\t\t\\begin{eqnarray}\n\t\t\t& & \\left\\{\n\t\t\t\\begin{array}{l}\n\t\t\t\t{\\left( L_{\\rm per}(\\theta^\\ast) - \\lambda_{\\rm per}(\\theta^\\ast) \\right)} {\\big[ \\psi^{(1)} \\big]} = \\mathcal{Y} {\\big[ \\psi_{\\rm per}(\\theta^\\ast) \\big]} \\;\\, \\text{in} \\;\\, [0,1)^n \\times \\Omega, \\\\ [6pt]\n\t\t\t\t\\hspace{1.5cm} \\psi^{(1)} \\; \\text{stationary},\n\t\t\t\\end{array}\n\t\t\t\\right. \\label{cvbnmdhfyhryfryhfiajbcjzx} \\\\ [7.5pt]\n\t\t\t& & \\left\\{\n\t\t\t\\begin{array}{l}   \n\t\t\t\t{\\left( L_{\\rm per}(\\theta^\\ast) - \\lambda_{\\rm per}(\\theta^\\ast) \\right)} {\\big[ \\xi_k^{(1)} \\big]} = \\mathcal{X}{\\big[ \\psi^{(1)} \\big]}  \\\\ [6pt]\n\t\t\t\t\\hspace{2cm} + \\, \\mathcal{Y}{\\big[ \\xi_{k,{\\rm per}}(\\theta^\\ast) \\big]} + \\mathcal{Z}_k{\\big[ \\psi_{\\rm per}(\\theta^\\ast) \\big]} \\;\\,  \\text{in} \\;\\, [0,1)^n \\times \\Omega, \\\\ [6pt]\n\t\t\t\t\\hspace{1.5cm} \\xi_k^{(1)} \\; \\text{stationary},\n\t\t\t\\end{array} \n\t\t\t\\right. \\label{cvbnmdhfdwadawdwayhryfryhfiajbcjzx}\n\t\t\\end{eqnarray}\n\t\twhere the operators ${ \\mathcal{Y} }$ and ${ \\mathcal{Z}_k }$, ${ k\\in\\{1,\\ldots,n\\} }$, are defined by\n\t\t\\begin{eqnarray*}\n\t\t\n\t\t\t\\mathcal{Y} {\\big[ f \\big]} & \\!\\! := \\!\\! & {\\left( {\\rm div}_{\\! y} + 2i\\pi \\theta^\\ast \\right)} {\\big\\{ A_{\\rm per} (y) {\\big( -[\\nabla_{\\!\\! y} Z](y,\\omega) \\nabla_{\\!\\! y} f + 2i\\pi \\theta^{(1)} f \\big)} \\big\\}} \\\\ [1pt]\n\t\t\t\t\t\t& & - \\, {\\rm div}_{\\! y} {\\big\\{ [\\nabla_{\\!\\! y} Z]^t(y,\\omega) A_{\\rm per} (y) {(\\nabla_{\\!\\! y} + 2i\\pi\\theta^\\ast)} f \\big\\}} \\\\ [1pt]\n\t\t\t\t\t\t& & + {\\left( 2i\\pi\\theta^{(1)} \\right)} {\\big\\{ A_{\\rm per}(y) {\\left( \\nabla_{\\!\\! y} + 2i\\pi\\theta^\\ast \\right)} f \\big\\}} \\\\ [1pt]\n\t\t\t\t\t\t& & + \\, {\\left( {\\rm div}_{\\! y} + 2i\\pi \\theta^\\ast \\right)}{\\big\\{ {\\left[ {\\rm div}_{\\! y} Z (y,\\omega) A_{\\rm per} (y) \\right]} {(\\nabla_{\\!\\! y} + 2i\\pi\\theta^\\ast)} f \\big\\}} + \\lambda^{(1)} f \\\\ [1pt]\n\t\t\t\t\t\t& & + \\, {\\big\\{ {\\rm div}_{\\! y} Z(y,\\omega) \\, {\\left[ \\lambda_{\\rm per}(\\theta^\\ast) - V_{\\rm per}(y) \\right]} \\big\\}} f, \\\\ [6.5pt]\n\t\t\t\\mathcal{Z}_k {\\big[ f \\big]} & \\!\\! := \\!\\! & {\\left( {\\rm div}_{\\! y} + 2i\\pi \\theta^\\ast \\right)} {\\big\\{ {\\left[ {\\rm div}_{\\! y} Z(y,\\omega) A_{\\rm per} (y) \\right]} {( e_k f )} \\big\\}} \\\\ [1pt]\n\t\t\t\t\t\t& & - \\, {\\rm div}_{\\! y} {\\big\\{ [\\nabla_{\\!\\! y} Z]^t(y,\\omega) A_{\\rm per} (y) {( e_k f )} \\big\\}} + {\\left( 2i\\pi\\theta^{(1)} \\right)} {\\left\\{ A_{\\rm per}(y) {( e_k f )} \\right\\}} \\\\ [1pt]\n\t\t\t\t\t\t& & + \\, {\\left( e_k \\right)} {\\big\\{ {\\big[ {\\rm div}_{\\! y} Z(y,\\omega) A_{\\rm per}(y) \\big]} {\\left( \\nabla_{\\!\\! y} + 2i\\pi\\theta^\\ast \\right)} f \\big\\}} \\\\ [1pt]\n\t\t\t\t\t\t& & - \\, {\\left( e_k \\right)} {\\left\\{ A_{\\rm per}(y) {\\left[ \\nabla_{\\!\\! y} Z \\right]}(y,\\omega) \\nabla_{\\!\\! y} f \\right\\}} + {\\left( e_k \\right)} {\\big\\{ A_{\\rm per}(y) {( 2i\\pi \\theta^{(1)} f )} \\big\\}},\n\t\t\\end{eqnarray*}\n\t\tfor ${ f \\in \\mathcal{H} }$.\n\t\\end{theorem}\nFor the proof of this theorem, we shall use essentially the structure of the spectral cell equation~\\eqref{92347828454trfhfd4rfghjls} and \n of the f.a.c. equation~\\eqref{8654873526rtgdrfdrfdrfrd4} with periodic coefficients accomplished by stochastic deformation of identity ${ \\Phi_\\eta }$ together with the identities~\\eqref{654367ytr6tfclmlml}.\n\\begin{proof}\n1. For begining, let us consider the set ${ \\mathcal{V} }$ as in \\eqref{563gdc}. Then, making change of variables in the spectral cell equation~\\eqref{92347828454trfhfd4rfghjls} adapted to this context, we find \n\t\t\\begin{eqnarray}\\label{8365287erdtfrewxzqzazaazzaaz}\n&&\\hspace{-0.5cm} \\int_{[0,1)^n} \\int_\\Omega \\Big\\{A_{\\rm per}(y) \\big( [\\nabla_{\\!\\! y} \\Phi_\\eta]^{-1} \\nabla_{\\!\\! y} \\psi_\\eta + 2i\\pi \\theta_\\eta \\psi_\\eta \\big) \\cdot \\overline{ \\big( [\\nabla_{\\!\\! y} \\Phi_\\eta]^{-1} \n\\nabla_{\\!\\! y} \\zeta + 2i\\pi \\theta_\\eta \\zeta \\big)}  \\, \\nonumber\\\\\n&& \\qquad\\qquad\\qquad+  {\\left( V_{\\rm per}(y) - \\lambda_\\eta \\right)} \\, \\psi_\\eta \\, \\overline{\\zeta} \\,  \\Big\\} {\\rm det} [\\nabla_{\\!\\! y} \\Phi_\\eta] \\, d\\mathbb{P}(\\omega) \\, dy = 0,\n\t\t\\end{eqnarray}\nfor all ${ \\eta \\in \\mathcal{V} }$ and ${ \\zeta \\in \\mathcal{H} }$. If we insert the equations~\\eqref{654367ytr6tfclmlml}, \\eqref{9789789794r6ttrtrtr}, \\eqref{6t8y365873586edtygc} and \\eqref{099uiuiyujhjchhtydfty} in equation~\\eqref{8365287erdtfrewxzqzazaazzaaz} and compute the term $\\eta$, we arrive at \n\n\\begin{equation*}\n\t\t\\begin{array}{l}\n\t\t\t\\displaystyle { \\int_{[0,1)^n} \\! \\int_\\Omega \\! \\Big\\{ A_{\\rm per}(y) {\\left( \\nabla_{\\!\\! y} \\psi^{(1)} + 2i\\pi \\theta^\\ast \\psi^{(1)} \\right)} \\cdot \\overline{{\\left( \\nabla_{\\!\\! y} \\zeta + 2i\\pi \\theta^\\ast \\zeta \\right)}}+{\\left( V_{\\rm per}(y) - \\lambda_{\\rm per}(\\theta^\\ast) \\right)} \\psi^{(1)} \\, \\overline{\\zeta}} \\\\ [15pt]\n\t\t\t\\displaystyle \\qquad\\qquad\\qquad+ \\, \\! A_{\\rm per}(y) {\\left( - [\\nabla_{\\!\\! y} Z] \\, \\nabla_{\\!\\! y} \\psi_{\\rm per}(\\theta^\\ast) + 2i\\pi\\theta^{(1)}\\psi_{\\rm per}(\\theta^\\ast) \\right)} \\! \\cdot \\! \\overline{{\\left( \\nabla_{\\!\\! y} \\zeta + 2i\\pi \\theta^\\ast \\zeta \\right)}}\\\\ [15pt]\n\t\t\t\\displaystyle\\qquad\\qquad\\qquad +  A_{\\rm per}(y) {\\left( \\nabla_{\\!\\! y} \\psi_{\\rm per}(\\theta^\\ast) + 2i\\pi \\theta^\\ast \\psi_{\\rm per}(\\theta^\\ast) \\right)} \\cdot \\overline{{\\left( -[\\nabla_{\\!\\! y} Z] \\nabla_{\\!\\! y} \\zeta + 2i\\pi\\theta^{(1)} \\zeta \\right)}} \\\\ [15pt]\n\t\t\t\\displaystyle\\qquad\\qquad\\qquad + A_{\\rm per}(y) {\\left( \\nabla_{\\!\\! y} \\psi_{\\rm per}(\\theta^\\ast) + 2i\\pi \\theta^\\ast \\psi_{\\rm per}(\\theta^\\ast) \\right)} \\cdot \\overline{{\\left( \\nabla_{\\!\\! y} \\zeta + 2i\\pi \\theta^\\ast \\zeta \\right)}} \\, {\\rm div}_{\\! y} Z \\\\ [15pt]\n\t\t\t\\displaystyle \\qquad\\qquad\\qquad- \\lambda^{(1)} \\, \\psi_{\\rm per}(\\theta^\\ast) \\, \\overline{\\zeta}+ {\\left( V_{\\rm per}(y) - \\lambda_{\\rm per}(\\theta^\\ast) \\right)} \\psi^{(0)} \\, \\overline{\\zeta} \\, {\\rm div}_{\\! y} Z\\Big\\}\\,d\\mathbb{P}(\\omega) \\, dy=0,\n\t\t\\end{array}\n\t\t\\end{equation*}\n\t\tfor all ${ \\eta \\in \\mathcal{V} }$ and ${ \\zeta \\in \\mathcal{H} }$. This equation is the variational formulation of the equation \\eqref{cvbnmdhfyhryfryhfiajbcjzx}, which concludes the first part of the proof. \t\n\n\\medskip\n2. For the second part of the proof, we proceed similarly with respect to the f.a.c. equation~\\eqref{8654873526rtgdrfdrfdrfrd4} and obtain\n\t\t\\begin{equation}\\label{8365wd}\n\t\t\\begin{split}\n\t\t\t& \\displaystyle \\int_{[0,1)^n} \\int_\\Omega\\Big\\{ A_{\\rm per}(y) \\big( [\\nabla_{\\!\\! y} \\Phi_\\eta]^{-1} \\nabla_{\\!\\! y} \\xi_{k,\\eta} + 2i\\pi \\theta_\\eta \\xi_{k,\\eta} \\big) \\cdot \\overline{\\big( [\\nabla_{\\!\\! y} \\Phi_\\eta]^{-1} \\nabla_{\\!\\! y} \\zeta + 2i\\pi \\theta_\\eta \\zeta \\big)} \\\\\n\t\t\t& \\displaystyle\\qquad\\qquad\\qquad + A_{\\rm per}(y) {\\left( e_k \\, \\psi_\\eta \\right)} \\cdot \\overline{\\big( [\\nabla_{\\!\\! y} \\Phi_\\eta]^{-1} \\nabla_{\\!\\! y} \\zeta + 2i\\pi \\theta_\\eta \\zeta \\big)} \\\\\n\t\t\t& \\displaystyle\\qquad\\qquad\\qquad - A_{\\rm per}(y) \\big( [\\nabla_{\\!\\! y} \\Phi_\\eta]^{-1} \\nabla_{\\!\\! y} \\psi_\\eta + 2i\\pi \\theta_\\eta \\psi_\\eta \\big) \\cdot \\overline{{\\left( e_k \\, \\zeta \\right)}} \\\\\n\t\t\t& \\displaystyle\\qquad\\quad + {\\left( V_{\\rm per}(y) - \\lambda_\\eta \\right)} \\, \\xi_{k,\\eta} \\, \\overline{\\zeta} - \\, \\frac{1}{2i\\pi} \\frac{\\partial \\lambda}{\\partial \\theta_k}(\\eta,\\theta(\\eta))\\,\n\t\t\t\\psi_\\eta \\, \\overline{\\zeta}\\Big\\}\\,{\\rm det} [\\nabla_{\\!\\! y} \\Phi_\\eta] \\, d\\mathbb{P}(\\omega) \\, dy = 0,\n\t\t\\end{split}\n\t\t\\end{equation}\n\t\tfor all ${ \\eta \\in \\mathcal{V} }$, ${ \\zeta \\in \\mathcal{H} }$ and ${ k \\in \\{1,\\ldots,n\\} }$. Hence, taking into account the Lemma \\ref{6487369847639gfhdghjdftrtrtfgcbvbv} and \ninserting the equations \\eqref{654367ytr6tfclmlml}, \\eqref{9789789794r6ttrtrtr}, \\eqref{6t8y365873586edtygc}, \\eqref{099uiuiyujhjchhtydfty} and \\eqref{67tyuxcvbdfgoikjjhbhb} \nin equation~\\eqref{8365wd}, a computation of the term of order ${ \\eta }$ lead us to\n\n\\begin{equation*}\n\t\t\\begin{array}{l} \n\t\t\t\\displaystyle\\hspace{-0.5cm} \\int_{[0,1)^n} \\int_\\Omega \\Big\\{A_{\\rm per}(y) \\big( \\nabla_{\\!\\! y}\\xi_k^{(1)} + 2i\\pi\\theta^\\ast \\xi_k^{(1)} \\big) \\cdot \\overline{ \\big( \\nabla_{\\!\\! y} \\zeta + 2i\\pi \\theta^\\ast \\zeta \\big)}+ {\\left( V_{\\rm per}(y) - \\lambda_{\\rm per}(\\theta^\\ast) \\right)} \\xi_k^{(1)} \\, \\overline{\\zeta} \\\\ [15pt]\n\t\t\t\\displaystyle\\qquad\\qquad + A_{\\rm per}(y) {\\left( e_k \\, \\psi^{(1)} \\right)} \\cdot \\overline{ \\big( \\nabla_{\\!\\! y} \\zeta + 2i\\pi \\theta^\\ast \\zeta \\big)} -\n\tA_{\\rm per}(y) \\big( \\nabla_{\\!\\! y} \\psi^{(1)} + 2i\\pi \\theta^\\ast \\psi^{(1)} \\big) \\cdot \\overline{{\\left( e_k \\, \\zeta \\right)}}\\\\ [15pt]\n\t\t\t\t\\displaystyle\\qquad\\qquad + A_{\\rm per}(y) \\big( - [\\nabla_{\\!\\! y} Z] \\nabla_{\\!\\! y} \\xi_{k,{\\rm per}}(\\theta^\\ast) + 2i\\pi\\theta^{(1)} \\xi_{k,{\\rm per}}(\\theta^\\ast) \\big) \\cdot \\overline{ \\big( \\nabla_{\\!\\! y} \\zeta + 2i\\pi \\theta^\\ast \\zeta \\big)} \\\\ [15pt]\n\t\t\t\\displaystyle\\qquad\\qquad + A_{\\rm per}(y) \\big( \\nabla_{\\!\\! y} \\xi_{k,{\\rm per}}(\\theta^\\ast) + 2i\\pi \\theta^\\ast \\xi_{k,{\\rm per}}(\\theta^\\ast) \\big) \\cdot \\overline{ \\big( -[\\nabla_{\\!\\! y} Z] \\nabla_{\\!\\! y} \\zeta + 2i\\pi\\theta^{(1)} \\zeta \\big)}  \\\\ [15pt]\n\t\t\t\\displaystyle\\qquad\\qquad +  A_{\\rm per}(y) \\big( \\nabla_{\\!\\! y} \\xi_{k,{\\rm per}}(\\theta^\\ast) + 2i\\pi \\theta^\\ast \\xi_{k,{\\rm per}}(\\theta^\\ast) \\big) \\cdot \\overline{\\big( \\nabla_{\\!\\! y} \\zeta + 2i\\pi \\theta^\\ast \\zeta \\big)} \\, {\\rm div}_{\\! y} Z \\\\ [15pt]\n\t\t\t\\displaystyle\\qquad\\qquad\\qquad\\qquad - \\lambda^{(1)} \\, \\xi_{k,{\\rm per}}(\\theta^\\ast) \\, \\overline{\\zeta} + {\\left( V_{\\rm per}(y) - \\lambda_{\\rm per}(\\theta^\\ast) \\right)} \\xi_{k,{\\rm per}}(\\theta^\\ast) \\, \\overline{\\zeta} \\, {\\rm div}_{\\! y} Z\\\\ [15pt]\n\t\t\t\\displaystyle\\qquad\\qquad\\qquad +\\, A_{\\rm per}(y) {\\left( e_k \\, \\psi_{\\rm per}(\\theta^\\ast) \\right)} \\cdot \\Big(\\overline{ \\big( \\nabla_{\\!\\! y} \\zeta + 2i\\pi \\theta^\\ast \\zeta \\big) \\, {\\rm div}_{\\! y} Z -[\\nabla_{\\!\\! y} Z] \\nabla_{\\!\\! y} \\zeta + 2i\\pi\\theta^{(1)} \\zeta}\\Big) \\\\ [15pt]\n\t\t\t\\displaystyle\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad -  A_{\\rm per}(y) \\big( \\nabla_{\\!\\! y} \\psi_{\\rm per}(\\theta^\\ast) + 2i\\pi \\theta^\\ast \\psi_{\\rm per}(\\theta^\\ast) \\big) \\cdot \\overline{{\\left( e_k \\, \\zeta \\right)}} \\, {\\rm div}_{\\! y} Z  \\\\ [15pt]\n\t\t\t\\displaystyle\\qquad\\qquad -\\, A_{\\rm per}(y) \\big( - [\\nabla_{\\!\\! y} Z] \\, \\nabla_{\\!\\! y} \\psi_{\\rm per}(\\theta^\\ast) + 2i\\pi\\theta^{(1)} \\psi_{\\rm per}(\\theta^\\ast) \\big) \\cdot \\overline{{\\left( e_k \\, \\zeta \\right)}}\\Big\\} \\, d\\mathbb{P}(\\omega) \\, dy =0,\n\t\t\\end{array}\n\t\t\\end{equation*}\n\t\tfor all ${ \\zeta \\in \\mathcal{H} }$. Noting that this is the variational formulation of the equation \\eqref{cvbnmdhfdwadawdwayhryfryhfiajbcjzx}, we conclude the proof. \n\n\\end{proof}\n\nWe remember the reader that if $f:\\mathbb R^n\\times\\Omega\\to\\mathbb R$ is a stationary function, then we shall use the following notation \n$$\n    \\mathbb{E}[f(x,\\cdot)]= \\int_\\Omega f(x,\\omega) \\,d\\mathbb{P}(\\omega),\n$$\nfor any $x\\in\\mathbb R^n$. Roughly speaking, the theorem below tell us that the homogenized matrix of the problem~\\eqref{765tdyyuty67tsss} can be obtained by solving periodic problems. \n\n\\begin{theorem}\n\\label{873627yuhfdd}\n\t\tLet ${ A_\\eta^\\ast }$ be the homogenized matrix as in \\eqref{askdjfhucomojkfdfd}. Then\n\t\t\\begin{equation*}\n\t\t\tA_\\eta^\\ast = A_{\\rm per}^\\ast + \\eta A^{(1)} + \\mathrm{O}(\\eta^2).\n\t\t\\end{equation*}\nMoreover, the term of order ${ \\eta^0 }$ is given by the homogenized matrix of the periodic case, that is, ${ A_{\\rm per}^\\ast = 2^{-1}{\\left( B^{(0)} + (B^{(0)})^t \\right)} }$, where the matrix ${ B^{(0)} }$ is the term of order ${ \\eta^0 }$ in \\eqref{678435} and it is defined b\n\t\t\\begin{equation*}\n\t\t\\begin{array}{r}\n\t\t\t\\displaystyle (B^{(0)})_{k\\ell} := \\int_{[0,1)^n} A_{\\rm per}(y) {(e_\\ell \\, \\psi_{\\rm per}(\\theta^\\ast))} \\cdot {(e_k \\, \\overline{\\psi_{\\rm per}(\\theta^\\ast)})} \\, dy \\hspace{4cm} \\\\ [10pt]\n\t\t\t\\displaystyle + \\, \\int_{[0,1)^n} A_{\\rm per}(y) {(e_\\ell \\, \\psi_{\\rm per}(\\theta^\\ast))} \\cdot \\overline{\\left( \\nabla_{\\!\\! y} \\xi_{k,{\\rm per}}(\\theta^\\ast) + 2i\\pi \\theta^\\ast \\xi_{k,{\\rm per}}(\\theta^\\ast) \\right)} \\, dy \\\\ [10pt]\n\t\t\t\\displaystyle - \\, \\int_{[0,1)^n} A_{\\rm per}(y) {\\Big( {\\left( \\nabla_{\\!\\! y} \\psi_{\\rm per}(\\theta^\\ast) + 2i\\pi \\theta^\\ast \\psi_{\\rm per}(\\theta^\\ast) \\right)} \\Big)} \\cdot \\overline{{(e_\\ell \\, \\xi_{k,{\\rm per}}(\\theta^\\ast))}} \\, dy .\n\t\t\\end{array}    \n\t\t\\end{equation*}\n\t\tThe term of order ${ \\eta }$ is given by ${ A^{(1)} = 2^{-1} {\\left( B^{(1)}+(B^{(1)})^t \\right)} }$, where the matrix ${ B^{(1)} }$ is the term of order ${ \\eta }$ in \\eqref{678435} and it is defined b\n\t\t\\begin{equation*}\n\t\t\\begin{array}{l}\n\t\t\t\\displaystyle (B^{(1)})_{k\\ell} = { {\\bigg[ \\int_{[0,1)^n} A_{\\rm per}(y) {(e_\\ell \\, \\psi_{\\rm per}(\\theta^\\ast))} \\cdot {(e_k \\, \\overline{\\psi_{\\rm per}(\\theta^\\ast)})} \\, \\mathbb{E}\\Big[{\\rm div}_{\\! y} Z(y,\\cdot)\\Big] \\, dy }} \\\\ [11pt]\n\t\t\t\\displaystyle + \\, \\int_{[0,1)^n} A_{\\rm per}(y) {(e_\\ell \\, \\psi_{\\rm per}(\\theta^\\ast))} \\cdot {( e_k \\, \\overline{ \\mathbb{E}\\big[ \\psi^{(1)}(y,\\cdot)\\big]} )} \\, dy \\\\ [11pt]\n\t\t\t\\displaystyle + \\int_{[0,1)^n} A_{\\rm per}(y) {\\left( e_\\ell \\, {\\mathbb{E}\\big[ \\psi^{(1)}(y,\\cdot)\\big]} \\right)} \\cdot {(e_k \\, \\overline{\\psi_{\\rm per}(\\theta^\\ast)})} \\, dy \\\\ [11pt]\n\t\t\t\\displaystyle + \\int_{[0,1)^n} A_{\\rm per}(y) {(e_\\ell \\, \\psi_{\\rm per}(\\theta^\\ast))} \\cdot \n\t\t\t\\overline{\\left( \\nabla_{\\!\\! y} \\xi_{k,{\\rm per}}(\\theta^\\ast) + 2i\\pi \\theta^\\ast \\xi_{k,{\\rm per}}(\\theta^\\ast) \\right)} \\, {\\mathbb{E}\\big[{\\rm div}_{\\! y} Z(y,\\cdot)\\big]} \\, dy \\\\ [11pt]\n\t\t\t\\displaystyle + \\int_{[0,1)^n} A_{\\rm per}(y) {(e_\\ell \\, \\psi_{\\rm per}(\\theta^\\ast))} \\cdot \\overline{( \\nabla_{\\!\\! y} {\\mathbb{E}\\big[ \\xi_k^{(1)}(y,\\cdot)\n\t\t\t\\big]} + 2i\\pi\\theta^\\ast {\\mathbb{E}\\Big[ \\xi_k^{(1)}(y,\\cdot)\\Big]}} \\\\ [11pt]\n\t\t\t\\hspace{4cm} \\overline{ + \\, 2i\\pi\\theta^{(1)} \\xi_{k,{\\rm per}}(\\theta^\\ast)-{\\mathbb{E}\\Big[[\\nabla_{\\!\\! y} Z](y,\\cdot)\\Big]} \\nabla_{\\!\\! y} \\xi_{k,{\\rm per}}(\\theta^\\ast)}) \\, dy \\\\ [11pt]\n\t\t\t\\displaystyle + \\int_{[0,1)^n} \\!\\!\\! A_{\\rm per}(y) {\\left( e_\\ell \\, {\\mathbb{E}\\Big[ \\psi^{(1)}(y,\\cdot)\\Big]} \\right)} \\cdot \\overline{\\left( \\nabla_{\\!\\! y} \\xi_{k,{\\rm per}}(\\theta^\\ast) + 2i\\pi \\theta^\\ast \\xi_{k,{\\rm per}}(\\theta^\\ast) \\right)} \\, dy \\\\ [11pt]\n\t\t\t\\displaystyle - \\int_{[0,1)^n} \\!\\!\\! A_{\\rm per}(y) {( {\\left( \\nabla_{\\!\\! y} \\psi_{\\rm per}(\\theta^\\ast) + 2i\\pi \\theta^\\ast \\psi_{\\rm per}(\\theta^\\ast) \\right)})} \\cdot \\overline{{(e_\\ell \\, \\xi_{k,{\\rm per}}(\\theta^\\ast))}} \\, {\\mathbb{E}\\big[{\\rm div}_{\\! y} Z(y,\\cdot)\\big]} \\, dy \\\\ [11pt]\n\t\t\t\\displaystyle - \\int_{[0,1)^n} \\!\\!\\! A_{\\rm per}(y) {\\Big( {\\left( \\nabla_{\\!\\! y} \\psi_{\\rm per}(\\theta^\\ast) + 2i\\pi \\theta^\\ast \\psi_{\\rm per}(\\theta^\\ast) \\right)} \\Big)} \\cdot \\overline{{\\Big(e_\\ell \\, {\\mathbb{E}\\Big[ \\xi_k^{(1)}(y,\\cdot)\\Big]} \\Big)}} \\, dy\n\t\t\\end{array}\n\t\t\\end{equation*}\n\t\t\\begin{equation*}\n\t\t\\begin{array}{l}\n\t\t\t\\displaystyle \\hspace{0.75cm} - \\int_{[0,1)^n} A_{\\rm per}(y) {\\left( \\nabla_{\\!\\! y} {\\mathbb{E}\\Big[ \\psi^{(1)}(y,\\cdot)\\Big]} + \n\t\t\t2i\\pi\\theta^\\ast {\\mathbb{E}\\Big[ \\psi^{(1)}(y,\\cdot)\\Big]}\\right.} \\\\ [10pt]\n\t\t\t\\hspace{2cm} {{\\left. + \\, 2i\\pi\\theta^{(1)} \\psi_{\\rm per}(\\theta^\\ast) -{\\mathbb{E}\\Big[[\\nabla_{\\!\\! y} Z](y,\\cdot)\\Big]} \\nabla_{\\!\\! y} \\psi_{\\rm per}(\\theta^\\ast)\\right)} \\cdot \\overline{{(e_\\ell \\, \\xi_{k,{\\rm per}}(\\theta^\\ast))}} \\, dy \\bigg]} \\\\ [11pt]\n\t\t\t\\displaystyle - {\\bigg[ \\int_{[0,1)^n} {\\vert \\psi_{\\rm per}(\\theta^\\ast) \\vert}^2 {\\mathbb{E}\\Big[ {\\rm div}_{\\! y} Z(y,\\cdot)\\Big]} \\, dy + \\int_{[0,1)^n} \\psi_{\\rm per}(\\theta^\\ast) \\, \\overline{ \\mathbb{E}\\Big[\\psi^{(1)}(y,\\cdot) \\Big]} } \\, dy  \\\\ [11pt]\n\t\t\t\\displaystyle \\hspace{0.75cm} { + \\int_{[0,1)^n} {\\mathbb{E}\\Big[\\psi^{(1)}(y,\\cdot) \\Big]} \\, \\overline{\\psi_{\\rm per}(\\theta^\\ast)} \\, dy \\bigg]} \\cdot {\\bigg[ \\int_{[0,1)^n}\n\t\t\tA_{\\rm per}(y) {(e_\\ell \\, \\psi_{\\rm per}(\\theta^\\ast))} \\cdot {(e_k \\, \\overline{\\psi_{\\rm per}(\\theta^\\ast)})} \\, dy } \\\\ [11pt]\n\t\t\t\\displaystyle \\hspace{0.75cm} + \\int_{[0,1)^n} A_{\\rm per}(y) {(e_\\ell \\, \\psi_{\\rm per}(\\theta^\\ast))} \\cdot \\overline{\\left( \\nabla_{\\!\\! y} \\xi_{k,{\\rm per}}(\\theta^\\ast) + 2i\\pi \\theta^\\ast \\xi_{k,{\\rm per}}(\\theta^\\ast) \\right)} \\, dy \\\\ [11pt]\n\t\t\t\\displaystyle \\hspace{0.75cm} - \\, {{ \\int_{[0,1)^n} A_{\\rm per}(y) {\\left[ {\\left( \\nabla_{\\!\\! y} \\psi_{\\rm per}(\\theta^\\ast) + 2i\\pi \\theta^\\ast \\psi_{\\rm per}(\\theta^\\ast) \\right)} \\right]} \\cdot \\overline{{(e_\\ell \\, \\xi_{k,{\\rm per}}(\\theta^\\ast))}} \\, dy \\bigg]}}.\n\t\t\\end{array}\n\t\t\\end{equation*}\n\t\\end{theorem}\n\\begin{proof}\n1.  Taking into account ${ \\mathcal{V} }$ as in \\eqref{563gdc}, we get from~\\eqref{786587tdyghs7rsdfxsdfsdf}, for ${ \\eta \\in \\mathcal{V} }$, that the homogenized matrix \nis given by ${ A_\\eta^\\ast = 2^{-1}(B_\\eta + B_\\eta^t) }$. Thus, in order to describe the terms of the expansion of ${ A_\\eta^\\ast }$, we only need to determine the terms in the expansion of ${ B_\\eta }$. \n\n2. Using the equations~\\eqref{654367ytr6tfclmlml} and \\eqref{099uiuiyujhjchhtydfty}, the map ${ \\eta \\mapsto c_\\eta \\in (0,+\\infty) }$ has an expansion about ${ \\eta=0 }$. Remembering \nthat ${ \\int_{[0,1)^n} {\\vert \\psi_{\\rm per}(\\theta^\\ast) \\vert}^2 dy = 1 }$, we have\n\n\\begin{equation*}\\label{47647342rfedrdrefrdfer} \n\t\t\\begin{split}\n\t\t\tc_\\eta^{-1} = & 1 - \\eta {\\left[ \\int_\\Omega \\int_{[0,1)^n} {\\vert \\psi_{\\rm per}(\\theta^\\ast) \\vert}^2 {\\rm div}_{\\! y} Z(y,\\omega) \\, dy \\, d\\mathbb{P} \\right.} \\\\\n\t\t\t& {\\left. + \\int_\\Omega \\int_{{[0,1)^n}} \\psi_{\\rm per}(\\theta^\\ast) \\overline{\\psi^{(1)}} \\, dy \\, d\\mathbb{P} + \\int_\\Omega \\int_{[0,1)^n} \\psi^{(1)} \\overline{\\psi_{\\rm per}(\\theta^\\ast)} \\, dy \\, d\\mathbb{P} \\right]} + \\; \\mathrm{O}(\\eta^2),\n\t\t\\end{split}\n\t\t\\end{equation*}\nin ${ \\mathbb{C} }$ as ${ \\eta \\to 0 }$. Thus, using the expansions \\eqref{654367ytr6tfclmlml}, \\eqref{9789789794r6ttrtrtr}, \\eqref{099uiuiyujhjchhtydfty} and \\eqref{67tyuxcvbdfgoikjjhbhb} in the formula \\eqref{786587tdyghs7rsdfxsdfsdf}, the computation of the resulting term of order ${ \\eta^0 }$ of ${ B_\\eta }$ give us the desired expression for $(B^{(0)})_{k,\\ell}$.  \nThe same reasoning with a little more computations, which is an exercise that we leave to the reader, allow us to obtain the expression for $(B^{(1)})_{k\\ell}$. \n\\end{proof}\n\n\\medskip\n\n\n\n\\begin{remark}\n\t\tWe next record the observation that the computation of the coefficients of ${ A_{\\rm per}^\\ast }$ is performed by solving the equations \\eqref{yhujtgvjnjnhnvfnvfshjbn} and \\eqref{yhfjsgfsfsdyhujtgvjnjnhnvfnvfshjbn}, which are equations with periodic boundary conditions. In order to compute the coefficients of ${ A^{(1)} }$, we need to know the functions \n${ \\psi^{(1)} }$ and ${ \\xi_k^{(1)} }$, ${ k\\in\\{1,\\ldots,n\\} }$, which are a priori stochastic in nature (see the equations~\\eqref{cvbnmdhfyhryfryhfiajbcjzx} and \\eqref{cvbnmdhfdwadawdwayhryfryhfiajbcjzx}, respectively). But in Theorem \\ref{873627yuhfdd}, it has seen that we only need their expectation values, \n${ \\mathbb{E}\\Big[ \\psi^{(1)}(y,\\cdot)\\Big] }$ and ${ \\mathbb{E}\\Big[ \\xi_k^{(1)}(y,\\cdot) \\Big] }$, ${ k \\in \\{1,\\ldots,n\\} }$, which are ${ [0,1)^n }$-periodic functions and, respectively, solutions of the following equations:\n\t\n\t\t\\begin{eqnarray*}\n\t\t\t& & \\left\\{\n\t\t\t\\begin{array}{l}\n\t\t\t\t{\\Big( L_{\\rm per}(\\theta^\\ast) - \\lambda_{\\rm per}(\\theta^\\ast) \\Big)}\\, {\\mathbb{E}\\Big[ \\psi^{(1)}(y,\\cdot)\\Big]} = \\mathcal{Y}_{\\rm per} {\\big[ \\psi_{\\rm per}(\\theta^\\ast) \\big]} \\;\\, \\text{in} \\;\\, [0,1)^n, \\\\ [6pt]\n\t\t\t\t\\hspace{2cm} {\\mathbb{E}\\Big[ \\psi^{(1)}(y,\\cdot)\\Big]} \\;\\text{is $[0,1)^n$-periodic},\n\t\t\t\\end{array}\n\t\t\t\\right. \\\\ [7.5pt]\n\t\t\t& & \\left\\{\n\t\t\t\\begin{array}{l}   \n\t\t\t\t{\\left( L_{\\rm per}(\\theta^\\ast) - \\lambda_{\\rm per}(\\theta^\\ast) \\right)}{\\mathbb{E}\\Big[ \\xi_k^{(1)}(y,\\cdot) \\Big]} = \n\t\t\t\t\\mathcal{X}\\Big[{\\mathbb{E}\\Big[ \\psi^{(1)}(y,\\cdot)\\Big]}\\Big] \\\\ [6pt]\n\t\t\t\t\\hspace{1.75cm} + \\, \\mathcal{Y}_{\\rm per} {\\big[ \\xi_{k,{\\rm per}}(\\theta^\\ast) \\big]} + \\mathcal{Z}_{k,{\\rm per}} {\\big[ \\psi_{\\rm per}(\\theta^\\ast) \\big]} \\;\\, \\text{in} \\;\\, \n\t\t\t\t[0,1)^n, \\\\ [6pt]\n\t\t\t\t\\hspace{2cm} {\\mathbb{E}\\Big[ \\xi_k^{(1)}(y,\\cdot) \\Big]} \\;\\text{is $[0,1)^n$-periodic},\n\t\t\t\\end{array}\n\t\t\t\\right.\n\t\t\\end{eqnarray*}\n\t\twhere \n\t\t\\begin{eqnarray*}\n\t\t\t\\mathcal{Y}_{\\rm per} {\\big[ f \\big]} & \\!\\! := \\!\\! & {\\left( {\\rm div}_{\\! y} + 2i\\pi \\theta^\\ast \\right)} {\\Big\\{ A_{\\rm per} (y)\n\t\t\t {\\big( - \\mathbb{E}\\Big[[\\nabla_{\\!\\! y} Z](y,\\cdot)\\Big] \\nabla_{\\!\\! y} f + 2i\\pi \\theta^{(1)} f \\big)} \\Big\\}} \\\\ [1pt]\n\t\t\t\t\t\t& & - \\, {\\rm div}_{\\! y} {\\Big\\{ \\mathbb{E}\\Big[[\\nabla_{\\!\\! y} Z](y,\\cdot)\\Big]^t A_{\\rm per} (y) {(\\nabla_{\\!\\! y} + 2i\\pi\\theta^\\ast)} f \\Big\\}} \\\\ [1pt]\n\t\t\t\t\t\t& & + {\\left( 2i\\pi\\theta^{(1)} \\right)} {\\big\\{ A_{\\rm per}(y) {\\left( \\nabla_{\\!\\! y} + 2i\\pi\\theta^\\ast \\right)} f \\big\\}} \\\\ [1pt]\n\t\t\t\t\t\t& & + \\, {\\left( {\\rm div}_{\\! y} + 2i\\pi \\theta^\\ast \\right)}{\\Big\\{ {\\left[ \\mathbb{E}\\Big[{\\rm div}_{\\! y} Z(y,\\cdot)\\Big] A_{\\rm per} (y) \\right]} {(\\nabla_{\\!\\! y} + 2i\\pi\\theta^\\ast)} f \\Big\\}} + \\lambda^{(1)} f \\\\ [1pt]\n\t\t\t\t\t\t& & + \\, {\\Big\\{ \\mathbb{E}\\Big[{\\rm div}_{\\! y} Z(y,\\cdot)\\Big] \\, {\\left[ \\lambda_{\\rm per}(\\theta^\\ast) - V_{\\rm per}(y) \\right]} \\Big\\}} f, \n\t\t\\end{eqnarray*}\n\t\t\\begin{eqnarray*}\n\t\t\t\\mathcal{Z}_{k,{\\rm per}} {\\big[ f \\big]} & \\!\\! := \\!\\! & {\\left( {\\rm div}_{\\! y} + 2i\\pi \\theta^\\ast \\right)} {\\Big\\{ {\\left[ \\mathbb{E}\\Big[{\\rm div}_{\\! y} Z(y,\\cdot)\\Big] A_{\\rm per} (y) \\right]} {( e_k f )} \\Big\\}} \\\\ [1pt]\n\t\t\t\t\t\t& & - \\, {\\rm div}_{\\! y} {\\Big\\{ \\mathbb{E}\\Big[[\\nabla_{\\!\\! y} Z](y,\\cdot)\\Big]^t A_{\\rm per} (y) {( e_k f )} \\Big\\}} + {\\left( 2i\\pi\\theta^{(1)} \\right)} {\\left\\{ A_{\\rm per}(y) {( e_k f )} \\right\\}} \\\\ [1pt]\n\t\t\t\t\t\t& & + \\, {\\left( e_k \\right)} {\\Big\\{ {\\Big[ \\mathbb{E}\\Big[{\\rm div}_{\\! y} Z(y,\\cdot)\\Big] A_{\\rm per}(y) \\Big]} {\\left( \\nabla_{\\!\\! y} + 2i\\pi\\theta^\\ast \\right)} f \n\t\t\t\t\t\t\\Big\\}} \\\\ [1pt]\n\t\t\t\t\t\t& & - \\, {\\left( e_k \\right)} {\\left\\{ A_{\\rm per}(y) \\mathbb{E}\\Big[[\\nabla_{\\!\\! y} Z](y,\\cdot)\\Big] \\nabla_{\\!\\! y} f \\right\\}} + {\\left( e_k \\right)} {\\big\\{ A_{\\rm per}(y) {( 2i\\pi \\theta^{(1)} f )} \\big\\}}.\n\t\t\\end{eqnarray*}\n\t\tfor ${ f\\in H^1_{\\rm per}([0,1)^n) }$.\n\t\n\t\\end{remark}\nSumming up, the determination of the homogenized coefficients for~\\eqref{jhjkhkjhkj765675233} is a stochastic problem in nature.  However, when we consider the \ninteresting context of materials which have small deviation from perfect ones (modeled by periodic functions), this problem, in the specific case~\\eqref{37285gdhddddddddddd} \nreduces, at the first two orders in $\\eta$, to the simpler solution to the two periodic problems above. Both of them are of the same nature. Importantly, note that \n$Z$ in~\\eqref{37285gdhddddddddddd} is only present through $\\mathbb{E}\\Big[{\\rm div}_{\\! y} Z(y,\\cdot)\\Big]$ and $\\mathbb{E}\\Big[[\\nabla_{\\!\\! y} Z](y,\\cdot)\\Big]$.\n\n\\medskip\nIn the theorem below, we assume that the homogenized matrix of the periodic case satisfies the uniform coercive condition, that is, \n$$\n A_{\\rm per}^\\ast \\xi \\cdot \\xi\\ge \\Lambda |\\xi|^2,\n$$\nfor some $\\Lambda>0$ and for all $\\xi\\in\\mathbb R^n$, which has experimental evidence for metals and semiconductors. \nTherefore, due to Theorem~\\ref{873627yuhfdd} the homogenized matrix of the perturbed case ${ A_\\eta^\\ast }$ has similar \nproperty for $\\eta\\sim 0$. \n\t\n\\begin{theorem}\n\\label{THM511}\nLet ${ v_\\eta }$ be the solution of homogenized equation \\eqref{askdjfhucomojkfdfd}. Then\n\t\t\\begin{equation*}\n\t\t\tv_\\eta\\Big(t,\\sqrt{A_\\eta^\\ast}\\,x\\Big) = v_{\\rm per}\\Big(t,\\sqrt{A_{\\rm per}^\\ast}\\,x\\Big) + \\eta\\, v^{(1)}\\Big(t,\\sqrt{A_{\\rm per}^\\ast}\\,x\\Big) + \\mathrm{O}(\\eta^2),\n\t\t\\end{equation*}\nweakly in ${ L^2(\\mathbb{R}^n_T) }$ as ${ \\eta \\to 0 }$, that means,\n\t\t\\begin{eqnarray*}\n\t\t\t&&  \\int_{\\mathbb{R}^n_T} {\\Bigg( v_\\eta\\Big(t,\\sqrt{A_\\eta^\\ast}\\,x\\Big)-v_{\\rm per}\\Big(t,\\sqrt{A_{\\rm per}^\\ast}\\,x\\Big) \n-\\eta\\, v^{(1)}\\Big(t,\\sqrt{A_{\\rm per}^\\ast}\\,x\\Big) \\Bigg)} \\, h(t,x) \\, dx \\, dt\\\\\n&&\\qquad\\qquad\\qquad= \\mathrm{O}(\\eta^2),\n\t\t\\end{eqnarray*}\nfor each ${ h \\in L^2(\\mathbb{R}^n_T) }$, where ${ v_{\\rm per} }$ is the solution of the periodic homogenized problem  \n\t\t\\begin{equation}\\label{987tr7tef76756g7rg5467g546r7g5}\n\t\t\t\\left\\{\n\t\t\t\\begin{array}{c}\n\t\t\t\ti \\displaystyle\\frac{\\partial v_{\\rm per}}{\\partial t} - {\\rm div} {\\left( A_{\\rm per}^\\ast \\nabla v_{\\rm per} \\right)} + U_{\\! \\rm per}^\\ast v_{\\rm per} = 0 , \\;\\, \\text{in} \\;\\, \\mathbb{R}^{n+1}_T, \\\\ [7,5pt]\n\t\t\t\tv_{\\rm per}(0,x) = v_0(x) \\, , \\;\\, x\\in \\mathbb{R}^n,\n\t\t\t\\end{array}\n\t\t\t\\right.\n\t\t\\end{equation}\n\t\tand ${ v^{(1)} }$ is the solution of\n\t\t\\begin{equation}\\label{73547tr764tr63tr4387tr8743tr463847}\n\t\t\t\\left\\{\n\t\t\t\\begin{array}{c}\n\t\t\t\ti \\displaystyle\\frac{\\partial v^{(1)}}{\\partial t} - {\\rm div} {\\left( A_{\\rm per}^\\ast \\nabla v^{(1)} \\right)} + U_{\\! \\rm per}^\\ast v^{(1)} = {\\rm div} {\\left( A_{\\rm per}^\\ast \\nabla v_{\\rm per} \\right)} - U^{(1)}v_{\\rm per}, \\;\\, \\text{in} \\;\\, \\mathbb{R}^{n+1}_T, \\\\ [7,5pt]\n\t\t\t\tv^{(1)}(0,x) = v^{1}_0(x) \\, , \\;\\, x\\in \\mathbb{R}^n,\n\t\t\t\\end{array}\n\t\t\t\\right.\n\t\t\\end{equation}\n\t\twhere ${ U^{(1)} }$ is the coefficient of the term of order ${ \\eta }$ of the expansion ${ U_\\eta^\\ast }$ and $v_0^{1}\\in C_c^{\\infty}(\\mathbb R^n)$ is given by the limit\n$$\nv_0^1\\Big(\\sqrt{A_{\\rm per}^\\ast}\\,x\\Big):= \\lim_{\\eta\\to 0}\\frac{v_0\\Big(\\sqrt{A_\\eta^\\ast}\\,x\\Big)-v_0\\Big(\\sqrt{A_{\\rm per}^\\ast}\\,x\\Big)}{\\eta}.\n$$\n\\end{theorem}\n\n\\begin{proof}\n1. Taking into account the set ${ \\mathcal{V} }$ as in \\eqref{563gdc}, we have for ${ \\eta \\in \\mathcal{V} }$ and from the conservation of energy of the homogenized Schr\\\"odinger equation \\eqref{askdjfhucomojkfdfd}, that the solution ${ v_\\eta : \\mathbb{R}^n_T \\to \\mathbb{C} }$ satisfies\n\n\t\t\\begin{equation*}\n\t\t\t{\\Vert v_\\eta \\Vert}_{L^2(\\mathbb{R}^{n+1}_T)} = T {\\Vert v_0 \\Vert}_{L^2(\\mathbb{R}^n)}, \\;\\, \\forall \\eta \\in \\mathcal{V}.\n\t\t\\end{equation*}\nThus, after possible extraction of a subsequence, we have the existence of a function ${ v^{(0)} \\in L^2(\\mathbb{R}^{n+1}_T) }$ such that \n\t\t\\begin{equation}\\label{tvdfvcfvfvfvfcdcdcdcdc}\n\t\t\tv_{\\eta} \\; \\xrightharpoonup[\\eta \\to 0]{} \\; v^{(0)} \\; \\text{em} \\; L^2(\\mathbb{R}^{n+1}_T).\n\t\t\\end{equation}\n\nBy the variational formulation of the equation \\eqref{askdjfhucomojkfdfd}, we find\n\t\t\\begin{equation}\\label{yhnygbtgbrvftgbdfervd}\n\t\t\\begin{array}{l}\n\t\t\t0 = \\displaystyle i \\int_{\\mathbb{R}^n} v_0(x) \\, \\overline{\\varphi}(0,x) \\, dx - i  \\int_{\\mathbb{R}^n_T} v_\\eta(t,x) \\, \\frac{\\partial \\overline{\\varphi}}{\\partial t} (t,x) \\, dx \\, dt \\\\ [15pt]\n\t\t\t\\displaystyle \\qquad\\qquad+ \\int_{\\mathbb{R}^n_T}\\Bigg\\{- {\\left\\langle A_\\eta^\\ast v_\\eta(t,x), D^2 {\\varphi}(t,x) \\right\\rangle} + U_\\eta^\\ast v_\\eta(t,x) \\, \\overline{\\varphi}(t,x)\\Bigg\\} \\, dx \\, dt,\n\t\t\\end{array}\n\t\t\\end{equation}\nfor all ${ \\varphi \\in C_{\\rm c}^1((-\\infty,T)) \\otimes C_{\\rm c}^2(\\mathbb{R}^n) }$. Recall that ${ {\\left\\langle P,Q \\right\\rangle} := {\\rm tr}(P \\overline{Q}^t) }$, for ${ P,Q }$ in \n${ \\mathbb{C}^{n \\times n} }$. Then, using~\\eqref{tvdfvcfvfvfvfcdcdcdcdc}, the Theorem~\\ref{jnchndhbvgfbdtegdferfer}, making ${ \\eta \\to 0 }$ and invoking the \nuniqueness property of the equation~\\eqref{987tr7tef76756g7rg5467g546r7g5}, we conclude that ${ v^{(0)}=v_{\\rm per} }$.\n\n2. Now, using that ${ U_\\eta^\\ast = U_{\\rm per}^\\ast + \\eta\\, U^{(1)} + \\mathrm{O}(\\eta^2) }$ as ${ \\eta \\to 0 }$, defining $V_{\\eta}(t,x):=v_\\eta\\Big(t,\\sqrt{A_\\eta^\\ast}\\,x\\Big)$ and \nusing the homogenized equation \\eqref{askdjfhucomojkfdfd}, we arrive at \n\\begin{equation}\\label{PertCase1}\n\t\t\\left\\{\n\t\t\\begin{array}{c}\n\t\t\ti \\displaystyle\\frac{\\partial V_\\eta}{\\partial t} - \\Delta V_{\\eta} + U_{\\rm per}^\\ast V_\\eta = -\\Big(\\eta\\,U^{(1)} + \\mathrm{O}(\\eta^2)\\Big)\\,V_{\\eta}\\, , \\;\\, \\text{in} \\;\\, \\mathbb{R}^{n+1}_T, \\\\ [7,5pt]\n\t\t\tV_\\eta(0,x) = v^0\\Big(\\sqrt{A_\\eta^\\ast}\\,x\\Big) \\, , \\;\\, x\\in \\mathbb{R}^n,\n\t\t\\end{array}\n\t\t\\right.\n\t\\end{equation}\nProceeding similarly with respect to $V(t,x):=v_{\\rm per}\\Big(t,\\sqrt{A_{\\rm per}^\\ast}\\,x\\Big)$, we obtain\n\n\\begin{equation}\\label{PertCase2}\n\t\t\\left\\{\n\t\t\\begin{array}{c}\n\t\t\ti \\displaystyle\\frac{\\partial V}{\\partial t} - \\Delta V + U_{\\rm per}^\\ast V_\\eta = 0 \\, , \\;\\, \\text{in} \\;\\, \\mathbb{R}^{n+1}_T, \\\\ [7,5pt]\n\t\t\tV(0,x) = v^0\\Big(\\sqrt{A_{\\rm per}^\\ast}\\,x\\Big) \\, , \\;\\, x\\in \\mathbb{R}^n.\n\t\t\\end{array}\n\t\t\\right.\n\t\\end{equation}\nNow, the difference between the equations~\\eqref{PertCase1} and~\\eqref{PertCase2} yields, \n\\begin{equation}\\label{PertCase3}\n\t\t\\left\\{\n\t\t\\begin{array}{c}\n\t\t\ti \\displaystyle\\frac{\\partial (V_\\eta-V)}{\\partial t} - \\Delta (V_{\\eta}-V) + U_{\\rm per}^\\ast (V_\\eta-V) = -\\Big(\\eta\\,U^{(1)} + \\mathrm{O}(\\eta^2)\\Big)\\,V_{\\eta}\\, , \\;\\, \\text{in} \\;\\, \\mathbb{R}^{n+1}_T, \\\\ [7,5pt]\n\t\t\t(V_\\eta-V)(0,x) = v^0\\Big(\\sqrt{A_\\eta^\\ast}\\,x\\Big)-v^0\\Big(\\sqrt{A_{\\rm per}^\\ast}\\,x\\Big) \\, , \\;\\, x\\in \\mathbb{R}^n.\n\t\t\\end{array}\n\t\t\\right.\n\t\\end{equation}\nHence, multiplying the last equation by $\\overline{V_{\\eta}-V}$, integrating over $\\mathbb R^n$ and taking the imaginary part yields\n$$\n\\frac{d}{dt}\\|V_{\\eta}-V{\\|}_{L^2(\\mathbb{R}^n)}\\le \\mathrm{O}(\\eta),\n$$\nfor $\\eta\\in \\mathcal{V}$. Defining \n\\begin{equation*}\n\t\t\tW_\\eta(t,x) := \\frac{V_\\eta(t,x)-V(t,x)}{\\eta}, \\;\\, \\eta \\in \\mathcal{V},\n\t\t\\end{equation*}\nthe last inequality provides \n$$\n\\sup_{\\eta\\in \\mathcal{V}} \\| W_{\\eta}{\\|}_{L^2(\\mathbb{R}^{n+1}_T)}< +\\infty.\n$$\nThus, taking a subsequence if necessary, there exists ${ v^{(1)} \\in L^2(\\mathbb{R}^{n+1}_T) }$ such that\n\t\t\\begin{equation}\\label{67dtguystrt6756456rt3yd}\n\t\t\tW_{\\eta}(t,x) \\; \\xrightharpoonup[\\eta \\to 0]{} \\; v^{(1)}\\Big(t,\\sqrt{A_{\\rm per}^\\ast}\\,x\\Big), \\; \\text{in} \\; L^2(\\mathbb{R}^{n+1}_T).\n\t\t\\end{equation}\n\t\t\nHence, multiplying the equation~\\eqref{PertCase3} by $\\eta^{-1}$, letting $\\eta\\to 0$ and performing a change of variables, we reach the \nequation~\\eqref{73547tr764tr63tr4387tr8743tr463847} finishing the proof of the theorem. \n\n\n\\end{proof}\n\n\n\\section*{Acknowledgements}\nConflict of Interest: Author Wladimir Neves has received research grants from CNPq\nthrough the grant  308064\/2019-4. Author Jean Silva has received research grants from CNPq through the Grant 302331\/2017-4.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nGravitationally unstable (GI) disks are expected in the early phases of star formation, when the disk mass can still\nbe an appreciable fraction of the stellar mass, during the Class 0\/Class I stage.\nWhether disk fragmentation is a common outcome of GI and results in long-lived objects that contract to become gas giant\nplanets (Mayer et al. 2004; Boss 2005), or even lower mass planets via tidal mass loss (Boley et al. 2010; Galvagni \\& Mayer 2014), \nis still debated  (Helled et al. 2014). Yet disk instability offers  a natural\nexplanation for the massive planets on wide orbits discovered via imaging surveys in the last decade\n(e.g. Marois et al. 2008) because the conditions required for disk fragmentation, namely a Toomre instability parameter $Q < 1.4$ \nand short radiative cooling timescales, should be satisfied in the disk at $R > 30$ AU (Durisen et al. 2007; Rafikov 2007; \nClarke 2009; Boley et al. 2010; Meru \\& Bate 2010;2012).\nYet direct evidence that disk fragmentation into planetary-sized objects can take place is still\nlacking. In disk instability when protoplanets form from condensations in overdense spiral arms they are massive and extended, spanning\n2-6 AU in size  for a typical mass of a few Jupiter masses (Boley et al. 2010). \nThe first phase of clump collapse should last  $10^3-10^4$ yr (Galvagni et al. 2012),\nafter which rapid contraction to Jupiter-like densities should occur owing to H$_2$ dissociation \n(Helled et al. 2006).\nThe initial slow phase of collapse, in which\nthe protoplanet is still very extended, should be the easiest to detect due to less stringent angular resolution constraints.\nThe huge step in sensitivity and angular resolution made\npossible with the advent of the ALMA observatory prompted us to consider the possible detection of such early stages of planet\nformation by disk instability.\n\nRecently several works have indeed focused on detecting spiral structure in gravitationally unstable disks using ALMA \n(Cossins et al. 2010; Douglas et al. 2013; Dipierro et al. 2014; Evans et al. 2015). Similar studies of marginally unstable\ndisks exhibiting strong spiral pattern have also been carried out  with near-infrared observations of scattered light (Dong et al. \n2015). These studies have been \nmotivated\nby the recent discovery of several disks with prominent spiral arms, such as MWC 758 (Benisty et al. 2015) and SAO 206462\n(Garufi et al. 2013). They do not focus on the detectability of the extreme outcome of GI, namely fragmentation into gas giant\nplanets or more massive sub-stellar companions, thus they do not address if ALMA could detect a \"smoking gun\" signature\nof planet formation by GI. \nIn addition, spiral arms can also be produced by migrating planets (Zhu et al. 2015; Dong et al. 2015; \nPohl et al. 2015) , or perturbations by nearby\nstellar companions, while extended clumps are a unique feature of disk instability. \n\nWith the exception of Douglas et al. (2013), previous works studying the detectability of spiral structure in GI disks\nhave employed simulations with simple radiative cooling prescriptions for the disk rather than coupling the hydro solver with radiative\ntransfer.\nThis applies also to the only previous study of the detectability of GI clumps, which employed  2D simulations of \nvery massive embedded protostellar disks fragmenting predominantly into brown-dwarf sized objects (Vorobyov, Zakhozhay \\& \nDunham 2013). The limited treatment of radiation\ncan strongly affect the resulting temperature structure in the disk (Durisen et al 2007; Boley et al. 2006; Evans et al. 2015), and hence any inference \nconcerning detectability.\n\nHere we report on the first study of the detectability of massive protoplanets formed by disk instability \nwith ALMA which employs state-of-the-art 3D radiation-hydro SPH simulations. The latter are used to generate \nALMA images by means of the  ray-tracing code RADMC-3D \\footnote { \nhttp:\/\/www.ita.uni-heidelberg.de\/$\\sim$dullemond\/software\/radmc-3d\/} combined with the ALMA simulator.\n\n\n\\section{Methods}\n\n\n\\subsection{The simulations}\n\nWe perform very high resolution 3D radiation hydrodynamics simulations using the\nGASOLINE SPH code (Wadsley et al. 2004) with the implementation of an implicit \nscheme for flux-limited diffusion and photospheric\ncooling described and thoroughly tested in Rogers \\& Wadsley (2011;2012). \nWe solve the mono-frequency radiation-hydro equations using\nRosseland mean and Planck mean opacities by means of a look-up table. We use the opacities\nof d'Alessio et al (2001).  Stellar irradiation is included in the computation of the initial equilibrium\nof the disk but not in the simulation. We adopt an equation of state with a variable adiabatic index\nas a function of temperature which includes the effect of hydrogen ionization and molecular dissociation\n(Boley et al. 2007; Galvagni et al. 2012).\n\nWe model self-gravitating protoplanetary disks without an embedding envelope.\nThe disk parameters are very similar to those adopted in Boley et al. (2010).\nThe central star has a mass of $1.35 M_{\\odot}$, comparable to that in the HR8799 \nexoplanetary system (Marois et al. 2006).\nThe disk mass is $0.69 M_{\\odot}$ out to a radius of $200$ AU. \nA high disk mass,\ncomprising a significant fraction of the mass of the host star, should be typical\nof Class 0-I disks (Greaves \\& Rice 2010; Dunham et al. 2014), although occasionally massive systems are found also in\nthe Class II\/T-Tauri stage (Miotello et al. 2014).\n\n\n\\begin{figure*}\n\\epsscale{0.8}\n\\plotone{fig1.pdf}\n\\caption{Disk surface density (top) and temperature (bottom) at two representative times, immediately\nbefore (early) and one rotation period after fragmentation (late). A slice with thickness equal to one grid cell, namely 0.16 AU,\nis shown}.\n\\end{figure*}\n\n\nThe initial conditions are constructed using an iterative procedure to ensure local balance between pressure, gravity\nand centrifugal forces, taking into account the actual gravitational potential of each gas\nparticle as determined by both the disk and the central star (Rogers \\& Wadsley 2011). \nThe temperature profile is determined by imposing an initial Toomre Q parameter that \nreaches a minimum of $Q_{min} \\sim 1.4$ at $R \\sim 60$ AU (see e.g. Durisen et al. 2007).\nThe simulations comprise 1 million particles in the disk,\nwith a fixed gravitational softening of $0.16$ AU and a variable SPH smoothing length\nwhich is comparable to the softening at the beginning but can become as small as $0.05$ AU in the highest density regions.\nThe simulation employed in this paper is part of a set of simulations with different disk masses, stellar\nmasses and opacities.  Here we focus on one particular simulation, which produces a few massive clumps, one of which is \ngravitationally  bound by the end of the simulation, thereby lending itself naturally for the analysis that we intend to carry out.\n\n\n\n\\subsection{Post-processing radiative transfer}\n\n\nWe map the SPH data to a homogeneous grid with dimensions $2500 \\times 2500 \\times\n1250$\nand a cell size of $0.16\\,$AU. We assume a constant gas-to-dust ratio of $100$ and\nthat gas and dust are collisionally coupled, so that the gas and dust temperatures\nare identical. We use the radiative-transfer code\nRADMC-3D\nto produce synthetic dust emission maps from these data. We follow Dipierro et al. (2014),\nwho used the opacity law adopted in Cossins et al. (2010):\n\n\\begin{equation}\n\\kappa_\\nu = 0.025 \\left( \\frac{\\nu}{10^{12}\\,\\mathrm{Hz}} \\right)\n\\,\\mathrm{cm}^2\\,\\mathrm{g}^{-1}\n\\end{equation}\nfor dust in solar metallicity gas. Note that in this step we adopt a frequency-dependent\nopacity law while the simulations simply employed frequency-integrated opacities\nto limit the computational burden of the radiative calculation (see previous section).\nWe produce images of thermal dust emission at four frequencies ($230\\,$GHz, $345\\,$GHz,\n$460\\,$GHz and $690\\,$GHz) and for five different inclination angles (face-on,\n$30^\\circ$,\n$45^\\circ$,  $60^\\circ$ and edge-on). \n\n\n\n\\subsection{ALMA synthetic observations}\n\nWe simulate the ALMA full array observations of the dust continuum emission using\ntasks \\verb+simobserve+ and \\verb+simanalyze+ in CASA 4.1.0 (McMullin et al. 2007).\nWe assume the disk is at the distance of the Ophiuchus star forming\nregion, 125 pc.\nWe simulate the continuum ALMA observations for a 10-minutes on-source time, with a\n2\\,GHz bandwidth, and using 5 different array configurations (alma.out01, alma.out07,\nalma.out14, alma.out21, alma.out28). We chose these parameters because they represent\nfeasible parameters of future snapshot surveys for young disks. The different array\nconfigurations allow for a clear display of the trade-off between sensitivity and angular\nresolution. \nNote that the chosen integration time is realistic and at the same time allows to achieve\na good signal-to-noise ratio. For comparison, Dipierro et al. (2014) have considered \nlonger integration times (typically 30-120 minutes) in their analysis of spiral structure\ndetection, while we preferred to be conservative. Clearly for systems located at significantly\nlarger distances (e.g. 400 pc for Orion) longer integration times will be required in order to approach\nthe quality of the results presented here.\nIn addition, we have also combined the synthetic observations of two array configurations:\nalma.out14 and alma.out28. The imaging of these combined datasets was done using\nstandard clean and multi-scale clean. The images created using multi-scale clean show an\nimproved image fidelity when compared to the standard clean, as we show in the next section.\n\n\n\\section{Results}\n\nThe disk quickly develops a prominent spiral pattern that grows in amplitude, quickly leading to overdensities\nalong the arms. \nThe disk fragments into two clumps with masses of several $M_J$ after a few orbits,\nat a radius of about 80 AU, where the orbital time is $\\sim 2000$ years (Figure 1). A third \noverdensity begins to  form along one of the spiral arms near the end of the simulation.\nThe simulation is stopped once the first clump that forms , seen at 10 o'clock in Figure 1 \n(right panel), \nbecomes gravitationally bound and collapses\nfurther, reaching extremely high central densities that render the time-integration prohibitively slow.\nThe bound mass of the latter clump is $\\sim 6.2 M_{J}$ at the last\nsnapshot, and it has been measured using the SKID group finder with unbinding procedure (see \n\\footnote{http:\/\/hpcforge.org\/projects\/skid\/}).\nThe strong spiral pattern, dominated by low-order modes, $m=2-4$, and the masses of the clumps, are fairly typical\nof GI-unstable disks undergoing fragmentation (Mayer et al. 2004; 2007; Durisen et al. \n2007; Boley et al. 2010). While the clump has a mass\nat the high end of the mass distribution of extrasolar gas giants, we note that small-scale\nsimulations with much higher resolution, capable of following the collapse of clumps to near-planetary densities, have found that \nthe planetary mass resulting at the end of the collapse is at least a factor of 2 lower since a significant fraction\nof the mass resides in an extended, loosely bound circumplanetary disk which can be easily stripped by stellar tides\nas the protoplanet migrates inward (Galvagni et al. 2012; Galvagni \\& Mayer 2014; Malik et al. 2015). \n\n\nOur most important result is that massive clumps formed by GI are detectable. Figure 2 shows \na comparison of the resulting ALMA images for the face-on disk projection. In general\nthe higher frequency channels, 460 GHz and 690 GHz, are those that best capture the\nactual substructure in the disk and its density contrast, separating correctly the clumps, even the more\ndiffuse ones, from the spiral arms.  At lower frequency the noisy map renders the interpretation\nmore uncertain, making it difficult to single out even the bound clump. Note that in all \nthese images multi-scale clean has been adopted. Its adoption as well as the combination of both\nhigh and mid resolution configuration are crucial, as shown by the comparison in Figure 3. Interestingly,\nFigure 3 shows that high resolution alone produces severe artifacts that prevent any identification\nof clumps or spiral structure.\nFinally Figure 4 shows the comparison of images\nobtained for different inclination angles for our best configuration and frequency band.\nIt is clear that\nsubstructure and its relative contrast can be identified for a range of inclinations.\n\n\n\n\\begin{figure*}\n\\epsscale{1.0}\n\\plotone{fig2.pdf}\n\\caption{Comparison of observed dust continuum emission using two ALMA configurations (C34-14 and C34-28) and multiscale clean. Left and \nright column\npresent the emission in the Early and Late stages of the disk evolution. From top to bottom, each panels shows the observation at different \nfrequency and covering the main bands\navailable with ALMA. The colorscale is the same for images at the same frequency using an arcsinh stretch, the color bar is shown at the \nright-hand edge. Beam size is shown at the\nbottom left corner.}.\n\\end{figure*}\n\n\n\nA key information that one would like to extract from the ALMA images is the mass of the clumps. This is\nfor two reasons. First, the inferred masses can be used to \nsupport or refute the hypothesis that the clumps are candidate gas giants rather than e.g. brown dwarf-sized \nobjects or simply false detections. Second, by combining the information on mass and size in \nthe ALMA images the comparison with simulations can allow to assess if clumps are \ngravitationally bound objects rather than transient over-densities. We thus measured fluxes on the \nmaps shown in Figure 2 with an aperture of radius $0.06\\arcsec$, corresponding to 7.5 AU at the distance\nof Ophiucus, which visually well identifies the (bound) clump in Figure 1. \nThis is about a factor of 2 larger than the radius of the bound clump estimated with SKID so it should yield an upper limit on its\nmass. Assuming optically-thin emission, a dust-to-gas ratio of 0.01, a temperature of 300 K, and\nthe Cossins (2010) opacities, we estimate a mass (in increasing frequency) of\n18.3, 17.7, 17.1, and 16.5 $M_J$ for this clump.\nUsing another popular choice of the opacity law,\n$\\kappa _{\\nu}=0.1 (\\nu\/1.2\\,THz) cm^2$\/g (Hildebrand, 1983), the estimated masses would be a factor of 3.3 lower,\nhence nearly identical to the actual bound mass of the clump.\n\n\nMass estimates are extremely sensitive to the assumed gas temperature. If we adopt 30K, \nwhich is close to the background disk temperature rather than to the temperature of the gas in the\nclump region, the inferred mass is largely overestimated, in the range 70-90 $M_J$ even for the\nlowest opacities. This would lead to the erroneous conclusion that the clump is a brown dwarf rather than a gas giant.\nTherefore the simulations are  instrumental in yielding a good guess on important parameters such as local temperature.\nThe temperature is however well constrained as spiral shocks in massive gravitationally unstable disks\nyield temperature of order 200-400 K quite irrespective of the details of the disk model, hydro code\nand radiation solver adopted (see e.g. Mayer et al. 2007; Podolak, Mayer \\& Quinn et al. 2011; Boley et al. 2006; Rogers\n\\& Wadsley 2011). Since spiral arms are the sites of clump formation this is the temperature expected for\nclumps soon after their formation. As a gravitationally bound clump collapses further its core temperature eventually approaches\nthe $H_2$ dissociation temperature of 2000 K eventually, but this occurs on scales of a few Jupiter radii (Helled et al.\n2014) that are not resolved with ALMA. Before that happens, however, high resolution simulations of clump\ncollapse show that the mean temperature rapidly increases to 500-600 K (Galvagni et al. 2012).\nUsing a temperature of 500 K would yield masses of\n$3-10 M_J$ depending on opacity choice, hence very close to the actual bound mass.\n\nFinally, we verified that the inferred mass estimates, for the apparent size of $~ \\sim 7.5$ AU\n and for the reference temperature of 300 K, automatically yield that the clump is \nvirialized (assuming the scalar virial theorem and spherical symmetry), hence bound. \nThe second clump at $\\sim 5$ o'clock in Figure 1 (right panel) is not\nbound according to  our SKID analysis, and less so is the over-density at 6 o'clock. Note that these\nother structures, while weaker in contrast, do show up quite well in the ALMA images at the \nhighest frequencies (690 GHz). They appear as extended as they are in the actual simulation at 690 \nGHz (and marginally at 480 GHz), while at lower frequencies they are less clear and blend with \nspiral arms, making it difficult to determine their presence as physical substructure in the disk \n(especially for the over-density at 6 o'clock). Note that marginally bound, transient \nover-densities that are easily dissolved by  shear are a recurrent feature of GI disks, hence \nrecovering their presence is almost as important as\nbeing able to identify a single bound clump as it is never observed in simulations that \na fragmenting disk produces only a single clump (Mayer et al. 2004 ; Meru 2015).\nApplying our flux-based\nmass estimates across the different frequency bands we obtain masses also in the gas\ngiant planet range for such two overdensities, varying in the range $1-4 M_J$,\ndepending on the assumed opacity.\n\n\n\n\\begin{figure}\n\\epsscale{1.2}\n\\plotone{fig3.pdf}\n\\caption{Comparison of the simulated disk with three mock observations from ALMA at 690GHz.\nBottom left: dust continuum emission at 690GHz obtained from the radiative transfer of the numerical simulation, this is used as input for \nthe ALMA simulated observations.\nTop left: ALMA simulated image of the highest angular resolution possible (C34-28) using standard clean. Notice the severe imaging artifacts \ndue to the limited uv-coverage, where the\ndisk emission is more extended.\nTop right: ALMA simulated imaged of the combined high- and medium-angular resolution configurations (C34-14 and C34-28) using standard clean.\nBottom right: ALMA simulated imaged of the combined high- and medium-angular resolution configurations (C34-14 and C34-28) using multiscale \nclean.\nIn all panels the color stretch is between 0 and the peak of the image using an arcsinh stretch. Beam size is shown in the bottom left \ncorner.}\n\\end{figure}\n\n\n\n\\section{Discussion and Conclusions}\n\nWe have reported a proof-of-concept study which combines high-resolution radiation hydro simulations of GI disks with synthetic observations. \nOur study shows that ALMA can detect GI clumps on the scale of gas giants in the early stages of their collapse.  This \nfinding extends the results of Dipierro et al. (2014), who showed how even fairly complex spiral structure can be detected by ALMA for a variety of \nfrequencies.\nWhile the possibility of clump detection by ALMA was already suggested by Vorobyov et al. (2013), we note that the 2D simulations\nin their work considered extremely massive disks, arising soon after the collapse of the molecular cloud core, that \nwere almost \nentirely shattered by fragmentation into very massive objects on the scale of brown dwarfs. Such a violent disk instability \nphase, if it ever happens, would last only a couple of rotations, after which the disk itself would disappear.\nIt would then be hardly observable, and it would not\nlead to long-lasting planetary-sized objects (Helled et al. 2014). Furthermore, their synthetic ALMA maps were obtained\nfrom SED modeling rather than by means of a ray-tracing radiative transfer calculation as we do here.\n\nBased on our results detection is possible not only for\nvery dense, gravitationally bound clumps, which yield the highest density contrast, but also for more loosely bound overdensities\nwhich, while transient, are expected during a GI phase.\nHigh-frequency, combination \nof two (or more) ALMA configurations, and imaging with multi-scale clean provide the optimal setup to capture \nvery closely the substructure in the disk at large radii, where the optical depth is relatively low. \nThis leads even to fairly accurate mass estimates for clumps, once these can be clearly identified and provided\nthat a sensible temperature is assumed. Radiation-hydro simulations are crucial in constraining the temperature.\n\nNote that\nwe find little dependence of the ability to detect clumps and spiral structure on the inclination angle, which is at variance\nwith Dipierro et al. (2014). However, in the latter work inclination was affecting the simulated ALMA maps when a high-order, tightly\nwound spiral pattern was present. In our case the disk, being very massive, develops  only global, large-scale, large pitch\nangle $m=2-4$ modes (see also Dong et al. 2015), whose morphology is inherently less affected by inclination, as found also\nby Douglas et al. (2013).\n\nOur simulations have some limitations that could have an impact on the ALMA mocks.\nWe do not include a gaseous envelope, which should still embed the disk in Class 0-I phases, and be the source\nof still significant accretion. In principle\nprotostellar collapse simulations in realistic turbulent cores should be considered.\nNote that the envelope could have a non-trivial temperature distribution, on average colder than\nthe spiral shocks and clumps (with temperatures of a few tens of K), but with hot spots where accretion shocks hit the disk. Since\naccretion is filamentary and patchy in a turbulent core (e.g. Hayfield et al. 2011) inhomogeneities in density and\ntemperature might arise in the outer disk that could render clump identification more difficult. However, using\nmocks for both continuum and molecular line emission\/absorption, Douglas et al. (2013) have shown that a strong spiral pattern \nshould be detectable with ALMA even in embedded disks.\n\nAnother important caveat is that here we assumed dust and gas to be distributed in the same\nway, both in the simulations and in the post-processing step with RADMC-3D. Recent observations of putative\nyoung protoplanets in HD100546 provide a striking example of how different the distributions of dust and\ngas can be (Pineda et al. 2014;  Walsh et al. 2014). In particular, in GI unstable disks dust\nwould tend to concentrate in spiral arms due to ensuing negative pressure gradients towards the overdensity peaks \n(Rice et al. 2005), and would do even more so after dense clumps have formed (Boley \\& Durisen 2010). Therefore \nopacities could be  significantly higher at clump sites, perhaps by an order of magnitude, and also to some extent in spiral arms, \nrelative to the background flow. In this case lower frequency observations may have to be considered. The\nresulting ALMA mocks will have to be investigated in detail by future work.\n \n\n\n\n\\begin{figure}\n\\epsscale{1.4}\n\\plotone{fig4.pdf}\n\\caption{Comparison of simulated observations of the dust continuum emission at 690GHz of the disk in a Late stage as observed by ALMA using \ntwo array configurations (C34-14 and\nC34-28) and imaged using multiscale clean. The inclination angles shown are 0 (face-on), 45 and 60 \ndegrees, and listed in the top left \ncorners. The beam size is shown in the bottom\nleft corner.}\n\\end{figure}\n\n\\bigskip\n\n\\acknowledgements\n\n\\smallskip\n\nThe authors thanh Patrick Rogers for deploying the new disk initial condition generator\nused, Marina Galvagni for running the GASOLINE simulations used in this paper, and Joachim Stadel\nfor improving the TIPGRID code employed to map particle datasets onto grids.\nWe thank Ravit Helled, Aaron Boley and Farzana Meru for useful comments during preparation of the \nfinal manuscript.\nL.M. thanks the Munich Institute for Astro and Particle Physics (MIAPP)\nfor hospitality during a crucial phase of this work during summer 2015. \nJ.E.P. was supported\nby the SINERGIA grant \"STARFORM\" of the Swiss National Science Foundation during the\nearly stages of this work, which also enabled the collaboration with L.M and T.P.\nJ.E.P. acknowledges the financial support of the European Research Council (ERC; project PALs 320620).\nT.P. acknowledges support  by a \"Forschungskredit\" grant of the\nUniversity of Z\\\"urich and by the DFG Priority Program 1573 {\\em\nPhysics of the Interstellar Medium}.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \n\\label{sec:intro}\nStrongly correlated electron systems, in which the Coulomb interaction between electrons plays an essential role, can exhibit a variety of phenomena such as ferromagnetism, antiferromagnetism, and superconductivity. The Hubbard model has been introduced as a minimum model to describe such systems \\cite{Kanamori1963, Gutzwiller1964, Hubbard1963}. \nDispite its apparent simplicity, the intricate competition between the kinetic and the \non-site Coulomb terms in the model is hard to deal with analytically. \nSo far exact\/rigorous results have been mostly limited to one dimension~\\cite{essler2005one} or systems with special hopping and filling~\\cite{Tasaki1992, Mielke1993, Mielke1993a, Tasaki1997, Tasaki2019mb, Derzhko2015}.\\par\n\n\nRecently, it has become possible to simulate the Hubbard model using ultracold atoms in optical lattices~\\cite{Kohl2005, Jordens2008, Schneider2008}. \nFurthermore, it was proposed theoretically~\\cite{Honerkamp2004} and demonstrated experimentally~\\cite{Taie2012} that multi-component fermionic systems with SU($n$) symmetry can be realized in cold-atom setups.\nThese systems are well described by the SU($n$) Fermi-Hubbard model, in which each atom carries $n$ internal degrees of freedom. When $n=2$, the model reduces to the original Hubbard model with spin-independent interaction. \nAlthough the SU($n$) ($n>2$) symmetry has been less explored in the condensed matter literature, there is a growing interest in recent years in studying the SU($n$) Hubbard model theoretically. \nFor example, it is argued that the SU($n$) Hubbard model can exhibit exotic \nphases that do not appear in the SU($2$) counterpart~\\cite{cazalilla2009ultracold, Honerkamp2004, Chung2019}.\nBesides, enlarged symmetry other than SU($n$), such as SO(5),  \nin higher-spin systems has also been discussed~\\cite{wu2003exact}.\n\\par\nThe SU($n$) Hubbard model is, in general, harder to theoretically study than the SU(2) Hubbard model.\nIt has been reported that the Nagaoka ferromagnetism~\\cite{Nagaoka1966, Tasaki1989}, which is the first rigorous result for the SU($2$) Hubbard model, can be generalized to the case of SU($n$)~\\cite{Katsura2013, Bobrow2018}.\nFlat-band ferromagnetism is another example of rigorous results for the SU(2) Hubbard model.\nHere, a flat band refers to a structure of single-particle energy spectrum which has a macroscopic degeneracy.\nA tight-binding model with a flat band can be constructed using standard methods such as the line-graph~\\cite{mielke1991ferromagnetic} and the cell constructions~\\cite{Tasaki1992}.\nIn the SU(2) case, it is known that if the system has a flat band at the bottom of the single-particle spectrum and the particle number is the same as the number of unit cells, the ground state of the model exhibits ferromagnetism~\\cite{Tasaki1992, Mielke1993, Tasaki1997, Li2004}.\nAn SU($n$) counterpart of the flat-band ferromagnetism has also been discussed recently~\\cite{Liu2019}.\\par\n\nIn this paper, \nwe consider the SU($n$) Hubbard model on a one-dimensional (1D) lattice called the railroad-trestle lattice\nand derive rigorous results. We first treat the model with a flat band at the bottom and prove that the model exhibits SU($n$) ferromagnetism in its ground states provided that the on-site interaction is repulsive and the total fermion number is the same as the number of unit cells. This is a slight generalization of the result obtained by Liu {\\it et al}. in~\\cite{Liu2019}, in the sense that our hopping Hamiltonian has one more parameter. \nWe then discuss SU($n$) ferromagnetism in a perturbed model obtained by adding extra hopping terms that make the flat band dispersive. We prove that this particular perturbation leaves the SU($n$) ferromagnetic ground states unchanged when the band width of the bottom band is sufficiently narrow. This is our main result and can be thought of as an SU($n$) extension of the previous theorem for the SU($2$) Hubbard model with nearly flat bands~\\cite{Tasaki1995}.\\par \nThe rest of this paper is organized as follows. In Sec.~\\ref{sec:thm1}, we introduce the SU($n$) Hubbard model with completely flat band and prove that its ground states exhibit SU($n$) ferromagnetism.\nIn Sec.~\\ref{sec:thm2}, we study a model with nearly flat band and prove that the ground states remain SU($n$) ferromagnetic when the repulsive interaction and the band gap are sufficiently large. \nWe present our conclusions in Sec.~\\ref{sec:conclusion}.\n\\section{Model with completely flat band}~\\label{sec:thm1}\nLet $M$ be an arbitrary positive integer and $\\Lambda = \\{1, 2, \\dots , 2M\\}$ be a set of $2M$ sites on the railroad-trestle lattice [Fig. \\ref{fig:deltachain}].\nWe impose periodic boundary condition, so that site $j$ and $j + 2M$ are identified. \nWe denote by $\\mathcal{E}$ and $\\mathcal{O}$ subsets of $\\Lambda$ consisting of even sites and odd sites, respectively.\nWe define creation and annihilation operators $c_{x, \\alpha}^{\\dag}$ and $c_{x, \\alpha}$ for a fermion at site $x \\in \\Lambda $ with color $\\alpha = 1, \\dots ,n$.\nThey satisfy $\\{c_{x, \\alpha}, c_{y, \\beta}^{\\dag}\\} = \\delta_{\\alpha, \\beta} \\delta_{x, y}$.\nThe number operator of fermion at site $x$ with color $\\alpha$ is denoted by $n_{x, \\alpha} = c_{x, \\alpha}^{\\dag} c_{x, \\alpha}$.\nWe consider the SU($n$) Hubbard Hamiltonian\n\\begin{align}\nH_{1} \n&=H_{\\mathrm{hop}} + H_{\\mathrm{int}} \\label{hamiltonian1}, \\\\ \nH_{\\mathrm{hop}}\n&= \\sum^n_{\\alpha=1} \\sum_{x,y \\in \\Lambda} t_{x, y} c_{x, \\alpha}^{\\dag} c_{y, \\alpha}, \\\\\nH_{\\mathrm{int}}\n&= U \\sum_{1 \\le \\alpha < \\beta \\le n} \\, \\sum_{x \\in \\Lambda} n_{x, \\alpha}n_{x, \\beta}, \\label{hint}\n\\end{align}\nwhere $t_{2x-1,2x-1} = t$, $t_{2x, 2x} = 2\\nu^{2} t$, $t_{2x-1,2x} = t_{2x, 2x-1}= \\nu t$, $t_{2x-2,2x}= t_{2x, 2x-2} = \\nu^{2} t$, and the remaining elements of $t_{x, y}$ are zero (see Fig. \\ref{fig:deltachain}).\nThe parameters $t, \\nu$ and $U$ are positive.\n\\begin{figure}[H]\n\t\\begin{tabular}{c}\n\t\t\\centering\n\t\t\\begin{minipage}{0.5 \\hsize}\n\t\t\\subcaption{}\\label{fig:deltachain}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=\\columnwidth]{fig1a.eps}\n\t\t\\end{minipage}\n\t\t\n\t\t\\begin{minipage}{0.5 \\hsize}\n\t\t\\subcaption{}\\label{fig:Energyband}\t\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=\\columnwidth]{fig1b.eps}\n\t\t\\end{minipage}\n\t\\end{tabular}\n\t\\caption{\\subref{fig:deltachain} The railroad-trestle lattice with hopping amplitudes $t_{1} = \\nu t$ and $t_{2} = \\nu^{2} t$. \n\tOdd (black) and even (white) sites have on-site potentials $t$ and $2 \\nu^{2} t$, respectively.\n\tThe shaded region indicates the unit cell.\n\t\\subref{fig:Energyband} The energy bands for $t = 1, \\nu = 1\/\\sqrt{2}$. The lowest band at zero energy is completely flat. }\n\t\\label{fig:flatband}\n\\end{figure}\nWhen $U = 0$, the model reduces a tight-binding model and we see that it has two bands with $\\epsilon_{1}(k) = 0, \\epsilon_{2}(k) = t(2\\nu^{2}+1) + 2\\nu^{2} t \\cos{k}$.\nClearly, the lowest band is dispersionless as shown in Fig. \\ref{fig:Energyband}.\\par\nWe define total number operators of fermion with color $\\alpha$ and color-raising and lowering operators as $ F^{\\alpha, \\beta} = \\sum_{x \\in \\Lambda} c_{x, \\alpha}^{\\dag} c_{x, \\beta} $.\nSince the Hamiltonian $H_{1}$ has SU($n$) symmetry, they commute with $H_{1}$.\nWe denote the eigenvalue of $F^{\\alpha, \\alpha}$ by $N_{\\alpha}$.\nSince $F^{\\alpha, \\alpha}$ commute with the Hamiltonian $H_{1}$, the eigenstates of $H_{1}$ are separated into different sectors labeled by $(N_{1}, \\dots, N_{n})$.\nIf the total fermion number $N_{\\mathrm{f}} = \\sum_{x \\in \\Lambda} \\sum_{\\alpha = 1}^{n} n_{x, \\alpha}$ is fixed, $N_{\\alpha}$ must satisfy $\\sum_{\\alpha=1}^{n} N_{\\alpha} = N_{\\mathrm{f}}$. \\par\nNow we define a new set of operators \n\\begin{align}\n&a_{x, \\alpha} := -\\nu c_{x-1, \\alpha} + c_{x, \\alpha} -\\nu c_{x+1, \\alpha} \\ \\ \\ &\\text{for} \\ x \\in \\mathcal{E},  \\\\\n&b_{x, \\alpha} := \\nu c_{x-1, \\alpha} + c_{x, \\alpha} + \\nu c_{x+1, \\alpha} \\ \\ \\ &\\text{for} \\  x \\in \\mathcal{O}, \n\\end{align}\nwhich satisfy \n\\begin{align}\n\\{a_{x, \\alpha}, b_{y, \\beta}^{\\dag}\\} &= 0,  \\label{ab}\\\\\n\\{a_{x, \\alpha}, a_{y, \\beta}^{\\dag}\\} &= \n\\begin{cases}\n\\delta_{\\alpha, \\beta} (\\nu^{2} + 2) \\ \\ &\\text{if}\\  x  = y , \\\\\n\\delta_{\\alpha, \\beta} \\ \\nu^{2} \\ \\ &\\text{if}\\ x = y \\pm 2, \\\\\n0 \\ \\ &\\text{otherwise},\n\\end{cases} \\\\\n\\{b_{x, \\alpha}, b_{y, \\beta}^{\\dag}\\} & = \n\\begin{cases}\n\\delta_{\\alpha, \\beta} (\\nu^{2} + 2) \\ \\ &\\text{if}\\ x = y, \\\\\n\\delta_{\\alpha, \\beta} \\ \\nu^{2}\\ \\ &\\text{if}\\ x = y \\pm 2, \\\\\n0 \\ \\ &\\text{otherwise}.\n\\end{cases}\n\\end{align} \nThe hopping Hamiltonian $H_{\\mathrm{hop}}$ is rewritten in terms of $b_{x, \\alpha}$ and $b_{x, \\alpha}^{\\dag}$ as \n\\begin{align}\nH_{\\mathrm{hop}} = t \\sum_{\\alpha=1}^{n} \\sum_{x \\in \\mathcal{O}} b_{x,\\alpha}^{\\dag} b_{x, \\alpha} \\label{hop1}\n\\end{align}\nand hence positive semi-definite.\nThe interaction term $H_{\\mathrm{int}}$ is also positive semi-definite because $n_{x, \\alpha} n_{x, \\beta} = \\left(c_{x, \\alpha} c_{x, \\beta}\\right)^{\\dag} c_{x, \\alpha} c_{x, \\beta}$.\nTherefore, the total Hamiltonian $H_{1} = H_{\\mathrm{hop}} + H_{\\mathrm{int}}$ is positive semi-definite as well.\nFrom now on, we fix the total fermion number as $N_{\\mathrm{f}} = |\\mathcal{E}| = M$ and define a fully polarized state as $\\ket{\\Phi_{\\mathrm{all}, \\alpha}} := \\prod_{x \\in \\mathcal{E}} a_{x, \\alpha}^{\\dag} \\ket{\\Phi_{\\mathrm{vac}}}$, where $\\ket{\\Phi_{\\mathrm{vac}}}$ is a vacuum state of $c_{x, \\alpha}$.\nFrom the anti-commutation relation (\\ref{ab}), we find that $\\ket{\\Phi_{\\mathrm{all}, \\alpha}}$ is an eigenstate of $H_{1}$ with eigenvalue zero.\nSince $H _{1}\\geq 0$, the fully polarized states are ground states of $H_{1}$.\nDue to the SU($n$) symmetry, one obtains a general form of degenerate ground states as\n\\begin{align}\n\\ket{\\Phi_{N_{1}, \\dots, N_{n}}} = \\left(F^{n, 1}\\right)^{N_{n}} \\dots \\left(F^{2, 1}\\right)^{N_{2}} \\ket{\\Phi_{\\mathrm{all},1}}, \\label{fully polarized}\n\\end{align}\nwhere $N_{1} = M -\\sum_{\\alpha=2}^{n}N_{\\alpha}$.\nWe also refer to states of the form Eq. (\\ref{fully polarized}) as fully polarized states~\\footnote{\nThe total number of such states is $\\frac{(M+n-1)!}{M! (n-1)!}$.\n}.\n\nThe first result of this paper is the following: \\par\n{\\it Theorem 1.}---Consider \nthe Hubbard Hamiltonian (\\ref{hamiltonian1}) with the total fermion number $N_{\\mathrm{f}} = M$.\nFor arbitrary  $t>0$ and $U>0$, the ground states of the Hamiltonian (\\ref{hamiltonian1}) are the fully polarized states and unique apart from trivial degeneracy due to the SU($n$) symmetry.\n\n\\renewcommand{\\proofname}{{\\indent \\it Proof of Theorem 1}}\n\\renewcommand{\\qedsymbol}{$\\blacksquare$}\n{\\it Proof of Theorem 1.}---Let $\\ket{\\Phi_{\\mathrm{GS}}}$ be an arbitrary ground state of $H_{1}$ with $N_{\\mathrm{f}} = M$.\nSince the ground state energy is zero, we have $H_{1} \\ket{\\Phi_{\\mathrm{GS}}} =0$.\nThe inequalities $H_{\\mathrm{hop}} \\geq 0$ and $H_{\\mathrm{int}} \\geq 0$ imply that $H_{\\mathrm{hop}} \\ket{\\Phi_{\\mathrm{GS}}} =0$ and $H_{\\mathrm{int}} \\ket{\\Phi_{\\mathrm{GS}}}=0$, which means that\n\\begin{align}\n&b_{x, \\alpha} \\ket{\\Phi_{\\mathrm{GS}}} \n=0\\  \\ \\text{for any $x \\in \\mathcal{O}$ and $\\alpha = 1, \\dots n$}, \\label{condition1}\\\\\n&c_{x,\\alpha} c_{x, \\beta} \\ket{\\Phi_{\\mathrm{GS}}} \n= 0\\ \\ \\text{for any $x \\in \\Lambda$ and $\\alpha \\neq \\beta$}. \\label{condition2}\n\\end{align}\nSince $a_{x, \\alpha}$ and $b_{x, \\alpha}$ obey the anti-commutation relation (\\ref{ab}), the condition (\\ref{condition1}) implies that $\\ket{\\Phi_{\\mathrm{GS}}}$ does not contain any $b_{x, \\alpha}^{\\dag}$ operator when it is constructed by acting with creation operators on the vacuum state.\nTherefore, it is written as\n\\begin{align}\n&\\ket{\\Phi_{\\mathrm{GS}}} \\nonumber \\\\\n& = \\! \\sum_{\\substack{A_{1}, A_{2} ,\\dots A_{n} \\subset \\mathcal{E}\\\\\n\\sum_{\\alpha =1}^{n} |A_{\\alpha}|= M}} \\!\nf(\\{A_{\\alpha}\\}) \\left( \\prod_{x \\in A_{1}} a_{x,1}^{\\dag}\\right) \\! \\!\n\\dots\n\\! \\left( \\prod_{x \\in A_{n}} a_{x,n}^{\\dag}\\right) \\! \\!\n\\ket{\\Phi_{\\mathrm{vac}}},\n\\end{align}\nwhere $A_{\\alpha}$ is a subset of $\\mathcal{E}$ and $f(\\{A_{\\alpha}\\})$ is a  certain coefficient.\\par\nNext, we make use of the condition (\\ref{condition2}).\nWe take an even site $x \\in \\mathcal{E}$. \nUsing the anti-commutation relation $\\{c_{x, \\alpha}, a_{y,\\beta}^{\\dag} \\} = \\delta_{\\alpha, \\beta} \\delta_{x,y}$ and\nEq. (\\ref{condition2}) we see that \n$f(\\{A_{\\alpha}\\})=0$ if there exist $A_{\\alpha}$ and $A_{\\beta}$ such that $A_{\\alpha} \\cap A_{\\beta} \\neq \\emptyset$.\nSince $\\sum_{\\alpha=1}^{n}|A_{\\alpha}| = M$ and $A_{\\alpha} \\cap A_{\\beta} = \\emptyset$ for $\\alpha \\neq \\beta$, we find that $\\cup_{\\alpha=1}^{n} A_{\\alpha} = \\mathcal{E}$.\nThis means that the ground state is rewritten as \n\\begin{align}\n\\ket{\\Phi_{\\mathrm{GS}}}\n= \\sum_{\\bm{\\alpha}}C(\\bm{\\alpha}) \\left(\\prod_{x \\in \\mathcal{E}} a_{x, \\alpha_{x}}^{\\dag}\\right)\n\\ket{\\Phi_{\\mathrm{vac}}}, \n\\end{align}\nwhere the sum is over all \npossible color configurations $\\bm{\\alpha} = (\\alpha_{x})_{x \\in \\mathcal{E}}$ with $\\alpha_{x} = 1, \\dots ,n$. \nThen we consider the condition (\\ref{condition2}) for $x \\in \\mathcal{O}$.\nBy using \n\\begin{align}\n\\{c_{x,\\alpha}, a_{y, \\beta}^{\\dag}\\} =\n\\begin{cases}\n-\\nu \\delta_{\\alpha, \\beta} \\ \\ &\\text{if $y = x\\pm1$}, \\\\\n0 \\ \\ &\\text{otherwise},\n\\end{cases} \n\\label{anticom1}\n\\end{align}\nwe get\n\\begin{align}\n&c_{x, \\alpha}c_{x, \\beta} \\ket{\\Phi_{\\mathrm{GS}}} \\nonumber \\\\\n&= \\sum_{\\substack{\\bm{\\alpha}\\\\\n\t\\mathrm{s.t.} \\alpha_{p} = \\beta, \\\\\n\t\\alpha_{q} = \\alpha}} \n\t\\nu^{2} \\left[C(\\bm{\\alpha} ) - C(\\bm{\\alpha}_{p\\leftrightarrow q})\\right]\n\\left(\\prod_{y \\in \\mathcal{E}\\backslash \\{x\\pm 1\\}} a_{y, \\alpha_{y}}^{\\dag}\\right)\\ket{\\Phi_{\\mathrm{vac}}}, \n\\end{align}\nwhere $p=x-1$ and $q = x+1$.\nThe color configuration $\\bm{\\alpha}_{p \\leftrightarrow q}$ is obtained from $\\bm{\\alpha}$ by swapping $\\alpha_{p}$ and $\\alpha_{q}$.\nSince all the states in the sum are linearly independent, we find from the condition (\\ref{condition2}) that $C(\\bm{\\alpha}) = C(\\bm{\\alpha}_{p\\leftrightarrow q})$ for all $\\bm{\\alpha}$ and all $x \\in \\mathcal{O}$.\nAs the two localized states on \nneighboring even sites share an odd site between them, we see that\n\\begin{align}\nC(\\bm{\\alpha}) = C(\\bm{\\alpha}_{x \\leftrightarrow y}), \\label{symmetric}\n\\end{align}\nwhere $x, y$ are arbitrary different sites in $\\mathcal{E}$.\\par\nTo show that \nstates satisfying Eq. (\\ref{symmetric}) are the fully polarized states, i.e., SU($n$) ferromagnetic, we introduce a concept of a word~\\cite{kitaev2011patterns}. \nA {\\it word} $w = (w_{1}, \\dots, w_{M})$ is a sequence of integers where $w_{i} \\in \\{1,\\dots,n\\}$ for all $i$. \nWe denote by $|w|_{\\alpha}$ the number of occurences of $\\alpha$ in $w$. \nWe define the set of words for which $|w|_{\\alpha} = N_{\\alpha}$ holds as follows: $W(N_{1}, \\dots , N_{n}) = \\{w | \\ |w|_{\\alpha} = N_{\\alpha},  \\ \\alpha = 1, \\dots , n \\}$.\nFor example, $W(2, 0, 1)$ consists of $(1, 1, 3), (1, 3, 1)$ and $(3, 1, 1)$.\nIt follows from \nEq. (\\ref{symmetric}) that the ground state of $H_{1}$ in the sector labeled by $(N_{1}, \\dots , N_{n})$ can be written as\n\\begin{align}\n\\ket{\\widetilde{\\Phi}_{N_{1}, \\dots , N_{n}}} = \\sum_{w \\in W(N_{1}, \\dots , N_{n})} a_{2, w_{1}}^{\\dag} a_{4, w_{2}}^{\\dag} \\dots a_{2M, w_{M}}^{\\dag} \\ket{\\Phi_{\\mathrm{vac}}}.\n\\end{align}\nNow using commutation relations $[F^{\\beta, \\alpha}, a_{x, \\gamma}^{\\dag}] = \\delta_{\\alpha, \\gamma} a_{x, \\beta}^{\\dag}$ for all $x \\in \\mathcal{E}$ , we see that \n\\begin{align}\n\\left(F^{2, 1}\\right)^{N_{2}} \\! \\ket{\\Phi_{\\mathrm{all}, 1}} \n= \\! \\! \\! \\! \\sum_{w \\in W(M-N_{2}, N_{2})} \\! \\! \\!  \\! a_{2, w_{1}}^{\\dag} a_{4, w_{2}}^{\\dag} \\dots a_{2M, w_{M}}^{\\dag} \\ket{\\Phi_{\\mathrm{vac}}}.\n\\end{align}\nBy repeating the procedure, we have the desired result $\\ket{\\widetilde{\\Phi}_{N_{1}, \\dots , N_{n}}} = \\ket{\\Phi_{N_{1}, \\dots, N_{n}}}$.\nThis proves that the ground states of $H_{1}$ are fully polarized states.\n\\hspace{\\fill}$\\blacksquare$\n\\section{Model with nearly flat band}\n\\label{sec:thm2}\nSo far, we have considered the flat-band model, but this is an idealized case in which the lowest energy band becomes completely dispersionless.\nAs a more realistic model, we consider a model with nearly flat band by adding a perturbation to the model in the previous section~\\footnote{\nIt was proposed that the hopping part of the Hamiltonian with a nearly flat band can be realized with ultracold atoms in a sawtooth lattice~\\cite{Zhang2015}.\n}.\nHere we define another Hubbard model on the same lattice as Theorem 1:\n\\begin{align}\nH_{2} \n&= H_{\\mathrm{hop}}' + H_{\\mathrm{int}}, \\label{ham2}\n\\end{align}\nwhere $H_{\\mathrm{hop}}'$ is defined as \n\\begin{align}\nH_{\\mathrm{hop}}' \n&= -s \\sum_{\\alpha=1}^{n} \\sum_{x \\in \\mathcal{E}} a_{x,\\alpha}^{\\dag} a_{x, \\alpha}\n+ t \\sum_{\\alpha=1}^{n} \\sum_{x \\in \\mathcal{O}} b_{x,\\alpha}^{\\dag} b_{x, \\alpha},  \\label{hop2}\n\\end{align}\nand $H_{\\mathrm{int}}$ is defined in Eq.(\\ref{hint}) with parameters $s, t, U > 0$.\nWhen the hopping Hamiltonian $H_{\\mathrm{hop}}'$ is written in terms of original fermion operator $c_{x, \\alpha}$,  \nit takes the form  $H_{\\mathrm{hop}}' = \\sum_{\\alpha}\\sum_{x,y \\in \\Lambda} t'_{xy} c_{x, \\alpha}^{\\dag} c_{y, \\alpha}$\n, where $t'_{2x-1, 2x-1} = t-2\\nu^{2}s$, $t'_{2x,2x} = -s + 2\\nu^{2}t$, $t'_{2x-1,2x} = t'_{2x,2x-1} = \\nu(t+s)$, $t'_{2x-2, 2x} = t'_{2x, 2x-2} = \\nu^{2}t$, $t'_{2x-1, 2x+1} = t'_{2x+1, 2x-1} = -\\nu^{2}s$, and the remaining elements of $t'_{x, y}$ are zero.\nWhen we consider the single particle problem, we obtain two bands with $\\epsilon_{1}(k) = -s(2\\nu^{2}+1) - 2\\nu^{2}s \\cos{k}$, $\\epsilon_{2}(k) = t(2\\nu^{2}+1) + 2\\nu^{2}t \\cos{k}$ (see Fig. \\ref{fig:Energy band2}).\nWe see that the lowest band is no longer flat, however, it can be regarded as a nearly flat band when $s$ is small enough.\nWe focus on this model in the following and prove a theorem on the ferromagnetism.\\par\n\\begin{figure}[H]\n\t\\begin{tabular}{c}\n\t\t\\centering\n\t\t\\begin{minipage}{0.5\\hsize}\n\t\t\\subcaption{}\\label{fig:deltachain2}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=\\columnwidth]{fig2a.eps}\n\t\t\\end{minipage}\n\t\t\\begin{minipage}{0.5\\hsize}\n\t\t\\subcaption{}\\label{fig:Energy band2}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=\\columnwidth]{fig2b.eps}\n\t\t\\end{minipage}\n\t\t\n\t\\end{tabular}  \n\t\\caption{\\subref{fig:deltachain2} The lattice geometry of $H_{\\mathrm{hop}}'$.\n\tThe hopping amplitudes are given by $t_{1} = \\nu(t+s)$, $t_{2} = \\nu^{2} t$, and $t_{2}' = -\\nu^{2} s$.\n\tOdd (black) and even (white) sites have on-site potentials $t-2\\nu^{2}s$ and $-s + 2\\nu^{2} t $, respectively.\n\tThe corresponding energy bands are shown in \\subref{fig:Energy band2} for $t = 1, \\nu = 1\/\\sqrt{2}, s = 1\/10$.\n\t}\\label{fig:flatband2}\n\\end{figure}\n{\\it Theorem 2.}---Consider the Hamiltonian (\\ref{ham2}) with the total fermion number $N_{\\mathrm{f}} = M$.\nFor sufficiently large $t\/s >0$ and $U\/s > 0$, the ground states are the fully polarized states and unique apart from the trivial degeneracy due to the SU($n$) symmetry.\n\n\\smallskip\n\n{\\it Proof of Theorem 2.}---\nFirst, we decompose the Hamiltonian (\\ref{ham2}) into the sum of local Hamiltonians as \n\\begin{align}\nH_{2} = -sM(2\\nu^{2}+1) + \\lambda H_{\\mathrm{flat}} + \\sum_{x\\in \\mathcal{E}} h_{x}, \\label{hamdecomposed}\n\\end{align}\nwhere \n\\begin{align}\nH_{\\mathrm{flat}} = \\sum_{\\alpha=1}^{n}\\sum_{x \\in \\mathcal{O}} b_{x, \\alpha}^{\\dag} b_{x, \\alpha} + \\sum_{x \\in \\Lambda} \\sum_{\\alpha < \\beta} n_{x,\\alpha}n_{x, \\beta}\n\\end{align}\nand \n\\begin{align}\n&h_{x} \\nonumber \\\\ \n&= \\sum_{\\alpha=1}^{n} \\left(-s a_{x, \\alpha}^{\\dag} a_{x, \\alpha} \n+ \\frac{t-\\lambda}{2} (b_{x-1, \\alpha}^{\\dag} b_{x-1, \\alpha} + b_{x+1, \\alpha}^{\\dag} b_{x+1, \\alpha})\n\\right) \\nonumber \\\\\n& + \\frac{\\kappa(U -\\lambda)}{4}n_{x-2}(n_{x-2}-1) + \\frac{U-\\lambda}{4} n_{x-1}(n_{x-1}-1) \\nonumber\\\\\n& + \\frac{(1-\\kappa)(U-\\lambda)}{2} n_{x}(n_{x}-1) + \\frac{U-\\lambda}{4} n_{x+1}(n_{x+1}-1) \\nonumber\\\\\n& + \\frac{\\kappa(U-\\lambda)}{4}n_{x+2}(n_{x+2}-1) + s(2\\nu^{2}+1), \n\\label{local_ham}\n\\end{align}\nwhere $n_{x}$ is defined as $n_{x} = \\sum_{\\alpha} n_{x, \\alpha}$.\nThe two parameters $\\lambda$ and $\\kappa$ satisfy $0 < \\lambda < \\min\\{t,U\\}$ and $0 \\leq \\kappa < 1$.\nTo prove Theorem 2, we use the following lemmas.\\par\n{\\it Lemma 1.}---Suppose the local Hamiltonian $h_{x}$ is positive semi-definite for any $x \\in \\mathcal{E}$. \nThen the ground states of the Hamiltonian (\\ref{hamdecomposed}), and hence Eq. (\\ref{ham2}),  are fully polarized states and unique apart from the trivial degeneracy due to the SU($n$) symmetry.\\par\n\n\n\n\n\n\n{\\it Lemma 2.}---Suppose that $t, U$ are infinitely large and $0 < \\kappa < 1$.\nThen the local Hamiltonian (\\ref{local_ham}) is positive semi-definite.\n(We take $\\lambda$ and $\\kappa$ to be proportional to $s$.)\\par\n\nWe note that $h_{x}$ can be regarded as a finite dimensional matrix independent of the system size since the local Hamiltonian $h_{x}$ acts nontrivially only on a finite number of sites.\nThis means that the energy levels of $h_{x}$ depend continuously on the parameters.\nTherefore, Lemma 2 guarantees that $h_{x}$ is positive semi-definite when $t, U$ are finite but sufficiently large.\nThen Lemma 1 implies that the ground states of the Hamiltonian (\\ref{ham2}) are fully polarized states, which proves Theorem 2.\n\\hspace{\\fill}$\\blacksquare$\n\\par\nBelow, we prove Lemmas 1 and 2.\n\\par\n{\\it Proof of Lemma 1.}---First, it is noted that a fully polarized state $\\ket{\\Phi_{\\mathrm{all},1}} = \\left(\\prod_{x \\in \\mathcal{E}} a_{x, 1}^{\\dag}\\right) \\ket{\\Phi_{\\mathrm{vac}}}$ satisfies $h_{x} \\ket{\\Phi_{\\mathrm{all},1}} = 0$ for each $h_{x}$.\nSince $h_{x}$ is SU($n$) invariant, all fully polarized states have zero energy.\nWe assume that $h_{x} \\geq 0$ for all $x \\in \\mathcal{E}$. \nLet $\\ket{\\Phi_{\\mathrm{GS}}^{\\mathrm{flat}}}$ be an arbitrary ground state of $H_{\\mathrm{flat}}$.\nSince $H_{\\mathrm{flat}} \\ket{\\Phi_{\\mathrm{GS}}^{\\mathrm{flat}}} = 0$ and $h_{x} \\ket{\\Phi_{\\mathrm{GS}}^{\\mathrm{flat}}} = 0$, we see that $H_{2} \\ket{\\Phi_{\\mathrm{GS}}^{\\mathrm{flat}}} = -s M(2\\nu^{2} +1) \\ket{\\Phi_{\\mathrm{GS}}^{\\mathrm{flat}}}$.\nFrom $h_{x} \\geq 0$, the ground energy of $H_{2}$ is $-s M (2\\nu^{2} +1)$.\nIf $\\ket{\\Phi_{\\mathrm{GS}}}$ is an arbitrary ground state of $H_{2}$, it satisfies $H_{2} \\ket{\\Phi_{\\mathrm{GS}}} = -s M(2\\nu^{2} + 1)$.\nFrom $H_{\\mathrm{flat}} \\geq 0$ and $h_{x} \\geq 0$, we find $H_{\\mathrm{flat}} \\ket{\\Phi_{\\mathrm{GS}}} = 0$ and $h_{x} \\ket{\\Phi_{\\mathrm{GS}}} = 0$.\nThis shows that any ground state of $H_{2}$ must be a ground state of $H_{\\mathrm{flat}}$.\nThe Hamiltonian $H_{\\mathrm{flat}}$ is nothing but the Hamiltonian $H_{1}$ with $t = U = 1$.\nThus, the ground states of $H_{2}$ are fully polarized and unique.\n\\hspace{\\fill} $\\blacksquare$\n\\par\n\nWe remark that one can check whether $h_{x}$ is positive semi-definite by numerically diagonalizing a finite dimensional matrix.\nThe result for the SU(4) case is shown in Fig.\\ref{fig:SU4boundary}.\n\\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[width=0.6\\columnwidth]{fig3.eps}\n\t\\caption{The positive semi-definiteness of $h_{x}$($n=4$) holds in the shaded region for $\\nu=1\/\\sqrt{2}, \\kappa = 0$.\n\tThe plot is obtained by diagonalizing $h_{x}$ numerically.\n\tLemma 1 says that the ground states of the full Hamiltonian $H_{2}$ are fully polarized states in the shaded region.\n\tFor example, the ferromagnetism is established if $t\/s \\geq 4.5$ when $U\/s = 25$.\n\t}\n\t\\label{fig:SU4boundary}\n\\end{figure}\n{\\it Proof of Lemma 2.}---Due to the translational invariance, it suffices to show the case for $h_{0}$.\nThe local Hamiltonian $h_{0}$ is regarded as an operator defined on five sites $\\{-2, -1, 0, 1, 2\\}$, where $x = -2, -1$, and $0$ are identified with $x = 2M-2, 2M-1$, and $2M$, respectively.\nOn these sites, we define operators \n\\begin{align}\n\\tilde{a}_{-2,\\alpha}&:= \\frac{1}{\\sqrt{\\nu^{2}+1}}(c_{-2, \\alpha} - \\nu c_{-1, \\alpha}), \\\\\n\\tilde{b}_{-1, \\alpha}&:= \\nu c_{-2, \\alpha} + c_{-1, \\alpha} + \\nu c_{0, \\alpha}, \\\\ \n\\tilde{a}_{0, \\alpha}&:= -\\nu c_{-1, \\alpha} + c_{0, \\alpha} + -\\nu c_{1, \\alpha}, \\\\ \n\\tilde{b}_{1, \\alpha}&:= \\nu c_{0, \\alpha} + c_{1, \\alpha} + \\nu c_{2, \\alpha}, \\\\ \n\\tilde{a}_{2, \\alpha}&:= \\frac{1}{\\sqrt{\\nu^{2}+1}}(-\\nu c_{1, \\alpha} + c_{2, \\alpha}).\n\\end{align}\nThese operators satisfy \n\\begin{align}\n\\{\\tilde{a}_{y, \\alpha}, \\tilde{a}_{y', \\beta}^{\\dag}\\}\n&= \\begin{cases}\n\\delta_{\\alpha,\\beta} (2\\nu^{2}+1) \\ \\ &\\text{for}\\ \\  y=y'=0, \\\\\n\\delta_{\\alpha,\\beta} \\frac{\\nu^{2}}{\\sqrt{\\nu^{2}+1}}\\ \\  &\\text{for}\\ \\ | y - y'| = 2, \\\\\n\\delta_{\\alpha,\\beta} \\delta_{y,y'}\\ \\  &\\text{for}\\ \\  y,y' = \\pm2, \n\\end{cases} \\label{anticom2}\\\\ \n\\{\\tilde{a}_{y,\\alpha}, \\tilde{b}_{y', \\alpha}^{\\dag}\\} &= 0.\n\\end{align}\nSingle-fermion states corresponding to these operators are linearly independent.\nTo show the lemma, we only need to consider states $\\ket{\\Phi}$ which have finite energy in this limit, i.e., $\\lim_{t,U \\rightarrow \\infty} \\bra{\\Phi} h_{0} \\ket{\\Phi} < \\infty$ .\nThe condition that $\\ket{\\Phi}$ has finite energy is equivalent to the following: \n\\begin{align}\n\\tilde{b}_{y, \\alpha} \\ket{\\Phi} &= 0 \\ \\ \\text{for $y= \\pm 1$}, \\label{cond_b}\\\\\nc_{y, \\alpha} c_{y, \\beta} \\ket{\\Phi} &= 0 \\ \\ \\text{for $y = 0, \\pm1, \\pm2$}. \\label{cond_c}\n\\end{align}\nLet $\\ket{\\Phi}$ be a state which has finite energy.\nFrom Eqs. (\\ref{cond_b}) and (\\ref{cond_c}) with $y=-2, 0, 2$, $\\ket{\\Phi}$ is written as \n\\begin{align}\n\\ket{\\Phi} = \\! \\! \\! \\sum_{\\substack{A_{1} , \\dots , A_{n} \\subset \\widetilde{\\mathcal{E}}\\\\\nA_{\\alpha }\\cap A_{\\beta} = \\emptyset\n}} \\! \\! f(\\{ A_{\\alpha}\\}) \n\\! \\left(\\prod_{y \\in A_{1}} \\tilde{a}_{y,1}^{\\dag}\\right)\n\\! \\dots \\!\n\\left(\\prod_{y \\in A_{n}} \\tilde{a}_{y,n}^{\\dag}\\right)\n\\! \\ket{\\Phi_{\\mathrm{vac}}},\n\\end{align}\nwhere $\\widetilde{\\mathcal{E}} = \\{-2, 0, 2\\}$ and $A_{\\alpha}$ is an arbitrary subset of $\\widetilde{\\mathcal{E}}$.\nSince $\\widetilde{\\mathcal{E}}$ contains three sites, the particle number of finite energy states must be less than or equal to three. \nUsing the condition Eq. (\\ref{cond_c}) with $y = \\pm 1$, we see that all the finite energy states $\\ket{\\Phi}$ must be a fully polarized state \nover the five sites and have zero energy when the particle number is three.\nFor one-particle states, all the eigenvalues of $h_{0}$ are non negative.\nThus we only need to verify the positive semi-definiteness for two-particle sectors labeled by $(N_{\\alpha}, N_{\\beta}) = (2,0)$ and $(N_{\\alpha}, N_{\\beta}) = (1,1)$.\nTo this end, we solve the eigenvalue problem for $Ph_{0}P$ where $P$ denotes the projection operator onto the space of finite energy states.\nIn the sector $(2, 0)$, we find that there are three eigenstates \n\\begin{align}\n\\ket{\\Phi_{1}} &= \\tilde{a}_{-2, \\alpha}^{\\dag} \\tilde{a}_{0, \\alpha}^{\\dag} \\ket{\\Phi_{\\mathrm{vac}}}, \\\\\n\\ket{\\Phi_{2}} &= \\tilde{a}_{0, \\alpha}^{\\dag} \\tilde{a}_{2, \\alpha}^{\\dag} \\ket{\\Phi_{\\mathrm{vac}}}, \\\\\n\\ket{\\Phi_{3}} \\! &= \\! \\left[\n\t\t\\frac{\\nu^{2}}{\\nu^{2}+1} \n(\\tilde{a}_{-2, \\alpha}^{\\dag}  \\!-\\! \\tilde{a}_{2, \\alpha}^{\\dag}) \\tilde{a}_{0, \\alpha}^{\\dag}\n\t\t\\! - \\! (2\\nu^{2}+1) \\tilde{a}_{-2, \\alpha}^{\\dag} \\tilde{a}_{2, \\alpha}^{\\dag} \n\t\t\\right] \\ket{\\Phi_{\\mathrm{vac}}}\n\\end{align}\nand their corresponding eigenenergies are 0, 0, and $s(2\\nu^{2} + 1)$, respectively.\nIn the sector $(1,1)$, there are four eigenstates.\nThree of them can be obtained by applying $F^{\\beta, \\alpha}$ to the states $\\ket{\\Phi_{1}}, \\ket{\\Phi_{2}}$ and $\\ket{\\Phi_{3}}$.\nAs a state orthogonal to them, we get a singlet state \n\\begin{align}\n\\ket{\\Phi_{4}} = \\left(\\tilde{a}_{-2, \\alpha}^{\\dag}\\tilde{a}_{2, \\beta}^{\\dag} - \\tilde{a}_{-2, \\beta}^{\\dag} \\tilde{a}_{2, \\alpha}^{\\dag} \\right) \\ket{\\Phi_{\\mathrm{vac}}},\n\\end{align}\nand we find that this state satisfies \n\\begin{align}\nP h_{0} P \\ket{\\Phi_{4}} = s(2\\nu^{2} + 1) \\ket{\\Phi_{4}}.\n\\end{align}\nClearly, the state $\\ket{\\Phi_{4}}$ has a positive energy.\nHence, we see that all the eigenvalues of $h_{0}$ are nonnegative.\nThus, we have proved Lemma 2.\n\\hspace{\\fill} $\\blacksquare$\n\\par\n\\section{Conclusion} \\label{sec:conclusion}\nWe have presented an extension of flat-band ferromagnetism to the SU($n$) Hubbard model on the railroad-trestle lattice.\nFurthermore, we proved that in the nearly flat-band case, all the ground states are fully polarized if $t$ and $U$ are sufficiently large. \nOne can similarly construct and analyze models in higher dimensions, in which \nthe ground states are fully polarized if the lowest band is completely flat. \nThe previous results for the SU(2) Hubbard models in higher dimensions suggest that the parameter $\\nu$ has to be larger than a threshold value $\\nu_{c}>0$ when the lowest band is nearly flat~\\cite{Shen1998, Tasaki2003}. The details will be discussed elsewhere. \n\n\nAlthough we have focused on models with a nonzero band gap, it would be interesting to see if the method developed in this paper can be extended to include SU($n$) Hubbard models with gapless flat or nearly flat bands~\\cite{tanaka2003stability}. \nIt would also be interesting to study SU($n$) ferromagnetism in systems with topological flat bands carryng nontrivial Chern number, as its SU($2$) counterpart has been discussed in~\\cite{katsura2010ferromagnetism}. \nAnother direction for future research is to explore ferromagnetism in multiorbital Hubbard models, including the one with SU($n$) symmetry. In such systems, rigorous~\\cite{li2014exact, li2015exact} and numerical results~\\cite{xu2015sign} about ferromagnetism, which are different from the flat-band scenario, have been obtained recently. It is thus interesting to see to what extent our results can be generalized to the multiorbital case. \n\n\\acknowledgments\nWe would like to thank Hal Tasaki and Akinori Tanaka for valuable discussions. H.K. was supported in part by JSPS Grant-in-Aid for Scientific Research on Innovative Areas: No. JP18H04478 and JSPS KAKENHI Grant No. JP18K03445.\n\\bibliographystyle{apsrev4-1}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\\label{sec:introduction}\n\nSpinning neutron stars with nonaxisymmetric deformations are expected to emit quasi-monochromatic and\nlong-lasting gravitational waves (GWs), commonly referred to as \\emph{continuous waves}\n(CWs)~\\cite{prix06:_cw_review, bildsten1998:_gwns, ushomirsky2000:_deformations, JMcD2013:_maxelastic}.\nOne of the main search methods for CWs was developed for ground-based detectors (such as LIGO~\\cite{LIGORef:2009},\nVirgo~\\cite{VirgoRef:2011}, GEO\\,600~\\cite{GEORef:2010}) and it is the so-called $\\F$-statistic.\nThe $\\F$-statistic was originally derived as a maximum-likelihood detection statistic\n\\cite{jks98:_data,cutler05:_gen_fstat}; it was later shown that it can also be derived as a Bayes\nfactor using somewhat unphysical  priors for the signal amplitude parameters \\cite{prix09:_bstat}.\n\nIn the context of CW searches, the GW data is reasonably well described by an underlying Gaussian noise\ndistribution with additional non-Gaussian disturbances (see, e.g., Fig.~3 in\nRef.~\\cite{abbott2004:_geoligo}, Fig.~3 in Ref.~\\cite{aasi13:_eathS5} and Sec.~5.8 in\nRef.~\\cite{behnke2013:_phdthesis}).\nThe $\\F$-statistic corresponds to a binary hypothesis test between a signal hypothesis and a Gaussian-noise\nhypothesis.\nAs a consequence, it is possible to obtain large $\\F$-statistic values due to non-Gaussian disturbances in the\ndata, even if they are not well matched to the signal model. Large $\\F$-statistic values only imply that the\nsignal hypothesis is a \\emph{better} fit to the data than pure Gaussian noise, but they do not imply a\n\\emph{good} fit.\n\nThe most problematic instrumental artifacts for any specific analysis of GW data are typically\nthose that resemble the signal family it searches for, i.e., disturbances with non-negligible projection\nonto the signal templates of a search.\nFor example, searches for short transient signals, such as bursts (e.g., from core-collapse\nsupernovae~\\cite{fryer2011:_collapsereview}) or compact binary coalescences\n(CBCs~\\cite{abadie2010:_cbcrates}),\nare most affected by ``glitches'' in the data, i.e., short broad-band\ndisturbances~\\cite{Blackburn2008:_glitch, Aasi2012:_virgochar, Slutsky2010:_falsealarms,\nprestegard2012:_transartifact}.\n\nOn the other hand, searches for CW signals are mainly affected by so-called ``lines'', i.e., narrow-band\ndisturbances that are present for a sizable fraction of the observation time.\nExamples include the so-called \\emph{mains} lines (i.e., lines at multiples of the 60\\,Hz electrical\npower system frequency for LIGO, or 50\\,Hz for Virgo and GEO600), the resonance frequencies of the\ndetector suspensions (different for each detector), and lines from digital components -- see\nRef.~\\cite{aasi13:_eathS5} for more details and a list of known instrumental lines identified in the data from\nthe fifth LIGO science run (S5), and Refs.~\\cite{Aasi2012:_virgochar, Accadia2012:_noemi} for line\nidentification in Virgo data.\n\nIn this article, we apply a Bayesian model-selection approach using an additional alternative noise hypothesis\nfor lines.\nSince the characteristics of the population of instrumental lines affecting CW searches are not well\nunderstood, we use a very simple line model. The model is based on an observed\ndistinguishing feature of many lines, namely that they do not affect all detectors in the same way.\nHence, we define our line model as any feature in the data that resembles the signal template \\emph{in only\none detector}.\n\nThis approach can also be seen as adding a coincidence criterion to the coherent multidetector\n$\\F$-statistic. Such a method is applicable only to multidetector CW searches, and in practice the most\nrecent $\\F$-statistic-based searches have all used data from multiple detectors.\nBy employing this approach, we obtain a new line-robust detection statistic, generalizing the\n$\\F$-statistic.\n\nThe plan of this paper is as follows: In Sec.~\\ref{sec:lit-review} we give a short review of existing\nmethods to deal with glitches and lines in GW data.\nIn Sec.~\\ref{sec:hypotheses} we describe signal-noise hypotheses relevant for the detection problem at hand:\nthe standard Gaussian-noise hypothesis, in Sec.~\\ref{sec:hypo_gauss}, the CW\nsignal hypothesis, in Sec.~\\ref{sec:hypo_signal} and a new simple line hypothesis in Sec.~\\ref{sec:hypo_line}.\nIn Sec.~\\ref{sec:line-veto-stats-coh} we use these hypotheses to construct two new detection statistics, a\n``line-veto'' statistic and a more general line-robust statistic.\nWe generalize the hypotheses and statistics to the case of semicoherent searches in\nSec.~\\ref{sec:semicoherent}. Next we discuss the choice of prior parameters for the line-robust statistic and we\npresent a simple method, albeit somewhat ad hoc, to choose decent priors in Sec.~\\ref{sec:tuning}. This\nconcludes the analytical part of the paper. In Sec.~\\ref{sec:tests} we assess the performance of the new\nstatistics through a series of numeric tests: on fully synthetic data in Sec.~\\ref{sec:tests_simdata} and on\nLIGO S5 data in\nSec.~\\ref{sec:tests_realdata}. We summarize our findings in Sec.~\\ref{sec:conclusions} and give a short\noutlook on applications and future generalizations of this approach.\nThe appendix~\\ref{sec:expect-f-stat} contains a short derivation of the expected $\\F$-statistic value under\nthe\nsimple line hypothesis.\n\n\\section{Existing methods to mitigate detector artifacts}\n\\label{sec:lit-review}\n\nThe problem of non-Gaussian artifacts in the data affects both searches for long-lived signals (e.g., the CWs\nwhich are the topic of this paper) as well as searches for short-lived signals.\nShort-lived signals are expected from the late phase of the inspiral of binaries of compact objects such as\nneutron stars and black holes as well as from catastrophic events such as supernovae.\nFor short-lived signal searches the artifacts that are responsible for an increase in the false alarm rates\nwith respect to purely Gaussian noise manifest themselves as loud glitches in the time-domain data.\nOn the other hand, for long-lived signal searches the most troublesome artifacts are broadly speaking those\nthat appear in the Fourier spectra on the typical time scales of the search.\n\nAn interesting distinction in search pipelines is the order in which multidetector coherence and coincidence\nare used. If the first step in the search is a coherent multidetector statistic (as\nfor the CW searches that we consider here), then the noise-artifact-mitigation strategy\nmay use subsequent consistency checks on the statistics from the individual detectors. If,\non the other hand, the first step was a single-detector search followed by a selection of triggers\nwhere only coincidence in the various detectors is required, an additional multidetector coherent statistic\ncan serve as an artifact-mitigation technique.\n\nA wide range of methods have been developed in order\nto deal with instrumental artifacts. In the following sub-sections, we give a short review of\nsuch methods for CBC, burst and CW searches.\n\nGenerally, we can distinguish between two fundamentally different approaches to artifact mitigation: Bayesian\nmodel-selection and heuristic methods.\nThe former is based on \\emph{explicit} alternative models whereas the latter consists in constructing\nad-hoc statistics to detect certain observed deviations from the GW signal model, which in the\nBayesian picture corresponds to a test against \\emph{implicit} (and often unknown) alternative hypotheses.\nA third ``hybrid'' approach uses Bayesian inference to directly construct empirical noise and\nsignal likelihoods \\cite{cannon2008:_bayescoinc} using the actual data and simulated GW signals,\nso-called \\emph{injections}.\n\n\n\\subsection{Instrumental glitches in burst and CBC searches}\n\\label{sec:glitch-short-trans}\n\nIn searches for short-lived GW signals popular ad-hoc glitch-veto methods are the\n$\\chi^2$-veto \\cite{allen2005:_chi2}, the null-stream veto \\cite{wen2005:_nullstream}, and signal amplitude\nconsistency vetoes \\cite{abbott2005:_cbcS2}.\nThese (among others) are commonly used in CBC searches (e.g., see Refs.~\\cite{babak2013:_searchcbc,\nharry2011:_targetcbc, abadie2012:_lowmasscbc}) and in burst searches (e.g., see\nRefs.~\\cite{sutton2010:_xpipeline, abbott2009:_s5burst, abadie2012:_allskyburst}).\n\nFor instance, in low-mass CBC searches the first step is a separate search in each detector.\nAfter a cut on single-detector $\\chi^2$ values, the glitch mitigation strategy consists in applying a\ncoincidence criterion and then\nconstructing on the surviving candidates a new multidetector statistic. This folds in the single-detector\nstatistics and $\\chi^2$ values. Significance thresholds are set based on Monte-Carlo studies on actual data and\ninjections.\n\nIn searches for signals for which we lack a waveform model (i.e., generic bursts), a main multidetector\nstatistic is constructed that accounts appropriately for time delays and antenna responses of the different\ndetectors to the same putative GW. This statistic is then augmented by other statistics\n(see Ref.~\\cite{sutton2010:_xpipeline} for details) specifically designed to\nfurther check for signal consistency across the detectors by means of appropriate veto conditions.\n\nVarious explicit glitch models have been considered, including Sine-Gaussians \\cite{clark2007:_ringdown,\ndalcanton2014:_sinegaussian} and wavelets \\cite{littenberg2010:_artifacts}, and Bayesian approaches have also\nbeen proposed to use these in constructing glitch-robust searches\n\\cite{clark2007:_ringdown, littenberg2010:_artifacts}. Notably, Veitch and Vecchio\n\\cite{veitch2010:_bayesian} have defined a glitch model describing coincident single-detector candidates with\nindependent amplitude parameters in different detectors.\nOn the other hand, the signal model requires candidates to be both coincident and coherent across all\ndetectors.\nBoth hypotheses would fit a true signal equally well, but the glitch hypothesis would be weighed down by its\nlarger prior volume (``Occam's razor''). In the case of glitches, however, the glitch hypothesis will\ngenerally provide a much better fit, allowing it to overcome its larger prior volume.\n\n\\subsection{Instrumental lines in CW searches}\n\\label{sec:deal-with-instr}\n\nThe most commonly used approaches to deal with instrumental lines in CW searches are all heuristic and can be\nsummarized as follows:\n\\begin{enumerate}[(i)] \\itemsep1pt \\parskip0pt\n\\item \\emph{Line cleaning} This is a widely used approach with many variants. It consists in effectively\n  excluding frequency bands from the search when they are known or believed to be affected by instrumental\n  lines.\n  This could be either as a result of previous detector characterization work or because the frequency-domain\n  data was flagged as particularly disturbed (referred to as {\\emph{line flagging}}).\n  Among the examples of this approach are the LIGO\/Virgo searches\n  \\cite{aasi13:_eathS5, abadie12:_powerflux,aasi2013:_gc-search, abbot2008:_s4cw, abadie2010:_casa}.\n\n  A downside of this method is the relatively large fraction of the total frequency band it\n  typically vetoes. For example, in Ref.~\\cite{abadie12:_powerflux} it vetoed a total of 270\\,Hz out of the\n  1140\\,Hz searched, i.e., $\\sim24\\%$ of the data.\n  Furthermore, this method is either limited to known instrumental lines or, when the line-flagging variant\n  is used, its efficacy is limited to strong  disturbances.\n  Weaker disturbances can only be identified with time baselines much longer than the ones typically\n  used by the line-flagging algorithms. Furthermore, the Fourier-transform-based line-flagging algorithm is not\n  optimally suited to detect lines with nonconstant frequency.\n\n\\item \\emph{S-veto} This is a method to remove candidates from a (frequency and spin-down dependent) region of\n  the sky.\n  This region is typically around the poles, where the corresponding signal templates are not well\n  distinguished from typical instrumental line artifacts. This method was initially developed in PowerFlux\n  \\cite{abbot2008:_s4cw} and subsequently adapted to $\\F$-statistic searches \\cite{abbott2009:_s4eath}.\n\n  The fraction of the total parameter space vetoed a priori through this approach can again be\n  quite large, for example about $\\sim 30\\%$ in Ref.~\\cite{abbott2009:_s4eath}.\n\n\\item \\emph{$\\F$-statistic consistency veto} If a candidate from a multidetector search has a\n  single-detector $\\F$-statistic value exceeding its multidetector $\\F$-statistic, then it is\n  vetoed as a likely instrumental line. This approach was described in more detail and tested in\n  Ref.~\\cite{aasi13:_eathS5} and Refs.~\\cite{behnke2013:_phdthesis, aasi2013:_gc-search}.\n\\end{enumerate}\n\nThe approach proposed in the present work is not a heuristic method as the ones described above. Instead, it\nshares some similarities to the glitch-robust method proposed in Ref.~\\cite{veitch2010:_bayesian}, but it\ndiffers in the following:\nIn the incoherent CBC pipeline, any candidate is already required to be \\emph{coincident} between detectors,\nso the method of Ref.~\\cite{veitch2010:_bayesian} adds the requirement of multidetector \\emph{coherence} to\ndistinguish GW signals from glitches.\nIn our case, instead, we start from the \\emph{coherent} multidetector $\\F$-statistic and add a\n\\emph{coincidence} requirement to distinguish CW signals from (noncoincident) lines.\n\nCurrently we do not include coincident lines in the alternative hypothesis, as we expect that this would\nsubstantially weaken the detection power of this method.\nMore work is required to deal with coincident lines that trigger the same templates in multiple detectors.\n\nHowever, the prevalence of coincident lines in detector data appears to be limited.\nFor example, the lines of known instrumental origin identified in the LIGO S5 data from the two detectors (see\nTables VI and VII in Ref.~\\cite{aasi13:_eathS5}) overlap by 1.6~Hz, corresponding to about 11\\% of the\ncontaminated bandwidth.\nFurthermore, in the $\\F$-statistic based analysis~\\cite{aasi13:_eathS5}, 0.46\\% of final high-significance\ncandidates passed the $\\F$-statistic consistency veto and therefore could be considered as caused by\ncoincident lines.\n\nThe approach taken here is that in a full CW search pipeline the noncoincident line model would serve as a\ncheap and simple ``first line of defense'' to reduce the number of spurious candidates, while more\nsophisticated steps can be applied to the surviving candidates in later steps.\n\n\\section{Hypotheses about the observed data}\n\\label{sec:hypotheses}\n\nLet $x^X(t)$ be the time series of GW strain measured in a detector $X$, where we use\n$X,Y,\\ldots$ as detector indices. Following the multidetector notation from\nRefs.~\\cite{cutler05:_gen_fstat,prix06:_searc}, using boldface indicates a multidetector vector, i.e.,\nwe write $\\detVec{x}(t)$ for the multidetector data vector with components $x^X(t)$.\n\nWe will consider three different\nhypotheses about the observed data $\\detVec{x}$ and derive their posterior probabilities: the Gaussian noise\nhypothesis $\\Hyp_\\Gauss$, the CW signal hypothesis $\\Hyp_\\Signal$ and a simple ``line'' hypothesis $\\Hyp_\\Line$.\n\n\\subsection{The Gaussian noise hypothesis \\texorpdfstring{$\\Hyp_\\Gauss$}{HG}}\n\\label{sec:hypo_gauss}\n\nThe Gaussian-noise hypothesis $\\Hyp_\\Gauss$ states that the measured multidetector time series $\\detVec{x}(t)$ only\ncontains stationary Gaussian noise, which we denote as $\\detVec{n}(t)$, i.e.,\n\\begin{equation}\n  \\label{eq:hypG}\n  \\Hyp_\\Gauss: \\detVec{x}(t) = \\detVec{n}(t)\\,,\n\\end{equation}\nwith a single-sided power-spectral density (PSD) $\\detVec{S}_n$ that is assumed to be known.\nThe corresponding likelihood for measuring the data $\\detVec{x}$ can therefore be written\nas\n\\begin{equation}\n  \\label{eq:gaussian}\n  \\prob{\\detVec{x}}{\\Hyp_\\Gauss} = \\kappa\\,\\eto{-\\frac{1}{2}\\scalar{\\detVec{x}}{\\detVec{x}}}\\,,\n\\end{equation}\nwhere $\\kappa$ is a data-independent normalization constant, and the scalar product is defined as\n\\begin{equation}\n  \\label{eq:scalarproduct}\n  \\scalar{\\detVec{x}}{\\detVec{y}} \\equiv \\sum_X \\frac{1}{\\SnX}\\int_0^T x^X(t)\\,y^X(t)\\,dt\\,,\n\\end{equation}\nassuming that the noise spectra $\\SnX$ are uncorrelated between different detectors $X$ and constant over the\n(narrow) frequency band of interest.\nFor simplicity of notation we omit the sometimes customary notation of a conditional ``$I$'' denoting all\nimplicit and explicit model assumptions, i.e., we write $\\prob{a}{b}$ as a shortcut for $\\prob{a}{b\\,,I}$, and\n$\\probI{a}$ as an abbreviation for $\\prob{a}{I}$.\n\nThe posterior probability for $\\Hyp_\\Gauss$ given the observed data $\\detVec{x}$ follows from Bayes' theorem as\n\\begin{equation}\n  \\label{eq:pHG}\n  \\prob{\\Hyp_\\Gauss}{\\detVec{x}} = \\frac{\\probI{\\Hyp_\\Gauss}}{\\probI{\\detVec{x}}}\\,\\kappa\\,\\eto{-\\frac{1}{2}\\scalar{\\detVec{x}}{\\detVec{x}}} \\,,\n\\end{equation}\nwhere $\\probI{\\Hyp_\\Gauss}$ is the prior probability for the Gaussian-noise hypothesis.\nThe normalization $\\probI{\\detVec{x}}$ depends on the full set of assumed hypotheses $\\{\\mathcal{H}_i\\}$, i.e.,\n\\mbox{$\\probI{\\detVec{x}} = \\sum_i \\prob{\\detVec{x}}{\\mathcal{H}_i}\\,\\probI{\\mathcal{H}_i}$}, but in the following we will only consider\nthe \\emph{odds} between different hypotheses, where this term drops out.\n\n\\subsection{The CW signal hypothesis \\texorpdfstring{$\\Hyp_\\Signal$}{HS}}\n\\label{sec:hypo_signal}\n\nThe hypothesis $\\Hyp_\\Signal$ for CW signals \\cite{jks98:_data,prix06:_cw_review} states that the data $\\detVec{x}$\ncontains a CW signal $\\detVec{h}$ in addition to Gaussian noise $\\detVec{n}$, namely $\\detVec{x} = \\detVec{n} + \\detVec{h}$.\n\nThe signal model $\\detVec{h}$ depends on a number of (generally unknown) signal parameters. For\npractical reasons, we usually distinguish between the set of four \\emph{amplitude parameters} $\\mathcal{A}$ and the\nremaining \\emph{phase-evolution parameters} $\\lambda$, i.e., we write the CW signal family as $\\detVec{h}(t;\\mathcal{A},\\lambda)$.\n\nTo fully specify the signal hypothesis, we therefore need a prior probability distribution\n$\\prob{\\mathcal{A},\\lambda}{\\Hyp_\\Signal}$ for the signal parameters, i.e.,\n\\begin{equation}\n  \\begin{split}\n  \\label{eq:hypS}\n  \\Hyp_\\Signal: \\detVec{x}(t) = \\detVec{n}(t) + \\detVec{h}(t;\\mathcal{A},\\lambda)\\\\\n  \\text{with prior}\\;\\;\\prob{\\mathcal{A},\\lambda}{\\Hyp_\\Signal}\\,.\n\\end{split}\n\\end{equation}\n\nThe amplitude parameters $\\mathcal{A}$ describe the signal amplitude $h_0$, the inclination angle $\\iota$, the\npolarization angle $\\psi$ and the initial phase $\\phi_0$. As first shown in Ref.~\\cite{jks98:_data}, a\nparticular parametrization $\\mathcal{A}^\\mu = \\mathcal{A}^\\mu ( h_0,\\cos\\iota,\\psi,\\phi_0)$, with $\\mu = 1\\ldots4$, allows one to\nwrite the signal model in the factorized form\n\\begin{equation}\n  \\label{eq:Amuhmu}\n  \\detVec{h}(t;\\mathcal{A},\\lambda) = \\mathcal{A}^\\mu\\,\\detVec{h}_\\mu(t;\\lambda)\\,,\n\\end{equation}\nin terms of four basis functions $\\detVec{h}_\\mu(t;\\lambda)$ and using the automatic summation convention over repeated\nindices.\n\nIn order to simplify the following discussion and notation, we follow the approach\nof Refs.~\\cite{prix09:_bstat,prix11:_transient} and formally restrict ourselves to a single-template statistic\nin $\\lambda$. This is equivalent to the assumption of known phase parameters, i.e., $\\lambda = \\lambda_\\mathrm{s}$. This can\nbe done without loss of generality, as for unknown $\\lambda\\in\\mathbb{P}$ this analysis would apply for each\ntemplate $\\lambda_i\\in\\mathbb{P}$, and one would then marginalize over the prior parameter space $\\mathbb{P}$.\nStudying this in further detail is outside the scope of the present work.\nWe will therefore assume a prior of the form\n\\begin{equation}\n  \\label{eq:priorAlambda}\n  \\prob{\\mathcal{A},\\lambda}{\\Hyp_\\Signal} = \\prob{\\mathcal{A}}{\\Hyp_\\Signal}\\,\\delta(\\lambda - \\lambda_\\mathrm{s})\\,,\n\\end{equation}\nand drop the phase-evolution parameters $\\lambda$ from the following expressions.\n\nWe can obtain the likelihood for \\emph{a particular} signal $\\detVec{h}(t;\\mathcal{A})$ by noting\nthat, according to $\\Hyp_\\Signal$, the combination $\\left[\\detVec{x} - \\detVec{h}(\\mathcal{A})\\right]$ is described by Gaussian noise.\nIn fact, by inserting the signal factorization from Eq.~\\eqref{eq:Amuhmu} and by factoring out terms\nequivalent to the Gaussian noise likelihood from Eq.~\\eqref{eq:pHG}, we obtain\n\\begin{align}\n  \\prob{\\detVec{x}}{\\Hyp_\\Signal,\\mathcal{A}}  &= \\kappa \\, \\eto{-\\frac{1}{2} \\scalar{\\detVec{x}-\\detVec{h}(\\mathcal{A})}{\\detVec{x}-\\detVec{h}(\\mathcal{A})}} \\notag\\\\\n                           &= \\kappa \\, \\eto{-\\frac{1}{2} \\scalar{\\detVec{x}}{\\detVec{x}}} \\,\n                              \\eto{\\scalar{\\detVec{x}}{\\mathcal{A}^\\mu\\detVec{h}_\\mu} - \\frac{1}{2}\n                              \\scalar{\\mathcal{A}^\\mu\\detVec{h}_\\mu}{\\mathcal{A}^\\nu\\detVec{h}_\\nu}}   \\label{eq:likeli_HSA}\\\\\n                           &= \\prob{\\detVec{x}}{\\Hyp_\\Gauss} \\exp\\left[\\mathcal{A}^\\mu \\, x_\\mu - \\frac{1}{2} \\mathcal{A}^\\mu\n                             \\mathcal{M}_{\\mu\\nu}\\mathcal{A}^\\nu\\right],\\notag\n\\end{align}\nwhere we introduced the four projections $x_\\mu$ of the data and the (symmetric positive-definite) matrix\n$\\mathcal{M}_{\\mu\\nu}$ as\n\\begin{equation}\n  \\label{eq:xmuMmunu}\n  x_\\mu \\equiv \\scalar{\\detVec{x}}{\\detVec{h}_\\mu} \\quad \\text{and} \\quad\n  \\mathcal{M}_{\\mu\\nu} \\equiv \\scalar{\\detVec{h}_\\mu}{\\detVec{h}_\\nu}\\,.\n\\end{equation}\n\nThe \\emph{marginal} likelihood $\\prob{\\detVec{x}}{\\Hyp_\\Signal}$ (sometimes referred to as ``evidence'') for the\nsignal hypothesis from Eq.~\\eqref{eq:hypS} can be obtained by marginalizing over the unknown\namplitudes $\\mathcal{A}$, namely\n\\begin{equation}\n \\prob{\\detVec{x}}{\\Hyp_\\Signal} = \\int \\prob{\\detVec{x}}{\\Hyp_\\Signal,\\mathcal{A}}\\,\\prob{\\mathcal{A}}{\\Hyp_\\Signal}\\,d\\mathcal{A}\\,. \\label{eq:likeli_HSmarg}\n\\end{equation}\nThis integral can be solved analytically for certain choices of amplitude priors $\\prob{\\mathcal{A}}{\\Hyp_\\Signal}$.\nIn particular, as discussed in Refs.~\\cite{prix09:_bstat,prix11:_transient}, for the (somewhat unphysical)\nprior that is uniform in $\\mathcal{A}^\\mu$, we can recover the standard $\\F$-statistic, namely, assuming\n\\begin{equation}\n  \\label{eq:priorA}\n  \\prob{\\{\\mathcal{A}^\\mu\\}}{\\Hyp_\\Signal} = \\left\\{\\begin{array}{ll}\n      C & \\text{for} \\quad h_0^4(\\mathcal{A}) < \\frac{70\\,c_*}{\\sqrt{|\\mathcal{M}|}}\\,.\\\\\n      0 & \\text{otherwise}\\,,\n    \\end{array}\\right.\n\\end{equation}\nwhere $|\\mathcal{M}|$ is the determinant of $\\mathcal{M}_{\\mu\\nu}$ and $c_*$ is an ad-hoc cutoff\\footnote{This\ntranslates to the notation of Ref.~\\cite{prix11:_transient} via $c_* = \\frac{{\\widehat{\\rho}_{\\mathrm{max}}}^4}{70}$.}\nused to normalize the prior, namely, \\mbox{$C = \\frac{\\sqrt{|\\mathcal{M}|}}{(2\\pi)^2}\\, c_*^{-1}$}.\n\nUsing this prior and taking the integration boundary to infinity, $c_*\\rightarrow\\infty$, we obtain\nthe (marginal) signal likelihood, from Eq.~\\eqref{eq:likeli_HSmarg}, in the form\n\\begin{equation}\n  \\label{eq:likeli_HS}\n  \\prob{\\detVec{x}}{\\Hyp_\\Signal} = \\prob{\\detVec{x}}{\\Hyp_\\Gauss} \\, c_*^{-1}\\,\\eto{\\F(\\detVec{x})}\\,,\n\\end{equation}\nwhere we define the (coherent) multidetector $\\F$-statistic as\n\\begin{equation}\n  \\label{eq:Fstat}\n  2\\F(\\detVec{x}) \\equiv x_\\mu\\,\\mathcal{M}^{\\mu\\nu}\\,x_\\nu\\,,\n\\end{equation}\nand $\\mathcal{M}^{\\mu\\nu}$ denotes the inverse matrix to $\\mathcal{M}_{\\mu\\nu}$, i.e.,\n$\\mathcal{M}_{\\mu\\alpha}\\mathcal{M}^{\\alpha\\nu} = \\delta_{\\mu}^{\\nu}$.\nWe obtain the posterior probability for the signal hypothesis as\n\\begin{equation}\n  \\label{eq:pHS_final}\n  \\prob{\\Hyp_\\Signal}{\\detVec{x}} = \\prior{\\OSG}\\,c_*^{-1}\\,\\prob{\\Hyp_\\Gauss}{\\detVec{x}}\\,\\eto{\\F(\\detVec{x})}\\,,\n\\end{equation}\nwhere $\\prior{\\OSG}\\equiv \\probI{\\Hyp_\\Signal}\/\\probI{\\Hyp_\\Gauss}$ denotes the prior odds between the signal- and\nGaussian-noise hypotheses.\n\nThe posterior odds between signal hypothesis $\\Hyp_\\Signal$ and Gaussian-noise hypothesis $\\Hyp_\\Gauss$ are therefore\nequivalent to the standard multidetector $\\F$-statistic\\footnote{\\emph{Equivalence} in the Neyman-Pearson\nsense: the same false-dismissal as a function of false-alarm probability.}, as we see by writing\n\\begin{equation}\n  \\label{eq:OSG}\n  \\OSG(\\detVec{x}) \\equiv \\frac{\\prob{\\Hyp_\\Signal}{\\detVec{x}}}{\\prob{\\Hyp_\\Gauss}{\\detVec{x}}}\n                  = \\prior{\\OSG}\\,c_*^{-1}\\, \\eto{\\F(\\detVec{x})}\\,.\n\\end{equation}\nNote that the corresponding (marginal) likelihood ratio\n\\begin{equation}\n  \\label{eq:2}\n  \\Bayes_{\\Signal\\Gauss}(\\detVec{x}) \\equiv \\frac{\\prob{\\detVec{x}}{\\Hyp_\\Signal}}{\\prob{\\detVec{x}}{\\Hyp_\\Gauss}} = c_*^{-1}\\, \\eto{\\F(\\detVec{x})}\\,,\n\\end{equation}\nis generally known as the \\emph{Bayes factor}, and is closely related to the odds via $\\OSG(\\detVec{x}) = \\prior{\\OSG}\\,\\Bayes_{\\Signal\\Gauss}(\\detVec{x})$.\n\nWhile this statistic is close to optimal for detecting signals in pure Gaussian noise \\cite{prix09:_bstat}, it\nis vulnerable to various signal-like instrumental artifacts in the data.\nAs discussed in Sec.~\\ref{sec:introduction}, we see from Eqs.~\\eqref{eq:OSG} and \\eqref{eq:2} that detector\nartifacts can trigger $\\OSG(\\detVec{x})$ or $\\Bayes_{\\Signal\\Gauss}(\\detVec{x})$, provided they resemble $\\Hyp_\\Signal$ \\emph{more} than $\\Hyp_\\Gauss$\neven if the agreement with $\\Hyp_\\Signal$ is poor.\nIn order to deal with this problem, we need to introduce an alternative hypothesis, which describes\ninstrumental lines \\emph{better} than $\\Hyp_\\Signal$.\n\n\\subsection{Simple line hypothesis: A CW-like disturbance in a single detector}\n\\label{sec:hypo_line}\n\nHere we introduce a simple line hypothesis designed to match one prominent feature of many instrumental lines,\ndistinguishing them from CW signals: the fact that they appear only in one detector.\nInspired by this, we reuse the signal hypothesis from Eq.~\\eqref{eq:hypS} in order to define a line in\ndetector $X$ :\n\\begin{equation}\n  \\begin{split}\n  \\label{eq:hypLX}\n  \\Hyp_\\Line^{X} :  x^{X}(t) = n^{X}(t) + h^{X}(t;\\mathcal{A}^{X})\\\\\n  \\text{with prior}\\quad\\prob{\\mathcal{A}^{X}}{\\Hyp_\\Line^{X}}\\,.\n\\end{split}\n\\end{equation}\nWe would expect lines to have a different amplitude distribution than real signals, but in the absence of any\nmore detailed knowledge on this point, we choose to reuse the signal amplitude prior given by\nEq.~\\eqref{eq:priorA} for $\\prob{\\mathcal{A}^X}{\\Hyp_\\Line^X}$. This choice simplifies the following\ncalculations. In analogy to Eq.~\\eqref{eq:pHS_final}, we directly obtain the probability for $\\Hyp_\\Line^X$:\n\\begin{equation}\n  \\label{eq:pHLX}\n  \\prob{\\Hyp_\\Line^X}{x^X} = c_*^{-1}\\,\\prob{\\Hyp_\\Gauss^X}{x^X}\\,\\prior{\\OLG}^X\\,\\eto{\\F^X(x^X)}\\,.\n\\end{equation}\nHere we define the per-detector prior line odds \\mbox{$\\prior{\\OLG}^X \\equiv {\\probI{\\Hyp_\\Line^X}}\/{\\probI{\\Hyp_\\Gauss^X}}$},\nwhich encode prior knowledge about how likely a line is, compared to pure Gaussian noise, in a given\ntemplate $\\lambda$ and detector $X$. The detector-specific $\\F$-statistic $\\F^X(x^X)$ is simply given by\nEq.~\\eqref{eq:Fstat} restricted to detector $X$.\n\nFor multiple detectors we can now formulate the simple line hypothesis $\\Hyp_\\Line$ as a CW-like disturbance\n$\\Hyp_\\Line^{X}$ in any one detector $X$ and data consistent with Gaussian noise $\\Hyp_\\Gauss^Y$ in all other detectors\n$Y\\not=X$:\n\\begin{equation}\n  \\begin{split}\n    \\Hyp_\\Line \\equiv& \\left( \\Hyp_\\Line^1 \\;\\mathrm{and}\\; \\Hyp_\\Gauss^2 \\;\\mathrm{and}\\; \\Hyp_\\Gauss^3 \\ldots\\right) \\;\\mathrm{or}\\; \\\\\n    & \\left( \\Hyp_\\Gauss^1 \\;\\mathrm{and}\\; \\Hyp_\\Line^2 \\;\\mathrm{and}\\; \\Hyp_\\Gauss^3 \\ldots \\right) \\;\\mathrm{or}\\; \\ldots\\,. \\label{eq:hypL}\n  \\end{split}\n\\end{equation}\nNote that in this approach $\\Hyp_\\Line$ does not include lines that are coincident across different detectors,\nwhich is postponed to future work.\n\nWe assume the different detectors to be independent to the extent that\nknowing $\\Hyp_\\Gauss^X$ or $\\Hyp_\\Line^X$ for detector $X$\ndoes not inform us about $\\Hyp_\\Gauss^Y$ or $\\Hyp_\\Line^Y$ for other detectors $Y\\not=X$. We also assume the different\nalternatives in Eq.~\\eqref{eq:hypL} to\nbe mutually exclusive. The laws of probability therefore yield\n\\begin{align}\n  \\prob{\\Hyp_\\Line}{\\detVec{x}} &= \\prob{\\Hyp_\\Line^1}{x^1} \\prob{\\Hyp_\\Gauss^2}{x^2} \\prob{\\Hyp_\\Gauss^3}{x^3}\\times \\ldots \\nonumber \\\\\n  & + \\prob{\\Hyp_\\Gauss^1}{x^1} \\prob{\\Hyp_\\Line^2}{x^2} \\prob{\\Hyp_\\Gauss^3}{x^3}\\times \\ldots \\nonumber \\\\\n  & + \\ldots \\nonumber \\\\\n  &= \\sum_{X} \\prob{\\Hyp_\\Line^{X}}{x^{X}} \\prod_{Y\\not=X} \\prob{\\Hyp_\\Gauss^Y}{x^Y}\\,. \\label{eq:pHL_initial}\n\\end{align}\n\nBy combining Eqs.~\\eqref{eq:hypL}, \\eqref{eq:pHLX} and the (per-detector) Gaussian-noise probability from\nEq.~\\eqref{eq:pHG}, we find the posterior probability for the line hypothesis $\\Hyp_\\Line$ as\n\\begin{equation}\n  \\label{eq:pHL_inserted}\n  \\prob{\\Hyp_\\Line}{\\detVec{x}} = c_*^{-1}\\,\\prob{\\Hyp_\\Gauss}{\\detVec{x}} \\, \\sum_X \\prior{\\OLG}^X\\,\\eto{\\F^X(x^X)}\\,,\n\\end{equation}\nwhere we used the fact that $\\prod_X \\prob{\\Hyp_\\Gauss^X}{x^X} = \\prob{\\Hyp_\\Gauss}{\\detVec{x}}$. Note that\n\\begin{equation}\n  \\label{eq:sumlX}\n  \\sum_X \\prior{\\OLG}^X = \\frac{\\probI{\\Hyp_\\Line}}{\\probI{\\Hyp_\\Gauss}} \\equiv \\prior{\\OLG}\\,,\n\\end{equation}\nwhere $\\prior{\\OLG}$ denotes the prior odds for a line versus Gaussian noise (in the present template $\\lambda$)\nincluding all detectors.\n\nIt will be convenient to define relative detector weights $r^X$ for the prior line odds, namely for\n${N_{\\mathrm{det}}}$ detectors:\n\\begin{equation}\n  \\label{eq:rX}\n  r^X \\equiv \\frac{\\prior{\\OLG}^X}{\\prior{\\OLG} \/ {N_{\\mathrm{det}}}}\\,,\\quad\\text{such that}\\quad \\sum_X r^X = {N_{\\mathrm{det}}}\\,.\n\\end{equation}\nIf all detectors are equally likely to contain a line, then $r^X = 1$ for all $X$.\nWe further denote the average of a quantity $Q^X$ over detectors as\n\\begin{equation}\n  \\label{eq:1}\n  \\avgX{Q^X} \\equiv \\frac{1}{{N_{\\mathrm{det}}}} \\sum_X Q^X\\,,\n\\end{equation}\nand hence $\\avgX{r^X} = 1$.\nBy using these definitions, we can write Eq.~\\eqref{eq:pHL_inserted} as follows:\n\\begin{equation}\n  \\label{eq:pHL_avg}\n  \\prob{\\Hyp_\\Line}{\\detVec{x}} = c_*^{-1} \\, \\prob{\\Hyp_\\Gauss}{\\detVec{x}} \\, \\prior{\\OLG}\\, \\avgX{r^X\\,\\eto{\\F^X(x^X)}}\\,.\n\\end{equation}\n\n\\section{Coherent line-robust statistics}\n\\label{sec:line-veto-stats-coh}\n\nWe use the posterior line probability of Eq.~\\eqref{eq:pHL_avg} to compute the odds for\nadditional model comparisons, thereby extending the standard multidetector $\\F$-statistic\ngiven by Eq.~\\eqref{eq:OSG}.\nIn particular, we consider two approaches:\n\\begin{enumerate}[(i)] \\itemsep1pt \\parskip0pt\n\\item Define a ``line-veto'' statistic as the odds between the signal hypothesis $\\Hyp_\\Signal$ and\n  the line hypothesis $\\Hyp_\\Line$.\n  This may be useful, for example, as a follow-up statistic for strong candidates from\n  an initial $\\F$-statistic search, which compared $\\Hyp_\\Signal$ versus Gaussian noise $\\Hyp_\\Gauss$.\n  In such a two-stage approach, one would test the signal hypothesis against the line-hypothesis\n  if the Gaussian-noise hypothesis has been ruled out with sufficient confidence.\n\n\\item \\emph{Extend} the standard signal-versus-Gaussian-noise odds $\\OSG(\\detVec{x})$ to a more\n  line-robust statistic $\\OSN(\\detVec{x})$ by allowing the noise hypothesis to include either pure Gaussian\n  noise $\\Hyp_\\Gauss$ or a line $\\Hyp_\\Line$.\n\\end{enumerate}\n\n\\subsection{Line-veto statistic \\texorpdfstring{$O_{\\Signal\\Line}(\\detVec{x})$}{OSL(x)}}\n\\label{sec:line-veto-stat}\n\nUsing the posterior probabilities given by Eqs.~\\eqref{eq:pHS_final} and \\eqref{eq:pHL_avg}, we obtain the\nposterior signal-versus-line odds as\n\\begin{equation}\n  \\label{eq:OSL}\n  O_{\\Signal\\Line}(\\detVec{x}) \\equiv \\frac{\\prob{\\Hyp_\\Signal}{\\detVec{x}}}{\\prob{\\Hyp_\\Line}{\\detVec{x}}} =\n  \\prior{\\OSL} \\; \\frac{ \\eto{\\F(\\detVec{x})} }{\\avgX{r^X\\,\\eto{\\F^X(x^X)}}}\\,,\n\\end{equation}\nwith the prior odds $\\prior{\\OSL} \\equiv \\probI{\\Hyp_\\Signal}\/\\probI{\\Hyp_\\Line} = \\prior{\\OSG}\/\\prior{\\OLG}$. Note\nthat the amplitude-prior cutoff $c_*$ has disappeared, as we have used the same amplitude\nprior on lines and signals.\n\nIn the following we will often neglect the dependency on $\\detVec{x}$ and $x^X$ to simplify notation.\nIt is instructive to consider the log-odds, which we can write as\n\\begin{align}\n \\lnO_{\\Signal\\Line} &= \\ln \\prior{\\OSL} + \\F - \\ln \\avgX{r^X\\eto{\\F^X}} \\nonumber \\\\\n               &= \\ln \\prior{\\OSL} + \\F - \\F'_{\\mathrm{max}} - \\ln \\avgX{r^X\\eto{\\left(\\F^X -\n                  \\F'_{\\mathrm{max}}\\right)}}\\,,\\label{eq:logOSL}\n\\end{align}\nwhere we define\n\\begin{equation}\n  \\label{eq:Fmaxp}\n  \\F'_{\\mathrm{max}} \\equiv \\max_X \\left( \\F^X + \\ln r^X \\right) \\,.\n\\end{equation}\nThe terms in the detector-average in Eq.~\\eqref{eq:logOSL} are bounded within $[0, 1]$, with at least one term\nbeing equal to $1$.\nHence, the logarithmic average $\\ln\\avgX{\\ldots}$ is bounded within\n$[-\\ln{N_{\\mathrm{det}}},0]$, i.e., of order 1.\n\nActually, for strong $\\F$-statistic candidates, i.e., $\\F\\gg1$, the logarithmic correction is\nnegligible, and therefore we can approximate\n\\begin{equation}\n  \\label{eq:logOSL_approx}\n  \\lnO_{\\Signal\\Line}(\\detVec{x}) \\approx \\F(\\detVec{x}) - \\F'_{\\mathrm{max}}(\\detVec{x}) + \\ln\\prior{\\OSL} \\,.\n\\end{equation}\nWithout prior knowledge about one detector being more affected by instrumental lines than others,\nwe would have $r^X = 1$ and therefore $\\F'_{\\mathrm{max}}(x) = \\max_X \\F^X(x^X)$.\nConsidered as a detection statistic, $O_{\\Signal\\Line}(\\detVec{x})$ is therefore approximately equivalent to the\ndifference between the multidetector $\\F$-statistic and the largest $\\F$-statistic value from the\nindividual detectors.\n\nBy choosing a special threshold of $O_{\\Signal\\Line}(\\detVec{x})=\\prior{\\OSL}$ and assuming equal prior line probabilities for all\ndetectors, we recover the well-known \\emph{$\\F$-statistic consistency veto}, namely\n\\begin{equation}\n  \\label{eq:F-veto}\n  \\textrm{If}\\quad \\F(\\detVec{x}) < \\max_{X}\\{\\F^X(x)\\}\\;\\;\\implies\\;\\textrm{veto the candidate}\\,,\n\\end{equation}\nwhich has been successfully used and tested in\nRefs.~\\cite{aasi13:_eathS5, behnke2013:_phdthesis, aasi2013:_gc-search}.\nCombining this veto with $\\F$-statistic ranking corresponds to defining a new statistic:\n\\begin{equation}\n  \\label{eq:Fveto}\n  \\F^{\\mathrm{+veto}}(\\detVec{x}) \\equiv \\left\\{\n    \\begin{array}{cc}\n      \\F(\\detVec{x})\\quad & \\textrm{if } \\F(\\detVec{x}) \\ge \\max_{X}\\{\\F^X(x)\\}\\,,\\\\\n      0        & \\textrm{otherwise}\\,,\n    \\end{array}\n  \\right.\n\\end{equation}\nwhich we will refer to as the $\\Fveto$-statistic.\n\n\n\\subsection{Line-robust detection statistic \\texorpdfstring{$\\OSN(\\detVec{x})$}{OSN(x)}}\n\\label{sec:extend-detect-stat}\n\nFrom the standpoint of probability theory it is more natural to use the line hypothesis to extend\nwhat we mean by ``noise'', namely, either pure Gaussian noise $\\Hyp_\\Gauss$ or a line $\\Hyp_\\Line$.\nHence, we introduce an extended noise hypothesis as\n\\begin{equation}\n  \\label{eq:hypN}\n  \\Hyp_\\Noise : \\left( \\Hyp_\\Gauss \\;\\mathrm{or}\\; \\Hyp_\\Line \\right) \\,.\n\\end{equation}\nSince we take $\\Hyp_\\Gauss$ and $\\Hyp_\\Line$ to be mutually exclusive, the posterior probability for $\\Hyp_\\Noise$ is\n\\begin{align}\n  \\prob{\\Hyp_\\Noise}{\\detVec{x}} &= \\prob{\\Hyp_\\Gauss}{\\detVec{x}} + \\prob{\\Hyp_\\Line}{\\detVec{x}} \\nonumber \\\\\n                     &= \\prob{\\Hyp_\\Gauss}{\\detVec{x}} \\left( 1 + c_*^{-1} \\,\\prior{\\OLG} \\,\\avgX{r^X \\eto{\\F^X(x^X)}} \\right) \\,,\n\\label{eq:pHN}\n\\end{align}\nwhere we have used Eq.~\\eqref{eq:pHL_avg} for the explicit line posterior.\n\nInterestingly, we can express the odds $\\OSN(\\detVec{x})$ of the signal versus extended noise hypotheses as\n\\begin{equation}\n  \\label{eq:OSN_initial}\n  \\OSN(\\detVec{x}) \\equiv \\frac{\\prob{\\Hyp_\\Signal}{\\detVec{x}}}{\\prob{\\Hyp_\\Noise}{\\detVec{x}}}\n  = \\left[ \\OSG^{-1}(\\detVec{x}) + O_{\\Signal\\Line}^{-1}(\\detVec{x}) \\right]^{-1}\\,.\n\\end{equation}\nWe can compare this result with the ad-hoc two-stage approach discussed previously, where one would\nset two independent thresholds on $\\OSG$ and on $O_{\\Signal\\Line}$. As we see from Eq.~\\eqref{eq:OSN_initial}, instead the\nlaws of probability tell us to compute the harmonic sum of $\\OSG$ and $O_{\\Signal\\Line}$ and to set a single threshold on\nthe resulting statistic.\n\nInserting the explicit expressions provided by Eqs.~\\eqref{eq:pHN} and \\eqref{eq:pHS_final}, we obtain\n\\begin{equation}\n  \\label{eq:OSN_cstar}\n  \\OSN(\\detVec{x}) = \\frac{\\prior{\\OSG} \\, \\eto{\\F(\\detVec{x})}}\n  {c_* + \\prior{\\OLG} \\avgX{r^X \\eto{\\F^X}}}\\,.\n\\end{equation}\nThe amplitude-prior cutoff parameter $c_*$ from Eq.~\\eqref{eq:priorA} is only a scale\nfactor in $\\OSG$ and thus not relevant for the performance as a detection statistic, and it is canceled\nout completely in $O_{\\Signal\\Line}$.\nHowever, in $\\OSN$ this parameter does affect the properties of the resulting statistic.\n\nWe can rewrite Eq.~\\eqref{eq:OSN_cstar} by introducing prior odds $\\prior{\\OSN} \\equiv \\probI{\\Hyp_\\Signal}\/\\probI{\\Hyp_\\Noise}$ and\n[noting that \\mbox{$\\prior{\\OSG} = \\prior{\\OSN}\\left( 1 + \\prior{\\OLG}\\right)$}] we obtain\n\\begin{equation}\n  \\label{eq:OSN_final}\n  \\OSN(\\detVec{x}) = \\prior{\\OSN} \\, \\frac{\\eto{\\F(\\detVec{x})}}\n  {(1-p_\\Line)\\,\\eto{\\Fth^{(0)}} + p_\\Line\\, \\avgX{r^X \\eto{\\F^X(x^X)}}}\\,,\n\\end{equation}\nwhere we define the prior line probability $p_\\Line$ as\n\\begin{equation}\n  \\label{eq:lineprob}\n  p_\\Line \\equiv \\frac{\\prior{\\OLG}}{1 + \\prior{\\OLG}} = \\frac{\\probI{\\Hyp_\\Line}}{\\probI{\\Hyp_\\Noise}} = \\prob{\\Hyp_\\Line}{\\Hyp_\\Noise}\\in [0,1]\\,,\n\\end{equation}\nand we used a more natural reparametrization of $c_*$ by defining\n\\begin{equation}\n  \\label{eq:Fth0}\n  \\Fth^{(0)} \\equiv \\ln c_*\\,.\n\\end{equation}\n\n\\subsubsection{Limiting cases of \\texorpdfstring{$\\OSN(\\detVec{x})$}{OSN}}\n\\label{sec:limiting-behavior}\n\nWe now consider the limiting behavior of $\\OSN$ as a function of the line prior $p_\\Line$ and of the\nsingle-detector $\\F^X(x)$ values.\nWe see from Eq.~\\eqref{eq:OSN_final} that $\\OSN(\\detVec{x})$ reduces to the $\\F$-statistic if we are certain that\nthere are no lines, i.e., $\\OSN(\\detVec{x}) \\rightarrow\\OSG(\\detVec{x})\\propto\\eto{\\F(\\detVec{x})}$ for $p_\\Line\\rightarrow0$.\nOn the other hand, it reduces to the pure line-veto statistic of Eq.~\\ref{eq:OSL} when we believe the noise to\nbe completely dominated by lines, i.e., $\\OSN(\\detVec{x}) \\rightarrow O_{\\Signal\\Line}(\\detVec{x})$ for $p_\\Line \\rightarrow 1$.\n\nFor fixed $p_\\Line$ we see that the transition between these two extremes depends on\nthe $\\F^X(x)$ values compared to the prior scale $\\Fth^{(0)}$. To illustrate this more clearly, we first rewrite\nEq.~\\eqref{eq:OSN_final} using the relations $\\prior{\\OSN}=p_\\Line\\,\\,\\prior{\\OSL}$ and $(1-p_\\Line)\/p_\\Line=\\prior{\\OLG}^{-1}$.\nIntroducing the ``transition scale'' $\\F_*$ as\n\\begin{equation}\n  \\label{eq:Fth}\n  \\F_* \\equiv \\Fth^{(0)} - \\ln {\\prior{\\OLG}}\\,,\n\\end{equation}\nwe obtain\n\\begin{equation}\n  \\label{eq:OSN_Fstar}\n  \\OSN(\\detVec{x}) = \\prior{\\OSL}\\,\\frac{\\eto{\\F(\\detVec{x})}}{\\eto{\\F_*} + \\avgX{r^X \\eto{\\F^X(x^X)}}}\\,.\n\\end{equation}\nFrom this reparametrization, we see that $\\F_*$ defines the scale of a smooth transition of $\\OSN(\\detVec{x})$\nbetween $\\OSG(\\detVec{x})\\propto\\eto{\\F(\\detVec{x})}$ and $O_{\\Signal\\Line}(\\detVec{x})$ depending on the values of $\\F^X$: namely the\n``line-veto term'' $\\avgX{r^X \\eto{\\F^X}}$ in Eq.~\\eqref{eq:OSN_Fstar} only starts to play a role when it is\ncomparable to $\\eto{\\F_*}$.\n\nTo see this more explicitly, we write the log-odds as\n\\begin{equation}\n  \\begin{split}\n  \\label{eq:logOSN}\n  \\ln &\\OSN(\\detVec{x}) = \\ln \\prior{\\OSL} + \\F(\\detVec{x}) - \\F''_{\\mathrm{max}}(x)\\\\\n   & - \\ln \\left( \\eto{\\F_*-\\F''_{\\mathrm{max}}} + \\avgX{r^X \\eto{\\F^X(x^X)-\\F''_{\\mathrm{max}}(\\detVec{x})} } \\right)\\,,\n  \\end{split}\n\\end{equation}\nwhere we define $\\F''_{\\mathrm{max}}(\\detVec{x}) \\equiv \\max\\left( \\F_*,\\, \\F^X(x^X) + \\lnr^X\\right)$.\nThe logarithmic correction is of order unity, therefore this effectively corresponds to\n$\\lnO_{\\Signal\\Line}(\\detVec{x})$ when \\mbox{$\\max( \\F^X(x) + \\ln r^X ) > \\F_*$}, and to $\\ln\\OSG(\\detVec{x})$ otherwise.\n\nIn practice it can be difficult to determine good prior values for $\\Fth^{(0)}$, due to the unphysical choice of\namplitude priors in Eq.~\\eqref{eq:priorA}. We will discuss this issue in more detail in\nSec.~\\ref{sec:choosing-prior-value}.\n\nThis transitioning behavior is reminiscent of the two-stage line-veto approach discussed in\nSec.~\\ref{sec:line-veto-stats-coh}. There one applies a line veto only to candidates that are ``strong'' in\nterms of $\\OSG(\\detVec{x})\\propto\\eto{\\F(\\detVec{x})}$, which means that the Gaussian-noise hypothesis\nis already considered sufficiently unlikely.\nNote, however, that for $\\OSN(\\detVec{x})$ the transition from $\\OSG(\\detVec{x})$ to $O_{\\Signal\\Line}(\\detVec{x})$ is smooth and depends on\nthe strength of the single-detector statistics $\\F^X(x^X)$ rather than the multidetector statistic $\\F(\\detVec{x})$.\n\n\n\\section{Semicoherent line-robust statistics}\n\\label{sec:semicoherent}\n\nFor unknown signal parameters $\\lambda$, the use of the fully coherent (in time) $\\F$-statistic is usually\nprohibitive in terms of computing cost.\nThus, \\emph{semicoherent} methods are typically used, being more sensitive at fixed computing cost\n\\cite{brady2000:_hierarchical,prix12:_optimal}.\nIn this approach the data $\\detVec{x}$ is divided into ${N_{\\mathrm{seg}}}$ segments of shorter duration, denoted as\n$\\{\\detVec{x}_\\segk\\}_{k=1}^{{N_{\\mathrm{seg}}}}$. The coherent statistic $\\F_\\segk(\\detVec{x}_\\segk;\\lambda)$ in a template $\\lambda$ is\ncomputed for each segment $\\segk$ separately and then combined \\emph{incoherently}, typically by summing over\nall data segments. This is often referred to as the ``StackSlide'' method. Other incoherent combinations\nsuch as the ``Hough transform'' method \\cite{krishnan04:_hough} will not be discussed here.\nThe following discussion refers to the statistic in a single template $\\lambda$, and we will therefore simplify\nthe notation again by dropping $\\lambda$.\n\nAs shown in Ref.~\\cite{prix11:_transient}, the semicoherent StackSlide $\\F$-statistic can be derived by\nrelaxing the requirement of consistent signal amplitudes $\\mathcal{A}$ across different segments, i.e., allowing for\na set of ${N_{\\mathrm{seg}}}$ independent amplitude parameters $\\mathcal{A}_\\segk$ in Eq.~\\eqref{eq:hypS}. This defines the\nsemicoherent signal hypothesis $\\sc{\\Hyp}_{\\Signal}$ as\n\\begin{equation}\n  \\label{eq:hypSsc}\n  \\sc{\\Hyp}_{\\Signal} : \\detVec{x}_\\segk = \\detVec{n}_\\segk + \\mathcal{A}_\\segk^\\mu\\,\\detVec{h}_\\mu\\,,\\quad\n  \\text{for } k = 1, \\ldots {N_{\\mathrm{seg}}}\\,,\n\\end{equation}\nwhere here and in the following the hat $\\sc{\\;\\;}$ notation refers to semicoherent quantities.\n\nFor the per-segment amplitude priors $\\prob{\\mathcal{A}_\\segk}{\\sc{\\Hyp}_{\\Signal}}$, we reuse the amplitude prior given by\nEq.~\\eqref{eq:priorA}. Hence, by marginalization as in Eq.~\\eqref{eq:likeli_HSmarg}, we obtain\nthe posterior\n\\begin{equation}\n  \\label{eq:pHSsc}\n  \\prob{\\sc{\\Hyp}_{\\Signal}}{\\detVec{x}} = \\prior{\\OSGsc}\\,\\prob{\\Hyp_\\Gauss}{\\detVec{x}}\\, c_*^{-{N_{\\mathrm{seg}}}}\\, \\eto{\\sc{\\F}(\\detVec{x})}\\,,\n\\end{equation}\nwhere we define the StackSlide $\\F$-statistic $\\sc{\\F}$ in parameter-space point $\\lambda$ as\n\\begin{equation}\n  \\label{eq:avFstat}\n  \\sc{\\F}(\\detVec{x};\\lambda) \\equiv \\sum\\limits_{k=1}^{{N_{\\mathrm{seg}}}} \\F_\\segk(\\detVec{x}_\\segk;\\lambda)\\,.\n\\end{equation}\nFor Gaussian noise we have $\\sc{\\Hyp}_{\\Gauss}=\\Hyp_\\Gauss$, but for consistency of notation we still write $\\sc{\\Hyp}_{\\Gauss}$ throughout\nthis section.\nThe posterior odds between the signal and Gaussian-noise hypotheses across the ${N_{\\mathrm{seg}}}$ segments is\n\\begin{equation}\n  \\label{eq:OSGsc}\n  \\OSGsc(\\detVec{x}) \\equiv \\frac{ \\prob{\\sc{\\Hyp}_{\\Signal}}{\\detVec{x}} }{ \\prob{\\sc{\\Hyp}_{\\Gauss}}{\\detVec{x}} } =\n  \\prior{\\OSGsc}\\,c_*^{-{N_{\\mathrm{seg}}}}\\, \\eto{\\sc{\\F}(\\detVec{x})}\\,.\n\\end{equation}\n\nWe can now generalize the single-detector line hypothesis of Eq.~\\eqref{eq:hypL} to the semicoherent case as\nwas done for the signal hypothesis in Eq.~\\eqref{eq:hypSsc}, namely,\n\\begin{equation}\n  \\begin{split}\n  \\label{eq:hypLsc}\n  \\sc{\\Hyp}_{\\Line} =& \\left(\\sc{\\Hyp}_{\\Line}^{1} \\;\\mathrm{and}\\; \\sc{\\Hyp}_{\\Gauss}^{2} \\;\\mathrm{and}\\; \\sc{\\Hyp}_{\\Gauss}^{3}\\ldots\\right) \\;\\mathrm{or}\\;\\\\\n  &\\left(\\sc{\\Hyp}_{\\Gauss}^{1} \\;\\mathrm{and}\\; \\sc{\\Hyp}_{\\Line}^{2} \\;\\mathrm{and}\\; \\sc{\\Hyp}_{\\Gauss}^{3} \\ldots\\right) \\;\\mathrm{or}\\; \\ldots\n  \\end{split}\n\\end{equation}\nThe probability of the line hypothesis in detector $X$ across all segments is\n\\begin{equation}\n  \\label{eq:pHLXsc}\n  \\prob{\\sc{\\Hyp}_{\\Line}^X}{x^X} = \\prob{\\sc{\\Hyp}_{\\Gauss}^X}{x^X}\\,c_*^{-{N_{\\mathrm{seg}}}} \\,\\prior{\\OLGsc}^X\\, \\eto{\\scF^X(x^X)}\\,,\n\\end{equation}\nwhere the semicoherent line-odds in detector $X$ is\n\\mbox{$\\prior{\\OLGsc}^X \\equiv \\probI{\\sc{\\Hyp}_{\\Line}^X}\/\\probI{\\sc{\\Hyp}_{\\Gauss}^X}$}.\nSimilarly to Eq.~\\eqref{eq:pHL_avg}, the posterior probability for the semicoherent\nline-hypothesis $\\sc{\\Hyp}_{\\Line}$ is obtained as\n\\begin{equation}\n  \\label{eq:pHLsc}\n  \\prob{\\sc{\\Hyp}_{\\Line}}{\\detVec{x}} = \\prob{\\sc{\\Hyp}_{\\Gauss}}{\\detVec{x}} \\, c_*^{-{N_{\\mathrm{seg}}}} \\,\\prior{\\OLGsc}\\, \\avgX{\\sc{r}^X\\,\\eto{\\scF^X(x^X)}}\\,,\n\\end{equation}\nwhere in analogy to Eqs.~\\eqref{eq:sumlX} and \\eqref{eq:rX} we define\n\\begin{align}\n  \\prior{\\OLGsc} &\\equiv \\frac{\\probI{\\sc{\\Hyp}_{\\Line}}}{\\probI{\\sc{\\Hyp}_{\\Gauss}}} = \\sum_X \\prior{\\OLGsc}^X\\,, \\label{eq:oLG_sc}\\\\\n  \\sc{r}^X &\\equiv \\frac{\\prior{\\OLGsc}^X}{\\prior{\\OLGsc}\/{N_{\\mathrm{det}}}}\\,.\\label{eq:rX_sc}\n\\end{align}\nThe posterior probability for the extended noise hypothesis,\n\\begin{equation}\n  \\label{eq:4}\n  \\sc{\\Hyp}_{\\Noise} \\equiv \\left( \\sc{\\Hyp}_{\\Gauss} \\;\\mathrm{or}\\; \\sc{\\Hyp}_{\\Line} \\right) \\,,\n\\end{equation}\nis therefore given by\n\\begin{equation}\n  \\label{eq:pHNsc}\n  \\prob{\\sc{\\Hyp}_{\\Noise}}{\\detVec{x}} = \\prob{\\Hyp_\\Gauss}{\\detVec{x}} \\left( 1 + c_*^{-{N_{\\mathrm{seg}}}} \\,\\prior{\\OLGsc} \\,\\avgX{\\sc{r}^X \\eto{\\scF^X(x^X)}}\n\\right).\n\\end{equation}\n\nWe can now define a semicoherent line-veto statistic, namely,\n\\begin{equation}\n  \\label{eq:OSLsc}\n  \\sc{O}_{{\\Signal\\Line}}(\\detVec{x}) \\equiv \\frac{\\prob{\\sc{\\Hyp}_{\\Signal}}{\\detVec{x}}}{\\prob{\\sc{\\Hyp}_{\\Line}}{\\detVec{x}}} =\n  \\prior{\\OSLsc}\\, \\frac{ \\eto{\\sc{\\F}(\\detVec{x})}}{\\avgX{\\sc{r}^X\\,\\eto{\\scF^X(x^X)} } }\\,,\n\\end{equation}\nand a semicoherent line-robust detection statistic as\n\\begin{equation}\n  \\label{eq:OSNsc_initial}\n  \\OSNsc(\\detVec{x}) \\equiv \\frac{\\prob{\\sc{\\Hyp}_{\\Signal}}{\\detVec{x}}}{\\prob{\\sc{\\Hyp}_{\\Noise}}{\\detVec{x}}} = \\left[\\OSGsc^{-1}(\\detVec{x}) +\n  \\sc{O}_{{\\Signal\\Line}}^{-1}(\\detVec{x})\\right]^{-1}\\!\\!.\n\\end{equation}\nThe latter can be written explicitly as\n\\begin{equation}\n  \\label{eq:OSNsc_final}\n  \\OSNsc(\\detVec{x}) = \\prior{\\OSNsc}\\,\\frac{  \\eto{\\sc{\\F}(\\detVec{x}) } }\n  {(1-\\sc{p}_{{\\Line}})\\,e^{\\scF_*^{(0)}} + \\sc{p}_{{\\Line}} \\avgX{\\sc{r}^X \\eto{\\scF^X(x^X)} } }\\,,\n\\end{equation}\nwith (semicoherent) line probability\n\\begin{equation}\n  \\label{eq:lineprob_sc}\n  \\sc{p}_{{\\Line}} \\equiv \\frac{\\prior{\\OLGsc}}{1 + \\prior{\\OLGsc}} = \\prob{\\sc{\\Hyp}_{\\Line}}{\\sc{\\Hyp}_{\\Noise}} \\;\\in\\; [0, 1]\n\\end{equation}\nand, in analogy to Eq.~\\eqref{eq:Fth0}, a prior cutoff parametrization of\n\\begin{equation}\n  \\label{eq:scFtho}\n  \\scF_*^{(0)} \\equiv \\ln c_*^{{N_{\\mathrm{seg}}}}\\,.\n\\end{equation}\nSimilarly to Eq.~\\eqref{eq:OSN_Fstar}, we can therefore write this equivalently as\n\\begin{equation}\n\\label{eq:OSNsc_Fstar}\n\\OSNsc(\\detVec{x}) = \\prior{\\OSLsc}\\,\\frac{\\eto{\\sc{\\F}(\\detVec{x})}}{\\eto{\\scF_*} + \\avgX{\\sc{r}^X \\eto{\\scF^X(x^X)}}}\\,,\n\\end{equation}\nwhere the semicoherent transition scale $\\scF_*$ is defined as\n\\begin{equation}\n  \\label{eq:scFth}\n  \\scF_* \\equiv \\scF_*^{(0)} - \\ln {\\prior{\\OLGsc}}\\,,\n\\end{equation}\nby generalizing Eq.~\\eqref{eq:Fth}. Hence, we find that $\\OSNsc(\\detVec{x})$ transitions from the standard\nsemicoherent statistic $\\OSGsc(\\detVec{x})\\propto\\eto{\\sc{\\F}}$ to the line-veto statistic $\\sc{O}_{{\\Signal\\Line}}(\\detVec{x})$ when\n\\begin{equation}\n  \\label{eq:denomTermsTransition_sc}\n  \\avgX{\\sc{r}^X \\eto{\\scF^X}} \\sim \\eto{\\scF_*}\\,.\n\\end{equation}\n\nWe can rewrite the log-odds as\n\\begin{equation}\n  \\begin{split}\n  \\label{eq:logOSNsc}\n  \\ln &\\OSNsc(\\detVec{x}) = \\ln \\prior{\\OSLsc} + \\sc{\\F}(\\detVec{x}) - \\scF''_{\\mathrm{max}}(\\detVec{x})\\\\\n  &- \\ln \\left( \\eto{\\scF_*-\\scF''_{\\mathrm{max}}(\\detVec{x})} + \\avgX{\\sc{r}^X \\eto{\\scF^X(x^X)-\\scF''_{\\mathrm{max}}(\\detVec{x})} } \\right)\\,,\n  \\end{split}\n\\end{equation}\nwith  $\\scF''_{\\mathrm{max}}(\\detVec{x}) \\equiv \\max\\left( \\scF_*,\\,\\scF^X(x^X) + \\ln \\sc{r}^X \\right)$.\nNote that in the semicoherent case we typically deal with much larger numerical values of $\\sc{\\F}$ [due to its\ndefinition as a sum over segments in Eq.~\\eqref{eq:avFstat}]. However, the\nlogarithmic correction term is still of order unity. This implies that the transition from\n$\\OSGsc(\\detVec{x})$ to the line-veto odds $\\sc{O}_{{\\Signal\\Line}}(\\detVec{x})$ is expected to be sharper than in the coherent case of\nEq.~\\eqref{eq:logOSN}.\n\nIncorporating the ad-hoc $\\F$-statistic consistency veto discussed in\nSec.~\\ref{sec:line-veto-stat}, we can define a semicoherent $\\scFveto$-statistic{} as\n\\begin{equation}\n  \\label{eq:scFveto}\n  \\sc{\\F}^{\\mathrm{+veto}}(\\detVec{x}) \\equiv \\left\\{\n    \\begin{array}{cc}\n      \\sc{\\F}(\\detVec{x})\\; & \\textrm{if } \\sc{\\F}(\\detVec{x}) \\ge \\max_{X}\\{\\sc{\\F}^X(x)\\}\\,,\\\\\n      0        & \\textrm{otherwise}\\,.\n    \\end{array}\n  \\right.\n\\end{equation}\n\n\\section{Choice of priors}\n\\label{sec:tuning}\n\nThe new line-veto and line-robust statistics derived in this paper depend on some prior\nparameters which need to be specified. We will now discuss a way to set their values.\n\nThe coherent statistics described in Sec.~\\ref{sec:line-veto-stats-coh} are simply special cases of the\nsemicoherent expressions given in Sec.~\\ref{sec:semicoherent} for ${N_{\\mathrm{seg}}}=1$. Hence, in the\nfollowing we can use the semicoherent notation without loss of generality.\n\nThe pure line-veto statistic $\\sc{O}_{{\\Signal\\Line}}(\\detVec{x})$ of Eq.~\\eqref{eq:OSLsc} seems, at first glance, to have ${N_{\\mathrm{det}}}$\nfree parameters.\nHowever, with the sum constraint \\eqref{eq:rX} on the line-probability weights $\\sc{r}^X$ and the fact that\nthe overall prior odds $\\prior{\\OLGsc}$ only enter through the proportionality factor $\\prior{\\OSLsc}$, this reduces to an\neffective ${N_{\\mathrm{det}}}-1$ parameters.\nHere we make use again of the fact that all monotonic functions of a test statistic are equivalent in the\nNeyman-Pearson sense.\n\nThe line-robust statistic $\\OSNsc(\\detVec{x})$ depends on the prior odds $\\prior{\\OLGsc}$ and on the amplitude-prior cutoff\nparameter $c_*$, not just as mere prefactors.\nHowever, these two prior parameters only appear in $\\OSNsc$ through the combination\n$\\scF_* \\equiv \\scF_*^{(0)} - \\ln {\\prior{\\OLGsc}}$ as defined in Eq.~\\eqref{eq:scFth}. Therefore, $\\OSNsc$ effectively has\n${N_{\\mathrm{det}}}$ free parameters.\n\nWhile the prior odds $\\prior{\\OLGsc}$ have a clear intuitive interpretation, this is not the case for the\nprior amplitude cutoff parameter $c_*$ and thus for $\\scF_*^{(0)} $, as defined in Eq.~\\eqref{eq:scFtho}.\nThis parameter results from the rather unphysical choice of the amplitude prior in Eq.~\\eqref{eq:priorA}, as\ndiscussed in more detail in Refs.~\\cite{prix09:_bstat,prix11:_transient}.\nHence, a certain amount of empirical ``tuning'' will be required to determine a reasonable value\nfor $\\scF_*^{(0)}$, which we will discuss in Sec.~\\ref{sec:choosing-prior-value}.\n\n\\subsection{Proxy estimate of prior line probabilities from the data}\n\\label{sec:estimate-line-probs}\n\nA maximally uninformative choice for the line-priors would be $\\sc{r}^X = 1$ and $\\prior{\\OLGsc}=1$, where the presence\nof lines is considered just as likely as pure Gaussian noise and all detectors are equally likely to be\naffected by lines.\nA more informed choice should be based on prior characterization of the detectors.\n\nA practical way to achieve this is to judiciously use the observed data $\\detVec{x}$ for a simple ``proxy'' estimate\nof $\\prior{\\OLGsc}^X$.\nEmpirically we find promising results when adopting the {line-flagging} method of Ref.~\\cite{wette09:_thesis}.\nWe use data from all frequency bins potentially contributing to the detection statistics in a given\nsearch band.\nWe compute the time-averaged normalized power over these bins and count how many exceed a\npredetermined threshold. The measured fraction of such outliers is used as a proxy estimate for the\nprior line probability.\n\nMore specifically, the data for $\\F$-statistic searches is usually prepared in the form of \\emph{Short Fourier\nTransforms} (SFTs) of the original time-domain data, conventionally spanning\nstretches of duration $T_\\sft = 1800\\,\\sec$ (e.g., see Ref.~\\cite{krishnan04:_hough}).\nWe compute the normalized average SFT power $\\Psft^X(f)$ for each detector $X$ as (e.g., see\nRef.~\\cite{abbott2004:_geoligo})\n\\begin{equation}\n  \\label{eq:Psft}\n  \\Psft^X(f) \\equiv \\frac{2}{N_\\sft\\,T_\\sft} \\sum_{\\alpha=1}^{N_\\sft}\n                    \\frac{ \\left| \\widetilde{x}_\\alpha^X(f)\\right|^2 }{\\SnXal(f)}\\,,\n\\end{equation}\nwhere the sum is over all $N_\\sft$ SFTs, $\\widetilde{x}_\\alpha^X(f)$ and $\\SnXal(f)$ denote the\nFourier-transformed data and the noise PSD in the $\\alpha$th SFT, respectively.\n\nWe estimate the prior line probability $\\sc{p}_{{\\Line}}^X$ for that frequency band as\n\\begin{equation}\n  \\label{eq:lineestimator}\n  \\sc{p}_{{\\Line}}^X = \\frac{N_{\\Psft > \\Psftthr}^X}{N_{\\mathrm{bins}}}\\,,\n\\end{equation}\nwhere $N_{\\Psft > \\Psftthr}^X$ is the number of bins $\\in [0, N_{\\mathrm{bins}}]$ for which $\\Psft^X(f)$ crossed the threshold\n$\\Psftthr^X$. A typical band is of the order of $100\\,\\mathrm{mHz}$ wide, corresponding to a few hundred bins.\nThe threshold $\\Psftthr^X$ is chosen empirically to be safely above the typical noise fluctuations in the data.\n\nFrom $\\sc{p}_{{\\Line}}^X$ the prior line odds may be computed as\n\\begin{equation}\n  \\label{eq:olg_estimate}\n  \\prior{\\OLGsc}^X = \\frac{\\sc{p}_{{\\Line}}^X}{1 - \\sc{p}_{{\\Line}}^X}\\,,\n\\end{equation}\nwhich also fully specifies $\\sc{r}^X$ and $\\prior{\\OLGsc}$ via Eqs.~\\eqref{eq:oLG_sc} and \\eqref{eq:rX_sc}.\n\nWe determine the threshold $\\Psftthr^X$ by fixing a certain false-alarm probability $p_{\\mathrm{FA},\\Psft}$.\nFor large $N_\\sft$ this can be computed approximately from a Gaussian distribution with unit mean and standard\ndeviation $\\sigma=1\/\\sqrt{N_\\sft}$.\n\nAs an illustrative example, Fig.~\\ref{fig:tuning_normSFTpower_example} shows $\\Psft^X(f)$ for a\n$\\sim60\\,\\mathrm{mHz}$ wide band of simulated Gaussian data consisting of 50 SFTs. The data is\ngenerated with a noise PSD of $\\Sn^X = 3 \\times 10^{-22}\\,\\mathrm{Hz}^{-1\/2}$ in two detectors\n$X\\in\\{\\textrm{H1},\\textrm{L1}\\}$, where $\\textrm{H1}$ and $\\textrm{L1}$ stand for the LIGO detectors at Hanford and Livingston,\nrespectively. A monochromatic stationary line of amplitude $h_0 = 2 \\times 10^{-23}\\,\\mathrm{Hz}^{-1\/2}$ at\n$50\\,\\mathrm{Hz}$ is injected in $\\textrm{H1}$ only. More examples from real data are presented in\nSec.~\\ref{sec:tests_realdata}.\n\nWe stress that the line-flagging procedure proposed here is not meant to yield a direct estimator of\n$\\sc{p}_{{\\Line}}^X$ but rather to provide an indication for the presence of lines based on spectral\nfeatures that can be robustly identified.\n\nFor instance, observing no threshold crossings in the average SFT power $\\Psft$ does not necessarily imply\nthat the $\\F$-statistic could not be affected by instrumental artifacts, while seeing many outliers in $\\Psft$\ndoes not always yield high values of $\\F$.\nHence we will not consider values of $\\prior{\\OLGsc}^X$ that suggest more confidence than seems justifiable, and\ntruncate its range to\n\\begin{equation}\n  \\label{eq:olg_trunc}\n  \\prior{\\OLGsc}^X \\in [0.001,\\; 1000]\\,.\n\\end{equation}\n\n\\begin{figure}[h!tbp]\n  \\includegraphics[width=\\columnwidth]{fake_50dot00Hz_1seg_normSFT_H1L1}\n  \\caption{\n   \\label{fig:tuning_normSFTpower_example}\n   Example of the normalized SFT power $\\Psft^X(f)$ as a function of frequency $f$ for LIGO $\\textrm{H1}$\n   (solid) and $\\textrm{L1}$ (dashed) for simulated Gaussian data containing a line in $\\textrm{H1}$.\n   The horizontal line shows the threshold $\\Psftthr$ at a false-alarm level of $p_{\\mathrm{FA},\\Psft}=10^{-9}$.}\n\\end{figure}\n\nFor the example simulated data set used in Fig.~\\ref{fig:tuning_normSFTpower_example} we can detail the\nmethod as follows: there are 127 frequency bins in the band considered, and for the threshold\n$\\Psftthr^\\textrm{H1}=\\Psftthr^\\textrm{L1}=\\Psftthr(p_{\\mathrm{FA},\\Psft}=10^{-9},N_\\sft=50)\\approx1.84$ there is a single crossing in $\\textrm{H1}$\nand none in $\\textrm{L1}$. Hence, we estimate the line priors as\n$\\prior{\\OLGsc}^\\textrm{H1}=\\mathrm{max}\\left(0.001,\\tfrac{1\/127}{1-1\/127}\\right)\\approx0.008$ and\n$\\prior{\\OLGsc}^\\textrm{L1}=\\mathrm{max}\\left(0.001,\\tfrac{0\/127}{1-0\/127}\\right)=0.001$.\n\nWe believe that this data-dependent prior estimation is not prone to the\n``sample reuse fallacy'' \\cite{jaynes:_logic_of_science}.\nThe reason is that the proxy estimate for $\\prior{\\OLGsc}^X$ is\nsufficiently \\emph{independent} from the posterior for the line\nhypothesis $\\Hyp_\\Line$, as they are derived from data sets with effectively very little data in common.\nThe line hypothesis $\\Hyp_\\Line$ (being based on the signal hypothesis $\\Hyp_\\Signal$) describes a\nnarrow-band signal, which in each half-hour SFT is confined to a few bins.\nIn fact the current $\\F$-statistic implementation \\cite{prix:_cfsv2} uses only $16$ frequency bins per SFT to\nconstruct the detection statistic, and they are very heavily weighted toward a few central ones.\nOn the other hand, the line-flagging prior estimate uses $\\sim\\Ord{100-200}$ frequency bins and each counts\nequally in the estimate.\nFurthermore, the results in Sec.~\\ref{sec:tests_realdata} show that this procedure appears to be ``safe'' also\nin the presence of (injected) signals.\n\n\n\\subsection{Empirical choice of transition scale \\texorpdfstring{$\\scF_*$}{F*}}\n\\label{sec:choosing-prior-value}\n\nAn additional free parameter in the line-robust statistic $\\OSNsc(\\detVec{x})$, as expressed in\nEq.~\\eqref{eq:OSNsc_Fstar}, is the transition scale $\\scF_*= \\scF_*^{(0)} - \\ln\\prior{\\OLGsc}$ of Eq.~\\eqref{eq:scFth}.\n\nAs discussed in Sec.~\\ref{sec:limiting-behavior}, $\\scF_*$ sets the scale (in terms of $\\scF^X$) for the\ntransition of $\\OSNsc$ from the signal-versus-Gaussian-noise odds $\\OSGsc\\propto\\eto{\\sc{\\F}}$ (for\n$\\scF^X \\ll \\scF_*$) to the signal-versus-line odds $\\sc{O}_{{\\Signal\\Line}}$ (for $\\scF^X \\gg \\scF_*$).\n\nThus we can interpret $\\scF_*^{(0)}$ as the transition scale in the case of even prior odds, i.e., $\\prior{\\OLGsc}=1$,\nbetween the line and Gaussian-noise hypotheses.\n\nThe effect of $\\prior{\\OLGsc}$, which we estimate with the method described in the previous section, is to shift the\ntransition scale up or down from this baseline, depending on whether prior knowledge gives lines lower or\nhigher odds, respectively.\n\nWe can also express $\\scF_*^{(0)}$ in terms of a Gaussian-noise false-alarm probability, denoted as\n$p_{\\mathrm{FA}*}^{(0)}$:\n  \\begin{equation}\n   \\label{eq:scFtho_pFA}\n   p_{\\mathrm{FA}*}^{(0)} = \\prob{\\sc{\\F}^X > \\scF_*^{(0)}}{\\Hyp_\\Gauss}\n  \\end{equation}\nThis follows a central $\\chi^2$-distribution with $4{N_{\\mathrm{seg}}}$ degrees of freedom.\nWe find it useful to fix a value for $p_{\\mathrm{FA}*}^{(0)}$ and use it to determine $\\scF_*^{(0)}\\left(p_{\\mathrm{FA}*}^{(0)},{N_{\\mathrm{seg}}}\\right)$.\n\nOn the one hand, we want $\\scF_*^{(0)}$ to be low enough ($p_{\\mathrm{FA}*}^{(0)}$ high enough) to suppress even weak lines, but\nnot so low as to compromise the performance in Gaussian noise.\nWhen most of the data is approximately Gaussian (as is typically the case for CW\nsearches,~\\cite{abbott2004:_geoligo,aasi13:_eathS5,behnke2013:_phdthesis}), a reasonable choice is to\nuse the lowest $\\scF_*^{(0)}$ (highest $p_{\\mathrm{FA}*}^{(0)}$) that does not yet adversely affect the detection power in\nGaussian noise.\nIn practice, we resort to an empirical choice of $p_{\\mathrm{FA}*}^{(0)}$ based on Monte-Carlo simulations on a small subset\nof Gaussian or near-Gaussian data.\n\n\n\\section{Performance tests}\n\\label{sec:tests}\n\nHere we will discuss the detection efficiency of the statistics introduced in\nthe previous sections for a population of signals embedded in different types of noise.\nIn order to do this we use two different and somewhat complementary approaches:\n(i) fully ``synthetic'' simulations, which allow for efficient large-scale explorations under idealized\nconditions, and (ii) injections of simulated signals into LIGO S5 data containing instrumental\nartifacts.\n\nWe compare the performance of the following statistics (the second equation always refers to the\ncorresponding semicoherent version):\n\\begin{enumerate}[(1)] \\itemsep1pt \\parskip0pt\n \\item Standard multidetector $\\F$-statistic, Eqs.~\\eqref{eq:Fstat} and \\eqref{eq:avFstat}\n \\item $\\Fveto$-statistic{}, Eqs.~\\eqref{eq:Fveto} and \\eqref{eq:scFveto}\n \\item Line-veto statistic $O_{\\Signal\\Line}$, Eqs.~\\eqref{eq:OSL} and \\eqref{eq:OSLsc}\n \\item Line-robust statistic $\\OSN$, Eqs.~\\eqref{eq:OSN_Fstar} and \\eqref{eq:OSNsc_Fstar}\n\\end{enumerate}\nIn the case of the line-robust statistic $\\OSN$ we use different transition scales $\\Fth^{(0)}$ corresponding to\nfalse-alarm levels $p_{\\mathrm{FA}*}^{(0)}$, which we denote as\n\\begin{equation}\n  \\label{eq:7}\n  \\OSNpFA{-n}(\\detVec{x}) \\equiv \\OSN(\\detVec{x};\\; p_{\\mathrm{FA}*}^{(0)}=10^{-n})\\,.\n\\end{equation}\nIn the following tests we use $\\OSNpFA{-1}$, $\\OSNpFA{-3}$, and $\\OSNpFA{-6}$, corresponding to\ntransition-scale false-alarm levels of $p_{\\mathrm{FA}*}^{(0)}=10^{-1},10^{-3},10^{-6}$, respectively.\n\nIn order to assess the importance of the choice of prior line odds $\\prior{\\OLG}^X$, we consider two cases:\n\\begin{enumerate}[(i)] \\itemsep1pt \\parskip0pt\n \\item Uninformative priors, i.e., $\\prior{\\OLG}^{X} = 1$ for all $X$: the  corresponding ``untuned'' statistics are\n       denoted as $\\OSL^{(0)}$ and $\\utOSNpFA{-n}$.\n \\item Line priors $\\prior{\\OLG}^X$ using prior information on the line population: the corresponding ``tuned''\n   statistics are denoted as $O_{\\Signal\\Line}$ and $\\OSNpFA{-n}$, respectively.\n\\end{enumerate}\n\n\\subsection{Tests using synthetic draws}\n\\label{sec:tests_simdata}\n\nIn this section, for simplicity, we consider only the coherent case (cf.~Sec.~\\ref{sec:line-veto-stats-coh}).\nUsing the synthesizing approach described in Refs.~\\cite{prix09:_bstat,prix11:_transient}, one can directly\ngenerate random draws of the various statistics of interest for pure noise and for noise containing a signal.\n\nThe synthesizing method consists in generating random draws of the $\\{x^X_\\mu\\}$ of Eq.~\\eqref{eq:xmuMmunu}\nusing their known (multivariate) Gaussian distribution.\nFrom these we compute the $\\F$- and $\\F^X$-statistics from Eq.~\\eqref{eq:Fstat}, $O_{\\Signal\\Line}$ from\nEq.~\\eqref{eq:OSL} and $\\OSN$ from Eq.~\\eqref{eq:OSN_Fstar}.\nIn the following we refer to each draw of $\\{x^X_\\mu\\}$ together with the resulting statistics as a\n\\emph{candidate}.\n\nWe generate the noise draws in such a way that a fraction $f_\\Line$ contains a line according to\n$\\Hyp_\\Line$ of Eq.~\\eqref{eq:hypL}, namely a CW signal in a single detector. The remaining fraction $1-f_\\Line$\nof noise draws follows the Gaussian-noise hypothesis $\\Hyp_\\Gauss$ of Eq.~\\eqref{eq:gaussian}.\nIn the following we refer to $f_\\Line$ as the \\emph{line contamination}.\n\nFrom the noise draws we estimate for each statistic a threshold corresponding to a particular false-alarm\nprobability $p_{\\mathrm{FA}}$. Applying this threshold to the signal candidates yields the detection\nprobability $\\pDet(p_{\\mathrm{FA}})$ for each statistic at the false-alarm level $p_{\\mathrm{FA}}$.\nThis is known as the \\emph{receiver operator characteristic} (ROC).\n\nThe strength of the injected signals is characterized by the (multidetector) \\emph{signal-to-noise ratio}\n$\\snr_{\\Signal}$, defined in the usual way \\cite{jks98:_data} as\n\\begin{equation}\n  \\label{eq:snrS}\n \\snr_{\\Signal}^2 \\equiv \\scalar{h}{h} = \\mathcal{A}^\\mu \\mathcal{M}_{\\mu\\nu} \\mathcal{A}^\\nu \\,.\n\\end{equation}\nThis is related to the expectation value of the $\\F$-statistic as $E[2\\F]_{\\Hyp_\\Signal}=4+\\snr_{\\Signal}^2$.\nAs shown in Appendix \\ref{sec:expect-f-stat}, for a line according to $\\Hyp_\\Line$ in detector $Y$, the\nexpectation value of the multidetector $\\F$-statistic is approximately\n\\begin{equation}\n  \\label{eq:exp_F_line}\n  \\expect{2\\F}_{\\Hyp_\\Line} \\approx 4 + \\frac{1}{{N_{\\mathrm{det}}}}\\,\\snr_{\\Line}^2\\;\\;\\text{with}\\;\\;\n  \\snr_{\\Line}^2 \\equiv \\mathcal{A}_Y^\\mu\\mathcal{M}^Y_{\\mu\\nu}\\mathcal{A}_Y^\\nu\\,,\n\\end{equation}\nwhere we refer to the (single-IFO) SNR $\\snr_{\\Line}$ as the ``line SNR''.\n\nThe signal candidates are generated for a fixed SNR of $\\snr_{\\Signal}=6$, and a data length of $T=25\\,\\mathrm{h}$ is\nassumed.\nThis signal strength is chosen to be representative of reasonably detectable signals in a wide-parameter-space\nsearch.\nIn such a search we would require a low (single-trial) false-alarm threshold $p_{\\mathrm{FA}}$ in order to consider a\ncandidate as significant.\nThe choice of $\\snr_{\\Signal}=6$ corresponds to a detection probability of $\\pDet\\approx70\\%$ at a false-alarm\nprobability of $p_{\\mathrm{FA}}=10^{-6}$ in Gaussian noise (for example, see Fig.~\\ref{fig:newSynth_Gauss}).\n\nThe signal amplitude parameters are drawn uniformly in $\\cos\\iota\\in[-1,1]$, $\\psi\\in[-\\pi\/4,\\pi\/4]$ and\n$\\phi_0\\in[0,2\\pi]$.\nThe sky position is drawn isotropically over the sky, and $(h_0\/\\sqrt{\\Sn})$ is determined by the fixed signal\nSNR of $\\snr_{\\Signal}=6$ according to Eq.~\\eqref{eq:snrS}.\nThe line draws use the same prior distributions, but the signal is added to only one detector, and\n$(h_0\/\\sqrt{\\Sn})$ is determined by fixing a (single-IFO) line SNR $\\snr_{\\Line}$ according to\nEq.~\\eqref{eq:exp_F_line}.\n\nIn each simulation we generate $10^7$ noise candidates and $10^7$ noise+signal candidates for two detectors,\nLIGO $\\textrm{H1}$ and $\\textrm{L1}$. These detectors are assumed here to have identical sensitivity.\nLines are only injected into $\\textrm{H1}$ without loss of generality.\nWe consider three examples of noise populations:\n\\begin{enumerate}[(i)] \\itemsep1pt \\parskip0pt\n  \\item pure Gaussian noise without lines ($f_\\Line=0$, $\\snr_{\\Line}=0$)\n  \\item 10\\% line contamination in H1 ($f_\\Line^{\\textrm{H1}}=0.1,\\;f_\\Line^{\\textrm{L1}}=0$) with line SNR of $\\snr_{\\Line}=9$,\n  \\item 10\\% line contamination in H1  ($f_\\Line^{\\textrm{H1}}=0.1,\\;f_\\Line^{\\textrm{L1}}=0$) with line SNR of $\\snr_{\\Line}=15$.\n\\end{enumerate}\nThe line SNR of $\\snr_{\\Line}=9$ corresponds to lines that are marginally stronger than the injected signals, namely,\n$\\expect{2\\F}_{\\Hyp_\\Line}\\approx 44.5$ from Eq.~\\eqref{eq:exp_F_line}), while $\\expect{2\\F}_{\\Hyp_\\Signal} = 40$ from\nEq.~\\eqref{eq:14}).\nThe lines with $\\snr_{\\Line}=15$ are substantially stronger (namely $\\expect{2\\F}_{\\Hyp_\\Line} \\approx 117$) than the\ninjected signals.\n\nNote that for the synthesized statistics we cannot use the line-prior estimation method for $\\prior{\\OLG}^X$ of\nSec.~\\ref{sec:estimate-line-probs}. Instead we assume ``perfect tuning'': in\nthe Gaussian-noise example we set $\\prior{\\OLG}^X=0.001$ for $X=\\textrm{H1},\\textrm{L1}$, and in the two line examples we use\n$p_\\Line^{\\textrm{H1}}=f_\\Line^{\\textrm{H1}}=0.1$ (therefore $\\prior{\\OLG}^\\textrm{H1}=1\/9$) and $\\prior{\\OLG}^\\textrm{L1}=0.001$ (no lines were\ninjected into L1).\n\n\\begin{figure}[b!]\n  \\includegraphics[width=1.05\\columnwidth,clip]{newSynth_snrS6_pL0_snrL0_N1e07-final}\n  \\caption{\n    \\label{fig:newSynth_Gauss}\n    Detection probability $\\pDet$ as a function of false-alarm $p_{\\mathrm{FA}}$ of different synthesized statistics,\n    for a signal population of fixed SNR of $\\snr_{\\Signal}=6$ in pure Gaussian noise ($f_\\Line=0$, $\\snr_{\\Line}=0$).\n    Statistical errors are similar to the line width.\n  }\n\\end{figure}\n\nIn Gaussian noise the coherent $\\F$-statistic is close to optimal \\cite{jks98:_data,prix09:_bstat}, and follows\na $\\chi^2$ distribution with 4 degrees of freedom and noncentrality parameter $\\snr_{\\Signal}^2$, which we denote as\n$\\chi^2_4(\\snr_{\\Signal})$. This is plotted as a thick solid line in Figs.~\\ref{fig:newSynth_Gauss} and\n\\ref{fig:newSynth_Lines} for the signal population of $\\snr_{\\Signal}=6$.\n\nIn the Gaussian-noise example shown in Fig.~\\ref{fig:newSynth_Gauss}, the $\\F$-statistic follows closely\nthe theoretical prediction, while the (untuned) line-veto statistic $\\OSL^{(0)}$ is notably less powerful.\nThe line-robust statistics $\\OSNpFA{-n}$ increasingly approach the $\\F$-statistic performance with decreasing\n$p_{\\mathrm{FA}*}^{(0)}$, i.e., increasing transition scale $\\Fth^{(0)}$.\nIn particular, starting from $\\OSNpFA{-3}$ (corresponding to a transition scale of $\\Fth^{(0)}\\approx 9.23$),\nthere are no appreciable losses in detection probability $\\pDet$ over the false-alarm range $p_{\\mathrm{FA}}\\in\n[10^{-6},1]$.\n\nAt low $p_{\\mathrm{FA}}$, the $\\Fveto$-statistic{} performs almost optimally, while\nthere are some losses above $p_{\\mathrm{FA}}\\gtrsim10^{-4}$. These are due to $\\F^{\\mathrm{+veto}}$ containing intrinsic upper\nbounds on the achievable $p_{\\mathrm{FA}}$ and $\\pDet$ as a result of vetoing a finite fraction of candidates.\nFor a practical GW analysis, where low $p_{\\mathrm{FA}}$ are required, this behavior is not particularly\nrelevant.\n\n\\begin{figure}[b]\n    \\raggedright (a)\\\\\\vspace*{-0.5cm}\n    \\includegraphics[width=\\columnwidth,clip]{newSynth_snrS6_pL0dot1_snrL9_N1e07-final}\n    \\raggedright(b)\\\\\\vspace*{-0.5cm}\n    \\includegraphics[width=\\columnwidth]{newSynth_snrS6_pL0dot1_snrL15_N1e07-final}\n  \\caption{\n    \\label{fig:newSynth_Lines}\n    Detection probability $\\pDet$ as a function of false-alarm $p_{\\mathrm{FA}}$ for different synthesized statistics,\n    for a signal population with fixed SNR of $\\snr_{\\Signal}=6$ in Gaussian noise with 10\\% line contamination,\n    with line-SNR of (a) $\\snr_{\\Line}=9$ and (b) $\\snr_{\\Line}=15$.\n    Statistical errors are similar to the line width.\n  }\n\\end{figure}\n\nThe performance in the two examples with $10\\%$ line contamination is shown in\nFig.~\\ref{fig:newSynth_Lines}.\nHere the $\\F$-statistic is found to perform substantially worse than in Gaussian noise at false-alarm\nprobabilities below $p_{\\mathrm{FA}}\\lesssim 0.1$. This is due to the fact that in $10\\%$ of the noise cases the\nfalse-alarm threshold is set by the line population, which is either difficult (for $\\snr_{\\Line}=9$, left plot) or\nalmost impossible (for $\\snr_{\\Line}=15$, right plot) for the $\\F$-statistic to cross for signals with SNR of\n$\\snr_{\\Signal}=6$.\n\n\\begin{figure}[b]\n    \\raggedright (a)\\\\\\vspace*{-0.5cm}\n    \\includegraphics[width=\\columnwidth,clip]{newSynth_snrS6_pL0dot1_snrL9_N1e07-finalAdaptive}\n    \\raggedright(b)\\\\\\vspace*{-0.5cm}\n    \\includegraphics[width=\\columnwidth]{newSynth_snrS6_pL0dot1_snrL15_N1e07-finalAdaptive}\n  \\caption{\n    \\label{fig:newSynth_LinesTuning}\n    Comparison of ``tuned'' statistics $\\OSNpFA{-n}$ (solid lines) using\n    ``perfect knowledge'' line-priors $\\{\\prior{\\OLG}^\\textrm{H1}=1\/9,\\,\\prior{\\OLG}^\\textrm{L1}=10^{-3}\\}$ versus ``untuned'' statistic\n    $\\utOSNpFA{-n}$ (dashed lines) using uninformative line-priors $\\prior{\\OLG}^X=1$.\n    Detection probability $\\pDet$ as a function of false-alarm $p_{\\mathrm{FA}}$ of different synthesized statistics,\n    for a signal population with fixed SNR of $\\snr_{\\Signal}=6$ in Gaussian noise with 10\\% line contamination,\n    with line-SNR of (a) $\\snr_{\\Line}=9$ and (b) $\\snr_{\\Line}=15$.\n    Statistical errors are similar to the line width.\n  }\n\\end{figure}\n\nWe observe that the $\\Fveto$-statistic{} starts to fail below false-alarm levels of\n$p_{\\mathrm{FA}}\\lesssim 10^{-4}$ in the case of weaker lines with $\\snr_{\\Line}=9$ (see Fig.~\\ref{fig:newSynth_Lines}(a)).\nThis can be understood as follows:\nFor the $\\snr_{\\Line}=9$ line population, we find that a fraction of $\\sim6\\times10^{-4}$ of line candidates survive\nthe veto.\nGiven that lines are present in 10\\% of the noise cases, this means that a fraction of\n$\\sim6\\times10^{-5}$ of total noise candidates are line candidates surviving the consistency veto.\nGiven that these have high $\\F$-statistic values, signal candidates can hardly surpass them, and thus the\ndetection probability drops toward zero at false-alarm probabilities below $\\sim6\\times10^{-5}$.\n\n\nThe same effect is also present for stronger lines, but the corresponding ``failure'' threshold is\npushed to lower values.\nFor example, for $\\snr_{\\Line}=12$ it would happen only below $p_{\\mathrm{FA}}\\lesssim10^{-6}$, while for $\\snr_{\\Line}=15$ it\nis too low to be resolvable by $10^7$ random draws.\n\nThe behavior of the line-robust statistics $\\OSNpFA{-n}$ depends on the choice of transition scale.\nIn the case of lines with $\\snr_{\\Line}=9$, shown in Fig.~\\ref{fig:newSynth_Lines}(a), the statistic $\\OSNpFA{-3}$\nperforms best, while using either lower or higher values of $p_{\\mathrm{FA}*}^{(0)}$ is less powerful at low false-alarm\nprobabilities.\nIn the case of stronger lines with $\\snr_{\\Line}=15$, shown in Fig.~\\ref{fig:newSynth_Lines}(b), the statistic\n$\\OSNpFA{-6}$ performs almost optimally, with $\\OSNpFA{-3}$ performing only slightly worse.\n\nThe line-veto statistic $\\OSL^{(0)}$ performs somewhat poorly in all three examples\nshown (Figs.~\\ref{fig:newSynth_Gauss} and \\ref{fig:newSynth_Lines}). This is not surprising, given that at\nmost $10\\%$ of noise draws contain a line, while $O_{\\Signal\\Line}$ would only be optimal for a noise population\nconsisting exclusively of lines.\n\nFigure~\\ref{fig:newSynth_LinesTuning} shows the effect of ``tuning'' the prior line odds\n$\\prior{\\OLG}^X$, using the same line populations as in Fig.~\\ref{fig:newSynth_Lines}. We see that the untuned\nstatistics $\\OSL^{(0)}$ and $\\utOSNpFA{-n}$ using uninformative line-odds $\\prior{\\OLG}^X=1$ perform reasonably well\ncompared to $O_{\\Signal\\Line}$ and $\\OSNpFA{-n}$, which are based on ``perfect-knowledge'' tuning.\nNote that tuning of $\\prior{\\OLG}^X$ can sometimes also {decrease} the detection power of a statistic,\nparticularly in cases where the choice of the transition scale $\\Fth^{(0)}(p_{\\mathrm{FA}*}^{(0)})$ is a poor fit to the actual\nline population.\nThis can be seen in the case of $\\OSNpFA{-6}$ with lines of SNR $\\snr_{\\Line}=9$, as shown in\nFig.~\\ref{fig:newSynth_LinesTuning}(a).\nIn cases where $\\Fth^{(0)}(p_{\\mathrm{FA}*}^{(0)})$ is a good match to the line population, the tuning of $\\prior{\\OLG}^X$ can yield gains\nin detection power of up to 5--10\\%.\n\n\\subsection{Tests using LIGO S5 data}\n\\label{sec:tests_realdata}\n\nHere we conduct a study using LIGO S5 data sets as noise.\nWe inject signals and search for them using methods similar to those of actual CW searches.\nInstead of ROC curves, i.e., $\\pDet(p_{\\mathrm{FA}})$, we present results in terms of the detection efficiency as\na function of signal strength scaled by the total multidetector noise PSD $\\Sn$, i.e.,\n$\\pDet(h_0\/\\sqrt{\\Sn})$.\nThis form is more suitable to assess improvements in sensitivity, which is typically expressed as the\nweakest signal $h_0$ detectable with a certain confidence $\\pDet$.\nTo compute an astrophysically motivated detection probability, these results could in principle be\nconvolved with an astrophysical prior on $h_0$, if available.\n\nThe injection and detection procedure used here is modeled after those commonly employed for estimating upper\nlimits on $h_0$ in CW searches such as Refs.~\\cite{aasi13:_eathS5, aasi2013:_gc-search}.\n\nSimulated CW signals are added to the data using the \\texttt{Makefakedata\\_v4} code \\cite{lalsuite}.\nThe resulting data set is analyzed both coherently and semicoherently using\n\\texttt{HierarchSearchGCT} \\cite{lalsuite}, a StackSlide implementation based on\nthe ``global correlations'' method of Ref.~\\cite{pletsch2009:_gct}. We have extended this code to also compute\nthe new statistics $\\sc{O}_{{\\Signal\\Line}}(\\detVec{x})$ and $\\OSNsc(\\detVec{x})$, in addition to $\\sc{\\F}(\\detVec{x})$.\nFor the coherent search we use shorter subsets of the data, and the coherent statistics are simply obtained as\nthe special case ${N_{\\mathrm{seg}}}=1$.\n\nThe tuning of the line priors $\\prior{\\OLGsc}^X$ in $\\sc{O}_{{\\Signal\\Line}}$ and $\\OSNscpFA{-n}$ is based on the method described in\nSec.~\\ref{sec:estimate-line-probs}, namely, Eqs.~\\eqref{eq:lineestimator},~\\eqref{eq:olg_estimate}, and\n\\eqref{eq:olg_trunc}.\nAs explained in Sec.~\\ref{sec:choosing-prior-value}, we fix the transition scale $\\scF_*^{(0)}$ of $\\OSNsc$\naccording to its performance in Gaussian noise.\nSpecifically, we perform injections on simulated Gaussian noise and analyze them as\ndescribed below for several values of $p_{\\mathrm{FA}*}^{(0)}$.\nWe then chose the highest $p_{\\mathrm{FA}*}^{(0)}$ value such that the achieved performance is indistinguishable within\nstatistical uncertainties from that of the $\\F$-statistic.\nAs a result of this we select $\\OSNpFA{-6}$,\nwith the false-alarm level of $p_{\\mathrm{FA}*}^{(0)}=10^{-6}$ corresponding to a transition scale of\n\\mbox{$\\Fth^{(0)}({N_{\\mathrm{seg}}}=1)\\approx16.7$} and \\mbox{$\\scF_*^{(0)}({N_{\\mathrm{seg}}}=84)\\approx237.0$}, respectively.\n\n\\subsubsection{Data selection}\n\\label{sec:data-selection}\nWe use four\\ narrow frequency bands of LIGO S5 data.\nThese bands are chosen depending on how severely they appear to be affected by lines:\n\\begin{enumerate}[(a)]\n\\item a ``quiet'' band where the distribution of the data is very close to Gaussian,\n\\item a band with a single line in $\\textrm{L1}$,\n\\item a band with a single line in $\\textrm{L1}$, narrower than in (b),\n\\item a band with multiple disturbances in $\\textrm{H1}$.\n\\end{enumerate}\nThe normalized SFT power $\\Psft^X(f)$ of Eq.~\\eqref{eq:Psft} for each of the four\\ bands is shown\nin Fig.~\\ref{fig:tests_realdata_normSFT_coh} for the coherent case, and in\nFig.~\\ref{fig:tests_realdata_normSFT_semicoh} for the semicoherent case.\nMore details about these sample frequency bands are given in Tables~\\ref{tbl:PsftthrCoh} and\n\\ref{tbl:PsftthrSC}, respectively.\n\nThe data sets are taken from the first year of the LIGO S5 science run.\nFor the semicoherent searches we use ${N_{\\mathrm{seg}}}=84$ data segments, spanning $T=25\\,\\mathrm{h}$ each, while the\ncoherent searches use only a single segment.\nThese segments were originally selected for the Einstein@Home~\\cite{EatH} search described in\nRef.~\\cite{aasi13:_eathS5}.\nSince CW searches on this data have not found any signals\n\\cite{abbott09:_earlyS5,abadie12:_powerflux,aasi13:_eathS5}, we consider it as a \\emph{pure noise} set\nfor the purpose of this study.\n\n\n\\begin{table*}[h!tbp]\n \\input{table_coherent_data_four.tex}\n \\caption{\n   \\label{tbl:PsftthrCoh}\n   Data used for tests of the coherent statistics in Sec.~\\ref{sec:tests_realdata}.\n  \n   All data is taken from the first year of the LIGO S5 run.\n  \n   CW signals are injected with frequencies $f\\in f_{\\mathrm{inj}}$, while $\\fsft$ denotes the SFT frequency\n   range used for the search and the prior line estimation.\n  \n   Each data set starts at a GPS time of $t_{\\mathrm{start}}$ and spans 25 hours, containing $N_\\sft^X$\n   SFTs of duration $T_\\sft=1800\\,\\sec$ from each detector.\n  \n   The multidetector noise PSD $\\Sn$ was obtained as the harmonic mean over SFTs and arithmetic\n   mean over frequency bins.\n  \n   The column labeled $\\maxnoise{2\\F}$ shows the corresponding highest multidetector $2\\F$ value without\n   injections.\n  \n   The noise PSD per detector is $\\SnX$.\n  \n   The column $\\Psftthr^X$ gives the threshold on the normalized SFT power $\\Psft^X$ at $p_{\\mathrm{FA},\\Psft}=10^{-9}$,\n   which is used to estimate the prior line-odds $\\prior{\\OLG}^X$ as described in Sec.~\\ref{sec:estimate-line-probs}.\n }\n\\end{table*}\n\n\n\\begin{table*}[h!tbp]\n \\input{table_semicoherent_data_four.tex}\n \\caption{\n   \\label{tbl:PsftthrSC}\n   Data used for tests of the semicoherent statistics in Sec.~\\ref{sec:tests_realdata}.\n   All data is taken from the first year of the LIGO S5 run, corresponding to the segment\n   selection used in an Einstein@Home search (S5R3) \\cite{aasi13:_eathS5}, spanning 381.04~days\n   starting from GPS epoch $t_{\\mathrm{start}}=818845553$, containing ${N_{\\mathrm{seg}}}=84$ segments, each 25 hours long.\n   The column labeled $\\maxnoise{2\\avgSeg{\\F}}$ refers to the highest average multidetector\n   $2\\avgSeg{\\F}$ value without injections (the average is over segments).\n   The remaining labels are identical to those in Table~\\ref{tbl:PsftthrCoh}.\n   }\n\\end{table*}\n\n\\subsubsection{Signal injection and detection criterion}\n\\label{sec:sign-inject-detect}\n\nThe search setup used here is different from that of Ref.~\\cite{aasi13:_eathS5}, and employs the\n\\texttt{HierarchSearchGCT} code instead of the Hough-transform~\\cite{krishnan04:_hough}.\nThis code is used in recent and ongoing wide-parameter-space searches such as\nRefs.~\\cite{aasi2013:_gc-search,EatH}.\n\nThe grid spacings in frequency and spin-down are $\\delta f \\approx 1.6 \\times 10^{-6}\\,\\mathrm{Hz}$ and $\\delta\\dot{f}\n\\approx 5.8 \\times 10^{-11}\\,\\mathrm{Hz}\/s$, respectively.\nThe angular sky-grid spacings are approximately $0.15\\,\\mathrm{rad}$ at $f = 54\\,\\mathrm{Hz}$, and scale with\nfrequency as $1\/f$.\n\nWe find that this template bank yields an average relative loss of SNR$^2$ (also known as mismatch) of $m\\sim\n0.6$ in the\nsemicoherent searches and of $m\\lesssim 0.05$ in the coherent searches.\n\n\nWe first perform searches on the data without any injections,\ncovering the whole sky in each of the four{} frequency bands of width $\\Delta f = 50\\,\\mathrm{mHz}$ (see\n$f_{\\mathrm{inj}}$ in Tables \\ref{tbl:PsftthrCoh} and \\ref{tbl:PsftthrSC}), and a fixed band $[-\\Delta\\dot{f},\\,0]$ in\nspin-down ${\\dot{\\Freq}}$, with $\\Delta \\dot{f} \\approx 2.6 \\times 10^{-9}\\,\\mathrm{Hz}\/\\sec$.\n\nFor each of the four statistics $\\{\\sc{\\F},\\sc{\\F}^{\\mathrm{+veto}},\\sc{O}_{{\\Signal\\Line}},\\OSNsc\\}$ we record the loudest noise candidate over\nthe whole template grid.\nA signal will be considered as detected with a given statistic if its highest value exceeds this noise value.\nThis definition of detection is equivalent to the common method of setting loudest-event upper\nlimits, employed for example in Ref.~\\cite{aasi13:_eathS5}.\n\nThe signals are injected using the \\texttt{Makefakedata\\_v4} code, with signal parameters randomly drawn\nfrom uniform distributions in the sky coordinates $\\{\\alpha,\\delta\\}$,\ninclination $\\cos\\iota$ and polarization angle $\\psi$, and at varying signal amplitude $h_0$.\nThe signal frequency and spin-down are drawn uniformly from the bands used in the noise search.\nFor each value of $h_0$ we perform 1000 injections.\nFor each injection we search a small parameter-space volume containing the signal.\nThis search region consists of a frequency band of $\\Delta f = 1\\,\\mathrm{mHz}$, a spin-down band of\n$\\Delta \\dot{f} \\approx\n2.3 \\times 10^{-10}\\,\\mathrm{Hz}\/\\sec$ and the 10 sky-grid points closest (in the metric sense \\cite{prix06:_searc})\nto the injection.\n\nNote that in (b) and (c) some of these injection searches do not use any data containing the narrow\ndisturbances. Hence, the statements in this section apply to \\emph{bands that contain disturbances}, and not\nonly to \\emph{sets of disturbed candidates}.\n\n\n\n\\begin{figure}[h!tbp]\n    \\raggedright ($\\widetilde{\\textrm{a}}$)\\\\\\vspace*{-1cm}\n    \\includegraphics[width=\\columnwidth]{S5R3_54dot20Hz_seg26_normSFT_H1L1} \\\\\n    \\raggedright ($\\widetilde{\\textrm{b}}$)\\\\\\vspace*{-0.5cm}\n    \\includegraphics[width=\\columnwidth]{S5R3_66dot50Hz_seg64_normSFT_H1L1}\\\\\n    \\raggedright ($\\widetilde{\\textrm{c}}$)\\\\\\vspace*{-0.5cm}\n    \\includegraphics[width=\\columnwidth,clip]{S5R3_69dot70Hz_seg9_normSFT_H1L1} \\\\\n    \\raggedright ($\\widetilde{\\textrm{d}}$)\\\\\\vspace*{-0.5cm}\n    \\includegraphics[width=\\columnwidth]{S5R3_58dot50Hz_seg13_normSFT_H1L1}\n  \\caption{\n    \\label{fig:tests_realdata_normSFT_coh}\n    Normalized average SFT power $\\Psft^X(f)$ of Eq.~\\eqref{eq:Psft} as a function of frequency $f$\n    for LIGO H1 (solid) and L1 (dashed) data used in the coherent searches.\n    The horizontal lines mark, for each detector, the threshold $\\Psftthr^X$ at $p_{\\mathrm{FA},\\Psft}=10^{-9}$ used in the\n    line prior estimation.\n    The panels show:\n    ($\\widetilde{\\textrm{a}}$) a quiet band,\n    ($\\widetilde{\\textrm{b}}$), ($\\widetilde{\\textrm{c}}$) two bands with lines,\n    ($\\widetilde{\\textrm{d}}$) a band with multiple disturbances. See Table~\\ref{tbl:PsftthrCoh} for more details on these data\n     sets.}\n\\end{figure}\n\\begin{figure}[h!tbp]\n    \\raggedright ($\\widetilde{\\textrm{a}}$)\\\\\\vspace*{-1cm}\n    \\includegraphics[width=\\columnwidth]{gct_injections_detprobs_54dot20Hz_seg26} \\\\\n    \\raggedright ($\\widetilde{\\textrm{b}}$)\\\\\\vspace*{-0.5cm}\n    \\includegraphics[width=\\columnwidth,clip]{gct_injections_detprobs_66dot50Hz_seg64} \\\\\n    \\raggedright ($\\widetilde{\\textrm{c}}$)\\\\\\vspace*{-0.5cm}\n    \\includegraphics[width=\\columnwidth]{gct_injections_detprobs_69dot70Hz_seg9}\n    \\raggedright ($\\widetilde{\\textrm{d}}$)\\\\\\vspace*{-0.5cm}\n    \\includegraphics[width=\\columnwidth]{gct_injections_detprobs_58dot50Hz_seg13}\n  \\caption{\n    \\label{fig:tests_realdata_detprobs_coherent}\n    Detection efficiency $\\pDet$ as a function of scaled signal amplitude $h_0\/\\sqrt{\\Sn}$ for four\n    different coherent statistics:\n    $\\F$,\n    $\\F^{\\mathrm{+veto}}$,\n    $\\OSL^{(0)}$,\n    and $\\OSNpFA{-6}$.\n    Statistical errors are similar to the size of the symbols.\n    The dashed horizontal line marks the $95\\%$ detection probability level.\n   \n    The panels show:\n    ($\\widetilde{\\textrm{a}}$) a quiet band,\n    ($\\widetilde{\\textrm{b}}$), ($\\widetilde{\\textrm{c}}$) two bands with lines,\n    ($\\widetilde{\\textrm{d}}$) a band with multiple disturbances. See Fig.~\\ref{fig:tests_realdata_normSFT_coh} and\n    Table~\\ref{tbl:PsftthrCoh} for more details on these data sets.\n  }\n\\end{figure}\n\n\n\\begin{figure}[h!tbp]\n    \\raggedright($\\sc{\\textrm{a}}$)\\\\\\vspace*{-1cm}\n    \\includegraphics[width=\\columnwidth]{S5R3_54dot20Hz_84seg_normSFT_H1L1} \\\\\n    \\raggedright ($\\sc{\\textrm{b}}$)\\\\\\vspace*{-0.5cm}\n    \\includegraphics[width=\\columnwidth]{S5R3_66dot50Hz_84seg_normSFT_H1L1} \\\\\n    \\raggedright (\\csc)\\\\\\vspace*{-0.5cm}\n    \\includegraphics[width=\\columnwidth,clip]{S5R3_69dot70Hz_84seg_normSFT_H1L1}\n    \\raggedright ($\\sc{\\textrm{d}}$)\\\\\\vspace*{-0.5cm}\n    \\includegraphics[width=\\columnwidth]{S5R3_58dot50Hz_84seg_normSFT_H1L1}\n  \\caption{\n    \\label{fig:tests_realdata_normSFT_semicoh}\n    Normalized average SFT power $\\Psft^X(f)$ of Eq.~\\eqref{eq:Psft} as a function of frequency $f$\n    for LIGO H1 (solid) and L1 (dashed) data used in the semicoherent searches.\n    The horizontal lines mark, for each detector, the threshold $\\Psftthr^X$ at $p_{\\mathrm{FA},\\Psft}=10^{-9}$ used in the\n    line prior estimation.\n    The panels show:\n    ($\\sc{\\textrm{a}}$) a quiet band,\n    ($\\sc{\\textrm{b}}$), (\\csc) two bands with lines,\n    ($\\sc{\\textrm{d}}$) a band with multiple disturbances.\n    See Table~\\ref{tbl:PsftthrSC} for more details on these data sets.\n  }\n\\end{figure}\n\\begin{figure}[h!tbp]\n    \\raggedright ($\\sc{\\textrm{a}}$)\\\\\\vspace*{-1cm}\n    \\includegraphics[width=\\columnwidth,clip]{gct_injections_detprobs_54dot20Hz_84seg} \\\\\n    \\raggedright ($\\sc{\\textrm{b}}$)\\\\\\vspace*{-0.5cm}\n    \\includegraphics[width=\\columnwidth,clip]{gct_injections_detprobs_66dot50Hz_84seg} \\\\\n    \\raggedright (\\csc)\\\\\\vspace*{-0.5cm}\n    \\includegraphics[width=\\columnwidth,clip]{gct_injections_detprobs_69dot70Hz_84seg}\n    \\raggedright ($\\sc{\\textrm{d}}$)\\\\\\vspace*{-0.5cm}\n    \\includegraphics[width=\\columnwidth,clip]{gct_injections_detprobs_58dot50Hz_84seg}\n  \\caption{\n    \\label{fig:tests_realdata_detprobs_semicoherent}\n    Detection efficiency $\\pDet$ as a function of scaled signal amplitude $h_0\/\\sqrt{\\Sn}$ for four\n    different semicoherent statistics:\n    $\\sc{\\F}$,\n    $\\sc{\\F}^{\\mathrm{+veto}}$,\n    $\\OSLsc^{(0)}$,\n    and $\\OSNscpFA{-6}$.\n    Statistical errors are similar to the size of the symbols.\n    The dashed horizontal line marks the $95\\%$ detection probability level.\n   \n    The panels show:\n    ($\\sc{\\textrm{a}}$) a quiet band,\n    ($\\sc{\\textrm{b}}$), (\\csc) two bands with lines,\n    ($\\sc{\\textrm{d}}$) a band with multiple disturbances. See Fig.~\\ref{fig:tests_realdata_normSFT_semicoh} and\n    Table~\\ref{tbl:PsftthrSC} for more details on these data sets.\n  }\n\\end{figure}\n\n\n\\subsubsection{Results for coherent statistics}\n\\label{sec:results-using-coher}\n\nFigure~\\ref{fig:tests_realdata_detprobs_coherent} shows the detection efficiency $\\pDet$ as a function of the\nscaled signal amplitude $h_0\/\\sqrt{\\Sn}$, for the single-segment coherent statistics.\n\nIn the quiet band, shown in Fig.~\\ref{fig:tests_realdata_detprobs_coherent}~($\\widetilde{\\textrm{a}}$),\nwe find that the line-veto statistic $\\OSL^{(0)}$ has less detection power than the\n$\\F$-statistic, as would be expected since it does not match the noise population.\nThe conventional $\\Fveto$-statistic{} is safer than $\\OSL^{(0)}$ and performs just as well as the pure\n$\\F$-statistic.\nThe line-robust statistic $\\OSN$ performs equally well as $\\F$ and $\\F^{\\mathrm{+veto}}$ on this line-free data set.\n\nIn the disturbed bands shown in\nFig.~\\ref{fig:tests_realdata_detprobs_coherent}~($\\widetilde{\\textrm{b}}$)-($\\widetilde{\\textrm{d}}$),\nall statistics lose detection power to varying degrees.\nWe find that the $\\Fveto$-statistic{} is often able to recover most of the losses of the pure $\\F$-statistic.\nThe line-veto statistic $\\OSL^{(0)}$ performs similarly in case ($\\widetilde{\\textrm{b}}$) and yields an improvement over $\\F^{\\mathrm{+veto}}$ in\ncases ($\\widetilde{\\textrm{c}}$) and ($\\widetilde{\\textrm{d}}$).\nHowever, these cases show $\\OSN$ to be more robust than either of the simpler vetoes.\n\nSummarizing these results, we see that the line-robust statistic $\\OSN$ consistently shows the best\nperformance over the different types of data: it is more robust to varying kinds of disturbances than $\\F^{\\mathrm{+veto}}$\nand safer in Gaussian noise than $\\OSL^{(0)}$.\n\n\\subsubsection{Results for semicoherent statistics}\n\\label{sec:results-using-semi}\n\nFigure~\\ref{fig:tests_realdata_detprobs_semicoherent} shows the detection efficiency $\\pDet$ as a function of\n$h_0\/\\sqrt{\\Sn}$ for the semicoherent statistics over the full data set.\nQualitatively, we find very similar results to the coherent case of\nFig.~\\ref{fig:tests_realdata_detprobs_coherent}.\n\nFor the quiet band, shown in Fig.~\\ref{fig:tests_realdata_detprobs_semicoherent}~($\\sc{\\textrm{a}}$),\nwe find that the simple line-veto $\\OSLsc^{(0)}$ loses a significant fraction of detection power compared to the\nsemicoherent $\\sc{\\F}$-statistic and to $\\sc{\\F}^{\\mathrm{+veto}}$, while the line-robust statistic $\\OSNsc$ does not show any\nsignificant degradation.\n\nIn the bands with noise disturbances\n(Fig.~\\ref{fig:tests_realdata_detprobs_coherent}~($\\sc{\\textrm{b}}$)-($\\sc{\\textrm{d}}$)),\nit is again the $\\sc{\\F}$-statistic which suffers the most.\nThese examples show the line-robust statistic $\\OSNsc$ consistently performing better than $\\sc{\\F}$ and as well\nas or better than either $\\OSLsc^{(0)}$ or $\\sc{\\F}^{\\mathrm{+veto}}$ in all the disturbed bands.\nThe largest improvement is found in the example shown in\nFig.~\\ref{fig:tests_realdata_detprobs_semicoherent}~(\\csc), where the signal amplitude at $95\\,\\%$ detection\nprobability is nearly two times smaller for $\\OSNsc$ compared to $\\sc{\\F}^{\\mathrm{+veto}}$.\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nWe have extended the standard derivation of the $\\F$-statistic by adding an explicit simple line hypothesis to\nthe standard Gaussian-noise hypothesis, namely a CW-signal-like disturbance in a single detector.\nMore work would be required to deal with coincident disturbances in multiple detectors.\n\nUsing the Bayesian framework we have derived two new detection statistics:\na ``line-veto'' statistic $O_{\\Signal\\Line}$, which complements the $\\F$-statistic and may be appropriate\nfor the follow-up of strong outliers, and a new line-robust detection statistic $\\OSN$, which\ncontains both $\\F$ and $O_{\\Signal\\Line}$ as limiting cases.\nWe have also generalized both statistics to semicoherent searches.\n\nThe line-robust $\\OSN$ requires choosing several prior parameters.\nWe have found in particular that the performance of $\\OSN$ is sensitive to $\\Fth^{(0)}$, which regulates the\ntransition scale between $\\F$ and $O_{\\Signal\\Line}$.\nThis parameter stems from a rather unphysical prior in the $\\F$-statistic derivation \\cite{prix09:_bstat}, and\nwe could therefore only provide an ad-hoc empirical prescription for choosing it.\nFurther work to improve on this prior could also result in increased robustness when the detectors are not\nequally sensitive.\n\nThe remaining parameters are more straightforward to interpret, as they encode the prior probability of line\nartifacts.\nFor these we have tested both an ignorance prior and a simple adaptive tuning method.\n\nWe have tested the detection power of the new statistics on synthetic candidates, where both signal and noise\nmatch our hypotheses, and on simulated signals injected into LIGO S5 data.\nIn both cases we have found that, with a reasonable choice of transition scale, $\\OSN$ is consistently the\nmost robust in the presence of various types of instrumental artifacts.\nIn particular, it consistently equals or surpasses the performance of the popular ad-hoc\n$\\F$-statistic consistency veto, reaching up to a factor of two improvement in detectable signal\nstrength at 95\\% confidence in example \\csc{} in Fig.~\\ref{fig:tests_realdata_detprobs_semicoherent}.\n\nCombined with its close-to-optimal performance in undisturbed data, this makes $\\OSN$ a promising statistic\nfor analyzing broadband, diverse data sets.\n\n\\section*{Acknowledgments}\nThis work has benefited from numerous discussions and comments from colleagues, in particular John T. Whelan,\nKarl Wette, Evan Goetz, Berit Behnke, Heinz-Bernd Eggenstein and Thomas Dent.\nWe acknowledge the LIGO Scientific Collaboration for providing the data from the LIGO S5 run.\nThe injection studies were carried out on the ATLAS cluster at AEI Hannover.\nP.L. and M.A.P. acknowledge support of the ``Sonderforschungsbereich'' Collaborative Research\nCentre (SFB\/TR7). D.K. was supported by the IMPRS on Gravitational Wave Astronomy.\nThis paper has been assigned LIGO document number LIGO-P1300167{} and AEI-preprint number AEI-2013-260{}.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{I. Introduction} \n\\label{sec1}\nThe fundamental work of\nEinstein-Podolsky-Rosen (EPR) \\cite{epr} on a distant entanglement of \na pair of non-interacting distinguished \nparticles and its effects on measurements\nis now at the foundations of long-distance quantum communications.\nThe entanglement concept coined by Schr\\\"odinger \\cite{schrodinger}\nwith a gedanken experiment of a cat, dead or alive,\nbecomes a resource of modern quantum computations \\cite{chuang,karol}.\nAn impressive modern progress of quantum information, \ncomputation and communication\nis described in \\cite{deutsch}. \n\nAn overview of various experimental\nrealizations of EPR pairs is given in \\cite{eprexprev}.\nVarious forms of propagating EPR pairs have been studied\nexperimentally but in its main aspect the propagation\nof EPR pairs was rather simple being similar\nto propagation on a line and always being integrable.\nHere we consider theoretically a  situation when \ntwo non-interacting but entangled  particles of an EPR pair propagate\nin a regime of quantum chaos \\cite{haake}.\nIn the classical limit a dynamics of these particles is chaotic \nbeing characterized by an exponential local divergence\nof trajectories with a positive Kolmogorov-Sinai entropy $h$\n\\cite{arnold,sinai,chirikov1979,lichtenberg}. The exponential instability\nof chaotic dynamics leads to exponential growth of round-off errors\nand breaking of time reversibility of classical evolution\ndescribed by reversible equations of motion.\nThus chaos resolves the famous Loschmidt-Boltzmann dispute\non time reversibility and emergence of statistical\nlaws from reversible dynamical equations \\cite{boltzmann1,loschmidt,boltzmann2}\n(see also \\cite{mayer}). \nPrior to classical chaos theory the problem of time reversal of \nlaws of nature was also discussed by such leading scientists as \nSchr\\\"odinger \\cite{schrodinger1931} \n(see English translation and overview in \\cite{schrodtrans}) and \nKolmogorov \\cite{kolmogorov}.\n\nHowever, in quantum mechanics a chaotic mixing in a phase-space cannot go \ndown to exponentially small scales being restricted by a quantum scale of the \nPlanck constant $\\hbar$. Thus in the regime of quantum chaos\nan exponential instability exists only during a logarithmically \nshort Ehrenfest time\nscale $\\tau_E \\sim |\\ln \\hbar|\/h$ \\cite{chi1981,dls1981,chi1988,ehrenfestime}\n(here $\\hbar$ is a dimensionless effective Planck constant \nrelated to typical quantum numbers). \nDue to the absence of exponential instability on\ntimes beyond $\\tau_E$ the quantum evolution remains reversible \nin presence of quantum errors in a drastic difference from\nthe classical dynamics as it was demonstrated\nin \\cite{dls1983} for the quantum Chirikov standard map,\nalso known as a kicked rotator \\cite{chi1981,chi1988,stmap}.\nThis system has been experimentally realized with cold atoms\nin kicked optical lattices and in particular \nthe quantum dynamical localization of\nchaotic diffusion has been observed in these experiments\n\\cite{raizen,garreau}.  This dynamical localization of chaotic diffusion\nappears due to quantum interference and is analogous \nto the Anderson localization \\cite{anderson}\nof electron diffusion in disordered solids (see e.g. \\cite{fishman1,fishman2,dls1987}).\n\nIn \\cite{martin} it was shown that the time evolution of cold atoms in \nkicked optical lattices, described by the quantum Chirikov standard map,\ncan be reversed in time in the regime of quantum chaos.\nThis proposal was indeed experimentally realized by the Hoogerland \ngroup \\cite{hoogerland}.\nThus this system represents an efficient experimental platform which\nallows to investigate nontrivial effects of quantum mechanics, localization,\nchaos and time reversal.\n\nIn this work we investigate the properties of chaotic EPR pairs\nevolving in this fundamental system of quantum chaos\nand show that a measurement of one of the entangled particles\nbreaks exact time reversal of the other particle but \npreserves its approximate time reversibility.\nWe explain this unusual effect on the basis of \nthe Schmidt decomposition \\cite{schmidt}\n(see also the review \\cite{fedorov} and Refs. therein)\nand the  Feynman path integral formulation of quantum mechanics \\cite{feynman}.\n\nThis article is composed as follows: the model is described in Section II,\nthe results are presented in Section III\nand the discussion and conclusion are given in Section IV;\nadditional Figures and data are given in Appendix.\n\n\\section{II. Model description} \n\\label{sec2}\n\nThe classical dynamics of one particle is described by\nthe Chirikov standard map \\cite{chirikov1979}:\n\\begin{equation}\n\\label{stmap}\n\\bar{p} = p + k \\sin{ x} \\; , \\;\\; \n\\bar{x} = x + T \\bar{p} \\; .\n\\end{equation}\nHere $x$ represents the position of an atom\nin an infinite $x-$axis of the kicked optical lattice,\nor  a cyclic variable $0 \\leq x < 2\\pi$\nfor the case of the kicked rotator; $p$ is the momentum of a particle.\nThe bars denote the new values of variables after one iteration of this \nsymplectic map. \nThe physical process described by this map corresponds to\na sharp change of momentum, generated \ne.g. by a kick  of the optical lattice \\cite{raizen,garreau},\nfollowed by a free particle propagation during a period $T$ between kicks.\nThe classical dynamics depends on a single chaos parameter\n$K=kT$ with a transition from integrability to unlimited chaotic\ndiffusion in momentum for $K > K_c =0.9715...$ \\cite{chirikov1979,lichtenberg}.\nThe system dynamics is reversible in time, e.g. by\ninverting all velocities in a middle of free rotation between two kicks.\n\nInside a chaotic component the dynamics is characterized by \nan exponential divergence of trajectories with the positive\nKolmogorov-Sinai entropy $h$. For $K>4$ the measure of stability\nislands is small and we have $h  \\approx \\ln(K\/2)$ \\cite{chirikov1979}.\nFor $K > K_c$ the dispersion of momentum\ngrows diffusively with time $\\langle(\\Delta p)^2\\rangle = D t$ \nwith a diffusion coefficient\n$ D \\approx k^2\/2$ (see more details in  \\cite{chirikov1979,dls1987}).\nHere and below the time $t$ is measured in number of map iterations.\nThe map captures a variety of universal features\nof dynamical chaos and appears in the description\nof various physical systems \\cite{stmap}. \n\nThe quantum evolution of the state $\\ket{\\psi}$ over a period is given \nby a unitary operator\n $\\hat{U}$ \\cite{chi1981,chi1988}:\n\\begin{eqnarray} \n\\label{qmap}\n\\ket{\\bar{\\psi}} = \\hat{U} \\ket{\\psi} = \ne^{-iT\\hat{p}^2\/2} e^{-ik\\cos{\\hat{x}}} \\ket{\\psi} \\; .\n\\end{eqnarray} \nHere the momentum $p$ is measured in recoil units of optical lattice with \n$\\hat{p}=-i \\partial \/ \\partial x $. Thus $T=\\hbar$ plays the role of\nan effective dimensionless Planck constant and the classical limit\ncorresponds to $T=\\hbar \\rightarrow 0$, $k \\rightarrow \\infty$,\n$K=kT = const$. Due to the periodicity of the optical lattice potential\nthe momentum operator $\\hat{p}=-i \\partial \/ \\partial x $ has eigenvalues\n$p=n+\\beta$ where $n$ is an integer and\n$\\beta$ is a quasimomentum conserved by the kick potential \n($0 \\leq \\beta < 1$). The value $\\beta=0$\ncorresponds to the case of a kicked rotator\nwith a wave function (in position representation) \n$\\psi(x)=\\langle x\\ket{\\psi}$ being \nperiodic on a circle $\\psi(x+2\\pi)=\\psi(x)$.\nIn this case the free rotation correspond (in momentum representation) \nto the phase shift\n${\\bar {\\psi}}_{n,0} = \\exp(-iTn^2\/2) \\psi_{n,0}$ with \n$\\psi_{n,\\beta}=\\langle p\\ket{\\psi}$ being the wave function \n(in momentum representation) at $p=n+\\beta$. \nIrrational values of $\\beta$ appear for\na particle propagation on an infinite $x$-axis;\nhere $\\beta$ is conserved\nand a free propagation of the momentum wave function $\\psi_{n,\\beta}$\ngives the phase shift ${\\bar {\\psi}}_{n,\\beta} = \n\\exp(-iT(n+\\beta)^2\/2)\\,\\psi_{n,\\beta}$.\nThe effects of quantum interference lead to dynamical localization\nof chaotic diffusion on a time scale $t_D \\approx D\/\\hbar^2 \\gg \\tau_E$\nand an exponential localization of quasienergy eigenstates\nwith a localization length $\\ell = D\/(2 \\hbar^2) \\approx k^2\/4$ \\cite{dls1987,chi1988}.\n\n\nIn \\cite{martin} it was pointed that the time reversal of a quantum evolution\nafter $t_r$ map iterations\ncan be realized by using a period between kicks\nbeing $T=4\\pi+\\epsilon$ for $t \\leq t_r$\nand $T'=4\\pi - \\epsilon$ for $t_r < t \\leq 2 t_r$.\nAlso the time reversal is done at the middle of the free propagation \nafter $t_r$ kicks (it is convenient to use a symmetrized scheme with a\nhalf-period of free  rotation then kick and then again \na half-period of free propagation).\nThe inversion of kick amplitude\n$k \\cos x \\rightarrow - k \\cos x$ can be realized \nby a $\\pi$-translational shift of the optical lattice potential.\nSuch a time reversal is exact for $\\beta=0$ (kicked rotator case)\nand it also works approximately for small $\\beta$ values\nin the case of the kicked particle \\cite{martin}. \nThe time reversal for cold atoms in a kicked optical lattice\nwas experimentally demonstrated in \\cite{hoogerland}.\n\nHere we consider the time reversal of two\nnon-interacting distinguished particles being in an initial\nentangled state. We concentrate our analysis on the case when\nboth particles evolve in the regime of quantum chaos.\nThus we have the new case of chaotic EPR pairs. Following (\\ref{qmap})\nthe evolution of the two particle state $\\ket{\\psi}$ \n(with wave function $\\psi(x_1,x_2)=\\langle x_1,x_2\\ket{\\psi}$) \nof such pairs is given by the quantum map\n\\begin{eqnarray} \n\\label{qmappair}\n\\ket{\\bar{\\psi}} = (\\hat{U}_1\\otimes \\hat{U}_2) \\ket{\\psi}   \\; ,\n\\end{eqnarray} \nwhere $\\hat{U}_1$ and  $ \\hat{U}_2$ are one time period\nevolution operators for the first and second particle. \nIn absence of interactions between particles\nthe entropy of entanglement $S$ is preserved during this time\nevolution. It is convenient to use the Schmidt decomposition \n\\cite{schmidt,fedorov} \nfor an initial entangled state\n\\begin{eqnarray} \n\\label{schmidt}\n\\ket{\\psi}=\\sum_{i=1}^m \\alpha_i \\ket{u_i}\\otimes\\ket{v_i}\n\\end{eqnarray} \nwhere $\\ket{u_i}$, $\\ket{v_i}$ are one-particle states satisfying \nthe orthogonality relations: \n$\\langle u_i\\ket{u_j}=\\langle v_i\\ket{v_j}=\\delta_{ij}$. \nThe number $m$ of Schmidt components can be up to $m=N$ if $N$ is the \ndimension of the one-particle Hilbert space. However, for ``less'' entangled \nstates $m$ may be smaller and in this work we will consider the case of \n$m=2$.\nThe entropy of entanglement is then \ngiven by (see e.g. \\cite{chuang,fedorov}):\n \\begin{eqnarray} \n\\label{entropy}\nS = -Tr(\\rho_1 \\log_2 \\rho_1) = - \\sum_i |\\alpha_i|^2 \\log_2  |\\alpha_i|^2 \\; ,\n\\end{eqnarray} \nwhere $\\rho_1$ is a reduced density matrix of first particle\nobtained by a trace taken over the second particle.\nDuring the time evolution of EPR pair given by (\\ref{qmappair})\nthe wave functions of each particle evolve independently\nwith $\\ket{u_i(t)} = {\\hat{U}_1^t} \\ket{u_i(t=0)}$\nand  $\\ket{v_i(t)} = {\\hat{U}_1^t} \\ket{v_i(t=0)}$.\nThus the coefficients $\\alpha_i$ of the Schmidt decomposition and the \nentropy of entanglement $S$ remain unchanged.\n\nHowever, since the particles are entangled\na measurement of the second particle after the time $t_r$ affects the \nwave function of first particle and thus the time reversal evolution of this \nparticle is modified so that the exact time reversibility\nis broken by the measurement. Nevertheless, we will see that still there is\nan approximate time reversal of the first particle.\nWe describe in detail this effect in the next section.\n\n\\section{III. Time evolution of chaotic EPR pairs} \n\\label{sec3}\n\nThe numerical simulations of the quantum map (\\ref{qmap}),~(\\ref{qmappair})\nare done in a usual way \\cite{chi1981,chi1988} \nby using the fact that the free propagation and the kick \nare diagonal in the momentum and coordinate representations respectively.\nConcerning the eigenphases $T n^2\/2$ of the free propagation operator \nwe mention an important technical detail: we compute these phases \nfor $n=-N\/2,\\,\\ldots,\\,N\/2-1$ \n(with $N$ being the dimension of the one-particle Hilbert space) \nand the values for $n<0$ are stored at the positions $N-n$ while the \nvalues for $n\\ge 0$ are stored at positions $n$. In this way \nif the initial states are localized close to small values of $n\\approx 0$ \n(or $n\\approx N$ which is topologically close to $n\\approx 0$ due \nto the periodic boundary conditions) and \nif during the time evolution the states do not touch the borders at \n$n\\approx \\pm N\/2$ the results are independent of the exact choice $N$ \nprovided $N$ is sufficiently large. In other words the momentum phases \nexhibit a smooth transition between $n\\approx 0$ and $n\\approx N$ according \nthe quadratic formula while at the ``system border'' $n\\approx N\/2$ \nthis transition is not smooth. Otherwise, if the phases were naively \ncomputed for $n=0,\\,\\ldots,N-1$ according to the quadratic formula the \nresults would depend in a sensitive way on $N$ even if the states remain \nlocalized close to $n\\approx 0$ since the eigenphases for $n\\approx N$ \nwould be very different.\n\nThe transitions from one representation (momentum or position) to another and\nback are done with the Fast Fourier Transform (FFT).\nFurthermore, we chose the quantum map to be directly symmetric in time and \ntherefore we present it as a half period of free propagation \n(using the operator $\\hat U_{\\rm half,free}=e^{-iT\\hat{p}^2\/4}$)\nfollowed by the kick (using $\\hat U_{\\rm kick}=e^{-ik\\cos{\\hat{x}}}$) \nand then again a half period of free propagation (using \n$\\hat U_{\\rm half,free}$).\nFurthermore, in order to have an exact mathematical equivalence \nbetween the two cases $T=4\\pi+\\epsilon$ and $T=\\epsilon$ \n(at $\\beta=0$) we also apply for \nthe first case to the initial states (given below for the different cases\nwe consider) \nan initial half period of free propagation \nwith $T=4\\pi$ (which provides an additional phase factor $(-1)^{n_1+n_2}$ \nin momentum representation). We have numerically verified that this \nequivalence is indeed valid. \n\nWe consider in detail 3 specific cases:\nA) kicked rotator case with a moderate dimensionless effective Planck constant\n$\\hbar_{\\rm eff}=\\epsilon = T -4\\pi < 1$ and a wavefunction periodic \non the $2\\pi$-circle (i.e. integer values of $p_i=n_i$ with $\\beta_i=0$ \nand $i=1,2$ for both particles);  \nB) same case but taken in the deep semiclassical regime with \n$\\hbar_{\\rm eff}  \\ll 1$; C) the case of kicked particles propagation\non an infinite (or quasi-infinite) line at moderate $\\hbar_{\\rm eff}$\nthat corresponds to the case of cold atoms in a kicked\noptical lattice \\cite{raizen,garreau,hoogerland} composed of $L$ periods \nsuch that $x\\in[0,\\,2\\pi L$. \nThe total computational basis size for one particle,\nused in the numerical simulations,\nwas changing from $N=1024$ up to $N=2^{22}$,\ndepending on the choice of A), B), C) and insuring \nthat the basis size does not affect the obtained results.\nFor two particles the size of the Hilbert space is\n$N_H=N^2$. For moderate values of $N$ (e.g. up to $N=2^{12}$ \nin cases A and B)\nwe used the whole basis with $N_H$ states\nusing two-dimensional (2D) FFT\ntransitions between momentum and coordinate \nrepresentations in (\\ref{qmappair}).\nFor larger $N$ values we used the fact that \nthe Schmidt decomposition (\\ref{schmidt}) has coefficients $\\alpha_i$\nbeing unchanged during the time evolution\nso that we propagate independently each particle\nand use the Schmidt entangled EPR wavefunction\nfor a measurement of the second particle\nat the time moment $t_r$ \nand backward propagating only the first particle\nafter measurement. \nWe checked, for $N \\leq 2^{12}$, that these two numerical \nmethods of time evolution simulation\ngive the same results up to the computer numerical accuracy.\nSome additional details about numerical simulations \nand Figures are given in the Appendix.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{fig1}\n\\end{center}\n\\caption{\\label{fig1} \nTime dependence of the average energy \nof the first particle $E_{1}(t) = \\langle n_{1}^2\/2\\rangle$\nfor the initial state (\\ref{eqinitialstate}) \nwith time evolution given by the quantum Chirikov standard map (\\ref{qmap})-(\\ref{qmappair}).\nThe measurement of second particle and time reversal are performed\nafter $t_r =40$ quantum map (\\ref{qmappair}) iterations.  \nThe black curves in both panels show the forward time evolution \nfor $0\\le t\\le t_r$; the blue curves show the \nbackward time evolution $t_r \\le t \\leq 2t_r=80$ \nwith the exact time reversal without measurement (using $T=4\\pi-\\epsilon$). \nIn panel (a) the curves of other colors show the backward time evolution\nafter measurement of the second particle at momentum states $n_2=8$\n(cyan), 12 (green), 20 (magenta) , 200 (red).\nIn panel (b) the red curve shows the backward time evolution \nafter the second particle measurement detection at $n_2$ and  \naveraging over all possible measurement results of $n_2$ values \n(black and red curves are shifted up by 50 units for a better visibility;\nred and blue curves coincide within numerical \nround-off errors ($\\sim 10^{-13}$)). The system parameters are:\n $N=1024$, $N_H=N^2$ and $\\hbar_{\\rm eff}=\\epsilon=5\/8$, $K_{\\rm eff}=5$, \n$k=K_{\\rm eff}\/\\hbar_{\\rm eff}=8$,\n$T=4\\pi \\pm \\epsilon$. We have verified that a further increase of $N$ \nto values of $2048$ and $4096$ provide identical results up to numerical \nround-off errors (provided the free propagation eigenphases are properly \ncomputed as explained in the text at the beginning of this section).\n}\n\\end{figure}\n\n\\subsection{IIIA. EPR pairs in kicked rotator at moderate $\\hbar_{\\rm eff}$ values}\n\\label{subsec3a}\n\nHere we present the results for a case with moderate effective value \nof the Planck constant.\nAs described above we use the values of parameter \n$T = 4\\pi +\\epsilon$ for forward\ntime propagation with $t_r$ quantum map iterations and $T=4\\pi - \\epsilon$\nfor next $t_r$ iterations corresponding to the time reversal. We remind that\nsince the phase shift $(4\\pi) n^2\/2$ is a multiple of $2\\pi$ for all integer\nvalues of the momentum $p=n$ the evolution is determined by an effective\nPlanck constant $\\hbar_{\\rm eff}=\\epsilon$. Thus the effective \nclassical chaos parameter \nis $K_{\\rm eff} = k \\epsilon = k \\hbar_{\\rm eff}$. The measurement is done \nfor the second particle after $t_r$ iterations.\nWe consider the case of projective measurement in the momentum basis\n$n_2$ of the second particle  performing the projection \nto a certain value of $n_2$ after $t_r$ iterations. After that the \nevolution of the first particle continues with $T=4\\pi - \\epsilon$\nand $k \\rightarrow -k$\nfor the next $t_r$ iterations. Without measurement the EPR wavefunction \nof two particles returns\nexactly to its initial state due to exact time reversibility\nof the quantum evolution. Also, in absence of entanglement of particles\nthe measurement of the second particle does not affect the reversibility\nof the first particle which would exactly return to its initial state.\nHowever, in presence of entanglement the measurement\nof the second particle affects the time reversibility of the first \nparticle in a nontrivial manner.\n\nTo illustrate the nontrivial features of\nmeasurements on time reversal \nof chaotic EPR pairs we use typical \nsystem parameters with $K=k \\epsilon = k\\hbar_{\\rm eff} =5$\nand $k=8$ (thus $\\hbar_{\\rm eff} =5\/8$). \nSuch a value of $k=8$ is \nnot very high being well accessible to the present experimental facilities\n(see e.g. \\cite{raizen,garreau,hoogerland}).\n\nIn this first part to characterize the quantum time evolution \nwe compute the one-particle probability (of the first particle) as: \n$w(n_1,t) = \\sum_{n_2}|\\psi(n_1,n_2,t)|^2$,\n(with the momentum wave function \n$\\psi(n_1,n_2,t)=\\langle n_1,n_2\\ket{\\psi(t)}$), \nand the one-particle energy (of the first particle):\n$E_{1}(t) = \\langle n_{1}^2\/2\\rangle=\\sum_{n_1} (n_1^2\/2)\\,w(n_1,t)$.\n\nAs initial state we take an entangled EPR pair \nwithout any symmetry and with \nmore or less arbitrary coefficients at two momentum values:\n\\begin{eqnarray}\n\\label{eqinitialstate}\n\\ket{\\psi(t=0)}&=&\\Bigl(\n\\ket{0}\\otimes\\ket{0}+\n0.7\\ket{0}\\otimes\\ket{1}+\\\\\n\\nonumber\n&&\\quad 0.3\\ket{1}\\otimes\\ket{0}-\n2\\ket{1}\\otimes\\ket{1}\\Bigr)\/\\sqrt{5.58} \\; ,\n\\end{eqnarray}\nwhere $\\ket{n_1}\\otimes\\ket{n_2}$ represents the momentum basis states.\nThus initially both particles are distributed over\nmomentum states at $n_{1,2}$ being $0$ or $1$.\n\nThis state can be rewritten in the Schmidt decomposition \\cite{schmidt} as~:\n\\begin{equation}\n\\label{eqschmidt}\n\\ket{\\psi(t=0)}=\\sum_{i=1,2} \\alpha_i \\ket{u_i}\\otimes\\ket{v_i}\n\\end{equation}\nwith\n\\begin{eqnarray}\n\\label{eqschmidt2}\n\\nonumber\n\\alpha_1&=&0.8973\\quad,\\quad\\alpha_2=0.4414,\\\\\n\\nonumber\n\\ket{u_1}&=& 0.3440\\ket{0}-0.9390\\ket{1},\\\\\n\\nonumber\n\\ket{u_2}&=& 0.9390\\ket{0}+0.3440\\ket{1},\\\\\n\\nonumber\n\\ket{v_1}&=& 0.0294\\ket{0}+0.9996\\ket{1},\\\\\n\\ket{v_2}&=& 0.9996\\ket{0}-0.0294\\ket{1},\n\\end{eqnarray}\nThe entropy of entanglement of this initial state is:\n\\begin{equation}\n\\label{eqentropy_log2}\nS=-\\sum_i \\alpha_i^2\\,\\log_2(\\alpha_i^2)=0.7114 \\; .\n\\end{equation}\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{fig2}\n\\end{center}\n\\caption{\\label{fig2} Panel (a) shows the time evolution \nof probability of the first particle $w(n_1,t)$ (color density plot) \nfor parameters of Fig.~\\ref{fig1} and $-128\\le n_1<128$ \n($y$-axis), $0\\le t\\le 80$ ($x$-axis), $t_r=40$. \nThe measurement and time reversal are done after $t_r$ map (\\ref{qmappair}) iterations\nwith the second particle detected at the momentum value $n_2=12$.\nThe thin white vertical \nline marks the time $t_r=40$ of measurement   \nand the beginning of backward iterations. \nThe numbers of the color bar correspond to \n$[w(n_1,t)\/w_{\\rm max}(t)]^{1\/4}$\nwith $w_{\\rm max}(t)=\\max_{n_1} w(n_1,t)$ being the density maximum \nat a given  value of $t$. \nPanels (b) and (c) provide a zoom \nfor $-32\\le n_1<32$ (both panels) \nand $0\\le t<10$ (b) or $70<t\\le 80$ (c). \n}\n\\end{figure}\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{fig3}\n\\end{center}\n\\caption{\\label{fig3} Same as Fig.~\\ref{fig2} but for  \nthe second particle  measured at $n_2=200$. \n}\n\\end{figure}\n\n\nThe time evolution of energy of the first particle\n$E_{1}(t) = \\langle n_{1}^2\/2\\rangle$ is shown in Fig.~\\ref{fig1}.\nAt short initial times $ t \\leq 12$ we have an\napproximately diffusive energy growth\n$E_1(t) \\approx Dt\/2 \\approx 16 t$ with the diffusion coefficient\n$D \\approx k^2\/2 \\approx 32$. For times $12 < t  \\leq t_r=40$ \nthe energy growth continues but its rate decreases due to the quantum interference effects\nbeing similar to the Anderson localization \\cite{chi1981,chi1988,dls1983}). \nAfter $t_r=40$ iterations of forward propagation in time,\nthe projective measurement is performed for the second particle\nand the evolution of the first particle is reversed in time\nwith a replacement $T= 4\\pi + \\epsilon \\rightarrow 4\\pi - \\epsilon$\nand $k \\rightarrow -k$ (effective backward propagation in time).\n\nIn the top panel Fig.~\\ref{fig1}(a) we show the energy\n$E_1(t)$ dependence on time for forward $0 \\leq t \\leq t_r$ \nand backward evolution $t_r < t \\leq 2t_r$\nfor different results of projective measurement, \ndone after $t_r=40$ iterations,\ngiving (projecting) the second particle at\ndifferent momentum states chosen as $n_2=8; 12; 20; 200$.\nFor each measured $n_2$ value we have a different curve \n$E_1(t)$ of the time reversal or backward branch $t_r< t \\leq 2t_r$\nbeing different from the forward branch  $0 \\leq t \\leq t_r$.\nHowever, all curves at different $n_2$ measured values\nhave an energy decrease with time and approximately \nreturn to the initial energy value.\nOf course, in absence of measurements there is the exact time\nreversal of evolution and energy $E_1(t)$ is exactly symmetric with \nrespect to the moment of time reversal and returns exactly to the initial\nvalue as it is shown in Fig.~\\ref{fig1}(b) (blue curve).\nIn the same panel we also present the result of\nbackward evolution of $E_1(t)$ averaged over \nall projective measurements of second particle \nfound at all possible momentum values $n_2$ (Fig.~\\ref{fig1}(b) red curve).\nThe red and blue curves coincide up to \nnumerical round-off errors being on a level of $10^{-13}$.\nSuch an exact coincidence of time reversal behavior \nwithout measurements and with  averaging over all\npossible measurement results \ncan be understood from the Feynman path integral formulation\nof quantum mechanics \\cite{feynman}. In this Feynman interpretation\na specific projective measurement of second particle at  $n_2$ value\nselects a specific entangled path of first particle\nwhich returns approximately to its initial state at $t=0$.\n\nExamples of the time evolution of the probability distribution\nof the first particle $w(n_1,t)$ are shown in Figs.~\\ref{fig2},~\\ref{fig3}\nfor two cases of projective measurements of the second particle\nat momentum values $n_2 = 12, 200$ respectively\n(see also Appendix Fig.~\\ref{figA1}).\nThe measurement is done after $t_r=40$ quantum map (\\ref{qmappair}) iterations.\nFor $0 \\leq t \\leq t_r$ the values of  $w(n_1,t)$\nare obtained by averaging the two-particle density over the second particle,\nafter measurement of second particle  $w(n_1,t)$\nrepresents the probability distribution \nover momentum states of the remaining first particle.\nThe results of Figs.~\\ref{fig2},~\\ref{fig3}\nshow a diffusive type spreading of probability $w(n_1,t)$\nduring the time range $0 \\leq t \\leq t_r$.\nThis corresponds to a diffusion produced \nby the underlined classical chaotic dynamics\n(quantum corrections give a certain  reduction of \nthe diffusion rate as discussed above\nand in \\cite{chi1981,chi1988}). After the projective measurement of the \nsecond particle and the time reversal of the propagation \nan inverse diffusion process takes \nplace where the \nprobability   $w(n_1,t)$ returns approximately\nto the initial state of the first particle. \nThis corresponds to a specific Feynman path selected \nby the projective measurement of the second particle\nimmediately after the time $t_r$. \nSome snapshots of the probability distribution $w(n_1,t)$ \ncorresponding to different results of measurements \nwith different $n_2$ values are shown for\nspecific time moments in Appendix Fig.~\\ref{figA2}.\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{fig4}\n\\end{center}\n\\caption{\\label{fig4} Probabilities $w(n_1,t)$ of the first particle\nat initial time $t=0$ (black lines)\nand final time $t=2t_r=80$ (red lines) for the cases when the second particle is measured \nat $n_2=8$ (a), $n_2=12$ (b), $n_2=20$ (c) and $n_2=200$ (d) after \nthe time reversal at $t_r=40$; system parameters are as in Fig.~\\ref{fig1}. \nThe blue stars provide for each case the semi-analytical theoretical prediction for the \nfinal density at $t=2t_r=80$ obtained from the Schmidt decomposition (see text).\n}\n\\end{figure}\n\nThe probability distribution $w(n_1,t=2t_r)$ \nat the reversal time $t=2t_r$ is shown in Fig.~\\ref{fig4}.\nAll probability is located at momentum states $n_1=0,\\,1$ \ncorresponding to the states populated by the first \nparticle in its initial entangled state (\\ref{eqinitialstate})\n(probability outside of these states\nis on a level of numerical round-off errors $10^{-13}$).\nHowever, the values of the two return probabilities $w(n_1,t=2t_r)$\nare affected by the measurement of the second particle and they are \nrather different from their values of \nthe initial state (\\ref{eqinitialstate}).\n\nThe values of $w(n_1,t=2t_r)$ can also be computed from a ``theoretical'' \nstate $\\ket{u_{\\rm \\,th}}$ for the first particle obtained \nfrom the assumption that \nonly the second particle is following the time evolution while \nthe first particle remains fixed and measuring the second particle \nat $t=t_r$. The theoretical state is then given by \n$\\ket{u_{\\rm \\,th}}=\\alpha_1\\,C_1\\ket{u_1}+\\alpha_2\\,C_2\\ket{u_2}$ \nwhere (for $j=1,2$) $\\alpha_j$ and $\\ket{u_j}$ are given by Eq. \n(\\ref{eqschmidt2}) used for \nthe Schmidt decomposition of the initial state (\\ref{eqschmidt}). \nThe coefficients $C_j$ are obtained from the measurement procedure as \n$C_j=C_g\\,\\langle n_2\\ket{v_j(t_r)}=C_g\\,v_j(n_2,t_r)$ \nwhere $\\ket{v_j(t_r)}$ are the second-particle states at the \nmoment of measurement (after $t_r$ iterations) \nand $C_g$ is the global normalization constant \nof the theoretical state $\\ket{u_{\\rm \\,th}}$. \nThe blue stars in Fig.~\\ref{fig4} show the values obtained from this \ntheoretical state which coincide numerically (up to usual round-off errors) \nwith the values of $w(n_1,t=2t_r)$ (also for the cases $n_1$ being different \nfrom 0 or 1 where are simple $w(n_1,t=2t_r)=0$).\n\nThe reason is that the Schmidt coefficients $\\alpha_i$ remain \nunchanged during the forward propagation in time till \nthe moment $t=t_r$ while the Schmidt vectors $u_1(n_1,t), v_1(n_2,t)$ and \n$u_2(n_1,t), v_2(n_2,t)$ evolve as one-particle wavefunctions\ncomputed numerically from (\\ref{qmap})\nwith the initial condition (\\ref{eqschmidt}),(\\ref{eqschmidt2}).\nThe measurement of the second particle, detected at momentum state $n_2$\nat $t=t_r$, gives the above coefficients $C_1$ and $C_2$ and \nafter the measurement the wavefunction of the first particle is:\n$\\psi(n_1,t) = \\alpha_1 C_1 u_1(n_1,t) + \\alpha_2 C_2 u_2(n_1,t)$.\nDuring the backward evolution the components $ u_1(n_1,t)$ and\n$ u_2(n_1,t)$ return to their initial values providing exactly \nthe theoretical state given above. \nHence, the probability to find the first particle\nat $n_1=0$ is $w(n_1=0)=(0.3440 \\alpha_1 C_1 + 0.9390 \\alpha_2 C_2)^2$\nand at $n_1=1$ it is \n$w(n_1=1) = (-0.9390 \\alpha_1 C_1 + 0.3440 \\alpha_2 C_2)^2$.\nAs it is shown in Fig.~\\ref{fig4}\nthe results of this semi-analytical theory\nreproduce the numerically obtained probabilities (up to \nusual round-off errors).\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{fig5}\n\\end{center}\n\\caption{\\label{fig5} Density plots of Husimi functions of the \nfirst particle at  certain time moments  $t$. \nColumns (a) and (b) show the Husimi functions \nof the time dependent Schmidt components $\\ket{u_1(t)}$ (a) and \n$\\ket{u_2(t)}$ (b) of the first particle after \n$t=0,\\,3,\\,7,\\,20$ iterations \nwith the horizontal axis corresponding to $x_1\\in[0,\\,2\\pi[$ and the \nvertical axis corresponding to $p_1 \\in[-\\pi,\\,\\pi[$ \n(or $n_1\\in[-N\/2,\\,N\/2[$ with  $p_1= \\hbar_{\\rm eff} n_1$\nfor $N=2^{10}$).\nThe initial conditions $\\ket{u_j(t=0)}$ (for $j=1,2$) \nare Gaussian coherent states (see  (\\ref{eqcoherent}) and text).\nThe initial Schmidt components $\\ket{v_j(t=0)}$ of \nthe second particle are also Gaussian coherent states of the same type with \nclose positions (see text). \nColumn (c)  shows the Husimi functions of the first particle after the \nsecond particle being \nmeasured at $n_2=8$  after $t=t_r=20$ iterations and \nfollowed by time reversal. The iteration times for the right column are \n$t_{(c)}=40-t$ with $t$ being the time values shown in the figure \nfor each row, i.e. $t_{(c)}=40,\\,37,\\,33,\\,20$ (top to bottom).\nThe color bar is the same as in Figs.~\\ref{fig2} and \\ref{fig3} where the \nnumbers correspond to $[H(x,p)\/H_{\\rm max}]^{1\/4}$ with \n$H(x,p)$ being the Husimi function \nof the first particle. System parameters are: \n$\\epsilon=\\hbar_{\\rm eff} = 2\\pi\/N = \\pi \\times 2^{-9}$,\n$K_{\\rm eff}=k \\hbar_{\\rm eff}=5$, $T=4\\pi \\pm \\epsilon$.\nThe pixel resolution of each panel corresponds to $P\\times P$ pixels with \n$P=8\\sqrt{N}=2^8$.\n}\n\\end{figure}\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{fig6}\n\\end{center}\n\\caption{\\label{fig6} Same as Fig.~\\ref{fig5} \nbut for $N=2^{14}$ and modified \niterations times $t=0,\\,4,\\,9,\\,20$ (a), (b) \nor $t_{(c)}=40-t=40,\\,36,\\,31,\\,20$ (c) (top to bottom);\n$\\epsilon = \\hbar_{\\rm eff}=2\\pi\/N= \\pi \\times 2^{-13}$, \n$K_{\\rm eff}= k \\hbar_{\\rm eff}=5$.\nThe number of pixels per direction is $P=8\\sqrt{N}=2^{10}$.\n}\n\\end{figure}\n\n\\subsection{IIIB. EPR pairs in kicked rotator at small $\\hbar_{\\rm eff}$ values}\n\\label{subsec3b}\n\nWe also study the behavior of chaotic EPR pair in a deep\nsemiclassical limit. For this we consider the quantum evolution on a torus\nof size $N$ with $\\epsilon = \\hbar_{\\rm eff} = 2\\pi\/N$ and \nperiodic conditions for the wavefunction in a momentum space\n$-\\pi \\leq p =\\hbar_{\\rm eff} n <\\pi$. Such a time evolution of \nquantum maps on a quantum torus\nhas been already studied in detail for the \none-particle case (see e.g. \\cite{chi1988,qcfrahm}).\nAs above, we choose the effective chaos parameter \n$K_{\\rm eff} =  \\hbar_{\\rm eff} k =5$\nsuch that the phase space of classical dynamics is mainly chaotic\nwith only small integrable islands embedded in the chaotic component\n(the measure of these islands is approximately 2\\% \\cite{chirikov1979}).\n\nThe initial state of the EPR pair is chosen as an entangled state \ngiven as the Schmidt decomposition of two pairs of coherent \nGaussian states and with equal coefficients $\\alpha_1=\\alpha_2=1\/\\sqrt{2}$. \nMore precisely, the initial components of the first particle are\n\\begin{eqnarray}\n\\label{eqcoherent}\n&&\\ket{u_j(t=0)} = \\ket{(x_0^{(j)},p_0^{(j)})_{\\rm coh.}}\\\\\n&&\\qquad = C\\sum_{p} \\exp[-G(p-p_0^{(j)})^2 \n\\nonumber\n-i p \\,x_0^{(j)}\/\\hbar_{\\rm eff}]\\ket{p} \n\\end{eqnarray}\nwith classical positions of the coherent wave packet at \n$x_0^{(1)}=0.3\\times (2\\pi)$, $p_0^{(1)}=0.1\\times (2\\pi)$ and \n$x_0^{(2)}=0.6\\times (2\\pi)$, $p_0^{(2)}=-0.05\\times (2\\pi)$. \n(Note that in (\\ref{eqcoherent}) the $p$-sum runs over the values \n$p=2\\pi n\/N$ with $n=0,\\,\\ldots,\\,N-1$.)\nThe parameter $G$ is related to the width \n$\\Delta p=\\langle (\\hat p-p_0^{(j)})^2\\rangle^{1\/2}$ of \nthe wave packet in momentum space by $G=1\/(4\\Delta p^2)$ \nand $C$ is the normalization constant. \nHere we choose $G=1\/(2\\hbar_{\\rm eff})=N\/(4\\pi)$ such that \n$\\Delta p=\\sqrt{\\pi\/N}$ and $\\Delta x=\\hbar_{\\rm eff}\/(2\\Delta p)=\n\\sqrt{\\pi\/N}$ are identical (note that \nhere for both variables $x$ and $p$ the system size is $2\\pi$). \n\nThe initial Schmidt components $\\ket{u_j(t=0)}$ of \nthe second particle are also Gaussian coherent states of the same type \n(\\ref{eqcoherent}) \nwith  close initial positions \n$\\ket{v_j(t=0)}=\\ket{(x_0^{(j)}+\\delta x,p_0^{(j)}+\\delta p)_{\\rm coh.}}$\nshifted by  $\\delta x=\\delta p=2\\pi\/\\sqrt{N}=2\\sqrt{\\pi}\\Delta p$. \n\nWe describe the probability distribution of the first particle \ncomponents (\\ref{eqcoherent})\nin the phase space $(x_1,p_1)$ by the Husimi function \ndefined as $H(x,p)=|\\langle (x,p)_{\\rm coh.}\\ket{u}|^2$ \nwhere $\\ket{u}$ is the first-particle state for which the \nHusimi function is computed and $\\ket{(x,p)_{\\rm coh.}}$ is the \ncoherent state (\\ref{eqcoherent}) with the above choice of \n$G=1\/(2\\hbar_{\\rm eff})$ at classical positions $(x,p)$. \nThe Husimi function corresponds to a smoothing of the \nWigner function over the above values of $\\Delta x$, $\\Delta p$ \nwith $\\Delta x \\Delta p = \\hbar_{\\rm eff}\/2$ \n(see e.g. \\cite{qcfrahm,husimi} for a description of Husimi functions). \nWe have numerically computed the Husimi function using an efficient algorithm \nbased on FFT and for a grid size $(\\delta x\/8)\\times (\\delta p\/8)$ (with the \nabove values of $\\delta x$ and $\\delta p$).\n\nThe Husimi functions of the first particle Schmidt components $u_1, u_2$\nare shown in Fig.~\\ref{fig5} at different iteration times \n$t=0,\\,3,\\,7,\\,20$ (in columns (a) and (b)) for\n$\\hbar_{\\rm eff}=2\\pi\/N$ and $N=1024$. \nWe see that due to the underlying classical chaos the initial coherent state\nspreads very rapidly over the whole available phase space,\nexcept the domain of integrable islands which can be occupied\nonly after very long tunneling times. In column (c) of Fig.~\\ref{fig5}\nwe show the backward time evolution of the first particle wave packet\nobtained from the measurement of the second particle \nat the momentum state $n_2=8$ performed after $t_r=20$ quantum map \n(\\ref{qmappair}) iterations. \nWe see that the backward evolution of the first particle has \ndifferent Husimi distributions at different return time moments.\nHowever, at $t=2t_r=40$ the first particle returns to its initial coherent\nstates of Schmidt components at $t=0$ but with different weights.\nIn Fig.~\\ref{fig6} we show also a similar time evolution with \nmeasurement and time\nreversal for the smaller value $\\hbar_{\\rm eff} =2\\pi\/N$, $N=10^{14}$ and \nthe measured state $n_2=8$.\nAs in Fig.~\\ref{fig5} there is an exponentially rapid spreading of initial \ncoherent wave packets which after measurement returns to the initial \ntwo coherent states but with different weights.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{fig7}\n\\end{center}\n\\caption{\\label{fig7} Panels show color \ndensity plot of $w(n_1,t)$ for the first particle  time evolution and \nparameters of Fig.~\\ref{fig5} with $-512\\le n_1<512$ \n($y$-axis) and $0\\le t\\le 40$ ($x$-axis). The time values \nfor $t\\le t_r=20$ correspond to the exact forward iteration \nand for backward iterations $ t_r < t \\leq 2t_r =40$,\nthe measurement, done\nafter $t_r$ iterations, detects the second particle at $n_2=8$ (a) or $n_2=200$ (b). \nThe thin vertical white \nline marks the time position of measurement  $t_r=20$ \nand the beginning of the time reversal backward iterations. The  \ncolor bar has the same meaning as in Fig.~\\ref{fig2}.\n}\n\\end{figure}\n\nIn Fig.~\\ref{fig7} we show the time evolution of \nprobability distribution $w(n_1,t)$ of first particle over\nmomentum states $n_1$. For $0 \\leq t \\leq t_r$ this probability is\naveraged ob the second particle. As in Fig.~\\ref{fig5}\nthe projective measurement\nis done after $t_r=20$ quantum map (\\ref{qmappair})  iterations\nwith the second particle detected at $n_2=8$.\nWe see that the backward probability distribution $w(n_1,t)$\nat $t>t_r$ is different from the forward one.\nHowever, at the return moment $t=2t_r$ we still\nhave two coherent wave packets for first particle\nwhich have the same shape as at the initial state\nbut with different coefficients. \n\nWe also show the initial $t=0$\nand final $t=2t_r=40$ probability distributions of the first particle\nin Fig.~\\ref{fig8} for different results of measurements\nof the second particle detected at $n_2=8, 12, 20, 200$. \nThe weights of each coherent state at $t=2t_r$ \nare determined from the Schmidt components of a theoretical state \nconstructed in the same way as in the case of Fig.~\\ref{fig4}.\nThe density of the theoretical state coincides with the final density \nat $t=2t_r$ up to usual numerical round-off errors (only the maximum of \neach theoretical state is shown in Fig.~\\ref{fig8} by a blue star).\n\nSimilar to the case of Fig.~\\ref{fig7}\nwith measured $n_2=8$ we show the time evolution $w(n_1,t)$\nfor other measured values $n_2 = 12, 20$ in Appendix Fig.~\\ref{figA3}.\nFor comparison we show in Appendix Fig.~\\ref{figA4}\nalso the case of exact time reversal without measurements\n(i.e. with average over all measured $n_2$ values):\nhere the distribution $w(n_1,t)$ is exactly symmetric with respect to\ntime reversal at the moment $t=t_r=20$.\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{fig8}\n\\end{center}\n\\caption{\\label{fig8} Densities $w(n_1,t)$ of the first particle\nfor the case of Fig.~\\ref{fig5} are\nshown at initial $t=0$ (black curves)\nand final $t=2t_r=40$ (red curves) time moments;\n the second particle  is measured \nat $n_2=8$ (a), $n_2=12$ (b), $n_2=20$ (c) and $n_2=200$ (d). \nThe blue stars provide for each case the maximum values of the \ndensity of the theoretical state obtained from the Schmidt decomposition \n(see text) and predicting the final density at $t=40$.\nThe theoretical density curves are identical to the red curves within \nnumerical precision $\\sim 10^{-13}$ but only their values at the two maximum \npositions are shown for a better visibility.\n}\n\\end{figure}\n\n\n\\subsection{IIIC. EPR pairs of cold atoms in a kicked optical lattice}\n\\label{subsec3c}\n\nAbove we studied the properties of measurements and time reversal of EPR pairs\nin the regime of kicked rotator when the evolution takes place on a ring \nof size $2\\pi$. However, the experiments with cold atoms \nin a kicked optical lattice \\cite{raizen,garreau,hoogerland}\ncorrespond to the situation when an EPR pair propagates on the \ninfinite $x$ axis containing many periods of size $2\\pi$. Due to \nthe periodicity of potential the wavefunction of each\nparticle is characterized by a quasimomentum  with irrational values\n$\\beta$ (with $p=n+\\beta$) that reduce the probability of the \nsingle atom time reversal as discussed in detail in \\cite{martin}.\nThus to model this experimental setup we consider the EPR propagation on an \n$x$ interval of size $2\\pi L$ containing $L$ periods $2\\pi$ of the \noptical lattice. We use periodic boundary conditions in $x$\nbut during the time evolution the wave packet \nis not reaching the boundaries such that the \nboundary conditions are not important.\nIn this case the free propagation of a particle between kicks\nis given by the same unitary operator\nas in (\\ref{qmap}) but now in numerical simulations\nthe momentum takes discrete values\n$p=m\/L$ with integers $m=-N\/2,\\,\\ldots,\\,, N\/2-1$\nand $N=L N_r$ where $L$ gives the number of different\nquasimomentum values $\\beta$ and $N_r$ gives the number of integer\nvalues of momentum $p$.\nThe integer $p$ values corresponds to the rotator case.\nThe kick operator remains the same as in (\\ref{qmap}) but the position \noperator now takes the discrete values $x=2\\pi m L\/N$ ($m$ having the \nsame integer values as above) corresponding to the interval $[-\\pi L,\\pi L[$. \nAs in the previous Sections the numerical simulations are done \nwith the propagation of the full wavefunction using its Schmidt components.\nThis allows to reach very high $N$ and $L$ values required to\neliminate boundary effects. \nAs it was shown in previous Sections this\ncomputational method gives the same results\nas  the full wavefunction propagation with 2D FFT\n(up to numerical precision).\nWe use as  maximal values $N=2^{22}$  with $L=2^{14}$, $N_r=2^8$.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{fig9}\n\\end{center}\n\\caption{\\label{fig9} Time dependence of rescaled IPR\n$\\xi_t\/\\xi_0$ (a) and peak probability (b) of first particle\nfor a chaotic EPR pair in a kicked optical lattice:\nthe time forward evolution is marked by black points;\nfull backward evolution is marked by blue stars\n(time reversal is done at $t=t_r=10$ without measurement),\nbackward evolution with measurement of the moment $p_2$ of the \nsecond particle done at $t_r$ is shown by different \ncolor symbols for different measured $p_2$ values\nwith $p_2=8$ (cyan full squares), $p_2=12$ (green crosses),\n$p_2=20$ (magenta open squares), $p_2=200$ (red pluses).\nSystem parameters are  $\\hbar_{\\rm eff}=\\epsilon=5\/8$, \n$K_{\\rm eff}=5$, $k=K_{\\rm eff}\/\\hbar_{\\rm eff}=8$,\n$T=4\\pi \\pm \\epsilon$ (as in Fig.~\\ref{fig1})\nand $N=L N_r =2^{22}$, $L=2^{14}$, $N_r=2^8$.\nThe initial state of the EPR pair is described in the text.\n}\n\\end{figure}\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{fig10}\n\\end{center}\n\\caption{\\label{fig10} (a) Husimi function of the first Schmidt component \n$\\ket{u_1(t)}$ of the first particle \nat the return time $t=2t_r=20$ without measurement at time reversal $t=t_r=10$;\n(b) Husimi function of the first particle at return time $t=2t_r=20$\nwith measurement of the second particle at $t=t_r=10$ with detected \nmomentum $p_2=12$. Parameters and initial state are as in Fig.~\\ref{fig9}.\nThe color bar is the same as in Figs.~\\ref{fig2} and \\ref{fig3} where the \nnumbers correspond to $[H(x,p)\/H_{\\rm max}]^{1\/8}$. Furthermore, the \ncontrast of the image files has been artificially enhanced to increase the \nvisibility of the regions with non-vanishing values of the Husimi function. \n$x$-axis shows the coordinate interval $-L\/2 \\leq x_1\/2\\pi < L\/2$ \nfor $L=10^{14}$;\n$y$-axis shows the momentum interval $-N_r\/2 \\leq p_1 < N_r\/2$ \nwith $N_r=2^8$. \n}\n\\end{figure}\n\nBelow we present results for time reversal \nfor chaotic EPR pair in a kicked optical lattice.\nThe momentum and energies are measured in recoil units \nas described in \\cite{martin} that corresponds\nto dimensionless units of $p$ used above.\nAs in the last subsection the initial state is given an entangled state \ngiven as the Schmidt decomposition of two pairs of coherent \nGaussian states and with equal coefficients $\\alpha_1=\\alpha_2=1\/\\sqrt{2}$. \nHowever, now the parameter $G$ in (\\ref{eqcoherent}) is given \nby $G=1\/(4\\Delta p)^2$ with $\\Delta p=0.01$ and due to notational reasons \nthe parameter $h_{\\rm eff}$ in (\\ref{eqcoherent}) is replaced with unity \n(not to be confused with $h_{\\rm eff}=5\/8$ mentioned below). \nThe corresponding width of the Gaussian packet in position representation \nis $\\Delta x=1\/(2\\Delta p)=50 \\approx 8\\times 2\\pi$ corresponding roughly \nto $8$ periods of the optical lattice.\nThe center and phase parameters of (\\ref{eqcoherent}) \nof the two Schmidt components for the \nfirst particle are $p_0^{(1)}=1$, $p_0^{(2)}=2$, $x_0^{(1)}=\\pi$ \n(in the middle of the cell of index $m=0$) and $x_0^{(2)}=3\\pi$ \n(in the middle of the cell of index $m=1$). The values for the two \ncorresponding Schmidt components of the second particle are \n$p_0^{(1)}=-1$, $p_0^{(2)}=-2$, $x_0^{(1)}=\\pi$ and $x_0^{(2)}=3\\pi$, \ni.e. negative $p_0^{(j)}$ values and same $x_0^{(j)}$ values with \nrespect to the first particle.\n\nConcerning the Chirikov map we use the same parameters of the first \nsubsection IIIA, i.e.: \n$\\hbar_{\\rm eff}=\\epsilon = 5\/8$, \n$K_{\\rm eff}=5$, $k=K_{\\rm eff}\/\\hbar_{\\rm eff}=8$, $T=4\\pi \\pm \\epsilon$.\nThe time reversal is done after $t_r=10$ followed by a measurement \nof the second particle and the observation of first\nparticle at the return moment $t=2t_r=20$.\n\nAs in \\cite{martin} we characterize the quantum evolution of the\nfirst particle \nby the  Inverse Participation Ratio (IPR) defined \nby $\\xi_t= [\\sum_{p_1} w(p_1,t)]^2\/\\sum_p w^2(p_1,t)=1\/\\sum_p w^2(p_1,t)$ \nwhere $w(p_1,t)=\\sum_{p_2} |\\langle p_1,p_2 |\\psi (t)\\rangle |^2$\nare the probabilities of the first particle in the momentum space at time $t$ \nand after summing over the second particle momentum $p_2$ (the second \nidentity in the expression of $\\xi_t$ \nholds if the probabilities $w(p_1,t)$ are properly normalized).\nIn addition we also compute the time variation of the relative peak probability\n$W_{\\rm peak}(0)\/W_{\\rm peak}(t)$ where \n$W_{\\rm peak}(t)=\\sum_{j=1,\\,2} w(p_0^{(j)},t)$ \nis the sum of the probabilities at the two initial peak positions $p_0^{(j)}=j$\n(with $j=1,\\,2$) in momentum space.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{fig11}\n\\end{center}\n\\caption{\\label{fig11} Zoom of Husimi functions shown \nin the range $-3.5\/256 \\leq x_{1,2}\/(2\\pi L) \\leq  3.5\/256$ (corresponding \nto $448$ periods of the optical lattice),\n$-3.5 \\leq p_{1.2} \\leq 3.5$; the top three rows show the Husimi functions of\nthe Schmidt components $\\ket{u_1(t)}$ (a), $\\ket{u_2(t)}$ (b), $\\ket{v_1(t)}$ \n(c) at $t=0$ (1st row), $t=10$ (2nd row) and the return time $t=20$ for the \ncase with no measurement (3rd row). \nThe last row $t=20,meas$ shows the Husimi functions of the first particle \nat the return time $t=20$ with the measured momentum of the second \nparticle at $t=10$ \nbeing $p_2=8$ (a),  $p_2=12$ (b), $p_2=20$ (c). \nThe color bar is the same as in Figs.~\\ref{fig2} and \\ref{fig3} where the \nnumbers correspond to $[H(x,p)\/H_{\\rm max}]^{1\/4}$.\nThe dashed white horizontal lines in top row mark integer momentum values.\n}\n\\end{figure}\n\nThe time dependence of the relative IPR value $\\xi_t\/\\xi_0$ is shown in \nFig.~\\ref{fig9}(a).\nUp to the reversal time $t_r=10$ we have an approximately diffusive growth\nof IPR $\\xi_t\/\\xi_0 \\propto \\sqrt{t}$ corresponding to the energy diffusion\nwell seen in Fig.~\\ref{fig1}. After the time reversal this growth is stopped\nbut at the return time $t=2t_r=20$ there is no real return to the \ninitial IPR value at $t=0$. The reason is that the time reversal \nis exact only for quasimomentum values $\\beta=0$ (integer $p$ values)\nand only approximate for rather small $\\beta$ close to zero or unity.\nThis point is discussed in detail in \\cite{martin}.\nIn fact the inversion of IPR is better for the case presented in \n\\cite{martin} (see Fig.1 there)\nsince the kick amplitude $k$ is significantly smaller ($k=4.5$ there vs. \n$k=8$ here). The new feature well seen in Fig.~\\ref{fig9}(a) is that \nthe measurement of the momentum\nof the second particle after $t=t_r=10$ map iterations significantly affects\nthe return behavior of IPR.\n\nTo demonstrate that certain characteristics have an exact return to \nthe initial value (up to numerical precision) we show in Fig.~\\ref{fig9}(b) \nthe time dependence of the probability ratio $W_{\\rm peak}(0)\/W_{\\rm peak}(t)$.\nDue to conservation of quasimomentum $\\beta$ the probability $W_{\\rm peak}(t)$\nis influenced only by the components of the wavefunction with $\\beta=0$\nwhich have an exact time reversal and the final value $W_{\\rm peak}(t=2t_r)$ \nis identical to its initial value $W_{\\rm peak}(0)$ \n(up to numerical precision).\nHowever, the measurement of the second particle at $t=t_r=10$ affects\nthe time evolution of $W_{\\rm peak}(0)\/W_{\\rm peak}(t)$ at intermediate times \n$11\\leq t\\leq 18$ as it is well seen in Fig.~\\ref{fig9}(b).\nNote that $W_{\\rm peak}(t)$ is given by the sum of probabilities over\nthe two initial peak probabilities of the first particle at integer\nvalues of $p$. Due to that we have the exact return of  $W_{\\rm peak}(t)$.\nHowever, at the return moment $t=2t_r=20$ the relative \ndistribution of the return probability over the two initial peak positions\nis strongly affected by the measurement of the second particle as we \nshow below.\n\nWe illustrate the global spreading of the initial wavefunction \nby showing the Husimi function in $(x,p)$ plane\nin Fig.~\\ref{fig10}. The top panel shows the Husimi function of\nthe first Schmidt component $v_1(p_1)$ at the return moment\n$t=2t_r=20$ (time reversal is done at $t_r=10$ without \nmeasurement of second particle).\nIn the bottom panel we show the Husimi function of the first particle\nat $t=2t_r=20$ for the case when a measurement \ndetected the second particle at $p_2=12$ at $t_r=10$.\nThis figure shows that the main part of probability is not affected by\ntime reversal and continues to spread in the phase space.\nDue to conservation of quasimomentum\n$\\beta$ the Husimi function is composed \nof narrow distributions (some kind of parallel lines)\nlocated at integer momentum values. This is a result\nof quasimomentum conservation and the narrow initial width \n$\\Delta p=0.01$ of the initial distribution in $\\beta$ at $t=0$.\n\nThis line-type  structure is better visible\nin the zoom of Fig.~\\ref{fig10} shown in Fig.~\\ref{fig11}. Here we show\ntime snapshots of the Husimi function of Schmidt components\n $\\ket{u_1(t)}$, $\\ket{u_2(t)}$ of the first particle \nand also $\\ket{v_1(t)}$ of the second particle at $t=0, 10, 20$\n(from left to right columns and top to down rows).\nIn the bottom row we show the  Husimi function\nof the first particle at return time $t=20$\nwith measured momentum of second particle\nbeing $p_2=8,12,20$ (left to right)\nat reversal time $t=t_r=10$.\nHere we see a part of probability which returns to the\ninitial distribution.\n\nHowever, in global we see that the main fraction of the wave packet \nis not affected by time reversal.\nIndeed, as it was shown in \\cite{martin}\nonly a relatively small fraction\nof the wave packet returns to the initial distribution\n(that was associated with the Loschmidt cooling).\nThe reason is that the described procedure\nof time reversal is exact only for\nthe quasimomentum value $\\beta=0$ and works approximately\nfor other values $|\\beta| \\ll 1$ and $|\\beta-1| \\ll 1$.  \n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{fig12}\n\\end{center}\n\\caption{\\label{fig12} Probability distribution $w(p_1+\\beta)$ of the \nfirst particle over quasimomentum $\\beta$ for two \ninteger offsets $p_1=1$ or $p_1=2$; the initial Gaussian probability \n(of width $\\Delta p=0.01$) $t=0$ is shown by the black dashed curve\nrepresenting the first initial peak at $p_1=1$ (curve for the \nsecond initial peak at $p_1=2$ is identical). \nAll shown distributions are rescaled by the maximum amplitude of the \ninitial Gaussian distribution at $\\beta=0$. The rescaled probabilities at \nreturn the time $t=2t_r=20$ are shown\nby red and blue curves for the initial peaks at $p_1=1$ and $p_1=2$ \nrespectively; the different panels correspond to: \n(a) time reversal at $t_r=10$ without measurement;\nmeasurement at $t_r=10$ detecting the second particle at $p_2=8$ (b),\n$p_2=12$ (c), $p_2=20$ (d). System parameters are as in Fig.~\\ref{fig9}.\n}\n\\end{figure}\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{fig13}\n\\end{center}\n\\caption{\\label{fig13} Similar as in Fig.~\\ref{fig12} but the results \nare shown on a larger momentum range of the \nfirst particle $-20 \\leq p_1 \\leq 20$;  \nblue curves show the probability at the moment of time reversal $t=t_r=10$, \nred curves show the probability at return time $t=2t_r=20$; \nthe different panels correspond to the same cases of Fig.~\\ref{fig12} \n(without or with measurement and detected $p_2$ values) and \nall densities are rescaled as in Fig.~\\ref{fig12}.\nThe shown curves also integrate the data for non-integer values of $p_1$ \nbut the density values at non-integer $p_1$ are essentially zero (in graphical \nprecision).\n}\n\\end{figure}\n\nTo see in a better way the fraction of the wave packet \nreturning\nto the initial distribution we show in Fig.~\\ref{fig12}\nthe probability distribution in quasimomentum $\\beta$\nof the first particle \nat $t=0$ and return time $t=2t_r=20$.\nIn panel Fig.~\\ref{fig12}(a) the time reversal is done\nwithout measurement of second particle.\nThe initial distribution has two peaks \nat $p_1=1,2$ and the return probability\nexactly returns to its initial values\nat $\\beta=0$. However, the width of \nreturn distribution in $\\beta$ is significantly narrowed\nsince the time reversal is only approximate for\n$\\beta$ different from (but close) zero. This effect, \ncalled Loschmidt cooling, \nis discussed in detail in \\cite{martin}.\nThe new feature present in Fig.~\\ref{fig12}\nis that a measurement of the second particle at $t=t_r=10$\nsignificantly affects the peak probabilities \nat two initial positions $p_1=1,2$ due to the entanglement\nof the EPR pair. At the same time the sum of probabilities\nof the two peaks at $p_1=1,2$ remains exactly equal to the initial \npeak probability sum at $t=0$ since the time reversal is exact for $\\beta=0$\n(see also Fig.~\\ref{fig9}(b)). As for the above case of the kicked rotator,\nwe interpret the fact that\na measurement of second particle drastically affects\nthe return path of the first particle with\na specific Feynman path \\cite{feynman} \nselected by measurement of the entangled second particle\nat the moment of time reversal. \n\nThe distribution of probabilities of the first particle\nat times $t=t_r=10$ and $t=2t_r=20$ is also shown in Fig.~\\ref{fig13}\non a larger scale of momentum $p_1$. We see that there is\na broad background of probability of the first particle\nwhich diffusively spreads in momentum due to quantum chaos\nand which is not significantly affected by the time reversal.\nHowever, we also see that at the return time $t=2t_r=20$ there \nappear two very high peaks near momentum positions of the initial\ndistribution. The amplitudes of these two peaks are\nstrongly affected by a measurement of the second particle at time \nreversal $t_r=10$.\nEven if the total probability in these two peaks at $t=20$ is \nsmall compared to the total probability, their very high peak amplitudes\nallow to detect them in a very robust way. In fact, as it was shown in \n\\cite{fink1,fink2}\nfor reversal of acoustic waves, the chaotic dynamics \nallows to enhance the time reversal signal making it much more visible\nin presence of chaotic background.\nHere we have a similar situation \nthat potentially allows to realize and detect the time reversal of\nentangled quantum cold atoms. The time reversal of cold atoms\nwithout measurement at the moment of time reversal has been \nrealized in \\cite{hoogerland}.   \n\nHere we presented results for measurements which\ndetect a specific momentum value of second particle.\nAdditional results for a measurement projection\non a broader distribution of momentum $p_2$ with\na certain width $\\Delta p_2$ are presented in \nAppendix Fig.~\\ref{figA5},  Fig.~\\ref{figA6}.\nIn this case the time reversal also reproduces the \nthe peaks of probability of first particle\nnear their initial positions. These results show that a measurement device,\nwhich is modeled by a width  $\\Delta p_2$,\naffects the probability distribution\nof first particle at the return moment $t=2t_r$.\n\nAbove we considered an initial entangled state\nwith a narrow probability distribution\nnear two integer momentum values of the \nEPR pair. We suppose that in an experimental \nsetup initially ultra cold atoms \ncan be trapped at very low temperatures\ncorresponding to $p$ values close to zero.\nThen a field pulse can move the momentum \nto higher $p$ values being close to their \ninteger values (in recoil units).\nThe entanglement between the atoms \ncan be created due to their initial interactions\nwhich is later switched off,\ne.g. with the help of the Feshbach resonance.\nIt is also possible that both atoms \nhave an initial momentum close to zero\nbut being entangled they may have a certain \nspacial separation.\nHere we consider the case of distinguishable atoms \nthat can be realized by taking two identical atoms but \nat different hyperfine states.\nSuch a difference of internal atomic structure\nallows to measure one atom without\naffecting the other one.\nOf course, such type of experiments are very \nchallenging but the technological progress allows now\nto perform operations with\ntwo entangled atoms (see e.g. \\cite{jorg})\nand we expect that the experimental\ninvestigation of chaotic EPR pairs\ncan be realized soon in cold atom experiments. \n\n\\section{IV. Discussion}\n\\label{sec4}\n\nIn this work we analyzed the case when the evolution of an \nEPR pair is chaotic in the classical limit of small Planck constant.\nAt the same time the system dynamics is reversible in time \nboth in classical and quantum cases. In the classical case\nthe errors grow exponentially with time due to\ndynamical chaos that breaks the time reversal \nof evolution is presence even of very small errors.\nIn contrast the quantum evolution remains relatively stable to\nquantum errors due to the existence of instability only during a \nlogarithmically short Ehrenfest time scale. Our main objective was to analyze \nhow measurements of one particle of a chaotic and entangled EPR pair\naffects the time reversal of the remaining particle.\nWe find that this particle retains an approximate time reversal\nreturning to one of all configurations\nrepresenting the initial entangled EPR state.\nWe explain such an approximate time reversal\non the basis of the Feynman path integral formulation\nof quantum mechanics according to which a measurement selects\na specific configuration which returns to its initial\nstate via time inverted specific pathway.\nWe show that the Schmidt decomposition of the initially entangled EPR state\nallows to identify the final quantum state at the return time.\n\nHere we considered the chaotic EPR pairs in the case of the quantum\nChirikov standard map. This system has been already realized \nin experiments with cold atoms in kicked optical lattices\n\\cite{raizen,garreau}. Moreover,  the time reversal, proposed in \n\\cite{martin}, has been realized experimentally by the Hoogerland \ngroup \\cite{hoogerland}. However, in this experiment the interplay \naspects of entanglement and measurement \nfor time reversal had not been studied. At present advanced \ncold atoms techniques allow to investigate various \nquantum correlations of entangled pairs of atoms (see e.g. \\cite{jorg})\nand we expect that experimental investigations \nof the time reversal of chaotic EPR pairs, discussed here, are possible.\nIt may also be interesting to consider the time reversal for \ntwo entangled Bose-Einstein condensates (BECs)\nwith  their chaotic evolution in a kicked optical lattice\nfollowing the proposal of time reversal for a single BEC\ndescribed in \\cite{martin2}.\n \n\n\\section{Acknowledgments}\nThis research was supported in part through the grant \nNANOX $N^o$ ANR-17-EURE-0009, (project MTDINA) in the frame of the Programme des Investissements d'Avenir, France;\nthe work is also done as a part of prospective ANR France project OCTAVES.\nThis work was granted access to the HPC resources of \nCALMIP (Toulouse) under the allocation 2021-P0110.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nTopological insulators are materials that are insulating in the bulk\nbut allow a current to flow on the boundary.  These boundary currents\nare protected by topological invariants and thus, in ideal cases, flow\nwithout dissipation.  The mathematical description of a topological\ninsulator uses a \\(\\Cst\\)\\nobreakdash-algebra~\\(\\mathcal{A}\\)\nthat contains the resolvent of the Hamiltonian~\\(H\\)\nof the system; this amounts to \\(H\\in\\mathcal{A}\\)\nif~\\(H\\)\nis bounded.  To describe an insulator, the spectrum of~\\(H\\)\nshould have a gap at the Fermi energy~\\(E\\).\nDepending on further symmetries of the system such as a time\nreversal, particle--hole or chiral symmetry, the topological phase\nof the material may be classified by a class in the \\(\\K\\)\\nobreakdash-theory\nof~\\(\\mathcal{A}\\)\nassociated to the spectral projection of~\\(H\\)\nat the Fermi energy (see, for instance,\n\\cites{Kellendonk:Cstar_phases, Prodan-Schulz-Baldes:Bulk_boundary}).  \nSo the observable algebra~\\(\\mathcal{A}\\)\nor rather its K\\nobreakdash-theory predicts the possible topological\nphases of a material.\n\nAt first, a material is often modelled without disorder and in a tight\nbinding approximation.  This gives a translation-invariant Hamiltonian\nacting on \\(\\ell^2(\\mathbb{Z}^d,\\mathbb{C}^N)\\)\n(see, for instance, \\cites{Bernevig-Hughes-Zhang:Quantum,\n  Fu-Kane-Mele:Insulators,\n  Liu-Qi-Zhang-Dai-Fang-Zhang:Model_Hamiltonian}).  Bloch--Floquet\ntheory describes the Fermi projection through a vector bundle over the\n\\(d\\)\\nobreakdash-torus,\nwith extra structure that reflects the symmetries of the system (see,\nfor instance, \\cites{De_Nittis-Gomi:Real_Bloch,\n  De_Nittis-Gomi:Quaternionic_Bloch, Kennedy-Zirnbauer:Bott_gapped}).\nThe K\\nobreakdash-theory of the \\(d\\)\\nobreakdash-torus\nis easily computed.  Once \\(d\\ge2\\),\nmany of the topological phases that are predicted this way are\nobtained by stacking a lower-dimensional topological insulator in some\ndirection.  Such topological phases are called ``weak'' by\nFu--Kane--Mele~\\cite{Fu-Kane-Mele:Insulators}.  They claim that\nweak topological phases are not robust under disorder.\n\nOther authors have claimed instead that weak topological insulators\nare also quite robust, see~\\cite{Ringel-Kraus-Stern:Strong_side}.\nTheir proof of robustness, however, is no longer topological.\nRoughly speaking, the idea is that, although disorder may destroy\nthe topological phase, it must be rather special to do this.\n\\emph{Random} disorder will rarely be so special.  So in a finite\nvolume approximation, the topological phase will remain intact in\nmost places, and the small area where the randomness destroys it\nwill become negligible in the limit of infinite volume.  Such an\nargument may also work for the Hamiltonian of an insulator that is\nhomotopic to a trivial one.  Our study is purely topological in\nnature and thus cannot see such phenomena.\n\nWe are going to explain the difference between strong and weak\ntopological phases and the robustness of the former through a\ndifference in the underlying observable algebras.  Namely, we shall\nmodel a material with disorder by the Roe \\(\\Cst\\)\\nobreakdash-algebra\nof \\(\\mathbb{R}^d\\)\nor~\\(\\mathbb{Z}^d\\),\nwhich is a central object of coarse geometry.\nRoe~\\cites{Roe:Index_open_I, Roe:Index_open_II} introduced them\nto get index theorems for elliptic operators on non-compact\nRiemannian manifolds.\n\nBefore choosing our observable algebra, we should ask: What is causing\ntopological phases?  At first sight, the answer seems to be the\ntranslation invariance of the Hamiltonian.  Translation-invariance\nalone is not enough, however.  And it is destroyed by disorder.  The\nsubalgebra of translation-invariant operators on the Hilbert space\n\\(\\ell^2(\\mathbb{Z}^d,\\mathbb{C}^N)\\)\nis the algebra of \\(N\\times N\\)-matrices\nover the group von Neumann algebra of~\\(\\mathbb{Z}^d\\),\nwhich is isomorphic to \\(L^\\infty(\\mathbb{T}^d, \\mathbb M_N)\\).\nIf topological phases were caused by translation invariance alone,\nthey should be governed by the K\\nobreakdash-theory of \\(L^\\infty(\\mathbb{T}^d, \\mathbb M_N)\\).\nThis is clearly not the case.  Instead, we need the group\n\\(\\Cst\\)\\nobreakdash-algebra,\nwhich is isomorphic to \\(\\Cont(\\mathbb{T}^d,\\mathbb M_N)\\).\nThe reason why the spectral projections of the Hamiltonian belong to\nthe group \\(\\Cst\\)\\nobreakdash-algebra\ninstead of the group von Neumann algebra is that the matrix\ncoefficients of the Hamiltonian for \\((x,y)\\in\\mathbb{Z}^d\\)\nare supported in the region \\(\\norm{x-y}\\le R\\)\nfor some \\(R>0\\);\nlet us call such operators \\emph{controlled}.  The controlled\noperators do not form a \\(\\Cst\\)\\nobreakdash-algebra,\nand it makes no difference for \\(\\K\\)\\nobreakdash-theory purposes to allow\nthe Hamiltonian to be a limit of controlled operators in the norm\ntopology.  We shall see below that this is equivalent to continuity\nwith respect to the action of~\\(\\mathbb{R}^d\\)\non \\(\\mathbb B(\\ell^2(\\mathbb{Z}^d,\\mathbb{C}^N))\\)\ngenerated by the position observables.  This action restricts to\nthe translation action of~\\(\\mathbb{R}^d\\)\non the group von Neumann algebra \\(L^\\infty(\\mathbb{T}^d, \\mathbb M_N)\\),\nso that its continuous elements are the functions in\n\\(\\Cont(\\mathbb{T}^d,\\mathbb M_N)\\).\nHence \\(\\Cont(\\mathbb{T}^d,\\mathbb M_N) \\subseteq \\mathbb B(\\ell^2(\\mathbb{Z}^d,\\mathbb{C}^N))\\)\nconsists of those operators that are both translation-invariant and\nnorm limits of controlled operators. \n\nSince disorder destroys translation invariance, we should drop this\nassumption to model systems with disorder.  The \\(\\Cst\\)\\nobreakdash-algebra\nof all operators on \\(\\ell^2(\\mathbb{Z}^d,\\mathbb{C}^N)\\)\nthat are norm limits of controlled operators is the algebra\nof \\(N\\times N\\)-matrices\nover the \\emph{uniform Roe \\(\\Cst\\)\\nobreakdash-algebra}\nof~\\(\\mathbb{Z}^d\\).\nTo get the Roe \\(\\Cst\\)\\nobreakdash-algebra,\nwe must work in \\(\\ell^2(\\mathbb{Z}^d,\\Hils)\\)\nfor a separable Hilbert space~\\(\\Hils\\)\nand add a local compactness property, namely, that the operators\n\\(\\braketop{x}{(H-\\lambda)^{-1}}{y} \\in\\mathbb B(\\Hils)\\)\nare compact for all \\(x,y\\in\\mathbb{Z}^d\\),\n\\(\\lambda\\in \\mathbb{C}\\setminus\\mathbb{R}\\).\nThis property is automatic for operators on \\(\\ell^2(\\mathbb{Z}^d,\\mathbb{C}^N)\\).\nWorking on \\(\\ell^2(\\mathbb{Z}^d,\\Hils)\\)\nand assuming local compactness means that we include infinitely many\nbands in our model and require only finitely many states with finite\nenergy in each finite volume.\nKubota~\\cite{Kubota:Controlled_bulk-edge} has already used the\nuniform Roe \\(\\Cst\\)\\nobreakdash-algebra\nand the Roe \\(\\Cst\\)\\nobreakdash-algebra\nin the context of topological insulators.  He prefers the uniform\nRoe \\(\\Cst\\)\\nobreakdash-algebra.  We explain why we consider this a\nmistake.\n\nWorking in the Hilbert space \\(\\ell^2(\\mathbb{Z}^d,\\mathbb{C}^N)\\)\nalready involves an approximation.  We ought to work in a continuum\nmodel, that is, in the Hilbert space \\(L^2(\\mathbb{R}^d,\\mathbb{C}^k)\\),\nwhere~\\(k\\)\nis the number of internal degrees of freedom.\nThis Hilbert space is isomorphic to\n\\[\nL^2(\\mathbb{R}^d,\\mathbb{C}^k)\n\\cong L^2(\\mathbb{Z}^d\\times (0,1]^d)\\otimes \\mathbb{C}^k\n\\cong \\ell^2(\\mathbb{Z}^d,\\Hils\\otimes \\mathbb{C}^k)\n\\]\nwhen we cover~\\(\\mathbb{R}^d\\)\nby the disjoint translates of the fundamental domain~\\((0,1]^d\\).\nThis identification preserves both controlled and locally compact\noperators.  Thus the Roe \\(\\Cst\\)\\nobreakdash-algebras\nof \\(\\mathbb{Z}^d\\)\nand~\\(\\mathbb{R}^d\\)\nare isomorphic.  For the Roe \\(\\Cst\\)\\nobreakdash-algebra\nof~\\(\\mathbb{R}^d\\),\nit makes no difference to replace \\(L^2(\\mathbb{R}^d)\\)\nby \\(L^2(\\mathbb{R}^d,\\mathbb{C}^k)\\)\nor \\(L^2(\\mathbb{R}^d,\\Hils)\\):\nall these Hilbert spaces give isomorphic \\(\\Cst\\)\\nobreakdash-algebras\nof locally compact, approximately controlled operators.  So\nthere is only one Roe \\(\\Cst\\)\\nobreakdash-algebra\nfor~\\(\\mathbb{R}^d\\),\nand it is isomorphic to the non-uniform Roe \\(\\Cst\\)\\nobreakdash-algebra\nof~\\(\\mathbb{Z}^d\\).\nWe view the appearance of the uniform Roe \\(\\Cst\\)\\nobreakdash-algebra\nfor~\\(\\mathbb{Z}^d\\)\nas an artefact of simplifying assumptions in tight binding models.\n\nWe describe some interesting elements of the Roe \\(\\Cst\\)\\nobreakdash-algebra\nof~\\(\\mathbb{R}^d\\)\nin Example~\\ref{exa:Roe_Rn}.  In particular, it contains all\n\\(\\Cont_0\\)\\nobreakdash-functions of the impulse operator~\\(P\\)\non \\(L^2(\\mathbb{R}^d)\\) or, equivalently,\n\\begin{equation}\n  \\label{eq:f_of_P}\n  \\int_{\\mathbb{R}^d} f(x) \\exp(\\ima x P) \\,\\diff x  \n\\end{equation}\nfor \\(f\\in \\Cst(\\mathbb{R}^d)\\);\nthis operator is controlled if and only if~\\(f\\)\nhas compact support.  If \\(V\\in L^\\infty(\\mathbb{R}^d)\\),\nthen the operator of multiplication by~\\(V\\) on~\\(L^2(\\mathbb{R}^d)\\)\nis controlled, but not locally compact.  Its product with an\noperator as in~\\eqref{eq:f_of_P} belongs to the Roe\n\\(\\Cst\\)\\nobreakdash-algebra.\n\nThe real and complex K\\nobreakdash-theory of the Roe \\(\\Cst\\)\\nobreakdash-algebra\nof~\\(\\mathbb{Z}^d\\)\nis well known: up to a dimension shift of~\\(d\\),\nit is the \\(\\K\\)\\nobreakdash-theory\nof \\(\\mathbb{R}\\)\nor~\\(\\mathbb{C}\\),\nrespectively.  In particular, the Roe \\(\\Cst\\)\\nobreakdash-algebra\nas an observable algebra is small enough to predict some distinct\ntopological phases.  These coincide with Kitaev's periodic\ntable~\\cite{Kitaev:Periodic_table}.  This corroborates the choice of\nthe Roe \\(\\Cst\\)\\nobreakdash-algebra\nas the observable algebra for disordered materials.\n\nWhen we disregard disorder, the Roe \\(\\Cst\\)\\nobreakdash-algebra\nmay be replaced by its translation-invariant subalgebra, which is\nisomorphic to\n\\[\n\\Cst(\\mathbb{Z}^d) \\otimes \\mathbb K(\\Hils) \\cong \\Cont(\\mathbb{T}^d,\\mathbb K(\\Hils)),\n\\]\nwhere \\(\\mathbb K(\\Hils)\\)\ndenotes the \\(\\Cst\\)\\nobreakdash-algebra\nof compact operators on an infinite-dimensional separable Hilbert\nspace~\\(\\Hils\\).\nIn the real case, the \\(d\\)\\nobreakdash-torus\nmust be given the real involution by the restriction of complex\nconjugation on \\(\\mathbb{C}^d\\supseteq\\mathbb{T}^d\\).\nThe real or complex \\(\\K\\)\\nobreakdash-theory\ngroups of the ``real'' \\(d\\)\\nobreakdash-torus describe both weak and strong\ntopological phases in the presence of different types of symmetries.\nWe show that the map\n\\begin{equation}\n  \\label{eq:comparison_group_Roe}\n  \\K_*(\\Cst(\\mathbb{Z}^d)_\\mathbb{F}) \\to \\K_*(\\Cst_\\Roe(\\mathbb{Z}^d)_\\mathbb{F})\n\\end{equation}\nfor \\(\\mathbb{F} = \\mathbb{R}\\)\nor \\(\\mathbb{F} = \\mathbb{C}\\)\nis split surjective and that its kernel is the subgroup generated by\nthe images of \\(\\K_*(\\Cst(\\mathbb{Z}^{d-1})_\\mathbb{F})\\)\nfor all coordinate embeddings \\(\\mathbb{Z}^{d-1} \\to \\mathbb{Z}^d\\).\nThat is, the kernel of the map in~\\eqref{eq:comparison_group_Roe}\nconsists exactly of the \\(\\K\\)\\nobreakdash-theory\nclasses of weak topological insulators as defined by\nFu--Kane--Mele~\\cite{Fu-Kane-Mele:Insulators}.  The strong topological\ninsulators are those that remain topologically protected even if the\nobservable algebra is enlarged to the Roe \\(\\Cst\\)\\nobreakdash-algebra,\nallowing rather general disorder.\n\nFollowing Bellissard \\cites{Bellissard:K-theory_solid,\n  Bellissard-Elst-Schulz-Baldes:Noncommutative_Hall}, disorder is\nusually modelled by crossed product \\(\\Cst\\)\\nobreakdash-algebras\n\\(\\mathcal{A} = \\Cont(\\Omega)\\rtimes\\mathbb{Z}^d\\),\nwhere~\\(\\Omega\\)\nis the space of disorder configurations.  It is more precise to say,\nhowever, that the space~\\(\\Omega\\)\ndescribes \\emph{restricted} disorder.  Uncountably many different\nchoices are possible.  Such models are only reasonable when the\nphysically relevant objects do not depend on the choice.  But the\n\\(\\K\\)\\nobreakdash-theory\nof the crossed product depends on the topology of the\nspace~\\(\\Omega\\).\nTo make it independent of~\\(\\Omega\\),\nthe space~\\(\\Omega\\)\nis assumed to be contractible\nin~\\cite{Prodan-Schulz-Baldes:Bulk_boundary}.\nThis fits well with standard choices of~\\(\\Omega\\)\nsuch as a product space \\(\\prod_{n\\in\\mathbb{Z}^d} [-1,1]\\)\nto model a random potential.  The resulting \\(\\K\\)\\nobreakdash-theory\nthen becomes the same as in the system without disorder.  So another\nargument must be used to explain the difference between weak and\nstrong topological phases, compare\n\\cite{Prodan-Schulz-Baldes:Bulk_boundary}*{Remark 5.3.5}.\nIf one allows non-metrisable~\\(\\Omega\\), then\nthere is a maximal choice for~\\(\\Omega\\), namely,\nthe Stone--\\v{C}ech compactification of~\\(\\mathbb{Z}^d\\).\nThe resulting crossed product \\(\\ell^\\infty(\\mathbb{Z}^d)\\rtimes\\mathbb{Z}^d\\)\nis isomorphic to the uniform Roe \\(\\Cst\\)\\nobreakdash-algebra\nof~\\(\\mathbb{Z}^d\\),\nsee~\\cite{Kubota:Controlled_bulk-edge}.\nNevertheless, even this maximal choice of~\\(\\Omega\\) still contains\na hidden restriction on disorder: the number of bands for a tight\nbinding model is fixed, and so the disorder is also limited to a\nfixed finite number of bands.  The Roe \\(\\Cst\\)\\nobreakdash-algebra\nof~\\(\\mathbb{Z}^d\\)\nalso removes this hidden restriction on the allowed disorder.\nIt is also a crossed product, namely,\n\\[\n\\Cst_\\Roe(\\mathbb{Z}^d) \\cong \\ell^\\infty(\\mathbb{Z}^d,\\mathbb K(\\Hils)) \\rtimes \\mathbb{Z}^d.\n\\]\nThe \\(\\Cst\\)\\nobreakdash-algebra \\(\\ell^\\infty(\\mathbb{Z}^d,\\mathbb K(\\Hils))\\)\nis not isomorphic to \\(\\ell^\\infty(\\mathbb{Z}^d) \\otimes \\mathbb K(\\Hils)\\):\nit even has different \\(\\K\\)\\nobreakdash-theory.\n\nSince the Roe \\(\\Cst\\)\\nobreakdash-algebra\nhas not been used much in the context of topological insulators, we\nrecall its main properties in Section~\\ref{sec:Roe_Cstar}.  We\nhighlight its robustness or even ``universality.''  Roughly\nspeaking, there is only one Roe\n\\(\\Cst\\)\\nobreakdash-algebra\nin each dimension, which describes all kinds of disordered materials\nin that dimension without symmetries.  The various symmetries\n(time-reversal, particle-hole, chiral) may be added by tensoring the\nreal or complex Roe \\(\\Cst\\)\\nobreakdash-algebra\nwith Clifford algebras, which replaces \\(\\K_0\\)\nby~\\(\\K_i\\)\nfor some \\(i\\in\\mathbb{Z}\\).\nWe shall not say much about this here.  The Roe \\(\\Cst\\)\\nobreakdash-algebra\nof a coarse space is a coarse invariant.  In particular, all coarsely\ndense subsets in~\\(\\mathbb{R}^d\\)\ngive isomorphic Roe \\(\\Cst\\)\\nobreakdash-algebras.\nFurthermore, the Roe \\(\\Cst\\)\\nobreakdash-algebras\nof~\\(\\mathbb{Z}^d\\)\nand other coarsely dense subsets of~\\(\\mathbb{R}^d\\)\nare isomorphic to that of~\\(\\mathbb{R}^d\\).\nThus it makes no difference whether we work in a continuum or lattice\nmodel.  We also consider the twists of the Roe \\(\\Cst\\)\\nobreakdash-algebra\ndefined by magnetic fields.  The resulting twisted Roe\n\\(\\Cst\\)\\nobreakdash-algebras\nare also isomorphic to the untwisted one.  This robustness of the Roe\n\\(\\Cst\\)\\nobreakdash-algebra\nmeans that the same \\emph{strong} topological phases occur for all\nmaterials of a given dimension and symmetry type, even for\nquasi-crystals and aperiodic materials.\n\nWe compute the \\(\\K\\)\\nobreakdash-theory\nof the Roe \\(\\Cst\\)\\nobreakdash-algebra\nin Section~\\ref{sec:coarse_MV} using the coarse Mayer--Vietoris\nprinciple introduced in~\\cite{Higson-Roe-Yu:Coarse_Mayer-Vietoris}.\nWe prove the Mayer--Vietoris exact sequence in the real and complex\ncase by reducing it to the \\(\\K\\)\\nobreakdash-theory\nexact sequence for \\(\\Cst\\)\\nobreakdash-algebra\nextensions.  The computation of \\(\\K_*(\\Cst_\\Roe(X)_\\mathbb{F})\\)\nfor \\(X=\\mathbb{Z}^d\\)\nis based on the vanishing of this invariant for half-spaces\n\\(\\mathbb{N}\\times\\mathbb{Z}^{d-1}\\).\nThis implies\n\\(\\K_{*+d}(\\Cst_\\Roe(X)_\\mathbb{F}) \\cong \\K_*(\\mathbb{F})\\).\nIt shows also that the inclusion\n\\(\\Cst_\\Roe(\\mathbb{Z}^{d-1})_\\mathbb{F} \\to \\Cst_\\Roe(\\mathbb{Z}^d)_\\mathbb{F}\\)\ninduces the zero map on \\(\\K\\)\\nobreakdash-theory.\nHence the map~\\eqref{eq:comparison_group_Roe} kills all\nelements of \\(\\K_*(\\Cst(\\mathbb{Z}^d)_\\mathbb{F})\\)\nthat come from inclusions \\(\\mathbb{Z}^{d-1} \\hookrightarrow \\mathbb{Z}^d\\).\n\nIn Section~\\ref{sec:weak_phases} we compute the real and complex\n\\(\\K\\)\\nobreakdash-theory for the group \\(\\Cst\\)\\nobreakdash-algebra\nof~\\(\\mathbb{Z}^d\\)\nand the map in~\\eqref{eq:comparison_group_Roe}.  More precisely, we\ndescribe the composite of the map in~\\eqref{eq:comparison_group_Roe}\nwith the isomorphism\n\\(\\K_{*+d}(\\Cst_\\Roe(X)_\\mathbb{F}) \\cong \\K_*(\\mathbb{F})\\):\nthis is the pairing with the fundamental class of the ``real''\n\\(d\\)\\nobreakdash-torus~\\(\\mathbb{T}^d\\).\nExcept for an adaptation to ``real'' manifolds, this fundamental\nclass is introduced in~\\cite{Kasparov:Novikov}.  We show that the\nfundamental class extends to a \\(\\K\\)\\nobreakdash-homology\nclass on the Roe \\(\\Cst\\)\\nobreakdash-algebra\nand that the pairing with this \\(\\K\\)\\nobreakdash-homology\nclass is an isomorphism\n\\(\\K_{*+d}(\\Cst_\\Roe(X)_\\mathbb{F}) \\cong \\K_*(\\mathbb{F})\\).\n\n\n\\section{Roe \\texorpdfstring{$\\Cst$}{C*}-algebras}\n\\label{sec:Roe_Cstar}\n\nIn this section, we define the real and complex Roe\n\\(\\Cst\\)\\nobreakdash-algebras\nof a proper metric space and prove that they are invariant under\npassing to a coarsely dense subspace and, more generally, under coarse\nequivalence.  We prove that the twists used to encode magnetic fields\ndo not change them.  And we describe elements of the Roe\n\\(\\Cst\\)\\nobreakdash-algebra\nof a subset of~\\(\\mathbb{R}^d\\)\nas those locally compact operators that are continuous for the\nrepresentation of~\\(\\mathbb{R}^d\\)\ngenerated by the position operators.  We describe the subalgebras of\nsmooth, real-analytic and holomorphic elements of the Roe\n\\(\\Cst\\)\\nobreakdash-algebra\nfor this action of~\\(\\mathbb{R}^d\\).\nWe show that Roe \\(\\Cst\\)\\nobreakdash-algebras\nhave approximate units of projections, which simplifies the definition\nof their \\(\\K\\)\\nobreakdash-theory.\n\nLet \\((X,d)\\)\nbe a locally compact, second countable, metric space.  We assume the\nmetric~\\(d\\)\nto be \\emph{proper}, that is, bounded subsets of~\\(X\\)\nare compact.  We shall be mainly interested in \\(\\mathbb{R}^d\\)\nor a discrete subset of~\\(\\mathbb{R}^d\\)\nwith the restriction of the Euclidean metric.  (All our results on\ngeneral proper metric spaces extend easily to the more general coarse\nspaces introduced in~\\cite{Roe:Lectures}.)  Let~\\(\\Hils\\)\nbe a real or complex separable Hilbert space and let\n\\(\\varrho\\colon \\Cont_0(X)\\to\\mathbb B(\\Hils)\\)\nbe a nondegenerate representation.  We are going to define the Roe\n\\(\\Cst\\)\\nobreakdash-algebra\nof~\\(X\\)\nwith respect to~\\(\\varrho\\),\nsee also \\cite{Higson-Roe:Analytic_K}*{Section 6.3}.  Depending on\nwhether~\\(\\Hils\\)\nis a real or complex Hilbert space, this gives a real or complex\nversion of the Roe \\(\\Cst\\)\\nobreakdash-algebra.\nBoth cases are completely analogous.\n\nLet \\(T\\in\\mathbb B(\\Hils)\\).\nWe call~\\(T\\)\n\\emph{locally compact} (on~\\(X\\))\nif the operators \\(\\varrho(f)T\\)\nand~\\(T\\varrho(f)\\)\nare compact for all \\(f\\in\\Cont_0(X)\\).\nThe \\emph{support} of~\\(T\\)\nis a subset \\(\\supp T\\subseteq X\\times X\\).\nIts complement consists of all \\((x,y)\\in X\\times X\\)\nfor which there are neighbourhoods \\(U_x\\),\n\\(U_y\\)\nin~\\(X\\)\nsuch that \\(\\varrho(f) T \\varrho(g)=0\\)\nfor all \\(f\\in\\Cont_0(U_x)\\),\n\\(g\\in\\Cont_0(U_y)\\).\nThe operator~\\(T\\)\nis \\emph{controlled} (or has \\emph{finite propagation}) if there is\n\\(R>0\\)\nsuch that \\(d(x,y)\\le R\\)\nfor all \\((x,y)\\in \\supp T\\).\nWe sometimes write ``\\(R\\)\\nobreakdash-controlled'' to highlight the control\nparameter~\\(R\\).\nThe locally compact, controlled operators on~\\(\\Hils\\)\nform a $^*$\\nobreakdash-{}algebra.  Its closure in \\(\\mathbb B(\\Hils)\\)\nis the \\emph{Roe \\(\\Cst\\)\\nobreakdash-algebra} \\(\\Cst_\\Roe(X,\\varrho)\\).\n\nThe representation~\\(\\varrho\\)\nis called \\emph{ample} if the operator \\(\\varrho(f)\\)\nfor \\(f\\in\\Cont_0(X)\\)\nis only compact for \\(f=0\\).\n\n\\begin{theorem}\n  \\label{the:Roe_ample_unique}\n  Let \\(\\varrho_i\\colon \\Cont_0(X)\\to\\mathbb B(\\Hils_i)\\)\n  for \\(i=1,2\\)\n  be ample representations, where \\(\\Hils_1\\)\n  and~\\(\\Hils_2\\)\n  are both complex or both real.  Then\n  \\(\\Cst_\\Roe(X,\\varrho_1) \\cong \\Cst_\\Roe(X,\\varrho_2)\\).\n  Even more, there is a unitary operator\n  \\(U\\colon \\Hils_1 \\xrightarrow\\sim \\Hils_2\\)\n  with\n  \\[\n  U \\Cst_\\Roe(X,\\varrho_1) U^* = \\Cst_\\Roe(X,\\varrho_2).\n  \\]\n\\end{theorem}\n\nMany references only assert the weaker statement that the Roe\n\\(\\Cst\\)\\nobreakdash-algebras\nfor all ample representations have canonically isomorphic\n\\(\\K\\)\\nobreakdash-theory,\ncompare \\cite{Higson-Roe:Analytic_K}*{Corollary 6.3.13}.  The\nstatement above is\n\\cite{Higson-Roe-Yu:Coarse_Mayer-Vietoris}*{Lemma~2}, and our proof is\nthe same.\n\n\\begin{proof}\n  If~\\(X\\)\n  is compact, then \\(\\Cst_\\Roe(X,\\varrho_i) = \\mathbb K(\\Hils_i)\\).\n  Since~\\(\\Hils_i\\)\n  for \\(i=1,2\\)\n  are assumed to be separable, there is a unitary\n  \\(U\\colon \\Hils_1 \\xrightarrow\\sim \\Hils_2\\),\n  and it will do the job.  So we may assume~\\(X\\)\n  to be non-compact.  Fix \\(R>0\\).\n  The open balls \\(B(x,R)\\)\n  for \\(x\\in X\\)\n  cover~\\(X\\).\n  Since~\\(X\\)\n  is second countable, there is a subordinate countable, locally\n  finite, open covering \\(X= \\bigcup_{n\\in\\mathbb{N}} U_n\\),\n  where each~\\(U_n\\)\n  is non-empty and has diameter at most~\\(R\\).\n  Then there is a countable covering of~\\(X\\)\n  by disjoint Borel sets, \\(X=\\bigsqcup_{n\\in\\mathbb{N}} B'_n\\),\n  where each~\\(B'_n\\)\n  has diameter at most~\\(R\\),\n  and such that any relatively compact subset is already covered by\n  finitely many of the~\\(B'_n\\):\n  simply take \\(B'_n \\mathrel{\\vcentcolon=} U_n \\setminus \\bigcup_{j< n} U_j\\).\n  Next we modify the subsets~\\(B'_n\\)\n  so that they all have non-empty interior.  Let \\(M\\subseteq\\mathbb{N}\\)\n  be the set of all \\(n\\in\\mathbb{N}\\)\n  for which~\\(B'_n\\)\n  has non-empty interior.  Let \\(m\\in M\\).\n  Let~\\(K_m\\)\n  be the set of all \\(k\\in\\mathbb{N}\\)\n  for which~\\(B'_k\\)\n  has empty interior and \\(U_m\\cap U_k\\neq \\emptyset\\).\n  The set~\\(K_m\\)\n  is finite because~\\(U_m\\)\n  is bounded and hence relatively compact.  Let\n  \\(K_m^\\circ \\mathrel{\\vcentcolon=} K_m \\setminus \\bigcup_{i\\in M, i<m} K_i\\).\n  Define\n  \\[\n  B_m \\mathrel{\\vcentcolon=} B'_m \\sqcup \\bigsqcup_{k\\in K_m^\\circ} B'_k.\n  \\]\n  This is a Borel set.  It has non-empty interior because~\\(B'_m\\)\n  has non-empty interior.  Its diameter is at most~\\(3R\\)\n  because all the~\\(U_k\\)\n  for \\(k\\in K_m\\)\n  intersect~\\(U_m\\).\n  The definition of~\\(K_m^\\circ\\)\n  ensures that the subsets~\\(B_m\\)\n  for \\(m\\in M\\)\n  are disjoint.  We claim that \\(\\bigsqcup_{m\\in M} B_m = X\\).\n  This is equivalent to\n  \\(\\bigcup_{m\\in M} K_m^\\circ = \\mathbb{N}\\setminus M\\)\n  because \\(B'_m\\subseteq B_m\\)\n  for all \\(m\\in M\\).\n  This is further equivalent to\n  \\(\\bigcup_{m\\in M} K_m = \\mathbb{N}\\setminus M\\).\n  Let \\(k_0\\in\\mathbb{N} \\setminus M\\),\n  that is, \\(B'_{k_0}\\)\n  has empty interior.  Then~\\(B'_{k_0}\\)\n  does not contain~\\(U_{k_0}\\).\n  So there is some \\(k_1<k_0\\)\n  with \\(U_{k_0} \\cap U_{k_1} \\neq \\emptyset\\).\n  If \\(k_1\\in \\mathbb{N}\\setminus M\\),\n  then~\\(B'_{k_1}\\)\n  does not contain \\(U_{k_0} \\cap U_{k_1}\\).\n  So there is \\(k_2<k_1\\)\n  with \\(U_{k_0} \\cap U_{k_1} \\cap U_{k_2} \\neq \\emptyset\\).\n  We continue like this and build a decreasing chain\n  \\(k_0>k_1>\\dotsc>k_\\ell\\)\n  such that \\(k_0,\\dotsc,k_{\\ell-1}\\in \\mathbb{N}\\setminus M\\) and\n  \\[\n  U_{k_0} \\cap U_{k_1} \\cap \\dotsb \\cap U_{k_\\ell}\n  \\neq \\emptyset.\n  \\]\n  We eventually reach \\(k_\\ell\\in M\\)\n  because \\(B'_1= U_1\\)\n  is open and so \\(1\\in M\\).\n  We have \\(k_0\\in K_{k_\\ell}\\).\n  So \\(\\bigcup_{m\\in M} K_m = \\mathbb{N}\\setminus M\\) as asserted.\n\n  We have built a covering of~\\(X\\)\n  by disjoint Borel sets \\(X = \\bigsqcup_{m\\in M} B_m\\)\n  of diameter at most~\\(3 R\\),\n  with non-empty interiors, and such that any relatively compact\n  subset is already covered by finitely many of the~\\(B_m\\).\n  The set~\\(M\\)\n  is at most countable, and it cannot be finite because then~\\(X\\)\n  would be bounded and hence compact.\n\n  Using the Borel functional calculus for the\n  representation~\\(\\varrho_i\\),\n  we may decompose the Hilbert space~\\(\\Hils_i\\)\n  as an orthogonal direct sum,\n  \\(\\Hils_i = \\bigoplus_{m\\in M} \\Hils_{i,m}\\),\n  where~\\(\\Hils_{i,m}\\)\n  is the image of the projection \\(\\varrho_i(1_{B_m})\\).\n  Since each~\\(B_m\\)\n  has non-empty interior and our representations are ample, there is a\n  non-compact operator on each~\\(\\Hils_{i,m}\\).\n  So no~\\(\\Hils_{i,m}\\)\n  has finite dimension.  Hence there is a unitary\n  \\(U_m\\colon \\Hils_{1,m}\\xrightarrow\\sim\\Hils_{2,m}\\)\n  for each \\(m\\in M\\).\n  We combine these into a unitary operator\n  \\(U= \\bigoplus_{m\\in M} U_m\\colon \\Hils_1 \\xrightarrow\\sim \\Hils_2\\).\n\n  Let \\(T\\in\\mathbb B(\\Hils_1)\\).\n  We claim that~\\(U T U^*\\)\n  is locally compact or controlled if and only if~\\(T\\)\n  is.  This implies\n  \\(U \\Cst_\\Roe(X,\\varrho_1) U^* = \\Cst_\\Roe(X,\\varrho_2)\\)\n  as asserted.  First, we claim that~\\(T\\)\n  is locally compact if and only if \\(T \\varrho_1(1_{B_m})\\)\n  and \\(\\varrho_1(1_{B_m}) T\\)\n  are compact for all \\(m\\in M\\).\n  In one direction, this uses that there is \\(g\\in \\Cont_0(X)\\)\n  with \\(1_{B_m} \\le g\\)\n  because~\\(B_m\\)\n  has finite diameter.  In the other direction, it uses that any\n  relatively compact subset of~\\(X\\)\n  is already covered by finitely many~\\(B_m\\).\n  Since \\(U(\\Hils_{1,m}) = \\Hils_{2,m}\\),\n  the criterion above shows that~\\(T\\)\n  is locally compact if and only if \\(U T U^*\\)\n  is so.  Since the diameter of~\\(B_m\\)\n  is at most~\\(3 R\\),\n  the operator~\\(U_m\\),\n  viewed as a partial isometry on \\(\\Hils_1 \\oplus \\Hils_2\\),\n  is \\(3 R\\)\\nobreakdash-controlled.\n  Thus~\\(U\\)\n  is also \\(3 R\\)\\nobreakdash-controlled.\n  So \\(U T U^*\\) is controlled if and only if~\\(T\\) is.\n\\end{proof}\n\n\\begin{corollary}\n  \\label{cor:Roe_matrix-stable}\n  Let~\\(\\varrho\\)\n  be an ample representation and let \\(m\\in\\mathbb{N}_{\\ge2}\\).  Then\n  \\[\n  \\Cst_\\Roe(X,\\varrho) \\cong \\mathbb M_m(\\Cst_\\Roe(X,\\varrho)).\n  \\]\n\\end{corollary}\n\n\\begin{proof}\n  The direct sum representation \\(m\\cdot \\varrho\\)\n  is still ample.  So\n  \\(\\Cst_\\Roe(X,m\\cdot \\varrho) \\cong \\Cst_\\Roe(X,\\varrho)\\).\n  An operator on~\\(\\Hils^m\\)\n  is locally compact or controlled if and only if its block\n  matrix entries in \\(\\mathbb B(\\Hils)\\)\n  are so.  Thus\n  \\(\\Cst_\\Roe(X,m\\cdot \\varrho) = \\mathbb M_m(\\Cst_\\Roe(X,\\varrho))\\).\n\\end{proof}\n\nThe stabilisation \\(\\Cst_\\Roe(X,\\varrho) \\otimes \\mathbb K(\\ell^2\\mathbb{N})\\),\nhowever, is usually not isomorphic to \\(\\Cst_\\Roe(X,\\varrho)\\).\n\n\\begin{example}\n  \\label{exa:Roe_discrete}\n  Let~\\(X\\)\n  be discrete, for instance, \\(X=\\mathbb{Z}^d\\).\n  The representation~\\(\\varrho\\)\n  of \\(\\Cont_0(X)\\)\n  on~\\(\\ell^2(X)\\)\n  by multiplication operators is not ample.  It defines the\n  \\emph{uniform Roe \\(\\Cst\\)\\nobreakdash-algebra}\n  of~\\(X\\).\n  To get the Roe \\(\\Cst\\)\\nobreakdash-algebra,\n  we may take the representation of \\(\\Cont_0(X)\\)\n  on \\(\\ell^2(X)\\otimes \\ell^2(\\mathbb{N})\\).\n\n  An operator~\\(T\\)\n  on \\(\\ell^2(X)\\otimes \\ell^2(\\mathbb{N})\\)\n  is determined by its matrix coefficients\n  \\(T_{x,y} = \\braketop{x}{T}{y} \\in \\mathbb B(\\ell^2(\\mathbb{N}))\\)\n  for \\(x,y\\in X\\).\n  It is locally compact if and only if all~\\(T_{x,y}\\)\n  are compact.  Its support is the set of all \\((x,y)\\in X^2\\)\n  with \\(T_{x,y}\\neq0\\).\n  So it is controlled if and only if there is \\(R>0\\)\n  so that \\(T_{x,y}=0\\)\n  for \\(d(x,y)>R\\).\n  The Roe \\(\\Cst\\)\\nobreakdash-algebra is the norm closure of these operators.\n\n  If~\\(X\\)\n  is a discrete group equipped with a translation-invariant metric,\n  then \\(\\Cst_\\Roe(X)\\)\n  is isomorphic to the reduced crossed product for the translation\n  action of~\\(X\\)\n  on~\\(\\ell^\\infty(X, \\mathbb K(\\ell^2\\mathbb{N}))\\)\n  (compare \\cite{Roe:Lectures}*{Theorem~4.28} for the uniform Roe\n  \\(\\Cst\\)\\nobreakdash-algebra).\n\\end{example}\n\n\\begin{example}\n  \\label{exa:Roe_Rn}\n  Let \\(X=\\mathbb{R}^d\\).\n  The representation~\\(\\varrho\\)\n  of \\(\\Cont_0(\\mathbb{R}^d)\\)\n  on \\(L^2(\\mathbb{R}^d,\\diff x)\\)\n  (real or complex) by multiplication operators is ample.  Actually,\n  all faithful representations of~\\(\\Cont_0(\\mathbb{R}^d)\\)\n  are ample.  So they all give isomorphic Roe \\(\\Cst\\)\\nobreakdash-algebras\n  by Theorem~\\ref{the:Roe_ample_unique}.\n\n  Let \\(T\\in\\Contc(\\mathbb{R}^d)\\)\n  (with real or complex values) act on~\\(L^2(\\mathbb{R}^d)\\)\n  by convolution.  Then \\(T\\cdot \\varrho(f)\\)\n  and \\(\\varrho(f)\\cdot T\\)\n  are compact because they have a compactly supported, continuous\n  integral kernel.  And \\(T\\)\n  is controlled by the supremum of~\\(\\norm{x}\\)\n  with \\(T(x)\\neq0\\).\n  So \\(T\\in\\Cst_\\Roe(\\mathbb{R}^d)\\).\n  Hence \\(\\Cst(\\mathbb{R}^d) \\subseteq \\Cst_\\Roe(\\mathbb{R}^d)\\).\n  In particular, the resolvent of the Laplace operator or another\n  translation-invariant elliptic differential operator on~\\(\\mathbb{R}^d\\)\n  belongs to\n  \\(\\Cst_\\Roe(\\mathbb{R}^d)\\).\n  Any multiplication operator is controlled.  Thus\n  multiplication operators are multipliers of \\(\\Cst_\\Roe(\\mathbb{R}^d)\\).\n  And \\(L^\\infty(\\mathbb{R}^d) \\cdot \\Cst(\\mathbb{R}^d) \\cdot L^\\infty(\\mathbb{R}^d)\\)\n  is contained in \\(\\Cst_\\Roe(\\mathbb{R}^d)\\).\n  (Since the translation action of~\\(\\mathbb{R}^d\\)\n  on \\(L^\\infty(\\mathbb{R}^d)\\)\n  is not continuous, there is no crossed product for this action and\n  it is unclear whether the closed linear spans of\n  \\(L^\\infty(\\mathbb{R}^d) \\cdot \\Cst(\\mathbb{R}^d)\\)\n  and \\(\\Cst(\\mathbb{R}^d) \\cdot L^\\infty(\\mathbb{R}^d)\\)\n  are equal and form a \\(\\Cst\\)\\nobreakdash-algebra.)\n\\end{example}\n\n\\begin{proposition}\n  \\label{pro:resolvent_in_Roe}\n  Let \\(V\\in L^\\infty(\\mathbb{R}^n)\\)\n  and let~\\(\\Delta\\)\n  be the Laplace operator on~\\(\\mathbb{R}^d\\).\n  Then the resolvent of \\(V+\\Delta\\)\n  belongs to the Roe \\(\\Cst\\)\\nobreakdash-algebra of~\\(\\mathbb{R}^d\\).\n\\end{proposition}\n\nWe are indebted to Detlev Buchholz for pointing out the following\nsimple proof.\n\n\\begin{proof}\n  View \\(V=V(Q)\\)\n  as an operator on \\(L^2(\\mathbb{R}^d)\\).\n  Then \\(\\norm{(\\ima c + \\Delta)^{-1} V}^2<1\\)\n  for sufficiently large \\(c\\in\\mathbb{R}_{>0}\\).\n  Hence the Neumann series \\(\\sum (-(\\ima c + \\Delta)^{-1} V)^n\\)\n  converges, and\n  \\[\n  \\sum_{n=0}^\\infty (-(\\ima c + \\Delta)^{-1} V)^n\n  \\cdot (\\ima c + \\Delta)^{-1}\n  = (1+(\\ima c + \\Delta)^{-1} V)^{-1} \\cdot (\\ima c + \\Delta)^{-1}\n  = (\\ima c + \\Delta+V)^{-1}.\n  \\]\n  We have already seen that \\((\\ima c + \\Delta)^{-1}\\)\n  and \\((\\ima c + \\Delta)^{-1} V\\)\n  belong to the Roe \\(\\Cst\\)\\nobreakdash-algebra.\n  Hence so does \\((\\ima c + \\Delta+V)^{-1}\\).\n\\end{proof}\n\nIf~\\(\\varrho\\)\nis ample, then we often leave out~\\(\\varrho\\)\nand briefly write \\(\\Cst_\\Roe(X)_\\mathbb{R}\\)\nor \\(\\Cst_\\Roe(X)_\\mathbb{C}\\),\ndepending on whether~\\(\\varrho\\)\nacts on a real or complex Hilbert space.\nTheorem~\\ref{the:Roe_ample_unique} justifies this.\n\n\\begin{definition}\n  \\label{def:coarsely_dense}\n  A closed subset \\(Y\\subseteq X\\)\n  is \\emph{coarsely dense} if there is \\(R>0\\)\n  such that for any \\(x\\in X\\) there is \\(y\\in Y\\) with \\(d(x,y)\\le R\\).\n\\end{definition}\n\n\\begin{theorem}\n  \\label{the:coarsely_dense}\n  Let \\(Y\\subseteq X\\)\n  be coarsely dense.  Then \\(\\Cst_\\Roe(Y)_\\mathbb{R} \\cong \\Cst_\\Roe(X)_\\mathbb{R}\\)\n  and \\(\\Cst_\\Roe(Y)_\\mathbb{C} \\cong \\Cst_\\Roe(X)_\\mathbb{C}\\).\n  Both isomorphisms are implemented by unitaries between the\n  underlying Hilbert spaces.\n\\end{theorem}\n\n\\begin{proof}\n  The proofs in the complex and real case are identical.  Let\n  \\(\\pi\\colon \\Cont_0(X) \\to \\Cont_0(Y)\\)\n  be the restriction homomorphism.  Let\n  \\(\\varrho_Y\\colon \\Cont_0(Y)\\to\\mathbb B(\\Hils_Y)\\)\n  and \\(\\varrho_X\\colon \\Cont_0(X)\\to\\mathbb B(\\Hils_X)\\)\n  be ample representations.  Then\n  \\(\\varrho' \\mathrel{\\vcentcolon=} \\varrho_X \\oplus \\varrho_Y\\circ\\pi\\)\n  is an ample representation of~\\(\\Cont_0(X)\\)\n  on \\(\\Hils'\\mathrel{\\vcentcolon=} \\Hils_X \\oplus \\Hils_Y\\).\n  By Theorem~\\ref{the:Roe_ample_unique}, we may use the particular\n  representations \\(\\varrho'\\)\n  and~\\(\\varrho_Y\\)\n  to define \\(\\Cst_\\Roe(X)\\)\n  and \\(\\Cst_\\Roe(Y)\\),\n  because different ample representations give Roe\n  \\(\\Cst\\)\\nobreakdash-algebras\n  that are isomorphic through conjugation with a unitary between the\n  underlying Hilbert spaces.\n\n  Pick \\(R>0\\)\n  such that for each \\(x\\in X\\)\n  there is \\(y\\in Y\\)\n  with \\(d(x,y) \\le R\\).\n  We build a Borel map \\(g\\colon X\\to Y\\)\n  with \\(g|_Y=\\Id_Y\\) and \\(d(g(x),x)\\le 2 R\\) for all \\(x\\in X\\),\n  First, there is a countable cover \\(X=\\bigsqcup_{m\\in\\mathbb{N}} B_m\\)\n  by non-empty, disjoint Borel sets of diameter at most~\\(R\\)\n  as in the proof of Theorem~\\ref{the:Roe_ample_unique}.  For each\n  \\(m\\in\\mathbb{N}\\),\n  pick \\(x_m\\in B_m\\)\n  and \\(y_m\\in Y\\)\n  with \\(d(x_m,y_m) \\le R\\).\n  Define \\(g(x) \\mathrel{\\vcentcolon=} x\\)\n  for \\(x\\in Y\\)\n  and \\(g(x) \\mathrel{\\vcentcolon=} y_m\\)\n  for \\(x\\in B_m\\setminus Y\\),\n  \\(m\\in\\mathbb{N}\\).  This map has all the required properties.\n\n  The representation\n  \\(\\varrho'\\circ g^*\\colon \\Cont_0(Y)\\to\\mathbb B(\\Hils')\\)\n  is ample because it contains\n  \\(\\varrho_Y\\circ \\pi\\circ g^* = \\varrho_Y\\)\n  as a direct summand.  An operator on~\\(\\Hils'\\)\n  is locally compact or controlled for~\\(\\varrho'\\)\n  if and only if it is so for \\(\\varrho'\\circ g^*\\).\n  Thus \\(\\Cst_\\Roe(X,\\varrho') = \\Cst_\\Roe(Y,\\varrho'\\circ g^*)\\).\n\\end{proof}\n\nIn particular, Theorem~\\ref{the:coarsely_dense} shows that all\ncoarsely dense subsets of~\\(\\mathbb{R}^d\\)\nhave isomorphic Roe \\(\\Cst\\)\\nobreakdash-algebras.\nThis applies, in particular, to \\(\\mathbb{Z}^d\\)\nand to all Delone subsets of~\\(\\mathbb{R}^d\\).\nThe latter are often used to model the atomic configurations of\nmaterials that are not crystals (see, for instance,\n\\cite{Bellissard-Herrmann-Zarrouati:Hull}).  So all kinds of materials\nlead to the same Roe \\(\\Cst\\)\\nobreakdash-algebra,\nwhich depends only on the dimension~\\(d\\).\n\nTheorem~\\ref{the:coarsely_dense} suffices for our purposes, but we\nmention that it extends to arbitrary coarse equivalences, see also\n\\cite{Higson-Roe:Analytic_K}*{Section 6.3}.\n\n\\begin{definition}\n  \\label{def:coarse_maps}\n  Let \\(X\\)\n  and~\\(Y\\)\n  be proper metric spaces as above.  Two maps \\(f_0,f_1\\colon X\\to Y\\)\n  are \\emph{close} if there is \\(R>0\\)\n  so that \\(d(f_0(x),f_1(x))<R\\)\n  for all \\(x\\in X\\).\n  A \\emph{coarse map} \\(f\\colon X\\to Y\\)\n  is a Borel map with two properties: for any \\(R>0\\)\n  there is \\(S>0\\)\n  such that \\(d(x,y)\\le R\\)\n  for \\(x,y\\in X\\)\n  implies \\(d(f(x),f(y))\\le S\\),\n  and \\(f^{-1}(B)\\)\n  is bounded in~\\(X\\)\n  if \\(B\\subseteq Y\\)\n  is bounded.  A \\emph{coarse equivalence} is a coarse map\n  \\(f\\colon X\\to Y\\)\n  for which there is another coarse map \\(g\\colon Y\\to X\\),\n  called the \\emph{coarse inverse} of~\\(f\\),\n  such that \\(g\\circ f\\)\n  and \\(f\\circ g\\)\n  are close to the identity maps on \\(X\\)\n  and~\\(Y\\),\n  respectively.  We call \\(X\\)\n  and~\\(Y\\)\n  \\emph{coarsely equivalent} if there is a coarse equivalence between\n  them.\n\\end{definition}\n\nFor instance, the inclusion of a coarsely dense subspace is a coarse\nequivalence: the proof of Theorem~\\ref{the:coarsely_dense} builds a\ncoarse inverse for the inclusion map.\n\n\\begin{theorem}\n  \\label{the:coarse_equivalence_Roe}\n  Let \\(X\\)\n  and~\\(Y\\)\n  be coarsely equivalent.  Then\n  \\(\\Cst_\\Roe(X)_\\mathbb{R} \\cong \\Cst_\\Roe(Y)_\\mathbb{R}\\)\n  and \\(\\Cst_\\Roe(X)_\\mathbb{C} \\cong \\Cst_\\Roe(Y)_\\mathbb{C}\\).\n\\end{theorem}\n\n\\begin{proof}\n  Here it is more convenient to work with coarse spaces.  Let\n  \\(f\\colon X\\to Y\\)\n  be the coarse equivalence.  We claim that there is a coarse\n  structure on the disjoint union \\(X\\sqcup Y\\)\n  such that both \\(X\\)\n  and~\\(Y\\)\n  are coarsely dense in \\(X\\sqcup Y\\).\n  This reduces the result to Theorem~\\ref{the:coarsely_dense}.  We\n  describe the desired coarse structure on \\(X\\sqcup Y\\).\n  A subset~\\(E\\)\n  of \\((X\\sqcup Y)^2\\)\n  is called controlled if its intersections with \\(X^2\\)\n  and~\\(Y^2\\)\n  and the set of all \\((f(x),y) \\in Y^2\\)\n  for \\((x,y)\\in E\\)\n  or \\((y,x)\\in E\\)\n  are controlled.  This is a coarse structure on \\(X\\sqcup Y\\)\n  because~\\(f\\)\n  is a coarse equivalence.  And the subspaces \\(X\\)\n  and~\\(Y\\) are coarsely dense for the same reason.\n\\end{proof}\n\n\n\\subsection{Twists}\n\\label{sec:twists}\n\nWe show that magnetic twists do not change the isomorphism class of\nthe Roe \\(\\Cst\\)\\nobreakdash-algebra.\nWe let~\\(X\\)\nbe a \\emph{discrete} metric space.  Let\n\\(\\varrho\\colon \\Cont_0(X)\\to \\mathbb B(\\Hils)\\)\nbe a representation.  This is equivalent to a direct sum decomposition\n\\(\\Hils = \\bigoplus_{x\\in X} \\Hils_x\\),\nsuch that \\(f\\in \\Cont_0(X)\\)\nacts by multiplication with \\(f(x)\\)\non the summand~\\(\\Hils_x\\).\nWe assume for simplicity that each~\\(\\Hils_x\\)\nis non-zero.  This is weaker than being ample, which means that\neach~\\(\\Hils_x\\)\nis infinite-dimensional.  So the following discussion also covers the\nuniform Roe \\(\\Cst\\)\\nobreakdash-algebra of~\\(X\\).\n\nWe describe an operator on~\\(\\Hils\\)\nby a block matrix \\((T_{x,y})_{x,y\\in X}\\)\nwith \\(T_{x,y}\\in \\mathbb B(\\Hils_y,\\Hils_x)\\).\nThese are multiplied by the usual formula,\n\\((S T)_{x,y} = \\sum_{z\\in X} S_{x,z} T_{z,y}\\).\nWe twist this multiplication by a scalar-valued function\n\\(w\\colon X\\times X \\times X\\to \\mathbb{T}\\):\n\\[\n(S *_w T)_{x,y} = \\sum_{z\\in X} w(x,z,y) S_{x,z} T_{z,y}.\n\\]\nThis defines a bounded bilinear map at least on the subalgebra\n\\(A(X,\\varrho)\\subseteq \\mathbb B(\\Hils)\\)\nof locally compact, controlled operators.\n\n\\begin{lemma}\n  \\label{lem:twisted_associative}\n  The multiplication~\\(*_w\\)\n  on \\(A(X,\\varrho)\\)\n  is associative if and only if\n  \\begin{equation}\n    \\label{eq:twist_cocycle_condition}\n    w(x,z,y) w(x,t,z) = w(x,t,y) w(t,z,y)\n  \\end{equation}\n  for all \\(x,t,z,y\\in X\\).\n\\end{lemma}\n\n\\begin{proof}\n  For \\(S,T,U\\in A(X,\\varrho)\\), we compute\n  \\begin{align*}\n    \\bigl((S *_w T) *_w U\\bigr)_{x,y}\n    \n      = \\sum_{z,t\\in X} w(x,z,y) w(x,t,z) S_{x,t} T_{t,z} U_{z,y},\\\\\n    \\bigl(S *_w (T *_w U)\\bigr)_{x,y}\n    \n      = \\sum_{z,t\\in X} w(x,t,y) w(t,z,y) S_{x,t} T_{t,z} U_{z,y}.\n  \\end{align*}\n  The condition~\\eqref{eq:twist_cocycle_condition} holds if and only if these are\n  equal for all \\(S,T,U\\in A(X,\\varrho)\\)\n  because all \\(\\Hils_x\\) are non-zero.\n\\end{proof}\n\n\\begin{proposition}\n  \\label{pro:all_twists_trivial}\n  If the function~\\(w\\)\n  satisfies the cocycle condition in the previous lemma, then there is\n  a function \\(v\\colon X\\times X \\to \\mathbb{T}\\) with\n  \\[\n  w(x,z,y) =  v(x,z) v(z,y) v(x,y)^{-1}.\n  \\]\n  The map \\(\\varphi\\colon (A(X, \\varrho),*_w) \\to (A(X,\\varrho),\\cdot)\\),\n  \\((T_{x,y})_{x,y\\in X} \\mapsto (v(x,y)\\cdot T_{x,y})_{x,y\\in X}\\),\n  is an algebra isomorphism.\n\\end{proposition}\n\n\\begin{proof}\n  Fix a ``base point'' \\(e\\in X\\)\n  and let \\(v(x,y) \\mathrel{\\vcentcolon=} w(x,y,e)\\).\n  The condition~\\eqref{eq:twist_cocycle_condition} for \\((x,z,y,e)\\)\n  says that\n  \\[\n  w(x,y,e) w(x,z,y) = w(x,z,e) w(z,y,e)\n  \\]\n  holds for all \\(x,z,y\\in X\\).  So\n  \\[\n  v(x,z) v(z,y) v(x,y)^{-1}\n  = w(x,z,e) w(z,y,e) w(x,y,e)^{-1}\n  = w(x,z,y).\n  \\]\n  The map~\\(\\varphi\\)\n  is a vector space isomorphism because \\(v(x,y)\\neq0\\)\n  for all \\(x,y\\in X\\).  The computation\n  \\begin{multline*}\n    \\varphi (S *_w  T)_{x,y}\n   \n    = v(x,y) \\sum_{z\\in X} w(x,z,y) S_{x,z} T_{z,y}\n    \\\\= \\sum_{z\\in X} w(x,z,y) v(x,y) v(x,z)^{-1} v(z,y)^{-1}\n    \\varphi(S)_{x,z} \\varphi(T)_{z,y}\n    = \\sum_{z\\in X} \\varphi(S)_{x,z} \\varphi(T)_{z,y}\n  \\end{multline*}\n  shows that it is an algebra isomorphism.\n\\end{proof}\n\nSo the twisted and untwisted versions of \\(A(X,\\varrho)\\)\nare isomorphic algebras.  Thus\na magnetic field does not change the isomorphism type of the Roe\n\\(\\Cst\\)\\nobreakdash-algebra.\n\nNow let~\\(X\\)\nbe no longer discrete.  Then controlled, locally compact\noperators are not given by matrices any more.  To write down the\ntwisted convolution as above, we use the smaller $^*$\\nobreakdash-{}algebra of\ncontrolled, locally \\emph{Hilbert--Schmidt} operators; it is\nstill dense in the Roe \\(\\Cst\\)\\nobreakdash-algebra.\nLet us assume for simplicity that the representation~\\(\\varrho\\)\nfor which we build the Roe \\(\\Cst\\)\\nobreakdash-algebra\nhas constant multiplicity, that is, it is the pointwise multiplication\nrepresentation on \\(L^2(X,\\mu) \\otimes \\Hils\\)\nfor some regular Borel measure~\\(\\mu\\)\non~\\(X\\)\nand some Hilbert space~\\(\\Hils\\).\nControlled, locally Hilbert--Schmidt operators on\n\\(L^2(X,\\mu) \\otimes \\Hils\\)\nare the convolution operators for measurable functions\n\\(T\\colon X\\times X\\to \\ell^2(\\Hils)\\)\nwith controlled support and such that\n\\[\n\\int_{K\\times K} \\norm{T(x,y)}^2 \\,\\diff \\mu(x)\\,\\diff\\mu(y)  < \\infty\n\\]\nfor all compact subsets \\(K\\subseteq X\\)\nand such that the resulting convolution operator is bounded.  The\nmultiplication of such operators is given by a convolution of their\nintegral kernels.  This may be twisted as above, using a Borel\nfunction \\(w\\colon X^3\\to\\mathbb{T}\\)\nthat satisfies the condition~\\eqref{eq:twist_cocycle_condition}.  The\nresulting function~\\(v\\)\nin Proposition~\\ref{pro:all_twists_trivial} is again Borel.  So the\nisomorphism in Proposition~\\ref{pro:all_twists_trivial} still works.\nThus the twist gives an isomorphic $^*$\\nobreakdash-{}algebra also in the\nnon-discrete case.\n\nFollowing Bellissard~\\cite{Bellissard:K-theory_solid}, a\n\\(d\\)\\nobreakdash-dimensional\nmaterial is often described through a crossed product\n\\(\\Cst\\)\\nobreakdash-algebra\n\\(\\Cont(\\Omega)\\rtimes\\mathbb{Z}^d\\)\nfor a compact space~\\(\\Omega\\)\nwith a \\(\\mathbb{Z}^d\\)\\nobreakdash-action\nby homeomorphisms and with an ergodic invariant measure\non~\\(\\Omega\\).\nTo encode a magnetic field, the crossed product is replaced by the\ncrossed product \\emph{twisted} by a \\(2\\)\\nobreakdash-cocycle\n\\(\\sigma\\colon \\mathbb{Z}^d \\times \\mathbb{Z}^d \\to \\Cont(\\Omega,\\mathbb{T})\\).\nThe space~\\(\\Omega\\)\nmay be built as the ``hull'' of a point set or a fixed Hamiltonian,\nsee~\\cite{Bellissard-Herrmann-Zarrouati:Hull}.  In this case, there is\na \\emph{dense} orbit \\(\\mathbb{Z}^d\\cdot\\omega\\)\nin~\\(\\Omega\\)\nby construction.  So assuming the existence of a dense orbit is a\nrather mild assumption in the context of Bellissard's theory.\n\nWe briefly explain why all twisted crossed products\n\\(\\Cont(\\Omega)\\rtimes_\\sigma \\mathbb{Z}^d\\)\nas above are ``contained'' in the uniform Roe \\(\\Cst\\)\\nobreakdash-algebra\nof~\\(\\mathbb{Z}^d\\)\nand hence also in the Roe \\(\\Cst\\)\\nobreakdash-algebra.\nThis observation is due to Kubota~\\cite{Kubota:Controlled_bulk-edge}.\n\nThe main point here is the description of the uniform Roe\n\\(\\Cst\\)\\nobreakdash-algebra\nas a crossed product \\(\\ell^\\infty(\\mathbb{Z}^d)\\rtimes\\mathbb{Z}^d\\),\nsee \\cite{Roe:Lectures}*{Theorem~4.28}.  Let \\(\\omega\\in\\Omega\\).\nThen we define a \\(\\mathbb{Z}^d\\)\\nobreakdash-equivariant\n$^*$\\nobreakdash-{}homomorphism\n\\(\\epsilon_\\omega\\colon \\Cont(\\Omega) \\to \\ell^\\infty(\\mathbb{Z}^d)\\)\nby \\((\\epsilon_\\omega f)(n) \\mathrel{\\vcentcolon=} f(n\\cdot \\omega)\\)\nfor all \\(n\\in\\mathbb{Z}^d\\),\n\\(f\\in \\Cont(\\Omega)\\).\nThis induces a $^*$\\nobreakdash-{}homomorphism\n\\(\\Cont(\\Omega)\\rtimes_\\sigma \\mathbb{Z}^d \\to \\ell^\\infty(\\mathbb{Z}^d)\n\\rtimes_{\\epsilon_\\omega\\circ \\sigma} \\mathbb{Z}^d\\),\nwhere~\\(\\rtimes_\\sigma\\)\ndenotes the crossed product twisted by a \\(2\\)\\nobreakdash-cocyle~\\(\\sigma\\).\nThe same argument that identifies the crossed product\n\\(\\ell^\\infty(\\mathbb{Z}^d) \\rtimes \\mathbb{Z}^d\\)\nwith the uniform Roe \\(\\Cst\\)\\nobreakdash-algebra\nof~\\(\\mathbb{Z}^d\\)\nidentifies\n\\(\\ell^\\infty(\\mathbb{Z}^d) \\rtimes_{\\epsilon_\\omega\\circ \\sigma} \\mathbb{Z}^d\\)\nwith a twist of the Roe \\(\\Cst\\)\\nobreakdash-algebra\nas above.  Since all these twists give isomorphic \\(\\Cst\\)\\nobreakdash-algebras\nby Proposition~\\ref{pro:all_twists_trivial}, we get a\n$^*$\\nobreakdash-{}homomorphism\n\\(\\Cont(\\Omega)\\rtimes_\\sigma \\mathbb{Z}^d \\to \\ell^\\infty(\\mathbb{Z}^d) \\rtimes\n\\mathbb{Z}^d\\).\nIf the orbit of~\\(\\omega\\)\nis dense, then the $^*$\\nobreakdash-{}homomorphism~\\(\\epsilon_\\omega\\)\nabove is injective.  Then the induced $^*$\\nobreakdash-{}homomorphism\n\\(\\Cont(\\Omega)\\rtimes_\\sigma \\mathbb{Z}^d \\to \\ell^\\infty(\\mathbb{Z}^d) \\rtimes\n\\mathbb{Z}^d\\)\nis also injective.  Hence the uniform Roe \\(\\Cst\\)\\nobreakdash-algebra\nreally contains the twisted crossed product algebra.\n\nNow we turn to the continuum version of the above theory.\nLet~\\(\\Omega\\)\nbe a compact space with a continuous action of~\\(\\mathbb{R}^d\\).\nThis leads to crossed products \\(\\Cont(\\Omega)\\rtimes_\\sigma \\mathbb{R}^d\\)\ntwisted, say, by Borel measurable \\(2\\)\\nobreakdash-cocycles\n\\(\\sigma\\colon \\mathbb{R}^d \\times \\mathbb{R}^d \\to \\Cont(\\Omega,\\mathbb{T})\\);\nonce again, the twist encodes a magnetic field.  Restricting to the\n\\(\\mathbb{R}^d\\)\\nobreakdash-orbit\nof some \\(\\omega\\in\\Omega\\)\nmaps \\(\\Cont(\\Omega)\\rtimes_\\sigma \\mathbb{R}^d\\)\nto\n\\(\\Contb^\\mathrm{u}(\\mathbb{R}^d)\\rtimes_{\\epsilon_\\omega\\circ\\sigma} \\mathbb{R}^d\\)\nfor the \\(\\Cst\\)\\nobreakdash-algebra\n\\(\\Contb^\\mathrm{u}(\\mathbb{R}^d)\\)\nof bounded, uniformly continuous functions on~\\(\\mathbb{R}^d\\).\nWe have seen in Example~\\ref{exa:Roe_Rn} that\n\\(\\Contb^\\mathrm{u}(\\mathbb{R}^d)\\rtimes \\mathbb{R}^d \\subseteq L^\\infty(\\mathbb{R}^d) \\cdot\n\\Cst(\\mathbb{R}^d)\\)\nis contained in the Roe \\(\\Cst\\)\\nobreakdash-algebra\nof~\\(\\mathbb{R}^d\\).\nThis remains the case also in the twisted case because a Borel\nmeasurable \\(2\\)\\nobreakdash-cocycle\n\\(\\sigma\\colon \\mathbb{R}^d \\times \\mathbb{R}^d \\to \\Cont(\\Omega,\\mathbb{T})\\)\ndefines a Borel function \\((\\mathbb{R}^d)^3 \\to\\mathbb{T}\\),\nwhich is untwisted by Proposition~\\ref{pro:all_twists_trivial}.  So\nall twisted crossed products \\(\\Cont(\\Omega)\\rtimes_\\sigma \\mathbb{R}^d\\)\nmap to the Roe \\(\\Cst\\)\\nobreakdash-algebra\nof~\\(\\mathbb{R}^d\\).\nAs above, this map is an embedding if the orbit of~\\(\\omega\\)\nis dense in~\\(\\Omega\\).\n\nThe Roe \\(\\Cst\\)\\nobreakdash-algebras\nfor \\(\\mathbb{R}^d\\)\nand~\\(\\mathbb{Z}^d\\)\nare isomorphic by Theorem~\\ref{the:coarsely_dense}.  So there is a unique\nRoe \\(\\Cst\\)\\nobreakdash-algebra\nin each dimension that contains all the twisted crossed product\nalgebras that are used as models for disordered materials, both in\ncontinuum models and tight binding models.  This fits interpreting\nthe twisted crossed products \\(\\Cont(\\Omega)\\rtimes_\\sigma \\mathbb{Z}^d\\)\nor \\(\\Cont(\\Omega)\\rtimes_\\sigma \\mathbb{R}^d\\)\nas models for disorder with built-in \\emph{a priori} restrictions,\nwhereas the Roe \\(\\Cst\\)\\nobreakdash-algebra describes general disorder.\n\n\n\\subsection{Approximation by controlled operators as a continuity property}\n\\label{sec:approximation_continuity}\n\nIn order to belong to the Roe \\(\\Cst\\)\\nobreakdash-algebra,\nan operator has to be a norm limit of locally compact, controlled\noperators.  Any such norm limit is again locally compact.  The\nproperty of being a norm limit of controlled operators may be hard to\ncheck.  A tool for this is Property~A, an approximation property for\ncoarse spaces that ensures that elements of the Roe\n\\(\\Cst\\)\\nobreakdash-algebra\nmay be approximated in a systematic way by controlled operators, see\n\\cites{Roe:Warped_A, Brodzki-Cave-Li:Exactness}.  We also mention the\nrelated Operator Norm Localization Property for subspaces of~\\(\\mathbb{R}^d\\).\n\nWe now specialise to the case where~\\(X\\)\nis a closed subset of~\\(\\mathbb{R}^d\\)\nwith the restriction of the Euclidean metric.  Such spaces have\nProperty~A.  We use it to define complex Roe \\(\\Cst\\)\\nobreakdash-algebras\nthrough continuity for a certain representation of~\\(\\mathbb{R}^d\\).\nWe fix a representation \\(\\varrho\\colon \\Cont_0(X) \\to \\mathbb B(\\Hils)\\)\non a complex Hilbert space~\\(\\Hils\\).\nLet \\(\\bar\\varrho\\colon \\Contb(X)\\to\\mathbb B(\\Hils)\\)\nbe its unique strictly continuous extension to the multiplier algebra.\nFor \\(t\\in\\mathbb{R}^d\\),\ndefine \\(\\Euler_t\\in\\Contb(X)\\)\nby \\(\\Euler_t(x) \\mathrel{\\vcentcolon=} \\Euler^{\\ima x\\cdot t}\\).\nThe map \\(t\\mapsto \\Euler_t\\)\nis continuous for the strict topology on~\\(\\Contb(X)\\).\nHence the representation~\\(\\sigma\\)\nof~\\(\\mathbb{R}^d\\)\non~\\(\\Hils\\)\ndefined by \\(\\sigma_t(\\xi) \\mathrel{\\vcentcolon=} \\bar\\varrho(\\Euler_t)(\\xi)\\)\nis continuous.  This representation is generated by the position\noperators.  If \\(X\\subseteq \\mathbb{Z}^d\\),\nthen \\(\\Euler_t=1\\)\nfor \\(t\\in 2\\pi\\mathbb{Z}^d\\),\nso that the representation~\\(\\sigma\\)\ndescends to the torus \\((\\mathbb{R}\/2\\pi \\mathbb{Z})^d\\).\n\nBy conjugation, \\(\\sigma\\)\ninduces an action \\(\\Ad \\sigma\\)\nof~\\(\\mathbb{R}^d\\)\nby automorphisms of \\(\\mathbb B(\\Hils)\\).\nWe call \\(S\\in\\mathbb B(\\Hils)\\)\n\\emph{continuous} with respect to \\(\\Ad \\sigma\\)\nif the map \\(\\mathbb{R}^d \\to \\mathbb B(\\Hils)\\),\n\\(t\\mapsto \\Ad \\sigma_t(S)\\),\nis continuous in the norm topology on~\\(\\mathbb B(\\Hils)\\).\nThe following theorem describes the Roe \\(\\Cst\\)\\nobreakdash-algebra\nthrough this continuity property:\n\n\\begin{theorem}\n  \\label{the:controlled_continuity}\n  An operator \\(S\\in\\mathbb B(\\Hils)\\)\n  is a norm limit of controlled operators if and only if it is\n  continuous with respect to \\(\\Ad \\sigma\\).\n  And \\(\\Cst_\\Roe(X,\\varrho)\\)\n  is the \\(\\Cst\\)\\nobreakdash-subalgebra\n  of all operators on~\\(\\Hils\\)\n  that are locally compact and continuous with respect to\n  \\(\\Ad \\sigma\\).\n\\end{theorem}\n\n\\begin{proof}\n  For each \\(S\\in\\mathbb B(\\Hils)\\),\n  the map \\(\\mathbb{R}^d\\to\\mathbb B(\\Hils)\\)\n  is continuous for the strong topology on \\(\\mathbb B(\\Hils)\\).\n  Therefore, the \\(\\mathbb B(\\Hils)\\)-valued integral\n  \\[\n  f*S \\mathrel{\\vcentcolon=} \\int_{\\mathbb{R}^d} f(t) \\Ad \\sigma_t(S) \\,\\diff t\n  \\]\n  makes sense for any \\(f\\in L^1(\\mathbb{R}^d)\\).\n  Let \\((f_n)_{n\\in\\mathbb{N}}\\)\n  be a bounded approximate unit in the Banach algebra \\(L^1(\\mathbb{R}^d)\\).\n  We claim that~\\(S\\)\n  is continuous if and only if \\((f_n*S)_{n\\in\\mathbb{N}}\\)\n  converges in the norm topology to~\\(S\\).\n  It is well known that any continuous representation of~\\(\\mathbb{R}^d\\)\n  becomes a nondegenerate module over \\(L^1(\\mathbb{R}^d)\\).\n  Thus \\((f_n*S)_{n\\in\\mathbb{N}}\\)\n  converges in norm to~\\(S\\)\n  if~\\(S\\)\n  is continuous.  Conversely, operators of the form \\(f*S\\)\n  are continuous because the action of~\\(\\mathbb{R}^d\\)\n  on \\(L^1(\\mathbb{R}^d)\\)\n  is continuous.  Since the set of continuous operators is closed in\n  the norm topology, \\(S\\)\n  is continuous if \\((f_n*S)_{n\\in\\mathbb{N}}\\) converges in norm to~\\(S\\).\n\n  There is an approximate unit~\\((f_n)_{n\\in\\mathbb{N}}\\)\n  for~\\(L^1(\\mathbb{R}^d)\\)\n  such that the Fourier transform of each~\\(f_n\\)\n  has compact support.  For instance, we may use the Fej\u00e9r kernel\n  \\[\n  \\Lambda(x_1,\\dotsc,x_n) \\mathrel{\\vcentcolon=} \\prod_{j=1}^d \\frac{\\sin^2(\\pi x_j)}{\\pi^2 x_j^2},\n  \\]\n  which has Fourier transform \\(\\prod_{j=1}^d (1-\\abs{x_j})_+\\),\n  and rescale it to produce an approximate unit for \\(L^1(\\mathbb{R}^d)\\).\n  We claim that \\(f_n*S\\)\n  is controlled for each \\(n\\in\\mathbb{N}\\).\n  More precisely, assume that \\(\\widehat{f_n}\\)\n  is supported in the ball of radius~\\(R\\)\n  in~\\(\\mathbb{R}^d\\).\n  We claim that~\\(f_n*S\\)\n  is \\(R\\)\\nobreakdash-controlled.\n\n  This is easy to prove if~\\(X\\)\n  is discrete.  Then we may describe operators on~\\(\\Hils\\)\n  using matrix coefficients \\(S_{x,y}\\in\\mathbb B(\\Hils_y,\\Hils_x)\\)\n  for \\(x,y\\in X\\).\n  A direct computation shows that the matrix coefficients of \\(f_n*S\\)\n  are \\(\\widehat{f_n}(x-y)\\cdot S_{x,y}\\).\n  This vanishes for \\(\\norm{x-y}>R\\).\n  So~\\(f_n*S\\)\n  is \\(R\\)\\nobreakdash-controlled\n  as asserted.  The following argument extends this result to the case\n  where~\\(X\\) is not discrete, such as \\(X=\\mathbb{R}^d\\).\n\n  Let \\(U,V\\subseteq \\mathbb{R}^d\\)\n  be two relatively compact, open subsets of distance at least~\\(R\\)\n  and let \\(g,h\\in\\Cont^\\infty(\\mathbb{R}^d)\\)\n  be smooth functions supported in \\(U\\)\n  and~\\(V\\), respectively.  Then\n  \\begin{equation}\n    \\label{eq:Roe_through_continuity_integral}\n    \\int_{\\mathbb{R}^d} g(y) \\Euler^{\\ima y\\cdot t}\\cdot h(x) \\Euler^{-\\ima x\\cdot t} f_n(t) \\,\\diff t\n   \n    = g(y) h(x) \\widehat{f_n}(y-x)\n    = 0\n  \\end{equation}\n  for all \\(x,y\\in \\mathbb{R}^d\\).  We restrict \\(g,h\\) to~\\(X\\) and compute\n  \\begin{align*}\n    \\varrho(g) (f_n* S) \\varrho(h)\n    &\\mathrel{\\vcentcolon=} \\int_{\\mathbb{R}^d} \\varrho(g) \\bar\\varrho(\\Euler_t) S\n    \\bar\\varrho(\\Euler_{-t}) \\varrho(h) f_n(t) \\,\\diff t\n    \\\\&= \\int_{\\mathbb{R}^d} \\varrho(g\\cdot \\Euler_t) S\n    \\varrho(h\\cdot \\Euler_{-t}) f_n(t) \\,\\diff t.\n  \\end{align*}\n  Let \\(\\Cont_0^\\infty(U\\times V)\\)\n  denote the Fr\u00e9chet space of smooth function on~\\(\\mathbb{R}^{2d}\\)\n  supported in \\(U\\times V\\).\n  We may identify this with the complete projective tensor product of\n  \\(\\Cont^\\infty_0(U)\\)\n  and \\(\\Cont^\\infty_0(V)\\).\n  Hence there is a continuous linear map\n  \\begin{equation}\n    \\label{eq:function_on_both_side}\n    \\Cont^\\infty_0(U\\times V)\\to \\mathbb B(\\Hils),\\qquad\n    g\\otimes h\\mapsto \\varrho(g) S \\varrho(h).\n  \\end{equation}\n  The integral in~\\eqref{eq:Roe_through_continuity_integral} converges\n  to~\\(0\\)\n  in the Fr\u00e9chet topology of \\(\\Cont^\\infty_0(U\\times V)\\).\n  Hence~\\eqref{eq:Roe_through_continuity_integral} implies\n  \\(\\varrho(g) (f_n*S) \\varrho(h)=0\\).\n\n  The continuous map~\\eqref{eq:function_on_both_side} still exists if\n  \\(U\\)\n  and~\\(V\\)\n  are not of distance~\\(R\\).\n  If~\\(S\\)\n  is locally compact, then \\(\\varrho(g) S \\varrho(h)\\)\n  is a compact operator on~\\(\\Hils\\)\n  for all \\(g\\in\\Cont^\\infty_0(U)\\),\n  \\(h\\in\\Cont^\\infty_0(V)\\).\n  This remains so for all operators in the image\n  of~\\eqref{eq:function_on_both_side} by continuity.  Therefore,\n  \\(\\varrho(g) (f_n*S)\\varrho(h)\\)\n  is compact for all \\(g,h\\)\n  as above.  Choosing~\\(U\\)\n  large enough, we may take~\\(g\\)\n  to be constant equal to~\\(1\\)\n  on the \\(R\\)\\nobreakdash-neighbourhood\n  of~\\(V\\).\n  Then \\(\\varrho(g) (f_n*S)\\varrho(h) =(f_n*S)\\varrho(h)\\)\n  because~\\(f_n*S\\)\n  is \\(R\\)\\nobreakdash-controlled.\n  So operators of the form \\((f_n*S)\\varrho(h)\\)\n  with smooth, compactly supported~\\(h\\)\n  are compact.  Since any continuous, compactly supported function is\n  dominated by a smooth, compactly supported function, we get the same\n  for all \\(h\\in\\Contc(X)\\).\n  A similar argument shows that \\(\\varrho(g)(f_n*S)\\)\n  is compact for all \\(g\\in\\Contc(X)\\).\n  Hence the operators \\(f_n*S\\)\n  are locally compact, controlled operators if~\\(S\\)\n  is locally compact.\n\\end{proof}\n\nProperty~A is equivalent to the ``Operator Norm Localization\nProperty'' for metric spaces with bounded geometry,\nsee~\\cite{Sako:A_operator_localization}.  Roughly speaking, this\nproperty says that the operator norm of a controlled operator may be\ncomputed using vectors in the Hilbert space with bounded support.  The\nsupport of a vector \\(\\xi\\in\\Hils\\)\nis the set of all \\(x\\in X\\)\nsuch that \\(f\\cdot \\xi\\neq0\\)\nfor all \\(f\\in\\Cont_0(X)\\)\nwith \\(f(x)\\neq0\\).\nWe formulate this property for subspaces of~\\(\\mathbb{R}^d\\):\n\n\\begin{theorem}\n  \\label{the:ONL}\n  Let \\(X\\subseteq \\mathbb{R}^d\\)\n  and let \\(\\varrho\\colon \\Cont_0(X)\\to\\mathbb B(\\Hils)\\)\n  be a representation.  Pick scalars \\(R>0\\)\n  and \\(c\\in(0,1)\\).\n  Then there is a scalar \\(S>0\\)\n  such that for any \\(R\\)\\nobreakdash-controlled\n  operator \\(T\\in\\mathbb B(\\Hils)\\),\n  there is \\(\\xi\\in\\Hils\\)\n  with \\(\\norm{\\xi}=1\\)\n  such that the support of~\\(\\xi\\)\n  has diameter at most~\\(S\\)\n  and \\(\\norm{T(\\xi)} \\ge \\norm{T} \\ge c\\cdot \\norm{T(\\xi)}\\).\n\\end{theorem}\n\n\\begin{proof}\n  The statement of the theorem is that the space~\\(X\\) has the\n  ``Operator Norm Localisation Property'' defined\n  in~\\cite{Chen-Tessera-Wang:Operator_localization}.  This property\n  is invariant under coarse equivalence and passes to subspaces by\n  \\cite{Chen-Tessera-Wang:Operator_localization}*{Propositions 2.5\n    and~2.6}.\n  \\cite{Chen-Tessera-Wang:Operator_localization}*{Theorem~3.11 and\n    Proposition~4.1} show that solvable Lie groups such as~\\(\\mathbb{R}^d\\)\n  have this property, and hence also all subspaces of~\\(\\mathbb{R}^d\\).\n\\end{proof}\n\n\n\\subsection{Dense subalgebras with isomorphic K-theory}\n\\label{sec:dense_same_K}\n\nLet~\\(A\\)\nbe a \\(\\Cst\\)\\nobreakdash-algebra\nwith a continuous \\(\\mathbb{R}^d\\)\\nobreakdash-action\n\\(\\alpha\\colon \\mathbb{R}^d\\to\\Aut(A)\\).\nThe action defines several canonical $^*$\\nobreakdash-{}subalgebras of~\\(A\\)\nwith the same \\(\\K\\)\\nobreakdash-theory.\nThe $^*$\\nobreakdash-{}subalgebra of \\emph{smooth elements} is\n\\[\nA^\\infty \\mathrel{\\vcentcolon=} \\setgiven{a\\in A}\n{t\\mapsto \\alpha_t(a) \\text{ is a smooth function } \\mathbb{R}^d\\to A}.\n\\]\nThis Fr\u00e9chet $^*$\\nobreakdash-{}subalgebra is closed under holomorphic functional\ncalculus and also under smooth functional calculus for normal\nelements, see~\\cite{Blackadar-Cuntz:Differential}.\n\nLet \\(F \\subseteq \\mathbb{R}^d\\)\nbe a compact convex subset with non-empty interior and\ncontaining~\\(0\\).\nLet \\(\\mathcal{O}(A,\\alpha,F) \\subseteq A\\)\nbe the set of all \\(a\\in A\\)\nfor which the function \\(\\mathbb{R}^d\\ni t\\mapsto \\alpha_t(a)\\)\nextends to a continuous function on \\(\\mathbb{R}^d + \\ima F\\)\nthat is holomorphic on the interior of \\(\\mathbb{R}^d + \\ima F\\).\nThis is a dense Banach subalgebra in~\\(A\\),\nand the inclusion \\(\\mathcal{O}(A,\\alpha,F) \\hookrightarrow A\\)\ninduces an isomorphism on topological \\(\\K\\)\\nobreakdash-theory\nby \\cite{Bost:Principe_Oka}*{Th\u00e9or\u00e8me~2.2.1}.  Let\n\\(\\mathcal{O}^\\infty(A,\\alpha,F) \\subseteq A\\)\nbe the set of those \\(a\\in A\\)\nfor which the function \\(\\mathbb{R}^d\\ni t\\mapsto \\alpha_t(a)\\)\nextends to a smooth function on \\(\\mathbb{R}^d + \\ima F\\)\nthat is holomorphic on the interior of \\(\\mathbb{R}^d + \\ima F\\).\nThe inclusion \\(\\mathcal{O}^\\infty(A,\\alpha,F) \\hookrightarrow A\\)\ninduces an isomorphism on topological \\(\\K\\)\\nobreakdash-theory\nas well.  If \\(F_1\\subseteq F_2\\),\nthen \\(\\mathcal{O}(A,\\alpha,F_2) \\hookrightarrow \\mathcal{O}(A,\\alpha,F_1)\\).\nThere are two important limiting cases of the subalgebras\n\\(\\mathcal{O}(A,\\alpha,F)\\).\n\nFirst, let~\\(F\\)\nrun through a neighbourhood basis of~\\(0\\)\nin~\\(\\mathbb{R}^d\\).\nThen the dense Banach subalgebras \\(\\mathcal{O}(A,\\alpha,F)\\)\nform an inductive system, whose colimit is the dense $^*$\\nobreakdash-{}subalgebra\n\\(A^\\omega \\subseteq A\\)\nof all \\emph{real-analytic elements} of~\\(A\\),\nthat is, those \\(a\\in A\\)\nwith the property that each \\(t\\in\\mathbb{R}^d\\)\nhas a neighbourhood on which \\(s\\mapsto \\alpha_s(a)\\)\nis given by a convergent power series with coefficients in~\\(A\\).\nThe subalgebra~\\(A^\\omega\\)\nis still closed under holomorphic functional calculus by\n\\cite{Meyer:HLHA}*{Proposition~3.46}.  This gives an easier\nexplanation than Bost's Oka principle why~\\(A^\\omega\\)\nhas the same topological \\(\\K\\)\\nobreakdash-theory as~\\(A\\).\n\nSecondly, let~\\(F\\)\nrun through an increasing sequence whose union is~\\(\\mathbb{R}^d\\).\nThen the dense Banach subalgebras \\(\\mathcal{O}(A,\\alpha,F)\\)\nform a projective system, whose limit is the dense $^*$\\nobreakdash-{}subalgebra\n\\(\\mathcal{O}(A,\\alpha)\\)\nof all \\emph{holomorphic elements} \\(a\\in A\\),\nthat is, those elements for which the map\n\\(\\mathbb{R}^d \\ni t\\mapsto \\alpha_t(a)\\)\nextends to a holomorphic function on~\\(\\mathbb{C}^d\\).\nThis is a locally multiplicatively convex Fr\u00e9chet algebra.\nPhillips~\\cite{Phillips:K_Frechet} has extended\ntopological \\(\\K\\)\\nobreakdash-theory\nto such algebras.  The Milnor \\(\\varprojlim^1\\)-sequence\nin \\cite{Phillips:K_Frechet}*{Theorem 6.5} shows that the\ninclusion \\(\\mathcal{O}(A,\\alpha)\\hookrightarrow A\\)\ninduces an isomorphism in topological \\(\\K\\)\\nobreakdash-theory.\n\nWe apply all this to the Roe \\(\\Cst\\)\\nobreakdash-algebra\nof~\\(\\mathbb{Z}^d\\)\nand the continuous \\(\\mathbb{R}^d\\)\\nobreakdash-action~\\(\\sigma\\)\ndefined in Section~\\ref{sec:approximation_continuity}.  Here this\naction descends to the torus~\\(\\mathbb{T}^d\\),\nwhich simplifies the study of the dense subalgebras above.  We\ndescribe the dense subalgebras of smooth, real-analytic and\nholomorphic elements in \\(\\Cst_\\Roe(\\mathbb{Z}^d)\\).\nAll these have the same topological \\(\\K\\)\\nobreakdash-theory.\nLet \\(\\varrho\\colon \\Cont_0(\\mathbb{Z}^d)\\to\\mathbb B(\\Hils)\\)\nbe a representation on a separable Hilbert space, not necessarily\nample.  Let~\\(\\Hils_x\\)\nfor \\(x\\in X\\)\nbe the fibres of~\\(\\Hils\\)\nwith respect to~\\(\\varrho\\).\nDescribe operators on~\\(\\Hils\\)\nby block matrices \\((T_{x,y})_{x,y\\in\\mathbb{Z}^d}\\)\nwith \\(T_{x,y}\\in\\mathbb B(\\Hils_y,\\Hils_x)\\)\nfor all \\(x,y\\in\\mathbb{Z}^d\\).  Then\n\\[\n\\sigma_t(T_{x,y}) = (\\exp(\\ima t\\cdot (x-y)) T_{x,y})_{x,y\\in\\mathbb{Z}^d}.\n\\]\n\n\\begin{proposition}\n  \\label{pro:smooth_analytic_in_Roe_Zd}\n  A block matrix \\((T_{x,y})_{x,y\\in\\mathbb{Z}^d}\\)\n  as above gives a smooth element for the\n  \\(\\mathbb{R}^d\\)\\nobreakdash-action~\\(\\sigma\\) on~\\(\\Cst_\\Roe(\\mathbb{Z}^d)\\) if and only if the function\n  \\[\n  \\mathbb{Z}^d \\ni k \\mapsto \\sup_{n\\in\\mathbb{Z}^d} \\{ \\norm{T_{n,n+k}} \\}\n  \\]\n  has rapid decay, that is, for each \\(a>0\\)\n  there is a constant~\\(C_a>0\\)\n  such that \\(\\norm{T_{n,n+k}} \\le C_a (1+ \\norm{k})^{-a}\\)\n  for all \\(n,k\\in\\mathbb{Z}^d\\).\n  It gives a real-analytic element for~\\(\\sigma\\)\n  if and only if there are \\(a>0\\) and \\(C_a>0\\) such that\n  \\[\n  \\norm{T_{n,n+k}} \\le C_a \\cdot \\exp(-a \\norm{k})\n  \\]\n  for all \\(n,k\\in\\mathbb{Z}^d\\).\n  It gives a holomorphic element for~\\(\\sigma\\)\n  if and only if for each \\(a>0\\) there is \\(C_a>0\\) such that\n  \\[\n  \\norm{T_{n,n+k}} \\le C_a \\cdot \\exp(-a \\norm{k}).\n  \\]\n\\end{proposition}\n\n\\begin{proof}\n  The \\(j\\)th\n  generator of the \\(\\mathbb{R}^d\\)\\nobreakdash-action~\\(\\sigma\\)\n  maps a block matrix~\\((T_{x,y})_{x,y\\in\\mathbb{Z}^d}\\)\n  to\n  \\[\n  \\lim_{t\\to0} \\frac{1}{t} (\\sigma_{t e_j}(T_{x,y}) - (T_{x,y}))_{x,y\\in\\mathbb{Z}^d}\n  = ((x_j-y_j)T_{x,y})_{x,y\\in\\mathbb{Z}^d}.\n  \\]\n  Hence polynomials in these generators multiply the\n  entries~\\(T_{x,y}\\)\n  with polynomials in \\(x-y\\in\\mathbb{Z}^d\\).\n  So~\\((T_{x,y})_{x,y\\in\\mathbb{Z}^d}\\)\n  belongs to a smooth element of~\\(\\Cst_\\Roe(\\mathbb{Z}^d)\\)\n  if and only if \\((p(x-y)\\cdot T_{x,y})_{x,y\\in\\mathbb{Z}^d}\\)\n  belongs to a bounded operator for each polynomial~\\(p\\)\n  in \\(d\\)~variables.\n  It suffices to consider the polynomials \\(1+\\norm{x-y}_2^{2 b}\\)\n  for \\(b\\in\\mathbb{N}\\).\n  Since the operator norm for diagonal block matrices is the supremum\n  of the operator norms of the entries, we see that the boundedness of\n  \\(((1+ \\norm{x-y}_2^{2 b})\\cdot T_{x,y})_{x,y\\in\\mathbb{Z}^d}\\)\n  for all \\(b\\in\\mathbb{N}\\)\n  is equivalent to the boundedness of\n  \\(\\sup_{k,n\\in\\mathbb{Z}^d} \\norm{T_{n,n+k}} (1+ \\norm{k}_2^{2 b})\\)\n  for all \\(b\\in\\mathbb{N}\\).\n  This proves the claim about the smooth elements.  The analytic\n  extension of~\\(\\sigma\\)\n  to \\(\\ima z\\in \\mathbb{C}^d\\)\n  must map~\\((T_{x,y})_{x,y\\in\\mathbb{Z}^d}\\)\n  to \\(((\\exp(z\\cdot (x-y))T_{x,y})_{x,y\\in\\mathbb{Z}^d}\\).\n  Thus~\\((T_{x,y})_{x,y\\in\\mathbb{Z}^d}\\)\n  describes an element of\n  \\(\\mathcal{O}^\\infty(\\Cst_\\Roe(\\mathbb{Z}^d),\\sigma,F)\\) if and only if\n  \\begin{equation}\n    \\label{eq:O_F_estimate}\n    \\sup_{k,n\\in\\mathbb{Z}^d} \\norm{T_{n,n+k}} (1+ \\norm{k}_2^{2 b}) \\exp(z\\cdot k)<\\infty    \n  \\end{equation}\n  for all \\(z\\in F\\),\n  \\(b\\in\\mathbb{N}\\).\n  When we let \\(F\\searrow\\{0\\}\\)\n  or \\(F\\nearrow \\mathbb{C}^d\\),\n  we may leave out the polynomial factors because they are dominated\n  by \\(\\exp(z\\cdot k)\\).\n  This proves the claims about the\n  real-analytic elements and holomorphic elements.\n\\end{proof}\n\nEstimates of the form\n\\(\\norm{T_{x,y}} \\le C_a \\exp(-a\\cdot \\norm{x-y})\\)\nfor some \\(a>0\\),\n\\(C_a>0\\)\nplay an important role in the study of Anderson localisation; see, for\ninstance,\n\\cite{Aizenman-Molchanov:Localization_elementary}*{Equation~(2.3)}.\n\n\n\\subsection{Approximate unit of projections}\n\\label{sec:apprid_projection}\n\nUnlike the uniform Roe \\(\\Cst\\)\\nobreakdash-algebra,\nthe Roe \\(\\Cst\\)\\nobreakdash-algebra\nof a proper metric space is never unital.  Instead, it has an\napproximate unit of projections:\n\n\\begin{proposition}\n  \\label{pro:Roe_apprid}\n  Let~\\(X\\)\n  be a proper metric space and let\n  \\(\\varrho\\colon \\Cont_0(X)\\to\\mathbb B(\\Hils)\\)\n  be a representation.  The Roe \\(\\Cst\\)\\nobreakdash-algebra\n  \\(\\Cst_\\Roe(X,\\varrho)\\) has an approximate unit of projections.\n\\end{proposition}\n\n\\begin{proof}\n  Any proper metric space contains a coarsely dense, discrete\n  subspace.  By Theorem~\\ref{the:coarsely_dense}, we may assume\n  that~\\(X\\)\n  itself is discrete.  By Theorem~\\ref{the:Roe_ample_unique}, we may\n  further assume that the Roe \\(\\Cst\\)\\nobreakdash-algebra\n  is built using the obvious representation of~\\(\\Cont_0(X)\\)\n  on \\(\\ell^2(X,\\ell^2(\\mathbb{N}))\\).\n  Then the Roe \\(\\Cst\\)\\nobreakdash-algebra\n  contains \\(\\ell^\\infty(X, \\Cont_0(\\mathbb{N}))\\)\n  as multiplication operators.  Any function \\(h\\colon X\\to\\mathbb{N}\\)\n  defines a projection in \\(\\ell^\\infty(X, \\Cont_0(\\mathbb{N}))\\),\n  namely, the characteristic function of\n  \\(\\setgiven{(x,n)\\in X\\times\\mathbb{N}}{n<h(x)}\\).\n  These projections form an approximate unit in the Roe\n  \\(\\Cst\\)\\nobreakdash-algebra.\n\\end{proof}\n\nLet \\((p_\\alpha)_{\\alpha \\in S}\\)\nbe an approximate unit of projections in \\(\\Cst_\\Roe(X,\\varrho)\\).\nThen \\(\\Cst_\\Roe(X,\\varrho)\\)\nis isomorphic to the inductive limit\n\\[\n\\Cst_\\Roe(X,\\varrho) = \\varinjlim p_\\alpha \\Cst_\\Roe(X,\\varrho) p_\\alpha.\n\\]\nSince \\(\\K\\)\\nobreakdash-theory commutes with inductive limits, we get\n\\begin{equation}\n  \\label{eq:K_Roe_indlim}\n  \\K_*(\\Cst_\\Roe(X,\\varrho)) \\cong\n  \\varinjlim \\K_*(p_\\alpha \\Cst_\\Roe(X,\\varrho) p_\\alpha).\n\\end{equation}\nEach of the corners \\(p_\\alpha \\Cst_\\Roe(X,\\varrho) p_\\alpha\\)\nis unital.  This simplifies the definition of the groups\n\\(\\K_*(p_\\alpha \\Cst_\\Roe(X,\\varrho) p_\\alpha)\\)\nfor fixed~\\(\\alpha\\).\n\n\\begin{corollary}\n  \\label{cor:K0_Roe_projections}\n  Any class in \\(\\K_0(\\Cst_\\Roe(X))\\)\n  is represented by a formal differences of projections in\n  \\(\\Cst_\\Roe(X)\\).\n  Two such differences \\([p_+]-[p_-]\\)\n  and \\([q_+]-[q_-]\\)\n  represent the same class in \\(\\K_0(\\Cst_\\Roe(X))\\)\n  if and only if there is a projection~\\(r\\)\n  in \\(\\Cst_\\Roe(X)\\)\n  such that the projections \\(p_+ \\oplus q_- \\oplus r\\)\n  and \\(p_- \\oplus q_+ \\oplus r\\)\n  in \\(\\mathbb M_3(\\Cst_\\Roe(X))\\) are Murray--von Neumann equivalent.\n\\end{corollary}\n\n\\begin{proof}\n  Any class in \\(\\K_0(\\Cst_\\Roe(X))\\)\n  is the image of a class in \\(\\K_0(p_\\alpha \\Cst_\\Roe(X) p_\\alpha)\\)\n  for some~\\(\\alpha\\)\n  by~\\eqref{eq:K_Roe_indlim}.  Since\n  \\(p_\\alpha \\Cst_\\Roe(X) p_\\alpha\\)\n  is unital, it is represented by a formal difference of two\n  projections in \\(\\mathbb M_k(p_\\alpha \\Cst_\\Roe(X) p_\\alpha)\\)\n  for some \\(k\\in\\mathbb{N}_{\\ge1}\\).\n  These projections also belong to \\(\\mathbb M_k(\\Cst_\\Roe(X))\\),\n  which is isomorphic to~\\(\\Cst_\\Roe(X)\\)\n  by Corollary~\\ref{cor:Roe_matrix-stable}.  Even more, the\n  isomorphism \\(\\mathbb M_k(\\Cst_\\Roe(X))\\cong\\Cst_\\Roe(X)\\)\n  is by conjugation with a unitary \\(k\\times1\\)-matrix\n  over the multiplier algebra of~\\(\\Cst_\\Roe(X)\\).\n  Hence any projection in \\(\\mathbb M_k(\\Cst_\\Roe(X))\\)\n  is Murray--von Neumann equivalent to one in~\\(\\Cst_\\Roe(X)\\).\n  Thus any class in \\(\\K_0(\\Cst_\\Roe(X))\\)\n  is represented by a formal difference of projections in\n  \\(\\Cst_\\Roe(X)\\).\n  Using~\\eqref{eq:K_Roe_indlim} and the definition of~\\(\\K_0\\)\n  for unital algebras once again, we see that\n  \\([p_+]-[p_-]=[q_+]-[q_-]\\)\n  holds if and only if there is a projection~\\(r\\)\n  in \\(\\mathbb M_k(p_\\alpha \\Cst_\\Roe(X) p_\\alpha)\\)\n  for some \\(k,\\alpha\\)\n  such that \\(p_+ \\oplus q_- \\oplus r\\)\n  and \\(p_- \\oplus q_+ \\oplus r\\)\n  are Murray--von Neumann equivalent in\n  \\(\\mathbb M_{k+2}(p_\\alpha \\Cst_\\Roe(X) p_\\alpha)\\).\n  Here we may replace~\\(r\\)\n  by an equivalent projection and so reduce to \\(k=1\\).\n  And since any projection in~\\(\\Cst_\\Roe(X)\\)\n  is equivalent to one in \\(p_\\alpha \\Cst_\\Roe(X) p_\\alpha\\)\n  for some~\\(\\alpha\\),\n  we may allow~\\(r\\) to be any projection in~\\(\\Cst_\\Roe(X)\\).\n\\end{proof}\n\n\n\\section{Coarse Mayer--Vietoris sequence}\n\\label{sec:coarse_MV}\n\nWe now recall how to compute the \\(\\K\\)\\nobreakdash-theory\nof the Roe \\(\\Cst\\)\\nobreakdash-algebra\nof \\(\\mathbb{R}^d\\)\nor~\\(\\mathbb{Z}^d\\)\nusing the coarse Mayer--Vietoris sequence.  This method is due to\nHigson--Roe--Yu~\\cite{Higson-Roe-Yu:Coarse_Mayer-Vietoris} and was\nalso explained by Kubota in\n\\cite{Kubota:Controlled_bulk-edge}*{Section 2.3.3}.  The Roe\n\\(\\Cst\\)\\nobreakdash-algebra\nis defined also for the half-space\n\\(\\mathbb{Z}^{d-1}\\times\\mathbb{N} \\subseteq \\mathbb{Z}^d\\),\nand its \\(\\K\\)\\nobreakdash-theory\nvanishes.  We identify the Roe \\(\\Cst\\)\\nobreakdash-algebra\nof a subspace with a corner in the larger Roe \\(\\Cst\\)\\nobreakdash-algebra\nand describe the ideal generated by this corner as a relative Roe\n\\(\\Cst\\)\\nobreakdash-algebra.\nThen we recall the Mayer--Vietoris sequence for two ideals in a\n\\(\\Cst\\)\\nobreakdash-algebra.\nWe give an elegant proof due to Wodzicki, which reduces its exactness\nto ordinary long exact sequences for \\(\\Cst\\)\\nobreakdash-algebra\nextensions.  This also describes the boundary map in the\nMayer--Vietoris sequence through the boundary map for a\n\\(\\Cst\\)\\nobreakdash-algebra\nextension.  Hence the many tools and formulas for the boundary maps\nfor extensions of \\(\\Cst\\)\\nobreakdash-algebras\nalso apply to the boundary map in the Mayer--Vietoris sequence.\n\n\n\\subsection{Subspaces and corners}\n\\label{sec:subspace_corner}\n\nTheorem~\\ref{the:coarsely_dense} shows that \\(\\Cst_\\Roe(Y)\\)\nand \\(\\Cst_\\Roe(X)\\)\nare isomorphic if \\(Y\\subseteq X\\)\nis coarsely dense.  Now let \\(Y\\subseteq X\\)\nbe an arbitrary closed subset, still with the restriction of the\nmetric from~\\(X\\).\nWe are going to relate the Roe \\(\\Cst\\)\\nobreakdash-algebras\nof \\(Y\\)\nand~\\(X\\).\nFirst, we are going to show that \\(\\Cst_\\Roe(Y)\\)\nis isomorphic to a corner in \\(\\Cst_\\Roe(X)\\);\nthis is an easy consequence of Theorem~\\ref{the:Roe_ample_unique}.\nFor the coarse Mayer--Vietoris sequence, we also need to know that the\nideal generated by this corner is the relative Roe\n\\(\\Cst\\)\\nobreakdash-algebra, defined as follows:\n\n\\begin{definition}\n  \\label{def:Roe_relative}\n  Let \\(Y\\subseteq X\\)\n  and let \\(\\varrho\\colon \\Cont_0(X)\\to\\mathbb B(\\Hils)\\)\n  be a representation.  An operator \\(T\\in\\mathbb B(\\Hils)\\)\n  is \\emph{supported near}~\\(Y\\) if there is \\(R>0\\) with\n  \\[\n  \\supp(T) \\subseteq \\setgiven{(x,y)\\in X \\times X}\n  {d(x,Y)<R\\text{ and }d(y,Y)<R}.\n  \\]\n  The \\emph{relative Roe algebra} \\(\\Cst_\\Roe(Y\\subseteq X,\\varrho)\\)\n  is the \\(\\Cst\\)\\nobreakdash-subalgebra\n  of~\\(\\mathbb B(\\Hils)\\)\n  generated by the controlled locally compact operators supported\n  near~\\(Y\\).\n\\end{definition}\n\nIf \\(T\\)\nis supported near~\\(Y\\)\nand~\\(S\\)\nis controlled, then \\(S T\\)\nand \\(T S\\)\nare again supported near~\\(Y\\),\nand~\\(T^*\\)\nis also supported near~\\(Y\\).\nThus the controlled operators supported near~\\(Y\\)\nform a (two-sided) $^*$\\nobreakdash-{}ideal in the $^*$\\nobreakdash-{}algebra of controlled\noperators.  Hence \\(\\Cst_\\Roe(Y\\subseteq X,\\varrho)\\)\nis the closure of this $^*$\\nobreakdash-{}algebra in \\(\\Cst_\\Roe(X,\\varrho)\\),\nand it is a closed two-sided $^*$\\nobreakdash-{}ideal in \\(\\Cst_\\Roe(X,\\varrho)\\).\n\nA \\emph{corner} in a \\(\\Cst\\)\\nobreakdash-algebra~\\(A\\)\nis a \\(\\Cst\\)\\nobreakdash-subalgebra\nof the form \\(P A P\\)\nfor a projection~\\(P\\)\nin the multiplier algebra of~\\(A\\).\nAny corner is a hereditary subalgebra.  It is canonically Morita\nequivalent to the ideal generated by~\\(P\\),\nwhich we denote by~\\(A P A\\)\nbecause it is the closed linear span of~\\(a_1 P a_2\\)\nfor \\(a_1,a_2\\in A\\).\nThe imprimitivity bimodule is~\\(A P\\)\nwith the obvious full Hilbert \\(A P A, P A P\\)-bimodule\nstructure, obtained by restricting the usual Hilbert \\(A,A\\)-bimodule\nstructure on~\\(A\\).\n\n\\begin{theorem}\n  \\label{the:subset_full_corner}\n  Let \\(Y\\subseteq X\\)\n  be a closed subspace with the subspace metric.  Then\n  \\(\\Cst_\\Roe(X)\\)\n  is isomorphic to a corner in \\(\\Cst_\\Roe(Y)\\).\n  The ideal generated by it is the relative Roe \\(\\Cst\\)\\nobreakdash-algebra\n  \\(\\Cst_\\Roe(Y\\subseteq X)\\).\n  This holds both in the complex and real case.\n\\end{theorem}\n\n\\begin{proof}\n  We prove the real case.  The complex case is analogous.  We choose\n  ample representations \\(\\varrho_X\\)\n  and~\\(\\varrho_Y\\)\n  of \\(\\Cont_0(X)\\)\n  and \\(\\Cont_0(Y)\\)\n  on separable real Hilbert spaces \\(\\Hils_X\\)\n  and~\\(\\Hils_Y\\),\n  respectively.  Using the restriction map\n  \\(p\\colon \\Cont_0(X)\\to\\Cont_0(Y)\\),\n  we build another ample representation\n  \\(\\varrho'\\mathrel{\\vcentcolon=} \\varrho_Y\\circ p\\oplus \\varrho_X\\)\n  of~\\(\\Cont_0(X)\\)\n  on the separable real Hilbert space \\(\\Hils_Y\\oplus\\Hils_X\\).\n  We use~\\(\\varrho'\\)\n  to build \\(\\Cst_\\Roe(X)\\),\n  which is allowed by Theorem~\\ref{the:Roe_ample_unique}.  The\n  projection~\\(P\\)\n  onto the summand \\(\\Hils_Y\\)\n  is a multiplier of \\(\\Cst_\\Roe(X)\\)\n  because it is \\(0\\)\\nobreakdash-controlled.\n  (It is only a multiplier because it is not locally compact.)  Since\n  the map \\(x\\mapsto P x P\\)\n  on \\(\\mathbb B(\\Hils_Y\\oplus\\Hils_X)\\)\n  is bounded, the corner \\(P \\Cst_\\Roe(X) P\\)\n  is the closure of the space of all controlled, locally compact\n  operators on~\\(\\Hils_Y\\).\n  Here we should use the representation \\(\\varrho_Y\\circ p\\)\n  of~\\(\\Cont_0(X)\\)\n  to define controlled operators and local compactness.  But\n  since~\\(Y\\)\n  carries the subspace metric from~\\(X\\)\n  and~\\(p\\)\n  is surjective, the representations \\(\\varrho_Y\\circ p\\)\n  of~\\(\\Cont_0(X)\\)\n  and~\\(\\varrho_Y\\)\n  of~\\(\\Cont_0(Y)\\)\n  define the same controlled or locally compact operators.  Hence\n  \\(P \\Cst_\\Roe(X,\\varrho') P = \\Cst_\\Roe(Y,\\varrho_Y)\\).\n  So~\\(\\Cst_\\Roe(Y)\\)\n  is isomorphic to a corner in~\\(\\Cst_\\Roe(X)\\).\n\n  The corner \\(P \\Cst_\\Roe(X) P\\)\n  is contained in \\(\\Cst_\\Roe(Y\\subseteq X)\\).\n  Since the latter is an ideal, the ideal\n  \\(\\Cst_\\Roe(X) P \\Cst_\\Roe(X)\\)\n  is also contained in \\(\\Cst_\\Roe(Y\\subseteq X)\\).\n  For the converse inclusion, we must show that any controlled,\n  locally compact operator~\\(T\\)\n  that is supported near~\\(Y\\)\n  belongs to \\(\\Cst_\\Roe(X) P \\Cst_\\Roe(X)\\).\n  Let \\(R>0\\)\n  and let~\\(T\\)\n  be supported in the \\(R\\)\\nobreakdash-neighbourhood\n  of~\\(Y\\).\n  This \\(R\\)\\nobreakdash-neighbourhood\n  is a closed subspace~\\(Y_R\\)\n  of~\\(X\\),\n  and \\(Y\\subseteq Y_R\\)\n  is coarsely dense by construction.  The restriction of~\\(\\varrho'\\)\n  to the Hilbert subspace\n  \\(\\Hils_{Y,R} \\mathrel{\\vcentcolon=} \\varrho'(1_{Y_R})(\\Hils_Y\\oplus \\Hils_X)\\)\n  is an ample representation of~\\(\\Cont_0(Y_R)\\).\n  Hence it defines \\(\\Cst_\\Roe(Y_R)\\)\n  by Theorem~\\ref{the:Roe_ample_unique}.  This \\(\\Cst\\)\\nobreakdash-algebra\n  is simply the corner in \\(\\Cst_\\Roe(X)\\)\n  generated by the projection onto~\\(\\Hils_{Y,R}\\).\n  Theorem~\\ref{the:coarsely_dense} gives a unitary\n  \\(U\\colon \\Hils_Y \\xrightarrow\\sim \\Hils_{Y,R}\\)\n  such that \\(U \\Cst_\\Roe(Y) U^* = \\Cst_\\Roe(Y_R)\\).\n  The unitary~\\(U\\)\n  is built in the proof of Theorem~\\ref{the:Roe_ample_unique}, and the\n  construction there shows that it is controlled as an operator on\n  \\(\\Hils_Y\\oplus \\Hils_X\\).\n  So it is a multiplier of~\\(\\Cst_\\Roe(X)\\),\n  where it is no longer unitary but a partial isometry.  The operator\n  \\(U^* T U\\)\n  belongs to \\(\\Cst_\\Roe(Y)\\)\n  because \\(T\\in\\Cst_\\Roe(Y_R)\\).\n  Since multipliers of \\(\\Cst_\\Roe(X)\\)\n  are also multipliers of any ideal in~\\(\\Cst_\\Roe(X)\\),\n  the operator \\(T = U (U^* T U) U^*\\)\n  belongs to the ideal in~\\(\\Cst_\\Roe(X)\\)\n  generated by \\(\\Cst_\\Roe(Y) = P \\Cst_\\Roe(X) P\\).\n  Thus \\(\\Cst_\\Roe(Y_R) \\subseteq \\Cst_\\Roe(X) P \\Cst_\\Roe(X)\\).\n\\end{proof}\n\n\n\\subsection{The coarse Mayer--Vietoris sequence}\n\\label{sec:coarse_MV_subsection}\n\n\\begin{proposition}[\\cite{Higson-Roe-Yu:Coarse_Mayer-Vietoris}]\n  \\label{pro:omega-excisiv}\n  Let~\\(X\\)\n  be a proper metric space and let \\(Y_1,Y_2\\subseteq X\\)\n  be closed subspaces with \\(Y_1 \\cup Y_2 = X\\).\n  Let \\(Z \\mathrel{\\vcentcolon=} Y_1\\cap Y_2\\).\n  Then\n  \\[\n  \\Cst_\\Roe(Y_1\\subseteq X) + \\Cst_\\Roe(Y_2\\subseteq X) = \\Cst_\\Roe(X).\n  \\]\n  We have\n  \\(\\Cst_\\Roe(Y_1\\subseteq X) \\cap \\Cst_\\Roe(Y_2\\subseteq X) =\n  \\Cst_\\Roe(Z\\subseteq X)\\)\n  if and only if the following coarse transversality condition holds:\n  for any \\(R>0\\)\n  there is \\(S(R)>0\\)\n  such that if \\(x\\in X\\)\n  satisfies \\(d(x,Y_1)<R\\)\n  and \\(d(x,Y_2)<R\\),\n  then \\(d(x,Z)<S(R)\\).\n  The statements above hold both for real and complex Roe\n  \\(\\Cst\\)\\nobreakdash-algebras.\n\\end{proposition}\n\n\\begin{proof}\n  The proof of Theorem~\\ref{the:subset_full_corner} identifies\n  \\(\\Cst_\\Roe(Y_j\\subseteq X)\\)\n  for \\(j=1,2\\)\n  with corners in \\(\\Cst_\\Roe(X)\\).\n  The ideal generated by these corners is all of \\(\\Cst_\\Roe(X)\\)\n  because \\(Y_1 \\cup Y_2 = X\\).\n  That is,\n  \\(\\Cst_\\Roe(Y_1\\subseteq X) + \\Cst_\\Roe(Y_2\\subseteq X) =\n  \\Cst_\\Roe(X)\\).\n\n  An operator that is supported near~\\(Z\\)\n  is also supported near~\\(Y_1\\) and near~\\(Y_2\\).  So\n  \\[\n  \\Cst_\\Roe(Z\\subseteq X)\n  \\subseteq \\Cst_\\Roe(Y_1\\subseteq X)\\cap \\Cst_\\Roe(Y_2\\subseteq X),\n  \\]\n  as these are closed ideals.  The other inclusion uses the coarse\n  transversality assumption above, which is called\n  ``\\(\\omega\\)-excisiveness''\n  in~\\cite{Higson-Roe-Yu:Coarse_Mayer-Vietoris}.\n\n  Let \\(T\\)\n  and~\\(U\\)\n  be locally compact operators that are \\(R_T\\)-\n  and \\(R_U\\)\\nobreakdash-controlled,\n  respectively, and such that~\\(T\\)\n  is supported within distance \\(P_T>0\\)\n  of~\\(Y_1\\)\n  and~\\(U\\)\n  within distance \\(P_U>0\\)\n  of~\\(Y_2\\).\n  Let \\(R \\mathrel{\\vcentcolon=} R_T+R_U+P_T+P_U\\).\n  If \\((x,y)\\in\\supp(T U)\\),\n  then there is \\(z\\in X\\)\n  with \\((x,z)\\in \\supp(T)\\) and \\((z,y)\\in \\supp(U)\\).  Then\n  \\begin{align*}\n    d(x,Y_1)&<P_T\\le R,\\\\\n    d(x,Y_2)&\\le d(x,z)+d(z,Y_2)< R_T+P_U\\le R.\n  \\end{align*}\n  Hence \\(d(x,Y_1\\cap Y_2)<S(R)\\).\n  A similar argument shows that \\(d(y,Y_1\\cap Y_2)<S(R)\\).\n  So \\(T U\\)\n  is supported near~\\(Y_1 \\cap Y_2\\).\n  Thus\n  \\(\\Cst_\\Roe(Y_1\\subseteq X)\\cdot \\Cst_\\Roe(Y_2\\subseteq X) \\subseteq\n  \\Cst_\\Roe(Z\\subseteq X)\\).\n  This implies the claim because \\(I\\cap J=I\\cdot J\\)\n  if \\(I,J\\)\n  are closed ideals in a (real) \\(\\Cst\\)\\nobreakdash-algebra;\n  the latter follows from the existence of approximate units.\n\\end{proof}\n\n\\begin{proposition}\n  \\label{pro:cstarmv}\n  Let \\(A\\)\n  be a real or complex \\(\\Cst\\)\\nobreakdash-algebra\n  and let \\(I,J\\subset A\\)\n  be closed ideals with \\(I+J=A\\).\n  Let \\(\\alpha_I\\colon I\\cap J\\hookrightarrow I\\),\n  \\(\\alpha_J\\colon I\\cap J\\hookrightarrow J\\),\n  \\(\\beta_I\\colon I\\hookrightarrow A\\),\n  and \\(\\beta_J\\colon J\\hookrightarrow A\\)\n  denote the inclusion maps and also the maps that they induce on\n  \\(\\K\\)\\nobreakdash-theory.\n  Then there is a long exact sequence in \\textup{(}real or complex\\textup{)}\n  \\(\\K\\)\\nobreakdash-theory\n  \\[\n  \\dotsb \\to \\K_j(I\\cap J)\n  \\xrightarrow{\\bigl(\\begin{smallmatrix} -\\alpha_I\\\\\\alpha_J \\end{smallmatrix}\\bigr)}\n  \\K_j(I)\\oplus \\K_j(J)\n  \\xrightarrow{\\bigl(\\begin{smallmatrix} \\beta_I& \\beta_J \\end{smallmatrix}\\bigr)}\n  \\K_j(A) \\xrightarrow{\\partial_\\mathrm{MV}}\n  \\K_{j-1}(I\\cap J) \\to \\dotsb.\n  \\]\n  The boundary map~\\(\\partial_\\mathrm{MV}\\)\n  is computed in~\\eqref{eq:MV_boundary} below.\n\\end{proposition}\n\n\\begin{proof}\n  There is a commuting diagram\n  \\[\n  \\begin{tikzcd}\n    I\\cap J \\arrow[d, \"\\alpha_J\"] \\arrow[r, rightarrowtail, \"\\alpha_I\"] &\n    I \\arrow[d, \"\\beta_I\"] \\arrow[r, twoheadrightarrow, \"\\pi_I\"] &\n    I\/(I\\cap J) \\arrow[d, \"\\beta_{I*}\", \"\\cong\"'] \\\\\n    J \\arrow[r, rightarrowtail, \"\\beta_J\"] &\n    A \\arrow[r, twoheadrightarrow, \"\\pi\"] &\n    A\/J\n  \\end{tikzcd}\n  \\]\n  whose rows are \\(\\Cst\\)\\nobreakdash-algebra\n  extensions.  The map between the quotients induced by~\\(\\beta_I\\)\n  is an isomorphism because \\(I\/(I\\cap J) \\cong (I+J)\/J = A\/J\\).\n  The rows in the above diagram generate \\(\\K\\)\\nobreakdash-theory\n  long exact sequences, which we view as exact chain complexes.  The\n  vertical maps generate a chain map between them.  Its mapping cone\n  is again exact.  So we get an exact sequence\n  \\begin{multline*}\n    \\dotsb \\to \\K_j(I\\cap J) \\oplus \\K_{j+1}(A\/J)\n    \\xrightarrow{\\bigl(\\begin{smallmatrix} -\\alpha_I&0\\\\\\alpha_J&\\delta\\end{smallmatrix}\\bigr)}\n    \\K_j(I) \\oplus \\K_j(J)\\\\\n    \\xrightarrow{\\bigl(\\begin{smallmatrix} -\\pi_I&0\\\\\\beta_I&\\beta_J\\end{smallmatrix}\\bigr)}\n    \\K_j(I\/(I\\cap J)) \\oplus \\K_j(A)\n    \\xrightarrow{\\bigl(\\begin{smallmatrix} -\\delta&0\\\\\\beta_{I*}&\\pi\\end{smallmatrix}\\bigr)}\n    \\K_{j-1}(I\\cap J) \\oplus \\K_j(A\/J)\n    \\to \\dotsb.\n  \\end{multline*}\n  Since~\\(\\beta_{I*}\\)\n  is invertible, the boundary map restricts to an injective map on the\n  summands \\(\\K_*(I\/(I\\cap J))\\).\n  So these summands and their images~\\(B\\)\n  under the boundary map form an exact subcomplex.  Dividing it out\n  gives another exact chain complex.  The\n  direct summand \\(\\K_{j-1}(I\\cap J)\\)\n  in \\(\\K_{j-1}(I\\cap J) \\oplus \\K_j(A\/J)\\)\n  is complementary to~\\(B\\),\n  and the projection to \\(\\K_{j-1}(I\\cap J)\\)\n  that kills~\\(B\\)\n  maps \\(x\\in \\K_j(A\/J)\\)\n  to \\(\\delta(\\beta_{I*}^{-1}(x))\\)\n  because the boundary map sends\n  \\(\\beta_{I*}^{-1}(x) \\in \\K_j(I\/(I\\cap J))\\)\n  to \\((-\\delta(\\beta_{I*}^{-1}(x)),x)\\).\n  Hence the quotient of the above complex by the exact subcomplex\n  \\(\\K_*(I\/(I\\cap J)) \\oplus B\\) becomes the exact chain complex\n  \\begin{multline*}\n    \\dotsb \\to \\K_j(I\\cap J)\n    \\xrightarrow{\\bigl(\\begin{smallmatrix} -\\alpha_I\\\\\\alpha_J\\end{smallmatrix}\\bigr)}\n    \\K_j(I) \\oplus \\K_j(J)\\\\\n    \\xrightarrow{\\bigl(\\begin{smallmatrix} \\beta_I&\\beta_J\\end{smallmatrix}\\bigr)}\n    \\K_j(A)\n    \\xrightarrow{\\delta(\\beta_{I*})^{-1}\\pi}\n    \\K_{j-1}(I\\cap J)\n    \\to \\dotsb.\n  \\end{multline*}\n  This is the desired long exact sequence.  We have also computed the\n  boundary map:\n  \\begin{equation}\n    \\label{eq:MV_boundary}\n    \\partial_\\mathrm{MV} = \\delta\\circ (\\beta_{I*})^{-1}\\circ \\pi\\colon\n    \\K_j(A) \\xrightarrow{\\pi} \\K_j(A\/J)\n    \\xrightarrow[\\cong]{\\beta_{I*}^{-1}} \\K_j(I\/(I\\cap J))\n    \\xrightarrow{\\delta} \\K_{j-1}(I\\cap J),\n  \\end{equation}\n  where~\\(\\delta\\)\n  is the boundary map for the \\(\\Cst\\)\\nobreakdash-extension\n  \\(I\\cap J \\rightarrowtail I \\twoheadrightarrow I\/(I\\cap J)\\).\n\\end{proof}\n\n\\begin{corollary}\n  \\label{cor:MV_coarse}\n  Let~\\(X\\)\n  be a proper metric space.  Let \\(X= Y_1 \\cup Y_2\\)\n  be a coarsely transverse decomposition as in\n  Proposition~\\textup{\\ref{pro:omega-excisiv}} and let\n  \\(Z\\mathrel{\\vcentcolon=} Y_1\\cap Y_2\\).  Then there is a long exact sequence\n  \\begin{multline*}\n    \\dotsb \\to \\K_j(\\Cst_\\Roe(Z))\n    \\xrightarrow{\\bigl(\\begin{smallmatrix} -\\alpha_1\\\\\\alpha_2 \\end{smallmatrix}\\bigr)}\n    \\K_j(\\Cst_\\Roe(Y_1))\\oplus \\K_j(\\Cst_\\Roe(Y_2))\n    \\\\ \\xrightarrow{\\bigl(\\begin{smallmatrix} \\beta_1& \\beta_2 \\end{smallmatrix}\\bigr)}\n    \\K_j(\\Cst_\\Roe(X)) \\xrightarrow{\\partial_\\mathrm{MV}}\n    \\K_{j-1}(\\Cst_\\Roe(Z)) \\to \\dotsb.\n  \\end{multline*}\n  Here \\(\\alpha_1,\\alpha_2,\\beta_1,\\beta_2\\)\n  are the maps on \\(\\K\\)\\nobreakdash-theory\n  induced by the $^*$\\nobreakdash-{}homomorphisms on Roe \\(\\Cst\\)\\nobreakdash-algebras\n  induced by the inclusion maps \\(Z\\to Y_1\\),\n  \\(Z\\to Y_2\\),\n  \\(Y_1\\to X\\)\n  and \\(Y_2\\to X\\),\n  respectively.  The above holds both for real and complex Roe\n  \\(\\Cst\\)\\nobreakdash-algebras.\n\\end{corollary}\n\n\\begin{proof}\n  Let \\(A\\mathrel{\\vcentcolon=} \\Cst_\\Roe(X)\\),\n  \\(I \\mathrel{\\vcentcolon=} \\Cst_\\Roe(Y_1 \\subseteq X)\\)\n  and \\(J \\mathrel{\\vcentcolon=} \\Cst_\\Roe(Y_2 \\subseteq X)\\).\n  Then \\(I+J=A\\)\n  and \\(I\\cap J = \\Cst_\\Roe(Z\\subseteq X)\\)\n  by Proposition~\\ref{pro:omega-excisiv}.  And the relative Roe\n  \\(\\Cst\\)\\nobreakdash-algebras\n  above are Morita equivalent to the absolute ones by\n  Theorem~\\ref{the:subset_full_corner}.  Thus\n  \\(\\K_*(\\Cst_\\Roe(Y_j \\subseteq X)) \\cong \\K_*(\\Cst_\\Roe(Y_j))\\)\n  for \\(j=1,2\\)\n  and \\(\\K_*(\\Cst_\\Roe(Z \\subseteq X)) \\cong \\K_*(\\Cst_\\Roe(Z))\\).\n  Plugging this into the Mayer--Vietoris sequence in\n  Proposition~\\ref{pro:cstarmv} gives the assertion.\n\\end{proof}\n\n\n\\subsection{Application to \\texorpdfstring{$\\mathbb{Z}^d$}{the plane}}\n\\label{sec:K-theory}\n\nWe apply the coarse Mayer--Vietoris sequence to the decomposition\n\\begin{equation}\n  \\label{eq:decompose_Zn}\n  \\mathbb{Z}^d = \\mathbb{Z}^{d-1} \\times \\mathbb{N} \\cup \\mathbb{Z}^{d-1} \\times (-\\mathbb{N})\n\\end{equation}\ninto two half-spaces, which intersect in \\(\\mathbb{Z}^{d-1} \\times \\{0\\}\\).\nIt is clearly coarsely transverse, so that\nCorollary~\\ref{cor:MV_coarse} applies to it.\n\n\\begin{proposition}[\\cite{Higson-Roe-Yu:Coarse_Mayer-Vietoris}*{Proposition~1}]\n  \\label{vanish}\n  The \\(\\K\\)\\nobreakdash-theory\n  of the real and complex Roe \\(\\Cst\\)\\nobreakdash-algebras\n  of \\(X\\times\\mathbb{N}\\) vanishes for any proper metric space~\\(X\\).\n\\end{proposition}\n\n\\begin{proof}\n  We sketch the proof in~\\cite{Higson-Roe-Yu:Coarse_Mayer-Vietoris}\n  for~\\(\\K_0\\)\n  and then explain briefly why this argument also works for all other\n  \\(\\K\\)\\nobreakdash-groups.\n  We realise the Roe \\(\\Cst\\)\\nobreakdash-algebra\n  on \\(\\ell^2(X\\times\\mathbb{N},\\Hils)\\)\n  for a separable Hilbert space~\\(\\Hils\\),\n  which may be real or complex.  The unilateral shift on\n  \\(\\ell^2(\\mathbb{N})\\)\n  is a \\(1\\)\\nobreakdash-controlled\n  isometry.  It also defines a controlled isometry~\\(S\\)\n  on \\(\\ell^2(X\\times\\mathbb{N},\\Hils)\\).\n  If \\(n\\in\\mathbb{N}\\),\n  then the map \\(T\\mapsto S^n T (S^*)^n\\)\n  on \\(\\Cst_\\Roe(X\\times\\mathbb{N})\\)\n  is a $^*$\\nobreakdash-{}homomorphism, and it maps \\(R\\)\\nobreakdash-controlled\n  operators to \\(R\\)\\nobreakdash-controlled\n  operators for the same~\\(R\\).  Hence the map\n  \\[\n  \\varphi\\colon \\mathbb B(\\ell^2(X\\times\\mathbb{N},\\Hils)) \\to\n  \\mathbb B(\\ell^2(X\\times\\mathbb{N},\\Hils^\\infty)),\\qquad\n  T\\mapsto \\bigoplus_{n=0}^\\infty S^n T (S^*)^n,\n  \\]\n  maps \\(R\\)\\nobreakdash-controlled\n  operators on~\\(\\Hils\\)\n  to \\(R\\)\\nobreakdash-controlled\n  operators on \\(\\Hils^\\infty \\mathrel{\\vcentcolon=} \\Hils\\otimes \\ell^2(\\mathbb{N})\\).\n  The matrix coefficients \\((S^n T (S^*)^n)_{x,y}\\)\n  vanish for \\(0\\le x,y<n\\).\n  Therefore, \\(\\varphi(T)\\)\n  is locally compact if~\\(T\\)\n  is locally compact.  So~\\(\\varphi\\)\n  restricts to a $^*$\\nobreakdash-{}homomorphism from the Roe \\(\\Cst\\)\\nobreakdash-algebra\n  of~\\(X\\times\\mathbb{N}\\)\n  realised on \\(\\ell^2(X\\times\\mathbb{N},\\Hils)\\)\n  to the isomorphic Roe \\(\\Cst\\)\\nobreakdash-algebra\n  of~\\(X\\times\\mathbb{N}\\)\n  realised on \\(\\ell^2(X\\times\\mathbb{N},\\Hils^\\infty)\\).\n  We may identify these using a unitary operator\n  \\(\\Hils \\cong \\Hils^\\infty\\).\n  We have \\(S\\varphi(T)S^* = \\bigoplus_{n=1}^\\infty S^n T (S^*)^n\\).\n  So~\\(\\varphi\\)\n  is equal to the direct sum of the canonical inclusion~\\(\\iota\\)\n  induced by the embedding \\(\\Hils\\to \\Hils^\\infty\\),\n  \\(\\xi\\mapsto \\xi\\otimes \\delta_0\\),\n  and the $^*$\\nobreakdash-{}homomorphism \\(T\\mapsto S\\varphi(T) S^*\\).\n\n  In particular, if \\(P\\in\\Cst_\\Roe(X\\times\\mathbb{N})\\)\n  is a projection, then \\(\\varphi(P)\\)\n  is another projection in \\(\\Cst_\\Roe(X\\times\\mathbb{N})\\).\n  And \\(\\varphi(P)\\)\n  is Murray--von Neumann equivalent to \\(\\iota(P) \\oplus \\varphi(P)\\).\n  Thus \\(\\iota(P)\\)\n  is stably equivalent to~\\(0\\).\n  That is, the inclusion~\\(\\iota\\)\n  induces the zero map on~\\(\\K_0\\).\n  We may also identify \\(\\Hils^\\infty \\cong \\Hils^2\\)\n  so that the inclusion~\\(\\iota\\) becomes the corner embedding\n  \\[\n  \\Cst_\\Roe(X\\times\\mathbb{N}) \\to \\mathbb M_2(\\Cst_\\Roe(X\\times\\mathbb{N})),\\qquad\n  T\\mapsto \\begin{pmatrix} T&0\\\\0&0 \\end{pmatrix}.\n  \\]\n  This induces an isomorphism on~\\(\\K_0\\).\n  So the zero map is an isomorphism on \\(\\K_0(\\Cst_\\Roe(X\\times\\mathbb{N}))\\),\n  which forces this group to vanish.  The argument above works for any\n  functor on the category of \\(\\Cst\\)\\nobreakdash-algebras\n  and $^*$\\nobreakdash-{}homomorphisms that is matrix-stable because inner\n  endomorphisms induced by isometries act by the identity on all such\n  functors (see \\cite{Cuntz-Meyer-Rosenberg}*{Proposition 3.16}).  In\n  particular, the proof above works for all real and complex\n  \\(\\K\\)\\nobreakdash-groups.\n\\end{proof}\n\nThe proof above breaks down for the uniform Roe \\(\\Cst\\)\\nobreakdash-algebra,\nand indeed the result is wrong in that case.\n\n\\begin{corollary}\n  \\label{cor:K_Roe_Zd}\n  Let \\(d\\in\\mathbb{N}\\).\n  Then the boundary map in the coarse Mayer--Vietoris sequence for the\n  decomposition~\\eqref{eq:decompose_Zn} is an isomorphism.  So\n  \\(\\K_{i+d}(\\Cst_\\Roe(\\mathbb{Z}^d)_\\mathbb{F}) \\cong \\K_i(\\mathbb{F})\\)\n  for \\(\\mathbb{F} \\in \\{\\mathbb{R},\\mathbb{C}\\}\\).\n\\end{corollary}\n\n\\begin{proof}\n  Apply the Mayer--Vietoris long exact sequence of\n  Corollary~\\ref{cor:MV_coarse} to the coarsely transverse\n  decomposition~\\eqref{eq:decompose_Zn}.  Plug in that the\n  \\(\\K\\)\\nobreakdash-theory\n  vanishes for the two half-spaces (Proposition~\\ref{vanish}).\n  Hence the boundary map is an\n  isomorphism.  Now an induction argument identifies\n  \\(\\K_{i+d}(\\Cst_\\Roe(\\mathbb{Z}^d)_\\mathbb{F})\\)\n  with \\(\\K_i(\\Cst_\\Roe(\\{0\\})_\\mathbb{F})\\).\n  Finally, \\(\\Cst_\\Roe(\\{0\\})_\\mathbb{F}\\)\n  is isomorphic to the \\(\\Cst\\)\\nobreakdash-algebra\n  of compact operators on a real or complex Hilbert space.  This gives\n  the statement because \\(\\K\\)\\nobreakdash-theory is \\(\\Cst\\)\\nobreakdash-stable.\n\\end{proof}\n\nThe well known \\(K\\)\\nobreakdash-theory computations for \\(\\mathbb{R}\\) and~\\(\\mathbb{C}\\) give\n\\begin{align*}\n  \\K_i(\\Cst_\\Roe(\\mathbb{Z}^d)_\\mathbb{C})&\\cong\n  \\begin{cases} \\mathbb{Z} &\\text{if }i-d\\equiv 0 \\mod 2,\\\\\n    0 &\\text{if }i-d \\equiv 1\\mod 2.\n  \\end{cases}\\\\\n  \\K_i(\\Cst_\\Roe(\\mathbb{Z}^d)_\\mathbb{R})&\\cong\n  \\begin{cases}\n    \\mathbb{Z} &\\text{if }i-d\\equiv 0 \\text{ or } 4 \\mod 8,\\\\\n    \\mathbb{Z}\/2&\\text{if }i-d\\equiv 1 \\text{ or } 2 \\mod 8,\\\\\n    0 &\\text{if } i-d\\equiv 3\\text{, }5\\text{, }6\\text{ or }7 \\mod 8.\n  \\end{cases}\n\\end{align*}\n\nIn contrast, the \\(\\K\\)\\nobreakdash-theory\nof the uniform \\(\\Cst\\)\\nobreakdash-Roe\nalgebra is far more complicated.  The \\(\\K_0\\)\\nobreakdash-group\nof the complex uniform Roe \\(\\Cst\\)\\nobreakdash-algebra\nof~\\(\\mathbb{Z}^d\\)\nis an uncountable Abelian group for all \\(d>1\\),\nsee \\cite{Spakula:Thesis}*{Example II.3.4}.\n\nWhen we consider Hamiltonians with symmetries, then we should tensor\nthe real Roe \\(\\Cst\\)\\nobreakdash-algebra\nof~\\(\\mathbb{Z}^d\\)\nwith a real or complex Clifford algebra.  This gives a\n\\(\\mathbb{Z}\/2\\)\\nobreakdash-graded\n\\(\\Cst\\)\\nobreakdash-algebra.\nUp to Morita equivalence, there are ten\ndifferent real or complex Clifford algebras.  So we get ten\ndifferent observable algebras in each dimension.  The resulting\nreal or complex \\(\\K\\)\\nobreakdash-groups\nagree with those in Kitaev's periodic\ntable~\\cite{Kitaev:Periodic_table}.  Hence the latter agrees with\nthe \\(\\K\\)\\nobreakdash-theory\nof the Roe \\(\\Cst\\)\\nobreakdash-algebra.\nWe interpret it as saying that Kitaev's table gives only the\n\\emph{strong} topological phases.\n\nThe real and complex \\(\\K\\)\\nobreakdash-groups\nof the point form a graded commutative, graded ring in a\nnatural way, and the \\(\\K\\)\\nobreakdash-theory\nof any real or complex \\(\\Cst\\)\\nobreakdash-algebra\nis a graded module over this ring.  The boundary map for an extension\nof real or complex \\(\\Cst\\)\\nobreakdash-algebras\nautomatically preserves this module structure.  In the\ncomplex case, the relevant ring is the ring of Laurent\npolynomials \\(\\mathbb{Z}[\\beta,\\beta^{-1}]\\)\nin \\(\\beta\\in \\K_2(\\mathbb{C})\\)\nthat describes Bott periodicity.  That a map on \\(\\K\\)\\nobreakdash-theory\nis a \\(\\K_*(\\mathbb{C})\\)-module\nhomomorphism only says that it is obtained by the maps on \\(\\K_0\\)\nand~\\(\\K_1\\)\nand Bott periodicity.  In other words, it is a homomorphism of\n\\(\\mathbb{Z}\/2\\)\\nobreakdash-graded groups.\nIn the real case, the relevant ring is more complicated, and so the\nmodule structure contains more useful information.\nOne way to get the \\(\\K_*(\\mathbb{R})\\)-module structure on \\(\\K_*(A)\\)\nfor a real \\(\\Cst\\)\\nobreakdash-algebra~\\(A\\)\nis to identify~\\(\\K_j(A)\\)\nwith the bivariant Kasparov groups\n\\(\\K_j(A) \\cong \\KK_0(\\mathbb{R},A \\otimes \\Cliff_j)\\).\nThe exterior product in Kasparov theory provides both the graded\ncommutative ring structure on\n\\(\\bigoplus_{j\\in\\mathbb{Z}} \\KK_0(\\mathbb{R},\\Cliff_j)\\)\nand the module structure on\n\\(\\bigoplus_{j\\in\\mathbb{Z}} \\KK_0(\\mathbb{R},A\\otimes \\Cliff_j)\\).\nThese structures are compatible with Kasparov products, and the\nboundary map in an extension may be written as such a Kasparov\nproduct.\n\nThe isomorphism\n\\(\\K_{*+d}(\\Cst_\\Roe(\\mathbb{Z}^d)_\\mathbb{R}) \\cong \\K_*(\\mathbb{R})\\)\nof \\(\\mathbb{Z}\\)\\nobreakdash-graded groups\nin Corollary~\\ref{cor:K_Roe_Zd} is a \\(\\K_*(\\mathbb{R})\\)-module\nisomorphism.  So \\(\\K_*(\\Cst_\\Roe(\\mathbb{Z}^d)_\\mathbb{R})\\)\nis a free \\(\\K_*(\\mathbb{R})\\)-module\nof rank~\\(1\\),\nshifted in degree by~\\(d\\).\nAnd the boundary map~\\(\\partial_\\mathrm{MV}\\)\nis a module isomorphism.  Thus it is determined by a single sign,\ndescribing whether the ``standard'' generator of\n\\(\\K_d(\\Cst_\\Roe(\\mathbb{Z}^d)_\\mathbb{R})\\)\ngoes to the ``standard'' generator of\n\\(\\K_{d-1}(\\Cst_\\Roe(\\mathbb{Z}^{d-1})_\\mathbb{R})\\)\nor its negative.  This sign is, in fact, a matter of convention: it\nchanges when we change the role of the left and right half-spaces in\nthe Mayer--Vietoris sequence.  So there is not much need to\n``compute'' the boundary map for the Roe \\(\\Cst\\)\\nobreakdash-algebras\nbecause the \\(\\K\\)\\nobreakdash-theory\ngroups in question are so small, even in the real case.\n\nThe boundary map~\\(\\partial_\\mathrm{MV}\\)\nis the incarnation of the bulk--edge correspondence in our\nRoe \\(\\Cst\\)\\nobreakdash-algebra\ncontext.  It is shown by Kubota~\\cite{Kubota:Controlled_bulk-edge}\nthat the boundary maps in the Toeplitz extension, which is used by\nmany authors to describe the bulk--edge correspondence, and the\ncoarse Mayer--Vietoris sequence are compatible.\n\n\\begin{proposition}\n  \\label{pro:low_dimensional_killed_Roe}\n  Let \\(\\varphi\\colon \\mathbb{Z}^{d-1}\\to \\mathbb{Z}^d\\)\n  be an injective group homomorphism.  Then the induced map\n  \\(\\varphi_*\\colon \\Cst_\\Roe(\\mathbb{Z}^{d-1}) \\to \\Cst_\\Roe(\\mathbb{Z}^d)\\)\n  induces the zero map in \\(\\K\\)\\nobreakdash-theory, both in the real and\n  complex cases.\n\\end{proposition}\n\n\\begin{proof}\n  Since~\\(\\varphi\\)\n  is an injective group homomorphism, it is a coarse equivalence\n  from~\\(\\mathbb{Z}^{d-1}\\)\n  onto a subspace of~\\(\\mathbb{Z}^d\\).\n  This explains the definition of\n  \\(\\varphi_*\\colon \\Cst_\\Roe(\\mathbb{Z}^{d-1}) \\to \\Cst_\\Roe(\\mathbb{Z}^d)\\).\n  There is \\(x\\in\\mathbb{Z}^d\\)\n  so that the map \\(\\mathbb{Z}^{d-1}\\times\\mathbb{Z} \\to \\mathbb{Z}^d\\),\n  \\((a,b) \\mapsto \\varphi(a)+b\\cdot x\\),\n  is injective.  So the map\n  \\(\\varphi_*\\colon \\Cst_\\Roe(\\mathbb{Z}^{d-1}) \\to \\Cst_\\Roe(\\mathbb{Z}^d)\\)\n  factors through \\(\\Cst_\\Roe(\\mathbb{Z}^{d-1}\\times\\mathbb{N})\\).\n  Since the \\(\\K\\)\\nobreakdash-theory\n  of \\(\\Cst_\\Roe(\\mathbb{Z}^{d-1}\\times\\mathbb{N})\\)\n  vanishes by Proposition~\\ref{vanish},\n  the map~\\(\\varphi\\)\n  induces the zero map on \\(\\K\\)\\nobreakdash-theory.\n\\end{proof}\n\n\n\\section{Comparison with the periodic case}\n\\label{sec:weak_phases}\n\nLet \\(\\mathbb{F} \\in \\{\\mathbb{R},\\mathbb{C}\\}\\).\nThe observable algebra \\(\\Cst(\\mathbb{Z}^d)_\\mathbb{F}\\)\nor a matrix algebra over it describes periodic observables in the\nlimiting case of no disorder, in the tight-binding approximation.\nThis is contained in the corresponding Roe \\(\\Cst\\)\\nobreakdash-algebra\n\\(\\Cst_\\Roe(\\mathbb{Z}^d)_\\mathbb{F}\\).\nIn this section, we recall how to compute the \\(\\K\\)\\nobreakdash-theory\nof \\(\\Cst_\\Roe(\\mathbb{Z}^d)_\\mathbb{F}\\)\nand we describe the map in \\(\\K\\)\\nobreakdash-theory\ninduced by the inclusion\n\\(\\Cst(\\mathbb{Z}^d)_\\mathbb{F} \\hookrightarrow \\Cst_\\Roe(\\mathbb{Z}^d)_\\mathbb{F}\\).\nIn particular, we show that this map is split surjective and that its\nkernel is generated by those elements that come from the\n\\(\\K\\)\\nobreakdash-theory\nof \\(\\Cst(\\mathbb{Z}^{d-1})_\\mathbb{F}\\)\nfor a coordinate embedding \\(\\mathbb{Z}^{d-1} \\to \\mathbb{Z}^d\\).\nSo its kernel consists of those topological phases that are obtained\nby stacking lower-dimensional topological insulators in a coordinate\ndirection.  These are called ``weak topological phases''\nin~\\cite{Fu-Kane-Mele:Insulators}.  In the end, we argue that stable\nhomotopy instead of homotopy is the physically reasonable equivalence\nrelation on Hamiltonians.\n\nThe following arguments are easier and more standard in the complex\ncase.  Hence we only discuss the real case.  It is convenient to\nreplace real \\(\\Cst\\)\\nobreakdash-algebras by ``real'' ones, that is, complex\n\\(\\Cst\\)\\nobreakdash-algebras equipped with a real involution.  We first\nrecall some basic facts and definitions about ``real'' and real\n\\(\\Cst\\)\\nobreakdash-algebras and then describe the relevant ``real''\n\\(d\\)\\nobreakdash-torus.\n\nA real \\(\\Cst\\)\\nobreakdash-algebra~\\(A\\)\ncorresponds to the ``real'' \\(\\Cst\\)\\nobreakdash-algebra\n\\(A\\otimes_\\mathbb{R} \\mathbb{C}\\)\nwith the real involution\n\\(\\conj{a\\otimes z} \\mathrel{\\vcentcolon=} a\\otimes \\conj{z}\\).\nA ``real'' \\(\\Cst\\)\\nobreakdash-algebra~\\(A\\)\ncorresponds to the real \\(\\Cst\\)\\nobreakdash-algebra\n\\[\nA_\\mathbb{R} \\mathrel{\\vcentcolon=} \\setgiven{a\\in A}{\\conj{a}=a}.\n\\]\nA ``real'' locally compact space~\\(X\\)\nis a locally compact space with an involutive homeomorphism\n\\(X\\to X\\),\n\\(x\\mapsto \\conj{x}\\).\nThen we turn \\(\\Cont_0(X)\\)\ninto a ``real'' \\(\\Cst\\)\\nobreakdash-algebra\nusing the real involution \\(\\conj{f}(x) \\mathrel{\\vcentcolon=} \\conj{f(\\conj{x})}\\)\nfor all \\(x\\in X\\), \\(f\\in\\Cont_0(X)\\).  So\n\\[\n\\Cont_0(X)_\\mathbb{R} = \\setgiven*{f\\in\\Cont_0(X)}\n{f(\\conj{x}) = \\conj{f(x)}\\text{ for all }x\\in X}.\n\\]\nFor a ``real'' \\(\\Cst\\)\\nobreakdash-algebra~\\(A\\), we define\n\\[\n\\KR_*(A) \\mathrel{\\vcentcolon=} \\K_*(A_\\mathbb{R}).\n\\]\nFor a ``real'' locally compact space~\\(X\\), we let\n\\[\n\\KR^*(X)\n\\mathrel{\\vcentcolon=} \\KR_{-*}(\\Cont_0(X))\n= \\K_{-*}(\\Cont_0(X)_\\mathbb{R}).\n\\]\nNote the grading convention here, which is analogous to the numbering\nconvention when a chain complex is treated as a cochain complex.\n\nFrom now on, \\(\\Cst(\\mathbb{Z}^d)\\)\ndenotes the ``real'' \\(\\Cst\\)\\nobreakdash-algebra\nthat corresponds to the real \\(\\Cst\\)\\nobreakdash-algebra\n\\(\\Cst(\\mathbb{Z}^d)_\\mathbb{R}\\).\nThat is, the real involution acts on \\(f\\colon \\mathbb{Z}^d\\to\\mathbb{C}\\)\nby pointwise complex conjugation.  We give the \\(d\\)\\nobreakdash-torus\n\\(\\mathbb{T}^d\\subseteq \\mathbb{C}^d\\)\nthe real involution by complex conjugation.  So \\(\\Cont(\\mathbb{T}^d)_\\mathbb{R}\\)\nis the closed \\(\\mathbb{R}\\)\\nobreakdash-linear\nspan of the functions \\(z^k \\mathrel{\\vcentcolon=} z_1^{k_1} \\dotsm z_d^{k_d}\\)\non~\\(\\mathbb{T}^d\\)\nfor \\(k_1,\\dotsc,k_d\\in\\mathbb{Z}\\).\nThis is the unique real structure on~\\(\\mathbb{T}^d\\)\nfor which the Fourier isomorphism \\(\\Cst(\\mathbb{Z}^d) \\cong \\Cont(\\mathbb{T}^d)\\)\nis an isomorphism of ``real'' \\(\\Cst\\)\\nobreakdash-algebras.  Thus\n\\[\n\\K_*(\\Cst(\\mathbb{Z}^d)_\\mathbb{R}) \\cong \\KR_*(\\Cont(\\mathbb{T}^d)) = \\KR^{-*}(\\mathbb{T}^d).\n\\]\nWe shall also use the ``real'' manifolds~\\(\\mathbb{R}^{p,q}\\)\nfor \\(p,q\\in\\mathbb{N}\\);\nthis is~\\(\\mathbb{R}^{p+q}\\)\nwith the real involution \\(\\conj{(x,y)} \\mathrel{\\vcentcolon=} (x,-y)\\)\nfor \\(x\\in\\mathbb{R}^p\\),\n\\(y\\in\\mathbb{R}^q\\).\nWe may also realise this as\n\\(\\mathbb{R}^p \\times (\\ima \\mathbb{R})^q \\subseteq \\mathbb{C}^{p+q}\\)\nwith complex conjugation as real involution.\n\n\\begin{proposition}\n  \\label{pro:Cst_Zd_in_KK}\n  The ``real'' \\(\\Cst\\)\\nobreakdash-algebra\n  \\(\\Cont(\\mathbb{T})\\)\n  is \\(\\KK\\)\\nobreakdash-equivalent\n  to \\(\\mathbb{C}\\oplus \\Cont_0(\\mathbb{R}^{0,1})\\).\n  And \\(\\Cont(\\mathbb{T}^d)\\)\n  is \\(\\KK\\)\\nobreakdash-equivalent\n  to a direct sum of copies of \\(\\Cont_0(\\mathbb{R}^{0,j})\\)\n  for \\(j=0,\\dotsc,d\\),\n  where the summand \\(\\Cont_0(\\mathbb{R}^{0,j})\\)\n  appears \\(\\binom{d}{j}\\) times.\n  The \\(\\K\\)\\nobreakdash-theory\n  of \\(\\Cst(\\mathbb{Z}^d)_\\mathbb{F}\\)\n  is a free \\(\\K_*(\\mathbb{F})\\)-module\n  of rank~\\(2^d\\),\n  with \\(\\binom{d}{j}\\) generators of degree \\(-j \\bmod 8\\).\n\\end{proposition}\n\n\\begin{proof}\n  The points \\(\\pm 1\\in\\mathbb{T}\\)\n  are real, that is, fixed by the real involution.  The complement\n  \\(\\mathbb{T}\\setminus\\{1\\}\\)\n  is diffeomorphic as a ``real'' manifold to~\\(\\mathbb{R}^{0,1}\\),\n  say, by stereographic projection at~\\(1\\).\n  Hence we get an extension of ``real'' \\(\\Cst\\)\\nobreakdash-algebras\n  \\[\n  \\Cont_0(\\mathbb{R}^{0,1}) \\rightarrowtail \\Cont(\\mathbb{T})\n  \\twoheadrightarrow \\mathbb{C},\n  \\]\n  where the quotient map is evaluation at~\\(1\\).\n  This extension splits by embedding~\\(\\mathbb{C}\\)\n  as constant functions in \\(\\Cont(\\mathbb{T})\\).\n  Since Kasparov theory is split-exact, also for ``real''\n  \\(\\Cst\\)\\nobreakdash-algebras,\n  \\(\\Cont(\\mathbb{T})\\)\n  is \\(\\KK\\)\\nobreakdash-equivalent to \\(\\Cont_0(\\mathbb{R}^{0,1}) \\oplus \\mathbb{C}\\).\n\n  We may get \\(\\Cst(\\mathbb{Z}^d)\\)\n  by tensoring \\(d\\)~copies\n  of \\(\\Cst(\\mathbb{Z})\\).\n  The tensor product of \\(\\Cst\\)\\nobreakdash-algebras\n  descends to a bifunctor in \\(\\KK\\)\\nobreakdash-theory,\n  also in the ``real'' case.  So \\(\\Cst(\\mathbb{Z}^d)\\)\n  is \\(\\KK\\)\\nobreakdash-equivalent\n  to the \\(d\\)\\nobreakdash-fold\n  tensor power of \\(\\mathbb{C} \\oplus \\Cont_0(\\mathbb{R}^{0,1})\\).\n  The tensor product of \\(\\Cst\\)\\nobreakdash-algebras\n  is additive in each variable, and\n  \\(\\Cont_0(\\mathbb{R}^{p,q})\\otimes \\Cont_0(\\mathbb{R}^{r,s}) \\cong\n  \\Cont_0(\\mathbb{R}^{p+r,q+s})\\).\n  A variant of the binomial formula now gives.\n  \\[\n  \\Cst(\\mathbb{Z}^d)\n  \\sim_\\KK (\\mathbb{R} \\oplus \\Cont_0(\\mathbb{R}^{0,1}))^{\\otimes d}\n  \\cong \\bigoplus_{j=0}^d \\binom{d}{j} \\Cont_0(\\mathbb{R}^{0,j}).\n  \\]\n  A Bott periodicity theorem by Kasparov shows that\n  \\(\\Cont_0(\\mathbb{R}^{p,q})\\)\n  is \\(\\KK\\)\\nobreakdash-equivalent\n  to~\\(\\Cont_0(\\mathbb{R}^{0,0})\\)\n  with a dimension shift of \\(p-q\\),\n  see \\cite{Kasparov:Operator_K}*{Theorem~7}.  This implies the claim\n  about \\(\\K\\)\\nobreakdash-theory.\n\\end{proof}\n\n\\begin{proposition}\n  \\label{pro:low_dimensional_killed}\n  Let \\(\\varphi\\colon \\mathbb{Z}^{d-1}\\to \\mathbb{Z}^d\\)\n  be an injective group homomorphism.  It induces an injective\n  $^*$\\nobreakdash-{}homomorphism\n  \\(\\varphi_*\\colon \\Cst(\\mathbb{Z}^{d-1})_\\mathbb{F} \\to\n  \\Cst(\\mathbb{Z}^d)_\\mathbb{F}\\)\n  and a grading-preserving \\(\\K_*(\\mathbb{F})\\)-module\n  homomorphism\n  \\(\\K_*(\\varphi_*)\\colon \\K_*(\\Cst(\\mathbb{Z}^{d-1})_\\mathbb{F}) \\to\n  \\K_*(\\Cst(\\mathbb{Z}^d)_\\mathbb{F})\\).\n  The map\n  \\(\\K_*(\\Cst(\\mathbb{Z}^d)_\\mathbb{F}) \\to \\K_*(\\Cst_\\Roe(\\mathbb{Z}^d)_\\mathbb{F})\\)\n  vanishes on the image of~\\(\\K_*(\\varphi_*)\\).\n\\end{proposition}\n\n\\begin{proof}\n  Proposition~\\ref{pro:low_dimensional_killed_Roe} shows\n  that~\\(\\varphi\\)\n  induces the zero map on the \\(\\K\\)\\nobreakdash-theory\n  of the Roe \\(\\Cst\\)\\nobreakdash-algebra.\n  The canonical map \\(\\Cred(G) \\hookrightarrow \\Cst_\\Roe(G)\\)\n  for a group~\\(G\\)\n  is a natural transformation with respect to injective group\n  homomorphisms.  So there is a commuting square\n  \\[\n  \\begin{tikzcd}\n    \\Cst(\\mathbb{Z}^{d-1})_\\mathbb{F}\n    \\arrow[r, hookrightarrow] \\arrow[d, hookrightarrow, \"\\varphi_*\"]&\n    \\Cst_\\Roe(\\mathbb{Z}^{d-1})_\\mathbb{F} \\arrow[d, hookrightarrow, \"\\varphi_*\"]\\\\\n    \\Cst(\\mathbb{Z}^d)_\\mathbb{F}\n    \\arrow[r, hookrightarrow]&\n    \\Cst_\\Roe(\\mathbb{Z}^d)_\\mathbb{F}\n  \\end{tikzcd}\n  \\]\n  This implies the statement.\n\\end{proof}\n\nThe coordinate embeddings\n\\[\n\\iota_k\\colon \\mathbb{Z}^{d-1}\\to \\mathbb{Z}^d,\\qquad\n(x_1,\\dotsc,x_{d-1}) \\mapsto\n(x_1,\\dotsc,x_{k-1},0,x_k,\\dotsc,x_{d-1}),\n\\]\nare injective group homomorphisms and induce injective\n$^*$\\nobreakdash-{}homomorphisms\n\\[\n\\iota_k\\colon \\Cst(\\mathbb{Z}^{d-1}) \\to \\Cst(\\mathbb{Z}^d).\n\\]\nThe Fourier transform maps \\(\\iota_k(\\Cst(\\mathbb{Z}^{d-1}))\\)\nonto the \\(\\Cst\\)\\nobreakdash-subalgebra\nof \\(\\Cont(\\mathbb{T}^d)\\)\nconsisting of all functions that are constant equal to~\\(1\\)\nin the \\(k\\)th\ncoordinate direction.  We have seen that\n\\(\\K_*(\\Cst(\\mathbb{Z}^d)_\\mathbb{F})\\)\nis a free \\(\\K_*(\\mathbb{F})\\)-module\nof rank~\\(2^d\\)\n(with generators in different degrees).  Now compute\n\\(\\K_*(\\Cst(\\mathbb{Z}^d)_\\mathbb{F})\\)\nas in Proposition~\\ref{pro:Cst_Zd_in_KK}.  The inclusion of functions\nthat are constant in the \\(k\\)th\ndirection corresponds in \\(\\K\\)\\nobreakdash-theory\nto the inclusion of those~\\(2^{d-1}\\)\nof the~\\(2^d\\)\nfree \\(\\K_*(\\mathbb{R})\\)-module\nsummands in \\(\\K_*(\\Cst(\\mathbb{Z}^d)_\\mathbb{F})\\)\nwhere we take the summand~\\(\\mathbb{R}\\)\nin the \\(k\\)th\nfactor.  The summands in the image of~\\(\\K_*(\\iota_k)\\)\ncorrespond to topological insulators that are built by stacking\ncopies of a \\(d-1\\)-dimensional\ninsulator in the \\(k\\)th\ndirection.  Such topological insulators are considered weak by\nFu--Kane--Mele~\\cite{Fu-Kane-Mele:Insulators}.  So the map\n\\(\\K_*(\\Cst(\\mathbb{Z}^d)_\\mathbb{F}) \\to \\K_*(\\Cst_\\Roe(\\mathbb{Z}^d)_\\mathbb{F})\\)\nkills the \\(\\K\\)\\nobreakdash-theory classes of weak topological insulators.\n\nIf~\\(k\\)\nvaries, then all but one of the~\\(2^d\\)\nsummands \\(\\K_{*-j}(\\mathbb{F})\\)\nin \\(\\K_*(\\Cst(\\mathbb{Z}^d)_\\mathbb{F})\\)\nare in the image of \\(\\K_*(\\iota_k)\\)\nfor some \\(k\\in\\{1,\\dotsc,d\\}\\).\nAll these summands are mapped to~\\(0\\)\nin \\(\\K_*(\\Cst_\\Roe(\\mathbb{Z}^d)_\\mathbb{F})\\)\nby Proposition~\\ref{pro:low_dimensional_killed}.  The remaining\nsummand is the \\(\\K\\)\\nobreakdash-theory\nof the ideal \\(\\Cont_0(\\mathbb{R}^{0,d}) \\idealin \\Cont(\\mathbb{T}^d)\\).\nHere we identify~\\(\\mathbb{R}^{0,d}\\)\nwith an open subset of~\\(\\mathbb{T}^d\\)\nusing the stereographic projection in each variable, compare the proof\nof Proposition~\\ref{pro:Cst_Zd_in_KK}.  Its ``real'' or complex\n\\(\\K\\)\\nobreakdash-theory\nis identified with \\(\\K_{*-d}(\\mathbb{R})\\)\nor \\(\\K_{*-d}(\\mathbb{C})\\)\nby Bott periodicity.  Kasparov proves Bott periodicity isomorphisms\n\\(\\KR_*(\\Cont_0(\\mathbb{R}^{p,q})) \\cong \\K_{*+p-q}(\\mathbb{R})\\)\nusing a canonical generator~\\(\\alpha_{p,q}\\)\nfor the \\(\\K\\)\\nobreakdash-homology group\n\\[\n\\KK_{q-p}^\\mathbb{R}(\\Cont_0(\\mathbb{R}^{p,q}),\\mathbb{C})\n\\cong \\KK_0^\\mathbb{R}(\\Cont_0(\\mathbb{R}^{p,q}, \\Cliff_{p,q}),\\mathbb{C});\n\\]\nhere~\\(\\Cliff_{p,q}\\)\nis the Clifford algebra with \\(p+q\\)~anti-commuting,\nodd, self-adjoint generators \\(\\gamma_1,\\dotsc,\\gamma_{p+q}\\)\nwith \\(\\conj{\\gamma_i} = \\gamma_i\\)\nfor \\(1\\le i\\le p\\)\nand \\(\\conj{\\gamma_i} = -\\gamma_i\\)\nfor \\(p+1\\le i \\le p+q\\).\nAnd we write~\\(\\KK^\\mathbb{R}\\)\nto highlight that the entries are treated as ``real''\n\\(\\Cst\\)\\nobreakdash-algebras.\n\nThe Bott generators~\\(\\alpha_{p,0}\\)\nare generalised by Kasparov in \\cite{Kasparov:Novikov}*{Definition\n  and Lemma 4.2} to build a ``fundamental class''\n\\[\n\\alpha_X \\in \\KK^\\mathbb{R}_0(\\Cont_0(X,\\Cliff X),\\mathbb{C})\n\\]\nfor any complete Riemannian manifold~\\(X\\)\n(without boundary).  Here~\\(\\Cliff X\\)\nis the bundle of ``real'' \\(\\Cst\\)\\nobreakdash-algebras\nover~\\(X\\)\nwhose fibre at \\(x\\in X\\)\nis the \\(\\mathbb{Z}\/2\\)\\nobreakdash-graded\n``real'' Clifford algebra of the cotangent space~\\(T_x^* X\\)\nfor the positive definite quadratic form induced by the Riemannian\nmetric, and \\(\\Cont_0(X,\\Cliff X)\\)\nmeans the \\(\\mathbb{Z}\/2\\)\\nobreakdash-graded\n``real'' \\(\\Cst\\)\\nobreakdash-algebra\nof \\(\\Cont_0\\)\\nobreakdash-sections\nof this Clifford algebra bundle.  We now adapt Kasparov's fundamental\nclass to the case where~\\(X\\)\nis a ``real'' complete Riemannian manifold, in such a way that the\nfundamental class for~\\(\\mathbb{R}^{p,q}\\)\nis the generator~\\(\\alpha_{p,q}\\)\nof Bott periodicity from~\\cite{Kasparov:Operator_K}.  The only changes\nare in the real structure.  In particular, all the analysis needed to\nproduce cycles for Kasparov theory is already done\nin~\\cite{Kasparov:Novikov}.\n\nRecall that the real involution on \\(\\Cont_0(X)\\)\nis defined by \\(\\conj{f}(x) \\mathrel{\\vcentcolon=} \\conj{f(\\conj{x})}\\)\nfor \\(f\\in\\Cont_0(X)\\).\nThere is a unique conjugate-linear involution on the space of\ncomplex \\(1\\)\\nobreakdash-forms on~\\(X\\)\nsuch that \\(\\conj{\\diff f} = \\diff{\\conj{f}}\\)\nfor all smooth \\(f\\in\\Cont_0(X)\\).\nThere is a unique conjugate-linear involution on\n\\(\\Cont_0(X,\\Cliff X)\\) with\n\\[\n\\conj{\\omega_1 \\dotsm \\omega_m}\n= \\conj{\\omega_1} \\dotsm \\conj{\\omega_m}\n\\]\nfor all sections \\(\\omega_1,\\dotsc,\\omega_m\\)\nof \\(T^* X \\otimes\\mathbb{C}\\).\nThis involution is also compatible with the multiplication and the\n\\(\\mathbb{Z}\/2\\)\\nobreakdash-grading.  So it turns \\(\\Cont_0(X,\\Cliff X)\\)\ninto a \\(\\mathbb{Z}\/2\\)\\nobreakdash-graded ``real'' \\(\\Cst\\)\\nobreakdash-algebra.\n\nLet~\\(L^2(\\Lambda^*(X))\\)\nbe the Hilbert space of square-integrable complex differential forms\non~\\(X\\).\nThis is the underlying Hilbert space of Kasparov's fundamental\nclass.  It is \\(\\mathbb{Z}\/2\\)\\nobreakdash-graded\nso that sections of~\\(\\Lambda^{2\\ell}(X)\\)\nare even and sections of~\\(\\Lambda^{2\\ell+1}(X)\\)\nare odd.  There is a unique conjugate-linear, isometric involution\non \\(L^2(\\Lambda^*(X))\\) with\n\\[\n\\conj{\\omega_1 \\wedge \\dotsb \\wedge \\omega_\\ell} =\n\\conj{\\omega_1} \\wedge \\dotsb \\wedge \\conj{\\omega_\\ell}\n\\]\nfor all complex \\(1\\)\\nobreakdash-forms \\(\\omega_1,\\dotsc,\\omega_\\ell\\).\nIt commutes with the \\(\\mathbb{Z}\/2\\)\\nobreakdash-grading,\nso that \\(L^2(\\Lambda^*(X))\\)\nbecomes a \\(\\mathbb{Z}\/2\\)\\nobreakdash-graded\n``real'' Hilbert space.\n\nGiven a complex \\(1\\)\\nobreakdash-form~\\(\\omega\\)\nand a differential form~\\(\\eta\\),\nlet \\(\\lambda_\\omega(\\eta) \\mathrel{\\vcentcolon=} \\omega \\wedge \\eta\\).\nThese operators satisfy the relations\n\\begin{equation}\n  \\label{eq:lambda_and_star}\n  \\lambda_\\omega \\lambda_\\eta + \\lambda_\\eta \\lambda_\\omega=0,\\qquad\n  \\lambda_\\omega^* \\lambda_\\eta + \\lambda_\\eta \\lambda_\\omega^*\n  = \\braket{\\omega}{\\eta}\n\\end{equation}\nfor all complex \\(1\\)\\nobreakdash-forms \\(\\omega,\\eta\\),\nwhere \\(\\braket{\\omega}{\\eta}\\in\\Cont_0(X)\\)\ndenotes the pointwise inner product, which acts on\n\\(L^2(\\Lambda^*(X))\\)\nby pointwise multiplication.\nThe representation of \\(\\Cont_0(X,\\Cliff X)\\)\non \\(L^2(\\Lambda^*(X))\\)\nis defined by letting a complex \\(1\\)\\nobreakdash-form~\\(\\omega\\),\nviewed as an element of \\(\\Cont_0(X,\\Cliff X)\\),\nact by \\(\\lambda_\\omega + \\lambda_{\\omega^*}^*\\).\nHere~\\(\\omega^*\\)\nis the adjoint of~\\(\\omega\\)\nin the \\(\\Cst\\)\\nobreakdash-algebra \\(\\Cont_0(X,\\Cliff X)\\),\nthat is,\n\\(\\omega^*(x) = \\omega(x)^*\\)\nfor all \\(x\\in X\\),\nwhere \\(\\omega(x)^* \\in T^*_x X \\otimes \\mathbb{C}\\)\nis the pointwise complex conjugation in the second tensor\nfactor~\\(\\mathbb{C}\\).\nThis defines a $^*$\\nobreakdash-{}representation of \\(\\Cont_0(X,\\Cliff X)\\)\nby~\\eqref{eq:lambda_and_star}.  It is grading-preserving and real as\nwell.\n\nLet~\\(d\\) be the de Rham differential, defined on smooth\nsections of~\\(\\Lambda^*(X)\\)\nwith compact support, and let~\\(d^*\\)\nbe its adjoint.  The unbounded operator \\(\\mathcal{D} \\mathrel{\\vcentcolon=} d+d^*\\)\nis essentially self-adjoint because~\\(X\\)\nis complete.  So\n\\[\nF \\mathrel{\\vcentcolon=} (1+\\mathcal{D}^2)^{-1\/2} \\mathcal{D}\n\\]\nis a well defined self-adjoint operator.  The operator~\\(d\\) is odd\nand real.  This is inherited by~\\(\\mathcal{D}\\)\nand~\\(F\\).\nKasparov shows that \\((1-F^2)\\cdot a\\)\nand \\([F,a]\\)\nare compact for all \\(a\\in \\Cont_0(X,\\Cliff X)\\).\nThus \\(\\alpha_X \\mathrel{\\vcentcolon=} (L^2(\\Lambda^*(X)),F)\\)\nis a cycle for the ``real'' Kasparov group\n\\(\\KK^\\mathbb{R}_0(\\Cont_0(X,\\Cliff X),\\mathbb{C})\\).\nWe call this the \\emph{fundamental class} of the ``real''\nmanifold~\\(X\\).\n(Kasparov calls it ``Dirac element'' instead.)\n\nIn particular, the fundamental class of the ``real''\nmanifold~\\(\\mathbb{R}^{p,q}\\)\nbecomes the Bott periodicity generator~\\(\\alpha_{p,q}\\)\nfrom~\\cite{Kasparov:Operator_K} when we trivialise the Clifford\nalgebra bundle on~\\(\\mathbb{R}^{p,q}\\)\nin the obvious way.  So\n\\(\\alpha_{\\mathbb{R}^{p,q}}\\in \\KK^\\mathbb{R}_0(\\Cont_0(\\mathbb{R}^{p,q}) \\otimes\n\\Cliff_{p,q},\\mathbb{C})\\)\nis invertible.\n\nWe give~\\(\\mathbb{T}^d\\)\nthe \\(\\mathbb{T}^d\\)\\nobreakdash-invariant\nRiemannian metric to build its fundamental class.\nThe torus~\\(\\mathbb{T}^d\\)\nis parallelisable as a ``real'' manifold: its tangent bundle is\nisomorphic to \\(\\mathbb{T}^d \\times \\mathbb{R}^{0,d}\\).\nThis induces an isomorphism\n\\(\\Cont(\\mathbb{T}^d, \\Cliff \\mathbb{T}^d) \\cong \\Cont(\\mathbb{T}^d) \\otimes \\Cliff_{0,d}\\).\nSo the fundamental class~\\(\\alpha_{\\mathbb{T}^d}\\)\nalso gives an element in \\(\\KK^\\mathbb{R}_d(\\Cont(\\mathbb{T}^d),\\mathbb{C})\\).\n\nLet~\\(\\Hils[L]\\)\nbe a separable ``real'' Hilbert space and build \\(\\Cst_\\Roe(\\mathbb{Z}^d)\\)\non the ``real'' Hilbert space \\(\\ell^2(\\mathbb{Z}^d,\\Hils[L])\\).\nThere is an obvious embedding\n\\(\\Cst(\\mathbb{Z}^d) \\otimes \\mathbb K(\\Hils[L]) \\subseteq \\Cst_\\Roe(\\mathbb{Z}^d)\\).  Let\n\\[\n\\alpha_{\\mathbb{T}^d}^{\\Hils[L]} \\in\n\\KK^\\mathbb{R}_0(\\Cont(\\mathbb{T}^d) \\otimes \\Cliff_{0,d} \\otimes \\mathbb K(\\Hils[L]), \\mathbb{C})\n\\cong\n\\KK^\\mathbb{R}_0(\\Cst(\\mathbb{Z}^d) \\otimes \\Cliff_{0,d} \\otimes \\mathbb K(\\Hils[L]), \\mathbb{C})\n\\]\nbe the exterior product of the fundamental class~\\(\\alpha_{\\mathbb{T}^d}\\)\nand the Morita equivalence \\(\\mathbb K(\\Hils[L]) \\sim \\mathbb{C}\\).\nThis is the Kasparov cycle with underlying \\(\\mathbb{Z}\/2\\)\\nobreakdash-graded\n``real'' Hilbert space\n\\(L^2(\\mathbb{T}^d, \\Lambda^*(\\mathbb{C}^d)) \\otimes \\Hils[L]\\)\nwith the operator \\(\\tilde{F} \\mathrel{\\vcentcolon=} F\\otimes 1\\)\nwith~\\(F\\)\nas above for the manifold \\(X=\\mathbb{T}^d\\).\nSo~\\(\\tilde{F}\\)\nis an odd, self-adjoint, real bounded operator with\n\\begin{equation}\n  \\label{eq:Kasparov_cycle}\n  [\\tilde{F},T],(1-\\tilde{F}^2)\\cdot T\\in\n  \\mathbb K(\\ell^2(\\mathbb{Z}) \\otimes \\Lambda^*(\\mathbb{C}^d) \\otimes \\Hils[L])\n\\end{equation}\nfor all\n\\(T\\in\\Cst(\\mathbb{Z}^d) \\otimes \\Cliff_{0,d} \\otimes \\mathbb K(\\Hils[L])\\)\n(the commutator is the graded one).\n\n\\begin{theorem}\n  \\label{the:fundamental_class}\n  Equation~\\eqref{eq:Kasparov_cycle} still holds for\n  \\(T\\in\\Cst_\\Roe(\\mathbb{Z}^d) \\otimes \\Cliff_{0,d}\\).\n  This gives\n  \\[\n  \\alpha'_{\\mathbb{T}^d}\n  \\mathrel{\\vcentcolon=} [(L^2(\\mathbb{T}^d, \\Lambda^*(\\mathbb{C}^d)) \\otimes \\Hils[L], \\tilde{F})]\n  \\in \\KK^\\mathbb{R}_0(\\Cst_\\Roe(\\mathbb{Z}^d) \\otimes \\Cliff_{0,d}, \\mathbb{C}).\n  \\]\n  The following diagram in~\\(\\KK^\\mathbb{R}\\) commutes:\n  \\[\n  \\begin{tikzcd}[column sep=2.3em]\n   \n    \\Cont_0(\\mathbb{R}^{0,d}, \\Cliff_{0,d}) \\arrow[r, \"\\textup{incl.}\"]\n    \\arrow[dr, \"\\alpha_{\\mathbb{R}^{0,d}}\", \"\\cong\"'] &\n    \\Cont(\\mathbb{T}^d, \\Cliff_{0,d}) \\arrow[r, \"\\textup{Fourier}\"]\n    \\arrow[d, \"\\alpha_{\\mathbb{T}^d}\"] &\n    \\Cst(\\mathbb{Z}^d) \\otimes \\Cliff_{0,d} \\arrow[r, \"\\textup{incl.}\"] &\n    \\Cst_\\Roe(\\mathbb{Z}^d) \\otimes \\Cliff_{0,d} \\arrow[dll, \"\\alpha'_{\\mathbb{T}^d}\"]\n    \\\\ &\\mathbb{C}\n  \\end{tikzcd}\n  \\]\n\\end{theorem}\n\n\\begin{corollary}\n  \\label{cor:K_periodic_split-injective}\n  The inclusion \\(\\Cont_0(\\mathbb{R}^{0,d}) \\to \\Cst_\\Roe(\\mathbb{Z}^d)\\)\n  induces a split injective map\n  \\(\\KR_*(\\Cont_0(\\mathbb{R}^{0,d})) \\to \\KR_*(\\Cst_\\Roe(\\mathbb{Z}^d))\\).\n  The map\n  \\(\\K_{*+d}(\\Cst_\\Roe(\\mathbb{Z}^d)_\\mathbb{F}) \\to \\K_*(\\mathbb{F})\\)\n  induced by~\\(\\alpha'_{\\mathbb{T}^d}\\)\n  is an isomorphism.  Analogous statements hold in complex\n  \\(\\K\\)\\nobreakdash-theory.\n\\end{corollary}\n\n\\begin{proof}[Proof of the corollary]\n  Both \\(\\KR_*(\\Cont_0(\\mathbb{R}^{0,d}))\\)\n  and \\(\\KR_*(\\Cst_\\Roe(\\mathbb{Z}^d))\\)\n  are isomorphic to free \\(\\K_*(\\mathbb{R})\\)-modules\n  with a generator in degree~\\(-d\\).\n  The Bott periodicity generator~\\(\\alpha_{\\mathbb{R}^{0,d}}\\)\n  maps the generator of \\(\\KR_{-d}(\\Cont_0(\\mathbb{R}^{0,d}))\\)\n  onto a generator of \\(\\K_0(\\mathbb{R})\\).\n  The commuting diagram in Theorem~\\ref{the:fundamental_class} shows\n  that its image in \\(\\KR_{-d}(\\Cst_\\Roe(\\mathbb{Z}^d))\\)\n  must be a generator as well.  So~\\(\\alpha'_{\\mathbb{T}^d}\\)\n  acts by multiplication with~\\(\\pm1\\)\n  on a generator.  Since~\\(\\alpha'_{\\mathbb{T}^d}\\)\n  is a \\(\\K\\)\\nobreakdash-homology\n  class, the map on \\(\\K\\)\\nobreakdash-theory\n  that it induces is a \\(\\K_*(\\mathbb{R})\\)-module\n  homomorphism.  Hence it is multiplication by~\\(\\pm1\\)\n  everywhere once this happens on a generator.  So the map\n  on~\\(\\KR_*\\)\n  induced by~\\(\\alpha'_{\\mathbb{T}^d}\\)\n  is invertible.  The same proof works for complex \\(\\K\\)\\nobreakdash-theory.\n\\end{proof}\n\nWe have already shown that all but one of the free\n\\(\\K_*(\\mathbb{F})\\)-module\nsummands in \\(\\K_*(\\Cst(\\mathbb{Z}^d)_\\mathbb{F})\\)\nare killed by the map to \\(\\K_*(\\Cst_\\Roe(\\mathbb{Z}^d)_\\mathbb{F})\\).\nWhen we combine this with the above corollary, it follows that the\nkernel of the map from \\(\\K_*(\\Cst(\\mathbb{Z}^d)_\\mathbb{F})\\)\nto \\(\\K_*(\\Cst_\\Roe(\\mathbb{Z}^d)_\\mathbb{F})\\)\nis exactly the sum of the images of \\(\\K_*(\\iota_k)\\)\nfor \\(k=1,\\dotsc,d\\),\nthat is, the subgroup generated by the \\(\\K\\)\\nobreakdash-theory\nclasses of weak topological insulators.\n\nWe still have to prove Theorem~\\ref{the:fundamental_class}.  The\nleft triangle in the diagram in Theorem~\\ref{the:fundamental_class}\ncommutes because of the following general fact:\n\n\\begin{proposition}\n  \\label{pro:fundamental_class_open_restrict}\n  Let \\(U\\subseteq X\\)\n  be an open subset of a ``real'' manifold~\\(X\\)\n  that is invariant under the real involution.\n  Give \\(U\\)\n  and~\\(X\\)\n  some complete Riemannian metrics.\n  The Kasparov product of the ideal inclusion\n  \\(j\\colon \\Cont_0(U,\\Cliff U) \\hookrightarrow \\Cont_0(X,\\Cliff X)\\)\n  and the fundamental class\n  \\(\\alpha_X\\in \\KK_0^\\mathbb{R}(\\Cont_0(X,\\Cliff X), \\mathbb{C})\\)\n  is the fundamental class\n  \\(\\alpha_U\\in \\KK_0^\\mathbb{R}(\\Cont_0(U,\\Cliff U), \\mathbb{C})\\).\n  In particular, the fundamental class does not depend on the choice\n  of the Riemannian metric.\n\\end{proposition}\n\n\\begin{proof}\n  Both Kasparov cycles \\(j^*(\\alpha_X)\\)\n  and~\\(\\alpha_U\\)\n  live on Hilbert spaces of \\(L^2\\)\\nobreakdash-differential\n  forms on~\\(U\\).\n  Here square-integrability is with respect to different metrics.\n  The resulting Hilbert spaces are isomorphic by pointwise\n  application of a suitable strictly positive, smooth function\n  \\(X\\to \\mathbb B(\\Lambda^* X)\\).\n  This isomorphism also respects the \\(\\mathbb{Z}\/2\\)\\nobreakdash-grading\n  and the real involution.  The operator~\\(\\mathcal{D}\\)\n  used to construct~\\(\\alpha_X\\)\n  is a first-order differential operator.  Hence \\(F \\mathrel{\\vcentcolon=}\n  (1+\\mathcal{D}^2)^{-1\/2} \\mathcal{D}\\)\n  is an order-zero pseudodifferential operator, and it has the same\n  symbol as~\\(\\mathcal{D}\\).\n  This is the function\n  \\[\n  S^* X \\to \\mathbb B(\\Lambda^*(X)),\\qquad\n  (x,\\xi) \\mapsto \\lambda_\\xi + \\lambda_\\xi^*.\n  \\]\n  The symbol of the operator~\\(F\\)\n  of~\\(\\alpha_U\\)\n  is given by the same formula, except that the adjoint is for\n  another Riemannian metric.  So the unitary\n  between the spaces of \\(L^2\\)\\nobreakdash-forms\n  will also identify these symbols.  The class of the Kasparov\n  cycle defined by an order-zero pseudodifferential operator~\\(F\\)\n  depends only on the symbol of~\\(F\\).\n  So \\(j^*(\\alpha_X)\\)\n  and~\\(\\alpha_U\\)\n  have the same class in \\(\\KK_0^\\mathbb{R}(\\Cont_0(U,\\Cliff U), \\mathbb{C})\\).\n  The last statement is the case \\(U=X\\)\n  of the proposition.\n\\end{proof}\n\nNow we build the Kasparov cycle\n\\(\\alpha'_{\\mathbb{T}^d}\\in \\KK_0^\\mathbb{R}(\\Cst_\\Roe(\\mathbb{Z}^d)\\otimes \\Cliff_{0,d},\\mathbb{C})\\).\nLet \\(z_j\\colon \\mathbb{T}^d\\to\\mathbb{C}\\)\nbe the \\(j\\)th coordinate function and let\n\\[\nz^k \\mathrel{\\vcentcolon=} z_1^{k_1} \\dotsm z_d^{k_d}\n\\qquad\\text{for }\nk=(k_1,\\dotsc,k_d)\\in\\mathbb{Z}^d.\n\\]\nThe real involution on~\\(\\mathbb{T}^d\\)\nis defined so that these are real elements of \\(\\Cont(\\mathbb{T}^d)\\).\nHence the \\(1\\)\\nobreakdash-forms \\(z_j^{-1} \\diff z_j\\)\nfor \\(j=1,\\dotsc,d\\)\nare real.  They form a basis of the space of \\(1\\)\\nobreakdash-forms\nas a \\(\\Cont(\\mathbb{T}^d)\\)-module.\nThe differential forms\n\\[\nz^k\\cdot (z_{i_1} \\dotsm z_{i_\\ell})^{-1}\n\\diff z_{i_1} \\wedge \\dotsb \\wedge \\diff z_{i_\\ell}\n\\]\nfor \\(k\\in\\mathbb{Z}^d\\)\nand \\(1\\le i_1 < i_2 < \\dotsb < i_\\ell\\le d\\)\nform a real, orthonormal basis of the Hilbert space\n\\(L^2(\\Lambda^*(\\mathbb{T}^d))\\).\nHence there is a unitary operator\n\\begin{multline*}\n  U\\colon\n  \\ell^2(\\mathbb{Z}^d) \\otimes \\Lambda^*(\\mathbb{C}^d) \\xrightarrow\\sim\n  L^2(\\Lambda^*(\\mathbb{T}^d)),\\\\\n  \\delta_k \\otimes e_{i_1} \\wedge \\dotsb \\wedge e_{i_\\ell} \\mapsto\n  z^k\\cdot (z_{i_1} \\dotsm z_{i_\\ell})^{-1}\n  \\diff z_{i_1} \\wedge \\dotsb \\wedge \\diff z_{i_\\ell}.\n\\end{multline*}\nThis unitary is grading-preserving and real for the\n\\(\\mathbb{Z}\/2\\)\\nobreakdash-grading and real structure on\n\\(\\ell^2(\\mathbb{Z}^d) \\otimes \\Lambda^\\ell(\\mathbb{C}^d)\\)\nwhere the standard basis vector\n\\(\\delta_k \\otimes e_{i_1} \\wedge \\dotsb \\wedge e_{i_\\ell}\\)\nis real and is even or odd depending on the parity of~\\(\\ell\\).\n\nThe above trivialisation of the cotangent bundle of~\\(\\mathbb{T}^d\\)\ngives the isomorphism\n\\[\n\\Cont(\\mathbb{T}^d) \\otimes \\Cliff_{0,d} \\xrightarrow\\sim \\Cont(\\mathbb{T}^d,\\Cliff \\mathbb{T}^d),\n\\qquad\n\\gamma_j\\mapsto \\ima z_j^{-1}\\,\\diff z_j;\n\\]\nrecall that \\(\\gamma_1,\\dotsc,\\gamma_d\\)\nare the odd, self-adjoint, anti-commuting unitaries that\ngenerate~\\(\\Cliff_{0,d}\\).\nThe action of \\(\\Cont(\\mathbb{T}^d,\\Cliff \\mathbb{T}^d)\\)\non \\(L^2(\\Lambda^* \\mathbb{T}^d)\\)\nnow translates to an action of\n\\(\\Cont(\\mathbb{T}^d) \\otimes \\Cliff_{0,d}\\)\non \\(\\ell^2(\\mathbb{Z}^d) \\otimes \\Lambda^*(\\mathbb{C}^d)\n\\cong \\ell^2(\\mathbb{Z}^d, \\Lambda^*(\\mathbb{C}^d))\\).\nNamely, the scalar-valued function \\(z^k\\in\\Cont(\\mathbb{T}^d)\\)\nacts by the shift \\((\\tau_k f)(n) \\mathrel{\\vcentcolon=} f(n-k)\\)\nfor all \\(k,n\\in\\mathbb{Z}^d\\),\n\\(f\\in \\ell^2(\\mathbb{Z}^d,\\Lambda^*(\\mathbb{C}^d))\\).\nAnd the Clifford generator \\(\\gamma_j\\in\\Cliff_{0,d}\\)\nacts by\n\\[\n(\\gamma_j f)(n)\n= \\ima \\lambda_{e_j}\\bigl(f(n)\\bigr)\n- \\ima \\lambda_{e_j}^*\\bigl(f(n)\\bigr).\n\\]\nThe unitary~\\(U^*\\)\nmaps the domain of~\\(d\\)\nto the space of rapidly decreasing functions\n\\(\\mathbb{Z}^d\\to\\Lambda^*(\\mathbb{C}^d)\\),\nwhere~\\(U^* d U\\)\nacts by pointwise application of the function\n\\[\nA\\colon \\mathbb{Z}^d\\to \\mathbb B(\\Lambda^*(\\mathbb{C}^d)),\\qquad\nn \\mapsto \\lambda_n\n= \\sum_{j=1}^d n_j\\cdot \\lambda_{e_j},\n\\]\nbecause\n\\[\nd\\left(z^k\\cdot \\frac{\\diff z_{i_1}}{z_{i_1}} \\wedge \\dotsc\n\\wedge \\frac{\\diff z_{i_\\ell}}{z_{i_\\ell}}\\right)\n= \\sum_{j=1}^d k_j z^k \\cdot \\frac{\\diff z_j}{z_j}\n\\wedge \\frac{\\diff z_{i_1}}{z_{i_1}} \\wedge \\dotsc\n\\wedge \\frac{\\diff z_{i_\\ell}}{z_{i_\\ell}}.\n\\]\nSo~\\(U^* \\mathcal{D} U\\)\nacts by pointwise application of the matrix-valued function \\(A+A^*\\)\non the space of rapidly decreasing functions\n\\(\\mathbb{Z}^d\\to\\Lambda^*(\\mathbb{C}^d)\\).\nWe compute\n\\[\n(A+A^*)^2(n)\n= \\lambda_n \\lambda_n^* + \\lambda_n^* \\lambda_n\n= \\norm{n}^2.\n\\]\nSo~\\(U^* F U\\)\nacts by pointwise application of the matrix-valued function\n\\[\n\\hat\\alpha_{\\mathbb{Z}^d}\\colon\n\\mathbb{Z}^d \\to \\mathbb B(\\Lambda^*(\\mathbb{C}^d)),\\qquad\nn \\mapsto (1+\\norm{n}^2)^{-1\/2} (\\lambda_n + \\lambda_n^*).\n\\]\n\nNext we take the exterior product with the Morita equivalence\nbetween \\(\\mathbb K(\\Hils[L])\\)\nand~\\(\\mathbb{C}\\).\nThis simply gives the Hilbert space\n\\(\\ell^2(\\mathbb{Z}^d,\\Lambda^*(\\mathbb{C}^d)) \\otimes \\Hils[L]\\)\nwith the induced \\(\\mathbb{Z}\/2\\)\\nobreakdash-grading\nand ``real'' structure, the exterior tensor product representation\nof \\(\\Cont(\\mathbb{T}^d)\\otimes\\Cliff_{0,d}\\otimes \\mathbb K(\\Hils[L])\\),\nand with the operator \\(F\\otimes 1_{\\Hils[L]}\\).\nThis is a Kasparov cycle for\n\\(\\KK^\\mathbb{R}_0(\\Cst(\\mathbb{Z}^d) \\otimes \\Cliff_{0,d} \\otimes \\mathbb K(\\Hils[L]),\n\\mathbb{C})\\).\nIn particular, the operator \\(\\tilde{F} \\mathrel{\\vcentcolon=} U^* F U\\otimes 1_{\\Hils[L]}\\)\nis real, odd, and self-adjoint.\nLet \\(T\\in \\Cst_\\Roe(\\mathbb{Z}^d) \\subseteq \\mathbb B(\\ell^2(\\mathbb{Z}^d,\\Hils[L]))\\)\nand \\(S\\in\\Cliff_{0,d}\\).\nWe must show that \\((1-\\tilde{F}^2) \\cdot (T\\otimes S)\\)\nand \\([\\tilde{F}^2, T\\otimes S]\\)\nare compact operators.\nThe operator \\(1 - \\tilde{F}^2\\)\nacts by pointwise multiplication with \\((1+\\norm{n}^2)^{-1}\\).\nSince~\\(T\\)\nis locally compact and~\\(\\Lambda^* \\mathbb{C}^d\\)\nhas finite dimension, the operator\n\\((1-\\tilde{F}^2) \\cdot (T\\otimes S)\\)\nis compact.  Describe~\\(T\\)\nas a block matrix \\((T_{x,y})_{x,y\\in\\mathbb{Z}^d}\\)\nwith \\(T_{x,y}\\in\\mathbb B(\\Hils[L])\\).\nThe operator~\\(\\tilde{F}\\)\nanti-commutes with \\(1\\otimes S\\).\nSo the graded commutator\n\\([A+A^*,T\\otimes S] = [A+A^*,T\\otimes 1]\\cdot (1\\otimes S)\\)\ncorresponds to the block matrix with \\((x,y)\\)-entry\n\\[\nT_{x,y} \\otimes (\\lambda_{x-y} + \\lambda_{x-y}^*) S\n\\in \\mathbb B(\\Hils[L]\\otimes \\Lambda^* \\mathbb{C}^d).\n\\]\nAssume that~\\(T\\)\nis \\(R\\)\\nobreakdash-controlled,\nthat is, \\(T_{x,y}=0\\)\nif \\(\\norm{x-y}>R\\),\nand that \\(\\sup_x \\sum_y \\norm{T_{x,y}}\\)\nand \\(\\sup_y \\sum_x \\norm{T_{x,y}}\\)\nare bounded; block matrices with these two properties give bounded\noperators, and these are dense in the Roe \\(\\Cst\\)\\nobreakdash-algebra.\nFor such~\\(T\\),\nthe commutator \\([A+A^*,T\\otimes S]\\)\nsatisfies analogous bounds because\n\\(\\norm{\\lambda_{x-y}+\\lambda_{x-y}^*} \\le 2 \\norm{x-y} \\le 2 R\\)\nwhenever \\(T_{x,y}\\neq0\\).\nSo the set of \\(T\\in \\Cst_\\Roe(X)\\)\nfor which \\([A+A^*,T \\otimes S]\\)\nis bounded is dense in \\(\\Cst_\\Roe(X)\\).\nThus~\\(A+A^*\\)\ndefines a spectral triple over\n\\(\\Cst_\\Roe(X) \\otimes \\Cliff_{0,d}\\).\nAs a consequence, \\([\\tilde{F}, T\\otimes 1]\\)\nis compact for all \\(T\\in\\Cst_\\Roe(X) \\otimes \\Cliff_{0,d}\\).\nThis finishes the proof of Theorem~\\ref{the:fundamental_class}.\n\n\n\\subsection{Another topological artefact of the tight binding\n  approximation}\n\\label{sec:artefact_tight_binding}\n\nWe already argued in the introduction that the tight binding\napproximation may produce topological artefacts.  Namely, it suggests\nto use the uniform Roe \\(\\Cst\\)\\nobreakdash-algebra\ninstead of the Roe \\(\\Cst\\)\\nobreakdash-algebra,\nwhose \\(\\K\\)\\nobreakdash-theory\nis much larger.  We briefly mention another artefact caused by the\ntight binding approximation.\n\nWe work in Bloch--Floquet theory for greater clarity.  The\nFermi projection of a Hamiltonian is described by a vector bundle\n\\(V\\twoheadrightarrow \\mathbb{T}^d\\)\nover the \\(d\\)\\nobreakdash-torus,\nmaybe with extra symmetries.  Here~\\(d\\)\nis the dimension of the material, which is \\(2\\)\nor~\\(3\\)\nin the most relevant cases.  In \\(\\K\\)\\nobreakdash-theory,\ntwo vector bundles \\(\\xi_1,\\xi_2\\)\nare identified if they are \\emph{stably isomorphic}, that is, there is\na trivial vector bundle~\\(\\vartheta\\)\nwith \\(\\xi_1\\oplus\\vartheta \\cong \\xi_2\\oplus\\vartheta\\).\nSeveral authors put in extra work to refine the classification of\nvector bundles (with symmetries) provided by \\(\\K\\)\\nobreakdash-theory\nto a classification up to isomorphism, see\n\\cites{De_Nittis-Gomi:Real_Bloch, De_Nittis-Gomi:Quaternionic_Bloch,\n  Kennedy:Thesis, Kennedy-Zirnbauer:Bott_gapped}.  Here we argue that\nsuch a refinement of the classification is of little physical\nsignificance.  The tight binding approximation leaves out energy bands\nthat are sufficiently far below the Fermi level.  Their inclusion only\nadds a trivial vector bundle -- but this is the difference\nbetween stable isomorphism and isomorphism.\n\n\\begin{theorem}[\\cite{Husemoller:Fibre_bundles}*{Chapter 8, Theorem 1.5}]\n  \\label{the:K_stable_range}\n  Let~\\(X\\)\n  be an \\(n\\)\\nobreakdash-dimensional\n  CW-complex and let \\(\\xi_1\\)\n  and~\\(\\xi_2\\)\n  be two \\(k\\)\\nobreakdash-dimensional\n  vector bundles.  Let \\(c=1,2,4\\)\n  depending on whether the vector bundles are real, complex or\n  quaternionic.  Assume \\(k \\ge \\lceil (n+2)\/c \\rceil - 1\\).\n  If \\(\\xi_1\\)\n  and~\\(\\xi_2\\) are stably isomorphic, then they are isomorphic.\n\\end{theorem}\n\nSo for a \\(3\\)\\nobreakdash-dimensional\nspace~\\(X\\),\nthe isomorphism and stable isomorphism classification agree for real\nvector bundles of dimension at least~\\(4\\),\nfor complex vector bundles of dimension at least~\\(2\\),\nand for all quaternionic vector bundles.\nFor instance, consider the material Bi$_2$Se$_3$ studied in\n\\cites{Zhang-Liu-Qi-Adi-Fang-Zhang:Insulators_BiSe,\n  Liu-Qi-Zhang-Dai-Fang-Zhang:Model_Hamiltonian}.  The model\nHamiltonian in \\cites{Zhang-Liu-Qi-Adi-Fang-Zhang:Insulators_BiSe,\n  Liu-Qi-Zhang-Dai-Fang-Zhang:Model_Hamiltonian} focuses on four\nbands, of which half are below and half above the Fermi energy.  But\nthe dimension of the physically relevant vector bundle is\n\\(2\\cdot 83 + 3\\cdot 34 = 268\\),\nthe number of electrons per unit cell of the crystal; each atom of\nBismuth has 83~electrons and each atom of Se has 34~electrons.\n\n\nThe theorem cited above does not take into account a real involution\non the space~\\(X\\).\nThe proof of Theorem~\\ref{the:K_stable_range} is elementary enough,\nhowever, to extend to ``real'' vector bundles over ``real'' manifolds.\nTo see this, one first describes a ``real'' manifold as a\n\\(\\mathbb{Z}\/2\\)\\nobreakdash-CW-complex.\nThe main step in the proof of Theorem~\\ref{the:K_stable_range} is to\nbuild nowhere vanishing sections of vector bundles, assuming that the\nfibre dimension is large enough.  This allows to split off a trivial\nrank-\\(1\\)\nvector bundle as a direct summand.  Similarly, if two vector bundles\nwith nowhere vanishing sections are homotopic, then there is a nowhere\nvanishing section for the homotopy if the dimension of the fibres is\nlarge enough.  The only change in the ``real'' case is that we need\na \\(\\mathbb{Z}\/2\\)\\nobreakdash-equivariant\nnowhere vanishing section of a ``real'' vector bundle to split off\ntrivial summands.  Such sections are built by induction over the\ncells of the \\(\\mathbb{Z}\/2\\)\\nobreakdash-CW-complex.\nThe \\(\\mathbb{Z}\/2\\)\\nobreakdash-action\non the interior of such a cell is either free or trivial.  In the\nfirst case, a \\(\\mathbb{Z}\/2\\)\\nobreakdash-equivariant\nsection is simply a section on one half of the cell.  In the second\ncase, the cell is contained in the fixed-point submanifold, and we\nneed a nowhere vanishing section of a real vector bundle in the usual\nsense.  So the argument in~\\cite{Husemoller:Fibre_bundles} allows to\nbuild nowhere vanishing real sections of ``real'' vector bundles\nunder the same assumptions on the dimension as for real vector bundles.\n\n\\begin{bibdiv}\n  \\begin{biblist}\n    \\bibselect{references}\n  \\end{biblist}\n\\end{bibdiv}\n\\end{document}\n\nWe propose the Roe C*-algebra from coarse geometry as a model for topological phases of disordered materials.  We explain the robustness of this C*-algebra and formulate the bulk-edge correspondence in this framework.  We describe the map from the K-theory of the group C*-algebra of Z^d to the K-theory of the Roe C*-algebra, both for real and complex K-theory.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nClustering of inertial particles in turbulent flows is relevant for meteorology\nand engineering, as well as fundamental research.  It is believed to play a\ncrucial role in rain-drop formation \\cite{falkovich_nature02}, as well as in the aggregation of\nproto-planetesimals in Keplerian accretion disks \\cite{bracco_pof99}. The physical mechanism which\noriginates such clustering is indeed rather simple: particles heavier than the\nfluid in which they are transported experience inertial forces which expel\nthem from vortices; particles lighter than the fluid are attracted into\nvortical structures, for similar reasons \\cite{Squires1991, cencini_jot06, Bec2005}.  In realistic flows, however,\nparticles are advected by the small scale vortical structures of turbulent\nflows: these have highly non-trivial statistical features, resulting in a\ncomplex clustering process which is still far from being completely understood.\nFrom the point of view of applications, the properties of concentration and\ndistribution of inertial particles play a crucial role in engineering and\nfor the design of industrial processes involving combustion and mixing \\cite{Warnatz2006,Rouson2001, Sbrizzai2006}. Suspensions of particles in viscoelastic fluids are used in many products of commercial and industrial relevance \\cite{barnes2003}.\n\n\nIn this paper we investigate, by means of direct numerical simulations \nof a turbulent flow, how the clustering properties of a dilute \nsuspension of inertial particles can be affected by the addition of \nsmall amounts of polymer additives. \nThe effects induced by polymers on turbulent flows \nare themselves of enormous relevance. It is enough to mention the celebrated \ndrag reduction effect which occurs in pipe flows \\cite{Lumley1969},\nor the recently discovered elastic turbulence regime \\cite{gs_nature00}.\nPolymers have striking effects also on Lagrangian properties of the flow. \nIn particular it has been shown that polymer addition in turbulent flows \nreduces the chaoticity of Lagrangian trajectories \\cite{bcm_prl03} \nand affects acceleration of fluid tracers \\cite{cmxb_njp08}.\nConversely in the elastic turbulence regime polymers are \nable to generate Lagrangian chaos in flows at vanishing \nReynolds number, which would be non chaotic in the Newtonian case\n\\cite{gs_nature01,bcm_prl03}.\n\nHere we show that the addition of polymers in a turbulent flow has \nimportant effects on the statistical properties of inertial particles \nwhich can result in both an increase or a decrease of the clustering.\nAn example of the effect of polymers on clustering \nis shown in Fig.~\\ref{fig1} which represents the distribution \nof an ensemble of inertial particles in a turbulent flow\nbefore and after the introduction of polymers. It is evident, already\nat the qualitative level of Fig.~\\ref{fig1}, that polymers are able\nto change the statistical distribution of particles.\nWe show that these effects can be understood and quantified \nin terms of the Lyapunov exponents of inertial particles, \nwhich are very sensitive to the presence of polymers.\nPrevious systematic investigations of inertial particle dynamics \nin Newtonian turbulent flows \\cite{calzavarini_jfm08} \nand stochastic flows \\cite{bec_pof03}\nhave shown that clustering (quantified by means of the Lyapunov Dimension of particle attractor) is maximum when the particle relaxation time\nis of the order of the shortest characteristic time of the flow. \n\n\n\n\\begin{figure}\n\\includegraphics[width=4.2cm]{fig1a.eps}\n\\includegraphics[width=4.2cm]{fig1b.eps}\n\\includegraphics[width=4.2cm]{fig1c.eps}\n\\includegraphics[width=4.2cm]{fig1d.eps}\n\\caption{Section on plane $z=0$ of the distribution of heavy particles\nwith $\\tau_S=0.035$ (upper panels) and light particles with $\\tau_S=0.03$ (lower panels) in statistically stationary conditions in a Newtonian \nflow (left) and a viscoelastic flow at $\\rm Wi=1$ (right).\nBoth flows are forced with the same \nforcing ${\\bf f}({\\bf x},t)$ $\\delta$-correlated in time and localized\non large scales. Numerical simulations are done by a pseudo-spectral,\nfully dealiased code at resolution $256^3$. For the viscoelastic simulations,\na small diffusive term is added to (\\ref{eq:4}) to prevent numerical\ninstabilities \\cite{sb_jnnfm95}.}\n\\label{fig1}\n\\end{figure}\n\n\nWe consider the case of a dilute suspension of small inertial particles, \nin which the effects of the disturbance flow induced by the particles \ncan be neglected. The dynamics of the suspension is hence \nmodeled by an ensemble of non-interacting point particles, \nwhich experience viscous drag and added mass forces. \nThe equation of motion of each particle reads \\cite{maxey_pof83}:  \n\\begin{eqnarray}\n{d {\\bm x} \\over dt}&=& \\bm v \n\\label{eq:1} \\\\\n{d {\\bm v} \\over dt}&=&-{1 \\over \\tau_S}\\left[\\bm v - \\bm u(\\bm x(t),t)\\right]+\n\\beta {d {\\bm u} \\over dt}\n\\label{eq:2}\n\\end{eqnarray}\nwhere $\\tau_S=a^2\/(3\\beta\\nu)$ is the Stokes relaxation time,\n$a$ is the particle radius, $\\beta=3\\rho_f\/(\\rho_f+2\\rho_p)$ \n($\\rho_p$ and $\\rho_f$ representing particle and fluid densities\nrespectively) and $\\nu$ is the kinematic viscosity of the fluid \n(replaced by the total viscosity $\\nu_T$ in a viscoelastic fluid, see below).\nLight (heavy) particles correspond to $\\beta>1$ ($\\beta<1$). \nIn this work we consider the two extreme cases of\nvery light particles (e.g. air bubble in water) for which $\\beta=3$\nand very heavy particles with $\\beta=0$. \nWe define the Stokes number as $\\rm St=\\tau_S \\lambda^ 0_1$, where\n$\\lambda^0_1$ is the maximum Lyapunov exponent of neutral Lagrangian tracers\n(i.e. $\\rm St=0$ particles) in the flow. With this definition, \nmaximum clustering is obtained for $\\rm St \\simeq 0.1$ \n\\cite{bec_pof03, calzavarini_jfm08}.\n\nThe viscoelastic flow ${\\bf u}({\\bf x},t)$ in which the particles are \nsuspended can be described by standard viscoelastic models,\nsuch as the Oldroyd-B model or the nonlinear FENE-P model, \nwhich accounts for the finite extensibility of polymers. \nIn spite of their simplicity, these models are able to reproduce\nmany relevant properties of dilute polymer solutions, including\nturbulent drag reduction~\\cite{sbh_pof97,bcm_pre05} and elastic turbulence\nphenomenology~\\cite{bbbcm_pre08}. \nHere we choose the Oldroyd-B model~\\cite{bird87}, in which the \ncoupled dynamics of the velocity field ${\\bf u}({\\bf x},t)$ \nand the polymer conformation tensor $\\sigma({\\bf x},t)$\n(which is proportional to local square polymer elongation) \nreads: \n\\begin{eqnarray}\n{\\partial \\bm u \\over \\partial t}+\\bm u\\cdot\\bm\\nabla\\bm u &=&\n-\\bm\\nabla p+\\nu\\nabla^2\\bm u+{2\\nu\\gamma \\over \\tau_p}\\bm\\nabla\\cdot\\sigma\n+{\\bm f}\n\\label{eq:3}\\\\\n{\\partial \\sigma \\over \\partial t}+\\bm u\\cdot\\nabla\\sigma &=&\n(\\nabla\\bm u)^T\\cdot\\sigma+\\sigma\\cdot(\\nabla\\bm u)-\n{2 \\over \\tau_p}(\\sigma-\\mathbb{I})\n\\label{eq:4}\n\\end{eqnarray}\nThe total viscosity of the solution $\\nu_T=\\nu (1+\\gamma)$ \nis written in terms of the kinematic viscosity of the solvent $\\nu$\nand the zero-shear contribution of the polymer $\\gamma$ which \nis proportional to the polymer concentration.\nThe polymer time $\\tau_p$ represents the longest relaxation time to the equilibrium configuration ($\\sigma=\\mathbb{I}$ in dimensionless units).\nViscoelasticity of the turbulent flow is parametrized by the \nWeissenberg number $\\rm Wi$, the ratio between $\\tau_p$ and a characteristic\ntime of the flow. Here we use $\\rm Wi=\\tau_p \\lambda_1^N$ \nwhere $\\lambda_1^N$ is the Lagrangian Lyapunov exponent of the Newtonian flow,\nbefore the addition of polymers (i.e. (\\ref{eq:3}) with $\\gamma=0$). We\nstress that $\\lambda_1^0$ introduced above refers instead to the specific flow\nthat carries the suspension and it clearly depends on $\\rm Wi$. \nTherefore $\\lambda_1^N\\equiv\\lambda_1^0|_{Wi=0}$.\n\n\\begin{table}[b]\n\\label{tab:1}\n\\begin{tabular}{c c c c c}\n$\\rm Wi$\t&$\\varepsilon_f$& $\\varepsilon_\\nu$&$u_{\\rm rms}$\t&$\\lambda^0_1$\t\\\\\n\\hline\n0\t&\t0.28\t&\t0.28\t&\t0.76\t&\t1.36\t\\\\\n0.5\t&\t0.28\t&\t0.18\t&\t0.73\t&\t1.08\t\\\\\n1\t&\t0.28\t&\t0.092\t&\t0.68\t&\t0.75\t\\\\\n\\end{tabular}\n\\caption{Parameters for the Newtonian and viscoelastic simulations. \nThe Weissenberg number $\\rm Wi$, energy input $\\varepsilon_f$, viscous\ndissipation rate $\\varepsilon_\\nu$, rms velocity $u_{rms}$ and Lagrangian \nLyapunov exponent $\\lambda_1^0$ of the carrier flow are shown. \nIn both viscoelastic runs an additional dissipative term was added on \npolymers (see text), with coefficient $\\nu_p=2.3\\times 10^3$ }\n\\label{table1}\n\\end{table}\n\nIn the following we discuss results obtained by integrating numerically\nthe viscoelastic model (\\ref{eq:3}-\\ref{eq:4}) at high resolution for different \nvalues of $\\rm Wi$ (see Table~\\ref{table1}). The flow is sustained by a \nstochastic Gaussian forcing ${\\bf f}({\\bf x},t)$ \n$\\delta$-correlated in time and localized on large scales. Fluid equations were integrated by means of a standard, fully dealiased, pseudo spectral code, on a cubic, triple-periodic domain with 256 grid points per side.\nWhen the flow reaches a turbulent, statistically stationary state, different \nfamilies (i.e. with different values of parameters $\\beta$ and $\\tau_S$) \nof inertial particles are injected, with initial homogeneous \ndistribution in space, and their motion integrated according to \n(\\ref{eq:1}-\\ref{eq:2}). For each value of $\\rm Wi$, we \nintegrated the motion of $1024$ particles for each of $21$ values\nof $\\tau_S$ and two values of $\\beta$, namely very heavy particles with \n$\\beta=0$ and \"bubbles\" with $\\beta=3$. \n\nAs an effect of inertia the distribution of particles does not remain \nhomogeneous and evolves to a fractal set dynamically evolving with \nthe flow, such as the examples shown in Fig.~\\ref{fig1}. In the language\nof dynamical systems, the equations (\\ref{eq:1}-\\ref{eq:2}) for particle motion\nrepresent a dissipative system whose chaotic trajectories evolve to \na fractal attractor (which evolves in time following the flow).\nA quantitative measure of clustering at small scales \nis therefore obtained by measuring the fractal dimension of the attractor \n(for each family of particles) using the Lyapunov dimension \n\\cite{bec_pof03,bec_pof06} defined in terms of Lyapunov exponents as \n$D_L=K+\\sum_{i=1}^{K} \\lambda_i\/|\\lambda_{K+1}|$ where $K$ is the \nlargest integer for which $\\sum_{i=1}^{K} \\lambda_i \\ge 0$ \\cite{ccv2010}.\nSince the space distribution of the particles is the projection \nof the attractor on the sub-space of particle positions, \nthe fractal dimension of clusters \nis given by $\\min(D_L, 3)$~\\cite{sy97,hk97}, \nprovided that the projection is generic \n(for a discussion on this issue see e.g. \\cite{bch07}). \nThis implies that $D_L<3$ gives fractal \ndistributions of dimension $D_L$, while $D_L>3$ corresponds to space-filling\nconfigurations, which however can be non-homogeneous. \n\n\\begin{figure}\n\\includegraphics[width=8.0cm]{fig2a.eps}\n\\includegraphics[width=8.0cm]{fig2b.eps}\n\\caption{\nLyapunov dimension for light (upper panel) and heavy (lower panel) particles\nplotted as a function of $\\tau_S$. Different lines correspond\nto the different Weissenberg numbers: $\\rm Wi=0$ (squares), $\\rm Wi=0.5$ (circles) and\n$\\rm Wi=1.0$ (triangles). \n}\n\\label{fig2}\n\\end{figure}\n\nIn Fig.~\\ref{fig2} we plot the fractal dimensions for both heavy\nand light particles as a function of $\\tau_S$ for the three simulations \nat different $\\rm Wi$.\nIt is evident that the addition of polymer changes substantially \nthe clustering properties of the particles, both increasing \n$D_L$ and reducing $D_L$ depending on value of $\\tau_S$. \nFigure~\\ref{fig1} shows examples of clustering reduction, for heavy and light particles respectively. The upper panels refer to heavy particles ($\\beta=0$) with $\\tau_S=0.035$, while the bottom ones are extracted from a simulation with $\\beta=3$ and $\\tau_S=0.03$. Both values of Stokes time are, for the Newtonian flow, on the left of the minimum in $D_L$. As a consequence, polymers produce a reduction of clustering. Such effect is more visible for light particles. A possible reason for this difference will be discussed further on.\n\nThe mechanism at the basis of this effect is not trivial and\nis a consequence of the change induced by the polymers on the \nsmall-scale properties of the turbulent flow. \nIn Fig.\\ref{fig3} we plot the energy spectra for the different $\\rm Wi$ numbers.\nThe effect of polymers is evident in the high-wavenumber range \nwhere velocity fluctuations are clearly suppressed, resulting in a \ndepletion of the energy spectrum, while large-scale fluctuations are \nunaffected.\n\nIndeed one can expect that only the fastest eddies of the flow, \ni.e. those whose eddy turn-over time $\\tau_\\ell$ is shorter that \nthe polymer relaxation time $\\tau_p$, \ncan produce a significant elongation of polymers. \nThe elastic feedback therefore affects only small scales $\\ell$ \nwith $\\tau_\\ell < \\tau_p$. \nConversely, large scales exhibit the same phenomenology of a \nNewtonian flow, characterized by a turbulent cascade with a \nconstant energy flux equal to the energy input rate $\\varepsilon_f$. \nThe turbulent cascade proceeds almost unaffected by the presence \nof polymers down to the Lumley scale $\\ell_L$,\nwhose eddy turn-over time equals the polymer relaxation time. \nA dimensional estimate, based on the Kolmogorov scaling for \nthe typical velocity $u_\\ell\\sim\\varepsilon_f^{1\/3}\\ell^{1\/3}$ \nand turn-over time $\\tau_\\ell=\\ell\/u_\\ell \\sim\\varepsilon_f^{-1\/3}\\ell^{2\/3}$ \nof an eddy of size $\\ell$, gives $\\ell_L=\\tau_p^{3\/2}\\varepsilon_f^{1\/2}$. \nPolymers would therefore affect only the small scales $\\ell < \\ell_L$.\nOur results are in qualitative agreement with this picture: the\n$\\rm Wi=0.5$ spectrum differs from the Newtonian \none only for $k\\gtrsim 8$, while\nat $\\rm Wi=1$ polymers are active over a larger range of scales.\nThe reduction of kinetic energy at small scales, due to the transfer of \nenergy to the polymers, is accompanied by a reduction of the viscous \ndissipation $\\varepsilon_\\nu=\\nu\\langle (\\nabla u)^2\\rangle$ \nat fixed energy input $\\varepsilon_f$, \nas can be seen from Table~\\ref{table1} and in the inset of \nFig.\\ref{fig3}. \nThis phenomenon has been previously observed both in forced and decaying\nsimulations of statistically homogeneous and isotropic turbulence\n(see, e.g., ~\\cite{dcbp05,pmp06}). \n\n\\begin{figure}\n\\includegraphics[width=9.0cm]{fig3.eps}\n\\caption{\nEnergy spectra for the Newtonian case ${\\rm Wi}=0$ (squares) and for the\nviscoelastic ones ${\\rm Wi}=0.5$ (circles) and ${\\rm Wi}=1$ (triangles). The\ndepletion due to polymer feed-back is evident on large wavenumbers, while the\nlarger scales are unaffected. The effect of polymers extends at lower\nwave-numbers as $\\rm Wi$ increases. Inset: viscous energy dissipation $\\varepsilon_\\nu$ during\na typical time interval in the stationary simulations, for the Newtonian (solid\nline), ${\\rm Wi}=0.5$ (dashed line) and ${\\rm Wi}=1$ (dash-dot) flows. The\ndecrease in $\\varepsilon_\\nu$ with ${\\rm Wi}$ is evident, as well as the reduction in\nfluctuations.\n}\n\\label{fig3}\n\\end{figure}\n\nThe suppression of small-scale motions caused by polymers \nhas major consequences also on the Lagrangian statistics. \nIt is responsible of the reduction of chaoticity \nof Lagrangian trajectories~\\cite{boffetta_prl03}.\nIndeed the chaoticity of the flow is directly related to its \nstretching efficiency via the Lyapunov exponents. \nWhen polymers are stretched, the elastic stress tensor produces a negative\nfeed-back on small scale stretching, thus reducing the degree of chaoticity of\nthe flow \\cite{boffetta_prl03,balkovsky_pre01}. This effect is clearly\nobservable in the decrease of the Lagrangian Lyapunov exponent of the flow \nat increasing polymer elasticity (see the inset of Fig.\\ref{fig4}). \n\nIt is worth to notice that, because of polymers counteraction, \nthe Lyapunov exponent of the resulting viscoelastic flow is smaller than $\\tau_p^{-1}$. \nIn other words, the $\\rm Wi$ number computed a posteriori (i.e. after polymer injection) \nis always smaller than unity. \nThis is not in contrast with the hypothesis that polymers have a strong active effect on the flow \nmainly when they are stretched, i.e. above the so-called coil-stretch transition, which is\nexpected to happen around $\\rm Wi\\simeq 1$ \\cite{balkovsky_prl00}.\nIndeed, the Lyapunov exponent simply provides a measure of the \naverage stretching in a chaotic flow. One should bear in mind that large fluctuations \nof the stretching rates (and therefore strong viscoelastic effects) \ncan occur also when $\\rm Wi \\lesssim 1$. \n\n\\begin{figure}\n\\includegraphics[clip=true,keepaspectratio,width=8.0cm]{fig4.eps}\n\\caption{Comparison between the Cram\\'er functions of the stretching rate\n$\\gamma_1$ computed at ${\\rm Wi}=0$ (solid line),${\\rm Wi}=0.5$ (dashed\nline),${\\rm Wi}=1$ (dash-dot). Inset: first Lagrangian Lyapunov exponent\n$\\lambda^0_1$ (circles) and width $\\mu$ (squares) of the Cram\\'er function (see\ntext) as a function of $\\rm Wi$. The Lyapunov exponents are compared with the\nNewtonian value $\\lambda_1^N$. \n}\n\\label{fig4}\n\\end{figure}\n\nDetailed information on the fluctuations of the stretching rates can be \nobtained from the statistics of the Finite Time Lyapunov Exponents (FTLE)\n$\\gamma_i$.  \nThe FTLE are defined via the exponential growth rate during a finite time\n$T$ of an infinitesimal $M$-dimensional volume as\n$\\sum_{i=1}^M\\gamma_i=(1\/T)\\ln[V^M(T)\/V^M(0)]$ \\cite{ccv2010}.\nFrom the definition of the Lyapunov exponents it follows that \n$\\lim_{T\\rightarrow\\infty}\\gamma^T_i=\\lambda_i$. A large deviation approach\nsuggests that the probability density function (PDF) of the largest \nstretching rate $\\gamma_1$ measured over a long time \n$T\\gg 1\/\\lambda_1$ takes the asymptotic form  \n$P_{T}(\\gamma_1)\\sim N(t)\\exp[-H(\\gamma_1)T]$ where the Cram\\'er function \n$H(\\gamma_1)$ is convex and obeys the conditions \n$H(\\lambda_1)=0$, $H^\\prime(\\lambda_1)=0$. \nWe computed the Cram\\'er function for the Lagrangian FTLE for the Newtonian case\nand the two viscoelastic cases. \nIn the inset of Fig.~\\ref{fig4} we plotted the average of the stretching rates\n(i.e., the first Lagrangian Lyapunov exponent of the flow $\\lambda_1^0$) and\nthe rescaled variance $\\mu=T\\langle\\gamma_1^2\\rangle$, for the three values of\n$\\rm Wi$ that we considered.  The decrease of the Lyapunov exponent (rescaled\nwith the Newtonian value $\\lambda_1^N$ for comparison) gives a measure of the\ndecrease in the chaoticity of the flow, due to the action of Polymers. On the\nother hand, we also observe a decrease in the relative variance\n$\\mu\/\\Lambda_1^0$, which implies that polymer feedback induces also a reduction\nof the fluctuations of of stretching rates. Inspection of the main panel of\nFig.~\\ref{fig4}, however, shows that fluctuations are not reduced uniformly.\nIndeed, the shape of $P(\\gamma_1)$ changes when polymers are added.\nAs is evident in Fig.\\ref{fig4}, elasticity has the effect of raising the \nright branch of the Cram\\'er function, while the left one is comparatively \nless affected. \nGiven the definition of $H(\\gamma_1)$, this amounts to a relative\nsuppression of positive fluctuations in the stretching rate: as one could\nexpect, polymers have a larger (negative) feedback on events of larger\nstretching.\n\n\\begin{figure}\n\\includegraphics[width=8.0cm]{fig5a.eps}\n\\includegraphics[width=8.0cm]{fig5b.eps}\n\\caption{\nLyapunov dimension for light (upper panel) and heavy (lower panel) particles\nplotted as a function of ${\\rm St=\\tau_S \\lambda^0_1}$. Different lines correspond\nto the different Weissenberg numbers with symbols as in Fig.~\\ref{fig2}.\n}\n\\label{fig5}\n\\end{figure}\n\nThe effect of polymers on Lyapunov exponents and the Lagrangian nature of the latter suggests to  introduce the dimensionless Stokes\nnumber defined as $\\rm St=\\tau_S \\lambda^0_1$ which depends on $\\rm Wi$ by the\ndependence of $\\lambda^0_1$ shown in Fig.~\\ref{fig4}. Figure~\\ref{fig5} \nshows the Lyapunov dimension $D_L$ for both heavy and light particles \nas a function of $\\rm St$.\nIt is evident that, with respect to Fig.~\\ref{fig2},\nthe collapse of the curves at different $\\rm Wi$ is improved. \nIn particular, the minimum \nof the fractal dimension (which corresponds to maximum clustering) occurs \nalmost for the same $\\rm St$ number. \nStill, some differences are observable, in particular for small $\\rm St$ in the case\nof light particles. \nThis can be understood by the following argument. \nBubbles, at variance with heavy particles, \nhave tendency to concentrate on filaments of high vorticity. \nIndeed, while the minimal dimension for heavy particles is about $2.5$\n(at $\\rm St \\simeq 0.1$), for light particles at maximal clustering \nit becomes as small as $1.26$.\nVortex filaments correspond to quasi-one-dimensional\nregions of intense stretching, in the direction\nlongitudinal to the vortex, which give major contributions to the right\ntail of the Cram\\'er function. \nAs shown in Fig.~\\ref{fig4}, the effects of polymers on the distribution\nof Lyapunov exponent is more evident in this region of strong fluctuations,\nwhere the distribution does not rescale with $\\lambda^0_1$. It is therefore\nnot surprising that also the effects on clustering of light particles\ncannot be completely absorbed in the rescaling of $\\tau_S$ with the mean\nstretching rate $\\lambda^0_1$.\n\nAs the fractal dimension is given by a combination of the Lyapunov exponents,\nin order to better understand the differences on light and heavy particles,\nin Fig.\\ref{fig6} we show the first three Lyapunov exponents as a function\nof $\\rm St$. The first observation is that bubbles, at variance with heavy \nparticles, exhibit negative values of $\\lambda_2$, consistently with the \nlower value of $D_L$ and the tendency of light particles to concentrate \ntowards vortex filaments.\n\nThe first Lyapunov exponent decreases with $\\rm Wi$ for any value of $\\rm St$,\nthus indicating that the phenomenon of chaos reduction, already discussed \nfor the case of Lagrangian tracers, is generic also for inertial particles.  \nOn the contrary, the second Lyapunov exponent shows a different behavior\nfor light and heavy particles: it increases for the former but slightly \ndecreases for the latter. Figure~\\ref{fig6} shows that the effect of \npolymers is not a simple rescaling of the Lyapunov spectrum, which \nwould trivially keep the dimension $D_L$ unchanged. From this point\nof view, the almost perfect rescaling of the Lyapunov dimensions\nshown in Fig.~\\ref{fig5} is quite surprising and arises as the result\nof compensations of different effects.\n\n\\begin{figure}\n\\includegraphics[width=8.0cm]{fig6a.eps}\n\\includegraphics[width=8.0cm]{fig6b.eps}\n\\caption{The first three Lyapunov exponents for light ($\\beta=3$, upper panel)\nand heavy ($\\beta=0$, lower panel) particles, at different $\\rm Wi$.\nContinuous, dashed and dotted lines represent the first, second and\nthird Lyapunov exponents, while symbols correspond to different $\\rm Wi$ \nas in Fig.~\\ref{fig2}.\n}\n\\label{fig6}\n\\end{figure}\n\nIn conclusion, we investigated the clustering properties of inertial \n(heavy and light) particles in a turbulent viscoelastic fluid. \nThe main effect of polymers on turbulent\nflows is to counteract small-scale fluctuations and to reduce its chaoticity.\nQuantitatively, this results in a decrease in the first Lyapunov exponent of \nthe flow, which, in turn, affects clustering of inertial particles. \nThe latter can be quantified by means of the fractal (Lyapunov) dimension of\nparticle distributions. Although the effects of polymers on the particle\nLyapunov exponents are complex and qualitatively different for light \nand heavy particles, the overall effect on fractal dimension is relatively\nsimple and can be rephrased in the rescaling of the characteristic\ntime of the flow. \nIndeed, when particle inertia is parametrized by the Stokes number $\\rm St$ \ndefined with the Lyapunov time of the flow, one can approximately rescale \nthe curves $D_L(\\rm St)$ at all $\\rm Wi$. \nIn contrast, as polymers do not affect large scale properties of the \nflow, a parametrization of particle inertia based on integral time scales\nwould not show a collapse of the curves $D_L(\\rm St)$ at different $\\rm Wi$. \nAs a consequence, any prediction of\nparticle clustering in turbulent polymeric solutions requires an accurate\nestimate of small scale stretching rates.\n\nWe acknowledge support from the the EU COST Action MP0806.\n\n\\input{paper.bbl}\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \nA $0\/1$-simplex is an $n$-dimensional $0\/1$-polytope \\cite{KaZi} with $n+1$ vertices. \nEquivalently, it is the convex hull of $n+1$ of the $2^n$ elements of the set $\\BB^n$ of \nvertices of the unit $n$-cube $I^n$ whenever this hull has dimension $n$. Throughout \nthis paper, we will study $0\/1$-simplices modulo the action of the hyperocthedral group \n$\\Bn$ of symmetries of $I^n$. As a consequence, we may assume without loss of \ngenerality that a $0\/1$-simplex $S$ has the origin as a vertex. This makes it possible \nto represent $S$ by a non-singular $n\\times n$ matrix $P$ whose columns are the \nremaining $n$ vertices of $S$. Of course, this representation is far from unique, as is \nillustrated by the $0\/1$-tetrahedron in Figure~\\ref{Nfigure1}. First of all, there is a \nchoice which vertex of $S$ is located at the origin. Secondly, column permutations of $P$ \ncorrespond to relabeling of the nonzero vertices of $S$, and thirdly, row \npermutations correspond to relabeling of the coordinate axis.\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tikzpicture}[scale=0.75, every node\/.style={scale=0.75}]\n\\begin{scope}[shift={(0,0)}]\n\\draw[fill=gray!20!white] (0,0)--(2,0)--(0.5,3)--cycle;\n\\draw (0,0)--(2,0)--(2,2)--(0,2)--cycle;\n\\draw (0.5,1)--(2.5,1)--(2.5,3)--(0.5,3)--cycle;\n\\draw (0,0)--(0.5,1);\n\\draw (2,0)--(2.5,1);\n\\draw (0,2)--(0.5,3);\n\\draw (2,2)--(2.5,3);\n\\draw (2,0)--(0.5,1);\n\\draw[fill=black] (0,0) circle [radius=0.05];\n\\draw[fill=black] (0.5,1) circle [radius=0.05];\n\\draw[fill=black] (2,0) circle [radius=0.05];\n\\draw[fill=black] (0.5,3) circle [radius=0.05];\n\\node[scale=0.8] at (3.4,0.5) {$\\left[\\begin{array}{rrr} 0 & 0 & 1 \\\\  1 & 1 & 0\\\\  1 & 0 & 0\\end{array}\\right]$};\n\\end{scope}\n\\begin{scope}[shift={(4.5,0)}]\n\\draw[fill=gray!20!white] (0,0)--(2,0)--(2.5,1)--(0,2)--cycle;\n\\draw (0,0)--(2,0)--(2,2)--(0,2)--cycle;\n\\draw (0.5,1)--(2.5,1)--(2.5,3)--(0.5,3)--cycle;\n\\draw (0,0)--(0.5,1);\n\\draw (2,0)--(2.5,1);\n\\draw (0,2)--(0.5,3);\n\\draw (2,2)--(2.5,3);\n\\draw (2,0)--(0,2);\n\\draw (0,0)--(2.5,1);\n\\draw[fill=black] (0,0) circle [radius=0.05];\n\\draw[fill=black] (2.5,1) circle [radius=0.05];\n\\draw[fill=black] (2,0) circle [radius=0.05];\n\\draw[fill=black] (0,2) circle [radius=0.05];\n\\node[scale=0.8] at (3.4,0.5) {$\\left[\\begin{array}{rrr} 1 & 1 & 0 \\\\  0 & 1 & 0\\\\  0 & 0 & 1\\end{array}\\right]$};\n\\end{scope}\n\\begin{scope}[shift={(9,0)}]\n\\draw[fill=gray!20!white] (0,0)--(2,0)--(2,2)--(0.5,1)--cycle;\n\\draw (0,0)--(2,0)--(2,2)--(0,2)--cycle;\n\\draw (0.5,1)--(2.5,1)--(2.5,3)--(0.5,3)--cycle;\n\\draw (0,0)--(0.5,1);\n\\draw (2,0)--(2.5,1);\n\\draw (0,2)--(0.5,3);\n\\draw (2,2)--(2.5,3);\n\\draw (2,0)--(0.5,1);\n\\draw (0,0)--(2,2);\n\\draw[fill=black] (0,0) circle [radius=0.05];\n\\draw[fill=black] (2,2) circle [radius=0.05];\n\\draw[fill=black] (2,0) circle [radius=0.05];\n\\draw[fill=black] (0.5,1) circle [radius=0.05];\n\\node[scale=0.8] at (3.4,0.5) {$\\left[\\begin{array}{rrr} 1 & 0 & 1 \\\\ 0 & 1 & 0\\\\  0 & 0 & 1\\end{array}\\right]$};\n\\end{scope}\n\\begin{scope}[shift={(13.5,0)}]\n\\draw[fill=gray!20!white] (0,0)--(0,2)--(2.5,3)--(2,2)--cycle;\n\\draw (0,0)--(2,0)--(2,2)--(0,2)--cycle;\n\\draw (0.5,1)--(2.5,1)--(2.5,3)--(0.5,3)--cycle;\n\\draw (0,0)--(0.5,1);\n\\draw (2,0)--(2.5,1);\n\\draw (0,2)--(0.5,3);\n\\draw (2,2)--(2.5,3);\n\\draw (0,0)--(2,2);\n\\draw (0,0)--(2.5,3);\n\\draw[fill=black] (0,0) circle [radius=0.05];\n\\draw[fill=black] (0,2) circle [radius=0.05];\n\\draw[fill=black] (2.5,3) circle [radius=0.05];\n\\draw[fill=black] (2,2) circle [radius=0.05];\n\\node[scale=0.8] at (3.4,0.5) {$\\left[\\begin{array}{rrr} 0 & 1 & 1 \\\\0 & 1 & 0\\\\ 1 & 1 & 1\\end{array}\\right]$};\n\\end{scope}\n\\end{tikzpicture}\n\\end{center}\n\\caption{\\small{Matrix representations of the same $0\/1$-tetrahedron modulo the action of $\\mathcal{B}_3$.}}\n\\label{Nfigure1}\n\\end{figure}  \n\n\\smallskip\n  \nWe will be studying $0\/1$-simplices with certain geometric properties. These will be \ninvariant under congruence, and in particular invariant under the action of $\\Bn$, \nwhich forms a subset of the congruences of $I^n$. Thus, each of the matrix representations \ncarries the required geometric information of the $0\/1$-simplex it represents. To be more \nspecific, we will study the set of {\\em acute} $0\/1$-simplices, whose dihedral angles are \nall acute, the {\\em nonobtuse} $0\/1$-simplices, none of whose dihedral angles is obtuse, \nand the set of {\\em orthogonal} simplices. An orthogonal simplex is a nonobtuse simplex \nwith exactly $n$ acute dihedral angles and $\\half n(n-1)$ right dihedral angles.\\\\[2mm]\nIt is not difficult to establish that a $0\/1$-simplex $S$ is nonobtuse if and only if the \ninverse $\\ptpi$ of the Gramian of any matrix representation $P$ of $S$ is a diagonally \ndominant Stieltjes matrix. This Gramian is {\\em strictly} diagonally dominant and has \neven negative off-diagonal entries if and only if $S$ is acute. See \\cite{BrCi,BrCi2,BrKoKr} \nfor details. In this paper we will study the properties of the $0\/1$-matrices \nthat represent acute, nonobtuse, and orthogonal simplices.\n\n\\subsection{Motivation} \nThe motivation to study nonobtuse simplices goes back to their appearance in \nfinite element methods \\cite{Bra,Bre} to approximate solutions of PDEs, in \nwhich triangulations consisting of nonobtuse simplices can be used to guarantee \ndiscrete maximum and comparison principles \\cite{BrKoKr2}. We then found that \nthey figure in other applications, see \\cite{BrKoKrSo} and the references therein. \nIn the context of $0\/1$-simplices and $0\/1$-matrices, it is well known \\cite{Gr} \nthat the {\\em Hadamard conjecture} \\cite{Hada} is equivalent to the existence \nof a {\\em regular $0\/1$-simplex} in each $n$ cube with $n-3$ divisible by $4$. \nNote that a regular simplex is always acute. Thus, studying acute $0\/1$-simplices \ncan be seen as an attempt to study the Hadamard conjecture in new context, which \nis wider, but not too wide. Indeed, acute $0\/1$-simplices, although present in any \ndimension, are still very rare in comparison to {\\em all} $0\/1$-simplices. \nSee \\cite{BrCi2}, in which we describe the computational generation of \nacute $0\/1$-simplices, as well as several mathematical properties. This \npaper can be seen as a continuation of \\cite{BrCi2}, in which some new \nresults on acute $0\/1$-simplices are presented, as well as on the again \nslightly larger class of nononbtuse $0\/1$-simplices.\n\n\n\\subsection{Outline}\nWe start in Section~\\ref{Sect-2} with some preliminaries related to the hyperoctahedral group \nof cube symmetries, to combinatorics, and to the linear algebra behind the geometry of \nnonobtuse and acute simplices. We refer to \\cite{BrCi2} for much more detailed information \non the hyperoctahedral group and combinatorical aspects, and to \\cite{BrKoKrSo} for \napplications of nonobtuse simplices. In Section~\\ref{Sect-3} we present our new results \nconcerning {\\em sign properties} of the {\\em transposed inverses} $P^{-\\top}$ of \nmatrix representations $P$ of acute $0\/1$-simplices $S$. These results imply that the \nmatrices $P$ are {\\em fully indecomposable} with {\\em doubly stochastic pattern} \\cite{Bru1}. \nFrom this follows the so-called {\\em one neighbor theorem}, which states that \nall $(\\nmo)$-facets $F$ of $S$ are {\\em interior} to the cube, and that each is \nshared by at most one other acute $0\/1$-simplex in $I^n$. See \\cite{BrDiHaKr} \nfor an alternative proof of that fact. If $S$ is merely a nonobtuse $0\/1$-simplex, \nthe support of $P$ only {\\em contains} a doubly stochastic pattern, and moreover, \n$P$ can be {\\em partly decomposable}. In Section~\\ref{Sect-4} we study the matrix \nrepresentations of such nonobtuse $0\/1$-simplices with partly decomposable matrix \nrepresentations. The main conclusion is that each of them consists of $p$ with \n$2\\leq p \\leq n$ mutually orthogonal facets $F_1,\\dots,F_p$ of respective \ndimensions $k_1,\\dots,k_p$ that add up to $n$. Moreover, each $k_j\\times k_j$ matrix \nrepresentation of each facet $F_j$ is fully indecomposable. In case all facets \n$F_1,\\dots,F_n$ are one-dimensional, the corresponding $0\/1$-simplex is a \nso-called {\\em orthogonal} simplex, as it has a spanning tree of mutually \northogonal edges. Orthogonal $0\/1$-simplices played an important role in \nthe nonobtuse cube triangulation problem, solved in \\cite{BrDiHaKr}. \nWe briefly recall them in Section~\\ref{Sect-5} and put them into the \nnovel context of Section~\\ref{Sect-4}. Finally, in Section~\\ref{Sect-6} \nwe use the insights developed so far to prove a one neighbor \ntheorem for a wider class of nonobtuse simplices.\n\n\\section{Preliminaries}\\label{Sect-2}\nLet $\\BB=\\{0,1\\}$, and write $\\BB^n=\\BB^{n\\times 1}$ for the \nset of vertices of the unit $n$-cube $I^n=[0,1]^n$, which also \ncontains the standard basis vectors $e_1^n,\\dots,e_n^n$ and their \nsum $e^n$, the {\\em all-ones vector}. The $0\/1$-matrices of size \n$n\\times k$ we denote by $\\BB^{n\\times k}$. For each \n$X\\in\\BB^{n\\times k}$, define its {\\em antipode} $\\ol{X}$ by\n\\be \\ol{X}=e^n(e^k)^\\top-X,\\ee\nand write\n\\be \\ones(X) = (e^n)^\\top X e^k \\und \\zeros(X) = \\ones(\\ol{X})\\ee\nfor the number of entries of $X$ equal to one, and equal to zero, \nrespectively. For any $n\\times k$ matrix $X$ define its {\\em support} $\\supp(X)$ by\n\\be \\supp(X) = \\{(i,j)\\in\\{1,\\dots,n\\}\\times\\{1,\\dots,k\\} \\sth (e_i^n)^\\top X e_j^k \\not=0\\}.\\ee\nWe now recall some concepts from combinatorial matrix theory \\cite{Bru1,BrRy}.\n\n\\begin{Def}{\\rm A nonnegative matrix $A$ has a {\\em doubly stochastic pattern} \nif there exists a doubly stochastic matrix $D=(d_{ij})$ such that $\\supp(D)=\\supp(A)$}.\n\\end{Def}\n\n\\begin{Def}{\\rm A matrix $A\\in\\Bnn$ is {\\em partly decomposable} if there \nexists a $k\\in\\{1,\\dots,n-1\\}$ and permutation matrices $\\Pi_1,\\Pi_2$ such that \n\\be\\label{one-1} \\Pi_1^\\top A \\Pi_2 = \\left[\\begin{array}{cc}A_{11} & A_{12} \\\\ 0 & A_{22}\\end{array}\\right], \\ee\nwhere $A_{11}$ is a $k\\times k$ matrix and $A_{22}$ an $(n-k)\\times(n-k)$ \nmatrix. If $\\Pi_1$ and $\\Pi_2$ can be taken equal in (\\ref{one-1}) then $A$ \nis called {\\em reducible}. If $A$ is not partly decomposable it is called \n{\\em fully indecomposable}. If $A$ is not reducible it is called {\\em irreducible}.}\n\\end{Def}  \nNote that $A\\in\\Bnn$ is partly decomposable if and only if there exist \nnonzero $v,w\\in\\BB^n$ with $\\ones(v)+\\ones(w)=n$ such that \n$v^\\top A w=0$, and that $A$ is reducible if additionally, $w=\\ol{v}$.\n \n\\begin{Le}\\label{lem-1} Let $X\\in\\Bnn$ be nonsingular and $X=[\\,X_1\\,|\\,X_2\\,]$ \na block partition of $X$ where $X_1\\in\\BB^{n\\times k}$ and $X_2\\in\\BB^{n\\times(n-k)}$ \nfor some $1\\leq k<n$. Suppose that $X_1^\\top X_2=0$, and hence,\n\\be X^\\top X = \\left[\\begin{array}{rl} X_1^\\top X_1 & 0 \\\\ 0 & X_2^\\top X_2\\end{array}\\right]. \\ee\nThen there exists a permutation $\\Pi$ such that\n\\[ \\Pi X = \\left[\\begin{array}{rl} X_{11} & 0 \\\\ 0 & X_{22}\\end{array}\\right], \\]\nwhere $X_{11}\\in \\BB^{k\\times k}$ and $X_{22}\\in\\BB^{(n-k)\\times(n-k)}$ are nonsingular.\n\\end{Le}\n{\\bf Proof. } If $X_1^\\top X_2=0$ then in particular $(e^k)^\\top X_1^\\top X_2 e^{n-k}=0$, hence\n\\be \\supp(X_1e^k) \\cap \\supp(X_2e^n) = \\emptyset. \\ee\nSince the rank of $X_1$ equals $k$, the support of $X_1e^k$ \nconsists of at least $k$ indices. Similarly, the support of $X_2e^{n-k}$ \nconsists of at least $n-k$ indices. Thus, they consist of exactly $k$ and \n$n-k$ indices. Now, let $\\Pi$ be a permutation that maps the support of \n$X_1e^k$ onto $\\{1,\\dots,k\\}$, then $\\Pi X$ has the required form. \\hfill $\\Box$\n  \n\\subsection{Matrix representations of $0\/1$-simplices modulo $0\/1$-equivalence}\nThe convex hull of any subset $\\PP\\subset\\BB^n$ is called a {\\em $0\/1$-polytope}. \nAs a first simple and intuitive result, we explicitly show that no two \ndistinct subsets of $\\BB^n$ define the same $0\/1$-polytope. \n\n\\begin{Le}\\label{lem-2} Suppose that $V\\in\\BB^{n\\times k}$ has distinct \ncolumns and that $v\\in\\BB^n$ is not a column of $V$. Then $v$ is not a convex combination of the columns of $V$.\n\\end{Le} \n{\\bf Proof. } Suppose that $v$ is not a column of $V$ and that $Vw=v$ for \nsome $w\\geq 0$. It suffices to show that $w^\\top e^k\\not=1$. If $v=0$ then \n$V$ has no zero column, hence $w=0$ is the only nonnegative vector solving \n$Vw=v$. If $v$ has an entry equal to one, say $(e_\\ell^n)^\\top v=1$, then \n$(e^n_\\ell)^\\top Vw=1$, hence a non-empty subset $\\II\\subset\\{1,\\dots,k\\}$ \nof the entries of $w$ sums to one. In order to obtain $w^\\top e^k=1$, it is \nnecessary that $\\supp(w)=\\II$. Since $Vw\\in\\BB^n$, this implies that each \nrow of $V$ has either only ones, or only zeros, at its entries at positions in \n$\\II$. If $\\II$ has more than one element, then $V$ has two or more columns \nthat are equal, contradicting its definition.\\hfill $\\Box$\\\\[2mm]\nThis lemma proves the bijective correspondence between the power set of the \n$2^n$ vertices of $I^n$ and the $0\/1$-polytopes. Since the cardinality $2^{(2^n)}$ \nof this power set is doubly exponential, the following concept makes sense. Two \n$0\/1$-polyoptes will be called {\\em $0\/1$-equivalent} if they can be transformed \ninto one another by the action of an element of the {\\em hyperoctahedral group} \n$\\Bn$ of affine isometries $\\BB^n\\rightarrow\\BB^n$. The group $\\Bn$ has \n$n!2^n$ elements and is generated by:\\\\[2mm]\n$\\bullet$ the $n$ reflections in the hyperplanes $2x_i=1$ for $i\\in\\{1,\\dots,n\\}$, and\\\\[2mm]\n$\\bullet$ the $n-1$ reflections in the hyperplanes $x_i=x_{i+1}$ for $i\\in\\{1,\\dots,n-1\\}$.\\\\[2mm]\nSee Figure~\\ref{Nfigure2} for the five reflection planes of $\\mathcal{B}_3$ in $I^3$.\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tikzpicture}[scale=0.94, every node\/.style={scale=0.94}]\n\\begin{scope}\n\\draw[fill=gray!20!white] (1,0)--(1.5,0.8)--(1.5,2.8)--(1,2)--cycle;\n\\draw (0,0)--(2,0)--(2,2)--(0,2)--cycle;\n\\draw (0.5,0.8)--(2.5,0.8)--(2.5,2.8)--(0.5,2.8)--cycle;\n\\draw (0,0)--(0.5,0.8);\n\\draw (2,0)--(2.5,0.8);\n\\draw (0,2)--(0.5,2.8);\n\\draw (2,2)--(2.5,2.8);\n\\end{scope}\n\\begin{scope}[shift={(3,0)}]\n\\draw[fill=gray!20!white] (0.25,0.4)--(2.25,0.4)--(2.25,2.4)--(0.25,2.4)--cycle;\n\\draw (0,0)--(2,0)--(2,2)--(0,2)--cycle;\n\\draw (0.5,0.8)--(2.5,0.8)--(2.5,2.8)--(0.5,2.8)--cycle;\n\\draw (0,0)--(0.5,0.8);\n\\draw (2,0)--(2.5,0.8);\n\\draw (0,2)--(0.5,2.8);\n\\draw (2,2)--(2.5,2.8); \n\\end{scope}\n\\begin{scope}[shift={(6,0)}]\n\\draw[fill=gray!20!white] (0,1)--(2,1)--(2.5,1.8)--(0.5,1.8)--cycle;\n\\draw (0,0)--(2,0)--(2,2)--(0,2)--cycle;\n\\draw (0.5,0.8)--(2.5,0.8)--(2.5,2.8)--(0.5,2.8)--cycle;\n\\draw (0,0)--(0.5,0.8);\n\\draw (2,0)--(2.5,0.8);\n\\draw (0,2)--(0.5,2.8);\n\\draw (2,2)--(2.5,2.8); \n\\end{scope}\n\\begin{scope}[shift={(9,0)}]\n\\draw[fill=gray!20!white] (0,0)--(2.5,0.8)--(2.5,2.8)--(0,2)--cycle;\n\\draw (0,0)--(2,0)--(2,2)--(0,2)--cycle;\n\\draw (0.5,0.8)--(2.5,0.8)--(2.5,2.8)--(0.5,2.8)--cycle;\n\\draw (0,0)--(0.5,0.8);\n\\draw (2,0)--(2.5,0.8);\n\\draw (0,2)--(0.5,2.8);\n\\draw (2,2)--(2.5,2.8); \n\\end{scope}\n\\begin{scope}[shift={(12,0)}]\n\\draw[fill=gray!20!white] (0,0)--(2,0)--(2.5,2.8)--(0.5,2.8)--cycle;\n\\draw (0,0)--(2,0)--(2,2)--(0,2)--cycle;\n\\draw (0.5,0.8)--(2.5,0.8)--(2.5,2.8)--(0.5,2.8)--cycle;\n\\draw (0,0)--(0.5,0.8);\n\\draw (2,0)--(2.5,0.8);\n\\draw (0,2)--(0.5,2.8);\n\\draw (2,2)--(2.5,2.8); \n\\end{scope}\n\\end{tikzpicture}\n\\end{center}\n\\caption{\\small{The five reflection planes corresponding to the octahedral group $\\mathcal{B}_3$.}}\n\\label{Nfigure2}\n\\end{figure}  \n\n\\smallskip\n\nNote that a reflection of the second type exchanges the values of the $i$th \nand  $(i+1)$st coordinate. Hence, products of this type can result in any permutation of the coordinates.\\\\[3mm]\nA $0\/1$-{\\em simplex} is a $0\/1$-polytope in $I^n$ with $n+1$ affinely independent vertices. We will study the set $\\SS^n$ of $0\/1$-\nsimplices in $I^n$ modulo the action of $\\Bn$. Therefore we can without loss of generality assume that one of its vertices is located at the \norigin, after which $S$ can be conveniently represented by any square matrix whose columns are its remaining $n$ vertices. \n\\begin{figure}[h]\n\\begin{center}\n\\begin{tikzpicture}[scale=0.94, every node\/.style={scale=0.94}]\n\\begin{scope}\n\\draw[fill=gray!20!white] (0,0)--(2.5,1)--(0.5,3)--cycle;\n\\draw (0,0)--(2,0)--(2,2)--(0,2)--cycle;\n\\draw (0.5,1)--(2.5,1)--(2.5,3)--(0.5,3)--cycle;\n\\draw (0,0)--(0.5,1);\n\\draw (2,0)--(2.5,1);\n\\draw (0,2)--(0.5,3);\n\\draw (2,2)--(2.5,3);\n\\draw[fill=black] (0,0) circle [radius=0.05];\n\\draw[fill=black] (0.5,1) circle [radius=0.05];\n\\draw[fill=black] (2.5,1) circle [radius=0.05];\n\\draw[fill=black] (0.5,3) circle [radius=0.05];\n\\end{scope}\n\\begin{scope}[shift={(3.5,0)}]\n\\draw[fill=gray!20!white] (0,0)--(2,0)--(0.5,3)--cycle;\n\\draw (0,0)--(2,0)--(2,2)--(0,2)--cycle;\n\\draw (0.5,1)--(2.5,1)--(2.5,3)--(0.5,3)--cycle;\n\\draw (0,0)--(0.5,1);\n\\draw (2,0)--(2.5,1);\n\\draw (0,2)--(0.5,3);\n\\draw (2,2)--(2.5,3);\n\\draw (2,0)--(0.5,1);\n\\draw[fill=black] (0,0) circle [radius=0.05];\n\\draw[fill=black] (0.5,1) circle [radius=0.05];\n\\draw[fill=black] (2,0) circle [radius=0.05];\n\\draw[fill=black] (0.5,3) circle [radius=0.05];\n\\end{scope}\n\\begin{scope}[shift={(7,0)}]\n\\draw[fill=gray!20!white] (0,0)--(0.5,1)--(2.5,3)--(2,0)--cycle;\n\\draw (0,0)--(2,0)--(2,2)--(0,2)--cycle;\n\\draw (0.5,1)--(2.5,1)--(2.5,3)--(0.5,3)--cycle;\n\\draw (0,0)--(0.5,1);\n\\draw (2,0)--(2.5,1);\n\\draw (0,2)--(0.5,3);\n\\draw (2,2)--(2.5,3);\n\\draw (0,0)--(2.5,3);\n\\draw (2,0)--(0.5,1);\n\\draw[fill=black] (0,0) circle [radius=0.05];\n\\draw[fill=black] (0.5,1) circle [radius=0.05];\n\\draw[fill=black] (2,0) circle [radius=0.05];\n\\draw[fill=black] (2.5,3) circle [radius=0.05];\n\\end{scope}\n\\begin{scope}[shift={(10.5,0)}]\n\\draw[fill=gray!20!white] (0,0)--(2.5,1)--(2,2)--(0.5,3)--cycle;\n\\draw (0,0)--(2,0)--(2,2)--(0,2)--cycle;\n\\draw (0.5,1)--(2.5,1)--(2.5,3)--(0.5,3)--cycle;\n\\draw (0,0)--(0.5,1);\n\\draw (2,0)--(2.5,1);\n\\draw (0,2)--(0.5,3);\n\\draw (2,2)--(2.5,3);\n\\draw (0,0)--(2,2);\n\\draw (2.5,1)--(0.5,3);\n\\draw[fill=black] (0,0) circle [radius=0.05];\n\\draw[fill=black] (0.5,3) circle [radius=0.05];\n\\draw[fill=black] (2.5,1) circle [radius=0.05];\n\\draw[fill=black] (2,2) circle [radius=0.05];\n\\end{scope}\n\\end{tikzpicture}\n\\end{center}\n\\caption{\\small{The only four $0\/1$-tetrahedra in $I^3$ modulo the action of $\\mathcal{B}_3$. In dimension $5$ and higher there exist \ncongruent simplices that are not equal modulo $\\Bn$.}}\n\\label{Nfigure3}\n\\end{figure}  \nModulo the action of $\\Bn$, two matrices describe the same $0\/1$-simplex if and only if one can be obtained from the other using any \ncombination of the following three operations,\\\\[2mm]\n(C) Column permutations;\\\\[2mm]\n(R) Row permutations;\\\\[2mm]\n(X) Select a column $c$ and replace each remaining column $d$ by $(c+d)(\\mod 2)$.\\\\[2mm]\nOperation (C) does not correspond to any action of $\\Bn$, it just relabels the vertices of $S$. Operations of type (R) correspond to products of \nreflections in hyperplanes $x_i=x_{i+1}$. The operation (X) corresponds to reflecting the vertex $c$ to the origin (and the origin to $c$). It is \nindicated by the letter X because the corresponding matrix operation can also be interpreted as taking the logical {\\em exclusive-or} operation \nbetween a fixed column and the remaining ones.\n\n\\subsection{Dihedral angles of $0\/1$-simplices}\nGiven a vertex $v$ of a simplex $S\\in\\SS^n$, the convex hull of the remaining vertices of $S$ is the {\\em facet} $F_v$ of $S$ opposite $v$. \nThe {\\em dihedral} angle $\\alpha$ between two given facets of $S$ is the angle supplementary to the angle $\\gamma$ between two normal \nvectors to those facets, both pointing into $S$ or both pointing out of $S$. In other words, $\\alpha+\\gamma=\\pi$.\n\n\\begin{Def}{\\rm  A simplex $S\\in\\SS^n$ will be called {\\em nonobtuse} if none of its dihedral angles is obtuse (greater than $\\pi\/2$), and \n{\\em acute} if all its dihedral angles are acute (less than $\\pi\/2$).}\n\\end{Def} \nIf $P$ is a matrix representation of $S$ with columns $p_1,\\dots,p_n$, then the columns $q_1,\\dots,q_n$ of the matrix $Q=P^{-\\top}$ are \ninward normals to the facets $F_{p_1},\\dots,F_{p_n}$, respectively, as $Q^\\top P=I$.\nThe vector $q$ satisfying $P^\\top q=e^n$ is orthogonal to each difference of two columns of $P$. It is an outward normal to the facet \n$F_{p_0}$ opposite the origin $p_0$ and equals $q=P^{-\\top}e^n = q_1+\\dots+q_n$. This proves the following proposition.\n\n\\begin{Pro}[\\cite{BrCi,BrKoKr}]\\label{pro-1} Let $F_1,\\dots,F_n$ be the facets of $S\\in\\SS^n$ meeting at the origin and $F_0$ its facet \nopposite the origin, and let $P$ be a matrix representation of $S$. Then $F_0$ makes nonobtuse dihedral angles with $F_1,\\dots,F_n$ if and \nonly if for all $i\\in\\{1,\\dots,n\\}$,\n\\be\\label{eq-1.1}  (e_i^n)^\\top(P^\\top P)^{-1}e^n \\geq 0. \\ee\nMoreover, each pair of facets $F_i$ and $F_j$ with $i\\not=0\\not=j$ makes a nonobtuse dihedral angle if and only if\n\\be\\label{eq-1.2} (e_i^n)^\\top(P^\\top P)^{-1}e_j^n \\leq 0,  \\ee\nfor all $i,j\\in\\{1,\\dots,n\\}$. Therefore, $S$ is nonobtuse if and only if both (\\ref{eq-1.1}) and (\\ref{eq-1.2}) hold.\n\\end{Pro}\nProperty (\\ref{eq-1.1}) translates as {\\em diagonal dominance} of $(P^\\top P)^{-1}$, and (\\ref{eq-1.2}) is called the {\\em Stieltjes property} \nof $(P^\\top P)^{-1}$, which is then called a {\\em Stieltjes matrix}.\n\n\\begin{rem}\\label{rem-1}{\\rm Condition (\\ref{eq-1.1}) is equivalent with the statement that the vertex $v$ of $S$ at the origin orthogonally \nprojects onto its opposite facet $F_v$. This proves that the following four statements are equivalent:\\\\[2mm]\n$\\bullet$ $S$ is nonobtuse;\\\\[2mm]\n$\\bullet$ each vertex $v$ of $S$ projects orthogonally onto its opposite facet $F_v$;\\\\[2mm]\n$\\bullet$ each matrix representation $P$ of $S$ satisfies $(P^\\top P)^{-1}e^n \\geq 0$;\\\\[2mm]\n$\\bullet$ each matrix representation $P$ of $S$ satisfies $(e_i^n)^\\top(P^\\top P)^{-1}e_j^n \\leq 0$ for all $i,j\\in\\{1,\\dots,n\\}$.}\n\\end{rem} \nIn fact, the second and third condition in Remark \\ref{rem-1} can be slightly relaxed. See Figure~\\ref{Nfigure4}.\n \n\\begin{figure}[h]\n\\begin{center}\n\\begin{tikzpicture}[scale=0.94, every node\/.style={scale=0.94}]\n\\begin{scope}[shift={(0,0)}]\n\\draw[gray!10!white, fill=gray!10!white] (2.5,0)--(0.5,1)--(0,2.5)--(3,3.5)--cycle;\n\\draw[gray!30!white, fill=gray!30!white] (0.5,1)--(0,2.5)--(3,3.5)--cycle;\n\\draw[->] (2.5,0)--(0.5,1);\n\\draw[->] (2.5,0)--(0,2.5);\n\\draw[->] (2.5,0)--(3,3.5);\n\\draw[fill] (2.5,0) circle [radius=0.07];\n\\node at (2.8,0.1) {$v_0$};\n\\node at (1.3,2.5) {$F_0$};\n\\end{scope}\n\\begin{scope}[shift={(3.7,0)}]\n\\draw[gray!10!white, fill=gray!10!white] (2.5,0)--(0.5,1)--(0,2.5)--(3,3.5)--cycle;\n\\draw[gray!30!white, fill=gray!30!white] (2.5,0)--(0,2.5)--(3,3.5)--cycle;\n\\draw[->] (0.5,1)--(2.5,0);\n\\draw[->] (0.5,1)--(0,2.5);\n\\draw[->] (0.5,1)--(3,3.5);\n\\draw[fill] (0.5,1) circle [radius=0.07];\n\\node at (0.3,0.7) {$v_1$};\n\\node at (1.8,1.5) {$F_1$};\n\\end{scope}\n\\begin{scope}[shift={(7.4,0)}]\n\\draw[gray!10!white, fill=gray!10!white] (2.5,0)--(0.5,1)--(0,2.5)--(3,3.5)--cycle;\\\n\\draw[gray!30!white, fill=gray!30!white] (0.5,1)--(2.5,0)--(3,3.5)--cycle;\n\\draw[->] (0,2.5)--(2.5,0);\n\\draw[->] (0,2.5)--(0.5,1);\n\\draw[->] (0,2.5)--(3,3.5);\n\\draw[fill] (0,2.5) circle [radius=0.07];\n\\node at (-0.3,2.2) {$v_2$};\n\\node at (2.1,1.2) {$F_2$};\n\\end{scope}\n\\begin{scope}[shift={(11.1,0)}]\n\\draw[gray!10!white, fill=gray!10!white] (2.5,0)--(0.5,1)--(0,2.5)--(3,3.5)--cycle;\n\\draw[gray!30!white, fill=gray!30!white] (0.5,1)--(0,2.5)--(2.5,0)--cycle;\n\\draw[->] (3,3.5)--(2.5,0);\n\\draw[->] (3,3.5)--(0.5,1);\n\\draw[->] (3,3.5)--(0,2.5);\n\\draw[fill] (3,3.5) circle [radius=0.07];\n\\node at (3.3,3.2) {$v_3$};\n\\node at (1,1) {$F_3$};\n\\end{scope}\n\\end{tikzpicture}\n\\end{center}\n\\caption{\\small{Dihedral angles are present in different ways in different matrix representations.}}\n\\label{Nfigure4}\n\\end{figure} \n\n\\smallskip    \n\nFor each $j\\in\\{0,1,2,3\\}$, let $P_j$ be a matrix representations of a tetrahedron $S\\in\\SS^3$ with its vertex $v_j$ located at the origin. If for \ninstance $(P_j^\\top P_j)^{-1}e^n \\geq 0$ for $j\\in\\{1,2,3\\}$ then $F_1,F_2$ and $F_3$ make only nonobtuse dihedral angles. These include \n{\\em all} the six dihedral angles of $S$.\n\n\\begin{rem}\\label{rem-2} {\\rm To characterize acute simplices similarly, replace $\\geq$ in (\\ref{eq-1.1}) by $>$ and $\\leq$ in (\\ref{eq-1.2}) by \n$<$. Moreover, replace {\\em onto} by {\\em into the interior} of its opposite facet.}\n\\end{rem}\nThe following two simple combinatorial lemmas will be used further on in this paper.\n \n\\begin{Le}\\label{lem-3} Let $P\\in\\Bnn$ represent a nonobtuse $0\/1$-simplex $S$. Then for all $v\\in\\BB^n$, \n\\be v^\\top(P^\\top P)^{-1}\\ol{v} \\leq 0.\\ee\nIf $S$ is even acute then \n\\be v^\\top(P^\\top P)^{-1}\\ol{v} < 0 \\ee\nfor all $v\\in\\BB^n$ with $v\\not\\in\\{0,e^n\\}$.\n\\end{Le} \n{\\bf Proof. } In fact, $v^\\top(P^\\top P)^{-1}\\ol{v}$ is the sum of $\\ones(v)\\ones(\\ol{v})$ of the off-diagonal entries of $(P^\\top P)^{-1}$. \nAccording to Proposition \\ref{pro-1} these are nonpositive if $S$ is nonobtuse. According to Remark \\ref{rem-2} they are negative if $S$ is \nacute. \\hfill $\\Box$ \n \n\\begin{Le}\\label{lem-4} Let $P\\in\\Bnn$ represent a nonobtuse $0\/1$-simplex $S$. If for some $v\\in\\BB^n$,\n\\be v^\\top (P^\\top P)^{-1} \\ol{v} = 0, \\ee\nthen $v=0$ or $\\ol{v}=0$ or $P^\\top P$ is reducible.\n\\end{Le}\n{\\bf Proof. } Let $v\\not=0\\not=\\ol{v}$ and write $k=\\zeros(v)$. Then $1\\leq k \\leq n-1$. Let $\\Pi$ be a permutation such that \n$\\supp(\\Pi\\ol{v})=\\{1,\\dots,k\\}$. Writing $w=\\Pi v$, we have that $\\ol{w}=\\Pi\\ol{v}$ and\n\\be 0 = v^\\top (P^\\top P)^{-1} \\ol{v} = v^\\top\\Pi^\\top\\Pi (P^\\top P)^{-1} \\Pi^\\top\\Pi \\ol{v} = w\\Pi (P^\\top P)^{-1} \\Pi^\\top\\ol{w},\\ee\nwhich is the sum of the entries of $\\Pi (P^\\top P)^{-1}\\Pi^\\top$ with indices $(i,j)$ with $k+1\\leq i\\leq n$ and $1\\leq j \\leq k$. Since these \nentries are non-positive and their sum equals zero, they are all zero, leading to an $(n-k)\\times k$ bottom left block of zeros in \n$\\Pi (P^\\top P)^{-1}\\Pi^\\top$. Thus, $(P^\\top P)^{-1}$ is reducible, and hence, so is its inverse $P^\\top P$. \\hfill $\\Box$\\\\[3mm]\nA final important observation is the following classical result by Fiedler.\n\n\\begin{Le}[\\cite{Fie}]\\label{lem-5} All $k$-facets of a nonobtuse simplex are nonobtuse and all $k$-facets of an acute simplex are acute.\n\\end{Le}\nIt is well known that the converse does not hold. Simplices whose facets are all nonobtuse or acute were studied recently in \\cite{BrCi}. \n \n\\section{Doubly stochastic patterns and full indecomposability}\\label{Sect-3}\nWe start our investigations with some results on matrix representations of acute $0\/1$-simplices. The first one gives a remarkable connection \nwith doubly stochastic matrices. It follows from the observation in Remark \\ref{rem-1} that for an acute $0\/1$-simplex, the altitude from each \nvertex points into the interior of $I^n$. This, in turn, defines the signs of the entries of the normals to its facets in terms of the supports of \ntheir corresponding vertices.\n\n\\begin{Th}\\label{th-1} Let $P\\in\\Bnn$ be a matrix representation of an acute $0\/1$-simplex $S\\in\\SS^n$, and write $Q=P^{-\\top}$. Then \n\\be\\label{eq-3.1} q_{ij}>0 \\Leftrightarrow p_{ij} = 1 \\und q_{ij}<0 \\Leftrightarrow p_{ij}=0. \\ee\nDefining $0\\leq C=(c_{ij})$ and $0\\leq D=(d_{ij})$ by \n\\be\\label{eq-3.2} C = \\half\\left( |Q|-Q\\right) \\und D = \\half\\left(|Q|+Q\\right), \\ee\nwhere $|Q|$ is the matrix whose entries are the moduli of the entries of $Q$, we have that \n\\be Q = D-C, \\ee\nwhere $D$ is doubly stochastic and $C$ row-substochastic. \n\\end{Th}  \n{\\bf Proof}. The $j$th column $q_j$ of $Q$ is an inward normal to the facet $F_j$ of $S$ opposite the $j$th column $p_j$ of $P$. Thus, \n$p_j-\\alpha q_j$ is an element of the interior of $I^n$ for $\\alpha>0$ small enough. From this, (\\ref{eq-3.1}) immediately follows. Combining \nthis with the fact that the inner product between $p_j$ and $q_j$ equals one, the positive elements in each column of $Q$ add to one. But \nsince $Q^\\top P=I$, also the inner products between corresponding rows of $P$ and $Q$ equals one, and thus also the positive elements in \neach row of $Q$ add to one. Finally, $Qe^n$ is the outward normal to the facet of $S$ opposite the origin and thus it points into the interior of \n$I^n$. Consequently, $Qe^n>0$, hence $De^n>Ce^n \\geq 0$, which shows that $C$ is row-substochastic. \\hfill $\\Box$ \n \n\\begin{rem}\\label{rem-3}{\\em For $n\\geq 7$ there exist matrix representations $P\\in\\BB^{n\\times n}$ of acute $0\/1$-simplices in $I^n$  for \nwhich the matrix $C$ in (\\ref{eq-3.2}) is not column-substochastic. This shows that Theorem \\ref{th-1} cannot be strengthened in this \ndirection. It also proves that if $P$ represents an acute $0\/1$-simplex, its transpose $P^\\top$ may not do the same. See Figure~\\ref{Nfigure5} for an \nexample.}\\end{rem}\n  \n\\begin{Co}\\label{co-1} Let $P\\in\\Bnn$ be a matrix representation of an acute $0\/1$-simplex $S\\in\\SS^n$. Then $P$ has a fully \nindecomposable doubly stochastic pattern. \n\\end{Co}\n{\\bf Proof. } Due to (\\ref{eq-3.1}), the matrix $D$ in (\\ref{eq-3.2}) has the same support as $P$, hence $P$ has a doubly stochastic pattern. \nNext, assume to the contrary that $P$ is partly decomposable, then there exist permutations $\\Pi_1,\\Pi_2$ such that\n\\[ \\Pi_1^\\top P \\Pi_2 = \\left[\\begin{array}{cc}P_{11} & P_{12} \\\\ 0 & P_{22}\\end{array}\\right], \\]\nwhere $P_{11}$ is a $k\\times k$ matrix and $P_{22}$ an $(n-k)\\times(n-k)$ matrix for some $k\\in\\{1,\\dots,n-1\\}$. But then $Q=P^{-\\top}$ \nhas entries equal to zero, which contradicts (\\ref{eq-3.1}) in Theorem \\ref{th-1}. \\hfill $\\Box$\\\\[3mm]  \nTheorem \\ref{th-1}, Corollary \\ref{co-1}, and Remark \\ref{rem-3} are all illustrated in Figure~\\ref{Nfigure5}.\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tikzpicture}[scale=0.94, every node\/.style={scale=0.94}]\n\\draw[fill=gray!20!white] (0,0)--(2.5,1)--(2,2)--cycle;\n\\draw[gray] (0,0)--(0.5,3)--(2,2);\n\\draw[thick,->] (1.5,1.2)--(0.6,2.75);\n\\node[scale=0.9] at (0.2,3) {$p$};\n\\node[scale=0.9] at (1,1.5) {$q$};\n\\node[scale=0.8] at (1.2,0.7) {$F_p$};\n\\node[scale=0.9] at (1.7,1.2) {$\\pi$};\n\\draw[fill=white] (1.5,1.2) circle [radius=0.05];\n\\draw (0,0)--(2,0)--(2,2)--(0,2)--cycle;\n\\draw (0.5,1)--(2.5,1)--(2.5,3)--(0.5,3)--cycle;\n\\draw (0,0)--(0.5,1);\n\\draw (2,0)--(2.5,1);\n\\draw (0,2)--(0.5,3);\n\\draw (2,2)--(2.5,3);\n\\draw (0,0)--(2,2);\n\\draw[gray] (2.5,1)--(0.5,3);\n\\draw[fill=black] (0,0) circle [radius=0.05];\n\\draw[fill=black] (0.5,3) circle [radius=0.05]; \n\\draw[fill=black] (2.5,1) circle [radius=0.05];\n\\draw[fill=black] (2,2) circle [radius=0.05];\n\\node[scale=0.73] at (8.5,1.5)  {$P=\\left[\\begin{array}{rrrrrrr}\n     \\fbox{1}  &   \\fbox{1}  &   \\fbox{1}  &   0  &   0  &   \\fbox{1}  &   \\fbox{1}\\\\\n     \\fbox{1}  &   0  &   0  &   \\fbox{1}  &   \\fbox{1}  &   0  &   0\\\\\n     0  &   \\fbox{1}  &   0  &   \\fbox{1}  &   \\fbox{1}  &   0  &   0\\\\\n     0  &   0  &   \\fbox{1}  &   \\fbox{1}  &   \\fbox{1}  &   \\fbox{1}  &   0\\\\\n     0  &   0  &   \\fbox{1}  &   \\fbox{1}  &   \\fbox{1}  &   0  &   \\fbox{1}\\\\\n     0  &   0  &   0  &   \\fbox{1}  &   0  &   \\fbox{1}  &   \\fbox{1}\\\\\n     0  &   0  &   0  &   0  &   \\fbox{1}  &   \\fbox{1}  &   \\fbox{1}\n     \\end{array}\\right], \\hspace{3mm} P^{-\\top}=\\dfrac{1}{13}\\left[\\begin{array}{ccccccc}     \n     \\fbox{4}  &   \\fbox{4}   &  \\fbox{3}  &  -2  &  -2  &   \\fbox{1}  &   \\fbox{1}\\\\\n     \\fbox{9}  &  -4   & -3  &   \\fbox{2}  &   \\fbox{2}  &  -1  &  -1\\\\\n    -4  &   \\fbox{9}   & -3  &   \\fbox{2}  &   \\fbox{2}  &  -1  &  -1\\\\\n    -2  &  -2   &  \\fbox{5}  &   \\fbox{1}  &   \\fbox{1}  &   \\fbox{6}  &  -7\\\\\n    -2  &  -2   &  \\fbox{5}  &   \\fbox{1}  &   \\fbox{1}  &  -7  &   \\fbox{6}\\\\\n    -1  &  -1   & -4  &   \\fbox{7}  &  -6  &   \\fbox{3}  &   \\fbox{3}\\\\\n    -1  &  -1   & -4  &  -6  &   \\fbox{7}  &   \\fbox{3}  &   \\fbox{3}\n    \\end{array}\\right]$};\n\\end{tikzpicture}\n\\end{center}\n\\caption{\\small{In an acute $0\/1$-simplex, each vertex $p$ following the inward normal $q$ to its opposite facet $F_p$ (in converse direction) \nprojects as $\\pi$ in the interior of $F_p$ and hence in the interior of $I^n$. This fixes the signs of the entries of $q$ in terms of those of $p$. \nThe matrices $P$ and $P^{-\\top}$ (not related to the depicted tetrahedron) constitute an example of the linear algebraic consequences. The \npositions of the positive entries (boxed) of $P^{-\\top}$ and $P$ coincide. The positive part $D$ of $P^{-\\top}$ is doubly-stochastic. The \nnegated negative part $C$ of $P^{-\\top}$ is a row-substochastic. It is not column-substochastic because the third column of $C$ adds to \n$\\frac{14}{13}$. Thus,  even though also $P^\\top$ has a doubly stochastic pattern and is fully indecomposable, it is not a matrix \nrepresentation of an acute binary simplex.}}\n\\label{Nfigure5}\n\\end{figure}  \n\\begin{rem}{\\rm Because {\\em each} matrix representation $P$ of an acute $0\/1$-simplex has a fully indecomposable doubly stochastic \npattern, applying to such a matrix $P$ any operation of type (X), as described below Figure~\\ref{Nfigure3}, results in another matrix with a fully \nindecomposable doubly stochastic pattern. From the linear algebraic point of view, this is remarkable because generally, both the $0\/1$-matrix \nproperties of full indecomposability and of having a doubly stochastic pattern are destroyed under operations of type (X). See for instance\n\\be \\small \\left[\\begin{array}{rrr} 1 & 1 & 1 \\\\ 1 & 1 & 1 \\\\ 1 & 1 & 1\\end{array}\\right] \\overset{\\mbox{\\rm(X)}}{\\longrightarrow} \\left[\\begin{array}{rrr} 1 & 0 & 0 \\\\ 1 & 0 & 0 \\\\ 1 & 0 & 0\\end{array}\\right] \\und \\left[\\begin{array}{rrr} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1\\end{array}\\right] \\overset{\\mbox{\\rm(X)}}{\\longrightarrow} \\left[\\begin{array}{rrr} 1 & 1 & 1 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1\\end{array}\\right]. \\ee\nFrom the geometric point of view, this is easy to understand, as the fact that each altitude from each vertex of $S$ points into the interior of \n$I^n$ is invariant under the action of $\\Bn$.}\n\\end{rem} \nThe geometric translation of Corollary \\ref{co-1} is that if $S\\in\\SS^n$ is acute, none of its $k$-dimensional facets is contained in a $k$-\ndimensional facet of $I^n$ for $k\\in\\{1,\\dots,n-1\\}$. The geometric {\\em proof} of this is to note that, given a $k$-facet $C$ of $I^n$, no \nvertex of $I^n$ orthogonally projects into {\\em the interior} of $C$. In fact, each vertex of $I^n$ projects on a {\\em vertex} of $C$. See \nFigure~\\ref{Nfigure6}. Thus, if an arbitrary $0\/1$-simplex $S$ has a $k$-facet $K$ contained in $C$, each remaining vertex of $S$ projects on a vertex of \n$C$. Remark \\ref{rem-2} now shows that $S$ cannot be acute.\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tikzpicture}[scale=0.94, every node\/.style={scale=0.94}]\n\\draw[fill=gray!20!white] (0,0)--(2,0)--(2.5,1)--(0.5,1)--cycle;\n\\draw (0,0)--(2,0)--(2,2)--(0,2)--cycle;\n\\draw (0.5,1)--(2.5,1)--(2.5,3)--(0.5,3)--cycle;\n\\draw (0,0)--(0.5,1);\n\\draw (2,0)--(2.5,1);\n\\draw (0,2)--(0.5,3);\n\\draw (2,2)--(2.5,3);\n\\node[scale=0.8] at (1.5,0.3) {$C$};\n\\draw[fill=black] (0,0) circle [radius=0.05];\n\\draw[fill=black] (2,0) circle [radius=0.05];\n\\draw[fill=black] (2.5,1) circle [radius=0.05];\n\\draw[fill=black] (0.5,1) circle [radius=0.05];\n\\draw[fill=white] (0,2) circle [radius=0.05];\n\\draw[fill=white] (2,2) circle [radius=0.05];\n\\draw[fill=white] (0.5,3) circle [radius=0.05];\n\\draw[fill=white] (2.5,3) circle [radius=0.05];\n\\draw[thick,->] (0,1.95)--(0,1.3);\n\\draw[thick,->] (2,1.95)--(2,1.3);\n\\draw[thick,->] (0.5,2.95)--(0.5,2.3);\n\\draw[thick,->] (2.5,2.95)--(2.5,2.3);\n\\begin{scope}[shift={(4,0)}]\n\\draw (0,0)--(2,0)--(2,2)--(0,2)--cycle;\n\\draw (0.5,1)--(2.5,1)--(2.5,3)--(0.5,3)--cycle;\n\\draw (0,0)--(0.5,1);\n\\draw[gray,very thick]  (2,0)--(2.5,1);\n\\draw (0,2)--(0.5,3);\n\\draw (2,2)--(2.5,3);\n\\draw[fill=white] (0,0) circle [radius=0.05];\n\\draw[fill=black] (2,0) circle [radius=0.05];\n\\draw[fill=black] (2.5,1) circle [radius=0.05];\n\\draw[fill=white] (0.5,1) circle [radius=0.05];\n\\draw[fill=white] (0,2) circle [radius=0.05];\n\\draw[fill=white] (2,2) circle [radius=0.05];\n\\draw[fill=white] (0.5,3) circle [radius=0.05];\n\\draw[fill=white] (2.5,3) circle [radius=0.05];\n\\draw[thick,->] (0,1.95)--(0.7,1.3);\n\\draw[thick,->] (2,1.95)--(2,1.3);\n\\draw[thick,->] (0.5,2.95)--(1.2,2.3);\n\\draw[thick,->] (2.5,2.95)--(2.5,2.3);\n\\draw[thick,->] (0.05,0)--(0.8,0);\n\\draw[thick,->] (0.55,1)--(1.2,1);\n\\node[scale=0.8] at (2.5,0.3) {$C$};\n\\end{scope}\n\\end{tikzpicture}\n\\end{center}\n\\caption{\\small{For each facet $C$ of $I^n$, each vertex of $I^n$ projects onto \na vertex of $C$. Consequently, no acute $0\/1$-simplex has a \nfacet contained in a facet of $I^n$.}}\n\\label{Nfigure6}\n\\end{figure}     \n\n\\smallskip\n\nContrary to an acute $0\/1$-simplex, a {\\em nonobtuse} $0\/1$-simplex $S$ may indeed have a $k$-facet $K$ that is contained in a cube facet \n$C$ of $I^n$. If this is the case, then each remaining vertex of $S$ projects on a vertex of $K$. Moreover, $S$ has a partly decomposable \nmatrix representation. Before discussing this structure, we first formulate the equivalent of Theorem \\ref{th-1} for nonobtuse simplices and \ndiscuss some of the differences with Theorem \\ref{th-1} using an example.\n\n\\begin{Th}\\label{th-2} Let $P\\in\\Bnn$ be a matrix representation of a nonobtuse $0\/1$-simplex $S\\in\\SS^n$, and write $Q=P^{-\\top}$. Then \n\\be\\label{eq-3.1-n} q_{ij}\\geq 0 \\Leftrightarrow p_{ij} = 1 \\und q_{ij}\\leq 0 \\Leftrightarrow p_{ij}=0. \\ee\nDefining $0\\leq C=(c_{ij})$ and $0\\leq D=(d_{ij})$ by \n\\be\\label{eq-3.2-n} C = \\half\\left( |Q|-Q\\right) \\und D = \\half\\left(|Q|+Q\\right), \\ee\nwhere $|Q|$ is the matrix whose entries are the moduli of the entries of $Q$, we have that \n\\be Q = D-C, \\ee\nwhere $D$ is doubly stochastic and $C$ row-substochastic. \n\\end{Th}  \n{\\bf Proof}. The proof only differs from the proof of Theorem \\ref{th-1} in the sense that $p_j-\\alpha q_j$ is now an element of $I^n$ \nincluding its boundary. This accounts for the $\\geq$ and $\\leq$ signs in (\\ref{eq-3.1-n}) in comparison to the $>$ and $<$ signs in \n(\\ref{eq-3.1}). \\hfill $\\Box$\\\\[3mm]\nTheorem \\ref{th-2} is rather weaker than Theorem \\ref{th-1}. First of all, the matrix $P^{-\\top}$ can have entries equal to zero. Moreover, it \ncannot anymore be concluded that $P$ has a doubly stochastic pattern, only that it {\\em contains} a doubly stochastic pattern,\n\\be \\supp(D)\\subset \\supp(P). \\ee\nA typical example of this is the following matrix representation $P$ of a nonobtuse $0\/1$-simplex,\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tikzpicture}[scale=0.94, every node\/.style={scale=0.94}]\n\\draw[fill=gray!20!white] (0,0)--(2.5,1)--(2.5,3)--cycle;\n\\draw[gray] (2,0)--(2.5,3);\n\\draw[thick,->] (1.25,0.5)--(1.9,0.1);\n\\node[scale=0.9] at (2.2,0) {$p$};\n\\node[scale=0.9] at (1.3,0.2) {$q$};\n\\node[scale=0.8] at (1.6,1.3) {$F_p$};\n\\node[scale=0.9] at (1.4,0.7) {$\\pi$};\n\\draw[fill=white] (1.25,0.5) circle [radius=0.05];\n\\draw (0,0)--(2,0)--(2,2)--(0,2)--cycle;\n\\draw (0.5,1)--(2.5,1)--(2.5,3)--(0.5,3)--cycle;\n\\draw (0,0)--(0.5,1);\n\\draw (2,0)--(2.5,1);\n\\draw (0,2)--(0.5,3);\n\\draw (2,2)--(2.5,3);\n\\draw[fill=black] (0,0) circle [radius=0.05];\n\\draw[fill=black] (2,0) circle [radius=0.05]; \n\\draw[fill=black] (2.5,1) circle [radius=0.05];\n\\draw[fill=black] (2.5,3) circle [radius=0.05];\n\\node[scale=0.73] at (8.5,1.5)  {$P=\\left[\\begin{array}{ccccccc}\n     \\fbox{1}  &   \\fbox{1}   &    0  &   0  &    \\fbox{1}   &    \\fbox{1}   &    \\fbox{1} \\\\\n     0  &    \\fbox{1}   &   0  &   0   &    \\fbox{1}   &    \\fbox{1}   &    \\fbox{1}\\\\\n     0  &   0  &    \\fbox{1}   &   \\fbox{1}   &    0   &   0   &    0 \\\\\n     0  &   0  &   0  &    \\fbox{1}   &   0  &   0  &   0\\\\\n     0  &   0  &   0  &   0  &    \\fbox{1}   &    \\fbox{1}   &   0\\\\ \n     0  &   0  &   0  &   0  &    \\fbox{1}   &   0  &    \\fbox{1} \\\\\n     0  &   0  &   0  &   0  &   0  &    \\fbox{1}   &    \\fbox{1} \n     \\end{array}\\right], \\hspace{3mm} P^{-\\top} = \\dfrac{1}{2}\n     \\left[\\begin{array}{rrrrrrr}\n      \\fbox{2}   &   0  &   0  &   0  &   0  &   0  &   0\\\\\n     -2  &    \\fbox{2}   &   0  &   0  &   0  &   0  &   0\\\\\n     0 &   0  &    \\fbox{2}   &   0  &   0  &   0  &   0\\\\\n     0  &  0  &   -2  &    \\fbox{2}   &   0  &   0  &   0\\\\\n     0  &   -1 &  0  &   0  &    \\fbox{1}   &    \\fbox{1}   &  -1\\\\\n     0  &   -1 &  0  &   0  &    \\fbox{1}   &  -1  &    \\fbox{1} \\\\\n     0  &   -1 &  0  &   0  &  -1  &    \\fbox{1}   &    \\fbox{1} \n     \\end{array}\\right]$};\n\\end{tikzpicture}\n\\end{center}\n\\caption{\\small{Analogue of Figure~\\ref{Nfigure5} for matrix representations of nonobtuse $0\/1$-simplices.}}\n\\label{Nfigure7}\n\\end{figure}   \n\n\\smallskip\n\nObviously, $P$ is partly decomposable and has no doubly stochastic pattern. It is only valid that the support of the doubly stochastic matrix $D$ \nis {\\em contained} in the support of $P$.    \n\\section{Partly decomposable matrix representations}\\label{Sect-4}\nWe will continue to study nonobtuse $0\/1$-simplices having a partly decomposable matrix representation $P$. Without loss of generality, we \nmay assume that $P$ is nontrivially block partitioned as\n\\be\\label{eq-5} P = \\left[\\begin{array}{r|r} N & R \\\\ \\hline 0 & A \\end{array}\\right], \\ee\nand that $A$ is fully indecomposable.\n\n\\begin{Th}\\label{th-3} Let $S\\in\\SS^n$ be nonobtuse with a matrix representation $P$ as in (\\ref{eq-5}) with $N\\in\\BB^{k\\times k}$ with \n$k\\in\\{1,\\dots,n-1\\}$ and with $A$ fully indecomposable. Then:\\\\[3mm]\n$\\bullet$ $N$ is a matrix representation of a nonobtuse simplex in $I^k$;\\\\[3mm]\n$\\bullet$ $A$ is a matrix representation of a nonobtuse simplex in $I^{n-k}$;\\\\[3mm]\n$\\bullet$ $R=\\nu(e^k)^\\top$, where $\\nu=0$ or $\\nu$ is a column of $N$.\n\\end{Th}\n{\\bf Proof.} Lemma \\ref{lem-5} proves that the first $k$ columns of $P$ together with the origin form a nonobtuse $k$-simplex, and obviously \nits vertices all lie in a $k$-facet of $I^n$. This proves the first item in the list of statements. Next, we compute \n\\be\\label{eq-6a} P^{-\\top} = \\left[\\begin{array}{rc} N^{-\\top} & 0 \\\\ -A^{-\\top}R^\\top N^{-\\top} & A^{-\\top}\\end{array}\\right]. \\ee \nand thus,\n\\be\\label{eq-6} (P^\\top P)^{-1} = \\left[\\begin{array}{rr} (N^\\top N)^{-1} + N^{-1}R(A^\\top A)^{-1}R^\\top N^{-\\top} &  -N^{-1}R(A^\\top A)^{-1}  \\\\-(A^\\top A)^{-1}R^\\top N^{-\\top}  & (A^\\top A)^{-1}\\end{array}\\right]. \\ee \nDue to Proposition \\ref{pro-1}, the matrix $(P^\\top P)^{-1}$ has nonpositive off-diagonal entries (\\ref{eq-1.2}) and nonnegative row sums \n(\\ref{eq-1.1}). Both properties are clearly inherited by its trailing submatrix $(A^\\top A)^{-1}$, possibly even with larger row sums. This \nproves the second statement of the theorem.  Next, due to (\\ref{eq-1.2}), the top-right block in (\\ref{eq-6}) satisfies \n\\be\\label{eq-7} -N^{-1}R(A^\\top A)^{-1} \\leq 0. \\ee\nMultiplication of this block from the left with $N\\geq 0$ and from the right with $\\ol{R}\\geq 0$ gives that\n\\be   R(A^\\top A)^{-1}\\ol{R}^\\top \\geq 0.\\ee\nHowever, by Lemma \\ref{lem-3}, the diagonal entries of $R(A^\\top A)^{-1}\\ol{R}^\\top$ are also non{\\em positive}, and thus, they all equal \nzero. We can therefore apply Lemma \\ref{lem-4}. Note that because $A$ is assumed fully indecomposable, $A^\\top A$ is irreducible by Lemma \n\\ref{lem-1}. Thus, row-by-row application of Lemma \\ref{lem-4} proves that each row of $R$ contains only zeros or only ones. This proves that \nthere exists an $r\\in\\BB^n$ such that\n\\be\\label{eq-8} R=r(e^{n-k})^\\top.\\ee\nWe will proceed to show that $r$ is a column of $N$, or zero. Substituting (\\ref{eq-8}) back into (\\ref{eq-7}) yields that  \n\\be\\label{eq-9} wu^\\top \\geq 0, \\hdrie \\mbox{\\rm where } w=N^{-1}r \\und u^\\top = (e^{n-k})^\\top (A^\\top A)^{-1}. \\ee \nDue to (\\ref{eq-1.1}) we have $u\\geq 0$. Because $A^\\top A$ is non-singular, $u$ has at least one positive entry. Thus, also $w$ is \nnonnegative. This turns $r=Nw$ into a nonnegative linear combination of columns of $N$. We continue to prove that it is a {\\em convex} \ncombination. For this, observe that the sums of the last $k$ rows of $(P^\\top P)^{-1}$ are nonnegative due to (\\ref{eq-1.1}). Thus,\n\\[ 0 \\leq -(A^\\top A)^{-1}R^\\top N^{-\\top}e^k + (A^\\top A)^{-1}e^{n-k} = u\\left(1-w^\\top e^k\\right) \\]\nwith $u,w$ as in (\\ref{eq-9}) and where we have used that $R^\\top N^{-\\top}=e^{n-k}r^\\top N^{-\\top} = e^{n-k}w^\\top$. As we showed \nalready that $u\\geq 0$ has at least one positive entry, we conclude that \\[  w^\\top e^k \\leq 1.\\] \nTherefore we now have that $Nw=r\\in\\BB^k$ for some $w\\geq 0$ with $w^\\top e^k \\leq 1$. Thus also\n\\be [0\\,|\\,N] \\left[\\begin{array}{c} 1-w^\\top e^k \\\\ w\\end{array}\\right] = r. \\ee\nAccording to Lemma \\ref{lem-2}, this implies that $r$ is a column of $[0\\,|\\,N]$. This proves the third item in the list of statements to \nprove.~\\hfill $\\Box$\\\\[3mm]\nIt is worthwhile to stress a number of facts concerning Theorem \\ref{th-3} and its nontrivial proof.\n\n\\begin{rem}\\label{rem-4}{\\rm The assumption that $A$ in (\\ref{eq-5}) is fully indecomposable is very natural, as each partly decomposable matrix can be put in the form (\\ref{eq-5}) using operations of type (C) and (R). But in the proof of Theorem \\ref{th-3} we only needed that $A^\\top A$ is irreducible. This is {\\em implied} by the full indecomposability of $A$ due to Lemma \\ref{lem-1}, but is not {\\em equivalent} to it. In fact, if $A$ is fully indecomposable, then $A^\\top A\\geq e^n(e^n)^\\top + I$. See Corollary \\ref{co-5}.}\n\\end{rem} \n\n\\begin{rem}\\label{rem-5}{\\rm The result proved in the third bullet of Theorem \\ref{th-3} that $R$ consists of $n-k$ copies of {\\em the same} column of $N$ is stronger than the geometrical observation that each of the last $n-k$ columns of $P$ should project on {\\em any} vertex of the $k$-simplex represented by $N$. It is the {\\em irreducibility} of $A^\\top A$ that forces the equality of all columns of $R$.}\n\\end{rem}\n\n\\begin{rem}\\label{rem-6}{\\rm Permuting rows and columns of the block upper triangular matrix in (\\ref{eq-5}) show that also for\n\\be  \\left[\\begin{array}{r|r} A & 0 \\\\ \\hline R & N \\end{array}\\right]  \\ee\nwith $A$ and $N$ as in Theorem \\ref{th-3}, similar conclusions can be drawn for $R$}.\n\\end{rem}\nSome details in the above remarks will turn out to be of central importance in Section \\ref{Sect-6}.\n\n\\begin{Co}\\label{co-3} Let $S\\in\\SS^n$ be a nonobtuse $0\/1$-simplex with matrix representation $P$. Then the following statements are \nequivalent:\\\\[3mm]\n$\\bullet$ $P$ is partly decomposable;\\\\[3mm]\n$\\bullet$ $S$ has a block diagonal matrix representation with at least one fully indecomposable block;\\\\[3mm]\n$\\bullet$ each matrix representation of $S$ is partly decomposable.\n\\end{Co} \n{\\bf Proof}. Suppose that $P$ is partly decomposable. Then Theorem \\ref{th-1} shows that $P$ is of the form\n\\be\\label{eq-10} P = \\left[\\begin{array}{r|c} N & \\nu \\left(e^{n-k}\\right)^\\top \\\\[1mm] \\hline 0 & A \\end{array}\\right],\\ee\nand $\\nu=0$ or $\\nu$ is a column of $N$. If $\\nu=0$ then $P$ itself is block diagonal. If $\\nu\\not=0$, apply to $P$ the operation of type (X) \nas described below Figure~\\ref{Nfigure3} with column $c$ equal to the column $(\\nu,0)^\\top$ of $P$. The simple observation that $\\nu+\\nu$ equal zero \nmodulo $2$ proves that the resulting matrix $\\tilde{P}$ is block diagonal. As the bottom right block of $\\tilde{P}$ equals $A$, this shows that \nat least one block is fully indecomposable. To show that each matrix representation of $S$ is partly decomposable, simply note that each \noperation of type (X) applied to the block-diagonal matrix representation will leave one of the two off-diagonal zero blocks \ninvariant.~\\hfill $\\Box$\n\n\\begin{rem}{\\rm The converse of Theorem \\ref{th-3} is also valid. Indeed, suppose that $N$ and $A$ are matrix representations of nonobtuse \nsimplices. Then it is trivially true that the block diagonal matrix $P$ having $N$ and $A$ as diagonal blocks represents a nonobtuse simplex. \nApplying operations of type (X) to $P$ proves that all matrices of the form (\\ref{eq-10}) then represent nonobtuse simplices. Note that this also \nholds without the assumption that $A$ is fully indecomposable.}\n\\end{rem}\nAnother corollary of Theorem \\ref{th-3} concerns its implications for the structure of the transposed inverse $P^{-\\top}$ of a partly \ndecomposable matrix representation of a nonobtuse $0\/1$-simplex.\n\n\\begin{Co}\\label{co-4} If $\\nu=Ne_j^k$ in (\\ref{eq-10}) for some $j\\in\\{1,\\dots,k\\}$, then\n\\be P^{-\\top} = \\left[\\begin{array}{c|c} N^{-\\top} & 0 \\\\\\hline\\\\[-4mm] ae_j^\\top & A^{-\\top}\\end{array}\\right],  \\ee\nwhere $a$ is the inward normal to the facet opposite the origin of the simplex represented by $A$. As a consequence, the sums of the \nlast $n-k$ rows of $P^{-\\top}$ all add to zero.\n\\end{Co}\n{\\bf Proof. } Substitute $R=\\nu e^{n-k}$ with $\\nu=Ne_j^k$ into the expression for $P^{-\\top}$ in (\\ref{eq-6a}). \\hfill $\\Box$\\\\[3mm]\nTo illustrate Corollary \\ref{co-3}, consider again the matrix $P\\in\\BB^{7\\times7}$ from Figure~\\ref{Nfigure7}, now displayed not as $0\/1$-matrix but as \ncheckerboard black-white pattern, at the top in Figure~\\ref{Nfigure8}. Applying an operation of type (X) with the second column results in the matrix to its \nright, which is block diagonal. Swapping the first two rows of that matrix yields one in which the top left block is now in its most reduced form. \nApplying an operation of type (X) with the sixth column results in a matrix in which the bottom left $3\\times 4$ zero block has been destroyed. \nFinally, swapping columns $2$ and $6$, and swapping rows $1$ and $2$, results in the matrix that could also have been obtained by applying \noperation (X) with column $6$ directly to $P$.\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tikzpicture}[scale=0.94, every node\/.style={scale=0.94}]\n\n\n\\node[xscale=0.6,yscale=0.73] at (5,1)  {$\\left[\\begin{array}{ccccccc}\n     \\blacksquare  &   \\blacksquare   &   \\square &  \\square &    \\blacksquare   &    \\blacksquare   &    \\blacksquare \\\\\n    \\square &    \\blacksquare   &  \\square &  \\square  &    \\blacksquare   &    \\blacksquare   &    \\blacksquare\\\\\n    \\square &  \\square &    \\blacksquare   &   \\blacksquare   &   \\square  &  \\square  &   \\square\\\\\n    \\square &  \\square &  \\square &    \\blacksquare   &  \\square &  \\square &  \\square\\\\\n    \\square &  \\square &  \\square &  \\square &    \\blacksquare   &    \\blacksquare   &   \\square\\\\ \n    \\square &  \\square &  \\square &  \\square &    \\blacksquare   &  \\square &    \\blacksquare \\\\\n    \\square &  \\square &  \\square &  \\square &  \\square &    \\blacksquare   &    \\blacksquare \n     \\end{array}\\right]$};\n     \n\\draw[<->] (6.8,1)--(7.8,1);\n\\draw[<->] (3.3,1)--(2.3,1);\n\\node[scale=0.9] at (7.3,1.3) {(X)};\n\\node[scale=0.9] at (2.8,1.3) {(X)};\n\\node[scale=0.9] at (7.3,0.7) {$c=2$};\n\\node[scale=0.9] at (2.8,0.7) {$c=6$};\n\n\\draw[<->] (10,-1.3)--(9,-2.3);\n\\draw[<->] (0,-1.3)--(1,-2.3);\n\\node[scale=0.9] at (10.5,-2) {(R), $1\\leftrightarrow 2$};\n\\node[scale=0.9] at (-0.5,-2) {(C), $2\\leftrightarrow 6$};\n\\node[scale=0.9] at (-0.5,-2.5) {(R), $1\\leftrightarrow 2$};\n\n\\draw[<->] (4.5,-3)--(5.5,-3);\n\\node[scale=0.9] at (5,-2.7) {(X)};\n\\node[scale=0.9] at (5,-3.3) {$c=6$};\n     \n\\node[xscale=0.6,yscale=0.73] at (9.5,0)  {$\\left[\\begin{array}{ccccccc}\n    \\square&    \\blacksquare   &   \\blacksquare  &   \\blacksquare   &  \\square  &   \\square  &  \\square\\\\\n       \\blacksquare &   \\blacksquare  &    \\blacksquare  &   \\blacksquare  &  \\square  &  \\square  &  \\square\\\\\n    \\square &  \\square &    \\blacksquare   &   \\blacksquare   &   \\square  &  \\square  &   \\square\\\\\n    \\square &  \\square &  \\square &    \\blacksquare   &  \\square &  \\square &   \\square\\\\\n    \\square &  \\square &  \\square &  \\square &    \\blacksquare   &    \\blacksquare   &   \\square\\\\ \n    \\square &  \\square &  \\square &  \\square &    \\blacksquare   &  \\square &    \\blacksquare \\\\\n    \\square &  \\square &  \\square &  \\square &  \\square &    \\blacksquare   &    \\blacksquare \n     \\end{array}\\right]$};\n     \n \\node[xscale=0.6,yscale=0.73] at (7.2,-3)  {$\\left[\\begin{array}{ccccccc}\n     \\blacksquare &   \\blacksquare  &    \\blacksquare  &   \\blacksquare  &  \\square  &  \\square  &  \\square\\\\\n    \\square&    \\blacksquare   &   \\blacksquare  &   \\blacksquare   &  \\square  &   \\square  &  \\square\\\\\n    \\square &  \\square &    \\blacksquare   &   \\blacksquare   &   \\square  &  \\square  &   \\square\\\\\n    \\square &  \\square &  \\square &    \\blacksquare   &  \\square &  \\square & \\square\\\\\n    \\square &  \\square &  \\square &  \\square &    \\blacksquare   &    \\blacksquare   &  \\square\\\\ \n    \\square &  \\square &  \\square &  \\square &    \\blacksquare   &  \\square &    \\blacksquare \\\\\n    \\square &  \\square &  \\square &  \\square &  \\square &    \\blacksquare   &    \\blacksquare \n     \\end{array}\\right]$};\n\n\\node[xscale=0.6,yscale=0.73] at (0.5,0)  {$\\left[\\begin{array}{ccccccc}\n     \\square  &   \\square   &   \\blacksquare &  \\blacksquare &   \\square  &   \\blacksquare &  \\square\\\\\n    \\blacksquare &    \\square   &  \\blacksquare &  \\blacksquare  &   \\square &  \\blacksquare  &  \\square\\\\\n    \\square &  \\square &    \\blacksquare   &   \\blacksquare   &   \\square  &  \\square  &   \\square\\\\\n    \\square &  \\square &  \\square &    \\blacksquare   &  \\square &  \\square & \\square\\\\\n     \\blacksquare  &   \\blacksquare  &   \\blacksquare  &   \\blacksquare  &   \\square  &    \\blacksquare   &   \\blacksquare\\\\ \n    \\square &  \\square &  \\square &  \\square &    \\blacksquare   &  \\square &    \\blacksquare \\\\\n     \\blacksquare &   \\blacksquare  &   \\blacksquare  &   \\blacksquare  &   \\blacksquare  &    \\blacksquare  &   \\square\n     \\end{array}\\right]$};\n     \n\\node[xscale=0.6,yscale=0.73] at (2.8,-3)  {$\\left[\\begin{array}{ccccccc}\n    \\blacksquare &    \\blacksquare   &  \\blacksquare &  \\blacksquare  &   \\square &  \\square  &  \\square\\\\\n       \\square  &   \\blacksquare   &   \\blacksquare &  \\blacksquare &   \\square  &   \\square &  \\square\\\\\n    \\square &  \\square &    \\blacksquare   &   \\blacksquare   &   \\square  &  \\square  &   \\square\\\\\n    \\square &  \\square &  \\square &    \\blacksquare   &  \\square &  \\square & \\square\\\\\n     \\blacksquare  &   \\blacksquare  &   \\blacksquare  &   \\blacksquare  &   \\square  &    \\blacksquare   &   \\blacksquare\\\\ \n    \\square &  \\square &  \\square &  \\square &    \\blacksquare   &  \\square &    \\blacksquare \\\\\n     \\blacksquare &   \\blacksquare  &   \\blacksquare  &   \\blacksquare  &   \\blacksquare  &    \\blacksquare  &   \\square\n     \\end{array}\\right]$};\n\n\\end{tikzpicture}\n\\end{center}\n\\caption{\\small{Illustration of Corollary \\ref{co-3} using the matrix $P\\in\\BB^{7\\times7}$ from Figure~\\ref{Nfigure7}.}}\n\\label{Nfigure8}\n\\end{figure}  \n\n\\smallskip\n\nCorollary \\ref{co-3} shows that the matrix representations of a nonobtuse $0\/1$-simplex are either all partly decomposable, or all fully \nindecomposable. This motivates to the following definition.\n \n\\begin{Def} {\\rm A nonobtuse simplex is called partly decomposable if it has a partly decomposable matrix representation, and fully \nindecomposable if it has not}.\n\\end{Def}\nWe will now investigate to what structure the recursive application of Theorem \\ref{th-3} leads. For this, assume again that $P$ is a partly \ndecomposable matrix representation of a nonobtuse $0\/1$-simplex $S\\in\\SS^n$. Then by Corollary \\ref{co-3}, $S$ has a matrix representation \nof the form\n\\be\\label{eq-11} P = \\left[\\begin{array}{r|c} N_1 & R_1 \\\\ \\hline 0 & A_1 \\end{array}\\right],\\ee\nin which $A_1$ is fully indecomposable. According to Theorem \\ref{th-3}, the $k\\times k$ matrix $N_1$ represents a nonobtuse $k$-simplex \n$K$ in $I^k$. If also $K$ is partly decomposable, we can block-partition $N_1$ using row (R) and column (C) permutations $\\Pi_1$ and \n$\\Pi_2$, such that \n\\be\\label{eq-12} \\tilde{P}=\\Pi_1 P\\Pi_2 = \\left[\\begin{array}{c|c|c} N_2 & R_{12} & R_{13} \\\\\\hline 0 & A_2 & R_{23}\\\\ \\hline 0 & 0 & A_1 \\end{array}\\right],\\ee\nwith $A_2$ fully indecomposable and $N_2$ possibly partly decomposable. Theorem \\ref{th-3} shows that\n\\[ R_{12} \\hdrie \\mbox{\\rm consists of copies of a column $\\nu$ of} \\hdrie [0|N_2], \\]\nand\n\\[ \\left[\\begin{array}{c}R_{13}\\\\\\hline R_{23}\\end{array}\\right]\\hdrie\\mbox{\\rm consists of copies of a column of} \\hdrie \\left[\\begin{array}{r|c} N_2 & R_{12} \\\\ \\hline 0 & A_2 \\end{array}\\right]. \\]\nThis means that if $R_{23}$ is nonzero, then $R_{13}$ consists of copies of the same column $\\nu$ of $N$ as does $R_{12}$. Thus the whole \nblock $[R_{12}\\,|\\,R_{23}]$ consists of copies of a column $\\nu$ of $[0|N_2]$. On the other hand, if $R_{23}$ is zero, then $R_{13}$ can \neither be zero, or consist of copies of {\\em any} column of $N_2$, including $\\nu$.\\\\[3mm]\nBy including operations of type (X) it is possible to map the entire strip above one of the fully indecomposable diagonal blocks to zero. Although \nthis will in general destroy the block upper triangular form of the square submatrix to the left of that strip, Corollary \\ref{co-4} shows that this \nsubmatrix remains partly decomposable. Therefore, its block upper triangular structure can be restored using only operations of type (R) and \n(C), which leave the zero strip intact.\n\n\\begin{rem}{\\rm It is generally not possible to transform $\\tilde{P}$ to block diagonal form with more than two diagonal blocks. See the \ntetrahedron $S$ in $I^3$ in Figure~\\ref{Nfigure9}. It is possible to put a vertex of $S$ at the origin such that facets $A$ and $N$ are orthogonal, and \nhence the corresponding matrix representation is block diagonal with two blocks. The triangular facet however requires a {\\em different} vertex \nat the origin for its $2\\times 2$ matrix representation to be diagonal.}\n\\end{rem}\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tikzpicture}[scale=0.94, every node\/.style={scale=0.94}] \n\\begin{scope}[shift={(0,0)}]\n\\draw[fill=gray!20!white] (0,0)--(2,0)--(2.5,1)--(0,0)--cycle;\n\\draw (2,0)--(0,2);\n\\draw (0,0)--(2.5,1);\n\\draw (0,2)--(2.5,1);\n\\draw[gray,very thick] (0,0)--(0,2);\n\\draw (0,0)--(2,0)--(2,2)--(0,2)--cycle;\n\\draw (0.5,1)--(2.5,1)--(2.5,3)--(0.5,3)--cycle;\n\\draw (0,0)--(0.5,1);\n\\draw (2,0)--(2.5,1);\n\\draw (0,2)--(0.5,3);\n\\draw (2,2)--(2.5,3);\n\\draw[fill=black] (0,0) circle [radius=0.05];\n\\draw[fill=black] (2,0) circle [radius=0.05]; \n\\draw[fill=black] (2.5,1) circle [radius=0.05];\n\\draw[fill=black] (0,2) circle [radius=0.05];\n\\node[scale=0.9] at (-0.3,1) {$A$};\n\\node[scale=0.9] at (1.2,0.2) {$N$};\n\\end{scope}\n\\node[scale=0.85] at (4.3,1.5) {$\\left[\\begin{array}{rr|r} 1 & 1 & 0 \\\\ 0 & 1 & 0 \\\\\\hline 0 & 0 & 1\\end{array}\\right]$};\n\\draw[<->] (5.5,1.5)--(6,1.5);\n\\node[scale=0.85] at (7.2,1.5) {$\\left[\\begin{array}{r|rr} 1 & 0 & 0 \\\\\\hline 0 & 1 & 1 \\\\ 0 & 0 & 1\\end{array}\\right]$};\n\\begin{scope}[shift={(9,0)}]\n\\draw[fill=gray!20!white] (0,0)--(0.5,1)--(0.5,3)--cycle;\n\\draw (2,0)--(0.5,1);\n\\draw (2,0)--(0.5,3);\n\\draw[gray,very thick] (0,0)--(2,0);\n\\draw (0,0)--(2,0)--(2,2)--(0,2)--cycle;\n\\draw (0.5,1)--(2.5,1)--(2.5,3)--(0.5,3)--cycle;\n\\draw (0,0)--(0.5,1);\n\\draw (2,0)--(2.5,1);\n\\draw (0,2)--(0.5,3);\n\\draw (2,2)--(2.5,3);\n\\draw[fill=black] (0,0) circle [radius=0.05];\n\\draw[fill=black] (2,0) circle [radius=0.05]; \n\\draw[fill=black] (0.5,3) circle [radius=0.05];\n\\draw[fill=black] (0.5,1) circle [radius=0.05];\n\\node[scale=0.9] at (0.7,0.3) {$A$};\n\\node[scale=0.9] at (0.35,1.3) {$N$};\n\\end{scope}\n\\end{tikzpicture}\n\\end{center}\n\\caption{\\small{Any simplex $S$ with a partly decomposable matrix representation has a pair of facets $A$ and $N$ of dimensions adding to \n$n$ are orthogonal to one another.}}\n\\label{Nfigure9}\n\\end{figure}      \n\n\\smallskip\n\nSummarizing, the above discussion shows that each nonobtuse $0\/1$-simplex $S$ has a matrix representation that is block upper triangular, \nwith fully indecomposable diagonal blocks (possibly only one). The strip above each diagonal block is of rank one and consists only of copies of \na column to the left of the strip. Any matrix representation $P$ of $S$ can be brought into this form using only operations of type (C) and (R). \nUsing an additional reflection of type (X), it is possible to transform an entire strip above one of the diagonal blocks to zero using a column to \nthe left of the strip. Although this may destroy the block upper triangular form to the left of the strip, this form can be restored using operations \nof type (R) and (C) only.\\\\[3mm]\nIn view of Corollary \\ref{co-4}, the transposed inverse $P^{-\\top}$ of a partly decomposable matrix representation $P$ of a nonobtuse $0\/1$-\nsimplex $S$ is perhaps even simpler in structure than $P$ itself. On the diagonal it has the transposed inverses of the fully indecomposable \ndiagonal blocks $A_1,\\dots,A_p$ of $P$, and each {\\em horizontal} strip to the left of such a diagonal block $(A_j)^{-\\top}$ has \n{\\em at most} one nonpositive column that is not identically zero. This columns has two interesting features. The first is that it nullifies the \nsums of the \nrows in its strip. The second is that its position clearly indicates to which vertex of which other block $A_i$ the block $A_j$ is related. To \nillustrate what we mean by this, see Figure~\\ref{Nfigure7} for an example. Above the bottom right $3\\times 3$ block $A$ of $P$ we see copies of the second \ncolumn of $P$. This fact can also be read from the position of the nonzero entries to the left of the corresponding block $A^{-\\top}$ in \n$P^{-\\top}$.\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tikzpicture}[scale=0.94, every node\/.style={scale=0.94}]\n\\draw[fill=gray!10!white] (0,4.5)--(4.5,0)--(4.5,4.5)--cycle;\n\\draw[fill=gray!90!white] (3,0)--(4.5,0)--(4.5,1.5)--(3,1.5)--cycle;\n\\draw[fill=gray!70!white] (2,1.5)--(3,1.5)--(3,2.5)--(2,2.5)--cycle;\n\\draw[fill=gray!40!white] (1,2.5)--(2,2.5)--(2,3.5)--(1,3.5)--cycle;\n\\draw[fill=gray!30!white] (0.5,3.5)--(1,3.5)--(1,4)--(0.5,4)--cycle;\n\\draw[fill=gray!20!white] (0,4)--(0.5,4)--(0.5,4.5)--(0,4.5)--cycle;\n\\node at (0.7,2.2) {$0$};\n\\node at (1.5,0.7) {$0$};\n\\node at (3.7,0.7) {$A$}; \n\\draw (0,4)--(0,0)--(3,0);\n\\draw (3,0)--(4.5,0);\n\\draw (0,1.5)--(4.5,1.5);\n\\draw (1,2.5)--(4.5,2.5);\n\\draw (0.5,3.5)--(4.5,3.5);\n\\draw (0,4)--(4.5,4);\n\\draw (0,4.5)--(4.5,4.5);\n\\draw (0,4)--(0,4.5);\n\\draw (0.5,3.5)--(0.5,4.5);\n\\draw (1,2.5)--(1,4.5);\n\\draw (2,1.5)--(2,4.5);\n\\draw (3,0)--(3,4.5);\n\\draw (4.5,0)--(4.5,4.5);\n\\node at (7,2.5) {\\fbox{$P$}};\n\\draw[->] (7,3.5)--(5,3.5);\n\\draw[->] (7,1.5)--(9.5,1.5);\n\\node at (6.3,4) {(C)+(R)};\n\\node at (8.2,1) {(C)+(R)+(X)};\n\\begin{scope}[shift={(10,0)}]\n\\draw[fill=gray!10!white] (0,4.5)--(3,1.5)--(3,4.5)--cycle;\n\\draw[fill=gray!90!white] (3,0)--(4.5,0)--(4.5,1.5)--(3,1.5)--cycle;\n\\draw[fill=gray!70!white] (2,1.5)--(3,1.5)--(3,2.5)--(2,2.5)--cycle;\n\\draw[fill=gray!40!white] (1,2.5)--(2,2.5)--(2,3.5)--(1,3.5)--cycle;\n\\draw[fill=gray!30!white] (0.5,3.5)--(1,3.5)--(1,4)--(0.5,4)--cycle;\n\\draw[fill=gray!20!white] (0,4)--(0.5,4)--(0.5,4.5)--(0,4.5)--cycle;\n\\node at (1.5,0.7) {$0$};\n\\node at (3.7,3) {$0$};\n\\node at (3.7,0.7) {$A$};\n\\node at (0.7,2.2) {$0$};\n\\draw (0,4)--(0,0)--(3,0);\n\\draw (3,0)--(4.5,0);\n\\draw (0,1.5)--(4.5,1.5);\n\\draw (1,2.5)--(3,2.5);\n\\draw (0.5,3.5)--(3,3.5);\n\\draw (0,4)--(3,4);\n\\draw (0,4.5)--(4.5,4.5);\n\\draw (0,4)--(0,4.5);\n\\draw (0.5,3.5)--(0.5,4.5);\n\\draw (1,2.5)--(1,4.5);\n\\draw (2,1.5)--(2,4.5);\n\\draw (3,0)--(3,4.5);\n\\draw (4.5,0)--(4.5,4.5);\n\\end{scope}\n\n\\end{tikzpicture}\n\\end{center}\n\\caption{\\small{A partly decomposable matrix representation $P$ of a nonobtuse $0\/1$-simplex $S$ can be brought in the left form using \noperations of type (C) and (R) only, and in the right form if using additional operations of type (X). The diagonal blocks are fully \nindecomposable.}}\n\\label{Nfigure10}\n\\end{figure}  \n\n\\begin{rem}\\label{rem-10}{\\rm The above shows that to each nonobtuse $0\/1$-simplex $S$ of dimension $n$ we can associate a special type \nof simplicial complex $C_p$ consisting of $p$ mutually orthogonal fully indecomposable simplicial facets $S_1,\\dots, S_p$ with respective \ndimensions $k_1,\\dots,k_p$ adding to $n$, where each facet $S_j$ lies in in its own $k_j$-facet of $I^n$. Explicitly, let $C_1=S_1$. The \ncomplex $C_{j+1}$ is obtained by attaching a vertex of $S_{j+1}$ to a vertex $v$ of $C_j$, such that the orthogonal projection of $S_{j+1}$ \nonto the $(k_1+\\dots+k_j)$-dimensional ambient space of $C_j$ equals $v$.}\n\\end{rem}\nRemark \\ref{rem-10} is illustrated by the tetrahedron in Figure~\\ref{Nfigure9}. It can be built from three $1$-simplices $S_1,S_2,S_3$ simply by first \nattaching $S_2$ with a vertex to a vertex of $S_1$ orthogonally to $S_1$, giving a right triangle $C_2$. Then attaching $S_3$ to the correct \nvertex $v$ of $C_2$ such that the projection of $S_3$ onto $C_2$ equals $v$ gives the tetrahedron. In Section~\\ref{Sect-5} we will pay special \nattention to the nonobtuse simplices whose fully indecomposable components are $n$-cube edges.\n\n\\begin{rem}{\\rm The $1$-simplex in $I^n$ has a fully indecomposable matrix representation with doubly stochastic pattern. It is formally an \nacute simplex. Indeed, the normals to its $0$-dimensional facets $0$ and $1$ point in opposite directions. Hence, its only dihedral angle equals \nzero. Since there does not exist a fully indecomposable triangle in $I^2$, matrix representations of a partly decomposable simplex do not have \n$2\\times 2$ fully indecomposable diagonal blocks.}\n\\end{rem}  \nWe would like to stress that although each partly decomposable matrix representation of a nonobtuse $0\/1$-simplex can be transformed into \nblock diagonal form by operations of types (C),(R) and (X) as depicted in the right in Figure~\\ref{Nfigure10}, the bottom right block cannot be {\\em any} of \nthe fully indecomposable diagonal blocks $A_j$. This is only possible if the corresponding simplex $S_j$ is attached to the remainder of the \ncomplex at exactly one vertex. For example, in Figure~\\ref{Nfigure11}, with $0$ as the origin, the matrix $A_3$ representing a regular tetrahedron in $I^3$ \nis a block of a block diagonal matrix. After mapping vertex $4$ to the origin with a reflection of type (X), the block $A_4$ representing a so-\ncalled antipodal $4$-simplex in $I^4$ is. However, the block $A_2$ representing the $1$-simplex in $I^1$ is never a block of a block diagonal \nmatrix representation of $S$. The only configurations of the three building blocks $S_1,S_2,S_3$ having a matrix representation that can be \ntransformed by operations of type (C),(R) and (X) onto block diagonal form with three diagonal blocks, are those in which $S_1,S_2$ and \n$S_3$ have a common vertex.\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tikzpicture}[scale=0.94, every node\/.style={scale=0.94}]\n\\begin{scope}[scale=1.1, every node\/.style={scale=1.1}]\n\\draw[fill=gray!30!white] (2,2)--(3.8,0.5)--(5,1.3)--(5,2.7)--(3.8,3.5)--cycle;\n\\draw[fill=gray!30!white] (1,2)--(-1,1)--(-1.5,2.5)--(-0.2,3.4)--cycle;\n\\draw (2,2)--(1,2);\n\\node[scale=0.9] at (3.2,2) {$S_1$};\n\\node[scale=0.9] at (-0.3,1.9) {$S_3$};\n\\node[scale=0.9] at (1.5,1.7) {$S_2$};\n\\draw (3.8,0.5)--(5,2.7)--(2,2); \n\\draw (2,2)--(5,1.3)--(3.8,3.5); \n\\draw (3.8,0.5)--(3.8,3.5);\n\\draw (-1,1)--(-0.2,3.4);\n\\draw[gray] (1,2)--(-1.5,2.5);\n\\draw[fill=black] (-1,1) circle [radius=0.05];\n\\draw[fill=white] (1,2) circle [radius=0.05];\n\\draw[fill=black] (2,2) circle [radius=0.05];\n\\draw[fill=black] (-1.5,2.5) circle [radius=0.05]; \n\\draw[fill=black] (-0.2,3.4) circle [radius=0.05];\n\\draw[fill=black] (3.8,0.5) circle [radius=0.05];\n\\draw[fill=black] (5,1.3) circle [radius=0.05];\n\\draw[fill=black] (5,2.7) circle [radius=0.05];\n\\draw[fill=black] (3.8,3.5) circle [radius=0.05];\n\\node[scale=0.8] at (0,3.6) {$1$};\n\\node[scale=0.8] at (1.2,2.2) {$0$};\n\\node[scale=0.8] at (1.8,2.2) {$4$};\n\\node[scale=0.8] at (-1,0.7) {$3$};\n\\node[scale=0.8] at (-1.7,2.5) {$2$};\n\\node[scale=0.8] at (4,0.4) {$5$};\n\\node[scale=0.8] at (5.2,1.4) {$6$};\n\\node[scale=0.8] at (5.2,2.6) {$7$};\n\\node[scale=0.8] at (4,3.6) {$8$};\n\\end{scope}\n\\begin{scope}[shift={(8.5,2.3)}]\n\\node[scale=0.9] at (0,-1.5) {$1\\,\\,\\,\\,2\\,\\,\\,\\,3\\,\\,\\,\\,4\\,\\,\\,\\,5\\,\\,\\,\\,6\\,\\,\\,\\,7\\,\\,\\,8$}; \n\\node[xscale=0.6,yscale=0.73] at (0,0.1)  {$\\left[\\begin{array}{ccc|c|cccc}\n     \\blacksquare  &   \\blacksquare   &   \\square &  \\square &    \\square   &    \\square   &    \\square &    \\square\\\\\n    \\blacksquare &    \\square   &  \\blacksquare &  \\square  &    \\square   &    \\square   &    \\square&    \\square\\\\\n    \\square &  \\blacksquare &    \\blacksquare   &   \\square   &   \\square  &  \\square  &   \\square&    \\square\\\\\\hline\n    \\square &  \\square &  \\square &    \\blacksquare   &  \\blacksquare &  \\blacksquare &  \\blacksquare&    \\blacksquare\\\\\\hline\n    \\square &  \\square &  \\square &  \\square &    \\blacksquare    &   \\blacksquare   &    \\blacksquare &   \\square\\\\ \n    \\square &  \\square &  \\square &  \\square  &   \\blacksquare     &  \\blacksquare         &     \\square     &  \\blacksquare \\\\\n    \\square &  \\square &  \\square &  \\square  &   \\blacksquare          &  \\square    &    \\blacksquare      & \\blacksquare \\\\\n     \\square &  \\square &  \\square &  \\square &   \\square          &  \\blacksquare         & \\blacksquare    & \\blacksquare \\\\\n     \\end{array}\\right]$};      \n     \\node[scale=0.9] at (3.3,0.1) {$=\\left[\\begin{array}{c|c|c} A_3 & 0 & 0 \\\\\\hline 0 & A_2 & R \\\\\\hline 0 & 0 & A_1\\end{array}\\right]$};\n\\end{scope}\n\\end{tikzpicture}\n\\end{center}\n\\caption{\\small{A simplicial complex of three mutually orthogonal simplices $S_1,S_2,S_3$. The given matrix representation corresponds to \nchoosing the vertex $0$ as the origin. Reflecting vertex $4$ to the origin decouples, alternatively, the bottom right $4\\times 4$ block. It is not \npossible to transform the matrix to block diagonal form with $A_2=[\\,1\\,]$ as one of the diagonal blocks. Reflecting any other vertex to the \norigin does not even lead to a block diagonal matrix representation.}}\n\\label{Nfigure11}\n\\end{figure}  \n\n\\smallskip\n\nIn general, the decomposability structure of a nonobtuse $0\/1$-simplex can be well visualized as a special type of planar graph, at the cost of \nthe geometrical structure. For this, assign to each $p\\times p$ fully indecomposable diagonal block a regular $p$-gon, and to attach these to \none another at the common vertex of the simplices they represent. See Figure~\\ref{Nfigure12} for an example. At the white vertices it is indicated how many \n$p$-gons meet. This number equals the number of diagonal blocks in the matrix representation when this vertex is reflected onto the origin. \n\\begin{figure}[h]\n\\begin{center}\n\\begin{tikzpicture}[scale=0.94, every node\/.style={scale=0.94}]\n\\draw[fill=gray!10!white] (0,0)--(1,-1)--(2,0)--(1,1)--cycle;\n\\draw (2,0)--(3,0);\n\\draw[fill=gray!30!white] (3,0)--(2,-0.7)--(2.4,-1.7)--(3.6,-1.7)--(4,-0.7)--cycle;\n\\draw (3,0)--(4,0);\n\\draw (4,0)--(5,1);\n\\draw (4,0)--(3,1);\n\\draw[fill=gray!50!white] (5,1)--(5.5,1.7)--(6.2,1.7)--(6.7,1)--(6.2,0.3)--(5.5,0.3)--cycle;\n\\draw[fill=gray!20!white] (5,0.3)--(6,-0.7)--(5,-1.7)--(4,-0.7)--cycle;\n\\draw[fill=black] (0,0) circle [radius=0.05];\n\\draw[fill=white] (2,0) circle [radius=0.05];\n\\draw[fill=white] (3,0) circle [radius=0.05];\n\\draw[fill=white] (4,0) circle [radius=0.05];\n\\draw[fill=white] (5,1) circle [radius=0.05];\n\\draw[fill=black] (3,1) circle [radius=0.05];\n\\draw[fill=black] (1,1) circle [radius=0.05];\n\\draw[fill=black] (1,0-1) circle [radius=0.05];\n\\draw[fill=black] (2,-0.7) circle [radius=0.05];\n\\draw[fill=black] (2.4,-1.7) circle [radius=0.05];\n\\draw[fill=black] (3.6,-1.7) circle [radius=0.05];\n\\draw[fill=black] (4,-0.7) circle [radius=0.05];\n\\draw[fill=black] (5.5,1.7) circle [radius=0.05];\n\\draw[fill=black] (6.2,1.7) circle [radius=0.05];\n\\draw[fill=black] (6.2,0.3) circle [radius=0.05];\n\\draw[fill=black] (5.5,0.3) circle [radius=0.05];\n\\draw[fill=black] (6.7,1) circle [radius=0.05];\n\\draw[fill=black] (6,-0.7) circle [radius=0.05];\n\\draw[fill=black] (5,-1.7) circle [radius=0.05];\n\\draw[fill=black] (5,0.3) circle [radius=0.05];\n\\draw[fill=white] (4,-0.7) circle [radius=0.05];\n\\node[scale=0.8] at (2.1,0.2) {$2$};\n\\node[scale=0.8] at (3,0.2) {$3$};\n\\node[scale=0.8] at (4,0.3) {$3$};\n\\node[scale=0.8] at (4.7,1) {$2$};\n\\node[scale=0.8] at (4,-0.4) {$2$};\n\\end{tikzpicture}\n\\end{center}\n\\caption{\\small{Schematic representation of a simplicial complex, built from a $5$-simplex, a $4$-simplex, two tetrahedra, and four edges, of \ntotal dimension $19$. With the origin at a white vertex, the matrix representation decouples into the indicated number of diagonal blocks. If the \nvertex is located at another vertex, the matrix representation does not decouple.}}\n\\label{Nfigure12}\n\\end{figure}  \n\n\\smallskip\n\nBefore studying further properties of partly decomposable nonobtuse $0\/1$-simplices in terms of their fully indecomposable components, we \nwill pay special attention to {\\em orthogonal} simplices.\n\\section{Orthogonal simplices and their matrix representations}\\label{Sect-5}\nThe simplest class of nonobtuse simplices is formed by the {\\em orthogonal simplices}. These are nonobtuse simplices with $\\binom{n}{2}-n$ \nright dihedral angles. Note that this is the {\\em maximum} number of right dihedral angles a simplex can have, as Fiedler proved in \\cite{Fie} \nthat any simplex has at least $n$ acute dihedral angles. Orthogonal simplices are useful in many applications, see \\cite{BrDiHaKr,BrKoKrSo} \nand the references therein. We will restrict our attention to orthogonal $0\/1$-simplices.\\\\[3mm]\nThe orthogonal $0\/1$-simplices can be defined recursively as follows \\cite{BrDiHaKr}. The cube edge $I^1$ is an orthogonal simplex. Now, a \nnonobtuse $0\/1$-simplex $S$ in $I^n$ is orthogonal if it has an $(\\nmo)$-facet $F$ with the properties that:\\\\[2mm]\n$\\bullet$ $F$ is contained in an $(\\nmo)$-facet of $I^n$;\\\\[2mm]\n$\\bullet$ $F$ is an orthogonal $(\\nmo)$-simplex.\\\\[2mm]\nClearly, this way to construct orthogonal $0\/1$-simplices is a special case of how nonobtuse $0\/1$-simplices were constructed from their fully \nindecomposable parts in Section~\\ref{Sect-4}. This is because a vertex $v$ forms an orthogonal simplex $S$ together with an $(\\nmo)$-facet \n$F$ that is contained in an $(\\nmo)$-facet of $I^n$ if and only if $v$ projects orthogonally on a vertex of $F$. This shows in particular that all \nfully indecomposable components of any matrix representation of $S$ equal $[\\,1\\,]$, which limits the number of their upper triangular matrix \nrepresentations.\n\n\\begin{Pro} There exist $n!$ distinct upper triangular $0\/1$ matrices that represent orthogonal $0\/1$-simplices in $I^n$. \n\\end{Pro}\n{\\bf Proof. } Let $P\\in\\BB^{n\\times n}$ be an upper triangular matrix representing an orthogonal $n$-simplex. Then the matrix\n\\[ \\tilde{P} = \\left[\\begin{array}{r|r} P & r \\\\\\hline 0 & 1\\end{array}\\right] \\]\nis upper triangular, and according to Theorem \\ref{th-3} it represents a nonobtuse simplex if and only if $r$ equals one of the $n+1$ distinct \ncolumns of $[0\\,|\\,P]$. Thus there are $\\npo$ times as many matrix representations of orthogonal $(\\npo)$-simplices in $I^{\\npo}$ as of \northogonal $n$-simplices in $I^n$, whereas $[\\,1\\,]$ is the only one in $I^1$.\\hfill $\\Box$\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tikzpicture}[scale=0.94, every node\/.style={scale=0.94}]\n\\draw[->] (6,-0.2)--(4,-0.8);\n\\draw[->] (6,-0.2)--(8,-0.8);\n\\draw[->] (3,-1.3)--(1.6,-2.3);\n\\draw[->] (3,-1.3)--(4.4,-2.3);\n\\draw[->] (3,-1.3)--(3,-1.8);\n\\draw[->] (9,-1.3)--(7.6,-2.3);\n\\draw[->] (9,-1.3)--(10.4,-2.3);\n\\draw[->] (9,-1.3)--(9,-1.8);\n\\draw[->] (7,-2.8)--(5,-3.8);\n\\draw[->] (7,-2.8)--(6.5,-3.8);\n\\draw[->] (7,-2.8)--(7.5,-3.8);\n\\draw[->] (7,-2.8)--(9,-3.8);\n\\node[xscale=0.4, yscale=0.5]  at (6,0) {$[\\,\\blacksquare\\,]$};\n\\node[xscale=0.4, yscale=0.5]  at (3,-1) {$\\left[\\begin{array}{c|c} \\blacksquare & \\square \\\\ \\hline \\square & \\blacksquare\\end{array}\\right]$};\n\\node[xscale=0.4, yscale=0.5]  at (9,-1) {$\\left[\\begin{array}{c|c} \\blacksquare & \\blacksquare \\\\ \\hline \\square & \\blacksquare\\end{array}\\right]$};\n\\node[xscale=0.4, yscale=0.5]  at (1,-2.3) {$\\left[\\begin{array}{cc|c} \\blacksquare & \\square & \\square \\\\ \\square & \\blacksquare & \\square \\\\\\hline  \\square & \\square & \\blacksquare \\end{array}\\right]$};\n\\node[xscale=0.4, yscale=0.5]  at (3,-2.3) {$\\left[\\begin{array}{cc|c} \\blacksquare & \\square & \\blacksquare \\\\ \\square & \\blacksquare & \\square \\\\ \\hline \\square & \\square & \\blacksquare \\end{array}\\right]$};\n\\node[xscale=0.4, yscale=0.5]  at (5,-2.3) {$\\left[\\begin{array}{cc|c} \\blacksquare & \\square & \\square \\\\ \\square & \\blacksquare & \\blacksquare \\\\\\hline  \\square & \\square & \\blacksquare \\end{array}\\right]$};\n\\node[xscale=0.4, yscale=0.5]  at (7,-2.3) {$\\left[\\begin{array}{cc|c} \\blacksquare & \\blacksquare &\\square\\\\ \\square & \\blacksquare &\\square \\\\\\hline \\square & \\square & \\blacksquare\\end{array}\\right]$};\n\\node[xscale=0.4, yscale=0.5]  at (9,-2.3) {$\\left[\\begin{array}{cc|c} \\blacksquare & \\blacksquare &\\blacksquare\\\\ \\square & \\blacksquare &\\square \\\\\\hline \\square & \\square & \\blacksquare\\end{array}\\right]$};\n\\node[xscale=0.4, yscale=0.5]  at (11,-2.3) {$\\left[\\begin{array}{cc|c} \\blacksquare & \\blacksquare & \\blacksquare\\\\ \\square & \\blacksquare &\\blacksquare \\\\\\hline \\square & \\square & \\blacksquare\\end{array}\\right]$};\n\\node[xscale=0.4, yscale=0.5]  at (4.8,-4.5) {$\\left[\\begin{array}{ccc|c} \\blacksquare & \\blacksquare & \\square& \\square\\\\ \\square & \\blacksquare &\\square & \\square\\\\ \\square & \\square & \\blacksquare& \\square \\\\\\hline \\square & \\square & \\square & \\blacksquare\\end{array}\\right]$};\n\\node[xscale=0.4, yscale=0.5]  at (6.3,-4.5) {$\\left[\\begin{array}{ccc|c} \\blacksquare & \\blacksquare &\\square& \\blacksquare\\\\ \\square & \\blacksquare &\\square & \\square\\\\\\square & \\square & \\blacksquare& \\square\\\\\\hline  \\square & \\square & \\square & \\blacksquare\\end{array}\\right]$};\n\\node[xscale=0.4, yscale=0.5]  at (7.7,-4.5) {$\\left[\\begin{array}{ccc|c} \\blacksquare & \\blacksquare &\\square& \\blacksquare\\\\ \\square & \\blacksquare &\\square & \\blacksquare\\\\ \\square & \\square & \\blacksquare& \\square\\\\ \\hline\\square & \\square & \\square & \\blacksquare\\end{array}\\right]$};\n\\node[xscale=0.4, yscale=0.5]  at (9.2,-4.5) {$\\left[\\begin{array}{ccc|c} \\blacksquare & \\blacksquare &\\square&\\square\\\\ \\square & \\blacksquare &\\square &\\square\\\\ \\square & \\square & \\blacksquare&\\blacksquare\\\\ \\hline\\square & \\square & \\square & \\blacksquare\\end{array}\\right]$};\n\\end{tikzpicture}\n\\end{center}\n\\caption{\\small{There are $n!$ upper triangular matrix representations of orthogonal $0\/1$-simplices.}}\n\\label{Nfigure13}\n\\end{figure}  \n\\begin{rem}{\\rm Modulo the action of the hyperoctahedral group, there remain as many as the number of unlabeled trees on $n+1$ vertices. \nIndeed, it is not hard to verify that two matrices $P$ and $R$ representing orthogonal $0\/1$-simplices can be transformed into one another \nusing operations of type (R),(C) and (X) if and only if the spanning trees of orthogonal edges of the simplices corresponding to $P$ and $R$ are \nisomorphic as graphs.}\n\\end{rem}\n\n \n\\section{One Neighbor Theorem for a class of nonobtuse simplices}\\label{Sect-6}\nIn this section we will discuss the one neighbor theorem for acute simplices \\cite{BrDiHaKr} and generalize it to a larger class of nonobtuse \nsimplices. This appears a very nontrivial matter, which can be compared with the complications that arise when generalizing the Perron-\nFrobenius theory for positive matrices to nonnegative matrices \\cite{BaRa,BePl}.\n\n\\subsection{The acute case revisited}\nThe one neighor theorem for acute $0\/1$-simplices reads as follows. We present a alternative proof to the proof in \\cite{BrDiHaKr}, based in \nTheorem \\ref{th-1}.\n\n\\begin{Th}[One Neighbor Theorem] \\label{th-5} Let $S$ be an acute $0\/1$-simplex in $I^n$, and $F$ an $(\\nmo)$-facet of $S$ opposite the \nvertex $v$ of $S$. Write $\\hat{S}$ for the convex hull of $F$ and $\\ol{v}$. Then:\\\\[3mm]\n$\\bullet$ $F$ does not lie in an $(\\nmo)$-facet of $I^n$;\\\\[3mm]\n$\\bullet$ $\\hat{S}$ is the only $0\/1$-simplex having $F$ as a facet that may be acute, too.\\\\[3mm]\nIn words, an acute $0\/1$-simplex has at most one acute face-to-face neighbor at each facet.\n\\end{Th}\n{\\bf Proof. } Let $q$ be a normal vector to a facet of $F$ opposite a vertex $p$ of an acute $0\/1$-simplex $S$. Then due to \nTheorem \\ref{th-1}, $q$ has no zero entries. Therefore, the line $v+\\alpha q$ parametrized by $\\alpha\\in\\RR$ intersect the interior of $I^n$ if \nand only if $v\\in\\{p,\\ol{p}\\}$. Thus, for other vertices $v$ of $I^n$, the altitude from $v$ to the ambient hyperplane of $F$ does not land in \n$I^n$ and thus not in $F$. This is however a necessary condition for the convex hull of $F$ and $v$ to be acute. \\hfill $\\Box$\\\\[3mm]\nSee the left picture in Figure~\\ref{Nfigure14} for an illustration of Theorem \\ref{th-5} in $I^3$. Only the pair of white vertices end up inside $I^3$ when \nfollowing the normal direction $q$ of the facet $F$. All six other vertices, when projected on the plane containing $F$ end up outside $I^3$, or \non themselves.\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tikzpicture}[scale=0.94, every node\/.style={scale=0.94}]\n\\draw[fill=gray!20!white] (0,0)--(2.5,1)--(2,2)--(0.5,3)--cycle;\n\\draw[fill=gray!50!white] (0,0)--(2.5,1)--(2,2)--cycle;\n\\draw[gray,thick,<->] (-0.35,2.8)--(0.35,1.2);\n\\draw[gray,thick,<->] (2.15,3.5)--(2.85,2.5);\n\\node at (2.3,0) {$\\overline{p}$};\n\\node at (0.2,3) {$p$};\n\\node at (0.9,0.6) {$F$};\n\\node at (0.9,1.7) {$q$};\n\\node at (1.5,0.3) {$-q$};\n\\draw (0,0)--(2,0)--(2,2)--(0,2)--cycle;\n\\draw (0.5,1)--(2.5,1)--(2.5,3)--(0.5,3)--cycle;\n\\draw (0,0)--(0.5,1);\n\\draw (2,0)--(2.5,1);\n\\draw (0,2)--(0.5,3);\n\\draw (2,2)--(2.5,3);\n\\draw (0,0)--(2,2);\n\\draw (2.5,1)--(0.5,3);\n\\draw[fill=black] (0,0) circle [radius=0.05];\n\\draw[fill=black] (2.5,1) circle [radius=0.05];\n\\draw[fill=black] (2,2) circle [radius=0.05];\n\\draw[fill=black] (2.5,3) circle [radius=0.05];\n\\draw[fill=black] (0,2) circle [radius=0.05];\n\\draw[fill=black] (0.5,1) circle [radius=0.05];\n\\draw[fill=white] (1.45,1.1) circle [radius=0.05];\n\\draw[thick,->] (2,0)--(1.55,0.9);\n\\draw[thick,->] (0.5,3)--(1.4,1.2);\n\\draw[fill=white] (0.5,3) circle [radius=0.07];\n\\draw[fill=white] (2,0) circle [radius=0.07];\n\\begin{scope}[shift={(5,0)}]\n\\node at (-0.3,0) {$p$};\n\\node at (2.8,1) {$\\hat{p}$};\n\\draw[fill=gray!20!white] (0,0)--(2,0)--(0.5,3)--cycle;\n\\draw[fill=gray!50!white] (2,0)--(0.5,3)--(0.5,1)--cycle;\n\\node at (0.6,0.5) {$q$};\n\\node at (0.9,1.5) {$F$}; \n\\draw (0,0)--(2,0)--(2,2)--(0,2)--cycle;\n\\draw (0.5,1)--(2.5,1)--(2.5,3)--(0.5,3)--cycle;\n\\draw (0,0)--(0.5,1);\n\\draw (2,0)--(2.5,1);\n\\draw (0,2)--(0.5,3);\n\\draw (2,2)--(2.5,3);\n\\draw (2,0)--(0.5,1);\n\\draw[fill=white] (0,0) circle [radius=0.07];\n\\draw[fill=black] (0.5,1) circle [radius=0.05];\n\\draw[fill=black] (2,0) circle [radius=0.05];\n\\draw[fill=black] (0.5,3) circle [radius=0.05];\n\\draw[fill=black] (2,2) circle [radius=0.05];\n\\draw[fill=white] (2.5,1) circle [radius=0.07];\n\\draw[fill=white] (0,2) circle [radius=0.07];\n\\draw[fill=white] (2.5,3) circle [radius=0.07];\n\\draw[fill=white] (1.25,0.5) circle [radius=0.05];\n\\draw[fill=white] (1.25,2.5) circle [radius=0.05];\n\\draw[thick,->] (0.1,0.05)--(1.2,0.47);\n\\draw[thick,->] (2.4,0.95)--(1.3,0.53);\n\\draw[thick,->] (0.1,2.05)--(1.2,2.47);\n\\draw[thick,->] (2.4,2.95)--(1.3,2.53);\n\\draw[gray,thick,<->] (1.3,1.7)--(2.7,2.3);\n\\end{scope}\n\\end{tikzpicture}\n\\end{center}\n\\caption{\\small{Left: for any facet $F$ of an acute $0\/1$-simplex, there is only one pair of antipodal vertices $p,\\ol{p}$ that may project onto \n$F$. All others end up outside $I^n$ when following the normal $q$ in either direction. Right: in a nonobtuse simplex, there can be more than \ntwo vertices that remain in $I^n$ when following the normal direction to an interior facet.}}\n\\label{Nfigure14}\n\\end{figure}  \n\n\\smallskip\n\nThe translation of Theorem \\ref{th-5} in terms of linear algebra is as follows. \n\\begin{Co} Let $P\\in\\BB^{n\\times(\\nmo)}$. The matrix $[P\\,|\\,v] \\in\\Bnn$ is a matrix representation of an acute $0\/1$-simplex for at most \none pair of antipodal points $v\\in\\{p,\\ol{p}\\}\\subset\\BB^n$.\n\\end{Co}\nThe one neighbor theorem dramatically restricts the number of $0\/1$-polytopes that can be face-to face triangulated by acute simplices. For \ninstance, only from dimension $n=7$ onwards there exists a pair of face-to-face acute simplices in $I^n$. In $I^7$ it is the Hadamard regular \nsimplex \\cite{Gr} and its face-to-face neighbor, which is unique modulo the action of the hyperoctahedral group, and which has the one-but-\nlargest volume in $I^7$ over all acute $0\/1$-simplices \\cite{BrCi2}. Also in \\cite{BrDiHaKr} the theorem turned out useful in constructing all \npossible face-to-face triangulations of $I^n$ consisting on nonobtuse simplices only, due to the following sharpening of the statement.\n\n\\begin{Co} Each acute $0\/1$-simplex $S$ in $I^n$ has at most one face-to-face nonobtuse neighbor at each of its facets.\n\\end{Co}\n{\\bf Proof.} This follows from the fact that in the proof of Theorem \\ref{th-5}, the altitudes from $v\\not=\\{p,\\ol{p}\\}$ intersect $I^n$ only in \n$v$ itself.\\hfill $\\Box$\\\\[3mm]\nA natural question is what can be proved for nonobtuse-$0\/1$ simplices. Theorem \\ref{th-2} showed that a normal to a facet of a nonobtuse \n$0\/1$-simplex $S$ can have entries equal to zero. Writing $\\zeros(q)$ for the number of entries of $q$ equal to zero, there \nare$2^{\\zeros(q)+1}$ vertices $v$ of $I^n$ from which the altitudes starting at $v$ onto the plane containing $F$ do not leave $I^n$. This is \nillustrated in the right picture in Figure~\\ref{Nfigure14}. The normal vector $q$ to the facet $F$ has one zero entry: $\\zeros(q)=1$. The altitudes from the \n$2^2$ white vertices of $I^3$ onto the plane containing $F$ lie in $I^3$.\\\\[3mm] \nNevertheless, only the altitudes from $p$ and $\\hat{p}$ land on $F$ itself, and we see that the interior facet $F$ of $S$ in $I^3$ has exactly \none nonobtuse neighbor. It is tempting to conjecture that the one neighbor theorem holds also for nonobtuse $0\/1$-simplices. The only \nadaption to make is then, based on the example in Figure~\\ref{Nfigure14}, that instead of the antipodal $\\ol{p}$ of $p$ in $I^n$, the second vertex \n$\\hat{p}$ such that the convex hull of $F$ with $\\hat{p}$ is a nonobtuse simplex would satisfy \n\\be\\label{restr-ant} \\hat{p}_j=1-p_j \\hdrie \\Leftrightarrow \\hdrie q_j\\not=0 \\und \\hat{p}_j = 0 \\hdrie \\Leftrightarrow \\hdrie q_j=0. \\ee\nIn words, $\\hat{p}$ is the antipodal of $p$ restricted to the $(n-\\zeros(q))$-dimensional $n$-cube facet that contains both $p$ and $q$. In \nFigure~\\ref{Nfigure14}, $\\hat{p}$ is the antipodal of $p$ in the bottom square facet of $I^3$.\\\\[3mm]\nAs an attempt to prove this conjecture, one may try to demonstrate that the remaining white vertices in the top square facet of $I^3$, although \ntheir altitudes lie in $I^3$, cannot fall onto $F$. Although we did not succeed in doing so, the nonobtusity of $S$ is a necessary condition. To \nsee this, consider the $0\/1$-simplex $S$ in $I^5$ with matrix representation\n\\be \\left[\\begin{array}{ccccc} 1 & 1 & 1 & 0 & 0\\\\ 1 & 1 & 0 & 1 & 0\\\\ 1 & 0 & 0 & 0 & 1\\\\ 0 & 0 & 1 & 1 & 0 \\\\ 0 & 1 & 1 & 0 & 1\\end{array}\\right] \\hdrie \\hdrie\\mbox{\\rm with} \\hdrie q = \\frac{1}{2}\\left[\\begin{array}{c} 0 \\\\ 1 \\\\ 1 \\\\ 1 \\\\ 1\\end{array}\\right],\\ee\nand $q$ is the normal to the facet $F$ of $S$ opposite the origin. Observe that the line from the origin to the vertex $2q$ in $I^5$ intersects \n$F$ in the midpoint of its edge between the two vertices of $S$ in the last two columns of $P$. This shows that both the origin and its \nantipodal in the bottom $4$-facet of $I^5$ as defined in (\\ref{restr-ant}) land in $F$ when following their respective altitudes. But also the line \nbetween $e_1^5$ and $e^5$ intersects $F$ in the midpoint of its edge between the two vertices in the first and third column of $P$. Thus, \nalso both the vertices $e_1^5$ and $e^5$ in the top $4$-facet of $I^5$ land in $F$ when following their altitudes. Thus, for a $0\/1$-simplex \nthat is not nonobtuse, it can occur that more than two vertices of $I^n$ project orthogonally onto $F$.\n\n\\subsection{More on fully indecomposable nonobtuse simplices}\nIn Section~\\ref{Sect-6.3} we will study the one neighbor theorem in the context of partly decomposable \nnonobtuse simplices. For this, but also for its own interest, we derive two auxiliary results on fully indecomposable nonobtuse simplices.\n\n\\begin{Le}\\label{lem-6} Each representation $P\\in\\Bnn$ of a fully indecomposable $0\/1$-simplex $S$ satisfies\n\\be P^\\top P  \\geq I+e^n\\left(e^n\\right)^\\top. \\ee\nIn geometric terms this implies that all triangular facets of $S$ are acute.\n\\end{Le}\n{\\bf Proof. } A standard type of argument is the following. Write $D$ for the diagonal matrix having the same diagonal \nentries as $B=(P^\\top P)^{-1}$, and let $C=D-B$. Then $C\\geq 0$, and\n\\be B = D-C = D(I-D^{-1}C) \\und P^\\top P =B^{-1} =  (I-D^{-1}C)^{-1}D^{-1}. \\ee\nBecause $B$ is an M-matrix \\cite{Joh,JoSm}, the spectral radius of $D^{-1}C$ is less than one, and the following Neumann series converges:\n\\be (I-D^{-1}C)^{-1} = \\sum_{j=0}^\\infty \\left(D^{-1}C\\right)^j. \\ee\nSince $P$ is fully indecomposable, $P^\\top P$ is irreducible, and thus $B$ is irreducible. But then, so are $C$ and $D^{-1}C$. \nBecause of the latter, for each pair $k,\\ell$ there is a $j$ such that $e_k^\\top(D^{-1}C)^j e_\\ell >0$. This proves that \n$B^{-1}=P^\\top P>0$. Since its entries are integers, $P^\\top P \\geq e^n(e^n)^\\top$. Thus, each pair of edges of $S$ that meet \nat the origin makes an acute angle. As by Theorem \\ref{th-1} all matrix representations of $S$ are fully indecomposable, we conclude \nthat all triangular facets of $S$ are acute. This implies that any diagonal entry of $P^\\top P$ is greater than the remaining entries in \nthe same row. Indeed, if two entries in the same row would be equal, then $p_j^\\top (p_j-p_i)=0$, which corresponds to two edges of $S$ making a right angle.\\hfill $\\Box$\n\n\\begin{Co}\\label{co-5} Let $P$ be a fully indecomposable matrix representation of a nonobtuse $0\/1$-simplex. \nIf $\\hat{P}$ equals $P$ with one column replaced by its antipodal, then $\\hat{P}^\\top\\hat{P}>0$.\n\\end{Co}\n{\\bf Proof. } Without loss of generality, assume that\n\\be P = [p\\,|\\,P_1] \\und \\hat{P} = [\\ol{p}\\,|\\,P_1] \\ee\nwith $p\\in\\BB^n$. Then\n\\be \\hat{P}^\\top\\hat{P} = \\left[\\begin{array}{cc} \\ol{p}^\\top\\ol{p} & \\ol{p}^\\top P_1 \\\\ P_1\\ol{p} & P_1^\\top P_1 \\end{array}\\right], \\ee\nand $P_1^\\top P_1>0$ because $P^\\top P>0$ as proved in Lemma \\ref{lem-5}. Due to the fact that for all $a,b\\in\\BB^n$,\n\\be a^\\top \\ol{b} = a^\\top (e^n-b) = a^\\top(a-b),\\ee\nwe see that also $P_1^\\top\\ol{p}>0$. Indeed, a zero entry would contradict that the diagonal entries of $P^\\top P$ are greater than its \noff-diagonal entries, as proved in Lemma \\ref{lem-6}.\\hfill $\\Box$\n  \n\\begin{rem}{\\rm The matrix $\\hat{P}$ in Corollary \\ref{co-5} is not always fully indecomposable. See\n\\be P=\\left[\\begin{array}{ccc} 0 & 1 & 1 \\\\ 1 & 0 & 1\\\\ 1 & 1 & 0\\end{array}\\right] \\und \\hat{P}=\\left[\\begin{array}{ccc} 1 & 1 & 1 \\\\ 0 & 0 & 1\\\\ 0 & 1 & 0\\end{array}\\right], \\ee \nwhere the first columns of both matrices are each other's antipodal. This example also shows that the top left diagonal entry of \n$\\hat{P}^\\top\\hat{P}$ need not be greater than the other entries in its row.}\n\\end{rem}\nWe end this section with a theorem which was proved by inspection of a finite number of cases. We refer to \\cite{BrCi2} for details \non how to computationally generate the necessary data.\n\n\\begin{Th}\\label{th-6} Each fully indecomposable nonobtuse $0\/1$-simplex in  $\\SS^n$ with $n\\leq 8$ is acute. There exist fully \nindecomposable nonobtuse $0\/1$-simplices in $\\SS^n$ with $n\\geq 9$ that are not acute.\n\\end{Th}\n{\\bf Proof. } See \\cite{BrCi2} for details on an algorithm to compute $0\/1$-matrix representations of $0\/1$-simplices modulo the action \nof the hyperoctahedral group. By inspection of all $0\/1$-simplices of dimensions less than or equal to $8$, we conclude the first \nstatement. For the second statement, we give an example. The $9\\times 9$ matrix in Figure~\\ref{Nfigure15} represents a nonobtuse simplex \nthat is not acute, but $P$ is fully indecomposable. \\hfill $\\Box$\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tikzpicture}[scale=0.94, every node\/.style={scale=0.94}]\n\\node[scale=0.7] at (0,0)  {$P = \\left[\\begin{array}{rrrrrrrrr}\n  1  & 1 &  0 &  0 &  1 &  1 &  1 &  1 &  0\\\\\n  1  & 0 &  1 &  1 &  1 &  0 &  0 &  1 &  1\\\\ \n  1  & 0 &  1 &  1 &  0 &  1 &  1 &  0 &  1\\\\\n  0  & 1 &  1 &  1 &  1 &  0 &  1 &  0 &  1\\\\\n  0  & 1 &  1 &  1 &  0 &  1 &  0 &  1 &  1\\\\\n  0  & 0 &  1 &  1 &  1 &  1 &  1 &  1 &  0\\\\\n  0  & 0 &  1 &  0 &  1 &  1 &  0 &  0 &  1\\\\\n  0  & 0 &  1 &  0 &  0 &  0 &  1 &  1 &  1\\\\\n  0  & 0 &  0 &  1 &  1 &  1 &  1 &  1 &  1\n  \\end{array}\\right], \\hspace{3mm}P^{-\\top}= \\dfrac{1}{20} \n   \\left[\\begin{array}{rrrrrrrrr}   \n     6  &   6  &  -2  &  -6  &   2  &   2  &   2  &   2  &  -2\\\\\n     7  &  -3  &   1  &   3  &   4  &  -6  &  -6  &   4  &   1\\\\\n     7  &  -3  &   1  &   3  &  -6  &   4  &   4  &  -6  &   1\\\\\n    -3  &   7  &   1  &   3  &   4  &  -6  &   4  &  -6  &   1\\\\\n    -3  &   7  &   1  &   3  &  -6  &   4  &  -6  &   4  &   1\\\\\n    -4  &  -4  &   8  &   4  &   2  &   2  &   2  &   2  & -12\\\\\n    -2  &  -2  &   4  &  -8  &   6  &   6  &  -4  &  -4  &   4\\\\\n    -2  &  -2  &   4  &  -8  &  -4  &  -4  &   6  &   6  &   4\\\\\n    -4  &  -4  & -12  &   4  &   2  &   2  &   2  &   2  &   8\n    \\end{array}\\right], \\hspace{3mm}\\dfrac{1}{20} \n\\left[\\begin{array}{r} 10 \\\\ 5 \\\\ 5 \\\\ 5 \\\\ 5 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 0 \\end{array}\\right]$};\n\\end{tikzpicture}\n\\end{center}\n\\caption{\\small{Example of a fully indecomposable matrix representation $P$ of a nonobtuse simplex $S$ that is not acute. The vector \non the right is its normal $q$ to the facet $F$ opposite the origin. Since $q$ has entries equal to zero, $S$ cannot be acute. But \n$(P^\\top P)^{-1}$ satisfies (\\ref{eq-1.1}) and (\\ref{eq-1.2}), hence $S$ is nonobtuse. Note that none of the other normals has a zero entry.}}\n\\label{Nfigure15}\n\\end{figure}  \n\n\\smallskip\n\nThus, Theorem \\ref{th-6} and Figure~\\ref{Nfigure14} prove that the {\\em indecomposability} of a matrix representation of a nonobtuse $0\/1$-simplex $S$ is, in \nfact, a {\\em weaker} property than the {\\em acuteness} of $S$. \\\\[3mm]\nEspecially since the two concepts coincide up to dimension eight, this came as a surprise. Citing G\\\"unther Ziegler in \nChapter 1 of {\\em Lectures on $0\/1$-Polytopes} \\cite{KaZi}: ``{\\em Low-dimensional intuition does not work!}\\,''. \nSee \\cite{BrCi2} for more such examples in the context of $0\/1$-simplices. \n \n\n\\subsection{A One Neighbor Theorem for partly decomposable simplices}\\label{Sect-6.3}\nLet $S$ be a partly decomposable nonobtuse simplex. Then according to Corollary \\ref{co-3}, $S$ has a matrix representation\n\\be P = \\left[\\begin{array}{cc} N & 0 \\\\ 0 & A\\end{array}\\right] \\ee \nin which $A$ is fully indecomposable. We will discuss some cases in which a modified version of the One Neighbor Theorem \\ref{th-5} \nholds also for nonobtuse simplices that are not acute.\\\\[3mm]\n{\\bf Case I.} To illustrate the main line of argumentation, consider first the simplest case, which is that $S\\in\\SS^n$ can be represented by\n\\be\\label{eq-21} P = \\left[\\begin{array}{cc} A_2 & 0 \\\\ 0 & A_1\\end{array}\\right],\\hdrie\\mbox{\\rm with} \\hdrie A_1\\in\\BB^{k\\times k} \\und A_2\\in\\BB^{(n-k)\\times(n-k)} \\ee\nand in which $A_1$ and $A_2$ represent acute simplices $S_1$ and $S_2$. Then $A_1$ and $A_2$ are fully indecomposable, \nand $S_1$ and $S_2$ satisfy the One Neighbor Theorem \\ref{th-5}. Assume first that neither $A_1$ or $A_2$ equals the $1\\times 1$ matrix $[\\,1\\,]$.\\\\[3mm] \n{\\bf Notation. } We will write $X^j(y)$ for the matrix $X$ with column $j$ replaced by $y$.\\\\[3mm]\nNow, let $v\\in\\BB^n$, partitioned as $v^\\top = (v_1^\\top \\,\\,v_2^\\top)$ with $v_1\\in\\BB^k$. Assume that the block lower \ntriangular matrix $P^1(v)$ represents a nonobtuse simplex. This implies that its top-left diagonal block $A_2^1(v_1)$ does \nso, too, hence $v_1=a_1=Ae_1^k$ or $v_1=\\ol{a_1}$ by Theorem \\ref{th-5}. Corollary \\ref{co-5} shows that in both cases \n$A_2^1(v_1)^\\top A_2^1(v_1) > 0$. But then Theorem \\ref{th-3} in combination with the observations in Remarks \\ref{rem-4} \nand \\ref{rem-6} proves that $v_2=0$, because all columns in the off-diagonal block must be copies of one and the same column \nof $A_1$. Hence, there is at most one $v\\in\\BB^n$ other than $Pe_1^n$ such that $P^1(v)$ is nonobtuse. See Figure~\\ref{Nfigure16} for a sketch of the proof.\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tikzpicture}[scale=0.94, every node\/.style={scale=0.94}]\n\\draw (0,0)--(3,0)--(3,3)--(0,3)--cycle;\n\n\\draw[fill=gray!20!white] (2,0)--(3,0)--(3,1)--(2,1)--cycle;\n\\draw[fill=gray!50!white] (0,1)--(2,1)--(2,3)--(0,3)--cycle;\n\n\\draw (0.5,1)--(0.5,3);\n\n\\node at (2.5,0.5) {$A_1$};\n\\node at (1,0.5) {$0$};\n\\node at (2.5,2) {$0$};\n\\node at (0.25,2) {$a_1$};\n\n\\draw[->] (3.2,1.5)--(3.8,1.5);\n\n\\begin{scope}[shift={(4,0))}]\n\n\\draw (0,0)--(3,0)--(3,3)--(0,3)--cycle;\n\n\\draw[fill=gray!20!white] (2,0)--(3,0)--(3,1)--(2,1)--cycle;\n\\draw[fill=gray!50!white] (0.5,1)--(2,1)--(2,3)--(0.5,3)--cycle;\n\n\\draw (0.5,0)--(0.5,3);\n\\draw (0,1)--(0.5,1);\n\n\\node at (2.5,0.5) {$A_1$};\n\\node at (1,0.5) {$0$};\n\\node at (2.5,2) {$0$};\n\n\\node at (0.25,2) {$v_1$};\n\\node at (0.25,0.5) {$v_2$};\n\n\\draw[->] (3.2,1.5)--(3.8,1.5);\n\n\\end{scope}\n\n\\begin{scope}[shift={(8,0))}]\n\n\\draw (0,0)--(3,0)--(3,3)--(0,3)--cycle;\n\n\\draw[fill=gray!20!white] (2,0)--(3,0)--(3,1)--(2,1)--cycle;\n\\draw[fill=gray!50!white] (0.5,1)--(2,1)--(2,3)--(0.5,3)--cycle;\n\\draw[fill=gray!30!white] (0,1)--(0.5,1)--(0.5,3)--(0,3)--cycle;\n\n\\draw (0.5,0)--(0.5,3);\n\\draw (0,1)--(0.5,1);\n\n\\node at (2.5,0.5) {$A_1$};\n\\node at (1,0.5) {$0$};\n\\node at (2.5,2) {$0$};\n\n\\node at (0.25,2.4) {$a_1$};\n\\node at (0.25,2) {or};\n\\node at (0.25,1.6) {$\\overline{a_1}$};\n\\node at (0.25,0.5) {$v_2$};\n\n\\draw[->] (3.2,1.5)--(3.8,1.5);\n\n\\end{scope}\n\n\\begin{scope}[shift={(12,0))}]\n\n\\draw (0,0)--(3,0)--(3,3)--(0,3)--cycle;\n\n\\draw[fill=gray!20!white] (2,0)--(3,0)--(3,1)--(2,1)--cycle;\n\\draw[fill=gray!50!white] (0.5,1)--(2,1)--(2,3)--(0.5,3)--cycle;\n\\draw[fill=gray!30!white] (0,1)--(0.5,1)--(0.5,3)--(0,3)--cycle;\n\n\\draw (0.5,0)--(0.5,3);\n\\draw (0,1)--(0.5,1);\n\n\\node at (2.5,0.5) {$A_1$};\n\\node at (1,0.5) {$0$};\n\\node at (2.5,2) {$0$};\n\n\\node at (0.25,2.4) {$a_1$};\n\\node at (0.25,2) {or};\n\\node at (0.25,1.6) {$\\overline{a_1}$};\n\\node at (0.25,0.5) {$0$};\n\n\\end{scope}\n\\end{tikzpicture}\n\\end{center}\n\\caption{\\small{Steps in proving a one neighbor theorem for partly decomposable nonobtuse simplices with two fully indecomposable \nblocks representing acute simplices.}}\n\\label{Nfigure16}\n\\end{figure}  \n\n\\smallskip\n\nClearly, the same argument can be applied to prove that for all $j\\in\\{1,\\dots,n\\}$, the matrix $P^j(v)$ represents a nonobtuse $0\/1$-simplex \nfor at most one $v\\in\\BB^n$ other than $Pe_j^n$. This proves that for each column $p$ of $P$, the facet $F_p$ of $S$ opposite $p$ has at \nmost one nonobtuse neighbor. It remains to prove the same for the facet $F_0$ of $S$ opposite the origin. But because $S_1$ and $S_2$ are \nby assumption acute, the normals $q_1$ and $q_2$ to their respective facets opposite the origin, which satisfy $A_1^\\top q_1 = e_k^k$ and \n$A_2^\\top q_2 = e_{n-k}^{n-k}$, are both positive. But then so is the normal $q$ of $F_0$, which satisfies $P^\\top q = e_n^n$, and \nhence $q^\\top = (q_1^\\top\\,\\,q_2^\\top)>0$. And thus, apart from the origin, only $e_n^n$ can form a nonobtuse simplex together with $F_0$.\n\n\\begin{rem}{\\rm Note that this last argument does not hold if $A_1$ and $A_2$ are merely assumed to represent fully indecomposable \nnonobtuse simplices: the $9\\times 9$ matrix in Figure~\\ref{Nfigure15} shows that the normal of the facet opposite the origin may contain entries equal to zero.}\n\\end{rem}\nTo finish the case in which $S$ has a matrix representation as in (\\ref{eq-21}), assume without loss of generality that $A_1=[\\,1\\,]$ and \n$A_2\\not=[\\,1\\,]$.  Then the facet $F$ of $S$ opposite the last column of $P$ lies in a cube facet, and thus it cannot have a nonobtuse \nface-to-face neighbor. For the remaining $n$ facets of $S$, arguments as above apply, and we conclude that $S$ has at most one nonobtuse \nneighbor at each of its facets. The remaining case that $A_1=A_2=[\\,1\\,]$ is trivial.\\\\[3mm]\nNote that if a nonobtuse $0\/1$-simplex has a block diagonal matrix representation with $p>2$ blocks, each representing an acute simplex, the \nresult remains valid, based on a similar proof.\\\\[3mm]\n{\\bf Case II.} Assume now that the matrix representation $P$ of a nonobtuse $0\/1$-simplex $S$ has the form\n\\be\\label{case-iii} P = \\left[\\begin{array}{cc} N_1 & 0 \\\\ 0 & A_1\\end{array}\\right], \\ee \nwhere $A_1\\in\\BB^{(n-k)\\times(n-k)}$ represents an acute simplex and $N_1$ a merely nonobtuse simplex. Using similar \narguments as in Case I it is easily seen that the only two choices of $v\\in\\BB^n$ such that $P^j(v)$ with $k+1\\leq j \\leq n$ is nonobtuse are\n\\be v = Pe_j^n = \\left[\\begin{array}{c} 0 \\\\ a_j \\end{array}\\right] \\und v = \\left[\\begin{array}{c} 0 \\\\ \\ol{a_j} \\end{array}\\right], \\ee \nas no additional properties of $N_1$ need to be known. This changes if we examine the matrix $P^j(v)$ with $1\\leq j\\leq k$, as it is generally not \ntrue that $N_1^\\top N_1>0$. A way out is the following. Assume that also $N_1$ is partly decomposable, then using only row and column permutations, \nwe can first transform $N_1$ into the form\n\\be N_1 \\overset{(C)+(R)}{\\longrightarrow } \\left[\\begin{array}{cc} N_2 & R \\\\ 0  & A_2\\end{array}\\right] \\ee \nwhere we assume that $A_2$ represents an acute simplex. Then reflecting the vertex to the origin such that the block above $A_2$ becomes zero, we find that\n\\be\\label{eq-22} P \\sim  \\tilde{P} = \\left[\\begin{array}{c|c|c}  N_2 & R & 0 \\\\ \\hline 0  & A_2 & 0 \\\\\\hline 0 & 0 & A_1\\end{array}\\right] \\sim  \\left[\\begin{array}{c|c|c}  \\tilde{N}_2 & 0 & \\tilde{R} \\\\\\hline 0  & A_2 & 0 \\\\\\hline 0 & 0 & A_1\\end{array}\\right] = \\hat{P},\\ee\nwhere $\\tilde {R}$ has the same columns as $R$ but possibly a different number of them. Now, select a column of $\\hat{P}$ that contains entries of $A_2$ and \nreplace it by $v$, partitioned as $v^\\top = (v_1^\\top \\,\\,v_2^\\top\\,\\,v_3^\\top)$. Because the bottom right $2\\times 2$ block part of $\\hat{P}$ is a matrix \nrepresentation of a nonobtuse simplex as considered in Case I, we conclude that $v_3=0$. Because the top left $2\\times 2$ block part of $\\tilde{P}$ is a \nmatrix representation as in (\\ref{case-iii}), we conclude that $v_3=0$ and $v_2$ is a column of $A_2$ or its antipodal. Thus, also the facets of the \nvertices of $S$ corresponding to its indecomposable part $A_2$ all have at most one nonobtuse neighbor.\\\\[3mm] \nNow, this process can be inductively repeated in case $\\tilde{N_2}$ is partly decomposable with a fully indecomposable part that represents an acute \nsimplex, and so on, until a fully indecomposable top left block $A_p$ remains. This block presents vertices for which it still needs to be proved that \ntheir opposite facets have at most one nonobtuse neighbor. To illustrate how to do this, consider the case $p=3$. Or, in other words, assume that \n$N_2$ in (\\ref{eq-22}) represents an acute simplex. Replace one of the corresponding columns of $\\tilde{P}$ by $v$ partitioned as \n$v^\\top = (v_1^\\top \\,\\,v_2^\\top\\,\\,v_3^\\top)$. Then $v_3=0$ because the $(1,3)$ block of $\\tilde{P}$ equals zero. Similarly, because \nthe $(1,2)$-block in $\\hat{P}$ equals zero, we find that $v_2=0$. And thus, $v_3$ is a column of $N_2$ or its antipodal. For $p>3$, we \ncan do the same one by one for the blocks at positions $(1,p),\\dots,(1,2)$.\\\\[3mm]\nThe analysis in this section can be summarized in the following theorem. \n\n\\begin{Th} Let $S$ be a nonobtuse $0\/1$-simplex whose fully indecomposable components are all acute. Then $S$ has at most one face-to-face neighbor at each  of its interior facets.\n\\end{Th}\n\n\n\\subsection*{Acknowledgments}\nJan Brandts and Apo Cihangir acknowledge the support by Research Project 613.001.019 of the Netherlands Organisation for Scientific Research (NWO), and are grateful to Michal K\\v{r}\\'{\\i}\\v{z}ek for comments and discussions on earlier versions of the manuscript.\n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe mechanical properties of amorphous solids have been harnessed\nextensively in designing materials which are ubiquitous in our everyday\nlife. However, a complete microscopic understanding of the mechanisms\nleading to the macroscopic response of these materials is still missing.\nIn order to develop materials with specific functions, it is necessary to\nhave an improved knowledge of these underlying processes.  This remains\na challenging task.\n\nIt is known that the material properties of amorphous solids, such as\ncolloidal or metallic glasses, depend on their history of production,\ne.g.~the cooling rate by which they were quenched from a fluid phase\n\\cite{glassbook}. This dependence on the history, i.e.~the age of the\namorphous solid, is an important issue in computer simulations of glasses,\nespecially because the accessible cooling rates in simulations are many\norders of magnitudes larger than those accessible in experiments of real\nsystems. With respect to the comparison between simulation and experiment,\nit is therefore crucial to systematically understand the dependence of\nstructural and dynamic properties on the age of the glassy solid.\n\nIt is thus expected that the response of a glass to an external mechanical\nloading is affected by the age of the glass.  If one shears an amorphous\nsolid under a constant strain rate, it is in general transformed\ninto a flowing fluid \\cite{rodneyrev2011,barratlemaitrerev}. While\nat sufficiently high strains, the flowing fluid reaches a steady state\nwithout any memory of the initial unsheared state, the transient response\nto the shear is affected by the history of the initial glass state. The\ncharacteristic stress-strain relation of a glassy system, in response\nto an externally applied shear rate, exhibits typically a maximum at a\nstrain of the order of 0.1 \\cite{zausch08}.  In numerical simulations\nof sheared thermal glasses, the amplitude of this maximum is observed\nto depend on the age of the material, typically growing logarithmically\nwith increasing age \\cite{varnik04,robbinsprl05}.\n\nMoreover, the transient response of glasses to an external shear\nfield is often associated with the occurrence of shear bands,\ni.e.~band-like structures with strain or mobility higher than other\nregions, observed both in experiments\n\\cite{schuhrev,mb08,divouxrev15,fs14,vp11,bp10,wilde11,divouxprl10} and\nnumerical simulations\n\\cite{vb03,ch13,ir14,gps15,sf06,bailey06,chboc12,ratul-procaccia-12}.  Such\nspatially localised structures are seen to emerge after the occurrence\nof the stress overshoot, as the stress relaxes to the steady state\nvalue.  Also, the formation of these transient shearbands have been\nobserved to be influenced by the thermal history of the glassy\nstate, with states which are obtained by faster cooling being less\nsusceptible to shearband formation. Further, it has been noted that\nsuch a spatially heterogeneous response is more likely to occur\nin the transient regime beyond the stress overshoot, at any given temperature.\n\nThe focus of our study are thermal glasses which are often characterized\nas simple yield stress fluids, e.g.~colloids, emulsions.  For such\nmaterials, the steady state flow curve (i.e.~stress vs.~imposed shear\nrate) is a monotonic function \\cite{bp10,nordstrom2010, BBDM15}.. Thus there are no persistent shear-bands,\nwhich would be the case for non-monotonic flow curves \\cite{fs14,coussot2010,mb12,ir14}. In the case of\nsimple yield stress fluids, the transient shearband that emerges are seen\nto broaden with time and eventually the entire material is fluidized, with\nthe timescale of fluidization depending on the imposed shear-rate or stress \n\\cite{divouxprl10,divoux12,ch13}. Such\nspatio-temporal fluctuations are also visible during steady flow, both\nin experiments and simulations \\cite{vb03,tsamados10,bp10}. \nHowever, in this work, our objective is\nto characterize the transient spatial heterogeneities, prior to onset of\nsteady flow. \n\nThe formation of transient shearbands has been addressed within\nthe scope of various theoretical models. Within the framework of\nspatially-resolved fluidity and soft-glass-rheology (SGR) models\n\\cite{fs14, moorcroft-cates-fielding-11, moorcroft-fielding-13},\nthe age-dependent spatially heterogeneous\nresponse has been obtained, with the occurrence of the stress-overshoot\nunder an applied shear-rate being associated with an instability\nleading to the formation of these transient heterogeneities. The\nmodel recovers the observation that more pronounced and long-lived\nshearbanding occurs for more aged glassy samples. Similarly, Manning et\nal.~\\cite{manningpre2007,manningpre2009} also observe various transient\nheterogeneous states, by analysing a shear-transformation-zone (STZ)\nmodel of glassy materials, which depend upon the initial state of the\nsystem (characterised by an initial effective temperature) and the\nimposed shear-rate. Further, they were able to map their results to\nthose obtained from numerical simulations \\cite{shiprl2007}. The same\nphenemenology has also been reproduced by other mesoscopic models\n\\cite{jaglajstat,damienroux2011}.\n\nIn this work, we address the question how the combination of ambient\ntemperature, applied shear-rate and age of the glass affects the\ntransient response, specifically the observation of spatio-temporal\nheterogeneities. This has not been systematically studied in earlier\nnumerical simulations. Consequently, we also compare our observations\nwith those from the theoretical models.\n\nTo this end, we perform molecular dynamics computer simulations of\nthe Kob-Andersen binary Lennard-Jones (KABLJ) model \\cite{ka94},\na well-studied glass former. Amorphous states are prepared by quenching\na supercooled liquid to different temperatures below the mode coupling\ntemperature, followed by a relaxation of the sample over a waiting time\n$t_{\\rm w}$.  Then, the resulting glass samples are sheared with different\nshear rates $\\dot{\\gamma}$.  The onset of plastic flow occurs around\nthe location of $\\sigma^{\\rm max}$, i.e.~at a strain $\\gamma^{\\rm max} =\n\\dot{\\gamma} t^\\star \\approx 0.1$ (with $t^\\star$ the time at which the\nmaximum is obtained), with the appearance of a peak in the stress-strain\nresponse. We demonstrate that at the different temperatures $T$, the\npeak height, $\\sigma_{\\rm max}$, for all ages and shear-rates, obeys the\nfunctional behavior $C(\\dot{\\gamma}, T) + A(T) {\\rm ln}(\\dot{\\gamma}t_{\\rm\nw})$ (with $C$ a function depending on $\\dot{\\gamma}$ and $T$ and $A$ a\ntemperature-dependent amplitude). Note that this finding is consistent\nwith earlier studies \\cite{varnik04,robbinsprl05}. Further, as we have\nshown recently \\cite{gps15}, transient (but long-lived) shear bands\nare formed for $\\gamma > \\gamma^{\\rm max}$, provided that shear rate\nis sufficiently low.  We quantify the contrast in spatial mobilities\nand demonstrate that the extent of spatial heterogeneities is not only\ndependent on the age of the glass, but also on the ambient temperature.\n\nThe rest of the paper is organized as follows. In Sec.~\\ref{sec2} we\ndescribe the KABLJ model and the details of the simulation.  Then, we\npresent the results for the stress-strain relations and the analysis in\nterms of mobility maps in Sec.~\\ref{sec3}. Finally, in Sec.~\\ref{sec4},\nwe summarize the results and draw some conclusions.\n\n\\section{Model and Methods}\n\\label{sec2}\nWe consider a binary mixture of Lennard-Jones (LJ) particles (say A and B)\nwith 80:20 ratio. This is a well-studied glass former. Particles interact\nvia LJ potential which is defined as:\n\\begin{eqnarray}\n\\label{LJ1}\n\\textrm{U}^{\\textrm{LJ}}_{\\alpha\\beta}(r) &=& \n\\phi_{\\alpha\\beta}(r)-\\phi_{\\alpha\\beta}(R_{c})-\\left(r-R_{c}\\right)\\left. \n\\frac{d\\phi_{\\alpha\\beta}}{dr}\\right|_{r=R_{c}},\\nonumber\\\\\n\\phi_{\\alpha\\beta}(r) &=& \n4\\epsilon_{\\alpha\\beta}\\left[\\left(\\sigma_{\\alpha\\beta}\/r\\right)^{12}-\n\\left(\\sigma_{\\alpha\\beta}\/r\\right)^{6}\\right]\\: r<R_{c},\n\\end{eqnarray}\nwhere $\\alpha, \\beta = \\textrm{A, B}$. The interaction parameters are\ngiven by $\\epsilon_{\\textrm{AA}} = 1.0$, $\\epsilon_{\\textrm{AB}}\n= 1.5\\epsilon_{\\textrm{AA}}$, $\\epsilon_{\\textrm{BB}} =\n0.5\\epsilon_{\\textrm{AA}}$, $\\sigma_{\\textrm{AA}} = 1.0$,\n$\\sigma_{\\textrm{AB}} = 0.8\\sigma_{\\textrm{AA}}$, $\\sigma_{\\textrm{BB}} =\n0.88\\sigma_{\\textrm{AA}}$, and $R_{c} = 2.5\\sigma_{\\textrm{AA}}$. Masses\nof both type of particles are equal, i.e., $m_{\\textrm{A}} =\nm_{\\textrm{B}} = m$. All quantities are expressed in LJ units in which\nthe unit of length is $\\sigma_{\\textrm{AA}}$, energy is expressed\nin the units of $\\epsilon_{\\textrm{AA}}$ and the unit of time is\n$\\sqrt{{m\\sigma_{\\textrm{AA}}^{2}}\/\\epsilon_{\\textrm{AA}}}$. More details\nabout the model and parameters can be found in Ref.~\\cite{ka94}.\n\nWe perform molecular dynamics (MD) simulation in the $NVT$ ensemble using\nthe package LAMMPS (``Large-scale Atomic\/Molecular Massively Parallel\nSimulator'') \\cite{plimpton95}.  The simulations are done for the box\ngeometry having the dimension $20\\times20\\times80$.  Temperature is kept\nconstant via a dissipative particle dynamics (DPD) thermostat \\cite{sk03}.\n\nOur method for the preparation of the glass samples is as follows: At a\ndensity $\\rho=1.2$, we first equilibrate the system at the temperature\n$T=0.45$, which is in the super-cooled regime. Then, we quench it to\nthe target temperatures $T = 0.2, 0.3, 0.4$, below the mode coupling\ntransition temperature \\cite{ka94}.  For exploring the effect of aging on\nthe mechanical response, we sample glassy states having different ages,\n$t_{\\rm w} = 10^{2}, 10^{3}, 3\\times10^{3}, 10^{4}, 3\\times10^{4}$\nand $10^{5}$, as the system evolves after the quench to each target temperature.\nUsing these initial states sampled at different $t_{\\rm w}$, we apply shear on\n$x$-$z$ plane in the direction of $x$ with different constant strain rates\n$\\dot{\\gamma} = 10^{-2}, 10^{-3}, 3\\times10^{-4}, 10^{-4}, 3\\times10^{-5}$\nand $10^{-5}$. To simulate a bulk glass under shear, we use Lees-Edwards\nperiodic boundary conditions \\cite{le72}.\n\n\\section{Results}\n\\label{sec3}\nIn order to characterize the mechanical response of the aged amorphous\nsolids, we measured different structural and dynamical properties, both\nat the macroscopic and local scales. We now discuss these observation\nin detail.\n\\subsection{Macroscopic response}\n\\subsubsection{Stress vs strain}\nWhen the externally applied shear is imposed on the aging quiescent glass,\nthe deformation response of the material is characterized by measuring\nthe stress, $\\sigma_{\\rm xz}$, generated in the system with increasing\nstrain ($\\dot{\\gamma}t$). In Fig.~\\ref{fig1}(a), we show how the stress\nevolves when we impose $\\dot{\\gamma}=10^{-4}$ on initial states sampled\nfrom different ages, at $T=0.2$ . We observe the characteristic stress\novershoot \\cite{varnik04}, with the peak height ($\\sigma^{\\rm max}$)\nincreasing for larger $t_{\\rm w}$. Across temperatures, within the glassy\nregime, this characteristic response does not change. However, as\nexpected the material becomes less rigid with increasing temperature.\nFor example,  for a fixed age, with increasing temperature, the steady\nstate value of $\\sigma_{\\rm xz}$ decreases, but still one observes the\nstress-overshoot, albeit with a lesser value of $\\sigma^{\\rm max}$;\nsee Fig.~\\ref{fig1}(b) for the corresponding data at $T=0.4,0.3,0.2$\nfor an imposed $\\dot{\\gamma}=10^{-4}$.\n\nIn Fig.~\\ref{fig1}(c), we illustrate the variation of $\\sigma^{\\rm max}$\nwith age, for a wide range of $\\dot{\\gamma}$ at $T=0.2$. Thereafter,\nwe demonstrate that at each temperature, the data for $\\sigma^{\\rm max}$\ncan be collapsed onto a master curve using the relation\n\\begin{equation}\n\\sigma^{\\rm max} = C(\\dot{\\gamma}, T) \n+ A(T) {\\rm ln}(\\dot{\\gamma}t_{\\rm w}),\n\\end{equation}\nwith $A(T)$ a temperature-dependent amplitude that is independent of\n$\\dot{\\gamma}$ and $t_{\\rm w}$ and $C(\\dot{\\gamma},T)$ a function that\nsolely depends on $\\dot{\\gamma}$ and $T$.\n\nSuch a logarithmic relationship is motivated by Ree-Eyring's viscosity\ntheory, modified appropriately to take into account the role of the\nsample's age, $t_{\\rm w}$, as noted by Varnik et al.~\\cite{varnik04}. A similar\ndependence was also proposed by Rottler et al.~\\cite{robbinsprl05}\nfor the same binary LJ mixture subjected to uniaxial strain at a\ndifferent state-point. Thus our scaling results are consistent with\nearlier observations. We note that such a logarithmic dependence\nis not observed in experiments involving other yield stress fluids like\ncolloidal pastes \\cite{derec2003} or carbopol \\cite{divouxsoftmatter2011}\nwhere a power-law increase is observed, which is also captured in the\nnumerical simulations of model gels \\cite{parksoft13,whittle97}. Thus,\nit seems that the response of dense thermal glasses are different from\nthe low density materials with more complex structures.\n\nHaving observed the scale of the stress overshoot for various ages,\nimposed shear-rates and ambient temperatures, we will later explore\nwhether such observations necessarily lead to the occurrence of transient\nshearbands.\n\\subsubsection{Potential Energy}\nTo have a measure of the structural changes in the system, both during\nthe aging process and the onset of flow, we monitor the potential energy\n($E_{\\rm {Pot}}$) of the system.  In Fig.~\\ref{fig2}(a), we show how the potential\nenergy decreases when the system is quenched from the supercooled state\nat $T=0.45$ to the lower temperatures $T=0.2, 0.3, 0.4$. After the\nfast decrease during the initial aging regime, the potential energy\ndecreases slowly, logarithmically depending on the age $t_{\\rm w}$ of the\nsystem, as is typically observed during aging.\n\nWhen the external shear is imposed on the system, the potential energy\nincreases, with the steady state value depending upon the magnitude\nof the imposed shear-rate; see Fig.~\\ref{fig2}(b),(c) for $T=0.2,\n0.4$, respectively. Further, we check whether the potential energy of\nthe system measured during the transient regime, prior to the onset of\nsteady flow, is dependent on the initial age of the quiescent glass. As\ncan be seen in Fig.~\\ref{fig2}(b), (c), for both the temperatures, the\ndependence on $t_{\\rm w}$ becomes more visible with decreasing shear-rate. This\nthus indicates that the system evolves through different intermediate\nstructures, dependent on the initial state having different $t_{\\rm w}$,\nbefore steady states structures start getting explored. We now explore\nhow these age-dependent transient structures are dynamically different. We\nalso note that for the applied shear-rates, the approach to steady state\nseems faster for $T=0.4$ than compared to $T=0.2$. While in the former\ncase, at long strain, the steady state potential energy, under applied\nshear, matches at large strains, this is not the case for $T=0.2$. This\ncombination of temperature, shear-rate and age of the sample will be\nfurther explored in later sections.\n\\subsection{Single particle dynamics}\n\n\\subsubsection{Average MSD}\nIn order to characterise the microscopic dynamical response of the glass,\nwhen yielding under shear, we monitor the single-particle dynamics.\nThis is quantified by measuring the non-affine mean-squared\ndisplacement (MSD), $\\Delta r_{z}^{2}$, in the direction transverse to\nthe applied shear; the data is shown for $\\dot{\\gamma}=10^{-4}$, at\n$T=0.2$ (Fig.~\\ref{fig3}(a)) and $T=0.4$ (in Fig.~\\ref{fig3}(c)). At\nboth temperatures, during early times, particles exhibit ballistic\nmotion, which is then followed by a caging regime.  Subsequently, the\nonset of flow is marked by a super-linear regime, and eventually diffusive\ndynamics is observed. The onset of the super-linear or ``super-diffusive''\nregime occurs around the stress overshoot in the stress-strain curve,\nwhen the stress, building up in the system, is released via the breaking of\nlocal cages leading to the subsequent diffusive motion of the particles\n\\cite{zausch08, koumakisprl12}. We check how the initial age of the glassy state\ninfluences the measured $\\Delta r_{z}^{2}$; the corresponding data is\nplotted in Fig.~\\ref{fig3}(a), (c) for the two temperatures.  At both\ntemperatures, both in the initial response and in the steady state,\nthere is no visible difference in the measured MSD, for the samples\nhaving different ages. Only during the transient regime, where the\nonset of super-diffusive dynamics occurs, we observe variation with\nage of the quenched glass, with the effect more visible for $T=0.4$;\nsee Fig.~\\ref{fig3}(c).  We also note that the onset timescale depends\nupon age, shifting to longer timescales with increasing $t_{\\rm w}$, as is\nclearly visible.\n\\subsubsection{Spatially resolved MSD}\nTo further probe the spatial features of the local dynamics during\nyielding, we divide the simulation box into eight layers of thickness $10\\sigma_{\\rm AA}$ along the\n$z$-direction. For each of these layers, we compute the averaged MSD\nfor the particles populating it at $t=0$, i.e.~before shear is imposed.\nIn Fig.~\\ref{fig3}(b), (d), we show the corresponding data, at $T=0.2,\n0.4$, respectively, for an imposed $\\dot{\\gamma}=10^{-4}$ and age\nof $t_{\\rm w}=10^5$. We see that there is variation in the dynamics across\nthe different layers, the variation being greater at $T=0.2$ than at\n$T=0.4$. While for $T=0.4$, eventually, the long time dynamics seems to\nconverge to the diffusive limit, within the timescale of observation,\nthat is not the case for $T=0.2$. In the latter case, while in one case,\nwe see the onset of diffusive dynamics at long times, in some of the\nother slices, the dynamics continues to be sub-diffusive. To summarise,\nthis demonstrates that, during yielding of the glass, at any temperature,\ndynamics is heterogeneous. However, the extent of heterogeneity and\npersistence increases with decreasing temperature.  We will quantify\nthat in more detail in the subsequent sections.\n\\subsubsection{MSD maps}\nTo visualise the dynamical heterogeneities, we construct three-dimensional\nmaps of local MSD \\cite{ch13}. These maps are constructed in the\nfollowing manner.  We divide the simulation box into small cubic\nsub-boxes having linear size of $\\sigma_{\\textrm{AA}}$.  At any time\n$t$, we calculate the average MSD of  the particles located in each\nsub-box at $t=0$ (unsheared glassy state), which provides a cumulative\npicture of how yielding proceeds locally starting from the quiescent\nglass. As discussed earlier, we measure the $z$-component of MSD of each\nparticle. The evolution of the three-dimensional maps with increasing\nstrain, is shown in Fig.~\\ref{fig4} and Fig.~\\ref{fig4b}, for different\nages of the glassy state at $T=0.2$ and $0.4$, respectively, corresponding\nto an imposed shear rate of $\\dot{\\gamma}=10^{-4}$.\n\nFirst, we focus on the situation at the lower temperature ($T=0.2$);\nsee Fig.~\\ref{fig4}. At early strain ($\\dot{\\gamma}=0.1$), we observe\nlocalised ``hot spots'' of large mobilities \\cite{clement12,sentja15}.\nAt a strain of $0.5$, more such local regions of faster dynamics have\nemerged. In the case of well aged initial state (e.g.~$t_{\\rm w}=10^5$), we see\nthat the hot spots are organised in the form of a shear band. On the other\nhand, for a younger initial state, i.e.~$t_{\\rm w}=10^2$, these hot spots are\nmuch more dispersed. At a later strain ($\\dot{\\gamma}=1$), this difference\nis further amplified.  For the younger sample, large regions of the system\nhave been fluidized, whereas for the older system, enhanced mobility\nis still localised in the form of a shear band. Thus, the age of the\ninitial glassy state, does influence the spatio-temporal organisation of\nregions of increased mobility.  This is consistent with earlier findings\nin numerical simulations using different model systems \\cite{sf06}\nor predictions from theoretical models \\cite{moorcroft-cates-fielding-11}.\n\nWe now contrast this with the situation at the higher temperature\n($T=0.4$); see Fig.~\\ref{fig4b}. At early strain ($\\dot{\\gamma}=0.1$),\nwe observe the hot spots, for all $t_{\\rm w}$; however, there are more in number\nas compared to the case of $T=0.2$, for any $t_{\\rm w}$. With increased strain\n$\\dot{\\gamma}=0.5$, we see that the hot spots have proliferated and cover\nalmost the entire domain, including for the most aged sample. Thus,\naccess to increased thermal fluctuations at higher temperatures lead\nto the tendency for a more homogenised and faster fluidisation of the\nglassy state under applied shear.\n\nWe will now carry out a more quantitative characterisation of this\nspatio-temporal response of the glassy states, sampled from different\ntemperatures, and aged over different timescales.\n\\subsubsection{Spatial profile of local mobility and fluctuations}\nSo far, by spatially resolving the single particle MSDs, we demonstrated\nthe existence of heterogeneous dynamics during yielding of the glass\nand how that depends on the temperature of the glassy state.\n\nUsing single particle MSDs, we construct the instantaneous spatial\nprofile $\\Delta r^{2}_{z}\\left(z\\right)\/\\langle{\\Delta r^{2}_{z}}\\rangle$, which gives us\na measure of fluctuations in local dynamics. The evolution of these\nprofiles, with increasing strain, is shown in Fig.~\\ref{fig6}, for two\ndifferent ages of the glass, viz.~$t_{\\rm w}=10^2, 10^5$, under an imposed\nshear-rate of $\\dot{\\gamma}=10^{-4}$, at $T=0.2$.  We observe that the\nspatial fluctuations are much larger and more persistent in the older\nsample, consistent with the qualitative discussion, above, involving\nMSD maps.\n\nUsing these spatial profiles, we can now quantify the\ncontrast in local mobilities in the deforming glass, under\napplied shear, and compare the response across temperatures.\nIn order to do that, we compute the fluctuations in local MSD,\n$\\chi=(\\Delta r^{2}_{z}\\left(z\\right)-\\langle{\\Delta r^{2}_{z}}\\rangle)^2\/\\langle{\\Delta r^{2}_{z}}\\rangle^2$\nwith increasing strain $\\dot{\\gamma}t$.  Such a quantity measures the\nextent of fluctuations in average mobilities, as characterised by the\nlocal MSD, across the regions parallel to the direction of flow. Defined\nin this manner, $\\chi$ would therefore easily capture the contrast in the\ndynamics during the formation of shear-bands. To note, such a quantity\nis different from the dynamical susceptibility, $\\chi_4$, which is a\nmeasure of fluctuations in single particle dynamics, measured within\nthe steady-state ensemble. Here, we are concerned with the fluctuations\nacross regions during the onset of flow from a quiescent state.\n\nThe corresponding data, for temperatures $T=0.2, 0.3, 0.4$, across ages\n$t_{\\rm w}=10^2, 10^3, 10^4, 10^5$, for a range of imposed shear-rates,\nviz.~$\\dot{\\gamma}=10^{-2}, 10^{-3}, {10^{-4}}$, are shown in\nFig.~\\ref{fig7}. Typically, $\\chi$ behaves non-monotonically with\nincreasing strain, with the maximal spatial fluctuations ($\\chi^{\\rm\nmax}$) occuring at a strain which corresponds to the super-linear regime\nin the MSD (see Fig.~\\ref{fig3}); i.e., once yielding of the glass\nhas occurred.  Further, for a fixed $\\dot{\\gamma}$, the magnitude of\n$\\chi^{\\rm max}$ increases with age, i.e.~the contrast in mobilities\nacross a sample becomes largest for a more aged sample. Importantly,\nwe also note that $\\chi^{\\rm max}$ also starts increasing when we\nshear the glasses with smaller and smaller shear-rates; e.g.~see\nFig.~\\ref{fig7}(a)-(c), for the variation of $\\chi^{\\rm max}$ with\nage and imposed shear-rate. With decreasing temperature, this spatial\nfluctuation increases even further.\n\nTo get a more complete picture of the variation of these spatial\nfluctuations, across age and temperature, we plot the corresponding\ncontour maps for $\\chi(t_{\\rm w},\\dot{\\gamma})$; these are shown in\nFig.~\\ref{fig8}. From the contour maps, one can infer how strong and\npersistent the spatial fluctuations are with varying temperature, age\nand imposed shear-rate.  It is evident that for large shear-rates\n($\\dot{\\gamma}=10^{-2}$), these fluctuations are negligible at\nany age of the glassy state sampled at any of the temperatures; see\nFig.~\\ref{fig8}(a),(d),(h). On the other hand, at smaller shear-rates\n($\\dot{\\gamma}=10^{-4}$), large fluctuations are visible at all\ntemperatures. However, the age of the sample then becomes a factor in\ndetermining the degree of fluctuations and persistence with increasing\nstrain.  At a lower temperature, $T=0.2$, these are prominent over the\nrange of ages we investigated and also persistent over large strains;\nsee Fig.~\\ref{fig8}(i). On the other hand, at higher temperatures, $T=0.4$\n(Fig.~\\ref{fig8}(c)), that scale of fluctuations is only slightly visible\nat very large ages over a small strain window.  The range of ages and\nstrain-interval over which similar fluctuations are seen broadens out at\nan intermediate temperature of $T=0.3$ (Fig.~\\ref{fig8}(f)) and, even more\nextensively, at $T=0.2$. This trend is also evident at an intermediate\nshear-rate of $\\dot{\\gamma}=10^{-3}$, with the scale of fluctuations\nbeing less but visible (Fig.~\\ref{fig8}(h)); howevever, it is important\nto note that the fluctuations are nearly negligible at the higher\ntemperature $T=0.4$ at this shear-rate (Fig.~\\ref{fig8}(b)).  Thus, we\ndemonstrate, that the scale of transient dynamical heterogeneities depend\non a combination of temperature of the glass, its age and the imposed\nshear-rate. Since the quantity we compute, $\\chi(t_{\\rm w},\\dot{\\gamma})$,\nmeasures the contrast in mobilities in layers parallel to the flow\ndirection, the conclusions drawn from the contour maps allow us to infer\nthat the propensity to shear-band during yielding is determined on how\nrigid the glass is and how small\/large the applied shear-rate is.\n\\subsubsection{Time evolution of shear-bands}\nFinally, we study how a transient shear-band, once formed, spatially\nevolves with time. Such analysis can only be done for the case where a\nwell-defined shear-band can be identified. Thus, we focus at the case of\n$T=0.2$ and small shear-rate ($\\dot{\\gamma}=10^{-4}$), where we observe\ntransient shear-bands for a range of $t_{\\rm w}$.\n\nIn order to further quantify and characterise the spatio-temporal\nevolution of the shear-band, we define a region to be mobile or not,\nby setting a threshold $\\mu_{th}=0.02$ on the local $\\Delta r^{2}_{z}$. As\ncan be seen in Fig.~\\ref{fig3}(a), such a choice of $\\mu_{th}$ is larger\nthan the plateau value in the MSD and thus corresponds to motions beyond\ncage-breaking. We then define the local mobility $\\psi$ as\n\\begin{eqnarray}\n\\label{orderparam}\n\\psi = \\begin{cases} 1 &\\mbox{if } \\mu \\ge \\mu_{th}, \\\\\n                     0 & \\mbox{otherwise},\\end{cases}\n\\end{eqnarray}\nwhere $\\mu$ is the average MSD of particles in a sub-box. Following\nthis convention, we digitize the whole system into mobile and immobile\nregions. An example of the mobility map, thus constructed, is shown\nin Fig.~\\ref{fig9}(a).\n\nWe quantify the extent of fluidisation in the system by measuring the\nfraction of mobile cells, $p$, at any given instant. The evolution of $p$\nas a function of strain, for an imposed shear of $\\dot{\\gamma}=10^{-4}$,\nis shown in Fig.~\\ref{fig9}(b). We observe that independent of age\n($t_{\\rm w}$), the sudden initial increase of $p$ occurs nearly at the same\nstrain. However, the subsequent increase in $p$ does depend upon the age\nand thus, the fraction of system which is mobilised at $\\dot{\\gamma}t=0.2$\nis substantially larger for $t_{\\rm w}=10^2$ as compared to $t_{\\rm w}=10^5$.\n\nFurther, we try to identify the spatial organisation of these mobile\ncells. By locating contiguous layers of mobile cells, we identify the\nformation of the shear band and, thereafter, by marking the interfaces\nof this band, we measure how the band-width, $\\xi_{\\textrm{b}}$, evolves\nwith time. For different ages of the initial glassy state, this time\nevolution is shown in Fig.~\\ref{fig9}(c), for a fixed\n$\\dot{\\gamma}=10^{-4}$. We see that $\\xi_{\\textrm{b}}$ initially grows\nquickly and then eventually it reaches a regime where the data can be\nfitted with $\\xi_{\\textrm{b}} \\sim t^{1\/2}$, implying that the propagating\ninterface of the shear band has a diffusive motion. This diffusive regime\nis not extensively dependent on $t_{\\rm w}$, although samples having largest\n$t_{\\rm w}$ do seem to display a slower motion of the spreading interface. \n\n\\section{Summary and conclusions}\n\\label{sec4}\n\nTo summarize, we have used numerical simulations of a model glass\nformer, to probe the mechanical response of amorphous solids, having\ndifferent aging histories, by imposing a wide range of external\nshear rates. The response is studied at different temperatures below\nthe mode coupling temperature, in order to ascertain how thermal\nfluctuations affect the spatio-temporal response. We underline,\nhere, that the numerical simulations are done by integrating the\nNewton's equations of motions using Lees-Edwards periodic boundary\nconditions and the local DPD thermostat. This, thus, allows for\nunhindered spatio-temporal fluctuations as the glassy state responds\nto the applied shear, without introducing any biases or suppressing\nany fluctuations, which would be the case if one were using walls\n\\cite{vb03} or integrating the SLLOD equations along with some\nglobal thermostat (like Nose-Hoover) \\cite{shiprl2007}. Therefore,\nqualitatively, the nature of dynamical heterogeneities observed\ncould be different from the earlier works.\n\nThe macroscopic response of the glass is characterized by the\noccurrence of a overshoot in the stress-strain curves, observed for\nthe range of shear-rates and ages that we have studied, at the\ndifferent temperatures. Consistent with earlier works, we find that,\nat each temperature, the peak height can be scaled on to a master-curve\nas a function of $\\dot{\\gamma}t_{\\rm w}$. The corresponding potential\nenergies of the system, measured during the deformation process,\nshow a dependence on age with decreasing shear rate. This implies\nthat the structures visited during the transient regime, prior to\nthe onset of steady flow, start becoming different, depending upon\nthe initial history. We also observe that for lower temperatures,\nit takes longer for potential energies to reach the steady state\nvalue at smaller shear-rates, reflecting the role of thermal\nfluctuations in affecting the duration of the transient regime for\nthese parameters.\n\nThis is further revealed, by measuring dynamical quantities at the\nmicroscopic scale. Our probe of choice is the time evolution of non-affine\nmean squared displacement (MSD) of the particles relative to the initial\nquiescent state.  Spatial profiles of the MSD show that for a fixed\nimposed shear and age, the transient dynamics is more heterogeneous\nand more long-lived at lower temperatures. This is illustrated by\nconstucting sptatially resolved three-dimensional maps which provide\na handle to visualize the extent of dynamical heterogeneity as the\nglassy states respond to the imposed shear. Such maps demonstrate that\nthe heterogeneities are spatially localised in the form of shearbands,\nfor old enough systems at low temperatures.\n\nWe quantify the extent of dynamical heterogeneity organised in the\nform of shear-bands, by measuring the degree of fluctuations, $\\chi$,\nin local mobilities across regions oriented parallel to the direction\nof imposed flow. Thus, large values in $\\chi$ reflect more propensity\nof the system to exhibit shear-banding. We do such measurements by\nscanning across entire range of shear-rates and thermal histories\nexplored by us. We observe that the fluctuations are maximum around\nthe yielding strain and the peak in fluctuations increase with\ndecreasing shear-rate as well as larger ages. Further by constructing\ncontour maps of $\\chi (t_{\\rm w}, \\dot{\\gamma}t)$ for varying\nshear-rates at different temperatures, we demonstrate that the\nemergence of significant shear-banding depends upon the combination\nof temperature, age and imposed shear-rate. The contrast in local\nmobilites is most prominent at low shear-rates and low temperatures,\nwhere the age of the system does not seem to matter, over the range\nexplored by us. With increasing temperature, the age of the sample\nstarts to determine whether transient shear-banding will be visible\nor not. For even higher temperatures, one has to go to lower\nshear-rates and even older systems in order for such largely localized\nheteroegeneities to be visible. Also, in the cases where shear-bands\ncan be identified, we observe that the spreading of the shear-band\nacross the system seems to depend on age, albeit weakly.\n\nThus, our study shows that the transient shearbanding, with sharply\ncontrasting mobilities across regions, only occur for specific\ncombinations of temperature, shear-rate and age, as discussed above.\nNote that, for most of the glassy samples that we study, the\nmacroscopic response under an imposed strain-rate is characterized\nby the stress overshoot. Yet, the transient shear-bands are only\nvisible in only a subset of the cases we study.  Thus, the occurrence\nof a stress overshoot does not necessarily lead to the transient\nshearbands, quantified via the mobilities relative to the initial\nquiescent state, as predicted by the spatially resolved fluidity\nand SGR models \\cite{fs14, moorcroft-cates-fielding-11, moorcroft-fielding-13}. \nOn the other hand, as predicted by these and other\nmodels \\cite{manningpre2007,manningpre2009, jaglajstat,damienroux2011}, \nwe do observe that the propensity to shearband is more in\nthe case of increasingly aged samples.  However, the choice of\ndisorder distributions in these models (e.g. distribution of yielding\nthresholds), with changing age, remains arbitrary, and it would be \nuseful to obtain these as inputs from atomistic simulations.\n\nThe transient shear-banding observed in our numerical simulations\nas well as other, bears resemblance to dynamical heterogeneities\nobserved in thermal glasses \\cite{ludormp11}, except that in the\ncase of a sheared glass, strong anisotropies come into play leading\nto the spatial localisation. In the case of supercooled liquids,\nthe scale of dynamical heterogeneities increases as the temperature\nis decreased towards the putative glass transition temperature.\nSimilarly, in the case of applied shear, the heterogeneous dynamics\nis enhanced as one approaches vanishingly small shear-rates, with\nthe yield stress being identified as a critical threshold\n\\cite{bocquet-prl-2009, divoux12, chenprl2016}. Also, it is possible\nthat the regime of imposed shear-rates over which the critical\nthreshold influences flow \\cite{ludo-jpcm-03, chboc12} could change\nwith temperature, with decreasing thermal fluctuations extending\nthis regime over longer scales. This would rationalise why with\nincreasing temperatures, the contrast in heterogeneities progressively\ndecreases, for a fixed shear-rate. Over and above, the aging effects\nare more prominent at higher temperatures, with the glass remembering\nthe history from where it was quenched over longer aging timescales\n\\cite{ludo-jpcm-03, ludormp11}.  At low temperature, thus, structural\narrest and therefore rigidity emerges at shorter timescale.  Hence,\nthe mechanical response of the glass, having the same age and same\nimposed shear-rate, would be different across temperatures.  This\ninterplay of different timescales therefore brings about the complex\nresponse as a function of all these control parameters.  Eventually,\nit would be useful to quantify the exact ranges in these control\nparameters, where we expect transient but prominent shearbanding\nto be visible.\n\nThe transient dynamical heterogeneities that we track and analyse\nare similar to measurements done via confocal microscopy in colloidal\nglasses \\cite{vp11} or scattering measurements in granular systems\n\\cite{clement12}.  This is different from using velocity profiles\nas a diagnostic tool, which capture plasticity over short timescales.\nFurther, in our case, because of periodic boundary conditions, the\nshear bands can emerge anywhere in the system, unlike many cases\nwhere initial mobile regions happen to occur near confining surfaces\n\\cite{divouxsoftmatter2011, vp11}.  However, similar to experiments,\nwe do observe the propagation of the mobile front to eventually\nfluidize the system.\n{\\bf Acknowledgement:}\\\\\nWe achnowledge financial support by the Deutsche Forschungsgemeinschaft\n(DFG) in the framework of the priority programme SPP 1594 (Grant No.~HO\n2231\/8-2). PC thanks FOR 1394 for financial support during visit to\nUniversity of D\\\"usseldorf.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nAI-based systems are increasingly being used \nto make or inform decisions that can greatly impact an individual's life.\nAlthough these systems can be beneficial, \nnumerous groups have demonstrated significant risks of these systems~\\cite{propublica,Selbst2017,amazon-recruiting-2018,UK-exam-2020,zillow-2021}.\nDue to these concerns, there has been strong demand for increased AI transparency from researchers, civil society advocates, and\ngovernments.  \nOne way to achieve this transparency is to have a standardized\ndocument, similar in spirit to a nutritional label, that would \ndescribe the relevant aspects of the AI system.\nSeveral research~\\cite{datasheets-cacm-2020,dataset-nut-label-2gen-2020,factsheets-2019,model-cards,aboutml-2020,co-designing-checklists-2020}, industry~\\cite{tm-forum-model-datasheet-2020},\nand government~\\cite{UK-transparency-proposal-2021,EuropeanCommission2020,canada-protocol-2019}\nefforts have proposed documentation templates containing questions to be answered by model or dataset developers.\nAlthough these templates provide a useful starting point, \nno single template can cover the needs of diverse documentation consumers.  \nThese include people not involved with the development of the AI model: e.g., end users, affected subjects, and regulators, as well as those\nwho have developed and deployed the model: e.g., data scientists, model validators, and model operations engineers. \n\nWhile a universal ``AI nutritional label'' may be appropriate for more general transparency goals, it will leave gaps due to the specifics of the use case, model implementation, problem domain, and regulatory context. To address these gaps, we can strive for a consistent, repeatable, flexible \\emph{process} for creating consumer-appropriate transparency documentation. \nRichards et al. ~\\cite{factsheet-methodology-IEEE} proposed\nsuch an approach, following the principles of user-centered design, \nto identify the needs of both producers and consumers of AI documentation in the form of a methodology to create AI FactSheets~\\cite{factsheets-2019}.  Although promising, it has yet to be evaluated with real AI systems.\n\nThe FactSheets methodology is a documentation process centered on three roles: content producers, content consumers, and a FactSheet team (FS team). The \\emph{FS team} follows the methodology and shepherds the creation of a FactSheet. During this process, they collaborate with both \\emph{producers}, who have the information to be captured, and \\emph{consumers}, who have specific documentation needs. The FS team determines consumers' needs and records them as fields to be completed in a FactSheet template. Then, the FS team iterates with producers and consumers to fill out and refine the template, producing a FactSheet.\n\nThe goal of this paper is to evaluate this methodology via a case study of an independent team that built FactSheets for multiple real AI models and systems over three months. We interviewed 16 participants to address the following research questions:\n\\begin{itemize}\n    \\item[RQ1] Is the methodology usable by FS team members not trained in human-centered practices?\n    \\item[RQ2] How well did the resulting FactSheets address the needs of different consumers?\n    \\item[RQ3] What did consumers and producers of FactSheets see as the benefits and costs?\n\\end{itemize}\n\n\\section{Background} \\label{sec-background}\n\nAI transparency requires the availability of model and system documentation that is understandable and trustworthy. Documenting any software artifact \\textit{well} requires effort. AI models, and the systems in which they are embedded, pose additional documentation challenges \\cite{experiences-2020}. First, AI development is highly collaborative. People with diverse skills, specialized vocabularies, and unique tools all contribute to the final deployed model or system. Second, AI development is highly iterative. Model development often begins with a series of lightweight experiments. Multiple threads are followed with frequent false starts and backtracking. Along the way, important decisions about training data and algorithm refinement are rarely captured. Finally, tooling to support traditional software development is not easily adapted to AI development and documentation. Proposals for new mechanisms for collecting important facts about AI development throughout the lifecycle~\\cite{experiences-2020}, may lead to improved tooling in the future. Currently, only the most disciplined teams can reliably capture information needed for complete and accurate documentation.\n\n\\subsection{FactSheets and FactSheets Methodology}\nTo tackle the complexity of creating AI documentation that affords effective transparency, Richards et al.~\\cite{factsheet-methodology-IEEE} have proposed a human-centered methodology for creating the form of documentation called AI FactSheets. FactSheets \\cite{factsheets-2019, experiences-2020, fs360} are collections of important facts about the development, testing, and deployment of AI models and systems. These include facts about model purpose, training data selection and cleaning, algorithm selection and tuning, testing for accuracy, bias, privacy risks, adversarial attacks, etc. Different \\textit{consumers} of FactSheets (data scientists, business owners, system integrators, deployment engineers, risk officers, regulators, end users, affected subjects, etc.) will have different skills and different information needs. By engaging with these consumers, it is possible to discover both common and unique needs and, thereby, create FactSheets that meet those needs. Those needs may include how to render the documentation, providing for example, a condensed table view for a quick overview or slides to aid presentation of the model~\\cite{fs360}. By engaging with fact \\textit{producers}, it is also possible to discover what facts they can create for consumption by others and how to help them create these facts efficiently. Earlier work~\\cite{factsheet-methodology-IEEE} has posited that executing the steps of this methodology will lead to useful FactSheets, contributing to model transparency. \n\nThe methodology consists of seven steps. The following list presents a somewhat idealized view insofar as there may be iterations within and between steps. In other cases, some of the steps may be collapsed due to informants playing both fact consumer and fact producer roles within the lifecycle~\\cite{experiences-2020}. Further details on these steps, including example  questions to explore with informants, can be found in \\cite{factsheet-methodology-IEEE}. \n\\begin{enumerate}\n    \\item \\textbf{Know Your FactSheet Consumers}. FactSheets (and by extension, any form of AI documentation) are produced so that they can be consumed. Understanding the information needs of FactSheet consumers is the first and most important task. This initial exploration need not be formal. Working with even one representative informant from each major process in the AI lifecycle will provide useful insights into the overall set of consumer needs.  \n    \\item \\textbf{Know Your FactSheet Producers}. Some facts can be automatically generated by tooling. Some facts can only be produced by a knowledgeable human. Both kinds of facts will be considered during this step. Again, working with even one representative producer of relevant facts from each major process in the AI lifecycle will provide the information needed to proceed to the next step.\n    \\item \\textbf{Create a FactSheet Template}. What is learned in the first two steps leads directly to the most important part of creating FactSheets, namely the creation of a FactSheet template. A FactSheet template will contain what can be thought of as questions or, alternatively, the fields in a form. Each individual FactSheet will contain the answers to these questions. For example a template may start with the question ``What is this model for?''. It may then expand on that question by asking where the model is well-suited and where the model is ill-suited.\n    \\item \\textbf{Fill In FactSheet Template}. This step is where the creator or creators of the FactSheet template attempt to fill it in for the first time. As this is done, it is important to informally assess the quality of the template itself by reflecting on what was learned about both consumers and producers, their skills, and information needs in the first two steps. While this assessment is not a substitute for further work with them (to follow), it may quickly highlight where improvements are needed.\n    \\item \\textbf{Have Actual Producers Create a FactSheet}. In this step, actual fact producers fill in the template for their part of the lifecycle. For example, if there is a question in the template about model purpose, find someone who would actually be providing that information and have them answer the question. Ask a data scientist to answer the questions related to the development and testing of the model. If this model was validated, the model validator would enter information about that process. Similarly, a person responsible for model deployment would answer questions related to deployment and ongoing monitoring. If the lifecycle is not that structured, the person responsible for most of the work could fill in this template.\n    \\item \\textbf{Evaluate Actual FactSheet with Consumers}. In this step an assessment is conducted of the quality and completeness of the actual FactSheet produced in the previous step. If the FactSheet is intended to be used by multiple roles (not uncommon), it should be evaluated separately for each role. To ground this assessment properly, each consumer should be asked to reflect and comment on how this FactSheet would actually help them perform their work or provide information to address their concerns.\n    \\item \\textbf{Devise Other Templates for Other Audiences and Purposes}. This step returns to the beginning and is the only \\textit{defined} iteration in the methodology. There may be other consumers that need to be supported. Or, if the work so far has focused on the process of creating and deploying AI models, it would be worthwhile to consider consumers beyond this such as internal or external review boards or regulators, sales personnel, and end users or others affected by the product or service.\n\\end{enumerate}\n\n\\section{Related Work} \\label{sec:related}\nThe drive to improve AI transparency via documentation has been motivated by the numerous high-profile examples of AI risk in society.  This has sparked research\ngroups~\\cite{datasheets-cacm-2020,dataset-nut-label-2gen-2020,factsheets-2019,model-cards,experiences-2020,aboutml-2020,co-designing-checklists-2020}\nand government agencies~\\cite{UK-transparency-proposal-2021,EuropeanCommission2020,canada-protocol-2019}\nto explore what form this documentation should take.\nThe main focus of these works has been on \\textit{what} should be included in this AI documentation for near universal use. \nSeveral of these efforts have included feedback loops, where society and industries have been able to comment on the proposals.\nEssential stakeholders for AI transparency are citizens who want to know more about the algorithmic decisions that are affecting them.  \nThis is challenging because the average citizen is not familiar with the technical details of AI.\nIn recent work,\nDomagala and Spiro~\\cite{UK-transparency-study-2021}\nengaged with citizens to better understand their needs regarding information about algorithmic systems deployed by the UK government. This was one way to approach Step 1 of the methodology described in Section~\\ref{sec-background}.  \nEarlier, Hind et al.~\\cite{experiences-2020} explored the needs of documentation producers \nvia interviews and a documentation creation exercise lasting a few hours. This paper differs from these works since it examines AI documentation practice in situ through a months-long case study involving both producers and consumers.\n\nSoftware documentation, in general, describes important characteristics of a system, application, or module. Despite its necessity, developers and software engineers find the task of \\textit{documenting} uninteresting and its creation often falls to technical writers who do not know all the details of the software they are documenting \\cite{parnas2011precise}. \nNot surprisingly, consumers of documentation generally dislike what is produced because such documentation is often incomplete, difficult to understand, or out of date \\cite{chomal2015software}. \nBut documentation is essential as it remains one of the few channels of communication between the developers of  software and its various users \\cite{kipyegen2013importance}. \n\nFor conventional software, researchers have developed sets of rules \\cite{parnas2011precise}, evaluation critieria \\cite{piorkowski2021ai}, and assessment methodologies \\cite{smart2002assessing,plosch2014value,ding2014knowledge}. Quality attributes such as accuracy, concreteness, writing style, and understandability have been offered as useful dimensions of quality \\cite{plosch2014value}. Other dimensions such as completeness, unambiguity, conciseness, and ease of access have been proposed \\cite{parnas2011precise}. Yet other dimensions include consistency, traceability, reusability, format, trustworthiness, and retrievability \\cite{ding2014knowledge,zhi2015cost}. This  paper draws on many of these dimensions to assess the quality of AI FactSheets.\n\n\\section{Evaluation Methodology} \\label{sec-eval-methodology}\nTo answer our research questions, we partnered with an AI organization in the healthcare domain that was piloting the FactSheets methodology~\\cite{factsheet-methodology-IEEE} for several models as a possible solution to their AI documentation and transparency needs. Data for the evaluation reported here consisted of two sets of interviews for each model: one for the FS team who followed the methodology to create the FactSheets, and one for FactSheet consumers to assess how the resulting FactSheets were meeting their needs.\nTogether, these two perspectives provide a holistic view of the usefulness of the methodology. \n\n\\subsection{Study Participants and Case Study Context}\nWe divided participants into two roles: \\textit{consumers} and \\textit{FactSheet (FS) team members}\\footnote{The FS team members were generally also the knowledgeable producers of facts in this case study. We ignore the distinction between FS team members and producers in what follows.}. As in the FactSheets methodology, \n\\textit{consumers} are the intended end users of the created FactSheet. FactSheet team members are the ones driving the methodology for each of the models, defining the FactSheet template, and iteratively making the FactSheet. \nIn total, we ran 17 interviews over 16 participants: seven FS team interviews and ten consumer interviews. One participant was interviewed twice. Participant details and which models each participant discussed are provided in Table \\ref{tbl:participants}. Throughout this paper, we identify participants using a three-part code. The first character indicates whether a participant is a FS team member (T) or consumer (C). The second part is a participant identification number. The last character notes the model that the participant produced or consumed. \nFor example, `C2-B' indicates the second consumer participant for model B. We interviewed T4 about two models: model B and D. We distinguish between the two using the identifier `T4-B' when the participant talks about model B and `T4-C' when talking about model C.\n\n\\begin{table}\n\\caption{The FS team (T\\#) and consumer (C\\#) participants and their roles for each model. The first character in each column header indicates the model under consideration, referred to as model A, model B, model C, and model D.}\n\\label{tbl:participants}\n\n\\hspace{-4em}\n\\begin{tabular}{c p{1.2in} c}\n\\cline{1-2} \n\\textbf{A} & \\textbf{Role}        \\\\ \n\\cmidrule{1-2} \\morecmidrules \\cmidrule{1-2} \nT1         & \\multicolumn{2}{l}{Data Scientist}   \\\\ \nT2         & \\multicolumn{2}{l}{Lead Software Architect}  \\\\\nC1         & \\multicolumn{2}{l}{Domain Expert}          \\\\\nC2         & \\multicolumn{2}{l}{Product Manager}        \\\\ \nC3         & Regulatory          \\\\ \nC4         & Customer        \\\\ \nC5         & \\multicolumn{2}{l}{Business Analyst}  \\\\\nC6         & Research                  \\\\\nC7         & \\multicolumn{2}{l}{Data Scientist}    \\\\ \\cline{1-2}\n\\end{tabular}\n\\hspace{2em}\n\\begin{tabular}{c p{1.2in} c}\n\\cline{1-2} \n\\textbf{B} & \\textbf{Role}             \\\\ \n\\cmidrule{1-2} \\morecmidrules \\cmidrule{1-2} \nT3         & \\multicolumn{2}{l}{Data Scientist}       \\\\ \nT4         & \\multicolumn{2}{l}{Data Scientist}      \\\\\nT5         & \\multicolumn{2}{l}{Data Scientist Manager} \\\\\nC8         & Consultant         \\\\ \nC9         & \\multicolumn{2}{l}{Database Administrator} \\\\ \nC10        & Customer       \\\\ \\cline{1-2}\n& \\\\\n& \\\\\n& \\\\\n\\end{tabular}\n\\hspace{2em}\n\\begin{tabular}{c p{1.2in} c}\n\\cline{1-2} \n\\textbf{C} & \\textbf{Role}  \\\\ \n\\cmidrule{1-2} \\morecmidrules \\cmidrule{1-2} \nT4         & \\multicolumn{2}{l}{Data Scientist}        \\\\  \n\\cline{1-2}\n& \\\\\n& \\\\\n\\cline{1-2}\n\\textbf{D} & \\textbf{Role}  \\\\ \n\\cmidrule{1-2} \\morecmidrules \\cmidrule{1-2} \nT6 & ML Researcher \\\\\n\\cline{1-2}\n& \\\\\n& \\\\\n& \\\\\n\\end{tabular}\n\\end{table}\n\nThe participants in this case study are part of an AI organization responsible for creating and maintaining several different kinds of models for a variety of health and medical use cases. During the case study period, the team worked on documentation for four models (Table \\ref{tbl:models}) and completed the FactSheets for models A and B. FactSheets for models C and D were in progress and had at least a first draft. Because of this, we were only able to interview consumers for models A and B.\nThe FS team used the publicly available resources on the AI FactSheets 360 site~\\cite{fs360} to help them follow the methodology. Resources included explanations of what FactSheets are and their intended applicability; a description of the methodology for creating FactSheets; example FactSheets; and links to research papers, videos and a Slack community.\n\n\\begin{table}\n\\caption{The four models documented during the case study}\n\\label{tbl:models}\n\\begin{tabular}{c p{5in}}\n\\hline\n\\textbf{Model ID} &  \\textbf{Model Description} \\\\ \\hline \\hline\nA            & \nA set of models to assist Medicaid Fraud, Waste, and Abuse investigators in retrieving relevant information from Medical Insurance policy\n                                                                     \\\\\nB            & \nA mature model (version 21) to predict relative mortality risk of a hospitalization stay  \\\\\nC            & \nA model (in development) to identify potential risks of surgical complications to proactively mitigate these potential issues\n\\\\\nD            & \nA model to identify populations that are likely to have improved health outcomes if their social factors are improved\n                                                       \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\\subsection{Interview Protocols}\n\nFor the FS team, we conducted approximately one-hour, semi-structured interviews that focused on how well the methodology worked or did not work in practice. More specifically, we focused on five main topics: (1) the context of, and motivations for, creating the FactSheet, \n(2) how they used the available resources, (3) how they implemented the FactSheets methodology within their actual context, (4) perceived benefits, and (5) perceived costs. \n\nInterviews with the consumers lasted approximately 30 minutes. Unlike the FS team interviews, that focused mainly on the process of creating a FactSheet, consumer interviews focused on whether the created FactSheet met their needs, and if not, where additional or different content was needed. Topics in consumer interviews focused on: (1) evaluating the quality of the created FactSheet and (2) how the FactSheet has changed the way they work compared to the past.\n\nBoth FS team and consumer interviews were similarly run. In each interview, participants were given access to the finished (or in progress) FactSheet that they could refer to as needed. Interview sessions were run remotely, recorded, and transcribed for later analysis. \n\n\\subsection{Data Analysis}\nTo analyze participants' responses, we used two approaches. For questions that expected more direct responses such as identifying missing content in the FactSheet, we identified and aggregated participants' answers. For more complex questions such as the ones around perceived benefits and costs, we conducted thematic analysis~\\cite{braun2006using}, focusing on themes that addressed our research questions. First, participants' responses were extracted from each interview transcript such that a complete response was available for each question, including any relevant context. One author extracted responses, annotating any references a participant made to the FactSheet as necessary.\n\nAll authors participated in the thematic analysis. The thematic analysis was an iterative and inductive process, with new themes emerging and collapsing as the authors worked through the data. Using the set of interview responses, each of the authors coded any sentences with a theme or topic relevant to the research or interview questions. Extracted codes and their definitions were iteratively added, consolidated, and removed as needed to form the codes. Upon completion, all three authors together discussed codes and consolidated codes into themes and subthemes. The emergent themes and subthemes contributed to our findings.\n\n\\section{Results} \\label{sec-results}\nOur results are structured by research question. We address if the methodology was usable by a real-world FactSheets team not trained in human-centered practices (RQ1), if the resulting FactSheets met the needs of their various consumers (RQ2),\nand what the benefits and costs of this approach were seen to be (RQ3).\n\n\\subsection{RQ1: Usability of Methodology by FS Team Members}\nPractitioners skilled in human-centered practices will recognize that the methodology applies these practices to the creation of FactSheet templates (and the resulting FactSheets). But is it reasonable to expect data scientists, engineers, and others \\textit{not} trained in these practices to execute the methodology and derive the expected benefits? In this section we report on how well FS team members were able to apply the methodology. Each subsection looks at a different aspect of the methodology's usability: the value of available educational resources; how well individual steps of the methodology could be followed and how useful they were; and the ways in which the methodology helped in documenting AI models.\n\n\\subsubsection{How FactSheet Teams Used Educational Resources} \\label{sec:education-resources}\nThe FactSheets 360 website \\cite{fs360} provides an overview of the methdology, a number of illustrative FactSheets, and links to more detailed instructional content. When we asked the four FS team members who used the website to identify the resources that were the most useful to them, responses included the example FactSheets on the website (T1-A and T5-B) and the methodology summary (T2-A). \nWhen asked for more details, T1-A described how one example showed how disparate kinds of documentation could be brought together into a single place saying \\say{It was self-contained. It provided all the info I needed to understand.} T5-B described how the different views in the example FactSheets, such as the table view and slide view, helped him understand how FactSheets could be tailored to the needs of different consumers saying\n\\say{Something clicked (for me) on one of the (example) pages about how the same facts can be represented in different ways, which was the key idea for me to really see the value of this approach. I haven't thought about documentation in that way before}.\nT6-D echoed this sentiment explaining how many of the discussions around what to document revolved around this perspective on documentation saying \\say{There were two critical things on my mind. It was understanding the players. What do we mean by producers? What do we mean by consumers?} By framing documentation as something to be used by others, FS teams changed their orientation from just reporting on what a model did to focusing on how to make the description useful for specific consumers. More details on this observation follow in Section \\ref{sec:methodology-pros-cons}.\n\nFS team members did not report that any resources were \\textit{un}helpful.  They did identify some gaps and stated that there was too much content to digest it all. T1-A identified procedural guidance that he would have liked to have such as a \\say{minimum set of fields} required for a FactSheet template.\nT2-A described a desire to have concrete guidance for how long the methodology should take and which facts to start with. \nTo address this, T2-A developed a schedule based on his team's first attempt at creating a FactSheet template, which provided guidance for subsequent efforts. \\say{We needed something to explain the process... Starting with the methodology of the site and then breaking it out into 'Here's a six-week schedule' basically. To give (the rest of the team) something concrete.} Similarly, to address T1-A's concern, the team shared the template (tailored through their work for the healthcare domain) and FactSheet for future teams to use as a starting point.\n\n\\subsubsection{Methodology Steps}\n\\label{sec:methodology-pros-cons}\nTo better understand how well the individual steps of the methodology could be applied, we asked FS team members to reflect on the steps one by one. FS team members responded most often that Step 1 of the methodology, \\textit{Know your FactSheet Consumers},\nwas the most useful of the seven steps.\nUnderstanding their consumers encouraged FS teams to consider who they would be writing for, and what\ntheir documentation for them should include. \nHalf the FS team members (T1-A, T2-A, T5-B) decided this step was the most important one. T1-A said, \\say{Knowing your consumers was a step that was particularly useful for me... Once I had a particular consumer identified I realize that there are things that I need to think more about, or I need to reach out to people who know about this. So, it was useful.} \nRevealingly, T1-A first tried skipping Step 1 to save time, opting to go directly to getting feedback from consumers on a FactSheet based on an example in the FactSheets 360 website. He recalled, \\say{\nWe started filling it out without the right [consumer needs]... and then we kind of realized a lot of this stuff isn't right. So we went back and rewrote quite a lot of it basically.} \nAfter this setback, the other FS team members decided to speak with potential consumers first.\n\nParticipants T2-A and T5-B noted the benefit of creating a first version of a FactSheet (Step 4) before gathering feedback from consumers on just the template (produced in Step 3). While creating this first version might be seen as adding unnecessary time, the conversations with consumers focused on making changes to this version instead of generating content from scratch. \nSince one of largest costs reported by participants in the FactSheets methodology was validating the FactSheet with consumers, FS teams appreciated this time-saver. \nT6-D noted the value of this step as well and added detail about how the act of filling out the FactSheet template spurred reflection on aspects of the model they previously had not considered saying, \\say{It's forcing us to answer questions about our model that we may have thought about, but never documented in any way.} \n\nAlthough FS team members did not consider any specific steps of the methodology to be \\textit{un}helpful, they did adapt the methodology to better fit their team's needs and timeline. \nFS teams chose, for example, to minimize the seeming seriality of Steps 5 (have actual producers create a FactSheet) and 6 (evaluate the FactSheet with actual consumers). They preferred a more iterative approach where FactSheet content was evaluated as soon as there was enough for a specific consumer role, even if the rest of the information was not yet ready. These changes were implemented during the first pilot. Since the subsequent pilots were informed from the first, T3-B, T4-B, T5-B, T4-C and T6-D all followed the altered protocol, further compressing the timeline by starting with the template from model A.\n\n\n\n\\begin{table}\n\\caption{FS team member responses to if the methodology helped them in various ways. A response of `n\/a' indicates that the FS team member was unsure or unable to give a response.}\n\\label{tbl:methodology-useful}\n\\begin{tabular}{llllllll|c}\n\\hline\n\\textbf{Did the methodology help...}                    & \\textbf{T1-A} & \\textbf{T2-A} & \\textbf{T3-B} & \\textbf{T4-B} & \\textbf{T5-B} & \\textbf{T4-C} & \\textbf{T6-D} & \\textbf{\\# Yes}\\\\ \\hline \\hline\n... Improve documentation consistency?        & Yes  & Yes  & Yes  & Yes  & Yes  & Yes  & Yes & 7 \\\\ \n... Evaluate documentation usefulness?         & Yes  & n\/a  & Yes  & Yes  & Yes  & Yes  & Yes & 6 \\\\\n... Identify new documentation needs?          & Yes  & Yes  & No   & No   & Yes  & Yes  & Yes & 5 \\\\\n... Facilitate creating useful documentation? & Yes  & No   & n\/a  & No   & Yes  & Yes  & Yes & 4 \\\\\n... Improve documentation practices?           & Yes  & n\/a  & Yes  & No   & No   & n\/a  & Yes & 3 \\\\\n... Identify new consumers?                    & Yes  & Yes  & No   & No   & No   & n\/a  & No  & 2 \\\\ \\hline\n\n\\end{tabular}\n\\end{table}\n\n\\subsubsection{How the Methodology Helped FactSheet Teams Document AI Models}\nFS team members noted how the methodology provided specific benefits over their previous documentation practices. Table \\ref{tbl:methodology-useful} summarizes the questions about specific benefits that we asked about and participants' responses. All FS team members agreed that the FactSheets methodology would likely enhance documentation consistency, primarily by consolidating what was previously scattered in multiple documents into a single place.\nT5-B summarized, \\say{We'll be able to consolidate in ways that make sense so that there's one place for facts, rather than, lots of places to keep things up to date.}\n\nT5-B and T6-D described how \ncompleting the FactSheet template \nhad caused them to reflect on the usefulness of what they were writing in the FactSheet fields. \nT5-B described how a consumer need for making the problem description \nreusable\nencouraged conciseness. He said, \\say{One of the best (consumer) feedbacks that we received was in the way we tried to condense the problem description in the FactSheet so that they could copy and paste it to engage with customers}.\n\nImportantly, five of the FS team members agreed that the methodology helped their team identify \\textit{new} documentation needs. They discovered several additional template fields as a direct result of \nfollowing the methodology. \nTable~\\ref{tbl:template} shows most of the fields from the FactSheet template for model A.  Newly-discovered fields are italicized.\nExamples included regulatory requirements, model maturity, usage considerations, \nand run time requirements. \nT6-D summarized the value stating, \\say{There was a lot of information here that we wouldn't perhaps otherwise made available or collected or even thought of.}\n\n\\begin{table}\n\\caption{The template created by the FS team for model A. Italicized fields indicate new fields that were added during the case study.} \n\\label{tbl:template}\n\n    \\begin{tabular}{p{2.2in} p{3.45in}}\n    \\hline\n    \\multicolumn{2}{c}{\\textbf{Overview}}  \\\\ \\hline \\hline\n    Model Name          & The  name of the model                                         \\\\\n    \\textit{Version and Dates}   & The model's version and date it was created                    \\\\\n    Problem Solved      & A description of the problem the model is designed to solve    \\\\\n    \\textit{Maturity Ratings}    & A evaluation of the model's maturity along multiple dimensions \\\\\n    \\textit{Contact Information} & Who to contact about this model                                \\\\ \\hline\n    \\multicolumn{2}{c}{\\textbf{Intended Use}}  \\\\ \\hline \\hline\n    Intended Domain    & The industry of functional area the model intends to support                             \\\\\n   \n    \\textit{Use Case}         & The use scenario, including who, what, when and how the model outputs intends to support in practice \\\\ \\hline\n    \\multicolumn{2}{c}{\\textbf{Data and Model}}  \\\\ \\hline \\hline\n    Training Data                        & Statistics and details about training data        \\\\\n    \\textit{Cohort \/ Therapeutic Area Definition} & Population cohorts of interest for this use case  \\\\\n    Model Information                    & Description of the model                          \\\\\n    Input\/Output                         & Description of expected model inputs and outputs  \\\\\n    Internal Test Data                   & Description of the internal model test data used  \\\\\n    External Validation Data             & Description of the external model validation data \\\\ \\hline\n    \\multicolumn{2}{c}{\\textbf{Performance}}  \\\\ \\hline \\hline\n    Performance Expectation \\& Robustness    & Expected performance metrics and evaluation of the model's adversarial robustness     \\\\\n    Model Performance                        & Description of model evaluation and actual performance metrics                        \\\\\n    Bias \\& Fairness                         & Description and evaluation of the model's fairness and bias                           \\\\\n    \\textit{Explainability}                           & Description of the model's ability to explain its predictions                         \\\\ \\hline\n    \\multicolumn{2}{c}{\\textbf{Usage Considerations}}  \\\\ \\hline \\hline\n    Optimal \\& Poor Conditions               & Conditions under where the model performs well and poorly                             \\\\\n    \\textit{Volume \\& Latency}                       & Expected number of predictions and time per prediction                                \\\\\n    \\textit{Discontinue Use If}                       & Conditions under which to stop using the model                                        \\\\\n    \\textit{Runtime Data Requirements}                & Data prerequisites necessary to run this model                                        \\\\\n    \\textit{Runtime Technology Requirements}          & Technology prerequisites necessary to run this model                                  \\\\\n    \\textit{Use-Case Tolerance of Error}              & Evaluation of the error tolerance of this model                                   \\\\\n    \\textit{Market Differentiator}                   & Description of market differentiators for this model                                  \\\\\n    \\textit{Related Areas Where Models May Be Useful} & Other potential use cases for this model              \\\\ \n    \\hline\n    \\end{tabular}\n\\end{table}\n\n\\subsection{RQ2: FactSheet Usefulness for Consumers}\n\nTo determine\nwhether the FactSheets generated by the FS teams were useful to consumers, we asked consumers to \n(1) evaluate the FactSheet's quality along several dimensions and describe any gaps\nand (2) to \ndescribe if and how the FactSheet differed from the AI documentation they had encountered previously.\n\n\\subsubsection{Evaluation of Usefulness}\n\nWe asked consumers to evaluate the FactSheets, along multiple quality dimensions, on a seven-point Likert scale ranging from \"strongly disagree\" to \"strongly agree\". The dimensions were drawn from work on assessing AI documentation quality \\cite{piorkowski2020towards} along with prior research on the pragmatics of effective communication. The dimensions and prompts are shown in Table~\\ref{tbl:quality-prompts}.\nWe found that the overall quality of the FactSheets produced was excellent, scoring high marks across all the quality dimensions as shown in Figure \\ref{fig:factsheet-quality}. These results suggest that the work of the FS teams, perhaps especially the iterative evaluation of FactSheet content with consumers, paid off and resulted in highly relevant, useful, and usable documentation.\n\n\\begin{table}\n\\caption{Likert-scale prompts given to consumers to assess FactSheet quality}\n\\label{tbl:quality-prompts}\n\\begin{tabular}{p{1.2in} p{4.05in}}\n\\hline\n\\textbf{Quality Dimension} & \\textbf{Prompt}                                                                                   \\\\ \\hline \\hline\nCompleteness      & The FactSheet has all the information that I require for my use case.                                          \\\\ \nEvidence          & The FactSheet's information is well supported with additional evidence provided where needed.                          \\\\ \nVocabulary        & The FactSheet is written using appropriate vocabulary and word choice.                                        \\\\ \nUnderstandability & The FactSheet's content and information is easy to understand.                                                    \\\\ \nLayout            & The FactSheet's structure and layout is intuitive.                                                                \\\\ \nRepresentation    & The FactSheet's information is presented in the expected way with text, tables, and figures appropriately chosen. \\\\\nOrganization      & The FactSheet's information was well organized and easy to locate.                                                \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=1.0\\textwidth]{factsheet-quality.pdf}\n    \\caption{Consumer responses for evaluating the FactSheet quality. The responses on the right side of the dark vertical line represent responses of `slightly agree' or higher.}\n    \\label{fig:factsheet-quality}\n\\end{figure}\n\n\\subsubsection{Things That Were Missing} \\label{sec:missing}\n\n\\begin{table}\n\\caption{Summary of missing documentation information per consumer. Italicized entries were mentioned by multiple consumers. Although market differentiation content existed in the FactSheets, participants wanted \\textit{more} detail than what was available, so it is included below.}\n\\label{tbl:missing-facts}\n\\begin{tabular}{c p{1.2in} p{3.7in}}\n\\hline\n\\textbf{Consumer} & \\textbf{Role}          & \\textbf{Missing documentation fields}                                                                                                                                                                         \\\\ \\hline \\hline\nC1-A                 & Domain Expert          & None                                                                                                                                                                                                          \\\\\nC2-A                 & Product Manager        & \\textit{Market differentiation}; \\textit{existing deployments}; customers; pricing   information; model maturity; links to marketing materials, presentations, and   demos                                                      \\\\\nC3-A                 & Regulatory             & \\textit{Market differentiation}; Process model is replacing; benefits and costs of deployment; access   information; model evaluation summary; known risks; applied risk mitigations;   \\textit{how to interpret model metrics}; productization details \\\\\nC4-A                 & Customer-Facing               & \\textit{Market differentiation}; Mapping between model metrics to business use case; \\textit{how to interpret model metrics}; \\textit{good\/bad ranges for model metrics}; business context                                                                     \\\\\nC5-A                 & Business Analyst       & \\textit{Good\/bad ranges for model metrics}                                                                                                                                                                             \\\\\nC6-A                 & Research               & Data sources; data cleaning details; data limitations and constraints;   model training details; background of domain experts; definitions for bias;   definitions for explainability                         \\\\\nC7-A                 & Data Scientist         & \\textit{Existing deployments}; opportunities for model improvement; customer   feedback; history of model evaluations                                                                                                  \\\\\nC8-B                 & Consultant             & None                                                                                                                                                                                                          \\\\\nC9-B                 & Database Administrator & None                                                                                                                                                                                                          \\\\\nC10-B                & Customer-Facing               & Customer-specific model evaluation                                                                                                                                                                            \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\nEven with the high quality-dimension scores, consumers were able to identify things that were still missing. For any response to a prompt with a score of less than `strongly agree', we asked the respondent what was missing or needed to be changed for that dimension. Table \\ref{tbl:missing-facts} summarizes their replies.\n\nA closer look at this reveals that missing content tends to be\nquite role-specific. \nFor example, C2-A's suggestions are related to how the model fits into the larger business context. He wanted to see more information about which customers are using the model along with pointers to additional materials useful in discussions with potential customers. He stated, \\say{It might be good to include \nsomething about where has the model been deployed?}\nThe requests from C3-A, whose role involved regulatory concerns, focused on risk: the model context, risk evaluations, and risk mitigation. \nC4-A and C10-B, both in customer-facing roles, focused on how to determine model quality and how the quality compares to competitors. \nC5-A, like C4-A, not having a deep data science background, wanted more guidance on how to interpret the model evaluation metrics, both in terms of how the model relates to the business, but also including the reasonable question of what makes a good metric score. P5-A said, \\say{I think I understand what these (metric) scores might tell me, but I have to do a lot of thinking to connect it back to the business problem... What do they mean for me and the situation that I'm trying to evaluate? ... What's a good range? What's a bad range?} \nC7-A, a data scientist, wanted to see information that would help him improve the model, along with a history of model evaluations over time. \nThese consumers' role-specific requests underscore the point that different consumers have markedly different documentation needs, and provide further evidence that a one-size-fits-all solution for AI documentation will likely not suffice.\n\nThe combination of high quality scores for the FactSheet along with several suggestions for further improvement tell a somewhat mixed story. On the one hand, as indicated by the high scores, participants seemed to be satisfied with the content of the FactSheets. On the other hand, they still had suggestions for how to improve them. There are potentially several reasons for this. FS teams may not have iterated enough with consumers to gather all the relevant documentation needs (on average going through only two iterations). Or FS teams intentionally did not address all these needs to keep FactSheet size manageable (a role-based filtering mechanism, see \\ref{sec:tools}, may help manage this trade off). Another interesting possibility is that by asking consumers in our interviews to reflect on a FactSheet from the perspective of the quality dimensions, gaps surfaced that they did not see before. If true, this latter interpretation may have implications for eliciting consumer needs more completely.\n\n\\subsection{RQ3: Methodology Benefits and Costs}\n\nTo better understand the value proposition of both the methodology and FactSheets themselves, we asked FS team members and consumers what specific pros and cons they experienced, further reflecting on how this compared with their documentation experiences from before. \n\n\\subsubsection{Methodology Benefits}\nAs mentioned earlier, one key benefit of FactSheets was to consolidate disparate documentation.\nT1-A expressed how bringing in the different viewpoints gave a more holistic picture of the model and its context. He said, \\say{When [you] have the document in front of [you] from all points of view, you start joining things together and asking questions... It gives a broader overall picture and fills gaps in your own knowledge.} This broader context would be difficult to capture without input from all the consumers.\nAnother benefit of consolidation was how the FactSheet facilitated additional exploration. T1-A, T3-B and T6-D discussed how the more broadly written content of the FactSheet coupled with its pointers to where more details can be found enabled such exploration. T1-A said, \\say{I think this is useful documentation for the reason that everyone can go in and find what they are looking for, maybe not at the level of detail that [they need], but there is links and hyperlinks to everything.} T6-D referred to the FactSheet as a \\say{one-stop shop for the model}. Seven consumers (C2-A, C3-A, C4-A, C5-A, C6-A, C8-B and C9-B) agreed that the FactSheet acted as an effective point of entry for understanding the aspects of the model relevant to their role.\n\nConsumers elaborated on the kinds of further exploration that the FactSheet enabled. For example, C2-A and C8-B explained how the FactSheet eased navigation to other relevant documents. C2-A said, \\say{It's a single document that makes it easy for me to navigate to the next level. (Previously), I'd have to hunt down multiple documents or ask people what was available... This seems to bring it together at least in a single launch point where I can read a bit of detail about it, and then I can understand where to go to find more information.} C3-A and C6-A explained that the FactSheet served as a pointer to the right people to contact for further follow up, potentially reducing the time spent gathering information about a model. C3-A summarized, \\say{I think it reduces the amount of time that will be spent of clarifying issues... It would allow me to do pre-work without having to do 3 meetings with a group of 5 to 6 people.} C5-A and C7-A likewise agreed that it reduced the number of meetings required to understand the current state of a model. C1-A, C4-A and C10-B echoed similar sentiments for discussions with customers.\n\nIn addition to serving as a jumping-off point, FactSheets acting as the sole source (or at least sole anchor) of truth, enabled FS teams and consumers to reduce work in several ways. \nPreviously,\nthere would be several sources of documentation that sometimes provided conflicting information. The FactSheet as the single source of truth allowed content to be reliably copied for other creations such as customer-facing descriptions, reports, or presentations. C8-B recalled a time that she was able to reuse content in the FactSheet for a client. She said, \\say{This is actually really nice that they've got this [documentation field] added in here because now I can lift this paragraph and put it into a client document... And in an approved definition or language for how to describe [it].} One FS team member, T3-B, likewise used the FactSheet as a text source for additional materials. Simply knowing where the most current documentation is also enabled more effective employee on boarding, especially when the people who worked on the model were no longer available. C6-A summarized, \\say{You would talk to [someone] who'd go talk to someone else who has a link somewhere, or sends you some old weird spreadsheet that says, `go talk to this person'. And then someone would have left the company and then they don't have access to something. So this is much much better, because it puts everything in one place.} Finally, FS team members also reported benefits centered around the theme of improved organization. They reported how FactSheets helped them get a better understanding of documentation needs earlier (T1-A, T2-A), reduce repetition (T2-A, T6-D), improve awareness of other documentation (T3-B), and help identify overlaps in existing documentation (T4-C, T5-B).\n\n\n\\subsubsection{Methodology Costs}\n\n\\begin{table}\n\\caption{The number of hours each FS team member spent working on the FactSheet}\n\\label{tbl:hours-spent}\n\\begin{tabular}{ccccccc}\n\\hline\n\\textbf{T1-A}     & \\textbf{T2-A}     & \\textbf{T3-B}     & \\textbf{T4-B}    & \\textbf{T5-B}     & \\textbf{T4-C}     & \\textbf{T6-D}      \\\\ \\hline \\hline\n15 hours & 24 hours & 10 hours & 8 hours & 15 hours & 24 hours & 6-8 hours \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\nThe most obvious cost faced by participants was simply the time required to create the template and FactSheet. FS team members reported times ranging from 6 to 24 hours (Table \\ref{tbl:hours-spent}). This estimate includes all the steps of the methodology, including meetings with others and time spent filling out the fields. Of course, creating documentation of any sort takes time, and it is not clear that this is outside the norm. Scheduling time with consumers was the most common cost mentioned by FS team members, specifically, T1-A, T2-A, T5-B, and T6-D.\n\nAside from the challenge of getting people together, some steps of the methodology required additional effort from the FS team. One mentioned by FS team members T1-A, T3-B and T6-D was Step 4, filling in the FactSheet template for the first round of consumer feedback.\nThis approach is meant to make the feedback sessions more streamlined as it asks consumers to critique something that exists instead of creating it on the spot. However, this approach passed the burden off to the FS team, leaving them to fill out fields they may be unfamiliar with.\n\nAnother challenge in preparing that first draft was locating some of the documentation that existed from past work on the model. Both T3-B and T6-D were working with a models that had scattered documentation. T6-D described how much of her time was spent just finding the source that had the information she needed for the FactSheet. She said, \\say{(The difficulty) is not knowing where to look, right? It's reading through documents that you potentially didn't have to read through.}\n\nOther than the missing fields, noted above, \nconsumers reported few specific costs to the methodology. The one exception was consumer C10-B who did not see the benefit of a FactSheet over the documentation that already existed for model B (with which she was already familiar). C10-B may be a bit of an outlier, however, as all nine of the other consumers found FactSheets to be better than their prior documentation.\n\n\n\\section{Discussion} \\label{sec-discussion}\nThe current case study looks at the early process of adopting the FactSheets methodology within an organization. It examines whether the methodology can be followed by those with no particular training in human-centered design, and whether the near-term benefits outweigh the costs. We believe this early picture is promising and expect even more gains from long-term adoption due to reuse of templates, growing awareness of what facts matter to consumers, and increased automation of fact collection. \n\n\\subsection{Enabling Reuse} \nDespite our belief that there is no universal template or checklist for AI transparency, there is a strong and quite understandable desire to reuse an existing template, for example, one from \\cite{fs360}, rather than developing one from scratch. What we have observed in the present work is that a core set of facts did apply in a very different domain from the ones considered to date. These included descriptions of model purpose, training data, and model inputs and outputs. As shown in Table \\ref{tbl:template}, this core set was augmented with domain-specific facts, such as therapeutic-area definition (for these models in the healthcare space) and market differentiator for models that reached a level of maturity where they are competing for sales within a segment. This kind of \"base plus extensions\" form of reuse is likely to become a dominant form.\n\nAn additional observation from the present work is that once a template has been tailored for a particular organization, it can be used with very little additional work by others within that organization. While the models in this study were all in the healthcare space, they had very different types, use cases, and maturities. Even so, the template developed for the first model was readily reused for subsequent models.\n\n\\subsection{Design Implications for Tools}\n\\label{sec:tools}\n\nOne of the key gaps identified by FS team participants was the lack of tool support for gathering facts. Facts needed to be found in existing documentation (large, dispersed, and created for a variety of purposes for a variety of audiences), then adapted for use within the much more compact FactSheet, or were obtained by tracking down and talking with others in the organization. We believe this situation will rapidly improve in several ways:\n\n\\begin{itemize}\n    \\item Fact automation: many kinds of facts can be captured automatically by suitably instrumented tools. Code repositories adapted for use in AI development can record key facts about training data and model versioning. Lightweight tools such as Python notebooks can include fact-capture mechanisms with very little overhead. As the field identifies the kinds of facts that best support transparency, this sort of automation will likely appear.\n    \\item Fact elicitation: some facts, such as model purpose or non-quantifiable aspects of potential bias, must be entered by a knowledgeable human. A good library of examples, wizards, and other elicitation tools will make this both more efficient and help make facts more consistent (supporting comparison) and consumable.\n    \\item Fact explanations: there will always be a wide range of expertise among the various consumers of FactSheets. Automatic mechanisms for tailoring fact content to skill level are not infeasible. In addition, a library of reusable links to definitions of terms and supporting material will allow consumers to better understand potentially complex aspects of AI without requiring fact producers to author this supplemental information repeatedly.\n    \\item Fact filtering: either through customized templates or filtering controls, consumers will eventually be able to customize documentation to their needs.\n\\end{itemize}\n\n\\subsection{A Broader View of Transparency}\n\nA key focus of the research on AI transparency views it from the perspective of AI's impacts on individual citizens and society. This is essential work. However, we must acknowledge that there are many stakeholders involved in the creation of facts about AI that collectively flow (or fail to flow) into documentation that may or may not be suitably transparent. AI transparency must grapple with this fairly untidy reality. The development of models is not a solitary activity. It involves cross-functional teams with different expertise, iteratively building, testing, refining, deploying, and monitoring AI models~\\cite{zhang2020data, piorkowski2021ai}. Critically, these roles drop in and out at different points of the AI model development lifecycle~\\cite{piorkowski2021ai}. Along the way, intermediate consumers of important facts about model development attempt to understand, and contribute to, the growing body of facts (the quality of their contributions being dependent on the quality of the facts they consume). \nProblematically, AI documentation is not designed to meet the needs of its various consumers. And while a single, high-level template may provide a level of basic transparency for some consumers, it will generally fail to generate the transparency needed by all.\n\nWe believe that documentation needs to be designed with a clear understanding of its various consumers. In the end, such documentation may have a multi-tiered structure with a near universal top-level view, supplemented by more detailed and specialized information in further tiers. \nThis is a common approach for structuring complex information, for example, in areas such as product comparisons where the top view may include capacities and energy ratings with additional views providing detailed information on modes of operation, compatibility with other products, and multi-dimensional ratings.\nOur study suggests that at these more detailed levels, consumers' documentation needs are going to be specific and unlikely to overlap across roles as shown in Table \\ref{tbl:missing-facts}. This requires a human-centered approach to understanding and capturing these needs in templates and, eventually, supporting tools.\n\n\\section{Conclusion} \\label{sec-conclusions}\nOur results suggest that the FactSheets methodology~\\cite{factsheet-methodology-IEEE} worked as intended, that is, FS team members were able to successfully use the methodology to create AI documentation that was useful to its intended audience. Specifically, we found that\n\\begin{enumerate}\n    \\item[RQ1] \\textbf{Methodology usable without human-centered training?} Even without training in human-centered design methodologies, FS team  members adopted the methodology, successfully elicited needs from their consumers, and encoded those needs into a FactSheet template. Furthermore, the template addressed needs specific to the AI healthcare domain, but were still general enough to be reused across several, quite different, models.\n    \\item[RQ2] \\textbf{FactSheets addressed needs of consumers?} Consumers found the resulting FactSheets met their needs, giving them high quality scores along several dimensions. Not all needs were met, however, and the identification of missing content further reinforced a core motivation for the FactSheets methodology: that AI documentation must be tailored to the specific needs of each consumer.\n    \\item[RQ3] \\textbf{Methodology benefits and costs?} FS team members and consumers benefited from documentation that was authoritative, in a single location, and supported additional detailed exploration for specific needs. In 16 of 17 interviews, participants agreed that the FactSheets were an improvement over earlier documentation practices.\n\\end{enumerate}\n\nIn summary, the benefits of following the FactSheets methodology to improve AI transparency seemed to outweigh the costs, and are best expressed by T2-A who closed their interview saying the following about the methodology, \\say{I think it's going to be very useful for us going forward and really it's becoming a core part of what we're doing.}\n\n\\ignore{\n\\begin{acks}\n\n\\end{acks}\n}\n\n\\bibliographystyle{ACM-Reference-Format}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\\label{sec:introduction}}\n\\label{sec:intro}\n\\IEEEPARstart{H}{uman} parsing \\cite{yamaguchi2012parsing, liang2018look, wang2019learning, wang2020hierarchical, li2022deep}, considered as the fundamental task of human-centric visual understanding \\cite{lin2020human}, aims to classify the human parts and clothing accessories in images or videos at pixel-level. Numerous studies have been conducted on human parsing due to its crucial role in widespread application areas, \\emph{e.g}., security monitoring, autonomous driving, social media, electronic commerce, visual special effects, artistic creation, giving birth to various excellent human parsing solutions and applications.\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=0.99\\linewidth]{figures\/taxonomy.pdf}\n\t\\end{center}\n\t\\vspace{-5pt}\n\t\\captionsetup{font=small}\n\t\\caption{\\small{\\textbf{Human parsing tasks reviewed in this survey:} (a) single human parsing (SHP) \\cite{cvpr2021l2id}; (b) multiple human parsing (MHP) \\cite{gong2018instance}; (c) video human parsing (VHP) \\cite{zhou2018adaptive}.}}\n\t\\label{fig:taxonomy}\n\t\\vspace{-10pt}\n\\end{figure}\n\nAs early as the beginning of this century, some studies tried to identify the level of upper body clothing \\cite{borras2003high}, the grammatical representations of clothing \\cite{chen2006composite} and the deformation of body contour \\cite{guan2010a} under very limited circumstances. These early studies facilitated the research on pixel-level human parts and clothing recognition, \\emph{i.e}., human parsing task. Immediately afterward, some traditional machine learning and computer vision techniques were utilized to solve human parsing problems, \\emph{e.g}., structured model \\cite{yang2011articulated, dong2013deformable, yamaguchi2012parsing}, clustering algorithm \\cite{caron2018deep}, grammar model \\cite{zhu2008max, dong2014towards}, conditional random field \\cite{kae2013augmenting, ladicky2013human, yamaguchi2013paper}, template matching \\cite{bo2011shape, liang2015deep} and super-pixel \\cite{fulkerson2009class, tighe2010super, liu2013fashion}. Afterward, the prosperity of deep learning and convolutional neural network \\cite{krizhevsky2012imagenet, girshick2014rich, jia2014caffe, lecun2015deep, szegedy2015going, shelhamer2016fully, he2016deep} has further promoted the vigorous development of human parsing. Attention mechanism \\cite{chen2016attenttion, liang2016object, yang2018attention, cheng2019spgnet}, scale-aware features \\cite{liang2015human, xiq2016zoom, zhang2020pcnet, yang2021quality}, tree structure \\cite{wang2019learning, ji2020learning}, graph structure \\cite{gong2019graphonomy, wang2020hierarchical, zhang2022human}, edge-aware learning \\cite{ruan2019devil, zhang2020correlating, liu2020hybrid}, pose-aware learning \\cite{liang2018look, nie2018mutual, zhao2022from} and other technologies \\cite{liu2018cross, luo2018macro, li2020self, li2020correction} greatly improved the performance of human parsing. However, some existing challenges and under-investigated issues make human parsing still a task worthy of further exploration.\n\n\n\n\\begin{figure*}\n\t\\begin{center}\n\t\t\\includegraphics[width=0.95\\linewidth]{figures\/overview.pdf}\n\t\\end{center}\n\t\\vspace{-5pt}\n\t\\captionsetup{font=small}\n\t\\caption{\\small{\\textbf{Outline of this survey.}}}\n\t\\label{fig:outline}\n\t\\vspace{-10pt}\n\\end{figure*}\n\nWith the rapid development of human parsing, several literature reviews have been produced. However, existing surveys are not precise and in-depth: some surveys only provide a superficial introduction of human parsing from a macro fashion\/social media perspective \\cite{mameli2021deep, cheng2021fashion}, or only review a sub-task of human parsing from a micro face parsing perspective \\cite{khan2020face}. In addition, due to the fuzziness of taxonomy and the diversity of methods, comprehensive and in-depth investigation is highly needed and helpful. In response, we provide the first review that systematically introduces background concepts, recent advances, and an outlook on human parsing.\n\n\n\\vspace{-6pt}\n\\subsection{Scope}\nThis survey reviews human parsing from a comprehensive perspective, including not only single human parsing (Figure~\\ref{fig:taxonomy} (a)) but also multiple human parsing (Figure~\\ref{fig:taxonomy} (b)) and video human parsing (Figure~\\ref{fig:taxonomy} (c)).\nAt the technical level, this survey focuses on the deep learning-based human parsing methods and datasets in recent ten years. To provide the necessary background, it also introduces some relevant literature from non-deep learning and other fields. At the practical level, the advantages and disadvantages of various methods are compared, and detailed performance comparisons are given. In addition to summarizing and analyzing the existing work, we also give an outlook for the future opportunities of human parsing and put forward a new transformer-based baseline to promote sustainable development of the community. A curated list of human parsing methods and datasets and the proposed transformer-based baseline can be found at \\url{https:\/\/github.com\/soeaver\/awesome-human-parsing}.\n\n\\vspace{-6pt}\n\\subsection{Organization}\nFigure~\\ref{fig:outline} shows the outline of this survey. \\S\\ref{sec:pre} gives some brief background on problem formulation and challenges (\\S\\ref{sec:problem}), human parsing taxonomy (\\S\\ref{sec:taxonomy}), relevant tasks (\\S\\ref{sec:relevant}), and applications of human parsing (\\S\\ref{sec:parsing-based}). \\S\\ref{sec:dl-base-hp} provides a detailed review of representative deep learning-based human parsing studies. Frequently used datasets and performance comparisons are reviewed in \\S\\ref{sec:hp-data} and \\S\\ref{sec:pc}. An outlook for the future opportunities of human parsing is presented in \\S\\ref{sec:outlook}, including a new transformer-based baseline (\\S\\ref{sec:new-baseline}), several under-investigated open issues (\\S\\ref{sec:open-issues}) and new directions (\\S\\ref{sec:new-direc}) for future study. Conclusions will be drawn in \\S\\ref{sec:concl}.\n\n\n\\section{Preliminaries}\n\\label{sec:pre}\n\n\\subsection{Problem Formulation and Challenges}\n\\label{sec:problem}\nFormally, we use \\bm{$x$} to represent input human-centric data, \\bm{$y$} to represent pixel-level supervision target, \\bm{$\\mathcal{X}$} and \\bm{$\\mathcal{Y}$} to denote the space of input data and supervision target. Human parsing is to map data \\bm{$x$} to target \\bm{$y$}: $\\bm{\\mathcal{X}}\\mapsto\\bm{\\mathcal{Y}}$. The problem formulation is consistent with image segmentation \\cite{minaee2021image}, but \\bm{$\\mathcal{X}$} is limited to the human-centric space. Therefore, in many literatures, human parsing is regarded as fine-grained image segmentation.\n\n\n\nThe central problem of human parsing is how to model human structures. As we all know, the human body presents a highly structured hierarchy, and all parts interact naturally. Most parsers hope to construct this interaction explicitly or implicitly. However, the following challenges make the problem more complicated:\n\n\\noindent$\\bullet$~\\textbf{Large Intra-class Variation.} In human parsing, objects with large visual appearance gaps may share the same semantic categories. For example, ``upper clothes\" is an abstract concept without strict visual constraints. Many kinds of objects of color, texture, and shape belong to this category, leading to significant intra-class variations. Further challenges may be added by illumination changes, different viewpoints, noise corruption, low-image resolution, and filtering distortion. Large intra-class variations will increase the difficulty of classifier learning decision boundaries, resulting in semantic inconsistency in prediction.\n\n\\noindent$\\bullet$~\\textbf{Unconstrained Poses.} In the earlier human parsing benchmarks \\cite{yamaguchi2012parsing, liu2013fashion, dong2013deformable, liang2015human}, the data is usually collected from fashion media. From them people often stand or have a limited number of simple pose. However, in the wild, human pose is unconstrained, showing great diversity. Therefore, more and more studies begin to pay attention to real-world human parsing. Unconstrained poses will increase the state space of target geometrically, which brings great challenges to the human semantic representations. Moreover, the left-right discrimination problem in human parsing is widespread (\\emph{e.g}., left-arm \\texttt{vs} right-arm, left-leg \\texttt{vs} right-leg), and it is also severely affected by unconstrained poses \\cite{liu2018cross, ruan2019devil, liu2019braidnet}.\n\n\\noindent$\\bullet$~\\textbf{Occlusion.} Occlusion mainly presents two modes: (1) occlusion between humans and objects; (2) occlusion between humans. The former will destroy the continuity of human parts or clothing, resulting in incomplete apparent information of the targets, forming local semantic loss, and easily causing ambiguity \\cite{liang2015human, zhang2020pcnet}. The latter is a more severe challenge. In addition to continuity destruction, it often causes foreground confusion. In human parsing, only the occluded target human is regarded as the foreground, while the others are regarded as the background. However, they have similar appearance, making it difficult to determine which part belongs to the foreground \\cite{yang2022part}.\n\n\n\\noindent\\textbf{Remark.} In addition to the above challenges, some scenario-based challenges also hinder the progress of human parsing, such as the trade-off between inference efficiency and accuracy in crowded scenes, motion blur, and camera position changes in movement scenes.\n\n\n\\vspace{-6pt}\n\\subsection{Human Parsing Taxonomy}\n\\label{sec:taxonomy}\nAccording to the characteristics (number of humans, data modal) of the input space \\bm{$\\mathcal{X}$}, human parsing can be categorized into three sub-tasks (see Figure~\\ref{fig:taxonomy}): single human parsing, multiple human parsing, and video human parsing.\n\n\\noindent$\\bullet$~\\textbf{Single Human Parsing (SHP).} SHP is the cornerstone of human parsing, which assumes that there is only one foreground human instance in the image. Therefore, \\bm{$y$} just contains corresponding semantic category supervision at the pixel-level. Simple and straightforward task definitions make most related research focus on how to model robust and generalized human parts relationship. In addition to being the cornerstone of human parsing, SHP is also often used as an auxiliary supervision for some tasks, \\emph{e.g}., person re-identification, human mesh reconstruction,  virtual try-on.\n\n\\noindent$\\bullet$~\\textbf{Multiple Human Parsing (MHP).} Multiple human parsing, also known as instance-level human parsing, aims to parse multiple human instances in a single pass. Besides category information, \\bm{$y$} also provides instance supervision in pixel-level, \\emph{i.e}., the person identity of each pixel. The core problems of MHP are how to discriminate different human instances and how to learn each human feature in crowded scenes comprehensively. In addition, inference efficiency is also an important concern of MHP. Ideally, inference should be real-time and independent of human instance numbers. Except as an independent task, MHP sometimes is jointed with other human visual understanding tasks in a multi-task learning manner, such as pose estimation \\cite{zhou2021differentiable, liu2021multi}, dense pose \\cite{yang2019parsing} or panoptic segmentation \\cite{de2021part}.\n\n\\noindent$\\bullet$~\\textbf{Video Human Parsing (VHP).} VHP needs to parse every human in the video data, which can be regarded as a complex visual task integrating video segmentation and image-level human parsing. The current VHP studies mainly adopt the unsupervised video object segmentation settings \\cite{wang2021survey}, \\emph{i.e}., \\bm{$y$} is unknown in the training stage, and the ground-truth of the first frame is given in the inference stage. The temporal correspondence will only be approximated according to \\bm{$x$}. Relative to SHP and MHP, VHP faces more challenges that are inevitable in video segmentation settings, \\emph{e.g}., motion blur and camera position changes. Benefitting by the gradual popularity of video data, VHP has a wide range of application potential, and the typical cases are intelligent monitoring and video editing.\n\n\n\\noindent\\textbf{Remark.} Over recent years, some potential research directions have also received attention, including weakly-supervised human parsing \\cite{fang2018weakly, li2020self, zhao2022from}, one-shot human parsing \\cite{he2021progressive, he2021end} and interactive human parsing \\cite{gao2022clicking}.\n\n\n\\vspace{-6pt}\n\\subsection{Relevant Tasks}\n\\label{sec:relevant}\nAmong the research in computer vision, there are some tasks with strong relevance to human parsing, which are briefly described in the following.\n\n\\noindent$\\bullet$~\\textbf{Pose Estimation.} The purpose of pose estimation is to locate human parts and build body representations (such as skeletons) from input data. \nHuman parsing and pose estimation share the same input space \\bm{$\\mathcal{X}$}, but there are some differences in the supervision targets. The most crucial difference is that human parsing is a dense prediction task, which needs to predict the category of each pixel. Meanwhile, pose estimation is a sparse prediction task, only focusing on the location of a limited number of keypoints. These two tasks are also often presented in multi-task learning, or one of them is used as a guiding condition for the other. For example, human parsing as a guide can help pose estimation to reduce the impact of clothing on human appearance \\cite{ladicky2013human}.\n\n\\noindent$\\bullet$~\\textbf{Image Segmentation.} Image segmentation is a fundamental topic in image processing and computer vision. It mainly includes semantic segmentation and instance segmentation.\nAs a basic visual task, there are many research directions can be regarded as branches, and human parsing is one of them. In the pre-deep learning era, image segmentation focuses on the continuity of color, texture, and edge, while human parsing pays more attention to the body topology modeling. In the deep learning era, the methods in two fields show more similarities. However, more and more human parsing literature choose to model the parts relationship as the goal, which is significantly different from the general goal of image segmentation. Therefore, human parsing and image segmentation are closely related but independent problems.\n\n\\noindent\\textbf{Remark.} Ordinarily, most human-centric dense prediction task show positively relevance with human parsing, \\emph{e.g}., human matting \\cite{chen2018semantic, liu2020boosting}, human mesh reconstruction \\cite{guler2019holopose, zheng2019deephuman} and face\/hand parsing \\cite{liang2014parsing, lin2019face}.\n\n\n\\begin{figure*}\n\t\\begin{center}\n\t\t\\includegraphics[width=0.95\\linewidth]{figures\/timeline.pdf}\n\t\\end{center}\n\t\\vspace{-5pt}\n\t\\captionsetup{font=small}\n\t\\caption{\\small{\\textbf{Timeline of representative human parsing works from 2012 to 2022.} The upper part represents the datasets of human parsing (\\S\\ref{sec:hp-data}), and the lower part represents the models of human parsing (\\S\\ref{sec:dl-base-hp}).}}\n\t\\label{fig:timeline}\n\t\\vspace{-10pt}\n\\end{figure*}\n\n\n\\vspace{-6pt}\n\\subsection{Applications of Human Parsing}\n\\label{sec:parsing-based}\nAs a crucial task in computer vision, there are a large number of applications based on human parsing. We will introduce some common ones below.\n\n\\noindent$\\bullet$~\\textbf{Dense Pose Estimation.} The goal of dense pose estimation is to map all human pixels in an RGB image to the 3D surface of the human body \\cite{guler2018densepose}. Human parsing is an important pre-condition that can constrain the mapping of dense points. At present, the mainstream dense pose estimation methods explicitly integrate human parsing supervision, such as DensePose R-CNN \\cite{guler2018densepose}, Parsing R-CNN \\cite{yang2019parsing}, \nand SimPose \\cite{zhu2020simpose}. Therefore, the performance of human parsing will directly affect dense pose estimation results.\n\n\\noindent$\\bullet$~\\textbf{Person Re-identification.} Person re-identification seeks to predict whether two images from different cameras belong to the same person. The apparent characteristics of human body is an important factor affecting the accuracy. Human parsing can provide pixel-level semantic information, helping re-identification models perceive the position and composition of human parts\/clothing. Various studies have introduced human parsing explicitly or implicitly into re-identification methods, which improves the model performance in multiple aspects, \\emph{e.g}., local visual cues \\cite{kalayeh2018human, yang2019towards}, spatial alignment \\cite{sun2019learning, huang2020improve, li2021person}, background-bias elimination \\cite{tian2018elimiating}, domain adaptation \\cite{chen2019instance}, clothes changing \\cite{yu2020cocas, qian2020long}.\n\n\\noindent$\\bullet$~\\textbf{Virtual Try-on.} Virtual try-on is a burgeoning and interesting application in the vision and graphic communities \\cite{han2018viton, wang2018toward, yu2019vtnfp, wu2019m2e, dong2019towards, liu2021toward, xie2021was, zhao2021m3d}. Most of the research follows the three processes: human parsing, appearance generation, and refinement. Therefore, human parsing is a necessary step to obtain clothing masks, appearance constraints and pose maintenance. Recently, some work began to study the parser-free virtual try-on \\cite{issenhuth2020do, chang2022pfvton, lin2022rmgn}. Through teacher-student learning, parsing-based pre-training, and other technologies, the virtual try-on can be realized without the human parsing map during inference. However, most of these works still introduced the parsing results during training, and the generation quality retains gap from parser-based methods.\n\n\\noindent$\\bullet$~\\textbf{Conditional Human Image Generation.} Image generation\/synthesis as a field has seen a lot of progress in recent years \\cite{goodfellow2014generative, karras2019style, niemeyer2021giraffe, nichol2021glide}. Non-existent but fidelity images can be created in large quantities. Among them, human image generation has attracted attention because of its rich downstream applications. Compared with unconditional generation, conditional generation can produce corresponding output as needed, and human parsing map is one of the most widely used pre-conditions. There have been a lot of excellent works on parsing-based conditional human image generation, \\emph{e.g}., CPFNet \\cite{wu2021image} and InsetGAN \\cite{fruhstuck2022insetgan}.\n\n\\noindent\\textbf{Remark.} Besides the above cases, in general, most of the human-centric generation applications can be built with the help of human parsing, \\emph{e.g}., deepfakes \\cite{chen2020simswap, yang2021attacks}, style transfer \\cite{liu2019swapgan, huo2021manifold, ma2022dual}, clothing editing \\cite{kim2019style, ntavelis2020sesame, tseng2020modeling}.\n\n\n\n\\section{Deep Learning Based Human Parsing}\n\\label{sec:dl-base-hp}\nThe existing human parsing can be categorized into three sub-tasks: single human parsing, multiple human parsing, and video human parsing, focusing on parts relationship modeling, human instance discrimination, and temporal correspondence learning, respectively. According to this taxonomy, we sort out the representative works (lower part of Figure~\\ref{fig:timeline}) and review them in detail below.\n\n\\vspace{-6pt}\n\\subsection{Single Human Parsing (SHP) Models}\n\\label{sec:shp-methods}\nSHP considers extracting human features through parts relationship modeling. According to the modeling strategy, SHP models can be divided into three main classes: context learning, structured representation, and multi-task learning. Moreover, considering some special but interesting methods, we will review them as ``other modeling models\". Table~\\ref{table:shp_taxonomy} summarizes the characteristics for reviewed SHP models.\n\n\\begin{table*}[htbp]\n\t\\centering\n\t\\captionsetup{font=small}\n\t\\caption{\\small{\\textbf{Summary of essential characteristics for reviewed SHP models} (\\S\\ref{sec:shp-methods}). The training datasets and whether it is open source are also listed. See \\S\\ref{sec:hp-data} for more detailed descriptions of datasets. These notes also apply to the other tables.}}\n\t\\label{table:shp_taxonomy}\n\t\\vspace{-5pt}\n\t\\begin{threeparttable}\n\t\\resizebox{0.88\\textwidth}{!}{\n\t\\setlength\\tabcolsep{2.5pt}\n\t\\renewcommand\\arraystretch{1.02}\n\t\\begin{tabular}{c|r|c||cc|cc|cc|cc||c||c}\n\t\\hline\\thickhline\n\t\\rowcolor{mygray1}\n\t                                  &                                        &                                    &  \\multicolumn{2}{c|}{Context} & \\multicolumn{2}{c|}{Structured} & \\multicolumn{2}{c|}{Multi-task} &          \\multicolumn{2}{c||}{Others}   &   & \\\\\n\t\\cline{4-5} \\cline{6-7} \\cline{8-9} \\cline{10-11}                                 \n\t\\rowcolor{mygray1}                                  \n\t\\multirow{-2}{*}{Year} & \\multirow{-2}{*}{Method} & \\multirow{-2}{*}{Pub.} &   Attention   &Scale-aware&      Tree     &     Graph      &      Edge      &      Pose      &  Denoising &  Adversarial  & \\multirow{-2}{*}{Datasets}   &  \\multirow{-2}{*}{Open Source}  \\\\\n\t\\hline\n\t\\hline\n\t\\multirow{1}{*}{{2012}} & Yamaguchi \\cite{yamaguchi2012parsing} &  CVPR  &    -         &       -             &      -              &           -         &        -            &\\checkmark   &       -             &        -            & FS    & - \\\\            \n\t\\hline\n\t\\multirow{3}{*}{{2013}} & DMPM \\cite{dong2013deformable} &  ICCV       &       -             &      -              &\\checkmark  &       -             &         -           &       -             &        -            &      -              & FS\/DP & - \\\\\n\t                                    & PaperDoll \\cite{yamaguchi2013paper}    & ICCV&     -               &       -             &      -              &        -            &      -              & \\checkmark  &    -                &     -               & FS & - \\\\\n\t                                    & CFPD \\cite{liu2013fashion}    &     TMM             &       -             &       -             &      -              &         -           &       -             & \\checkmark  &     -               &     -               & CFPD  & -\\\\\n\t\\hline\n\t\\multirow{1}{*}{{2014}} & HPM \\cite{dong2014towards} &  CVPR              &      -              &      -              & \\checkmark &          -          &          -          & \\checkmark &     -               &     -               & FS\/DP  & -\\\\\n\t\\hline\n\t\\multirow{4}{*}{{2015}} & M-CNN \\cite{liu2015matching} &  CVPR            &        -            &       -              &        -            & \\checkmark &       -             &       -             &  -                  &     -              & ATR & - \\\\\n\t                                    & Co-CNN \\cite{liang2015human} &  ICCV            &       -             & \\checkmark &          -          &           -         &         -           &         -           &         -           &        -            & FS\/ATR & - \\\\\n\t                                    & FPVC \\cite{liu2015fashion} &  TMM                    &      -              &       -             &       -              &         -           &       -             & \\checkmark &       -             &    -                & FS\/DP & - \\\\\n\t                                    & ATR \\cite{liang2015deep} &  TPAMI                   &        -            &        -            & \\checkmark &         -           &       -             &          -            &      -              &      -              & FS\/DP& -  \\\\  \n\t\\hline\n\t\\multirow{6}{*}{{2016}} & AOG \\cite{xia2016pose} &      AAAI                     &       -              &      -              & \\checkmark &         -          &   -                 & \\checkmark  &    -                &     -               & -    & - \\\\\n\t                                    & Attention \\cite{chen2016attenttion} &  CVPR      & \\checkmark &\\checkmark   &         -            &          -          &    -                &    -                &     -               &     -               & PPP   & \\checkmark\\\\     \n\t                                    & LG-LSTM \\cite{liang2016object} &      CVPR      &\\checkmark  &         -           &         -            &         -          &    -                &     -                 &    -                &   -                  &FS\/ATR\/PPP & - \\\\                        \n\t                                    & Graph-LSTM \\cite{liang2016semantic} & ECCV  &\\checkmark &          -          &         -            & \\checkmark&       -             &       -               &      -              &      -              & FS\/ATR\/PPP  & -\\\\     \n\t                                    & HAZN \\cite{xiq2016zoom} & ECCV                     &     -               & \\checkmark &          -           &        -           &      -              &     -                 &    -                &     -               & PPP  & -\\\\  \n\t                                    & SYSU-Clothes \\cite{liang2016clothes} &  TMM   &     -               &       -             &         -            &\\checkmark &       -             &     -                 &     -               &   -                 &  SYSU-Clothes  & -\\\\  \n\t\\hline\n\t\\multirow{3}{*}{{2017}} & Struc-LSTM \\cite{liang2017interpretable} &CVPR&\\checkmark&         -           & \\checkmark & \\checkmark&      -              &         -             &        -            &        -            & ATR\/PPP& -  \\\\ \n\t                                    & SSL \\cite{gong2017look}  &            CVPR           &     -               &       -             &         -             &    -               &-                    & \\checkmark  &    -                &    -                & LIP\/PPP  & \\checkmark\\\\ \n\t                                    & Joint \\cite{xia2017joint}    &            CVPR           &      -              &         -           &          -            &     -              &   -                 & \\checkmark  &     -               &      -              & PPP & - \\\\ \n\t                                   \n\t                                   \n\t                                   \n\t\\hline\n\t\\multirow{7}{*}{{2018}} & ProCNet \\cite{zhu2018progressive} &   AAAI      &      -               &        -           & \\checkmark  &         -          &       -              &        -            &      -              &     -               & PPP & - \\\\\n\t                                    & AFLA \\cite{liu2018cross} &               AAAI           &       -              &         -          &         -             &           -        &    -                 &      -              &    -                &\\checkmark & LIP  & -\\\\\n\t                                    & WSHP \\cite{fang2018weakly} &  CVPR              &        -              &        -            &        -             &        -            &   -                 & \\checkmark&      -              &      -               &PPP & - \\\\ \n\t                                    & TGPNet \\cite{luo2018trusted} & MM                   &        -              & \\checkmark &           -          &           -        &      -              &        -            &      -              &     -               & ATR & \\checkmark \\\\            \n\t                                   \n\t                                    & MuLA \\cite{nie2018mutual}   &          ECCV        &       -             &         -           &      -                &     -              &     -               & \\checkmark  &   -                 &       -             & LIP\/PPP & - \\\\ \n\t                                    & MMAN \\cite{luo2018macro}  &          ECCV        &       -             &         -           &       -               &       -            &     -               &                     &      -              & \\checkmark& LIP\/PPP\/PPSS & \\checkmark  \\\\ \n\t                                    & JPPNet \\cite{liang2018look} &   TPAMI               &       -             &         -           &       -               &         -          &      -              & \\checkmark &      -              &        -             &LIP\/PPP  & \\checkmark \\\\ \n\t\\hline\n\t\\multirow{6}{*}{{2019}} &   CE2P \\cite{ruan2019devil}  &          AAAI         &        -            & \\checkmark &         -            &         -           & \\checkmark &       -              &-                    &      -              & LIP  & \\checkmark\\\\ \n\t                                    & Graphonomy \\cite{gong2019graphonomy} &CVPR&   -             &       -             & \\checkmark &\\checkmark &       -             &           -           &   -                 &         -           & ATR\/PPP & \\checkmark  \\\\ \n\t                                    & CNIF \\cite{wang2019learning}  &       ICCV        &       -             &        -            & \\checkmark  &         -          &      -               &          -           &   -                 &         -           & ATR\/LIP\/PPP & \\checkmark \\\\                         \n\t                                    & BSANet \\cite{zhao2019multi} &        ICCV          &       -              & \\checkmark &       -              &        -           & \\checkmark &              -      &        -            &              -       &   PPP & -\\\\                                        \n\t                                    & SPGNet \\cite{cheng2019spgnet} &     ICCV       & \\checkmark  &           -          &       -              &          -         &        -            &              -      &       -             &        -             &   PPP & -\\\\    \n\t                                    & BraidNet \\cite{liu2019braidnet}     &       MM       &        -             & \\checkmark &          -           &             -       &        -            &                -    &         -           &     -             & LIP  & -\\\\  \n\t                                   \n\t                                   \n\t\\hline\n\t\\multirow{11}{*}{{2020}} & Grapy-ML \\cite{he2020grapyml} &     AAAI      & \\checkmark  &        -             & \\checkmark &\\checkmark &  -                  &          -           &         -           &        -             & ATR\/PPP & \\checkmark  \\\\  \n\t                                    & HHP \\cite{wang2020hierarchical}  &     CVPR    &        -            &      -              & \\checkmark  &\\checkmark &   -                  &       -              &        -            &       -             & ATR\/LIP\/PPP\/PPSS & \\checkmark \\\\\n\t                                    & SLRS \\cite{li2020self}     &            CVPR           &         -            &      -              &        -              &\\checkmark& \\checkmark &        -              & \\checkmark &        -             & ATR\/LIP  & - \\\\ \n\t                                    & PCNet \\cite{zhang2020pcnet}     &    CVPR      &         -            & \\checkmark &          -            &\\checkmark &       -             &         -             &            -        &       -              & LIP\/PPP   & -\\\\  \n\t                                    & CorrPM \\cite{zhang2020correlating}  &  CVPR   &       -            &       -             &       -               &      -              & \\checkmark & \\checkmark  &         -           &      -              & ATR\/LIP  & \\checkmark \\\\   \n\t                                    & DTCF \\cite{liu2020hybrid} &           MM              &       -              & \\checkmark &         -            &        -           & \\checkmark &         -             &          -          &       -              &  LIP\/PPP & - \\\\    \n\t                                   \n\t                                    & SemaTree \\cite{ji2020learning} &    ECCV         & \\checkmark &       -             & \\checkmark  &         -          &        -             &              -        &        -            &         -            &   LIP & \\checkmark  \\\\                   \n\t                                    & OCR \\cite{yuan2020object} &        ECCV           & \\checkmark & \\checkmark &            -         &        -           &        -             &           -           &        -            &         -            &   LIP  & \\checkmark \\\\          \n\t                                    & BGNet \\cite{zhang2020blended} &       ECCV    &         -            &        -            & \\checkmark  &\\checkmark&      -               &             -         &        -            &          -           &   LIP\/PPP\/PPSS  & -\\\\        \n\t                                   \n\t                                   \n\t                                   \n\t                                   \n\t                                   \n\t                                   \n                                       \t \n\t                                    & HRNet \\cite{wang2020deep}  &       TPAMI        &        -             & \\checkmark &          -           &       -            &         -            &         -             &          -          &          -           &  LIP  & \\checkmark\\\\        \n\t                                    & SCHP \\cite{li2020correction}    &     TPAMI        &       -              &          -          &        -             &         -           &\\checkmark  &         -             & \\checkmark &          -           & \\tabincell{c}{ATR\/LIP\/PPP}  & \\checkmark \\\\ \n        \\hline\n\t\\multirow{7}{*}{{2021}} & HIPN \\cite{liu2021hier} &     AAAI                    &        -              &       -            &         -             &         -          &         -           &          -             & \\checkmark &       -              & LIP\/PPP & - \\\\  \n\t                                    & POPNet \\cite{he2021progressive}  &  AAAI       &\\checkmark   &        -            &         -             &         -           &        -            &        -             &       -             &       -             & ATR-OS & \\checkmark  \\\\\n\t                                    & MCIBI \\cite{jin2021mining}     &        ICCV         &\\checkmark   &        -             &        -             &        -            &         -          &         -             &        -            &       -              & LIP & \\checkmark  \\\\ \n\t                                    & ISNet \\cite{jin2021isnet}     &          ICCV           &\\checkmark   &        -             &         -            &        -            &        -           &         -             &        -            &      -               & LIP & \\checkmark  \\\\ \n\t                                    & NPPNet \\cite{zeng2021neural}     &    ICCV      &         -            & \\checkmark  &         -            &        -            &      -             & \\checkmark   &           -         &        -             & LIP\/PPP  & \\checkmark \\\\ \n\t                                   \n\t                                   \n\t                                   \n\t                                   \n\t                                   \n\t                                   \n\t                                    & HTCorrM \\cite{zhang2021on}  &        TPAMI     & \\checkmark &         -            &            -         &           -        & \\checkmark & \\checkmark    &         -           &          -           &   ATR\/LIP & - \\\\   \n\t                                    & PRHP \\cite{wang2021hierarchical}  &     TPAMI&        -            &       -             & \\checkmark  &\\checkmark &         -            &         -              &      -              &        -            & ATR\/LIP\/PPP\/PPSS & \\checkmark \\\\\n\t                                   \n\t                                   \n\t\\hline\n\t\\multirow{4}{*}{{2022}} & CDGNet \\cite{liu2022cdgnet}    &  CVPR            & \\checkmark  &        -            &           -           &        -            & \\checkmark &          -         &            -        &     -               & ATR\/LIP  & \\checkmark \\\\       \n\t                                    & HSSN \\cite{li2022deep}             &  CVPR            &         -            &  \\checkmark &\\checkmark   &        -            &       -             &          -         &            -        &     -               & LIP\/PPP  & \\checkmark \\\\     \n\t                                   \n\t                                    & PRM \\cite{zhang2022human} &  TMM                &          -          &        -            &           -           &\\checkmark &         -           &          -            &         -           &             -       & LIP\/PPP& -  \\\\\n\t                                   \n\t                                    & PADNet \\cite{zhao2022from}  &        TPAMI     &           -           &        -             &          -           &          -         &     -                 & \\checkmark  &   -                 &        -             &   PPP & - \\\\                                                                     \n\t                                                           \n\t\\hline                                    \n\t\\end{tabular}\n\t}\n\t\\end{threeparttable}\n\t\\vspace{-5pt}\n\\end{table*}\n\n\\vspace{-6pt}\n\\subsubsection{Context Learning}\nContext learning, a mainstream paradigm for single human parsing, seeks to learn the connection between local and global features to model human parts relationship. Recent studies have developed various context learning methods to handle single human parsing, including attention mechanism and scale-aware features. \n\n\\noindent$\\bullet$~\\textbf{Attention Mechanism.} The first initiative was proposed in \\cite{chen2016attenttion} that applies an attention mechanism for parts relationship modeling. Specifically, soft weights, learned by attention mechanism, are used to weight different scale features and merge them. At almost the same time, LG-LSTM \\cite{liang2016object}, Graph-LSTM \\cite{liang2016semantic} and Struc-LSTM \\cite{liang2017interpretable} exploit complex local and global context information through Long Short-Term Memory (LSTM) \\cite{hochreiter1997long} and achieve very competitive results. Then, \\cite{cheng2019spgnet} proposes a Semantic Prediction Guidance (SPG) module that learns to re-weight the local features through the guidance from pixel-wise semantic prediction. With the rise of graph model, researchers realized that attention mechanism is able to establish the correlation between graph model nodes. For example, \\cite{he2020grapyml} introduces Graph Pyramid Mutual Learning (Grapy-ML) to address the cross-dataset human parsing problem, in which the self-attention is used to model the correlations between context nodes. Although attention mechanisms have achieved great results in previous work, global context dependency cannot be fully understood due to the lack of explicit prior supervision. CDGNet \\cite{liu2022cdgnet} adopts the human parsing labels accumulated in the horizontal and vertical directions as the supervisions, aiming to learn the position distribution of human parts, and weighting them to the global features through attention mechanism to achieve accurate parts relationship modeling.\n\n\\noindent$\\bullet$~\\textbf{Scale-aware Features.} The most intuitive context learning method is to directly use scale-aware features (\\emph{e.g}. multi-scale features \\cite{zhao2017pyramid, chen2017deeplab}, features pyramid networks \\cite{lin2017feature, kirillov2019pfpn}), which has been widely verified in semantic segmentation \\cite{minaee2021image}. The earliest effort can be tracked back to CoCNN \\cite{liang2015human}. It integrates cross layer context, global image-level context, super-pixel context, and cross super-pixel neighborhood context into a unified architecture, which solves the obstacle of low-resolution features in FCN \\cite{shelhamer2016fully} for modeling parts relationship. Subsequently, \\cite{xiq2016zoom} proposes Hierarchical Auto-Zoom Net (HAZN), which adaptively zooms predicted image regions into their proper scales to refine the parsing. TGPNet \\cite{luo2018trusted} considers that the label fragmentation and complex annotation in human parsing datasets is a non-negligible problem to hinder accurate parts relationship modeling, \ntrying to alleviate this limitation by supervising multi-scale context information. PCNet \\cite{zhang2020pcnet} further studies the adaptive contextual features, and captures the representative global context by mining the associated semantics of human parts through proposed part class module, relational aggregation module, and relational dispersion module. \n\n\n\\vspace{-6pt}\n\\subsubsection{Structured Representation}\nThe purpose of structured representation is to learn the inherent combination or decomposition mode of human parts, so as to model parts relationship. Research efforts in this field are mainly made along two directions: using a tree structure to represent the hierarchical relationship between body and parts, and using a graph structure to represent the connectivity relationship between different parts. These two ideas are complementary to each other, so they have often been adopted simultaneously in some recent work.\n\n\\noindent$\\bullet$~\\textbf{Tree Structure.} DMPM \\cite{dong2013deformable} and HPM \\cite{dong2014towards} solve the single human parsing issue by using the parselets representation, which construct a group of parsable segments by low-level over-segmentation algorithms, and represent these segments as leaf nodes, then search for the best graph configuration to obtain semantic human parsing results. Similarly, \\cite{liang2015deep} formulates human parsing as an Active Template Regression (ATR) problem, where each human part is represented as the linear combination of learned mask templates and morphed to a more precise mask with the active shape parameters. Then the human parsing results are generated from the mask template coefficients and the active shape parameters. In the same line of work, ProCNet \\cite{zhu2018progressive} deals with human parsing as a progressive recognition task, modeling structured parts relationship by locating the whole body and then segmenting hierarchical components gradually. CNIF \\cite{wang2019learning} further extends the human tree structure and represents human body as a hierarchy of multi-level semantic parts, treating human parsing as a multi-source information fusion process. A more efficient solution is developed in \\cite{ji2020learning}, which uses a tree structure to encode human physiological composition, then designs a coarse to fine process in a cascade manner to generate accurate parsing results. \n\n\\noindent$\\bullet$~\\textbf{Graph Structure.} Graph structure is an excellent relationship modeling method. Some researchers consider introducing it into human parsing networks for part-relation reasoning. A clothing co-parsing system is designed by \\cite{liang2016clothes}, which takes the segmented regions as the vertices. It incorporates several contexts of clothing configuration to build a multi-image graphical model. To address the cross-dataset human parsing problem, Graphonomy \\cite{gong2019graphonomy} proposes a universal human parsing agent, introducing hierarchical graph transfer learning to encode the underlying label semantic elements and propagate relevant semantic information. BGNet \\cite{zhang2020blended} hopes to improve the accuracy of human parsing in similar or cluttered scenes through graph structure. It exploits the human inherent hierarchical structure and the relationship between different human parts employing grammar rules in both cascaded and paralleled manner to correct the segmentation performance of easily confused human parts. A landmark work on this line was proposed by Wang \\emph{et al}.\\cite{wang2020hierarchical, wang2021hierarchical}. A hierarchical human parser (HHP) is constructed, representing the hierarchical human structure by three kinds of part relations: decomposition, composition, and dependency. Besides, HHP uses the prism of a message-passing, feed-back inference scheme to reason the human structure effectively. Following this idea, \\cite{zhang2022human} proposes Part-aware Relation Modeling (PRM) to handle human parsing, generating features with adaptive context for various sizes and shapes of human parts.\n\n\n\\vspace{-6pt}\n\\subsubsection{Multi-task Learning}\nThe auxiliary supervisions can help the parser better understand the relationship between parts, such as part edges or human pose. Therefore, multi-task learning has become an essential paradigm for single human parsing.\n\n\\noindent$\\bullet$~\\textbf{Edge-aware Learning.} Edge information is implicit in the human parsing dataset. Thus edge-aware supervision or feature can be introduced into the human parser without additional labeling costs. In particular, edge-aware learning can enhance the model's ability to discriminate adjacent parts and improve the fineness of part boundaries. The typical work is \\cite{ruan2019devil}, which proposes a Context Embedding with Edge Perceiving (CE2P) framework, using an edge perceiving module to integrate the characteristic of object contour to refine the part boundaries. Because of its excellent performance and scalability, CE2P has become the baseline for many subsequent works. CorrPM \\cite{zhang2020correlating} and HTCorrM \\cite{zhang2021on}  are built on CE2P, and further use part edges to help model the parts relationship. They construct a heterogeneous non-local module to mix the edge, pose and semantic features into a hybrid representation, and explore the spatial affinity between the hybrid representation and the parsing feature map at all positions. BSANet \\cite{zhao2019multi} considers that edge information is helpful to eliminate the part-level ambiguities and proposes a joint parsing framework with boundary and semantic awareness to address this issue. Specifically, a boundary-aware module is employed to make intermediate-level features focus on part boundaries for accurate localization, which is then fused with high-level features for efficient part recognition. To further enrich the edge-aware features, a dual-task cascaded framework (DTCF) is developed in \\cite{liu2020hybrid}, which implicitly integrates parsing and edge features to refine the human parsing results progressively. \n\n\\noindent$\\bullet$~\\textbf{Pose-aware Learning.} Both human parsing and pose estimation seek to predict dense and structured human representation. There is a high intrinsic relationship between them. Therefore, some studies have tried to use pose-aware learning to assist in parts relationship modeling. As early as 2012, Yamaguchi \\emph{et al}. \\cite{yamaguchi2012parsing, yamaguchi2013paper} exploited the relationship between clothing and the underlying body pose, exploring techniques to accurately parse person wearing clothing into their constituent garment pieces. Almost immediately, Liu \\emph{et al}. \\cite{liu2013fashion} combined the human pose estimation module with an MRF-based color\/category inference module and a super-pixel category classifier module to parse fashion items in images. Subsequently, Liu \\emph{et al}. \\cite{liu2015fashion} extends this idea to semi-supervised human parsing, collecting a large number of unlabeled videos, using cross-frame context for human pose co-estimation, and then performs video joint human parsing. SSL \\cite{gong2017look} and JPPNet \\cite{liang2018look} choose to impose human pose structures into parsing results without resorting to extra supervision, and adopt the multi-task learning manner to explore efficient human parts relationship modeling. A similar work is developed by \\cite{nie2018mutual}, which presents a Mutual Learning to Adapt model (MuLA) for joint human parsing and pose estimation. MuLA can fast adjust the parsing and pose models to provide more robust and accurate results by incorporating information from corresponding models. Different from the above work, Zeng \\emph{et al}. \\cite{zeng2021neural}. focus on how to automatically design a unified model and perform two tasks simultaneously to benefit each other. Inspired by NAS \\cite{fang2020densely}, they propose to search for an efficient network architecture (NPPNet), searching the encoder-decoder architectures respectively, and embed NAS units in both multi-scale feature interaction and high-level feature fusion. To get rid of annotating pixel-wise human parts masks, a weakly-supervised human parsing approach is proposed by PADNet \\cite{zhao2022from}. They develop an iterative training framework to transform pose knowledge into part priors, so that only pose annotations are required during training, greatly alleviating the annotation burdens.\n\n\\begin{table}\n\t\\centering\n\t\\captionsetup{font=small}\n\t\\caption{\\small{\\textbf{Highlights of parts relationship modeling methods for SHP models} (\\S\\ref{sec:shp-methods}). Representative Works of each method are also give.}}\n\t\\label{table:shp_highlights}\n\t\\vspace{-5pt}\n\t\\begin{threeparttable}\n\t\\resizebox{0.49\\textwidth}{!}{\n\t\\setlength\\tabcolsep{2.5pt}\n\t\\renewcommand\\arraystretch{1.02}\n\t\\begin{tabular}{c|c|c}\n\t\\hline\\thickhline\n\t\\rowcolor{mygray1}\n\tMethod                & \\tabincell{c}{Representative\\\\ Works} & Highlights \\\\\n\t\\hline\n\t\\hline\n         Attention              & \\tabincell{c}{\\cite{chen2016attenttion, liang2016object, liang2016semantic}\\\\ \\cite{liang2017interpretable, cheng2019spgnet, he2020grapyml}\\\\ \\cite{ liu2022cdgnet}}    & \\tabincell{c}{It is helpful to locate interested human parts,\\\\ suppress useless background information.} \\\\\n         \\hline\n\tScale-aware         & \\tabincell{c}{\\cite{liang2015human, xiq2016zoom, luo2018trusted}\\\\ \\cite{zhang2020pcnet}}   & \\tabincell{c}{Fusion low-level texture and high-level semantic\\\\ features, help to parse small human parts.} \\\\\n\t\\hline\n\t\\hline\n\tTree                     & \\tabincell{c}{\\cite{dong2013deformable, dong2014towards, liang2015deep}\\\\ \\cite{zhu2018progressive, wang2019learning, ji2020learning}}    & \\tabincell{c}{Simulate the composition and decomposition\\\\ relationship between human parts and body.} \\\\\n         \\hline\n\tGraph                  & \\tabincell{c}{\\cite{liang2016clothes, gong2019graphonomy, zhang2020blended}\\\\ \\cite{wang2020hierarchical, wang2021hierarchical, zhang2022human}}   & \\tabincell{c}{Modeling the correlation and difference between\\\\ human parts.} \\\\\n\t\\hline\n\t\\hline\n\tEdge                     & \\tabincell{c}{\\cite{ruan2019devil, zhang2020correlating, zhang2021on}\\\\ \\cite{zhao2019multi, liu2020hybrid}}  & \\tabincell{c}{Solve the pixel confusion problem on the boundary\\\\ of adjacent parts, generating finer boundary.} \\\\\n         \\hline\n\tPose                     & \\tabincell{c}{\\cite{yamaguchi2012parsing, yamaguchi2013paper, liu2013fashion}\\\\ \\cite{liu2015fashion, gong2017look, liang2018look}\\\\ \\cite{nie2018mutual, zeng2021neural, zhao2022from}}   & \\tabincell{c}{As context clues to improve semantic consistency\\\\ between parsing results and body structure.} \\\\\n\t\\hline\n\t\\hline\n\tDenoising             & \\tabincell{c}{\\cite{li2020self, li2020correction, liu2021hier}}   & \\tabincell{c}{Alleviate the impact of super-pixel or annotation\\\\ errors, improving the robustness.} \\\\\n         \\hline\n\tAdversarial           & \\tabincell{c}{\\cite{liu2018cross, luo2018macro}}   & \\tabincell{c}{Reduce the domain differences between training\\\\ data and testing data, improving the generalization.} \\\\\n         \\hline\n\t\\end{tabular}\n\t}\n\t\\end{threeparttable}\n\t\\vspace{-5pt}\n\\end{table}\n\n\\vspace{-6pt}\n\\subsubsection{Other Modeling Models}\nOther works attempt to employ techniques outside of the above taxonomy, such as denoising and adversarial learning, which also make specific contributions to the human parts relationship modeling and deserve a separate look.\n\n\\noindent$\\bullet$~\\textbf{Denoising.} To reduce the labeling cost, there is a large amount of noise in the mainstream SHP datasets \\cite{liang2015deep, gong2017look}, so denoising learning for accurate human parts relationship modeling has also received some attention. SCHP \\cite{li2020correction} is the most representative work. It starts with using inaccurate parsing labels as the initialization and designs a cyclically learning scheduler to infer more reliable pseudo labels\nIn the same period, Li \\emph{et al}. \\cite{li2020self} attempt to combine denoising learning and semi-supervised learning, proposing Self-Learning with Rectification (SLR) strategy for human parsing. SLR generates pseudo labels for unlabeled data to retrain the parsing model and introduces a trainable graph reasoning method to correct typical errors in pseudo labels. Based on SLR, HIPN \\cite{liu2021hier} further explores to combine denoising learning with semi-supervised learning, which develops the noise-tolerant hybrid learning, taking advantage of positive and negative learning to better handle noisy pseudo labels.\n\n\\noindent$\\bullet$~\\textbf{Adversarial Learning.} Earlier, inspired by the Generative Adversarial Nets (GAN) \\cite{goodfellow2014generative}, a few works use adversarial learning to solve problems in parts relationship modeling. For example, to solve the domain adaptation problem, AFLA \\cite{liu2018cross} proposes a cross-domain human parsing network, introducing a discriminative feature adversarial network and a structured label adversarial network to eliminate cross-domain differences in visual appearance and environment conditions. MMAN \\cite{luo2018macro} hopes to solve the problem of low-level local and high-level semantic inconsistency in pixel-wise classification loss. It contains two discriminators: Macro D, acting on low-resolution label map and penalizing semantic inconsistency; Micro D, focusing on high-resolution label map and restraining local inconsistency.\n\n\\noindent\\textbf{Remark.} In fact, many single human parsing models use a variety of parts relationship modeling methods. Therefore, our above taxonomy only introduces the core methods of each model. Table~\\ref{table:shp_highlights} summarizes the highlights of each parts relationship modeling method.\n\n\n\\vspace{-6pt}\n\\subsection{Multiple Human Parsing (MHP) Models}\n\\label{sec:mhp-methods}\nMHP seeks to locate and parse each human in the image plane. The task setting is similar to instance segmentation, so it is also called instance-level human parsing. We divide MHP into three paradigms: bottom-up, one-stage top-down, and two-stage top-down, according to its pipeline of discriminating human instances. The essential characteristics of reviewed MHP models are illustrated in Table~\\ref{table:mhp_taxonomy}.\n\n\n\\begin{table}\n\t\\centering\n\t\\captionsetup{font=small}\n\t\\caption{\\small{\\textbf{Summary of essential characteristics for reviewed MHP models} (\\S\\ref{sec:mhp-methods}). ``BU\" indicates bottom-up; ``1S-TD\" indicates one-stage top-down; ``2S-TD\" indicates two-stage top-down.}}\n\t\\label{table:mhp_taxonomy}\n\t\\vspace{-5pt}\n\t\\begin{threeparttable}\n\t\\resizebox{0.49\\textwidth}{!}{\n\t\\setlength\\tabcolsep{2.5pt}\n\t\\renewcommand\\arraystretch{1.02}\n\t\\begin{tabular}{c|r|c||c||c||c}\n\t\\hline\\thickhline\n\t\\rowcolor{mygray1}\n\tYear & Method & Pub. & Pipeline &  Datasets   & Open Source\\\\\n\t\\hline\n\t\\hline\n\t\\multirow{1}{*}{{2017}} & Holistic \\cite{li2017holistic} & BMVC & 1S-TD & PPP &  -   \\\\\n\t\\hline\n\t\\multirow{1}{*}{{2018}} & PGN \\cite{gong2018instance} & ECCV & BU & PPP\/CIHP &  \\checkmark \\\\\n\t\\hline\n\t\\multirow{4}{*}{{2019}} & CE2P \\cite{ruan2019devil} & AAAI & 2S-TD & CIHP\/MHP-v2.0 &  \\checkmark \\\\\n\t\t                            & Parsing R-CNN \\cite{yang2019parsing} & CVPR & 1S-TD & CIHP\/MHP-v2.0 &  \\checkmark  \\\\\n\t\t                            & BraidNet \\cite{liu2019braidnet} & MM & 2S-TD & CIHP &    -  \\\\\n\t\t                            & Unified \\cite{qin2019top} & BMVC & 1S-TD & PPP\/CIHP &  -    \\\\\n\t\\hline\t                            \n\t\\multirow{5}{*}{{2020}} & RP R-CNN \\cite{yang2020renovating} & ECCV & 1S-TD & CIHP\/MHP-v2.0 &  \\checkmark  \\\\\n\t\t                            & SemaTree \\cite{ji2020learning} & ECCV & 2S-TD & CIHP\/MHP-v2.0 &  \\checkmark \\\\\n\t\t\t                   \n\t\t\t                    & NAN \\cite{zhao2020fine} & IJCV & BU & MHP-v1.0\/MHP-v2.0 &  \\checkmark  \\\\\n\t\t                            & SCHP \\cite{li2020correction} & TPAMI & 2S-TD & CIHP\/MHP-v2.0\/VIP &  \\checkmark \\\\\n\t\\hline\n        \\multirow{1}{*}{{2021}}\n                                             & MGHR \\cite{zhou2021differentiable} & CVPR & BU & \\tabincell{c}{PPP\/MHP-v2.0\\\\\/COCO-DP} &  \\checkmark   \\\\\n\t\t                           \n\t\\hline\n\t\\multirow{1}{*}{{2022}} & AIParsing \\cite{zhang2022aiparsing} & TIP & 1S-TD & CIHP\/MHP-v2.0\/VIP & - \\\\\n\t\t                           \n\t\t                           \n\t\n\t\\hline\n\t\\end{tabular}\n\t}\n\t\\end{threeparttable}\n\t\\vspace{-5pt}\n\\end{table}\n\n\\noindent$\\bullet$~\\textbf{Bottom-up.} Bottom-up paradigm regards multiple human parsing as a fine-grained semantic segmentation task, which predicts the category of each pixel and grouping them into corresponding human instance. In a seminal work \\cite{gong2018instance}, Gong \\emph{et al}. propose a detection-free Part Grouping Network (PGN) that reformulates multiple human parsing as two twinned sub-tasks (semantic part segmentation and instance-aware edge detection) that can be jointly learned and mutually refined via a unified network. Among them, instance-aware edge detection task can group semantic parts into distinct human instances. Then, NAN \\cite{zhao2020fine} proposes a deep Nested Adversarial Network for multiple human parsing. NAN consists of three GAN-like sub-nets, performing semantic saliency prediction, instance-agnostic parsing, and instance-aware clustering, respectively. Recently, Zhou \\emph{et al}. \\cite{zhou2021differentiable} propose a new bottom-up regime to learn category-level multiple human parsing as well as pose estimation in a joint and end-to-end manner, called Multi-Granularity Human Representation (MGHR) learning. MGHR exploits structural information over different human granularities, transforming the difficult pixel grouping problem into an easier multi human joint assembling task to simplify the difficulty of human instances discrimination.\n\n\\noindent$\\bullet$~\\textbf{One-stage Top-down.} One-stage top-down is the mainstream paradigm of multiple human parsing. It first locates each human instance in the image plane, then segments each human part in an end-to-end manner. An early attempt is Holistic \\cite{li2017holistic}, which consists of a human detection network and a part semantic segmentation network, then passing the results of both networks to an instance CRF \\cite{kirfel2014human} to perform multiple human parsing. Inspired by Mask R-CNN \\cite{he2017mask}, Qin \\emph{et al}. \\cite{qin2019top} propose a top-down unified framework that simultaneously performs human detection and single human parsing, identifying instances and parsing human parts in crowded scenes. A milestone one-stage top-down multiple human parsing model is proposed by Yang \\emph{et al}., that enhances Mask R-CNN in all aspects, and proposes Parsing R-CNN \\cite{yang2019parsing} network, greatly improving the accuracy of multiple human parsing concisely. Subsequently, Yang \\emph{et al}. propose an improved version of Parsing R-CNN, called RP R-CNN \\cite{yang2020renovating}, which introduces a global semantic enhanced feature pyramid network and a parsing re-scoring network into the high-performance pipeline, achieving better performance. Later, AIParsing \\cite{zhang2022aiparsing} introduces the anchor-free detector \\cite{tian2020fcos} into the one-stage top-down paradigm for discriminating human instances, avoiding the hyper-parameters  sensitivity caused by anchors. \n\n\\noindent$\\bullet$~\\textbf{Two-stage Top-down.} One-stage top-down and two-stage top-down paradigms are basically the same in operation flow. The difference between them is whether the detector is trained together with the segmentation sub-network in an end-to-end manner. All the two-stage bottom-up multiple human parsing methods consist of a human detector and a single human parser. The earliest attempt is CE2P \\cite{ruan2019devil}, which designs a framework called M-CE2P on CE2P and Mask R-CNN, cropping the detected human instances, then sending them to the single human parser, finally combining the parsing results of all instances into a multiple human parsing prediction. Subsequent works, \\emph{e.g}., BraidNet \\cite{liu2019braidnet}, SemaTree \\cite{ji2020learning}, and SCHP \\cite{li2020correction}, basically inherit this pipeline.\n\n\\noindent\\textbf{Remark.} The advantage of bottom-up and one-stage top-down is efficiency, and the advantage of two-stage top-down is accuracy. But as a non-end-to-end pipeline, the inference speed of two-stage top-down is positively correlated with the number of human instances, which also limits its practical application value. The detailed highlights of three human instances discrimination methods are summarized in Table~\\ref{table:mhp_highlights}.\n\n\n\\begin{table}\n\t\\centering\n\t\\captionsetup{font=small}\n\t\\caption{\\small{\\textbf{Highlights of human instances discrimination methods for MHP models} (\\S\\ref{sec:mhp-methods}). Representative Works of each method are also give.}}\n\t\\label{table:mhp_highlights}\n\t\\vspace{-5pt}\n\t\\begin{threeparttable}\n\t\\resizebox{0.49\\textwidth}{!}{\n\t\\setlength\\tabcolsep{2.5pt}\n\t\\renewcommand\\arraystretch{1.02}\n\t\\begin{tabular}{c|c|c}\n\t\\hline\\thickhline\n\t\\rowcolor{mygray1}\n\tMethod                & \\tabincell{c}{Representative\\\\ Works} & Highlights \\\\\n\t\\hline\n\t\\hline\n         Bottom-up              & \\tabincell{c}{\\cite{gong2018instance, zhao2020fine, zhou2021differentiable}}    & \\tabincell{c}{Good model efficiency, good accuracy on pixel-wise\\\\ segmentation, and poor accuracy on instances\\\\ discrimination.} \\\\\n         \\hline\n\t\\tabincell{c}{One-stage\\\\ Top-down}         & \\tabincell{c}{\\cite{li2017holistic, yang2019parsing, qin2019top}\\\\ \\cite{yang2020renovating, zhang2022aiparsing}}   & \\tabincell{c}{Better trade-off between model efficiency and accuracy. \\\\But pixel-wise segmentation, especially the part\\\\ boundary is not fine enough.} \\\\\n\t\\hline\n\t\\tabincell{c}{Two-stage\\\\ Top-down}                     & \\tabincell{c}{\\cite{ruan2019devil, liu2019braidnet, ji2020learning}\\\\ \\cite{li2020correction}}    & \\tabincell{c}{Good accuracy and poor efficiency, the model inference\\\\ time is proportional to human instances number.} \\\\\n         \\hline\n\t\\end{tabular}\n\t}\n\t\\end{threeparttable}\n\t\\vspace{-5pt}\n\\end{table}\n\n\n\n\\vspace{-6pt}\n\\subsection{Video Human Parsing (VHP) Models}\n\\label{sec:vhp-methods}\nExisting VHP studies mainly focus to propagate the first frame into the entire video by the affinity matrix, which represents the temporal correspondences learnt from raw video data. Considering the unsupervised learning paradigms, we can group them into three classes: cycle-tracking, reconstructive learning, and contrastive learning. We summarize the essential characteristics of reviewed VHP models in Table~\\ref{table:vhp_taxonomy}.\n\n\\begin{table}\n\t\\centering\n\t\\captionsetup{font=small}\n\t\\caption{\\small{\\textbf{Summary of essential characteristics for reviewed VHP models} (\\S\\ref{sec:vhp-methods}). ``Cycle.\" indicates cycle-tracking; ``Recons.\" indicates reconstructive learning; ``Contra.\" indicates contrastive learning. All models are test on the VIP dataset.}}\n\t\\label{table:vhp_taxonomy}\n\t\\vspace{-5pt}\n\t\\begin{threeparttable}\n\t\\resizebox{0.49\\textwidth}{!}{\n\t\\setlength\\tabcolsep{2.5pt}\n\t\\renewcommand\\arraystretch{1.02}\n\t\\begin{tabular}{c|r|c||ccc||c}\n\t\\hline\\thickhline\n\t\\rowcolor{mygray1}\n\tYear & Method & Pub.  & \\tabincell{c}{Cycle.} & \\tabincell{c}{Recons.}  & \\tabincell{c}{Contra.}    & Open Source \\\\\n\t\\hline\n\t\\hline\n\t\\multirow{1}{*}{{2018}} & ATEN \\cite{zhou2018adaptive} & MM &  \\checkmark  &  -  &   -  &  \\checkmark   \\\\\n\n         \\hline\t                         \n\t\\multirow{2}{*}{{2019}} & TimeCycle \\cite{wang2019corres} & CVPR &  \\checkmark   &  - &  -&  \\checkmark   \\\\\n\t\t                            & UVC \\cite{li2019joint} & NeurIPS & \\checkmark  & \\checkmark &- &  \\checkmark    \\\\\n\t\\hline\n\t\\multirow{1}{*}{{2020}} & CRW \\cite{jabri2020space} & NeurIPS & \\checkmark & -& -&  \\checkmark  \\\\\n\t\\hline\n\t\\multirow{4}{*}{{2021}} & ContrastCorr \\cite{wang2021contrasive} & AAAI & \\checkmark & \\checkmark & & \\checkmark    \\\\\n\t                                    & CLTC \\cite{jeon2021mining} & CVPR & - & - &\\checkmark &  -   \\\\\n\t\t                            & VFS \\cite{xu2021rethinking} & ICCV &  -&  -& \\checkmark &  \\checkmark    \\\\\n\t\t                            & JSTG \\cite{zhao2021modelling} & ICCV & \\checkmark    &  -  & \\checkmark & -   \\\\\n\t\\hline\n\t\\multirow{3}{*}{{2022}} & LIIR \\cite{li2022locality} & CVPR &    -    & \\checkmark  &    -   &  \\checkmark   \\\\\n\t                                     & SCC \\cite{son2022contrastive} & CVPR &  \\checkmark  &   -   & \\checkmark & -    \\\\\n\t\t                            & UVC+ \\cite{mckee2022transfer} & ArXiv &  \\checkmark & \\checkmark & \\checkmark &  -    \\\\\n\t\\hline\n\t\\end{tabular}\n\t}\n\t\\end{threeparttable}\n\\end{table}\n\n\\noindent$\\bullet$~\\textbf{Cycle-tracking.}  Early VHP methods model the unsupervised learning target mainly by the cycle-consistency of video frames, \\emph{i.e.}, pixels\/patches are expected to fall into the same locations after a cycle of forward-backward tracking. ATEN\\cite{zhou2018adaptive} first leverages convolutional gated recurrent units to encode temporal feature-level changes, optical flow of non-key frames is wrapped with the temporal memory to generate their features. TimeCycle \\cite{wang2019corres} tracks the reference patch backward-forward in the video. The reference and the tracked patch at the end of the tracking cycle are considered to be consistent both in spatial coordinates and feature representation. Meanwhile, UVC \\cite{li2019joint} performs the region-level tracking and pixel-level corresponding with a shared affinity matrix, the tracked patch feature and the region-corresponding sub-affinity matrix are used to reconstruct the reference patch. Roles of the target and reference patches are then switched to regularizing the affinity matrix as orthogonal, which satisfies the cycle-consistency constraint. Its later version, UVC+ \\cite{mckee2022transfer} combines features learned by image-based tasks with video-based counterparts to further boost the performance. Lately, CRW \\cite{jabri2020space} represents video as a graph, where nodes are patches and edges are affinities between nodes in adjacent frames. A cross-entropy loss guides a graph walk to track the initial node bi-directionally in feature space, which is considered the target node after a bunch of cycle paths. However, the cycle-consistency in \\cite{wang2019corres}, \\cite{jabri2020space} strictly assumes that the target patch preserves visible in consecutive frames. Once it is occluded or disappears, the correspondences will be incorrectly assigned, thus leaving an optimal transport problem between video frames.\n\n\\begin{table}\n\t\\centering\n\t\\captionsetup{font=small}\n\t\\caption{\\small{\\textbf{Highlights of temporal correspondences learning methods for VHP models} (\\S\\ref{sec:vhp-methods}). Representative Works of each method are also give.}}\n\t\\label{table:vhp_highlights}\n\t\\vspace{-5pt}\n\t\\begin{threeparttable}\n\t\\resizebox{0.49\\textwidth}{!}{\n\t\\setlength\\tabcolsep{2.5pt}\n\t\\renewcommand\\arraystretch{1.02}\n\t\\begin{tabular}{c|c|c}\n\t\\hline\\thickhline\n\t\\rowcolor{mygray1}\n\tMethod                & \\tabincell{c}{Representative\\\\ Works} & Highlights \\\\\n\t\\hline\n\t\\hline\n\t\\tabincell{c}{Cycle-\\\\ tracking}         & \\tabincell{c}{\\cite{wang2019corres, li2019joint}\\\\ \\cite{mckee2022transfer, jabri2020space}}   & \\tabincell{c}{Capturing temporal variations, may produce\\\\ wrong correspondences when occlusion occurs.} \\\\\n\t\\hline\n\t\\tabincell{c}{Reconstructive\\\\ Learning}         & \\tabincell{c}{\\cite{wang2021contrasive, li2022locality}}   & \\tabincell{c}{Modelling fine-grained temporal correspondence\\\\ and guiding focus on part details.} \\\\\n\t\\hline\n\t\\tabincell{c}{Contrastive\\\\ Learning}                     & \\tabincell{c}{\\cite{jeon2021mining, xu2021rethinking}\\\\ \\cite{zhao2021modelling, son2022contrastive}}    & \\tabincell{c}{Search for discriminative features to segment\\\\ similar or position-transformed human instances.} \\\\\n         \\hline\n\t\\end{tabular}\n\t}\n\t\\end{threeparttable}\n\t\\vspace{-5pt}\n\\end{table}\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=0.98\\linewidth]{figures\/method-rele.pdf}\n\t\\end{center}\n\t\\vspace{-5pt}\n\t\\captionsetup{font=small}\n\t\\caption{\\small{\\textbf{Correlations of different SHP, MHP and VHP methods} (\\S\\ref{sec:hp-summary}). We use the connections between the arc edges to summary the correlation between human parsing methods, each connecting line stands for a study that uses both methods. The longer the arc, the more methods of this kind, same for the width of connecting lines. This correlation summary reveals the prevalence of various human parsing methods.}}\n\t\\label{fig:corre-mothods}\n\t\\vspace{-10pt}\n\\end{figure}\n\n\\noindent$\\bullet$~\\textbf{Reconstructive Learning.} As video contents smoothly shift in time, pixels in a ``query\" frame can be considered as copies from a set of pixels in other reference frames \\cite{vondrick2018tracking, liu2018switchable}. Following UVC \\cite{li2019joint} to establish pixel-level correspondence, several methods \\cite{wang2021contrasive, li2022locality} are proposed to learn temporal correspondence completely by reconstructing correlating frames. Subsequently, ContrastCorr \\cite{wang2021contrasive} not only learns from intra-video self-supervision, but also steps further to introduce inter-video transformation as negative correspondence. The inter-video distinction enforces the feature extractor to learn discriminations between videos while preserving the fine-grained matching characteristic among intra-video frame pairs. Based on the intra-inter video correlation, LIIR \\cite{li2022locality} introduces a locality-aware reconstruction framework, which encodes position information and involves spatial compactness into intra-video correspondence learning, for locality-aware and efficient visual tracking.  \n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=0.98\\linewidth]{figures\/citation-rele.pdf}\n\t\\end{center}\n\t\\vspace{-5pt}\n\t\\captionsetup{font=small}\n\t\\caption{\\small{\\textbf{Correlations of different SHP, MHP and VHP studies} (\\S\\ref{sec:hp-summary}). We list out all the involved human parsing studies by dots and use connecting lines to represent their citing relations. The citing relation here refers to the citation appears in experimental comparisons, to avoid citations of low correlation in background introduction. As each line represents a citation between two studies, so the larger the dot, the more times cited. These correlations highlight the relatively prominent studies.}}\n\t\\label{fig:corre-studies}\n\t\\vspace{-10pt}\n\\end{figure}\n\n\\noindent$\\bullet$~\\textbf{Contrastive Learning.} Following the idea of pulling positive pairs close together and pushing negative pairs away from each other, considerable VHP algorithms adopt contrastive learning as a training objective. To solve the optimal transport problem, CLTC \\cite{jeon2021mining} proposes to mine positive and semi-hard negative correspondences via consistency estimation and dynamic hardness discrimination, respectively. The well-defined positive and negative pixel pairs prevent side-effects from non-consistent positive and too hard\/easy negative samples to contrastive learning. Specifically, unlike most methods that perform patch-level contrastive learning, VFS \\cite{xu2021rethinking} learns visual correspondences at frame level. Following data augmentation of image-level contrastive learning \\cite{he2020momentum} and a well-designed temporal sampling strategy, VFS encourages convolutional features to find correspondences between similar objects and parts. Lately, \\cite{zhao2021modelling, son2022contrastive} extend the video graph with space relations of neighbor nodes, which determine the aggregation strength from intra-frame neighbors. The proposed space-time graph draws more attention to the association of center-neighbor pairs, thus explicitly helping learning correspondence between part instances. SCC \\cite{son2022contrastive} mixes sequential Bayesian filters to formulate the optimal paths that track nodes from one frame to others, to alleviate the correspondence missing caused by random occlusion.\n\n\\noindent\\textbf{Remark.} To our investigation scope, the current VHP research essentially follows an unsupervised semi-automatic video object segmentation setup. But considering the potential demand, it is more expectant to fully utilize the annotations and solve the VHP problem through an instance-discriminative manner, \\emph{i.e}., a fine-grained video instance segmentation task. The highlights of temporal correspondences learning methods for VHP are shown in Table~\\ref{table:vhp_highlights}.\n\n\n\n\n\\vspace{-6pt}\n\\subsection{Summary}\n\\label{sec:hp-summary}\n\nThrough the detailed review, we have subdivided SHP, MHP, and VHP studies into multiple methods and discussed their characteristics. To further investigate the development picture of the human parsing community, we summarize the correlations of the methods in Figure~\\ref{fig:corre-mothods} and correlations of the involved studies in Figure~\\ref{fig:corre-studies}, respectively.\n\nFigure~\\ref{fig:corre-mothods} presents correlations between research methods, \\emph{i.e}., two methods are connected if a study uses both as its technical components, making the length of arcs represent the number of studies using them. The connecting line distribution first obviously shows that Graph (Structure), Attention (Mechanism), and Edge(-aware Learning) of SHP are more correlated with multiple other methods, which indicates their compatibility with others and prevalence in the community. It is worth noting that though Tree (Structure) has many correlations with others, a large proportion of them are with Graph method. This phenomenon indicates that Tree method is much less generalizable compared to Graph, Attention, and Edge methods. Regrettably, negligible relations between VHP and other methods show that current VHP studies have not yet gone deep into parts relationship modeling or human instance discrimination. \n\nThe correlations of human parsing studies are presented in form of citing relations as Figure~\\ref{fig:corre-studies}, each line represents a citation between two studies. For reliable statistics, we only consider citations that appear in experimental comparisons for all studies. From the citing relations, we can easily observe that Attention \\cite{chen2016attenttion}, JPPNet \\cite{liang2018look}, CE2P \\cite{ruan2019devil}, CNIF \\cite{wang2019learning} and PGN \\cite{gong2018instance} have the largest dots, \\emph{i.e}., they are experimental compared by most other studies, this indicates they are recognized as baseline studies of great prominence by the community. Additionally, since CE2P proposed to handle MHP sub-task by 2S-TD pipeline and make a milestone, lots of SHP studies start to compare their algorithms with MHP studies, this trend breaks down the barriers between the two sub-tasks of human parsing. Lastly, similar to the method correlation, VHP studies form citations strictly along with the proposed order among their own, which once again shows that VHP studies have not focused on human-centric data.\n\nSynthesizing detailed review and correlation analysis, we can draw some conclusions about the historical evolution of human parsing models. First, the research focus has gradually shifted from SHP to MHP and VHP. As more challenging tasks, the latter two also have greater application potential. With the emergence of high-quality annotated datasets and the improvement of computing power, they have received increasing attention. Secondly, the technical diversity is insufficient, and the achievements of representation learning in recent years have not fully benefited the human parsing field. Finally, the number of open source work has increased significantly, but still insufficient. It is hoped that subsequent researchers will open source code and models as much as possible to benefit the follow-up researchers.\n\n\n\\section{Human Parsing Datasets}\n\\label{sec:hp-data}\nIn the past decades, a variety of visual datasets have been released for human parsing (upper part of Figure~\\ref{fig:timeline}). We summarize the classical and commonly used datasets in Table~\\ref{table:hp_datasets}, and give a detailed review from multiple angles.\n\n\\begin{table*}[t]\n\t\\centering\n\t\\captionsetup{font=small}\n\t\\caption{\\small{\\textbf{Statistics of existing human parsing datasets.} See \\S\\ref{sec:shp-data} - \\S\\ref{sec:vhp-data} for more detailed descriptions. The 14 datasets are divided into 3 groups according to the human parsing taxonomy. ``Instance\" indicates that instance-level human labels are provided; ``Temporal\" indicates that video-level labels are provided; ``Super-pixel\" indicates that super-pixels are used for labeling.}}\n\t\\label{table:hp_datasets}\n\t\\vspace{-5pt}\n\t\\begin{threeparttable}\n\t\\resizebox{0.99\\textwidth}{!}{\n\t\\setlength\\tabcolsep{2.5pt}\n\t\\renewcommand\\arraystretch{1.02}\n\t\\begin{tabular}{r||c|c|c|c|c|c|c|c|c|c}\n\t\\hline\\thickhline\n\t\\rowcolor{mygray1}\n\tDataset & Year & Pub. & \\#Images & \\#Train\/Val\/Test\/ & \\#Class & Purpose & Instance & Temporal & Super-pixel  & Other Annotations \\\\\n\t\\hline\n\t\\hline\n        Fashionista \\cite{yamaguchi2012parsing} & 2012 & CVPR & 685 & 456\/-\/299  & 56 &  Clothing & -      &    -   &\\checkmark&     Clothing-tag \\\\\n       \n        CFPD \\cite{liu2013fashion} & 2013 & TMM & 2,682 & 1,341\/-\/1,341  & 23 &  Clothing &   -    &    -   & \\checkmark &     Color-seg. \\\\\n        DailyPhotos \\cite{dong2013deformable} & 2013 & ICCV & 2,500 & 2,500\/-\/-  & 19 &  Clothing & -      &   -    &\\checkmark&     Clothing-tag\\\\\n       \n        PPSS \\cite{luo2013pedestrian} & 2013 & ICCV & 3,673 & 1,781\/-\/1,892  & 6 &  Human &   -    &   -    &   -    & -      \\\\\n        ATR \\cite{liang2015deep} & 2015 & TPAMI & 7,700 & 6,000\/700\/1,000  & 18 &  Human &   -    &   -    &   -    & -      \\\\\n       \n        Chictopia10k \\cite{liang2015human} & 2015 & ICCV & 10,000 & 10,000\/-\/-  & 18 &  Clothing &   -    &  -     &  -     &    Clothing-tag   \\\\\n        SYSU-Clothes \\cite{liang2016clothes} & 2016 & TMM & 2,682 & 2,682\/-\/-  & 57 &  Clothing &   -    &  -     &\\checkmark&     Clothing-tag  \\\\\n       \n        LIP \\cite{gong2017look} & 2017 & CVPR &  50,462 & 30,462\/10,000\/10,000 & 20 &  Human &   -    &    -   &\\checkmark&   -     \\\\\n       \n       \n       \n        HRHP \\cite{cvpr2021l2id} & 2021 & CVPRW & 7,500 & 6,000\/500\/1,000  & 20 &  Human &  -     &   -    &  -     &   -    \\\\\n        \\hline\n       \n        PASCAL-Person-Part \\cite{chen2014detect} & 2014 & CVPR & 3,533 & 1,716\/-\/1,817  & 7 &  Human &\\checkmark&  -     &-   &     Human-box    \\\\\n        MHP-v1.0 \\cite{li2017multiple} & 2017 & ArXiv & 4,980 & 3,000\/1,000\/980  & 19 &  Human &\\checkmark&    -   &-   &     Human-box    \\\\\n       \n        MHP-v2.0 \\cite{zhao2018understanding} & 2018 & MM & 25,403 & 15,403\/5,000\/5,000  & 59 &  Human &\\checkmark&    -   & -  &    Human-box    \\\\\n        COCO-DensePose \\cite{guler2018densepose} & 2018 & CVPR & 27,659 & 26,151\/-\/1,508  & 15 &  Human &\\checkmark&   -    &  - &    \\tabincell{c}{Human-box\/\\\\keypoints\/densepoints}     \\\\\n       \n        CIHP \\cite{gong2018instance} & 2018 & ECCV & 38,280 & 28,280\/5,000\/5,000  & 20 &  Human &\\checkmark&   -    & -  &    Human-box    \\\\\n       \n        \\hline\n       \n        VIP \\cite{zhou2018adaptive} & 2018 & MM & 21,246 & 18,468\/-\/2,778  & 20 &  Human &\\checkmark& \\checkmark & -  &    Human-box\/identity    \\\\\n       \n       \n       \n       \n       \n       \n       \n       \n       \n       \n       \n       \n\t\\hline\n\t\\end{tabular}\n\t}\n\t\\end{threeparttable}\n\t\\vspace{-5pt}\n\\end{table*}\n\n\n\\vspace{-6pt}\n\\subsection{Single Human Parsing (SHP) Datasets}\n\\label{sec:shp-data}\n\n\\noindent$\\bullet$~\\textbf{Fashionista (FS)} \\cite{yamaguchi2012parsing} consists of 685 photographs collected from \\texttt{Chictopia.com}, a social networking website for fashion bloggers. There are 456 training images and 299 testing images annotated with 56-class semantic labels, and text tags of garment items and styling are also provided. Fashionista was once the main single human\/clothing parsing dataset but was limited by its scale. It is rarely used now.\n\n\n\\noindent$\\bullet$~\\textbf{Colorful Fashion Parsing Data (CFPD)} \\cite{liu2013fashion} is also collected from \\texttt{Chictopia.com}, which provides 23-class noisy semantic labels and 13-class color labels. The annotated images are usually grouped into 1,341\/1,341 for \\texttt{train}\/\\texttt{test}.\n\n\\noindent$\\bullet$~\\textbf{DailyPhotos (DP)} \\cite{dong2013deformable} contains 2,500 high resolution images, which are crawled following the same strategy as the Fashionista dataset and thoroughly annotated with 19 categories.\n\n\\noindent$\\bullet$~\\textbf{PPSS} \\cite{luo2013pedestrian} includes 3,673 annotated samples collected from 171 videos of different surveillance scenes and provides pixel-wise annotations for hair, face, upper-\/lower-clothes, arm, and leg. It presents diverse real-word challenges, \\emph{e.g}. pose variations, illumination changes, and occlusions. There are 1,781 and 1,892 images for training and testing, respectively.\n\n\\noindent$\\bullet$~\\textbf{ATR} \\cite{liang2015deep} contains data which combined from three small benchmark datasets: the Fashionista \\cite{yamaguchi2012parsing} containing 685 images, the CFPD \\cite{liu2013fashion} containing 2,682 images, and the DailyPhotos \\cite{dong2013deformable} containing 2,500 images. The labels are merged of Fashionista and CFPD datasets to 18 categories. To enlarge the diversity, another 1,833 challenging images are collected and annotated to construct the Human Parsing in the Wild (HPW) dataset. The final combined dataset contains 7,700 images, which consists of 6,000 images for training, 1,000 for testing, and 700 as the validation set. \n\n\\noindent$\\bullet$~\\textbf{Chictopia10k} \\cite{liang2015human} contains 10,000 real-world human pictures from \\texttt{Chictopia.com}, annotating pixel-wise labels following \\cite{liang2015deep}. The dataset mainly contains images in the wild (\\emph{e.g}., more challenging poses, occlusion, and clothes).\n\n\\noindent$\\bullet$~\\textbf{SYSU-Clothes} \\cite{liang2016clothes} consists of 2,098 high resolution fashion photos in high-resolution (about 800$\\times$500 on average) from the shopping website. In this dataset, six categories of clothing attributes (\\emph{e.g}., clothing category, clothing color, clothing length, clothing shape, collar shape, and sleeve length) and 124 attribute types of all categories are collected. \n\n\\noindent$\\bullet$~\\textbf{Look into Person (LIP)} \\cite{gong2017look} is the most popular single human parsing dataset, which is annotated with pixel-wise annotations with 19 semantic human part labels and one background label. LIP contains 50,462 annotated images and be grouped into 30,462\/10,000\/10,000 for \\texttt{train}\/\\texttt{val}\/\\texttt{test}. The images in the LIP dataset are cropped person instances from COCO \\cite{lin2014microsoft} training and validation sets.\n\n\n\n\n\\noindent\\textbf{Remark.} ATR and LIP are the mainstream benchmarks among these single human parsing datasets. In recent years, the research purpose has changed from ``clothing\" to ``human\", and the data scale and annotation quality have also been significantly improved.\n\n\n\\vspace{-6pt}\n\\subsection{Multiple Human Parsing (MHP) Datasets}\n\\label{sec:mhp-data}\n\n\\noindent$\\bullet$~\\textbf{PASCAL-Person-Part (PPP)} \\cite{chen2014detect} is annotated from the PASCAL-VOC-2010 \\cite{everingham2010pascal}, which contains 3,533 multi-person images with challenging poses and splits into 1,716 training images and 1,817 test images. Each image is pixel-wise annotated with 7 classes, namely head, torso, upper\/lower arms, upper\/lower legs, and a background category.\n\n\\noindent$\\bullet$~\\textbf{MHP-v1.0} \\cite{li2017multiple} contains 4,980 multi-person images with fine-grained annotations at pixel-level. For each person, it defines 7 body parts, 11 clothing\/accessory categories, and one background label. The \\texttt{train}\/\\texttt{val}\/\\texttt{test} sets contain 3,000\/1,000\/980 images, respectively.\n\n\\noindent$\\bullet$~\\textbf{MHP-v2.0} \\cite{zhao2018understanding} is an extend version of MHP-v1.0 \\cite{li2017multiple}, which provides more images and richer categories. MHP-v2.0 contains 25,403 images and has great diversity in image resolution (from 85$\\times$100 to 4,511$\\times$6,919) and human instance number (from 2 to 26 persons). These images are split into 15,403\/5,000\/5,000 for \\texttt{train}\/\\texttt{val}\/\\texttt{test} with 59 categories.\n\n\\noindent$\\bullet$~\\textbf{COCO-DensePose (COCO-DP)} \\cite{guler2018densepose} aims at establishing the mapping between all human pixels of an RGB image and the 3D surface of the human body, and has 27,659 images (26,151\/1,508 for \\texttt{train}\/\\texttt{test} splits) gathered from COCO \\cite{lin2014microsoft}. The dataset provides 15 pixel-wise human parts with dense keypoints annotations.\n\n\\noindent$\\bullet$~\\textbf{Crowd Instance-level Human Parsing (CIHP)} \\cite{gong2018instance} is the largest multiple human parsing dataset to date. With 38,280 diverse real-world images, the persons are labelled with pixel-wise annotations on 20 categories.  It consists of 28,280 training and 5,000 validation images with publicly available annotations, as well as 5,000 test images with annotations withheld for benchmarking purposes. All images of the CIHP dataset contain two or more instances with an average of 3.4.\n\n\n\\noindent\\textbf{Remark.} So far, several multiple human parsing datasets have high-quality annotation and considerable data scale. In addition to pixel-wise parsing annotations, many datasets provide other rich annotations, such as box, keypoints\/landmark and style. PPP, CIHP and MHP-v2.0 are widely studied datasets, and most classical multiple human parsing methods have been verified on them.\n\n\n\\vspace{-6pt}\n\\subsection{Video Human Parsing (VHP) Datasets}\n\\label{sec:vhp-data}\n\n\\noindent$\\bullet$~\\textbf{Video Instance-level Parsing (VIP)} \\cite{zhou2018adaptive} is the first video human parsing dataset. VIP contains 404 multi-person Full HD sequences, which are collected from \\texttt{Youtube} with great diversity. For every 25 consecutive frames in each sequence, one frame is densely annotated with 20 classes and identities. All the sequences are grouped into 354\/50 for \\texttt{train}\/\\texttt{test}, containing 18,468\/2,778 annotated frames respectively.\n\n\n\\noindent\\textbf{Remark.} Since video human parsing has only attracted attention in recent years, there are few publicly available datasets, and its data scale and richness still need to be continuously invested by the community.\n\n\n\n\n\n\n\n\n\n\n\n\\vspace{-6pt}\n\\subsection{Summary}\n\\label{sec:data-summary}\nThrough Table~\\ref{table:hp_datasets}, we can observe that the human parsing datasets show several development trends. Firstly, the scale of datasets continues to increase, from hundreds in the early years \\cite{yamaguchi2012parsing} to a tens of thousands now \\cite{gong2017look, gong2018instance}. Secondly, the quality of annotation is constantly improving. Some early datasets use super-pixel \\cite{yamaguchi2012parsing, liang2016clothes, gong2017look} to reduce the annotation cost, while in recent years, pixel-wise accurate annotation has been adopted. Finally, the annotation dimensions are becoming increasingly diverse, \\emph{e.g}., COCO-DensePose \\cite{guler2018densepose} provides boxes, keypoints, and UVs annotation in addition to parsing. \n\n\n\\section{Performance Comparisons}\n\\label{sec:pc}\nTo provide a more intuitive comparison, we tabulate the performance of several previously discussed models. It should be noted that the experimental settings of each study are not entirely consistent (\\emph{e.g}., backbone, input size, training epochs).\nTherefore, we suggest only taking these comparisons as references, and a more specific analysis needs to study the original articles deeply.\n\n\n\\vspace{-6pt}\n\\subsection{SHP Performance Benchmarking}\n\\label{sec:shp-bench}\nWe select ATR \\cite{liang2015deep} and LIP \\cite{gong2017look} as the benchmark for single human parsing performance comparison, and compared 14 and 26 models, respectively.\n\n\\vspace{-6pt}\n\\subsubsection{Evaluation Metrics}\nThe evaluation metrics of single human parsing are basically consistent with semantic segmentation \\cite{shelhamer2016fully}, including pixel accuracy, mean pixel accuracy, and mean IoU. In addition, foreground pixel accuracy and F-1 score are also commonly used metrics on the ATR dataset.\n\n\\noindent$\\bullet$~\\textbf{Pixel accuracy (pixAcc)} is the simplest and intuitive metric, which expresses the proportion of pixels with correct prediction in the overall pixel.\n\n\\noindent$\\bullet$~\\textbf{Foreground pixel accuracy (FGAcc)} only calculates the pixel accuracy of foreground human parts.\n\n\\noindent$\\bullet$~\\textbf{Mean pixel accuracy (meanAcc)} is a simple improvement of pixel accuracy, which calculates the proportion of correctly predicted pixels in each category.\n\n\\noindent$\\bullet$~\\textbf{Mean IoU (mIoU)} is short for mean intersection over union, which calculates the ratio of the intersection and union of two sets. The two sets are the ground-truth and predicted results of each category respectively.\n\n\\noindent$\\bullet$~\\textbf{F-1 score (F-1)} is the harmonic average of precision and recall, which is a common evaluation metric.\n\n\\begin{table}\n\t\\centering\n\t\\captionsetup{font=small}\n\t\\caption{\\small{\\textbf{Quantitative SHP results on ATR} \\texttt{test} (\\S\\ref{sec:shp-bench}) in terms of pixel accuracy (pixAcc), foreground pixel accuracy (FGAcc) and F-1 score (F-1). The three best scores are marked in \\textcolor{scorered}{\\textbf{red}}, \\textcolor{scoreblue}{\\textbf{blue}}, and \\textcolor{scoregreen}{\\textbf{green}}, respectively.}}\n\t\\label{table:atrbench}\n\t\\vspace{-5pt}\n\t\\begin{threeparttable}\n\t\\resizebox{0.49\\textwidth}{!}{\n\t\\setlength\\tabcolsep{2.5pt}\n\t\\renewcommand\\arraystretch{1.02}\n\t\\begin{tabular}{c|r|c||c|c|c||c|c|c}\n\t\\hline\\thickhline\n\t\\rowcolor{mygray1}\n\tYear & Method & Pub. & Backbone  & \\#Input Size & \\#Epoch  & pixAcc & FGAcc & F-1   \\\\\n\t\\hline\n\t\\hline\n\t2012 & Yamaguchi \\cite{yamaguchi2012parsing} & CVPR  &  - & - & - &  84.38 & 55.59 & 41.80  \\\\\n\t\\hline\n\t2013 & Paperdoll \\cite{yamaguchi2013paper} & ICCV  &  - & - & - &  88.96 & 62.18 & 44.76  \\\\\n\t\\hline\n\t\\multirow{3}{*}{{2015}} & M-CNN \\cite{liu2015matching} & CVPR  &  - & - & 50 &  89.57 & 73.98 & 62.81  \\\\\n\t                                    & Co-CNN \\cite{liang2015human} & ICCV  &  - & 150$\\times$100  & 90 &  95.23 & 80.90 & 76.95  \\\\\n\t                                    & ATR \\cite{liang2015deep} & TPAMI  &  - & 227$\\times$227 & 120 &  91.11 & 71.04 & 64.38  \\\\\n\t\\hline                                    \n\t\\multirow{2}{*}{{2016}} & LG-LSTM \\cite{liang2016object} & CVPR  &  VGG16 & 321$\\times$321  & 60 &  96.18 & 84.79 & 80.97  \\\\\n\t                                    & Graph-LSTM \\cite{liang2016semantic} & ECCV  &  VGG16 & 321$\\times$321  & 60 &  \\textcolor{scoreblue}{\\textbf{97.60}} &  \\textcolor{scoreblue}{\\textbf{91.42}} & 83.76  \\\\\n\t\\hline                                    \n\t2017 & Struc-LSTM \\cite{liang2017interpretable} & CVPR  &  VGG16 & 321$\\times$321  & 60 &  \\textcolor{scorered}{\\textbf{97.71}} & \\textcolor{scorered}{\\textbf{91.76}} &  \\textcolor{scorered}{\\textbf{87.88}}  \\\\\n\t\\hline\n\t2018 & TGPNet \\cite{luo2018trusted} & MM  & VGG16 & 321$\\times$321 & 35 &  96.45 & 87.91 & 81.76  \\\\\n\t\\hline\n\t2019 & CNIF \\cite{wang2019learning} & ICCV  & ResNet101 & 473$\\times$473 & 150 &  96.26 & 87.91 & 85.51  \\\\\n\t\\hline\n\t\\multirow{3}{*}{{2020}} & CorrPM \\cite{zhang2020correlating}  & CVPR  & ResNet101 & 384$\\times$384 & 150 &  97.12 &  \\textcolor{scoregreen}{\\textbf{90.40}} & 86.12  \\\\\n\t                                    & HHP \\cite{wang2020hierarchical}   & CVPR  & ResNet101 & 473$\\times$473 & 150 &  96.84 & 89.23 &  \\textcolor{scoreblue}{\\textbf{87.25}}  \\\\\n\t                                   \n\t                                   \n\t                                  \n\t                                   \n\t                                    & SCHP \\cite{li2020correction}  & TPAMI  & ResNet101 & 473$\\times$473 & 150 &  96.25 & 87.97 & 85.55  \\\\\n\n\n\t                                   \n\t\\hline                                    \n\t\\multirow{1}{*}{{2022}} & CDGNet \\cite{liu2022cdgnet}  & CVPR  & ResNet101 & 512$\\times$512 & 250 &  \\textcolor{scoregreen}{\\textbf{97.39}} & 90.19 & \\textcolor{scoregreen}{\\textbf{87.16}}  \\\\\n\t                                   \n\n\t\\hline\n\t\\end{tabular}\n\t}\n\t\\end{threeparttable}\n\t\\vspace{-5pt}\n\\end{table}\n\n\\begin{table}\n\t\\centering\n\t\\captionsetup{font=small}\n\t\\caption{\\small{\\textbf{Quantitative SHP results on LIP} \\texttt{val} (\\S\\ref{sec:shp-bench}) in terms of pixel accuracy (pixAcc), mean pixel accuracy (meanAcc) and mean IoU (mIoU). The three best scores are marked in \\textcolor{scorered}{\\textbf{red}}, \\textcolor{scoreblue}{\\textbf{blue}}, and \\textcolor{scoregreen}{\\textbf{green}}, respectively.}}\n\t\\label{table:lipbench}\n\t\\vspace{-5pt}\n\t\\begin{threeparttable}\n\t\\resizebox{0.49\\textwidth}{!}{\n\t\\setlength\\tabcolsep{2.5pt}\n\t\\renewcommand\\arraystretch{1.02}\n\t\\begin{tabular}{c|r|c||c|c|c||c|c|c}\n\t\\hline\\thickhline\n\t\\rowcolor{mygray1}\n\tYear & Method & Pub. & Backbone  & \\#Input Size & \\#Epoch  & pixAcc & meanAcc & mIoU   \\\\\n\t\\hline\n\t\\hline\n\t\\multirow{1}{*}{{2017}} & SSL \\cite{gong2017look} & CVPR  &  VGG16 & 321$\\times$321 & 50 &  - & - & 46.19  \\\\\n\t                                   \n\t\\hline                                    \n\t\\multirow{4}{*}{{2018}} & HSP-PRI \\cite{kalayeh2018human} & CVPR  &  InceptionV3 & - & - &  85.07 & 60.54 & 48.16  \\\\\n\t                                    & MMAN \\cite{luo2018macro} & ECCV  &  ResNet101 & 256$\\times$256 & 30 &  85.24 & 57.60 & 46.93  \\\\\n\t                                    & MuLA \\cite{nie2018mutual} & ECCV  &  Hourglass & 256$\\times$256 & 250 &  88.50 & 60.50 & 49.30  \\\\\n\t                                    & JPPNet \\cite{liang2018look} & TPAMI  &  ResNet101 & 384$\\times$384 & 60 &  86.39 & 62.32 & 51.37  \\\\\n\t\\hline                                             \n\t\\multirow{3}{*}{{2019}} & CE2P \\cite{ruan2019devil} & AAAI  &  ResNet101 & 473$\\times$473 & 150 &  87.37 & 63.20 & 53.10  \\\\\n\t\t                            & CNIF \\cite{wang2019learning} & ICCV  &  ResNet101 & 473$\\times$473 & 150 &  88.03 & 68.80 & 57.74  \\\\\n\t\t                            & BraidNet \\cite{liu2019braidnet} & MM  &  ResNet101 & 384$\\times$384 & 150 &  87.60 & 66.09 & 54.42  \\\\\n\t\t                           \n\t\\hline          \t                            \t\n\t\\multirow{10}{*}{{2020}} & CorrPM \\cite{zhang2020correlating} & CVPR  &  ResNet101 & 384$\\times$384 & 150 &  - & - & 55.33  \\\\\n\t\t                              & SLRS \\cite{li2020self} & CVPR  &  ResNet101 & 384$\\times$384 & 150 &  88.33 & 66.53 & 56.34  \\\\                      \n\t\t                              & PCNet \\cite{zhang2020pcnet} & CVPR  &  ResNet101 & 473$\\times$473 & 120 &  - & - & 57.03  \\\\\n\t\t                              & HHP \\cite{wang2020hierarchical} & CVPR  &  ResNet101 & 473$\\times$473 & 150 &   \\textcolor{scoreblue}{\\textbf{89.05}} & \\textcolor{scoregreen}{\\textbf{70.58}} & 59.25  \\\\\n\t\t                              & DTCF \\cite{liu2020hybrid} & MM  &  ResNet101 & 473$\\times$473 & 200 &  88.61 & 68.89 & 57.82  \\\\\n\t\t                             \n\t\t                              & SemaTree \\cite{ji2020learning} & ECCV  &  ResNet101 & 384$\\times$384 & 200 &  88.05 & 66.42 & 54.73  \\\\\n\t\t                              & OCR \\cite{yuan2020object} & ECCV  &  HRNetW48 & 473$\\times$473 &  \\app100 &  - & - & 56.65  \\\\\n\t\t                              & BGNet \\cite{zhang2020blended} & ECCV  &  ResNet101 & 473$\\times$473 & 120 &  - & - & 56.82  \\\\\n\t\t                             \n\t\t                             \n\t\t                             \n\t\t                             \n\t\t                             \n\t\t                              & HRNet \\cite{wang2020deep} & TPAMI  &  HRNetW48 & 473$\\times$473 & \\app150 &  88.21 & 67.43 & 55.90  \\\\\n\t\t                              & SCHP \\cite{li2020correction} & TPAMI  &  ResNet101 & 473$\\times$473 & 150 &  - & - & 59.36  \\\\\n\t\\hline          \t\t                              \n\t\\multirow{5}{*}{{2021}} & HIPN \\cite{liu2021hier} & AAAI  &  ResNet101 & 473$\\times$473 & 150 &   \\textcolor{scorered}{\\textbf{89.14}} & \\textcolor{scoreblue}{\\textbf{71.09}} &  \\textcolor{scoregreen}{\\textbf{59.61}}  \\\\\n\t\t                              & MCIBI \\cite{jin2021mining} & ICCV  &  ResNet101 & 473$\\times$473 & 150 &  - & - & 55.42  \\\\\n\t\t                              & ISNet \\cite{jin2021isnet} & ICCV  &  ResNet101 & 473$\\times$473 & 160 &  - & - & 56.96  \\\\\n\t\t                              & NPPNet \\cite{zeng2021neural} & ICCV  &  NAS & 384$\\times$384 & 120 &  - & - & 58.56  \\\\\n\t\t                             \n\t                                     \n\t\t                             \n\t\t                             \n\t\t                             \n\t\t                             \n\t\t                              & HTCorrM \\cite{zhang2021on} & TPAMI  &  HRNetW48 & 384$\\times$384 & 180 &  - & - & 56.85  \\\\\n\t\t                             \n\t\\hline          \t\t                             \n\t\\multirow{3}{*}{{2022}}   & CDGNet \\cite{liu2022cdgnet} & CVPR  &  ResNet101 & 473$\\times$473 & 150 &  \\textcolor{scoregreen}{\\textbf{88.86}} &  \\textcolor{scorered}{\\textbf{71.49}} &  \\textcolor{scoreblue}{\\textbf{60.30}}  \\\\\n\t                                      & HSSN \\cite{li2022deep} & CVPR  &  ResNet101 & 480$\\times$480 & \\app84 &  - &  - &  \\textcolor{scorered}{\\textbf{60.37}}  \\\\\n\t\t                             \n\t\t                              & PRM \\cite{zhang2022human} & TMM  &  ResNet101 & 473$\\times$473 & 120 &  - & - & 58.86  \\\\\n\t\t\t                    \n\n\t\\hline\n\t\\end{tabular}\n\t}\n\t\\end{threeparttable}\n\\end{table}\n\n\\vspace{-6pt}\n\\subsubsection{Results}\nTable~\\ref{table:atrbench} presents the performance of the reviewed SHP methods on ATR \\texttt{test} set. Struc-LSTM \\cite{liang2017interpretable} achieves the best performance, scoring 91.71\\% pixAcc. and 87.88\\% F-1 score, which greatly surpassed other methods. Table~\\ref{table:lipbench} shows the method results on the LIP benchmark since 2017. Overall, HIPN \\cite{liu2021hier} and HSSN \\cite{li2022deep} achieve remarkable results in various metrics, in which HIPN scored 89.14\\% pixelAcc and HSSN scored 60.37\\% mIoU.\n\n\n\\vspace{-6pt}\n\\subsection{MHP Performance Benchmarking}\n\\label{sec:multi-bench}\nWe select 7 models experimented on PASCAL-Person-Part \\cite{chen2014detect}, 9 models experimented on CIHP \\cite{gong2018instance} and 8 models experimented on MHP-v2 \\cite{zhao2018understanding} to compare the performance of multiple human parsing.\n\n\n\\begin{table}\n\t\\centering\n\t\\captionsetup{font=small}\n\t\\caption{\\small{\\textbf{Quantitative MHP results on PASCAL-Person-Part} \\texttt{test} (\\S\\ref{sec:multi-bench}) in terms of mIoU, AP$^\\text{r}_\\text{vol}$ and AP$^\\text{r}_\\text{50}$. We only mark the best score in \\textcolor{scorered}{\\textbf{red}} color.}}\n\n\t\\label{table:pppbench}\n\t\\vspace{-5pt}\n\t\\begin{threeparttable}\n\t\\resizebox{0.49\\textwidth}{!}{\n\t\\setlength\\tabcolsep{2.5pt}\n\t\\renewcommand\\arraystretch{1.02}\n\t\\begin{tabular}{c|r|c||c|c|c||c|c|c}\n\t\\hline\\thickhline\n\t\\rowcolor{mygray1}\n\tYear & Method & Pub. & Pipeline &  Backbone  & \\#Epoch  & mIoU & AP$^\\text{r}_\\text{vol}$ & AP$^\\text{r}_\\text{50}$   \\\\\n\t\\hline\n\t\\hline\n\n\n\n\n\t\\hline\t                            \n\t\\multirow{1}{*}{{2017}}\n\t\t                           \n\t\t                           \n\t\t                           \n\t\t                            & Holistic \\cite{li2017holistic} & BMVC & 1S-TD & ResNet101 &  100 & 66.34 & 38.40   &  40.60  \\\\\n\t\\hline\t                            \n\t\\multirow{1}{*}{{2018}}\n\t\t                           \n\t\t                           \n\t\t                           \n\t\t                           \n\t\t                            & PGN \\cite{gong2018instance} & ECCV & BU & ResNet101 &  \\app80 & \\textcolor{scorered}{\\textbf{68.40}} & 39.20   &  39.60  \\\\\n\t\t                           \n\t\t                           \n\t\\hline\t                            \n\t\\multirow{2}{*}{{2019}} & Parsing R-CNN \\cite{yang2019parsing} & CVPR & 1S-TD & ResNet50 &  75 & 62.70 & 40.40   &  43.70  \\\\\n\t\t                           \n\t\t                           \n\t\t                           \n\t\t                           \n\t\t                            & Unified \\cite{qin2019top} & BMVC & 1S-TD & ResNet101 &  \\app600 & - & 43.10  &  48.10  \\\\\n\t\t                           \n\t\\hline\t                            \n\t\\multirow{2}{*}{{2020}}\n\t\t\t                     \n\t\t\t                     \n\t\t\t                     \n\t\t\t                     \n\t\t\t                     \n\t\t\t                      & RP R-CNN \\cite{yang2020renovating} & ECCV & 1S-TD & ResNet50 &  75 & 63.30 & 40.90  &  44.10  \\\\\n\t\t\t                     \n\t\t\t                     \n\t\t\t                      & NAN \\cite{zhao2020fine} & IJCV & BU & - &  80 & - & 52.20   &   \\textcolor{scorered}{\\textbf{59.70}}   \\\\\n\t\t\t                    \n\t\\hline\t\t                      \n\t\\multirow{1}{*}{{2021}}\n\t\t                            & MGHR \\cite{zhou2021differentiable} & CVPR & BU & ResNet101 &  150 & - & \\textcolor{scorered}{\\textbf{55.90}}   &  59.00  \\\\\n\t\t                           \n\t\t                           \n\t                                   \n\t\t                           \n\t\t                           \n\t\t                           \n\t\t                           \n\t\t                           \n\t\t                           \n\n\n\n\n\t\\hline\n\t\\end{tabular}\n\t}\n\t\\end{threeparttable}\n\t\\vspace{-5pt}\n\\end{table}\n\n\n\\vspace{-6pt}\n\\subsubsection{Evaluation Metrics}\nGenerally speaking, multiple human parsing uses mIoU to measure the semantic segmentation performance, and AP$^\\text{r}_\\text{vol}$\/AP$^\\text{r}_\\text{50}$ or AP$^\\text{p}_\\text{vol}$\/AP$^\\text{p}_\\text{50}$ to measure the performance of instance discrimination.\n\n\\noindent$\\bullet$~\\textbf{Average precision based on region (AP$^\\text{r}_\\text{vol}$\/AP$^\\text{r}_\\text{50}$)} \\cite{hariharan2014simu} is similar to AP metrics in object detection \\cite{lin2014microsoft}. If the IoU between the predicted part and ground-truth part is higher than a certain threshold, the prediction is considered to be correct, and the mean Average Precision is calculated. The defined AP$^\\text{r}_\\text{vol}$ is the mean of the AP score for overlap thresholds varying from 0.1 to 0.9 in increments of 0.1 and AP$^\\text{r}_\\text{50}$ is the AP score for threshold equals 0.5.\n\n\\noindent$\\bullet$~\\textbf{Average precision based on part (AP$^\\text{p}_\\text{vol}$\/AP$^\\text{p}_\\text{50}$)} \\cite{li2017multiple, zhao2020fine} is adopted to evaluate the instance-level human parsing performance. AP$^\\text{p}$ is very similar to AP$^\\text{r}$ in calculation mode, except that it calculates mIoU with the whole human body.\n\n\\vspace{-6pt}\n\\subsubsection{Results}\nPASCAL-Person-Part benchmark is the classical benchmark in multiple human parsing. Table~\\ref{table:pppbench} gathers the results of 7 models on PASCAL-Person-Part \\texttt{test} set. PGN \\cite{gong2018instance} is the top one in mIoU metric. In AP$^\\text{r}_\\text{vol}$\/AP$^\\text{r}_\\text{50}$ metrics, MGHR \\cite{zhou2021differentiable}, and NAN \\cite{zhao2020fine} are the best two methods at present. The results on CIHP \\texttt{val} set are summarized in Table~\\ref{table:cihpbench}. As seen, SCHP \\cite{li2020correction} performs the best on all metrics, which yields 67.67\\% mIoU, 52.74\\% AP$^\\text{r}_\\text{vol}$, and 58.95\\% AP$^\\text{r}_\\text{50}$. Table~\\ref{table:mhpv2bench} summarizes 8 models on MHP-v2 \\texttt{val} set. SCHP achieves the best mIoU again. In terms of AP$^\\text{p}_\\text{vol}$\/AP$^\\text{p}_\\text{50}$, RP R-CNN \\cite{yang2020renovating} has won the best results so far.\n\n\\begin{table}\n\t\\centering\n\t\\captionsetup{font=small}\n\t\\caption{\\small{\\textbf{Quantitative MHP results on CIHP} \\texttt{val} (\\S\\ref{sec:multi-bench}) in terms of mIoU, AP$^\\text{r}_\\text{vol}$ and AP$^\\text{r}_\\text{50}$. We only mark the best score in \\textcolor{scorered}{\\textbf{red}} color.}}\n\t\\label{table:cihpbench}\n\t\\vspace{-5pt}\n\t\\begin{threeparttable}\n\t\\resizebox{0.49\\textwidth}{!}{\n\t\\setlength\\tabcolsep{2.5pt}\n\t\\renewcommand\\arraystretch{1.02}\n\t\\begin{tabular}{c|r|c||c|c|c||c|c|c}\n\t\\hline\\thickhline\n\t\\rowcolor{mygray1}\n\tYear & Method & Pub. & Pipeline &  Backbone  & \\#Epoch  & mIoU & AP$^\\text{r}_\\text{vol}$ & AP$^\\text{r}_\\text{50}$   \\\\\n\t\\hline\n\t\\hline\n\t2018 & PGN \\cite{gong2018instance} & ECCV & BU & ResNet101 &  \\app80 & 55.80 & 33.60   &  35.80  \\\\\n\t\\hline\n\t\\multirow{4}{*}{{2019}} & CE2P \\cite{ruan2019devil} & AAAI & 2S-TD & ResNet101 &  150 & 59.50 & 42.80  &  48.70  \\\\\n\t\t                            & Parsing R-CNN \\cite{yang2019parsing} & CVPR & 1S-TD & ResNet50 &  75 & 56.30 & 36.50   &  40.90  \\\\\n\t\t                           \n\t\t                            & BraidNet \\cite{liu2019braidnet} & MM & 2S-TD & ResNet101 &  150 & 60.62 & 43.59  &  48.99  \\\\\n\t\t                            & Unified \\cite{qin2019top} & BMVC & 1S-TD & ResNet101 &  \\app36 & 53.50 & 37.00  &  41.80  \\\\\n\t\\hline\t                            \n\n\t\\multirow{3}{*}{{2020}}\n\t\t                           \n\t\t                           \n\t\t                           \n\t\t                            & RP R-CNN \\cite{yang2020renovating} & ECCV & 1S-TD & ResNet50 &  150 & 60.20 & 42.30  &  48.20  \\\\\n\t\t                            & SemaTree \\cite{ji2020learning} & ECCV & 2S-TD & ResNet101 &  200 & 60.87 & 43.96  &  49.27  \\\\\n\t\t                           \n\t\t\t                   \n\t\t                            & SCHP \\cite{li2020correction} & TPAMI & 2S-TD & ResNet101 &  150 &  \\textcolor{scorered}{\\textbf{67.47}} &  \\textcolor{scorered}{\\textbf{52.74}}    &  \\textcolor{scorered}{\\textbf{58.94}}   \\\\\n\n       \n                                            \n                                            \n\t\t                           \n\t\t                           \n\t\t                           \n\t\\hline\n\t\\multirow{1}{*}{{2022}}\n\t\t                           \n\t\t                           \n\t\t                            & AIParsing \\cite{zhang2022aiparsing} & TIP & 1S-TD & ResNet101 &  75 & 60.70 & -   &  -  \\\\\n\t\t                           \n\t\t                           \n\t\n\t\\hline\n\t\\end{tabular}\n\t}\n\t\\end{threeparttable}\n\\end{table}\n\n\n\n\\begin{table}\n\t\\centering\n\t\\captionsetup{font=small}\n\t\\caption{\\small{\\textbf{Quantitative MHP results on MHP-v2} \\texttt{val} (\\S\\ref{sec:multi-bench}) in terms of mIoU, AP$^\\text{p}_\\text{vol}$ and AP$^\\text{p}_\\text{50}$. We only mark the best score in \\textcolor{scorered}{\\textbf{red}} color.}}\n\t\\label{table:mhpv2bench}\n\t\\vspace{-5pt}\n\t\\begin{threeparttable}\n\t\\resizebox{0.49\\textwidth}{!}{\n\t\\setlength\\tabcolsep{2.5pt}\n\t\\renewcommand\\arraystretch{1.02}\n\t\\begin{tabular}{c|r|c||c|c|c||c|c|c}\n\t\\hline\\thickhline\n\t\\rowcolor{mygray1}\n\tYear & Method & Pub. & Pipeline &  Backbone  & \\#Epoch  & mIoU & AP$^\\text{p}_\\text{vol}$ & AP$^\\text{p}_\\text{50}$   \\\\\n\t\\hline\n\t\\hline\n\t\\multirow{2}{*}{{2019}} & CE2P \\cite{ruan2019devil} & AAAI & 2S-TD & ResNet101 &  150 & 41.11 & 42.70  &  34.47  \\\\\n\t\t                            & Parsing R-CNN \\cite{yang2019parsing} & CVPR & 1S-TD & ResNet50 &  75 & 36.20 & 38.50   &  24.50  \\\\\n\t\\hline\t                            \n\n\t\\multirow{4}{*}{{2020}} & RP R-CNN \\cite{yang2020renovating} & ECCV & 1S-TD & ResNet50 &  150 & 38.60 & \\textcolor{scorered}{\\textbf{46.80}}  &   \\textcolor{scorered}{\\textbf{45.30}}   \\\\\n\t\t                            & SemaTree \\cite{ji2020learning} & ECCV & 2S-TD & ResNet101 &  200 & - & 42.51  &  34.36  \\\\\n\t\t                           \n\t\t\t                    & NAN \\cite{zhao2020fine} & IJCV & BU & - &  80 & - & 41.78   &   25.14   \\\\\n\t\t                            & SCHP \\cite{li2020correction} & TPAMI & 2S-TD & ResNet101 &  150 &  \\textcolor{scorered}{\\textbf{45.21}} &  45.25    &  35.10   \\\\\n\t\\hline\n        \\multirow{1}{*}{{2021}}\n                                            \n                                            \n\t\t                           \n\t\t                            & MGHR \\cite{zhou2021differentiable} & CVPR & BU & ResNet101 &  150 & 41.40 & 44.30   &  39.00  \\\\\n\t\t                           \n\t\\hline\n\t\\multirow{1}{*}{{2022}} & AIParsing \\cite{zhang2022aiparsing} & TIP & 1S-TD & ResNet101 &  75 & 40.10 & 46.60   &  43.20 \\\\\n\t\t                           \n\t\t                           \n\t\n\t\\hline\n\t\\end{tabular}\n\t}\n\t\\end{threeparttable}\n\t\\vspace{-5pt}\n\\end{table}\n\n\n\\vspace{-6pt}\n\\subsection{VHP Performance Benchmarking}\n\\label{sec:video-bench}\nVIP datasets is widely used to benchmark video human parsing. We selected 11 models since 2018.\n\n\\vspace{-6pt}\n\\subsubsection{Evaluation Metrics}\nSimilar to multiple human parsing, mIoU and  AP$^\\text{r}_\\text{vol}$ are also adopted for video human parsing performance evaluation.\n\n\\vspace{-6pt}\n\\subsubsection{Results}\nTable~\\ref{table:vipbench} gives the results of recent methods on VIP \\texttt{val} set. It is clear that LIIR \\cite{zhou2021differentiable} and UVC+ \\cite{mckee2022transfer} have achieved the best performance in mIoU and AP$^\\text{r}_\\text{vol}$ metrics respectively.\n\n\n\n\n\n\n\n\\vspace{-6pt}\n\\subsection{Summary}\n\\label{sec:per-summary}\nThrough the above performance comparison, we can observe several apparent phenomena. The first and most important is the fairness of the experimental setting. For single human parsing and multiple human parsing, many studies have not given detailed experimental settings, or there are great differences in several essential hyper-parameters, resulting fair comparison impossible. The second is that most methods do not give the parameters number and the inference time, which makes some methods occupy an advantage in comparison by increasing the model capacity, and also brings trouble to some computationally sensitive application scenarios, such as social media and automatic driving.\n\nIn addition to the above phenomena, we can also summarize some positive signals. Firstly, in recent years, human parsing research has shown an upward trend, especially from 2020. Secondly, although some studies have achieved high performance on LIP, CIHP and VIP, these benchmarks are still not saturated. Thus the community still needs to continue its efforts. Thirdly, some specific issues and hotspots of human parsing are gradually attracting people's attention, which will further promote the progress of the whole field.\n\n\\begin{table}\n\t\\centering\n\t\\captionsetup{font=small}\n\t\\caption{\\small{\\textbf{Quantitative VHP results on VIP} \\texttt{val} (\\S\\ref{sec:multi-bench}) in terms of mIoU and AP$^\\text{r}_\\text{vol}$. The three best scores are marked in \\textcolor{scorered}{\\textbf{red}}, \\textcolor{scoreblue}{\\textbf{blue}}, and \\textcolor{scoregreen}{\\textbf{green}}, respectively.}}\n\t\\label{table:vipbench}\n\t\\vspace{-5pt}\n\t\\begin{threeparttable}\n\t\\resizebox{0.42\\textwidth}{!}{\n\t\\setlength\\tabcolsep{2.5pt}\n\t\\renewcommand\\arraystretch{1.02}\n\t\\begin{tabular}{c|r|c||c||c|c}\n\t\\hline\\thickhline\n\t\\rowcolor{mygray1}\n\tYear & Method & Pub.   &  Backbone   & mIoU & AP$^\\text{r}_\\text{vol}$  \\\\\n\t\\hline\n\t\\hline\n\n\n\n\t\\multirow{2}{*}{{2019}} & TimeCycle \\cite{wang2019corres} & CVPR  & ResNet50  & 28.9 & 15.6    \\\\\n\t\t                            & UVC \\cite{li2019joint} & NeurIPS & ResNet18  & 34.1 & 17.7    \\\\\n\t\\hline\n\t\\multirow{1}{*}{{2020}} & CRW \\cite{jabri2020space} & NeurIPS & ResNet18  & 38.6 & -    \\\\\n\t\\hline\n\t\\multirow{4}{*}{{2021}} & ContrastCorr \\cite{wang2021contrasive} & AAAI  & ResNet18  & 37.4 &  \\textcolor{scoregreen}{\\textbf{21.6}}    \\\\\n\t                                    & CLTC \\cite{jeon2021mining} & CVPR & ResNet18  & 37.8 & 19.1    \\\\\n\t\t                            & VFS \\cite{xu2021rethinking} & ICCV & ResNet18  &  39.9 & -    \\\\\n\t\t                            & JSTG \\cite{zhao2021modelling} & ICCV & ResNet18  &  \\textcolor{scoregreen}{\\textbf{40.2}} & -    \\\\\n\t\\hline\n\t\\multirow{3}{*}{{2022}} & LIIR \\cite{li2022locality} & CVPR & ResNet18  & \\textcolor{scorered}{\\textbf{41.2}}  &  \\textcolor{scoreblue}{\\textbf{22.1}}    \\\\\n\t                                     & SCC \\cite{son2022contrastive} & CVPR & ResNet18  & \\textcolor{scoreblue}{\\textbf{40.8}}  & -    \\\\\n\t\t                            & UVC+ \\cite{mckee2022transfer} & ArXiv & ResNet18  & 38.3 & \\textcolor{scorered}{\\textbf{22.2}}     \\\\\n\t\\hline\n\t\\end{tabular}\n\t}\n\t\\end{threeparttable}\n\t\\vspace{-5pt}\n\\end{table}\n\n\\section{An Outlook: Future Opportunities of Human Parsing}\n\\label{sec:outlook}\nAfter ten years of long development, with the whole community's efforts, human parsing has made remarkable achievements, but it has also encountered a bottleneck. In this section, we will discuss the opportunities of human parsing in the next era from multiple perspectives, hoping to promote progress in the field.\n\n\n\\begin{figure*}\n\t\\begin{center}\n\t\t\\includegraphics[width=0.90\\linewidth]{figures\/m2fp.pdf}\n\t\\end{center}\n\t\\vspace{-5pt}\n\t\\captionsetup{font=small}\n\t\\caption{\\small{\\textbf{Architecture of the proposed M2FP} (\\S\\ref{sec:new-baseline}). Through the explicit construction of background, part and human queries, we can model the relationship between humans and parts, and predict high-quality masks.}}\n\t\\label{fig:m2fp_arch}\n\t\\vspace{-10pt}\n\\end{figure*}\n\n\\vspace{-6pt}\n\\subsection{A Transformer-based Baseline for Human Parsing}\n\\label{sec:new-baseline}\nAlthough several mainstream benchmarks of human parsing have not been saturated, the accuracy growth has slowed down. The reason for this, we believe, is that some advances in deep learning have not yet benefited the human parsing task (\\emph{e.g}., transformer \\cite{vaswani2017attention, dosovitskiy2020an, carion2020end}, unsupervised representation learning \\cite{devlin2019bert, he2020momentum, bao2022beit, he2022masked}), and the lack of a concise and easily extensible code base for researchers. Therefore, the community urgently needs a new and strong baseline.\n\nWe consider that a new human parsing baseline should have the following four characteristics: a) \\textbf{Universality}, which can be applied to all mainstream human parsing tasks, including SHP, MHP, and VIP; b) \\textbf{Conciseness}, the baseline method should not be too complex; c) \\textbf{Extensibility}, complete code base, easy to modify or expand other modules or methods; d) \\textbf{High performance}, state-of-the-arts or at least comparable performance can be achieved on the mainstream benchmarks under the fair experimental setting. Based on the above views, we design a new transformer-based baseline for human parsing. The proposed new baseline is based on the \\texttt{Mask2Former} \\cite{cheng2022masked} architecture, with a few improvements adapted to human parsing, called \\texttt{Mask2Former for Parsing (M2FP)}. M2FP can adapt to almost all human parsing tasks and yield amazing performances.\\footnote{Code and models are publicly available at \\fontsize{7pt}{1em}\\url{https:\/\/github.com\/soeaver\/M2FP}}\n\n\n\\vspace{-6pt}\n\\subsubsection{Mask2Former for Parsing}\n\\noindent$\\bullet$~\\textbf{Modeling Human as Group Queries.} To solve the three human parsing sub-tasks, we need to simultaneously model the parts relationship and distinguish human instances. DETR series work \\cite{carion2020end, zhu2021deformable, cheng2021perpixel, cheng2022masked} regard objects as queries, and transform object detection or instance segmentation task into a direct set prediction problem. A naive idea is to regard human parts as queries, then use mask classification to predict the category and mask of each part. However, this creates two problems that cannot be ignored. Firstly, only modeling parts will make it difficult to learn the global relationship between parts and humans; Secondly, the subordination between part and human instance is unknown, resulting in the inadaptability for MHP task. Thus, we introduce the body hierarchy into the queries and use the powerful sequence encoding ability of transformer to build multiple hierarchical relationships between parts and humans. Specifically, we explicitly divide the queries into three groups: background queries, part queries and human queries. Through the relationship modeling ability of self-attention mechanism, besides the basic part-part relationship, the part-human, human-human, and part\/human-background relationships are also modeled. Thanks to the direct modeling of parts and the introduction of multiple hierarchical granularities, M2FP can be applied to all supervised human parsing tasks.\n\n\n\\noindent$\\bullet$~\\textbf{Architecture and Pipeline.} The architecture of proposed M2FP is illustrated in Figure~\\ref{fig:m2fp_arch}. We try to make the smallest modification to the Mask2Former. An encoder is used to extract image or video features, which is composed of a backbone and a pixel decoder \\cite{zhu2021deformable}. Then the features are flattened and sent into a transformer decoder. The transformer decoder consists of multiple repeated units, each containing a masked attention module, a self-attention module, and a shared feed-forward network (FFN) in turn. The grouped queries and flattened features conduct sufficient information exchange through the transformer decoder, and finally use bipartite matcher to match between queries and ground-truths uniquely. For SHP, in the inference stage, the background and part masks are combined with their class predictions to compute the final semantic segmentation prediction through matrix multiplication. For MHP, the intersection ratio of semantic segmentation prediction and human masks is calculated to obtain the final instance-level human parsing prediction. M2FP can also be extended to supervised VHP task. Follow \\cite{cheng2021m2forvis}, the background, parts, and humans in the video can be regarded as 3D spatial-temporal masks, and using the sequence encoding ability of transformer to make an end-to-end prediction.\n\n\\vspace{-6pt}\n\\subsubsection{Experiments}\n\\noindent$\\bullet$~\\textbf{Experimental Setup.} We validate M2FP on several mainstream benchmarks, including LIP, PASCAL-Person-Part, CIHP, and MHP-v2. All models are trained with nearly identical hyper-parameters under 8 NVIDIA V100 GPUs. Specifically, we use AdamW \\cite{loshchilov2018decoupled} optimizer with a mini-batch size of 16, an initial learning rate of 0.0004 with poly (LIP) or step (PASCAL-Person-Part, CIHP, and MHP-v2) learning rate schedule, then train each model for 150 epochs. Large scale jittering in the range of [0.1, 2.0] and typical data augmentation techniques, \\emph{e.g}., fixed size random crop (512$\\times$384 for LIP, 800$\\times$800 for PASCAL-Person-Part, CIHP, and MHP-v2), random rotation from [-40\\degree, +40\\degree], random color jittering and horizontal flip, are also used. For fair comparison, horizontal flipping is adopted during testing, and multi-scale test is used for LIP. The default backbone is ResNet-101 with pre-training on ImageNet-1K \\cite{russakovsky2015imagenet}.\n\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=0.99\\linewidth]{figures\/m2fp_performance.pdf}\n\t\\end{center}\n\t\\vspace{-5pt}\n\t\\captionsetup{font=small}\n\t\\caption{\\small{\\textbf{Comparison of M2FP with previous human parsing state-of-the-art models}. M2FP achieves state-of-the-art (PPP, CIHP and MHP-v2) or comparable performance (LIP) on all human parsing sub-tasks.}}\n\t\\label{fig:m2fp_performance}\n\t\\vspace{-5pt}\n\\end{figure}\n\n\n\\noindent$\\bullet$~\\textbf{Results.} As shown in Table~\\ref{table:m2fp_compare} and Figure~\\ref{fig:m2fp_performance}, M2FP achieves state-of-the-art or comparable performance across a broad range of human parsing benchmarks. For SHP, M2FP only falls behind HIPN \\cite{liu2021hier} and CDGNet \\cite{liu2022cdgnet}, obtaining 88.93\\% pixAcc. and 59.86\\% mIoU, showing great potential in the parts relationship modeling. For MHP, M2FP shows amazing performance, greatly surpassing the existing methods on all metrics and even exceeding the state-of-the-art two-stage top-down method, \\emph{i.e}., SCHP \\cite{li2020correction}. Specifically, M2FP outperforms PGN \\cite{gong2018instance} with 4.14 point mIoU and MGHR \\cite{zhou2021differentiable} with 0.56 point AP$^\\text{r}_\\text{vol}$ on PASCAL-Person-Part. On the more challenging CIHP and MHP-v2, M2FP beats SCHP in terms of mIoU while running in an end-to-end manner. Meanwhile, M2FP is also 7.96 points ahead of SCHP in AP$^\\text{r}_\\text{vol}$ (CIHP) and 5.97 points ahead of RP R-CNN \\cite{yang2020renovating} in AP$^\\text{p}_\\text{vol}$ (MHP-v2). These results demonstrate that M2FP surpasses almost all human parsing methods in a concise, effective and universal way, and can be regarded as a new baseline in the next era.\n\n\n\\begin{table}\n\t\\centering\n\t\\captionsetup{font=small}\n\t\\caption{\\small{\\textbf{Overview of M2FP results on various human parsing benchmarks}. \\underline{~~~~} denotes the previous state-of-the-art results. \\textbf{Bold} results denote M2FP achieve new state-of-the-art.}}\n\t\\label{table:m2fp_compare}\n\t\\vspace{-5pt}\n\t\\begin{threeparttable}\n\t\\resizebox{0.49\\textwidth}{!}{\n\t\\setlength\\tabcolsep{2.5pt}\n\t\\renewcommand\\arraystretch{1.02}\n\t\\begin{tabular}{r||c||c||c|c||c|c||c|c}\n\t\\hline\\thickhline\n\t\\rowcolor{mygray1}\n\t                                                                &   \\multicolumn{2}{c||}{LIP}                                           & \\multicolumn{2}{c||}{PPP}                             & \\multicolumn{2}{c||}{CIHP}                              &     \\multicolumn{2}{c}{MHP-v2}    \\\\\n\t\\cline{2-9}                               \n\t\\rowcolor{mygray1}                                  \n\t \\multirow{-2}{*}{Method}                         &   pixAcc.                             &   mIoU                             &      mIoU     &     AP$^\\text{r}_\\text{vol}$         &      mIoU               &      AP$^\\text{r}_\\text{vol}$      &  mIoU &  AP$^\\text{p}_\\text{vol}$   \\\\\n\t\\hline\n\t\\hline\n        HIPN \\cite{liu2021hier}                           &        \\underline{89.14}          &    59.61                            &      -                                  &      -                        &       -                       &     -                                           &        -                   &            -                       \\\\\n\n\tHSSN \\cite{li2022deep}                           &       -                                     &     \\underline{60.37}        &       -                                 &       -                       &        -                     &          -                                      &       -                    &             -                      \\\\\n\n\n\tPGN \\cite{gong2018instance}                 &          -                                  &     -                                    &     \\underline{68.40}        &  39.20                   &   55.80                  &       33.60                               &        -                    &               -                    \\\\\n\tMGHR \\cite{zhou2021differentiable}       &         -                                   &    -                                     &       -                                 &  \\underline{55.90} &    -                         &             -                                   &   41.40                &         44.30                 \\\\\n\tSCHP                   \\cite{li2020correction} &           -                                 &    -                                     &       -                                 &     -                         & \\underline{67.47}  &        \\underline{52.74}            &\\underline{45.21}&         45.25                 \\\\\n\tRP R-CNN \\cite{yang2020renovating}    &              -                              &    -                                     &   63.30                            &     40.90                & 60.20                     &        42.30                              &  38.60                &         \\underline{46.80} \\\\\n\t\\hline   \n\tM2FP (ours)                                            &      88.93                             &      59.86                          &      \\textbf{72.54}             &  \\textbf{56.46}    &\\textbf{69.15}&        \\textbf{60.47}                  &\\textbf{47.64} &         \\textbf{53.36}    \\\\  \n\t\\hline                               \n\t\\end{tabular}\n\t}\n\t\\end{threeparttable}\n\t\\vspace{-5pt}\n\\end{table}\n\n\n\n\\vspace{-6pt}\n\\subsection{Under-Investigated Open Issues}\n\\label{sec:open-issues}\nBased on the reviewed research, we list several under-investigated open issues that we believe should be pursued.\n\n\\noindent$\\bullet$~\\textbf{Efficient Inference.} In practical applications, human parsing models generally need real-time or even faster inference speed. The current research has not paid enough attention to this issue, especially the multiple human parsing research. Although some literature \\cite{zhou2021differentiable, zhang2022aiparsing} has discussed the model efficiency, it can not achieve real-time inference, and there is no human parser designed for this purpose. Therefore, from the perspective of practical application, it is an under-investigated open issue to design an efficient inference human parsing model.\n\n\\noindent$\\bullet$~\\textbf{Synthetic Dataset.} It is a common practice in many fields to use synthetic datasets to train models and transfer them to real scenes. Through CG technology (\\emph{e.g}., NVIDIA Omniverse\\footnote{\\fontsize{7pt}{1em}\\url{https:\/\/developer.nvidia.com\/nvidia-omniverse}}), we can obtain almost unlimited synthetic human data at a very low cost, as well as parsing annotations. Considering the labeling cost of human parsing dataset, this is a very attractive scheme. Wood \\emph{et al}. have made a preliminary attempt on the face parsing task and achieved very excellent performance \\cite{wood2021fake}, but at present, there is a lack of research on the human parsing field.\n\n\\noindent$\\bullet$~\\textbf{Long-tailed Phenomenon.} The long-tailed distribution is the most common phenomenon in the real world, and also exists in the human parsing field. For example, the Gini coefficient of MHP-v2.0 is as high as 0.747 \\cite{yang2022longtailed}, exceeding some artificially created long-tailed datasets, but this problem is currently ignored. Therefore, the existing methods are often brittle once exposed to the real world, where they are unable to adapt and robustly deal with tail categories effectively. This calls for a more general human parsing model, with the ability to adapt to long-tailed distributions in the real world.\n\n\n\n\\vspace{-6pt}\n\\subsection{New Directions}\n\\label{sec:new-direc}\nConsidering some potential applications, we shed light on several possible research directions.\n\n\\noindent$\\bullet$~\\textbf{Video Instance-level Human Parsing.} The current VHP research basically follows an unsupervised semi-automatic video object segmentation setting, which reduces the labeling cost in a way that greatly loses accuracy. However, most of the practical requirements of video human parsing require extremely high precision. Therefore, making full use of annotations and solving the VHP issue through an instance-discriminative manner, \\emph{i.e}., a fine-grained video instance segmentation task, has great research prospects.\n\n\n\\noindent$\\bullet$~\\textbf{Whole-body Human Parsing.} Besides human parsing, face parsing and hand parsing \\cite{liang2014parsing, lin2019face} are also important issues. To fully understand the pixel-wise temporal-spatial attributes of human in the wild, it is necessary to parse body, face, and hands simultaneously, which implies a new direction to end-to-end parse the whole body: Whole-body Human Parsing. Natural hierarchical annotation and large-scale variation bring new challenges to existing parsing techniques. Thus the targeted datasets and whole-body parsers are necessary.\n\n\\noindent$\\bullet$~\\textbf{Cooperation across Different Human-centric Directions.} Some human-centric visual tasks (\\emph{e.g}., human attribute recognition \\cite{yang2020hier}, pose estimation \\cite{zheng2020deep}, human mesh reconstruction \\cite{guler2019holopose}) face similar challenges to human parsing. Different tasks can play a positive role in promoting each other, although developments of these fields are independent. Moreover, the settings of different human-centric visual tasks are related, while there are no precedents for modeling these tasks in a unified framework. Thus, we call for closer collaboration across different human-centric visual tasks.\n\n\n\n\\section{Conclusions}\n\\label{sec:concl}\n\nAs far as we know, this is the first survey to comprehensively review deep learning techniques in human parsing, covering three sub-tasks: SHP, MHP, and VHP. We first provided the readers with the necessary knowledge, including task settings, background concepts, relevant problems, and applications. Afterward, we summarized the mainstream deep learning methods based on human parsing taxonomy, and analyzing them according to the theoretical background, technical contributions, and solving strategies. We also reviewed 14 popular human parsing datasets, benchmarking results on the 6 most widely-used ones. To promote sustainable community development, we discussed the under-investigated open issues and provided insight into new directions. We also put forward a new transformer-based human parsing framework, servicing a high-performance baseline for follow-up research through universal, concise, and extensible solutions. In summary, we hope this survey to provide an effective way to understand the current state-of-the-art human parsing models and promote the sustainable development of this research field.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\ifCLASSOPTIONcaptionsoff\n  \\newpage\n\\fi\n\n\n\n{\\small\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe {\\it IJCAI--20 Proceedings} will be printed from electronic\nmanuscripts submitted by the authors. These must be PDF ({\\em Portable\nDocument Format}) files formatted for 8-1\/2$''$ $\\times$ 11$''$ paper.\n\n\\subsection{Length of Papers}\n\nAll paper {\\em submissions} must have a maximum of six pages, plus at most one for references. The seventh page cannot contain {\\bf anything} other than references.\n\nThe length rules may change for final camera-ready versions of accepted papers and will differ between tracks. Some tracks may include only references in the last page, whereas others allow for any content in all pages. Similarly, some tracks allow you to buy a few extra pages should you want to, whereas others don't.\n\nIf your paper is accepted, please carefully read the notifications you receive, and check the proceedings submission information website\\footnote{\\url{https:\/\/proceedings.ijcai.org\/info}} to know how many pages you can finally use. That website holds the most up-to-date information regarding paper length limits at all times. Please notice that if your track allows for a special references-only page, the {\\bf references-only page(s) cannot contain anything else than references} (i.e.: do not write your acknowledgments on that page or you will be charged for it).\n\n\\subsection{Word Processing Software}\n\nAs detailed below, IJCAI has prepared and made available a set of\n\\LaTeX{} macros and a Microsoft Word template for use in formatting\nyour paper. If you are using some other word processing software, please follow the format instructions given below and ensure that your final paper looks as much like this sample as possible.\n\n\\section{Style and Format}\n\n\\LaTeX{} and Word style files that implement these instructions\ncan be retrieved electronically. (See Appendix~\\ref{stylefiles} for\ninstructions on how to obtain these files.)\n\n\\subsection{Layout}\n\nPrint manuscripts two columns to a page, in the manner in which these\ninstructions are printed. The exact dimensions for pages are:\n\\begin{itemize}\n\\item left and right margins: .75$''$\n\\item column width: 3.375$''$\n\\item gap between columns: .25$''$\n\\item top margin---first page: 1.375$''$\n\\item top margin---other pages: .75$''$\n\\item bottom margin: 1.25$''$\n\\item column height---first page: 6.625$''$\n\\item column height---other pages: 9$''$\n\\end{itemize}\n\nAll measurements assume an 8-1\/2$''$ $\\times$ 11$''$ page size. For\nA4-size paper, use the given top and left margins, column width,\nheight, and gap, and modify the bottom and right margins as necessary.\n\n\\subsection{Format of Electronic Manuscript}\n\nFor the production of the electronic manuscript, you must use Adobe's\n{\\em Portable Document Format} (PDF). A PDF file can be generated, for\ninstance, on Unix systems using {\\tt ps2pdf} or on Windows systems\nusing Adobe's Distiller. There is also a website with free software\nand conversion services: \\url{http:\/\/www.ps2pdf.com}. For reasons of\nuniformity, use of Adobe's {\\em Times Roman} font is strongly suggested. \nIn \\LaTeX2e{} this is accomplished by writing\n\\begin{quote} \n\\mbox{\\tt $\\backslash$usepackage\\{times\\}}\n\\end{quote}\nin the preamble.\\footnote{You may want also to use the package {\\tt\nlatexsym}, which defines all symbols known from the old \\LaTeX{}\nversion.}\n\nAdditionally, it is of utmost importance to specify the {\\bf\nletter} format (corresponding to 8-1\/2$''$ $\\times$ 11$''$) when\nformatting the paper. When working with {\\tt dvips}, for instance, one\nshould specify {\\tt -t letter}.\n\n\\subsection{Title and Author Information}\n\nCenter the title on the entire width of the page in a 14-point bold\nfont. The title must be capitalized using Title Case. Below it, center author name(s) in 12-point bold font. On the following line(s) place the affiliations, each affiliation on its own line using 12-point regular font. 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Such information can be added to the final camera-ready\nversion for publication.\n\n\\subsection{Abstract}\n\nPlace the abstract at the beginning of the first column 3$''$ from the\ntop of the page, unless that does not leave enough room for the title\nand author information. Use a slightly smaller width than in the body\nof the paper. Head the abstract with ``Abstract'' centered above the\nbody of the abstract in a 12-point bold font. The body of the abstract\nshould be in the same font as the body of the paper.\n\nThe abstract should be a concise, one-paragraph summary describing the\ngeneral thesis and conclusion of your paper. A reader should be able\nto learn the purpose of the paper and the reason for its importance\nfrom the abstract. The abstract should be no more than 200 words long.\n\n\\subsection{Text}\n\nThe main body of the text immediately follows the abstract. Use\n10-point type in a clear, readable font with 1-point leading (10 on\n11).\n\nIndent when starting a new paragraph, except after major headings.\n\n\\subsection{Headings and Sections}\n\nWhen necessary, headings should be used to separate major sections of\nyour paper. (These instructions use many headings to demonstrate their\nappearance; your paper should have fewer headings.). All headings should be capitalized using Title Case.\n\n\\subsubsection{Section Headings}\n\nPrint section headings in 12-point bold type in the style shown in\nthese instructions. Leave a blank space of approximately 10 points\nabove and 4 points below section headings.  Number sections with\narabic numerals.\n\n\\subsubsection{Subsection Headings}\n\nPrint subsection headings in 11-point bold type. Leave a blank space\nof approximately 8 points above and 3 points below subsection\nheadings. Number subsections with the section number and the\nsubsection number (in arabic numerals) separated by a\nperiod.\n\n\\subsubsection{Subsubsection Headings}\n\nPrint subsubsection headings in 10-point bold type. Leave a blank\nspace of approximately 6 points above subsubsection headings. Do not\nnumber subsubsections.\n\n\\paragraph{Titled paragraphs.} You should use titled paragraphs if and \nonly if the title covers exactly one paragraph. Such paragraphs should be\nseparated from the preceding content by at least 3pt, and no more than\n6pt. The title should be in 10pt bold font and ended with a period. \nAfter that, a 1em horizontal space should follow the title before \nthe paragraph's text.\n\nIn \\LaTeX{} titled paragraphs should be typeset using\n\\begin{quote}\n{\\tt \\textbackslash{}paragraph\\{Title.\\} text} .\n\\end{quote}\n\n\\subsubsection{Acknowledgements}\n\nYou may include an unnumbered acknowledgments section, including\nacknowledgments of help from colleagues, financial support, and\npermission to publish. If present, acknowledgements must be in a dedicated,\nunnumbered section appearing after all regular sections but before any\nappendices or references.\n\nUse \n\\begin{quote}\n    {\\tt \\textbackslash{}section*\\{Acknowledgements\\}})\n\\end{quote}\nto typeset the acknowledgements section in \\LaTeX{}.\n\n\\subsubsection{Appendices}\n\nAny appendices directly follow the text and look like sections, except\nthat they are numbered with capital letters instead of arabic\nnumerals. See this document for an example.\n\n\\subsubsection{References}\n\nThe references section is headed ``References'', printed in the same\nstyle as a section heading but without a number. A sample list of\nreferences is given at the end of these instructions. Use a consistent\nformat for references. The reference list should not include unpublished\nwork.\n\n\\subsection{Citations}\n\nCitations within the text should include the author's last name and\nthe year of publication, for example~\\cite{gottlob:nonmon}.  Append\nlowercase letters to the year in cases of ambiguity.  Treat multiple\nauthors as in the following examples:~\\cite{abelson-et-al:scheme}\nor~\\cite{bgf:Lixto} (for more than two authors) and\n\\cite{brachman-schmolze:kl-one} (for two authors).  If the author\nportion of a citation is obvious, omit it, e.g.,\nNebel~\\shortcite{nebel:jair-2000}.  Collapse multiple citations as\nfollows:~\\cite{gls:hypertrees,levesque:functional-foundations}.\n\\nocite{abelson-et-al:scheme}\n\\nocite{bgf:Lixto}\n\\nocite{brachman-schmolze:kl-one}\n\\nocite{gottlob:nonmon}\n\\nocite{gls:hypertrees}\n\\nocite{levesque:functional-foundations}\n\\nocite{levesque:belief}\n\\nocite{nebel:jair-2000}\n\n\\subsection{Footnotes}\n\nPlace footnotes at the bottom of the page in a 9-point font.  Refer to\nthem with superscript numbers.\\footnote{This is how your footnotes\nshould appear.} Separate them from the text by a short\nline.\\footnote{Note the line separating these footnotes from the\ntext.} Avoid footnotes as much as possible; they interrupt the flow of\nthe text.\n\n\\section{Illustrations}\n\nPlace all illustrations (figures, drawings, tables, and photographs)\nthroughout the paper at the places where they are first discussed,\nrather than at the end of the paper.\n\nThey should be floated to the top (preferred) or bottom of the page, \nunless they are an integral part \nof your narrative flow. When placed at the bottom or top of\na page, illustrations may run across both columns, but not when they\nappear inline.\n\nIllustrations must be rendered electronically or scanned and placed\ndirectly in your document. They should be cropped outside latex, otherwise portions of the image could reappear during the post-processing of your paper. All illustrations should be understandable when printed in black and\nwhite, albeit you can use colors to enhance them. Line weights should\nbe 1\/2-point or thicker. Avoid screens and superimposing type on\npatterns, as these effects may not reproduce well.\n\nNumber illustrations sequentially. Use references of the following\nform: Figure 1, Table 2, etc. Place illustration numbers and captions\nunder illustrations. Leave a margin of 1\/4-inch around the area\ncovered by the illustration and caption.  Use 9-point type for\ncaptions, labels, and other text in illustrations. Captions should always appear below the illustration.\n\n\\section{Tables}\n\nTables are considered illustrations containing data. Therefore, they should also appear floated to the top (preferably) or bottom of the page, and with the captions below them.\n\n\\begin{table}\n\\centering\n\\begin{tabular}{lll}\n\\hline\nScenario  & $\\delta$ & Runtime \\\\\n\\hline\nParis       & 0.1s  & 13.65ms     \\\\\nParis       & 0.2s  & 0.01ms      \\\\\nNew York    & 0.1s  & 92.50ms     \\\\\nSingapore   & 0.1s  & 33.33ms     \\\\\nSingapore   & 0.2s  & 23.01ms     \\\\\n\\hline\n\\end{tabular}\n\\caption{Latex default table}\n\\label{tab:plain}\n\\end{table}\n\n\\begin{table}\n\\centering\n\\begin{tabular}{lrr}  \n\\toprule\nScenario  & $\\delta$ (s) & Runtime (ms) \\\\\n\\midrule\nParis       & 0.1  & 13.65      \\\\\n            & 0.2  & 0.01       \\\\\nNew York    & 0.1  & 92.50      \\\\\nSingapore   & 0.1  & 33.33      \\\\\n            & 0.2  & 23.01      \\\\\n\\bottomrule\n\\end{tabular}\n\\caption{Booktabs table}\n\\label{tab:booktabs}\n\\end{table}\n\nIf you are using \\LaTeX, you should use the {\\tt booktabs} package, because it produces better tables than the standard ones. Compare Tables \\ref{tab:plain} and~\\ref{tab:booktabs}. The latter is clearly more readable for three reasons:\n\n\\begin{enumerate}\n    \\item The styling is better thanks to using the {\\tt booktabs} rulers instead of the default ones.\n    \\item Numeric columns are right-aligned, making it easier to compare the numbers. Make sure to also right-align the corresponding headers, and to use the same precision for all numbers.\n    \\item We avoid unnecessary repetition, both between lines (no need to repeat the scenario name in this case) as well as in the content (units can be shown in the column header).\n\\end{enumerate}\n\n\\section{Formulas}\n\nIJCAI's two-column format makes it difficult to typeset long formulas. A usual temptation is to reduce the size of the formula by using the {\\tt small} or {\\tt tiny} sizes. This doesn't work correctly with the current \\LaTeX{} versions, breaking the line spacing of the preceding paragraphs and title, as well as the equation number sizes. The following equation demonstrates the effects (notice that this entire paragraph looks badly formatted):\n\\begin{tiny}\n\\begin{equation}\n    x = \\prod_{i=1}^n \\sum_{j=1}^n j_i + \\prod_{i=1}^n \\sum_{j=1}^n i_j + \\prod_{i=1}^n \\sum_{j=1}^n j_i + \\prod_{i=1}^n \\sum_{j=1}^n i_j + \\prod_{i=1}^n \\sum_{j=1}^n j_i\n\\end{equation}\n\\end{tiny}%\n\nReducing formula sizes this way is strictly forbidden. We {\\bf strongly} recommend authors to split formulas in multiple lines when they don't fit in a single line. This is the easiest approach to typeset those formulas and provides the most readable output%\n\\begin{align}\n    x =& \\prod_{i=1}^n \\sum_{j=1}^n j_i + \\prod_{i=1}^n \\sum_{j=1}^n i_j + \\prod_{i=1}^n \\sum_{j=1}^n j_i + \\prod_{i=1}^n \\sum_{j=1}^n i_j + \\nonumber\\\\\n    + & \\prod_{i=1}^n \\sum_{j=1}^n j_i\n\\end{align}%\n\nIf a line is just slightly longer than the column width, you may use the {\\tt resizebox} environment on that equation. The result looks better and doesn't interfere with the paragraph's line spacing: %\n\\begin{equation}\n\\resizebox{.91\\linewidth}{!}{$\n    \\displaystyle\n    x = \\prod_{i=1}^n \\sum_{j=1}^n j_i + \\prod_{i=1}^n \\sum_{j=1}^n i_j + \\prod_{i=1}^n \\sum_{j=1}^n j_i + \\prod_{i=1}^n \\sum_{j=1}^n i_j + \\prod_{i=1}^n \\sum_{j=1}^n j_i\n$}\n\\end{equation}%\n\nThis last solution may have to be adapted if you use different equation environments, but it can generally be made to work. Please notice that in any case:\n\n\\begin{itemize}\n    \\item Equation numbers must be in the same font and size than the main text (10pt).\n    \\item Your formula's main symbols should not be smaller than {\\small small} text (9pt).\n\\end{itemize}\n\nFor instance, the formula\n\\begin{equation}\n    \\resizebox{.91\\linewidth}{!}{$\n    \\displaystyle\n    x = \\prod_{i=1}^n \\sum_{j=1}^n j_i + \\prod_{i=1}^n \\sum_{j=1}^n i_j + \\prod_{i=1}^n \\sum_{j=1}^n j_i + \\prod_{i=1}^n \\sum_{j=1}^n i_j + \\prod_{i=1}^n \\sum_{j=1}^n j_i + \\prod_{i=1}^n \\sum_{j=1}^n i_j\n$}\n\\end{equation}\nwould not be acceptable because the text is too small.\n\n\\section{Examples, Definitions, Theorems and Similar}\n\nExamples, definitions, theorems, corollaries and similar must be written in their own paragraph. The paragraph must be separated by at least 2pt and no more than 5pt from the preceding and succeeding paragraphs. They must begin with the kind of item written in 10pt bold font followed by their number (e.g.: Theorem 1), optionally followed by a title\/summary between parentheses in non-bold font and ended with a period. After that the main body of the item follows, written in 10 pt italics font (see below for examples). \n\nIn \\LaTeX{} We strongly recommend you to define environments for your examples, definitions, propositions, lemmas, corollaries and similar. This can be done in your \\LaTeX{} preamble using \\texttt{\\textbackslash{newtheorem}} -- see the source of this document for examples. Numbering for these items must be global, not per-section (e.g.: Theorem 1 instead of Theorem 6.1).\n\n\\begin{example}[How to write an example]\nExamples should be written using the example environment defined in this template.\n\\end{example}\n\n\\begin{theorem}\nThis is an example of an untitled theorem.\n\\end{theorem}\n\nYou may also include a title or description using these environments as shown in the following theorem.\n\n\\begin{theorem}[A titled theorem]\nThis is an example of a titled theorem.\n\\end{theorem}\n\n\\section{Proofs}\n\nProofs must be written in their own paragraph separated by at least 2pt and no more than 5pt from the preceding and succeeding paragraphs. Proof paragraphs should start with the keyword ``Proof.\" in 10pt italics font. After that the proof follows in regular 10pt font. At the end of the proof, an unfilled square symbol (qed) marks the end of the proof.\n\nIn \\LaTeX{} proofs should be typeset using the \\texttt{\\textbackslash{proof}} environment.\n\n\\begin{proof}\nThis paragraph is an example of how a proof looks like using the \\texttt{\\textbackslash{proof}} environment. \n\\end{proof}\n\n\n\\section{Algorithms and Listings}\n\nAlgorithms and listings are a special kind of figures. Like all illustrations, they should appear floated to the top (preferably) or bottom of the page. However, their caption should appear in the header, left-justified and enclosed between horizontal lines, as shown in Algorithm~\\ref{alg:algorithm}. The algorithm body should be terminated with another horizontal line. It is up to the authors to decide whether to show line numbers or not, how to format comments, etc.\n\nIn \\LaTeX{} algorithms may be typeset using the {\\tt algorithm} and {\\tt algorithmic} packages, but you can also use one of the many other packages for the task.  \n\n\\begin{algorithm}[tb]\n\\caption{Example algorithm}\n\\label{alg:algorithm}\n\\textbf{Input}: Your algorithm's input\\\\\n\\textbf{Parameter}: Optional list of parameters\\\\\n\\textbf{Output}: Your algorithm's output\n\\begin{algorithmic}[1]\n\\STATE Let $t=0$.\n\\WHILE{condition}\n\\STATE Do some action.\n\\IF {conditional}\n\\STATE Perform task A.\n\\ELSE\n\\STATE Perform task B.\n\\ENDIF\n\\ENDWHILE\n\\STATE \\textbf{return} solution\n\\end{algorithmic}\n\\end{algorithm}\n\n\\section*{Acknowledgments}\n\nThe preparation of these instructions and the \\LaTeX{} and Bib\\TeX{}\nfiles that implement them was supported by Schlumberger Palo Alto\nResearch, AT\\&T Bell Laboratories, and Morgan Kaufmann Publishers.\nPreparation of the Microsoft Word file was supported by IJCAI.  An\nearly version of this document was created by Shirley Jowell and Peter\nF. Patel-Schneider.  It was subsequently modified by Jennifer\nBallentine and Thomas Dean, Bernhard Nebel, Daniel Pagenstecher,\nKurt Steinkraus, Toby Walsh and Carles Sierra. The current version \nhas been prepared by Marc Pujol-Gonzalez and Francisco Cruz-Mencia.\n\n\n\\section{Introduction}\n\nReinforcement Learning (RL) is a dynamical learning paradigm, in which the algorithm (also known as the `agent') learns through sequential interaction with the environment. On each round, the agent performs an action, which transitions it into a new state, and is provided with a state-dependent reward. The goal of the agent is to maximize the cumulative reward it observes throughout these interactions with the environment. When considering tabular RL, in which there is a finite number of states and actions, there exist efficient algorithms with theoretical convergence and stability guarantees \\citep{sutton1998reinforcement,jaksch2010near,jin2018q,zanette2019tighter,efroni2019tight,dann2018policy}. However, this is not the case when considering deep RL approaches -- when a neural network is used to cope with large state spaces, such as images \\citep{mnih2015human,hessel2017rainbow}, and\/or large action spaces, for example in continuous control \\citep{lillicrap2015continuous,schulman2017proximal}.\n\nTwo issues hinder the stability of deep RL algorithms: (i) the optimization process is finite, and as such does not follow many of the requirements for the process to converge, such as decaying learning rates \\citep{borkar2008stochastic}, and (ii) the non-linearity of the function approximators (e.g., neural networks) results in a highly non-convex optimization landscape. Due to this, as can be seen in all empirical works \\citep{prashanth2014policy,schulman2015trust,lillicrap2015continuous,fujimoto2018addressing,haarnoja2018soft}, the learning process exhibits high variance; during a short time of training, a near-optimal policy may become arbitrarily bad. As a result, in recent years, the stability of RL algorithms has become a major concern of the research community \\citep{henderson2018deep}.\n\nInstead of tackling the stability problems in the optimization process, many previous works tackled sub-optimal performance by focusing on different aspects of the loss function \\citep{fujimoto2018addressing,haarnoja2018soft,tessler2019distributional}. While these methods are capable of finding relatively good policies, they are strictly inferior when compared to their tabular counterparts. The practical variants exhibit instability and high variance, whereas the classic algorithms ensure convergence and stability.\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=0.49\\linewidth]{figures\/hopper_td3}\n    \\includegraphics[width=0.49\\linewidth]{figures\/inverted_td3}\n    \\caption{Performance plots of the TD3 algorithm \\citep{fujimoto2018addressing}.}\n    \\label{fig:td3 issues}\n\\end{figure}\n\nAs the policy is represented using a non-linear neural network, a small change in the parameters $\\theta$ may result in a very large change in the output. This issue is illustrated in \\Cref{fig:td3 issues}, where a state-of-the-art off-policy approach learns to control the Hopper and InvertedPendulum robots.\n\nThese figures show the individual learning curves over several seeds, without temporal smoothing\\footnote{Temporal smoothing is a common practice in empirical studies, in which each data-point in the graph represents a moving average over the past N evaluations. While this makes the graphs easier to read, it hides a lot of important information such as the variance of each seed during training.}, whereas the bold line represents the mean performance across runs. These results paint a clear picture -- while the agent is capable of learning a policy which attains a score of approximately 3500 in Hopper and the optimal score of 1000 in InvertedPendulum, it is \\emph{highly unstable}, i.e., a policy which was near-optimal at time $t$ might become arbitrarily bad at time $t+1$.\n\nIn this work, we draw the connection between deep RL approaches and their tabular counterparts and highlight the issues inherent in deep RL algorithms. Through this connection, we propose a solution, in the form of a hierarchical training procedure. Specifically, we propose to replace the end-to-end training scheme with an ``evaluate $\\Longleftrightarrow$ improve\" cycle, similar to policy iteration. As opposed to the tabular case, our approach has no convergence guarantees, but rather is stable, i.e., \nwith high probability, the performance of the policy improves and does not suffer high degradation. \n\nWe validate our proposal empirically\\footnote{Anonymous code repository accompanying this submission \\href{https:\/\/bit.ly\/37bnd9A}{bit.ly\/37bnd9A}} across a vast range of continuous control tasks in MuJoCo. Specifically, we consider Risk and Reliability metrics proposed by \\cite{rl_reliability_metrics}. These results show that our proposed method indeed results in improved behavior both in terms of stability and overall performance.\n\n\\section{Background and Notation}\\label{sec: background}\nWe consider an infinite-horizon discounted Markov Decision Process \\citep[MDP]{puterman1994markov}. An MDP is defined as the 5-tuple $(\\mathcal{S}, \\mathcal{A},P,r,\\gamma)$, where ${\\mathcal S}$ is the state space, $\\mathcal{A}$ is the action space, ${P : \\mathcal{S} \\times \\mathcal{S} \\times \\mathcal{A} \\mapsto [0,1]}$ is a transition kernel, ${r : \\mathcal{S} \\times A \\to [r_\\text{min}, r_\\text{max}]}$ is a reward function and $\\gamma \\in (0, 1)$ is the discount factor.\n\nThe goal of an RL agent is to learn a policy $\\pi(\\state, \\action)$ that maps states into distributions over actions. A well known result in this setting is that there exists a deterministic optimal policy. \nThe quality of a policy is measured by its value, ${v^\\pi (\\state) = \\mathbb{E}^\\pi \\brs*{\\sum_{t=0}^\\infty \\gamma^t r(\\state_t, \\action_t) \\mid \\state_0 = \\state}}$, which is the average cumulative reward when starting from state $\\state$ and acting according to $\\pi$. Another important quantity is the Q-function, ${Q^\\pi (\\state, \\action) = \\mathbb{E}^\\pi \\brs*{\\sum_{t=0}^\\infty \\gamma^t r(\\state_t, \\action_t) \\mid \\state_0 = \\state, \\action_0 = \\action}}$, which is strongly connected to the value via the relation $v^\\pi(\\state) = \\int_{\\action} Q^\\pi (\\state, \\action) \\pi(\\state, \\action) d\\action$. We denote an optimal policy that maximizes the value simultaneously across all of the states by $\\pi^* \\in \\argmax_\\pi v^\\pi$ and the optimal value function by $v^*=v^{\\pi^*}$. \nWhen the $Q$-function and the policy $\\pi$ are represented by a parametric function, we denote them by $Q_\\phi$ and $\\pi_\\theta$, where $\\phi$ and $\\theta$ are the parameters of the function (e.g. the parameters of a deep neural network).\n\nWhile our goal is to maximize $v^\\pi$, in practice we measure an empirical estimation of $\\mathbb{E}_{\\state \\sim \\rho} [\\sum_t r_t$], where $\\rho$ is the initial state distribution. To accommodate for this fact, we define the cumulative reward of a sampled trajectory $i$ by\n\\begin{equation}\n    J_i^\\pi = \\sum_t r_t \\,.\n\\end{equation}\nThis is a random variable, as randomness may occur due to (i) the initial state distribution, (ii) the transition kernel and reward function and (iii) the policy.\n\n\n\n\nIn this work we focus on off-policy continuous control tasks, however our approach can be trivially extended to discrete action spaces.\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=0.65\\linewidth]{figures\/diagram-v2}\n    \\caption{Conservative Policy Gradients.}\n    \\label{fig: diagram}\n\\end{figure*}\n\n\\subsection{Off-Policy Deep RL for Continuous Control}\\label{sec: deep rl}\n\nAs opposed to discrete actions, in which the agent selects a single action index (scalar) at each step, in \\textbf{continuous control}, the agent provides the environment with a continuous vector. For instance, a robot with $d$ motors is provided an action vector $\\action \\in \\mathbb{R}^d$ where each dimension $a^{(i)}$ represents the amount of torque to apply to each motor. In addition, and in contrast to on-policy RL, off-policy methods utilize one policy for data collection and another for optimization.\n\nDDPG \\citep{lillicrap2015continuous} and its latest improvement TD3 \\citep{fujimoto2018addressing} use neural networks as function approximators in order to find the optimal continuous-control policy in an off-policy manner.\n\n        \n        \n                \n                \n                \n                \n\n\\textbf{Deep Deterministic Policy Gradients} \\citep[DDPG]{lillicrap2015continuous} aims to directly learn a deterministic policy in a continuous action space. Similar to Policy Iteration. The $Q$-function, also known as the critic, is trained to estimate the utility of the \\emph{target policy} $\\pi_{\\theta_{-}}$, whereas the online policy (the actor) is trained to find the 1-step greedy update w.r.t. this estimation. DDPG and Policy Iteration (PI) \\citep{howard1960dynamic} are connected through the target network update procedure. In PI, the previous policy $\\pi_k$ is evaluated by $v^{\\pi_k}$, followed by an improvement step. Thus, $\\pi_{k+1}$ becomes the one-step greedy policy w.r.t. $v^{\\pi_k}$. In this work, we build on-top of TD3 \\citep{fujimoto2018addressing}, an improved version of DDPG.\n\n\\textbf{Twin Delayed DDPG} \\citep[TD3]{fujimoto2018addressing} improves DDPG by tackling the problem of overestimation. Specifically, an additional critic is introduced, similarly to Double-DQN \\citep{van2016deep}. Theoretical analysis and empirical evaulation both show that by bootstrapping the minimal value between both estimators $Q_i (\\state, \\action) \\gets r + \\min_i Q_i (\\state', \\pi(\\state'))$, TD3 overcomes the overestimation problem.\n\n\nIn order to focus our analysis, and as TD3 is an improvement over DDPG, it will serve as the baseline to which we compare.\n\n\\subsection{Measuring Stability in RL}\nIn this work we focus on improving the stability and reliability of each seed during training. While it is of great importance to improve the reproducability of each run, i.e., that all seeds attain similar performance; in this work we will focus on the inner-seed metrics, which aim to measure the stability and reliability of each individual run.\n\nWe compare the various approaches using methods recently proposed by \\cite{rl_reliability_metrics}. Specifically, we will consider (i) Dispersion and (ii) Risk. Dispersion measures the width of the distribution, the inter-quantile range over a sliding window; whereas the Risk is measured using the Conditional-Value-at-Risk (CVaR) on the drawdown, which is the expectation over the $\\alpha$ worst performance downtrends during training.\n\nAn algorithm which performs well on these measures, is less likely to suddenly suffer from performance degradation (e.g., forget previously learned behaviors) and thus has a higher correlation with the theoretical methods which are montonically improving.\n\n\n\n\\section{Conservative Policy Gradients}\\label{sec: theory}\nWhile there exists a connection between the tabular approach and that which is used with non-linear function approximators, due to the non-convex optimization landscape, a small gradient step can lead to an arbitrarily bad solution. As such, current policy-based methods with non-linear function approximation (neural networks) behave poorly when compared to their tabular counterparts.\n\nWe propose a simple, conservative, yet intuitive approach for tackling these issues:\nevery $T$ time-steps, evaluate both the target network\\footnote{The target network needs to be re-evaluated in order to ensure an unbiased estimate of its performance.} and the online network. If with confidence of $1-\\delta$ the online network is better than the target, perform the switch $\\theta_- = \\theta$ (\\Cref{fig: diagram}). As opposed to the standard approach\\citep{lillicrap2015continuous,fujimoto2018addressing}, which performs Polyak averaging, we keep the target network fixed between updates.\n\nThis method introduces the ``evaluate $\\Longleftrightarrow$ improve\" cycle to deep RL; since in DDPG\/TD3 the critic \\textbf{evaluates} the utility of the target network $\\pi_k$, and as the target network is kept static $\\rightarrow$ the online network is \\textbf{optimizing the 1-step greedy} policy $\\pi_{k+1}$ with respect to the target. However, as the optimization landscape is non-convex, the online policy is not ensured to be monotonically improving -- thus we evaluate it periodically, until an improving policy has been found.\n\n\\subsection{Evaluating the Policies}\n\nOur approach requires determining whether or not an online policy is better, with probability $1 - \\delta$. Assuming $J^\\pi$, the performance of a sampled trajectory for any deterministic policy $\\pi$, is normally distributed, we may consider the $t$-test for evaluating improvement.\n\nWe formulate the null hypothesis as $\\mu_{\\pi_\\theta} = \\mathbb{E}_\\rho J^{\\pi_\\theta} \\le \\mathbb{E}_\\rho J^{\\pi_{\\theta_{-}}} = \\mu_{\\pi_{\\theta{-}}}$. In other words, we hypothesize that the online policy has not improved over the target policy, and use the Welch's one-tailed t-test \\citep{welch1947generalization}. The $t$-statistic is then defined as\n\\begin{equation}\n    t = \\left(\\hat{\\mu}_{\\pi_{\\theta}} - \\hat{\\mu}_{\\pi_{\\theta_{-}}}\\right) \/ \\left(\\sqrt{{\\hat{\\sigma}^2_{\\mu_{\\theta}}}\/{n_{\\pi_{\\theta}}} + {\\hat{\\sigma}^2_{\\mu_{\\theta_{-}}}}\/{n_{\\pi_{\\theta_{-}}}}}\\right) \\,,\n\\end{equation}\nwhere $\\hat{\\mu}_\\pi$ is the empirical estimate of $\\mathbb{E}_\\rho J^\\pi$, $\\hat{\\sigma}_\\pi$ is its standard deviation and $n_\\pi$ the number of sampled trajectories. \n\nThe $t$-statistic enables us to find the $p$-value, which represents the confidence in the null hypothesis. Under the Gaussian assumption, the $p$-value can be directly calculated by $p=1-\\Phi(t)$, where $\\Phi(x)$ is the cumulative distribution function of a standard normal random variable. Hence, when $p > 1 - \\delta$ we have a high enough confidence that the null hypothesis is incorrect, meaning the new policy is indeed better, in which case we update the target policy ($\\theta_\\text{target} = \\theta_\\text{online}$).\n\n\n\\begin{figure*}\n    \\centering\n    \\begin{subfigure}[b]{0.6\\linewidth}\n        \\centering\n        \\includegraphics[width=\\linewidth]{figures\/median}\n        \\caption{\\label{fig: median}}\n    \\end{subfigure}\n    \\\\\n    \\begin{subfigure}[b]{0.6\\linewidth}\n        \\centering\n        \\includegraphics[width=\\linewidth]{figures\/iqr}\n        \\caption{\\label{fig: iqr}}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.18\\linewidth}\n        \\centering\n        \\includegraphics[width=\\linewidth]{figures\/srt}\n        \\caption{\\label{fig: srt}}\n    \\end{subfigure}\n    \\caption{Reliability metrics and median performance for TD3 (Baseline), CNSV (Conservative), MWE (Max w\/ re-evaluation), and MNE (Max w\/o re-evaluation). Rank 1 always indicates ``best\" reliability \\citep{rl_reliability_metrics}.}\n    \\label{fig: comparisons2}\n\\end{figure*}\n\n\\subsection{Parameter Selection}\n\nThere are three parameters: (i) how often we evaluate the online policy, (ii) how many episodes are used for evaluation and (iii) the minimal confidence required for policy improvement. The confidence level and the number of evaluations are strongly connected; \nas the number of evaluations grows, the confidence itself increases. Hence, in environments which are characterized by high variance, a small number of evaluations will result in a very low confidence, even when the empirical mean is higher. On the other hand, the evaluation frequency controls how fast the process converges.\n\nWe observed good performance when the confidence is set to $\\delta \\in [0.1, 0.2]$, the evaluation episode number is $K = 10$ and evaluation is performed every $N$ (Hopper, InvertedPendulum and InvertedDoublePendulum $10,000$; and Walker2d, Ant and HalfCheetah $50,000$) training steps.\n\n\\section{Experiments}\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=0.4\\linewidth,height=0.355\\linewidth]{figures\/walker_img}\\hfill\n    \\includegraphics[width=0.4\\linewidth]{figures\/hopper_img}\n    \\caption{Illustration of MuJoCo tasks.}\n    \\label{fig:mujoco}\n\\end{figure}\n\nFor all the experiments, we plot 5 different random seeds for each configuration. Each seed is trained for a total of $1e6$ interactions with the environment (when evaluation is used for network update, these samples are included in the total interactions) and evaluated every $1,000$ training steps (regardless of how often the evaluation for policy selection is performed). Each seed is evaluated for 100 episodes, for which we present the average performance -- this high number of evaluations is aimed to ensure a high confidence in the reported mean, thus any observed instability is likely due to performance degradation in the policy selection procedure. We plot both the individual seeds (light curves) and the average results across seeds (dark curves). As opposed to previous works, we do not smooth the per-seed graphs using a moving average procedure, thus enabling a better analysis of the experiments' variance and stability.\n\n\n\\subsection{Results}\n\nWe follow the same hyper-parameter scheme as is used for the TD3 algorithm and perform 4 evaluations:\n\\begin{enumerate}\n    \\item \\textbf{Baseline:} We report the performance of the baseline TD3 algorithm which updates the target network at each step in a Polyak averaging procedure.\n    \n    \\item \\textbf{Conservative update:} This is our proposed conservative method which periodically evaluates both the target and online policy. If with high probability the online policy improves upon the target -- the target is updated.\n    \n    \\item \\textbf{Max with re-evaluation:} Similar to the conservative approach, the max over means evaluates both the target and online network, however it does not consider the variance. This method simply updates the target based on the empirical means.\n    \n    \\item \\textbf{Max without re-evaluation:} This method is more sample efficient than (3), however, it results in a biased estimator of the performance. This in turn is expected to result in a higher variance.\n\\end{enumerate}\n\n\\begin{figure*\n\\centering\n\\begin{tabular}{>{\\centering\\arraybackslash}m{.22\\linewidth} >{\\centering\\arraybackslash}m{.22\\linewidth} >{\\centering\\arraybackslash}m{.22\\linewidth} >{\\centering\\arraybackslash}m{.22\\linewidth}}\n ~~Conservative & ~~Max w\/ re-evaluation & ~~Max w\/o re-evalution & ~~Baseline \\\\\n \\includegraphics[width=\\linewidth]{figures\/hopper_prob} & \n \\includegraphics[width=\\linewidth]{figures\/hopper_max} & \\includegraphics[width=\\linewidth]{figures\/hopper_max_noreeval} &\n \\includegraphics[width=\\linewidth]{figures\/hopper_td3} \\\\\n \\includegraphics[width=\\linewidth]{figures\/inverted_prob} & \\includegraphics[width=\\linewidth]{figures\/inverted_max} &\n \\includegraphics[width=\\linewidth]{figures\/inverted_max_noreeval} & \n \\includegraphics[width=\\linewidth]{figures\/inverted_td3}\\\\\n \\includegraphics[width=\\linewidth]{figures\/inverteddouble_prob} & \n \\includegraphics[width=\\linewidth]{figures\/inverteddouble_max} &\n \\includegraphics[width=\\linewidth]{figures\/inverteddouble_max_noreeval} &\n \\includegraphics[width=\\linewidth]{figures\/inverteddouble_td3} \\\\\n \\includegraphics[width=\\linewidth]{figures\/walker_prob} &\n \\includegraphics[width=\\linewidth]{figures\/walker_max} &\n \\includegraphics[width=\\linewidth]{figures\/walker_max_noreeval} &\n \\includegraphics[width=\\linewidth]{figures\/walker_td3} \\\\\n \\includegraphics[width=\\linewidth]{figures\/cheetah_prob} &\n \\includegraphics[width=\\linewidth]{figures\/cheetah_max} &\n \\includegraphics[width=\\linewidth]{figures\/cheetah_max_noreeval} &\n \\includegraphics[width=\\linewidth]{figures\/cheetah_td3}  \\\\\n \\includegraphics[width=\\linewidth]{figures\/ant_prob} &\n \\includegraphics[width=\\linewidth]{figures\/ant_max} &\n \\includegraphics[width=\\linewidth]{figures\/ant_max_noreeval} &\n \\includegraphics[width=\\linewidth]{figures\/ant_td3}\n\\end{tabular}\n\\caption{x-axis denotes the total number of environment interactions.}\n\\label{fig: results}\n\\end{figure*}\n\nThe experiments were performed in the MuJoCo control suite, a continuous control benchmark in which the goal is to learn to control a robotic agent in a forward movement task, and the results are presented in \\Cref{fig: results}. These results show that, across all domains, the conservative approach (Conservative-TD3) reduces the variance of individual seeds, also seen in \\Cref{fig: comparisons2,fig: comparisons}, and results in increased stability across most domains.\n\nInvertedPendulum and InvertedDoublePendulum serve as perfect examples. While TD3 is capable of finding an optimal policy - e.g., the maximal score - subsequent policies may be arbitrarily bad. By adding the hierarchical selection scheme, our approach is capable of mitigating this behavior. That is, once an optimal policy is found, it is not likely to be replaced with an inferior one.\n\n\\begin{figure*}\n    \\centering\n   \n   \n        \\includegraphics[width=0.7\\linewidth,height=0.37\\linewidth]{figures\/srt_per_domain}\n       \n   \n    \\caption{Per-domain stability metrics for TD3 (Baseline), CNSV (Conservative), MWE (Max w\/ re-evaluation), and MNE (Max w\/o re-evaluation).}\n    \\label{fig: comparisons}\n\\end{figure*}\n\n\\subsection{Performance Analysis:}\nIn addition to providing the full learning curves (\\Cref{fig: results}), we perform an in-depth analysis of the learning process based on recent reliability measures proposed by \\cite{rl_reliability_metrics} in \\Cref{fig: comparisons2,fig: comparisons}. Specifically, we analyze:\n\n\\paragraph{\\Cref{fig: median}, Mean performance during training:} We observe that while the baseline (TD3 with Polyak averaging) has better initial performance; overtime the Conservative and Max methods surpass it, where the best performing (in terms of raw performance) is the Conservative update rule.\n\n\\paragraph{\\Cref{fig: iqr}, Inner Quantile Range across time:} Dispersion (width of distribution) across Time aims to measure reliability by isolating higher-frequency variability, rather than capturing longer-term trends. This is achieved by calculating the IQR over a sliding window, after detrending the data. We observe that based on this metric, the Conservative method outperforms the rest throughout the entire learning process.\n\n\\paragraph{\\Cref{fig: srt,fig: comparisons}, Lower CVaR on Differences:} This measure aims to measure the most extreme short-term drop over time. This is done by applying CVaR to the changes in performance from one evaluation point to the next. This method provides the worst-case expected drop in performance during training, from one point of evaluation to the next. In the top right, we present the aggregated score across environments, a somewhat surprising result. While the Conservative method outperforms the baseline, the biased Max without re-evaluation estimator attains a similar score. Additionally, in the bottom we provide a more detailed per-domain comparison; e.g., how stable each method behaves for each domain. Although we are unable to find a single method which performs optimally on all domains, these results provide additional insights which can't be observed in \\Cref{fig: results}. For instance, the Max w\/ and w\/o re-evaluation are less stable than the Conservative in the InvertedPendulum domain.\n\n\n\n\n\n\\subsection{Potential Extensions}\nIn this work, we presented an approach for providing probabilistic policy improvement. The performance of a trajectory is a random variable, as the initial state, the transition kernel and sometimes the policy -- may all be stochastic. To determine which policy $\\pi_\\text{target}$ or $\\pi$ (the target or online) is better, we performed $K$ evaluations, assumed that the samples are normally distributed and performed a $t$-test. Below we present several intuitive extensions to our work:\n\n\\paragraph{Unknown distribution:} While the Gaussian assumption worked well in practice, it may not always be correct. In such a scenario, we see two possible routes: (i) known range, in which the minimal and maximal possible values are known, where one may use the Empirical Bernstein \\citep{maurer2009empirical}, or (ii) when the range is unknown, one may consider a bootstrap-based approach for estimating the confidence intervals around the mean \\citep{abhishek2017nonparametric} or non-parametric methods such as the Mann\u2013Whitney U-test \\citep{mann1947test}.\n    \n\n\n\\paragraph{Adaptive Sampling:} In our experiments, we succeeded to run the method with a rather small number of evaluations, and thus resorted to an equal number of evaluations of each policy -- 10 evaluations for each policy. However, when provided with a larger budget, it is of great interest to combine adaptive sampling procedures. \\cite{gabillon2011multi} propose such an approach, which takes into account the empirical mean and variance of each policy evaluation in order to determine which policy to evaluate. At the end of this process, the policy with the highest empirical mean is returned.\n\n\n\n\\section{Related Work}\n\n\n\\paragraph{Meta-RL:} In Meta-RL, similar to our method, an additional algorithm runs on top of the RL agent and adapts various algorithmic parameters (e.g., ``interferes with the end-to-end nature). RL\\^2 \\citep{duan2016rl} adaptively learns the learning algorithm, Meta-gradients \\citep{xu2018meta} optimize $\\gamma$, and LIRPG \\citep{zheng2018learning} learn an intrinsic reward.\n\n\\paragraph{Safe Policy Improvement:} Given a baseline policy, safe policy improvement (SPI) algorithms aim to find a new policy that is better than the baseline with high probability \\citep{thomas2015high,laroche2017safe}. \nAlthough this may seem similar to our update stage, there are two major differences: (i) most SPI algorithms directly calculate the improving policy, which is much less efficient than continuously updating the policy and performing the switch when improvement is achieved, and (ii) due to safety constraints, SPI algorithms usually perform off-policy evaluation for the suggested policy, which is still an open problem in high dimensional problems. In contrast, we evaluate it directly and thus enjoy a better sample complexity.\n\n\n\n\\section{Conclusions}\nIn this work, we tackled the stability problems of policy-based reinforcement learning approaches. Even though the empirical gradient (SGD) is a noisy estimate of the ideal descent direction, the policy's performance varies greatly during training. This is due to the structure of the optimization landscape and the large step sizes used in practice.\n\nDrawing from the connection between actor-critic approaches and approximate policy iteration schemes, we introduced a hierarchical training method. Our method replaces the per-step target network update using Polyak averaging, with a periodic conservative update. This scheme results in a deep RL method which learns in an ``evaluate $\\Longleftrightarrow$ improve\" cycle, similar to policy iteration schemes.\n    \nWe validate our proposal on several continuous control tasks in MuJoCo and analyze it using recent reliability metrics proposed by \\cite{rl_reliability_metrics}. The empirical evidence suggests that our method is indeed more reliable, stable and improves overall performance across most domains. This raises the question as to what other improvements we may obtain by providing a higher correlation between the deep RL algorithms and the dynamic programming methods, with theoretical guarantees, upon which they have been built.\n\n\n\\section{Introduction}\n\n\nReinforcement Learning (RL) is a dynamical learning paradigm, in which the algorithm (also known as the `agent') learns through sequential interaction with the environment. On each round, the agent performs an action, which transitions it into a new state, and is provided with a state-dependent reward. The goal of the agent is to maximize the cumulative reward it observes throughout these interactions with the environment. When considering tabular RL, in which there is a finite number of states and actions, there exist efficient algorithms with theoretical convergence and stability guarantees \\citep{sutton1998reinforcement,jaksch2010near,jin2018q,zanette2019tighter,efroni2019tight,dann2018policy}. However, this is not the case when considering deep RL approaches -- when a neural network is used to cope with large state spaces, such as images \\citep{mnih2015human,hessel2017rainbow}, and\/or large action spaces, for example in continuous control \\citep{lillicrap2015continuous,schulman2017proximal}.\n\nThe issue with deep RL is twofold: (i) the optimization process is finite, and as such does not follow many of the requirements for the process to converge, such as decaying learning rates \\citep{borkar2008stochastic}, and (ii) the non-linearity of the function approximators (e.g., neural networks) results in a highly non-convex optimization landscape. Due to this, as can be seen in all empirical works \\citep{prashanth2014policy,schulman2015trust,lillicrap2015continuous,fujimoto2018addressing,haarnoja2018soft}, the learning process exhibits high variance; during a short time of training, a near-optimal policy may become arbitrarily bad. As a result, in recent years, the stability of RL algorithms has become a major concern of the research community \\citep{henderson2018deep}.\n\nInstead of tackling the stability problems in the optimization process, many previous works focused on different aspects of the loss function \\citep{fujimoto2018addressing,haarnoja2018soft,tessler2019distributional}. Then, they usually apply a standard variant of gradient descent algorithm on the modified loss function. While these approaches are capable of finding relatively good policies, they are strictly inferior when compared to their tabular counterparts -- the tabular approaches ensure convergence and stability, whereas the practical variants exhibit instability and high variance.\n\n\n\nIn this work, we suggest a novel approach that improves the stability of actor-critic methods. Specifically, we draw the connection between deep RL approaches and their tabular counterparts and highlight the issues inherent in deep RL algorithms. Through this connection, we propose a solution, in the form of a hierarchical training procedure. As opposed to the tabular case, our approach is not ensured to converge, but rather has stability guarantees, i.e., the stationary distribution of the policies is shown to be stable, meaning that with high probability the performance of the policy improves and does not suffer high degradation. We show, empirically across a range of continuous control tasks in MuJoCo, that our approach indeed results in improved behavior both in terms of stability and overall performance.\n\n\\section{Background and Notation}\\label{sec: background}\nWe consider an infinite-horizon discounted Markov Decision Process \\citep[MDP]{puterman1994markov}. An MDP is defined as the 5-tuple $(\\mathcal{S}, \\mathcal{A},P,r,\\gamma)$, where ${\\mathcal S}$ is the state space, $\\mathcal{A}$ the action space, ${P : \\mathcal{S} \\times \\mathcal{S} \\times \\mathcal{A} \\mapsto [0,1]}$ is a transition kernel, ${r : \\mathcal{S} \\times A \\to [r_\\text{min}, r_\\text{max}]}$ is a reward function and $\\gamma \\in (0, 1)$ is the discount factor.\n\n\nThe goal of an RL agent is to learn a policy $\\pi(\\state, \\action)$ that maps states into a distribution over actions. \nThe quality of a policy is measured by its value, ${v^\\pi (\\state) = \\mathbb{E}^\\pi \\brs*{\\sum_{t=0}^\\infty \\gamma^t r(\\state_t, \\action_t) \\mid \\state_0 = \\state}}$, which is the average cumulative reward when starting from state $\\state$ and acting according to $\\pi$. Another important quantity is the Q-function, ${Q^\\pi (\\state, \\action) = \\mathbb{E}^\\pi \\brs*{\\sum_{t=0}^\\infty \\gamma^t r(\\state_t, \\action_t) \\mid \\state_0 = \\state, \\action_0 = \\action}}$, which is strongly connected to the value through the relation $v^\\pi(\\state) = \\int_{\\action} Q^\\pi (\\state, \\action) \\pi(\\state, \\action) d\\action$. We denote an optimal policy that maximizes the value simultaneously across all of the states by $\\pi^* \\in \\argmax_\\pi v^\\pi$ and the optimal value function by $v^*=v^{\\pi^*}$. \nWhen the $Q$-function and the policy $\\pi$ are represented by a parametric function, we denote them by $Q_\\phi$ and $\\pi_\\theta$, where $\\phi$ and $\\theta$ are the parameters of the function (e.g. the parameters of a deep neural network).\n\nWhile our goal is to maximize $v^\\pi$, in practice we measure an empirical estimation of $\\mathbb{E}_{\\state \\sim \\rho} v^\\pi (\\state)$, where $\\rho$ is the initial state distribution. To accommodate for this fact, we define the cumulative reward of a sampled trajectory by $J^\\pi$. This is a random variable, as randomness may occur due to (i) the initial state distribution, (ii) the transition kernel and reward function and (iii) the policy.\n\n\\begin{figure*}\n    \\begin{center}\n    \\begin{minipage}[t]{\\textwidth}\n        \\begin{algorithm}[H]\n        \\caption{Value Iteration}\\label{alg:vi}\n        \\begin{algorithmic}[1]\n        \\State \\textbf{Input:} stopping criteria $\\epsilon$\n        \\State Initialize $v_0$\n        \\State $k = 0$\n        \n        \\While{$||v_k - v_{k-1}||_\\infty > \\epsilon$}\n            \\For{$\\state \\in \\mathcal{S}$}\n                \\State \\textcolor{darkgreen}{$v_{k+1} (\\state) = \\max_{\\action \\in \\mathcal{A}} r(\\state, \\action) + \\gamma \\sum_{\\state' \\in \\mathcal{S}} P(\\state, \\action, \\state') v_k (\\state')$}\n            \\EndFor\n            \\State $k = k + 1$\n        \\EndWhile\n        \\State \\textbf{return} value $v_k$\n        \\end{algorithmic}\n        \\end{algorithm}\n    \\end{minipage} \\vspace{-0.1cm}\n    \\begin{minipage}[t]{\\textwidth}\n        \\begin{algorithm}[H]\n        \\caption{Deep Q Learning}\\label{alg:dqn}\n        \\begin{algorithmic}[1]\n        \\State \\textbf{Input:} learning rate $\\alpha$, target update interval $\\tau$ and steps $T$.\n        \\State Initialize action-value function $\\phi$ and replay buffer $R$\n        \\State Set $\\phi_{-} = \\phi, t = 0$\n        \n        \\While{$t < T$}\n           \n           \n               \n               \n               \n                \n                \\State Collect data and append to replay buffer $R$\n                \n                \\State $t = t + 1$\n                \n                \\State $\\{\\state_i, \\action_i, r_i, \\state_{i+1}\\} = R.\\text{sample}()$ \\Comment{Training phase}\n                \\State \\textcolor{darkgreen}{$\\phi = \\phi - \\alpha \\nabla_\\phi ||r_i + \\gamma \\max_{\\action} Q_{\\phi_{-}} (\\state_{i+1}, \\action) - Q_{\\phi} (\\state_i, \\action_i)||_2^2$} \\label{algline:dqn update}\n                \n                \\If{$(t \\;\\;\\text{mod}\\;\\; \\tau) == 0$}\n                    \\State $\\phi_{-} = \\phi$ \\Comment{Update target network}\n                \\EndIf\n           \n        \\EndWhile\n        \\end{algorithmic}\n        \\end{algorithm}\n    \\end{minipage}\n    \\end{center}\n    \\centering\n    \\emph{Colored lines represent similarities between the classic and modern approach.} \\vspace{-0.2cm}\n\\end{figure*}\n\n\\subsection{Classic Reinforcement Learning Algorithms}\nValue Iteration and Policy Iteration \\citep[VI, PI]{howard1960dynamic} are algorithms for solving tabular RL tasks with convergence guarantees.\n\n\\textbf{Value Iteration} (\\Cref{alg:vi}) follows a simple iterative scheme, in which at each step, the value is updated using the Bellman Operator (\\Cref{algline:bellman}). That is, the value on each state is estimated using the value on future states, which is taken as a bootstrap from the previous estimation. This approach has a contraction property and is proven to converge asymptotically to the fixed point -- the value of the optimal policy $v^*$. While this approach converges asymptotically, it converges at an exponential rate of $\\bigO(\\gamma^k)$, i.e., $|| v^k - v^* ||_\\infty \\leq \\frac{\\gamma^k}{1 - \\gamma} r_\\text{max}$.\n\n\\begin{figure*}\n    \\begin{center}\n    \\begin{minipage}[t]{\\textwidth}\n        \\begin{algorithm}[H]\n        \\caption{Policy Iteration}\\label{alg:pi}\n        \\begin{algorithmic}[1]\n        \\State Initialize $\\pi_0$\n        \\State $k = 0$\n        \n        \\While{not converged}\n            \\While{not converged} \\Comment{Policy Evaluation}\n                \\For{$\\state \\in \\mathcal{S}$}\n                    \\State \\textcolor{darkgreen}{$v^{\\pi_k} (\\state) = r(\\state, \\pi_k(\\state)) + \\gamma \\sum_{\\state' \\in \\mathcal{S}} P(\\state, \\pi_k(\\state), \\state') v^{\\pi_k} (\\state')$} \\label{algline:bellman}\n                \\EndFor\n            \\EndWhile\n            \\For{$\\state \\in \\mathcal{S}$} \\Comment{Policy Improvement}\n                \\State \\textcolor{blue}{$\\pi_{k+1} (\\state) \\in \\argmax_{\\action \\in \\mathcal{A}} r(\\state, \\action) + \\gamma \\sum_{\\state' \\in \\mathcal{S}} P(\\state, \\action, \\state') v^{\\pi_k} (\\state')$}\n            \\EndFor\n            \\State $k = k + 1$\n        \\EndWhile\n        \\State \\textbf{return} policy $\\pi_k$\n        \\end{algorithmic}\n        \\end{algorithm}\n    \\end{minipage} \\vspace{-0.1cm}\n    \\begin{minipage}[t]{\\textwidth}\n        \\begin{algorithm}[H]\n        \\caption{Deep Deterministic Policy Gradients}\\label{alg:ddpg}\n        \\begin{algorithmic}[1]\n        \\State \\textbf{Input:} learning rate $\\alpha$, averaging parameter $\\tau$, steps $T$ and exploration $\\mathcal{N}$.\n        \\State Initialize policy $\\theta$, critic $\\phi$ and replay buffer $R$\n        \\State Set $\\theta_{-} = \\theta, \\phi_{-} = \\phi, t = 0$\n        \n        \\While{$t < T$}\n           \n           \n               \n               \n               \n                \n                \\State Collect data and append to replay buffer $R$\n                \\State $t = t + 1$\n                \n                \\State $\\{\\state_i, \\action_i, r_i, \\state_{i+1}\\} = R.\\text{sample}()$ \\Comment{Training phase}\n                \\State \\textcolor{darkgreen}{$\\phi = \\phi - \\alpha \\nabla_\\phi ||r_i + \\gamma Q_{\\phi_{-}} (\\state_{i+1}, \\pi_{\\theta_{-}} (\\state_{i+1})) - Q_{\\phi} (\\state_i, \\action_i)||_2^2$}\n                \\State \\textcolor{blue}{$\\theta = \\theta + \\alpha \\nabla_\\theta Q_{\\phi_{-}} (\\state_i, \\pi_\\theta (\\state_i))$}\n                \n                \\State $\\theta_{-} = (1 - \\tau) \\theta_{-} + \\tau \\theta$, $\\phi_{-} = (1 - \\tau) \\phi_{-} + \\tau \\phi$ \\Comment{Update target networks}\n           \n        \\EndWhile\n        \\end{algorithmic}\n        \\end{algorithm}\n    \\end{minipage}\n    \\end{center}\n    \\centering\n    \\emph{Colored lines represent similarities between the classic and modern approach.}\n    \\vspace{-0.2cm}\n\\end{figure*}\n\n\n\\textbf{Policy Iteration} (\\Cref{alg:pi}) iterates between two steps: (i) \\emph{policy evaluation} in which the current policy $\\pi_k$ is evaluated, producing the value $v^{\\pi_k}$ and (ii) \\emph{policy improvement} in which the policy $\\pi_{k+1}$ is updated greedily w.r.t. $v^{\\pi_k}$. This iterative scheme is proven to converge to the optimal policy.\n\nPolicy Iteration schemes have also been extended to the approximate case. \\citet{scherrer2014approximate} provides an overview of such approaches, which are called Approximate Policy Iteration (API). In API we consider the scenario in which $||\\hat v^{\\hat \\pi_k} - v^{\\pi_k}||_\\infty \\leq \\epsilon$. This approximation error can be caused due to two factors, the first being the inability to estimate the \\emph{value} and the second being the inability to find the \\emph{greedy policy}, i.e., the $\\argmax$ at each state. For instance, such errors may occur due to the functional class being used, e.g., neural networks. Most importantly, the sub-optimality of these approaches is bounded by $||v^{\\hat \\pi_\\infty} - v^*||_\\infty \\leq \\frac{\\epsilon}{(1 - \\gamma)^2}$ \\citep{scherrer2014approximate}, i.e., the sub-optimality of the final policy is proportional to the approximation error $\\epsilon$.\n\n\\subsection{Off-Policy Deep Reinforcement Learning}\\label{sec: deep rl}\nTwo fundamental Deep RL (DRL) approaches are the DQN \\citep{mnih2015human} and DDPG \\citep{lillicrap2015continuous}. Both approaches utilize a target network, which is crucial for the success of the methods. While in their original work, this was merely considered as a design parameter, we argue that the importance of these target networks lies in the connection to classical RL -- namely Value and Policy Iteration.\n\n\\textbf{Deep Q Networks} \\citep[DQN]{mnih2015human} (\\Cref{alg:dqn}) is a gradient-based approach for learning the action-value function $Q(\\state, \\action)$. In their work, \\citeauthor{mnih2015human} combine two concepts which are crucial for the approach to converge -- the experience replay and the target network. The \\emph{target network} is an additional neural network which is kept fixed for a pre-defined period of time, from which the next-state value is bootstrapped. The gradient steps of \\Cref{algline:dqn update} estimate the 1-step greedy update w.r.t. the target network (the previous value function), and thus try to approximate the update stage of VI.\n\nNotice that the convergence guarantees in VI are based on the contraction property, which is similar to saying that the estimation error is decreasing over time. \\citet{schaul2015prioritized} proposed a DRL variant based on prioritized sweeping \\citep{moore1993prioritized}, which prioritizes states during learning based on the TD-error. The Prioritized Experience Replay \\citep{hessel2017rainbow}, which has been observed to be one of the major performance boosters in value-based DRL, attempts to enforce this contractive behavior in a DRL setting, thus reducing the gap between the theoretical and practical behavior.\n\n\\textbf{Deep Deterministic Policy Gradients} \\citep[DDPG]{lillicrap2015continuous} (\\Cref{alg:ddpg}) aims to directly learn a deterministic policy in a continuous action space. DDPG has both policy evaluation and policy improvement steps. The $Q$-function, also known as the critic, is trained to estimate the utility of the \\emph{target policy} $\\pi_{\\theta_{-}}$, whereas the online policy (the actor) is trained to find the 1-step greedy update w.r.t. this estimation. Similarly to the connection between DQN and VI, DDPG and PI are connected through the target network update procedure. In PI, the previous policy $\\pi_k$ is evaluated by $v^{\\pi_k}$, followed by an improvement step in which $\\pi_{k+1}$ becomes the one-step greedy policy w.r.t. this estimate. DDPG follows a similar scheme, in which the critic $Q_\\phi$ estimates the utility of the policy $\\pi_{\\theta_{-}}$, followed by the online policy $\\pi_\\theta$ which is updated w.r.t. this estimate. In this work, we compare to TD3 \\citep{fujimoto2018addressing}, an improved version of DDPG.\n\n\nAs opposed to VI, practical PI approaches (such as DDPG) do not exhibit the theoretical behavior of PI. While the expected behavior for value-based methods is a minimization of the TD-error (contraction); in policy-based methods, the desired behavior corresponds to the improvement of the policy between iterations.\n\n\n\n\n\n\\section{Instability in Policy-based Deep Reinforcement Learning}\nIn the previous section, we provided an overview of two classical algorithms, VI and PI, which have convergence guarantees, and two off-policy DRL variants. While the prioritized experience replay \\citep{schaul2015prioritized} attempts to provide DQN with the required contractive behavior, there is no such approach for policy-based learning.\n\nAs the policy is represented using a non-linear neural network, a small change in the parameters $\\theta$ may result in a very large change in the output. We illustrate this issue in \\Cref{fig:td3 issues}, where a state of the art off-policy approach learns to control the Hopper and InvertedPendulum robots.\n\n\\begin{figure}[H]\n    \\centering\n    \\vspace{-0.1cm}\n    \\includegraphics[width=50mm]{figures\/hopper_td3_actor}~~~~~~~~~~~~~\n    \\includegraphics[width=50mm]{figures\/inverted_td3_actor}\n    \\caption{Performance plots of the TD3 \\citep{fujimoto2018addressing} algorithm.}\n    \\label{fig:td3 issues} \\vspace{-0.15cm}\n\\end{figure}\n\nIn these figures, we show the individual learning curves over several seeds, without performing temporal smoothing\\footnote{Temporal smoothing is a common practice in empirical studies, in which each data-point in the graph represents a moving average over the past N evaluations. While this makes the graphs easier to read, it hides a lot of important information such as the variance of each seed during training.}, whereas the bold line represents the mean performance across runs. These results paint a clear picture -- while the agent is capable of learning a policy which attains a score of approximately 3500 in Hopper and the optimal score of 1000 in InvertedPendulum, it is \\emph{highly unstable}, i.e., a policy which was near-optimal at time $t$ might become arbitrarily bad at time $t+1$.\n\n\\section{Conservative Policy Gradients\n}\\label{sec: theory}\nWhile there exists a connection between the tabular approach and that which is used with non-linear function approximators, due to the non-convex optimization landscape, a small gradient step can lead to an arbitrarily bad solution. As such, current policy-based methods with non-linear function approximation (neural networks) behave poorly when compared to their tabular counterparts.\n\nWe propose a simple, conservative, yet theoretically grounded approach for tackling these issues. Taking inspiration from the classic approaches, we observe that the gradient-based algorithms lack the improvement guarantees. To overcome this issue, we propose a hierarchical learning procedure: while the online network is trained as before (as seen in \\Cref{alg:ddpg}), the target network is updated periodically if, with high probability, the online policy is better.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{figures\/diagram-v2} \\vspace{-0.1cm}\n    \\caption{Conservative Policy Gradients.}\n    \\label{fig: diagram} \\vspace{-0.15cm}\n\\end{figure}\n\nIn simple terms, \\emph{our proposal is as follows:} every $T$ time-steps, evaluate both the target network\\footnote{The target network needs to be re-evaluated in order to ensure an unbiased estimate of its performance.} and the online network. If with confidence of $1-\\delta$ the online network is better than the target, perform the switch $\\theta_- = \\theta$. In addition, between updates, the target network is kept fixed, as opposed to the standard approach \\citep{lillicrap2015continuous,fujimoto2018addressing} which performs Polyak averaging \\citep{polyak1990new}. We provide a pseudo-code example, similar to \\cref{alg:ddpg}, in \\cref{apdnx: empirical details and analysis}.\n\nThe following proposition analyzes this model. We discretize the range $\\brs*{v_{\\min}, v_{\\max}}$ into bins of size $\\Delta$ and formulate the learning process as a random-walk, in which with probability $1 - \\delta$, the policy improves (1 step to the right) and decreases otherwise. The random-walk is bounded at both ends by $v_{\\min}$ and $v_{\\max}$, such that the value is projected back into this range.\n\n\\begin{proposition}\nLet $v^\\pi_t$ be a random walk admitting values in the set  $V=\\brc*{v_{\\min}, v_{\\min} + \\Delta, \\hdots, v_{\\max} - \\Delta, v_{\\max}}$. Also, assume that w.p. $\\delta$: $v^\\pi_{t+1} = \\min\\brc*{v^\\pi_t + \\Delta,v_{\\max}}$ and $v^\\pi_{t+1} = \\max\\brc*{v^\\pi_t - \\Delta,v_{\\min}}$ otherwise. Then, the stationary distribution $v$ of $v^\\pi_t$ equals $\\mathbb{P}(v=x) = r^x \\frac{1 - r}{1 - r^{(v_\\text{max} - v_\\text{min})\/\\Delta + 1}}$ for any $x\\in V$, where $r = \\frac{\\delta}{1 - \\delta}$.\n\\end{proposition}\n\n\n\n\\begin{proof}[Proof Sketch]\n    We can write the recurrence relations by:\n    \\begin{align*}\n        &\\mathbb{P}(v = v_\\text{min}) = (1 - \\delta) \\mathbb{P}(v = v_\\text{min}) + \\delta \\mathbb{P}(v = v_\\text{min} + \\Delta) \\\\ \n        &\\mathbb{P}(v = x) = (1 - \\delta) \\mathbb{P}(v = x - \\Delta) + \\delta \\mathbb{P}(v = x + \\Delta) \\\\\n        &\\mathbb{P}(v = v_\\text{max}) = (1 - \\delta) \\mathbb{P}(v = v_\\text{max} - \\Delta) + \\delta \\mathbb{P}(v = v_\\text{max})\n    \\end{align*}\n    the stationary distribution follows.\n\\end{proof}\n\nAn illustration of this process is provided in \\cref{apndx: random walk}. When the probability of improvement is 50\\%, i.e., we are unable to determine which policy is better, the stationary distribution over the value may be arbitrarily bad. However, as the confidence in the improvement of the policy increases, the distribution shifts towards a delta function at the optimal value function.\n\n\n\\begin{remark}\n    In practice, the optimal attainable policy is a function of the initial starting parameters. Due to the non-convexity of the optimization landscape, not all initializations are ensured to be capable of attaining an optimal policy.\n\\end{remark}\n\n\\section{Experiments}\n\nImplementing our approach in an off-policy actor-critic scheme is straightforward. While current approaches \\citep{lillicrap2015continuous,fujimoto2018addressing} hold two policies, an online and a target policy, such that the target policy is updated each step using a Polyak averaging procedure \\citep{polyak1990new} -- our approach, illustrated in \\cref{fig: diagram}, requires the target policy to be updated only when the online policy is superior, with high probability.\n\nSince in our framework, the online policy is aimed at finding the 1-step greedy policy, and due to the continuous nature of the action space, we add a `trust region' element to the loss $\\lambda || \\pi_\\theta (\\state) - \\pi_{\\theta_{-}} (\\state) ||_2^2$ such that the online policy is kept close to the target. In addition, we found that collecting data interchangeably from the online and the target network leads to a further improvement in stability.\n\nFor all the experiments, we plot 5 different random seeds for each configuration, discarding failure seeds\\footnote{We observed that some seeds in deep RL result in utter failure, for instance, a score of 40 in Hopper. This issue is relatively rare, and also happens in the baselines, but hinders the proper evaluation of the policy.}. At each evaluation step, each seed is evaluated for 100 episodes, and the average performance is presented. We plot both the individual seeds (light curves) and the average results across seeds (dark curves). As opposed to previous works, we do not smooth the graphs using a moving average procedure, which enables a better analysis of the experiments' variance and stability.\n\n\\subsection{Evaluating the Policies}\n\nOur approach requires determining whether or not an online policy is better, with probability $1 - \\delta$. Assuming $J^\\pi$, the performance of a sampled trajectory for any deterministic policy $\\pi$, is normally distributed, we may consider the $t$-test for evaluating improvement. We formulate the null hypothesis as $\\mu_{\\pi_\\theta}\\! =\\! \\mathbb{E}_\\rho J^{\\pi_\\theta}\\! >\\! \\mathbb{E}_\\rho J^{\\pi_{\\theta_{-}}}\\! =\\! \\mu_{\\pi_{\\theta{-}}}$. In other words, we hypothesize that the online policy has improved over the target policy. The $t$-statistic is then defined as\n    $t = \\left(\\hat{\\mu}_{\\pi_{\\theta}} - \\hat{\\mu}_{\\pi_{\\theta_{-}}}\\right) \/ \\left(\\sqrt{{\\hat{\\sigma}^2_{\\mu_{\\theta}}}\/{n_{\\pi_{\\theta}}} - {\\hat{\\sigma}^2_{\\mu_{\\theta_{-}}}}\/{n_{\\pi_{\\theta_{-}}}}}\\right)$,\nwhere $\\hat{\\mu}_\\pi$ is the empirical estimate of $\\mathbb{E}_\\rho J^\\pi$, $\\hat{\\sigma}_\\pi$ is its standard deviation and $n_\\pi$ the number of sampled trajectories. The $t$-statistic enables us to find the $p$-value, which represents the confidence in the null hypothesis. Hence, when $p < \\delta$ we have a high enough confidence that the new policy is better, in which case we update the target policy.\n\n\\subsection{Parameter Selection}\n\nFor this approach, there are three parameters which require tuning. (i) how often we evaluate the online policy, (ii) how many episodes are used for evaluation and (iii) the minimal confidence required for policy improvement. Notice that there is a connection between the confidence and the number of evaluation episodes; as the number of evaluations grows, the confidence itself increases. Hence, when there is a relatively high variance in the evaluations, a low number of samples will result in a low confidence, even when the empirical mean is higher. On the other hand, the evaluation frequency controls how fast the process converges. We evaluate the effect of these parameters and provide comprehensive results in \\cref{apndx: c-td3 ablation}.\n\n\n\\paragraph{Confidence Value $\\delta$:} The confidence level is $1-\\delta$, i.e., $\\mathbb{P}(\\mathbb{E}_\\rho J^{\\pi_{\\theta}} > \\mathbb{E}_\\rho J^{\\pi_{\\theta_{-}}}) \\geq 1 - \\delta$. While a natural approach would be to attempt a high level of confidence, our empirical results show otherwise. Demanding a confidence which is too high, makes it much harder to find an improving policy, resulting in a stable yet strictly sub-optimal policy. On the other hand, when very low, the approach becomes unstable. We found that $\\delta \\in [0.1, 0.2]$ work well in practice.\n\n\\paragraph{Evaluation Episodes $K$:} The number of evaluations is correlated to the confidence. A higher number of evaluations increases the confidence in the empirical mean, enabling the algorithm to determine that a policy is better even when the difference is small. However, this comes at a cost -- evaluating the policy takes time and a multitude of evaluations may drastically lengthen training.\n\n\\paragraph{Evaluation Frequency $N$:} When comparing evaluation frequency, we see that when the evaluation is performed every $10,000$ steps, it requires a high probability of improvement to successfully converge. However, when the evaluation is performed more often, the process behaves well even for a lower improvement probability (fewer evaluation episodes).\n\nBased on these insights and the results in \\cref{apndx: c-td3 ablation}, we opted to run with a minimal confidence for swapping of $90\\%$, evaluation every $1,000$ steps over $10$ episodes (sampled trajectories).\n\n\\subsection{Results}\n\n\\begin{figure}[t\n\\centering\n\\begin{tabular}{>{\\centering\\arraybackslash}m{.48\\linewidth} >{\\centering\\arraybackslash}m{.48\\linewidth}}\n ~~~~~~~TD3 & ~~C-TD3 \\\\\n \\includegraphics[width=33mm]{figures\/hopper_td3_target} \\includegraphics[width=33mm]{figures\/inverted_td3_target} & \\includegraphics[width=33mm]{figures\/hopper_t_test_swap_1000_eval_10_conf_0_1} \\includegraphics[width=33mm]{figures\/inverted_t_test_swap_1000_eval_10_conf_0_1} \\\\\n \\includegraphics[width=33mm]{figures\/inverteddouble_td3_target} \\includegraphics[width=33mm]{figures\/walker_td3_target} & \\includegraphics[width=33mm]{figures\/inverteddouble_t_test_swap_1000_eval_10_conf_0_1} \\includegraphics[width=33mm]{figures\/walker_t_test_swap_1000_eval_10_conf_0_1} \\\\\n \\includegraphics[width=33mm]{figures\/halfcheetah_td3_target} \\includegraphics[width=33mm]{figures\/ant_td3_target} & \\includegraphics[width=33mm]{figures\/halfcheetah_t_test_swap_1000_eval_10_conf_0_1} \\includegraphics[width=33mm]{figures\/ant_t_test_swap_1000_eval_10_conf_0_1}\n\\end{tabular}\n\\caption{Evaluation without (TD3) and with the Conservative (C-TD3) update rule.}\n\\label{fig: results}\n\\end{figure}\n\nWe compared our approach to the baseline -- TD3 without our policy selection scheme. The results are presented in \\Cref{fig: results}. These results show a high resemblance to the theoretical model, showing that, across all domains, the conservative approach (Conservative-TD3) reduces the variance of individual seeds, also seen in \\cref{apndx: stability}, and results in increased stability across most domains.\n\nInvertedPendulum and InvertedDoublePendulum serve as perfect examples. While TD3 is capable of finding an optimal policy - e.g., the maximal score - subsequent policies may be arbitrarily bad. By adding the hierarchical selection scheme, our approach is capable of reducing this behavior. That is, once an optimal policy is found, it is not likely to be replaced with an inferior one.\n\nAn interesting result can be seen in the Walker2d domain. While some seeds attain near-optimal performance, others converge to a sub-optimal policy. As our approach searches for a high-confidence improvement, it may sometimes get stuck. However, the inner-process variance, i.e., the variance of individual seeds during training, is much lower across all domains.\n\nFinally, we observe sub-optimal behavior in the Ant domain, both in terms of individual seeds and average performance across seeds. The parameters (confidence, number of evaluation episodes and evaluation frequency) were selected based on the Hopper domain, however it is important to note that these parameters are not optimal for each domain.\n\n\\begin{figure}[t]\n    \\centering\n    \\begin{subfigure}{33mm}\n        \\includegraphics[width=\\linewidth]{figures\/ant_online_std}\n        \\caption{~}\\label{fig: ant std online}\n    \\end{subfigure}\n    \\begin{subfigure}{33mm}\n        \\includegraphics[width=\\linewidth]{figures\/ant_target_std}\n        \\caption{~}\\label{fig: ant std target}\n    \\end{subfigure}\n    \\begin{subfigure}{33mm}\n        \\includegraphics[width=\\linewidth]{figures\/ant_delta_r}\n        \\caption{~}\\label{fig: ant delta r}\n    \\end{subfigure}\n    \\begin{subfigure}{33mm}\n        \\includegraphics[width=\\linewidth]{figures\/ant_t_test_swap_1000_eval_10_conf_0_8}\n        \\caption{~}\\label{fig: ant eval new}\n    \\end{subfigure}\n    \\caption{Failure analysis of the Ant-v2 domain.}\n    \\label{fig:ant analysis}\n    \\vspace{-0.2cm}\n\\end{figure}\n\n\\Cref{fig:ant analysis} presents the standard deviation (STD) of both the online (\\Cref{fig: ant std online}) and target (\\Cref{fig: ant std target}) networks, and the difference between the empirical means $|\\hat{\\mu}_\\pi - \\hat{\\mu}_{\\pi_{-}}|$ (\\Cref{fig: ant delta r}). This clearly shows that in Ant, there is a very large variance, compared the difference between the means, which results in a low confidence. To overcome this issue, we provide an additional experiment, for the Ant domain based on these insights (\\Cref{fig: ant eval new}), which considered a minimal confidence of $20\\%$ ($90\\%$ in \\Cref{fig: results}). These results show that while the parameters are transferable across domains, the optimal parameters found on one domain are not necessarily optimal for another.\n\n\\subsection{Potential Extensions}\nIn this work, we presented an approach for providing probabilistic improvement guarantees. The performance of a trajectory is a random variable, as the initial state, the transition kernel and sometimes the policy -- may all be stochastic. To determine which policy $\\pi_{-}$ or $\\pi$ (the target or online) is better, we performed $N$ evaluations, assumed that the samples are normally distributed and performed a $t$-test. Below we present several intuitive extensions to our work:\n\n\\textbf{Unknown distribution:} While the Gaussian assumption worked well in practice, it may not always be correct. In such a scenario, we see two possible routes: (i) known range, in which the minimal and maximal possible values are known, where one may use the Empirical Bernstein \\citep{maurer2009empirical}, or (ii) when the range is unknown, one may consider a bootstrap-based approach for estimating the confidence intervals around the mean \\citep{abhishek2017nonparametric}.\n    \n\\textbf{Upper\/Lower Confidence Bounds:} Another approach is to consider improvement by a minimal margin. For instance, we may set the update rule to $\\mathbb{P}(v^{\\pi} - \\epsilon \\geq v^{\\pi_{-}}) \\geq 1-\\delta$, so that the statistical test ensures that with high confidence, the online policy has improved by at least $\\epsilon$ over the target.\n\n\n\\textbf{Adaptive Sampling:} In our approach, we considered a rather small number of evaluations, and thus resorted to an equal number of evaluations of each policy -- up to 100 evaluations for each policy. However, when provided with a larger budget, it is of great interest to combine adaptive sampling procedures. \\citet{gabillon2011multi} propose such an approach, which takes into account the empirical mean and variance of each policy evaluation in order to determine which policy to evaluate. At the end of this process, the policy with the highest empirical mean is returned.\n\n\nIn our initial experiments, we tested the UCB\/LCB approach. However, we found that $\\epsilon$ is domain-specific. While in some domains the attained rewards are relatively high, and the policies improve greatly between iterations, in others the improvement is rather small. This led to an early convergence to sub-optimal policies. Nevertheless, this approach may prove beneficial in certain settings and should be considered. Additionally, we tested policy swap based on the empirical estimate of the mean, without ensuring a proper confidence. While this approach worked relatively well in practice, as it lacks proper confidence bounds, we decided to focus on the $t$-test based approach.\n\n\\section{Related Work}\n\n\\paragraph{Trust Region Optimization:} Recent works in on-policy RL have also taken inspiration from Conservative Policy Iteration \\citep[CPI]{kakade2002approximately}. \\citet{schulman2015trust} formulated learning through a trust region optimization scheme, which was later optimized in \\citet{schulman2017proximal}. These approaches bound the KL-divergence between the policy before and after the update. While TRPO greatly improved the stability compared to previous policy gradient approaches, such as A2C \\citep{mnih2016asynchronous}, it lacks the theoretical guarantees.\n\n\\paragraph{Safe Policy Improvement:} Given a baseline policy, safe policy improvement (SPI) algorithms aim to find a new policy that is better than the baseline with high probability \\citep{thomas2015high,ghavamzadeh2016safe,laroche2017safe}. Although this may seem similar to our update stage, there are two major differences: (i) most SPI algorithms directly calculate the improving policy, which is much less efficient than continuously updating the policy and performing the switch when improvement is achieved, and (ii) due to safety constraints, SPI algorithms usually perform off-policy evaluation for the suggested policy, which is still an open problem in high dimensional problems. In contrast, we evaluate it directly and thus enjoy a better sample complexity.\n\n\n\\paragraph{Evolution Strategies:} ES \\citep{salimans2017evolution,such2017deep,conti2018improving} is a gradient-free approach for solving reinforcement learning tasks. At each iteration, given the previous policy $\\pi_{\\theta_i}$, a set of augmented policies is created $\\{ \\pi_{\\theta_i + \\xi_i} \\}_i$ and evaluated, where $\\xi_i$ is some random noise. The ES update rule can be seen as a form of numerical gradient estimation (i.e., finite differences) \\citep{zhang2017relationship}. There are some similarities between our approach and ES. While ES is a gradient-free method, it uses empirical evaluations of the policies to determine the parameter update direction. Our approach uses these evaluations in order to determine whether or not the policy has improved, and in contrast to ES, will only change the policy if there is a high enough confidence for improvement.\n\n\\section{Conclusions}\nIn this work, we tackled the stability problems of policy-based reinforcement learning approaches. Even though the empirical gradient (SGD) is a noisy estimate of the ideal descent direction, and due to the structure of the optimization landscape and the large step sizes used in practice, the policy's performance varies greatly during training.\n\nDrawing from the connection between actor-critic approaches and approximate policy iteration schemes, we introduced a hierarchical training scheme, which we analyzed by comparing this approach to a bounded random-walk. While this scheme is not an exact representation of the optimization process, it provides an intuition on the behavior of such a probabilistic improvement scheme, i.e., as the probability of improvement increases, the stationary distribution over the performance of the policy shifts towards the optimal solution.\n    \nFinally, we proposed the Conservative-TD3, which performs a periodic evaluation of both the online and target policies such that it updates the target network only when there is a higher enough confidence that the online network has surpassed its performance. We validate this on several continuous control tasks in MuJoCo; empirical evidence shows that indeed this approach behaves similarly to the theoretical analysis. When compared to the baseline, we observe a dramatic reduction in variance, increased stability and an overall improvement in the performance itself.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\subsection{The Candidate Comparison Graph \\COMPG}\nA key analysis tool in this section is a directed graph \\COMPG on the\nset of all candidates \\ALLCANDS, which we call the \\emph{\\CandCompGr}.\n\\COMPG contains the directed edge $(y,x)$ if and only if the graph\n\\Bip{x}{y} does \\emph{not} have a perfect matching.\nIn a sense, $y$ is a witness that $x$ would be a dangerous choice as\nwinner, since Corollary~\\ref{cor:bipartite} would not apply\nto bound the cost of $x$ when $y$ is the optimal candidate.%\n\\footnote{Of course, Corollary~\\ref{cor:bipartite} is only a\n  \\emph{sufficient} condition, not a necessary one.\n  So even when Corollary~\\ref{cor:bipartite} cannot be applied, it is\n  conceivable that \\WINNER would achieve a distortion of 3.\n  However, it is not clear which tool we could use to bound the\n  distortion, which is why we focus only on the implications of\n  Corollary~\\ref{cor:bipartite} here.}\nConversely, any candidate \\WINNER without incoming edges in \\COMPG is a\nsafe choice as a winner, because Corollary~\\ref{cor:bipartite} implies\na bound of 3 on its cost ratio to \\OPT.\n\nThe following straightforward lemma captures that if we remove some\ncandidates, and leave each voter's ranking of the remaining candidates\nunchanged, then the edges of the resulting graph \\COMPGP\nare a superset of the edges of the subgraph of \\COMPG\ninduced by the remaining candidates.\nWe write $\\ALLPREFS[\\ALLCANDSP]$ for the vector of rankings\n\\PREFP[v], where each \\PREFP[v] is the ranking \\PREF[v],\nrestricted to candidates in \\ALLCANDSP.\nIn other words, \\PrefP[v]{x}{y} if and only if \\Pref[v]{x}{y},\nfor all $x, y \\in \\ALLCANDSP$.\n\n\\begin{lemma} \\label{lem:induced-graph}\n  Let $(\\ALLCANDS, \\ALLPREFS)$ be a social choice instance,\n  and $\\ALLCANDSP \\subseteq \\ALLCANDS$.\n  Let $\\COMPG = \\CompG{\\ALLCANDS}{\\ALLPREFS}$ and\n  $\\COMPGP = \\CompG{\\ALLCANDSP}{\\ALLPREFS[\\ALLCANDSP]}$.\n  Then, $\\COMPGP \\supseteq \\COMPG[\\ALLCANDSP]$; \n  that is, \\COMPGP contains all edges of the induced subgraph\n  $\\COMPG[\\ALLCANDSP]$.\n\\end{lemma}\n\n\\begin{proof}\nConsider two candidates $x,y \\in \\ALLCANDSP$ and their corresponding\nbipartite graph \\BipP{x}{y} on voters under the restricted instance.\nBy definition, \\BipP{x}{y} contains the edge $(v,v')$ iff there exists\na candidate $z \\in \\ALLCANDSP$ such that \\WPref[v]{x}{z} and \\WPref[v']{z}{y}.\nThus, if \\BipP{x}{y} contains the edge $(v,v')$, then so does the\nbipartite graph \\Bip{x}{y} for the larger\/original instance\n$(\\ALLCANDS,\\ALLPREFS)$.\nIn other words, the edges of \\BipP{x}{y} are a subset of the edges of\n\\Bip{x}{y}.\nTherefore, whenever \\BipP{x}{y} contains a perfect matching,\nso does \\Bip{x}{y}.\nBecause edges in \\COMPG (and \\COMPGP) correspond to pairs that do\n\\emph{not} have bipartite matchings, the graph \\COMPGP is a supergraph\nof the induced subgraph $\\COMPG[\\ALLCANDSP]$.\n\\end{proof}\n\n\n\n\\subsection{Acyclicity of \\COMPG} \\label{sec:reformulation}\nOne sufficient condition for the existence of a source node in \\COMPG\n(i.e., a node without incoming edges) is for \\COMPG to be acyclic.\nThis gives rise to our first conjecture,\nwhich was also given as Conjecture~4.8 in\n\\cite{munagala:wang:improved}\n(though it is expressed slightly differently there):\n\n\\begin{conjecture} \\label{conj:acyclic}\n  For every instance $(\\ALLCANDS,\\ALLPREFS)$\n  the graph $\\COMPG = \\CompG{\\ALLCANDS}{\\ALLPREFS}$ is non-Hamiltonian.%\n  \\footnote{Recall that a directed graph is \\emph{Hamiltonian} if it\n    contains a directed cycle of all nodes.}\n\\end{conjecture}\n\nSince our goal here is merely to show the existence of a source node\nin \\COMPG, it appears like overkill to aim for the ``stronger''\nconjecture of being non-Hamiltonian\/acyclic.\nDespite appearances, Conjecture~\\ref{conj:acyclic} is not in fact\nstronger than the existence of a source node,\nas we show in the following proposition:\n\n\\begin{proposition} \\label{prop:acyclic-enough}\n  \\ABM succeeds \\emph{on all inputs} if and only if\n  Conjecture~\\ref{conj:acyclic} is true.\n\\end{proposition}\n\nNotice that the proposition does not say that whenever a \\emph{specific}\ninstance violates Conjecture~\\ref{conj:acyclic},\nthe algorithm will fail on \\emph{that} instance.\nAs will be evident in the proof, it only implies that the algorithm\nfails on \\emph{some} (potentially different) instance. \n\n\\begin{emptyproof}\n\\begin{enumerate}\n\\item For the first direction,\n  if Conjecture~\\ref{conj:acyclic} is true,\n  then \\COMPG is non-Hamiltonian for all inputs.\n  We claim that this implies that in fact,\n  \\COMPG is \\emph{acyclic} for all inputs.\n  Suppose that we had an instance $(\\ALLCANDS, \\ALLPREFS)$\n  for which \\COMPG contains a directed cycle, say,\n  $C = (x_1, x_2, \\ldots, x_k, x_1)$ for some $k$.\n  Let $\\ALLCANDSP = \\SET{x_1, \\ldots, x_k}$.\n  Consider the instance $(\\ALLCANDSP, \\ALLPREFS[\\ALLCANDSP])$.\n  By Lemma~\\ref{lem:induced-graph},\n  the graph \\CompG{\\ALLCANDSP}{\\ALLPREFS[\\ALLCANDSP]}\n  is a supergraph of the induced subgraph\n  $\\COMPG[\\ALLCANDSP]$, and must therefore contain a cycle including\n  all vertices \\ALLCANDSP, i.e.,\n  \\CompG{\\ALLCANDSP}{\\ALLPREFS[\\ALLCANDSP]} is Hamiltonian.\n\n  We have thus shown that for all instances, \\COMPG is acyclic, meaning that\n  for all inputs, \\COMPG has a source node, which is a safe output for\n  \\ABM.\n\\item For the converse direction, assume that\n  Conjecture~\\ref{conj:acyclic} is false, and consider an instance\n  $(\\ALLCANDS, \\ALLPREFS)$ for which \\CompG{\\ALLCANDS}{\\ALLPREFS}\n  is Hamiltonian.\n  Because each node has at least one incoming edge,\n  \\CompG{\\ALLCANDS}{\\ALLPREFS} cannot have a source node. \\QED\n\\end{enumerate}\n\\end{emptyproof}\n\n\\subsection{Distributions of Permutations}\n\nWe next derive a much simpler-looking --- but actually equivalent ---\nconjecture, which is phrased only in terms of distributions of\npermutations.\nThe key lemma for deriving this equivalent conjecture is the\nfollowing:\n\n\\begin{lemma} \\label{lem:hall-application}\n  \\COMPG contains the edge $(y,x)$ if and only if\n  there exists a set \\EWit{x}{y} of candidates with\n  $x \\in \\EWit{x}{y}$ and $y \\notin \\EWit{x}{y}$ such that\n\n  \\begin{align}\n    \\SetCard{\\CandFirst{y}{\\EWit{x}{y}}} +\n    \\SetCard{\\CandLast{x}{\\Compl{\\EWit{x}{y}}}}\n  & > m. \\label{eqn:general-hall}\n  \\end{align}\n\\end{lemma}\n\n\\begin{emptyproof}\nThe proof relies on the well-known Hall Theorem:\n\\begin{theorem}[Hall's Bipartite Matching Theorem]\n  Let $G = (X \\cup Y, E)$ be a bipartite graph\n  with $\\SetCard{X} = \\SetCard{Y}$.\n  For any vertex set $S$, let \\Neigh{S} denote the neighbors of $S$.\n  $G$ has a perfect matching if and only if there is no contracting\n  vertex set, i.e., no set $X' \\subseteq X$ with\n  $\\SetCard{\\Neigh{X'}} < \\SetCard{X'}$.\n\\end{theorem}\n\n  Fix a pair $x,y$ of candidates.\n  By Hall's Theorem, $(y,x) \\in \\COMPG$\n  (i.e., \\Bip{x}{y} has no perfect matching)\n  if and only if there is a contracting set of voters $V_{x,y}$.\n  By definition of \\COMPG, for every voter $v$ on the left,\n  the neighborhood $\\Neigh{v}$ consists of all voters $v'$ on the\n  right such that there exists a candidate $z$\n  with \\WPref[v]{x}{z} and \\WPref[v']{z}{y}. \n\n\\begin{enumerate}\n\\item For the first direction,\n  we assume that \\COMPG contains the edge $(y,x)$.\n  Let $V_{x,y}$ be a maximal contracting set.\n  Let \\EWit{x}{y} be the set of all candidates $z$ such that\n  at least one voter $v \\in V_{x,y}$ has \\WPref[v]{x}{z}.\n  Then, $\\Neigh{V_{x,y}} = \\Set{v'}{\\mbox{there exists a } z \\in\n  \\EWit{x}{y} \\mbox{ with } \\WPref[v']{z}{y}}$.\n  This implies two things:\n  \\begin{itemize}\n  \\item $\\Compl{\\Neigh{V_{x,y}}} = \\CandFirst{y}{\\EWit{x}{y}}$\n    is the set of all voters who rank $y$ strictly ahead of all of \\EWit{x}{y};\n    this follows directly from the preceding characterization.\n  \\item $V_{x,y} = \\CandLast{x}{\\Compl{\\EWit{x}{y}}}$.\n    The reason is that every voter $v \\in V_{x,y}$ \n    ranks only candidates in \\EWit{x}{y} (weakly) after $x$,\n    so $V_{x,y} \\subseteq \\CandLast{x}{\\Compl{\\EWit{x}{y}}}$.\n    Since $\\Neigh{V_{x,y}} = \\Neigh{\\CandLast{x}{\\Compl{\\EWit{x}{y}}}}$,\n    the set \\CandLast{x}{\\Compl{\\EWit{x}{y}}} is also a candidate for\n    a contracting set, and must equal $V_{x,y}$ by maximality of $V_{x,y}$.\n  \\end{itemize}\n  By definition of \\EWit{x}{y}, we always have $x \\in \\EWit{x}{y}$.\n  Also, we always have $y \\notin \\EWit{x}{y}$,\n  because any voter $v$ (on the left) with \\Pref[v]{x}{y}\n  has edges to all voters $v'$ on the right,\n  and can therefore not be in a contracting set $V_{x,y}$.\n\n  Next, we consider the cardinalities of the sets involved. Because\n  \\[\n    m - \\SetCard{\\CandFirst{y}{\\EWit{x}{y}}}\n    \\; = \\; m - \\SetCard{\\Compl{\\Neigh{V_{x,y}}}}\n    \\; = \\; \\SetCard{\\Neigh{V_{x,y}}}\n    \\; \\stackrel{\\text{Hall}}{<} \\; \\SetCard{V_{x,y}}\n    \\; = \\; \\SetCard{\\CandLast{x}{\\Compl{\\EWit{x}{y}}}},\n  \\]\n  we have shown that there exists a set \\EWit{x}{y} with\n  $x \\in \\EWit{x}{y}$ and $y \\notin \\EWit{x}{y}$ such that\n  $\\SetCard{\\CandFirst{y}{\\EWit{x}{y}}}\n    + \\SetCard{\\CandLast{x}{\\Compl{\\EWit{x}{y}}}}\n    > m$.\n\n  \\item For the converse direction, assume that \n  there exists a set \\EWit{x}{y} of candidates with\n  $x \\in \\EWit{x}{y}$ and $y \\notin \\EWit{x}{y}$ such that\n  $\\SetCard{\\CandFirst{y}{\\EWit{x}{y}}}\n  + \\SetCard{\\CandLast{x}{\\Compl{\\EWit{x}{y}}}}\n  > m$.\n\n  Let $V_{x,y} = \\CandLast{x}{\\Compl{\\EWit{x}{y}}}$ be the set of all\n  voters who rank $x$ behind all of $\\Compl{\\EWit{x}{y}}$.\n  We will show that $V_{x,y}$ is contracting.\n  Thereto, the important step is to characterize the neighborhood\n  $\\Neigh{V_{x,y}}$.\n  By definition, it consists of all voters $v'$ such that\n  there exists a voter $v \\in V_{x,y}$ and a candidate $z$\n  with \\WPref[v]{x}{z} and \\WPref[v']{z}{y}.\n  Because each voter $v \\in V_{x,y}$ ranks $x$ behind all of\n  $\\Compl{\\EWit{x}{y}}$, the only potential candidates for $z$ are\n  candidates in \\EWit{x}{y}.\n  In particular, no voter $v' \\in \\CandFirst{y}{\\EWit{x}{y}}$\n  can be in $\\Neigh{V_{x,y}}$, implying that \n  $\\CandFirst{y}{\\EWit{x}{y}} \\subseteq \\Compl{\\Neigh{V_{x,y}}}$.\n\n  This implies that\n  $\\SetCard{\\CandFirst{y}{\\EWit{x}{y}}} \\leq \\SetCard{\\Compl{\\Neigh{V_{x,y}}}}$,\n  so $\\SetCard{\\Neigh{V_{x,y}}} \\leq m - \\SetCard{\\CandFirst{y}{\\EWit{x}{y}}}$.\n  And by definition of $V_{x,y}$, we also have\n  $\\SetCard{V_{x,y}} = \\SetCard{\\CandLast{x}{\\Compl{\\EWit{x}{y}}}}$.\n  Together with the assumption that \n  $\\SetCard{\\CandFirst{y}{\\EWit{x}{y}}}\n  + \\SetCard{\\CandLast{x}{\\Compl{\\EWit{x}{y}}}}\n  > m$, we get that\n  $m < \\SetCard{V_{x,y}} + (m - \\SetCard{\\Neigh{V_{x,y}}})$,\n  implying that $V_{x,y}$ is contracting.\n  By Hall's Theorem, \\Bip{x}{y} has no perfect matching,\n  so \\COMPG contains the edge $(y,x)$. \\QED\n\\end{enumerate}\n\\end{emptyproof}\n\n\\begin{remark} \\label{rem:compg-subgraph}\nAs an easy corollary of Lemma~\\ref{lem:hall-application}, notice that\nthe constraint \\eqref{eqn:general-hall} implies that strictly more\nthan half of the voters prefer $y$ over $x$.\nThis is because $x \\in \\EWit{x}{y}$ and $y \\notin \\EWit{x}{y}$ implies that\nall voters in \\CandFirst{y}{\\EWit{x}{y}} and all voters in \n\\CandLast{x}{\\Compl{\\EWit{x}{y}}} rank $y$ ahead of $x$.\nBecause the combined cardinalities of the two sets add up to more than $m$,\nby the Pigeon Hole Principle, at least one of the two sets\n\\CandFirst{y}{\\EWit{x}{y}}, \\CandLast{x}{\\Compl{\\EWit{x}{y}}} must contain\nmore than half of all voters.\nThis proves that \\COMPG is a subgraph of the weak comparison graph.\n\\end{remark}\n\nBased on Lemma~\\ref{lem:hall-application}, we formulate the following\nconjecture, and prove it equivalent to Conjecture~\\ref{conj:acyclic}.\n\n\\begin{conjecture} \\label{conj:permutation-distribution}\nLet $\\EWitS{1}, \\EWitS{2}, \\ldots, \\EWitS{n} \\subseteq \\SET{1, \\ldots, n}$\nbe arbitrary sets with $i \\in \\EWitS{i}$.\nDefine the following two indicator functions over elements $i$ and\ntotal orders \\PREF:\n\\begin{align}\n  \\IndPos{i}{\\PREF} & = \\begin{cases}\n    1 & \\text{if } \\Pref{i+1}{\\EWitS{i}} \\\\\n    0 & \\text{otherwise};\n                \\end{cases}\n  & \\IndNeg{i}{\\PREF} & = \\begin{cases}\n    1 & \\text{if } \\Pref{\\Compl{\\EWitS{i}}}{i}\\\\\n    0 & \\text{otherwise};\n                \\end{cases}\n\\end{align}\nhere and for the rest of this proof, all additions\/subtractions are\nmodulo $n$; that is, $n+1 := 1$ and $1-1 := n$.\n\nLet \\PREFDIST be any distribution over total orders \\PREF of\n$\\SET{1, \\ldots, n}$.\nThen, there exists an $i$ such that\n\\begin{align}\n  \\Expect[\\PREF \\sim \\PREFDIST]{\\IndPos{i}{\\PREF} + \\IndNeg{i}{\\PREF}} & \\leq 1.\n\\end{align}\n\\end{conjecture}\n\n\\begin{proposition} \\label{lem:first-two-conjectures}\nConjecture~\\ref{conj:permutation-distribution} is true if and only if\nConjecture~\\ref{conj:acyclic} is true.\n\\end{proposition}\n\n\\begin{proof}\nWe first show that Conjecture~\\ref{conj:permutation-distribution}\nimplies Conjecture~\\ref{conj:acyclic}, by proving the contrapositive.\nAssume that \\COMPG is Hamiltonian, containing a directed cycle\n$x_n \\to x_{n-1} \\to \\cdots \\to x_1 \\to x_n$ comprising all $n$ candidates.\nBy Lemma~\\ref{lem:hall-application},\nthere exist sets $\\EWitS{i} = \\EWit{x_i}{x_{i+1}}$\n(with $\\EWitS{n} = \\EWit{x_n}{x_1}$)\nwith $x_i \\in \\EWitS{i}, x_{i+1} \\notin \\EWitS{i}$,\nsuch that for all $i=1, \\ldots, n$, we have\n  \\begin{align}\n    \\SetCard{\\CandFirst{x_{i+1}}{\\EWitS{i}}}\n    + \\SetCard{\\CandLast{x_i}{\\Compl{\\EWitS{i}}}}\n  \\; > \\; m. \\label{eqn:hall}\n  \\end{align}\n\nWe define a distribution \\PREFDIST over rankings by drawing a\nuniformly random voter from $\\SET{1, \\ldots, m}$,\nand choosing this voter's ranking.\nBecause the distribution is uniform, we obtain that\n\\begin{align}\n  \\Expect[\\PREF \\sim \\PREFDIST]{\\IndPos{i}{\\PREF}}\n  & = \\frac{1}{m} \\cdot \\SetCard{\\CandFirst{x_{i+1}}{\\EWitS{i}}},\n  & \\Expect[\\PREF \\sim \\PREFDIST]{\\IndNeg{i}{\\PREF}}\n  & = \\frac{1}{m} \\cdot \\SetCard{\\CandLast{x_i}{\\Compl{\\EWitS{i}}}}.\n    \\label{eqn:expectation-uniform}\n\\end{align}\nThe inequality~\\eqref{eqn:hall} then implies that\n$\\Expect[\\PREF \\sim \\PREFDIST]{\\IndPos{i}{\\PREF} + \\IndNeg{i}{\\PREF}} > 1$\nfor all $i$,\nshowing that Conjecture~\\ref{conj:permutation-distribution} is\nviolated.\n\nFor the converse direction, assume that \\PREFDIST is a distribution\nover total orders of $\\SET{1, \\ldots, n}$ such that\n\\begin{align} \n  \\Expect[\\PREF \\sim \\PREFDIST]{\\IndPos{i}{\\PREF} + \\IndNeg{i}{\\PREF}}\n  & > 1, \\text{ for all $i$}. \\label{eqn:permutation-violation}\n\\end{align}\nDefine $\\delta := \\min_i\n\\Expect[\\PREF \\sim \\PREFDIST]{\\IndPos{i}{\\PREF} + \\IndNeg{i}{\\PREF}}$;\nbecause the number of candidates is finite, $\\delta$ is well-defined,\nand $\\delta > 1$.\nTherefore, with sufficiently small changes to \\PREFDIST,\nwe can ensure that the probability for each total order \\PREF is a\nrational number, while preserving all (strict)\ninequalities~\\eqref{eqn:permutation-violation}.\nOnce the probabilities are all rational, we can write them with a\ncommon denominator, meaning that we can define \\PREFDIST as a uniform\ndistribution over a finite multi-set of rankings;\nin turn, we can consider these rankings as voters.\n\nBecause the distribution is uniform over voters,\nwe can apply the characterization~\\eqref{eqn:expectation-uniform} to\nconclude that \n$\\SetCard{\\CandFirst{x_{i+1}}{\\EWitS{i}}}\n+ \\SetCard{\\CandLast{x_i}{\\Compl{\\EWitS{i}}}}\n> m$ for all candidates $i$.\nBy Lemma~\\ref{lem:hall-application},\napplied to each pair $(x_i, x_{i+1})$,\nthe graph \\COMPG contains each edge $(x_{i+1}, x_i)$,\nso \\COMPG is Hamiltonian.\n\\end{proof}\n\n\\subsection{A Graph-Theoretic Reformulation}\n\nOur attempts to prove Conjecture~\\ref{conj:permutation-distribution}\n(so far unsuccessful) have been based on proofs by contradiction.\nThe assumed constraints \\eqref{eqn:permutation-violation} prescribe\nseveral constraints on rankings that must hold simultaneously;\nusing transitivity, this leads to a contradiction by forcing\npreferences to contain cycles.\nThe essence of this approach is captured by another conjecture.\nTo formulate it, we define the following class of directed graphs,\nwhich we term \\emph{Constraint-Choice Graphs}.\n\n\\begin{definition}[Constraint-Choice Graph]\n  \\label{def:constraint-choice-graph}\n  Let $\\CandNodes{n} = \\SET{\\CandNode{1}, \\ldots, \\CandNode{n}},\n  \\SetNodes{n} = \\SET{\\SetNode{1}, \\ldots, \\SetNode{n}},\n  \\ComplNodes{n} = \\SET{\\ComplNode{1}, \\ldots, \\ComplNode{n}}$ \n  be three disjoint sets of nodes.\n  A \\emph{constraint-choice graph} for a given $n$ contains $3n$ nodes\n  $U_n = \\CandNodes{n} \\cup \\SetNodes{n} \\cup \\ComplNodes{n}$, \n  and the following edges:\n  \\begin{itemize}\n  \\item For each $i$, it contains the directed edges%\n\\footnote{As before, we define $1-1:= n$ and $n+1 := 1$.}\n  $(\\CandNode{i}, \\SetNode{i-1}), (\\CandNode{i}, \\ComplNode{i-1}), \n   (\\SetNode{i}, \\CandNode{i}), (\\ComplNode{i}, \\CandNode{i})$.\n  \\item For each $i, j$ with $j \\neq i, j \\neq i-1$, it contains\n    exactly one of the two directed edges \n    $(\\SetNode{j}, \\CandNode{i}), (\\CandNode{i}, \\ComplNode{j})$.\n  \\end{itemize}\n\\end{definition}\n\nAn example of a constraint choice graph is shown in\nFigure~\\ref{fig:constraint-choice-graph}.\nThe edges listed first in Definition~\\ref{def:constraint-choice-graph}\nare shown in solid black, while the edges listed second are shown in\ndashed red lines.\n\n\\begin{figure}[htb]\n\\begin{center}\n\\begin{tikzpicture}[auto,thick,->,active\/.style={circle,draw=black}]\n\n  \\node[active] (y1) at (1,0) {\\CandNode{1}};\n  \\node[active] (y2) at (3,0) {\\CandNode{2}};\n  \\node[active] (y3) at (5,0) {\\CandNode{3}};\n  \\node[active] (y4) at (7,0) {\\CandNode{4}};\n\n  \\node[active] (a1) at (2,1.5) {\\SetNode{1}};\n  \\node[active] (a2) at (4,1.5) {\\SetNode{2}};\n  \\node[active] (a3) at (6,1.5) {\\SetNode{3}};\n  \\node[active] (a4) at (8,1.5) {\\SetNode{4}};\n\n  \\node[active] (b1) at (2,-1.5) {\\ComplNode{1}};\n  \\node[active] (b2) at (4,-1.5) {\\ComplNode{2}};\n  \\node[active] (b3) at (6,-1.5) {\\ComplNode{3}};\n  \\node[active] (b4) at (8,-1.5) {\\ComplNode{4}};\n\n  \\draw (a1) to (y1);\n  \\draw (b1) to (y1);\n  \\draw (y1) to [bend left = 90] (a4);\n  \\draw (y1) to [bend right = 90] (b4);\n  \\draw (a2) to (y2);\n  \\draw (b2) to (y2);\n  \\draw (y2) to (a1);\n  \\draw (y2) to (b1);\n  \\draw (a3) to (y3);\n  \\draw (b3) to (y3);\n  \\draw (y3) to (a2);\n  \\draw (y3) to (b2);\n  \\draw (a4) to (y4);\n  \\draw (b4) to (y4);\n  \\draw (y4) to (a3);\n  \\draw (y4) to (b3);\n\n  \\begin{scope}[red,dashed]\n    \\draw (y3) to (b1);\n    \\draw (y4) to (b1);\n    \\draw (y4) to (b2);\n    \\draw (a2) to (y1);\n    \\draw (a3) to (y1);\n    \\draw (y2) to (b3);\n    \\draw (a4) to (y3);\n    \\draw (a4) to (y2);\n  \\end{scope}\n\\end{tikzpicture}\n\\end{center}\n\\caption{An illustration of a constraint-choice graph for $n=4$ candidates.\n  The graph depicted here corresponds to the sets\n  $\\EWitS{1} = \\SET{1}, \\EWitS{2} = \\SET{1,2}, \\EWitS{3} = \\SET{1,3},\n  \\EWitS{4} = \\SET{2,3,4}$ in the construction of the proof of\n  Proposition~\\ref{prop:choice-graph}.\n\\label{fig:constraint-choice-graph}}\n\\end{figure}\n\n\n\n\n\n\n\n\n\\begin{conjecture} \\label{conj:constraint-choice-graph}\n  For every $n$ and every constraint choice graph $G_n$ of $3n$ nodes,\n  there exists a non-empty index set $S \\subseteq \\SET{1, \\ldots, n}$\n  with the following property:\n  For every vertex set \n  $T \\subseteq \\Set{\\SetNode{i}, \\ComplNode{i}}{i \\in S}$ \n  of size $\\SetCard{T} > \\SetCard{S}$, the induced subgraph\n  $G_n[\\CandNodes{n} \\cup T]$ contains a directed cycle.\n\\end{conjecture}  \n\n\\begin{remark}\n  Notice that the conjecture indeed talks about the subgraph induced\n  by \\emph{all} nodes \\CandNode{i} (not just those with indices in $T$),\n  in addition to at least $\\SetCard{S} + 1$ nodes from among the\n  \\SetNode{i}, \\ComplNode{i} with $i \\in S$.\n\\end{remark}\n\n\\begin{proposition} \\label{prop:choice-graph}\n  If Conjecture~\\ref{conj:constraint-choice-graph} is true,\n  then Conjecture~\\ref{conj:permutation-distribution} is true.\n\\end{proposition}\n\n\\begin{proof}\nWe prove the contrapositive, and assume that \nConjecture~\\ref{conj:permutation-distribution} is false; that is,\nInequality~\\eqref{eqn:permutation-violation} holds for all $i$.\n\nGiven the (assumed) sets\n$\\EWitS{1}, \\EWitS{2}, \\ldots, \\EWitS{n} \\subseteq \\SET{1, \\ldots, n}$,\nwe define the following constraint-choice graph $G_n$.\nIt contains the $3n$ nodes\n$\\CandNodes{n} \\cup \\SetNodes{n} \\cup \\ComplNodes{n}$,\nand all the edges that are prescribed by\nDefinition~\\ref{def:constraint-choice-graph};\nin addition, if $i \\in \\EWitS{j}$, then $G_n$ contains the edge\n$(\\SetNode{j}, \\CandNode{i})$;\notherwise, it contains the edge $(\\CandNode{i}, \\ComplNode{j})$.\nThis completes the definition of $G_n$.\nAn example for specific sets \\EWitS{j} is shown in\nFigure~\\ref{fig:constraint-choice-graph}.\nTo gain intuition for the following proof, the reader is encouraged to\nthink of \\CandNode{i} as corresponding to candidate $x_i$,\nof \\SetNode{i} as corresponding to \\EWitS{i},\nand of \\ComplNode{i} as corresponding to \\Compl{\\EWitS{i}}.\n\nTo prove that $G_n$ violates\nConjecture~\\ref{conj:constraint-choice-graph},\nconsider an arbitrary non-empty set of indices\n$S \\subseteq \\SET{1, \\ldots, n}$.\nBy linearity of expectations,\n\\eqref{eqn:permutation-violation} implies that \n$\\Expect[\\PREF \\sim \\PREFDIST]{\\sum_{i \\in S} (\\IndPos{i}{\\PREF} + \\IndNeg{i}{\\PREF})}\n> \\SetCard{S}$. \nBecause the maximum is at least the average,\nthis implies that there exists some ranking \\PREF[S] with \n$\\sum_{i \\in S} (\\IndPos{i}{\\PREF[S]} + \\IndNeg{i}{\\PREF[S]}) > \\SetCard{S}$,\nand because the quantity on the left-hand side is integral,\nwe can strengthen this inequality to\n\\begin{align}\n  \\sum_{i \\in S} (\\IndPos{i}{\\PREF[S]} + \\IndNeg{i}{\\PREF[S]})\n  & \\geq \\SetCard{S} + 1.\n    \\label{eqn:permutation-violation-set}\n\\end{align}\n\nDefine the node set\n$T_S := \\Set{\\SetNode{i}}{\\IndPos{i}{\\PREF[S]} = 1} \\cup\n     \\Set{\\ComplNode{i}}{\\IndNeg{i}{\\PREF[S]} = 1}$.\nBy Inequality~\\eqref{eqn:permutation-violation-set},\n$T_S$ contains at least $\\SetCard{S} + 1$ nodes from\n$\\Set{\\SetNode{i}, \\ComplNode{i}}{i \\in S}$.\nWe will show that the induced subgraph $G_n[T_S \\cup \\CandNodes{n}]$ is acyclic;\nsince we show this for every $S$, it proves that $G_n$ violates\nConjecture~\\ref{conj:constraint-choice-graph}.\n\nTo show that $G_n[T_S \\cup \\CandNodes{n}]$ is acyclic,\nwe use a proof by contradiction,\nand assume that $G_n[T_S \\cup \\CandNodes{n}]$ contains a cycle $C$.\nBecause edges only go between nodes \\CandNode{i} and either\n\\SetNode{j} or \\ComplNode{j}, this cycle must alternate nodes\n\\CandNode{i} with nodes \\SetNode{j} or \\ComplNode{j}.\nLet $\\SetCard{C} = 2k$, and let $i_1, i_2, \\ldots, i_k$ be such that\nthe order of nodes \\CandNode{i} in $C$ is\n$\\CandNode{i_1}, \\CandNode{i_2}, \\ldots, \\CandNode{i_k}, \\CandNode{i_1}$.\nFix some $\\ell \\in \\SET{1, \\ldots, k}$.\nBetween \\CandNode{i_\\ell} and \\CandNode{i_{\\ell+1}}\n(here, $k+1 := 1$), the cycle must visit either some node\n$\\SetNode{j} \\in T_S$ or some node $\\ComplNode{j} \\in T_S$.\nWe distinguish two cases:\n\n\\begin{itemize}\n\\item If the intermediate node is \\SetNode{j},\n  observe first that by Definition~\\ref{def:constraint-choice-graph},\n  the only incoming edge to \\SetNode{j} is\n  $(\\CandNode{j+1}, \\SetNode{j})$, so $i_\\ell = j+1$.\n  The specific constraint-choice graph $G_n$ defined in this proof\n  includes outgoing edges from \\SetNode{j} to exactly\n  the \\CandNode{i} with $i \\in \\EWitS{j}$;\n  notice that this includes the case of the edge\n  $(\\SetNode{j}, \\CandNode{j})$, because $j \\in \\EWitS{j}$.\n  In particular, it applies to the edge\n  $(\\SetNode{j}, \\CandNode{i_{\\ell+1}})$,\n  implying that $i_{\\ell+1} \\in \\EWitS{j}$.\n\n  Because $\\SetNode{j} \\in T_s$, we have that\n  $\\IndPos{i}{\\PREF[S]} = 1$, meaning that under \\PREF[S],\n  candidate $i_{\\ell} = j+1$ precedes all candidates in \\EWitS{j};\n  this includes, in particular, the candidate $i_{\\ell+1}$.\n  In summary, we have inferred that\n  \\Pref[S]{i_{\\ell}}{i_{\\ell+1}}.\n\\item If the intermediate node is \\ComplNode{j},\n  observe that by Definition~\\ref{def:constraint-choice-graph},\n  the only outgoing edge from \\ComplNode{j} is\n  $(\\ComplNode{j}, \\CandNode{j})$, implying that $i_{\\ell+1} = j$.\n  For the particular graph $G_n$ defined in this proof, the incoming edge\n  $(\\CandNode{i_{\\ell}}, \\ComplNode{j})$ exists exactly when\n  $i_{\\ell} \\notin \\EWitS{j}$.\n\n  Because $\\ComplNode{j} \\in T_s$, we have that\n  $\\IndNeg{j}{\\PREF[S]} = 1$, so under \\PREF[S],\n  candidate $j = i_{\\ell+1}$ is ranked after all candidates in\n  $\\Compl{\\EWitS{j}}$.\n  By the argument of the preceding paragraph,\n  the set \\EWitS{j} includes, in particular, the candidate $i_{\\ell}$.\n  In summary, we have again inferred that\n  \\Pref[S]{i_{\\ell}}{i_{\\ell+1}}.\n\\end{itemize}\n\nThus, we have derived that \\Pref[S]{i_{\\ell}}{i_{\\ell+1}}\nfor each $\\ell = 1, \\ldots, k$.\nBy transitivity, this results in a cycle in \\PREF[S],\na contradiction to it being a ranking.\n\\end{proof}\n\n\\begin{remark}\nConjecture~\\ref{conj:constraint-choice-graph} is sufficiently clean\nand combinatorial that it can be verified by hand for $n \\leq 4$.\nAn exhaustive computer search for $n \\leq 7$ has verified\nthe conjecture for all such $n$, recovering and extending\ncomputer-assisted results in Theorem~4.11 in\n\\cite{munagala:wang:improved} (although we did not include some ranges\nfor $n > 7$ which \\cite{munagala:wang:improved} handle with a\nrestricted number of voters).\nUnfortunately, because it basically involves a search over all\npossibilities of $n$ subsets \\EWitS{i} of $n$ elements\n(represented as graph edge choices), the running time scales roughly\nas $2^{n^2}$; from $n=6$ to $n=7$, the running time increases from\nless than a minute to roughly a day.\nHence, even $n=8$ is likely out of reach.\nBut the computer search is encouraging in terms of trying to prove the\nconjecture (rather than disproving it).\n\\end{remark}\n\n\n\\subsection{Our Contribution}\nOur main contribution, presented in Section~\\ref{sec:primal-dual},\nis an analysis framework based on LP duality and flows\nfor proving upper bounds on the metric distortion of voting mechanisms.\nOur point of departure is a well-known linear program for the\nfollowing problem:\ngiven the rankings of all voters, a winning candidate\n(presumably selected by a mechanism) and an ``optimum'' candidate,\nfind a metric space maximizing the distortion of this choice;\nthat is, find a metric that makes the selected winner as expensive as\npossible, subject to the ``optimum'' candidate having cost 1\n\\footnote{This approach can of course immediately be leveraged into an\n  optimal polynomial-time voting mechanism;\n  we discuss this more in Section~\\ref{sec:optimal}.}\nWe show that the dual of the cost minimization LP can be interpreted\nas a flow problem with an unusual objective function.\nUsing this framework, in order to show an upper bound on the metric\ndistortion of a particular mechanism,\nrather than having to explicitly consider all possible metric spaces,\nit is enough to exhibit a flow of small cost meeting certain demands.\nWe illustrate the power of this analysis framework with three applications.\n\nFirst, in Section~\\ref{sec:uncovered},\nwe give a strong generalization of the key lemmas from\n\\cite{anshelevich:bhardwaj:postl} (Theorem~7)\nand \\cite{munagala:wang:improved} (Lemma~3.7),\nused to prove distortions of 5 and $2+\\sqrt{5}$ for the respective\nmechanisms under consideration.\nThe common idea of both is that when a large enough fraction of voters\nprefer $x$ to $y$, and a large enough fraction prefer $y$ to $z$,\nthen the cost of $x$ can be bounded in terms of the cost of $z$.\nTheorem~7 of \\cite{anshelevich:bhardwaj:postl} is the special case\nwhere both fractions are \\half,\nwhile Lemma~3.7 of \\cite{munagala:wang:improved} is the case\nwhen the first fraction is $\\frac{3-\\sqrt{5}}{2}$,\nand the second is $\\frac{\\sqrt{5}-1}{2}$.\nThese bounds immediately imply the upper bounds on the distortion \nfor any candidate in the uncovered set of a suitably defined\ntournament graph.\nWe give a generalization to arbitrary chains of preferences,\nand upper-bound the cost of $x_1$ in terms of the cost of $x_{\\ell}$\nwhen a \\PrefFrac{i} fraction of voters prefer $x_i$ over $x_{i+1}$,\nfor each $i=1, \\ldots, \\ell-1$.\nFor the specific case when all $\\PrefFrac{i} = \\PREFFRAC$,\nthe bound can be stated very cleanly:\nthe cost of $x_1$ is at most $\\frac{\\ell}{\\PREFFRAC} - 1$ times that\nof $x_{\\ell}$ if $\\ell$ is even, \nand at most $\\frac{\\ell-1}{\\PREFFRAC} + 1$ times that of $x_{\\ell}$ if\n$\\ell$ is odd.\nOur results fully recover and generalize the bounds of\n\\cite{anshelevich:bhardwaj:postl} and \\cite{munagala:wang:improved}.\nThe generalization to longer path lengths can be useful in\nanalyzing voting mechanisms that are missing information.\nThis can happen if the environment restricts the\ncommunication between voters and the mechanism,\nso that parts of the rankings remain unknown,\nas in \\cite{DistortionCommunication}.\nIn fact, the results of Section~\\ref{sec:uncovered} can be used to\nsignificantly improve the upper bounds on the performance of\n``Copeland-like'' mechanisms with missing information,\ncompared to the bounds in \\cite{DistortionCommunication}.\n\nAs a direct application of this generalized bound,\nin Section~\\ref{sec:rp-schulze},\nwe resolve the distortion of the Ranked Pairs and Schulze rules\n(defined in Section~\\ref{sec:preliminaries}):\nwe show that both have distortion $\\Theta(\\sqrt{n})$.\nThe upper bound is a clean application of the lemma bounding\ndistortion via longer chains of preferences,\nwhile the lower bound is obtained with a generalization of the\nexample which \\cite{goel:krishnaswamy:munagala} used to\nlower-bound the distortion of both rules by 5.\nThe distortion of both rules is thus significantly higher than\nthe distortions of 5 and $2+\\sqrt{5}$ achieved by the uncovered set\nmechanisms.\nUnderstanding the distortion of the Schulze rule in\nparticular is of importance because it is widely used in practice.\n\nAs a third application, the flow interpretation naturally\nsuggests a candidate mechanism that might achieve \ndistortion 3, which we present in Section~\\ref{sec:combinatorial}.\nThe analysis points to a sufficient condition for distortion 3:\nthat for every given preference profile of the voters,\nthere be a candidate $x$ such that for all other candidates $y$,\na certain bipartite graph on the voters have a perfect matching.\nIn fact, the mechanism itself can be phrased in this terminology,\nleading to a purely combinatorial polynomial-time mechanism.\n\nThis mechanism was independently discovered and presented in\n\\cite{munagala:wang:improved}.\nIn \\cite{munagala:wang:improved}, it is also shown --- again with a\ncase distinction proof over metric spaces --- that if such a\ncandidate $x$ exists, the mechanism guarantees distortion 3.\nOur duality framework gives a cleaner and simpler proof\nof this fact.\nThe main question is then whether the desired candidate $x$ always\nexists.\n\nMunagala and Wang \\cite{munagala:wang:improved} conjecture --- as do\nwe --- that it does.\nThey phrase a conjecture which is essentially a restatement of the\nfact that the algorithm succeeds in finding a candidate $x$.\nIn Section~\\ref{sec:combinatorial}, we present a slight rephrasing of\nthis conjecture, along with two more very different-looking (in fact,\nmuch more self-contained) conjectures, each of which would resolve the\nquestion positively, i.e., establish a distortion of 3.\nOne of the two new conjectures is phrased in terms of certain preferences\nbetween candidates and sets under randomly drawn preference orders,\nwhile another talks about cycles in certain induced subsets of a type\nof graph we define.\nThe fact that they are sufficient to establish distortion 3 is based\non Hall's Marriage Theorem for bipartite graphs.\nWe have verified the conjecture by hand for $n \\leq 4$ candidates,\nand using exhaustive computer search for $n \\leq 7$.\nResolving any of the three conjectures positively would answer the key\nopen question of the field of metric voting,\nclosing the gap between the upper bound of $2+\\sqrt{5}$\non the best distortion of any deterministic mechanism,\nand the lower bound of 3. \n\n\\subsection{Additional Related Work}\n\nThe observation that mechanisms may have to optimize a cardinal\nobjective function while only given ordinal information (i.e.,\nrankings) extends beyond just voting mechanisms, to more general\nproblems.\nSee, e.g., \\cite{anshelevich:sekar:blind,anshelevich:ordinal} for\nresults on other optimization problems under ordinal information.\n\nThe lower bound of 3 on the distortion of any mechanism is based on\nworst-case input instances.\nBetter bounds can be obtained when additional assumptions are placed\non the instances.\nAs one example, \n\\cite{anshelevich:postl:randomized,gross:anshelevich:xia:agree}\nshow that when instances are \\emph{decisive}, in the sense that each\nvoter has a candidate she strongly prefers over all others,\nbetter upper bounds on the distortion are obtained.\nAs another example, when the candidates are drawn i.i.d.~from the set\nof all voters, \\cite{OfThePeople} gives improved constant distortion\nbounds in the case of two candidates, while\n\\cite{BordaRepresentative} shows that many position-based scoring\nrules now achieve constant distortion (instead of linear).\n\nThe lower bound of 3 on the distortion of voting mechanisms only\napplies to deterministic mechanisms.\nRandomization can lead to lower distortion \\cite{anshelevich:postl:randomized}.\nFor example, it is known that the Randomized Dictatorship mechanism,\nwhich outputs the first choice of a uniformly random voter,\nhas distortion strictly smaller than 3.\n\nOur work ignores the issue of incentives, i.e., whether voters\ntruthfully report their preferences.\nThe connection between strategy proofness and distortion in metric\nvoting is studied in \\cite{feldman:fiat:golomb}.\n\nThe use of LP duality for analyzing the performance of optimization\nalgorithms has a long history, e.g., in approximation algorithms\n(see \\cite{vazirani:approximation-algorithms}).\nAnother more recent example is the duality framework of Cai, Devanur,\nand Weinberg \\cite{cai:devanur:weinberg:duality}\n(see also references in \\cite{cai:devanur:weinberg:duality} to prior,\nless general, work) for analyzing the revenue of Bayesian Incentive\nCompatible mechanisms.\nIn their case as well, dual solutions can be interpreted as flows,\nand Cai et al.~obtain performance guarantees by exhibiting particular\ntypes of ``canonical'' flows that can be interpreted as witnesses for\nthe revenue guarantees.\nWhile this work and ours have the use of duality, and the\ninterpretation as flows, in common, the specific technical details are\nvery different.\n\n\n\\section{Introduction} \\label{sec:introduction}\n\\input{introduction}\n\n\\section{Preliminaries} \\label{sec:preliminaries}\n\\input{preliminaries}\n\n\\section{The LP Duality Approach and Flows} \\label{sec:primal-dual}\n\\input{primal-dual}\n\n\\section{Generalization of Distortion Bounds for Undominated Nodes}\n\\label{sec:uncovered}\n\\input{uncovered}\n\n\\section{Distortion of Ranked Pairs and Schulze} \\label{sec:rp-schulze}\n\\input{RP-Schulze}\n\n\\section{A Candidate Algorithm for Distortion 3} \\label{sec:combinatorial}\n\\input{combinatorial}\n\n\\section{Conclusions} \\label{sec:conclusions}\n\\input{conclusions}\n\n\\subsubsection*{Acknowledgements}\nThe author would like to thank\nElliot Anshelevich, Sid Banerjee, Shaddin Dughmi, Bobby Kleinberg and Kai Wang\nfor useful conversations and pointers, and anonymous reviewers for\nuseful feedback.\n\n\\bibliographystyle{plain}\n\n\\subsection{Voters, Candidates, and Social Choice Rules}\n\nAn \\emph{instance} $(\\ALLCANDS, \\ALLPREFS)$ consists of a set\nof $n$ candidates \\ALLCANDS, and the voters' preferences \\ALLPREFS\namong these candidates.\nCandidates will always be denoted by lowercase letters $\\WINNER, x, y,\nz$ (and their variations), with \\WINNER \nspecifically reserved for a candidate chosen as winner by a mechanism\n(which will be clear from the context).\nSets of candidates are denoted by uppercase letters $X, Y, Z$.\nThe $m$ voters are denoted by $v,v'$ and variants thereof,\nand the set of all voters is \\ALLVOTERS.\n\nEach voter $v$ has a \\emph{total order} (or \\emph{preference order} or\n\\emph{ranking} --- we use the three terms interchangeably) \\PREF[v]\nover the $n$ candidates.\nWe write \\Pref[v]{x}{y} to denote that $v$ (strictly) prefers $x$ over $y$,\nand \\WPref[v]{x}{y} to denote that $v$ weakly prefers $x$ over $y$;\nthe difference is that the latter allows $x=y$.\nWe extend this notation to sets, writing, for instance,\n\\Pref[v]{Y}{Z} to denote that $v$ (strictly) prefers all candidates in\n$Y$ over all candidates in $Z$.\nWe write $\\CandFirst{x}{Y} = \\Set{v \\in \\ALLVOTERS}{\\Pref[v]{x}{Y}}$\nfor the set of voters who rank $x$ strictly ahead of all candidates in\n$Y$,\nand $\\CandLast{x}{Y} = \\Set{v \\in \\ALLVOTERS}{\\Pref[v]{Y}{x}}$\nfor the set of voters who rank $x$ strictly behind all candidates in $Y$.\n\nA \\emph{vote profile} \\ALLPREFS is the vector of the rankings of all\nvoters $\\ALLPREFS = (\\PREF[v])_{v \\in \\ALLVOTERS}$.\nA \\emph{social choice rule}\n(we use the term \\emph{mechanism} interchangeably) \n$\\SCRULE : (\\ALLCANDS, \\ALLPREFS) \\mapsto \\WINNER$\nis given the rankings of all voters, i.e., \\ALLPREFS,\nand deterministically produces as output one \\emph{winning} candidate\n$\\WINNER = \\SCRule{\\ALLCANDS, \\ALLPREFS} \\in \\ALLCANDS$.\n\n\\subsection{(Pseudo-)Metric Space and Distortion}\n\nThe voter preferences are assumed to be derived from distances between\nvoters and candidates.\nThe distance \\Dist{v}{x} between voter $v$ and candidate $x$ captures\nhow similar their positions on key issues are.\nThe distances \\DIST form a \\emph{pseudo-metric}, i.e.,\nthey are non-negative and satisfy the triangle inequality%\n\\footnote{Distances between pairs of voters, or between pairs of\n  candidates, could be defined using shortest-path distances;\n  however, they are irrelevant for our analysis.\n  \\emph{Symmetry}, another defining property of a pseudo-metric,\n  would arise automatically when using this definition.}\n$\\Dist{v}{x} \\leq \\Dist{v}{y} + \\Dist{v'}{y} + \\Dist{v'}{x}$\nfor all voters $v, v'$ and candidates $x, y$.\n\nA vote profile \\ALLPREFS is \\emph{consistent} with the\npseudo-metric \\DIST if and only if each voter ranks the candidates by \nnon-decreasing distance from herself;\nthat is, if \\Pref[v]{x}{y} whenever $\\Dist{v}{x} < \\Dist{v}{y}$.\nWhen \\ALLPREFS is consistent with \\DIST, we write \\Legal{\\DIST}{\\ALLPREFS}.\nIf there are ties among distances,\nseveral vote profiles will be consistent with \\DIST.\n\n\\begin{definition}[Social Cost, Distortion]\n\\begin{enumerate}\n\\item The \\emph{social cost} of candidate $x$ is the sum of distances\n  from $x$ to all voters: $\\Cost{x} = \\sum_{v} \\Dist{v}{x}$.\n\\item A candidate is an \\emph{optimum candidate} iff he minimizes%\n  \\footnote{There could be multiple optimum candidates --- for our\n    analysis, it will never matter which of them is designated as\n    ``the'' optimum.}\n  the social cost: $\\OPT[\\DIST] \\in \\argmin_{x \\in \\ALLCANDS} \\Cost{x}$.\n\\item The \\emph{distortion} of a mechanism \\SCRULE is the largest\n  possible ratio between the cost of the candidate chosen by \\SCRULE,\n  and the optimal\n  (with respect to the pseudo-metric \\DIST, which \\SCRULE does not know)\n  candidate \\OPT[\\DIST]:\n\\[\n  \\Distortion{\\SCRULE} \\; = \\;\n  \\max_{\\ALLPREFS} \\sup_{\\DIST: \\Legal{\\DIST}{\\ALLPREFS}}\n  \\frac{\\Cost{\\SCRule{\\ALLCANDS, \\ALLPREFS}}}{\\Cost{\\OPT[\\DIST]}}.\n\\]\n\\end{enumerate}\n\\end{definition}\n\nThe main cause for (large) distortion is that while\nthe social choice rule knows the voter preferences \\ALLPREFS,\nit does not know the pseudo-metric \\DIST.\nWe can think of the pseudo-metric \\DIST as being chosen\nadversarially,\nbased on the winning candidate \\WINNER = \\SCRule{\\ALLPREFS} chosen by\nthe mechanism.\nHowever, the adversary is constrained by having to ensure that \\DIST\nis consistent with \\ALLPREFS.\n\n\\subsection{Ranked Pairs and the Schulze Rule}\nIn Section~\\ref{sec:rp-schulze}, we will characterize the distortion\nof the Ranked Pairs and Schulze Rules.\nWe briefly review these rules here.\nBoth are based on a weighted directed graph on the set of candidates\n\\ALLCANDS.\nThe weight $p_{x,y}$ of the edge from candidate $x$ to $y$\nis the fraction of voters who have \\Pref{x}{y}.\nAs a result, $p_{x,y} + p_{y,x} = 1$ for all $x, y$.\n\nIn Ranked Pairs \\cite{tideman:independence-of-clones},\nthe (ordered) pairs $(x,y)$ are considered in\nnon-increasing order of $p_{x,y}$.\nWhen the pair $(x,y)$ is considered,\nthe directed edge $(x,y)$ is inserted into the graph if and only if\ndoing so creates no cycle.\nWhen the insertion process terminates, the graph has a unique source\nnode, which is returned as the winner.\n\nIn the Schulze Method \\cite{schulze:single-winner-election-method},\na directed weighted graph is created in which\neach ordered pair $(x,y)$ has an edge with weight $p_{x,y}$.\nThen, for each pair $(x,y)$, let $s_{x,y}$ be the width of the widest\npath from $x$ to $y$, that is, the largest $p$ such that there is a\npath from $x$ to $y$ on which all edges $(x',y')$ have\n$p_{x',y'} \\geq p$.\nIt has been shown \\cite{schulze:single-winner-election-method}\nthat there is a candidate node $x$ such\nthat $s_{x,y} \\geq s_{y,x}$ for all other candidates $y$.\nAny such candidate $x$ is returned as the winner.\n\nFor the purposes of our analysis, the only property of these methods\nthat matters is captured by the following lemma, which is well known.\n(We prove it only for completeness.)\n\n\\begin{lemma} \\label{lem:rp-schulze-basic}\n  Let \\WINNER be the candidate selected by the rule\n  (either Ranked Pairs or Schulze),\n  and $y$ any other candidate.\n  Then, there exists a $p$ and a sequence of (distinct) candidates\n  $x_1 = \\WINNER, x_2, \\ldots, x_{\\ell} = y$ with the property that\n  at least a $p$ fraction of voters prefer $x_i$ over $x_{i+1}$\n  (for each $i$),\n  and at most a $p$ fraction of voters prefer $y$ over \\WINNER.\n\\end{lemma}\n\n\\begin{proof}\n  For the Ranked Pairs rule, because \\WINNER was selected, it\n  has no incoming edges in the DAG that is constructed.\n  In particular, this means that Ranked Pairs did not insert the\n  edge $(y, \\WINNER)$, so when it was considered for insertion,\n  it would have caused a cycle, meaning that there was a path from\n  \\WINNER to $y$ all of whose edges had been inserted earlier.\n  By the definition of the Ranked Pairs insertion order, this\n  means that all edges on this path had a higher fraction of voters\n  agreeing with them, giving us the path claimed above.\n\n  For the Schulze rule, recall that the winner has the\n  property that $s_{\\WINNER,x} \\geq s_{x,\\WINNER}$ for all\n  candidates $x$. Let $p=s_{\\WINNER,y}$.\n  Then, there is a path from \\WINNER to $y$ in which each edge\n  corresponds to a preference by at least a $p$ fraction of voters.\n  On the other hand, because $(y, \\WINNER)$ is a path from $y$\n  to \\WINNER, at most an $s_{y,\\WINNER} \\leq s_{\\WINNER,y}$\n  fraction of voters can prefer $y$ to \\WINNER.\n\\end{proof}\n\n\\subsection{An Efficient Optimal Mechanism} \\label{sec:optimal}\n\nAs already observed in\n\\cite{anshelevich:bhardwaj:elkind:postl:skowron,goel:krishnaswamy:munagala},\nthe LP \\eqref{eqn:primal-lp} can be leveraged to immediately yield an\ninstance-optimal polynomial-time mechanism for minimizing distortion,\nas follows.\nGiven the voter preferences \\PREF[v],\nlet \\OptDist{\\WINNER}{\\LPC} denote the maximum LP objective of the\nLP~\\eqref{eqn:primal-lp} for the winner \\WINNER and putative optimum\n\\LPC.\nThe distortion for \\WINNER as a winner is then\n$\\WOptDist{\\WINNER} = \\max_{\\LPC} \\OptDist{\\WINNER}{\\LPC}$.\nThe mechanism returns as winner any candidate in\n$\\argmin_{\\WINNER} \\WOptDist{\\WINNER}$.\n\nBecause the algorithm only involves solving $n^2$ linear programs,\nit runs in polynomial time.\nBy definition (and correctness of the LP~\\eqref{eqn:primal-lp}),\nthe distortion for a given vote profile \\ALLPREFS\nand winner \\WINNER is \\WOptDist{\\WINNER};\nthus, the mechanism does indeed minimize distortion.\nUnfortunately, as also observed in\n\\cite{anshelevich:bhardwaj:elkind:postl:skowron},\nit is not immediate from the mechanism and the LP formulation how to\nbound the distortion for all vote profiles;\nthough \\cite{goel:krishnaswamy:munagala} conjecture that the LP-based\nalgorithm guarantees distortion at most 3.\n\nThe dual program provides a very useful tool towards making the\nLP-based algorithm combinatorial,\nand for reducing an analysis of its distortion to simpler\ncombinatorial conjectures.\nMore generally (and perhaps importantly),\nthe dual program provides a general approach for bounding the metric\ndistortion of other voting rules, too.\n\n\\subsection{The Dual Linear Program} \\label{sec:dual}\nRearranging the primal LP into normal form, taking the dual,\nand switching the signs of the \\DNo{x} variables (for clarity)\nyields the following dual LP~\\eqref{eqn:dual-lp}.\nIn this LP, the \\DTe{v}{v'}{x}{y} are the dual variables for the\ntriangle inequality constraints,\n\\DCo{v}{x}{y} are the dual variables for the consistency constraints,\nand the \\DNo{x} are the dual variables for the\nnormalization\/optimality constraints.\n\n\\begin{LP}[eqn:dual-lp]{Minimize}{\\sum_x \\DNo{x}}\n\\multicolumn{2}{l}{  \\DNo{x}\n  + \\sum_{y: \\Pref[v]{x}{y}} \\DCo{v}{x}{y}\n  - \\sum_{y: \\Pref[v]{y}{x}} \\DCo{v}{y}{x}\n  + \\sum_{y, v'}\n    \\left(  \\DTe{v}{v'}{x}{y} - \\DTe{v}{v'}{y}{x}\n          - \\DTe{v'}{v}{x}{y} - \\DTe{v'}{v}{y}{x} \\right)}\n  \\\\ \\qquad \\geq \\;\n  \\begin{cases} 1 \\mbox{ if } x = \\WINNER \\\\\n                0 \\mbox{ if } x \\neq \\WINNER \\end{cases} \n        & \\mbox{ for all } v, x \\\\\n  \\DTe{v}{v'}{x}{y} \\geq 0 & \\mbox{ for all } v, v', x, y \\\\\n  \\DCo{v}{x}{y} \\geq 0 & \\mbox{ for all } v, x, y \\\\\n  \\DNo{x} \\leq 0 & \\mbox{ for all } x \\neq \\LPC.\n\\end{LP}\n\nNotice that \\DNo{\\LPC} is in fact unconstrained,\ndue to the equality constraint in the normalization.\n\nThe advantage of studying the dual linear program instead of the\nprimal (or reasoning about the distortion directly) is that\nit omits any reference to any metric space.\nRather than having to reason about all candidate metric spaces\nconsistent with given voting patterns, by weak duality,\nwe only have to exhibit one setting of the dual variables that yields\na small dual objective value.\nThus, our goal in analyzing a mechanism will be to show that for any\nvoter preferences \\ALLPREFS, \nwith a suitably chosen winner \\WINNER, there is a setting of dual\nvariables giving a small objective value.\n\n\\subsection{Using the Dual by Exhibiting Flows}\n\nThe LP~\\eqref{eqn:dual-lp} looks rather unwieldy,\nmostly due to the terms involving the \\DTe{v}{v'}{x}{y} variables.\nHowever, by making some specific choices for these variables,\nit can be interpreted as a flow\\footnote{%\nSome sources use the word ``flow'' only when there is a single source\nand a single sink; here, we will have multiple sources and sinks.\nWe will still use the word ``flow'' in a generic sense.}\nproblem on a suitably defined graph,\nwith a somewhat unusual objective function.\nThis is captured by the following lemma:\n\n\\begin{lemma} \\label{lem:dual-flow}\n  Let $H = (U, E)$ be a directed graph with vertex set\n  $U = \\ALLVOTERS \\times \\ALLCANDS$, and edges defined as follows:\n  \\begin{itemize}\n  \\item Whenever \\Pref[v]{x}{y},\n    $E$ contains the directed edge $(v,x) \\to (v,y)$.\n    We call such edges \\emph{preference edges}.\n  \\item For all $x$ and $v \\neq v'$,\n    $E$ contains the directed edge $(v,x) \\to (v',x)$.\n    We call such edges \\emph{sideways edges}.\n  \\end{itemize}\n\n  Let $f$ be a flow on $H$ such that exactly one unit of flow\n  originates at the node $(v,\\WINNER)$ for each voter $v$,\n  and flow is only absorbed at nodes $(v,\\LPC)$ for voters $v$.\n  Define the cost of $f$ at voter $v$ to be\n  $\\FlCost{f}{v} = \\sum_{e \\text{ into } (v,\\LPC)} f_e\n  + \\sum_{x \\neq \\LPC} \\sum_{v' \\neq v}\n  (f_{(v',x) \\to (v,x)} + f_{(v,x) \\to (v',x)})$.\n\n  Then, $\\Cost{\\WINNER} \\leq \\Cost{\\LPC} \\cdot \\max_v \\FlCost{f}{v}$.\n\\end{lemma}\n\nThe graph $H$ has two types of edges.\nFor any fixed voter $v$, the preference edges\n$(v,x) \\to (v,y)$ (over all candidate pairs $x, y$) exactly correspond\nto $v$'s preference order.\nFor any fixed candidate $x$, the sideways edges\n$(v,x) \\to (v',x)$ (over all voter pairs $v, v'$) form a complete\ndirected graph.\n\nThe flow's cost function has two terms for each voter $v$.\nThe first is fairly standard in the study of multi-commodity flows:\nthe capacity required at the sink node $(v,\\LPC)$ to be able\nto absorb all of the flow.\nThe second one is rather non-standard: for each voter $v$,\nthere is an additional penalty term for all incoming and outgoing\nflows of nodes $(v,x)$ for $x \\neq \\LPC$ along sideways edges.\nIn other words, using preference edges is much\nless costly than using sideways edges:\nthe former just route flow, while the latter route the flow,\nbut also incur a cost penalty at both endpoints.\n\n\\begin{proof}\n  Let $f$ be a flow with one unit of flow originating at each\n  node $(v,\\WINNER)$,\n  such that flow is only absorbed at nodes $(v,\\LPC)$.\n  We define dual variables, and show that these dual variables are\n  feasible.\n  Then, we will obtain the statement of the lemma by weak LP duality. \n\n  For each triple $v,x,x'$, we set\n  $\\DCo{v}{x}{x'} = f_{(v,x) \\to (v,x')}$.\n  For each triple $v,v',x$, we set\n  $\\DTe{v}{v'}{x}{\\LPC} = f_{(v,x) \\to (v',x)}$;\n  notice that we carefully choose \\LPC as the additional candidate for\n  the dual variable.\n  Finally, we set $\\DNo{\\LPC} = \\max_v \\FlCost{f}{v}$.\n  All other dual variables are set to 0.\n\n  We now verify that this assignment satisfies all dual constraints.\n  First, because $\\DTe{v}{v'}{x}{y} = 0$ and $\\DNo{y} = 0$\n  whenever $y \\neq \\LPC$, \n  the dual constraints for all $x \\neq \\LPC$ are exactly circulation\n  constraints; that is, they require that (at least) one unit of flow\n  originate with $(v,\\WINNER)$, and that flow be conserved\n  (or appear) at each node $x \\neq \\LPC, x \\neq \\WINNER$.\n  Thus, all of these constraints are satisfied for the given dual\n  variable assignment.\n\n  For pairs $(v,\\LPC)$, the left-hand side of the dual constraint\n  totals the flow into $(v,\\LPC)$ along any edges\n  (these are the \\DCo{v}{x}{\\LPC} variables and the\n  \\DTe{v'}{v}{\\LPC}{\\LPC} variables),\n  as well as all the \\DTe{v}{v'}{x}{\\LPC} and \\DTe{v'}{v}{x}{\\LPC}\n  variables, for all $v' \\neq v$.\n  By definition of the dual variables, this is exactly the flow\n  into $(v,\\LPC)$,\n  plus the flow into and out of all nodes $(v,x)$ for $x \\neq \\LPC$\n  along edges of the form $(v,x) \\to (v',x)$ and $(v',x) \\to (v,x)$.\n  Thus, it is exactly \\FlCost{f}{v}.\n  Because we set $\\DNo{\\LPC} = \\max_v \\FlCost{f}{v}$,\n  the dual constraints for all pairs $(v,\\LPC)$ are also satisfied.\n\n  Since we have a dual feasible solution of objective value\n  $\\DNo{\\LPC} = \\max_v \\FlCost{f}{v}$,\n  by weak duality, for every metric, the cost of the primal\n  is at most $\\max_v \\FlCost{f}{v}$.\n  This completes the proof.\n\\end{proof}\n\n\n\n\\subsection{Special Cases}\nLemma~3.7 of \\cite{munagala:wang:improved} is the special case\nof Corollary~\\ref{cor:paths} with $\\ell=3, x_1 = \\WINNER, x_3 = \\OPT$,\nand $\\PrefFrac{1} = \\frac{3-\\sqrt{5}}{2}, \\PrefFrac{2} = \\frac{\\sqrt{5}-1}{2}$.\nOur Corollary~\\ref{cor:paths} then exactly recovers the bound of\n$2+\\sqrt{5}$.\n\nWhen we have a uniform lower bound on the $\\PrefFrac{i}$,\nCorollary~\\ref{cor:paths} can be simplified significantly.\n(A direct proof of Corollary~\\ref{cor:paths-uniform} would\nalso be simpler than the proof of the more general\nCorollary~\\ref{cor:paths}.)\n\n\\begin{corollary} \\label{cor:paths-uniform}\n  Let $x_1, x_2, \\ldots, x_\\ell$ be (distinct) candidates such that\n  for each $i = 2, \\ldots, \\ell$, at least a $\\PREFFRAC > 0$ fraction of\n  the voters prefer candidate $x_{i-1}$ over candidate $x_{i}$.\n  Then, if $\\ell$ is even, \n  $\\Cost{x_1} \\leq (\\frac{\\ell}{\\PREFFRAC} - 1) \\cdot \\Cost{x_\\ell}$;\n  if $\\ell$ is odd,\n  $\\Cost{x_1} \\leq (\\frac{\\ell-1}{\\PREFFRAC} + 1) \\cdot \\Cost{x_\\ell}$.\n\\end{corollary}\n\n\\begin{proof}\n  We substitute $\\PrefFrac{i} = \\PREFFRAC$ for all $i$ in\n  Corollary~\\ref{cor:paths};\n  then, we observe that for even $\\ell$, the independent set of\n  integers giving the largest sum is $\\SET{2, 4, \\ldots, \\ell}$,\n  while for odd $\\ell$, it is $\\SET{1, 3, \\ldots, \\ell}$.\n\\end{proof}\n\nThe asymmetry between even and odd $\\ell$ disappears when\n$\\PREFFRAC = \\half$ (i.e., in the case of the majority graph),\nwhere the bound simply becomes $2\\ell-1$.\nThe result thus strongly generalizes Theorem~7 in\n\\cite{anshelevich:bhardwaj:postl},\nwhich is the special case of $\\PREFFRAC = \\half$ and $\\ell=3$.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Results}\n\\subsection{Experimental approach}\n\nCapillary leveling of stepped films is used to measure the slip length of polystyrene (PS) on Teflon\\texttrademark $\\,$  fluoropolymer (AF) substrates (see Methods), a combination of materials which has been previously shown to exhibit interfacial slip~\\cite{Baumchen2009,Baumchen2012,Baumchen2014,Haefner2015}. As a calibration, identical films of PS are prepared on silicon (Si) substrates since the Si\/PS interface has no interfacial slip~\\cite{McGraw2012}. Both types of samples are annealed simultaneously and side-by-side as outlined schematically in Fig.~\\ref{fig:1}a, and surface profiles are obtained with atomic force microscopy (AFM). The self-similar profile of the Si\/PS sample provides a calibration measurement of the PS capillary velocity, $v_c=\\gamma\/\\eta$, where $\\gamma$ and $ \\eta$ are the surface tension and viscosity, respectively. Note that the value of $v_c$ depends on temperature and molecular weight, which are identical for the simultaneously studied Si\/PS and AF\/PS samples. The protocol thus allows the unambiguous and quantitative determination of the slip length of the solid\/liquid (AF\/PS) interface, the only differing quantity between the two simultaneously annealed samples. The measured film thickness profiles are self-similar in the reduced variable $x\/t^{1\/4}$, where $x$ is the horizontal coordinate and $t$ is the annealing time, for PS stepped films leveling on both substrates (Fig.~\\ref{fig:1}b). We find that PS films broaden more rapidly on the AF substrates than on the Si calibration substrates (Fig.~\\ref{fig:1}b) for all investigated molecular weights, and this faster broadening is more significant at higher PS molecular weight. In order to demonstrate that we can resolve even the smaller slip lengths, we show a zoom on the dip region of the lower molecular weight film as an inset. There, it can clearly be seen that the film on AF has also leveled further than that on Si. As it will be shown below (Fig.~\\ref{fig:3}), we resolve slip lengths at the level of tens of nm. Capillary leveling thus provides one advantage over dewetting, for which small slip lengths have comparatively larger measurement error (Fig.~\\ref{fig:3}).\n\n\\subsection{Theoretical approach}\n\nTo extract quantitatively the slip length at the solid\/liquid interface, we employ a continuum hydrodynamic model for the thin liquid film. Using the incompressible Stokes' equations in the lubrication approximation~\\cite{Oron1997}, and allowing for weak slip\\footnote{We have also analyzed our experiments using the intermediate-slip thin-film equation outlined in Ref.~\\cite{Munch2005}. In the worst case, this results in a small (on the order of $30\\%$) increase in the measured slip length, which does not affect any of the conclusions of this work. Besides, we stress that strong-slip~\\cite{Munch2005} or infinite-slip~\\cite{Ilton2016} descriptions would be incompatible with the observed self-similarity (see Fig.~\\ref{fig:1}b).} (slip length much smaller than the characteristic film thickness) at the solid\/liquid interface, leads to a partial differential equation describing the evolution of the film thickness profile $h(x,t)$~\\cite{Munch2005}:\n\\begin{equation}\n\\label{eq:WSTFE}\n\\frac{\\partial h}{\\partial t} = -\\frac{v_c}{3}\\frac{\\partial}{\\partial x}\\left[ \\left(h^3+3bh^2\\right) \\frac{\\partial^3 h}{\\partial x^3}\\right].\n\\end{equation}\n\nOne can nondimensionalize this equation by introducing an arbitrary reference length scale $h_0\\!=\\!h_1+h_2\/2$, and the associated time scale $3h_0\/v_c$. Furthermore, for a given stepped initial profile (Fig.~\\ref{fig:1}a), the rescaled solution $(h-h_1)\/h_2$ of Eq.~(\\ref{eq:WSTFE}) is self-similar in the variable~\\cite{Munch2005,Salez2012a}:\n\\begin{equation}\nU_0=\\left(\\frac{3x^4}{h_0^3v_ct}\\right)^{1\/4},\n\\end{equation}\nbut depends intrinsically on two parametric ratios, $h_2\/h_1$ and $b\/h_1$. As a particular case, for a stepped initial profile with $h_2\/h_1\\ll1$ one can linearize Eq.~(\\ref{eq:WSTFE}). Nondimensionalizing the obtained equation by introducing the previous length scale $h_0$, but a different time scale $3h_0\/[v_c(1+3b\/h_0)]$, one obtains the result that the rescaled solution $(h-h_1)\/h_2$ is now a single universal function of only the following generalized variable: \n\\begin{equation}\nU_b=\\left[\\frac{3x^4}{h_0^3v_c(1+3b\/h_0)t}\\right]^{1\/4}.\n\\end{equation}\n\n\\begin{figure}[tp]\n\t\\includegraphics[width=0.95\\columnwidth]{Fig2}\n\t\\caption{\\label{fig:2} Capillary leveling is a robust experimental probe to measure slip length. For three different sample geometries, the rescaled self-similar theoretical profiles (dashed lines) fit the experimentally measured ones (solid lines) with one free parameter, the slip length $b$. Parameters are indicated in legends, and theoretical details are provided in main text. In (c), the position $x$ is replaced by the radial coordinate $r$.}\n\\end{figure}\nIn general, numerical solutions of Eq.~(\\ref{eq:WSTFE})~\\cite{Salez2012a} can be used to fit the data (Fig.~\\ref{fig:2}a). For the particular case of $h_2\\ll h_1$, analytical solutions of the linearized version of Eq.~(\\ref{eq:WSTFE})~\\cite{Salez2012d} can also be used to fit the data (Fig.~\\ref{fig:2}b). Since $v_c$ is fixed by the simultaneous no-slip calibration experiment, and the sample geometry is directly measured using AFM, the slip length $b$ is the only free parameter in fitting the theory to experimentally measured AF\/PS profiles (Fig.~\\ref{fig:2}). The slip length is found to be independent of temperature (Fig.~\\ref{fig:2}a) in the considered range, and is not sensitive to changes in the sample geometry through $h_1$ and $h_2$ (Fig.~\\ref{fig:2}b). \n\nComplementary experiments were performed in a different geometry, in which the PS film was created with a cylindrical hole at the top~\\cite{Backholm2014} (see Methods) instead of a step. The result is shown in Fig.~\\ref{fig:2}c. The slip length is determined by fitting the radially averaged normalized profile to the analytical asymptotic solution of the linearized axisymmetric thin-film equation~\\cite{Backholm2014} -- including weak slip through the variable $U_b$ above, where $x$ becomes the radial coordinate here.  \n\n\n\\subsection{Effect of molecular weight on slip}\n\nThe effect of chain length on interfacial slip was studied using a series of 13 different PS molecular weights $8 \\leq M_w \\leq 373$\\,kg\/mol spanning the range between unentangled and well-entangled PS~\\cite{rubinstein2003polymer, Fetters1999}. Results are shown in Fig.~\\ref{fig:3}a (blue circles). At low molecular weight, the slip length increases with increasing PS molecular weight, but becomes approximately constant for molecular weights greater than $\\sim100$\\,kg\/mol.\n\n\\begin{figure}[tp]\n\t\\includegraphics[width=1\\columnwidth]{Fig3}\n\t\\caption{\\label{fig:3} Slip on ideal substrates is inhibited at low shear rates due to adsorbing polymer chains. (a) Results from PS leveling experiments (blue circles) on AF substrates. Each data point consists of 2-12 individual measurements. For comparison, results from PS dewetting experiments (orange squares, data from~\\cite{Baumchen2009}) on AF substrates are also shown. Two equations with one free parameter (solid lines) describe both sets of experiments: it assumes adsorption of chains in the low-shear-rate leveling experiments and no chain adsorption in the high-shear-rate dewetting experiments. (b) The difference in the measured slip length between leveling (blue) and dewetting (orange) experiments is confirmed using a different substrate (SAM) for PS (9\\,kg\/mol at $110\\,^\\circ$C and 65\\,kg\/mol at $135\\,^\\circ$C).}\n\\end{figure} \n\nThe molecular weight dependence of the AF\/PS slip length has previously been found in dewetting studies (Fig.~\\ref{fig:3}a, orange squares)  to increase sharply at large molecular weight~~\\cite{Baumchen2009,Baumchen2012,Baumchen2014}, which contrasts with the leveling results (Fig.~\\ref{fig:3}a, blue circles) in the current work. Although the results from the two techniques agree at low PS molecular weight, the leveling results exhibit slip lengths that are reduced by two orders of magnitude at the highest molecular weights. \n\nTo determine if the difference in slip length at high molecular weight is specific to AF\/PS, we performed a set of experiments with PS on self-assembled monolayer (SAM) substrates which are known to provide a slip boundary condition for PS~\\cite{Fetzer2005,Baumchen2014}. In the SAM\/PS experiments, both leveling and dewetting measurements were performed for two different molecular weights (9\\,kg\/mol and 65\\,kg\/mol). Results are shown in Fig.~\\ref{fig:3}b. As for AF\/PS, both molecular weights show a discrepancy between the slip length accessed with leveling and dewetting. Furthermore, the difference grows with molecular weight. Therefore, the observed difference in slip length between dewetting and leveling experiments exists also in the SAM\/PS system, and is thus not specific to AF\/PS.\n\n\\section{Discussion}\n\nUsing lubrication theory~\\cite{Oron1997} applied to the leveling experiments, typical shear rates at the substrate can be estimated through $\\partial_zu|_{z=0} =v_c h\\partial_{x}^3h$. Using the experimental data (Fig.~\\ref{fig:1}), we find strain rates of order $10^{-5}-10^{-6}\\,\\mathrm{s}^{-1}$ for the molecular weights used. This range is three orders of magnitude lower than the average shear rates calculated for dewetting with the same molecular weights~\\cite{Baumchen2012}, and even lower if the maximum shear rate at the dewetting contact line is used. \n\nA quantitative analysis of the residence time for polymer molecules under flow illustrates the effects of the different shear rates between leveling and dewetting experiments (see Supplemental Information for details). The energy associated with the external force acting on an adsorbed polymer chain is estimated as $R_\\mathrm{g}^{\\,4}\\partial_xP$, where $P$ denotes the (Laplace) pressure and $R_\\mathrm{g}$ the radius of gyration of the chain. Typical values from the leveling experiments ($\\partial_xP\\approx 0.5\\,$kPa\/$\\mu$m), with $R_\\mathrm{g}$ = 29\\,nm \\cite{rubinstein2003polymer}, provide $R_\\mathrm{g}^{\\,4}\\partial_xP\\approx 3.5\\times10^{-22}$\\,J, which is substantially smaller than thermal energy $k_{\\mathrm{B}}T\\approx 5\\times10^{-21}$J at $T = 150\\,^\\circ$C, or the van der Waals interaction energies. Conversely, the energy associated with dewetting experiments is larger than $k_{\\mathrm{B}}T$ or van der Waals interaction energies by about one order of magnitude, thus crossing the threshold energy scale for chain desorption. The low shear rates for leveling thus lead to residence times for polymer chains at the solid\/liquid interface that are long compared to the typical polymer relaxation time, and these adsorbed chains are expected to significantly contribute to the solid\/liquid friction. This analysis is supported by studies which find a shear dependence of polymer adsorption~\\cite{Cohen1982,McGlinn1988} and a recent work demonstrating that dewetting processes are faster when chain adsorption becomes weaker~\\cite{Wang2017}. Assuming that such an adsorption scenario is operative in leveling but not in dewetting, the large difference in measured slip lengths at high molecular weights between the low-shear-rate leveling experiments and high-shear-rate dewetting experiments (Fig.~\\ref{fig:3}) can be rationalized, as detailed below.\n\nThe Navier-de Gennes model~\\cite{DeGennes1979} predicts that under ideal conditions of no adsorption, where the polymer melt slips along a smooth passive surface, the slip length follows the form:\n\\begin{equation}\n\\label{eq:b_ideal}\n b_\\mathrm{ideal} = a \\frac{\\eta}{\\eta_0} = a \\frac{M_w}{M_0}\\left[1+\\left(\\frac{M_w}{M_e}\\right)^2\\right], \n\\end{equation}\nwhere  $a$ is the monomer size, $\\eta$ is the polymer-melt viscosity, $\\eta_0$ is the viscosity of a melt of monomers, $M_0$ is the monomeric molecular weight, and $M_e$ is the entanglement molecular weight. The right-hand side of Eq.~(\\ref{eq:b_ideal}) corresponds to a smooth interpolation between the Rouse and mean-field reptation regimes for the polymer-melt viscosity~\\cite{rubinstein2003polymer}. To be consistent with the reference dewetting measurements~\\cite{Baumchen2009,Henot2017}, we have chosen the  scaling of Eq.~(\\ref{eq:b_ideal}) to match the large-molecular-weight limit ($b_\\mathrm{ideal}\\sim M_w^{\\,3}$) found previously. We stress that the choice of a different power law (namely, the empirical reptation scaling $b_\\mathrm{ideal}\\sim M_w^{\\,3.4}$) does not alter the conclusions of the present work. Using the parameters and data from Ref.~\\cite{Baumchen2009} (\\emph{i.e.} $a = 0.3$~nm, $M_0 = 104\\,\\mathrm{g\/mol}$, $M_e\/M_0 = 517$), we recall on Fig.~\\ref{fig:3}a (orange line) that Eq.(\\ref{eq:b_ideal}) agrees with the dewetting data over the entire molecular-weight range used. \n\nWe now turn to the case of low-shear-rate experiments, and we describe the influence of transient physically adsorbed chains in an analogous fashion to the case of permanent chemically grafted chains~\\cite{Brochard-Wyart1996}. In the dilute-adsorption regime, adding the adsorption-induced frictional stress of Eq.~(10b) from Ref.~\\cite{Brochard-Wyart1996} to the previous ideal frictional stress $ \\eta u|_{z=0}\/b_\\mathrm{ideal}$, within the Navier-de Gennes construction~\\cite{DeGennes1979}, leads to the dilute-adsorption prediction for the slip length:\n\\begin{equation}\n\\label{eq:b_ads}\nb_\\mathrm{ads} =  \\frac{b_\\mathrm{ideal}}{1 + b_\\mathrm{ideal}\/b^\\star}\\ , \n\\end{equation}\nwhere $b^\\star =  a M_e\/(n M_0)$ and $n$ is the number of adsorbed chains in one cross-sectional chain area $\\sim M_wa^2\/M_0$. Invoking the parameters from Ref.~\\cite{Baumchen2009} as above, the dimensionless number $n$ is thus the only unknown quantity, and we now make the assumption (justified from the fit below) that $n$ does not vary (or varies weakly) with $M_w$. Stated differently, the density of physically adsorbed chains per unit surface is assumed to scale inversely with the cross-sectional chain area.\n \nBy fitting Eq.~(\\ref{eq:b_ads}) to the leveling experimental data in Fig.~\\ref{fig:3}a, we find an excellent agreement (blue line) and extract\n$n=0.45\\pm0.02$. Therefore, with a single free parameter we are able to reconcile the two very different experimental measurements of the slip length on the same AF\/PS system. The saturation value of the slip length at high molecular weights and for low shear rates appears to be set by $b^\\star\\sim N_e a$, where $N_e$ is the number of monomers between entanglements (omitting the numerical prefactor $1\/n$ in front). The prefactor $1\/n$ is expected to increase with shear rate, and to eventually diverge, thus allowing for a continuum of curves in between the two shown in Fig.~\\ref{fig:3}a. In addition, the leveling data appears to have a sufficient resolution to observe for the first time the low-$M_w$ Rouse limit of the Navier-de Gennes prediction. We add two remarks: i) we self-consistently find $n<1$ which validates the dilute-adsorption hypothesis; ii) $n$ is indeed nearly constant, as having a variation of $n$ with $M_w$ would correspond to not having a plateau for $b$ at large $M_w$.\n\n\nAlthough the substrates we use are very smooth (see Methods), it is reasonable to expect that chain adsorption may occur at least temporarily at low shear stresses. First, even ultra-smooth surfaces show contact-angle hysteresis: if a contact line can be pinned on atomic-scale roughness, then so too can polymer chains. Secondly, unfavorable wetting does not imply repulsive interaction between the solid and the liquid, as wetting is rather controlled by a balance between this interaction and the solid-air interaction. Finally, molecular dynamics simulations have shown that adsorbed groups of connected monomers can occur at unfavorable interfaces, and the length of these adsorbed groups increases with molecular weight~\\cite{Smith2005}. Chains which are adsorbed for long enough durations of time to affect the interfacial fluid dynamics are likely to have multiple attached monomers. Therefore, larger adsorbed chains should exclude other chains from adsorbing to the substrate. The fact that $n$ is a constant smaller than 1 could be a signature of this exclusion mechanism. \n\nIn conclusion, we have demonstrated that capillary leveling can quantitatively probe interfacial dynamics at low shear rates. The use of simultaneously-annealed measurement samples on AF substrates and calibration samples on no-slip Si substrates, combined with weak-slip lubrication theory, allow for a robust one-parameter-fit of the slip length to the experimental data. For the case of PS films on an AF substrate, we find the slip length to increase with PS molecular weight before reaching a plateau value at large molecular weights. This contrasts with previous dewetting measurements on the same AF\/PS system, which showed a strong increase in slip length at large PS molecular weights, consistent with the Navier-de Gennes model. Inspired by previous results for grafted chains, we propose an extension of the Navier-de Gennes model which takes into account a dilute physical adsorption of polymer chains in the low-shear-rate leveling experiments, and no adsorption in the high-shear-rate dewetting experiments. With one free parameter, the proposed extension of the Navier-de Gennes model  is able to capture the molecular-weight dependence of the slip length for both sets of experiments. Beyond providing new fundamental insights on the actively-studied problem of hydrodynamic slip, these results demonstrate that even ultra-smooth low-energy surfaces such as Teflon cannot always be considered as ideal substrates.\n\n\\section{Methods}\n\n\\subsection{Substrate preparation and characterization}\nSilicon (Si) wafers (obtained from University Wafer and Si-Mat) were cleaved into 1 cm $\\times$ 1 cm squares. To create the calibration samples, the silicon wafers were  rinsed with ultra-pure water (18.2 M$\\mathrm{\\Omega}\\,$cm, Pall), methanol, and toluene (Fisher Scientific, Optima grade). To create a slip substrate, the wafers were coated with a thin film of the amorphous fluoropolymer AF (AF1600\/AF2400, obtained from Sigma Aldrich) by dip coating from a dilute solution (solvent FC-72, obtained from Acros Organics, 0.5\\% w\/w concentration solution, 0.5 $\\,$mm\/s retraction speed). Following the manufacturer's recommended procedure, the AF substrates were annealed for 20 minutes at $5\\,^\\circ\\mathrm{C}$ above the glass-transition temperature of AF ($160\\,^\\circ\\mathrm{C}$ for AF1600 or $240\\,^\\circ\\mathrm{C}$ for AF2400) to remove residual solvent. The AF film thickness was 10-15$\\,$nm, measured using ellipsometry (EP3, Accurion). Atomic force microscopy (AFM, Caliber, Veeco; Dimension and Multimode, Bruker) measurements showed that the AF substrates have a $0.3\\,\\mathrm{nm}$ RMS surface roughness, and that PS droplets have a Young's contact angle of $88^\\circ$ on these substrates. As a second set of ultra-smooth, low-energy substrates, we decorated Si wafers with a dense self-assembled monolayer (SAM) of octadecyltrichlorosilane (OTS) and dodecyltrichlorosilane (DTS, both purchased from Sigma-Aldrich). The SAM was composed of a mixture of equal parts OTS and DTS, providing the largest slip length for low-molecular-weight PS, see~\\cite{McGraw2017} for details. Silane molecules covalently bind to the native oxide layer of the Si wafer during the established procedure~\\cite{Lessel2015, McGraw2017} for fabrication. These substrates have an RMS roughness of $0.2\\,\\mathrm{nm}$ and PS droplets have a long-time, receding contact angle of $63\\,^\\circ$ on these mixed OTS\/DTS SAMs~\\cite{McGraw2017}.\n\n\\subsection{Polymer film preparation}\nPolystyrene (PS) with molecular weight ($M_w$) ranging between 8--373 kg\/mol and polydispersity less than 1.1 was obtained from Polymer Source and PSS. Films with an initially stepped thickness profile (as in Fig.~\\ref{fig:1}a) were made using a previously-described technique~\\cite{McGraw2012}, with only minor modification. A bottom PS film (thickness range $h_1\\!=\\!100-800\\,\\mathrm{nm}$) and a top PS film (thickness range $h_2\\!=\\!40-400\\,\\mathrm{nm}$) were spun cast from a dilute toluene solution (liquid chromatography grade) onto freshly cleaved mica substrates (Ted Pella). The PS films were pre-annealed on mica in a home-built vacuum oven for at least ten times longer than the calculated longest relaxation time of the PS~\\cite{rubinstein2003polymer} (pre-annealing temperature 140-180$\\,^\\circ\\mathrm{C}$, pre-annealing time 4-72 hours; depending on the PS molecular weight). After annealing, the bottom PS film was floated onto an ultra-pure water bath (18.2 M$\\mathrm{\\Omega}\\,$cm, Pall), and picked up onto either a silicon substrate (calibration) or AF substrate (measurement). The bottom film was then allowed to dry for at least 2 hours before undergoing another annealing (annealing for at least two times the calculated longest relaxation time) to relax residual stress.  The top PS film was then floated off its mica substrate onto the water bath. Sharp edges in the top PS film were created by the floating process for low $M_w$ PS~\\cite{Baumchen2013}, or for high $M_w$ PS by a procedure which involved floating onto Si, cleaving, and refloating onto the water bath~\\cite{McGraw2012}. The sharp-edged top film was then picked up off the water bath with the previously-prepared bottom PS film on a substrate. A final drying of the film at room temperature concluded the sample preparation procedure. Identical procedures were applied for the experiments on the SAM substrates.\nAdditional samples where the second film was prepared with a hole (as in Fig.~\\ref{fig:2}c) were made in the same manner as the stepped films described above, except for the creation of sharp edges. For the hole geometry, a top film was floated onto the water bath and picked up using a metal washer with a millimetric circular hole to create a freestanding film. The top film was then heated above the PS glass-transition temperature in the freestanding state until small holes were nucleated with a diameter between $\\sim 3$ and $10\\,\\mathrm{\\mu m}$. After quenching to room temperature, the top film was transferred onto the bottom film supported by a solid substrate. Full details on the hole-geometry sample preparation are presented in Ref.~\\cite{Backholm2014}. \n\n\\subsection{Experimental setup}\nPairs of otherwise identically-prepared samples were used with only the substrate being different (Si or AF). The film thickness profile of each sample was determined by measuring the surface topography of the film using AFM, and averaging the 3D topography along the direction of translational quasi-invariance of the sample to obtain a 2D thickness profile. The pairs of samples were then placed side-by-side for simultaneous annealing in either the home-built vacuum oven or on a hot stage (Linkam, UK). For a given pair of samples, the annealing temperature was held constant (between 120 and 160$\\,^\\circ{}\\mathrm{C}$), and chosen such that the PS was in its liquid melt state inducing the capillary-driven leveling of the thickness profiles. After a chosen duration of annealing $t$, the samples were rapidly quenched to room temperature, deep into the glassy state of PS, where the leveling process was temporarily halted. The broadening of the thickness profiles were measured using AFM. The samples were then further annealed, quenched, and measured again using AFM. The process of alternate annealing and AFM measurements was repeated until the measured thickness profiles became self-similar (Fig.~\\ref{fig:1}b).\n\n\\section{Acknowledgements}\nThe authors thank Vincent Bertin, Pascal Damman, Fr\\'ed\\'eric Restagno, Barbara Wagner, Andreas M\\\"unch and Dirk Peschka for interesting discussions. They acknowledge financial support from the German Research Foundation (DFG) under grant BA3406\/2 and NSERC (Canada). T.S. acknowledges financial support from the Global Station for Soft Matter, a project of Global Institution for Collaborative Research and Education at Hokkaido University. O.B. acknowledges financial support from the Joliot ESPCI Paris Chair and the Total-ESPCI Paris Chair. J.D.M. and M.A. were supported by LabEX ENS-ICFP: No. ANR-10- LABX-0010\/ANR-10-IDEX-0001-02 PSL.\n\n\\section{Author Contributions}\nM.I., T.S., J.D.M., E.R., K.D.-V. and O.B. conceived the project and designed research. M.I., P.D.F., M.R., M.A., J.D.M. and O.B. performed experiments. T.S., M.B., J.D.M. and E.R. developed the theory. All authors contributed to the analysis and interpretation of the results. M.I., T.S., J.D.M., E.R., K.D.-V. and O.B. wrote the manuscript.\n\n\\section{Additional information}\n\n\\noindent Supplementary Information accompanies this paper at [URL].\\\\\n\n\\noindent Competing financial interests: The authors declare no competing financial\ninterest.\\\\\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nRare-earth nickelates \\emph{R}NiO$_3$ (\\emph{R} = rare-earth element) are a class of materials exhibiting a particularly rich phase diagram~\\cite{Medarde1997}. Except LaNiO$_3$, these materials undergo a metal-insulator phase transition (MIT) coinciding with a symmetry-lowering distortion of the crystal structure~\\cite{Alonso1999, Alonso2000}. \nThis distortion changes the space group from $Pbnm$, in which all Ni sites are crystallographically equivalent, to $P2_1\/n$ -- which possesses two inequivalent Ni sites -- because of the activation of a breathing mode, that contracts and expands the NiO$_6$ octahedra in a rock-salt pattern. The MIT temperature, $T_\\mathrm{MIT}$, is a monotonically decreasing function of the size of the rare-earth cation ($\\text{Lu}\\rightarrow\\text{La}$), that coincides with the N\u00e8el temperature $T_{\\text{N\u00e8el}}$ for $R$ = Nd, Pr, and vanishes for $R$ = La. In the last years rare-earth nickelates have attracted a great deal of attention, especially for their application within epitaxial heterostructures~\\cite{Middey2016}, which have been shown to possess crossovers between different magnetic orders~\\cite{Hepting2018}, the possibility to manipulate $T_\\mathrm{MIT}$ through electric fields~\\cite{Scherwitzl2014}, and exotic types of phases transitions~\\cite{Post2018,Liu2013}. \nRecently, superconductivity has been induced via apical oxygen deintercalation together with Sr doping in $R$NiO$_3$ with $R$ = Nd, Pr~\\cite{Li2019,Osada2020}. Finally, recent experiments have suggested a low-temperature multiferroic phase signaled by additional Raman-active modes compatible with the loss of the inversion center in the antiferromagnetic (AFM) phase~\\cite{Ardizzone2021}.\n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=17cm]{.\/figures\/magnet.pdf}\n\\caption{(\\textit{a}) Unit cell of $R$NiO$_3$ in the $P2_1\/c$ space group doubled along the direction of the lattice vector $\\mathbf{a}$ in order to accommodate the different AFM orderings studied here. Blue (orange) octahedra contain the short (long)-bond nickels, Ni$_\\mathrm{S}$ (Ni$_\\mathrm{L}$), respectively. The grey and the green atoms are the rare-earth elements ($R$ = Pr or Y) and the O atoms, respectively. The other panels show the three  AFM orderings investigated: (\\textit{b})~$T$-type, (\\textit{c})~$B$-type, and (\\textit{d})~$S$-type; here, for clarity, only the Ni atoms are shown. The arrows represent the directions of the magnetic moments on the Ni atoms and their lengths are proportional to their modulus. In the $S\/T$-type orderings all of the Ni atoms have a magnetic moment, while in the $B$-type ordering the nickels with shorter Ni$-$O bond lengths are nonmagnetic. The green and red planes are guides for the eyes to visualize planes of spin-up and spin-down moments, respectively. Rendered using \\textsc{VESTA}~\\cite{Momma2008}.}\n\\label{magn_exp}\n\\end{figure*} \n\nBesides the MIT, the nature of the low-temperature insulating phase has been the subject of intense study. An orbital ordering induced by a Jahn-Teller distortion was ruled out in favour of charge disproportionation~\\cite{Mazin2007}; this picture had also reached experimental consensus~\\cite{Alonso1999, Medarde2008}. However, such charge disproportionation has not manifested in first-principles calculations~\\cite{Park2012}, which show a negligible difference in the $3d$ electronic occupations of the two crystallographically inequivalent Ni$_\\mathrm{S}$ and Ni$_\\mathrm{L}$ (defined below), both of which have approximately 2 electrons in the $e_g$ states, and therefore 8 electrons in the $3d$ manifold -- an effect clearly explained within the negative-feedback charge regulation mechanism discussed in Ref. \\cite{Raebiger2008}. Hereafter we use the subscripts S\/L, to indicate the Ni ion in the smaller\/larger NiO$_6$ octahedrons, which are contracted\/expanded by the breathing-mode distortion. As rare-earth nickelates are considered to be negative charge-transfer insulators~\\cite{Zaanen1985}, it has been proposed that, starting from a configuration with one ligand hole ($\\underline{L}$) on every nickel, the transition $(3d^8\\underline{L})\\,(3d^8\\underline{L})\\rightarrow (3d^8\\underline{L}^2)_\\mathrm{S}\\,(3d^8)_\\mathrm{L}$ takes place at the MIT~\\cite{Johnston2014}. According to this picture, one NiO$_6$ octahedron contracts and the two $e_g$ electrons on Ni$_\\mathrm{S}$ couple with the two oxygen holes, forming a low-spin state. The other NiO$_6$ octahedron expands and ends up in a high-spin state owing to the Hund's coupling. Similar conclusions have been drawn from an analysis based on density-functional theory (DFT) + dynamical mean field theory (DMFT)~\\cite{Park2012, Haule2017} (that named the state a ``site-selective\" Mott state), DFT+$U$ studies~\\cite{Varignon2017, Hampel2017}, multi-band many-body Hamiltonians solved with Hartree-Fock~\\cite{Mizokawa2000, Johnston2014} and exact diagonalization methods~\\cite{Green2016}. \n\nDespite such intense theoretical effort, several open questions still remain. In particular, the precise magnetic order remains elusive. Indeed, there is wide experimental agreement only on the magnetic propagation vector $\\mathbf{Q}$, which encodes the periodicity of the magnetic order with respect to the primitive cell of the space group. However, the precise values of the magnetic moments on the Ni atoms have not yet been unambiguously determined~\\cite{Garcia-Munoz1994, Alonso1999}. From first-principles calculations it has been reported that only small values of Hubbard $U\\lesssim 2$ eV successfully stabilize an AFM order that is both compatible with the experimental \\textbf{Q} and energetically favourable, with respect to the ferromagnetic (FM) order~\\cite{Varignon2017, Hampel2017, Mercy2017}. Still, the values for the Hubbard parameter were tuned by scanning different values, and were not calculated rigorously with a well-determined theoretical protocol, thus preventing predictions to be entirely nonempirical. More importantly, these calculated magnetic orders have systematically zero magnetic moments on Ni$_\\mathrm{S}$~\\cite{Badrtdinov2021}. As it is shown here, this feature is not compatible with a possible multiferroic nature for $R$NiO$_3$, which was first computationally predicted~\\cite{Giovannetti2009} and more recently experimentally highlighted by Raman spectroscopy measurements, which revealed the emergence of additional phonon modes in the low-temperature phase, compliant with an inversion symmetry breaking required for a type II multiferroic phase~\\cite{Ardizzone2021}.  In order to computationally demonstrate such multiferroic phase, it is necessary to establish a coupling between a nonvanishing electric polarization \\textbf{P} and the magnetic degrees of freedom, i.e. to prove a dependence of \\textbf{P} on the simulated magnetic order. A group theory analysis  \\cite{PerezMato2016} has shown that a nonzero \\textbf{P} can be obtained if the magnetic moments on the Ni$_\\mathrm{S}$ atoms are different from zero, for instance within the $S$- and\/or $T$-types magnetic orderings~\\cite{Giovannetti2009} (see Fig.~\\ref{magn_exp}). Unfortunately, for $U\\simeq 2$ eV these latter magnetic patterns are stable only in a range of amplitudes for the breathing mode distortion that are much smaller than the experimental ones~\\cite{Badrtdinov2021}. Larger values of $U$ may lead to converged $S\/T$-types orderings, but they could also cause a FM instability~\\cite{Varignon2017, Hampel2017}. \n\nIn this paper we use a fully first-principles DFT+$U$+$V$ approach that includes both on-site ($U$) interactions to capture the electronic localization and inter-site ($V$) interactions to account for hybridization effects~\\cite{Campo2010}. We calculate the Hubbard parameters self-consistently using density-functional perturbation theory (DFPT)~\\cite{Timrov2018, Timrov2021}, thus resulting in a parameter-free theory. Our calculated Hubbard $U$ parameters range from 8 to 9~eV and are numerically different for the two crystallographically inequivalent $\\mathrm{Ni}_\\mathrm{S}\/\\mathrm{Ni}_\\mathrm{L}$. Using these high values of $U$, after an iterative approach aimed to determine the self-consistent crystal structure and Hubbard parameters, we find that within DFT+$U$~\\cite{Anisimov1991, Dudarev1998} -- having only on-site electronic contributions -- the investigated $R$NiO$_3$ structures would be found in a FM ground state, in agreement with other theoretical studies \\cite{Varignon2017,Hampel2017} and in contrast with experimental observations. Moreover, the DFT+$U$ calculated AFM orders of the $S\/T$-types are also found to have a vanishing breathing mode distortion and zero band gap, in disagreement with experiments. Conversely, by including the inter-site Hubbard $V$ interactions between the Ni and O sites, the results change considerably: in particular, we find qualitatively different final states for the $S\/T$ orderings. \nIn addition to the fact that some of them exhibit the expected AFM instability against FM ordering, with DFT+$U$+$V$ we always recover an insulating character, consistently with experiments. This feature allows us to evaluate the electric polarization \\textbf{P}, which is found to be magnetization-dependent, thus providing additional support for the emergence of multiferroicity \\cite{Giovannetti2009,Ardizzone2021}. We also simulate the magnetic order with zero magnetic moments on Ni$_\\mathrm{S}$ (i.e. the $B$-type ordering, see Fig.~\\ref{magn_exp}) and show that this is not able to generate a polar instability, thus keeping the materials in a non-polar phase with a vanishing \\textbf{P}. Our results for PrNiO$_3$ and YNiO$_3$ -- sitting on opposite ends of the $R$NiO$_3$ phase diagram -- are qualitatively similar, suggesting that these numerical findings could also be extended to other members of the rare-earth nickelate series.\n\n\\begin{table*}[]\n\\centering\n\\caption{\\label{table1} Structural and electronic properties of PrNiO$_3$ and YNiO$_3$ in the low-temperature phase for different magnetic orderings computed with different DFT+Hubbard schemes and as measured in experiments. Positive $\\Delta E$ means that FM is lower in energy than AFM, and vice versa. Three types of the AFM order are considered: $T$, $S$, and $B$ types (see Fig.~\\ref{magn_exp}). The $V$ interaction parameters are different within a given Ni$-$O octahedron because of the different Ni$-$O bond lengths yielding three inequivalent sites (O$_1$, O$_2$ and O$_3$). However, the differences between the different values of $V$ within a given octahedron are very small and for conciseness we report their average values; The full results are available in the Materials Cloud Archive~\\cite{MaterialsCloudArchive2022}. The metallic (M) or insulating (I) character for each case is also indicated.}\n\\bgroup\n\\def1.0{1.2}\n\\setlength\\tabcolsep{0.1in}\n\\begin{tabular}{lccccccc}\n\\toprule\n\\toprule\nMethod&AFM&$U_\\mathrm{S}\/U_\\mathrm{L}$ (eV)&$\\langle V_\\mathrm{S}\/V_\\mathrm{L}\\rangle_\\mathrm{avg}$ (eV)&$m_\\mathrm{S}\/m_\\mathrm{L}$ ($\\mu_\\mathrm{B}$)&$\\langle\\ell_{\\mathrm{S}}\/\\ell_{\\mathrm{L}}\\rangle_\\mathrm{avg}$ (\\AA)&$\\Delta E$ (meV\/f.u.)& Met\/Ins\\\\\n\\midrule\n\\multicolumn{8}{c}{PrNiO$_3$}\\\\\n\\midrule\nDFT+$U$&$S$&9.40\/9.40&&1.54\/1.53&2.00\/2.00&156&M\\\\\n&$T$&9.32\/9.33&&1.52\/1.53&1.99\/2.00&137&M\\\\\n&$B$&9.19\/8.19&&0.00\/1.68&1.88\/2.03&200&I\\\\\nDFT+$U$+$V$&\\strue&8.76\/9.61&0.76\/0.90&0.64\/1.65&1.89\/2.03&$-26$&I\\\\\n&$T_{(a)}$\\xspace&8.77\/9.60&0.76\/0.90&0.69\/1.68&1.90\/2.03&$-23$&I\\\\\n&$S_{(b)}$\\xspace&9.78\/9.33&1.02\/0.95&1.14\/1.61&1.94\/2.02&71&I\\\\\n&$T_{(b)}$\\xspace&9.78\/9.42&1.01\/0.96&1.15\/1.61&1.94\/2.02&75&I\\\\\n&$B$&9.60\/8.49&0.84\/0.71&0.00\/1.67&1.88\/2.03&90&I\\\\\nExpt. \\cite{Gawryluk2019}&&&&&1.92\/1.97& & I\\\\\n\\midrule\n\\multicolumn{8}{c}{YNiO$_3$}\\\\\n\\midrule\nDFT+$U$&$S$&9.38\/9.38&&1.51\/1.51&2.00\/2.00&115&M\\\\\n&$T$&9.21\/9.20&&1.49\/1.49&2.00\/2.00&72&M\\\\\n&$B$&9.17\/8.06&&0.00\/1.68&1.88\/2.04&143&I\\\\\nDFT+$U$+$V$&\\strue&8.38\/9.49&0.65\/0.80&0.15\/1.70&1.88\/2.05&$-94$&I\\\\\n&$T_{(a)}$\\xspace&8.40\/9.50&0.66\/0.81&0.18\/1.70&1.88\/2.05&$-91 $&I\\\\\n&$S_{(c)}$\\xspace&9.31\/9.25&0.82\/0.79&1.32\/1.32&1.99\/1.99&7&I\\\\\n&$T_{(c)}$\\xspace&9.35\/9.30&0.84\/0.82&1.33\/1.33&1.99\/1.99&6&I \\\\\n&$B$&9.54\/8.35&0.80\/0.68&0.00\/1.67&1.88\/2.04&61&I\\\\\nExpt. \\cite{Alonso2001}&&&&&1.92\/2.01& & I\\\\\n\\bottomrule\n\\bottomrule\n\\end{tabular}\n\\egroup\n\\end{table*}\n\n\\section{Results}\nThe AFM magnetic orderings studied in this paper are shown in Fig.~\\ref{magn_exp}. All of these require a supercell of the 20-atom unit cell of the monoclinic $P2_1\/n$ structure, which in turn is a $\\sqrt{2}\\times\\sqrt{2}\\times2$ supercell of the 5-atom cubic perovskite structure.\nPreviously these orderings have been investigated using a 80-atom 2$\\times$1$\\times$2 supercell of the $P2_1\/n$ space group, in order to reproduce the experimentally observed AFM propagation vector $\\mathbf{Q}_{P2_1\/n}=(\\nicefrac{1}{2},0,\\nicefrac{1}{2})$ \\cite{Hampel2017,Varignon2017}. \nTo reduce the computational cost\\LB{s}, we have used the $P2_1\/c$ setting of the space group 14, which has the same $\\mathbf{a}$ and $\\mathbf{b}$ primitive lattice vectors  as $P2_1\/n$, but $\\mathbf{c}_{P2_1\/c}=\\mathbf{a}+\\mathbf{c}_{P2_1\/n}$~\\cite{Ardizzone2021}. In this setting the magnetic propagation vector becomes $\\mathbf{Q}_{P2_1\/c}=(\\nicefrac{1}{2},0,0)$, only requiring a doubling of the 20-atoms $P2_1\/c$ unit cell (instead of quadrupling it, as for the 20-atoms $P2_1\/n$). \n\nIn Table \\ref{table1}, we report for each of the AFM configurations the results of the calculated magnetic moments $m$, total energy difference with respect to the FM configuration $\\Delta E$, the average bond lengths for the Ni$-$O octahedra $\\langle \\ell\\rangle_\\mathrm{avg}$, and the computed values of the Hubbard interaction parameters $U$ and $V$. \nAs shown in Table \\ref{table1}, DFT+$U$+$V$ yields two states for each of the two $S\/T$ types of orderings. Both of them are energetically degenerate in pairs (within numerical accuracy): the lowest-energy ones, that we denote as $(S\/T)_{(a)}$\\xspace, and the higher-energy (metastable) states named $(S\/T)_{(b)}$\\xspace and $(S\/T)_{(c)}$\\xspace. As described in the Methods section these states are found with an iterative self-consistent loop alternating calculations of Hubbard parameters and crystal structure optimizations until convergence of the two is reached simultaneously. In general we find stable points within this workflow, except for the $(S\/T)_{(a)}$\\xspace states. For these latter, at each iteration the protocol produces a periodic reversal of the breathing mode amplitude, to which equivalent switches of the Hubbard parameters and magnetic moments are associated. Since the numerical values of $U$, $V$, Ni--O octahedral volumes and magnetic moments associated to Ni$_\\mathrm{S}$\/Ni$_\\mathrm{L}$ are numerically always the same, together with the fact that the total energy does not exhibit any variation but a smooth convergence with the iterations, we argue that this does not create physical ambiguities, as all the main features of the final result are well-defined (for more information, see Supplementary Information~\\cite{SupplementaryInformation}). This fact, however, underlines the need of a formulation for Hubbard-corrected DFT functionals able to take into account the variation of the Hubbard parameters with respect to atomic positions and strains, inasmuch as this dependence can be quite significant \\cite{Kulik2011}.\n\n\\begin{figure*}[]\n\\centering\n\\includegraphics[width=17.5cm]{.\/figures\/pdos_egFULL.pdf}\n\\caption{pDOS of the relaxed crystal structure for each magnetic ordering. (\\textit{a}), (\\textit{b}):~\\textit{B}-type and $T$-type, respectively, for PrNiO$_3$. (\\textit{c}), (\\textit{d}):~$T_{(a)}$\\xspace-type and (\\textit{e}), (\\textit{f}):~$T_{(b)}$\\xspace-type and $T_{(c)}$\\xspace-type, for both PrNiO$_3$ and YNiO$_3$. The Hubbard functionals employed to produce (\\textit{a}), (\\textit{c})--(\\textit{f}) include both the $U$ and the $V$ interaction parameters, while in (\\textit{b}) only the on-site $U$ is present. In the plots the Ni--$e_g$ states and the full O--$2p$ projection are shown. We averaged over inequivalent O atoms and over the two Ni($e_g$) states for each spin channel. Given the symmetry between spin-up\/spin-down channels for AFM orderings, the notation for majority\/minority (maj\/min) spin channels is: for a Ni with positive magnetic moments ($m>0$ $\\mu_\\mathrm{B}$), $\\sigma_\\mathrm{maj}=\\,\\uparrow$ and $\\sigma_\\mathrm{min}=\\,\\downarrow$; vice versa for Ni with $m<0$ $\\mu_\\mathrm{B}$. The zero of energy corresponds to the valence band maximum.}\n\\label{pDOS}\n\\end{figure*} \n\n\nThe total energies reported in the last column of Table \\ref{table1} are calculated at the relaxed crystal structure with the corresponding self-consistent, site-dependent $U$ and $V$ Hubbard parameters, both for the AFM and the FM configurations. Since the final $U$ and $V$ are not numerically the same, it follows that the reference energies are different. This could in principle create ambiguities when comparing total energies calculated with functionals containing numerically different parameters. However, we regard the Hubbard term as a correction to the self-interaction error of the DFT base functional, selectively applied to the Ni--$3d$ (and O--$2p$) manifold of the corresponding ground state. According to this point of view it is natural that different electronic ground states will be characterized by different self-interaction errors (i.e. different curvatures with respect to the $3d$ electron occupancy, see Ref.~\\cite{Cococcioni2005}). As a consequence we follow Ref.~\\cite{Cococcioni2019} and argue that, for each of electronic state, the corresponding $U$ and $V$, aimed to remove such errors, must necessarily be different. \n\nWe highlight that we use orthogonalized atomic states taken from the pseudopotential as Hubbard projectors (see Methods), while the calculations with $U\\simeq 2$~eV by other works mentioned above in the text were carried out with orbital projectors derived from the projector augmented-wave (PAW) formalism~\\cite{Bengone2000}. In general same values for the Hubbard parameters can have a different impact on the electronic structure when applied to different Hubbard projectors~\\cite{Wang2016}. Therefore, to establish the consistency of  two computational setups in the same range of $U$ parameters, we have numerically verified for PrNiO$_3$ that with the empirical $U\\simeq 2$~eV we stabilize a $B$-type AFM order -- even starting from the $S\/T$-type orderings -- that is lower in energy than the FM configuration, in compliance with previous studies carried out with PAW projectors. \n\nIn Fig.~\\ref{pDOS} we show the projected density of states (pDOS) of the $T_{(a)}$\\xspace and $T_{(b)}$\\xspace types of orderings with and without the inter-site $V$, and the $B$-type ordering computed using DFT+$U$+$V$. Equivalent results for the $S$-type ordering are almost identical to the $T$-type ordering. Furthermore, the results for the $B$-type ordering obtained with and without inter-site $V$ present no qualitative differences and, thus for sake of conciseness we report only the DFT+$U$+$V$ results. \nIn particular we plot the Ni--$e_g$ and the O--$2p$ states. The Ni $t_{2g}$ bands, corresponding to localized, fully-occupied states, yield narrow peaks located below $-5$~eV with respect to the highest occupied states and for clarity are not reported in the same figure (they can be found in SI~\\cite{SupplementaryInformation}).\n\nIn Table \\ref{table2} we show the results of the calculated electric polarization \\textbf{P} using the Berry phase approach \\cite{King-Smith1993,Umari2002}. Because of the polar instability, during the structural optimization the crystal structure lowers its symmetry, and thus the space group changes. The newly obtained space groups are reported in the same table. For the $S\/T$ orderings, we mention that we obtained a nonzero \\textbf{P} also at the experimental crystal structure, in agreement with Ref. \\cite{Giovannetti2009}, as the inversion symmetry-breaking is induced at the level of the {magnetic} space group.\n\n\n\n\\section{Discussion}\n\\subsection{Electronic, crystal and magnetic structures}\n\\paragraph{DFT+$U$}\nThe $B$-type AFM ordering presents no relevant differences when examined with the two methods, i.e. with or without the inter-site parameter $V$: they show a band gap of $\\simeq 1.4$ eV, very similar magnetic moments of $|m|\\simeq1.7$ $\\mu_\\mathrm{B}$ on Ni$_\\mathrm{L}$ and it is always found to be higher in energy than the FM configuration. As shown in Table~\\ref{table1}, the breathing mode of the $S$- and $T$-type orderings  obtained with on-site interactions only (DFT+$U$) disappears, as the Ni$-$O bond lengths of the two structurally inequivalent  NiO$_6$ octahedra become equal. In addition, as shown in Fig.~\\ref{pDOS}, the electronic states associated with these orderings display a semimetallic behaviour, with the $3d$ states of the two Ni atoms becoming equivalent. This state also exhibits almost the same values of $U$ interaction parameters and magnetic moments for the two inequivalent Ni sites. For this reason we conclude that within this \\emph{ab initio} model, at the self-consistent crystal structure and Hubbard parameters, purely on-site interactions can not sustain the breathing-mode-based structural distortion. \n\n\\paragraph{DFT+$U$+$V$}\nUpon the inclusion of the inter-site Hubbard interactions between the nickels and the oxygens, the materials recover the experimentally observed insulating nature. The lowest-energy, semi-degenerate states $(S\/T)_{(a)}$\\xspace share similar features for PrNiO$_3$ and YNiO$_3$: they display significantly different magnetic moments and Ni$-$O bond lengths for the two inequivalent Ni sites. This difference in $m_\\mathrm{S}\/m_\\mathrm{L}$ and $\\langle\\ell_\\mathrm{S}\/\\ell_\\mathrm{L}\\rangle$ is larger for YNiO$_3$ than for PrNiO$_3$ mirroring the larger disproportionation observed for $R$NiO$_3$ with a smaller rare-earth ionic radius. Importantly for these states, the numerically larger (smaller) Hubbard parameters are applied to the larger (smaller) Ni--O octahedra. The opposite applies to the $B$ and $(S\/T)_{(b)}$\\xspace--types orderings. This difference turned out to be essential for stabilize $(S\/T)_{(a)}$\\xspace states as lower in energy than the FM state.  \nThe metastable $(S\/T)_{(b)}$\\xspace for PrNiO$_3$ states exhibit a substantially smaller $\\mathrm{Ni}_\\mathrm{S}\/\\mathrm{Ni}_\\mathrm{L}$ inequivalence because of a significant reduction of the breathing mode. They are different from the corresponding $(S\/T)_{(c)}$\\xspace states obtained for YNiO$_3$, as in these latter the breathing mode amplitude shrinks to zero (see Table~\\ref{table1}). In addition, both $(S\/T)_{(c)}$\\xspace display different electronic occupations in the $e_g$ bands (see Supplementary Information) and different polar distortions with respect to $(S\/T)_{(a)}$\\xspace states, exemplified as opposite signs of some components of \\textbf{P} between $S_{(c)}$\\xspace and \\strue (see Table \\ref{table2}). \n\nThe pDOS in Fig. \\ref{pDOS} shows that the materials are negative charge-transfer insulators, since -- according to  definition -- the $2p$ electronic states of the oxygens stand between the upper\/lower Hubbard bands of the $3d$ states of the two nickels. The negative character of charge-transfer is due to the presence of residual O$-{2p}$ states in the lowest conduction bands (the so-called hole-doped ligands)~\\cite{Bisogni2016}. In all the panels in Fig.~\\ref{pDOS}, the $e_g$ states of Ni$_\\mathrm{L}$ [$e_g(\\mathrm{Ni}_\\mathrm{L})$] present concentrated spectral weights 5~eV above (minority-spin channel, $\\sigma_\\mathrm{min}$) and 7~eV below (majority-spin channel, $\\sigma_\\mathrm{maj}$) the valence-band maximum (VBM). The situation is similar for $e_g(\\mathrm{Ni}_\\mathrm{S})$ in DFT+$U$, as the two nickel sites become structurally equivalent [see Fig.~\\ref{pDOS}~(b)]. A reduction in the magnetization on Ni$_\\mathrm{S}$ \nproduces a gradual downshift of the $e_g(\\mathrm{Ni}_\\mathrm{S})$ $\\sigma_\\mathrm{min}$ states, until they finally merge with the $e_g(\\mathrm{Ni}_\\mathrm{S})$ $\\sigma_\\mathrm{maj}$ in a unique group of electronic bands centered 2~eV above the VBM in the limit of vanishing magnetic moments ($B$-type ordering). The $T_{(a)}$\\xspace states of PrNiO$_3$ and YNiO$_3$ exhibit different conduction states: for PrNiO$_3$ two separate peaks in the pDOS can be distinguished at 1 and 3~eV above VBM, while in YNiO$_3$ they partially overlap at 2~eV above the same level. This effect is sharply reflected in the magnetization of Ni$_\\mathrm{S}$, that decreases substantially from YNiO$_3$ to PrNiO$_3$. We attribute this change in the electronic structure to the larger structural disproportion for the two Ni that is present in the YNiO$_3$ compared to PrNiO$_3$. The $T_{(b)}$\\xspace and $T_{(c)}$\\xspace states display a strongly reduced band gap. Interestingly in $T_{(c)}$\\xspace for YNiO$_3$, the system still remains insulating despite the vanishing breathing mode, in contrast with DFT+$U$ where instead the gap closes. \n\n\\subsection{Magnetism-induced electric polarization\\label{sec4}} \n\\begin{table}[]\n\\centering\n\\caption{\\label{table2}Electric polarization $\\mathbf{P}=\\mathbf{P}_\\mathrm{e}+\\mathbf{P}_\\mathrm{I}$ (electronic + ionic) in cartesian axes (the $x$ direction is aligned along $\\hat{\\textbf{a}}$) in units of $\\mu $C\/cm$^2$ within the different AFM phases calculated using the DFT+$U$+$V$ method for PrNiO$_3$ and YNiO$_3$, together with the space group of the relaxed structures. The polarization quanta are $(38.3,19.4,27.2)$ $\\mu$C\/cm$^2$ for PrNiO$_3$ and $(38.8, 21.0,27.6)$ $\\mu$C\/cm$^2$ for YNiO$_3$. Nonzero components are highlighted in bold.} \n\\bgroup\n\\def1.0{1}\n\\setlength\\tabcolsep{0.09in}\n\\begin{tabular}{lcccc}\n\\toprule\n\\toprule\nState&$P_{\\mathrm{e},x}$&$P_{\\mathrm{e},y}$&$P_{\\mathrm{e},z}$&Space group (\\#)\\\\\n&$P_{\\mathrm{I},x}$&$P_{\\mathrm{I},y}$&$P_{\\mathrm{I},z}$&\\\\\n\\midrule\n\\multicolumn{5}{c}{PrNiO$_3$}\\\\\n\\midrule\n$B$&$0.00$&$0.00$&$0.00$&$P2_1\/c\\,(14)$\\\\\n&$0.00$&$0.00$&$0.00$&\\\\\n\\strue&0.00&\\textbf{0.21}&$0.00$&$P2_1\\,(4)$\\\\\n&0.00&\\textbf{0.90}&$0.00$&\\\\\n$T_{(a)}$\\xspace&\\textbf{4.25}&0.00&$\\textbf{2.74}$&$Pc\\,(7)$\\\\\n&$-\\textbf{0.86}$&0.00&$-\\textbf{2.68}$&\\\\\n$S_{(b)}$\\xspace&0.00 & \\textbf{1.72}&0.00&$P2_1\\,(4)$\\\\\n&0.00 & \\textbf{1.58}&0.00&\\\\\n$T_{(b)}$\\xspace&\\textbf{11.7}& 0.00&  \\textbf{6.14}&$Pc\\,(7)$\\\\\n&$-\\textbf{1.15}$& 0.00& $-\\textbf{4.40}$&\\\\\n\\midrule\n\\multicolumn{5}{c}{YNiO$_3$}\\\\\n\\midrule\n$B$&$0.00$&$0.00$&$0.00$&$P2_1\/c\\,(14)$\\\\\n&$0.00$&$0.00$&$0.00$&\\\\\n\\strue&0.00&  \\textbf{0.07}& 0.00&$P2_1\\,(4)$\\\\\n&0.00&  \\textbf{0.77}& 0.00&\\\\\n$T_{(a)}$\\xspace&$\\textbf{1.27}$&  0.00&  \\textbf{0.68}&$Pc\\,(7)$\\\\\n&$\\textbf{0.13}$&  0.00&  \\textbf{0.99}&\\\\\n$S_{(c)}$\\xspace&0.00& $-\\textbf{5.57}$& 0.00&$P2_1\\,(4)$\\\\\n&0.00& $\\textbf{3.18}$& 0.00&\\\\\n$T_{(c)}$\\xspace&$-\\textbf{38.0}$& 0.00&  \\textbf{1.02}&$Pc\\,(7)$\\\\\n&$-\\textbf{0.17}$& 0.00&  \\textbf{1.70}&\\\\\n\\bottomrule\n\\bottomrule\n\\end{tabular}\n\\egroup\n\\end{table}\nThe $P2_1\/c$ crystal structure of rare-earth nickelates is centrosymmetric. However, a group theory analysis shows that the magnetic space group of the $S\/T$-type magnetic orderings breaks inversion symmetry, thus allowing symmetry-lowering \\emph{polar} crystal distortions to occur, resulting in space groups $Pc$ for $T$-type and $P2_1$ for the $S$-type \\cite{Giovannetti2009,Ardizzone2021,PerezMato2016}. As described in detail in Ref.~\\cite{Ardizzone2021}, the loss of centrosymmetry in the $T$-type ordering is induced by breaking the symmetry of the two-fold screw-axis parallel to $\\mathbf{b}$, while for the $S$-type ordering it is due to the loss of a glide plane consisting of a reflection in the plane perpendicular to $\\mathbf{b}$ plus a translation along $\\mathbf{c}$ of $|\\mathbf{c}|\/2$.\nIt has been predicted that the $T$-type ordering would induce an electric polarization $\\mathbf{P}$ with nonzero component along $\\mathbf{a}$ and $\\mathbf{c}$, while the $S$-type ordering an electric polarization predominantly along $\\mathbf{b}$~\\cite{Giovannetti2009}. The proposed microscopic mechanism consists in a partial charge disproportionation of the oxygens O$_i$ ($i=1,2,3$) forming the octahedra (thus creating two inequivalent O$_i$ and $\\mathrm{O}_i'$), which produces a shift from the Ni-site-centered charge disproportionation to a Ni-O-bond-centered charge disproportionation, and therefore allowing a nonzero \\textbf{P} to develop. This charge disproportionation of the oxygens is caused by the presence of neighboring nickels with different magnetic moments: either with parallel $\\mathrm{Ni}_\\sigma$--$\\mathrm{O}_i$--$\\mathrm{Ni}_\\sigma$ or antiparallel $\\mathrm{Ni}_\\sigma$--$\\mathrm{O}_i'$--$\\mathrm{Ni}_{-\\sigma}$ spins $\\sigma=\\,\\uparrow,\\downarrow$. Applying the same argument to the $B$-type ordering, the charge disproportionation of the oxygens can not occur because every O is surrounded by a magnetic and a nonmagnetic Ni atom and this results in a zero electric polarization. \n\nWe analyze the space groups of the relaxed crystal structures and calculate from first-principles the electric polarization of the magnetic orders investigated using DFT+$U$+$V$ method (the $+U$ functional, having vanishing band gaps for $S$ and $T$ types, would yield an ill-defined polarization). As shown in Table \\ref{table2}, the relaxed crystal structure of the $B$-type ordering presents a centrosymmetric $P2_1\/c$ space group and its electronic ground state does not exhibit any electric polarization, in agreement with our previous considerations. On the contrary, the $S\/T$-type AFM orderings display a nonvanishing \\textbf{P}, whose magnitude increases with $m(\\mathrm{Ni}_\\mathrm{S})$. This trend is correlated to the proximity of a (semi)metallic state in the limit of $m(\\mathrm{Ni}_\\mathrm{S})\\rightarrow m(\\mathrm{Ni}_\\mathrm{L})$. Indeed, an increase of the magnetic moment $m$ is caused by an increase of the difference of the $e_g(\\mathrm{Ni})$ occupations for the two spin channels, and thus an enhancement of the energy separation of the associated $e_g(\\sigma_\\mathrm{min})$ and $e_g(\\sigma_\\mathrm{maj})$ peaks in the pDOS. Therefore, upon enlarging $m_\\mathrm{S}$, the $e_g(\\mathrm{Ni}_\\mathrm{S})\\,\\sigma_\\mathrm{min}$ states downshift toward the VBM reducing the band gap.\n\n\\subsection{Summary}\nWe studied the structural, electronic and magnetic properties of two members of the rare-earth nickelates series: PrNiO$_3$ and YNiO$_3$. We used DFT functionals augmented with on-site $U$ and inter-site $V$ Hubbard corrections, and we determined the crystal and electronic structures via an iterative self-consistent procedure alternating structural optimizations and evaluations of the Hubbard parameters, until convergence between the two is reached. Calculations were performed without any adjustable parameters, as both Hubbard $U$ and $V$ were calculated from first-principles using density-functional perturbation theory; and we showed that the results change qualitatively upon the inclusion of inter-site Hubbard interactions. We analysed three different antiferromagnetic orderings: \\textit{B}, \\textit{S} and \\textit{T}-types. When employing purely on-site electronic interactions (DFT+$U$), the investigated antiferromagnetic patterns are found to be metastable with respect a ferromagnetic configuration, and some of these (\\textit{S}-type and \\textit{T}-type orders) yield a semimetallic electronic structure and lose the crystallographic inequivalence (bond disproportionation) of the two Ni sites. Within DFT+$U$+$V$, the results for \\textit{B}-type order are very similar to the ones obtained with DFT+\\textit{U}. On the other hand, for the $S\/T$ antiferromagnetic orderings two different self-consistent states for each magnetic pattern are reached: two metastable states with respect the ferromagnetic solution [$(S\/T)_{(b)}$\\xspace and $(S\/T)_{(c)}$\\xspace for PrNiO$_3$ and YNiO$_3$, respectively] with very similar magnetic moments on the inequivalent Ni sites, reduced breathing mode distortion and small band gaps; and two lowest-energy states with respect to all the magnetic patterns analysed [$(S\/T)_{(a)}$\\xspace] with substantially different Ni--O bond lengths and magnetic moments for the crystallographic inequivalent nickel atoms, and larger band gaps. In the iterative loop over the calculation of Hubbard parameters and structural optimization, these latter states oscillate between two identical crystal structures with inverted breathing mode amplitudes; we believe that this behaviour might be due to the neglect of the derivative of the Hubbard parameters with respect to the atomic positions and strains \\cite{Kulik2011}, and in general for the lack of a functional formulation of the method involving the stationarity with respect to the Hubbard parameters themselves. Finally, without resorting to any empirical parameter, within the DFT+\\textit{U}+\\textit{V} approach we naturally predict the occurrence of a magnetically-induced electric polarization, thus supporting recent experimental observations pointing to the emergence of a multiferroic phase in this class of materials.\n\n\n\\section{methods}\nIn our first-principles calculations we employ both the rotationally-invariant DFT+$U$ formulation of Dudarev \\emph{et al.}~\\cite{Dudarev1998}, and the extended DFT+$U$+$V$ one~\\cite{Campo2010}. The parameters $U$ and $V$ are calculated from first principles  using DFPT~\\cite{Baroni2001,Timrov2018,Timrov2021} as implemented in the \\textsc{HP} code~\\cite{Timrov2022}. The localized atomic-like set of functions defining the occupation matrices (defined in Ref.~\\cite{Cococcioni2005}) employed in our calculations consists of atomic orbitals as found in the pseudopotentials orthogonalized through the L\\\"{o}wdin algorithm~\\cite{Lowdin1950}. For the calculation of the Hubbard contribution to forces and stresses we employ a recently-developed method that uses the solution of the Lyapunov equation to evaluate the derivative of the inverse square root of the overlap matrix~\\cite{Timrov2020}. \n\nAll the first-principles calculations are carried out using the Quantum ESPRESSO distribution \\cite{Giannozzi2009, Giannozzi2017,Giannozzi2020} . We use a spin-polarized generalized-gradient approximation (GGA) with the PBEsol prescription for the exchange-correlation functional~\\cite{Perdew2008}, as well as a PAW pseudopotential (PP) ~\\cite{Blochl1994_PAW} for O from the Pslibrary v0.3.1~\\cite{DalCorso2014} and ultrasoft (US) PPs ~\\cite{Vanderbilt1990} from the GBRV (v1.2) and GBRV (v1.4) libraries~\\cite{Garrity2014} for Y and Ni, respectively, as recommended from the SSSP (efficiency) library ~\\cite{Prandini2018}. For the Pr we used the US PP  from the Pslibrary1.0.0 (\\texttt{Pr.pbesol-spdn-rrkjus\\_psl.1.0.0.UPF}) having the $4f$ states frozen in the core. In all our calculations we use 50~Ry as the plane-wave kinetic-energy cutoff for the wavefunctions, 400~Ry kinetic-energy cutoff for the charge density, Gaussian smearing of 0.005~Ry to converge narrow-gap magnetic states and a uniform $\\Gamma$-centered 5$\\times$8$\\times$6 grid of $\\mathbf{k}$ points to sample the Brillouin zone. Calculations of the electric polarization are performed using the Berry-phase approach~\\cite{King-Smith1993,Umari2002} and using a finer 7$\\times$12$\\times$8 \\textbf{k} mesh. The calculations of the Hubbard parameters are performed using 1$\\times$2$\\times$2 \\textbf{q} points, which corresponds to a supercell of an equivalent size \\cite{Timrov2018}. We adopt the iterative procedure detailed in \\cite{Cococcioni2019,Timrov2021} to incorporate in the parameters $U$ and $V$ the dependence of both the electronic structure and the crystal environment: this consists in a series of alternating self-consistent structural optimizations and linear-response evaluations of the Hubbard parameters until both the crystal structure and the $U$ and $V$ parameters are converged self-consistently.  \n\n\\section{data availability statement}\nThe data used to produce the results of this paper are available on the Materials Cloud Archive~\\cite{MaterialsCloudArchive2022}.\n\n\\section{acknowledgements}\nWe gratefully acknowledge Marisa Medarde for fruitful discussions. This research was supported by the NCCR MARVEL, a National Centre of Competence in Research, funded by the Swiss National Science Foundation (grant number 182892). Computer time was provided by CSCS (Piz Daint) through project No.~s1073.\n\n\\section{competing interests}\nThe authors declare no competing interests.\n\n\\section{author contributions}\nL.B. performed and analyzed the first-principles calculations,  M.K., I.T. and N.M. helped to interpret and understand the data, while N.M. conceived project. L.B. wrote the initial manuscript and all the authors participated to the final version of it.  \n\n\n\n\\section{Spin-resolved projected density of states for Nickel $\\bm{t_{2g}}$ states}\n\\begin{figure*}[h]\n\\centering\n\\includegraphics[width=16cm]{.\/figures\/pdos_t2gFULL.pdf}\n\\caption{pDOS of the relaxed crystal structure for each magnetic ordering. (\\textit{a}), (\\textit{b}):~\\textit{B}-type and $T$-type, respectively, for PrNiO$_3$. (\\textit{c}), (\\textit{d}):~$T_{(a)}$\\xspace-type and (\\textit{e}), (\\textit{f}):~$T_{(b)}$\\xspace-type and $T_{(c)}$\\xspace-type, for both PrNiO$_3$ and YNiO$_3$. The Hubbard functionals employed to produce (\\textit{a}), (\\textit{c})--(\\textit{f}) include both the $U$ and the $V$ interaction parameters, while in (\\textit{b}) only the on-site $U$ is present. In the plots the valence Ni--$t_{2g}$ states and the full O--$2p$ projection are shown. We averaged over inequivalent O atoms and over the two Ni($t_{2g}$) states for each spin channel. Given the symmetry between spin-up\/spin-down channels for AFM orderings, the notation for majority\/minority (maj\/min) spin channels is: for a Ni with positive magnetic moments ($m>0$ $\\mu_\\mathrm{B}$), $\\sigma_\\mathrm{maj}=\\,\\uparrow$ and $\\sigma_\\mathrm{min}=\\,\\downarrow$; vice versa for Ni with $m<0$ $\\mu_\\mathrm{B}$. The zero of energy corresponds to the valence band maximum.}\n\\label{pDOS}\n\\end{figure*} \n\nIn this section we discuss the pDOS displaying the Ni--$t_{2g}$ states. As shown in Fig.~\\ref{pDOS}, the $t_{2g}$ states are located in the range between $-5$ to $-7$~eV with respect to the valence band maximum. These states have a weak hybridization with O-$2p$ states, and since they lie relatively deeply in energy, they are not responsible for the formation of the band gap. This is a consequence of the fact that they are fully occupied, as it can be verified by looking at the eigenvalues $\\lambda_{t_{2g},\\sigma}$ of the occupation matrices in Table~\\ref{table::os}. \n\n\n\n\\section{Convergence of Hubbard parameters and total energy differences}\n\\begin{figure*}[h]\n\\centering\n\\includegraphics[width=17cm]{.\/figures\/swapHubbard.pdf}\n\\caption{Convergence of the Hubbard $U$ parameter (within DFT+$U$+$V$) for the two types of Ni atoms (Ni$_1$ and Ni$_2$) in PrNiO$_3$ for the three types of the AFM magnetic ordering: (\\textit{a})~$T_{(b)}$\\xspace, (\\textit{b})~$T_{(a)}$\\xspace, and (\\textit{c})~$B$. Each iteration of the plot corresponds to a DFPT calculation of $U$ on top of a ground state obtained after a variable-cell structural optimization according to a self-consistent protocol of Ref.~\\cite{Timrov2021}. (\\textit{d})~Total energy difference $\\Delta E = E_\\mathrm{AFM} - E_\\mathrm{FM}$, where $E_\\mathrm{AFM}$ and $E_\\mathrm{FM}$ are the total energies of the AFM and FM ground state. On top of panels (\\textit{a}) and (\\textit{b}), a schematic illustration of the NiO$_6$ volume changes with iterations are shown (blue and yellow octahedra correspond to larger and smaller volumes, respectively).}\n\\label{fig:conv}\n\\end{figure*} \nIn this section we show the convergence of the Hubbard parameters and total energy differences for the different types of the magnetic ordering discussed in the main text for the PrNiO$_3$ material. The convergence for YNiO$_3$ follows similar trends. The Hubbard parameters are computed using the self-consistent protocol that is described in detail in Refs.~\\cite{Cococcioni2019,Timrov2021}. This workflow consists in the alternate evaluation of Hubbard parameters and structural optimization until convergence is achieved within a certain accuracy. \n\nFig.~\\ref{fig:conv}~(\\textit{a})--(\\textit{c}) show the convergence of Hubbard $U$ during the iterations of the self-consistent cycle for PrNiO$_3$. The $T_{(b)}$\\xspace-type (+$U$+$V$) and the $T$-type (+$U$) orderings display converged values of $U$ and Ni--O octahedral volumes. On the contrary, the $T_{(a)}$\\xspace-type exhibit oscillations of the Hubbard $U$ values, that is associated to a similar behaviour of the octahedral volumes of the crystallographically inequivalent Ni$_1$\/Ni$_2$, as schematically shown in the cartoon above panel (\\textit{b}) in the same figure. Similar trends occur also for the magnetic moments in the Ni atoms and on the Hubbard $V$ parameters. A possible cause for this swapping could be due to the neglect of the derivative of the Hubbard parameters with respect to the atomic displacements \\cite{Kulik2011}. Indeed, focusing on the leading on-site term, such contribution to the force would be $\\sum_{I}\\frac{dU^I}{d\\mathbf{R}}\\mathrm{tr}\\,(\\mathbf{n}^I(\\mathbf{1}-\\mathbf{n}^I))$. Since the trace gives a positive contribution while $U^I$ increases (decreases) upon contraction (expansion) of the Ni-O octahedra, the net result is that this term opposes to the mechanism of swapping during the structural optimization. A quantification of this component to the Hubbard force, although possibly significant, is however beyond the scope of this study. We remark that, although the breathing mode and the Hubbard parameters swap, the total energy of the $T_{(a)}$\\xspace state smoothly converges and -- after some initial iterations -- it stabilizes as the lowest energy magnetic state, as displayed in panel (\\textit{d}) of Figure~\\ref{fig:conv}. In addition, together with the Hubbard parameters, also the magnetic moments and the Ni--O volumes octahedra associated to the Ni$^\\mathrm{S}_i$ and Ni$^\\mathrm{L}_i$ ($i=1,2$) are numerically always the same. We therefore conclude that although the apparently oscillating behaviour, the final result is unambiguous.     \n\n\n\n\\section{Band gaps}\nIn Table~\\ref{tab:gaps} we show the band gap values for PrNiO$_3$ and YNiO$_3$ computed using DFT+$U$ and DFT+$U$+$V$ at the self-consistent electronic and crystal structures. Experimentally, both materials are known to be insulating in the investigated phase \\cite{Medarde1997}. The insulating character is obtained for all types of the magnetic ordering when the DFT+$U$+$V$ functional is used, while only the AFM-$B$ type is insulating within DFT+$U$. \n\n\\begin{table*}[h]\n\\centering\n\\caption{Values of the band gaps for different magnetic orderings calculated with the two different methods DFT+$U$ and DFT+$U$+$V$ for $R$NiO$_3$ ($R=$ Pr and Y).}\n\\bgroup\n\\def1.0{1}\n\\setlength\\tabcolsep{0.18in}\n\\begin{tabular}{lccccccccc}\n\\toprule\n\\toprule\n&\\multicolumn{3}{c}{DFT+$U$}&&\\multicolumn{5}{c}{DFT+$U$+$V$}\\\\\n\\midrule\nPrNiO$_3$&$S$&$T$&$B$&&\\strue&$T_{(a)}$\\xspace&$S_{(b)}$\\xspace&$T_{(b)}$\\xspace&$B$\\\\\n$E_\\mathrm{gap}$ (eV)&0.00&0.00&1.37&&0.92&0.89&0.36&0.35&1.64\\\\\n\\midrule\nYNiO$_3$&$S$&$T$&$B$&&\\strue&$T_{(a)}$\\xspace&$S_{(c)}$\\xspace&$T_{(c)}$\\xspace&$B$\\\\\n$E_\\mathrm{gap}$ (eV)&0.00&0.00&1.41&&1.42&1.33&0.27&0.34&1.65\\\\\n\\bottomrule\n\\bottomrule\n\\end{tabular}\n\\egroup\n\\label{tab:gaps}\n\\end{table*}\n\n\\section{L\\\"owdin occupation numbers and oxidation states}\n\nIn table~\\ref{table::os} we show the eigenvalues $\\lambda^I_{m,\\sigma}$ of the occupation matrices $n^{I\\sigma}_{mm'}=\\sum_i\\,f_i\\times\\langle\\psi_{i\\sigma}|\\phi_{m'}^I\\rangle\\langle\\phi_m^I|\\psi_{i\\sigma}\\rangle$ \\cite{Cococcioni2005}. We also provide the values of the oxidation states following the prescription of Ref.~\\cite{Sit2011}. This is based on considerations about occupations of bonding\/antibonding states when the atomic orbitals of a transition-metal (TM) and a ligand mix; according to~\\cite{Sit2011}, only full $d$-occupancies ($\\lambda_{m,\\sigma}^I\\gtrsim0.9$) of the TM orbitals provide evidence that an electron is effectively bound to the ion, whereas considering only the total occupation $n^I=\\sum_{m,\\sigma}n^{I\\sigma}_{mm}$ could lead to misleading conclusions. \n\n\\begin{table*}[h]\n\\centering\n\\label{table::os}\n\\caption{Eigenvalues ($\\lambda_{i,\\sigma}$, $i=1,\\dots,5$) of the occupation matrices for the majority\/minority spin channels $(\\sigma_\\mathrm{maj}\/\\sigma_\\mathrm{min})$ of the Ni--$3d$ manifold, both for the long and short bonds (L\/S) Ni atoms, for different magnetic orderings. All the results refers to DFT+$U$+$V$ calculations, except for $T^{(+U)}$ that is obtained with DFT+$U$. In the table, $n=\\sum_{i,\\sigma}\\lambda_{i,\\sigma}$ and OS refers  to the oxidation state.} \n\\bgroup\n\\def1.0{1.0}\n\\setlength\\tabcolsep{0.07in}\n\\begin{tabular}{lcccccccccccccccc}\n\\toprule\n\\toprule\n&&\\multicolumn{5}{c}{$\\lambda_{i,{\\sigma_\\mathrm{maj}}}$}&&\\multicolumn{5}{c}{$\\lambda_{i,{\\sigma_\\mathrm{min}}}$}\\\\\nState&Ni&&$e_g$ &$e_g$ &$t_{2g}$ &$t_{2g}$ &$t_{2g}$ &&$e_g$ &$e_g$ &$t_{2g}$ &$t_{2g}$ &$t_{2g}$&&$n$&OS \\\\\n\\midrule\n\\multicolumn{17}{c}{PrNiO$_3$}\\\\\n\\midrule\n$B$&S&&0.54&  0.55&  0.99&  1.00&  1.00&&0.54& 0.55&  0.99&  1.00&  1.00&&8.16&Ni(IV)\\\\\n&L&&0.98&  0.98&  0.99&  1.00&  1.00&&0.13&  0.14&  0.99&  0.99&  0.99&&8.18&Ni(II)\\\\\n$T^{(+U)}$&S&&0.91&  0.95&  0.99&  1.00&  1.00&&0.14&  0.16&  0.99&  0.99&  0.99&&8.12&Ni(II)\\\\\n&L&&0.92&  0.95&  0.99&  1.00&  1.00&&0.14&  0.16&  0.99&  0.99&  0.99&&8.13&Ni(II)\\\\\n$T_{(b)}$\\xspace&S&&0.83&  0.86&  1.00&  1.00&  1.00&&0.24&  0.25&  0.99&  0.99&  0.99&&8.13&Ni(IV)\\\\\n&L&&0.96&  0.97&  0.99&  1.00&  1.00&&0.14&  0.15&  0.99& 0.99&  0.99&&8.16&Ni(II)\\\\\n$T_{(a)}$\\xspace&S&&0.72&  0.73&  0.99&  1.00&  1.00&&0.36&  0.37&  0.99&  0.99&  1.00&&8.14&Ni(IV)\\\\\n&L&&0.98&  0.98&  0.99&  1.00&  1.00&&0.13&  0.13&  0.99&  0.99&  0.99&&8.16&Ni(II)\\\\\n\\midrule\n\\multicolumn{17}{c}{YNiO$_3$}\\\\\n\\midrule\n$T_{(c)}$\\xspace&S&&0.80&  0.98&  0.99&  1.00&  1.00&&0.14&  0.26&  0.99&  0.99&  0.99&&8.14&Ni(III)\\\\\n&L&&0.80&  0.98&  0.99& 1.00&  1.00&&0.14&  0.26& 0.99&  0.99&  0.99&&8.14&Ni(III)\\\\\n$T_{(a)}$\\xspace&S&&0.58&  0.60&  0.99& 1.00&  1.00&&0.49&  0.51&  0.99&  1.00&  1.00&&8.15&Ni(IV)\\\\\n&L&&0.98&  0.99&  0.99&  1.00&  1.00&&0.12&  0.13&  0.99&  0.99&  0.99&&8.17&Ni(II)\\\\\n\\bottomrule\n\\bottomrule\n\\end{tabular}\n\\egroup\n\\end{table*}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section*{Abstract}\nThe symport of lactose and $\\rm H^+$ is an important physiological process in $E.\\ coli$, for it is closely related to cellular energy supply. In this paper, the symport of $\\rm H^+$ and lactose by $E.\\ coli$ LacY protein is computationally simulated using a newly proposed cotransport model that takes the ``leakage\" phenomenon (uncoupled sugar translocation) into account and also satisfies the static head equilibrium condition. Then, we study the equilibrium properties of this cotransport process including equilibrium solution and the time required to reach equilibrium state, when varying the parameters of the initial state of cotransport system. It can be found that in the presence of leakage, $\\rm H^+$ and lactose will reach their equilibrium state separately, but the intensity of ``leakage\" has almost no effect on the equilibrium solution, while the stronger leakage is, the shorter the time required for $\\rm H^+$ and lactose to reach equilibrium. For $E.\\ coli$ cells with different periplasm and cytoplasm volumes, when cotransport is performed at constant initial particle concentration, the time for cytoplasm pH to be stabilized increases monotonically with the periplasm to cytoplasm volume ratio. For a certain $E.\\ coli$ cell, as it continues to lose water and contract, the time for cytoplasm pH to be stabilized by cotransport also increases monotonically when the cell survives. The above phenomena and other findings in this paper may help us to not only further validate or improve the model, but also deepen our understanding of the cotransport process of $E.\\ coli$ LacY protein.\n\n\n\n\n\n\\section*{Introduction}\n\tLacY protein (lactose permease) of $Escherichia\\ coli$ is a kind of transport protein involved in the secondary active transport of hydrogen ions and lactose molecules. The protein uses the energy stored in the $\\rm H^+$ electrochemical potential to cotransport hydrogen ions into the cytoplasm along with $\\rm\\beta-galactosides$, such as lactose\\cite{guan2006lessons}. Since large amounts of lactose are required to sustain life activities in $E.\\ coli$ cells, the intracellular lactose concentration is usually higher than the environment, and the above cotransport process is usually inverse to the lactose concentration gradient\\cite{abramson2003structure}. Possible mechanisms and mathematical models for the cotransport process have been extensively studied, starting roughly from Cohen and Rickenberg's report in 1955\\cite{kramer2014membrane}. Among the numerous research results so far, one of the most significant achievement, and also a mechanism now generally accepted, is the ``six-state\" mechanism described by Kaback $et\\ al.$ in their 2001 article\\cite{kaback2001kamikaze}, which is baesd on a more universal cotransport mechanism proposed by Jardetzky in 1966\\cite{jardetzky1966simple}. The so-called ``six-state\" mechanism refers to the fact that the cotransporter lactose permease have six different functional conformations (states) in the cotransport process. The LacY protein begins the reaction cycle at a outward-facing state 1, quickly binds a hydrogen ion, turning to state 2, then continues to trap a lactose molecule and turns to state 3, during which the cotransporter remains in the outward-facing state. After the combination of particles, the cotransporter makes a rapid conformational change to inward-facing, followed by detachment of the lactose molecule to state 5, then to state 6 by shedding the hydrogen ion, and finally returns to state 1 by another rapid conformational change. All above processes are naturally reversible. However, the former conventional mechanism is challenged by some subsequent experimental results\\cite{andrini2008leak}, where uncoupled transport is observed and determined, $i.e.$ , the two kinds of particles involved in cotransport do not follow the previous stoichiometric, and one kind has a uniport-like``leakage\" phenomenon. It also poses a problem for how to modify previous existing mathematical models that simulate this ``six-state\" mechanism. For the symport of sodium ions with glucose by SGLT1 protein, which is similar to the cotransport process of $E.\\ coli$ LacY protein, several possible mechanisms leading to the leakage phenomenon have been proposed in the literature\\cite{centelles1991energetic}, among which the simpler one that has also been used many times to modify mathematical models, is to allow the transition between cotransporter states 2 and 5. Nevertheless, some literatures such as\\cite{naftalin2010reassessment} propose that any determinate modified cotransport model by allowing the transition between states 2 and 5 cannot satisfy a certain thermodynamic condition (static head equilibrium condition). Then Barreto $et\\ al.$ propose a statistical mechanical model based on the above way of modifying conventional mechanism on the work of paper\\cite{barreto2019transport}, which is a random-walk model and satisfies the static head equilibrium condition in the case of leakage\\cite{barreto2020random}.\n\n\tBased on such a model, we are finally able to computationally simulate the transport process of $E.\\ coli$ LacY protein in the presence of leakage, and find some distinct equilibrium state properties in the presence and absence of leakage, such as the equilibrium solution and the convergence rate to the equilibrium solution. In this paper, we first briefly restate and analyze the model in article\\cite{barreto2020random} and predict the results of the computational simulations in the following. Next, we vary the intensity of the leakage, the volume of $E.\\ coli$ periplasm and cytoplasm, the initial concentration of the particles and the number (density) of cotransporter to study the variation of the corresponding equilibrium solution and the time required to reach equilibrium. Finally we find some very similar or different phenomena in the presence and absence of leakage.\n\n\\section*{A brief statement and analysis of the model}\n\tThe model in article\\cite{barreto2020random} considers a closed system of periplasm and cytoplasm, the volumes of which are $V_p$ and $V_c$ respectively. Periplasm and cytoplasm are separated by a cell membrane with cotransporters embedded, across which there is a membrane potential $\\Delta\\Psi$ that we assume a constant (cytoplasm lower). The total number of cotransporters on the cell membrane is fixed to $n$, and they are all assumed to be independent of each other. Let $n_k,k=1,2\\cdots6,$ represent the number of cotransporters in the current state $k$, respectively, and the sum of these six terms is clearly the fixed value $n$. Here I use an illustration  Fig~\\ref{1} to graphically show the six-state mechanism described in the introduction section for the readers to understand it more visually.\n\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[scale=0.7]{Fig_1_new.jpg}\n\\caption{{\\bf The modified six-state mechanism of cotransport.}\nAn alternating access model of $E.\\ coli$ $\\rm lactose-H^+$ cotransport by Jardetzky modified with the transition between cotransporter states 2 and 5. A simple schematic of the ``six-state\" mechanism of cotransporter LacY protein of $E.\\ coli$ during cotransport is shown in the figure above. Numbers 1 to 6 indicate state 1 to state 6, which are the six effective conformations of LacY protein. The blue double-head arrows refer to the six states of cotransporter in relation to each other, and the yellow dashed arrow refers to the way to modify the present model\u2014allowing the transition between cotransporter states 2 and 5.}\n\\label{1}\n\\end{figure}\n\n\t$N_c^{\\rm H^+}(t)$ and $N_c^{\\rm L}(t)$ represent the current number of hydrogen ions and lactose molecules in cytoplasm, and $N_p^{\\rm H^+}(t)$ and $N_p^{\\rm L}(t)$ similarly for periplasm. When it is not necessary to specify one kind of particle or one of periplasm and cytoplasm, we will later use $S$ to refer to either of the two kinds of particles involved in cotransport and $l$ refer to periplasm or cytoplasm. Since the system is closed, the total number of $S$ particles in all states and positions is always equal to the number we put in at the beginning, as a constant value, denoted $N_S$. $\\xi$ represents the intensity of leakage (similar to the ratio of leakage current to cotransport current excluding leakage) and takes the value in $[0,1]$. $\\xi=0$ means no leakage occurs, $\\xi=1$ means the cotransporter in state 2 has an equal probability to take a conformation change to state 1,3 or 5, and the larger the value of $\\xi$, the more obvious the leakage phenomenon is. The master equations of the model consists of ten differential equations related to each other, the exact form of which is not repeated here; please move to the original \\cite{barreto2020random}. Here we perform a brief analysis of the model.\n\n\tThe original \\cite{barreto2020random} first makes the left-hand side quotient of the master equation tend to 0 by making $t\\rightarrow\\infty$, which causes all variables to take equilibrium solutions (or asymptotic solutions), and the solutions when $\\xi=0$ and $\\xi\\neq0$ have the following form,\n\\begin{displaymath}\nN_{l,\\xi=0}^S=C_0V_l\\left(\\frac{\\alpha_{K+1K}}{\\alpha_{KK+1}}\\frac{n_{K+1}}{n_K}\\right)^{h\/\\nu_S} ,\n\\end{displaymath}\t\n\\begin{displaymath}\nN_{l,\\xi\\neq0}^S=\\left(1-\\frac{\\xi}{3}\\right)^{\\tilde{h}\/\\nu_S}\\!\\!\\!\\!\\!\\!\\!\\!\\!C_0V_l\\left(\\frac{\\alpha_{K+1K}}{\\alpha_{KK+1}}\\frac{n_{K+1}}{n_K}\\right)^{h\/\\nu_S} .\n\\end{displaymath}\n\t$\\nu_S$ is the number of particle $S$ transported in one cycle of cotransporter in figure\\ref{1}, here equals 1 for $\\rm H^+$ and lactose. $h, K, \\tilde{h}, \\alpha_{pq}$ are all constants decided by $S$ and $l$, and the details are shown in article\\cite{barreto2020random}. After obtaining the above results, the paper \\cite{barreto2020random} states that $N_{l,\\xi\\neq0}^S=\\left(1-\\frac{\\xi}{3}\\right)^{\\tilde{h}\/\\nu_S}N_{l,\\xi=0}^S$, which leads to the equilibrium solution for all values of $\\xi$. However, a careful observation shows that the above two equations are essentially the relationship between the equilibrium solution of the particle numbers $N_{l,\\xi}^S$ and the cotransporter numbers $n_K$. The equilibrium solutions of the cotransporters are not necessarily the same for different values of $\\xi$ (in practice, it is clear from the calculations below that they are indeed not the same), so the conclusion of the original equilibrium solution in \\cite{barreto2020random} does not hold.\n\n\tIn fact, we can get new information about the equilibrium solution by the Gibbs free energy change as well as the electrochemical gradient. The original\\cite{barreto2020random} proves that the equilibrium solution satisfies the following static head equilibrium condition when $\\xi=0$, \n\\begin{displaymath}\n\\frac{N_c^A\/V_c}{N_p^A\/V_p}=\\left(\\frac{N_p^B\/V_p}{N_c^B\/V_c}\\right)^{(2f-1)\\nu_B\/\\nu_A}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\times\\exp\\left(-\\frac{1}{k_BT}\\left[Q_A+(2f-1)\\frac{\\nu_A}{\\nu_B}Q_B\\right]\\Delta\\Psi\\right).\n\\end{displaymath}\n\tHere $f=1$ for lactose permease and $A,B$ are particles transported by cotransporter. And the following scaling relationships between equilibrium solutions are obtained in the same way when $\\xi\\neq0$ (also satisfying the static head equilibrium),\n\\begin{displaymath}\n\\frac{N_c^S\/V_c}{N_p^S\/V_p}=\\exp\\left(-\\frac{1}{k_BT}Q_S\\Delta\\Psi\\right) .\n\\end{displaymath}\n\t$Q_S$ is the charge of a particle $S$, $k_B$ is the Boltzmann constant, and $T$ is the is the ambient temperature of $E.\\ coli$. The above two relations are important constraints on the equilibrium solution, since in reality the number of cotransporters is much smaller than the number of particles involved in cotransport, and the total number of particles is constant during the transport, $i.e.$ the following two equations hold at all times during the transport, \n\\begin{displaymath}\nN_A-N_c^A-N_p^A=\\nu_A[n_3+n_4+(n_2+n_5)f],\n\\end{displaymath}\n\\begin{displaymath}\nN_B-N_c^B-N_p^B=\\nu_B[(n_3+n_4)f+(1-f)(n_1+n_6)].\n\\end{displaymath}\n\tThen we can see that under the general condition that the total number of cotransporters is very small, the equilibrium solution changes quite little with the parameter $\\xi$ when $\\xi\\neq0$. Only if the total number of cotransporters is not small relative to the total number of particles and also the distribution of cotransporters in each state varies considerably when $\\xi$ changes does the equilibrium solution change significantly. The computational verification of the above theoretical descriptions will be carried out below, while the above conclusions give directional hints for our later calculations.\n\\section*{Computational simulations and equilibrium property studies using the above model}\n\tIn the following of this paper, computational simulations of the $E.\\ coli$ LacY protein 1:1 symport processes of $\\rm H^+$ and lactose are performed with the aforementioned model. The values used in the calculation are from the article \\cite{barreto2020random}\\cite{barreto2019transport} or selected according to the reality, as detailed in the calculation section later. We use the finite difference method to solve the differential master equations of the model numerically\\cite{bathe2006finite}, that is, we use the relation ${dN(t)}\/{dt}\\approx{(N(t+\\Delta t)-N(t))}\/{\\Delta t}$ to transform the differential equation into a difference equation for calculation, and in subsequent calculations in this paper we take the time step $\\Delta t=0.1$\n\t\\subsection*{Effect of parameter $\\xi$ (leakage intensity) on equilibrium properties}\n\tThe $\\xi=0$ and $\\xi\\neq0$ cases are too different to be studied together, so we focus on the $\\xi\\neq0$ case. First, the correlation between the equilibrium solution and $\\xi$ will be verified. The following Fig~\\ref{2} shows the correlation between $\\xi$ and the equilibrium solutions as well as that between $\\xi$ and the time to reach equilibrium states for $\\rm H^+$ and lactose at a fixed initial condition, respectively.\n\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[scale=0.8]{Fig_2.jpg}\n\\caption{{\\bf Equilibrium properties of lactose permease cotransport in $E.\\ coli$ as $\\xi$ changes.}\nOnly the $\\xi\\neq0$ cases are shown with the following initial conditions: $n_1=10^3, n_{k\\neq1}=0, N_p^{\\rm H^+}=10^5, N_p^{\\rm L}=2.5\\times10^4, N_c^{\\rm L}=7.5\\times10^4, N_c^{\\rm H^+}=10^3$. And also, we set $V_c=V_p=10^6\/C_0$, $C_0$ is a constant with the unit of $\\rm nm^{-3}$. The meaning of $C_0$ is the same as which in \\cite{barreto2020random}, similarly hereinafter. (a) The equilibium solution of $\\rm H^+$ in periplasm and cytoplasm. (b) The equilibium solution of lactose in periplasm and cytoplasm. (c) Time used to reach equilbrium state for $\\rm H^+$ and lactose with the unit of $\\Delta t$.}\n\\label{2}\n\\end{figure}\n\n\tThe reason why the initial conditions being so is that, according to the article \\cite{wilks2007ph}, $E.\\ coli$ cells are generally in an environment that causes the pH of periplasm about 1.8 to 2.1 higher than that of cytoplasm. From the above Fig~\\ref{2}(a) and Fig~\\ref{2}(b), it can be found that in the case that $\\xi$ is not very small ($\\geq0.1$), the above general conclusions about the equilibrium solution (insignificant variation of the equilibrium solution with $\\xi$ and general proportionality between the equilibrium solutions) are correct within error permissibility. In fact, when the total number of cotransporters is increased (in this example we just need to increase $n_1$), the error will be significantly reduced and the proportional relationship between the equilibrium solutions will become more obvious and precise.\n\n\tAnother important feature about cotransport is the time required to reach the equilibrium solution. Although the equilibrium solution is asymptotic and theoretically not achievable in finite time, we can still consider that equilibrium has been reached when the change rate or the first-order backward difference quotient of the solution is small enough in our simulation. When there is no leakage ($\\xi=0$), it is easy to find that $\\rm H^+$ and lactose reach equilibrium at the same time by calculation, which is also easy to see from the above theoretical analysis. However, when leakage does exist ($\\xi\\neq0$), the time for the two kinds of particles to reach equilibrium is seperated, which can also be inferred by splitting the full reaction cycle into two independent transport process. The above Fig~\\ref{2}(c) shows the relationship between the time of two kinds of particles to reach equilibrium and $\\xi$ when $\\xi\\in(0,1]$.\n\n\tFig~\\ref{2}(c) is calculated by considering that when $[N_c^S(t)-N_c^S(t-\\Delta t)]\/N_c^S(t)\\leq10^{-9}$ holds for almost every discrete time moments from some t (for over $99.9999\\%$ of time moments), the particle $S$ reach the equilibrium. Subsequent calculations of the time to reach the equilibrium are similar, but accuracy may be adjusted as needed. The reasons for not using the first-order backward difference quotient and instead using change rate are the poor estimation of the range of the difference quotient and the fact that the difference quotient in this model is not guaranteed to be consistently smaller than the required accuracy after a certain time. As can be seen in the figure, the time for $\\rm H^+$ to reach equilibrium is always shorter than the time for lactose when $\\xi\\neq0$. Because we can formally split the overall transort process in the presence of leakage, the full reaction process can be viewed as a combination of a leakage-free cotransport process and a separate uniport process of $\\rm H^+$, with little interference between them. Then $\\rm H^+$ has two transport processes transporting it, while lactose has only one. So $\\rm H^+$ can reach equilibrium with the net transport of the two processes canceling each other, while lactose has not reached equilibrium at that point. It can also be observed that the time for the two kinds of particles to reach equilibrium do not have a consistent trend with the change of $\\xi$. Both of them are large when $\\xi$ is small, after that it decreases rapidly with the increase of $\\xi$, and the change is not obvious after $\\xi>0.2$. But the equilibrium time of lactose has a small rebound after $\\xi>0.5$, while $\\rm H^+$ continues to remain monotonically decreasing. The reason for this phenomenon is not clear, and it is speculated that there may be some calculation errors. Another point worth mentioning is that particle numbers converge much faster when $\\xi=0$ than $\\xi\\neq0$. The addition of the transition between cotransporter states 2 and 5 ($i.e.$ , making $\\xi\\neq0$) makes the convergence of the system much slower.\n\\subsection*{Effect of periplasm and cytoplasm volumes $V_p,V_c$ on equilibrium properties}\n\tThe previous arithmetic examples have been calculated assuming $V_p=V_c$, and as seen before, the proportionality satisfied by the equilibrium solution when $\\xi\\neq0$ is actually a proportionality between the particle concentrations in the two reaction chambers (periplasm and cytoplasm), so we consider changing the periplasm and cytoplasm volume ratios to verify the conclusion and observe if there are new phenomena. Also due to the previous conclusions on the equilibrium solution and the convergence rate in the case of $\\xi\\neq0$, we can only study the $\\xi=0$ and $\\xi=1$ cases. The equilibrium solutions of the $\\xi=0$ case are shown in the following Fig~\\ref{3}. \n\\begin{figure}[!h]\n\\centering\n\\includegraphics[scale=0.7]{Fig_3.jpg}\n\\caption{{\\bf The equilibium solutions for $\\xi=0$ when $V_c$ or $V_p$ is fixed and the other one varies.}\n(a) The equilibium solutions of $\\rm H^+$ and lactose when $V_c$ is fixed to $10^6\/C_0$ and $V_p\/V_c>1$, with initial conditions $n_1=10^3, n_{k\\neq1}=0, N_p^{\\rm H^+} =10^5$, $N_p^{\\rm L}=2.5\\times10^4$, $N_c^{\\rm L}=7.5\\times10^4, N_c^{\\rm H^+}=10^3$. (b) The equilibium solutions of $\\rm H^+$ and lactose when $V_c$ is fixed to $10^6\/C_0$ and $V_p\/V_c<1$, with the same initial conditions before.}\n\\label{3}\n\\end{figure}\n\n\tFrom Fig~\\ref{3}, the following phenomenon can be found that under the condition of fixing the number of particles involved in cotransport ($i.e.$ fixing $N_c^S,N_p^S$ in initial conditions), the change of $V_c$ seems to have less effect on the equilibrium solution than $V_p$, and the system equilibrium solution is more insensitive to the change of $V_c$. When changing other initial conditions and making $V_p$ a constant value to change $V_c\/V_p$, it is found that this phnomenon is actually more accurately formulated as that changing $V_c$ or $V_p$ only under the condition that $V_p\/V_c>1$ has a more pronounced effect on the equilibrium solution when $\\xi=0$, thus showing an insensitivity to the increase in the denominator $V_c$. After adjusting the parameters and performing a lot of calculations, it is found that this property is somewhat related to the membrane potential. In this case the direction of potential decrease is from periplasm to cytoplasm, and the membrane potential is $-100mV$. However, when decreasing this potential difference, this phenomenon becomes less and less obvious, especially when reversing the membrane potential, $i.e.$ making the potential of cytoplasm higher than that of periplasm, the significances of $V_c$ and $V_p$ in the above properties are switched, and $V_p$ becomes the one that has less influence on the equilibrium solution of the system. \n\n\tThe above phenomena can be explained from the theoretical analysis, for which we consider the Gibbs free energy variation at the starting moment $\\Delta G$. For this cotransport process of the $E.\\ coli$ LacY protein, we can rewrite the free energy variation in terms of the sum of the chemical potential multiplying the stoichiometry of the two particles \\cite{nelson2008lehninger}, bringing the data and slightly transforming then we have, \n\\begin{displaymath}\n\\Delta G=k_BT\\ln\\left(\\frac{N_c^{\\rm H^+}N_c^{\\rm L}}{N_p^{\\rm H^+}N_p^{\\rm L}}\\right)+Q_{\\rm H^+}\\Delta\\Psi+2k_BT\\ln(V_p\/V_c).\n\\end{displaymath}\n\tFor a fixed initial condition, the first of the three terms on the right-hand side of the above equation is constant, and based on the data we use in calculation and the data corresponding to the environment in which $E.\\ coli$ is usually found, this term is usually in the $(-4k_BT,k_BT)$ interval, and the second term is usually around $-3.9k_BT$ \\cite{lo2007nonequivalence}. When changing $V_p$ or $V_c$, the absolute value of $\\Delta G$ can be reduced only if $V_p\/V_c>1$, and making $\\left|\\Delta G\\right|$ decrease produces a larger rate of change than making it increase when the amount of change in $V_p\/V_c$ is the same. The equilibrium solution is reached when $\\Delta G$ is 0, and it is easy to find that $\\Delta G$ varies monotonically during the cotransport process. So $\\Delta G$ can actually represent the \"gap\" between the initial state and the equilibrium state, and the gap between the equilibrium solutions is larger when the difference between $\\Delta G$ is larger. And since the negative membrane potential in this process contributes positively (also not small proportion) to $\\Delta G$, it produces a very different property when the membrane potential decreases or even reverses.\n\n\tNext, according to common biological sense, we will observe the relationship between $V_p\/V_c$ or $V_c\/V_p$ and the time required for the system to reach equilibrium when $V_p\/V_c<1$. In the following, the case of $\\xi=0$ comes first. Inspired by the above phenomena, we fix $V_c$ and $V_p$ separately and change the other one to calculate. For practical reasons, particle numbers should not be fixed as in the case of equilibrium solution just probed before, but the concentration of particles in periplasm and cytoplasm in initial conditions should be fixed, so that the results are more realistic. The following Fig~\\ref{4}(a) and Fig~\\ref{4}(b) are plotted with $V_p\/V_c$ as the horizontal coordinate (with $V_c$ fixed) and $V_c\/V_p$ as the horizontal coordinate (with $V_p$ fixed), respectively, and the time to reach the equilibrium solution as the vertical coordinate.\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[scale=0.9]{Fig_4_new.jpg}\n\\caption{{\\bf Time to reach equilibrium for $\\xi=0$ when $V_c$ or $V_p$ is fixed and the other one varies.}\nTime used to reach equilbrium state with the unit of $\\Delta t$ when the initial concentrations of $\\rm H^+$ and lactose are fixed in periplasm and cytoplasm, which has the initial conditions $n_1=10^3, n_{k\\neq1}=0, N_p^{\\rm H^+}\/V_p=C_0\/10, N_c^{\\rm H^+}\/V_c=C_0\/1000, N_p^{\\rm L}\/V_p=C_0\/40, N_c^{\\rm L}\/V_c=3C_0\/40$, $\\xi=0$. (a) $V_c$ fixed to $10^6\/C_0$. (b) $V_p$ fixed to $10^6\/C_0$.}\n\\label{4}\n\\end{figure}\n\n\tWhen the change rate of $N_c^S$ is always less than $10^{-9}$, the particle $S$ is considered to reach equilibrium in the above Fig~\\ref{4}. It is still clear to find different effects of $V_p$ and $V_c$ on the time required to reach equilibrium, and the effect of $V_c$ is still much smaller than that of $V_p$. Increasing $V_p$ when $V_c$ is fixed can monotonically increase the time for the system to reach equilibrium. However, changing $V_c$ when $V_p$ is fixed has a less significant effect on the convergence rate. This phenomenon can still be explained using the free energy variation $\\Delta G$, because although $\\Delta G$ is constant as particle concentration is constant, the net change of particles transported through the cotransporter is different as the number of particles involved in cotransport expands with $V_p$ or $V_c$ increasing. Based on the initial conditions we give and the general conditions in practice, the $\\rm H^+$ concentration is much higher in periplasm than in cytoplasm (possibly by an order of magnitude or more), and lactose concentration is generally higher in cytoplasm than in periplasm. In the case of $\\xi=0$, the net transport is a 1:1 cotransport for both kinds of particles from periplasm to cytoplasm, so when periplasm volume $V_p$ fixed and cytoplasm volume $V_c$ varied, the net transport does not change much and therefore the time to reach equilibrium does not change significantly. But in contrast the net transport increases almost linearly when $V_c$ fixed and $V_p$ varied, so that the time required to reach equilibrium also increases monotonically with the volume ratio $V_p\/V_c$. \n\n\tThe above is the case of $\\xi=0$. And for the case of $\\xi=1$, it is verified that the conclusion about the ratio of equilibrium solutions, $i.e.$ , ${(N_c^S\/V_c)}\/{(N_p^S\/V_p)}=\\rm const.$ , holds for both $\\rm H^+$ and lactose particles within error permissibility when fixing the number of each particle in the initial condition. Then the next concern is the time required to reach the equilibrium solution when $\\xi=1$, and we still require a fixed concentration of the particles in periplasm and cyptoplasm in the initial conditions. The following figures Fig~\\ref{5}(a) and Fig~\\ref{5}(b) show the relations between $V_p\/V_c$ ($V_c$ fixed) or $V_c\/V_p$ ($V_p$ fixed), and the time to reach the equilibrium solution, respectively, and we consider that the particle $S$ reaches equilibrium when the rate of change of $N_c^S$ is always less than $10^{-8}$. It is easy to see that in the case of $\\xi=1$ there are some phenomena that do not match our expectations. For example, with $V_c$ fixed, the time to reach equilibrium for both kinds of particles increase monotonically with the increase of $V_p$; however, with $V_p$ fixed, the equilibrium time for $\\rm H^+$ decreases monotonically with $V_c$ increasing, but the equilibrium time for lactose shows an increase followed by a decrease and takes a maximum around $V_c\/V_p=4$. \n\\begin{figure}[!h]\n\\centering\n\\includegraphics[scale=0.9]{Fig_5_new.jpg}\n\\caption{{\\bf Time to reach equilibrium for $\\xi=1$ when $V_c$ or $V_p$ is fixed and the other one varies.}\n Different time for $\\rm H^+$ and lactose to reach equilbrium state with the unit of $\\Delta t$ when the initial concentrations of $\\rm H^+$ and lactose are fixed in periplasm and cytoplasm, which has the initial conditions $n_1=10^3, n_{k\\neq1}=0, N_p^{\\rm H^+}\/V_p=C_0\/10, N_c^{\\rm H^+}\/V_c=C_0\/1000, N_p^{\\rm L}\/V_p=C_0\/40, N_c^{\\rm L}\/V_c=3C_0\/40$, $\\xi=1$. (a) $V_c$ fixed to $10^6\/C_0$. (b) $V_p$ fixed to $10^6\/C_0$.}\n\\label{5}\n\\end{figure}\n\n\tThe above two phenomena are quite anomalous, but they can still be broadly explained as follows. Since the concentration of $\\rm H^+$ and lactose on both sides of the cell membrane and the number of cotransporters are all constant, the transport rate of the cotransporter changes little when varying volume ratio, so the main difference in the equilibrium time comes from the number of particles by net cotransport. The $\\rm H^+$ concentration is considerably higher in periplasm than in cytoplasm, with lactose concentration the opposite. Therefore, according to the proportional relationship that needs to be satisfied by the equilibrium solution, the net transport of $\\rm H^+$ is from periplasm to cytoplasm, which eventually makes the $\\rm H^+$ concentration in cytoplasm much higher than in periplasm. Then increasing $V_p$ causes the net transport of $\\rm H^+$ to increase significantly, so the equilibrium time also increases significantly and monotonically with $V_p\/V_c$ up. Meanwhile, the net transport of lactose is from cytoplasm to periplasm, so the effect of changing $V_p$ on the net transport of lactose is positive, and therefore the equilibrium time increases with $V_p\/V_c$ up, but asymptotic phenomena occur as $V_p\/V_c$ continues to increase. Thus, the increasing rate of the time for lactose to reach equilibrium slows down rapidly as $V_p\/V_c$ increases, but is still greater than the $\\rm H^+$ equilibrium time.\n\t\n\tWhen varying $V_c$, the case of $\\rm H^+$ is similar to the earlier discussion and easily explained as monotonically decreasing with $V_c\/V_p$, but the case of lactose is more complicated. Due to the proportionality of the equilibrium solution, when $V_c$ is very large or small, the concentration of lactose in cytoplasm at equilibrium will be close to the initial concentration in cytoplasm or periplasm, respectively, so that the net transport is similar and both are small, but $V_c\/V_p$ of moderate size may produce a large net transport. This may lead to a phenomenon that equilibrium time of lactose increases and then decreases with $V_c\/V_p$ up as shown in Fig~\\ref{5}(b), but the exact situation still needs to be supported by experimental data. From the above discussion we can find that in the case of $\\xi\\neq0$, the correlation between the equilibrium of the two particles is very weak, not only the equilibrium solution itself is irrelevant, but also the correlation of the time required to reach the equilibrium solution is very low, which is exactly an important property of this model. This leads a slightly questioning of the model when $\\xi\\neq0$.\n\n\tAbove we set the volume ratio of periplasm and cytoplasm as the horizontal coordinate to calculate, but for a fixed $E.\\ coli$ cell (or cells from the same population), periplasm is the portion between the cell wall (cytoderm) and the cell membrane, and cyptoplasm is the portion within the cell membrane other than the nuclear region, and the sum of the volumes $V_c+V_p$ varies very little due to the rigidity of the cytoderm. Therefore, we continue to investigate its effect on the convergence rate by taking the percentage of periplasm to the sum of periplasm and cyptoplasm volumes as the horizontal coordinate while fixing the sum of the two volumes. In this case, we choose to fix the initial concentration and initial number of particles in periplasm and cyptoplasm, respectively, and calculate the time required for different particles to reach equilibrium. The following Fig~\\ref{6} is the image of the time required to reach equilibrium as the periplasm fraction changes for the case $\\xi=0$, where it is still considered that the particle $S$ reaches the equilibrium solution when the change rate of $N_c^S$ is always less than $10^{-9}$. As seen from the experimental data in article \\cite{stock1977periplasmic}\\cite{graham1991periplasmic}, the range of horizontal ordinates in Fig~\\ref{6} already includes the volume fraction of periplasm (8\\%-40\\%) in which $E.\\ coli$ can survive.\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[scale=0.9]{Fig_6.jpg}\n\\caption{{\\bf Time to reach equilibrium with the variation of periplasm fraction $\\frac{V_p}{V_p+V_c}$, $\\xi=0$.}\nThe sum of volumes of periplasm and cytoplasm is fixed, $V_c+V_p=2\\times10^6\/C_0$. (a) Initial particle concentration fixed, and the initial conditions are like $n_1=10^3, n_{k\\neq1}=0, N_p^{\\rm H^+}\/V_p=C_0\/10, N_c^{\\rm H^+}\/V_c=C_0\/1000, N_p^{\\rm L}\/V_p=C_0\/40, N_c^{\\rm L}\/V_c=3C_0\/40$. (b) Initial particle population fixed, and the initial conditions are like $n_1=10^3, n_{k\\neq1}=0, N_p^{\\rm H^+} =10^5, N_p^{\\rm L}=2.5\\times10^4, N_c^{\\rm L}=7.5\\times10^4, N_c^{\\rm H^+}=10^3$.}\n\\label{6}\n\\end{figure}\n\n\tIt can be found that the two graphs have some similarity, both appearing to be approximately linearly increasing followed by monotonically decreasing. Although it may seem strange, the above case of $\\xi=0$ can actually be roughly explained by the standard theoretical methods. With a fixed initial concentration, the $\\rm H^+$ concentration in cytoplasm increases continuously as the volume fraction of periplasm increases due to the higher concentration of $\\rm H^+$ in periplasm than in cyplasm in the initial conditions .However, the increase does not exceed the initial $\\rm H^+$ concentration in periplasm, meanwhile the volume of cyptoplasm decreases accordingly. If the volume difference between periplasm and cytoplasm is too large, and the $\\rm H^+$ concentrations in the smaller part of the volume at equilibrium will be close to the initial values of the larger part, then the net transport will be neither large, so the net transport of hydrogen ions is likely to increase and then decrease as the volume fraction of periplasm increases. Therefore, the time to reach equilibrium is likely to increase and then decrease at constant initial concentration and constant cotransport rate.\n\n\tWith a fixed initial particle number, $\\left|\\Delta G\\right|$ decreases as the fraction of periplasm increases, resulting in a decrease in the net amount of particles transported, but the change in volume also decreases the transport rate of cotransporters from periplasm to cytoplasm but increases the rate of reverse direction, with a decrease in the (initial) net transport rate. A qualitative analysis of the form reveals a more pronounced change in the net transport rate (than in the net amount of particles transported), but the change rate of the net transport rate gradually decreases as the fraction of periplasm increases, thus possibly leading to an increase and then a decrease in the time required to reach equilibrium. The above analyses and explanations are qualitative and still need to be verified or denied by experimental data. The above is the case of $\\xi=0$, while the following Fig~\\ref{7}(a) and Fig~\\ref{7}(b) are the case of $\\xi=1$, which are surprisingly not similar, unlike the similarity of the previous Fig~\\ref{6}(a) and Fig~\\ref{6}(b).\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[scale=0.9]{Fig_7.jpg}\n\\caption{{\\bf Time to reach equilibrium with the variation of periplasm fraction $\\frac{V_p}{V_p+V_c}$, $\\xi=1$.}\n Different time for $\\rm H^+$ and lactose to reach equilbrium state with the unit of $\\Delta t$. The sum of volumes of periplasm and cytoplasm is fixed, $V_c+V_p=2\\times10^6\/C_0$. (a) Initial particle concentration fixed, and the initial conditions are like $n_1=10^3, n_{k\\neq1}=0, N_p^{\\rm H^+}\/V_p=C_0\/10, N_c^{\\rm H^+}\/V_c=C_0\/1000, N_p^{\\rm L}\/V_p=C_0\/40, N_c^{\\rm L}\/V_c=3C_0\/40$. (b) Initial particle population fixed, and the initial conditions are like $n_1=10^3, n_{k\\neq1}=0, N_p^{\\rm H^+} =10^5, N_p^{\\rm L}=2.5\\times10^4, N_c^{\\rm L}=7.5\\times10^4, N_c^{\\rm H^+}=10^3$.}\n\\label{7}\n\\end{figure}\n\n\tA closer observation reveals that although the equilibrium time images of lactose are disparate, the equilibrium time images of $\\rm H^+$ are still similar in shape. This is primarily due to the fact that in the presence of leakage, as mentioned earlier, the equilibrium of the two kinds of particles has little influence on each other. And no matter initial concentration or initial particle numbers is fixed, in the range of periplasm volume fraction in our calculation (${V_p}\/{(V_p+V_c)}\\in[0.1,0.9]$), the net transport direction of hydrogen ions is from periplasm to cytoplasm according to $\\Delta G$. Increasing the periplasm volume proportion in the presence of leakage, when the initial concentration is fixed, the (initial) net transport rate remains constant and the net transport amount increases, while when the initial particle number is fixed, the (initial) net transport rate decreases and the net transport volume does not change much from the aforementioned proportionality of the equilibrium solution, so the time required to reach equilibrium in both cases shows a monotonically increasing state. \n\n\tThe case of lactose is more complicated, especially in the condition of fixed initial particle number, where there are three inflection points and also the order for $\\rm H^+$ and lactose to reach equilibrium changes with different volume fraction of periplasm. We can give the following explanation for the minimal value point generated near ${V_p}\/{(V_p+V_c)}=0.25$ under the condition of fixed initial particle numbers. Since the initial value of $N_c^{\\rm L}\/N_p^{\\rm L}={1}\/{3}$, when $V_c\/V_p$ is around 3, the initial condition of lactose is actually very close to the equilibrium state because the total number of cotransporters is small compared to the number of particles. $N_c^{\\rm L}$ and $N_p^{\\rm L}$ change very little during the cotransport process and can be considered as reaching equilibrium very early. Changing the initial value of lactose and computationally verifying, it is found that this phenomenon is practically universal. When the volume ratio is adjusted so that the initial value of lactose is close to the equilibrium solution, there is always a significant reduction in the time for lactose to reach equilibrium, causing lactose's reaching equilibrium before $\\rm H^+$. More generally, a similar phenomenon occurs for both kinds of particles. Nevertheless, it should still be noted that for the general realistic initial values, as in the previous analysis, $\\rm H^+$ usually reaches equilibrium before lactose because of the additional leakage current regulation, which also complements and explains the above inference. Regarding the remaining phenomena embodied in the two figures, it is difficult for the author to give a reasonable explanation here, and I can only leave it to the experimental data to verify or deny.\n\t\\subsection*{Effect of changing the initial concentration and initial distribution of particles on equilibrium properties}\n\tNow we will explore the effect of the initial state on the equilibrium properties. Inspired by the previous section, we can observe the effect of changing the initial state on $\\Delta G$ to determine the effect on the equilibrium solution and the time required to reach equilibrium. We still study only the $\\xi=0,1$ cases in the following, as in the previous section. If the number of particles in the initial state is expanded linearly, and $\\Delta G$ does not change in this case, then it can be presumed that the equilibrium solution and the time to reach equilibrium should also increase roughly linearly, which is verified to be correct when $\\xi=0,1$ after calculation, and we will not elaborate it here. In the following we will consider the effect of changing the initial distribution of particles in periplasm and cytoplasm. Since both volumes $V_c,V_p$ are kept constant, we do not need to care about the difference between the two conditions of initial concentrations and initial particle numbers. The following Fig~\\ref{8} shows the change of $\\rm H^+$ and lactose equilibrium fractions in periplasm with the variation of $\\rm H^+$ initial fraction in periplasm. \n\\begin{figure}[!h]\n\\centering\n\\includegraphics[scale=1.0]{Fig_8.jpg}\n\\caption{{\\bf Equilibrium fractions of particles in periplasm as initial fraction of $\\rm H^+$ in periplasm changes.}\nThe initial conditions are $n_1=10^3, n_{k\\neq1}=0, N_p^{\\rm H^+}+N_c^{\\rm H^+}=10^5, N_p^{\\rm L}=2.5\\times10^4, N_c^{\\rm L}=7.5\\times10^4$. And also, we set $V_c=V_p=10^6\/C_0$, $\\xi=0,1$. There is always enough time for particles to stabilize so the final fractions equals the equilibrium fractions. (a) Final $\\rm H^+$ fraction. (b) Final lactose fraction.}\n\\label{8}\n\\end{figure}\n\n\tIt can be seen that the case of $\\xi=1$ conforms well to our expectation, and the equilibrium solution scale relation hardly changes with the initial conditions. In the case of $\\xi=0$, the equilibrium proportion of $\\rm H^+$ in periplasm increases monotonically with the initial proportion of $\\rm H^+$ in periplasm, and it can be observed that equilibrium value is always lower than the initial value. Meanwhile, the proportion of lactose in periplasm in the equilibrium solution decreases monotonically with the proportion of $\\rm H^+$ in periplasm in the initial condition. Both phenomena are consistent with common biological sense. When the $\\rm H^+$ concentration in periplasm is higher than in cytoplasm, $\\rm H^+$ provides a sufficient electrochemical potential for the transportation of lactose to cytoplasm, which allows $E.\\ coli$ to take up lactose against the concentration gradient of it. But doing so depletes $\\rm H^+$ in periplasm, making the $\\rm H^+$ concentration gap between periplasm and cytoplasm smaller when equilibrium is reached. The larger the initial $\\rm H^+$ concentration difference between periplasm and cyptoplasm, the more $\\rm H^+$ can be used and the more lactose is transported to cyptoplasm. Also, because of the need to maintain a high concentration of lactose in the cytoplasm, the periplasm must maintain a sufficiently high $\\rm H^+$ concentration at equilibrium to counteract the chemical potential of lactose. Due to the fact that some of the $\\rm H^+ $ are transported into cytoplasm with lactose, the $\\rm H^+$ concentration difference at equilibrium is lower than in the initial condition. The above two phenomena are thus explained. In the following, we consider the change in the initial proportion of lactose in periplasm, corresponding to the change in the proportion of two particles in periplasm at equilibrium, and the figure \\ref{11} is shown below. The following Fig~\\ref{9} show the variations in $\\rm H^+$ and lactose equilibrium fraction as the initial fraction of lactose in periplasm changes.\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[scale=1.0]{Fig_9.jpg}\n\\caption{{\\bf Equilibrium fractions of particles in periplasm as initial fraction of lactose in periplasm changes.}\nThe initial conditions are $n_1=10^3, n_{k\\neq1}=0, N_p^{\\rm H^+}=9\\times10^4, N_c^{\\rm H^+}=10^4, N_p^{\\rm L}+N_c^{\\rm L}=10^5$. And also, we set $V_c=V_p=10^6\/C_0$, $\\xi=0,1$. There is always enough time for particles to stabilize so the final fractions equals the equilibrium fractions. (a) Final $\\rm H^+$ fraction. (b) Final lactose fraction.}\n\\label{9}\n\\end{figure}\n\n\tThe above two plots obtained by varying the initial distribution of lactose are also fully interpretable in a similar way as above. When $\\xi=0$, as the initial lactose molecules in periplasm keep increasing, the absolute value of the free energy change of cotransport in the initial state $\\left|\\Delta G\\right|$ increases, and more lactose molecules are transported into cytoplasm. As the two particles are transported into cytoplasm in strictly equal amounts when $\\xi=0$, there will be a monotonic decrease in the amount of hydrogen ions and a slow monotonic increase in the amount of remaining lactose in periplasm at equilibrium with initial fraction of lactose in periplasm increasing. The $\\xi=1$ cases still meet the expectation and we do not elaborate on it.\n\t\\subsection*{Effect of changing the total number of cotransporters on equilibrium properties}\n\tFrom the master equations of the system, we know that the first order derivatives of both particle numbers and cotransporter numbers in different states are linearly related to the current values of some states of the cotransporters. So we guess that the convergence rate should be linearly related to the total number of cotransporters with constant initial values, and then the time required to reach the equilibrium state should be inversely proportional to cotransporter numbers.  After the calculation we have Fig~\\ref{10} for $\\xi=0,1$ cases.\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[scale=1.0]{Fig_10_new.jpg}\n\\caption{{\\bf Time to reach equilibium with the variation of the total number of cotransporters.}\nThe initial conditions are $N_p^{\\rm H^+}=10^5, N_c^{\\rm H^+}=10^3, N_p^{\\rm L}=2.5\\times10^4, N_c^{\\rm L}=7.5\\times10^4, V_c=V_p=10^6\/C_0$. All cotransporters are supposed to be in state 1 initially. (a) The $\\xi=0$ case. (b) The $\\xi=1$ case.}\n\\label{10}\n\\end{figure}\n\n\tAlthough the negative correlation is evident in both plots, a closer observation does not show the inverse proportion relationship as we would expect. The reason for this is unclear and needs to be verified or disproved by experimental data.\n\\section*{Conclusions and further discussions of the above results}\n\tFor the cotransport process of $E.\\ coli$ LacY protein symporting $\\rm H^+$ and lactose, we apply the random-walk model proposed in article \\cite{barreto2020random} to computationally simulate and find, validate or predict the following phenomena. \\begin{itemize}\n\t\\item In the absence of leakage, the concentrations of the two particles $\\rm H^+$ and lactose reach stability simultaneously; in the presence of leakage, the time required for the two particles to reach the equilibrium state is separated. In the realistic situation where the number of cotranporters is small relative to the number of transported particles, the intensity of leakage, if leakage exists, hardly affects the equilibrium state of cotransport, but affects the time for both particles to reach the equilibrium state, and generally the more pronounced leakage is, the shorter the time to reach equilibrium. \n\t\\item In the absence of leakage, fixing the initial number or concentration of $\\rm H^+$ and lactose in periplasm and cytoplasm, for homogenous $E.\\ coli$ cells with different periplasm and cytoplasm volumes, the periplasm volume but not the cytoplasm volume has a greater effect on the equilibrium state and the time required to reach equilibrium. In other words, when periplasm volumes are similar and cytoplasm volumes are different, cells have similar equilibrium states and need similar time to reach equilibrium by this cotransport process, but not vice versa. \n\t\\item In the presence of leakage, fixing the initial concentration of $\\rm H^+$ and lactose in periplasm and cytoplasm, for homogenous $E.\\ coli$ cells with different periplasm and cytoplasm volumes, the time required for the pH of cytoplasm to stabilize increases monotonically with the periplasm to cytoplasm volume ratio increasing. Meanwhile,  the time for the lactose concentration to reach equilibrium has a more complex relationship with the volume ratio of periplasm and cytoplasm as shown in Fig~\\ref{5}.\n\t\\item For a certain $E.\\ coli$ cell (or cells with similar size from the same population), fixing the initial number or concentration of $\\rm H^+$ and lactose in periplasm and cytoplasm, the time for the pH of cytoplasm to stabilize increases monotonically as the cell loses water regardless of the presence or absence of leakage (under the condition of cell survival), whereas the time for the concentration of lactose to stabilize varies more complexly, as seen in Fig~\\ref{6} and Fig~\\ref{7}.\n\t\\item For $E.\\ coli$ cells from the same population, the concentration ratio of particles in periplasm and cyptoplasm in equilibrium state does not vary with the initial state concentration ratio if leakage exists.\n\t\\item For different subpecies of $E.\\ coli$, the time for the cotransport process to reach equilibrium is negatively but not inversely correlated with the amount of the cotransporter LacY with initial concentrations of $\\rm H^+$ and lactose in periplasm and cytoplasm fixed.\n\\end{itemize}\n\tMany of the above phenomena can be qualitatively explained, but some of them still can not be well explained and need to be verified or negated by experimental data.\n\n\tAfter summarizing the results above, it is easy to notice that in the case of the parameter $\\xi=0$, $i.e.$ , no leakage, the results are mostly consistent with biological or chemical intuition, but the appearance of some properties for $\\xi\\neq0$ case is quite anomalous. The biggest problem is the equilibrium solution. When $\\xi\\neq0$, the equilibrium solution almost strictly satisfies the concentration proportionality under the general condition that the number of cotransporters is much smaller than the number of particles to be transported. For the cotransport of $E.\\ coli$ LacY protein, the concentration of lactose in periplasm and cytoplasm at equilibrium is almost equal because the lactose molecule is neutral, and this relationship does not change with other conditions such as the initial value. This phenomenon may be somewhat different from the reality. Actually for this model in $\\xi\\neq0$ cases, any neutral particles involved in cotransport has equal concentration in periplasm and cytoplasm, which is no different from ordinary diffusion and far from the energy-consuming active transport. Next, from a kinetic point of view, there are still some parts of the model that need discussions when the parameter $\\xi\\neq0$. Although the essence of the model is cotransport, we find that the correlation of the time required for two particles to reach equilibrium is not strong. If the initial state of a certain particle happens to satisfy the proportionality relation that the equilibrium state should satisfy, it will soon equilibrate and is hardly controlled by the other particle. From this point of view, the two kinds of particles are almost independent of each other. At this point, the effect of  on lactose transport is more like a facilitated transport, which is also used to explain the uncoupled cotransport in recent years, such as literature \\cite{kaback2015chemiosmotic} and \\cite{\nkaback2019takes}. In contrast, there is no case where some particle reaches equilibrium by itself when $\\xi=0$. Therefore, although changing the parameter $\\xi$ gives the model good kinetic properties, such as adjusting $\\xi$ to control the speed of convergence, the model still has some defects. The parameter $\\xi$ is introduced to solve the problem of leakage current, but this solution is not perfect, and we may still need to think about other solutions. And furthermore, actually there is no experimental evidence showing this uncoupled sugar translocation (``leakage\") could happen before the concentrations of lactose across the cytoplasmic membrane reach equilibrium. In other words, all the above calculations and explanations are based on a shared assumption, and the specific mechanism of cotransporter LacY still lacks knowledge.\n\\section*{Acknowledgement}\n\tI would like to give my sincere gratitude to Prof. Yunxin Zhang from the School of Mathematical Sciences, Fudan University. From my freshman year in Fudan, I have been in contact with Professor Zhang, who is although not my tutor but still gives me much selfless guidance and help in my academic learning and undergraduate studies. It was with his advice and help that I was able to complete this paper as an undergraduate. He taught me almost all the basics about academic writing, read my drafts and made many valuable suggestions. Once again, my sincere thanks to all the people who helped me during the completion of this paper.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\bigskip We are interested in the regularity of weak solutions to the\nviscous incompressible magnetohydrodynamics (MHD) equations in $\\mathbb{R}%\n^{3}$\n\\begin{equation}\n\\left\\{\n\\begin{array}{c}\n\\partial _{t}u+(u\\cdot \\nabla )u-\\left( b\\cdot \\nabla \\right) b-\\Delta\nu+\\nabla \\pi =0, \\\\\n\\partial _{t}b+(u\\cdot \\nabla )b-(b\\cdot \\nabla )u-\\Delta b=0, \\\\\n\\nabla \\cdot u=\\nabla \\cdot b=0, \\\\\nu(x,0)=u_{0}(x),\\text{ \\ }b(x,0)=b_{0}(x),%\n\\end{array}%\n\\right.  \\label{eq1.1}\n\\end{equation}%\nwhere $u=(u_{1},u_{2},u_{3})$ is the velocity field, $b=(b_{1},b_{2},b_{3})$\nis the magnetic field, and $\\pi $ is the scalar pressure, while $u_{0}$ and $%\nb_{0}$ are the corresponding initial data satisfying $\\nabla \\cdot\nu_{0}=\\nabla \\cdot b_{0}=0$ in the sense of distribution.\n\nLocal existence and uniqueness theories of solutions to the MHD equations\nhave been studied by many mathematicians and physicists (see, e.g., \\cite%\n{CW, DL, ST}). But due to the presence of Navier-Stokes equations in the\nsystem (\\ref{eq1.1}) whether this unique local solution can exist globally\nis an outstanding challenge problem. For this reason, there are many\nregularity criteria of weak solutions for the MHD equations has been\ninvestigated by many authors over past years (see e.g., \\cite{DJZ, D, FJNZ,\nG1, GR1, GR2, GRZ, LD, NGZ, Z1, Z2} and references therein). Note that the\nliteratures listed here are far from being complete, we refer the readers to\nsee for example \\cite{GR20, JZ1, JZ2, JZ3, JZ4} for expositions and more\nreferences.\n\nMore recently, Beir\\~{a}o and Yang \\cite{BY} proved the following regularity\ncriterion for the mixed pressure-velocity in Lorentz spaces for Leray-Hopf\nweak solutions to 3D Navier-Stokes equations\n\\begin{equation}\n{\\frac{\\pi }{\\left( e^{-\\left\\vert x\\right\\vert ^{2}}+\\left\\vert\nu\\right\\vert \\right) ^{\\theta }}\\in L}^{p}(0,T;L^{q,\\infty }(\\mathbb{R}%\n^{3})),\\text{ \\ where \\ }0\\leq \\theta \\leq 1\\text{ and }\\frac{2}{p}+\\frac{3}{%\nq}=2-\\theta ,  \\label{eq7}\n\\end{equation}%\nwhere $L^{q,\\infty }(\\mathbb{R}^{3})$ denotes the Lorentz space (c.f. \\cite%\n{Tri}).\n\nMotivated by the recent work of \\cite{BY}, the purpose of this note is to\nestablish the regularity for the MHD equations (\\ref{eq1.1}) with the mixed\npressure-velocity-magnetic in Lorentz spaces. Our main result can be stated\nas follows:\n\n\\begin{thm}\n\\label{th1}Suppose that $(u_{0},b_{0})\\in L^{2}(\\mathbb{R}^{3})\\cap L^{4}(%\n\\mathbb{R}^{3})$ with $\\nabla \\cdot u_{0}=\\nabla \\cdot b_{0}=0$ in the sense\nof distribution.\\ Let $\\left( u,b\\right) $ be a weak solution to the MHD\nequations on some interval $\\left[ 0,T\\right] $ with $0<T\\leq \\infty $.\\\nAssume that $0\\leq \\theta \\leq 1$ and that\n\\begin{equation}\n{\\frac{\\pi }{\\left( e^{-\\left\\vert x\\right\\vert ^{2}}+\\left\\vert\nu\\right\\vert +\\left\\vert b\\right\\vert \\right) ^{\\theta }}\\in L}%\n^{p}(0,T;L^{q,\\infty }(\\mathbb{R}^{3})),\\text{ \\ where }\\frac{2}{p}+\\frac{3}{%\nq}=2-\\theta  \\label{eq15}\n\\end{equation}%\nthen the weak $\\left( u,b\\right) $ is regular on $(0,T].$\n\\end{thm}\n\n\\begin{re}\nA special consequence of Theorem \\ref{th1} and its proof is the regularity\ncriterion of the 3D Navier-Stokes equations with the mixed pressure-velocity\nin Lorentz spaces. This generalizes those of \\cite{BY}.\n\\end{re}\n\nIn order to derive the regularity criterion of weak solutions to the MHD\nequations (\\ref{eq1.1}), we introduce the definition of weak solution.\n\nNext, let us writing\n\\begin{equation*}\nw^{\\pm }=u\\pm b,\\ \\ \\text{\\ }w_{0}^{\\pm }=u_{0}\\pm b_{0}.\n\\end{equation*}%\nWe reformulate equation (\\ref{eq1.1}) as follows. Formally, if the first\nequation of MHD equations (\\ref{eq1.1}) plus and minus the second one,\nrespectively, then MHD equations (\\ref{eq1.1}) can be re-written as:\n\\begin{equation}\n\\left\\{\n\\begin{array}{l}\n\\partial _{t}w^{+}-\\Delta w^{+}+(w^{-}\\cdot \\nabla )w^{+}+\\nabla \\pi =0, \\\\%\n[3mm]\n\\partial _{t}w^{-}-\\Delta w^{-}+(w^{+}\\cdot \\nabla )w^{-}+\\nabla \\pi =0, \\\\%\n[2mm]\n\\mathrm{div}~w^{+}=0,~~~~\\mathrm{div}~w^{-}=0, \\\\\nw^{+}(x,0)=w_{0}^{+}(x),\\text{ \\ \\ }w^{-}(x,0)=w_{0}^{-}(x).%\n\\end{array}%\n\\right.  \\label{eq1.3}\n\\end{equation}%\nThe advantage is that the equations becomes symmetric.\n\n\\section{Proof of Theorem \\protect\\ref{th1}}\n\nThis section is devoted to the proof of Theorem \\ref{th1}. In order to do\nit, we first recall the following estimates for the pressure in terms of $u$\nand $b$ (see e.g., \\cite{GR20}) :\n\\begin{equation}\n\\left\\Vert \\pi \\right\\Vert _{L^{q}}\\leq C\\left( \\left\\Vert u\\right\\Vert\n_{L^{2q}}^{2}+\\left\\Vert b\\right\\Vert _{L^{2q}}^{2}\\right) ,\\text{ \\ \\textrm{%\nwith} \\ }1<q<\\infty .  \\label{eq120}\n\\end{equation}\n\nWe are now in position to prove our main result.\n\n\\begin{pf}\nMultiplying the first and the second equations of (\\ref{eq1.3}) by $%\n\\left\\vert w^{+}\\right\\vert ^{2}w^{+}$ and $\\left\\vert w^{-}\\right\\vert\n^{2}w^{-}$ , respectively, integrating by parts and summing up, we have\n\\begin{eqnarray*}\n&&\\frac{1}{4}\\frac{d}{dt}(\\left\\Vert w^{+}\\right\\Vert\n_{L^{4}}^{4}+\\left\\Vert w^{-}\\right\\Vert _{L^{4}}^{4})+\\int_{\\mathbb{R}%\n^{3}}(\\left\\vert \\nabla w^{+}\\right\\vert ^{2}\\left\\vert w^{+}\\right\\vert\n^{2}+\\left\\vert \\nabla w^{-}\\right\\vert ^{2}\\left\\vert w^{-}\\right\\vert\n^{2})dx+\\frac{1}{2}\\int_{\\mathbb{R}^{3}}(\\left\\vert \\nabla \\left\\vert\nw^{+}\\right\\vert ^{2}\\right\\vert ^{2}+\\left\\vert \\nabla \\left\\vert\nw^{-}\\right\\vert ^{2}\\right\\vert ^{2})dx \\\\\n&=&-\\int_{\\mathbb{R}^{3}}\\nabla \\pi \\cdot (w^{+}\\left\\vert w^{+}\\right\\vert\n^{2}+w^{-}\\left\\vert w^{-}\\right\\vert ^{2})dx \\\\\n&=&\\int_{\\mathbb{R}^{3}}\\pi \\cdot \\mathrm{div}(w^{+}\\left\\vert\nw^{+}\\right\\vert ^{2}+w^{-}\\left\\vert w^{-}\\right\\vert ^{2})dx \\\\\n&\\leq &\\int_{\\mathbb{R}^{3}}\\left\\vert \\pi \\right\\vert (\\left\\vert\nw^{+}\\right\\vert +\\left\\vert w^{-}\\right\\vert )(\\nabla \\left\\vert\nw^{+}\\right\\vert ^{2}+\\nabla \\left\\vert w^{-}\\right\\vert ^{2})dx \\\\\n&\\leq &C\\int_{\\mathbb{R}^{3}}\\left\\vert \\pi \\right\\vert ^{2}(\\left\\vert\nw^{+}\\right\\vert +\\left\\vert w^{-}\\right\\vert )^{2}dx+\\frac{1}{4}\\int_{%\n\\mathbb{R}^{3}}(\\left\\vert \\nabla \\left\\vert w^{+}\\right\\vert\n^{2}\\right\\vert ^{2}+\\left\\vert \\nabla \\left\\vert w^{-}\\right\\vert\n^{2}\\right\\vert ^{2})dx.\n\\end{eqnarray*}%\nNotice that $u=\\frac{1}{2}(w^{+}+w^{-})$ and $b=\\frac{1}{2}(w^{+}-w^{-})$,\nthen the above inequality means that\n\\begin{eqnarray}\n&&\\frac{d}{dt}(\\left\\Vert u\\right\\Vert _{L^{4}}^{4}+\\left\\Vert b\\right\\Vert\n_{L^{4}}^{4})+2\\left\\Vert \\nabla \\left\\vert u\\right\\vert ^{2}\\right\\Vert\n_{L^{2}}^{2}+2\\left\\Vert \\nabla \\left\\vert b\\right\\vert ^{2}\\right\\Vert\n_{L^{2}}^{2}  \\notag \\\\\n&&+2\\left\\Vert \\left\\vert u\\right\\vert \\left\\vert \\nabla u\\right\\vert\n\\right\\Vert _{L^{2}}^{2}+2\\left\\Vert \\left\\vert b\\right\\vert \\left\\vert\n\\nabla b\\right\\vert \\right\\Vert _{L^{2}}^{2}+2\\left\\Vert \\left\\vert\nu\\right\\vert \\left\\vert \\nabla b\\right\\vert \\right\\Vert\n_{L^{2}}^{2}+2\\left\\Vert \\left\\vert b\\right\\vert \\left\\vert \\nabla\nu\\right\\vert \\right\\Vert _{L^{2}}^{2}  \\notag \\\\\n&\\leq &C\\int_{\\mathbb{R}^{3}}\\left\\vert \\pi \\right\\vert ^{2}(\\left\\vert\nu\\right\\vert +\\left\\vert b\\right\\vert )^{2}dx=K,  \\label{eq21}\n\\end{eqnarray}%\nwhere we have used\n\\begin{equation*}\n\\left\\vert w^{+}\\right\\vert +\\left\\vert w^{-}\\right\\vert \\leq \\left\\vert\nw^{+}+w^{-}\\right\\vert +\\left\\vert w^{+}-w^{-}\\right\\vert .\n\\end{equation*}%\nFor $K$, borrowing the arguments in \\cite{BY}, we set%\n\\begin{equation*}\nV=e^{-\\left\\vert x\\right\\vert ^{2}}+\\left\\vert u\\right\\vert +\\left\\vert\nb\\right\\vert \\text{ \\ \\ and \\ \\ }\\widetilde{\\pi }={\\frac{\\pi }{\\left(\ne^{-\\left\\vert x\\right\\vert ^{2}}+\\left\\vert u\\right\\vert +\\left\\vert\nb\\right\\vert \\right) ^{\\theta }}.}\n\\end{equation*}%\nBy the H\\\"{o}lder inequality and the following interpolation in Lorentz\nspace (see \\cite{Tri})\n\\begin{equation*}\n\\left\\Vert f^{\\alpha }\\right\\Vert _{L^{p,q}(\\mathbb{R}^{3})}\\leq C\\left\\Vert\nf\\right\\Vert _{L^{\\alpha p,\\alpha q}(\\mathbb{R}^{3})}^{\\alpha }\\text{ \\ \\\nfor \\ }\\alpha >0,\\text{ }p>0,\\text{ }q>0,\n\\end{equation*}\nwe have%\n\\begin{eqnarray*}\nK &=&\\int_{\\mathbb{R}^{3}}\\left\\vert \\pi \\right\\vert ^{\\lambda }V^{-\\lambda\n\\theta }\\left\\vert \\pi \\right\\vert ^{2-\\lambda }V^{\\lambda \\theta\n}(\\left\\vert u\\right\\vert +\\left\\vert b\\right\\vert )^{2}dx \\\\\n&\\leq &\\int_{\\mathbb{R}^{3}}\\left\\vert \\widetilde{\\pi }\\right\\vert ^{\\lambda\n}\\left\\vert \\pi \\right\\vert ^{2-\\lambda }V^{2+\\lambda \\theta }dx \\\\\n&\\leq &\\left\\Vert \\left\\vert \\widetilde{\\pi }\\right\\vert ^{\\lambda\n}\\right\\Vert _{L^{\\frac{q}{\\lambda },\\infty }}\\left\\Vert \\left\\vert \\pi\n\\right\\vert ^{2-\\lambda }\\right\\Vert _{L^{s,\\frac{2}{2-\\lambda }}}\\left\\Vert\nV^{2\\lambda }\\right\\Vert _{L^{r,\\frac{2}{\\lambda }}} \\\\\n&=&\\left\\Vert \\widetilde{\\pi }\\right\\Vert _{L^{q,\\infty }}^{\\lambda\n}\\left\\Vert \\pi \\right\\Vert _{L^{s(2-\\lambda ),2}}^{2-\\lambda }\\left\\Vert\nV^{2}\\right\\Vert _{L^{\\lambda r,2}}^{\\lambda },\n\\end{eqnarray*}%\nwhere\n\\begin{equation*}\n\\frac{\\lambda }{q}+\\frac{1}{s}+\\frac{1}{r}=1\\text{ \\ and \\ }\\lambda =\\frac{2%\n}{2-\\theta }.\n\\end{equation*}%\nBy (\\ref{eq120}), we have%\n\\begin{eqnarray*}\nK &\\leq &\\left\\Vert \\widetilde{\\pi }\\right\\Vert _{L^{q,\\infty }}^{\\lambda\n}\\left( \\left\\Vert \\left\\vert u\\right\\vert ^{2}\\right\\Vert _{L^{s(2-\\lambda\n),2}}+\\left\\Vert \\left\\vert b\\right\\vert ^{2}\\right\\Vert _{L^{s(2-\\lambda\n),2}}\\right) ^{2-\\lambda }\\left\\Vert V^{2}\\right\\Vert _{L^{\\lambda\nr,2}}^{\\lambda } \\\\\n&\\leq &C\\left\\Vert \\widetilde{\\pi }\\right\\Vert _{L^{q,\\infty }}^{\\lambda\n}\\left\\Vert V^{2}\\right\\Vert _{L^{s(2-\\lambda ),2}}^{2-\\lambda }\\left\\Vert\nV^{2}\\right\\Vert _{L^{\\lambda r,2}}^{\\lambda }.\n\\end{eqnarray*}%\nBy the interpolation and Sobolev inequalities in Lorentz spaces, it follows\nthat%\n\\begin{equation}\n\\left\\{\n\\begin{array}{c}\n\\left\\Vert V^{2}\\right\\Vert _{L^{s(2-\\lambda ),2}}\\leq C\\left\\Vert\nV^{2}\\right\\Vert _{L^{2,2}}^{1-\\delta _{1}}\\left\\Vert V^{2}\\right\\Vert\n_{L^{6,2}}^{\\delta _{1}}\\leq C\\left\\Vert V^{2}\\right\\Vert _{L^{2}}^{1-\\delta\n_{1}}\\left\\Vert \\nabla V^{2}\\right\\Vert _{L^{2}}^{\\delta _{1}}, \\\\\n\\left\\Vert V^{2}\\right\\Vert _{L^{\\lambda r,2}}\\leq C\\left\\Vert\nV^{2}\\right\\Vert _{L^{2,2}}^{1-\\delta _{2}}\\left\\Vert V^{2}\\right\\Vert\n_{L^{6,2}}^{\\delta _{2}}\\leq C\\left\\Vert V^{2}\\right\\Vert _{L^{2}}^{1-\\delta\n_{2}}\\left\\Vert \\nabla V^{2}\\right\\Vert _{L^{2}}^{\\delta _{2}},%\n\\end{array}%\n\\right.   \\label{eq6.6}\n\\end{equation}%\nwhere $0<\\delta _{1},\\delta _{2}<1$ and\n\\begin{equation*}\n\\frac{1}{s(2-\\lambda )}=\\frac{1-\\delta _{1}}{2}+\\frac{\\delta _{1}}{6},\\text{\n\\ }\\frac{1}{\\lambda r}=\\frac{1-\\delta _{2}}{2}+\\frac{\\delta _{2}}{6}.\n\\end{equation*}%\nHence from (\\ref{eq6.6}) and Young inequality, it follows that%\n\\begin{eqnarray*}\nK &\\leq &C\\left\\Vert \\widetilde{\\pi }\\right\\Vert _{L^{q,\\infty }}^{\\lambda\n}\\left\\Vert V^{2}\\right\\Vert _{L^{2}}^{(2-\\lambda )(1-\\delta _{1})+\\lambda\n(1-\\delta _{2})}\\left\\Vert \\nabla V^{2}\\right\\Vert _{L^{2}}^{(2-\\lambda\n)\\delta _{1}+\\lambda \\delta _{2}} \\\\\n&\\leq &C\\left\\Vert \\widetilde{\\pi }\\right\\Vert _{L^{q,\\infty }}^{\\frac{%\n2\\lambda }{2-(2-\\lambda )\\delta _{1}-\\lambda \\delta _{2}}}\\left\\Vert\nV^{2}\\right\\Vert _{L^{2}}^{2}+\\frac{1}{2}\\left\\Vert \\nabla V^{2}\\right\\Vert\n_{L^{2}}^{2}.\n\\end{eqnarray*}%\nDue to the definition of $V$, we see that\n\\begin{equation*}\n\\left\\Vert V^{2}\\right\\Vert _{L^{2}}^{2}\\leq C(1+\\left\\Vert \\left\\vert\nu\\right\\vert +\\left\\vert b\\right\\vert \\right\\Vert _{L^{2}}^{2}+\\left\\Vert\n\\left\\vert u\\right\\vert ^{2}+\\left\\vert b\\right\\vert ^{2}\\right\\Vert\n_{L^{2}}^{2}),\n\\end{equation*}%\nand%\n\\begin{equation*}\n\\left\\Vert \\nabla V^{2}\\right\\Vert _{L^{2}}^{2}\\leq C(1+\\left\\Vert\n\\left\\vert u\\right\\vert +\\left\\vert b\\right\\vert \\right\\Vert\n_{L^{2}}^{2}+\\left\\Vert \\nabla (\\left\\vert u\\right\\vert +\\left\\vert\nb\\right\\vert )\\right\\Vert _{L^{2}}^{2}+\\left\\Vert \\nabla (\\left\\vert\nu\\right\\vert ^{2}+\\left\\vert b\\right\\vert ^{2})\\right\\Vert _{L^{2}}^{2}).\n\\end{equation*}%\nConsequently, we get%\n\\begin{eqnarray*}\nK &\\leq &C\\left\\Vert \\widetilde{\\pi }\\right\\Vert _{L^{q,\\infty }}^{\\frac{%\n2\\lambda }{2-(2-\\lambda )\\delta _{1}-\\lambda \\delta _{2}}}(1+\\left\\Vert\n\\left\\vert u\\right\\vert +\\left\\vert b\\right\\vert \\right\\Vert\n_{L^{2}}^{2}+\\left\\Vert \\left\\vert u\\right\\vert ^{2}+\\left\\vert b\\right\\vert\n^{2}\\right\\Vert _{L^{2}}^{2}) \\\\\n&&+C(1+\\left\\Vert \\left\\vert u\\right\\vert +\\left\\vert b\\right\\vert\n\\right\\Vert _{L^{2}}^{2}+\\left\\Vert \\nabla (\\left\\vert u\\right\\vert\n+\\left\\vert b\\right\\vert )\\right\\Vert _{L^{2}}^{2})+\\frac{1}{2}\\left\\Vert\n\\nabla (\\left\\vert u\\right\\vert ^{2}+\\left\\vert b\\right\\vert\n^{2})\\right\\Vert _{L^{2}}^{2} \\\\\n&\\leq &C\\left\\Vert \\widetilde{\\pi }\\right\\Vert _{L^{q,\\infty }}^{\\frac{%\n2\\lambda }{2-(2-\\lambda )\\delta _{1}-\\lambda \\delta _{2}}}(1+\\left\\Vert\nu\\right\\Vert _{L^{2}}^{2}+\\left\\Vert b\\right\\Vert _{L^{2}}^{2}+\\left\\Vert\nu\\right\\Vert _{L^{4}}^{4}+\\left\\Vert b\\right\\Vert _{L^{4}}^{4}) \\\\\n&&+C(1+\\left\\Vert u\\right\\Vert _{L^{2}}^{2}+\\left\\Vert b\\right\\Vert\n_{L^{2}}^{2}+\\left\\Vert \\nabla u\\right\\Vert _{L^{2}}^{2}+\\left\\Vert \\nabla\nb\\right\\Vert _{L^{2}}^{2})+\\frac{1}{2}\\left\\Vert \\nabla \\left\\vert\nu\\right\\vert ^{2}\\right\\Vert _{L^{2}}^{2}+\\frac{1}{2}\\left\\Vert \\nabla\n\\left\\vert b\\right\\vert ^{2}\\right\\Vert _{L^{2}}^{2}.\n\\end{eqnarray*}%\nSince $(u,b)$ is a weak solution to (\\ref{eq1.1}), then $(u,b)$ satisfies%\n\\begin{equation*}\n(u,b)\\in L^{\\infty }(0,T;L^{2}(\\mathbb{R}^{3}))\\cap L^{2}(0,T;H^{1}(\\mathbb{R%\n}^{3})).\n\\end{equation*}%\nInserting the above estimates into (\\ref{eq21}), we obtain\n\\begin{eqnarray*}\n&&\\frac{d}{dt}(\\left\\Vert u\\right\\Vert _{L^{4}}^{4}+\\left\\Vert b\\right\\Vert\n_{L^{4}}^{4})+\\left\\Vert \\nabla \\left\\vert u\\right\\vert ^{2}\\right\\Vert\n_{L^{2}}^{2}+\\left\\Vert \\nabla \\left\\vert b\\right\\vert ^{2}\\right\\Vert\n_{L^{2}}^{2} \\\\\n&&+2\\left\\Vert \\left\\vert u\\right\\vert \\left\\vert \\nabla u\\right\\vert\n\\right\\Vert _{L^{2}}^{2}+2\\left\\Vert \\left\\vert b\\right\\vert \\left\\vert\n\\nabla b\\right\\vert \\right\\Vert _{L^{2}}^{2}+2\\left\\Vert \\left\\vert\nu\\right\\vert \\left\\vert \\nabla b\\right\\vert \\right\\Vert\n_{L^{2}}^{2}+2\\left\\Vert \\left\\vert b\\right\\vert \\left\\vert \\nabla\nu\\right\\vert \\right\\Vert _{L^{2}}^{2} \\\\\n&\\leq &C\\left\\Vert \\widetilde{\\pi }\\right\\Vert _{L^{q,\\infty }}^{\\frac{%\n2\\lambda }{2-(2-\\lambda )\\delta _{1}-\\lambda \\delta _{2}}}(1+\\left\\Vert\nu\\right\\Vert _{L^{2}}^{2}+\\left\\Vert b\\right\\Vert _{L^{2}}^{2}+\\left\\Vert\nu\\right\\Vert _{L^{4}}^{4}+\\left\\Vert b\\right\\Vert _{L^{4}}^{4}) \\\\\n&&+C(1+\\left\\Vert u\\right\\Vert _{L^{2}}^{2}+\\left\\Vert b\\right\\Vert\n_{L^{2}}^{2}+\\left\\Vert \\nabla u\\right\\Vert _{L^{2}}^{2}+\\left\\Vert \\nabla\nb\\right\\Vert _{L^{2}}^{2}) \\\\\n&\\leq &C\\left\\Vert \\widetilde{\\pi }\\right\\Vert _{L^{q,\\infty }}^{\\frac{%\n2\\lambda }{2-(2-\\lambda )\\delta _{1}-\\lambda \\delta _{2}}}(1+\\left\\Vert\nu\\right\\Vert _{L^{4}}^{4}+\\left\\Vert b\\right\\Vert\n_{L^{4}}^{4})+C(1+\\left\\Vert \\nabla u\\right\\Vert _{L^{2}}^{2}+\\left\\Vert\n\\nabla b\\right\\Vert _{L^{2}}^{2}),\n\\end{eqnarray*}%\nUsing Gronwall's inequality with the assumption (\\ref{eq15}), we deduce that%\n\\begin{equation*}\n(u,b)\\in L^{\\infty }(0,T;L^{4}(\\mathbb{R}^{3}))\\subset L^{8}(0,T;L^{4}(%\n\\mathbb{R}^{3})).\n\\end{equation*}%\nWe complete the proof of Theorem \\ref{th1}.\n\\end{pf}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec1}\n Recently, CDF-II collaboration published a new result in  W boson mass with increased precision  $M^W_{CDF} =80.4335 \\pm 0.0094 ~\\rm{GeV}$ which deviates from SM prediction by $7\\sigma$\\cite{CDF:2022hxs}. The SM prediction for the W boson mass is $ M^W_{SM}=80.357 \\pm 0.006 ~\\rm {GeV}$ \\cite{ParticleDataGroup:2020ssz}.\nNeedless to say a better understanding of SM calculations, and also more accurate  measurements are needed. Nevertheless, a lot of new suggestions have been proposed to explain  this anomaly \\cite{Ghorbani:2022vtv,Cheng:2022aau,Borah:2022zim,Arcadi:2022dmt,Nagao:2022oin,Kawamura:2022uft,Fan:2022dck,Lu:2022bgw,Athron:2022qpo,Yuan:2022cpw,Strumia:2022qkt,Yang:2022gvz,deBlas:2022hdk,Du:2022pbp,Tang:2022pxh,Cacciapaglia:2022xih,Blennow:2022yfm,Sakurai:2022hwh,Fan:2022yly,Liu:2022jdq,Lee:2022nqz,Bagnaschi:2022whn,Paul:2022dds,Bahl:2022xzi,Asadi:2022xiy,DiLuzio:2022xns,Athron:2022isz,Gu:2022htv,Babu:2022pdn,Heo:2022dey,Du:2022brr,Cheung:2022zsb,Crivellin:2022fdf,Endo:2022kiw,Biekotter:2022abc,Balkin:2022glu,Han:2022juu,Ahn:2022xeq,Zheng:2022irz,Ghoshal:2022vzo,FileviezPerez:2022lxp,Mondal:2022xdy,Borah:2022obi,Chowdhury:2022moc}. In\\cite{Zhang:2022nnh}, the extra $U(1)$ gauge field mix with the $U(1$) hypercharge via gauge kinetic term,  and this kinetic mixing can generate an enhancement of the W boson mass. Another approach for the $U(1)$ extension of SM is to consider an additional scalar field in which the new field can enhance W-boson mass via the loop corrections.\n\nIt is known that the SM is an incomplete theory in part  due to its inability to explain phenomena such as DM, neutrino mass, and the baryon asymmetry of the Universe. In this paper, we consider a $U(1)$ extension of SM in which the vector field does not mix with the SM gauge field and consider the scalar field  that plays the role of mediator between the dark sector and SM. This field shift W-boson mass in loop corrections. We study this simple extension of the SM to explain the W boson mass enhancement and also offer a viable DM candidate with mass ranging from $1~\\rm GeV$ to $2~\\rm TeV$. In the following, we apply various phenomenological constraints such as invisible Higgs decay mode and direct detection experiment in our analysis.\n\n\nIn the context of SM, the vacuum stability and perturbativity have resulted in theoretical bounds on the Higgs mass. The Higgs mass value, together with other relevant parameters such as the top quark mass, affects on the behaviour of  Higgs potential at very high energy scales, in particular for the sake of electroweak vacuum stability \\cite{Bezrukov:2012sa},\\cite{Buttazzo:2013uya}.\nThis is because, for Higgs and top mass values, the Higgs quartic coupling can be very small or even negative. Since the top Yukawa coupling dependency is strong and very subtle, there are different views on an explanation of this issue in literature, some of them favouring\\cite{Bezrukov:2012sa} and some others disfavouring \\cite{Degrassi:2012ry}. In the following, we also ask how the model behaves at a high energy scale and is it computationally reliable?  Standard treatment is the study of the running of the coupling constants in terms of the mass scale $\\Lambda$ via the RGE. The three standard problems to consider are the positivity, perturbativity of the coupling constants and the vacuum stability  of the model. These issues have been studied in the literature in the presence of a scalar extension of SM \\cite{Ghorbani:2017qwf,Gonderinger:2012rd} and it was shown that the vacuum stability requirement can affect the DM relic density. Here, we discuss the requirement of vacuum stability and RGE of the model.\n\nThis work is organized as follows: After the introduction, we introduce the model.\nIn section~3, we study conditions of vacuum stability and calculate the RGEs of the model. In section~4,  we study the contribution of the model to W-boson mass and probe the consistent parameter space of the model with CDF-II measurement. Invisible Higgs decay constraint on the model study in section.~5. In section.~6, we find the allowed regions in the parameter space which will give rise to the correct DM relic density. We describe combined results  consistent with constraints provided and examine the RGEs numerically in section~7 . Section 8 contains our conclusions.\n\n\n\n\n\n\\section{The Model} \\label{sec2}\nIn our model, beyond the SM, we employ two new fields to furnish the model: a  scalar field $S$ which has a unit charge under a dark $U(1)$ gauge symmetry with a dark photon vector field $ V_{\\mu} $. The model has a $Z_2$ discrete flavor symmetry, under which $V_{\\mu}$ is odd and all the other fields are even.  $Z_2$ symmetry forbids the kinetic mixing between the vector field $ V_{\\mu} $ and SM $ U_{Y}(1) $ gauge boson $ B_{\\mu} $, i.e., $ V_{\\mu \\nu} B_{\\mu \\nu} $. Therefore,  the vector field $ V_{\\mu} $ is stable and can be considered  a DM candidate. The Lagrangian one can write with assumption is:\n\\begin{equation}\n {\\cal L} ={\\cal L}_{SM} + (D'_{\\mu} S)^{*} (D'^{\\mu} S) - V(H,S) - \\frac{1}{4} V_{\\mu \\nu} V^{\\mu \\nu} , \\label{2-2}\n\\end{equation}\nwhere $ {\\cal L} _{SM} $ is the SM Lagrangian without the Higgs potential term and\n\\begin{align}\n& D'_{\\mu} S= (\\partial_{\\mu} + i g_v V_{\\mu}) S,\\nonumber \\\\\n& V_{\\mu \\nu}= \\partial_{\\mu} V_{\\nu} - \\partial_{\\nu} V_{\\mu},\\nonumber \\end{align}\nand the potential which is renormalizable and invariant\nunder gauge and $ Z_{2} $ symmetry is:\n\\begin{equation}\nV(H,S) = -\\mu_{H}^2 H^{\\dagger}H-\\mu_{S}^2 S^*S+\\lambda_{H} (H^{\\dagger}H)^{2} + \\lambda_{S} (S^*S)^{2} +  \\lambda_{S H} (S^*S) (H^{\\dagger}H). \\label{2-3}\n\\end{equation}\nNote that the quartic portal interaction, $ \\lambda_{SH} (S^*S) (H^{\\dagger}H) $, is the only connection between the dark sector and the SM.\n\nSM Higgs field $ H $, as well as dark scalar $S$, can receive VEVs breaking respectively the electroweak and $ U'_{D}(1) $ symmetries.\nIn the unitary gauge, the imaginary component of $S$ can be absorbed as the longitudinal component of $ V_{\\mu} $.\nIn this gauge, we can write\n\\begin{equation}\nH = \\frac{1}{\\sqrt{2}} \\begin{pmatrix}\n0 \\\\ h_{1} \\end{pmatrix} \\, \\, \\, {\\rm and} \\, \\, \\, S = \\frac{1}{\\sqrt{2}} h_{2} , \\label{2-4}\n\\end{equation}\nwhere $ h_{1} $ and $ h_{2} $ are real scalar fields which can get VEVs.\nThe tree level potential in unitary gauge is:\n\\begin{equation}\nV_{\\text{tree}}(h_{1},h_{2})=-\\frac{1}{2} \\mu _H^2 h_1^2-\\frac{1}{2} \\mu _S^2 h_2^2 +\\frac{1}{4} \\lambda _H h_1^4 +\\frac{1}{4} \\lambda _S h_2^4+\\frac{1}{4} \\lambda _{SH} h_1^2 h_2^2.\n\\end{equation}\nGiven differentiable $ V_{\\text{tree}} $, one can obtain\nthe Hessian matrix, $ {\\cal{H}}_{ij}(h_{1},h_{2})=\\frac{\\partial^{2}V_{\\text{tree}}}{\\partial h_{i} \\partial h_{j} }$.\nIn order to get the mass spectrum of the model, it is necessary to consider the sufficient conditions for a local minimum:\n\\begin{align}\n& \\nabla V_{\\text{tree}} = 0 \\label{minimum1} \\\\\n&  \\det {\\cal{H}} > 0 \\label{minimum2} \\\\\n& {\\cal{H}}_{11} > 0 \\label{minimum3}\n\\end{align}\nto occur at a point $ (\\nu_{1},\\nu_{2}) $. Note that Eq. (\\ref{minimum2}) and Eq. (\\ref{minimum3}) also imply that $ {\\cal{H}}_{22} > 0 $.\nEq. (\\ref{minimum1}) leads to\n\\begin{align}\n& \\mu _H^2= \\lambda _H \\nu _1^2+ \\frac{1}{2} \\lambda _{SH} \\nu _2^2 , \\nonumber\\\\\n& \\mu _S^2=\\lambda _S \\nu _2^2+ \\frac{1}{2} \\lambda _{SH} \\nu _1^2  \\label{mini}\n\\end{align}\nEq. (\\ref{mini}) leads to the non-diagonal mass matrix $\\cal{H}$ as follows:\n\\begin{equation}\n{\\cal{H}}(\\nu_{1},\\nu_{2})= \\left(\n\\begin{array}{cc}\n 2 \\lambda _H \\nu _1^2 & \\lambda _{SH} \\nu _1 \\nu _2 \\\\\n \\lambda _{SH} \\nu _1 \\nu _2 & 2 \\lambda _S \\nu _2^2 \\\\\n\\end{array}\n\\right) \\label{hess}\n\\end{equation}\nTherefore, according to the conditions $ {\\cal{H}}_{11} > 0 $ and $ {\\cal{H}}_{22} > 0 $ and Eq. (\\ref{minimum2}) we should have\n\\begin{equation}\n\\lambda_{H} > 0 \\, \\, \\, , \\, \\, \\,  \\lambda_{S} > 0 \\, \\, \\, , \\, \\, \\, \\lambda_{SH}^{2} < 4 \\lambda_{H} \\lambda_{S}\n\\end{equation}\nNow by substituting $ h_1 \\rightarrow \\nu_1 + h_1 $ and $ h_2 \\rightarrow \\nu_2 + h_2 $, the fields $ h_1 $ and $ h_1 $ mix with each other and they can be rewritten by the mass eigenstates $ H_1 $ and $ H_1 $ as\n\\begin{equation}\n\\begin{pmatrix}\nh_{1}\\\\h_{2}\\end{pmatrix}\n =\\begin{pmatrix} cos \\alpha~~~  sin \\alpha \\\\-sin \\alpha  ~~~~~cos \\alpha\n \\end{pmatrix}\\begin{pmatrix}\nH_1 \\\\  H_{2}\n\\end{pmatrix}, \\label{matri}\n\\end{equation}\nwhere $ \\alpha $ is the mixing angle. After symmetry breaking, we have\n\\begin{align}\n& \\nu _2=\\frac{M_V}{g_v} \\nonumber,~~~~~~~~~~ \\sin\\alpha=\\frac{\\nu_1}{\\sqrt{\\nu _1^2+\\nu_2^2}} \\\\\n& \\lambda _H=\\frac{\\cos ^2\\alpha M_{H_1}^2+\\sin ^2\\alpha  M_{H_2}^2}{2 \\nu _1^2}  \\nonumber \\\\\n& \\lambda _S=\\frac{\\sin ^2\\alpha M_{H_1}^2+\\cos ^2\\alpha  M_{H_2}^2}{2 \\nu _2^2}  \\nonumber \\\\\n&  \\lambda _{SH}=\\frac{ \\left(M_{H_2}^2-M_{H_1}^2\\right) \\sin \\alpha  \\cos \\alpha}{\\nu _1 \\nu _2} \\label{cons}\n\\end{align}\nThe mass eigenstates of scalar fields can be written as following:\n\\begin{align}\nM^2_{H_{2},H_{1}}=\\lambda_H \\nu_1^2+\\lambda_S \\nu_2^2 \\pm \\sqrt{(\\lambda_H \\nu_1^2-\\lambda_S \\nu_2^2)^2+\\lambda_{SH}^2\\nu_1^2\\nu_2^2},\n\\label{mas}\n\\end{align}\nwhere we take $ M_{H_1} = 125 $ GeV and $ \\nu _1 = 246 $ GeV. Note that, beside of SM parameters, the model has only three free parameters $ g_v $, $ M_{H_2} $ and $ M_V $.\n\n\\section{Vacuum stability and RGE}\n\nA prominent feature of the study of high energy physics is the evolution of the coupling constants with energy. This has become an incentive to further strengthen theories such as the GUT and supersymmetry by merging couplings at high energies\\cite{Wulzer:2019max}. Renormalization Group Equation(RGE) describes the behavior of quantities with energy. After the discovery of the Higgs particle by ATLAS and CMS experiments at the LHC in 2012, the vacuum stability study has been done more clearly\\cite{Ghorbani:2017qwf,Gonderinger:2012rd,Abada:2013pca,Baek:2012uj,Ghorbani:2021rgs,Duch:2015jta}. In the SM, the Higgs quartic coupling becomes negative at the scale $10^{10}$ GeV, and the Higgs non-zero VEV is no longer a minimum of the theory. The reason for this is that the top quark has a large negative contribution to the RGE for $\\lambda_{H}$.  We  will show the running of $\\lambda_{H}$ in SM in the next sections.\n\n\n\nThere are three types of theoretical constraints on the couplings in the model. The first one is related to the perturbativity condition for couplings, which is satisfied when $\\lambda_i< 4\\pi$. The second is vacuum stability of the model dictates some other constraints on the couplings such that for self-coupling constants. In this regard, we should have $\\lambda_i$ and $\\lambda_H>0$. On the other hand, by adding a new scalar mediator the vacuum structure of the model will be modified. The third condition is positivity where potential must be well-defined and positive at all scales. The requirement of positivity for the potential implies the following relations :\n\\begin{equation}\n\\lambda_H>0   ,  \\lambda_S>0  , \\lambda_{SH}>-2 \\sqrt{\\lambda_H \\lambda_S }\n\\end{equation}\nAlso, the common investigation for vacuum stability analyses in the literature begins with the RGE improved  potential and choice of the renormalization scale to minimize  the one-loop potential. In this light, we  consider the running of the coupling constants with energy. The Model is implemented in SARAH \\cite{Staub:2015kfa} to compute $\\beta$ functions and their runnings. We calculate the one-loop RGE and one-loop $\\beta$ functions for scalar couplings and dark coupling including the following relationships:\n\n\\begin{align}\n& (16\\pi^2)\\beta_{\\lambda_{S}} = -20\\lambda_{S}^2 -2\\lambda_{SH}^2 -6g_v^4 -12g_v^2 \\lambda_{S} ,\\nonumber \\\\\n& (16\\pi^2)\\beta_{\\lambda_{SH}}= -\\frac{3}{2}g_{1}^2 \\lambda_{SH} -\\frac{9}{2}g_{2}^2 \\lambda_{SH} -12\\lambda_{SH}\\lambda_{H} -8\\lambda_{SH}\\lambda_{S} -4\\lambda_{SH}^2 + 6\\lambda_{SH}\\lambda_{t}^2 -6g_v^2 \\lambda_{SH}  \\nonumber ,\\\\\n& (16\\pi^2)\\beta_{\\lambda_{H}}= -\\frac{3}{8}g_{1}^4 -\\frac{3}{4}g_{1}^2 g_{2}^2 -\\frac{9}{8}g_{2}^4 -3g_{1}^2 \\lambda_{H} -9g_{2}^2 \\lambda_{H} -24\\lambda_{H}^2 -\\lambda_{SH}^2 +12\\lambda_{H}\\lambda_{t}^2 +6\\lambda_{t}^4, \\nonumber \\\\\n& (16\\pi^2)\\beta_{g_v}= \\frac{1}{3}g_v^3.\n\\label{RGE}\n\\end{align}\nwhere $\\beta_a\\equiv \\mu \\frac{da}{d\\mu}$ that $\\mu$ is the renormalization scale with initial value $\\mu_0=100~\\rm GeV$. The $\\beta$ functions of couplings, $g_1$, $g_2$ and $g_3$ are given to one-loop order by:\n\\begin{align}\n& (16\\pi^2)\\beta_{g_{1}}= \\frac{41}{6} g_{1}^3 , \\nonumber \\\\\n& (16\\pi^2)\\beta_{g_{2}}= -\\frac{19}{6} g_{2}^3 , \\nonumber \\\\\n& (16\\pi^2)\\beta_{g_{3}}= -7 g_{3}^3 .\n\\end{align}\nAmong the Yukawa couplings of SM, the top quark has the largest contribution compared with other fermions in the SM. Therefore, we set all the SM Yukawa couplings equal to zero and consider only the top quark coupling. The RGE of top quark Yukawa coupling is given to one-loop order by\n\\begin{align}\n(16\\pi^2)\\beta_{\\lambda_t}= -\\frac{17}{12}g_{1}^2 \\lambda_t -\\frac{9}{4}g_2^2 \\lambda_{t} -8g_3^2 \\lambda_t +\\frac{9}{2}\\lambda_t^3 .\n\\end{align}\n\nIn the following, we first study experimental constraints on the model, and then in the final section, we investigate conditions of the Higgs stability and RGE of coupling parameters of the model.\n\n\n\n\n\n\\section{$\\rm W$-Mass Anomaly}\nTo explain the CDF-II anomaly, we study W-mass correction in the context of the model. The corrections of new physics to the W-boson mass can be written in terms of the Peskin-Takeuchi oblique parameters $S, T,$ and $U$. The Peskin\u2013Takeuchi parameters are only sensitive to new physics that contribute to the oblique corrections, i.e., the vacuum polarization corrections to four-fermion scattering processes. In general, the SM contribution to an oblique parameter  is subtracted from the new physics contribution  to define the oblique parameter. The effects of $S, T,$ and $U$ on $W$ boson mass can be expressed as follows\n \\cite{Peskin:1991sw}:\n\\begin{equation}\n\\Delta M_W^2 = \\frac{ M_{SM}^2}{c_W^2 - s_W^2} \\left(\n-\\frac{ \\alpha S}{2}  + c_W^2  \\alpha T +\\frac{c_W^2-s_W^2}{4s_W^2}   \\alpha U \\right)\\,.\n\\label{WMco}\n\\end{equation}\nwhere $M_{SM}$ is SM of W-boson mass, $c_W$ and $s_W$ are cosine and sine of Weinberg angle. Oblique parameters $S, T,$ and $U$ for our model are as follows \\cite{Grimus:2008nb} :\n\\begin{equation}\n\\alpha S=\\frac{g^2 sin ^2\\alpha}{96 {\\pi}^2 }[(ln {M_{H_2}}^2 + G( {M_{H_2}}^2 ,{M_{Z}}^2 ))-(ln {M_{H_1}}^2 + G({M_{H_1}}^2 ,{M_{Z}}^2 )) ] ,\n\\end{equation}\n\\begin{equation}\n\\alpha T=\\frac{3g^2 sin ^2\\alpha}{64\\pi^2 s_W^2 M_W^2 }[(F(M_Z^2 ,M^2_{H_2}) - F(M_W^2 ,M_{H_2}^2)) - (F(M_Z^2 ,M_{H1}^2)-F(M_W^2 ,M_{H_1}^2))] ,\n\\end{equation}\n\\begin{equation}\n\\alpha U=\\frac{g^2  sin ^2\\alpha}{96\\pi^2 }[(G( M_{H_2}^2 ,M_W^2 )-G( M_{H_2}^2 ,M_{Z}^2 ))-(G(M_{H_1}^2 ,M_{W}^2 )-G(M_{H_1}^2 ,M_{Z}^2))] ,\n\\end{equation}\nwhere\n\\begin{equation}\ng^2= 4\\pi \\alpha_{QED}\n\\end{equation}\n\\begin{eqnarray}\nF(x,y)=\\bigg\\lbrace \\begin{array}{cc}\n{\\frac{x+y}{2}-\\frac{xy}{x-y}ln\\frac{x}{y}}&{for~x\\neq y ,}\\\\{0}&{for~ x=y ,}\n\\end{array} ,\n\\end{eqnarray}\n\\begin{equation}\nG(x,y)=-\\frac{79}{3}+ 9\\frac{x}{y} -2\\frac{x^2}{y^2} +(-10+18\\frac{x}{y}-6\\frac{x^2}{y^2}+\\frac{x^3}{y^3}-9\\frac{x+y}{x-y}) ln\\frac{x}{y} +(12-4\\frac{x}{y}+\\frac{x^2}{y^2}) \\frac{f(x,x^2 -4xy)}{y} ,\n\\end{equation}\n\\begin{eqnarray}\nf(a,b)=\\bigg\\lbrace \\begin{array}{cc}\n{\\sqrt b ln\\vert \\frac{a-\\sqrt b}{a+\\sqrt b}\\vert}&{for~ b>0}\\\\{0}&{for~ b=0}\\\\{2\\sqrt{-b}~ Arctan \\frac{\\sqrt{-b}}{a}}&{for~ b<0} .\n\\end{array} ,\n\\end{eqnarray}\n\n\n\n\\begin{figure\n\t\\begin{center}\n\t\t\\centerline{\\hspace{0cm}\\epsfig{figure=ScaterWmass.eps,width=12cm}}\n\t\t\\centerline{\\vspace{-0.2cm}}\n\t\t\\caption{Scatter points depict allowed range of parameters space of the model consistent with W-boson mass measurement.} \\label{Wmass}\n\t\\end{center}\n\\end{figure}\n\n To study the model parameter space, other than theoretical constraints (such as perturbativity condition and vacuum stability), it is necessary to consider the experimental upper limit on mixing angle $\\alpha$. For low masses, $M_{H_2} < 5~\\rm GeV$, the strongest limit comes from  decay $B\\rightarrow K\\ell\\ell$\\cite{LHCb:2012juf,Belle:2009zue}. It was shown that for this range of parameters, $\\sin\\alpha$ should be smaller than $10^{-3}$. Between $5-12~\\rm GeV$, the constraint on $sin \\alpha< 0.5$ is imposed by the decay of a low-mass Higgs boson in radiative decay of the Y and the DELPHI searches for a light Higgs in Z-decay\\cite{DELPHI:1990vtb,BaBar:2012wey}. For above this mass range  to about $65~\\rm GeV$, an overall result of the Higgs signal strength measured by ATLAS and CMS \\cite{ATLAS:2016neq} severely constrains the mixing angle  to values smaller than $sin\\alpha<0.12$. For the larger values of $M_{H_2}$, the lower limit is set by the LHC constraints on the mixing angle in which $sin \\alpha\\leq0.44$\\cite{Farzinnia:2013pga,Farzinnia:2014xia}. In the following, for low mass $M_{H_2}$ in which decay of SM Higgs like to $H_2$ is kinematically  possible, we consider severe upper limit on mixing angle and for large $M_{H_2}$ mass, we relax this bound. As was discussed in the previous section, the present model has three free parameters, $ g_v $, $ M_{H_2} $ and $ M_V$.  In addition to the mixing angle constraints, we also make the following choices for the mass parameters:\n \\begin{itemize}\n \t\\item The DM mass $M_V$ is between $1-2000~\\rm GeV$;\n \t\\item The mediator scalar mass ($M_{H_2}$) is between $1-500~\\rm GeV$;\n \\end{itemize}\nWe scan over the three-dimensional parameters $ g_v $, $ M_{H_2} $, and $ M_V $ to probe a consistent range of parameters space with observables.\n  In figure.~\\ref{Wmass}, we depict the allowed range of parameters of the model which is consistent with CDF measurement for W-boson mass. Note that, for a large value of $M_{H_2}$, we choose $sin\\alpha\\leq0.44$ on the mixing angle, and for $M_{H_2}<65~\\rm GeV$, we suppose $\\alpha<6.9^{~\\circ}$ to satisfy ATLAS and CMS upper limit on the mixing angle. As is seen in the figure, the $\\rm W$-Mass measurement, for $M_{H_2}\\lesssim 4.5~\\rm GeV$ and $M_{H_2}\\gtrsim 124~\\rm GeV$, excludes  the parameters space of the model. However, for $M_{H_2}$ between $4.5-124~\\rm GeV$ and $M_V$  between $1-2000~\\rm GeV$, the model is consistent with the W-mass anomaly.\n\n\\section{Invisible Higgs decay}\nIn the model, SM Higgs-like can decay invisibly into a pair of DM if kinematically allowed. Also, it can decay to another Higgs boson for $M_{H_2}<1\/2M_{H_1}$. Therefore, $H_1$ can contribute to the invisible decay mode with a branching ratio:\n\\begin{eqnarray}\nBr(H_1\\rightarrow \\rm Invisible)& =\\frac{\\Gamma(H_1\\rightarrow 2VDM)+\\Gamma(H_1\\rightarrow 2H_2)}{\\Gamma(h)_{SM}+\\Gamma(H_1\\rightarrow 2VDM)+\\Gamma(H_1\\rightarrow 2H_2)},\n\\label{decayinv1}\n\\end{eqnarray}\n\n\nwhere $\\Gamma(h)_{SM}=4.15 ~ \\rm [MeV]$ is total width of Higgs boson \\cite{LHCHiggsCrossSectionWorkingGroup:2011wcg}. The partial width for processes $H_1\\rightarrow 2VDM$ and $H_1\\rightarrow 2H_2$ are given by:\n\\begin{eqnarray}\n\\Gamma(H_1\\rightarrow 2VDM)& =\\frac{g_v^4v^2_2 sin^2{\\alpha}}{8\\pi M_{H_1}}\\sqrt{1-\\frac{4M^2_{V}}{M^2_{H_1}}}.\n\\label{decayinv1}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\Gamma(H_1\\rightarrow 2H_2)& =\\frac{a^2}{8\\pi M_{H_1}}\\sqrt{1-\\frac{4M^2_{H_2}}{M^2_{H_1}}}.\n\\label{decayinv1}\n\\end{eqnarray}\n\nwhere $a=(1\/2cos^3\\alpha -sin^2\\alpha cos\\alpha)v_1$.\nThe SM prediction for the branching ratio of the Higgs boson decaying to invisible particles which coming from process $h\\rightarrow ZZ^*\\rightarrow 4\\nu$ \\cite{Denner:2011mq},\\cite{Dittmaier:2012vm},\\cite{Brein:2003wg},\\cite{LHCHiggsCrossSectionWorkingGroup:2013rie} is, $1.2\\times10^{-3}.$\nCMS Collaboration has  reported the observed (expected) upper limit\non the invisible branching fraction of the Higgs boson to be $0.18 (0.10)$ at the $95\\%$ confidence level, by assuming the SM production cross section \\cite{CMS:2022qva}. A Similar analysis was performed by ATLAS collaboration in which an observed upper limit of $0.145$ is placed on the branching fraction of its decay into invisible particles at a $95\\%$ confidence level\\cite{ATLAS:2022yvh}.\n\n\\begin{figure\n\t\\begin{center}\n\t\t\\centerline{\\hspace{0cm}\\epsfig{figure=Invisible.eps,width=12cm}}\n\t\t\\centerline{\\vspace{-0.2cm}}\n\t\t\\caption{ The cross points depict allowed region which is consistent with invisible Higgs decay at \\cite{CMS:2022qva}.\n\t\t} \\label{Invisible}\n\t\\end{center}\n\\end{figure}\n\nFigure.~\\ref{Invisible}, shows the allowed range of parameters  by considering CMS\\cite{CMS:2022qva} upper limit for invisible Higgs mode. In this figure, we consider LHC bound on the mixing angle $\\sin\\alpha<0.12$. For $M_{H_2}<1\/2M_{SM}$  CMS upper limit excludes the parameters space. Note that this upper limit  practically can not constraint the model for low mass DM and also part of the parameters space in which invisible Higgs decay is forbidden.\n\n\n\\section{Relic Density}\nThe evolution of the number density of DM particles with time is governed by the Boltzmann equation. In this regard, we calculate the relic density\nnumerically for the VDM particle by implementing  the model into micrOMEGAs \\cite{Belanger:2014vza}. We investigate viable parameter space which satisfies constraints from observed DM relic density ( according to the data of Planck collaboration \\cite{Planck:2014egr}):\n\\begin{equation} \\label{44}\n\\Omega_{DM} h^{2} = 0.1199 \\pm 0.0027.\n\\end{equation}\n\n\\begin{figure\n\t\\begin{center}\n\t\t\\centerline{\\hspace{0cm}\\epsfig{figure=Relic.eps,width=12cm}}\n\t\t\\centerline{\\vspace{-0.2cm}}\n\t\t\\caption{The allowed range of parameter space consistent with DM relic density.} \\label{Relic}\n\t\\end{center}\n\\end{figure}\n\nThe allowed range of parameter space corresponding to this constraint is depicted in figure \\ref{Relic}. As seen in the figure, for small values of VDM mass ($ M_V\\lesssim 22~ \\rm GeV$), relic density measurement excludes the model.\n\n\n\n\n\\section{Final Results}\nBefore, we present a combined analysis of all constraints, let us turn our attention to the direct detection of VDM in the model. In the model, at the tree level, a VDM particle can interact elastically with a nucleon either through $H_1$ or via $H_2$ exchange\\cite{YaserAyazi:2019caf,YaserAyazi:2018lrv}. Presently, the XENON1T experiment \\cite{XENON:2018voc} excludes new parameter space for the WIMP-nucleon spin-independent elastic scatter cross-section above 6 GeV with a minimum of $ 4.1\\times10^{-47} cm^{2}$ at 30 GeV.  We restrict the model with these results. The direct detection restrictions and constraints discussed in the previous sections are summarized in figure.~\\ref{final}.\nThe cross points show allowed region consistent with relic density, W-mass anomaly, direct detection as well as the invisible decay rate.\n\nThe outcome of imposing these experimental constraints on the model is for a large portion of VDM mass values, narrow region of scalar mediator $H_2$ ($100~\\rm GeV \\lesssim M_{H_2}\\lesssim 124~\\rm GeV$), and $0<g_v\\leqslant 1$ all observational constraints are satisfied. The point is that  for $M_{H_2}$ and $M_V<1\/2M_{SM}$  (in which $H_1$ can decay to $H_2$ and VDM), and $M_{H_2}>124~\\rm GeV$ the model is respectively excluded by invisible Higgs upper limit and W-mass CDF-II measurement.\n\n\\begin{figure\n\t\\begin{center}\n\t\t\\centerline{\\hspace{0cm}\\epsfig{figure=final.eps,width=12cm}}\n\t\t\\centerline{\\vspace{-0.2cm}}\n\t\t\\caption{Final results for allowed ranges of parameters of the model. The cross points depict allowed region which is consistent with relic density, direct detection, W mass anomaly and invisible decay rate.} \\label{final}\n\t\\end{center}\n\\end{figure}\n\nIn section.~3, we analyse RGEs of coupling of the model. We show that adding a new VDM and scalar mediator, the RGEs will be modified. It is interesting to see behaviour of RGEs of couplings in Planck scale. We solve the RG equations numerically and determine the RG evolution of the couplings of the models. For input parameters, we pick benchmark points for parameters of the model that are  consistent with all constraints considered previously in the paper. Similar analyses with different input parameters have been performed for $U(1)$ extension of SM in Ref\\cite{Duch:2015jta}.\tThe running of couplings up to the Planck scale have been shown in Figures.(\\ref{RGE}~.a-c). In Figure.~(\\ref{RGE}~.d), we compared running of $\\lambda_H$ in the model with SM Higgs coupling. As it is known, SM Higgs coupling will be negative for $\\mu> 10^{10}~\\rm GeV$. This means, it is not possible to establish all three conditions (perturbativity, vacuum stability and positivity) simultaneously in any scale. It is remarkable that  the SM stability problem (positivity of  $\\lambda_H$) is solved in the model. This issue arises that  $\\lambda_{SH}$ changes very little in our model. This leads small changes for $\\lambda_H$ and as a result, $\\lambda_H$ remains positive until the Planck scale.\n\n\n\\begin{figure\n\t\\begin{center}\n\t\t\\centerline{\\hspace{0cm}\\epsfig{figure=RGE-Plot.eps,width=8cm}\\hspace{0.5cm}\\hspace{0cm}\\epsfig{figure=RGE-Plot1.eps,width=8cm}}\n\t\t\\centerline{\\vspace{0.2cm}\\hspace{1.5cm}(a)\\hspace{8cm}(b)}\n\t\t\\centerline{\\hspace{0cm}\\epsfig{figure=RGE-Plot2.eps,width=8cm}\\hspace{0.5cm}\\hspace{0cm}\\epsfig{figure=plpt.eps,width=8cm}}\t\t\n\t\t\\centerline{\\vspace{0.2cm}\\hspace{1.5cm}(c)\\hspace{8cm}(d)}\n\t\t\\centerline{\\vspace{-0.2cm}}\n\t\t\\caption{Running of couplings of the model up to Planck scale. We select sample point in which all the experimental constraints considered in the paper are satisfied.} \\label{RGE}\n\t\\end{center}\n\\end{figure}\n\n\\section{Conclusions} \\label{sec7}\n\nWe proposed a model to explain the W boson mass anomaly reported by the CDF-II collaboration. We studied an $U(1)$ extension of the SM including a VDM candidate and a scalar mediator. In the model, there is no kinetic mixing between the VDM field and SM Z-boson, but scalar field exchange between SM and dark side. To explain the W mass anomaly one needs extra degrees of freedom that affects on W-boson mass. In the model, the one-loop corrections induced by the new\nscalar can shift the W boson mass. We have also imposed constraints on the Higgs mixing angle and other parameters of the model by investigating of relic density of DM, invisible Higgs decay mode at LHC and direct detection of DM. We have shown that the model for a large part of VDM mass values, scalar mediator mass range $100~\\rm GeV \\lesssim M_{H_2}\\lesssim 124~\\rm GeV$ and $0<g_v\\leqslant 1$ can explain W-mass anomaly and satisfy all experimental constraints.\n\nWe have also investigated the effects and consequences of the RGE at one-loop order on the model. It was shown that adding of new fields to SM, changes  positivity and stability conditions of SM Higgs as it will be stable up to Planck scale.\n\n\n\n\n\n\n\n\\section*{Acknowledgments}\nWe would like to thank Dr. Ahmad Mohamadnejad for helping with micrOMEGAs code issues.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and preliminaries}\\label{intro}\n\nWe begin with the well-known Riemann-Liouville (R-L) fractional derivative of order $\\mu$ is defined  (see \\cite{Samko1933},\\cite{Kilbas}) by\n\\begin{eqnarray}\\label{eq1}\n\\mathfrak{D}_{x}^{\\mu}\\{f(x)\\}=\\frac{1}{\\Gamma(-\\mu)}\\int_0^xf(t)(x-t)^{-\\mu-1}dt, \\Re(\\mu)<0.\n\\end{eqnarray}\nFor the case $m-1<\\Re(\\mu)<m$ where $m=1,2,\\cdots$, it follows\n\\begin{align}\\label{eq2}\n\\mathfrak{D}_{x}^{\\mu}\\{f(x)\\}&=\\frac{d^m}{dx^m}\\mathfrak{D}_{x}^{\\mu-m}\\Big\\{f(x)\\Big\\}\\notag\\\\\n&=\\frac{d^m}{dx^m}\\Big\\{\\frac{1}{\\Gamma(-\\mu+m)}\\int_0^xf(t)(x-t)^{-\\mu+m-1}dt\\Big\\}\n\\end{align}\nand\n\\begin{eqnarray}\\label{eq3}\n\\mathfrak{D}_{x}^{\\mu}\\{x^\\sigma\\}=\\frac{\\Gamma(\\sigma+1)}{\\Gamma(\\sigma-\\mu+1)}x^{\\sigma-\\mu}.\n\\end{eqnarray}\nThe researchers (see \\cite{Kiymaz,Luo,Olver,Parmar,Srivastava2012}) investigated the various extensions and generalization of fractional derivative operators.\n\nThe extended R-L fractional derivative of order $\\mu$ is defined in \\cite{Ozerslan} by\n\\begin{eqnarray}\\label{Ofrac}\n\\mathfrak{D}_{x}^{\\mu}\\{f(x);p\\}=\\frac{1}{\\Gamma(-\\mu)}\\int_0^xf(t)(x-t)^{-\\mu-1}\\exp\\Big(-\\frac{px^2}{t(x-t)}\\Big) dt, \\Re(\\mu)<0.\n\\end{eqnarray}\nFor the case $m-1<\\Re(\\mu)<m$ where $m=1,2,\\cdots$, it follows\n\\begin{align}\\label{Ofrac1}\n\\mathfrak{D}_{x}^{\\mu}\\{f(x);p\\}&=\\frac{d^m}{dx^m}\\mathfrak{D}_{x}^{\\mu-m}\\Big\\{f(x);p\\Big\\}\\notag\\\\\n&=\\frac{d^m}{dx^m}\\Big\\{\\frac{1}{\\Gamma(-\\mu+m)}\\int_0^xf(t)(x-t)^{-\\mu+m-1}\\exp\\Big(-\\frac{px^2}{t(x-t)}\\Big) dt\\Big\\}.\n\\end{align}\nAn extension of fractional derivative operator established in \\cite{Baleanu} is given by\n\\begin{eqnarray}\\label{Efrac}\n\\mathfrak{D}_{x}^{\\mu}\\{f(x);p,q\\}=\\frac{1}{\\Gamma(-\\mu)}\\int_0^xf(t)(x-t)^{-\\mu-1}.\n\\exp\\Big(-\\frac{px}{t}-\\frac{qx}{(x-t)}\\Big) dt, \\Re(\\mu)<0.\n\\end{eqnarray}\nFor example,\n\\begin{align*}\n\\mathfrak{D}_{x}^{\\mu}\\left\\{x^\\nu;p,q\\right\\}_{x=1}=\\frac{B_{p,q}(\\nu+1,-\\mu)}{\\Gamma(-\\mu)},\n\\end{align*}\nwhere $B_{p,q}(x,y)$ is the extended Beta function (see \\cite{Mehrez}) defined by\n\\begin{align*}\nB_{p,q}(x,y)=\\int_{0}^{1}t^{x-1}(1-t)^{y-1}{\\mathcal{E}}_{p,q}(t)dt, x,y,p,q\\in \\mathbb{C}, \\Re(p),\\Re(q)>0,\n\\end{align*}\nwhere ${\\mathcal{E}}_{p,q}(t)=e^{-\\frac{p}{t}-\\frac{q}{1-t}}$.\nFor $p=q$ we denote $B_{p,q}$ by $B_{p}$ and for $p=q=0$, we get classical Beta function defined by\n\\begin{eqnarray}\\label{beta}\nB(x,y)=\\int\\limits_{0}^{1}t^{x-1}(1-t)^{y-1}dt, (\\Re(x)>0, \\Re(y)>0).\n\\end{eqnarray}\n\nFor the case $m-1<\\Re(\\mu)<m$ where $m=1,2,\\cdots$, it follows\n\\begin{align}\n\\mathfrak{D}_{x}^{\\mu}\\{f(x);p,q\\}&=\\frac{d^m}{dx^m}\\mathfrak{D}_{x}^{\\mu-m}\\Big\\{f(x);p,q\\Big\\}\\notag\\\\\n&=\\frac{d^m}{dx^m}\\Big\\{\\frac{1}{\\Gamma(-\\mu+m)}\\int_0^xf(t)(x-t)^{-\\mu+m-1}\\notag\\\\\n&\\times\\exp\\Big(-\\frac{px}{t}-\\frac{qx}{(x-t)}\\Big) dt\\Big\\}.\n\\end{align}\nRecently, Rahman et al. \\cite{Rahman} define an extension of extended R-L fractional derivative of order $\\mu$ as\n\\begin{align}\\label{FEfrac}\n\\mathfrak{D}_{x}^{\\mu}\\{f(x);p,q,\\lambda,\\rho\\}&=\\frac{1}{\\Gamma(-\\mu)}\\int_0^xf(t)(x-t)^{-\\mu-1}\\notag\\\\\n&\\times_1F_1\\Big[\\lambda;\\rho;-\\frac{px}{t}\\Big]\\,_1F_1\\Big[\\lambda;\\rho;-\\frac{qx}{(x-t)}\\Big] dt, \\Re(\\mu)<0.\n\\end{align}\nFor the case $m-1<\\Re(\\mu)<m$ where $m=1,2,\\cdots$, it follows\n\\begin{align}\n&\\mathfrak{D}_{x}^{\\mu}\\{f(x);p,q,\\lambda,\\rho\\}=\\frac{d^m}{dx^m}\\mathfrak{D}_{x}^{\\mu-m}\\Big\\{f(x);p,q,\\lambda,\\rho\\Big\\}\\notag\\\\\n&=\\frac{d^m}{dx^m}\\Big\\{\\frac{1}{\\Gamma(-\\mu+m)}\\int_0^xf(t)(x-t)^{-\\mu+m-1}\n\\,_1F_1\\Big[\\lambda;\\rho;-\\frac{px}{t}\\Big]\\,_1F_1\\Big[\\lambda;\\rho;-\\frac{qx}{(x-t)}\\Big] dt\\Big\\},\n\\end{align}\nwhere  $\\Re(p)>0$ and $\\Re(q)>0$. It is clear that when $\\lambda=\\rho$, then (\\ref{FEfrac}) reduce to (\\ref{Efrac}).\\\\\n\nThe Gauss hypergeometric  function which is defined (see \\cite{Rainville1960}) as\n\\begin{eqnarray}\\label{Hyper}\n_2F_1(\\sigma_1,\\sigma_2;\\sigma_3;z)=\\sum\\limits_{n=0}^{\\infty}\\frac{(\\sigma_1)_n(\\sigma_2)_n}{(\\sigma_3)_n}\\frac{z^n}{n!}, (|z|<1),\n\\end{eqnarray}\n $\\Big(\\sigma_1, \\sigma_2, \\sigma_3\\in\\mathbb{C}$ and $\\sigma_3\\neq0,-1,-2,-3,\\cdots\\Big)$.\n The integral representation of hypergeometric  hypergeometric function is defined by\n\\begin{eqnarray}\\label{Ihyper}\n_2F_1(\\sigma_1,\\sigma_2;\\sigma_3;z)=\\frac{\\Gamma(\\sigma_3)}{\\Gamma(\\sigma_2)\\Gamma(\\sigma_3-\\sigma_2)}\n\\int_0^1t^{\\sigma_2-1}(1-t)^{\\sigma_3-\\sigma_2-1}(1-zt)^{-\\sigma_1}dt,\n\\end{eqnarray}\n$\\Big(\\Re(\\sigma_3)>\\Re(\\sigma_2)>0, |\\arg(1-z)|<\\pi\\Big)$.\\\\\nThe Appell series or bivariate hypergeometric series and its integral representation  is respectively defined by\n\\begin{eqnarray}\\label{CAppell}\nF_{1}(\\sigma_1,\\sigma_2,\\sigma_3;\\sigma_4;x, y)=\\sum\\limits_{m,n=0}^{\\infty}\\frac{(\\sigma_1)_{m+n}(\\sigma_2)_{m}(\\sigma_3)_{n}x^{m}y^{n}}{(\\sigma_4)_{m+n}m!n!};\n\\end{eqnarray}\n for all $ \\sigma_1,\\sigma_2,\\sigma_3,\\sigma_4\\in \\mathbb{C},  \\sigma_4\\neq 0,-1,-2,-3,\\cdots, \\quad |x|<1, |y|<1$.\\\\\n\\begin{align}\\label{FIAppell}\nF_{1}\\Big(\\sigma_1,\\sigma_2, \\sigma_3,\\sigma_4;x,y\\Big)&=\\frac{\\Gamma(\\sigma_4)}{\\Gamma(\\sigma_1)\\Gamma(\\sigma_4-\\sigma_1)}\\notag\\\\\n&\\times\\int\\limits_{0}^{1}t^{\\sigma_1-1}(1-t)^{\\sigma_4-\\sigma_1-1}(1-xt)^{-\\sigma_2}(1-yt)^{-\\sigma_3}dt\n\\end{align}\n$\\Re(\\sigma_4)>\\Re(\\sigma_1)>0$, $|\\arg(1-x)|<\\pi$ and $|\\arg(1-y)|<\\pi$.\\\\\n\nChaudhry et al. \\cite{Chaudhry1997} introduced the  extended Beta function is defined by\n\\begin{eqnarray}\\label{Ebeta}\nB(\\sigma_1,\\sigma_2;p)=B_p(\\sigma_1,\\sigma_2)=\\int\\limits_{0}^{1}t^{\\sigma_1-1}(1-t)^{\\sigma_2-1}\ne^{-\\frac{p}{t(1-t)}}dt\n\\end{eqnarray}\n(where $\\Re(p)>0, \\Re(\\sigma_1)>0, \\Re(\\sigma_2)>0$) respectively. When $p=0$, then $B(\\sigma_1,\\sigma_2;0)=B(\\sigma_1,\\sigma_2)$.\\\\\nThe extended  hypergeometric  function introduced in \\cite{Chaudhrya2004} by using the definition of extended Beta function $B_p(\\delta_1,\\delta_2)$ as follows:\n\\begin{eqnarray}\\label{Ehyper}\nF_p(\\sigma_1,\\sigma_2;\\sigma_3;z)=\\sum\\limits_{n=0}^{\\infty}\\frac{B_p(\\sigma_2+n, \\sigma_3-\\sigma_2)}{B(\\sigma_2, \\sigma_3-\\sigma_2)}(\\sigma_1)_n\\frac{z^n}{n!},\n\\end{eqnarray}\nwhere $p\\geq0$ and $\\Re(\\sigma_3)>\\Re(\\sigma_2)>0$, $|z|<1$.\\\\\nIn the same paper, they defined the following integral representations of extended hypergeometric and confluent hypergeometric functions as\n\\begin{align}\\label{IEhyper}\nF_p(\\sigma_1,\\sigma_2;\\sigma_3;z)&=\\frac{1}{B(\\sigma_2, \\sigma_3-\\sigma_2)}\\notag\\\\\n&\\times\\int_0^1t^{\\sigma_2-1}(1-t)^{\\sigma_3-\\sigma_2-1}(1-zt)^{-\\sigma_1}\\exp\\Big(\\frac{-p}{t(1-t)}\\Big)dt,\n\\end{align}\n$$\\Big(p\\geq0, \\Re(\\sigma_3)>\\Re(\\sigma_2)>0, |\\arg(1-z)|<\\pi\\Big).$$\nThe extended Appell's function is defined by (see \\cite{Ozerslan})\n\\begin{eqnarray}\\label{EAppell}\nF_1(\\sigma_1,\\sigma_2,\\sigma_3;\\sigma_4;x,y;p)=\\sum\\limits_{m,n=0}^{\\infty}\\frac{B_p(\\sigma_1+m+n, \\sigma_4-\\sigma_1)}{B(\\sigma_1, \\sigma_4-\\sigma_1)}(\\sigma_2)_m(\\sigma_3)_n\\frac{x^my^n}{m!n!}\n\\end{eqnarray}\nwhere $p\\geq0$ and $\\Re(\\sigma_4)>\\Re(\\sigma_1)>0$ and $|x|,|y|<1$.\\\\\n \\\"{O}zerslan and  \\\"{O}zergin \\cite{Ozerslan} defined its integral representation by\n\\begin{align}\\label{IEAppell}\nF_1(\\sigma_1,\\sigma_2,\\sigma_3;\\sigma_4;z;p)&=\\frac{1}{B(\\sigma_1, \\sigma_4-\\sigma_1)}\\notag\\\\\n&\\times\\int_0^1t^{\\sigma_1-1}(1-t)^{\\sigma_4-\\sigma_1-1}(1-xt)^{-\\sigma_2}(1-yt)^{-\\sigma_3}\n\\exp\\Big(\\frac{-p}{t(1-t)}\\Big)dt,\\\\\n&\\Big(\\Re(p)>0, \\Re(\\sigma_4)>\\Re(\\sigma_1)>0, |\\arg(1-x)|<\\pi, |\\arg(1-y)|<\\pi\\Big)\\notag.\n\\end{align}\n\nIt is clear that when $p=0$, then the equations (\\ref{Ehyper})-(\\ref{IEAppell}) reduce to the well known hypergeometric, confluent hypergeometric and Appell's series and their integral representation respectively.\\\\\nVery recently Shadab et al.  \\cite{Choi2017} introduced a new and modified extension of Beta function as:\n\\begin{eqnarray}\\label{Cbeta}\nB^{\\alpha}_{p}(\\sigma_1,\\sigma_2)=\\int_0^1t^{\\sigma_1-1}(1-t)^{\\sigma_2-1}E_{\\alpha}\\Big(-\\frac{p}{t(1-t)}\\Big)dt,\n\\end{eqnarray}\nwhere $\\Re(\\sigma_1)>0$, $\\Re(\\sigma_2)>0$ and $E_{\\alpha}\\Big(.\\Big)$ is the Mittag-Leffler function defined by\n\\begin{eqnarray}\nE_{\\alpha}\\Big(z\\Big)=\\sum_{n=0}^\\infty\\frac{z^n}{\\Gamma(\\alpha n+1)}.\n\\end{eqnarray}\nObviously, when $\\alpha=1$  then $B^{1}_{p}(x,y)=B_p(x,y)$ is the extended Beta function (see\\cite{Chaudhry1997}). Similarly, when when $\\alpha=1$ and $p=0$,  then $B^{1}_{0}(x,y)=B_0(x,y)$ is the classical Beta function.\\\\\nShadab et al.  \\cite{Choi2017} also defined extended hypergeometric function and its integral representation\n\\begin{align}\\label{pqhyper}\nF_{p}^{\\alpha}(\\sigma_1,\\sigma_2;\\sigma_3;z)={}_2F_{1}\\Big(\\sigma_1,\\sigma_2;\\sigma_3;z;p,\\alpha\\Big)\\notag&=\n\\sum_{n=0}^\\infty(\\sigma_1)_n\\frac{B_{p}^{\\alpha}(\\sigma_2+n,\\sigma_3-\\sigma_2)}{B(\\sigma_2,\\sigma_3-\\sigma_2)}\n\\frac{z^n}{n!}\\notag\\\\\n=&\\sum_{n=0}^\\infty(\\sigma_1)_n\\frac{B(\\sigma_2+n,\\sigma_3-\\sigma_2; p,\\alpha)}{B(\\sigma_2,\\sigma_3-\\sigma_2)}\n\\frac{z^n}{n!}\n\\end{align}\nwhere $p,\\alpha\\geq0$, $\\sigma_1,\\sigma_2,\\sigma_3\\in\\mathbb{C}$  and $|z|<1$.\n \\begin{align}\\label{pqIhyper}\nF_{p}^{\\alpha}(\\sigma_1,\\sigma_2;\\sigma_3;z)&=\\frac{1}{\\beta(\\sigma_2;\\sigma_3-\\sigma_2)}\\notag\\\\\n&\\times\\int_0^1t^{\\sigma_2-1}\n(1-t)^{\\sigma_3-\\sigma_2-1}(1-tz)^{-\\sigma_1}E_{\\alpha}\\Big(-\\frac{p}{t(1-t)}\\Big)dt,\n\\end{align}\nwhere $\\Re(p)>0$, $\\Re(\\alpha)>0$,  $\\Re(\\sigma_3)>\\Re(\\sigma_2)>0$. Obviously when $\\alpha=1$, then the hypergeometric function (\\ref{pqhyper}) will reduce to the extended hypergeometric function (\\ref{Ehyper}) and similarly when $\\alpha=1$ and $p=0$ then the hypergeometric function (\\ref{pqhyper}) will reduce to the  hypergeometric function (\\ref{Hyper}).\\\\\n\nFor various extensions and generalizations of Beta function and hypergeometric functions the interested readers may refer to the recent work of researchers (see e. g., \\cite{Choi2014,Mubeen2017,Ozerslan,Ozergin}).\n\\section{ extension of Appell's functions and its integral representations}\n\nWe start the section by deriving the relation of \\eqref{Cbeta} with multi-index Mittag-Leffler function \\cite{Kir1} as follows:\n\n\n\\begin{proposition}\nFor $\\Re(p), \\Re(\\sigma_1), \\Re(\\sigma_2), \\Re(\\alpha) > 0$ the following relation holds true:\n\\begin{align}\nB^{\\alpha}_{p}(\\sigma_1,\\sigma_2)=\\frac{\\pi sin\\pi(\\sigma_1+\\sigma_2)}{(sin\\pi\\sigma_1)(sin\\pi\\sigma_2)}E_{(\\alpha,1), (1, 1-\\sigma_1), (1,1-\\sigma_2)}^{(2,1-\\sigma_1-\\sigma_2)}(-p)\n\\end{align}\nwhere $E_{(\\alpha,1), (1, 1-\\sigma_1), (1,1-\\sigma_2)}^{(2,1-\\sigma_1-\\sigma_2)}(-p)$ is the multi-index Mittag-Leffler function \\cite{Kir1}.\n\\end{proposition}\n\n{\\bf Proof}\nUsing the definition of Beta function and reduction theorem of Gamma function, we get\n\\begin{align*}\nB^{\\alpha}_{p}(\\sigma_1,\\sigma_2)&=\\sum_{n=0}^{\\infty}\\frac{(-p)^n}{\\Gamma(\\alpha n+1)}\\int_{0}^{1}t^{\\sigma_1-1}(1-t)^{\\sigma_2-1}\\frac{1}{t^n(1-t)^n}dt\\\\\n&=\\sum_{n=0}^{\\infty}\\frac{(-p)^n}{\\Gamma(\\alpha n+1)}\\int_{0}^{1}t^{\\sigma_1-n-1}(1-t)^{\\sigma_2-n-1}dt\\\\\n&=\\sum_{n=0}^{\\infty}\\frac{(-p)^n}{\\Gamma(\\alpha n+1)}B(\\sigma_1-n, \\sigma_2-n)\\\\\n&=\\sum_{n=0}^{\\infty}\\frac{(-p)^n}{\\Gamma(\\alpha n+1)}\\frac{\\Gamma(\\sigma_1-n)\\Gamma(\\sigma_2-n)}{\\Gamma(\\sigma_1+\\sigma_2-2n)}\\\\\n&=\\frac{\\Gamma(-\\sigma_1)\\Gamma(1+\\sigma_1)\\Gamma(-\\sigma_2)\\Gamma(1+\\sigma_2)}{\\Gamma(-\\sigma_1-\\sigma_2)\\Gamma(1+\\sigma_1+\\sigma_2)}\\\\\n&\\times \\sum_{n=0}^{\\infty} \\frac{\\Gamma(2n+1-\\sigma_1-\\sigma_2)(-p)^n}{\\Gamma(\\alpha n+1)\\Gamma(n+1-\\sigma_1)\\Gamma(n+1-\\sigma_2)}\n\\end{align*}\nNow, using the Euler's reflection formula on Gamma function,\n\\begin{align}\\label{rel1}\n{\\Gamma(r)\\Gamma(1-r)}=\\frac{\\pi}{\\sin(\\pi r)},\n\\end{align}\nwe get the desired result.\n\nNext, we used the definition (\\ref{Cbeta}) and consider the following modified extension Appell's  functions.\n\\begin{definition}\\label{def2}\nThe modified extended Appell's function $F_1$ is defined by\n\\begin{align}\\label{Appell}\nF_{1,p}^{\\alpha}(\\sigma_1,\\sigma_2,\\sigma_3;\\sigma_4;x,y)&=\nF_{1}(\\sigma_1,\\sigma_2,\\sigma_3;\\sigma_4;x,y;p,\\alpha)\\notag\\\\&=\\sum_{m,n=0}^\\infty(\\sigma_2)_m(\\sigma_3)_n\n\\frac{B_{p}^{\\alpha}(\\sigma_1+m+n,\\sigma_4-\\sigma_1)}{B(\\sigma_1,\\sigma_4-\\sigma_1)}\n\\frac{x^m}{m!}\\frac{y^n}{n!}\\notag\\\\\n=&\\sum_{m,n=0}^\\infty(\\sigma_2)_m(\\sigma_3)_n\\frac{B(\\sigma_1+m+n,\\sigma_4-\\sigma_1; p,\\alpha)}{B(\\sigma_1,\\sigma_4-\\sigma_1)}\n\\frac{x^m}{m!}\\frac{y^n}{n!}\n\\end{align}\nwhere $p,\\alpha\\geq0$, $\\sigma_1,\\sigma_2,\\sigma_3,\\sigma_4\\in\\mathbb{C}$ and  $|x|<1$, $|y|<1$.\n\\end{definition}\n\\begin{remark}\nSetting  $\\alpha=1$ in (\\ref{Appell}), then we get the extended Appell's functions (see \\cite{Ozerslan}).\n\\end{remark}\nNow, we derive the following proposition\n\\begin{proposition} For $p,q >0$, $0<x, y<\\frac{1}{2}$, the following inequality holds true:\n\\begin{align}\nB_{p,q}(x,y) \\leq (2p)^{\\frac{2x-1}{2}}(2q)^{\\frac{2y-1}{2}}\\sqrt{\\Gamma(-2x+1,2p)\\Gamma(-2y+1,2q)}.\n\\end{align}\n\\end{proposition}\n\\begin{proof}\nApplying the Cauchy-Schwarz integral inequality, we obtain:\n\\begin{align*}\nB_{p,q}(x,y)&=\\int_{0}^{1}t^{x-1}(1-t)^{y-1}e^{-\\frac{p}{t}}e^{-\\frac{q}{t}} dt\\\\\n&\\leq\t\\left(\\int_{0}^{1}(t^{x-1}e^{-p\/t})^2dt\\right)^{1\/2}\\left(\\int_{0}^{1}[(1-t)^{y-1}e^{-q\/1-t}]^2dt\\right)^{1\/2}\\\\\n&=\\left(\\int_{1}^{\\infty}t^{-2x}e^{-2 pt}dt\\right)^{1\/2}\\left(\\int_{1}^{\\infty}(1-t)^{-2y}e^{-2 qt}dt\\right)^{1\/2}\\\\\n&=\\left[(2p)^{2x-1}\\Gamma(-2x+1, 2p)\\right]^{1\/2}\\left[(2q)^{2y-1}\\Gamma(-2y+1, 2q)\\right]^{1\/2}\\\\\n&=(2p)^{\\frac{2x-1}{2}}(2q)^{\\frac{2y-1}{2}}\\sqrt{\\Gamma(-2x+1,2p)\\Gamma(-2y+1,2q)},\n\\end{align*}\nwhere $\\Gamma (x,y)$ is the incomplete Gamma function.\n\\end{proof}\n\n\\begin{theorem}\\label{th2}\nThe following integral representation holds true for (\\ref{Appell})\n\\begin{align}\\label{pqIhyper2}\nF_1\\Big(\\sigma_1,\\sigma_2,\\sigma_3;\\sigma_4;x,y;p,\\alpha\\Big)&=\\frac{1}{B(\\sigma_1;\\sigma_4-\\sigma_1)}\n\\int_0^1t^{\\sigma_1-1}(1-t)^{\\sigma_4-\\sigma_1-1}\n(1-tx)^{-\\sigma_2}(1-ty)^{-\\sigma_3}\\notag\\\\&\\times\nE_{\\alpha}\\Big(-\\frac{p}{t(1-t)}\\Big)dt,\n\\end{align}\nwhere $\\Re(p)>0$, $\\Re(\\alpha)>0$,  $\\Re(\\sigma_4)>\\Re(\\sigma_1)>0$.\n\\end{theorem}\n\\begin{proof}\nUsing the definition (\\ref{Cbeta}) in (\\ref{Appell}), we have\n\\begin{align}\\label{PS2}\nF_1\\Big(\\sigma_1,\\sigma_2,\\sigma_3;\\sigma_4;x,y;p,\\alpha\\Big)&=\\frac{1}{B(\\sigma_1;\\sigma_4-\\sigma_1)}\n\\int_0^1t^{\\sigma_1-1}(1-t)^{\\sigma_4-\\sigma_1-1}\\notag\\\\\n\\times&\\Big(\\sum_{m,n=0}^\\infty\\frac{(\\sigma_2)_m(\\sigma_3)_n(tx)^m(ty)^n}{m!n!}\\Big) dt.\n\\end{align}\n Since\n\\begin{align}\\label{PS}\n\\sum_{m,n=0}^\\infty\\frac{(\\sigma_2)_m(\\sigma_3)_n(x)^m(y)^n}{m!n!}=(1-tx)^{-\\sigma_2}(1-ty)^{-\\sigma_3}.\n\\end{align}\n\nThus by using (\\ref{PS}) in (\\ref{PS2}),   we get the desired result.\n\\end{proof}\n\\section{Extension of fractional derivative operator}\nIn this section, we define a new and modified extension of   Riemann-Liouville fractional derivative and obtain its related results.\n\\begin{definition}\\label{frac}\n\\begin{eqnarray}\\label{eq7}\n\\mathfrak{D}_{z;p}^{\\mu;\\alpha}\\{f(z)\\}=\\frac{1}{\\Gamma(-\\mu)}\\int_0^zf(t)(z-t)^{-\\mu-1}E_\\alpha\\Big(-\\frac{pz^2}{t(z-t)}\\Big)dt, \\Re(\\mu)<0.\n\\end{eqnarray}\nFor the case $m-1<\\Re(\\mu)<m$ where $m=1,2,\\cdots$, it follows\n\\begin{align}\\label{eq8}\n\\mathfrak{D}_{z;p}^{\\mu;\\alpha}\\{f(z)\\}&=\\frac{d^m}{dz^m}\\mathfrak{D}_{z;p}^{\\mu-m;\\alpha}\\Big\\{f(z)\\Big\\}\\notag\\\\\n=&\\frac{d^m}{dz^m}\\Big\\{\\frac{1}{\\Gamma(-\\mu+m)}\\int_0^zf(t)(z-t)^{-\\mu+m-1}E_\\alpha\\Big(-\\frac{pz^2}{t(z-t)}\\Big)dt\\Big\\}.\n\\end{align}\n\\end{definition}\n\\begin{remark}\nObviously if  $\\alpha=1$, then (\\ref{eq7}) and (\\ref{eq8}) respectively reduces to the extended fractional derivative (\\ref{Ofrac}) and (\\ref{Ofrac1}). Similarly, if we set $\\alpha=1$ and $p=0$ we get (\\ref{eq1}) and (\\ref{eq2}).\n\\end{remark}\nNow, we prove some theorems involving the modified extension of fractional derivative operator.\n\\begin{theorem} \\label{th1} The following formula hold true,\n\\begin{eqnarray}\\label{fd1}\n\\mathfrak{D}_{z}^{\\mu}\\{z^{\\eta};p,\\alpha\\}=\\frac{B_{p}^{\\alpha}(\\eta+1,-\\mu)}{\\Gamma(-\\mu)}z^{\\eta-\\mu},\n\\Re(\\mu)<0.\n\\end{eqnarray}\n\\end{theorem}\n\\begin{proof}\nFrom (\\ref{frac}), we have\n\\begin{eqnarray}\\label{fd2}\n\\mathfrak{D}_{z}^{\\mu}\\{z^\\eta;p,\\alpha\\}=\\frac{1}{\\Gamma(-\\mu)}\\int_0^zt^\\eta(z-t)^{-\\mu-1}\\,\nE_\\alpha\\Big(-\\frac{pz^2}{t(z-t)}\\Big) dt.\n\\end{eqnarray}\nSubstituting $t=uz$ in (\\ref{fd2}), we get\n\\begin{eqnarray*}\n\\mathfrak{D}_{z}^{\\mu}\\{z^\\eta;p,\\alpha\\}&=&\\frac{1}{\\Gamma(-\\mu)}\\int_0^1(uz)^\\eta(z-uz)^{-\\mu-1}\\,\nE_\\alpha\\Big(-\\frac{pz^2}{uz(z-uz)}\\Big) dt\\\\\n&=&\\frac{z^{\\eta-\\mu}}{\\Gamma(-\\mu)}\\int_0^1u^\\eta(1-u)^{-\\mu-1}\\,\nE_\\alpha\\Big(-\\frac{p}{u(1-u)}\\Big) dt,\n\\end{eqnarray*}\nIn view of (\\ref{Cbeta}) to the above equation, we get the required result.\n\\end{proof}\n\\begin{theorem}\\label{th2a}\nLet $\\Re(\\mu)>0$ and assume that the function $f(z)$ is analytic at the origin with its Maclaurin expansion given by\n$f(z)=\\sum_{n=0}^{\\infty} a_n z^n$ where $|z|<\\delta$ for some $\\delta\\in \\mathbb{R^+}$. Then\n\\begin{align}\n\\mathfrak{D}_{z;p}^{\\mu;\\alpha}\\{f(z)\\}=\\mathfrak{D}_{z}^{\\mu}\\{f(z);p,\\alpha\\}&=\\sum_{n=0}^\\infty a_n\\mathfrak{D}_{z}^{\\mu}\\{z^n;p,\\alpha\\}\\notag\\\\\n=&\\frac{1}{\\Gamma(-\\mu)}\\sum_{n=0}^\\infty a_nB_p^\\alpha(n+1,-\\mu)z^{n-\\mu}.\n\\end{align}\n\\end{theorem}\n\n\\begin{proof}\nUsing the series expansion of the function $f(z)$ in (\\ref{eq7}) gives\n\\begin{eqnarray*}\n\\mathfrak{D}_{z}^{\\mu}\\{f(z);p,\\alpha\\}=\\frac{1}{\\Gamma(-\\mu)}\\int_0^z\\sum_{n=0}^\\infty a_nt^n(z-t)^{-\\mu-1}\\,\nE_\\alpha\\Big(-\\frac{pz^2}{t(z-t)}\\Big) dt.\n\\end{eqnarray*}\nThe series is uniformly convergent on any closed disk centered at the origin with its radius smaller than $\\delta$, so does on the line segment from $0$ to a fixed $z$ for $|z|<\\delta$. Thus it guarantee terms by terms integration as follows\n\\begin{eqnarray*}\n\\mathfrak{D}_{z}^{\\mu}\\{f(z);p,\\alpha\\}&=&\\sum_{n=0}^\\infty a_n\\Big\\{\\frac{1}{\\Gamma(-\\mu)}\\int_0^z t^n(z-t)^{-\\mu-1}\\,\nE_\\alpha\\Big(-\\frac{pz^2}{uz(z-uz)}\\Big) dt\\\\\n&=&\\sum_{n=0}^\\infty a_n\\mathfrak{D}_{z}^{\\mu}\\{z^n;p,\\alpha\\}.\n\\end{eqnarray*}\nNow, applying Theorem \\ref{th1}, we get\n\\begin{eqnarray*}\n\\mathfrak{D}_{z}^{\\mu}\\{f(z);p,\\alpha\\}\n&=&\\frac{1}{\\Gamma(-\\mu)}\\sum_{n=0}^\\infty a_nB_p^\\alpha(n+1,-\\mu)z^{n-\\mu}, \\Re(\\mu)<0.\n\\end{eqnarray*}\nwhich is the required proof.\n\\end{proof}\n\\begin{example}\nThe following result holds true:\n\\begin{align}\n\\mathfrak{D}_{z}^{\\mu}\\{e^z;p,\\alpha\\}=\\frac{z^{-\\mu}}{\\Gamma(-\\mu)}\\sum_{n=0}^\\infty B_p^\\alpha(n+1,-\\mu)\\frac{z^n}{n!}.\n\\end{align}\nUsing the power series of $\\exp(z)$ and applying Theorem \\ref{th2a}, we have\n\\begin{align}\n\\mathfrak{D}_{z}^{\\mu}\\{e^z;p,\\alpha\\}=\\sum_{n=0}^\\infty \\frac{1}{n!}\\mathfrak{D}_{z}^{\\mu}\\{z^{n};p,\\alpha\\}.\n\\end{align}\nNow, applying Theorem \\ref{th1}, we get the desired result.\n\\end{example}\n\\begin{theorem}\\label{th3}\nThe following formula holds true:\n\\begin{eqnarray}\n\\mathfrak{D}_{z}^{\\eta-\\mu}\\{z^{\\eta-1}(1-z)^{-\\beta};p,\\alpha\\}=\n\\frac{\\Gamma(\\eta)}{\\Gamma(\\mu)}z^{\\mu-1}\\,_2F_{1;p}^{\\alpha}\n\\Big(\\beta, \\eta;\\mu;z\\Big),\n\\end{eqnarray}\nwhere $\\Re(\\mu)> \\Re(\\eta)>0$ and $|z|<1$.\n\\end{theorem}\n\\begin{proof}\nBy direct calculation, we have\n\\begin{align*}\n\\mathfrak{D}_{z}^{\\eta-\\mu}\\{z^{\\eta-1}(1-z)^{-\\beta};p,\\alpha\\}\n&=\\frac{1}{\\Gamma(\\mu-\\eta)}\\int_0^zt^{\\eta-1}(1-t)^{-\\beta}(z-t)^{\\mu-\\eta-1}E_\\alpha\\Big(-\\frac{pz^2}{t(z-t)}\\Big) dt\\\\\n&=\\frac{z^{\\mu-\\eta-1}}{\\Gamma(\\mu-\\eta)}\\int_0^zt^{\\eta-1}(1-t)^{-\\beta}(1-\\frac{t}{z})^{\\mu-\\eta-1}\nE_\\alpha\\Big(-\\frac{pz^2}{t(z-t)}\\Big) dt.\n\\end{align*}\nSubstituting $t=zu$ in the above equation, we get\n \\begin{align*}\n\\mathfrak{D}_{z}^{\\eta-\\mu}\\{z^{\\eta-1}(1-z)^{-\\beta};p,\\alpha\\}\n&=\\frac{z^{\\mu-1}}{\\Gamma(\\mu-\\eta)}\\int_0^1u^{\\eta-1}(1-uz)^{-\\beta}(1-u)^{\\mu-\\eta-1}E_\\alpha\\Big(-\\frac{pz^2}{uz(z-uz)}\\Big) du\\\\\n=&\\frac{z^{\\mu-1}}{\\Gamma(\\mu-\\eta)}\\int_0^1u^{\\eta-1}(1-uz)^{-\\beta}(1-u)^{\\mu-\\eta-1}E_\\alpha\\Big(-\\frac{p}{u(1-u)}\\Big) du\n\\end{align*}\nUsing (\\ref{pqIhyper}) and after simplification we get the required proof.\n\\end{proof}\n\\begin{theorem}\\label{th4}\nThe following formula holds true:\n\\begin{align}\\label{fd3}\n\\mathfrak{D}_{z}^{\\eta-\\mu}\\{z^{\\eta-1}(1-az)^{-\\alpha}(1-bz)^{-\\beta};p,\\alpha\\}\n=\\frac{\\Gamma(\\eta)}{\\Gamma(\\mu)}z^{\\mu-1}F_{1}\\Big(\\eta,\\alpha,\\beta;\\mu;az,bz;p,\\alpha\\Big),\n\\end{align}\nwhere $\\Re(\\mu)> \\Re(\\eta)>0$, $\\Re(\\alpha)>0$, $\\Re(\\beta)>0$, $|az|<1$ and $|bz|<1$.\n\\end{theorem}\n\\begin{proof}\nConsider the following power series expansion\n\\begin{eqnarray*}\n(1-az)^{-\\alpha}(1-bz)^{-\\beta}=\\sum_{m=0}^\\infty\\sum_{n=0}^\\infty(\\alpha)_m(\\beta)_n\\frac{(az)^m}{m!}\\frac{(bz)^n}{n!}.\n\\end{eqnarray*}\nNow, applying Theorem \\ref{th3}, we obtain\n\\begin{align*}\n&\\mathfrak{D}_{z}^{\\eta-\\mu}\\{z^{\\eta-1}(1-az)^{-\\alpha}(1-bz)^{-\\beta};p,\\alpha\\}\\\\\n&=\\sum_{m=0}^\\infty\\sum_{n=0}^\\infty(\\alpha)_m(\\beta)_n\\frac{(a)^m}{m!}\\frac{(b)^n}{n!}\n\\mathfrak{D}_{z}^{\\eta-\\mu}\\{z^{\\eta+m+n-1};p,\\alpha\\}.\n\\end{align*}\nUsing Theorem \\ref{th1}, we have\n\\begin{align*}\n&\\mathfrak{D}_{z}^{\\eta-\\mu}\\{z^{\\eta-1}(1-az)^{-\\alpha}(1-bz)^{-\\beta};p,\\alpha\\}\\\\\n&=\\sum_{m=0}^\\infty\\sum_{n=0}^\\infty(\\alpha)_m(\\beta)_n\\frac{(a)^m}{m!}\\frac{(b)^n}{n!}\n\\frac{B_{p}^{\\alpha}(\\eta+m+n,\\mu-\\eta)}\n{\\Gamma(\\mu-\\eta)}z^{\\mu+m+n-1}.\n\\end{align*}\nNow, applying (\\ref{Appell}), we get\n\\begin{align*}\n\\mathfrak{D}_{z}^{\\eta-\\mu}\\{z^{\\eta-1}(1-az)^{-\\alpha}(1-bz)^{-\\beta};p,\\alpha\\}\n=\\frac{\\Gamma(\\eta)}{\\Gamma(\\mu)}z^{\\mu-1}F_{1}\\Big(\\eta,\\alpha,\\beta;\\mu;az,bz;p,\\alpha\\Big).\n\\end{align*}\n\\end{proof}\n\\begin{theorem}\\label{th5}\nThe following Mellin transform formula holds true:\n\\begin{eqnarray}\\label{Mellin}\nM\\Big\\{\\mathfrak{D}_{z;p}^{\\mu;\\alpha}(z^{\\eta});p\\rightarrow r,\\Big\\}=\\frac{\\pi}{\\sin(\\pi r)}\\frac{z^{\\eta-\\mu}}{\\Gamma(-\\mu)\\Gamma(1-r\\alpha)}B(\\eta+r+1,-\\mu+r),\n\\end{eqnarray}\nwhere $\\Re(\\eta)>-1$, $\\Re(\\mu)<0$,  $\\Re(r)>0$.\n\\end{theorem}\n\\begin{proof}\nApplying the Mellin transform on definition (\\ref{eq7}), we have\n\\begin{align*}\n&M\\Big\\{\\mathfrak{D}_{z;p}^{\\mu;\\alpha}(z^{\\eta});p\\rightarrow r\\Big\\}=\\int_0^\\infty p^{r-1}\\mathfrak{D}_{z;p}^{\\mu;\\alpha}(z^{\\eta})dp\\\\\n&=\\frac{1}{\\Gamma(-\\mu)}\\int_0^\\infty p^{r-1}\\Big\\{\\int_0^z t^\\eta(z-t)^{-\\mu-1}\\,\nE_\\alpha\\Big(-\\frac{pz^2}{t(z-t)}\\Big)dt\\Big\\}dp \\\\\n&=\\frac{z^{-\\mu-1}}{\\Gamma(-\\mu)}\\int_0^\\infty p^{r-1}\\Big\\{\\int_0^z t^\\eta(1-\\frac{t}{z})^{-\\mu-1}\\,\nE_\\alpha\\Big(-\\frac{pz^2}{t(z-t)}\\Big)dt\\Big\\}dp \\\\\n&=\\frac{z^{\\eta-\\mu}}{\\Gamma(-\\mu)}\\int_0^\\infty p^{r-1}\\Big\\{\\int_0^1 u^\\eta(1-u)^{-\\mu-1}\\,\nE_\\alpha\\Big(-\\frac{p}{u(1-u)}\\Big)du\\Big\\}dp\n\\end{align*}\nFrom the uniform convergence of the integral, the order of integration can be interchanged. Thus, we have\n\\begin{align}\\label{Th-eqn-Mellin}\nM\\Big\\{\\mathfrak{D}_{z;p}^{\\mu;\\alpha}(z^{\\eta});p\\rightarrow r\\Big\\}&=\\frac{z^{\\eta-\\mu}}{\\Gamma(-\\mu)}\\notag\\\\\n&\\times\\int_0^1 u^\\eta(1-u)^{-\\mu-1}\\Big(\\int_0^\\infty p^{r-1}\\,E_\\alpha\\Big(-\\frac{p}{u(1-u)}\\Big)dp\\Big)du.\n\\end{align}\nLetting $v=\\frac{p}{u(1-u)}$, \\eqref{Th-eqn-Mellin} reduces to\n\\begin{align*}\n&M\\Big\\{\\mathfrak{D}_{z;p}^{\\mu;\\alpha}(z^{\\eta});p\\rightarrow r\\Big\\}=\\frac{z^{\\eta-\\mu}}{\\Gamma(-\\mu)}\\int_0^1 u^{\\eta+r}(1-u)^{-\\mu+r-1}\\Big(\\int_0^\\infty v^{r-1}\\,E_\\alpha\\Big(-v\\Big)dv\\Big)du.\n\\end{align*}\nBy using the following formula,\n\\begin{align}\\label{gam}\n\\int_{0}^{\\infty}v^{r-1}E_{\\alpha,\\gamma}^{\\delta}(-wv)dv=\\frac{\\Gamma(r)\\Gamma(\\delta-r)}{\\Gamma(\\delta)w^{r}\\Gamma(\\gamma-r\\alpha)},\n\\end{align}\n for $\\gamma=\\delta=1$ and $w=1$, we have\n\\begin{align*}\nM\\Big\\{\\mathfrak{D}_{z;p}^{\\mu;\\alpha}(z^{\\eta});p\\rightarrow r\\Big\\}&=\\frac{z^{\\eta-\\mu}\\Gamma(r)\\Gamma(1-r)}{\\Gamma(-\\mu)\\Gamma(1-r\\alpha)}\\int_0^1 u^{\\eta+r}(1-u)^{-\\mu+r-1}du\\\\\n&=\\frac{z^{\\eta-\\mu}\\Gamma(r)\\Gamma(1-r)}{\\Gamma(-\\mu)\\Gamma(1-r\\alpha)}B(\\eta+r+1,-\\mu+r),\n\\end{align*}\nNow, using \\eqref{rel1} we get the desired result.\n\\end{proof}\n\\begin{theorem}\\label{th6}\nThe following Mellin transform formula holds true:\n\\begin{align}\\label{Mellin1}\nM\\Big\\{\\mathfrak{D}_{z;p}^{\\mu;\\alpha}((1-z)^{\\alpha});p\\rightarrow r\\Big\\}&=z^{-\\mu}\\frac{\\pi}{\\sin(\\pi r)}\\frac{B(1+r,-\\mu+r)}{\\Gamma(-\\mu)\\Gamma(1-r\\alpha)}\\notag\\\\\n&\\times_2F_1\\Big(\\lambda,r+1;1-\\mu+2r;z\\Big),\n\\end{align}\nwhere $\\Re(p)>0$, $\\Re(\\mu)<0$, $\\Re(r)>0$.\n\\end{theorem}\n\\begin{proof}\nApplying Theorem \\ref{th5} with $\\eta=n$, we can write\n\\begin{align*}\nM\\Big\\{\\mathfrak{D}_{z;p}^{\\mu;\\alpha}((1-z)^{\\lambda});p\\rightarrow r\\Big\\}&=\n\\sum_{n=0}^\\infty\\frac{(\\lambda)_n}{n!}\nM\\Big\\{\\mathfrak{D}_{z;p}^{\\mu;\\alpha}(z^{n});p\\rightarrow r\\Big\\}\\\\\n&=\\frac{\\Gamma(r)\\Gamma(1-r)}{\\Gamma(-\\mu)\\Gamma(1-r\\alpha)}\\sum_{n=0}^\\infty\\frac{(\\lambda)_n}{n!}\nB(n+r+1,-\\mu+r)z^{n-\\mu}\\\\\n&=z^{-\\mu}\\frac{\\Gamma(r)\\Gamma(1-r)}{\\Gamma(-\\mu)\\Gamma(1-r\\alpha)}\\sum_{n=0}^\\infty\nB(n+r+1,-\\mu+r)\\frac{(\\lambda)_nz^{n}}{n!}.\n\\end{align*}\nIn view of \\eqref{rel1}, we obtain the required result.\n\\end{proof}\n\\section{Generating relations}\nIn this section, we derive generating relations of linear and bilinear type for the extended hypergeometric functions.\n\\begin{theorem}\nThe following generating relation holds true:\n\\begin{align}\\label{fd4}\n\\sum_{n=0}^\\infty\\frac{(\\lambda)_n}{n!}\\,_2F_{1;p}^{\\alpha}\\Big(\\lambda+n,\\beta;\\gamma;z\\Big)t^n\n=(1-t)^{-\\lambda}\\,_2F_{1;p}^{\\alpha}\\Big(\\lambda,\\beta;\\gamma;\\frac{z}{1-t}\\Big),\n\\end{align}\nwhere $|z|<\\min(1, |1-t|)$, $\\Re(\\lambda)>0$, $\\Re(\\alpha)>0$, $\\Re(\\gamma)>\\Re(\\beta)>0$.\n\n\\end{theorem}\n\\begin{proof}\nConsider the following series identity\n\\begin{eqnarray*}\n[(1-z)-t]^{-\\lambda}=(1-t)^{-\\lambda}[1-\\frac{z}{1-t}]^{-\\lambda}.\n\\end{eqnarray*}\nThus, the power series expansion yields\n\\begin{eqnarray}\\label{fd5}\n\\sum_{n=0}^\\infty\\frac{(\\lambda)_n}{n!}(1-z)^{-\\lambda}\\Big(\\frac{t}{1-z}\\Big)^n=(1-t)^{-\\lambda}[1-\\frac{z}{1-t}]^{-\\lambda}.\n\\end{eqnarray}\nMultiplying both sides of (\\ref{fd5}) by $z^{\\beta-1}$ and then applying the operator $\\mathfrak{D}_{z;p}^{\\beta-\\gamma;\\alpha}$ on both sides, we have\n\\begin{align*}\n\\mathfrak{D}_{z;p}^{\\beta-\\gamma;\\alpha}\\Big[\\sum_{n=0}^\\infty\\frac{(\\lambda)_n}{n!}\n(1-z)^{-\\lambda}\\Big(\\frac{t}{1-z}\\Big)^nz^{\\beta-1}\\Big]=(1-t)^{-\\lambda}\\mathfrak{D}_{z;p}^{\\beta-\\gamma;\\alpha}\n\\Big[z^{\\beta-1}\\Big(1-\\frac{z}{1-t}\\Big)^{-\\lambda}\\Big].\n\\end{align*}\nInterchanging the order of summation and the operator $\\mathfrak{D}_{z;p}^{\\beta-\\gamma;\\alpha}$, we have\n\\begin{align*}\n\\sum_{n=0}^\\infty\\frac{(\\lambda)_n}{n!}\\mathfrak{D}_{z;p}^{\\beta-\\gamma;\\alpha}\\Big[\nz^{\\beta-1}(1-z)^{-\\lambda-n}\\Big]t^n=(1-t)^{-\\lambda}\\mathfrak{D}_{z;p}^{\\beta-\\gamma;\\alpha}\n\\Big[z^{\\beta-1}\\Big(1-\\frac{z}{1-t}\\Big)^{-\\lambda}\\Big].\n\\end{align*}\nThus by applying Theorem \\ref{th3}, we obtain the required result.\n\\end{proof}\n\\begin{theorem}\nThe following generating relation holds true:\n\\begin{align}\\label{fd6}\n\\sum_{n=0}^\\infty\\frac{(\\beta)_n}{n!}\\,_2F_{1;p}^{\\alpha}\\Big(\\delta-n,\\beta;\\gamma;z\\Big)t^n\n=(1-t)^{-\\beta}F_{1}\\Big(\\lambda,\\delta,\\beta;\\gamma;-\\frac{zt}{1-t};p,\\alpha\\Big),\n\\end{align}\nwhere $|z|<\\frac{1}{1+|t|}$, $\\Re(\\delta)>0$, $\\Re(\\beta)>0$, $\\Re(\\gamma)>\\Re(\\lambda)>0$.\n\n\\end{theorem}\n\\begin{proof}\nConsider the series identity\n\\begin{eqnarray*}\n[1-(1-z)t]^{-\\beta}=(1-t)^{-\\beta}\\Big[1+\\frac{zt}{1-t}\\Big]^{-\\beta}.\n\\end{eqnarray*}\nUsing the power series expansion to the left sides, we have\n\\begin{eqnarray}\\label{fd7}\n\\sum_{n=0}^\\infty\\frac{(\\beta)_n}{n!}(1-z)^nt^n=(1-t)^{-\\beta}\\Big[1-\\frac{-zt}{1-t}\\Big]^{-\\beta}.\n\\end{eqnarray}\nMultiplying both sides of (\\ref{fd7}) by $z^{\\alpha-1}(1-z)^{-\\delta}$ and applying the operator $\\mathfrak{D}_{z;p}^{\\lambda-\\gamma;\\alpha}$ on both sides, we have\n\\begin{eqnarray*}\n\\mathfrak{D}_{z;p}^{\\lambda-\\gamma;\\alpha}\\Big[\\sum_{n=0}^\\infty\\frac{(\\beta)_n}{n!}z^{\\alpha-1}(1-z)^{-\\delta+n}t^n\\Big]\n=(1-t)^{-\\beta}\\mathfrak{D}_{z;p}^{\\lambda-\\gamma;\\alpha}\\Big[z^{\\lambda-1}(1-z)^{-\\delta}\\Big(1-\\frac{-zt}{1-t}\\Big)^{-\\beta}\n\\Big],\n\\end{eqnarray*}\nwhere $\\Re(\\lambda)>0$ and $|zt|<|1-t|$,\nthus by Theorem \\ref{th2}, we have\n\\begin{eqnarray*}\n\\sum_{n=0}^\\infty\\frac{(\\beta)_n}{n!}\\mathfrak{D}_{z;p}^{\\lambda-\\gamma;\\alpha}\\Big[z^{\\lambda-1}(1-z)^{-\\delta+n}\\Big]t^n\n=(1-t)^{-\\beta}\\mathfrak{D}_{z;p}^{\\lambda-\\gamma;\\alpha}\\Big[z^{\\lambda-1}(1-z)^{-\\delta}\\Big(1-\\frac{-zt}{1-t}\\Big)^{-\\beta}\n\\Big].\n\\end{eqnarray*}\nApplying Theorem \\ref{th4} on both sides, we get the required result.\n\\end{proof}\n\n\\begin{theorem}\\label{th7}\nThe following result holds true:\n\\begin{eqnarray}\\label{fd8}\n\\mathfrak{D}_{z;p}^{\\eta-\\mu;\\alpha}\\Big[z^{\\eta-1}E_{\\gamma,\\delta}^{\\mu}(z)\\Big]=\\frac{z^{\\mu-1}}{\\Gamma(\\mu-\\eta)}\n\\sum_{n=0}^\\infty\\frac{(\\mu)_n}{\\Gamma(\\gamma n+\\delta)}B_{p}^{\\alpha}(\\eta+n,\\mu-\\eta)\\frac{z^n}{n!},\n\\end{eqnarray}\nwhere $\\gamma,\\delta,\\mu, \\alpha\\in\\mathbb{C}$, $\\Re(p)>0$,  $\\Re(\\mu)>\\Re(\\eta)>0$ and $E_{\\gamma,\\delta}^{\\mu}(z)$ is Mittag-Leffler function (see \\cite{Prabhakar}) defined as:\n\\begin{eqnarray}\\label{fd9}\nE_{\\gamma,\\delta}^{\\mu}(z)=\\sum_{n=0}^\\infty\\frac{(\\mu)_n}{\\Gamma(\\gamma n+\\delta)}\\frac{z^n}{n!}, \\Re(\\gamma)>0.\n\\end{eqnarray}\n\\begin{proof}\nUsing (\\ref{fd9}) in (\\ref{fd8}), we have\n\\begin{eqnarray*}\n\\mathfrak{D}_{z;p}^{\\eta-\\mu;\\alpha}\\Big[z^{\\eta-1}E_{\\gamma,\\delta}^{\\mu}(z)\\Big]=\n\\mathfrak{D}_{z;p}^{\\eta-\\mu;\\alpha}\\Big[z^{\\eta-1}\\Big\\{\\sum_{n=0}^\\infty\\frac{(\\mu)_n}{\\Gamma(\\gamma n+\\delta)}\\frac{z^n}{n!}\\Big\\}\\Big].\n\\end{eqnarray*}\nBy Theorem \\ref{th2}, we have\n\\begin{eqnarray*}\n\\mathfrak{D}_{z;p}^{\\eta-\\mu;\\alpha}\\Big[z^{\\eta-1}E_{\\gamma,\\delta}^{\\mu}(z)\\Big]=\n\\sum_{n=0}^\\infty\\frac{(\\mu)_n}{\\Gamma(\\gamma n+\\delta)}\\Big\\{\\mathfrak{D}_{z;p}^{\\eta-\\mu;\\alpha}\\Big[z^{\\eta+n-1}\\Big]\\Big\\}.\n\\end{eqnarray*}\nApplying Theorem \\ref{th1}, we get the required proof.\n\\end{proof}\n\\end{theorem}\n\\begin{remark}\nIn \\cite{Mehrez} Mehrez and Tomovski introduced a new $p$-Mittag-Leffler function defined by\n\\begin{align}\\label{MT}\nE_{\\lambda,\\gamma,\\delta;p}^{(\\mu,\\eta,\\omega)}(z)=\\sum_{k=0}^\\infty\\frac{(\\mu)_k}{[\\Gamma(\\gamma k+\\delta)]^\\lambda}\n\\frac{B_p(\\eta+k,\\omega-\\eta)}{B(\\eta,\\omega-\\eta)}\\frac{z^k}{k!}, (z \\in \\mathbb{C})\n\\end{align}\n$$(\\mu,\\,\\eta,\\,\\omega,\\,\\gamma,\\,\\delta,\\,\\lambda>0,\\, \\Re(p)>0).$$\n\n\\end{remark}\nHence we get the following corollary.\n\\begin{corollary} The following formula holds true for $E_{\\gamma,\\delta}^{\\mu}(z)$:\n\\begin{align}\n\\mathfrak{D}_{z}^{\\eta-\\mu}\\Big\\{z^{\\eta-1}E_{\\gamma,\\delta}^{\\mu}(z);p\\Big\\}=\nz^{\\mu-1}\\frac{\\Gamma(\\eta)}{\\Gamma(\\mu)}E_{1,\\gamma,\\delta;p}^{(\\mu,\\eta,\\mu)}(z),\n\\end{align}\n$$(\\mu,\\eta,\\,\\gamma,\\,\\delta>0,\\, \\Re(p)>0).$$\n\\end{corollary}\n\n\\begin{theorem}\nThe following result holds true:\n\\begin{align}\\label{fd10}\n\\mathfrak{D}_{z;p}^{\\eta-\\mu;\\alpha}\\Big\\{z^{\\eta-1}\\,_m\\Psi_n\\left[\n                                                                        \\begin{array}{cc}\n                                                                          (\\alpha_i,A_i)_{1,m}; &  \\\\\n                                                                           & |z \\\\\n                                                                          (\\beta_j,B_j)_{1,n}; &  \\\\\n                                                                        \\end{array}\n                                                                      \\right]\n\\Big\\}&=\\frac{z^{\\mu-1}}{\\Gamma(\\mu-\\eta)}\\notag\\\\\n&\\times\\sum_{k=0}^\\infty\\frac{\\prod_{i=1}^{m}\\Gamma(\\alpha_i+A_ik)}{\\prod_{j=1}^{n}\\Gamma(\\beta_j+B_jk)}\nB_{p}^{\\alpha}(\\eta+k,\\mu-\\eta)\\frac{z^k}{k!},\n\\end{align}\nwhere  $\\Re(p)>0$, $\\Re(\\alpha)>0$, $\\Re(\\mu)>\\Re(\\eta)>0$ and $_m\\Psi_n(z)$ represents the Fox-Wright function (see \\cite{Kilbas}, pp. 56-58)\n\\begin{eqnarray}\\label{fd11}\n_m\\Psi_n(z)=\\,_m\\Psi_n\\left[\n \\begin{array}{cc}\n (\\alpha_i,A_i)_{1,m}; &  \\\\\n & |z \\\\\n (\\beta_j,B_j)_{1,n}; &  \\\\\n\\end{array}\n\\right]=\\sum_{k=0}^\\infty\\frac{\\prod_{i=1}^{m}\\Gamma(\\alpha_i+A_ik)}{\\prod_{j=1}^{n}\\Gamma(\\beta_j+B_jk)}\\frac{z^k}{k!}.\n\\end{eqnarray}\n\\end{theorem}\n\\begin{proof}\nApplying Theorem \\ref{th1} and followed the same procedure used in Theorem \\ref{th7}, we get the desired result.\n\\end{proof}\n\\begin{remark}\nIn \\cite{Sharma} Sharma and Devi introduced extended Wright generalized\nhypergeometric function defined by\n\\begin{align}\\label{fd12}\n_{m+1}\\Psi_{n+1}(z;p)&=\\,_{m+1}\\Psi_{n+1}\\left[\n \\begin{array}{cc}\n (\\alpha_i,A_i)_{1,m},(\\gamma,1); &  \\\\\n & |(z;p) \\\\\n (\\beta_j,B_j)_{1,n}; (c,1) &  \\\\\n\\end{array}\n\\right]\\notag\\\\=&\\frac{1}{\\Gamma(c-\\gamma)}\\sum_{k=0}^\\infty\\frac{\\prod_{i=1}^{m}\\Gamma(\\alpha_i+A_ik)}{\\prod_{j=1}^{n}\n\\Gamma(\\beta_j+B_jk)}\\frac{B_p(\\gamma+k,c-\\gamma)z^k}{k!},\n\\end{align}\n$$(\\Re(p)>0,\\, \\Re(c)>\\Re(\\gamma)>0).$$\n\\end{remark}\nHence we get the following corollary.\n\\begin{corollary} The following results holds true for the Fox-Wright\nhypergeometric function:\n\\begin{align}\n\\mathfrak{D}_{z}^{\\eta-\\mu}\\Big\\{z^{\\eta-1}\\,_{m}\\Psi_{n}\\left[\n                                                                        \\begin{array}{cc}\n                                                                          (\\alpha_i,A_i)_{1,m}; &  \\\\\n                                                                           & |z \\\\\n                                                                          (\\beta_j,B_j)_{1,n}; &  \\\\\n                                                                        \\end{array}\n                                                                      \\right]\n;p\\Big\\}&=z^{\\mu-1}{}_{m+1}\\Psi_{n+1}\\left[\n \\begin{array}{cc}\n (\\alpha_i,A_i)_{1,m},(\\eta,1); &  \\\\\n & |(z;p) \\\\\n (\\beta_j,B_j)_{1,n}; (\\mu,1) &  \\\\\n\\end{array}\n\\right],\n\\end{align}\n$$(\\Re(p)>0,\\, \\Re(\\mu)>\\Re(\\eta)>0).$$\n\\end{corollary}\n\\section{Concluding remarks}\nIn this paper, we  established a modified extension of Riemnn-Liouville fractional derivative operator.  We conclude that when $\\alpha=1$, then all the results established in this paper will reduce to the  results associated with classical Riemnn-Liouville derivative operator (see \\cite{Ozerslan}).\nSimilarly, if we letting $\\alpha=1$ and $p=0$ then all the results established in this paper will reduce to the  results associated with classical Riemnn-Liouville fractional derivative operator (see \\cite{Kilbas}).\n\n\\begin{remark}\nThe preprint of this paper is available at 'https:\/\/arxiv.org\/abs\/1801.05001'.\n\\end{remark}\n\n\\textbf{Acknowledgments}\\\\\nThe authors would like to express profound gratitude to Prof. Virginia Kiryakova for valuable comments and remarks which improved the final version of this paper\\\\\n\n\n\n\\vskip 20pt\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nCataclysmic Variables are semi-detached binaries built of a white dwarf (WD) which is accreting mass from a Roche-lobe filling main sequence (MS) star. Mass transfer is driven by the loss of angular momentum in absence of strong magnetic fields, and the transferred material forms an accretion disc surrounding the central WD (see e.g., \\citealt{warner95-1}, \\citealt{hellier_book} and \\citealt{kniggeetal11-1} for comprehensive reviews). Since the structure of both components is relatively simple, CVs are one of the best sources to test our understanding of many astrophysical phenomena involving evolution of compact, interacting binaries and accretion phenomena. Their study helps to resolve standing discrepancies between current population models and observations in many present and complex topics including black hole binaries, short gamma-ray bursts, X-ray transients, milli-second pulsars and Supernovae Ia. \n\nThe orbital period distribution is the main tool to study the evolution of CVs, as it presents features in key points that allow us to understand their behaviour. As a consequence of the angular momentum loss and the mechanisms driving it, CVs move from long orbital periods and high mass transfer rates to short orbital periods and low mass transfer rates \\citealt{paczynski+sienkiewicz83-1}; \\citealt{Townsley09}; \\citealt{GoliaschNelson15}; \\citealt{palaetal7}). The evolution proceeds in this way until the system reaches the ``period minimum'' at $\\sim$ 76-80 minutes (\\citealt{knigge06-1}; \\citealt{gaensickeetal09}) in which the donor turns into a brown dwarf. Consequently, the orbital separation and period now increases as the mass transfer continues, becoming in the so-called period bouncers, faint systems with short orbital periods. On their way to the period minimum, observations show an abrupt drop in the number of systems with periods between 2 and 3h, referred to as the period gap. Below this range (Porb $<$ 2\\,h) systems have low mass-transfer rates governed by gravitational radiation (GR) \\citep{patterson84-1} while the higher mass-transfer rates above the gap (Porb $>$ 3\\,h) are a consequence of the stronger magnetic braking (MB) (\\citealt{rappaportetal83-1}; \\citealt{spruit+ritter83-1}; \\citealt{hameuryetal88-2}; \\citealt{davis08}). The standard explanation suggests that MB switches off when a CV has evolved down to 3h, the secondary contracts to its thermal equilibrium and detaches from its Roche lobe. Such systems crossing the gap are known as detached Cataclysmic Variables (dCVs). The continuing angular momentum loss by GR shrinks the orbit until at a period of about 2h, the Roche lobe makes contact with the stellar surface again and mass transfer is re-established albeit at a lower level. \n\nDuring this evolution, the CV will change appearance: The relative contribution of the WD, the secondary star and the accretion disc or stream makes for unique colours \\citep[e.g.][]{szkodyetal02-2}. The systems thus occupy distinct locations in colour-colour diagrams with respect to single stars. Due to the relatively small sample of CVs and the inherent difficulty of any source in obtaining its distance, it has not been possible so far to perform an analysis of the CV absolute magnitude distribution. \nWith the arrival of {\\it Gaia},\nthis has changed. Already \\cite{Pala19} show the advances that {\\it Gaia} parallaxes bring to the understanding of CVs, and we now have the data to study CVs in the HR-diagram.\n\nThe paper is organised as such: In section \\ref{sec:gaia}, we explain how we use {\\it Gaia} to define our CV sample. In section \\ref{sec:HRdiagram} the results are presented for all CVs and the trends with the orbital periods and the subtypes are discussed. Finally, we present our summary in \\ref{sec:Summary}.\n\n\n\\section{The Catalogue of {\\it Gaia} DR2 and the cross-match with CV catalogue}\\label{sec:gaia}\n\nThe goal of the {\\it Gaia} \nspace mission is to make  the largest, most precise three-dimensional map of the Milky Way to-date by detecting and measuring the motion and parallax of each star in its orbit around the centre of the Galaxy. To this means, the three filters $G$, $G_{BP}$ and $G_{RP}$ are observed at several epochs over a period of about 670 days of mission operations. For details, see \\cite{GaiaDR2}.\n\nThe second data release (GDR2 hereafter) is based on 22 months of observations and provides positions, parallaxes and proper motions for 1.3 billion sources up to G $\\sim$ 20 magnitudes. This kind of data allows the derivation of distances and absolute magnitudes to study the position of all objects in the global HR-diagram.\n\n\\subsection{Deriving absolute magnitudes}\\label{subsubsec:Abs_mag}\n\nOne of the aims of this paper is finding the CV locus in the HR-diagram. We make use of GDR2 data to compute their absolute magnitudes. \nThe absolute magnitude $M$ of an object is given by:\n\\begin{equation}\nM = m + 5 - 5\\log(d) + A,\n\\label{eq:M}\n\\end{equation}\n\nwhere $m$ is the apparent magnitude, $A$ is the interstellar extinction and $d$, the distance to the source which can be obtained by the GDR2 data. \nGDR2 provides weighted mean fluxes\\footnote{Weighted means are used because flux errors on different epochs may vary depending on the configuration of each observation, see \\cite{CarrascoGDR1} and \\cite{RielloGDR2} for detailed information.} and, as CVs are variable stars, this procedure has an effect on their $G\n, $G_{BP}$ and $G_{RP}$ values. The degree of impact might be determined by comparing the 670-days length of the {\\it Gaia}-mission and the cycle of variation length for every CV subtype. The highest impact is on Dwarf Novae systems as they can have outbursts even on a weekly base (\\citealt{SterkenBook} and references therein). The low recurrence in Novae, Nova-like and Magnetic CVs should have no significant impact on the overall sample.\n\nInferring the distance from the {\\it Gaia} DR2 parallax is not a trivial issue. Distance can be derived as the inverse of the parallax, only if the parallax error is lower than 20\\% and by doing so we would discard 80\\% of the sources (\\citealt{Luri18}). We used instead the distances inferred by \\citet{Bailer-Jones18} who compute distances and their uncertainties through a probabilistic analysis based on the Bayes theorem and adopting an exponentially decreasing space density\nprior\\footnote{For a detailed explanation of this approach and an analysis of applying this technique refer to \\citet{Bailer-Jones18}. For a discussion of the use of different priors, see\n\\citet{Bailer-Jones18},\\citet{Luri18},\\citet{igoshevetal2016},\\citet{astraatmadjaetal16}.\n} \nfor the 1.33 billion sources from GDR2. These are available using ADQL\\footnote{http:\/\/gaia.ari.uni-heidelberg.de\/tap.html} and we here use them to derive  the absolute magnitudes through Equation~\\ref{eq:M}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{The CV sample}\\label{sec:CVsample}\n\n\nThe Catalog and Atlas of Cataclysmic Variables \\citep{DownesCat} includes all objects which have been classified as a CV at some point in time. Although it was frozen on February 1st, 2006, it is one of the main references among the community, providing coordinates, proper motion, type, chart, spectral and period references for all 1830 sources when available. In order to obtain the purest sample, we discarded from this catalogue the objects designated as ``NON-CV'', which are stars that have been previously identified as CVs but later confuted, and those with the extensions ``:'' and ``::'' because their classification is not conclusive.\n\nThe Catalog of Cataclysmic Binaries, Low-Mass X-Ray Binaries and Related Objects \\citep{RKCat} which only contains objects with a measured period, is updated up to December 31st, 2015 and it provides coordinates, apparent magnitudes, orbital parameters, stellar parameters of the components and other characteristic properties for 1429 CVs. In this case uncertain values are followed by only one ``:'' and have been discarded as well. \n\nBoth catalogues have been merged into a final sample of 1920 CVs, out of which 1187 are contained in the GDR2 footprint. The density studies of CVs in the HR-diagram\nwere done using this full sample. For 839 of these systems, the orbital period is known, and for 1130 systems, the subtype is unambiguously known (see Tab.\\ref{tab:subtypes_sample}).\n\n\n\\begin{table}\n\\scriptsize\n\t\\centering\n\t\\caption{\\label{tab:subtypes_sample} Distribution of the CV sample utilised by subtype.}\n\t\\begin{tabular}{lcccc} \n\t\t\\hline\nCV subtype & Periods &  Main & \\multicolumn{2}{c}{Centroid position in HRD}\\\\\n& sample & sample & $G_{BP}-G_{RP}$ & $G_{abs}$\\\\\n\t\t\\hline\n\t\tNovalike & 76 & 119 & 0.37 & 5.63 \\\\\n\t\tDwarf Novae & 484 & 688 & 0.64 & 9.49 \\\\\n\t\tOld Novae & 77 & 119 & 0.79 & 5.58 \\\\\n\t\tPolar & 75 & 135 & 0.83 & 9.67 \\\\\n\t\tIntermediate Polar & 51 & 69 & 0.59 & 5.61 \\\\\n\t\t\\hline\n\t\tTotal Sample & 839 & 1130 & &\\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table}\n\n\n\n\n\n\\section{CVs in the HR-diagram}\\label{sec:HRdiagram}\n\n\n\n\n\\subsection{The impact of the orbital period}\n\n\n\n\n\n\n\n\n\n\n\\begin{figure*}\n    \\centering\n\t    \\begin{tabular}{c c}\n\t        \\includegraphics[width=0.45\\textwidth]{H-Rdiag_periods_rast.pdf}&\t\\includegraphics[width=0.42\\textwidth]{H-R_density_rast.pdf}\t\n\t    \\end{tabular}\n    \\caption{In grey, all stars from {\\it Gaia}'s 2nd data release are plotted in the HR-diagrams. On the left side, the CVs period distribution, the CVs from our sample with known orbital periods (see Section \\ref{sec:CVsample}) are plotted in larger dots. The colour of each dot refers to the orbital period as given in the bar at the right of the panel. CVs with larger periods lie close to the main sequence path getting shorter while approaching the white dwarfs area. On the right, the density distribution of our whole sample of CVs (brown dots), surfaces with different tones of blue represent areas of equal density. On the x- and y-axis the marginal distributions are shown. CVs lie on average between the MS path and the WDs with a high density area peaking at $G_{BP}-G_{RP} \\sim 0.56$ and $G_{abs} \\sim 10.15$. Such area corresponds to the overpopulation below the period gap as reflected in the left panel by black dots, CVs with orbital period below 2h\n    \\label{fig:HRper_densityMap}}\n\\end{figure*}\n\nLeft panel of Fig. \\ref{fig:HRper_densityMap} displays the CV locus in the HR-diagram of all CVs for which an orbital period is known (839 systems).  The orbital period of each system is represented by the colour of the symbol as defined in the auxiliary axis. The CVs lie on average between the main sequence stars and the WDs.\nA clear trend is seen on their position with the orbital period: CVs with longer periods fall close to the main sequence path, while, as the orbital period decreases, they approach the WDs region.\nThis behaviour can be understood from the contribution of the secondary star. On average, a Roche-lobe filling secondary star is larger and brighter for longer orbital periods, while the WD does not change much during the secular CV evolution. Hence, the contribution of the secondary should be more dominant for longer orbital periods. Systems below the period gap, are instead dominated by their WD, as the secondary becomes only visible in the near infrared and does not contribute to the {\\it Gaia} colour. The contribution of the accretion disc should change colour and magnitude depending on the sub-type and will be discussed in the next subsection.\n\nThe right panel of Fig. \\ref{fig:HRper_densityMap} shows the locus of all CVs of our sample defined in Section \\ref{sec:CVsample} within {\\it Gaia's} HR-diagram. On the x- and y-axis, the respective projected density is plotted. A high density area is well distinguishable at $G_{BP}-G_{RP} \\sim 0.56$ and $G_{abs} \\sim 10.15$ (values obtained from the mode of the marginal distributions) which corresponds to the population below the period gap.\n\n\n\\subsection{The locus depending on the subtype}\nFigure \\ref{fig:HR_per_types} exhibits the distribution of every CV subtype on the HR-diagram. Bivariate Gaussian distributions are computed for 1 and 3 $\\sigma$ given by\n\n\\begin{equation}\np(x, y | \\mu_x, \\mu_y, \\sigma_x, \\sigma_y, \\sigma_{xy}) = \\frac{1}{2 \\pi \\sigma_x \\sigma_y \\sqrt{1-\\rho^2}}exp \\left(\\frac{-z^2}{2(1-\\rho^2)}\\right),\n\\label{eq:bivariate1}\n\\end{equation}\n\nwhere\n\n\\begin{equation}\nz^2 = \\frac{(x-\\mu_x)^2}{\\sigma_x^2}+\\frac{(y-\\mu_y)^2}{\\sigma_y^2}-2\\rho \\frac{(x-\\mu_x)(y-\\mu_y)}{\\sigma_x \\sigma_y},      \n\\label{eq:bivariate2}\n\\end{equation}\n\n\\begin{equation}\n\\rho = \\frac{\\sigma_{xy}}{\\sigma_x \\sigma_y},\n\\label{eq:bivariate3}\n\\end{equation}\n\nusing the median instead of the mean and the interquartile range to estimate variances in order to avoid the impact of outliers. The results are given in Table \\ref{tab:subtypes_sample}.\n\n\\begin{figure*\n\\center\n\t\\begin{tabular}{c c c}\n\t\\includegraphics[width=0.3\\textwidth]{H-Rdiag_types_rast.pdf}\t&\n\t\\includegraphics[width=0.3\\textwidth]{H-Rdiag_Nova_like_rast.pdf}\t&\n\t\\includegraphics[width=0.3\\textwidth]{H-Rdiag_Dwarf_Novae_rast.pdf}\t\\\\\n\t\\includegraphics[width=0.3\\textwidth]{H-Rdiag_Novae_rast.pdf}\t&\n\t\\includegraphics[width=0.3\\textwidth]{H-Rdiag_Polars_rast.pdf}\t&\n\t\\includegraphics[width=0.3\\textwidth]{H-Rdiag_Intermediate_Polars_rast.pdf}\t\\\\\n\t\\includegraphics[width=0.3\\textwidth]{NL_N_hist_short.pdf}\t&\n\t\\includegraphics[width=0.3\\textwidth]{AM_IP_hist_short.pdf}\t&\n\t\\includegraphics[width=0.3\\textwidth]{DN_WZSge_hist_short.pdf}\t\n\t\\end{tabular}\n\\caption{The distribution of CV subtypes in the HR-diagram. Top-left panel shows all subtypes together, after that every subtype separately. The dashed ellipses represent 1 and 3 $\\sigma$ of each subtype bivariate Gaussian distribution. The sample utilized here is composed by all CVs in the Ritter\\&Kolb and Downes catalogues (see Section \\ref{sec:CVsample}) whose subtype is unambiguously known and are included in the {\\it Gaia} footprint; 119 Nova-likes, 688 Dwarf Novae, 119 Novae, 135 Polars and 69 Intermediate Polars. On the bottom, the period histograms for each subtype.  \\label{fig:HR_per_types}}\n\\end{figure*}\n\nNova-likes are dominated by a high mass-transfer accretion disc, that usually \novershines\nthe WD and the secondary star at optical and even infrared wavelengths. Their colour and final absolute magnitude mainly depends on the inclination with respect to the line of sight. In the HR-diagram, they concentrate around $G_{abs}=5.63$ and $G_{BP}-G_{RP}=0.37$, i.e. on the blue and bright corner of all CVs. A similar locus but with a much higher scatter is occupied by the old novae and by intermediate polars. This can be explained by the eclectic composition of these two sub-groups which also contain a large fraction of novalike stars. \n\nIn contrast, polars which do not accrete mass through a disc, are much fainter and their colour and magnitude will depend on the nature of the secondary. In the HR-diagram they scatter around $G_{abs}=9.67$ and $G_{BP}-G_{RP}=0.83$\nrepresenting the reddest and faintest of all the CV subgroups.\n\nDwarf novae (DNe) occupy the whole region between MS stars and WDs with the centroid being at $G_{abs}=9.49$ and $G_{BP}-G_{RP}=0.64$. Since the secondary star in these systems can be anything from an early K-type star down to a brown dwarf, the range in colours and magnitude is not surprising. In addition, these sources are characterised by undergoing \nregular outbursts increasing their brightness and blueness. \nAs discussed in Section \\ref{sec:gaia}, the given magnitude is a weighted mean of several epochs and thus also increases the spread of this distribution. A detailed study of the DNe locus depending on their subtype and outburst state can be done following the next {\\it Gaia} release when individual measurements and epochs become available\n\n\nWZ Sge-type objects deserve a separate mention, a class of DNe characterised by great outburst amplitudes, slow declines and long intervals between outbursts compared with ordinary DNe. These kind of systems have been considered to be period bouncer candidates \\citep{Patterson11}, some of them extensively investigated in this regard (QZ Lib, \\citealt{Pala18}; J122221 \\citealt{vitaly17} and \\citealt{kato13b}; J184228 \\citealt{kato13b}; J075418 and J230425 \\citealt{Nakata14}). We have plotted a sample of 71 of such systems in the upper-right panel of Fig. \\ref{fig:HR_per_types}, along with the rest of DNe, and it can be seen that they concentrate near the WDs area. This is consistent with them being period bouncers or similar systems, as these are the CVs with the lowest mass transfer and faintest secondary stars. The disc is only visible in some emission lines, the secondary does not contribute to the optical range at all.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Detached CVs}\n\n\\begin{figure\n\t\\includegraphics[width=0.425\\textwidth]{HR_WDM4-6.pdf}\n\\caption{\\label{fig:dCVs} Systems comprised by a WD and an M4-6. The figure contains the 39 objects from the sample \nby \\citep{Zorotovic16} present in {\\it Gaia} (see text). In green are those within the period gap and in brown the rest. All of them follow a trend but the objects within the gap are on average fainter.\n}\n\\end{figure}\n\nAnother question we can address is finding the locus occupied by the so-called detached cataclysmic variables (dCVs) crossing the orbital period gap. A first approach could be made by finding the area with boundaries in 2 and 3h in the left panel of Fig. \\ref{fig:HRper_densityMap} using\nregression techniques. However, since dCVs no longer \ncontain an accretion disc, they should appear fainter than regular CVs of the same period. \n\nDue to the continuous mass loss, the donor is being driven out of equilibrium and secondaries in CVs just above the period gap are bloated up to 30\\% with respect to regular MS stars \\citep{kniggeetal11-1}. When the mass transfer stops, the secondary shrinks towards its thermal equilibrium radius to nearly its equivalent for MS stars \\citep{stevehowell-01} and hence we expect secondaries in dCVs to be comparable to single MS stars of the same type.\n\nSince the mass transfer ceases, the mass and spectral type of the donor star stays constant during the interval in which the binary is detached. In regular CVs this happens at $M_{sec} = 0.2 \\pm 0.02$ $M\\odot$ \\citep{knigge06-1} and spectral type $\\sim M6$ \\citep{rebassaetal2007}, though variations occur depending on the moment in which the CV started the mass transfer and the time passed as CV until the secondary becomes fully convective. In the extreme case of a CV starting the mass transfer within the period gap range, the donor star type will be that of a fully convective isolated M star, which, according to \\cite{Chabrier-97}, occurs at $M_{sec} \\sim0.35$ $M\\odot$ and spectral type M4. We thus assume that the secondary of dCVs is in the range M4-M6.\n\nSo far the only observational evidence for the existence of dCVs come from \\cite{Zorotovic16}, who show that the orbital period distribution of detached close binaries consisting of a WD and an M4-M6 secondary star cannot be produced by Post Common Envelope Binaries alone, but a contribution of dCVs is needed to explain the peak between 2 and 3h. They also show that the systems inside this peak have a higher average mass than would be expected for normal WDMS systems. Still, with only 52 such systems known in total (WDMS systems with secondary spectral types in the range M4-6 and orbital periods below 10h) and 12 between 2 and 3h, the significance is not very high.\n\n\nWe distinguish two groups, the sources with orbital periods corresponding to those of the period gap (2-3h) and therefore, more likely to be dCVs, and the rest with periods outside this range. In Fig. \\ref{fig:dCVs} they are plotted in the HR-diagram, the former appear\nfainter compared to the latter. This can be explained by the higher WD masses in CVs, and consequently in dCVs, than in PCEBs, making them smaller in size and surface and contributing in a lesser extent on the brightness\nof the whole system.\n\n\n\\section{Summary}\\label{sec:Summary}\nWe have analyzed the evolutionary cycle of CVs from a statistical perspective using {\\it Gaia} DR2 data in conjunction with the HR-diagram tool. We have reported the discovery of a trend of the period and mass accretion with colour and absolute magnitude. We have also investigated their density distribution as a whole population, peaking at $G_{BP}-G_{RP} \\sim 0.56$ and $G_{abs} \\sim 10.15$, and the contribution of the main CV subtypes to this regard, highlighting the location of WZ Sge systems, which are period bouncer candidates. Finally, we have identified the location and a trend among systems comprised of a WD and secondary in the range M4-M6, which correspond with dCVs, CVs going through the orbital period gap.\n\n\\section*{Acknowledgements}\nThis research has made use of the VizieR catalogue access tool, CDS, Strasbourg, France (DOI: 10.26093\/cds\/vizier). The original description of the VizieR service was published in 2000, A\\&AS 143, 23.\n\n\n\n\n\\bibliographystyle{mnras}\n\n\\section{Introduction}\n\nThis is a simple template for authors to write new MNRAS papers.\nSee \\texttt{mnras\\_sample.tex} for a more complex example, and \\texttt{mnras\\_guide.tex}\nfor a full user guide.\n\nAll papers should start with an Introduction section, which sets the work\nin context, cites relevant earlier studies in the field by \\citet{Others2013},\nand describes the problem the authors aim to solve \\citep[e.g.][]{Author2012}.\n\n\\section{Methods, Observations, Simulations etc.}\n\nNormally the next section describes the techniques the authors used.\nIt is frequently split into subsections, such as Section~\\ref{sec:maths} below.\n\n\\subsection{Maths}\n\\label{sec:maths}\n\nSimple mathematics can be inserted into the flow of the text e.g. $2\\times3=6$\nor $v=220$\\,km\\,s$^{-1}$, but more complicated expressions should be entered\nas a numbered equation:\n\n\\begin{equation}\n    x=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}.\n\t\\label{eq:quadratic}\n\\end{equation}\n\nRefer back to them as e.g. equation~(\\ref{eq:quadratic}).\n\n\\subsection{Figures and tables}\n\nFigures and tables should be placed at logical positions in the text. Don't\nworry about the exact layout, which will be handled by the publishers.\n\nFigures are referred to as e.g. Fig.~\\ref{fig:example_figure}, and tables as\ne.g. Table~\\ref{tab:example_table}.\n\n\\begin{figure}\n\n\n\n\t\\includegraphics[width=\\columnwidth]{example}\n    \\caption{This is an example figure. Captions appear below each figure.\n\tGive enough detail for the reader to understand what they're looking at,\n\tbut leave detailed discussion to the main body of the text.}\n    \\label{fig:example_figure}\n\\end{figure}\n\n\\begin{table}\n\t\\centering\n\t\\caption{This is an example table. Captions appear above each table.\n\tRemember to define the quantities, symbols and units used.}\n\t\\label{tab:example_table}\n\t\\begin{tabular}{lccr}\n\t\t\\hline\n\t\tA & B & C & D\\\\\n\t\t\\hline\n\t\t1 & 2 & 3 & 4\\\\\n\t\t2 & 4 & 6 & 8\\\\\n\t\t3 & 5 & 7 & 9\\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table}\n\n\n\\section{Conclusions}\n\nThe last numbered section should briefly summarise what has been done, and describe\nthe final conclusions which the authors draw from their work.\n\n\\section*{Acknowledgements}\n\nThe Acknowledgements section is not numbered. 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Only these three are included, if you require other symbols you will need to include the \\verb'amsmath' or \\verb'amsymb' packages (see section~\\ref{sec:packages}).\n\\item \\verb\"usedcolumn\" -- includes the package \\verb\"dcolumn\", which includes two new types of column alignment for use in tables.\n\\end{itemize}\n\nSome of these options are deprecated and retained for backwards compatibility only.\nOthers are used in almost all papers, but again are retained as options to ensure that papers written decades ago will continue to compile without problems.\nIf you want to include any other packages, see section~\\ref{sec:packages}.\n\n\\section{Title page}\n\nIf you are using \\texttt{mnras\\_template.tex} the necessary code for generating the title page, headers and footers is already present.\nSimply edit the title, author list, institutions, abstract and keywords as described below.\n\n\\subsection{Title}\nThere are two forms of the title: the full version used on the first page, and a short version which is used in the header of other odd-numbered pages (the `running head').\nEnter them with \\verb'\\title[]{}' like this:\n\\begin{verbatim}\n\\title[Running head]{Full title of the paper}\n\\end{verbatim}\nThe full title can be multiple lines (use \\verb'\\\\' to start a new line) and may be as long as necessary, although we encourage authors to use concise titles. The running head must be $\\le~45$ characters on a single line.\n\nSee appendix~\\ref{sec:advanced} for more complicated examples.\n\n\\subsection{Authors and institutions}\n\nLike the title, there are two forms of author list: the full version which appears on the title page, and a short form which appears in the header of the even-numbered pages. Enter them using the \\verb'\\author[]{}' command.\n\nIf the author list is more than one line long, start a new line using \\verb'\\newauthor'. Use \\verb'\\\\' to start the institution list. Affiliations for each author should be indicated with a superscript number, and correspond to the list of institutions below the author list.\n\nFor example, if I were to write a paper with two coauthors at another institution, one of whom also works at a third location:\n\\begin{verbatim}\n\\author[K. T. Smith et al.]{\nKeith T. Smith,$^{1}$\nA. N. Other,$^{2}$\nand Third Author$^{2,3}$\n\\\\\n$^{1}$Affiliation 1\\\\\n$^{2}$Affiliation 2\\\\\n$^{3}$Affiliation 3}\n\\end{verbatim}\nAffiliations should be in the format `Department, Institution, Street Address, City and Postal Code, Country'.\n\nEmail addresses can be inserted with the \\verb'\\thanks{}' command which adds a title page footnote.\nIf you want to list more than one email, put them all in the same \\verb'\\thanks' and use \\verb'\\footnotemark[]' to refer to the same footnote multiple times.\nPresent addresses (if different to those where the work was performed) can also be added with a \\verb'\\thanks' command.\n\n\\subsection{Abstract and keywords}\n\nThe abstract is entered in an \\verb'abstract' environment:\n\\begin{verbatim}\n\\begin{abstract}\nThe abstract of the paper.\n\\end{abstract}\n\\end{verbatim}\n\\noindent Note that there is a word limit on the length of abstracts.\nFor the current word limit, see the journal instructions to authors$^{\\ref{foot:itas}}$.\n\nImmediately following the abstract, a set of keywords is entered in a \\verb'keywords' environment:\n\\begin{verbatim}\n\\begin{keywords}\nkeyword 1 -- keyword 2 -- keyword 3\n\\end{keywords}\n\\end{verbatim}\n\\noindent There is a list of permitted keywords, which is agreed between all the major astronomy journals and revised every few years.\nDo \\emph{not} make up new keywords!\nFor the current list of allowed keywords, see the journal's instructions to authors$^{\\ref{foot:itas}}$.\n\n\\section{Sections and lists}\n\nSections and lists are generally the same as in the standard \\LaTeX\\ classes.\n\n\\subsection{Sections}\n\\label{sec:sections}\nSections are entered in the usual way, using \\verb'\\section{}' and its variants. It is possible to nest up to four section levels:\n\\begin{verbatim}\n\\section{Main section}\n \\subsection{Subsection}\n  \\subsubsection{Subsubsection}\n   \\paragraph{Lowest level section}\n\\end{verbatim}\n\\noindent The other \\LaTeX\\ sectioning commands \\verb'\\part', \\verb'\\chapter' and \\verb'\\subparagraph{}' are deprecated and should not be used.\n\nSome sections are not numbered as part of journal style (e.g. the Acknowledgements).\nTo insert an unnumbered section use the `starred' version of the command: \\verb'\\section*{}'.\n\nSee appendix~\\ref{sec:advanced} for more complicated examples.\n\n\\subsection{Lists}\n\nTwo forms of lists can be used in MNRAS -- numbered and unnumbered.\n\nFor a numbered list, use the \\verb'enumerate' environment:\n\\begin{verbatim}\n\\begin{enumerate}\n \\item First item\n \\item Second item\n \\item etc.\n\\end{enumerate}\n\\end{verbatim}\n\\noindent which produces\n\\begin{enumerate}\n \\item First item\n \\item Second item\n \\item etc.\n\\end{enumerate}\nNote that the list uses lowercase Roman numerals, rather than the \\LaTeX\\ default Arabic numerals.\n\nFor an unnumbered list, use the \\verb'description' environment without the optional argument:\n\\begin{verbatim}\n\\begin{description}\n \\item First item\n \\item Second item\n \\item etc.\n\\end{description}\n\\end{verbatim}\n\\noindent which produces\n\\begin{description}\n \\item First item\n \\item Second item\n \\item etc.\n\\end{description}\n\nBulleted lists using the \\verb'itemize' environment should not be used in MNRAS; it is retained for backwards compatibility only.\n\n\\section{Mathematics and symbols}\n\nThe MNRAS class mostly adopts standard \\LaTeX\\ handling of mathematics, which is briefly summarised here.\nSee also section~\\ref{sec:packages} for packages that support more advanced mathematics.\n\nMathematics can be inserted into the running text using the syntax \\verb'$1+1=2$', which produces $1+1=2$.\nUse this only for short expressions or when referring to mathematical quantities; equations should be entered as described below.\n\n\\subsection{Equations}\nEquations should be entered using the \\verb'equation' environment, which automatically numbers them:\n\n\\begin{verbatim}\n\\begin{equation}\n a^2=b^2+c^2\n\\end{equation}\n\\end{verbatim}\n\\noindent which produces\n\\begin{equation}\n a^2=b^2+c^2\n\\end{equation}\n\nBy default, the equations are numbered sequentially throughout the whole paper. If a paper has a large number of equations, it may be better to number them by section (2.1, 2.2 etc.). To do this, add the command \\verb'\\numberwithin{equation}{section}' to the preamble.\n\nIt is also possible to produce un-numbered equations by using the \\LaTeX\\ built-in \\verb'\\['\\textellipsis\\verb'\\]' and \\verb'$$'\\textellipsis\\verb'$$' commands; however MNRAS requires that all equations are numbered, so these commands should be avoided.\n\n\\subsection{Special symbols}\n\n\n\\begin{table}\n \\caption{Additional commands for special symbols commonly used in astronomy. These can be used anywhere.}\n \\label{tab:anysymbols}\n \\begin{tabular}{lll}\n  \\hline\n  Command & Output & Meaning\\\\\n  \\hline\n  \\verb'\\sun' & \\sun & Sun, solar\\\\[2pt]\n  \\verb'\\earth' & \\earth & Earth, terrestrial\\\\[2pt]\n  \\verb'\\micron' & \\micron & microns\\\\[2pt]\n  \\verb'\\degr' & \\degr & degrees\\\\[2pt]\n  \\verb'\\arcmin' & \\arcmin & arcminutes\\\\[2pt]\n  \\verb'\\arcsec' & \\arcsec & arcseconds\\\\[2pt]\n  \\verb'\\fdg' & \\fdg & fraction of a degree\\\\[2pt]\n  \\verb'\\farcm' & \\farcm & fraction of an arcminute\\\\[2pt]\n  \\verb'\\farcs' & \\farcs & fraction of an arcsecond\\\\[2pt]\n  \\verb'\\fd' & \\fd & fraction of a day\\\\[2pt]\n  \\verb'\\fh' & \\fh & fraction of an hour\\\\[2pt]\n  \\verb'\\fm' & \\fm & fraction of a minute\\\\[2pt]\n  \\verb'\\fs' & \\fs & fraction of a second\\\\[2pt]\n  \\verb'\\fp' & \\fp & fraction of a period\\\\[2pt]\n  \\verb'\\diameter' & \\diameter & diameter\\\\[2pt]\n  \\verb'\\sq' & \\sq & square, Q.E.D.\\\\[2pt]\n  \\hline\n \\end{tabular}\n\\end{table}\n\n\\begin{table}\n \\caption{Additional commands for mathematical symbols. These can only be used in maths mode.}\n \\label{tab:mathssymbols}\n \\begin{tabular}{lll}\n  \\hline\n  Command & Output & Meaning\\\\\n  \\hline\n  \\verb'\\upi' & $\\upi$ & upright pi\\\\[2pt]\n  \\verb'\\umu' & $\\umu$ & upright mu\\\\[2pt]\n  \\verb'\\upartial' & $\\upartial$ & upright partial derivative\\\\[2pt]\n  \\verb'\\lid' & $\\lid$ & less than or equal to\\\\[2pt]\n  \\verb'\\gid' & $\\gid$ & greater than or equal to\\\\[2pt]\n  \\verb'\\la' & $\\la$ & less than of order\\\\[2pt]\n  \\verb'\\ga' & $\\ga$ & greater than of order\\\\[2pt]\n  \\verb'\\loa' & $\\loa$ & less than approximately\\\\[2pt]\n  \\verb'\\goa' & $\\goa$ & greater than approximately\\\\[2pt]\n  \\verb'\\cor' & $\\cor$ & corresponds to\\\\[2pt]\n  \\verb'\\sol' & $\\sol$ & similar to or less than\\\\[2pt]\n  \\verb'\\sog' & $\\sog$ & similar to or greater than\\\\[2pt]\n  \\verb'\\lse' & $\\lse$ & less than or homotopic to \\\\[2pt]\n  \\verb'\\gse' & $\\gse$ & greater than or homotopic to\\\\[2pt]\n  \\verb'\\getsto' & $\\getsto$ & from over to\\\\[2pt]\n  \\verb'\\grole' & $\\grole$ & greater over less\\\\[2pt]\n  \\verb'\\leogr' & $\\leogr$ & less over greater\\\\\n  \\hline\n \\end{tabular}\n\\end{table}\n\nSome additional symbols of common use in astronomy have been added in the MNRAS class. These are shown in tables~\\ref{tab:anysymbols}--\\ref{tab:mathssymbols}. The command names are -- as far as possible -- the same as those used in other major astronomy journals.\n\nMany other mathematical symbols are also available, either built into \\LaTeX\\ or via additional packages. If you want to insert a specific symbol but don't know the \\LaTeX\\ command, we recommend using the Detexify website\\footnote{\\url{http:\/\/detexify.kirelabs.org}}.\n\nSometimes font or coding limitations mean a symbol may not get smaller when used in sub- or superscripts, and will therefore be displayed at the wrong size. There is no need to worry about this as it will be corrected by the typesetter during production.\n\nTo produce bold symbols in mathematics, use \\verb'\\bmath' for simple variables, and the \\verb'bm' package for more complex symbols (see section~\\ref{sec:packages}). Vectors are set in bold italic, using \\verb'\\mathbfit{}'.\n\nFor matrices, use \\verb'\\mathbfss{}' to produce a bold sans-serif font e.g. \\mathbfss{H}; this works even outside maths mode, but not all symbols are available (e.g. Greek). For $\\nabla$ (del, used in gradients, divergence etc.) use \\verb'$\\nabla$'.\n\n\\subsection{Ions}\n\nA new \\verb'\\ion{}{}' command has been added to the class file, for the correct typesetting of ionisation states.\nFor example, to typeset singly ionised calcium use \\verb'\\ion{Ca}{ii}', which produces \\ion{Ca}{ii}.\n\n\\section{Figures and tables}\n\\label{sec:fig_table}\nFigures and tables (collectively called `floats') are mostly the same as built into \\LaTeX.\n\n\\subsection{Basic examples}\n\\begin{figure}\n \\includegraphics[width=\\columnwidth]{example}\n \\caption{An example figure.}\n \\label{fig:example}\n\\end{figure}\nFigures are inserted in the usual way using a \\verb'figure' environment and \\verb'\\includegraphics'. The example Figure~\\ref{fig:example} was generated using the code:\n\\begin{verbatim}\n\\begin{figure}\n \\includegraphics[width=\\columnwidth]{example}\n \\caption{An example figure.}\n \\label{fig:example}\n\\end{figure}\n\\end{verbatim}\n\n\\begin{table}\n \\caption{An example table.}\n \\label{tab:example}\n \\begin{tabular}{lcc}\n  \\hline\n  Star & Mass & Luminosity\\\\\n   & $M_{\\sun}$ & $L_{\\sun}$\\\\\n  \\hline\n  Sun & 1.00 & 1.00\\\\\n  $\\alpha$~Cen~A & 1.10 & 1.52\\\\\n  $\\epsilon$~Eri & 0.82 & 0.34\\\\\n  \\hline\n \\end{tabular}\n\\end{table}\nThe example Table~\\ref{tab:example} was generated using the code:\n\\begin{verbatim}\n\\begin{table}\n \\caption{An example table.}\n \\label{tab:example}\n \\begin{tabular}{lcc}\n  \\hline\n  Star & Mass & Luminosity\\\\\n   & $M_{\\sun}$ & $L_{\\sun}$\\\\\n  \\hline\n  Sun & 1.00 & 1.00\\\\\n  $\\alpha$~Cen~A & 1.10 & 1.52\\\\\n  $\\epsilon$~Eri & 0.82 & 0.34\\\\\n  \\hline\n \\end{tabular}\n\\end{table}\n\\end{verbatim}\n\n\\subsection{Captions and placement}\nCaptions go \\emph{above} tables but \\emph{below} figures, as in the examples above.\n\nThe \\LaTeX\\ float placement commands \\verb'[htbp]' are intentionally disabled.\nLayout of figures and tables will be adjusted by the publisher during the production process, so authors should not concern themselves with placement to avoid disappointment and wasted effort.\nSimply place the \\LaTeX\\ code close to where the figure or table is first mentioned in the text and leave exact placement to the publishers.\n\nBy default a figure or table will occupy one column of the page.\nTo produce a wider version which covers both columns, use the \\verb'figure*' or \\verb'table*'  environment.\n\nIf a figure or table is too long to fit on a single page it can be split it into several parts.\nCreate an additional figure or table which uses \\verb'\\contcaption{}' instead of \\verb'\\caption{}'.\nThis will automatically correct the numbering and add `\\emph{continued}' at the start of the caption.\n\\begin{table}\n \\contcaption{A table continued from the previous one.}\n \\label{tab:continued}\n \\begin{tabular}{lcc}\n  \\hline\n  Star & Mass & Luminosity\\\\\n   & $M_{\\sun}$ & $L_{\\sun}$\\\\\n  \\hline\n  $\\tau$~Cet & 0.78 & 0.52\\\\\n  $\\delta$~Pav & 0.99 & 1.22\\\\\n  $\\sigma$~Dra & 0.87 & 0.43\\\\\n  \\hline\n \\end{tabular}\n\\end{table}\nTable~\\ref{tab:continued} was generated using the code:\n\n\\begin{verbatim}\n\\begin{table}\n \\contcaption{A table continued from the previous one.}\n \\label{tab:continued}\n \\begin{tabular}{lcc}\n  \\hline\n  Star & Mass & Luminosity\\\\\n   & $M_{\\sun}$ & $L_{\\sun}$\\\\\n  \\hline\n  $\\tau$~Cet & 0.78 & 0.52\\\\\n  $\\delta$~Pav & 0.99 & 1.22\\\\\n  $\\sigma$~Dra & 0.87 & 0.43\\\\\n  \\hline\n \\end{tabular}\n\\end{table}\n\\end{verbatim}\n\nTo produce a landscape figure or table, use the \\verb'pdflscape' package and the \\verb'landscape' environment.\nThe landscape Table~\\ref{tab:landscape} was produced using the code:\n\\begin{verbatim}\n\\begin{landscape}\n \\begin{table}\n  \\caption{An example landscape table.}\n  \\label{tab:landscape}\n  \\begin{tabular}{cccccccccc}\n    \\hline\n    Header & Header & ...\\\\\n    Unit & Unit & ...\\\\\n    \\hline\n    Data & Data & ...\\\\\n    Data & Data & ...\\\\\n    ...\\\\\n    \\hline\n  \\end{tabular}\n \\end{table}\n\\end{landscape}\n\\end{verbatim}\nUnfortunately this method will force a page break before the table appears.\nMore complicated solutions are possible, but authors shouldn't worry about this.\n\n\\begin{landscape}\n \\begin{table}\n  \\caption{An example landscape table.}\n  \\label{tab:landscape}\n  \\begin{tabular}{cccccccccc}\n    \\hline\n    Header & Header & Header & Header & Header & Header & Header & Header & Header & Header\\\\\n    Unit & Unit & Unit & Unit & Unit & Unit & Unit & Unit & Unit & Unit \\\\\n    \\hline\n    Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\\\\n    Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\\\\n    Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\\\\n    Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\\\\n    Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\\\\n    Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\\\\n    Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\\\\n    Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\\\\n    \\hline\n  \\end{tabular}\n \\end{table}\n\\end{landscape}\n\n\\section{References and citations}\n\n\\subsection{Cross-referencing}\n\nThe usual \\LaTeX\\ commands \\verb'\\label{}' and \\verb'\\ref{}' can be used for cross-referencing within the same paper.\nWe recommend that you use these whenever relevant, rather than writing out the section or figure numbers explicitly.\nThis ensures that cross-references are updated whenever the numbering changes (e.g. during revision) and provides clickable links (if available in your compiler).\n\nIt is best to give each section, figure and table a logical label.\nFor example, Table~\\ref{tab:mathssymbols} has the label \\verb'tab:mathssymbols', whilst section~\\ref{sec:packages} has the label \\verb'sec:packages'.\nAdd the label \\emph{after} the section or caption command, as in the examples in sections~\\ref{sec:sections} and \\ref{sec:fig_table}.\nEnter the cross-reference with a non-breaking space between the type of object and the number, like this: \\verb'see Figure~\\ref{fig:example}'.\n\nThe \\verb'\\autoref{}' command can be used to automatically fill out the type of object, saving on typing.\nIt also causes the link to cover the whole phrase rather than just the number, but for that reason is only suitable for single cross-references rather than ranges.\nFor example, \\verb'\\autoref{tab:journal_abbr}' produces \\autoref{tab:journal_abbr}.\n\n\\subsection{Citations}\n\\label{sec:cite}\n\nMNRAS uses the Harvard -- author (year) -- citation style, e.g. \\citet{author2013}.\nThis is implemented in \\LaTeX\\ via the \\verb'natbib' package, which in turn is included via the \\verb'usenatbib' package option (see section~\\ref{sec:options}), which should be used in all papers.\n\nEach entry in the reference list has a `key' (see section~\\ref{sec:ref_list}) which is used to generate citations.\nThere are two basic \\verb'natbib' commands:\n\\begin{description}\n \\item \\verb'\\citet{key}' produces an in-text citation: \\citet{author2013}\n \\item \\verb'\\citep{key}' produces a bracketed (parenthetical) citation: \\citep{author2013}\n\\end{description}\nCitations will include clickable links to the relevant entry in the reference list, if supported by your \\LaTeX\\ compiler.\n\n\\defcitealias{smith2014}{Paper~I}\n\\begin{table*}\n \\caption{Common citation commands, provided by the \\texttt{natbib} package.}\n \\label{tab:natbib}\n \\begin{tabular}{lll}\n  \\hline\n  Command & Ouput & Note\\\\\n  \\hline\n  \\verb'\\citet{key}' & \\citet{smith2014} & \\\\\n  \\verb'\\citep{key}' & \\citep{smith2014} & \\\\\n  \\verb'\\citep{key,key2}' & \\citep{smith2014,jones2015} & Multiple papers\\\\\n  \\verb'\\citet[table 4]{key}' & \\citet[table 4]{smith2014} & \\\\\n  \\verb'\\citep[see][figure 7]{key}' & \\citep[see][figure 7]{smith2014} & \\\\\n  \\verb'\\citealt{key}' & \\citealt{smith2014} & For use with manual brackets\\\\\n  \\verb'\\citeauthor{key}' & \\citeauthor{smith2014} & If already cited in close proximity\\\\\n  \\verb'\\defcitealias{key}{Paper~I}' &  & Define an alias (doesn't work in floats)\\\\\n  \\verb'\\citetalias{key}' & \\citetalias{smith2014} & \\\\\n  \\verb'\\citepalias{key}' & \\citepalias{smith2014} & \\\\\n  \\hline\n \\end{tabular}\n\\end{table*}\n\nThere are a number of other \\verb'natbib' commands which can be used for more complicated citations.\nThe most commonly used ones are listed in Table~\\ref{tab:natbib}.\nFor full guidance on their use, consult the \\verb'natbib' documentation\\footnote{\\url{http:\/\/www.ctan.org\/pkg\/natbib}}.\n\nIf a reference has several authors, \\verb'natbib' will automatically use `et al.' if there are more than two authors. However, if a paper has exactly three authors, MNRAS style is to list all three on the first citation and use `et al.' thereafter. If you are using \\bibtex\\ (see section~\\ref{sec:ref_list}) then this is handled automatically. If not, the \\verb'\\citet*{}' and \\verb'\\citep*{}' commands can be used at the first citation to include all of the authors.\n\n\\subsection{The list of references}\n\\label{sec:ref_list}\n\nIt is possible to enter references manually using the usual \\LaTeX\\ commands, but we strongly encourage authors to use \\bibtex\\ instead.\n\\bibtex\\ ensures that the reference list is updated automatically as references are added or removed from the paper, puts them in the correct format, saves on typing, and the same reference file can be used for many different papers -- saving time hunting down reference details.\nAn MNRAS \\bibtex\\ style file, \\verb'mnras.bst', is distributed as part of this package.\nThe rest of this section will assume you are using \\bibtex.\n\nReferences are entered into a separate \\verb'.bib' file in standard \\bibtex\\ formatting.\nThis can be done manually, or there are several software packages which make editing the \\verb'.bib' file much easier.\nWe particularly recommend \\textsc{JabRef}\\footnote{\\url{http:\/\/jabref.sourceforge.net\/}}, which works on all major operating systems.\n\\bibtex\\ entries can be obtained from the NASA Astrophysics Data System\\footnote{\\label{foot:ads}\\url{http:\/\/adsabs.harvard.edu}} (ADS) by clicking on `Bibtex entry for this abstract' on any entry.\nSimply copy this into your \\verb'.bib' file or into the `BibTeX source' tab in \\textsc{JabRef}.\n\nEach entry in the \\verb'.bib' file must specify a unique `key' to identify the paper, the format of which is up to the author.\nSimply cite it in the usual way, as described in section~\\ref{sec:cite}, using the specified key.\nCompile the paper as usual, but add an extra step to run the \\texttt{bibtex} command.\nConsult the documentation for your compiler or latex distribution.\n\nCorrect formatting of the reference list will be handled by \\bibtex\\ in almost all cases, provided that the correct information was entered into the \\verb'.bib' file.\nNote that ADS entries are not always correct, particularly for older papers and conference proceedings, so may need to be edited.\nIf in doubt, or if you are producing the reference list manually, see the MNRAS instructions to authors$^{\\ref{foot:itas}}$ for the current guidelines on how to format the list of references.\n\n\\section{Appendices and online material}\n\nTo start an appendix, simply place the \\verb'\n\\section{Introduction}\n\nCataclysmic Variables are semi-detached binaries built of a white dwarf (WD) which is accreting mass from a Roche-lobe filling main sequence (MS) star. Mass transfer is driven by the loss of angular momentum in absence of strong magnetic fields, and the transferred material forms an accretion disc surrounding the central WD (see e.g., \\citealt{warner95-1}, \\citealt{hellier_book} and \\citealt{kniggeetal11-1} for comprehensive reviews). Since the structure of both components is relatively simple, CVs are one of the best sources to test our understanding of many astrophysical phenomena involving evolution of compact, interacting binaries and accretion phenomena. Their study helps to resolve standing discrepancies between current population models and observations in many present and complex topics including black hole binaries, short gamma-ray bursts, X-ray transients, milli-second pulsars and Supernovae Ia. \n\nThe orbital period distribution is the main tool to study the evolution of CVs, as it presents features in key points that allow us to understand their behaviour. As a consequence of the angular momentum loss and the mechanisms driving it, CVs move from long orbital periods and high mass transfer rates to short orbital periods and low mass transfer rates \\citealt{paczynski+sienkiewicz83-1}; \\citealt{Townsley09}; \\citealt{GoliaschNelson15}; \\citealt{palaetal7}). The evolution proceeds in this way until the system reaches the ``period minimum'' at $\\sim$ 76-80 minutes (\\citealt{knigge06-1}; \\citealt{gaensickeetal09}) in which the donor turns into a brown dwarf. Consequently, the orbital separation and period now increases as the mass transfer continues, becoming in the so-called period bouncers, faint systems with short orbital periods. On their way to the period minimum, observations show an abrupt drop in the number of systems with periods between 2 and 3h, referred to as the period gap. Below this range (Porb $<$ 2\\,h) systems have low mass-transfer rates governed by gravitational radiation (GR) \\citep{patterson84-1} while the higher mass-transfer rates above the gap (Porb $>$ 3\\,h) are a consequence of the stronger magnetic braking (MB) (\\citealt{rappaportetal83-1}; \\citealt{spruit+ritter83-1}; \\citealt{hameuryetal88-2}; \\citealt{davis08}). The standard explanation suggests that MB switches off when a CV has evolved down to 3h, the secondary contracts to its thermal equilibrium and detaches from its Roche lobe. Such systems crossing the gap are known as detached Cataclysmic Variables (dCVs). The continuing angular momentum loss by GR shrinks the orbit until at a period of about 2h, the Roche lobe makes contact with the stellar surface again and mass transfer is re-established albeit at a lower level. \n\nDuring this evolution, the CV will change appearance: The relative contribution of the WD, the secondary star and the accretion disc or stream makes for unique colours \\citep[e.g.][]{szkodyetal02-2}. The systems thus occupy distinct locations in colour-colour diagrams with respect to single stars. Due to the relatively small sample of CVs and the inherent difficulty of any source in obtaining its distance, it has not been possible so far to perform an analysis of the CV absolute magnitude distribution. \nWith the arrival of {\\it Gaia},\nthis has changed. Already \\cite{Pala19} show the advances that {\\it Gaia} parallaxes bring to the understanding of CVs, and we now have the data to study CVs in the HR-diagram.\n\nThe paper is organised as such: In section \\ref{sec:gaia}, we explain how we use {\\it Gaia} to define our CV sample. In section \\ref{sec:HRdiagram} the results are presented for all CVs and the trends with the orbital periods and the subtypes are discussed. Finally, we present our summary in \\ref{sec:Summary}.\n\n\n\\section{The Catalogue of {\\it Gaia} DR2 and the cross-match with CV catalogue}\\label{sec:gaia}\n\nThe goal of the {\\it Gaia} \nspace mission is to make  the largest, most precise three-dimensional map of the Milky Way to-date by detecting and measuring the motion and parallax of each star in its orbit around the centre of the Galaxy. To this means, the three filters $G$, $G_{BP}$ and $G_{RP}$ are observed at several epochs over a period of about 670 days of mission operations. For details, see \\cite{GaiaDR2}.\n\nThe second data release (GDR2 hereafter) is based on 22 months of observations and provides positions, parallaxes and proper motions for 1.3 billion sources up to G $\\sim$ 20 magnitudes. This kind of data allows the derivation of distances and absolute magnitudes to study the position of all objects in the global HR-diagram.\n\n\\subsection{Deriving absolute magnitudes}\\label{subsubsec:Abs_mag}\n\nOne of the aims of this paper is finding the CV locus in the HR-diagram. We make use of GDR2 data to compute their absolute magnitudes. \nThe absolute magnitude $M$ of an object is given by:\n\\begin{equation}\nM = m + 5 - 5\\log(d) + A,\n\\label{eq:M}\n\\end{equation}\n\nwhere $m$ is the apparent magnitude, $A$ is the interstellar extinction and $d$, the distance to the source which can be obtained by the GDR2 data. \nGDR2 provides weighted mean fluxes\\footnote{Weighted means are used because flux errors on different epochs may vary depending on the configuration of each observation, see \\cite{CarrascoGDR1} and \\cite{RielloGDR2} for detailed information.} and, as CVs are variable stars, this procedure has an effect on their $G\n, $G_{BP}$ and $G_{RP}$ values. The degree of impact might be determined by comparing the 670-days length of the {\\it Gaia}-mission and the cycle of variation length for every CV subtype. The highest impact is on Dwarf Novae systems as they can have outbursts even on a weekly base (\\citealt{SterkenBook} and references therein). The low recurrence in Novae, Nova-like and Magnetic CVs should have no significant impact on the overall sample.\n\nInferring the distance from the {\\it Gaia} DR2 parallax is not a trivial issue. Distance can be derived as the inverse of the parallax, only if the parallax error is lower than 20\\% and by doing so we would discard 80\\% of the sources (\\citealt{Luri18}). We used instead the distances inferred by \\citet{Bailer-Jones18} who compute distances and their uncertainties through a probabilistic analysis based on the Bayes theorem and adopting an exponentially decreasing space density\nprior\\footnote{For a detailed explanation of this approach and an analysis of applying this technique refer to \\citet{Bailer-Jones18}. For a discussion of the use of different priors, see\n\\citet{Bailer-Jones18},\\citet{Luri18},\\citet{igoshevetal2016},\\citet{astraatmadjaetal16}.\n} \nfor the 1.33 billion sources from GDR2. These are available using ADQL\\footnote{http:\/\/gaia.ari.uni-heidelberg.de\/tap.html} and we here use them to derive  the absolute magnitudes through Equation~\\ref{eq:M}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{The CV sample}\\label{sec:CVsample}\n\n\nThe Catalog and Atlas of Cataclysmic Variables \\citep{DownesCat} includes all objects which have been classified as a CV at some point in time. Although it was frozen on February 1st, 2006, it is one of the main references among the community, providing coordinates, proper motion, type, chart, spectral and period references for all 1830 sources when available. In order to obtain the purest sample, we discarded from this catalogue the objects designated as ``NON-CV'', which are stars that have been previously identified as CVs but later confuted, and those with the extensions ``:'' and ``::'' because their classification is not conclusive.\n\nThe Catalog of Cataclysmic Binaries, Low-Mass X-Ray Binaries and Related Objects \\citep{RKCat} which only contains objects with a measured period, is updated up to December 31st, 2015 and it provides coordinates, apparent magnitudes, orbital parameters, stellar parameters of the components and other characteristic properties for 1429 CVs. In this case uncertain values are followed by only one ``:'' and have been discarded as well. \n\nBoth catalogues have been merged into a final sample of 1920 CVs, out of which 1187 are contained in the GDR2 footprint. The density studies of CVs in the HR-diagram\nwere done using this full sample. For 839 of these systems, the orbital period is known, and for 1130 systems, the subtype is unambiguously known (see Tab.\\ref{tab:subtypes_sample}).\n\n\n\\begin{table}\n\\scriptsize\n\t\\centering\n\t\\caption{\\label{tab:subtypes_sample} Distribution of the CV sample utilised by subtype.}\n\t\\begin{tabular}{lcccc} \n\t\t\\hline\nCV subtype & Periods &  Main & \\multicolumn{2}{c}{Centroid position in HRD}\\\\\n& sample & sample & $G_{BP}-G_{RP}$ & $G_{abs}$\\\\\n\t\t\\hline\n\t\tNovalike & 76 & 119 & 0.37 & 5.63 \\\\\n\t\tDwarf Novae & 484 & 688 & 0.64 & 9.49 \\\\\n\t\tOld Novae & 77 & 119 & 0.79 & 5.58 \\\\\n\t\tPolar & 75 & 135 & 0.83 & 9.67 \\\\\n\t\tIntermediate Polar & 51 & 69 & 0.59 & 5.61 \\\\\n\t\t\\hline\n\t\tTotal Sample & 839 & 1130 & &\\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table}\n\n\n\n\n\n\\section{CVs in the HR-diagram}\\label{sec:HRdiagram}\n\n\n\n\n\\subsection{The impact of the orbital period}\n\n\n\n\n\n\n\n\n\n\n\\begin{figure*}\n    \\centering\n\t    \\begin{tabular}{c c}\n\t        \\includegraphics[width=0.45\\textwidth]{H-Rdiag_periods_rast.pdf}&\t\\includegraphics[width=0.42\\textwidth]{H-R_density_rast.pdf}\t\n\t    \\end{tabular}\n    \\caption{In grey, all stars from {\\it Gaia}'s 2nd data release are plotted in the HR-diagrams. On the left side, the CVs period distribution, the CVs from our sample with known orbital periods (see Section \\ref{sec:CVsample}) are plotted in larger dots. The colour of each dot refers to the orbital period as given in the bar at the right of the panel. CVs with larger periods lie close to the main sequence path getting shorter while approaching the white dwarfs area. On the right, the density distribution of our whole sample of CVs (brown dots), surfaces with different tones of blue represent areas of equal density. On the x- and y-axis the marginal distributions are shown. CVs lie on average between the MS path and the WDs with a high density area peaking at $G_{BP}-G_{RP} \\sim 0.56$ and $G_{abs} \\sim 10.15$. Such area corresponds to the overpopulation below the period gap as reflected in the left panel by black dots, CVs with orbital period below 2h\n    \\label{fig:HRper_densityMap}}\n\\end{figure*}\n\nLeft panel of Fig. \\ref{fig:HRper_densityMap} displays the CV locus in the HR-diagram of all CVs for which an orbital period is known (839 systems).  The orbital period of each system is represented by the colour of the symbol as defined in the auxiliary axis. The CVs lie on average between the main sequence stars and the WDs.\nA clear trend is seen on their position with the orbital period: CVs with longer periods fall close to the main sequence path, while, as the orbital period decreases, they approach the WDs region.\nThis behaviour can be understood from the contribution of the secondary star. On average, a Roche-lobe filling secondary star is larger and brighter for longer orbital periods, while the WD does not change much during the secular CV evolution. Hence, the contribution of the secondary should be more dominant for longer orbital periods. Systems below the period gap, are instead dominated by their WD, as the secondary becomes only visible in the near infrared and does not contribute to the {\\it Gaia} colour. The contribution of the accretion disc should change colour and magnitude depending on the sub-type and will be discussed in the next subsection.\n\nThe right panel of Fig. \\ref{fig:HRper_densityMap} shows the locus of all CVs of our sample defined in Section \\ref{sec:CVsample} within {\\it Gaia's} HR-diagram. On the x- and y-axis, the respective projected density is plotted. A high density area is well distinguishable at $G_{BP}-G_{RP} \\sim 0.56$ and $G_{abs} \\sim 10.15$ (values obtained from the mode of the marginal distributions) which corresponds to the population below the period gap.\n\n\n\\subsection{The locus depending on the subtype}\nFigure \\ref{fig:HR_per_types} exhibits the distribution of every CV subtype on the HR-diagram. Bivariate Gaussian distributions are computed for 1 and 3 $\\sigma$ given by\n\n\\begin{equation}\np(x, y | \\mu_x, \\mu_y, \\sigma_x, \\sigma_y, \\sigma_{xy}) = \\frac{1}{2 \\pi \\sigma_x \\sigma_y \\sqrt{1-\\rho^2}}exp \\left(\\frac{-z^2}{2(1-\\rho^2)}\\right),\n\\label{eq:bivariate1}\n\\end{equation}\n\nwhere\n\n\\begin{equation}\nz^2 = \\frac{(x-\\mu_x)^2}{\\sigma_x^2}+\\frac{(y-\\mu_y)^2}{\\sigma_y^2}-2\\rho \\frac{(x-\\mu_x)(y-\\mu_y)}{\\sigma_x \\sigma_y},      \n\\label{eq:bivariate2}\n\\end{equation}\n\n\\begin{equation}\n\\rho = \\frac{\\sigma_{xy}}{\\sigma_x \\sigma_y},\n\\label{eq:bivariate3}\n\\end{equation}\n\nusing the median instead of the mean and the interquartile range to estimate variances in order to avoid the impact of outliers. The results are given in Table \\ref{tab:subtypes_sample}.\n\n\\begin{figure*\n\\center\n\t\\begin{tabular}{c c c}\n\t\\includegraphics[width=0.3\\textwidth]{H-Rdiag_types_rast.pdf}\t&\n\t\\includegraphics[width=0.3\\textwidth]{H-Rdiag_Nova_like_rast.pdf}\t&\n\t\\includegraphics[width=0.3\\textwidth]{H-Rdiag_Dwarf_Novae_rast.pdf}\t\\\\\n\t\\includegraphics[width=0.3\\textwidth]{H-Rdiag_Novae_rast.pdf}\t&\n\t\\includegraphics[width=0.3\\textwidth]{H-Rdiag_Polars_rast.pdf}\t&\n\t\\includegraphics[width=0.3\\textwidth]{H-Rdiag_Intermediate_Polars_rast.pdf}\t\\\\\n\t\\includegraphics[width=0.3\\textwidth]{NL_N_hist_short.pdf}\t&\n\t\\includegraphics[width=0.3\\textwidth]{AM_IP_hist_short.pdf}\t&\n\t\\includegraphics[width=0.3\\textwidth]{DN_WZSge_hist_short.pdf}\t\n\t\\end{tabular}\n\\caption{The distribution of CV subtypes in the HR-diagram. Top-left panel shows all subtypes together, after that every subtype separately. The dashed ellipses represent 1 and 3 $\\sigma$ of each subtype bivariate Gaussian distribution. The sample utilized here is composed by all CVs in the Ritter\\&Kolb and Downes catalogues (see Section \\ref{sec:CVsample}) whose subtype is unambiguously known and are included in the {\\it Gaia} footprint; 119 Nova-likes, 688 Dwarf Novae, 119 Novae, 135 Polars and 69 Intermediate Polars. On the bottom, the period histograms for each subtype.  \\label{fig:HR_per_types}}\n\\end{figure*}\n\nNova-likes are dominated by a high mass-transfer accretion disc, that usually \novershines\nthe WD and the secondary star at optical and even infrared wavelengths. Their colour and final absolute magnitude mainly depends on the inclination with respect to the line of sight. In the HR-diagram, they concentrate around $G_{abs}=5.63$ and $G_{BP}-G_{RP}=0.37$, i.e. on the blue and bright corner of all CVs. A similar locus but with a much higher scatter is occupied by the old novae and by intermediate polars. This can be explained by the eclectic composition of these two sub-groups which also contain a large fraction of novalike stars. \n\nIn contrast, polars which do not accrete mass through a disc, are much fainter and their colour and magnitude will depend on the nature of the secondary. In the HR-diagram they scatter around $G_{abs}=9.67$ and $G_{BP}-G_{RP}=0.83$\nrepresenting the reddest and faintest of all the CV subgroups.\n\nDwarf novae (DNe) occupy the whole region between MS stars and WDs with the centroid being at $G_{abs}=9.49$ and $G_{BP}-G_{RP}=0.64$. Since the secondary star in these systems can be anything from an early K-type star down to a brown dwarf, the range in colours and magnitude is not surprising. In addition, these sources are characterised by undergoing \nregular outbursts increasing their brightness and blueness. \nAs discussed in Section \\ref{sec:gaia}, the given magnitude is a weighted mean of several epochs and thus also increases the spread of this distribution. A detailed study of the DNe locus depending on their subtype and outburst state can be done following the next {\\it Gaia} release when individual measurements and epochs become available\n\n\nWZ Sge-type objects deserve a separate mention, a class of DNe characterised by great outburst amplitudes, slow declines and long intervals between outbursts compared with ordinary DNe. These kind of systems have been considered to be period bouncer candidates \\citep{Patterson11}, some of them extensively investigated in this regard (QZ Lib, \\citealt{Pala18}; J122221 \\citealt{vitaly17} and \\citealt{kato13b}; J184228 \\citealt{kato13b}; J075418 and J230425 \\citealt{Nakata14}). We have plotted a sample of 71 of such systems in the upper-right panel of Fig. \\ref{fig:HR_per_types}, along with the rest of DNe, and it can be seen that they concentrate near the WDs area. This is consistent with them being period bouncers or similar systems, as these are the CVs with the lowest mass transfer and faintest secondary stars. The disc is only visible in some emission lines, the secondary does not contribute to the optical range at all.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Detached CVs}\n\n\\begin{figure\n\t\\includegraphics[width=0.425\\textwidth]{HR_WDM4-6.pdf}\n\\caption{\\label{fig:dCVs} Systems comprised by a WD and an M4-6. The figure contains the 39 objects from the sample \nby \\citep{Zorotovic16} present in {\\it Gaia} (see text). In green are those within the period gap and in brown the rest. All of them follow a trend but the objects within the gap are on average fainter.\n}\n\\end{figure}\n\nAnother question we can address is finding the locus occupied by the so-called detached cataclysmic variables (dCVs) crossing the orbital period gap. A first approach could be made by finding the area with boundaries in 2 and 3h in the left panel of Fig. \\ref{fig:HRper_densityMap} using\nregression techniques. However, since dCVs no longer \ncontain an accretion disc, they should appear fainter than regular CVs of the same period. \n\nDue to the continuous mass loss, the donor is being driven out of equilibrium and secondaries in CVs just above the period gap are bloated up to 30\\% with respect to regular MS stars \\citep{kniggeetal11-1}. When the mass transfer stops, the secondary shrinks towards its thermal equilibrium radius to nearly its equivalent for MS stars \\citep{stevehowell-01} and hence we expect secondaries in dCVs to be comparable to single MS stars of the same type.\n\nSince the mass transfer ceases, the mass and spectral type of the donor star stays constant during the interval in which the binary is detached. In regular CVs this happens at $M_{sec} = 0.2 \\pm 0.02$ $M\\odot$ \\citep{knigge06-1} and spectral type $\\sim M6$ \\citep{rebassaetal2007}, though variations occur depending on the moment in which the CV started the mass transfer and the time passed as CV until the secondary becomes fully convective. In the extreme case of a CV starting the mass transfer within the period gap range, the donor star type will be that of a fully convective isolated M star, which, according to \\cite{Chabrier-97}, occurs at $M_{sec} \\sim0.35$ $M\\odot$ and spectral type M4. We thus assume that the secondary of dCVs is in the range M4-M6.\n\nSo far the only observational evidence for the existence of dCVs come from \\cite{Zorotovic16}, who show that the orbital period distribution of detached close binaries consisting of a WD and an M4-M6 secondary star cannot be produced by Post Common Envelope Binaries alone, but a contribution of dCVs is needed to explain the peak between 2 and 3h. They also show that the systems inside this peak have a higher average mass than would be expected for normal WDMS systems. Still, with only 52 such systems known in total (WDMS systems with secondary spectral types in the range M4-6 and orbital periods below 10h) and 12 between 2 and 3h, the significance is not very high.\n\n\nWe distinguish two groups, the sources with orbital periods corresponding to those of the period gap (2-3h) and therefore, more likely to be dCVs, and the rest with periods outside this range. In Fig. \\ref{fig:dCVs} they are plotted in the HR-diagram, the former appear\nfainter compared to the latter. This can be explained by the higher WD masses in CVs, and consequently in dCVs, than in PCEBs, making them smaller in size and surface and contributing in a lesser extent on the brightness\nof the whole system.\n\n\n\\section{Summary}\\label{sec:Summary}\nWe have analyzed the evolutionary cycle of CVs from a statistical perspective using {\\it Gaia} DR2 data in conjunction with the HR-diagram tool. We have reported the discovery of a trend of the period and mass accretion with colour and absolute magnitude. We have also investigated their density distribution as a whole population, peaking at $G_{BP}-G_{RP} \\sim 0.56$ and $G_{abs} \\sim 10.15$, and the contribution of the main CV subtypes to this regard, highlighting the location of WZ Sge systems, which are period bouncer candidates. Finally, we have identified the location and a trend among systems comprised of a WD and secondary in the range M4-M6, which correspond with dCVs, CVs going through the orbital period gap.\n\n\\section*{Acknowledgements}\nThis research has made use of the VizieR catalogue access tool, CDS, Strasbourg, France (DOI: 10.26093\/cds\/vizier). The original description of the VizieR service was published in 2000, A\\&AS 143, 23.\n\n\n\n\n\\bibliographystyle{mnras}\n\n\\section{Introduction}\n\nThe journal \\textit{Monthly Notices of the Royal Astronomical Society} (MNRAS) encourages authors to prepare their papers using \\LaTeX.\nThe style file \\verb'mnras.cls' can be used to approximate the final appearance of the journal, and provides numerous features to simplify the preparation of papers.\nThis document, \\verb'mnras_guide.tex', provides guidance on using that style file and the features it enables.\n\nThis is not a general guide on how to use \\LaTeX, of which many excellent examples already exist.\nWe particularly recommend \\textit{Wikibooks \\LaTeX}\\footnote{\\url{https:\/\/en.wikibooks.org\/wiki\/LaTeX}}, a collaborative online textbook which is of use to both beginners and experts.\nAlternatively there are several other online resources, and most academic libraries also hold suitable beginner's guides.\n\nFor guidance on the contents of papers, journal style, and how to submit a paper, see the MNRAS Instructions to Authors\\footnote{\\label{foot:itas}\\url{http:\/\/www.oxfordjournals.org\/our_journals\/mnras\/for_authors\/}}.\nOnly technical issues with the \\LaTeX\\ class are considered here.\n\n\n\\section{Obtaining and installing the MNRAS package}\nSome \\LaTeX\\ distributions come with the MNRAS package by default.\nIf yours does not, you can either install it using your distribution's package manager, or download it from the Comprehensive \\TeX\\ Archive Network\\footnote{\\url{http:\/\/www.ctan.org\/tex-archive\/macros\/latex\/contrib\/mnras}} (CTAN).\n\nThe files can either be installed permanently by placing them in the appropriate directory (consult the documentation for your \\LaTeX\\ distribution), or used temporarily by placing them in the working directory for your paper.\n\nTo use the MNRAS package, simply specify \\verb'mnras' as the document class at the start of a \\verb'.tex' file:\n\n\\begin{verbatim}\n\\documentclass{mnras}\n\\end{verbatim}\nThen compile \\LaTeX\\ (and if necessary \\bibtex) in the usual way.\n\n\\section{Preparing and submitting a paper}\nWe recommend that you start with a copy of the \\texttt{mnras\\_template.tex} file.\nRename the file, update the information on the title page, and then work on the text of your paper.\nGuidelines for content, style etc. are given in the instructions to authors on the journal's website$^{\\ref{foot:itas}}$.\nNote that this document does not follow all the aspects of MNRAS journal style (e.g. it has a table of contents).\n\nIf a paper is accepted, it is professionally typeset and copyedited by the publishers.\nIt is therefore likely that minor changes to presentation will occur.\nFor this reason, we ask authors to ignore minor details such as slightly long lines, extra blank spaces, or misplaced figures, because these details will be dealt with during the production process.\n\nPapers must be submitted electronically via the online submission system; paper submissions are not permitted.\nFor full guidance on how to submit a paper, see the instructions to authors.\n\n\\section{Class options}\n\\label{sec:options}\nThere are several options which can be added to the document class line like this:\n\n\\begin{verbatim}\n\\documentclass[option1,option2]{mnras}\n\\end{verbatim}\nThe available options are:\n\\begin{itemize}\n\\item \\verb'letters' -- used for papers in the journal's Letters section.\n\\item \\verb'onecolumn' -- single column, instead of the default two columns. This should be used {\\it only} if necessary for the display of numerous very long equations.\n\\item \\verb'doublespacing' -- text has double line spacing. Please don't submit papers in this format.\n\\item \\verb'referee' -- \\textit{(deprecated)} single column, double spaced, larger text, bigger margins. Please don't submit papers in this format.\n\\item \\verb'galley' -- \\textit{(deprecated)} no running headers, no attempt to align the bottom of columns.\n\\item \\verb'landscape' -- \\textit{(deprecated)} sets the whole document on landscape paper.\n\\item \\verb\"usenatbib\" -- \\textit{(all papers should use this)} this uses Patrick Daly's \\verb\"natbib.sty\" package for citations.\n\\item \\verb\"usegraphicx\" -- \\textit{(most papers will need this)} includes the \\verb'graphicx' package, for inclusion of figures and images.\n\\item \\verb'useAMS' -- adds support for upright Greek characters \\verb'\\upi', \\verb'\\umu' and \\verb'\\upartial' ($\\upi$, $\\umu$ and $\\upartial$). Only these three are included, if you require other symbols you will need to include the \\verb'amsmath' or \\verb'amsymb' packages (see section~\\ref{sec:packages}).\n\\item \\verb\"usedcolumn\" -- includes the package \\verb\"dcolumn\", which includes two new types of column alignment for use in tables.\n\\end{itemize}\n\nSome of these options are deprecated and retained for backwards compatibility only.\nOthers are used in almost all papers, but again are retained as options to ensure that papers written decades ago will continue to compile without problems.\nIf you want to include any other packages, see section~\\ref{sec:packages}.\n\n\\section{Title page}\n\nIf you are using \\texttt{mnras\\_template.tex} the necessary code for generating the title page, headers and footers is already present.\nSimply edit the title, author list, institutions, abstract and keywords as described below.\n\n\\subsection{Title}\nThere are two forms of the title: the full version used on the first page, and a short version which is used in the header of other odd-numbered pages (the `running head').\nEnter them with \\verb'\\title[]{}' like this:\n\\begin{verbatim}\n\\title[Running head]{Full title of the paper}\n\\end{verbatim}\nThe full title can be multiple lines (use \\verb'\\\\' to start a new line) and may be as long as necessary, although we encourage authors to use concise titles. The running head must be $\\le~45$ characters on a single line.\n\nSee appendix~\\ref{sec:advanced} for more complicated examples.\n\n\\subsection{Authors and institutions}\n\nLike the title, there are two forms of author list: the full version which appears on the title page, and a short form which appears in the header of the even-numbered pages. Enter them using the \\verb'\\author[]{}' command.\n\nIf the author list is more than one line long, start a new line using \\verb'\\newauthor'. Use \\verb'\\\\' to start the institution list. Affiliations for each author should be indicated with a superscript number, and correspond to the list of institutions below the author list.\n\nFor example, if I were to write a paper with two coauthors at another institution, one of whom also works at a third location:\n\\begin{verbatim}\n\\author[K. T. Smith et al.]{\nKeith T. Smith,$^{1}$\nA. N. Other,$^{2}$\nand Third Author$^{2,3}$\n\\\\\n$^{1}$Affiliation 1\\\\\n$^{2}$Affiliation 2\\\\\n$^{3}$Affiliation 3}\n\\end{verbatim}\nAffiliations should be in the format `Department, Institution, Street Address, City and Postal Code, Country'.\n\nEmail addresses can be inserted with the \\verb'\\thanks{}' command which adds a title page footnote.\nIf you want to list more than one email, put them all in the same \\verb'\\thanks' and use \\verb'\\footnotemark[]' to refer to the same footnote multiple times.\nPresent addresses (if different to those where the work was performed) can also be added with a \\verb'\\thanks' command.\n\n\\subsection{Abstract and keywords}\n\nThe abstract is entered in an \\verb'abstract' environment:\n\\begin{verbatim}\n\\begin{abstract}\nThe abstract of the paper.\n\\end{abstract}\n\\end{verbatim}\n\\noindent Note that there is a word limit on the length of abstracts.\nFor the current word limit, see the journal instructions to authors$^{\\ref{foot:itas}}$.\n\nImmediately following the abstract, a set of keywords is entered in a \\verb'keywords' environment:\n\\begin{verbatim}\n\\begin{keywords}\nkeyword 1 -- keyword 2 -- keyword 3\n\\end{keywords}\n\\end{verbatim}\n\\noindent There is a list of permitted keywords, which is agreed between all the major astronomy journals and revised every few years.\nDo \\emph{not} make up new keywords!\nFor the current list of allowed keywords, see the journal's instructions to authors$^{\\ref{foot:itas}}$.\n\n\\section{Sections and lists}\n\nSections and lists are generally the same as in the standard \\LaTeX\\ classes.\n\n\\subsection{Sections}\n\\label{sec:sections}\nSections are entered in the usual way, using \\verb'\\section{}' and its variants. It is possible to nest up to four section levels:\n\\begin{verbatim}\n\\section{Main section}\n \\subsection{Subsection}\n  \\subsubsection{Subsubsection}\n   \\paragraph{Lowest level section}\n\\end{verbatim}\n\\noindent The other \\LaTeX\\ sectioning commands \\verb'\\part', \\verb'\\chapter' and \\verb'\\subparagraph{}' are deprecated and should not be used.\n\nSome sections are not numbered as part of journal style (e.g. the Acknowledgements).\nTo insert an unnumbered section use the `starred' version of the command: \\verb'\\section*{}'.\n\nSee appendix~\\ref{sec:advanced} for more complicated examples.\n\n\\subsection{Lists}\n\nTwo forms of lists can be used in MNRAS -- numbered and unnumbered.\n\nFor a numbered list, use the \\verb'enumerate' environment:\n\\begin{verbatim}\n\\begin{enumerate}\n \\item First item\n \\item Second item\n \\item etc.\n\\end{enumerate}\n\\end{verbatim}\n\\noindent which produces\n\\begin{enumerate}\n \\item First item\n \\item Second item\n \\item etc.\n\\end{enumerate}\nNote that the list uses lowercase Roman numerals, rather than the \\LaTeX\\ default Arabic numerals.\n\nFor an unnumbered list, use the \\verb'description' environment without the optional argument:\n\\begin{verbatim}\n\\begin{description}\n \\item First item\n \\item Second item\n \\item etc.\n\\end{description}\n\\end{verbatim}\n\\noindent which produces\n\\begin{description}\n \\item First item\n \\item Second item\n \\item etc.\n\\end{description}\n\nBulleted lists using the \\verb'itemize' environment should not be used in MNRAS; it is retained for backwards compatibility only.\n\n\\section{Mathematics and symbols}\n\nThe MNRAS class mostly adopts standard \\LaTeX\\ handling of mathematics, which is briefly summarised here.\nSee also section~\\ref{sec:packages} for packages that support more advanced mathematics.\n\nMathematics can be inserted into the running text using the syntax \\verb'$1+1=2$', which produces $1+1=2$.\nUse this only for short expressions or when referring to mathematical quantities; equations should be entered as described below.\n\n\\subsection{Equations}\nEquations should be entered using the \\verb'equation' environment, which automatically numbers them:\n\n\\begin{verbatim}\n\\begin{equation}\n a^2=b^2+c^2\n\\end{equation}\n\\end{verbatim}\n\\noindent which produces\n\\begin{equation}\n a^2=b^2+c^2\n\\end{equation}\n\nBy default, the equations are numbered sequentially throughout the whole paper. If a paper has a large number of equations, it may be better to number them by section (2.1, 2.2 etc.). To do this, add the command \\verb'\\numberwithin{equation}{section}' to the preamble.\n\nIt is also possible to produce un-numbered equations by using the \\LaTeX\\ built-in \\verb'\\['\\textellipsis\\verb'\\]' and \\verb'$$'\\textellipsis\\verb'$$' commands; however MNRAS requires that all equations are numbered, so these commands should be avoided.\n\n\\subsection{Special symbols}\n\n\n\\begin{table}\n \\caption{Additional commands for special symbols commonly used in astronomy. These can be used anywhere.}\n \\label{tab:anysymbols}\n \\begin{tabular}{lll}\n  \\hline\n  Command & Output & Meaning\\\\\n  \\hline\n  \\verb'\\sun' & \\sun & Sun, solar\\\\[2pt]\n  \\verb'\\earth' & \\earth & Earth, terrestrial\\\\[2pt]\n  \\verb'\\micron' & \\micron & microns\\\\[2pt]\n  \\verb'\\degr' & \\degr & degrees\\\\[2pt]\n  \\verb'\\arcmin' & \\arcmin & arcminutes\\\\[2pt]\n  \\verb'\\arcsec' & \\arcsec & arcseconds\\\\[2pt]\n  \\verb'\\fdg' & \\fdg & fraction of a degree\\\\[2pt]\n  \\verb'\\farcm' & \\farcm & fraction of an arcminute\\\\[2pt]\n  \\verb'\\farcs' & \\farcs & fraction of an arcsecond\\\\[2pt]\n  \\verb'\\fd' & \\fd & fraction of a day\\\\[2pt]\n  \\verb'\\fh' & \\fh & fraction of an hour\\\\[2pt]\n  \\verb'\\fm' & \\fm & fraction of a minute\\\\[2pt]\n  \\verb'\\fs' & \\fs & fraction of a second\\\\[2pt]\n  \\verb'\\fp' & \\fp & fraction of a period\\\\[2pt]\n  \\verb'\\diameter' & \\diameter & diameter\\\\[2pt]\n  \\verb'\\sq' & \\sq & square, Q.E.D.\\\\[2pt]\n  \\hline\n \\end{tabular}\n\\end{table}\n\n\\begin{table}\n \\caption{Additional commands for mathematical symbols. These can only be used in maths mode.}\n \\label{tab:mathssymbols}\n \\begin{tabular}{lll}\n  \\hline\n  Command & Output & Meaning\\\\\n  \\hline\n  \\verb'\\upi' & $\\upi$ & upright pi\\\\[2pt]\n  \\verb'\\umu' & $\\umu$ & upright mu\\\\[2pt]\n  \\verb'\\upartial' & $\\upartial$ & upright partial derivative\\\\[2pt]\n  \\verb'\\lid' & $\\lid$ & less than or equal to\\\\[2pt]\n  \\verb'\\gid' & $\\gid$ & greater than or equal to\\\\[2pt]\n  \\verb'\\la' & $\\la$ & less than of order\\\\[2pt]\n  \\verb'\\ga' & $\\ga$ & greater than of order\\\\[2pt]\n  \\verb'\\loa' & $\\loa$ & less than approximately\\\\[2pt]\n  \\verb'\\goa' & $\\goa$ & greater than approximately\\\\[2pt]\n  \\verb'\\cor' & $\\cor$ & corresponds to\\\\[2pt]\n  \\verb'\\sol' & $\\sol$ & similar to or less than\\\\[2pt]\n  \\verb'\\sog' & $\\sog$ & similar to or greater than\\\\[2pt]\n  \\verb'\\lse' & $\\lse$ & less than or homotopic to \\\\[2pt]\n  \\verb'\\gse' & $\\gse$ & greater than or homotopic to\\\\[2pt]\n  \\verb'\\getsto' & $\\getsto$ & from over to\\\\[2pt]\n  \\verb'\\grole' & $\\grole$ & greater over less\\\\[2pt]\n  \\verb'\\leogr' & $\\leogr$ & less over greater\\\\\n  \\hline\n \\end{tabular}\n\\end{table}\n\nSome additional symbols of common use in astronomy have been added in the MNRAS class. These are shown in tables~\\ref{tab:anysymbols}--\\ref{tab:mathssymbols}. The command names are -- as far as possible -- the same as those used in other major astronomy journals.\n\nMany other mathematical symbols are also available, either built into \\LaTeX\\ or via additional packages. If you want to insert a specific symbol but don't know the \\LaTeX\\ command, we recommend using the Detexify website\\footnote{\\url{http:\/\/detexify.kirelabs.org}}.\n\nSometimes font or coding limitations mean a symbol may not get smaller when used in sub- or superscripts, and will therefore be displayed at the wrong size. There is no need to worry about this as it will be corrected by the typesetter during production.\n\nTo produce bold symbols in mathematics, use \\verb'\\bmath' for simple variables, and the \\verb'bm' package for more complex symbols (see section~\\ref{sec:packages}). Vectors are set in bold italic, using \\verb'\\mathbfit{}'.\n\nFor matrices, use \\verb'\\mathbfss{}' to produce a bold sans-serif font e.g. \\mathbfss{H}; this works even outside maths mode, but not all symbols are available (e.g. Greek). For $\\nabla$ (del, used in gradients, divergence etc.) use \\verb'$\\nabla$'.\n\n\\subsection{Ions}\n\nA new \\verb'\\ion{}{}' command has been added to the class file, for the correct typesetting of ionisation states.\nFor example, to typeset singly ionised calcium use \\verb'\\ion{Ca}{ii}', which produces \\ion{Ca}{ii}.\n\n\\section{Figures and tables}\n\\label{sec:fig_table}\nFigures and tables (collectively called `floats') are mostly the same as built into \\LaTeX.\n\n\\subsection{Basic examples}\n\\begin{figure}\n \\includegraphics[width=\\columnwidth]{example}\n \\caption{An example figure.}\n \\label{fig:example}\n\\end{figure}\nFigures are inserted in the usual way using a \\verb'figure' environment and \\verb'\\includegraphics'. The example Figure~\\ref{fig:example} was generated using the code:\n\\begin{verbatim}\n\\begin{figure}\n \\includegraphics[width=\\columnwidth]{example}\n \\caption{An example figure.}\n \\label{fig:example}\n\\end{figure}\n\\end{verbatim}\n\n\\begin{table}\n \\caption{An example table.}\n \\label{tab:example}\n \\begin{tabular}{lcc}\n  \\hline\n  Star & Mass & Luminosity\\\\\n   & $M_{\\sun}$ & $L_{\\sun}$\\\\\n  \\hline\n  Sun & 1.00 & 1.00\\\\\n  $\\alpha$~Cen~A & 1.10 & 1.52\\\\\n  $\\epsilon$~Eri & 0.82 & 0.34\\\\\n  \\hline\n \\end{tabular}\n\\end{table}\nThe example Table~\\ref{tab:example} was generated using the code:\n\\begin{verbatim}\n\\begin{table}\n \\caption{An example table.}\n \\label{tab:example}\n \\begin{tabular}{lcc}\n  \\hline\n  Star & Mass & Luminosity\\\\\n   & $M_{\\sun}$ & $L_{\\sun}$\\\\\n  \\hline\n  Sun & 1.00 & 1.00\\\\\n  $\\alpha$~Cen~A & 1.10 & 1.52\\\\\n  $\\epsilon$~Eri & 0.82 & 0.34\\\\\n  \\hline\n \\end{tabular}\n\\end{table}\n\\end{verbatim}\n\n\\subsection{Captions and placement}\nCaptions go \\emph{above} tables but \\emph{below} figures, as in the examples above.\n\nThe \\LaTeX\\ float placement commands \\verb'[htbp]' are intentionally disabled.\nLayout of figures and tables will be adjusted by the publisher during the production process, so authors should not concern themselves with placement to avoid disappointment and wasted effort.\nSimply place the \\LaTeX\\ code close to where the figure or table is first mentioned in the text and leave exact placement to the publishers.\n\nBy default a figure or table will occupy one column of the page.\nTo produce a wider version which covers both columns, use the \\verb'figure*' or \\verb'table*'  environment.\n\nIf a figure or table is too long to fit on a single page it can be split it into several parts.\nCreate an additional figure or table which uses \\verb'\\contcaption{}' instead of \\verb'\\caption{}'.\nThis will automatically correct the numbering and add `\\emph{continued}' at the start of the caption.\n\\begin{table}\n \\contcaption{A table continued from the previous one.}\n \\label{tab:continued}\n \\begin{tabular}{lcc}\n  \\hline\n  Star & Mass & Luminosity\\\\\n   & $M_{\\sun}$ & $L_{\\sun}$\\\\\n  \\hline\n  $\\tau$~Cet & 0.78 & 0.52\\\\\n  $\\delta$~Pav & 0.99 & 1.22\\\\\n  $\\sigma$~Dra & 0.87 & 0.43\\\\\n  \\hline\n \\end{tabular}\n\\end{table}\nTable~\\ref{tab:continued} was generated using the code:\n\n\\begin{verbatim}\n\\begin{table}\n \\contcaption{A table continued from the previous one.}\n \\label{tab:continued}\n \\begin{tabular}{lcc}\n  \\hline\n  Star & Mass & Luminosity\\\\\n   & $M_{\\sun}$ & $L_{\\sun}$\\\\\n  \\hline\n  $\\tau$~Cet & 0.78 & 0.52\\\\\n  $\\delta$~Pav & 0.99 & 1.22\\\\\n  $\\sigma$~Dra & 0.87 & 0.43\\\\\n  \\hline\n \\end{tabular}\n\\end{table}\n\\end{verbatim}\n\nTo produce a landscape figure or table, use the \\verb'pdflscape' package and the \\verb'landscape' environment.\nThe landscape Table~\\ref{tab:landscape} was produced using the code:\n\\begin{verbatim}\n\\begin{landscape}\n \\begin{table}\n  \\caption{An example landscape table.}\n  \\label{tab:landscape}\n  \\begin{tabular}{cccccccccc}\n    \\hline\n    Header & Header & ...\\\\\n    Unit & Unit & ...\\\\\n    \\hline\n    Data & Data & ...\\\\\n    Data & Data & ...\\\\\n    ...\\\\\n    \\hline\n  \\end{tabular}\n \\end{table}\n\\end{landscape}\n\\end{verbatim}\nUnfortunately this method will force a page break before the table appears.\nMore complicated solutions are possible, but authors shouldn't worry about this.\n\n\\begin{landscape}\n \\begin{table}\n  \\caption{An example landscape table.}\n  \\label{tab:landscape}\n  \\begin{tabular}{cccccccccc}\n    \\hline\n    Header & Header & Header & Header & Header & Header & Header & Header & Header & Header\\\\\n    Unit & Unit & Unit & Unit & Unit & Unit & Unit & Unit & Unit & Unit \\\\\n    \\hline\n    Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\\\\n    Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\\\\n    Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\\\\n    Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\\\\n    Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\\\\n    Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\\\\n    Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\\\\n    Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\\\\n    \\hline\n  \\end{tabular}\n \\end{table}\n\\end{landscape}\n\n\\section{References and citations}\n\n\\subsection{Cross-referencing}\n\nThe usual \\LaTeX\\ commands \\verb'\\label{}' and \\verb'\\ref{}' can be used for cross-referencing within the same paper.\nWe recommend that you use these whenever relevant, rather than writing out the section or figure numbers explicitly.\nThis ensures that cross-references are updated whenever the numbering changes (e.g. during revision) and provides clickable links (if available in your compiler).\n\nIt is best to give each section, figure and table a logical label.\nFor example, Table~\\ref{tab:mathssymbols} has the label \\verb'tab:mathssymbols', whilst section~\\ref{sec:packages} has the label \\verb'sec:packages'.\nAdd the label \\emph{after} the section or caption command, as in the examples in sections~\\ref{sec:sections} and \\ref{sec:fig_table}.\nEnter the cross-reference with a non-breaking space between the type of object and the number, like this: \\verb'see Figure~\\ref{fig:example}'.\n\nThe \\verb'\\autoref{}' command can be used to automatically fill out the type of object, saving on typing.\nIt also causes the link to cover the whole phrase rather than just the number, but for that reason is only suitable for single cross-references rather than ranges.\nFor example, \\verb'\\autoref{tab:journal_abbr}' produces \\autoref{tab:journal_abbr}.\n\n\\subsection{Citations}\n\\label{sec:cite}\n\nMNRAS uses the Harvard -- author (year) -- citation style, e.g. \\citet{author2013}.\nThis is implemented in \\LaTeX\\ via the \\verb'natbib' package, which in turn is included via the \\verb'usenatbib' package option (see section~\\ref{sec:options}), which should be used in all papers.\n\nEach entry in the reference list has a `key' (see section~\\ref{sec:ref_list}) which is used to generate citations.\nThere are two basic \\verb'natbib' commands:\n\\begin{description}\n \\item \\verb'\\citet{key}' produces an in-text citation: \\citet{author2013}\n \\item \\verb'\\citep{key}' produces a bracketed (parenthetical) citation: \\citep{author2013}\n\\end{description}\nCitations will include clickable links to the relevant entry in the reference list, if supported by your \\LaTeX\\ compiler.\n\n\\defcitealias{smith2014}{Paper~I}\n\\begin{table*}\n \\caption{Common citation commands, provided by the \\texttt{natbib} package.}\n \\label{tab:natbib}\n \\begin{tabular}{lll}\n  \\hline\n  Command & Ouput & Note\\\\\n  \\hline\n  \\verb'\\citet{key}' & \\citet{smith2014} & \\\\\n  \\verb'\\citep{key}' & \\citep{smith2014} & \\\\\n  \\verb'\\citep{key,key2}' & \\citep{smith2014,jones2015} & Multiple papers\\\\\n  \\verb'\\citet[table 4]{key}' & \\citet[table 4]{smith2014} & \\\\\n  \\verb'\\citep[see][figure 7]{key}' & \\citep[see][figure 7]{smith2014} & \\\\\n  \\verb'\\citealt{key}' & \\citealt{smith2014} & For use with manual brackets\\\\\n  \\verb'\\citeauthor{key}' & \\citeauthor{smith2014} & If already cited in close proximity\\\\\n  \\verb'\\defcitealias{key}{Paper~I}' &  & Define an alias (doesn't work in floats)\\\\\n  \\verb'\\citetalias{key}' & \\citetalias{smith2014} & \\\\\n  \\verb'\\citepalias{key}' & \\citepalias{smith2014} & \\\\\n  \\hline\n \\end{tabular}\n\\end{table*}\n\nThere are a number of other \\verb'natbib' commands which can be used for more complicated citations.\nThe most commonly used ones are listed in Table~\\ref{tab:natbib}.\nFor full guidance on their use, consult the \\verb'natbib' documentation\\footnote{\\url{http:\/\/www.ctan.org\/pkg\/natbib}}.\n\nIf a reference has several authors, \\verb'natbib' will automatically use `et al.' if there are more than two authors. However, if a paper has exactly three authors, MNRAS style is to list all three on the first citation and use `et al.' thereafter. If you are using \\bibtex\\ (see section~\\ref{sec:ref_list}) then this is handled automatically. If not, the \\verb'\\citet*{}' and \\verb'\\citep*{}' commands can be used at the first citation to include all of the authors.\n\n\\subsection{The list of references}\n\\label{sec:ref_list}\n\nIt is possible to enter references manually using the usual \\LaTeX\\ commands, but we strongly encourage authors to use \\bibtex\\ instead.\n\\bibtex\\ ensures that the reference list is updated automatically as references are added or removed from the paper, puts them in the correct format, saves on typing, and the same reference file can be used for many different papers -- saving time hunting down reference details.\nAn MNRAS \\bibtex\\ style file, \\verb'mnras.bst', is distributed as part of this package.\nThe rest of this section will assume you are using \\bibtex.\n\nReferences are entered into a separate \\verb'.bib' file in standard \\bibtex\\ formatting.\nThis can be done manually, or there are several software packages which make editing the \\verb'.bib' file much easier.\nWe particularly recommend \\textsc{JabRef}\\footnote{\\url{http:\/\/jabref.sourceforge.net\/}}, which works on all major operating systems.\n\\bibtex\\ entries can be obtained from the NASA Astrophysics Data System\\footnote{\\label{foot:ads}\\url{http:\/\/adsabs.harvard.edu}} (ADS) by clicking on `Bibtex entry for this abstract' on any entry.\nSimply copy this into your \\verb'.bib' file or into the `BibTeX source' tab in \\textsc{JabRef}.\n\nEach entry in the \\verb'.bib' file must specify a unique `key' to identify the paper, the format of which is up to the author.\nSimply cite it in the usual way, as described in section~\\ref{sec:cite}, using the specified key.\nCompile the paper as usual, but add an extra step to run the \\texttt{bibtex} command.\nConsult the documentation for your compiler or latex distribution.\n\nCorrect formatting of the reference list will be handled by \\bibtex\\ in almost all cases, provided that the correct information was entered into the \\verb'.bib' file.\nNote that ADS entries are not always correct, particularly for older papers and conference proceedings, so may need to be edited.\nIf in doubt, or if you are producing the reference list manually, see the MNRAS instructions to authors$^{\\ref{foot:itas}}$ for the current guidelines on how to format the list of references.\n\n\\section{Appendices and online material}\n\nTo start an appendix, simply place the \\verb'\n\\section{Introduction}\n\nThis is a simple template for authors to write new MNRAS papers.\nSee \\texttt{mnras\\_sample.tex} for a more complex example, and \\texttt{mnras\\_guide.tex}\nfor a full user guide.\n\nAll papers should start with an Introduction section, which sets the work\nin context, cites relevant earlier studies in the field by \\citet{Others2013},\nand describes the problem the authors aim to solve \\citep[e.g.][]{Author2012}.\n\n\\section{Methods, Observations, Simulations etc.}\n\nNormally the next section describes the techniques the authors used.\nIt is frequently split into subsections, such as Section~\\ref{sec:maths} below.\n\n\\subsection{Maths}\n\\label{sec:maths}\n\nSimple mathematics can be inserted into the flow of the text e.g. $2\\times3=6$\nor $v=220$\\,km\\,s$^{-1}$, but more complicated expressions should be entered\nas a numbered equation:\n\n\\begin{equation}\n    x=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}.\n\t\\label{eq:quadratic}\n\\end{equation}\n\nRefer back to them as e.g. equation~(\\ref{eq:quadratic}).\n\n\\subsection{Figures and tables}\n\nFigures and tables should be placed at logical positions in the text. Don't\nworry about the exact layout, which will be handled by the publishers.\n\nFigures are referred to as e.g. Fig.~\\ref{fig:example_figure}, and tables as\ne.g. Table~\\ref{tab:example_table}.\n\n\\begin{figure}\n\n\n\n\t\\includegraphics[width=\\columnwidth]{example}\n    \\caption{This is an example figure. Captions appear below each figure.\n\tGive enough detail for the reader to understand what they're looking at,\n\tbut leave detailed discussion to the main body of the text.}\n    \\label{fig:example_figure}\n\\end{figure}\n\n\\begin{table}\n\t\\centering\n\t\\caption{This is an example table. Captions appear above each table.\n\tRemember to define the quantities, symbols and units used.}\n\t\\label{tab:example_table}\n\t\\begin{tabular}{lccr}\n\t\t\\hline\n\t\tA & B & C & D\\\\\n\t\t\\hline\n\t\t1 & 2 & 3 & 4\\\\\n\t\t2 & 4 & 6 & 8\\\\\n\t\t3 & 5 & 7 & 9\\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table}\n\n\n\\section{Conclusions}\n\nThe last numbered section should briefly summarise what has been done, and describe\nthe final conclusions which the authors draw from their work.\n\n\\section*{Acknowledgements}\n\nThe Acknowledgements section is not numbered. Here you can thank helpful\ncolleagues, acknowledge funding agencies, telescopes and facilities used etc.\nTry to keep it short.\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\\label{introduction}\n\n\nIn papers of Allcock, Carlson, Toledo \\cite{ACT1, ACT2}, resp. of Looijenga, Swierstra \\cite{LS1, LS2}, resp. of \nKondo \\cite{K1, K2},  \"hidden\" period maps are constructed in certain  cases. The target spaces of these maps \nare certain {\\it arithmetic quotients of complex unit balls}. The basic \nobservation, which is the starting  point of this paper, is that these arithmetic quotients can be interpreted as the complex points of certain {\\it moduli spaces of abelian varieties \nof Picard type}, of the kind  considered in our paper \\cite{KR2}. Consequently, the purpose in this paper is to interpret these hidden period maps in moduli-theoretic terms.  \nThe pay-off of this exercise is that we can raise and partially answer some {\\it descent problems} which seem natural from our view point, and which are related to a  similar \ndescent problem addressed by Deligne in \\cite{De1} in his theory of {\\it complete intersections of Hodge level one}.\n\nWhy do we speak of \"hidden\", or \"occult\" period maps in this context? This is done in order to make the distinction \nwith the usual period maps which associate to a family of smooth projective complex varieties (over some \nbase scheme $S$)  the (polarized) Hodge structures of its fibers, which \nthen induces a map from $S$ to a quotient by a discrete group of a period domain. Let us recall three examples of classical period maps:\n\n(1) {\\bf Case of quartic surfaces.} In this case the period map is a holomorphic map of orbifolds \n\\begin{equation*}\n\\varphi: {\\it Quartics}_{2, {\\mathbb C}}^\\circ\\to \\big[\\Gamma\\backslash V(2, 19)\\big].\n\\end{equation*}\n\\noindent{\\rm Here ${\\it Quartics}_{2, {\\mathbb C}}^\\circ$ denotes the stack parametrizing smooth quartic surfaces up to projective \nequivalence,\n\\begin{equation*}\n\\text{\\it Quartics}_{2, {\\mathbb C}}^\\circ = \\big[{\\rm PGL}_4\\backslash \\mathbb P{\\text{\\rm Sym}}^4 ({\\mathbb C}^4)^\\circ \\big]\n\\end{equation*}\n{\\rm (stack quotient in the orbifold sense).} The target space is the orbifold quotient of the space of \noriented positive $2$-planes in a quadratic space $V$ of signature $(2, 19)$ by the automorphism group $\\Gamma$ of a lattice in $V$.  \n\n(2) {\\bf Case of cubic threefolds.} In this case the period map is a holomorphic map of orbifolds \n\\begin{equation*}\n\\varphi: {\\it Cubics}_{3, {\\mathbb C}}^\\circ\\to \\big[\\Gamma\\backslash\\mathfrak H_5\\big].\n\\end{equation*}\n\\noindent{\\rm Here ${\\it Cubics}_{3, {\\mathbb C}}^\\circ$ denotes the stack parametrizing smooth cubic threefolds up to projective \nequivalence.\nThe target space is the orbifold quotient of the Siegel upper half space of genus $5$ by the Siegel group $\\Gamma={\\rm Sp}_5(\\mathbb Z)$. \n\n(3) {\\bf Case of cubic fourfolds.} In this case the period map is a holomorphic map of orbifolds \n\\begin{equation*}\n\\varphi: {\\it Cubics}_{4, {\\mathbb C}}^\\circ\\to \\big[\\Gamma\\backslash V(2, 20)\\big].\n\\end{equation*}\n\\noindent{\\rm Here ${\\it Cubics}_{4, {\\mathbb C}}^\\circ$ denotes the stack parametrizing smooth cubic fourfolds up to projective \nequivalence.\nThe target space is the orbifold quotient of the space of oriented positive $2$-planes in a quadratic space \n$V$ of signature $(2, 20)$ by the automorphism group $\\Gamma$ of a lattice in $V$.  \n\nIn the first case, by the Torelli theorem of Piatetskii-Shapiro\/Shafarevich, the induced map $|\\varphi|$ on coarse moduli spaces is an open embedding. In the second case, \nby the Torelli theorem of Clemens\/Griffiths, the map $|\\varphi|$ is a locally closed embedding  (it is not an open embedding since the \nsource of $\\varphi$ has dimension $10$, and the target has dimension $15$).   In the third case, by the Torelli \ntheorem of Voisin, the map $|\\varphi|$ is an open embedding.\n\n\n\n\nThe construction of the occult period maps is quite different, although it does use the classical period maps indirectly. \nFor instance, the construction of Allcock, Carlson, Toledo attaches a certain Hodge structure to any smooth cubic \nsurface which allows one to distinguish between non-isomorphic ones, even though the natural Hodge structures on the \ncohomology in the middle dimension of all cubic surfaces are isomorphic. \nAlso, in one dimension higher, their construction allows them to define an {\\it open embedding} of the space of cubic threefolds into \nan arithmetic quotient of the complex unit ball of dimension $10$. \n\n\nOur second aim in this paper is to  identify the complements of the images of occult period maps \nwith  {\\it special divisors} considered in \\cite{KR2}. \n\nThe lay-out of the paper is as follows. In sections \\ref{subsectionpic}, \\ref{cu}  and \\ref{sc} we recall \nsome of the theory and notation of \\cite{KR2}. In sections \\ref{cubicsurfaces},  \\ref{cubicthreefolds}, \n \\ref{curvesgenus3}, and \\ref{curvesgenus4}, respectively, \n we explain in turn the case of cubic surfaces,  cubic threefolds,  curves of genus $3$, and curves of genus $4$. \n In section \\ref{descent}, we explain the descent problem, and solve it in zero characteristic. In the final section, \nwe make a few supplementary  remarks.\n\nWe stress that the proofs of our statements are all contained in the papers mentioned above, and that our work only consists in interpreting  these results.\n\nWe thank B.~van Geemen, D.~Huybrechts and E.~Looijenga for very helpful discussions. \nWe also thank J.~Achter for keeping us informed about his progress in proving our conjecture in section \\ref{descent} in some cases. Finally, we thank the referee who alerted us to a mistake concerning the stacks aspect of period maps. \n\n\n\n\n\n\n\n\\section{Moduli spaces of Picard type}\\label{subsectionpic} \nLet ${\\text{\\cute k}}={\\mathbb Q}(\\sqrt{\\Delta})$ be an imaginary-quadratic field with discriminant $\\Delta$, ring of integers $O_{\\smallkay}$, and a \nfixed complex embedding.   We write $\\aa\\mapsto{\\aa^\\sigma}$ for the\nnon-trivial automorphism of $O_{\\smallkay}$.\n\nFor integers $n\\geq 1$ and $r$, $0\\leq r\\leq n$, we consider the groupoid $\\Cal M = \\Cal M (n-r, r) = \\Cal M ({\\text{\\cute k}}; n-r, r)$ fibered over $({\\rm Sch} \/ O_{\\smallkay})$ which associates to an \n$O_{\\smallkay}$-scheme $S$ the groupoid of triples $(A, \\iota, \\lambda)$. Here $A$ is an abelian scheme over $S$, \n$\\lambda$ is a principal polarization, and $\\iota : O_{\\smallkay}\\to\\text{\\rm End} (A)$ is a homomorphism such that\n\\begin{equation*}\n\\iota (\\aa)^\\ast = \\iota ({\\aa^\\sigma})\\ ,\n\\end{equation*}\nfor the Rosati involution $\\ast$ corresponding to $\\lambda$.   In addition, the following signature condition is imposed\n\\begin{equation}\n{\\rm char} (T, \\iota (\\aa)\\mid\\text{\\rm Lie}\\,  A) = (T-i(\\aa))^{n-r}\\cdot (T-i({\\aa^\\sigma}))^r\\ ,\\quad\\forall\\,\\aa\\inO_{\\smallkay}\\ ,\n\\end{equation}\nwhere $i:O_{\\smallkay}\\rightarrow \\mathcal O_S$ is the structure map. \n\nWe will mostly consider the complex fiber $\\Cal M_{\\mathbb C} = \\Cal M\\times_{\\text{\\rm Spec}\\, O_{\\smallkay}}\\text{\\rm Spec}\\, {\\mathbb C}$ of $\\Cal M$. In any\ncase, $\\Cal M$ is a Deligne-Mumford stack and $\\Cal M_{\\mathbb C}$ is smooth. We denote by $|\\Cal M_{\\mathbb C}|$ the coarse moduli scheme. \n\nWe will also have to consider the following variant,  defined  by modifying the requirement above that \nthe polarization $\\lambda$ be principal. Let $d>1$ be  a square free divisor of $\\vert \\Delta\\vert$. \nThen $\\Cal M ({\\text{\\cute k}}, d; n-r, r)^*=\\Cal M ({\\text{\\cute k}}; n-r, r)^*$ parametrizes triples $(A, \\iota, \\lambda)$ as in the case \nof  $\\Cal M ({\\text{\\cute k}}; n-r, r)$, except that we impose  the following condition on $\\lambda$. We require first of all \nthat $ \\ker\\lambda\\subset A[d]$, so that $O_{\\smallkay}\/(d)$ acts on $\\ker\\lambda$. In addition, we require \nthat this action factor  through the quotient ring $\\prod_{p\\mid d}{\\mathbb F}_p$ of $O_{\\smallkay}\/(d)$, and that $\\lambda$ \nbe of degree $d^{n-1}$, if $n$ is odd, resp.\\ $d^{n-2}$, if $n$ is even.  In the notation introduced in section 13 of \\cite{KR2}, we have \n$\\Cal M ({\\text{\\cute k}}, d; n-r, r)^*=\\Cal M ({\\text{\\cute k}}, {\\bf t}; n-r, r)^{*, {\\rm naive}}$, where  the function ${\\bf t}$ on the set of \nprimes $p$ with  $p\\mid \\Delta$  assigns to $p$ the  integer $2[(n-1)\/2]$ if $p\\mid d$, and $0$ \nif $p\\nmid d$. Note that if ${\\text{\\cute k}}$ is the Gaussian field ${\\text{\\cute k}}={\\mathbb Q}(\\sqrt{-1})$, then necessarily $d=2$; if ${\\text{\\cute k}}$ is the Eisenstein field  ${\\text{\\cute k}}={\\mathbb Q}(\\sqrt{-3})$, then $d=3$. We denote by $|\\Cal M_{\\mathbb C}^*|$ the corresponding coarse moduli scheme. \n\n\\section{Complex uniformization}\\label{cu} \nLet us recall from \\cite{KR2} the complex uniformization of $\\Cal M ({\\text{\\cute k}}; n-1, 1)({\\mathbb C})$ in the special case that ${\\text{\\cute k}}$ has class number one. \n For $n>2$, let $(V, (\\ ,\\  ))$ be a hermitian vector space over ${\\text{\\cute k}}$ of signature\n$(n-1, 1)$ which contains a self-dual $O_{\\smallkay}$-lattice $L$. By the class number hypothesis, $V$ is unique up to isomorphism.  \nWhen $n$ is odd,  or when $n$ is even and $\\Delta$ is odd, the lattice $L$ is also unique up to isomorphism. We assume \nthat one of these conditions is satisfied. \n Let $\\mathcal D$ be the space of negative lines in the\n${\\mathbb C}$-vector space $(V_{\\mathbb R}, \\mathbb I_0)$, where the complex structure $\\mathbb I_0$ is defined in terms of the discriminant \nof ${\\text{\\cute k}}$, as $\\mathbb I_0 = \\sqrt{\\Delta}\/{|\\sqrt{\\Delta}|}$.\nLet $\\Gamma$  be the isometry group of $L$. \nThen the complex uniformization is the isomorphism of orbifolds, \n\\begin{equation*}\n\\Cal M ({\\text{\\cute k}}; n-1, 1)({\\mathbb C})\\simeq [\\Gamma\\backslash \\mathcal D] .\n\\end{equation*} \nThere is an obvious $*$-variant of this uniformization, which gives \n\\begin{equation*}\n\\Cal M ({\\text{\\cute k}}; n-1, 1)^*({\\mathbb C})\\simeq [\\Gamma^*\\backslash \\mathcal D] ,\n\\end{equation*} \nwhere $\\Gamma^*$ is  the automorphism group  of the  ({\\it parahoric}) lattice $L^*$ corresponding to the $*$-moduli problem.  \nThe lattice $L^*$  is  uniquely determined up to isomorphism  by the condition \nthat there is a chain of inclusions of $O_{\\smallkay}$-lattices \n$L^*\\subset (L^*)^\\vee\\subset (\\sqrt{d})^{-1}L^*$, with quotient $(L^*)^\\vee \/L^*$ of dimension $n-1$ if $n$ is odd and $n-2$ if $n$ is even, when localized \nat any prime ideal $\\mathfrak p$ dividing $d$.  Here, for an $O_{\\smallkay}$-lattice $M$ in $V$, \nwe write \n$$M^\\vee = \\{ \\ x\\in V\\ \\mid \\ h(x,L) \\subset O_{\\smallkay}\\ \\}$$\nfor the dual lattice. \n\n\n\n\\section{ Special cycles (KM-cycles)}\\label{sc}\n We continue to assume that the class number of ${\\text{\\cute k}}$ is one, and recall from \\cite{KR2} the definition of special cycles over \n ${\\mathbb C}$. Let $(E, \\iota_0)$ be an elliptic curve with $CM$ by $O_{\\smallkay}$ over ${\\mathbb C}$, which we fix in what follows. \n Note that, due to our class number hypothesis, $(E, \\iota_0)$\nis unique up to isomorphism. We denote its canonical  principal polarization by $\\lambda_0$. \nFor any ${\\mathbb C}$-scheme $S$, and $(A, \\iota, \\lambda)\\in\\Cal M ({\\text{\\cute k}}; n-1, 1) (S)$, let\n\\begin{equation*}\nV' (A, E) = \\text{\\rm Hom}_{O_{\\smallkay}} (E_S, A)\\ ,\n\\end{equation*}\nwhere $E_S = E\\times_{\\mathbb C} S$ is the constant elliptic scheme over $S$ defined by $E$. \nThen $V' (A, E)$ is a projective $O_{\\smallkay}$-module of finite rank  with a positive definite $O_{\\smallkay}$-valued hermitian form given by\n\\begin{equation*}\nh' (x, y) = \\lambda_0^{-1}\\circ y^\\vee\\circ\\lambda\\circ x\\in\\text{\\rm End}_{O_{\\smallkay}} (E_S) = O_{\\smallkay}\\ .\n\\end{equation*}\nFor a positive integer $t$, we define the DM-stack\\footnote{This notation \ndiffers from that in \\cite{KR2}, in that here the special cycles are defined over ${\\mathbb C}$, and are considered as lying over $\\Cal M({\\text{\\smallcute k}}; n-1, 1)_{\\mathbb C}$.}\n$\\Cal Z(t)$ by \n\\begin{equation*}\n\\Cal Z(t)(S) = \\{(A, \\iota, \\lambda; x)\\mid (A, \\iota, \\lambda) \\in \\Cal M ({\\text{\\cute k}}; n-1, 1)(S), \\  x\\in V' (A, E), \\ \\ h' (x, x) = t\\}\\ .\n\\end{equation*}\nThen $\\Cal Z(t)$ maps by a finite unramified morphism to $\\Cal M ({\\text{\\cute k}}; n-1, 1)_{\\mathbb C}$,  and its image is a divisor in the sense that, locally \nfor the \\'etale topology,  it is defined by a non-zero equation. \n\nThe cycles $\\Cal Z(t)$ also admit a complex uniformization. More precisely, under the assumption of the triviality of the class group of ${\\text{\\cute k}}$, we have\n\\begin{equation*}\n\\Cal Z(t)({\\mathbb C}) \\simeq \\Big[\\Gamma\\backslash\\Big(\\coprod_{\\substack{x\\in L\\\\ h(x, x)=t}}\\mathcal D_x\\Big)\\Big] ,\n\\end{equation*} \nwhere $\\mathcal D_x$ is the set of lines in $\\mathcal D$ which are perpendicular to $x$. \n\nAgain, there is a $*$-variant of these definitions and a corresponding DM-stack $\\Cal Z(t)^*$ above $\\Cal M({\\text{\\cute k}}; n-1, 1)^*$. \n\n\n\\section{ Cubic surfaces} \\label{cubicsurfaces}\nIn this paper we  consider  four occult  period mappings. We start with the case of cubic surfaces, following Allcock, Carlson, Toledo \\cite{ACT1}, comp.\\ also \\cite{B}. \nAs explained in the introduction, in these sources, the results are formulated in terms of arithmetic ball quotients;\nhere we use the complex uniformization of the previous two sections to\nexpress these results in terms of moduli spaces of Picard type.\n\nLet $S\\subset\\mathbb P^3$ be a smooth cubic surface. Let $V$ be a cyclic covering of degree $3$ of $\\mathbb P^3$, \nramified along $S$. Explicitly, if $S$ is defined by the homogeneous equation of degree $3$ in $4$\nvariables\n\\begin{equation*}\nF (X_0, \\ldots , X_3) = 0\\ ,\n\\end{equation*}\nthen $V$ is defined by the homogeneous equation of degree $3$\nin $5$ variables,\n\\begin{equation*}\nX_4^3 - F(X_0, \\ldots , X_3) = 0\\ .\n\\end{equation*}\n\nLet ${\\text{\\cute k}} = {\\mathbb Q} (\\o)$, $\\o = e^{2\\pi i \/3}$. Then the obvious $\\mu_3$-action on $V$ determines an action of $O_{\\smallkay}={\\mathbb Z}[\\o]$ on $H^3 (V, {\\mathbb Z})$. \nFor  the (alternating) cup product pairing $\\langle \\ , \\ \\rangle$,\n\\begin{equation*}\n\\langle \\o x, \\o y \\rangle = \\langle  x,  y \\rangle  ,\n\\end{equation*}\nwhich implies that\n\\begin{equation*}\n\\langle \\aa x,  y \\rangle = \\langle  x,  {\\aa^\\sigma} y \\rangle ,\\quad \\forall\\,\\aa \\in O_{\\smallkay} . \n\\end{equation*}\nHence there is a unique $O_{\\smallkay}$-valued hermitian form $h$ on $H^3 (V, {\\mathbb Z})$ such that \n\\begin{equation}\\label{altherm}\n\\langle  x,  y \\rangle = {\\rm tr} \\big(\\frac{1}{\\sqrt \\Delta} h( x,  y)\\big) ,\n\\end{equation}\nwhere the discriminant $\\Delta$   of ${\\text{\\cute k}}$ is equal to $-3$ in the case at hand. Explicitly,\n\\begin{equation}\nh(x, y)=\\frac{1}{2}\\big(\\langle \\sqrt{\\Delta} x,  y \\rangle +\\langle  x,  y \\rangle \\sqrt{\\Delta}\\big) .  \n\\end{equation} \n\nFurthermore, an $O_{\\smallkay}$-lattice is self-dual wrt $\\langle\\ ,\\ \\rangle$ if and only if it is self-dual wrt $h(\\ ,\\ )$. \n\\smallskip\n\n{\\bf Fact:} {\\it $H^3 (V, {\\mathbb Z})$ is a self-dual hermitian $O_{\\smallkay}$-module of signature $(4, 1)$.}\n\\smallskip\n\nAs noted above, such a lattice is unique up to isomorphism.\n\nLet\n\\begin{equation*}\nA = A(V) = H^3 (V, {\\mathbb Z})\\backslash H^3 (V, {\\mathbb C}) \/ H^{2, 1} (V)\n\\end{equation*}\nbe the intermediate Jacobian of $V$. Then $A$ is an abelian variety of dimension $5$ which is principally\npolarized by the intersection form. Since the association $V\\mapsto (A(V), \\lambda)$ is functorial, we obtain\nan action $\\iota$ of $O_{\\smallkay}$ on $A(V)$.\n\\begin{theo}\\label{cubicsurf}\n(i) The object $(A, \\iota, \\lambda )$ lies in $\\Cal M ({\\text{\\cute k}}; 4,1) ({\\mathbb C})$.\n\\smallskip\n\n\\noindent (ii)This construction is functorial and compatible with families,   \nand defines a morphism of\nDM-stacks,\n\\begin{equation*}\n\\varphi: {\\it Cubics}_{2, {\\mathbb C}}^\\circ\\to\\Cal M ({\\text{\\cute k}}; 4,1)_{\\mathbb C}\\ .\n\\end{equation*}\n{Here ${\\it Cubics}_{2, {\\mathbb C}}^\\circ$ denotes the stack parametrizing smooth cubic surfaces up to projective \nequivalence,\n\\begin{equation*}\n\\text{\\it Cubics}_{2, {\\mathbb C}}^\\circ = [ {\\rm PGL}_4\\backslash \\mathbb P{\\text{\\rm Sym}}^3 ({\\mathbb C}^4)^\\circ ]\n\\end{equation*}\n{\\rm [stack quotient in the orbifold sense].}}\n\\smallskip\n\n\\noindent (iii) The induced morphism on coarse moduli spaces $|\\varphi|: |{\\it Cubics}_{2, {\\mathbb C}}^\\circ|\\to|\\Cal M ({\\text{\\cute k}}; 4,1)_{\\mathbb C}|$ is an open embedding. Its image is the complement of the image of the \nKM-cycle $\\Cal Z(1)$ in $|\\Cal M ({\\text{\\cute k}}; 4,1)_{\\mathbb C}|$.\n\\end{theo}\n\\begin{proof}\nWe only comment  on  the assertions in (ii) and (iii). In (ii), the compatibility with families is always true of Griffiths'  \nintermediate jacobians (which however are abelian varieties only when the Hodge structure is of type \n$(m+1, m)+(m, m+1)$). This constructs $\\varphi$ as a complex-analytic morphism. \nThe algebraicity of $\\varphi$ then follows from Borel's theorem that any analytic family of abelian varieties over a ${\\mathbb C}$-scheme is automatically algebraic \\cite{Bo}. \nThe fact that the image is contained in the complement of $\\Cal Z(1)$ is true because, by the Clemens-Griffiths \ntheory, intermediate Jacobians of cubic threefolds  are simple as polarized abelian varieties, whereas,  over $\\Cal Z(1)$ the \npolarized abelian varieties split off an elliptic curve. However, the fact that $\\Cal Z(1)$ makes up the whole complement \nis surprising and results from the fact that the morphism $\\varphi$ extends to an isomorphism from a \npartial compactification $|\\text{\\it Cubics}_{2, {\\mathbb C}}^{\\rm s}|$ of $|\\text{\\it Cubics}_{2, {\\mathbb C}}^\\circ|$ (obtained by \nadding {\\it stable} cubics) to  $|\\Cal M ({\\text{\\cute k}}; 4,1)_{\\mathbb C}|$, such that the complement of \n$|\\text{\\it Cubics}_{2, {\\mathbb C}}^\\circ|$ in $|\\text{\\it Cubics}_{2, {\\mathbb C}}^{\\rm s}|$  is an irreducible divisor, cf.\\ \\cite{B}, Prop.\\ 6.7, Prop.\\ 8.2. \n\\end{proof}\n\\begin{rem} {\\rm Let us comment on the stacks aspect of Theorem \\ref{cubicsurf}. Any automorphism of $S$ is induced by an automorphism of $\\mathbb P^3$, which in turn induces an automorphism of $V$. We therefore obtain a homomorphism\n$ {\\rm Aut}(S)\\to {\\rm Aut}(A(V), \\iota, \\lambda).$\nThe statement of \n\\cite{ACT1}, Thm.~2.20. implies that this homomorphism induces an isomorphism\n \\begin{equation}\\label{stackiso}\n  {\\rm Aut}(S)\\overset{\\sim}{\\longrightarrow} {\\rm Aut}(A(V), \\iota, \\lambda)\/O_{\\smallkay}^\\times\\, ,\n\\end{equation} \nwhere the units $O_{\\smallkay}^\\times\\simeq\\mu_6$ act via $\\iota$  on $A(V)$. Indeed, in loc. cit. it is asserted that $\\varphi$ is an open immersion of orbifolds ${\\it Cubics}_{2, {\\mathbb C}}^\\circ\\to [P\\Gamma\\backslash \\mathcal D]$, where $P\\Gamma=\\Gamma\/O_{\\smallkay}^\\times$; however, we were not able to follow the argument. Note that the orbifold  $[P\\Gamma\\backslash \\mathcal D]$ is different from $[\\Gamma\\backslash \\mathcal D]$, which occurs in \\S 3. \n}\n\\end{rem}\n\\section{ Cubic threefolds}\\label{cubicthreefolds} \nOur next example concerns cubic threefolds, following Allcock, Carlson, Toledo  \\cite{ACT2} and Looijenga, Swierstra \\cite{LS1}.\n\nLet $T\\subset\\mathbb P^4$ be a cubic threefold. Let $V$ be the cyclic covering of degree $3$ of $\\mathbb P^4$, \nramified in $T$. Then $V$ is a cubic hypersurface in $\\mathbb P^5$ and we define the primitive cohomology as \n\\begin{equation}\nL=H_0^4 (V, {\\mathbb Z}) = \\{x\\in H^4 (V, {\\mathbb Z})\\mid ( x, \\rho) = 0\\}\\ ,\n\\end{equation}\nwhere $\\rho$ is the square of the hyperplane section class. Note that  ${\\rm rk}_{{\\mathbb Z}} L = 22$. \nAgain, let ${\\text{\\cute k}} ={\\mathbb Q} (\\o)$, with $\\o= e^{2\\pi i\/3}$, so that  $L$\nbecomes an $O_{\\smallkay}$-module. Now the cup-product $(\\ , \\  )$ on $H^4 (V, {\\mathbb Z})$  is a perfect {\\it symmetric} \npairing satisfying $(a x, y)= (x, {a^\\sigma} y)$ for $a \\inO_{\\smallkay}$. It induces on\n$L$ a symmetric bilinear form $(\\  ,\\  )$ of discriminant $3$. We wish to define an {\\it alternating} pairing \n$\\langle \\ , \\ \\rangle$ on $L$ satisfying $\\langle \\aa x, y\\rangle=\\langle x, {\\aa^\\sigma} y\\rangle$ for $\\aa\\in O_{\\smallkay}$. \nWe do this by giving the associated $O_{\\smallkay}$-valued hermitian pairing $h(\\ , \\ )$, \nin the sense of (\\ref{altherm}) defined by \n\\begin{equation}\\label{hermit1}\nh(x, y) = \\frac{3}{2}  \\big(( x, y) +  ( x, \\sqrt{\\Delta} y) \\frac{1}{\\sqrt{\\Delta}}\\big). \n\\end{equation}\nHere the factor $3\/2$ is used instead of $1\/2$  to have better integrality properties. Set $\\pi=\\sqrt{\\Delta}$. \n\n\n\n{\\bf Fact:} {\\it For the pairing \\eqref{hermit1},  $L^{\\vee}$ contains $\\pi^{-1}L$ with \n$L^\\vee\/\\pi^{-1}L\\simeq {\\mathbb Z}\/3{\\mathbb Z}$.} \\hfill\\break \nFor this result, see \\cite{ACT2}, Theorem\\ 2.6 and its proof, as well as \\cite{LS1}, the passage below (2.1). \n\n\nNow consider the eigenspace decomposition of $H_0^4 (V, {\\mathbb C})$ under ${\\text{\\cute k}}\\otimes{\\mathbb C} = {\\mathbb C}\\oplus{\\mathbb C}$.\n\n\n{\\bf Fact:} {\\it The Hodge structure of $H_0^4 (V, {\\mathbb R})$ is of type\n\\begin{equation*}\nH_0^4 (V, {\\mathbb C}) = H^{3, 1}\\oplus H_0^{2, 2}\\oplus H^{1, 3}\\ ,\n\\end{equation*}\nwith $\\dim H^{3, 1} = \\dim H^{1, 3} = 1$. Furthermore, the only nontrivial eigenspaces of the generator $\\o$ of $\\mu_3$ are\n\\begin{equation*}\n\\begin{aligned}\nH_0^4 (V, {\\mathbb C})_\\o & = & H^{3, 1}\\oplus (H_0^{2, 2})_\\o\\ , & \\text{ with}\\  \\dim (H_0^{2, 2})_\\o = 10\\\\\nH_0^4 (V, {\\mathbb C})_{\\overline{\\o}} & = & (H_0^{2, 2})_{\\overline{\\o}}\\oplus H^{1, 3}\\ , & \\text{ with}\\ \\dim (H_0^{2, 2})_{\\overline{\\o}} = 10\\ ,\n\\end{aligned}\n\\end{equation*}}\nsee \\cite{ACT2}, \\S 2, resp.\\ \\cite{LS1} \\S 4.\n\n\\smallskip\n\nNow set $\\Lambda=\\pi L^\\vee$. Then we have the chain of inclusions of $O_{\\smallkay}$-lattices\n\\begin{equation*}\n\\Lambda\\subset \\Lambda^\\vee\\subset \\pi^{-1}\\Lambda\\  ,\n\\end{equation*}\nwhere the quotient $\\Lambda^\\vee\/\\Lambda$ is isomorphic to $({\\mathbb Z}\/3{\\mathbb Z})^{10}$, and where $\\pi^{-1}\\Lambda\/\\Lambda^\\vee$ is isomorphic to ${\\mathbb Z}\/3{\\mathbb Z}$. \nLet\n\\begin{equation*}\nA = \\Lambda\\backslash H_0^4 (V, {\\mathbb C}) \/ H^-\\ ,\n\\end{equation*}\nwhere\n\\begin{equation*}\nH^- = H^{3, 1}\\oplus (H_0^{2, 2})_{\\overline{\\o}}\\ .\n\\end{equation*}\nNote that the map $\\Lambda\\to H_0^4 (V, {\\mathbb C}) \/ H^-$ is an $O_{\\smallkay}$-linear injection, hence $A$ is \na complex torus. In fact, the hermitian form $h$ and its associated alternating form $\\langle \\ , \\ \\rangle$ define a polarization $\\lambda$ on $A$. Hence $A$ is an abelian variety of dimension $11$, with an action of $O_{\\smallkay}$ and a polarization of degree\n$3^{10}$. In fact, we obtain in this way an object $(A, \\iota, \\lambda )$ of $\\Cal M ({\\text{\\cute k}}; 10, 1)^\\ast ({\\mathbb C})$ (see section \\ref{subsectionpic} for the definition of the $*$-variants of our moduli stacks).\n\\begin{theo}\\label{thmcubic3}\n\n\\noindent (i) The construction which associates to a smooth cubic $T$ in $\\mathbb P^4$ the object $(A, \\iota, \\lambda )$\nof $\\Cal M ({\\text{\\cute k}}; 10, 1)^\\ast({\\mathbb C})$ is functorial and compatible with families,  and defines a morphism of DM-stacks,\n\\begin{equation*}\n\\varphi : \\text{Cubics}_{3, {\\mathbb C}}^\\circ\\to\\Cal M ({\\text{\\cute k}}; 10,1)_{\\mathbb C}^\\ast\\ .\n\\end{equation*}\n\n\\noindent (ii) The induced morphism on coarse moduli spaces $|\\varphi| : |\\text{Cubics}_{3, {\\mathbb C}}^\\circ|\\to|\\Cal M ({\\text{\\cute k}}; 10,1)_{\\mathbb C}^\\ast| $ is an open embedding. Its image is the complement of the  image of the KM-cycle $\\Cal Z (3)^*$ in $|\\Cal M ({\\text{\\cute k}}; 10,1)_{\\mathbb C}^\\ast|$.\n\\end{theo}\n\\begin{proof}\n The compatibility with families is due to the fact \nthat the eigenspaces for the $\\mu_3$-action and the Hodge filtration both vary in a \nholomorphic way. The point (ii)  is \\cite{ACT2}, Thm.\\ 1.1, resp.\\ \\cite{LS1}, Thm.\\ 3.1.  \n\\end{proof}\n\\begin{rem}{\\rm  The stack aspect is not treated in these sources. However, it seems reasonable to conjecture that the analogue of \\eqref{stackiso} is also true in this case, i.e., that there is an isomorphism \n \\begin{equation}\\label{stackiso2}\n  {\\rm Aut}(T)\\overset{\\sim}{\\longrightarrow} {\\rm Aut}(A, \\iota, \\lambda)\/O_{\\smallkay}^\\times\\, ,\n\\end{equation} \nwhere $(A, \\iota, \\lambda)$ is the object of $\\Cal M ({\\text{\\cute k}}; 10,1)_{\\mathbb C}^\\ast$ attached to $T$. }\\end{rem}\n\\begin{rem}\n{\\rm The construction of the rational Hodge structure $H^1 (A, {\\mathbb Q})$ from $H_0^4 (V, {\\mathbb Q})$ is a very special case\nof a general construction due to van Geemen \\cite{G}. More precisely, it arises (up to Tate twist) as the  {\\it inverse\nhalf-twist} in the sense of loc.\\ cit.\\  of the Hodge structure $H_0^4 (V, {\\mathbb Q})$ with complex \nmultiplication by ${\\text{\\cute k}}$. The {\\it half twist} construction attaches to a rational Hodge structure \n$V$ of weight $w$ with complex multiplication  by a CM-field ${\\text{\\cute k}}$ a rational Hodge \nstructure of weight $w+1$. More precisely, if $\\Sigma$ is a fixed half system of complex \nembeddings of ${\\text{\\cute k}}$, then van Geemen defines a new Hodge structure on $V$ by setting\n\\begin{equation*}\nV_{\\rm new}^{r, s}=V_{\\Sigma}^{r-1, s}\\oplus V_{\\overline{\\Sigma}}^{r, s-1} , \n\\end{equation*}  \nwhere $V_{\\Sigma}$, resp.\\ $V_{\\overline{\\Sigma}}$ denotes the sum of the \neigenspaces for the ${\\text{\\cute k}}$-action corresponding to the complex embeddings in ${\\Sigma}$, resp. in ${\\overline{\\Sigma}}$. }\n\\end{rem}\n\n\n\n\n\\section{Curves of genus three} \\label{curvesgenus3}\n\nOur third example concerns the moduli space of curves of genus $3$ following Kondo \\cite{K1}. \n\nLet $C$ be a non-hyperelliptic smooth projective curve of genus 3. The canonical system embeds $C$ as a\nquartic curve in $\\mathbb P^2$. Let $X(C)$ be the $\\mu_4$-covering of $\\mathbb P^2$ ramified in $C$. Then the quartic $X(C)\\subset \\mathbb P^3$ is a\n$K3$-surface with an automorphism $\\tau$ of order $4$ and hence an action of $\\mu_4$. Let\n\\begin{equation*}\nL= \\{x\\in H^2 (X (C), {\\mathbb Z})\\mid\\tau^2 (x) = -x\\}\\ .\n\\end{equation*}\nLet ${\\text{\\cute k}} = {\\mathbb Q} (i)$ be the Gaussian field.\n\n\\smallskip\n\n{\\bf Fact:} {\\it $L$ is a free ${\\mathbb Z}$-module of rank $14$. The restriction $(\\ , \\ )$ of the symmetric\ncup product pairing to $L$ has discriminant $2^8$; more precisely, for the dual lattice $L^*$ for the symmetric pairing, \n\\begin{equation*}\nL^* \/ L\\cong ({\\mathbb Z}\/2)^8\\ ,\n\\end{equation*}}\nsee \\cite{K1}, top of p.\\ 222.\n\n\\smallskip\n\nNow consider the eigenspace decomposition of $L_{ {\\mathbb C}}=L\\otimes{\\mathbb C}$ under ${\\text{\\cute k}}\\otimes{\\mathbb C} = {\\mathbb C}\\oplus{\\mathbb C}$, where $i\\otimes 1$ acts via $\\tau$.\n\n\\smallskip\n\n{\\bf Fact:} {\\it The induced Hodge structure on $L_{{\\mathbb C}}$ is of type\n\\begin{equation*}\nL_{{\\mathbb C}} = L^{2,0}\\oplus L^{1,1}\\oplus L^{0,2}\\ ,\n\\end{equation*}\nwith $\\dim L^{2,0} = \\dim L^{0,2} = 1$. Furthermore the only nontrivial eigenspaces of $\\tau$ are\n\\begin{equation*}\n\\begin{aligned}\n(L_{{\\mathbb C}})_i & = & L^{2,0}\\oplus (L^{1,1})_i\\ , & \\text{ with}\\ \\dim (L^{1,1})_i = 6\\\\\n(L_{{\\mathbb C}})_{-i} & = & (L^{1,1})_{-i}\\oplus L^{0,2}\\ , & \\text{ with}\\ \\dim (L^{1,1})_{-i} = 6\\ .\n\\end{aligned}\n\\end{equation*}}\n\n\\smallskip\n\nWe define an $O_{\\smallkay}$-valued  hermitian pairing $h$ on $L_{\\mathbb Q}$ by setting \n\\begin{equation}\nh(x, y)= (x, y) +(x, \\tau y)\\ i\\  .\n\\end{equation}\nThen it is easy to see that the dual lattice $L^\\vee$ of $L$ for the hermitian form $h$ is the same as the dual lattice $L^*$ for the symmetric form. \n\nNow set $\\Lambda=\\pi L^\\vee$, where $\\pi=1+ i$. Then we obtain a chain of inclusions of $O_{\\smallkay}$-lattices\n\\begin{equation*}\n\\Lambda\\subset \\Lambda^\\vee\\subset \\pi^{-1}\\Lambda\\  ,\n\\end{equation*}\nwhere the quotient $\\Lambda^\\vee\/\\Lambda$ is isomorphic to $({\\mathbb Z}\/2{\\mathbb Z})^{6}$, and where $\\pi^{-1}\\Lambda\/\\Lambda^\\vee$ is isomorphic to ${\\mathbb Z}\/2{\\mathbb Z}$. \n\n\nLet\n\\begin{equation*}\nA = \\Lambda\\backslash L_{{\\mathbb C}} \/ L^-\\ ,\n\\end{equation*}\nwhere\n\\begin{equation*}\nL^-= L^{2,0}\\oplus (L^{1,1})_{-i}\\ .\n\\end{equation*}\nNote that the map $\\Lambda\\to L_{{\\mathbb C}} \/ L^-$ is a $O_{\\smallkay}$-linear injection, hence\n$A$ is a complex torus. In fact, the hermitian form $h$ and its associated \nalternating form $\\langle \\ , \\ \\rangle$ define a polarization $\\lambda$ \non $A$. Hence $A$ is an abelian variety of dimension $7$, with an action of $O_{\\smallkay}$ and a polarization of degree\n$2^{6}$. In fact, we obtain in this way an object $(A, \\iota, \\lambda )$ \nof $\\Cal M ({\\text{\\cute k}}; 6, 1)^\\ast ({\\mathbb C})$.\nNow \\cite{K1}, Thm.\\ 2.5  implies the following theorem. \n\n\\begin{theo}\n\n\\noindent (i) The construction which asssociates to a non-hyperelliptic curve of genus $3$ the object $(A, \\iota, \\lambda )$\nof $\\Cal M ({\\text{\\cute k}}; 6,1)^\\ast({\\mathbb C})$ is functorial and compatible with families,  and defines a morphism of DM-stacks,\n\\begin{equation*}\n\\varphi : \\Cal N_{3, {\\mathbb C}}^\\circ\\to\\Cal M (k; 6,1)^\\ast_{\\mathbb C}\\ .\n\\end{equation*}\n{ Here $\\Cal N_{3,{\\mathbb C}}^\\circ$ denotes the stack of smooth non-hyperelliptic curves of genus $3$, i.e., of smooth non-hyperelliptic quartics in $\\mathbb P^2$ up to projective equivalence.}\n\\smallskip\n\n\\noindent (ii) The induced morphism on coarse moduli schemes $|\\varphi| : |\\Cal N_{3, {\\mathbb C}}^\\circ|\\to|\\Cal M (k; 6,1)^\\ast_{\\mathbb C}| $ is an open embedding. Its image is the complement of the image of the KM-cycle $\\Cal Z (2)^*$ in $|\\Cal M (k; 6,1)^\\ast_{\\mathbb C}|$.\\qed\n\\end{theo}\n\\begin{rem}{\\rm  Again, the stack aspect is not treated in \\cite{K1}. It seems reasonable to conjecture that the analogue of \\eqref{stackiso} is also true in this case, i.e., that there is an isomorphism \n \\begin{equation}\\label{stackiso2}\n  {\\rm Aut}(C)\\overset{\\sim}{\\longrightarrow} {\\rm Aut}(A, \\iota, \\lambda)\/O_{\\smallkay}^\\times\\, ,\n\\end{equation} \nwhere $(A, \\iota, \\lambda)$ is the object of $\\Cal M ({\\text{\\cute k}}; 6,1)_{\\mathbb C}^\\ast$ attached to $C$, and where $O_{\\smallkay}^\\times=\\mu_4$. }\\end{rem}\n\n\\section{Curves of genus four} \\label{curvesgenus4}\nOur final example concerns the moduli space of curves of genus four and is also due to Kondo  \\cite{K2}. \n\nLet $C$ be a non-hyperelliptic curve of genus $4$. The canonical system embeds $C$ into\n$\\mathbb P^3$. More precisely, $C$ is the intersection of a smooth cubic surface $S$ and a\nquartic $Q$ which is either smooth or a quadratic cone. Furthermore, $Q$ is uniquely determined by $C$. Let $X$ be a cyclic cover of\ndegree $3$ over $Q$ branched along $C$ (in case $Q$ is singular, we take the minimal\nresolution of the singularities, cf.\\ loc.cit.). Then $X$ is a $K3$-surface with an action of $\\mu_3$. Let\n\\begin{equation*}\nL = (H^2 (X, {\\mathbb Z})^{\\mu_3})^\\perp\n\\end{equation*}\nbe the orthogonal complement of the invariants of this action in $H^2 (X, {\\mathbb Z})$, equipped with the symmetric form $(\\ , \\ )$ obtained by restriction. \n\n\\smallskip\n\n{\\bf Fact:} {\\it $L$ is a free ${\\mathbb Z}$-module of rank $20$, with dual $L^*$ for the symmetric form satisfying \n\\begin{equation*}\nL^* \/ L \\simeq ({\\mathbb Z} \/3{\\mathbb Z})^2\\ ,\n\\end{equation*}}\ncf. \\cite{K2}, top of p.\\ 386. \n\n\\smallskip\n\nFor ${\\text{\\cute k}}={\\mathbb Q}(\\o)$, $\\o= e^{2\\pi i\/3}$, we again define an alternating form $\\langle \\ , \\ \\rangle$ through its associated \n$O_{\\smallkay}$-valued\nhermitian form $h$. Using the action of $O_{\\smallkay}$ on $L$,  we set\n\\begin{equation}\\label{hermit2}\nh(x, y) = \\frac{3}{2}  \\big(( x, y) +  ( x, \\sqrt{\\Delta} y) \\frac{1}{\\sqrt{\\Delta}}\\big)\\ .\n\\end{equation}\nSet $\\pi=\\sqrt{\\Delta}$. \n\n\\smallskip\n\n{\\bf Fact:} {\\it For the hermitian pairing \\eqref{hermit2},  $L^{\\vee}$ is  an over-lattice of $\\pi^{-1}L$ with $L^\\vee\/\\pi^{-1}L\\simeq ({\\mathbb Z}\/3{\\mathbb Z})^2$.}\n\n\\smallskip\n\nNow consider the eigenspace decomposition of $L\\otimes{\\mathbb C}$ under ${\\text{\\cute k}}\\otimes{\\mathbb C} = {\\mathbb C}\\oplus{\\mathbb C}$.\n\n\\smallskip\n\n{\\bf Fact:} {\\it The induced Hodge structure on $L_{\\mathbb C}$ is of type\n\\begin{equation*}\nL_{\\mathbb C} = L^{2,0}\\oplus L^{1,1}\\oplus L^{0,2}\\ ,\n\\end{equation*}\nwith $\\dim L^{2,0} = \\dim L^{0,2} = 1$. Furthermore the only nontrivial eigenspaces of $\\mu_3$ are\n\\begin{equation*}\n\\begin{aligned}\n(L_{\\mathbb C})_\\omega = L^{2,0}\\oplus (L^{1,1})_\\o\\ ,\\ \\text{with}\\ \\dim (L^{1,1})_\\o &= 9\\\\\n(L_{\\mathbb C})_{\\overline{\\omega}} = (L^{1,1})_{\\overline{\\o}}\\oplus L^{0,2}\\ ,\\ \\text{with}\\ \\dim (L^{1,1})_{\\overline{\\o}} &= 9\\ .\n\\end{aligned}\n\\end{equation*}}\n\n\\smallskip\n\nNow set $\\Lambda=\\pi L^\\vee$. Then we have the chain of inclusions of $O_{\\smallkay}$-lattices\n\\begin{equation*}\n\\Lambda\\subset \\Lambda^\\vee\\subset \\pi^{-1}\\Lambda\\  ,\n\\end{equation*}\nwhere the quotient $\\Lambda^\\vee\/\\Lambda$ is isomorphic to $({\\mathbb Z}\/3{\\mathbb Z})^{8}$, and where $\\pi^{-1}\\Lambda\/\\Lambda^\\vee$ is isomorphic to $({\\mathbb Z}\/3{\\mathbb Z})^2$. \n\n\nLet\n\\begin{equation*}\nA = \\Lambda\\backslash L_{\\mathbb C} \/ L^-\\ ,\n\\end{equation*}\nwhere \n\\begin{equation*}\nL^- = L^{2,0}\\oplus (L^{1,1})_{\\overline{\\o}}\\ .\n\\end{equation*}\n\n Then the map\n$\\Lambda\\to L_{\\mathbb C} \/ L^-$ is a $O_{\\smallkay}$-linear injection, hence $A$ is a complex torus. \nIn fact, the hermitian form $h$ and its associated alternating form $\\langle \\ , \\ \\rangle$ \ndefine a polarization $\\lambda$ on $A$. Hence $A$ is an abelian variety of dimension $10$, with an action of $O_{\\smallkay}$ and a polarization of degree\n$3^{8}$. In fact, we obtain in this way an object $(A, \\iota, \\lambda )$ of $\\Cal M ({\\text{\\cute k}}; 9, 1)^\\ast ({\\mathbb C})$, \n\\begin{theo}\n\n\\noindent (i) The construction which associates to a non-hyperelliptic curve of genus $4$ the object\n$(A,\\iota, \\lambda )$ of $\\Cal M ({\\text{\\cute k}}; 9,1)^\\ast({\\mathbb C})$ is functorial and compatible with families,  and defines a morphism of\nDM-stacks,\n\\begin{equation*}\n\\varphi :\\Cal N^\\circ_{4, {\\mathbb C}}\\to\\Cal M ({\\text{\\cute k}}; 9,1)^\\ast_{\\mathbb C}\\ .\n\\end{equation*}\n{\\rm Here $\\Cal N^\\circ_{4, {\\mathbb C}}$ denotes the stack of smooth non-hyperelliptic curves of genus $4$.}\n\n\\smallskip\n\n\\noindent (ii)  The induced morphism on coarse moduli schemes $|\\varphi| :|\\Cal N^\\circ_{4, {\\mathbb C}}|\\to|\\Cal M ({\\text{\\cute k}}; 9,1)^\\ast_{\\mathbb C}| $ is an open embedding. Its image is the complement of the image of the KM-cycle\n$\\Cal Z (2)^*$ in $|\\Cal M ({\\text{\\cute k}}; 9,1)^\\ast_{\\mathbb C}|$.\\qed\n\\end{theo}\n\\begin{rem}{\\rm  Again, the stack aspect is not treated  in \\cite{K2}. It seems reasonable to conjecture that the analogue of \\eqref{stackiso} is also true in this case, i.e., that there is an isomorphism \n \\begin{equation}\\label{stackiso2}\n  {\\rm Aut}(C)\\overset{\\sim}{\\longrightarrow} {\\rm Aut}(A, \\iota, \\lambda)\/O_{\\smallkay}^\\times\\, ,\n\\end{equation} \nwhere $(A, \\iota, \\lambda)$ is the object of $\\Cal M ({\\text{\\cute k}}; 9,1)_{\\mathbb C}^\\ast$ attached to $C$, and where $O_{\\smallkay}^\\times=\\mu_6$. }\\end{rem}\n\n\\section{Descent} \\label{descent}\nIn all four cases discussed above, we obtain  morphisms over ${\\mathbb C}$ between \nDM-stacks defined over ${\\text{\\cute k}}$. These \nmorphisms are constructed using transcendental methods. In this section \nwe will show that  these morphisms are in fact defined over ${\\text{\\cute k}}$.  The argument is \nmodelled on Deligne's solution of the analogous problem for complete intersections of \nHodge level one \\cite{De1}, where he shows that the corresponding family of intermediate \njacobians is an abelian scheme over the moduli scheme over ${\\mathbb Q}$ of complete intersections of given multi-degree. \n\nIn our discussion below, to simplify notations,  we will deal with  the case of cubic threefolds, as explained in \nsection \\ref{cubicthreefolds}; the other cases are completely analogous. Below we will shorten the \nnotation $Cubics^\\circ_3$ to $\\mathcal C$, and consider this as a DM-stack over $\\text{\\rm Spec}\\,  {\\text{\\cute k}}$. \nLet $v:V\\to \\mathcal C$ be the universal family of cubic threefolds, and let $a: A\\to \\mathcal C_{\\mathbb C}$ \nbe the polarized family of abelian varieties constructed from $V$ in section \\ref{cubicthreefolds}. \nHence $A$ is the pullback  of the universal abelian scheme over $\\mathcal M({\\text{\\cute k}}; 10, 1)^*_{\\mathbb C}$ under the \nmorphism  $\\varphi: \\mathcal C_{\\mathbb C}\\to \\mathcal M({\\text{\\cute k}}; 10, 1)^*_{\\mathbb C}$.  \n\\begin{lem}\\label{elladic}\n Let $b: B\\to \\mathcal C_{\\mathbb C}$ be a polarized abelian scheme with $O_{\\smallkay}$-action, which is the pullback \n under a morphism $\\psi: \\mathcal C_{\\mathbb C}\\to \\Cal M({\\text{\\cute k}}; 10, 1)_{\\mathbb C}^*$ of the universal abelian scheme, \n and such that there exists $\\ell$ and an $O_{\\smallkay}$-linear isomorphism of lisse $\\ell$-adic sheaves on $\\mathcal C_\\mathbb C$, \n$$\n\\alpha_{\\ell}: R^1a_*{\\mathbb Z}_{\\ell}\\simeq R^1b_*{\\mathbb Z}_{\\ell}\n$$\ncompatible with the Riemann forms on source and target. Then there exists a unique \nisomorphism $\\alpha: A\\to B$ that induces $\\alpha_{\\ell}$. This isomorphism is compatible with polarizations. \n\\end{lem}\n\nTo prove this,  we are going to use the following lemma. In it, we denote by $\\Lambda$ the \nhermitian $O_{\\smallkay}$-module $H^1(A_s, {\\mathbb Z})$, for $s\\in\\mathcal C_{\\mathbb C}$  a fixed base point. \nRecall from section \\ref{cubicthreefolds} that there is a chain \nof inclusions $\\Lambda\\subset \\Lambda^\\vee\\subset \\pi^{-1}\\Lambda$, where $\\pi=\\sqrt {-3}$ is a generator of the unique prime ideal \nof $O_{\\smallkay}$ dividing $3$. \n\\begin{lem}\\label{monodr}\nLet  $s\\in\\mathcal C_{\\mathbb C}$ be the chosen base point.\n\n\\noindent (i) The monodromy representation  $\\rho_A:\\pi_1(\\mathcal C_{\\mathbb C}, s)\\to {\\rm GL}_{{\\text{\\smallcute k}}}\\big(\\Lambda\\otimes_{O_{\\smallkay}}{\\text{\\cute k}}\\big)$ is absolutely irreducible.\n\n\\noindent (ii) For every prime ideal $\\mathfrak p$ prime to  $3$, the monodromy \nrepresentation  $\\pi_1(\\mathcal C_{\\mathbb C}, s)\\to {\\rm GL}_{\\kappa(\\mathfrak p)}\\big(\n\\Lambda\/\\mathfrak p \\Lambda\\big)$ is absolutely irreducible.  \n\n\\noindent (iii) For the unique prime ideal $\\mathfrak p=(\\pi)$ lying over $3$, the \nmonodromy representation $\\pi_1(\\mathcal C_{\\mathbb C}, s)\\to {\\rm GL}_{\\kappa(\\mathfrak p)}\\big(\n\\Lambda\/\\mathfrak p \\Lambda\\big)$  is not absolutely irreducible, but there is  a \nunique non-trivial stable subspace, namely, the $10$-dimensional image of $\\pi \\Lambda^\\vee$ in $\\Lambda\/\\pi \\Lambda$.    \n\\end{lem}\n\\begin{proof} The monodromy representations in question are induced by the composition of homomorphisms\n\\begin{equation}\\label{monodromy}\n\\pi_1(\\mathcal C_{\\mathbb C}, s)\\longrightarrow \\pi_1(\\mathcal M({\\text{\\cute k}}; 10, 1)^*_{\\mathbb C}, \\varphi(s))\\longrightarrow \\text{\\rm GL}_{O_{\\smallkay}}\\big(H^1(A_s, {\\mathbb Z})\\big) .\n\\end{equation}\nHere by Theorem \\ref{thmcubic3}, and using complex uniformization (cf. section \\ref{cu}),  the first \nhomomorphism is induced by the inclusion of connected spaces \n$$\\iota: \\mathcal D\\setminus \\Big(\\bigcup_{\\substack{x\\in L\\\\ h(x, x)=3}}\\mathcal D_x\\Big) \\hookrightarrow \\mathcal D\\ , \n$$ followed by quotienting out by the free action of $\\Gamma^*$. Since $\\mathcal D$ is simply-connected,  it follows \nthat $\\pi_1(\\mathcal M({\\text{\\cute k}}; 10, 1)^*_{\\mathbb C}, \\varphi(s))=\\Gamma^*$ and that the first homomorphism \nin (\\ref{monodromy}) is surjective. Now, $\\Gamma^*$ can \nbe identified with the  group  of unitary automorphisms of the {\\it parahoric}  lattice $\\Lambda$, \nand it is elementary that the representations of $\\Gamma^*$ on $\\Lambda\\otimes_{O_{\\smallkay}}{\\text{\\cute k}}$ \nand on $\\Lambda\/\\mathfrak p \\Lambda$ for $\\mathfrak p$ prime to $ 3$ \nare absolutely irreducible (the latter since $\\Lambda^\\vee\\otimes {{\\mathbb Z}}_\\ell=\\Lambda\\otimes {{\\mathbb Z}}_\\ell$ for $\\ell\\neq 3$). \nThe statement (iii) is proved in the same way. \n\\end{proof}\n\\begin{proof}(of Lemma \\ref{elladic}) \nLet us compare the monodromy representations, \n\\begin{equation}\n\\begin{aligned}\n\\rho_A:& \\pi_1(\\mathcal C_{\\mathbb C}, s)\\to \\text{\\rm GL}_{O_{\\smallkay}}\\big(H^1(A_s, {\\mathbb Z})\\big)\\\\\n\\rho_B:& \\pi_1(\\mathcal C_{\\mathbb C}, s)\\to \\text{\\rm GL}_{O_{\\smallkay}}\\big(H^1(B_s, {\\mathbb Z})\\big) .\n\\end{aligned}\n\\end{equation}\nBy hypothesis, these representations are isomorphic after tensoring with ${\\mathbb Z}_{\\ell}$. \nHence, they are also isomorphic after tensoring with  ${\\text{\\cute k}}$. Hence there \nexists a $\\pi_1(\\mathcal C_{\\mathbb C}, s)$-equivariant ${\\text{\\cute k}}$-linear isomorphism \n\\begin{equation*}\n\\beta: H^1(A_s, {\\mathbb Q})\\simeq H^1(B_s, {\\mathbb Q})\\ .\n\\end{equation*}\nBy the irreducibility of the representation of $\\pi_1(\\mathcal C_{\\mathbb C}, s)$ in $H^1(A_s, {\\mathbb Q})$, $\\beta$ is unique \nup to a scalar in ${\\text{\\cute k}}^\\times$. Let us compare the $O_{\\smallkay}$-lattices \n$\\beta^{-1}\\big(H^1(B_s, {\\mathbb Z})\\big)$ and $H^1(A_s, {\\mathbb Z})$. Since we are assuming that $O_{\\smallkay}$ is a PID, after replacing $\\beta$ by a multiple $\\beta_{\\text{\\rm O}}=c\\beta$, \nwe may assume that $L_B= \\beta_{\\text{\\rm O}}^{-1}\\big(H^1(B_s, {\\mathbb Z})\\big)$ \nis a primitive $O_{\\smallkay}$-sublattice in $\\Lambda=H^1(A_s, {\\mathbb Z})$. Let $\\mathfrak p$ be a prime ideal in $O_{\\smallkay}$, \nand let us consider the image of $L_B$ in $\\Lambda\/\\mathfrak p \\Lambda$. Since $L_B$ is \nprimitive in $\\Lambda$, this image is non-zero. If $\\mathfrak p$ is prime to $3$, the irreducibility \nstatement in (ii) of Lemma \\ref{monodr} implies that this image is everything, and hence\n$L_B\\otimes O_{{\\text{\\smallcute k}}, \\mathfrak p}=\\Lambda\\otimes O_{{\\text{\\smallcute k}}, \\mathfrak p}$ in this case. \n\nTo handle the prime ideal $\\mathfrak p=(\\pi)$ over $3$, we use the polarizations. By the irreducibility \nstatement in (i) of Lemma \\ref{monodr}, the polarization forms on $H^1(A_s, {\\mathbb Q})$ and on $H^1(B_s, {\\mathbb Q})$ \ndiffer by a scalar in ${\\mathbb Q}^\\times$ under the isomorphism $\\beta_{\\text{\\rm O}}$. Now, by hypothesis on $B$, with respect to the polarization form \non $H^1(B_s, {\\mathbb Q})$, we have a chain of inclusions $L_B\\subset L_B^\\vee\\subset \\pi^{-1}L_B$ \nwith respective quotients of dimension $10$ and $1$ over ${\\mathbb F}_p$, just as for $\\Lambda$. Since the \ntwo polarization forms differ by a scalar, this excludes the possibility that the image \nof $L_B$ in $\\Lambda\/\\pi \\Lambda$ be non-trivial. It follows that $L_B=\\Lambda$. \n\nFurthermore, the isomorphism $\\beta_{\\text{\\rm O}}$ is unique up to a unit in $O_{\\smallkay}^\\times$, and it is \nan isometry with respect to both polarization forms. Now, by \\cite{De2}, 4.4.11 and 4.4.12, $\\beta_\\text{\\rm O}$ is induced \nby an isomorphism of polarized abelian schemes. Finally, $\\beta_\\text{\\rm O}\\otimes_{\\mathbb Z}\\Z_\\ell=\\alpha_\\ell$ up to a unit, \nsince these homomorphisms differ by a scalar and both preserve the Riemann forms.\\hfill\\break \nThe uniqueness of $\\alpha$ follows from Serre's Lemma. \n\\end{proof}\nNow Lemma \\ref{elladic} implies  that over any  field extension $k'$ of ${\\text{\\cute k}}$ inside ${\\mathbb C}$, there exists  \nat most one  polarized abelian variety $b: B\\to \\mathcal C_{k'}$ obtained by pull-back from the universal \nabelian variety over $\\Cal M({\\text{\\cute k}}; 10, 1)^*$,  equipped with an $O_{\\smallkay}$-linear isomorphism of lisse $\\ell$-adic sheaves over $\\mathcal C_{\\mathbb C}$\n$$ R^1a_{*}{\\mathbb Z}_{\\ell}\\simeq R^1b_{{\\mathbb C} *}{\\mathbb Z}_{\\ell}\\ ,$$\npreserving the Riemann forms. By the argument in \\cite{De1}, 2.2 this implies that, in fact, $B$ \nexists (since it does for $k'={\\mathbb C}$). Hence the morphism $\\varphi$ is defined over ${\\text{\\cute k}}$. Put otherwise, \nfor any ${\\text{\\cute k}}$-automorphism $\\tau$ of ${\\mathbb C}$, the conjugate embedding $\\varphi^\\tau$, which corresponds \nto the conjugate $(A, \\iota, \\l)^\\tau$, is equal to $\\varphi$;  hence $\\varphi$ is defined over ${\\text{\\cute k}}$.   \n\n\\begin{comment}\\begin{rem}{\\rm In the case of cubic surfaces, the original argument of Deligne \\cite{De1} applies directly.  Indeed,  \nconsider the following commutative diagram, in which the lower row is defined by associating to \na cubic threefold its intermediate jacobian, a principally polarized abelian variety of dimension $5$, \n$$\\xymatrix{& Cubics^\\circ_{2 {\\mathbb C}}\\ar[d]\\ar[r] & \\Cal M({\\text{\\cute k}}; 4, 1)_{\\mathbb C}\\ar[d]\\\\\n& Cubics^\\circ_{3 {\\mathbb C}}\\ar[r]\\ar[r]&\\mathcal A_{5 {\\mathbb C} } \\ .}$$\nBy the Torelli theorem for cubic threefolds, this diagram is cartesian. The horizontal morphisms \nare open embeddings and the vertical morphisms are unramified. By \\cite{De1}, the lower horizontal morphism \nis defined over ${\\text{\\cute k}}$ (even over ${\\mathbb Q}$). Hence also the upper horizontal morphism is defined over ${\\text{\\cute k}}$. }\n\\end{rem}\n\\end{comment}\n \n\\begin{conj}\nIn all four cases above, the  morphisms $\\varphi$ can  be extended over $O_{\\smallkay}[\\Delta^{-1}]$.\n\\end{conj}\n\nSince we circulated a first version of our paper, this has been proved by J.~Achter \\cite{Ach} in the case of cubic surfaces. \n\n\\section{Concluding remarks} \n\nWe end this paper with a few remarks. \n\\begin{rem}\n{\\rm In all four cases, the complement of ${\\rm Im}(|\\varphi|)$ is identified with a certain KM-divisor. In fact,\n for other KM-divisors, the intersection with ${\\rm Im}(|\\varphi|)$ sometimes has a geometric interpretation. For example,\n in the case of cubic surfaces, the intersection of ${\\rm Im}(|\\varphi|)$ with the image of the KM-divisor $\\Cal Z(2)$ in $|\\Cal M({\\text{\\cute k}}; 4, 1)_{{\\mathbb C}}|$\n can be identified with the locus of cubic surfaces admitting Eckardt points, cf. \\cite{DGK}, Thm.~8.10. Similarly, in the case of curves of genus $3$,\n the intersection of ${\\rm Im}(|\\varphi|)$ with the image of $\\Cal Z(t)^*$ in $|\\Cal M({\\text{\\cute k}}; 6, 1)^*_{{\\mathbb C}}|$ can be identified with the locus of  curves $C$ where the K3-surface $X(C)$ admits a ``splitting curve\" of a certain degree depending on $t$, cf.  \\cite{Art}, Thm.~4.6. }\n\\end{rem}\n\n\\begin{rem}\n{\\rm In \\cite{DK, DGK, MSY}, occult period morphisms  are often set in comparison with \nthe Deligne-Mostow theory which establishes a relation between  configuration spaces \n(e.g., of points in the projective line) and quotients of the complex unit ball by complex \nreflection groups, via monodromy groups of hypergeometric equations. This aspect of \nthese examples has been suppressed entirely here. Also, it should be mentioned that there \nare other ways of constructing the period map for cubic surfaces, e.g., \\cite{DK, DGK}.  }\n\\end{rem}\n\\begin{rem}\\label{invar}\n{\\rm Let us return to the section \\ref{cu}. There we had fixed an hermitian vector space $(V, (\\ , \\ ))$ \nover ${\\text{\\cute k}}$ of signature $(n-1, 1)$. Let $V_0$ be the  underlying ${\\mathbb Q}$-vector space, with the symmetric pairing defined by\n\\begin{equation*}\ns( x, y)= {\\rm tr}(h( x, y)) .\n\\end{equation*}\nThen $s$ has signature $(2(n-1), 2)$, and we obtain an embedding of ${\\rm U}(V)$ into ${\\rm O}(V_0)$. This \nalso induces an embedding of  symmetric spaces,\n\\begin{equation}\\label{embed}\n\\mathcal D\\hookrightarrow \\mathcal D_{\\rm O} ,\n\\end{equation}\nwhere, as before, $\\mathcal D$ is the space of negative (complex) lines in $(V_{{\\mathbb R}}, \\mathbb I_0)$, \nand where $\\mathcal D_{\\rm O}$ is the space of oriented negative $2$-planes in $V_{{\\mathbb R}}$. The \nimage of \\eqref{embed} is precisely the set of negative $2$-planes that are stable by $\\mathbb I_0$. In the \ncases of the Gauss field resp.\\ the Eisenstein field, this invariance is equivalent to being stable \nunder the action of $\\mu_4$, resp.\\ $\\mu_6$. Hence in these two cases, the image of \\eqref{embed} \ncan also be identified with the fixed point locus of $\\mu_4$ resp.\\ $\\mu_6$ in $\\mathcal D_{\\rm O}$. }\n\\end{rem}\n\\begin{rem}{\\rm By going through the tables in \\cite{rapo'}, \\S 2, one sees that there is no further example of \nan occult period map of the type above which embeds the moduli stack of {\\it hypersurfaces} of suitable degree and \ndimension into a Picard type moduli stack of abelian varieties. Note, however, that, in the case of curves of genus 4, \nthe source of the hidden period morphism is a moduli stack of {\\it complete intersections} of a certain multi-degree of dimension one, and there may be more examples of this type. }\n\\end{rem}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:intro}\n\nThe Cox proportional hazards model (hereafter, the Cox model) introduced by \\citet{cox72}\nis one of the most popular regression models for survival analysis. \nIt is a {\\em semiparametric} model with two sets of unknown parameters: the regression coefficients, which measure the correlation between the covariates and the\nexplanatory variables, and the baseline hazard, a nonparametric quantity, which describes the risk of\nevents happening within given time intervals at baseline levels conditional on the covariates.\nA commonly used approach to estimate the two parameters takes two steps: first estimate the regression coefficients from the Cox partial likelihood \\citep{cox72} and then derive   \nthe estimated cumulative hazard function (known as the Breslow estimator \\citep{breslow72}) through maximizing the full likelihood via plugging-in the estimated value of the regression coefficients. \n\nIn the past few decades, Bayesian methods for the Cox model have been widely applied for analyzing datasets in, e.g., astronomy \\citep{isobe86}, medical and genetics studies \\citep{jialiang13}, and engineering \\citep{equeter20}.\nAn advantage of the Bayesian approach is that uncertainty quantification for the parameters of interest is in principle straightforward to obtain once posterior samples are available.  \nContrary to the standard two-step procedure mentioned above, the Bayesian approach provides estimates for the joint distribution of all parameters, which enables to capture dependencies: in particular, as one of the practical applications considered below, one can derive meaningful uncertainty quantification simultaneously for the Cox model parameter and functionals of the hazard rate (e.g. its mean, or the value of the survival function at a point) from corresponding credible sets, in particular automatically capturing the (optimal  `efficient') dependence structure.\n\nThe prior for the hazard function needs to be chosen carefully, as it is a nonparametric quantity. \nTwo main common approaches to place a prior on hazards have been considered in the literature.\nA first approach puts a prior on the cumulative hazard function, modeling this quantity rather than the hazard itself. \nA prominent example is the family of {\\it neutral to the right process} priors, which includes the Beta process prior \n\\citep{hjort90, damien96},\nthe Gamma process prior \\citep{kalbfleisch78, burridge81}, \nand the Dirichlet process prior \\citep{florens99} as special cases.\nA second approach, which is the one we follow here, is to put a prior on the baseline hazard function. A commonly used family  is that of piecewise constant priors \\citep{ibrahim01}. \nThe latter approach is particularly attractive, first because it allows for inference on the hazard rate (assuming it exists), and second because in practice the follow-up period is often split into several intervals, with  \nthe hazard rate taking a distinct constant value on each sub-interval, making the output of the method easy to interpret for practitioners.\n\nOne primary goal of the paper is to validate the practical use of Bayesian credible sets for inference on the Cox model unknown parameters, for instance credible bands for the conditional survival function. Indeed, practitioners often treat Bayesian credible sets as confidence sets. Before discussing the possible mathematical validity of this practice, let us conduct a simple illustrative simulation study (see Section \\ref{sec:sim} for a detailed description of the simulation setting). \n\nFrom data simulated from the Cox model, suppose we want to make inference on the conditional survival function, which gives the probability that a patient survives past a certain time $t$ given a covariate $z \\in \\mathbb{R}^p$, a useful quantity for practitioners. Let us compare a simple $95\\%$ credible band of the posterior distribution (with the piecewise constant prior; see Section \\ref{sec:sim} for its construction) induced on the conditional survival function and a certain 95\\% confidence band---which requires estimation of the covariance structure---of the same function obtained by a commonly used frequentist approach (see Section \\ref{sec:sim} for a precise description of how the band is obtained).  \nIn Figure \\ref{fig:intro}, we plot the credible band (blue) and the confidence band (orange).    \nThe sample size is $n = 200$, and we let $p=1$ and $z = 1$.\nOne first notes that the true function (black) is contained in both bands, which suggests that both provide a reasonable uncertainty assessment for the conditional survival function. Second, we compared the total area of the two bands; interestingly, we found that the area of the credible band is smaller than that of the second band: the area of the credible band is 0.163 and that of the confidence band is 0.183 ($12\\%$ larger). \nA thorough Monte Carlo study, carried out in Section \\ref{sec:sim}, confirms that the area of the Bayesian credible band is indeed consistently smaller on average than the size of the frequentist confidence band when the sample size is 200. \nIt may be noted that these results hold for the in a sense `simplest possible' Bayesian credible band: as can be seen in Figure \\ref{fig:intro}, its width is fixed through the time interval, and even better results are expected for bands that become thinner close to $t=0$; see Section \\ref{sec:diss} for more discussion on this.   \nThis simulation study suggests that, aside from not having to estimating covariances, \nusing the Bayesian credible set can be particularly advantageous, especially for \nsmall sample size datasets. \n \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{plots\/credibleband-200.pdf}\n\\caption{\nThe 95\\% Bayesian credible band (blue) using the random histogram prior and the 95\\% frequentist confidence band (orange).\nThe bolded black line is the true conditional survival function. \nSample size is $n = 200$. \n}\n\\label{fig:intro}\n\\end{figure}\n\nThe observations from Figure \\ref{fig:intro} raise some interesting questions: can one validate and generalize our findings in the figure, that is, can one provide theory explaining why the credible band is a confidence band, and will the two bands become more similar as sample size increases? \nWhat can be said in terms of the hazard function: does the Bayesian procedure estimate it in a possibly `optimal' way? We now discuss the existing literature on these questions and the main contributions of the paper. \n\nIn smooth parametric models, taking certain quantile credible sets as confidence sets is justified mathematically by the celebrated Bernstein--von Mises theorem (henceforth BvM, see e.g. \\cite{vdvAS}, Chapter 10): a direct consequence thereof is that taking, in dimension $1$ say, the $\\alpha\/2$ and $1-\\alpha\/2$ posterior quantiles provides a credible set (by definition) whose frequentist coverage asymptotically goes to $1-\\alpha$. Its diameter also asymptotically matches the information bound so is optimal in the frequentist sense from the efficiency perspective. For more complex models, such as the\nCox model, obtaining a semiparametric BvM theorem for the regression parameter is possible, as we see below, but requires non-trivial work. Obtaining analogous results at the level of the {\\em survival} function itself is even more challenging. We now review recent advances in the area for such semi- and non-parametric models.\n \nSemiparametric BvM theorems where obtained in \\citet{cast12} under general conditions on the statistical model using Gaussian process priors on the nuisance parameter. \n\\citet{cast15BvM} considered an even more  general framework, allowing for BvMs for linear and non-linear functionals (also generalizing some early results of \\cite{RR12} for density estimation).  A multiscale approach was introduced in \\citet{cast13, cast14a} in order to derive nonparametric BvM theorems for families of possibly non-conjugate priors, as well as Donsker--type theorems. Yet, the first applicative examples of these works were mostly confined to relatively simple models and\/or priors.\n\nTheory for convergence of Bayesian posterior distributions in survival models has mostly followed two directions, which we briefly review now (see also \\citet{ghosal17}, Section 12.3.3 and Chapter 13). A first series of influential results has been concerned with classes of neutral to the right priors: in the standard nonparametric survival analysis model, \\citet{hjort90} showed in particular conjugacy for Beta process priors in the context of cumulative hazard estimation; \\citet{kimlee01} obtained posterior consistency, while \\citet{Kim2004} derived Donsker theorems for the posterior survival function. In \\citet{kim06},  the Cox model was considered and the joint posterior distribution of parameter and survival function was shown to satisfy the Bernstein--von Mises theorem. These results share two common features: they model the cumulative baseline hazard (equivalently, the survival function), not the baseline hazard itself---which can be desirable to model for practitioners---, and they rely on conjugacy of the class of neutral to the right priors, which provides fairly explicit characterisations of the posterior distributions.\n\nA second series of results, closer in spirit to ours, considers priors on the baseline hazard function. \\citet{DeBlasi2009a} used a kernel mixture  with respect to a completely random measure as a prior, and obtained both posterior consistency for the hazard and limit results for linear and nonlinear functionals thereof. The work \\cite{DeBlasi2009b} derived  a semiparametric BvM theorem in competing risk models. The present work can be seen as following the footsteps of \\cite{cast20survival}, where the simple nonparametric model with right-censoring is treated, and for which  results for the hazard and cumulative hazard are derived. However, the latter model is much simpler than the Cox model, which features both regression coefficients and random covariates: this requires several more delicate bounding of both (semiparametric) bias terms and remainder terms, see Section \\ref{sec:support-lemma-thm1} in \\citet{ning22supp} for more details.    \n In \\citet{cast12}, the Cox model is treated as an application of the general results, which yield the semiparametric BvM theorem for the Cox model parameter for (transformed) Gaussian process priors on the hazard. Although this result has a general flavor, and can be adapted to handle other prior families, it requires a fast enough posterior contraction rate for the baseline hazard, and thus cannot be applied for histogram priors on the hazard (see  Section \\ref{sec:hellinger} for more on this). Perhaps more importantly, the later result is confined to the Cox regression parameter, and says nothing about uncertainty quantification for the cumulative hazard, for the survival function, or even simply for linear functionals of the hazard.\n\n \nThe present work obtains the first results for Bayesian uncertainty quantification jointly on regression parameters and survival function in the Cox model using non-conjugate priors. In particular, our results demonstrate that the popular and broadly used histogram priors on the baseline hazard provide not only contraction of the posterior distribution around the true unknown parameters, but  also optimal and efficient uncertainty quantification on those. We adopt the multiscale analysis approach, which is motivated by \\citet{cast20survival}'s study of the survival model.\nMore precisely, we derive \n\\begin{itemize}\n\\item[(a)] a joint Bernstein--von Mises (BvM) theorem for linear functionals of the regression coefficients and of the baseline hazard function;\n\\item[(b)] a Bayesian Donsker theorem for the conditional cumulative hazard and survival functions;\n\\item[(c)] a minimax optimal contraction rate for the conditional hazard function in supremum-norm distance.\n\\end{itemize}\nThe Bayesian Donsker theorem, in particular, implies that certain $(1-\\alpha)\\%$ credible bands for the conditional survival function are asymptotically  $(1-\\alpha)\\%$ confidence bands.\nIn addition to these results, a nonparametric BvM theorem for the conditional baseline hazard function \n is obtained in the Supplemental Material \\citep[see][]{ning22supp}. All these results are completely new for non-conjugate (in particular, histogram) priors; also, we derive the first supremum-norm posterior contraction rates for the hazard in the Cox model; we also show the corresponding matching minimax lower bound, which to the best of our knowledge was not yet available in the literature. \n\nFinally, the techniques we introduce have a general flavor. First, they do not rely on conjugacy of the priors considered, so that they can virtually be applied to a wide variety of families, as long as a certain change-of-measure condition is met.  Second, the specific form of the model (here the Cox model) comes in through its local asymptotic normality (LAN) expansion, so similar techniques can be used in more complex settings, as long as a form of local asymptotic normality of the model holds. In particular, the techniques developed herein can serve as a useful tool for future studies of other semiparametric and nonparametric models, in survival analysis and beyond: \n as further discussed in Section \\ref{sec:diss}. \n\n\nThe paper is organized as follows. \nSection \\ref{sec:setup} presents the model, the prior families, and key assumptions. \nMain results are presented in Section \\ref{sec:main-results}.\nSimulation studies are conducted in Section \\ref{sec:sim}. \nSection \\ref{sec:diss} concludes the paper and discuss a variety extensions for future studies. \nAll the relevant proofs for the main results and auxiliary lemmas are left to the Supplementary Material. \n\n{\\em Notation.} For any two real numbers $a$ and $b$, let $a \\vee b = \\max(a, b)$ and $a \\wedge b = \\min(a, b)$;\nalso, let $a \\lesssim b$ as $a \\leq C b$ for some constant $C$.  \nLet us denote by $o(1)$ a deterministic sequence going to \n$0$ with $n$ and $o_P(1)$ a sequence of random variables going to $0$ in probability under the distribution $P$. \nFor a vector $x$, denote $\\|x\\|_q$ as the $\\ell_q$-norm of $x$ ($q \\geq 1$), i.e., $\\|x\\|_q = (\\sum_i |x_i|^q)^{1\/q}$. When $q = \\infty$, $\\|x\\|_\\infty = \\max_i |x_i|$ is the infinity norm of $x$. \nFor a matrix $A$, denote $\\|A\\|_{(\\infty, \\infty)} = \\max_{ij} |a_{ij}|$.\n \nFor a function $f \\in L^{p}[a,b]$ ($p \\geq 1$), where $L^p[a,b]$ is the space of functions whose $p$-th power is Lebesgue integrable on $[a,b]$, we define $\\|f\\|_{p} = (\\int_a^b |f|^{p})^{1\/p}$ the $L^{p}$-norm of $f$. \nIf $p = 2$, $\\|f\\|_{{2}} = \\sqrt{\\langle f, f \\rangle}$ is the $L^2$-norm\nand if $p = \\infty$, $\\|f \\|_{{\\infty}}= \\sup_t |f(t)|$ is the supremum norm.\nThe associate inner product between any two functions, $f, g \\in L^p[0, 1]$, is denoted as $\\langle f, g\\rangle = \\int_0^1 fg$ and the space of continuous functions on $[a,b]$ is given by $\\mathcal{C}[a, b]$ (resp. $L^{\\infty}[a, b]$), which is equipped with the supremum norm $\\|\\cdot\\|_\\infty$.\n\n\nFor $\\beta, D > 0$, let $l = \\lfloor \\beta \\rfloor$ be the largest integer smaller than $\\beta$, a standard H\\\"older-ball on $[a,b]$ can be defined as \n$$\n\\mathcal{H}(\\beta, D) = \\{f: |f^{(l)}(x) - f^{(l)}(y)| \\leq D|x-y|^{\\beta - l}, \\  \\|f\\|_\\infty \\leq D, \\ x, y\\in [a, b]\\}.\n$$\nLet $(\\mathcal{S}, d)$ be a metric space and $\\mu, \\nu$ be probability measures of $\\mathcal{S}$.\nFor $F: \\mathcal{S} \\to \\mathbb{R}$, set\n$$\n\\|F\\|_{BL} = \\sup_{x \\in \\mathcal{S}} |F(x)| + \\sup_{x \\neq y} \\frac{|F(x) - F(y)|}{d(x,y)},\n$$\nand denote the bounded Lipschitz metric $\\mathcal{B}_{\\mathcal{S}}$ as\n$$\n\\mathcal{B}_{\\mathcal{S}}(\\mu, \\nu) = \\sup_{F: \\|F\\|_{BL} \\leq 1} \n\\left|\n\\int_{\\mathcal{S}} F(x) (d \\mu - d \\nu ) (x)\n\\right|.\n$$\nFor two densities $f$ and $g$, denote $h^2(f, g) = \\int (\\sqrt{f} - \\sqrt{g})^2 d\\mu$ as their squared Hellinger distance.\n\n\\section{Model, prior families and structural assumptions}\n\\label{sec:setup}\n\n\\subsection{The Cox model with random right censoring}\n\nThe observations $X=X^n$ are $n$ independent identically distributed (i.i.d.) triplets given by \n$X = ((Y_1, \\delta_1, Z_1), \\dots, (Y_n, \\delta_n, Z_n))$. The observed $Y_i\\in\\mathbb{R}^+$ are censored versions of (unobserved) survival times $T_i\\in\\mathbb{R}^+$,\nwith $\\delta_i$ indicator variables informing on whether $T_i$ has been observed or not: that is, $Y_i = T_i \\wedge C_i$ and $\\delta_i = \\mathbbm{1}(T_i \\leq C_i)$, where $C_i$'s are i.i.d. censoring times.  The  variables $Z_1, \\dots, Z_n \\in \\mathbb{R}^p$, $p$ fixed, are called covariates. \n\nFor a fixed covariate vector $z\\in\\mathbb{R}^p$ and $t>0$, define the {\\it conditional hazard rate}  $\\lambda(t\\,|\\, z) = \\text{lim}_{h\\to0} h^{-1} P(t \\leq T \\leq t+h \\,|\\, T \\geq t, Z = z)$. \nThe Cox model assumes, for some $\\theta\\in\\mathbb{R}^p$ and denoting by $\\theta'z$ the standard inner product in $\\mathbb{R}^p$, \n\\[ \\lambda(t \\,|\\, z) = e^{\\theta'z} \\lambda(t),\\]\nwhere $\\lambda(t)$ is the {\\it baseline hazard function}. \nThe {\\it conditional cumulative hazard function} is defined as $\\Lambda(\\cdot \\,|\\, z) = \\int_0^\\cdot \\lambda(u \\,|\\, z) du = e^{\\theta'z} \\int_0^\\cdot \\lambda(u) du = e^{\\theta'z} \\Lambda(\\cdot)$ and the {\\it conditional survival function} is denoted by \n$S(\\cdot \\,|\\, z) =  \n\\exp(-\\Lambda(\\cdot \\,|\\, z)) = \n\\exp(- e^{\\theta'z} \\Lambda(\\cdot) )$. \n\nAssuming the baseline hazard rate is positive, one can alternatively make inference on the log-hazard. The unknown parameters of the Cox model are then \n\\[ \\eta = (\\theta, r),\\qquad \\text{where } r = \\log \\lambda.\\] \nThe goal is to estimate the pair $\\eta=(\\theta,r)$. We denote by $\\eta_0=(\\theta_0,r_0)$ the true values of the parameters (and similarly for the related quantities $\\lambda_0, \\Lambda_0$).\n\nWe now give a set of standard assumptions used in this paper. \nFirst, we assume both $T$ and $Z$ admit a continuous density function, $f_T(\\cdot)$ and $f_Z(\\cdot)$ respectively. Given $Z$, the survival time $T$ and the censoring time $C$ are independent. \nAt the end of the follow-up, some individuals are still event free and uncensored such that \n$P(T > \\varrho \\,|\\, Z = z) > 0$ and $P(C > \\varrho \\,|\\, Z = z) = P(C = \\varrho \\,|\\, Z = z) = 0$ for some fixed time $\\varrho$. \nThe censoring $C$ is assumed to follow a distribution $G$ and admit a density such that\n    \\begin{align*}\n        p_C(u) = g_z(u) \\mathbbm{1}\\{0\\leq u < \\varrho\\} + \\bar G_z(\\varrho) \\mathbbm{1}\\{u = \\varrho\\}\n    \\end{align*}\n    with respect to $\\text{Leb}([0, \\varrho]) + \\delta_\\varrho$, where $\\text{Leb}(I)$ is the the Lebesgue measure on $I$.\nWithout loss of generality, we assume $\\varrho = 1$ throughout the paper.\n\n\nBased on this set-up, the joint density function of the triple $(y, \\delta, z)$ is given by\n\\begin{equation}\n\\begin{split}\n\\label{eqn:density-function}\n    f_\\eta (y, \\delta, z) = \n   & \\left(\n        g_z(y) e^{- \\Lambda(y) e^{\\theta'z}}\n    \\right)^{1-\\delta} \n    \\left(\n        \\bar{G}_z (y) \\lambda(y) e^{\\theta' z - \\Lambda(y) e^{\\theta'z}}\n    \\right)^\\delta \n    f_Z(z)\n    \\mathbbm{1}(y < t)\\\\\n    & + \n    \\left(\n        \\bar{G}_z(t) e^{-\\Lambda(t) e^{\\theta'z}}\n    \\right) \n    f_Z(z)\n    \\mathbbm{1}(\\delta = 0, y = t),\n\\end{split}\n\\end{equation}\nwhere $g_z(\\cdot)$ is a continuous density of $C$ given $Z = z$,\n$\\bar{G}_z(\\cdot) = 1 - G_z(\\cdot -)$, where $G_z(\\cdot)$ is the cumulative distribution function of $g_z(\\cdot)$. \n\n\nLet $\\ell_n(\\eta) = \\sum_{i=1}^n \\log f_\\eta (X_i)$ be the log-likelihood function,\nthe likelihood ratio is given by \n\\begin{align}\n\\label{log-likelihood}\n\\ell_n(\\eta) - \\ell_n(\\eta_0)\n= \n\\sum_{i=1}^n \\left(\\delta_i \\{(\\theta-\\theta_0)'Z_i + (r-r_0)(Y_i) \\} - \\Lambda(Y_i) e^{\\theta'Z_i} + \\Lambda_0(Y_i) e^{\\theta_0'Z_i}\\right).\n\\end{align}\nFrom (\\ref{log-likelihood}), one sees that the log-likelihood ratio does not depend on $g_z(y)$ and $\\bar G_z(\\cdot)$, thus one does not need to model $g_z(y)$ in order to make inference on $\\eta$. \n\n\\subsection{Information--related quantities} \\label{sec:infonot}\n\nWe now introduce some of the key quantities arising in the study of the Cox model: these are all related to the information operator (extending the usual Fisher information in parametric models) arising from the LAN--expansion in the model (see also Section \\ref{sec:bg}).  \nFor a bounded function $b(\\cdot)$ on $[0, 1]$ and a cumulative hazard function $\\Lambda(\\cdot)$, \nwe denote \n$\\Lambda\\{b\\}(\\cdot) = \\int_0^\\cdot b(u) d\\Lambda(u)$ and, in slight abuse of notation, we set $\\Lambda \\{b\\} = \\Lambda \\{b\\}(1)$. \nThe following notations are commonly used in the literature for the Cox model (see e.g., Section VIII 4.3 of \\citet{andersen93} and Section 12.3.3 of \\citet{ghosal17}):\n\\begin{align}\n\\label{M0}\n    M_0(u) &= \\mathbb{E}_{\\eta_0} \\left(e^{\\theta_0'Z} \\mathbbm{1}_{u \\leq T}\\right) = \n    \\int \\bar{G}_z(u) e^{\\theta_0'z - \\Lambda_0(u) e^{\\theta_0'z}} f_Z(z) dz,\\\\\n    \\label{M1}\n    M_1(u) & = \\mathbb{E}_{\\eta_0} \\left(Z e^{\\theta_0'Z} \\mathbbm{1}_{u \\leq T}\\right) \n    = \\int z \\bar{G}_z(u) e^{\\theta_0'z - \\Lambda_0(u) e^{\\theta_0'z}} f_Z(z) dz,\\\\\n    \\label{M2}\n\tM_2(u) & = \\mathbb{E}_{\\eta_0} \\left(ZZ' e^{\\theta_0'Z} \\mathbbm{1}_{u \\leq T}\\right) \n    = \\int zz' \\bar{G}_z(u) e^{\\theta_0'z - \\Lambda_0(u) e^{\\theta_0'z}} f_Z(z) dz.\n\\end{align}\nThe least favorable direction is defined as $\\gamma_{M_1}(\\cdot) = (M_1\/M_0)(\\cdot)$. \nFor a bounded function $b(\\cdot)$ on $[0,1]$, we let $\\gamma_b(\\cdot) = (b\/M_0)(\\cdot)$.\nThe efficient information matrix of the Cox model is denoted by $\\tilde I_{\\eta_0}$ and is given by\n\\begin{align}\n\\tilde I_{\\eta_0} = & \\Lambda_0\\{M_2(\\cdot) - \\gamm(\\cdot) \\gamm'(\\cdot) M_0(\\cdot)\\}.\n\\label{eff-info-mtx}\n\\end{align}\nFor any $\\vartheta \\in \\mathbb{R}^p$ and $g \\in L^2\\{\\Lambda_0\\}$ (i.e., $\\int g^2d\\Lambda_0<\\infty$), we define\n\\begin{align}\n\\label{Wn}\nW_n(\\vartheta, g)\n    =\n    \\frac{1}{\\sqrt{n}}\n    \\sum_{i=1}^n \\left\\{\n    \\delta_i \\left(\n    \\vartheta'Z_i + g(Y_i)\n    \\right)\n    - e^{\\theta_0'Z_i}\n    \\left(\n    \\vartheta'Z_i \\Lambda_0(Y_i) + (\\Lambda_0 g)(Y_i)\n    \\right)\n    \\right\\}.\n\\end{align}\nThis quantity corresponds to the empirical process part of the LAN expansion of the Cox model (see (\\ref{lan}) in Section \\ref{sec:bg} of \\citet{ning22supp}).\n\n\\subsection{Structural assumptions}\n\\label{sec:assumption}\n\nThe following fairly mild conditions are assumed on the unknown quantities of the model.  For some positive constants $c_1, \\dots, c_8$,\n\\begin{enumerate}[label=\\textbf{(i)}]\n    \\item \\label{asp:i} \\text{the random variable }$Z\\in\\mathbb{R}^p$ is bounded (i.e. $\\|Z\\|_\\infty\\le c_1$ a.s.);\n\n\\end{enumerate}    \n\\begin{enumerate}[label=\\textbf{(ii)}]    \n\t\\item \\label{asp:ii} \n         $\\|\\theta_0\\|_\\infty \\le c_2$;\n\\end{enumerate}\t\nNote that from \\ref{asp:i} and \\ref{asp:ii}, one can bound $e^{\\theta_0'z} \\le e^{pc_1c_2}$. Also, suppose\n\\begin{enumerate}[label=\\textbf{(iii)}]    \n    \\item \\label{asp:iii} $c_3 \\leq \\inf_{t \\in [0, \\varrho]} \\lambda_0(t) \\leq \\sup_{t \\in [0, \\varrho]} \\lambda_0(t) \\leq c_4$ and $r_0 = \\log \\lambda_0 \\in \\mathcal{H}(\\beta, D)$, where $\\beta, D > 0$;\n\\end{enumerate}    \n\\begin{enumerate}[label=\\textbf{(iv)}]\n    \\item \\label{asp:iv} $g_z$ is a continuous density and\n    $c_5 \\leq \\inf_{t \\in [0, \\varrho]} g_z(t) \\leq \\sup_{t \\in [0, \\varrho]} g_z(t) \\leq c_6$;\n\\end{enumerate}\n\nAssumptions \\ref{asp:i}--\\ref{asp:iv} are common in the related literature; e.g., see \\citet{cast12} (p. 17). \nAs $\\theta_0, \\lambda_0, g_z(u)$ are all assumed to be bounded from above and below, \\ref{asp:i}--\\ref{asp:iv} imply\n \\begin{enumerate}[label=\\textbf{(v)}]    \n\t\\item \\label{asp:v} For $M_2(u)$ and $\\tilde I_{\\eta_0}^{-1}$ in (\\ref{M2}) and (\\ref{eff-info-mtx}) respectively,\n\t$\\Lambda_0\\{M_2(\\cdot)\\} \\geq c_7 I_p$ and $\\|\\tilde I_{\\eta_0}^{-1}\\|_{(\\infty,\\infty)} \\leq c_8$.\n\\end{enumerate} \n\nThe above conditions are mostly assumed for technical simplicity: boundedness of the true vector $\\theta$ enables one to take a prior with bounded support, which is particularly helpful in order to carry out likelihood expansions. Attempting to remove this condition would lead to delicate questions on likelihood remainder terms and is beyond the scope of this work. The smoothness condition on the (log--) hazard is quite mild for smooth hazards: it includes for instance the case of Lipschitz hazards. Another interesting setting would be the one of piecewise constant hazards. It could be treated with the techniques developed of this paper (a given histogram can always be approximated arbitrarily well in the $L^2$--sense by a Haar-histogram, and then the problem becomes --nearly-- parametric) although it would require a somewhat separate treatment: we refrain from providing theory here for this case, although we consider it as one of the examples of the simulations study in Section \\ref{sec:sim}, where simulations show very good behavior in this setting as well.  Henceforth we assume that \\ref{asp:i}--\\ref{asp:iv}  hold without explicit reference. \n\n\\subsection{Prior distributions}\n\\label{priors}\n\nPriors for $\\theta$ and $\\lambda$ are chosen independently. \nThe prior for $\\theta$ is chosen as follows: \n\\begin{enumerate}[label=\\textbf{(T)}]\n\\item \\label{prior:T} \nLet $\\theta_j$ be the $j$-th coordinate of $\\theta$, \n$\\pi(\\theta) = \\bigotimes_{j=1}^p \\pi(\\theta_j) = \\bigotimes_{j=1}^p f(\\theta_j)\\mathbbm{1}_{[-C, C]}$ for some constant $C > 0$.\nExamples for $f(\\theta_j)$ include the uniform density, i.e., $f(\\theta_j) = 1$, the {\\em truncated} (to $[-C,C]$) $\\tau$--Subbotin density \\citep{subb23}, which includes the {\\em truncated} Laplace (when $\\tau = 1$) density and {\\em truncated} normal (when $\\tau = 2$) density as special cases. \n\\end{enumerate}\n\nImposing the truncation does not seem necessary in practice, as we found in the simulation study. However, for deriving our theoretical results, and similar to \\citet{cast12} and \\citet{ghosal17}, we assume that at least some upper-bound on $\\|\\theta\\|_\\infty$\n is known, as discussed in the previous subsection (so that one can take e.g. $C=c_2$ in \\ref{asp:ii}). \n\nFor the prior on $\\lambda$, two classes of piecewise constant priors are considered throughout the paper:\n\\begin{enumerate}[label=\\textbf{(H)}]\n\\item \\label{prior:H} {\\it Random histogram prior.} \nFor $k \\geq 1, L' = L+1$, and $L$ an $n$-dependent deterministic value to be specified below,\nlet\n\\begin{align}\n\\label{histogram-prior}\n\\lambda_H = \\sum_{k=0}^{2^{L+1} - 1} \\lambda_k \\mathbbm{1}_{I_k^{L+1}}, \n\\end{align}\nwhere \n$I_0^{L'} = [0, 2^{-L'}], I_k^{L'} = (k 2^{-L'}, (k+1)2^{-L'}]$,\nand ($\\lambda_k$) are independent random variables. We consider putting the following two types of priors on the $k$-th histogram height, $\\lambda_k$:\n\\end{enumerate}\n\n\\begin{itemize}\n\\item[(i)] An independent Gamma prior: $\\lambda_k \\sim \\text{Gamma}(\\alpha_0, \\beta_0)$  are i.i.d. variables, for some fixed positive $\\alpha_0,\\beta_0$.\n\\item[(ii)] A dependent Gamma prior: let $\\lambda_0 \\sim \\text{Gamma}(\\alpha_0, \\beta_0)$ and $\\lambda_k| \\lambda_{k-1} \\sim \\text{Gamma}(\\alpha, \\alpha\/\\lambda_{k-1})$ for some positive constant $\\alpha$. \nThen for $k \\geq 1$, $E(\\lambda_k \\,|\\, \\lambda_{k-1}) = \\lambda_{k-1}$ and $\\text{Var}(\\lambda_k \\,|\\, \\lambda_{k-1}) = \\lambda_{k-1}^2\/\\alpha$. \n\\end{itemize}\n\n\\begin{enumerate}[label=\\textbf{(W)}]\n\\item \\label{prior:W} {\\it Haar wavelet prior.} \nLet $r_S = \\log \\lambda_S$ and again for $L$ to be specified below, let us set\n\\begin{align} \n\\label{wavelet-prior}\nr_S = \\sum_{l = -1}^{L} \\sum_{k=0}^{2^l - 1} \\sigma_l Z_{lk} \\psi_{lk},\n\\end{align}\nwhere $Z_{lk}$ are random variables, $\\sigma_l = 1$, $0 \\leq l \\leq L$, and $(\\psi_{lk})$ are Haar wavelet basis. \n\\end{enumerate}\nAlthough a variety of densities can be considered for $Z_{lk}$, we specifically consider for simplicity the standard normal density and the standard Laplace density (other choices of $\\sigma_l$ e.g. $2^{-l\/2}$ are possible). Note that both densities give a non-conjugate prior for $r_S$. \n \nAlso note that the random histogram prior can be viewed as a special case of the Haar wavelet prior if one allows for possibly dependent variables $Z_{lk}$ (and possibly different values for $\\sigma_l$). \n\n{\\em Choice of the parameter $L$}. For results on the specific priors as above, we consider the choice of cut-off $L=L_n$ defined as, for $\\beta>0$ the assumed regularity of $r_0=\\log\\lambda_0$ (see \\ref{asp:iii}),\n\\begin{align} \\label{choicel}\n2^{L_n}=2^{L_n(\\beta)} & \\circeq \\left(\\frac{n}{\\log{n}}\\right)^{\\frac{1}{2\\beta+1}},\n\\end{align}\nwhere $\\circeq$ means that one picks a closest integer solution in $L_n$ of the equation. If the regularity of the true $r_0$ is not known in advance, as is usually the case in practice, then all the limiting shape (Bernstein--von Mises) results below go through if one replaces $\\beta$ by $1\/2$ in \\eqref{choicel} (note also that all the main results, except the Hellinger rate which requires no minimal smoothness, require a regularity $\\beta>1\/2$). In other words, for the semiparametric results, it is enough to `undersmooth'. The strict knowledge of $\\beta$ is only required if one wishes to obtain an optimal minimax supremum-norm contraction rate (see Section \\ref{sec:sup-norm} for more on this).\n\n\\section{Main results}\n\\label{sec:main-results}\n\nLet us give a brief outline of our results.\nSection \\ref{sec:hellinger} provides a preliminary contraction result in Hellinger distance.\nSection \\ref{sec:jointBvM} presents a joint Bernstein--von Mises (BvM) theorem for linear functionals of $\\theta$ and $\\lambda$. \nSection \\ref{sec:npBvM} derives a Donsker theorem for the joint posterior distribution of $\\theta$ and the cumulative hazard function $\\Lambda$. \nThis result leads to the Donsker theorem for the posterior of the conditional cumulative hazard and survival functions.\nA supremum-norm convergence rate for the conditional hazard function is obtained in Section \\ref{sec:sup-norm}.\nIn Sections \\ref{sec:jointBvM}-\\ref{sec:sup-norm}, we provide generic conditions that are suitable for a wide range of priors. In Section \\ref{sec:application}, we verify those conditions for those specific choices of priors listed in Section \\ref{priors}. \n\n\n\\subsection{A key preliminary contraction result}\n\\label{sec:hellinger}\n\nWe start by obtaining a preliminary Hellinger contraction rate for the posterior distribution for the priors considered above.  \n\nDefine the rate, for $\\beta\\in(0,1]$,\n\\begin{equation} \\label{rateveps}\n\\varepsilon_n=\\varepsilon_{n,\\beta}=\\left(\\frac{\\log{n}}{n}\\right)^{\\frac{\\beta}{2\\beta+1}}.\n\\end{equation}\n\nLet $\\epsilon_n=o(1)$ be a sequence such that $n\\epsilon_n^2\\to\\infty$ as $n \\to \\infty$. Define $\\zeta_n=\\zeta_n(\\epsilon_n)= 2^{L_n\/2} \\epsilon_n + 2^{-\\beta L_n}$ and \n\\begin{equation}\\label{defan}\nA_n = \\{\\eta:\\  \\|\\theta - \\theta_0\\| \\leq \\epsilon_n, \\|\\lambda - \\lambda_0\\|_{1} \\leq \\epsilon_n, \\|\\lambda - \\lambda_0\\|_\\infty \\leq \\zeta_n\\}\n\\end{equation}\nLet us consider the following condition:\n\\begin{enumerate}[label=\\textbf{(P)}]\n\\item \\label{cond:P} \nThe sequences $L_n,\\epsilon_n$ verify $L_n = o(\\sqrt{n} \\epsilon_n)$, $L_n^2 = o(1\/\\epsilon_n)$, and $\\sqrt{n} \\epsilon_n^2 L_n = o(1)$ and, for $A_n$ as in \\eqref{defan},\n\\[ \\Pi(A_n^c \\,|\\, X) = o_{P_{\\eta_0}}(1).\\]\n\\end{enumerate}\nCondition \\ref{cond:P} requires the posterior distribution to contract in a certain sense around the true pair $(\\theta_0,r_0=\\log{\\lambda_0})$. In order to derive such a result, one may first apply the general contraction rate theorem of \\cite{ghosal00}. This, however, entails a rate for the overall density $f_\\eta$ in the Cox model only, not the parameters themselves. The main difficulty here is then to derive results on $\\theta$ and $\\lambda$ separately. The rate $\\epsilon_n$ can be thought of as a typical (possibly optimal) nonparametric rate. We call condition  \\ref{cond:P} a preliminary contraction result, because faster  rates both for $\\theta$ and for $\\lambda$ in the supremum norm can be derived, as will be seen below. In fact, for $\\theta$, it is expected that the posterior contracts at parametric, near $1\/\\sqrt{n}$ rate; a much more precise result is obtained in Section \\ref{sec:jointBvM} below in the form of a BvM theorem. \n\nThe next lemma shows that condition \\ref{cond:P} is indeed satisfied for the examples of priors introduced in Section \\ref{priors}. \n\\begin{lemma}\n\\label{lemma1}\nConsider the Cox model \nwith priors as specified in {\\rm \\ref{prior:T}}, and {\\rm \\ref{prior:H}} or {\\rm \\ref{prior:W}} with $L=L_n$ as in \\eqref{choicel}. \nThen for any $\\beta\\in(0,1]$ and $\\varepsilon_n$ as in \\eqref{rateveps},\n\\begin{align*}\n& \\Pi[\\{\\eta:\\ h^2(f_{\\eta_0}, f_\\eta) \\gtrsim \\varepsilon_{n}^2\\} \\,|\\, X]=o_{P_{\\eta_0}}(1).\n\\end{align*}\nFurther, for any $\\beta\\in(1\/2,1]$, condition {\\rm \\ref{cond:P}} is satisfied for these priors for $\\epsilon_n=\\varepsilon_n$ as in \\eqref{rateveps}.\n\\end{lemma}\n\nLet us now briefly comment on the preliminary supremum-norm rate $\\zeta_n$ for $\\lambda$ entailed by \\ref{cond:P}. For some cut--offs $L_n$, the rate can be slow. However, it is a $o(1)$ as soon as $\\epsilon_n=o(2^{-L_n\/2})$. For the typical choice of $L_n$ in \\eqref{choicel}, this only requires that $\\beta>1\/2$, which corresponds to a preliminary rate faster than $n^{-1\/4}$. This is much less than what is required for Theorem 5 in \\cite{cast12} or Theorem 12.12 in \\cite{ghosal17}, where a preliminary rate faster than $n^{-3\/8}$ is needed (note that the latter rate rules out the use of regular histograms as priors, since these can get only a rate $n^{-1\/3}$ at best). In Section \\ref{sec:sup-norm}, we show the rate $\\zeta_n$ can be improved by adopting a multiscale analysis approach. For this, a BvM theorem for linear functionals of $\\lambda$ will be needed: it is a consequence of the joint BvM derived in the next section. \n\nFrom Section \\ref{sec:jointBvM}  to Section \\ref{sec:sup-norm}, we work with a generic histogram prior of the form \\eqref{wavelet-prior} with cut--off $L=L_n$ (which includes both {\\rm \\ref{prior:H}} and {\\rm \\ref{prior:W}} as special cases), under the above condition \\ref{cond:P}. This way, the reader can directly see what generic conditions underpin our results, and adapt these to other relevant families of priors not considered here for the sake of brevity. For instance, smoother wavelet bases $(\\psi_{lk})$ can be used in the prior definition and require only minor adaptations of the proofs (in a similar way as in  \\cite{cast20survival} for the simple nonparametric survival model); although we do not prove this here for brevity, using these priors would enable one to derive optimal contraction rates in the supremum norm for arbitrary regularities $\\beta>1\/2$. We come back to the specific examples of priors {\\rm \\ref{prior:H}} and {\\rm \\ref{prior:W}} in Section \\ref{sec:application}. \n\n\\subsection{The joint BvM theorem for the linear functionals of $\\theta$ and $\\lambda$} \n\\label{sec:jointBvM}\n\nLet us consider the joint estimation of the two linear functionals defined by\n\\begin{align}\n\\label{linear-fns}\n\\varphi_a(\\theta) = \\theta'a, \\quad \n\\varphi_b(\\lambda) = \\langle b, \\lambda \\rangle = \\int_0^1 b(u) \\lambda(u) du = \\Lambda\\{b\\},\n\\end{align}\nfor fixed $a \\in \\mathbb{R}^p$ and $b \\in L^2(\\Lambda)$.\n Let us recall that we work under the generic form of prior  \\eqref{wavelet-prior} with cut--off $L=L_n$ and generic condition \\ref{cond:P}. Let us also recall the notation from Section \\ref{sec:infonot}: $M_0, M_1, M_2$, $\\gamma_b(\\cdot) = (b\/M_0)(\\cdot)$ for a bounded function $b$, and $\\tilde I_{\\eta_0}$. \n\nConsider the following conditions:\n\\begin{enumerate}[label=\\textbf{(B)}]\n\\item \\label{cond:B} \nLet $P_{L_n}(\\cdot)$ be the orthogonal projection onto $\\mathcal{V}_{L_n} := \\text{Vect}\\{\\psi_{lk}, \\ l \\leq L_n, \\ 0 \\leq k < 2^l\\}$, the subspace of $L^2[0,1]$ spanned by the first $L_n$ wavelet levels, and denote $\\gamma_{b,L_n} = P_{L_n}(\\gamma_b)$ and $\\gamma_{M_1, L_n} = P_{L_n}(\\gamm)$.\nFor any fixed $b \\in L^\\infty[0,1]$ and $\\epsilon_n$ in {\\ref{cond:P}}, \n$$\n\\sqrt{n} \\epsilon_n \\|\\gamma_b - \\gamma_{b,L_n}\\|_{\\infty} = o(1).\n$$\n\\end{enumerate}\n\nCondition \\ref{cond:B} is sometimes called {\\it no-bias} condition, and holds true if $b$ is sufficiently smooth and (or) the preliminary contraction rate $\\epsilon_n$ is fast enough. \nNext, let $h = (t, s) \\in \\mathbb{R}^2$,\nfor fixed $a \\in \\mathbb{R}^p$ and $b \\in L^2(\\Lambda)$, consider the two local paths:\n\\begin{align}\n&\\theta_h = \\theta - \\frac{t \\tilde I_{\\eta_0}^{-1} a}{\\sqrt{n}} + \\frac{s \\tilde I_{\\eta_0}^{-1} \\Lambda_0\\{b \\gamm\\}}{\\sqrt{n}},\n\\label{path-theta}\\\\\nr_h = r +& \\frac{t \\gamma_{M_1, L_n}' \\tilde I_{\\eta_0}^{-1} a}{\\sqrt{n}} - \\frac{s\\gamma_{b,L_n}}{\\sqrt{n}} - \\frac{s\\gamma_{M_1, L_n}' \\tilde I_{\\eta_0}^{-1} \\Lambda_0\\{b\\gamm\\}}{\\sqrt{n}}.\n\\label{path-r}\n\\end{align}\n\\begin{enumerate}[label=\\textbf{(C1)}]\n\\item \\label{cond:C1} ({\\it Change of variables condition}) with $r = \\log \\lambda$ and $r_0 = \\log \\lambda_0$, let $\\eta_0 = (\\theta_0, r_0)$ and $\\eta_h = (\\theta_h, r_h)$, \nwhere $\\theta_h$ in (\\ref{path-theta}) and $r_h$ in (\\ref{path-r}) with $a, b$ to be specified below\nand $A_n$ as in \\ref{cond:P}, suppose \n$$\n\\frac{\n\\int_{A_n} e^{\\ell_n(\\eta_h) - \\ell_n(\\eta_0)} d\\Pi(\\eta)\n}{\n\\int e^{\\ell_n(\\eta) - \\ell_n(\\eta_0)} d\\Pi(\\eta)\n} \n= 1 + o_{P_{\\eta_0}}(1).\n$$\n\\end{enumerate}\n\nCondition \\ref{cond:C1} is often called {\\em change of variables condition}: indeed, one natural way to check it is via controlling the change in distribution from $\\eta\\sim \\Pi$, to the distribution induced on $\\eta_h$. For priors such as \\ref{prior:H} and \\ref{prior:W}, this can be checked by posing a change of variables with respect to the Lebesgue measure on $\\mathbb{R}^{L_n}$. \nThe verification of these conditions for these priors is given in  Section \\ref{verify-cov}. \n\nFor $\\eta \\sim \\Pi(\\cdot \\,|\\, X)$, $\\mu = (\\mu_1, \\mu_2) \\in \\mathbb{R}^2$, $a \\in \\mathbb{R}^p$, and $b \\in L^2(\\Lambda)$,\nlet us define the map\n$$\n\\tau_\\mu: \\eta \\to \\sqrt{n} \n(\\theta'a - \\mu_1,\n\\langle \\lambda, b \\rangle - \\mu_2),\n$$\nand let $\\Pi(\\cdot \\,|\\, X) \\circ \\tau_\\mu^{-1}$ be the distribution induced on $\\sqrt{n} (\\theta'a - \\mu_1,\n\\langle \\lambda, b \\rangle - \\mu_2)$.\nWe are ready to present the joint BvM theorem for the bivariate functions $\\varphi_a(\\theta)$ and $\\varphi_b(\\lambda)$. \n\n\\begin{theorem}\n\\label{thm:bvm-linear-fns}\nLet $a$ and $b$ be fixed elements of $\\mathbb{R}^p$ and $L^2(\\Lambda_0)$ respectively, for any $b$ that satisfies {\\rm \\ref{cond:B}},\nsuppose the prior for $\\eta = (\\theta, \\lambda)$ is chosen such that {\\rm \\ref{cond:P}} and {\\rm \\ref{cond:C1}} hold. Then\n\\begin{align}\n\\label{thm1-1}\n\\mathcal{B}_{\\mathbb{R}^2} \\left( \\Pi(\\cdot \\,|\\, X) \\circ \\tau_{\\hat \\varphi}^{-1}, \n\\mathcal{L}(a'\\mathbb{V}, \\Upsilon_b - \\mathbb{V}\\Lambda_0\\{b\\gamm\\} )\n\\right) \n\\stackrel{P_{\\eta_0}}{\\to} 0,\n\\end{align}\nwhere $\\mathbb{V}$ and $\\Upsilon_b$ are independent, \n$\\mathbb{V} \\sim N\\left(0, \\tilde I_{\\eta_0}^{-1}\\right)$ and $\\Upsilon_b \\sim N(0, \\Lambda_0\\{b \\gamma_b\\})$, \n$\\mathcal{B}_{\\mathbb{R}^2}$ is the bounded Lipschitz metric between distributions on $\\mathbb{R}^2$, and\n$\\hat\\varphi = (\\hat\\varphi_a(\\theta), \\hat\\varphi_b(\\lambda))$ is given by \n\\begin{align*}\n\\hat \\varphi_a(\\theta) & := \\varphi_a(\\hat \\theta) = \\varphi_a(\\theta_0) \n+ \\frac{1}{\\sqrt{n}}W_n\\left(\\tilde I_{\\eta_0}^{-1} a, \\ - \\gamm' \\tilde I_{\\eta_0}^{-1} a\\right),\\\\\n\\hat \\varphi_b(\\lambda) : = \\varphi_b(\\hat \\lambda) & = \\varphi_b(\\lambda_0) + \n\\frac{1}{\\sqrt{n}} W_n \\left(- \\tilde I_{\\eta_0}^{-1} \\Lambda_0\\{b \\gamm\\}, \\gamma_b + \\gamm' \\tilde I_{\\eta_0}^{-1} \\Lambda_0 \\{b\\gamm\\}\\right).\n\\end{align*}\n\\end{theorem}\n\nThe centering sequences in the last display of the statement can be seen to be `efficient' ones from the semiparametric perspective (see e.g. \\cite{vdvAS}, Chapter 25). An important added value to the {\\em joint} BvM (in contrast to individual limiting statement for marginal coordinates) is that it captures the dependence between $\\theta$ and $\\lambda$: a practical application is given in Figure \\ref{fig:joint-density}.    \nThe result enables to consider many combinations of functionals by choosing specific $a \\in \\mathbb{R}^p$ and $b \\in L^2(\\Lambda_0)$. For example, let $a = (1, 0, \\dots, 0)$ and $b = \\mathbbm{1}_{[0,1]}$: Theorem \\ref{thm:bvm-linear-fns} implies a joint joint BvM for $(\\theta_1, \\Lambda(1))$, where $\\theta_1$ is the first coordinate of $\\theta$ and $\\Lambda(1) = \\int_0^1 \\lambda$ is the cumulative hazard function at time one.\nThe limiting distribution is given in the next corollary, where, in addition, we center the joint posterior at efficient frequentist estimators for $\\theta$ and $\\Lambda(1)$. \n\n\n\\begin{corollary}[Joint BvM for $\\theta_1$ and $\\Lambda(1)$]\n\\label{cor-1}\nConsider the Cox model with the density function in (\\ref{eqn:density-function}), \nlet $a = (1, 0, \\dots, 0)$ and $b = 1$, and $\\tau_{(\\hat \\theta_1, \\hat \\Lambda(1))}$ be the map such that $$\n\\tau_{(\\hat \\theta_1, \\hat \\Lambda(1))}: \\eta \\rightarrow \\sqrt{n} \\left(\\theta_1 - \\hat \\theta_1, \\Lambda(1) - \\hat \\Lambda(1)\\right),\n$$\nwhere $\\hat \\theta$ is the maximum Cox partial likelihood estimator and $\\hat \\Lambda(1)$ is the Breslow estimator. \nDenote $\\Pi(\\cdot \\,|\\, X) \\circ \\tau_{(\\hat\\theta_1, \\hat \\Lambda(1))}^{-1}$ as the distribution induced on \n$\\sqrt{n} (\\theta_1 - \\hat \\theta_1, \\Lambda(1) - \\hat \\Lambda(1))$, then \nunder the same conditions as in Theorem \\ref{thm:bvm-linear-fns},  \n\\begin{align}\n\\label{BvM-2}\n\\mathcal{B}_{\\mathbb{R}^2}\n\\left(\n\\Pi(\\cdot \\,|\\, X) \\circ \\tau_{(\\hat \\theta_1, \\hat \\Lambda(1))}^{-1}, \\\n\\mathcal{L}(a' \\mathbb{V}, \\Upsilon_1 - \\mathbb{V}\\Lambda_0\\{\\gamm\\})\n\\right) \\stackrel{P_{\\eta_0}}{\\to} 0,\n\\end{align}\nwhere \n$\\Upsilon_1 \\sim N(0, \\Lambda_0\\{M_0^{-1}\\})$ and $\\mathbb{V} \\sim N(0, \\tilde I_{\\eta_0}^{-1})$ are independent. \n\\end{corollary}\n\nAn immediate practical implication of the BvM theorem in Corollary \\ref{cor-1} is that two-sided quantile credible sets for $\\hat\\theta_1$ (or more generally for any given coordinate $\\theta_j$, $j = 1, \\dots, p$) are asymptotically optimal confidence sets from the perspective. Results in this vein can also be derived for the survival function in the functional sense: this is the object of the next section.\n\n\\subsection{Joint Bayesian Donsker theorems}\n\\label{sec:npBvM}\n\nWe now present the second main result in this paper, the Bayesian Donsker theorem for the joint posterior distribution of $\\theta$ and the cumulative hazard function $\\Lambda(\\cdot)$. \n\nLet us denote\n\\begin{align} \nW_n^{(1)} = W_n^\\star \\left(\\tilde I_{\\eta_0}^{-1}, \\ -\\gamm' \\tilde I_{\\eta_0}^{-1}\\right),\n\\label{Wn-1}\n\\end{align}\nwhere $W_n^{(1)}$ is a $p$-dimensional vector and\n\\begin{equation*}\n\\begin{split}\nW_n^\\star \\left(\\tilde I_{\\eta_0}^{-1},\\ -\\gamm' \\tilde I_{\\eta_0}^{-1}\\right) \n= \n\\frac{1}{\\sqrt{n}} \\sum_{i=1}^n\n\\left\\{\n\\delta_i \\tilde I_{\\eta_0}^{-1} \\left(Z_i - \\gamm \\right)\n- e^{\\theta_0'Z_i} \\tilde I_{\\eta_0}^{-1} \\left(Z_i \\Lambda_0(Y_i) - (\\Lambda_0{\\gamm})(Y_i) \n\\right)\n\\right\\},\n\\end{split}\n\\end{equation*}\nand given $b\\in L^2(\\Lambda_0)$,\n\\begin{align} \nW_n^{(2)}(b) = W_n \\left(-\\tilde I_{\\eta_0}^{-1} \\Lambda_0\\{b \\gamm\\}, \\gamma_b + \\gamm' \\tilde I_{\\eta_0}^{-1} \\Lambda_0 \\{b \\gamm\\} \\right).\n\\label{Wn-2}\n\\end{align}\n\nDefine the centering sequences for $\\theta$ and $\\lambda$ as follows:\n$$\nT_n^\\theta = \\theta_0 + W_n^{(1)}\/\\sqrt{n},\n$$\nand, for a given sequence $L_n$, \n\\begin{align*}\n\\langle T_{n}^\\lambda, \\psi_{lk} \\rangle \n= \n\\begin{cases}\n\\langle \\lambda_0, \\psi_{lk} \\rangle \n+\nW_n^{(2)}(\\psi_{lk})\/\\sqrt{n} & \\quad \\text{if} \\ l\\leq L_n,\\\\\n\\ 0 & \\quad \\text{if} \\ l > L_n.\n\\end{cases}\n\\end{align*}\n\nThe Donsker theorem requires, in addition to \\ref{cond:C1}, a similar condition,  where $t$ and $s$ are allowed to increase with $n$. This condition is stated as follows:\n\n\\begin{enumerate}[label=\\textbf{(C2)}]\n\\item \\label{cond:C2} (Change of variables condition, version 2) with the same notation as in \\ref{cond:C1}, let $\\eta_h = (\\theta_h, r_h)$, $\\theta_h$ and $r_h$ be the local paths in \\eqref{path-theta} and \\eqref{path-r} respectively with $a, b$ to be specified below,\nfor $n$ large enough and any $|t|, |s| \\leq \\log n$, one assumes, for $A_n$ as in \\ref{cond:P} and some constant $C_1 > 0$,\n\n$$\n\\frac{\n\\int_{A_n} e^{\\ell_n(\\eta_h) - \\ell_n(\\eta_0)} d\\Pi(\\eta)\n}\n{\n\\int e^{\\ell_n(\\eta) - \\ell_n(\\eta_0)} d\\Pi(\\eta)\n}\n\\leq e^{C_1(1 +t^2 + s^2)}.\n$$\n\\end{enumerate}\n\nCondition \\ref{cond:C2} is similar to \\ref{cond:C1}. A major difference between the two conditions is that in \\ref{cond:C2}, $t$ and $s$ are allowed to increase with $n$; however, in \\ref{cond:C1}, $t$ and $s$ are fixed. \n\nWe further require the rates $\\epsilon_n$ and $\\zeta_n$ and the cut-off $L_n$ in \\ref{cond:P} to satisfy\n\\begin{align}\n\\label{ez-rate-condition}\n\\sqrt{n} \\epsilon_n 2^{-L_n} = o(L_n^{-5\/2}), \\quad\n\\zeta_n L_n^2 = o(1),\n\\end{align}\n\n\\begin{theorem}[Joint Bayesian Donsker theorem]\n\\label{thm:donsker-joint}\n Suppose the prior for $\\eta = (\\theta, \\eta)$ is chosen such that both\n{\\rm  \\ref{cond:P}} and (\\ref{ez-rate-condition}) hold. Suppose  {\\rm \\ref{cond:C1}} holds for $a = z$, any fixed $z \\in \\mathbb{R}^p$, and any $b \\in \\mathcal{V}_{\\mathcal{L}} = \\text{Vect}\\{\\psi_{lk}, \\ l \\leq L_n, \\ 0 \\leq k < 2^l \\}$ \nfor a fixed $\\mathcal{L}$ and {\\rm \\ref{cond:C2}} holds uniformly for $a = z$, any fixed $z \\in \\mathbb{R}^p$, and any $b = \\psi_{LK}$ with $0 \\leq L \\leq L_n$ and $0 \\leq K < 2^L$.\n\nLet $\\mathcal{L}((\\theta, \\Lambda(\\cdot)) \\in \\cdot \\,|\\, X)$ be the distribution induced on $\\theta$ and $\\Lambda(\\cdot) = \\int_0^\\cdot \\lambda$ and $\\mathbb{T}_n^\\lambda(\\cdot) = \\int_0^\\cdot T_n^\\lambda$ be the centering for $\\Lambda(\\cdot)$. \nDenote $\\mathbb{B}(\\cdot)$ as standard Brownian motion and set $U_0(\\cdot) = \\int_0^\\cdot (\\lambda_0\/M_0)(u) du$.\nLet $\\mathbb{V} \\sim N(0, \\tilde I_{\\eta_0}^{-1})$ that is independent of $\\mathbb{B}(\\cdot)$.\nThen, as $n \\to \\infty$,\n\\begin{align}\n\\label{eqn:donsker-joint}\n\\mathcal{B}_{\\mathbb{R}^p \\times \\mathcal{C}([0,1])} \\left(\n\\mathcal{L} \\left(\\sqrt{n} (\\theta - T_n^\\theta, \\Lambda(\\cdot) - \\mathbb{T}_n^\\lambda(\\cdot) ) \\,|\\, X \\right), \\\n\\mathcal{L} \\left( \\mathbb{V}, \\mathbb{B}(U_0(\\cdot)) - \\mathbb{V}' \\Lambda_0\\{\\gamm\\}(\\cdot) \\right)\n\\right) \\stackrel{P_{\\eta_0}}{\\to} 0,\n\\end{align}\nwhere $\\mathcal{B}_{\\mathbb{R}^p \\times \\mathcal{C}([0,1])}$ is the bounded-Lipschitz metric on $\\mathbb{R}^p \\times \\mathcal{C}([0,1])$.\n\\end{theorem}\n\n\\begin{remark}\nWhile the proof is left to the supplemental material \\citep{ning22supp}, a key step for obtaining the joint Bayesian Donsker theorem, following ideas from \\cite{cast14a}, is to establish first a BvM for $(\\theta,\\lambda)$ in an appropriate space. However, unlike in \\cite{cast14a} where one can work directly on the nonparametric quantity of interest, here due to the split semiparametric model at hand, one needs to prove a {\\em joint} nonparametric BvM for the pair $(\\theta,\\lambda)$, see Proposition \\ref{prop-tightness}.\nThis result is new in this context and is of independent interest for proving similar results in other semiparametric models.  \n\\end{remark}\n\nThe centerings $T_n^\\theta$ and $\\mathbb{T}_n^\\lambda$ in Theorem \\ref{thm:donsker-joint} can be replaced with any efficient estimators for $\\theta$ and $\\Lambda$: the next corollary formalizes this with centering at standard  frequentist estimators.\n\n\\begin{corollary}\n\\label{cor-donsker}\nLet $\\hat \\theta$ be the maximum Cox partial likelihood estimator and $\\hat \\Lambda(\\cdot)$ be the Breslow estimator,\nthen,\nunder the same conditions as in Theorem \\ref{thm:donsker-joint}, \nas $n \\to \\infty$,\n\\begin{align}\n\\label{cor-donsker-1}\n& \\mathcal{B}_{\\mathbb{R}^p \\times \\mathcal{D}([0,1])} \\left(\n\\mathcal{L}\\left(\\sqrt{n} (\\theta - \\hat \\theta, \\Lambda(\\cdot) - \\hat \\Lambda(\\cdot)) \\,|\\, X \\right), \\\n\\mathcal{L} \\left( \\mathbb{V}, \\mathbb{B}(U_0(\\cdot)) - \\mathbb{V}' \\Lambda_0\\{\\gamm\\}(\\cdot) \\right)\n\\right) \\stackrel{P_{\\eta_0}}{\\to} 0,\n\\end{align}\nwhere $\\mathcal{D}([0,1])$ is the Skorokhod space on $[0,1]$. \n\\end{corollary}\n\nAs an application of Corollary \\ref{cor-donsker}, one obtains the Bayesian Donsker theorem for the conditional hazard and survival functions by simply applying the functional delta method \\citep[Chapter 20 in][]{vdvAS}. \nLet $\\tz$ be a fixed element in $\\mathbb{R}^p$, and recall we define $S(\\cdot \\,|\\, \\tz) = \\exp(-\\Lambda(\\cdot) e^{\\theta' \\tz})$, the survival function conditional on $\\tz$. \nDenote $\\hat S(\\cdot \\,|\\, \\tz) = \\exp(-\\hat \\Lambda(\\cdot) e^{\\hat \\theta' \\tz})$ with $\\hat\\theta$ and $\\hat \\Lambda(\\cdot)$  the frequentist estimators as above, then, as $n \\to \\infty$,\n\\begin{align}\n& \\mathcal{B}_{\\mathcal{D}([0,1])} \n\\left(\n\t \\mathcal{L}\\left(\\sqrt{n} (\\Lambda(\\cdot) e^{\\theta'\\tz} - \\hat \\Lambda(\\cdot) e^{\\hat \\theta'\\tz}) \\,|\\, X \\right), \n\t\\ \\mathcal{L} (\\mathbb{H}_1)\n\\right) \n\\stackrel{P_{\\eta_0}}{\\to} 0,\n\\label{donsker-cond-hazard}\n\\\\\n& \\mathcal{B}_{\\mathcal{D}([0,1])} \n\\left(\n\t \\mathcal{L}\\left(\\sqrt{n} (S(\\cdot \\,|\\, \\tz) - \\hat S(\\cdot \\,|\\, \\tz)) \\,|\\, X \\right), \\\n\t\\mathcal{L}(\\mathbb{H}_2)\n\\right) \n\\stackrel{P_{\\eta_0}}{\\to} 0,\n\\label{donsker-cond-survival}\n\\end{align}\nwhere $\\mathbb{H}_1$ and $\\mathbb{H}_2$ are the transformed processes obtained after applying the functional delta method from (\\ref{cor-donsker-1}).\nMoreover, by applying the continuous mapping theorem and noting that the map for any function $f \\to \\|f\\|_\\infty$ is continuous from $\\mathcal{D}([0,1])$, equipped with the supremum norm, to $\\mathbb{R}^+$, (\\ref{donsker-cond-hazard}) and (\\ref{donsker-cond-survival}) imply \n\\begin{align*}\n&\t\\mathcal{B}_{\\mathbb{R}} \n\\left(\n\t \\mathcal{L} \\left(\\sqrt{n} \\|\\Lambda(\\cdot) e^{\\theta'\\tz} - \\hat \\Lambda(\\cdot) e^{\\hat \\theta'\\tz}\\|_{\\infty} \\,|\\, X \\right), \n\t\\ \\mathcal{L} \\left(\\|\\mathbb{H}_1\\|_{\\infty}\\right)\n\\right) \n\\stackrel{P_{\\eta_0}}{\\to} 0, \\\\\n&\t\\mathcal{B}_{\\mathbb{R}} \n\\left(\n\t \\mathcal{L} \\left(\\sqrt{n} \\|S(\\cdot \\,|\\, \\tz) - \\hat S(\\cdot \\,|\\, \\tz) \\|_{\\infty} \\,|\\, X \\right), \\\n\t\\mathcal{L} \\left(\\|\\mathbb{H}_2\\|_{\\infty}\\right)\n\\right) \n\\stackrel{P_{\\eta_0}}{\\to} 0.\n\\end{align*}\nA simple consequence of the last display is that the two-sided $(1-\\alpha)\\%$ quantile credible band for the conditional hazard (resp. the conditional survival) function is asymptotically a two-sided $(1-\\alpha)\\%$ confidence band (see \\cite{cast14a}, Corollary 2).\n\n\\subsection{The supremum-norm convergence rate for the conditional hazard function}\n\\label{sec:sup-norm}\n\nIn this section, we present the third main result: a faster supremum-norm posterior contraction rate for the conditional hazard function than the rate $\\zeta_n$ in \\ref{cond:P}. We denote this rate as \n$\\xi_n$, which depends on $L_n$, a diverging sequence, such that $L_n 2^{L_n} \\lesssim \\sqrt{n}$,  where\n$$\n\\xi_n: = \\xi_n(\\beta, L_n, \\epsilon_n) = \\sqrt{\\frac{L_n 2^{L_n}}{n}} + 2^{-\\beta L_n} + \\epsilon_n.\n$$\n\n\\begin{theorem}\n\\label{thm:supnorm-rate}\nSuppose $r_0 \\in \\mathcal{H}(\\beta, D)$ with $1\/2 < \\beta \\leq 1$. \nLet $L_n$ be a diverging sequence such that $L_n 2^{L_n} \\lesssim \\sqrt{n}$ and let $\\tz \\in \\mathbb{R}^p$ be fixed.\nFor the prior of $\\eta = (\\theta, \\lambda)$ chosen such that both {\\rm \\ref{cond:P}} and (\\ref{ez-rate-condition}) hold, and {\\rm \\ref{cond:C2}} also holds uniformly for $a = z$ and any $b = \\psi_{LK}$, with $(\\psi_{LK})$ the Haar wavelet basis, $0 \\leq L \\leq L_n$ and $0 \\leq K < 2^L$,  \nthen for $\\xi_n = o(1)$ and $n \\xi_n^2 \\to \\infty$, and an arbitrary sequence $M_n \\to \\infty$, \n\\begin{align}\n\\label{supnorm-rate}\n\\Pi \\left(\\eta: \\|\\lambda e^{\\theta'\\tz} - \\lambda_0 e^{\\theta_0'\\tz} \\|_{\\infty} > M_n \\xi_n \\,|\\, X \\right) = o_{P_{\\eta_0}}(1).\n\\end{align}\n\\end{theorem}\n\nWe will show that in the next section, with a specific choice of the value of $L_n$, the rate $\\xi_n$\nis within the same order of the Hellinger rate $\\varepsilon_n$ in \\eqref{rateveps}. \n \n\\subsection{Results for specific priors}\n\\label{sec:application}\n\nIn this section, we apply the generic results in Sections \\ref{sec:jointBvM}, \\ref{sec:npBvM}, and \\ref{sec:sup-norm} to study the specific priors considered in Section \\ref{priors}.\nThe result is stated in the following theorem.\n\\begin{theorem}\n\\label{thm:specific-sup-rate}\nConsider the Cox model with the priors as specified in {\\rm \\ref{prior:T}, and \\ref{prior:H}} or {\\rm \\ref{prior:W}} with $L = L_n$ in (\\ref{choicel}) and $\\varepsilon_n$ given in \\eqref{rateveps}. For any $\\beta \\in (1\/2, 1]$,\n\\begin{enumerate}\n\\item conditions {\\rm \\ref{cond:P}} and {\\rm \\ref{cond:C1}} hold, {\\rm \\ref{cond:B}} holds for any $b \\in \\mathcal{H}(\\mu, D)$ with $\\mu > 1\/2$ and $D > 0$, then, (\\ref{thm1-1}) in Theorem \\ref{thm:bvm-linear-fns} holds;\n\\item condition {\\rm \\ref{cond:C2}} also holds, thus (\\ref{eqn:donsker-joint}) in Theorem \\ref{thm:donsker-joint} holds;\n\\item the supremum-norm rate $\\xi_n$ in \\eqref{supnorm-rate} can be taken to be $\\xi_n = \\varepsilon_n$ as in \\eqref{rateveps}. \n\\end{enumerate}\n\\end{theorem}\n\nThe proof of this result, implying that conditions \\ref{cond:P}, \\ref{cond:B}, \\ref{cond:C1}, and \\ref{cond:C2} hold for priors given in Section \\ref{priors}, can be found in \\citet{ning22supp}.\n\n\\begin{remark}\nIf $\\beta > 1$, the first and second points in Theorem \\ref{thm:specific-sup-rate} still hold. The third point also holds but the supremum-norm rate becomes $(\\log n\/n)^{1\/3}$.\n\\end{remark}\n\n\nLet us compare the supremum rate $\\varepsilon_n$ in the third point of the theorem and the rate $\\zeta_n$ obtained in Lemma \\ref{lemma1}. Obviously, $\\varepsilon_n < \\zeta_n$, as $\\zeta_n \\geq 2^{L_n\/2} \\varepsilon_n$, and $L_n \\to \\infty$ as $n\\to \\infty$. In fact, by plugging-in the value of $L_n$ in (\\ref{choicel}), \none obtains $\\zeta_n = (\\log n\/n)^{\\frac{2\\beta - 1}{2(2\\beta+1)}}$ which can become extremely slow when $\\beta$ is close to $1\/2$. In Lemma \\ref{lower-bound} in \\citet{ning22supp}, we derive a lower bound for the minimax rate in the supremum norm for the hazard which shows that the rate $\\varepsilon_n$ is sharp. \nTo our best knowledge, this is the first sharp supremum-norm result for the hazard obtained for the Cox model. \n\n\\section{Simulation studies}\n\\label{sec:sim}\n\nTwo simulation studies are conducted in this section. \nThe first study, described in Section \\ref{study-I}, compares the limiting distribution given in Corollary \\ref{cor-1} with the empirical distributions obtained from the MCMC algorithm, which is given in Section \\ref{generate-data}.\nThe second study compares the coverage and the area of the 95\\% credible bands for the MCMC algorithm to the 95\\% confidence bands for a commonly used frequentist method by varying the sample size and changing the censoring distribution.\nWe choose the standard normal distribution for $\\theta$ and the random histogram prior for $\\lambda$ with the independent gamma and dependent priors in Section \\ref{priors} as those are often used in practice. \nIn Section \\ref{generate-data}, we describe how we generate the simulated data and the MCMC sampler. Section \\ref{study-I} presents results for the first study, and Section \\ref{study-II} summarizes results for the second study.\n\n\\subsection{Generating the data and the MCMC sampler}\n\\label{generate-data}\n\nThe data are generated from the ``true'' conditional hazard function $\\lambda_0(t) e^{\\theta_0'z}$, which is chosen as follows:\nlet $\\theta_0 = -0.5$, a scalar, and generate the covariate $z$ randomly from the standard normal distribution.\nTwo different baseline hazard functions are used to generate the data:\n\\begin{enumerate}[label=\\textbf{(1)}]\n\\item \\label{chzf} $\\lambda_0(t) = 6 \\times ((t + 0.05)^3 - 2(t + 0.05)^2 + t + 0.05) + 0.7, \\ t \\in [0,1]$,\n\\end{enumerate}\n\\begin{enumerate}[label=\\textbf{(2)}]\n\\item \\label{pwhzf} $\\lambda_0(t) = 3 \\times \\mathbbm{1}_{[0,0.4)}(t) + 1.5 \\times \\mathbbm{1}_{[0.4, 0.6)}(t) + 2 \\times \\mathbbm{1}_{[0.6, 1]}(t)$,\n\\end{enumerate}\n\n\n\n\\begin{figure}[!h]\n\\centering\n\\begin{minipage}{0.45\\textwidth}\n\\centering\n\\includegraphics[width=1\\textwidth]{plots\/chzf.pdf}\n\\caption{Plot of the continuous baseline hazard function in {\\rm \\ref{chzf}}.}\n\\label{fig:chzf}\n\\end{minipage}%\n\\hspace{0.04\\textwidth}\n\\begin{minipage}{0.45\\textwidth}\n\\includegraphics[width = 1\\textwidth]{plots\/pwhzf.pdf}\n\\caption{Plot of the piecewise constant baseline hazard function in {\\rm \\ref{pwhzf}}.}\n\\label{fig:pwhzf}\n\\end{minipage}\n\\end{figure}\n\nThe first one is a smooth function and the second one is piecewise constant. \nPlots of the two functions are given in Figures \\ref{fig:pwhzf} and \\ref{fig:chzf} respectively. These numerical choices are for illustration purposes and otherwise fairly arbitrary. Similar simulation results would hold if choosing other either smoothly varying or piecewise constant functions (we note once again that, although our theoretical results assume H\\\"older smoothness of the true log-hazard, the techniques go through for histogram true hazards as well).\n\n\nThe ``observations'' $X^n = (Y^n, \\delta^n)$ are generated using the ``{\\it simsurv}'' function in $\\mathsf{R}$. \nWe consider the following two types of censoring: \n\\begin{enumerate}\n\\item {\\it Administrative censoring only}. Time points are censored at a fixed time point $t = 1$;\n\\item {\\it Administrative censoring $+$ uniform censoring}. The censoring time is generated from the uniform distribution on $[0,1]$. Any time point beyond $t = 1$ is also censored. \n\\end{enumerate}\n\nAlthough the first type of censoring violates our assumption in Section \\ref{sec:assumption}, as we assumed the censoring follows a random distribution, it is interesting to find out in the next two sections that the empirical results still match with our theoretical results quite well.\n\nPosterior draws are obtained using the MCMC algorithm given as follows:\n\\begin{enumerate}\n\\item For the {\\it independent gamma prior,} since it is conjugate with the posterior distribution given $\\theta$, we sample each $\\lambda_k \\sim \\text{Gamma}(d_k + \\alpha, T_k(\\theta) + \\beta)$,\nwhere $d_k = \\sum_{i=1}^n \\delta_{ki}$ is the number of events in $k$-th interval and \n$T_k(\\theta) = \\sum_{i=1}^n Y_{ik} e^{\\theta Z_i}$, $\\alpha$ and $\\beta$ are the hyperparameters, and we chose them to 1.\nAfter obtaining samples for $(\\lambda_{lk})$, we draw $\\theta$. Since $\\pi(\\theta)$ is not conjugate, we first draw a candidate from the proposal density, i.e., $\\theta^{\\text{prop}} \\sim N(\\theta^{\\text{prev}}, 1)$, where $\\theta^{\\text{prev}}$ stands for the draw from the previous iteration, and then use the metropolis algorithm to accept or reject this candidate.\n\n\\item For the {\\it dependent gamma prior}, as it is non-conjugate, we thus draw each $\\lambda_k$ from the proposal density as follows: $\\lambda_1^{\\text{prop}} \\sim \\text{Gamma}(d_1 + \\alpha_0 - \\alpha, \\ T_1(\\theta) + \\beta_0)$ and\n$\\lambda_k^{\\text{prop}} \\sim \\text{Gamma}(d_k + \\varepsilon, \\ \\alpha\/\\lambda_{k-1} + T_k(\\theta))$ for $k = 2, \\dots, L_n-1$. The last interval $\\lambda_K \\sim \\text{Gamma}(d_K + \\alpha,\\  \\alpha\\lambda_{K-1} + T_K(\\theta))$ for $K=L_n$. In practice, we choose $\\varepsilon = 10^{-6}$, $\\alpha_0 = 1.5$ and $\\alpha = \\beta_0 = 1$. \nThe proposal density for $\\theta$ is the same.\n\\end{enumerate}\n\nTo initialize the MCMC algorithm, we choose the initial values for $\\theta$ and $\\lambda_k$ as their frequentist estimators (the same as in Corollary \\ref{cor-donsker}). We choose $L_n$ as in (\\ref{choicel}) and $\\beta = 1\/2$.\nFor each simulation, we run 10,000 iterations and discard the first 2,000 draws as burn-in. \n\nLet us now discuss in more detail the simulation in Figure \\ref{fig:intro}. \nBoth datasets (in the left plot and the right plot) are generated by choosing \\ref{chzf}.\nOnly administrative censoring is considered. \nThe prior is chosen to be the independent gamma prior (choosing the dependent gamma prior won't change the result dramatically, as can be seen in Table \\ref{tab:coverage-1} below).\nThe 95\\% credible band is a fixed width band whose width is constant with the time. \nThe width is determined such that the posterior probability is 95\\%.\nThe 95\\% confidence band, on the other hand, is obtained using the $\\mathsf{predictCox}$ function of the `riskRegression' package in $\\mathsf{R}$. Its width varies with time. \nHere we briefly describe the approach used in their package. \nWe refer the interested readers to read \\citet{lin94} and \\citet{scheike08} for more details. \nUsing the fact that the frequentist estimator $(\\hat \\theta, \\hat \\Lambda)$ converges to the same limiting process as the joint distribution $(W_n^{(1)}, \\int_0^t W_n^{(2)}(\\mathbbm{1}_{u \\leq t}) du)$, where \n$W_n^{(1)}$ and $W_n^{(2)}$ are given in \\eqref{Wn-1} and \\eqref{Wn-2}, their approach first defines another process, depending on $W_n^{(1)}$ and $W_n^{(2)}$, that is asymptotically equivalent to $\\sqrt{n} (\\hat \\Lambda(t)e^{\\hat\\theta'z} - \\Lambda(t) e^{\\theta' z})$; see equation 2.1 in \\citet{lin94} for the exact expression of that process.\nTheir approach then further approximates that process by a summation of independent normal variables whose distribution can be easily generated through Monte Carlo simulation and replaces other unknown quantities with their sample estimators.\nAfter large enough samples are generated, the last step is to obtain the size of the 95\\% confidence band for the conditional cumulative hazard function by choosing the 95th quantile from those samples. \nThe confidence band for the conditional survival function can be obtained similarly, except that one first needs to apply the functional delta method to obtain the limiting process for the conditional survival function. The remaining parts are the same. \n \n\\subsection{Study I: Comparing the empirical posterior distributions and the limiting distributions of $\\theta$ and $\\Lambda(1)$}\n\\label{study-I}\n\nWe compare the limiting distribution given in Corollary \\ref{cor-1} with the empirical distribution obtained using the MCMC sampler in Section \\ref{generate-data}.\nWe generate the hazard rate with a sample of 1,000 and choose $\\lambda_0$ as \\ref{chzf}.\nThe prior is chosen as the independent gamma prior. Results for choosing the dependent gamma prior are similar. To obtain draws, we run the MCMC algorithms in parallel for 1,000 times. \nEach time we only record the last pair draw for $\\theta$ and $\\Lambda(1)$, where $\\Lambda(1) = \\sum_{k=1}^{L_n} \\lambda_k$.  Therefore, the 1,000 draws we obtained are independent.\n\nWe first study the marginal posterior distributions for $\\theta$ and $\\Lambda(1)$. In \nFigures \\ref{fig:marginal-theta} and \\ref{fig:marginal-Lambda}, \nwe first draw their empirical histogram from the 1,000 independent draws. \nWe then draw a normal density with blue color centered at the posterior mean, and its variance is estimated from those draws. \nLast, we draw another normal density with red color, which has the same centering as the blue one, but its variance is chosen as the theoretical value from the limiting distribution in Corollary \\ref{cor-1}. \nWe observe that in either the left plot (i.e., for $\\theta$) or the right plot (i.e., for $\\Lambda(1)$), \nthe density with blue color is well aligned with the one with red color. This finding suggests that empirical variances are close to their theoretical variances obtained from the corollary.\nWe also found that both empirical histograms show similar shapes as their corresponding normal density, which verifies their limiting distributions should be normal. \nLast, both the true values of $\\theta_0$ and $\\Lambda_0(1)$, $-0.5$ and $1.2$ respectively, are contained inside the corresponding 95\\% credible intervals.\nFor $\\theta$, the interval is $[-0.54, -0.39]$, and for $\\Lambda(1)$, it is $[1.14, 1.34]$.\n\n\\begin{figure}\n\\centering\n\\begin{minipage}{0.45\\textwidth}\n\\centering\n\\includegraphics[width = 1\\textwidth]{plots\/theta.pdf}\n\\caption{Plot of the empirical histogram, the empirical distribution (blue), and the limiting distribution (red) for the marginal posterior distribution of $\\theta$. True value of $\\theta$ is $-0.5$.}\n\\label{fig:marginal-theta}\n\\end{minipage}%\n\\hspace{0.04\\textwidth}\n\\begin{minipage}{0.45\\textwidth}\n\\includegraphics[width=1\\textwidth]{plots\/Lambda1.pdf}\n\\caption{Plot of the empirical histogram, the empirical distribution (blue), and the limiting distribution (red) for the marginal posterior distribution of $\\Lambda(1)$. True value of $\\Lambda(1)$ is $1.2$.}\n\\label{fig:marginal-Lambda}\n\\end{minipage}\n\\end{figure}\n\n\nNext, we study the joint posterior distribution of $\\theta$ and $\\Lambda(1)$, which involves the correlation between the two quantities.\nIn Figure \\ref{fig:joint-density}, we give three plots. \nIn (a), we plot the 86\\%, 90\\%, 95\\%, and 99\\% contour plots of the limiting joint distribution in Corollary \\ref{cor-1}. In (b), we plot the contours with the same four quantiles for a bivariate normal distribution, which its mean, variances, and correlations are estimated from the 1,000 draws. \nIn (c), we found that the two sets of contour plots in (a) and (b) indeed align quite well, \nwhich suggests that the empirical distribution matches with the theoretical limiting distribution in the corollary. \nOur calculation reveals that the correlation between $\\theta$ and $\\Lambda(1)$ in (a) is 0.15 and that in (b) is 0.10. \nA benefit of studying the joint posterior distribution with the correlation is that one can obtain the elliptical credible sets instead of rectangular credible sets. The length and the width of the rectangular credible sets are the 97.5\\% credible intervals of $\\theta$ and $\\Lambda(1)$ respectively. \nTherefore, the area of a rectangular credible set is typically larger than that of an elliptical credible set. For example, in (b), the area of the 95\\% elliptical credible set is 1.07 and that of the 95\\% rectangular credible set is 1.76, which is 64\\% bigger.\n\n\\begin{figure}[!]\n\\centering\n\\subfigure[ ]{\\includegraphics[width=0.34\\textwidth]{plots\/jointBvM-th.pdf}}%\n\\subfigure[ ]{\\includegraphics[width=0.34\\textwidth]{plots\/jointBvM-ep.pdf}}%\n\\subfigure[ ]{\\includegraphics[width=0.34\\textwidth]{plots\/jointBvM-overlay.pdf}}\n\\caption{Contour plots of the elliptical credible sets at the 68\\%, 90\\%, 95\\%, and 99\\% quantiles. (a) is obtained using the joint limiting distribution given in Corollary \\ref{cor-1}, (b) is obtained using the 1,000 independent draws of the pair $(\\theta, \\Lambda(1))$ from the MCMC output. In (c), we overlay the credible sets in (a) and (b). The 1,000 draws are plotted with gray color.}\n\\label{fig:joint-density}\n\\end{figure}\n\n\\subsection{Study II: Comparing the coverage and the area between the credible bands and the confidence bands}\n\\label{study-II}\n\nThe study in the last section is based on a single simulated dataset. We now provide a more thorough study to compare the coverage and the area of the credible (or confidence) bands under various settings. Specially, we want to compare: 1) the two Bayesian methods using the independent gamma  and the dependent gamma priors; 2) datasets with two different sample sizes $n = 200$ and $n = 1,000$; 3) data with two different types of censoring: administrative censoring only and administrative censoring with additional uniform censoring; 4) coverages of the baseline survival and conditional survival functions; and 5) data are generated from the continuous function in \\ref{chzf} and that from the piecewise constant function in \\ref{pwhzf}.\n\nFor each setting, we generate 1,000 datasets. For each dataset, we run the MCMC sampler to obtain the 95\\% credible (or confidence) band. \nThe coverage is the percentage of the credible (or confidence) bands encompassing the true function. The area estimated by taking the average of 1,000 areas of the credible (or confidence) bands. Results using the continuous baseline hazard function are given in Table \\ref{tab:coverage-1} and those using the piecewise constant baseline hazard function are given in Table \\ref{tab:coverage-2}.\n\n\n\\begin{table}[ht]\n\\caption{Coverages and areas of the 95\\% Bayesian credible bands and of the 95\\% confidence bands with $\\lambda_0$ is chosen as the continuous function in {\\rm \\ref{chzf}}.}\n\\centering \n\\begin{tabular}{c|cccc||cccc} \n\\hline\\hline \n & \\multicolumn{4}{c||}{\\bf Adm. censoring only} & \\multicolumn{4}{c}{\\bf Adm. $+$ Unif. censoring}\\\\\n\\hline\n& \\multicolumn{2}{c}{baseline survival} & \\multicolumn{2}{c||}{cond. survival} \n& \\multicolumn{2}{c}{baseline survival} & \\multicolumn{2}{c}{cond. survival}\\\\\n & coverage & area & coverage & area & coverage & area & coverage & area\\\\[2pt]\n\\hline\nind. 200 & 0.96 & 0.16 & 0.94 & 0.16 & 0.95 & 0.21 & 0.94 & 0.21\\\\\ndep. 200 & 0.94 & 0.16 & 0.93 & 0.16 & 0.93 & 0.22 & 0.93 & 0.22  \\\\\nfreq. 200 & 0.82 & 0.18 & 0.83 & 0.18 & 0.81 & 0.23 & 0.80 & 0.23 \\\\[2pt]\n\\hline \nind. 1000 & 0.95 & 0.08 & 0.94 & 0.08 & 0.95 & 0.11 & 0.94 & 0.11 \\\\\ndep. 1000 & 0.95 & 0.08  & 0.94 & 0.08 & 0.94 & 0.11 & 0.94 & 0.11 \\\\\nfreq. 1000 & 0.95 & 0.08 & 0.95 & 0.08 & 0.94 & 0.11 & 0.94 & 0.11 \\\\[2pt]\n\\hline\\hline\n\\end{tabular}\n\\label{tab:coverage-1}\n\\end{table}\n\n\nFrom Table \\ref{tab:coverage-1}, in which $\\lambda_0$ is chosen to be a continuous function, first, we found that the two Bayesian methods, either using the independent or the dependent gamma prior, produce similar converge results and areas for the credible bands. \nSecond, when $n = 200$, not only the coverage of the two Bayesian methods is better than that of the frequentist method, but the area is also smaller.  The coverages and the areas of the three methods become similar when $n = 1,000$.\nAs approximation becomes more accurate when the sample size increases, this leads to higher coverage and a smaller area of the confidence band. \nThird, we found that when data are administratively and uniformly censored, the area of the credible bands is larger than those only administratively censored. \nSuch a result is expected, as we found that in a typical simulated dataset, a former has $\\sim$45\\% data are censored, and the latter only has $\\sim$10\\% data are censored.  \nLast, there is no significant difference between the coverage and the area of the baseline survival function and the conditional survival function, even though the latter accounts for the uncertainty for estimating the regression coefficients. \n\n\\begin{table}[ht]\n\\caption{Coverages and areas of the 95\\% Bayesian credible bands and of the 95\\% confidence bands with $\\lambda_0$ is chosen as the piecewise constant function in {\\rm \\ref{pwhzf}}.}\n\n\\centering \n\\begin{tabular}{c|cccc||cccc} \n\\hline\\hline \n & \\multicolumn{4}{c||}{\\bf Adm. censoring only} & \\multicolumn{4}{c}{\\bf Adm. $+$ Unif. censoring}\\\\\n\\hline\n& \\multicolumn{2}{c}{baseline survival} & \\multicolumn{2}{c||}{cond. survival} \n& \\multicolumn{2}{c}{baseline survival} & \\multicolumn{2}{c}{cond. survival}\\\\\n & coverage & area & coverage & area & coverage & area & coverage & area\\\\[2pt]\n\\hline\nind. 200 & 0.95 & 0.15 & 0.93 & 0.15 & 0.98 & 0.17 & 0.96 & 0.17\\\\\ndep. 200 & 0.93 & 0.15 & 0.92 & 0.15 & 0.94 & 0.18 & 0.94 & 0.18  \\\\\nfreq. 200 & 0.94 & 0.17 & 0.94 & 0.17 & 0.90 & 0.21 & 0.90 & 0.21 \\\\[2pt]\n\\hline \nind. 1000 & 0.93 & 0.07 & 0.92 & 0.07 & 0.94 & 0.09 & 0.93 & 0.09 \\\\\ndep. 1000 & 0.92 & 0.07  & 0.91 & 0.07 & 0.91 & 0.09 & 0.91 & 0.09 \\\\\nfreq. 1000 & 0.93 & 0.07 & 0.93 & 0.08 & 0.92 & 0.10 & 0.92 & 0.10 \\\\[2pt]\n\\hline\\hline\n\\end{tabular}\n\\label{tab:coverage-2} \n\\end{table}\n\n\nTable \\ref{tab:coverage-2} gives the results for data are generated from the baseline hazard that is the piecewise constant function in \\ref{pwhzf}. \nWe notice that the coverage of the frequentist confidence bands improved significantly when $n = 200$ compared to that in Table \\ref{tab:coverage-1}. This suggests that the frequentist approach is more accurate for estimating a piecewise constant function. Indeed, the Breslow estimator treats the $\\lambda_0$ as piecewise constant between uncensored failure times \\citep[see Page 473 of][]{lin07}.\nHowever, we again observe that the area of the credible bands provided by the two Bayesian methods is smaller than that of the confidence bands. \nWe also found that both of the two Bayesian methods provide similar coverage results whether $\\lambda_0$ is chosen to be the continuous function or the piecewise constant function. \n\n\n\nIn summary, using the Bayesian method can be attractive for estimating data with a relatively small sample size, especially when the true baseline hazard function is not piecewise constant. \nThe frequentist method needs to apply an asymptotical approximation to obtain the confidence band, and the approximation can perform poorly when the sample size is relatively small. On the other hand, the proposed Bayesian method provides a credible band without using any asymptotical approximation. Notice that the width for the credible band is constant over time. It should be possible to build a varying-width credible band whose overall area is smaller, both in small samples and asymptotically, but \nthe construction and analysis of such a band is outside the scope of the paper. \nYet, the considered fixed--width credible band performs already remarkably well, in particular in finite samples, and achieves the asymptotic limits expected from the Donsker theorem for large sample sizes.\n\n\n\n\\section{Discussion}\n\\label{sec:diss}\n\nWe provide three new exciting results for the study of the Bayesian Cox model: 1) a joint Bernstein--von Mises theorem for the linear functionals of $\\theta$ and $\\lambda$; in particular, the correlation between the two functionals is captured by the results; 2) a Bayesian Donsker theorem for the conditional hazard function and the conditional survival functions; 3) a supremum-norm posterior contraction rate for the conditional hazard function. \n\nThe paper makes major advances on two fronts: on the one hand, it provides new results on optimal posterior convergence rates both in $L^1$-- and $L^\\infty$--sense for the hazard; uncertainty quantification is considered for finite dimensional functionals as well as for the posterior cumulative hazard process: those are the first results of this kind for non--conjugate priors (in particular priors for which explicit posterior expressions are not accessible) in this model. On the other hand, the paper provides validation for several classes of practically used histogram priors (see e.g. \\cite{ibrahim01}), both for dependent and independent histogram heights.\n\nAs a comparison, the results from \\cite{cast12} (Theorem 5) and \\cite{ghosal17} (Theorem 12.12) require a fast enough preliminary posterior contraction rate of $n^{-3\/8}$ in terms of the Hellinger distance. This effectively rules out the use of regular histogram priors, which are limited in terms of rate by $n^{-1\/3}$ (corresponding to the optimal minimax rate for Lipschitz functions). Two key novelties here are that {\\em a)} we only require a preliminary rate of an order faster than $n^{-1\/4}$ (corresponding to $\\beta=1\/2$ in \\eqref{rateveps}) {\\em b)} the use of the multiscale approach introduced in \\cite{cast14a} enables one to provide both optimal supremum norm contraction rates for the conditional hazard, justifying practically the visual closeness of estimated hazard curves to the true curve, and uncertainty quantification for the conditional cumulative hazard, which follows a BvM for $\\Lambda(\\cdot) e^{\\theta'z}$.\n \nWe also underline that although not investigated here in details for reasons of space, the results extend to smoother dictionaries than histograms: for instance, if the true hazard is very smooth, one can derive correspondingly very fast posterior rates (obtaining optimal rates $n^{-\\beta\/(2\\beta+1)}$ for any $\\beta>1\/2$, up to log factors) if one chooses the basis $(\\psi_{lk})$ to be a suitably smooth wavelet basis. We refer the interested reader to \\cite{cast20survival} for more on how to effectively obtain this.   \n\nThe present work studies the classical Cox model.\nMany extensions of the model have been proposed, such as the Cox model with time-varying covariates \\citep{fisher99}, the nonproportional hazards model \\citep{schemper02}, the Cox-Aalan model \\citep{scheike02} to name a few. The Bayesian nonparametric perspective is particularly appealing in these more complex settings; let us cite two recent practical success stories of the approach in settings going beyond the Cox model (in particular enabling more complex dependencies in terms of covariates and hazard, and\/or time dependence):  one is the use of  BART ({Bayesian} additive regression trees) priors in \\citet{sparapani16}, another is the use of dependent Dirichlet process priors in \\citet{xu19}. \nIt would be very desirable to obtain theory and validation for these more complex settings: the present work can be seen as a first step towards this aim. \n\n\\section*{Acknowledgement}\n\nThe authors would like to thank St\\'ephanie van der Pas for helpful discussions  with the $\\mathsf{R}$ code for simulation studies. \n\n\\begin{supplement}\nThe supplement \\citet{ning22supp} includes the proofs of the results stated in this paper. \n\\end{supplement}\n\n\\bibliographystyle{chicago}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:1}\n\\IEEEPARstart{N}{etwork} mining is the basis for many network analysis tasks, such as classification and link prediction. The dimensionality of traditional node representations is proportional to the network scale, which requires large amount of storage and computation resources for network analysis tasks. Thus, it is necessary to learn low-dimensional representations of nodes to capture and preserve the network features. Network embedding, also known as network representation learning, is a way of learning low-dimensional representations and preserving useful features to commonly support subsequent network analysis tasks.\n\nExisting network embedding methods \\cite{perozzi2014deepwalk,grover2016node2vec,cao2015grarep} that emphasize preserving network structural features have achieved promising performance in several network analysis tasks. However, nodes in real-world networks have rich attribute information beyond the structural details, such as text information in citation networks and user profiles in social networks. Attribute features are essential to network analysis applications \\cite{hu2013exploiting,tang2013exploiting}and it is insufficient to learn network representations only based on preserving structural features. Node attributes carry semantic information that largely alleviates the link sparsity problem and supplement the incompleteness structure information. The strong correlations between structures and attributes enable them to be integrated to learn network representations according to the principles of homophily \\cite{mcpherson2001birds} and social influence theory \\cite{marsden1988homogeneity}. Therefore, we integrate the topological structures and node attributes to perform network embedding to preserve both structural and attribute features of the network. Differing from some task-oriented network embedding methods that learn network representations for the specific task, we aim to learn the network representations generally applying to various advanced network analysis tasks. We face three challenges: (1) Both the underlying network structures \\cite{luo2011cauchy} and the complex interactions between attributes and structures \\cite{cui2017survey} are highly non-linear. Thus, designing a model to capture these non-linear relationships is difficult. (2) The structures and attributes are information from different sources, which make it difficult to find direct correlations between originally observed information due to sparsity and noise. Modeling the correlations between network structures and attribute information is a tough problem. (3) The nodes with coherent links and similar attributes in the original network have strong proximity. They are supposed to be close to each other in the embedding space as well. Thus, mapping the proximity of nodes from both structure and attribute perspectives to the embedding space is critically important.\n\nTo address the above challenges, a Multimodal Deep Network Embedding method named MDNE is proposed in this paper. Most of existing shallow models have limited ability to represent complex non-linear relationships \\cite{bengio2009learning}. A deep model comprising of multiple layers of non-linear functions, using each layer to capture the non-linear relationships of units in the lower layer, is able to extract the non-linear relationships of data progressively during training \\cite{hinton2006reducing}. Moreover, deep learning has been demonstrated to have powerful non-linear representation and generalization ability \\cite{bengio2009learning}. In order to capture the highly non-linear network structures and the complex interactions between structures and attributes, a deep model comprising multiple layers of non-linear functions is proposed to learn compact representations of nodes.The original structure and attribute information, which are represented by an adjacency matrix and attribute matrix, respectively, are usually sparse and noisy, making it difficult for the deep model to extract the correlations between them directly. In this paper, a multimodal learning method \\cite{ngiam2011multimodal} is adopted to pre-process the structure and attribute information to obtain their high-order features. High-order features are condensed and less noisy, so concatenating the two high-order features facilitates the deep model to extract the high-order correlations between the network structures and node attributes. To ensure the obtained representations preserve both structural and attribute features of the original network, we use the structural proximity and attribute proximity to define the loss function for the new model. We preserve the structural features by taking the advantage of the first-order proximity and second-order proximity, which capture the local and global network structures \\cite{wang2016structural}. The attribute proximity, which indicates the similarity of node attributes, is also utilized in the learning process to preserve the attribute features of the network. Thus, the learned representations preserve both the structural and attribute features of nodes in the embedding space.\n\nTo evaluate the effectiveness and generality of the proposed method in a variety of scenes, we conduct experiments to analyze network representations obtained by different network embedding methods from four real-world network datasets in three analysis tasks including link prediction, attribute prediction, and classification. The results show that the network representations obtained by MDNE offer better performance on different tasks compared to other methods. This demonstrates that the proposed method effectively preserves the topological structure and attribute features of nodes in the embedding space, which improves the performance on diverse network analysis tasks.\n\nThe rest of the paper is organized as follows. Section 2 discusses the related works. The proposed method MDNE is described in details in Section 3. Experimental results of different network analysis tasks on various real-world datasets are presented in Section 4. Finally, Section 5 concludes the paper.\n\n\\section{Related Works}\n\\label{sec:2}\nThe early works of network embedding are related to graph embedding \\cite{roweis2000nonlinear,belkin2002laplacian}, which aims to embed an affinity graph into a low-dimensional vector space. The affinity graph is obtained by calculating the proximity between feature vectors of nodes. Recent network embedding aims to embed naturally formed networks into a low-dimensional space, such as social networks, citation networks, etc. Most of the existing works \\cite{jacob2014learning,hofmann2001unsupervised,blei2003latent} focused on reducing the dimensions of structure information while preserving the structural features of nodes. GraRep \\cite{cao2015grarep} defined different loss functions of models to preserve high-order proximity among nodes and optimized each model by matrix factorization techniques. The final representations of nodes combined representations learned from different models. M-NMF \\cite{wang2017community} proposed a novel modularized nonnegative matrix factorization model to incorporate the community structure into network embedding. The above shallow models have applied in various network analysis tasks, but have limited ability to represent the highly non-linear structure of networks. Thus, techniques with deep models were introduced to deal with the problem. LINE \\cite{tang2015line} designed the objective function based on the first-order proximity and second-order proximity and adopted negative sampling approach to minimize the objective function to get low-dimensional representations which preserve the local and global structure of the network. DeepWalk \\cite{perozzi2014deepwalk} utilized random walks in the network to sample the neighbors of nodes. By regarding the path generated as sentences, it adopted Skip-Gram, a general word representation learning model, to learn the node representations. Node2vec \\cite{grover2016node2vec} modified the way of generating node sequences and proposed a flexible notion of node's neighborhood. A biased random walk procedure was designed, which explored diverse neighborhood. SDNE \\cite{wang2016structural} designed a clear objective function to preserve the first-order proximity and second-order proximity of nodes and mapped the network into a highly non-linear latent space through an autoencoder-based model.\n\nBesides structure information, most of the recently obtained network datasets often carry a large amount of attribute information. However, it is difficult for pure structure-based methods to compress attribute information and obtain the representations combining the structure and attribute information. Therefore, efforts have been done to jointly exploit structure and attribute information in network embedding, and the representations integrating structure and attribute information have been demonstrated to improve the performance in network analysis tasks \\cite{hu2013exploiting,tang2013exploiting,li2016robust}. TADW \\cite{yang2015network} proved DeepWalk to be equivalent to matrix factorization and incorporated text features into network representation learning under the framework of matrix factorization. It can only handle the text attributes. AANE \\cite{huang2017accelerated} modeled and incorporated node attribute proximity into network embedding in a distributed way. The above matrix factorization methods did not preserve the attribute features directly, but performed the learning based on the attribute affinity matrix calculated by a specific affinity metric, which limited the attribute feature preservation ability of the obtained representations. UPP-SNE \\cite{zhang2017user} learned joint embedding representations by performing a non-linear mapping on user profiles guided by network structure. It mainly dealt with user profile information. TriDNR \\cite{pan2016tri} separately learned embedding from a coupled neural network architecture and linearly combined them in an iterative way. It lacked sufficient knowledge interactions between the two separate models. ASNE \\cite{Liao2017Attributed} proposed a multilayer perceptron framework to integrate the structural and attribute features of nodes. It preserved the structural proximity and attribute proximity by maximizing the likelihood function defined based on random walks. Its model lacked a non-linear pre-processing of the structure and attribute information, which could facilitate to extract the high-order correlations between attribute and structural features in the later learning. In this paper, a multimodal learning method is adopted to pre-process the original data.\n\nMultimodal learning methods which have aroused considerable research interests aim to project data from multiple modalities into a latent space. The classical methods CCA, PLS, BLM and their variants \\cite{hardoon2004canonical,rosipal2005overview,tenenbaum2000separating} were widely applied in previous time. Recent decades have seen great power of deep learning method to generate integrated representations for multimodal data. \\cite{ngiam2011multimodal} proposed an autoencoder-based method to learn features over multiple modalities (video and audio) and achieved in speech recognition. \\cite{srivastava2012learning} proposed a Deep Belief Network (DBN) architecture for learning a joint representation of multimodal data, which made it possible to create representations when some data modalities are missing. The multimodal Deep Boltzmann Machine (DBM) model proposed in  \\cite{srivastava2014multimodal} fused modalities (image and tag) together and extracted unified representations which were useful for classification and information retrieval tasks. \\cite{kang2015learning} learned consistent representations for two modalities and facilitated the cross-matching problem. \\cite{xu2017learning} proposed a cross-modal hashing method to learn unified binary representations for multimodal data. Following these successful works, we introduced multimodal learning method into network embedding. The structure and attribute information of the network are regarded as different modalities. An autoencoder-based multimodal model \\cite{ngiam2011multimodal} is adopted to pre-process the bimodal data and forms high-order features, which facilitate the fused representations to be learnt.\n\nThere are also methods learning network representations for specific applications. PinSage \\cite{ying2018graph} combined the recent Graph Convolutional Network (GCN) algorithm \\cite{defferrard2016convolutional,kipf2016semi} with efficient random walks to generate representations applying in web-scale recommender systems. However, our MDNE learns integrated representations generally applying for various network analysis tasks.\n\n\\section{Multimodal Deep Network Embedding Method}\n\\label{sec:3}\n\\subsection{Problem Definition}\nAn attributed network is defined as $G=(U,E,A)$, where $U=\\{u_1,\\dots,u_n\\}$ represents a set of $n$ nodes, $E=\\{e_{i,j}\\}$ represents a set of $l$ edges, and $A=\\{\\bf{a}_i\\}_{i=1}^n$ represents the attribute matrix. Edge information is represented by the adjacency matrix $S=\\{\\bf{s}_i\\}_{i=1}^n$.\n\nThe adjacency vector $\\bf{s}_i$ and attribute vector $\\bf{a}_i$ of node $i$ represent the structure and attribute information, respectively. Thus, the goal of network embedding is to compress the two vectors into a low-dimensional vector, and preserving the structure and attribute features in the low-dimensional space (embedding space).\n\nThe first-order proximity and second-order proximity capture the local and global network structural features, respectively \\cite{wang2016structural}.\n\\newtheorem{definition}{Definition}\n\\begin{definition}[First-Order Structural Proximity]\n  The first-order proximity describes the local pairwise proximity between two nodes. For each pair of nodes, the edge weight, $s_{i,j}$ indicates the first-order proximity between $u_i$ and $u_j$.\n\\end{definition}\n\\begin{definition}[Second-Order Structural Proximity]\n  The second-order proximity between a pair of nodes $(u_i,u_j)$ in a network describes the similarity between their neighborhood structures which are represented by the adjacency vectors.\n\\end{definition}\n The first-order proximity and second-order proximity jointly compose the structural proximity between nodes. The attribute proximity captures the attribute feature of nodes.\n\\begin{definition}[Attribute Proximity]\n  The attribute proximity between a pair of nodes $(u_i,u_j)$ describes the proximity of their attributes information. It is determined by the similarity between their attribute vectors, i.e., $a_i$ and $a_j$.\n\\end{definition}\nThe attribute proximity and structural proximity between nodes are the basis of many network analysis tasks. For example, community detection on social networks clusters nodes based on the structural proximity and attribute proximity \\cite{wu2018mining}. In recommendation on citation networks, papers having strong structural and attribute proximity are most likely to be reference papers of the given manuscript \\cite{cai2018three}. In user alignment across social networks, users are aligned based on their structure and attribute proximity on each network \\cite{zhao2018learning}. These applications benefit from utilizing both structural proximity and attribute proximity, which lead us to vestigate the problem of learning the low-dimensional representations of the network in the condition of preserving the two proximities. The problem is defined as follows.\n\\begin{definition}[Attributed Network Embedding]\n  Given an attributed network denoted as $G=(U,E,A)$ with $n$ nodes and $m$ attributes, attributed network embedding aims to learn a mapping function $f:(\\bf{s}_i,\\bf{a}_i)\\mapsto \\bf{y}_i\\in \\mathbb{R}^d$, where $d \\ll \\min (n,m)$. The objective of the function is to make the similarity between ${{\\bf{y}}_{i}}$ and ${{\\bf{y}}_{j}}$ explicitly preserve the attribute proximity and structural proximity of ${u_i}$ and ${u_j}$.\n\\end{definition}\n\n\\subsection{Framework}\n\\label{sec:3.2}\nIn order to address the attributed network embedding problem, a Multimodal Deep Network Embedding (MDNE) method is proposed. Figure 1 shows the MDNE framework. The parameters marked with $\\hat{}$ are parameters of the reconstruction component. Table 1 lists the  terms and notations. Note that the attributes pre-processing layer and structures pre-processing layer have different weight matrices ${W^{{a^{(1)}}}}$ and ${W^{{s^{(1)}}}}$, respectively. For simplicity, we denote ${W^{{a^{(1)}}}}$ and ${W^{{s^{(1)}}}}$ as ${W^{(1)}}$.\n\nThe strong interactions and complex dependencies between nodes in real-world networks result in the high non-linearity of the network structures. The interactions between structure and attribute features are non-linear as well. Deep neural networks have demonstrably strong representation and generalization abilities for such non-linear relationships \\cite{he2016deep}. Therefore, the proposed model is established based on a deep autoencoder, one of the most common deep neural network architectures. Autoencoder is an unsupervised learning model that performs well in data dimensionality reduction and feature extraction \\cite{hinton2006reducing}. An autoencoder consists of two parts, the encoder and decoder. The encoder consists of one or multiple layers of non-linear functions that map the input data into the representation space and obtain its feature vector; the decoder reconstructs the data in the representation space to obtain its original input form by an inverse process. A shallow autoencoder has three layers (input, encoding and output), where the encoder has only one layer of non-linear functions. The deep autoencoder of our implementation has more hidden layer and is able to learn higher-order features of data. Given the input data vector $\\bf{x}_i$, the output feature vectors for each layer are\n\\begin{equation*}\n\\begin{array}{l}\n{\\bf{y}_i}^{(1)} = \\sigma \\left( {{W^{(1)}}{\\bf{x}_i} + {\\bf{b}^{(1)}}} \\right)\\\\\n{\\bf{y}_i}^{\\left( k \\right)} = \\sigma \\left( {{W^{(k)}}{\\bf{y}_i}^{(k - 1)} + {\\bf{b}^{(k)}}} \\right),k = 2,\\dots,K\n\\end{array}\n\\end{equation*}\n, where $\\sigma$ denotes the non-linear activation function for bringing the non-linearity into the models. The activation functions must be chosen according to the loss function \\cite{charte2018practical}, the requirements of the applied representations, and the datasets. In practice, we can choose them based on their test performance. In this work, the sigmoid function $\\sigma (x) = \\frac{1}{{1 + \\exp ( - x)}}$ is adopted as it provided the best performance in the experiments\\footnote{Regarding the choice of activation function, we have tried sigmoid, Rectified Linear Unit (ReLU), Scaled Exponential Linear Unit (SELU), and hyperbolic tangent function (tanh). Empirically, the sigmoid function leads to the best performance in general.}. After obtaining the mid-layer representation, i.e., the encoding result ${\\bf{y}_i}^{(K)}$, we can obtain the decoding result through an inverse calculation process. The autoencoder optimizes the parameters by minimizing the reconstruction error between the input data and the reconstructed data. A typical loss function is the mean squared error (MSE)\n\\begin{equation*}\n{\\cal L} = \\sum\\limits_{i = 1}^n {\\left\\| {{{\\bf{\\hat x}}_i} - {\\bf{x}_i}} \\right\\|_2^2}\n\\end{equation*}\n. To alleviate the noise and redundant information in the input feature vectors, an undercomplete autoencoder is adopted to learn compact low-dimensional representations. The undercomplete autoencoder has a tower structure, with each upper layer having a smaller number of neurons than the layer below it. A smaller number of neurons restricts the dimensionality of the learned features, so that the autoencoder is forced to learn more abstract features of data during training \\cite{charte2018practical}. A layer-by-layer pre-training algorithm, such as Restricted Boltzmann Machine (RBM) enables each upper layer of the encoder to capture the high-order correlations between the feature units in the lower layer, which is an efficient way to extract non-linear structures  progressively \\cite{hinton2006reducing}. Thus, the tower structure with stacked multiple layers of non-linear functions is able to map the data into a compressive latent space, and capture the highly non-linear structures of the network, as along with the complex interactions between the structures and attributes during training. The basic undercomplete autoencoder is chosen in our framework because of its generality and simplicity. Variants of the autoencoder can replace the basic autoencoder with slight modifications to accommodate specific scenarios, such as denoising autoencoder, contractive autoencoder, etc. \\cite{charte2018practical}.\n\\begin{figure}[!t]\n\\centering\n\\includegraphics{Fig1}\n\\caption{The framework of MDNE}\n\\label{F1}\n\\end{figure}\n\\begin{table}[!t]\n\\renewcommand{\\arraystretch}{1.3}\n\\caption{Terms and notations}\n\\label{T1}\n\\centering\n\\begin{tabular}{|c|c|}\n\\hline\nSymbol & Definition   \\\\\n\\hline\n$K$ & Number of layers of the encoder\/decoder \\\\\n\\hline\n${W^{(k)}},{\\hat W^{(k)}}$ & Weight matrix of the ${k^{th}}$ layer \\\\\n\\hline\n${{\\bf{b}}^{(k)}},{{\\bf{\\hat b}}^{(k)}}$ & Biases of the ${k^{th}}$ layer \\\\\n\\hline\n${Y^{(k)}} = \\{ {{\\bf{y}}_i}^{(k)}\\} _{i = 1}^n$ & Representations of the ${k^{th}}$ layer \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nAn intuitive way to integrate both structure and attribute information in the representations is to concatenate the two feature vectors separately learned from both modalities. The way of learning individual modalities separately is limited in its ability to extract the correlations between structures and attributes. Alternatively, two kinds of information can be concatenated first at the input and the integrated representations are learned by a unified model. The inputs of the unified model are the adjacency vectors describing network structure and the attribute vectors describing node attributes. Since the adjacency vectors and attribute vectors of nodes are sparse and noisy, inputting the concatenated adjacency vector and attribute vector to the deep autoencoder directly, as shown in Figure 2(a), increases the difficulty in training the model to capture the correlations between structure and attribute information. We have also found that, in practice, learning in this way results in hidden units have strong connections of either structure or attribute variables, but few units connect across the two modalities \\cite{ngiam2011multimodal}.\n\nTo enable the deep model to better capture the correlations between structure and attribute information, multimodal learning method is introduced into the proposed model. The autoencoder-based multimodal learning model \\cite{ngiam2011multimodal} is adopted to pre-process the original structure and attribute data. The pre-processing reduces the dimensionality of data from different modalities, specifically removing noise and redundant information to obtain compact high-order features. The correlations across modalities are strengthened between their high-order features. As shown in Figure 2(b), the structure information (adjacency vector) and attribute information (attribute vector) are input separately to a one-layer neural network serving as a pre-processing layer. The use of a pre-training algorithm such as a single-layer RBM enables the pre-processing layer to extract high-order features of each modality. Then, the structure and attribute feature vectors are concatenated and input to the deep autoencoder for further learning. The high-order correlations between structure and attribute will be more facilely learned by deep autoencoder using high-order features obtained by the pre-processing layer. With the subsequent fine-tuning algorithm, the deep autoencoder provides a unified framework to integrate structure and attribute information.\n\nThe training goal of the model is preserving the structural and attribute features in the embedding space. The structural features and attribute features are captured by the structural and attribute proximities, respectively. Thus, the model loss function is defined based on the two proximities, as detailed in the next subsection. By fine-tuning the model based on the optimization of the loss function, the obtained representations preserve both the structure and attribute features of the original network.\n\\begin{figure}[!t]\n\\centering\n\\includegraphics{Fig2}\n\\caption{Concatenated model and Pre-processing model}\n\\label{F2}\n\\end{figure}\n\nIn comparison with SDNE \\cite{wang2016structural}, which adopts a basic autoencoder to directly reconstruct the input structure information data, the proposed MDNE pre-processes the original adjacency matrix and attribute matrix using a multimodal learning method respectively, and concatenates the resulting high-order structure and attribute features for input to the deep model. The loss function is defined based on the structural and attribute proximities to preserve the structural and attribute features of nodes in the embedding space.\n\n\\subsection{Loss Functions}\n\\label{sec:3.3}\nThe structural proximity includes the first-order proximity describing the local network structure and the second-order proximity describing the global network structure \\cite{wang2016structural}. They are preserved in the loss function to preserve the local and global structural features in low-dimensional embedding space. With the first-order proximity indicating the proximity between directly connected nodes, a corresponding loss function is defined to guarantee that connected nodes with larger weight have a shorter distance in the embedding space, i.e., \n\\begin{equation}\\label{E1}\n{{\\cal L}_{1st}} = \\sum\\limits_{i,j = 1}^n {{s_{ij}}\\left\\| {{\\bf{y}}_i^{(K)} - {\\bf{y}}_j^{(K)}} \\right\\|_2^2}\n\\end{equation}\n. Minimizing ${{\\cal L}_{1st}}$ forces the model to preserve the first-order proximity in the embedding space. The second-order proximity represents the similarity of the neighborhood structure between nodes. The neighborhood structure of each node can be described by its adjacency vector. Thus the second-order proximity between two nodes is determined by the similarity of their adjacency vectors, and the goal of the corresponding loss function is to guarantee that nodes with similar adjacency vectors have a short distance in the embedding space. Minimizing the reconstruction error of the input data amounts to maximizing the mutual information between input data and learnt representations \\cite{vincent2010stacked}. Intuitively, if the representation allows a good reconstruction of the input data, it means that it has retained much of the information that was present in the input. That is, the MSE-based loss function prompts the basic autoencoder to latently preserve the similarity between input vectors in the embedding space during training. Since the adjacency vector describes the neighborhood structure of each node, minimizing of the reconstruction error of the adjacency vectors preserves the similarity of neighborhood structure (i.e., the second-order proximity) between nodes in the embedding space. Thus, the  loss function based on the second-order proximity is as follows:\n\\begin{equation}\\label{E2}\n{{\\cal L}_{2nd}} = \\sum\\limits_{i = 1}^n {\\left\\| {\\left( {{{\\bf{\\hat s}}_i} - {\\bf{s}_i}} \\right) \\odot \\bf{r}_i^s} \\right\\|_2^2 = \\left\\| {\\left( {\\hat S - S} \\right) \\odot {R^s}} \\right\\|_F^2} \n\\end{equation}\n, where $ \\odot $ means the Hadamard product, and ${R^s} = \\left\\{ {\\bf{r}_i^s} \\right\\}_{i = 1}^n$ are the penalty parameters for non-zero adjacency elements. If ${s_{ij}} = 0$, $r_{ij}^s = 1$, else $r_{ij}^s = {\\gamma _1} > 1$. Increasing the penalty for the reconstruction error of non-zero elements avoids the reconstruction process's tendency to reconstruct zero elements, making the model robust to sparse networks. Minimizing ${{\\cal L}_{1st}}$ and ${{\\cal L}_{2nd}}$ imposes a restriction to force the model to preserve the first-order and second-order proximities between nodes.\n\nThe attribute proximity of nodes is determined by the similarity of their attribute vectors. The similarity metric of attribute vectors depends on whether the attributes are symmetric or asymmetric. In real-world networks, most of the attributes are highly asymmetric, such as word-counts on citation networks. Moreover, symmetric attributes can also be transformed into asymmetric ones by regarding each ${a_{ij}}$ in node $i$'s attribute vector ${{\\bf{a}}_{\\bf{i}}}$ as an asymmetric attribute indicating whether node $i$ has attribute value $j$. Therefore, the attribute vectors are treated as highly asymmetric to match real-world circumstances. The asymmetry of both attribute vectors and adjacency vectors results in the same similarity metric of the two data forms. Training the autoencoder to minimize reconstruction error enables the model to preserve the similarity between input vectors in the embedding space \\cite{vincent2010stacked}. Meanwhile, experiments in \\cite{belkin2003laplacian} shows that minimizing the reconstruction error of the word-count vectors, a kind of highly asymmetric attribute vectors, with a deep autoencoder makes the similar input word-count vectors close to each other in the embedding space. Thus, to preserve the attribute proximity between nodes in the embedding space, the autoencoder is trained to minimize the reconstruction error of the attribute vectors. The corresponding loss function is\n\\begin{equation}\\label{E3}\n{{\\cal L}_{att}} = \\sum\\limits_{i = 1}^n {\\left\\| {\\left( {{{\\bf{\\hat a}}_i} - {\\bf{a}_i}} \\right) \\odot \\bf{r}_i^a} \\right\\|_2^2 = \\left\\| {\\left( {\\hat A - A} \\right) \\odot {R^a}} \\right\\|_F^2} \n\\end{equation}\n, where ${R^a} = \\left\\{ {\\bf{r}_i^a} \\right\\}_{i = 1}^n$ are the penalty parameters for non-zero attribute elements. If ${a_{ij}} = 0$, $r_{ij}^a = 1$, and $r_{ij}^a = {\\gamma _2} > 1$ otherwise. The penalty for the reconstruction error of non-zero attribute values reflects that the reconstruction of non-zero elements is more meaningful than the reconstruction of zero ones. This is because there are significantly fewer non-zero elements than zero ones in highly asymmetrical attribute vectors, with non-zero elements much more important in determining the similarity.\n\nThe final loss function combines the above structural and attribute proximity loss functions and preserves the structural and attribute proximities between nodes in the embedding space:\n\\begin{equation}\\label{E4}\n\\begin{array}{rl}\n{{\\cal L}_{mix}} = &\\lambda {{\\cal L}_{att}} + \\alpha {{\\cal L}_{2nd}} + {{\\cal L}_{1st}} + \\upsilon {{\\cal L}_{reg}}\\\\\n = &\\!\\lambda \\left\\| {(\\hat A - A) \\odot {R^a}} \\right\\|_F^2 + \\alpha \\left\\| {(\\hat S - S) \\odot {R^s}} \\right\\|_F^2\\\\\n &\\!+ \\sum\\limits_{i,j = 1}^n {{s_{ij}}\\left\\| {{\\bf{y}}_i^{(K)} - {\\bf{y}}_j^{(K)}} \\right\\|_2^2}  + \\upsilon {{\\cal L}_{reg}}\n\\end{array}\n\\end{equation}\n, where ${{\\cal L}_{reg}}$ is an ${\\cal L}2$-norm regularization term to prevent overfitting, and $\\lambda $, $\\alpha $, and $\\upsilon $ are the weight of the attribute proximity loss, second-order proximity loss and regularization term in the loss function. ${{\\cal L}_{reg}}$ is defined as:\n\\begin{equation*}\n{{\\cal L}_{reg}} = \\frac{1}{2}\\sum\\limits_{k = 1}^K {(\\left\\| {{W^{(k)}}} \\right\\|_F^2 + \\left\\| {{{\\hat W}^{(k)}}} \\right\\|_F^2)}\n\\end{equation*}\n, where ${W^{(k)}},{\\hat W^{(k)}},k = 1,\\dots,K$ are the weight matrices of the ${k^{th}}$ layer of the encoder and decoder, respectively.\n\n\\subsection{Optimization}\n\\label{sec:3.4}\nAs presented so far, we seek to minimize the loss function to preserve the structural proximity and attribute proximity in the embedding space. Stochastic gradient descent is a general way to optimize the deep model. However, it is difficult to obtain the optimal result of the model when using stochastic gradient descent directly over randomized weights due to the existence of many local optima \\cite{hinton2006reducing}. Otherwise, the gradient descent works well when the initial weights are close to a good solution. Therefore, Deep Belief Network \\cite{hinton2006fast} is adopted to pre-train the model and obtain the initial weights, which have been proved to be close to the optimal weights \\cite{erhan2010does}. Then, the model is optimized using stochastic gradient descent and the initial weights.\n\nBy iterating and updating the parameters until model converges, we obtain the optimal model. Experimental results show that the model optimization converges quickly after the first 10 iterations, and slowly approaches the optimum in the later iterations. Approximately 400 iterations produce the satisfactory results. After proper optimization, informative representations are learned based on the trained model. Algorithm 1 presents the pseudo-code of the proposed method. All the parameters ${W^{(k)}},{\\hat W^{(k)}},{\\bf{b}^{(k)}},{\\bf{\\hat b}^{(k)}}$ are signed as $\\theta $.\n\\renewcommand{\\algorithmicrequire}{\\textbf{Input:}}\n\\renewcommand{\\algorithmicensure}{\\textbf{Output:}}\n\\begin{algorithm}\n\\caption{MDNE}\n\\label{A1}\n\\begin{algorithmic}[1]\n\\REQUIRE the adjacency matrix $S$, the attribute matrix $A$.\n\\ENSURE network representation ${Y^{(K)}}$, updated parameters $\\theta $.\n \\STATE Build pre-processing layer($PPL$), encoder($EC$) and decoder($DC$), pre-train them through Deep Belief Network to obtain the initialized parameters $\\theta $;\n \\REPEAT\n \\STATE ${Y^{(K)}} = EC(PPL(\\left[ {S\\left. A \\right]} \\right.),\\theta )$, $\\left[ {\\hat S\\left. {\\hat A} \\right]} \\right. = DC({Y^{(K)}},\\theta )$;\n \\STATE Obtain ${{\\cal L}_{mix}}$ based on Eq. (4);\n \\STATE Updated parameters $\\theta $ through back-propagate algorithm;\n \\UNTIL {converge}\n \\STATE Obtain the network representations ${Y^{(K)}}$ based on the optimal parameters $\\theta $.\n \\end{algorithmic}\n\\end{algorithm}\n\n\\subsection{Analysis and Discussions}\n\\label{sec:3.5}\nIn this section, we discuss and analyze the proposed model of MDNE.\n\n\\textbf{Time Complexity:} \nThe time complexity of MDNE is $O((l + f)hr)$, where $l$ is the number of edges, $f$ is the total number of the attributes carried by all the nodes, $h$ is the maximum number of dimensions of the hidden layer, and $r$ is the number of iterations.\nSince $h$ and $r$ are independent of the other parameters, the overall training complexity of the model is linear to the sum of the number of edges and attributes carried by all the nodes.\n\n\\textbf{New nodes:} A practical issue for network embedding is how to capture evolving networks. Many researches \\cite{huang2017accelerated,Liao2017Attributed} have shown interest in dealing with dynamic topological structures and node attributes. Since newly arriving nodes are an important factor for evolving networks, the proposed method provides a possible way to represent them. If new nodes have observable links connecting to existing nodes and bringing attribute information as well, their representations can be obtained by feeding their adjacency vectors and attribute vectors into the finely trained model. If the new nodes lack structure or attribute information, most existing methods cannot handle them \\cite{wang2016structural}. However, MDNE can learn the representations of the new nodes lacking one modality of information by replacing the missing vectors with zero vectors and inputting the existing vectors together with the zero vectors to the trained model.\n\n\\section{Experimental Results}\n\\label{sec:4}\nIn this section, we empirically evaluate the effectiveness and generality of the proposed algorithm. First, the experimental setup is introduced, including datasets, baseline methods and parameter settings. We also investigate the convergence of MDNE, and verify the ability of all methods to reconstruct the network structure. Then, the comparisons of the proposed method and baselines are conducted on three real-world network analysis tasks, i.e., link prediction, attribute prediction and classification, to verify the ability of the obtained representations. Finally, the parameter sensitivity and the impact of pre-processing are discussed. Experiments run on a Dell Precision Tower 5810 with an Intel Xeon CPU E5-1620 v3 at 3.50 GHz and 16 GB of RAM.\n\n\\subsection{Experiment Setup}\n\\label{sec:4.1}\n\\subsubsection{Datasets}\nFour real-world network datasets are used in this work, including two citation networks and two social networks. Considering the characteristics of these datasets, one or more datasets are chosen to evaluate the performances on each network analysis task. Four datasets are described as follows.\n\n\\textbf{cora:} cora\\footnote{http:\/\/linqs.cs.umd.edu\/projects\/\/projects\/lbc\/index.html} is a citation network which contains 2,708 nodes and 5,278 edges. Each node indicates a machine learning paper, and the edge indicates the citation relation between papers. After stemming and removing stop-words, a vocabulary of 1433 unique words is regarded as the attribute information of papers. Each attribute indicates the absence\/presence of the corresponding word in papers. These papers are classified into one of the following seven classes: Case Based, Genetic Algorithms, Neural Networks, Probabilistic Methods, Reinforcement Learning, Rule Learning, and Theory.\n\n\\textbf{citesee:} citeseer is a citation network which consists of 3,312 nodes and 4,551 edges. Similarly, nodes and edges represent scientific publications and their citations, respectively. The vocabulary of size 3,703 words is extracted and set as the attributes. These papers are classified into one of the following six classes: Agents, AI, DB, IR, ML, HCI.\n\n\\textbf{UNC, Oklahoma:} They are two Facebook sub-networks, which respectively contains 18,163 students from the University of North Carolina and 17,425 students from University of Oklahoma, and also with their seven anonymized attributes: status, gender, major, second major, dorm\/house, high school, class year. Note that not all of the students have the seven attributes available.\n\\begin{table}[!t]\n\\renewcommand{\\arraystretch}{1.3}\n\\caption{Dataset statistics}\n\\label{T2}\n\\centering\n\\begin{tabular}{|c|c|c|c|}\n\\hline\nDataset & \\# nodes & \\# edges & \\# attributes \\\\\n\\hline\nUNC &\t 18163 &\t766800 &\t2788 \\\\\n\\hline\nOklahoma &\t17425 &\t892528 &\t2305 \\\\\n\\hline\nciteseer &\t3312 &\t4551 &\t3703 \\\\\n\\hline\ncora &\t 2708 &\t5278 &\t 1433 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nThe statistics of the four datasets are summarized in Table 2. Experiments are conducted on both weighted and unweighted, small and large networks. Diverse datasets allow us to evaluate whether the proposed network embedding method has a better performance on networks with different characteristics.\n\n\\subsubsection{Baseline Methods}\nFive typical methods are chosen to be baselines.\n\n\\textbf{LE \\cite{belkin2002laplacian}:} It provides Laplacian Eigenmaps and spectral techniques to embed the data into a latent low-dimensional space. The solution reflects the features of the network structure.\n\n\\textbf{node2vec \\cite{grover2016node2vec}:} It samples the network structure by the biased random walk. By regarding the paths as sentences, it adopts the natural language processing model to generate network embedding. The hyper-parameters $p$ and $q$ introduce breadth-first sampling and depth-first sampling in the random walk. It can recover DeepWalk when $p$ and $q$ are set to 1.\n\n\\textbf{SDNE \\cite{wang2016structural}:} It exploits the first-order proximity and second-order proximity to preserve the local and global network structure. A deep model is adopted to address the highly non-linear structure and sparsity problem of networks.\n\n\\textbf{AANE \\cite{huang2017accelerated}:} It proposes a scalable and efficient framework which incorporates node attribute proximity into network embedding. It processes each node efficiently by decomposing the complex modeling and optimization into many sub-problems.\n\n\\textbf{ASNE \\cite{Liao2017Attributed}:} It adopts a multilayer neural network to capture the complex interactions between features which denote the ID and attributes of nodes, and the proposed framework performs network embedding by preserving the structural proximity and attribute proximity of nodes in the paths generated by the random walk.\n\nThe first three methods are pure structure-based methods, and the others integrate attribute and structure information into network embedding.\n\n\\subsubsection{Parameter Settings}\nThe depth of neural networks and the number of neurons are essential factors in learning effect. Recent evidences \\cite{wang2016structural,simonyan2014very,szegedy2015going} reveal that the number of stacked layers (depth) and neurons should be neither too large nor too small. Large numbers of layers and neurons increase the difficulty of training the model, and bring over-fitting problem. However, too few layers and neurons fail to extract effective low-dimensional representations \\cite{zeiler2014visualizing}, especially for large-scale datasets. Therefore, we vary MDNE's neural network structure according to different datasets, as shown in Table 3. Two numbers in the first layer and second layer indicate the dimensions of the vectors related to the structure and attribute data, respectively.\n\nWe implemented of MDNE using TensorFlow\\footnote{https:\/\/www.tensorflow.org\/}. We fine-tuned the loss function hyper-parameters $\\lambda ,\\alpha ,\\upsilon ,{\\gamma _1},{\\gamma _2}$ using grid search based on the performance of the network reconstruction \\cite{wang2016structural}, which is introduced as a basic quality criterion of the proposed method in Section 4.3. We first perform a parameter sweep setting $\\lambda ,\\alpha ,\\upsilon ,{\\gamma _1},{\\gamma _2} = \\{0, 0.01, 0.1, 1, 10, 100, 1000\\}$ on each dataset. They are tuned one by one iteratively until all of them are converged. Then every hyper-parameter is further fine-tuned by grid search on a smaller space around optimal value got in previous search for each dataset.\n\\begin{table}[!t]\n\\renewcommand{\\arraystretch}{1.3}\n\\caption{Neural Network Structures}\n\\label{T3}\n\\centering\n\\begin{tabular}{|c|c|}\n\\hline\nDataset & \\# nodes in each layer \\\\\n\\hline\ncora &\t (2708,1433)-(300,200)-128 \\\\\n\\hline\nciteseer &\t(3312,3703)-(250,250)-128 \\\\\n\\hline\nUNC &\t(18163,2788)-(3000,500)-128 \\\\\n\\hline\nOklahoma &\t(17425,2305)-(3600,650)-128 \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nThe parameters of the baseline methods are adjusted to the optimal values as given in their researches. For the sake of fairness, we set the embedding dimensions of all the methods $d=128$ on different tasks.\n\n\\subsection{Convergence}\nExperiments are conducted to investigate the convergence property of MDNE. We vary the number of iterations from 0 to 800 and plot the corresponding value of loss function on a citation network cora and a social network UNC. The learning curves are shown in Figure 3. The result indicates that MDNE convergences at about 400 iterations on different datasets. Although the performance may be better with more iterations, 400 iterations have achieved the best result among baselines. To balance the effectiveness and efficiency of MDNE, the model is trained about 400 iterations in experiments.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics{Fig3}\n\\caption{Convergence of MDNE on cora and UNC datasets.}\n\\label{F3}\n\\end{figure}\n\n\\subsection{Network Reconstruction}\nNetwork reconstruction verifies the ability of the method to reconstruct the network structure, which is also a basic requirement for network embedding methods. Given the learned network representations, all links in the original network need to be predicted. The way to predict the links is ranking all node pairs based on their similarity and predicting that a certain number of top pairs are linked by edges. The cosine distance of learned vectors measures the similarities between nodes. The higher-ranking node pairs are more likely to have links in the original network. The evaluation indicator is $precision@k$ \\cite{wang2016structural}, referring to the ratio of the top $k$ node pairs to be connected in the original network. A larger $precision@k$ indicates the better performance of the reconstruction.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics{Fig4}\n\\caption{Network Reconstruction performance on different datasets. Node2vec can't obtain results on UNC and Oklahoma which have more than 10,000 nodes, due to an out of memory problem. $k$ is set based on the network scale.}\n\\label{F4}\n\\end{figure}\n\n\\begin{table}[!t]\n\\renewcommand{\\arraystretch}{1.3}\n\\caption{$precision@k$ of Network Reconstruction on cora and citeseer datasets}\n\\label{T4}\n\\centering\n\\tiny\n\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\\hline\n     &Algorithm & $P$@1000 & $P$@3000 & $P$@5000 & $P$@7000 & $P$@9000 & $P$@10000 \\\\\n\\hline\n               & LE &  0.661          & 0.481          & 0.408           & 0.353           & 0.316           & 0.300 \\\\\n  & node2vec & \\textbf{1.000} & \\textbf{0.903}& 0.542           & 0.388           & 0.302           & 0.272 \\\\\ncora  & SDNE &  0.924          & 0.703          & 0.543           & 0.432           & 0.353           & 0.323 \\\\\n       & AANE &  0.792          & 0.465          & 0.318           & 0.239           & 0.194           & 0.179 \\\\\n       & ASNE  &  0.954          & 0.796          & 0.514           & 0.383           & 0.307           & 0.281 \\\\\n      & MDNE & 0.996           & 0.871          & \\textbf{0.701} & \\textbf{0.581} & \\textbf{0.491} & \\textbf{0.455} \\\\\n\\hline\n              & LE &  0.480          & 0.376          & 0.334           & 0.307           & 0.280           & 0.269 \\\\\n  & node2vec & \\textbf{1.000}& \\textbf{1.000} & 0.654           & 0.467           & 0.364           & 0.327 \\\\\nciteseer  & SDNE &  0.869  & 0.787          & 0.658           & 0.530           & 0.430           & 0.390 \\\\\n       & AANE &  0.774          & 0.586          & 0.424           & 0.323           & 0.262           & 0.239 \\\\\n       & ASNE  &  0.962          & 0.908          & 0.713           & 0.543           & 0.438           & 0.400 \\\\\n      & MDNE & 0.994           & 0.951          & \\textbf{0.798} & \\textbf{0.637} & \\textbf{0.530} & \\textbf{0.488} \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table}[!t]\n\\renewcommand{\\arraystretch}{1.3}\n\\caption{$precision@k$ of Network Reconstruction on UNC and Oklahoma datasets}\n\\label{T5}\n\\centering\n\\tiny\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n\\hline\n     &Algorithm & $P$@5000 & $P$@10000 & $P$@15000 & $P$@20000 & $P$@25000 \\\\\n\\hline\n               & LE &  0.942          & 0.915            & 0.901           & 0.894           & 0.885  \\\\\nUNC & SDNE &  0.997          & 0.988          & 0.968           & 0.943           & 0.915  \\\\\n       & AANE &  0.012          & 0.010          & 0.008           & 0.008           & 0.008  \\\\\n       & ASNE  &  \\textbf{0.999} & 0.989          & 0.922           & 0.765           & 0.635  \\\\\n      & MDNE & 0.998 & \\textbf{0.998} & \\textbf{0.997} & \\textbf{0.982} & \\textbf{0.963} \\\\\n\\hline\n              & LE &  0.952          & 0.938          & 0.925           & 0.916           & 0.907   \\\\\nOklahoma & SDNE &  0.998  & 0.986          & 0.981           & 0.978           & 0.976  \\\\\n       & AANE &  0.022          & 0.018          & 0.015           & 0.014           & 0.013    \\\\\n       & ASNE  &  0.995          & 0.974          & 0.943           & 0.914           & 0.888    \\\\\n      & MDNE & \\textbf{0.999}& \\textbf{0.996} & \\textbf{0.993} & \\textbf{0.983} & \\textbf{0.969} \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nNetwork reconstruction has been performed on the four datasets, the results of which are shown in Figure 4. Also, Table 4 and Table 5 provide the numeric results helping to compare the close curves. Numbers in bold represent the best result in each column. Compared to UNC and Oklahoma, the performance of all the methods visibly decrease on cora and citeseer. This is because cora and citeseer have sparsity problem, as their average degree is much smaller than that of UNC and Oklahoma.\n\nLE, which is a shallow model-based method, has poor performance. It indicates that going deep enhance the model's generalization ability, and helps to capture the high non-linearity of network structures. SDNE adopts deep autoencoder model but only uses structure information. Its inferior performance demonstrates the usefulness of attribute information in learning better node representations. Node2vec is slightly better than MDNE on cora and citeseer network when $k=1000$ to $k=3000$. The reason might be that node2vec can capture the higher-order proximity between nodes by random walks in the network.\n\nAANE has relatively poor performance, especially on UNC and Oklahoma. This is because AANE only considers the first-order proximity, and its performance largely depends on the computation of attribute similarity under full attribute space, while attribute similarity of nodes computed under high-dimensional attribute space explicitly on certain networks has little discriminability. ASNE has slightly inferior performance because it is hard to capture the non-linear correlations of structure and attribute information, as it pre-processes structure and attribute data linearly before concatenating them. MDNE has the best performance on four datasets in most cases. The good performance of MDNE is because it adopts a deep model to learn non-linear features, and uses multimodal learning method to better capture the correlations of attribute and structure, and preserves the attribute proximity by minimizing the reconstruction error instead of computing the attribute similarity explicitly.\n\n\\subsection{Link Prediction and Attribute Prediction}\nIn this section, we evaluate the ability of the learned representations to predict missing links and attributes in the network, which is a practical task in real-world applications.\n\n\\subsubsection{Link Prediction}\nLink prediction is the prediction of missing links based on the existing information. After hiding $5\\% \\sim 45\\%$ of links randomly, the left network is utilized as a sub-dataset to perform network embedding. The test set consisted of positive instances and negative instances. The hidden links are taken as positive instances and the same ratio of unconnected node pairs in the original network are randomly selected to be negative instances. Similarities between the learned representations of nodes in the test set are calculated and are sorted in descending order. A higher ranking of a node pair corresponds to a greater possibility for them to be connected. Area Under the ROC Curve (AUC) is adopted as the evaluation metric as it is commonly used to measure the quality of classification based on ranking. A large AUC indicates good performance. If an algorithm ranks all positive instances higher than all negative instances, the AUC is 1. The above steps are repeated 10 times and the average AUC is taken as the final result. All methods had extremely poor performance on the cora and citeseer networks, as the low average degrees of the two networks make link prediction very hard. Thus we only show the results on the UNC and Oklahoma networks in Figure 5 and Table 6. Numbers in bold represent the highest performance in each column.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics{Fig5}\n\\caption{Link prediction performance on UNC and Oklahoma datasets.}\n\\label{F5}\n\\end{figure}\n\n\\begin{table}[!t]\n\\renewcommand{\\arraystretch}{1.3}\n\\caption{AUC of Link Prediction on UNC and Oklahoma datasets}\n\\label{T6}\n\\centering\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n\\hline\n     &Test Ratio & 0.05    & 0.15    & 0.25    & 0.35     & 0.45    \\\\\n\\hline\n               & LE & 0.670 & 0.668  & 0.644  & 0.616   & 0.588   \\\\\nUNC & SDNE & \\textbf{0.915} & 0.902   & 0.896  & 0.880  & 0.873  \\\\\n       & AANE &  0.501  & 0.500  & 0.500  & 0.500   & 0.499  \\\\\n       & ASNE  &  0.711  & 0.683  & 0.653  & 0.629   & 0.605  \\\\\n      & MDNE & 0.912  & \\textbf{0.907} & \\textbf{0.901} & \\textbf{0.900} & \\textbf{0.906} \\\\\n\\hline\n               & LE & 0.682 & 0.685  & 0.685  & 0.670   & 0.637   \\\\\nOklahoma & SDNE &0.891 & 0.885   & 0.882  & 0.873  & 0.852  \\\\\n       & AANE &  0.498  & 0.499  & 0.500  & 0.501   & 0.500  \\\\\n       & ASNE  &  0.899  & 0.889  & 0.879  & 0.868   & 0.855  \\\\\n      & MDNE & 0.937  & \\textbf{0.935} & \\textbf{0.933} & \\textbf{0.931} & \\textbf{0.924} \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table}[!t]\n\\renewcommand{\\arraystretch}{1.3}\n\\caption{$p$-value of Friedman Test on MDNE with baselines for Link Prediction}\n\\label{T7}\n\\centering\n\\begin{tabular}{|c|c|c|}\n\\hline\nBaseline & UNC & Oklahoma \\\\\n\\hline\n LE      & 1.125$e-5$ & 1.125$e-5$  \\\\\n\\hline\nSDNE & 0.0084 & 1.125$e-5$ \\\\\n\\hline\nAANE & 1.125$e-5$ & 1.125$e-5$  \\\\\n\\hline\nASNE &  1.125$e-5$ & 1.125$e-5$  \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nCompared with the shallow model-based methods LE and AANE, the deep model-based methods MDNE, SDNE and ASNE perform significantly better. This is because the deep model can better capture highly non-linear network structures. The reason for the extremely poor performance of AANE is similar to that on network reconstruction task. SDNE and MDNE have good performance since preserve both the first-order and second-order proximities between nodes in the embedding space. MDNE is slightly better than SDNE which does not preserve attribute features in the learned representations. This result justifies the usefulness of attribute information in link prediction.\n\nFriedman test is conducted to better endorse the superiority of MDNE with respect to other methods. The $p$-values are computed based on the ranking for the AUC value of MDNE with each method on different sub-datasets with different test ratios. In Table 7, all the $p$-values are less than 0.05. The results show that the performance of MDNE is significantly different from the compared methods on link prediction task. The $p$-value of MDNE with SDNE on UNC is slightly higher than others because SDNE is slightly better than MDNE when the ratio of links for test is 5\\%.\n\nThe network becomes sparse with the ratio of links for test increasing, and the AUC of MDNE is stable while that of other methods dropped. It indicates that the penalty for non-zero elements in the loss function improves MDNE's performance in dealing with sparse networks. Such an advantage is pivotal for downstream applications since links are often sparse, especially in large-scale real-world networks. Despite the link prediction task's favoring pure structure-based methods, our MDNE outperforms the others. This demonstrates the effectiveness of the learned representations in predicting missing links.\n\n\\subsubsection{Attribute Prediction} \nAttribute prediction refers to predicting unknown attribute values of nodes based on the obtained information. It has enjoyed increasing interest in network analysis tasks. For example, in social network recommendation, predicting attribute features is essential to help users to locate their interested information \\cite{cai2018three}.\n\nIn attribute prediction experiments, $5\\%\\sim45\\%$ of the attribute values (including value 1 and value 0) in the original network are hidden randomly, i.e., $5\\%\\sim45\\%$ of ${a_{jk}}$ in the attribute matrix $A$ are hidden. They are set as the test set. The left attribute information and structure information is trained to learn the representations of nodes. The obtained representations are used to predict the attributes in the test set. Assuming that the attribute $k$ of node $j$ is hidden, here is the way to predict. Similarities between node $j$ with all the other nodes in the embedding space are calculated, denoting as $sim_{ij},i = 1,\\dots,n$. We denote the top 10 nodes with the highest similarity to $j$ as set $N_S$. ${N_p} = \\left\\{ {i|{a_{ik}} = 1,i \\in {N_S}} \\right\\}$, is the set of nodes with the attribute value $k=1$ in the above set. Similarly, ${N_n} = \\left\\{ {i|{a_{ik}} = 0,i \\in {N_S}} \\right\\}$, is the set of the rest of nodes with the attribute value $k=0$ in $N_S$. $p = {{\\sum\\limits_{i \\in {N_p}} {si{m_{ij}}} } \/ {\\sum\\limits_{i \\in {N_n}} {si{m_{ij}}} }}$ is calculated, which indicates the possibility of that the ${a_{jk}}$, i.e., attribute $k$ of node $j$, is 1. All ${a_{jk}}$ in the test set are sorted in descending order of $p$. AUC is the metric to evaluate the $p$ ranking list. A high AUC value indicates the high accuracy of the prediction. The result shows as in Figure 6.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics{Fig6}\n\\caption{Attribute prediction performance on different datasets.}\n\\label{F6}\n\\end{figure}\n\n\\begin{table}[!t]\n\\renewcommand{\\arraystretch}{1.3}\n\\caption{$p$-value of Friedman Test on MDNE with baselines for Attribute Prediction}\n\\label{T8}\n\\centering\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\nBaseline & cora & citeseer & UNC & Oklahoma \\\\\n\\hline\n LE      & 1.125$e-5$ & 1.125$e-5$ & 1.125$e-5$ & 1.125$e-5$  \\\\\n\\hline\nSDNE & 1.125$e-5$ & 1.125$e-5$  & 1.125$e-5$ & 1.125$e-5$\\\\\n\\hline\nAANE & 1.125$e-5$ & 1.125$e-5$ & 1.125$e-5$ & 1.125$e-5$  \\\\\n\\hline\nASNE &  1.125$e-5$ & 1.125$e-5$ &    \/    &   \/    \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nThe performances of LE, node2vec and SDNE are worse than that of MDNE. The reason is their lacking of consideration of preserving the attribute features in the embedding space, which is important for predicting missing attributes of nodes. AANE still has poor performance. It is because the attribute affinity matrix adopted by AANE is calculated based on the full attribute space, which decreases the discriminability of the representations. Compared with ASNE, the superior performance of our MDNE credits the pre-processing of the original attribute and structure information based on multimodal learning method. The high-order features of the attribute vectors and adjacency vectors obtained by the pre-processing layer help the successive layers better extract the high-order correlations between the structure and attribute feature of nodes.\n\nAlso, the Friedman test is conducted on MDNE with others. The $p$-values are listed in Table 8, all of which are less than 0.05. The results show that the performance of MDNE is significantly different from baselines on attribute prediction task.\n\nThe attribute sparsity of different datasets is quite different, as the average number of attributes of each node is 34.3, 31.7, 5.4 and 5.3 on cora, citeseer, UNC and Oklahoma respectively. Moreover, the attributes of each network become sparse with the ratio of the test set increasing. The proposed method has good performance in all cases. This demonstrates that MDNE is effective in attribute prediction tasks and is robust to networks with different extent of attribute sparseness.\n\n\\subsection{Classification}\n\nClassification is one of the important tasks in network analysis. It classifies nodes based on their features. The representations generated are used as features. The widely used classifier LIBLINEAR \\cite{fan2008liblinear} is adopted. A portion of node representations and their labels are taken as the training set, and the rest to be the test set. For a fair comparison, the test ratio varies from $10\\%\\sim90\\%$ by an increment of $10\\%$. F-measure is a commonly adopted metric for binary classification. Micro-F1 \\& Macro-F1 are employed to judge the classification quality. Macro-average is defined as an arithmetic average of F-measure of all the label categories, and Micro-average is the harmonic mean of average precision and average recall. For both metrics, the higher values indicate better performance. For each training ratio, we randomly split the training set and the test set for 10 times and report the average result as Figure 7. The experiment is conducted on citeseer and cora, since they are the only datasets containing class labels for nodes.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics{Fig7}\n\\caption{Classification performance on cora and citeseer datasets.}\n\\label{F7}\n\\end{figure}\n\nMDNE always has the best performance in all cases. Although node2vec has satisfactory performance on network reconstruction task, it returns the disappointing result on classification task. This shows the representations learned by node2vec have task preference. AANE still has poor performance. We replay the classification experiments with the non-linear kernel SVM classifier and 5-fold cross validation. The performance of AANE is improved. This is because AANE is hard to capture the non-linear correlations between structure and attribute features, and the representations learned by AANE are non-linear. The SVM with the non-linear kernel has the classification ability with non-linear representations, which is difficult for the linear classifier LIBLINEAR to deal with. MDNE has good performance on both LIBLINEAR and SVM. Considering LIBLINEAR has advantages in time complexity, it is beneficial that the learned representations are suitable for linear classifiers. The poor performance of ASNE is due to its lacking of non-linear pre-processing of the original structure and attribute information. The non-linear pre-processing of the adjacency vector and attribute vector can help the model to capture the high-order correlations between the two information in the subsequent learning. SDNE and LE are worse than MDNE, as they do not consider attribute information when embedding networks. The significant improvement of MDNE over baselines proves that adopting multimodal deep model and optimizing the loss function defined based on the structural proximity and attribute proximity are able to learn effective representations for classification tasks.\n\n\\subsection{Parameters Sensitivity and the Impact of Pre-Processing}\nIn this section, we investigate how different choices of $\\lambda$ and embedding dimensions, along with the consideration of pre-processing affect the performance of MDNE on the cora dataset. The results of classification tasks with different test ratios are reported. The results from other tasks on other datasets are omitted as they are similar.\n\\begin{figure}[!t]\n\\centering\n\\includegraphics{Fig8}\n\\caption{Classification performance of MDNE on cora dataset with different weight of the attribute proximity loss $\\lambda$.}\n\\end{figure}\n\n\\subsubsection{The weight of the attribute proximity loss $\\lambda$}\nThe hyper-parameter $\\lambda$ adjusts the importance of attribute proximity loss in the loss function. The weight of structural proximity loss is fixed as $\\alpha  = 0.5$. Then the $\\lambda$ will determine the relative importance between the attribute proximity loss and the structural proximity loss.\n\nFigure 8 shows the impact of $\\lambda$ on the range of [0, 0.04] at an interval of 0.005. The slightly improving performance on $\\lambda  = [0, 0.02]$ shows that attribute proximity loss plays an important role in learning network representations. The performance relatively stabilized on $\\lambda  = [0.02, 0.04]$ indicates that the performance of MDNE is not sensitive to values on this range, which means the value of $\\lambda$ is suitable for the proposed model in a wide range in real-world applications. The great difference between $\\lambda$ and the weight of structural proximity  loss is due to the inherent characteristics of dataset cora. The total number of edges is 5278, and the total number of attribute values is 49216. That is, the attribute proximity loss ${{\\cal L}_{att}}$ is much larger than the structural proximity loss. To balance the different effect from them, the smaller weight of ${{\\cal L}_{att}}$ is necessary. Besides, compared with Figure 7, when $\\lambda  = 0$, which means the attribute proximity loss is ignored in loss function, MDNE still outperforms baselines. The observation indicates that the structure of the proposed multimodal deep autoencoder with the pre-training algorithm is able to capture the highly non-linear relationship between structure and attribute features even without attribute proximity loss in the loss function.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics{Fig9}\n\\caption{Classification performance of MDNE on cora dataset with different embedding dimensions.}\n\\label{F9}\n\\end{figure}\n\n\\subsubsection{Embedding dimensions}\nThe effect of the embedding dimensions on classification performance is shown in Figure 9. The performance gets better as the number of dimensions increasing initially. When the number of dimensions is larger than a threshold, the performance becomes stable. The reason is twofold. When the number of dimensions is small, more useful information is incorporated into representations with the number of dimensions increasing and the performance also increase. However, the too large number of dimensions also bring noise and redundant information which weaken the classification ability of the representations. Thus, it is important to select a reasonable embedding dimension. It is observed from Figure 9 that the proposed method is not very sensitive to embedding dimensions when the number of dimensions is larger than 60. Taking into account the accuracy and complexity of nodes, the embedding dimensions of MDNE is set as 128 in our experiments.\n\\begin{figure}[!t]\n\\centering\n\\includegraphics{Fig10}\n\\caption{Classification performance of MDNE on cora dataset with and without preprocessing.}\n\\label{F10}\n\\end{figure}\n\n\\subsubsection{Pre-processing}\nFigure 10 shows the results of MDNE with and without the pre-processing procedure. The model with the pre-processing procedure, which corresponds to Figure 2(b), has the structure of \\{(2708,1433)-(300,200)-128\\}. Except for the pre-processing layer, the subsequent deep model has an input layer with the concatenated high-order features and an output layer. The model without the pre-processing procedure, which corresponds to Figure 2(a), has the structure of \\{(2708,1433)-500-128\\}. The corresponding deep model has an input layer with concatenated original vectors, a hidden layer, and an output layer. The total number of weight parameters in the model without the pre-processing procedure is larger than that in the model with the pre-processing procedure. As Figure 10 shows, although the pre-processing model has smaller computation complexity, its result is slightly better. Moreover, compared with Figure 7, the proposed method without the pre-processing procedure is still better than baselines. It demonstrates that besides the pre-processing procedure, both the deep model and loss function of the proposed method contribute to the good performance of MDNE.\n\n\\section{Conclusion}\nIn this paper, a Multimodal Deep Network Embedding method is proposed for learning informative network representations by integrating the structure and attribute information of nodes. Specifically, the deep model comprising of multiple layers of non-linear functions is adopted to capture the non-linear network structure and the complex interactions with node attributes. In order to better extract the high-order correlations between the topological structures and attributes of nodes, the multimodal learning method is adopted to pre-process the original structure and attribute data. The structural proximity and attribute proximity are utilized to describe the structure and attribute features of the network, respectively. The model loss function is defined based on the two proximities. Minimizing the loss function preserves both proximities in the embedding space. Experiments are conducted on four real-world networks to evaluate the performance of the representations obtained. Compared with baselines, the result demonstrates that MDNE offers superior performance on various real-world applications. In the future, we will consider improving the efficiency of MDNE through a parallel processing framework and expanding the model to learn task-oriented representations combined with the requirements from specific applications.\n\n\n\n\n\n\n\n\n\\section*{Acknowledgments}\nThe authors would like to thank the anonymous referees for their critical appraisals and useful suggestions.\n\n\n\\ifCLASSOPTIONcaptionsoff\n  \\newpage\n\\fi\n\n\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nInclusive and semi-inclusive deep inelastic scatterings (SIDIS) are important tools to understand\nthe structure of nucleon and nucleus governed by the\nQuantum Chromodynamics (QCD) for the strong interaction.\nThe azimuthal asymmetries and their spin and\/or nuclear dependences\nof the SIDIS cross sections are directly related to the parton distribution and polarization inside nucleon\nor nuclei and therefore are the subjects of intense studies both \ntheoretically\\cite{Georgi:1977tv,Cahn:1978se,Berger:1979kz,Liang:1993re,\nMulders:1995dh,Oganesian:1997jq,Chay:1997qy,Nadolsky:2000kz,\nAnselmino:2006rv,Anselmino:2005nn,Liang:2006wp,Gao:2010mj}\nand experimentally\\cite{Aubert:1983cz,Arneodo:1986cf,Adams:1993hs,\nBreitweg:2000qh,Chekanov:2002sz,Airapetian:2001eg,Airapetian:2002mf,Airapetian:2004tw,\nAlexakhin:2005iw,Webb:2005cd,:2008rv,Alekseev:2010dm}.\nThey provide us with a glimpse into the dynamics of strong interaction within nucleons or nuclei \nand a baseline for the study of parton dynamics in other extreme conditions at high temperature and\nbaryon density.\n\n\nIn the unpolarized SIDIS experiments, the azimuthal angle $\\phi$  of the final hadrons is defined with respect to \nthe leptonic plane and is directly related to the transverse momentum of the hadron from either parton \nfragmentation or the initial and final state interaction of the parton before hadronization.\nIn this paper we will restrict our study to SIDIS $e^-+N (A) \\to e^-+q+X$ of quark jet production so that we\ndon't need to deal with the azimuthal asymmetry resulting from parton fragmentation \nand have no need to consider Boer-Mulders effect \\cite{Boer:1997nt}. \nWe instead focus primarily on the effect of initial and final state interaction.\nIn the large transverse momentum region, the azimuthal asymmetries arise predominately\nfrom hard gluon bremsstrahlung that can be calculated using perturbative QCD (pQCD) \\cite{Georgi:1977tv},\nand are clearly observed in experiments\\cite{Aubert:1983cz,Arneodo:1986cf,Adams:1993hs,Breitweg:2000qh,Chekanov:2002sz}.\nOn the other hand, in the small transverse momentum region  $p_{h\\perp}\\sim k_\\perp \\le 1$GeV\/c,\nthe asymmetry was shown\\cite{Cahn:1978se} to arise mainly from the intrinsic\ntransverse momentum of quarks in nucleon and is\na higher twist effect proportional to $k_\\perp\/Q$ for $\\langle\\cos\\phi\\rangle$\nand to $k_\\perp^2\/Q^2$ for $\\langle\\cos 2\\phi\\rangle$.\n(Here, $p_{h\\perp}$ denotes the transverse momentum of the hadron produced,\n$k_\\perp$ is the intrinsic transverse momentum of quark in nucleon,\n$Q^2=-q^2$ and $q$ is the four-momentum transfer from the lepton).\nThe calculations in \\cite{Cahn:1978se} are based on a generalization\nof the naive parton model to include intrinsic transverse momentum.\nTo go beyond the naive parton model, one has to consider multiple soft\ngluon interaction between the struck quark and the remanent of the target nucleon or nucleus.\nInclusion of such soft gluon interaction ensures the gauge invariance of the final\nresults and relate the azimuthal asymmetry to the transverse momentum\ndependent (TMD) parton matrix elements of the nucleon or nucleus.\n\nWithin the framework of TMD parton distributions and correlations, the intrinsic\ntransverse momentum of partons arises naturally from multiple soft\ngluon interaction inside the nucleon or nucleus. The TMD parton distributions and correlations\ncan be in fact expressed in terms of the expectation values of matrix elements\nrelated to the accumulated total transverse momentum as a result of the\ncolor Lorentz force enforced upon the parton through soft gluon exchange \\cite{Liang:2008vz}.\nThese soft gluon interactions are responsible for the single-spin asymmetries\nobserved in SIDIS,  $pp$ and $\\bar pp$ collisions. They also lead to the \ntransverse momentum broadening \\cite{Liang:2008vz} of hadron production in \ndeep-inelastic lepton-nucleus scattering\\cite{VanHaarlem:2007kj,VanHaarlem:2009zz,Domdey:2008aq}\nas well as the jet quenching observed at the Relativistic Heavy Ion Collider (RHIC) \\cite{Gyulassy:1993hr,Baier:1996sk,\nWiedemann:2000za,Gyulassy:2000er,Guo:2000nz,Wang:2001ifa}.  Such transverse\nmomentum broadening inside nucleus is directly related to the gluon \nsaturation scale \\cite{Liang:2008vz,McLerran:1993ka} and can be studied directly through the nuclear \ndependence of the azimuthal asymmetry in SIDIS.\n\n\nHigher twist contributions in inclusive DIS have been studied systematically using the\ncollinear expansion technique \\cite{Ellis:1982wd,Qiu:1988dn,Qiu:1990xxa} which not only provides \na useful tool to study the higher twist contributions but also is a necessary procedure to ensure\ngauge invariance of the parton distribution and\/or correlation functions.\nIn Ref.\\cite{Liang:2006wp}, such collinear expansion is extended to the SIDIS process $e^-+N\\to e^-+q+X$\nand calculation of the TMD differential cross section and the azimuthal asymmetries up to twist-3.\nTaking into account of multiple gluon scattering, the study found the azimuthal  asymmetry $\\langle \\cos\\phi\\rangle$\nproportional to a twist-3 TMD parton correlation \nfunction $f_{q\\perp}(x,k_\\perp)$ defined as,\n\\begin{eqnarray}\nf_{q\\perp}^N(x,k_\\perp) &&\n=\\int \\frac{p^+dy^- d^2y_{\\perp}}{(2\\pi)^3}\ne^{ix p^+ y^- -i\\vec{k}_{\\perp}\\cdot \\vec{y}_{\\perp}} \\nonumber \\\\\n&& \\times\\langle N|\\bar{\\psi}(0)\\frac{\\slash{\\hspace{-5pt}k_\\perp}}{2k_\\perp^2}{\\cal{L}}(0;y)\\psi(y)|N\\rangle,\n\\label{eq:fqperp}\n\\end{eqnarray}\nwhere ${\\cal{L}}(0;y)$ is the gauge link,\n\\begin{eqnarray}\n\\label{CollinearGL}\n&&{\\cal{L}}(0;y)=\\mathcal{L}^\\dag_{\\parallel}(\\infty,\\vec{0}_\\perp; 0,\\vec{0})\n\\mathcal{L}_\\perp(\\infty,\\vec 0_\\perp;\\infty,\\vec{y}_{\\perp}) \\nonumber\\\\\n&&\\phantom{{\\cal{L}}(0;y)=}\n\\mathcal{L}_{\\parallel}(\\infty,\\vec{y}_{\\perp}; y^-,\\vec{y}_{\\perp},\\vec{y}_\\perp),\\nonumber\\\\\n\\label{TMDGL}\n&&{\\cal{L}}_{\\parallel}(\\infty,\\vec{y}_{\\perp}; y^-,\\vec y_\\perp)\\equiv \nP e^{- i g \\int_{y^-}^\\infty d \\xi^{-} A^+ ( \\xi^-, \\vec{y}_{\\perp})} , \\nonumber \\\\\n&&\\mathcal{L}_\\perp(\\infty,\\vec 0_\\perp;\\infty,\\vec{y}_{\\perp}) \\equiv  \nPe^{-ig\\int_{\\vec 0_\\perp}^{\\vec y_\\perp} d\\vec\\xi_\\perp\\cdot\n\\vec A_\\perp(\\infty,\\vec\\xi_\\perp)},\n\\end{eqnarray}\nfrom the resummation of multiple soft gluon interaction that ensures the gauge invariance of the\ntwist-3 parton correlation function in Eq.~(\\ref{eq:fqperp}) under any gauge transformation.\nThe asymmetry obtained within this generalized collinear expansion method  reduces to that in the\nnaive parton model \\cite{Cahn:1978se} if and only if one neglects the soft gluon interaction as contained\nin the gauge link or equivalently by setting the strong coupling constant $g=0$ in the final result.\nMeasurements of $\\langle \\cos\\phi\\rangle$ in $e^-+N\\to e^-+q+X$ and its $k_\\perp$-dependence\ntherefore provide an unique determination of this new parton correlation function in Eq.~(\\ref{eq:fqperp}).\nFurthermore, the nuclear dependence of the asymmetry \\cite{Liang:2008vz} from multiple soft gluon\ninteraction within the target nucleus can probe the transverse momentum broadening or the jet\nquenching parameter in cold nuclear matter \\cite{Gao:2010mj} which also determines the gluon\nsaturation scale in cold nuclei.\n\nIn this paper, we present a complete calculation of the hadronic tensor and the differential cross section\nfor $e^-+N\\to e^-+q+X$ up to twist-4. We study in particular the azimuthal asymmetry  $\\langle\\cos2\\phi\\rangle$\nin terms of the corresponding TMD quark correlation functions and its nuclear dependence.\nFor completeness, in Sec. II, we present the formulae for calculating the hadronic tensor and\ndifferential cross sections within the framework of generalized collinear expansion.\nIn Sec. III,  we present the cross section and discuss azimuthal asymmetry  $\\langle\\cos2\\phi\\rangle$\nincluding its nuclear dependence with a Gaussian ansatz for the TMD correlation functions. \nA summary is given in Sec. IV.\n\n\\section{Hadronic tensor $W_{\\mu\\nu}$ in $e^-+N\\to e^-+q+X$ up to twist-4}\n\nWe consider the SIDIS process $e^-+N\\to e^-+q+X$ with unpolarized beam and target.\nThe differential cross section is given by,\n\\begin{equation}\nd\\sigma=\\frac{\\alpha_{em}^2e_q^2}{sQ^4}L^{\\mu\\nu}(l,l')\\frac{d^2W_{\\mu\\nu}}{d^2k'_\\perp}\n\\frac{d^3l'd^2k'_\\perp}{ 2E_{l'}},\n\\end{equation}\nwhere $l$ and $l'$ are respectively the four-momenta of the incoming and outgoing leptons,\n$p$ is the four-momentum of the incoming nucleon $N$,\n$k'$ is the four-momentum of the outgoing quark.\nWe neglect the masses and use the light-cone coordinates.\nThe unit vectors are taken as,\n$\\bar n^\\mu=(1,0,0,0)$, $n^\\mu=(0,1,0,0)$, $n_{\\perp 1}^\\mu=(0,0,1,0)$,\n$n_{\\perp 2}^\\mu=(0,0,0,1)$.\nWe chose the coordinate system in the way so that,\n$p=p^+\\bar n$, $q=-x_Bp+nQ^2\/(2x_Bp^+)$, $l_\\perp=|\\vec l_\\perp|n_{\\perp 1}$,\nand $k_\\perp=(0,0,\\vec k_\\perp)$;\nwhere $x_B=Q^2\/2p\\cdot q$ is the Bjorken-$x$ and $y=p\\cdot q\/p\\cdot l$.\nThe leptonic tensor $L^{\\mu\\nu}$ is defined as usual ,\n\\begin{equation}\nL^{\\mu\\nu}(l,l')=4[l^\\mu{l'}^\\nu+l^\\nu{l'}^\\mu-(l\\cdot l')g^{\\mu\\nu}],\n\\end{equation}\nand the differential hadronic tensor is,\n\\begin{equation}\n\\frac{d^2W_{\\mu\\nu}}{d^2k'_\\perp}=\\int \\frac{dk'_z}{(2\\pi)^3 2E_{k'}}W_{\\mu\\nu}^{(si)}(q,p,k'),\n\\end{equation}\n\\begin{eqnarray}\nW_{\\mu\\nu}^{(si)}(q,p,k')&=&\n\\frac{1}{2\\pi}\\sum_X \\langle N| J_\\mu(0)|k',X\\rangle \\langle k',X| J_\\nu(0)|N\\rangle \\nonumber\\\\\n&&\\times (2\\pi)^4\\delta^4(p+q-k'-p_X),\n\\end{eqnarray}\nwhere the superscript $(si)$ denotes SIDIS.\nIt has been shown\\cite{Liang:2006wp} that, after  collinear expansion,\nthe hadronic tensor can be expressed in an expansion series characterized by the number of\ncovariant derivatives in the parton matrix elements in each term,\n\\begin{equation}\n\\frac{d^2W_{\\mu \\nu}}{d^2{k}_\\perp}=\\sum_{j=0}^\\infty\n\\frac{d^2\\tilde W^{(j)}_{\\mu \\nu}}{d^2{k}_\\perp},\n\\end{equation}\n\\begin{widetext}\n\\begin{eqnarray}\n\\label{eq:Wsi0}\n&&\\frac{d\\tilde W_{\\mu\\nu}^{(0)}}{d^2k'_\\perp}\n=\\frac{1}{2\\pi} \\int dx d^2k_\\perp\n{\\rm Tr}[\\hat H_{\\mu\\nu}^{(0)}(x)\\ \\hat \\Phi^{(0)N}(x,k_\\perp)] \\delta^{(2)}(\\vec k_\\perp-\\vec{k'}_\\perp);\\\\\n\\label{eq:Wsi1}\n&&\\frac{d\\tilde W_{\\mu\\nu}^{(1)}}{d^2k'_\\perp}\n=\\frac{1}{2\\pi}\n\\int dx_1d^2k_{1\\perp} dx_2d^2k_{2\\perp} \\sum_{c=L,R}\n{\\rm Tr}\\bigl[\\hat H_{\\mu\\nu}^{(1,c)\\rho}(x_1,x_2) \\omega_\\rho^{\\ \\rho'}\n\\hat \\Phi^{(1)N}_{\\rho'}(x_1,k_{1\\perp},x_2,k_{2\\perp})\\bigr] \\delta^{(2)}(\\vec k_{c\\perp}-\\vec{k'}_\\perp); \\phantom{XX}\\\\\n\\label{eq:Wsi2}\n&&\\frac{d\\tilde W_{\\mu\\nu}^{(2)}}{d^2k'_\\perp}=\\frac{1}{2\\pi}\n\\int dx_1d^2k_{1\\perp}dx_2d^2k_{2\\perp}dxd^2k_{\\perp}\\nonumber\\\\\n&&\\phantom{\\frac{d\\tilde W_{\\mu\\nu}^{(2)}}{d^2k'_\\perp}=\\frac{1}{2\\pi}}\n\\sum_{c=L,R,M}{\\rm Tr}\\bigl[\\hat H_{\\mu\\nu}^{(2,c)\\rho\\sigma}(x_1,x_2,x)\n\\omega_\\rho^{\\ \\rho'}\\omega_\\sigma^{\\ \\sigma'} \\hat\\Phi^{(2)N}_{\\rho'\\sigma'}(x_1,k_{1\\perp},x_2,k_{2\\perp},x,k_\\perp)\\bigr]\n\\delta^{(2)}(\\vec k_{c\\perp}-\\vec{k'}_\\perp).\\\n\\end{eqnarray}\nwhere, for different cuts $c=L$, $R$ or $M$, $\\vec k_{c\\perp}$ denotes\n$\\vec k_{L\\perp}=\\vec k_{1\\perp}$, $\\vec k_{R\\perp}=\\vec k_{2\\perp}$,  and\n$\\vec k_{M\\perp}=\\vec k_{\\perp}$;\n$\\omega_\\rho^{\\ \\rho'}=g_\\rho^{\\ \\rho'}-\\bar n_\\rho n^{\\rho'}$ is a projection operator.\nThe matrix elements are defined as,\n\\begin{eqnarray}\n\\hat\\Phi^{(0)N}(x,k_\\perp)&=&\\int \\frac{p^+dy^- d^2y_\\perp}{(2\\pi)^3}\ne^{ix p^+ y^- -i\\vec{k}_{\\perp}\\cdot \\vec{y}_{\\perp}}\n\\langle N|\\bar{\\psi}(0){\\cal{L}}(0;y)\\psi(y)|N\\rangle\n\\label{eq:Phi0def}\\\\\n\\hat\\Phi^{(1)N}_\\rho(x_1,k_{1\\perp},x_2,k_{2\\perp})\n&=&\\int \\frac{p^+dy^- d^2y_\\perp}{(2\\pi)^3}\\frac{p^+dz^-d^2z_\\perp}{(2\\pi)^3}\ne^{ix_2p^+z^- -i\\vec{k}_{2\\perp}\\cdot \\vec{z}_{\\perp}+\nix_1p^+(y^--z^-)-i\\vec{k}_{1\\perp}\\cdot (\\vec y_\\perp-\\vec z_\\perp)} \\nonumber\\\\\n&&\\langle N|\\bar\\psi(0) {\\cal L}(0;z)D_\\rho(z){\\cal L}(z;y)\\psi(y)|N\\rangle,\n\\label{eq:Phi1def}\\\\\n\\hat\\Phi^{(2)N}_{\\rho\\sigma}(x_1,k_{1\\perp},x_2,k_{2\\perp},x,k_\\perp)&=&\n\\int \\frac{p^+dz^-d^2z_\\perp}{(2\\pi)^3} \\frac{p^+dy^- d^2y_\\perp}{(2\\pi)^3} \\frac{p^+d{y'}^- d^2{y'}_\\perp}{(2\\pi)^3}\\nonumber\\\\\n&&e^{ix_2p^+z^- -i\\vec{k}_{2\\perp}\\cdot \\vec{z}_{\\perp}+\nixp^+({z'}^--z^-)-i\\vec{k}_{\\perp}\\cdot (\\vec {z'}_\\perp-\\vec {z}_\\perp)+\nix_1p^+({y}^--{z'}^-)-i\\vec{k}_{1\\perp}\\cdot (\\vec {y}_\\perp-\\vec {z'}_\\perp)} \\nonumber\\\\\n&&\\langle N|\\bar\\psi(0){\\cal L}(0;z) D_\\rho(z) {\\cal L}(z;z')D_\\sigma(z'){\\cal L}(z';y)\\psi(y)|N\\rangle,\n\\label{eq:Phi2def}\n\\end{eqnarray}\nwhere ${\\cal{L}}(0;y)$ is the gauge link as defined in Eq.~(\\ref{eq:fqperp}), and also in the remainder of this paper, for brevity, \nunless explicitly specified, the coordinate $y$ in the field operator denotes $(0,y^-,\\vec{y}_{\\perp})$.\n\n\n\n\nThe hard parts after the collinear expansion are given as \\cite{Liang:2006wp},\n\\begin{eqnarray}\n\\hat H_{\\mu\\nu}^{(0)}(x)&=&\\frac{2\\pi}{2q\\cdot p}\n\\gamma_\\mu(\\slash{\\hspace{-5pt}q}+x\\slash{\\hspace{-5pt}p})\\gamma_\\nu\\delta(x-x_B),\\\\\n\\hat H_{\\mu\\nu}^{(1,L)\\rho}(x_1,x_2)&=&\\frac{2\\pi}{(2q\\cdot p)^2}\n\\frac{\\gamma_\\mu(\\slash{\\hspace{-5pt}q}+x_2\\slash{\\hspace{-5pt}p})\\gamma^\\rho(\\slash{\\hspace{-5pt}q}+x_1\\slash{\\hspace{-5pt}p})\n\\gamma_\\nu}{x_2-x_B-i\\varepsilon}\\delta(x_1-x_B),\\\\\n\\hat H_{\\mu\\nu}^{(2,L)\\rho\\sigma}(x_1,x_2,x)&=&\\frac{2\\pi}{(2q\\cdot p)^3}\n\\frac{\\gamma_\\mu(\\slash{\\hspace{-5pt}q}+x_2\\slash{\\hspace{-5pt}p})\\gamma^\\rho(\\slash{\\hspace{-5pt}q}+x\\slash{\\hspace{-5pt}p})\n\\gamma^\\sigma(\\slash{\\hspace{-5pt}q}+x_1\\slash{\\hspace{-5pt}p})\\gamma_\\nu}\n{(x-x_B-i\\varepsilon)(x_2-x_B-i\\varepsilon)}\\delta(x_1-x_B).\n\\end{eqnarray}\nThese equations form the basis for calculating the hadronic tensor in $e^-+N\\to e^-+q+X$.\nDue to the existence of the projection operators $\\omega_\\rho^{\\ \\rho'}$ and $\\omega_\\sigma^{\\ \\sigma'}$,\nthe hard parts can be simplified to,\n\\begin{eqnarray}\n&&\\hat H^{(0)}_{\\mu\\nu}(x)=\\pi\\hat h^{(0)}_{\\mu\\nu}\\delta(x-x_B),\n\\label{eq:H0simple}\\\\\n&&\\hat H^{(1,L)\\rho}_{\\mu\\nu}(x_1,x_2)\\omega_\\rho^{\\ \\rho'}\n=\\frac{\\pi}{2q\\cdot p}\\hat h^{(1)\\rho}_{\\mu\\nu}\\omega_\\rho^{\\ \\rho'}\\delta(x_1-x_B),\n\\label{eq:H1Lsimple}\\\\\n&&\\hat H^{(2,L)\\rho\\sigma}_{\\mu\\nu}(x_1,x_2,x)\\omega_\\rho^{\\ \\rho'}\\omega_\\sigma^{\\ \\sigma'}=\n\\frac{2\\pi}{(2q\\cdot p)^2}\\bigl[\\bar n^\\rho\\hat h^{(1)\\sigma}_{\\mu\\nu}+\\frac{\\hat N^{(2)\\rho\\sigma}_{\\mu\\nu}}{x_2-x_B-i\\varepsilon}\\bigr]\n\\omega_\\rho^{\\ \\rho'}\\omega_\\sigma^{\\ \\sigma'}\\delta(x_1-x_B),\n\\label{eq:H2Lsimple}\\\\\n&& \\hat H^{(2,M)\\rho\\sigma}_{\\mu\\nu}(x_1,x_2,x)\\omega_\\rho^{\\ \\rho'}\\omega_\\sigma^{\\ \\sigma'}=\n\\frac{2\\pi }{(2q\\cdot p)^2}\\hat h^{(2)\\rho\\sigma}_{\\mu\\nu}\n\\omega_\\rho^{\\ \\rho'}\\omega_\\sigma^{\\ \\sigma'}\\delta(x-x_B),\n\\label{eq:H2Msimple}\n\\end{eqnarray}\nwhere $\\hat h^{(0)}_{\\mu\\nu}=\\gamma_\\mu\\slash{\\hspace{-5pt}n}\\gamma_\\nu\/p^+$,\n$\\hat h^{(1)\\rho}_{\\mu\\nu}=\\gamma_\\mu\\slash{\\hspace{-5pt}\\bar n}\\gamma^\\rho\\slash{\\hspace{-5pt}n}\\gamma_\\nu$,\n$\\hat h^{(2)\\rho\\sigma}_{\\mu\\nu}=p^+\\gamma_\\mu\\slash{\\hspace{-5pt}\\bar n}\\gamma^\\rho\n\\slash{\\hspace{-5pt} n}\\gamma^\\sigma\\slash{\\hspace{-5pt}\\bar n}\\gamma_\\nu\/2$  and\n$\\hat N^{(2)\\rho\\sigma}_{\\mu\\nu}=q^-\\gamma_\\mu\\gamma^\\rho\\slash{\\hspace{-5pt}n}\\gamma^\\sigma\\gamma_\\nu$.\nWe insert them into Eqs.(\\ref{eq:Wsi0}-\\ref{eq:Wsi2}) and obtain,\n\\begin{eqnarray}\n\\frac{d^2\\tilde W^{(0)}_{\\mu\\nu}}{d^2k_\\perp} &=&\n\\frac{1}{2}{\\rm Tr}\\bigl[\\hat h^{(0)}_{\\mu\\nu}\\hat\\Phi^{(0)N}(x_B,k_\\perp)\\bigr],\n\\label{eq:W0} \\\\\n\\frac{d^2\\tilde W^{(1,L)}_{\\mu\\nu}}{d^2k_\\perp} &=&\n\\frac{1}{4q\\cdot p}{\\rm Tr}\\bigl[\\hat h^{(1)\\rho}_{\\mu\\nu}\\omega_\\rho^{\\ \\rho'}\\hat\\varphi^{(1,L)N}_{\\rho'}(x_B,k_\\perp)\\bigr],\n\\label{eq:W1L} \\\\\n\\frac{d^2\\tilde W^{(2,L)}_{\\mu\\nu}}{d^2k_\\perp} &=&\n\\frac{1}{(2q\\cdot p)^2}\\left\\{{\\rm Tr}\\bigl[\\hat h^{(1)\\rho}_{\\mu\\nu}\n\\omega_\\rho^{\\ \\rho'}\\hat\\phi^{(2,L)N}_{\\rho'}(x_B,k_\\perp)\\bigr]\n+\n{\\rm Tr}\\bigl[\\hat N^{(2)\\rho\\sigma}_{\\mu\\nu}\\omega_\\rho^{\\ \\rho'}\\omega_\\sigma^{\\ \\sigma'}\n\\hat\\varphi^{(2,L)N}_{\\rho'\\sigma'}(x_B,k_\\perp)\\bigr]\\right\\},\n\\label{eq:W2L} \\\\\n\\frac{d^2\\tilde W^{(2,M)}_{\\mu\\nu}}{d^2k_\\perp} &=&\n\\frac{1}{(2q\\cdot p)^2}\n{\\rm Tr}\\bigl[\\hat h^{(2)\\rho\\sigma}_{\\mu\\nu}\\omega_\\rho^{\\ \\rho'}\\omega_\\sigma^{\\ \\sigma'}\n\\hat\\varphi^{(2,M)N}_{\\rho'\\sigma'}(x_B,k_\\perp)\\bigr].\n\\label{eq:W2M}\n\\end{eqnarray}\nThe correlation matrices are defined as,\n\\begin{eqnarray}\n\\hat\\varphi^{(1,L)N}_\\rho(x_1,k_{1\\perp})&\\equiv&\\int dx_2d^2k_{2\\perp}\\hat\\Phi^{(1)N}_\\rho(x_1,k_{1\\perp},x_2,k_{2\\perp}),\\\\\n\\hat\\varphi^{(2,L)N}_{\\rho\\sigma}(x_1,k_{1\\perp})&\\equiv&\\int dxd^2k_{\\perp}\n\\frac{dx_2d^2k_{2\\perp}}{x_2-x_1-i\\varepsilon}\n\\hat\\Phi^{(2)N}_{\\rho\\sigma}(x_1,k_{1\\perp},x_2,k_{2\\perp},x,k_\\perp),\\\\\n\\hat\\varphi^{(2,M)N}_{\\rho\\sigma}(x,k_{\\perp})&\\equiv&\\int dx_1d^2k_{1\\perp}dx_2d^2k_{2\\perp}\n\\hat\\Phi^{(2)N}_{\\rho\\sigma}(x_1,k_{1\\perp},x_2,k_{2\\perp},x,k_\\perp),\\\\\n\\hat\\phi^{(2,L)N}_\\sigma(x_1,k_{1\\perp})&\\equiv&\\int dxd^2k_\\perp dx_2d^2k_{2\\perp}\n\\hat\\Phi^{(2)N}_{\\rho\\sigma}(x_1,k_{1\\perp},x_2,k_{2\\perp},x,k_\\perp).\n\\end{eqnarray}\nThey are given by,\n\\begin{eqnarray}\n&& \\hat\\varphi^{(1,L)N}_\\rho(x,k_\\perp)\n=\\int \\frac{p^+dy^- d^2y_\\perp}{(2\\pi)^3}e^{ixp^+y^- -i\\vec{k}_{\\perp}\\cdot \\vec{y}_{\\perp}}\n\\langle N|\\bar\\psi(0) D_\\rho(0){\\cal L}(0;y)\\psi(y)|N\\rangle,\\\\\n&&\\hat\\varphi^{(2,L)N}_{\\rho\\sigma}(x,k_{\\perp})\n=\\int \\frac{dx_2}{x_2-x-i\\varepsilon}\\frac{p^+dy^- d^2y_\\perp}{(2\\pi)^3}\\frac{p^+dz^-}{2\\pi}\ne^{ix_2p^+z^-+ixp^+(y^--z^-)-i\\vec{k}_{\\perp}\\cdot \\vec{y}_\\perp} \\nonumber\\\\\n&&\\phantom{XXXXXXXXXXXXX}\n\\langle N|\\bar\\psi(0){\\cal L}(0;z^-,y_\\perp) D_\\rho(z^-,y_\\perp)D_\\sigma(z^-,y_\\perp){\\cal L}(z^-,\\vec y_\\perp;y)\\psi(y)|N\\rangle,\\\\\n&&\\hat\\varphi^{(2,M)N}_{\\rho\\sigma}(x,k_{\\perp})\n=\\int \\frac{p^+dy^- d^2y_\\perp}{(2\\pi)^3}e^{ixp^+y^- -i\\vec{k}_\\perp\\cdot \\vec{y}_{\\perp}}\n\\langle N|\\bar\\psi(0) D_\\rho(0){\\cal L}(0;y)D_\\sigma(y)\\psi(y)|N\\rangle,\\\\\n&&\\hat\\phi^{(2,L)N}_\\sigma(x,k_{\\perp})\n=\\int \\frac{p^+dy^- d^2y_\\perp}{(2\\pi)^3}e^{ixp^+y^- -i\\vec{k}_\\perp\\cdot \\vec{y}_{\\perp}}\n\\langle N|\\bar\\psi(0) D^-(0)D_\\sigma(0){\\cal L}(0;y)\\psi(y)|N\\rangle.\n\\end{eqnarray}\n\nWe note that,\n$\\tilde W^{(0)*}_{\\mu\\nu}=\\tilde W^{(0)}_{\\nu\\mu}$,\n$\\tilde W^{(2,M)*}_{\\mu\\nu}=\\tilde W^{(2,M)}_{\\nu\\mu}$,\n$\\tilde W^{(1,R)}_{\\mu\\nu}=\\tilde W^{(1,L)*}_{\\nu\\mu}$,\nand $\\tilde W^{(2,R)}_{\\mu\\nu}=\\tilde W^{(2,L)*}_{\\nu\\mu}$.\nHence, if we divide $W_{\\mu\\nu}$ into a $\\mu\\leftrightarrow\\nu$ symmetric part and an anti-symmetric part,\nand denote $W_{\\mu\\nu}=W_{S,\\mu\\nu}+iW_{A,\\mu\\nu}$, we obtain,\n\\begin{equation}\n\\frac{d^2W_{S,\\mu\\nu}}{d^2k_\\perp}=\n\\frac{d^2\\tilde W^{(0)}_{S,\\mu\\nu}}{d^2k_\\perp}+\n2{\\rm Re}\\frac{d^2\\tilde W^{(1,L)}_{S,\\mu\\nu}}{d^2k_\\perp}+\n2{\\rm Re}\\frac{d^2\\tilde W^{(2,L)}_{S,\\mu\\nu}}{d^2k_\\perp}+\\frac{d^2\\tilde W^{(2,M)}_{S,\\mu\\nu}}{d^2k_\\perp},\n\\end{equation}\n\\begin{equation}\n\\frac{d^2W_{A,\\mu\\nu}}{d^2k_\\perp}=\n\\frac{d^2\\tilde W^{(0)}_{A,\\mu\\nu}}{d^2k_\\perp}+\n2{\\rm Im}\\frac{d^2\\tilde W^{(1,L)}_{S,\\mu\\nu}}{d^2k_\\perp}+\n2{\\rm Im}\\frac{d^2\\tilde W^{(2,L)}_{S,\\mu\\nu}}{d^2k_\\perp}+\\frac{d^2\\tilde W^{(2,M)}_{A,\\mu\\nu}}{d^2k_\\perp}.\n\\end{equation}\n\nThe  anti-symmetric part contributes only in reactions with polarized lepton.\nIn this paper, we concentrate on the unpolarized reactions and calculate the symmetric part in the following.\n\n\nNow, we continue with a complete calculation of the hadronic tensor $d^2 W_{\\mu \\nu}\/d^2k_\\perp$\nin the unpolarized $e^-+N\\to e^-+q+X$ up to twist-4 level.\nFor this purpose, we need to calculate $d^2 W_{\\mu \\nu}\/d^2k_\\perp$ up to\n$d^2\\tilde W_{\\mu \\nu}^{(2)}\/d^2k_\\perp$ and we now present\nthe calculations of each term in the following.\n\nThe contribution from $d^2\\tilde W_{\\mu \\nu}^{(0)}\/d^2k_\\perp$ is the easiest one to calculate.\nBecause $\\hat H^{(0)}_{\\mu\\nu}(x)$ contains 3 $\\gamma$-matrices, only $\\gamma^\\alpha$\nterm of $\\hat\\Phi^{(0)}(x,k_\\perp)$ contributes in the unpolarized case so we need only to consider\n$\\hat\\Phi^{(0)N}(x,k_\\perp)=\\gamma^\\alpha\\Phi^{(0)N}_\\alpha(x,k_\\perp)\/2$,\n\\begin{equation}\n\\Phi^{(0)N}_{\\alpha}(x,k_{\\perp})\n=\\int \\frac{p^+dy^- d^{2}y_{\\perp}}{(2\\pi)^3}\ne^{ix p^+ y^- -i\\vec{k}_{\\perp}\\cdot \\vec{y}_{\\perp}}\n\\langle N|\\bar{\\psi}(0)\\frac{\\gamma_\\alpha}{2}{\\cal{L}}(0;y)\\psi(y)|N\\rangle\n=p_\\alpha f_q^N+k_{\\perp\\alpha} f_{q\\perp}^N+\\frac{M^2}{p^+}n_\\alpha f_{q(-)}^N.\n\\label{eq:Phi0alpha}\n\\end{equation}\nand obtain the result for $d^2\\tilde W_{\\mu \\nu}^{(0)}\/d^2k_\\perp$ as,\n\\begin{equation}\n\\frac{d^2\\tilde W_{\\mu \\nu}^{(0)}}{d^2k_\\perp}=-d_{\\mu\\nu}f_q^N(x_B,k_\\perp)+\n\\frac{1}{q\\cdot p}k_{\\perp\\{\\mu}(q+x_Bp)_{\\nu\\}}f_{q\\perp}^N(x_B,k_\\perp)+\n2(\\frac{M}{q\\cdot p})^2(q+x_Bp)_\\mu (q+x_Bp)_\\nu f_{q(-)}^N(x_B,k_\\perp),\n\\end{equation}\nwhere $d^{\\mu\\nu}=g^{\\mu\\nu}-\\bar{n}^{\\mu}n^{\\nu}-\\bar{n}^{\\nu}n^{\\mu}$ and\n$A_{\\{\\mu}B_{\\nu\\}}\\equiv A_\\mu B_\\nu+A_\\nu B_\\mu$,\n$A_{[\\mu}B_{\\nu]}\\equiv A_\\mu B_\\nu-A_\\nu B_\\mu$.\nThe TMD quark distribution\/correlation functions are given by,\n\\begin{eqnarray}\n&&f_q^N(x,k_\\perp)=\\frac{n^\\alpha}{p^+}\\Phi^{(0)N}_\\alpha(x,k_\\perp)=\\int \\frac{dy^- d^2y_{\\perp}}{(2\\pi)^3}\ne^{ix p^+ y^- -i\\vec{k}_{\\perp}\\cdot \\vec{y}_{\\perp}}\n\\langle N|\\bar{\\psi}(0)\\frac{\\gamma^+}{2}{\\cal{L}}(0;y)\\psi(y)|N\\rangle,\n\\label{eq:fqn} \\\\\n&& k_\\perp^\\alpha f_{q\\perp}^N(x_B,k_\\perp)=d^{\\alpha\\beta}\\Phi^{(0)N}_\\beta(x,k_\\perp)\n=\\int \\frac{p^+dy^- d^2y_{\\perp}}{(2\\pi)^3}\ne^{ix p^+ y^- -i\\vec{k}_{\\perp}\\cdot \\vec{y}_{\\perp}}\n\\langle N|\\bar{\\psi}(0)\\frac{\\gamma^\\alpha_\\perp}{2}{\\cal{L}}(0;y)\\psi(y)|N\\rangle,\n\\label{eq:fqnperp} \\\\\n&&f_{q(-)}^N(x_B,k_\\perp)=\\frac{p^+}{M^2}\\bar n^\\alpha\\Phi^{(0)N}_\\alpha(x,k_\\perp)\n=\\frac{p^+}{M^2}\\int \\frac{p^+dy^- d^2y_{\\perp}}{(2\\pi)^3}\ne^{ix p^+ y^- -i\\vec{k}_{\\perp}\\cdot \\vec{y}_{\\perp}}\n\\langle N|\\bar{\\psi}(0)\\frac{\\gamma^-}{2}{\\cal{L}}(0;y)\\psi(y)|N\\rangle.\n\\label{eq:fqnminus}\n\\end{eqnarray}\n\nBecause $\\hat h^{(1)\\rho}_{\\mu\\nu}$ contains 5 $\\gamma$-matrices,\nwe have contributions from $\\gamma_\\alpha$ and $\\gamma_5\\gamma_\\alpha$ terms of $\\varphi^{(1,L)N}_\\rho$,\ni.e., we need to consider\n$\\hat\\varphi^{(1,L)N}_\\rho(x,k_\\perp)=[\\gamma^\\alpha\\varphi^{(1)N}_{\\rho\\alpha}(x,k_\\perp)-\n\\gamma_5\\gamma^\\alpha\\tilde\\varphi^{(1)N}_{\\rho\\alpha}(x,k_\\perp)]\/2\n$ \nand obtain,\n\\begin{equation}\n\\frac{d^2\\tilde W_{\\mu\\nu}^{(1,L)}}{d^2k_\\perp}\n=\\frac{1}{2p\\cdot q} \\Bigl[h^{(1)\\rho\\alpha}_{\\mu\\nu}\\omega_\\rho^{\\ \\rho'}\\varphi^{(1)N}_{\\rho'\\alpha}(x_B,k_\\perp)\n-\\tilde h^{(1)\\rho\\alpha}_{\\mu\\nu}\\omega_\\rho^{\\ \\rho'}\\tilde\\varphi^{(1)N}_{\\rho'\\alpha}(x_B,k_\\perp)\\Bigr],\n\\end{equation}\nwhere $h^{(1)\\rho\\alpha}_{\\mu\\nu}\\equiv {\\rm Tr}[\\gamma^\\alpha \\hat h^{(1)\\rho}_{\\mu\\nu}]\/4$,\n$\\tilde h^{(1)\\rho\\alpha}_{\\mu\\nu}\\equiv {\\rm Tr}[\\gamma_5\\gamma^\\alpha \\hat h^{(1)\\rho}_{\\mu\\nu}]\/4$ and,\n\\begin{eqnarray}\n&&\\varphi^{(1)N}_{\\rho\\alpha}(x, k_\\perp)\n=\\int \\frac{p^+dy^-d^2y_\\perp}{(2\\pi)^3}e^{ixp^+y^--i\\vec y_\\perp\\cdot\\vec k_\\perp}\n\\langle N|\\bar\\psi(0)\\frac{\\gamma_\\alpha}{2}{\\cal L}(0;y)D_\\rho(y)\\psi(y)|N\\rangle,\n\\label{eq:varphi1}\\\\\n&&\\tilde\\varphi^{(1)N}_{\\rho\\alpha}(x, k_\\perp)\n=\\int \\frac{p^+dy^-d^2y_\\perp}{(2\\pi)^3}e^{ixp^+y^--i\\vec y_\\perp\\cdot\\vec k_\\perp}\n\\langle N|\\bar\\psi(0)\\frac{\\gamma_5\\gamma_\\alpha}{2}{\\cal L}(0;y)D_\\rho(y)\\psi(y)|N\\rangle.\\\n\\label{eq:tildevarphi1}\n\\end{eqnarray}\nAfter evaluating the two traces in $h^{(1)\\rho\\alpha}_{\\mu\\nu}$ and $\\tilde h^{(1)\\rho\\alpha}_{\\mu\\nu}$, we\nobtain the symmetric parts as,\n\\begin{eqnarray}\n&&h^{(1)\\rho\\alpha}_{S,\\mu\\nu}=-g^\\alpha_{\\ \\mu}d^\\rho_{\\ \\nu}-g^\\alpha_{\\ \\nu}d^\\rho_{\\ \\mu}+g_{\\mu\\nu}d^{\\rho\\alpha},\\\\\n&& \\tilde h^{(1)\\rho\\alpha}_{S,\\mu\\nu}=ig^\\alpha_{\\ \\mu}\\varepsilon_{\\perp\\nu}^\\rho\n+ig^\\alpha_{\\ \\nu}\\varepsilon_{\\perp\\mu}^\\rho-ig_{\\mu\\nu}\\varepsilon_{\\perp}^{\\rho\\alpha},\n\\end{eqnarray}\nwhere $\\epsilon_{\\perp\\rho\\gamma}\\equiv\\epsilon_{\\alpha\\beta\\rho\\gamma}\\bar n^\\alpha n^\\beta$.\nUp to twist-4, the contributing terms of $\\varphi^{(1)N}_{\\rho\\alpha}(x,k_\\perp)$\nand  $\\tilde\\varphi^{(1)N}_{\\rho\\alpha}(x,k_\\perp)$\nare respectively,\n\\begin{eqnarray}\n\\varphi^{(1)N}_{\\rho\\alpha}(x,k_\\perp)&=&\np_\\alpha k_{\\perp\\rho}\\varphi^{(1)N}_\\perp(x,k_\\perp)+\n(k_{\\perp\\alpha}k_{\\perp\\rho}-\\frac{k_\\perp^2}{2}d_{\\rho\\alpha})\\varphi^{(1)N}_{\\perp 2}(x,k_\\perp)+\n\\frac{k_\\perp^2}{2}(\\bar n_{\\{\\alpha}n_{\\rho\\}}-d_{\\rho\\alpha})\\varphi^{(1)N}_{\\perp 3}(x,k_\\perp),\\\\\n\\tilde\\varphi^{(1)N}_{\\rho\\alpha}(x,k_\\perp)&=&\nip_\\alpha\\varepsilon_{\\perp\\rho\\gamma}k_\\perp^\\gamma \\tilde\\varphi^{(1)N}_\\perp(x,k_\\perp)+\n\\frac{i}{2}k_{\\perp\\{\\alpha}\\varepsilon_{\\perp\\rho\\}\\gamma}k_\\perp^\\gamma \\tilde\\varphi^{(1)N}_{\\perp2}(x,k_\\perp)+\n\\frac{i}{2}k_{\\perp[\\alpha}\\varepsilon_{\\perp\\rho]\\gamma}k_\\perp^\\gamma\\tilde\\varphi^{(1)N}_{\\perp3}(x,k_\\perp).\n\\end{eqnarray}\nThe result for $d^2\\tilde W_{S,\\mu\\nu}^{(1,L)}\/d^2k_\\perp$ is,\n\\begin{eqnarray}\n&&\\frac{d^2\\tilde W_{S,\\mu\\nu}^{(1,L)}}{d^2k_\\perp}\n=-\\frac{1}{2q\\cdot p}\\bigl\\{ (p_\\mu k_{\\perp\\nu}+p_\\nu k_{\\perp\\mu})\n[\\varphi^{(1)N}_{\\perp}(x_B,k_\\perp)-\\tilde\\varphi^{(1)N}_{\\perp}(x_B,k_\\perp)]\\nonumber\\\\\n&&\\phantom{XXXXXXXX}\n+(2k_{\\perp\\mu}k_{\\perp\\nu}-k_\\perp^2d_{\\mu\\nu})\n  [\\varphi^{(1)N}_{\\perp2}(x_B,k_\\perp)-\\tilde\\varphi^{(1)N}_{\\perp2}(x_B,k_\\perp)]\\nonumber\\\\\n&&\\phantom{XXXXXXXX}\n+k_\\perp^2(g_{\\mu\\nu}-d_{\\mu\\nu})\n[\\varphi^{(1)N}_{\\perp3}(x_B,k_\\perp)-\\tilde\\varphi^{(1)N}_{\\perp3}(x_B,k_\\perp)]\\bigr\\}.\n\\label{eq:W1Lres}\n\\end{eqnarray}\n\nUp to twist-4 level, we need only to consider $\\slash{\\hspace{-5pt}p}$ and the $\\gamma_5\\slash{\\hspace{-5pt}p}$-term\nin the calculations of $d\\tilde W^{(2)}_{\\mu \\nu}\/d^2k_\\perp$.\nFor the first term in Eq.(\\ref{eq:W2L}), because of $\\omega_\\rho^{\\ \\rho'}$ and\n$n_\\rho\\hat h^{(1)\\rho}_{\\mu\\nu}=0$, we need only to consider the $k_{\\perp\\rho}$ terms and\nwe found out that they contribute only at twist-5 or higher level.\nFor the second term, because $n_\\rho\\hat N^{(2)\\rho\\sigma}_{\\mu\\nu}=n_\\sigma\\hat N^{(2)\\rho\\sigma}_{\\mu\\nu}=0$ and\n$\\hat\\varphi^{(2,L)N}_{\\rho\\sigma}=\\hat\\varphi^{(2,L)N}_{\\sigma\\rho}$,\nwe need to consider only $k_{\\perp\\rho}k_{\\perp\\sigma}$ and $k_\\perp^2d_{\\rho\\alpha}$\nfor the tensor term and $k_{\\perp\\{\\rho}\\varepsilon_{\\perp\\sigma\\}\\gamma}k_\\perp^\\gamma$\nfor the pseudo-tensor term.\nFurthermore,\n\\begin{eqnarray}\n&&k^2_\\perp\\hat N^{(2)\\rho\\sigma}_{\\mu\\nu}d_{\\rho\\sigma}=2\\hat N^{(2)\\rho\\sigma}_{\\mu\\nu}k_{\\perp\\rho}k_{\\perp\\sigma}\n=-2k_\\perp^2\\gamma_\\mu\\slash{\\hspace{-5pt}n}\\gamma_\\nu,\\\\\n&&\\hat N^{(2)\\rho\\sigma}_{\\mu\\nu}k_{\\perp\\rho}\\varepsilon_{\\perp\\sigma\\gamma}k_\\perp^\\gamma=\n-\\hat N^{(2)\\rho\\sigma}_{\\mu\\nu}k_{\\perp\\sigma}\\varepsilon_{\\perp\\rho\\gamma}k_\\perp^\\gamma\n=k_\\perp^2\\gamma_\\mu\\slash{\\hspace{-5pt}n}\\slash{\\hspace{-5pt}n_{\\perp1}}\\slash{\\hspace{-5pt}n_{\\perp2}}\\gamma_\\nu,\n\\end{eqnarray}\nwe need only to consider,\n\\begin{equation}\n\\hat\\varphi^{(2,L)N}_{\\rho\\sigma}(x,k_\\perp)=\\frac{\\slash{\\hspace{-5pt}p}}{2}\n\\left(-\\frac{1}{2}k^2_\\perp d_{\\rho\\sigma}\\right)\\varphi^{(2,L)N}_\\perp(x,k_\\perp)+...\n\\end{equation}\nand obtain the results for $d^2\\tilde W^{(2,L)}_{\\mu\\nu}\/d^2k_\\perp$ up to $1\/Q^2$ as,\n\\begin{equation}\n\\frac{d^2\\tilde W^{(2,L)}_{\\mu\\nu}}{d^2k_\\perp}=\n-\\frac{1}{2q\\cdot p}k_\\perp^2d_{\\mu\\nu}\\varphi^{(2,L)N}_\\perp(x_B,k_\\perp)+... .\n\\end{equation}\n\nSimilarly, to calculate $d^2\\tilde W^{(2,M)}_{\\mu\\nu}\/d^2k_\\perp$ up to $1\/Q^2$ level, we need to consider\n\\begin{equation}\n\\hat\\varphi^{(2,M)N}_{\\rho\\sigma}(x,k_\\perp)=\n\\frac{\\slash{\\hspace{-5pt}p}}{2}\\left(-\\frac{1}{2}k^2_\\perp d_{\\rho\\sigma}\\right)\\varphi^{(2,M)N}_\\perp(x,k_\\perp)-\n\\frac{i}{4}\\gamma_5\\slash{\\hspace{-5pt}p}\nk_{\\perp[\\rho}\\varepsilon_{\\perp\\sigma]\\gamma}k_\\perp^\\gamma\\tilde\\varphi^{(2,M)N}_\\perp(x,k_\\perp),\n\\end{equation}\nand the results for  $d^2\\tilde W^{(2,M)}_{\\mu\\nu}\/d^2k_\\perp$ are given by,\n\\begin{equation}\n\\frac{d^2\\tilde W^{(2,M)}_{\\mu\\nu}}{d^2k_\\perp}=\n\\frac{k_\\perp^2}{(q\\cdot p)^2}p_\\mu p_\\nu[\\varphi^{(2,M)N}_\\perp(x_B,k_\\perp)-\\tilde\\varphi^{(2,M)N}_\\perp(x_B,k_\\perp)].\n\\end{equation}\n\nQCD equation of motion relates matrix elements with different number of $D_\\rho$ and gives\n\\begin{eqnarray}\n&&xf_{q\\perp}^N(x,k_\\perp)=-[\\varphi_\\perp^{(1)N}(x,k_\\perp)-\\tilde\\varphi_\\perp^{(1)N}(x,k_\\perp)],\\\\\n&&2(xM)^2f_{q(-)}^N(x,k_\\perp)=k_\\perp^2[\\varphi_\\perp^{(2,M)N}(x,k_\\perp)-\\tilde\\varphi_\\perp^{(2,M)N}(x,k_\\perp)],\\\\\n&&x[\\varphi^{(1)N}_{\\perp 3}(x,k_\\perp)-\\tilde \\varphi^{(1)N}_{\\perp 3}(x,k_\\perp)]\n=-[\\varphi^{(2,M)N}_{\\perp }(x,k_\\perp)-\\tilde \\varphi^{(2,M)N}_{\\perp }(x,k_\\perp)],\n\\end{eqnarray}\nwhere, as well as in the following of this paper,\nall the correlation functions in the results of the hadronic tensors and\/or cross section\nstand for their real parts. %\nThe final results for $d^2W_{\\mu\\nu}\/d^2k_\\perp$ up to twist-4 level are given by,\n \\begin{eqnarray}\n\\frac{d^2W_{\\mu \\nu}}{d^2k_\\perp}&=&\n-\\frac{1}{q\\cdot p}\\Bigl\\{ (q\\cdot p)d_{\\mu\\nu}f_q^N(x_B,k_\\perp)\n+\\frac{2M^2}{q\\cdot p}(q+2x_Bp)_\\mu(q+2x_Bp)_\\nu f_{q(-)}^N(x_B,k_\\perp)\\nonumber\\\\\n&&-(q+2x_Bp)_{\\{\\mu} k_{\\perp\\nu\\}} f_{q\\perp}^N(x_B,k_\\perp) +(2k_{\\perp\\mu}k_{\\perp\\nu}-k_\\perp^2d_{\\mu\\nu})\n  [\\varphi^{(1)N}_{\\perp2}(x_B,k_\\perp)-\\tilde\\varphi^{(1)N}_{\\perp2}(x_B,k_\\perp)]\\nonumber\\\\\n&&+k_\\perp^2d_{\\mu\\nu} \\varphi^{(2,L)N}_{\\perp2}(x_B,k_\\perp)\\Bigr\\}.\n\\end{eqnarray}\n\n\n\\section{Differential cross section and $\\langle \\cos 2\\phi\\rangle$ up to the $1\/Q^2$}\n\n\nMaking the Lorentz contraction of the result for $d^2W_{\\mu\\nu}\/d^2k_\\perp$\nwith the leptonic tensor $L_{\\mu\\nu}$ given in Eq.(3),\nwe obtain the differential cross section as,\n\\begin{eqnarray}\n\\frac{d\\sigma}{dx_Bdyd^2k_\\perp}&=&\\frac{2\\pi\\alpha_{em}^2e_q^2}{Q^2y}\n\\bigg\\{ [1+(1-y)^2] f_q^N(x_B,k_\\perp)\n-4(2-y)\\sqrt{1-y} \\frac{|\\vec k_\\perp|}{Q} x_Bf_{q\\perp}^{(1)N}(x_B,k_\\perp)\\cos\\phi \\nonumber\\\\\n\\nonumber &&-4(1-y)\\frac{|\\vec k_\\perp|^2}{Q^2}x_B\n[\\varphi^{(1)N}_{\\perp 2}(x_B,k_\\perp)-\\tilde\\varphi^{(1)N}_{\\perp 2} (x_B,k_\\perp)]\\cos 2\\phi\\\\\n\\nonumber &&+8(1-y)\\left(\\frac{|\\vec k_\\perp|^2}{Q^2}\nx_B[\\varphi^{(1)N}_{\\perp 2}(x_B,k_\\perp)-\\tilde\\varphi^{(1)N}_{\\perp 2} (x_B,k_\\perp)]\n+\\frac{2x_B^2M^2}{Q^2}f_{q(-)}^N (x_B,k_\\perp)\\right)\\\\\n&&-2\\left[1+(1-y)^2\\right]\\frac{|\\vec k_\\perp|^2}{Q^2}x_B\\varphi_{\\perp 2}^{(2,L)N} (x_B,k_\\perp)\\bigg\\}.\n\\label{eq:CSres}\n\\end{eqnarray}\n\n\\end{widetext}\n\nFrom Eq.(\\ref{eq:CSres}), we can calculate the azimuthal asymmetries\n$\\langle\\cos\\phi\\rangle$ and $\\langle\\cos 2\\phi\\rangle$.\nThe result for $\\langle\\cos\\phi\\rangle$ and its nuclear dependence are\ndiscussed in \\cite{Gao:2010mj}.\nWe now discuss the result for $\\langle\\cos 2\\phi\\rangle$.\nAt fixed $k_\\perp$, it is given by,\n\\begin{eqnarray}\n&&\\langle \\cos2\\phi\\rangle_{eN}=-\\frac{2(1-y)}{1+(1-y)^2}\\frac{|\\vec k_\\perp|^2}{Q^2}\\times\\phantom{XXXXXXXX}\\nonumber\\\\\n&&\\phantom{XXXX}\\frac{x_B[\\varphi_{\\perp 2}^{(1)N}(x_B,k_\\perp)-\\tilde\\varphi_{\\perp 2}^{(1)N}(x_B,k_\\perp)]}{f_q^N(x_B,k_\\perp)}.\\ \\\n\\label{eq:cos2phires}\n\\end{eqnarray}\nIntegrating over the magnitude of $\\vec k_\\perp$, we obtain,\n\\begin{eqnarray}\n&&\\langle\\langle \\cos 2\\phi\\rangle\\rangle_{eN}=\n-\\frac{2(1-y)}{1+(1-y)^2} \\times\\phantom{XXXX}\\nonumber\\\\\n&&\\phantom{XX}\\frac{\\int |\\vec k_\\perp|^2d^2k_\\perp\nx_B[\\varphi_{\\perp 2}^{(1)N}(x_B,k_\\perp)-\\tilde\\varphi_{\\perp 2}^{(1)N}(x_B,k_\\perp)]}{Q^2f_q^N(x_B)},\\nonumber\n\\end{eqnarray}\nwhere $f_q^N(x)=\\int d^2k_\\perp f_q^N(x,k_\\perp)$ is the usual quark distribution in nucleon.\nThe new quark correlation functions involved are given by,\n\\begin{eqnarray}\n|\\vec k_\\perp|^2\\varphi_{\\perp 2}^{(1)N}(x,k_\\perp)&=(2\\hat k_\\perp^\\alpha\\hat k_\\perp^\\rho+d^{\\alpha\\rho})\n\\varphi^{(1)N}_{\\rho\\alpha}(x,k_\\perp), \\\\\n|\\vec k_\\perp|^2\\tilde\\varphi_{\\perp 2}^{(1)N}(x,k_\\perp)&=\n-i\\hat k_\\perp^{\\{\\alpha}\\varepsilon_\\perp^{\\rho\\}\\sigma}\\hat k_{\\perp\\sigma} \\tilde\\varphi^{(1)N}_{\\rho\\alpha}(x,k_\\perp),\n\\end{eqnarray}\nwhere $\\hat k_\\perp=k_\\perp\/|\\vec k_\\perp|$ denotes the unit vector.\nIf we consider only ``free parton with intrinsic transverse momentum\",\ni.e., the same case as considered in \\cite{Cahn:1978se},\nwe need to just set $g=0$ in the results mentioned above.\nIn this case, ${\\cal L}=1$ and\n$x[\\varphi_{\\perp 2}^{(1)N}(x,k_\\perp)-\\tilde\\varphi_{\\perp 2}^{(1)N}(x,k_\\perp)]=f_q^N(x,k_\\perp)$,\nso that,\n\\begin{equation}\n\\langle \\cos 2\\phi\\rangle_{eN}|_{g=0}=-\\frac{2(1-y)}{1+(1-y)^2}\\frac{|\\vec k_\\perp|^2}{Q^2},\n\\label{eq:cos2phig=0}\n\\end{equation}\nwhich is just the result obtained in \\cite{Cahn:1978se}.\n\nIn general, we need to take QCD multiple parton scattering into account thus\n$\\langle \\cos 2\\phi\\rangle_{eN}$ is given by Eq.~(\\ref{eq:cos2phires}) where new\nquark correlation functions are involved.\nMeasurements of $\\langle \\cos 2\\phi\\rangle_{eN}$, in particular whether the\nresults deviate from  Eq.~(\\ref{eq:cos2phig=0}),  can provide useful information\non the new parton correlation functions and on multiple parton scattering as well.\n\n\n\nIf we consider $e^-+A\\to e^-+q+X$, i.e. instead of a nucleon but a nucleus target,\nall the calculations given above apply and we obtain similar results\nwith only a replacement of  the state $|N\\rangle$\nby $|A\\rangle$ in the definitions of the matrix elements and\/or parton distribution\/correlation functions.\nThe multiple gluon scattering now can be connected to different nucleons in the nucleus $A$\nthus give rise to nuclear dependence.\nIt has been shown that,  under the ``maximal two gluon approximation\",\na TMD quark distribution $\\Phi^A_\\alpha(x,k_\\perp)$ in nucleus defined in the form,\n\\begin{eqnarray}\n\\Phi^A_\\alpha(x,k_\\perp)&\\equiv &\\int \\frac{p^+dy^-d^2y_\\perp}{(2\\pi)^3}\ne^{ixp^+y^- -i\\vec  k_\\perp\\cdot \\vec y_\\perp}\\times\\nonumber\\\\\n&&\\langle A \\mid \\bar\\psi(0)\\Gamma_\\alpha{\\cal L}(0;y)\\Psi(y)\\mid A \\rangle,\n\\label{form}\n\\end{eqnarray}\nis given by a convolution of the corresponding distribution $\\Phi^N_\\alpha(x,k_\\perp)$ in nucleon\nand a Gaussian broadening,\n\\begin{equation}\n\\Phi^A_\\alpha(x,k_\\perp)\\approx\\frac{A}{\\pi \\Delta_{2F}}\n\\int d^2\\ell_\\perp e^{-(\\vec k_\\perp -\\vec\\ell_\\perp)^2\/\\Delta_{2F}}\\Phi^N_\\alpha(x,\\ell_\\perp),\n\\label{tmdgeneral}\n\\end{equation}\nwhere $\\Gamma_\\alpha$ is any gamma matrix,  $\\Psi(y)$ is a field operator;\n$\\Delta_{2F}$  is the broadening width given by,\n\\begin{equation}\n\\Delta_{2F}=\\int d\\xi^-_N \\hat q_F(\\xi_N)=\\frac{2\\pi^2\\alpha_s}{N_c}\\int d\\xi^-_N\\rho_N^A(\\xi_N)[xf^N_g(x)]_{x=0},\n\\label{eq:Delta2F}\n\\end{equation}\nwhere $\\rho_N^A(\\xi_N)$ is the spatial nucleon number density inside the nucleus\nand $f^N_g(x)$ is the gluon distribution function in nucleon.\n\nWe note that both $\\varphi_{\\rho\\alpha}(x,k_\\perp)$ and  $\\tilde\\varphi_{\\rho\\alpha}(x,k_\\perp)$\nhave the form of $\\Phi^A_\\alpha(x,k_\\perp)$.\nHence,\n\\begin{equation}\n\\varphi^{(1)A}_{\\rho\\alpha}(x,k_\\perp)\n\\approx\\frac{A}{\\pi \\Delta_{2F}}\n\\int d^2\\ell_\\perp e^{-(\\vec k_\\perp -\\vec\\ell_\\perp)^2\/\\Delta_{2F}}\\varphi_{\\rho\\alpha}^{(1)N}(x,\\ell_\\perp),\n\\label{tmdphi}\n\\end{equation}\n\\begin{equation}\n\\tilde\\varphi^{(1)A}_{\\rho\\alpha}(x,k_\\perp)\n\\approx\\frac{A}{\\pi \\Delta_{2F}}\n\\int d^2\\ell_\\perp e^{-(\\vec k_\\perp -\\vec\\ell_\\perp)^2\/\\Delta_{2F}}\\tilde\\varphi_{\\rho\\alpha}^{(1)N}(x,\\ell_\\perp),\n\\label{tmdtildephi}\n\\end{equation}\nMaking the Lorentz contration of both sides of these two equations\nwith $2\\hat k_\\perp^\\rho \\hat k_\\perp^\\alpha+d^{\\rho\\alpha}$ and $\\hat k_\\perp^{\\{\\alpha}\\varepsilon_\\perp^{\\rho\\}\\sigma}$\nrespectively, we obtain that,\n\\begin{eqnarray}\n&&|\\vec k_\\perp|^2\\varphi^{(1)A}_{\\perp2}(x,k_\\perp)\n\\approx \\frac{A}{\\pi \\Delta_{2F}}\n\\int d^2\\ell_\\perp e^{-(\\vec k_\\perp -\\vec\\ell_\\perp)^2\/\\Delta_{2F}}\\times\\nonumber\\\\\n&&\\phantom{XXXXX}\n\\left[2(\\ell_\\perp\\cdot\\hat k_\\perp)^2+\\ell_\\perp^2\\right]\\varphi_{\\perp2}^{(1)N}(x,\\ell_\\perp),\\\\\n\\label{eq:phi1NA}\n&&|\\vec k_\\perp|^2\\tilde\\varphi^{(1)A}_{\\perp2}(x,k_\\perp)\n\\approx \\frac{A}{\\pi \\Delta_{2F}}\n\\int d^2\\ell_\\perp e^{-(\\vec k_\\perp -\\vec\\ell_\\perp)^2\/\\Delta_{2F}}\\times\\nonumber\\\\\n&&\\phantom{XXXXX}\n\\left[2(\\ell_\\perp\\cdot\\hat k_\\perp)^2+\\ell_\\perp^2\\right]\\tilde\\varphi_{\\perp2}^{(1)N}(x,\\ell_\\perp).\n\\label{eq:tildephi1NA}\n\\end{eqnarray}\nAdopting a Gaussian ansatz, i.e.,\n\\begin{eqnarray}\nf_q^N(x,k_\\perp)&=&\\frac{1}{\\pi\\alpha}f_q^N(x)e^{-\\vec k_\\perp^2\/\\alpha},\\\\\n\\varphi_{\\perp2}^{(1)N}(x,k_\\perp)&=&\\frac{1}{\\pi\\beta}\\varphi_{\\perp2}^{(1)N}(x)e^{-\\vec k_\\perp^2\/\\beta},\\\\\n\\tilde\\varphi_{\\perp2}^{(1)N}(x,k_\\perp)&=&\\frac{1}{\\pi\\tilde\\beta}\\tilde\\varphi_{\\perp2}^{(1)N}(x)e^{-\\vec k_\\perp^2\/\\tilde\\beta},\n\\end{eqnarray}\nwe obtain, for those functions in nucleus,\n\\begin{equation}\nf_q^A(x,k_\\perp)\\approx\\frac{A}{\\pi\\alpha_A}f_q^N(x)e^{-\\vec k_\\perp^2\/\\alpha_A},\n\\end{equation}\n\\begin{equation}\n\\varphi_{\\perp2}^{(1)A}(x,k_\\perp)\\approx\n\\frac{A}{\\pi\\beta_A}\\Bigl(\\frac{\\beta}{\\beta_A}\\Bigr)^2\n\\varphi_{\\perp2}^{(1)N}(x)e^{-\\vec k_\\perp^2\/\\beta_A},\n\\end{equation}\n\\begin{equation}\n\\tilde\\varphi_{\\perp2}^{(1)A}(x,k_\\perp)\\approx\n\\frac{A}{\\pi\\tilde\\beta_A}\\Bigl(\\frac{\\tilde\\beta}{\\tilde\\beta_A}\\Bigr)^2\n\\tilde\\varphi_{\\perp2}^{(1)N}(x)e^{-\\vec k_\\perp^2\/\\tilde\\beta_A},\n\\end{equation}\nwhere $\\alpha_A=\\alpha+\\Delta_{2F}$, $\\beta_A=\\beta+\\Delta_{2F}$ and $\\tilde\\beta_A=\\tilde\\beta+\\Delta_{2F}$.\nThe azimuthal asymmetry is given by,\n\\begin{eqnarray}\n&&\\frac{\\langle \\cos 2\\phi\\rangle_{eA}}{\\langle \\cos 2\\phi\\rangle_{eN}}\\approx\n\\frac{\\alpha}{\\alpha_A}e^{-{\\vec k_\\perp}^2\/\\alpha_A+\\vec k_\\perp^2\/\\alpha}\\times \\nonumber\\\\\n&&\\phantom{XX}\\frac{\n\\frac{\\beta^2}{\\beta_A^3}\\varphi_{\\perp2}^{(1)N}(x_B)e^{-\\vec k_\\perp^2\/\\beta_A}\n-\\frac{\\tilde\\beta^2}{\\tilde\\beta_A^3}\n\\tilde\\varphi_{\\perp2}^{(1)N}(x_B)e^{-\\vec k_\\perp^2\/\\tilde\\beta_A}  }\n{\\Bigl[\\frac{1}{\\beta}\\varphi_{\\perp2}^{(1)N}(x_B)e^{-\\vec k_\\perp^2\/\\beta}\n-\\frac{1}{\\tilde\\beta}\\tilde\\varphi_{\\perp2}^{(1)N}(x_B)e^{-\\vec k_\\perp^2\/\\tilde\\beta}\\Bigr]}, \\nonumber\n\\end{eqnarray}\nwhich reduces to\n\\begin{equation}\n\\frac{\\langle \\cos 2\\phi\\rangle_{eA}}{\\langle \\cos 2\\phi\\rangle_{eN}}\n=(\\frac{\\beta}{\\beta+\\Delta_{2F}})^2,\n\\end{equation}\nin the case that $\\alpha=\\beta=\\tilde\\beta$.\nWe see that, in this case, for given $x_B$, $Q^2$ and $|\\vec k_\\perp|$,\n$\\langle\\cos2\\phi\\rangle_{eA}$ in deep inelastic $eA$ scattering\nis suppressed compared to that in $eN$ scattering with a\nsuppression factor $\\beta^2\/(\\beta+\\Delta_{2F})^2$.\nComparing with result of ~\\cite{Gao:2010mj}, we can see that $\\langle\\cos2\\phi\\rangle_{eA}$ is more suppressed\nthan $\\langle\\cos\\phi\\rangle_{eA}$.\nIn general, $\\beta,\\tilde \\beta$ can be different from $\\alpha$, and the ratio can also be different\nat different $k_\\perp$ and $\\Delta_{2F}$.\nAs example, we show the results for a few cases in Figs. 1a and 1b with $\\beta=\\tilde \\beta$.\n \\begin{figure} [htb\n \\resizebox{0.45\\textwidth}{!}{\\includegraphics{ratio2.eps}}\n \\resizebox{0.45\\textwidth}{!}{\\includegraphics{ratio5.eps}}\n    \\caption{(color online) Ratio $\\langle \\cos2\\phi\\rangle_{eA}\/\\langle \\cos2\\phi\\rangle_{eN}$ as a function of $\\Delta_{2F}$ for different $k_\\perp$ and $\\beta$.  }\n    \\label{fig:AfH}      \n    \\end{figure}\n\nWe see that  the asymmetry can be suppressed or enhanced depending on the values of  $k_\\perp$ and $\\Delta_{2F}$,\nand the magnitude is smaller than $\\langle\\cos\\phi\\rangle$case.\n\nIf we integrate over the magnitude of $\\vec k_\\perp$, we obtain,\n\\begin{equation}\n\\frac{\\langle\\langle \\cos\\phi\\rangle\\rangle_{eA}}{\\langle\\langle \\cos\\phi\\rangle\\rangle_{eN}}\n\\approx\\frac{(\\frac{\\beta}{\\beta_A})^2\\beta_A\\varphi_{\\perp2}^{(1)N}(x_B)\n-(\\frac{\\tilde\\beta}{\\tilde\\beta_A})^2\\tilde\\beta_A\\tilde\\varphi_{\\perp2}^{(1)N}(x_B)}\n{\\beta\\varphi_{\\perp2}^{(1)N}(x_B)-\\tilde\\beta\\tilde\\varphi_{\\perp2}^{(1)N}(x_B)},\n\\end{equation}\nwhich reduces to $\\beta\/(\\beta+\\Delta_{2F})$ for the special case $\\beta=\\tilde\\beta$.\n\n\n\n\n\\section{Summary and discussions}\n\nWe calculated the hadronic tensor and differential cross section for unpolarized SIDIS process\n$e^-+N\\to e^-+q+X$ in LO pQCD and up to twist-4 contributions\nThe results depend on a number of new TMD parton correlation functions.\nWe showed that measurements of the azimuthal asymmetry $\\langle \\cos2\\phi\\rangle$\nand its $k_\\perp$-dependence provides information on these TMD\ncorrelation  functions which in turn can shed light on the properties of multiple gluon interaction\nin hadronic processes. Under two-gluon correlation approximation, we also show the relationship between these TMD\ncorrelation functions inside large nuclei and that of a nucleon. One can therefore study the\nnuclear dependence of the azimuthal asymmetry $\\langle \\cos2\\phi\\rangle$ which\nis determined by the jet transport parameter $\\hat q$ inside nuclei. With \na Gaussian ansatz for the TMD parton correlation functions inside the nucleon, we also illustrate \nnumerically that the asymmetry $\\langle \\cos2\\phi\\rangle$ is suppressed \nin the corresponding SIDIS with nuclear target.\n\nThere exist experimental measurements of the azimuthal asymmetries in both unpolarized and polarized \nDIS \\cite{Aubert:1983cz,Arneodo:1986cf,Adams:1993hs,Breitweg:2000qh,Chekanov:2002sz,\nAirapetian:2001eg,Airapetian:2002mf,Airapetian:2004tw,Alexakhin:2005iw,Webb:2005cd,:2008rv,Alekseev:2010dm}.\nMore results are expected from CLAS at JLab and COMPASS at CERN.\nThe available data seem to be consistent with the Gaussian ansatz for the \ntransverse momentum dependence of the TMD matrix elements\\cite{Schweitzer:2010tt}.\nHowever these data are still not adequate enough to provide any precise  constraints on the form of \nthe higher twist matrix elements.  Our calculations of the azimuthal asymmetries are\nmost valid in the small transverse momentum region where NLO pQCD corrections are\nnot dominant. The high twist effects are also most accessible in intermediate region\nof $Q^{2}$.  One expects that future experiments such as those at the proposed Electron Ion \nCollider (EIC) \\cite{EIC} will be better equipped to study these high twist effects in detail.\n\n\nThis work was supported in part by the National Natural Science Foundation of China\nunder the approval Nos. 10975092 and 11035003,\nthe China Postdoctoral Science Foundation funded project under\nContract No.20090460736,\nOffice of Energy\nResearch, Office of High Energy and Nuclear Physics, Division of\nNuclear Physics, of the U.S. Department of Energy under Contract No.\nDE-AC02-05CH11231.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n\n\nWe prove a conjecture of M\\'esz\\'aros and Morales \\cite{MM} on the\nvolume of a flow polytope, which is a type $D$ analog of the\nChan-Robbins-Yuen polytope \\cite{CRY2000}. Independently from our\nwork, Zeilberger also proved the conjecture and sketched his proof in\n\\cite{Z}. In fact, our proof is the same as Zeilberger's proof. The\npurpose of this note is to give a more detailed proof of the\nconjecture.\n\nFor a function $f(z)$ with a Laurent series expansion at $z$, we\ndenote by $\\CT_z f(z)$ the constant term of the Laurent expansion of\n$f(z)$ at $0$. In other words, if $f(z)=\\sum_{n=-\\infty}^\\infty a_n\nz^n$, then $\\CT_z f(z) = a_0$.\n\nThe conjecture of M\\'esz\\'aros and Morales can be stated as the\nfollowing constant term identity.\n\n\\begin{conj} \\cite[Conjecture~7.6]{MM} \\label{MMconj}\nFor an integer $n\\ge2$, we have\n\\[\n  \\CT_{x_n}\\CT_{x_{n-1}}\\cdots \\CT_{x_1}\n\\prod_{j=1}^n x_j^{-1}(1-x_j)^{-2}\n\\prod_{1\\le j<k\\le  n} (x_k-x_j)^{-1} (1-x_k-x_j)^{-1} \n= 2^{n^2} \\prod_{k=1}^n \\Cat(k),\n\\]  \nwhere $\\Cat(k)=\\frac1{k+1}\\binom{2k}{k}$. \n\\end{conj}\n\nConjecture~\\ref{MMconj} is a type $D_n$ analog of the constant term\nidentity\n\\begin{equation}\n  \\label{eq:1}\n  \\CT_{x_n}\\CT_{x_{n-1}}\\cdots \\CT_{x_1}\n\\prod_{j=1}^n (1-x_j)^{-2}\n\\prod_{1\\le j<k\\le  n} (x_k-x_j)^{-1} \n=\\prod_{k=1}^n \\Cat(k),\n\\end{equation}\nwhich was conjectured by Chan, Robbins, and Yuen \\cite{CRY2000} and\nproved by Zeilberger \\cite{Zeilberger1999}.\n\nIn Section~\\ref{sec:zeilb-proof-chan} we recall Zeilberger's\nreformulation of Morris' constant term identity. In\nSection~\\ref{sec:proof-conjecture} we prove Conjecture~\\ref{MMconj}.\n\n\\section{Zeilberger's reformulation of Morris' constant term identity}\n\\label{sec:zeilb-proof-chan}\n\nZeilberger \\cite{Zeilberger1999} rewrote Morris' identity\n\\cite{Morris} as follows: For nonnegative integers $a$ and $b$ and a\npositive half integer $c$, we have\n\\begin{equation}\n  \\label{eq:zeilberger}\n\\CT_{x_n}\\CT_{x_{n-1}}\\cdots \\CT_{x_1} \\prod_{i=1}^n (1-x_i)^{-a} x_i^{-b} \n\\prod_{1\\le i<j\\le  n}(x_j-x_i)^{-2c} \n= \\frac{1}{n!} \\prod_{j=0}^{n-1} \\frac{\\Gamma(a+b+(n-1+j)c)\\Gamma(c)}\n{\\Gamma(a+jc)\\Gamma(c+jc)\\Gamma(b+jc+1)}.\n\\end{equation}\nZeilberger \\cite{Zeilberger1999} showed that, when $a=2,b=0,c=1\/2$,\nthe right hand side of \\eqref{eq:zeilberger} is\n\\begin{equation}\n  \\label{eq:cat}\n\\frac{1}{n!} \\prod_{j=0}^{n-1}\n\\frac{\\Gamma(\\frac{n+3+j}2)\\Gamma(\\frac 12)}\n{\\Gamma(\\frac{4+j}2)\\Gamma(\\frac{1+j}2)\\Gamma(\\frac{2+j}2)}=  \n\\prod_{k=1}^n \\Cat(k),\n\\end{equation}\nwhich together with \\eqref{eq:zeilberger} implies \\eqref{eq:1}. \n\nBy Cauchy's integral formula, if $f(z)$ has a Laurent series expansion\nat $0$, we have\n\\begin{equation}\n  \\label{eq:CT}\n\\CT_z f(z) = \\frac{1}{2\\pi i} \\oint_C \\frac{f(z)} z dz,\n\\end{equation}\nwhere $C$ is the circle $\\{z: |z|=\\epsilon\\}$ oriented\ncounterclockwise for a real number $\\epsilon >0$ such that $f(z)$ is\nholomorphic inside $C$ except $0$.  Thus \\eqref{eq:zeilberger} can be rewritten\nas\n\\begin{multline}\n  \\label{eq:zeilberger2}\n\\frac{1}{(2\\pi i)^n}\\oint_{C_n}\\dots\\oint_{C_1}  \\prod_{j=1}^n (1-x_j)^{-a} x_j^{-b-1} \n\\prod_{1\\le j<k\\le  n}(x_k-x_j)^{-2c}  dx_1\\cdots dx_n\\\\\n= \\frac{1}{n!} \\prod_{j=0}^{n-1} \\frac{\\Gamma(a+b+(n-1+j)c)\\Gamma(c)}\n{\\Gamma(a+jc)\\Gamma(c+jc)\\Gamma(b+jc+1)},\n\\end{multline}\nwhere $C_j$ is the circle $\\{z: |z|=j\\epsilon\\}$ oriented\ncounterclockwise for a number $0<\\epsilon<\\frac 1n$.\n\n\n\\section{Proof of the conjecture of M\\'esz\\'aros and Morales}\n\\label{sec:proof-conjecture}\n\nWe will prove the following theorem. \n\\begin{thm}\\label{thm} For a nonnegative integer $a$ and a positive half integer\n  $c$, we have\n\\begin{multline*}\n\\CT_{x_n}\\CT_{x_{n-1}}\\cdots \\CT_{x_1}  \\prod_{j=1}^n x_j^{-a+1}(1-x_j)^{-a}\n\\prod_{1\\le j<k\\le  n} (x_j-x_k)^{-2c} (1-x_j-x_k)^{-2c}  \n\\\\= 2^{2an+4c\\binom n2-2n} \n\\frac{1}{n!} \\prod_{j=0}^{n-1} \\frac{\\Gamma(a-\\frac12+(n-1+j)c)\\Gamma(c)}\n{\\Gamma(\\frac12+jc)\\Gamma(c+jc)\\Gamma(a+jc)}.\n\\end{multline*}\n\\end{thm}\n\nWe can get Conjecture~\\ref{MMconj} as the special case $a=2,c=1\/2$ of\nTheorem~\\ref{thm} as follows. When $a=2,c=1\/2$, the right hand side of\nthe formula in Theorem~\\ref{thm} is\n\\[\n\\frac{2^{n^2+n}}{n!}\\prod_{j=0}^{n-1}\n\\frac{\\Gamma(\\frac{n+2+j}2)\\Gamma(\\frac 12)}\n{\\Gamma(\\frac{1+j}2)\\Gamma(\\frac{1+j}2)\\Gamma(\\frac{4+j}2)}.\n\\]\nSince\n\\[\n\\prod_{j=0}^{n-1}\n\\frac{\\Gamma(\\frac{n+3+j}2)\\Gamma(\\frac 12)}\n{\\Gamma(\\frac{4+j}2)\\Gamma(\\frac{1+j}2)\\Gamma(\\frac{2+j}2)}\n\\left\/\n\\prod_{j=0}^{n-1}\n\\frac{\\Gamma(\\frac{n+2+j}2)\\Gamma(\\frac 12)}\n{\\Gamma(\\frac{1+j}2)\\Gamma(\\frac{1+j}2)\\Gamma(\\frac{4+j}2)}=  \n\\right.\n\\frac{\\Gamma(\\frac{2n+2}2)\\Gamma(\\frac{1}2)}\n{\\Gamma(\\frac{n+2}2)\\Gamma(\\frac{n+1}2)}\n=2^n, \n\\]\nwe obtain Conjecture~\\ref{MMconj} by \\eqref{eq:cat}.\n\nFor the rest of this section we prove Theorem~\\ref{thm}.  From now on\nwe assume that $\\epsilon>0$ is a very small number.\n\nBy \\eqref{eq:CT} we have\n\\begin{align*}\n&  \\CT_{x_n}\\CT_{x_{n-1}}\\cdots \\CT_{x_1}  \\prod_{j=1}^n x_j^{-a+1}(1-x_j)^{-a}\n\\prod_{1\\le j<k\\le  n} (x_k-x_j)^{-2c} (1-x_k-x_j)^{-2c} \\\\\n&= \\frac{1}{(2\\pi i)^n}\\oint_{C_n}\\cdots\\oint_{C_1}\n\\prod_{j=1}^n x_j^{-a}(1-x_j)^{-a}\n\\prod_{1\\le j<k\\le  n} (x_k-x_j)^{-2c} (1-x_k-x_j)^{-2c}\ndx_1\\cdots dx_n,\n\\end{align*}\nwhere $C_j$ is the circle $\\{z: |z|=j\\epsilon\\}$ oriented\ncounterclockwise. \n\nUsing the change of variables $x_j = \\frac{1-z_j}2$ or $z_j=1-2x_j$,\nthe above is equal to\n\\begin{multline*}\n\\frac{1}{(2\\pi i)^n}\\oint_{C'_n}\\cdots\\oint_{C'_1}\n\\prod_{j=1}^n \\left(\\frac{1-z_j}2\\right)^{-a}\n\\left(\\frac{1+z_j}2\\right)^{-a} \\\\\n\\times \\prod_{1\\le j<k\\le  n} \\left(\\frac{-z_k+z_j}2\\right)^{-2c}\n\\left(\\frac{z_k+z_j}2\\right)^{-2c} (-2)^{-n} dz_1\\cdots dz_n\\\\\n=\\frac{(-1)^n2^{2an+4c\\binom n2-n}}{(2\\pi i)^n}\\oint_{C'_n}\\cdots\\oint_{C'_1}\n\\prod_{j=1}^n (1-z_j^2)^{-a}\n\\prod_{1\\le j<k\\le  n} (z_j^2-z_k^2)^{-2c}\ndz_1\\cdots dz_n,\n\\end{multline*}\nwhere $C'_j$ is the circle $\\{z: |z-1|=2j\\epsilon\\}$ oriented\ncounterclockwise.\n\nUsing the change of variables $z_j^2 = y_j$ or $z_j=y_j^{1\/2}$, the above is\nequal to\n\\begin{align*}\n&\\frac{(-1)^n2^{2an+4c\\binom n2-2n}}{(2\\pi i)^n}\\oint_{C''_n}\\cdots\\oint_{C''_1}\n\\prod_{j=1}^n (1-y_j)^{-a} y_j^{-1\/2}\n\\prod_{1\\le j<k\\le  n} (y_j-y_k)^{-2c}\ndy_1\\cdots dy_n,\n\\end{align*}\nwhere $C''_j$ is the circle $\\{z: |z-1|=4j\\epsilon\\}$ oriented\ncounterclockwise. This is because if $C_j'$ is parametrized by\n$1+2j\\epsilon e^{i\\theta}$ for $0\\le \\theta\\le 2\\pi$, then the image\nof $C_j'$ under the map $z\\mapsto z^2$ can be parametrized by\n$1+4j\\epsilon e^{i\\theta} + 4j^2\\epsilon^2 e^{2i\\theta}$ for $0\\le\n\\theta\\le 2\\pi$. Since $\\epsilon$ is very small we can deform this\nimage to the circle $C_j''$ without changing the contour integral.\n\nUsing the change of variables $t_j = 1-y_j$, the above is\nequal to\n\\begin{align*}\n\\frac{2^{2an+4c\\binom n2-2n}}{(2\\pi i)^n} \\oint_{C'''_n}\\cdots\\oint_{C'''_1}\n\\prod_{j=1}^n t_j^{-a} (1-t_j)^{-1\/2}\n\\prod_{1\\le j<k\\le  n} (t_k-t_j)^{-2c}\ndt_1\\cdots dt_n,\n\\end{align*}\nwhere $C'''_j$ is the circle $\\{z: |z|=4j\\epsilon\\}$ oriented\ncounterclockwise.  \n\nUsing \\eqref{eq:zeilberger2}, we finish the proof of\nTheorem~\\ref{thm}. \n\n\\section*{Acknowledgement}\nThe author would like to thank Dennis Stanton for helpful discussion. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nIn his seminal paper~\\cite{Foellmer_1981},\nF{\\\"o}llmer presented a new perspective on\nIt{\\^o}'s stochastic calculus.\nThe main theorem of F{\\\"o}llmer~\\cite{Foellmer_1981}\nstates that a deterministic c\\`{a}dl\\`{a}g path satisfies the It{\\^o} formula\nprovided it has quadratic variation\nalong a given sequence of partitions. \nThis theorem enables us to construct the It{\\^o} integral \n$\\int_0^t f(X_{s-}) \\mathrm{d}X_s$ for\na sufficiently nice function $f$ and a path $X$ of quadratic variation.\nThis suggests the possibility \nof developing an analogue of the It{\\^o} calculus\nin completely analytic, probability-free situations.\nWe call this framework F{\\\"o}llmer's pathwise It{\\^o} calculus\nor, more simply, the It{\\^o}-F{\\\"o}llmer calculus.\nIt can be regarded as a deterministic counterpart of the classical It{\\^o} calculus.\n\nRecently, the It{\\^o}-F{\\\"o}llmer calculus has been receiving increasing attention \nfrom the viewpoint of its financial applications.\nIt is regarded as a useful tool to study financial theory under probability-free settings and has been used to construct financial strategies in a strictly pathwise manner \n(see, e.g., F{\\\"o}llmer and Schied~\\cite{Foellmer_Schied_2013},\nSchied~\\cite{Schied_2014}, \nDavis, Ob{\\l}{\\'o}j and Raval~\\cite{Davis_Obloj_Raval_2014},\nand Schied, Speiser, and Voloshchenko~\\cite{Schied_Speiser_Voloshchenko_2018}).\nWe expect that the It{\\^o}-F{\\\"o}llmer calculus will\nhave a growing presence\nin financial applications.\n\nThe It{\\^o}-F{\\\"o}llmer calculus can be applied to\na stochastic process having quadratic variation.\nA standard example of such a process is a semimartingale.\nHowever, it is known that \nthe class of processes possessing quadratic variation \nis strictly larger than that of semimartingales\n(see, e.g., F{\\\"o}llmer~\\cite{Foellmer_1981,Foellmer_1981b}).\nIn this sense, the It{\\^o}-F{\\\"o}llmer calculus\nenables us to extend stochastic integration theory beyond semimartingales.\n\nThere are several approaches to pathwise construction of stochastic integration.\nFirst, we mention classic studies by\nBichteler~\\cite{Bichteler_1981},\nKarandikar~\\cite{Karandikar_1995},\nand Willinger and Taqqu~\\cite{Willinger_Taqqu_1988,Willinger_Taqqu_1989}, but\nsee also Nutz~\\cite{Nutz_2012}.\nThe theory of Vovk's outer measure and typical\npaths was pioneered by Vovk~\\cite{Vovk_2008,Vovk_2011,Vovk_2012,Vovk_2015,Vovk_2016},\nand further developed by several authors, including \nPerkowski and Pr{\\\"o}mel~\\cite{Perkowski_Proemel_2016,Perkowski_Proemel_2015b},\n{\\L}ochowski~\\cite{Lochowski_2016},\n{\\L}ochowski, Perkowski, and Pr{\\\"o}mel~\\cite{Lochowski_Perkowski_Proemel_2018},\nand Bartl, Kupper, and Neufeld~\\cite{Bartl_Kupper_Neufeld_2019}.\nRusso and Vallois~\\cite{Russo_Vallois_1991,\n    Russo_Vallois_1993,\n    Russo_Vallois_1993b,\n    Russo_Vallois_1995,\n    Russo_Vallois_1996,\n    Russo_Vallois_2000,\n    Russo_Vallois_2007}\ndeveloped a theory called stochastic calculus via regularization.\nThe rough path theory,\npioneered by Lyons~\\cite{Lyons_1998},\nand its generalization have become \nimportant in stochastic calculus and its applications.\nIn addition, we refer to\nGubinelli~\\cite{Gubinelli_2004},\nGubinelli and Tindel~\\cite{Gubinelli_Tindel_2010},\nFriz and Shekhar~\\cite{Friz_Shekhar_2017}, \nand Friz and Zhang~\\cite{Friz_Zhang_2018}.\nSome studies have investigated the relation between the It{\\^o}-F{\\\"o}llmer\ncalculus and rough path theory (see, e.g.,\nPerkowski and Pr{\\\"o}mel~\\cite{Perkowski_Proemel_2016},\nand Friz and Hairer~\\cite{Friz_Hairer_2014}).\n\nAmong the various pathwise methods,\nwe consider F{\\\"o}llmer's approach to be\nsimplest and most intuitively clear.\nIt needs only elementary arguments to establish \ncalculation rules such as It{\\^o}'s formula \nwithin this framework.\nMoreover, the It{\\^o}-F{\\\"o}llmer calculus requires only a minimal \nassumption that the integrator has a quadratic variation.\nWe believe that these are advantages of this theory,\nand also that careful observation of F{\\\"o}llmer's theory\nallows us to know phenomena occurring on paths of processes\nwhen we consider semimartingales and stochastic integration.\n\nIncreasingly many works related to It{\\^o}-F{\\\"o}llmer calculus\nhave been appearing recently.\nFirst, we refer to Sondermann~\\cite{Sondermann_2006},\nSchied~\\cite{Schied_2014}, Hirai~\\cite{Hirai_2019}, and\nCont and Perkowski~\\cite{Cont_Perkowski_2019}.\nSchied~\\cite{Schied_2016},\nand Mishura and Schied~\\cite{Mishura_Schied_2016}\nconstruct deterministic continuous\npaths with nontrivial quadratic variation.\nSee also Chiu and Cont~\\cite{Chiu_Cont_2018}.\nFunctional extensions of the It{\\^o}-F{\\\"o}llmer calculus \nhave been developed by Dupire~\\cite{Dupire_2009}, \nCont and Fourni{\\'e}~\\cite{Cont_Fournie_2010,\nCont_Fournie_2010b,\nCont_Fournie_2013}, and \nAnanova and Cont~\\cite{Ananova_Cont_2017}, for example.\nExtension of the It{\\^o}-F{\\\"o}llmer calculus\nin terms of local times has been investigated in\nDavis, Ob{\\l}{\\'o}j, and Raval~\\cite{Davis_Obloj_Raval_2014},\nDavis, Ob{\\l}{\\'o}j, and Siorpaes~\\cite{Davis_Obloj_Siorpaes_2018},\n{\\L}ochowski et al.~\\cite{Lochowski_Obloj_Promel_Siorpaes_2021},\nand Hirai~\\cite{Hirai_2016,Hirai_2021}.\n\nTo our knowledge, however,\nthe It{\\^o}-F{\\\"o}llmer calculus in an infinite-dimensional setting\nhas not yet been sufficiently studied.\nStochastic integration in infinite dimensions naturally appears\nwhen we treat stochastic partial differential equations\n(see, e.g. Da Prato and Zabczyk~\\cite{DaPrato_Zabczyk_2014}).\nThese have played an important role\nin modelling term structures of interest rates or forward variances in mathematical finance,\nand also in models of statistical mechanics and quantum field theories.\nThen we aim to extend F{\\\"o}llmer's theory \nto Banach space-valued paths.\nIn this paper, we prove the It{\\^o} formula for \npaths in Banach spaces\nwith suitably defined quadratic variation.\nWe will study relations between various quadratic variations\nand prove some transformation formulae for quadratic variations \nin our second paper in this series~\\cite{Hirai_2021c}.\nWe not only generalize the state space of paths,\nbut also relax the assumption on the sequence of\npartitions along which we consider the quadratic variation.\nIn the context of the It{\\^o}-F{\\\"o}llmer calculus,\ntwo types of assumptions about a sequence of partitions are \nfrequently used.\nOne is $\\lvert \\pi_n \\rvert \\to 0$, as used in F{\\\"o}llmer~\\cite{Foellmer_1981},\nand the other is the condition $O_t^{-}(X;\\pi_n) \\to 0$,\nwhich is used in many papers handling continuous paths \nand some dealing with discontinuous paths\nsuch as Vovk~\\cite{Vovk_2015}.\nIn this paper, we introduce new conditions\nto a sequence of partitions and a c\\`{a}dl\\`{a}g path\n(Definition~\\ref{2d}),\nwhich gives a unified approach.\n\nThere have been many attempts to extend classical stochastic \ncalculus to Banach or Hilbert space-valued processes.\nExamples include Kunita~\\cite{Kunita_1970},\nMetivier~\\cite{Metivier_1972},\nPellaumail~\\cite{Pellaumail_1973},\nYor~\\cite{Yor_1974},\nGravereaux and Pellaumail~\\cite{Gravereaux_Pellaumail_1974},\nMetivier and Pistone~\\cite{Metivier_Pistone_1975},\nMeyer~\\cite{Meyer_1977a},\nMetivier and Pellaumail~\\cite{Metivier_Pellaumail_1980b},\nGy{\\\"o}ngy and Krylov~\\cite{Gyongy_Krylov_1980,Gyongy_Krylov_1982},\nGy{\\\"o}ngy~\\cite{Gyongy_1982},\nMetivier~\\cite{Metivier_1982},\nPratelli~\\cite{Pratelli_1988},\nBrooks and Dinculeanu~\\cite{Brooks_Dinculeanu_1990},\nMikulevicius and Rozovskii~\\cite{Mikulevicius_Rozovskii_1998,Mikulevicius_Rozovskii_1999},\nDinculeanu~\\cite{Dinculeanu_2000}, and\nDe Donno and Pratelli~\\cite{DeDonno_Pratelli_2006}.\nNote that Di Girolami, Fabbri, and Russo~\\cite{DiGirolami_Fabbri_Russo_2014}\ntreats quadratic covariation of Banach space-valued processes\nwithin the framework of stochastic calculus via regularization,\nwith F{\\\"o}llmer's calculus in mind.\n\nOur method can be interpreted as a deterministic counterpart\nof these stochastic integration theories in Banach spaces.\nSome of the works listed above,\nsuch as Metivier and Pellaumail~\\cite{Metivier_Pellaumail_1980b}\nand Dinculeanu~\\cite{Dinculeanu_2000},\ngive a proof of It{\\^o}'s formula\nin a similar manner to F{\\\"o}llmer's calculus.\nOne of the advantages of our approach \nappears in the statement of the It{\\^o} formula.\nFor a function $f$ to satisfy the It{\\^o} formula,\nwe require $f$ to be just $C^2$-class,\nwhile a stochastic approach needs some additional assumptions about \nthe boundedness of $f$ and its derivatives. \n\nNow we summarize the main result of F{\\\"o}llmer~\\cite{Foellmer_1981}. \nLet $\\Pi = (\\pi_n)_{n \\in \\mathbb{N}}$ be a sequence of partitions of\n$\\mathbb{R}_{\\geq 0}$ such that $\\lvert \\pi_n \\rvert \\to 0$ as $n \\to \\infty$.\nWe say that a {c\\`{a}dl\\`{a}g} path $X \\colon \\mathbb{R}_{\\geq 0} \\to \\mathbb{R}$ has quadratic variation along $\\Pi$\nif there exists a {c\\`{a}dl\\`{a}g} increasing function $[X,X]$ such that for all $t \\in \\mathbb{R}_{\\geq 0}$,\n\\begin{enumerate}\n    \\item $\\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n} (X_{s \\wedge t} - X_{r \\wedge t})^2$\n        converges to $[X,X]_t$ as $n \\to \\infty$, and\n    \\item $\\Delta [X,X]_{t} = (\\Delta X_t)^2$.\n\\end{enumerate}\nAn $\\mathbb{R}^d$-valued {c\\`{a}dl\\`{a}g} path $X = (X^1,\\dots,X^d)$\nhas quadratic variation along $\\Pi$ if\nthe real-valued path $X^i + X^j$ has quadratic variation along the same sequence\nfor each $i$ and $j$.\nF{\\\"o}llmer~\\cite{Foellmer_1981} proved that if $X$ has quadratic variation, then for any $f \\in C^2(\\mathbb{R}^d)$\nthe path $t \\mapsto f(X_t)$ satisfies It{\\^o}'s formula.\nThat is,\n\\begin{align} \\label{1.1c}\n    f(X_t)\n    & = \n    f(X_0)\n    + \\int_0^t \\langle Df (X_{s-}), \\mathrm{d}X_s \\rangle\n    + \\frac{1}{2} \\sum_{1 \\leq i,j \\leq d} \\int_0^t \\frac{\\partial^2 f}{\\partial x_i \\partial x_j}(X_{s-}) \\mathrm{d}[X^i,X^j]_s    \\\\\n    & \\quad\n    + \\sum_{0 < s \\leq t}\n        \\left\\{ \n            \\Delta f(X_s) \n            - \\sum_{1 \\leq i \\leq d} \\frac{\\partial f}{\\partial x_i}(X_{s-})\\Delta X^i_s\n            - \\frac{1}{2} \\sum_{1 \\leq i,j \\leq d}\n                \\frac{\\partial^2 f}{\\partial x_i \\partial x_j}(X_{s-})\n                \\Delta X^{i}_s \\Delta X^{j}_s  \\notag\n        \\right\\}\n\\end{align}\nholds for all $t \\in \\mathbb{R}_{\\geq 0}$.\nHere, the first term on the right-hand side,\nwhich we call the It{\\^o}-F{\\\"o}llmer integral along $\\Pi$,\nis defined as the limit\n\\begin{equation*}\n    \\int_0^t \\langle Df (X_{s-}), \\mathrm{d}X_s \\rangle\n    =\n    \\lim_{n \\to \\infty} \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n}\n        \\langle D f(X_r), X_{s \\wedge t} - X_{r \\wedge t} \\rangle.\n\\end{equation*}\nF{\\\"o}llmer's theorem claims that if $X$ has quadratic variation along $\\Pi$,\nthen the It{\\^o}-F{\\\"o}llmer integral above\nexists and it satisfies equation~\\eqref{1.1c}.\n\nAs stated above, \nour aim is to extend F{\\\"o}llmer's pathwise It{\\^o} formula to Banach space-valued paths.\nLet us give a brief summary of the result.\nLet $E$ be a Banach space.\nWe say that an $E$-valued c\\`{a}dl\\`{a}g path $X$\nhas tensor quadratic covariation along $(\\pi_n)$\nif there is a c\\`{a}dl\\`{a}g path $[X,X] \\colon \\mathbb{R}_{\\geq 0} \\to E \\widehat{\\otimes}_{\\pi} E$\nof finite variation such that \n\\begin{enumerate}\n    \\item the sequence $\\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n} (X_{s \\wedge t} - X_{r \\wedge t})^{\\otimes 2}$ \n        converges to $[X,X]_t$ in the norm topology of $E \\widehat{\\otimes}_{\\pi} E$\n        for all $t \\geq 0$, and\n    \\item the equation $\\Delta [X,Y]_t = \\Delta X_t^{\\otimes 2}$ holds for all $t \\geq 0$.\n\\end{enumerate}\nHere $E \\widehat{\\otimes}_{\\pi} E$ denotes the projective tensor product of the Banach space $E$.\nMoreover, we say that $X$ has upper scalar quadratic variation if there is an \nincreasing c\\`{a}dl\\`{a}g path $\\overline{Q}(X) \\colon \\mathbb{R}_{\\geq 0} \\to \\mathbb{R}_{\\geq 0}$\nsuch that \n\\begin{equation*}\n    \\varlimsup_{n \\to \\infty} \\sum_{\\mathopen{\\rbrack} r,s \\rbrack} \\lVert X_{s \\wedge t} - X_{r \\wedge t} \\rVert^2\n    \\leq \\overline{Q}(X)_t.\n\\end{equation*} \nWe say that a sequence of partitions $(\\pi_n)$\ncontrols a path $X \\colon \\lbrack 0,\\infty \\mathclose{\\lbrack} \\to E$\nin a Banach space if it satisfies Conditions~(C1)--(C3) of Definition~\\ref{2d}.\nRoughly speaking, Conditions~(C1) and (C2) state that \n$(\\pi_n)$ reconstructs the information of the jumps of $X$.\nCondition~(C3) means that \n$(\\pi_n)$ controls the oscillation of $X$ in some sense.\nUnder these settings, our main result (Theorem~\\ref{2e}) is stated as follows.\nLet a c\\`{a}dl\\`{a}g path $X \\colon \\mathbb{R}_{\\geq 0} \\to E$ have\nboth tensor and upper scalar quadratic variations\nalong a sequence of partitions $(\\pi_n)_{n \\in \\mathbb{N}}$,\nand let $A \\colon \\mathbb{R}_{\\geq 0} \\to F$ be a c\\`{a}dl\\`{a}g path of finite variation. \nSuppose that $(\\pi_n)$ controls $(X,A)$ and \nleft-side discretization of $(X,A)$ along $(\\pi_n)$\napproximates $(X_{-},A_{-})$ pointwise\n(see Definition~\\ref{2b} for the exact definition).\nIf $f \\in C^{1,2}(E \\times F,G)$,\nthen the composite function $f \\circ (A,X)$ satisfies\n\\begin{align*} \n& \n    f(A_t,X_t) - f(A_0,X_0)  \\\\\n    & \\qquad = \n    \\int_{0}^{t} \\langle D_a f(A_{s-},X_{s-}), \\mathrm{d}A^{\\mathrm{c}}_s \\rangle \n    + \\int_0^t \\langle D_x f (A_{s-},X_{s-}),\\mathrm{d}X_s \\rangle   \\notag\\\\\n    & \\qquad\\quad \n    + \\frac{1}{2} \\int_{0}^{t} \\langle D_x^2 f(A_{s-},X_{s-}), \\mathrm{d}[X,X]^{\\mathrm{c}}_s \\rangle \n    + \\sum_{0 < s \\leq t} \\left\\{ \\Delta f(A_{s},X_{s}) - \\langle D_x f(A_{s-},X_{s-}), \\Delta X_s \\rangle \\right\\}. \n\\end{align*}\n\nTo conclude this section,\nwe give an outline of the remainder of this paper.\nIn Section~2, we set up the basic notation and terminology \nand state the main result of the paper.\nAs a preliminary, we recall basic properties of \nc\\`{a}dl\\`{a}g paths in a Banach space\nand those of c\\`{a}dl\\`{a}g paths of finite variation in Section~3.\nIn Section~4, we study conditions\non the sequence of partitions and the relation between them and \nc\\`{a}dl\\`{a}g paths.\nFundamental properties of quadratic variations are studied in Section~5.\nSection~6 shows the existence of \nquadratic variations for paths of finite variation. \nIn Section~7, we prove a lemma \n(Lemma~\\ref{2g}) that is essentially used in the main theorems of this paper.\nIn the last section, Section~8, \nwe finally prove the It{\\^o} formula \nfor a Banach space-valued path with quadratic variation.\nWe give a $C^{1,2}$-type It{\\^o} formula (Theorem~\\ref{2e}),\nwhich immediately leads to the standard $C^2$ formulation.\nIn that theorem, we can see an advantage of a strictly pathwise approach\nto It{\\^o} integration.\n\n\\section{Settings and the main result}\n\nIn this section, \nwe introduce the main theorem of this paper,\nnamely, the It{\\^o} formula\nwithin the framework of It{\\^o}-F{\\\"o}llmer calculus in Banach spaces.\nThe statement of the theorem will be found in Theorem~\\ref{2e}.\n\nFirst, we introduce the basic notation and terminology used in this paper.\nThe symbol $\\mathbb{N}$ denotes the set of natural numbers $\\{ 0,1,2,\\dots \\}$\nand $\\mathbb{R}$ denotes the real numbers.\nIf $A$ is a subset of $\\mathbb{R}$ and $a \\in \\mathbb{R}$, we define \n$A_{\\geq a} = \\{ x \\in A \\mid x \\geq a \\}$.\nIf $E$ and $F$ are two real Banach spaces,\n$\\mathcal{L}(E,F)$ denotes the space of bounded linear maps from $E$ to $F$.\nIn addition, given another Banach space $G$,\nwe define $\\mathcal{L}^{(2)}(E,F;G)$ as the \nspace of bounded bilinear maps from $E \\times F$ to $G$.\n\nLet $[0,\\infty \\mathclose{\\lbrack} = \\mathbb{R}_{\\geq 0} = \\{ r \\in \\mathbb{R} \\mid t \\geq 0 \\}$\nand let $E$ be a Banach space.\nA \\emph{c\\`{a}dl\\`{a}g} path in $E$\nis a function $X \\colon [0,\\infty \\mathclose{\\lbrack} \\to E$,\nwhich is right-continuous at every $t \\geq 0$\nand has the left limit at every $t > 0$.\nThe term \\emph{right regular} is also used\nto stand for the same property.\nThe symbol $D([0,\\infty \\mathclose{\\lbrack}, E)$\nor $D(\\mathbb{R}_{\\geq 0},E)$ denotes the set of \nall c\\`{a}dl\\`{a}g paths in $E$.\nIf $X$ is an element of $D([0,\\infty \\mathclose{\\lbrack}, E)$, we define\n\\begin{equation*}\n    X(t-) = \\lim_{s \\uparrow\\uparrow t} X(s) = \\lim_{s \\to t, s < t} X_s, \\qquad\n    \\Delta X(t) = X(t) - X(t-).\n\\end{equation*}\nWe also use $X_t$, $X_{t-}$, and $\\Delta X_t$ \nto indicate the values $X(t)$, $X(t-)$, and $\\Delta X(t)$, respectively.\nNext, set\n\\begin{gather*}\n    D(X) = \\{ t \\in \\mathbb{R}_{\\geq 0} \\mid \\lVert \\Delta X_t \\rVert \\neq 0 \\}  \\\\\n    D_{\\varepsilon}(X) = \\{ t \\in \\mathbb{R}_{\\geq 0} \\mid \\lVert \\Delta X_t \\rVert \\geq \\varepsilon \\}  \\\\\n    D^{\\varepsilon}(X) = D(X) \\setminus D_{\\varepsilon}(X)\n        = \\{ t \\in \\mathbb{R}_{\\geq 0} \\mid 0 < \\lVert \\Delta X_s \\rVert < \\varepsilon \\}.\n\\end{gather*}\nWe simply write $D$, $D_{\\varepsilon}$, and $D^{\\varepsilon}$ if there is no ambiguity.\nGiven a discrete set $D \\subset [0,\\infty[$ and a c\\`{a}dl\\`{a}g path $X$, we define\n\\begin{equation*}\n    J_D(X)_t = J(D;X)_t = \\sum_{0 < s \\leq t} \\Delta X_s 1_{D}(s).\n\\end{equation*}\nThen $J_D(X)$ is a c\\`{a}dl\\`{a}g path of finite variation.\nFor abbreviation, we often write\n$J_\\varepsilon(X)$ instead of $J(D_{\\varepsilon}(X);X)$.\nThe difference of a path $X$ along an interval $I = \\mathopen{\\rbrack} r,s \\rbrack$\nis defined as $\\delta_I X = \\delta_r^s X = X_s - X_r$.\nIf we are given an additional point $t$,\nlet $\\delta_I X_t = \\delta_r^s X_t = X_{s \\wedge t} - X_{r \\wedge t}$.\n\nThroughout this paper,\nthe term `partition of $\\mathbb{R}_{\\geq 0}$' always \nmeans the set of intervals of the form\n$\\pi = \\{ \\mathopen{\\rbrack} t_i,t_{i+1} ; i \\in \\mathbb{N} \\}$\nwhich satisfies $0=t_0 < t_1 < \\dots \\to \\infty$. \nThe set of all partitions of $\\mathbb{R}_{\\geq 0}$ is \ndenoted by $\\mathop{\\mathrm{Par}}(\\mathbb{R}_{\\geq 0})$ or $\\mathop{\\mathrm{Par}} (\\lbrack 0,\\infty \\mathclose{\\lbrack} )$.\nSimilarly, $\\mathop{\\mathrm{Par}}([a,b])$ indicates the set of partitions of the \nform $\\pi = \\{ \\mathopen{\\rbrack} t_i,t_{i+1} \\rbrack ; 0 \\leq i \\leq n-1 \\}$\nwith $a=t_0 < t_1 < \\dots < t_n = b$.\n\n\\begin{Def} \\label{2b}\nLet $E,F,G$ be Banach spaces, $b \\colon E \\times F \\to G$ a bounded bilinear map,\nand $(X,Y) \\in D(\\mathbb{R}_{\\geq 0},E \\times F)$.\nTwo paths $X$ and $Y$ have\n\\emph{quadratic covariation\nalong $\\Pi = (\\pi_n)_{n \\in \\mathbb{N}}$ with respect to $b$}\nif there exists a $G$-valued c\\`{a}dl\\`{a}g path $Q_b(X,Y)$ of \nfinite variation such that\n\\begin{enumerate}\n    \\item for all $t \\in \\mathbb{R}_{\\geq 0}$\n    \\begin{equation*}\n        \\sum_{I \\in \\pi_n} b(\\delta_I X_t, \\delta_I Y_t) \\xrightarrow[n \\to \\infty]{\\text{in $G$}} Q_b(X,Y)_t,\n    \\end{equation*} and\n    \\item for all $t \\in \\mathbb{R}_{\\geq 0}$, the jump of $b \\circ (X,Y)$ is given by\n        \\begin{equation*}\n            \\Delta Q_b(X,Y)_t = b(\\Delta X_t,\\Delta Y_t).\n        \\end{equation*}\n\\end{enumerate}\nThen the path $Q_b(X,Y)$ is called\nthe quadratic covariation of $X$ and $Y$ with respect to $b$.\nIf $E=F$ and $X=Y$,\nwe call $Q_b(X,X)$ the quadratic variation of $X$ with respect to $b$.\n\\end{Def}\n\nThe quadratic covariation $Q_b(X,Y)$ depends\non the sequence of partition $\\Pi$.\nIf there is the possibility of confusion, we also use $Q_b^{\\Pi}(X,Y)$\nto indicate the quadratic covariation $Q_b(X,Y)$.\n\nFor convenience, we often write\n\\begin{equation*}\n    Q_b^{\\pi}(X,Y)_t = \\sum_{I \\in \\pi} b(\\delta_I X_t,\\delta_I Y_t).\n\\end{equation*}\nThe continuity of the map $(s,t) \\mapsto s \\wedge t$\nand the c\\`{a}dl\\`{a}g property of $X$ and $Y$\nimply that the map $t \\mapsto Q^{\\pi}_b (X,Y)_t$\nis c\\`{a}dl\\`{a}g. It is not, however, of finite variation\nunless $X$ and $Y$ are of finite variation.\n\nThe quadratic covariation with respect to the \nthe canonical bilinear map $\\otimes \\colon E \\times F \\to E \\widehat{\\otimes}_{\\pi} F$\nis denoted by $[X,Y]$ or $[X,Y]^{\\Pi}$,\nand it is called the \\emph{tensor quadratic covariation} of $X$ and $Y$.\nHere, $E \\widehat{\\otimes}_{\\pi} F$ is the projective tensor product \nof two Banach spaces $E$ and $F$.\nGiven two c\\`{a}dl\\`{a}g paths $X$ and $Y$,\nwe can consider two tensor quadratic covariations\n$[X,Y]$ and $[Y,X]$. \nThough they are equivalent under the canonical isomorphism\n$E \\widehat{\\otimes}_{\\pi} F \\cong F \\widehat{\\otimes}_{\\pi} E$,\nthey are not equal even if $E = F$.\nWe also write $[X,Y]^{\\pi} = Q_{\\otimes}^{\\pi}(X,Y)$\nand call the path $[X,Y]^{\\pi} \\colon \\mathbb{R}_{\\geq 0} \\to E \\widehat{\\otimes}_{\\pi} E$\nthe discrete tensor quadratic covariation of $X$ and $Y$ along $\\pi$.\nThe path $[X,X]$ is simply called the \\emph{tensor quadratic variation}.\n\nThe tensor quadratic variation of an $\\mathbb{R}^d$-valued path\n$X= (X^1,\\dots,X^d)$ has the following matrix representation\n\\begin{equation*}\n    [X,X]_t\n    =\n    \\begin{pmatrix}\n    [X^1,X^1]_t & \\cdots & [X^1,X^d]_t    \\\\\n    \\vdots & \\ddots & \\vdots   \\\\\n    [X^d,X^1]_t & \\cdots & [X^d,X^d]_t\n    \\end{pmatrix}\n\\in M_{d}(\\mathbb{R}) \\cong \\mathbb{R}^d \\otimes \\mathbb{R}^d = \\mathbb{R}^d \\widehat{\\otimes}_{\\pi} \\mathbb{R}^d.\n\\end{equation*}\nA c\\`{a}dl\\`{a}g path $X \\colon \\lbrack0,\\infty \\mathclose{\\lbrack} \\to \\mathbb{R}^d$\nhas tensor quadratic variation if and only if\nit has quadratic variation in the sense of \nDefinition~2.3 of Hirai~\\cite{Hirai_2019}.\n\nNow we introduce a different type of quadratic variation,\nnamely, scalar quadratic variation.\nAgain we assume that $\\Pi = (\\pi_n)$ is a sequence of partitions of $[0,\\infty \\mathclose{\\lbrack}$.\n\n\\begin{Def} \\label{2c}\nLet $E$ be a Banach space and $X$ be an $E$-valued c\\`{a}dl\\`{a}g path.\n\\begin{enumerate}\n    \\item The path $X$ has \\emph{finite upper scalar quadratic variation along $(\\pi_n)$}\n        if there is an increasing c\\`{a}dl\\`{a}g path\n        $\\overline{Q}(X) \\colon \\mathbb{R}_{\\geq 0} \\to \\mathbb{R}_{\\geq 0}$\n        such that\n        \\begin{equation*}\n            \\varlimsup_{n \\to \\infty} \\sum_{I \\in \\pi_n} \\lVert \\delta_I X_t \\rVert^2 \\leq \\overline{Q}(X)_t\n        \\end{equation*}\n        holds for all $t \\in \\mathbb{R}_{\\geq 0}$.\n        We call such a path $\\overline{Q}(X)$ an upper scalar quadratic variation of $X$.\n    \\item A c\\`{a}dl\\`{a}g path $X \\colon \\mathbb{R}_{\\geq 0} \\to E$ has\n        \\emph{scalar quadratic variation}\n        if there exists a real-valued c\\`{a}dl\\`{a}g increasing path\n        $Q(X)$ such that\n        \\begin{enumerate}\n            \\item for all $t \\in \\mathbb{R}_{\\geq 0}$,\n                \\begin{equation*}\n                    \\sum_{I \\in \\pi_n} \\lVert \\delta_I X_t \\rVert^2 \n                    \\xrightarrow[n \\to \\infty]{} Q(X)_t,\n                \\end{equation*}\n            \\item for all $t \\in \\mathbb{R}_{\\geq 0}$, the jump of $Q(X)$ at $t$ is given by\n                $\\Delta Q(X)_t = \\lVert \\Delta X_t \\rVert_E^2$.\n        \\end{enumerate}\n        We call the increasing path $Q(X)$ the scalar quadratic variation of $X$\n        along $(\\pi_n)$.\n\\end{enumerate}\n\\end{Def}\n\nClearly, the scalar quadratic variation $Q(X)$ is an upper scalar quadratic variation of $X$.\nIf $E$ is a Hilbert space,\nthe scalar quadratic variation coincides with the quadratic variation\n$Q_{\\langle\\phantom{x},\\phantom{x}\\rangle}(X,X)$,\nwhere $\\langle\\phantom{x},\\phantom{x}\\rangle \\colon E \\times E \\to \\mathbb{R}$\nis the inner product of $E$.\n\nIf a c\\`{a}dl\\`{a}g path\n$X = (X^1,\\dots,X^d) \\colon \\mathbb{R}_{\\geq 0} \\to \\mathbb{R}^d$\nhas tensor quadratic variation along $(\\pi_n)$,\nthen it has scalar quadratic variation given by\n\\begin{equation*}\n    Q(X)_t = \\mathop{\\mathrm{Trace}} [X,X]_t = \\sum_{1 \\leq i \\leq d} [X^i,X^i]_t.\n\\end{equation*}\nThis trace representation is still valid for \nHilbert space-valued c\\`{a}dl\\`{a}g paths.\nThis result will be proved in Hirai~\\cite{Hirai_2021c}.\n\nNext, we introduce some conditions to \na sequence of partitions and a c\\`{a}dl\\`{a}g path.\nLet $\\pi \\in \\mathop{\\mathrm{Par}}[0,\\infty[$ and $t \\in \\mathopen{\\rbrack} 0,\\infty \\mathclose{[}$.\nThe symbol $\\pi(t)$ denotes the element of $\\pi$ that contains $t$.\nBy definition, there exists only one such interval.\nThen we set\n\\begin{equation*}\n    \\overline{\\pi}(t) = \\sup \\pi(t), \\qquad \n    \\underline{\\pi}(t) = \\inf \\pi(t).\n\\end{equation*}\nHere, note that $\\pi(t) = \\mathopen{\\rbrack} \\underline{\\pi}(t),\\overline{\\pi}(t) \\rbrack$ and \n\\begin{gather*}\n    \\delta_{\\pi(s)}X_t = X(\\overline{\\pi}(s) \\wedge t) - X(\\underline{\\pi}(s) \\wedge t), \\qquad\n    \\delta_{\\pi(s)}X = X(\\overline{\\pi}(s)) - X(\\underline{\\pi}(s))\n\\end{gather*}\nhold for all $s$ and $t$ in $\\mathopen{\\rbrack} 0,\\infty \\mathclose{\\lbrack}$.\n\nLet $f \\colon S \\to E$ be a function into a Banach space and $A$ be a subset of $S$.\nThe oscillation of $f$ on the set $A$ is defined as \n\\begin{equation*}\n    \\omega(f;A) = \\sup_{x,y \\in A} \\lVert f(x) - f(y) \\rVert_E.\n\\end{equation*}\nUsing this notation, we define two kinds of oscillation of\na path $X \\in D(\\mathbb{R}_{\\geq 0},E)$ \nalong a partition $\\pi \\in \\mathop{\\mathrm{Par}}(\\mathbb{R}_{\\geq 0})$ as follows.\n\\begin{gather*} \\label{2cb}\n    O^+_{t}(X;\\pi) = \\sup_{\\mathopen{]}r,s] \\in \\pi} \\omega(X;\\mathopen{\\rbrack} r,s \\rbrack \\cap [0,t]),  \\\\\n    O^{-}_t(X;\\pi)\n    =\n    \\sup_{\\mathopen{]}r,s] \\in \\pi} \\omega(X;\\mathopen{]}r,s\\mathclose{[} \\cap [0,t])\n    =\n    \\sup_{\\mathopen{]}r,s] \\in \\pi} \\omega(X;\\mathopen{[}r,s\\mathclose{[} \\cap [0,t]).\n\\end{gather*}\nThe second equality in the definition of $O^{-}_t(X;\\pi)$\nis valid because $X$ is supposed to be right continuous.\nClearly we have $O_t^-(X;\\pi) \\leq O_t^+(X;\\pi)$\nfor all $t \\geq 0$.\nIf $X$ is continuous, these two quantities coincide.\n\n\\begin{Def} \\label{2d}\nLet $E$ be a Banach space, $X \\in D(\\mathbb{R}_{\\geq 0},E)$, and $(\\pi_n)_{n \\in \\mathbb{N}}$\nbe a sequence of partitions of $\\mathbb{R}_{\\geq 0}$.\n\\begin{enumerate}\n\\item We say that $(\\pi_n)$\n    \\emph{approximates the jumps of $X$}\n    if it satisfies the following two conditions:\n\t\\begin{enumerate}\n\t\\item[(C1)] Let $t \\in [0,\\infty[$ and $\\varepsilon > 0$. \n\t\tThen there exists an $N \\in \\mathbb{N}$ such that \n        for all $n \\geq N$ and for all $I \\in \\pi_n$,\n        the set $I \\cap [0,t] \\cap D_{\\varepsilon}(X)$\n        has at most one element.\n    \\item[(C2)] Let $s \\in D(X)$ and $t \\in [s,\\infty \\mathclose{\\lbrack}$. Then\n\t\t\\begin{equation*}\n            \\lim_{n \\to \\infty} \\delta_{\\pi_n(s)} X_t \n            =\n            \\lim_{n \\to \\infty} \\left\\{ X(\\overline{\\pi_n}(s) \\wedge t) - X(\\underline{\\pi_n}(s) \\wedge t) \\right\\}\n            =\n            \\Delta X_s.\n        \\end{equation*}\n    \\end{enumerate}\n\\item The sequence $(\\pi_n)$ in $\\mathop{\\mathrm{Par}} \\lbrack 0,\\infty \\mathclose{\\lbrack}$\n    \\emph{weakly controls the oscillation of $X$}\n    if it satisfies Condition~(C3).\n    \\begin{enumerate}\n    \\item[(C3)] For all $t \\in \\mathbb{R}_{\\geq 0}$,\n        \\begin{equation*}\n        \\varlimsup_{\\varepsilon \\downarrow\\downarrow 0} \\varlimsup_{n \\to \\infty} O^{+}_t(X-J_{\\varepsilon}(X);\\pi_n) = 0.\n        \\end{equation*}\n    \\end{enumerate}\n\\item The sequence $(\\pi_n)$ in $\\mathop{\\mathrm{Par}} \\lbrack 0,\\infty \\mathclose{\\lbrack}$\n    \\emph{controls $X$}\n    if it satisfies Conditions~(C1)--(C3).\n\\item The sequence \\emph{$(\\pi_n)$ approximates $X \\colon \\mathbb{R}_{\\geq 0} \\to E$ from the left}\n    if $\\lim_{n \\to \\infty} X(\\underline{\\pi_n}(t)) = X(t-)$ holds for all $t > 0$.\n    Then we call $(\\pi_n)$ a \\emph{left approximation sequence}\n    for $X$.\n\\end{enumerate}\n\\end{Def}\n\nUnder these assumptions, we have the following $C^{1,2}$-type It{\\^o}\nformula for Banach space-valued paths.\n\n\\begin{Th}[It{\\^o} formula] \\label{2e}\nLet $(\\pi_n)$ be a sequence in $\\mathop{\\mathrm{Par}}[0,\\infty[$.\nSuppose that a c\\`{a}dl\\`{a}g path $X$ in $E$ has tensor and finite upper quadratic variations, \nand that $A$ is a c\\`{a}dl\\`{a}g path of finite variation in $F$.\nIf $(\\pi_n)$ is both a controlling and\na left approximation sequence for $(X,A)$,\nthen for any $f \\in C^{1,2}(F \\times E, G)$ and for any $t \\in [0,\\infty[$,\nthe It{\\^o}-F{\\\"o}llmer integral\n$\\int_0^t \\langle D_x f (A_{s-},X_{s-}), \\mathrm{d}X_s \\rangle$ exists,\nand it satisfies\n\\begin{align} \\label{2f}\n    &\n    f(A_t,X_t) - f(A_0,X_0)  \\\\\n    & \\qquad = \n    \\int_{0}^{t} \\langle D_a f (A_{s-},X_{s-}), \\mathrm{d}A^{\\mathrm{c}}_s \\rangle \n    + \\int_0^t \\langle D_x f(A_{s-},X_{s-}),\\mathrm{d}X_s \\rangle   \\notag\\\\\n    & \\qquad\\quad \n    + \\frac{1}{2} \\int_{0}^{t} \\langle D_x^2 f(A_{s-},X_{s-}), \\mathrm{d}[X,X]^{\\mathrm{c}}_s \\rangle\n    + \\sum_{0 < s \\leq t} \\left\\{ \\Delta f(A_s,X_s) - \\langle D_x f(A_{s-},X_{s-}), \\Delta X_s \\rangle \\right\\}. \\notag\n\\end{align}\n\\end{Th}\n\nHere, note that \\eqref{2f} is an equation in the Banach space $G$.\nThe It{\\^o}-F{\\\"o}llmer integral in Theorem~\\ref{2e} is defined by \n\\begin{equation*}\n    \\int_0^t \\langle D_x f(A_{s-},X_{s-}),\\mathrm{d}X_s \\rangle\n    =\n    \\lim_{n \\to \\infty} \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n}\n        \\langle f(A_{r},X_{r}),\\delta_r^s X_t \\rangle.\n\\end{equation*}\n(See Definition~\\ref{5.9b}.)\nDifferentiability of the function in Theorem~\\ref{2e}\nis understood in the sense of the Fr{\\'e}chet derivative.\nRecall that the derivatives of $f \\in C^2(U;G)$\non an open subset $U \\subset E$ satisfy\n$D f \\in C(U, \\mathcal{L}(E,G))$ and\n$D^2 f \\in C(U, \\mathcal{L}(E,\\mathcal{L}(E,G)))$.\nIn the remainder of this paper, we use the identification\n$\n    \\mathcal{L}(E,\\mathcal{L}(E,G)) \n    \\simeq \\mathcal{L}(E \\widehat{\\otimes}_{\\pi} E,G) \n    \\simeq \\mathcal{L}^{(2)}(E,E;G)\n$\nwithout mention.\n\nThe following lemma is essentially used to prove Theorem~\\ref{2e}.\n\n\\begin{Lem} \\label{2g}\nLet $(\\pi_n)$ be a sequence in $\\mathop{\\mathrm{Par}}[0,\\infty[$, and suppose that\n$X \\in D(\\mathbb{R}_{\\geq 0},E)$ has tensor and finite upper quadratic variations.\nMoreover, assume that $(\\pi_n)$ \ncontrols $X \\in D([0,\\infty\\mathclose{[},E)$\nand approximates $\\xi \\in D([0,\\infty\\mathclose{[},\\mathcal{L}(E \\widehat{\\otimes}_{\\pi} E,G))$\nfrom the left.\nThen for all $t \\in [0,\\infty[$, we have\n\\begin{equation*}\n    \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n} \\left\\langle \\xi_r, \\left( \\delta_r^s X_t \\right)^{\\otimes 2} \\right\\rangle\n    \\xrightarrow[n \\to \\infty]{}\n    \\int_{\\mathopen{\\rbrack} 0,t \\rbrack} \\left\\langle \\xi_{u-}, \\mathrm{d}[X,X]_u \\right\\rangle.\n\\end{equation*}\n\\end{Lem}\n\n\\section{Remarks on Banach space-valued c\\`{a}dl\\`{a}g paths}\n\nIn this section, we review properties of c\\`{a}dl\\`{a}g paths that will be \nreferred to later.\n\nRight- and a left-continuous step functions on $[0,\\infty[$ in a Banach space $E$\nare functions of the form\n\\begin{equation*}\n    \\sum_{i \\in \\mathbb{N}} 1_{[t_i,t_{i+1}\\mathclose{[}} a_{i}, \\qquad\n    1_{\\{ 0 \\}} b_0  + \\sum_{i \\in \\mathbb{N}} 1_{\\mathopen{\\rbrack} t_i,t_{i+1} \\rbrack} b_{i+1}, \n\\end{equation*}\nrespectively, where $0 = t_0 < t_1 < \\dots < t_n < \\dots \\to \\infty$\nand all $a_i,b_i$ are elements of $E$.\nAll right-continuous step functions are clearly c\\`{a}dl\\`{a}g\nand left-continuous step functions are c\\`{a}gl\\`{a}d.\nEvery right-continuous step function $f= \\sum_{i \\in \\mathbb{N}} 1_{[t_i,t_{i+1}\\mathclose{[}} a_{i}$\nis strongly $\\mathcal{B}([0,\\infty \\mathclose{\\lbrack})\/\\mathcal{B}(E)$ measurable,\nbecause it is the pointwise limit of the sequence $(f_n)$ defined by\n$f_n = \\sum_{0 \\leq i \\leq n} 1_{[t_i,t_{i+1}\\mathclose{[}} a_{i}$.\n\nA c\\`{a}dl\\`{a}g path in a Banach space satisfies the following properties.\n\n\\begin{Lem} \\label{3b}\nLet $f$ be a c\\`{a}dl\\`{a}g path in a Banach space $E$.\n\\begin{enumerate}\n    \\item For every $C \\in \\mathbb{R}_{> 0}$, there are only finitely many $s$\n        satisfying $\\lVert \\Delta f_{s} \\rVert_E > C$\n        in each compact interval of $[0,\\infty \\mathclose{\\lbrack}$.\n    \\item The image $f(I)$ of any compact interval $I \\subset [0,\\infty \\mathclose{\\lbrack}$ is relatively compact in $E$.\n    \\item Suppose that every jump of $f$ is smaller than $C > 0$\n        on a compact interval $I \\subset [0,\\infty \\mathclose{\\lbrack}$.\n        Then for all $\\varepsilon > 0$, there exists a $\\delta > 0$\n        such that $\\lVert f(s) - f(u) \\rVert_E < C + \\varepsilon$ holds\n        for any $s,u \\in I$ satisfying $\\lvert s - u \\rvert < \\delta$.\n    \\item The path $f$ is the uniform limit of some sequence\n        of right-continuous step functions on each compact interval.\n    \\item For any $t > 0$ and $\\varepsilon > 0$,\n        there is a partition $\\pi \\in \\mathop{\\mathrm{Par}}[0,t]$ that satisfies\n        $O^-(f,\\pi) < \\varepsilon$.\n\\end{enumerate}\n\\end{Lem}\n\nFor an analogue of Proposition~\\ref{3b}\nabout c\\`{a}dl\\`{a}g paths in arbitrary separable complete metric spaces, see Billingsley~\\cite[122]{Billingsley_1999}.\n\nNext, recall that a function $f \\colon [0,\\infty \\mathclose{\\lbrack} \\to E$ is\nof bounded variation on a compact interval $I \\subset [0,\\infty \\mathclose{\\lbrack}$ if\n\\begin{equation*}\n    V(f;I)\n    :=\n    \\sup_{\\pi \\in \\mathop{\\mathrm{Par}} I}\n        \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi} \\lVert f(s) - f(r) \\rVert_E < \\infty.\n\\end{equation*}\nFor convenience, set $V(f;\\emptyset) = 0$ and $V(f;[a,a]) = 0$\nfor every $a \\in [0,\\infty \\mathclose{\\lbrack}$.\nThe function $f$ is of finite variation \nif it has bounded variation on every compact subinterval of $[0,\\infty \\mathclose{\\lbrack}$.\nThe set of the paths of finite variation \nin $E$ is denoted by $FV(\\mathbb{R}_{\\geq 0},E)$\nor $FV(\\lbrack 0,\\infty \\mathclose{\\lbrack},E)$.\nWe define the total variation path $V(f)$ of a function $f$\nof $FV(\\mathbb{R}_{\\geq 0},E)$ by $V(f)_t = V(f;[0,t])$.\nClearly, $V(f)$ is increasing and satisfies $V(f)_0 = 0$.\n\nWe list basic properties of a path of finite variation below.\nSee Dinculeanu~\\cite[{\\S}18]{Dinculeanu_2000} for proofs.\n\n\\begin{Lem} \\label{3c}\nLet $f \\colon [0,\\infty \\mathclose{\\lbrack} \\to E$ be a c\\`{a}dl\\`{a}g path of finite variation in a Banach space.\n\\begin{enumerate}\n    \\item If three numbers $a,b,c \\in [0,\\infty \\mathclose{\\lbrack}$ satisfy $a \\leq b \\leq c$, then\n        \\begin{equation*}\n            V(f,[a,c]) = V(f,[a,b]) + V(f,[b,c]).\n        \\end{equation*}\n    \\item The total variation path $V(f)$ is c\\`{a}dl\\`{a}g.\n    \\item The jump of $V(f)$ at $t \\geq 0$ is given by $\\Delta V(f)(t) = \\lVert \\Delta f(t) \\rVert$.\n    \\item The family $(\\Delta f(s))_{s \\in [0,t]}$ is absolutely summable for all $t \\geq 0$.\n    \\item The function $f^{\\mathrm{d}}$ defined by\n        \\begin{equation*}\n            f^{\\mathrm{d}}(t) = \\sum_{0 < s \\leq t} \\Delta f(s).\n        \\end{equation*}\n        is again a c\\`{a}dl\\`{a}g path of finite variation.\n\\end{enumerate}\n\\end{Lem}\n\nNote that the summation in (v) of Proposition~\\ref{3c}\nis defined in the following sense.\nLet $D$ be the set of all finite subsets of $\\mathopen{\\rbrack} 0,t \\rbrack$.\nWe regard $D$ as a directed set with the order defined by inclusion.\nThen the net $(\\sum_{s \\in d} \\Delta f(s))_{d \\in D}$ \nconverges in $E$, and we define\n\\begin{equation*}\n    \\sum_{0 < s \\leq t} \\Delta f(s) = \\lim_{d} \\sum_{s \\in d} \\Delta f(s).\n\\end{equation*}\nThe function $f^{\\mathrm{d}}$ defined in Proposition~\\ref{3c}\nis called the discontinuous part of $f$.\nWe also define $f^{\\mathrm{c}} = f - f^{\\mathrm{d}}$\nand call this the continuous part of $f$.\n\nLet $\\mathcal{I}$ be the set of all bounded intervals of \nthe form $\\mathopen{\\rbrack} a,b \\rbrack$ or $[0,a]$,\nwhere $a$ are $b$ are nonnegative real numbers. \nThen $\\mathcal{I}$ is a semiring of subsets of $[0,\\infty \\mathclose{\\lbrack}$\ngenerating the Borel $\\sigma$-algebra \n$\\mathcal{B}([0,\\infty \\mathclose{\\lbrack})$.\nGiven an $f \\in D(\\mathbb{R}_{\\geq 0},E)$,\ndefine\n\\begin{equation*}\n    \\mu_{f}(\\mathopen{\\rbrack} a,b \\rbrack) = f(b) - f(a), \\quad \n    \\mu_{f}([0,a]) = f(a)\n\\end{equation*}\nfor any two real numbers satisfying $0 \\leq a \\leq b$.\nIf $f$ has finite variation,\nthe function $\\mu_f \\colon \\mathcal{I} \\to E$\ncan be uniquely extended to a $\\sigma$-additive \nmeasure defined on the $\\delta$-ring\ngenerated by $\\mathcal{I}$.\nRefer to Theorem~18.19 of Dinculeanu~\\cite[208]{Dinculeanu_2000}\nfor a proof.\n\nBecause there is a measure $\\mu_f$ associated with $f$,\nwe can consider the Stieltjes integral with respect to $f$.\nLet $b \\colon F \\times E \\to G$ a continuous bilinear map\nbetween Banach spaces.\nSet $L^{1}_{\\mathrm{loc}}(\\mu_f) = L^{1}_{\\mathrm{loc}}(\\lvert \\mu_f \\rvert)$,\nwhere $\\lvert \\mu_f \\rvert$ denotes the variation measure of $\\mu_f$.\nThen for each $g \\in L^{1}_{\\mathrm{loc}}(\\mu_f)$ and\nthe compact interval $I$, define the Stieltjes integral through the formula\n\\begin{equation*}\n    \\int_{\\mathopen{\\rbrack} 0,t \\rbrack} b(g(s), \\mathrm{d}f(s))\n    =\n    \\int_{\\mathopen{\\rbrack} 0,t \\rbrack} (g(s), \\mu_f(\\mathrm{d}s)).\n\\end{equation*}\nNote that the integral of the left-hand side is well-defined because \n$g$ belongs to $L^{1}_{\\mathrm{loc}}(\\mu_f)$.\n\n\\section{Auxiliary results regarding sequences of partitions}\n\nIn this section, we investigate conditions on a sequence of partitions \nalong which we deal with quadratic variations and the It{\\^o}-F{\\\"o}llmer integral.\nRecall that basic notions were defined in Definition~\\ref{2d}.\n\n\\begin{Def} \\label{4b}\nLet $E$ be a Banach space, $X \\in D(\\mathbb{R}_{\\geq 0},E)$, and $(\\pi_n)_{n \\in \\mathbb{N}}$\nbe a sequence of partitions of $\\mathbb{R}_{\\geq 0}$.\n\\begin{enumerate}\n\\item We say that $(\\pi_n)$\n    \\emph{controls the oscillation of $X$}\n    if $\\lim_{n \\to \\infty}O^-_t(X;\\pi_n) = 0$ holds for all $t$.\n\\item A sequence $(\\pi_n)$ \\emph{exhausts the jumps of $X$} if\n    $D(X) \\subset \\bigcup_{n \\in \\mathbb{N}} \\bigcap_{k \\geq n} \\pi^{\\mathrm{p}}_k$.\n\\end{enumerate}\n\\end{Def}\n\n\\begin{Exm} \\label{4c}\n\\begin{enumerate}\n    \\item Let $r$ be an irrational number and $X = 1_{\\lbrack r,\\infty \\mathclose{\\lbrack}}$.\n        For each $n \\in \\mathbb{N}$, we set $\\pi_n = \\{ \\mathopen{\\rbrack} k2^{-n},(k+1)2^{-n} \\rbrack ; k \\in \\mathbb{N} \\}$.\n        Then the sequence $(\\pi_n)$ clearly satisfies $\\lvert \\pi_n \\rvert \\to 0$ as $n \\to \\infty$.\n        The sequence, however, does not control the oscillation of $X$.\n    \\item Let $X = 1_{\\lbrack 1,\\infty \\mathclose{\\lbrack}}$\n        and $\\pi_n = \\{ \\mathopen{\\rbrack} k,k+1 \\rbrack; k \\in \\mathbb{N} \\}$.\n        Though the sequence $(\\pi_n)$ controls the oscillation of $X$,\n        it does not satisfy $\\lvert \\pi_n \\rvert \\to 0$.\n    \\item Let $X = 1_{\\lbrack 1\/2, \\infty \\mathclose{\\lbrack}}$\n        and $\\pi_n = \\{ \\mathopen{\\rbrack} k,k+1 \\rbrack; k \\in \\mathbb{N} \\}$.\n        The sequence $(\\pi_n)$ neither controls the oscillation of $X$\n        nor satisfies $\\lvert \\pi_n \\rvert \\to 0$.\n        However, it controls $X$ in the sense of Definition~\\ref{2d}.\n\\end{enumerate}\n\\end{Exm}\n\nCondition~(i) of Definition~\\ref{4b} is characterized as follows.\n\n\\begin{Lem} \\label{4e}\nLet $X \\in D([0,\\infty\\mathclose{[},E)$, and $(\\pi_n) \\in (\\mathop{\\mathrm{Par}}[0,\\infty\\mathclose{[})^{\\mathbb{N}}$.\nThen there is an equivalence between\n\\begin{enumerate}\n    \\item the sequence $(\\pi_n)$ controls the oscillation of $X$, and\n    \\item the sequence $(\\pi_n)$ satisfies the conditions:\n        \\begin{enumerate}\n            \\item The sequence $(\\pi_n)$ exhausts the jumps of $X$.\n            \\item If $X$ is not constant on $\\mathopen{]}s,t\\mathclose{[}$, \n                then $]s,t[$ contains at least one element of $\\pi_n^{\\mathrm{p}}$ for sufficiently large $n$.\n        \\end{enumerate}\n\\end{enumerate}\n\\end{Lem}\n\nLemma~\\ref{4e} is a generalization of\nLemma~1 of Vovk~\\cite[272]{Vovk_2015}.\nNote that the condition\n`\\emph{$]s,t[$ contains at least one point of $\\pi_n^{\\mathrm{p}}$}'\nis equivalent to\n`\\emph{there is no $I \\in \\pi_n$ including $\\mathopen{\\rbrack} s,t \\mathclose{\\lbrack}$}.'\n\n\\begin{proof}\n\\emph{Step~1.1: (i) $\\Longrightarrow$ (ii)-(a).}\nLet $s \\in D(X)$ and set $\\varepsilon = \\lVert \\Delta X(s) \\rVert_{E}$.\nMoreover, fix $T > s$ arbitrarily.\nThen, by assumption, we can choose an $N \\in \\mathbb{N}$ such that\n$O_T^-(X,\\pi_n) < \\varepsilon\/2$ holds for all $n$.\nWe will show that $s = \\overline{\\pi_n}(s)$ holds for any $n \\geq N$.\nIf we had $s < \\overline{\\pi_n}(s)$, we could take an $s'$\nfrom the interval $\\mathopen{]}\\underline{\\pi_n}(s),s \\mathclose{[}$\nsuch that\n\\begin{equation*}\n    \\lVert X_{s'} - X_{s-} \\rVert_{E} \\leq O^{-}_T(X,\\pi_n) < \\frac{\\varepsilon}{2}.\n\\end{equation*}\nThen\n\\begin{equation*}\n    \\lVert X_s - X_{s'} \\rVert_{E} \n    \\geq \n    \\lVert \\Delta X_s \\rVert_{E} - \\lVert X_{s-} - X_{s'} \\rVert_{E} \n    > \\varepsilon - \\frac{\\varepsilon}{2} \n    = \n    \\frac{\\varepsilon}{2},\n\\end{equation*}\nwhich contradicts the assumption.\nThus $(\\pi_n)$ exhausts the jumps of $X$.\n\n\\emph{Step~1.2: (i) $\\Longrightarrow$ (ii)-(b).}\nAssume that $\\varepsilon := \\omega(X;\\mathopen{]} s,t \\mathclose{[}) > 0$.\nChoose an $N \\in \\mathbb{N}$ that satisfies \n$O^{-}_t(X,\\pi_n) < \\varepsilon$ for all $n \\geq N$.\nThen, for arbitrarily fixed $n \\geq N$, pick an $i$ satisfying $s \\in [t^n_i,t^n_{i+1}[$.\n$t^n_{i+1} \\in \\mathopen{]}s,t \\mathclose{[}$ remains to be proven.\nIf $t^n_{i+1} \\geq t$, we have \n\\begin{equation*}\n    O^{-}_t(X,\\pi_n) \n    \\geq\n    \\sup_{u,v \\in [t^n_i,t^n_{i+1}\\mathclose{[} \\cap [0,t]} \\lVert X_u - X_v \\rVert\n    \\geq\n    \\sup_{u,v \\in [s,t[} \\lVert X_u - X_v \\rVert\n    =\n    \\varepsilon.\n\\end{equation*}\nThis contradicts the assumption, and hence we obtain $t^n_{i+1} < t$.\n\n\\emph{Step~2: (ii) $\\Longrightarrow$ (i).}\nSuppose that $(\\pi_n)$ satisfies the conditions (ii)-(a) and (b).\nFix an $\\varepsilon > 0$ and a $t > 0$ arbitrarily.\nBecause $X$ is c\\`{a}dl\\`{a}g, we can take a sequence $0 = s_0 < s_1 < \\dots < s_N = t$\nsuch that $\\omega(X; \\mathopen{]} s_i,s_{i+1}\\mathclose{[}) < \\varepsilon\/2$\nfor all $i$ (see Lemma~\\ref{3b}).\n\nBy assumption, we can choose an $N \\in \\mathbb{N}$ satisfying\nthe following conditions.\n\\begin{enumerate}\n    \\item[1.] If $n \\geq N$, there are no $I \\in \\pi_n$ and $i \\in \\{ 0,\\dots, N \\}$\n        satisfying $\\mathopen{\\rbrack} s_i,s_{i+1} \\mathclose{\\lbrack} \\subset I$ and\n        $\\omega(X;\\mathopen{\\rbrack} s_i,s_{i+1} \\mathclose{\\lbrack}) > 0$.\n    \\item[2.] $\n        \\{ s_0,\\dots, s_N \\} \\cap D(X) \n        \\subset \\bigcap_{n \\geq N} \\pi_n^{\\mathrm{p}}\n        $.\n\\end{enumerate}\n\nLet $n \\geq N$ and $\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n$.\nFirst, assume that $\\omega(X; \\mathopen{\\rbrack} u,v \\mathclose{\\lbrack}) > 0$.\nBy Condition~1, we see that there are only two cases\nfor the relationship between $\\mathopen{\\rbrack} u,v \\rbrack$ and $(s_i)_{0 \\leq i \\leq N}$, as follows.\n\\begin{enumerate}\n    \\item[A.] There is a unique $i$ such that $\\mathopen{\\rbrack} u,v \\rbrack \\subset \\mathopen{\\rbrack} s_i,s_{i+1}\\mathclose{\\lbrack}$.\n    \\item[B.] There is a unique $i$ such that $s_i \\in \\mathopen{\\rbrack} u,v \\mathclose{\\lbrack}$.\n\\end{enumerate}\nIn Case~A, the oscillation of $X$ on $\\mathopen{\\rbrack} u,v \\mathclose{\\lbrack}$ is estimated as\n\\begin{equation*}\n    \\omega(X;\\mathopen{\\rbrack} u,v \\mathclose{\\lbrack} \\cap [0,t]) \\leq \\omega(X; \\mathopen{\\rbrack} s_i,s_{i+1}\\mathclose{\\lbrack}) < \\frac{\\varepsilon}{2}.\n\\end{equation*}\nOn the other hand, in Case~B, \n$X$ is continuous at $s_i \\in \\mathopen{\\rbrack} u,v \\mathclose{\\lbrack}$\nbecause of Condition~2. Therefore,\n\\begin{align*}\n    \\omega(X;\\mathopen{\\rbrack} u,v \\mathclose{\\lbrack} \\cap [0,t])\n    \\leq\n    \\omega(X; \\mathopen{\\rbrack} s_{i-1},s_{i}\\mathclose{\\lbrack}) + \\omega(X; \\mathopen{\\rbrack} s_i,s_{i+1}\\mathclose{\\lbrack}) \n    <\n    \\varepsilon.\n\\end{align*}\nIf $\\omega(X; \\mathopen{\\rbrack} u,v \\mathclose{\\lbrack}) > 0$, we clearly have the same estimate.\nBy the discussion above,\nwe find that $\\omega(X;\\mathopen{\\rbrack} u,v \\mathclose{\\lbrack} \\cap [0,t]) < \\varepsilon$ holds for all\n$\\mathopen{\\rbrack} u,v, \\rbrack \\in \\pi_n$, and consequently\n\\begin{equation*}\n    O^{-}_t(X;\\pi_n) = \\sup_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n} \\omega(X;\\mathopen{\\rbrack} r,s \\mathclose{\\lbrack} \\cap [0,t]) \\leq \\varepsilon\n\\end{equation*}\nfor every $n \\geq N$. This completes the proof.\n\\end{proof}\n\nWe next define the left discretization of\n$\\xi \\colon \\mathbb{R}_{\\geq 0} \\to E$ along a partition $\\pi$ by \n\\begin{equation*}\n    {^\\pi\\xi} = \\sum_{\\mathopen{]}r,s] \\in \\pi} \\xi(r) 1_{]r,s]}.\n\\end{equation*}\nIf $\\xi$ is c\\`{a}dl\\`{a}g, then the sequence $(\\pi_n)$ approximate $\\xi$ from the left\nin the sense of Definition~\\ref{2d}\nif and only if $({^{\\pi_n}\\xi})_{n \\in \\mathbb{N}}$ converges\nto $\\xi_{-}$ pointwise.\nConsider the following example.\n\n\\begin{Exm} \\label{4d}\nLet $X = 1_{\\lbrack 1\/2, \\infty \\mathclose{\\lbrack}}$\nand $\\pi_n = \\{ \\mathopen{\\rbrack} k,k+1 \\rbrack; k \\in \\mathbb{N} \\}$.\nAs we have seen in Example~\\ref{4c},\n$(\\pi_n)$ controls the path $X$.\nFor each $n \\in \\mathbb{N}$,\nthe left discretization of $X$ is given by \n${^{\\pi_n} X} = 1_{\\mathopen{\\rbrack} 1,\\infty \\mathclose{\\lbrack}}$.\nThe sequence $({^{\\pi_n} X})$ does not converge to $X$ pointwise,\nand hence $(\\pi_n)$ does not approximate $X$ from the left.\n\\end{Exm}\n\nAs we mentioned in Section~1, \ntwo types of assumptions about a sequence of partitions are \nfrequently used in the context of the It{\\^o}-F{\\\"o}llmer calculus.\nOne is `$\\lvert \\pi_n \\rvert \\to 0$' and the other is `$O_t^{-}(X;\\pi_n) \\to 0$'.\nIn the next proposition, we show that both conditions \nimply `$(\\pi_n)$ controls $X$'.\n\n\\begin{Prop} \\label{4f}\nLet $(\\pi_n)$ be a sequence of partitions of $\\mathbb{R}_{\\geq 0}$ and let $E$ be a Banach space.\n\\begin{enumerate}\n    \\item Suppose that $(\\pi_n)$ satisfies $\\lvert \\pi_n \\rvert \\to 0$. \n        Then it controls every c\\`{a}dl\\`{a}g path in $E$.\n        Moreover, it approximates every c\\`{a}dl\\`{a}g path in $E$ from the left.\n    \\item Suppose that $(\\pi_n)$ controls the oscillation of $X \\in D([0,\\infty\\mathclose{[},E)$. \n        Then it controls $X$ and approximates $X$ from the left.\n\\end{enumerate}\n\\end{Prop}\n\n\\begin{proof}\n(i)\nIf $\\lvert \\pi_n \\rvert \\to 0$, then $\\overline{\\pi_n}(t) \\to t$\nand $\\underline{\\pi_n}(t) \\to t$ hold for every $t \\geq 0$.\nThis directly implies that $(\\pi_n)$ approximates $X$ from the left.\nMoreover, we have $\\delta_{\\pi_n(t)} X_u \\to X_{t}$ for every $t,u > 0$\nwith $t \\leq u$. Hence, $(\\pi_n)$ satisfies Condition~(C2).\nCondition~(C3) follows from (iii) of Lemma~\\ref{3b}.\nCondition~(C1) remains to be shown.\nGiven an $\\varepsilon > 0$, define\n\\begin{equation*}\n    r := \\inf \\{ \\lvert u-v \\rvert \\mid u,v \\in D_{\\varepsilon}(X) \\cap [0,t], u \\neq v \\} > 0.\n\\end{equation*}\nIf $D_{\\varepsilon}(X) \\cap [0,t]$ has only one element,\nthere is nothing to do.\nOtherwise, $r$ is not zero, \nbecause $D_{\\varepsilon}(X) \\cap [0,t]$ has at most finitely many elements\n(see (i) of Lemma~\\ref{3b}).\nNow we take an $N$ satisfying $\\lvert \\pi_n \\rvert < r$ for all $n \\geq N$.\nThen for each $n \\geq N$ and $\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n$, \nthe set $\\mathopen{\\rbrack} u,v \\rbrack \\cap [0,t]$ contains at most one element of $D_{\\varepsilon}$.\n\n(ii)\nAssume that $(\\pi_n)$ controls the oscillation of $X$.\nThen we see that $(\\pi_n)$ approximates $X$ from the left\nby the following estimate.\n\\begin{equation*}\n    \\lVert X_{\\underline{\\pi_n}(t)} - X_{t-} \\rVert_{E}\n    \\leq\n    \\omega(X;\\lbrack \\underline{\\pi_n}(t), \\overline{\\pi_n}(t)\\mathclose{\\lbrack} \\cap [0,t] )\n    \\leq\n    O^{-}_t(X,\\pi_n).\n\\end{equation*}\n\nNow, let us show that $(\\pi_n)$ controls $X$.\nTo obtain (C2), take a $t \\in D(X)$.\nBecause $(\\pi_n)$ exhausts the jumps of $X$ (Lemma~\\ref{4e}), \nwe have $\\overline{\\pi_n}(t) = t$ for sufficiently large $n$.\nThis combined with the fact that $(\\pi_n)$ is a left-approximation sequence\nimplies (C2).\nNext, consider (C1). \nLet $\\varepsilon > 0$ and fix an $N_{\\varepsilon} \\in \\mathbb{N}$\nso that $O^{-}_{t}(X,\\pi_n) < \\varepsilon$ holds for any $n \\geq N_{\\varepsilon}$.\nThen for every $n \\geq N_{\\varepsilon}$\nand $\\mathopen{\\rbrack} r,s] \\in \\pi_n$, the interval $\\mathopen{\\rbrack} r,s \\mathclose{\\lbrack} \\cap [0,t]$ does not \ncontain any jump of $X$ that is greater than $\\varepsilon$.\nTherefore, $\\mathopen{\\rbrack} r,s \\rbrack$ possesses at most one element of $D_{\\varepsilon}(X)$.\nThis means that $(\\pi_n)$ satisfies (C1).\nAll that is left is to check Condition~(C2).\nChoose an $M_{\\varepsilon}$\nthat satisfies $O^{-}_t(X;\\pi_n) < \\varepsilon\/2$ for all $I \\in \\pi_n$ and $n \\geq M$.\nAs we just have shown, $J_{\\varepsilon\/2}(X)$ is zero on the interior of each \n$I \\in \\pi_n$ and $n \\geq M_{\\varepsilon}$. Hence, \n\\begin{align*}\n    \\omega(X-J_{\\varepsilon\/2}(X);\\mathopen{\\rbrack} r,s \\rbrack \\cap [0,t])\n& \\leq\n    \\omega(X-J_{\\varepsilon\/2}(X);\\mathopen{\\rbrack} r,s \\mathclose{\\lbrack} \\cap [0,t]) + \\lVert \\Delta (X-J_{\\varepsilon\/2}(X))_s \\rVert_{E}  \\\\\n& \\leq \n    O^{-}_t(X;\\pi_n)\n    + \\lVert \\Delta (X-J_{\\varepsilon\/2}(X))_s \\rVert_{E}\n<\n    \\varepsilon\n\\end{align*}\nholds for all $\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n$ and $n \\geq M_{\\varepsilon}$,\nwhich implies (C3).\n\\end{proof}\n\nIn the last part of this section,\nwe give an additional lemma about a sequence of partitions.\n\n\\begin{Lem} \\label{4g}\n\\begin{enumerate}\n    \\item Let $X$ be a c\\`{a}dl\\`{a}g path in a Banach space $E$.\n        If $(\\pi_n)$ approximates $\\xi$ from the left,\n        then $(\\pi_n)$ also approximates $f \\circ \\xi$ from the left for every\n        continuous function $f \\colon E \\to E'$ to an arbitrary Banach space.\n    \\item Let $X$ and $Y$ be c\\`{a}dl\\`{a}g paths in Banach spaces $E$ and $F$,\n        respectively. If $(\\pi_n)$ controls the path $(X,Y)$ in $E \\times F$,\n        then $(\\pi_n)$ controls both $X$ and $Y$.\n        Here, we regard $E \\times F$ as a Banach space endowed with the direct sum norm.\n\\end{enumerate}\n\\end{Lem}\n\n\\begin{proof}\n(i) immediately follows from the continuity of $f$.\n\nTo show (ii), suppose that $(\\pi_n)$ controls $(X,Y)$.\nIt suffices to show that $(\\pi_n)$ controls $X$.\nFirst fix $t \\in \\mathbb{R}_{\\geq 0}$ and $\\varepsilon > 0$ arbitrarily,\nand then choose an $N$ so that $D_\\varepsilon(X,Y) \\cap I \\cap [0,t]$\nhas at most one element for all $n \\geq N$ and $I \\in \\pi_n$.\nThe inclusion\n$I \\cap [0,t] \\cap D_\\varepsilon(X) \\subset D_\\varepsilon(X,Y) \\cap I \\cap [0,t]$\nimplies that the cardinality of $I \\cap [0,t] \\cap D_\\varepsilon(X)$ is \nno greater than 1.\nCondition~(C2) obviously follows from the definition of product topology.\nCondition~(C3) remains to be shown.\nFor an arbitrary $\\delta > 0$, \nchoose an $\\varepsilon_0 > 0$ satisfying\n\\begin{equation*}\n    \\sup_{\\varepsilon \\leq \\varepsilon_0} \\varlimsup_{n \\to \\infty} O^+_t((X,Y)-J_\\varepsilon(X,Y);\\pi_n) < \\frac{\\delta}{2}.\n\\end{equation*}\nSet $\\varepsilon_1 = \\varepsilon_0 \\wedge (\\delta\/2)$.\nGiven $\\varepsilon \\leq \\varepsilon_1$,\nwe can take $M_{\\varepsilon} > 0$ such that\n\\begin{enumerate}\n    \\item[(a)] $I \\cap [0,t] \\cap D_{\\varepsilon}(X,Y)$ has at most one element\n        for all $I \\in \\pi_n$ and $n \\geq M_{\\varepsilon}$.\n    \\item[(b)] $\\sup_{n \\geq M_{\\varepsilon}} O^+_t((X,Y)-J_{\\varepsilon}(X,Y);\\pi_n) < \\delta\/2$.\n\\end{enumerate}\nIf $n \\geq M_{\\varepsilon}$ and $I \\in \\pi_n$, then for any $u,v \\in I$, we have\n\\begin{align*}\n    & \n    \\lVert (X-J_\\varepsilon(X))_u - (X-J_\\varepsilon(X))_v \\rVert_{E}  \\\\\n    & \\qquad \\leq \n    \\lVert ( X-J(D_\\varepsilon(X,Y),X) )_u - ( X-J (D_\\varepsilon(X,Y),X) )_v \\rVert_{E}  \\\\\n    & \\qquad\\quad \n    + \\lVert ( J(D_\\varepsilon(X,Y),X) - J(D_\\varepsilon(X),X) )_u - ( J(D_\\varepsilon(X,Y),X) - J (D_\\varepsilon(X),X) )_v \\rVert_{E}  \\\\\n    & \\qquad \\leq\n    \\sup_{n \\geq M_{\\varepsilon}} O^+_t((X,Y)-J_{\\varepsilon}(X,Y);\\pi_n) + \\varepsilon   \\\\\n    & \\qquad \\leq\n    \\frac{\\delta}{2} + \\frac{\\delta}{2} = \\delta.\n\\end{align*}\nHere, note that the second inequality holds by Condition~(a) above.\nThus, we get\n\\begin{equation*}\n    \\varlimsup_{n \\to \\infty} O^+_t(X-J_{\\varepsilon}(X);\\pi_n)\n    \\leq\n    \\sup_{n \\geq M_{\\varepsilon}} O^+_t(X-J_{\\varepsilon}(X);\\pi_n)\n    \\leq\n    \\delta.\n\\end{equation*}\nfor arbitrary $\\varepsilon \\leq \\varepsilon_1$.\nThis implies (C3) for $X$.\n\\end{proof}\n\n\\section{Properties of quadratic variations}\nWe defined quadratic variations treated in this paper \nin Section~2.\nThis section is devoted to studying their basic properties.\nThroughout this section, suppose that\nwe are given a sequence $\\Pi = (\\pi_n)_{n \\in \\mathbb{N}}$\nof elements of $\\mathop{\\mathrm{Par}} \\lbrack 0,\\infty \\mathclose{\\lbrack}$.\n\nFirst, we give some examples of quadratic variations.\n\n\\begin{Exm} \\label{5b}\nLet $A$ be a path of finite variation in a Banach space $E$.\nIf $(\\pi_n)$ satisfies $\\lvert \\pi_n \\rvert \\to \\to 0$,\nthen $A$ has tensor and scalar quadratic variations given by \n\\begin{equation*}\n    [A,A]_t = \\sum_{0 < s \\leq t} (\\Delta A_s)^{\\otimes 2}, \\qquad \n    Q(A)_t = \\sum_{0 < s \\leq t} \\lVert \\Delta A_s \\rVert^2.\n\\end{equation*}\nThis result will be proved later in this section.\n\\end{Exm}\n\n\\begin{Exm} \\label{5c}\nSuppose that $x \\colon [0,\\infty \\mathclose{\\lbrack} \\to \\mathbb{R}$\nhave quadratic variation along $(\\pi_n)$.\nLet $a$ be any element of a Banach space $E$ and \nset $X(t) = x(t) a$.\nThen $X$ have tensor and scalar quadratic variation\nand they have the following expression.\n\\begin{equation*}\n    [X,X]_t = [x,x]_t a \\otimes a, \\qquad \n    Q(A)_t = [x,x]_t \\lVert a \\rVert^2.\n\\end{equation*}\nFor the construction of \na real continuous path of nontrivial quadratic variation,\nrefer to Schied~\\cite{Schied_2016} and Mishura and Schied~\\cite{Mishura_Schied_2016}.\n\\end{Exm}\n\nNext, we treat an example from the theory of stochastic processes.\n\n\\begin{Exm} \\label{5d}\nLet $(\\Omega,\\mathcal{F},(\\mathcal{F}_t)_{t \\geq 0},P)$ be\na filtered probability space satisfying the usual condition.\nConsider a semimartingale $X = (X_t)_{t \\geq 0}$ \nin a separable Hilbert space $H$.\nMoreover, let $\\pi = (\\tau^n_k)_{n \\in \\mathbb{N}}$ be an \nincreasing sequence of bounded stopping times such that \n$\\tau^n_k \\to \\infty$ as $k \\to \\infty$\nand $\\sup_{k} (\\tau^{n+1}_k - \\tau^n_k) \\to 0$ as $n \\to \\infty$\nalmost surely.\nThen the process $[X,X]^{\\pi}_t$ converges to \nthe quadratic variation process $[X,X]$ uniformly in probability.\nBy passing to an appropriate subsequence, \nwe see that almost all paths have quadratic variation \nalong the subsequence\n(see Metivier and Pellaumail~\\cite{Metivier_Pellaumail_1980b} for details). \n\\end{Exm}\n\nNow we consider the transpose of quadratic covariation.\nLet $b \\in \\mathcal{L}^{(2)}(E,F;G)$ and define the transpose $^tb \\colon F \\times E \\to G$\nof $b$ as $^tb(y,x) = b(x,y)$.\nThen $X \\in D(\\mathbb{R}_{\\geq 0},E)$\nand $Y \\in D(\\mathbb{R}_{\\geq 0},F)$ have quadratic covariation with respect to $b$\nif and only if $Y$ and $X$ do with respect to the transpose $^tb$.\n\nRecall that a $d$-dimensional c\\`{a}dl\\`{a}g path \n$X = (X_1,\\dots, X_d)$ has tensor quadratic variation \nif and only if $X_i$ and $X_j$ have quadratic covariation \nfor each $i$ and $j$.\nThis characterization is generalized to quadratic covariation \nwith respect to a bilinear map in Banach spaces.\n\n\\begin{Prop} \\label{5.4c}\nGiven a family of bilinear maps\n$(b_{ij})_{1 \\leq i,j \\leq 2} \\in \\prod_{1 \\leq i,j \\leq 2} \\mathcal{L}^{(2)}(E_i,F_j;G_{ij})$,\ndefine a continuous bilinear map $\\mathbf{b} \\colon (E_1 \\times E_2) \n\\times (F_1 \\times F_2) \\to \\prod_{1 \\leq i,j \\leq 2} G_{i,j}$ as\n\\begin{equation*}\n    \\mathbf{b}((x_1,x_2),(y_1,y_2)) = (b_{ij}(x_i,y_i))_{1 \\leq i,j \\leq 2}.\n\\end{equation*}\nThen, $(X_1,X_2) \\in D(\\mathbb{R}_{\\geq 0}, E_1 \\times E_2)$\nand $(Y_1,Y_2) \\in D(\\mathbb{R}_{\\geq 0}, F_1 \\times F_2)$\nhave quadratic covariation with respect to $\\mathbf{b}$\nif and only if $X_i$ and $Y_j$ have quadratic covariation\nwith respect to $b_{ij}$ for every $i$ and $j$.\nIn this case, we have\n\\begin{equation} \\label{5.4cb}\n    Q_{\\mathbf{b}}((X_1,X_2),(Y_1,Y_2)) = (Q_{b_{ij}}(X_i,X_j))_{i,j \\in \\{1, 2 \\}}.\n\\end{equation}\n\\end{Prop}\n\nUsing the matrix notation, we can also express equation~\\eqref{5.4cb} as follows.\n\\begin{equation*}\n    Q_{\\mathbf{b}}((X_1,X_2),(Y_1,Y_2)) =\n    \\begin{pmatrix}\n        Q_{b_{11}}(X_1,Y_1) & Q_{b_{12}}(X_1,Y_2)  \\\\\n        Q_{b_{21}}(X_2,Y_1) & Q_{b_{22}}(X_2,Y_2)\n    \\end{pmatrix}.\n\\end{equation*}\n\n\\begin{proof}\nThe equation\n\\begin{align*}\n    \\sum_{\\mathopen{]}r,s] \\in \\pi_n} \\mathbf{b} \\biggl( \\Bigl( \\delta_r^s X_1(t), \\delta_r^s X_2(t) \\Bigr), \\Bigl( \\delta_r^s Y_1(t), \\delta_r^s Y_2(t) \\Bigr) \\biggr)\n    & = \\left( \\sum_{\\mathopen{]}r,s] \\in \\pi_n}  b_{ij} \\bigl( \\delta_r^s X_i(t),\\delta_r^s Y_i(t) \\bigr) \\right)_{i,j \\in \\{1,2 \\}}.\n\\end{align*}\nimplies the equivalence of convergence of discrete quadratic covariations.\nMoreover, the equation\n\\begin{align*}\n    \\mathbf{b} \\left( \\Delta (X_1,X_2)_t,\\Delta (Y_1,Y_2)_t \\right)\n    & = \\left( b_{ij}( \\Delta X_i(t),\\Delta Y_j(t) ) \\right)_{i,j \\in \\{1,2\\}}.\n\\end{align*}\nguarantees the equivalence of the jump conditions.\n\\end{proof}\n\nApplying Proposition~\\ref{5.4c} to\nthe canonical bilinear map $\\otimes \\colon E_i \\times E_j \\to E_i \\widehat{\\otimes}_{\\pi} E_j$,\nwe obtain the following corollary.\n\n\\begin{Cor} \\label{5.4d}\nLet $(X_1,X_2) \\in D(\\mathbb{R}_{\\geq 0},E_1 \\times E_2)$.\nThen $(X_1,X_2)$ has tensor quadratic variation if and only if\n$X_i$ and $X_j$ have tensor quadratic covariation\nwith respect to the canonical bilinear map\n$\\otimes \\colon E_i \\times E_j \\to E_i \\widehat{\\otimes}_{\\pi} E_j$ for each $i,j \\in \\{1,2 \\}$.\n\\end{Cor}\n\nNext, we show the bilinear property of quadratic covariation.\n\n\\begin{Prop} \\label{5.4e}\nLet $X_1,X_2 \\in D(\\mathbb{R}_{\\geq 0},E)$,\n$Y_1,Y_2 \\in D(\\mathbb{R}_{\\geq 0},F)$.\nSuppose that $X_i$ and $Y_j$ have quadratic covariation with respect to a bounded bilinear\nmap $b \\colon E \\times F \\to G$ for each $i,j \\in \\{ 1,2 \\}$.\nThen, $X_1 + X_2$ and $Y_1 + Y_2$ have\nquadratic covariation with respect to $b$, and\n$Q_b(X_1+X_2,Y_1+Y_2)$ satisfies \n\\begin{equation*}\n    Q_b(X_1+X_2,Y_1+Y_2) = \\sum_{ i,j \\in \\{1,2\\} } Q_b(X_i,Y_j).\n\\end{equation*}\n\\end{Prop}\n\n\\begin{proof}\nAccording to the bilinear property of $b$, we see that\n\\begin{equation*}\n    \\sum_{I \\in \\pi_n} b(\\delta_I (X_1+X_2)_t,\\delta_I (Y_1+Y_2)_t)\n    =\n    \\sum_{1 \\leq i,j \\leq 2} \\sum_{I \\in \\pi_n} b(\\delta_I(X_i)_t,\\delta_I (Y_j)_t).\n\\end{equation*}\nEach term on the right-hand side converges to $Q_b(X_i,Y_j)$.\nTherefore, the left-hand side converges to $\\sum_{ij} Q_b(X_i,Y_j)$.\nAgain by bilinearity, we obtain\n\\begin{align*}\n    \\Delta \\left( \\sum_{i,j \\in \\{1,2\\}} Q_b(X_i,Y_j) \\right)_t\n    & = \\sum_{i,j \\in \\{1,2\\}} \\Delta Q_b(X_i,Y_j)_t   \\\\\n    & = \\sum_{i,j \\in \\{1,2\\}} b(\\Delta (X_i)_t, \\Delta (Y_j)_t)   \\\\\n    & = b\\left( \\Delta (X_1 + X_2)_t, \\Delta (Y_1 + Y_2)_t \\right).\n\\end{align*}\nHence, $\\sum_{ij} Q_b(X_i,Y_j)$ is the quadratic covariation of $X_1 + X_2$ and $Y_1 + Y_2$\nwith respect to $b$.\n\\end{proof}\n\n\\begin{Cor} \\label{5.4eb} \nLet $X_1,X_2 \\in D(\\mathbb{R}_{\\geq 0},E)$\nand $Y_1,Y_2 \\in D(\\mathbb{R}_{\\geq 0},F)$.\nSuppose that $X_i$ and $Y_j$ have tensor quadratic covariation\nfor every $i,j \\in \\{ 1,2 \\}$.\nThen, $X_1 + X_2$ and $Y_1 + Y_2$ have\ntensor quadratic covariation and satisfy\n\\begin{equation*}\n    [X_1+X_2,Y_1+Y_2] = [X_1,Y_1] + [X_1,Y_2] + [X_2,Y_1] + [X_2,Y_2].\n\\end{equation*}\n\\end{Cor}\n\nNext, we investigate quadratic variations of a path of finite variation.\nFor convenience, we introduce the following notation.\nLet $D \\subset \\mathbb{R}_{\\geq 0}$.\nThen we define functions\n$e^1_D$ and $e^2_D$ from $\\mathcal{P}(\\mathbb{R}_{\\geq 0})$ to $\\{ 0,1 \\}$ as \n\\begin{equation*}\n    e^1_D(A) =\n    \\begin{cases}\n        1 & \\text{if} \\quad A \\cap D \\neq \\emptyset  \\\\\n        0 & \\text{if} \\quad A \\cap D = \\emptyset\n    \\end{cases}\n\\end{equation*}\nand $e^2_D = 1 - e^1_D$.\nHere, $\\mathcal{P}(\\mathbb{R}_{\\geq 0})$ denotes the \npower set of $\\mathbb{R}_{\\geq 0}$.\n\n\\begin{Prop} \\label{5.6b}\nLet $A \\in FV(\\mathbb{R}_{\\geq 0},E)$\nand $X \\in D(\\mathbb{R}_{\\geq 0},F)$.\nIf $(\\pi_n) \\in \\mathop{\\mathrm{Par}}[0,\\infty \\mathclose{\\lbrack}^{\\mathbb{N}}$ controls both $X$ and $A$,\nthen they have quadratic covariation with respect to every $b \\in \\mathcal{L}^{(2)}(E,F;G)$.\nMoreover, the quadratic covariation is \n\\begin{equation*}\n    Q_b(A,X)_t = \\sum_{0 < s \\leq t} b(\\Delta A_s,\\Delta X_s).\n\\end{equation*}\n\\end{Prop}\n\n\\begin{proof}\nLet $b \\in \\mathcal{L}^{(2)}(E,F;G)$,\nand set $D = D(X)$, $D_{\\varepsilon} = D_{\\varepsilon}(X)$,\nand $D^{\\varepsilon} = D^{\\varepsilon}(X)$.\nMoreover, fix $t \\in \\mathbb{R}_{\\geq 0}$ arbitrarily.\nThen we have\n\\begin{align} \\label{5.6c}\n    & \\left\\lVert \\sum_{I \\in \\pi_n} b(\\delta_I A_t,\\delta_I X_t) - \\sum_{0 < u \\leq t} b \\left( \\Delta A_u,\\Delta X_u \\right) \\right\\rVert_{G}   \\\\\n    & \\qquad \\leq \\left\\lVert \\sum_{I \\in \\pi_n}\n        b(\\delta_I A_t,\\delta_I X_t)e^1_{D_{\\varepsilon}}(I) \n        - \\sum_{0 < u \\leq t} b\\left( \\Delta A_u, \\Delta X_u \\right) \\right\\rVert_{G}\n        + \\left\\lVert \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n} b(\\delta_I A_t,\\delta_I X_t)e^2_{D_{ \\varepsilon}}(I) \\right\\rVert_{G}  \\notag\n\\end{align}\nfor any $t \\in \\mathbb{R}_{\\geq 0}$.\nWe will observe the behaviour of each term of the right-hand side of equation~\\eqref{5.6c}.\n\nBecause $(\\pi_n)$ controls $X$,\nthere exists an $N_1$ such that $D_{\\varepsilon} \\cap [0,t] \\cap I$\ncontains at most one point for all $n \\geq N_1$ and $I \\in \\pi_n$.\nIf $n \\geq N_1$, we have\n\\begin{align*}\n    \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n} b(\\delta_I A_t,\\delta_I X_t)e^1_{D_{\\varepsilon}}(I)\n    & = \\sum_{u \\in D_{\\varepsilon}} b\\left( \\delta_{\\pi_n(u)} A_t, \\delta_{\\pi_n(u)} X_t \\right).\n\\end{align*}\nNote that the c\\`{a}dl\\`{a}g property of $X$ implies that $D_{\\varepsilon}$ is a discrete set.\nTherefore, combining the above equality with Condition~(C2), we obtain\n\\begin{equation*}\n    \\lim_{n \\to \\infty}\n        \\sum_{u \\in D_{\\varepsilon}}\n            b \\left( \\delta_{\\pi_n(u)} A_t, \\delta_{\\pi_n(u)} X_t \\right)\n    =\n    \\sum_{u \\in D_{\\varepsilon} \\cap [0,t]}\n        b\\left( \\Delta A_u,\\Delta X_u \\right).\n\\end{equation*}\nThis observation allows us to deduce that\n\\begin{align*}\n    & \n    \\lim_{n \\to \\infty} \\left\\lVert \\sum_{I \\in \\pi_n} b(\\delta_I A_t,\\delta_I X_t)e_{D_{\\varepsilon}}(I) \n        - \\sum_{0 < u \\leq t} b\\left( \\Delta A_u,\\Delta X_u \\right)\\right\\rVert_G  \\\\\n    & \\qquad = \n    \\left\\lVert \\sum_{u \\in D_{\\varepsilon} \\cap [0,t]} b\\left( \\Delta A_u,\\Delta X_u \\right)\n        - \\sum_{0 < u \\leq t} b\\left( \\Delta A_u,\\Delta X_u \\right) \\right\\rVert_G   \\\\\n    & \\qquad \\leq \n    \\lVert b \\rVert \\sup_{u \\in [0,t]} \\lVert \\Delta X_u \\rVert_F \\sum_{u \\in D^{\\varepsilon} \\cap [0,t]} \\lVert \\Delta A_u \\rVert_E .\n\\end{align*}\n\nNext, we consider the second norm of the right-hand side of equation~\\eqref{5.6c},\nwhich is estimated as\n\\begin{equation*}\n    \\left\\lVert \\sum_{I \\in \\pi_n} b(\\delta_I A_t,\\delta_I X_t)\n        e^2_{D_{ \\varepsilon}}(I) \\right\\rVert_{G} \n    \\leq\n    \\lVert b \\rVert \\sum_{I \\in \\pi_n} \\lVert \\delta_I A_t \\rVert_E \\lVert \n        \\delta_I X_t \\rVert_F e^2_{D_{ \\varepsilon}}(\\mathopen{\\rbrack} r,s \\rbrack).\n\\end{equation*}\nIf $e^2_{D_{ \\varepsilon}}(I) = 1$, \n$X$ has no jumps greater than $\\varepsilon$ on $I$.\nThen\n\\begin{equation*}\n    \\left\\lVert \\delta_I X_t \\right\\rVert_F e^2_{D_{ \\varepsilon}}(I)\n= \n    \\lVert \\delta_I (X - J_{D_\\varepsilon}(X))_t \\rVert\n        e^2_{D_{ \\varepsilon}}(I)\n\\leq \n    O_t^{+}(X-J_{D_{\\varepsilon}}(X);\\pi_n) e^2_{D_{ \\varepsilon}}(I).\n\\end{equation*}\nTherefore,\n\\begin{equation*}\n    \\left\\lVert \\sum_{I \\in \\pi_n} b(\\delta_I A_t,\\delta_I X_t) \n        e^2_{D_{ \\varepsilon}}(I) \\right\\rVert_{G}  \\\\\n    \\leq \\lVert b \\rVert O_t^{+}(X-J_{D_{\\varepsilon}}(X);\\pi_n) V(A)_t.\n\\end{equation*}\n\nBy the discussion above, we can deduce that\n\\begin{align*}\n    & \n    \\varlimsup_{n \\to \\infty} \\left\\lVert \\sum_{I \\in \\pi_n} \n        b(\\delta_I A_t,\\delta_I X_t ) \n        - \\sum_{0 < u \\leq t} b\\left( \\Delta A_u,\\Delta X_u \\right) \\right\\rVert_{G}  \\\\\n    & \\qquad \n    \\leq \n    \\lVert b \\rVert \\sup_{u \\in [0,t]} \\lVert \\Delta X_u \\rVert_F \n        \\sum_{u \\in D^{\\varepsilon} \\cap [0,t]} \\lVert \\Delta A_u \\rVert_E\n    + \\lVert b \\rVert V(A)_t \\varlimsup_{n \\to \\infty} O_t^{+}(X-J_{D_{\\varepsilon}}(X);\\pi_n).\n\\end{align*}\nLetting $\\varepsilon$ go to $0$, we obtain \n\\begin{equation*}\n    \\limsup_{n \\to \\infty} \\left\\lVert \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n} \n        b(\\delta_I A_t,\\delta_I X_t)\n        - \\sum_{0 < u \\leq t} b\\left( \\Delta A_u,\\Delta X_u \\right) \\right\\rVert_{G}  \\\\\n    = 0,\n\\end{equation*}\nwhich is the desired conclusion.\n\\end{proof}\n\nApplying Proposition~\\ref{5.6b} to the canonical \nbilinear maps $\\otimes \\colon E \\times F \\to E \\widehat{\\otimes}_{\\pi} F$\nand $\\otimes \\colon F \\times E \\to F \\widehat{\\otimes}_{\\pi} E$,\nwe get the following corollary.\n\n\\begin{Cor} \\label{5.6d}\nLet $A \\in FV(\\mathbb{R}_{\\geq 0},E)$\nand $X \\in D(\\mathbb{R}_{\\geq 0},F)$.\nIf $(\\pi_n)$ controls both $A$ and $X$,\nthey have tensor quadratic covariations\n$[A,X]$ and $[X,A]$ given by\n\\begin{equation*}\n    [A,X]_t = \\sum_{0 < s \\leq t} \\Delta A_s \\otimes \\Delta X_s, \\qquad\n    [X,A]_t = \\sum_{0 < s \\leq t} \\Delta X_s \\otimes \\Delta A_s.\n\\end{equation*}\n\\end{Cor}\n\nUsing Corollaries~\\ref{5.4eb} and \\ref{5.6d}, we obtain the following.\n\n\\begin{Cor} \\label{5.6e}\nLet $(X,A) \\in D(\\mathbb{R}_{\\geq 0},E) \\times FV(\\mathbb{R}_{\\geq 0},E)$\nand suppose that $(\\pi_n)$ controls both $X$ and $A$.\nIf $X \\colon \\mathbb{R}_{\\geq 0} \\to E$ has\ntensor quadratic variation along $(\\pi_n)$,\nthen $X+A$ has tensor quadratic variation along $(\\pi_n)$.\nThe tensor quadratic variation is expressed as\n\\begin{equation*}\n    [X+A,X+A] = [X,X] + [X,A] + [A,X] + [A,A].\n\\end{equation*} \n\\end{Cor}\n\nBy a discussion similar to the proof of Proposition~\\ref{5.6b},\nwe see that a path of finite variation has scalar quadratic variation.\n\n\\begin{Prop} \\label{5.6f}\nLet $A$ be a c\\`{a}dl\\`{a}g path of finite variation in a Banach space $E$.\nIf $(\\pi_n)$ controls $A$, then $A$ has the scalar quadratic covariation given by \n\\begin{equation*}\n    Q(A)_t = \\sum_{0 < s \\leq t} \\lVert \\Delta A_s \\rVert^2.\n\\end{equation*}\n\\end{Prop}\n\nIn the proceeding part of this paper,\nwe have used the summation\n\\begin{equation*}\n    \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n} b(X_{s \\wedge t}-X_{r \\wedge t},Y_{s \\wedge t}-Y_{r \\wedge t})\n\\end{equation*}\nto define the quadratic covariation.\nWe can also consider a different form of summation\n\\begin{equation*}\n    \\sum_{\\substack{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n \\\\ r < t}} b(X_{s}-X_{r},Y_{s}-Y_{r}),\n\\end{equation*}\nwhich is a slightly modified version the summation \nused in the original paper by F{\\\"o}llmer~\\cite{Foellmer_1981}.\nLet us investigate the relation between\nthese two summations.\n\n\\begin{Prop} \\label{5.4f}\nLet $(X,Y) \\in D(\\mathbb{R}_{\\geq 0}, E \\times F)$\nand $b \\in \\mathcal{L}^{(2)}(E,F;G)$.\nSuppose that $(X,Y)_{\\overline{\\pi_n}(t)} \\to (X,Y)_t$\nholds for all $t \\in \\mathbb{R}_{\\geq 0}$.\nThen the following two conditions are equivalent:\n\\begin{enumerate}\n    \\item The paths $X$ and $Y$ have quadratic covariation along $(\\pi_n)$ with respect to $b$.\n    \\item There exists a c\\`{a}dl\\`{a}g path $V \\in FV(\\mathbb{R}_{\\geq 0},G)$ such that\n        \\begin{enumerate}\n            \\item for all $t \\in \\mathbb{R}_{\\geq 0}$\n                \\begin{equation*}\n                    \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n}\n                    1_{\\mathopen{\\rbrack} r,\\infty\\mathclose{\\lbrack}}(t) b(X_{s}-X_{r},Y_{s}-Y_{r})\n                    \\xrightarrow[n \\to \\infty]{} V_t \\quad \\text{in $G$,}\n                \\end{equation*}\n            \\item for all $t \\in \\mathbb{R}_{\\geq 0}$\n                \\begin{equation*}\n                    \\Delta V_t = b(\\Delta X_t,\\Delta Y_t).\n                \\end{equation*}\n        \\end{enumerate}\n\\end{enumerate}\nIf these conditions are satisfied, the path $V$ coincides with the quadratic covariation\n$Q_b(X,Y)$.\n\\end{Prop}\n\n\\begin{proof}\nLet $t \\in \\mathbb{R}_{> 0}$.\nThen the difference of the two summations is estimated as follows.\n\\begin{align*}\n    & \n    \\left\\lVert\n        \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n} b( \\delta_r^s X_t,\\delta_r^s Y_t )\n        - \\sum_{\\substack{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n \\\\ r < t}} b(\\delta_r^s X,\\delta_r^s Y)\n    \\right\\rVert_G  \\\\\n    & \\qquad =\n    \\left\\lVert \n        b \\left( \\delta_{\\pi_n(t)} X_t,\\delta_{\\pi_n(t)} Y_t \\right) \n        - b\\left( \\delta_{\\pi_n(t)} X,\\delta_{\\pi_n(t)} Y \\right) \n    \\right\\rVert_G    \\\\\n    & \\qquad \\leq\n    \\left\\lVert\n        b \\left( \\delta_{\\pi_n(t)} X_t-\\delta_{\\pi_n(t)} X, \\delta_{\\pi_n(t)}Y_t \\right)\n    \\right\\rVert_G\n    + \\left\\lVert\n        b \\left( \\delta_{\\pi_n(t)} X,\\delta_{\\pi_n(t)} Y_t-\\delta_{\\pi_n(t)} Y \\right) \n    \\right\\rVert_G \\\\\n    & \\qquad =\n    \\left\\lVert\n        b \\left( X_{t} - X_{\\overline{\\pi_n}(t)}, \\delta_{\\pi_n(t)}Y_t \\right)\n    \\right\\rVert_G\n    + \\left\\lVert\n        b \\left( \\delta_{\\pi_n(t)} X, Y_{t} - Y_{\\overline{\\pi_n}(t) } \\right) \n    \\right\\rVert_G \\\\\n    & \\qquad \\leq \n    \\lVert b \\rVert\n        \\left\\lVert X_{\\overline{\\pi_n}(t) \\wedge t} - X_t \\right\\rVert_{E}\n        \\left\\lVert \\delta_{\\pi_n(t)} Y_t \\right\\rVert_F\n    + \\lVert b \\rVert\n        \\left\\lVert \\delta_{\\pi_n(t)} X \\right\\lVert_E\n        \\left\\lVert Y_{\\overline{\\pi_n}(t) \\wedge t} - Y_t \\right\\rVert_F.\n\\end{align*}\nThis combined with the assumption implies\nthe equivalence of the conditions.\n\\end{proof}\n\nAccording to Proposition~\\ref{5.4f},\nwe see that the two definitions of quadratic covariation are equivalent\nprovided that $(\\pi_n)$ satisfies the assumption in the proposition.\nThe first definition,\nwhich is given in Definition~\\ref{2b}, is more intuitive.\nThe second one has some technical advantages because the path\n$t \\mapsto \\sum_{\\pi_n} 1_{\\mathopen{\\rbrack} r,\\infty \\mathclose{\\lbrack}}(t) b(\\delta_r^s X,\\delta_r^s Y)$\nis of finite variation\\footnote{Note that the path $t \\mapsto \\sum_{\\pi_n} 1_{\\mathopen{\\rbrack} r,\\infty \\mathclose{\\lbrack}}(t) b(\\delta_r^s X,\\delta_r^s Y)$ is \\emph{c\\`{a}gl\\`{a}d} but not c\\`{a}dl\\`{a}g.}.\n\n\n\\begin{Rem} \\label{5.5c}\nFollowing a discussion similar to that in Proposition~\\ref{5.4f},\nwe can obtain an equivalent definition of scalar quadratic variation using the summation\n$\\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n} \\left\\lVert X_{s}-X_{r} \\right\\rVert^2 1_{\\mathopen{\\rbrack} r,\\infty \\mathclose{\\lbrack}}(t)$\nif $(\\pi_n)$ satisfies the same condition as Proposition~\\ref{5.4f}.\n\\end{Rem}\n\n\\section{Proof of Lemma \\ref{2g}}\n\nIn this section, we prove Lemma~\\ref{2g},\nwhich is essentially used to show the main theorems of this paper.\nTo prove that lemma, we prepare some additional lemmas.\n\n\\begin{Lem} \\label{5.7c}\nSuppose that $X \\in D(\\mathbb{R}_{\\geq 0},E)$ has\ntensor quadratic variation along\na controlling sequence $(\\pi_n)$.\nLet $r,s$ be two real numbers satisfying $0 \\leq r < s$.\nIf $(\\pi_n)$ approximates $1_{\\lbrack r,s \\mathclose{\\lbrack}}$\nfrom the left, then \n\\begin{equation*}\n    \\lim_{n \\to \\infty} \\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n} 1_{\\lbrack r,s \\mathclose{\\lbrack}}(u)\\left( \\delta_u^v X_t \\right)^{\\otimes 2} = [X,X]_{s \\wedge t } - [X,X]_{r \\wedge t}\n\\end{equation*}\nholds for all $t \\in \\mathbb{R}_{\\geq 0}$.\n\\end{Lem}\n\n\\begin{proof}\nBy considering the decomposition \n\\begin{equation*}\n\\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n} 1_{\\lbrack r,s \\mathclose{\\lbrack}}(u)\\left( \\delta_u^v X_t \\right)^{\\otimes 2}\n= \\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n} 1_{\\lbrack 0,s \\mathclose{\\lbrack}}(u)\\left( \\delta_u^v X_t \\right)^{\\otimes 2}\n    - \\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n} 1_{\\lbrack 0,r \\mathclose{\\lbrack}}(u)\\left( \\delta_u^v X_t \\right)^{\\otimes 2},\n\\end{equation*}\nwe see that it suffices to show that the convergence\n\\begin{gather*}\n    \\lim_{n \\to \\infty} \\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n} 1_{\\lbrack 0,r \\mathclose{\\lbrack}}(u) \\left( \\delta_u^v X_t \\right)^{\\otimes 2}\n    = [X,X]_{t \\wedge r}  \\\\\n    \\lim_{n \\to \\infty} \\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n} 1_{\\lbrack 0,s \\mathclose{\\lbrack}}(u) \\left( \\delta_u^v X_t \\right)^{\\otimes 2}\n    = [X,X]_{t \\wedge s}\n\\end{gather*}\nholds for all $t \\in \\mathbb{R}_{\\geq 0}$.\n\nIf $t \\leq s$, we see that\n\\begin{align*}\n    \\lim_{n \\to \\infty}\n        \\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n} \n        1_{\\lbrack 0,s \\mathclose{\\lbrack}}(u)\n        \\left( \\delta_u^v X_t \\right)^{\\otimes 2}\n    =\n    \\lim_{n \\to \\infty}\n        \\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n}\n        \\left( \\delta_u^v X_t \\right)^{\\otimes 2} \n    =\n    [X,X]_{t}\n    =\n    [X,X]_{t \\wedge s}.\n\\end{align*}\n\nNext, assume $s < t$.\nTake an arbitrary $\\varepsilon > 0$.\nThen $s \\notin \\pi_n(s+\\varepsilon)$ holds\nfor sufficiently large $n$,\nbecause $(\\pi_n)$ approximates $1_{\\lbrack r,s \\mathclose{\\lbrack}}$\nfrom the left.\nThis leads to $\\overline{\\pi_n}(s) \\leq s + \\varepsilon$\nfor sufficiently large $n$,\nand hence $\\overline{\\pi_n}(s) \\to s$.\nTherefore, we obtain\n\\begin{equation*}\n    \\lim_{n \\to \\infty}\n        \\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n} \n        1_{\\lbrack 0,s \\mathclose{\\lbrack}}(u)\n        \\left( \\delta_u^v X_t \\right)^{\\otimes 2}\n=\n    \\lim_{n \\to \\infty}\n        \\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n} \n        1_{\\lbrack 0,s \\mathclose{\\lbrack}}(u)\n        \\left( \\delta_u^v X \\right)^{\\otimes 2}\n=\n    [X,X]_s\n=\n    [X,X]_{s \\wedge t}.\n\\end{equation*}\n\nThe convergence about $1_{\\lbrack 0,r \\mathclose{\\lbrack}}$ is shown\nby a similar discussion.\n\\end{proof}\n\n\\begin{Lem} \\label{5.7d}\nLet $X \\in D(\\mathbb{R}_{\\geq 0},E)$ be a path of\ntensor quadratic variation along a controlling sequence $(\\pi_n)$.\nSuppose that $\\xi \\in D(\\mathbb{R}_{\\geq 0}, \\mathcal{L}(E \\widehat{\\otimes}_{\\pi} E,G))$\nhas the representation\n\\begin{equation} \\label{5.7e}\n    \\xi = \\sum_{i \\geq 1} 1_{\\lbrack \\tau_{i-1},\\tau_{i} \\mathclose{\\lbrack}} a_{i},\n\\end{equation}\nwhere $0 = \\tau_0 < \\tau_1 < \\dots < \\tau_i < \\tau_{i+1} < \\dots \\to \\infty$\nand $(a_i) \\in \\mathcal{L}(E \\widehat{\\otimes}_{\\pi} E,G)^{\\mathbb{N}}$.\nIf $(\\pi_n)$ approximates $\\xi$ from the left, the Stieltjes integral of $\\xi_{-}$ with respect\nto $[X,X]$ is approximated as\n\\begin{equation} \\label{5.7f}\n    \\lim_{n \\to \\infty} \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n} \n        \\langle \\xi_{r}, (\\delta_r^s X_t)^{\\otimes 2} \\rangle\n    = \\int_{]0,t]} \\langle \\xi_{s-},\\mathrm{d}[X,X]_s \\rangle.\n\\end{equation}\n\\end{Lem}\n\n\\begin{proof}\nFirst, note that the Stieltjes integral on the right-hand side \nof equation~\\eqref{5.7f} has the representation\n\\begin{equation*}\n    \\int_{]0,t]} \\langle \\xi_{s-},\\mathrm{d}[X,X]_s \\rangle\n    = \\sum_{i \\geq 1} \\langle a_{i}, [X,X]_{\\tau_i \\wedge t} - [X,X]_{\\tau_{i-1} \\wedge t} \\rangle.\n\\end{equation*}\nBy the bilinearity of the canonical pairing, \nwe can calculate this as\n\\begin{align*}\n    \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n} \\langle \\xi_{r}, (\\delta_r^s X_t)^{\\otimes 2} \\rangle\n    = \n    \\sum_{i \\geq 1} \\left\\langle a_{i},\n        \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n} 1_{\\lbrack \\tau_{i-1},\\tau_{i} \\mathclose{\\lbrack}}(r)\n        (\\delta_r^s X_t)^{\\otimes 2} \\right\\rangle.\n\\end{align*}\nHence, it remains to show that\n\\begin{equation*}\n    \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n} 1_{\\lbrack \\tau_{i-1},\\tau_{i} \\mathclose{\\lbrack}}(r)\n        (\\delta_r^s X_t)^{\\otimes 2}\n    \\xrightarrow[n \\to \\infty]{}\n    [X,X]_{\\tau_i \\wedge t} - [X,X]_{\\tau_{i-1} \\wedge t}\n\\end{equation*}\nholds in the sense of the norm topology of $E \\widehat{\\otimes}_{\\pi} E$.\nThis has already been proved in Lemma~\\ref{5.7c}.\n\\end{proof}\n\nFinally, we start dealing with the proof of Lemma~\\ref{2g}.\n\n\\begin{proof}[Proof of Lemma \\ref{2g}]\nFix $t > 0$ and $\\varepsilon > 0$ arbitrarily.\nWe can choose\nan $h \\in D(\\mathbb{R}_{\\geq 0},L(E \\widehat{\\otimes}_{\\pi} E,G))$\nof the form \\eqref{5.7e}\nso that $\\lVert h(s) - \\xi(s) \\rVert \\leq \\varepsilon$ holds for all $s$\nand $(\\pi_n)$ controls $h$ from the left\\footnote{\n    In particular, one can take an appropriate partition\n    $\\tau = \\{ \\mathopen{\\rbrack} t_i, t_{i+1} \\rbrack ; i \\in \\mathbb{N} \\} \\in \\mathop{\\mathrm{Par}}(\\mathbb{R}_{\\geq 0})$\n    such that $I \\subset [t_i, t_{i+1}]$\n    for some $i$ if $\\xi$ is constant on $I$.\n    Now set \n    \\begin{equation*}\n        h = \\sum_{i} \\xi(t_i) \\lbrack t_i,t_{i+1} \\mathclose{\\lbrack}.\n    \\end{equation*}\n    Then the sequence $(\\pi_n)$ approximates $h$ from the left.\n}.\n\nThen,\n\\begin{align*}\n    &\n    \\left\\lVert \n        \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n}\n            \\left\\langle\n                \\xi(r), \\delta_r^s X_t \\otimes \\delta_r^s X_t \n            \\right\\rangle \n        - \\int_{\\mathopen{\\rbrack} 0,t \\rbrack}\n            \\left\\langle \n                \\xi_{u-}, \\mathrm{d}\\!\\left[ X,X \\right]_u\n            \\right\\rangle\n    \\right\\rVert_G   \\notag \\\\\n    & \\qquad \\leq \n    \\left\\lVert\n        \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n}\n            \\left\\langle \n                \\xi(r), \\delta_r^s X_t \\otimes \\delta_r^s X_t\n            \\right\\rangle\n        - \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n} \n            \\left\\langle\n                h(r), \\delta_r^s X_t \\otimes \\delta_r^s X_t\n            \\right\\rangle \n    \\right\\rVert_G \\\\\n    & \\qquad\\quad +\n    \\left\\lVert\n        \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n}\n            \\left\\langle \n                h(r), \\delta_r^s X_t \\otimes \\delta_r^s X_t\n            \\right\\rangle\n        - \\int_{\\mathopen{\\rbrack} 0,t \\rbrack}\n            \\left\\langle\n                h(u-), \\mathrm{d}[X, X]_u\n            \\right\\rangle \n    \\right\\rVert_G \\\\\n    & \\qquad\\quad +\n    \\left\\lVert\n        \\int_{\\mathopen{\\rbrack} 0,t \\rbrack}\n            \\left\\langle\n                h(u-), \\mathrm{d}[X, X]_u\n            \\right\\rangle \n        - \\int_{\\mathopen{\\rbrack} 0,t \\rbrack}\n            \\left\\langle\n                \\xi(u-), \\mathrm{d}[ X,X ]_u \n            \\right\\rangle\n    \\right\\rVert_G.\n\\end{align*}\nWe will observe the behaviour of each part of the last side.\n\nWe know from Lemma~\\ref{5.7d} that the second term converges to $0$ as $n \\to \\infty$.\nThat is,\n\\begin{equation*}\n    \\left\\lVert\n        \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n}\n            \\left\\langle \n                h(r), \\delta_r^s X_t \\otimes \\delta_r^s X_t\n            \\right\\rangle\n        - \\int_{\\mathopen{\\rbrack} 0,t \\rbrack}\n            \\left\\langle\n                h(u-), \\mathrm{d}[X, X]_u\n            \\right\\rangle \n    \\right\\rVert_G\n    \\xrightarrow[n \\to \\infty]{} 0.\n\\end{equation*}\nFrom the choice of $h$, we find that\n\\begin{align*}\n    \\left\\lVert\n    \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n}\n        \\left\\langle \n            \\xi(r), \\delta_r^s X_t \\otimes \\delta_r^s X_t\n        \\right\\rangle\n    - \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n} \n        \\left\\langle\n            h(r), \\delta_r^s X_t \\otimes \\delta_r^s X_t\n        \\right\\rangle \n    \\right\\rVert_G\n& \\leq\n    \\varepsilon\n    \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n}\n        \\left\\lVert \n            \\delta_r^s X_t\n        \\right\\rVert_E^2.\n\\end{align*}\nTherefore,\n\\begin{equation*}\n    \\varlimsup_{n \\to \\infty} \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n}\n        \\left\\lVert \\xi(r)-h(r) \\right\\rVert\n        \\left\\lVert \n            \\delta_r^s X_t \\otimes \\delta_r^s X_t\n        \\right\\rVert\n    \\leq \\varepsilon \\overline{Q}(X)_t.\n\\end{equation*}\nOn the other hand, we have\n\\begin{align*}\n    \\left\\lVert\n        \\int_{\\mathopen{\\rbrack} 0,t \\rbrack}\n            \\left\\langle\n                h(u-)-\\xi(u-), \\mathrm{d}[ X,X ]_u \n            \\right\\rangle\n    \\right\\rVert_G\n& \\leq \n    \\int_{\\mathopen{\\rbrack} 0,t \\rbrack}\n        \\left\\lVert h(u-)-\\xi(u-) \\right\\rVert\n        \\mathrm{d}V([X,X])_t  \\\\\n& \\leq\n    \\varepsilon V([X,X])_t.\n\\end{align*}\n\nConsequently, we see that\n\\begin{equation*}\n    \\varlimsup_{n \\to \\infty}\n    \\left\\lVert \n        \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n}\n            \\left\\langle\n                \\xi(r), \\delta_r^s X_t \\otimes \\delta_r^s X_t \n            \\right\\rangle \n        - \\int_{\\mathopen{\\rbrack} 0,t \\rbrack}\n            \\left\\langle \n                \\xi_{u-}, \\mathrm{d}[ X,X ]_u\n            \\right\\rangle\n    \\right\\rVert_G  \\\\\n\\leq\n    \\varepsilon \\{ \\overline{Q}(X)_t + V([X,X])_t \\}.\n\\end{equation*}\nBecause $\\varepsilon$ is chosen arbitrarily,\nwe get the desired conclusion.\n\\end{proof}\n\n\\section{The It{\\^o} formula}\n\nThis section is devoted to showing the It{\\^o} formula\nwithin our framework of the It{\\^o}-F{\\\"o}llmer calculus \nin Banach space.\nLet us begin with the definition of It{\\^o}-F{\\\"o}llmer integrals.\n\n\\begin{Def} \\label{5.9b}\nLet $(H,X) \\in D([0,\\infty\\mathclose{[},E \\times F)$\nand $b \\in \\mathcal{L}^{(2)}(E,F;G)$.\nSuppose that a sequence of partitions $(\\pi_n)$ approximates\n$H$ from the left. We call the limit\n\\begin{equation*}\n    \\int_0^t b(H_{s-}, \\mathrm{d}X_{s})\n    = \\int_{\\mathopen{\\rbrack} 0,t \\rbrack} b(H_{s-}, \\mathrm{d}X_{s})\n    := \\lim_{n \\to \\infty} \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n} b\\left( H_r, \\delta_r^s X_t \\right) \\in G\n\\end{equation*}\nthe \\emph{It{\\^o}-F{\\\"o}llmer integral\nof $H$ with respect to $X$ through $b$\nalong the sequence $(\\pi_n)$}, if it exists.\nSimilarly,\n\\begin{equation*}\n    \\int_{\\mathopen{\\rbrack} 0,t \\rbrack} b'(\\mathrm{d}X_s, H_{s-})\n    = \\lim_{n \\to \\infty} \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n} b'\\left( \\delta_r^s X_t,H_{r} \\right) \\in G\n\\end{equation*}\nis defined for a $b' \\in \\mathcal{L}^{(2)}(F,E;G)$.\n\\end{Def}\n\nIf $b$ is the canonical bilinear map\n$\\otimes \\colon E \\times F \\to E \\widehat{\\otimes}_{\\pi} F$,\nwe write\n\\begin{equation*}\n    \\int_{\\mathopen{\\rbrack} 0,t \\rbrack} b(H_{s-},\\mathrm{d}X_s)\n    = \\int_{\\mathopen{\\rbrack} 0,t \\rbrack} H_{s-} \\otimes \\mathrm{d}X_s.\n\\end{equation*}\n\n\\begin{Rem} \\label{5.9c}\nThe It{\\^o}-F{\\\"o}llmer integral of Definition~\\ref{5.9b}\ninherits the bilinear property from\n$b \\in \\mathcal{L}^{(2)}(E,F;G)$ in the following sense. \n\n\\begin{enumerate}\n    \\item Suppose that the following two It{\\^o}-F{\\\"o}llmer integrals exist:\n        \\begin{equation*}\n            \\int_{\\mathopen{\\rbrack} 0,t \\rbrack} b(H_{s-}, \\mathrm{d}X_{s}), \\qquad \n            \\int_{\\mathopen{\\rbrack} 0,t \\rbrack} b(K_{s-}, \\mathrm{d}X_{s}).\n        \\end{equation*}\n        Then, for every $\\alpha$, $\\beta \\in \\mathbb{R}$,\n        the It{\\^o}-F{\\\"o}llmer integral of $\\alpha H + \\beta K$\n        with respect to $X$ satisfies\n        \\begin{equation*}\n            \\int_{\\mathopen{\\rbrack} 0,t \\rbrack} b(H_{s-} + K_{s-}, \\mathrm{d}X_{s})\n            =\n            \\alpha \\int_{\\mathopen{\\rbrack} 0,t \\rbrack} b(H_{s-}, \\mathrm{d}X_{s}) \n            + \\beta \\int_{\\mathopen{\\rbrack} 0,t \\rbrack} b(K_{s-}, \\mathrm{d}X_{s}).\n        \\end{equation*}\n    \\item Suppose that the following two It{\\^o}-F{\\\"o}llmer integrals exist:\n        \\begin{equation*}\n            \\int_{\\mathopen{\\rbrack} 0,t \\rbrack} b(H_{s-}, \\mathrm{d}X_{s}), \\qquad \n            \\int_{\\mathopen{\\rbrack} 0,t \\rbrack} b(H_{s-}, \\mathrm{d}Y_{s}).\n        \\end{equation*}\n        Then, for every $\\alpha$, $\\beta \\in \\mathbb{R}$,\n        the It{\\^o}-F{\\\"o}llmer integral of $H$ with respect to \n        $\\alpha X + \\beta Y$ exists and satisfies\n        \\begin{equation*}\n            \\int_{\\mathopen{\\rbrack} 0,t \\rbrack} b(H_{s-}, \\mathrm{d}(\\alpha X+ \\beta Y)_{s})\n            =\n            \\alpha \\int_{\\mathopen{\\rbrack} 0,t \\rbrack} b(H_{s-}, \\mathrm{d}X_{s}) \n            + \\beta \\int_{\\mathopen{\\rbrack} 0,t \\rbrack} b(H_{s-}, \\mathrm{d}Y_{s}).\n        \\end{equation*}\n\\end{enumerate}\n\\end{Rem}\n\nFirst, we consider the case where\nthe integrator is a path of finite variation.\nFrom the dominated convergence theorem,\nwe can easily deduce the following proposition.\n\n\\begin{Prop} \\label{5.9d}\nLet $H \\in D(\\mathbb{R}_{\\geq 0},E)$, $A \\in FV(\\mathbb{R}_{\\geq 0},F)$,\nand $b \\in \\mathcal{L}^{(2)}(E,F;G)$.\nIf a sequence of partition $(\\pi_n)$ approximates $H$ from the left,\nwe have\n\\begin{equation*}\n    (\\mathrm{IF}) \\int_{\\mathopen{\\rbrack} 0,t \\rbrack} b(H_{s-},\\mathrm{d}A_s)\n    =\n    (\\mathrm{S}) \\int_{\\mathopen{\\rbrack} 0,t \\rbrack} b(H_{s-},\\mathrm{d}A_s).\n\\end{equation*}\nHere, the integral of the left-hand side is the It{\\^o}-F{\\\"o}llmer integral\nby Definition~\\ref{5.9b},\nand that of the right-hand side is the usual Stieltjes integral.\n\\end{Prop}\n\nNow we start to prove our main theorem.\n\n\\begin{proof}[Proof of Theorem \\ref{2e}]\nFix $t>0$ arbitrarily and pick a compact convex set $K \\subset F \\times E$\nso that the image $A[0,t] \\times X[0,t]$ is contained in $K$\\footnote{\n    For example, we can take $K$ as the convex closed hull of $A[0,t] \\times X[0,t]$.\n}.\nWe know by Proposition~\\ref{3b} that such a $K$ actually exists.\n\n\\emph{Step 1: Convergence of the summation in the formula~\\eqref{2f}.}\nWe first confirm that the summation of jump terms converges absolutely in $G$.\nBecause $X$ has quadratic variation, we have\n\\begin{equation*}\n    \\sum_{0 < s \\leq t}\n        \\left\\lVert\n            \\langle D_x^2 f(A_{s-},X_{s-}), \\Delta X_s \\otimes \\Delta X_s \\rangle\n        \\right\\rVert_{G}\n\\leq\n    \\sup_{(a,x) \\in K}\n        \\left\\lVert D_x^2 f(a,x) \\right\\rVert\n        \\sum_{0 < s \\leq t} \\left\\lVert \\Delta X_s \\right \\rVert_{E}^2\n< \\infty.\n\\end{equation*}\nBy Taylor's theorem, we also find that\n\\begin{align*}\n    &\n    \\sum_{0 < s \\leq t}\n    \\left\\lVert\n        f(A_s,X_s) - f(A_{s-},X_{s})  - \\langle D_a f(A_{s-},X_{s-}), \\Delta A_{s} \\rangle\n    \\right\\rVert_G  \\\\\n    & \\qquad \\leq \n    \\sum_{0 < s \\leq t}\n    \\left\\lVert\n        f(A_s,X_s) - f(A_{s-},X_{s})  - \\langle D_a f(A_{s-},X_{s}), \\Delta A_{s} \\rangle\n    \\right\\rVert_G  \\\\\n    & \\qquad \\quad +\n    \\sum_{0 < s \\leq t}\n    \\left\\lVert\n        \\langle D_a f(A_{s-},X_{s}), \\Delta A_{s} \\rangle - \\langle D_a f(A_{s-},X_{s-}), \\Delta A_{s} \\rangle\n    \\right\\rVert_G  \\\\\n    & \\qquad \\leq\n    2\\, \\omega(D_af(a,x);K)\n    V(A)_t < \\infty.\n\\end{align*}\nAgain by Taylor's theorem, we obtain\n\\begin{align*}\n    &\n    \\sum_{0 < s \\leq t}\n        \\left\\lVert\n            f(A_{s-},X_s) - f(A_{s-},X_{s-})  - \\langle D_x f(A_{s-},X_{s-}), \\Delta X_{s} \\rangle\n        \\right\\rVert_G  \\\\\n    & \\qquad \\leq\n    \\sup_{(a,x) \\in K} \\lVert D^2_x f(a,x) \\rVert V([X,X])_t\n    <\n    \\infty.\n\\end{align*}\nTherefore,\n\\begin{align*}\n    &\n    \\sum_{0 < s \\leq t}\n        \\lVert\n            f(A_s,X_s) - f(A_{s-},X_{s-})\n            - \\langle D_x f(A_{s-},X_{s-}),\\Delta X_s \\rangle\n            - \\langle D_a f(A_{s-},X_{s-}),\\Delta A_s \\rangle\n        \\rVert_{G}  \\\\\n    & \\qquad \\leq\n    \\sum_{0 < s \\leq t}\n        \\lVert\n            f(A_s,X_s) - f(A_{s-},X_{s})\n            - \\langle D_a f(A_{s-},X_{s-}),\\Delta A_s \\rangle\n        \\rVert_{G}  \\\\\n    & \\qquad\\quad +\n    \\sum_{0 < s \\leq t}\n        \\lVert\n            f(A_{s-},X_{s}) - f(A_{s-},X_{s-})\n            - \\langle D_x f(A_{s-},X_{s-}),\\Delta X_s \\rangle\n        \\rVert_{G}  \\\\\n    & \\qquad \\leq \n    2 \\,\\omega(D_af(a,x);K) \\, \n        V(A)_t\n    +\n    \\sup_{(a,x)\\in K} \\lVert D^2_x f(a,x) \\rVert V([X,X])_t,\n\\end{align*}\nand that estimate implies\n\\begin{align*}\n    &\n    \\sum_{0 < s \\leq t}\n    \\lVert\n        f(A_s,X_s) - f(A_{s-},X_{s-})\n        - \\langle D_x f(A_{s-},X_{s-}),\\Delta X_s \\rangle\n    \\rVert_{G}  \\\\\n    & \\qquad \\leq\n    \\sum_{0 < s \\leq t}\n    \\lVert\n        f(A_s,X_s) - f(A_{s-},X_{s-})\n        - \\langle D_x f(A_{s-},X_{s-}),\\Delta X_s \\rangle\n        - \\langle D_a f(A_{s-},X_{s-}),\\Delta A_s \\rangle\n    \\rVert_{G}  \\\\\n    & \\qquad\\quad +\n    \\sum_{0 < s \\leq t}\n        \\lVert \\langle D_a f(A_{s-},X_{s-}),\\Delta A_s \\rangle \\rVert_G \\\\\n    & \\qquad \\leq \n    3 \\,\\omega(D_af(a,x);K) \\, \n        V(A)_t\n    +\n    \\sup_{(a,x)\\in K} \\lVert D^2_x f(a,x) \\rVert V([X,X])_t \\\\\n    & \\qquad < \\infty.\n\\end{align*}\n\nFrom these absolute convergence results,\nwe can see that equation~\\eqref{2f}\nis equivalent to the following:\n\\begin{align} \\label{5.10d}\n    & \n    f(A_t,X_t) - f(A_0,X_0) \n    -\n    \\int_{\\mathopen{\\rbrack} 0,t \\rbrack}\n        \\langle D_x f (A_{s-},X_{s-}), \\mathrm{d}X_s \\rangle   \\\\\n    & \\qquad = \n    \\int_{\\mathopen{\\rbrack} 0,t \\rbrack} \\langle D_a f (A_{s-},X_{s-}), \\mathrm{d}A_{s} \\rangle \n    +\n    \\frac{1}{2} \\int_{\\mathopen{\\rbrack} 0,t \\rbrack} \n        \\langle D_x^2 f (A_{s-},X_{s-}), \\mathrm{d}[X,X]_s \\rangle   \\notag \\\\\n    & \\qquad\\quad +\n    \\sum_{0 < s \\leq t} \n        \\left\\{ \\Delta f (A_s,X_s) - \\langle D_x f (A_{s-},X_{s-}), \\Delta X_s \\rangle \\right\\}  \\notag \\\\\n    & \\qquad\\quad - \n    \\sum_{ 0 < s \\leq t} \n        \\langle D_a f (A_{s-},X_{s-}), \\Delta A_s \\rangle\n        - \n        \\frac{1}{2} \\sum_{0 < s \\leq t} \n            \\langle D_x^2 f (A_{s-},X_{s-}), \\Delta X_s \\otimes \\Delta X_s \\rangle. \\notag\n\\end{align}\nWe will therefore prove equation~\\eqref{5.10d} instead of \\eqref{2f}.\n\n\\emph{Step 2: The Taylor expansion.}\nNow we write $D = D(A,X)$,\n$D_{\\varepsilon} = D_{\\varepsilon}(A,X)$,\nand $D^{\\varepsilon}(A,X)$.\n\nLet $\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n$ and \nconsider the first-order Taylor expansion with respect to the variable $a$\nbetween $A_{u \\wedge t}$ and $A_{v \\wedge t}$.\nThen we have \n\\begin{equation} \\label{5.10e}\n    f(A_{v \\wedge t},X_{v \\wedge t}) - f(A_{u \\wedge t},X_{v \\wedge t})\n=\n    \\langle \n        D_a f( A_{u \\wedge t}, X_{u \\wedge t} ),\\delta_u^v A_t\n    \\rangle\n    +\n    \\langle\n        r_u^v, \\delta_u^v A_t\n    \\rangle,\n\\end{equation}\nwhere\n\\begin{equation*}\n    r_u^v\n=\n    \\int_{[0,1]}\n        \\left\\{\n            D_a f \\left( A_{u \\wedge t} + \\theta_u^v A_t,X_{v \\wedge t} \\right)\n            -\n            D_a f\\left( A_{u \\wedge t},X_{u \\wedge t} \\right)\n        \\right\\}\n    \\mathrm{d}\\theta.\n\\end{equation*}\n\nNext, we consider the second-order Taylor expansion\n\\begin{align} \\label{5.10f}\n    &\n        f(A_{u \\wedge t},X_{v \\wedge t}) - f(A_{u \\wedge t},X_{u \\wedge t})  \\\\\n    & \\qquad =\n        \\langle D_x f(A_{u \\wedge t},X_{u \\wedge t}), \\delta_u^v X_t \\rangle\n        + \n        \\frac{1}{2}\n        \\langle\n            D_x^2 f(A_{u \\wedge t},X_{u \\wedge t}), (\\delta_u^v X_t)^{\\otimes 2}\n        \\rangle\n        +\n        \\langle \n            R_u^v, (\\delta_u^v X_t)^{\\otimes 2}\n        \\rangle, \\notag\n\\end{align}\nwith $R_u^v$ in the residual term given by \n\\begin{equation*}\n    R_u^v\n=\n    \\frac{1}{2}\n    \\int_{[0,1]}\n        (1-\\theta)\n        \\left\\{ \n            D^2_x f(A_{u \\wedge t},X_{u \\wedge t} + \\theta \\delta_u^v X_t)\n            -\n            D_x^2 f(A_{u \\wedge t},X_{u \\wedge t})\n        \\right\\}\n    \\mathrm{d}\\theta.\n\\end{equation*}\n\nCombining equations~\\eqref{5.10e} and \\eqref{5.10f}, we obtain\n\\begin{align*}\n    \\delta_u^v (f \\circ (A,X))_t\n& =\n    \\langle D_a f(A_{u},X_{u}), \\delta_{u}^v A_t \\rangle\n    +\n    \\langle r_u^v, \\delta_u^v A_t \\rangle\n    +\n    \\langle D_x f(A_{u},X_{u}), \\delta_u^v X_t \\rangle     \\\\\n    & \\quad +\n        \\frac{1}{2}\n        \\langle\n            D_x^2 f \\left( A_{u},X_{u} \\right), (\\delta_u^v X_t)^{\\otimes 2}\n        \\rangle\n    +\n    \\langle R_u^v, (\\delta_u^v X_t)^{\\otimes 2} \\rangle.\n\\end{align*}\nMoreover, by summing up this equality through $\\pi_n$, \nwe see that\n\\begin{align*}\n    &\n    f(X_t)- f(X_0)\n    -\n    \\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n}\n        \\langle \n            D_x f(A_{u},X_{u}),\\delta_u^v X_t\n        \\rangle    \\\\\n    & \\qquad =\n    \\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n}\n        \\langle D_a f(A_{u},X_{u}), \\delta_{u}^v A_t \\rangle\n    +\n    \\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n}\n        \\langle r_u^v, \\delta_u^v A_t \\rangle  \\\\\n    & \\qquad\\quad +\n    \\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n}\n        \\frac{1}{2}\n        \\langle\n            D_x^2 f \\left( A_{u},X_{u} \\right), (\\delta_u^v X_t)^{\\otimes 2}\n        \\rangle\n    +\n    \\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n}\n        \\langle R_u^v, (\\delta_u^v X_t)^{\\otimes 2} \\rangle.\n\\end{align*}\nUsing the notation $e^1_{D_{\\varepsilon}}$ and $e^2_{D_{\\varepsilon}}$,\nwe can transform the previous equality as\n\\begin{align} \\label{5.10g}\n    &\n    f(X_t)- f(X_0)\n    -\n    \\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n}\n        \\langle \n            D_x f(A_{u},X_{u}),\\delta_u^v X_t\n        \\rangle   \\\\\n    & \\qquad =\n    \\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n}\n        \\delta_u^v (f \\circ (A,X))_t e^1_{D_\\varepsilon}(\\mathopen{\\rbrack} u,v \\rbrack)\n    -\n    \\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n}\n        \\langle D_x f(A_u,X_u), \\delta_u^v X_t \\rangle \n        e^1_{D_{\\varepsilon}}(\\mathopen{\\rbrack} u,v \\rbrack)  \\notag \\\\\n    & \\qquad\\quad +\n    \\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n}\n        \\langle D_a f(A_{u},X_{u}), \\delta_{u}^v A_t \\rangle\n    -\n    \\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n}\n        \\langle D_a f(A_{u},X_{u}), \\delta_{u}^v A_t \\rangle e^1_{D_\\varepsilon}(\\mathopen{\\rbrack} u,v \\rbrack)  \\notag \\\\\n    & \\qquad\\quad +\n    \\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n}\n        \\frac{1}{2}\n        \\langle\n            D_x^2 f \\left( A_{u},X_{u} \\right),\n            (\\delta_u^v X_t)^{\\otimes 2}\n        \\rangle\n    -\n    \\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n}\n        \\frac{1}{2}\n        \\langle\n            D_x^2 f \\left( A_{u},X_{u} \\right),\n            (\\delta_u^v X_t)^{\\otimes 2}\n        \\rangle\n        e^1_{D_\\varepsilon}(\\mathopen{\\rbrack} u,v \\rbrack) \\notag \\\\\n    & \\qquad\\quad +\n    \\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n}\n        \\langle r_u^v, \\delta_u^v A_t \\rangle e^2_{D_{\\varepsilon}}(\\mathopen{\\rbrack} u,v \\rbrack)\n    +\n    \\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n}\n        \\langle R_u^v, (\\delta_u^v X_t)^{\\otimes 2} \\rangle\n        e^2_{D_{\\varepsilon}}(\\mathopen{\\rbrack} u,v \\rbrack)  \\notag \\\\\n    & \\qquad =\n    I_1^{(n)}(t) - I_2^{(n)}(t) + I_3^{(n)}(t) - I_4^{(n)}(t)\n    + I_5^{(n)}(t) - I_6^{(n)}(t) + I_7^{(n)}(t) + I_8^{(n)}(t). \\notag\n\\end{align}\n\n\\emph{Step 3: Behaviour of $I_1^{(n)}(t),\\dots , I_8^{(n)}(t)$ of \\eqref{5.10g}.}\nBecause $D_{\\varepsilon}$ is discrete and $(\\pi_n)$ controls $(X,A)$,\nwe can deduce that\n\\begin{align*}\n    \\lim_{n \\to \\infty} I_4^{(n)}(t)\n& = \n    \\sum_{s \\in D_{\\varepsilon} \\cap [0,t]} \n        \\langle \n            D_a f(A_{s-},X_{s-}), \\Delta A_s\n        \\rangle   \\\\\n    \\lim_{n \\to \\infty} I_2^{(n)}(t)\n& =\n    \\sum_{s \\in D_\\varepsilon \\cap [0,t]} \n        \\langle D_x f(A_{s-},X_{s-}), \\Delta X_s \\rangle    \\\\\n    \\lim_{n \\to \\infty} I_6^{(n)}(t)\n& = \n    \\frac{1}{2} \\sum_{s \\in D_\\varepsilon \\cap [0,t]} \n        \\langle \n            D_x^2 f(A_{s-},X_{s-}), (\\Delta X_s)^{\\otimes 2}\n        \\rangle.\n\\end{align*}\nIf $s \\in D$ and $s \\leq t$, we see that \n\\begin{align*}\n    \\delta_{\\pi_n(s)}f(A,X)_t\n    & =\n    \\int_0^1 \\langle D_{a,x}f((A,X)_{\\underline{\\pi_n}(s)} + \\theta \\delta_{\\pi_n(s)} (A,X)_t), \n        \\delta_{\\pi_n(s)} (A,X)_t \\rangle \\,\\mathrm{d} \\theta   \\\\\n    & \\xrightarrow[n \\to \\infty]{}\n    \\int_0^1 \\langle D_{a,x}f((A,X)_{s-} + \\theta \\Delta (A,X)_s), \n        \\Delta (A,X)_s \\rangle \\,\\mathrm{d} \\theta  \n    = \\Delta f(A,X)_s.\n\\end{align*}\nfrom the assumption and the dominated convergence theorem. Hence, \n\\begin{equation*}\n    \\lim_{n \\to \\infty} I_1^{(n)}(t)\n    = \n    \\sum_{s \\in D_{\\varepsilon} \\cap [0,t]} \\Delta f (A_s,X_s).\n\\end{equation*}\n\nBy Lemma~\\ref{2g}, we have\n\\begin{equation*}\n    \\lim_{n \\to \\infty} I_5^{(n)}(t)\n= \n    \\frac{1}{2} \\int_{\\mathopen{\\rbrack} 0,t \\rbrack} \n        \\langle \n            D_x^2 f(A_{s-},X_{s-}),\\mathrm{d}[X,X]_s\n        \\rangle.\n\\end{equation*}\n\nThe dominated convergence theorem gives\n\\begin{equation*}\n    \\lim_{n \\to \\infty} I_3^{(n)}(t)\n=\n    \\int_{\\mathopen{\\rbrack} 0,t \\rbrack} \\langle D_a f(A_{s-},X_{s-}) , \\mathrm{d}A_s \\rangle.\n\\end{equation*}\n\nIt remains to estimate the residual terms.\nIf $\\mathopen{\\rbrack} u,v \\rbrack \\cap D_{\\varepsilon} = \\emptyset$,\noscillation of the paths on $\\mathopen{\\rbrack} u,v \\rbrack$ is well controlled as\n\\begin{gather*}\n\\omega(X, \\mathopen{\\rbrack} u,v \\rbrack \\cap [0,t])\n    \\leq \n        O^{+}_t(X-J(D_{\\varepsilon}(X);X);\\pi_n) + \\varepsilon,   \\\\\n\\omega(A, \\mathopen{\\rbrack} u,v \\rbrack \\cap [0,t])\n    \\leq \n        O^{+}_t(A-J(D_{\\varepsilon}(A);A);\\pi_n) + \\varepsilon.\n\\end{gather*}\nNow we write, for convenience,\n\\begin{gather*}\n    \\alpha(\\varepsilon,n) = O^{+}_t(X-J(D_{\\varepsilon}(X);X);\\pi_n) + \\varepsilon,   \\\\\n    \\beta(\\varepsilon,n) = O^{+}_t(A-J(D_{\\varepsilon}(A);A);\\pi_n) + \\varepsilon.\n\\end{gather*}\nBy the assumption that $(\\pi_n)$ controls $(A,X)$---and hence so does both $A$ and $X$---we see that\n\\begin{equation*}\n    \\varlimsup_{\\varepsilon \\downarrow\\downarrow 0} \\varlimsup_{n \\to \\infty} \\alpha(\\varepsilon,n) = 0,  \\qquad\n    \\varlimsup_{\\varepsilon \\downarrow\\downarrow 0} \\varlimsup_{n \\to \\infty} \\beta(\\varepsilon,n) = 0.\n\\end{equation*}\nThis implies\n\\begin{gather*}\n    I_8^{(n)}(t)\n\\leq \n    \\sup_{\\substack{z,w \\in K  \\\\ \\lvert z-w \\rvert \\leq \\alpha(\\varepsilon,n)}}\n        \\lVert D_x^2 f(z) - D_x^2 f(w) \\rVert\n    \\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n} \\lVert \\delta_u^v X_t \\rVert_{E}^2,   \\\\\n    I_7^{(n)}(t)\n\\leq \n    \\sup_{\\substack{z,w \\in K  \\\\ \\lvert z-w \\rvert \\leq \\alpha(\\varepsilon,n) + \\beta(\\varepsilon,n)}}\n        \\lVert D_a f(z) - D_a f(w) \\rVert\n    \\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n} \\lVert \\delta_u^v A_t \\rVert_{E}.\n\\end{gather*}\n\nConsequently,\n\\begin{align*}\n    &\n    \\varlimsup_{n \\to \\infty}\n        \\left\\lVert\n            (\\text{RHS of \\eqref{5.10d}}) - (\\text{RHS of \\eqref{5.10g}})\n        \\right\\rVert_{G}     \\\\\n    & \\qquad \\leq\n    \\left\\lVert \n        \\sum_{s \\in D^{\\varepsilon} \\cap [0,t]}\n            \\left\\{ \n                \\Delta f(A_s,X_s) - \\langle D_x f(A_{s-},X_{s-}), \\Delta X_s \\rangle\n            \\right\\}\n    \\right\\rVert       \\\\\n    & \\qquad\\quad +\n    \\left\\lVert\n        \\sum_{ s \\in D^{\\varepsilon} \\cap [0,t] } \n            \\langle D_a f(A_{s-},X_{s-}), \\Delta A_s \\rangle\n    \\right\\rVert\n    + \n    \\left\\lVert\n        \\sum_{ s \\in D^{\\varepsilon} \\cap [0,t] } \n            \\langle D_x^2 f(A_{s-},X_{s-}),  \\Delta X_s \\otimes \\Delta X_s \\rangle \n    \\right\\rVert     \\\\\n    & \\qquad\\quad +\n    \\varlimsup_{n \\to \\infty}\n        \\sup_{\\substack{z,w \\in K  \\\\ \\lvert z-w \\rvert \\leq \\alpha(\\varepsilon,n)}}\n            \\lVert D_x^2 f(z) - D_x^2 f(w) \\rVert\n        \\overline{Q}(X)_t,   \\\\\n    & \\qquad\\quad +\n    \\varlimsup_{n \\to \\infty}\n        \\sup_{\\substack{z,w \\in K  \\\\ \\lvert z-w \\rvert \\leq \\alpha(\\varepsilon,n) + \\beta(\\varepsilon,n)}}\n            \\lVert D_a f(z) - D_a f(w) \\rVert\n        V(A)_t.\n\\end{align*}\nFinally, by letting $\\varepsilon \\downarrow\\downarrow 0$,\nwe obtain\n\\begin{equation*}\n    \\varlimsup_{n \\to \\infty}\n        \\left\\lVert\n            (\\text{ RHS of \\eqref{5.10d} }) - (\\text{ RHS of \\eqref{5.10g} })\n        \\right\\rVert_{G}\n    = 0.\n\\end{equation*}\nThis immediately leads the claim of the theorem.\n\\end{proof}\n\nCombining Theorem~\\ref{2e} and Corollary~\\ref{5.6d},\nwe obtain the integration by parts formula.\nNote that the existence of the It{\\^o}-F{\\\"o}llmer\nintegral $\\int_{0}^{t} A_{s-} \\otimes \\mathrm{d}X_{s}$\nfollows from Theorem \\ref{2e} and Proposition \\ref{5.9d}.\n\n\\begin{Cor}\nLet $(\\pi_n)$, $X$, and $A$ satisfy the same assumption \nas Theorem~\\ref{2e}. Then,\n\\begin{equation*}\n    A_t \\otimes X_t\n    = \n    \\int_{0}^{t} \\mathrm{d}A_s \\otimes X_{s-}\n    + \\int_{0}^{t} A_{s-} \\otimes \\mathrm{d}X_{s}\n    + [A,X]_t.\n\\end{equation*}\n\\end{Cor}\n\n\\paragraph{Acknowledgments}\nThe author thanks his supervisor Professor Jun Sekine for helpful support and encouragement.\nThe author also thanks Professor Masaaki Fukasawa for his helpful comments and discussion. \n\n\\printbibliography[heading=bibintoc]\n\nGraduate School of Engineering Science,\nOsaka University, Osaka, Japan\n\n\\emph{E-mail}: hirai@sigmath.es.osaka-u.ac.jp\n\\end{document}","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec-intro}\n\nWe address the problem of learning representations for event sequences generated by real-world users which we call \\emph{lifestream} data or lifestreams. Event sequence data is produced in many business applications, some examples being credit card transactions and click-stream data of internet site visits, and the event sequence analysis is a very common machine learning problem~\\cite{laxman2008stream},~\\cite{wiese2009credit},~\\cite{zhang2017credit},~\\cite{bigon2019prediction}. Lifestream is an event sequence that is attributed to a person and captures his\/her regular and routine actions of certain type, e.g., transactions, search queries, phone calls and messages.\n\n\n\n\n\n\nIn this paper, we present a novel \\emph{Metric Learning for Event Sequences (MeLES)} method for learning low-dimensional representations of event sequences, which copes with specific properties of lifestreams such as their discrete nature. In a broad sense, MeLES method adopts metric learning techniques~\\cite{xing2003distance},~\\cite{Hadsell:2006:DRL:1153171.1153654}. Metric learning is often used in a supervised manner for mapping high-dimensional objects to a low-dimensional embedding space. The aim of metric learning is to represent semantically similar objects (images, video, audio, etc.) closer to each other, while dissimilar ones further. Most metric learning methods are used in such applications as speech recognition~\\cite{wan2017generalized}, computer vision~\\cite{Schroff2015FaceNetAU},~\\cite{Mao2019AdvRobust} and text analysis~\\cite{reimers-2019-sentence-bert}. In these domains, metric learning is successfully applied in a supervised manner to datasets, where pairs of high-dimensional instances are labeled as the same object or different ones. \nUnlike all the previous metric learning methods, MeLES is fully self-supervised and does not require any labels. It is based on the observation that lifestream data obeys periodicity and repeatability of events in a sequence. Therefore, one can consider some convenient sub-sequences of the same lifestream as auxiliary high-dimensional representations of the same person. The idea of MeLES is that low-dimensional embeddings of such sub-sequences should be closer to each other.\n\nSelf-supervised learning approach allows us to train rich models using the internal structure of large unlabelled or partially labeled training datasets. Self-supervised learning have demonstrated effectiveness in different machine learning domains, such as Natural Language Processing (e. g. ELMO \\cite{ELMO2018}, BERT \\cite{Devlin2019BERTPO}) and computer vision \\cite{doersch2015unsupervised}.\n\n\nMeLES model trained in self-supervised manner can be used in two ways. Representations, produced by the model can be directly used as a fixed vector of features in some supervised downstream task (e. g. classification task) similarly to~\\cite{word2vec}. Alternatively, trained model can be fine-tuned~\\cite{Devlin2019BERTPO} for the specific downstream task.\n\n\n\nWe conducted experiments on two public bank transaction datasets and evaluated performance of the method on downstream classification tasks. When MeLES representations is directly used as features the method achieve strong performance comparable to the baseline methods. The fine-tuned representations achieve state-of-the-art performance on downsteam classification tasks, outperforming several other supervised methods and methods with unsupervised pre-training by a significant margin.\n\nMoreover, we show superiority of MeLES embeddings over supervised approach applied to partially labeled raw data due to insufficient amount of the target to learn a sufficiently complex model from scratch.\n\nEmbedding generation is a one-way transformation, hence it is impossible to restore exact event sequence from its representation. Therefore, the usage of embeddings leads to better privacy and data security for the end users than when working directly with the raw event data, and all this is achieved without sacrificing valuable modelling power.\n\nIn this paper, we make the following contributions. We\n\\begin{enumerate}\n    \\item adopted the ideas of metric learning to the analysis of the lifestream data in a novel, self-supervised, manner;\n    \\item proposed a specific method, called Metric Learning for Event Sequences (MeLES), to accomplish this task; \n    \\item demonstrated that the proposed MeLES method significantly outperforms other baselines for both the supervised and the semi-supervised learning scenarios on lifestream data.\n\\end{enumerate}\n\n\n\\section{Related work} \\label{sec-rel-work}\n\n\nThe metric learning approach, which underlies our MeLES method, has been widely used in different domains, including computer vision, NLP and audio domains. \n\nIn particular, metric learning approach for face recognition was initially proposed in \\cite{chopra2005learning}, where contrastive loss function was used to learn a mapping of the input data to a low-dimensional manifold using some prior knowledge of neighborhood relationships between training samples or manual labeling. Further, in~\\cite{Schroff2015FaceNetAU}, authors introduced FaceNet, a method which learns a mapping from face images to 128-dimensional embeddings using a triplet loss function based on large margin nearest neighbor classification (LMNN)~\\cite{weinberger2006distance}. In FaceNet, authors also introduced online triplet selection and hard-positive and hard-negative mining technique for training procedure.\n\n\nAlso, metric learning has been used for the speaker verification task \\cite{wan2017generalized}, where the contrast loss is defined as embedding of each utterance being similar to the centroid of all that speaker's embeddings (positive pair) and far from other speaker's centroids with the highest similarity among all false speakers (hard negative pair).\n\nFinally, in~\\cite{reimers-2019-sentence-bert}, authors proposed a fine-tuned BERT model~\\cite{Devlin2019BERTPO} that use metric learning in the form of siamese and triplet networks to train sentence embeddings for semantic textual similarity tasks using semantic proximity annotation of sentence pairs.\n\nAlthough metric learning was used in all these domains, it has not been applied to the analysis of the lifestream problems involving transactional, click-stream and other types of lifestream data, which is the focus of this paper.\n\nImportantly, previous literature applied metric learning to their domains in a supervised manner, while our MeLES method adopts the ideas of metric learning in a novel fully self-supervised manner to the event sequence domain.  \n\nThe another idea of applying self-supervised learning to sequential data has been previously proposed in Contrastive Predictive Coding (CPC) method \\cite{DBLP:journals\/corr\/abs-1807-03748}, where meaningful representations are extracted by predicting future in the latent space by using autoregressive methods. CPC representations demonstrated strong performance on four distinct domains: audio, computer vision, natural language and reinforcement learning. \n\nIn computer vision domain, there are many other different approaches to self-supervised learning that are nicely summarized in \\cite{jing2019selfsupervised}. There are several ways to define a self-supervision task (pretext task) for an image. One option is to somehow change an image and then try to restore the original image. The examples of this approach are super-resolution, image colorization and corrupted image restoration. Another option is to predict context information from the local features e. g. predict the place of image patch on the image with several missing patches.\n\nNote that almost every self-supervised learning approach can be reused for the representation learning in the form of embeddings. There are several examples of using a single set of embeddings for several downstream tasks \\cite{Song2017LearningUE}, \\cite{Zhai:2019:LUE:3292500.3330739}.\n\n\nOne of the common approaches to learn self-supervised representations is either traditional autoencoder (\\cite{rumelhart1985learning}) or variational autoencoder (\\cite{kingma2013auto}). It is widely used for images, text and audio or aggregated lifestream data (\\cite{mancisidor2019learning}). Although autoencoders has been successfully used in several domains listed above, they has not been applied to the raw lifestream data in the form of event sequences, mainly due to the challenges of defining distances between the input and the reconstructed input sequences.\n\nIn the next section, we describe how the ideas of metric learning are applied to the event sequences in a self-supervised manner.\n\n\\section{The Method} \\label{sec-method}\n\n\\subsection{Lifestream data}\n\nWe designed the method specially for the lifestream data. Lifestream data consists of discrete events per person in continuous time, for example,  behavior on websites, credit card transactions, etc. \n\nConsidering credit card transactions, each transaction have a set of attributes, either categorical or numerical including the timestamp of the transaction. An example of the sequence of three transactions with their attributes is presented in the Table \\ref{tab-tr-data}.\nMerchant type field represents the category of a merchant, such as \"airline\", \"hotel\", \"restaurant\", etc.\n\n\\begin{table}[h]\n\\caption{Data structure for a single credit card}\n\\begin{tabular}{ | m{7em} |  m{5em} m{5em} m{5em}| }\n\\hline\n\\textbf{Amount} & 230 & 5 & 40 \\\\\n\\textbf{Currency} & EUR & USD & USD \\\\\n\\textbf{Country} & France & US & US \\\\\n\\textbf{Time} & 16:40 & 20:15 & 09:30 \\\\\n\\textbf{Date} & Jun 21 & Jun 21 & Jun 22 \\\\\n\\textbf{Merchant Type} & Restaurant & Transport\\-ation & Household Appliance \\\\\n\\hline\n\\end{tabular}\n\\label{tab-tr-data}\n\\end{table}\n\nAnother example of lifestream data is click-stream: the log of internet pages  visits. The example of a click-stream log of a single user is presented in Table \\ref{tab-cs-data}.\n\n\\begin{table}[h]\n\\caption{Click-stream structure for a single user}\n\\begin{tabular}{ | m{2em} m{3em} m{7em} m{10em} | }\n\\hline\n\\textbf{Time} & \\textbf{Date} & \\textbf{Domain} & \\textbf{Referrer Domain} \\\\\n\\hline\n17:40 & Jun 21 & amazon.com & google.com \\\\\n17:41 & Jun 21 & amazon.com & amazon.com \\\\\n17:45 & Jun 21 & en.wikipedia.org & google.com \\\\\n\\hline\n\\end{tabular}\n\\label{tab-cs-data}\n\\end{table}\n\n\\subsection{General framework}\n\n\\begin{figure*}[ht]\n  \\caption{General framework}\n  \\includegraphics[scale=0.85]{figures\/arch-v2.pdf}\n  \\label{fig-arch}\n\\end{figure*}\n\nThe overview of the method is presented at Figure \\ref{fig-arch}. Given a sequence of discrete events $\\{x_t \\}^T_{t=1}$ in a given observation interval [1, T] the ultimate goal is to obtain a sequence embedding $c_t$ for the timestamp $T$ in the latent space $R^d$. To train the encoder to generate meaningful embedding $c_t$ from $\\{x_t \\}^T_{t=1}$ we apply a metric learning approach such that the distance between embeddings of the same person is small, whereas embeddings of the different persons (negative pairs) is large.\n\nOne of the difficulties with applying metric learning approach to the lifestream data is that the notion of semantic similarity as well as dissimilarity requires underlying domain knowledge and human labor-intensive labeling process to constrain positive and negative examples. \nThe key property of the lifestream data domain is periodicity and repeatability of the events in the sequence which allows us to reformulate the metric learning task in a self-supervised manner. MeLES learns low-dimensional embeddings from person sequential data, sampling positive pairs as sub-sequences of the same person sequence and negative pairs as sub-sequences from different person sequences. See section \\ref{sec-pos-pairs} for details of the positive pairs generation.\n\nEmbedding $c_t$ is generated by encoder neural network which is described in section \\ref{sec-enc-arch}. Metric learning losses are described in section \\ref{sec-ml-loss}. Positive pairs generation strategies are described in section \\ref{sec-pos-pairs}. Negative pairs sampling strategies are described in section \\ref{sec-neg-samples}.\n\nSequence embedding $c_t$ obtained by the metric learning approach is then used in various donwstream machine learning tasks as a feature vector. Also, a possible way to improve the downstream task performance is to feed a pre-trained embedding $c_t$ (e. g. the last layer of RNN) to a task-specific classification subnetwork and then jointly fine-tune the model parameters of the encoder and classifier subnetworks.\n\n\\subsection{Encoder architecture} \\label{sec-enc-arch}\n\nTo embed a sequence of events to the fixed-size vector $c_t \\in R^d$ we use approach similar to the E.T.-RNN card transaction encoder proposed in \\cite{10.1145\/3292500.3330693}. The whole encoder network consists of two conceptual parts: the event encoder and the sequence encoder subnetworks.\n\nThe event encoder $e$ takes the set of attributes of a single event $x_t$ and outputs its representation in the latent space $Z \\in R^m$: $z_t = e(x_t)$. The sequence encoder $s$ takes latent representations of the sequence of events: $ z_{1:T} = z_1, z_2, \\cdots z_T $ and outputs the representation of the whole sequence $c_t$ in the time-step $t$: $ c_t = s(z_{1:t}) $.\n\nThe event encoder consists of the several embedding layers and batch normalization\\cite{10.5555\/3045118.3045167} layer. Each embedding layer is used to encode each categorical attribute of the event. Batch normalization is applied to numerical attributes of the event. Finally, outputs of every embedding layer and batch normalization layer are concatenated to produce the latent representation $z_t$ of the single event.\n\nThe sequence of latent representations of event representations $z_{1:t}$ is passed to sequence encoder $s$ to obtain a fixed-size vector $c_t$. Several approaches can be used to encode a sequence. One possible approach is to use the recurrent network (RNN) as in \\cite{Sutskever:2014:SSL:2969033.2969173}. The other approach is to use the encoder part of the Transformer architecture presented in \\cite{DBLP:journals\/corr\/VaswaniSPUJGKP17}. In both cases the output produced for the last event can be used to represent the whole sequence of events. In case of RNN the last output $h_t$ is a representation of the sequence.\n\nEncoder, based on RNN-type architecture like GRU\\cite{cho2014learning}, allows to calculate embedding $c_{t+k}$ by updating embedding $c_t$ instead of  calculating embedding $c_{t+k}$ from the whole sequence of past events $z_{1:t}$: $c_k = rnn(c_t, z_{t+1:k})$. This option allows to reduce inference time to update already existing person embeddings with new events, occurred after the calculation of embeddings. This is possible due to the recurrent nature of RNN-like networks.\n\n\\subsection{Metric learning losses} \\label{sec-ml-loss}\n\nMetric learning loss discriminates embeddings in a way that embeddings from same class are moved closer together and embeddings from the different class are moved further. Several metric learning losses have been considered - contrastive~loss~\\cite{Hadsell:2006:DRL:1153171.1153654}, binomial deviance loss \\cite{Yi:2014:LUE:1407.4979}, triplet loss \\cite{Hoffer:2015:LUE:1412.6622}, histogram~loss~\\cite{histogram-loss} and margin~loss~\\cite{wu2017sampling}. All of this losses address the following challenge of the metric learning approach: using all pairs of samples is inefficient, for example, some of the negative pairs are already distant enough thus this pairs are not valuable for the training~(\\cite{simo2015discriminative},~\\cite{wu2017sampling},~\\cite{Schroff2015FaceNetAU}).\n\nIn the next paragraphs we will consider two kinds of losses, which are conceptually simple, and yet demonstrated strong performance on validation set in our experiments (see Table \\ref{tab-loss-type}), namely contrastive loss and margin loss.\n\nContrastive loss has a contrastive term for the negative pair of embeddings which penalizes the model only if the negative pair is not distant enough and the distance between embeddings is less than a margin $m$:  \n\\begin{equation}\n \\mathcal{L} = \\sum_{i=1}^P \\left[ (1-Y)\\dfrac{1}{2}(D_W^i)^2 +Y*\\dfrac{1}{2}\\{max(0,m-D_W^i)\\}^2 \\right],\n\\end{equation}\nwhere $P$ is the count of all pairs in a batch, $D_W^i$ - is a distance function between a i-th labeled sample pair of embeddings $X_1$ and $X_2$, \n$Y$ is a binary label assigned to a pair: $Y = 0$ means a similar pair, $Y = 1$ means dissimilar pair, $m > 0$ is a margin.\nAs proposed in \\cite{Hadsell:2006:DRL:1153171.1153654} we use euclidean distance as the distance function: $D_W^i = D(A,B) = \\sqrt{\\sum_i(A_i - B_i)^2}$.\n\nMargin loss is similar to the contrastive loss, the main difference is that there is no penalty for positive pairs which are closer than threshold in a margin loss.\n\\begin{equation}\n \\mathcal{L} = \\sum_{i=1}^P \\left[ (1-Y)max(0, D_W^i - b + m) + Y*max(0, b-D_W^i + m) \\right],\n\\end{equation}\nwhere $P$ is the count of all pairs in a batch, $D_W^i$ - is a distance function between a i-th labeled sample pair of embeddings $X_1$ and $X_2$,\n$Y$ is a binary label assigned to a pair: $Y = 0$ means a similar pair, $Y = 1$ means dissimilar pair, $m > 0$ and $b > 0$ define the bounds of a margin.\n\n\\subsection{Negative sampling} \\label{sec-neg-samples}\n\nNegative sampling is another way to address the issue that some of the negative pairs are already distant enough thus this pairs are not valuable for the training (\\cite{simo2015discriminative}, \\cite{wu2017sampling}, \\cite{Schroff2015FaceNetAU}). Hence, only part of possible negative pairs are considered during loss calculation. Note, that only current batch samples are considered. There are several possible strategies of selecting most relevant negative pairs.\n\n\\begin{enumerate}\n    \\item Random sample of negative pairs\n    \\item Hard negative mining: generate k hardest negative pairs for each positive pair.\n    \\item Distance weighted sampling, where negative samples are drawn uniformly according to their relative distance from the anchor. \\cite{wu2017sampling}\n    \\item Semi-hard sampling, where we choose the nearest to anchor negative example, from samples which further away from the anchor than the positive exemplar (\\cite{Schroff2015FaceNetAU}).\n\\end{enumerate}\n\nIn order to select negative samples, we need to compute pair-wise distance between all possible pairs of embedding vectors of a batch. For the purpose of making this procedure more computationally effective we perform normalization of the embedding vectors, i.e. project them on a hyper-sphere of unit radius. Since $D(A,B) = \\sqrt{\\sum_i(A_i - B_i)^2} = \\sqrt{\\sum_i A_i^2 + \\sum_i B_i^2 - 2\\sum_i A_i B_i} $ and $||A||= ||B||=1$, to compute the the euclidean distance we only need to compute: $\\sqrt{2 - 2(A \\cdot B)}$.\n\nTo compute the dot product between all pairs in a batch we just need to multiply the matrix of all embedding vectors of a batch by itself transposed, which is a highly optimized computational procedure in most modern deep learning frameworks. Hence, the computational complexity of the negative pair selection is $O(n^2h)$ where $h$ is the size of the output embeddings and $n$ is the size of the batch.\n\n\\subsection{Positive pairs generation} \\label{sec-pos-pairs}\n\nThe following procedure is used to create a batch during MeLES training. $N$ initial sequences are taken for batch generation. Then, $K$ sub-sequences are produced for each initial sequence.\n\nPairs of sub-sequences produced from the same sequence are considered as positive samples and pairs from different sequences are considered as negative samples. Hence, after positive pair generation each batch contains $N \\times K$ sub-sequences used as training samples. There are $K-1$ positive pairs and $(N - 1) \\times K$ negative pairs per sample in batch.\n\nThere are several possible strategies of sub-sequence generation. The simplest strategy is the random sampling without replacement. Another strategy is to produce a sub-sequence from random splitting sequence to several sub-sequences without intersection between them (see Algorithm \\ref{alg-disj-ss}). The third option is to use randomly selected slices of events with possible intersection between slices (see Algorithm \\ref{alg-slce-ss}). \n\nNote, that the order of events in generated sub-sequences is always preserved.\n\n\\begin{algorithm}\n\\SetAlgoLined\n\\textbf{hyperparameters:} $k$ - amount of sub-sequences to be produced. \\\\\n\\textbf{input:} A sequence $S$ of length $l$. \\\\\n\\textbf{output:} $S_1,...,S_k$ - sub-sequences from $S$. \\\\\n\n\\BlankLine\nGenerate vector $inds$ of length $l$ with random integers from [1,k].\\\\\n \\For{$i\\leftarrow 1$ \\KwTo $k$}{\n $S_i = S[inds == i]$\n }\n \\caption{Disjointed sub-sequences generation strategy}\n\\label{alg-disj-ss}\n\n\\end{algorithm}\n\n\\begin{algorithm}\n\\SetAlgoLined\n\\textbf{hyperparameters:} $m, M$ - minimal and maximal possible length of sub-sequence. $k$ - amount of sub-sequences to be produced. \\\\\n\\textbf{input:} A sequence $S$ of length $l$. \\\\\n\\textbf{output:} $S_1,...,S_k$ - sub-sequences from $S$. \\\\\n\\BlankLine\n \\For{$i\\leftarrow 1$ \\KwTo $k$}{\n Generate random integer $l_i$, $m \\leqslant l_i \\leqslant \\min(M,l).$\\\\\n Generate random integer $s$, $0 \\leqslant s \\leqslant l-l_i.$\\\\\n $S_i = S[s: s + l_i]$\n }\n \\caption{Random slices sub-sample generation strategy}\n\\label{alg-slce-ss}\n\\end{algorithm}\n\n\\section{Experiments} \\label{sec-exp}\n\n\\subsection{Datasets} \\label{sec-datasets}\nIn our research we used several publicly available datasets of bank transactions.\n\\begin{enumerate}\n    \\item \\textbf{Age group prediction competition}\\footnote{https:\/\/onti.ai-academy.ru\/competition} - the task is to predict the age group of a person within 4 classes target and accuracy is used as a performance metric.\n    The dataset consists of 44M anonymized transactions representing 50k persons with a target labeled for only 30k of them (27M out of 44M transactions), for the other 20k persons (17M out of 44M transactions) label is unknown. Each transaction includes date, type (for example, grocery store, clothes, gas station, children's goods, etc.) and amount. We use all available 44M transactions for metric learning, excluding 10\\% - for the test part of the dataset, and  5\\% for the metric learning validation.\n        \n    \\item \\textbf{Gender prediction competition}\\footnote{https:\/\/www.kaggle.com\/c\/python-and-analyze-data-final-project\/} - the task is a binary classification problem of predicting the gender of a person and ROC-AUC metric is used.\n    The dataset consists of 6,8M anonymized transactions representing 15k persons, where only 8,4k of them are labeled. Each transaction is characterized by date, type (for ex. \"ATM cash deposit\"), amount and Merchant Category Code (also known as MCC).\n\\end{enumerate}\n\n\\subsection{Experiment setup}\n\nFor each dataset, we set apart 10\\% persons from the labeled part of data as the test set that we used for comparing different models.\n\nIf we do not explicitly mention alternative, in our experiments we use contrastive loss and random slices pair generation strategy.\n\nFor all methods random search on 5-fold cross-validation over the train set is used for hyper-parameter selection. The hyper-parameters with the best out-of-fold performance on train set is then chosen.\n\nThe final set of hyper-parameters used for MeLES is shown in the Table\\ref{tab-hyper}.\n\n\\begin{table}[h]\n\\caption{Hyper-parameters for MeLES training}\n\\begin{tabular}{ | m{18em} |  m{3em} | m{3em} | }\n\\hline\n& \\textbf{Age task} & \\textbf{Gender task} \\\\\n\\hline\n\\textbf{Learning rate} & 0.002 & 0.002 \\\\\n\\textbf{Number of samples in batch} & 64 & 128 \\\\\n\\textbf{Number of epochs} & 100 & 150 \\\\\n\\textbf{Number of generated sub-samples} (see Section \\ref{sec-pos-pairs}) & 5 & 5 \\\\\n\\hline\n\\end{tabular}\n\\label{tab-hyper}\n\\end{table}\n\nFor evaluation of semi-supervised\/self-supervised techniques (including MeLES), we used all transactions including unlabeled data, except for the test set, as far as those methods are suitable for partially labeled datasets, or does not require labels at all.\n\n\\subsubsection{Performance}\n\nNeural network training was performed on a single Tesla P-100 GPU card. For the training part of MeLES, the single training batch is processed in 142 millisecond. For age prediction dataset, the single training batch contains 64 unique persons with 5 sub-samples per person, i.e. 320 training samples in total, the mean number of transactions per sample is 90, hence each batch contains around 28800 transactions.\n\n\\subsubsection{Baselines} \\label{sec-baselines}\n\nWe compare our MeLES method with the following two baselines. First, we consider the Gradient Boosting Machine (GBM) method~\\cite{friedman2001} on hand-crafted features. GBM can be considered as a strong baseline in cases of tabular data with heterogeneous features. In particular, GBM-based approaches achieve state-of-the-art results in a variety of practical tasks including web search, weather forecasting, fraud detection, and many others~\\cite{wu2010adapting, Vorobev2019filter, zhang2015gradient, niu2019comparison}.\n\nSecond, we apply recently proposed Contrastive Predictive Coding (CPC)~\\cite{DBLP:journals\/corr\/abs-1807-03748}, a self-supervised learning method, which has shown an excellent performance for sequential data of such traditional domains as audio, computer vision, natural language, and reinforcement learning.\n\nGBM based model requires a large number of hand-crafted aggregate features produced from the raw transactional data. An example of an aggregate feature would be an average spending amount in some category of merchants, such as hotels of the entire transaction history.\nWe used LightGBM\\cite{NIPS2017_6907} implementation of GBM algorithm with nearly 1k hand-crafted features for the application. Please see the companion code for the details of producing hand-crafted features.\n\n\nIn addition to the mentioned baselines we compare our method with supervised learning approach where the encoder sub-network and with classification sub-network are jointly trained on the downstream task target. Note, that no pre-training is used in this case.\n\n\\subsubsection{Design choices}\n\nIn the Table\\ref{tab-enc-type}, Table\\ref{tab-loss-type}, Table\\ref{tab-pair-gen} and Table\\ref{tab-neg-sampl} we present the results of experiments on different design choices of our method.\n\nAs shown in Table\\ref{tab-enc-type}, different choices of encoder architectures show comparable performance on the downstream tasks.\n\nIt is interesting to observe that even contrastive loss that can be considered as the basic variant of metric learning loss allows to get strong results on the downstream tasks (see Table \\ref{tab-loss-type}). Our hypothesis is that an increase in the model performance on metric learning task does not always lead to an increase in performance on downstream tasks.\n\nAlso observe that hard negative mining leads to significant increase in quality on downstream tasks in comparison to random negative sampling (see Table\\ref{tab-neg-sampl}).\n\nAnother observation is that a more complex sub-sequence generation strategy (e. g. random slices) shows slightly lower performance on the downstream tasks in comparison to the random sampling of events (see Table\\ref{tab-pair-gen}).\n\n\\begin{table}[h]\n\\caption{Comparison of encoder types}\n\\begin{tabular}{ | m{10em} |  m{7em} | m{7em} | }\n\\hline\n\\textbf{Econder type} & \\makecell{\\textbf{Age,} \\\\ \\textbf{Accuracy $\\pm 95\\%$}} & \\makecell{\\textbf{Gender,} \\\\ \\textbf{AUROC $\\pm 95\\%$}} \\\\\n\\hline\n\\textbf{LSTM} & $0.620 \\pm 0.003$ & $0.870 \\pm 0.005$ \\\\\n\\textbf{GRU} & \\pmb{$0.639 \\pm 0.006$} & \\pmb{$0.871 \\pm 0.004$}  \\\\\n\\textbf{Transformer} & $0.621 \\pm 0.001$ & $0.848 \\pm 0.002$  \\\\\n\\hline\n\\end{tabular}\n\\label{tab-enc-type}\n\\end{table}\n\n\\begin{table}[h]\n\\caption{Comparison of metric learning losses}\n\\begin{tabular}{ | m{10em} |  m{7em} | m{7em} |}\n\\hline\n\\textbf{Loss type} & \\makecell{\\textbf{Age,} \\\\ \\textbf{Accuracy $\\pm 95\\%$}} & \\makecell{\\textbf{Gender,} \\\\ \\textbf{AUROC $\\pm 95\\%$}} \\\\\n\\hline\n\\textbf{Contrastive loss} & $0.639 \\pm 0.006$ & \\pmb{$0.871 \\pm 0.003$} \\\\\n\\textbf{Binomial deviance loss} & $0.535 \\pm 0.005$ & $0.853 \\pm 0.005$ \\\\\n\\textbf{Histogram loss} & \\pmb{$0.642 \\pm 0.002$} & $0.851 \\pm 0.004$ \\\\\n\\textbf{Margin loss} & $0.631 \\pm 0.003$ & \\pmb{$0.871 \\pm 0.004$} \\\\\n\\textbf{Triplet loss} & $0.610 \\pm 0.006$ & $0.855 \\pm 0.003$ \\\\\n\\hline\n\\end{tabular}\n\\label{tab-loss-type}\n\\end{table}\n\n\\begin{table}[h]\n\\caption{Comparison of pair generation strategies}\n\\begin{tabular}{ | m{10em} |  m{7em} |  m{7em} | }\n\\hline\n\\textbf{Pair generation method} & \\makecell{\\textbf{Age,} \\\\ \\textbf{Accuracy $\\pm 95\\%$}} & \\makecell{\\textbf{Gender,} \\\\ \\textbf{AUROC $\\pm 95\\%$}} \\\\\n\\hline\n\\textbf{Random samples} & $0.628 \\pm 0.003$ & $0.851 \\pm 0.004$ \\\\\n\\textbf{Random disjoint samples} & $0.608 \\pm 0.004$ & $0.836 \\pm 0.008$ \\\\\n\\textbf{Random slices} & \\pmb{$0.639 \\pm 0.006$} & \\pmb{$0.872 \\pm 0.005$}  \\\\\n\\hline\n\\end{tabular}\n\\label{tab-pair-gen}\n\\end{table}\n\n\\begin{table}[h]\n\\caption{Comparison of negative sampling strategies}\n\\begin{tabular}{ | m{10em} |  m{7em} |  m{7em} | }\n\\hline\n\\textbf{Negative sampling strategy} & \\makecell{\\textbf{Age,} \\\\ \\textbf{Accuracy $\\pm 95\\%$}} & \\makecell{\\textbf{Gender,} \\\\ \\textbf{AUROC $\\pm 95\\%$}} \\\\\n\\hline\n\\textbf{Hard negative mining} & \\pmb{$0.637 \\pm 0.005$} & \\pmb{$0.872 \\pm 0.004$} \\\\\n\\textbf{Random negative sampling} & $0.615 \\pm 0.005$ & $0.826 \\pm 0.004$ \\\\\n\\textbf{Distance weighted sampling} & $0.620 \\pm 0.003$ & $0.867 \\pm 0.003$ \\\\\n\\hline\n\\end{tabular}\n\\label{tab-neg-sampl}\n\\end{table}\n\n\\begin{figure}[h]\n  \\caption{Embedding dimensionality vs. quality for age prediction task}\n  \\includegraphics[width=0.46\\textwidth]{figures\/age-pred-hidden-size.png}\n  \\label{fig-emb-dim-age}\n\\end{figure}\n\n\\begin{figure}[h]\n  \\caption{Embedding dimensionality vs. quality for gender prediciton task}\n  \\includegraphics[width=0.46\\textwidth]{figures\/gender-hidden-size.png}\n  \\label{fig-emb-dim-gender}\n\\end{figure}\n\nFigure \\ref{fig-emb-dim-age} shows that the quality on downstream task increases with the dimensionality of embedding. The best quality is achieved at size 800. Further increase in the dimensionality of embedding reduces quality.\nThe results can be interpreted as bias-variance trade-off. When embedding dimensionality is too small, too much information can be discarded (high bias). On the other hand, when embedding dimensionality is too large, too much noise is added (high variance).\n\nAt Figure \\ref{fig-emb-dim-gender} we see a similar dependency. We can find a plateau between 256 and 2048, when quality on downstream tasks does not increase. The final embedding size used in the other experiments is 256.\n\nNote, that increasing embedding size will also linearly increase the training time and the volume of consumed memory on the GPU.\n\n\\subsubsection{Embedding visualization}\n\nIn order to visualize MeLES embeddings in 2-dimensional space, we applied tSNE transformation~\\cite{maaten2008visualizing} on them. tSNE transforms high-dimensional space to low-dimensional based on local relationships between points, so neighbour vectors in high-dimensional embedding space are pushed to be close in 2-dimensional space. We colorized 2-dimensional vectors using the target values of the datasets.\n\nNote, that embeddings was learned in a fully self-supervised way from raw user transactions without any target information. Sequence of transactions represent user' behavior, thus the MeLES model captures behavioral patterns and outputs embeddings of users with similar patterns nearby.\n\ntSNE vectors from the age prediction dataset are presented in the Figure \\ref{fig-tsne-age}. We can observe 4 clusters: clusters for group '1' and '2' are on the opposite side of the cloud, clusters for groups '2' and '3' are in the middle.\n\n\n\n\\begin{figure}[ht]\n  \\caption{2D tSNE mapping of MeLES embeddings trained on age prediction task dataset, colored by age group labels}\n  \\includegraphics[width=0.46\\textwidth]{figures\/age-pred-tsne.png}\n  \\label{fig-tsne-age}\n\\end{figure}\n\n\n\\subsection{Results} \\label{sec-res}\n\n\\subsubsection{Comparison with baselines} \\label{sec-res-baselines}\n\nAs shown in Table \\ref{tab-downstream-res} our method generates sequence embeddings of lifestream data that achieve strong performance, comparable to performance on manually crafted features when used on downstream tasks. Moreover fine-tuned representations obtained by our method achieve state-of-the-art performance on both bank transactions datasets, outperforming all previous learning methods by a significant margin.\n\nFurthermore note that the usage of sequence embedding together with hand-crafted aggregate features leads to better performance than usage of only hand-crafted features or sequence embeddings, i.e. it is possible to combine different approaches to get even better model.\n\n\\begin{table*}[ht]\n\\caption{Final results on the downstream tasks}\n\\begin{tabular}{ | l |  c | c | }\n\\hline\n\\textbf{Method} & \\makecell{\\textbf{Age,} \\\\ \\textbf{Accuracy $\\pm 95\\%$}} & \\makecell{\\textbf{Gender,} \\\\ \\textbf{AUROC $\\pm 95\\%$}} \\\\\n\\hline\n\\textbf{LightGBM on hand-crafted features} & $0.626 \\pm 0.004$ & $0.875 \\pm 0.004$ \\\\\n\\textbf{LightGBM on MeLES embeddings} & $0.639 \\pm 0.006$ & $0.872 \\pm 0.005$ \\\\\n\\textbf{LightGBM on both hand-crafted features and MeLES embeddings} & \\pmb{$0.643 \\pm 0.009$} & $0.882 \\pm 0.003$ \\\\\n\\textbf{Supervised learning} & $0.631 \\pm 0.010$ & $0.871 \\pm 0.007$ \\\\\n\\textbf{MeLES fine-tuning} & \\pmb{$0.643 \\pm 0.007$} & \\pmb{$0.888 \\pm 0.002$}  \\\\\n\\textbf{LightGBM on CPC embeddings} & $0.595 \\pm 0.004$ & $0.848 \\pm 0.004$ \\\\\n\\textbf{Fine-tuned Contrastive Predictive Coding} & $0.621 \\pm 0.007$ & $0.873 \\pm 0.007$  \\\\\n\\hline\n\\end{tabular}\n\\label{tab-downstream-res}\n\\end{table*}\n\n\\subsubsection{Semi-supervised setup} \\label{sec-semi}\n\nTo evaluate our method in condition of a restricted amount of labeled data we use only part of available target labels for the semi-supervised experiment.\nAs well as in the supervised setup we compare proposed method with ligthGBM over hand-crafted features and Contrastive Predictive Coding (see Section \\ref{sec-baselines}).\nFor both embedding generation methods (MeLES and CPC) we evaluate both performance of the lightGBM on embeddings and performance of fine-tuned models.\nIn addition to this baselines we compare our method with supervised learning on the available part of the data.\n\nIn figures \\ref{fig-semi-age-0} and \\ref{fig-semi-gender-0} we compare the quality of hand-crafted features and embeddings by learning the lightGBM on top of them. Moreover, in figures \\ref{fig-semi-age-1} and \\ref{fig-semi-gender-1} one can find comparison of a single models trained on downstream tasks considered in the paper. As you can see in figures, if labeled data is limited, MeLES significantly outperforms supervised and other approaches. Also MeLES consistently outperforms CPC for different volumes of labeled data.\n\n\\begin{figure}[h]\n  \\caption{Age group prediction task quality on features for different dataset sizes}\n  \\includegraphics[width=0.46\\textwidth]{figures\/ss_age_0.png}\n  \\small{The rightmost point correspond to all labels and supervised setup. X-axis is shown on a logarithmic scale.}\n  \\label{fig-semi-age-0}\n\\end{figure}\n\n\\begin{figure}[h]\n  \\caption{Gender prediction task quality on features for different dataset sizes}\n  \\includegraphics[width=0.46\\textwidth]{figures\/ss_gen_0.png}\n  \\small{The rightmost point correspond to all labels and supervised setup. X-axis is shown on a logarithmic scale.}\n  \\label{fig-semi-gender-0}\n\\end{figure}\n\n\\begin{figure}[h]\n  \\caption{Age group prediction task quality of single model for different dataset sizes}\n  \\includegraphics[width=0.46\\textwidth]{figures\/ss_age_1_wopl.png}\n  \\small{The rightmost point correspond to all labels and supervised setup. X-axis is shown on a logarithmic scale.}\n  \\label{fig-semi-age-1}\n\\end{figure}\n\n\\begin{figure}[h]\n  \\caption{Gender prediction task quality of single model for different dataset sizes}\n  \\includegraphics[width=0.46\\textwidth]{figures\/ss_gen_1.png}\n  \\small{The rightmost point correspond to all labels and supervised setup. X-axis is shown on a logarithmic scale.}\n  \\label{fig-semi-gender-1}\n\\end{figure}\n\n\\section{Conclusions} \\label{sec-conclusions}\n\nIn this paper, we adopted the ideas of metric learning to the analysis of the lifestream data in a novel, self-supervised, manner. As a part of this proposal, we developed the Metric Learning for Event Sequences (MeLES) method that is based on self-supervised learning. \nIn particular, the MeLES method can be used to produce embeddings of complex event sequences that can be effectively used in various downstream tasks. Also, our method can be used for pre-training in semi-supervised settings.\n\nWe also empirically demonstrate that our approach achieves strong performance results on several downstream tasks by significantly (see Section \\ref{sec-res}) outperforming both classical machine learning baselines on hand-crafted features and neural network based approaches.\nIn the semi-supervised setting, where the number of labelled data is limited, our method demonstrates even stronger results: it outperforms supervised methods by significant margins.\n\nThe proposed method of generating embeddings is convenient for production usage since almost no pre-processing is needed for complex event streams to get their compact embeddings. The pre-calculated embeddings can be easily used for different downstream tasks without performing complex and time-consuming computations on the raw event data. For some encoder architectures, such as those presented in Section~\\ref{sec-enc-arch}, it is even possible to incrementally update the already calculated embeddings when additional new lifestream data arrives.\n\n\nAnother advantage of using event sequence based embeddings, instead of the raw explicit event data, is that it is impossible to restore the exact input sequence from its embeddings. Therefore, the usage of embeddings leads to better privacy and data security for the end users than when working directly with the raw event data, and all this is achieved without sacrificing valuable information for downstream tasks.\n\n\n\n\\bibliographystyle{ACM-Reference-Format}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\noindent\nEven though the general structure of the configuration space of the\nheterotic\nstring remains to a large extent terra incognita, some of its important\nproperties have been uncovered. Perhaps the most interesting of these\nis the\nrecently discovered mirror symmetry of the space of (2,2)--supersymmetric\nvacua \\cite{cls}\\cite{gp} .\nIdeally questions about the space of ground states should be analyzed\nstarting from first principles, given an appropriate parametrization of\nthis manifold. Not too much progress however has\nbeen made along this avenue. Instead one proceeds somewhat indirectly.\nThe symmetry principles of string theory are used to formulate a set of\nconsistency conditions which\nare solved explicitly. Unfortunately this\nintroduces some uncertainty as to whether the part of the space of vacua\nthat has been uncovered via these constructions represents a typical slice\nof the whole space. Properties that are generic in specific\nconstructions\nmay not at all be features of the total space one is interested in\nbut instead could merely be artefacts of the techniques employed.\n\nAn example of such an artefact is furnished by the class of\nheterotic string vacua described by\n{\\bf c}omplete {\\bf i}ntersection {\\bf C}alabi--{\\bf Y}au\nmanifolds embedded in products of projective spaces (CICYs). In this class\nthe number of generations and antigenerations of the models are\nparametrized\n by the only two\nindependent Hodge numbers $(h^{(1,1)},h^{(2,1)})$ that exist on such\nmanifolds and the number of light\ngenerations of these theories is measured by the Euler number\n$\\chi=2(h^{(1,1)}-h^{(2,1)})$.\nThe results for the latter turn out to lie in the range\n$-200 \\leq \\chi \\leq 0$ \\cite{cdls}.\nIn Fig.1 the Euler number of all CICY vacua is plotted versus the\nsum of the two independent Hodge numbers \\cite{ghl}.\n\n\\vskip .1truein\n\n\\plot{2.5truein}{\\scriptsize{$\\bullet$}}\n\\nobreak\n\\Place{-960}{50}{\\vbox{\\hrule width5pt}~~50}\n\\Place{-960}{100}{\\vbox{\\hrule width5pt}~~100}\n\\Place{-960}{150}{\\vbox{\\hrule width5pt}~~150}\n\\Place{-960}{200}{\\vbox{\\hrule width5pt}~~200}\n\\Place{-960}{250}{\\vbox{\\hrule width5pt}~~250}\n\\Place{-960}{300}{\\vbox{\\hrule width5pt}~~300}\n\\Place{-960}{350}{\\vbox{\\hrule width5pt}~~350}\n\\Place{-960}{400}{\\vbox{\\hrule width5pt}~~400}\n\\Place{-960}{450}{\\vbox{\\hrule width5pt}~~450}\n\\Place{-960}{500}{\\vbox{\\hrule width5pt}~~500}\n\\Place{960}{50}{\\kern-5pt\\vbox{\\hrule width5pt}\\vphantom{0}}\n\\Place{960}{100}{\\kern-5pt\\vbox{\\hrule width5pt}\\vphantom{0}}\n\\Place{960}{150}{\\kern-5pt\\vbox{\\hrule width5pt}\\vphantom{0}}\n\\Place{960}{200}{\\kern-5pt\\vbox{\\hrule width5pt}\\vphantom{0}}\n\\Place{960}{250}{\\kern-5pt\\vbox{\\hrule width5pt}\\vphantom{0}}\n\\Place{960}{300}{\\kern-5pt\\vbox{\\hrule width5pt}\\vphantom{0}}\n\\Place{960}{350}{\\kern-5pt\\vbox{\\hrule width5pt}\\vphantom{0}}\n\\Place{960}{400}{\\kern-5pt\\vbox{\\hrule width5pt}\\vphantom{0}}\n\\Place{960}{450}{\\kern-5pt\\vbox{\\hrule width5pt}\\vphantom{0}}\n\\Place{960}{500}{\\kern-5pt\\vbox{\\hrule width5pt}\\vphantom{0}}\n\\Place{-960}{0}{\\kern-0.2pt\\lower10pt\\hbox{\\vrule height 5pt}\\lower18pt\\hbox{-960}}\n\\Place{-720}{0}{\\kern-0.2pt\\lower10pt\\hbox{\\vrule height 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This conclusion is\n reinforced by the complete construction \\cite{ls3,fkss1} of the set\nof heterotic vacua based on tensor products of minimal $N=2$ superconformal\nfield theories \\cite{g}.\nThe resulting space is again rather asymmetric even though {\\it some}\nmirror pairs appear in this construction. Fig.2 contains again a plot\nof the Euler numbers versus the sum of generations and antigenerations for\nthose theories.\n\n\\vskip .1truein\n\n\\plot{3.5truein}{\\tiny{$\\bullet$}}\n\\nobreak\n\\Place{-960}{50}{\\vbox{\\hrule width5pt}~~50}\n\\Place{-960}{100}{\\vbox{\\hrule width5pt}~~100}\n\\Place{-960}{150}{\\vbox{\\hrule width5pt}~~150}\n\\Place{-960}{200}{\\vbox{\\hrule width5pt}~~200}\n\\Place{-960}{250}{\\vbox{\\hrule width5pt}~~250}\n\\Place{-960}{300}{\\vbox{\\hrule width5pt}~~300}\n\\Place{-960}{350}{\\vbox{\\hrule width5pt}~~350}\n\\Place{-960}{400}{\\vbox{\\hrule width5pt}~~400}\n\\Place{-960}{450}{\\vbox{\\hrule width5pt}~~450}\n\\Place{-960}{500}{\\vbox{\\hrule width5pt}~~500}\n\\Place{960}{50}{\\kern-5pt\\vbox{\\hrule width5pt}\\vphantom{0}}\n\\Place{960}{100}{\\kern-5pt\\vbox{\\hrule width5pt}\\vphantom{0}}\n\\Place{960}{150}{\\kern-5pt\\vbox{\\hrule width5pt}\\vphantom{0}}\n\\Place{960}{200}{\\kern-5pt\\vbox{\\hrule 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394}\n\\begin{center}\n\\parbox{6.4truein}{\\noindent {\\bf Fig. 2}~~{\\it A plot of the Euler\nnumber\nversus $(h^{(1,1)} +h^{(2,1)})$ for all ADE tensor models.}}\n\\end{center}\n\nAs it turns out, however, that the idea of an asymmetric space of ground\nstates is not correct.\nThe purpose of this review is to first describe in Part I a class of\nstring vacua whose spectra are almost\nsymmetrically distributed in a range of positive and negative Euler\nnumbers,\nsecondly to show in Part II that this is not an accident and thirdly to\npresent evidence, in Part III, that mirror symmetry is a `robust'\nproperty in that it is not an artefact of any one construction.\n\nSections 2 and 3 are devoted to the construction of a class of\nCalabi--Yau\nmanifolds which may be realized by polynomials in weighted\n$\\relax{\\rm I\\kern-.18em P}_4's$. The result of this investigation \\cite{cls} are some\n6,500 examples\n\\fnote{1}{It was shown in \\cite{ls4} and will be discussed in later\nsections\n that not all these spaces are distinct.}.\nThis class of vacua is of considerable interest because it\ninterpolates between the previously studied class the CICYs \\cite{cdls}\nmentioned above,\nwhich have negative Euler numbers and the orbifolds of tori which have\npositive Euler number.\n\nMore recently the construction of all Calabi--Yau manifolds embedded in\nweighted $\\relax{\\rm I\\kern-.18em P}_4$ and, more generally, of all Landau--Ginzburg vacua with\nan arbitrary number of fields was completed in \\cite{ks}\\cite{krsk}. The\ntotal\nclass consists of some 10,000 Landau--Ginzburg configurations, the\nresulting spectra of which have been plotted in Fig.3.\n\nThe most remarkable feature of this class of manifolds is\nimmediately apparent from Fig~3.\nIt is evident that the manifolds are very\nevenly divided between positive and negative Euler numbers the\ndistribution\nexhibitting an approximate but compelling symmetry under\n$\\chi\\to -\\chi$. This\nresonates with the observation made with regard to conformal field\ntheories\nthat the distinction between particles and antiparticles is purely one of\nconvention and the suggestion that for every Calabi--Yau manifold\\ with Euler number\n$\\chi$ there should be one with Euler number $-\\chi$.\n\n\n\\begin{center}\n\n\\plot{5.6truein}{\\tiny{$\\bullet$}}\n\\nobreak\n\\Place{-960}{50}{\\vbox{\\hrule width5pt}~~50}\n\\Place{-960}{100}{\\vbox{\\hrule width5pt}~~100}\n\\Place{-960}{150}{\\vbox{\\hrule width5pt}~~150}\n\\Place{-960}{200}{\\vbox{\\hrule width5pt}~~200}\n\\Place{-960}{250}{\\vbox{\\hrule width5pt}~~250}\n\\Place{-960}{300}{\\vbox{\\hrule width5pt}~~300}\n\\Place{-960}{350}{\\vbox{\\hrule width5pt}~~350}\n\\Place{-960}{400}{\\vbox{\\hrule width5pt}~~400}\n\\Place{-960}{450}{\\vbox{\\hrule width5pt}~~450}\n\\Place{-960}{500}{\\vbox{\\hrule width5pt}~~500}\n\\Place{960}{50}{\\kern-5pt\\vbox{\\hrule width5pt}\\vphantom{0}}\n\\Place{960}{100}{\\kern-5pt\\vbox{\\hrule 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262}\n\\datum{ -456}{ 264}\n\\datum{ -456}{ 272}\n\\datum{ -468}{ 286}\n\\datum{ -480}{ 246}\n\\datum{ -480}{ 262}\n\\datum{ -480}{ 278}\n\\datum{ -480}{ 286}\n\\datum{ -480}{ 306}\n\\datum{ -480}{ 334}\n\\datum{ -492}{ 256}\n\\datum{ -504}{ 276}\n\\datum{ -504}{ 312}\n\\datum{ -516}{ 302}\n\\datum{ -528}{ 278}\n\\datum{ -528}{ 286}\n\\datum{ -528}{ 334}\n\\datum{ -540}{ 274}\n\\datum{ -540}{ 334}\n\\datum{ -552}{ 306}\n\\datum{ -564}{ 322}\n\\datum{ -564}{ 330}\n\\datum{ -564}{ 340}\n\\datum{ -576}{ 314}\n\\datum{ -588}{ 346}\n\\datum{ -612}{ 330}\n\\datum{ -624}{ 330}\n\\datum{ -624}{ 358}\n\\datum{ -636}{ 342}\n\\datum{ -660}{ 366}\n\\datum{ -672}{ 374}\n\\datum{ -720}{ 394}\n\\datum{ -732}{ 386}\n\\datum{ -744}{ 402}\n\\datum{ -804}{ 430}\n\\datum{ -900}{ 474}\n\\datum{ -960}{ 502}\n\\parbox{6.4truein}{\\noindent {\\bf Fig. 3}~~{\\it A plot of Euler numbers\nagainst\n ${\\bar n}_g+n_g$ for the 2997 spectra of all the LG potentials.\n}}\n\\end{center}\n\\end{center}\n\nThe richness of the list\ncan be appreciated from the fact that the number of distinct spectra in\nthis class\nis about an order of magnitude larger than in the previous constructions.\nMirror\nsymmetry emerges only when a sufficiently large subspace of the\nconfiguration space\nof the heterotic string is probed. Furthermore this class is also of\npotential\nphenomenological interest. Whereas it had previously proved very difficult\nto find Calabi--Yau manifolds\\ with $\\chi=\\pm 6$ the construction of weighted CICYs leads to\nsome tens such manifolds. These spaces are not of immediate phenomenological\ninterest since they are all simply connected. Hence the best place to seek\ninteresting models may well be among manifolds with Euler number\n$\\pm 6k$, with\n$k>1$, although it may also be possible to construct interesting\n(2,0)--models\n\\cite{dg} from the manifolds of Euler number $\\pm 6$ which have reduced\ngauge groups\ndue to an enlarged background gauge group. This however is a question for\nfuture work.\n\nWith benefit of hindsight it seems odd that no systematic construction of\nsuch\nmanifolds was previously attempted. Hypersurfaces in weighted projective\nspaces\nwere first discussed in the physics literature by Yau \\cite{yau} and\nStrominger and Witten\\cite{hitnrun} who constructed a number of examples\nwith\nlarge negative Euler number\n$-200>\\chi>-300$. Subsequently Kim, Koh and Yoon \\cite{kky} constructed a\nfew further\nexamples with Euler numbers in a similar range. A complicating feature\nof these\nexamples was the assumed need to avoid the singular sets of the embedding\n$\\relax{\\rm I\\kern-.18em P}_4$, coupled with the large values of $|\\chi|$ achieved, this led to the\nimpression that the construction was contrived and the resulting\nmanifolds very\ncomplicated. To those elucidating the connections between exactly soluble\nconformal theories and Calabi--Yau manifolds \\cite{m}\\cite{gvw} however\nit was\napparent that avoiding the\nsingular sets of the embedding space was unnecessary since the\nsingularities\non the resulting hypersurface can be resolved while maintaining the\ncondition\n$c_1=0$. This greatly increases the number of manifolds that can be\nconstructed\nin this way. In fact of the 10,000 odd examples that have been constructed\nin refs. \\cite{cls}\\cite{ks} only a small subset consisting of spaces\ndescribed by\npolynomials of Fermat\ntype do not intersect the singular sets. Resolution of singularities has the\neffect of raising the Euler number and in this way the many Euler numbers\nplotted in Fig.~3 are achieved including many values of moderate size that\nmay be of interest for model building.\n\nEven though the high degree of symmetry of this space of groundstates\nunder the flip of the sign of the Euler numbers strongly suggests a\nrelation between dual pairs there is a priori no reason as far as the\ndifferent Landau--Ginzburg potentials are concerned why this should\nbe the case. Looking more closely at the Hodge pairs of \\cite{cls} seems to\nconfuse the issue since generically one finds for a given dual pair of\nHodge numbers not two manifolds but {\\it many} and it is unclear which\nof the possible combinations (if any) are in fact related.\n\nIt is therefore of interest to ask whether a systematic procedure can be\nestablished which relates pairs of vacua as mirror partners. This is indeed\nthe case. The construction, introduced in ref. \\cite{ls4}, consists\nof a combination of an orbifolding and a nonlinear transformation with\nsomewhat unusual properties in that it involves fractional powers. Even\nthough this procedure can be formulated in the manifold picture it is\nmost easily motivated in Landau--Ginzburg language. By applying this\nconstruction to a certain class of mirror candidates it is possible to\nestablish a close connection between dual vacua. The description of this\nconstruction comprises Part II of this review.\n\nThe starting point\nis the well known equivalence of the affine D--invariant in the\nN=2 superconformal minimal series to the $\\relax{\\sf Z\\kern-.4em Z}_2$--orbifold of the diagonal\ninvariant at the same level. In section 4 this equivalence will be lifted to\nthe full vacuum described by a tensor product of $N=2$ minimal theories.\nIt is necessary to reformulate this equivalence in terms of the\nLandau--Ginzburg (LG) potential and its projectivization as a\nCalabi--Yau (CY) manifold since for most of our models\nno exactly solvable theory is known.\nThe computations in this framework then either involve orbifoldizing\nLG--potentials \\cite{v} or modding out discrete groups of CY--manifolds and\nresolving singularities \\cite{ry}\\cite{s3}.\nOnce this procedure has been formulated in the simple framework of the\ntensor series of $N=2$ minimal models it will be clear how to proceed in\ngeneral. Sections 5 and 6 extend the discussion to more general potentials\nand illustrate to what extent our LG--potentials can be viewed as orbifolds.\nAs already mentioned a general technique to find mirror pairs will emerge.\nAnother, more mathematical, aspect which lends itself to an analysis via\nfractional transformation is the strange duality of Arnold.\n\nFinally, in Part III, the emphasis is shifted from Landau--Ginzburg theories\nto their orbifolds. Even though it is possible to map via the fractional\ntransformations of Part II many classes of orbifolds of the models described\nin Part I to complete intersections it will become clear that not all\norbifolds can be understood this way, at least with the techniques\npresently available. Hence\nit is useful to consider orbifolds in their own right firstly as a\npotential pool of new models and also in order to pursue the question\nraised at the beginning of the introduction regarding the `robustness'\n the mirror\nproperty against a change of technique\n\\fnote{2}{An orbifold analysis of minimal exactly solvable models has been\n performed in ref. \\cite{gp}. The interesting result is that the\n set of all orbifolds emerging from a given diagonal theory is\n selfdual (see also ref. \\cite{alr})}.\n In ref. \\cite{kss} we have\nconstructed the complete set of Landau--Ginzburg potentials with an\narbitrary number of fields involving\nonly couplings between at most two fields (i.e. arbitrary combinations\nof Fermat, 1--Tadpole and 1--Loop polynomials) and implemented some 40 odd\nactions of phase symmetries. It turns out that for this class\nof orbifolds the mirror property improves from about eighty percent for\nthe LG--potentials\nto about 94\\%. Figure 4 contains the plot of the resulting spectra.\n\n\n\n\\begin{center}\n\\plot{6truein}{\\tiny{$\\bullet$}}\n\\nobreak\n\\Place{-960}{50}{\\vbox{\\hrule width5pt}~~50}\n\\Place{-960}{100}{\\vbox{\\hrule width5pt}~~100}\n\\Place{-960}{150}{\\vbox{\\hrule width5pt}~~150}\n\\Place{-960}{200}{\\vbox{\\hrule width5pt}~~200}\n\\Place{-960}{250}{\\vbox{\\hrule width5pt}~~250}\n\\Place{-960}{300}{\\vbox{\\hrule width5pt}~~300}\n\\Place{-960}{350}{\\vbox{\\hrule width5pt}~~350}\n\\Place{-960}{400}{\\vbox{\\hrule width5pt}~~400}\n\\Place{-960}{450}{\\vbox{\\hrule width5pt}~~450}\n\\Place{-960}{500}{\\vbox{\\hrule width5pt}~~500}\n\\Place{960}{50}{\\kern-5pt\\vbox{\\hrule width5pt}\\vphantom{0}}\n\\Place{960}{100}{\\kern-5pt\\vbox{\\hrule width5pt}\\vphantom{0}}\n\\Place{960}{150}{\\kern-5pt\\vbox{\\hrule width5pt}\\vphantom{0}}\n\\Place{960}{200}{\\kern-5pt\\vbox{\\hrule width5pt}\\vphantom{0}}\n\\Place{960}{250}{\\kern-5pt\\vbox{\\hrule 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216}{ 124}\n\\datum{ 204}{ 130}\n\\datum{ 168}{ 148}\n\\datum{ 156}{ 154}\n\\datum{ 108}{ 178}\n\\datum{ 84}{ 190}\n\\datum{ 36}{ 214}\n\\datum{ -228}{ 346}\n\\datum{ 216}{ 126}\n\\datum{ 200}{ 134}\n\\datum{ 192}{ 138}\n\\datum{ 96}{ 186}\n\\datum{ 216}{ 128}\n\\datum{ 204}{ 134}\n\\datum{ 168}{ 152}\n\\datum{ 160}{ 156}\n\\datum{ 144}{ 164}\n\\datum{ 192}{ 142}\n\\datum{ 96}{ 190}\n\\datum{ 0}{ 238}\n\\datum{ -96}{ 286}\n\\datum{ 228}{ 126}\n\\datum{ 216}{ 132}\n\\datum{ 36}{ 222}\n\\datum{ 232}{ 126}\n\\datum{ 216}{ 134}\n\\datum{ 56}{ 214}\n\\datum{ 0}{ 242}\n\\datum{ -48}{ 266}\n\\datum{ 240}{ 124}\n\\datum{ 228}{ 130}\n\\datum{ 144}{ 172}\n\\datum{ 132}{ 178}\n\\datum{ 108}{ 190}\n\\datum{ 240}{ 126}\n\\datum{ 228}{ 132}\n\\datum{ 192}{ 150}\n\\datum{ 144}{ 174}\n\\datum{ 0}{ 246}\n\\datum{ 180}{ 158}\n\\datum{ 72}{ 212}\n\\datum{ 60}{ 218}\n\\datum{ 232}{ 134}\n\\datum{ 216}{ 142}\n\\datum{ 200}{ 150}\n\\datum{ 192}{ 154}\n\\datum{ 120}{ 190}\n\\datum{ 228}{ 138}\n\\datum{ 216}{ 144}\n\\datum{ 96}{ 204}\n\\datum{ 240}{ 134}\n\\datum{ 228}{ 140}\n\\datum{ 224}{ 142}\n\\datum{ 144}{ 182}\n\\datum{ 48}{ 230}\n\\datum{ 252}{ 130}\n\\datum{ 120}{ 196}\n\\datum{ 240}{ 138}\n\\datum{ 216}{ 150}\n\\datum{ 180}{ 168}\n\\datum{ 168}{ 174}\n\\datum{ 228}{ 146}\n\\datum{ 216}{ 152}\n\\datum{ 96}{ 212}\n\\datum{ 84}{ 218}\n\\datum{ 240}{ 142}\n\\datum{ 192}{ 166}\n\\datum{ 176}{ 174}\n\\datum{ 168}{ 178}\n\\datum{ 144}{ 190}\n\\datum{ 0}{ 262}\n\\datum{ 260}{ 134}\n\\datum{ 240}{ 144}\n\\datum{ 180}{ 174}\n\\datum{ 216}{ 158}\n\\datum{ 192}{ 170}\n\\datum{ 108}{ 212}\n\\datum{ 252}{ 142}\n\\datum{ 168}{ 184}\n\\datum{ 132}{ 202}\n\\datum{ 240}{ 150}\n\\datum{ 196}{ 172}\n\\datum{ 216}{ 164}\n\\datum{ 120}{ 212}\n\\datum{ 232}{ 158}\n\\datum{ 192}{ 178}\n\\datum{ 180}{ 184}\n\\datum{ -240}{ 394}\n\\datum{ 264}{ 144}\n\\datum{ 252}{ 150}\n\\datum{ -180}{ 366}\n\\datum{ 240}{ 158}\n\\datum{ 144}{ 206}\n\\datum{ 264}{ 148}\n\\datum{ 260}{ 150}\n\\datum{ 236}{ 162}\n\\datum{ 192}{ 184}\n\\datum{ -84}{ 322}\n\\datum{ 240}{ 162}\n\\datum{ 216}{ 174}\n\\datum{ 228}{ 170}\n\\datum{ 272}{ 150}\n\\datum{ 240}{ 166}\n\\datum{ 192}{ 190}\n\\datum{ 144}{ 214}\n\\datum{ 0}{ 286}\n\\datum{ 280}{ 148}\n\\datum{ 276}{ 150}\n\\datum{ 252}{ 162}\n\\datum{ 216}{ 180}\n\\datum{ 156}{ 210}\n\\datum{ 288}{ 146}\n\\datum{ 264}{ 158}\n\\datum{ 228}{ 176}\n\\datum{ 216}{ 182}\n\\datum{ 48}{ 266}\n\\datum{ 276}{ 154}\n\\datum{ 204}{ 190}\n\\datum{ 264}{ 162}\n\\datum{ 256}{ 166}\n\\datum{ 240}{ 174}\n\\datum{ 192}{ 198}\n\\datum{ 288}{ 152}\n\\datum{ 264}{ 164}\n\\datum{ 296}{ 150}\n\\datum{ 240}{ 178}\n\\datum{ 96}{ 250}\n\\datum{ 216}{ 192}\n\\datum{ 288}{ 158}\n\\datum{ 256}{ 174}\n\\datum{ 192}{ 206}\n\\datum{ 0}{ 302}\n\\datum{ 264}{ 172}\n\\datum{ 252}{ 178}\n\\datum{ 132}{ 238}\n\\datum{ 288}{ 162}\n\\datum{ 240}{ 188}\n\\datum{ 120}{ 248}\n\\datum{ 288}{ 166}\n\\datum{ 276}{ 172}\n\\datum{ 228}{ 196}\n\\datum{ 156}{ 232}\n\\datum{ 276}{ 174}\n\\datum{ 296}{ 166}\n\\datum{ 288}{ 170}\n\\datum{ 272}{ 178}\n\\datum{ 264}{ 184}\n\\datum{ 300}{ 168}\n\\datum{ 264}{ 188}\n\\datum{ 216}{ 212}\n\\datum{ 312}{ 166}\n\\datum{ 192}{ 226}\n\\datum{ 288}{ 182}\n\\datum{ 240}{ 206}\n\\datum{ 312}{ 172}\n\\datum{ 304}{ 176}\n\\datum{ 252}{ 202}\n\\datum{ -60}{ 358}\n\\datum{ 324}{ 168}\n\\datum{ 300}{ 180}\n\\datum{ 288}{ 186}\n\\datum{ 276}{ 192}\n\\datum{ 288}{ 188}\n\\datum{ 252}{ 206}\n\\datum{ 312}{ 178}\n\\datum{ 288}{ 190}\n\\datum{ 96}{ 286}\n\\datum{ 120}{ 276}\n\\datum{ 120}{ 278}\n\\datum{ 168}{ 256}\n\\datum{ 336}{ 178}\n\\datum{ 324}{ 184}\n\\datum{ 312}{ 190}\n\\datum{ 240}{ 226}\n\\datum{ 312}{ 192}\n\\datum{ 300}{ 198}\n\\datum{ 216}{ 240}\n\\datum{ 288}{ 206}\n\\datum{ 312}{ 196}\n\\datum{ 216}{ 244}\n\\datum{ -156}{ 430}\n\\datum{ 336}{ 188}\n\\datum{ 288}{ 214}\n\\datum{ 0}{ 358}\n\\datum{ 348}{ 186}\n\\datum{ 264}{ 228}\n\\datum{ 336}{ 194}\n\\datum{ 84}{ 322}\n\\datum{ 336}{ 198}\n\\datum{ 288}{ 222}\n\\datum{ 340}{ 198}\n\\datum{ 360}{ 190}\n\\datum{ 336}{ 202}\n\\datum{ 288}{ 226}\n\\datum{ 348}{ 198}\n\\datum{ 276}{ 234}\n\\datum{ 336}{ 206}\n\\datum{ 300}{ 226}\n\\datum{ 180}{ 286}\n\\datum{ 372}{ 194}\n\\datum{ 312}{ 224}\n\\datum{ 300}{ 230}\n\\datum{ 372}{ 202}\n\\datum{ 368}{ 204}\n\\datum{ 60}{ 358}\n\\datum{ 360}{ 212}\n\\datum{ 324}{ 232}\n\\datum{ 264}{ 262}\n\\datum{ 384}{ 204}\n\\datum{ 336}{ 230}\n\\datum{ 216}{ 292}\n\\datum{ 376}{ 214}\n\\datum{ 360}{ 228}\n\\datum{ 384}{ 218}\n\\datum{ 372}{ 226}\n\\datum{ 348}{ 238}\n\\datum{ 288}{ 270}\n\\datum{ 408}{ 212}\n\\datum{ 396}{ 222}\n\\datum{ 324}{ 262}\n\\datum{ 420}{ 218}\n\\datum{ 408}{ 224}\n\\datum{ 384}{ 242}\n\\datum{ 408}{ 232}\n\\datum{ 420}{ 230}\n\\datum{ 384}{ 250}\n\\datum{ 408}{ 240}\n\\datum{ 372}{ 258}\n\\datum{ -60}{ 474}\n\\datum{ 0}{ 446}\n\\datum{ 372}{ 262}\n\\datum{ 432}{ 238}\n\\datum{ 180}{ 366}\n\\datum{ 432}{ 242}\n\\datum{ 396}{ 262}\n\\datum{ 228}{ 346}\n\\datum{ 456}{ 234}\n\\datum{ 276}{ 330}\n\\datum{ 408}{ 268}\n\\datum{ 312}{ 318}\n\\datum{ 456}{ 248}\n\\datum{ 432}{ 266}\n\\datum{ 456}{ 256}\n\\datum{ 480}{ 246}\n\\datum{ 456}{ 264}\n\\datum{ 456}{ 272}\n\\datum{ 492}{ 256}\n\\datum{ 480}{ 262}\n\\datum{ 0}{ 502}\n\\datum{ 60}{ 474}\n\\datum{ 156}{ 430}\n\\datum{ 240}{ 394}\n\\datum{ 420}{ 306}\n\\datum{ 288}{ 374}\n\\datum{ 468}{ 286}\n\\datum{ 480}{ 286}\n\\datum{ 336}{ 358}\n\\datum{ 372}{ 346}\n\\datum{ 396}{ 340}\n\\datum{ 528}{ 278}\n\\datum{ 540}{ 274}\n\\datum{ 528}{ 286}\n\\datum{ 432}{ 334}\n\\datum{ 516}{ 302}\n\\datum{ 480}{ 334}\n\\datum{ 552}{ 306}\n\\datum{ 528}{ 334}\n\\datum{ 564}{ 322}\n\\datum{ 540}{ 334}\n\\datum{ 564}{ 330}\n\\datum{ 564}{ 340}\n\\datum{ 612}{ 330}\n\\datum{ 588}{ 346}\n\\datum{ 624}{ 330}\n\\datum{ 624}{ 358}\n\\datum{ 660}{ 366}\n\\datum{ 672}{ 374}\n\\datum{ 732}{ 386}\n\\datum{ 720}{ 394}\n\\datum{ 804}{ 430}\n\\datum{ 900}{ 474}\n\\datum{ 960}{ 502}\n\n\\begin{center}\n\\parbox{6.4truein}{\\noindent {\\bf Fig. 4}~~{\\it A plot of Euler\nnumbers against\n ${\\bar n}_g+n_g$ for the 1900 odd spectra of all the LG potentials\nand phase\norbifolds constructed.}}\n\\end{center}\n\\end{center}\n\n\nIt is obvious from Fig. 4 that the upper boundary of the distribution of\nspectra is the same for the orbifolds as for the complete intersection\nmanifolds. Very likely this boundary is in fact a\nproperty of the total moduli space of all\nthree--dimensional Calabi--Yau manifolds.\nIt would be interesting\nto see whether all Calabi--Yau Hodge numbers fall into the limits defined\nby the Figures 3 \\& 4. As expected the lower part of the plot has shifted\nsince\nnew theories with smaller numbers for the total number of fields have\nappeared.\nIt is to be expected that, as the construction becomes more complete, the\nstructure\nof the lower part will change again. In fact there are well known manifolds\nthat lie below the models presented here; these however involve permutation\ngroups.\n\n\n\n\\vfill \\eject\n\\part{ Construction of Landau--Ginzburg Theories and Weighted CICYs.}\n\n\\noindent\nEven though there is some overlap between the sets of string vacua\ndescribed by Landau--Ginzburg vacua on the one hand and Calabi--Yau\nmanifolds on the other it is not, at present, clear whether the former\nis contained in the latter. It is appropriate\njustified to separate the discussion of these two classes somewhat.\nIn Sections 2 and 3 the emphasis will be on the explicit construction\nof a set of CY manifolds embedded in weighted $\\relax{\\rm I\\kern-.18em P}_4$ whereas\nthe remaining sections of Part I will be concerned\n with the complete class of LG vacua.\n\n\\vskip .1truein\n\\noindent\n\\section{Complete Intersection Manifolds in Weighted $\\relax{\\rm I\\kern-.18em P}_4$.}\n\n\\noindent\nThis section contains some elements of the theory of hypersurfaces defined\nby polynomials in weighted projective spaces. An extensive discussion\nof these\nspaces can be found in \\cite{wproj}.\n\nA weighted $\\relax{\\rm I\\kern-.18em P}_4$ with weights $(k_1,k_2,k_3,k_4,k_5)$, which will be\ndenoted by\n$\\relax{\\rm I\\kern-.18em P}_{(k_1,k_2,k_3,k_4,k_5)}$, is most easily described in terms of 5\ncomplex `homogeneous coordinates' $(z_1,z_2,z_3,z_4,z_5)$, not all zero,\nwhich are subject to the identification\n\\begin{equation}\n(z_1,...,z_5) \\simeq (\\l^{k_1}z_1,...,\\l^{k_5}z_5)\n\\lleq{pro5}\nfor all nonzero $\\l \\in \\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}$. Thus a weighted projective space is a\ngeneralization\nof ordinary projective space and $\\relax{\\rm I\\kern-.18em P}_4 = \\relax{\\rm I\\kern-.18em P}_{(1,1,1,1,1)}$ in\nthis notation.\nIn the following, when referring to a generic weighted $\\relax{\\rm I\\kern-.18em P}_4$, we shall\nfrequently consider the weights to be understood and write $\\relax{\\rm I\\kern-.18em P}_4$ for\n$\\relax{\\rm I\\kern-.18em P}_{(k_1,k_2,...,k_5)}$.\n\nThe first point to note concerning these spaces\nis the fact that weighted projective spaces have\norbifold singularities owing to the identification (\\ref{pro5}), except for\nthe case that the weights are all unity. This is most easily seen by setting\n$z_j = \\left ( \\zeta_j \\right)^{k_j}$ so that (\\ref{pro5}) becomes\n$(\\zeta_1,...,\\zeta_5) \\simeq \\l (\\zeta_1,...,\\zeta_5)$. However in virtue\nof the definition of $\\zeta_i$ we must also identify\n$\\zeta_j \\simeq e^{2\\pi i\/k_j} \\zeta_j$. So we see that\n\\begin{equation}\n\\relax{\\rm I\\kern-.18em P}_{(k_1,k_2,k_3,k_4,k_5)} = \\frac{\\relax{\\rm I\\kern-.18em P}_4}\n {\\relax{\\sf Z\\kern-.4em Z}_{k_1}\\times \\cdots \\times \\relax{\\sf Z\\kern-.4em Z}_{k_5}}\n\\end{equation}\nThese identifications lead to singular sets. A simple example is\n$\\relax{\\rm I\\kern-.18em P}_{(1,1,1,2,5)}$ which has weights that are mutually prime. If we\ntake in turn $\\l=-1$ and $\\l=\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma$ with $\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma$ a fifth root of unity in\n(\\ref{pro5}) then we see that\n\\begin{eqnarray}\n (z_1,z_2,z_3,z_4,z_5) &\\simeq & (-z_1,-z_2,-z_3,z_4,-z_5) \\nonumber \\\\\n &\\simeq &(\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma z_1,\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma z_2,\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma z_3,\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma^2 z_4,z_5).\\llea{ex}\nConsider now a neighborhood of the point $(0,0,0,1,0)$. We can take\ncoordinates on the neighborhood by setting $\\l = z_4^{-1\/2}$ and\nwriting $u_j = z_j\/z_4^{1\/2}$ for $j=1,2,3$ and $u_5 = z_5\/z_4^{5\/2}$.\nWe can therefore think of points in the neighborhood as corresponding\nto $(u_1,u_2,u_3,1,u_5)$. However from the first of identifications\n(\\ref{ex})\nit follows that\n\\begin{equation}\n(u_1,u_2,u_3,1,u_5) \\simeq (-u_1,-u_2,-u_3,1,-u_5)\n\\end{equation}\nso that there is a $\\relax{\\sf Z\\kern-.4em Z}_2$ identification on the space and hence on a\nneighborhood of the point and the action fixes $(0,0,0,1,0)$. In the same\nway there is a $\\relax{\\sf Z\\kern-.4em Z}_5$ action which fixes $(0,0,0,0,1)$. In this case\nthe singular set consists of points owing to the fact that the weights\nare mutually prime. Consider now $\\relax{\\rm I\\kern-.18em P}_{(1,1,1,2,4)}$ then the\nidentification\nis\n\\begin{equation}\n(z_1,z_2,z_3,z_4,z_5) \\simeq (\\l z_1,\\l z_2,\\l z_3,\\l^2 z_4,\\l^4 z_5)\n\\end{equation}\nIf we set $\\l = z_5^{-1\/4}$ and\nchoose coordinates $w_j = z_j\/z_5^{1\/4}$, $w_4 = z_4\/z_5^{1\/2}$ then due\nto the freedom to set $\\l = -1$ in (5) we have\n\\begin{equation}\n(w_1,w_2,w_3,w_4,1) \\simeq (-w_1,-w_2,-w_3,w_4,1)\n\\end{equation}\nand we see that we have a $\\relax{\\sf Z\\kern-.4em Z}_2$ action which fixes the curve\n$(0,0,0,z_4,z_5)$.\nThere is in addition a $\\relax{\\sf Z\\kern-.4em Z}_4$ action with fixed point $(0,0,0,0,1)$ that\nlies within this curve. In general there is a fixed point for each\nweight that\nis greater than unity, a fixed curve for every pair of weights $k_i,k_j$\nwhose greatest common factor, which we denote by $(k_i,k_j)$, is\ngreater than\nunity, a fixed surface for each triple with $(k_i,k_j,k_l) > 1$ and so on.\n\nWe wish to study Calabi--Yau\\ hypersurfaces defined by polynomials in the\nhomogeneous coordinates. We require the polynomials to be transverse,\nthat is\n$p=0$ and $dp=0$ have no common solution. Given the weights of the ambient\nspace the requirement of a vanishing first Chern class fixes the degree of\nthe polynomial as in the case of CICYs. To derive the explicit\ncondition we digress briefly on the Chern classes of the submanifold\n${\\cal M}$ defined by the equation $p=0$. We denote by ${\\cal T}_{\\relax{\\rm I\\kern-.18em P}_4}$\nand\n${\\cal T}_{\\cal M}$ the tangent spaces to the $\\relax{\\rm I\\kern-.18em P}_4$ and ${\\cal M}$ and by\n${\\cal N}$ the normal bundle of ${\\cal M}$ in $\\relax{\\rm I\\kern-.18em P}_4$. Proceeding in a\nstandard manner we have\n\\begin{equation}\n{\\cal T}_{\\relax{\\rm I\\kern-.18em P}_4} = {\\cal T}_{\\cal M} \\oplus {\\cal N}\n\\end{equation}\nwhich allows us to calculate the Chern polynomial of ${\\cal M}$ once\nthose for\n$\\relax{\\rm I\\kern-.18em P}_4$ and ${\\cal N}$ are known. The tangent space of $\\relax{\\rm I\\kern-.18em P}_4$ may be\nthought of\nas the set of vectors\n\\begin{equation}\n{\\cal V} = {\\cal V}^j \\frac{\\partial}{\\partial z^j}\n\\end{equation}\nwhich act on bona fide functions of the homogeneous coordinates, i.e.\nfunctions\nof degree zero. From Euler's theorem for homogeneous functions of\ndegree $m$\nwe have that\n\\begin{equation}\n\\sum_j k_j z^j \\frac{\\partial}{\\partial z^j} f = m f.\\lleq{euler}\nSo when acting on functions of degree zero we see that we may regard the\n${\\cal V}^j$ as independent apart from the identification\n${\\cal V}^j \\simeq {\\cal V}^j + k_jz^j$. This reduces the dimension of the\nspace of ${\\cal V}^j$ to 4 as is necessary. It follows that\n\\begin{equation}\n{\\cal T}_{\\relax{\\rm I\\kern-.18em P}_4} = {\\cal O}(k_1) \\oplus \\cdots \\oplus {\\cal O}(k_5)\/{\\cal O}\n\\end{equation}\nwhere ${\\cal O}(k)$ is a line bundle with $c_1=kJ$ and $J$ is the K\\\"ahler\nclass. It is also the line\nbundle whose fibre coordinates transform like a polynomial of degree $k$.\n${\\cal O}$ is the trivial bundle. Since ${\\cal O}(k)$ is one dimensional\nwe have $c({\\cal O}(k))= 1 +kJ$ and hence\n\\begin{equation}\nc({\\cal T}_{\\relax{\\rm I\\kern-.18em P}_4}) = \\prod_{j=1}^5 \\left (1 + k_j J\\right ).\n\\end{equation}\nThe defining polynomial $p$ can be regarded as a fibre coordinate on\n${\\cal N}$ so if $p$ is of degree $d$ we have ${\\cal N} = {\\cal O}(d)$ and\n$c({\\cal N}) = 1 + dJ$. Hence\n\\begin{equation}\nc({\\cal T}_{\\cal M}) = \\frac{ \\prod_{j=1}^5 \\left (1 + k_jJ \\right )}\n { \\left (1 + dJ \\right )}\n\\lleq{ctot}\nIt follows that $c_1=0$ is the condition\n\\begin{equation}\nd = \\sum_j k_j~ .\\lleq{c1}\nWe record here also an expression for $c_3$, obtained by expanding\n(\\ref{ctot})\nto third order, which is useful for the\ncomputation of the Euler number of these spaces\n\\begin{equation}\nc_3 = - \\frac{1}{3} \\left (d^3 - \\sum_{j=1}^5 k_j^3 \\right )J^3~.\n\\end{equation}\nThere are, roughly speaking, two sorts of singularities that can arise\nfor a hypersurface defined by the vanishing of a polynomial $p$. The first\nis that the locus $p=0$ intersects the $\\relax{\\sf Z\\kern-.4em Z}_n$--singularities of the ambient\nspace. This does not pose a difficulty however as these can in general be\nresolved.\n\nThe second is that the weights of a given $\\relax{\\rm I\\kern-.18em P}_4$ may prevent all\npolynomials\nof the required degree from being transverse. This is in contradistinction\nto CICYs where every configuration admits a smooth\nrepresentative (in fact almost all representatives are smooth \\cite{gh1}).\nTo see this let\n\\begin{equation}\nd_j:= d-k_j~,~j=1,\\dots,5\n\\end{equation}\nand expand $p$ in powers of $z_1$, say,\n\\begin{equation}\np=\\sum_{r=0}^{a_1} C_r(z_m)\\,z_1^r~,~~~m\\neq 1~.\\lleq{pexp}\nIt follows from Euler's Theorem (\\ref{euler}) that $p=0$ and $dp=0$ if and\nonly if there\nis a simultaneous solution of the equations\n\\begin{equation}\n\\frac{\\partial p}{\\partial z_j}=0~,~j=1,\\dots,5~.\n\\lleq{transv5}\nApplying this to (\\ref{pexp}) we have\n\\begin{eqnarray}\n\\frac{\\partial p}{\\partial z_1}&=& \\sum_{r=1}^{a_1} rC_r(z_m)\\,z_1^{r-1} \\nonumber \\\\\n\\frac{\\partial p}{\\partial z_m}&=& \\sum_{r=0}^{a_1} \\frac{\\partial C_r}{\\partial z_m}\\,z_1^r~\n\\llea{partexp}\nNow the degrees of the $C_r$ are fixed by (\\ref{pexp})\n\\begin{eqnarray}\n\\deg\\,(C_r) &=& d-rk_1=d_1-(r-1)k_1~,~{\\rm for\\ } r\\ge 1~,\\nonumber \\\\\n\\deg\\left(\\frac{\\partial C_r}{\\partial z^m}\\right) &=&d_m-rk_1~.\n\\end{eqnarray}\nand a polynomial of negative degree is understood to vanish identically.\nUnless\nat least one of the coefficients in (\\ref{partexp}) has degree\nprecisely zero\nthen equations (\\ref{transv5}) will be satisfied for $z_1=1$ and $z_m=0$\nand $p$\n will not be transverse.\nThis leads to the following neccessary condition on the weights if\n $p$ is to be\ntransverse:\n\n\\noindent\n{\\sl For each $i$ there must exist a $j$ such that $k_i|d_j$\n ($k_i$ divides $d_j$).}\n\nThis condition is quite restrictive. For example it is immediate that\nthe only\nmanifolds of the form $\\relax{\\rm I\\kern-.18em P}_{(1,1,1,1,k)}[k+4]$ that it allows are those\nfor the\nfour values $k=1,2,3,4$, and the assiduous reader may care to check\nthat apart\nfrom these there only eleven other allowed cases of the form\n$\\relax{\\rm I\\kern-.18em P}_{(1,1,1,k,l)}[k+l+3]$, where $k+l+3$ indicates the degree of the\npolynomial.\n\nThis criterion however is not a sufficient condition. A\ncounterexample being furnished by the configuration\n$\\relax{\\rm I\\kern-.18em P}_{(1,2,2,2,2)}[9]$ whose most general polynomial is of the form\n\\begin{equation}\np(z_1,z_2,z_3,z_4,z_5) = z_1 \\tilde{p}(z_1,z_2,z_3,z_4,z_5).\n\\end{equation}\nTransversality\nrequires that the equations $p=0$ and $dp=0$ have no common solution.\nConsider\nthen a neighborhood $U_5$ say, on which we can take $z_5=1$. Taking the\ndifferential gives\n$dp =\\nobreak \\tilde{p}dz_1 + z_1 d\\tilde{p}$ which has a zero for $z_1=0$\nand $\\tilde{p}=0$. These equations\ngive two constraints in 4 variables and therefore always have a solution.\nSince the polynomial constraint is identically satisfied it follows that\nthis configuration cannot admit a transverse realization.\nThus there are further criteria which must be satisfied in order for a\npolynomial to be transverse. In refs. \\cite{cls} a list of\ntransverse polynomials was constructed. A little thought shows that\npolynomials of\nFermat type for which $k_i|d_i$ for each $i$ {\\sl are} transverse. These\nare of the form\n\\begin{equation}\nz_1^a + z_2^b + z_3^c + z_4^d + z_5^e~~~~~~~~~~~~~~~~~~~~~~~~~\n{\\thicklines \\begin{picture}(150,30)\n \\put(0,0){\\circle*{5}}\n \\put(0,7){\\circle{12}}\n \\put(26,0){\\circle*{5}}\n \\put(26,7){\\circle{12}}\n \\put(52,0){\\circle*{5}}\n \\put(52,7){\\circle{12}}\n \\put(78,0){\\circle*{5}}\n \\put(78,7){\\circle{12}}\n \\put(104,0){\\circle*{5}}\n \\put(104,7){\\circle{12}}\n \\end{picture}}\n\\end{equation}\n\n\\noindent\nto which we have appended a diagrammatic shorthand.\n\nThere are other types which are also always transverse such as\n\\begin{eqnarray}\nz_1^az_2 + z_2^b + z_3^c + z_4^d + z_5^e\n& &~~~~~~~~~~~~{\\thicklines \\begin{picture}(150,30)\n \\put(0,0){\\circle*{5}}\n \\put(0,0){\\line(1,0){26}}\n \\put(26,0){\\circle*{5}}\n \\put(26,7){\\circle{12}}\n \\put(52,0){\\circle*{5}}\n \\put(52,7){\\circle{12}}\n \\put(78,0){\\circle*{5}}\n \\put(78,7){\\circle{12}}\n \\put(104,0){\\circle*{5}}\n \\put(104,7){\\circle{12}}\n \\end{picture}}\n\\\\\nz_1^az_2 + z_2^bz_3 + z_3^cz_1 + z_4^d + z_5^e\n& &~~~~~~~~~~~~{\\thicklines {\\begin{picture}(150,30)\n \\put(0,0){\\circle*{5}}\n \\put(0,0){\\line(1,0){26}}\n \\put(26,0){\\circle*{5}}\n \\put(26,0){\\line(1,0){26}}\n \\put(52,0){\\circle*{5}}\n \\put(26,0){\\oval(52,26)[t]}\n \\put(78,0){\\circle*{5}}\n \\put(78,7){\\circle{12}}\n \\put(104,0){\\circle*{5}}\n \\put(104,7){\\circle{12}}\n \\end{picture}}}\n\\end{eqnarray}\nParts of these expressions corresponding to connected subdiagrams can also\nbe combined together to produce yet other transverse polynomials such as\n\\begin{equation}\n{\\thicklines \\begin{picture}(150,30)\n \\put(0,0){\\circle*{5}}\n \\put(0,0){\\line(1,0){26}}\n \\put(26,0){\\circle*{5}}\n \\put(26,7){\\circle{12}}\n \\put(52,0){\\circle*{5}}\n \\put(52,0){\\line(1,0){26}}\n \\put(78,0){\\circle*{5}}\n \\put(78,0){\\oval(52,26)[t]}\n \\put(78,0){\\line(1,0){26}}\n \\put(104,0){\\circle*{5}}\n \\end{picture}}\n{\\rm or}~~~~~~~~~~~~~\n{\\thicklines \\begin{picture}(150,30)\n \\put(0,0){\\circle*{5}}\n \\put(0,0){\\line(1,0){26}}\n \\put(26,0){\\circle*{5}}\n \\put(26,7){\\circle{12}}\n \\put(52,0){\\circle*{5}}\n \\put(52,0){\\line(1,0){26}}\n \\put(78,0){\\circle*{5}}\n \\put(78,7){\\circle{12}}\n \\put(104,0){\\circle*{5}}\n \\put(104,7){\\circle{12}}\n \\end{picture}}\n\\end{equation}\n\nWhat is needed then is a classification of nondegenerate weighted\nhomogeneous\npolynomials in five variables. Unfortunately, there exists as yet no such\nclassification and indeed its formulation seems to be a hard problem.\n\nThere {\\it does} exist a classification of smooth polynomials in three\nvariables \\cite{agzv} and what has been done in \\cite{cls} is to extend\nthis to a partial classification of polynomials in five variables.\nThese constructions do not describe all possible\npolynomials but they do respresent a minimal extension of Arnold's\nclassification to the case of five variables.\nTable 1 contains the polynomial types implemented in \\cite{cls}\nto construct nondegenerate polynomials in five variables. By combining\nthe nineteen types listed below in the way described above one obtains\nthirty different five dimensional catastrophes.\n\n{\\scriptsize\n{\\begin{center}\n\\begin{tabular}{|| l | l | l ||}\n\\hline\n$\\#$ & Polynomial Type & Diagram \\hbox to0pt{\\phantom{\\Huge A}\\hss} \\\\\n\\hline\n1 &$z_1^a$\n&~~~~~{\\thicklines \\begin{picture}(150,20)\n \\put(0,0){\\circle*{5}}\n \\put(0,7){\\circle{12}}\n \\end{picture}\n } \\\\ [.5mm]\n\\hline\n2 &$z_1^az_2 + z_2^b$\n&~~~~~{\\thicklines \\begin{picture}(150,20)\n \\put(0,0){\\circle*{5}}\n \\put(0,0){\\line(1,0){26}}\n \\put(26,0){\\circle*{5}}\n \\put(26,7){\\circle{12}}\n \\end{picture}\n } \\\\ [.5mm]\n\\hline\n3 &$z_1^az_2 + z_2^bz_1$\n&~~~~~{\\thicklines \\begin{picture}(150,20)\n \\put(0,0){\\circle*{5}}\n \\put(0,0){\\line(1,0){26}}\n \\put(26,0){\\circle*{5}}\n \\put(13,0){\\oval(26,26)[t]}\n \\end{picture}\n } \\\\ [.5mm]\n\\hline\n4 &$z_1^az_2 + z_2^bz_3 + z_3^c$\n&~~~~~{\\thicklines {\\begin{picture}(150,20)\n \\put(0,0){\\circle*{5}}\n \\put(0,0){\\line(1,0){26}}\n \\put(26,0){\\circle*{5}}\n \\put(26,0){\\line(1,0){26}}\n \\put(52,0){\\circle*{5}}\n \\put(52,7){\\circle{12}}\n \\end{picture}}} \\\\ [.5mm]\n\\hline\n5 &$z_1^az_2 + z_2^b + z_3^cz_2 + z_1^pz_3^q$\n&~~~~~{\\thicklines {\\begin{picture}(150,20)\n \\put(0,0){\\circle*{5}}\n \\put(0,0){\\line(1,0){26}}\n \\put(26,0){\\circle*{5}}\n \\put(26,7){\\circle{12}}\n \\put(26,0){\\line(1,0){26}}\n \\put(52,0){\\circle*{6}}\n \\end{picture}}} \\\\ [.5mm]\n\\hline\n6 &$z_1^az_2 + z_2^bz_3 + z_3^cz_2 + z_1^pz_3^q$\n&~~~~~{\\thicklines {\\begin{picture}(150,20)\n \\put(0,0){\\circle*{5}}\n \\put(0,0){\\line(1,0){26}}\n \\put(26,0){\\circle*{5}}\n \\put(26,0){\\line(1,0){26}}\n \\put(52,0){\\circle*{5}}\n \\put(39,0){\\oval(26,26)[t]}\n \\end{picture}}} \\\\ [.5mm]\n\\hline\n7 &$z_1^az_2 + z_2^bz_1 + z_3^cz_1$\n&~~~~~{\\thicklines {\\begin{picture}(150,20)\n \\put(0,0){\\circle*{5}}\n \\put(0,0){\\line(1,0){26}}\n \\put(26,0){\\circle*{5}}\n \\put(26,0){\\line(1,0){26}}\n \\put(52,0){\\circle*{5}}\n \\put(26,0){\\oval(52,26)[t]}\n \\end{picture}}} \\\\ [.5mm]\n\\hline\n8 &$z_1^az_2 + z_2^bz_3 + z_3^cz_4 + z_4^d$\n&~~~~~{\\thicklines {\\begin{picture}(150,20)\n \\put(0,0){\\circle*{5}}\n \\put(0,0){\\line(1,0){26}}\n \\put(26,0){\\circle*{5}}\n \\put(26,0){\\line(1,0){26}}\n \\put(52,0){\\circle*{5}}\n \\put(52,0){\\line(1,0){26}}\n \\put(78,0){\\circle*{5}}\n \\put(78,7){\\circle{12}}\n \\end{picture}}} \\\\ [.5mm]\n\\hline\n9 &$z_1^az_2 + z_2^bz_3 + z_3^c + z_4^dz_3 + z_2^pz_4^q$\n&~~~~~{\\thicklines {\\begin{picture}(150,20)\n \\put(0,0){\\circle*{5}}\n \\put(0,0){\\line(1,0){26}}\n \\put(26,0){\\circle*{5}}\n \\put(26,0){\\line(1,0){26}}\n \\put(52,0){\\circle*{5}}\n \\put(52,7){\\circle{12}}\n \\put(52,0){\\line(1,0){26}}\n \\put(78,0){\\circle*{5}}\n \\end{picture}}} \\\\ [.5mm]\n\\hline\n10 &$z_1^az_2 + z_2^bz_3 + z_3^cz_4 + z_4^dz_3 + z_2^pz_4^q$\n&~~~~~{\\thicklines {\\begin{picture}(150,20)\n \\put(0,0){\\circle*{5}}\n \\put(0,0){\\line(1,0){26}}\n \\put(26,0){\\circle*{5}}\n \\put(26,0){\\line(1,0){26}}\n \\put(52,0){\\circle*{5}}\n \\put(52,0){\\line(1,0){26}}\n \\put(78,0){\\circle*{5}}\n \\put(65,0){\\oval(26,26)[t]}\n \\end{picture}}} \\\\ [.5mm]\n\\hline\n11 &$z_1^az_2 + z_2^bz_3 + z_3^cz_4 + z_4^dz_2 + z_1^pz_4^q$\n&~~~~~{\\thicklines {\\begin{picture}(150,20)\n \\put(0,0){\\circle*{5}}\n \\put(0,0){\\line(1,0){26}}\n \\put(26,0){\\circle*{5}}\n \\put(26,0){\\line(1,0){26}}\n \\put(52,0){\\circle*{5}}\n \\put(52,0){\\oval(52,26)[t]}\n \\put(52,0){\\line(1,0){26}}\n \\put(78,0){\\circle*{5}}\n \\end{picture}}} \\\\ [.5mm]\n\\hline\n12 &$z_1^az_2 + z_2^bz_3 + z_3^cz_4 + z_4^dz_1$\n&~~~~~{\\thicklines {\\begin{picture}(150,20)\n \\put(0,0){\\circle*{5}}\n \\put(0,0){\\line(1,0){26}}\n \\put(26,0){\\circle*{5}}\n \\put(26,0){\\line(1,0){26}}\n \\put(39,0){\\oval(78,26)[t]}\n \\put(52,0){\\circle*{5}}\n \\put(52,0){\\line(1,0){26}}\n \\put(78,0){\\circle*{5}}\n \\end{picture}}} \\\\ [.5mm]\n\\hline\n13 &$z_1^az_2 + z_2^bz_3 + z_3^cz_4 + z_4^dz_5 + z_5^e$\n&~~~~~{\\thicklines {\\begin{picture}(150,20)\n \\put(0,0){\\circle*{5}}\n \\put(0,0){\\line(1,0){26}}\n \\put(26,0){\\circle*{5}}\n \\put(26,0){\\line(1,0){26}}\n \\put(52,0){\\circle*{5}}\n \\put(52,0){\\line(1,0){26}}\n \\put(78,0){\\circle*{5}}\n \\put(78,0){\\line(1,0){25}}\n \\put(104,0){\\circle*{5}}\n \\put(104,7){\\circle{12}}\n \\end{picture}}} \\\\ [.5mm]\n\\hline\n14 &$z_1^az_2 + z_2^bz_3 + z_3^cz_4 + z_4^d + z_5^ez_4 + z_3^pz_5^q$\n&~~~~~{\\thicklines {\\begin{picture}(150,20)\n \\put(0,0){\\circle*{5}}\n \\put(0,0){\\line(1,0){26}}\n \\put(26,0){\\circle*{5}}\n \\put(26,0){\\line(1,0){26}}\n \\put(52,0){\\circle*{5}}\n \\put(52,0){\\line(1,0){26}}\n \\put(78,0){\\circle*{5}}\n \\put(78,7){\\circle{12}}\n \\put(78,0){\\line(1,0){25}}\n \\put(104,0){\\circle*{5}}\n \\end{picture}}} \\\\ [.5mm]\n\\hline\n15 &$z_1^az_2 + z_2^bz_3 + z_3^c + z_4^dz_3 + z_5^ez_4 + z_2^pz_4^q$\n&~~~~~{\\thicklines {\\begin{picture}(150,20)\n \\put(0,0){\\circle*{5}}\n \\put(0,0){\\line(1,0){26}}\n \\put(26,0){\\circle*{5}}\n \\put(26,0){\\line(1,0){26}}\n \\put(52,0){\\circle*{5}}\n \\put(52,7){\\circle{12}}\n \\put(52,0){\\line(1,0){26}}\n \\put(78,0){\\circle*{5}}\n \\put(78,0){\\line(1,0){25}}\n \\put(104,0){\\circle*{5}}\n \\end{picture}}} \\\\ [.5mm]\n\\hline\n16 &$z_1^az_2 + z_2^bz_3 + z_3^cz_4 + z_4^dz_5 + z_5^ez_4 + z_3^pz_5^q$\n&~~~~~{\\thicklines {\\begin{picture}(150,20)\n \\put(0,0){\\circle*{5}}\n \\put(0,0){\\line(1,0){26}}\n \\put(26,0){\\circle*{5}}\n \\put(26,0){\\line(1,0){26}}\n \\put(52,0){\\circle*{5}}\n \\put(52,0){\\line(1,0){26}}\n \\put(78,0){\\circle*{5}}\n \\put(78,0){\\line(1,0){25}}\n \\put(91,0){\\oval(26,26)[t]}\n \\put(104,0){\\circle*{5}}\n \\end{picture}}} \\\\ [.5mm]\n\\hline\n17 &$z_1^az_2 + z_2^bz_3 + z_3^cz_4 + z_4^dz_5 + z_5^ez_3 + z_2^pz_5^q$\n&~~~~~{\\thicklines {\\begin{picture}(150,20)\n \\put(0,0){\\circle*{5}}\n \\put(0,0){\\line(1,0){26}}\n \\put(26,0){\\circle*{5}}\n \\put(26,0){\\line(1,0){26}}\n \\put(52,0){\\circle*{5}}\n \\put(52,0){\\line(1,0){26}}\n \\put(78,0){\\circle*{5}}\n \\put(78,0){\\oval(52,26)[t]}\n \\put(78,0){\\line(1,0){25}}\n \\put(104,0){\\circle*{5}}\n \\end{picture}}} \\\\ [.5mm]\n\\hline\n18 &$z_1^az_2 + z_2^bz_3 + z_3^cz_4 + z_4^dz_5 + z_5^ez_2 + z_1^pz_5^q$\n&~~~~~{\\thicklines {\\begin{picture}(150,20)\n \\put(0,0){\\circle*{5}}\n \\put(0,0){\\line(1,0){26}}\n \\put(26,0){\\circle*{5}}\n \\put(26,0){\\line(1,0){26}}\n \\put(52,0){\\circle*{5}}\n \\put(52,0){\\line(1,0){26}}\n \\put(65,0){\\oval(78,26)[t]}\n \\put(78,0){\\circle*{5}}\n \\put(78,0){\\line(1,0){25}}\n \\put(104,0){\\circle*{5}}\n \\end{picture}}} \\\\ [.5mm]\n\\hline\n19 &$z_1^az_2 + z_2^bz_3 + z_3^cz_4 + z_4^dz_5 + z_5^ez_1$\n&~~~~~{\\thicklines {\\begin{picture}(150,20)\n \\put(0,0){\\circle*{5}}\n \\put(0,0){\\line(1,0){26}}\n \\put(26,0){\\circle*{5}}\n \\put(26,0){\\line(1,0){26}}\n \\put(52,0){\\circle*{5}}\n \\put(52,0){\\oval(104,26)[t]}\n \\put(52,0){\\line(1,0){26}}\n \\put(78,0){\\circle*{5}}\n \\put(78,0){\\line(1,0){25}}\n \\put(104,0){\\circle*{5}}\n \\end{picture}}} \\\\ [.5mm]\n\\hline\n\\end{tabular}\n\\end{center}\n}}\n\n\\begin{center}\n\\parbox{6.4truein}\n{\\noindent {\\bf Table 1.}~~\n {\\it The polynomial types that have been implemented}.}\n\\end{center}\n\nThe last problem that confronts the construction of all Calabi--Yau\\ manifolds\nin weighted projective space is the question whether the list is finite.\nAgain we compare the new situation with the one in the case\nof CICYs. There the condition of vanishing first Chern class leads to just\none configuration that can be found in the case\nof one projective space and one polynomial, namely the quintic\n$\\relax{\\rm I\\kern-.18em P}_4[5]$. For a weighted $\\relax{\\rm I\\kern-.18em P}_4$ however this condition leads to\nthe equation (\\ref{c1}).\nIt seems at first that this equation has an infinite number of solutions.\nIn fact a little thought shows that for each of the thirty polynomial types\nthere are restrictions on the range of possible weights. For the polynomials\nof Fermat type this is already known because of their relation\n\\cite{kms}\\cite{m}\\cite{gvw} to\ncertain types of conformal field theories which have all been constructed\n\\cite{ls12}\\cite{ls3}\\cite{lr}\\cite{fkss1}. Consider a Fermat polynomial\n\\begin{equation}\np = z_1^{a_1} + z_2^{a_2} + z_3^{a_3} + z_4^{a_4} + z_5^{a_5}\n\\end{equation}\nwith weights $(k_1,....,k_5)$ and degree\n\\begin{equation}\nd= k_1a_1 = k_2a_2 = k_3a_3 = k_4a_4 = k_4a_5~.\n\\end{equation}\nThe condition of vanishing first Chern class becomes\n\\begin{equation}\n1 = \\sum_{i=1}^5 \\frac{1}{a_i} \\lleq{cycon1}\nIt is possible to iteratively bound the $a_i$, which we take to be\nordered such that $a_i\\leq a_{i+1}$, in virtue of the fact that\n$\\sum_i \\frac{1}{a_i}$ is a decreasing function of all its arguments.\nThe smallest possible value for $a_1$ is 2.\nThe next step consists in finding the smallest possible\nvalue for $a_2$. Using the lower bound on $a_1$ condition (\\ref{cycon1})\nbecomes\n\\begin{equation}\n\\frac{1}{2} = \\sum_{i=2}^5 \\frac{1}{a_i},\n\\end{equation}\nfrom which it follows that the smallest possible value for\n$a_2$ is 3. Proceeding in this manner we end up with\nthe condition\n\\begin{equation}\n\\frac{1}{42} - \\frac{1}{43} = \\frac{1}{a_5}\n\\end{equation}\nfrom which we find $a_5 = 1806$ which turns out to be the highest power\nthat arises in our polynomials.\n\nAs a second example consider polynomials of type 2,\n\\begin{equation}\np = z_1^{a_1} + z_2^{a_2} + z_3^{a_3} + z_4^{a_4} + z_5^{a_5}z_4.\n\\end{equation}\nIn this case we have\n\\begin{equation}\nd= k_1a_1 = k_2a_2 = k_3a_3 = k_4a_4 = k_4 + k_5a_5.\n\\end{equation}\nand the condition of vanishing first Chern class is now\n\\begin{equation}\n1 = \\sum_{i=1}^5 \\frac{1}{a_i} - \\frac{1}{a_4a_5}\n\\lleq{c1con2}\n\nTaking again $a_1=2$ we proceed as above. Using the lower bound on $a_1$\ncondition (\\ref{c1con2}) becomes\n\\begin{equation}\n\\frac{1}{2} = \\sum_{i=2}^5 \\frac{1}{a_i} - \\frac{1}{a_4a_5}.\n\\end{equation}\n{}From this equation it follows that the smallest possible value for\n$a_2$ is 3. At the next stage we find $a_3=7$ and then $a_4=43$. Finally\nwe find the condition\n\\begin{equation}\n\\frac{1}{42\\cdot 43} = \\frac{1}{a_5} - \\frac{1}{43\\cdot a_5}\n\\end{equation}\nwhence $a_5 = 42^2 = 1764$. In a similar way we find constraints\non all other types of polynomials.\n\n\\vskip .1truein\n\\noindent\n\\section{Computation of the Spectrum}\n\nHaving constructed these 6,500 odd spaces one wants, of course, to know\n about the spectrum, especially about the number of light\ngenerations. There are several methods available to compute\nthese numbers. First there is of course the geometrical analysis that can\nbe used to compute the independent Hodge numbers of a Calabi--Yau manifold.\nAnother method that is more useful for the class of spaces at hand are\ntechniques for computing the spectrum in the framework of Landau--Ginzburg\ntheories. In those cases for which an exactly solvable theory corresponding\nto the model is known, techniques from conformal field theory are available\nas well. Unfortunately for most of the theories constructed in the previous\nsection no exactly solvable model is known and hence the tools from\nconformal field theory, even though most powerful, are not available here.\n\nThe very first step in a systematic analysis is, of course,\nthe determination\nof the number of light generations, i.e. the computation of\nthe Euler number.\n This can be done by computing the integral\nof the third Chern class using the fact that\n$\\int J^3 = 1\/\\prod k_j$ and taking into\naccount the contributions from the singularities \\cite{yau}\n\\begin{equation}\n\\chi = -\\frac{\\frac{1}{3}\\left (d^3 - \\sum k_j^3 \\right)d}{\\prod k_j}\n - \\sum_i \\frac{\\chi (S_i)}{n_i} + \\sum_i n_i \\chi (S_i)\n\\end{equation}\nwhere $\\chi (S_i)$ is the Euler number of the singular set $S_i$ and\n$n_i$ is the order of the cyclic symmetry group $\\relax{\\sf Z\\kern-.4em Z}_{n_i}$.\n\nThis can be illustrated with an example. Consider the manifold\n\\begin{equation}\n\\relax{\\rm I\\kern-.18em P}_{(4,4,5,5,7)}[25] ~~~~~~~~~\n{\\thicklines \\begin{picture}(150,20)\n \\put(0,0){\\circle*{5}}\n \\put(0,0){\\line(1,0){26}}\n \\put(26,0){\\circle*{5}}\n \\put(26,7){\\circle{12}}\n \\put(52,0){\\circle*{5}}\n \\put(52,0){\\line(1,0){26}}\n \\put(78,0){\\circle*{5}}\n \\put(78,0){\\line(1,0){26}}\n \\put(104,0){\\circle*{5}}\n \\put(104,7){\\circle{12}}\n\n \\end{picture}}.\n\\lleq{examp}\nIn this example we have three types of singular sets; first there is a\n$\\relax{\\sf Z\\kern-.4em Z}_4$--curve $C$ with Euler number $\\chi (C) = 2$. Next there are five\n$\\relax{\\sf Z\\kern-.4em Z}_5$--points and finally the $\\relax{\\sf Z\\kern-.4em Z}_7$ leads to one additional fixed point.\nPut together with $\\chi_s = -44\\frac{5}{14}$ for the Euler number of the\nsingular space this gives $\\chi = -6$ for this space.\n\nIt is clear from this example that the geometrical technique is rather\nawkward to apply systematically as it entails a detailed analysis of the\nsingular sets $S_i$ for each manifold. These not only depend on the\ndivisibility\nproperty of the weights but also on the type of the individual catastrophe\ninvolved and therefore need to be determined on a case by case basis. This\nis not easily automated.\n\nIt is much easier to use a result of Vafa \\cite{v} on the Euler number of\n$c=9$, $N=2$ Landau--Ginzburg models. His formula specialized to the\ncase at hand yields\n\\begin{equation}\n\\chi = \\frac{1}{d} \\sum_{l,r = 0}^{d-1}~~\n (-1)^{r+l+d}\\prod_{lq_i,rq_i \\in \\relax{\\sf Z\\kern-.4em Z}} \\frac{d-k_i}{k_i},\n\\end{equation}\nwhere $q_i = k_i\/d$.\nComputing the Euler number for all 10,839 odd spaces leads to the results\nof Figure 3. As already mentioned the Euler number $-960\\leq \\chi \\leq 960$.\nAmong these\nspaces are many that lead to 2, 3 and 4 light generations.\n\nThe next question then is how the Euler number actually splits up\ninto generations and antigenerations. Again it is possible to use both,\nmanifold techniques and Landau--Ginzburg type methods. As we have already\neasy methods to compute the Euler number $\\chi = 2(h^{(1,1)} - h^{(2,1)})$\nwe only need to compute either the number of generations of the number of\nantigenerations to have the complete Hodge diamond.\n\nWe consider first the geometrical techniques and compute the number of\nantigenerations. In a projective space there would be nothing to compute since\nin this case the dimensions of this cohomology group is always one,\nits only contribution coming from the K\\\"ahler form. In the case of weighted\nprojective spaces the K\\\"ahler form of course is not the only contribution\nbecause the blow--ups of the singular sets introduce new (1,1)--forms.\nThe singular sets consist of points and\/or curves. The techniques for blowing\nup points have been discussed in \\cite{ry} and the contributions\ncoming from blowing up curves have been discussed in \\cite{s3}.\nThese are in fact the\nonly types of singularity that can arise if $p$ is transverse. In other words\nthat singular subsets of $M$ cannot have dimension 2 or 3.\nFirst we show that the embedding $\\relax{\\rm I\\kern-.18em P}_4$ cannot have singular sets of\ndimension\ngreater than or equal to 3. Recall that the $\\relax{\\rm I\\kern-.18em P}_4$ has a singular point for\neach\n$k_i>1$, a singular curve for each pair with $(k_i,k_j)>1$, a singular subset\nof dimension 2 for each triple with $(k_i,k_j,k_l)>1$ {\\it etc\\\/}. In the definition\nof $\\relax{\\rm I\\kern-.18em P}_4$ we have assumed that the weights have no common factor so there\nare no\nsingular sets of dimension 4. Consider the possibility of a singular set of\ndimension 3. Such a set would correspond to weights such that\n$$(k_2,k_3,k_4,k_5)=m>1~~{\\rm but}~~m\\not|k_1~.$$\nSince $m$ does not divide $k_1$ but does divide the other $k$'s it cannot\ndivide\nthe degree $d=\\sum_{j=1}^5k_j$. Every monomial of degree $d$ must\ntherefore contain at least one factor of $z_1$. Thus\n$p(z)=z_1\\tilde{p}(z)$ and\nso is not transverse. On the other hand singular sets of dimension 2\ncan occur\nand these will generically intersect $M$ in subsets of dimension 1.\nIt remains\nto\nshow that a singular subset of dimension 2 cannot lie within $M$. To\nthis end\nsuppose there is a fixed point set of dimension 2 for the identification\n(\\ref{pro5})\nand that this subset lies within $M$. We may choose coordinates $(x,y,z)$\nsuch that, locally, the fixed point set is $(x,y,0)$. Suppose the\nidentification is represented by a matrix $A$. Since $A$ fixes $(x,y,0)$ it\nhas the form $$A=\\pmatrix{1&0&a\\cr 0&1&b\\cr 0&0&c}~.$$\nThe three--form $dx\\wedge dy\\wedge dz$ transforms as\n$$dx\\wedge dy\\wedge dz\\longrightarrow\\det A\\,dx\\wedge dy\\wedge dz$$\nwe must have $\\det\\,A=1$ and hence $c=1$. Since $A$ is contained in a finite\ngroup $A^n=1$ for some $n$, however\n$$A^n=\\pmatrix{1&0&na\\cr 0&1&nb\\cr 0&0&1}$$\nso $a$ and $b$ must, in fact, vanish. Thus the only identification that\nfixes a two--dimensional subset is the identity.\n\nWhen resolving the singularities it needs to be checked that the blown up\nmanifolds are still Calabi--Yau manifolds. For the case of singular points\nthis\nhas been discussed in ref. \\cite{ry} whereas the case of singular curves\nwas first analyzed in ref. \\cite{s3}.\n\nIn order to resolve curves consider the action of a $\\relax{\\sf Z\\kern-.4em Z}_n$ on a weighted\nCICY leaving a curve invariant. In the three--dimensional Calabi--Yau manifold\nthe normal bundle of this curve has fibres $\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_2$ and therefore this discrete\ngroup induces an action on the fibres described by the matrix\n\\begin{equation}\n\\left (\\matrix{~~\\alpha^{mq}&0\\cr 0&~~\\alpha^m\\cr} \\right ) \\lleq{normact}\nwhere $\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma$ is an n$^{th}$ root of unity, $0\\leq m \\leq n$, and $q,n$ have\nno common divisor. This action has an isolated singularity which needs\nto be resolved. The essential point is that the singularity of\n$\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_2\/\\relax{\\sf Z\\kern-.4em Z}_n$ can be described as the singular set of the surface\n\\begin{equation}\nS :~~ z_3^n = z_1z_2^{n-q}\n\\end{equation}\nin $\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_3$ and therefore can be resolved by a construction that\nis completely determined by the type $(n,q)$ of the action through the method\nof continued fractions.\nThe expansion of $\\frac{n}{q}$ in a continued fraction\n\\begin{eqnarray}\n\\frac{n}{q} &=& [b_1,...,b_s] \\nonumber \\\\\n &=& b_1 - \\frac{1}{b_2 - \\frac{\\Large 1}{ \\ddots -\n \\frac{\\Large 1}{\\Large b_s} }}\n\\end{eqnarray}\n\n\\noindent\ndetermines the numbers $b_i$ which specify uniquely the plumbing process\nwhich replaces the singularity. Furthermore, it also determines the additional\ngenerators of the cohomology, since the number\nof $\\relax{\\rm I\\kern-.18em P}_1$'s necessary to resolve the singularity is precisely the number of\nsteps needed in the evaluation of $\\frac{n}{q}= [b_1,..,b_s]$.\nThe reason for this is that the singularity is replaced by a bundle which\nis constructed of $s+1$ patches with $s$ transition functions that are\nspecified by the $b_i$'s. Each of these glueing steps introduces a sphere\nwhich in turn\nsupports a (1,1)--form. A standard shorthand notation for the geometry of\nthe blow--up is through what is called a Hirzebruch--Jung\ntree \\cite{hirz} which in the case of the blow--up of a $\\relax{\\sf Z\\kern-.4em Z}_n$ action is\njust an\nSU($s+1$) Dynkin diagram with the negative values of the $b_i's$ attached to\nthe nodes. Each node in the diagram corresponds to a sphere and the diagram\nshows which spheres intersect each other (in the case of the $\\relax{\\sf Z\\kern-.4em Z}_n$--action\nonly the neighboring spheres intersect) whereas the $b_i$ determine the\nintersection numbers.\n\nIn order to show that these blow--up procedures can be applied, we have to\nshow that the determinant (\\ref{normact}) is always 1. This can be done by\nchecking the invariance of the holomorphic threeform $\\Omega} \\def\\s{\\sigma} \\def\\th{\\theta$ under $\\relax{\\sf Z\\kern-.4em Z}_n$.\n\nConsider first manifolds of Fermat type. A singular curve in such a manifold\nis signalled by three weights of the ambient space that are not coprime\n\\begin{equation}\n(k_1,k_2,k_3) =n > 1\n\\end{equation}\n($n$ does not divide $k_4,k_5$). The integer $n$ defines a $\\relax{\\sf Z\\kern-.4em Z}_n$ discrete\ngroup. The action of $\\relax{\\sf Z\\kern-.4em Z}_n$ is given by\n\\begin{equation}\n(z_1,z_2,z_3,z_4,z_5) \\mapsto (z_1,z_2,z_3,\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma^{k_4} z_4, \\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma^{k_5} z_5),\n\\end{equation}\nwhere $\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma$ is an $n$th root of unity. In order to show that the blown--up\nmanifold is still of Calabi--Yau type one needs to show that $\\Omega} \\def\\s{\\sigma} \\def\\th{\\theta$\nis invariant. In this case the action of $\\relax{\\sf Z\\kern-.4em Z}_n$ on $\\Omega} \\def\\s{\\sigma} \\def\\th{\\theta$ is\n\\begin{equation}\n\\Omega} \\def\\s{\\sigma} \\def\\th{\\theta \\mapsto \\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma^{k_4+k_5} \\Omega} \\def\\s{\\sigma} \\def\\th{\\theta.\n\\end{equation}\nThe condition for $\\Omega} \\def\\s{\\sigma} \\def\\th{\\theta$ to be invariant is, therefore, that\n\\begin{equation}\n\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma^{k_4+k_5}=1,\n\\end{equation}\nwhich is always true since for Fermat type polynomials\n\\begin{equation}\n1=\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma^d = \\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma^{\\sum k_i} =\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma^{k_4+k_5}.\n\\end{equation}\n\nNow suppose the curve is embedded in a non--Fermat Calabi--Yau manifold.\nThen\nthe possibility exists that there are only two weights for which\n\\begin{equation}\n(k_1,k_2) =n > 1\n\\end{equation}\n($n$ does not divide $k_3,k_4,k_5$), i.e. $k_1,k_2$ are not constrained by\nthe polynomial. In this case the coordinates parametrizing the curve occur\nas\n\\begin{equation}\nz_1^{l_1}z_p,~~~z_2^{l_2}z_q.\n\\end{equation}\nThe action of $\\relax{\\sf Z\\kern-.4em Z}_n$ is given by\n\\begin{equation}\n(z_1,z_2,z_3,z_4,z_5) \\mapsto (z_1,z_2,\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma^{k_2} z_3,\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma^{k_4} z_4, \\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma^{k_5}\nz_5),\n\\end{equation}\nand the condition for $\\Omega} \\def\\s{\\sigma} \\def\\th{\\theta$ to be invariant becomes\n\\begin{equation}\n\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma^{k_3+k_4+k_5}=\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma^{k_p}=\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma^{k_q}.\n\\end{equation}\nAs before\n\\begin{equation}\nd=k_1l_1+k_p=k_2l_2+k_q=nr_1l_1+k+_p=nr_2l_2+k_q\n\\end{equation}\nhence\n\\begin{equation}\n\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma^d = \\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma^{ k_3+k_4+k_5} =\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma^{k_p}=\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma^{k_q}\n\\end{equation}\nwhich was to be shown.\n\nReturning now to the discussion of the example $\\relax{\\rm I\\kern-.18em P}_{(4,4,5,5,7)}[25]$, we\nfind from the results of ref. \\cite{ry} that a $\\relax{\\sf Z\\kern-.4em Z}_5$--point blow--up\nof this\nmanifold contributes two (1,1)--forms, whereas a $\\relax{\\sf Z\\kern-.4em Z}_7$--point blow--up\nleads\nto three additional generators of the second cohomology. Since\nthe $\\relax{\\sf Z\\kern-.4em Z}_5$\nsingular set $\\relax{\\rm I\\kern-.18em P}_1[5]$ consists of five points and the $\\relax{\\sf Z\\kern-.4em Z}_7$ singular\nset\nconsists just of one point all point blow--ups together contribute a total\nof\nthirteen (1,1)--forms. To find out how many (1,1)--forms the blow--up of\nthe\n$\\relax{\\sf Z\\kern-.4em Z}_4$--curve contributes we need to check the induced action of this\ndiscrete group action on the normal bundle of the curve \\cite{s3}.\n\nIn our example (\\ref{examp}) this induced action\n\\begin{equation}\n\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_2 \\ni (z_1,z_2) \\longmapsto (\\alpha z_1, \\alpha^3 z_2)\n\\end{equation}\nis of type $(n,q)=(4,3)$ and therefore the singularity of $\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_2\/\\relax{\\sf Z\\kern-.4em Z}_4$\nis equivalent to the singular set of the surface\n\\begin{equation}\n z_3^4 = z_1z_2 .\n\\end{equation}\nThe resolution is specified by the Hirzebruch--Jung tree\n\\begin{center}\n{\\thicklines {\\begin{picture}(60,15)\n \\put(0,6){--2}\n \\put(8,0){\\circle*{5}}\n \\put(8,0){\\line(1,0){26}}\n \\put(26,6){--2}\n \\put(34,0){\\circle*{5}}\n \\put(34,0){\\line(1,0){26}}\n \\put(52,6){--2}\n \\put(60,0){\\circle*{5}}\n \\end{picture}}}\n\\end{center}\nwhich in turn is determined completely by the continued fraction\\begin{equation}\n\\frac{4}{3} = 2 - \\frac{1}{2-\\frac{1}{2}}.\n\\end{equation}\nTherefore the blow--up of the curve contributes three more (1,1)--forms.\nTaking into account the K\\\"ahler form of the ambient space then leads\nto a total of 17 (1,1)--forms for this manifold.\n\nAs is evident however the manifold technique is awkward to implement in a\nsystematic\nway and is better used as an independent check of the Landau--Ginzburg\ntype formulation of this problem by Vafa who constructs a Poincar\\'e--type\npolynomial for the $l^{\\rm th}$ twisted sector\n\\begin{equation}\nTr_l ((t\\bt)^{dJ_0})\n= t^{d\\left (Q_l+\\frac{1}{6}c_T \\right )}\n \\bt^{d\\left (-Q_l+\\frac{1}{6}c_T \\right )}\n \\prod_{lq_i\\in \\relax{\\sf Z\\kern-.4em Z}}\n \\left ( \\frac{1- (t\\bt)^{d-k_i}}{1-(t\\bt)^{k_i}}\\right)\n\\end{equation}\nwith\n\\begin{eqnarray}\n Q_l &=& \\sum_{lq_i {\\footnotesize{\\not \\in}} \\relax{\\sf Z\\kern-.4em Z}}\n \\left (lq_i - [lq_i] - \\frac{1}{2} \\right )~,\\nonumber \\\\\n \\frac{1}{6}c_T &=& \\sum_{lq_i {\\footnotesize{\\not \\in}} \\relax{\\sf Z\\kern-.4em Z}}\n \\left (\\frac{1}{2} - q_i\\right )\n\\end{eqnarray}\nwhere $t$ and $\\bt$ are formal variables, $d$ is the degree of the\nLandau--Ginzburg\npotential, the $q_i=k_i\/d$ are the normalized weights of the fields and\n$[lq_i]$ is the integer part of $lq_i$.\nExpanding this polynomial in powers in $t$ and $\\bt$ it is possible to read\noff the contributions to the various cohomology groups from the different\nsectors of the twisted LG--theory. The (2,1) forms for example are given by\nthe number of fields with charge (1,1), i.e. the coefficient of $(t\\bt)^d$.\nIn general, the number of $(p,q)$ forms are given by the coefficients of\n$t^{(3-p)d}\\bt^{qd}$ in the Poincar\\'e polynomials summed over all sectors\n$l=0,1,\\dots,d-1$.\n\n\n\\vskip .1truein\n\\noindent\n\\section{Landau--Ginzburg Vacua}\n\n\\noindent\nIn this section we briefly review the construction of all\n Landau--Ginzburg vacua based on superpotentials with an isolated\nsingularity at the origin.\nConsider a string ground state based on a Landau--Ginzburg theory\nwhich we\nassume to be $N=2$ supersymmetric since we demand $N=1$ spacetime\nsupersymmetry.\nUsing a superspace formulation in terms of the coordinates\n$(z,\\bz,\\th^+,\\bth^+,\\th^-,\\bth^-)$ the action takes the form\n\\begin{equation}\n{\\cal A}} \\def\\cB{{\\cal B}} \\def\\cC{{\\cal C}} \\def\\cD{{\\cal D} = \\int d^2zd^4\\th~K(\\Phi_i,\\bar \\Phi} \\def\\bth{\\bar \\theta_i) + \\int d^2zd^2\\th^- ~W(\\Phi_i)\n + \\int d^2zd^2\\th^+ ~W(\\bar \\Phi} \\def\\bth{\\bar \\theta_i)\n\\end{equation}\nwhere $K$ is the K\\\"ahler potential and the superpotential $W$ is a\nholomorphic function of the chiral superfields $\\Phi_i$.\nSince the ground states of the bosonic potential are the critical points\nof the superpotential of the LG theory\n we demand their existence. The type of critical points we\nneed is determined by the fact that we wish to keep the fermions in\nthe theory massless; hence we assume that the critical points are\ncompletely degenerate. Furthermore we require that all critical points\nbe isolated, since we wish to relate the finite dimensional ring of\nmonomials associated to such a singularity with the chiral ring of\nphysical states in the Landau--Ginzburg theory, in order to construct\nthe spectrum of the corresponding string vacuum.\nFinally we demand that the theory is conformally invariant; from this\nfollows, relying on some assumptions regarding the renormalization\nproperties of the theory, that the Landau--Ginzburg potential is\nquasihomogeneous. In other words, we require to be able to assign,\nto each field\n$\\Phi_i$, a weight $q_i$ such that for any non--zero complex number\n$\\l \\in \\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}^{\\star}$\n\\begin{equation}\nW(\\l^{q_1}\\Phi_1,\\dots,\\l^{q_n}\\Phi_n) =\\l W(\\Phi_1,\\dots,\\Phi_n).\n\\end{equation}\nThus we have formulated the class of potentials that we need to consider:\nquasihomogeneous polynomials that have an isolated, completely\ndegenerate singularity (which we can always shift to the origin).\n\nAssociated to each of the superpotentials, $W(\\Phi_i)$ is a\nso--called catastrophe which is obtained by first truncating the\nsuperfield $\\Phi_i$ to its lowest bosonic component\n$\\phi_i(z,\\bz)$, and then going to the\nfield theoretic limit of the string by assuming $\\phi_i$ to be constant\n$\\phi_i=z_i$. Writing the weights as $q_i = k_i\/d$, we will denote by\n\\begin{equation}\n\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_{(k_1,k_2,\\dots,k_n)}[d]\n\\end{equation}\nthe set of all catastrophes described by the zero locus of polynomials\nof degree $d$ in variables $z_i$ of weight $k_i$.\n\nThe affine varieties described by these polynomials are not compact\nand hence it is necessary to implement a projection in order to\ncompactify these spaces. In Landau--Ginzburg language, this amounts to an\norbifolding of the theory with respect to a discrete group $\\relax{\\sf Z\\kern-.4em Z}_d$ the\norder\nof which is the degree of the LG potential \\cite{v}. The spectrum of the\norbifold theory will contain twisted states which, together with\nthe monomial ring of the potential, describe the complete spectrum of the\ncorresponding Calabi--Yau manifold. We will denote the orbifold\nof a Landau--Ginzburg theory by\n\\begin{equation}\n\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}^{\\star}_{(k_1,k_2,\\dots,k_n)}[d]\n\\end{equation}\nand call it a configuration.\n\nIn manifold speak the projection should lead to a three--dimensional\nK\\\"ahler manifold, with vanishing first Chern class. For a general\nLandau--Ginzburg theory no unambiguous universal prescription for doing\nso has been found and, as we will describe in Section 4, none can exist.\nOne way to compactify amounts to simply imposing projective\nequivalence\n\\begin{equation}\n(z_1,....,z_n) \\equiv (\\l^{k_1} z_1,.....,\\l^{k_n} z_n)\n\\lleq{projn}\nwhich embeds the hypersurface described by the zero locus of the\npolynomial into a weighted projective space $\\relax{\\rm I\\kern-.18em P}_{(k_1,k_2,\\dots,k_n)}$\nwith weights $k_i$. The set of hypersurfaces of degree $d$ embedded\nin weighted projective space will be denoted by\n\\begin{equation}\n\\relax{\\rm I\\kern-.18em P}_{(k_1,k_2,\\dots,k_n)}[d].\n\\end{equation}\nFor a potential with five scaling variables this\nconstruction is completely sufficient in order to pass from the\nLandau--Ginzburg theory to a string vacuum \\cite{m}\\cite{gvw} provided\n$d=\\sum_{i=1}^5 k_i$, which is the condition that these hypersurfaces\nhave vanishing first Chern class. For more than five variables, however,\nthis type of compactification does not lead to a string vacuum.\n\nEven though the precise relation between LG theories and CY manifolds\nis not known for the most general case certain facts are known.\nSince LG theories with five variables describe a CY manifold embedded in\na 4 complex dimensional weighted projective space one might expect\ne.g. that LG potentials with 6, 7, etc., variables describe manifolds\nembedded in 5, 6, etc., dimensional weighted projective spaces. This is\nnot correct.\n\nIn fact none of the models with more than five variables is related\nto manifolds embedded in one weighted projective space. Instead they\ndescribe Calabi--Yau manifolds embedded in products of weighted\nprojective space. A simple example is\nfurnished by the LG potential in six variables\n\\begin{equation}\nW=\\Phi_1 \\Psi_1^2+\\Phi_2 \\Psi_2^2+\\sum_{i=1}^3 \\Phi_i^{12}+\n\\Phi_4^3\n\\end{equation}\nwhich corresponds to the exactly solvable model described by the\ntensor product of $N=2$ minimal theories at the levels\n\\begin{equation}\n(22^2 \\cdot 10\\cdot 1)_{D^2\\cdot A^2},\n\\end{equation}\nwhere the subscripts indicate the affine invariants chosen for the\nindividual factors. This theory belongs to the LG configuration\n\\begin{equation}\n\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}^{\\star}_{(2,11,2,11,2,8)}[24]^{(3,243)}_{-480}\n\\lleq{lgform}\nand is equivalent to the weighted complete intersection Calabi--Yau (CICY)\n manifold in the configuration\n\\begin{equation}\n\\matrix{\\relax{\\rm I\\kern-.18em P}_{(1,1,1,4,6)}\\cr \\relax{\\rm I\\kern-.18em P}_{(1,1)}\\hfill\\cr}\n \\left [\\matrix{1&12\\cr 2&0\\cr}\\right]^{(3,243)}_{-480}\n\\lleq{cyform}\ndescribed by the intersection of the zero locus of the two potentials\n\\begin{eqnarray}\np_1 &=& x_1^2y_1+x_2^2y_2 \\nonumber \\\\\np_2 &=& y_1^{12}+y_2^{12}+y_3^{12}+y_4^3+y_5^2.\n\\end{eqnarray}\nHere we have added a trivial factor $\\Phi_5^2$ to the potential and\ntaken the field theory limit via $\\phi_i(z,\\bz)=y_i$, where $\\phi_i$\n is the lowest component of the chiral superfield $\\Phi_i$. The first\ncolumn in the degree matrix (\\ref{cyform}) indicates that the first\npolynomial is of bidegree (2,1) in the coordinates $(x_i,y_j)$\nof the product of the projective line $\\relax{\\rm I\\kern-.18em P}_1$ and the weighted\nprojective space $\\relax{\\rm I\\kern-.18em P}_{(1,1,1,4,6)}$ respectively, whereas the second\ncolumn shows that the second polynomial is independent of the\nprojective line and of degree 12 in the coordinates of the\nweighted $\\relax{\\rm I\\kern-.18em P}_4$. The superscripts in (\\ref{lgform}) and (\\ref{cyform})\ndescribe the dimensions $(h^{(1,1)},h^{(2,1)})$ of the fields\ncorresponding to the cohomology groups $(H^{(1,1)},H^{(2,1)})$,\nwhereas the subscript is the Euler number of the configuration.\n\nIt should be noted however that Landau--Ginzburg potentials in six\nvariables do not describe the most general complete intersection in\nproducts of weighted spaces, simply because not all of these manifolds\ninvolve trivial factors, or put differently, quadratic monomials.\nA simple example is the manifold that corresponds to the\nLandau--Ginzburg theory\n$(1\\cdot 16^3)_{A\\cdot E_7^3}$ with LG potential\n\\begin{equation}\nW= \\sum_{i=1}^3 \\left(\\Phi_i^3 + \\Phi_i \\Psi_i^3\\right) + \\Phi_4^3.\n\\end{equation}\nThis theory describes a\ncodimension--2 Calabi--Yau manifold embedded in\n\\begin{equation}\n\\matrix{\\relax{\\rm I\\kern-.18em P}_2\\cr \\relax{\\rm I\\kern-.18em P}_3\\cr} \\left[\\matrix{3 &0\\cr\n 1 &3\\cr}\\right].\n\\end{equation}\nThis space has 8 (1,1)--forms and 35 (2,1)--forms which correspond to\nthe possible complex deformations in the two polynomials\n$p_1,p_2$ \\cite{s1}.\n\nAssociated to this Calabi--Yau manifold in a product of ordinary\nprojective\nspaces is an auxiliary algebraic manifold in a weighted six--dimensional\nprojective space\n\\begin{equation}\n\\relax{\\rm I\\kern-.18em P}_{(2,2,2,3,3,3,3)}[9].\n\\end{equation}\nobtained via the na\\\"{\\i}ve compactification (\\ref{projn}) where the first\nthree\ncoordinates come from the fields $\\Psi_i$ and the last four come from\nthe\n$\\Phi_i$. We want to compute the number of complex deformations of this\nmanifold, i.e. we want to compute the number of monomials of charge 1.\n\nThe most general monomial is of the form $\\prod_i \\Phi_i \\prod \\Psi_j$.\nIt is easy to show explicitly that there are precisely 35 monomials by\nwriting them down but the following remarks may suffice. There are\nfour different types of possible monomials, depending on whether they\ncontain the fields $\\Phi_i$ not at all (I), linearly (II),\nquadratically (III) or cubically (IV). First note\nthat monomials of type (I) do not contribute to the marginal operators.\nMonomials of type (II) have to contain cubic monomials in the $\\Psi$.\nSince there are three fields $\\Psi_i$ available, we obtain a total of\n$ 40$ marginal operators of this type.\nBecause of the equation of motion nine of these are in fact in the ideal\nand we are left with 31 complex deformations of this type. Monomials\nquadratic in the $\\Phi_i's$ again do not contribute, whereas those cubic\nin the $\\Phi_i$ fields contribute the remaining 4. Indeed, there are\n20 cubic monomials in terms of the four $\\Phi_i$ but using the equations\nof motion one finds that 16 of those are in the ideal.\n\nIn other words, for the total of $60=40+20$ monomials of degree\n9 (or charge 1) it is possible to fix the coefficients of 9 of the\n40 by allowing linear field redefinitions of the first three coordinates\nand also to fix the coefficients of 16 of the 20 via linear field\nredefinitions of the last four coordinates. Hence even though the ambient\nspace is singular and the manifold hits these singularities in a $\\relax{\\rm I\\kern-.18em P}_2$\nand in a cubic surface $\\relax{\\rm I\\kern-.18em P}_3[3]$ the resolution does not contribute any\ncomplex deformations because these surfaces are simply connected.\n\nIt should be emphasized that this manifold is not the physical internal\npart\nof a string ground state, but that it plays an auxiliary role, which\nallows to\ndiscuss just one particular sector of the string vacuum, namely the\ncomplex deformations.\n\n Before turning to the problem of constructing LG configurations\nsatisfying the constraints described above, we wish to make some remarks\nregarding the validity of the\nrequirements formulated in the previous paragraph.\n\nEven though the assumptions formulated in refs. \\cite{m}\\cite{vw} and\nreviewed above seem rather reasonable, and previous work shows that the\nset of such Landau--Ginzburg theories certainly is an interesting and\nquite extensive class of models, it is clear that it is not the most\ngeneral class of (2,2) vacua. Although it provides a rather large set\nof different models\n\\fnote{2}{The rather extravagant values that have been\n mentioned in the literature as the number of possible (2,2)\n vacua are based on extrapolations that do not take into account\n the problem of overcounting that is generic to all of these\n different constructions.},\n which contains many classes of previously constructed vacua\n\\fnote{3}{Such as vacua constructed tensor models based on the ADE minimal\n models \\cite{m}\\cite{gvw}\\cite{ls12}\\cite{ls3}\\cite{fkss1}\\cite{sy}\n and \\break\n level--1 Kazama--Suzuki models \\cite{lvwg}\\cite{fiq2}\\cite{s},\n as well as G\/H LG theories \\cite{ls5} related to Kazama--Suzuki\n models of higher levels.},\nthere are known vacua which cannot be described in this framework.\n\nPerhaps the simplest example that does not fit into the classification\n above is that of the Calabi--Yau manifolds in\n\\begin{equation}\n\\relax{\\rm I\\kern-.18em P}_5[4~~2],\n\\end{equation}\ndescribed by the intersection of two hypersurfaces defined by a quartic\nand a quadratic\npolynomial in a five--dimensional\nprojective space $\\relax{\\rm I\\kern-.18em P}_5$ because of the purely quadratic polynomial\nthat appears as one of the constraints defining the hypersurface.\n The requirement that the singularity\nbe completely degenerate seems, in fact, to exclude a\ngreat many of the CICY manifolds,\nthe complete class of which was constructed in ref. \\cite{cdls}.\nAn important set of manifolds in that class that does not fit into\nthe present framework\neither is defined by\npolynomials of bidegree (1,4) and (1,1)\n\\begin{equation}\n\\matrix{\\relax{\\rm I\\kern-.18em P}_1\\cr \\relax{\\rm I\\kern-.18em P}_4\\cr}\\left[\\matrix{1&1\\cr 1&4\\cr}\\right].\n\\end{equation}\nThe superpotential $W=p_1+p_2$ is not quasihomogeneous, nor does it\nhave an isolated singularity\n\\fnote{4}{These manifolds are important in the context of possible\n phase transitions between Calabi--Yau string vacua \\cite{cgh}\n via the process of splitting and contraction introduced in\n \\cite{cdls}.}.\nThus it appears that there ought to be a generalization of the framework\ndescribed above, which allows a modified LG description of these and other\nstring vacua. This however we leave for future work.\n\n\\vskip .1truein\n\\noindent\n\\section{Transversality of Catastrophes}\n\n\\noindent\nThe most explicit way of constructing a Landau--Ginzburg vacuum is, of\ncourse,\nto exhibit a specific potential that satisfies all the conditions\nimposed\nby the requirement that it ought to describe a consistent ground state\n of the string.\n Even though much effort has\ngone into the classification of singularities of the type described in\nthe\nprevious section,\nsuch polynomials have not been classified yet. As already mentioned above\nthe mathematicians have classified\npolynomials with at most three variables \\cite{agzv}, which is two short\nof\nthe lowest number of variables\nthat is needed in order to construct a vacuum that allows a formulation of\na four--dimensional low--energy effective theory\n\\fnote{5}{The complete list of K3 representations embedded in\n$dim_{\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}} =3$ weighted projective space $\\relax{\\rm I\\kern-.18em P}_{(k_1,k_2,k_3,k_4)}$ has\nbeen obtained in \\cite{reid}.}.\n\nIn Sections 1--3 a set of potentials in five variables was described\n which represents an obvious\ngeneralization of the polynomials that appear in two--dimensional\ncatastrophes.\nAfter imposing the conditions for these theories to describe\nstring vacua, only a finite number of the infinite number of LG theories\nsurvive and all these solutions were constructed. It is clear that this\n classification of singularities is\nnot complete, even after restricting to five variables.\nIt is indeed easy to construct polynomials that are not contained in the\nclassification of \\cite{cls}, a simple one being furnished by the\nexample \\cite{orlik}\n \\begin{equation}\n\\relax{\\rm I\\kern-.18em P}_{(15,10,6,1)}[45]\\ni\n\\left\\{z_1(z_1^2+z_2^3+z_3^5)+z_4^{45}+z_2^2z_3^4z_4=0\\right\\}.\n\\end{equation}\nThis polynomial is not of\nany of the types analysed discussed above but it is nevertheless transverse.\n\nKnowledge of the explicit form of the potential of a LG theory is very\nuseful information when it comes to the detailed analysis of such a\nmodel.\nIt is however not necessary if only limited knowledge, such as the\ncomputation of the spectrum of the theory, is required. In fact\nthe only ingredients necessary for the computation of the spectrum\nof a LG vacuum \\cite{v} are the anomalous dimensions of the scaling\nfields\nas well as the fact that in a configuration of weights\nthere exists a polynomial of appropriate degree with an\nisolated singularity. However, it is much easier to check whether\nthere exists such a polynomial\nin a configuration than to actually construct\nsuch a potential. The reason is a theorem by Bertini \\cite{algeom},\nwhich asserts that\nif a polynomial does have an isolated singularity on the base locus\nthen,\neven though this potential may have worse singularities away from the\nbase locus, there exists a deformation of the original polynomial that\nonly\nadmits an isolated singularity anywhere. Hence we only have to find\ncriteria\nthat guarantee at worst an isolated singularity on the base locus.\nIt is precisely this problem that was addressed in the mathematics\nliterature \\cite{f} at the same time as the explicit construction of LG\nvacua was started in ref. \\cite{cls}.\nThe main point of the argument in \\cite{f} is the following.\n\nSuppose we\nwish to check whether a polynomial in $n$ variables $z_i$ with weights\n$k_i$ has an isolated singularity, i.e. whether the condition\n\\begin{equation}\ndp=\\sum_i \\frac{\\partial p}{\\partial z_i} dz_i = 0\n\\lleq{transv}\ncan be solved at the origin $z_1=\\cdots = z_n=0$. According to\nBertini's theorem,\nthe singularities of a general element in $\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_{(k_1,...,k_n)}[d]$\nwill lie\non the base locus, i.e., the intersection of the hypersurface and\nall the\ncomponents of the base locus, described by coordinate planes of\ndimension\n$k=1, ..., n$. Let $\\cP_k$ such a $k$--plane, which we may assume to be\ndescribed by setting to zero the coordinates $z_{k+1}=\\cdots = z_n$.\nExpand\nthe polynomials around the non--vanishing coordinates $z_1,...,z_k$\n\\begin{equation}\np(z_1,...,z_n) =\nq_0(z_1,...,z_k)~ +~ \\sum_{j=k+1}^n q_j(z_1,...,z_k)z_j + h.o.t.\n\\end{equation}\nClearly, if $q_0\\neq 0$ then $\\cP_k$ is not part of the base locus\nand hence\nthe hypersurface is transverse. If on the other hand $q_j=0$, then\n$\\cP_k$ is part of the base locus and singular points\ncan occur on the intersection of the hypersurfaces defined by\n$\\cH_j=\\{q_j=0\\}$.\nIf, however, we can arrange this intersection to be empty, then the\npotential is smooth on the base locus.\n\nThus we have found that the conditions for transversality in any\nnumber of variables is the existence for any index set\n$\\cI_k=\\{1,...,k\\}$ of\n\\begin{itemize}\n\\begin{enumerate}\n\\item{either a monomial $z_1^{a_1}\\cdots z_k^{a_k}$ of degree $d$}\n\\item{or of at least $k$ monomials $z_1^{a_1}\\cdots z_k^{a_k}z_{e_i}$\nwith distinct $e_i$.}\n\\end{enumerate}\n\\end{itemize}\n\nAssume on the other hand that neither of these conditions\nholds for\nall index sets and let $\\cI_k$ be the subset for which they fail. Then\nthe potential has the form\n\\begin{equation}\np(z_1,...,z_n) = \\sum_{j=k+1}^n q_j(z_1,...,z_k)z_j ~+~ \\cdots\n\\end{equation}\nwith at most $k-1$ non--vanishing $q_j$. In this case the intersection\nof the hypersurfaces $\\cH_j$ will be positive and hence the polynomial\n$p$ will not be transverse.\n\nAs an example for the considerable ease with which one can check whether\na given configuration allows for the existence of a\npotential with an isolated singularity, consider the polynomial of Orlik\nand Randall\n\\begin{equation}\np=z_1^3+z_1z_2^3+z_1z_3^5+z_4^{45}+z_2^2z_3^4z_4.\n\\end{equation}\nCondition (\\ref{transv}) is equivalent to the system of equations\n\\begin{equation}\n\\begin{array}{r l r l}\n 0 &=~ 3z_1^2+z_2^3+z_3^5 , & 0 &=~ 3z_1z_2^2+2z_2z_3^4z_4 \\\\\n 0 &=~ 5z_1z_3^4 + 4z_2^2z_3^3z_4, & 0 &=~ z_2^2z_3^4+45z_4^{44} .\n\\end{array}\n\\end{equation}\nwhich, on the base locus, collapses to the trivial pair of\nequations $z_2z_3=0=z_2^3+z_5^5$. Hence this configuration allows for\nsuch a polynomial. To check the system away from the base locus\nclearly is much more complicated.\n\nBy adding a fifth variable $z_5$ of weight 13 it is possible to define\na Calabi--Yau deformation class\n$\\relax{\\rm I\\kern-.18em P}_{(1,6,10,13,15)}[45]_{-72}^{(17,53)}$, a configuration not\nconsidered in \\cite{cls}.\n\n\\vskip .1truein\n\\noindent\n\\section{Finiteness Considerations}\n\n \\noindent\nThe problem of finiteness has two parts: first one has to put a\nconstraint\non the number of scaling fields that can appear in the LG theory\nand then one has to determine limits on the exponents with which the\nvariables occur in the superpotential. Both of these constraints follow\nfrom the fact that the central charge of a Landau--Ginzburg theory with\nfields of charge $q_i$\n\\begin{equation}\nc=3\\sum \\left(1-2 q_i\\right)=:\\sum c_i\n\\lleq{cc}\nhas to be $c=9$ in order to describe a string vacuum.\n\nIt should be clear that without any additional input the number of\nLandau--Ginzburg vacuum configurations that can be exhibited is infinite.\nThis is to be\nexpected simply because it is known from the construction of CICYs\n\\cite{cls} that it is often possible to rewrite a manifold in an infinite\nnumber of ways and we ought to encounter similar things in the LG\nframework. A trivial way to do this is to simply add mass terms that\ndo not contribute to the central charge. Even though trivial such\nmass terms\nare important and necessary for LG theories, not only in orbifold\nconstructions \\cite{kss} but also in order to relate them to CY\nmanifolds. Consider e.g. the codimension--four Calabi--Yau manifold\n\\begin{equation}\n\\matrix{\\relax{\\rm I\\kern-.18em P}_1 \\cr \\relax{\\rm I\\kern-.18em P}_2\\cr \\relax{\\rm I\\kern-.18em P}_2 \\cr \\relax{\\rm I\\kern-.18em P}_2\\cr}\n\\left[\\matrix{2 &0 &0 &0\\cr\n 1 &2 &0 &0\\cr\n 0 &1 &2 &0\\cr\n 0 &0 &1 &2\\cr}\\right]\n\\lleq{boggle}\nwith the defining polynomials\n\\begin{equation}\n\\begin{array}{r l r l}\np_1 &=~ \\sum_{i=1}^2 u_i^2v_i, & p_2 &=~ \\sum_{i=1}^3 v_i^2w_i \\\\\np_3 &=~ \\sum_{i=1}^3 w_i^2x_i, & p_4 &=~ \\sum_{i=1}^3 x_i^2\n\\end{array}\n\\lleq{bogglepollies}\nthe superpotential $W=\\sum p_i$ of which lives in\n\\begin{equation}\n\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}^{\\star}_{(5,5,6,6,6,4,4,4,8,8,8)}[16]_{-32}^{10}\n\\end{equation}\nand has an isolated singularity at the origin. All eleven variables\nare coupled and hence\nthis example appears to involve three fields with zero central charge\nin a nontrivial way.\n\nIt turns out, however, that the manifold (\\ref{boggle}) is equivalent\nto a manifold with nine variables. One way to see this is by making\n use of some topological identities introduced in \\cite{cdls}. First\nconsider the well--known isomorphism\n\\begin{equation}\n\\relax{\\rm I\\kern-.18em P}_2[2] \\equiv \\relax{\\rm I\\kern-.18em P}_1,\n\\end{equation}\nwhich allows us to rewrite the manifold above as\n\\begin{equation}\n\\matrix{\\relax{\\rm I\\kern-.18em P}_1 \\cr \\relax{\\rm I\\kern-.18em P}_2\\cr \\relax{\\rm I\\kern-.18em P}_2 \\cr \\relax{\\rm I\\kern-.18em P}_1\\cr}\n\\left[\\matrix{2 &0 &0\\cr\n 1 &2 &0\\cr\n 0 &1 &2\\cr\n 0 &0 &2\\cr}\\right].\n\\end{equation}\nUsing the surface identity \\cite{cdls}\n\\begin{equation}\n\\matrix{\\relax{\\rm I\\kern-.18em P}_1 \\cr \\relax{\\rm I\\kern-.18em P}_2\\cr} \\left[\\matrix{ 2\\cr 1\\cr}\\right]=\n\\matrix{\\relax{\\rm I\\kern-.18em P}_1 \\cr \\relax{\\rm I\\kern-.18em P}_1\\cr}\n\\end{equation}\napplied via the rule\n\\begin{equation}\n\\matrix{\\relax{\\rm I\\kern-.18em P}_1 \\cr \\relax{\\rm I\\kern-.18em P}_2\\cr X\\cr} \\left[\\matrix{ 2 &0\\cr\n 1 &a\\cr\n 0 &M\\cr}\\right]=\n\\matrix{\\relax{\\rm I\\kern-.18em P}_1 \\cr \\relax{\\rm I\\kern-.18em P}_1\\cr X\\cr} \\left[\\matrix{ a\\cr\n a\\cr\n M\\cr}\\right]\n\\end{equation}\nshows that this space is in turn equivalent to\n\\begin{equation}\n\\matrix{\\relax{\\rm I\\kern-.18em P}_1 \\cr \\relax{\\rm I\\kern-.18em P}_1\\cr \\relax{\\rm I\\kern-.18em P}_1 \\cr\\relax{\\rm I\\kern-.18em P}_2\\cr}\n\\left[\\matrix{2 &0 \\cr\n 2 &0 \\cr\n 0 &2 \\cr\n 1 &2 \\cr}\\right],\n\\lleq{nonlg}\na manifold with only nine homogeneous coordinates.\nIt should be noted that the LG potential of (\\ref{nonlg}), defined\nby the sum of the two polynomials, certainly does not\nhave an isolated singularity. Furthermore it is not possible to\n even assign weights to the fields such that the\ncentral charge comes out to be nine!\nIt is thus possible, by applying topological identities, to extend\nthe applicability of Landau--Ginzburg theories to types of Calabi--Yau\nmanifolds that were hitherto completely inaccessible by the standard\nformulation.\n\nFurther insight into the problem of redundancy in the construction of\nLG potentials can be gained by an LG theoretic analysis of this\nexample. From the weights of the scaling variables in the LG configuration\nabove it is clear that the spectrum of this LG configuration remains\nthe same if the last three coordinates are set to zero. In the\npotential\n\\begin{equation}\nW=\\sum_{i=1}^2 \\left(u_i^2v_i+v_i^2w_i+w_i^2x_i+x_i^2\\right) +\n (v_3^2w_3+w_3^2x_3+x_3^2)\n\\lleq{bogglelg}\n described by the CICY polynomials (\\ref{bogglepollies}), these variables\ncannot be set to zero because they are coupled to other fields; hence it\nseems impossible to reduce the number of fields. Consider however the\nfollowing change of variables\n\\begin{equation}\ny_i=x_i+\\frac{1}{2}w_i^2,~~i=1,2,3.\n\\end{equation}\n It follows from these transformations that the potential defined by\nby (\\ref{bogglelg}) is equivalent to\n\\begin{equation}\nW=\\sum_{i=1}^2\\left(u_i^2v_i+v_i^2w_i-\n \\frac{1}{4}w_i^4\\right)+v_3^2w_3-\\frac{1}{4}w_3^4.\n\\end{equation}\n Adding a trivial factor and splitting this potential into three\nseparate polynomials\n\\begin{equation}\np_1=u_1^2v_1+u_2^2v_2,~~~p_2=v_1^2w_1+v_2^2w_2+v_3^2w_3,~~~\np_3=-\\frac{1}{4}\\left(w_1^4+w_2^4+w_3^4\\right)+x^2\n\\end{equation}\nwe see that the original model is equivalent to a weighted\n configuration\n\\begin{equation}\n\\matrix{\\relax{\\rm I\\kern-.18em P}_1 \\hfill \\cr \\relax{\\rm I\\kern-.18em P}_2 \\hfill \\cr \\relax{\\rm I\\kern-.18em P}_{(1,1,1,2)}\\cr}\n\\left[\\matrix{2 &0 &0\\cr\n 1 &2 &0\\cr\n 0 &1 &4\\cr}\\right],\n\\end{equation}\nwhich again describes manifolds with nine variables.\nThus the original configuration (\\ref{boggle}) is in fact equivalent\nto two different (weighted) CICY representations\n\\begin{equation}\n\\matrix{\\relax{\\rm I\\kern-.18em P}_1 \\cr \\relax{\\rm I\\kern-.18em P}_1\\cr \\relax{\\rm I\\kern-.18em P}_1 \\cr\\relax{\\rm I\\kern-.18em P}_2\\cr}\n\\left[\\matrix{2 &0 \\cr\n 2 &0 \\cr\n 0 &2 \\cr\n 1 &2 \\cr}\\right]\n{}~~\\equiv ~~\n\\matrix{\\relax{\\rm I\\kern-.18em P}_1 \\cr \\relax{\\rm I\\kern-.18em P}_2\\cr \\relax{\\rm I\\kern-.18em P}_2 \\cr \\relax{\\rm I\\kern-.18em P}_2\\cr}\n\\left[\\matrix{2 &0 &0 &0\\cr\n 1 &2 &0 &0\\cr\n 0 &1 &2 &0\\cr\n 0 &0 &1 &2\\cr}\\right]\n{}~~\\equiv ~~\n\\matrix{\\relax{\\rm I\\kern-.18em P}_1 \\hfill \\cr \\relax{\\rm I\\kern-.18em P}_2 \\hfill \\cr \\relax{\\rm I\\kern-.18em P}_{(1,1,1,2)}\\cr}\n\\left[\\matrix{2 &0 &0\\cr\n 1 &2 &0\\cr\n 0 &1 &4\\cr}\\right].\n\\end{equation}\n\nTo summarize the last few paragraphs, we have shown two things: first\nthat by adding trivial factors and coupling them to the remaining fields\nwe can give a Landau--Ginzburg description of a\nlarger class of Calabi--Yau manifolds than previously thought\npossible. Furthermore we can use topological identities to obtain\nan LG formulation of CY manifolds, which do not admit a\ncanonical LG potential at all.\nIncidentally we have also shown that it is possible to relate complete\nintersection manifolds embedded in products of projective spaces to\n weighted complete intersection manifolds\nembedded in products of weighted projective space.\n\nSimilarly the number of fields can grow without bound if we not only\nallow\nfields that do not contribute to the central charge but also fields\nwith\na negative contribution. Again such fields provide redundant\ndescriptions\nof simpler LG theories; they are nevertheless important for the LG\/CY\nrelation and\noccur in the constructions of splitting and contraction introduced in\nref. \\cite{cdls}. Even though these\nconstructions were discussed in \\cite{cdls} only in the context of\nCalabi--Yau\nmanifolds embedded in products of ordinary projective spaces they\nreadily generalize to the more general framework of weighted projective\nspaces.\n\nIn special circumstances, the splitting or contraction process does not\nchange\nthe spectrum of the theory and hence it provides another tool to relate\n LG potentials with at most nine variables manifolds with more than\nnine\nhomogeneous coordinates. Consider e.g. the manifold embedded in\n\\begin{equation}\n\\matrix{\\relax{\\rm I\\kern-.18em P}_{(1,1)} \\hfill \\cr \\relax{\\rm I\\kern-.18em P}_{(1,1,1,1,3)}\\cr}\n\\left[\\matrix{2&0\\cr 1&6\\cr}\\right]\n_{-252}^{(2,128)}\n\\end{equation}\nwhich is described by the zero locus of the two polynomials\n\\begin{eqnarray}\np_1&=&x_1^2y_1+x_2^2y_2 \\nonumber \\\\\np_2&=&y_1^6+y_2^6+y_3^6+y_4^6+y_5^2\n\\end{eqnarray}\nthat can be described by a superpotential $W=p_1+p_2$ defining a\nLandau--Ginzburg theory in\n\\begin{equation}\n\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}^{\\star}_{(5,5,2,2,2,2,6)}[12]^{(2,128)}_{-252}.\n\\end{equation}\nThis manifold can be rewritten via an ineffective split in an infinite\nsequence of different representations as\n\\begin{equation}\n\\matrix{\\relax{\\rm I\\kern-.18em P}_{(1,1)} \\hfill \\cr \\relax{\\rm I\\kern-.18em P}_{(1,1,1,1,3)}\\cr}\n\\left[\\matrix{2&0\\cr 1&6\\cr}\\right]\n\\longrightarrow\n\\matrix{\\relax{\\rm I\\kern-.18em P}_{(1,1)}\\hfill\\cr \\relax{\\rm I\\kern-.18em P}_{(1,1)}\\hfill\\cr \\relax{\\rm I\\kern-.18em P}_{(1,1,1,1,3)}\\cr}\n\\left[\\matrix{2&0&0\\cr 1&1&0\\cr 0&1&6\\cr}\\right]\n\\longrightarrow\n\\matrix{\\relax{\\rm I\\kern-.18em P}_{(1,1)}\\hfill\\cr \\relax{\\rm I\\kern-.18em P}_{(1,1)}\\hfill\\cr\\relax{\\rm I\\kern-.18em P}_{(1,1)}\\hfill \\cr\n \\relax{\\rm I\\kern-.18em P}_{(1,1,1,1,3)}\\cr}\n\\left[\\matrix{2&0&0&0\\cr 1&1&0&0\\cr 0&1&1&0\\cr 0&0&1&6}\\right]\n\\longrightarrow\n\\cdots\n\\end{equation}\nwhich are described by LG potentials in the alternating classes\n\\begin{equation}\n\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}^{\\star 7}_{(5,5,2,2,2,2,6)}[12] \\longrightarrow\n\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}^{\\star 9}_{(1,1,10,10,2,2,2,2,6)}[12] \\longrightarrow\n\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}^{\\star 11}_{(5,5,2,2,10,10,2,2,2,2,6)}[12] \\longrightarrow \\cdots\n\\end{equation}\ni.e. the infinite sequence belongs to the configurations\n\\begin{equation}\n\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}^{\\star 7+4k}_{(5,5,2,2,10,10,2,2,10,10,...,2,2,10,10,2,2,6)}[12]\n\\end{equation}\nwhere the part $(2,2,10,10)$ occurs $k$ times, and\n\\begin{equation}\n\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}^{\\star 5+4k}_{(1,1,10,10,2,2,10,10,2,2,...10,10,2,2,2,2,6)}[12].\n\\end{equation}\nwhere $(10,10,2,2)$ occurs $k$ times.\n\nThe construction above easily generalizes to a number of examples\nwhich all belong to a class of spaces discussed in ref. \\cite{s3}.\nConsider manifolds embedded in\n\\begin{equation}\n\\matrix{\\relax{\\rm I\\kern-.18em P}_{(1,1)} \\hfill\\cr \\relax{\\rm I\\kern-.18em P}_{(k_1,k_1,k_3,k_4,k_5)}\\cr}\n\\left[\\matrix{2&0\\cr k_1&k\\cr}\\right],\n\\end{equation}\nwhere $k=k_1+k_3+k_4+k_5$. These spaces can be split\ninto the infinite sequences\n\\begin{equation}\n\\matrix{\\relax{\\rm I\\kern-.18em P}_{(1,1)} \\hfill \\cr \\relax{\\rm I\\kern-.18em P}_{(k_1,k_1,k_3,k_4,k_5)}\\cr}\n\\left[\\matrix{2&0\\cr k_1&k\\cr}\\right]\n\\longrightarrow\n\\matrix{\\relax{\\rm I\\kern-.18em P}_{(1,1)} \\hfill \\cr \\relax{\\rm I\\kern-.18em P}_{(1,1)}\\hfill \\cr\n \\relax{\\rm I\\kern-.18em P}_{(k_1,k_1,k_3,k_4,k_5)}\\cr}\n\\left[\\matrix{2&0&0\\cr 1&1&0\\cr 0&k_1&k\\cr}\\right]\n\\longrightarrow\n\\matrix{\\relax{\\rm I\\kern-.18em P}_{(1,1)} \\hfill \\cr \\relax{\\rm I\\kern-.18em P}_{(1,1)}\\hfill\\cr\\relax{\\rm I\\kern-.18em P}_{(1,1)}\\hfill \\cr\n \\relax{\\rm I\\kern-.18em P}_{(k_1,k_1,k_3,k_4,k_5)}\\cr}\n\\left[\\matrix{2&0&0&0\\cr 1&1&0&0\\cr 0&1&1&0\\cr 0&0&k_1&k}\\right]\n\\longrightarrow\n\\cdots\n\\end{equation}\nIf the weights are such that $k\/k_i$ is an integer,\nthen it is easy to write down the tensor model that corresponds to\nit (but this is not essential).\nIn such models the levels $l_i$ of the tensor model\n $l_1^2\\cdot l_3\\cdot l_4\\cdot l_5$\nin terms of the weights are given by\n\\begin{equation}\nl_1=l_2=\\frac{2k}{k_1}-2,~~~l_i=\\frac{k}{k_i}-2,~i=3,4,5\n\\end{equation}\nand the corresponding LG potentials live in\n\\begin{eqnarray}\n\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}^{\\star 7}_{(k-k_1,k-k_1,2k_1,2k_1,2k_3,2k_4,2k_5)}[2k]\n\\longrightarrow\n\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}^{\\star 9}_{(k_1,k_1,2(k-k_1),2(k-k_1),2k_1,2k_1,2k_3,2k_4,2k_5)}[2k]\n\\longrightarrow \\cdots\n\\end{eqnarray}\ni.e. they belong to the sequences\n\\begin{equation}\n\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}^{\\star 7+4p}_{((k-k_1),(k-k_1),2k_1,2k_1,2(k-k_1),2(k-k_1),....,\n 2k_1,2k_1,2k_3,2k_4,2k_5)}[2k]\n\\end{equation}\nwhere the part $(2k_1,2k_1,2(k-k_1),2(k-k_1))$ occurs $p$ times, and\n\\begin{equation}\n\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}^{\\star 9+4k}_{(k_1,k_1,2(k-k_1),2(k-k_1),\n2k_1,2k_1,...,2k_3,2k_4,2k_5)}[2k].\n\\end{equation}\nwhere $(2(k-k_1),2(k-k_1),2k_1,2k_1)$ occurs $p$ times.\n\nAll these models are constructed in such a way that they have central\ncharge\nnine, but in contrast to the example discussed previously,\nthere now appear fields with negative central charge.\nIn the case at hand, however, these dangerous fields\nonly occur in a {\\it coupled} subpart of the theory; the smallest\nsubsystem which involves these fields and which one can isolate is in\nfact a theory with positive central charge. In the series of splits\njust described e.g., the fields with negative central charge that occur\nin the\nfirst split always appear in the subsystem described by the\nconfigurations\n\\begin{equation}\n\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}^{\\star}_{(k_1,2(k-k_1),2k_1)}[2k]\n\\end{equation}\nwith potentials of the form\n\\begin{equation}\nx^2y +yz+z^{2k\/k_1}.\n\\end{equation}\nThus the contribution to the central charge of this sector becomes\n\\begin{equation}\nc=3\\left[\\left(1-\\frac{k_1}{k}\\right)+\\left(1-\\frac{2(k-k_1)}{k}\\right)+\n \\left(1-\\frac{2k_1}{k}\\right)\\right],\n\\lleq{ccc}\nwhich is always positive. This formula suggests that it ought\nto be possible to dispense with the variables $y,z$ altogether, as\ntheir total contribution to the central charge adds up to zero, and\nthat this theory is equivalent to that of a single monomial of degree\n$k\/k_1$.\n\nMore generally one may consider the Landau--Ginzburg theory defined by\nthe potential\n\\begin{equation}\nx_1^ax_2+x_2x_3+\\cdots + x_{n-1}x_n + x_n^b.\n\\lleq{triv}\n From the central charge\n\\begin{equation}\nc= \\left\\{ \\begin{array}{l l}\n 6\\left(1-\\frac{1}{a}\\right)\\left(1-\\frac{1}{b}\\right),\n & n~{\\rm even} \\\\\n 3\\left(1-\\frac{2}{ab}\\right), &n~{\\rm odd}\n \\end{array} \\right\\},\n\\end{equation} as well as from the dimension of the chiral ring\n\\begin{equation}\n{\\rm dim}~R_n = \\prod \\left(\\frac{1}{q_i}-1\\right)\n = \\left\\{ \\begin{array}{l l}\n ab-b-1,&n~{\\rm even} \\\\\n ab-1, &n~{\\rm odd}\n \\end{array} \\right\\},\n\\end{equation}\nwe expect this theory to be equivalent to\n\\begin{equation}\nx_1^ax_2+x_2x_3+\\cdots x_{n-1}x_n + x_n^b =\n\\left\\{ \\begin{array}{l l}\n x_1^ax_n+x_n^b, &n~{\\rm even} \\\\\n x_1^{ab}, &n~{\\rm odd}\n \\end{array} \\right\\}.\n\\lleq{ident}\n\nSuch relations are supported by the identification of rather different\n LG configurations such as\n\\begin{equation}\n\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}^{\\star}_{(1,1,1,6,6,6,3,3,3,3,3)}[9] \\equiv \\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}^{\\star}_{(1,1,1,3,3)}[9]\n\\end{equation}\nas well as many other identifications which are rather nontrivial\nin the context of the associated manifolds.\n\nIt follows from the above considerations that we have to assume,\n in order to avoid redundant\nreconstructions of LG theories, that the central charge\nof all scaling fields of the potential should be positive.\n In order to relate the potentials to manifolds, we may then add\none or several trivial factors or more complicated theories with zero\ncentral charge.\n\nUsing the above results we derive in the following more detailed\nfiniteness conditions. Observe first that from eq. (\\ref{cc})\nwritten as\n\\begin{equation}\n\\sum_{i=1}^r q_i=\\left( {r\\over 2}-{c\\over 6}\\right):=\\hat c\\label{cbed}\n\\end{equation}\nwe obtain $r>c\/3$.\n\nNow let $p$ be a polynomial of degree $d$ in $r$ variables.\nFor the index set $\\cI_1$ the conditions for transversality\nimply the existence of $n_i\\in \\relax{\\rm I\\kern-.18em N}^+$, $i=1,\\ldots,r$ and of a map\n$j:\\cI_r \\rightarrow \\cI_r$ such that for all $i$ there exists\n$j(i)$ such that\n\\begin{equation}\nq_i={{1-q_{j(i)}}\\over n_i}.\n\\lleq{qs1}\nLet us first see how many non-trivial fields can occur at most.\nFields which have charge $q_i\\le 1\/3$\ncontribute $c_i\\ge 1$ to the conformal anomaly.\nNow consider fields with larger charge. Since we assume $c_i>0$, they are\nin the range $1\/3<q_i<1\/2$.\nAmong these fields the transversality condition (1.) cannot hold,\nbecause two of them are not enough and three of them are too many\nfields in order to form a monomial of charge one. Transversality\ncondition (2.) implies that each of them has to occur together with a\npartner field $z_{j(i)}$. These pairs contribute to the conformal\nanomaly according to\n(\\ref{qs1},\\ref{cc}) $c_i+c_{j(i)}>2$, so we\ncan conclude that $r\\le c$.\n\nIn order to construct all transversal\nLG potentials for a given $c$,\nwe choose a specific $r$ in the range obtained above and consider\nall possible maps $j$ of which there are $r^r$.\nWithout restricting on the generality, we may\n{\\sl then} assume the $n_i$ to be ordered\n$n_1\\le \\cdots \\le n_r$.\nStarting with (\\ref{cbed}) we obtain via eq. (\\ref{qs1}) and the\npositivity of the charges a bound $n_1<r\/c$. Now we choose $n_1$ in the\nallowed range and use (\\ref{qs1}) in order to eliminate the $q_1$, if\nnecessary in favour of the $q_2,\\ldots,q_r$. This yields\nan equation of the general form $(p=2)$\n\\begin{equation}\n\\sum_{i=p}^r w^{(p)}_i q_i=\\hat c^{(p)}.\\label{cbed2}\n\\end{equation}\nIf $\\hat c^{(p)}\\neq 0$, eq. (\\ref{cbed2}) either is in contradiction\nwith the positivity of the charges or one can derive a finite bound for\n$n_{p}$. Assume $\\hat c^{(p)}>0$ and let $\\cI_+$ be the\nindices of the positive $w_i^{(p)}$; then one has\n$n_{p}<(\\sum_{i\\in \\cI_+} w_i^{(p)})\/\\hat c^{(p)}$; likewise if\n$\\hat c^{(p)}<0$ we have\n$n_{p}<(\\sum_{i\\in\\cI_-} w_i^{(p)})\/\\hat c^{(p)}$.\n\nConsider the case $\\hat c^{(p)}=0$.\nIf the $w_i^{(p)}$ are indefinite\nwe get no bound from (\\ref{cbed2}). However we will show that the\nexistence of certain monomials, which are required by the transversality\nconditions, implies a bound for $n_p$.\nLet $\\cI_a$ denote the indices of the already bounded\n$n_i$ and $\\cI_b$ the others. The charge of the field $z_a$ with\n$a\\in \\cI_a$\nwill depend on the unknown charge of a field $z_{b(a)}$ with\n$b\\in \\cI_b$ if\nthere is a chain of indices\n$a_0=a,a_1=j(a),\\dots,a_l=j(\\ldots j(a)\\ldots)=:b(a)$ linked by the\nmap $j$. The charge of $z_a$ is given by\n\\begin{equation}\nq_a={1\\over n_a}-{1\\over n_a n_{a_1}}+\n\\cdots -{(-1)^l\\over n_a\\ldots n_{l-1}}+\n{(-1)^l q_{b(a)}\\over n_a\\ldots n_{l-1}}.\n\\lleq{qform}\nIndefiniteness of the $w_i^{(p)}$ can only occur if there are fields\n$z_a$, $a\\in \\cI_a$, with odd $l$, i.e. the last term in (\\ref{qform})\n$s_a q_{b(a)}:=(-1)^l\/(n_a\\ldots n_{a_{l-1}} q_{b(a)})$ is negative.\nCall the index set of these fields $\\cI_a^-$.\nAssume first that the transversality condition (1.) holds.\nThis implies the existence of $m_i\\in \\relax{\\rm I\\kern-.18em N}^+$ ($m_i<2 n_i$)\nsuch that $\\sum_{i\\in \\cI^-_a}\nm_i q_i=1$. For the unknown $q_i$, $i\\in \\cI_b$, we get an equation\nof the form $\\sum w_i q_i=\\epsilon$, which yields a bound for the lowest\n$n_i$, $i\\in \\cI_b$, since $w_i>0$.\nThe lowest possible value for $\\epsilon>0$ can be readily calculated from\nthe denominators occurring in (\\ref{qform}).\nIf transversality condition (2.) applies, we have $|\\cI_a^-|$ equations\nof the form $\\sum_{i\\in \\cI_a^-} m^{(j)}_i q_i=1-q_{e_j}$ which can be\nrewritten as $\\sum_{i\\in \\cI_b} w_i^{(j)} q_i=\\epsilon^{(j)}$.\nOnly if all $w_i^{(j)}$ happen to be indefinite and all\n$\\epsilon^{(j)}$ are zero we get {\\sl no bound} from this condition.\n Assuming this to be true we have\n\\begin{equation}\n\\sum_{i\\in \\cI_b} m^{(j)} s_i q_i-s_{e_j} q_{b(e_j)}=0,\n\\lleq{g1}\nwhere $s_{e_j}:=1$ and $b(e_j):=b$ if $e_j\\in \\cI_b$. Note that\n$\\sum_i m^{(j)}_i\\ge 2$ in order to avoid quadratic mass terms.\nNow we can rewrite (\\ref{cbed2}) in the form\n\\begin{equation}\n\\sum s_i q_{b(i)}=0.\n\\lleq{g2}\nIf one uses now (\\ref{g1}) and $\\sum_i m^{(j)}_i\\ge 2$\nin order to eliminate the negative $s_i$, one finds\n $\\sum_i w_i q_i\\le 0$ with $w_i>0$, which is in contradiction with\nthe positivity of the charges, hence we get a bound in any case.\n\nThis procedure of restricting the bound for $n_{p}$, given\n$n_i,\\dots,n_{p-1}$,\nwas implemented in a computer program. It allows all\nconfigurations to be found without testing unnecessarily many combinations of\nthe $n_i$.\nThe actual upper bounds for the $(n_i,\\dots,n_r)$\nin the four--variable case are $(7,17,83,1805)$, and we have found $2390$\nconfigurations which allow for transversal polynomials.\nIn the five--variable case the bounds are $(4,6,14,62,923)$ and\n$5165$ configurations exist.\nBy adding a trivial mass term $z_5^2$ in the four--variable case, the\nconfigurations mentioned so far lead to three--dimensional\nCalabi--Yau manifolds described by a one polynomial constraint\nin a four--dimensional weighted projective space.\n\nThe same figures for the six--variable case and seven--variable case are\n$(3,3,5,11,41,482)$, $2567$ and $(2,2,3,4,8,26,242)$, $669$ respectively,\nleading to a total of $3236$ combinations.\nLikewise for eight-- and nine--variable potentials the bounds\nbecome $(2,2,2,2,2,3,3,5,14)$ with $47$ examples and $(2,2,2,2,2,2,2,2,2)$\nwith\n$1$ example respectively. The lists with all models can be found in\n\\cite{ks}.\n\n\n\\vskip .1truein\n\\noindent\n\\section{Results and Comparisons}\n\n\\noindent\nWe have constructed 10,839 Landau--Ginzburg theories with\n2997 different spectra, i.e. pairs of generations and antigenerations.\nThe massless spectrum is very rough information about a theory and\nit is likely that the degeneracy is lifted to a large degree when\nadditional information, such as the number of singlets and\/or the\nYukawa couplings, becomes available. We expect the situation to be\nvery similar to the class of CICYs \\cite{cdls}, which only leads to\nsome 250\ndifferent spectra \\cite{ghl}, but for which a detailed analysis of the\nYukawa couplings \\cite{ch} shows that it contains several thousand\ndistinct theories.\n\nIt is clear however that there is in fact some redundancy in this\nclass of Landau--Ginzburg theories even beyond the one discussed in\nthe previous sections. In the list there appear, for instance,\n two theories with\nspectrum $(h^{(1,1)},h^{(2,1)},\\chi)=(2,122,-240)$ involving five variables\n\\begin{equation}\n\\relax{\\rm I\\kern-.18em P}_{(2,2,2,1,7)}[14] \\ni \\{z_1^7+z_2^7+z_3^7+z_4^{14}+z_5^2=0\\}\n\\end{equation}\nand\n\\begin{equation}\n\\relax{\\rm I\\kern-.18em P}_{(1,1,1,1,3)}[7] \\ni \\{y_1^7+y_2^7+y_3^7+y_4^7+y_4y_5^2=0\\}.\n\\end{equation}\nUsing the fractional transformations introduced in ref. \\cite{ls4}\nit is\neasy to show that these two models are equivalent, even though\nthis is not obvious by just looking at the potentials. To see this\nconsider first the orbifold\n\\begin{equation}\n\\relax{\\rm I\\kern-.18em P}_{(2,2,2,1,7)}[14] {\\large \/} \\relax{\\sf Z\\kern-.4em Z}_2: [~0~0~0~1~1]\n\\end{equation}\nof the first model where the action indicated by\n$\\relax{\\sf Z\\kern-.4em Z}_2:~[~0~0~0~1~1] $\nmeans that the first three coordinates remain invariant, whereas the\nlatter\ntwo variables are to be multiplied with the generator of the cyclic\n$\\relax{\\sf Z\\kern-.4em Z}_2$, $\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma=-1$.\nSince the action of this $\\relax{\\sf Z\\kern-.4em Z}_2$ on the weighted projective space\nis part\nof the projective equivalence, the orbifold is of course isomorphic\nto the\noriginal model. On the other hand the fractional transformation\n \\begin{equation}\nz_i=y_i,~i=1,2,3;~~~ z_4=y_4^{1\/2},~~~ z_5=y_4^{1\/2}y_5\n\\end{equation}\nassociated with this symmetry \\cite{ls4} defines a 1--1 coordinate\ntransformation on the orbifold, which transforms the first theory into\nthe second; these are therefore equivalent as well.\n\nSimilarly the equivalences\n\\begin{eqnarray}\n\\relax{\\rm I\\kern-.18em P}_{(2,2,2,3,9)}[18]_{-216}^{(4,112)}&=&\\relax{\\rm I\\kern-.18em P}_{(1,1,1,3,3)}[9] \\nonumber\n\\\\\n\\relax{\\rm I\\kern-.18em P}_{(2,6,6,7,21)}[42]_{-96}^{(17,65)}&=&\\relax{\\rm I\\kern-.18em P}_{(1,3,3,7,7)}[21]\\nonumber\n\\\\\n\\relax{\\rm I\\kern-.18em P}_{(2,5,14,14,35)}[70]_{-64}^{(27,59)} &=&\\relax{\\rm I\\kern-.18em P}_{(1,5,7,7,15)}[35]\n\\end{eqnarray}\ncan be shown, as well as a number of others.\n\nIt should be noted that even though we now have constructed\nLandau--Ginzburg potentials with an arbitrary number of scaling fields,\nthe basic range of the spectrum has not changed as compared with the\nresults of \\cite{cls} where it was found that the spectra of all\n6000 odd\ntheories constructed there lead to Euler numbers which fall in the\nrange\n\\fnote{6}{This should be compared with the result for the complete\n intersection Calabi--Yau manifolds where\n\\break $-200 \\leq \\chi \\leq 0$ \\cite{cdls}. See Figure 1.}\n\\begin{equation}\n-960 \\leq \\chi \\leq 960.\n\\end{equation}\nIn fact not only\ndo all the LG spectra fall into this range, all known Calabi--Yau\nspectra\nand all the spectra from exactly solvable tensor models are contained\nin this range as well! This suggests that perhaps the spectra of all\nstring vacua based on $c=9$\nwill be found within this range. To put it differently, we conjecture that\nthe Euler numbers of all Calabi--Yau manifolds are contained in the\nrange $-960 \\leq \\chi \\leq 960$.\n\nSimilarly to the results in \\cite{cls}, the Hodge pairs do not pair up\ncompletely.\nIn fact the mirror symmetry of the space of Landau--Ginzburg vacua is just\nabout $77\\%$.\nIt thus appears that orbifolding is an essential ingredient\nin the construction of a mirror--symmetric slice of the configuration space\nof the heterotic string. It is in fact easy to produce examples of\norbifolds whose spectrum does not appear in our list of LG vacua.\nAn example of a mirror pair is furnished by the orbifold\n\\begin{equation}\n\\relax{\\rm I\\kern-.18em P}_{(1,1,1,1,2)}[6]^{(1,103)}_{-204} {\\LARGE \/} \\relax{\\sf Z\\kern-.4em Z}_6:\n\\left[\\matrix{3&2&1&0&0\\cr}\\right]\n\\end{equation}\nwhich has the spectrum $(11,23,-24)$ and the space\n\\begin{equation}\n\\relax{\\rm I\\kern-.18em P}_{(1,1,1,1,2)}[6]^{(1,103)}_{-204} {\\LARGE \/} \\relax{\\sf Z\\kern-.4em Z}_6\\times \\relax{\\sf Z\\kern-.4em Z}_3 :\n\\left[\\matrix{3&2&1&0&0\\cr\n 0&0&1&0&2\\cr}\\right]\n\\end{equation}\nwhich has the mirror--flipped spectrum $(23,11,24)$.\nThe space of LG orbifolds that has been constructed so far \\cite{kss}\nis indeed much closer to being mirror--symmetric than the space of\nLG theories itself. Even though the construction in \\cite{kss} is\nincomplete, about 94\\% of the Hodge numbers pair up.\n\nBy now there exist many different prescriptions to construct string\nvacua and we would like to put the LG framework into the context of\nother left--right--symmetric constructions.\nAmong the more prominent ones other than Calabi--Yau manifolds, which\nare obviously closely related to LG theories, there are constructions which\nhave traditionally been called, somewhat misleadingly,\norbifolds (what is meant are\norbifolds of tori) \\cite{dhvw}, free--fermion constructions\n\\cite{freeferm}, lattice constructions \\cite{lls}, and interacting\nexactly solvable models \\cite{g}\\cite{kasu}.\n\nNone of these classes are known completely, even though much effort\nhas gone into the exploration of some of them. Because powerful\ncomputational tools are available, toroidal orbifolds have been analysed\nin some detail \\cite{orbis}\\cite{orbiyuk} and much attention has\nfocused on the explicit construction of exactly solvable theories in\nthe\ncontext of tensor models via $N=2$ superconformal minimal theories\n\\cite{ls12}\\cite{ls3}\\cite{fkss1} and Kazama--Suzuki models\n\\cite{fiq2}\\cite{lvwg}\\cite{ls5}\\cite{s}\\cite{fksv}\n\\cite{aafn} as well\nas their in depth analysis \\cite{lvwg}\\cite{s2}\\cite{fks}\\cite{exyuk}.\nIn the context of\n(2,2) vacua, orbifolds lead only to some tens of distinct models, whereas\nthe known\nclasses of exactly solvable theories lead only to a few hundred models\nwith distinct spectra. Similar results have been obtained so far via the\ncovariant--lattice approach \\cite{lsw} and hence it is obvious that these\nconstructions do\nnot exhaust the configuration space of heterotic string by far. The\nclass of Landau--Ginzburg string vacua thus appears as a rather\nextensive source of (2,2)--symmetric models.\n\nFinally we should remark that the (2,2) Landau--Ginzburg theories\nwe have constructed here can be used to build a probably much\nlarger class\nof (2,0) models along the lines described in \\cite{dg}, using an\nappropriate adaptation of the work of \\cite{cdgp}\\cite{dave}\n\\cite{klth}\\cite{anamaria} in order to determine the instantons\non which the existence of a certain split of a vector bundle\nhas to be checked.\n\n\n\\vfill \\eject\n\n\\part{ Mirror Pairs via Fractional Transformations.}\n\nEven though the emerging mirror symmetry of the construction discussed in\nPart I is clearly very suggestive there is a priori no reason why models\nwith mirror flipped spectra should be related at all. After all,\nHodge numbers do not classify manifolds topologically and knowing\nthe spectrum of a physical theory is only a very first step in the analysis\nof its properties. It turns out that the detailed analysis of the models\nrests both mathematically and physically on the computation of the Yukawa\ncouplings. This, in general, is a difficult business. It is therefore\ngratifying that it is possible to proceed via a different line of thought\nto relate different models, namely via a direct map between different\ntheories. In particular cases this rather general map turns out to be the\nmirror map.\n\n\\vskip .1truein\n\\noindent\n\\section{ From A--Models to D--Models via Orbit Construction}\n\n\\noindent\nIt is well known from the ADE--classification of partition functions of\n($N=2$)minimal models that the ${\\rm D}$--type models are equivalent to\n${\\rm A}$--type models modded out by a $\\relax{\\sf Z\\kern-.4em Z}_2$ discrete symmetry.\nAs a consequence subsets of vacua of the Heterotic String involving the\n$N=2$ minimal series are also related via orbifold constructions\n\\fnote {3}{In this article orbifold construction will be understood\n to include twisted modes in the conformal field theory or\n Landau--Ginzburg construction and blow--up modes in the\n geometric formulation.}.\nIn terms of the corresponding Landau-Ginzburg potentials of the exact\nmodels\nthis can be seen as follows. The potential corresponding to a model\nat level\n$k$ with a diagonal affine invariant is \\cite{kms}\n\\begin{equation}\n{\\rm A}_{k+1}~~~\\sim~~~\\Phi_1^{k+2} + \\Phi_2^2 \\lleq{diag}\nwhere, with hindsight, a trivial factor has been added.\nThe LG potential corresponding to the exact model at the same level\n\\fnote{4}{The ${\\rm D}$--models only exist for even level $k=2n$.}\nbut with the ${\\rm D}$--invariant is described by \\cite{m}\\cite{vw}\n\n\\begin{equation}\n{\\rm D}_{\\frac{k}{2}+2}~~\\sim~~~{\\tilde {\\Phi}}_1^{(k+2)\/2} +\n \\tilde {\\Phi}_1 \\tilde {\\Phi}_2^2. \\lleq{daff}\nBy defining the transformation of the scaling variables\n\\begin{equation}\n\\Phi_1 = \\tilde {\\Phi}_1^{1\/2}~~~ {\\rm and}~~~\\Phi_2 = \\tilde {\\Phi}_1^{1\/2}\\tilde {\\Phi}_2\n\\lleq{fraccy}\none can map the diagonal theory of (\\ref{diag}) into the nondiagonal\nmodel of\n(\\ref{daff}).\nMoreover this transformation has a constant Jacobian and therefore one\nmight naively expect that\nthey are in fact equivalent descriptions of a given theory since\ntheir path integrals are equivalent. It is clear however that this is not\nthe case as the two theories have a different spectrum:\nthe local ring of the diagonal theory consists of states\n\\begin{equation}\n{\\cal R}} \\def\\cV{{\\cal V}_{{\\rm A}_{k+1}}=\\{1,\\Phi_1, \\Phi_1^2,\\cdots,\\Phi_1^k\\}\n\\end{equation}\ni.e. the theory has $k+1$ states. The ${\\rm D}$--theory however has only\n$\\frac{k}{2}+2$ states in its spectrum\n\\begin{equation}\n{\\cal R}} \\def\\cV{{\\cal V}_{{\\rm D}_{\\frac{k}{2}+2}}=\n\\{1,\\tilde {\\Phi}_2,\\tilde {\\Phi}_1,\\cdots,\\tilde {\\Phi}_1^{\\frac{k}{2}}\\}.\n\\end{equation}\nHence these two theories are not the same. The resolution of this puzzle\ncomes\nfrom the fact that the transformation of (\\ref{fraccy}) is not a coordinate\ntransformation since it is not a bijection but is 2--1 as it stands.\nTo make it 1--1 we should identify\n\\begin{equation}\n\\Phi_i \\sim -\\Phi_i,~~~i=1,2.\n\\end{equation}\nOf the $k+1$ states of ${\\rm A}_{k+1}$--models $\\frac{k}{2}+1$ are\ninvariant with respect to the action of this $\\relax{\\sf Z\\kern-.4em Z}_2$. By including the one\ntwisted state we find precisely the states of the non--diagonal\n${\\rm D}$--theory.\n\nThis construction can immediately be applied to string compactification\nproper.\nIt may seem unlikely at first that the simple modding of a $\\relax{\\sf Z\\kern-.4em Z}_2$ should\naccount for the different behaviour that the various tensor models\nexhibit under the exchange\nof a diagonal invariant by a D--invariant. A quick look at the results\nof refs.\n\\cite{ls3}\\cite{fkss1} shows that this exchange generically changes the\nspectrum\n in some\narbitrary way without any obvious systematics. However in some special\ncases\nthe spectrum is flipped, exchanging generations and anti--generations while\nin other models the spectrum does not change at all. The reason for this\nerratic behaviour can be traced to the fact that the other coordinates\ninvolved in the construction determine the fixed point structure of the\ndiscrete group in an essential way.\n\nConsider the following example where the exchange of the affine\ninvariant does\nnot change the spectrum\n\\fnote{5}{The notation of ref. \\cite{ls3} is used.}\n\\begin{equation}\n\\relax{\\rm I\\kern-.18em P}_{(1,7,2,2,2)}[14]~~\\ni~~(12\\cdot 5^3)_{A_{13}\\otimes A_6^3}~~\n\\longrightarrow\n{}~~(12\\cdot 5^3)_{D_{8}\\otimes A_6^3}~~\\in~~\\relax{\\rm I\\kern-.18em P}_{(1,3,1,1,1)}[7].\n\\end{equation}\nwhere we have added one trivial factor\n\\fnote{6}{An explanation for the necessity of this trivial\n factor and when it is to be added can be found in\n \\cite{ls3}.}.\nThe reason that the spectrum does not change after the replacement of the\ndiagonal affine invariant by the D--invariant is rather obscure from the\npoint\nof view of the conformal field theory described by the tensor model\n$(12\\cdot 5^3)$ with different affine invariants but can be understood\neasily\nfrom the orbifold point of view. In the manifold picture it simply follows\nfrom the fact that the $\\relax{\\sf Z\\kern-.4em Z}_2$--action $[1,1,0,0,0]$, generated by\n\\begin{equation}\n(\\Phi_1,\\Phi_2,\\Phi_3,\\Phi_4,\\Phi_5)\\longrightarrow\n(\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma \\Phi_1,\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma \\Phi_2,\\Phi_3,\\Phi_4,\\Phi_5),\n\\end{equation}\n(where $\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma^2=1, \\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma\\neq1$) is not an action at all on the Calabi--Yau\nmanifold embedded in the ambient\nweighted projective space but it is part of the projective equivalence\ntransformation because we have\n\\begin{equation}\n\\relax{\\rm I\\kern-.18em P}_{(1,7,2,2,2)} = \\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_5\/\\sim\n\\end{equation}\nwith $(\\Phi_1,\\Phi_2,\\Phi_3,\\Phi_4,\\Phi_5)\\sim\n(\\l \\Phi_1,\\l^7 \\Phi_2,\\l^2 \\Phi_3,\\l^2 \\Phi_4,\\l^2 \\Phi_5)$,\nwhere $\\l\\in \\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}^*$. Therefore the $\\relax{\\sf Z\\kern-.4em Z}_2$ does not transform the\nprojective\ncoordinates at all but is trivial. In the conformal field theory picture\nthis translates into the fact that the action is actually part of the\nmodding that has to be done to implement the GSO projection. This example\nshows that not all our models are independent but that some models appear\nin different representations.\n\nA more complicated example is furnished by the pair of minimal models\n\\begin{equation}\n\\relax{\\rm I\\kern-.18em P}_{(1,5,1,1,2)}[10]^1_{-288}~~=~~(8^3\\cdot 3)_{A_{9}^3\\otimes A_4}\n\\longrightarrow\n(8^3\\cdot 3)_{D_{6}\\otimes A_9^2\\otimes\nA_4}~~=~~\\relax{\\rm I\\kern-.18em P}_{(2,4,1,1,2)}[10]^3_{-192}.\n\\lleq{minmod}\nThe superscript denotes the dimension of the\nsecond cohomology group $h^{(1,1)}={\\rm dim}~{\\rm H}^{(1,1)}(\\cM)$ whereas\nthe subscript denotes the Euler number. It is again not obvious from the\npoint of view of the affine invariants involved why the spectrum changes the\nway it does but the result is clear from the orbifold construction. Defining\nthe $\\relax{\\sf Z\\kern-.4em Z}_2$--action as $[1,1,0,0,0]$ it follows that in this case the map is\nnontrivial. Its fixed point set consists of two curves\n$C_1=\\relax{\\rm I\\kern-.18em P}_{(1,1,2)}[10]$ with Euler number $\\chi_{C_1}=-30$ and\n$C_2=\\relax{\\rm I\\kern-.18em P}_{(1,5,2)}[10]$ with $\\chi_{C_2}=-2$. The Euler number of the\nresolved orbit manifold is therefore\n\\begin{equation}\n\\chi=-\\frac{288}{2}-\\frac{(-30-2)}{2}+2(-30-2)=-192.\n\\end{equation}\nThe cohomology can be computed as well. Given the Euler number we only need\nto compute either the second cohomology group or the number of (2,1)--forms.\nUsing the results of \\cite{s3} it is clear that each of the curves\ncontributes\none additional generator to the second cohomology, i.e. for the resolved\nmanifold we have $h^{(1,1)}=3$ and therefore the result of (\\ref{minmod}).\n\nThese simple identifications of a class of Landau-Ginzburg potentials\nwith orbifolds of other potentials via fractional transformations can be\nconsidered as a generalization\nof the strange duality known from the exceptional singularities\nof modality one \\cite{agzv}. One of the `strangely dual pairs' appearing\nin the\nclassification of Landau--Ginzburg potentials in 3 variables with an\nisolated singularity at the origin is described by the two polynomials\n\\begin{eqnarray}\nK_{14}~&:&\\relax{\\rm I\\kern-.18em P}_{(3,12,8)}[24]~\\ni~ \\{\\Phi_1^8+\\Phi_2^2+\\Phi_3^3=0\\}\\\\\nQ_{10}~&:&\\relax{\\rm I\\kern-.18em P}_{(6,9,8)}[24]~\\ni~\\{\\tilde {\\Phi}_1^4+\\tilde {\\Phi}_1\\tilde {\\Phi}_2^2+\\tilde {\\Phi}_3^3=0\\}.\n\\end{eqnarray}\nThe dimension of the chiral ring of these singularities is 10 and 14,\nrespectively, as indicated by the subscript of $Q$ and $K$. Each of these\ncatastrophes is characterized by two triplets of numbers, the Dolgachev\nnumbers\nand the Gabrielov numbers. For the above singularity $Q_{10}$ the\ncorresponding pair of triplets is $\\cD(Q_{10})=(2,3,9)$ and\n$\\cG(Q_{10})=(3,3,4)$ for the Dolgachev and Gabrielov numbers\nrespectively whereas the singularity $K_{14}$ leads to the triplets\n$(3,3,4)$ and $(2,3,9)$ For the above pair $\\cD(Q_{10})=\\cG(K_{14})$ and\n$\\cG(Q_{10})=\\cD(K_{14})$. It is in this sense that the two polynomials are\ncalled dual\n\\fnote{7}{The precise nature of these numbers are not of\n concern here. More details can be found in\n ref. \\cite{agzv}}.\n\n{}From our previous results it is clear that an alternative way to relate\nthese two singularities flows from the description of affine\n${\\rm D}$--invariants as orbifolds of diagonal invariants.\nThe above polynomials can be viewed as the Landau--Ginzburg potentials\nof the tensor product $(1\\cdot 6)$\n\\begin{equation}\nQ_{10}\\sim (1\\cdot 6)_{A_2\\otimes D_5}~~{\\rm and}~~\nK_{14}\\sim (1\\cdot 6)_{A_2\\otimes A_7}\n\\end{equation}\nwhere we have added a trivial factor for the $K_{14}$ singularity.\nTherefore it follows that\n\\begin{equation}\nQ_{10}=K_{14}\/\\relax{\\sf Z\\kern-.4em Z}_2.\n\\end{equation}\nMore explicitly consider the chiral ring of $K_{14}$\n\\begin{equation}\n{\\cal R}_K=\\{1,\\Phi_1,\\Phi_1^2,...,\\Phi_1^6,\n \\Phi_3,\\Phi_1\\Phi_3,...,\\Phi_1^6\\Phi_3\\}.\n\\end{equation}\nThe action of the $\\relax{\\sf Z\\kern-.4em Z}_2$ is defined as $[1,0,1]$ on the fields\n$(\\Phi_1,\\Phi_2,\\Phi_3)$. The two sectors of the orbifold consist of the\neight invariant states of ${\\cal R}} \\def\\cV{{\\cal V}_K$ and of two twisted states giving the\ntotal of ten states necessary for $Q$.\n\nThis strangely dual pair can again be used as building block of\nstring compactifications. In the models constructed\nin \\cite{ls3}\\cite{fkss1}\nthis involves vacua which have $(1\\cdot 6)_{A_1\\otimes A_7}$ or\n$(1\\cdot 6)_{A_1\\otimes D_5}$ as subfactors.\n\nThe ADE models make up only a very small part of the string\ncompactifications constructed in \\cite{cls} and it is natural to ask\nwhether it is\npossible to relate spaces in this set to one another in a similar way\nand if so\nwhether it is possible to construct {\\it all} of these models as\norbifolds of\nsome basic set of polynomial types, e.g. Fermat type spaces. This is the\nquestion to which will be addressed in the following section.\n\n\\vskip .1truein\n\\noindent\n\\section{Fractional Transformations}\n\n\\noindent\nIn the notation of \\cite{cls} the transition from the diagonal affine\ninvariant to the\nnon--diagonal D--invariant can be formulated as the transition from\na Fermat\ntype polynomial with diagram\n\\begin{equation}\n{\\thicklines \\begin{picture}(120,20)\n \\put(0,0){\\circle*{5}}\n \\put(0,7){\\circle{12}}\n \\put(26,0){\\circle*{5}}\n \\put(26,7){\\circle{12}}\n \\end{picture}\n }\n\\Phi_1^a + \\Phi_2^b\n\\lleq{gendiag}\nto particular polynomials of the type of a tadpole\n\\begin{equation}\n{\\thicklines \\begin{picture}(120,20)\n \\put(0,0){\\circle*{5}}\n \\put(0,0){\\line(1,0){26}}\n \\put(26,0){\\circle*{5}}\n \\put(26,7){\\circle{12}}\n \\end{picture}\n }\n\\tilde {\\Phi}_1^c\\tilde {\\Phi}_2 + \\tilde {\\Phi}_2^d\n\\lleq{gend}\nEquations (\\ref{diag},\\ref{daff}) clearly describe only a very small\nsubset of these polynomials and one can ask whether transformations\ngeneralizing (\\ref{fraccy}) are possible. Indeed they are.\n\nConsider the transformation\n\\begin{equation}\n\\Phi_1=\\tilde {\\Phi}_1^{c\/a} \\tilde {\\Phi}_2^{1\/a},~~~~~~~~\n\\Phi_2 = \\tilde {\\Phi}_2^{d\/b}\n\\lleq{genfraccy}\nwhich transforms the polynomial in (\\ref{gendiag}) into the one of eq.\n(\\ref{gend}).\nThe next step is to find the constraints on the weights of the fields that\nmake the Jacobian of this transformation constant. They are given by\n\\begin{equation}\nc=a~~~{\\rm and}~~~ d=b\\left[1-\\frac{1}{a}\\right].\n\\end{equation}\nAs in the case of the previous transformation however this transformation\nis not well defined yet and we have to find the discrete group which makes\nthe transformation of variables well defined. Suppose that we have a map\n\\begin{equation}\n\\Phi_1 \\longrightarrow \\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma \\Phi_1 ~~~{\\rm and}~~~\n\\Phi_2 \\longrightarrow \\b \\Phi_2\n\\end{equation}\nwhere $\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma, \\b$ are roots of unity. The condition that the change of variables\nin eq. (\\ref{genfraccy}) is invariant determines $\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma$ as the generator of\n$\\relax{\\sf Z\\kern-.4em Z}_a$ and\n$\\b=\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma^{a-1}$. Similarly observation regarding the inverse transformation\nleads to the group that one has to mod out by in the nondiagonal theory.\nOnly after modding out these cyclic groups does\ntransformation (\\ref{genfraccy}) become well defined.\n\nThe isomorphism can be summarized concisely with the following diagram\n\n\\vfill\n\\eject\n\\begin{eqnarray}\n& &\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_{\\left(\\frac{b}{g_{ab}},\\frac{a}{g_{ab}}\\right)}\n \\left[\\frac{ab}{g_{ab}}\\right]\n \\ni \\left \\{z_1^a+z_2^b=0\\right \\}\n ~{\\Big \/}~ \\relax{\\sf Z\\kern-.4em Z}_b: \\left[\\matrix{(b-1)&1}\\right] ~~\n \\nonumber \\\\ [3ex]\n&\\sim & \\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_{\\left(\\frac{b^2}{h_{ab}},\\frac{a(b-1)-b}{h_{ab}}\\right)}\n \\left[\\frac{ab(b-1)}{h_{ab}}\\right]\n \\ni \\left \\{y_1^{a(b-1)\/b}+y_1y_2^b=0\\right\\}\n ~{\\Big \/}~ \\relax{\\sf Z\\kern-.4em Z}_{b-1}: \\left[\\matrix{1&(b-2)}\\right].\n\\llea{iso}\n\\vskip .1truein\n\n\\noindent\nHere $g_{ab}$ is the greatest common divisor of $a$ and $b$ and\n$h_{ab}$ is the greatest common divisor of $b^2$ and $(ab-a-b)$.\nThe action of a cyclic group $\\relax{\\sf Z\\kern-.4em Z}_b$ of order $b$ denoted by\n$[m~~n]$ indicates that the symmetry acts like\n$(z_1,z_2) \\mapsto (\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma^m z_1, \\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma^n z_2)$ where $\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma$ is the $b^{th}$ root\nof unity.\n\n\nIt is easy to generalize this analysis to general transformations\nfrom Fermat--type polynomials to tadpole--type polynomials with $N$ points\nattached. Consider the transformation\n\\begin{equation}\n\\sum_{i=1}^N \\Phi_i^{l_i} \\longrightarrow\n\\left( \\sum_{j=1}^{N-1} \\tilde {\\Phi}_j^{n_j} \\tilde {\\Phi}_{j+1}\\right) + \\tilde {\\Phi}_N^{n_N}\n\\end{equation}\ndefined by\n\\begin{eqnarray}\n\\Phi_i &=& \\tilde {\\Phi}_i^{n_i\/l_i} \\tilde {\\Phi}_{i+1}^{1\/l_i} ~~~~~~~~i=1,..,N-1;\n \\nonumber \\\\\n\\Phi_N &=& \\tilde {\\Phi}_N^{n_N\/l_n}.\n\\llea{multifrac}\nThe condition that the Jacobian of this transformation is constant\nagain leads to constraints on the exponents:\n\\begin{eqnarray}\nn_1 &=&l_1;\\\\\nn_i &=&l_i\\left[1 - \\frac{1}{l_{i-1}}\\right]~,~i=2,...,N.\n\\end{eqnarray}\nTransformation (\\ref{multifrac}) then becomes\n\\begin{eqnarray}\n\\Phi_1 &=& \\tilde {\\Phi}_1 \\tilde {\\Phi}_2^{1\/l_1}; \\\\\n\\Phi_i &=& \\tilde {\\Phi}_i^{{(l_{i-1}-1)}\/l_{i-1}} \\tilde {\\Phi}_{i+1}^{1\/l_i}\n ~~~~~~~~~~i=2,..,N-1; \\\\\n\\Phi_N &=& \\tilde {\\Phi}_N^{{(l_{N-1}-1)}\/l_{N-1}}.\n\\llea{multiexp}\nAs before, this transformation is not well defined, but one can find\na discrete group under which it is invariant. Let\n\\begin{equation}\n\\Phi_i \\longrightarrow \\alpha_i \\Phi_i;\n\\lleq{scaling}\nit then follows that under this transformation\n\\begin{equation}\n\\tilde {\\Phi}_N \\longrightarrow \\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma_N^{l_{N-1}\/{(l_{N-1}-1)}} \\tilde {\\Phi}_N,\n\\end{equation}\ni.e. we find the constraint\n\\begin{equation}\n\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma_N^{l_{N-1}\/{(l_{N-1}-1)}}=1\n\\end{equation}\nfor the generator $\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma_N$. Plugging the solutions of this constraint\niteratively into eqs. (\\ref{multiexp}) one finds\nthat all generators $\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma_i$ have to satisfy $\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma_i^{l_i} = 1~,i=1,..,N-1$\nfor the transformation of variables to be invariant under the\nidentification (\\ref{scaling}). The group by which one has to mod out is\n$\\prod_{i=1}^{N-1} \\relax{\\sf Z\\kern-.4em Z}_{l_i}$. Again one has to similarly analyze the\ninverse map. It is obvious that this generalization just amounts to\nrepeated application of the original isomorphism (\\ref{iso}).\n\nConsider e.g. the manifold\n\\begin{equation}\n\\relax{\\rm I\\kern-.18em P}_{(1,3,3,3,5)} [15]^3_{-144}=\n\\{\\Phi_1^{15}+\\Phi_2^5+\\Phi_3^5+\\Phi_4^5+\\Phi_5^3=0\\}.\n\\end{equation}\nWe will show now that the orbifold of this model with respect to two\ncyclic groups $\\relax{\\sf Z\\kern-.4em Z}_5$, the first generated by $[0,4,1,0,0]$\nthe second one by $\\relax{\\sf Z\\kern-.4em Z}_5:[0,0,4,1,0]$, is isomorphic to the manifold\n\\begin{equation}\n\\relax{\\rm I\\kern-.18em P}_{(16,60,45,39,80)}[240]^{75}_{144}~=\n \\{\\tilde {\\Phi}_1^{15}+\\tilde {\\Phi}_2^4+\\tilde {\\Phi}_2\\tilde {\\Phi}_3^4\n +\\tilde {\\Phi}_3\\tilde {\\Phi}_4^5+\\tilde {\\Phi}_5^3=0\\}.\n\\end{equation}\nConsider the first $\\relax{\\sf Z\\kern-.4em Z}_5$. The fixed points of this action are determined\nby the requirement that\n\\begin{equation}\n(\\Phi_1,\\alpha^4 \\Phi_2,\\alpha \\Phi_3,\\Phi_4,\\Phi_5)\n= c(\\Phi_1,\\Phi_2,\\Phi_3,\\Phi_4,\\Phi_5)\n\\end{equation}\nwhere $c\\in \\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}^*$. The fixed point set consists of one curve\n$\\relax{\\rm I\\kern-.18em P}_{(1,3,5)}[15]$\nwith Euler number $ \\chi= -6$ and one further fixed point $\\relax{\\rm I\\kern-.18em P}_{(3,5)}[15]$.\nThe Euler number of the resolved orbifold therefore is\n$\\chi= \\frac{-144}{5} - \\frac{1}{5}(-6+1) + 5(-6+1) - \\frac{1}{5} +5 =-48$.\nThe corresponding weighted CY--manifold of this space is described by\n\\begin{equation}\n\\relax{\\rm I\\kern-.18em P}_{(4,15,9,12,20)}[60]~=~\\{\\phi_1^{15}+\\phi_2^4+\\phi_2\\phi_3^5+\n \\phi_4^5+\\phi_5^3=0\\}.\n\\end{equation}\nand the coordinate transformation is defined as follows\n\\begin{equation}\n(\\Phi_1,\\Phi_2,...,\\Phi_5) =\n(\\phi_1,\\phi_2^{4\/5},\\phi_2^{1\/5}\\phi_3,\\phi_4,\\phi_5).\n\\end{equation}\n\nModding out by the other $\\relax{\\sf Z\\kern-.4em Z}_5$ leads to a fixed point set of three curves\n\\begin{equation}\n\\relax{\\rm I\\kern-.18em P}_{(4,15,20)}[60]_{15},~~~\\relax{\\rm I\\kern-.18em P}_{(15,9,20)}[60]_{16},\n ~~~\\relax{\\rm I\\kern-.18em P}_{(15,12,20)}[60]_{11}.\n\\end{equation}\nThese three curves intersect in the point $\\relax{\\rm I\\kern-.18em P}_{(15,20)}[60]=\\relax{\\rm I\\kern-.18em P}_{(3,4)}[12]$\nand the Euler number of the resolved manifold is given by\n\\begin{equation}\n\\chi = -\\frac{48}{5} - \\frac{1}{5}\\left(15 + 11 + 16 -2 \\times 5\\right) +\n 5\\left(15 +11 + 16 -2\\times 5\\right) = 144.\n\\end{equation}\nThe term $-2\\times 5$ comes from the fact that in all three curves we have\nblown up the $\\relax{\\sf Z\\kern-.4em Z}_5$ fixed point when in fact we only should have done\nso once.\n\nWe have thus established that the discrete groups of the\nLG--CY theories play a role comparable to the discrete symmetries of the\nfusion rules in conformal field theories even though for most of the models\nat hand the corresponding exact theory is not known. We have seen that\nin general there are constraints on the weights of the potentials that\ncan be identified with with orbifolds of other models. Nevertheless, in the\ntypes of potentials discussed above there was enough freedom to have\nnontrivial identifications. This is not true in general as we will show in\nthe next section.\n\n\\vskip .1truein\n\\noindent\n\\section{ Loop Potentials}\n\n\\noindent\nConsider potentials of the type\n\\begin{equation}\n{\\thicklines \\begin{picture}(80,20)\n \\put(0,0){\\circle*{5}}\n \\put(0,0){\\line(1,0){26}}\n \\put(26,0){\\circle*{5}}\n \\put(13,0){\\oval(26,26)[t]}\n \\end{picture}\n }\n\\tilde {\\Phi}_1^c\\tilde {\\Phi}_2 + \\tilde {\\Phi}_2^d\\tilde {\\Phi}_1\n\\end{equation}\nand more general polynomials of this type with an arbitrary number of\nfields. For these polynomials it is possible to find a 1--1 coordinate\ntransformation from Fermat--type polynomials, but the\ncondition that the Jacobian is constant leads to the conclusion that\nonly deformations of the original space can be obtained in this way.\n\nThe transformation of a Fermat type polynomial into a loop--type\npolynomial\n\\begin{equation}\n\\sum_{i=1}^{N} \\Phi_i^{l_i} \\longrightarrow\n \\left( \\sum_{i=1}^{N-1} \\tilde {\\Phi}_i^{n_i} \\tilde {\\Phi}_{i+1} \\right) +\n \\tilde {\\Phi}_N^{n_N}\\tilde {\\Phi}_1.\n\\end{equation}\nleads to the transformation of variables\n\\begin{eqnarray}\n\\Phi_i &=& \\tilde {\\Phi}_i^{n_i\/l_i} \\tilde {\\Phi}_{i+1}^{1\/l_i}~~~~~~~~i=1,..,N-1;\n\\nonumber \\\\\n\\Phi_N &=& \\tilde {\\Phi}_N^{n_N\/l_N} \\tilde {\\Phi}_1^{1\/l_N}.\n\\llea{loopfrac}\nThe condition that the Jacobian be constant\nleads to the constraints\n\\begin{eqnarray}\nn_1 &=& l_1\\left[1 -\\frac{1}{l_N}\\right];\\\\\nn_i &=& l_i\\left[1 -\\frac{1}{l_{i-1}}\\right],~~~~~~~~~~~~~i=2,..,N;\n\\end{eqnarray}\nso that the coordinate transformation (\\ref{loopfrac}) becomes\n\\begin{eqnarray}\n\\Phi_1 &=& \\tilde {\\Phi}_1^{{(l_N-1)}\/l_N} \\tilde {\\Phi}_2^{1\/l_1}\\\\\n\\Phi_i &=& \\tilde {\\Phi}_i^{{(l_{i-1}-1)}\/l_{i-1}} \\tilde {\\Phi}_{i+1}^{1\/l_i},\n{}~~~~~~i=2,..,N.\n\\end{eqnarray}\n{}From this it follows however that the charges of the loop fields are\nprecisely those of the fields of the Fermat type polynomial, i.e.\nthe loop--type model is a trivial deformation of the original one.\n\n\n\n\\vfill \\eject\n\n\\part{ Landau--Ginzburg Orbifolds.}\n\nEven though the class of Heterotic Vacua described in Part I is the largest\nconstructed so far it clearly does not encompass all spectra that are known.\nConsider e.g. the orbifold of the Fermat quintic in $\\relax{\\rm I\\kern-.18em P}_4[5]$ with respect\nto the action\n\\begin{equation}\n\\relax{\\sf Z\\kern-.4em Z}_5: \\left[\\matrix{ 3 &1 &1 &0 &0}\\right]\n\\end{equation}\nwhich leads to a space with spectrum $(h^{(1,1)},h^{(2,1)},\\chi)=(21,17,8)$.\nNo complete intersection model with such Hodge numbers appears among the\nresults of \\cite{ks}.\nAn obvious question of course is whether one could use the fractional\ntransformation discussed in Part II to {\\it construct} the complete\nintersection representation of this orbifold. Unfortunately this yields\na singular space.\nThus it is unclear at this point whether a complete intersection of this\norbifold exist. There are many examples of this type, some of which will\nbe mentioned below.\n\nOrbifolding is important for the construction of mirrors as well because\nin many examples the weighted CICY representation of a mirror is not known\nwhereas it is easy to construct the mirror as an orbifold. A simple example\nis furnished by the manifold\n\\begin{equation}\n\\matrix{\\relax{\\rm I\\kern-.18em P}_2\\cr \\relax{\\rm I\\kern-.18em P}_3\\cr} \\left[\\matrix{3&0\\cr 1&3\\cr}\\right].\n\\end{equation}\nthe mirror of which can easily be constructed as the orbifold with respect\nto the action\n\\begin{equation}\n\\relax{\\sf Z\\kern-.4em Z}_9\\times \\relax{\\sf Z\\kern-.4em Z}_3 ~:~ \\left[\\matrix{6~&1~&3~&2~&0~&0~&6\\cr\n 0~&0~&0~&0~&0~&2~&1\\cr}\\right]\n\\end{equation}\nwhen viewed as a symmetry on the corresponding LG theory embedded in\n$\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_{(3,2,3,2,3,2,3)}[9]$.\nIt is unclear however what the CICY representation of this mirror is.\n\nA further motivation for the construction of orbifolds has been mentioned\nalready in the introduction. Namely in order to get some idea of how\n`robust' the mirror property of the configuration space is it is useful\nto implement different types of constructions. Until the proper framework\nfor mirror symmetry and its explicit form has been found this appears to be\n one feasible avenue for collecting support for the existence\n\\fnote{8}{There are of course also other reasons to look for\n new models but this is another story.}.\n\nThis last part of the review contains a brief description of the work\ndone in \\cite{kss}\nin which some 40odd types of actions have been considered\non all Landau--Ginzburg potentials that can be build from Fermat monomials,\nsingle Tadpole potentials and single Loop potentials as well as arbitrary\ncombinations thereof.\n\n\\vskip .1truein\n\\noindent\n\\section{Actions of Symmetries: General Considerations}\n\n\\noindent\nIt is useful to first discuss some general aspects that are important\nfor group actions on Landau--Ginzburg theories that have been\norbifolded with respect to the U(1)--symmetry in order to describe\nstring vacua with $N=1$ spacetime supersymmetry.\n\nAn obvious question when considering orbifolds is whether\nthere is any a priori insight into\nwhat spectra are possible for the orbifolds of a given model with respect\nto a particular set of symmetries.\nThis question is of particular interest if the goal is to produce\norbifolds with presribed spectra, say models with a small number\nof fields where the difference between the number of generations and\nantigenerations is three.\n\nEven though it is possible to formulate constraints on the\norbifold spectrum for\nparticular types of actions, we know of no constraints that hold\nin full generality, or even for arbitrary cyclic actions.\nOne very simple class of symmetries are those without fixed points.\nFor such actions there are no twisted sectors and hence\nthere exists a simple formula expressing the Euler number $\\chi_{orb}$ of the\norbifold in terms of the Euler characteristic $\\chi$ of the covering space\nand the order $|G|$ of the group\n\\begin{equation}\n\\chi_{orb} = \\frac{\\chi}{|G|}.\n\\end{equation}\nThe vast majority of actions however do have fixed points and hence the\nresult above does not apply very often.\n\nFor orbifolds with respect to cyclic groups of prime order there\nexists a generalization of this result. For such group actions it was shown\nin the first reference in \\cite{orbis} that\n\\begin{equation}\n\\bar n} \\def\\bt{\\bar t} \\def\\bz{\\bar z^g_{orb} - n^g_{orb} =\n\\left( |G|+1\\right)\\left(\\bar n} \\def\\bt{\\bar t} \\def\\bz{\\bar z^g_{inv} - n^g_{inv}\\right)\n-\\left(\\bar n} \\def\\bt{\\bar t} \\def\\bz{\\bar z^g - n^g\\right),\n\\end{equation}\nwhere $n^g_{orb}$, $n^g_{inv}$, $n^g$ are the numbers of generations\nof the orbifold theory, the invariant sector and the original LG theory,\nrespectively.\n\nConsider then the problem of constructing an orbifold with a prescribed Euler\nnumber $\\chi_{orb}$ from a given theory.\nOnly for fixed point free actions will the order of the group be completely\nspecified as $|G|=\\chi\/\\chi_{orb}$. It is important to realize that in\ngeneral the order of the group by which a theory is orbifolded does\n{\\it not} determine its spectrum -- the precise form of the action of the\nsymmetry is important.\n\nNevertheless we can derive {\\it some} constraints on the order of the\naction that we are looking for. Even though we don't know a priori what\nthe invariant\nsector of the orbifold will be we do know that its associated Euler number\nmust be an integer\n\\begin{equation}\n\\chi_{inv} = \\frac{\\chi + \\chi_{orbi}}{|G|+1}~~\\in \\relax{\\rm I\\kern-.18em N}.\n\\end{equation}\nThis simple condition does lead to restrictions for the order of the group.\nSuppose, e.g., that we wish to check whether the quintic threefold admits\na three--generation orbifold: For the deformation class of the quintic\n\\begin{equation}\n\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_{(1,1,1,1,1)}[5]: ~\\chi=-200\n\\end{equation}\nthe order of the discrete group in question must satisfy the constraint\n$-206\/(|G|+1) \\in \\relax{\\sf Z\\kern-.4em Z}$, implying $|G|=102$.\nHence there exists no three--generation orbifold of the quintic with respect\nto a discrete group with prime order.\nA counterexample for nonprime orders is furnished by the following theory\n\\begin{equation}\n\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_{(2,2,2,3,3,3,3)}[9]:~ (\\bar n} \\def\\bt{\\bar t} \\def\\bz{\\bar z^g,n^g,\\c)=(8, 35,-54),\n\\lleq{3gen}\nwhich corresponds to a CY theory embedded in a product of two projective\nspaces by two polynomials of bidegree $(0,3)$ and $(3,1)$ \\cite{s1}\\cite{s2}\ni.e. the Calabi--Yau manifold of this model is embedded in\nan ambient space consisting of a product of two projective spaces\n\\begin{equation}\n\\matrix{\\relax{\\rm I\\kern-.18em P}_2\\cr \\relax{\\rm I\\kern-.18em P}_3\\cr} \\left[\\matrix{3&0\\cr 1&3\\cr}\\right].\n\\end{equation}\nSuppose we are searching for three--generation orbifolds of this space\nwith $\\chi_{orb}=\\pm 6$ . If $\\chi=-6$ the constraint is not very restrictive\nand allows a number of possible groups $|G|\\in \\{2,3,5,11,19,29\\}$.\nEven though it is not known whether any of these groups lead to a\nthree--generation model it {\\it is} known that at a particular\npoint in the configuration space of (\\ref{3gen}) described by the\nsuperpotential\n\\begin{equation} W=\\sum_{i=1}^3 (\\Phi_i^3+\\Phi_i \\Psi_i^3)+\\Phi_4^3 \\end{equation}\na symmetry of order nine exists that leads to a three--generation model\n\\cite{s1}.\n\nOur interest however is not restricted to models with particular spectra\nfor reasons explicated in the introduction. Hence we wish to implement\ngeneral types of actions regardless of their fixed point structure and\norder. A general analysis of symmetries for an arbitrary Landau--Ginzburg\npotential is beyond the scope of this paper; instead we restrict our\nattention to the types of potentials that we have constructed explicitly.\nBefore we discuss these types we should remark upon a number of aspects\nconcerning actions on string vacua defined by LG--theories.\n\nIt is important to note that depending on the weights (or charges) of the\noriginal LG theory it can and does happen that actions that take rather\ndifferent forms when considered as actions\non the LG theory actually are isomorphic when viewed as action of the\nstring vacuum proper because of the U(1) projection. It is easiest to\nexplain this with an example.\nConsider the superpotential\n\\begin{equation}\nW=\\Phi_1^{18}+\\Phi_2^{18}+\\Phi_3^3+\\Phi_4^3+\\Phi_4\\Phi_5^3\n\\end{equation}\nwhich belongs to the configuration\n$\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_{(1,1,6,6,4)}[18]_{-204}^9$ (here the superscript denotes the\nnumber of antigenerations and the subscript denotes the Euler number of the\nconfiguration). At this\nparticular point in moduli space we can, e.g., consider the orbifolds with\nrespect to the actions\n\\begin{eqnarray}\n\\relax{\\sf Z\\kern-.4em Z}_3&:&\\left[\\matrix{0&0&1&0&2\\cr}\\right],~~~(13, 79, -132) \\nonumber \\\\\n\\relax{\\sf Z\\kern-.4em Z}_3&:&\\left[\\matrix{1&1&1&0&0\\cr}\\right],~~~(13, 79, -132) \\nonumber \\\\\n\\relax{\\sf Z\\kern-.4em Z}_3&:&\\left[\\matrix{1&0&1&0&1\\cr}\\right],~~~(14, 44, -60) ,\\label{z3}\n\\end{eqnarray}\nwhere the notation $\\relax{\\sf Z\\kern-.4em Z}_a: [p_1~\\ldots~p_n]$ indicates that the\nfields $\\Phi_i$ transform with phases $(2\\pi i p_i\/a)$ under the\ngenerator of the $\\relax{\\sf Z\\kern-.4em Z}_a$ symmetry.\nIt is clear from the last action in (\\ref{z3}) that the order of a group\nis, in general, not suffient to determine the resulting orbifold spectrum\nbut that the specific form of the way the symmetry acts is essential.\n\nSince the first two actions lead to the same spectrum we are led to ask\nwhether the two resulting orbifolds are equivalent.\nTheories with the\nsame number of light fields need, of course, not be equivalent and to\nshow whether they are is, in general a rather involved analysis,\nentailing the transformation behaviour of the fields and the computation\nof the Yukawa couplings.\n\nIn the case at hand it is, however, very easy to check this question.\nThe first two actions only differ by the $6^{\\rm th}$ power of the\ncanonical $\\relax{\\sf Z\\kern-.4em Z}_{18}$ which is given by $\\relax{\\sf Z\\kern-.4em Z}_3:\\,[1~1~0~0~1]$.\nSince the orbifolding with respect to this group is always present\nin the construction of a LG vacuum the fist two orbifolds in\n$eq.$~(\\ref{z3}) are trivially equivalent.\n\nAnother important point is the role of trivial factors in the LG theories.\nGiven a superpotential $W_0$ with the correct central charge to define a\nHeterotic String vacuum we always have the freedom to add trivial factors\nto it\n\\begin{equation}\nW=W_0 + \\sum_i \\Phi_i^2,\n\\end{equation}\nsince neither the central charge nor the chiral ring are changed by this\noperation.\n As we restrict our attention to symmetries with unit determinant,\nwe gain, however, the possibility to cancel a negative sign of the determinant\nby giving some $\\Phi_i$ a nontrivial transformation property under a\n$\\relax{\\sf Z\\kern-.4em Z}_{2n}$. Adding a trivial factor hence changes the symmetry properties\nof the LG--potential with regards to this class of symmetries.\n\\fnote{9}{In LG theories the determinant restriction is necessary for modular\n invariance and can be avoided by introducing discrete torsion \\cite{iv}.}\nIf we wish to relate the vacuum described by the potential to a\nCalabi--Yau manifold, consideration of trivial factors becomes essential\n\\cite{ls3}.\nConsider e.g. the LG--potential\n\\begin{equation}\nW_0=\\Phi_1^{12}+\\Phi_2^{12}+\\Phi_3^6+\\Phi_4^6\n\\end{equation}\nwhich has $c=9$ and charges\n$(\\frac{1}{12},\\frac{1}{12},\\frac{1}{6},\\frac{1}{6})$ and hence is a member\nof the configuration $\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_{(1,1,2,2)}[12]$. Only after adding the necessary\ntrivial factor\nthis theory can be orbifolded with an action defined by\n$\\relax{\\sf Z\\kern-.4em Z}_2:[~1~0~0~0~1~]$ acting on the Fermat polynomial in\n$\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_{(1,1,2,2,6)}[12]$; this action leads to the orbifold spectrum\n$(4,94,-180)$ and is not equivalent to any symmetry that\nacts only on the first four variables with determinant 1. Neglecting the\naddition of the quadratic term to the LG potential $W_0$\nwould have meant missing the above spectrum as one of the possible\norbifold results.\n\nFinally it should be noted that obviously we have to make {\\it some}\nchoice about which points in moduli space we wish to consider. Different\nmembers of a moduli space have, in general, drastically different\nsymmetry properties. An example is the well known quintic theory\nwhich we already mentioned. The most symmetric\npoint in the 101 dimensional space of complex deformations of the quintic is\ndescribed by the Fermat polynomial\n\\begin{equation} W= \\sum_i \\Phi_i^5,\n\\end{equation}\nwhich has a discrete symmetry group of order $5!\\cdot 5^4$. Any\ndeformation\nbreaks most of these symmetries but in some cases new symmetries appear\nwhich\nturn out to lead to new spectra. An example is furnished by the quintic\ndescribed by a combination of two 1--Tadpole polynomials and one Fermat\nmonomial\n\\begin{equation}\nW=\\Phi_1^5+\\Phi_2^5+\\Phi_2\\Phi_3^4+\\Phi_4^5+\\Phi_4\\Phi_5^4\n\\end{equation}\nwhich, when orbifolded with respect to the symmetry\n$\\relax{\\sf Z\\kern-.4em Z}_2: [0~0~1~0~1]$ leads to a model with spectrum $(3,59,-112)$. This\nspectrum cannot be obtained via orbifolding the Fermat quintic by any of\nits supersymmetry preserving symmetries.\n\n\\vskip .1truein\n\\noindent\n\\section{Phase Actions: Implementation and Results}\n\nConsider then a potential $W$ with $n$ order parameters\nnormalized such that the degree $d$ takes the lowest value such that\nall order\nparameters have integer weight.\nIn the following we discuss potentials of the type\n\\begin{equation}\nW= \\sum_i \\Phi_i^{a_i} + \\sum_j \\left(\\Phi_j^{e_j} + \\Phi_j \\Psi_j^{f_j}\\right)\n + \\sum_k \\left(\\Phi_k^{e_k}\\Psi_k + \\Phi_k \\Psi_k^{f_k} \\right)\n\\end{equation}\nwhich consist of Fermat parts, tadpole parts and loop parts.\n\n\\noindent\n{\\bf FERMAT POTENTIALS}:\nClearly the potential $W=\\sum_{i=1}^n \\Phi_i^{a_i}$ is\ninvariant under $\\prod_i \\relax{\\sf Z\\kern-.4em Z}_{a_i}$, i.e. the phases of the individual fields,\nacting like\n\\begin{equation}\n\\Phi_i \\longrightarrow e^{2\\pi i \\frac{m_i}{a_i}} \\Phi_i.\n\\end{equation}\nFor some divisor $a$ of ${\\rm lcm}(a_1,\\dots, a_n)$ and\n$\\frac{m_i}{a_i}=\\frac{p_i}{a}$\nwe denote such an action by\n\\begin{equation}\n\\relax{\\sf Z\\kern-.4em Z}_a:~\\left[\\matrix{p_1&p_2 &\\cdots &p_n\\cr}\\right],~~~0\\leq~ p_i \\leq~ a-1.\n\\end{equation}\nand require that $a$ divides $\\sum p_i$ in order to have determinant 1.\n\nWe have implemented such symmetries in the form\n\\begin{equation}\n\\relax{\\sf Z\\kern-.4em Z}_a:~\\left[\\matrix{(a-\\sum_l i_l)&i_1&\\cdots&i_p\n &(a-\\sum_m j_m)&j_1&\\cdots&j_q&\\cdots} \\right] \\end{equation}\nwith the obvious divisibility conditions. For small $p$ and $q$\nthese symmetries can act on a large number of spaces and therefore lead to\nmany different orbifolds, but as $p,q$ get larger\nthe number of resulting orbifolds decreases rapidly.\nWe have stopped implementation of more complicated actions when the number of\nresults for the different orbifold Hodge pairs was of the order of a few tens.\nAs already mentioned above, the precise form of the action is very important\nwhen considering symmetries with fixed points since the order itself is\nnot sufficient to determine the orbifold spectrum.\n\nMore complicated symmetries can be constructed via multiple actions\nby multiplying single actions of the type described above\n\\begin{equation}\n\\prod_c \\relax{\\sf Z\\kern-.4em Z}_{a_c} :~~\n\\left[\\matrix{(a_c-\\sum_l i_{c,l})&i_{c,1}&\\cdots&i_{c,p}\n &(a_c-\\sum_m j_{c,m})&j_{c,1}&\\cdots&j_{c,q}&\\cdots} \\right].\n\\end{equation}\nWe have considered (an incomplete set of) actions of this type with up to\nsix twists (i.e. six $\\relax{\\sf Z\\kern-.4em Z}_a$ factors). Again the precise form of the action\nis rather important.\n\n\\vskip .2truein\n\n\\noindent\n{\\bf TADPOLE AND LOOP POLYNOMIALS}:~~\nThe action of the generator of the maximal phase symmetry within a tadpole\nor loop sector is\n\\begin{equation} \\relax{\\sf Z\\kern-.4em Z}_{\\cO}: \\left[\\matrix{-f&1}\\right], \\end{equation}\nwhere $\\cO = ef$ or $ef-1$, respectively. If we want unit determinant within\none sector, we must take our generator to the $n^{\\rm th}$ power with some\n$n$ fulfilling $n(f-1)\/\\cO\\in\\relax{\\sf Z\\kern-.4em Z}$. With $\\om =gcd(f-1,\\cO)$ the action\nof the resulting subgroup can be chosen to be\n\\begin{equation}\n\\relax{\\sf Z\\kern-.4em Z}_{\\om}:~~ \\left[\\matrix{(\\om-1)&1}\\right].\n\\end{equation}\n\nOther types of actions that we have considered for superpotentials consisting\nof Fermat parts and tadpole\/loop parts involve phases acting both on the\ntadpole\/loop part as well as on a number of Fermat monomials.\nAs was the case with pure Fermat polynomials we have also implemented\nmultiple actions of the type considered above.\n\n\nWe have implemented some forty different actions of the types\ndescribed in the previous paragraphs. These symmetries lead to a large\nnumber of orbifolds not all of which are distinct however for reasons\nexplained in the previous section.\nOur computations have concentrated on the number of generations and\nanti--generations of these models and we have found some 1900 distinct\nHodge pairs. This set of spectra shows a mirror symmetry that is even\nhigher than the one exhibited by the complete intersection vacua: whereas\nabout 8\nspectra have mirror partners!\n\n\nIt is obvious from this plot that there is a large overlap between the\nresults of \\cite{cls} and the orbifolds constructed here. This might indicate\nthat the relation established in \\cite{ls4} between orbifolds of\nLandau--Ginzburg\ntheories and other Landau--Ginzburg theories is a general phenomenon and\nnot restricted to the particular classes of actions which were analysed in\n\\cite{ls4}.\n\nModels with a low number of fields are clearly\nof particular interest. There are two aspects to this question,\nas mentioned in the introduction -- low numbers for the {\\it difference}\nof\ngenerations and anti--generations\n(more precisely one wants the number 3 here)\nand low values for the {\\it total} number of generations and\nanti--generations.\nAs far as the latter are concerned the following `low--points' are the\n`highlights' among the results for phase symmetry orbifolds.\n\nThe lowest models have $\\chi=0$, more precisely the spectra (9,9,0)\nand (11,11,0). These spectra appear many times in different\norbifolds of Fermat type; an example for the first one being\n\\begin{equation}\n\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_{(1,\\ldots,1)}[9]\/\\relax{\\sf Z\\kern-.4em Z}_3^2: \\left[\\matrix{1&1&1&0&0&0&0&0&0\\cr\n 0&0&0&1&1&1&0&0&0\\cr}\\right] \\end{equation}\nor, even simpler,\n\\begin{equation}\n\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_{(4,4,4,4,4,4,3,3)}[12]\/\\relax{\\sf Z\\kern-.4em Z}_3: [1~1~1~0~0~0~0~0].\n\\end{equation}\nThe second one can be constructed e.g. as\n\\begin{equation}\n\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_{(4,3,3,3,3,2)}[12]\/\\relax{\\sf Z\\kern-.4em Z}_4^2: \\left[\\matrix{0&1&1&2&0&0\\cr\n 0&0&2&1&1&0\\cr}\\right].\n\\end{equation}\n\nOther examples with a total of 22 generations and anti--generations are\nthe following orbifolds of the Fermat quintic:\n\\begin{equation}\n\\relax{\\sf Z\\kern-.4em Z}_5: \\left[\\matrix{0&1&2&3&4}\\right],~~\\qqd (1,21,-40)\n\\end{equation}\nand\n\\begin{equation}\n\\relax{\\sf Z\\kern-.4em Z}_5^2: \\left[\\matrix{3&1&1&0&0\\cr\n 0&3&1&1&0\\cr}\\right],~~\\qqd (21,1,40)\n\\end{equation}\n\n\\noindent\nOf particular interest, of course, are three--generation models.\nIn the list of 3112 models there are no new such models aside from the\nknown three--generation models \\cite{ks}.\n\nVia orbifolding a number of such models can be found, which all, however,\nhave\na fairly large number of generations and antigenerations. We list those\nin Table 2.\n\n\n\\begin{small}\n\\begin{center}\n\\begin{tabular}{||l|l l l r||}\n\\hline\n\\hline\n$\\#$ &Configuration &Potential &Action\n&Spectrum \\hbox to0pt{\\phantom{\\Huge A}\\hss} \\\\\n\\hline\n\\hline\n1 &$\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_{(9,2,5,9,2)}[27]_{-66}^{16}$\n&$\\Phi_1^3+\\Phi_2^{11}\\Phi_3+\\Phi_2\\Phi_3^5+\\Phi_4^3+\\Phi_4\\Phi_5^9$\n&$\\relax{\\sf Z\\kern-.4em Z}_3~:[1~0~0~0~2]$ &$(18, 21, -6)$ \\hbox to0pt{\\phantom{\\Huge A}\\hss} \\\\ [2ex]\n2 &$\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_{(9,2,5,3,8)}[27]_{-54}^{10}$\n&$\\Phi_1^3+\\Phi_2^{11}\\Phi_3+\\Phi_2\\Phi_3^5+\\Phi_4^9+\\Phi_4\\Phi_5^3$\n&$\\relax{\\sf Z\\kern-.4em Z}_2~:[0~1~1~0~2]$ &$(21, 18, 6)$ \\\\ [2ex]\n3 &$\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_{(17,6,9,3,16)}[51]_{-102}^{15}$\n&$\\Phi_1^3+\\Phi_2^7\\Phi_3+\\Phi_2\\Phi_3^5+\\Phi_4^{17}+\\Phi_4\\Phi_5^3$\n&$\\relax{\\sf Z\\kern-.4em Z}_2~:[0~1~1~0~0]$ &$(31, 34, -6)$ \\\\ [2ex]\n4 &$\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_{(15,15,2,9,4)}[45]_{-30}^{23}$\n& $\\Phi_1^3+\\Phi_2^3+\\Phi_2\\Phi_3^{15}+\\Phi_4^5+\\Phi_4\\Phi_5^9$\n&$\\relax{\\sf Z\\kern-.4em Z}_3~:[1~0~2~0~0]$ &$(23, 20, 6)$\\\\ [2ex]\n5 &$\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_{(15,15,10,3,2)}[45]_{-54}^{22}$\n&$\\Phi_1^3+\\Phi_2^3+\\Phi_2\\Phi_3^3+\\Phi_4^{15}+\\Phi_4\\Phi_5^{21}$\n&$\\relax{\\sf Z\\kern-.4em Z}_3~:[1~0~2~0~0]$ &$(35, 32, 6)$ \\\\ [2ex]\n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{small}\n\n{\\bf Table 2.} {\\it Three--generation orbifold models; models which are\nequivalent up to the U(1) projection are not listed separately.}\n\n\n\\noindent\nBy using the relation established in \\cite{ls4} between LG\/CY--theories\nvia fractional transformations it can be shown that the orbifold $\\#1$ in\nTable 2,\n\\begin{equation}\n\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_{(2,5,9,2,9)}[27]_{-66}^{16}\/\\relax{\\sf Z\\kern-.4em Z}_3:[~0~0~0~2~1],\n\\end{equation}\nfor which the covering model is described by the polynomial\n\\begin{equation}\nW= \\Phi_1^{11}\\Phi_2+\\Phi_1\\Phi_2^5+\\Phi_3^3+\\Phi_3\\Phi_4^9+\\Phi_5^3,\n\\end{equation}\nis isomorphic to the orbifold\n\\begin{equation}\n\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_{(2,5,9,3,8)}[27]_{-54}^{10}\/\\relax{\\sf Z\\kern-.4em Z}_2:[~0~0~0~1~1]\n\\end{equation}\nwhere the covering theory is described by the polynomial\n\\begin{equation}\nW= \\Phi_1^{11}\\Phi_2+\\Phi_1\\Phi_2^5+\\Phi_3^3+\\Phi_3\\Phi_4^6\n +\\Phi_4\\Phi_5^3.\n\\end{equation}\nThe latter is a theory involving a subtheory with couplings among three\nscaling fields and hence goes beyond the types of potentials we have\nimplemented.\nThis example indicates that more complicated examples than the ones\ninvestigated here are likely to yield more (perhaps more realistic)\nthree generation models.\n\nThe covering spaces of all the three generation models are described by\neither tadpole or loop type polynomials, and with our actions none of the\nFermat type polynomials leads to a three generation model.\nIt should be noted that these orbifolds exist only at particular points in\nmoduli space.\n\n\\vskip .1truein\n\\noindent\n\\section{ Conclusion}\n\n\\noindent\nIt is clear that the structure of the configuration space of the Heterotic\nString is not particularly well understood and that much remains to be done;\nit is apparent from\nthe results described above that what has been achieved so far at best is\n little more than scratching the surface. It might be hoped that once\na completely mirror symmetric part of the moduli space has been constructed\na representative part of the complete space has been uncovered.\nPursueing work\nalong the lines described above is certainly promising in this regard;\nvery likely it is possible to generate a mirror symmetric subspace by\nall orbifolds of the Landau--Ginzburg theories listed in \\cite{ks}\nor, more generally, to construct all weighted complete intersection\nCalabi--Yau\nmanifolds and their orbifolds.\n\nHaving done that it still remains to show that all potential mirror\npartners\nin this symmmetric subspace of ground states are in fact related.\nEven though many types of LG--potentials constructed in \\cite{cls} admit,\nvia fractional transformations, an interpretation as orbifolds not every\nmirror potential can be constructed in this way at present. It is therefore\n clear that a generalization of this type of mirror map is necessary.\n\nAside from the question of mirror symmetry the orbifold technique is\nextremely useful to get insight into both, the detailed structure\nof the vacua constructed via such a classification of LG--potentials and\nthe relation between these vacua.\nProperties of the ground states that are obscure from the point of view\nof the\nsuperpotentials alone or from the point of view of the partition function\nof the underlying conformal field theory (if known) become rather obvious\nin the orbifold picture.\n\n\\vskip .2truein\n\n\\noindent\n\\section*{Acknowledgement}\n\n\\noindent\nMost of the work described here has been the result of the joint efforts\nof several collaborations. I'm grateful to all the people involved, in\nparticular\nPhilip Candelas, Albrecht Klemm, Max Kreuzer and Monika Lynker.\n\n\\vfill \\eject\n\n\\noindent\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nAs is well known the r\\^ole of symmetry in Physics is fundamental. Symmetries are usually divided in two types: spacetime and internal symmetries. \n\nIf we ignore the effect of gravity that provides spacetime curvature, Special Relativity tells us that our spacetime is the flat Minkowski space and its symmetries are given by the Poincar\\'e group. However, if no massive particle is present, the symmetry group can be larger. The largest transformation group compatible with locality is the conformal group. The conformal group is a finite-dimensional Lie group; its symmetry action is so stringent that one expects only few conformal Quantum Field Theory models to exist in four dimensions, see \\cite{BRNT} and references therein.\n\nA much richer structure comes out in lower dimensional spacetimes. On the two-dimensional Minkowski spacetime the conformal group is infinite dimensional and factors as a product of the diffeomorphism groups of the light-like (chiral) lines $x\\pm t=0$ and acts on the their compactification, namely one gets an action of ${\\mathrm {Diff}}(S^1)\\times{\\mathrm {Diff}}(S^1)$. Restricting a quantum field to a chiral line gives rise to a quantum field theory on $S^1$. \n\nWe do not dwell here on crucial r\\^ole played by Conformal Quantum Field Theory in various physical contexts, nor on its deep mathematical connections with other subjects.\nWe mention however a less emphasised r\\^ole of the subject as a laboratory for more general analysis. This is due to the variety of constructed models and the powerful methods of analysis. We are interested here in particular in Operator Algebraic methods whose effectiveness has  also been shown by recent classification results and new model constructions \\cite{KL1,KLPR,X6}.\n\nInternal symmetries are also fundamental. In particular, in Quantum Field Theory, they are basically expressed by a gauge group. Concerning the particle-antiparticle symmetry, it showed up with the discovery of the Dirac equation and is, in a sense, a derived symmetry. There are general arguments to link the particle-antiparticle symmetry with the spacetime symmetries, see \\cite{GL1}.\n\nNow there is an important higher level symmetry relating spacetime and internal symmetries: supersymmetry, see \\cite{WB}. Supersymmetry sets up a correspondence between Bose and Fermi particles. Leaving aside important physical aspects of supersymmetry, e.g. in relation to the Higgs particle, we wish to mention that the related mathematical structure has profound noncommutative geometrical implications, see \\cite{C,JLO}.\n\nIf one considers a quantum field theory which is both chiral conformal and supersymmetric, one then expects a very stringent interesting structure to show up. Superconformal models have indeed been studied by different approaches. For example, the tricritical Ising model is a basic model with a remarkable structure \\cite{FQS}.\n\nThe purpose of this paper is to initiate a general, model independent, operator algebraic study of superconformal quantum field theory on the circle and pursue this analysis to the classification of the superconformal nets in the discrete series. \n\nIn the first sections we describe the general property of a Fermi conformal net on $S^1$. Basic properties, already considered in the non-local case \\cite{DLR,LR2}, are here described in our context (Bisognano-Wichmann property, twisted Haag duality,\\dots).\n\nA Fermi net ${\\mathcal A}$ contains a local subnet ${\\mathcal A}_b$, the Bose subnet (the fixed-point under the grading symmetry) which is automatically equipped with an involutive sector ${\\sigma}$, dual to the grading. This splits the sectors of ${\\mathcal A}_b$ in two subsets, ${\\sigma}$-Bose and ${\\sigma}$-Fermi sectors, that can be studied in the standard way as ${\\mathcal A}_b$ is local.\n\nFor a general model analysis one would like to consider representations of ${\\mathcal A}$. There is an obvious extension of the notion of DHR representation to Fermi nets, that we study at the beginning. However, as we shall see, there is more general natural class of representations for Fermi net: the \\emph{general representations}. These are representations of the promotion ${\\mathcal A}^{(n)}$ of ${\\mathcal A}$ to the $n$-cover $S^{1(n)}$ of $S^1$, that restrict to DHR representations of the Bose subnet ${\\mathcal A}_b$. We shall see that a general representation is indeed  a representation of the double cover net ${\\mathcal A}^{(2)}$ and can be equivalently described as a \\emph{general soliton}. This splits the general representations in two classes: Neveu-Schwarz (DHR) and Ramond (properly associated with ${\\mathcal A}^{(2)}$). We shall later see that a representation is Neveu-Schwarz (resp. Ramond) iff it comes by $\\alpha$-extension by a ${\\sigma}$-Bose (resp. ${\\sigma}$-Fermi) representation of ${\\mathcal A}_b$.\n\nHaving clarified the structure of the representations of a Fermi net, we consider the case where supersymmetry is present. In particular we consider a \\emph{supersymmetric representation} of a conformal Fermi net ${\\mathcal A}$, namely a graded general representation $\\lambda$ of ${\\mathcal A}$ where \n\\[\n\\tilde H_\\lambda \\equiv H_\\lambda - \\frac{c}{24} = Q_\\lambda^2 \\ ;\n\\]\nhere $H_\\lambda$ is the conformal Hamiltonian and $Q$ is an odd selfadjoint operator (the supercharge). In this case the McKean-Singer lemma (Appendix \\ref{MKS}) gives\n\\[\n\\Str(e^{-t\\tilde H_\\lambda}) = \\text{Fredholm index of}\\ Q_{\\lambda +}\\ , \n\\]\nfor all $t>0$, where $\\Str$ denotes the supertrace. In particular the left hand side, also called the Witten index, is an integer, the Fredholm index $\\ind(Q_{\\lambda +})$ of upper off diagonal part of $Q_\\lambda$.\n\nIf $\\lambda$ is irreducible, its restriction $\\lambda_b$ to ${\\mathcal A}_b$ has two irreducible components $\\lambda_b = \\rho\\oplus\\rho'$ and we can consider the Jones index of $\\rho$, which is the square of the Doplicher-Haag-Roberts statistical dimension $d(\\rho)$ \\cite{L5}.\n\nIf ${\\mathcal A}_b$ is modular, we can express $d(\\rho)$ by the Kac-Peterson Verlinde matrix $S$ as this is equal to the Rehren matrix $S$. By combining all this information and an argument in \\cite{KL05}, we then get the formula\n\\[\n\\ind(Q_{\\lambda +}) = \n\\frac{d(\\rho)}{\\sqrt{\\mu_{\\mathcal A}}}\\sum_{\\nu\\in\\mathfrak R}\nK(\\rho,\\nu)d(\\nu) \n\\]\nwhere $K$ is a matrix that is related to $S$, $\\mu_{\\mathcal A}$ is the $\\mu$-index \\cite{KLM} and the sum is over all Ramond irreducible sectors. This formula thus involves both the Fredholm index and the Jones index.\n\nWe then put our attention towards model analysis and aiming firstly to illustrate concrete situations where our general setting is realised.\n\nThe infinite-dimensional super-Lie algebra that governs the superconformal symmetries is the super-Virasoro algebra. It is a central extension by a central element $c$ of the Lie algebra generated by the Fourier modes $L_n$ and $G_r$ of the Bose and the Fermi stress-energy tensor. Clearly the $L_n$ generate the Virasoro algebra, thus the super-Virasoro algebra contains the Virasoro algebra; the commutation relation are given by the equations \\eqref{svirdef}. It has been studied in \\cite{GKO, FQS}.\n\nThe super-Virasoro algebra plays for superconformal fields the universal same r\\^ole that the Virasoro algebra plays for local conformal fields. Our initial task is then to construct and analyse the super-Virasoro nets. Following the work \\cite{GKO} we describe the nets; if the central charge $c$ is less than $3\/2$ we identify the Bose subnet as a coset, study the representation structure and show the modularity of the net. There is a distinguished representation with lowest weight $c\/24$ which is supersymmetric and thus provides an interesting example for our index formula. \n\nNow, if $c<3\/2$ we are in the discrete series \\cite{FQS}; analogously to the local conformal case \\cite{KL1} every superconformal net is an irreducible finite-index extension of a super-Virasoro net. We classify all such extensions. There are two series of extensions, the one given by the trivial extension and a second one given by index 2 extensions. Beside these there are six exceptional extensions that we describe explicitly.\n\\section{Fermi nets on $S^1$}\nIn this section we discuss the basic notions for a Fermi conformal net of von Neumann algebras on $S^1\\equiv \\{z\\in\\mathbb C: |z|=1\\}$, and their first implications. It is convenient to start by analysing M\\\"obius covariant nets. Note that a general discussion of M\\\"obius covariant net is contained in \\cite{DLR}, here however we focus on the Fermi case.\n\\subsection{M\\\"obius covariant Fermi nets}\n\\label{MobFermiNets}\nWe shall denote by ${\\rm\\textsf{M\\\"ob}}$ the M\\\"obius group, which is isomorphic to $SL(2,\\mathbb R)\/\\mathbb Z_2$ and acts naturally and faithfully on the circle $S^1$. The $n$-cover of ${\\rm\\textsf{M\\\"ob}}$ is denoted by ${\\rm\\textsf{M\\\"ob}}^{(n)}$, $n\\in\\mathbb N\\cup\\{\\infty\\}$. Thus ${\\rm\\textsf{M\\\"ob}}^{(2)}\\simeq SL(2,\\mathbb R)$ and ${\\rm\\textsf{M\\\"ob}}^{(\\infty)}$ is the universal cover of ${\\rm\\textsf{M\\\"ob}}$.\n\nBy an interval of $S^1$ we mean, as usual, a non-empty, non-dense, open, connected subset of $S^1$ and we denote by ${\\mathcal I}$ the set of all intervals. If $I\\in{\\mathcal I}$, then also $I'\\in{\\mathcal I}$ where $I'$ is the interior of the complement of $I$. \n\nA {\\it  net ${\\mathcal A}$ of von Neumann algebras on $S^1$} is a map\n\\[\nI\\in{\\mathcal I}\\mapsto{\\mathcal A}(I)\n\\]\nfrom the set of intervals to the set of von Neumann algebras on a\n(fixed) Hilbert space ${\\mathcal H}$ which verifies the \\emph{isotony property}:\n\\[\nI_1\\subset I_2\\Rightarrow  {\\mathcal A}(I_1)\\subset{\\mathcal A}(I_2)\n\\]\nwhere $I_1 , I_2\\in{\\mathcal I}$.\n\nA \\emph{M\\\"obius covariant net} ${\\mathcal A}$ of von Neumann algebras on $S^1$ is a net of von Neumann algebras on $S^1$ such that the following properties $1-4$ hold:\n\\begin{description}\n\\item[\\textnormal{\\textsc{1. M\\\"obius covariance}}:] {\\it There is a\nstrongly continuous unitary representation $U$ of} ${\\rm\\textsf{M\\\"ob}}^{(\\infty)}$ {\\it on ${\\mathcal H}$ such that}\n\\[\nU(g){\\mathcal A}(I)U(g)^*={\\mathcal A}(\\dot{g}I)\\ ,\n\\qquad g\\in {\\rm\\textsf{M\\\"ob}}^{(\\infty)},\\ I\\in{\\mathcal I} \\ ,\n\\]\n{\\it where $\\dot{g}$ is the image of $g$ in} ${\\rm\\textsf{M\\\"ob}}$ {\\it under the quotient map.}\n\\end{description}\n\\begin{description}\n\\item[$\\textnormal{\\textsc{2. Positivity of the energy}}:$]\n{\\it The generator of the rotation one-para\\-meter subgroup \n$\\theta\\mapsto U({\\rm rot}^{(\\infty)}(\\theta))$ \n(conformal Hamiltonian) is positive}, na\\-me\\-ly $U$ is a positive energy representation\n\\footnote{Here ${\\rm rot}^{(n)}(\\theta)$ is the lift to ${\\rm\\textsf{M\\\"ob}}^{(n)}$ of the $\\theta$-rotation ${\\rm rot}(\\theta)$ of ${\\rm\\textsf{M\\\"ob}}$. For shortness we sometime write ${\\rm rot}^{(n)}(\\theta)$ simply by ${\\rm rot}(\\theta)$ and $U(\\theta) = U({\\rm rot}(\\theta))$.}. \n\\end{description}\n\\begin{description}\n\\item[\\textnormal{\\textsc{3. Existence and uniqueness of the vacuum}}:]\n{\\it There exists a unit $U$-invariant vector $\\Omega$ \n(vacuum vector), unique up to a phase, and $\\Omega$ is\ncyclic for the von~Neumann algebra $\\vee_{I\\in{\\mathcal I}}{\\mathcal A}(I)$}\n\\end{description}\nGiven a M\\\"obius covariant net ${\\mathcal A}$ in $S^1$, a unitary $\\Gamma$ such that $\\Gamma\\Omega = \\Omega$, $\\Gamma{\\mathcal A}(I)\\Gamma^* ={\\mathcal A}(I)$ for all $I\\in{\\mathcal I}$ is called a gauge unitary and the adjoint net automorphism $\\gamma\\equiv{\\hbox{\\rm Ad}}\\Gamma$ a \\emph{gauge} automorphism\\footnote{The requirement $\\Gamma\\Omega =\\Omega$ (up to a phase that can be put equal to one) is equivalent to $\\Gamma U(g) = U(g)\\Gamma$ by the uniqueness of the representation $U$ in Cor. \\ref{unique}; we shall use the second formulation in the non-vacuum case.}. \n\nA $\\mathbb Z_2$-grading on ${\\mathcal A}$ is an involutive gauge automorphism $\\gamma$ of ${\\mathcal A}$. Given the grading $\\gamma$, an element $x$ of ${\\mathcal A}$ such that $\\gamma(x)=\\pm x$ is called homogeneous, indeed a Bose or Fermi element according to the $\\pm$ alternative. \nWe shall say that the degree $\\partial x$ of the homogeneous element $x$ is $0$ in the Bose case and $1$ in the Fermi case.\n\nEvery element $x$ of ${\\mathcal A}$ is uniquely the sum $x = x_0 + x_1$ with $\\partial x_k=k$, indeed \n$x_k = \\big(x  + (-1)^k \\gamma(x)\\big)\/2$.\n\nA  {\\em M\\\"obius covariant Fermi net} ${\\mathcal A}$ on $S^1$ is a $\\mathbb Z_2$-graded M\\\"obius covariant net satisfying graded locality, namely a  M\\\"obius covariant net of \nvon Neumann algebras on $S^1$ such that the following holds:\n\\begin{description}\n\\item[\\textnormal{\\textsc{4. Graded locality}}:]\n{\\it There exists a grading automorphism $\\gamma$ of ${\\mathcal A}$ such that,  if $I_1$ and $I_2$ are disjoint intervals,}\n\\[\n[x,y] = 0,\\quad x\\in{\\mathcal A}(I_1), y\\in{\\mathcal A}(I_2) \\ .\n\\]\n\\end{description}\nHere $[x,y]$ is the graded commutator with respect to the grading automorphism $\\gamma$ defined as follows: if $x,y$ are homogeneous then\n\\[\n[x,y]\\equiv xy - (-1)^{\\partial x \\cdot \\partial y}yx\n\\]\nand, for the general elements $x,y$, is  extended by linearity.\n\nNote the \\emph{Bose subnet} ${\\mathcal A}_b$, namely the $\\gamma$-fixed point subnet ${\\mathcal A}^\\gamma$ of degree zero elements, is local. Moreover, setting\n\\[\nZ\\equiv \\frac{1 - i\\Gamma}{1 - i}\n\\]\nwe have that the unitary $Z$ fixes $\\Omega$ and\n\\[\n{\\mathcal A}(I')\\subset Z{\\mathcal A}(I)'Z^*,\n\\]\n(twisted localiy w.r.t. $Z$), that is indeed equivalent to graded locality.\n\\medskip\n\n\\noindent\n{\\it Remark.} Strictly speaking a M\\\"obius covariant net is a pair $({\\mathcal A}, U)$ where ${\\mathcal A}$ is a net and $U$ is a unitary representation of ${\\rm\\textsf{M\\\"ob}}$ that satisfy the above properties. In most cases it is convenient to denote the M\\\"obius covariant net simply by ${\\mathcal A}$ and then consider the unitary representation $U$ (note that $U$ is unique once we fix the vacuum vector). However, later in this paper, it will be convenient to use the notation $({\\mathcal A}, U)$ (also in the diffeomorphism covariant case).\n\\subsection{First consequences}\nWe collect here a few first consequences of the axioms of a M\\\"obius covariant Fermi net on $S^1$. They can be mostly derived by a simple extension of the proofs in the local case, cf. \\cite{DLR,LR2}.\n\\subsubsection*{Reeh-Schlieder theorem}\n\\begin{theorem}\\label{Reeh-Schlieder} Let ${\\mathcal A}$ be a M\\\"obius covariant Fermi net on\n$S^1$.  Then $\\Omega$ is cyclic and separating for each von Neumann\nalgebra ${\\mathcal A}(I)$, $I\\in{\\mathcal I}$.\n \\end{theorem}\n \\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }]\nThe cyclicity of $\\Omega$ follows exactly as in the local case. By twisted locality, then $\\Omega$ is separating too.\n\\null\\hfill\\qed\\endtrivlist\\noindent\n\\subsubsection*{Bisognano-Wichmann property}\nIf $I\\in {\\mathcal I}$, we shall denote by $\\Lambda_I$ the one parameter subgroup of ${\\rm\\textsf{M\\\"ob}}$ of ``dilation associated with $I$\\ \\!'', see \\cite{BGL}. We shall use the same symbol also to denote the unique one parameter subgroup of ${\\rm\\textsf{M\\\"ob}}^{(n)}$  ($n$ finite or infinite) that project onto $\\Lambda_I$ under the quotient map. The following theorem and its corollary are proved as usual, see \\cite{DLR}.\n\\begin{theorem}\nLet $I\\in{\\mathcal I}$ and $\\Delta_I$, $J_I$ be the modular operator and the modular \nconjugation of $({\\mathcal A}(I),\\Omega)$. Then we have:\n\n$(i)$: \n\\begin{equation}\\label{BW}\n\\Delta_{I}^{it} = U(\\Lambda_I(-2\\pi t)), \\ t\\in\\mathbb R,\n\\end{equation}\n\n$(ii)$: \n$U$ extends to an (anti-)unitary representation of {\\rm ${\\rm\\textsf{M\\\"ob}}^{(\\infty)}\\ltimes\\mathbb Z_2$}\ndetermined by\n\\[\nU(r_I)=ZJ_I,\\ I\\in{\\mathcal I},\n\\]\nacting covariantly on ${\\mathcal A}$, namely\n\\[\nU(g){\\mathcal A}(I)U(g)^*={\\mathcal A}(\\dot{g}I)\\quad g\\in\\text{\\rm ${\\rm\\textsf{M\\\"ob}}^{(\\infty)}$}\\ltimes\\mathbb Z_2\\ I\\in {\\mathcal I}\\ .\n\\]\nHere $r_I:S^1\\to S^1$ is the reflection mapping $I$ onto $I'$, see \\cite{BGL}.\n\\end{theorem}\n\\begin{corollary}\\label{unique} {\\em (Uniqueness of the M\\\"obius representation)}\nThe representation $U$ is unique.\n\\end{corollary}\n\\begin{corollary} {\\em (Additivity)}\n  Let $I$ and $I_i$ be intervals with $I\\subset\\cup_i I_i$. Then \n  ${\\mathcal A}(I)\\subset\\vee_i{\\mathcal A}(I_i)$.\n\\end{corollary}\n\\medskip\n\n\\noindent\n{\\it Remark.} If $E\\subset S^1$ is any set, we denote by ${\\mathcal A}(E)$ the von Neumann algebra generated by the ${\\mathcal A}(I)$'s as $I$ varies in the intervals $I\\in{\\mathcal I}$, $I\\subset E$. It follows easily by M\\\"obius covariance that if $I_0\\in{\\mathcal I}$ has closure $\\bar I_0$, then ${\\mathcal A}(\\bar I_0)=\\bigcap_{I\\supset {\\bar I_0}, I\\in{\\mathcal I}}{\\mathcal A}(I)$.\n\\subsubsection*{Twisted  Haag duality}\n\\begin{theorem}\nFor every  $I\\in{\\mathcal I}$, we have:\n\\[\n{\\mathcal A}(I')=Z{\\mathcal A}(I)'Z^*\n\\]\n\\end{theorem}\n\\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }] As already noted twisted locality ${\\mathcal A}(I)' \\supset Z{\\mathcal A}(I')Z^*$ holds as a consequence of the Bose-Fermi commutation relations. By the Bisognano-Wichmann property, $Z{\\mathcal A}(I')Z^*$ is a von Neumann subalgebra of ${\\mathcal A}(I)'$ globally invariant under the vacuum modular group. As the vacuum vector is cyclic for ${\\mathcal A}(I')$ by the Reeh-Schlieder theorem, we have ${\\mathcal A}(I)' = Z{\\mathcal A}(I')Z^*$ by the Tomita-Takesaki modular theory.\n\\null\\hfill\\qed\\endtrivlist\\noindent\nIn the following corollary, the grading and the graded commutator is considered on $B({\\mathcal H})$ w.r.t. ${\\hbox{\\rm Ad}}\\Gamma$.\n\\begin{corollary}\\label{td}\n${\\mathcal A}(I')= \\big\\{x\\in B({\\mathcal H}):\\ [x,y]=0\\ \\forall y\\in{\\mathcal A}(I)\\big\\}$.\n\\end{corollary}\n\\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }]\nSet $X\\equiv\\{x\\in B({\\mathcal H}):\\ [x,y]=0\\ \\forall y\\in{\\mathcal A}(I)\\}$. We have to show that $X\\subset{\\mathcal A}(I')$. \nIf $x = x = x_0 + x_1\\in X$ then $Z^*xZ =  x_0 - ix_1\\Gamma$. As $[x,y]=0$ for all $y\\in{\\mathcal A}(I)$ it follows that $Z^*xZ\\in{\\mathcal A}(I)'$ so $x\\in Z{\\mathcal A}(I)'Z^* = {\\mathcal A}(I')$.\n\\null\\hfill\\qed\\endtrivlist\\noindent\n\\subsubsection*{Irreducibility}\nWe shall say that ${\\mathcal A}$ is \\emph{irreducible} if the von Neumann algebra $\\vee{\\mathcal A}(I)$\ngenerated by all local algebras coincides with $B({\\mathcal H})$.\n\nThe irreducibility property is indeed equivalent to several other requirements.\n\\begin{proposition}\\label{irr} Assume all properties $1-4$ for ${\\mathcal A}$ except the uniqueness of the vacuum. The following are equivalent:\n\\begin{itemize}     \n\\item[{$(i)$}] $\\mathbb C\\Omega$ are the only $U$-invariant vectors.\n\\item[{$(ii)$}] The algebras ${\\mathcal A}(I)$, $I\\in{\\mathcal I}$, are\nfactors. In particular they are type III$_1$ factors (or ${\\rm dim}\\, {\\mathcal H} = 1$).\n\\item[{$(iii)$}] The net ${\\mathcal A}$ is irreducible.\n\\end{itemize}\n\\end{proposition}\n\\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }] See \\cite{DLR}. \n\\null\\hfill\\qed\\endtrivlist\\noindent\n\\subsubsection*{Vacuum spin-statistics relation}\n\\label{vacss} \nThe relation $U(2\\pi)=1$ holds in the local case \\cite{GL2}; in the graded local case it generalizes to\n\\[\nU(4\\pi)=1\\ ,\n\\]\nwhere $U(s) \\equiv U({\\rm rot}(s)) = e^{isL_0}$ is the rotation one-parameter unitary group, see \\cite{DLR}. Indeed we have the following.\n\\begin{proposition}\\label{vss}\nLet ${\\mathcal A}$ be M\\\"obius covariant Fermi net. Then:\n\\[\nU(2\\pi) =\\Gamma\\ .\n\\]\n\\end{proposition}\n\\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }]\nBy restricting to ${\\mathcal A}_b$ the identity representation of ${\\mathcal A}$ we get the direct sum $\\iota\\oplus{\\sigma}$ of the identity and an automorphism. Clearly ${\\sigma}$ has Fermi statistics because ${\\mathcal A}$ has Bose-Fermi commutation relations. Thus $U = U_\\iota \\oplus U_{\\sigma}$; by the conformal spin-statistics theorem \\cite{GL2} we then have $U(2\\pi)= U_\\iota (2\\pi) \\oplus U_{\\sigma} (2\\pi) = 1\\oplus -1 = \\Gamma$.\n\\null\\hfill\\qed\\endtrivlist\\noindent\n\\begin{corollary} {\\em (Uniqueness of the grading)}\nThe grading automorphism is unique.\n\\end{corollary}\n\\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }]\nImmediate from the uniqueness of the M\\\"obius unitary representation and the spin-statistics relation $U(2\\pi) =\\Gamma$.\n\\null\\hfill\\qed\\endtrivlist\\noindent\n\\subsection{Fermi conformal nets on $S^1$}\nIf ${\\mathcal A}$ is a M\\\"obius covariant Fermi net on $S^1$ then, by the spin-statistics relation, the unitary representation of ${\\rm\\textsf{M\\\"ob}}^{(\\infty)}$ is indeed a representation of ${\\rm\\textsf{M\\\"ob}}^{(2)}$. As ${\\rm\\textsf{M\\\"ob}}$ is naturally a subgroup of the group ${\\mathrm {Diff}}(S^1)$ of orientation preserving diffeomorphisms of $S^1$, clearly ${\\rm\\textsf{M\\\"ob}}^{(2)}$  is naturally a subgroup of ${\\mathrm {Diff}}^{(2)}(S^1)$, the 2-cover of ${\\mathrm {Diff}}(S^1)$.\n\nGiven an interval $I\\in{\\mathcal I}$, we shall denote by ${\\mathrm {Diff}}_I(S^1)$ the subgroup of diffeomorphisms $g$ of $S^1$ localised in $I$, namely $g(t)=t$ for all $t\\in I'$, and by\n${\\mathrm {Diff}}_I^{(2)}(S^1)$ the connected component of the identity of the pre-image of \n${\\mathrm {Diff}}_I(S^1)$ in ${\\mathrm {Diff}}^{(2)}(S^1)$.\n\nA \\emph{Fermi conformal net} ${\\mathcal A}$ (of von Neumann algebras) on $S^1$ is a M\\\"obius covariant Fermi net of von Neumann algebras on $S^1$ such that the following holds:\n\\begin{description}\n\\item[\\textnormal{\\textsc{5. Diffeomorphism covariance}}:]\n{\\it There exists a projective unitary representation $U$ of ${\\mathrm {Diff}}^{(2)}(S^1)$ on ${\\mathcal H}$, extending the unitary representation of} ${\\rm\\textsf{M\\\"ob}}^{(2)}$, {\\it such that\n\\[\nU(g){\\mathcal A}(I)U(g)^* = {\\mathcal A}(\\dot{g}I),\\ g\\in{\\mathrm {Diff}}^{(2)}(S^1),\\ I\\in{\\mathcal I},\n\\]\nand\n\\[\nU(g)xU(g)^* = x, \\ x\\in{\\mathcal A}(I'),\\ g\\in{\\mathrm {Diff}}_I^{(2)}(S^1), \\ I\\in{\\mathcal I}\\ .\n\\]}\n\\end{description}\nHere $\\dot{g}$ denotes the image of $g$ in ${\\mathrm {Diff}}(S^1)$ under the quotient map.\n\\begin{lemma}\nFor every $g\\in{\\mathrm {Diff}}^{(2)}(S^1)$ we have $U(g)\\Gamma = \\Gamma U(g)$. In particular,\nif $g\\in{\\mathrm {Diff}}_I^{(2)}(S^1)$, then $U(g)\\in{\\mathcal A}_b(I)$.\n\\end{lemma}\n\\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }]\nAs $\\Gamma= U(2\\pi)$, we have\n$\n\\Gamma U(g) \\Gamma^* = U(2\\pi)U(g)U(2\\pi)^*=\\chi(g) U(g)\n$ for all  $g\\in{\\mathrm {Diff}}^{(2)}(S^1)$,\nwhere $\\chi$ is a continuos scalar function on ${\\mathrm {Diff}}^{(2)}(S^1)$. As $\\Gamma^2 =1$, we have $\\chi(g)^2 =1$ for all $g$, thus $\\chi(g) =1$ because ${\\mathrm {Diff}}^{(2)}(S^1)$ is connected and $\\chi(g) =1$ is $g$ is the identity.\n\nWith $g\\in{\\mathrm {Diff}}_I^{(2)}(S^1)$, then $U(g)$ commutes with $\\Gamma$, hence with $Z$. By the covariance condition $U(g)$ commutes with ${\\mathcal A}(I')$, hence with $Z{\\mathcal A}(I')Z^*$, so $U(g)\\in{\\mathcal A}(I)$ by twisted Haag duality.\n\\null\\hfill\\qed\\endtrivlist\\noindent\nBy the above lemma, the representation of the diffeomorphism group belongs to the Bose subnet, so we may apply the uniqueness result in the local case in \\cite{CW} and get the following:\n\\begin{corollary} \\emph{(Uniqueness of the diffeomorphism representation)}\nThe projective unitary representation $U$ of ${\\mathrm {Diff}}^{(2)}(S^1)$ is unique (up to a projective phase).\n\\end{corollary}\n\\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }]\nWith $E = (1 +\\Gamma)\/2$ the orthogonal projection onto $\\overline{{\\mathcal A}_b \\Omega}$, clearly $U|_{E{\\mathcal H}}$ is the projective unitary representation of ${\\mathrm {Diff}}(S^1)$ associated with ${\\mathcal A}_b$, unique by \\cite{CW,W2}. As $\\Omega$ is separating for the local algebras, $U(g)E$ determines $U(g)$ if \n$g\\in{\\mathrm {Diff}}_I^{(2)}(S^1)$ for any interval $I\\in{\\mathcal I}$. By Prop. \\ref{locdiff}, as $I$ varies, ${\\mathrm {Diff}}_I(S^1)$ algebraically generates all ${\\mathrm {Diff}}(S^1)$, and $U$ is so determined up to a phase.\n\\null\\hfill\\qed\\endtrivlist\\noindent\n\\subsection{DHR representations}\nThere is a natural notion of representation for a Fermi net which is the \nstraightforward extension of the notion of representation for a local net, see \\eqref{rep} here below. We shall see in later sections that is important to consider also more general representations when dealing with a Fermi net. In this section, however, we deal with the most obvious notion.\n\nIn the following we assume that ${\\mathcal A}$ be a Fermi conformal net, although certain notions and results are obviously valid in the M\\\"obius covariant case.\n\nA \\emph{representation} $\\lambda$ of ${\\mathcal A}$ is a map $I\\to\\lambda_I$ that \nassociates to an interval $I$ of $S^1$ a normal\\footnote{The normality of $\\lambda_I$ is automatic if ${\\mathcal H}_\\lambda$ is separable because ${\\mathcal A}(I)$ is a type $III$ factor.}\nrepresentation $\\lambda_I$ of ${\\mathcal A}(I)$ on a fixed Hilbert space ${\\mathcal H}_{\\lambda}$ such that\n\\begin{equation}\n\\label{rep}\n\\lambda_{\\tilde I}|_{{\\mathcal A}(I)} = \\lambda_I,\\quad I\\subset \\tilde I\\ .\n\\end{equation}\nWe shall say that a representation $\\lambda$ on ${\\mathcal H}_{\\lambda}$ \nis {\\em diffeomorphism covariant} \nif there exists a projective unitary representation $U_\\lambda$ of the universal cover \n${\\mathrm {Diff}}^{(\\infty)}(S^1)$ of ${\\mathrm {Diff}}(S^1)$ on ${\\mathcal H}_{\\lambda}$ such that\n\\[\n\\lambda_{\\dot{g}I}\\big(U(g)xU(g)^*\\big) = U_\\lambda(g)\\lambda_I(x)U_\\lambda(g)^*\\ ,\\  x\\in{\\mathcal A}(I) , \n\\forall g\\in {\\mathrm {Diff}}^{(\\infty)}(S^1)\\ .\n\\]\nHere $\\dot g$ denotes the image of $g$ in ${\\mathrm {Diff}}(S^1)$ under the quotient map.\nA M\\\"obius covariant representation is analogously defined.\n\nAs we shall later deal with more general representations, we may sometime emphasise that we are considering a representation $\\lambda$ as above, by saying that $\\lambda$ is a DHR representation, the `DHR' being however pleonastic.\n\nWe shall say that a representation $\\lambda$ on ${\\mathcal H}_{\\lambda}$ is \\emph{graded}\nif there exists a unitary $\\Gamma_{\\lambda}$ on ${\\mathcal H}_{\\lambda}$ such that\n\\[\n\\lambda_I(\\gamma(x)) = \\Gamma_{\\lambda}\\lambda_I(x)\\Gamma^*_{\\lambda},\\quad x\\in{\\mathcal A}(I) ,\n\\]\nfor all $I\\in{\\mathcal I}$. As $\\gamma$ is involutive one may then also choose a selfadjoint $\\Gamma_\\lambda$.\n\\begin{proposition}\\label{DHRgraded}\nLet $\\lambda$ be an irreducible DHR representation of the Fermi conformal net ${\\mathcal A}$. Then $\\lambda$ is graded and diffeomorphism covariant with positive energy. Moreover we may take $\\Gamma_\\lambda = U_\\lambda(2\\pi)$.\n\\end{proposition}\n\\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }]\nWe first use an argument in \\cite{KL1}. Let $I\\in{\\mathcal I}$ and $g\\in{\\mathrm {Diff}}_I^{(2)}(S^1)$. For every $x\\in{\\mathcal A}(\\tilde I)$ with $\\tilde I\\supset I$ we have $\\lambda_I(U(g))\\lambda_{\\tilde I}(x)\\lambda_I(U(g))^* = \\lambda_{\\tilde I}(U(g)xU(g))^* = \\lambda_{gI}(U(g)xU(g))^*$. As the group generated by ${\\mathrm {Diff}}_I^{(2)}(S^1)$ as $I$ varies in ${\\mathcal I}$ is the entire ${\\mathrm {Diff}}^{(2)}(S^1)$, we see every diffeomorphism is implemented in the representation $\\lambda$ by a unitary $U_\\lambda(g)$, which is equal to $\\lambda_I(U(g))$ if $g$ is localised in $I$.\n\nIt remains to show that, by multiplying $U_\\lambda(g)$ by a phase factor, we can get a continuous projective representation with positive energy.\nIn the local case, the automatic diffeomorphism covariance is proved in \\cite{DFK} and the automatic positivity of the energy in \\cite{W}. Let $U_{\\lambda_b}$ be the covariance unitary representation associated with $\\lambda_b\\equiv \\lambda |_{{\\mathcal A}_b}$. If $g$ is localised in $I$ then $U_\\lambda(g)U_{\\lambda_b}(g)^*\\in\\lambda_I\\big({\\mathcal A}_b(I)'\\cap{\\mathcal A}(I)\\big)=\\mathbb C$ (cf. Lemma \\ref{outer}), so we are done by replacing $U_\\lambda$ with $U_{\\lambda_b}$.\n\nFinally notice that, by diffeomorphism covariance, it follows that \n\\[\nU_\\lambda(2\\pi)\\lambda_I(x)U_\\lambda(2\\pi)^* = \\lambda_I(\\gamma(x))\n\\]\ndue to Prop. \\ref{vss}. So we may take $\\Gamma_\\lambda = U_\\lambda(2\\pi)$.\n\\null\\hfill\\qed\\endtrivlist\\noindent\nA \\emph{localised endomorphism} of ${\\mathcal A}$ is an endomorphism $\\rho$ of the universal $C^*$-algebra $C^*({\\mathcal A})$ such that $\\rho|_{{\\mathcal A}(I')}$ is the identity for some $I\\in{\\mathcal I}$ (then we say that $\\rho$ is localised in the interval $I$). In other words, $\\rho$ is a representation such that $\\rho_{I'}= \\iota $ and $\\rho_{\\tilde I}$ maps ${\\mathcal A}(\\tilde I)$ into itself if $\\tilde I\\supset I$. This last property is automatic by Haag duality in the local case, and we now see to hold also in the Fermi case.\n\nNext lemma shows that the grading is locally outer. This applies indeed to every gauge automorphism, \nsee \\cite{Car99,X01}.\n\\begin{lemma}\\label{outer}\nGiven any interval $I$, $\\gamma|_{{\\mathcal A}(I)}$ is an outer automorphism (unless the grading is trivial). As a consequence ${\\mathcal A}_b(I)'\\cap{\\mathcal A}(I)=\\mathbb C$.\n\\end{lemma}\n\\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }]\nSuppose $\\gamma|_{{\\mathcal A}(I)}$ is inner; then there exist unitaries $u\\in{\\mathcal A}(I)$ and $u'\\in{\\mathcal A}(I)'$ such that $\\Gamma =  u'u$. In fact $u$ and $u'$ are unique up to a phase factor because ${\\mathcal A}(I)$ is a factor. Now $\\Gamma$ commutes with the M\\\"obius unitary group, so $\\Delta_I^{it}u\\Omega = e^{iat}u\\Omega$ for some $a\\in\\mathbb R$. But $\\log\\Delta_I$ has no non-zero eigenvalue (see \\cite{DLR}), so $u\\Omega \\in\\mathbb C\\Omega$, thus $u$ is a scalar $\\gamma$ is the identity.\n\\null\\hfill\\qed\\endtrivlist\\noindent\nThe following proposition generalizes to the Fermi case the well known DHR argument for the correspondence between representations and localised endomorphisms on the Minkowski space. \n\\begin{proposition}\\label{dhrendo}\nLet $\\pi$ be a representation of the Fermi conformal net ${\\mathcal A}$ on $S^1$, and suppose the Hilbert spaces ${\\mathcal H}$ and ${\\mathcal H}_\\pi$ to be separable. Given an interval $I$, there exists a localised endomorphism of ${\\mathcal A}$ unitarily equivalent to $\\pi$.\n\\end{proposition}\n\\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }]\nLet $\\Gamma$ be the unitary, $\\Omega$-fixing implementation of $\\gamma$. As $\\gamma|_{{\\mathcal A}(I')}$ is outer, the algebra $\\langle {\\mathcal A}(I'),\\Gamma\\rangle$ generated by ${\\mathcal A}(I')$ and $\\Gamma$ is the von Neumann algebra crossed product of ${\\mathcal A}(I')$ by $\\gamma|_{{\\mathcal A}(I')}$.\n\nNow $\\pi$ is graded and normal so, by choosing $\\Gamma_\\pi$ selfadjoint, also the algebra $\\langle \\pi_{I'}({\\mathcal A}(I')),\\Gamma_\\pi\\rangle$ generated by $\\pi_{I'}({\\mathcal A}(I'))$ and $\\Gamma_\\pi$ is naturally isomorphic to the von Neumann algebra crossed product of ${\\mathcal A}(I')$ by $\\gamma|_{{\\mathcal A}(I')}$.\n\nSo there exists an isomorphism $\\Phi:\\langle {\\mathcal A}(I'),\\Gamma\\rangle\\to\\langle \\pi_{I'}({\\mathcal A}(I')),\\Gamma_\\pi\\rangle$ such that $\\Phi|_{{\\mathcal A}(I')} =\\pi_{I'}$ and $\\Phi(\\Gamma)=\\Gamma_\\pi$, namely\n\\[\n\\Phi: x + y\\Gamma \\mapsto \\pi_{I'}(x) + \\pi_{I'}(y)\\Gamma_\\pi\\ ,\\quad x,y\\in{\\mathcal A}(I')\\ .\n\\]\nAs ${\\mathcal A}(I')$ is a type III factor, also $\\langle {\\mathcal A}(I'),\\Gamma\\rangle$ is a type III factor. Thus $\\Phi$ is spatial and there exists a unitary $U:{\\mathcal H}\\to{\\mathcal H}_\\pi$ such that $\\Phi(x) = U x U^*$ for all $x\\in{\\mathcal A}(I')$.\n\nSet $\\rho \\equiv U^*\\pi(\\cdot) U$. Clearly $\\rho$ is a representation of ${\\mathcal A}$ on ${\\mathcal H}$ that is unitarily equivalent to $\\pi$ and such that $\\rho_{I'}$ acts identically on ${\\mathcal A}(I')$. \n\nMoreover $\\rho\\cdot\\gamma = {\\hbox{\\rm Ad}}\\Gamma\\cdot \\rho$. Indeed, since $ U^*\\Gamma_\\pi= \\Gamma U^*$, we have\n\\[\n\\rho\\cdot\\gamma = {\\hbox{\\rm Ad}} U^*\\cdot\\pi\\cdot{\\hbox{\\rm Ad}}\\Gamma \n={\\hbox{\\rm Ad}} U^*\\Gamma_\\pi\\cdot\\pi  = {\\hbox{\\rm Ad}}\\Gamma U^*\\cdot\\pi ={\\hbox{\\rm Ad}}\\Gamma\\cdot\\rho \\ .\n\\]\nWith $x\\in{\\mathcal A}(I)$, we now want to show that $\\rho_I(x)\\in{\\mathcal A}(I)$. Indeed for all $y\\in{\\mathcal A}(I_0)$ with $\\bar I_0\\subset I'$ we have $[x,y]=0$, where the brackets denote the graded commutator. Therefore, choosing an interval $\\tilde I\\supset I'\\cup I_0$, we have\n\\begin{equation*}\n[\\rho_I(x),y] = [\\rho_{\\tilde I}(x),\\rho_{\\tilde I}(y)] = \\rho_{\\tilde I}([x,y]) = 0\\ .\n\\end{equation*}\nIt then follows by Cor. \\ref{td} that $\\rho_I(x)\\in{\\mathcal A}(I)$.\n\\null\\hfill\\qed\\endtrivlist\\noindent\n\\subsection{${\\sigma}$-Bose and ${\\sigma}$-Fermi sectors of the Bosonic subnet}\n\\label{sfb}\nAs above, let $\\gamma$ be the vacuum preserving, involutive grading automorphism of the Fermi net\n${\\mathcal A}$ on $S^1$ as above and ${\\mathcal A}_b$ the fixed-point subnet. \nWe denote by ${\\sigma}$ a representative of the sector of ${\\mathcal A}_b$ dual to $\\gamma$. Choosing \nan interval $I_0\\subset\\mathbb R$, there is a unitary \n\\[\nv\\in{\\mathcal A}(I),\\: v^*=v,\\: \\gamma(v)= -v \\ .\n\\]\nThen  we may take  ${\\sigma}\\equiv{\\hbox{\\rm Ad}} v |_{{\\mathcal A}_b}$, so ${\\sigma}$ is an automorphism of ${\\mathcal A}_b$ localised in \n$I_0$. We have $d({\\sigma})=1$ and ${\\sigma}^2 = 1$.\n\nGiven DHR endomorphisms $\\mu$ and $\\nu$ of ${\\mathcal A}_b$ \nwe denote by $\\varepsilon(\\mu,\\nu)$ the right (clockwise) statistics operator (see \\cite{R1,GL2}).\n\nThe \\emph{monodromy operator}  is given by\n\\[\nm(\\mu,\\nu)\\equiv\\varepsilon(\\mu,\\nu)\\varepsilon(\\nu,\\mu).\n\\]\nNote that if $\\mu$ is localised left to $\\nu$, then $\\varepsilon(\\nu,\\mu)=1$ \nand thus $m(\\mu,\\nu)=\\varepsilon(\\mu,\\nu)$.\n\nWe shall also set \n\\[\nK(\\mu,\\nu)\\equiv \\Phi_{\\nu}(m(\\nu,\\mu)^*)=\\Phi_{\\nu}(\\varepsilon(\\mu,\\nu)^*\\varepsilon(\\nu,\\mu)^*)\n\\]\nwhere $\\Phi_{\\nu}$ is the left inverse of $\\nu$. \nAs $\\varepsilon(\\mu,\\nu)\\in{\\hbox{Hom}}(\\mu\\nu,\\nu\\mu)$, we have $m(\\mu,\\nu)\\in {\\hbox{Hom}}(\\nu\\mu,\\nu\\mu)$, \ntherefore $K(\\mu,\\nu)\\in {\\hbox{Hom}}(\\mu,\\mu)$ and so, if $\\mu$ is irreducible, \n$K(\\mu,\\nu)$ is a complex number with modulus $\\leq 1$.\n \nLet $\\mu$ be an irreducible endomorphism localised left to ${\\sigma}$. \nAs $\\varepsilon(\\mu,{\\sigma})\\in{\\hbox{Hom}}(\\mu{\\sigma},{\\sigma}\\mu)$ and ${\\sigma}$ and $\\mu$ commute, it \nfollows that $\\varepsilon(\\mu,{\\sigma})$ is scalar. Denoting by $\\iota$ the identity \nsector, by the braiding fusion relation we have\n\\[\n1=\\varepsilon(\\mu,\\iota)=\\varepsilon(\\mu,{\\sigma}^2)\n={\\sigma}(\\varepsilon(\\mu,{\\sigma}))\\varepsilon(\\mu,{\\sigma})=\\varepsilon(\\mu,{\\sigma})\\varepsilon(\\mu,{\\sigma})\\ ,\n\\]\nthus $m(\\mu,{\\sigma})=\\varepsilon(\\mu,{\\sigma})=\\pm 1$.\n\nIf $\\mu$ is not necessarily irreducible, we shall say that $\\mu$ is \n\\emph{${\\sigma}$-Bose} \nif $m(\\mu,{\\sigma})= 1$ and that $\\mu$ is \\emph{${\\sigma}$-Fermi} \nif $m(\\mu,{\\sigma})= -1$. As we have seen, if $\\mu$ is irreducible then \n$\\mu$ is either ${\\sigma}$-Bose or ${\\sigma}$-Fermi. With $S$ Rehren matrix \\eqref{matS} we have:\n\\begin{proposition}\\label{DS}\nLet $\\rho,\\nu$ be irreducible, localized endomorphisms of ${\\mathcal A}_b$ and $\\rho'\\equiv \\rho{\\sigma}$. We have\n$S_{\\rho',\\nu} =\\pm S_{\\rho,\\nu}$ according with $\\rho$ is ${\\sigma}$-Bose or \n${\\sigma}$-Fermi. \n\\end{proposition}\n\\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }]\nBy definition\n\\[\nS_{\\rho',\\nu}=S_{\\rho{\\sigma},\\nu}=\n\\frac{d(\\rho')d(\\nu)}{\\sqrt{\\mu_{\\mathcal A}}}\\Phi_{\\nu}(\\varepsilon(\\rho{\\sigma},\\nu)^*\\varepsilon(\\nu,\\rho{\\sigma})^*)\n\\]\nand we have $d(\\rho')=d(\\rho)$. We may also assume that $\\rho$, ${\\sigma}$ and $\\nu$ are localised one left to the next. We have\n\\[\nm(\\rho',\\nu)=\\varepsilon(\\rho{\\sigma},\\nu)=\\rho(\\varepsilon({\\sigma},\\nu))\\varepsilon(\\rho,\\nu) =\\pm \\varepsilon(\\rho,\\nu) = \\pm m(\\rho,\\nu)\n\\]\nas $\\varepsilon({\\sigma},\\nu)=m({\\sigma},\\nu)=\\pm 1$; thus $S_{\\rho',\\nu} =\\pm S_{\\rho,\\nu}$ where the sign depends on the ${\\sigma}$-Bose\/${\\sigma}$-Fermi alternative for $\\nu$.\n\\null\\hfill\\qed\\endtrivlist\\noindent\nThus\n\\[\nD_{\\rho,\\nu} \\equiv S_{\\rho,\\nu} - S_{\\rho',\\nu}= \n2\\frac{d(\\rho)}{\\sqrt{\\mu_{{\\mathcal A}_b}}}\\cdot\\begin{cases} 0\\qquad &\\text{if $\\nu$ is ${\\sigma}$-Bose}\\\\\nK(\\rho,\\nu)d(\\nu)\\qquad &\\text{if $\\nu$ is ${\\sigma}$-Fermi}\n\\end{cases}\n\\]\n\\subsection{Graded tensor product} \nWe briefly recall the notion of graded tensor product of Fermi nets. With ${\\mathcal A}$ a Fermi net on $S^1$, denote by ${\\mathcal A}_f(I)$ the Fermi (i.e. degree one) subspace of ${\\mathcal A}(I)$. If $v\\in{\\mathcal A}_f(I)$ is a selfadjoint unitary then ${\\mathcal A}(I)= \\langle{\\mathcal A}_b(I),v\\rangle$ is the crossed product ${\\mathcal A}_b(I)\\rtimes \\mathbb Z_2$ with respect to the action ${\\sigma} ={\\hbox{\\rm Ad}} v$ on ${\\mathcal A}_b(I)$. If $W\\in{\\mathcal A}(I)'$ is a selfadjoint unitary, $W v$ also implements ${\\sigma}$ on ${\\mathcal A}_b(I)$ and the von Neumann algebra $\\langle{\\mathcal A}_b(I),W v\\rangle$ is also isomorphic to ${\\mathcal A}_b(I)\\rtimes \\mathbb Z_2$, namely we have an isomorphism\n\\[\na\\to a,\\quad f\\to W f, \\qquad \\partial a = 0, \\partial f = 1\\ .\n\\]\nLet now, for $i=1,2$, ${\\mathcal A}_i$ be a Fermi net on $S^1$ on the Hilbert \nspace ${\\mathcal H}_i$, and let $\\Gamma_i$ be the associated grading unitary. \nGiven an interval $I\\in{\\mathcal I}$, define the von Neumann algebra on ${\\mathcal H}_1\\otimes{\\mathcal H}_2$\n\\[\n{\\mathcal A}_1(I)\\hat\\otimes{\\mathcal A}_2(I)\\equiv\n\\{a_1\\otimes a_2,\\  f_1\\otimes 1,\\  \\Gamma_1\\otimes f_2\\}''\n\\]\nwhere $a_i,f_i\\in{\\mathcal A}_i(I)$, $\\partial a_i =0$ and $\\partial f_i =1$. Namely ${\\mathcal A}_1(I)\\hat\\otimes{\\mathcal A}_2(I) $ is the direct sum\n\\[\n\\underbrace{{\\mathcal A}_{1b}(I)\\otimes{\\mathcal A}_{2b}(I) + \\Gamma_1{\\mathcal A}_{1f}(I)\\otimes{\\mathcal A}_{2f}(I)}_{\\rm Bosons}\n+ \\underbrace{{\\mathcal A}_{1f}(I)\\otimes {\\mathcal A}_{2b}(I) + \\Gamma_1 {\\mathcal A}_{1b}(I)\\otimes{\\mathcal A}_{2f}(I)}_{\\rm Fermions}\n\\]\nClearly the map $I\\to {\\mathcal A}_1(I)\\hat\\otimes{\\mathcal A}_2(I)$ is isotone and satisfies graded\nlocality with respect to the grading induced by $\\Gamma_1 \\otimes \\Gamma_2$. \nThus it defines a Fermi net ${\\mathcal A}_1 \\hat\\otimes {\\mathcal A}_2$ on ${\\mathcal H}_1\\otimes {\\mathcal H}_2$. \n\nBy the previous comments, with ${\\mathcal A}(I) = 1 \n\\otimes {\\mathcal A}_2(I)$ and $W=\\Gamma_1\\otimes 1$, the von Neumann algebra $\\hat{\\mathcal A}_2(I)$ on \n${\\mathcal H}_1\\otimes{\\mathcal H}_2$ generated by $1\\otimes{\\mathcal A}_{2b}(I)$ and $\\Gamma_1\\otimes {\\mathcal A}_{2f}(I)$ \nis isomorphic to $1\\otimes{\\mathcal A}_2(I)$ and hence to ${\\mathcal A}_2(I)$. Actually, an easy \ncalculation show that \nif $a, f \\in {\\mathcal A}_2(I)$, $\\partial a = 0, \\partial f = 1$, \n$Z\\equiv \\frac{1-i\\Gamma_1\\otimes\\Gamma_2}{1-i}$ and \n $Z_2\\equiv \\frac{1-i\\Gamma_2}{1-i}$ then\n$$(1\\otimes Z_2)Z^*\\big(1\\otimes (a+f)\\big)Z(1\\otimes Z_2)^*= 1\\otimes a + \n\\Gamma_1\\otimes f.$$\nHence the unitary operator $(1\\otimes Z_2)Z^*$ on ${\\mathcal H}_1\\otimes{\\mathcal H}_2$ implements \nthe isomorphism of $1\\otimes{\\mathcal A}_2(I)$ onto $\\hat{\\mathcal A}_2(I)$ for every interval $I$. \n  \nObviously $\\hat{\\mathcal A}_1(I)\\equiv {\\mathcal A}_1(I)\\otimes 1$ is isomorphic to ${\\mathcal A}_1(I)$. Moreover \nwe have \n\\begin{equation}\n\\label{gradedtensor}\n{\\mathcal A}_1(I)\\hat\\otimes{\\mathcal A}_2(I) = \\hat{\\mathcal A}_1(I)\\vee\\hat{\\mathcal A}_2(I),\\quad [\\hat{\\mathcal A}_1(I),\\hat{\\mathcal A}_2(I)] = 0\\ ,\n\\end{equation}\nwhere the brackets denote the graded commutator corresponding.\n\nOne can check that, up to isomorphism, the graded tensor product \n${\\mathcal A}_1(I)\\hat\\otimes{\\mathcal A}_2(I)$ is the unique von Neumann algebra generated by copies\n $\\hat{\\mathcal A}_1(I)$ of ${\\mathcal A}_1(I)$ and $\\hat{\\mathcal A}_2(I)$ of ${\\mathcal A}_2(I)$, having \na grading that restricts to the grading of $\\hat{\\mathcal A}_i(I)$, $i=1,2$,\nsatisfying the relations \\eqref{gradedtensor}, and such that that ${\\mathcal A}_{1b}(I)\\vee{\\mathcal A}_{2b}(I)$ is the usual tensor product of von Neumann algebras.\n\nWe can then define the graded tensor product of DHR representations. \nIf $\\lambda_1$ and $\\lambda_2$ are DHR representations of ${\\mathcal A}_1$ and ${\\mathcal A}_2$ \nrespectively and if $\\lambda_1$ is graded, then it can be shown that there \nexists a (necessarily unique) representation $\\lambda_1 \\hat\\otimes \\lambda_2$ on \n${\\mathcal H}_{\\lambda_1} \\otimes {\\mathcal H}_{\\lambda_2}$ such that, for every interval \n$I\\subset S^1$,  \n\\[\n(\\lambda_1 \\hat\\otimes \\lambda_2)_I (x_1\\otimes a_2+\n\\Gamma_1x_1 \\otimes f_2) = {\\lambda_1}_I(x_1)\\otimes {\\lambda_2}_I(a_2)\n+\\Gamma_{\\lambda_1}{\\lambda_1}_I(x_1)\\otimes {\\lambda_2}_I(f_2)\n\\]\nwhere $x_1\\in {\\mathcal A}_1(I)$, $a_2,f_2\\in{\\mathcal A}_2(I)$, $\\partial a_2 =0$ and $\\partial \nf_2=1$. \nAnalogously we can \ndefine the graded tensor product of general representations defined below, \nprovided one of them is graded.\n\\section{Nets on $\\mathbb R$ and on a cover of $S^1$}\nBesides nets on $S^1$, it will be natural to consider nets on $\\mathbb R$ and nets on topological covers of $S^1$. Indeed the two notions are related as we shall see.\n\\subsection{Nets on $\\mathbb R$}\n\\label{netR}\nDenote by ${\\mathcal I}_{\\mathbb R}$ the family of open intervals of $\\mathbb R$, \ni.e. of open, non-empty, connected, bounded subsets of $\\mathbb R$. \n\nThe M\\\"obius group ${\\rm\\textsf{M\\\"ob}}$ can be naturally viewed as a subgroup of ${\\mathrm {Diff}}(S^1)$. We then have a corresponding inclusion ${\\rm\\textsf{M\\\"ob}}^{(n)}\\subset{\\mathrm {Diff}}^{(n)}(S^1)$ of covering groups. In the following we denote by $G$ a group that can be either the M\\\"obius group ${\\rm\\textsf{M\\\"ob}}$  or the diffeomorphism group ${\\mathrm {Diff}}(S^1)$. Analogously, we have $G^{(n)}={\\rm\\textsf{M\\\"ob}}^{(n)}$ or\n$G^{(n)}={\\mathrm {Diff}}^{(n)}(S^1)$.\nBy identifying $\\mathbb R$ with $S^1\\!\\smallsetminus\\!\\{-1\\}$ via the stereographic \nmap, we have a local action of $G$ on $\\mathbb R$\n(where $SL(2,\\mathbb R)\/\\{1,-1\\}\\simeq{\\rm\\textsf{M\\\"ob}}$ acts on $\\mathbb R$ by linear fractional transformations). See \\cite{BGL,GL5} for the definition and  a discussion about local actions. \n\nA \\emph{ $G$-covariant net on $\\mathbb R$} (of von Neumann algebras) ${\\mathcal A}$ is a isotone map \n\\[\nI\\in{\\mathcal I}_{\\mathbb R}\\mapsto{\\mathcal A}(I)\n\\]\nthat associates to each $I\\in{\\mathcal I}_{\\mathbb R}$ a von Neumann algebra ${\\mathcal A}(I)$ on a fixed Hilbert space ${\\mathcal H}$ and there exists a projective, positive energy, unitary representation $U$ of $G^{(\\infty)}$ on \n${\\mathcal H}$ with\n\\[\nU(g){\\mathcal A}(I)U(g)^{-1} = {\\mathcal A}(\\dot{g} I), \\quad g\\in{\\mathcal U}_I,\n\\]\nfor every $I\\in{\\mathcal I}_{\\mathbb R}$ where ${\\mathcal U}_I$ is the connected component of the identity in $G^{(\\infty)}$ of the open set $\\{g\\in G^{(\\infty)}: gI\\in{\\mathcal I}_{\\mathbb R}\\}$.\n\nWe will further assume the irreducibility of ${\\mathcal A}$ and the existence of a vacuum vector $\\Omega$ for $U$, although this is not always necessary.\n\nNote that it would be enough to require the existence of the projective unitary representation $U$ only in a neighbourhood of the identity of $G^{(\\infty)}$, then $U$ would extend to all $G^{(\\infty)}$ by multiplicativity.\n\nSince the cohomology of the Lie algebra of ${\\rm\\textsf{M\\\"ob}}$ is trivial, \nby multiplying $U(g)$ by a phase factor (in a unique fashion), we may remove the \nprojectiveness of $U|_{{\\rm\\textsf{M\\\"ob}}^{(\\infty)}}$ and assume that the restriction of $U$ to ${\\rm\\textsf{M\\\"ob}}^{(\\infty)}$\nis a unitary representation of ${\\rm\\textsf{M\\\"ob}}^{(\\infty)}$. \n\\subsection{Nets on a cover of $S^1$}\n\\label{coverNets}\nThe group $G^{(n)}$ has a natural action on $S^{1(n)}$, $n$ finite or infinite, the one obtained by promoting the action of $G$ on $S^1$, see Sect. \\ref{top}. Here $S^{1(n)}$ denotes the $n$-cover of $S^1$. Denote by ${\\mathcal I}^{(n)}$ the family of intervals of $S^{1(n)}$, i.e. $I\\in{\\mathcal I}^{(n)}$ iff $I$ is a connected subset of $S^{1(n)}$ that projects onto a (proper) interval of the base $S^1$.\n\nA \\emph{$G$-covariant net ${\\mathcal A}$ on $S^{1(n)}$} is a isotone map \n\\[\nI\\in{\\mathcal I}^{(n)}\\mapsto{\\mathcal A}(I)\n\\]\nthat associates with each $I\\in{\\mathcal I}^{(n)}$ a von Neumann algebra ${\\mathcal A}(I)$ on a fixed Hilbert space ${\\mathcal H}$, and there exists a projective\nunitary, positive energy representation $U$ of $G^{(\\infty)}$ on \n${\\mathcal H}$, implementing a  covariant action on ${\\mathcal A}$, namely \n\\[\nU(g){\\mathcal A}(I)U(g)^{-1} = {\\mathcal A}(\\dot{g}I),\\quad  I\\in {\\mathcal I}^{(n)},\\ \ng\\in G^{(\\infty)}\n\\]\nHere $\\dot{g}$ denotes the image of $g$ in $G^{(n)}$ under the quotient map.\n\nAs above, we may also assume irreducibility of the net and the existence of a vacuum vector $\\Omega$ for $U$.\n\nOf course a $G$-covariant net $\\tilde{\\mathcal A}$ on $S^{1(n)}$ determines a $G$-covariant net $\\tilde{\\mathcal A}$ on $\\mathbb R$. Indeed, with $p:S^{1(\\infty)}\\to S^1$ the covering map, every connected component of \n$p^{-1}(S^{1}\\!\\smallsetminus\\!\\{{\\rm point}\\})$\n is a copy of \n$\\mathbb R$ in $S^{1(n)}$ and we may restrict $\\tilde{\\mathcal A}$ to any of this copy; by $G$-covariance we get always the same $G$-covariant net on $\\mathbb R$, up to unitary equivalence.\n\nConversely, a $G$-covariant net ${\\mathcal A}$ on $\\mathbb R$ can be extended to a net\n$\\tilde{\\mathcal A}$ on $S^{1(\\infty)}$ by defining, for any given $I\\in{\\mathcal I}$,\n\\begin{equation}\\label{gext}\n\\tilde{\\mathcal A}(I)\\equiv U(g){\\mathcal A}(I_1)U(g)^{-1}\n\\end{equation}\nwhere $I_1\\in{\\mathcal I}_{\\mathbb R}$, $g\\in G^{(\\infty)}$, and $gI_1 = I$. Here \nthe action of $G^{(\\infty)}$ on $S^{1(\\infty)}$ is the one obtained by \npromoting the action of $G^{(\\infty)}$ on $S^1$ (see above). \nTherefore we have:\n\\begin{proposition}\\label{rtos}\nThere is a 1-1 correspondence (up to unitary equivalence) between $G$-covariant nets on $\\mathbb R$ and $G$-covariant  nets on $S^{1(\\infty)}$\n\\end{proposition}\n\\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }]\nWe only show that if ${\\mathcal A}$ is a $G$-covariant net on $\\mathbb R$, then formula \\eqref{gext} well defines $\\tilde{\\mathcal A}(I)$. If $g'\\in G^{(\\infty)}$ also satisfies $g'I_1 = I$, then $g$ and $g'$ have ``the same degree\", namely $h\\equiv g^{-1}g'$ maps $I_1$ onto $I_1$ and is in ${\\mathcal U}_{I_1}$. \nThen $U(h){\\mathcal A}(I_1)U(h)^{-1}= {\\mathcal A}(I_1)$, so $U(g){\\mathcal A}(I_1)U(g)^{-1}= U(g'){\\mathcal A}(I_1)U(g')^{-1}$. \nThe rest is clear.\n\\null\\hfill\\qed\\endtrivlist\\noindent\nWith ${\\mathcal A}$ a be a $G$-covariant net on $\\mathbb R$ as above, we shall say that a unitary operator $V$ on ${\\mathcal H}$ is a gauge unitary if \n\\begin{equation}\n\\label{gauge}\nV{\\mathcal A}(I)V^* ={\\mathcal A}(I), \\ VU(g)=U(g)V,\n\\end{equation}\nfor all $I\\in{\\mathcal I}_{\\mathbb R}$ and $g$ in a suitable neighbourhood of \nthe identity of ${\\rm\\textsf{M\\\"ob}}$. Clearly the same relations \\eqref{gauge} then \nhold true for all $I\\inS^{1(\\infty)}$, $g\\in{\\rm\\textsf{M\\\"ob}}^{(\\infty)}$ for the corresponding net \non $S^{1(\\infty)}$. Now ${\\rm rot}_{2\\pi}$ is a \ncentral element of $G^{(\\infty)}$ and it is immediate to check that if $n\\in\\mathbb \nN$: \n\\[\nU(2\\pi)^n\\ \\text{is a gauge unitary}\\Leftrightarrow\\text{${\\mathcal A}$ extends to a \n$G$-covariant net on $S^{1(n)}$} ;\n\\]\nthen ${\\mathcal A}$ is covariant with respect to the corresponding action of \n$G^{(\\infty)}$ on $S^{1(n)}$. Clearly the net on $S^{1(n)}$ is the projection of the \nnet on $S^{1(\\infty)}$ by the covering map from $S^{1(\\infty)}$ to $S^{1(n)}$. In other words we have the following where $n\\in\\mathbb N$:\n\\begin{corollary}\nA $G$-covariant net ${\\mathcal A}$ on $\\mathbb R$ is the restriction of a $G$-covariant net on $S^{1(n)}$ if and only if $U(2n\\pi)$ is a gauge unitary for ${\\mathcal A}$. This is the case, in particular, if the representation $U$ of $G^{(\\infty)}$ factors through a representation of $G^{(n)}$ (i.e. $U(2n\\pi) = 1$).\n\\end{corollary}\n\\noindent\nIf ${\\mathcal A}$ is a $G$-covariant net on $\\mathbb R$ we shall denote by ${\\mathcal A}^{(\\infty)}$ its promotion to $S^{1(\\infty)}$. If $U(2\\pi)^n$ is a gauge unitary for some $n\\in\\mathbb N$, we shall denote by ${\\mathcal A}^{(n)}$ the promotion of ${\\mathcal A}$ to $S^{1(n)}$.\n\nWe now define the \\emph{promotion} of a net ${\\mathcal A}$ on $S^1$ to a net ${\\mathcal A}^{(n)}$ on $S^{1(n)}$. If ${\\mathcal A}$ is a $G$-covariant net on $S^1$, then its restriction ${\\mathcal A}_0$ to $\\mathbb R$ is a $G$-covariant net on $\\mathbb R$, and we set\n\\[\n{\\mathcal A}^{(n)}\\equiv {\\mathcal A}_0^{(n)}\\ ;\n\\]\nhere, if $n$ is finite, we have to assume that $U(2\\pi n)$ is a gauge unitary.\nWe shall be mainly interested in the case of a $G$-covariant Fermi net ${\\mathcal A}$ on $S^1$. As $U(4\\pi)=1$ in this case, we have a natural net ${\\mathcal A}^{(2)}$ on $S^{1(2)}$ associated with ${\\mathcal A}$. Of course if ${\\mathcal A}$ is local then $U(2\\pi) =1$ the net ${\\mathcal A}^{(2)}$ on $S^{1(2)}$ is defined in this case. \n\nOf course if ${\\mathcal A}^{(n)}$  is the promotion to $S^{1(n)}$ of a net ${\\mathcal A}$ on $S^1$, then  ${\\mathcal A}^{(n)}(I)={\\mathcal A}(pI)$ for any interval $I$ of $S^{1(n)}$.\n\n\n\\section{Solitons and representations of cover nets}\nWe now consider more general representations associated with a Fermi conformal net. The point is that a Fermi conformal nets lives naturally in a double cover of $S^1$ rather than on $S^1$ itself  because the $2\\pi$ rotation unitary $U(2\\pi)$ is not the identity ($U(2\\pi)=\\Gamma$, see Sect. \\ref{vacss}) but $U(4\\pi)=1$. So representations as a net on $S^{1(2)}$ come naturally into play. These representations can be equivalently viewed as a natural class of solitions.\n\\subsection{Representations of a net on $S^{1(n)}$}\nWe begin by giving the notion of representation for a net on a cover of $S^1$.\n\nLet ${\\mathcal A}$ be a $G$-covariant net of von Neumann algebras on $S^{1(n)}$ and $U$ the associated covariance unitary representation of of $G^{(\\infty)}$ (thus $U(2\\pi n)$ is a gauge unitary). A \\emph{representation} of ${\\mathcal A}$ is a map\n\\[\nI\\in{\\mathcal I}^{(n)}\\mapsto \\lambda_I\n\\]\nwhere $\\lambda_I$ is a normal representation of ${\\mathcal A}(I)$ on a fixed Hilbert space ${\\mathcal H}_\\lambda$ with the usual consistency condition $\\lambda_{\\tilde I}|_{{\\mathcal A}(I)} = \\lambda_I$ if $\\tilde I\\supset I$. \n\nWe shall say that $\\lambda$ is \\emph{$G$-covariant} if there exists a projective unitary, positive energy representation $U_\\lambda$ of $G^{(\\infty)}$ on ${\\mathcal H}_\\lambda$ such that\n\\[\nU_\\lambda(g)\\lambda_I(x)U_\\lambda(g)^* = \\lambda_{\\dot{g}I}(U(g)xU(g)^*),\\ g\\in G^{(\\infty)},\\ I\\in{\\mathcal I}^{(n)}\\ .\n\\]\nHere $\\dot g$ denotes the image of $g\\in G^{(\\infty)}$ in $G^{(n)}$ under the quotient map, thus $\\dot{g}I$ is the projection of $gI\\inS^{1(\\infty)}$ onto $S^{1(n)}$.\n \nLet now ${\\mathcal A}$ be a $G$-covariant net on $S^1$. If $\\lambda$ is a representation of ${\\mathcal A}$, then $\\lambda$ promotes to a representation $\\lambda^{(n)}$ of ${\\mathcal A}^{(n)}$  given by\n\\[\n\\lambda^{(n)}_I \\equiv \\lambda_{p(I)},\\quad I\\in{\\mathcal I}^{(n)}\\ .\n\\]\nIf $\\lambda$ is $G$-covariant and $U_\\lambda$ is the associated unitary representation of $G^{(n)}$, then $\\lambda^{(n)}$ is also $G$-covariant with the same unitary representation $U_\\lambda$ of $G^{(n)}$.\n\\begin{lemma}\\label{prom}\nLet ${\\mathcal A}$ a $G$-covariant Fermi (resp. local) net on $S^1$ and $\\nu$ a $G$-covariant representation of ${\\mathcal A}^{(n)}$, with $U_\\nu$ the associated projective unitary representation of $G^{(n)}$.\n\nThen $\\nu$ is the promotion $\\lambda^{(n)}$ of a $G$-covariant representation $\\lambda$ of ${\\mathcal A}$ iff $U_\\nu(2\\pi)$ implements the grading (resp. the identity) in the representation $\\nu$. \n\\end{lemma}\n\\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }]\nWe assume ${\\mathcal A}$ to be a Fermi net (the local case is simpler and follows by the same argument). If $\\lambda$ is a $G$-covariant representation of ${\\mathcal A}$, then $U_\\lambda(2\\pi)$ implements the grading by the spin-statistics theorem \\cite{GL2} (applied to $\\lambda |_{{\\mathcal A}_b}$). Conversely, if $\\nu$ is a $G$ covariant representation of ${\\mathcal A}^{(n)}$ and $U_\\nu(2\\pi)$ implements the grading, let $\\lambda_0$ be the restriction of $\\nu$ to a copy of $\\mathbb R$ in $S^{1(n)}$. Then $\\lambda_0$ extends by $G$-covariance to a representation of ${\\mathcal A}$ (Prop. \\ref{rtos}). \n\\null\\hfill\\qed\\endtrivlist\\noindent\n\\subsection{Solitons}\nLet ${\\mathcal A}_0$ be a net of von Neumann algebras on $\\mathbb R$ and denote by $\\bar{\\mathcal I}_{\\mathbb R}$ the family of intervals\/half-lines of $\\mathbb R$ (open connected subsets of $\\mathbb R$ different from $\\varnothing , \\mathbb R$). If $I$ is a half-line, we define ${\\mathcal A}(I)$ as the von Neumann algebras generated by the von Neumann algebras associated to the intervals contained in $I$.\n\nA \\emph{soliton} $\\lambda$ of ${\\mathcal A}_0$ a map\n\\[\nI\\in\\bar{\\mathcal I}_{\\mathbb R}\\mapsto \\lambda_I\n\\]\nwhere $\\lambda_I$ is a normal representation of the von Neumann algebra ${\\mathcal A}_0(I)$ on a fixed Hilbert space ${\\mathcal H}_\\lambda$ with the usual isotony property $\\lambda_{\\tilde I}|_{{\\mathcal A}(I)} =\\lambda_I$, $I\\subset\\tilde I$.\n\nIf ${\\mathcal A}$ is a net on $S^1$, by a soliton of ${\\mathcal A}$ we shall mean a soliton of the restriction of ${\\mathcal A}$ to $\\mathbb R$. \n\nLet ${\\mathcal A}$ be a conformal net on $S^1$. We set\n\\[\n\\text{${\\mathcal A}_0\\ \\equiv\\ $ restriction of ${\\mathcal A}$ to $\\mathbb R$}. \n\\]\nIf $\\lambda$ is a DHR representation of ${\\mathcal A}$ then, obviously, $\\lambda |_{{\\mathcal A}_0}$ is a soliton of ${\\mathcal A}_0$. We shall say that a soliton $\\lambda_0$ of ${\\mathcal A}$ is a DHR representation of ${\\mathcal A}$ if it arises in this way, namely $\\lambda_0=\\lambda |_{{\\mathcal A}_0}$ with $\\lambda$ a DHR representation of ${\\mathcal A}$ (in other words, $\\lambda |_{{\\mathcal A}_0}$ extends to a DHR representation of ${\\mathcal A}$, note that the extension is unique if strong additivity is assumed).\n\nMore generally, we get solitons of ${\\mathcal A}$ by restricting to ${\\mathcal A}_0$ representations of the cover nets ${\\mathcal A}^{(n)}$.\n\nLet ${\\mathcal A}$ be a $G$-covariant net on $S^1$ with covariance unitary representation $U$. A \\emph{$G$-covariant soliton} $\\lambda$ of ${\\mathcal A}$ is a soliton of ${\\mathcal A}_0$ such that there exists a projective unitary representation $U_\\lambda$ of $G^{(\\infty)}$ such that for every bounded interval $I$ of $\\mathbb R$ we have\n\\[\n\\lambda_{\\dot g I}(U(g)xU(g)^*) = U_\\lambda (g)\\lambda_I(x)U_\\lambda (g)^*\\ ,\\quad  g\\in{\\mathcal U}_I ,\\ x\\in{\\mathcal A}(I)\\ ,\n\\]\nwhere ${\\mathcal U}_I$ is a connected neighborhood of the identity of $G^{(\\infty)}$ as in Sect. \\ref{netR} and $\\dot g$ is the image of $g$ in $G$.\n\\begin{proposition}\\label{r-si}\nLet ${\\mathcal A}$ be a $G$-covariant net on $S^1$.\nThere is a one-to-one correspondence between\n\\begin{itemize}\n\\item[$(a)$] $G$-covariant representations of ${\\mathcal A}^{(\\infty)}$,\n\\item[$(b)$] $G$-covariant solitons of ${\\mathcal A}$.\n\\end{itemize}\nThe correspondence is given by restricting a representation of ${\\mathcal A}^{(\\infty)}$ to a copy of ${\\mathcal A}_0$.\n\nSuppose ${\\mathcal A}$ is local. Then the above restricts to a one-to-one correspondence between $(a)$: $G$-covariant representations of ${\\mathcal A}^{(n)}$ and $(b)$: $G$-covariant solitons of ${\\mathcal A}$ with $U_\\lambda(2\\pi n)$ commuting with $\\lambda$.\n\nSuppose ${\\mathcal A}$ is Fermi. Then the above restricts to a one-to-one correspondence between $(a)$: $G$-covariant representations of ${\\mathcal A}^{(n)}$ and $(b)$: $G$-covariant solitons of ${\\mathcal A}$ with $U_\\lambda(2\\pi n)$ implementing the grading ($n$ even) or commuting with $\\lambda$ ($n$ odd).\n\\end{proposition}\n\\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }]\nClearly if $\\lambda$ is a $G$-covariant representation of ${\\mathcal A}^{(\\infty)}$ then $\\lambda_0\\equiv\\lambda |_{{\\mathcal A}_0}$ is a $G$-covariant solitons of ${\\mathcal A}$. Conversely, if $\\lambda_0$ is $G$-covariant solitons of ${\\mathcal A}$ then we set\n\\[\n\\lambda_{g I}(U(g)xU(g)^*) = U_\\lambda (g)\\lambda_I(x)U_\\lambda (g)^*\\ ,\\quad  g\\in G^{(\\infty)} ,\\ x\\in{\\mathcal A}(I)\\ ,\n\\]\nwhere $I\\subset\\mathbb R$ is a bounded interval for any given copy of $\\mathbb R$ is in $S^{1(\\infty)}$ and $G^{(\\infty)}$ acts on $S^{1(\\infty)}$ as usual . By  $G$-covariance the above formula well-define a representation of ${\\mathcal A}^{(\\infty)}$.\n\nThe second part follows because by the vacuum spin-statistics relation we have $U(2\\pi)=1$ in the local case and $U(2\\pi)=\\Gamma$ in the Fermi case.\n\\null\\hfill\\qed\\endtrivlist\\noindent\nA \\emph{graded soliton} of the Fermi net ${\\mathcal A}$ is, of course, a soliton such that the grading is unitarily implemented, namely the soliton $\\lambda$ is graded iff there exists a unitary $\\Gamma_\\lambda$ on ${\\mathcal H}_\\lambda$ such that\n\\[\n\\lambda_I(\\gamma(x)) = \\Gamma_\\lambda \\lambda(x)\\Gamma^*_\\lambda,\\quad I\\in{\\mathcal I}_{\\mathbb R},\\ x\\in{\\mathcal A}(I).\n\\]\n\\subsection{Neveu-Schwarz and Ramond representations}\nWe give now the general definition for a representation of a Fermi net. We have two formulations for this notion: as a representation of the cover net and as a soliton.\n\nLet ${\\mathcal A}$ be a Fermi net on $S^1$. A \\emph{general representation} $\\lambda$ of ${\\mathcal A}$ is a representation the cover net of ${\\mathcal A}^{(\\infty)}$ such that $\\lambda$ restricts to a DHR representation $\\lambda_b$ of the Bose subnet ${\\mathcal A}_b$.\n\nWe shall see here later on that a general representation is indeed a representation of ${\\mathcal A}^{(2)}$. We begin by noticing an automatic covariance for a general representation $\\lambda$. In this case $\\lambda$ is not always graded.\n\\begin{proposition}\\label{diffcov0}\nLet ${\\mathcal A}$ be a Fermi conformal net on $S^1$. Every general representation $\\lambda$ of ${\\mathcal A}$ is diffeomorphism covariant with positive energy. Indeed $U_\\lambda = U_{\\lambda_b}$.\n\\end{proposition}\n\\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }]\nThe proof follows by an immediate extension of the argument proving Prop. \\ref{DHRgraded}.\n\\null\\hfill\\qed\\endtrivlist\\noindent\nLet ${\\mathcal A}$ be a Fermi net on $S^1$. A \\emph{general soliton} $\\lambda$ of ${\\mathcal A}$ is a  diffeomorphism covariant soliton of ${\\mathcal A}$ such that $\\lambda$ restricts to a DHR representation $\\lambda_b$ of the Bose subnet ${\\mathcal A}_b$. (In other words $\\lambda|_{{\\mathcal A}_{b,0}}$ extends to a DHR representation $\\lambda_b$ of ${\\mathcal A}_b$.)\n\nBy Prop. \\ref{r-si} we have a one-to-one correspondence between\n\\begin{gather*}\n\\text{General Representations}\\\\\n\\Updownarrow\\\\\n\\text{General Solitons}\n\\end{gather*}\nWe prove now the diffeomorphism covariance of general solitons in the strong additive case. Notice that strong additivity is automatic in the completely rational case \\cite{LX}. The general proof follows by Prop. \\ref{acov} below.\n\\begin{proposition}\\label{diffcov}\nLet ${\\mathcal A}$ be a  Fermi conformal net on $S^1$. Every soliton $\\lambda$ of ${\\mathcal A}$ such that $\\lambda_b$ is a DHR representation is diffeomorphism covariant with positive energy, namely $\\lambda$ is a general soliton. Indeed $U_\\lambda = U_{\\lambda_b}$.\n\\end{proposition}\n\\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }]\n(Assuming strongly additive). Note that if $g\\in{\\mathrm {Diff}}(S^1)$ is localized in an interval $I$ then, up to a phase, $U_{\\lambda_b}(g) = \\lambda_b(U(g))\\in{\\mathcal A}_b(I)$.\n\nLet $I$ be an interval contained in $\\mathbb R = S^1\\smallsetminus\\{-1\\}$ a suppose that $g\\in{\\mathrm {Diff}}(S^1)$ is localised in $I$; then\n\\begin{multline*}\nU_{\\lambda_b}(g)\\lambda_{\\tilde I}(x)U^*_{\\lambda_b}(g) = {\\lambda_b}_{\\tilde I}(U(g))\\lambda_{\\tilde I}(x){\\lambda_b}_{\\tilde I}(U^*(g))\n= {\\lambda}_{\\tilde I}(U(g))\\lambda_{\\tilde I}(x){\\lambda}_{\\tilde I}(U^*(g))\\\\\n= \\lambda_{\\tilde I}\\big(U(g)xU(g)^*\\big)\n= \\lambda_{g\\tilde I}\\big(U(g)xU(g)^*\\big)\\ ,\n\\quad x\\in{\\mathcal A}(\\tilde I)\\ ,\n\\end{multline*}\nfor any interval $\\tilde I\\supset I$ of $\\mathbb R$. \n\nNotice at this point that if $x\\in{\\mathcal A}(\\tilde I)$ and $v\\in{\\mathcal A}_b(\\tilde I')$ then ${\\lambda_b}_{\\tilde I'}(v)$ commutes with $\\lambda_{\\tilde I}(x)$:\n\\[\n[{\\lambda_b}_{\\tilde I'}(v),\\lambda_{\\tilde I}(x)]=0\\ .\n\\]\nIndeed, as ${\\mathcal A}$ is assumed to be strongly additive, also ${\\mathcal A}_b$ is strongly additive \\cite{X01}, see also \\cite{L03}, so it is sufficient to show the above relation with $v\\in{\\mathcal A}_b(I_0)$,  where $I_0$ is any interval with closure contained in $\\tilde I'\\smallsetminus \\{-1\\}$. Then\n\\begin{equation*}\n[{\\lambda_b}_{\\tilde I'}(v),\\lambda_{\\tilde I}(x)] =[{\\lambda_b}_{I_0}(v),\\lambda_{\\tilde I}(x)]\n= [{\\lambda}_{I_0}(v),\\lambda_{\\tilde I}(x)] \n= [{\\lambda}_{I_1}(v),\\lambda_{I_1}(x)] \n= {\\lambda}_{I_1}([v,x]) = 0\\ ,\n\\end{equation*}\nwhere we have taken an interval $I_1$ of $\\mathbb R$ containing $I_0\\cup\\tilde I$.\n\nLet now ${\\mathcal U}_I$ be a connected neighbourhood of the identity in ${\\mathrm {Diff}}(S^1)$ such that $gI$ is an interval of $\\mathbb R$ for all $g\\in{\\mathcal U}_I$. Take $g\\in{\\mathcal U}_I$ and let $\\tilde I$ be an interval of $\\mathbb R$ with $\\tilde I\\supset I\\cup gI$. Let $h\\in {\\mathrm {Diff}}(S^1)$ be a diffeomorphism localised in an interval of $\\mathbb R$ containing $\\tilde I$ such that $h|_{\\tilde I} = g|_{\\tilde I}$. Then\n\\[\nU(g) = U(h)v\n\\]\nwhere $v\\equiv U(h^{-1}g )\\in {\\mathcal A}_b(\\tilde I')$ (the Virasoro subnet is contained in the Bose subnet).\n\nWe then have with $x\\in{\\mathcal A}(\\tilde I)$:\n\\begin{multline*}\nU_{\\lambda_b}(g)\\lambda_{\\tilde I}(x)U^*_{\\lambda_b}(g) =\nU_{\\lambda_b}(h){\\lambda_b}_{\\tilde I'}(v)\\lambda_{\\tilde I}(x){\\lambda_b}_{\\tilde I'}(v^*)U^*_{\\lambda_b}(h) \\\\\n= U_{\\lambda_b}(h)\\lambda_{\\tilde I}(x)U^*_{\\lambda_b}(h)\\ .\n=\\lambda_{h\\tilde I}\\big(U(h)xU(h)^*\\big)\n=\\lambda_{g\\tilde I}\\big(U(g)xU(g)^*\\big)\n\\end{multline*}\n\\null\\hfill\\qed\\endtrivlist\\noindent\n\\begin{proposition}\\label{n1}\nLet ${\\mathcal A}$ be a Fermi conformal net on $S^1$ and $\\lambda$ an irreducible \ngeneral soliton of ${\\mathcal A}$. With $\\lambda_{b,0}\\equiv\\lambda |_{{\\mathcal A}_{b,0}}$, the following are equivalent:\n\\begin{itemize}\n\\item[$(i)$] $\\lambda$ is graded,\n\\item[$(ii)$] $\\lambda_{b,0}$ is not irreducible,\n\\item[$(iii)$] $\\lambda_{b,0}$ is direct sum of two inequivalent irreducible representations \nof ${\\mathcal A}_{b,0}$:\n\\[\n\\lambda_{b,0} \\simeq \\rho\\oplus \\rho'\n\\]\nwhere $\\rho$ is an irreducible DHR representation of ${\\mathcal A}_{b,0}$ and \n$\\rho'\\equiv \\rho{\\sigma}$ with ${\\sigma}$ is the dual involutive localised automorphism of ${\\mathcal A}_b$ as above. \\end{itemize}\nIf the above holds and $\\lambda$ (i.e. $\\rho$) has finite index\nwe then have equation for the statistics phases\n\\begin{equation}\\label{mp1}\n\\omega_{\\rho'} = - m(\\rho,{\\sigma})\\omega_\\rho\n\\end{equation}\nThus\n\\begin{equation}\\label{mp2}\nU_{\\rho'}(2\\pi) = \\mp U_\\rho(2\\pi)\n\\end{equation}\naccording to $\\rho$ is ${\\sigma}$-Bose\/${\\sigma}$-Fermi.    \n\\end{proposition}\n\\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }]\n$(i)\\Rightarrow (iii)$: If $\\lambda$ is graded, then the unitary $\\Gamma_\\lambda$ implementing the grading in the representation commutes with $\\lambda({\\mathcal A}_{b,0})$, so $\\lambda_{b,0}$ is reducible. Moreover, as $\\lambda({\\mathcal A}_{b,0})''= \\lambda({\\mathcal A}_0)''\\cap\\{\\Gamma_\\lambda\\}'$ , we have\n$\\lambda({\\mathcal A}_{b,0})'= \\lambda({\\mathcal A}_0)'\\vee\\{\\Gamma_\\lambda\\}''=\\{\\Gamma_\\lambda\\}''$, so $\\lambda({\\mathcal A}_{b,0})'$ is 2-dimensional. \nLet $\\rho$ one of the two irreducible components of $\\lambda_{b,0}$. As the dual canonical endomorphism associated with ${\\mathcal A}_{b,0}\\subset{\\mathcal A}_0$ is $\\iota\\oplus{\\sigma}$ (see \\cite{LR1}), the other component must be $\\rho'\\equiv\\rho{\\sigma}$, namely $\\lambda_{b,0}=\\rho\\oplus\\rho'$.\n\nIn order to show eq. \\eqref{mp2} we may assume that $\\rho$ is localized left to ${\\sigma}$, so $\\rho$ and ${\\sigma}$ commute. By using the cocycle equations for the statistics operators, we then have\n\\begin{multline*}\n\\varepsilon(\\rho',\\rho') = \\varepsilon(\\rho,\\rho')\\rho(\\varepsilon({\\sigma},\\rho')) \n= \\varepsilon(\\rho,\\rho')\\varepsilon({\\sigma},\\rho') \n= \\rho(\\varepsilon(\\rho,{\\sigma}))\\varepsilon(\\rho,\\rho)\\rho(\\varepsilon({\\sigma},{\\sigma}))\\varepsilon({\\sigma},\\rho) \\\\\n= \\varepsilon(\\rho,{\\sigma})\\varepsilon(\\rho,\\rho)\\varepsilon({\\sigma},{\\sigma})\\varepsilon({\\sigma},\\rho)\n= - m(\\rho,{\\sigma})\\varepsilon(\\rho,\\rho)\\ ,\n\\end{multline*}\nwhere we have used that $\\varepsilon({\\sigma},\\rho')$ and $\\varepsilon(\\rho,{\\sigma})$ are scalars and  $\\varepsilon({\\sigma},{\\sigma})= -1$. So, if $\\rho$ has finite index, we infer the equation \\eqref{mp1} for the statistics phases and eq. \\eqref{mp2} follows by the conformal spin-statistics theorem \\cite{GL2}.\n\nThe implication $(iii)\\Rightarrow (ii)$ is obvious. For the implication $(ii)\\Rightarrow (i)$ assume that $\\lambda$ is not graded, namely $\\gamma$ is not unitarily implemented in the representation $\\lambda$; then $\\lambda$ is irreducible by known arguments, see \\cite{Ro} (the fixed point of an irreducible $C^*$-algebra with respect to a period two outer automorphism is still irreducible).\n\\null\\hfill\\qed\\endtrivlist\\noindent\n\\noindent\n{\\it Remark.} In the above proposition the diffeomorphism covariance of $\\lambda$ is unnecessary, only the M\\\"obius covariance of $\\lambda_b$ has been used.\n\\begin{corollary}\\label{extrep}\nLet ${\\mathcal A}$ be a Fermi conformal net on $S^1$ and $\\lambda$ an irreducible general soliton of ${\\mathcal A}$ with finite index.\nThe following alternative holds:\n\\begin{itemize}\n\\item[$(a)$] $\\lambda$ is a DHR representation of ${\\mathcal A}$. Equivalently $U_{\\lambda_b}(2\\pi)$ is not a scalar.\n\\item[$(b)$] $\\lambda$ is the restriction of a representation of ${\\mathcal A}^{(2)}$ and $\\lambda$ is not a DHR representation of ${\\mathcal A}$. Equivalently $U_{\\lambda_b}(2\\pi)$ is a scalar.\n\\end{itemize}\n\\end{corollary}\n\\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }]\nFirst note that $U_{\\lambda_b}(4\\pi) = U_{\\rho}(4\\pi)\\oplus U_{\\rho'}(4\\pi)$ is a scalar by eq. \\eqref{mp2}, so $\\lambda$ is always representation of ${\\mathcal A}^{(2)}$ by Prop. \\ref{r-si}.\n\nIf $\\lambda$ is a DHR representation of ${\\mathcal A}$ then by, Prop. \\ref{DHRgraded}, $U_\\lambda(2\\pi) =\\Gamma$ is not a scalar.\n\nAssume now that $\\lambda$ is not a DHR representation of ${\\mathcal A}$. It remains to show that if $U_\\lambda(2\\pi)$ is a scalar. \n\nIf $\\lambda$ is not graded, then by Prop. \\ref{n1} $\\lambda_b$ is irreducible; as $U_\\lambda(2\\pi)$ commutes with $\\lambda_b$, must then be a scalar.\n\nFinally, if $\\lambda$ is graded, then $\\rho$ must be a ${\\sigma}$-Fermi sector, as otherwise its $\\alpha$-induction $\\alpha^+_\\rho$ to ${\\mathcal A}$ would be a DHR representation of ${\\mathcal A}$, while $\\alpha^+_\\rho= \\lambda$ by Frobenious reciprocity. Then by eq.  \\eqref{mp2} we have $U_{\\rho'}(2\\pi) = U_\\rho(2\\pi)$, namely $U_\\lambda(2\\pi)$ is a scalar.\n\\null\\hfill\\qed\\endtrivlist\\noindent\n{\\it Remark.} The finite index assumption in the above proposition is probably unnecessary (it has been used to make use of eq.  \\eqref{mp2}).\n\\medskip\n\n\\noindent\nIt is now convenient to use the following terminology concerning the general soliton in Corollary \\ref{extrep}. In the first case, namely if $\\lambda$ is a DHR representation of ${\\mathcal A}$, we say that $\\lambda$ is a \\emph{Neveu-Schwarz representation} of ${\\mathcal A}$. In the second case, namely if $\\lambda$ is a soliton (and not a representation of ${\\mathcal A}$) we say that $\\lambda$ is a \\emph{Ramond representation} of ${\\mathcal A}$. \n\nIf $\\lambda$ is reducible we shall say that $\\lambda$ is a Neveu-Schwarz representation of ${\\mathcal A}$ if $\\lambda$ is a DHR representation of ${\\mathcal A}$; we shall say that $\\lambda$ is a Ramond representation of ${\\mathcal A}$ if no subrepresentation of $\\lambda$ is a DHR representation of ${\\mathcal A}$.\n\nLet as above ${\\mathcal A}$ be a Fermi conformal net on $S^1$ and ${\\mathcal A}_b$ the Bose subnet. We shall now study the representations of ${\\mathcal A}_b$ in relation to the representations of ${\\mathcal A}$.\n\nGiven a representation $\\nu$ of ${\\mathcal A}_b$ we consider its $\\alpha$-induction $\\alpha_\\nu$ to ${\\mathcal A}$ (say right $\\alpha$-induction, so $\\alpha_\\nu\\equiv\\alpha^+_\\nu$). More precisely we first restrict $\\nu$ to the net ${\\mathcal A}_{b,0}$ on the real line and then consider its $\\alpha$-induction $\\alpha_\\nu$ which is a soliton of ${\\mathcal A}_0$. Note that $\\alpha_\\nu$ is defined in \\cite{LR1} assuming Haag duality on the real line (equivalent to strong additivity in the conformal case), but this assumption is unnecessary here because we are considering nets on $S^1$ that satisfy Haag duality on $S^1$, see \\cite[Sect. 3.1]{LX}.\n\\begin{proposition}\\label{acov}\nIf $\\nu$ is a DHR irreducible representation of ${\\mathcal A}_b$ then $\\alpha_\\nu$ is a general soliton of ${\\mathcal A}$, namely $\\alpha_\\nu$ is diffeomorphism covariant with positive energy. We have:\n\\begin{gather*}\n\\text{$\\nu$ is ${\\sigma}$-Bose} \\Leftrightarrow \\text{$\\alpha_\\nu$ is Neveu-Schwarz}\\\\\n\\text{$\\nu$ is ${\\sigma}$-Fermi} \\Leftrightarrow \\text{$\\alpha_\\nu$ is Ramond}\n\\end{gather*}\n\\end{proposition}\n\\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }]\nWe only sketch the proof.\nWe use the extension method by Roberts (see \\cite{Ro}), but in a covariant way.\nAs $\\nu$ is diffeomorphism covariant, there is a localized unitary cocycle $w$ with value in ${\\mathcal A}_b$ such that\n\\begin{equation}\\label{ce}\n {\\hbox{\\rm Ad}} w_g\\cdot \\nu = {\\hbox{\\rm Ad}} U(g)\\cdot \\nu\\cdot{\\hbox{\\rm Ad}} U(g^{-1}),\\quad g\\in{\\mathrm {Diff}}(S^1),\n\\end{equation}\nsee \\cite[Sect. 8]{GL1}. The cocycle $w$ reconstructs $\\nu$. Now $w$ can be seen as a cocycle with values in ${\\mathcal A}$ but with less localization properties (if $w$ is bi-localized in $I\\cup gI$ in ${\\mathcal A}_b$, it is in general only localized in $\\tilde I$ in ${\\mathcal A}$ where $\\tilde I$ is an interval containing $I\\cup gI$). Thus, in general,  $w$ can be associated with a soliton of ${\\mathcal A}$, which is $\\alpha_\\nu$. Now the covariance equation still remains true for $g$ in a neighborhood of the identity, and gives the diffeomorphism covariance of $\\alpha_\\nu$.\n\nIf $\\nu$ a DHR irreducible representation of ${\\mathcal A}_b$, its $\\alpha$-induction to ${\\mathcal A}$ is a DHR representation iff $\\nu$ has trivial monodromy with the dual canonical endomorphism $\\iota\\oplus{\\sigma}$ of ${\\mathcal A}_b$ (the restriction to ${\\mathcal A}_b$ of the identity representation of ${\\mathcal A}$), thus iff $\\nu$ has trivial monodromy with ${\\sigma}$. By definition this means that $\\nu$ is a ${\\sigma}$-Bose representation.\n\\null\\hfill\\qed\\endtrivlist\\noindent\nAn immediate corollary is the following.\n\\begin{corollary}\nA  DHR representation $\\nu$ of ${\\mathcal A}_b$ is ${\\sigma}$-Bose (resp. ${\\sigma}$-Fermi) if and only if $\\nu$ is the restriction of a Neveu-Schwarz (resp. Ramond) representation of ${\\mathcal A}$.\n\\end{corollary}\n\\noindent\nBecause of the above corollary, we shall also call Neveu-Schwarz (resp. Ramond) representation of ${\\mathcal A}_b$ a ${\\sigma}$-Bose (resp. ${\\sigma}$-Fermi) representation of ${\\mathcal A}_b$.\n\\subsection{Tensor categorical structure. Jones index}\nIn the local case, the DHR argument shows that every representation is equivalent to a localized endomorphism and a first basic consequence of this fact is that representations give rise to a tensor category because one can indeed compose localized endomorphisms and get the monoidal structure. As we have seen in Prop. \\ref{dhrendo}, we can extend the DHR argument to the case of representations of a Fermi net; this argument however makes use that DHR representations are automatically graded. In order to have a tensor structure for general representations of a Fermi net, we consider graded representations only. Note however that if $\\lambda$ is any soliton, setting where $\\lambda_\\gamma\\equiv\\gamma\\lambda\\gamma^{-1}$, then $\\lambda\\oplus\\lambda_\\gamma$ is graded.\nWe obviously have\n\\[\n{\\rm Sol}_\\gamma({\\mathcal A})\\subset{\\rm Sol}({\\mathcal A})\\ .\n\\]\nwhere ${\\rm Sol}({\\mathcal A})$ denotes the general solitons of ${\\mathcal A}$ that are localized, say, in a right half line of $\\mathbb R$ and ${\\rm Sol}_\\gamma({\\mathcal A})$ are the general solitons commuting with the grading, namely $\\lambda\\in{\\rm Sol}({\\mathcal A})$ belongs to  ${\\rm Sol}_\\gamma({\\mathcal A})$ iff $\\lambda=\\lambda^\\gamma$.\n\nNow every general representation of ${\\mathcal A}$ is equivalent to an element of ${\\rm Sol}({\\mathcal A})$. Thus, if $\\lambda_1,\\lambda_2\\in{\\rm Sol}({\\mathcal A})$ an intertwiner $T:\\lambda_1\\to\\lambda_2$ is a bounded linear operator such that\n\\[\nT\\lambda_1(x)=\\lambda_2(x)T,\\quad x\\in{\\mathcal A}_0\\ .\n\\]\nNote that $T$ is not necessarily localized in a half-line, but by twisted duality $T\\in Z{\\mathcal A}(I)Z^*$ where $I$ is a half-line where $\\lambda_1$ and $\\lambda_2$ are both localized. By further assuming that $T$ commutes with $\\Gamma$, we also have that  $T$ commutes with $Z$, so $T\\in{\\mathcal A}(I)$ and $\\partial T = 0$, thus $T\\in{\\mathcal A}_b(I)$.\n\nIt follows that ${\\rm Sol}_\\gamma({\\mathcal A})$ is a tensor category where the arrows are defined by\n\\[\n{\\rm Hom}(\\lambda_1,\\lambda_2)\\equiv T:\\lambda_1\\to\\lambda_2,\\ T\\Gamma =\\Gamma T\\ .\n\\]\n\\begin{proposition}\n$\\alpha$-induction is a surjective tensor functor \n\\[\n{\\rm Rep}({\\mathcal A}_b)\\overset{\\alpha}{\\longrightarrow}{\\rm Sol}_\\gamma({\\mathcal A})\\ .\n\\]\nwhere ${\\rm Rep}({\\mathcal A}_b)$ is the tensor category of endomorphisms of ${\\mathcal A}_b$ localized in intervals of $\\mathbb R$. The kernel of $\\alpha$ is $\\{\\iota,{\\sigma}\\}$.\n\\end{proposition}\n\\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }]\nFirst of all note that if $\\nu\\in{\\rm Rep}({\\mathcal A}_b)$ is localized in the interval $I$, then $\\alpha_\\nu$ commutes with $\\gamma$. Indeed both $\\alpha_\\nu$ and $\\alpha^\\gamma_\\nu= \\gamma\\alpha_\\nu\\gamma$ have the same restriction to ${\\mathcal A}_b$. If $v\\in{\\mathcal A}(I)$ is a selfadjoint unitary with $\\partial v = 1$, then $\\alpha_\\nu(v) = uvu^*$ for a Bose unitary $u$ in the covariance cocycle for $\\nu$, therefore \n\\begin{equation*}\n\\alpha^\\gamma_\\nu(v) =\\gamma(\\alpha_\\nu(\\gamma(v)))= -\\gamma(\\alpha_\\nu(v))\n= -\\gamma(uvu)  = -\\gamma(u)\\gamma(v)\\gamma(u) = uvu =\\alpha_\\nu(v)\\ ,\n\\end{equation*}\nso $\\alpha=\\alpha^\\gamma_\\nu$ because ${\\mathcal A}$ is generated by ${\\mathcal A}_b$ and $v$.\n\nConversely, if $\\lambda\\in{\\rm Sol}_\\gamma({\\mathcal A})$, then $\\lambda_b$ is a localized endomorphism of ${\\mathcal A}_b$ because $\\lambda$ commutes with $\\gamma$. Now the covariance cocycle $w_g$ for $\\lambda$ has degree $0$; indeed $\\gamma(w_g)$ is also a covariance cocycle for $\\lambda$ thus $\\gamma(w_g)=w_g$ up to a phase that must be 1 by the cocycle property. Thus $w$ belongs to the Bose subnet and is the covariance cocycle for $\\lambda_b$; a calculation as above then shows that $\\lambda = \\alpha_{\\lambda_b}$.\n\nIt is now rather immediate to show that $\\alpha_\\nu$ is the identity iff $\\nu =\\iota$ or $\\nu={\\sigma}$ and that we have a corresponding surjective map of the arrows.\n\\null\\hfill\\qed\\endtrivlist\\noindent\nSo we have a braided tensor category ${\\rm Sol}_\\gamma({\\mathcal A})$. In particular, the \\emph{Jones index} of $\\lambda$ is defined, as in the local case, see \\cite{L5,L5'}. \n\nNote however that an irreducible element of ${\\rm Sol}_\\gamma({\\mathcal A})$ may be reducible as a general representation of ${\\mathcal A}$, namely it can decomposed into non-graded subrepresentations.\n\\section{Modular and superconformal invariance}\nWe now study Fermi nets whose Bose subnets are modular and get some consequences in the supersymmetric case.\n\\subsection{Modular nets}\nWe recall here a discussion made in \\cite{KL05}.\nLet ${\\mathcal B}$ be a completely rational local conformal net of von Neumann algebras on $S^1$. Then the tensor category or representations of ${\\mathcal B}$ is modular, i.e. rational with non-degenerate unitary braiding  \\cite{KLM}.\n\nWe then have Rehren \\cite{R1} matrices defining\na unitary representation of the group $SL(2,{\\mathbb Z})$ on the space spanned by the irreducible sectors (i.e. unitary equivalence classes of representations) $\\rho$'s , in particular we have the matrices $T \\equiv \\{T_{\\lambda,\\nu}\\}$ and $S \\equiv \\{S_{\\lambda,\\nu}\\}$ where\n\\begin{equation}\\label{matS}\nS_{\\lambda,\\nu}\\equiv \n\\frac{d(\\lambda)d(\\nu)}{\\sqrt{\\sum_i d(\\rho_i)^2}}\\Phi_{\\nu}(\\varepsilon(\\nu,\\lambda)^*\\varepsilon(\\lambda,\\nu)^*) \\ .\n\\end{equation}\nHere $\\varepsilon(\\nu,\\lambda)$ is the right statistics operators between $\\lambda$\nand $\\nu$, $\\Phi_{\\nu}$ is left inverse of $\\nu$ and $d(\\lambda)$ is the \nstatistical dimension of $\\lambda$.  \n\nAssume that ${\\mathcal B}$ is diffeomorphism covariant with central charge $c$.\nFor a sector $\\rho$, we consider the specialised character\n$\\chi_\\rho(\\tau)$ for complex numbers $\\tau$ with ${\\rm Im}\\; \\tau >0$\nas follows:\n\\[\n\\chi_\\rho(\\tau)=\\Tr\\!\\big(e^{2\\pi i\\tau (L_{0,\\rho}-c\/24)}\\big).\n\\]\nHere the operator $L_{0,\\rho}$ is the conformal Hamiltonian in \nthe representation $\\rho$.  We also assume that the above trace converges, and so each eigenspace \nof $L_{0,\\rho}$ is finite dimensional.  \n\nIn many cases the $T$ and $S$ matrices give an action\nof $SL(2,{\\mathbb Z})$ on the linear span of these specialised\ncharacters through change of variables $\\tau$ as follows:\n\\begin{equation}\\label{modularity}\n\\begin{split}\n\\chi_\\rho(-1\/\\tau)&= \\sum_\\nu S_{\\rho,\\nu} \\chi_\\nu(\\tau),\\\\\n\\chi_\\rho(\\tau+1)&= \\sum_\\nu T_{\\rho,\\nu} \\chi_\\nu(\\tau).\n\\end{split}\n\\end{equation}\nThis is always the case if the Rehren matrices $S$ and $T$ coincide with the  \nso called Kac-Peterson or Verlinde matrices.\nThe Kac-Peterson-Verlinde matrix $T$ and Rehren matrix $T$ always coincide\nup to a phase by the spin-statistics theorem \\cite{GL2}.\n\nWe shall say that ${\\mathcal B}$ is \\emph{modular} if  the Rheren matrices give the modular \ntransformations of specialized character as in Eq. (\\ref{modularity}) .  \nNote that we are assuming that ${\\mathcal B}$ is completely rational, namely the $\\mu$-index is finite \n(see below) and the split property holds. However, as we are assuming\n$\\Tr(e^{-tL_{0}})<\\infty$ the split property follows, see \\cite{BDL}. \n(Strong additivity follows from  \\cite{LX}.)\n\nModularity holds in all computed rational cases, cf. \\cite{X3,X4}. \nThe $SU(N)_k$ nets and the Virasoro nets ${\\rm Vir}_c$ with $c<1$ are\nboth modular. We expect all local conformal completely rational \nnets to be modular (see \\cite{Hu} for results of similar kind). \nFurthermore, if ${\\mathcal B}\n$ is a modular local conformal net and ${\\mathcal C}\n$ an irreducible extension of ${\\mathcal B}\n$, then ${\\mathcal C}\n$ is also modular \\cite{KL05}, that allows to check \nthe modularity property in several cases.\n\nIf ${\\mathcal B}\n$ is modular, the Kac-Wakimoto formula holds\n\\begin{equation}\\label{KW1}\nd(\\rho)=\\frac{S_{\\rho,0}}{S_{0,0}}=\n\\lim_{\\tau\\to i\\infty}\\frac{\\sum_\\nu S_{\\rho,\\nu}\\chi_\\nu(\\tau)}\n{\\sum_\\nu S_{0,\\nu}\\chi_\\nu(\\tau)}=\n\\lim_{\\tau\\to 0}\\frac{\\chi_\\rho(\\tau)}{\\chi_0(\\tau)}.\n\\end{equation}\nHere we denote the vacuum sector $\\iota$ also by $0$.\n\nNow, if ${\\mathcal B}$ is a modular net, then ${\\mathcal B}$ is two-dimensional \nlog-elliptic with noncommutative area $a_0 = 2\\pi c\/24$ \\cite{KL05}, \nindeed the following asymptotic formula holds:\n\\[\n\\log\\Tr(e^{-2\\pi tL_{0,\\rho}})\\sim \\frac{\\pi c}{12}\\frac1t + \n\\frac{1}{2}\\log\\frac{d(\\rho)^2}{\\mu_{{\\mathcal B}\n}}-\n\\frac{\\pi c}{12}t\\qquad {\\rm as }\\ t\\to 0^+ \\ .\n\\]\nwhere $\\rho$ a representation of ${\\mathcal B}$ and $L_{0,\\rho}$ is the conformal \nHamiltonian in the representation $\\rho$.\n\n\\subsection{Fermi nets and modularity}\nAssume that $\\lambda$ is a graded irreducible general soliton of the Fermi conformal net ${\\mathcal A}$ \non $S^1$. Then $\\lambda$ is the restriction of a representation of the double cover ${\\mathcal A}^{(2)}$ of \n${\\mathcal A}$ and is diffeomorphism covariant. We denote by $H_{\\lambda}$ the conformal Hamiltonian of \n${\\mathcal A}$ in the representation $\\lambda$. \n\nThen $H_{\\lambda}$ is affiliated to the Virasoro subnet in the representation $\\lambda$, which is \ncontained in the Bosonic subnet, so $H_{\\lambda}$\ncommutes with $\\Gamma_{\\lambda}$ and thus respects the graded decomposition \n${\\mathcal H}_\\lambda = {\\mathcal H}_{\\lambda,+}\\oplus{\\mathcal H}_{\\lambda,-}$ given by $\\Gamma_\\lambda$; we then have a unitary equivalence\n\\[\nH_{\\lambda} \\simeq L_{0,\\rho}\\oplus L_{0,\\rho'}\n\\]\nwhere $L_{0,\\rho}$ and $L_{0,\\rho'}$ are the conformal Hamiltonians of ${\\mathcal A}_b$ \nin the  representations $\\rho$ and $\\rho'$, where $\\lambda_b = \\rho\\oplus\\rho'$. \nConsequently\n\\begin{equation}\\label{strp}\n\\Str(e^{-tH_{\\lambda}}) = \\Tr(e^{-tL_{0,\\rho}}) - \\Tr(e^{-tL_{0,\\rho'}})\n\\end{equation}\nWe also set\n\\[\n\\tilde H_{\\lambda}\\equiv H_{\\lambda} - c\/24,\\quad\\tilde L_{0,\\rho} \\equiv \nL_{0,\\rho} - c\/24\\dots\n\\]\nIf ${\\mathcal A}_b$ is modular we then have by formula\n\\begin{align}\\label{m1}\n\\Str(e^{-2\\pi t{\\tilde H_{\\lambda}}}) =&\\sum_\\nu\nS_{\\rho,\\nu}\\Tr(e^{-2\\pi\\tilde L_{\\rho,\\nu}\/t}) -\\sum_\\nu \nS_{\\rho',\\nu}\\Tr(e^{-2\\pi\\tilde L_{\\rho',\\nu}\/t})\\\\\n=& \\sum_\\nu\n(S_{\\rho,\\nu} - S_{\\rho',\\nu})\\Tr(e^{-2\\pi\\tilde L_{0,\\nu}\/t})\\\\\n=& \\sum_\\nu\nD_{\\rho,\\nu}\\Tr(e^{-2\\pi\\tilde L_{0,\\nu}\/t})\n\\end{align}\nwhere $D_{\\rho,\\nu}\\equiv S_{\\rho,\\nu} - S_{\\rho',\\nu}$ as before and, by Prop. \\ref{DS},  $D_{\\rho,\\nu}= 2S_{\\rho,\\nu}$ if $\\nu$ is ${\\sigma}$-Fermi, and zero otherwise.\n\n\\subsection{Fredholm and Jones index within superconformal invariance}\nWe shall say that a general representation $\\lambda$ of the Fermi local conformal net ${\\mathcal A}$ is \n\\emph{supersymmetric}\\footnote{A characteristic feature of supersymmetry is also that the graded derivation implemented by $Q_\\lambda$ is densely defined in a appropriate sense. We shall not need this property in this paper.}\nif $\\lambda$ is graded and the conformal Hamiltonian $H_{\\lambda}$ satisfies\n\\[\n\\tilde H_{\\lambda} \\equiv H_{\\lambda} - c\/24 = Q_{\\lambda}^2\n\\]\nwhere $Q_{\\lambda}$ is a selfadjoint on ${\\mathcal H}_{\\lambda}$ which is odd w.r.t. the grading unitary $\\Gamma_{\\lambda}$. \n\nLet then $\\lambda$ be supersymmetric. An immediate important consequence is a positive lower bound for the energy:\n\\[\nH_{\\lambda}\\geq c\/24\\ .\n\\]\nNote that by McKean-Singer lemma $\\Str(e^{-t(H_{\\lambda} - c\/24)})$ is constant thus, taking the limit as $t\\to\\infty$, we have\n\\begin{equation}\\label{Windex}\n\\Str(e^{-t(H_{\\lambda} - c\/24)}) = {\\hbox{dim}}\\ker(H_{\\lambda} - c\/24)\\ ,\n\\end{equation}\nthe multiplicity of the lowest eigenvalue $c\/24$ of $H_{\\lambda}$.\n\nLet $\\rho$ be one of the two irreducible components of $\\lambda_b$ as above. We denote by $\\mathfrak R$ the set of ${\\sigma}$-Fermi irreducible sectors of ${\\mathcal A}_b$, that correspond to the Ramond sectors of ${\\mathcal A}$, see Prop. \\ref{acov}. Note that $\\mathfrak R \\neq\\varnothing$ as otherwise ${\\sigma}$ would be a degenerate sector, which is not possible because the braided tensor category of DHR \nsectors in modular as a consequence of the complete rationality assumption.\n\nAssume now that ${\\mathcal A}_b$ is modular. We notice that formula \\eqref{m1} can be written as\n\\begin{equation}\\label{expmin}\n\\Str(e^{-2\\pi t\\tilde H_{\\lambda}}) = 2\\sum_{\\nu\\in\\mathfrak R}\nS_{\\rho,\\nu}\\Tr(e^{-2\\pi\\tilde L_{0,\\nu}\/t}) \\ .\n\\end{equation}\n\\begin{corollary} We have\n\\[\n\\sum_{\\nu\\in\\mathfrak R } S_{\\rho,\\nu}d(\\nu) = 0\n\\]\n\\end{corollary}\n\\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }]\nDivide by $\\Tr(e^{-2\\pi tL_0})$ both members of (\\ref{expmin}); \nthe statement then follows by the Kac-Wakimoto formula \\eqref{KW1}.\n\\null\\hfill\\qed\\endtrivlist\\noindent\n\\begin{sublemma}\nLet $A_{1}, A_{2},\\dots A_{n}$ be selfadjoint operators such that $\\Tr(e^{-sA_k})<\\infty$  and $\\sum_{k}c_{k}\\Tr(e^{-sA_k})$ is a constant function of $s>0$, for some scalars $c_{k}$. Then \n\\[\n\\lim_{s\\to +\\infty}\\sum_{k}c_{k}\\Tr(e^{-sA_k})=\\sum_{k}c_{k}\\Dim\\ker A_k\n\\]\n\\end{sublemma}\n\\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }]\nClearly $\\lim_{s\\to +\\infty}\\Tr(e^{-sA})=\\Dim\\ker A$ if $A$ is trace class and non-negative. The finite trace condition implies that $A_k$ bounded below.\nLet $A_k = A^{+}_k + A^-_k$ with $A^{+}_k\\geq 0$ and $A^{-}_k <0$. Then the function $\\sum_{k}c_{k}\\Tr(e^{-sA^{-}_k})$ is a linear combination of exponentials of the form $e^{as}$ with $a>0$, and vanishes at infinity, thus it must be identically zero. It follows that\n\\[\n\\sum_{k}c_{k}\\Tr(e^{-sA_k})=\\sum_{k}c_{k}\\Tr(e^{-sA^+_k})\\underset{s=\\infty}{\\longrightarrow}\n\\sum_{k}c_{k}\\Dim\\ker A^+_k = \\sum_{k}c_{k}\\Dim\\ker A_k\n\\]\n\\null\\hfill\\qed\\endtrivlist\\noindent\n\\begin{lemma}\\label{ex1}\nLet ${\\mathcal A}$ be a Fermi modular net. If $\\lambda$ is supersymmetric then \n\\begin{equation}\\label{lim}\n\\Str(e^{-2\\pi t\\tilde H_\\lambda}) = 2\\sum_{\\nu\\in\\mathfrak R} S_{\\rho,\\nu}\\fn(\\nu,c\/24)\n\\end{equation}\nwhere $\\fn(\\nu,h)\\equiv\\Dim\\ker(L_{0,\\nu}- h)$.\n\\end{lemma}\n\\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }]\nThe left hand side of  \\eqref{expmin} is constant by McKean-Singer lemma, so we have that\n$\\sum_{\\nu\\in\\mathfrak R}S_{\\rho,\\nu}\\Tr(e^{-2\\pi s\\tilde L_{0,\\nu}})$ is a constant function of \n$s>0$. \nThus  \\eqref{lim} holds by the sublemma.\n\\null\\hfill\\qed\\endtrivlist\\noindent\nWith $\\lambda$ as in the above lemma, by eq. \\eqref{Windex} we have\n\\[\n\\Str(e^{-2\\pi t\\tilde H_{\\lambda}}) = \\ind(Q_{\\lambda +}\n)\\ .\n\\]\nTherefore by Lemma \\ref{ex1} we have\n\\[\n\\ind(Q_{\\lambda +}\n) = 2\\sum_{\\nu\\in\\mathfrak R}  S_{\\rho,\\nu}\\fn(\\nu,c\/24)\n\\]\nthen, writing Rehren definition of the $S$ matrix, we have\n\\[\n\\ind(Q_{\\lambda +}\n) = 2\\frac{d(\\rho)}{\\sqrt{\\mu_{{\\mathcal A}_b}}}\\sum_{\\nu\\in\\mathfrak R}K(\\rho,\\nu)\nd(\\nu)\\fn(\\nu,c\/24)\n\\]\nwhere $\\mu_{{\\mathcal A}_b}$ is the $\\mu$-index of ${\\mathcal A}_b$. By \\cite{KLM} we have $\\mu_{{\\mathcal A}_b} = 4\\mu_{{\\mathcal A}}$  therefore:\n\\begin{theorem}\\label{FJ}\nLet ${\\mathcal A}$ be a Fermi conformal net as above and $\\lambda$ \na supersymmetric irreducible representation of ${\\mathcal A}$. Then\n\\[\n\\ind(Q_{\\lambda +}\n) = \n\\frac{d(\\rho)}{\\sqrt{\\mu_{\\mathcal A}}}\\sum_{\\nu\\in\\mathfrak R}K(\\rho,\\nu)\nd(\\nu)\\fn(\\nu,c\/24)\n\\]\nwhere $\\rho$ is one of the two irreducible components of $\\lambda_b$ and $\\mu_{\\mathcal A}$ is the $\\mu$-index of ${\\mathcal A}$.\n\\end{theorem}\n\\noindent\nIn the above formula the Fredholm index of the supercharge operator $Q_{\\lambda +}\n$ is expressed \nby a formula involving the Jones index of the Ramond representations whose lowest eigenvalue $c\/24$ modulo integers.\n\\begin{corollary} If $\\ind(Q_{\\lambda})\\neq 0$ there exists a ${\\sigma}$-Fermi sector $\\nu$ such that  $c\/24$  is an eigenvalue of $L_{0,\\nu}$.\n\\end{corollary}\n\\begin{corollary} Suppose that, in Th. \\ref{FJ}, $\\rho$ is the only Ramond sector with lowest eigenvalue $c\/24$ modulo integers. Then \n\\[\nS_{\\rho,\\rho} = \\frac{d(\\rho)^2}{\\sqrt{\\mu_{{\\mathcal A}_b}}}K(\\rho,\\rho) = \\frac12 \\ .\n\\]\n\\end{corollary}\n\\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }]\nBy Lemma \\ref{ex1} we have:\n\\[\n\\ind(Q_{\\lambda +}\n) = 2S_{\\rho,\\rho}\\fn(\\rho,c\/24)\\ .\n\\]\nOn the other hand by formula \\eqref{Windex} \n\\[\n\\ind(Q_{\\lambda +}\n) = {\\hbox{dim}}\\, {\\rm ker}(H_\\lambda - c\/24) = \\fn(\\rho,c\/24) + \\fn(\\rho',c\/24)=  \\fn(\\rho,c\/24) \n\\]\nbecause $H_\\lambda = L_{0,\\rho}\\oplus L_{0,\\rho'}$ and $c\/24$ is not in the spectrum of $L_{0,\\rho'}$, so we get our formula.\n\\null\\hfill\\qed\\endtrivlist\\noindent\n\\section{Super-Virasoro algebra and super-Virasoro nets}\nWe now focus on model analysis and shall consider the most basic superconformal nets, namely the ones associated with the super-Virasoro algebra.\n\\subsection{Super-Virasoro algebra}\nThe super-Virasoro algebra governs the superconformal invariance \\cite{FQS,GKO}. It plays in the supersymmetric context the same r\\^ole that the Virasoro algebra  plays in the local conformal case.\n\nStrictly speaking, there are two super-Virasoro algebras. They are the super-Lie algebras generated by even \nelements $L_n$, $n\\in\\mathbb Z$, odd elements $G_r$, and a central even element $c$, \nsatisfying the relations\n\\begin{gather}\\label{svirdef}\n    [L_m , L_n] = (m-n)L_{m+n} + \\frac{c}{12}(m^3 - m)\\delta_{m+n, 0}\\\\\n    \\nonumber\n    [L_m, G_r] = (\\frac{m}{2} - r)G_{m+r}\\\\\n    \\nonumber\n    [G_r, G_s] = 2L_{r+s} + \\frac{c}{3}(r^2 - \\frac14)\\delta_{r+s,0}\n    \\end{gather}\nHere the brackets denote the super-commutator. \nIn the \\emph{Neveu-Schwarz} case $r\\in\\mathbb Z + 1\/2$, while in the \n\\emph{Ramond} case $r\\in\\mathbb Z$. We shall sometime use the term \n\\emph{super-Virasoro algebra} to indicate either the Neveu-Schwarz \nalgebra or the Ramond algebra. \n\nThe point is that, although the Neveu-Schwarz algebra and the  Ramond algebra are not isomorphic graded Lie algebras, they are representations of a same object, in a sense that we shall later see (they have the same ``isomorphic completion\").\n\nBy definition, the Neveu-Schwarz algebra and the Ramond algebra are both\nextensions of the Virasoro algebra. \n\nThe super-Virasoro algebra is equipped with the involution $L^*_n = \nL_{-n}$, $G^*_r = G_{-r}$, $c^* = c$ and we will be only interested in unitary \nrepresentations on a Hilbert space,\ni.e. representations preserving the involution. Note that unitary representations have automatically positive energy, namely $L_0\\geq 0$. Indeed we have \n\\begin{gather*}\n  L_0 =  \\frac12[G_{\\frac12}, G_{-\\frac12}] = \\frac12\\left(G_{\\frac12}G_{\\frac12}^* +\n  G_{\\frac12}^*G_{\\frac12}\\right)\\geq 0\\qquad (\\text{NS case})\\\\\n  L_0 =  \\frac12[G_0, G_0] = G_0^2 + c\/24 \\geq c\/24\n\\qquad (\\text{R case})    \n\\end{gather*}\nThe unitary, lowest weight representations of the super-Virasoro \nalgebra, namely the unitary representations of the super-Virasoro \nalgebra on a Hilbert space ${\\mathcal H}$ with a cyclic vector \n$\\xi\\in {\\mathcal H}$ satisfying\n\\[\nL_0\\xi = h\\xi, \\quad L_n\\xi = 0, \\ n>0, \\quad G_r\\xi = 0,\\ r>0,\n\\]\nare studied in \\cite{FQS,GKO}; in the NS case they are irreducible and uniquely determined \nby the values of $c$ and $h$. In the Ramond case one has to further specify the action of $G_0$ on the lowest energy subspace. It turns out that for a possible value of $c$ and $h$ there two inequivalent irreducible lowest weight representations (but for the case $c = h\/24$ when the representation is unique and graded). Not also that for $c \\neq h\/24$ the direct sum of the two inequivalent irreducible representations has a cyclic lowest energy vector and it is the unique Ramond $(c,h)$ lowest weight representation where grading is implemented.\n\nThe possible values are either $c\\geq 3\/2$, $h\\geq 0$ or\n\\begin{equation}\\label{c-values}\nc = \\frac32 \\left( 1 - \\frac{8}{m(m+2)}\\right), \\ m=2,3,\\ldots\n\\end{equation}\nand\n\\[\nh= h_{p,q}(c)\\equiv\\frac{[(m+2)p - mq]^2 - 4}{8m(m+2)} + \\frac{\\varepsilon}{8}\n\\]\nwhere $p= 1,2,\\ldots,m-1$, $q=1,2,\\ldots, m+1$ and $p-q$ is even or \nodd corresponding to the Neveu-Schwarz case ($\\varepsilon= 0$) or Ramond \ncase ($\\varepsilon = 1\/2$).\n\nNote that the Neveu-Schwarz algebra has a vacuum representation, \nnamely a irreducible representation with $0$ as eigenvalue of $L_0$, the Ramond \nalgebra has no vacuum representation. \n\\subsection{Stress-energy tensor}\n\\label{set}\nLet $c$ be an admissible value as above with $L_n$ $(n\\in\\mathbb Z)$, \n$n\\in\\mathbb Z$, $G_r$, the operators in \nthe corresponding to a Neveu-Schwarz ($r\\in\\mathbb Z + \\frac12$) or Ramond ($r\\in\\mathbb Z$)\nrepresentation. The Bose and Fermi stress-energy \ntensors are defined by\n\\begin{align}\nT_B(z) =& \\sum_{n} z^{-n-2} L_n\\\\\nT_F(z) =& \\frac12 \\sum_r z^{-r-3\/2} G_r\n\\end{align}\nnamely\n\\begin{equation}\n  \\text{Neveu-Schwarz case:}\\begin{cases}\nT_{B}(z) =& \\sum_{n\\in\\mathbb Z} z^{-n-2} L_n\\\\\nT_{F}(z) =& \\frac12 \\sum_{m\\in\\mathbb Z} z^{-m-2} G_{m+\\frac12}\n\\end{cases}\n\\end{equation}\n\\begin{equation}  \\text{Ramond case:}\\begin{cases}\nT_{B}(z) =& \\sum_{n} z^{-n-2} L_n\\\\\nT_{F}(z) =& \\frac12 \\sum_{m\\in\\mathbb Z} z^{-m-2} \\sqrt{z} G_m\n\\end{cases}\n\\end{equation}\nLet's now make a formal calculation for the (anti-)commutation relations of the Fermi stress energy tensor $T_F$. We want to show that, setting $w\\equiv z_2 \/z_1$, we have\n\\begin{equation}\\label{comrel}\n[T_F(z_1), T_F(z_2)]= \\frac12 z_1^{-1}T_B(z_1) \\delta(w)\n+ z_1^{-3}w^{-\\frac32}\\frac{c}{12}\n\\big(w^2\\delta''(w)  + \\frac34\\delta(w)\\big)\n\\end{equation}\nboth in the Neveu-Schwarz and in the Ramond case.\n\nSetting $k= r=s\\in\\mathbb Z$ we have, say in the Ramond case, \n\\begin{align*}\n[T_F&(z_1), T_F(z_2)]= \\frac14\\sum_{r,s}[G_r, G_s]z_1^{-r-3\/2}z_2^{-s-3\/2}\\\\ \n&= \\frac12\\sum_{r,s}L_{r+s}z_1^{-r-3\/2}z_2^{-s-3\/2} \n+  \\sum_r \\frac{c}{12}\\big(r^2 - \\frac14\\big)z_1^{-r-3\/2}z_2^{r-3\/2}\\\\\n&=\\frac12\\sum_{r,k}L_{k}z_1^{-r-3\/2}z_2^{-k+r-3\/2}\n + z_1^{-3\/2}z_2^{-3\/2}\\sum_r\\frac{c}{12}\\big(r^2 - \\frac14\\big)w^r\\\\\n&= \\frac12 z_1^{-1}\\Big(\\sum_k L_{k}z_2^{-k-2}\\Big) w^{\\frac12}\\sum_r w^{r} + z_1^{-3}w^{-\\frac32}\\sum_r\\frac{c}{12}\\big(r^2 - \\frac14\\big)w^r\\\\\n&=\\frac12 z_1^{-1}T_B(z_2) w^{\\frac12}\n\\sum_r w^r + z_1^{-3} w^{-\\frac32}\\sum_r\\frac{c}{12}\\big(r^2 - \\frac14\\big)w^r\\\\\n&= \\frac12 z_1^{-1}T_B(z_2) w^{\\frac12}\\delta(w)\n+ z_1^{-3}w^{-\\frac32}\\frac{c}{12}\n\\Big(w^2\\delta''(w)  + \\big(w-\\frac14\\big)\\delta(w)\\Big)\\\\\n&= \\frac12 z_1^{-1}T_B(z_1) \\delta(w)\n+ z_1^{-3}w^{-\\frac32}\\frac{c}{12}\n\\Big(w^2\\delta''(w)  + \\frac34\\delta(w)\\Big)\n\\end{align*}\nAs\n\\[\n\\sum_{r\\in\\mathbb Z} \\big(r^2 - \\frac14\\big)w^r \n=\\sum_{r\\in\\mathbb Z +\\frac12} \\big(r^2 - \\frac14\\big)w^r\n= w^2\\delta''(w)  + \\frac34\\delta(w)\n\\]\nthe above calculation (by using equalities as $\\delta(w)\\sqrt{w}=\\delta(w)$ and similar ones) shows that the commutation relations for the Fermi stress energy tensor $T_F$ and, analogously, the commutation relations for $T_B$ and $T_F$ are indeed the same in the Neveu-Schwarz case and in the Ramond case, namely they are representations of the same (anti-)commutation relations. This is basic reason to view the Neveu-Schwarz and Ramond algebras as different types of representations of a unique algebra. \n\nHowever the above calculation is only formal. To give it a rigorous meaning, \nand have convergent series, we have to smear the stress energy tensor with a \nsmooth test function with support in an interval. We then arrive naturally to \nconsider the net of von Neumann algebras of operators localised in intervals. \nIn the case of central charge $c<3\/2$ we shall see that Neveu-Schwarz and \nRamond representations correspond to DHR and general \nsolitons of the associated super-Virasoro net.\n\n\\subsection{Super-Virasoro nets}\nWe give here the definition of the super-Virasoro nets for all the \nallowed values of the central charge. \nWe follow the strategy adopted in \\cite{BS-M} for the case of Virasoro nets, \ncf. also \\cite{Car04,loke}. An alternative construction in the case of the \ndiscrete series ($c<3\/2)$ is outlined in Sect. \\ref{SVirnet}. \n\nLet $\\lambda$ be a unitary positive energy representation of the \nsuper-Virasoro algebra on a Hilbert space ${\\mathcal H}_\\lambda$. The \ncorresponding conformal Hamiltonian $L_0$ is the self-adjoint operator on ${\\mathcal H}_\\lambda$ \nand we denote ${\\mathcal H}_\\lambda^\\infty$ the dense subspace of smooth vectors for $L_0$, \nnamely the subspace of vectors belonging to the domain of $L_0^n$ for all \npositive integers $n$.\n\nThe operators $L_n$,\n$n\\in {\\mathbb Z}$ satisfy the linear energy bounds \n\\begin{equation}\n\\label{e-boundsB}\n\\|L_n v\\|\\leq M (1+|n|^{\\frac{3}{2}})\\|(1+L_0)v\\|, \\quad v\\in {\\mathcal \nH}_\\lambda^\\infty, \n\\end{equation}\nfor suitable constant $M>0$ depending on the central charge $c$, cf. \n\\cite{BS-M,CW}. \nMoreover from the relations \n$[G_{-r}, G_r] = 2L_{0} + \\frac{c}{3}(r^2 - \\frac14)$, \nwe find the energy bounds\n\\begin{equation}\n\\label{e-boundsF}\n\\|G_r v\\|\\leq (2+ \\frac{c}{3}r^2)^{\\frac12}\\|(1+L_0)^{\\frac12}v\\|, \\quad v\\in \n{\\mathcal H}_\\lambda^\\infty,\n\\end{equation}\nwhere $r \\in {\\mathbb Z}+1\/2$ (resp. $r \\in {\\mathbb Z}$) if $\\lambda$ is a Neveu-Schwarz \n(resp. Ramond) representation. \nWe now consider the vacuum representation of the super-Virasoro algebra with \ncentral charge $c$ and denote by ${\\mathcal H}$ the corresponding Hilbert space and \nby $\\Omega$ the vacuum vector, namely the unique (up to a phase) unit \nvector such that $L_0\\Omega=0$.  \n\nLet $f$ be a smooth function on $S^1$. It follows from the linear \nenergy bounds in Eq. (\\ref{e-boundsB}) and the fact that the Fourier \ncoefficients\n\\begin{equation}\n\\hat{f}_n=\\int_{-\\pi}^\\pi f(e^{i\\theta})e^{-in\\theta}\\frac{{\\rm d}\\theta}{2\\pi},\\;\nn\\in {\\mathbb Z},\n\\end{equation}\nare rapidly decreasing, that the smeared Bose stress-energy tensor  \n\\begin{equation}\nT_B(f)= \\sum_{n \\in {\\mathbb Z}}\\hat{f}_nL_n\n\\end{equation}\nis a well defined operator with invariant domain ${\\mathcal H}^\\infty$. Moreover, for \n$f$ real,  $T_B(f)$ is essentially self-adjoint on ${\\mathcal H}^\\infty$ (cf. \\cite{BS-M})\nand we shall denote again $T_B(f)$ its self-adjoint closure.  \n\nNow let $f$ be a smooth function on $S^1$ whose support do not contains \n$-1$. Then also the coefficients  \n\\begin{equation}\n\\hat{f}_r=\\int_{-\\pi}^\\pi f(e^{i\\theta})e^{-ir\\theta}\\frac{{\\rm d}\\theta}{2\\pi},\\;\nr\\in {\\mathbb Z} + \\frac{1}{2},\n\\end{equation}\nare rapidly decreasing and it follows from the energy bounds \nin Eq. (\\ref{e-boundsF}) that the corresponding smeared Fermi stress-energy \ntensor \n\\begin{equation}\nT_F(f)=\\frac{1}{2}\\sum_{r \\in {\\mathbb Z}+\\frac{1}{2}}\\hat{f}_rG_r\n\\end{equation}\nis also a well defined operator with invariant domain ${\\mathcal H}^\\infty$. Again, \nfor $f$ real, $T_F(f)$ is essentially self-adjoint on ${\\mathcal H}^\\infty$ \n(cf. \\cite{BS-M}) and we denote its self-adjoint closure by the same \nsymbol. \n\nAs in Sect. \\ref{netR} we identify $\\mathbb R$ with $S^1\/ \\{-1\\}$ and\nconsider the family ${\\mathcal I}_{\\mathbb R}$ of nonempty, bounded, open intervals of\n${\\mathbb R}$ as a subset of the family ${\\mathcal I}$ of intervals of $S^1$. We define\na net ${\\rm SVir}_{c}$ of von Neumann algebras on ${\\mathbb R}$ by \n\\begin{equation}\n\\label{gener1} \n{\\rm SVir}_{c}(I) \\equiv \\{ e^{iT_B(f)}, e^{iT_F(f)}:f\\in C^{\\infty}(S^1) \n\\;{\\rm real},\\, {\\rm supp}f, \\subset I\\}'', \\; I \\in {\\mathcal I}_{\\mathbb R}. \n\\end{equation} \nIsotony is clear from the definition and we have to show that the net is graded \nlocal and covariant.  \n\nWe first consider graded locality. The spectrum of $L_0$ is \ncontained in ${\\mathbb Z}\/2$ and hence the unitary operator $\\Gamma = e^{i2\\pi L_0}$ \nis an involution such that $\\Gamma \\Omega$.  \nIt is straightforward to check that if $f_1, f_2$ \nare smooth functions on $S^1$ and the support of $f_2$ does not contain $-1$ \nthen $\\Gamma T_B(f_1) = T_B(f_1)\\Gamma$ and \n$\\Gamma T_F(f_1) = -T_F(f_1) \\Gamma$.  \nHence, $\\gamma = {\\rm Ad}\\Gamma$ is a ${\\mathbb Z}_2$-grading on the net \n${\\rm SVir}_{c}$, namely \n$\\Gamma {\\rm SVir}_{c}(I) \\Gamma^* ={\\rm SVir}_{c}(I)$, for all $I\\in \n{\\mathcal I}_{\\mathbb R}$. Now let $I_1, I_2 \\in {\\mathcal I}_{\\mathbb R}$ be disjoint intervals and \nlet $f_1, f_2$ be real smooth functions on $S^1$ with support in $I_1, \nI_2$ respectively. Then the operators $T_B(f_1)$ and $T_F(f_1)$ commute\nwith $ZT_B(f_2)Z^*$ and $ZT_F(f_2)Z^*$, $Z=\\frac{1-i\\Gamma}{1-i}$, on \n${\\mathcal H}^\\infty$, cf. the (anti-)commutation relations in Sect. \\ref{set}\n(note that $Z{\\mathcal H}^\\infty={\\mathcal H}^\\infty)$. Using the energy  bounds in \nEq. (\\ref{e-boundsB}) and Eq. (\\ref{e-boundsF}) and the fact that \n$Z$ commutes with $L_0$ one can apply the argument in \n\\cite[Sect. 2]{BS-M} to show that $e^{iT_B(f_1)}$ and $e^{iT_F(f_1)}$ \ncommute with $Ze^{iT_B(f_2)}Z^*$ and $Ze^{iT_F(f_2)}Z^*$. It follows that\nthe net ${\\rm SVir}_c$ is graded local, namely \n\\begin{equation} \n{\\rm SVir}_c(I_1) \\subset Z{\\rm SVir}_c(I_2)Z^*\n\\end{equation}  \nwhenever $I_1, I_2$ are disjoint interval in ${\\mathcal I}_{\\mathbb R}$. \n\nWe now discuss covariance. The crucial fact here is that the representation \nof the Virasoro algebra on ${\\mathcal H}$ integrates to a strongly continuous \nunitary projective positive-energy representation of ${\\mathrm {Diff}}^{(\\infty)}(S^1)$ \non ${\\mathcal H}$ by \\cite{GoWa,Tol99} which factors through ${\\mathrm {Diff}}^{(2)}(S^1)$ because\n$e^{i4\\pi L_0}=1$. Hence there is a strongly continuous projective \nunitary representation $U$ of ${\\mathrm {Diff}}^{(2)}(S^1)$ on ${\\mathcal H}$ such that, \nfor all real $f \\in C^\\infty(S^1)$ and all $x \\in B({\\mathcal H})$,  \n\\begin{equation}\nU(\\exp^{(2)}(tf))xU(\\exp^{(2)}(tf))^*= \ne^{itT_B(f)}xe^{-itT_B(f)}, \n\\end{equation}\nwhere, $\\exp^{(2)}(tf)$ denotes the lift to ${\\mathrm {Diff}}^{(2)}(S^1)$ of the\none-parameter subgroup $\\exp(tf)$ of ${\\mathrm {Diff}}(S^1)$ generated by the \n(real) smooth vector field $f(e^{i\\theta})\\frac{{\\rm d}}{{\\rm d}\\theta}$. Moreover, \nif $\\theta \\to r^{(2)}(\\theta)$ is the lift to ${\\mathrm {Diff}}^{(2)}(S^1)$ of the \none-parameter subgroup of rotations in ${\\mathrm {Diff}}(S^1)$ we have  \n\\begin{equation}\nU(r^{(2)}(\\theta))= e^{i\\theta L_0},\n\\end{equation}\nfor all $\\theta$ in ${\\mathbb R}$. \nThe following properties of $U$ follow rather straightforwardly. \n\\medskip\n\n\\noindent $(1)$ The restriction of $U$ to the subgroup \n${\\rm\\textsf{M\\\"ob}}^{(2)} \\subset {\\mathrm {Diff}}^{(2)}(S^1)$ lifts to a unique strongly continuous \nunitary representation which we again denote by $U$. \nIf $\\exp^{(2)}(tf) \\in {\\rm\\textsf{M\\\"ob}}^{(2)}$ for all $t\\in {\\mathbb R}$, this \nunitary representation satisfies \n\\begin{equation}\nU(\\exp^{(2)}(tf))=e^{itT_B(f)},\\; U(\\exp^{(2)}(tf))\\Omega=\\Omega,\n\\end{equation}\nfor all $t\\in {\\mathbb R}$.   \n\n\\medskip\n\n\\noindent $(2)$  If the support of the real smooth function $f$ is contained \nin $I \\in {\\mathcal I}_{\\mathbb R}$ then \n\\begin{equation}\n\\label{covTB}\nU(g)e^{iT_B(f)}U(g)^*\\in {\\rm SVir}_c(\\dot{g}I)\n\\end{equation}\nfor all $g \\in {\\mathrm {Diff}}^{(2)}(S^1)$ such that $\\dot{g}I \\in {\\mathcal I}_{\\mathbb R}$. \n\\medskip\n\n\\noindent $(3)$ For all $g \\in {\\mathrm {Diff}}^{(2)}(S^1)$ we have \n\\begin{equation}\nU(g)\\Gamma U(g)^* = \\Gamma\n\\end{equation}\nfor all $g \\in {\\mathrm {Diff}}^{(2)}(S^1)$.\n\n\\medskip\n\nNote that property $(3)$ follows from the fact that ${\\mathrm {Diff}}^{(2)}(S^1)$ is \nconnected and $\\Gamma^2=1$. \n\nWe now consider the covariance properties of the Fermi stress-energy\ntensor $T_F$. From the commutation relations in equations (\\ref{svirdef})\nwe find,  \n\\begin{equation}\n\\label{smearedcommutator}\ni[T_B(f_1), T_F(f_2)]v = T_F(\\frac{1}{2}f'_1f_2-f'_2f_1)v, \\; v\\in \n{\\mathcal H}^\\infty\n\\end{equation}\nwhere $f_1, f_2$ are real smoot functions such that \n${\\rm supp} f_2 \\subset I$ for some $I\\in {\\mathcal I}_{\\mathbb R}$ and, for any\n$f\\in C^{\\infty}(S^1)$, \n$f'$ is defined by $f'(e^{i\\theta})=\\frac{d}{d\\theta}f(e^{\\theta})$.  \nFor any $g \\in {\\mathrm {Diff}}(S^1)$ consider the function \n$X_g:S_1 \\to {\\mathbb R}$ defined by\n\\begin{equation}\nX_g(e^{i\\theta})= -i\\frac{{\\rm d}}{{\\rm d}\\theta}\\log (ge^{i\\theta}).\n\\end{equation}\nSince $g$ is a diffeomorfism of $S^1$ preserving the orientation \nthen $X_g(z)>0$ for all $z \\in S^1$. Moreover $X_g \\in C^\\infty(S^1)$. \nAnother straightforward consequence of the definition is that \n\\begin{equation}\nX_{g_1g_2}(z)=X_{g_1}(g_2z)X_{g_2}(z). \n\\end{equation}\nAs a consequence the the family of continuous linear operators \n$\\beta(g)$, $g\\in {\\mathrm {Diff}} (S^1)$  on the Fr\\'echet space $C^\\infty(S^1)$ \ndefined by \n\\begin{equation}\n(\\beta(g)f)(z)= X_g(g^{-1}z)^{\\frac12}f(g^{-1}z)\n\\end{equation}\ngives a strongly continuous representation of ${\\mathrm {Diff}} (S^1)$ leaving \nthe real subspace of real functions invariant. Moreover if \n$f_1, f_2\\in C^\\infty(S^1)$ are real then vector valued function \n$t \\to \\beta(\\exp(tf_1))f_2$ is differentiable in $C^\\infty(S^1)$ and\n\\begin{equation}\n\\label{actionderivative}\n\\frac{d}{dt}\\beta(\\exp(tf_1))f_2|_{t=0}=\\frac{1}{2}f'_1f_2-f_1f'_2.\n\\end{equation}  \nNow let ${\\rm supp} f_2$ be a subset of some interval $I \\in I_{\\mathbb R}$ \nand let $J_I\\subset {\\mathbb R}$ be the connected component of $0$ in ${\\mathbb R}$ \nof the open set $\\{t\\in {\\mathbb R}:\\exp(tf_1)I \\in I_{\\mathbb R} \\}$. Then, for any \n$v\\in {\\mathcal H}^\\infty$ the function $J_I \\ni t \\to T_F(\\beta(\\exp(tf_1))f_2)v$ \nis differentiable in ${\\mathcal H}$ and it follows from Eq. \n(\\ref{smearedcommutator}) and Eq. (\\ref{actionderivative}) \nthat \n\\begin{equation}\n\\label{dTF} \n\\frac{d}{dt}T_F(\\beta(\\exp(tf_1))f_2)v|_{t=0}=i[T_B(f_1), T_F(f_2)]v.\n\\end{equation}\nWe now specialize to the case of M\\\"{o}bius transformations i.e. \nwe assume that $\\exp(tf_1) \\in {\\rm\\textsf{M\\\"ob}}$ for all $t\\in {\\mathbb R}$.   \nThe map $J_I \\ni t \\to v(t) \\in {\\mathcal H}$ given by \n\\begin{equation}\nv(t)=T_F(\\beta(\\exp(tf_1))f_2)U(\\exp^{(2)}(tf_1))v\n\\end{equation} \nis well defined because, $U(\\exp^{(2)}(-tf_1)){\\mathcal H}^\\infty = {\\mathcal H}^\\infty$ \nfor all $t\\in {\\mathbb R}$. Note also that $v(t)\\in {\\mathcal H}^\\infty$ for all $t\\in J_I$.\nNow using Eq. (\\ref{dTF}) and the energy bounds in Eq. \n(\\ref{e-boundsF}) it can be shown that $v(t)$ is differentiable (in the \nstrong topology of ${\\mathcal H}$) and that it satisfy the following differential \nequation on ${\\mathcal H}$ \n\\begin{equation}\n\\frac{d}{dt}v(t)=iT_B(f_1)v(t).\n\\end{equation}\nIf follows that \n\\begin{equation}\nv(t)=U(\\exp^{(2)}(tf_1))T_F(f_2)v\n\\end{equation}\nand since $v \\in {\\mathcal H}^\\infty$ was arbitrary we get, for all $t\\in J_I$, \nthe following equality of self-adjoint operators\n\\begin{equation}\nU(\\exp^{(2)}(tf_1))T_F(f_2)U(\\exp^{(2)}(-tf_1))=T_F(\\beta(\\exp(tf_1))f_2). \n\\end{equation}\nNow, if we denote by ${\\mathcal U}^{(2)}_I$ the connected component of the identity \nin ${\\rm\\textsf{M\\\"ob}}^{(2)}$\nof the open set $\\{g\\in {\\rm\\textsf{M\\\"ob}}^{(2)}: gI \\in {\\mathcal I}_{\\mathbb R} \\}$ it follows that \n\\begin{equation}\n\\label{covTF}\nU(g)T_F(f)U(g)^*=T_F(\\beta(\\dot{g})f),\n\\end{equation}\nfor any real smoot function on $S^1$ with ${\\rm supp}f \\subset I$\nand all $g\\in {\\mathcal U}^{(2)}_I$. \n\nFrom Eq. (\\ref{covTF}) and Eq. (\\ref{covTB}) we have \n\\begin{equation}\nU(g){\\rm SVir}_c (I)U(g)^* = {\\rm SVir}_c(\\dot{g}I)\\; I\\in {\\mathcal I}_{\\mathbb R},\n\\; g \\in {\\mathcal U}^{(2)}_I. \n\\end{equation} \nHence ${\\rm SVir}_c$ extends to a M\\\"{o}bius covariant  net on \n$S^1$ satisfying graded locality, see Sect. \\ref{coverNets}. Note that we \nhave not yet shown that the vacuum vector $\\Omega$ is cyclic and \nhence we still don't knew if the net satisfy all the requirements of \nProperty 3 in the definition of M\\\"{o}bius covariant Fermi nets on\n$S^1$ given in Sect. \\ref{MobFermiNets}. \nWe shall however prove the cyclicity of the vacuum as a part of the following \ntheorem. \n\n\\begin{theorem}    \n${\\rm SVir}_c$ is an irreducible Fermi conformal on $S^1$ for any of the \nallowed values \nof the central charge $c$.\n\\end{theorem}\n\\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }] Since $\\Omega$ is the unique (up to a phase) unit vector in \nthe kernel of $L_0$, we only have to show that $\\Omega$ is cyclic and that \nthe strongly continuous positive-energy projective representation $U$\nof ${\\mathrm {Diff}}^{(2)}(S^1)$ defined above makes the net diffeomorphism \ncovariant in the appropriate sense. We first show that $\\Omega$ \nis cyclic for the net. Let ${\\mathcal K} \\subset {\\mathcal H}$ be the closure \nof $\\bigvee_{I\\in {\\mathcal I}} {\\rm SVir}_c(I) \\Omega$. \nWe have to show that ${\\mathcal K} ={\\mathcal H}$.\nClearly $U(g){\\mathcal K}={\\mathcal K}$\nfor all $g \\in {\\rm\\textsf{M\\\"ob}}^{(2)}$. It follows that if $j\\in {\\mathbb Z} \/2$ \n$P_j$ is the orthogonal projection of ${\\mathcal H}$ onto the kernel of $L_0-k1$ \nthen $P_j {\\mathcal K} \\subset {\\mathcal K}\\cap {\\mathcal H}^\\infty$. Now let $r\\in {\\mathbb Z}+1\/2$. Since the \nsmooth functions on $S^1$ wose support does not contain the point $-1$ is \ndense in $L^2(S^1)$ we can find an interval $I\\in {\\mathcal I}_{\\mathbb R}$ and \na real smooth function $f$ with ${\\rm supp}f \\subset I$ such that \n$\\hat{f}_r \\neq 0$. Since $T_F(f)P_j {\\mathcal K} \\subset {\\mathcal K}$ we find \n\\begin{equation}\nG_rP_j{\\mathcal K}=\\frac{1}{\\hat{f}_r }P_{j-r}T_F(f)P_j{\\mathcal K} \\subset {\\mathcal K}. \n\\end{equation}\nA similar argument applies to the operators $L_n$, $n\\in {\\mathbb Z}$ \nand hence the linear span ${\\cal L}$ of the subspaces \n$P_j{\\mathcal K}$, $j\\in {\\mathbb Z} \/2$ is invariant for the \nrepresentation of the super-Virasoro algebra. Since $\\Omega \\in {\\cal L}$ \nis cyclic for the latter representation it follows that ${\\cal L}$ is \ndense in ${\\mathcal H}$. Hence ${\\mathcal K} = {\\mathcal H}$ because ${\\cal L} \\subset {\\mathcal K}$.  Hence \n${\\rm SVir}_c$ is an irreducible M\\\"{o}bius covariant Fermi net \non ${S^1}$.  \n\nTo show that ${\\rm SVir}_c$ is diffeomorphism covariant we first observe \nthat by \\cite[Sect. V.2]{loke}  for any $I\\in {\\mathcal I}$ the group generated by \ndiffeomorphisms of the form \n$\\exp(f)$ with ${\\rm supp}f \\subset I$ is dense in ${\\mathrm {Diff}}_I(S^1)$. It \nfollows that the group generated by elements of the form \n$\\exp^{(2)}(f)$ with ${\\rm supp}f \\subset I$ is dense in \n${\\mathrm {Diff}}^{(2)}_I(S^1)$. Hence, for any $I \\in {\\mathcal I}$ and any \n$g\\in \\in {\\mathrm {Diff}}^{(2)}_I(S^1)$,  $U(g) \\in {\\rm SVir}_c(I)$ and, \nby graded locality, $U(g) \\in {\\rm SVir}(I')'$ because. Now, an \nadaptation of the argument in the proof of \\cite[Proposition 3.7]{Car04}\nshows that ${\\rm SVir}_c$ is diffeomorphism covariant and the proof is \ncomplete. \n\\null\\hfill\\qed\\endtrivlist\\noindent\n\\subsection{The discrete series of super-Virasoro nets}\\label{SVirnet}\nWe shall now use the construction in \\cite{GKO} to study ${\\rm SVir}_{c}$ with $c < 3\/2$ an admissible value.\nFirst consider three real free Fermi fields in the NS representation. \nThey define a graded-local net on $S^1$.  \nThis net coincides with ${\\mathcal F}^{\\hat{\\otimes}3}={\\mathcal F}\\hat{\\otimes}{\\mathcal F}\\hat{\\otimes}{\\mathcal F}$ \nwhere ${\\mathcal F}$ is the net generated by a single real free Fermi field in the NS representation \n(cf. \\cite{Bock}) and $\\hat{\\otimes}$ denotes the graded tensor product. The net \n${\\mathcal A}_{{{\\rm SU}(2)}_2}$ embeds as a subnet of ${\\mathcal F}^{\\hat{\\otimes}3}$. Actually, from the discussion in \n\\cite[page 115]{GKO} we have\n\\[\n{\\mathcal A}_{{{\\rm SU}(2)}_2}={\\mathcal F}^{\\hat{\\otimes}3}_b\\ .\n\\] \nNow consider the conformal net ${\\mathcal F}_N$ (on the Hilbert space \n${\\mathcal H}_N$) given by ${\\mathcal F}^{\\hat{\\otimes}3}\\otimes {\\mathcal A}_{{{\\rm SU}(2)}_N}$, $N$ positive integer.  \n\nConsider now the the representation of the \nsuper-Virasoro algebra on ${\\mathcal H}_N$ with central \ncharge \n\\begin{equation}\nc_N=\\frac{3}{2}\\left(1-\\frac{8}{(N+2)(N+4)}\\right),\n\\end{equation} \nconstructed in \\cite[Sect. 3]{GKO} (coset construction). Then the corresponding \nstress energy-tensors $T_B$ and $T_F$ generate a family of von Neumann algebras on ${\\mathcal H}_N$ \nas in eq. \\eqref{gener1}. Using the energy bounds in Eq. (\\ref{e-boundsB}) and Eq. \n(\\ref{e-boundsF}) it can be shown that this family defines \na Fermi subnet of ${\\mathcal F}_N$ as in eq. \\eqref{gener1} which can be identified with \nthe super-Virasoro net ${\\rm SVir}_{c_N}$.\nIn this way we obtain all the super-Virasoro nets corresponding to the discrete series.\n\n \nUsing \\cite{GKO} we can identify these super-Virasoro nets as \ncoset subnets. From the embedding  \n\\begin{equation}\n{\\mathcal A}_{{{\\rm SU}(2)}_2}\\otimes {\\mathcal A}_{{{\\rm SU}(2)}_N} \\subset {\\mathcal F}_N\n\\end{equation}\nwe have the embedding\n\\begin{equation}\n{\\mathcal A}_{{{\\rm SU}(2)}_{N+2}} \\subset {\\mathcal F}_N.\n\\end{equation}\nIt follows from Eq. (3.13) and the claim at the end of \npage 114 in \\cite[Sect. 3]{GKO} that ${\\rm SVir}_{c_N}$ is \ncontained in the coset \n\\begin{equation}\n({\\mathcal A}_{{{\\rm SU}(2)}_{N+2}})^c = ({\\mathcal A}_{{{\\rm SU}(2)}_{N+2}})'\\cap {\\mathcal F}_N.\n\\end{equation}\nMoreover it follows from the branching rules in \n\\cite[Eq. 4.15]{GKO} that these nets coincide (cf. \\cite{KL1})\nnamely \n\\begin{equation} \n{\\rm SVir}_{c_N} = ({\\mathcal A}_{{{\\rm SU}(2)}_{N+2}})^c. \n\\end{equation}\nAs a consequence the Bose subnet ${\\rm SVir}^0_{c_N}\\equiv ({\\rm SVir}_{c_N})_b$ of \nthe super-Virasoro net ${\\rm SVir}_{c_N}$ is equal to the coset\n\n\\begin{equation}\n\\label{BoseCoset}\n({\\mathcal A}_{{{\\rm SU}(2)}_{N+2}})'\\cap \\left( {\\mathcal A}_{{{\\rm SU}(2)}_2}\\otimes {\\mathcal A}_{{{\\rm SU}(2)}_N} \\right)\n\\end{equation}\nand hence, by \\cite[Corollary 3.4]{X2} and \\cite[Theorem 24]{L03} \n${\\rm SVir}^0_{c_N}\n$ is completely rational, see also \\cite[Corollary 28]{L03}. \n\nNow we look at representations. We denote $(NS)$ and $(R)$ the \nNeveu-Schwartz and Ramond representations for three Fermion fields \nrespectively. In $(NS)$ the lowest energy eigenspace is one-dimensional\n(``nondegenerate vacuum ''), whilst \nin $(R)$ it is two-dimensional (``2-fold degenerate vacuum'').\\footnote{Different Ramond \nrepresentations could be defined corresponding to different choices of the corresponding representation of the Dirac algebra of the 0-modes on the subspace of  lowest energy vectors, \ncf. page 113 and page 115 of \\cite{GKO}.} \n\nIt is almost obvious that $(NS)$ corresponds to the vacuum \nrepresentation $\\pi_{NS}$ of ${\\mathcal F}^{\\hat{\\otimes}3}$ and, \narguing as in the proof of \\cite[Lemma 4.3]{Bock}, it can be \nshown that $(R)$ corresponds to a general soliton  \n$\\pi_R$ of the latter net. \nClearly $(NS)$ and $(R)$ restrict to positive-energy \nrepresentations of ${{\\rm SU}(2)}_2$. We denote by \n$\\pi_{(N,l)}$ the representation of ${\\mathcal A}_{{{\\rm SU}(2)}_N}$ with spin $l$. \nAt level $N$ the possible values of the spin are those satisfying \n$0\\leq 2l \\leq N$. Then the following identities hold (see \\cite[page \n116]{GKO}): \n\n\\begin{eqnarray}\n\\label{NSrest} \n\\pi_{NS}|_{{\\mathcal A}_{{{\\rm SU}(2)}_2}} = \\pi_{(2,0)} \\oplus \\pi_{(2,1)} \\\\\n\\label{Rrest}\n\\pi_R|_{{\\mathcal A}_{{{\\rm SU}(2)}_2}} = \\pi_{(2,\\frac{1}{2})}.\n\\end{eqnarray}   \nNote that the restriction of $(R)$ remains irreducible because the \ngrading automorphism is not unitarily implemented, cf. Prop. \\ref{n1}. \n\nDenote by $(c_N,h_{p,q})_{NS}$, resp. $(c_N,h_{p,q})_{R}$, a \nNS, resp. R, irreducible representation of super-Virasoro algebra with central \ncharge $c_N$ and lowest energy \n$$h_{p,q}=\\frac{\\left[(N+4)p-(N+2)q\\right]^2 -4}{8(N+2)(N+4)},$$\nresp. \n$$h_{p,q}=\\frac{\\left[(N+4)p-(N+2)q\\right]^2 -4}{8(N+2)(N+4)} + \n\\frac{1}{16},$$\nwhere $p=1,2,\\dots,N+1$, $q=1,2, \\dots N+3$ and $p-q$ is even in the \nNS case and odd in the R case. \n\nAs already mentioned, in the NS case, for every value of the central charge, the lowest energy\n$h_{p,q}$ completely determines the (equivalence class of) the \nrepresentation. In contrast for a given values of the central charge \nand of the lowest energy $h_{p,q}$ there are two Ramond representations \none with \n\\[\nG_0\\Psi_{h_{p,q}}= \\sqrt{h_{p,q} - \\frac{c_N}{24}}\\Psi_{h_{p,q}}\n\\]\nand the other with\n\\[\nG_0\\Psi_{h_{p,q}}= -\\sqrt{h_{p,q} - \\frac{c_N}{24}}\\Psi_{h_{p,q}},\n\\]\nwhere $\\Psi_{h_{p,q}}$ is the lowest energy vector. These two \nrepresentations are connected by the automorphism \n$G_r \\to - G_r$ and become equivalent when restricted to the even \n(Bose) subalgebra. Accordingly $(c_N,h_{p,q})_{R}$ denotes indifferently these \ntwo representations which are clearly inequivalent when \n$h_{p,q} \\neq \\frac{c_N}{24}$. \n\nFor a given $N$ the equality $h_{p,q} = h_{p',q'}$ when \n$p-q$ and $p'-q'$ are both even or odd hold if and only \nif $p'=N+2-p$ and $q'= N+4 - q$. Note also that it may happen \nthat $h_{p,q} = h_{p',q'}$ when $p-q$ is even and $p'-q$ \nis odd. For example, if $N=2$ then $h_{2,2} = h_{1,2}= 1\/16$. \nAccordingly there are values of $N$ for which a given value \nof the lowest energy corresponds to three distinct irreducible \nrepresentations of super-Virasoro algebra: one NS representation and two\nR representations.  \n\nFrom \\cite[Section 4]{GKO} we can conclude that there exist DHR representations \n$\\pi_{h_{p,q}}^{NS}$, $p-q$ even, and general solitons \n$\\pi_{h_{p,q}}^{R}$, $p-q$ odd, of ${\\rm SVir}_{c_N}$ (associated to the representations \n$(c_N,h_{p,q})_{NS}$, resp. $(c_N,h_{p,q})_{R}$ of the of super-Virasoro algebra)\nsuch that  \n\\begin{equation}\n\\label{GKOnetsNS}\n\\left( \\pi_{NS} \\otimes \\pi_{(N,\\frac{1}{2}[p-1])}\\right)\n|_{{\\mathcal A}_{{{\\rm SU}(2)}_{N+2}}\\otimes {\\rm SVir}_{c_N}} =\n\\bigoplus_q \\pi_{(N+2, \\frac{1}{2}[q-1])} \\otimes \\pi_{h_{p,q}}^{NS},\n\\end{equation}\n$1\\leq q \\leq N+3$, $p-q$ even, and\n\\begin{equation}\n\\label{GKOnetsR}\n\\left( \\pi_{R} \\otimes \\pi_{(N,\\frac{1}{2}[p-1])}\\right)\n|_{{\\mathcal A}_{{{\\rm SU}(2)}_{N+2}}\\otimes {\\rm SVir}_{c_N}} =\n\\bigoplus_q \\pi_{(N+2, \\frac{1}{2}[q-1])} \\otimes \\pi_{h_{p,q}}^{R},\n\\end{equation}\n$1\\leq q \\leq N+3$, $p-q$ odd. \n                                                                                \nWe now denote $\\rho_{h_{p,q}}^{NS}$,  resp. $\\rho_{h_{p,q}}^{R}$, \nthe restriction of $\\pi_{h_{p,q}}^{NS}$, resp \n$\\pi_{h_{p,q}}^{R}$, to ${\\rm SVir}^0_{c_N}$. \n\nIn the representation space of $\\pi_{h_{p,q}}^{NS}$ the grading is always \nunitarily implemented and hence we have the direct sum \n$$\\rho_{h_{p,q}}^{NS}= \\rho_{h_{p,q}}^{NS+} \\oplus\\rho_{h_{p,q}}^{NS-}$$ \nof two (inequivalent) irreducible representations corresponding to the \neigenspaces with eigenvalues 1 and -1 of the grading operator \nrespectively. \n\nIn contrast in the case of $\\pi_{h_{p,q}}^{R}$ the grading automorphism \nis unitarily implemented only if $h_{p,q}=c_N\/24$. This happens  \nif and only if $N$ is even and $p=(N+2)\/2$, $q=(N+4)\/2$. \nIn this case $\\pi_{\\frac{c_N}{24}}^R$ is a supersymmetric \ngeneral representation of the Fermi conformal net ${\\rm SVir}_{c_N}$.\nMoreover we have the decomposition into irreducible (inequivalent) \nsubrepresentations $$\\rho_{\\frac{c_N}{24}}^R= \\rho_{\\frac{c_N}{24}}^{R+}\\oplus \n\\rho_{\\frac{c_N}{24}}^{R-}.$$ In the remaining cases \n$\\rho_{h_{p,q}}^{R}$ is irreducible. \n\nRestricting Eq. (\\ref{GKOnetsNS} ) and Eq. (\\ref{GKOnetsR} ) to the Bose \nelements and using Equations (\\ref{NSrest}), (\\ref{Rrest})  we get \n\\begin{equation}\n\\label{GKOnetsNSb+}\n\\left( \\pi_{(2,0)} \\otimes \\pi_{(N,\\frac{1}{2}[p-1])}\\right)\n|_{{\\mathcal A}_{{{\\rm SU}(2)}_{N+2}}\\otimes {\\rm SVir}^0_{c_N}\n} =\n\\bigoplus_q \\pi_{(N+2, \\frac{1}{2}[q-1])} \\otimes \\rho_{h_{p,q}}^{NS+},\n\\end{equation}\n\\begin{equation}\n\\label{GKOnetsNSb-}\n\\left( \\pi_{(2,1)} \\otimes \\pi_{(N,\\frac{1}{2}[p-1])}\\right)\n|_{{\\mathcal A}_{{{\\rm SU}(2)}_{N+2}}\\otimes {\\rm SVir}^0_{c_N}} =\n\\bigoplus_q \\pi_{(N+2, \\frac{1}{2}[q-1])} \\otimes \\rho_{h_{p,q}}^{NS-},\n\\end{equation}\n$1\\leq q \\leq N+3$, $p-q$ even, and \n\\begin{equation}\n\\label{GKOnetsRb}\n\\left( \\pi_{(2,\\frac{1}{2})} \\otimes \\pi_{(N,\\frac{1}{2}[p-1])}\\right)\n|_{{\\mathcal A}_{{{\\rm SU}(2)}_{N+2}}\\otimes {\\rm SVir}^0_{c_N}} =\n\\bigoplus_q \\pi_{(N+2, \\frac{1}{2}[q-1])} \\otimes \\rho_{h_{p,q}}^{R},\n\\end{equation}\n$1\\leq q \\leq N+3$, $p-q$ odd.     \n\nNow, recalling the identification of ${\\rm SVir}^0_{c_N}$ as a \ncoset in Eq. (\\ref{BoseCoset} ), it \nfollows from \\cite[Corollary 3.2]{X2} that every irreducible \nDHR representation of this net is equivalent to one of those considered \nbefore, namely $\\rho_{h_{p,q}}^{NS+}$, $\\rho_{h_{p,q}}^{NS-}$\nand $\\rho_{h_{p,q}}^{R}$ ($h_{p,q}\\neq c_N\/24$), $\\rho_{\\frac{c_N}{24}}^{R+}$\nand $\\rho_{\\frac{c_N}{24}}^{R-}$.  \n\\subsection{Modularity of local super-Virasoro nets}\nWe state here explicitly the modularity of the Bose subnet super-Virasoro nets for $c<3\/2$. \nIn this case the Bose super-Virasoro net can be obtained as the coset \\eqref{BoseCoset}. \nThen the Rehren $S$ and $T$ matrices as been computed by Xu in \\cite[Sect.2.2.]{X3} \n(see also Sect. \\ref{classification} below). These matrices agree with those in \\cite{GW} and \n\\cite{FSS} giving modular transformations of specialised characters. Accordingly we have \nthe following. \n\\begin{theorem} \nFor a positive even integer $N$ then ${\\rm SVir}^0_{c_N}$ is a modular conformal net. \n\\end{theorem}\n\\section{Classification of superconformal nets in the discrete series}\n\\label{classification}\nBy a \\emph{superconformal net} (of von Neumann algebras on $S^1$) \nwe shall mean a Fermi net on $S^1$ that contains\na super-Virasoro net as irreducible subnet.\nIf the central charge $c$ of a superconformal net\nis less than $3\/2$, it is of the form\n$c=\\displaystyle\\frac{3}{2}\\left(1-\\frac{8}{m(m+2)}\\right)$\nfor some $m=3,4,5,\\dots$ \\cite{FQS}.  We classify all such superconformal\nnets.\n\n\\subsection{Outline of classification}\nAs above, we denote the super Virasoro net with central charge $c$\nand its Bosonic part by ${\\mathrm {SVir}}_c$ and ${\\mathrm {SVir}}^0_c$, respectively.\nWe are interested in the case $c<3\/2$.  In this case, we have\n$c=\\displaystyle\\frac{3}{2}\\left(1-\\frac{8}{m(m+2)}\\right)$\nfor some $m=3,4,5,\\dots$, and we have already seen that in this case\nthe local conformal net ${\\mathrm {SVir}}^0_c$ is realised as a coset\nnet for the inclusion $SU(2)_m\\subset SU(2)_{m-2}\\otimes SU(2)_2$.\nThis net is completely rational in the sense of \\cite{KLM}\nby \\cite{X2}.  The DHR sectors of the local conformal net ${\\mathrm {SVir}}^0_c$ is\ndescribed as follows by \\cite[Section 2.2]{X3}.\nLabel the DHR sectors of the local conformal\nnets $SU(2)_m$, $SU(2)_{m-2}$, and $SU(2)_2$ by $k=0,1,\\dots, m$,\n$j=0,1,\\dots,m-2$, and $l=0,1,2$, respectively.  Then we consider the\ntriples $(j,k,l)$ with $j-k+l$ being even.\nFor $l=0,2$, we have identification\n$$(j,k,l)\\leftrightarrow(m-2-j,m-k,2-l),$$\nthus it is enough to consider the triples $(j,k,0)$ with $j-k$\nbeing even.  Each such triple labels an irreducible DHR sector\nof the coset net ${\\mathrm {SVir}}^0_c$.  For the case $l=1$, we also have\nidentification\n$$(j,k,l)\\leftrightarrow(m-2-j,m-k,2-l),$$\nbut if we have a fixed point for this symmetry, that is,\nif $m$ is even, then the fixed point $(2\/m-1, m\/2,1)$ splits\ninto two pieces, $(2\/m-1, m\/2,1)_+$ and $(2\/m-1, m\/2,1)_-$.\nAll of these triples, with this identification\nand splitting, label all the irreducible DHR sectors of\nthe coset net ${\\mathrm {SVir}}^0_c$.  The sectors with $l=0$ and $l=1$\nare called Neveu-Schwarz and Ramond sectors, respectively.\n\nThe conformal spin of the sector $(j,k,l)$ is given by\n$$\\exp\\left(\\frac{\\pi i}{2}\\left(\\frac{j(j+2)}{m}-\n\\frac{k(k+2)}{m+2}+\\frac{l(l+2)}{4}\\right)\\right).$$\n(This also works for the case $(j,k,l)=(2\/m-1, m\/2,1)$.) \n\nFor example, if $m=3$, we have six\nirreducible DHR sectors and they are labelled with triples\n$(0,0,0)$, $(0,2,0)$, $(1,1,0)$, $(1,3,0)$, $(0,3,1)$, $(1,2,1)$.\n(This local conformal net is equal to the Virasoro net with\n$c=7\/10$.)\nFor $m=4$, we have 13 irreducible DHR sectors and they are\nlabelled with\n$(0,0,0)$, $(0,2,0)$, $(0,4,0)$, $(1,1,0)$, $(1,3,0)$,\n$(2,0,0)$, $(2,2,0)$, $(2,4,0)$, $(0,3,1)$, $(1,4,1)$,\n$(2,3,1)$, $(1,2,1)_+$, $(1,2,1)_-$, where\nthe two labels $(1,2,1)_+$, $(1,2,1)_-$ arise from the fixed\npoint $(1,2,1)$ of the symmetry of order 2.\n\nFor all $m$, the irreducible DHR sector $(m-2,m,0)$ has a\ndimension 1 and a spin $-1$.  The superconformal net\n${\\mathrm {SVir}}_c$ arises as a non-local extension of ${\\mathrm {SVir}}^0_c$\nas a crossed product by ${\\mathbb Z}_2$ using identity and this sector.\n\nLet ${\\mathcal A}$ be any superconformal net on the circle with $c<3\/2$.\nThen let ${\\mathcal B}$ be its Bosonic part.  By a similar argument to that\nin \\cite[Proposition 3.5]{KL1}, we know that the local conformal\nnet ${\\mathcal B}$ is an irreducible extension of the local conformal\nnet ${\\mathrm {SVir}}^0_c$, where $c$ is the central charge of ${\\mathcal A}$.\nBy the strategy in \\cite{KL1} based on \\cite{BEK1},\nwe know that the dual canonical endomorphism $\\theta$ of an extension\nis of the form $\\theta=\\sum_{\\lambda}Z_{0,\\lambda}\\lambda$, where $Z$ is the\nmodular invariant arising from the extension as in \\cite{BEK1}.\nCappelli \\cite{Ca} gave a list of type I modular invariants and\nconjectured that it is a complete list.  From his list,\nit is easy to guess that the dual canonical endomorphisms\nwe use for obtaining extensions are those listed\nin Table \\ref{dual-can}.\n(Cappelli also considered type II modular invariants, but they\ndo not correspond to local extensions, so we ignore them here.)\n\\begin{table}[htbp]\n\\begin{center}\n\\begin{tabular}{|c|l|l|c|}\\hline\n& $m$ & $\\theta$ & Label\n\\\\ \\hline\n(1) & any $m$ & $\\theta=(0,0,0)$ & $(A_{m-1},A_{m+1})$ \\\\ \\hline\n(2) & $m=4m'$ & $\\theta=(0,0,0)\\oplus(0,m,0)$ & $(A_{4m'-1}, D_{2m'+2})$ \n\\\\ \\hline\n(3) & $m=4m'+2$ & $\\theta=(0,0,0)\\oplus(m-2,0,0)$ & $(D_{2m'+2}, A_{4m'+3})$ \n\\\\ \\hline\n(4) & $m=10$ & $\\theta=(0,0,0)\\oplus(0,6,0)$ & $(A_9, E_6)$ \\\\ \\hline\n(5) & $m=12$ & $\\theta=(0,0,0)\\oplus(6,0,0)$ & $(E_6, A_{13})$ \\\\ \\hline\n(6) & $m=28$ & $\\theta=(0,0,0)\\oplus(0,10,0)\\oplus(0,18,0)\\oplus(0,28,0)$\n& $(A_{27},E_8)$ \\\\ \\hline\n(7) & $m=30$ & $\\theta=(0,0,0)\\oplus(10,0,0)\\oplus(18,0,0)\\oplus(28,0,0)$\n& $(E_8, A_{31})$ \\\\ \\hline\n(8) & $m=10$ & $\\theta=(0,0,0)\\oplus(0,6,0)\\oplus(8,6,0)\\oplus(8,6,0)$\n& $(D_6, E_6)$ \\\\ \\hline\n(9) & $m=12$ & $\\theta=(0,0,0)\\oplus(6,0,0)\\oplus(0,12,0)\\oplus(6,12,0)$\n& $(E_6, D_8)$ \\\\ \\hline\n\\end{tabular}\n\\caption{List of candidates of the dual canonical endomorphisms}\n\\label{dual-can}\n\\end{center}\n\\end{table}\nWe will prove that each of the dual canonical endomorphisms \nin Table \\ref{dual-can} gives a local extension of ${\\mathrm {SVir}}^0_c$ \nin a unique way and that an arbitrary such local extension of\n${\\mathrm {SVir}}^0_c$ gives one of the dual canonical endomorphisms \nin Table \\ref{dual-can}.\n\n\\subsection{Study of type I modular invariants} \nWe study type I modular invariants for the coset\nnets for the inclusions $SU(2)_m\\subset SU(2)_{m-2}\\otimes SU(2)_2$.\n\nFirst we recall the $S$ and $T$ matrices for $SU(2)_m$.\nFor $j,k=0,1,2,\\dots,m$, we have the following.\n\\begin{eqnarray*}\nS^{(m)}_{jk}&=&\\sqrt{\\frac{2}{m+2}}\\sin \\pi\\frac{(j+1)(k+1)}{m+2},\\\\\nT^{(m)}_{jk}&=&\\delta_{jk}\\exp\\frac{\\pi i}{2}\n\\left(\\frac{(j+1)^2}{m+2}-\\frac12\\right).\n\\end{eqnarray*}\nFor odd $m$, we have no problem arising from a fixed point of\nthe order two symmetry, and in this case, the modular invariants\nhave been already classified by Gannon-Walton \\cite{GW}, which\nshows that the identity matrix is the only modular invariant.  So\nwe have no non-trivial extensions in these cases.\n\nSo we now deal with the case of even $m$ in the rest of this\nsection and put $m=2m_0$.  In this case, the\n$S$-matrix of the modular tensor category of the irreducible\nDHR-sectors of the coset net is already not so easy to\nobtain, and it has been computed by Xu \\cite{X3}.\n\nAs in the previous section, we label the irreducible DHR\nsectors of the coset net for the inclusion\n$SU(2)_m\\subset SU(2)_{m-2}\\otimes SU(2)_2$ with\ntriples $(j,k,l)$ with $j-k+l$ being even, except for the\ncase $(m\/2-1, m\/2, 1)$.  For this fixed point of the order\ntwo symmetry, we use labels $\\varphi_1=(m\/2-1, m\/2, 1)_+$, \n$\\varphi_2=(m\/2-1, m\/2, 1)_-$ to denote the two\nirreducible DHR sectors.  We also use the symbols\n$S^{(m-2)}$, $S^{(m)}$, $S^{(2)}$ and $S$ for the $S$-matrices\nfor the nets $SU(2)_{m-2}$, $SU(2)_m$, $SU(2)_2$ and the coset\nfor $SU(2)_m\\subset SU(2)_{m-2}\\otimes SU(2)_2$, respectively.\nThen Xu's computation for the matrix\n$S$ in \\cite{X3} gives the following, where $n=1,2$.\n\\begin{enumerate}\n\\item $S_{(j,k,l),(j',k',l')}=2S^{(m-2)}_{jj'} \\overline{S^{(m)}_{kk'}}\nS^{(2)}_{ll'}$ for $(j,k,l), (j',k',l')\\neq \\varphi_{1,2}$.\n\\item $S_{(j,k,l),(j',k',l'),\\varphi_n}=S_{\\varphi_n,(j,k,l),(j',k',l')}=\nS^{(m-2)}_{j,m\/2-1} \\overline{S^{(m)}_{k,m\/2}}\nS^{(2)}_{l1}$ for $(j,k,l)\\neq \\varphi_{1,2}$.\n\\item $S_{\\varphi_n,\\varphi_{n'}}=\\delta_{nn'}+(S^{(m-2)}_{m\/2-1,m\/2-1}\n\\overline{S^{(m)}_{m\/2,m\/2}}S^{(2)}_{11}-1)\/2$.\n\\end{enumerate}\nThe $T$-matrix of the coset is described as follows more easily.\n\\begin{enumerate}\n\\item $T_{(j,k,l),(j',k',l')}=T^{(m-2)}_{jj'} \\overline{T^{(m)}_{kk'}}\nT^{(2)}_{ll'}$ for $(j,k,l), (j',k',l')\\neq \\varphi_{1,2}$.\n\\item $T_{(j,k,l),\\varphi_n}=T_{\\varphi_n,(j,k,l)}=\nT^{(m-2)}_{j,m\/2-1} \\overline{T^{(m)}_{k,m\/2}}\nT^{(2)}_{l1}$ for $(j,k,l)\\neq \\varphi_{1,2}$.\n\\item $T_{\\varphi_n,\\varphi_n'}=\\delta_{nn'}\nT^{(m-2)}_{m\/2-1,m\/2-1} \\overline{T^{(m)}_{m\/2,m\/2}}\nT^{(2)}_{11}$.\n\\end{enumerate}\nSuppose we have a modular invariant $Z$ for the coset\nfor $SU(2)_m\\subset SU(2)_{m-2}\\otimes SU(2)_2$.\nWe define a new matrix  $\\tilde Z_{(j,k,l),(j',k',l')}$ where\nthe triples $(j,k,l)$ and $(j',k',l')$ satisfy\n$j,j'\\in \\{0,1,\\dots,m-2\\}$,\n$k,k'\\in \\{0,1,\\dots,m\\}$,\n$l,l'\\in \\{0,1,2\\}$.  Note that we have no identification\nor splitting for the triples $(j,k,l)$ and $(j',k',l')$ here.\n\\begin{enumerate}\n\\item If $j-k+l, j'-k'+l'\\in 2{\\mathbb Z}$ and\n$(j,k,l), (j',k',l')\\neq(m\/2-1,m\/2,1)$, then\nwe set $\\tilde Z_{(j,k,l),(j',k',l')}=Z_{(j,k,l),(j',k',l')}$.\n\\item If $j-k+l\\in 2{\\mathbb Z}$ and\n$(j,k,l)\\neq(m\/2-1,m\/2,1)$, then\nwe set $\\tilde Z_{(j,k,l),(m\/2-1,m\/2,1)}=\nZ_{(j,k,l),\\varphi_1}+Z_{(j,k,l),\\varphi_2}$.\n\\item If $j'-k'+l'\\in 2{\\mathbb Z}$ and\n$(j',k',l')\\neq(m\/2-1,m\/2,1)$, then\nwe set $\\tilde Z_{(m\/2-1,m\/2,1),j'k'l'}=\nZ_{\\varphi_1,(j',k',l')}+Z_{\\varphi_2,j'k'l'}$.\n\\item $\\tilde Z_{(m\/2-1,m\/2,1),(m\/2-1,m\/2,1)}=\nZ_{\\varphi_1,\\varphi_1}+Z_{\\varphi_1,\\varphi_2}+Z_{\\varphi_2,\\varphi_1}+Z_{\\varphi_2,\\varphi_2}$.\n\\item If $j-k+l\\notin 2{\\mathbb Z}$ or $j'-k'+l'\\notin 2{\\mathbb Z}$, then\nwe set $\\tilde Z_{(j,k,l),(j',k',l')}=0$.\n\\end{enumerate}\nThis construction is analogous to the one of ${\\cal L}^{cw}$\nin \\cite[page 178]{GW}, but now\nunlike in \\cite{GW}, our map $Z\\mapsto \\tilde Z$ may not be \ninjective because of the definition of $\\tilde Z$ involving\nthe row\/column labelled with $(m\/2-1,m\/2,1)$.\n\nFor triples $(j,k,l)$ and $(j',k',l')$ satisfying\n$j,j'\\in \\{0,1,\\dots,m-2\\}$,\n$k,k'\\in \\{0,1,\\dots,m\\}$,\n$l,l'\\in \\{0,1,2\\}$, we set as follows.\n(We do not impose the conditions $j-k+l, j'-k'+l'\\in 2{\\mathbb Z}$ here.)\n\\begin{enumerate}\n\\item $\\tilde S_{(j,k,l),(j',k',l')}=S^{(m-2)}_{jj'}\n\\overline{S^{(m)}_{kk'}} S^{(2)}_{ll'}$.\n\\item $\\tilde T_{(j,k,l),(j',k',l')}=T^{(m-2)}_{jj'}\n\\overline{T^{(m)}_{kk'}} T^{(2)}_{ll'}$.\n\\end{enumerate}\nWe also write $\\varphi=(m\/2-1,m\/2,1)$ and set\n$$I=\\{(j,k,0)\\mid j+k\\in2{\\mathbb Z}\\}\\cup\n\\{(j,k,1)\\mid j+k\\notin2{\\mathbb Z}, j+k<m-1\\},$$\nwhich gives representatives of\n$\\{(j,k,l)\\mid j-k+l\\in2{\\mathbb Z}\\}\\smallsetminus\\{\\varphi\\}$ under the\norder 2 symmetry.\n\nWe now have the following proposition.\n\\begin{proposition}\nIf the matrix $Z$ is a modular invariant for $S, T$, then\nthe matrix $\\tilde Z$ is a modular invariant for $\\tilde S, \\tilde T$.\n\\end{proposition}\n\\begin{proof}\nIt is clear that the entries of the matrix $\\tilde Z$ are\nnonnegative integers and we have $\\tilde Z_{(0,0,0), (0,0,0)}=1$.\nWe thus have to prove $\\tilde Z \\tilde S=\\tilde S \\tilde Z$ and\n$\\tilde Z \\tilde T=\\tilde T \\tilde Z$.\nThe latter is clear from the definition, so we have to prove\nthe former identity at $(j,k,l)$, $(j',k',l')$.\nWe deal with seven cases separately.\n\nCase (1).  $j-k+l\\notin2{\\mathbb Z}$, $j'-k'+l'\\notin2{\\mathbb Z}$.\nThe identity is trivial because the both hand sides are zero.\n\nCase (2).  $j-k+l\\in2{\\mathbb Z}$, $j'-k'+l'\\notin2{\\mathbb Z}$.\nThe right hand side is obviously zero by the definition of $\\tilde Z$,\nand the following three identities now prove that the left hand\nside is also zero.\n\\begin{eqnarray*}\n\\tilde S_{\\varphi,(j',k',l')}&=&0,\\\\\n\\tilde Z_{(j,k,l),(j'',k'',l'')}&=&\n\\tilde Z_{(j,k,l),(m-2-j'',m-k'',2-l'')},\\\\\n\\tilde S_{(j'',k'',l''),(j',k',l')}&=&\n-\\tilde S_{(m-2-j'',m-k'',2-l''),(j',k',l')}.\n\\end{eqnarray*}\nNote that the third identity holds because $j'-k'+l'\\notin2{\\mathbb Z}$.\n\nCase (3).  $j-k+l\\notin2{\\mathbb Z}$, $j'-k'+l'\\in2{\\mathbb Z}$.\nThis case can be proved in the same way as above.\n\nCase (4).  $j-k+l\\in2{\\mathbb Z}$, $j'-k'+l'\\in2{\\mathbb Z}$,\n$(j,k,l)\\neq \\varphi$, $(j',k',l')\\neq \\varphi$.\nThe identity $ZS=SZ$ gives the following identity.\n\\begin{eqnarray*}\n&&\\sum_{(j'',k'',l'')\\in I} 2Z_{(j,k,l),(j'',k'',l'')}\n\\tilde S_{(j'',k'',l''),(j',k',l')}+\nZ_{(j,k,l),\\varphi_1}\\tilde S_{\\varphi_1,(j',k',l')}+\nZ_{(j,k,l),\\varphi_2}\\tilde S_{\\varphi_2,(j',k',l')}\\\\\n&=&\\sum_{(j'',k'',l'')\\in I} 2S_{(j,k,l),(j'',k'',l'')}\nZ_{(j'',k'',l''),(j',k',l')}+\n\\tilde S_{(j,k,l),\\varphi} Z_{\\varphi_1,(j',k',l')}+\n\\tilde S_{(j,k,l),\\varphi} Z_{\\varphi_2,(j',k',l')}.\n\\end{eqnarray*}\nSince we now have\n\\begin{eqnarray*}\n\\tilde S_{(j'',k'',l''),(j',k',l')}&=&\n\\tilde S_{(m-2-j'',m-k'',2-l''),(j',k',l')},\\\\\n\\tilde S_{(j,k,l),(j'',k'',l'')}&=&\n\\tilde S_{(j,k,l),(m-2-j'',m-k'',2-l'')}\n\\end{eqnarray*}\nwe conclude $(\\tilde Z\\tilde S)_{(j,k,l),(j',k',l')}\n=(\\tilde Z\\tilde Z)_{(j,k,l),(j',k',l')}$.\n\nCase (5).  $j-k+l\\in2{\\mathbb Z}$, $(j,k,l)\\neq \\varphi$, $(j',k',l')=\\varphi$.\nThe identity $ZS=SZ$ gives the following identity for $n=1,2$.\n\\begin{eqnarray*}\n&&\\sum_{(j'',k'',l'')\\in I} Z_{(j,k,l),(j'',k'',l'')}\n\\tilde S_{(j'',k'',l''),\\varphi}+\nZ_{(j,k,l),\\varphi_1} S_{\\varphi_1,\\varphi_n}+\nZ_{(j,k,l),\\varphi_2} S_{\\varphi_2,\\varphi_n}\\\\\n&=&\\sum_{(j'',k'',l'')\\in I} 2\\tilde S_{(j,k,l),(j'',k'',l'')}\nZ_{(j'',k'',l''),\\varphi_n}+\n\\tilde S_{(j,k,l),\\varphi} Z_{\\varphi_1,\\varphi_n}+\n\\tilde S_{(j,k,l),\\varphi} Z_{\\varphi_2,\\varphi_n}.\n\\end{eqnarray*}\n\nAdding these two equalities for $n=1,2$, we obtain the following\nidentity, which gives $(\\tilde Z\\tilde S)_{(j,k,l),\\varphi}\n=(\\tilde Z\\tilde Z)_{(j,k,l),\\varphi}$.\n\\begin{eqnarray*}\n&&\\sum_{(j'',k'',l'')\\in I} Z_{(j,k,l),(j'',k'',l'')}\n\\tilde S_{(j'',k'',l''),\\varphi}+\n(Z_{(j,k,l),\\varphi_1}+Z_{(j,k,l),\\varphi_2}) \\tilde S_{\\varphi,\\varphi}\\\\\n&=&\\sum_{(j'',k'',l'')\\in I} 2\\tilde S_{(j,k,l),(j'',k'',l'')}\n(Z_{(j'',k'',l''),\\varphi_1}+Z_{(j'',k'',l''),\\varphi_2})+\\\\\n&&\\quad\\quad\\quad\\quad\\tilde S_{(j,k,l),\\varphi}\n(Z_{\\varphi_1,\\varphi_1}+Z_{\\varphi_1,\\varphi_2}+Z_{\\varphi_2,\\varphi_1}+Z_{\\varphi_2,\\varphi_2}).\n\\end{eqnarray*}\n\nCase (6).  $(j,k,l)=\\varphi$, $j'-k'+l'\\in2{\\mathbb Z}$, $(j',k',l')\\neq \\varphi$.\nThis case can be proved in the same way as above.\n\nCase (7).  $(j,k,l)=(j',k',l')=\\varphi$.\nThe identity $ZS=SZ$ gives the following identity for $n=1,2$, $n'=1,2$.\n\\begin{eqnarray*}\n&&\\sum_{(j'',k'',l'')\\in I} Z_{\\varphi_n,(j'',k'',l'')}\n\\tilde S_{(j'',k'',l''),\\varphi}+\nZ_{\\varphi_n,\\varphi_1} S_{\\varphi_1,\\varphi_{n'}}+\nZ_{\\varphi_n,\\varphi_2} S_{\\varphi_2,\\varphi_{n'}}\\\\\n&=&\\sum_{(j'',k'',l'')\\in I} \\tilde S_{\\varphi,(j'',k'',l'')}\nZ_{(j'',k'',l''),\\varphi_{n'}}+\nS_{\\varphi_n,\\varphi_1} Z_{\\varphi_1,\\varphi_{n'}}+\nS_{\\varphi_n,\\varphi_2} Z_{\\varphi_2,\\varphi_{n'}}.\n\\end{eqnarray*}\nAdding these four identities for $n,n'=1,2$, we obtain the following\nidentity, which gives $(\\tilde Z\\tilde S)_{\\varphi,\\varphi}\n=(\\tilde Z\\tilde Z)_{\\varphi,\\varphi}$.\n\\begin{eqnarray*}\n&&\\sum_{(j'',k'',l'')\\in I} \n2(Z_{\\varphi_1,(j'',k'',l'')}+Z_{\\varphi_2,(j'',k'',l'')})\n\\tilde S_{(j'',k'',l''),\\varphi}+\\\\\n&&\\quad\\quad\\quad\\quad\n(Z_{\\varphi_1,\\varphi_1}+Z_{\\varphi_1,\\varphi_2}+Z_{\\varphi_2,\\varphi_1}+Z_{\\varphi_2,\\varphi_2})\n\\tilde S_{\\varphi,\\varphi}\\\\\n&=&\\sum_{(j'',k'',l'')\\in I} 2\\tilde S_{\\varphi,(j'',k'',l'')}\n(Z_{(j'',k'',l''),\\varphi_1}+Z_{(j'',k'',l''),\\varphi_2})+\\\\\n&&\\quad\\quad\\quad\\quad\\tilde S_{\\varphi,\\varphi}\n(Z_{\\varphi_1,\\varphi_1}+Z_{\\varphi_1,\\varphi_2}+Z_{\\varphi_2,\\varphi_1}+Z_{\\varphi_2,\\varphi_2})\n\\end{eqnarray*}\n\\end{proof}\nWe now recall Gannon's parity rule \\cite[page 696]{G}.\nFor $n$ and $j$, we set $\\varepsilon_{2n}(j)$ as follows.\n\\begin{enumerate}\n\\item If $j\\equiv 0,n$ mod $2n$, then $\\varepsilon_{2n}(j)=0$.\n\\item If $j\\equiv 1,2,\\dots,n-1$ mod $2n$, then $\\varepsilon_{2n}(j)=1$.\n\\item If $j\\equiv n+1,n+2,\\dots,2n-1$ mod $2n$, then $\\varepsilon_{2n}(j)=-1$.\n\\end{enumerate}\nSet $$L=\\{n\\mid (n,8m(m+2))=1\\}.$$\nThen the parity rule is the following.\nIf we have a modular invariant $\\tilde Z$ for $\\tilde S$ and $\\tilde T$\nwith $tilde Z_{000,jkl}\\neq0$, then\nthe parity rule says that we have\n$$\\varepsilon_{2m}(n)\\varepsilon_{2m+4}(n)\\varepsilon_{8}(n)=\n\\varepsilon_{2m}(n(j+1))\\varepsilon_{2m+4}(n(k+1))\\varepsilon_{8}(n(l+1))$$\nfor all $n\\in L$.\n\nNow suppose $m_0=m\/2$ is odd.\nFor any $n_1$ with $(n_1,4m_0)=1$, choose $p\\in\\{1,-1\\}$ so\nthat $n_1 \\equiv p$ mod $4$.  Then it is easy to see that\nthere exists $n\\in L$ satisfying\n$n\\equiv n_1$ mod $4m_0$,\n$n\\equiv p$ mod $4(m_0+1)$,\n$n\\equiv p$ mod $8$, since we have $(m_0, m_0+1)=1$.\nSuppose $tilde Z_{000,jkl}\\neq0$.  Then the parity rule says\n$$\\varepsilon_{2m}(n_1)\\varepsilon_{2m+4}(p)\\varepsilon_{8}(p)=\n\\varepsilon_{2m}(n_1(j+1))\\varepsilon_{2m+4}(p(k+1))\\varepsilon_{8}(p(l+1)),$$\nwhich gives $\\varepsilon_{2m}(n_1)=\\varepsilon_{2m}(n_1(j+1))$\nfor all $n_1$ with $(n_1,2m)=1$.  If $m_0\\neq 3,5,15$, then\nwe have $j=0,m-2$ by \\cite{G}.  Using the order two symmetry\n$(j,k,l)\\leftrightarrow(m-2-j,m-k,2-l)$ and the\ndefinition of $\\tilde Z$,\nwe conclude that our dual canonical endomorphism $\\theta$ has a \ndecomposition within $\\{(0,k,l)\\mid -k+l\\in2{\\mathbb Z}\\}.$\nLet ${\\cal T}_1$ be the tensor category of the representations of\n$SU(2)_m$ with the opposite braiding and ${\\cal T}_2$ be the\ntensor category of the representations of $SU(2)_2$ with the\nusual braiding.  If $\\theta$ gives a local extension for the\ntensor category generated by $\\{(0,k,l)\\mid -k+l\\in2{\\mathbb Z}\\}$, then\nit also gives a local extension for the tensor category\n${\\cal T}_1 \\times {\\cal T}_2$ by the description of the\nbraided tensor category having\nthe simple objects $\\{(0,k,l)\\mid -k+l\\in2{\\mathbb Z}\\}$ by\n\\cite[Section 4.3]{X1},\n\\cite[Proposition 2.3.1, Proof of Theorem B]{X5}.\n(Note that we have the notion\nof a local extension as a local $Q$-system in \n\\cite{LR1} for an abstract tensor category.)  Then this extension\nfor ${\\cal T}_1 \\times {\\cal T}_2$ gives a type I modular invariant\nfor the $S$- and $T$-matrices arising from ${\\cal T}_1 \\times {\\cal T}_2$.\nThen Gannon's classification of modular invariants for\n$SU(2)_m\\otimes SU(2)_2$ in \\cite[Section 7]{G} shows that\nour $\\theta$ must be one of Table \\ref{dual-can}.\n(Here we have an opposite braiding for ${\\cal T}_1$, but this\ndoes not matter since we can consider a modular invariant\n$Z_{k'l,kl'}$ instead of $Z_{kl,k'l'}$ as in \\cite[page 178]{GW}.)\n\nNext suppose $m_0=m\/2$ is even.\nFor any $n_2$ with $(n_2,4(m_0+1))=1$, choose $p\\in\\{1,-1\\}$ so\nthat $n_2 \\equiv p$ mod $4$.  Then it is easy to see that\nthere exists $n\\in L$ satisfying\n$n\\equiv p$ mod $4m_0$,\n$n\\equiv n_2$ mod $4(m_0+1)$,\n$n\\equiv p$ mod $8$, since we have $(m_0, m_0+1)=1$.\nSuppose $\\tilde Z_{(0,0,0),(j,k,l)}\\neq0$.  Then the parity rule says\n$$\\varepsilon_{2m}(p)\\varepsilon_{2m+4}(n_2)\\varepsilon_{8}(p)=\n\\varepsilon_{2m}(p(j+1))\\varepsilon_{2m+4}(n_2(k+1))\\varepsilon_{8}(p(l+1)),$$\nwhich gives $\\varepsilon_{2m+4}(n_2)=\\varepsilon_{2m+4}(n_2(k+1))$\nfor all $n_2$ with $(n_2,4(m_0+1))=1$.  If $m_0\\neq 2,4,14$, then\nwe have $k=0,m$ by \\cite{G}.  Using the order two symmetry\n$(j,k,l)\\leftrightarrow(m-2-j,m-k,2-l)$ and the\ndefinition of $\\tilde Z$ again,\nwe conclude that our dual canonical endomorphism $\\theta$ has a \ndecomposition within $\\{(j,0,l)\\mid j+l\\in2{\\mathbb Z}\\}$.\nIf $\\theta$ gives a local extension for the\ntensor category generated by $\\{(0,k,l)\\mid -k+l\\in2{\\mathbb Z}\\}$, then\nit also gives a local extension for the tensor category\nof the representations $SU(2)_{m-2}\\otimes SU(2)_2$ by\nthe description of the braided tensor category having\nthe simple objects $\\{(j,0,l)\\mid j+l\\in2{\\mathbb Z}\\}$ by\n\\cite[Section 4.3]{X1},\n\\cite[Proposition 2.3.1]{X6}.  Then this extension\ngives a type I modular invariant\nfor the $S$- and $T$-matrices arising from $SU(2)_{m-2}\\otimes SU(2)_2$,\nso Gannon's classification of modular invariants for\n$SU(2)_{m-2}\\otimes SU(2)_2$ in \\cite[Section 7]{G} again shows that\nour $\\theta$ must be one of Table \\ref{dual-can}.\n\nWe now deal with the remaining exceptional cases $m=4,6,8,10,28,30$.\nIf $\\tilde Z_{(0,0,0),(j,k,l)}\\neq0$ and $j-k+l\\in2{\\mathbb Z}$, then the\nconformal spin of $(j,k.l)$ is 1.  This condition already\ngives the restriction that the dual canonical endomorphisms\n$\\theta$ has a decomposition within\n$$\\{(j,k,0)\\mid j+l\\in2{\\mathbb Z}\\}$$\nfor $m=4,6,8,10,28$.  Again Gannon's classification of modular invariants\nin \\cite{G} shows that our $\\theta$ must be one of Table \\ref{dual-can}.\n\nThe final remaining case is $m=30$.  In this case, \nthe only sectors $(j,k,l)$ with $l=1$ and $j-k+l\\in2{\\mathbb Z}$\nhaving conformal spins 1 are\n$(3,12,1)$ and $(13,18,1)$.\nThe parity rules with $n=13, 7$\nexclude these two sectors from $\\theta$, respectively.  Thus again\nthe dual canonical endomorphism $\\theta$ has a decomposition within\n$$\\{(j,k,0)\\mid j+l\\in2{\\mathbb Z}\\}$$\nand Gannon's classification in \\cite{G} shows that our $\\theta$ must\nbe one of Table \\ref{dual-can}.\n\nWe have thus proved that if we have a local extension of\n${\\mathrm {SVir}}^0_c$, then its dual canonical endomorphism must be one of\nthose listed in Table \\ref{dual-can}.\n\n\\subsection{Classification of the Fermi extensions of the \nsuper-Vi\\-ra\\-so\\-ro net, $c<3\/2$}\n\nWe now show that each endomorphism in Table \\ref{dual-can}\nuniquely produces a superconformal net extending ${\\mathrm {SVir}}_c$.\n\n(1) We have the label $(A_{m-1}, A_{m+1})$.\nThe identity matrix obviously gives the coset net for\n$SU(2)_m\\subset SU(2)_{m-2}\\otimes SU(2)_2$ itself.  This has\na unique Fermionic extension.\n\n(2) Now we have the label $(A_{4m'-1}, D_{2m'+2})$ with $m=4m'$.\nThe irreducible DHR sectors $(0,0,0)$, $(0,4m',0)$ give the\ngroup ${\\mathbb Z}_2$.  The crossed product of the coset net by this group\ngives a local extension.  The irreducible DHR sector $(4m'-2,0,0)$\nhas spin $-1$, and this sector still gives an irreducible DHR sector\nof index 1 and spin $-1$ after the $\\alpha$-induction applied to\nthe crossed product construction.\nThis $\\alpha$-induced sector gives a unique Fermionic extension\nthrough a crossed product by  ${\\mathbb Z}_2$.\n\n(3) Now we have the label $(D_{2m'+2}, A_{4m'+3})$ with $m=4m'+2$.\nThe irreducible DHR sectors $(0,0,0)$, $(4m',0,0)$ give the\ngroup ${\\mathbb Z}_2$.  The crossed product of the coset net by this group\ngives a local extension.  The irreducible DHR sector $(0,4m'+2,0)$\nhas spin $-1$, and this sector still gives an irreducible DHR sector\nof index 1 and spin $-1$ after the $\\alpha$-induction applied to\nthe crossed product construction.\nThis again gives a unique Fermionic extension in a similar way to\ncase (2).\n\n(4) Now we have the label $(A_9, E_6)$ for $m=10$.\nThe DHR sector  $(0,0,0)\\oplus(0,6,0)$ gives a local\n$Q$-system as in \\cite[Subsection 4.2]{KL1}\nby the description of the braided tensor category having\nthe simple objects $\\{(0,k,0)\\mid k\\in2{\\mathbb Z}\\}$ as in\n\\cite[Section 4.3]{X1},\n\\cite[Proposition 2.3.1, Proof of Theorem B]{X5}.\n\nUniqueness also follows as in\n\\cite[Remark 2.5]{KL1}, using the cohomology vanishing result\nin \\cite{KL2}.\n(This is also a mirror extension in the sense of \\cite{X6}, so\nlocality of the extension also follows from \\cite{X6}.)\nThe irreducible DHR sector $(0,10,0)$\nhas spin $-1$, and this sector still gives an irreducible DHR sector\nof index 1 and spin $-1$ after the $\\alpha$-induction applied to\nthe local extension.  This gives a unique Fermionic extension\nas above.\n\n(5) Now we have the label $(E_6, A_{13})$ for $m=12$.\nThe irreducible DHR sectors  $(0,0,0)$, $(6,0,0)$ give a local\n$Q$-system as in (4) by the description of the braided tensor\ncategory having the simple objects $\\{(j,0,0)\\mid j\\in2{\\mathbb Z}\\}$\nas in \\cite[Section 4.3]{X1}, \\cite[Proposition 2.3.1]{X5}.\nUniqueness also follows as above.\n(This is also a coset model as in \\cite{KL1}.)\nThe irreducible DHR sector $(10,0,0)$\nhas spin $-1$, and this sector still gives an irreducible DHR sector\nof index 1 and spin $-1$ after the $\\alpha$-induction applied to\nthe local extension.  This gives a unique Fermionic extension\nas above.\n\n(6) Now we have the label $(A_{27}, E_8)$ for $m=28$.\nThe irreducible DHR sectors  $(0,0,0)$, $(0,10,0)$, $(0,18,0)$,\n$(0,28,0)$ give a local\n$Q$-system as in case (4).  Uniqueness also follows as above,\nusing the cohomology vanishing result in \\cite{KL2}.\n(This is again also a mirror extension in the sense of \\cite{X6}.)\nThe irreducible DHR sector $(26,0,0)$\nhas spin $-1$, and this sector still gives an irreducible DHR sector\nof index 1 and spin $-1$ after the $\\alpha$-induction applied to\nthe local extension.  This gives a unique Fermionic extension\nas above.\n\n(7) Now we have the label $(E_8, A_{31})$ for $m=30$.\nThe irreducible DHR sectors  $(0,0,0)$, $(10,0,0)$, $(18,0,0)$,\n$(28,0,0)$ give a local\n$Q$-system as in (5).  Uniqueness also follows as above.\n(This is again also a coset model as in \\cite{KL1}.)\nThe irreducible DHR sector $(0,30,0)$\nhas spin $-1$, and this sector still gives an irreducible DHR sector\nof index 1 and spin $-1$ after the $\\alpha$-induction applied to\nthe local extension.  This gives a unique Fermionic extension\nas above.\n\n(8) Now we have the label $(D_6, E_6)$ for $m=10$.\nThe irreducible DHR sectors  $(0,0,0)$, $(0,6,0)$,\n$(8,0,0)$, $(8,6,0)$ give a local\n$Q$-system, since this is a further index 2 extension of the\nlocal extension given in the above (4).\nUniqueness holds for this second extension.\nThe irreducible DHR sector $(0,10,0)$\nhas spin $-1$, and this sector still gives an irreducible DHR sector\nof index 1 and spin $-1$ after the $\\alpha$-induction applied to\nthe local extension.  This gives a unique Fermionic extension\nas above.\n\n(9) Now we have the label $(E_6, D_8)$ for $m=12$.\nThe irreducible DHR sectors  $(0,0,0)$, $(6,0,0)$,\n$(0,12,0)$, $(6,12,0)$ give a local\n$Q$-system, since this is a further index 2 extension of the\nlocal extension given in the above (5).\nUniqueness holds for this second extension.\nThe irreducible DHR sector $(10,0,0)$\nhas spin $-1$, and this sector still gives an irreducible DHR sector\nof index 1 and spin $-1$ after the $\\alpha$-induction applied to\nthe local extension.  This gives a unique Fermionic extension\nas above.\n\nAll the above give the following classification theorem.\n\\begin{theorem}\\label{main}\nThe following gives a complete list of superconformal nets with\n$c<3\/2$ and the dual canonical endomorphisms for the Bosonic\npart extending ${\\mathrm {SVir}}^0_c$ in each case is given as in\nTable \\ref{dual-can}.\n\\begin{enumerate}\n\\item The super Virasoro net with\n$c=\\displaystyle\\frac{3}{2}\\left(1-\\frac{8}{m(m+2)}\\right)$,\nlabelled with $(A_{m-1}, A_{m+1})$.\n\\item Index 2 extensions of the above (1), labeled with\n$(A_{4m'-1}, D_{2m'+2})$, $(D_{2m'+2}, A_{4m'+3})$.\n\\item Six exceptionals labelled with\n$(A_9, E_6)$, $(E_6, A_{13})$, $(A_{27}, E_8)$, $(E_8, A_{31})$,\n$(D_6, E_6)$, $(E_6, D_8)$.\n\\end{enumerate}\n\\end{theorem}\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nMany analytical and practical problems require a solution to an underdetermined\nor overdetermined system of equations in the form of a generalized matrix \ninverse of a singular matrix $\\Amn$ as $\\Atinv$ satisfying\n\\begin{eqnarray}\n      \\Amn\\Atinv\\Amn & = & \\Amn,\\\\\n      \\Atinv\\Amn\\Atinv\\!\\! & = & \\Atinv.\n\\end{eqnarray}\nIn addition to these properties, a practical application may require\nthe system to be consistent with respect to a general class of \ntransformations. For example, if the behavior of the system should\nbe invariant with respect to arbitrary rotations of the Cartesian\ncoordinate frame in which it is defined then the appropriate \ngeneralized inverse should also satisfy\n\\begin{equation}\n     \\tinv{(\\Um\\Amn\\Vm)} ~ = ~ \\Vmi\\Atinv\\Umi \\label{udist}\n\\end{equation}\nwhere $\\Um$ and $\\Vm$ are arbitrary rotation matrices (or, more generally,\narbitrary orthonormal\/unitary matrices) such that $\\Um\\Umi=\\Imat$, where\n$\\Umi$ is the conjugate-transpose of $\\Um$. This condition is satisfied\nby the Moore-Penrose inverse (MP inverse), which we will denote as\n$\\Apinv$. In other applications the behavior of the system of interest is \nexpected to be invariant with respect to changes of units defined for \nthe its key state variables\/parameters. For example, if all variables\nare defined consistently in imperial units then a change to metric\nunits should have no effect on the behavior of the system, i.e., the\nsystem should be invariant to the arbitrary choice of units. This condition can\nbe expressed as \n\\begin{eqnarray}\n      \\tinv{(\\Dm\\Amn\\Em)}  & = & \\Emi\\Atinv\\Dmi\n\\end{eqnarray}\nwhere $\\Dm$ and $\\Em$ are arbitrary nonsingular diagonal\nmatrices. This condition is satisfied by a recently-derived\nunit-consistent (UC) generalized inverse~\\cite{uhlmann0}.\n\nAn unfortunate fact is that in many scientific and engineering\ndomains the MP inverse is widely used \nas the default generalized inverse without consideration\nfor the properties it preserves -- {\\em and does not preserve}.\nExamples include some forms of Channel Estimation and \nsome applications of the Relative Gain Array (RGA).\nIn particular, the RGA is used to\nmeasure process variable interactions defined by a matrix $\\Amn$\nand is defined as~\\cite{bristol}\n\\begin{equation}\n   \\mbox{RGA}(\\Amn) ~ \\doteq ~ \\Amn \\circ (\\inv{\\Amn})^{\\mbox{\\scriptsize T}}.\n\\end{equation}\nwhere $\\circ$ represents the elementwise Hadamard matrix product.\nWhen $\\Amn$ is singular (or non-square) the RGA is often \nevaluated in practice as~\\cite{changyu}\n\\begin{equation}\n    \\Amn \\circ (\\Apinv)^{\\mbox{\\scriptsize T}}\n\\end{equation}\nwithout regard for whether the relative magnitudes of interactions\nshould be invariant to the choice of units on input\/output\nvariables. In many applications involving singular system\nmatrices (likely most) unit consistency is needed \nand thus the RGA should be evaluated using the UC inverse as\n\\begin{equation}\n    \\Amn \\circ (\\Aginv)^{\\mbox{\\scriptsize T}}\n\\end{equation}\nFailure to do so can lead to erroneous pairings of controlled \nand manipulated variables due to inappropriate dependencies\non the choice of units. \n\nIn other applications there may be need for an\n$n\\times n$ generalized\ninverse to be consistent with respect to an arbitrary matrix\nsimilarity transformation:\n\\begin{equation}\n     \\tinv{(\\Sm\\Amn\\Smi)} ~ = ~ \\Sm\\Atinv\\Smi \\label{dinvcnst}\n\\end{equation}\nThe need for this condition can be seen from a standard linear model\n\\begin{equation}\n     \\yv ~ = ~ \\Amn\\cdot\\av \\label{linmod}\n\\end{equation}\nwhere the objective is to identify a vector $\\av$ of parameter values\nsatisfying the above equation for a data matrix $\\Amn$ and\na known\/desired state vector $\\yv$ defined in the same state\nspace as $\\av$. If $\\Amn$ is nonsingular\nthen there exists a unique $\\Amni$ which gives the solution\n\\begin{equation}\n     \\av ~ = ~ \\Amni\\cdot\\yv\n\\end{equation}\nIf $\\Amn$ is singular, however, then the\nMoore-Penrose inverse could be applied as\n\\begin{equation}\n     \\av ~ = ~ \\Apinv\\cdot\\yv\n\\end{equation}\nor the UC inverse could be applied as\n\\begin{equation}\n     \\av ~ = ~ \\Aginv\\cdot\\yv\n\\end{equation}\nto obtain a solution. Now \nsuppose the state space of $\\yv$ and $\\av$ is\nlinearly transformed by a nonsingular matrix $\\Sm$ as \n\\begin{eqnarray}\n     \\yyv & = & \\Sm\\yv \\\\\n     \\aav & = & \\Sm\\av\n\\end{eqnarray}\nwhere the transformation $\\Sm$ could, for example, define \ntangent-space coordinates for a robot moving on a non-planar manifold\nor, similarly, for gradient descent in a given function space.\nEq.(\\ref{linmod}) can then \nbe rewritten in the new coordinates as\n\\begin{equation}\n     \\yyv ~ = ~ \\Sm\\yv  ~ = ~ (\\Sm\\Amn\\Smi)\\cdot\\Sm\\av\n     ~ = ~ (\\Sm\\Amn\\Smi)\\cdot\\aav\n\\end{equation}\nbut for which  \n\\begin{eqnarray}\n      \\Sm\\av  &  \\neq & \\pinv{(\\Sm\\Amn\\Smi)}\\cdot\\yyv \\\\\n      \\Sm\\av  &  \\neq & \\ginv{(\\Sm\\Amn\\Smi)}\\cdot\\yyv \n\\end{eqnarray}\nIn other words, the solution in the transformed space is not\nthe same as the original solution transformed to the new\nspace -- a situation that represents failure of a fundamental\nsanity check. This is because the MP inverse and UC inverse\ndo not satisfy similarity consistency. \n\nAlthough the similarity-consistency (SC) condition of Eq.\\!~(\\ref{dinvcnst}) can be \nsatisfied using the {\\em Drazin}  inverse~\\cite{drazin}, $\\Adinv$, its use is not an option\nfor most engineering-related systems. Specifically, although the Drazin inverse\n$\\Adinv$ does have some practical applications~(\\cite{cmr76,drazinApp}), \nits use is severely limited because $\\Adinv$ may\nhave rank less than that of $\\Amn$, which is often problematic\nbecause it implies subspace information can be progressively\nand irretrievably lost. In this paper we address this limitation of\nthe Drazin inverse by defining a new generalized inverse which satisfies\nthe condition of Eq.\\!~(\\ref{dinvcnst}) while also preserving\nthe rank of the original matrix.\n\n\\section{The SC Generalized Matrix Inverse}\n\nOne way to derive a generalized inverse that satisfies\na particular consistency condition is to identify a normal\nform relative to that condition. For example, the singular\nvalue decomposition (SVD) decomposes an arbitrary \nmatrix $\\Amn$ as \n\\begin{equation}\n      \\Amn ~=~ \\Um\\Dm\\Vmi\n\\end{equation}\nwhere $\\Um$ and $\\Vm$ are unitary and $\\Dm$ is a nonnegative\ndiagonal matrix of the singular values of $\\Amn$. Critically, $\\Dm$\nis unique up to permutation, so all matrices with the same set of\nsingular values are equivalent up to left and right unitary\ntransformations. This allows the MP inverse to be derived as\n\\begin{equation}\n   \\Apinv ~ = ~ \\Vm\\tinv{\\Dm}\\Umi\n\\end{equation}\nwhere $\\tinv{\\Dm}$ simply inverts the nonzero elements of $\\Dm$.\nFrom this the UC inverse can be similarly derived from a decomposition of the form\n\\begin{equation}\n   \\Amn ~ = ~ \\Dm\\Sm\\Em\n\\end{equation}\nwhere $\\Dm$ and $\\Em$ are nonnegative diagonal\nmatrices and $\\Sm$ is unique up to left and right transformations\nby unitary diagonal matrices~\\cite{uhlmann}. The UC inverse can then be\nobtained as\n\\begin{equation}\n   \\Aginv ~ = ~ \\Emi\\pinv{\\Sm}\\Dmi\n\\end{equation}\nwhere the MP inverse can be applied because $\\Sm$ is \nunique up to unitary diagonal transformations and\nthe MP inverse is consistent with respect to general\nunitary transformations. \n\nFollowing a similar strategy, it can be recognized that the \nJordan normal form of a square matrix $\\Amn$ is\n\\begin{equation}\n   \\Amn ~ = ~ \\Pm\\Jm\\inv{\\Pm}\n\\end{equation}\nwhere $\\Pm$ is an arbitrary nonsingular matrix of the same\nsize as $\\Amn$, and $\\Jm$ is block-diagonal and unique up\nto permutation of the blocks. Thus we can derive from this\ndecomposition the following similarity-consistent (SC) inverse:\n\\begin{equation}\n   \\Asinv ~ = ~ \\Pm\\pinv{\\Jm}\\inv{\\Pm}\n\\end{equation}\nwhere the MP inverse can be applied because $\\Jm$ is \nunique up to block-diagonal permutation transformations,\nwhich are of course unitary and thus permit the MP\ninverse to be consistently applied.\n\nWhat remains is to formally show that this definition of\n$\\Asinv$ is in fact consistent with respect to similarity\ntransformations, i.e., \n\\begin{equation}\n     \\sinv{(\\Sm\\Amn\\Smi)} ~ = ~ \\Sm\\Asinv\\Smi .\n\\end{equation}\nThe Jordan decomposition of $\\Amn$ is\n\\begin{equation}\n   \\Amn ~ = ~ \\Pm\\Jm\\inv{\\Pm}\n\\end{equation}\nso the Jordan decomposition of $\\Sm\\Amn\\Smi$ must\nbe\n\\begin{eqnarray}\n   \\Sm\\Amn\\Smi & = & \\Sm(\\Pm\\Jm\\inv{\\Pm})\\Smi\\\\\n   ~ & = & (\\Sm\\Pm)\\Jm(\\inv{\\Pm}\\Smi)\\\\\n   ~ & = & (\\Sm\\Pm)\\Jm\\inv{(\\Sm\\Pm)}\n\\end{eqnarray}\nand thus the SC inverse of $\\Sm\\Amn\\Smi$ is\n\\begin{eqnarray}\n   \\sinv{(\\Sm\\Amn\\Smi)} & = & \\sinv{\\left((\\Sm\\Pm)\\Jm\\inv{(\\Sm\\Pm)}\\right)}\\\\\n   ~ & = &  (\\Sm\\Pm)\\pinv{\\Jm}(\\inv{\\Pm}\\Smi)\\\\\n   ~ & = & \\Sm(\\Pm\\pinv{\\Jm}\\inv{\\Pm})\\Smi\\\\\n   ~ & = & \\Sm\\Asinv\\Smi.\n\\end{eqnarray}\nThis establishes similarity consistency, and because the Moore-Penrose\ninverse preserves rank, the SC inverse does as well. The key difference\nbetween the SC inverse and the Drazin inverse is that the latter \nenjoys the analytically important property that:\n\\begin{equation}\n   \\Amn\\Adinv ~=~ \\Adinv\\Amn\n\\end{equation}\nwhereas product commutativity of $\\Amn$ and $\\Asinv$ \nis not a property of the SC inverse because $\\pinv{\\Jm}\\Jm$\ndoes not generally equal $\\Jm\\pinv{\\Jm}$.\n\n\\section{Example and Practical Considerations}\n\nHere we consider an illustrative numerical example\ninvolving a nilpotent matrix\n\\begin{equation}\n   \\Amn ~=~ \\left[\\!\\begin{tabular}{rrr}\n                             4 & -1 & 2\\\\\n                             7 & -2 & 3\\\\\n                             -4 & 1 & -2\n                     \\end{tabular}\\right]\n\\end{equation}\nwith Jordan form $\\Amn=\\Pm\\Jm\\inv{\\Pm}$ where\n\\begin{equation}\n   \\Pm ~=~ \\left[\\!\\begin{tabular}{rrr}\n                             1 & -2 & -3\\\\\n                             2 & -5 & 10\\\\\n                            -1 & 2 & -2\n                     \\end{tabular}\\right]\n~ \\mbox{and} ~ ~\n   \\Jm ~=~ \\left[\\!\\begin{tabular}{rrr}\n                             0 & 1 & 0\\\\\n                             0 & 0 & 1\\\\\n                             0 & 0 &  0\n                     \\end{tabular}\\right].\n\\end{equation}\nIt can be verified that \n\\begin{equation}\n   \\Asinv ~=~ \\left[\\!\\begin{tabular}{rrr}\n                             -2 & 1 & 2\\\\\n                             10 & 0 & 15\\\\\n                             8 & -2 & 2\n                     \\end{tabular}\\right]\n~ \\mbox{and} ~ ~\n   \\Adinv ~=~ \\left[\\!\\begin{tabular}{rrr}\n                             0 & 0 & 0\\\\\n                             0 & 0 & 0\\\\\n                             0 & 0 &  0\n                     \\end{tabular}\\right].\n\\end{equation}\nwhere the Drazin inverse of a nilpotent matrix is always\nzero, which graphically demonstrates complete loss of \nrank\\footnote{The integer matrices in this section were constructed\nusing the method of~\\cite{hanson} so that inverses are also \ninteger matrices. This allows the results to be more conveniently\nverified by direct hand calculations.}.\nWe now consider a matrix $\\Mm=\\Sm\\Amn\\Smi$ similar to $\\Amn$\nwith\n\\begin{equation}\n   \\Sm ~=~ \\left[\\!\\begin{tabular}{rrr}\n                             -1 & 4 & 0\\\\\n                             -2 & 9 & 3\\\\\n                              1 & -4 & 1\n                     \\end{tabular}\\right].\n\\end{equation}\nSimilarity consistency for the Drazin inverse is trivially\nmaintained because \n$\\dinv{\\Mm}=\\dinv{(\\Sm\\Amn\\inv{\\Sm})}=\\Sm\\dinv{\\Amn}\\inv{\\Sm}={\\bf 0}$.\nBy contrast, it can be verified that the rank of the SC inverse\n\\begin{equation}\n\\sinv{\\Mm} ~=~ \\left[\\!\\begin{tabular}{rrr}\n                             -819 & 167 & -443\\\\\n                             -2301 & 464 & -1255\\\\\n                              663 & -137 & 355\n                     \\end{tabular}\\right]\n\\end{equation}\nis the same as $\\Mm$ and that the similarity consistency condition\n$\\sinv{(\\Sm\\Amn\\inv{\\Sm})}=\\Sm\\sinv{\\Amn}\\inv{\\Sm}=\\sinv{\\Mm}$\nis satisfied. And of couse it can also be verified\nthat $\\sinv{\\Mm}$ satisfies the generalized inverse\nconditions $\\Mm\\sinv{\\Mm}\\Mm=\\Mm$ and $\\sinv{\\Mm}\\Mm\\sinv{\\Mm}=\\sinv{\\Mm}$.\nWhen this example is numerically evaluated using the Matlab {\\sf jordan(M)} \nand {\\sf pinv(M)} functions the SC inverse is found to be accurate to machine precision.\nHowever, when applied in two different high data-rate robotic control applications \nthe observed errors using the SC inverse were sufficiently large to cause the \ncontrol process to diverge rapidly\\footnote{These simulation experiments \nwere performed using variations on models for a robot arm and a \nrover examined in~\\cite{zhang}. Unit consistency was the focus of that\nwork and the two failed tests of the SC inverse were neither analyzed nor reported.}. \n\nThe numerical instability of the Jordan decomposition is well known~\\cite{golub}, \nand this problem is exacerbated when applied with the MP inverse because large errors \ncan be introduced into singular values that should be zero, which can lead to huge\nspike errors when inverted. It is not clear to what \nextent this issue can be mitigated when these two operations are used in\nsuccession to compute the SC inverse. In many practical dynamical-system applications \nthe system matrix changes relatively slowly over time, so it may be possible\nto track the evolution of its distribution of singular values to actively avoid \ncatastrophic discontinuities in the case of a singular value near the \nthreshold between zero and nonzero when the MP inverse is applied\nduring evaluation of the SC inverse. More specifically, a tracked presumed-zero \nsingular value that drifts above the MP threshold can be maintained as zero while \na tracked presumed-nonzero singular value can be maintained as nonzero even if it \ndrifts below the threshold.\n\nA potentially more general approach for mitigating some of the numerical fragility is \nto either jointly solve the Jordan+MP calculation or to relax the form of the \ndecomposition, e.g., so that the central matrix is unique only up to orthonormal\nsimilarity to a complex symmetric matrix rather than\npermutation similarity to a Jordan matrix. For example, any matrix $\\Amn$\ncan be decomposed\\!~\\cite{HJ1} as\n\\begin{equation}\n   \\Amn ~ = ~ \\Sm\\Xm\\Smi  \\label{symdecomp}\n\\end{equation}\nwhere $\\Xm$ is complex symmetric and thus\nthe SC inverse can be defined as:\n\\begin{equation}\n   \\Asinv ~ = ~ \\Sm\\pinv{\\Xm}\\Smi   \\label{symsc}.\n\\end{equation}\nThat the decomposition of\nEq.\\!~(\\ref{symsc}) is only unique up to unitary\/orthonormal \nsimilarity can be inferred from the fact that for\ncomplex symmetric $\\Ym$ and orthonormal $\\Qm$ ($\\inv{\\Qm}=\\Qmt$),\n$\\Ym=\\Qm\\Xm\\Qmt$ implies\n\\begin{eqnarray}\n   \\Asinv & = & \\Sm\\pinv{\\Xm}\\Smi\\\\\n    ~ & = & \\Sm\\pinv{(\\Qm\\Ym\\Qmt)}\\Smi \\\\\n     ~ & = &      (\\Sm\\Qm)\\pinv{\\Ym}\\inv{(\\Sm\\Qm)} \n\\end{eqnarray}\nwhere the last step is obtained by exploiting the unitary consistency property of the MP inverse.\nIt is reasonable to expect that this less rigid definition will be more amenable\nto numerically-robust evaluation than is possible using a construction requiring\nan explicit Jordan decomposition.\n\n\\section{Fixed-Rank Assumption}\n\nIn some applications it is known that the rank of the\ntime-evolving system matrix can potentially change, \ne.g., during transitions through gimbal-lock \nconfigurations, while in others\nit can be expected to remain constant. In the latter case\na fixed-rank assumption can be exploited to mitigate\nthe problem of spurious singular values when applying\nthe MP inverse to the Jordan matrix during evaluation of\nthe SC inverse. Specifically, if a fixed rank of $k$ is \nknown\/assumed then the MP inverse can be evaluated\nexplicitly using the SVD with all but the $k$\nlargest singular values set identically to zero instead of having\nto discriminate with a threshold.  \n\nThe efficacy of exploiting the fixed-rank assumption \ncan be demonstrated in a carefully controlled experiment\ninvolving a dynamical 3-dimensional system that is\nconfined to a 2-dimensional subspace such that the\n$3\\times 3$ system matrix can be assumed to have a fixed\nrank of $2$. The testing model can be further tailored \nso that the system matrix is guaranteed to never be\ndiagonalizable and thus is maximally susceptible to the \nnumerical issues discussed in the previous section. We\nachieve this by defining the Jordan form of the system\nas a function of time as\n\\begin{equation}\n   (\\Qm(t)\\Sm)\\,\\Jm\\,\\inv{(\\Qm(t)\\Sm)}\n\\end{equation}\nwith\n\\begin{equation}\n\\Jm = \\begin{array}{|ccc|}\n             1 & 1 & 0\\\\\n             0 & 1 & 0\\\\\n             0 & 0 & 0\n          \\end{array}\n~,~ ~\n\\Qm(t) = \\begin{array}{|ccc|}\n                    \\cos(t) & -\\sin(t) & 0\\\\\n                    \\sin(t)  & ~\\cos(t) & 0\\\\ \n                    0 & 0 & 1\n               \\end{array}\n\\end{equation}\nand $\\Sm$ is an arbitrary fixed nonsingular matrix. Thus\nthe system matrix is singular and has an orbit period of $2\\pi$ \nduring which it is never diagonalizable. The plot of Figure\\!~1 \nshows the maximum absolute elementwise deviation between the \ncomputed SC inverse and the ground-truth known SC inverse \nfor each of 300 uniform timesteps $t$, $0\\leq t<2\\pi$. It\nis seen that spike errors occur at a significant fraction of the \ntimesteps. These spikes are actually of magnitude $>\\!10^{14}$ \nand result from inverting spurious singular values \nthat exceed {\\sf pinv}'s zero threshold. In tests of many\nrandomly-generated $\\Sm$ matrices the number of\nspike errors over the entire 300-timestep simulation ranged from\nzero to more than $50\\%$, with Figure\\!~1 showing\nan example of the latter. \n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=3.5in]{SCINV-RESULT-Raw-300}\n\t\\caption{Error magnitudes (vertical axis) resulting from standard evaluation of the\n                         SC inverse from the Jordan form over 300 timesteps (horizontal axis).\n                         Clipped magnitudes are actually of size $\\approx 10^{14}$.}        \n\\end{figure}\n\n\nFigure\\!~2 shows the results of the same\nscenario but with the smallest singular value of the Jordan matrix\nalways treated as identically zero for computation of the SC inverse at\neach timestep. The results shown in Figure\\!~2 are typical of those\nobserved for the fixed-rank method in the battery of tests over many\nrandomly-generated $\\Sm$ matrices. By design it is likely that the model \nused for these tests overestimates the frequency of spike errors that can be \nexpected from standard evaluation of the SC inverse in real-world \napplications, but because any such errors will be time-correlated it likely will\nnot be practical to simply identify and discard them (e.g., apply some sort of \nheuristic outlier filter) on the assumption that they will only occur sporadically.\nIn other words, the intrinsic numerical instability of the SC inverse cannot be\nmitigated in the general case. However, the results of Figure\\!~2 suggest\nthat the problem may be effectively mitigated in a large class of applications.\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=3.5in]{SCINV-RESULT-Fixed-300}\n\t\\caption{Error magnitudes (vertical axis) resulting from fixed-rank evaluation of the\n                         SC inverse from the Jordan form over 300 timesteps (horizontal axis). The mean\n                         error magnitude is $1.4420e$-$07$ and the maximum magnitude is $2.3101e$-$06$.}        \n\\end{figure}\n\n\n\n\\section{Discussion}\n\nSimilarity transformations capture the most general linear transformations that can\nbe applied to the common coordinate frame of a set of input and output \nvariables, e.g., as can arise in machine learning applications in which little is \nknown about the best space in which to represent an input-output mapping.\nIn this paper we defined a new similarity-consistent generalized matrix\ninverse that preserves\/maintains the rank of the original matrix, a property\nwhich is not preserved by the Drazin inverse. It has been noted that the\nnumerical evaluation of the SC inverse can be a challenging practical\nlimitation. One possible approach for addressing this problem is to \nrelax the structure of the decomposition from requiring similarity to a Jordan\nmatrix to similarity to a complex symmetric matrix for which uniqueness\nis only required up to orthonormal similarity, rather than permutation \nsimilarity, so the MP inverse is still applicable. This less-rigid\nformulation may be more amenable to numerically-robust evaluation,\nespecially in conjunction with the fixed-rank method.\nRegardless of the choice of decomposition, \nit would be interesting to examine whether recent\nmethods developed for evaluating the Drazin inverse can be applied\nto the SC inverse~\\cite{drazinalg,pan}. \n\nDespite its computational challenges, the SC inverse is the only \ngeneral option available for applications that demand consistency with\nrespect to arbitrary linear transformations of a common coordinate\nframe. We have provided evidence that these challenges can be \neffectively addressed in applications in which the rank of the system\ncan be assumed fixed during its time evolution, but we have also\nemphasized that intrinsic numerical instabilities represent a \nfundamental obstacle in the general case. \nThis motivates continued work toward identifying special properties\nof particular applications that can be exploited to permit the SC\ninverse to be practically applied.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section*{References}}\n\n\n\n\n\\usepackage{amssymb}\n\\usepackage{amsthm}\n\\usepackage{framed}\n\\usepackage{amsmath,color}\n\\usepackage{mathrsfs}\n\\usepackage{graphicx}\n\\usepackage{epstopdf}\n\\usepackage{float}\n\\usepackage{caption}\n\\usepackage{subcaption}\n\\usepackage{bm}\n\\usepackage{bbm}\n\\usepackage{mathrsfs}\n\\usepackage{cleveref}\n\\usepackage{soul}\n\\usepackage{accents}\n\\usepackage{color,soul}\n\\usepackage{color}\n\\usepackage{tabu}\n\\usepackage{longtable}\n\\usepackage{nomencl}\n\\makenomenclature\n\\setlength{\\nomitemsep}{-\\parskip}\n\\biboptions{sort&compress}\n\\soulregister\\citep7\n\\soulregister\\citet7\n\\soulregister\\citealp7\n\\newcommand{\\highlight}[1]{\\colorbox{yellow}{$\\displaystyle #1$}}\n\\newsavebox{\\measurebox}\n\\newcommand{\\vect}[1]{\\boldsymbol{\\mathbf{#1}}}\n\n\n\\journal{Engineering Fracture Mechanics}\n\n\\makeatletter\n\\def\\@author#1{\\g@addto@macro\\elsauthors{\\normalsize%\n    \\def\\baselinestretch{1}%\n    \\upshape\\authorsep#1\\unskip\\textsuperscript{%\n      \\ifx\\@fnmark\\@empty\\else\\unskip\\sep\\@fnmark\\let\\sep=,\\fi\n      \\ifx\\@corref\\@empty\\else\\unskip\\sep\\@corref\\let\\sep=,\\fi\n      }%\n    \\def\\authorsep{\\unskip,\\space}%\n    \\global\\let\\@fnmark\\@empty\n    \\global\\let\\@corref\\@empty \n    \\global\\let\\sep\\@empty}%\n    \\@eadauthor={#1}\n}\n\\makeatother\n\n\\begin{document}\n\n\\begin{frontmatter}\n\n\n\n\\title{A cohesive zone framework for environmentally assisted fatigue}\n\n\n\\author{Susana {del Busto}\\fnref{Uniovi}}\n\n\\author{Covadonga Beteg\\'on\\fnref{Uniovi}}\n\n\\author{Emilio Mart\\'{\\i}nez-Pa\\~neda\\corref{cor1}\\fnref{DTU}}\n\\ead{mail@empaneda.com}\n\n\\address[Uniovi]{Department of Construction and Manufacturing Engineering, University of Oviedo, Gij\\'on 33203, Spain}\n\n\\address[DTU]{Department of Mechanical Engineering, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark}\n\n\\cortext[cor1]{Corresponding author. Tel: +45 45 25 42 71; fax: +45 25 19 61.}\n\n\n\\begin{abstract}\nWe present a compelling finite element framework to model hydrogen assisted fatigue by means of a hydrogen- and cycle-dependent cohesive zone formulation. The model builds upon: (i) appropriate environmental boundary conditions, (ii) a coupled mechanical and hydrogen diffusion response, driven by chemical potential gradients, (iii) a mechanical behavior characterized by finite deformation J2 plasticity, (iv) a phenomenological trapping model, (v) an irreversible cohesive zone formulation for fatigue, grounded on continuum damage mechanics, and (vi) a traction-separation law dependent on hydrogen coverage calculated from first principles. The computations show that the present scheme appropriately captures the main experimental trends; namely, the sensitivity of fatigue crack growth rates to the loading frequency and the environment. The role of yield strength, work hardening, and constraint conditions in enhancing crack growth rates as a function of the frequency is thoroughly investigated. The results reveal the need to incorporate additional sources of stress elevation, such as gradient-enhanced dislocation hardening, to attain a quantitative agreement with the experiments.\n\\end{abstract}\n\n\\begin{keyword}\n\nHydrogen embrittlement \\sep Cohesive zone models \\sep Hydrogen diffusion \\sep Finite element analysis \\sep Fatigue crack growth\n\n\n\n\\end{keyword}\n\n\\end{frontmatter}\n\n\n\n\\begin{framed}\n\\nomenclature{$a$}{crack length}\n\\nomenclature{$b, \\, b_0$}{current and initial crack opening displacement}\n\\nomenclature{$\\vect{B}$}{standard strain-displacement matrix}\n\\nomenclature{$\\vect{B_c}$}{global cohesive displacement-separation matrix}\n\\nomenclature{$C$}{total hydrogen concentration}\n\\nomenclature{$C_L, \\, C_T$}{hydrogen concentration in lattice and trapping sites}\n\\nomenclature{$c_q$}{specific heat capacity}\n\\nomenclature{$\\mathcal{C}, \\, m$}{Paris law coefficients}\n\\nomenclature{$\\mathcal{D}, \\, \\mathcal{D}_e$}{standard and effective diffusion coefficients}\n\\nomenclature{$D, \\, D_c, \\, D_m$}{damage variable: total, cyclic and monotonic}\n\\nomenclature{$E$}{Young's modulus}\n\\nomenclature{$f$}{load frequency}\n\\nomenclature{$\\vect{f_c}$}{cohesive internal force vector}\n\\nomenclature{$\\Delta g_b^0$}{Gibbs free energy difference}\n\\nomenclature{$\\vect{J}$}{hydrogen flux vector}\n\\nomenclature{$\\vect{K_c}$}{cohesive tangent stiffness matrix}\n\\nomenclature{$K_T$}{trap equilibrium constant}\n\\nomenclature{$K, \\, K_0$}{remote and reference stress intensity factor}\n\\nomenclature{$\\vect{L}$}{local displacement-separation matrix}\n\\nomenclature{$\\vect{\\mathscr{L}}$}{elastoplastic constitutive matrix}\n\\nomenclature{$\\vect{N}$}{shape functions matrix}\n\\nomenclature{$N$}{number of cycles}\n\\nomenclature{$\\mathcal{N}$}{strain hardening exponent}\n\\nomenclature{$N_A$}{Avogadro's number}\n\\nomenclature{$N_L, \\, N_T$}{number of lattice and trapping sites per unit volume}\n\\nomenclature{$q$}{heat flux per unit area}\n\\nomenclature{$\\mathcal{R}$}{universal gas constant}\n\\nomenclature{$\\vect{R}$}{rotational matrix}\n\\nomenclature{$R$}{load ratio}\n\\nomenclature{$R_0$}{reference plastic length}\n\\nomenclature{$T$}{elastic T-stress}\n\\nomenclature{$\\mathcal{T}$}{absolute temperature}\n\\nomenclature{$\\vect{T}, \\, \\tilde{\\vect{T}}$}{standard and effective cohesive traction vectors}\n\\nomenclature{$\\vect{t}$}{external traction vector}\n\\nomenclature{$T_n$}{normal cohesive traction}\n\\nomenclature{$U$}{internal energy per unit mass}\n\\nomenclature{$\\vect{U}$}{global nodal displacement vector}\n\\nomenclature{$\\vect{u}, \\, \\tilde{\\vect{u}}$}{field and local nodal displacement vectors}\n\\nomenclature{$\\bar{V}_H$}{partial molar volume of hydrogen}\n\\nomenclature{$V_M$}{molar volume of the host lattice}\n\\nomenclature{$W_B$}{trap binding energy}\n\\nomenclature{$\\alpha$}{compression penalty factor}\n\\nomenclature{$\\beta$}{number of lattice sites per solvent atom}\n\\nomenclature{$\\vect{\\Delta}, \\, \\tilde{\\vect{\\Delta}}$}{local field and nodal separation vectors}\n\\nomenclature{$\\Delta_n$}{normal cohesive separation}\n\\nomenclature{$\\delta_n$}{characteristic normal cohesive length}\n\\nomenclature{$\\delta_{\\Sigma}$}{accumulated cohesive length}\n\\nomenclature{$\\varepsilon_p$}{equivalent plastic strain}\n\\nomenclature{$\\vect{\\varepsilon}$}{Cauchy strain tensor}\n\\nomenclature{$\\theta_H$}{hydrogen coverage}\n\\nomenclature{$\\theta_L, \\, \\theta_T$}{occupancy of lattice and trapping sites}\n\\nomenclature{$\\mu_L$}{lattice chemical potential}\n\\nomenclature{$\\rho$}{density}\n\\nomenclature{$\\sigma_f$}{cohesive endurance limit}\n\\nomenclature{$\\sigma_H$}{hydrostatic stress}\n\\nomenclature{$\\sigma_Y$}{initial yield stress}\n\\nomenclature{$\\vect{\\sigma}$}{Cauchy stress tensor}\n\\nomenclature{$\\sigma_{max}, \\, \\sigma_{max,0}$}{current and original cohesive strength}\n\\nomenclature{$\\phi_n$}{normal cohesive energy}\n\\printnomenclature\n\\end{framed}\n\n\\section{Introduction}\n\\label{Sec:Introduction}\n\nMetallic materials play a predominant role in structures and industrial components because of their strength, stiffness, toughness and tolerance of high temperatures. However, hydrogen has been known for over a hundred years to severely degrade the fracture resistance of advanced alloys, with cracking being observed in modern steels at one-tenth of the expected fracture toughness. With current engineering approaches being mainly empirical and highly conservative, there is a strong need to understand the mechanisms of such hydrogen-induced degradation and to develop mechanistic-based models able to reproduce the microstructure-dependent mechanical response at scales relevant to engineering practice.\\\\\n\nModels based on the hydrogen enhanced decohesion (HEDE) mechanism have proven to capture the main experimental trends depicted by high-strength steels in aqueous solutions and hydrogen-containing gaseous environments \\cite{G03}. The use of cohesive zone formulations is particularly appealing in this regard, as they constitute a suitable tool to characterize the sensitivity of the fracture energy to hydrogen coverage. The cohesive traction separation law can be derived from first principles quantum mechanics \\cite{S04} or calibrated with experiments \\cite{S08,Y16}. The statistical distribution of relevant microstructural features has also fostered the use of weakest-link approaches \\cite{N10,A14}. Very recently, Mart\\'{\\i}nez-Pa\\~neda \\textit{et al.} \\cite{M16} integrated strain gradient plasticity simulations and electrochemical assessment of hydrogen solubility in Gerberich \\cite{G12} model. The investigation of a Ni-Cu superalloy and a modern ultra-high-strength steel revealed an encouraging quantitative agreement with experimental data for the threshold stress intensity factor and the stage II crack growth rate. However, and despite the fact that most industrial components experience periodic loading, modeling efforts have been mainly restricted to monotonic conditions. Recently, Moriconi \\textit{et al.} \\cite{M14} conducted experiments and simulations to investigate the role of hydrogen on a 15-5PH martensitic steel intended for gaseous hydrogen storage. Model predictions provided a very good agreement with experimental data for low hydrogen pressures but failed to capture the deleterious effect of hydrogen on the fatigue crack propagation under high pressures. Understanding the role of hydrogen in accelerating crack growth rates under cyclic loading could be crucial to enable the use of high-strength steels in the energy sector and to develop reliable transport and storage infrastructure for future energy systems.\\\\\n\nIn this work, we present a general numerical framework for hydrogen-assisted fatigue. The main ingredients of the model are: (i) realistic Dirichlet type conditions to account for stress-assisted diffusion at the boundaries, (ii)  an extended hydrogen transport equation governed by hydrostatic stresses and plastic straining through trapping, (iii) higher order elements incorporating a coupled mechanical-diffusion response, (iv) continuum large strains elastoplasticity, (v) a hydrogen coverage dependent cohesive strength, and (vi) a Lemaitre-type damage response for an irreversible traction-separation law. The influence of diffusible hydrogen in fatigue crack growth is systematically investigated, the main experimental trends captured and valuable insight achieved.\n\n\\section{Numerical framework}\n\\label{Sec:NumModel}\n\nHydrogen transport towards the fracture process zone and subsequent cracking under cyclic loading conditions are investigated by means of a coupled mechanical-diffusion-cohesive finite element framework. Section \\ref{Sec:UMATHT} describes the mechanical-diffusion coupling that builds upon the analogy with heat transfer, Section \\ref{Sec:CZM} provides details of the cyclic and hydrogen dependent cohesive zone formulation employed and finally Section \\ref{Sec:FEM} outlines the general assemblage and implementation. \n\n\\subsection{Coupled mechanical-diffusion through the analogy with heat transfer}\n\\label{Sec:UMATHT}\n\nThe hydrogen transport model follows the pioneering work by Sofronis and McMeeking \\cite{SM89}. Hence, hydrogen transport is governed by hydrostatic stress and plastic straining through trapping. Hydrogen moves through normal interstitial lattice site diffusion and the diffusible concentration of hydrogen $C$ is defined as the sum of the hydrogen concentrations at reversible traps $C_T$ and lattice sites $C_L$. The latter is given by,\n\\begin{equation}\\label{Eq:CL}\nC_L=N_L \\theta_L\n\\end{equation}\n\n\\noindent where $N_L$ is the number of sites per unit volume and $\\theta_L$ the occupancy of lattice sites. The former can be expressed as a function of $V_M$, the molar volume of the host lattice, as:\n\\begin{equation}\\label{Eq:NLVA}\nN_L=\\frac{\\beta N_A}{V_M}\n\\end{equation} \n\n\\noindent with $N_A$ being Avogadro's number and $\\beta$ the number of interstitial lattice sites per solvent atom. On the other hand, the hydrogen concentration trapped at microstructural defects is given by,\n\\begin{equation}\\label{Eq:CT}\nC_T=N_T \\theta_T \n\\end{equation}\n\n\\noindent where $N_T$ denotes the number of traps per unit volume and $\\theta_T$ the occupancy of the trap sites. Here, focus will be placed on reversible trapping sites at microstructural defects generated by plastic straining - dislocations; a key ingredient in the mechanics of hydrogen diffusion \\cite{O14,D15}. A phenomenological relation between the trap density and the equivalent plastic strain is established based on the permeation tests by Kumnick and Johnson \\cite{KJ80},\n\\begin{equation}\\label{Eq:NTKJ}\n\\log N_T = 23.26 - 2.33 \\exp \\left( -5.5 \\varepsilon_p \\right)\n\\end{equation}   \n\nOriani's equilibrium theory \\cite{O70} is adopted, resulting in a Fermi-Dirac relation between the occupancy of trap and lattice sites,\n\\begin{equation}\\label{Eq:Oriani}\n\\frac{\\theta_T}{1-\\theta_T}=\\frac{\\theta_L}{1-\\theta_L} K_T \n\\end{equation}\n\n\\noindent with $K_T$ being the trap equilibrium constant,\n\\begin{equation}\nK_T=\\textnormal{exp}\\left( \\frac{-W_B}{\\mathcal{R}\\mathcal{T}} \\right)\n\\end{equation}\n\nHere, $W_B$ is the trap binding energy, $\\mathcal{R}$ the gas constant and $\\mathcal{T}$ the absolute temperature. Under the common assumption of low occupancy conditions ($\\theta_L << 1$), the equilibrium relationship between $C_T$ and $C_L$ becomes,\n\\begin{equation}\\label{Eq:CT2}\nC_T=\\frac{N_T K_T C_L}{K_T C_L + N_L}\n\\end{equation}\n\nIn a volume, $V$, bounded by a surface, $S$, with outward normal, $\\vect{n}$, mass conservation requirements relate the rate of change of $C$ with the hydrogen flux through $S$,\n\\begin{equation}\\label{Eq:MassBal}\n\\frac{\\textnormal{d}}{\\textnormal{d}t} \\int_V C \\,\\, \\textnormal{d}V + \\int_S \\vect{J} \\cdot \\vect{n} \\,\\, \\textnormal{d} S=0\n\\end{equation}\n\nFick's law relates the hydrogen flux with the gradient of the chemical potential $\\nabla \\mu_L$,\n\\begin{equation}\\label{Eq:J1}\n\\vect{J} = - \\frac{\\mathcal{D} C_L}{\\mathcal{R}\\mathcal{T}} \\nabla \\mu_L\n\\end{equation}\n\n\\noindent with $\\mathcal{D}$ being the diffusion coefficient. The chemical potential of hydrogen in lattice sites is given by,\n\\begin{equation}\\label{Eq:muL}\n\\mu_L = \\mu_L^0 + \\mathcal{R}\\mathcal{T} \\, \\textnormal{ln} \\frac{\\theta_L}{1-\\theta_L} - \\bar{V}_H \\sigma_H\n\\end{equation}\n\nHere, $\\mu_L^0$ denotes the chemical potential in the standard state and the last term corresponds to the so-called stress-dependent part of the chemical potential $\\mu_{\\sigma}$, with $\\bar{V}_H$ being the partial molar volume of hydrogen in solid solution. Assuming a constant interstitial sites concentration and substituting (\\ref{Eq:muL}) into (\\ref{Eq:J1}), one reaches\n\\begin{equation}\\label{Eq:JL2}\n\\vect{J}=-\\mathcal{D} \\nabla C_L + \\frac{\\mathcal{D}}{\\mathcal{R}\\mathcal{T}} C_L \\bar{V}_H \\nabla \\sigma_H\n\\end{equation}\n\nReplacing $\\vect{J}$ in the mass balance equation (\\ref{Eq:MassBal}), using the divergence theorem and considering the arbitrariness of $V$ renders,\n\\begin{equation}\\label{Eq:Bal2}\n\\frac{dC_L}{dt}+\\frac{dC_T}{dt} = \\mathcal{D} \\nabla^2 C_L  - \\nabla \\cdot \\left( \\frac{\\mathcal{D} C_L \\bar{V}_H}{\\mathcal{R}\\mathcal{T}}  \\nabla \\sigma_H \\right)\n\\end{equation}\n\nIt is possible to phrase the left-hand side of (\\ref{Eq:Bal2}) in terms of $C_L$ by making use of Oriani's theory of equilibrium,\n\\begin{equation}\n\\frac{\\mathcal{D}}{\\mathcal{D}_e} \\frac{d C_L}{dt}= \\mathcal{D} \\nabla^2 C_L - \\nabla \\cdot \\left( \\frac{\\mathcal{D} C_L \\bar{V}_H}{\\mathcal{R}\\mathcal{T}}  \\nabla \\sigma_H \\right)\n\\end{equation}\n\n\\noindent where an effective diffusion constant has been defined,\n\\begin{equation}\\label{Eq:Deff}\n\\mathcal{D}_e=\\mathcal{D} \\frac{C_L}{C_L + C_T (1 - \\theta_T)}\n\\end{equation}\n\nRegarding the boundary conditions, a constant hydrogen concentration $C_b$ is prescribed at the crack faces in the vast majority of hydrogen embrittlement studies. However, as noted by Turnbull \\cite{T15}, such scheme may oversimplify the electrochemistry-diffusion interface and the use of generalized boundary conditions is particularly recommended for materials with high hydrogen diffusivity. Here, we follow Mart\\'{\\i}nez-Pa\\~neda \\textit{et al.} \\cite{M16b} and adopt Dirichlet-type boundary conditions where the lattice hydrogen concentration at the crack faces depends on the hydrostatic stress. Hence, the lattice hydrogen concentration at the crack faces equals,\n\\begin{equation}\\label{eq:DISP}\nC_L=C_b \\exp \\left( \\frac{\\bar{V}_H \\sigma_H}{\\mathcal{R}\\mathcal{T}} \\right)\n\\end{equation}\n\n\\noindent which is equivalent to prescribing a constant chemical potential. To this end, a user subroutine DISP is employed in ABAQUS to relate the magnitude of $C_L$ to a nodal averaged value of the hydrostatic stress. Also, the domain where the boundary conditions are enforced changes with crack advance. Consequently, a multi-point constraint (MPC) subroutine is defined to update the boundary region throughout the analysis - see Section \\ref{Sec:FEM}.\\\\\n\nFinite deformation J2 plasticity theory is used to compute the two mechanical ingredients of the present hydrogen transport scheme, $\\varepsilon_p$ and $\\sigma_H$. We develop a fully coupled mass transport - continuum elastoplastic finite element framework that is solved in a monolithic way. Higher order elements are used, with nodal displacements and lattice hydrogen concentration being the primary variables. The numerical implementation is carried out in the well-known finite element package ABAQUS. To this end, a UMATHT subroutine is developed to exploit the analogy with heat transfer \\cite{B16,D16}. Thus, the energy balance for a stationary solid in the absence of heat sources is given by,\n\\begin{equation}\n\\int_V \\rho \\, \\dot{U} \\, \\textnormal{d}V - \\int_S q \\, \\textnormal{d}S = 0\n\\end{equation}\n\n\\noindent where $\\rho$ is the density, $q$ the heat flux per unit area of the solid and $\\dot{U}$ the material time rate of the internal energy, the latter being related to the temperature change through the specific heat capacity: $\\dot{U}=c_q \\dot{\\mathcal{T}}$. The similitude with (\\ref{Eq:MassBal}) is clear and an appropriate analogy can be easily established (see Table \\ref{Tab:DiffusionHeat}), enabling the use of the coupled temperature-displacement capabilities already available in ABAQUS.\n\n\\begin{table}[H]\n\\centering\n\\caption{Analogy between heat transfer and mass diffusion.}\n\\label{Tab:DiffusionHeat}\n   {\\tabulinesep=1.2mm\n   \\begin{tabu} {cc}\n       \\hline\n Heat transfer & Mass diffusion \\\\ \\hline\n $\\rho c_p \\frac{\\partial \\mathcal{T}}{\\partial t} + \\nabla q = 0$  &   $\\frac{\\partial C_i}{\\partial t} + \\nabla \\vect{J}=0$\\\\\n $\\dot{U}=c_p \\dot{\\mathcal{T}}$ & $\\frac{\\partial C}{\\partial t}=\\frac{\\partial (C_L + C_T)}{\\partial t}$  \\\\\n $\\mathcal{T}$ & $C_L$  \\\\\n $c_p$ & $\\mathcal{D}\/\\mathcal{D}_{e}$ \\\\\n $\\rho$ & 1  \\\\\\hline\n   \\end{tabu}}\n\\end{table}\n\n\\subsection{Cohesive zone model}\n\\label{Sec:CZM}\n\nA cohesive zone formulation will be employed to model crack initiation and subsequent growth. Based on the pioneering works by Dugdale \\cite{D60} and Barenblatt \\cite{B62}, cohesive zone models introduce the notion of a cohesive force ahead of the crack that prevents propagation. The micromechanisms of material degradation and failure are thus embedded into the constitutive law that relates the cohesive traction with the local separation. Damage is restricted to evolve along the pre-defined cohesive interface, and consequently, the numerical implementation is generally conducted by inserting cohesive finite elements in potential crack propagation paths. Hence, in the absence of body forces, the weak form of the equilibrium equations for a body of volume $V$ and external surface $S$ renders,\n\\begin{equation}\\label{Eq:CoheWeak}\n\\int_V \\vect{\\sigma} : \\delta \\vect{\\varepsilon}  \\, \\textnormal{d}V + \\int_{S_c} \\vect{T} \\cdot \\delta \\vect{\\Delta} \\, \\textnormal{d}S= \\int_S \\vect{t} \\cdot \\delta \\vect{u} \\, \\textnormal{d}S\n\\end{equation}\n\nHere, $\\vect{T}$ are the cohesive tractions and $S_c$ is the surface across which these tractions operate. The standard part of the mechanical equilibrium statement is characterized by the Cauchy stress tensor $\\vect{\\sigma}$, the work-conjugate strain tensor $\\vect{\\varepsilon}$, the external tractions $\\vect{t}$ and the displacement vector $\\vect{u}$; the latter being obtained by interpolating the global nodal displacement $\\vect{u}=\\vect{N}\\vect{U}$. The local nodal separation $\\tilde{\\vect{\\Delta}}$ is related to the local nodal displacement $\\tilde{\\vect{u}}$ by\n\\begin{equation}\n\\tilde{\\vect{\\Delta}} = \\vect{L} \\tilde{\\vect{u}}\n\\end{equation}\n\n\\noindent where $\\vect{L}$ is a local displacement-separation relation matrix. The separation along a cohesive surface element is interpolated from the nodal separation by means of standard shape functions,\n\\begin{equation}\n\\vect{\\Delta} = \\vect{N} \\tilde{\\vect{\\Delta}}\n\\end{equation}\n\n\\noindent and the global nodal displacement is related to the local nodal displacement by means of a rotational matrix:\n\\begin{equation}\n\\tilde{\\vect{u}} = \\vect{R} \\vect{U}\n\\end{equation}\n\nThe relationship between the local separation and the global nodal displacement can be then obtained by combining the previous equations,\n\\begin{equation}\n\\vect{\\Delta} = \\vect{B}_c \\vect{U}\n\\end{equation}\n\n\\noindent where $\\vect{B}_c$ is a global displacement-separation relation matrix: $\\vect{B}_c=\\vect{N} \\vect{R} \\vect{L}$. Thus, accounting for the classic finite element discretization in (\\ref{Eq:CoheWeak}) and requiring the variational statement to hold for any admissible field, it renders\n\\begin{equation}\n\\int_V \\vect{B}^T \\vect{\\mathscr{L}} \\vect{\\varepsilon}  \\, \\textnormal{d}V + \\int_{S_c} \\vect{B}_c^T \\vect{T} \\, \\textnormal{d}S= \\int_S \\vect{N}^T \\vect{t} \\, \\textnormal{d}S\n\\end{equation}\n\n\\noindent where $\\vect{\\mathscr{L}}$ is the elastoplastic constitutive matrix and $\\vect{B}$ the standard strain-displacement matrix. Considering the dependence of $\\vect{\\varepsilon}$ and $\\vect{T}$ on $\\vect{U}$,\n\\begin{equation}\n\\vect{U} \\left( \\int_V \\vect{B}^T \\vect{\\mathscr{L}} \\vect{B} \\, \\textnormal{d}V + \\int_{S_c} \\vect{B}_c^T \\frac{\\partial \\vect{T}}{\\partial \\vect{\\Delta}} \\vect{B}_c \\, \\textnormal{d}S \\right) = \\int_S \\vect{N}^T \\vect{t} \\, \\textnormal{d}S\n\\end{equation}\n\n\\noindent and the components of the classic finite element global system of equations can be readily identified. The stiffness matrix of the cohesive elements is therefore given by,\n\\begin{equation}\n\\vect{K}_c = \\int_{S_c} \\vect{B}_c^T \\frac{\\partial \\vect{T}}{\\partial \\vect{\\Delta}} \\vect{B}_c \\, \\textnormal{d}S\n\\end{equation}\n\n\\noindent which corresponds to the gradient of the internal cohesive force vector,\n\\begin{equation}\n\\vect{f}_c=\\int_{S_0} \\vect{B}_c^T \\vect{T} dS\n\\end{equation} \n\nThe pivotal ingredient of cohesive zone models is the traction-separation law that governs material degradation and separation. The exponentially decaying cohesive law proposed by Xu and Needleman \\cite{XN93} is here adopted. Focus will be placed on pure mode I problems and consequently, the constitutive equations related to the tangential separation will be omitted for the sake of brevity. As depicted in Fig. \\ref{fig:CoheLaw}, for a given shape of the traction-separation curve, the cohesive response can be fully characterized by two parameters, the cohesive energy $\\phi_n$ and the critical cohesive strength $\\sigma_{max,0}$.\\\\\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[scale=0.8]{CoheLaw.eps}\n\\caption{Traction-separation law characterizing the cohesive zone model in the absence of cyclic loading and hydrogen degradation.}\n\\label{fig:CoheLaw}\n\\end{figure}\n\nThe cohesive response is therefore characterized by the relation between the normal tractions ($T_n$) and the corresponding displacement jump ($\\Delta_n$) as,\n\\begin{equation}\nT_n=\\frac{\\phi_n}{\\delta_n} \\left( \\frac{\\Delta_n}{\\delta_n} \\right) \\exp \\left( - \\frac{\\Delta_n}{\\delta_n} \\right)\n\\end{equation}\n\n\\noindent with the normal work of separation $\\phi_n$ being given by,\n\\begin{equation}\n\\phi_n= \\exp(1) \\sigma_{max,0} \\delta_n\n\\end{equation}\n\n\\noindent where $\\delta_N$ is the characteristic cohesive length under normal separation. The effect of hydrogen in lowering the cohesive strength, and subsequently the fracture toughness, is captured here by employing the impurity-dependent cohesive law proposed by Serebrinsky et al. \\cite{S04}. Hence, a first-principles-based relation between the \\emph{current} cohesive strength $\\sigma_{max}$ and the original cohesive strength in the absence of hydrogen $\\sigma_{max,0}$ is defined,\n\\begin{equation}\n\\frac{\\sigma_{max}(\\theta_H)}{\\sigma_{max,0}}=1 - 1.0467 \\theta_H + 0.1687 \\theta_H^2\n\\end{equation}\n\n\\noindent where $\\theta_H$ is the hydrogen coverage, which is defined as a function of hydrogen concentration and Gibbs free energy difference between the interface and the surrounding material, as expressed in the Langmuir-McLean isotherm:\n\\begin{equation}\n\\theta_H=\\frac{C}{C+\\exp \\left( \\frac{- \\Delta g_b^0}{\\mathcal{R}\\mathcal{T}} \\right)}\n\\end{equation}\n\nA value of 30 kJ\/mol is assigned to the trapping energy $\\Delta g_b^0$ in \\citep{S04} from the spectrum of experimental data available. Thus, from first principles calculations of hydrogen atoms in bcc Fe, a quantum-mechanically informed traction-separation law can be defined as a function of the hydrogen coverage \\cite{S04} (see Fig. \\ref{fig:CoheLawH}).\\\\ \n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[scale=0.9]{CoheLawH.eps}\n\\caption{Effect of hydrogen coverage $\\theta_H$ on the traction-separation law characterizing the cohesive response.}\n\\label{fig:CoheLawH}\n\\end{figure}\n\nOn the other hand, cyclic damage is incorporated by means of the irreversible cohesive zone model proposed by Roe and Siegmund \\cite{RS03}. The model incorporates (i) loading-unloading conditions, (ii) accumulation of damage during subcritical cyclic loading, and (iii) crack surface contact. A damage mechanics approach is adopted to capture the cohesive properties degradation as a function of the number of cycles. A damage variable $D$ is defined so that it represents the effective surface density of micro defects in the interface. Consequently, an effective cohesive zone traction can be formulated: $\\tilde{\\vect{T}}=\\vect{T}\/(1-D)$. Subsequently, the current or effective cohesive strength $\\sigma_{max}$ is related to the initial cohesive strength $\\sigma_{max,0}$ as,\n\\begin{equation}\n\\sigma_{max}=\\sigma_{max,0} (1 - D)\n\\end{equation}\n\nA damage evolution law is defined so that it incorporates the relevant features of continuum damage approaches, namely: (i) damage accumulation starts if a deformation measure is greater than a critical magnitude, (ii) the increment of damage is related to the increment of deformation, and (iii) an endurance limit exists, bellow which cyclic loading can proceed infinitely without failure. From these considerations, cyclic damage evolution is defined as,\n\\begin{equation}\\label{Eq:Damage}\n\\dot{D}_c= \\frac{|\\dot{\\Delta}_n|}{\\delta_{\\Sigma}} \\left[ \\frac{T_n}{\\sigma_{max}} - \\frac{\\sigma_f}{\\sigma_{max,0}} \\right] H \\left( \\bar{\\Delta}_n -  \\delta_n \\right)\n\\end{equation}\n\n\\noindent with $\\bar{\\Delta}_n=\\int |\\dot{\\Delta}_n| dt$ and $H$ denoting the Heaviside function. Two new parameters have been introduced: $\\sigma_f$, the cohesive endurance limit and $\\delta_{\\Sigma}$, the accumulated cohesive length - used to scale the normalized increment of the effective material separation. The modeling framework must also incorporate damage due to monotonic loading; as a consequence, the damage state is defined as the maximum of the cyclic and monotonic contributions,\n\\begin{equation}\nD= \\int \\textnormal{max} \\left( \\dot{D}_c, \\dot{D}_m \\right) dt\n\\end{equation}\n\n\\noindent being the latter characterized as:\n\\begin{equation}\n\\dot{D}_m = \\frac{ \\left. \\textnormal{max} \\left( \\Delta_n \\right) \\right|_{t_i} -  \\left. \\textnormal{max} \\left( \\Delta_n \\right) \\right|_{t_{i-1}}}{4 \\delta_n}\n\\end{equation}\n\n\\noindent and updated only when the largest stored value of $\\Delta_n$ is greater than $\\delta_N$. Here, $t_{i-1}$ denotes the previous time increment and $t_i$ the current one. In addition to damage evolution, the cohesive response must be defined for the cases of unloading\/reloading, compression, and contact between the crack faces. Unloading is defined based on the analogy with an elastic-plastic material undergoing damage. Thereby, unloading takes place with the stiffness of the cohesive zone at zero separation, such that\n\\begin{equation}\nT_n=T_{max} + \\left( \\frac{\\exp(1) \\sigma_{max}}{\\delta_n} \\right) \\left( \\Delta_n - \\Delta_{max} \\right)\n\\end{equation}\n\n\\noindent where $\\Delta_{max}$ is the maximum separation value that has been attained and $T_{max}$ its associated traction quantity. Compression behavior applies when the unloading path reaches $\\Delta_n=0$ at $T_n < 0$. In such circumstances, the cohesive response is given by,\n\\begin{align}\nT_n = & \\frac{\\phi_n}{\\delta_n} \\left( \\frac{\\Delta_n}{\\delta_n} \\right) \\exp \\left( - \\frac{\\Delta_n}{\\delta_n} \\right) + T_{max} - \\sigma_{max} \\exp(1) \\frac{\\Delta_{max}}{\\delta_n} \\nonumber \\\\\n& + \\alpha \\sigma_{max,0} \\exp(1) \\frac{\\Delta_n}{\\delta_n} \\exp \\left( - \\frac{\\Delta_n}{\\delta_n} \\right)\n\\end{align}\n\n\\noindent being $\\alpha$ a penalty factor that is taken to be equal to 10, following \\cite{RS03}. Contact conditions are enforced if $\\Delta_n$ is negative and the cohesive element has failed completely ($D=1$). At this instance the cohesive law renders,\n\\begin{equation}\nT_n= \\alpha \\sigma_{max,0} \\exp(1) \\exp \\left( - \\frac{\\Delta_n}{\\delta_n} \\right) \\frac{\\Delta_n}{\\delta_n} \n\\end{equation}\n\n\\noindent where friction effects have been neglected. Fig. \\ref{fig:CoheLawF} shows a representative response obtained by applying a stress-controlled cyclic loading $\\Delta \\sigma \/ \\sigma_{max,0}=1$ with a zero stress ratio. The accumulated separation increases with the number of loading cycles, so that it becomes larger than $\\delta_n$ and damage starts to play a role, lowering the stiffness and the cohesive strength.\\\\\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[scale=0.9]{CoheLawF.eps}\n\\caption{Cohesive response under stress-controlled cyclic loading conditions.}\n\\label{fig:CoheLawF}\n\\end{figure}\n\nThis novel cyclic- and hydrogen concentration-dependent cohesive zone framework is implemented in ABAQUS by means of a user element UEL subroutine. The code can be downloaded from www.empaneda.com\/codes and is expected to be helpful to both academic researchers and industry practitioners.\\\\\n\nIn some computations, numerical convergence is facilitated by employing the viscous regularization technique proposed by Gao and Bower \\cite{GB04}. Such scheme leads to accurate results if the viscosity coefficient, $\\xi$, is sufficiently small \\cite{Y16}. A sensitivity study has been conducted in the few cases where viscous regularization was needed; values of $\\xi$ on the order of $10^{-6}$ have proven to be appropriate for the boundary value problem under consideration. Other approaches to overcome snap-back instabilities, less suitable for cyclic loading, include the use of explicit finite element solution schemes \\cite{S16} or determining the equilibrium path for a specified crack tip opening by means of control algorithms \\cite{T76,SL04,M17b}.\n\n\\subsection{Finite element implementation}\n\\label{Sec:FEM}\n\nThe aforementioned mechanical-diffusion-cohesive numerical framework is implemented in the commercial finite element package ABAQUS. Fortran modules are widely employed to transfer information between the different user subroutines. Thus, as described in Fig. \\ref{fig:Abaqus}, a user material UMAT subroutine is developed to characterize the mechanical response by means of a finite strain version of conventional von Mises plasticity. The nodal averaged value of the hydrostatic stress at the crack faces is then provided to a DISP subroutine, so as to prescribe a more realistic $\\sigma_H$-dependent lattice hydrogen concentration. The hydrostatic stress gradient is computed by means of linear shape functions and, together with the equivalent plastic strain, is afterward given as input to the UMATHT subroutine to capture the effects of chemical expansion and trapping. The UMATHT subroutine provides the cohesive elements with the diffusible concentration of hydrogen in their adjacent continuum element. The damage variable is then transferred from the user elements to the MPC subroutine to keep track of the crack extension. Multi-point constraints have been defined between the nodes ahead of the crack and a set of associated dummy nodes that are activated as the crack advances. Hydrogen diffusion is assumed to be instantaneous, such that the lattice hydrogen concentration at the boundary is immediately prescribed when a new portion of crack surface is available.\\\\\n\n\\begin{figure}[H]\n  \\makebox[\\textwidth][c]{\\includegraphics[width=1.2\\textwidth]{Abaqus.eps}}%\n  \\caption{Schematic overview of the relations between the different Abaqus subroutines.}\n  \\label{fig:Abaqus}\n\\end{figure}\n\nHigher order elements are used in all cases: 8-node quadrilateral elements with reduced integration are employed to model the bulk response, and crack initiation and growth are captured by 6-node quadrilateral cohesive elements with 12 integration points. Results post-processing is carried out in MATLAB by making use of \\emph{Abaqus2Matlab} \\cite{P17}, a novel tool that connects the two aforementioned well-known software suites. \n\n\\section{Results}\n\\label{Sec:Results}\n\nWe investigate the pernicious effect of hydrogen in fatigue crack growth, of great relevance in both energy storage and transport. The synergistic interaction of cyclic plastic deformation and local hydrogen uptake is particularly detrimental, with catastrophic failure being observed in cases where hydrogen-assisted cracking is negligible under monotonic loading \\cite{G90}.\\\\\n\nThe boundary layer model employed by Sofronis and McMeeking \\cite{SM89} is taken as a benchmark. Hence, hydrogen transport and subsequent cracking are investigated in an iron-based material with a diffusion coefficient of $\\mathcal{D}=0.0127$ mm$^2$\/s, Young's modulus of $E=207$ GPa, Poisson's ratio of $\\nu=0.3$ and initial yield stress of $\\sigma_Y=250$ MPa. Work hardening is captured by means of the following isotropic power law,\n\\begin{equation}\n\\sigma = \\sigma_Y \\left(1 + \\frac{E \\varepsilon_p}{\\sigma_Y} \\right)^\\mathcal{N}\n\\end{equation}\n\n\\noindent with the strain hardening exponent being equal to $\\mathcal{N}=0.2$. Isotropic hardening has been adopted to reproduce the conditions of \\cite{SM89}, but one should note that other plastic flow models can be easily incorporated; the use of non-linear kinematic hardening laws is particularly convenient to appropriately capture the Bauschinger effect under low load ratios. As described in Fig. \\ref{fig:Mesh1}, the crack region is contained within a circular zone and a remote Mode I load is applied by prescribing the displacements of the nodes at the outer boundary,\n\\begin{equation}\nu \\left( r, \\theta \\right) = K_I \\frac{1+\\nu}{E} \\sqrt{\\frac{r}{2 \\pi}} \\cos \\left( \\frac{\\theta}{2} \\right) \\left(3 - 4 \\nu - \\cos \\theta \\right)\n\\end{equation}\n\\begin{equation}\nv \\left( r, \\theta \\right) = K_I \\frac{1+\\nu}{E} \\sqrt{\\frac{r}{2 \\pi}} \\sin \\left( \\frac{\\theta}{2} \\right) \\left(3 - 4 \\nu - \\cos \\theta \\right)\n\\end{equation}\n\n\\noindent where $u$ and $v$ are the horizontal and vertical components of the displacement boundary condition, $r$ and $\\theta$ the radial and angular coordinates of each boundary node in a polar coordinate system centered at the crack tip, and $K_I$ is the applied stress intensity factor that quantifies the remote load in small scale yielding conditions. The lattice hydrogen concentration is prescribed in the crack surface as a function of $\\sigma_H$ and the boundary concentration in the absence of hydrostatic stresses $C_b$. Following \\cite{SM89}, an initial bulk concentration equal to $C_b$ is also defined in the entire specimen at the beginning of the analysis. Only the upper half of the circular domain is modeled due to symmetry and the outer radius is chosen to be significantly larger than the initial crack tip blunting radius. As shown in Fig. \\ref{fig:Mesh1}, a very refined mesh is used, with the characteristic element size in the vicinity of the crack, $l_e$, being significantly smaller than a reference plastic length,\n\\begin{equation}\nR_0 = \\frac{1}{3 \\pi \\left( 1 - \\nu^2 \\right)} \\frac{E \\phi_n}{\\sigma_Y^2}\n\\end{equation}\n\n\\noindent ($l_e<2000R_0$). A sensitivity study is conducted to ensure that the mesh resolves the cohesive zone size - approximately 14000 quadrilateral 8-node plane strain elements are employed. The modeling framework is suitable for both low and high cycle fatigue, with computations of $10^4$ cycles (with at least 10 load increments per cycle) running overnight on a single core.\\\\\n\n\\begin{figure}[H]\n  \\makebox[\\textwidth][c]{\\includegraphics[width=1.2\\textwidth]{Mesh1.eps}}%\n  \\caption{General and detailed representation of the finite element mesh employed for\nthe boundary layer model. Mechanical and diffusion boundary conditions are shown superimposed.}\n  \\label{fig:Mesh1}\n\\end{figure}\n\nWe first validate the coupled mechanical-diffusion implementation by computing crack tip fields under monotonic loading conditions in the absence of crack propagation. Thus, the load is increased from zero at a rate of 21.82 MPa$\\sqrt{mm} \\,$s$^{-1}$ for 130 s and held fixed afterward, when the crack opening displacement is approximately 10 times the initial blunting radius $b=5b_0=10r_0$. Fig. \\ref{fig:SH} shows the estimated hydrostatic stress distribution along with the predictions by Sofronis and McMeeking \\cite{SM89} (symbols); results are shown along the extended crack plane with the distance to the crack tip normalized by the current crack tip opening $b$. A very good agreement is observed, verifying the finite strains J2 plasticity implementation. Fig. \\ref{fig:CLCT} shows the results obtained for the lattice and trapped hydrogen concentrations for a boundary concentration of $C_b=2.08 \\cdot 10^{12}$ H atoms\/mm$^3$ at 130 s and after reaching steady-state conditions. The quantitative response described by the lattice hydrogen concentration when accounting for the dilatation of the lattice significantly differs to that obtained prescribing a constant $C_L$, as highlighted by Di Leo and Anand \\cite{DA13} in the context of their constant lattice chemical potential implementation. The results achieved by means of the present $\\sigma_H$-dependent Dirichlet scheme accurately follow the analytical steady-state solution for the distribution of the lattice hydrogen concentration ahead of the crack. On the other hand, $C_T$ shows a high peak at the crack tip and negligible sensitivity to the diffusion time (the curves for 130 s and steady state fall on top of each other); this is due to the governing role of plastic deformation as a result of the direct proportional relationship between $C_T$ and $N_T$.\\\\\n\n\\begin{figure}[H]\n        \\centering\n        \\begin{subfigure}[h]{1\\textwidth}\n                \\centering\n                \\includegraphics[scale=0.8]{SH.eps}\n                \\caption{}\n                \\label{fig:SH}\n        \\end{subfigure}\n        \n        \\begin{subfigure}[h]{1\\textwidth}\n                \\centering\n                \\includegraphics[scale=0.8]{CLCT.eps}\n                \\caption{}\n                \\label{fig:CLCT}\n        \\end{subfigure}\n       \n        \\caption{Crack tip fields for a stationary crack in an iron-based material under monotonic loading conditions, (a) normalized hydrostatic stress distribution for $K_I=2836.7$ MPa$\\sqrt{mm}$ and (b) lattice and trap sites hydrogen concentrations at steady state and after 130 s.}\\label{fig:CrackTipFields}\n\\end{figure}\n\nEnvironmentally assisted fatigue is subsequently investigated by scaling in time the external load by a sinusoidal function. The cyclic boundary conditions prescribed are characterized by the load amplitude $\\Delta K = K_{max} - K_{min}$ and the load ratio $R=K_{min}\/K_{max}$. An initial prestressing is defined, such that the mean load equals the load amplitude, and both $R$ and $\\Delta K$ remain constant through the analysis.\\\\\n\nFollowing \\cite{RS03}, the accumulated cohesive length in (\\ref{Eq:Damage}) is chosen to be $\\delta_{\\Sigma}=4 \\delta_n$ and the endurance coefficient $\\sigma_f \/ \\sigma_{max,0}=0.25$. The initial cohesive strength is assumed to be equal to $\\sigma_{max,0}=3.5 \\sigma_Y$ based on the seminal work by Tvergaard and Hutchinson \\cite{TH92}. One should, however, note that such magnitude is intrinsically associated with the stress bounds of conventional plasticity - more realistic values can be obtained if the role of the increased dislocation density associated with large gradients in plastic strain near the crack tip is accounted for \\cite{M17b,MB15}. A reference stress intensity factor,\n\\begin{equation}\nK_0 = \\sqrt{ \\frac{E \\phi_n}{(1-\\nu^2)}}\n\\end{equation}\n\n\\noindent is defined to present the results.\\\\\n\nThe capacity of the model to capture the sensitivity of fatigue crack growth rates to a hydrogenous environment is first investigated by computing the crack extension $\\Delta a$ as a function of the number of cycles for different values of $C_L$ at the boundary. Figure \\ref{fig:InfluenceH} shows the results obtained for a load ratio of $R=0.1$ and frequency of 1 Hz. The magnitude of the load ratio is appropriate for applications in the context of hydrogen-fueled vehicles, where load ratios between $R=0.1$ and $R=0.2$ accurately characterize the mechanical stresses resulting from filling cycles. Results reveal a strong influence of the environment, with crack propagation rates increasing significantly with the hydrogen content; the model appropriately captures the deleterious effect of hydrogen on crack growth resistance under cyclic loading conditions.\\\\\n\n\\begin{figure}[H]\n        \\centering\n        \\begin{subfigure}[h]{1\\textwidth}\n                \\centering\n                \\includegraphics[scale=0.8]{FigHa.eps}\n                \\caption{}\n                \\label{fig:InfluenceHa}\n        \\end{subfigure}\n        \n        \\begin{subfigure}[h]{1\\textwidth}\n                \\centering\n                \\includegraphics[scale=0.8]{FigHb.eps}\n                \\caption{}\n                \\label{fig:InfluenceHb}\n        \\end{subfigure}\n       \n        \\caption{Predicted influence of the environment on (a) crack extension versus number of cycles for $\\Delta K\/K_0=0.1$ and (b) fatigue crack growth rate versus load amplitude. Results have been obtained for an iron-based material under a load ratio of $R=0.1$ and frequency of $f=1$ Hz}\\label{fig:InfluenceH}\n\\end{figure}\n\nLattice hydrogen concentrations at the boundary range from 1 wppm ($4.68 \\cdot 10^{15}$ H atoms\/mm$^3$), which corresponds to a 3\\% NaCl aqueous solution \\cite{G86}, to 0.1 wppm. The important role of hydrogen in the fatigue crack growth behavior can be clearly observed in the crack growth versus crack amplitude curves (Fig. \\ref{fig:InfluenceHb}). By making use of the well-known Paris equation \\cite{P61}:\n\\begin{equation}\n\\frac{da}{dN}=\\mathcal{C} \\Delta K^m\n\\end{equation}\n\n\\noindent one can easily see that $\\mathcal{C}$ significantly increases with the environmental hydrogen content, in agreement with the experimental trends. On the other hand, results render a Paris exponent that shows little sensitivity to the hydrogen concentration, falling in all cases within the range of experimentally reported values for metals in inert environments ($m \\approx 4$). The cycle-dependent contribution of the environment manifests significantly, while the influence of $C_b$ as $\\Delta K$ increases is governed by a trade-off between larger levels of equivalent plastic strain (increasing $C_T$ and subsequently $C$) and shorter diffusion times due to greater crack growth rates. Thus, for a given frequency, the effect of hydrogen on the slope of the $da\/dN$ versus $\\Delta K$ curve depends heavily on the diffusion and mechanical properties of the material under consideration. The sensitivity of a steel to hydrogen embrittlement is therefore bounded between two limit cases: (a) \\emph{slow tests}, where the testing time significantly exceeds the diffusion time for hydrogen within the specimen, and (b) \\emph{fast tests}, where the testing time is much less than the diffusion time. In the former bound, hydrogen atoms accumulate in the fracture process zone and the distribution of lattice hydrogen concentration is governed by Eq. (\\ref{eq:DISP}). For sufficiently rapid tests the initial (pre-charged) hydrogen concentration and the contributions from reversible microstructural traps dominate the response. As a consequence, experiments reveal a relevant increase in $da\/dN$ with decreasing frequency until an upper bound is reached where the load-cycle duration is sufficient to allow hydrogen to diffuse and fully saturate the crack tip fracture process zone \\cite{M12}.\\\\\n\nWe subsequently investigate the influence of the loading frequency. A normalized frequency is defined as,\n\\begin{equation}\n\\bar{f}=\\frac{f R_0^2}{\\mathcal{D}}\n\\end{equation}\n\n\\noindent so as to quantify the competing influence of test and diffusion times. Fig. \\ref{fig:InfluenceFreqIronA} shows crack growth resistance curves obtained for the iron-based material under consideration in the aforementioned asymptotic limits - \\emph{slow tests} ($\\bar{f} \\to 0$) and \\emph{fast tests} ($\\bar{f} \\to \\infty$). In agreement with expectations, crack propagation is enhanced by larger testing times but results reveal very little sensitivity to the loading frequency, as opposed to experimental observations. Fig. \\ref{fig:InfluenceFreqIronB} provides the basis for the understanding of the small susceptibility of crack growth rates to the loading frequency; the $C_L$ elevation in the $\\sigma_H$-dominated case is less than 10\\% of the lattice hydrogen concentration in the \\emph{fast test}.\\\\\n\n\\begin{figure}[H]\n        \\centering\n        \\begin{subfigure}[h]{1\\textwidth}\n                \\centering\n                \\includegraphics[scale=0.8]{FigFreqIron.eps}\n                \\caption{}\n                \\label{fig:InfluenceFreqIronA}\n        \\end{subfigure}\n        \n        \\begin{subfigure}[h]{1\\textwidth}\n                \\centering\n                \\includegraphics[scale=0.8]{CLIronFreq.eps}\n                \\caption{}\n                \\label{fig:InfluenceFreqIronB}\n        \\end{subfigure}\n       \n        \\caption{Influence of the frequency in an iron-based material: (a) crack extension versus number of cycles, and (b) lattice hydrogen distribution ahead of the crack at the maximum $\\Delta K$ and for a crack extension of $\\Delta a \/R_0=0.8$. Results have been obtained for $\\Delta K\/K_0=0.2$, under a load ratio of $R=0.1$ and an external hydrogen concentration of $C_b=1$ wppm.}\\label{fig:InfluenceFreqIron}\n\\end{figure}\n\nThe difference between the two limiting cases is governed by the exponential dependence of the lattice hydrogen concentration to hydrostatic stresses ahead of the crack, given the independence of the trap density to the loading frequency in Sofronis and McMeeking's \\cite{SM89} framework. Since the maximum level of $\\sigma_H$ is load-independent in finite strain J2 plasticity \\cite{M77}, we investigate the influence of yield strength, strain hardening and triaxiality conditions in providing a response closer to the experimental observations.\\\\\n\nThe role of the yield strength is first investigated by considering a high-strength steel with $\\sigma_Y=1200$ MPa and otherwise identical properties as the iron-based material assessed so far. As shown in Fig. \\ref{fig:InfluenceFreqHSA}, a considerably larger effect of the loading frequency is observed, even without the need of considering the two limiting cases. The lattice hydrogen concentration ahead of the crack tip is shown in Fig. \\ref{fig:InfluenceFreqHSB}; results reveal a much larger stress elevation when compared to the low-strength case (Fig. \\ref{fig:InfluenceFreqIronB}).\\\\ \n\n\\begin{figure}[H]\n        \\centering\n        \\begin{subfigure}[h]{1\\textwidth}\n                \\centering\n                \\includegraphics[scale=0.8]{FigFreqHS.eps}\n                \\caption{}\n                \\label{fig:InfluenceFreqHSA}\n        \\end{subfigure}\n        \n        \\begin{subfigure}[h]{1\\textwidth}\n                \\centering\n                \\includegraphics[scale=0.8]{CLHSFreq.eps}\n                \\caption{}\n                \\label{fig:InfluenceFreqHSB}\n        \\end{subfigure}\n       \n        \\caption{Influence of the frequency in a high-strength steel: (a) crack extension versus number of cycles, and (b) lattice hydrogen distribution ahead of the crack at the maximum $\\Delta K$ and for a crack extension of $\\Delta a \/R_0=0.6$. Results have been obtained for $\\Delta K\/K_0=0.2$, under a load ratio of $R=0.1$ and an external hydrogen concentration of $C_b=1$ wppm.}\\label{fig:InfluenceFreqHstrength}\n\\end{figure}\n\nThe increase in fatigue crack growth rates with decreasing frequency is quantified in Fig. \\ref{fig:FreqHS}. Again, the model qualitatively captures the main experimental trends; low loading frequencies enable hydrogen transport to the fracture process zone, augmenting crack propagation rates. This $da\/dN$-dependence with frequency reaches a plateau when approaching the two limiting responses, a clear transition between the upper and lower bounds can be observed. However, crack growth rates on the lower frequency bound are less than 1.5 times the values attained when $da\/dN$ levels out at high loading frequencies; these quantitative estimations fall significantly short of reaching the experimentally reported differences. A 5-10 times crack growth rate elevation has been observed when decreasing frequency in a mid-strength martensitic SCM435 steel \\cite{T07}, and similar data have been obtained for a 2.25Cr\u20131Mo (SA542-3) pressure vessel steel \\cite{SR82} and an age-hardened 6061 aluminum alloy \\cite{K07}, among many other (see, e.g., the review by Murakami \\cite{M12}).\\\\\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[scale=0.9]{FreqHS.eps}\n\\caption{Fatigue crack growth rate versus normalized frequency in a high-strength steel. Results have been obtained for $\\Delta K\/K_0=0.2$, under a load ratio of $R=0.1$ and an external hydrogen concentration of $C_b=1$ wppm.}\n\\label{fig:FreqHS}\n\\end{figure}\n\nThe gap between the maximum and minimum $da\/dN$ levels can also be affected by the strain hardening of the material under consideration. We, therefore, estimate the fatigue crack growth rates as a function of the loading frequency for three different strain hardening exponents. As shown in Fig. \\ref{fig:N}, higher values of $\\mathcal{N}$ lead to higher crack propagation rates. This comes as no surprise as larger strain hardening exponents translate into higher stresses. However, the $da\/dN$-elevation is not very sensitive to the range of loading frequencies examined. The effect of the stress elevation due to larger $\\mathcal{N}$ values could be attenuated by the intrinsic reduction of the plastic strain contribution to $N_T$. One should, however, note that, for the present choice of cohesive parameters, cracking takes place without significant plastic deformation. A different choice will probably increase the differences between the two limiting cases but highly unlikely to the level required to attain a quantitative agreement with the experiments.\\\\\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[scale=0.9]{N.eps}\n\\caption{Fatigue crack growth rate versus normalized frequency in high-strength steel for different strain hardening exponents. Results have been obtained for $\\Delta K\/K_0=0.2$, under a load ratio of $R=0.1$ and an external hydrogen concentration of $C_b=1$ wppm.}\n\\label{fig:N}\n\\end{figure}\n\nCrack tip constraint conditions are also expected to play a role in augmenting crack growth rates in sufficiently slow tests. Here, we make use of the elastic $T$-stress \\cite{BH91} to prescribe different triaxiality conditions by means of what is usually referred to as a modified boundary layer. Hence, the displacements at the remote boundary now read,\n\\begin{equation}\nu \\left( r, \\theta \\right) = K_I \\frac{1+\\nu}{E} \\sqrt{\\frac{r}{2 \\pi}} \\cos \\left( \\frac{\\theta}{2} \\right) \\left(3 - 4 \\nu - \\cos \\theta \\right) + T \\frac{1-\\nu^2}{E} r \\cos \\theta\n\\end{equation}\n\\begin{equation}\nv \\left( r, \\theta \\right) = K_I \\frac{1+\\nu}{E} \\sqrt{\\frac{r}{2 \\pi}} \\sin \\left( \\frac{\\theta}{2} \\right) \\left(3 - 4 \\nu - \\cos \\theta \\right) - T \\frac{\\nu (1+\\nu)}{E}  r \\sin \\theta\n\\end{equation}\n\nFig. \\ref{fig:Tstress} shows the sensitivity of $da\/dN$ to the loading frequency under different constraint conditions. We restrict our attention to positive values of the $T$-stress, as lower triaxialities will not contribute to increasing crack growth rates in the lower frequency bound. Results reveal a substantial increase of $da\/dN$ with increasing crack tip constraint. However, the influence on the ratio between the crack propagation rates for \\emph{slow} and \\emph{fast} tests is almost negligible.\\\\\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[scale=0.9]{Tstress.eps}\n\\caption{Fatigue crack growth rate versus normalized frequency in high-strength steel for different constraint conditions. Results have been obtained for $\\Delta K\/K_0=0.2$, under a load ratio of $R=0.1$ and an external hydrogen concentration of $C_b=1$ wppm.}\n\\label{fig:Tstress}\n\\end{figure}\n\nResults provide a mechanistic interpretation of the reduction in fatigue crack resistance with decreasing frequency observed in the experiments. By properly incorporating the kinetics of hydrogen uptake into the fracture process zone, model predictions can be employed to identify the critical frequency above which the time per load cycle is insufficient for diffusible hydrogen to degrade the crack growth resistance. Accurate estimations are however hindered by the lack of quantitative agreement with experimental data regarding the impact of loading frequency on $da\/dN$. Tests conducted at the low-frequency bound lead to crack growth rates that are 5-10 times larger than the values attained in experiments with a duration much shorter than the diffusion time. Such differences cannot be attained in the framework of conventional J2 plasticity, where the peak $\\sigma_H$ (on the order of $5 \\sigma_Y$) is insufficient to draw in sufficient levels of hydrogen to cause a 5-fold increase in $da\/dN$. Crack growth rates at low loading frequencies increase with yield strength, material hardening and constraint conditions, but not even the most critical combination of these parameters appears to provide a quantitative agreement with a phenomenon that is observed in a wide range of metallic alloys and testing configurations. Additional sources of stress elevation are therefore needed to provide a reliable characterization of environmentally assisted fatigue for different frequencies. One possibility lies on the large gradients of plastic strain present in the vicinity of the crack, which exacerbate dislocation density and material strength. Geometrically necessary dislocations (GNDs) arise in large numbers to accommodate lattice curvature due to non-uniform plastic deformation, and act as obstacles to the motion of \\emph{conventional} statistically stored dislocations. Strain gradient plasticity theories have been developed to extend plasticity theory to the small scales by incorporating this dislocation storage phenomenon that significantly contributes to material hardening (see \\cite{M16c} and references therein). Gradient plasticity models have been consistently used to characterize behavior at the small scales involved in crack tip deformation, predicting a much higher stress level than classic plasticity formulations (see, e.g., \\cite{MN16,M17}). This stress elevation due to dislocation hardening has proven to play a fundamental role in fatigue \\cite{BS08,BS08b} and hydrogen-assisted cracking \\cite{M16,MB17}.\n\n\\section{Conclusions}\n\\label{Concluding remarks}\n\nWe propose a predictive cohesive modeling framework for corrosion fatigue. The model is grounded on the mechanism of hydrogen embrittlement, which governs fatigue crack initiation and subsequent propagation in a wide range of metallic alloys exposed to gasses and electrolytes. Mechanical loading and hydrogen transport are coupled through lattice dilatation due to hydrostatic stresses and the generation of traps by plastic straining. An irreversible cohesive zone model is employed to capture material degradation and failure due to cyclic loads. The impact of the hydrogen coverage in the cohesive traction is established from first principles quantum mechanics. Finite element analysis of a propagating crack reveals a relevant increase in crack growth rates with (i) hydrogen content in the surrounding environment and (ii) decreasing load frequency; in agreement with experimental observations. A robust and appropriate numerical model for hydrogen-assisted fatigue opens up many possibilities, enabling rapid predictions that could be key to risk quantification in industrial components. Moreover, important insight can be gained into the mechanisms at play, identifying the relevant variables and their critical magnitudes for a given material, environment and loading scenario.\\\\\n\nThe influence of the yield strength, work hardening and constraint conditions is extensively investigated aiming to quantitatively reproduce the relation between the loading frequency and the crack growth rates observed in the experiments. Results reveal the need to incorporate additional sources of stress elevation to sufficiently enhance hydrogen uptake into the fracture process zone. Future work will focus on extending the present scheme to encompass the role of geometrically necessary dislocations through strain gradient plasticity formulations. \n \n\\section{Acknowledgments}\n\\label{Acknowledge of funding}\n\nThe authors gratefully acknowledge financial support from the Ministry of Economy and Competitiveness of Spain through grant MAT2014-58738-C3. E. Mart\\'{\\i}nez-Pa\\~neda additionally acknowledges financial support from the People Programme (Marie Curie Actions) of the European Union's Seventh Framework Programme (FP7\/2007-2013) under REA grant agreement n$^{\\circ}$ 609405 (COFUNDPostdocDTU).\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nDeep inelastic production of heavy quarks at HERA is important for \nseveral reasons:\n\\begin{itemize}\n\\item At small $x$ ($x \\approx 10^{-3}$) the charm contribution \n$F_2^c$ to $F_2^{ep}$ amounts to $20 \\%$--$30 \\%$ \\cite{adloff96} \nmaking a reliable theoretical treatment of $F_2^c$ necessary for \na precision measurement of $F_2^{ep}$.\n\\item The bulk of heavy quarks is produced via the photon gluon \nfusion mechanism \\cite{adloff96,zeus97} providing the possibility to \nconstrain the gluon distribution $g(y,\\mu^2)$ in the proton \n\\cite{vogt96,laenen96,daum96}.\n\\item By measuring differential distributions, the charm production \nmechanism can be tested, and it can be studied whether and when the \ncharm quark behaves like a massless parton \\cite{laenen96,daum96}.\n\\end{itemize}\n\nIn order to estimate radiative corrections to heavy quark production, \nwe employ the well-known leading log approximation (LLA) \n\\cite{km88,kms91,jbl90,jbl95}.\nSee also ref.~\\cite{jbl91} where radiative corrections to heavy\nflavour production have been studied in the Weisz\\\"acker-Williams\napproximation (WWA) using hadronic variables.\nThe ${\\cal O}(\\alpha)$ (QED-)corrections to \nthe process $e+g\\longrightarrow e+c+\\bar{c}$ are shown in \nfig.~\\ref{Feynman} (not shown are crossed and virtual diagrams).\nThe main part of the corrections comes from bremsstrahlung from the \nelectron line (plus corresponding virtual corrections), see a) and b). \nThese two contributions are enhanced by a large collinear logarithm \n$L_e=\\ln(Q^2\/m_e^2)$. In the same way, calculating  c), d), and e), \none gets logarithms $L_c=\\ln(Q^2\/m_c^2)$ which are much smaller than \n$L_e$ because of the large charm mass $m_c$, compared to the electron \nmass $m_e$. For example, for $Q^2=10$ GeV$^2$, one finds \n$L_e \\approx 17.5>>L_c\\approx 1.5$ ($m_c=1.5$ GeV).\nDiagrams d) and e) do not contribute for another reason: the photons \nare not resolved from the final state jets in general.\n(Only a light intermediary quark in diagram c) could give rise for a \ncollinear singularity, which would have to be handled the same way as \nthe corresponding collinear gluon emission, leading to a negligible \n(at least in LLA) modification of the parton distributions.)\nThus, we only take the leptonic corrections a) and b) \n(in ${\\cal O}(\\alpha)$-LLA) into consideration.\n\nThe paper is organized as follows: In Sect.~2 the formulae needed to \ncalculate radiative corrections to neutral current heavy quark \nproduction in ${\\cal O}(\\alpha)$-LLA are collected. \nIn Sect.~3 numerical results are presented.\nFinally, the main results are briefly summarized in Sect.~4.\n\\section{Formalism}\nWe calculate the ${\\cal O}(\\alpha)$ QED-corrections to deep inelastic \nproduction of heavy quarks in leading log approximation (LLA) \n\\cite{kms91, jbl90}.\nFollowing the formalism of ref.~\\cite{kms91}, the cross section for \ndeep inelastic production of charm quarks via the photon gluon fusion \n(PGF) mechanism ($e+g\\rightarrow e+c+\\bar{c}$) reads:\n\\begin{equation}\nd\\sigma(ep\\rightarrow ec\\bar{c}X)=\\int_0^1 dz_1\\int_0^1dz_2\n\\int_0^1dz_3 D_{e\/e}(z_1,Q^2)\ng(z_2,\\mu^2)\\bar{D}_{e\/e}(z_3,Q^2)\nd\\hat{\\sigma}(e+g\\rightarrow e+c+\\bar{c})\\ .\n\\label{wq}\n\\end{equation}\nThe functions $D_{e\/e}$, $\\bar{D}_{e\/e}$ contain the factorized mass \nsingularities from the electron line and are, up to ${\\cal O}(\\alpha)$, \ngiven by \\cite{kms91}\n\\begin{eqnarray}\nD_{e\/e}(x,Q^2) = \\bar{D}_{e\/e}(x,Q^2)= \\delta(1-x) + \n\\left(\\frac{\\alpha L_e}{2\\pi}\\right)P_{ee}(x)\n\\label{dee}\n\\end{eqnarray}\nwith \n\\begin{equation}\nL_e\\equiv \\ln \\frac{Q^2}{m_e^2}\n\\end{equation}\nand the splitting function\n\\begin{equation}\nP_{ee}(x) =  \\left[\\frac{1+x^2}{1-x}\\right]_{+}= \n\\frac{1+x^2}{(1-x)_{+}}+\\frac{3}{2}\\delta(1-x).\n\\end{equation}\nThe $(+)$-distribution is defined by\n\\begin{equation}\n\\int_x^1\\ dz\\ P_{ee}(z)\\ f(z)=\\int_x^1\\ dz\\ \\frac{1+z^2}{1-z}\\ \n(f(z)-f(1))-\\int_0^x\\ dz\\ \\frac{1+z^2}{1-z}\\ f(1)\\ .\n\\end{equation}\n$D_{e\/e}(z_1,Q^2)$ can be interpreted as the probability of finding an \nelectron with momentum fraction $z_1$ \ninside the incoming electron, $\\bar{D}_{e\/e}(z_3,Q^2)$ describes the \nfragmentation of the scattered electron \ninto the observed outgoing electron with momentum fraction $z_3$, and \n$g(z_2,\\mu^2)$ is the gluon density inside the proton. Finally, \n$d\\hat{\\sigma}$ ist the cross section of the underlying hard scattering\nsubprocess.\n\nThe variables $x$, $y$ and $Q^2$ are reconstructed by the 4-momentum \n$p'_e$ of the observed electron\nand the 4-momenta $p_e$, $p$ of the initial state electron and proton:\n\\begin{eqnarray}\nq_{l}\\equiv p_e-{p^{\\prime}_e},\\qquad Q_{l}^{2}\\equiv -q_{l}^{2},\n\\qquad y_{l}\n=\\frac{p\\cdot q_{l}}{p\\cdot p_e},\n\\qquad x_{l}\\equiv \\frac{Q_{l}^{2}}{2p\\cdot q_{l}}\n=\\frac{Q_l^2}{S y_l}\\ ,\n\\label{outerkin}\n\\end{eqnarray}\nwhere $S\\equiv (p+p_e)^2$. In the same way, one \ndefines for the hard scattering process\n$e(\\hat{p}_e=z_1 p_e)+g(\\hat{p}=z_2 p)\\longrightarrow e(\\hat{p}^\\prime_e\n=p_{e}^{\\prime}\/z_3)+c(p_c)+X$: \n\\begin{eqnarray}\n\\label{subkin}\n\\hat{q}&=&{\\hat{p}_e}-{{\\hat{p}^\\prime_e}},\\qquad \n\\hat{Q}^{2}=-\\hat{q}^{2}=\\frac{z_1}{z_3}Q_l^2,\\qquad\n\\hat{y}=\\frac{\\hat{p}\\hat{q}}{\\hat{p}{\\hat{p}_e}}\n=\\frac{z_1z_3-1+y_l}{z_1z_3},\n\\\\\n\\hat{x}&=&\\frac{\\hat{Q}^{2}}{2\\hat{p}\\cdot \\hat{q}}\n=\\frac{\\hat{Q}^2}{\\hat{S} \\hat{y}}\n=\\frac{z_1x_ly_l}{z_2(z_1z_3-1+y_l)},\\qquad\n\\hat{S}=(\\hat{p}_e +\\hat{p})^2=z_1 z_2 S\\ .\n\\end{eqnarray}\nBecause we only use leptonic variables $x_l$, $y_l$, etc.~, the index\n$l$ will be suppressed from now on.\n\nIn the following, we calculate the radiative corrections to the photon \ngluon fusion (PGF) subprocess \n\\begin{eqnarray}\ne(\\hat{p_e})+g(\\hat{p})\\longrightarrow e(\\hat{p}_e^\\prime)+c(p_c)\n+\\bar{c}(p_{\\bar{c}}).\n\\label{pgf}\n\\end{eqnarray}\nand compare the results with the radiative corrections to the charm \nexcitation (CE) subprocess\n\\begin{eqnarray}\ne(\\hat{p_e})+c(\\hat{p})\\longrightarrow e(\\hat{p}_e^\\prime)+c(p_c) \n\\label{eqeq}\n\\end{eqnarray}\nwhere, in contrast to the PGF, the charm quark is treated as an\nintrinsic (massless) parton of the proton.\nThe latter process has been used by the H1 Collab.~\\cite{glazov} in a \nrecent analysis of leptoproduction of D-mesons \\cite{adloff96},\nemploying some charm density $c(x,\\mu^2)$. \n\\subsection{Inclusive case}\n\\subsubsection{Photon gluon fusion (PGF)}\nThe hard cross section for the process (\\ref{pgf}) can be written as:\n\\begin{equation}\n\\frac{d^2\\hat{\\sigma}}{d\\hat{x}d\\hat{y}}=\n\\frac{4\\pi\\alpha^{2} \\hat{S}}{{\\hat{Q}}^4}\n\\left[(1-{\\hat{y}})f_2\\left({\\hat{x}},{\\hat{Q}}^{2}\\right)\n+{\\hat{x}}{\\hat{y}}^{2}f_1\\left({\\hat{x}},{\\hat{Q}}^{2}\\right)\\right]\n\\label{partonwq}\n\\end{equation}\nwith the partonic structure functions\n\\cite{witten76,gr79,leveille79,lrsn93} \n\\begin{eqnarray}\nf_k({\\hat{x}},{\\hat{Q}}^2,m_c^2,\\mu^2) & = & \n\\frac{\\alpha_s(\\mu^2)}{\\pi}\\ \\frac{\\hat{\\xi}}{4\\pi}\n\\ e_c^2\\ c_{k,g}^{(0)}({\\hat{x}},{\\hat{Q}}^2)\\ \n\\theta(\\hat{W}^2-4m_c^2),\\ \\: k=2,L \\: ,\n\\\\\nf_1({\\hat{x}},{\\hat{Q}}^2,m_c^2,\\mu^2) & = & \\frac{1}{2{\\hat{x}}}\n(f_2-f_L)\\: ,\n\\\\\n\\frac{\\hat{\\xi}}{4\\pi}c_{2,g}^{(0)} & = & \n\\left[\\frac{{\\hat{x}}}{2} -{\\hat{x}}^2(1-{\\hat{x}})+\n2\\frac{m_c^2}{{\\hat{Q}}^2}{\\hat{x}}^2(1-3{\\hat{x}})\n-4\\frac{m_c^4}{{\\hat{Q}}^4}{\\hat{x}}^3\\right]\n\\ln \\frac{1+\\hat{\\beta}}{1-\\hat{\\beta}}\\nonumber \\\\\n& & +\\hat{\\beta}\\left[4{\\hat{x}}^2(1-{\\hat{x}})\n-\\frac{{\\hat{x}}}{2}-2\\frac{m_c^2}{{\\hat{Q}}^2}{\\hat{x}}^2\n(1-{\\hat{x}})\\right],\n\\\\\n\\frac{\\hat{\\xi}}{4\\pi}c_{L,g}^{(0)} & = & -4\n\\frac{m_c^2}{{\\hat{Q}}^2}{\\hat{x}}^3\n\\ln \\frac{1+\\hat{\\beta}}{1-\\hat{\\beta}}+2\\hat{\\beta} {\\hat{x}}^2\n(1-{\\hat{x}}), \n\\label{HQProd}\n\\end{eqnarray}\nwhere $e_c$, $m_c$ are the charm quark charge and mass, respectively, \n$\\displaystyle \\hat{\\xi}\\equiv \\xi(\\hat{Q}^2) = \n\\frac{{\\hat{Q}}^2}{m_c^2}$\\,,\\,$\\displaystyle \n\\hat{W}^2\\equiv W^2(\\hat{x},\\hat{Q}^2)=\\frac{\\hat{Q}^2\n(1-\\hat{x})}{\\hat{x}}$, and\n$\\hat{\\beta}^2\\equiv \\beta^2(\\hat{x},\\hat{Q}^2)=1-4m_c^2\/\\hat{W}^2$.\n\nBy insertion of eq.~(\\ref{dee}), (\\ref{partonwq}) into eq.~(\\ref{wq}),\none obtains the cross section \n$d\\sigma=d\\sigma ^0+d\\sigma^{i}+ d\\sigma^{f}$.\nThe Born cross section $d\\sigma^0$ is given by\n\\begin{equation}\n\\frac{d^{2} \\sigma^0}{dx dy}=\\frac{4\\pi\\alpha^{2} S}{Q^4}\n\\left[(1-y)F_2\\left(x,Q^{2}\\right)+xy^{2}F_1\\left(x,Q^{2}\\right)\\right]\n\\label{wqem}\n\\end{equation}\nwith the leading order structure functions\n\\begin{eqnarray}\nF_1(x,Q^2,\\mu^2,m_c^2)=\\int_{ax}^1\\ \\frac{dz_2}{z_2}\\ g(z_2,\\mu^2)\n\\ f_1(x\/z_2,Q^2,m_c^2,\\mu^2),\n\\\\\nF_2(x,Q^2,\\mu^2,m_c^2)=\\int_{ax}^1\\ \\frac{dz_2}{z_2} z_2\\ g(z_2,\\mu^2)\n\\ f_2(x\/z_2,Q^2,m_c^2,\\mu^2) ,\n\\end{eqnarray} \nwhere $a=1+4m_c^2\/Q^2$.\n$d\\sigma^{i,f}$ contain the contribution due to radiation of a collinear\nphoton from the incoming (initial state radiation, ISR) and outgoing \nelectron (final state radiation, FSR),\nrespectively:\n\\begin{eqnarray}\n\\frac{d^2\\sigma^i}{dxdy}& = & \\frac{\\alpha L_{e}}{2\\pi}\n\\int^1_{z_{1}^{min}}dz_1\\left[\\frac{1+z_1^2}{1-z_1}\n(\\sigma_0(z_1,1)-\\sigma_0(1,1))\\right] \\nonumber\n\\\\\n&& +\\ \\frac{\\alpha L_{e}}{2\\pi} H(z_{1}^{min})\\sigma_0(1,1),\n\\label{initial}\n\\end{eqnarray}\n\\begin{eqnarray}\n\\frac{d^2\\sigma^f}{dxdy} & = & \\frac{\\alpha L_{e}}{2\\pi}\n\\int^1_{z_{3}^{min}}dz_3\\left[\\frac{1+z_3^2}{1-z_3}\n(\\sigma_0(1,z_3)-\\sigma_0(1,1))\\right] \\nonumber\n\\\\\n&& +\\ \\frac{\\alpha L_{e}}{2\\pi} H(z_{3}^{min})\\sigma_0(1,1),\n\\label{final}\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\n\\label{pgfs0}\n\\sigma_0(z_1,z_3) & = &\\int^1_{z_{2}^{min}}dz_2\\  g\\left(z_2,\\mu ^2\\right)\n\\ \\frac{\\partial{(\\hat{x},\\hat{y})}}{\\partial{(x,y)}}\n\\ \\frac{d^2\\hat{\\sigma}}{d\\hat{x}d\\hat{y}}, \n\\\\\nH(z) & = & -\\int_0^z dx \\frac{1+x^2}{1-x}=2\\ln(1-z)+z+\\frac{1}{2}z^2,\n\\end{eqnarray}\nand the Jacobian\n\\begin{equation}\n \\frac{\\partial{(\\hat{x},\\hat{y})}}{\\partial{(x,y)}}\n=\\frac{y}{z_2z_3(z_1z_3-1+y)}.\n\\label{jacobi1}\n\\end{equation}\nThe electromagnetic coupling is taken to be \n$\\alpha\\equiv \\alpha(Q^2=m_e^2)=1\/137.036$\\footnote{\nUsing a running coupling $\\alpha(Q^2)$ according to \neq.~(14) in \\protect\\cite{kms91},  \ntaking effective quark masses \\protect\\cite{hector}\n$m_u=m_d=0.041$, $m_s=.15$, $m_c=1.5$, and $m_b=4.5\\ {\\rm GeV}$, \ninstead of a constant $\\alpha$, leads to negligible differences.}.\nThe integration bounds\n\\begin{eqnarray}\n\\label{bounds}\nz_{1}^{min} & = & \\frac{1-y}{1-xy}+\\frac{4m_c^2}{S(1-xy)},\n\\nonumber\\\\\nz_{3}^{min} & = & \\frac{1-y+xy}{1-4m_c^2\/S},\n\\\\\nz_{2}^{min}(z_1,z_3)&=&\\left(1+\\frac{4m_c^2}{Q^2}\\frac{z_3}{z_1}\\right)\nz_1xy\\frac{1}{z_1z_3+y-1}\n\\nonumber\n\\end{eqnarray}\nfollow from the conditions $\\hat{W}^2 \\ge 4 m_c^2\\ , \n\\ 0 \\le z_{2}^{min}(z_1,z_3) \\le 1$. \nIf no photon is radiated ($z_1=1$, $z_3=1$), one finds the well-known\nexpression $z_{2}^{min}(1,1)=(1+4m_c^2\/Q^2)x\\equiv a x$.\nIn the kinematical region of small $x$, the bounds read approximately: \n$z_{1}^{min}= z_{3}^{min}= 1-y+{\\cal O}(x)+{\\cal O}(m_c^2\/S)$.\nIt should be remarked that $\\sigma_0$ is related to the Born \ncross section by $d^2\\sigma^0\/dxdy=\\sigma_0(1,1)$.\n\\subsubsection{Charm excitation (CE)}\nIn the case of electron quark scattering, the radiative corrections \nin LLA are known up to\n${\\cal O}(\\alpha^2)$ \\cite{km88,kms91,jbl90,jbl95,hector}.\nFor comparison with eq.~(\\ref{initial}), (\\ref{final}), we use \nequations (22)--(29) in \\cite{kms91},\nwith the charm quark as the only massless parton (MP) in the \ninitial state. \nFurthermore, contributions due to $Z$-exchange can be neglected \nin the realm of $Q^2 \\le 50$ GeV$^2$, i.~e.~, we make the\nreplacements $A^c\\to e_c^2$, $B^c\\to 0$ \n(in eq.~(22)--(29) in \\cite{kms91}).\nFor the partonic structure functions, the simple relations\n\\begin{eqnarray}\nf_2(\\hat{x},\\hat{Q}^2) & = & e_c^2\\ \\delta(1-\\hat{x})\\ ,\n \\\\\nf_1(\\hat{x},\\hat{Q}^2) & = & \\frac{1}{2\\hat{x}}\\ f_2(\\hat{x},\\hat{Q}^2)\n\\label{mpstrukt}\n\\end{eqnarray}\nhold, and one easily finds \\cite{kms91}\n\\begin{equation}\n\\sigma_0(z_1,z_3)=\\frac{2\\pi\\alpha^2 y}{z_1 z_3^2 \\hat{y}^3 \\hat{S}}\n[1+(1-\\hat{y})^2]\ne_c^2\\left(c(\\bar{z_2},\\hat{Q}^2)+\\bar{c}(\\bar{z_2},\\hat{Q}^2)\\right)\n\\label{mps0}\n\\end{equation}\nwith\n\\begin{equation}\n\\bar{z}_2=\\frac{z_1 x y}{z_1 z_3 + y -1}\\ , \\ z_{1}^{min}\n=\\frac{1-y}{1-xy}\\ , \\ z_{3}^{min}=1-y(1-x)\\ .\n\\label{mpbounds}\n\\end{equation}\n(Of course, eq.~(\\ref{mps0}) and (\\ref{mpbounds}) have to be \ninserted into eq.~(\\ref{initial}) and (\\ref{final}).)\n\nFinally, we briefly discuss the Compton contribution $d\\sigma^C$ \nto the radiative corrections\nwhich can be obtained from eq.~(21) ($Q_0=200\\ {\\rm MeV}$) \nand (29) in \\cite{kms91}, only \nallowing for the charm (or anti-charm) quark in the initial state.\nFig.~\\ref{compton} displays the Compton \ncontribution $\\displaystyle \\delta^C=\\frac{d^2\\sigma^C}{dxdy}\n\/\\frac{d^2\\sigma^0}{dxdy}$ for $x=10^{-2}$ (dashed line) and $x=10^{-3}$\n(full line) using the CTEQ4L parton distributions \\cite{cteq4}.\nFor $y\\lesssim 0.7$, $\\delta^C$ is small and can be safely neglected for \nexperimentally relevant values of $y$ \\cite{adloff96}.\n\\subsection{$z$-differential case}\nIn order to calculate radiative corrections to the $z$-differential \ncross section \n$\\displaystyle d^3\\sigma^D\/dx dy dz$, where $z=p\\cdot p_D\/p\\cdot q$, \nthe fragmentation of the charm quark into \nthe observed $D$-meson must be taken into consideration.\nThus, it is necessary to extend eq.~(\\ref{wq}):\n\\begin{equation}\nd\\sigma(ep\\rightarrow eDX)=\\int_0^1 dz_1\\int_0^1dz_2\n\\int_0^1dz_3 D_{e\/e}(z_1,Q^2)\ng(z_2,\\mu^2)\\bar{D}_{e\/e}(z_3,Q^2)\n\\int_0^1dz_4 D_c(z_4)d\\hat{\\sigma} .\n\\label{fragds}\n\\end{equation}\nThe hadronization of the outgoing charm quark with momentum $p_c$ \ninto the observed $D$-meson \nwith momentum $p_D=z_4p_c$ is modeled by the fragmentation \nfunction $D_c(z_4)$.\nWriting the partonic cross section differential in \n$\\hat{z}_c=\\hat{p}\\cdot p_c\/\\hat{p}\\cdot \\hat{q}$, the relations in\neq.~(\\ref{subkin}) have to be accomplished with\n\\begin{equation}\n\\hat{z}_c=\\frac{z y z_3}{z_4(z_1z_3+y-1)}\\equiv \\frac{z}{z_4}r,\n\\label{subkin2}\n\\end{equation}\nwith $r$ defined by \n\\begin{equation}\nr\\equiv \\frac{y\\ z_3}{z_1z_3+y-1},\n\\label{r}\n\\end{equation}\nwhich can be derived from the definitions of $\\hat{z}_c$, $z$ \nand $\\hat{q}$, using $p_D=z_4 p_c$.\n\nIn analogy to the above discussion,\nwe compare the ``massive'' (PGF) with the ``massless'' (CE) \ncorrections, using a massless charm parton (MP).\n\\subsubsection{Photon gluon fusion}\nThe partonic cross section reads\n\\begin{equation}\n\\frac{d^3\\hat{\\sigma}}{d\\hat{x}d\\hat{y}d\\hat{z}_c}=\n\\frac{4\\pi\\alpha^{2} \\hat{S}}{{\\hat{Q}}^4}\n\\left[(1-{\\hat{y}})f_2\\left({\\hat{x}},{\\hat{Q}}^{2};\\hat{z}_c\\right)\n+{\\hat{x}}{\\hat{y}}^{2}f_1\\left({\\hat{x}},{\\hat{Q}}^{2};\n\\hat{z}_c\\right)\\right],\n\\end{equation}\nwhere the structure functions are given by\n\\footnote{The corresponding expressions for\ngeneral boson gluon fusion can be found in \\protect\\cite{ks97}.}\n\\begin{eqnarray}\nf_2(x,Q^2;{\\zeta}) &=& \\frac{\\alpha_s}{\\pi}e_{c}^2x\n\\left[\\frac{1}{4}{\\rm BG}\\left(x,Q^2,{\\zeta},{m_c}\\right)\n+\\frac{3}{4}{\\rm BL}\\left(x,Q^2,{\\zeta},{m_c}\\right)\\right],\n\\label{dpgfdz1}\\\\\nf_L(x,Q^2;{\\zeta})&=&\\frac{\\alpha_s}{\\pi}e_{c}^2x\\frac{1}{2}\n{\\rm BL}\\left(x,Q^2,{\\zeta},{m_c}\\right),\n\\label{dpgfdz2}\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\n{\\rm BL}\\left(x,Q^2,{\\zeta},{m_c}\\right)&=&\n4x\\left[1-x-x\\frac{{m_c}^2}{Q^2}\\frac{1}{{\\zeta}(1-{\\zeta})}\\right],\n\\nonumber\\\\\n{\\rm BG}\\left(x,Q^2,{\\zeta},{m_c}\\right)&=&-2+\n\\left[1-2x+2x^2+4\\frac{{m_c}^2}{Q^2}x(1-x)\\right]\n\\frac{1}{{\\zeta}(1-{\\zeta})}\n\\nonumber\\\\\n&&+2x^2\\frac{{m_c}^2}{Q^2} \\left(1-2\\frac{{m_c}^2}{Q^2}\\right)\n\\frac{1}{(1-{\\zeta})^2{\\zeta}^2},\n\\nonumber\n\\end{eqnarray}\n\\cite{lrsn93,schuler88} and\n$\\displaystyle f_1(\\hat{x},\\hat{Q}^2;\\hat{z}_c)= \\frac{1}{2\\hat{x}}\n\\left[f_2(\\hat{x},\\hat{Q}^2;\\hat{z}_c)-\nf_L(\\hat{x},\\hat{Q}^2;\\hat{z}_c)\\right]$.\n\nThe cross sections for ISR and FSR have the same structure\nas in eq.~(\\ref{initial}) and (\\ref{final}) with a $z$ dependent \nfunction $\\sigma_0$:\n\\begin{eqnarray}\n\\frac{d^3\\sigma^i}{dxdydz} & = & \\frac{\\alpha L_{e}}{2\\pi}\n\\int^1_{z_{1}^{min}}dz_1\\left[\\frac{1+z_1^2}{1-z_1}\n(\\sigma_0(z_1,1;z)-\\sigma_0(1,1;z))\\right] \\nonumber\n\\\\\n&& +\\ \\frac{\\alpha L_{e}}{2\\pi}H(z_{1}^{min})\\sigma_0(1,1;z),\n\\label{fragini}\\\\\n\\frac{d^3\\sigma^f}{dxdydz} & = & \\frac{\\alpha L_{e}}{2\\pi}\n\\int^1_{z_{3}^{min}}dz_3\\left[\\frac{1+z_3^2}{1-z_3}\n(\\sigma_0(1,z_3;z)-\\sigma_0(1,1;z))\\right] \\nonumber\n\\\\\n&& +\\frac{\\alpha L_{e}}{2\\pi}H(z_{3}^{min})\\sigma_0(1,1;z),\n\\label{fragfin}\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\n\\sigma_0(z_1,z_3;z) & = &\\int^1_{z_{2}^{min}}dz_2 g\n\\left(z_2,\\mu ^2\\right)\\int_{z_{4}^{min}}^{z_{4}^{max}}dz_4\n\\frac{\\partial{(\\hat{x},\\hat{y},\\hat{z_c})}}{\\partial{(x,y,z)}}\n\\frac{d^3\\hat{\\sigma}}{d\\hat{x}d\\hat{y}d\\hat{z_c}}D_c(z_4)\\ .\n\\label{frags0}\n\\end{eqnarray}\nThe Jacobian in eq.~(\\ref{frags0}) can easily be calculated as\n\\begin{eqnarray}\n\\frac{\\partial{(\\hat{x},\\hat{y},\\hat{z_c})}}{\\partial{(x,y,z)}}=\n\\frac{\\partial{(\\hat{x},\\hat{y})}}{\\partial{(x,y)}}\n\\frac{\\partial \\hat{z}_c}{\\partial z}=\n\\frac{y^2}{z_2z_4(z_1z_3-1+y)^2}.\n\\end{eqnarray}\nThe integration bounds can be derived from the conditions\n$\\displaystyle \\hat{W}^2=\\frac{\\hat{Q}^2(1-\\hat{x})}{\\hat{x}} \n\\ge 4m_c^2$, leading to eq.~(\\ref{bounds}), and \n$\\displaystyle \\hat{z}_{c}^{min}\\le \\hat{z}_c \\le \n\\hat{z}_{c}^{max}$ with\n$\\displaystyle \\hat{z}_{c}^{max \\atop min}=\n\\frac{1\\pm \\beta(\\hat{x},\\hat{Q}^2)}{2}$ \\cite{schuler88}, where\n$\\beta^2(\\hat{x},\\hat{Q}^2)=1-4m_c^2\/{\\hat{W}}^2$, implying\n\\begin{eqnarray}\nz_{4}^{max}&=&\\min \\left[1,\\frac{z r}{\\hat{z}_{c}^{min}}\\right],\n\\\\\nz_{4}^{min}&=&\\min \\left[z_{4}^{max},\\frac{z r}{\\hat{z}_{c}^{max}}\n\\right]\n\\end{eqnarray}\nwith $r$ from eq.~(\\ref{r}).\nFor $D_c(z)$, we use a Peterson et al.\\ fragmentation function \n\\cite{peterson83}\n\\begin{equation}\nD_c(z) = N \\left\\{ z \\left[ 1-z^{-1}-\\varepsilon_c\/(1-z)\n\\right]^2\\right\\}^{-1}\\ ,\n\\label{peterson}\n\\end{equation}\nnormalized to $\\int_0^1 dz D_c(z) = 1$, i.~e.~,\n\\begin{equation}\nN^{-1} =\n\\frac{({\\varepsilon_c}^2-6{\\varepsilon_c}+4)}{(4-{\\varepsilon_c}) \n\\sqrt{4{\\varepsilon_c}-{\\varepsilon_c}^2}}\n\\left\\{\n\\arctan\\frac{{\\varepsilon_c}}{\\sqrt{4{\\varepsilon_c}\n-{\\varepsilon_c}^2}}\n+ \\arctan\\frac{2-{\\varepsilon_c}}{\\sqrt{4{\\varepsilon_c}\n-{\\varepsilon_c}^2}} \\right\\}\n+ \\frac{1}{2} \\ln {\\varepsilon_c} + \\frac{1}{4-{\\varepsilon_c}}\\ .\n\\end{equation} \n\n\\subsubsection{Flavor excitation}\nIn the subprocess $e+c\\longrightarrow e+c$ we have\n$\\displaystyle \\hat{z}_c=\\hat{p}\\cdot p_c\/\\hat{p}\\cdot \\hat{q}=1$, \nfollowing from $p_c=\\hat{q}+\\hat{p}$.\nThus, one obtains the $\\hat{z}_c$-differential cross section \nby inserting the structure functions\n\\begin{eqnarray}\nf_2(\\hat{x},\\hat{Q}^2,\\hat{z}_c) & = & \ne_c^2\\ \\delta(1-\\hat{x})\\delta(1-\\hat{z}_c)\n \\\\\nf_1(\\hat{x},\\hat{Q}^2,\\hat{z}_c) & = & \n\\frac{1}{2\\hat{x}}\\ f_2(\\hat{x},\\hat{Q}^2,\\hat{z}_c)\n\\label{mpstruktdz}\n\\end{eqnarray}\ninto eq.~(\\ref{partonwq}).\nBecause of $\\displaystyle \\delta(1-\\hat{z}_c)=z_4\\delta(z_4-r z)$ and\n$\\displaystyle \\partial\\hat{z}_c\/\\partial z=r\/z_4$ one finds\n\\begin{equation}\n\\sigma_0(z_1,z_3;z)=\\sigma_0(z_1,z_3)\\ r D_c(rz)\n\\label{mps0dz}\n\\end{equation}\nwith $\\sigma_0(z_1,z_3)$ from eq.~(\\ref{mps0}) and $r$ \nfrom eq.~(\\ref{r}).\n\nFinally the $z$-differential corrections are given by \neq.~(\\ref{fragini}) and (\\ref{fragfin}) \nwith $\\sigma_0$ from eq.~(\\ref{mps0dz}) and the integration bounds\n\\begin{eqnarray}\nz_{1}^{min}&=&\\max\\left[\\frac{1-y}{1-xy},1-y(1-z)\\right],\n\\\\\nz_{3}^{min}&=&\\max\\left[1-y(1-x),\\frac{1-y}{1-yz}\\right] .\n\\label{mpboundsdz}\n\\end{eqnarray}\nThe second boundary in $\\max[ \\ldots , \\ldots ]$ can be deduced \nfrom $0 \\le rz \\le 1$.\n\nEq.~(\\ref{fragini}), (\\ref{fragfin}) can easily be tested:\n\\begin{equation}\n\\int_0^1 dz\\frac{d^3\\sigma^{i,f}}{dxdydz}\\stackrel{!}{=}\n\\frac{d^2\\sigma^{i,f}}{dxdy}\\ ,\n\\end{equation}\nwith $d^2\\sigma^{i,f}\/dxdy$ from (\\ref{initial}), (\\ref{final}).\nThe Compton contribution will be neglected for the same reasons as \nin the inclusive case.\n\\section{Numerical results}\nAs usual, the radiative corrections will be shown in form of a \ncorrection factor $\\delta$ defined by \n$d\\sigma=d\\sigma^0(1+\\delta)$, i.~e., \n$\\delta=\\delta^i+\\delta^f+\\ldots$\nwith \n\\begin{displaymath}\n\\delta^{i,f}(x,y)=\\frac{d^2 \\sigma^{i,f}}{dx dy}\/\n\\frac{d^2 \\sigma^0}{dx dy}\n\\end{displaymath}\nor, in the $z$-differential case, \n\\begin{displaymath}\n\\delta^{i,f}(x,y,z)=\\frac{d^3 \\sigma^{i,f}}{dx dy dz}\/\n\\frac{d^3 \\sigma^0}{dx dy dz}.\n\\end{displaymath}\nIn all figures we use HERA centre-of-mass energies \n$S=4\\cdot27.5\\cdot 820$ GeV$^2$.\n\nFig.~\\ref{rcpgf} shows the radiative corrections to heavy quark \nproduction (PGF) for experimentally relevant values\\footnote{For \na clean extraction of the \ngluon density it is necessary to extend the $x$-range \nto $x\\lesssim 5 \\cdot 10^{-4}$ \\cite{daum96}.} \nof Bjorken-$x$ \\cite{adloff96} as a function of $y$ according to\neq.~(\\ref{initial}) and (\\ref{final}) using \nthe GRV94(LO) parton distributions \\protect\\cite{grv94}.\nThe factorization scale has been chosen to be \n$\\mu^2=Q^2+4m_c^2$ \\cite{vogt96} \nwith $m_c=1.5\\ {\\rm GeV}$.\nFor $x= 10^{-2}$, the typical shape of radiative corrections \nin leptonic variables can be seen, with large corrections \nfor $y\\to 1$, whereas for smaller $x\\le 10^{-3}$ the curves \nbecome more and more flat for $y \\to 1$.\n\nThe theoretical uncertainties due to different choices of \nparton distributions, factorization scales, and charm masses turn \nout to be small, as can be seen in fig.~\\ref{pdfcomp} \nand fig.~\\ref{compare}.\nOne finds $\\delta({\\rm CTEQ})-\\delta({\\rm GRV})< 0.02$,\n$\\delta(m_c=1.3)-\\delta(m_c=1.7)< 0.03$ for relevant $y\\le 0.7$, and\n$\\delta(\\mu^2=4m_c^2)-\\delta(\\mu^2=Q^2+4m_c^2)< 0.03$, where the \nscale $\\mu^2=4m_c^2$ has been favored in \\cite{grs94}.\nThis could be expected because variations of $\\mu$, $m_c$, or \nthe parton distributions lead to rather similar changes in \n$d\\sigma^0$ and $d\\sigma^{i,f}$, so that the quotient $\\delta$\ndoes not change too much.\n\nIn the recent analysis of deep inelastic charm production by the \nH1 Collab.~\\cite{adloff96}, the radiative corrections have been \ncalculated in ${\\cal O}(\\alpha)$-LLA with the \nHECTOR package \\cite{hector} using the charm excitation \nsubprocess ($F_L^c=0$) and the GRV92 parton distributions \\cite{grv92}.\nFurthermore, only initial state radiation has been taken into \nconsideration, assuming the collinear final state photon not to be \nseparated from the outgoing electron \\cite{glazov},\nleading to small corrections. Because PGF has been measured to be \nthe dominant \n($>95 \\%$) charm production mechanism in the small\n$x$ (and $Q^2$) range \\cite{adloff96}, it is necessary to check \nif the ``massless'' corrections ($\\delta^{MP}$) agree with the \n``massive'' ones ($\\delta^{PGF}$).\nIn fig.~\\ref{mpcpgfgrvi} we compare the massive \n($\\mu^2=Q^2+4m_c^2$; $m_c=1.5$ GeV) \nwith the massless corrections due to initial state radiation.\nThe experimentally relevant values of $Q^2$ \\cite{adloff96} are\nindicated by dotted vertical lines. \nThe conversion of ``masslessly corrected'' ($\\delta^{MP}$) data to \n``massively corrected'' ($\\delta^{PGF}$) data\ncan be performed by applying the factor \n$R\\equiv (1+\\delta^{MP})\/(1+\\delta^{PGF})$ because of\n\\begin{eqnarray*}\nd\\sigma({\\rm PGF})&=&d\\sigma({\\rm exp})\\frac{1}{1+\\delta^{\\rm PGF}}\n\\\\\n&=&d\\sigma({\\rm MP})\\frac{1+\\delta^{\\rm MP}}{1+\\delta^{\\rm PGF}}\n\\equiv d\\sigma({\\rm MP})R\\ .\n\\end{eqnarray*}\nFor the $(x,Q^2)$ data points one finds $|R-1|=|\\delta^{MP}-\n\\delta^{PGF}|\/(1+\\delta^{PGF})< 3\\%$,i.~e.~, the differences \nbetween massless and massive radiative corrections lead to a small\nincrease ($\\delta^{MP}>\\delta^{PGF}$) of the data.\nHowever, this difference is small enough\nto use the simpler charm excitation subprocess for calculating \nthe radiative corrections in the inclusive case.\n\nOf course, heavy quark production processes are exclusive in\nthe heavy quark momentum and on this more differential level the\nphoton gluon fusion and the charm excitation processes are not \ncompatible which can be seen, e.g., from the different shapes\nof the $x_D=|\\vec{p}_D^*|\/|\\vec{p}_p^{\\, *}|=2|\\vec{p}_D^{\\, *}|\/W$ \ndistributions  shown in the experimental analyses\n\\cite{adloff96,zeus97} (fig.~6, fig.~1 resp.).\nWe prefer to employ the lorentz-invariant variable\n$z=p\\cdot p_D\/p\\cdot q$ which approximately transforms\ninto $x_D$ in the $\\gamma^{\\, *}$-$p$ centre-of-mass \nsystem, following the argumentation in \\cite{adloff96}.\nThis can be easily seen by calculating $z$\nin the $\\gamma^{\\, *}$-$p$-CMS. One finds $z=x_D \\sin^2 \\theta^{\\, *}\/2$\nwith\\footnote{$\\vec{p}_{\\gamma}^{\\, *}+\\vec{p}_{p}^{\\, *}=0$, \n$\\vec{p}_{g,c}^{\\, *}=z_2\\vec{p}_{p}^{\\, *} \\Rightarrow \n\\vec{p}_{\\gamma}^{\\, *}+\\vec{p}_{g,c}^{\\, *}=-(1-z_2)\\vec{p}_{p}^{\\, *}$.\nThis means for the\nCE-subprocess: $\\vec{p}_{\\gamma}^{\\, *}+\\vec{p}_{c}^{\\, *}=\\vec{p'}_{c}^{\\, *}\n=z_4 \\vec{p}_D^{\\, *} \\Rightarrow \\theta^{\\, *}=\\pi$\nand for the PGF: $\\vec{p}_{\\gamma}^{\\, *}+\\vec{p}_{g}^{\\, *}=\\vec{p}_{c}^{\\, *}\n+\\vec{p}_{\\bar{c}}^{\\, *}\n\\Rightarrow \\theta^{\\, *}\\approx \\pi$ \n(the collinear configuration is dominant). Note that an \nangle $\\theta^{\\, *}=160^{\\circ}$ still yields \n$\\sin^2 \\theta^{\\, *}\/2=0.97$.} \n $\\theta^{\\, *}=\\angle (\\vec{p}_p^{\\, *},\\vec{p}_D^{\\, *})\n\\approx \\pi$. \n\nIn fig.~\\ref{mpcpgffrag} we show the  $z$-differential radiative \ncorrections according to eq.~(\\ref{fragini}), (\\ref{fragfin}), \n(\\ref{frags0}) (PGF, full line) and\n(\\ref{mps0dz}) (MP, dashed line), where it has been integrated \nover the kinematical range \n$10\\ {\\rm GeV}^2\\le Q^2\\le 100\\ {\\rm GeV}^2,\\ 0.01 \\le y\\le 0.7$.\nThe left hand side shows the correction factor \n$\\delta^{i}=d\\sigma^{i}\/d\\sigma^0$ as well as\nthe sum of initial and final state radiation $\\delta^i+\\delta^f$. \nIn all cases, we employed the GRV92(LO) (massless) \n\\protect\\cite{grv92} and GRV94(LO) (massive) \\protect\\cite{grv94} \nparton distributions, in the massive case\nwith $m_c=1.5$ GeV and $\\mu^2=Q^2+4m_c^2$. For $D_c$ \na Peterson et al.\\ fragmentation function \\protect\\cite{peterson83} \nwith $\\varepsilon=0.15$ has been taken.\nTo study the dependence of $\\delta$ on $\\varepsilon$, we \ncompare the radiative corrections for two\ndifferent choices of $\\varepsilon$ in fig.~\\ref{epscomp}.\nIn the massive case the $\\varepsilon$-dependence is obviously small, \nwhereas for massless corrections\none finds big differences in the steep region $z\\lesssim 0.4$.\nFig.~\\ref{mpcpgffrag} reveals a rather big difference between \nmassive and massless corrections for $z\\lesssim 0.5$.\nFor $z\\to 1$ only soft photon radiation is allowed, so that the \ncorrections factorize and become independent of the \nunderlying subprocess.\nOn this more differential level it is necessary to\ncalculate the radiative corrections using the photon gluon fusion \nsubprocess because of the big deviations of the massless\nfrom the massive corrections for $z\\lesssim 0.5$.\nOn the right side of fig.~\\ref{mpcpgffrag} the $z$-dependence of the \ncross sections $d\\sigma^{0,i,f}$ is displayed.\nIntegration over $0\\le z\\le 1$ leads to the observed small \ndifferences between massless and massive corrections because \npositive and negative contributions compensate each \nother. To get an impression of the effect of $z$-cuts, \nwe show in fig.~\\ref{zcut1} \nthe correction factor $\\delta^{i,f}(x,y;z^{min},z^{max})=\n\\int_{z^{min}}^{z^{max}}dz \\frac{d\\sigma^{i,f}}{dx dy dz}\/\n\\int_{z^{min}}^{z^{max}}dz \\frac{d\\sigma^{0}}{dx dy dz}$ \nfor three $z$-integration ranges, from top to bottom \n$z^{min}=0 \\le z\\le z^{max}=1$, $0.2 \\le z\\le 0.8$, and\n$0.3 \\le z \\le 0.9$ for $x=10^{-3}$.\nFor completeness, we used the GRV94\/92(LO) parton distributions \n\\cite{grv94}, \\cite{grv92} for the massive ($m_c=1.5$ GeV, \n$\\mu^2=Q^2+4m_c^2$) and massless corrections\nand a Peterson et al.\\ fragmentation function \\cite{peterson83} with \n$\\varepsilon=0.15$.\nAs can be seen from fig.~\\ref{mpcpgffrag}, a cut $z\\gtrsim 0.2$\nleads to the exclusion of positive contributions of $d\\sigma^{i,f}$, \ndiminishing the radiative corrections.\n\\section{Summary}\nThe ${\\cal O}(\\alpha)$-QED corrections to inclusive and \n$z$-differential deep inelastic electroproduction of heavy\nquarks have been calculated in the leading log approximation,\nusing electron variables.\nThe results have been compared to the radiative corrections in the \nMP scheme, where the charm quark is assumed to be a massless parton\nin the proton.\nIn the inclusive case, the differences between these two approaches \nturned out to be negligible.\nHowever, the measurement of heavy quark production is of course\ndifferential in the momentum of the (observed) heavy quark, \nrecommending to perform the radiative corrections on the same\ndifferential level.\nThus, we have considered the semi-inclusive $z$-differential \ncase, in which the massive \ncorrections have to be applied, i.e., using the photon gluon fusion \nsubprocess, because for $0.2 \\lesssim z \\lesssim  0.5$ the \nmassless corrections differ from the massive ones by about\n$\\approx 40 \\%$--$10 \\%$.\nFurthermore, we studied the effect of cuts on the $z$-integration range. \nA cut $z \\gtrsim 0.2$, e.g., \nexcludes positive contributions of $d\\sigma^{i,f}$, so that\nthe ($z^{min}\\le z\\le z^{max}$)-integrated corrections are smaller as \nin the fully inclusive, i.e., ($0\\le z\\le 1$)-integrated case.\n\\section*{Acknowledgements}\nWe thank E.\\ Reya and M.\\ Gl\\\"{u}ck for advice and useful discussions.\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{Introduction}\n\nSince Macdonald \\cite{Mac88} defined Macdonald polynomials $\\widetilde{H}_\\mu({\\mathbf {x}};q,t)$ and conjectured the Schur positivity of them, the Macdonald polynomial has been one of the central objects in algebraic combinatorics. Even though a combinatorial formula for Macdonald polynomials is given \\cite{HHL04} and the Schur positivity of Macdonald polynomials is proved \\cite{Hai01}, not much is known about an explicit combinatorial formula for the $(q,t)$-Kostka polynomials, which are the Schur coefficients of the Macdonald polynomial. In this paper, we discuss some enumerative results involving the $(q,t)$-Kostka polynomials in the words of cyclic sieving phenomena. This will provide a series of identities between the number of matrices with certain cyclic symmetries and evaluation of $(q,t)$-Kostka polynomials at a root of unity, uncovering a part of the mystery of the $(q,t)$-Kostka polynomials.\n\nTo begin, we define the cyclic seiving phenomena. Let $X$ be a set with an action of a cyclic group $C$. Fix a generator $c$ of $C$ and let $\\zeta$ be a root of unity having the same multiplicative order as $c$. Let $X(q)\\in{\\mathbb {Z}}[q]$ be a polynomial in $q$. We say that the triple $(X, C, X(q))$ exhibits the \\emph{cyclic sieving phenomenon (CSP)} \\cite{RSW04} if for any integer $r$ the number of fixed points of $c^r$ in $X$ is equal to the evalation of $X(q)$ at $q=\\zeta^r$, i.e.\n\\begin{equation*}\n\\left|X^{c^r}\\right| = \\left|\\{ x \\in X \\,:\\, c^r \\cdot x = x \\}\\right| = X(\\zeta^r).\n\\end{equation*}\n\nMore generally, let $X$ be a set with an action of a direct product of $k$ cyclic groups $C_1\\times C_2\\times \\cdots \\times C_k$. For each $i=1,2,\\dots,k$, fix a generator $c_i$ for $C_i$. Let $X(q_1,q_2,\\dots,q_k)\\in{\\mathbb {Z}}[q_1,q_2,\\dots,q_k]$ be a polynomial in $k$ variables. Following \\cite{BRS08}, we say the triple $\\left(X, C_1\\times C_2\\times \\cdots \\times C_k, X(q_1,q_2,\\dots,q_k)\\right)$ exhibits the \\emph{k-ary-cyclic sieving phenomenon (k-ari-CSP)} if for any integers $r_1, r_2, \\dots, r_k$ the number of fixed points of $(c_1^{r_1},c_2^{r_2},\\dots, c_k^{r_k})$ in $X$ is equal to the evalation of $X(q_1,q_2,\\dots,q_k)$ at $(q_1,q_2,\\dots,q_k)=(\\zeta_1^{r_1}, \\zeta_2^{r_2},\\dots, \\zeta_k^{r_k})$, i.e.\n\\begin{equation*}\n\\left|X^{(c_1^{r_1},c_2^{r_2},\\dots,c_k^{r_k})}\\right| = \\left|\\{ x \\in X \\,:\\, (c_1^{r_1},c_2^{r_2},\\dots,c_k^{r_k}) \\cdot x = x \\}\\right| = X(\\zeta_1^{r_1}, \\zeta_2^{r_2}, \\dots, \\zeta_k^{r_k}),\n\\end{equation*}\nwhere $\\zeta_i$ is a root of unity having the same multiplicative order as $c_i$. In this paper, we provide instances of tricyclic sieving and quadracyclic sieving phenomena, i.e. $k$-ary-CSP for $k=3$ and $k=4$.\n\nThe main tool we used to prove our results is the theory of orbit harmonics. The orbit harmonics is a tool in combinatorial representation theory that promotes an (ungraded) action of $\\mathfrak{S}_n$ on a finite set $X$ to a graded action of $\\mathfrak{S}_n$ on a polynomial ring quotient, by viewing $X$ as an $\\mathfrak{S}_n$-stable point locus in $\\mathbb{C}^n$. The idea goes as follows. Let $X\\subseteq\\mathbb{C}^n$ be a finite set which is closed under the action of $\\mathfrak{S}_n\\times C$, where\n\\begin{itemize}\n    \\item a symmetric group $\\mathfrak{S}_n$ acts on $\\mathbb{C}^n$ by permuting coordinates, and\n    \\item $C$ is a finite cyclic group acting on $\\mathbb{C}^n$ by scaling a root of unity.\n\\end{itemize}\nLet $\\mathbf{I}(X)$ be the ideal of polynomials in $\\mathbb{C}[{\\mathbf {x}}_n]:=\\mathbb{C}[x_1,\\dots,x_n]$ which vanish on $X$. Then we have an isomorphism\n\\begin{equation}\\label{isomorphism I-ideal}\n    \\mathbb{C}[X]\\cong\\mathbb{C}[{\\mathbf {x}}_n]\/\\mathbf{I}(X),\n\\end{equation}\nwhere $\\mathbb{C}[X]$ is the algebra of all functions $X\\rightarrow \\mathbb{C}$. We further define a homogeneous ideal\n\\begin{equation*}\n    \\mathbf{T}(X):=\\langle\\tau(f):f\\in\\mathbf{I}(X)\\setminus\\{0\\}\\rangle\\subseteq\\mathbb{C}[{\\mathbf {x}}_n],\n\\end{equation*}\nwhere $\\tau(f)$ denote the top degree homogeneous part of $f$. Then the isomorphism~\\eqref{isomorphism I-ideal} extends to an isomorphism \n\\begin{equation}\\label{isomorphism T-ideal}\n    \\mathbb{C}[X]\\cong\\mathbb{C}[{\\mathbf {x}}_n]\/\\mathbf{I}(X)\\cong\\mathbb{C}[{\\mathbf {x}}_n]\/\\mathbf{T}(X).\n\\end{equation}\nNote that  $\\mathbb{C}[{\\mathbf {x}}_n]\/\\mathbf{T}(X)$ admits an additional structure of a graded $\\mathfrak{S}_n$-module.\n\nUsing the isomorphism~\\eqref{isomorphism T-ideal}, the author and Rhoades provided a `generating theorem' for sieving results \\cite[Theorem 3.4]{OR20}. By varying the choice of combinatorial locus $X$, one can obtain various sieving results concerning $X$. The associated polynomial is given by a variant of the graded Frobenius image ${\\mathrm {grFrob}}({\\mathbb {C}}[{\\mathbf {x}}_n]\/\\mathbf{T}(X);q)$.\n\nIn this paper, we adopt orbit harmonics to the diagonal orbit harmonics to obtain a `generating theorem' (Theorem~\\ref{sieving-generator}) for sieving results of the combinatorial locus $X\\subseteq\\mathbb{C}^{2n}$ with a diagonal action of $\\mathfrak{S}_n$ on $X$. A precise explanation of the diagonal orbit harmonics is given in Section~\\ref{subsection Orbit harmonics and cyclic sieving}. \n\nTo prove the Macdonald positivity conjecture, Garsia and Haiman \\cite{GH93} suggested a bigraded ${\\mathfrak{S}}_n$-module ${\\mathbf {H}}_\\mu$ (called the Garsia--Haiman module) for a partition $\\mu$ whose Frobenius image is the Macdonald polynomial of $\\mu$. This module is defined as the $\\mathbb{C}$-span of a variant of the Vandermonde determinant and its partial derivatives (See Section~\\ref{modules of garsia--Haiman} for detail). Garsia and Haiman defined another module ${\\mathbf {R}}_\\mu$ via orbit harmonics which is later shown to be isomorphic to the original Garsia--Haiman module ${\\mathbf {H}}_\\mu$.  Therefore by taking $\\mu=(m^n)$, we can apply Theorem~\\ref{sieving-generator} to this module ${\\mathbf {R}}_\\mu$ to obtain triCSP for $n\\times m$ matrices of content $(1^{mn})$ (each of $1,2,\\dots,mn$ appears once) and biCSP for $n\\times m$ matrices of given content $\\nu$ ($i$ appears $\\nu_i$ times). This relates roots of unity specializations of $(q,t)$-Kostka polynomials with enumerations of matrices invariant under certain rotation of row index and column index and translation of entries.\n\n\\begin{theorem}\\label{Main theorem 1}\nLet $X_{(m^n)}$ be the set of $n\\times m$ matrices of content $(1^{mn})$. A product of cyclic groups ${\\mathbb {Z}}_n\\times{\\mathbb {Z}}_m\\times{\\mathbb {Z}}_{mn}$ acts on $X_{(m^n)}$ by row rotation, column rotation, and adding 1 modulo $mn$ to each entry. In addition for a composition $\\nu\\models mn$, let $X_{(m^n),\\nu}$ be the set of $n\\times m$ matrices of content $\\nu$ where a product of cyclic groups ${\\mathbb {Z}}_n\\times{\\mathbb {Z}}_m$ acts on $X_{(m^n),\\nu}$ by row and column rotation. Then we have the followings.\n\\begin{itemize}\n    \\item $\\left(X_{(m^n)},{\\mathbb {Z}}_n\\times{\\mathbb {Z}}_m\\times{\\mathbb {Z}}_{mn},X_{(m^n)}(q,t,z)\\right)$ exhibits triCSP, where $$X_{(m^n)}(q,t,z)=\\sum_{\\lambda\\vdash mn} \\widetilde{K}_{\\lambda,(m^n)}(q,t)f^{\\lambda}(z).$$\n    \\item $\\left(X_{(m^n),\\nu},{\\mathbb {Z}}_n\\times{\\mathbb {Z}}_m,X_{(m^n),\\nu}(q,t)\\right)$ exhibits biCSP, where $$X_{(m^n),\\nu}(q,t)=\\sum_{\\lambda\\vdash mn} \\widetilde{K}_{\\lambda,(m^n)}(q,t)K_{\\lambda,\\nu}.$$\n\\end{itemize}\nHere, $\\widetilde{K}_{\\lambda,\\mu}(q,t)$ ($K_{\\lambda,\\mu}$, respectively) denotes the modified $(q,t)$-Kostka polynomial (Kostka number, respectively) and $f^{\\lambda}(z):=\\sum_{T\\in{{\\mathrm {SYT}}}(\\lambda)}z^{\\operatorname{maj}(T)}$ is the fake degree polynomial, where ${{\\mathrm {SYT}}}(\\lambda)$ is the set of standard tableaux of shape $\\lambda$ and $\\operatorname{maj}$ is the major index.\n\\end{theorem}\n\n\nWe generalize Theorem~\\ref{Main theorem 1} in two directions. First direction is to generalize biCSP in the second bullet point of Theorem~\\ref{Main theorem 1} to triCSP. We say a composition $\\nu$ has a cyclic symmetry of order $a$ if $\\nu_i=\\nu_{i+a}$ always, where the subscripts are interpreted modulo the length $l(\\nu)$ of $\\nu$. In the second bullet point of the above theorem, if $\\nu$ has a cyclic symmetry of order $a$ dividing $l(\\nu)$, the set $X_{(m^n),\\nu}$ possesses an additional action of a cyclic group ${\\mathbb {Z}}_{l(\\nu)\/a}$ by adding $a$ modulo $l(\\nu)$ to each entry. Then one might ask if there is natural $z$-analogue of $X_{(m^n),\\nu}(q,t)$ to give triCSP for $X_{(m^n),\\nu}$. We give an answer of this question in the following theorem.\n\n\\begin{theorem}\\label{Main theorem 2}\nLet $\\nu\\models mn$ be a composition with a cyclic symmetry of order $a$ dividing $l(\\nu)$. Let $X_{(m^n),\\nu}$ be the set of $n\\times m$ matrices of content $\\nu$. A product of cyclic groups ${\\mathbb {Z}}_n\\times{\\mathbb {Z}}_m\\times{\\mathbb {Z}}_{l(\\nu)\/a}$ acts on $X_{(m^n),\\nu}$ by row rotation, column rotation, and adding $a$ modulo $l(\\nu)$ to each entry. Then the triple $\\left(X_{(m^n),\\nu},{\\mathbb {Z}}_n\\times{\\mathbb {Z}}_m\\times{\\mathbb {Z}}_{l(\\nu)\/a},X_{(m^n),\\nu}(q,t,z)\\right)$ exhibits the triCSP, where\n$$X_{(m^n),\\nu}(q,t,z)=\\sum_{\\lambda\\vdash mn} \\widetilde{K}_{\\lambda,(m^n)}(q,t)\\widetilde{K}_{\\lambda,\\nu}(z).$$\nHere, $\\widetilde{K}_{\\lambda,\\mu}(q,t)$ ($\\widetilde{K}_{\\lambda,\\mu}(z)$, respectively) denotes the modified $(q,t)$-Kostka polynomial ($z$-Kostka polynomial, respectively).\n\\end{theorem}\n\nThe second generalization starts from rephrasing Theorem~\\ref{Main theorem 1}. For a $mn \\times mn$ permutation matrix $M$, we associate a $n \\times m$ matrix $\\phi(M)$ of content $(1^{mn})$ by letting $k$ be the $(i,j)$-entry if the $(n(i-1)+j, k)$-entry of $M$ is 1. For example, if we set $m=2, n=2$, then the following permutation matrix $M$ in the left corresponds to a $2 \\times 2$ matrix $\\phi(M)$ in the right.\n\\begin{figure}[h]\n  $M=\\begin{pmatrix}\n0 & 0 & 1 & 0 \\\\\n1 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0 \\\\\n0 & 0 & 0 & 1\n  \\end{pmatrix}$\n  \\qquad$\\phi(M)=\\begin{pmatrix}\n  3 & 1\\\\\n  2 & 4\n  \\end{pmatrix}$\n  \\end{figure}\\\\\nUnder this correspondence, the row rotation, column rotation, and adding 1 modulo $mn$ to each entry of $\\phi(M)$ corresponds to `external' row rotation (of order $m$), `internal' row rotation (of order $n$), and column rotation of $M$. Here, by external row rotation, we mean sending the $k$th $n$ rows (from $n(k-1)+1$-th to $nk$-th row) to the next $n$ rows (from $nk+1$-th to $n(k+1)$-th row). Here the row numbers are interpreted modulo $mn$. By internal row rotation, we mean sending each row to the next row except for the $nk$-th row for $k=1,2,\\dots,m$. For the $nk$-th row we send this row to the $n(k-1)+1$-th row. In our running example, if we apply a row rotation and a column rotation to $\\phi(M)$, we get $\\begin{pmatrix} 2&4 \\\\ 3&1\\end{pmatrix}$ and $\\begin{pmatrix} 1&3 \\\\ 4&2\\end{pmatrix}$. Each of these corresponds to the matrix \n\\[\n\\begin{pmatrix}\n0&1&0&0\\\\0&0&0&1 \\\\ 0&0&1&0\\\\ 1&0&0&0 \\end{pmatrix} \n\\text{ and } \n\\begin{pmatrix}\n1&0&0&0\\\\0&0&1&0 \\\\ 0&0&0&1\\\\ 0&1&0&0 \\end{pmatrix}.\n\\] \nThe first matrix can also be obtained from $M$ by sending the first two rows to the third and the fourth row and sending the third and the fourth row to the first and the second row which is an external row rotation (of order 2). Similarly, the second matrix can be obtained from $M$ by applying an internal row rotation (of order 2). Now we can understand Theorem~\\ref{Main theorem 1} as a tricyclic sieving result concerning $mn \\times mn$ matrices under external row rotation, internal row rotation, and column rotation. One might ask if we could get a quadracyclic sieving result concerning external and internal rotation to both columns and rows. The following theorem gives a positive answer.\n\n\\begin{theorem}\\label{Main theorem 3}\nLet $l=mn=ab$ be a positive integer with two factorizations. Let $\\mathfrak{S}_{l}$ be the set of $l\\times l$ permutation matrices. A product of cyclic groups ${\\mathbb {Z}}_n\\times{\\mathbb {Z}}_m\\times{\\mathbb {Z}}_{b}\\times {\\mathbb {Z}}_{a}$ acts on $\\mathfrak{S}_l$ by external row rotation, internal row rotation, external column rotation, and internal column rotation. Then the triple $\\left(\\mathfrak{S}_l, {\\mathbb {Z}}_n\\times{\\mathbb {Z}}_m\\times{\\mathbb {Z}}_{b}\\times {\\mathbb {Z}}_{a}, \\mathfrak{S}_l(q,t,z,w)\\right)$ exhibits the quadraCSP, where\n\n$$\\mathfrak{S}_l(q,t,z,w)=\\sum_{\\lambda\\vdash l} \\widetilde{K}_{\\lambda,(m^n)}(q,t)\\widetilde{K}_{\\lambda,(a^b)}(z,w).$$\n\nHere, $\\widetilde{K}_{\\lambda,\\mu}(q,t)$ denotes the modified $(q,t)$-Kostka polynomial.\n\\end{theorem}\n\n\nThere have been similar results discovered which relate root of unity specializations of $q$-Kostka polynomials or Macdonald polynomials and fixed point enumerations of matrices or fillings of tableaux (see \\cite{Rho10, AU19, BRS08} for example). We remark some results which are especially close to our results. First of all, the set $X_{(m^n),\\nu}$ in Theorem~\\ref{Main theorem 2} bijects with the set of $0,1$-matrices with column content $\\mu$ and row content $(1^{mn})$. If we specialize $n=1$, then Theorem~\\ref{Main theorem 2} recovers \\cite[Theorem 1.2]{Rho10}. In addition, Barcelo, Reiner and Stanton considered biCSP concerning row and column rotation of permutation matrices (\\cite[Theorem 1.4]{BRS08}, in the case $W=\\mathfrak{S}_n$). Theorem~\\ref{Main theorem 3} specializes to their result if we take $n=1, b=1$.\n\nIn \\cite[Theorem 1.3]{Rho10}, using Hall--Littlewood polynomial, Rhoades showed that ${\\mathbb{N}}$-matrices with fixed column content $\\mu$ and row content $\\nu$ exhibits biCSP. It should be mentioned that we modify the argument of Rhoades to prove Theorem~\\ref{Main theorem 2} and Theorem~\\ref{Main theorem 3} in Section~\\ref{subsection: proof of the main theorem 2}. \n\nThe remainder of this paper is organized as follows. In  Section~\\ref{Preliminaries} we give background on combinatorics, symmetric functions, the representation theory of ${\\mathfrak{S}}_n$, and the Garsia--Haiman modules. In Section~\\ref{Section: Sieving generating theorem and orbit harmonics}, we illustrate the diagonal orbit harmonics and how to obtain sieving results from orbit harmonics. We then present a combinatorial locus that gives a graded module isomorphic to Garsia--Haiman module via orbit harmonics. In Section~\\ref{Section: proofs}, we provide proofs of Theorem~\\ref{Main theorem 1}, Theorem~\\ref{Main theorem 2} and Theorem~\\ref{Main theorem 3}. We conclude this paper with some remarks in Section~\\ref{concluding remarks}.\n\n\\section{Preliminaries}\n\\label{Preliminaries}\n\n\\subsection{Combinatorics}\\label{subsection Combinatorics}\nA \\emph{weak composition} of $n$ is a sequence of non-negative integers which sum to $n$. A \\emph{composition} is a weak composition which consists of positive integers. A \\emph{partition} of $n$ is a composition of $n$ which is weakly decreasing. We denote $\\mu\\models n$ and $\\lambda\\vdash n$ for a composition $\\mu$ and a partition $\\lambda$ of $n$. For a composition $\\mu=(\\mu_1,\\dots,\\mu_k)$, the \\emph{length} $l(\\mu)$ of $\\mu$ is $k$.\n\nFor a partition $\\lambda\\vdash n$ we abbuse our notation so that a partition $\\lambda$ also denotes for its \\emph{Young diagram}. We draw Young diagrams in French style:\n$$\\lambda=\\{(i,j)\\in{\\mathbb {Z}}_{\\ge0}\\times{\\mathbb {Z}}_{\\ge0}:i<\\lambda_{j+1}\\}.$$ The elements of Young diagram are called cells. For example,\n\\begin{align*}\n    \\lambda &= (4,3,1)\\\\\n            &=\\{(0,0),(1,0),(2,0),(3,0),(0,1),(1,1),(2,1),(0,2)\\}\n\\end{align*}\n\n\\quad\\qquad\\qquad\\qquad\\qquad = \\quad\\begin{young}\n   \\cr\n & & \\cr\n & & & \\cr\n\\end{young}\\\\\nWe define the \\emph{conjugate} $\\lambda'$ to be the partition obtained by reflecting $\\lambda$ with respect to the diagonal line $x=y$ in the plane. \n\nA \\emph{tableau} of a partition $\\lambda$ is a filling $T:\\lambda\\rightarrow{\\mathbb {Z}}_{>0}$. In this case, we call the \\emph{shape} of $T$ is $\\lambda$. The \\emph{content} of a tableau $T$ of $\\lambda$ is a weak composition $(T_1,T_2,\\dots)$ of $n$, where $T_i$ is the number of $i$'s appearing in $T$. A tableau $T$ is called \\emph{semistandard} if the entries in each row are weakly increasing (left to right) and the entries in each column are strictly increasing (bottom to top). The \\emph{Kostka number} $K_{\\lambda,\\mu}$ is the number of semistandard tableaux of shape $\\lambda$ and content $\\mu$. A semistandard tableau is called \\emph{standard} if its content is $(1,1,\\dots)$. The set of standard tableaux of shape $\\lambda$ is denoted by ${\\mathrm {SYT}}(\\lambda)$. Examples of semistandard tableau and standard tableau of shape $(4,3,1)$ are shown in the left and the right below, respectively.\n\n\\begin{center}\n\\begin{young}\n 4  \\cr\n 2& 2& 4\\cr\n 1& 1& 2& 3\\cr\n\\end{young}\\qquad \\qquad\n\\begin{young}\n 5  \\cr\n 2 & $\\mathbf{4}$ & 7\\cr\n $\\mathbf{1}$ & $\\mathbf{3}$ & $\\mathbf{6}$ & 8\\cr\n\\end{young}\n\n\\end{center}\n\nFor a standard tableau $T$, a \\emph{descent} is an index $i$ such that $i+1$ is in the upper row than $i$. The \\emph{major index} ${\\mathrm {maj}}(T)$ of $T$ is defined as the sum of all descents in $T$. For the standard tableau given in the right above, $1, 3, 4$ and $6$ are descents (descents of the tableau are written in bold), so the major index of the tableau is $1+3+4+6=14$. The \\emph{fake degree polynomial} of a partition $\\lambda$ is defined by the major index generating function for the standard tableaux of shape $\\lambda$, i.e.,\n\\begin{equation*}\n    f^\\lambda(q):=\\sum_{T\\in {\\mathrm {SYT}}(\\lambda)}q^{{\\mathrm {maj}}(T)}.\n\\end{equation*}\n\\subsection{Symmetric functions}\n\\label{subsection Symmetric functions}\nLet $\\Lambda=\\bigoplus_{d\\ge0}\\Lambda_d$ be the ring of symmetric functions in an infinite number of variables ${\\mathbf {x}}=(x_1,x_2,\\dots)$ over $\\mathbb{C}(q,t)$. Here $\\Lambda_d$ denotes the subspace of $\\Lambda$ consisting of symmetric functions of homogeneous degree $d$. \n\nBases of $\\Lambda_n$ are indexed by partitions $\\lambda\\vdash n$. Among various bases of $\\Lambda_n$, one of the most important basis is given by Schur functions. The \\emph{Schur function} $s_\\lambda$ of a partition $\\lambda$ is defined by\n\\begin{equation*}\n    s_\\lambda({\\mathbf {x}}):=\\sum_{T}{\\mathbf {x}}^T,\n\\end{equation*}\nwhere the sum is over all semistandard tableaux of shape $\\lambda$ and ${\\mathbf {x}}^T=x_1^{T_1}x_2^{T_2}\\cdots$.\n\nWe write $\\langle \\cdot, \\cdot\\rangle$ for the Hall inner product \n\\begin{equation*}\n    \\langle s_\\lambda, s_\\mu \\rangle =\\delta_{\\lambda,\\mu},\n\\end{equation*}\nwhere $\\delta_{\\lambda,\\mu}$ is the Kronecker delta. The (modified) \\emph{Macdonald polynomials} $\\widetilde{H}_\\lambda({\\mathbf {x}};q,t)$ indexed by partitions $\\lambda\\vdash n$ form another basis of $\\Lambda_n$. They are defined by the unique family satisfying the following \\emph{triangulation} and \\emph{normalization} axioms \\cite{HHL04},\n\\begin{itemize}\n    \\item $\\widetilde{H}_\\lambda[{\\mathbf {x}}(1-q);q,t]=\\sum_{\\lambda\\ge\\mu} a_{\\lambda,\\mu}(q,t)s_\\lambda$,\n    \\item $\\widetilde{H}_\\lambda[{\\mathbf {x}}(1-t);q,t]=\\sum_{\\lambda\\ge\\mu'} b_{\\lambda,\\mu}(q,t)s_\\lambda$, \n    \\item $\\langle \\widetilde{H}_\\mu, s_{(n)}\\rangle=1$,\n\\end{itemize}\nfor suitable coefficients $a_{\\lambda,\\mu}, b_{\\lambda,\\mu'}\\in{\\mathbb {Q}}(q,t)$. Here, a partial order $\\le$ called \\emph{dominance order} of partitions of $n$ is defined by \n\\begin{equation*}\n    \\lambda\\le\\mu \\text{ if } \\lambda_1+\\cdots+\\lambda_k\\le\\mu_1+\\cdots\\mu_k \\text{ for all } k,\n\\end{equation*} and $[\\cdot]$ denotes the plethystic substitution. These axioms are equivalent with Macdonald's triangularity and orthogonality axioms.\n\nExpanding the Macdonald polynomial with Schur functions, we may write \n\\begin{equation*}\n    \\widetilde{H}_\\mu({\\mathbf {x}};q,t)=\\sum_\\lambda \\widetilde{K}_{\\lambda,\\mu}(q,t)s_{\\lambda}({\\mathbf {x}}),\n\\end{equation*}\nwhere the sum is over partitions $\\lambda$ of $n$. The coefficients $\\widetilde{K}_{\\lambda,\\mu}(q,t)$ are called the (modified) $(q,t)$-\\emph{Kostka polynomials}. A combinatorial description of general $(q,t)$-Kostka polynomials is unknown, and it is one of the most important open problems in algebraic combinatorics. Since $\\widetilde{K}_{\\lambda,\\mu}(1,1)=|{\\mathrm {SYT}}(\\lambda)|$, the most desirable form of the combinatorial formula would be a generating function for the standard tabelaux.\n\nThe (modified) \\emph{Hall--Littlewood polynomial} $\\widetilde{Q}_\\mu(X;q)$ and $q$-\\emph{Kostka polynomial} $\\widetilde{K}_{\\lambda,\\mu}(q)$ can be obtained by specializing $t=0$ to the Macdonald polynomial $\\widetilde{H}_\\mu(X;q,t)$ and the $(q,t)$-Kostka polynomial $\\widetilde{K}_{\\lambda,\\mu}(q,t)$. The $q$-Kostka polynomial $\\widetilde{K}_{\\lambda,\\mu}(q)$ can also be defined as the generating function of the cocharge statistics for the semistandard tableaux of shape $\\lambda$ and content $\\mu$ (see \\cite{Rho10} for a definition of cocharge statistics). \n\n\\subsection{Representation theory of $\\mathfrak{S}_n$}\\label{subsection rep of Sn}\n\nIrreducible representations of the symmetric group $\\mathfrak{S}_n$ are in one to one correspondence with partitions $\\lambda$ of $n$. We let $S^\\lambda$ be the corresponding irreducible representation. If $V$ is a finite dimensional $\\mathfrak{S}_n$-module, there is a unique way of decomposing $V$ into irreducibles as $V=\\bigoplus_{\\lambda\\vdash n} c_\\lambda S^\\lambda$. The \\emph{Frobenius image} of $V$ is the symmetric function defined by \n\\begin{equation*}\n    {\\mathrm {Frob}}(V):=\\sum_{\\lambda\\vdash n} c_\\lambda s_\\lambda.\n\\end{equation*}\nIf $V$ is graded (or bigraded) $\\mathfrak{S}_n$-module as $V=\\bigoplus_{d\\ge0} V_d$ (or $V=\\bigoplus_{d,e\\ge0} V_{d,e}$) the graded Frobenius image is the symmetric function over ${\\mathbb {C}}(q)$ (or ${\\mathbb {C}}(q,t)$) given by\n\n\\begin{equation*}\n{\\mathrm {grFrob}}(V;q) :=\\sum_{d\\ge0}{\\mathrm {Frob}}(V_d)q^d\n\\end{equation*}\n\\begin{equation*}\n    (\\text{or }{\\mathrm {grFrob}}(V;q,t):=\\sum_{d,e\\ge0} {\\mathrm {Frob}}(V_{d,e})q^d t^e).\n\\end{equation*}\nIf $V = \\bigoplus_{d \\geq 0} V_d$ (or $V=\\bigoplus_{d,e\\ge 0} V_{d,e}$ is any graded (or bigraded) vector space, its {\\em Hilbert series} is\n \\begin{equation*}\n {\\mathrm {Hilb}}(V;q) = \\sum_{d \\geq 0} \\dim (V_d) q^d\n \\end{equation*}\n \\begin{equation*}\n    (\\text{or }{\\mathrm {Hilb}}(V;q,t):=\\sum_{d,e\\ge0} \\dim(V_{d,e})q^d t^e).\n\\end{equation*}\n\n\nWe recall two facts that we use in the proof of the main results. The first one is the theorem of Springer. Note that the original result of Springer deals with the action of a regular element of an arbitrary complex reflection group. For simplicity, we focus only on the action of a long cycle in a symmetric group ${\\mathfrak{S}}_n$.\n\n\\begin{theorem}(\\cite{Spr74})\n\\label{springer-theorem}\nLet $c=(1,2,\\dots,n)$ be a long cycle of ${\\mathfrak{S}}_n$ and $\\chi^\\lambda:{\\mathfrak{S}}_n \\rightarrow {\\mathbb {C}}$ be the irreducible character associated to the $\\mathfrak{S}_n$-representation $S^\\lambda$. Then we have\n\\begin{equation*}\n\\chi^\\lambda(c^r) = f^{\\lambda}(\\zeta^r),\n\\end{equation*}\nwhere $f^\\lambda(q) \\in {\\mathbb {C}}[q]$ is the fake degree polynomial and $\\zeta$ is a (primitive) $n$-th root of unity.\n\\end{theorem}\n\nThe second fact we need is about the Kronecker coefficients. For ${\\mathfrak{S}}_n$-modulues $V$ and $W$, define the \\emph{inner tensor product} $V\\otimes W$ to be the the usual tensor product of vector spaces with $\\mathfrak{S}_n$-module structure given by $$\\sigma\\cdot(v\\otimes w)=(\\sigma \\cdot v)\\otimes (\\sigma \\cdot w).$$ For given partitions $\\lambda, \\mu$ and $\\nu$ of $n$, the \\emph{Kronecker coefficients} $g^\\lambda_{\\mu,\\nu}$ is the multiplicity of $S^\\lambda$ in the inner tensor product $S^\\mu\\otimes S^\\nu$, i.e. $$S^\\mu\\otimes S^\\nu\\cong\\bigoplus g^\\lambda_{\\mu,\\nu}S^\\lambda.$$ Although giving an explicit combinatorial description of general Kronecker coefficients is difficult in general (it is one of the major open problems in algebraic combinatorics), we have the following identity for the special case when $\\lambda$ is a single row.\n\n\\begin{proposition}\\label{prop:Kronecker} For partitions $\\mu,\\nu$ of $n$, the Kronecker coefficient\n$g^{(n)}_{\\mu,\\nu}=\\delta_{\\mu,\\nu},$\nwhere $\\delta_{\\mu,\\nu}$ is the Kronecker delta.\n\\end{proposition}\n\n\n\n\n\\subsection{Modules of Garsia and Haiman}\\label{modules of garsia--Haiman}\n\nTo each partition $\\mu$ of $n$, let $(a_1, b_1), \\dots, (a_n, b_n)$ be the cells of $\\mu$, taken in some arbitrary order. Then we define\n\\[\n\\Delta_\\mu:=\\mathrm{det}(x_i^{a_j} y_i^{b_j})_{1\\le i,j\\le n}.\n\\]\nThe ${\\mathfrak{S}}_n$-module $\\mathbf{H}_\\mu$ is the smallest vector space over ${\\mathbb {C}}$ that contains the determinant $\\Delta_\\mu$ and its partial derivatives with respect to any of the variables $x_i$'s and $y_i$'s for $1\\le i \\le n$. For example, for a partition $\\mu=(3,2)$, the cells of $\\mu$ are $(0,0), (1,0), (2,0), (0,1)$ and $(1,1)$ and the corresponding determinant is given by \n\\begin{equation*}\n\\Delta_\\mu:=\\mathrm{det}\\left[\\begin{array}{ccccc}\n1 & x_1 & x_{1}^{2} & y_1 & x_{1} y_1 \\\\\n1 & x_2 & x_{2}^{2} & y_2 & x_{3} y_3 \\\\\n1 & x_3 & x_{3}^{2} & y_3 & x_{3}y_3 \\\\\n1 & x_4 & x_4^2 & y_4 & x_{4}y_4 \\\\\n1 & x_5 & x_5^{2} & y_5 & x_{5}y_5\n\\end{array}\\right].\n\\end{equation*}\nThen the module $\\mathbf{H}_\\mu$ is given by the $\\mathbb{C}$-span \n\\begin{equation*}\n\\mathbb{C}\\{\\partial_{{\\mathbf {x}}_I}\\partial_{{\\mathbf {y}}_J}\\Delta_\\mu\\}_{I,J}\n\\end{equation*}\nwhere $\\partial_{{\\mathbf {x}}_I}:=\\partial_{x_{i_1}}\\dots\\partial_{x_{i_k}}$ for a multiset $I=\\{i_1,\\cdots,i_k\\}$ and $\\partial_{{\\mathbf {y}}_J}$ is defined similarly. The symmetric group ${\\mathfrak{S}}_n$ acts on ${\\mathbf {H}}_\\mu$ diagonally i.e. permuting $x$ and $y$ coordinates simultaneously. Since the action of ${\\mathfrak{S}}_n$ preserves the degree of ${\\mathbf {x}}_n$ and ${\\mathbf {y}}_n$, the module ${\\mathbf {H}}_\\mu$ is bigraded ${\\mathfrak{S}}_n$-module, where the bigrade is given by degree of ${\\mathbf {x}}_n$ and ${\\mathbf {y}}_n$. This is called the \\emph{Garsia--Haiman module}. Haiman \\cite{Hai01} proved the \\emph{$n!$ conjecture} which asserts that this module is of dimension $n!$ regardless of $\\mu$, and moreover, the graded Frobenius image of ${\\mathbf {H}}_\\mu$ is the Macdonald polynomial of $\\mu$:\n\\begin{equation}\\label{graded Frob=Macdonald}\n    {\\mathrm {grFrob}}({\\mathbf {H}}_\\mu;q,t)=\\widetilde{H}_\\mu({\\mathbf {x}};q,t)=\\sum_\\lambda \\widetilde{K}_{\\lambda,\\mu}(q,t)s_\\lambda({\\mathbf {x}}),\n\\end{equation}\nwhich proves the Schur positivity of the Macdonald polynomials.\n\n\\section{Sieving generating theorem and diagonal orbit harmonics}\n\\label{Section: Sieving generating theorem and orbit harmonics}\n\n\n\\subsection{Orbit harmonics and cyclic sieving}\n\\label{subsection Orbit harmonics and cyclic sieving}\nIn this section, we introduce a systematic way to generate sieving results using orbit harmonics. The author and Rhoades provided a `generating theorem' for sieving results \\cite[Theorem 3.4]{OR20} by exploiting orbit harmonics applied to a locus $X\\subseteq \\mathbb{C}^n$ with ${\\mathfrak{S}}_n$ acting on $X$ by permuting coordinates. To modify this idea for our purpose, we first explain the \\emph{diagonal orbit harmonics} (see \\cite{GH96} for more details). Consider a finite set $X\\subseteq {\\mathbb {C}}^{2n}$ which is closed under $\\mathfrak{S}_n\\times C_1\\times C_2$-action where \n\\begin{itemize}\n    \\item a symmetric group $\\mathfrak{S}_n$ acts on $\\mathbb{C}^{2n}$ by permuting coordinates diagonally, i.e. $$\\sigma(x_1,\\dots,x_n,y_1,\\dots,y_n)=(x_{\\sigma(1)},\\dots,x_{\\sigma(n)},y_{\\sigma(1)},\\dots,y_{\\sigma(n)}),$$\n    \\item a finite cyclic group $C_1$ is acts on $\\mathbb{C}^n$ by scaling $x$-coordinates by a root of unity, and\n    \\item a finite cyclic group $C_2$ is acts on $\\mathbb{C}^n$ by scaling $y$-coordinates by a root of unity.\n\\end{itemize}\nThen the method of orbit harmonics gives us an isomorphism of $\\mathfrak{S}_n\\times C_1 \\times C_2$-modules:\n\\begin{equation}\\label{isomorphism T-ideal xy}\n    \\mathbb{C}[X]\\cong\\mathbb{C}[{\\mathbf {x}}_n,{\\mathbf {y}}_n]\/\\mathbf{I}(X).\n\\end{equation}\nWe further define homogeneous ideal\n\\begin{equation*}\n    {\\mathbf {T}}(X):=\\langle \\tau_x\\circ\\tau_y(f): f\\in {\\mathbf {I}}(X)\\setminus \\{0\\}\\rangle \\subseteq {\\mathbb {C}}[{\\mathbf {x}}_n],\n\\end{equation*}\nwhere $\\tau_x$ and $\\tau_y$ is the map taking top degree homogeneous part with respect to ${\\mathbf {x}}_n$ and ${\\mathbf {y}}_n$, respectively. Then the isomorphism~\\eqref{isomorphism T-ideal xy} extends to an isomorphism\n\\begin{equation*}\n    \\mathbb{C}[X]\\cong\\mathbb{C}[{\\mathbf {x}}_n,{\\mathbf {y}}_n]\/\\mathbf{I}(X)\\cong\\mathbb{C}[{\\mathbf {x}}_n,{\\mathbf {y}}_n]\/\\mathbf{T}(X),\n\\end{equation*}\nwhere the last item $\\mathbb{C}[{\\mathbf {x}}_n,{\\mathbf {y}}_n]\/\\mathbf{T}(X)$ has an additional structure of graded $\\mathfrak{S}_n\\times C_1\\times C_2$-module on which $C_1$ and $C_2$ acts by scaling in each fixed (bi)degree. Thanks to this isomorphism, we can provide a generating theorem for sieving results in diagonal orbit harmonics whose proof is analogous to the proof of Theorem 3.4 in \\cite{OR20}.\n\n\\begin{theorem}\n\\label{sieving-generator}\nLet $C$ be the subgroup of $\\mathfrak{S}_n$ generated by a long cycle $c=(1,2,\\dots,n)$. Fix positive integers $k_1$ and $k_2$. For $j=1,2$, let\n$\\zeta_j := \\exp(2 \\pi i \/ k_j) \\in {\\mathbb {C}}^{\\times}$ and $C_j = \\langle c_j \\rangle \\cong {\\mathbb {Z}}_{k_j}$ be a cyclic group of order $k_j$. Consider the action of $\\mathfrak{S}_n \\times C_1\\times C_2$ on ${\\mathbb {C}}^{2n}$ where $c_1$ scales $x$-coordinates by $\\zeta_1$, $c_2$ scales $y$-coordinates by $\\zeta_2$ and $\\mathfrak{S}_n$ acts by permuting coordinates diagonally.\n\nLet $X \\subseteq {\\mathbb {C}}^{2n}$ be a finite point set which is closed under the action of $\\mathfrak{S}_n\\times C_1\\times C_2$.\n\\begin{enumerate}\n\\item  Suppose that for $d,e \\geq 0$, the isomorphism type of the degree $(d,e)$-piece of \n${\\mathbb {C}}[{\\mathbf {x}}_n,{\\mathbf {y}}_n]\/{\\mathbf {T}}(X)$ is given by\n\\begin{equation*}\n( {\\mathbb {C}}[{\\mathbf {x}}_n,{\\mathbf {y}}_n]\/{\\mathbf {T}}(X) )_{d,e} \\cong \\bigoplus_{\\lambda\\vdash n} c_{\\lambda,d,e}S^\\lambda.\n\\end{equation*}\nThe triple $(X, C_1\\times C_2 \\times C, X(q,t,z))$ exhibits the tricyclic sieving phenomenon where \n\\begin{equation*}\nX(q,t,z) = \\sum_{\\lambda\\vdash n}  c_{\\lambda}(q,t) f^{\\lambda}(z).\n\\end{equation*}\nwhere $c_{\\lambda}(q,t) := \\sum_{d,e \\geq 0} c_{\\lambda,d,e} q^d t^e$.\n\\item Let $G \\subseteq \\mathfrak{S}_n$ be a subgroup. The set $X\/G$ of $G$-orbits in $X$ carries a natural\n$C_1\\times C_2$-action and the triple $(X\/G, C_1\\times C_2, X\/G(q,t))$ exhibits the bicyclic sieving phenomenon where\n\\begin{equation*}\nX\/G(q,t) = {\\mathrm {Hilb}}( ({\\mathbb {C}}[{\\mathbf {x}}_n,{\\mathbf {y}}_n]\/{\\mathbf {T}}(X))^G; q,t).\n\\end{equation*}\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof} Applying orbit harmonics to the action of ${\\mathfrak{S}}_n \\times C_1 \\times C_2$ on $X$ \nyields an isomorphism of ungraded ${\\mathfrak{S}}_n \\times C_1 \\times C_2$-modules\n\\begin{equation}\n\\label{ungraded-isomorphism}\n{\\mathbb {C}}[X] \\cong {\\mathbb {C}}[{\\mathbf {x}}_n,{\\mathbf {y}}_n]\/{\\mathbf {T}}(X).\n\\end{equation}\nLet $\\zeta := \\exp(2 \\pi i \/ n)$.\nTo prove (1), apply Theorem~\\ref{springer-theorem} to obtain that \nfor any integers $r, s, k$, the trace of $(c_1^r, c_2^s, c^k) \\in C_1\\times C_2 \\times C'$ acting on ${\\mathbb {C}}[{\\mathbf {x}}_n,{\\mathbf {y}}_n]\/{\\mathbf {T}}(X)$ is given by $$\\sum_{\\lambda\\vdash n} c_\\lambda(\\zeta_1^r, \\zeta_2^s) f^{\\lambda}(\\zeta^k) = X(\\zeta^r, \\zeta_2^s, \\zeta^k).$$\nBy the isomorphism \\eqref{ungraded-isomorphism}, this coincides with the trace of $(c_1^r, c_2^s, c^k)$ which is the number of fixed\npoints of $(c_1^r, c_2^s, c^k)$ acting on $X$, completing the proof of (1).\n\nFor (2), we take $G$-invariants of both sides of the isomorphism \\eqref{ungraded-isomorphism}\nto get an isomorphism of $C_1\\times C_2$-modules\n\\begin{equation}\n\\label{g-invariants}\n{\\mathbb {C}}[X\/G] \\cong ({\\mathbb {C}}[{\\mathbf {x}}_n,{\\mathbf {y}}_n]\/{\\mathbf {T}}(X))^G.\n\\end{equation}\nSince $C_1\\times C_2$ acts on the graded vector space $({\\mathbb {C}}[{\\mathbf {x}}_n,{\\mathbf {y}}_n]\/{\\mathbf {T}}(X))^G$ by a root of unity scaling for each $x$ and $y$ variables, the trace of $(c_1^r, c_2^s)$ on the right hand side of the isomorphism \\eqref{g-invariants} is given by\n$$[{\\mathrm {Hilb}}( ({\\mathbb {C}}[{\\mathbf {x}}_n,{\\mathbf {y}}_n]\/{\\mathbf {T}}(X))^G; q,t)]_{q = \\zeta_1^r, t=\\zeta_2^s} = X\/G(\\zeta_1^r,\\zeta_2^s).$$ \nThe trace of $(c_1^r, c_2^s)$ on the left hand side of the isomorphism \\eqref{g-invariants} coincides with the number \nof $G$-orbits in $X\/G$ fixed by $(c_1^r, c_2^s)$.\n\\end{proof}\n\n\\begin{remark}\\label{remark sieving generator}\nIn order to obtain a sieving result involving a combinatorial set $X$ with a cyclic group action using Theorem~\\ref{sieving-generator}, we must \n\\begin{itemize}\n    \\item realize $X$ (or its quotient $X\/G$) and the relevant action on it as a point locus in ${\\mathbb {C}}^{2n}$ and the compatible action,\n    \\item calculate the graded Frobenius image of ${\\mathbb {C}}[{\\mathbf {x}}_n,{\\mathbf {y}}_n]\/{\\mathbf {T}}(X)$ or the Hilbert series of the quotient ${\\mathrm {Hilb}}\\left(({\\mathbb {C}}[{\\mathbf {x}}_n,{\\mathbf {y}}_n]\/{\\mathbf {T}}(X))^G; q,t\\right)$.\n\\end{itemize}\n\n\\end{remark}\n\n\\subsection{Orbit harmonics and Garsia--Haiman module}\\label{Modules of Garsia--Haiman and Garsia--Procesi}\nThere is a way to understand the Garsia--Haiman module ${\\mathbf {H}}_\\mu$ via orbit harmonics. Let $\\mu$ be a partition of $n$ with $l(\\mu)=l$ and $l(\\mu')=l'$. Let $\\{\\alpha_0,\\dots,\\alpha_{l-1}\\}$ and $\\{\\beta_0,\\dots,\\beta_{l'-1}\\}$ be two sets of distinct complex numbers. Recall that a \\emph{injective tableau} $T$ of shape $\\mu\\vdash n$ is a filling of cells of $\\mu$ by integers $1,2,\\dots,n$ without repetition. The collection of such tableaux will be denoted by $\\mathbf{IT}(\\mu)$. For each $T\\in\\mathbf{IT}(\\mu)$, we assign a point $p_T\\in{\\mathbb {C}}^{2n}$ by letting the $i$-th and the $(n+i)$-th coordinates of $p_T$ record the position of $i$ in $T$:\n\\begin{equation*}\n    p_T=(\\alpha_{y_T(1)},\\dots,\\alpha_{y_T(n)},\\beta_{x_T(1)},\\dots,\\beta_{x_T(n)}),\n\\end{equation*}\nwhere $x_T(i)$ and $y_T(i)$ are $x$ and $y$ coordinates of the cell which contains $i$ in $T$. For example, for a partition $\\mu=(2,1)$ and an injective tableau $T=$\\begin{young}\n 2 \\cr\n 3& 1\\cr\n\\end{young} of shape $\\mu$, the point assigned for $T$ is $p_T=(\\alpha_0, \\alpha_1, \\alpha_0, \\beta_1, \\beta_0, \\beta_0)$. Let us denote the collection of points associated to the injective tableaux by\n\\begin{equation*}\n    X_\\mu=\\{p_T\\in{\\mathbb {C}}^{2n}:T\\in\\mathbf{IT}(\\mu)\\}.\n\\end{equation*}\nNote that there are exactly $n!$ points in $X_\\mu$. The point locus $X_\\mu$ possesses a natural diagonal action of ${\\mathfrak{S}}_n$: For $\\sigma\\in{\\mathfrak{S}}_n$,\n\\begin{equation*}\n    \\sigma (x_1,\\dots,x_n,y_1,\\dots,y_n)=(x_{\\sigma(1)},\\dots,x_{\\sigma(n)},y_{\\sigma(1)},\\dots,y_{\\sigma(n)}).\n\\end{equation*}\nUsing orbit harmonics, one can promote the ungraded ${\\mathfrak{S}}_n$-module ${\\mathbb {C}}[X_\\mu]$ to the bigraded ${\\mathfrak{S}}_n$-module. As usual, let $\\mathbf{I}(X_\\mu)$ be the ideal of polynomials in $\\mathbb{C}[{\\mathbf {x}}_n,{\\mathbf {y}}_n]$ which vanish on $X$ and define a homogeneous ideal \n\\begin{equation*}\n    \\mathbf{T}(X_\\mu):=\\langle\\tau_x\\circ\\tau_y(f):f\\in\\mathbf{I}(X_\\mu)\\setminus\\{0\\}\\rangle\\subseteq\\mathbb{C}[{\\mathbf {x}}_n,{\\mathbf {y}}_n].\n\\end{equation*}\nThen the module ${\\mathbf {R}}_\\mu:=\\mathbb{C}[{\\mathbf {x}}_n,{\\mathbf {y}}_n]\/\\mathbf{T}(X_\\mu)$ has an additional structure of (bi)graded $\\mathfrak{S}_n$-module.\n\nGarsia and Haiman \\cite{GH96} proved that the Garsia--Haiman module ${\\mathbf {H}}_\\mu$ embedds into this graded module ${\\mathbf {R}}_\\mu$. Thanks to the $n!$-conjecture, we can conclude the following isomorphism between ${\\mathbf {R}}_\\mu$ and ${\\mathbf {H}}_\\mu$.\n\\begin{theorem}\n\\label{theorem:Garsia--Haiman module} We have an isomorphism as bigraded ${\\mathfrak{S}}_n$-modules:\n$${\\mathbf {R}}_\\mu\\cong{\\mathbb {C}}\\left[X_\\mu\\right]\\cong {\\mathbf {H}}_\\mu.$$\n\\end{theorem}\n\n\\section{Proofs of main theorems}\n\\label{Section: proofs}\n\\subsection{A proof of Theorem~\\ref{Main theorem 1}}\n\\label{subsection: proof of the main theorem}\n\nWe first construct a point locus $X_{(m^n)}$ associated to a rectangular partition $\\mu=(m^n)$. Following Section~\\ref{Modules of Garsia--Haiman and Garsia--Procesi}, to consider $X_{(m^n)}$ as a point locus in $\\mathbb{C}^{2n}$, we choose two sets of distinct complex numbers $\\{\\alpha_0,\\dots,\\alpha_{n-1}\\}$ and $\\{\\beta_0,\\dots,\\beta_{m-1}\\}$. For our purpose, let $\\zeta_1=\\exp(\\frac{2\\pi i}{n})$ and $\\zeta_2=\\exp(\\frac{2\\pi i}{m})$, then set $\\alpha_j=\\zeta_1^{j}$ for $0\\le j \\le n-1$ and $\\beta_k=\\zeta_2^{k}$ for $0\\le k \\le m-1$. Then the corresponding locus $X_\\mu$ possesses\n\\begin{itemize}\n    \\item diagonal action of ${\\mathfrak{S}}_n$,\n    \\item action of a cyclic group $C_1$ of order $n$ acting by scaling a root of unity $\\zeta_1$ to each $x$-coordinates, and \n    \\item action of a cyclic group $C_2$ of order $m$ acting by scaling a root of unity $\\zeta_2$ to each $y$-coordinates.\n\\end{itemize}\n\nNow we can present a proof of Theorem~\\ref{Main theorem 1}. By the construction above, $X_{(m^n)}$ has ${\\mathfrak{S}}_n\\times C_1\\times C_2$ action which corresponds to permutation of letters, row rotation, and column rotation on $\\mathbf{IT}(\\mu)$, respectively. Combining an isomorphism between ${\\mathbf {R}}_\\mu$ and ${\\mathbf {H}}_\\mu$ (Theorem~\\ref{theorem:Garsia--Haiman module}), the fraded Frobenius image of ${\\mathbf {H}}_\\mu$ (Equation~\\eqref{graded Frob=Macdonald}) and the sieving generating theorem (Theorem~\\ref{sieving-generator}), the first bullet point of Theorem~\\ref{Main theorem 1} immediately follows.\n\nTo proceed to the second bullet point, consider a composition $\\nu$ of $mn$. For the Young subgroup $G={\\mathfrak{S}}_\\nu={\\mathfrak{S}}_{\\nu_1}\\times{\\mathfrak{S}}_{\\nu_2}\\cdots$ of $\\nu$, the $G$-orbits of $X_{(m^n)}$ are in one-to-one correspondence with the set of $n\\times m$ matrices with content equal to $\\nu$. \n\nTo obtain a sieving result for $X_{(m^n)}\/G$, we must calculate the Hilbert series of $G$-fixed subspace of ${\\mathbf {R}}_\\mu$. Let ${\\bf 1}$ be the trivial representation of ${\\mathfrak{S}}_\\nu$. It is a standard fact that the induction of ${\\bf 1}$ from ${\\mathfrak{S}}_\\nu$ to ${\\mathfrak{S}}_n$\ncan be written as\n\\begin{equation*}\n{\\bf 1} \\uparrow_{{\\mathfrak{S}}_\\nu}^{{\\mathfrak{S}}_n} \\cong \\bigoplus_\\lambda K_{\\lambda,\\nu} S^\\lambda,\n\\end{equation*}\nwhere $K_{\\lambda,\\nu}$ denotes a Kostka number. Applying Frobenius reciprocity, it follows that the dimension of the ${\\mathfrak{S}}_\\nu$-fixed subspace of \nthe ${\\mathfrak{S}}_n$-irreducible $S^{\\lambda}$ is given by the character inner product:\n\\begin{equation*}\n\\dim (S^{\\lambda})^{{\\mathfrak{S}}_\\nu} = \\langle {\\bf 1}, S^{\\lambda} \\downarrow^{{\\mathfrak{S}}_n}_{{\\mathfrak{S}}_\\nu} \\rangle_{{\\mathfrak{S}}_\\nu} = \n\\langle {\\bf 1} \\uparrow_{{\\mathfrak{S}}_\\nu}^{{\\mathfrak{S}}_n}, S^{\\lambda} \\rangle_{{\\mathfrak{S}}_n} = K_{\\lambda,\\nu}.\n\\end{equation*}\nCorrespondingly, if $V$ is any bigraded ${\\mathfrak{S}}_n$-module with Frobenius image\n\\begin{equation*}\n    {\\mathrm {grFrob}}(V;q,t)=\\sum_{\\lambda\\vdash n}c_{\\lambda}(q,t)S^\\lambda,\n\\end{equation*}\nthe Hilbert series of ${\\mathfrak{S}}_\\nu$ fixed subspace will be\n\\begin{equation*}\n    {\\mathrm {Hilb}}(V^{{\\mathfrak{S}}_\\nu};q,t)=\\sum_{\\lambda\\vdash n} c_{\\lambda}(q,t)K_{\\lambda,\\nu}.\n\\end{equation*}\nBy (2) of Theorem~\\ref{sieving-generator}, this concludes the second bullet point.\n\n\\subsection{A proof of Theorem~\\ref{Main theorem 2} and Theorem~\\ref{Main theorem 3}}\n\\label{subsection: proof of the main theorem 2}\n\nIn previous section, we proved that for a composition $\\nu$ of $mn$, the triple $\\left(X_{(m^n),\\nu}, {\\mathbb {Z}}_n\\times {\\mathbb {Z}}_m, \\sum_\\lambda \\widetilde{K}_{\\lambda,\\mu}(q,t)K_{\\lambda,\\nu}\\right)$ exhibits biCSP. Suppose, furthermore, $\\nu$ has a cyclic symmetry of order $a$. Then the set $X_{(m^n),\\nu}$ possesses another cyclic group action by adding $a$ modulo $l(\\nu)$ to each entry. Therefore, it is natural to seek for a sieving result that reflects this additional cyclic group action. \n\n\nBefore we begin, we recall the Tanisaki locus. For a composition $\\nu\\models d$, let $W_\\nu$ be the set of length $d$ words $w=(w_1,\\dots,w_d)$ of content $\\nu$ ($i$ appears $\\nu_i$ times). Let $\\zeta=\\exp\\left(\\frac{2\\pi i}{l(\\nu)}\\right)$. We assign a point $p_w$ in ${\\mathbb {C}}^d$ so that we can realize $W_\\nu$ as a point locus $Y_\\nu$ (called the Tanisaki locus) in ${\\mathbb {C}}^d$ as follows:\n\\begin{equation*}\n    p_w=(\\zeta^{w_1},\\dots,\\zeta^{w_d}).\n\\end{equation*}\nGarsia and Procesi \\cite{GP92} proved that the T-ideal corresponding to the Tanisaki locus is given by the ideal generated by elementary symmetric polynomials with extra conditions (for precise definition of this `Tanisaki ideal', we refer \\cite{GP92}). By orbit harmonics, there is an isomorphism\n\\begin{equation}\\label{Garsia--Procesi module orbit harmonics}\n    {\\mathbb {C}}[Y_\\nu]\\cong {\\mathbf {L}}_\\nu:={\\mathbb {C}}[{\\mathbf {x}}_d]\/{\\mathbf {T}}(Y_\\nu).\n\\end{equation}\nMoreover, they showed that the graded Frobenius image coincides with the Hall--Littlewood symmetric function, $${\\mathrm {grFrob}}({\\mathbf {L}}_\\nu;q)=\\widetilde{Q}_\\nu({\\mathbf {x}};q).$$\nFurthermore, if a composition $\\nu$ has a cyclic symmetry of order $a$, the set $W_\\nu$ has additional cyclic group action given by adding $a$ modulo $l(\\nu)$ to each letter. This action corresponds with the action of scaling a root of unity $\\zeta^a$ in each coordinates in $Y_\\nu$. In this setting, the isomorphism~\\eqref{Garsia--Procesi module orbit harmonics} extends to an isomorphism as graded ${\\mathfrak{S}}_d\\times C$-modules, where $C$ is a cyclic group of order $l(\\nu)\/a$.\n\nNow let $\\mu=(m^n)$ be a rectangular partition and $\\nu$ be a composition of $mn$ with a cyclic symmetry of order $a$. Then the product $X_\\mu\\times Y_\\nu$ carries an $\\mathfrak{S}_{mn}\\times {\\mathbb {Z}}_n\\times {\\mathbb {Z}}_m \\times {\\mathbb {Z}}_{l(\\nu)\/a}$-action, where $\\mathfrak{S}_{mn}$ acts diagonally on $X_\\mu$ and $Y_\\nu$ and the cyclic groups ${\\mathbb {Z}}_n$, ${\\mathbb {Z}}_m$ and ${\\mathbb {Z}}_{l(\\nu)\/a}$ acts by row rotation on $X_\\mu$, column rotation on $X_\\mu$ and translation on the entries on $Y_\\nu$, respectively. By Theorem~\\ref{theorem:Garsia--Haiman module} and the isomorphism~\\eqref{Garsia--Procesi module orbit harmonics}, we have an isomorphism\n\\begin{equation}\\label{equation: isomorphism between product}\n{\\mathbb {C}}[X_\\mu\\times Y_\\nu]\\cong {\\mathbf {R}}_\\mu \\otimes {\\mathbf {L}}_\\nu\n\\end{equation}\nas $\\mathfrak{S}_{mn}\\times{\\mathbb {Z}}_n\\times{\\mathbb {Z}}_m\\times{\\mathbb {Z}}_{l(\\nu)\/a}$-modules. Since the graded Frobenius image of the module ${\\mathbf {R}}_\\mu$ is given by the Macdonald polynomial and the graded Frobenius image of the Garsia--Procesi module ${\\mathbf {L}}_\\nu$ is given by the Hall--Littlewood polynomial, the Frobenius image is given by\n\\begin{equation*}\n\\label{Frob Xmu times Ynu}\n    {\\mathrm {grFrob}}\\left({\\mathbb {C}}[X_\\mu\\times Y_\\nu];q,t,z\\right)=\\sum_{\\rho,\\lambda,\\lambda'\\vdash mn} \\widetilde{K}_{\\lambda,\\mu}(q,t)\\widetilde{K}_{\\lambda',\\nu}(z)g^{\\rho}_{\\lambda,\\lambda'}s_\\rho,\n\\end{equation*}\nwhere $g^\\rho_{\\lambda,\\lambda'}$ denotes a Kronecker coefficient.\nBy taking isotypic components for a trivial representation $S^{(mn)}$ of $\\mathfrak{S}_{mn}$ on both sides of equation~\\eqref{equation: isomorphism between product}, we have\n\\begin{equation}\\label{equation: isomorphism isotypic components}\n{\\mathbb {C}}[X_\\mu\\times Y_\\nu]^{{\\mathfrak{S}}_{mn}}\\cong [{\\mathbf {R}}_\\mu\\otimes {\\mathbf {L}}_\\nu]^{{\\mathfrak{S}}_{mn}}.\n\\end{equation}\n\nThere is a natural basis of ${\\mathbb {C}}[X_\\mu\\times Y_\\nu]^{\\mathfrak{S}_{mn}}$ indexed by $\\mathfrak{S}_{mn}$-orbits of $X_\\mu\\times Y_\\nu$, given by the sum of elements in each orbit. Note that each of these orbits corresponds to a $n\\times m$ matrix with content equal to $\\nu$. It is clear that the cyclic groups ${\\mathbb {Z}}_n$, ${\\mathbb {Z}}_{m}$ and ${\\mathbb {Z}}_{l(\\nu)\/a}$ act on these matrices by row rotation, column rotation, and translation of the entries. For an element $g\\in {\\mathbb {Z}}_n\\times {\\mathbb {Z}}_m \\times {\\mathbb {Z}}_{l(\\nu)\/a}$, we can count the number of fixed points of $g$ in $(X_\\mu\\times Y_\\nu)\/{\\mathfrak{S}_{mn}}$ is given by the trace of $g$ acting on the left hand side of the isomorphism~\\eqref{equation: isomorphism isotypic components}. On the other hand, this can be calculated by trigraded Hilbert series \n\\begin{equation}\\label{Hilbert poly of S_mn invariant of R_mu times L_nu}\n    {\\mathrm {Hilb}}\\left(\\left[{\\mathbf {R}}_\\mu\\otimes {\\mathbf {L}}_\\nu\\right]^{{\\mathfrak{S}}_{mn}};q,t,z\\right)=\\sum_{\\lambda, \\lambda'\\vdash mn} \\widetilde{K}_{\\lambda,(m^n)}(q,t)\\widetilde{K}_{\\lambda',\\nu}(z)g^{(mn)}_{\\lambda,\\lambda'},\n\\end{equation}\nof $[{\\mathbf {R}}_\\mu\\otimes_{{\\mathbb {C}}}{\\mathbf {L}}_\\nu]^{\\mathfrak{S}_{mn}}$ at roots of unity. By Proposition~\\ref{prop:Kronecker}, taking the coefficient of the Schur function $s_{(mn)}$ in Equation~\\eqref{Hilbert poly of S_mn invariant of R_mu times L_nu}, we have the following polynomial\n$$ {\\mathrm {Hilb}}\\left([{\\mathbf {R}}_\\mu\\otimes {\\mathbf {L}}_\\nu]^{{\\mathfrak{S}}_{mn}};q,t,z\\right)=X_{\\mu,\\nu}(q,t,z):=\\sum_{\\lambda\\vdash mn} \\widetilde{K}_{\\lambda,(m^n)}(q,t)\\widetilde{K}_{\\lambda,\\nu}(z)$$\nfor sieving result. This proves Theorem~\\ref{Main theorem 2}.\n\\begin{example}\nTake $\\mu=(2,2)$ and $\\nu=(2,2)$. We have six $2\\times2$ matrices with content equal to $\\nu$ listed in the following.\n\n\\begin{figure}[h]\n  $\n    \\begin{pmatrix}\n    1 & 1 \\\\\n    2 & 2\n    \\end{pmatrix}$\n    \\qquad$\n    \\begin{pmatrix}\n    1 & 2\\\\\n    1 & 2\n    \\end{pmatrix}$\n    \\qquad$\n    \\begin{pmatrix}\n    1 & 2\\\\\n    2 & 1\n    \\end{pmatrix}$\n    \\qquad$\n    \\begin{pmatrix}\n    2 & 1\\\\\n    1 & 2\n    \\end{pmatrix}$\n    \\qquad$\n    \\begin{pmatrix}\n    2 & 1\\\\\n    2 & 1\n    \\end{pmatrix}$\n    \\qquad$\n    \\begin{pmatrix}\n    2 & 2\\\\\n    1 & 1\n    \\end{pmatrix}$\n\n  \\end{figure}\n\nFixed points of $(0,1,1)\\in{\\mathbb {Z}}_2\\times{\\mathbb {Z}}_2\\times{\\mathbb {Z}}_2$ correspond to the following four matrices and there is no fixed point of $(0,0,1)\\in{\\mathbb {Z}}_2\\times{\\mathbb {Z}}_2\\times{\\mathbb {Z}}_2$\n\n\\begin{figure}[h]\n  $\n    \\begin{pmatrix}\n    1 & 2\\\\\n    1 & 2\n    \\end{pmatrix}$\n    \\qquad$\n    \\begin{pmatrix}\n    1 & 2\\\\\n    2 & 1\n    \\end{pmatrix}$\n    \\qquad$\n    \\begin{pmatrix}\n    2 & 1\\\\\n    1 & 2\n    \\end{pmatrix}$\n    \\qquad$\n    \\begin{pmatrix}\n    2 & 1\\\\\n    2 & 1\n    \\end{pmatrix}$\n  \\end{figure}\n\nThe polynomial $X(q,t,z)$ is given by\n\\begin{align*}\nX(q,t,z)\n    &=\\sum_{\\lambda\\vdash 4} \\widetilde{K}_{\\lambda,(2,2)}(q,t)\\widetilde{K}_{\\lambda,(2,2)}(z)\\\\\n    &=\\widetilde{K}_{(4),(2,2)}(q,t)\\widetilde{K}_{(4),(2,2)}(z)+\\widetilde{K}_{(3,1),(2,2)}(q,t)\\widetilde{K}_{(3,1),(2,2)}(z)+\\widetilde{K}_{(2,2),(2,2)}(q,t)\\widetilde{K}_{(2,2),(2,2)}(z)\\\\\n    &\\equiv 3+qz+tz+qtz \\qquad \\operatorname{mod} \\quad(q^2-1, t^2-1, z^2 -1).\n\\end{align*}\nNote that $X(-1,1,-1)=4$ and $X(1,1,-1)=0$, which agrees with Theorem~\\ref{Main theorem 2}.\n\\end{example}\n\nTo keep this paper concise, instead of providing a precise proof of Theorem~\\ref{Main theorem 3}, we rather sketch a proof that is very similar to the proof of Theorem~\\ref{Main theorem 2}. Let $l=mn=ab$ be a positive integer with two factorizations. Following the argument in the proof of Theorem~\\ref{Main theorem 2}, we consider point locus $X_{(m^n)}$ to the $l\\times l$ permutation matrices by the map $\\phi$ defined in Section~\\ref{Introduction}. We also consider $X_{(a^b)}$ as the set of $l\\times l$ permutation matrices in the same way.\n\nA product of groups ${\\mathfrak{S}}_l\\times \\mathbb{Z}_n\\times \\mathbb{Z}_m$ acts on $X_{(m^n)}$ by permuting columns, external row rotation and internal row rotation, while a product of groups ${\\mathfrak{S}}_l \\times\\mathbb{Z}_b\\times \\mathbb{Z}_a$ acts on $X_{(a^b)}$ by permuting columns, external row rotation and internal column rotation. Each $\\mathfrak{S}_l$-orbits of $X_{(m^n)}\\times X_{(a^b)}$ corresponds to a $l\\times l$ permutation matrix. One can see that there is a natural action of $\\mathbb{Z}_n\\times \\mathbb{Z}_m\\times\\mathbb{Z}_b\\times \\mathbb{Z}_a$ on orbits in $[X_{(m^n)}\\times X_{(a^b)}]\/\\mathfrak{S}_l$ by external and internal row rotation, and external and internal column rotation.\n\nOn the other hand, let ${\\mathbf {R}}_{(m^n)}$ and ${\\mathbf {R}}_{(a^b)}$ be two bigraded ring obtained from point loci $X_{(m^n)}$ and $X_{(a^b)}$ by orbit harmonics. Since their graded Frobenius images are Macdonald polynomials, by Proposition~\\ref{prop:Kronecker}, we have\n\\begin{align*}\\label{Hilbert poly of S_mn invariant of R_mu times R_nu}\n    {\\mathrm {Hilb}}\\left(\\left[{\\mathbf {R}}_{(m^n)}\\otimes {\\mathbf {R}}_{(a^b)}\\right]^{{\\mathfrak{S}}_{l}};q,t,z,w\\right)&=\\sum_{\\lambda, \\lambda'\\vdash l} \\widetilde{K}_{\\lambda,(m^n)}(q,t)\\widetilde{K}_{\\lambda',(a^b)}(z,w)g^{(mn)}_{\\lambda,\\lambda'}\\\\\n    &=\\sum_{\\lambda\\vdash l} \\widetilde{K}_{\\lambda,(m^n)}(q,t)\\widetilde{K}_{\\lambda,(a^b)}(z,w).\n\\end{align*}\nBy applying orbit harmonics, we have an isomorphism\n\\[\n\\mathbb{C}\\left[X_{(m^n)}\\times X_{(a^b)}\\right]^{\\mathfrak{S}_l}\\cong \\left[{\\mathbf {R}}_{(m^n)}\\otimes {\\mathbf {R}}_{(a^b)}\\right]^{{\\mathfrak{S}}_{l}}.\n\\]\nTherefore, we can conclude that the triple $\\left(\\mathfrak{S}_l, {\\mathbb {Z}}_n\\times{\\mathbb {Z}}_m\\times{\\mathbb {Z}}_{b}\\times {\\mathbb {Z}}_{a}, \\mathfrak{S}_l(q,t,z,w)\\right)$ exhibits the quadracyclic sieving phenomenon, where\n$$\\mathfrak{S}_l(q,t,z,w)=\\sum_{\\lambda\\vdash l} \\widetilde{K}_{\\lambda,(m^n)}(q,t)\\widetilde{K}_{\\lambda,(a^b)}(z,w).$$\n\n\\section{Concluding remarks}\\label{concluding remarks}\n\n\\subsection{Other combinatorial loci}\nRecall that we obtained Theorem~\\ref{Main theorem 2} and Theorem~\\ref{Main theorem 3} by taking the tensor product of modules of Garsia--Haiman and Garsia--Procesi, and then taking ${\\mathfrak{S}}_{mn}$-invariant part. We could replace one of those modules to obtain various sieving results. One way to do this is replacing one of the modules with the module ${\\mathbf {R}}_{n,k}$ defined in \\cite{HRS18}. They defined this module to construct a graded ${\\mathfrak{S}}_n$-module for the Delta conjecture at $t=0$. The module ${\\mathbf {R}}_{n,k}$ can also be obtained by applying orbit harmonics to the locus corresponding to the set of surjective functions from $[k]$ to $[n]$. For $mn\\le k$ by taking ${\\mathfrak{S}}_{mn}$ invariant part of ${\\mathbf {R}}_{(m^n)}\\otimes {\\mathbf {R}}_{mn,k}$, we could obtain a triCSP for $n$ times $m$ matrices (or fillings of a rectangular partition) with entries given by nonempty set partitions of $[k]$ into $[mn]$ parts (which may be called the surjective tableaux).\n\nThis process can be applied to a broad class of modules obtained via orbit harmonics. One of the interesting modules which we did not consider in this paper is the module ${\\mathbf {R}}_{\\mu,k}$ defined by Griffin \\cite{Gri21}. This module is a common generalization of the Garsia--Procesi module and the module ${\\mathbf {R}}_{n,k}$ of Haglund--Rhoades--Shimozono and it is possible to obtain this module via orbit harmonics. \n\n\\subsection{Combinatorial proof of main theorems}\nRhoades used two facts to prove cyclic sieving results involving $q$-Kostka polynomials \\cite{Rho10}. The first one is about the evaluation of the Hall--Littlewood polynomial at a root of unity due to Lascoux, Leclerc and Thibbon \\cite{LLT94, LLT97}. The second one is the rim hook correspondence of Stanton and White \\cite{SW85}.\n\nFor the evaluation of the Macdonald polynomials of a rectangular partition at a root of unity, Descouens, and Morita gave a formula \\cite{DM08}. If one can provide a counterpart for rim hook correspondence of Stanton--White in the setting of Theorem~\\ref{Main theorem 1}, Theorem~\\ref{Main theorem 2} or Theorem~\\ref{Main theorem 3}, it would give more combinatorial proof of those.\n\n\\section{Acknowledgements}\n\nThe author is grateful to Brendon Rhoades for helpful conversations about orbit harmonics. The author also thanks anonymous referees for their careful reading and valuable comments. In particular, the author thanks a referee's suggestion to clearly spell out the connections and differences between the results in this paper and known CSP.\n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\subsection{Mondrian Processes \\& Mondrian Forest}\n\\textbf{Mondrian Processes}\nare families of (potentially infinite) hierarchical binary\npartitions of a subdomain ${\\mathcal{D}} \\subseteq \\mathbb{R}^D$;\nthey can be thought of as a family of $k$d trees with height $h$, which\nsequentially refine the partition of ${\\mathcal{D}}$ as $h$ increases \n\\cite{roy2008mondrian}.\nA Mondrian Tree can be defined as a restriction of the underlying \nMondrian Process to an observed set of data points \n\\cite{lakshminarayanan2014mondrian}.\nUnlike the Mondrian Process, it allows for the online sampling of the \nstored tree as more data is observed.\nSpecifically, a \\textbf{Mondrian Tree} $T$ can be represented by the tuple\n$(\\mathsf{T}, {\\boldsymbol \\delta}, {\\boldsymbol \\xi}, {\\boldsymbol \\tau})$ for a decision tree\n$(\\mathsf{T}, {\\boldsymbol \\delta}, {\\boldsymbol \\xi} )$ whose cut dimensions ${\\boldsymbol \\delta}$ are \nchosen with probability proportional to the feature lengths of data \nstored in a node\nand ${\\boldsymbol \\tau}$ is a \nsequence of cut times\n${\\boldsymbol \\tau} = (\\tau_j)_{j \\in \\mathsf{T}}$\nwhich begin from 0 at the root ($\\tau_{\\epsilon}$) while monotonically increasing up to a \\textbf{lifetime budget} $\\lambda > 0$. \nFor any node $j$, the \\textbf{time} or \\textbf{weighted depth} is the value\n$\\tau_j$, whereas  the \\textbf{(absolute) depth} is the length of the\n(unweighted) path from the root to $j$.\nGiven \\emph{observations} ${\\mathbf{X}}$, the generative process for sampling Mondrian\nTrees is denoted $\\MTree{{\\mathbf{X}}}{\\lambda}$.\nFor every node $j \\in \\mathsf{T}$, the indices of the data stored at $j$ is \ndenoted $N(j)$ (so we clearly have $N(\\epsilon) = \\{1,\\ldots, n\\}$) and the \nregions of space a every node $B_j$ are the \\emph{minimal axis-aligned box}\ncontaining the data ${\\mathbf{X}}_{N(j)}$.\nAdditionally, the dimension-wise minima and maxima over \n${\\mathbf{X}}_{N(j)}$ are stored in the vectors ${\\mathbf{l}}_j^{{\\mathbf{X}}}$ and \n${\\mathbf{u}}_j^{{\\mathbf{X}}}$.\nAn example implementation is given in \\Cref{alg: mondrian-forest}.\n\nMondrian Trees are attractive models as they can be sampled \\emph{online}\nas new data is observed. The key principle for this is \\textbf{projectivity}, meaning that if \n$T \\sim \\MTree{{\\mathbf{X}}}{\\lambda}$ and ${\\mathbf{X}}'$ is a subset of the data from ${\\mathbf{X}}$,\nthen the tree restricted to the datapoints ${\\mathbf{X}}'$  is drawn from  \n$\\MTree{{\\mathbf{X}}'}{\\lambda}$ \\cite{lakshminarayanan2014mondrian}.\nCrucially, this enables the sequential building of Mondrian Trees:\n\n\\begin{lemma}[Projectivity] \\label{lem:projectivity}\nLet ${\\mathbf{X}} =  \\{{\\mathbf{x}}_i\\}_{i=1}^n, {\\mathbf{X}}' = {\\mathbf{X}} \\cup {\\mathbf{x}}_{n+1}$.\nSuppose $\\textsf{MTx}({\\mathbf{X}}',\\lambda)$ is a random\nfunction to extend the tree $T$.\n  If $T \\sim \\MTree{{\\mathbf{X}}}{\\lambda}$ and \n   \n  $T' | T, {\\mathbf{X}}' \\sim \\textsf{MTx}({\\mathbf{X}}',\\lambda)$ then \n  $T' \\sim \\MTree{{\\mathbf{X}}'}{\\lambda}$.\n\\end{lemma}\n\nHence, Mondrian \\emph{Trees} are essentially, finite, truncated versions of \nMondrian Processes in the regions of $\\mathbb{R}^D$ where data is observed.\nAn ensemble of trees each independently sampled from $\\MTree{{\\mathbf{X}}}{\\lambda}$ is referred\nto as a \\textbf{Mondrian Forest}.\n\n\n\n\n\n\\subsection{\\Polya{} Tree}  \\label{sec: ptree-intro}\nThe \\Polya{} Tree{} is a nonparametric model for estimating the density function \nover a nested binary partition of a bounded input domain, ${\\mathcal{D}}$.\nWe require the \\Polya{} Tree{} to decide how to distribute mass about the space\nrepresented according to a random binary partition that we will sample.\nFirst we will introduce the infinite version of the \\Polya{} Tree{} and then\ndemonstrate a restricted, finite \\Polya{} Tree{}\n(further details can be found in \\cite{muller2013polya}).\n\n\nSuppose $\\Pi_m =  \\{ A_{b(j)}: j = 1,\\dots,2^m\\} $ is the depth $m$ partition of ${\\mathcal{D}}$\ninto $2^m$ disjoint subsets $j$, indexed by the binary\nstring $b(j) = e_0 e_1 \\dots e_{m-1}$.\nIf we refine $\\Pi_{m}$ to $\\Pi_{m+1}$ by splitting every \n$A_{b(j)} = A_{b(j)0} \\cup A_{b(j)1} $ with \n$A_{b(j)0} \\cap A_{b(j)1} = \\emptyset$ to generate \nthen it remains to understand how mass is allocated to all subsets in \n$\\Pi_{m+1}$.\nThe \\Polya{} Tree{} treats probability mass as a random variable which is \ndistributed throughout $\\Pi_m$ through split \nprobabilities $\\pi_{b(j)} \\sim \\BetaDist{\\alpha_{b(j)0}}{\\alpha_{b(j)1}}$,\neach $\\pi_{b(j)}$ being sampled independently across all levels of \nrefinement, $m$.\nThe probability $\\pi_{b(j)}$ is the probability of reaching the \n``right-hand side'' of the split: that is, choosing a point that is \nin $A_{b(j)1}$ given that the point is in $A_{b(j)}$.\nOverall, the \\Polya{} Tree{} has two sets of parameters: the nested partition\n$\\Pi = \\{ \\Pi_m : m \\ge 0 \\}$ and the Beta distribution parameters\n${\\mathcal{A}} = \\{ (\\alpha_{b(j)0}, \\alpha_{b(j)1} ) : j = 1,\\dots, 2^m \\}$.\n\n\nA \\textbf{\\Polya{} Tree{} over infinite depth partition} allows \n$m \\rightarrow \\infty$ and \nis capable of modelling absolutely continuous functions if the\n$\\alpha_{b(j)} = \\Theta(m^2)$ or discrete functions if \n$\\alpha_{b(j)} = \\Theta(2^{-m})$.\nRather than let $m \\rightarrow \\infty$ \na \\textbf{\\Polya{} Tree{} over a finite depth partition} assumes the partition\nis truncated at some fixed $m$.\nProbability mass is then assumed to be distributed uniformly \nwithin the final $2^m$ bins.\nAn implementation of the \\Polya{} Tree{} is given in \\Cref{alg: polya-tree-sampling} when the \npartition $\\Pi_m$ is defined by a binary tree of height $m$,\nas opposed to the online setting.\nThe predicitve distribution for density estimation over a finite \npartition is the product of expectations of the Beta distributions along the \nleaf-to-root path \\cite{muller2013polya}.\n\n\\subsection{Streaming Mondrian \\Polya{} Tree} \\label{sec: mpt-intro}\n\\input{MondrianPolyaTreeExample}\nThe standard Mondrian Tree \\emph{only} considers (sub-)regions \nwhere data is observed: information about space \nwithout observations is discarded.\nWe decouple the splitting method used to generate Mondrian Trees into\na two-step procedure which will allow a tree sampled\nin the Mondrian Tree $T \\sim  \\MTree{{\\mathbf{X}}}{\\lambda}$ to implicitly \nrepresent the entirety of a given input domain so we can appeal to the \n\\Polya{} Tree{} model.\n\n\nOur modified Mondrian Tree is the \n\\textbf{streaming Mondrian P{\\'o}lya{} Tree}\n(\\acrshort{mpt}) and draws from this structure are \ndenoted $T \\sim \\MondPolyaTree{{\\mathbf{X}}}{\\lambda}$.\nWe generate an \\acrshort{mpt} by drawing $T \\sim \\MTree{{\\mathbf{X}}}{\\lambda}$\nand introducing `pseudosplits' in $T$ to generate a new, implicitly \ndefined $T_s$.\nThese pseudosplits distinguish between the regions of space where data is\nobserved, and those which do not contain any data so the only extra\nspace cost we incur is that of storing the parameters for the Beta \ndistributions necessary for the \\Polya{} Tree.\nAn example is illustrated in \\Cref{fig:mpt-example-plot} and is \nimplemented in \\Cref{alg: mpt}.\n\n\n\nLet $T = (\\mathsf{T}, {\\boldsymbol \\delta}, {\\boldsymbol \\xi}, {\\boldsymbol \\tau}) \\sim \\MTree{{\\mathbf{X}}}{\\lambda}$\nbe a Mondrian Tree.\nWe define two functions over nodes $j \\in \\mathsf{T}$ to decouple the \ncutting of space from the restriction to bounding boxes.\nFirst recall that every node $j$ has a minimal axis-aligned bounding box $B_j$:\n(i) $\\Cut{j}$ samples a cut dimension $\\delta_j$ and cut location $\\xi_j$ splitting\n$j$ into disjoint sets $R_{j0}, R_{j1}$ with $R_{j0} \\cup R_{j1} = B_j$, and \n$R_{j0}, R_{j1}$ representing the \\emph{region} of space less than cut $\\xi_j$ and greater \nthan $\\xi_j$, respectively.\nFor $k=0,1$, the set $R_{j k}$ is a bounded region of space which is \\emph{not  \nrestricted} to the observations located at the child nodes ${\\mathbf{X}}_{N(\\Child{j})}$.\nThis motivates the subsequent `split' to maintain the Mondrian Tree \nstructure of only performing $\\Cut{\\cdot}$ on bounding boxes of \nobserved data; \n(ii) $\\Restrict{j}$ acts on the pair $R_{j0}, R_{j1}$, returning \n$\\left\\{ \\left( B_{\\lChild{j}}, B_{\\lChild{j}}^C \\right), \n\\left( B_{\\rChild{j}}, B_{\\rChild{j}}^C \\right) \\right\\}$ such that \n$B_{\\lChild{j}} \\cup B_{\\lChild{j}}^C = R_{j0}$ and are pairwise disjoint.\nThe same property holds for $R_{j1}$ with the right-hand child nodes.\nWe refer to $B_{\\Child{j}}$ as the \\textbf{observed region} and \n$B_{\\Child{j}}^C$ as the \\textbf{complementary region} where $\\Child{j}$ can be \neither $\\lChild{j}$ or $\\rChild{j}$.\nWe term this as a `pseudosplit' because all of the information \nrequired to perform $\\Restrict{\\Cut{j}}$ is already defined in the generation \nof the Mondrian Tree.\n\n\n\\textbf{Combining \\acrshort{mpt} with Finite \\Polya{} Tree{}.}\nDecoupling the `cut-then-restrict' allows the Mondrian Tree $T$ \nto encode a valid hierarchical partition over the entirety of the input domain ${\\mathcal{D}}$.\nAdditionally, we only ever store the $T$ and the extra Beta parameters\nfrom the \\Polya{} Tree{} structure as $T$\nimplicitly defines the \\acrshort{mpt} $T_s$.\nThese Beta parameters are \\textbf{cut parameters}: $\\chi_{j0}, \\chi_{j1}$, index 0 for less than $\\xi_j$, 1 otherwise; \n\\textbf{restriction parameters} \n$\\rho_{j0 \\in}, \\rho_{j0 \\neg}$ indexed by $j0\\in$ for the observed region\n$B_{\\lChild{j}}$ and $j0\\neg$ for the complementary region $B_{\\lChild{j}}^C$\n(and similarly for $\\rho_{j1 \\in}, \\rho_{j1 \\neg}$) at node $j$.\n\n\nThe following distinctions are necessary to ensure all volume\n comparisons for the \\Polya{} Tree{} construction are on $D$-dimensional \n hypervolumes while the final distinction is necessary to account for \n the mass associated to regions with no observations.\n \\begin {enumerate*} [label=(\\roman*)]\n   \\item{\\textbf{Observation leaves (Type I)}:\n   Any leaf $l$ for which the bounding box \n   at $l$, $B_l = \\bbox{X_{N(l)}}$ has at least one of the dimensions with zero\n   length.\n   Note that this includes the case when only one datapoint is stored\n   in $l$; \n   }\n\n   \\item{\\textbf{Observation leaves (Type II)}:\n   Any leaf $l$ formed from a cut at node $j$ which contains two or more datapoints and $B_l = \\bbox{X_{N(l)}}$ has all $D$ lengths positive; \n  }\n\n   \\item{\\textbf{Complementary leaves}: Leaves formed in a region where no\n   observations are made.\n    }\n \\end{enumerate*}\n\nPseudosplits in the Mondrian Tree $T$ are used to generate the \n\\acrshort{mpt}, so it is necessary to revise the indexing scheme of the\nnested partition over domain ${\\mathcal{D}}$.\nFor every node $j \\in \\mathsf{T}$ of (absolute) depth $k$ in\n$T$, we generate a set of encodings for the spaces represented in $T_s$:\n$b(j) = c_0 r_0 \\dots c_k r_k$.\nThe length of $b(j)$ is at most twice the maximum absolute depth \nof $T$ and indexes all nodes in the (implicitly defined) $T_s$.\nThe symbol $c_i \\in \\{ 0,1 \\}$ indicates ``less than'' or ``greater than'' the cut at level\n$i$, and $r_i \\in \\{ \\in, \\neg, \\emptyset \\}$ indicates whether the node represents\nthe observed region ($\\in$), complementary region ($\\neg$), or can be $\\emptyset$\nif the leaf is a Type I observed leaf as no restriction is performed.\n\n\\subsubsection{Model Parameters for the \\acrshort{mpt}} \\label{sec: mpt-params}\nFor a Mondrian Tree $T = (\\mathsf{T}, {\\boldsymbol \\delta}, {\\boldsymbol \\xi}, {\\boldsymbol \\tau})$ we now\nshow how to set the parameters for the \\Polya{} Tree{} over the induced \n\\acrshort{mpt}, $T_s$.\n\\begin{itemize}\n\n\\item{Let $P_j$ denote the probability mass associated with node $j$.\nWe define the probability mass associated with the root $\\epsilon$ to\nbe $P_{\\epsilon} = 1$.}\n\n\\item{\\textbf{Setting the prior.}\nAt every internal node $j \\in \\mathsf{T}$, the probability of a point\nbeing greater than the cut $\\xi_j$ is given by a Bernoulli \nwith parameter $\\kappa_j$ whose prior is \n$\\BetaDist{\\chi_{j0}}{\\chi_{j1}}$.\nLikewise, the probability of a point being in the observed \nregion after cut $\\xi_j$ follows a\\footnote{Note that we choose this\nordering so the expected\nvalue of the Beta distribution is associated with being in \nthe observed region after every cut.\nSee example in \\Cref{fig:mpt-example-tree}.}  $\\text{Bernoulli}(1-\\theta_j)$ whose prior is \n$\\BetaDist{\\rho_{j k \\in}}{\\rho_{j k \\neg}}$\nfor $k=0,1$, depending on which side \nof the cut the point lies.\nTo set the \\Polya{} Tree{} parameters, we need to evaluate the volumes of \nvarious parts of the space, this will be denoted $V_X = \\vol{X}$ \nfor $X \\in \\{R_{\\cdot}, B_{\\cdot}, B_{\\cdot}^C\\}$.\nThe \\acrshort{mpt} $T_s$ is defined from a two-stage split so we correct\nthe `depth' of nodes in $T_s$ from the usual \\Polya{} Tree{} construction by\na simple translation: if $j \\in \\mathsf{T}$ has depth $d_j$ then\n$\\PolyaTreeDepth{j}  = (2d_j, 2d_j+1)$.\nThe prior strength is controlled by hyperparameter $\\gamma > 0$.\nParameters for cut ($\\chi_{\\cdot}$)  and \nrestriction ($\\rho_{\\cdot}$) are then:\n\\begin{equation} \\label{eq: mpt-prior}\n  \\begin{split}\n    \\chi_{j0} &= \\gamma (2d_j + 1)^2 \n    \\nicefrac{ V_{R_{j0}}}{ ( V_{R_{j0}} +  V_{R_{j1}} ) }\\\\\n    \\chi_{j1} &= \\gamma (2d_j + 1)^2 \n    \\nicefrac{ V_{R_{j1}}}{ ( V_{R_{j0}} +  V_{R_{j1}} ) }\\\\\n  \\end{split}\n\\quad\n  \\begin{split}\n    \\rho_{j k \\in} &= \\gamma (2d_j + 2)^2 \n    \\nicefrac{ V_{B_{jk\\in}}}{ ( V_{B_{jk\\in}} +  V_{B_{jk\\neg}} ) }\\\\\n    \\rho_{j k \\neg} &= \\gamma (2d_j + 2)^2 \n    \\nicefrac{ V_{B_{jk\\neg}}}{ ( V_{B_{jk\\in}} +  V_{B_{jk\\neg}} ) }\\\\\n  \\end{split}\n\\end{equation}\n\n}\n\n\\item{\\textbf{Distributing Mass.}\nThe predictive distribution of the \\Polya{} Tree{} over a finite \ndepth partition is the product of expected value of Beta distributions on the leaf-root path.\nThere can be maintained exactly over all nodes for both cutting \\& restricting.\nWe allocate a \n$\\mu_{\\chi_j} = \\mathbb{E}(\\BetaDist{\\chi_{j0}}{\\chi_{j1}})$ fraction of $j$'s mass to $R_{j0}$ so $P_{j0} = P_j \\mu_{\\chi_j}$ \\& \n$P_{j1} = P_j (1 - \\mu_{\\chi_j})$.\nNext, we repeat for the restriction step which allots \n$\\mu_{\\rho_{j0}} = \\mathbb{E}(\\BetaDist{\\rho_{j0\\in}}{\\rho_{j0\\neg}})$ to $B_{j0}$ so  $P_{j0\\in} = P_{j0} \\mu_{\\rho_{j0}}$ and \n$P_{j0\\neg} = P_{j0} (1 - \\mu_{\\rho_{j0}})$; likewise for \nabove the cut $\\xi_j$.\n\n}\n\n\\item{\\textbf{The Posterior Distribution.}\nBy Beta-Binomial conjugacy, on inserting data, the parameters of any given\nBeta distribution can be updated by the number of datapoints observed at a node.\nIn the Mondrian Tree, $n_0$ points are passed from $j$ to $\\lChild{j}$\n and $n_1$ points to $\\rChild{j}$.\n Hence, all of the $n_0$ points in $\\lChild{j}$ are both at most the \n cut value $\\xi_j$ and present in the bounding box $B_{\\lChild{j}}$ \n while the opposite is true for $n_1$ and $\\rChild{j}$.\n Therefore, we obtain the simple posterior update procedure:\n\n\\begin{equation} \\label{eq: mpt-posterior}\n  \\chi_{jk}^* = \\chi_{jk} + n_k, \\quad\n  \\rho_{j k \\in}^* = \\rho_{j k \\in} + n_k, \\quad\n    \\rho_{j k \\neg}^* = \\rho_{j k \\neg}, \\quad \\mbox{for }k=0,1\n\\end{equation}\n}\n\\item{Mass in the leaves of a finite \\Polya{} Tree{} is assumed to be \ndistributed uniformly;  if a point \nfalls into a leaf $j$, then the mass associated to that point is \nsimply the product of the expected Beta distributions on the path from \n$\\epsilon$ to $j$, and its density is the mass divided by the volume.}\n\n\\end{itemize}\nIt is necessary to retain the volumes of both observed and complementary regions for the restriction parameters.\nHowever, this is straightforward given the cut and volume at node $j$ (see \\Cref{sec: mpt-appendix}). \n\n\n\n\\textbf{Complexity: Instantiating the \\acrshort{mpt}.}\nThe complexity of combining the \\Polya{} Tree{} with the Mondrian Tree incurs only mild overhead.\nLet $T = (\\mathsf{T}, {\\boldsymbol \\delta}, {\\boldsymbol \\xi}, {\\boldsymbol \\tau})$ denote the stored Mondrian Tree which \ngenerates the \\acrshort{mpt}.\nThe extra space necessary to use \\acrshort{mpt} for density estimation is  $ 7 |\\mathsf{T}|$ due to the extra counters needed for every Beta distribution (i.e. $\\chi_{j0},\\chi_{j1},\n\\rho_{jk\\in},\\rho_{jk\\neg}, k=0,1$) and the probability mass float $P_j$. \nAt every node we must compute the volume at a cost\nof $O(D)$ which is $O(D |\\mathsf{T}|)$ over the entire tree but this can be done on-the-fly as the tree is constructed.\n\n\\textbf{\\acrshort{mpt}: Insertions and Deletions.}\nFor a \\acrshort{mpt}, we provide efficient algorithms to insert and \ndelete points over the data stream.\nA full treatment is given in \\Cref{sec: mpt-appendix}: the pertinent points\nbeing that we retain \\emph{projectivity} due to the underlying Mondrian \nTree which generates the \\acrshort{mpt}.\nDeletions require a little more work as removal points could lie on a \nbounding box, so it is necessary to check how this interacts with the\nlifetime of the stored tree.\n\n\n\n\n\n\\textbf{Example.}\nIn \\Cref{fig:mpt-example-tree} we present an instantiation of the \\acrshort{mpt}.\nObserve that the \\emph{Mondrian Tree} which is used to generate the partition\nin \\Cref{fig:mpt-example-plot} splits the entire input domain into \ndisjoint subsets of Type I\/II observed leaves and complementary regions.\nIt also implicitly encodes the associated \\acrshort{mpt} as given in \n\\Cref{fig:mpt-binary-tree}.\nThe calculations to obtain density estimates over this tree are given in \\Cref{sec:mpt-example-calcs}.\n\n \n \\subsection{Mondrian P{\\'o}lya{} Forest for Density Estimation\n \\& Anomaly Detection}\n \\label{sec: mpf-anomaly}\n Recall that an independently sampled ensemble of \n batch\/streaming Mondrian \\Polya{} Tree{s} is \n referred to as a batch\/streaming Mondrian P{\\'o}lya{} Forest\n (\\acrshort{bmpf}), (\\acrshort{smpf}),\n $F = \\cup_i T_i$.\n Each $T_i$ defines a function over its leaves $p_i({\\mathbf{x}})$\n which is a noisy estimate of the true underlying \n density function $p({\\mathbf{x}})$.\n \n \\begin{definition}[Density Estimation]\nLet $p({\\mathbf{x}})$ be a density function and suppose \n $F = \\cup_i T_i$ is a \\acrshort{bmpf} or \\acrshort{smpf}.\n Let $l$ denote the leaf in $T_i$ which contains ${\\mathbf{x}}$\n and whose mass is $P^{(i)}_l$.\nThe \\emph{density estimate} of ${\\mathbf{x}}$ in $T_i$ is \n$p_i({\\mathbf{x}}) = P_l \/ \\vol{l}$ while the density estimate over\nthe forest is $\\hat{p}({\\mathbf{x}}) = \\frac1n \\sum_{i=1}^{|F|}p_i({\\mathbf{x}})$.\n \\end{definition}\nRather than using density estimates, we adopt the following simple approach to declare anomalies while \nremaining in probability space; using simply\nthe $P^{(i)}_l$ rather than $p_i({\\mathbf{x}})$.\nThis alteration is to prevent a small number of trees from \ncorrupting the `score' if they are not good trees.\n\nThe simplicity of this approach is one of the strengths of \nour work.\nWhile previous works add an extra scoring mechanism over \nthe forest, ours is an inherent property of the underlying\nprobabilistic framework.\nWe can threshold exactly in probability space which makes \nthese `scores' more interpretable than prior work.\nSynthetic density estimation \\& anomaly detection examples\nare in \\Cref{sec: synthetic-examples}.\n\n\n\\begin{definition}[${\\varepsilon}$-anomaly \\& $({\\varepsilon},\\phi)$-anomaly]\n\\label{def: anomaly}\nLet $F = \\cup_i T_i$ be a \\acrshort{bmpf} or \\acrshort{smpf} and\n${\\varepsilon}, \\phi \\in [0,1]$.\nA point ${\\mathbf{x}} \\in \\mathbb{R}^d$ is an \\emph{${\\varepsilon}$-anomaly} in tree $T$ \nif the probability mass of the leaf in which ${\\mathbf{x}}$ is stored is at most\n${\\varepsilon}$.\nA point ${\\mathbf{x}} \\in \\mathbb{R}^d$ is an \\emph{$({\\varepsilon},\\phi)$-anomaly}\nif ${\\mathbf{x}}$ is and ${\\varepsilon}$-anomaly in at least $\\phi |F|$\ntrees from $F$.\n\\end{definition}\n\n\n\n\\section{Sampling Mondrian Trees, \\Polya{} Tree{s} and Mondrian \\Polya{} Tree{s}}\n\n\\input{tab_lay_summary}\n\nFor clarity we describe the structures necessary to introduce our Mondrian \\Polya{} Tree{s} which \nare summarised in \\Cref{tab:lay-summary}.\\footnote{\nPlease note that between paper submission and supplementary submission\nwe added \\Cref{tab:lay-summary} so the table indexing has been incremented\nby 1 from the paper originally submitted.}\nWe will begin with the \\textbf{Mondrian Process} which can be succinctly described as:\ngiven an input domain ${\\mathcal{D}}$ and a lifetime $\\lambda > 0$, choose a direction (feature)\nto cut with probability proportional to length.\nNext, choose a cut location uniformly at random on the selected feature and \nsplit into two sets less than and greater than the cut location.\nThis cut procedure has a random cost associated to the ``linear dimension''\n(sum of the lengths) of the region at a given node and the process is \nrepeated until the lifetime is exhausted by accumulating the random costs.\nAn implementation is given in \\cite{roy2008mondrian}.\n\nThe \\textbf{Mondrian Tree} builds on the Mondrian Process by building the trees in a more\ndata-aware fashion.\nAt a high-level this process is similar to the Mondrian Process except every cut\ntakes place on a restriction of space to the bounding box on which observations \nare made.\nThe advantage of this is that cuts are guaranteed to pass through observations\nwhich in high dimensions could result in substantially shortened trees.\nMondrian Trees can also be sampled online which makes them highly efficient.\nHowever, the price to pay for these efficiency gains is that behaviour outside of the \nbounding boxes cannot be modelled.\n\nWhile the previous two methods are useful for partitioning the data into clusters,\nthey make no statements about the underlying density of the dataset.\nTo accommodate this we introduce  the \\textbf{\\Polya{} Tree{}} which is a Bayesian nonparametric\nmodel for estimating the underlying density function generating the data.\nThe \\Polya{} Tree{} model takes as input a binary nested partition of an input space ${\\mathcal{D}}$,\n(represented by a binary tree) and assigns probability to each of the bins (nodes in \nthe tree).\nGiven a point in a bin indexed $A_{b(j)}$, the presence of a point in the bins \n$A_{b(j)1}$ is modelled by a Bernoulli distribution with parameter $p$.\nLet $d_j$ denote the depth of $A_{b(j)}$ and \n$V_0, V_1$ denote the volumes of the the bins $A_{b(j)0}, A_{b(j)1}$, \nrespectively.\nThe prior distribution for $p$ is a Beta distribution which has parameters:\n\\begin{align}\n    \\alpha_{j0} &= \\gamma \\left(d_j + 1\\right)^2 \\frac{V_0}{V_0 + V_1} \\\\\n    \\alpha_{j1} &= \\gamma \\left(d_j + 1\\right)^2 \\frac{V_1}{V_0 + V_1}\n\\end{align}\nfor a hyperparameter $\\gamma > 0$ denoting the strength of the prior distribution.\nThe posterior parameters for the $\\alpha_{jk}$ are then incremented by the \nnumber of points observed in the $A_{b(j)k}$ bin for $k=0,1$.\nAn implementation is given in \\Cref{alg: polya-tree-sampling} which takes \nas input the partition of space ${\\mathcal{D}}$, thus requiring an extra pass through the tree.\nHowever, for our applications as defined in \\Cref{sec: mpf-anomaly}, \nwe will be able to implement this in an online fashion.\n\nOur \\textbf{Mondrian \\Polya{} Tree{}} can be implemented in either a batch or streaming fashion.\nFor a batch computation, we can adapt the Mondrian Process and easily combine this \nwith the \\Polya{} Tree{}.\nHowever, for streaming computation, the `empty space' caused by restricting to \nbounding boxes in the Mondrian Tree procedure is highly problematic and this \nmotivated our revised construction, the \\acrshort{mpt} as described in \n\\Cref{sec: mpt-intro}.\nWe describe this revision in \\Cref{alg: mpt} while the parameter \nupdate algorithms are presented in \\Cref{alg: mpt-parameters}.\n\n\\textbf{Generating Mondrian \\Polya{} Tree{s}: Computational Complexity.}\nCombining the \\Polya{} Tree{} with either the Mondrian Process or Mondrian Tree \nincurs only a mild overhead in both time and space as all that needs to be stored is an\nextra set of parameters.\nFor the \\textbf{batch Mondrian \\Polya{} Tree{}} (\\Cref{sec: mondrian-polya-forest}) this is simply\n3 counters per node ($\\alpha_{j0},\\alpha_{j1}, P_j$).\nIn \\Cref{sec: mpt-intro} we showed that a two-stage split was necessary for the \n\\textbf{streaming Mondrian \\Polya{} Tree{}} and this \nslightly increases the number of parameters to at most 7 per node (see \\ref{sec: mpt-params})\nwhich come from the 2 cut parameters, at most 4 restriction parameters, and the mass \nfloat $P_j$.\nOverall, both methods need $O(|\\mathsf{T}|)$ extra space which, nevertheless, is only a\nconstant factor more space than is required to build the partitioning tree.\n\nThe time cost to evaluate these parameters is $O(d | \\mathsf{T} |)$ as computing the volume \nof every node costs $O(d)$.\nSince we make the distinction between \\textbf{type I\/II observation \\& complementary\nleaves}, volume comparisons are made over nonzero $D$-dimensional hypervolumes.\nThis permits the following distinctions\nat every node to avoid incurring complex volume computations of the complementary regions.\n\n\\textbf{Volume Computation for \\acrshort{mpt}.}\nRecall that for a node $j$ we sample a cut dimension $\\delta_j$ and in that dimension \na cut location $\\xi_j$.\nThe node $j$ contains the restriction to bounding box $B_j$ which is split into two \nregions $R_{j0}$ and $R_{j1}$ either side of $\\xi_j$.\nNode $j$ has volume $V_{B_j} = \\vol{j}$\nand let $h_j$ denote the length of the sampled dimension $\\delta_j$;\nthe volumes associated \nwith $R_{j0}$ and $R_{j1}$ are:\n\\begin{align}\n    V_{R_{j0}} &= \\frac{V_{B_j}}{h_j} |\\min_{{\\mathbf{x}} \\in j} {\\mathbf{x}}_{\\delta_j} - \\xi_j | \\\\\n    V_{R_{j1}} &= \\frac{V_{B_j}}{h_j}|\\max_{{\\mathbf{x}} \\in j} {\\mathbf{x}}_{\\delta_j} - \\xi_j |.\n\\end{align}\n\n\n\n\n\nWe obtain the volume of the observed region when \n computing $\\Restrict{j}$ for the restriction to bounding boxes either side of \n the cut $\\xi_j$ at $j$.\n Recall that \n $V_{\\lChild{j}} = \\vol{ B_{\\lChild{j}} }$, so the subtraction \n $V_{R_{j0}} - V_{\\lChild{j}} = V_{B_{j0}^C}$ \n yields the complementary volume necessary for setting the restriction\n P{\\'o}lya{} parameters $\\rho_{\\cdot \\in}, \\rho_{\\cdot \\neg}$.\nAll volumes being supported on $D$-dimensional boxes ensures that none of these\nquantities trivially collapse to zero.\nIf one of the feature lengths is zero then we simply treat such a node as a Type I\n observation leaf.\n \n\n\n\n\n\n\n\n\n\\begin{algorithm}[htb]\n\\KwIn{Training data ${\\mathbf{X}} \\in \\mathbb{R}^{n \\times D}$, lifetime $\\lambda > 0$ }\n\\SetKwFunction{FMain}{SampleMondrianTree}\n\\SetKwFunction{FSub}{SampleMondrianBlock}\n\\SetKwProg{Fn}{Function}{:}{}\n\\Fn{\\FMain{${\\mathbf{X}}, \\lambda$}}{\nInitialise $\\mathsf{T} = \\emptyset, \\leaves{\\mathsf{T}} = \\emptyset,\n{\\boldsymbol \\delta} = \\emptyset, {\\boldsymbol \\xi} = \\emptyset, {\\boldsymbol \\tau} = \\emptyset,\nN(\\epsilon) = \\{1,2,\\dots,n\\}.$\\\\\n    \\FSub{$\\epsilon, {\\mathbf{X}}_{N(\\epsilon)}, \\lambda$}  \\\\\n}\n\\SetKwProg{Pn}{Function}{:}{\\KwRet}\n\\Pn{\\FSub{$j, {\\mathbf{X}}_{N(j)}, \\lambda$}}{\n      $\\mathsf{T} \\leftarrow \\mathsf{T} \\cup \\{ j \\} $ \\\\\n      For all $d \\in [D]$ set\n      ${\\mathbf{l}}_{jd}^{{\\mathbf{X}}} = \\min_d {\\mathbf{X}}_{N(j)},{\\mathbf{u}}_{jd}^{{\\mathbf{X}}} = \\max_d {\\mathbf{X}}_{N(j)}$\n      to be the dimension-wise minima and maxima of the observations in $j$ \\\\\n      Let $L = \\sum_d (u_{jd}^{{\\mathbf{X}}} - l_{jd}^{{\\mathbf{X}}})$ denote the\n      \\emph{linear dimension} of the data in $j$ \\\\\n      Sample $E \\sim \\ExpDist{L}$ \\\\\n      \\uIf{$\\tau_{\\parent{j}} + E < \\lambda$}{\n          Set $\\tau_j = \\tau_{\\parent{j}} + E$ \\\\\n          Sample cut dimension $\\delta_j$ with probability proportional to\n          $u_{jd}^{{\\mathbf{X}}} -  l_{jd}^{{\\mathbf{X}}}$ \\\\\n          Sample cut location uniformly on the interval\n          $[ l_{j \\delta_j }^{{\\mathbf{X}}}, u_{j \\delta_j}^{{\\mathbf{X}}}]$ \\\\\n\n          Set $N(\\lChild{j}) = \\{ n \\in N(j) : X_{n \\delta_j} \\le \\xi_j \\}$ and\n          $N(\\rChild{j}) = \\{ n \\in N(j) : X_{n \\delta_j} > \\xi_j \\}$ \\\\\n\n          \\FSub{$\\lChild{j}, {\\mathbf{X}}_{N(\\lChild{j})}, \\lambda$}\\\\\n          \\FSub{$\\rChild{j}, {\\mathbf{X}}_{N(\\rChild{j})}, \\lambda$}\n      }\n      \\Else{\n      $\\tau_j \\leftarrow \\lambda$ and\n      $\\leaves{\\mathsf{T}} \\leftarrow \\leaves{\\mathsf{T}} \\cup \\{j\\}$\n  }\n}\n \\caption{Mondrian Forest Sampling \\cite{lakshminarayanan2014mondrian}}\n \\label{alg: mondrian-forest}\n\\end{algorithm}\n\n\\begin{algorithm}[htb]\n\\KwIn{Input domain ${\\mathcal{D}} \\subset \\mathbb{R}^{D}$,\n      a decision tree $T$ which partitions ${\\mathcal{D}}$,\n      hyperparameter $\\gamma > 0$}\n\\KwOut{Probability distribution ${\\mathcal{P}} = (P_l)_{l \\in \\leaves{T}}$}\n\\SetKwFunction{FMain}{SampleP{\\'o}lya{}Tree}\n\\SetKwFunction{FSub}{UpdateP{\\'o}lya{}Parameters}\n\\SetKwProg{Fn}{Function}{:}{}\n\\Fn{\\FMain{${\\mathcal{D}}, T, \\gamma$}}{\n$\\epsilon = \\text{root}\\left( T \\right)$ \\\\\n$P_{\\epsilon} = 1$ \\Comment{Assume all mass is located in the region ${\\mathcal{D}}$}\\\\\n \\FSub{$\\epsilon, \\gamma$} \\\\\n }\n\n \\SetKwProg{Pn}{Function}{:}{\\KwRet}\n \\Pn{\\FSub{$j, \\gamma$}}{\n    \\uIf{$j \\in \\leaves{T}$}{\n    $\\text{ProbDensity}(j) = P_j \/ \\vol{j}$\n    \\Comment{$P_j$ was defined at the preceeding level.}\n    }\n    \\uElse{\n    $V_j = \\vol{j}$ \\\\\n    Let $R_j = [l_1,u_1] \\times \\dots \\times [l_d,u_d]$ define the region in\n    as a product of intervals from the minimum in dimension $i$, $l_i$, to\n    the maximum in dimension $i$, $u_i$. \\\\\n    $L_j = \\sum_{j=1}^d (u_j - l_j)$ is the linear dimension of the region.\\\\\n    $V_0 = \\frac{V_j}{u_{\\delta_j} - l_{\\delta_j}} \\cdot | l_{\\delta_j} - \\xi_j |$\n    \\Comment{Volume less than cut $\\xi_j$}\\\\\n    $V_1 = \\frac{V_j}{u_{\\delta_j} - l_{\\delta_j}} \\cdot | u_{\\delta_j} - \\xi_j |$\n    \\Comment{Volume greater than cut $\\xi_j$}\\\\\n    $n_0 = \\text{NumberOfPoints}(\\lChild{j}),\n    n_1 = \\text{NumberOfPoints}(\\rChild{j})$\n    \\Comment{Number of points in children nodes}\\\\\n    $\\alpha_0 = \\gamma (d+1)^2 \\frac{V_0}{V_0 + V_1} + n_0,\n    \\alpha_1 = \\gamma (d+1)^2 \\frac{V_1}{V_0 + V_1} + n_1$\n    \\Comment{Set prior parameters using \\Polya{} Tree~ and then increment using\n    Beta-Binomial conjugacy}\\\\\n    $\\mu_j = \\frac{\\alpha_0}{\\alpha_0 + \\alpha_1}$\n    \\Comment{$\\mathbb{E}(\\BetaDist{\\alpha_0}{\\alpha_1})$}\\\\\n    $P_{\\lChild{j}} = \\mu_j P_j,  P_{\\rChild{j}} = (1 - \\mu_j) P_j$\\\\\n    \\FSub{$\\lChild{j}, \\gamma$}, \\FSub{$\\rChild{j}, \\gamma$}\n    }\n}\n \\caption{\\Polya{} Tree{} Sampling. Sets probability (density) for all nodes in the\n given random partition $T$.}\n \\label{alg: polya-tree-sampling}\n\\end{algorithm}\n\n\\begin{algorithm}[p]\n \\KwIn{Training data ${\\mathbf{X}} \\in \\mathbb{R}^{n \\times D}$, at least one of lifetime\n $\\lambda > 0$ or maximum tree height $m$, \\Polya{} Tree{} hyperparameter $\\gamma > 0$}\n \\KwOut{A classical Mondrian Tree data structure $T$;\n Partition $\\Pi$ over $T$ such that\n ${\\mathcal{P}} = (P_l)_{l \\in \\leaves{T}}$\n is a probability distribution over the leaves of $T$ induced by \\Polya{} Tree{} prior}\n \\SetKwFunction{FMain}{SampleMondrianP{\\'o}lya{}Tree}\n \\SetKwFunction{FSub}{SampleMondrianP{\\'o}lya{}Block}\n \\SetKwFunction{FSubCut}{SetCutParameters}\n \\SetKwFunction{FSubObs}{SetRestrictionParameters}\n\\SetKwFunction{FRestrict}{Restrict}\n\\SetKwFunction{FCut}{Cut}\n \\SetKwProg{Fn}{Function}{:}{}\n \\Fn{\\FMain{${\\mathbf{X}}, \\lambda$}}{\n Initialise $\\mathsf{T} = \\emptyset, \\leaves{\\mathsf{T}} = \\emptyset,\n {\\boldsymbol \\delta} = \\emptyset, {\\boldsymbol \\xi} = \\emptyset, {\\boldsymbol \\tau} = \\emptyset,\n N(\\epsilon) = \\{1,2,\\dots,n\\}.$\\\\\n $\\epsilon.\\text{ObservedVolume}  = \\vol{\\bbox{{\\mathbf{X}}_{N(\\epsilon)}}}$ \\\\\n     \\FSub{$\\epsilon, {\\mathbf{X}}_{N(\\epsilon)}, \\lambda$}  \\\\\n }\n \\SetKwProg{Pn}{Function}{:}{\\KwRet}\n \\Pn{\\FSub{$j, {\\mathbf{X}}_{N(j)}, \\lambda$}}{\n       $\\mathsf{T} \\leftarrow \\mathsf{T} \\cup \\{ j \\} $ \\\\\n       $B_j \\leftarrow \\bbox{ {\\mathbf{X}}_{N(j)}}, L = \\LinDim{B_j}$\n       \\Comment{\\emph{linear dimension} of the bounding box for $j$ }\\\\\n      \n      \n       Sample $E \\sim \\ExpDist{L}$ \\\\\n       \\uIf{$\\tau_{\\parent{j}} + E < \\lambda \\text{~and all feature lengths are positive}$}{\n           Set $\\tau_j = \\tau_{\\parent{j}} + E$ \\\\\n           \\FCut{$j, {\\mathbf{X}}_{N(j)}, B_j$} \\\\\n\n           Set $N(\\lChild{j}) = \\{ n \\in N(j) : {\\mathbf{X}}_{n \\delta_j} \\le \\xi_j \\}$ and\n           $N(\\rChild{j}) = \\{ n \\in N(j) : {\\mathbf{X}}_{n \\delta_j} > \\xi_j \\}$ \\\\\n\n           \\FRestrict{$j, N(\\lChild{j})$} \\\\\n           \\FRestrict{$j, N(\\rChild{j})$}\n\n           \\FSub{$\\lChild{j}, {\\mathbf{X}}_{N(\\lChild{j})}, \\lambda$}\\\\\n           \\FSub{$\\rChild{j}, {\\mathbf{X}}_{N(\\rChild{j})}, \\lambda$}\n       }\n       \\Else{\n            \\uIf{any feature length is 0}{\n                $\\mathsf{T} \\leftarrow \\mathsf{T} \\cup \\{ j \\}$\n                \\Comment{Bounding box supported on $ < d$ dimensions: Type I Observed leaf}\n            }\n            \\Else{\n             \\FRestrict{$j, N(j)$}\n             \\Comment{Restrict once more to generate a \\emph{complementary leaf}} \\\\\n            }\n       $\\tau_j \\leftarrow \\lambda$ and\n       $\\leaves{\\mathsf{T}} \\leftarrow \\leaves{\\mathsf{T}} \\cup \\{j\\}$\n       \\Comment{Type II Observed leaf (see \\Cref{sec: mpt-intro})}\n   }\n }\n\n\\SetKwProg{Pn}{Function}{:}{\\KwRet}\n\\Pn{\\FCut{$j, {\\mathbf{X}}_{N(j)}, B_j$}}{\n    Sample cut dimension $\\delta_j$ with probability proportional to\n    $u_{jd}^{{\\mathbf{X}}} -  l_{jd}^{{\\mathbf{X}}}$ \\\\\n    Sample cut location $\\xi_j$ uniformly on the interval\n    $[ l_{j \\delta_j }^{{\\mathbf{X}}}, u_{j \\delta_j}^{{\\mathbf{X}}}]$ \\\\\n    ${R_{\\textsf{left}}} = \\{{\\mathbf{z}} \\in B_j : {\\mathbf{z}}_{\\delta_j}  \\le \\xi_j\\}$ \\\\\n    ${R_{\\textsf{right}}} = \\{{\\mathbf{z}} \\in B_j : {\\mathbf{z}}_{\\delta_j}  > \\xi_j\\}$ \\\\\n    $n_{\\textsf{left}} = |{\\mathbf{X}}_{N(j)} \\cap {R_{\\textsf{left}}}|$ \\\\\n    $n_{\\textsf{right}} = |{\\mathbf{X}}_{N(j)} \\cap {R_{\\textsf{right}}}|$ \\\\\n    $V_{\\textsf{left}} = \\vol{{R_{\\textsf{left}}}},\n    V_{\\textsf{right}} = \\vol{{R_{\\textsf{right}}}}$ \\\\\n    $d_j = \\TreeDepth{j}$ \\Comment{Absolute depth in Mondrian Tree} \\\\\n    \\FSubCut{$2d_j, n_{\\textsf{left}}, n_{\\textsf{right}},V_{\\textsf{left}}, V_{\\textsf{right}}$}\n}\n\n\\SetKwProg{Pn}{Function}{:}{\\KwRet}\n\\Pn{\\FRestrict{$j, {\\mathbf{X}}_{N(j)}$}}{\n    $d = j.\\text{depth}$\n    \\Comment{Get absolute depth in Mondrian tree} \\\\\n    $n_{\\textsf{obs}} = |N(j)|$ \\Comment{Num. points in node} \\\\\n    $V_p = \\parent{j}.\\text{ObservedVolume}$\n    \\Comment{Parent volume} \\\\\n    $V_o = \\vol{B({\\mathbf{X}}_{N(j)})}$\n    \\Comment{Observed volume} \\\\\n    $V_c = V_p - V_o$\n    \\Comment{Complementary volume} \\\\\n    $\\rho_0^*, \\rho_1^*$ = \\FSubObs{$2d+1, n_{\\textsf{obs}}, V_o, V_c$}\n    \\Comment{Set Beta parameters.}\\\\\n    \\Return{$\\rho_0^*, \\rho_1^*$}\n}\n\n  \\caption{Mondrian \\Polya{} Tree{} Sampling.  Subroutines:\n  \\Cref{alg: mpt-parameters}}\n  \\label{alg: mpt}\n \\end{algorithm}\n \n \\begin{algorithm}[htb]\n \\SetKwFunction{FSubCut}{SetCutParameters}\n \\SetKwFunction{FSubObs}{SetRestrictionParameters}\n \\SetKwProg{Pn}{Function}{:}{\\KwRet}\n \\Pn{\\FSubCut{$\\text{depth}, n_{\\textsf{left}}, n_{\\textsf{right}},\n             V_{\\textsf{left}}, V_{\\textsf{right}}$}}{\n     $d = \\text{depth}$ \\\\\n     $j.\\chi_0^* = \\gamma (d + 1)^2\n     \\frac{ V_{\\textsf{left}} }{ V_{\\textsf{left}} + V_{\\textsf{right}}} + n_{\\textsf{left}},\n     j.\\chi_1^* = \\gamma (d + 1)^2\n     \\frac{ V_{\\textsf{right}} }{ V_{\\textsf{left}} + V_{\\textsf{right}}} + n_{\\textsf{right}}$ \\\\\n }\n %\n \\SetKwProg{Pn}{Function}{:}{\\KwRet}\n \\Pn{\\FSubObs{$\\text{depth},\\text{NodeSize}, \\text{ObservedVolume}, \\text{ComplemetaryVolume}$}}{\n     $d = \\text{depth}; \\qquad$ \n     $n = \\text{NodeSize}$ \\Comment{Number of points in observed bounding box} \\\\\n     $V_{obs} = \\text{ObservedVolume}, V_{comp} = \\text{ComplemetaryVolume}$ \\\\\n     $\\rho_0^* = \\gamma (d+1)^2 \\frac{V_{obs}}{V_{obs} + V_{comp}} + n$,\n     $\\rho_1^* = \\gamma (d+1)^2 \\frac{V_{comp}}{V_{obs} + V_{comp}}$ \\\\\n     \\Return{$\\rho_0^*, \\rho_1^*$}\n }\n \\caption{Subroutines for setting Beta Distribution parameters for the\n Mondrian \\Polya{} Tree.\n Note that the depth parameters in these subroutines refer to depth in \\Polya{} Tree, not absolute\n depth in Mondrian Tree!}\n \\label{alg: mpt-parameters}\n \\end{algorithm}\n\\newpage\n\\section{\\acrshort{mpt}: Insertions and Deletions} \n\\label{sec: mpt-appendix}\n\nA substantial benefit of the Mondrian Tree construction is that it can \nbe built online as new data is seen.\nThe key idea underpinning this is\n\\emph{projectivity} (\\Cref{lem:projectivity} \\Cref{sec: preliminary}), \nwhich asserts that if a Mondrian tree \n$T \\sim \\MTree{{\\mathbf{X}}}{\\lambda}$ is sampled and a new point ${\\mathbf{z}}$ is \nobserved, then inserting ${\\mathbf{z}}$ into $T$ to generate $T'$ yields\n$T' \\sim \\MTree{{\\mathbf{X}} \\cup {\\mathbf{z}}}{\\lambda}$ \\cite{lakshminarayanan2014mondrian}; \nmoreover, this process is efficient.\nThis is where the restriction of a cut $\\xi_j$ to the \nbounding box $B_j$ is critical, because it permits the sequential\naddition of ${\\mathbf{z}}$ into $T$ while preserving the distribution over which \n$T$ was sampled had ${\\mathbf{z}}$ been seen prior to sampling $T$!\nWe adapt the online update procedures from Mondrian Trees\nto streaming Mondrian \\Polya{} Tree{s} by invoking projectivity and then recognising that \nthe necessary parameters can be easily incremented as the \ndata is observed.\nInserting a point ${\\mathbf{z}}$ into tree $T$ is denoted \n$T' \\sim \\MPTreeInsert{T}{{\\mathbf{z}}}$.\n\n\n\nHowever, for data streams we also need the capability to delete from the tree;\nthis is where the link with the RRCF{} work becomes \nnecessary, as we can adapt their deletion mechanism for the \nMondrian \\Polya{} Tree{} setting.\nOur alteration is necessary for the Mondrian Tree setting as\nnodes have an associated time which cannot\nexceed the lifetime budget $\\lambda$ and deleting a point on the bounding box\ncan affect the times of all nodes in the subtree rooted at that node.\nIn this setting, the point to delete, ${\\mathbf{z}}$, is chosen\nahead of time, hence the algorithm is deterministic which is why we will\nwrite $T' = \\MPTreeDelete{T}{{\\mathbf{z}}}$ (in contrast to \n$T' \\sim \\MPTreeInsert{T}{{\\mathbf{z}}}$) for deleting ${\\mathbf{z}}$ from $T$.\nThe following lemmas summarise the insertion and deletion \nprocedures from \\cite{lakshminarayanan2014mondrian} and \\cite{guha2016robust}\nto account for the additional \\Polya{} Tree{} parameters that we need\nwhen using the Mondrian \\Polya{} Tree.\nThe insertions procedure is described in \\Cref{alg: mpt-insertion}, \nand \\Cref{alg: mpt-deletion} illustrates the deletion mechanism.\n\n\n\n\n\n\n\\begin{lemma}[Insertions] \\label{lem:mpt-insertion}\nLet $T \\sim \\MondPolyaTree{{\\mathbf{X}}}{\\lambda}$ be a Mondrian \\Polya{} Tree sampled over \ndata ${\\mathbf{X}}$ with lifetime $\\lambda > 0$.\nIf ${\\mathbf{z}}$ is a new observation and $T' \\sim \\MPTreeInsert{T}{{\\mathbf{z}}}$ then \n$T' \\sim \\MondPolyaTree{{\\mathbf{X}} \\cup {\\mathbf{z}}}{\\lambda}$.\n\\end{lemma}\n\n\\begin{proof} \nThe tree that we sample and store is exactly a Mondrian Tree, hence we \ninvoke projectivity so that $T'$ is a valid Mondrian Tree over \n${\\mathbf{X}} \\cup {\\mathbf{z}}$.\nSince the Mondrian Tree $T$ implicitly but uniquely defines a Mondrian \\Polya{} Tree{}\nwhich partitions the input space, projectivity also applies to the \nMondrian \\Polya{} Tree{} structure as a random partition.\nAdditionally, we need to alter the (cut and restrict) Beta parameters for\nevery node which are affected by the insertion of ${\\mathbf{z}}$ in tree $T$. \nHowever, this amounts to simply incrementing counts over the subtree: \nupdating the parameters is sufficient as we only need the expected value\nof every Beta distribution.\n\n\n\\end{proof}\n\n\\begin{lemma}[Deletions] \\label{lem:mpt-deletion}\nLet $T \\sim \\MondPolyaTree{{\\mathbf{X}}}{\\lambda}$ and \nlet ${\\mathbf{z}}$ be the point to be removed from ${\\mathbf{X}}$ and $T$.\nIf $T' = \\MPTreeDelete{T}{{\\mathbf{z}}}$ then \n$T' \\sim \\MondPolyaTree{{\\mathbf{X}} \\setminus {\\mathbf{z}}}{\\lambda}$.\n\\end{lemma}\n\n\\begin{proof}\n\nFirst, locate the deepest node $j$ containing ${\\mathbf{z}}$, there are two cases:\n(i) ${\\mathbf{z}}$ is \\emph{internal} to the box $B_j$\n(ii) ${\\mathbf{z}}$ is a \\emph{boundary point} defining part of the bounding box\n$B_j$ (i.e. it is maximal or minimal at $j$ in one dimension).\nIf ${\\mathbf{z}}$ is internal to $B_j$ then we are free to simply remove it from \n$j$ and decrement the necessary counts.\nOtherwise, deleting ${\\mathbf{z}}$ causes a change to the bounding box: let \n$B_j'$ denote the new bounding box for $j$ under the removal of ${\\mathbf{z}}$.\nNow, it must be the case that $L_j' = \\LinDim{B_j'}$ is \\emph{at most}\n$L_j = \\LinDim{B_j}$.\nHowever, if there is a ${\\mathbf{u}} \\ne {\\mathbf{z}}$ \nin $j$ but is equal to ${\\mathbf{z}}$ in \\emph{all} dimensions on which ${\\mathbf{z}}$ lies \non the boundary, then we could treat ${\\mathbf{z}}$ as an internal point and \nremove then decrement.\nSo assume ${\\mathbf{z}}$ uniquely defines $B_j$ in the required dimensions, hence\n$L_j' < L_j$ so the exponential distribution used to generate the \nnode time $\\tau_j$ is different under the absence of ${\\mathbf{z}}$.\nLet $F(t) = \\cdf{\\ExpDist{L_j}}(t)$ and $G(t) = \\cdf{\\ExpDist{L_j'}}(t)$ be the\nCDF functions of the exponential distributions $\\ExpDist{L_j}$ and \n$\\ExpDist{L_j'}$, respectively as functions of time $t$.\nThe mass associated to time $\\tau_j$ is $\\psi = F(\\tau_j)$ hence, the time \nwith the same mass in $G(t)$ is\n$\\tau_j' = G^{-1}(\\psi)$ (these are straightforward for the exponential\ndistribution since $\\cdf{\\ExpDist{\\zeta}}(t) = 1 - \\exp(-\\zeta t)$).\nFinally, since $L' < L$, we must have $\\tau_j' > \\tau_j$ so the time \nhas increased, meaning we must check whether $\\tau_j' < \\lambda$.\nIf so, then keep $j$, else contract $j$ and its descendants into $\\parent{j}$.\nThis approach must be done for every node on the path from $\\epsilon$ to \n $j$ which contains ${\\mathbf{z}}$ so in the worst case is \n $O(d \\cdot \\TreeDepth{T})$.\n Finally, it remains to decrement all necessary counts which were affected \n by the presence of ${\\mathbf{z}}$ on the path from $\\epsilon$ to $j$ \n (or the contracted ancestor of $j$).\n\\end{proof}\n\n \\textbf{Complexity: Insertions \\& Deletions}\n Both procedures are efficient and are dominated by the time it takes to locate\nthe locate the node which stores query point and requires checking inclusion in a\nbounding box at $O(D)$ cost a maximum of $\\TreeDepth{T}$ times, hence  $O(D \\TreeDepth{T})$ overall.\n Note that this is the absolute depth measured in the Mondrian\n Tree sense, not the adjusted depth to account for the \\Polya{} Tree{}\n construction as defined prior to \\Cref{eq: mpt-prior},\n nor the lifetime $\\lambda$ which could potentially be large.\n Since we only store the Mondrian Tree which generates the Mondrian \\Polya{} Tree{}\n which, in expectation, should be balanced and hence \n $\\TreeDepth{T} = \\Theta(\\log n)$.\n In the random forest literature (\\cite{liu2008isolation}, \n \\cite{guha2016robust}, \\cite{gopalan2019pidforest}), the depth is \n typically a parameter of small magnitude relative to \n the size of input data, usually 10.\n Hence, the presence of the maximum tree depth term in the above\n time complexity bounds is not problematic.\n \n\n\n\n\n\\begin{algorithm}[h]\n \\KwIn{Mondrian Tree $T = (\\mathsf{T}, {\\boldsymbol \\delta}, {\\boldsymbol \\xi}, {\\boldsymbol \\tau})$}\n \\KwOut{Mondrian Tree $T$ sampled over ${\\mathbf{X}} \\cup {\\mathbf{z}}$}\n\\SetKwFunction{FMain}{$\\textsf{MT}_+$}\n\\SetKwFunction{FSub}{MTx}\n\\SetKwProg{Fn}{Function}{:}{}\n\\Fn{\\FMain{$T, {\\mathbf{X}}, \\lambda, {\\mathbf{z}}$}}{\n$\\epsilon = \\textsf{root}(T)$ \\\\ \n \\FSub{$T, {\\mathbf{X}}, \\lambda, {\\mathbf{z}}, \\epsilon$} \\\\\n }\n \\SetKwProg{Pn}{Function}{:}{\\KwRet}\n \\Pn{\\FSub{$T, {\\mathbf{X}}, \\lambda, {\\mathbf{z}}, j$}}{\n    ${\\mathbf{e}}^l = \\max(l_j^{{\\mathbf{X}}} - {\\mathbf{z}},0), {\\mathbf{e}}^u = \\max({\\mathbf{z}} - u_j^{{\\mathbf{X}}},0)$ \n    \\Comment{${\\mathbf{e}}^l, {\\mathbf{e}}^u  = {\\mathbf{0}}_d$ iff $z \\in B_j$}\\\\\n    Increment the \\emph{observed restriction} parameter $\\rho_0$ by 1 \\\\ \n    Sample $E \\sim \\ExpDist{ \\sum_{i=1}^d \\left({\\mathbf{e}}^u  + {\\mathbf{e}}^l\\right)_i}$ \\\\ \n    \\uIf{$\\tau_{\\parent{j}} + E< \\tau_j$}{\n        Sample $\\delta$ with probability proportional to ${\\mathbf{e}}^u  + {\\mathbf{e}}^l$ \\\\\n        Sample a cut $\\chi \\sim \\textsf{Uniform}\\left(a,b\\right)$ with \n        $a = u_{j\\delta}^{{\\mathbf{X}}}, b = {\\mathbf{z}}_{\\delta}$ if ${\\mathbf{z}}_{\\delta} > u_{j\\delta}$,\n        else $a = {\\mathbf{z}}_{\\delta}, b = l_{j\\delta}^{{\\mathbf{X}}}$ \\\\\n        Insert $j'$ ($j$ but below $\\parent{j}$) to $j$ with: $N(j') = N(j) \\cup \\{{\\mathbf{z}}\\}$,\n        $\\delta_{j'} = \\delta, \\xi_{j'} = \\chi, \\tau_{j'} = \\tau_{\\parent{j}} + E,\n        l_{j'}^{{\\mathbf{X}}} =  \\min (l_j^{{\\mathbf{X}}}, {\\mathbf{z}}),\n        u_{j'}^{{\\mathbf{X}}} = \\max (u_j^{{\\mathbf{X}}}, {\\mathbf{z}})$ \\\\\n        Insert sibling $\\sib{j}$ containing ${\\mathbf{z}}$ such that \n        $\\lChild{j'} = j, \\rChild{j'} = \\sib{j}$ if ${\\mathbf{z}}_{\\delta_j} > \\xi_j$ or \n         $\\rChild{j'} = j, \\lChild{j'} = \\sib{j}$, otherwise \\\\ \n        Set the Beta parameters according to number of points either side of $\\xi_j$\n    }\n    \\uElse{\n        Update $l_{j}^{{\\mathbf{X}}} \\leftarrow  \\min (l_j^{{\\mathbf{X}}}, {\\mathbf{z}}),\n        u_{j'}^{{\\mathbf{X}}} \\leftarrow \\max (u_j^{{\\mathbf{X}}}, {\\mathbf{z}})$ \\\\ \n        \\uIf{$j \\in \\leaves{\\mathsf{T}}$}{\n            \\Return{}\n        }\n        \\uElse{\n            \\uIf{${\\mathbf{z}}_{\\delta_j} < \\xi_j$}{\n                $\\Child{j} = \\lChild{j}$\n            }\n            \\uElse{\n                $\\Child{j} = \\rChild{j}$\n            }\n            Increment $\\chi_{\\Child{j}}$ by 1 \\\\ \n            \\FSub{$T, {\\mathbf{X}}, \\lambda, {\\mathbf{z}}, \\Child{j}$}\n        }\n    }\n    }\n \\caption{ Mondrian \\Polya{} Tree{} Insertion: $\\MPTreeInsert{T}{{\\mathbf{z}}}$ }\n \\label{alg: mpt-insertion}\n\\end{algorithm}\n \n\\begin{algorithm}[!htb]\n \\KwIn{Mondrian Tree $T = (\\mathsf{T}, {\\boldsymbol \\delta}, {\\boldsymbol \\xi}, {\\boldsymbol \\tau})$}\n \\KwOut{Mondrian Tree $T$ sampled over ${\\mathbf{X}} \\setminus {\\mathbf{z}}$}\n\\SetKwFunction{FMain}{$\\textsf{MT}_-$}\n\\SetKwFunction{FSub}{MTd}\n\\SetKwProg{Fn}{Function}{:}{}\n\\Fn{\\FMain{$T, {\\mathbf{X}}, \\lambda, {\\mathbf{z}}$}}{\n$\\epsilon = \\textsf{root}(T), \\textsf{path} = \\{ \\epsilon \\} $\\\\\n \\FSub{$T, {\\mathbf{X}}, \\lambda, {\\mathbf{z}}, \\epsilon$} \\\\\n }\n \\SetKwProg{Pn}{Function}{:}{\\KwRet}\n \\Pn{\\FSub{$T, {\\mathbf{X}}, \\lambda, {\\mathbf{z}}, j, \\textsf{path}$}}{\n Find the deepest node $j$ containing ${\\mathbf{z}}$ \\\\ \n Let $\\textsf{path} = \\{\\epsilon, u_1, u_2, \\dots, u_k\\}$ be the set of nodes from $\\epsilon$ to $j$ \\\\ \n \\For{$j \\in \\textsf{path}$}{\n    Check if ${\\mathbf{z}}$ is \\emph{internal} to the bounding box, or a point which defines\n    the bounding box in one of dimensions $i \\in \\{1,2,\\dots,d\\}$ \\\\ \n    \\uIf{${\\mathbf{z}}$ is \\emph{internal}}{\n        $\\Child{j} = \\lChild{j}$ iff ${\\mathbf{z}}_{\\delta_j} \\le \\xi_{j}$\n        Decrement the cut and restriction counter corresponding to ${\\mathbf{z}}$ at $v$ by 1 \\\\\n        Decrement all counts by 1 at the subtree rooted at $\\Child{j}$ \n        \\Comment{This only decrements on the side of the cut that ${\\mathbf{z}}$ should go.}\\\\\n        \\Return\n    }\n    \\uElse{\n        Let $B_j'$ be the bounding box at $j$ with ${\\mathbf{z}}$ ignored which has linear \n        dimension $L' = \\LinDim{B_j'}$ \\\\\n        Evaluate new node time $\\tau_j'$ through inverse CDF \\\\\n        \\uIf{$\\tau_j' \\ge \\lambda$}{\n            Contract the entire subtree rooted at $j$ into $j$\\\\\n            Set $\\tau_j' = \\lambda$ \\\\ \n            \\Return\n        }\n    }\n }\n \\Return{$T$}\n    }\n \\caption{Inplace deletions for the Mondrian \\Polya{} Tree{}, $\\MPTreeDelete{T}{{\\mathbf{z}}}$.}\n \\label{alg: mpt-deletion}\n\\end{algorithm}\n\n\\newpage\n\\newpage\n\n\\section{Calculations for \\Cref{fig:mpt-example-tree}} \\label{sec:mpt-example-calcs}\n\nLet us consider the generative process for sampling a \n\\emph{streaming Mondrian \\Polya{} Tree} (\\acrshort{mpt}) to clarify the interplay between the underlying\nMondrian and \\Polya{} Tree{s}.\nLet ${\\mathbf{X}} = \\{{\\mathbf{x}}_1,{\\mathbf{x}}_2,{\\mathbf{x}}_3,{\\mathbf{x}}_4\\}$ with ${\\mathbf{x}}_1 = (0,0), {\\mathbf{x}}_2=(1\/4,1\/4),\n{\\mathbf{x}}_3=(2\/5,4\/5),{\\mathbf{x}}_4=(1,1)$.\nSet ${\\mathcal{D}} = \\bbox{{\\mathbf{X}}}$ to be the bounding box of the region containing ${\\mathbf{X}}$ and denote \nthe two directions which span $\\mathbb{R}^2$ be $x$ and $y$.\nWe show how sampling a depth 2 \\emph{Mondrian Tree}\nencodes a depth 4 \\acrshort{mpt} which can be used to estimate the \ndensity over ${\\mathcal{D}}$.\nNote that the only tree we store is the Mondrian Tree (with\nlifetime $\\lambda = \\infty$), albeit with the extra parameters necessary\nfor \\Polya{} Tree{} density estimation.\nRecall that the root of the tree is the node $\\epsilon$ which has an\nempty index bitstring $b(\\epsilon) = \\emptyset$ so it can be ignored\nfrom node\/parameter index strings.\nThe following example is illustrated in \\Cref{fig:mpt-example-tree}.\n\nSuppose the prior strength hyperparameter is $\\gamma=1$\nso it can be ignored from the \\Polya{} Tree{} calculations.\nThe first cut, $\\xi_{\\epsilon}$ occurs at $x = 0.5$ and traverses the\nentire bounding box ${\\mathcal{D}}$ in direction $x$.\nThis splits ${\\mathcal{D}}$ into two regions \n$R_{ 0 } \\supset \\{{\\mathbf{x}}_1,{\\mathbf{x}}_2,{\\mathbf{x}}_3\\}$\nand \n$R_{ 1} \\supset {\\mathbf{x}}_4$,\neach with volume $V_0,V_1 = 0.5$.\nHence, the posterior cut parameters are $\\chi_0^* = 7\/2$ \\& $\\chi_1^* = 3\/2$\nwhich results in $\\mu_{\\chi_{\\epsilon}} = 7\/10$.\n\nNext, call $\\Restrict{\\epsilon}$  which\ncomputes $\\Restrict{R_{0}}$\n\\& $\\Restrict{R_{ 1}}$ (see \\Cref{alg: mpt}).\nSince ${\\mathbf{x}}_4$ is isolated in $R_{ 1}$, the bounding box containing ${\\mathbf{x}}_4$ is supported on only 1 dimension;\n$\\Restrict{R_{ 1}} = (R_{1}, \\emptyset)$ and the node \nstoring ${\\mathbf{x}}_4$ is a \\emph{Type I observation leaf} with volume $1\/2$.\nHence, the \\emph{mass} associated to this leaf is \n$P_{ 1} =  3\/10$ and dividing\nout the volume of the leaf yields the \\emph{density} as $6\/10$.\n\nLet $B_{ 0} = \\bbox{R_{ 0}}$ be the bounding box\ncontaining the points ${\\mathbf{x}}_1,{\\mathbf{x}}_2,{\\mathbf{x}}_3$ from the region $R_{ 0}$.\nThen, $\\Restrict{R_{ 0}} = (B_{ 0}, B_{ 0}^C)$ which have volumes: \n$V_{B_{0}}= 32\/100$ and $ V_{B_{ 0}^C} = 18\/100$. \nThe \\text{P{\\'o}lya} depth of this node is now 1 \n(use restriction parameters \\eqref{eq: mpt-prior} with $d_{\\epsilon} = 0$)\nso the\n(posterior) parameters for the split at this level are, for inclusion in $B_{0}$\n(encoded with a $\\in$) and exclusion (encoded with a $\\neg$), respectively:\n\\begin{equation}\n    \\rho_{0\\in}^* = 2^2 \\cdot \\nicefrac{32\/100}{50\/100} + 3 \n    \\qquad\n    \\text{and}\n    \\qquad\n     \\rho_{0\\neg}^* = 2^2 \\cdot \\nicefrac{18\/100}{50\/100}.\n\\end{equation}\nAccordingly, we obtain $\\mu_{\\rho_{0}} = 36\/175$, `generate' an\n\\emph{internal node} with bitstring $0\\in$ and a \n\\emph{complementary leaf} with bitstring $0 \\neg$.\nNote that neither of these nodes is ever materialised as they \nare wholly defined by the node with index $b(j) = 0$ in the \nMondrian Tree.\nThe masses allocated are $\\mu_{\\chi_0} \\mu_{\\rho_{0}}$ \nfor the node $0 \\in$ \\&  $\\mu_{\\chi_0} (1 - \\mu_{\\rho_{0}})$\nfor the node $0 \\neg$.\n\nSince we have fixed a maximum depth of 2 for the Mondrian \nTree, we perform one subsequent cut, $\\xi_{ 0}$, to \nseparate $\\{{\\mathbf{x}}_1,{\\mathbf{x}}_2\\}$ from $\\{{\\mathbf{x}}_3\\}$, and perform a final \nrestriction procedure.\nHence, we have used the Mondrian Tree to correctly define\na \\Polya{} Tree{} over the partition of ${\\mathcal{D}}$.\nFurther details and calculations\ncan be found in \\Cref{sec:mpt-example-calcs}.\n\nNext, we deal with the points ${\\mathbf{x}}_1,{\\mathbf{x}}_2,{\\mathbf{x}}_3$ and the region $R_0$\ngenerated left of the cut $\\xi_{\\epsilon}$ by performing the first $\\Restrict{\\epsilon}$ step.\nNote that $\\Restrict{\\epsilon}$ separately computes the restriction to \nbounding boxes either side of $\\xi_{\\epsilon}$ noting that \n$\\Restrict{R_0} = (B_0, B_0^C)$ and $\\Restrict{R_1} = (R_1, \\emptyset$).\nRecall that $R_1$ contains a bounding box supported only on one dimension\nso we set this to be a Type I observation leaf so the restriction simply\nreturns the set $R_1$.\nOn the other hand, consider $R_{{\\mathbf{x}}_1,{\\mathbf{x}}_2,{\\mathbf{x}}_3}$ which has a\nvolume of 1\/2 and is decomposed into the subregions $B_0$ and\n$R_{0} \\setminus  B_0 = B_0^C$ (here the $B_0^C$\nnotation denotes the set complement in the universe $R_0$).\nWe thus obtain $V_{0\\in} = \\vol{B_0} = 32\/100$ and \n$V_{0\\neg} = 18\/100$.\nThe \\text{P{\\'o}lya} depth of this node is now 1 so the\n(posterior) parameters for the split at this level are, for inclusion in $B_0$\n(encoded with a $\\in$) and exclusion (encoded with a $\\neg$), respectively:\n\\begin{align*}\n    \\rho_{0\\in}^* &= (1+1)^2 \\cdot \\frac{32\/100}{50\/100} + 3 \\\\\n    \\rho_{0\\neg}^* &= (1+1)^2 \\cdot \\frac{18\/100}{50\/100}.\n\\end{align*}\nOverall, this results in $\\mu_{\\rho_{0}} = 36\/175$, the internal node \nwhose bitstring is $0\\in$ and a \\emph{complementary leaf} encoded by \n$0 \\neg$.\nThe mass assigned to each of these nodes is \n$\\mu_{\\chi_0} \\mu_{\\rho_{0}}$ and  \n$\\mu_{\\chi_0} (1 - \\mu_{\\rho_{0}})$, respectively.\n\nFollowing this restriction, we complete one more cut: $\\xi_{0\\in}$\nat $y=0.4$ defined only on the box $B_0$ and generate\nthe two regions $R_{00}, R_{01}$.\nSince $|R_{01} \\cap {\\mathbf{X}}| = 1$, we terminate the process and treat this leaf as an observed leaf of type I with mass \n$\\mu_{\\chi_0} \\mu_{\\rho_{0}} (1 - \\mu_{\\chi_{0\\in}})$.\nOn the other hand, $|R_{00} \\cap {\\mathbf{X}}| = 2$ so we again perform\n$\\Restrict{R_{0\\in0}} = (B_{00}, B_{00}^C)$\nwhich returns the bounding box $B_{00} = \\bbox{ {{\\mathbf{x}}_1, {\\mathbf{x}}_2} }$\nin an observation leaf of type II, along with its complementary region\nwhich is added to the set of complementary leaves.\nAt this point we terminate the process, so there are 5 leaves generated which partition the entire input domain as defined by the input data.\n\n\\paragraph{Numerics.}\nGiven data ${\\mathbf{X}}$ and the cuts $\\xi_{\\epsilon}, \\xi_{0}$ the following \nquantities are used to evaluate the density in each of the 5 leaves:\n\\begin{itemize}\n    \\item{Cut at root node P{\\'o}lya{} depth = 0: \n    $\\chi_0^* = \\frac{1\/2}{1\/2}+3$ and \n    $\\chi_0^* = \\frac{1\/2}{1\/2}+1$ so that \n    $\\mu_{\\chi_{\\epsilon}} = 7\/10$ and nodes $0,1$ are created.\n    They are internal and observation leaf Type I, respectively.}\n    \n    \\item{Restrict at node $0$, P{\\'o}lya{} depth = 1:\n    $ \\rho_{0\\in}^* = (1+1)^2 \\cdot \\frac{32\/100}{50\/100} + 3$,\n    $ \\rho_{0\\neg}^* = (1+1)^2 \\cdot \\frac{18\/100}{50\/100}$ so that \n    $ \\mu_{\\rho_{0}} = 36\/175$.\n    Internal node $0\\in$ and complementary leaf $0\\neg$ are created.}\n    \n    \\item{Cut at node $0\\in$, P{\\'o}lya{} depth = 2:\n    $\\chi_{0\\in0}^* = 9 \\frac{16}{32} + 2 = 13\/2$,\n    $\\chi_{0\\in1}^* = 9 \\frac{16}{32} + 1 = 11\/2$, so that \n    $\\mu_{\\chi_{0 \\in}} = 13 \/ 24$.\n    Internal node $0\\in 0 $ and $0\\in 1$ are created, however, $0\\in 1$\n    has exactly one datapoint in so is a Type I observation leaf.\n    }\n    \n    \\item{Restrict at node $0\\in$ with P{\\'o}lya{} depth=3:\n    $\\rho_{0\\in0\\in} = 4^2 \\frac{1\/16}{16\/100} + 2$,\n    $\\rho_{0\\in0\\neg} = 4^2 \\frac{16\/100 - 1\/16}{16\/100}$ so that \n    $\\mu_{\\rho_{0 \\in}} = 11\/24$ to get the Type II observation leaf\n    containing $B({\\mathbf{x}}_1, {\\mathbf{x}}_2)$ and the complementary leaf.}\n\\end{itemize}\n\n\n\\section{Further Details: \\Cref{sec: anomaly-detection}} \n\\label{sec: anom-appendix}\n\nThe dataset details are given in \\Cref{table: data-details}. \nWe then present the numeric results corresponding to \\Cref{tab:mean_ranks} in \n\\Cref{tab:pidforest-baseline} and a discussion in the subsequent section.\nWe also briefly present some results on the running time as well as an\ninitial statistical analysis.\n\n\\input{tab_data_summary}\n\n\n\\subsection{Experimental Results}\nThe experimental setup is as in Section \\ref{sec: anomaly-detection} and the \n\\acrshort{auc} is recorded for each dataset.\nWe separately test the batch (iForest{}, PIDForest{} and \\acrshort{bmpf}) and \nstreaming methods (\\acrshort{smpf}, RRCF{}).\nThe results are given in \\Cref{tab:pidforest-baseline}: 5 independent trials are \nperformed for each dataset with the mean and standard deviation being reported.\nWe boldface the winner for every dataset and this is used to evaluate the \nmean rank and number of wins from \\Cref{tab:mean_ranks}. \nNote that \\Cref{tab:mean_ranks} is evaluated for every trial over all datasets, \nwhereas \\Cref{tab:pidforest-baseline} simply records the winner for the best \nreported mean \\acrshort{auc}.\nThe general behaviour is that both of the Mondrian P{\\'o}lya{} Forests behave comparably\nprior state-of-the-art methods.\n\n\\input{tab_results}\n\n\n\n\n\n\\subsection{Classical Batch Methods}\n\nAnomaly detection is a classification problem with\nimbalanced classes consisting of a (large) `normal' subset of data, and\na small subset containing anomalies.\nOne could adapt supervised learning techniques (e.g a One-Class \nSupport Vector Machines (\\acrshort{1csvm}) \n\\cite{scholkopf2001estimating}) but\nlabelling anomalies is time-consuming \\& expensive so\nsupervised learning is incompatible with the large-scale streaming model.\nFor instance, \ntraining a \\acrshort{1csvm} takes time between\n$O(n^2)$ and $O(n^3)$ depending on\nthe sizes of $n$ and $D$ \\cite{bottou2007support}.\nUnsupervised methods have also been proposed which rely on some\nnotion of local or global clustering.\nFor example, Local Outlier Factor\n(\\acrshort{lof}) \\cite{breunig2000lof};\n$k$-Nearest Neighbours (\\acrshort{knn}) (\\cite{ramaswamy2000efficient}, \\cite{angiulli2002fast});\nor Principal Components Analysis (\\acrshort{pca}),\n(\\cite{shyu2003novel}, \\cite{aggarwal2015outlier}).\nHowever, the time complexity of these methods can scale\nquadratically with $n$ or $D$ so are unsuitable in the \nlarge-scale or high-dimensional setting.\n\n\nWe are interested in \\emph{unsupervised} methods: typically, these \napproaches rely on some notion of local or global clustering, for example Local Outlier Factor \n(\\acrshort{lof}) \\cite{breunig2000lof}, $k$-Nearest Neighbours (\\acrshort{knn}) (\\cite{ramaswamy2000efficient}, \\cite{angiulli2002fast}),\nor Principal Components Analysis (\\acrshort{pca}),\n(\\cite{shyu2003novel}, \\cite{aggarwal2015outlier}).\nThese solutions do not scale for large-scale and high-dimensional datasets in the offline \nsetting, let alone when we are constrained to the data stream model;\nconsider input data ${\\mathbf{X}} \\in \\mathbb{R}^{n \\times D}$,\n\\acrshort{lof} requires time at least $\\Omega(n)$, \nbut for high dimensions requires $\\Theta(n^2)$ \ntime \\cite{breunig2000lof}.\nAdditionally, \\acrshort{pca} requires \na singular value decomposition (\\acrshort{svd}) which takes\ntime $O(nD^2)$.\nUsing these datasets in the large-scale batch setting is problematic because of \nthe overhead incurred, let alone when we are further constrained to the \nstreaming environment.\nDue to the scalability of the batch offline methods, we only present the \nresults on a small subset of the datasets tested: these are given in \n\\Cref{tab:non-rf-baseline}.\n\n\\input{tab_offline_methods}\n\n \n\n \n \\subsection{Running Time}\n \\input{tab_runtime}\nAlthough not the focus of this investigation, we present an interesting \ncontrast between our method and PIDForest~ in terms of running time.\nThese results are summarised in \\Cref{table:runtime-comparison} in which the wall \nclock time necessary to perform the forest sampling from the previous\nexperiment (\\Cref{tab:pidforest-baseline}) is recorded.\nWe compare only \\acrshort{smpf} and PIDForest~ as both RRCF{} and iForest{}\nare heavily optimised and the other methods are not suitable for \nstreaming data.\nRecall that our algorithm uses all datapoints in ${\\mathbf{X}}$ to \n(i) cut the data at random,\n(ii) update model parameters for probability mass estimation.\nWhile the cutting is cheap, it is likely that the cuts may not be \ninformative which is why the second corrective step is required.\n\nPIDForest{} takes a complementary approach by optimising for the \ncut at every level rather than cutting at random, using only a small \nsubset of the data to build the tree.\nOur findings suggest that it is more efficient to make random cuts and \nupdate the parameters of the density model than solving the optimisation\nproblem for PIDForest{}.\nThis is borne out in \\Cref{table:runtime-comparison}, \\Cref{sec: anom-appendix} where our streaming \nimplementation of \\acrshort{mpf} is at least a (small) constant factor\nquicker than\nPIDForest{}, but can reach almost 50x (approximate) speedup over the time it \ntakes to fit a PIDForest{}.\nOf further interest is the fact that we use \\emph{all} datapoints per \ntree, whereas PIDForest{} uses only 100 points per tree meaning that,\nin aggregate, our method is substantially faster.\nWhile both implementations of \\acrshort{smpf} and PIDForest{} are \nproof-of-concept, the similarity of our proposed \\acrshort{bmpf} and \n\\acrshort{smpf} to the iForest{} and RRCF{} suggests that it should be\nsubstantial room for improvement, achieving runtime comparable to the best\nimplementations of each.\n\n\n\n\n\n\n\n\n\\subsection{Statistical Analysis}\n\n\nWe use repeated measures ANOVA as an omnibus test to determine if there are any significant differences between the mean values of the populations, shown in \\Cref{tab:anova}. We reject the null hypothesis ($F=104.844$, $p<0.001$) of the repeated measures ANOVA that there is a difference between the mean values of the for the independent variable of algorithm (the dataset and interaction were also significant). Therefore, we assume that there is a statistically significant difference between the mean values of the populations. Given that the results of the ANOVA test are significant, for post-hoc testing we use the paired two-way t-tests to infer which differences are significant.\nThe results are shown in \\Cref{tab:post_hoc}. The results at the $p<0.01$ level that show that bMPF and iForest both significantly outperform sMPF, all methods significantly outperform RRCF. All other comparisons failed to reach significance, indicating that based on these experiments, these methods cannot be separated from one another.\n\n\\input{tab_statistics}\n\n\\subsection{NAB Datasets}\nThe result in \\Cref{tab:pidforest-baseline} often suggest that the \\acrshort{auc} for the NAB datasets can be relatively low.\nAdditionally, sometimes our streaming method appears to lose out to the \nRRCF{} approach.\nWe suggest that part of the reason here for the slightly diminished \n\\acrshort{auc} performance could be to do with the labelling of the \nNAB datasets.\nThe anomalies are not labelled as specific datapoints, but rather windows\nor intervals which contain an anomaly.\nThis can clearly hurt the performance of a detector as not detecting an\nanomaly at the start of a window (which may well be \\emph{normal}\nbehaviour) would be recorded as incorrect predictions in the NAB labelling\nscheme.\nLikewise, the same applies if a detector quickly returns to normal \nbehaviour after the anomaly despite the labelling suggesting that the data\nindex still lies in an anomalous window.\nBoth of these behaviours are observed in Figures \\ref{fig:trace-rogue} and\n\\ref{fig:trace-adexch}.\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.9\\linewidth]{figures\/trace_rogue_.pdf}\n    \\caption{``Rogue\\_hold'' trace denoted by the blue curve in each panel.\n    The top panel illustrates the ground truth anomalies with their \n    associated window in green.\n    Flagged anomalies are in the grey shading and the red dashed line\n    is the threshold which achieves the optimum \\acrshort{auc}.\n    }\n    \\label{fig:trace-rogue}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.9\\linewidth]{figures\/trace_ad_exch.pdf}\n    \\caption{``Ad\\_exchange'' trace.\n    Plots as described in Figure \\ref{fig:trace-rogue}}\n    \\label{fig:trace-adexch}\n\\end{figure}\n\n\\newpage\n\\section{Illustrative Examples}\n\\label{sec: synthetic-examples}\n\n\n\\subsection{2d Toy Datasets}\nWe provide a simple comparison of the methods on all of the baseline\nsynthetic examples taken from the scikit-learn outlier detection \npage \\cite{outlier-sklearn} which contains unimodal and bimodal data.\nThe data is of size $n=500$ which is split between \n$n_{\\text{inliers}} = 425$ inlier points and the remaining \n$n_{\\text{outliers}} =75$ being planted outliers chosen uniformly over \nthe input domain.\nFor visual comparison, we plot the resulting classification induced by \neach of the random forest methods at the optimum threshold.\nThe results are illustrated in Figure \\ref{fig:sklearn-outlier}.\nWe additionally record the area under the ROC curve (AUC) and \narea under the\nprecision-recall-gain (PRG) curve in Table \\ref{table:synthetic-auprg}\n\\cite{flach2015precision}.\nArea under a precision-recall curve is not justified, instead use area under the PRG curve.\nWe use \\cite{prg} to evaluate the Precision-Recall-Gain and observe that\nagain our methods perform well compared to other random forests.\nThese results are presented in \\Cref{table:synthetic-auprg} but a more\nin-depth study is deferred for future work.\n\\input{appendix-synthetic-figures}\n\n\n\\section{Density Estimation}\nWe provide 4 synthetic examples to illustrate the use of our proposed models.\nA more significant experimental study will be necessary to evaluate the \nefficacy of the models in this context.\nThe synthetic datasets given below are used to generate an initial sample of \n5000 points, after which a grid is placed over the domain to estimate the \ndensity.\nFurther investigation is necessary to understand the efficicacy of both \nMondrian P{\\'o}lya{} Forests as density estimators along with a comparison to \npopular methods.\n\n\\begin{enumerate}\n    \\item{Standard Normal: Figure \\ref{fig:univariate-density} (left)}\n    \\item{Univariate Gaussian Mixture: \n    Figure \\ref{fig:univariate-density} (right)\n    taken from \n    \\url{https:\/\/scikit-learn.org\/stable\/auto_examples\/neighbors\/plot_kde_1d.html}}\n    \\item{Standard Bivariate Normal: Figure \\ref{fig:bivariate-density}\n    (left)}\n    \n    \\item{Bivariate Bimodal Mixture: \n    Figure \\ref{fig:bivariate-density} (right).\n    As in the univariate case, except the \n    covariances are adjusted to alter the shape of the clusters.\n    Also the dataset used in \\Cref{fig:all-mondrians}.} \n\\end{enumerate}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.9\\linewidth]{figures\/univariate_density_estimation.pdf}\n    \\caption{Density estimation on univariate Gaussians}\n    \\label{fig:univariate-density}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.9\\linewidth]{figures\/bivariate_density_estimation.pdf}\n    \\caption{Density estimation on bivariate Gaussians.\n    Left-right: True density function, \\acrshort{bmpf}, \\acrshort{smpf}.}\n    \\label{fig:bivariate-density}\n\\end{figure}\n\n\n\n\n\n\\section{Introduction} \\label{sec: intro}\n\n\\input{01_n_intro}\n\n\\section{Preliminaries} \\label{sec: preliminary}\n\n\\input{02_preliminaries}\n\n\\section{Mondrian P{\\'o}lya{} Forest{}} \\label{sec: mondrian-polya-forest}\n\\input{03_mondrian-polya-forest}\n\n\\section{Related Work} \\label{sec: related-work}\n\\input{04_related-work}\n\n\n\n\\section{Anomaly Detection Experiments} \n\\label{sec: anomaly-detection}\n\\input{05_anomaly-detection}\n\n\\paragraph{Conclusion.}\nWe have introduced the random forest consisting of Mondrian \\Polya{} Tree{}s.\nThese trees have natural interpretations as density estimators of the underlying\ndistribution of data.\nOur approach relates open questions concerning anomaly detection in \\cite{lakshminarayanan2016decision} through the lens of density\nestimation, thus resolving the open question in \\cite{balog2015mondrian}.\nOur method enables interpretable anomaly detection as we can threshold in the probability\ndomain and use masses rather than densities.\n\nIn addition, our random forest can be maintained on a dynamic data stream with\ninsertions and deletions, thus allowing the scalability required for large-data.\nIn future work, we plan a more in-depth analysis of the performance on data\nstreams and a rigorous study of the Mondrian \\Polya{} Tree{} as a density estimator and change-point\ndetector, rather than simply an anomaly detector.\n\nFinally, there are several directions in which this work could be\nextended to allow scalability to higher dimensions by applying random\nrotations and\/or projections after cuts.\nThis has the effect of introducing oblique cuts into the space as opposed to\naxis-aligned cuts, and could be of further benefit.\nAnother area for investigation would be to study the effect of \\emph{approximate\ncounting} for the \\Polya{} Tree{} parameters using sketches such as,\nfor example, the \\textsf{CountMin}{} sketch.\n\n\n\n\\newpage\n\n\\begin{ack}\n  CD is supported by European Research Council grant ERC-2014-CoG 647557.\n  We thank Shuai Tang for helpful discussions concerning the experiments and \n  maunscript preparation.\n\\end{ack}\n\n\\section*{Broader Impact}\n\n\nImportant applications of anomaly detection include cybersecurity intrusion detection, operational metrics monitoring, IoT signals (such as detecting broken sensors), and fraud detection. \nThus, our contribution can have impact across all these domains. \nWhile there are many applications of varying ethical value that use anomaly detection, such as the possibility for misuse by a surveillance state to \"detect anomalous citizen behavior\", we believe that by focusing on the addition of interpretability to this solution helps to mitigate the misuses possible and allow better auditing of systems that do make use of anomaly detection \\cite{Borradaile2020}.\n\nOne application in this domain where there are fairness concerns is regarding the rate of \"anomalies\" triggered by certain subgroups in fraud detection. \nA poorly calibrated or heuristic measure of anomalous behavior in this setting has the potential to discriminate against subgroups, where the data may be more sparse and thus more likely to appear anomalous.\nIn this case, additional interpretability of how the model chooses anomalies is extremely important, as it allows the system operator to properly calibrate, using existing probabilistic fairness techniques, to remove or otherwise mitigate discrimination \\cite{Davidson2019AFF}.\n\nWe present a method that enhances the state of the art for streaming anomaly detection by casting the problem as one of probabilistic density estimation.\nModeling the problem in this way brings the immediate benefit of interpretability in the anomaly space:\ntypical approaches such as thresholding at say 3 standard deviations away from the mean or median is a standard way of declaring outliers\nin applications\nbut may not be suitable in settings when arbitrary scoring metrics are\nproposed\nImportantly however, the reframing of this into probability space allows future work to integrate other important socio-technical properties such as privacy and fairness into the same solution, for which there is much research in the field.\n\n\nDeveloping accurate, efficient methods for dealing with or summarizing streaming data has the potential to reduce environmental impact significantly, as summarized data is less expensive to send and dealing with data in a localized manner (i.e. on device) removes the need to send data into the cloud for further computation.\nThis enhancement of downstream analytics also inherently allows for more privacy, by aggregating less raw data together in the cloud. Additional research into how streaming summary methods can be applied in such cases is an exciting area in the preservation of user privacy. Privacy, differential privacy in particular, in the regime of anomaly detection involves a trade-off between knowing enough about a particular data point to determine its anomaly status and the plausible deniability of that data-point. \nImproving the capabilities of private, useful models for anomaly \ndetection could be an important area for future work;\nfor example, integrating existing differential privacy models for kd-trees \\cite{cormode2012differentially} with the interpretable\nanomaly detectors we have proposed.\n\n\\newpage\n\n\\bibliographystyle{plain}\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nIn 2009,   the CDF collaboration   observed   a narrow structure\n($Y(4140)$) near the $J\/\\psi\\phi$ threshold with statistical\nsignificance in excess of $3.8 \\sigma$ in exclusive\n$B^+\\to J\/\\psi\\phi K^+$ decays produced in $\\bar{p} p $ collisions\nat $\\sqrt{s}=1.96 \\,\\rm{TeV}$ \\cite{CDF0903}. The measured mass and  width   are   $\\left(4143.0\\pm2.9\\pm1.2\\right)\\,\\rm{ MeV}$ and\n$\\left(11.7^{+8.3}_{-5.0}\\pm3.7\\right)\\, \\rm{MeV}$, respectively \\cite{CDF0903}. There have been several assignments, such as the molecular state \\cite{Y4140-molecule-1,Y4140-molecule-2,Y4140-molecule-3,Y4140-molecule-4,Y4140-molecule-5,Y4140-molecule-6,Y4140-molecule-7,Wang2014-4140}, charmonium hybrid \\cite{Y4140-hybrid}, rescattering effect \\cite{Y4140-rescattering}, tetraquark state \\cite{Y4140-tetraquark}, etc.\n\n\nLater, the Belle collaboration measured the process $\\gamma \\gamma \\to\n\\phi J\/\\psi$ for the $\\phi J\/\\psi$ invariant mass distributions\nbetween the threshold and $5\\,\\rm{GeV}$, and observed no signal for\n the decay $Y(4140)\\to \\phi J\/\\psi$, however, they observed a\nnarrow peak ($X(4350)$) of $8.8^{+4.2}_{-3.2}$ events with an\nsignificance of $3.2\\,\\sigma$ \\cite{Belle4350}. The measured mass\nand width  are $(4350.6^{+4.6}_{-5.1}\\pm 0.7)\\,\\rm{MeV}$ and\n$(13.3^{+17.9}_{-9.1}\\pm 4.1)\\,\\rm{MeV}$,  respectively\n\\cite{Belle4350}.  There also have been several assignments, such as the molecular state \\cite{X4350-molecule-1,X4350-molecule-2,X4350-molecule-3,X4350-molecule-4}, conventional charmonium \\cite{X4350-charmonium-1,X4350-charmonium-2}, charmonium-molecule mixing state \\cite{X4350-mixing}, etc.\n\n\n In 2011, the CDF collaboration confirmed the\n$Y(4140)$ in the $B^\\pm\\rightarrow J\/\\psi\\,\\phi K^\\pm$ decays  with\na  statistical significance greater  than $5\\sigma$, the measured mass and width are $\\left(4143.4^{+2.9}_{-3.0} \\pm0.6\n\\right)\\, \\rm{MeV}$ and\n$\\left(15.3^{+10.4}_{-6.1}\\pm2.5\\right)\\,\\rm{MeV}$, respectively\n\\cite{CDF1101}. Furthermore, the CDF\ncollaboration  observed an evidence for a second structure ($Y(4274)$) with approximate significance of $3.1\\,\\sigma$. The\nmeasured mass and width\n are $\\left(4274.4^{+8.4}_{-6.7}\\pm1.9\\right)\\,\\rm{MeV}$ and\n$\\left(32.3^{+21.9}_{-15.3}\\pm7.6\\right)\\,\\rm{MeV}$, respectively\n\\cite{CDF1101}. The $Y(4274)$ maybe (or maybe not) a molecular state \\cite{Y4274-molecule-1,Y4274-molecule-2,Y4274-molecule-3} or  a $0^{-+}$ tetraquark state \\cite{Y4274cscs}.\n\n\nIn 2013, the CMS collaboration   confirmed the $Y(4140)$ in the $J\/\\psi\\phi$  mass spectrum in the $B^\\pm \\to J\/\\psi \\phi K^\\pm$ decays produced in $pp$ collisions at $\\sqrt{s} = 7\\,\\rm{ TeV}$ collected with the CMS detector at the Large Hadron Collider,  and fitted the structure to a $S$-wave relativistic Breit-Wigner line-shape with the statistical significance exceeding $5 \\sigma$ \\cite{CMS1309}.\nAlso in 2013,  the D0 collaboration confirmed  the  $Y(4140)$  in the  $B^+ \\to J\/\\psi \\phi K^+$ decays in $p\\bar{p}$ collisions at $\\sqrt{s} = 1.96\\,\\rm{ TeV}$ collected by the D0 experiment at the Fermilab Tevatron collider with the statistical significance of $3.1\\sigma$  \\cite{D0-1309}.\nThe $X(4350)$ and $Y(4274)$ have not been confirmed yet.\n For detailed discussions on this subject, one can consult Ref.\\cite{Review-Jpsiphi}.\n\nThe S-wave   $J\/\\psi\\phi$ systems  have the quantum numbers $J^{PC}=0^{++}$, $1^{++}$, $2^{++}$, while the P-wave $ J\/\\psi\\phi$ systems have the quantum numbers $0^{-+}$, $1^{-+}$, $2^{-+}$, $3^{-+}$. The $X(4350)$ is observed in the $\\gamma\\gamma$ fusion, the $J^{PC}=1^{++}$, $1^{-+}$, $3^{-+}$ assignments are excluded due to Yang's Theorem \\cite{Review-Jpsiphi}. The possible assignments are $J^{PC}=0^{++}$, $0^{-+}$, $2^{++}$, $2^{-+}$. In  the scenario of tetraquark states, the masses of the $0^{-+}$ and $2^{-+}$ states are much larger than that of the $0^{++}$ and $2^{++}$ states \\cite{EFG-2008}. The $Y(4140)$, $X(4350)$ and $Y(4274)$ are observed in the $J\/\\psi\\phi$ invariant mass distribution, if they are tetraquark states,  their quark constituents must be $cs\\bar{c}\\bar{s}$.   So in this article, we study the masses of the $0^{++}$ and $2^{++}$ $cs\\bar{c}\\bar{s}$ tetraquark states with the QCD sum rules,  and try to identify the $Y(4140)$, $X(4350)$ and $Y(4274)$.\n\n\n\nThe article is arranged as follows:  we derive the QCD sum rules for the masses and pole residues of  the scalar and tensor tetraquark states  in section 2; in section 3, we present the numerical results and discussions; section 4 is reserved for our conclusion.\n\n\n\\section{QCD sum rules for  the  scalar  and tensor tetraquark states }\nIn the following, we write down  the two-point correlation functions $\\Pi_{\\mu\\nu\\alpha\\beta}(p)$ and $\\Pi(p)$ in the QCD sum rules,\n\\begin{eqnarray}\n\\Pi_{\\mu\\nu\\alpha\\beta}(p)&=&i\\int d^4x e^{ip \\cdot x} \\langle0|T\\left\\{J_{\\mu\\nu}(x)J_{\\alpha\\beta}^{\\dagger}(0)\\right\\}|0\\rangle \\, , \\\\\n\\Pi(p)&=&i\\int d^4x e^{ip \\cdot x} \\langle0|T\\left\\{J(x)J^{\\dagger}(0)\\right\\}|0\\rangle \\, ,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n J_{\\mu\\nu}(x)&=&\\frac{\\epsilon^{ijk}\\epsilon^{imn}}{\\sqrt{2}}\\left\\{s^j(x)C\\gamma_\\mu c^k(x) \\bar{s}^m(x)\\gamma_\\nu C \\bar{c}^n(x)+s^j(x)C\\gamma_\\nu c^k(x)\\bar{s}^m(x)\\gamma_\\mu C \\bar{c}^n(x) \\right\\} \\, , \\\\\n J(x)&=&\\epsilon^{ijk}\\epsilon^{imn}s^j(x)C\\gamma_\\mu c^k(x) \\bar{s}^m(x)\\gamma^\\mu C \\bar{c}^n(x) \\, ,\n\\end{eqnarray}\nthe $i$, $j$, $k$, $m$, $n$ are color indexes, the $C$ is the charge conjugation matrix. The\n   currents $J_{\\mu\\nu}(x)$ and $J(x)$ have positive parity and charge conjugation. We  take the   currents $J(x)$ and $J_{\\mu\\nu}(x)$  to interpolate the scalar  and tensor tetraquark states, respectively.\n\n\n\nAt the hadronic side, we can insert  a complete set of intermediate hadronic states with\nthe same quantum numbers as the current operators $J_{\\mu\\nu}(x)$ and $J(x)$ into the\ncorrelation functions $\\Pi_{\\mu\\nu\\alpha\\beta}(p)$ and $\\Pi(p)$ to obtain the hadronic representation\n\\cite{SVZ79,Reinders85}. After isolating the ground state\ncontributions of the scalar  and tensor tetraquark states (denoted by $X$, $Y$ and $Z$), we get the following results,\n\\begin{eqnarray}\n\\Pi_{\\mu\\nu\\alpha\\beta} (p) &=&\\frac{\\lambda_{X\/Y\/ Z}^2}{M_{X\/Y\/Z}^2-p^2}\\left( \\frac{\\widetilde{g}_{\\mu\\alpha}\\widetilde{g}_{\\nu\\beta}+\\widetilde{g}_{\\mu\\beta}\\widetilde{g}_{\\nu\\alpha}}{2}-\\frac{\\widetilde{g}_{\\mu\\nu}\\widetilde{g}_{\\alpha\\beta}}{3}\\right) +\\cdots \\, \\, , \\\\\n\\Pi (p) &=&\\frac{\\lambda_{X\/Y\/ Z}^2}{M_{X\/Y\/Z}^2-p^2} +\\cdots \\, \\, ,\n\\end{eqnarray}\nwhere $\\widetilde{g}_{\\mu\\nu}=g_{\\mu\\nu}-\\frac{p_{\\mu}p_{\\nu}}{p^2}$, the pole residues  $\\lambda_{X\/Y\/Z}$ are defined by\n\\begin{eqnarray}\n \\langle 0|J_{\\mu\\nu}(0)|X\/Y\/Z (p)\\rangle &=& \\lambda_{X\/Y\/Z} \\, \\varepsilon_{\\mu\\nu}   \\, , \\nonumber\\\\\n \\langle 0|J (0)|X\/Y\/Z (p)\\rangle &=& \\lambda_{X\/Y\/Z}     \\, ,\n\\end{eqnarray}\nthe summation of the polarization vector $\\varepsilon_{\\mu\\nu}$\n results in the following formula,\n \\begin{eqnarray}\n \\sum_{\\lambda}\\varepsilon^*_{\\alpha\\beta}(\\lambda,p)\\varepsilon_{\\mu\\nu}(\\lambda,p)\n &=&\\frac{\\widetilde{g}_{\\alpha\\mu}\\widetilde{g}_{\\beta\\nu}+\\widetilde{g}_{\\alpha\\nu}\\widetilde{g}_{\\beta\\mu}}{2}-\\frac{\\widetilde{g}_{\\alpha\\beta}\\widetilde{g}_{\\mu\\nu}}{3}\\,.\n \\end{eqnarray}\n\n\n\n In the following,  we briefly outline  the operator product expansion for the correlation functions $\\Pi_{\\mu\\nu\\alpha\\beta}(p)$ and $\\Pi(p)$ in perturbative QCD.  We contract the $s$ and $c$ quark fields in the correlation functions\n$\\Pi_{\\mu\\nu\\alpha\\beta}(p)$ and $\\Pi(p)$ with Wick theorem, and obtain the results:\n\\begin{eqnarray}\n\\Pi_{\\mu\\nu\\alpha\\beta}(p)&=&\\frac{i\\epsilon^{ijk}\\epsilon^{imn}\\epsilon^{i^{\\prime}j^{\\prime}k^{\\prime}}\\epsilon^{i^{\\prime}m^{\\prime}n^{\\prime}}}{2}\\int d^4x e^{ip \\cdot x}   \\nonumber\\\\\n&&\\left\\{{\\rm Tr}\\left[ \\gamma_{\\mu}C^{kk^{\\prime}}(x)\\gamma_{\\alpha} CS^{jj^{\\prime}T}(x)C\\right] {\\rm Tr}\\left[ \\gamma_{\\beta} C^{n^{\\prime}n}(-x)\\gamma_{\\nu} C S^{m^{\\prime}mT}(-x)C\\right] \\right. \\nonumber\\\\\n&&+{\\rm Tr}\\left[ \\gamma_{\\nu} C^{kk^{\\prime}}(x)\\gamma_{\\beta} CS^{jj^{\\prime}T}(x)C\\right] {\\rm Tr}\\left[ \\gamma_{\\alpha} C^{n^{\\prime}n}(-x)\\gamma_{\\mu} C S^{m^{\\prime}mT}(-x)C\\right] \\nonumber\\\\\n&&+{\\rm Tr}\\left[ \\gamma_{\\mu} C^{kk^{\\prime}}(x) \\gamma_{\\beta} CS^{jj^{\\prime}T}(x)C\\right] {\\rm Tr}\\left[ \\gamma_{\\alpha} C^{n^{\\prime}n}(-x) \\gamma_{\\nu}C S^{m^{\\prime}mT}(-x)C\\right] \\nonumber\\\\\n &&\\left.+{\\rm Tr}\\left[ \\gamma_{\\nu} C^{kk^{\\prime}}(x)\\gamma_{\\alpha} CS^{jj^{\\prime}T}(x)C\\right] {\\rm Tr}\\left[ \\gamma_{\\beta} C^{n^{\\prime}n}(-x)\\gamma_{\\mu} C S^{m^{\\prime}mT}(-x)C\\right] \\right\\} \\, , \\nonumber\\\\\n \\Pi(p)&=&i\\epsilon^{ijk}\\epsilon^{imn}\\epsilon^{i^{\\prime}j^{\\prime}k^{\\prime}}\\epsilon^{i^{\\prime}m^{\\prime}n^{\\prime}}\\int d^4x e^{ip \\cdot x}   \\nonumber\\\\\n&&{\\rm Tr}\\left[ \\gamma_{\\mu}C^{kk^{\\prime}}(x)\\gamma_{\\alpha} CS^{jj^{\\prime}T}(x)C\\right] {\\rm Tr}\\left[ \\gamma^{\\alpha} C^{n^{\\prime}n}(-x)\\gamma^{\\mu} C S^{m^{\\prime}mT}(-x)C\\right]   \\, ,\n\\end{eqnarray}\nwhere the   $S_{ij}(x)$   and $C_{ij}(x)$ are the full $s$ and $c$ quark propagators respectively,\n \\begin{eqnarray}\nS_{ij}(x)&=& \\frac{i\\delta_{ij}\\!\\not\\!{x}}{ 2\\pi^2x^4}\n-\\frac{\\delta_{ij}m_s}{4\\pi^2x^2}-\\frac{\\delta_{ij}\\langle\n\\bar{s}s\\rangle}{12} +\\frac{i\\delta_{ij}\\!\\not\\!{x}m_s\n\\langle\\bar{s}s\\rangle}{48}-\\frac{\\delta_{ij}x^2\\langle \\bar{s}g_s\\sigma Gs\\rangle}{192}+\\frac{i\\delta_{ij}x^2\\!\\not\\!{x} m_s\\langle \\bar{s}g_s\\sigma\n Gs\\rangle }{1152}\\nonumber\\\\\n&& -\\frac{ig_s G^{a}_{\\alpha\\beta}t^a_{ij}(\\!\\not\\!{x}\n\\sigma^{\\alpha\\beta}+\\sigma^{\\alpha\\beta} \\!\\not\\!{x})}{32\\pi^2x^2} -\\frac{i\\delta_{ij}x^2\\!\\not\\!{x}g_s^2\\langle \\bar{s} s\\rangle^2}{7776} -\\frac{\\delta_{ij}x^4\\langle \\bar{s}s \\rangle\\langle g_s^2 GG\\rangle}{27648}-\\frac{1}{8}\\langle\\bar{s}_j\\sigma^{\\mu\\nu}s_i \\rangle \\sigma_{\\mu\\nu} \\nonumber\\\\\n&&   -\\frac{1}{4}\\langle\\bar{s}_j\\gamma^{\\mu}s_i\\rangle \\gamma_{\\mu }+\\cdots \\, ,\n\\end{eqnarray}\n\\begin{eqnarray}\nC_{ij}(x)&=&\\frac{i}{(2\\pi)^4}\\int d^4k e^{-ik \\cdot x} \\left\\{\n\\frac{\\delta_{ij}}{\\!\\not\\!{k}-m_c}\n-\\frac{g_sG^n_{\\alpha\\beta}t^n_{ij}}{4}\\frac{\\sigma^{\\alpha\\beta}(\\!\\not\\!{k}+m_c)+(\\!\\not\\!{k}+m_c)\n\\sigma^{\\alpha\\beta}}{(k^2-m_c^2)^2}\\right.\\nonumber\\\\\n&&\\left. +\\frac{g_s D_\\alpha G^n_{\\beta\\lambda}t^n_{ij}(f^{\\lambda\\beta\\alpha}+f^{\\lambda\\alpha\\beta}) }{3(k^2-m_c^2)^4}-\\frac{g_s^2 (t^at^b)_{ij} G^a_{\\alpha\\beta}G^b_{\\mu\\nu}(f^{\\alpha\\beta\\mu\\nu}+f^{\\alpha\\mu\\beta\\nu}+f^{\\alpha\\mu\\nu\\beta}) }{4(k^2-m_c^2)^5}+\\cdots\\right\\} \\, ,\\nonumber\\\\\nf^{\\lambda\\alpha\\beta}&=&(\\!\\not\\!{k}+m_c)\\gamma^\\lambda(\\!\\not\\!{k}+m_c)\\gamma^\\alpha(\\!\\not\\!{k}+m_c)\\gamma^\\beta(\\!\\not\\!{k}+m_c)\\, ,\\nonumber\\\\\nf^{\\alpha\\beta\\mu\\nu}&=&(\\!\\not\\!{k}+m_c)\\gamma^\\alpha(\\!\\not\\!{k}+m_c)\\gamma^\\beta(\\!\\not\\!{k}+m_c)\\gamma^\\mu(\\!\\not\\!{k}+m_c)\\gamma^\\nu(\\!\\not\\!{k}+m_c)\\, ,\n\\end{eqnarray}\nand  $t^n=\\frac{\\lambda^n}{2}$, the $\\lambda^n$ is the Gell-Mann matrix,  $D_\\alpha=\\partial_\\alpha-ig_sG^n_\\alpha t^n$ \\cite{Reinders85}. Then we compute  the integrals both in the coordinate and momentum spaces to obtain the correlation functions $\\Pi_{\\mu\\nu\\alpha\\beta}(p)$ and $\\Pi(p)$ therefore the QCD spectral densities.\nIn Eq.(10), we retain the terms $\\langle\\bar{s}_j\\sigma_{\\mu\\nu}s_i \\rangle$ and $\\langle\\bar{s}_j\\gamma_{\\mu}s_i\\rangle$ originate from the Fierz re-arrangement of the $\\langle s_i \\bar{s}_j\\rangle$ to  absorb the gluons  emitted from the heavy quark lines   to extract the mixed condensate $\\langle\\bar{s}g_s\\sigma G s\\rangle$ and four-quark condensates  $g_s^2\\langle\\bar{s}s\\rangle^2$, respectively.\n\n Finally  we can take the\nquark-hadron duality below the continuum thresholds  $s_0$ and perform Borel transform  with respect to\nthe variable $P^2=-p^2$ to obtain  the QCD sum rules:\n\\begin{eqnarray}\n\\lambda^2_{X\/Y\/Z}\\, \\exp\\left(-\\frac{M^2_{X\/Y\/Z}}{T^2}\\right)= \\int_{4m_c^2}^{s_0} ds\\, \\rho(s) \\, \\exp\\left(-\\frac{s}{T^2}\\right) \\, ,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\rho(s)&=&\\rho_{0}(s)+\\rho_{3}(s) +\\rho_{4}(s)+\\rho_{5}(s)+\\rho_{6}(s)+\\rho_{7}(s) +\\rho_{8}(s)+\\rho_{10}(s)\\, ,\n\\end{eqnarray}\nthe explicit expressions of the $\\rho_i(s)$ are given in the appendix.\n\n\n\n\n\n\n We differentiate   Eq.(12) with respect to  $\\frac{1}{T^2}$, then eliminate the\n pole residues $\\lambda_{X\/Y\/Z}$, and  obtain the QCD sum rules for\n the masses of the scalar   and tensor tetraquark states,\n \\begin{eqnarray}\n M^2_{X\/Y\/Z}= \\frac{\\int_{4m_c^2}^{s_0} ds\\frac{d}{d \\left(-1\/T^2\\right)}\\rho(s)\\exp\\left(-\\frac{s}{T^2}\\right)}{\\int_{4m_c^2}^{s_0} ds \\rho(s)\\exp\\left(-\\frac{s}{T^2}\\right)}\\, .\n\\end{eqnarray}\n\n\n\n\n\n\\section{Numerical results and discussions}\nThe vacuum condensates are taken to be the standard values\n$\\langle\\bar{q}q \\rangle=-(0.24\\pm 0.01\\, \\rm{GeV})^3$, $\\langle\\bar{s}s \\rangle=(0.8\\pm0.1)\\langle\\bar{q}q \\rangle$,\n$\\langle\\bar{s}g_s\\sigma G s \\rangle=m_0^2\\langle \\bar{s}s \\rangle$,\n$m_0^2=(0.8 \\pm 0.1)\\,\\rm{GeV}^2$, $\\langle \\frac{\\alpha_s\nGG}{\\pi}\\rangle=(0.33\\,\\rm{GeV})^4 $    at the energy scale  $\\mu=1\\, \\rm{GeV}$\n\\cite{SVZ79,Reinders85,Ioffe2005-1,Ioffe2005-2}.\nThe quark condensates and mixed quark condensates evolve with the   renormalization group equation,\n$\\langle\\bar{q}q \\rangle(\\mu)=\\langle\\bar{q}q \\rangle(Q)\\left[\\frac{\\alpha_{s}(Q)}{\\alpha_{s}(\\mu)}\\right]^{\\frac{4}{9}}$,\n$\\langle\\bar{s}s \\rangle(\\mu)=\\langle\\bar{s}s \\rangle(Q)\\left[\\frac{\\alpha_{s}(Q)}{\\alpha_{s}(\\mu)}\\right]^{\\frac{4}{9}}$,\n  $\\langle\\bar{s}g_s \\sigma Gs \\rangle(\\mu)=\\langle\\bar{s}g_s \\sigma Gs \\rangle(Q)\\left[\\frac{\\alpha_{s}(Q)}{\\alpha_{s}(\\mu)}\\right]^{\\frac{2}{27}}$, we take into account the energy scale dependence.\n\nIn the article, we take the $\\overline{MS}$ masses $m_{c}(m_c)=(1.275\\pm0.025)\\,\\rm{GeV}$  and $m_s(\\mu=2\\,\\rm{GeV})=(0.095\\pm0.005)\\,\\rm{GeV}$\n from the Particle Data Group \\cite{PDG}, and take into account\nthe energy-scale dependence of  the $\\overline{MS}$ masses from the renormalization group equation,\n\\begin{eqnarray}\nm_s(\\mu)&=&m_s({\\rm 2GeV} )\\left[\\frac{\\alpha_{s}(\\mu)}{\\alpha_{s}({\\rm 2GeV})}\\right]^{\\frac{4}{9}} \\, ,\\nonumber\\\\\nm_c(\\mu)&=&m_c(m_c)\\left[\\frac{\\alpha_{s}(\\mu)}{\\alpha_{s}(m_c)}\\right]^{\\frac{12}{25}} \\, ,\\nonumber\\\\\n\\alpha_s(\\mu)&=&\\frac{1}{b_0t}\\left[1-\\frac{b_1}{b_0^2}\\frac{\\log t}{t} +\\frac{b_1^2(\\log^2{t}-\\log{t}-1)+b_0b_2}{b_0^4t^2}\\right]\\, ,\n\\end{eqnarray}\n  where $t=\\log \\frac{\\mu^2}{\\Lambda^2}$, $b_0=\\frac{33-2n_f}{12\\pi}$, $b_1=\\frac{153-19n_f}{24\\pi^2}$, $b_2=\\frac{2857-\\frac{5033}{9}n_f+\\frac{325}{27}n_f^2}{128\\pi^3}$,  $\\Lambda=213\\,\\rm{MeV}$, $296\\,\\rm{MeV}$  and  $339\\,\\rm{MeV}$ for the flavors  $n_f=5$, $4$ and $3$, respectively  \\cite{PDG}.\n\nIn Refs.\\cite{Wang2014-4140,WangHuangTao-1,WangHuangTao-2,WangHuangTao-3,Wang4430,Wang-Cu-Cu,WangHuang-molecule}, we study the acceptable energy scales of the QCD spectral densities for the hidden  charmed (bottom) tetraquark states and molecular states in the QCD sum rules  in details for the first time,  and suggest a  formula,\n\\begin{eqnarray}\n\\mu&=&\\sqrt{M^2_{X\/Y\/Z}-(2{\\mathbb{M}}_Q)^2} \\, ,\n \\end{eqnarray}\nwith the effective $Q$-quark masses ${\\mathbb{M}}_Q$ to determine  the energy scales of the QCD spectral densities.\nIn Refs.\\cite{WangHuangTao-1,WangHuangTao-2,WangHuangTao-3,Wang4430,Wang-Cu-Cu}, we focus on the scenario of tetraquark states, study the diquark-antidiquark type scalar, vector, axial-vector, tensor hidden charmed tetraquark states and\naxial-vector hidden bottom tetraquark states systematically  with the QCD sum rules,\nand try to make possible   assignments of the $X(3872)$,\n$Z_c(3900)$, $Z_c(3885)$, $Z_c(4020)$, $Z_c(4025)$, $Z(4050)$, $Z(4250)$, $Y(4360)$, $Z(4430)$, $Y(4630)$, $Y(4660)$, $Z_b(10610)$  and $Z_b(10650)$.\nIn the operator product expansion, we calculate the  vacuum condensates up to dimension-10, just like in the present case;\n the energy scale  formula  works very well.\n\n\nIn the conventional QCD sum rules \\cite{SVZ79,Reinders85}, we\n  usually take  the energy gap between the ground\nstates and the first radial excited states to be $(0.4-0.6)\\,\\rm{GeV}$. Such relation survives in the tetraquark sector,\n for example,\nthe  $Z(4430)$ is tentatively assigned to be the first radial excitation of the $Z_c(3900)$ according to the\nanalogous decays,\n$Z_c(3900)^\\pm\\to J\/\\psi\\pi^\\pm$, $Z(4430)^\\pm\\to\\psi^\\prime\\pi^\\pm$,\nand the mass differences $M_{Z(4430)}-M_{Z_c(3900)}=576\\,\\rm{MeV}$, $M_{\\psi^\\prime}-M_{J\/\\psi}=589\\,\\rm{MeV}$ \\cite{Wang4430,Maiani-2014,Nielsen-1401}.\n\nFirstly, we take the $Y(4140)$, $Y(4274)$ and $X(4350)$ as the scalar and tensor $cs\\bar{c}\\bar{s}$ tetraquark states, respectively, and choose the continuum threshold parameters\nas $s^0_{Y(4140)}=(4.70\\,\\rm{GeV})^2$,  $s^0_{Y(4274)}=(4.80\\,\\rm{GeV})^2$ and $s^0_{X(4350)}=(4.85\\,\\rm{GeV})^2$.\nIn Fig.1,\n the masses of the scalar  and tensor  tetraquark states  are plotted  with variations of the  Borel parameters $T^2$ and energy scales $\\mu$. From the figure, we can see that the masses decrease monotonously with increase of the energy scales, and we can also obtain the allowed energy scales to reproduce the experimental values of the masses.\n\n In Table 1, we denote the allowed energy scales which can reproduce the experimental values of the masses  as $\\mu_A$, and denote the resulting energy scales from the energy scale formula as $\\mu_T$.\n From the table, we can see that the $\\mu_A$ and $\\mu_T$ are compatible only in the case of the $Y(4140)$ with the assignment $J^{PC}=2^{++}$.\n\nNow, we assume the $Y(4140)$ to be the tensor tetraquark state, take the continuum threshold parameter as $s^0_{Y(4140)}=(4.7\\pm 0.1)^2\\,\\rm{GeV}^2$ and the energy scale as $\\mu=2.0\\,\\rm{GeV}$ to search for\nthe Borel parameter $T^2$ to satisfy the\ntwo criteria (pole dominance and convergence of the operator product\nexpansion) of the QCD sum rules. Furthermore, we study the scalar tetraquark state in the same way, i.e.\nwe search for the optimal Borel parameter $T^2$ and threshold\nparameter $s_0$ to satisfy the\ntwo criteria   of the QCD sum rules and the energy scale formula of the QCD spectral densities.\nThe resulting Borel parameters, continuum threshold parameters and the pole contributions are shown explicitly in Table 2.\n\nIn Fig.2,  we plot the contributions of different terms in the\noperator product expansion   with variations of the Borel parameters  $T^2$ for the threshold parameters $s^0_{J=2}=(4.7\\,\\rm{GeV})^2$  and $s^0_{J=0}=(4.5\\,\\rm{GeV})^2$, respectively.  In the Borel windows,  the $D_0$, $D_3$ and $D_5$\nplay an important  role,   the $D_6$  and $D_{8}$ play a minor important role, while  the $D_4$, $D_7$ and $D_{10}$ are tiny, where the $D_i$ denote the contributions of the vacuum condensates of dimensions $D=i$. The operator product expansion is well convergent. It is obvious that the two criteria  of the QCD sum rules are fully satisfied, so we expect to make reasonable predictions.\n\n\nWe take into account all uncertainties of the input parameters,\nand obtain the values of the masses and pole residues of\n the   scalar and tensor tetraquark states, which are  shown explicitly in Figs.3-4 and Table 2.\nThe prediction   $M_{J=2} =\\left(4.13^{+0.08}_{-0.08}\\right)\\,\\rm{GeV}$ is consistent with the experimental value $M_{Y(4140)}=(4143.0\\pm 2.9\\pm1.2)\\,\\rm{MeV}$ \\cite{PDG}.   The present predictions favor assigning the $Y(4140)$   to be the $J^{PC}=2^{++}$   diquark-antidiquark type tetraquark states, and disfavor assigning the $Y(4274)$ and $X(4350)$   to be the $J^{PC}=0^{++}$ or $2^{++}$   diquark-antidiquark type tetraquark states.\nAt the present time, there is no experimental candidate for the scalar $cs\\bar{c}\\bar{s}$ tetraquark state,\n we can search for the scalar  tetraquark state  at the BESIII, LHCb and Belle-II in the futures.\n\n\n Recently, Mo et al study  the $X(4350)$ as a   $ c s\\bar{c}\\bar{s}$ tetraquark state with the assignment $J^{PC}=1^{-+}$ using the QCD sum rules, and obtain the mass $M_{J=1}=(4.82\\pm 0.19)\\,\\rm{GeV}$, which is not compatible with the $X(4350)$   as a $1^{-+}$ tetraquark state \\cite{X4350-Huang}.  So the $X(4350)$ is unlikely to be a $ c s\\bar{c}\\bar{s}$ tetraquark state. Furthermore, the $X(4350)$ and $Y(4274)$ are still need confirmation.\n\n\\begin{figure}\n\\centering\n\\includegraphics[totalheight=6cm,width=7cm]{Y4140-J0-miu.EPS}\n\\includegraphics[totalheight=6cm,width=7cm]{Y4140-J2-miu.EPS}\n\\includegraphics[totalheight=6cm,width=7cm]{Y4274-J0-miu.EPS}\n\\includegraphics[totalheight=6cm,width=7cm]{Y4274-J2-miu.EPS}\n\\includegraphics[totalheight=6cm,width=7cm]{Y4350-J0-miu.EPS}\n\\includegraphics[totalheight=6cm,width=7cm]{Y4350-J2-miu.EPS}\n  \\caption{ The masses of the $Y(4140)$, $Y(4274)$ and $X(4350)$ with the assignments $J^{PC}=0^{++}$ and $2^{++}$ respectively  vary with the\n  Borel parameters $T^2$ and the energy scales $\\mu$, where the horizontal lines denote the experimental values of the masses of the $Y(4140)$, $Y(4274)$ and $X(4350)$, respectively.   }\n\\end{figure}\n\n\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\\hline\\hline\n                  &      $J^{PC}$    & $\\sqrt{s_0} (\\rm{GeV})$    & $\\mu_A(\\rm{GeV})$     & $\\mu_{T}(\\rm{GeV})$   &       \\\\ \\hline\n $Y(4140)$        &      $0^{++}$    & 4.70                       & $1.4-1.7$             & 2.0                   & $\\times$  \\\\   \\hline\n $Y(4140)$        &      $2^{++}$    & 4.70                       & $1.8-2.1$             & 2.0                   & $\\surd$   \\\\   \\hline   $Y(4274)$        &      $0^{++}$    & 4.80                       & $1.2-1.4$             & 2.3                   & $\\times$  \\\\   \\hline\n $Y(4274)$        &      $2^{++}$    & 4.80                       & $1.4-1.6$             & 2.3                   &  $\\times$ \\\\   \\hline\n $X(4350)$        &      $0^{++}$    & 4.85                       & $1.1-1.2$             & 2.4                   &  $\\times$  \\\\   \\hline\n $X(4350)$        &      $2^{++}$    & 4.85                       & $1.2-1.3$             & 2.4                   &  $\\times$  \\\\   \\hline\n \\hline\n\\end{tabular}\n\\end{center}\n\\caption{ The continuum threshold parameters $s_0$, allowed energy scales $\\mu_A$, theoretical energy scales $\\mu_T$ for the $Y(4140)$, $Y(4274)$ and $X(4350)$\nwith the possible assignments $J^{PC}$, where the $\\times$ and $\\surd$ denote the compatibility between the $\\mu_A$ and $\\mu_T$. }\n\\end{table}\n\n\n\n\n\n\n\n\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\\hline\\hline\n$J^{PC}$ &$T^2(\\rm{GeV}^2)$ &$\\sqrt{s_0} (\\rm{GeV})$ &$\\mu(\\rm{GeV})$ &pole        &$M_{X\/Y\/Z}(\\rm{GeV})$  &$\\lambda_{X\/Y\/Z}$ \\\\ \\hline\n$2^{++}$ &$3.0-3.4$         &$4.7\\pm0.1$             &2.0             &$(49-69)\\%$ &$4.13^{+0.08}_{-0.08}$ &$5.34^{+0.76}_{-0.68}\\times10^{-2}\\rm{GeV}^5$\\\\ \\hline\n$0^{++}$ &$2.5-2.9$         &$4.5\\pm0.1$             &1.7             &$(46-70)\\%$ &$3.98^{+0.08}_{-0.08}$ &$4.87^{+0.81}_{-0.68}\\times10^{-2}\\rm{GeV}^5$ \\\\ \\hline\n \\hline\n\\end{tabular}\n\\end{center}\n\\caption{ The Borel parameters, continuum threshold parameters, energy scales of the QCD spectral densities, pole contributions, masses and pole residues of the scalar  and tensor  tetraquark states. }\n\\end{table}\n\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[totalheight=6cm,width=7cm]{fraction-0.EPS}\n \\includegraphics[totalheight=6cm,width=7cm]{fraction-2.EPS}\n  \\caption{ The contributions of different terms in the operator product expansion for the $J^{PC}=0^{++}$ and $2^{++}$ tetraquark states  with variations of the\n  Borel parameters $T^2$, where the 0, 3, 4, 5, 6, 7, 8,  10 denote  the dimensions of the vacuum condensates.   }\n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[totalheight=6cm,width=7cm]{mass-J0.EPS}\n \\includegraphics[totalheight=6cm,width=7cm]{mass-J2.EPS}\n  \\caption{ The masses   of the   $J^{PC}=0^{++}$ and $2^{++}$ tetraquark states  with variations of  the\n  Borel parameters $T^2$, where the horizontal lines denote the experimental value of the mass of the $Y(4140)$. }\n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[totalheight=6cm,width=7cm]{residue-J0.EPS}\n \\includegraphics[totalheight=6cm,width=7cm]{residue-J2.EPS}\n  \\caption{ The pole residues of the   $J^{PC}=0^{++}$ and $2^{++}$ tetraquark states  with variations of  the\n  Borel parameters $T^2$.  }\n\\end{figure}\n\n\n\n\\section{Conclusion}\nIn this article, we tentatively assign the $Y(4140)$, $Y(4274)$ and $X(4350)$ to be the scalar and tensor $cs\\bar{c}\\bar{s}$ tetraquark states, respectively, and study them with the QCD sum rules. In the operator product expansion, we  calculate the contributions of the vacuum condensates up to dimension-10. Furthermore,  we use the  formula $\\mu=\\sqrt{M^2_{X\/Y\/Z}-(2{\\mathbb{M}}_c)^2}$  to determine  the energy scales of the QCD spectral densities.  The numerical results of the masses $M_{X\/Y\/Z}$ favor assigning the $Y(4140)$ to be the $J^{PC}=2^{++}$   $cs\\bar{c}\\bar{s}$ tetraquark state, and disfavor assigning the $Y(4274)$ and $X(4350)$ to be  the $0^{++}$ or $2^{++}$  tetraquark states.  There is no candidate for the scalar $cs\\bar{c}\\bar{s}$ tetraquark state, we can search for it at the BESIII, LHCb and Belle-II  in the futures.\n\n\\section*{Appendix}\nThe spectral densities $\\rho_i(s)$ with $i=0$, 3, 4, 5, 6, 7, 8, 10  at the level of the quark-gluon degrees of\nfreedom,\n\n\n\\begin{eqnarray}\n\\rho^{2}_{0}(s)&=&\\frac{1}{15360\\pi^6}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, yz\\, (1-y-z)^3\\left(s-\\overline{m}_c^2\\right)^2\\left(293s^2-190s\\overline{m}_c^2+17\\overline{m}_c^4 \\right)  \\nonumber\\\\\n&&+\\frac{1}{5120\\pi^6} \\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, yz \\,(1-y-z)^2\\left(s-\\overline{m}_c^2\\right)^4    \\nonumber\\\\\n&&+\\frac{m_sm_c}{128\\pi^6}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, (y+z)\\, (1-y-z)^2 \\left(s-\\overline{m}_c^2\\right)^2\\left(4s-\\overline{m}_c^2 \\right)  \\, ,\n\\end{eqnarray}\n\n\n\\begin{eqnarray}\n\\rho_{3}^{2}(s)&=&-\\frac{m_c\\langle \\bar{s}s\\rangle}{16\\pi^4}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, (y+z)(1-y-z)\\left(s-\\overline{m}_c^2\\right)\\left(3s-\\overline{m}_c^2\\right)  \\nonumber\\\\\n&&+\\frac{m_s\\langle \\bar{s}s\\rangle}{160\\pi^4}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, yz(1-y-z)\\left(115s^2-112s\\overline{m}_c^2+17\\overline{m}_c^4 \\right)  \\nonumber\\\\\n&&+\\frac{m_s\\langle \\bar{s}s\\rangle}{160\\pi^4}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, yz \\left( s - \\overline{m}_c^2  \\right)^2 \\nonumber\\\\\n&&-\\frac{m_sm_c^2\\langle \\bar{s}s\\rangle}{4\\pi^4}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz   \\left( s - \\overline{m}_c^2  \\right)   \\, ,\n\\end{eqnarray}\n\n\n\\begin{eqnarray}\n\\rho_{4}^{2}(s)&=&-\\frac{m_c^2}{11520\\pi^4} \\langle\\frac{\\alpha_s GG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\left( \\frac{z}{y^2}+\\frac{y}{z^2}\\right)(1-y-z)^3 \\nonumber\\\\\n&&\\left\\{ 56s-17\\overline{m}_c^2+10\\overline{m}_c^4\\delta\\left(s-\\overline{m}_c^2\\right)\\right\\} \\nonumber\\\\\n&&-\\frac{m_c^2}{3840\\pi^4}\\langle\\frac{\\alpha_s GG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\left(\\frac{z}{y^2}+\\frac{y}{z^2} \\right) (1-y-z)^2 \\left(s-\\overline{m}_c^2\\right) \\nonumber\\\\\n&&-\\frac{1}{15360\\pi^4} \\langle\\frac{\\alpha_s GG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\left( y+z\\right)(1-y-z)^2 \\left( 185s^2-208s\\overline{m}_c^2+43\\overline{m}_c^4\\right) \\nonumber\\\\\n&&+\\frac{1}{7680\\pi^4} \\langle\\frac{\\alpha_s GG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\left( y+z\\right)(1-y-z) \\left( s-\\overline{m}_c^2\\right)^2 \\nonumber\\\\\n&&-\\frac{1}{2304\\pi^4} \\langle\\frac{\\alpha_s GG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\left( y+z\\right)(1-y-z)^2 \\left( 15s^2-16s\\overline{m}_c^2+3\\overline{m}_c^4\\right) \\nonumber\\\\\n&&-\\frac{1}{13824\\pi^4} \\langle\\frac{\\alpha_s GG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, (1-y-z)^3 \\left( 25s^2-24s\\overline{m}_c^2+3\\overline{m}_c^4\\right)  \\nonumber\\\\\n&&-\\frac{1}{6912\\pi^4} \\langle\\frac{\\alpha_s GG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz  \\, yz\\,(1-y-z) \\left( 25s^2-24s\\overline{m}_c^2+3\\overline{m}_c^4\\right) \\nonumber\\\\\n&&-\\frac{1}{4608\\pi^4} \\langle\\frac{\\alpha_s GG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, (1-y-z)^2 \\left( s-\\overline{m}_c^2\\right)^2 \\nonumber\\\\\n&&-\\frac{1}{6912\\pi^4} \\langle\\frac{\\alpha_s GG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, yz \\left( s-\\overline{m}_c^2\\right)\\left( 13s-5\\overline{m}_c^2\\right) \\, ,\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\rho^{2}_{5}(s)&=&\\frac{m_c\\langle \\bar{s}g_s\\sigma Gs\\rangle}{32\\pi^4}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz  \\, (y+z) \\left(2s-\\overline{m}_c^2 \\right) \\nonumber\\\\\n&&+\\frac{m_c\\langle \\bar{s}g_s\\sigma Gs\\rangle}{144\\pi^4}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz  \\,  (1-y-z) \\left(2s-\\overline{m}_c^2 \\right)    \\nonumber\\\\\n&&-\\frac{m_s\\langle \\bar{s}g_s\\sigma Gs\\rangle}{480\\pi^4}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, yz \\left\\{56s - 17\\overline{m}_c^2 +10\\overline{m}_c^4 \\delta(s-\\overline{m}_c^2 )\\right\\}\\nonumber\\\\\n&&-\\frac{m_s\\langle \\bar{s}g_s\\sigma Gs\\rangle}{480\\pi^4}\\int_{y_i}^{y_f}dy   \\, y(1-y) \\left(s - \\widetilde{m}_c^2 \\right)+\\frac{m_sm_c^2\\langle \\bar{s}g_s\\sigma Gs\\rangle}{16\\pi^4}\\int_{y_i}^{y_f}dy  \\nonumber\\\\\n&&+\\frac{m_sm_c^2\\langle \\bar{s}g_s\\sigma Gs\\rangle}{288\\pi^4}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz  \\left(\\frac{1}{y}+\\frac{1}{z} \\right) \\, ,\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\rho_{6}^{2}(s)&=&\\frac{m_c^2\\langle\\bar{s}s\\rangle^2}{6\\pi^2}\\int_{y_i}^{y_f}dy   +\\frac{g_s^2\\langle\\bar{s}s\\rangle^2}{3240\\pi^4}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz\\, yz \\left\\{56s-17\\overline{m}_c^2 +10\\overline{m}_c^4\\delta\\left(s-\\overline{m}_c^2 \\right)\\right\\}\\nonumber\\\\\n&&+\\frac{g_s^2\\langle\\bar{s}s\\rangle^2}{3240\\pi^4}\\int_{y_i}^{y_f}dy \\,y(1-y)\\left(s-\\widetilde{m}_c^2 \\right)  \\nonumber\\\\\n&&-\\frac{g_s^2\\langle\\bar{s}s\\rangle^2}{9720\\pi^4}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, (1-y-z)\\left\\{ 45\\left(\\frac{z}{y}+\\frac{y}{z} \\right)\\left(2s-\\overline{m}_c^2 \\right)+\\left(\\frac{z}{y^2}+\\frac{y}{z^2} \\right)\\right.\\nonumber\\\\\n&&\\left.m_c^2\\left[ 19+20\\overline{m}_c^2\\delta\\left(s-\\overline{m}_c^2 \\right)\\right]+(y+z)\\left[18\\left(3s-\\overline{m}_c^2\\right)+10\\overline{m}_c^4\\delta\\left(s-\\overline{m}_c^2\\right) \\right] \\right\\} \\nonumber\\\\\n&&-\\frac{g_s^2\\langle\\bar{s}s\\rangle^2}{9720\\pi^4}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, (1-y-z)\\left\\{  15\\left(\\frac{z}{y}+\\frac{y}{z} \\right)\\left(2s-\\overline{m}_c^2 \\right)+\\left(\\frac{z}{y^2}+\\frac{y}{z^2} \\right)\\right. \\nonumber\\\\\n&&\\left.m_c^2\\left[ 6+5\\overline{m}_c^2\\delta\\left(s-\\overline{m}_c^2\\right)\\right]+(y+z)\\left[56s-17\\overline{m}_c^2 +10\\overline{m}_c^4\\delta\\left(s-\\overline{m}_c^2\\right)\\right] \\right\\}\\nonumber\\\\\n&&-\\frac{m_sm_c \\langle\\bar{s}s\\rangle^2}{12\\pi^2}\\int_{y_i}^{y_f}dy \\left\\{ 1+\\widetilde{m}_c^2\\delta(s-\\widetilde{m}_c^2)\\right\\}\\, ,\n\\end{eqnarray}\n\n\n\n\\begin{eqnarray}\n\\rho_7^{2}(s)&=&\\frac{m_c^3\\langle\\bar{s}s\\rangle}{144\\pi^2 T^2}\\langle\\frac{\\alpha_sGG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\left(\\frac{y}{z^3}+\\frac{z}{y^3}+\\frac{1}{y^2}+\\frac{1}{z^2}\\right)(1-y-z)\\, \\overline{m}_c^2 \\, \\delta\\left(s-\\overline{m}_c^2\\right)\\nonumber\\\\\n&&-\\frac{m_c\\langle\\bar{s}s\\rangle}{48\\pi^2}\\langle\\frac{\\alpha_sGG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\left(\\frac{y}{z^2}+\\frac{z}{y^2}\\right)(1-y-z)  \\left\\{1+\\overline{m}_c^2\\delta\\left(s-\\overline{m}_c^2\\right) \\right\\}\\nonumber\\\\\n&&+\\frac{m_c\\langle\\bar{s}s\\rangle}{48\\pi^2}\\langle\\frac{\\alpha_sGG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz\\left\\{1+\\frac{\\overline{m}_c^2}{3}\\delta\\left(s-\\overline{m}_c^2\\right) \\right\\} \\nonumber\\\\\n&&+\\frac{m_c\\langle\\bar{s}s\\rangle}{432\\pi^2}\\langle\\frac{\\alpha_sGG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz\\left(\\frac{1-y}{y}+\\frac{1-z}{z}\\right)\n\\left\\{1+\\overline{m}_c^2\\delta\\left(s-\\overline{m}_c^2\\right) \\right\\}\\nonumber \\\\\n&&-\\frac{m_c\\langle\\bar{s}s\\rangle}{288\\pi^2}\\langle\\frac{\\alpha_sGG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\left\\{1+ \\widetilde{m}_c^2 \\, \\delta \\left(s-\\widetilde{m}_c^2\\right) \\right\\}\\, ,\n\\end{eqnarray}\n\n\n\\begin{eqnarray}\n\\rho_8^{2}(s)&=&-\\frac{m_c^2\\langle\\bar{s}s\\rangle\\langle\\bar{s}g_s\\sigma Gs\\rangle}{12\\pi^2}\\int_0^1 dy \\left(1+\\frac{\\widetilde{m}_c^2}{T^2} \\right)\\delta\\left(s-\\widetilde{m}_c^2\\right)\\nonumber \\\\\n&&-\\frac{ m_c^2\\langle\\bar{s}s\\rangle\\langle\\bar{s}g_s\\sigma Gs\\rangle}{216\\pi^2}\\int_{0}^{1} dy \\frac{1}{y(1-y)}\\delta\\left(s-\\widetilde{m}_c^2\\right)\n \\, ,\n\\end{eqnarray}\n\n\n\\begin{eqnarray}\n\\rho_{10}^{2}(s)&=&\\frac{m_c^2\\langle\\bar{s}g_s\\sigma Gs\\rangle^2}{96\\pi^2T^6}\\int_0^1 dy \\, \\widetilde{m}_c^4 \\, \\delta \\left( s-\\widetilde{m}_c^2\\right)\n\\nonumber \\\\\n&&-\\frac{m_c^4\\langle\\bar{s}s\\rangle^2}{108T^4}\\langle\\frac{\\alpha_sGG}{\\pi}\\rangle\\int_0^1 dy  \\left\\{ \\frac{1}{y^3}+\\frac{1}{(1-y)^3}\\right\\} \\delta\\left( s-\\widetilde{m}_c^2\\right)\\nonumber\\\\\n&&+\\frac{m_c^2\\langle\\bar{s}s\\rangle^2}{36T^2}\\langle\\frac{\\alpha_sGG}{\\pi}\\rangle\\int_0^1 dy  \\left\\{ \\frac{1}{y^2}+\\frac{1}{(1-y)^2}\\right\\} \\delta\\left( s-\\widetilde{m}_c^2\\right)\\nonumber\\\\\n&&-\\frac{m_c^2\\langle\\bar{s}s\\rangle^2}{324T^2}\\langle\\frac{\\alpha_sGG}{\\pi}\\rangle\\int_0^1 dy   \\frac{1}{y(1-y)} \\delta\\left( s-\\widetilde{m}_c^2\\right)\\nonumber \\\\\n&&+\\frac{m_c^2\\langle\\bar{s}g_s\\sigma Gs\\rangle^2}{864 \\pi^2T^4} \\int_0^1 dy   \\frac{1}{y(1-y)}  \\widetilde{m}_c^2 \\, \\delta\\left( s-\\widetilde{m}_c^2\\right)\\nonumber\\\\\n&&+\\frac{m_c^2\\langle\\bar{s}g_s\\sigma Gs\\rangle^2}{576 \\pi^2T^2} \\int_0^1 dy   \\frac{1}{y(1-y)}   \\delta\\left( s-\\widetilde{m}_c^2\\right)\\nonumber \\\\\n&&+\\frac{m_c^2\\langle\\bar{s} s\\rangle^2}{108 T^6}\\langle\\frac{\\alpha_sGG}{\\pi}\\rangle\\int_0^1 dy \\, \\widetilde{m}_c^4 \\, \\delta \\left( s-\\widetilde{m}_c^2\\right) \\, ,\n\\end{eqnarray}\n\n\n\n\\begin{eqnarray}\n\\rho^{0}_{0}(s)&=&\\frac{1}{256\\pi^6}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, yz\\, (1-y-z)^3\\left(s-\\overline{m}_c^2\\right)^2\\left(7s^2-6s\\overline{m}_c^2+\\overline{m}_c^4 \\right)  \\nonumber\\\\\n&&+\\frac{1}{256\\pi^6} \\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, yz \\,(1-y-z)^2\\left(s-\\overline{m}_c^2\\right)^3 \\left(3s-\\overline{m}_c^2\\right)  \\nonumber\\\\\n&&+\\frac{m_sm_c}{128\\pi^6}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, (y+z)\\, (1-y-z)^2 \\left(s-\\overline{m}_c^2\\right)^2\\left(5s-2\\overline{m}_c^2 \\right)  \\, ,\n\\end{eqnarray}\n\n\n\\begin{eqnarray}\n\\rho_{3}^{0}(s)&=&-\\frac{m_c\\langle \\bar{s}s\\rangle}{8\\pi^4}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, (y+z)(1-y-z)\\left(s-\\overline{m}_c^2\\right)\\left(2s-\\overline{m}_c^2\\right)  \\nonumber\\\\\n&&+\\frac{m_s\\langle \\bar{s}s\\rangle}{8\\pi^4}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, yz(1-y-z)\\left(10s^2-12s\\overline{m}_c^2+3\\overline{m}_c^4 \\right)   \\nonumber\\\\\n&&+\\frac{m_s\\langle \\bar{s}s\\rangle}{8\\pi^4}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, yz\\left(s-\\overline{m}_c^2\\right)\\left(2s-\\overline{m}_c^2\\right)  \\nonumber\\\\\n&&-\\frac{m_sm_c^2\\langle \\bar{s}s\\rangle}{2\\pi^4}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz  \\left(s-\\overline{m}_c^2\\right)   \\, ,\n\\end{eqnarray}\n\n\n\\begin{eqnarray}\n\\rho_{4}^{0}(s)&=&-\\frac{m_c^2}{192\\pi^4} \\langle\\frac{\\alpha_s GG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\left( \\frac{z}{y^2}+\\frac{y}{z^2}\\right)(1-y-z)^3 \\nonumber\\\\\n&&\\left\\{ 2s-\\overline{m}_c^2+\\frac{\\overline{m}_c^4}{6}\\delta\\left(s-\\overline{m}_c^2\\right)\\right\\} \\nonumber\\\\\n&&-\\frac{m_c^2}{384\\pi^4}\\langle\\frac{\\alpha_s GG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\left(\\frac{z}{y^2}+\\frac{y}{z^2} \\right) (1-y-z)^2 \\left(3s-2\\overline{m}_c^2\\right) \\nonumber\\\\\n&&-\\frac{1}{768\\pi^4} \\langle\\frac{\\alpha_s GG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\left( y+z\\right)(1-y-z)^2 \\left( 10s^2-12s\\overline{m}_c^2+3\\overline{m}_c^4\\right) \\nonumber\\\\\n&&+\\frac{1}{384\\pi^4} \\langle\\frac{\\alpha_s GG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\left( y+z\\right)(1-y-z) \\left( s-\\overline{m}_c^2\\right)\\left( 2s-\\overline{m}_c^2\\right) \\nonumber\\\\\n&&+\\frac{1}{384\\pi^4} \\langle\\frac{\\alpha_s GG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\left( y+z\\right)(1-y-z)^2 \\left( 10s^2-12s\\overline{m}_c^2+3\\overline{m}_c^4\\right) \\nonumber\\\\\n&&+\\frac{1}{3456\\pi^4} \\langle\\frac{\\alpha_s GG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, (1-y-z)^3 \\left( 10s^2-12s\\overline{m}_c^2+3\\overline{m}_c^4\\right)  \\nonumber\\\\\n&&+\\frac{1}{576\\pi^4} \\langle\\frac{\\alpha_s GG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz  \\, yz\\,(1-y-z) \\left( 10s^2-12s\\overline{m}_c^2+3\\overline{m}_c^4\\right) \\nonumber\\\\\n&&+\\frac{1}{576\\pi^4} \\langle\\frac{\\alpha_s GG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, (1-y-z)^2 \\left( s-\\overline{m}_c^2\\right)\n\\left( 2s-\\overline{m}_c^2\\right) \\nonumber\\\\\n&&+\\frac{1}{288\\pi^4} \\langle\\frac{\\alpha_s GG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, yz \\left( s-\\overline{m}_c^2\\right)\\left( 2s-\\overline{m}_c^2\\right) \\, ,\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\rho^{0}_{5}(s)&=&\\frac{m_c\\langle \\bar{s}g_s\\sigma Gs\\rangle}{32\\pi^4}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz  \\, (y+z) \\left(3s-2\\overline{m}_c^2 \\right) \\nonumber\\\\\n&&-\\frac{m_c\\langle \\bar{s}g_s\\sigma Gs\\rangle}{48\\pi^4}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz  \\,  (1-y-z) \\left(3s-2\\overline{m}_c^2 \\right)     \\nonumber\\\\\n&&-\\frac{m_s\\langle \\bar{s}g_s\\sigma Gs\\rangle}{8\\pi^4}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz  \\,  yz\\left\\{2s-\\overline{m}_c^2+\\frac{\\overline{m}_c^2}{6} \\delta\\left(s-\\overline{m}_c^2 \\right) \\right\\}    \\nonumber\\\\\n&&-\\frac{m_s\\langle \\bar{s}g_s\\sigma Gs\\rangle}{48\\pi^4}\\int_{y_i}^{y_f}dy    \\,  y(1-y)\\left(3s-2\\widetilde{m}_c^2 \\right)     \\nonumber\\\\\n&&+\\frac{m_sm_c^2\\langle \\bar{s}g_s\\sigma Gs\\rangle}{8\\pi^4}\\int_{y_i}^{y_f}dy        \\nonumber\\\\\n&&-\\frac{m_sm_c^2\\langle \\bar{s}g_s\\sigma Gs\\rangle}{48\\pi^4}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz  \\left(\\frac{1}{y}+\\frac{1}{z} \\right) \\, ,\n\\end{eqnarray}\n\n\\begin{eqnarray}\n\\rho_{6}^{0}(s)&=&\\frac{m_c^2\\langle\\bar{s}s\\rangle^2}{3\\pi^2}\\int_{y_i}^{y_f}dy   +\\frac{g_s^2\\langle\\bar{s}s\\rangle^2}{54\\pi^4}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz\\, yz \\left\\{2s-\\overline{m}_c^2 +\\frac{\\overline{m}_c^4}{6}\\delta\\left(s-\\overline{m}_c^2 \\right)\\right\\}\\nonumber\\\\\n&&+\\frac{g_s^2\\langle\\bar{s}s\\rangle^2}{324\\pi^4}\\int_{y_i}^{y_f}dy \\,y(1-y)\\left(3s-2\\widetilde{m}_c^2 \\right)  \\nonumber\\\\\n&&-\\frac{g_s^2\\langle\\bar{s}s\\rangle^2}{648\\pi^4}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, (1-y-z)\\left\\{ 3\\left(\\frac{z}{y}+\\frac{y}{z} \\right)\\left(3s-2\\overline{m}_c^2 \\right)+\\left(\\frac{z}{y^2}+\\frac{y}{z^2} \\right)\\right.\\nonumber\\\\\n&&\\left.m_c^2\\left[ 2+ \\overline{m}_c^2\\delta\\left(s-\\overline{m}_c^2 \\right)\\right]+(y+z)\\left[12\\left(2s-\\overline{m}_c^2\\right)+2\\overline{m}_c^4\\delta\\left(s-\\overline{m}_c^2\\right) \\right] \\right\\} \\nonumber\\\\\n&&-\\frac{g_s^2\\langle\\bar{s}s\\rangle^2}{1944\\pi^4}\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\, (1-y-z)\\left\\{  15\\left(\\frac{z}{y}+\\frac{y}{z} \\right)\\left(3s-2\\overline{m}_c^2 \\right)+7\\left(\\frac{z}{y^2}+\\frac{y}{z^2} \\right)\\right. \\nonumber\\\\\n&&\\left.m_c^2\\left[ 2+\\overline{m}_c^2\\delta\\left(s-\\overline{m}_c^2\\right)\\right]+(y+z)\\left[12\\left(2s-\\overline{m}_c^2\\right) +2\\overline{m}_c^4\\delta\\left(s-\\overline{m}_c^2\\right)\\right] \\right\\} \\nonumber\\\\\n&&-\\frac{m_sm_c \\langle\\bar{s}s\\rangle^2}{12\\pi^2}\\int_{y_i}^{y_f}dy \\left\\{ 2+\\widetilde{m}_c^2\\delta(s-\\widetilde{m}_c^2)\\right\\}\\, ,\n\\end{eqnarray}\n\n\n\n\\begin{eqnarray}\n\\rho_7^{0}(s)&=&\\frac{m_c^3\\langle\\bar{s}s\\rangle}{144\\pi^2  }\\langle\\frac{\\alpha_sGG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\left(\\frac{y}{z^3}+\\frac{z}{y^3}+\\frac{1}{y^2}+\\frac{1}{z^2}\\right)(1-y-z)\\nonumber\\\\\n&&\\left(1+\\frac{ \\overline{m}_c^2}{T^2}\\right) \\delta\\left(s-\\overline{m}_c^2\\right)\\nonumber\\\\\n&&-\\frac{m_c\\langle\\bar{s}s\\rangle}{48\\pi^2}\\langle\\frac{\\alpha_sGG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz \\left(\\frac{y}{z^2}+\\frac{z}{y^2}\\right)(1-y-z)  \\left\\{2+\\overline{m}_c^2\\delta\\left(s-\\overline{m}_c^2\\right) \\right\\}\\nonumber\\\\\n&&+\\frac{m_c\\langle\\bar{s}s\\rangle}{48\\pi^2}\\langle\\frac{\\alpha_sGG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz\\left\\{2+ \\overline{m}_c^2 \\delta\\left(s-\\overline{m}_c^2\\right) \\right\\} \\nonumber\\\\\n&&-\\frac{m_c\\langle\\bar{s}s\\rangle}{144\\pi^2}\\langle\\frac{\\alpha_sGG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\int_{z_i}^{1-y}dz\\left(\\frac{1-y}{y}+\\frac{1-z}{z}\\right)\n\\left\\{2+\\overline{m}_c^2\\delta\\left(s-\\overline{m}_c^2\\right) \\right\\}\\nonumber \\\\\n&&-\\frac{m_c\\langle\\bar{s}s\\rangle}{288\\pi^2}\\langle\\frac{\\alpha_sGG}{\\pi}\\rangle\\int_{y_i}^{y_f}dy \\left\\{2+ \\widetilde{m}_c^2 \\, \\delta \\left(s-\\widetilde{m}_c^2\\right) \\right\\}\\, ,\n\\end{eqnarray}\n\n\n\\begin{eqnarray}\n\\rho_8^{0}(s)&=&-\\frac{m_c^2\\langle\\bar{s}s\\rangle\\langle\\bar{s}g_s\\sigma Gs\\rangle}{6\\pi^2}\\int_0^1 dy \\left(1+\\frac{\\widetilde{m}_c^2}{T^2} \\right)\\delta\\left(s-\\widetilde{m}_c^2\\right)\\nonumber \\\\\n&&+\\frac{ m_c^2\\langle\\bar{s}s\\rangle\\langle\\bar{s}g_s\\sigma Gs\\rangle}{36\\pi^2}\\int_{0}^{1} dy \\frac{1}{y(1-y)}\\delta\\left(s-\\widetilde{m}_c^2\\right)\n \\, ,\n\\end{eqnarray}\n\n\n\\begin{eqnarray}\n\\rho_{10}^{0}(s)&=&\\frac{m_c^2\\langle\\bar{s}g_s\\sigma Gs\\rangle^2}{48\\pi^2T^6}\\int_0^1 dy \\, \\widetilde{m}_c^4 \\, \\delta \\left( s-\\widetilde{m}_c^2\\right)\n\\nonumber \\\\\n&&-\\frac{m_c^4\\langle\\bar{s}s\\rangle^2}{54T^4}\\langle\\frac{\\alpha_sGG}{\\pi}\\rangle\\int_0^1 dy  \\left\\{ \\frac{1}{y^3}+\\frac{1}{(1-y)^3}\\right\\} \\delta\\left( s-\\widetilde{m}_c^2\\right)\\nonumber\\\\\n&&+\\frac{m_c^2\\langle\\bar{s}s\\rangle^2}{18T^2}\\langle\\frac{\\alpha_sGG}{\\pi}\\rangle\\int_0^1 dy  \\left\\{ \\frac{1}{y^2}+\\frac{1}{(1-y)^2}\\right\\} \\delta\\left( s-\\widetilde{m}_c^2\\right)\\nonumber\\\\\n&&+\\frac{m_c^2\\langle\\bar{s}s\\rangle^2}{54T^2}\\langle\\frac{\\alpha_sGG}{\\pi}\\rangle\\int_0^1 dy   \\frac{1}{y(1-y)} \\delta\\left( s-\\widetilde{m}_c^2\\right)\\nonumber \\\\\n&&-\\frac{m_c^2\\langle\\bar{s}g_s\\sigma Gs\\rangle^2}{144 \\pi^2T^4} \\int_0^1 dy   \\frac{1}{y(1-y)}  \\widetilde{m}_c^2 \\, \\delta\\left( s-\\widetilde{m}_c^2\\right)\\nonumber\\\\\n&&+\\frac{m_c^2\\langle\\bar{s}g_s\\sigma Gs\\rangle^2}{32 \\pi^2T^2} \\int_0^1 dy   \\frac{1}{y(1-y)}   \\delta\\left( s-\\widetilde{m}_c^2\\right)\\nonumber \\\\\n&&+\\frac{m_c^2\\langle\\bar{s} s\\rangle^2}{54 T^6}\\langle\\frac{\\alpha_sGG}{\\pi}\\rangle\\int_0^1 dy \\, \\widetilde{m}_c^4 \\, \\delta \\left( s-\\widetilde{m}_c^2\\right) \\, ,\n\\end{eqnarray}\n the subscripts $0$, $3$, $4$, $5$, $6$, $7$, $8$ and $10$ denote the dimensions of the vacuum condensates, the superscripts $0$ and $2$ denote the spin the tetraquark states, the $T^2$ denotes the Borel parameter; $y_{f}=\\frac{1+\\sqrt{1-4m_c^2\/s}}{2}$,\n$y_{i}=\\frac{1-\\sqrt{1-4m_c^2\/s}}{2}$, $z_{i}=\\frac{y\nm_c^2}{y s -m_c^2}$, $\\overline{m}_c^2=\\frac{(y+z)m_c^2}{yz}$,\n$ \\widetilde{m}_c^2=\\frac{m_c^2}{y(1-y)}$, $\\int_{y_i}^{y_f}dy \\to \\int_{0}^{1}dy$, $\\int_{z_i}^{1-y}dz \\to \\int_{0}^{1-y}dz$,\n when the $\\delta$ functions $\\delta\\left(s-\\overline{m}_c^2\\right)$ and $\\delta\\left(s-\\widetilde{m}_c^2\\right)$ appear.\n\n\n\n\\section*{Acknowledgements}\nThis  work is supported by National Natural Science Foundation,\nGrant Numbers 11375063,  and Natural Science Foundation of Hebei province, Grant Number A2014502017.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{intro}\nQuantum steering was first proposed by Schr\\\"odinger in 1936 \\cite{Schrodinge1936} in response to the EPR paradox \\cite{einstein1935can}. However, it did not attract much attentions until 2007, when Wiseman \\textit{et al.} reinterpreted quantum steering strictly from the operational view and even proposed some experimental criteria \\cite{wiseman2007steering}. Ever since, the research of quantum steering has made great progress both in theory \\cite{piani2015necessary,sun2017exploration} and experiment \\cite{kocsis2015experimental,cavalcanti2016quantum,deng2017demonstration,zhao2020experimental,wollmann2020experimental}. In Wiseman's definition, quantum steering that logically intermediates between quantum entanglement and Bell nonlocality, describes the ability of one party, Alice, to nonlocally control the state of another party, Bob, even when Bob does not trust Alice's measurement apparatus, exhibiting unique asymmetric behavior  \\cite{gallego2015resource,he2013genuine,xiao2017demonstration,uola2020quantum}. As an essential type of quantum correlations, quantum steering has great applications in quantum key distribution \\cite{gehring2015implementation,walk2016experimental}, subchannel discrimination \\cite{sun2018demonstration}, asymmetric quantum network \\cite{cavalcanti2015detection}, randomness generation \\cite{skrzypczyk2018maximal,guo2019experimental} and randomness certification \\cite{curchod2017unbounded}.   \nIn the standard EPR steering tasks, $N$ entangled particles are separately distributed to $N$ different observers and each observer performs some projective (sharp) measurements to demonstrate her or his steerability. Since each observer is spatially separated, the non-signaling condition is strictly satisfied between different observers, i.e., the marginal probability distribution of each observer does not depend on the measurements of any other observers \\cite{masanes2006general}. Due to the monogamy constraints, the number of observers who share quantum correlation via sharp measurement is limited \\cite{coffman2000distributed, toner2006monogamy,reid2013monogamy,mal2017necessary}. Recently, a surprising result was reported by Silva \\textit{et al.} that the number of observers sharing non-locality can be increased if the sequential weak (unsharp) measurement was employed, where the non-signaling condition is dropped \\cite{silva2015multiple}. Their result later is confirmed by theoretical \\cite{mal2016sharing,das2019facets,brown2020arbitrarily} as well as experimental works \\cite{schiavon2017three,hu2018observation} and the sequential unsharp measurement strategy has been extended to study other types of quantum correlation \\cite{bera2018witnessing,datta2018sharing,saha2019sharing}. It has shown that the maximum number of Alices who can simultaneously share steering with a single Bob can also beat the steering monogamy limits \\cite{sasmal2018steering,shenoy2019unbounded,choi2020demonstration}.\n\nHowever, all the steering sharing scenarios \\cite{sasmal2018steering,choi2020demonstration} investigated till now have the following commonalities: the initial shared state is restricted to be the maximum entangled state, the sequential unsharp measurement is only adopted by one side, and the number of measurement settings is not more than 3. Thus some interesting questions raise: whether or not the steering correlation can be kept when the shared state is not pure any more? If there exist multiple observers on both sides, can multiple Alices steer multiple Bobs simultaneously? Compared to the single Bob case, do multiple Bobs make a difference?  And how many observers can share steering simultaneously if the number of measurement settings increases? \n\nIn this work, we consider a more general sequential steering scenario featuring that unsharp measurements are sequentially performed on both sides. We investigate how many pairs of Alice and Bob can sequentially demonstrate steering in the above scenario when each party performs $N$-setting equal sharpness measurements. With the $N$-setting linear steering criterion  \\cite{cavalcanti2009experimental}, we find no more than 5 Alices can steer a single Bob for a Werner state when $N$ increases from 2 to 16. Then we show how such sequential steering sharing scenarios tolerate the environmental noise and experimental imperfections by analyzing the useful sharpness measurement range of each observer and the minimum purity bound of the initial state. Furthermore, we explore the case when multiple Bobs involved in, reporting a counter-intuitive result that at most 2 Alices can steer 2 Bobs even the number of steering sharing observers larger than 4 in the single Bob case. Finally, we show that our scenario can be used to simulate quantum decoherence channels to effectively change the ability and direction of quantum steering. Our results not only reveal the rich structure of steering sharing but also can be applied to more general scenarios involving high dimension or genuine multipartite quantum steering  \\cite{he2013genuine}.\n\n\\section{The both-sides sequential steering sharing scenario}\n\\begin{figure}[htbp]\n\t\\begin{center}\n\t\t\\includegraphics[width=10cm]{Fig_1.pdf}\n\t\t\\caption{The scenario of steering sharing with multiple observers on both sides. A two-qubit entangled state is initially shared between a sequence of Alices and Bobs. Multiple Alices implement unsharp measurements on one part of the state successively, and multiple Bobs do similar operations on the other part.}\n\t\t\\label{Fig 1}\n\t\\end{center}\n\\end{figure}\nA schematic of steering sharing scenario with both sides sequential unsharp measurements is shown in Fig.~\\ref{Fig 1}. A pair of two-qubit entangled state  $\\rho _{AB} $  is sent to multiple pairs of spatially separated observers. One of the qubits is accessed by $m$ Alices, say, $A_{1}$, $A_{2}$,..., $A_{m}$, while the other qubit is possessed by $n$ Bobs, say, $B_{1} $, $B_{2}  $,..., $B_{n}$.\nTo demonstrate the steering between multiple Alices and Bobs at the same time, all observers except the last Alice and Bob should perform unsharp measurements, otherwise, the steerability will be completely destroyed. For convenience, we assume that the sharpness of the $N$-setting measurements that each observer used is equal, which is denoted as $\\lambda_{i}$ and $\\eta_{p}$ for the $ i $-th Alice and the $ p $-th Bob respectively. Thus their corresponding $N$-setting measurements can be represented by  $\\lbrace \\hat{\\Pi}^{\\lambda_{i}}_{\\vec{m}_{1}}, \\hat{\\Pi}^{\\lambda_{i}}_{\\vec{m}_{2}}$,..., $ \\hat{\\Pi}^{\\lambda_{i}}_{\\vec{m}_{N}}\\rbrace $ and $\\lbrace \\hat{\\Lambda}^{\\eta_{p}}_{\\vec{n}_{1}}, \\hat{\\Lambda}^{\\eta_{p}}_{\\vec{n}_{2}}$,..., $ \\hat{\\Lambda}^{\\eta_{p}}_{\\vec{n}_{N}}\\rbrace $, where $\\vec{m}_{k}$ and $\\vec{n}_{k}$ represent the measurement directions with $k\\in\\lbrace 1 ,...,\\, N \\rbrace $, $i\\in\\lbrace 1 ,...,\\, m \\rbrace $, $p\\in\\lbrace 1 ,..., \\, n \\rbrace $, $\\lambda_{i} \\in [0,1]$ and $\\eta_{p} \\in [0,1]$. $\\lambda_{i} (\\eta_{p})\\!=\\!0$ corresponds to no measurement, $\\lambda_{i} (\\eta_{p})\\!=\\!1$ implies the measurement is sharp, and $\\lambda_{i} (\\eta_{p})\\in(0,1)$ means it is unsharp. It has been demonstrated that an unsharp measurement is optimal when quality factor $ F $ and the precision $ G $ of the measurement satisfy the trade-off relation $ F^{2}+G^{2}=1 $ \\cite{silva2015multiple}.\nHere, each observer adopts the optimal measurement strategy.\n\nIn the first step, suppose $A_{1}$ wants to convince $B_{1}$ that she can remotely affect his state through local measurements. However, $B_{1}$ does not trust her, so he asks $A_{1}$ to perform a measurement along $\\vec{m}_{k}$. After each run of experiment, $A_{1}$ sends $B_{1}$ the corresponding outcome $ a_{k}\\in \\lbrace 0,1 \\rbrace$ and sends $A_{2}$ the post-measurement state, which can be described by the L\\\"uders ruler \\cite{busch1986unsharp}\n\\begin{center}\n\t\\begin{equation}\n\t\\rho_{AB}\\rightarrow(\\!K^{\\lambda_{1}}_{a_{k}\\mid\\vec{m}_{k}} \\otimes I_{B} )\\rho_{AB}(K^{\\lambda_{1}\\dagger}_{a_{k}\\mid\\vec{m}_{k}} \\otimes I_{B}),\n\t\\end{equation}\n\\end{center}\nwhere $K^{\\lambda_{1}}_{a_{k}\\mid\\vec{m}_{k}} K^{\\lambda_{1}\\dagger}_{a_{k}\\mid\\vec{m}_{k}}\\!=\\!\\hat{\\Pi}^{\\lambda_{1}}_{a_{k}\\mid\\vec{m}_{k}}\\!=\\!(I_{A}+(-1)^{a_{k}} \\lambda_{1}\\,\\vec{m}_{k}\\cdot\\vec{\\sigma})\/2$, $\\vec{\\sigma}\\!=\\!\\{\\sigma_x, \\sigma_y, \\sigma_z\\}$ is the Pauli matrix, and $I_{A}$  $(I_{B})$ is the identity matrix. Repeating the process many times, when $A_{1}$ finishes all the $N$-setting measurements, $B_{1}$ will obtain $2N$ conditional states. Then $B_{1}$ performs some measurements along $\\lbrace \\vec{n}_{1}, \\vec{n}_{2},...,\\vec{n}_{N} \\rbrace$ to analyze whether these conditional states can be described by a local hidden variable state (LHS) model. If they can not, $B_{1}$ is convinced that $A_{1}$ can steer his state, and vice versa. Here, we certify the steering sharing by violating the most widely used $N$-setting linear steering inequality \\cite{cavalcanti2009experimental}, which is defined as $S_N^{1,1} \\leq C_N$ for $A_{1}$ and $B_{1}$, where $S_N^{1,1}\\equiv\\frac{1}{N}\\sum_{k=1}^N\\langle \\hat{\\Pi}^{\\lambda_{1}}_{\\vec{m}_{k}}\\hat{\\Lambda}^{\\eta_{1}}_{\\vec{n}_{k}}\\rangle=\\mathrm{Tr}  [\\rho_{AB}\\,( \\hat{\\Pi}^{\\lambda_{1}}_{a_{k}\\mid\\vec{m}_{k}}\\otimes \\hat{\\Lambda}^{\\eta_{1}}_{b_{k}\\mid\\vec{n}_{k}})] $,  $b_k$ is $B_{1}$'s measurement result, $C_N$ is the maximum value of $S_N$, which can have if the LHS model exists. On the other hand, as $A_{1}$ is assumed to act independently, thus the state shared between  $A_{2}$ and $B_{1}$ should be averaged over $A_{1}$'s outputs, i.e.,\n\\begin{equation}\n\\rho^{2,1}_{N}\\!=\\!\\sum_{k=1}^{N} \\sum_{a_{k}=0}^{1}(\\!K^{\\lambda_{1}}_{a_{k}\\mid\\vec{m}_{k}} \\otimes I_{B} )\\rho_{AB}(K^{\\lambda_{1}\\dagger}_{a_{k}\\mid\\vec{m}_{k}} \\otimes I_{B}).\n\\end{equation}\nSimilarly, the state shared between $A_{1}$ and $B_{2}$ can be described by\n\\begin{equation}\n\\rho^{1,2}_{N}\\!=\\!\\sum_{k=1}^{N} \\sum_{b_{k}=0}^{1}(\\! I_{A} \\otimes K^{\\eta_{1}}_{b_{k}\\mid\\vec{n}_{k}})\\rho_{AB}(\\! I_{A}\\otimes  K^{\\eta_{1}\\dagger}_{b_{k}\\mid\\vec{n}_{k}}).\n\\end{equation}\n\nSuppose $B_{1}$ wants to show steering with $A_{2}$ in the next step. They can verify it by calculating the steering parameter\n$S_N^{2,1}\\!\\equiv\\!\\frac{1}{N}\\sum_{k=1}^N\\langle \\hat{\\Pi}^{\\lambda_{2}}_{\\vec{m}_{k}}\\hat{\\Lambda}^{\\eta_{1}}_{\\vec{n}_{k}}\\rangle\\!=\\!\\mathrm{Tr} [\\rho^{2,1}\\,( \\hat{\\Pi}^{\\lambda_{2}}_{a_{k}\\mid\\vec{m}_{k}}\\otimes \\hat{\\Lambda}^{\\eta_{1}}_{b_{k}\\mid\\vec{n}_{k}})]$. If it is larger than $C_N$, then $B_{1}$ succeeds; otherwise, he fails. Considering the first pair Alice and Bob ($A_{1}$ and $B_{1}$) have implemented the matched measurement (when $A_{1}$ performs a measurement along $\\vec{m}_{k}$, and $B_{1}$ should measure along the $\\vec{n}_{k}$), the average state shared between $A_{2}$ and $B_{2}$ can be expressed as\n\\begin{equation}\n\\begin{split}\n\\rho^{2,2}_{N}=\\!\\sum_{k=1}^{N} \\sum_{a_{k}=0 \\atop b_{k}=0}^{1}\\!(K^{\\lambda_{1}}_{a_{k}\\mid\\vec{m}_{k}}\\!\\otimes\\!K^{\\eta_{1}}_{b_{k}\\mid\\vec{n}_{k}}) \\rho_{AB}(K^{\\lambda_{1}\\dagger}_{a_{k}\\mid\\vec{m}_{k}}\\!\\otimes\\!K^{\\eta_{1}\\dagger}_{b_{k}\\mid\\vec{n}_{k}}).\n\\end{split}\n\\end{equation}\n\nActing in analogy with the above process, at any step the state $ \\rho^{i,p}_{N} $ shared between the $ i $-th Alice and the $ p $-th Bob can be obtained by averaging over the previous observers' measurements with the help of the L\\\"uders transformation rule. The corresponding steering criterion can be written as\n\\begin{equation}\\label{eq:SAB}\nS_N^{i,p}\\equiv\\frac{1}{N}\\sum_{k=1}^N\\langle \\hat{\\Pi}^{\\lambda_{i}}_{\\vec{m}_{k}}\\hat{\\Lambda}^{\\eta_{p}}_{\\vec{n}_{k}}\\rangle\\leq C_N,\\\\\t\n\\end{equation}\nwhere $ \\langle \\hat{\\Pi}^{\\lambda_{i}}_{\\vec{m}_{k}}\\hat{\\Lambda}^{\\eta_{p}}_{\\vec{n}_{k}}\\rangle=\\mathrm{Tr}[\\rho^{i,p }\\,( \\hat{\\Pi}^{\\lambda_{i}}_{a_{k}\\mid\\vec{m}_{k}}\\otimes \\hat{\\Lambda}^{\\eta_{p}}_{b_{k}\\mid\\vec{n}_{k}})]$. Thus we can investigate the behavior of quantum steering under sequential measurements by comparing the steering parameter with the classical bound in the same measurement setting.\n\n\\section{Sharing the steering of an initial Werner state}\n\\label{Sharing the steering}\nNoting that the environmental effects may turn a pure state into a mixed one and considering the imperfection of the experimental device, one can not prepare a maximum entangled pure state. Here, we take Werner state, the best-known class of mixed entangled states, as an example to investigate the steering sharing among multiple Alices and Bobs with the aid of steering criterion shown in Eq.~(\\ref{eq:SAB}). For qubits, the Werner state is given by \\cite{werner1989quantum}\n\\begin{equation}\n\\rho\\left(\\mu\\right)=\\mu\\vert\\psi\\rangle\n\\langle\\psi\\vert+(1-\\mu)\n\\frac{I}{4},\n\\label{eq:Werner}\n\\end{equation}\nwhere $\\vert\\psi\\rangle\\!=\\!\\frac{1}{\\sqrt{2}}(\\vert01\\rangle-\\vert10\\rangle)$ is the singlet state, $I$ is the identity matrix and $\\mu\\in[0,1]$.\n\nAccording to the symmetrical property of the state, it has been demonstrated that the optimal measurement settings for any Alice and Bob are defined by the directions through antipodal pairs of vertices of a regular polyhedron \\cite{saunders2010experimental}. Thus, we can get 2, 3, 4, 6, 10 measurement settings from the square, octahedron, cube, icosahedron, and dodecahedron, respectively. And it can be further increased by combining the measurement directions from above five regular polyhedrons. In combination with the measurement directions of the icosahedron and dodecahedron, the 16 measurement settings can be obtained \\cite{bennet2012arbitrarily}.\n\nFor the case of multiple Alices and a single Bob, the state sharing among the $ i $-th Alice and the single Bob in the case of $N=2$ settings becomes    \n\\begin{equation}\\label{eq:symmetry state for two-settings}\n\\rho_2^{i,1}=\\left( \\begin{array}{cccc}\\frac{1-x}{4} & 0 & 0 & \\frac{-x+z}{4}\\\\\n0 & \\frac{1+x}{4} & -\\frac{x+z}{4} & 0\\\\\n0 & -\\frac{x+z}{4} & \\frac{1+x}{4} & 0\\\\\n\\frac{-x+z}{4} & 0 & 0 & \\frac{1-x}{4}\\end{array} \\right),\n\\end{equation}\nwhere $x\\!=\\!\\frac{1}{2^{i\\!\\!-\\!1}}\\mu\\!\\prod\\limits_{1\\leq j\\leq i\\!-\\!1}(1\\!+\\!\\sqrt{1\\!-\\!{\\lambda_j}^2})\\in[0,1]$ and $z\\!=\\!\\mu\\prod\\limits_{1 \\leq j\\leq i\\!-\\!1}(\\sqrt{1\\!-\\!{\\lambda_j}^2})\\in[0,1]$, $ i\\in\\lbrace 1,2,...,m\\rbrace $. The $j$ is positive integer, and its minimum value is 1.\nWhile the shared state for $N\\geq3$ settings keeps the Werner state's form \n\\begin{equation}\\label{eq:Werner state for three-settings}\n\\rho_N^{i,1}=\\mu'\\vert\\psi\\rangle\n\\langle\\psi\\vert+(1-\\mu')\n\\frac{I}{4},\n\\end{equation}\nwhere $\\mu'\\!=\\!\\frac{1}{3^{i\\!-\\!1}}\\mu\\prod\\limits_{1\\leq j\\leq i\\!-\\!1}(1\\!+2\\!\\sqrt{1\\!-\\!{\\lambda_j}^2})$ $\\in[0,1]$. Obviously, the shared state of each step remains symmetrical, thus the $N$-setting steering inequality Eq. (\\ref{eq:SAB}) is a sufficient and necessary criterion, which can be rewritten as\n\\begin{equation}\\label{S2AB}\nS_2^{i, 1}\\!=\\!\\frac{1}{2^{i\\!-\\!1}}\\mu\\lambda_i\\eta_1\\prod\\limits_{1\\leq j\\leq i\\!-\\!1}(1\\!+\\!F_{\\lambda_j}),\n\\end{equation}\nfor $N\\!=\\!2$ settings and\n\\begin{equation}\\label{SNAB}\nS_N^{i,1}\\!=\\!\\frac{1}{3^{i\\!-\\!1}}\\mu\\lambda_i\\eta_1\\!\\prod\\limits_{1\\leq j\\leq i\\!-\\!1}(1\\!+2F_{\\lambda_j})\\!,\n\\end{equation}\t\nfor $N\\!\\geq\\!3$ settings. $F_{\\lambda_j}$=$\\sqrt{1\\!-\\!{\\lambda_j}^2}$ represents the quality factors of related measurements. Similarly, the case of a single Alice and multiple Bobs can also be calculated.\n\nFor the case of multiple Alices and Bobs, considering the previous pair of observers adopting the matched measurements (if the Alice performs a measurement along $\\vec{m}_{k}$, the Bob should measure along the $\\vec{n}_{k}$) to verify their state's steering ability, here we choose the optimal method to calculate the state shared by the $ i $-th Alice and the $ p $-th Bob, which can maximize the value of the steering parameter. We take the case that $ i\\geq p>1$ for example, the shared state among the current $ i $-th Alice and $ p $-th Bob for $N\\!=\\!2$ settings is same as the form of Eq. (\\ref{eq:symmetry state for two-settings}), while the value of $x, z$ change to $\\!\\frac{1}{2^{i\\!-\\!1}}\\mu\\prod\\limits_{1\\leq j \\leq p\\!-\\!1}(1\\!+\\!F_{\\lambda_{j\\!+\\!i\\!-\\!p}}\\!F_{\\eta_j})\\prod\\limits_{1\\leq l \\leq i\\!\\!-p}(1\\!+\\!F_{\\lambda_l})$$\\in[0,1]$, $\\!\\mu\\prod\\limits_{1\\leq j \\leq p\\!-\\!1}F_{\\lambda_{j\\!+\\!i\\!-\\!p}}\\!F_{\\eta_j}\\prod\\limits_{1\\leq l \\leq i\\!-\\!p}F_{\\lambda_l}$$\\in[0,1]$ respectively. Then the steering parameter can be written as\n\\begin{equation}\\label{S2ABs} \nS_2^{i,p}\\!=\\!\\frac{1}{2^{i\\!-\\!1}}\\mu\\lambda_i\\eta_p\\!\\prod\\limits_{1\\leq j \\leq p\\!-\\!1}(1\\!+\\!F_{\\lambda_{j\\!+\\!i\\!-\\!p}}\\!F_{\\eta_j})\\!\\prod\\limits_{1\\leq l \\leq i\\!-\\!p}(1\\!+\\!F_{\\lambda_l}), \n\\end{equation} \nwhere the $l$ is positive integer.\n\nWhen $N\\!\\geq\\!3$, their shared state still follows the Werner state's form of Eq. (\\ref{eq:Werner state for three-settings}), where $\\mu'\\!=\\!\\frac{1}{3^{i\\!-\\!1}}\\mu\\prod\\limits_{1\\leq j \\leq p\\!-\\!1}(1\\!+\\!2\\!F_{\\lambda_{j\\!+\\!i\\!-\\!p}}\\!F_{\\eta_j})\\prod\\limits_{1\\leq l \\leq i\\!-\\!p}(1\\!+\\!2\\!F_{\\lambda_l})$ $\\in[0,1]$. And the steering parameter becomes\n\\begin{equation}\\label{SNABs} \nS_N^{i,p}\\!=\\!\\frac{1}{3^{i\\!-\\!1}}\\mu\\lambda_i\\eta_p\\prod\\limits_{1\\leq j \\leq p\\!-\\!1}(1\\!+\\!2\\!F_{\\lambda_{j\\!+\\!i\\!-\\!p}}\\!F_{\\eta_j})\\prod\\limits_{1\\leq l \\leq i\\!-\\!p}(1\\!+\\!2\\!F_{\\lambda_l}).\n\\end{equation}\nThe other case that $ p\\geq i>1$ can be obtained with the same method.\n\nObviously, the unsharp measurement strategy used here is optimal. For Werner state, the classical bound $C_N\\!=\\!\\lbrace1\/\\sqrt{2},\\,1\/\\sqrt{3},\\,1\/\\sqrt{3},\\,0.5393,\\,0.5236,\\,0.503,\\,0.5\\rbrace$ when $N\\!=\\!\\lbrace 2,3,4,6,10,16,\\infty \\rbrace$ respectively \\cite{saunders2010experimental,pramanik2019nonlocal}. \nSince the classical bound of $N\\!=\\!16$ is very close to that of infinite measurement settings, we implement $N\\!=\\!\\lbrace 2,3,4,6,10,16 \\rbrace$ to investigate the behavior of quantum steering in this work.\n\t\n\\subsection{Multiple Alices and a single Bob}\n\\begin{figure}[H]\n\t\\begin{center}\n\t\t\\includegraphics[width=9cm, height=6cm]{Fig_2.pdf}\n\t\t\\caption{The relationship between the maximum number of Alices  $N_A^\\mathrm{max}$ who can share steering with a single Bob and the number of measurement settings $N$.}\n\t\t\\label{Fig 2}\t\n\t\\end{center}\n\\end{figure}\n\\begin{table}[]\n\t\\renewcommand\\arraystretch{1.5}\n\t\\caption{The range of different measurement sharpness,  $\\lambda_1$, $\\lambda_2$, $\\lambda_3$ and $\\lambda_4$, when steering sharing is achieved for the pure initial shared state ($\\mu=1$), where $N$ is the number of measurement settings, and $N_A$, $N_B$ represent the number of Alices and Bob respectively. $\\lambda_1$, $\\lambda_2$, $\\lambda_3$ and $\\lambda_4$ denote the measurement sharpness of $A_1$, $A_2$, $A_3$ and $A_4$ respectively, and similarly, $\\eta_1$ is the sharpness of $B_1$'s measurement.  $\\mu_\\mathrm{min}$ indicates the purity infimum of the initial Werner state, which reflects the noise robustness. As long as $\\mu$ is greater than $\\mu_\\mathrm{min}$, steering sharing happened in the case $\\mu\\!=\\!1$ can still be realized.}\\label{Table 1}\n    \\centering  \n    \\scriptsize\n\t\\begin{tabular}{|p{4pt}<{\\centering}|p{3pt}<{\\centering}|p{3pt}<{\\centering}|p{51pt}<{\\centering}|p{51pt}<{\\centering}|p{51pt}<{\\centering}|p{10pt}<{\\centering}|p{30pt}<{\\centering}|p{20pt}<{\\centering}|}\n\t\t\\hline\t\t\n\t\t$N$&$N_A$&$N_B$&$\\lambda_1$&$\\lambda_2$&$\\lambda_3$& $\\lambda_4$&$\\eta_1$&$\\mu_\\mathrm{min}$ \\\\\n\t\t\\hline\n\t\t2&1&1&$[0.707(2),1]$&--&--&--&$[0.707(2),1]$&$0.707(2)$\\\\\n\t\t\\cline{1-9}\\cline{2-9} \\cline{3-9} \\cline{4-9} \\cline{5-9} \\cline{6-9} \\cline{7-9}\\cline{8-9}\\cline{9-9}\n\t\t2&2&1&$[0.707(2),0.910(1)]$&1&--&--&$[0.884(1),1]$&$0.891(9)$\\\\\n\t\t\\cline{1-9}\\cline{2-9} \\cline{3-9} \\cline{4-9} \\cline{5-9} \\cline{6-9} \\cline{7-9}\\cline{8-9}\\cline{9-9}\n\t\t3\/4&1&1&$[0.577(4),1]$&--&--&--&$[0.577(4),1]$&$0.577(4)$\\\\\n\t\t\\cline{1-9}\\cline{2-9} \\cline{3-9} \\cline{4-9} \\cline{5-9} \\cline{6-9} \\cline{7-9}\\cline{8-9}\\cline{9-9}\n\t\t3\/4&2&1& $[0.577(4),0.930(6)]$&1&--&--&$[0.756(0),1]$&$0.759(8)$\\\\\n\t\t\\cline{1-9}\\cline{2-9} \\cline{3-9} \\cline{4-9} \\cline{5-9} \\cline{6-9} \\cline{7-9}\\cline{8-9}\\cline{9-9}\n\t\t3\/4&3&1&$[0.577(4),0.773(3)]$&$[0.657(9),0.873(5)]$&1&--&$1$&$0.909(4)$\\\\\n\t\t\\cline{1-9}\\cline{2-9} \\cline{3-9} \\cline{4-9} \\cline{5-9} \\cline{6-9} \\cline{7-9}\\cline{8-9}\\cline{9-9}\n\t\t6&1&1&$[0.539(4),1]$&--&--&--&$[0.539(4),1]$&$0.539(4)$\\\\\n\t\t\\cline{1-9}\\cline{2-9} \\cline{3-9} \\cline{4-9} \\cline{5-9} \\cline{6-9} \\cline{7-9}\\cline{8-9}\\cline{9-9}\n\t\t6&2&1&$[0.539(4),0.951(0)]$&1&--&--&$[0.706(2),1]$&$0.706(7)$\\\\\n\t\t\\cline{1-9}\\cline{2-9} \\cline{3-9} \\cline{4-9} \\cline{5-9} \\cline{6-9} \\cline{7-9}\\cline{8-9}\\cline{9-9}\n\t\t6&3&1&$[0.539(4),0.828(9)]$&$[0.602(8),0.914(7)]$&1&--&1&$0.846(4)$\\\\\n\t\t\\cline{1-9}\\cline{2-9} \\cline{3-9} \\cline{4-9} \\cline{5-9} \\cline{6-9} \\cline{7-9}\\cline{8-9}\\cline{9-9}\n\t\t6&4&1&$[0.539(4),0.643(7)]$&$[0.602(8),0.710(2)]$&$[0.707(5),0.829(1)]$&1&1&$0.965(5)$\\\\\n\t\t\\cline{1-9}\\cline{2-9} \\cline{3-9} \\cline{4-9} \\cline{5-9} \\cline{6-9} \\cline{7-9}\\cline{8-9}\\cline{9-9}\n\t\t10&1&1&$[0.523(7),1]$&--& --&--&$[0.523(7),1]$&$0.523(7)$\\\\\n\t\t\\cline{1-9}\\cline{2-9} \\cline{3-9} \\cline{4-9} \\cline{5-9} \\cline{6-9} \\cline{7-9}\\cline{8-9}\\cline{9-9}\n\t\t10&2&1&$[0.523(7),0.958(4)]$&1&--&--&$[0.685(7),1]$&$0.686(1)$\\\\\n\t\t\\cline{1-9}\\cline{2-9} \\cline{3-9} \\cline{4-9} \\cline{5-9} \\cline{6-9} \\cline{7-9}\\cline{8-9}\\cline{9-9}\n\t\t10&3&1&$[0.523(7),0.848(9)]$&$[0.581(0),0.928(4)]$&1&--&1&$0.816(0)$\\\\\n\t\t\\cline{1-9}\\cline{2-9} \\cline{3-9} \\cline{4-9} \\cline{5-9} \\cline{6-9} \\cline{7-9}\\cline{8-9}\\cline{9-9}\n\t\t10&4&1&$[0.523(7),0.677(5)]$&$[0.581(0),0.749(6)]$&$[0.674(3),0.859(3)]$&1&1&$0.930(2)$\\\\\n\t\t\\cline{1-9}\\cline{2-9} \\cline{3-9} \\cline{4-9} \\cline{5-9} \\cline{6-9} \\cline{7-9}\\cline{8-9}\\cline{9-9}\n\t\t16&1&1&$[0.503(1),1]$&--&--&--&$[0.503(1),1]$&$0.503(1)$\\\\\n\t\t\\cline{1-9}\\cline{2-9} \\cline{3-9} \\cline{4-9} \\cline{5-9} \\cline{6-9} \\cline{7-9}\\cline{8-9}\\cline{9-9}\n\t\t16&2&1&$[0.503(1),0.967(0)]$&1&--&--&$[0.658(7),1]$&$0.658(7)$\\\\\n\t\t\\cline{1-9}\\cline{2-9} \\cline{3-9} \\cline{4-9} \\cline{5-9} \\cline{6-9} \\cline{7-9}\\cline{8-9}\\cline{9-9}\n\t\t16&3&1& $[0.503(1),0.872(8)]$&$[0.553(1),0.944(1)]$&1&--&1&$0.783(3)$\\\\\n\t\t\\cline{1-9}\\cline{2-9} \\cline{3-9} \\cline{4-9} \\cline{5-9} \\cline{6-9} \\cline{7-9}\\cline{8-9}\\cline{9-9}\n\t\t16&4&1& $[0.503(1),0.727(0)]$&$[0.553(1),0.795(4)]$&$[0.626(3),0.898(3)]$&1&1&$0.888(9)$\\\\\n\t\t\\cline{1-9}\\cline{2-9} \\cline{3-9} \\cline{4-9} \\cline{5-9} \\cline{6-9} \\cline{7-9}\\cline{8-9}\\cline{9-9}\n\t\t16&5&1&$[0.503(1),0.545(4)]$&$[0.553(1),0.612(5)]$&$[0.626(6),0.679(6)]$&0.766&1&$0.979(5)$\\\\\n\t\t\\cline{1-9}\\cline{2-9} \\cline{3-9} \\cline{4-9} \\cline{5-9} \\cline{6-9} \\cline{7-9}\\cline{8-9}\\cline{9-9}\n\t\t\\hline\n\t\\end{tabular}\n\t\\normalsize\n\\end{table}  \nFirstly, we take multiple Alices and a single Bob as an example to explore how many observers in one part can simultaneously steer the state of a single observer in the other part in different settings. It is obvious that the steering parameter increases with the increasing of current measurement sharpness and the decreasing of previous measurement sharpness. Since the previous measurement may decrease the steerability of the current shared state, the measurement sharpness of the latter Alices would be increased to obtain enough information to show their steerability, i.e., $\\lambda_1<\\lambda_2<$,...,$<\\lambda_m$. And the steering sharing process can continue, with each latter Alice and Bob being able to violate steering inequality with the average shared state obtained from the previous stage, as long as $\\lambda_i\\!<\\!1$ and $\\eta_1\\!<\\!1$. From this condition, one can obtain the maximum number of Alices $N_A^\\mathrm{max}$ who can share steering simultaneously with a single Bob. The result is presented in Fig.~\\ref{Fig 2}. It is obvious that as the number of measurement settings $N$ increases, the overall tendency of $N_A^\\mathrm{max}$ rises.  We find at most 5 Alices can simultaneously steer Bob's state when the number of measurement setting reaches 16. Interestingly, for some special case $N_A^\\mathrm{max}$ remains the same even if $N$ increases (such as $N\\!=\\!3,4$, or $N\\!=\\!6,10$). Note that it was conjectured in Ref.~\\cite{sasmal2018steering} that at most $N$ Alices can exhibit steering with a single Bob by the violation of $N$-setting linear inequality. From our results, it seems that this conjecture is not true. \n\nWe further calculate the useful sharpness parameter regions for all possible sharing scenarios with the maximally entangled initial state. The results are summarized in Table~\\ref{Table 1}. Here, we assume Bob performs sharp measurements when $N_A\\geq 3$ and the final Alice also performs sharp measurements when $N_A\\geq 2$, while in other cases, the observer's measurements are unsharp.  It clearly indicates that the useful measurement sharpness interval of these observers decreases with the number of Alices increasing and the number of the measurement settings decreasing. For the case of 2 Alices and a single Bob, we find the ranges of the first Alice's sharpness $\\lambda_1$ and the first Bob's sharpness $\\eta_1$ are respectively expanded about 2.3 and 2.9 times, as the number of measurement settings increases from 2 to 16, making it easier to apply directly to experiments.\n\nIt should be noted that all the above results are restricted to a pure initial shared state. Considering the decoherence effect of environmental noise and the imperfection of the experimental device, we further investigate whether or not the steering correlation can be kept when the shared state is not pure any more. We find that it can be kept indeed. The minimum purity bound of the initial state $\\mu_\\mathrm{min}$ is presented in the last column of Table \\ref{Table 1}. Obviously, for any possible steering sharing scenarios, there exist a finite continuous range of purity such that these Alices can share steering with Bob. And for a fixed number of observers, the more measurement settings, the greater the purity range and the stronger the robustness.\n\n\\subsection{Multiple Alices and Bobs}\n\\begin{figure}[H]\n\t\\begin{center}\n\t\t\n\t\t\\subfigure[ ]{\\includegraphics[width=3.75cm]{Fig_3_a.pdf}}\\\\\n\t\t\\subfigure[ ]{\\includegraphics[width=5.5cm]{Fig_3_b.pdf}}\n\t\t\\subfigure[ ]{\\includegraphics[width=5.5cm]{Fig_3_c.pdf}}\n\t\t\\caption{(a) The schematic diagram for the 2 Alices and 2 Bobs is displayed via 3 or more settings measurement. Here, the arrow indicates the steering direction. (b) Steering parameters $S_3^{i,p}$ ($i\\!=\\!1,2$, and $p\\!=\\!1,2$) are presented for 3-setting measurements as a function of $\\lambda_{1}$ and $\\eta_{1}$.  $\\lambda_{2}=1$ and $\\eta_{2}=1$ indicates that $A_2$ and $B_2$ implement the sharp measurements. The green, blue, red, and purple lines correspond to $S_3^{1,1}\\!=\\!C_3$, $S_3^{1,2}\\!=\\!C_3$, $S_3^{2,1}\\!=\\!C_3$, and $S_3^{2,2}\\!=\\!C_3$ respectively. The regions where the corresponding colored arrows point to indicate that the $S_3^{1,1}$, $S_3^{1,2}$, $S_3^{2,1}$, and $S_3^{2,2}$ exceed $C_3$, respectively. (c) Displaying the steering parameters $S_{16}^{i,p}$ ($i\\!=\\!1,2$, and $p\\!=\\!1,2$) for 16-setting measurements versus $\\lambda_{1}$ and $\\eta_{1}$. Similarly, The regions where the corresponding colored arrows point to mean that the violation of linear inequality between $A_1$-$B_1$, $A_1$-$B_2$, $A_2$-$B_1$, $A_2$-$B_2$, respectively. The overlapping regions in (b) and (c) colored in yellow demonstrate the steering sharing among 2 Alices and 2 Bobs.}\n\t\t\\label{Fig 3}\n\t\\end{center}\n\\end{figure}\t\nIn the previous section, we get the number limitation of Alice who can demonstrate steering with a single Bob. Now, we address the question of whether these Alices can further share steering with more Bobs. We find that it is not possible to increase the number of Bobs when the number of Alices reaches the maximum value in the single Bob scenario. \nHowever, we can reduce the number of observers on one side to increase that on the other side, and then make it possible that multiple Alices show steering with multiple Bobs. Counterintuitively, we find that at most 2 Alices can be simultaneously steered by 2 Bobs in the multiple Alices and Bobs scenario even if the total number of steering shared observers in the single Bob scenario is greater than 4 (see Appendix for more details).\n\nTaking the first two Alices and Bobs as an example, the  2 Alices and 2 Bobs successful steering sharing scenario is depicted Fig. \\ref{Fig 3}(a) and the relationship between the steering parameters and sharpness parameters in the case of 3-setting measurements and 16-setting measurements is presented Fig. \\ref{Fig 3}(b) and Fig. \\ref{Fig 3}(c). The yellow region represents the valid ranges of $\\lambda_{1}$ and $\\eta_{1}$ where 2 Alices and 2 Bobs share steering at the same time. Due to the symmetrical property of the state, $\\lambda_{1}$ and $\\eta_{1}$ have the same ranges. For the 3-setting measurements, they both are $[0.756(1),0.802(5)]$ which is much smaller than the valid ranges of $\\lambda_{1}$ and $\\lambda_{2}$ in the case of 3 Alices and a single Bob with the same measurement settings, indicating it is harder to sharing quantum steering between multiple Alices and Bobs. However, one can improve its robustness by adding the measurement settings. Fig. \\ref{Fig 3}(b) and Fig. \\ref{Fig 3}(c) show that the useful ranges of $\\lambda_{1}$ and $\\lambda_{2}$ can be expanded more than 10 times as the measurement settings increase from 3 to 16. \n\n\\section{Application}\n\\label{Application}\nNote that if the disturbance caused by the former observer's measurement is regarded as noise, our steering sharing protocol can also be applied to investigate the dynamic of steering in the presence of decoherence \\cite{pramanik2019nonlocal}, such as steering sudden death and revival \\cite{sun2017recovering}. Especially, our 3 settings unsharp measurement strategy is essentially equivalent to the depolarizing channel. By changing the former observer's measurement sharpness, the steering ability and direction of the current observers can be controlled. For example, in our 2 Alices and 2 Bobs steering sharing scenario, $A_2$ and $B_2$ can share steering if $\\lbrace \\lambda_{1}, \\eta_{1} \\rbrace$ locates in left side of purple line ($S_3^{2,2}=C_3$ ) in Fig. \\ref{Fig 3}(b), otherwise they cannot. If the initial Werner state is replaced by an asymmetric state \\cite{xiao2017demonstration} or $A_1$ and $B_1$ adopt some asymmetric measurements, a tunable $\\lbrace \\lambda_{1}, \\eta_{1} \\rbrace$ further allows  $A_2$ and $B_2$ to exhibit their steerabilities from both directions to only one direction.   \n\n\\section{Conclusion and Discussion}\n\\label{Conclusion and Discussion}\nIn this work, we discuss a new steering sharing scenario, where half of an entangled pair is accessed by a sequence of Alices, and the other half is distributed to multiple Bobs. We address the question of how many pairs of Alice and Bob can demonstrate quantum steering by violating the $ N $-setting steering inequality where $N=2,3,4,6,10,16$. Contrary to the conjectured proposed by Sasmal \\textit{et al.} \\cite{sasmal2018steering}, we find at most 5 Alices can steer a single Bob and no more than 2 observers can be steered in the multiple Alices and Bobs scenario when the sharpness of the $N$-setting measurements that each Alice and Bob used is equal. We also provide the useful sharpness parameter ranges and the minimum purity of the initial state for different steering sharing cases and give evidence that they increase as the number of observers decreases and the number of measurement settings increases. The noise robustness of our sharing scenario makes our results applicable to the experimental demonstration. On the other hand, we show that our protocol can also be applied to investigate the dynamic of steering in a noise channel and even control the steering direction.\n\nThe shareable steering is a primary resource for some practical and commercial quantum information processing tasks where the general consumers may not want to trust their providers, such as, in the context of quantum internet \\cite{Kimble2008internet}, secret sharing \\cite{Armstrong2015secret}, and random number generation \\cite{Cavalcanti2015random}.  It is thus of importance to further increase the shareable observers to utilize it for many times which could be realized by adopting multipartite entangled states \\cite{gupta2021genuine} or allow the sequential observers in the above scenario to share some classical information. We will carry out some researches in these directions in the near future.\n\n\\begin{acknowledgements}\t\t\nThis work was supported by the National Natural Science Foundation Regional Innovation and Development Joint Fund (Grant No. 932021070), the National Natural Science Foundation of China (Grant No. 912122020), the China Postdoctoral Science Foundation (Grant No. 861905020051), the Fundamental Research Funds for the Central Universities (Grants No. 841912027, and 842041012), the Applied Research Project of Postdoctoral Fellows in Qingdao (Grant No. 861905040045), and the Young Talents Project at Ocean University of China (Grant No. 861901013107). The authors thank Jie Zhu for fruitful discussions.\n\\end{acknowledgements}\n\\newpage\n\\section{Appendix steering sharing with three Alices and two Bobs}\n\\begin{figure}[H]\n\t\\begin{center}\n\t\n\t\t\\subfigure[ ]{\\includegraphics[width=5.2cm]{Fig_4_a.pdf}}\n\t\t\\subfigure[ ]{\\includegraphics[width=6.3cm]{Fig_4_b.pdf}}\n\t\t\\caption{(a) The schematic diagram of steering sharing with 3 Alices and 2 Bobs. The arrow indicates the steering direction, and the lines of the same type (solid or dotted) indicate that the steering can demonstrate simultaneously, and vice versa. (b) The region of the violation of 16-setting steering inequality for the maximum entangled state ($\\mu\\!=1$) with $\\lambda_{3}=1$ and $\\eta_{2}=1$. The yellow region and dark purple area represent that the three Alices can steer the state of the first Bob and the second Bob, respectively. There is no overlap indicates the first three Alices can not share steering with the first two Bob at the same time.}\n\t\t\\label{Fig 4}\n\t\\end{center}\n\\end{figure}\nIn the third part of the main text, we explore the steering sharing scenario with multiple Alices and Bobs. In this case, we find that only 2 Alices can detect steering with 2 Bobs at the same time. Here, we take the first three Alices ($A_1,A_2,A_3 $) and the first two Bobs ($B_1,B_2 $) as an example to illustrate why it is impossible to further increase the number of observers as the number of measurement settings increases to 16. As shown in Fig. \\ref{Fig 4}(b), the yellow region represents the case of $A_1,A_2,A_3 $ can steer the $B_1$'s state. The dark purple region represents the case of $A_1,A_2,A_3 $ can steer the $B_2$'s state. It clearly shows that these Alices and Bobs can not share steering at the same time, because there is no overlap in the two regions even though they initially share a maximum entangled state ($\\mu\\!=\\!1$) and the last observer at each side performs sharp measurements ($\\lambda_{3}\\!=\\!1$ and $\\eta_{2}\\!=\\!1$). Thus, for other fewer measurement settings or mixed initial state, it certainly doesn't exist steering sharing among 3 Alices and 2 Bobs.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\subsection{Introduction}\nThe astonishing experimental accomplishments in the optical control and manipulation of cold atomic gases in the past decade now allow to deterministically load them in extremely well controlled optical traps of almost any shape down to effectively zero temperature. A particular fruitful example are periodic optical lattices in which many intriguing phenomena of solid state physics can be studied with unprecedented control\\cite{bloch2008many}.\n\nIn contrast to conventional solids, however, the spatial order and lattice geometry is fixed by the external lasers and does not appear from a self consistent dynamics of particle interactions. As the back-action of the atoms onto the confining light fields is generally negligible\\cite{asboth2008optomechanical}, local lattice perturbations do not propagate and long range interactions, which appear as phonons in solids are absent. This changes for spatially tightly confined fields as in small optical resonators or optical micro-structures\\cite{domokos2002quantum}. \n\nExperimentally it is very challenging to implement and load microtraps close to optical microstructures, where such atom field coupling is enhanced\\cite{colombe2007strong, alton2010strong, schleier2011optomechanical}. In an important step Rauschenbeutel and coworkers, however, recently managed to trap atoms in an array of optical dipole traps generated by two color evanescent light fields alongside a tapered optical fiber\\cite{vetsch2010optical} where the backaction of even a single atom on the propagating fiber field is surprisingly strong\\cite{domokos2002quantum,chang2012cavity}. This setup was improved with higher control and coupling by other groups recently\\cite{goban2012demonstration}. With the atoms firmly trapped within the evanescent field of the fiber modes, field mediated atom-atom interaction and collective coupling to the light modes play a decisive role in this a setup\\cite{zoubi2010hybrid}.\n\nAlready a decade ago it was theoretically predicted\\cite{domokos2002collective} and experimentally confirmed\\cite{black2003observation,baumann2010dicke,arnold2012self} that light scattering within optical resonators induces self-ordering of atoms in regular patterns maximizing collective coupling to the cavity mode\\cite{ritsch2012cold,gopalakrishnan2009emergent}. This transition can be directly monitored from the super-radiant light scattering \\cite{black2003observation,arnold2012self} and appears also at zero temperature in the quantum regime as phase transition from a superfluid to a supersolid\\cite{baumann2010dicke}. In an recent proposal Chang and coworkers predicted, that nanofiber-mediated infinite range dipole-dipole coupling can also induce stable regular patterns of laser illuminated atoms trapped along the fiber\\cite{chang2012self}. The stable configurations, characterized by minimal dipole interaction energy, assume surprising configurations and exhibit characteristic collective light scattering. \n\nHere we develop a generalized many particle model to study the properties of such a light scattering induced crystallization of an laser illuminated ultracold gas in an elongated 1D trap at finite temperature as schematically drawn in Fig.\\ref{fig1}. The atoms trapped parallel to the fiber are illuminated at right angle by a Gaussian laser beam of frequency $\\omega_L$, sufficiently detuned from any atomic resonance so that spontaneous emission plays only a minor role and their polarizability $\\alpha$ has only a negligible imaginary part.\n\n\\begin{figure}[h!]\n\n\\includegraphics[width=8cm]{Bild1.jpg}\n\\caption{Cigar-shaped atomic gas alongside optical nanoguide}\\label{fig1}\n\\end{figure}\nThe laser field $\\mathbf E_L=(\\mathcal E_L(\\mathbf x)e^{-i\\omega_Lt}+c.c.)\\mathbf e_L$ gives rise to the field $\\mathbf E_s$ scattered by the atoms. Neglecting polarization effects and retardation, it can be written as $\\mathbf E_s=(\\mathcal E_s(\\mathbf x,t)e^{-i\\omega_Lt}+c.c.)\\mathbf e_L$, its envelope satisfying Helmholtz's equation\n\\begin{subequations}\\label{VlasovHelmholtz}\n\\eq{\\label{FullHelmholtz}\\nabla^2\\mathcal{E}_s +k_L^2\\big(n_F^2+\\chi\\big)\\mathcal{E}_s=-k_L^2\\chi\\mathcal{E}_L.}\nHere, $n_F(\\mathbf x)$ denotes the refractive index profile of the optical fiber and $\\chi(\\mathbf x,t) ={\\alpha}\\rho(\\mathbf x,t)\/{\\epsilon_0}$ is the susceptibility of the particles. The particle density distribution $\\rho$ is obtained from the momentum integral \n$\\rho(\\mathbf x,t)=\\int F(\\mathbf x,\\mathbf p,t)d^3p$ \nover the one-body distribution function $F(\\mathbf x,\\mathbf p,t)$ proper to the gas. \nIn the classical mean-field limit it satisfies Vlasov's equation\n\\eq{\\label{FullVlasov}\n\\frac{\\partial F}{\\partial t}+\\frac{\\mathbf p}{m}\\!\\cdot\\!\\frac{\\partial F}{\\partial \\mathbf  x}-\\frac{\\partial }{\\partial \\mathbf x}\\left(\\Phi_d+U_T\\right)\\!\\cdot\\!\\frac{\\partial F}{\\partial \\mathbf  p} =0.}\n\\end{subequations}\nHere, $U_T$ denotes the elongated dipole trap potential and \\eq{\\Phi_d=-\\alpha \\left|\\mathcal{E}_s+\\mathcal E_L\\right|^2} the optical potential due to pump laser and fiber field.  For strong radial confinement of the gas the one-body distribution approximately factorizes into a longitudinal and a transverse part,\n$F(\\mathbf x,\\mathbf p,t)\\simeq f(z,p_z,t) F_\\perp(\\mathbf x_\\perp,\\mathbf p_\\perp)$, where $F_\\perp$ is the Maxwell-Boltzmann distribution for the transverse degrees of freedom: $F_\\perp:= Z_\\perp^{-1}\\exp\\left[-\\beta\\left(\\frac{\\mathbf p_\\perp^2}{2m}+U_\\perp(\\mathbf x_\\perp)\\right)\\right]$. $\\beta$ denotes the inverse thermal energy and $U_\\perp$ the radial confining potential.\n\nIn the absence of atoms the fiber supports only a single relevant TE mode, which -- within the framework of scalar theory -- is supposed to possess a propagation constant $\\beta_m$ and a normalized transverse mode function $u(x,y)$ extending outside the fiber \\cite{vetsch2010optical,chang2012self}. The dominant forces on the particles along $z$  are due to photon scattering into and out of the fiber. As long as the total atomic susceptibility stays small even for large particle numbers, the radial fiber mode function is only weakly perturbed and we can set $\\mathcal E_s(\\mathbf x,t)\\simeq\\sqrt{A}\\,E(z,t)u(x,y)$ with the cross section $A:=\\left(\\int d^2x_\\perp d^2p_\\perp u^2 F_\\perp \\right)^{-1}$. Integrating over the transverse degrees of freedom we finally arrive at an effective description of the longitudinal dynamics\n\\begin{subequations}\\label{HV}\n\\eq{\\label{Helmholtz}\\frac{\\partial^2E}{\\partial z^2}+\\left(\\beta_m^2+k_L^2\\tilde\\chi\\right)E=-k_L^2 \\tilde\\chi E_L,}\n\\eq{\\label{Vlasov}\\frac{\\partial f}{\\partial t}+\\frac{p_z}{m}\\frac{\\partial f}{\\partial z}-\\frac{\\partial }{\\partial z}\\!\\left(U-\\alpha\\!\\left[|E|^2+2E_LE_r\\right]\\right)\\!\\frac{\\partial f}{\\partial p_z} =0,}\n\\end{subequations}\nwhere $E_r$ is the real part of $E$. The effective laser field is given by $E_L:=\\sqrt{A}\\,\\int d^2x_\\perp d^2p_\\perp\\mathcal{E}_L u F_\\perp$ and the local atom-fiber coupling is governed by the effective susceptibility\n\\eq{\\label{Susc}\\tilde\\chi(z,t):=\\frac{\\alpha}{\\epsilon_0 A}\\int^{\\infty}_{-\\infty} f(z,p_z,t)dp_z}\nproportional to the atomic line density. To ensure physically consistent solutions, equation \\eqref{Helmholtz} for the electric field has to fulfill Sommerfeld's radiation conditions\n\\begin{equation}\n\\label{BC}\\frac{\\partial E}{\\partial z}=\\pm i\\beta_m E(z,t),~~z\\rightarrow \\pm\\infty,\n\\end{equation}\nwhich ensure outgoing waves at infinity. \n\nEquations \\eqref{HV} can at least numerically be solved directly. At this point we nevertheless restrict ourselves to such stationary solutions, which correspond to a gas in thermal equilibrium\\cite{asboth2005self}, where the particles are distributed according to the Maxwell-Boltzmann distribution\n\\eq{\\label{ThermEq}f(z,p_z)=Z^{-1} e^{-\\beta(p_z^2\/2m+U)}\\,e^{\\beta\\alpha\\left(|E|^2+2E_pE_r\\right)}.} \n\nSubstituting the effective susceptibility \\eqref{Susc} obtained from a thermal distribution \\eqref{ThermEq} into the effective Helmholtz equation \\eqref{Helmholtz} leads to a highly nonlinear equation, which determines the selfconsistent electric field. We assume here that the external part of the potential, $U(z)$, can be harmonically approximated by $U(z)=\\frac{1}{2}m\\omega_z^2 z^2$ within the (very large) thermal extension of the gas $l_z:=2k_\\mathrm{B} T\/m\\omega_z^2\\gg\\beta_m^{-1}$, and we will ignore the dependence of the driving laser amplitude on position, i.e. $E_L(z)=E_L$.\n \nWith these assumptions one sees that the zero field solution $E(z)\\approx 0$ and\n\\eq{\\label{ThermEq0}f(z,p_z)\\equiv f_0(z,p_z)=Z^{-1} e^{-\\beta(p_z^2\/2m+U)}}\nalways solves Helmholtz's equation, which we will call the normal phase.\n\n\\begin{figure}\n\t\\centering\n\t\t \\includegraphics[width=8cm]{Bild3.jpg}\n\t\\caption{Properties of three coexisting selfordered solutions on different branches $n=0$ ($(1a)-(1c)$), $n=1$ ($(2a)-(2c)$) and $n=2$ ($(3a)-(3c)$) in the strong collective coupling regime for $\\zeta_0=150\/\\pi$ and $\\varepsilon=9.5\\varepsilon_c$.  The upper plots (a) show the atomic density along the fiber, while the plots labeled by (c) depict the local fraction of right $N_+$ (blue curve) and left  $N_-$  (red curve) traveling photons, the number of zeros being equal to the branch number $n$. The plots labeled (b) depict the envelopes of the two parts of the optical potential. Blue indicates the potential from scattering between pump laser and fiber, the brown area shows the part due to photon redistribution within the fiber. For $n=0$ the latter dominates in the center of the cloud causing a density modulation with a period of one half of $\\lambda_m=2\\pi\/\\beta_m$. This region is broken up into three for $n=1$ and into five, for $n=2$.}\\label{Bild2}\n\\end{figure}\nBefore we investigate the possibility of selfordered solutions, let us examine the dynamical stability of the normal phase, using the Helmholtz-Vlasov equations \\eqref{HV}, linearized around the unordered state\n\\begin{subequations}\\label{linVlas}\n\\eq{\\frac{\\partial^2E}{\\partial z^2}+\\beta_m^2E=-E_L\\frac{k_L^2\\alpha}{\\epsilon_0A}\\int^{\\infty}_{-\\infty}f_1(z,p_z,t)dp_z ,}\n\\eq{\\frac{\\partial f_1}{\\partial t}+\\frac{p_z}{m}\\frac{\\partial f_1}{\\partial z}-\\frac{d U}{d z}\\frac{\\partial f_1}{\\partial p_z}=-2\\alpha \\frac{\\partial E_L E_r}{\\partial z}\\frac{\\partial f_0}{\\partial p_z}.}\n\\end{subequations}\nHere, $f_1(z,p_z,t)$ represents a small deviation from the normal phase, such that the complete distribution function is given by $f(z,p_z,t)=f_0(z,p_z)+f_1(z,p_z,t)$.\nWe seek solutions to \\eqref{linVlas}, which are of the form\n\\begin{subequations}\\label{NormalMode}\n\\eq{f_1(z,p_z,t)=\\psi(z,p_z)e^{st}+\\psi^*(z,p_z)e^{s^*t},}\n\\eq{E(z,t)=a(z)e^{st}+b^*(z)e^{s^*t}.}\n\\end{subequations}\nIn this Ansatz, $(\\psi,a,b)$ will be referred to as normal mode and $s=\\gamma+i\\omega$ as the complex valued mode parameter.\nWhenever $\\gamma>0$, the deviation from equilibrium defined by the normal mode increases exponentially in time, causing the eventual destruction of the normal phase. The existence of such a mode implies dynamical instability. With the help of asymptotic expansions, we find that if \\eqref{NormalMode} is to solve \\eqref{linVlas} and satisfy the radiation conditions, the mode parameter must satisfy $D_n(s)=0$ for some $n\\in\\mathbb{Z}$, where\n\\eq{D_n(s)=(2n+1)\\pi+i\\frac{k_L^2\\alpha^2}{\\epsilon_0A}\\iint^{\\infty}_{-\\infty}\\frac{E_L^2\\,\\partial f_0\/\\partial p_z}{s+i\\beta_m v_z} dp_z dz.}\n\nThis condition first demands $\\omega=0$. Introducing the collective coupling parameter $\\zeta_0$ and effective pump strength $\\varepsilon$\n\\eq{\\zeta_0:=\\frac{k_p}{\\beta_m}\\frac{N\\alpha}{A\\lambda_L\\epsilon_0}, \\; \\varepsilon:=\\frac{\\alpha E_L^2}{k_BT},}\nwe then see that there exists at least one normal mode with a positive growth rate $\\gamma>0$, if and only if\n\\eq{\\label{CC1}\\varepsilon>\\frac{1}{2\\zeta_0}=:\\varepsilon_c.}\n\nHence as our first important result we see that beyond this pump threshold  \\eqref{CC1} the normal phase ceases to be stable and no longer represents a physical state of the system. We therefore study other solutions of \\eqref{HV} for $\\varepsilon>\\varepsilon_c$.\nFortunately, we can obtain approximate solutions with the help of perturbation theory in the weak collective coupling regime, $\\zeta_0\\leq1$, as well as in the strong collective coupling regime $\\zeta_0\\gg1$. The real and imaginary parts of the electric field envelope $E_r, E_i$ \\eq{E_{r,i}(z)=a_{r,i}(z)\\cos(\\beta_m z+\\phi_{r,i}(z))E_L,} can be assumed to behave almost harmonically with phases $\\phi_r, \\phi_i$ and amplitudes $a_r, a_i$ that vary slowly on a scale defined by $\\beta_m^{-1}$.  \nThis amplitudes and phases are directly related to the complex amplitudes of the more familiar decomposition into left and right running waves, $E=(E_+e^{i\\beta_mz}+E_-e^{-i\\beta_mz})E_L$ via $E_\\pm=a_r e^{\\pm i\\phi_r}+ia_i e^{\\pm i\\phi_i}$.\nA perturbative analysis of the steady state Helmholtz equation reveals, that \n$\\Theta:=\\frac{a_r^2+a_i^2}{2}=\\frac{1}{2}(|E_+|^2+|E_-|^2)$, proportional to the sum of locally right going and left going photons, is spatially constant and will thus serve us as order parameter.\n\nLet us first consider the weak collective coupling regime, where scattering from the laser into the fiber and vice versa dominates over scattering within the fiber.  The demand to satisfy the radiation conditions \\eqref{BC} leads to the equation determining this order parameter\n\n\\eq{\\label{OP}2\\zeta_0 I_1\\left(2\\varepsilon\\sqrt{\\Theta}\\right)=(1+2n)\\sqrt{\\Theta} I_0\\left(2\\varepsilon\\sqrt{\\Theta}\\right),~~n\\in\\mathbb{N}_0,}\nwhere $I_k$ denotes the $k$th modified Bessel function of the first kind.\n\nAs quite a surprise one observes that the solution is not unique and for\n$\\varepsilon_c<\\varepsilon<\\frac{1+2n}{2\\zeta_0}$,\nthere exist $n$ different solutions ${\\Theta_n}$ as shown in the example of figure \\ref{Bild3}. As the phase of the outgoing scattered light is not determined, each solution of \\eqref{OP} corresponds to an infinite family of solutions of \\eqref{HV} with the periodic density modulation slightly displaced under a slow envelope. The appearance of this degeneracy of course corresponds to breaking of a continuous symmetry.  It is important to note that the point of first appearance of an ordered family of solutions, $\\varepsilon=\\varepsilon_c$, exactly coincides with the point where the normal phase becomes unstable. All this is confirmed by a numerical solution of the underlying equations \\eqref{HV}.\nFocusing on the region close to the first branching point, we find the behavior of the order parameter to be given by \\eq{\\Theta\\sim\\varepsilon-\\varepsilon_c,~~\\zeta_0<1}\nand\n\\eq{\\Theta\\sim\\left(\\varepsilon-\\varepsilon_c\\right)^{1\/2},~~\\zeta_0=1.}\nCuriously, if $\\zeta_0>1$, the order parameter is discontinuous, exhibiting a jump of magnitude\n\\eq{\\Delta\\Theta=16(\\zeta_0-1).}\nThe results of the perturbation theory hold only as long as $\\Theta\\ll1$ and the predicted jump soon violates this assumption.\nLet us therefore turn to the strong collective coupling regime. Again, each possible value of the order parameter corresponds to an infinite family of stationary solutions and\n\\begin{figure}\n\t\\centering\n\t\t\\includegraphics[width=8cm]{Bild2.jpg}\n\n\t\\caption{Branches of the order parameter vs energy ratio according to \\eqref{OP} and \\eqref{OP1} for $n=0,\\ldots,3$ in the weak collective coupling regime $(a)$, with $\\zeta_0=0.05$ and in the strong collective coupling regime $(b)$, with $\\zeta_0=75\/\\pi$. Note that in the latter regime all branches converge and there may be even more than $n$ families of solutions for $\\varepsilon_c<\\varepsilon<(1+2n)\\varepsilon_c$.}\\label{Bild3}\n\\end{figure}\nany nonzero value for $\\Theta$ satisfies\n\\eq{\\label{OP1}\\zeta_0 \\varepsilon\\pi=4(1+2n)P(\\Theta)K[P^2(\\Theta)\\Theta^2],~~n\\in\\mathbb{N}_0,}\nwhere $P(\\Theta):=\\frac{1+2\\varepsilon \\Theta}{4+6\\varepsilon \\Theta}$ and $K$ denotes the complete elliptic integral of the first kind.\nIn this regime, there are at least $n$ families of ordered solutions whenever $1<2\\zeta_0\\varepsilon<1+2n$. For an example see figure \\ref{Bild3}.\nFurthermore, we find that according to perturbation theory, the order parameter is bounded by $\\Theta<4$. Figure \\ref{Bild2} depicts some properties of the solutions corresponding to possible values of the order parameter. It should be noted that while we have here shown the existence of multiple families of selfordered thermal solutions of the Vlasov-Helmholtz equations, nothing has been said about their dynamical stability properties, which remains an open problem.\n\\par\nCertain phenomena called optical binding, similar to the ordering discussed in this work, have been observed with laser illuminated small beads in liquids\\cite{burns1990optical} in 1D and even 2D geometries as well. More interestingly, in analogy with cavity induced selfordering we expect a corresponding phase transition also in the zero temperature limit, where the threshold is determined by the recoil energy and effective particle interaction strength replacing thermal energy. This should have distinctly different properties as compared to a 1D prescribed optical lattice. Obviously, even without the presence of the fiber, the interference of the scattered field by two distant atoms can induced long range forces and ordering. As part of the scattered field is lost, this leads to a higher threshold, but to some extend the particles itself can guide the scattered light. In the limit of sufficient initial particle density the trapped atoms themselves could form a fiber like guide for the scattered light enhancing long range interactions. One could speculate that the trapping field for the 1D confinement could be unnecessary and be provided by the light guided by the atoms. The predicted ordering phenomena could also be related to recent observations of collective scattering in dense atomic vapors\\cite{greenberg2011bunching,schmittberger2012free} where light induced selfordering mechanisms play an important role.\n\\par\nWe thank A. Rauschenbeutel, J. Kimble and I.Cirac for stimulating discussions and acknowledge support by the Austrian Science Fund FWF through the projects SFB FoQuS P13.\n\n\\bibliographystyle{apsrev}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \nOff-shell currents are ordinarily used in efficient recursive evaluation of scattering amplitudes. These are computed from \n off-shell currents by setting \nthe external legs on-shell. Off-shell currents were famously employed by  Berends and Giele   \\cite{Berends:1987me} to recursively calculate tree-level gluon scattering and they are routinely used in  matrix element generators \\cite{Gleisberg:2008fv, Cafarella:2007pc, Lifson:2022mxf}.  Despite their success in practical calculations many of the appealing structures  of on-shell scattering amplitudes are  either not manifest \nor difficult to expose \\cite{Britto:2012qi, Mastrolia:2015maa, Jurado:2017xut} in off-shell currents. Similarly, in the classical limit of scattering amplitudes\\footnote{The classical of amplitudes has been received considerable attention in the last few years. The subject has been reviewed extensively in Refs.\\cite{Kosower:2022yvp,Buonanno:2022pgc}.}  some of the on-shell features present before taking classical limit are not inherited.  Off-shell currents in the classical limit have appeared recently in Refs.\\cite{Goldberger:2016iau,Goldberger:2017frp,Goldberger:2017ogt,Shen:2018ebu,Ben-Shahar:2021zww,Cristofoli:2020hnk}. \n\nIn this paper we are concerned with off-shell \ncurrents in the classical limit from the perspective of  Kosower-Maybee-O'Connell (KMOC) \\cite{Kosower:2018adc, Maybee:2019jus, delaCruz:2020bbn, Cristofoli:2021vyo, Aoude:2021oqj}, \nthe Worldline Quantum Field Theory\n\\cite{Mogull:2020sak,Jakobsen:2021smu,Jakobsen:2021lvp, Jakobsen:2021zvh,Shi:2021qsb, Bastianelli:2021nbs,Wang:2022ntx} and their connection. \nThroughout,  we will focus on the study of the classical limit of the tree-level off-shell current\n\\begin{align}\n\\mathcal{A}_n\n(p_1, p_2, k_1, \\dots, k_n)\\Big|_{k_i^2\\ne 0,\\, p_i^2=m^2} \\, ,\n\\end{align} \nwhich, assuming that  $p_1$, $p_2$ are the momenta of two massive particles and $k_1, \\dots, k_n$ are the momenta of massless gauge bosons, can be fused into a $n+2$ scattering amplitude. Unlike typical currents in Berends-Giele recursions here more than one leg is off-shell.  Similar objects   have appeared previously in QCD, see e.g. Ref.\\cite{Schwinn:2005pi}. The  \n$n=2$ current with spinning massive particles  is relevant in the context of ongoing discussions of higher spin gravitational Compton amplitudes in the classical limit\n\\cite{Chiodaroli:2021eug,Johansson:2019dnu, Bautista:2021wfy, Saketh:2022wap, Bern:2022kto, Kosmopoulos:2021zoq,Jakobsen:2022zsx,Cangemi:2022abk,Bautista:2022wjf,Bern:2020buy, Cristofoli:2021jas,Aoude:2022thd}.  \n\n\nThe remaining of the paper is organized as follows. In Section \\ref{off-shell-classical}  we  start with a general definition of a current and  show how its classical limit is obtained in WQFT. In Section \\ref{computational-part} we calculate currents in QED and gravity.\nIn Section \\ref{currents-and-HTLs} we apply our methods to compute Hard Thermal Loops, which can be understood as classical objects in the high temperature regime. Our conclusions are presented in Section \\ref{conclusions}.\n\n\n\\section{Off-shell currents in the classical limit}\n\\label{off-shell-classical}\nLet us consider\na theory which models massive scalar particles coupled with massless gauge bosons. Let $p=(p_1, p_2)$ and $k=(k_1, \\dots, k_n)$ denote  tuples of outgoing momenta  of  massive and massless external legs, respectively.  We define the  $n+2$ off-shell current by\n\\begin{align}\n\\mathcal{A}_n^{I_1, \\dots, I_n}\n(p, k):= \\hat{\\delta}^4(p_1+p_2+\\sum_{i=1}^n k_i)\nA_n^{I_1, \\dots, I_n}\n(p,k),\n\\label{general-def}\n\\end{align}\nwhere,  without loss of generality \nthe  massive scalars obey $p_i^2=m^2$, while for the massless particles $k_i^2\\ne 0$. We have introduced the notation $\\hat{\\delta}^n (x):=(2\\pi)^n (x)$ and for later use we define $\\hat{\\mathrm{d}}^n x:= (2\\pi)^n \\mathrm{d}^n x$. \n The upper indices denote collectively color and\/or Lorentz indices associated with the massless particles. \n\\begin{figure}\n\t\\centering\n\t\\begin{tikzpicture}[thick, transform shape]\n\t\\node[circle, fill=lightgray, draw] (c) at (0,0){$\\mathcal A$};\n\t\\draw[draw, snake it] (c.north west)--(-1.,1)node[left]\n\t{$\\qquad k_1$};\n\t\\draw[draw, snake it] (c.north east)--(1.,1)node[right]\n\t{$k_n$};\n\t\\draw[draw](1.5,-0.30)-- (c.south east)node[right,    currarrow,\n\tpos=0.5, \n\txscale=1,\n\tsloped,\n\tscale=1]\n\t{};\n\t\\draw[draw] (-1.5,-0.30)--(c.south west)node[left,    currarrow,\n\tpos=0.5, \n\txscale=-1,\n\tscale=1]{};\n\t\\node(A) at (0,1){$\\dots$};\n\t\t\\node(B) at (-1.5,0){$p_1$};\n\t \\node(C) at (1.5,0){$p_2$};\n\t\\end{tikzpicture}\n\t\\caption{Off-shell current.  The blob represents a sum over tree-level Feynman diagrams where diagrams.  Wavy lines represent off-shell outgoing massless gauge bosons.}\n\t\\label{current-off-shell}\n\\end{figure}\nFor our purposes the evaluation\nof this current will be through Feynman diagrams (see Fig.\\ref{current-off-shell}).  The \ncurrent can be fused into an on-shell amplitude by\ndressing  it with appropriate polarization vectors, imposing physical constraints such as transverse polarization, and setting all external legs on-shell,\nnamely\n\\begin{align}\n\\ii \\mathcal T  \\sim \\mathcal \\xi_{I_1} \\cdots \\xi_{I_n}\\mathcal{A}_n^{I_1, \\dots, I_n}\n(p_1, p_2, k_1, \\dots, k_n)\\Big|_{k_i^2=0,p_i^2=m^2} \\,.\n\\end{align}\n\\subsection{Classical limit \\`a  la KMOC}\nSuppose  that we are interested in the classical limit of Eq.\\eqref{general-def}  understood as the limit $\\hbar \\to 0$ or more precisely as a Laurent expansion in powers of some dimensionless parameter $\\xi$.  \n In the spirit of KMOC one\n   would restore $\\hbar$ into the current and perform dimensional analysis. One should also distinguish  the momenta of  massive scalar particles and massless gauge bosons. The latter being described by wavenumbers through the rescaling $k \\to \\hbar k$, while the former obeys\n   $p_i^\\mu=m_i u_i^\\mu$, where $u_i^2=1$. To achieve definite classical momenta the initial states must be dressed with appropriate coherent\\footnote{ Here we think of coherent states in the sense of Perelemov \\cite{perelomov:1972}.   We \n   \trefer the interested reader to \\cite{Perelomov:1986tf} for details of the Perelomov formalism.} wavefunctions, giving us the notion of sharply peaked position and momenta. The most general coherent relativistic wavefunctions associated with the restricted Poincare group have the form \\cite{Kaiser:1977ys}\n %\n \\begin{align}\n f_z (p):=\\braket{e_z|f}=\n \\int \\mathrm{d} \\Phi(p) e^{-\\ii z\\cdot p} f(p)  \\,,\n \\end{align}     \n where $\\mathrm{d} \\Phi(p):= (2 \\pi)^{d-1}\\mathrm{d}^ d p \\delta (p^2-m^2)\\Theta(p_0)$ and  \n $\\braket {e_z|p}:= \\mathcal C_z e^{-\\ii p \\cdot z}$. Here  $z=x-\\ii y$ is a complex vector, which in general is time-dependent. \n The classical phase space  is obtained by setting $t=0$ \\cite{Kowalski:2018xsw}. The normalization of states can be derived from\n\\begin{align}\n\\braket{e_z|e_w}= \\int \\mathrm{d} \\Phi (p) e^{-\\ii p\\cdot (z-\\bar w)}=\n \\mathcal C_ z \n\\mathcal C^{*}_w\n \\left(\\frac{m}{2\\pi \\eta}\\right)^{d\/2-1} \nK_{d\/2-1}(\\eta m),  \n\\end{align}\nwhere $\\eta=\\sqrt{-(z-\\bar w)^2}$ and  $K_\\nu (x)$ is the modified Bessel function.\n For $z=w$,  $\\eta=2 |y|$,  one obtains\n\\begin{align}\n\\mathcal C_z= \\left[{\\left(\\frac{4 \\pi |y|}{m} \\right)^{d\/2-1}\\frac{1}{K_{d\/2-1}(2m |y|)}}\\right]^{1\/2}.\n\\end{align}\n Wavefunctions employed by KMOC  correspond to the case where one chooses the complex vector to be\n\\begin{align}\nz_i^\\mu=-b_i^\\mu+\\ii \\frac{ u_i^\\mu}{m \\xi}, \n\\end{align}\nwhere $\\xi$ is a dimensionless parameter, which can be thought as the square of the ratio of the Compton wavelength to the intrinsic spread of the wavepacket.  Here $u_i$ is the classical four velocity of the particle of mass $m_i$.\nTherefore,  it is natural to consider the current\\footnote{We will keep\n\tthe indices explicit for calculations but otherwise suppress them to avoid cluttered expressions.} weighted with coherent wavefunctions as \n\\begin{align}\n\\label{current1}\n&\\mathcal{C}_n (p, k):=\\int \\mathrm{d} \\Phi (p_1, p_2) \\phi_{z_1}(p_1) \\phi_{z_2} (p_2)  \\mathcal{A}_n\n(p_1, p_2, k_1, \\dots, k_n) \\, ,\n\\end{align} \nwhere $ \\mathrm{d} \\Phi (p_1, p_2):=\\mathrm{d} \\Phi(p_1) \\mathrm{d} \\Phi(p_2)$. Now the classical limit of the current can\n be computed from \n a Laurent expansion of the formal expression\n\\begin{align}\n\\label{current}\n&\\mathcal{C}_n (p, k)=\\int \\mathrm{d} \\Phi (p_1, p_2) \\phi (p_1) \\phi (p_2) e^{\\ii b \\cdot \\sum_{i=1}^n k_i} \\mathcal{A}_n\n(p_1, p_2, \\hbar k_1, \\dots, \\hbar k_n) \\, ,\n\\end{align} \nwhere we have used momentum conservation, $k\\to\\hbar k$ and $b \\to b\/\\hbar$.  The rescaling has an important effect into the structure of \\eqref{current}. Indeed writing explicitly the momentum conservation Dirac-delta  we have\n\\begin{align}\n&\\mathcal{C}_n (p, k)= \n\\int \\mathrm{d} \\Phi (p_1, p_2)  \\phi (p_1) \\phi(p_2) e^{\\ii b \\cdot \\sum_{i=1}^n k_i} \\hat{\\delta}^4(p_1+p_2+\\hbar \\sum_{i=1}^nk_i ) { A}_{n} (p,\\hbar k)\\\\ \n& = \n\\int \\mathrm{d} \\Phi(p_1) \\ \\phi (p_1)\n\\phi(-p_1 - \\hbar \\sum_{i=1}^n k_i )\n\\hat{\\delta}( \\hbar^{2} \\sum_{i,j=1}^{n} k_{i}\\cdot k_{j} +2 \\hbar p_1 \\cdot \\sum_{i=1}^n k_i ) e^{\\ii  b \\cdot \\sum_{i=1}^n k_i} {A}_{n}(p,\\hbar k), \\nonumber\n\\end{align}\nwhere we have used the momentum conservation Dirac-delta to perform the phase-space integral over $p_2$.\nMaking the identifications $q \\to \\sum_{i=1}^n k_i$ the remaining integral has the form\n\\begin{align}\n\\int  \\mathrm{d} \\Phi(p) \\phi (p)\n\\phi (-p- \\hbar  q) \\hat{\\delta}(2\\hbar p \\cdot q+\\hbar^2 q^2) f(p, q),\n\\end{align} \nwhich is sharply peaked around $p^\\mu=m u^\\mu$, where $u^\\mu$ is the classical velocity of the particle with mass $m$. The analysis of the above integral by KMOC (see Appendix B of Ref.\\cite{Kosower:2018adc}) does not depend on the on-shell properties of \n$q$, which plays the role of the momentum mismatch in KMOC,  so we can apply it here as well. The current then simplifies to\n\\begin{align}\n\\mathcal{C}_n (p,  k)= \\frac{1}{2}\\hat{\\delta}\\left( p \\cdot \\sum_{i=1}^n k_i\\right)\ne^{\\ii b \\cdot \\sum_{i=1}^n k_i}\n\\bar { A}_{n} (p,k) \\,, \n\\label{final-current-KMOC}\n\\end{align}\nwhere $\\bar A(p,k)$ denotes the non-vanishing term in the Laurent expansion, where by abuse of notation we have set $p_{1}=p$.  From a practical point of view we are done.  We can already perform calculations following the KMOC algorithm and reach   Eq.\\eqref{final-current-KMOC} for the theory under study.\n\nA few comments are in order. Strictly speaking  Eq.\\eqref{final-current-KMOC} should be understood as the average of the RHS\nover wavefunction of $p$, which in KMOC is denoted by a double bracket. \n The net effect of weighting over coherent wavefunctions is producing an overall factor depending on the external soft momenta, which is analogous to the momentum mismatch in classical observables.  \n The attentive reader might ask about the presence of singular terms produced by the series expansion.\nThe current is not an observable so one might expect such terms. However, the current is an off-shell tree-level object so we may safely ignore Feynman's $\\ii \\epsilon$ prescription in calculations, thus leading to cancellation of those singular terms. In QED we have checked this up to 7-points. In the worldline formulation of the\ncurrent the absence of those singular terms will become clear as we discuss now.\n  \n\\subsection{Relation to WQFT }\\label{sec-proof-WQFT}\n\nWe wish to relate the classical current \\eqref{final-current-KMOC} to a worldline path integral. Although we will consider scalar QED as an illustration,  the following discussion is also applicable to other  theories where the WQFT is known. The field theory \naction for scalar electrodynamics \n\\begin{align}\nS_{\\text{sQED}}[\\varphi,\\varphi^{\\dagger},A] =\\int \\mathrm{d}^4 x\\left[(D^\\mu\\varphi)^\\dagger D_\\mu \\varphi -m^2 \\varphi^\\dagger \\varphi\\right] \\, \n\\end{align}\nwith the gauge covariant derivative $D_\\mu=\\partial_\\mu-\\ii e A_\\mu$. The gauge-fixed Maxwell action is\n\\begin{align}\nS_{\\text{gf}}[A]=- \\int \\mathrm{d}^4 x \\left[\\frac 1 4 F_{\\mu\\nu}F^{\\mu\\nu}+\\frac 1 {2\\xi } (\\partial^\\mu A_\\mu)^2\\right],\n\\end{align}\nwhere we will use Feynman gauge ($\\xi=1$) in  in calculations. We know  from standard QFT that the current \\eqref{general-def} can be written off-shell as a Fourier transform of a time ordered correlation function of the quantum fields describing the external states after LSZ reduction, which  can be represented as a path integral. Moreover,  we are interested in the classical limit so it will be convenient to perform the LSZ reduction in two steps, first on the photon legs and then on the scalar ones. \n In momentum space the amputation of the photon external legs leads to\n\\begin{align} \\label{P-off}\n{\\cal P}_n^{\\text{sQED}}(p,k )\n=\\frac{1}{\\cal N}\\int {\\cal D} A \\ e^{\\ii S_{\\text{gf}}[A]} \\int& {\\cal D}\\varphi {\\cal D} \\varphi^{\\dagger } e^{\\ii S_{\\text{sQED}}[\\varphi,\\varphi^{\\dagger},A]}\\\\\n& \\ii k_1^2 \\cdots \\ii k_n^2\\ \\varphi(p_{1})\\varphi^{\\dagger}(p_{2}) A_{\\mu_{1}}(k_{1})\\cdots A_{\\mu_{n}}(k_{n})\n\\Big|_{k^2\\ne 0} \\nonumber\n \\,.\n\\end{align}\n\nBefore LSZ let us consider  the classical limit of Eq.\\eqref{P-off} following Ref.~\\cite{Mogull:2020sak}. In the classical approximation we can neglect loops mediated by scalars and replace the path integral over scalar fields  by the photon-dressed scalar propagator, which we briefly review now\\footnote{Dressed propagators have been developed\n\tin a  worldline representation for a variety of models, see e.g. \\cite{Ahmadiniaz:2015kfq,\n\t\tAhmadiniaz:2015xoa, Edwards:2017bte, Ahmadiniaz:2017rrk, Ahmadiniaz:2020wlm, Corradini:2020prz,\n\t\tAhmadiniaz:2021gsd}. A standard reference for worldline methods is Ref.\\cite{Schubert:2001he}.}.\n\tIn position space the photon-dressed scalar propagator reads\n\\begin{equation}\n\\mathcal G(x_1,x_2)[A] = \\frac{1}{{\\bar{ N}}}\\int {\\cal D} \\varphi {\\cal D}\\varphi^{\\dagger} e^{\\ii S_{\\text{sQED}}[\\varphi,\\varphi^{\\dagger},A]}\n\\varphi(x_{1})\\varphi^{\\dagger}(x_{2})\\,.\n\\end{equation}\nThe dressed propagator admits  a worldline path integral representation of the form\n\\begin{align}\n\\mathcal G(x_1,x_2)[A] &= \\langle x_{2}|\\frac{1}{D^{2} +m^{2} - \\ii\\epsilon }| x_{1}\\rangle \\\\\\nonumber\n&= \\ii \\int_{0}^{\\infty}\\mathrm{d} T \\,\\langle x_{2}| e^{-\\ii T (D^{2} +m^{2} + \\ii\\epsilon )}| x_{1}\\rangle\n= \\int_{0}^{\\infty } \\mathrm{d} T \\int_{x(0)=x_{1}}^{x(1)=x_{2}} {\\cal D} x \\, e^{-\\ii \\int_{0}^{1}d\\tau \\left( \\frac{1}{2T}\\dot{x}^{2} +e \\dot{x}_{\\mu}A^{\\mu}\\right)}\\,,\n\\end{align} \nwhere $T$ is the so-called Schwinger proper time while the $\\ii\\epsilon$ prescription is understood in the path integral. The dressed propagator in momentum space can be written as \\cite{Daikouji:1995dz}\n\\begin{align}\n\\mathcal G(p,k)[A]=\n\\hat{\\delta}^4 \\left(p_1+p_2+\\sum_{i=1}^n k_i\\right) G (p,k)[A]\\,,\n\\label{dressed-gen}\n\\end{align} \nwhose explicit form is not required for our purposes. Hence the path integral with no matter loops is\n\\begin{align} \\label{P-off2}\n{\\cal P}_n^{\\text{sQED}}(p,k )\n=&\\frac{1}{\\tilde {\\cal N}}\n\\hat{\\delta}^4 \\left(p_1+p_2+\\sum_{i=1}^n k_i\\right)\\\\\n&\n\\int {\\cal D} A \\ e^{\\ii S_{\\text{gf}}[A]}   G (p,k)[A]\\ \\ii k_1^2 \\cdots \\ii k_n^2\\ A_{\\mu_{1}}(k_{1})\\cdots A_{\\mu_{n}}(k_{n})\\Big|_{k^2\\ne 0}^{\\text{trees}} \\nonumber\n\\,,\n\\end{align}\nwhere we have absorbed factors related to our definition of dressed propagator into  $\\tilde {\\mathcal N}$. This is not yet the classical limit of \nthe current we are looking for because we have not performed the LSZ reduction on the external scalar legs, namely \t\\cite{Fabbrichesi:1993kz}\n \\begin{align}\n G^{\\text{c}}(p)[A]= \\lim\\limits_{p_1^2, p_2^2 \\to m^2} \\ii(p_1^2-m^2) \\ \\ii (p_2^2-m^2) \\int \\mathrm{d}^4 x \\mathrm{d}^4y \\\n e^{\\ii p_1\\cdot x+\\ii p_2\\cdot y}G(x, y)[A] \\;.\n \\label{Gconnected}\n \\end{align}\n  At the level of the worldline integral we can also achieve the LSZ reduction by\n Fradkin's prescription of exchanging the limit of integration in the worldline action to $(-\\infty, +\\infty)$ \n as a consequence of performing the Schwinger proper time integration after amputating the external scalar propagators. \n The latter step fixes the boundary conditions needed to perform the perturbative expansion of the amputated dressed propagator. The equivalence between both procedures  is encoded in the relation\n \\begin{align} \\frac{\\Sigma(b,p;A)}{\\Sigma_{0}}=\n e^{\\ii b \\cdot \\sum_{i=1}^n k_i} \\hat{\\delta}\\left( p\\cdot \\sum_{i=1}^{n}k_{i}\t\\right)G^c(p;A) \\,, \n \\label{factors-MPS}\n \\end{align}\n which was \n explicitly demonstrated for the graviton-dressed scalar propagator in Ref.\\cite{Mogull:2020sak}. Here $\\Sigma_{0}$ is some overall factor that we can absorb into the normalization of the correlation function. Notice that both sides depend only on $p_1=p$ on the support of the Dirac-delta in Eq.\\eqref{dressed-gen}. The left hand side of Eq.\\eqref{factors-MPS} is given by\n \\begin{align}\n \\Sigma(b,p;A) = \\int {\\cal D} x \\exp \\left[ \n -\\ii \\int_{-\\infty}^{\\infty}d\\tau \\left( \\frac{1}{2}\\dot{x}^{2} + e \\dot{x}_{\\mu}A^{\\mu}(x(\\tau))\t\t\t\t\\right)\\right], \n \\label{sigma-path}\n \\end{align} \nwhere  $b$ and $p$ arise\n from  the background expansion $x^{\\mu}(\\tau)= b^{\\mu} + p^{\\mu}\\tau +q^{\\mu}(\\tau)$. We will see in later Sections how to evaluate \\eqref{sigma-path} from Worldline Feynman Rules (WFRs). \n \n Going back to Eq.\\eqref{Gconnected} and restricting the calculation to tree-level, the current can be written as\n \\begin{align} \\label{P-off3}\n {\\cal A}_n^{\\text{sQED}}(p,k )\n =&\\frac{1}{\\tilde {\\cal N}}\n \\hat{\\delta}^4 \\left(p_1+p_2+\\sum_i k_i\\right)\\\\\n &\n \\int {\\cal D} A \\ e^{\\ii S_{\\text{gf}}[A]}   G^c(p,k)[A]\\ \\ii k_1^2 \\cdots \\ii k_n^2\\ A_{\\mu_{1}}(k_{1})\\cdots A_{\\mu_{n}}(k_{n})\n \\Big|_{k^2\\ne 0, p_i^2 =m^2}^{\\text{trees}} \\nonumber\n \\,,\n \\end{align} \n which brings us closer to the WQFT representation of $\\mathcal C^{\\text{sQED}}_n (p,k)$. Recalling  Eq.\\eqref{general-def} and imposing momentum conservation we have\n\\begin{align}\n\\bar A(p,k)=\\frac{1}{\\tilde {\\cal N}}\n\\int {\\cal D} A \\ e^{\\ii S_{\\text{gf}}[A]}   G^c(p,k)[A]\\ \\ii k_1^2 \\cdots \\ii k_n^2\\ A_{\\mu_{1}}(k_{1})\\cdots A_{\\mu_{n}}(k_{n})\\Big|_{p_2=-p_1-\\sum_i k_i} \\, ,\n\\end{align}\nwhich is a purely tree-level object in the classical approximation which we can identify \nwith  $\\bar A(p,k)$ in\nEq.\\eqref{final-current-KMOC}. Thus, inserting this equation into Eq.\\eqref{final-current-KMOC} we have\n\\begin{align}\n\\mathcal{C}^{\\text{sQED}}_n (p,  k)=&\n\\frac{1}{\\tilde {\\cal N}}\ne^{\\ii b \\cdot \\sum_{i=1}^n k_i} \\hat{\\delta}\\left( p\\cdot \\sum_{i=1}^{n}k_{i}\t\\right)\\\\\n&\\int {\\cal D} A \\ e^{\\ii S_{\\text{gf}}[A]}   G^c(p,k)[A]\\ \\ii k_1^2 \\cdots \\ii k_n^2\\ A_{\\mu_{1}}(k_{1})\\cdots A_{\\mu_{n}}(k_{n})\\Big|_{p_2=-p_1-\\sum_i k_i} \\, .\\nonumber\n\\end{align}\nFinally, applying the relation \\eqref{sigma-path} \nwe arrive at the WQFT representation of the off-shell current\n\\begin{align}\n\\mathcal{C}^{\\text{sQED}}_n (p,  k)=\n\\frac{1}{\\cal Z} \n\\int {\\cal D} A \\, e^{\\ii S_{\\text{gf}}[A]} \\,\n\\int {\\cal D} x e^{\n\t-\\ii \\int_{-\\infty}^{\\infty}d\\tau \\left[ \\frac{1}{2}\\dot{x}^{2} + e \\dot{x}_{\\mu}A^{\\mu}(x(\\tau))\t\t\t\t\\right]}   \\, \\ii k_1^2 A_{\\mu_{1}}(k_{1})\\cdots \\ii k_n^2A_{\\mu_{n}}(k_{n}) ,\n\\label{path-integral-exp}\n\\end{align}\nwhich generalizes WQFT for an arbitrary number of off-shell photons\\footnote{Another alternative to generate photon insertions is to consider the functional with a factor $\\exp(JA)$ and take derivatives over $J$.}.  \nThe factor\n${\\cal Z}$ ensures that the current is normalized to one when there are  no gauge fields in the path integral and defines the WQFT partition function\n\\begin{equation}\\label{zqed}\n{\\cal Z} = \\int {\\cal D} A \\, e^{\\ii S_{\\text{gf}}[A]} \\, \\int {\\cal D} x \\, e^{-\\ii \\int_{-\\infty}^{\\infty} \\mathrm{d}\\tau \\left(\t\\frac{1}{2}\\dot{x}^{2} + e \\dot{x}\\cdot A\t\\right) } \\, .\n\\end{equation}\nThe  boundary conditions on the worldline variables depend on the model under consideration.  Collecting all of the integration constants into $\\mathcal Z$ the classical off-shell current in WQFT can be succinctly written as\n\\ba \n\\mathcal{C}^{\\text{sQED}}_n (p, k)&= \\ii k_1^2 \\cdots \\ii k_n^2\\left \\langle A_{\\mu_{1}}(k_{1})\\hdots A_{\\mu_{n}}(k_{n})\\right\\rangle_{\\text{WQFT}}\\, .\n\\label{current-classical-worldlined-QED}\n\\ea \nCalculations based on Eq.\\eqref{path-integral-exp} will produce precisely the factors in  Eq.\\eqref{final-current-KMOC} from which we can\nextract $\\bar A_n(p,k)$. In doing so one must keep in mind the overall factor of two in Eq.\\eqref{final-current-KMOC}. Moreover, the presence of the Dirac-delta is required to match currents in both approaches but can be dropped at the end of the computation.   The worldline path integral \\eqref{current-classical-worldlined-QED} can be treated perturbatively deriving Feynman rules from the worldline action leading directly to $\\bar A_n(p,k)$ without any Laurent expansion. \n\nOther theories can be treated along the same lines. For gravity we will consider this in Section \\ref{computational-part}. \nGauge theories with color can  be treated along the same lines with the usual caveats surrounding properties of color charges in the classical limit. On the amplitudes side one can follow\nRef.\\cite{delaCruz:2020bbn} while in the WQFT front one can apply the methods of Ref.\\cite{Shi:2021qsb}.  \n \n\n\\section{Computing off-shell currents}\n\\label{computational-part}\nThe evaluation of Eq.\\eqref{current-classical-worldlined-QED} can be done by deriving worldline Feynman rules (WFRs), which take care of the perturbative evaluation of the Gaussian path integral. There are two types of Wick contractions: those related with the worldline path integral and  Wick contractions among gauge fields.\nThe net effect of the contraction among fields will be to generate  permutations. \n\nThe partition functions considered in the rest of the paper have the schematic form\n %\n \\begin{align}\n \\mathcal Z = \\int {\\mathcal D} B\\, e^{\\ii S[B]}\n \\int {\\mathcal D} x \\int \\mathcal D \\chi e^{\\ii S\\left[x, \\chi; B (x)\\right]}\\, ,\n \\end{align}\n where $B$ represents a background field and  $\\chi$ are auxiliary worldline variables. The action $S[B]$ is the gauge fixed action of the background field and   \n $S\\left[x, \\chi; B (x)\\right]$ is the worldline action in WQFT, from which we can derive WFRs. \n We rescale the worldline parameter  to derive WFRs written in terms of the worldline momentum $p^{\\mu}$ instead of $u^\\mu$.   The relevant  wordline actions are\n \\begin{align}\n S_{\\text{BA}}[x,\\lambda,\\bar{\\lambda},\\gamma,\\bar{\\gamma}; \\Phi]=&- \\int_{-\\infty}^\\infty \\mathrm{d} \\sigma\\left(\\frac 1 2  \\dot x ^2+\\ii \\bar \\lambda_a \\dot \\lambda^a +\\ii \\bar \\gamma_\\alpha \\dot \\gamma^\\alpha - \\frac {y} 2  Q^a\\Phi^{a\\alpha} \\tilde Q^\\alpha \\right)\\, , \\label{BAction}\\\\\n S_{\\text{YM}}[x, \\lambda; A] =& -\\int_{-\\infty}^{\\infty}\\mathrm{d}\\sigma\\left(\n \\frac{1}{2}\\dot{x}^{2}+\\ii \\bar{\\lambda}_{a}\\dot{\\lambda}^{a}+g \\dot{x}^{\\mu}A_{\\mu}^{a}Q^{a} \\right)\\,, \\label{YMC}\\\\\n\tS_{\\text{GR}}[x, \\psi, \\bar \\psi;A]=& -\\int_{-\\infty}^{\\infty} \\mathrm{d}\\sigma \\left( \\frac{1}{2}g_{\\mu\\nu}\\dot{x}^{\\mu}\\dot{x}^{\\nu} + \\ii \\bar{\\psi}_{a} \\frac{D \\psi^{a}}{D\\tau} - \\frac{1}{8}R_{abcd}S^{ab}S^{cd}\t\t\t\t\\right)\\, ,\\label{GRSpin}\n\\end{align}\nfor Bi-Adjoints, Yang-Mills, and GR, respectively.\nThe covariant derivative is $\\frac{D\\psi^{a}}{D\\tau} = \\dot{\\psi}^{a} + \\dot{x}^{\\mu}\\omega_{\\mu a b}\\psi^{b}$ and $\\omega_{\\mu a b}= e_{a\\nu}(\\partial_{\\mu}e^{\\nu}_{b} + \\Gamma^{\\nu}_{\\mu\\lambda} e^{\\lambda}_{b} )$ is the spin connection. The translation between curved and flat indices is done by using the tetrad fields $e^{\\mu}_{a}$.\nThe spin tensor of the particle is defined by $S^{\\mu\\nu} = -2\\ii e^{\\mu}_{a} e^{\\nu}_{b}\\ \\bar{\\psi}^{[a}\\psi^{b]}$.\nThe derivation of WFRs is now well established and described at length\nin Refs. \\cite{Mogull:2020sak, Jakobsen:2021zvh, Shi:2021qsb, Bastianelli:2021nbs, Wang:2022ntx} so here we only  consider briefly EM. Our treatment of color variables is equivalent to Ref.\\cite{Shi:2021qsb} but differs on the introduction of auxiliary color variables\\footnote{A pedagogical description of auxiliary variables is in Ref.\\cite{Bastianelli:2021rbt}.}.\n  The interested reader can see the details of the derivation of WFRs with color in Appendices \\ref{bi-adjoint-WQFT}  and \\ref{colors-appendix}.\n\n  \\subsection{Gauge} \n %\n \\label{gauge-examples}\nWFRs are easy to derive  for the colorless version of \n\\eqref{YMC}. \n  Inserting the background expansion  $x^\\mu (\\tau)= b^{\\mu} + p^{\\mu}\\tau + q^{\\mu}(\\tau)$ into the action\n\\begin{align}\nS_{\\text{sQED}}[x; A] =& -\\int_{-\\infty}^{\\infty}\\mathrm{d}\\sigma\\left(\n\\frac{1}{2}\\dot{x}^{2}+e \\dot{x}^{\\mu}A_{\\mu} \n\\right)\\,,\n\\end{align}\nwhere \n\\begin{align}\nq^{\\mu}(\\tau) = \\int_{-\\infty}^{\\infty}\\hat{\\mathrm{d}}\\omega \\, e^{\\ii \\omega \\tau}q^{\\mu}(-\\omega), \\  \\hskip.5cm A^\\mu (x) =  \\int_{-\\infty}^{\\infty}\\hat{\\mathrm{d}}^4 k \\, e^{\\ii k \\cdot x }\nA(-k),\n\\end{align} \n  the propagators of the worldline and the gauge field (inf Feynman gauge) are\n\\begin{align} \\label{wprop}\n\\raisebox{-1.7mm}{\n\t\\begin{tikzpicture}[thick]\n\t\\coordinate (A) at (-1,-0);\n\t\\coordinate (B) at (1,-0);\n\t\\filldraw (A) circle (2pt) node[left] {$q^\\mu$};\n\t\\filldraw (B) circle (2pt) node[right] {$q^\\nu$};\n\t\\draw[] (0,.5) node[]{$\\omega$};\n\t\\path[ very thick,draw] (-1,0)--(1,0); \t\\path[ thick,draw,->] (-0.5,0.3)--(0.5,0.3) node[pos=.5] {}; \n\t\\end{tikzpicture}\n}\n= - \\ii \\frac{\\eta^{\\mu\\nu}}{(\\omega+\\ii\\epsilon)^2},\n\\hskip.4cm \n\\raisebox{-1.7mm}{\n\t\\begin{tikzpicture}[thick]\n\t\\coordinate (A) at (-1,-0);\n\t\\coordinate (B) at (1,-0);\n\t\\filldraw (A) circle (2pt) node[left] {$A^\\mu$};\n\t\\filldraw (B) circle (2pt) node[right] {$A^\\nu$};\n\t\\draw[] (0,.6) node[]{$k$};\n\n\t\\path[ thick,draw,->] (-0.5,0.3)--(0.5,0.3) node[pos=.5] {}; \n\t\\path[snake it,draw] (-1,0)--(1,0); \n\t\\end{tikzpicture}\n}\n= - \\ii \\frac{\\eta^{\\mu\\nu}}{(k^{2}+\\ii\\epsilon)} \\, . \n\\end{align}\n  In addition,  the interaction vertices propagating $n$-quantum fluctuations of the worldline variables can be compactly written as (see also Ref.\\cite{Wang:2022ntx})\n\\begin{equation}\n\\raisebox{-10mm}{\\vnQED } = e\\,\\ii^{n-1} e^{\\ii b\\cdot k}\\hat{\\delta}\\left(k\\cdot p +\\sum_{i=1}^{n} \\omega_{i}\\right) \\left(\np^{\\mu}\\prod_{i=1}^{n}k^{\\alpha_{i}} + \\sum_{i=1}^{n}\\omega_{i}\\eta^{\\mu \\alpha_{i}}\\prod_{j\\neq i}^{n}k^{\\alpha_{j}}\n\\right)\\,, \n\\end{equation}\nwhere the solid lines are outgoing. Contractions of gauge fields generate topologies of worldline diagrams, in which external propagators must be amputated in the sense of LSZ. For instance at lower orders  it is easy to see that the Wick contractions among gauge fields lead us to evaluate the diagrams shown in Fig.\\ref{main-topologies-off-shell}.\n\n\n\\subsubsection*{Examples}\n\n \\begin{figure}\n\t\\centering\n\t\\begin{subfigure}[t]{0.3\\textwidth}\n\t\t\\centering\t\\lotop{$A_1$}{$A_2$}{black}\n\t\t\\caption{}\n\t\t\\label{lo-gen}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.5\\textwidth}\n\t\t\\centering\n\t\t\\nlotop{$A_1$}{$A_2$}{$A_3$}{black}{black}\n\t\t\\caption{}\n\t\t\\label{nlo-gen}\n\t\\end{subfigure}\n\t\\caption{Worldline diagrams required for the calculation of 2-point (a) and 3-point (b) currents. The dashed lines are drawn only for illustration and represent the worldline. Solid lines represent fluctuations of matter lines and later also auxiliary variables. Wavy lines represent off-shell massless particles and $A$ its associated field. }\n\t\\label{main-topologies-off-shell}\n\\end{figure}\nLet us consider the  $n=2$ off-shell current for scalar electrodynamics.\nTwo equivalent diagrams with symmetry factor $\\frac 12$ are generated by the wordline path integration. Hence it is enough to consider the one in Fig.\\ref{lo-gen}  whose symmetry factor is unity.\nAn easy calculation gives\n\\ba \n\\mathcal{C}_{\\text{sQED}}^{\\mu\\nu} (p, k) &=e^2 \\, e^{\\ii b\\cdot (k_1+k_2)} \n \\hat{\\delta}\\left(p\\cdot (k_{1}+k_{2})\\right) \\bar A_{\\text{sQED}}^{\\mu\\nu}(k_{1},k_{2})\\\\\n&=e^2 \\, e^{\\ii b\\cdot (k_1+k_2)}\\hat{\\delta}\\left(p\\cdot (k_{1}+k_{2}) \\right) \\ii \\left(\n \\eta ^{\\mu  \\nu }+\\frac{k_2^{\\mu } p^{\\nu }}{p\\cdot k_{1} }-\\frac{ k_1^{\\nu } p^{\\mu }}{p\\cdot k_{1} }-\\frac{ k_1\\cdot k_2\n   p^{\\mu } p^{\\nu }}{(p\\cdot k_{1})^2}\n   \\right),\n   \\label{2ptsQED}\n\\ea \nwhich satisfies the Ward identity $k_{i,\\mu} \\mathcal{C}_{\\text{sQED}}^{\\mu\\nu} (p, k) = 0$ and matches what one calculates using Eq.\\eqref{final-current-KMOC}. A  nontrivial example is the $n=5$ case. Some topologies involved are shown in Fig.\\ref{topologies-five-photons}. Its Feynman diagrammatic calculation requires 450 diagrams but  the worldline calculation effectively requires only 12 diagrams summed over the permutations of the external photons with no need to perform any Laurent expansion in $\\hbar$, so its calculation is simpler in WQFT. The result obtained with WQFT is lengthy\\footnote{The full expression for $\\bar A_{\\text{sQED}}^{\\mu_1\\dots\\mu_5}$ is attached to this submission in FeynCalc notation.}. Schematically it can written as \n %\n \\begin{align}\n \\mathcal{C}_{\\text{sQED}}^{\\mu_1 \\dots \\mu_5}=e^5\\, e^{\\ii b\\cdot (k_1+\\dots +k_5)} \\hat{\\delta} \\left(p\\cdot \\sum _{i=1}^5 k_i\\right) \\bar A_{\\text{sQED}}^{\\mu_1\\dots\\mu_5}=e^5\\, e^{\\ii b\\cdot (k_1+\\dots +k_5)} \\hat{\\delta} \\left(p\\cdot \\sum _{i=1}^5 k_i\\right) \\sum_{i=1}^{2451} a_i \\mathsf{T}_i^{\\mu_1 \\dots \\mu_5},\n \\label{eq-5-pt}\n \\end{align}\n where the sum runs over independent tensor structures. We have compared this result against  the Feynman diagram calculation and found agreement. \n \n \n \\begin{figure}[h]\n\t\\centering\n\t\\begin{subfigure}[t]{0.4\\textwidth}\n\t\t\\centering\t\\lotopfivePHone{$A^{\\mu_1}$}{$A^{\\mu_2}$}{$A^{\\mu_3}$}{$A^{\\mu_4}$}{$A^{\\mu_5}$}\n\t\t\\caption{$S=4!$}\n\t\\end{subfigure}\n\t\\begin{subfigure}[t]{0.5\\textwidth}\n\t\t\\centering\t\\lotopfivePHtwo{$A^{\\mu_1}$}{$A^{\\mu_2}$}{$A^{\\mu_3}$}{$A^{\\mu_4}$}{$A^{\\mu_5}$}\n\t\t\\caption{$S= 3!$}\n\t\\end{subfigure}\\\\\n\t\\begin{subfigure}[t]{0.5\\textwidth}\n\t\t\\centering\t\\lotopfivePHthree{$A^{\\mu_1}$}{$A^{\\mu_2}$}{$A^{\\mu_3}$}{$A^{\\mu_4}$}{$A^{\\mu_5}$}\n\t\t\\caption{$S=2 \\times 3!$}\n\t\\end{subfigure}\n\t\\caption{Examples of worldline topologies required for the calculation of 7-point sQED current and their symmetry factors $S$. }\n\t\\label{topologies-five-photons}\n\\end{figure}\n\n\n\n \n\\subsection{Gravity}\n\\label{gr-off-shell}\n The worldline QFT allows us to introduce classical spin effects at almost no cost so we will do so. Let us briefly\nsummarize the WQFT approach of Ref.\\cite{Jakobsen:2021zvh}, which employs\n the so called ${\\cal N} = 2$ model. This model captures quadratic in spin effects in classical scattering through the introduction of complex Grassman variables $\\psi,\\bar{\\psi}$ allowing to gauge two super-symmetries on the worldline. The relevant actions to construct the partition function are \n %\n \\begin{equation}\n S_{\\textrm{EH}}[g] = -\\frac {2}{\\kappa^{2}}\\int \\mathrm{d}^{4}x\\, \\sqrt{-g}\\, R, \\quad S_{\\text{gf}} = \\int \\mathrm{d}^{4}x \\left(\t\\partial^{\\nu}h_{\\nu\\mu}-\\frac 1 2 \\partial_{\\mu} h^{\\nu}_{\\nu}\\right)^{2},\n \\end{equation}\nwhere the space time metric is perturbatively expanded as $g_{\\mu\\nu}= \\eta_{\\mu\\nu}+\\kappa h_{\\mu\\nu}$. The gauge-fixing term\n imposes a weighted version of the de Donder gauge $\\partial^{\\nu}h_{\\nu\\mu} = \\frac 12 \\partial_{\\mu} h^{\\nu}_{\\nu}$, which leads to the graviton\n propagator\n \\begin{align}\n \\label{eq:propq}\n \\raisebox{-2.8mm}{\n \t\\begin{tikzpicture}[thick]\n \t\\coordinate (A) at (-1,-0);\n \t\\coordinate (B) at (1,-0);\n \t\\filldraw (A) circle (2pt) node[left] {$h_{\\mu\\nu}$};\n \t\\filldraw (B) circle (2pt) node[right] {$h_{\\rho\\sigma}$};\n \t\\draw[] (0,.6) node[]{$k$};\n \n \t\\path[ thick,draw,->] (-0.5,0.3)--(0.5,0.3) node[pos=.5] {}; \n \t\\path[double,snake it,draw] (-1,0)--(1,0); \n \t\\end{tikzpicture}\n }\n = \\frac{\\ii}{2k^{2}} \\left(\t \\eta_{\\mu \\rho}\\eta_{\\nu \\sigma} +  \\eta_{\\mu \\sigma}\\eta_{\\rho\\nu} -  \\eta_{\\mu\\nu}\\eta_{\\rho\\sigma} \t\\right) \\;.\n \\end{align}\n %\nThe partition function can be written as follows\n\\begin{equation} \\label{zgr}\n{\\cal Z} = \\int {\\cal D} h_{\\mu\\nu}\\, e^{\\ii (S_{\\textrm{EH}} +S_{\\textrm{gf}} ) } \\int {\\cal D} x {\\cal D} \\psi {\\cal D} \\bar{\\psi} \\, e^{\\ii S_{\\text{GR}}},\\end{equation}\nwith the worldline action given in \\eqref{GRSpin}. The worldline path integral measure includes Lee-Yang ghost fields\n\\begin{equation}\n{\\cal D} x = \\int  Dx DaDbDc\\, \\exp\\left(\t-\\frac{\\ii}{2}\\int_{-\\infty}^{\\infty}\\mathrm{d}\\sigma\\,  g_{\\mu\\nu}\\left(a^{\\mu}a^{\\nu} + b^{\\mu}c^{\\nu}\t\t\\right)\t\t\\right)\n\\end{equation}\nwith $a$ bosonic and $b,c $ fermionic, which make the measure translational invariant. For details on the model we refer the reader to \\cite{Jakobsen:2021zvh}.\nTo derive WFRs in such a case one also needs to background expand $\\psi^{a}(\\tau) = \\theta^{a}+ \\Psi^{a}(\\tau)$ where complex conjugation leads to the expansion for the barred partner. This yields  the classical value of the spin tensor ${\\cal S}^{ab} = -2\\ii \\bar{\\theta}^{[a}\\theta^{b]}$, which is related to the Pauli-Lubanski spin vector $s^{a} = -\\frac{1}{2}\\epsilon^{abcd}p_{b}{\\cal S}_{cd}$.  The required WFRs to perform calculations are given in the Appendix \\ref{WFR-gr}. Generalizing the results in Sec.\\ref{sec-proof-WQFT}, we write the off-shell current in gravity as\n\\begin{equation}\n{\\cal C}_n^{\\text{GR}}(p,k) = (- \\ii)^{n} k_{1}^{2} k_{2}^{2} \\cdots  k_{n}^{2}\\braket{ h_{\\mu_{1} \\nu_{1}}(k_{1}) h_{\\mu_{2} \\nu_{2}}(k_{2}) \\cdots h_{\\mu_{n} \\nu_{n}}(k_{n})\n}_{\\text{WQFT}}\n\\label{current-scalar-GR}\n\\end{equation} \nunderstood as an expectation value over the partition function \\eqref{zgr}. \n\nAs an example we consider the classical on-shell Compton amplitude\\footnote{In Ref.\\cite{Saketh:2022wap} a direct classical calculation of the Compton amplitude has been done using the worldline effective theory including cubic in spin corrections.}, which describes the scattering of linearized gravitational waves off a black hole. \n\\begin{figure}\n\t\\centering\n\t\\begin{subfigure}[t]{0.3\\textwidth}\n\t\t\t\\centering \\lotopGR{$h_{\\mu\\nu}$}{$h_{\\alpha\\beta}$}{black}\n\t\t\\caption{}\n\t\t\\label{tu} \t\t\n\t\t\\end{subfigure}\n\t\t\\begin{subfigure}[t]{0.3\\textwidth}\n\t\t\t\\centering \\lotopGRspin{$h_{\\mu\\nu}$}{$h_{\\rho\\sigma}$}{black}\n\t\t\t\\caption{}\n\t\t\\label{tu1} \n\t\t\\end{subfigure}\\\\[1em]\n\t\t\\begin{subfigure}[t]{.3\\textwidth}\n\t\t \\centering  \\hhh\n\t\t \\caption{} \n\t\t \\label{s}\n\t\t \\end{subfigure}\n\t\t \\begin{subfigure}[t]{.3\\textwidth} \n\t\t\\centering \\hhvertex \n\t\t\\caption{}\n\t\t\\label{seagull} \n\t\\end{subfigure}\n\t\\caption{Worldline diagrams contributing to the on-shell Compton amplitude. \\ref{tu},\\ref{tu1}, with crossed topologies are associated with fluctuations  of kinematic and spin variables, respectively.\n\t }\n\t\\label{compton-amplitude}\n\\end{figure}\nThe on-shell Compton amplitude  is given by\n\\begin{equation}\\label{sGR-Compton}\n{\\cal M}^{h_{1}h_{2}}(p,k_{1},k_{2}) = \n{\\bar M}_{\\text{GR}}^{\\mu\\nu\\alpha\\beta} (p, k_{1},k_{2}) \\epsilon^{h_{1}}_{\\mu\\nu}(k_{1})\\epsilon^{h_{2}}_{\\alpha\\beta}(k_{2})\\Big|_{k_i^2=0}\\,,\n\\end{equation}\nwhere we can extract the amplitude from $\\mathcal{C}_{\\text{sGR}}^{\\mu\\nu\\alpha\\beta}=\\frac{1}{2}\\kappa^2 e^{\\ii b\\cdot (k_1+k_2)} \\hat{\\delta}(p\\cdot (k_1+k_2)) \\bar M_\\text{GR}^{\\mu\\nu\\alpha\\beta}$, which receives contributions from the diagrams shown in Fig.\\ref{compton-amplitude}. \nIn the spinless case, we can apply and off-shell version of the the Kawai-Lewellen-Tye \\cite{Kawai:1985xq} relation\n\\begin{align}\n{\\mathcal C}_{\\text{sGR}}^{\\mu_1\\nu_1,\\mu_2\\nu_2}=-\\kappa^2 \\ii \\, e^{\\ii b\\cdot(k_1+k_2)}\\hat{\\delta}( p \\cdot(k_1+k_2)) \\frac{k_1\\cdot p \\, k_2\\cdot p}{k_1\\cdot k_2} \\bar A_{\\text{sQED}}^{\\mu_1\\mu_2}\n\\bar A_{\\text{sQED}}^{\\nu_1\\nu_2} \\,,\n\\end{align}\nwith the QED current given in Eq.\\eqref{2ptsQED}, after replacing $e^{2}\\to \\kappa^{2}\/4$.\nIn order to fuse  the full current into the amplitude \\eqref{sGR-Compton}, we use physical polarization tensors $\\epsilon^{\\mu\\nu}_{\\pm \\pm}(k_{i}) = \\epsilon^{\\mu}_{\\pm}(k_{i})\\epsilon^{\\nu}_{\\pm}(k_{i})$ written as a product of null transverse  photon polarizations. We set $k_{1}$ as incoming momentum and $k_{2}$ as outgoing and choose the rest frame of the worldline, i.e.,\n\\begin{equation}\np^{\\mu}=m u^{\\mu} = (m,0,0,0),\n\\quad k^{\\mu}_{1} =E(1,0,0,1), \\quad\nk^{\\mu}_{2} = E(1,\\sin\\theta,0,\\cos\\theta),\n\\end{equation} \n where $E$ is the energy of the graviton.  Explicit polarization vectors $\\epsilon^{\\mu}_{\\pm}$ follow from the transversality and traceless conditions. Therefore, we  can evaluate \\eqref{sGR-Compton} for the independent set of helicity configurations $(++),(+-)$. The scalar contributions are given by \n\\begin{equation}\n|{\\cal M}_{++}|_{_{0}} = |{\\cal M}_{--}|_{_{0}}= \\frac{\\kappa ^{2}m^{2}}{4}\\frac{\\cos^{4}\\frac{\\theta }{2}}{ \\sin^{2}\\frac{\\theta }{2} },\n\\hskip.4cm \n|{\\cal M}_{+-}|_{_{0}} = |{\\cal M}_{-+}|_{_{0}} = \\frac{\\kappa ^{2} m^{2}}{4} \\sin^{2}\\frac{\\theta }{2}\n\\,.\n\\end{equation}\nSimilarly, using the full current the linear in spin terms read \n\\begin{equation}\n|{\\cal M}_{++}|_{_{O({\\cal S})}} = |{\\cal M}_{++}|_{_{0}}\\Big|\t\t\t\ns\\cdot(k_{1}+k_{2})\\tan^{2}\\frac{\\theta}{2} +\\ii \\frac{k_{1}\\cdot {\\cal S}\\cdot k_{2}}{mE \\cos^{2}\\frac{\\theta}{2}}\n\\Big|,\n\\hskip.4cm |{\\cal M}_{+-}|_{_{O({\\cal S})}} = |{\\cal M}_{+-}|_{_{0}} \\, |s\\cdot (k_{1}-k_{2})|  \\,, \n\\end{equation}\nwhile the quadratic in spin contributions can be recast in the suggestive way\n\\begin{equation}\n|{\\cal M}_{++}|_{_{O({\\cal S}^{2})}} = \\frac{1}{2} \\frac{\\hskip.4cm |{\\cal M}_{++}|^{2}_{_{O({\\cal S})}}  }{ |{\\cal M}_{++}|_{_{0}}},\n\\hskip.4cm |{\\cal M}_{+-}|_{_{O({\\cal S}^{2})}} = \\frac{1}{2} \\frac{ \\hskip.4cm |{\\cal M}_{+-}|^{2}_{_{O({\\cal S})}} }{ |{\\cal M}_{+-}|_{_{0}}} \\,.\n\\end{equation}\nThe remaining helicity configurations can be obtained by replacing ${\\cal S}^{a b }\\to -{\\cal S}^{a b},s^{a}\\to -s^{a}$.\nThe above results are in agreement with \\cite{Saketh:2022wap} up to quadratic in spin\nupon matching with our conventions.\n\n\\section{Application to Hard Thermal loops}\n\\label{currents-and-HTLs}\nA nice application of the ideas presented in the past sections is the case of Hard Thermal Loops (HTLs). These are  currents in the high temperature limit which can be resumed and incorporated into an effective theory known as HTL effective theory~\\cite{Braaten:1989mz,  Frenkel:1989br, Braaten:1990az, Taylor:1990ia,\n\tFrenkel:1991ts}.  It is well known that the high temperature regime is equivalent to a classical regime. Using a KMOC-like approach this was explicitly demonstrated in  Ref.\\cite{delaCruz:2020cpc}, where HTLs were computed  as the limit $\\hbar \\to 0$. Schematically HTL currents can be written as \n\\begin{align} \n\\Pi_n(k)=\\int \\mathrm{d}\\Phi (p) f(p_{0})\n \\bar A_n(p, k) \\, ,\n\\label{htl}\n\\end{align} \nwhere $f (p_{0})$ is a distribution function at equilibrium and $\\bar A_n(p, k)$ is the classical limit of the  current  in the regularized forward limit. The regularization is required  since the same diagrams that contribute to the currents contribute to amplitudes so in general the forward limit is singular.\n Let $\\mathcal F$ be the set of all Feynman diagrams contributing to the current \\eqref{general-def}. Diagrammatically the regularization consists on dropping the set of all diagrams\nproducing zero momentum internal edges $\\mathcal X$ in the forward limit\\footnote{\n\tThese problematic diagrams appear e.g.,  Eq.\\eqref{current-classical-worldlined-QED} and \\eqref{current-scalar-GR}\n\tso the regularization should also be implemented  in the worldline formalism.}.  It is defined by\n\\begin{align}\t\nA_n (p,k):=\\sum_{G\\in \\mathcal F   \\backslash \\mathcal X}\nd(G)\\, ,\\label{reg-current}\n\\end{align}\nwhere $d(G)$ is a rational  expression of the form $N(G)\/D(G)$.   In the forward limit $p_1=-p_2$ so momentum conservation becomes\n\\begin{align}\n\\sum_{i=1}^{n}k_{i} =0.\n\\label{momentum-conservation}\n\\end{align}\nThe classical limit of Eq.\\eqref{reg-current} is obtained through Eq.\\eqref{final-current-KMOC}. These currents have been considered in Refs.\\cite{delaCruz:2020cpc, delaCruz:2022nlj}.   \n\nThe WQFT approach gives us a new way of obtaining these currents. In QED the equivalence between\n\\eqref{final-current-KMOC} and \\eqref{current-classical-worldlined-QED} implies that the  $n-$point HTL can be read off from\n\\begin{equation}\n \\hat{\\delta} \\left(p\\cdot \\sum\\limits_{i=1}^n k_i \\right)\\frac{1}{2}{\\bar A}_n (p, k) = \\ii k_{1}^{2}\\ii k_{2}^{2}\\cdots \\ii k_{n}^{2} \\langle\nA_{\\mu_{1}}(k_{1})A_{\\mu_{2}}(k_{2})\\cdots A_{\\mu_{n}}(k_{n}) \\rangle_{\\text{WQFT}}, \n\\end{equation}\nwhere the regularization is  understood in  both sides. Since we are interested in $\\bar A_n(p,k)$ we will strip-off the Dirac-delta  produced by WQFT.\nInserting\nthe RHS of this equation  side into Eq.\\eqref{htl} gives and alternative worldline path integral representation of the HTL resumed current.  \nA similar matching can be used to obtain HTLs in other theories. \n\n\nDiagrams that contribute at each order in perturbation can be determined from dimensional analysis.\nThe treatment of colour in the classical limit requires special care due to  nontrivial cancellations between color and kinematics.\n In the worldline context this is translated into the dynamics of auxiliary variables which are present in the worldline action and give additional interactions. Their role is to encode non commutativity of color factors in general. The fact that classical color factors commute in the classical limit is recovered after integration over  the auxiliary variables, a property that requires representations of the gauge group to be large\n  (See Appendix \\ref{colors-appendix}). \n \n  When classical color factors are recovered,  thermal currents are obtained after phase-space integration over color.  Phase space integration over classical color factor is defined by\n \\begin{align}\n \\mathrm{d} c:=&  \\mathrm{d}^8 c\\, c_R \\delta(c^a c^b \\delta^{ab}-q_2 ) \\delta(d^{abc}c^a c^bc^c-q_3 ), \\label{DIPSC}\n \\end{align}\n %\n where $q_2$ and $q_3$ are Casimir invariants. The factor $c_R$ ensure that the color measure is normalized to unity and we have set the gauge group to be $SU(3)$.  For bi-adjoint scalars we will take two copies of the phase-space integration measure.\n\n\\subsection{Scalars}\n\\label{Bi-adjoint}\nAs our first application let us consider  scalar theories with cubic interactions. For instance, the bi-adjoint field theory Lagrangian  is given by \\cite{Luna:2015paa,White:2016jzc}\n\\begin{align} \nS_{\\text {BA}}[\\varphi]=\\int\\mathrm{d}^4 x \\left[\\frac 12 \\partial_\\mu \\varphi^{a \\alpha} \\partial^\\mu \\varphi^{a \\alpha} \n-\\frac{m^2}{2} \\varphi^{a \\alpha}\\varphi^{a \\alpha} \n+\\frac{y}{3!} f^{abc} \\tilde f^{\\alpha \\beta \\gamma}\n\\varphi^{a \\alpha} \\varphi^{b \\beta} \\varphi^{c \\gamma}\\right] \\:, \n\\label{lag-phi3}\n\\end{align}\nwhere $m$ is the mass and $y$ the coupling constant. The bi-adjoint scalar field  $\\varphi^{a\\alpha}$ transforms under the adjoint representation for each factor of its globally symmetry group $G \\times \\tilde G$. The Lie algebra for each factor has the form $[T^a, T^b]=f^{abc} T^c$ and the adjoint representation is given by its structure constants, i.e.\n$(T_A^a)^{b}_{\\ c}=-  f^{abc}$. Throughout we  use Greek indices   for the group $\\tilde G$.  We briefly outline the derivation of WFRs for this theory in Appendix \\ref{bi-adjoint-WQFT}. It\nis instructive to present the colorless and colorful cases separately to see how the appearance of color affect the final results. \n %\n\\subsubsection*{Cubic scalars}\nLet's consider the simplest example, where we have two soft scalars. Two equivalent diagrams with symmetry factor $\\frac 12$ are generated by the path integration over scalar fields. Hence it is enough to consider the one in Fig.\\ref{lo-gen}  whose symmetry factor is unity. Using WFRs we obtain \n\\begin{align} \\label{tp1}\n  \\raisebox{-.9cm}{\n\\lotop{$\\varphi(k_1)$}{$\\varphi(k_2)$}{black}}=\n \\frac{1}{2} \n \\bar A_2 (p,k_1)\n=& \\ii  \n\\frac{1}{4}y^2 \\frac{k_1^2}{(k_1\\cdot p)^2}\\, ,\n\\end{align}\nwhich we can use to extract $A (p,k_1)$. \nHence using momentum conservation, relabeling the independent momentum $k_1=k$ \nwe obtain\n\\begin{align}\n\\Pi_2 (k)= \\frac{y^2}{2} \\int\\mathrm{d} \\Phi (p) f(p_0)\n \\frac{k^2}{(k\\cdot p)^2}.\n\\end{align}\nThe same procedure can be carried on in order to evaluate the three point thermal current. Notice however that a diagram analogous to Fig.\\ref{compton-amplitude} also contributes to the current but is singular in the forward limit so we discard it due to the regularization.  The result now depends on permutations of the \ndiagrams in Fig.\\ref{nlo-gen}, which can be easily evaluated\n\\begin{align}\n\\bar A_3(k_1,k_2,k_3)= \\frac{y^3}{4} \n\\sum\\limits_{\\sigma \\in \\text{Cyclic}} \\frac{k_{\\sigma_1}\\cdot k_{\\sigma_2} k_{\\sigma_2}\\cdot k_{\\sigma_3}}{\\left(p\\cdot\n\tk_{\\sigma_1}\\right){}^2 \\left(p\\cdot k_{\\sigma_3}\\right){}^2},\n\\end{align}\nwhere the sum runs only over cyclic permutations of $\\{1,2,3\\}$. This equation is in agreement with Ref.\\cite{delaCruz:2022nlj} obtained from semi-classical kinetic theory.\n\n\\subsubsection*{Bi-adjoint}\nThe bi-adjoint scalar also generates Feynman rules which involve color-color fluctuations and color-matter fluctuations, as explained in Appendix \\ref{bi-adjoint-WQFT}. It receives the contributions from the diagrams\\footnote{In the colored case, color fluctuations are of the same order in coupling as those with matter. However upon restoring $\\hbar$ one must also take into account that color factorization  brings an additional power of $\\hbar$. See Appendix \\ref{colors-appendix}.} with kinematic fluctuations of the worldlines\n  %\n \\begin{align}\n\\raisebox{-0.7cm}{ \\lotop{}{}{black} } \n   = \\ii \\frac{y^2}{4} c^{a_1} c^{a_2} \\tilde{c}^{\\alpha_1} \\tilde{c}^{\\alpha_2} \\frac{k_1\\cdot k_2}{(k_1\\cdot p)^2}\\,,\n   \\end{align} \n   and  from the diagrams propagating color fluctuations\n   \\begin{align}\n \\raisebox{-0.7cm}{ \\lotopcol{}{}{red}} +\n \\raisebox{-0.7cm}{ \\lotopcolCross{}{}{red}} \n =&\\, \\ii \\frac{y^2}{4} \\frac{ \\tilde{c}^{\\alpha_1} \\tilde{c}^{\\alpha_2}}{k_1\\cdot p}\\bar{u}\\cdot(T^{a_1}\\cdot T^{a_2}-T^{a_2} \\cdot T^{a_1}  )\\cdot u\\, , \\label{cont2-bi}\\\\\n\\raisebox{-0.7cm}{ \\lotopcol{}{}{blue}}\n+\n \\raisebox{-0.7cm}{ \\lotopcolCross{}{}{blue}} \n=&\\, \\ii \\frac{y^2}{4}    \\frac{c^{a_1} c ^{a_2}}{k_1\\cdot p}\\bar{v}\\cdot( \\tilde T^{\\alpha_1}\\cdot \\tilde T^{\\alpha_2}-\\tilde T^{\\alpha_2}\\cdot \\tilde T^{\\alpha_1})\\cdot v\\label{cont3-bi}\\, ,\n\\end{align}\nIt should be noticed how adding up the two topologies in each of Eq.\\eqref{cont2-bi} and \\eqref{cont3-bi} generates the structure constants of the Lie algebra \n\\begin{align}\n\\bar{u}\\cdot \\left( T^{a_1}\\cdot T^{a_2}-T^{a_2}T^{a_1} \\right)\\cdot u = f^{a_1 a_2 a_3}c^{a_3}\\, \n\\label{Lie-bracket}\n\\end{align}\nwhere $c^a=\\bar u\\cdot T^a \\cdot u$ and\nthe same for the tilded partner.\nThen, the current simplifies to \n\\begin{align}\n\\bar A_2^{a_1 \\alpha_1, a_2 \\alpha_2}=  \\frac{y^2}{2}\\left( c^{a_1} c^{a_2} \\tilde c^{\\alpha_1} \\tilde c^{\\alpha_2} \\frac{ k_1^2}{(k_1 \\cdot p)^2}+\n\\frac{\\tilde c^{\\alpha_1} \\tilde c^{\\alpha_2}f^{a_1 a_2 a_3} c^{a_3} +c^{a_1} c^{a_2}\\tilde f^{\\alpha_1 \\alpha_2 \\alpha_3} \\tilde c^{\\alpha_3}}{k_1\\cdot p} \n\\right)\\, .\\end{align} \nThe phase-space integration over color can be done using the identities\n\\begin{align}\\label{psint}\n\\int \\mathrm{d} c\\ c^{a}=0, \\qquad \\int \\mathrm{d} c\\ c^{a} c^{b}= \\delta^{ab},\n\\end{align}\nwhich follow from Eq.\\eqref{DIPSC}. \nHence after some relabeling\n\\begin{align}\n\\Pi^ {a_1 \\alpha_1, a_2 \\alpha_2}(k)= \\delta^{a_1 a_2}\\delta^{\\alpha_1\\alpha_2 } \\frac{q_{2}^{2}y^{2}}{2}\\int\\mathrm{d} \\Phi(p) \\frac{ k^2}{(k \\cdot p)^2},\n \\end{align}\n %\nwhich \n is in agreement with kinetic theory of Ref.\\cite{delaCruz:2022nlj}.\n\n\\subsection{Gauge and gravity}\n\\label{gauge}\nAs our final example, we \nmove directly to scalar QCD since \n scalar QED HTLs can be recovered \nfrom the off-shell currents in \nSection \\ref{gauge-examples} using momentum conservation. In QED no singular diagrams arise in the forward limit.  \n For instance, the 5-pt HTL it is straightforward to obtain from Eq.\\eqref{eq-5-pt} and Eq.\\eqref{momentum-conservation}. \n\nLet us consider the 3-point HTL. \nFrom the WFRs in Appendix \n\\ref{colors-appendix} there are two\ndiagrams we need to calculate. For example, \nfor the permutation $\\sigma=(1,2,3)$ for the external gluons one has \n\\begin{align}\n\\raisebox{-4mm}{ \\nlotopC } &= g^{3}\\ \\bar u\\cdot T^{a_{1}}\\cdot T^{a_{3}}u \\ c^{a_{2}}\n\\frac{p^{\\mu_{3} }}{p\\cdot k_{3} }\\bar A_{\\text{sQED}}^{\\mu_{2}\\mu_{1}} (k_{2},k_{1})\\,, \n\\\\\n\\raisebox{-4mm}{ \\nlotopD } &=  -g^{3}\\ \\bar u\\cdot T^{a_{3}} \\cdot T^{a_{1}}u\\ c^{a_{2}}\n\\frac{p^{\\mu_{3} }}{p\\cdot k_{3} }\\bar A_{\\text{sQED}}^{\\mu_{2}\\mu_{1}} (k_{2},k_{1})\\,,\n\\end{align}\nwhich  generate the $SU(N)$ structure constants as in Eq.\\eqref{Lie-bracket}. Performing the phase space integration \\eqref{psint} and summing over all permutations\n the final answer can be written as \n\\begin{equation}\n\\bar A^{\\mu_{1} \\mu_{2} \\mu_{3}}_{a_{1} a_{2} a_{3}}=- 2g^{3}  \\sum_{\\sigma \\in S_3}\nf^{a_{\\sigma_1 } a_{\\sigma_2} a_{\\sigma_3}} \n\\frac{p^{\\mu_{\\sigma_3}}}{p\\cdot k_{\\sigma_3}} \\bar A_{\\text{sQED}}^{\\mu_{\\sigma_2}\\mu_{\\sigma_1}} (k_{\\sigma_2},k_{\\sigma_1})\\label{current-3pt}\n\\end{equation}\nwhere $S_3$ is the set of all permutations of $\\{1,2,3\\}$. \nThe above result satisfies the identity \n\\begin{equation}\n k_{3\\mu_{3}} \\, {\\bar A}^{\\mu_{1} \\mu_{2} \\mu_{3}}_{a_{1} a_{2} a_{3}} = 2g^{3} f^{a_{1} a_{2} a_{3}} \\left(\n\\bar A_{\\text{sQED}}^{\\mu_{1}\\mu_{2}} (k_{1},-k_{1}) -\\bar A_{\\text{sQED}}^{\\mu_{1}\\mu_{2}} (k_{2},-k_{2})\n\\right)\n\\end{equation}\nand can be\nstraightforwardly brought into the form given in Ref.\\cite{delaCruz:2020cpc}. The form of Eq.\\eqref{current-3pt} shows the direct connection between QED and QCD in the high temperature regime.\n\nWe have have checked that the WQFT  approach  reproduces the higher-point \nexamples considered in Refs.\\cite{delaCruz:2020cpc, delaCruz:2022nlj}\nincluding the  \ngravitational case. It is then safe to  conjecture that the remarkable simple expression\n\\begin{equation}\n k_{1}^{2}k_{2}^{2}\\cdots k_{n}^{2} \\langle\nA^{I_1}(k_{1})A^{I_{2}}(k_{2})\\cdots A^{I_{n}}(k_{n}) \\rangle_{\\text{WQFT}}\n\\end{equation}\nencodes  the resumed HTL expansion for any theory where scalars interact with other scalar,  gauge bosons or gravitons. \n\n\\section{Conclusions}\n\\label{conclusions}\nWe have exposed a relation between the  WQFT path integral and the classical limit of an off-shell current. The latter\nobtained through dressing up an off-shell current with first quantized coherent states and its classical limit obtained through a KMOC-like\nprocedure. Using a localized wave-packet to\ncompute classical  currents is equivalent to pick up the background expansion $x^{\\mu} = b^{\\mu }+p^{\\mu}\\tau +q^{\\mu}(\\tau)$ and setting $\\tau\\in\\, (-\\infty,\\infty)$ in the worldline action as we have shown in Sec.\\ref{off-shell-classical}.\nWe then constructed  the path integral representation of the Green functions associated with the off-shell current and found a link to the worldline theory in the classical limit, thus \nrecasting the current as an expectation value of soft fields inside a worldline path integral. \n\nWe have applied our methods\nin various settings. In QED we have tested the validity of our approach up to  7-point, while in gravity we have verified that the \n${\\cal N}=2$ susy model correctly reproduces the classical Compton amplitude up to quadrupole finding agreement with  Ref.\\cite{Saketh:2022wap}. For colored scalars and QCD, we have applied our methods to obtain HTLs, which are related to the classical limit of off-shell currents \\cite{delaCruz:2020cpc}. \nIt is straightforward to extend our approach to include more particles either massive or massless. For instance one could easily include photons using the dressed propagator of Ref.\\cite{Bastianelli:2021nbs}, while on the KMOC side massless particles  were studied in Ref.\\cite{Cristofoli:2021vyo}. \n \nOur work shows how to map computations of the classical\ncontributions of the off-shell  (amplitude-like) current and the\ncalculation based \non worldline path integration, thus providing a route to test future WQFT developments. \nFor instance a possible\napplication of our currents would be the case of  \nmassive higher-spin particles, where one  could compare the off-shell current in the classical limit and the output produced by WQFT. \n   \n\n\nOff-shell currents  are usually  computed recursively in high-energy-physics applications  so it would be interesting to  recursively compute currents WQFT too. Hints of their recursive structure have already appeared in Ref.\\cite{Jakobsen:2021zvh}. Although we have considered only tree-level currents our construction can be relaxed to include  loops, which can be used in the calculation of loop-level\ncorrections to the background metric in general relativity  \\cite{DOnofrio:2022cvn,Mougiakakos:2020laz,Jakobsen:2020ksu}. On the other hand, cuts that contribute classically \n  might arise from sewing  tree-level currents as it is usual in the generalized unitarity method \n  \\cite{Bern:1994cg, Bern:1994zx}. \n   \n\\addsec{Acknowledgements}\nFC would like to thank Canxin Shi for helpful discussions. \n LDLC acknowledges financial support from the Open Physics Hub at the Physics and Astronomy Department in Bologna. His work is also supported by \nthe European Research Council, under grant ERC-AdG-885414. \n Some of the \ncalculations in this paper were done with Feyncalc \\cite{Mertig:1990an, Shtabovenko:2016sxi, Shtabovenko:2020gxv}.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:intro}\n\n\\subsection{Motivation}\n\n\nOne of the first phase transition results  for random graphs  is the celebrated result of Erd\\H{o}s and R\\'enyi~\\cite{ErdosRenyi60} on the emergence of a \\emph{giant component} of linear\norder when the number of edges passes $\\frac{n}{2}$, or from the viewpoint that is now more common,\nwhen the edge probability passes $\\frac{1}{n}$. This result has since been strengthened and generalised\nin a number of directions. In particular in hypergraphs it has been extended to vertex-components\n(e.g.~\\cite{BCOK10,BCOK14,BollobasRiordan12,BollobasRiordan17,KaronskiLuczak02,Poole15,SPS85})\nas well as to high-order components (e.g.~\\cite{CFDGK20,CKK18,CKK19,CKP18}).\n\n\nThe $k$-core of a graph $G$, defined as the maximal subgraph of minimum degree at least~$k$, has been studied extensively in the literature (e.g.~\\cite{Cooper2004, JansonLuczak07, kimcore, LuczakCore1991, PittelSpencerWormald96}). \nIn random graphs, the $k$-core may be seen as a natural generalisation of the largest component: \nin the case $k=2$, whp\\footnote{short for \\emph{with high probability}, i.e.\\ with probability tending to one as the number of vertices $n\\to \\infty$.}\na linear-sized $2$-core emerges at the same time as the giant component, and indeed\nlies almost entirely within the giant component, while for $k\\ge 3$, whp the $k$-core\nis identical to the largest $k$-connected subgraph~\\cite{LuczakCore1991,Luczak92}. \nIn~\\cite{LuczakCore1991} {\\L}uczak estimated the order of (i.e.\\ the number of vertices in)\nthe $k$-core of $G(n,p)$ and the asymptotic probability that the $k$-core is $k$-connected.\n{\\L}uczak also showed in~\\cite{Luczak92} that in the random graph process, in which edges are added to an empty graph one by one in\na uniformly random order, whp\\ at the moment the $k$-core first becomes non-empty, its order\nis already linear in~$n$. \nA crucial milestone was achieved by Pittel, Spencer and Wormald~\\cite{PittelSpencerWormald96},\nwho for $k\\geq 3$ determined the threshold probability at which the non-empty $k$-core appears whp\\ and\ndetermined its asymptotic order and size (i.e.\\ number of edges).\nThis was strengthened by Janson and Luczak~\\cite{JansonLuczak08}, who proved a bivariate\ncentral limit theorem for the order and size of the $k$-core.\nCain and Wormald~\\cite{CainWormald06} determined the asymptotic distribution\nof vertex degrees within the $k$-core.\nFurther research has focussed for example on the robustness of the core against edge deletion~\\cite{Sato14}\nand how quickly the peeling process arrives at the core~\\cite{AchlioptasMolloy15,Gao18,GaoMolloy18,JMT14}.\nThere are many more results in the literature for cores in random graphs, see e.g. \\cite{Cooper2004, JansonLuczak07, kimcore}. \n\n\nPaths and cycles in random graphs have been investigated at least since 1979 by de la Vega~\\cite{delavega79} and somewhat later by Ajtai, Koml\\'os,\nand Szemer\\'edi~\\cite{AjtaiKomlosSzemeredi81}.\nRegarding the length of the longest path in the random graph $G(n,p)$,\na standard ``sprinkling'' argument (see Lemma~\\ref{lem:standardsprinkling} with $r=2$)\nshows that in the supercritical regime the length of the longest path and cycle are very similar.\nThus it follows from the results of {\\L}uczak~\\cite{Luczak91} on the length of the longest cycle\nthat for $\\varepsilon=\\varepsilon(n)=o(1)$ and $p=\\frac{1+\\varepsilon}{n}$ (i.e.\\ shortly after the phase transition),\nunder the assumption $\\varepsilon^5n\\to\\infty$ whp\\ the longest path has length $\\Theta(\\varepsilon^2 n)$, where explicit constants can be given.\nThe best-known upper bounds derive from a careful analysis of the $2$-core\nand the simple observation that any cycle must lie within the $2$-core.\n\n\n\n \nThere are many different ways of generalising the concept of a $k$-core to hypergraphs; some results for these cores can be found in e.g. Molloy~\\cite{Molloy2005} and Kim~\\cite{kimcore}. However,\nin the case $k=2$,\nall $k$-cores which have been studied so far do not fully capture the nice connection between the $2$-core and cycles in graphs.\nIn~\\cite{Molloy2005} Molloy determined the threshold for the appearance of a non-trivial $k$-core\n(in that paper defined as a subhypergraph where every vertex has degree at least $k$)\nin the $r$-uniform binomial random hypergraph $H^r(n,p)$ for all $r,k\\geq 2$ such that $r+k\\geq 5$.\nThe proof relied on a clever heuristic argument which was first introduced by Pittel, Spencer and Wormald in~\\cite{PittelSpencerWormald96} and has been adapted by many other authors, see e.g.~\\cite{JansonLuczak07, kimcore,Riordan08,skubch2015core}. It turns out that the proofs in~\\cite{Molloy2005} can be extended to a wide range of core-type structures.\nIn the case $k=2$, Dembo and Montanari~\\cite{DemboMontanari08} strengthened this by determining\nthe width of, and examining the behaviour within, the critical window.\n\n\nOne of the most natural  concepts of paths and cycles in hypergraphs\nis \\emph{loose paths} and \\emph{loose cycles} (see Definition~\\ref{def:loosecycle}).\nA special case of a recent result of Cooley, Garbe, Hng, Kang, Sanhueza-Matamala and Zalla~\\cite{cooley2020longest}\nshows that the length of the longest loose path in an $r$-uniform binomial random hypergraph undergoes\na phase transition from logarithmic length to linear, and they also determined the critical threshold,\nas well as proving upper and lower bounds on the length in the subcritical and supercritical ranges.\n\nInspired by the substantial body of research on loose cycles, in this paper we introduce the\n\\emph{loose core} (see Definition~\\ref{def:loosecore}), a structure which does indeed capture the connection between cores and cycles in hypergraphs.\nOur first main result concerns the degree distribution of vertices in the loose core (see Theorem~\\ref{thm:mainresultdegrees}). In fact we prove a stronger result regarding degree distributions of both vertices and edges (see Theorem~\\ref{thm:factor:reducedcoredegs}).\nAs a consequence we can deduce both the asymptotic numbers of vertices and edges in the loose core (see Theorem~\\ref{thm:mainresultorder}) and an improved upper bound on the length of the longest loose cycle in an $r$-uniform binomial random hypergraph  (see Theorem~\\ref{thm:mainresultcycle}). \n\nBefore stating our main results, in the next section we introduce some definitions and notations which we will use throughout the paper.\n\n\n\\subsection{Setup}\\label{def:basicdefinitions}\nGiven a natural number $r\\geq 3$,\nan \\emph{$r$-uniform hypergraph} consists of a vertex set $V$ and an edge set\n$E\\subset\\binom{V}{r}$, where $\\binom{V}{r}$ denotes the set of all $r$-element subsets of $V$. Let $H^r(n,p)$ denote the \\emph{$r$-uniform binomial random hypergraph}\non vertex set $[n]$ in which each set of $r$ distinct vertices forms an edge with probability $p$ independently. For any positive integer $k$ we write $[k]\\coloneqq\\{1,\\ldots,k\\}$ and $\\ [k]_0\\coloneqq\\{0,\\ldots,k\\}$.\nWe also include $0$ in the natural numbers, so we write $\\mathbb{N}=\\{0,1,\\ldots\\}$ and $\\mathbb{N}_{\\geq k}\\coloneqq\\{k,k+1,\\ldots\\}$.\nThroughout the paper, unless otherwise stated any asymptotics are taken as $n\\to\\infty$. \nIn particular, we use the standard\nLandau notations $o(\\cdot)$, $O(\\cdot),\\Theta(\\cdot),\\omega(\\cdot)$\nwith respect to these asymptotics.\n\nThe loose core will be defined in terms of two parameters, namely the standard notion of (vertex-)degree\nand a notion we call the connection number.\n\n\\begin{definition}\nLet $H$ be an $r$-uniform hypergraph. Let $d_H(v)$ be the \\emph{degree} of a vertex $v$ in $H$ (i.e.\\ the number of edges which contain it) and let $\\delta(H)$ denote the \\emph{minimum (vertex-)degree} of $H$, i.e.\\ the smallest degree of any vertex of $H$. For any edge $e\\in E(H)$, define the \\textit{connection number} $\\kappa(e)\\in[r]_0$ of $e$ as \\[\\kappa(e)=\\kappa_H(e)\\coloneqq|\\{v\\in e: d_H(v)\\geq 2\\}|\\] and let $\\kappa(H)\\coloneqq\\min\\limits_{e\\in E(H)}\\kappa(e)$.\n\\end{definition}\nWe are now ready to define the loose core.\n\\begin{definition}[Loose core]\\label{def:loosecore}\nThe \\textit{loose core} of an $r$-uniform hypergraph $H$ is the maximal subhypergraph $H'$ of $H$ such that \n\\begin{enumerate}[label=\\textnormal{\\textbf{(C\\arabic*)}}]\n    \\item\\label{item:loosecorefirstcond} $\\delta(H')\\geq 1$, \n    \\item\\label{item:loosecoresecondcond} $\\kappa(H')\\geq 2$.\n\\end{enumerate}\nIf such a subhypergraph does not exist, then we define the loose core to be the empty hypergraph (i.e.\\ the hypergraph with no vertices and no edges).\n\\end{definition}\nNote that the loose core is unique, since the union of two hypergraphs each with properties \\ref{item:loosecorefirstcond} and \\ref{item:loosecoresecondcond} again has these properties. The first condition in Definition~\\ref{def:loosecore} simply states that the loose core contains no isolated vertices and the second condition specifies how edges are connected to each other in the loose core. Note that for $r\\ge 3$ the loose core might contain vertices of degree $1$, in contrast to the graph case. For $r=2$, Definition~\\ref{def:loosecore} coincides with the $2$-core of a graph.\n\nOur motivation to study loose cores arises from the study of loose cycles in hypergraphs which are closely related to loose paths.\n\\begin{definition}[Loose path\/cycle]\\label{def:loosecycle}\nA \\textit{loose path of length $\\ell$} in an $r$-uniform hypergraph is a sequence of distinct vertices $v_1,\\ldots,v_{\\ell(r-1)+1}$ and a sequence of edges $e_1,\\ldots,e_{\\ell}$, where $e_i=\\{v_{(i-1)(r-1)+1},\\ldots,v_{(i-1)(r-1)+r}\\}$ for $i\\in[\\ell]$. A \\textit{loose cycle of length $\\ell$} in an $r$-uniform hypergraph is defined similarly except that $v_{\\ell(r-1)+1}=v_1$ (and otherwise all vertices are distinct).\n\\end{definition}\n\nNote that for $i \\in [\\ell-1]$ we have $e_i \\cap e_{i+1}=\\{v_{i(r-1)+1}\\}$ (and in the case of a loose cycle,\n$e_\\ell \\cap e_1 = \\{v_1\\}$), so in particular two consecutive edges\nintersect in precisely one vertex.\nObserve that a loose cycle satisfies conditions~\\ref{item:loosecorefirstcond} and~\\ref{item:loosecoresecondcond} of a loose core (Definition~\\ref{def:loosecore}) and hence it must be contained in the maximal subhypergraph with these properties,\ni.e.\\ in the loose core.\n\nWe will now define various parameters which will occur often in this paper. Some of these\ndefinitions may seem arbitrary and unmotivated initially, but their meaning will become clearer over the course of the paper.\nGiven $d>0$, consider a sequence $(d_n)_{n \\in \\mathbb{N}}$ of real numbers such that $d_n\\to d$.\nThen for\n$r\\in\\mathbb{N}_{\\geq 3}$ and $n\\in\\mathbb{N}$, set\n\\[\np=p(r,n)\\coloneqq\\frac{d_n}{\\binom{n-1}{r-1}}, \\qquad d^*=d^*(r)\\coloneqq\\frac{1}{r-1}.\n\\]\nIn addition we define a function $F:[0,\\infty) \\to\\mathbb{R}$ by setting\n\\begin{equation*}\\label{eq:gfunctiondefinition}\n    F(x)=F_{r,d}(x)\\coloneqq\\exp{\\left(-d\\left(1-x^{r-1}\\right)\\right)}\n\\end{equation*}\nand let $\\rho_*=\\rho_*(r,d)$ be the largest solution\\footnote{$\\rho_*$ is well-defined since $0$ is certainly a solution and the set of solutions is closed by continuity.} of the fixed-point equation\n\\begin{equation}\\label{eq:fixedpointequation}\n    1-\\rho=F(1-\\rho).  \n\\end{equation}\nSince the function $F$ is dependent on $d$, so too are the solutions to this equation.\nIt turns out that $d^*$ is a threshold at which the solution set changes its behaviour from only containing the trivial solution $0$ to containing a unique positive solution (see Claim~\\ref{claim:behaviouroffixedpointsol}). \n\n\nWe define\n\\begin{equation}\\label{eq:relatingtherhos}\n\\hat \\rho_*=\\hat \\rho_*(r,d)\\coloneqq1-(1-\\rho_*)^{r-1}\n\\end{equation}\nand\n\\begin{equation}\\label{eq:eta}\n\\eta=\\eta(r,d)\\coloneqq1-\\frac{(r-1)\\rho_*(1-\\rho_*)^{r-2}}{\\hat \\rho_*}.\n\\end{equation}\n Furthermore let \n\\begin{equation*}\\label{eq:alphadefinition}\n    \\alpha=\\alpha(r,d)\\coloneqq\\rho_*\\left(1-d(r-1)(1-\\rho_*)^{r-1}\\right),\n\\end{equation*}\n\\begin{equation}\\label{eq:definitionofeta}\n\\beta=\\beta(r,d)\\coloneqq\\frac{d}{r}\\left(1-(1-\\rho_*)^ r-r\\rho_*(1-\\rho_*)^ {r-1}\\right),\n\\end{equation}\nand\n\\begin{equation}\\label{eq:definitionofgamma}\n    \\gamma=\\gamma(r,d)\\coloneqq1-\\exp(-d\\hat \\rho_*)-d\\hat \\rho_* \\exp(-d\\hat \\rho_*).\n\\end{equation}\n\n\n\\subsection{Main results: loose cores and cycles in hypergraphs}\\label{sec:mainresults}\n\n\nFor any $j\\in\\mathbb{N}_{\\ge 1}$, let $\\numvsdeg{j}{C_H}$ be the number of vertices of $H= H^r(n,p)$\nwith degree~$j$ in the loose core $C_H$ of $H$\nand let \n\\[\\mu_j\\coloneqq\\numvsdeg{j}{C_H}\\cdot n^{-1}.\\]\nLet $\\numvs{C_H} = \\sum_{j \\ge 1}\\numvsdeg{j}{C_H}$ denote the number of vertices and $\\numedg{C_H}$ the number of edges in the loose core  $C_H$ in $H$.\nWe also define $\\numvsdeg{0}{C_H}$ to be the number of vertices of $H$ which are not in the loose core of $H$ (so $v_0(C_H)=n-v(C_H)$),\nand $\\mu_0\\coloneqq \\numvsdeg{0}{C_H}\\cdot n^{-1}$. (Observe that this notation is consistent if, with a slight abuse of terminology,\nwe view vertices which are not in the loose core as having degree $0$ in the loose core.) We interpret a $\\mathrm{Po}(0)$ variable as being deterministically $0$. \n\nOur first main result describes the asymptotic degree distribution of vertices in the loose core $C_H$  of $H= H^r(n,p)$.\n\\begin{theorem}\\label{thm:mainresultdegrees}\nLet $r,d,p,\\hat \\rho_*$ and $\\eta$  be as in Section~\\ref{def:basicdefinitions} and  let $H=H^r(n,p)$. Let $Y$ be a random variable with distribution $\\mathrm{Po}(d\\hat \\rho_*)$ and define\n\\[\n    Z \\coloneqq \\bigg\\{\\begin{array}{lr}\n        Y & \\text{if } Y\\neq 1,\\\\\n        \\mathrm{Ber}\\left(\\eta\\right) & \\text{if } Y=1.\n        \\end{array}\\] \nThen there exists $\\varepsilon = \\varepsilon(n) = o(1)$ such that whp\\ for any constant $j\\in\\mathbb{N}$ we have\n\\[\\mu_j=\\prob(Z=j)\\pm \\varepsilon.\\]\n\\end{theorem}\nOur second main result describes the asymptotic numbers of vertices and edges in the loose core  $C_H$  of $H= H^r(n,p)$.\n\n\\begin{theorem}\\label{thm:mainresultorder}\nLet $r,p, \\alpha$ and $\\beta$  be as in Section~\\ref{def:basicdefinitions} and  let $H=H^r(n,p)$.\nThen whp\n\t\t\\[\\numvs{C_H}=(\\alpha+o(1)) n\\]\n\t\tand\n\t\t\\[\\numedg{C_H}=(\\beta+o(1))n.\\]\n\\end{theorem}\n\nBy analysing the loose core we obtain an upper bound on the length of the longest loose cycle in $H^r(n,p)$.\n\n\\begin{theorem}\\label{thm:mainresultcycle} \nLet $r,p, \\beta$ and $\\gamma$  be as in Section~\\ref{def:basicdefinitions} and  let $H=H^r(n,p)$.\nLet $L_C$ be the length of the longest loose cycle in $H$. Then whp\n\\[L_C\\leq \\left(\\min\\{\\beta,\\gamma\\}+o(1)\\right)\\cdot n.\\]\n\\end{theorem}\n\nIn fact, a standard ``sprinkling'' argument shows that whp the longest loose path in $H^r(n,p)$ is not significantly longer than the longest loose cycle and therefore we obtain the following corollary.\n\n\\begin{corollary}\\label{cor:upperboundlongestpath} \nLet $r,p, \\beta$ and $\\gamma$  be as in Section~\\ref{def:basicdefinitions} and  let $H=H^r(n,p)$. \nLet $L_P$ be the length of the longest loose path in $H$. Then whp\n\\[L_P\\leq \\left(\\min\\{\\beta,\\gamma\\}+o(1)\\right)\\cdot n.\\]\n\\end{corollary}\n\nAs mentioned previously, Claim~\\ref{claim:behaviouroffixedpointsol} will state that $d^*=\\frac{1}{r-1}$ is a threshold at which the solution set of~\\eqref{eq:fixedpointequation} changes its behaviour from only containing $0$ to containing a unique positive solution. Together with Theorem~\\ref{thm:mainresultcycle} and Corollary~\\ref{cor:upperboundlongestpath} (and recalling the definitions of $\\beta,\\gamma$\nin~\\eqref{eq:definitionofeta} and~\\eqref{eq:definitionofgamma}) this implies that $d^*=\\frac{1}{r-1}$ is a threshold for the existence of a loose path\/cycle of linear order,\nand it is interesting in particular to examine the behaviour shortly after the phase transition.\n\\begin{corollary}\\label{cor:shortlyafterphasetransition}\nLet $r\\in\\mathbb{N}_{\\geq 3}$, let $\\varepsilon>0$ be constant and let $p=\\frac{1+\\varepsilon}{(r-1)\\binom{n-1}{r-1}}$. Let $L_C$ and $L_P$ be the length of the longest loose cycle and  the longest loose  path in $H^r(n,p)$.  Then whp\n\\[L_C\\leq L_P+1\\leq \\left(\\frac{2\\varepsilon^2}{(r-1)^2}+O(\\varepsilon^3)\\right)\\cdot n.\\]\n\\end{corollary}\n\nIn other words, we have an upper bound on $L_C$ and $L_P$ in the barely supercritical regime.\nFor a corresponding lower bound, we will quote (a special case of) a result from~\\cite{cooley2020longest}\n(which we later state formally as Theorem~\\ref{thm:pathsresult}) which gives a lower bound on $L_P$.\nBy applying the sprinkling argument again we also obtain a lower bound on $L_C$,\nand together with Corollary~\\ref{cor:shortlyafterphasetransition} we obtain the following.\n\n\n\\begin{theorem}\\label{thm:bestknownresultcycles}\nLet $r\\in\\mathbb{N}_{\\geq 3}$, let $\\varepsilon>0$ be constant and let $p=\\frac{1+\\varepsilon}{(r-1)\\binom{n-1}{r-1}}$.  Let $L_C$ and $L_P$ be the length of the longest loose cycle and  the longest loose  path in $H^r(n,p)$.   Then whp\\ \n\\[\n\\left(\\frac{\\varepsilon^2 }{4(r-1)^2}+O(\\varepsilon^3)\\right)\\cdot n\\leq L_C\\leq L_P+1\\leq\\left(\\frac{2\\varepsilon^2}{(r-1)^2}+O(\\varepsilon^3)\\right)\\cdot n .\n\\]\n\\end{theorem}\nTheorem~\\ref{thm:bestknownresultcycles} provides the best known upper and lower bounds on $L_P,L_C$ in the regime when\n$p= \\frac{1+\\varepsilon}{(r-1)\\binom{n-1}{r-1}}$, \nbut there is a multiplicative factor of~$8$ between these two bounds and the correct asymptotic value is still unknown.\nIndeed it is not even clear that the random variables $L_P,L_C$ are concentrated around a single value.\n\nWe note that even for cycles in  $G(n,p)$ in the barely supercritical regime the correct asymptotic value is not known\ndespite considerable efforts in this direction. In particular, the best known bounds\nwhen $0<\\varepsilon=\\varepsilon(n)=o(1)$ satisfies $\\varepsilon^3n\\to\\infty$ and $p=\\frac{1+\\varepsilon}{n}$\nare due to {\\L}uczak~\\cite{Luczak91} (lower bound) and Kemkes and Wormald~\\cite{KemkesWormald13} (upper bound),\nand state that whp\\ the length $L_C$ of the longest cycle satisfies\n\\[\n\\left(\\frac{4}{3}+o(1)\\right)\\varepsilon^2n\\leq L_C\\leq(1.7395+o(1))\\varepsilon^2n.\n\\]\n\nThe proofs of all the results of this section appear in Section~\\ref{sec:proofofmainresults} as a consequence of a single, more general result (Theorem~\\ref{thm:factor:reducedcoredegs}).\n\n\n\n\n\n\n\\subsection{Key proof techniques}\nIn order to prove our main results, we transfer the problem from $H^r(n,p)$ to the \\emph{factor graph} $G\\coloneqq G(H^r(n,p))$\nwhich will be formally defined in Section~\\ref{sec:factorgraphs}. In the factor graph we define the \\emph{reduced core} $R_G$,\nwhich is closely related to the $2$-core of $G$ and from which we can reconstruct the loose core of $H^r(n,p)$, but which is easier to analyse. We use a peeling process (Definition~\\ref{def:peeling}) and an auxiliary algorithm called $\\coredetector $ to determine the asymptotic proportion of variable and factor nodes of $G$ with fixed degree in the reduced core (Theorem~\\ref{thm:factor:reducedcoredegs}). We also need martingale techniques, in particular an Azuma-Hoeffding inequality and an associated vertex-exposure martingale to show concentration of the numbers of vertices and edges of fixed degree around the respective expectations. \n\\subsection{Paper overview}\nThe rest of the paper is structured as follows. \n\nIn Section~\\ref{sec:prelim} we set basic notation and state some standard probabilistic lemmas which we will use later.\nIn Section~\\ref{sec:factorgraphs} we switch our focus to factor graphs,\ndefine the reduced core and state Theorem~\\ref{thm:factor:reducedcoredegs}\nwhich describes degree distributions in the reduced core and which\nimplies all of our main results, as we prove in Section~\\ref{sec:proofofmainresults}.\n\nSubsequently, Section~\\ref{sec:peeling} describes a standard peeling process to obtain the reduced core and contains two main lemmas\nwhich together imply Theorem~\\ref{thm:factor:reducedcoredegs}.\nThe first of these (Lemma~\\ref{lem:mainlemma1}) describes the degree distribution after a sufficiently large number of steps of the peeling process,\nand will be proved in Section~\\ref{sec:algorithm}.\nThe second main lemma (Lemma~\\ref{lem:mainlemma2}) states that subsequently, very few further vertices will be deleted in the remainder of the peeling process,\nand therefore this degree distribution is also a good approximation for the degree distribution in the reduced core.\nLemma~\\ref{lem:mainlemma2} will be proved in Section~\\ref{sec:prooflowerbound}.\n\nIn Section~\\ref{sec:concluding}, we conclude with some discussion and open questions.\nWe omit from the main body of the paper many proofs which simply involve technical calculations\nor standard applications of common methods,\nbut include them in the appendices for completeness.\nAppendix~\\ref{app:analysisfixedpoint} contains an analysis of the fixed-point equation~\\eqref{eq:fixedpointequation},\nwhile Appendix~\\ref{app:problemmas} contains the proofs of some basic probabilistic lemmas which are needed throughout the paper.\nFinally, Appendix~\\ref{app:eventE} and Appendix~\\ref{sec:proofofuniformity} constitute the proofs of Lemma~\\ref{lem:eventE}\nand Lemma~\\ref{claim:uniformity}, respectively.\n\n\\section{Preliminaries and Notation}\\label{sec:prelim}\n\nFor the rest of the paper, $r\\in\\mathbb{N}_{\\geq 3}$ and $d>0$ will be fixed.\nIn particular, we consider these to be constant, so if we say, for example,\nthat $x = O(n)$, we mean that there exists a constant $C=C(r,d)$\nsuch that $x \\le Cn$.\nBy the notation $x=a\\pm b$ we mean that $a-b\\leq x\\leq a+b$.\nSimilarly, the notation $x = (a\\pm b)c$ means that $(a-b)c \\le x \\le (a+b)c$.\nWe will omit floors and ceilings whenever these do not significantly affect the argument. \\\\\n\nAs mentioned in Section~\\ref{def:basicdefinitions}, the solution set of the fixed-point equation~\\eqref{eq:fixedpointequation}\nchanges its behaviour at $d=d^*$. More precisely we have the following. \n\n\\begin{claim}\\label{claim:behaviouroffixedpointsol}\\mbox{ }\n\\begin{enumerate}[label=\\textnormal{\\textbf{(F\\arabic*)}}]\n\\item If $d<d^*$, then $\\rho_*=0$.\n\\item If $d>d^*$, then there is a unique positive solution to~\\eqref{eq:fixedpointequation}.\n\\end{enumerate}\n\\end{claim}\nWe defer the (rather technical) proof of this claim to Appendix~\\ref{app:analysisfixedpoint}.\n\nFurthermore we will often use the following alternative relation between $\\rho_*$ and $\\hat \\rho_*$.\n\\begin{equation}\\label{eq:relatingrhoswithexpfunction}\n1-\\rho_*\\numeq{\\eqref{eq:fixedpointequation}}F(1-\\rho_*)=\\exp\\left(-d\\left(1-(1-\\rho_*)^ {r-1}\\right)\\right)\\numeq{\\eqref{eq:relatingtherhos}}\\exp(-d\\hat \\rho_*).\n\\end{equation}\n\n\\subsection{Large deviation bounds}\n\nIn this section, we collect some standard large deviation results which will be needed later. We will use the following Chernoff bound\n(see e.g.~\\cite[Theorem 2.1]{JansonLuczakRucinskiBook}).\n\\begin{lemma}[Chernoff]\\label{lem:chernoffbounds}\n\tIf $X\\sim \\mathrm{Bi}(N,p)$, then for any $s> 0$\n\t\\begin{equation*} \\label{eqn:chernoffstd}\n\t\\mathbb{P}(|X-Np|\\geq s)\\leq 2\\cdot\\exp\\left(-\\frac{s^2}{2\\left(Np+\\frac{s}{3}\\right)}\\right).\n\t\\end{equation*}\n\t\n\\end{lemma}\nThis bound is less precise than the form in~\\cite{JansonLuczakRucinskiBook} since we have combined the upper and lower tail bounds for simplicity.\nWe will also need an Azuma-Hoeffding inequality---the following lemma is taken from~\\cite{JansonLuczakRucinskiBook}, Theorem $2.25$.\n\\begin{lemma}[Azuma-Hoeffding]\\label{lem:Azuma}\nLet $X$ be a real-valued random variable.\nIf $(X_i)_{1\\leq i\\leq n}$ is a martingale with $X_n=X$ and $X_0=\\expec(X)$,\nand for every $1\\leq i\\leq n$ there exists a constant $c_i>0$ such that\n\\[|X_i-X_{i-1}|\\leq c_i\\]then, for any $s>0$,\n\\begin{align}\n    \\prob(|X-\\expec(X)|\\geq s) & \\leq 2\\cdot\\exp\\left(-\\frac{s^2}{2\\sum_{i=1}^nc_i^2}\\right).\\nonumber\n\\end{align}\n\\end{lemma}\nAgain, the bound in~\\cite{JansonLuczakRucinskiBook} is a bit stronger than what we state here since for simplicity we have combined upper and lower tail bounds.\n\n\\section{Factor graphs}\\label{sec:factorgraphs}\n\nThere is a natural representation of a hypergraph as a bipartite graph known as a \\emph{factor graph},\nwhich is a well-known concept in literature (see e.g.~\\cite{MezardMontanariBook}).\n\n\\begin{definition}[Factor graph]\nGiven a hypergraph $H$, the \\emph{factor graph} $G=G(H)$ of $H$ is a bipartite graph on vertex classes $\\mathcal{V}\\coloneqq V(H)$ and $\\mathcal{F}\\coloneqq E(H)$,\nwhere $v \\in \\mathcal{V}$ and $a \\in \\mathcal{F}$ are joined by an edge in $G$ if and only if $v \\in a$. In other words, the vertices of $G$ are the vertices and edges of $H$,\nand the edges of $G$ represent incidences.\n\nTo avoid confusion, we refer to the vertices of a factor graph as \\emph{nodes}. In particular, the nodes in $\\mathcal{V}$ are called \\emph{variable nodes}\nand the nodes in $\\mathcal{F}$ are called \\emph{factor nodes}.\\footnote{In some contexts in the literature,\nfactor nodes may be called \\emph{functional nodes} or \\emph{constraint nodes}.} We define\n$$\nG^r(n,p)\\coloneqq G(H^r(n,p)),\n$$\ni.e.\\ the factor graph of the $r$-uniform binomial random hypergraph $H^r(n,p)$.\n\\end{definition}\n\nNote that if $H$ is an $r$-uniform hypergraph, then the factor nodes of $G(H)$ all have degree~$r$.\nWe will need the following basic fact about the number of factor nodes in $G^r(n,p)$. The proof is a simple application of a Chernoff bound,\nand appears in Appendix~\\ref{app:numberfactornodes} for completeness.\n\n\\begin{proposition}\\label{prop:numberfactornodes}\nLet $d>0$ be a constant and let $p=\\frac{(1+o(1))d}{\\binom{n}{r-1}}$. Then there exists a function $\\omega_0=\\omega_0(n)$ with\n$\\omega_0\\xrightarrow{n\\to\\infty}\\infty$ such that whp\\ the number $m$ of factor nodes\nin $G^r(n,p)$ satisfies\n\\[\nm=\\left(1\\pm \\frac{1}{\\omega_0}\\right)\\frac{dn}{r}.\n\\]\n\\end{proposition}\n\n\n\n\nIt will be more convenient to study the factor graph than the original hypergraph---in order to do this,\nwe need to understand what the structure corresponding to the loose core looks like in the factor graph.\nWe first define the loose core\nof a factor graph and subsequently observe that it does indeed correspond to the loose core of the hypergraph (Definition~\\ref{def:loosecore}).\n\n\n\\begin{definition}[Loose core]\\label{def:loosecorefactorgraph}\nThe \\emph{loose core} $C=C_G$ of a factor graph $G$ is the maximal subgraph of $G$ such that each factor node of $C$\nhas degree $r$ in $C$ and furthermore:\n\\begin{enumerate}[label=\\textnormal{\\textbf{(C\\arabic*')}}]\n\\item\\label{item:isolatednodesfactorgraph} $C$ contains no isolated variable nodes;\n\\item\\label{item:degreeconditionfactorgraphs} Each factor node in $C$ is adjacent to at least two variable nodes of degree at least two in $C$.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{proposition}\\label{prop:loosecorecorrespondence}\nGiven an $r$-uniform hypergraph $H$, the loose core $C_G$ of the factor graph $G=G(H)$ of $H$ is\nidentical to the factor graph of the loose core $C_H$ of $H$.\n\\end{proposition}\n\n\\begin{proof}\nThe condition that each factor node of $C=C_G$ has degree $r$ in $C$\nmeans that $C$ corresponds to a subhypergraph of $H$ (i.e.\\ no edge of\n$H$ has a vertex removed from it without itself being removed).\nSince variable nodes of $G$ correspond to vertices of $H$, condition \\ref{item:isolatednodesfactorgraph} in Definition~\\ref{def:loosecorefactorgraph}\ncorresponds precisely to \\ref{item:loosecorefirstcond} in Definition~\\ref{def:loosecore}.\nFurthermore, condition~\\ref{item:degreeconditionfactorgraphs} in Definition~\\ref{def:loosecorefactorgraph} is directly analogous\nto condition~\\ref{item:loosecoresecondcond} in Definition~\\ref{def:loosecore}.\n\\end{proof}\n\n\nIn view of Proposition~\\ref{prop:loosecorecorrespondence},\nrather than studying the loose core of the hypergraph, we can study the loose core of the corresponding factor graph instead.\nIn fact, even more convenient than this is a slightly different structure.\n\n\\begin{definition}[Reduced core]\\label{def:reducedcore}\nThe \\emph{reduced core} $R=R_G$ of a factor graph $G$ is the maximal subgraph of $G$\nwith no nodes of degree~$1$.\n\\end{definition}\n\nNote that the reduced core is very similar to the $2$-core of $G$---the only difference\nis that we do not delete isolated nodes, so all original nodes are still present. This will be convenient\nsince it means that all nodes have a well-defined degree within the reduced core\n(and have degree zero if and only if they are not in the $2$-core of $G$).\nSimilarly we will want to describe degree distributions within the loose core,\nbut also incorporating nodes which are in fact not contained in the loose core.\nTo avoid confusion and abuse of terminology, we define the \\emph{padded core}.\n\n\\begin{definition}[Padded core]\\label{def:paddedcore}\nThe \\emph{padded core} $P=P_G$ of a factor graph $G$ is the subgraph of $G$ whose\nnodes are the nodes of $G$ and whose edges are the edges of $C_G$.\n\\end{definition}\nIn other words, the padded core\\ $P_G$ is identical to the loose core $C_G$ except that\nall nodes of $G$ are still present. Equivalently, $P_G$ is the maximal subgraph of $G$\nin which each non-isolated factor node has degree~$r$ and is adjacent to at least two\nvariable nodes of degree at least~$2$. The following observation motivates both our definition\nof the padded core\\ and the interpretation of $\\mu_0$ as the proportion of vertices\nof $H^r(n,p)$ which do not lie in the loose core.\n\n\\begin{remark}\\label{rem:paddedcoredegs}\nFor each $j\\in \\mathbb{N}$,\nthe proportion of variable nodes of $G=G^r(n,p)$ which have degree $j$ in the padded core\\ $P_G$ is $\\mu_j$.\n\\end{remark}\n\n\nIt is important to observe that, if we have found the reduced core, it is very easy to reconstruct the padded core,\nand hence also the loose core. \nLet $\\mathcal{F}_R$ be the set of non-isolated factor nodes of the reduced core $R_G$ and let $P^{'}_G$\nbe the factor graph whose nodes are the nodes of $G$ and whose edges are all edges of $G$ incident to $\\mathcal{F}_R$.\nIn other words, $P^{'}_G$ is the factor graph obtained from $R_G$ by adding back in all\nedges of $G$ attached to non-isolated factor nodes of $R_G$.\n\n\\begin{proposition}\\label{prop:reducedtoloose}\n$P^{'}_G = P_G$.\n\\end{proposition}\n\n\\begin{proof}\nLet $R_1$ denote the graph obtained from the padded core $P_G$ of $G$ by removing\nall edges incident to leaves (which must be variable nodes).\nNote that, since any non-isolated factor node in $P_G$ has at least two neighbours of\ndegree at least two, the same is still true in $R_1$.\nFor the sake of intuitive notation, we also denote\n$$\nR_2 \\coloneqq R_G, \\qquad\nP_1 \\coloneqq P_G, \\qquad P_2 \\coloneqq P^{'}_G.\n$$\nOur goal is to show that $P_1=P_2$.\n\nLet us observe that  $P_1$ can be obtained from $R_1$ by the same operation\nwith which $P_2$ is obtained from $R_2$, namely by adding in\nedges of $G$ incident to non-isolated factor nodes.\n\nWe next observe that $R_1$ is a subgraph of $G$ with no nodes of degree $1$,\nand therefore $R_1 \\subseteq R_2=R_G$, by the maximality of $R_G$.\nSince the operation constructing $P_i$ from $R_i$ is inclusion-preserving, $R_1 \\subseteq R_2$ implies that $P_1 \\subseteq P_2$. \n\nIt therefore remains to prove that $P_2 \\subseteq P_1$. \nTo this end, we observe that certainly $P_2=P^{'}_G$ is a subgraph of $G$ in which each non-isolated factor node is\nin the $2$-core of $G$, and therefore\nadjacent to at least two variable nodes of degree at least two. Furthermore each non-isolated\nfactor node of $P_2$ has degree $r$ in $C_2$, and since\n$P_1=P_G$ is the maximal subgraph with these two properties, we have $P_2 \\subseteq P_1$, as required.\n\\end{proof}\n\n\nLet us observe one further fact about the transformation from the reduced core $R_G$ to the padded core $P_G=P^{'}_G$:\nalthough this seemed to be dependent on the initial factor graph $G$, in fact the operation\nsimply involves connecting non-isolated factor nodes of $R_G$ to (distinct) isolated variable nodes until each factor node has degree precisely~$r$.\nThis means that given $R_G$, by Proposition~\\ref{prop:reducedtoloose} we can describe $P_G$ (and therefore also the loose core $C_G$)\nentirely, up to the assignment of which\nnodes are leaves. In other words, $R_G$ already contains all of the ``essential'' information\nof both $P_G$ and $C_G$. It will therefore be enough to study $R_G$ rather than $P_G$ or $C_G$, and this turns out to be simpler.\n\n\nNow the main results of this paper are implied by the following theorem about the reduced core $R_G$\nof the factor graph $G= G^r(n,p)$ of the $r$-uniform binomial random hypergraph.\n\n\nFor a non-negative real number $\\lambda$,\nlet us denote by $\\widetilde \\mathrm{Po}(\\lambda)$ the distribution of a random variable $X$ satisfying\n$$\n\\Pr(X=j) = \\begin{cases}\n\\Pr(\\mathrm{Po}(\\lambda)\\le 1) & \\mbox{if }j=0, \\\\\n0 & \\mbox{if }j=1,\\\\\n\\Pr(\\mathrm{Po}(\\lambda)=j) & \\mbox{if } j \\ge 2.\n\\end{cases}\n$$\nIn other words, the $\\widetilde\\mathrm{Po}$ distribution is identical to the $\\mathrm{Po}$ distribution except\nthat values of $1$ are replaced by $0$. We define the $\\widetilde\\mathrm{Bi}$ distribution analogously.\n\n\\begin{theorem}\\label{thm:factor:reducedcoredegs}\nLet $r,d,p,\\rho_*,\\hat \\rho_*$ be as in Section~\\ref{def:basicdefinitions} and let $G= G^r(n,p)$, i.e.\\ the factor graph of $H^r(n,p)$.\nFor each $j\\in\\mathbb{N}$, let $\\varprop{j}$ and $\\factprop{j}$ be the proportion\nof variable nodes and factor nodes of $G$ respectively which have degree $j$ in the reduced core $R_G$ of $G$.\nThen there exists a function $\\eps = \\eps(n) =o(1)$\nsuch that\nwhp\\ for any constant $j \\in \\mathbb{N}$ we have\n$$\n\\varprop{j} = \\Pr(\\widetilde\\mathrm{Po}(d\\hat \\rho_*)=j)\\pm\\eps\n$$\nand\n$$\n\\factprop{j} = \\Pr(\\widetilde\\mathrm{Bi}(r,\\rho_*)=j)\\pm\\eps.\n$$\n\\end{theorem}\nIn other words, in terms of their degrees in $R_G$, variable nodes and factor nodes\nhave degree distributions which are asymptotically those of a $\\widetilde \\mathrm{Po}(d \\hat \\rho_*)$\nand a $\\widetilde \\mathrm{Bi}(r,\\rho_*)$ distribution respectively.\n\nThe proof of this theorem will form the main body of the paper. In Section~\\ref{sec:peeling} we will prove how Theorem~\\ref{thm:factor:reducedcoredegs} follows from two auxiliary statements,\nstating that for some large integer $\\ell$\nthe proportions of variable and factor nodes of degree $j$ in the graph obtained after $\\ell$ rounds of a peeling process\nare approximately the values given in Theorem~\\ref{thm:factor:reducedcoredegs}\n(Lemma~\\ref{lem:mainlemma1}),\nand furthermore not many nodes are deleted after round $\\ell$ (Lemma~\\ref{lem:mainlemma2}). \n\n\\section{Back to hypergraphs: Proofs of main results}\\label{sec:proofofmainresults}\n\\noindent We now show how all of the results of Section~\\ref{sec:mainresults} follow from Theorem~\\ref{thm:factor:reducedcoredegs}. First we deduce our result on the asymptotic degree distribution of vertices in the loose core of $H^r(n,p)$.\n\\begin{proof}[Proof of Theorem~\\ref{thm:mainresultdegrees}]\nWe will apply Theorem~\\ref{thm:factor:reducedcoredegs} to provide us with a function $\\varepsilon$,\nand we will prove Theorem~\\ref{thm:mainresultdegrees} with $\\varepsilon'\\coloneqq\\sqrt{\\varepsilon} + \\frac{1}{\\sqrt{\\omega_0}}$, where\n$\\omega_0=\\omega_0(n)$ is the function given by \nProposition~\\ref{prop:numberfactornodes}.\n\nFor convenience, for any $j\\in\\mathbb{N}$, let us define\n$$\n\\mu_j' \\coloneqq \\begin{cases}\n\\Pr(\\mathrm{Po}(d\\hat \\rho_*)=j) & \\mbox{if } j \\ge 2;\\\\\n\\eta \\cdot \\Pr(\\mathrm{Po}(d\\hat \\rho_*)=j) & \\mbox{if } j=1;\\\\\n\\Pr(\\mathrm{Po}(d\\hat \\rho_*)=0) + (1-\\eta)\\cdot \\Pr(\\mathrm{Po}(d\\hat \\rho_*)=1) & \\mbox{if } j=0.\n\\end{cases}\n$$\nIn other words, $\\mu_j'$ is the ``idealised version'' of $\\mu_j$,\nand our goal is simply to prove that whp, for each $j\\in\\mathbb{N}$ we have $\\mu_j = \\mu_j' \\pm \\varepsilon$.\nSimilarly we also define\n$$\n\\varpropp{j} \\coloneqq \\Pr(\\widetilde{\\mathrm{Po}}(d\\hat \\rho_*)=j),\n$$\nso by Theorem~\\ref{thm:factor:reducedcoredegs} we have $\\varprop{j} = \\varpropp{j} \\pm \\varepsilon$\nwhp for each $j\\in\\mathbb{N}$. The proof of Theorem~\\ref{thm:mainresultdegrees}\nnow simply consists of relating the $\\mu_j$ to the $\\varprop{j}$,\nrelating the $\\mu_j'$ to the $\\varpropp{j}$ and applying Theorem~\\ref{thm:factor:reducedcoredegs}.\nNote that it follows instantly from the definitions\nthat $\\mu_j' = \\varpropp{j}$ for $j\\in\\mathbb{N}_{\\geq 2}$. We will split the proof into three cases.\n\n\\vspace{0.2cm}\n\\noindent \\textbf{Case 1: $j\\ge 2$.}\\\\\nWe start by showing that $\\mu_j=\\varprop{j}$. Observe that by Remark~\\ref{rem:paddedcoredegs},\n$\\mu_j$ is simply\nthe proportion of variable nodes with degree $j$ in the padded core\\ $P_G$ of $G=G^r(n,p)$.\nTheorem~\\ref{thm:factor:reducedcoredegs} tells us the degrees of variable and factor\nnodes in the reduced core $R_G$ of $G$. By Proposition~\\ref{prop:reducedtoloose},\nmoving from $R_G$ to $P_G$ means that \nwe connect all non-isolated factor nodes of $R_G$ to their original neighbours in $G$,\nand any variable nodes which receive additional incident edges in this process have their degrees changed from~$0$ to~$1$.\nIt follows that for $j\\ge 2$, the proportion $\\mu_j$\nof variable nodes in $G$ with degree $j$ in the padded core\\ $P_G$ is precisely equal to $\\varprop{j}$,\nthe proportion of variable nodes in $G$ with degree $j$ in the reduced core $R_G$.\nTherefore \n\\[\n\\mu_j=\\varprop{j}\\numeq{\\text{Th.}~\\ref{thm:factor:reducedcoredegs}}\\varpropp{j} \\pm \\varepsilon=\\mu_j'\\pm \\varepsilon,\n\\]\nand the statement of Theorem~\\ref{thm:mainresultdegrees} is certainly true for $j\\ge 2$ (indeed, we have proved something stronger since $\\varepsilon<\\varepsilon'$).\n\n\\vspace{0.2cm}\n\\noindent \\textbf{Case 2: $j=1$.}\\\\\nTo prove the case $j=1$, we need to check how many isolated variable nodes become leaves when moving from $R_G$ to $P_G$.\nSince by Proposition~\\ref{prop:reducedtoloose} every factor node of $R_G$ with degree $j\\ge 2$ has $r-j$ leaves connected to it, and since whp\nthe number $m$ of factor nodes in total is $m=\\left(1\\pm\\frac{1}{\\omega_0}\\right)\\frac{dn}{r}$ for some growing function\n$\\omega_0\\xrightarrow{n\\to\\infty}\\infty$ by Proposition~\\ref{prop:numberfactornodes}, whp\nthe number of leaves added, which is simply $\\mu_1 n$, satisfies\n\\begin{align}\\label{eq:leaves1}\n\\mu_1 n=\\sum_{j=2}^r (r-j) \\factprop{j}  m\n& = \\left(1\\pm\\frac{1}{\\omega_0}\\right)\\frac{dn}{r} \\sum_{j=2}^r (r-j)\\left(\\Pr(\\widetilde\\mathrm{Bi}(r,\\rho_*)=j)\\pm \\varepsilon\\right)\\nonumber \\\\\n& = \\frac{dn}{r} \\sum_{j=2}^r (r-j)\\Pr(\\widetilde\\mathrm{Bi}(r,\\rho_*)=j) \\pm \\frac{\\varepsilon'n}{2},\n\\end{align} \nwhere the last line follows since $\\frac{1}{\\omega_0},\\varepsilon = o\\left(\\sqrt{\\varepsilon}+\\frac{1}{\\sqrt{\\omega_0}}\\right)=o(\\varepsilon')$.\nThe sum can be estimated using the definition of the $\\widetilde\\mathrm{Bi}$ distribution\nand equations~\\eqref{eq:relatingtherhos} and~\\eqref{eq:relatingrhoswithexpfunction}:\n\\begin{align*}\n&\\sum\\nolimits_{j=2}^r (r-j)\\Pr(\\widetilde\\mathrm{Bi}(r,\\rho_*)=j) \\\\\n& \\hspace{2cm} \\numeq{\\phantom{\\eqref{eq:relatingtherhos},\\eqref{eq:relatingrhoswithexpfunction}}} \\sum\\nolimits_{j=0}^r (r-j)\\Pr(\\mathrm{Bi}(r,\\rho_*)=j)  - r(1-\\rho_*)^r - (r-1)r\\rho_*(1-\\rho_*)^{r-1}\\\\\n& \\hspace{2cm} \\numeq{\\phantom{\\eqref{eq:relatingtherhos},\\eqref{eq:relatingrhoswithexpfunction}}} r (1-\\rho_*)\\left( 1 - (1-\\rho_*)^{r-1} - (r-1)\\rho_*(1-\\rho_*)^{r-2} \\right)\\\\\n& \\hspace{2cm} \\numeq{\\eqref{eq:relatingtherhos},\\eqref{eq:relatingrhoswithexpfunction}} r \\exp(-d\\hat \\rho_*)\\left(\\hat \\rho_* - (r-1)\\rho_*(1-\\rho_*)^{r-2} \\right).\n\\end{align*}\nSubstituting this into~\\eqref{eq:leaves1} gives\n\\begin{align}\\label{eq:leaves2}\n\\mu_1\n&  = d\\exp(-d\\hat \\rho_*) \\left(\\hat \\rho_* - (r-1)\\rho_*(1-\\rho_*)^{r-2}\\right) \\pm \\varepsilon'\/2.\n\\end{align}\nOn the other hand, we have\n\\begin{align*}\n\\mu_1' = \\eta \\cdot\\Pr(\\mathrm{Po}(d\\hat \\rho_*)=1) & = \\left(1- \\frac{(r-1)\\rho_* (1-\\rho_*)^{r-2}}{\\hat \\rho_*}\\right) d\\hat \\rho_* \\exp(-d\\hat \\rho_*)\\\\\n& = d\\exp(-d\\hat \\rho_*)\\left(\\hat \\rho_* - (r-1)\\rho_* (1-\\rho_*)^{r-2}\\right),\n\\end{align*}\nwhich combined with~\\eqref{eq:leaves2} tells us that\n\\begin{equation}\\label{eq:deg1proportion}\n\\mu_1 = \\mu_1' \\pm \\varepsilon'\/2,\n\\end{equation}\nwhich is in fact slightly stronger than required.\n\n\\vspace{0.2cm}\n\\noindent \\textbf{Case 3: $j=0$.}\\\\\nFinally to prove the statement for $j=0$, note that\n$\\mu_0 = \\varprop{0} - \\mu_1$ (deterministically).\nFurthermore, we have $\\sum_{j=0}^\\infty \\mu_j' = \\sum_{j=0}^\\infty \\varpropp{j} =1$,\nand we have already observed that $\\mu_j'=\\varpropp{j}$ if $j\\ge 2$,\nand therefore $\\mu_0' + \\mu_1' = \\varpropp{0} + \\varpropp{1}$.\nObserving also that $\\varpropp{1}=0$, we deduce that\n$\\mu_0' = \\varpropp{0} - \\mu_1'$.\nTherefore, applying Theorem~\\ref{thm:factor:reducedcoredegs} (for $j=0$) and~\\eqref{eq:deg1proportion},\nwe obtain\n$$\n\\mu_0 = \\varprop{0}-\\mu_1 = \\varpropp{0} \\pm \\varepsilon - (\\mu_1'\\pm \\varepsilon'\/2) = \\mu_0'\\pm \\varepsilon'\n$$\nas required.\n\\end{proof}\n\n\nWith a little more calculation we can also determine\nthe number of vertices and edges in the loose core, and therefore also prove Theorem~\\ref{thm:mainresultorder}.\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:mainresultorder}]\nObserve that the number of vertices in the loose core of $H=H^r(n,p)$ is simply the number of variable nodes\nof $G=G(H)$ which have degree at least one in the padded core\\ of $G$,\nand thus the proportion of such vertices is $1-\\mu_0$ (see Remark~\\ref{rem:paddedcoredegs}). By Theorem~\\ref{thm:mainresultdegrees}, whp\n\\begin{align*}\n1-\\mu_0 & \\numeq{\\phantom{\\eqref{eq:relatingtherhos},\\eqref{eq:eta},\\eqref{eq:relatingrhoswithexpfunction}}} 1-\\exp(-d\\hat \\rho_*) - d\\hat \\rho_*\\exp(-d\\hat \\rho_*)(1-\\eta)+o(1) \\\\\n& \\numeq{\\eqref{eq:relatingtherhos},\\eqref{eq:eta},\\eqref{eq:relatingrhoswithexpfunction}}\\rho_* - d(1-\\rho_*)(r-1) \\rho_*(1-\\rho_*)^{r-2}+o(1)\\\\\n& \\numeq{\\phantom{\\eqref{eq:relatingtherhos},\\eqref{eq:eta},\\eqref{eq:relatingrhoswithexpfunction}}}\\rho_* (1 - d(r-1)(1-\\rho_*)^{r-1})+o(1)=\\alpha+o(1),\n\\end{align*}\nprecisely as stated in Theorem~\\ref{thm:mainresultorder}.\n\nThe number of edges in the loose core of $H$ is the number of factor nodes with degree at least $1$ in $R_G$,\nwhich is $(1-\\factprop{0})m$, where recall that $m$ denotes the total number of factor nodes of $G$.\nApplying Theorem~\\ref{thm:factor:reducedcoredegs} to estimate $\\factprop{0}$\nand Proposition~\\ref{prop:numberfactornodes} to estimate $m$,\nwe deduce that\nwhp\\ the number of edges in the loose core is\n\\begin{align*}\n\\left(1-\\factprop{0}\\right)m\n&=\\left(1-(1-\\rho_*)^r-r\\rho_*(1-\\rho_*)^{r-1}\\pm o(1)\\right)\\frac{(1+o(1))dn}{r}\n\\numeq{\\eqref{eq:definitionofeta}}\\left(\\beta+o(1)\\right)n,\n\\end{align*} \nas claimed.\n\\end{proof}\n\nNow we can also prove the bound on the length of the longest loose cycle in Theorem~\\ref{thm:mainresultcycle}.\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:mainresultcycle}]\nLet us observe that for any loose cycle in $H=H^r(n,p)$, the edges and the vertices which lie in two edges form\na cycle in the factor graph $G=G(H)$, which must clearly lie within the reduced core $R_G$ of $G$. Thus the\nlength of the loose cycle is bounded both by the number of variable nodes and the number of factor nodes\nwhich are not isolated\nin $R_G$. In other words, the length $L_C$ of the longest loose cycle (deterministically) satisfies\n\\begin{equation}\\label{eq:cyclelengthmin}\nL_C\\le \\min\\left\\{(1-\\varprop{0})n \\; , \\; (1-\\factprop{0}) m\\right\\}.\n\\end{equation}\nBy Proposition~\\ref{prop:numberfactornodes} we have that whp\\ $m=\\left(1+o(1)\\right)\\frac{dn}{r}$.\nObserve also that by Theorem~\\ref{thm:factor:reducedcoredegs}, whp\\ $\\varprop{0}$ is asymptotically\n\\begin{align*}\n\\Pr(\\widetilde{\\mathrm{Po}}(d\\hat \\rho_*)=0) = \\prob(\\mathrm{Po}(d\\hat \\rho_*)\\leq 1) =  \\exp(-d\\hat \\rho_*)(1+d\\hat \\rho_*)=1-\\gamma,\n\\end{align*}\nwhile whp $\\factprop{0}$ is asymptotically\n\\begin{align*}\n\\Pr(\\widetilde{\\mathrm{Bi}}(r,\\rho_*)=0) = \\prob(\\mathrm{Bi}(r,\\rho_*)\\leq 1) = (1-\\rho_*)^r + r\\rho_*(1-\\rho_*)^{r-1}=1-\\frac{\\beta r}{d}.\n\\end{align*}\nSubstituting these values into~\\eqref{eq:cyclelengthmin} gives the bound in Theorem~\\ref{thm:mainresultcycle}.\n\\end{proof}\n\nOur next goal is to prove the remaining results of Section~\\ref{sec:mainresults},\nfor which we will need to relate $L_P$ and $L_C$. To do this, we use a standard sprinkling argument.\n\n\\begin{lemma}\\label{lem:standardsprinkling}\nLet $\\omega = \\omega(n)$ be any function and\n$p_1=p_1(n)$ and $p_2=p_2(n)$ be any probabilities satisfying\n\\begin{enumerate}\n\\item $p_1 \\le \\left(1+\\frac{1}{\\omega}\\right)p_1 \\le p_2;$\n\\item $p_1 n^r \/\\omega \\to \\infty$.\n\\end{enumerate}\nSuppose that $H_1 \\sim H^r(n,p_1)$ and $H_2 \\sim H^r(n,p_2)$ are coupled in such a way that $H_1 \\subset H_2$.\nFor $i=1,2$, let $L_P^{(i)},L_C^{(i)}$ denote the length of the longest loose path and loose cycle, respectively, in $H_i$. Then whp\\ \\[L_C^{(2)} \\ge L_P^{(1)}+o(n).\\]\n\\end{lemma}\nWe defer the proof of this lemma to Appendix~\\ref{app:sprinkling}.\nThe following slightly different form will be a little more convenient to apply.\nWe omit the proof, which is elementary given Lemma~\\ref{lem:standardsprinkling}. \\pagebreak\n\\begin{corollary}\\label{cor:sprinkling}\nGiven the setup of Lemma~\\ref{lem:standardsprinkling}, the following hold.\n\\begin{enumerate}\n\\item If there exists a constant $\\zeta_1$ such that whp\\ $L_P^{(1)} \\ge (\\zeta_1 +o(1))n$, then whp\\ $L_C^{(2)} \\ge (\\zeta_1 +o(1))n$.\n\\item If there exists a constant $\\zeta_2$ such that whp\\ $L_C^{(2)} \\le (\\zeta_2 +o(1))n$, then whp\\ $L_P^{(1)} \\le (\\zeta_2 +o(1))n$.\n\\end{enumerate}\n\\end{corollary}\n\n\n\nWe also need a further technical result which states that the parameters $\\beta,\\gamma$,\nwith which we bound $L_C$ in Theorem~\\ref{thm:mainresultcycle}, are continuous in $p$\n(except at the threshold $p=d^*\/\\binom{n-1}{r-1}$).\nLet $r,d,p$ be as in Section~\\ref{def:basicdefinitions}. \nLet $p'=(1+1\/\\omega)p$ for a function $\\omega=\\omega(n)\\to\\infty$ but $\\omega=o(\\log n)$.\nThe following lemma shows that if we replace $p$ by $p'$,\nthe parameters $\\beta,\\gamma$ remain essentially the same. The (technical) proof can be found in Appendix~\\ref{app:analysisfixedpoint}.\n\\begin{lemma}\\label{lem:continuity}\nLet $\\beta,\\gamma$ be defined as in~\\eqref{eq:definitionofeta} and~\\eqref{eq:definitionofgamma},\nand let $\\beta',\\gamma'$ be defined similarly but with $p'$ in place of $p$. If $d\\neq d^*,$ then \n\\[\\min\\{\\beta',\\gamma'\\}=\\min\\{\\beta,\\gamma\\}+o(1).\\]\n\\end{lemma}\n\n\n\nWe can now bound the length of the longest loose path in $H^r(n,p)$.\n\\begin{proof}[Proof of Corollary~\\ref{cor:upperboundlongestpath}]\nLet us set $\\omega = 1\/(\\log n)$, set $p_1=p$ and set $p_2 = \\left(1+\\frac{1}{\\omega}\\right)p_1$.\nIt is easy to check that these parameters satisfy the assumptions of Lemma~\\ref{lem:standardsprinkling},\nand therefore also of Corollary~\\ref{cor:sprinkling}.\nTheorem~\\ref{thm:mainresultcycle} applied to $H_2 \\sim H^r(n,p_2)$ implies that whp\\ $L_C^{(2)} \\le (\\min\\{\\beta_2,\\gamma_2\\} + o(1))n$,\nwhere $\\beta_2,\\gamma_2$ are defined analogously to $\\beta,\\gamma$, but with $p_2 = (1+1\/\\omega)p$ in place of $p$.\nFurthermore, Lemma~\\ref{lem:continuity} implies that $\\min\\{\\beta_2,\\gamma_2\\} = \\min\\{\\beta,\\gamma\\}+o(1)$,\nso we deduce that whp\\ $L_C^{(2)} \\le (\\min\\{\\beta,\\gamma\\} + o(1))n$.\nFinally, Corollary~\\ref{cor:sprinkling} then implies that whp\\ $L_P=L_P^{(1)} \\le (\\min\\{\\beta,\\gamma\\} + o(1))n$,\nas required.\n\\end{proof}\n\nBy applying Corollary~\\ref{cor:upperboundlongestpath} shortly beyond the phase transition\nthreshold, we are able to prove Corollary~\\ref{cor:shortlyafterphasetransition}.\n\\begin{proof}[Proof of Corollary~\\ref{cor:shortlyafterphasetransition}]\nSince $\\min\\{\\beta,\\gamma\\}\\leq \\gamma$ it suffices to show that \n\\[\\gamma\\leq \\frac{2\\varepsilon^2}{(r-1)^2}+O(\\varepsilon^3).\\] (In fact a similar computation for $\\beta$ gives exactly the same result.)\nBy definition\n\\begin{align}\n    \\gamma &\\numeq{\\eqref{eq:definitionofgamma}} 1-\\exp(-d\\hat \\rho_*)-d\\hat \\rho_* \\exp(-d\\hat \\rho_*) \\nonumber \\\\\n    & = 1-\\left(1-d\\hat \\rho_*+\\frac{d^2\\hat \\rho_*^2}{2}+O\\left(\\hat \\rho_*^3\\right)\\right)-d\\hat \\rho_*\\left(1-d\\hat \\rho_*+O\\left(\\hat \\rho_*^2\\right)\\right) \\nonumber \\\\\n    &= \\frac{d^2\\hat \\rho_*^2}{2}+O\\left(\\hat \\rho_*^3\\right) \\label{eq:gammadefapprox}\n\\end{align}\nRecall from~\\eqref{eq:relatingtherhos} that $\\hat \\rho_*$ was defined as a function of $\\rho_*$,\nwhich itself was defined as the largest solution of the fixed-point equation~\\eqref{eq:fixedpointequation}.\nWe therefore need to estimate $\\rho_*$.\nFrom~\\eqref{eq:fixedpointequation} we obtain\n\\begin{align*}\n    \\rho &= \\frac{-d(r-1)+1}{(-\\frac{1}{2}-\\frac{d}{2}(r-1)(r-2))}+O\\left(\\rho^2\\right).\\label{eq:rhoapproximation}\n\\end{align*}\n\n\\noindent Substituting $d=\\frac{1+\\varepsilon}{r-1}$ gives\n\\begin{equation*}\n    \\rho=\\frac{2\\varepsilon}{1+(1+\\varepsilon)(r-2)}+O\\left(\\rho^2\\right)=\\frac{2\\varepsilon}{r-1+O(\\varepsilon)}+O\\left(\\rho^2\\right)=\\frac{2\\varepsilon}{r-1}+O\\left(\\rho^2\\right).\n\\end{equation*}\nIn particular this implies that there exists a solution $\\rho=\\frac{2\\varepsilon}{r-1}+O(\\varepsilon^2)$\nof the fixed point equation~\\eqref{eq:fixedpointequation},\nand by Claim~\\ref{claim:behaviouroffixedpointsol} this is the unique positive solution\nand therefore\n$\\rho_*=\\frac{2\\varepsilon}{r-1}+O(\\varepsilon^2)$.\nSubstituting this into~\\eqref{eq:relatingtherhos} we obtain\n\\begin{equation*}\n\\hat \\rho_*=1-\\left(1-\\frac{2\\varepsilon}{r-1}+O(\\varepsilon^2)\\right)^{r-1}=2\\varepsilon+O(\\varepsilon^2).\n\\end{equation*}\nSubstituting this into~\\eqref{eq:gammadefapprox}, we obtain\n\\begin{align*}\n    \\gamma &= \\frac{2\\varepsilon^2}{(r-1)^2}+O\\left(\\varepsilon^3\\right).\\qedhere\n\\end{align*}\n\\end{proof}\n\nIn order to prove Theorem~\\ref{thm:bestknownresultcycles},\nwe also need a lower bound on $L_C$.\nWe will use a result of~\\cite{cooley2020longest},\nwhich provides a lower bound on $L_P$ together with Lemma~\\ref{lem:standardsprinkling} to relate $L_P$ and $L_C$.\nMore precisely, one special case (the supercritical regime for $j=1$) of~\\cite[Theorem~4]{cooley2020longest} can be\nreformulated (in a slightly weakened but much simplified way) as follows.\n\n\n\n\\begin{theorem}[\\!\\!\\cite{cooley2020longest}]\\label{thm:pathsresult}\nLet  $L_P$ denote the length of the longest loose  path in $H^r(n,p)$. \nFor all $r\\in \\mathbb{N}_{\\ge 3}$ there exists $\\varepsilon_0 \\in (0,1]$ such that for any function\n$\\varepsilon=\\varepsilon(n)<\\varepsilon_0$ which satisfies\n$\\varepsilon^5 n \\xrightarrow{n\\to \\infty} \\infty$, setting $\\delta=\\varepsilon\/\\sqrt{\\varepsilon_0}$\nthe following holds. \nIf $p=\\frac{1+\\varepsilon}{(r-1)\\binom{n-1}{r-1}}$,\nthen whp\n \\[\n (1 - \\delta)\\frac{\\varepsilon^2 n}{4(r-1)^2} \\leq L_P \\leq (1 + \\delta)\\frac{2 \\varepsilon n}{(r-1)^2}.\n \\]\n\\end{theorem}\n\nNote that Theorem~\\ref{thm:pathsresult} allows for a wider range of $\\varepsilon$ than we consider in this paper,\nin particular allowing $\\varepsilon$ to tend to zero sufficiently slowly. However, there is a $\\Theta(\\varepsilon)$ gap\nbetween the upper and lower bounds. Theorem~\\ref{thm:bestknownresultcycles} improves the upper bound\nand thus narrows the gap to just a constant factor.\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:bestknownresultcycles}]\nThe second and third inequalities are simply the statement of Corollary~\\ref{cor:upperboundlongestpath},\nso it remains to show that whp\n\\[\n\\left(\\frac{\\varepsilon^2}{4(r-1)^2}+O(\\varepsilon^3)\\right)\\cdot n\\leq L_C.\n\\]\nNote that we may assume that $\\varepsilon<\\varepsilon_0$, where $\\varepsilon_0$ is the parameter from\nTheorem~\\ref{thm:pathsresult},\nsince otherwise the $O(\\varepsilon^3)$ error term may in fact be the dominant term, and the result is trivial.\n\nLet us set $p_2 = p$ and $p_1 = \\left(1-\\frac{1}{\\log n}\\right)p$. It is easy to check that these parameters\nsatsify the assumptions of Lemma~\\ref{lem:standardsprinkling}, and therefore also of Corollary~\\ref{cor:sprinkling}.\n\nIt is also clear that $p_1 = \\frac{1+\\varepsilon_1}{(r-1)\\binom{n-1}{r-1}}$, where $\\varepsilon_1 = \\varepsilon-\\frac{1}{\\log n}-\\frac{\\varepsilon}{\\log n}=\\varepsilon + O(\\varepsilon^2)$,\nand\ntherefore the lower bound in Theorem~\\ref{thm:pathsresult} (together with the observation that $\\varepsilon_1\/\\sqrt{\\varepsilon_0} = O(\\varepsilon_1)$)\nstates that whp\\ \n$$\nL_P^{(1)} \\ge \\left(\\frac{\\varepsilon_1^2}{4(r-1)^2} + O(\\varepsilon_1^3)\\right)\\cdot n = \\left(\\frac{\\varepsilon^2}{4(r-1)^2} + O(\\varepsilon^3)\\right)\\cdot n,\n$$\nand an application of Corollary~\\ref{cor:sprinkling} completes the proof.\n\\end{proof}\n\n\n\n\n\\section{Peeling process}\\label{sec:peeling}\nRecall that for a given hypergraph $H$ the reduced core of the factor graph $G=G(H)$\nis defined as the maximum subgraph with no nodes of degree one, which is similar to the $2$-core of $G$\nexcept that isolated nodes are not deleted.\nThere is a simple peeling process to obtain the $2$-core of $G$ which is a standard procedure and has been used and analysed extensively in the literature.\nWe will consider the obvious adaptation of this process which obtains the reduced core rather than the $2$-core.\n\n\\begin{definition}[Peeling Process]\\label{def:peeling}\nIn every round we check whether the factor graph has any nodes of degree one\nand delete edges incident to such nodes. More precisely, we recursively define a sequence of graphs $(G_i)_{i\\in\\mathbb{N}}$\nwhere $G_0$ is the input graph and for $i\\in\\mathbb{N}_{\\geq 1}$, $G_i$ is the graph obtained from $G_{i-1}$\nby removing all edges incident to nodes of degree one. We say that we \\emph{disable} a node\nif we delete its incident edges.\n\n\n\\end{definition} \n\\noindent Note that there exists an $i_0$ such that $G_{i_0}=G_{i_0+k}=R_{G_0}$ for any $k\\in\\mathbb{N}$. \\pagebreak\n\nWe recall the definition of $\\varprop{j}$ and $\\factprop{j}$ in Theorem~\\ref{thm:factor:reducedcoredegs} and observe that \\[\\varprop{j}\\coloneqq\\lim\\limits_{\\ell\\to\\infty}\\varpropl{j}{\\ell} \\text{ \\ and \\ } \\factprop{j}\\coloneqq\\lim\\limits_{\\ell\\to\\infty}\\factpropl{j}{\\ell},\\] \nwhere $\\varpropl{j}{\\ell}, \\factpropl{j}{\\ell}$ are the proportions of variable nodes and factor nodes respectively which have degree $j$ in $G_{\\ell}$ for $\\ell\\in\\mathbb{N}$. These limits exist since both $\\left(\\varpropl{j}{\\ell}\\right)_{\\ell}$ and $\\left(\\factpropl{j}{\\ell}\\right)_{\\ell}$ remain constant after a finite number of steps.\n\n\nWe will prove Theorem~\\ref{thm:factor:reducedcoredegs} with the help of two lemmas. The first describes the asymptotic distribution of $\\varpropl{j}{\\ell}$ and $\\factpropl{j}{\\ell}$ for large $\\ell$.\n\\begin{lemma}\\label{lem:mainlemma1}\nLet $r,d,\\rho_*,\\hat \\rho_*$ be as in Section~\\ref{def:basicdefinitions}.\nThere exist an integer $\\ell= \\ell(n) \\in \\mathbb{N}$ and a real number $\\varepsilon_1=\\varepsilon_1(n)=o(1)$\nsuch that whp, for any constant $j\\in\\mathbb{N}$ \n\\[\n\\varpropl{j}{\\ell}=\\prob(\\widetilde \\mathrm{Po}(d\\hat \\rho_*)=j)\\pm\\varepsilon_1\n\\]\nand\n\\[\n\\factpropl{j}{\\ell}=\\prob(\\widetilde \\mathrm{Bi}(r,\\rho_*)=j)\\pm\\varepsilon_1.\n\\]\n\\end{lemma}\n\nThe second lemma states that $\\varpropl{j}{\\ell}$ and $\\factpropl{j}{\\ell}$ approximate $\\varprop{j}$ and $\\factprop{j}$, respectively.\n\\begin{lemma}\\label{lem:mainlemma2}\nLet $r,d$ be as in Section~\\ref{def:basicdefinitions}.\nFor each $j\\in \\mathbb{N}$,\nlet $\\varprop{j},\\factprop{j}$ be as defined in Theorem~\\ref{thm:factor:reducedcoredegs},\nlet $\\ell,\\varepsilon_1$ be as in Lemma~\\ref{lem:mainlemma1} and set $\\varepsilon_2\\coloneqq\\sqrt{\\varepsilon_1}.$ Then whp\\ the peeling process will disable at most $\\varepsilon_2 n$ nodes after round $\\ell$. In particular whp, for any constant $j\\in\\mathbb{N}$\n\\[\n\\varprop{j}=\\varpropl{j}{\\ell}\\pm\\varepsilon_2\n\\]\nand\n\\[\n\\factprop{j}=\\factpropl{j}{\\ell}\\pm \\frac{2\\varepsilon_2r}{d} .\n\\]\n\\end{lemma}\nBefore proving these two lemmas, we show how together they imply our main result. \n\\begin{proof}[Proof of Theorem~\\ref{thm:factor:reducedcoredegs}]\nLet $\\ell,\\varepsilon_1,\\varepsilon_2$ be as in Lemmas~\\ref{lem:mainlemma1} and~\\ref{lem:mainlemma2}.\nApplying these two lemmas, whp\\ we have\n$$\n\\varprop{j} \\stackrel{\\mbox{\\tiny L.\\ref{lem:mainlemma2}}}{=} \\varpropl{j}{\\ell} \\pm \\varepsilon_2\n\\stackrel{\\mbox{\\tiny L.\\ref{lem:mainlemma1}}}{=} \\Pr(\\widetilde \\mathrm{Po}(d\\hat \\rho_*)=j) \\pm (\\varepsilon_1 + \\varepsilon_2).\n$$\nSimilarly, whp\\ we have\n$$\n\\factprop{j} \\stackrel{\\mbox{\\tiny L.\\ref{lem:mainlemma2}}}{=} \\factpropl{j}{\\ell} \\pm \\frac{2\\varepsilon_2 r}{d}\n\\stackrel{\\mbox{\\tiny L.\\ref{lem:mainlemma1}}}{=} \\prob(\\widetilde \\mathrm{Bi}(r,\\rho_*\\, )=j)\\pm \\left(\\varepsilon_1 + \\frac{2\\varepsilon_2 r}{d}\\right).\n$$\nThe statement of Theorem~\\ref{thm:factor:reducedcoredegs} follows by setting $\\varepsilon = \\varepsilon_1+\\varepsilon_2\\max\\{1,2r\/d\\}$.\n\\end{proof}\n\n\n\n\n\n\n\\section{\\coredetector \\ Algorithm: Proof of Lemma~\\ref{lem:mainlemma1}}\\label{sec:algorithm}\n\\subsection{Main algorithm}\nIn this section we will introduce the \\coredetector \\ algorithm,\nwhich is related to the peeling process. To do so, we need to define some notation---this\nnotation could apply to any graph, but since we will need it specifically for factor graphs, we introduce it in this\n(slightly restrictive) setting for clarity.\n\\begin{definition}\\label{def:algorithmdefs}\nLet $G$ be a factor graph with variable node set $\\mathcal{V}$ and factor node set $\\mathcal{F}$.\nWe denote by $d_G(u,v)$ the distance between two nodes $u,v\\in \\mathcal{V} \\cup \\mathcal{F}$, i.e.\\ the number of edges in a shortest path between them.\nFor each $\\ell\\in\\mathbb{N}$ and each $w\\in \\mathcal{V}\\cup\\mathcal{F}$, we define\n\\[\nD_{\\ell}(w)\\coloneqq\\{u\\in\\mathcal{V}\\cup\\mathcal{F}:d_{G}(w,u)=\\ell\\}\n\\]\nand\n\\[\nd_{\\ell}(w)\\coloneqq|D_{\\ell}(w)|.\n\\]\n\\pagebreak \n\n\\noindent Let \n\\[\nD_{\\leq\\ell}(w)=\\bigcup_{i=0}^\\ell D_i(w)\n\\]\nand\n\\[N_{\\leq\\ell}(w)\\coloneqq G[D_{\\leq\\ell}(w)],\\]  i.e.\\ the subgraph of $G$ induced on $D_{\\leq\\ell}(w)$.\n\\end{definition}\n\nWe consider a procedure called \\coredetector .\nGiven a factor graph $G$ on node set $\\mathcal{V} \\cup \\mathcal{F}$\nand a node $w \\in \\mathcal{V} \\cup \\mathcal{F}$,\nwe consider the factor graph as being rooted at $w$. In particular, neighbours of a node $v$ which are at distance $d_G(v,w)+1$ from $w$\nare called \\emph{children} of $v$.\nStarting at distance $\\ell \\in \\mathbb{N}$ and moving up towards the root $w$, we recursively delete any node with no (remaining) children;\nAlgorithm~\\ref{algorithm:coredetector 2} gives a formal description of this procedure.\nWe will denote by $D^*_{\\ell-i}(w)$ the set of nodes in $D_{\\ell-i}(w)$ which survive round $i$ and let $d^*_{i}(w)\\coloneqq|D^*_{i}(w)|$. \n\\begin{algorithm}\n\t\\DontPrintSemicolon\n\t\\KwIn{Integer $\\ell\\in\\mathbb{N}$, node $w\\in\\mathcal{V}\\cup\\mathcal{F}$, factor graph $N_{\\leq\\ell+1}(w)$}\n\t\\KwOut{$d^*_1(w)$}\n    $D^*_{\\ell+1}(w)=D_{\\ell+1}(w)$\\;\n\t\\For{$1\\leq i\\leq \\ell$}{\n    $D^*_{\\ell-i+1}(w)\\gets D_{\\ell-i+1}(w)\\setminus\\ \\Big\\{v:N(v)\\capD^*_{\\ell-i+2}(w)=\\emptyset\\Big\\}$\\;\n\t$d^*_{\\ell-i+1}(w)\\gets|D^*_{\\ell-i+1}(w)|$}\n\t\t\\caption{\\textnormal{\\texttt{CoreConstruct}} }\n\t\\label{algorithm:coredetector 2}\n\\end{algorithm}\n\nIt is rather difficult to analyse the peeling process directly and it turns out that \\coredetector \\ is easier to analyse\nwhile also being closely related.\n\\coredetector \\ is intended to model the effect of the peeling process on the degree of $w$ after $\\ell$ steps\n(although note that \\coredetector \\ does delete nodes rather than merely disabling them).\nNote, however, that it does not mirror the peeling process precisely; some nodes may be disabled in the peeling process much\nearlier than they are deleted in \\coredetector ,\nand some nodes may be deleted in \\coredetector \\ even\nthough they are actually in the reduced core, and are therefore never disabled in the peeling process.\nNevertheless, we obtain the following important relation.\nRecall that $G_{\\ell}$ is the graph obtained after the $\\ell$-th round of the peeling process (see Definition~\\ref{def:peeling})\nand that $d_{G_\\ell}(w)$ is the degree of the node $w$ in $G_\\ell$.\n\n\\begin{lemma}\\label{lem:algvspeeling}\nLet $\\ell\\in\\mathbb{N}_{\\geq 1}$ and $w\\in\\mathcal{V}\\cup\\mathcal{F}$. If there are no cycles in $N_{\\leq\\ell+1}(w)$, then\nthe output $d^*_1(w)$ of \\coredetector \\ \nwith input $\\ell,w$ and $N_{\\le \\ell+1}(w)$ satisfies\n\\[d_{G_{\\ell}}(w) \\begin{cases}\n=d^*_1(w) & \\text{if  }d^*_1(w)\\neq 1, \\\\\n\\leq d^*_1(w) & \\text{if  }d^*_1(w)=1.\n\\end{cases}\n\\]\n\\end{lemma}\n\\begin{proof}\nFor an upper bound,\nwe will show that if a given node $v\\in\\mathcal{V}\\cup\\mathcal{F}$ is deleted in round $i$ of \\coredetector , \nit must have been disabled at some round $i'\\leq i$ in the peeling process for the reduced core.\nIn particular, by setting $i=\\ell$ we immediately obtain\n$d_{G_{\\ell}}(w)\\leqd^*_1(w).$\nWe prove the statement by induction on $i$.\n\nFor $i=1$, if a node $v$ is deleted in round one of \\coredetector , then $v$ had no children,\nand therefore it has only one neighbour in $G=G_0$ (its parent in $N_{\\le \\ell+1}(w)$).\nThus $v$ will be disabled in round one\nof the peeling process.\nNow suppose $v$ is deleted in round $i\\ge 2$ of \\textnormal{\\texttt{CoreConstruct}}, which must mean that all its children\n(if it had any) are deleted in step $i-1$ of \\textnormal{\\texttt{CoreConstruct}}.\nBy the induction hypothesis, all its children are disabled by step at most $i-1$ of the peeling process\nand so have degree $0$ in $G_{i-1}$.\nTherefore $v$ itself has degree at most one in $G_{i-1}$ (from its parent)\nand so will be disabled in round $i$ of the peeling process if it has not been disabled already.\n\n\nIt remains to prove that \n$d_{G_{\\ell}}(w)\\geqd^*_1(w)$\nif $d^*_1(w)\\geq 2$. Let $j\\coloneqqd^*_1(w)\\ge 2$ be the number of children of the root $w$ which survive \\coredetector .\nEach such child must have a descendant in $D_{\\ell+1}(w)$, otherwise it would not survive \\coredetector .\nThus we have $j$ paths of length $\\ell+1$ which all meet at $w$, but are otherwise disjoint (since $N_{\\le \\ell+1}(w)$ contains no cycles).\nBy induction on $i$, we deduce that after $i$ rounds of the peeling process,\nthere are $j$ paths of length $\\ell+1-i$ which meet only in $w$,\nand in particular after $\\ell$ rounds of the peeling process, $d_{G_\\ell}(w) \\ge j$, as required.\n\\end{proof}\n\nFrom now on for the rest of this section, we will always have $G=G^r(n,p)$.\nObserve that if $d_{G_\\ell}(w)\\in \\{0,1\\}$, then $d_{R_G}(w)=0$. However, if $\\ell$ is sufficiently large\nwe can even say that\nin $G_\\ell$ the degree will almost always be $0$.\n\n\\begin{proposition}\\label{prop:1rare}\nFor any integer-valued function $\\ell=\\ell(n) \\xrightarrow{n\\to \\infty} \\infty$ and node $w$ we have\n$\\Pr\\left(d_{G_\\ell}(w)=1\\right) = o(1)$.\n\\end{proposition}\n\n\\begin{proof}\nLet us first assume that $w$ is a variable node.\nFor any integer $i \\ge 1$, let $\\mathcal{V}_i$ and $\\mathcal{F}_i$ be the set of variable nodes and factor nodes respectively which are disabled in round~$i$\nof the peeling process.\nIt is an elementary fact about the peeling process that for any integer $i \\ge 2$\nwe have\n$|\\mathcal{V}_i| \\le |\\mathcal{F}_{i-1}|$ and $|\\mathcal{F}_i| \\le |\\mathcal{V}_{i-1}|$ (deterministically),\nfrom which it follows that $|\\mathcal{V}_i|\\le |\\mathcal{V}_{i-2}|$ for $i \\ge 3$.\nTherefore we have\n$$\n|\\mathcal{V}_\\ell| \\le |\\mathcal{V}_{\\ell-2}| \\le \\ldots \\le |\\mathcal{V}_{\\ell -2\\lfloor \\frac{\\ell-1}{2}\\rfloor}|.\n$$\nFurthermore, the $\\mathcal{V}_i$ are all disjoint, and so we have\n$$|\\mathcal{V}_\\ell| \\le \\frac{n}{1+\\lfloor \\frac{\\ell-1}{2}\\rfloor} = O(n\\ell^{-1}) = o(n)$$\ndeterministically. Therefore for large enough $n$ we have\n$|\\mathcal{V}_\\ell| \\in K \\coloneqq \\left[n\/\\sqrt{\\ell}\\right]_0$ and so by symmetry we have\n$$\n\\Pr\\left(w \\in \\mathcal{V}_\\ell\\right)  = \\sum_{k \\in K} \\left( \\Pr\\left(|\\mathcal{V}_\\ell|=k\\right) \\cdot \\frac{k}{n} \\right) \\le \\frac{1}{\\sqrt{\\ell}} \\cdot \\sum_{k \\in K} \\Pr\\left(|\\mathcal{V}_\\ell|=k\\right) = o(1),\n$$\nas required.\nThe proof when $w$ is a factor node is essentially identical.\n\\end{proof}\n\n\n\\subsection{Analysis of \\coredetector }\n\nWe proceed with the analysis of \\coredetector \\ and we will choose the parity of $\\ell$ such that $D_{\\ell+1}(w)$ consists of variable nodes,\ni.e.\\ if $w\\in \\mathcal{V}$, then we will choose $\\ell$ odd, and if $w \\in \\mathcal{F}$ we will choose $\\ell$ to be even.\nThis convention is merely for technical convenience since it ensures that we\nknow which type of nodes are being considered in round $i$ of \\textnormal{\\texttt{CoreConstruct}}\\ and thus avoid\na case distinction.\n\nLet $w\\in\\mathcal{V}\\cup\\mathcal{F}$ and $\\ell\\in\\mathbb{N}_{\\geq 1}$ be given. We say the event $E_w(\\ell)$ holds if \\emph{neither} of the following two events occur.  \n\\begin{enumerate}[label=\\textnormal{\\textbf{(E\\arabic*)}}]\n    \\item $|D_{\\leq \\ell+1}(w)|\\geq(\\log n)^2$;\n    \\item $D_{\\leq \\ell+1}(w)$ contains a node which lies on a cycle of length at most $2\\ell$.\n\\end{enumerate}\nWe will later condition on the event $E_{w}(\\ell)$ holding, and therefore need to know that it is very likely.\n\n\\begin{lemma}\\label{lem:eventE}\nFor any function $\\ell=o(\\log\\log n)$ and node $w \\in \\mathcal{V} \\cup \\mathcal{F}$,\n$$\n\\Pr\\left(E_{w}(\\ell)\\right) \\ge 1-\\exp\\left(-\\Theta\\left(\\sqrt{\\log n}\\right)\\right).\n$$\nFurthermore, whp\\ all but $o(n)$ nodes $w\\in\\mathcal{V}\\cup\\mathcal{F}$ satisfy $E_{w}(\\ell)$.\n\\end{lemma}\n\nThe (standard) proof appears in Appendix~\\ref{app:eventE}.\n\nNow let us define\n$$\n\\tilde d_1(w):=\n\\begin{cases}\nd^*_1(w) & \\mbox{if }d^*_1(w) \\neq 1\\\\\n0 & \\mbox{if }d^*_1(w) =1.\n\\end{cases}\n$$\nIn other words, $\\tilde d_1(w)$ is identical to $d^*_1(w)$ except that values of $1$ are replaced by $0$\n(similar to the $\\widetilde\\mathrm{Po}$ and $\\widetilde\\mathrm{Bi}$ distributions compared to the $\\mathrm{Po}$ and $\\mathrm{Bi}$ distributions).\nWe can combine Lemmas~\\ref{lem:eventE} and~\\ref{lem:algvspeeling} and Proposition~\\ref{prop:1rare}\nto obtain the following.\n\n\\begin{corollary}\\label{cor:Gldegalmosteq}\nFor any integer-valued function $\\ell=\\ell(n) \\xrightarrow{n\\to \\infty} \\infty$ which also satisfies $\\ell = o(\\log \\log n)$ and any node $w$ we have\n$$\n\\Pr\\left(d_{G_\\ell}(w) \\neq \\tilde d_1(w)\\right) = o(1).\n$$\n\\end{corollary}\n\n\\begin{proof}\nBy Lemma~\\ref{lem:algvspeeling}, the only cases in which $d_{G_\\ell}(w)$ and $\\tilde d_1(w)$ can differ are if $E_{w}(\\ell)$ does not hold or if $d_{G_\\ell}(w)=1$.\nThus by applying Lemma~\\ref{lem:eventE} and Proposition~\\ref{prop:1rare}, we obtain\n\\begin{align*}\n\\Pr\\left(d_{G_\\ell}(w) \\neq \\tilde d_1(w)\\right) & \\le \\Pr\\left(\\bar{E}_w(\\ell)\\right) + \\Pr\\left(d_{G_\\ell}(w) =1\\right)\\\\\n& \\le \\exp\\left(-\\Theta\\left(\\sqrt{\\log n}\\right)\\right) + o(1) = o(1).\\qedhere\n\\end{align*}\n\\end{proof}\n\n\nWe next describe the survival probabilities of internal (i.e.\\ non-root) variable and factor nodes in each round of \\textnormal{\\texttt{CoreConstruct}}.\nRecall that for any $i\\in[\\ell]$\nthe set $D^*_{\\ell+1-i}(w)$ consists of nodes within $D_{\\ell+1-i}(w)$ which survive the $i$-th round of \\coredetector . We define the recursions\n\\begin{align*}\n    \\intvarsurv{0} &= 1,\\\\\n    \\intfactsurv{t} &= \\prob(\\mathrm{Bi}(r-1,\\intvarsurv{t-1})\\geq 1),\\\\\n    \\intvarsurv{t} &= \\prob(\\mathrm{Po}(d\\intfactsurv{t})\\geq 1).\n\\end{align*}\n\\begin{lemma}\\label{lem:survivalprobs}\nLet $w\\in\\mathcal{V}\\cup\\mathcal{F}$ and $\\ell$ be odd if $w\\in\\mathcal{V}$ and even if $w\\in\\mathcal{F}$. Let $t\\in\\mathbb{N}$ with $\\ 0\\leq t\\leq \\frac{\\ell+1 }{2}$ be given.\nConditioned on the event $E_{w}(\\ell)$:\n\\begin{enumerate}\n\\item\nFor each $u\\in D_{\\ell+1-2t}(w)$ independently of each other we have\n\\[\n\\prob[u\\inD^*_{\\ell+1-2t}(w)]=\\intvarsurv{t}+o(1);\n\\]\n\\item\nFor each $a\\in D_{\\ell-2t}(w)$ independently of each other we have\n\\[\n\\prob[a\\inD^*_{\\ell-2t}(w)]=\\intfactsurv{t+1}+o(1).\n\\]\n\\end{enumerate}\nIn particular:\n\\begin{enumerate}[(i)]\n\\item If $w \\in \\mathcal{F}$ and $t_1=\\ell\/2$, then for each $u \\in D_1(w)$ independently of each other,\n$$\n    \\prob[u\\inD^*_{1}(w)]=\\intvarsurv{t_1}+o(1);\n$$\n\n\\item\\label{eq:probfactnodesurvives} If $w \\in \\mathcal{V}$ and $t_2=(\\ell+1)\/2$, then for each $a\\in D_1(w)$ independently of each other,\n$$\n    \\prob[a\\inD^*_{1}(w)]=\\intfactsurv{t_2}+o(1).\n$$\n\\end{enumerate}\n\\end{lemma}\n\nTo prove this lemma, \nwe will need the asymptotic degree distribution of a variable node in $N_{\\leq\\ell}(w)$, which is a standard result.\n\\begin{proposition}\\label{prop:distributionofnochildren}\n Let $w\\in\\mathcal{V}\\cup \\mathcal{F}$ and an integer $\\ell=o(\\log\\log n)$ be given. Conditioned on the event $E_{w}(\\ell)$,\n for each $u \\in D_{\\le \\ell}(w)\\cap \\mathcal{V}$ independently, the number of children of $u$ in $N_{\\leq\\ell+1}(w)$\n is asymptotically distributed as $\\mathrm{Po}(d)$.\n\\end{proposition}\n\nWe defer the proof to Appendix~\\ref{app:offspringdist}. We can now prove Lemma~\\ref{lem:survivalprobs}.\n \n\\begin{proof}[Proof of Lemma~\\ref{lem:survivalprobs}]\nWe will prove both statements (1) and (2) by a common induction on $t$.\nFor $t=0$ the statements are clear since we have $\\intvarsurv{0}=1$,\nwhich corresponds to the fact that nothing ever gets deleted from $D_{\\ell+1}(w)$,\nwhile $\\intfactsurv{1}=\\prob(\\mathrm{Bi}(r-1,1)\\geq 1)=1$,\ncorresponding to the fact that internal factor nodes have $r-1 \\ge 1$ children in the input graph\nand therefore also no nodes of $D_\\ell(w)$ will be deleted.\n\nWe assume both statements are true for~$t-1$ and aim to prove that they also hold for~$t$.\nWe first consider $u\\in D_{\\ell+1-2t}$ and let $X_u$ be the number of children of $u$ in $D^*_{\\ell+2-2t}$.\nObserve that the probability that $u$ survives in \\textnormal{\\texttt{CoreConstruct}}\\ is simply $\\prob(X_u\\geq 1)$.\n\nBy Proposition~\\ref{prop:distributionofnochildren},\nthe number of children of $u$ is asymptotically $\\mathrm{Po}(d)$, \nand by the induction hypothesis each child survives with probability $\\intfactsurv{t}+o(1)$ independently of each other.\nTherefore the asymptotic survival probability of $u$ is given by \n\\[\n\\prob\\left(\\mathrm{Bi}(\\mathrm{Po}(d),\\intfactsurv{t}+o(1))\\geq 1\\right)=\\prob\\left(\\mathrm{Po}(d(\\intfactsurv{t}+o(1)))\\geq 1\\right)=\\intvarsurv{t}+o(1),\n\\]\nby definition of $\\intvarsurv{t}$, as required for statement~(1).\nIndependence simply follows from the conditioning on $E_{w}(\\ell)$, which\nin particular means that $N_{\\le \\ell+1}(w)$ is a tree.\n\n\nSimilarly for $a\\in D_{\\ell-2t}(w)$ we define $X_a$ to be the number of children of $a$ which survive.\nSince $a$ is an internal factor node it has precisely $r-1$ children,\nand by statement~1 which we have just proved, each child survives with probability $\\intvarsurv{t}+o(1)$ independently.\nTherefore the probability that $a$ survives the \\textnormal{\\texttt{CoreConstruct}}\\ is\n$$\n\\Pr(\\mathrm{Bi}(r-1,\\intvarsurv{t}+o(1))\\ge 1) = \\intfactsurv{t+1}+o(1),\n$$\nas required for statement~(2). Again, independence simply follows from the conditioning on $E_{w}(\\ell)$.\n \\end{proof}\n\nA consequence of \nLemma~\\ref{lem:survivalprobs} is that, if $\\ell$ is large,\nthe distribution of the number of children of the root $w$ which survive $\\coredetector $ is almost identical\nto the claimed distributions in Theorem~\\ref{thm:factor:reducedcoredegs}.\nIn order to quantify this,\nfor two discrete random variables $X,Y$ taking values in $\\mathbb{N}$ we define the \\textit{total variation distance} as\n\\[\nd_{\\mathrm{TV}}(X,Y)\\coloneqq\\sum_{m\\in\\mathbb{N}}\\Big|\\prob(X=m)-\\prob(Y=m)\\Big|.\n\\]\n \n\n\\begin{corollary}\\label{cor:totalvariationdistance}\nThere exist $\\varepsilon=o(1)$ and $\\ell=\\ell(\\varepsilon)$ such that if we run $\\coredetector $ with input $\\ell$ and root $w\\in\\mathcal{V}\\cup\\mathcal{F}$, then:\n\\begin{enumerate}\n    \\item If $w \\in \\mathcal{V}$, then\n    \\[\n    d_{\\mathrm{TV}}\\left(\\tilde d_1(w),\\widetilde \\mathrm{Po}(d\\hat \\rho_*)\\right) \\le d_{\\mathrm{TV}}\\left(d^*_1(w), \\mathrm{Po}(d\\hat \\rho_*)\\right)<\\varepsilon;\n    \\]\n    \\item If $w\\in\\mathcal{F}$, then\n    \\[\n    d_{\\mathrm{TV}}\\left(\\tilde d_1(w),\\widetilde \\mathrm{Bi}(r,\\rho_*)\\right)\\le d_{\\mathrm{TV}}\\left(d^*_1(w),\\mathrm{Bi}(r,\\rho_*)\\right)<\\varepsilon.\n    \\]\n\\end{enumerate}\n\\end{corollary}\n\nLet us note that the second statement involves the $\\mathrm{Bi}(r,\\rho_*)$ distribution\nrather than $\\mathrm{Bi}(r-1,\\rho_*)$ since $w$ is the root, and therefore all $r$ of its\nneighbours are children rather than just $r-1$ children and one parent.\n\n\\begin{proof}\nWe will prove only the first statement; the proof of the second is very similar.\n\nNote that the first inequality follows directly from the definitions of $d^*_1(w)$ and the $\\widetilde\\mathrm{Po}$ distributions.\nThere are $2$ reasons why the second inequality is not quite immediate from Lemma~\\ref{lem:survivalprobs}:\n\\begin{enumerate}[label=\\textnormal{\\textbf{(R\\arabic*)}}]\n    \\item\\label{item:eventereasonone}Lemma~\\ref{lem:survivalprobs} has the conditioning on the event $E_{w}(\\ell)$;\n    \n    \\item\\label{item:eventereasontwo} $\\intfactsurv{(\\ell+1)\/2}$ in Lemma~\\ref{lem:survivalprobs} has been replaced by $\\hat \\rho_*$.\n\n\\end{enumerate}\nTo overcome \\ref{item:eventereasonone},\nfor ease of notation we set $q_j\\coloneqq \\Pr\\left(d^*_1(w)=j\\right)$ and\\linebreak[4] $q_j'\\coloneqq\\Pr\\left(d^*_1(w)=j|E_{w}(\\ell)\\right)$ for each $j\\in \\mathbb{N}$,\nand define\n\\begin{align*}\nJ^+ & \\coloneqq\\left\\{j:q_j > q_j'\\right\\} \\qquad  \\text{and}\\qquad \nJ^-  \\coloneqq\\left\\{j:q_j < q_j'\\right\\}.\n\\end{align*}\nSince for any $w \\in \\mathcal{V} \\cup \\mathcal{F}$ we have\n$$\n\\sum_{j=0}^\\infty (q_j-q_j') = \\sum_{j=0}^\\infty q_j - \\sum_{j=0}^\\infty q_j' = 1-1=0,\n$$\nwe deduce that\n$$\n\\sum_{j=0}^\\infty \\left|q_j-q_j'\\right|\n = \\sum_{j \\in J^+} \\left(q_j-q_j'\\right) - \\sum_{j \\in J^-}\\left(q_j-q_j'\\right)\n= 2\\sum_{j \\in J^+}\\left(q_j-q_j'\\right).\n$$\nTherefore we have\n\\begin{align}\nd_{\\mathrm{TV}}\\left(d^*_1(w),d^*_1(w)|_{E_{w}(\\ell)}\\right)\n& = 2\\sum_{j\\in J^+} \\left(q_j-q_j'\\right) \\nonumber\\\\\n& \\leq 2\\cdot \\sum_{j \\in J^+} \\Pr\\left(d^*_1(w)=j\\right) - \\Pr\\left( (d^*_1(w)=j)\\capE_{w}(\\ell)\\right) \\nonumber\\\\\n& \\leq 2 \\cdot\\Pr\\left(\\overline{E_{w}(\\ell)}\\right) \\nonumber\\\\\n& \\leq 2 \\exp\\left(-\\Theta\\left(\\sqrt{\\log n}\\right)\\right) \\nonumber\\\\\n& \\le \\varepsilon\/3, \\label{eq:totalvariationconditioning}\n\\end{align}\nwhere we applied Lemma~\\ref{lem:eventE} in the penultimate line, and where the last line holds\nfor $\\varepsilon$ tending to $0$ sufficiently slowly.\n\nTo address \\ref{item:eventereasontwo} we note that if $\\ell = \\ell(\\varepsilon)$ is sufficiently large, then\n\\begin{equation}\\label{eq:totalvariationpoissons}\n    d_{\\mathrm{TV}}\\left(\\mathrm{Po}(d\\intfactsurv{(\\ell+1)\/2}),\\mathrm{Po}(d\\hat \\rho_*)\\right)<\\varepsilon\/3\n\\end{equation}\nbecause $\\intfactsurv{t} \\xrightarrow{t\\to \\infty} \\hat \\rho_*$ by definition.\n\nTo complete the proof, observe that Lemma~\\ref{lem:survivalprobs}~\\ref{eq:probfactnodesurvives} implies that\n\\begin{align*}\nd_{\\mathrm{TV}}\\left(d^*_1(w)|_{E_{w}(\\ell)} \\, , \\, \\mathrm{Po}(d\\intfactsurv{(\\ell+1)\/2})\\right) =\nd_{\\mathrm{TV}}\\left(d^*_1(w)|_{E_{w}(\\ell)} \\, , \\, \\mathrm{Bi}\\left(\\mathrm{Po}(d),\\intfactsurv{(\\ell+1)\/2} \\right)\\right) \\le \\varepsilon\/3,    \n\\end{align*}\nand combining this with~\\eqref{eq:totalvariationconditioning} and~\\eqref{eq:totalvariationpoissons},\nwe obtain\n\\begin{align*}\nd_{\\mathrm{TV}}\\left(d^*_1(w),\\mathrm{Po}(d\\hat \\rho_*)\\right)\n& \\le d_{\\mathrm{TV}}\\left(d^*_1(w),d^*_1(w)|_{E_{w}(\\ell)}\\right)\n\t+ d_{\\mathrm{TV}}\\left(d^*_1(w)|_{E_{w}(\\ell)},\\mathrm{Po}(d\\intfactsurv{(\\ell+1)\/2})\\right)\\\\\n\t& \\hspace{3cm}+ d_{\\mathrm{TV}}\\left(\\mathrm{Po}(d\\intfactsurv{(\\ell+1)\/2},\\mathrm{Po}(d\\hat \\rho_*)\\right)\\\\\n\t& \\le \\varepsilon\n\\end{align*}\nas required.\n\\end{proof}\nAs a further consequence of Corollary~\\ref{cor:totalvariationdistance}, we can asymptotically determine the expected degree distribution in $G_{\\ell}$.\n\\begin{corollary}\\label{cor:expectationvarfac}\nThere exist $\\varepsilon = o(1)$ and $\\ell=\\ell(\\varepsilon)$ such that for all $j \\in \\mathbb{N}$, \n\\[\\expec\\left(\\varpropl{j}{\\ell}\\right)=\\prob\\left(\\widetilde\\mathrm{Po}(d\\hat \\rho_*)=j\\right)\\pm\\varepsilon,\\]\nand\n\\[\\expec\\left(\\factpropl{j}{\\ell}\\right)=\\prob\\left(\\widetilde\\mathrm{Bi}(r,\\rho_*)=j\\right)\\pm\\varepsilon.\\]\n\\end{corollary}\n\n\n\n\\begin{proof}\nObserve that for any $j \\in \\mathbb{N}$ and any variable node $w$, by linearity of expectation and Corollary~\\ref{cor:Gldegalmosteq},\nfor some $\\varepsilon_1=o(1)$ we have\n$$\n\\expec\\left(\\varpropl{j}{\\ell}\\right)=\\prob(d_{G_{\\ell}}(w)=j) = \n\\prob\\left(\\tilde d_1(w)=j\\right) \\pm \\varepsilon_1\n$$\nFurthermore, Corollary~\\ref{cor:totalvariationdistance} implies that for some $\\varepsilon_2=o(1)$ we have\n$$\n\\prob\\left(\\tilde d_1(w)=j\\right) = \\prob\\left(\\widetilde\\mathrm{Po}(d\\hat \\rho_*)=j\\right)\\pm\\varepsilon_2.\n$$\nCombining these two approximations and setting $\\varepsilon = \\varepsilon_1+\\varepsilon_2 = o(1)$ completes the proof of the first statement.\nThe second statement is proven similarly.\n\\end{proof}\nWe will show concentration of $\\varpropl{j}{\\ell}$ and $\\factpropl{j}{\\ell}$ around their expectations\nby an application of an Azuma-Hoeffding inequality (Lemma~\\ref{lem:Azuma}).\nWe will therefore define a vertex-exposure martingale\n(or, using the terminology of factor graphs, a variable-exposure martingale)\n $X_0,\\ldots,X_n$ on $G$, where\n$X_0=\\expec(\\varpropl{j}{\\ell})$ and $X_n=\\varpropl{j}{\\ell}$. This means that we reveal the variable nodes of $G$\nand their incident edges one by one,\nsuccessively revealing more and more information about the input graph.\nWe would like to show that this martingale satisfies a Lipschitz property so that we can apply Lemma~\\ref{lem:Azuma}.\nHowever, this Lipschitz property cannot be guaranteed in general.\nWe therefore restrict attention to a class of factor graphs on which the Lipschitz property does indeed hold.\n\n\\begin{definition}\\label{def:graphclass}\nLet $\\mathcal{G}$ be the class of factor graphs $G=G(H)$ of $r$-uniform hypergraphs $H$ on vertex set $[n]$\nsuch that no node has degree greater than $\\log n$ in $G$.\n\\end{definition}\nIt is important to observe that for fixed $r$ and for $p$ in the range we are considering\n(i.e. $\\Theta\\left(\\binom{n-1}{r-1}^{-1}\\right)$), the factor graph\n $G^r(n,p)$ is very likely to lie in $\\mathcal{G}$, so the restriction to this class is reasonable.\n\\begin{claim}\\label{claim:maxdeg}\nWe have \n\\[\\prob(G^r(n,p)\\in\\mathcal{G})=1-o(1).\\]\n\\end{claim}\nThe (standard) proof appears in Appendix~\\ref{app:maxdeg}.\n\nNow let $\\overline{G}=G^r(n,p)|_{\\mathcal{G}}$ be the factor graph of the $r$-uniform binomial random hypergraph $H^r(n,p)$\nconditioned on being in $\\mathcal{G}$. For each $i\\in\\mathbb{N}$ let $Z_i\\in\\{0,1\\}^{\\binom{[i-1]}{r-1}}$\nbe the sequence $(Y_{i,A})_{A \\in \\binom{[i-1]}{r-1}}$ of indicator random variables of the events\nthat there is a factor node of $\\overline{G}$ whose neighbours are $\\{i\\}\\cup A$\n(or equivalently that $\\{i\\}\\cup A$ forms an edge of $H^r(n,p)$).\nLet $f$ be a graph theoretic function.\nSince there is a one-to-one correspondence between the factor graph $\\overline{G}$ and its associated sequence $(Z_i)=(Z_i)_{i \\in [n]}$,\nwe can view the \nrandom variable $X\\coloneqq f(\\overline{G})$ as a function of the sequence $(Z_i)$ and we let $X_i\\coloneqq\\expec(X|Z_1,\\ldots,Z_i).$\nThis martingale can be seen as revealing variable nodes of $\\overline{G}$ one by one, and $X_i$ is\nthe expected value of $X$ conditioned on the information revealed after $i$ steps.\n\n\\begin{definition}\\label{lem:lipschitzproperty}\nA graph function $f$ is \\emph{$c$-variable-Lipschitz} if whenever two factor graphs $G$ and $G'$ differ at exactly one variable node\n(but any number of factor nodes),\nthen $|f(G)-f(G')|\\leq c$ holds.\n\\end{definition}\n\nThis is very similar to the standard notion of being $c$-vertex-Lipschitz -- indeed, it corresponds\nprecisely to the original hypergraph being $c$-vertex-Lipschitz -- and it is\na standard observation that being $c$-variable-Lipschitz\nimplies that the corresponding variable-exposure martingale satisfies $|X_i-X_{i-1}|\\leq c$,\nsee e.g.\\ Corollary 2.27 in~\\cite{JansonLuczakRucinskiBook}.\n\nFor the purposes of the Azuma-Hoeffding argument,\nwe will view $\\varpropl{j}{\\ell},\\factpropl{j}{\\ell}$ as graph functions---thus by the original\ndefinition we have e.g.\\ $\\varpropl{j}{\\ell} = \\varpropl{j}{\\ell}(G^r(n,p))$,\ni.e.\\ the graph function applied to the factor graph of the $r$-uniform binomial random hypergraph.\n\n\n \\begin{lemma}\nFor $\\ell\\in\\mathbb{N}$, the graph functions $\\varpropl{j}{\\ell},\\factpropl{j}{\\ell}$ are $c_{\\ell}$-variable-Lipschitz within the class $\\mathcal{G}$, where $c_{\\ell}=\\frac{2(\\log n)^{\\ell+1}}{n}$.\n \\end{lemma}\n \n \\begin{proof}\n We will show that $\\varpropl{j}{\\ell}$ is $c_\\ell$-variable-Lipschitz---the corresponding proof for $\\factpropl{j}{\\ell}$ is completely analogous.\n \n Suppose that $G,G' \\in \\mathcal{G}$ are two factor graphs which differ at exactly one variable node, say $v$,\n and let $A_j = A_j(G,\\ell)$ and $A_j'=A_j'(G',\\ell)$ be the sets of variable nodes of degree~$j$ in $G_\\ell$ and $G_{\\ell}'$, respectively.\n Since $G,G'$ differ only at $v$, any edges in the symmetric difference $G_\\ell \\Delta G_{\\ell}'$ must lie within\n distance $\\ell+1$ of $v$ in either $G$ or $G'$,\n and therefore any nodes in the symmetric difference $A_j \\Delta A_j'$ must lie within distance $\\ell+1$ of $v$ in either $G$ or $G'$.\nSince $G,G' \\in \\mathcal{G}$, and therefore have maximum degree at most $\\log n$,\n there are at most $(\\log n)^{\\ell+1}$ such nodes in each of $G,G'$, and\ntherefore $|A_j \\Delta A_j'| \\le 2(\\log n)^{\\ell+1}$. Since $\\varpropl{j}{\\ell}(G)=|A_j|\/n$ and $\\varpropl{j}{\\ell}(G')=|A_j'|\/n$,\nwe deduce that\n$$\n\\left|\\varpropl{j}{\\ell}(G)-\\varpropl{j}{\\ell}(G')\\right| \\le \\frac{2(\\log n)^{\\ell+1}}{n}\n$$\nas required.\n \\end{proof}\n \nNow for $j,\\ell \\in \\mathbb{N}$, let us define the events $B_j=B_j(\\ell) \\coloneqq \\left\\{\\left|\\varpropl{j}{\\ell} - \\expec\\left(\\varpropl{j}{\\ell}\\right)\\right| \\ge n^{-1\/3}\\right\\}$\nand $\\hat B_j=\\hat B_j(\\ell) \\coloneqq \\left\\{\\left|\\factpropl{j}{\\ell} - \\expec\\left(\\factpropl{j}{\\ell}\\right)\\right| \\ge n^{-1\/3}\\right\\}$.\n \n\\begin{lemma}\\label{lem:concentration}\nLet $G=G^r(n,p)$ be a random factor graph and $\\ell = o(\\log \\log n)$. Then\n\\[\n\\Pr\\left(\\bigcup_{j \\in \\mathbb{N}}\\left(B_j \\cup \\hat B_j\\right) \\right) = o(1).\n\\]\n\\end{lemma} \n\n\\begin{proof}\nFor convenience, we will prove that $\\Pr(\\bigcup_{j \\in \\mathbb{N}}B_j)=o(1)$---the proof of the corresponding\nstatement for the $\\hat B_j$ is almost identical.\n\nLet us define $B\\coloneqq \\bigcup_{j\\in \\mathbb{N}}B_j$, and observe that\n\\begin{align}\\label{eq:azumaconditioning}\n\\Pr(B) & = \\Pr(B \\cap \\{G \\in \\mathcal{G}\\}) + \\Pr(B \\cap \\{G \\notin \\mathcal{G}\\}) \\nonumber \\\\\n& \\le \\Pr(B | G \\in \\mathcal{G}) + \\Pr(G \\notin \\mathcal{G}) \\nonumber \\\\\n& \\le \\sum_{j \\in \\mathbb{N}} \\Pr(B_j | G \\in \\mathcal{G}) + o(1),\n\\end{align}\nwhere the last line follows by a union bound and Claim~\\ref{claim:maxdeg}.\n\nNow in order to bound the sum, let us first fix $j \\in \\mathbb{N}$.\nSince $\\ell=o(\\log\\log n)$, we have \n$(\\log n)^ {\\ell+1}=n^ {o(1)}$,\nand by Lemma~\\ref{lem:Azuma} with $c_{\\ell}=\\frac{2(\\log n)^ {\\ell+1}}{n}$ and $s=n^ {-1\/3}$,\nwe have\n\\begin{align*}\n    \\prob\\left(\\left|\\varpropl{j}{\\ell}-\\expec\\left(\\varpropl{j}{\\ell}\\right)\\right|\\geq n^{-1\/3} \\; \\Bigg| \\; G \\in \\mathcal{G}\\right)\n    &\\leq 2\\cdot \\exp\\left(-\\frac{n^{-2\/3}}{2\\cdot 2(\\log n)^{\\ell+1}\/n}\\right)< \\exp\\left(-n^{1\/2}\\right).\n\\end{align*}\nObserving that the degree of any node is (deterministically) bounded by $\\binom{n}{r-1}$,\nwe have\n$$\n\\sum_{j\\in \\mathbb{N}}\\Pr(B_j | G \\in \\mathcal{G}) \\le \\left(\\binom{n}{r-1}+1\\right)\\exp(-n^{1\/2}) = o(1).\n$$\nSubstituting this into~\\eqref{eq:azumaconditioning} completes the proof.\n\\end{proof}\nWe are now able to give the proof of Lemma~\\ref{lem:mainlemma1}.\n\\begin{proof}[Proof of Lemma~\\ref{lem:mainlemma1}]\nBy Corollary~\\ref{cor:expectationvarfac} (with $\\varepsilon\/2$ in place of $\\varepsilon$),\nif $\\ell = \\ell(\\varepsilon)$ is sufficiently large  we have\n\\[\n\\expec\\left(\\varpropl{j}{\\ell}\\right)=\\prob\\left(\\widetilde\\mathrm{Po}\\left(d\\hat \\rho_*\\right)=j\\right)\\pm\\varepsilon\/2\n\\]\nfor any $j \\in \\mathbb{N}$.\nFurthermore, by\nLemma~\\ref{lem:concentration}, we have that whp\\ for all $j\\in\\mathbb{N}$\n\\[\n\\varpropl{j}{\\ell}=\\expec\\left(\\varpropl{j}{\\ell}\\right)+o(1) = \\expec\\left(\\varpropl{j}{\\ell}\\right) \\pm \\varepsilon\/2\n\\]\nfor $\\varepsilon$ tending to $0$ sufficiently slowly,\nand combining these two facts proves the lemma for variable nodes. The proof for factor nodes is essentially identical.\n\\end{proof}\n\n\n\n\n\n\n\\section{Subcriticality: Proof of Lemma~\\ref{lem:mainlemma2}}\\label{sec:prooflowerbound}\nOur goal in this section is to show that after some large number $\\ell$ rounds of the peeling process on $G^r(n,p)$ have been completed,\nwhp\\ very few nodes will be disabled in subsequent rounds (at most $\\varepsilon n$ for some $\\varepsilon=\\varepsilon(n) =o(1)$),\nthus proving Lemma~\\ref{lem:mainlemma2}.\n\n\\emph{\nLet us fix $\\ell,\\varepsilon_1$ as in Lemma~\\ref{lem:mainlemma1}, and for the rest of this section we will assume that\nthe high probability events of Lemma~\\ref{lem:mainlemma1} and Proposition~\\ref{prop:numberfactornodes}\nboth hold.\n}\n\nTo help intuitively describe the argument, let us suppose for simplicity that in round $\\ell$ exactly \\emph{one}\nnode $x_0$ is disabled and we consider the future effects of such a disabling.\nSince $x_0$ was disabled, it had degree at most one,\nand therefore there is at most one neighbour $x_1$ whose degree is decreased as a result.\nIf $x_1$ originally had degree two, it now has degree one and will therefore be disabled in round $\\ell+1$.\nContinuing in this way, we observe that we will never be disabling more than one node in any subsequent round.\nFurthermore, if we reach a node $x_i$ whose original degree was not exactly two, the peeling process stops\n(either without disabling this node, or once it has been disabled and no further nodes' degrees are decreased).\nHeuristically, it will not take long before we reach a node whose original degree was not exactly two---this\nis because Lemma~\\ref{lem:mainlemma1} implies in particular that a constant proportion\nof the nodes have degree at least three.\n\nOf course, in reality there may be more than one node disabled in round $\\ell$.\nThis slightly complicates matters because some node may receive the knock-on effects\nof more than one disabling in round $\\ell$, and therefore have its degree decreased by more than one.\nHowever, this will turn out to be no more than a technical nuisance.\n\nWe first need a result which states that almost any graph with a fixed (reasonable) degree sequence\nis approximately equally likely to be $G_\\ell$, the graph obtained from $G=G^r(n,p)$ after $\\ell$ rounds of the peeling process.\nTo introduce this result, we need some definitions.\n\n\\begin{definition}\nAn \\emph{$r$-duplicate} in a factor graph consists of two factor nodes and $r$ variable nodes which together\nform a copy of $K_{2,r}$.\n\\end{definition}\n\nObserve that an $r$-duplicate would correspond to a double-edge in an $r$-uniform hypergraph,\nwhich in our model cannot occur since the hypergraph must be simple.\nTherefore our factor graphs may not contain any $r$-duplicates.\nOn the other hand, a ``loop'', in the sense of an edge which contains the same vertex more than once,\nmust involve a double-edge in the corresponding factor graph. This motivates the following definition,\nwhich (roughly) describes when a factor graph corresponds to a simple hypergraph.\n\n\\begin{definition}\\label{def:rplain}\nWe say that a factor graph is \\emph{$r$-plain} if:\n\\begin{enumerate}\n\\item it contains no double-edge, i.e.\\ two edges between the same variable node and factor node;\n\\item it contains no $r$-duplicates.\n\\end{enumerate}\n\\end{definition}\n\n\n\n\\begin{claim}\\label{claim:uniformity}\nSuppose $H_1,H_2$ are two $r$-plain factor graphs with common variable node set $\\mathcal{V}=\\mathcal{V}(H_1)=\\mathcal{V}(H_2)=[n]$\nand with factor node sets $\\mathcal{F}_1=\\mathcal{F}(H_1)$ and $\\mathcal{F}_2=\\mathcal{F}(H_2)$.\nSuppose further that there is a bijection $\\phi: \\mathcal{V} \\cup \\mathcal{F}(H_1) \\to \\mathcal{V} \\cup \\mathcal{F}(H_2)$\nsuch that\n\\begin{itemize}\n\\item $\\phi(\\mathcal{V})=\\mathcal{V}$;\n\\item $d_{H_2}(\\phi(v))=d_{H_1}(v)$ for all $v \\in \\mathcal{V} \\cup \\mathcal{F}(H_1)$.\n\\end{itemize}\nLet $G=G^r(n,p)$.\nThen \n\\[\n\\prob\\left(G_{\\ell}=H_1\\right)=\\prob\\left(G_{\\ell}=H_2\\right).\n\\]\n\\end{claim} \n\nThis claim is very similar to standard results for simple graphs or hypergraphs\n(see e.g.~\\cite{Molloy2005})\nand we defer the proof to Appendix~\\ref{sec:proofofuniformity}. However, we note that in our setting there \nis one subtle technical difficulty to overcome which does not appear in many other cases,\nnamely that given a factor graph $G$ such that $G_\\ell=H_1$,\nif we transform $G$ by changing $H_1$ to $H_2$ but otherwise leaving $G$ unchanged,\nwe need to show that\nthe resulting graph is indeed the factor graph of an $r$-uniform hypergraph, and in particular is $r$-plain.\n\n\nClaim~\\ref{claim:uniformity} tells us that the factor graph $G_\\ell$ after $\\ell$ rounds\nof the peeling process is uniformly random conditioned on its degree sequence and being $r$-plain.\nSince Lemma~\\ref{lem:mainlemma1} tells us the degree sequence quite precisely,\nthis is very helpful. We can change our point of view by saying that we first reveal\nthe degree sequence of $G_\\ell$ without revealing any of its edges,\nand subsequently we reveal edges only as required. More precisely, we consider the\n\\emph{configuration model}, in which each node is given half-edges based on its degree,\nand we generate a uniformly random perfect matching between the two classes of half-edges\n(at variable and factor nodes) conditioned on the\nresulting factor graph being $r$-plain.\nWe need to know that this conditioning is not too restrictive, i.e.\\ that the probability\nthat the resulting factor graph is $r$-plain is not too small. This will be stated\nin Proposition~\\ref{prop:simpleprob}, for which we first need some preliminaries.\nWe begin by observing that\n$G^r(n,p)$ does not have too many nodes of high degree.\n\n\\begin{definition}\\label{def:largedegsquare}\nGiven a function $\\omega=\\omega(n) \\to \\infty$, we say that a factor graph $H$\nhas property $\\widetilde{\\mathcal{D}}=\\widetilde{\\mathcal{D}}(\\omega,n)$\nif\n$$\n\\sum_{\\substack{v \\in \\mathcal{V}(H):\\  d(v)>\\omega}}\\  d(v)^2 = o(n).\n$$\nFurthermore given $\\varepsilon>0$ we say that $H$ has property\n$\\mathcal{D}=\\mathcal{D}(\\varepsilon,\\omega,n)$ if it satisfies property $\\widetilde{\\mathcal{D}}(\\omega,n)$\nand also satisfies the conclusion of Lemma~\\ref{lem:mainlemma1} (with this $\\varepsilon$).\n\\end{definition}\n\n\\begin{claim}\\label{claim:largedegsquare}\nFor any $\\omega\\xrightarrow{n \\to \\infty} \\infty$,\nwith high probability $G^r(n,p)$ has property $\\widetilde{\\mathcal{D}}(\\omega,n)$.\n\\end{claim}\n\\begin{proof}\nFor $k \\in \\mathbb{N}$, let us define $X_k$ to be the number of variable nodes of degree $k$\nand $X_{\\ge k}\\coloneqq \\sum_{j \\in \\mathbb{N}_{\\ge k}} X_j$.\nObserve that the expected degree of a vertex is $\\binom{n-1}{r-1}p = (1+o(1))d$.\nIt is a standard fact that the degrees of vertices are approximately\ndistributed as independent $\\mathrm{Po}(d)$ variables. More formally (though much weaker),\nit is an easy exercise in the second-moment method\nto prove that whp, for any $k\\in \\mathbb{N}$ we have $X_{\\ge k} \\le n\\cdot \\Pr(\\mathrm{Po}(2d)\\ge k)$ (we omit the details).\nWe therefore have\n\\begin{align*}\n\\sum_{v \\in \\mathcal{V}(H): d(v) \\ge \\omega} d(v)^2 = \\sum_{k \\ge \\omega} k^2 X_k\n& \\le n\\sum_{k \\ge \\omega} k^2 \\frac{e^{-2d}(2d)^k}{k!} = n \\cdot (1+o(1))\\omega^2\\frac{e^{-2d}(2d)^{\\omega}}{\\omega!} = o(n),\n\\end{align*}\nas required.\n\\end{proof}\n\nNote that if $\\widetilde{\\mathcal{D}}$ holds in a factor graph $G$, then it also holds\nin any subgraph of $G$, and in particular in $G_{\\ell}$, the factor graph obtained\nafter $\\ell$ steps of the peeling process. Together with Lemma~\\ref{lem:mainlemma1},\nthis shows that, with $\\ell$ and $\\varepsilon$ as in given in that lemma and any $\\omega \\to \\infty$,\nsetting $G = G^r(n,p)$,\nwith high probability $G_\\ell$ satisfies $\\mathcal{D}(\\varepsilon,\\omega,n)$.\n\nLet us observe further that $\\mathcal{D}$ is a property that depends only on the\ndegree sequences of variable and factor nodes of the graph, and therefore with a slight\nabuse of terminology we may also say that it holds in a factor graph with half-edges,\nwhere we have not yet determined which half-edges will be matched together.\n\n\n\n\n\\begin{proposition}\\label{prop:simpleprob}\nLet $\\varepsilon = o(1)$ and\nsuppose that $n$ variable nodes and $m=(1+o(1))\\frac{dn}{r}$ factor nodes are given half-edges\nin such a way that property $\\mathcal{D}(\\varepsilon,\\varepsilon^{-1\/4},n)$ holds.\nSuppose also that the total numbers of half-edges at factor nodes and at variable nodes are equal,\nand that we construct a uniformly random perfect matching between these two sets of half-edges.\nThen there exists a constant $c_0>0$ (independent of $\\varepsilon,n$)\nsuch that for sufficiently large $n$ the probability that the resulting factor graph is\n$r$-plain is at least $c_0$.\n\\end{proposition}\n\n\n\nThe proof of Proposition~\\ref{prop:simpleprob} is a standard exercise\nin applying the method of moments to determine the asymptotic distribution\nof the number of double edges and $r$-duplicates -- we omit the details.\nThe proposition states in particular that we may condition on the\nresulting graph being $r$-plain without skewing the distribution of the matching too much.\nMore precisely, any statements that are true with high probability for the uniformly random perfect matching\nare also true with high probability under the condition that the resulting graph is simple.\nTherefore in what follows, for simplicity we will suppress this conditioning.\n\n\\begin{definition}[Change process]\\label{def:changeprocess}\nWe will track the changes that the peeling process makes after reaching round $\\ell$\nby revealing information a little at a time as follows.\n\\begin{itemize}\n\\item Reveal the degrees of all nodes.\n\\item While there are still nodes of degree one, pick one such node $x_0$.\n\\begin{itemize}\n\\item Reveal its neighbour $x_1$, delete the edge $x_0x_1$ and update the degrees of $x_0,x_1$.\n\\item If $x_1$ now has degree one, continue from $x_1$; otherwise find a new $x_0$ (if there is one).\n\\end{itemize}\n\\end{itemize}\n\\end{definition}\n\n\n\\noindent In other words, we track the changes in a depth-first search manner (rather than the breadth-first\nview of considering rounds of the peeling process). We call this the \\emph{change process}.\n\nObserve that we only reveal edges one at a time (just before deleting them).\nThe following claim is simple given Lemma~\\ref{lem:mainlemma1}, but is the essential heart of our proof.\nRecall that $\\varepsilon_2 \\coloneqq \\sqrt{\\varepsilon_1}$ as defined in Lemma~\\ref{lem:mainlemma2}.\n\n\n\\begin{claim}\\label{claim:terminationprob}\nLet $G'$ be any graph obtained from $G_\\ell$ by deleting at most $\\varepsilon_2 n$ edges.\nThen when revealing the second endpoint of any half-edge, the probability of revealing a node of degree at least three\nis at least\n$$\n\\min\\left\\{\\frac{(d\\hat \\rho_*)^2 \\exp(-d\\hat \\rho_*)}{2} , \\frac{(r-1)(r-2)\\rho_*^2(1-\\rho_*)^{r-3}}{2}\\right\\} - 20\\varepsilon_2.\n$$\nIn particular there exists a constant $c=c(r,d)>0$ such that this probability is at least $c$.\n\\end{claim}\nWe defer the proof to Appendix~\\ref{sec:proofofterminationprob}.\n\n\\noindent This claim tells us that, provided we have not deleted too many edges so far, there is a reasonable probability\nof revealing a node of degree at least three, which blocks the continued propagation of any deletions.\n\nLet $c=c(r,d)$ be as in Claim~\\ref{claim:terminationprob}\nand let us set $\\delta_1\\coloneqq\\varepsilon_1^{3\/4}$.\nWe now define an abstract branching process\nwhich will provide an upper coupling on the change process starting from $G_\\ell$.\n\n\\begin{definition}\nLet $\\mathcal{T}$ be a branching process which begins with $\\delta_1n$ vertices in generation~$0$,\nand in which each vertex independently has a child with probability $1-c$, and otherwise has no children.\n\\end{definition}\n\n\n\n\\begin{proposition}\\label{prop:changecoupling}\nThe process $\\mathcal{T}$ can be coupled with the change process in such a way that,\nif both processes are run until one of the stopping conditions\n\\begin{itemize}\n\\item $\\mathcal{T}$ has reached size at least $\\varepsilon_2 n$;\n\\item $\\mathcal{T}$ has died out,\n\\end{itemize}\nis satisfied, then $\\mathcal{T}$ forms an upper coupling on the change process.\n\\end{proposition}\n\n\\begin{proof}\nWe first need to show that whp\\ we make at most $\\delta_1 n$ changes in round $\\ell+1$ of the peeling process.\nSince we have assumed that the high probability statement of Lemma~\\ref{lem:mainlemma1} holds, we have\n$\\varpropl{j}{\\ell}=\\prob(\\widetilde\\mathrm{Po}(d\\hat \\rho_*)=j)\\pm\\varepsilon_1$ and $\\factpropl{j}{\\ell}=\\prob(\\widetilde\\mathrm{Bi}(r,\\rho_*)=j)\\pm\\varepsilon_1$.\nBy the definition of the peeling process (Definition~\\ref{def:peeling}) the only change we make when moving from $G_{\\ell}$ to $G_{\\ell+1}$ is that we disable all nodes of degree one.\nThe proportion of such variable and factor nodes in $G_{\\ell}$ is $\\varpropl{1}{\\ell}$ and $\\factpropl{1}{\\ell}$ respectively.\nRecalling that $\\prob(\\widetilde\\mathrm{Po}(d\\hat \\rho_*)=1) = \\prob(\\widetilde\\mathrm{Bi}(r,\\rho_*)=1) =0$,\nthis immediately implies that at most\n$\\varepsilon_1(m+n) \\le \\delta_1 n$ nodes are disabled in round $\\ell+1$ of the peeling process,\nand these disablings represent the first nodes of the change process.\n\n\nNow the proposition follows directly from the observation that in the change process,\na node only has at most one incident edge deleted (if it has degree one), and\ntherefore at most one neighbour is revealed, along with Claim~\\ref{claim:terminationprob},\nwhich implies that the probability of not causing any further changes is at least $c$.\nThe first stopping condition ensures that we have deleted at most $\\varepsilon_2 n$ edges,\nand therefore the assumptions of Claim~\\ref{claim:terminationprob} are indeed satisfied.\n\\end{proof}\n\nIn view of Proposition~\\ref{prop:changecoupling}, it is enough to prove that whp\\ the\nbranching process $\\mathcal{T}$ dies out (i.e.\\ fulfills the second stopping condition)\nbefore reaching size $\\varepsilon_2 n$.\n\n\\begin{proposition}\\label{prop:changebranchsmall}\nWhp\\ $\\mathcal{T}$ contains at most $\\varepsilon_2 n$ vertices.\n\\end{proposition}\n\n\\begin{proof}\nIn order to reach size $\\varepsilon_2 n$, the first (at most) $\\varepsilon_2 n$ vertices of the process\nwould have to have a total of at least $(\\varepsilon_2-\\delta_1)n$ children,\nwhich, by Lemma~\\ref{lem:chernoffbounds}, occurs with probability at most\n\\begin{align*}\n\\Pr\\left(\\mathrm{Bi}(\\varepsilon_2 n,1-c)\\ge (\\varepsilon_2-\\delta_1)n\\right) & \\le 2\\cdot \\exp\\left(-\\frac{(c\\varepsilon_2-\\delta_1)^2 n^2}{2\\varepsilon_2 n + 2(c\\varepsilon_2-\\delta_1)n\/3}\\right)\\\\\n& \\le 2\\cdot \\exp\\left(-\\frac{(c\\varepsilon_2\/2)^2n}{3\\varepsilon_2}\\right) =o(1)\n\\end{align*}\n(since $\\varepsilon_2 n = \\sqrt{\\varepsilon_1}n \\to \\infty$)\nas required.\n\\end{proof}\n\nWe can now complete the proof of Lemma~\\ref{lem:mainlemma2}.\n\n\\begin{proof}[Proof of Lemma~\\ref{lem:mainlemma2}]\nProposition~\\ref{prop:changecoupling} implies that the probability that at least $\\varepsilon_2 n$\nfurther nodes are disabled after round $\\ell$ in the peeling process is at most the probability\nthat $\\mathcal{T}$ reaches size $\\varepsilon_2 n$. But this event has probability $o(1)$ by\nProposition~\\ref{prop:changebranchsmall}.\n\nThis proves that whp\\ at most $\\varepsilon_2 n$ further nodes are disabled\nafter round $\\ell$ in the peeling process. To show the remaining two statements,\nobserve that\nsince disabling a node deletes at most one edge, and therefore changes the degree of at most one variable node\nand at most one factor node, at most $\\varepsilon_2 n$ nodes of each type will have their degree changed.\nIt follows immediately that for any $j \\in \\mathbb{N}$ we have $\\varprop{j} = \\varpropl{j}{\\ell} \\pm \\varepsilon_2$,\nwhile similarly\n$$\n\\factprop{j} = \\factpropl{j}{\\ell} \\pm \\frac{\\varepsilon_2 n}{m} = \\factpropl{j}{\\ell} \\pm \\frac{2\\varepsilon_2 r}{d},\n$$\nwhere we have used the fact that whp\\ $\\frac{n}{m} \\le \\frac{2r}{d}$ by Proposition~\\ref{prop:numberfactornodes}.\n\\end{proof}\n\n\n\n\n\\section{Concluding remarks}\\label{sec:concluding}\n\n\\subsection{Upper bound on $L_P,L_C$}\nTheorem~\\ref{thm:mainresultcycle} and Corollary~\\ref{cor:upperboundlongestpath} state that whp\\ $L_P$ and $L_C$,\nthe length of the longest loose path and longest loose cycle respectively in $H^r(n,p)$,\nsatisfy $L_P,L_C\\leq(\\min\\{\\beta,\\gamma\\}+o(1))\\cdot n$, but which of $\\beta,\\gamma$ is smaller?\nRecall that both $\\beta$ and $\\gamma$ are functions of $d$ and $r$, with $\\beta=\\gamma =0$ for $d<d^*$.\nNumerical approximations and plots with Mathematica suggest that \nfor $r=2,3$, we have\n$\\beta \\le \\gamma$\nfor all $d$,\nbut that for $r\\ge 4$, for values of $d$ not too much larger than $d^*$ we have $\\gamma<\\beta$,\nand there is a crossing point after which (for larger $d$) we have $\\beta <\\gamma$.\nIt would be interesting to investigate this behaviour more closely to determine whether this is indeed true,\nand to determine the crossing point precisely as a function of $r$.\n\n\\subsection{High-order cores}\nThe methods used in this paper are amenable to more general definitions of cores in hypergraphs.\nMore precisely, vertex degrees are far from the only type of degrees that have been\nextensively studied in hypergraphs---one can consider the degrees of $j$-sets of vertices for each $1\\le j \\le r-1$,\nand for each $j$ there is a natural associated definition of a core. So far only the case $j=1$ has been studied,\nbut it would also be interesting to consider ``high-order cores'', i.e.\\ $j\\ge 2$.\n\n\n\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Preliminaries}\\label{sec:preli}\n\n\n\n\n\n\n\nWe begin with some notations. Let $C$ be a generic constant that may vary from one place to another, and set $H := L^{2}((0,1);\\mathbb{R})$, equipped with usual product $\\langle \\cdot,\\cdot \\rangle$ and norm $|\\cdot|$. We denote by $(\\mathcal{L}(H),|\\cdot|_{\\mathcal{L}(H)})$ the Banach space of all bounded linear operators from $H$ to $H$. Besides, let $(\\mathcal{L}_{2}(H),\\langle\\cdot,\n\\cdot\\rangle_{\\mathcal{L}_{2}(H)},|\\cdot|_{\\mathcal{L}_{2}(H)})$ be the real separable Hilbert space  of all Hilbert--Schmidt operators from $H$ to $H$. Throughout this paper, let  $A \\colon \\mathcal{D}(A) \\subset H \\to H$ be the Laplacian operator with homogeneous Dirichlet boundary conditions, i.e.,  $Au = \\Delta u, u \\in \\mathcal{D}(A) = H_{0}^{1} \\cap H^{2}$.  Then $A$ generates a $C_{0}$-semigroup $\\{E(t) := e^{tA}\\}_{t \\geq 0}$ on $H$ and $-Ae_i=\\lambda_{i}e_i, i \\in \\mathbb{N}^{+}$ with $\\{\\lambda_{i} := \\pi^2i^2\\}_{i \\in \\mathbb{N}^{+}}$ and $\\{e_{i}(\\xi) := \\sqrt{2}\\sin(i \\pi \\xi),\\xi \\in (0,1)\\}_{i \\in \\mathbb{N}^{+}}$ being an orthonormal basis of $H$. Further, one can define the fractional powers $(-A)^{\\gamma}$ of $-A$ and the Hilbert space\n     $\n         \\dot{H}^{\\gamma}\n     :=\n         \\mathcal{D}((-A)^{\\frac{\\gamma}{2}})\n     $\nfor every $\\gamma \\in \\mathbb{R}$, equipped with the inner product\n     $\n         \\langle u,v \\rangle_{\\dot{H}^{\\gamma}}\n     =\n         \\langle\n             (-A)^{\\frac{\\gamma}{2}}u\n             ,\n             (-A)^{\\frac{\\gamma}{2}}v\n         \\rangle\n     =\n         \\sum_{i=1}^{\\infty}\n         \\lambda_{i}^{\\gamma}\n         \\langle u,e_{i} \\rangle\n         \\langle v,e_{i} \\rangle\n     $\n     and the corresponding norm\n     $\n         |u|_{\\dot{H}^{\\gamma}}\n     =\n         \\sqrt{\\langle u,u \\rangle_{\\dot{H}^{\\gamma}}},\n     u,v \\in \\dot{H}^{\\gamma}$; see \\cite[Appendix~B.2]{kruse2014strong}.\n\n\n\n\n\n\n\n\nAssume that the  cylindrical Wiener process $\\{W(t)\\}_{t \\geq 0}$ admits a formal series\n$$\nW(t) = \\sum_{i=1}^{\\infty}\\beta_{i}(t) e_{i},\n\\quad t \\geq 0,\n$$\nwhere $\\{\\beta_{i}(t)\\}_{i \\in \\mathbb{N}^{+}}, t \\geq 0$ is a sequence of independent real-valued standard Brownian motions.\nMoreover, assume that $Q \\colon H \\to H$ is a self-adjoint, positive definite bounded linear operator, and satisfies $Qe_i=q_ie_i$, $i \\in \\mathbb{N}^{+}$ for a sequence of positive numbers $\\{q_i\\}_{i \\in \\mathbb{N}^{+}}$. Denote by $Q^{-\\frac{1}{2}} \\colon Q^{\\frac{1}{2}}(H) \\to H$ the inverse of $Q^{\\frac{1}{2}}$. To proceed, we make the following assumptions.\n\n\n\n\n\n\\begin{Assumption}\\label{ass:AQ\n      Assume that  $\\mathcal{D}(A) \\subset \\mathcal{D}(Q^{-\\frac{1}{2}})$ and\n      \\begin{equation}\\label{eq:hybridAQ\n            |(-A)^{\\frac{1}{2}}Q^{\\frac{1}{2}}|_{\\mathcal{L}_{2}(H)}\n            <\n            \\infty,\n            \\quad\n            |Q^{-\\frac{1}{2}}(-A)^{-1}|_{\\mathcal{L}(H)}\n            <\n            \\infty.\n      \\end{equation}\n\\end{Assumption}\n\n\n\n\n\n\n\n\\begin{Assumption}\\label{ass:F\n      Let $F \\colon H \\to H$ be Fr\\'{e}chet differentiable and there exist $L_{F}, L > 0$ such that\n      \\begin{align}\n      \\label{eq:globalLF\n            |F(u)-F(v)|\n            \\leq&~\n            L_{F}|u-v|,\n            \\quad u,v \\in H,\n      \\\\\n      \\label{eq:auxiliaryassumptionFuH1\n            |F(u)|_{\\dot{H}^{1}}\n            \\leq&~\n            L(1+|u|_{\\dot{H}^{1}}),\n            \\quad u \\in \\dot{H}^{1},\n      \\\\\n      \\label{eq:auxiliaryassumption\n            |F'(u)v|_{\\dot{H}^{2}}\n            \\leq&~\n            L\\big(1+|u|_{\\dot{H}^{2}}^{2}\\big)|v|_{\\dot{H}^{2}},\n            \\quad u,v \\in \\dot{H}^{2}.\n      \\end{align}\n\\end{Assumption}\n\n      Concerning Assumptions \\ref{ass:AQ} and \\ref{ass:F}, for example,  one may take $q_{i} = \\lambda_{i}^{-\\delta},i \\in \\mathbb{N}^{+}$\n      for some $\\delta \\in (\\frac{3}{2},2]$\n      and $F \\colon H \\to H$ a Nemytskij operator defined by\n      $\n            F(u)(\\xi) :=  f(u(\\xi)),\\xi \\in (0,1), u \\in H     $\n      with $f \\in C_{b}^{3}(\\mathbb{R})$.\n\n\n\n\n\nFrom \\cite[Theorem 7.2]{da2014stochastic}, we know that \\eqref{eq:SPDE} admits a unique mild solution.\n\\begin{Proposition}\n      Suppose that Assumptions \\ref{ass:AQ} and \\ref{ass:F} hold. Then \\eqref{eq:SPDE} admits a unique mild solution $\\{X_{x}^{\\varepsilon}(t)\\}_{t \\geq 0}$ given by\n     \n      \\begin{equation}\\label{eq:SPDEmildsolution\n        X_{x}^{\\varepsilon}(t)\n        =\n        E(t)x\n        +\n        \\int_{0}^{t} E(t-s) F(X_{x}^{\\varepsilon}(s)) \\diff{s}\n        +\n        \\varepsilon \\int_{0}^{t} E(t-s)Q^{\\frac12} \\diff{W(s)},\n        \\quad t \\geq 0,\\quad\\P\\text{-a.s.}\n        \\end{equation}\n      Moreover, it belongs to $L^{p}(\\Omega;C([0,T];H))$ for any $p \\geq 1$ and $T > 0$.\n\\end{Proposition}\n\n\n\n\n\n\n\n\n\n\n\nTo formulate the LDPs for the solution of \\eqref{eq:SPDE}, we introduce some preliminaries upon the theory of large deviations; see \\cite[Chapter 1]{dembo2009large}. In what follows, let $(\\mathcal{U},\\rho^{\\mathcal{U}})$ be a Polish space and $\\mathcal{B}(\\mathcal{U})$ its Borel $\\sigma$-field.\n\n\n\n\n\n\\begin{Definition\n      A rate function $I$ is a lower semi-continuous mapping $I \\colon \\mathcal{U} \\to [0,+\\infty]$ (such that for all $\\alpha \\geq 0$, the level set $K_{I}(\\alpha) := \\{u \\in \\mathcal{U} : I(u) \\leq \\alpha\\}$ is a closed subset of $\\mathcal{U}$). A good rate function is a rate function for which all the level sets $K_{I}(\\alpha)$ are compact subsets of $\\mathcal{U}$.\n\\end{Definition}\n\n\n\n\n\n\n\\begin{Definition\n\\label{def:generaldefLDP\n      Let $\\{\\mu^{\\varepsilon}\\}_{\\varepsilon > 0}$ be a family of probability measures on $(\\mathcal{U},\\mathcal{B}(\\mathcal{U}))$. We say that $\\{\\mu^{\\varepsilon}\\}_{\\varepsilon > 0}$ satisfies an LDP on $\\mathcal{U}$ with the rate $\\frac{1}{\\varepsilon}$ and the rate function $I$ if\n      \\begin{equation}\\label{eq:LDPclassical\n            -\\inf_{u \\in U^{o}} I(u)\n            \\leq\n            \\liminf_{\\varepsilon \\to 0} \\varepsilon \\log \\mu^{\\varepsilon}(U)\n            \\leq\n            \\limsup_{\\varepsilon \\to 0} \\varepsilon \\log \\mu^{\\varepsilon}(U)\n            \\leq\n            -\\inf_{u \\in \\overline{U}} I(u),\n            \\quad U \\in \\mathcal{B}(\\mathcal{U}),\n      \\end{equation}\n      where $U^{o}$ denotes the interior of $U$,\n      and $\\overline{U}$ the closure of $U$.\n\\end{Definition}\n\n\n\n\n\n\n\n\nThe following proposition about Freidlin--Wentzell exponential estimates gives an equivalent characterization of LDP; see \\cite[Chapter 3]{freidlin1984random}. \n\n\n\n\n\n\\begin{Proposition}\n      If $(\\mathcal{U},\\rho^{\\mathcal{U}})$ is a Polish space and $\\{\\mu^{\\varepsilon}\\}_{\\varepsilon > 0}$ is a family of probability measures on $(\\mathcal{U},\\mathcal{B}(\\mathcal{U}))$, then $\\{\\mu^{\\varepsilon}\\}_{\\varepsilon > 0}$ satisfying an LDP on $\\mathcal{U}$ with the rate $\\frac{1}{\\varepsilon}$ and the good rate function $I$ is equivalent to the following statements:\n      \\begin{enumerate}\n        \\item [(i)] compact level set: the level set $K_{I}(\\alpha) :=  \\{u \\in \\mathcal{U} : I(u) \\leq \\alpha\\}$ is compact for every $\\alpha \\geq 0$;\n\n        \\item [(ii)] lower bound: for any $u \\in \\mathcal{U}$, $\\delta > 0$ and $\\gamma > 0$, there exists $\\varepsilon_{0} > 0$ such that\n            $$\\mu^{\\varepsilon}(\\{v \\in \\mathcal{U}:\\rho^{\\mathcal{U}}(u,v) < \\delta\\})\n            \\geq \\exp\\Big(-\\frac{I(u)+\\gamma}{\\varepsilon}\\Big),\n            \\quad \\varepsilon \\leq \\varepsilon_{0};$$\n\n        \\item [(iii)] upper bound: for any $\\alpha > 0$, $\\delta > 0$ and $\\gamma > 0$, there exists $\\varepsilon_{0} > 0$ such that\n            $$\\mu^{\\varepsilon}(\\{v \\in \\mathcal{U}:\n              \\rho^{\\mathcal{U}}(v,K_{I}(\\alpha)) \\geq \\delta\\})\n              \\leq \\exp\\Big(-\\frac{\\alpha-\\gamma}{\\varepsilon}\\Big),\n              \\quad \\varepsilon \\leq \\varepsilon_{0}.$$\n      \\end{enumerate}\n\\end{Proposition}\n\n\n\n\n\n\nRegarding $X_{x}^{\\varepsilon}$ as a $C([0,T];H)$-value random variable for each $\\varepsilon > 0$, the uniform LDP of sample paths of $\\{X_{x}^{\\varepsilon}\\}_{\\varepsilon > 0}$ can be summarized as follows; see \\cite{peszat1994large} for more details.\n\\begin{Theorem}\n\\label{th:exactsoluLDP\n      Suppose that Assumptions \\ref{ass:AQ} and \\ref{ass:F} hold. Then $\\{X_{x}^{\\varepsilon}\\}_{\\varepsilon > 0}$ satisfies a uniform LDP on $C([0,T];H)$ with the rate $\\frac{1}{\\varepsilon^{2}}$ and the good rate function $I_{0,T}^{x}$ defined by\n      \\begin{equation}\\label{eq:I0TxzI0Txz\n            I_{0,T}^{x}(z)\n            =\n            \\frac{1}{2}\n            \\inf\\{|\\varphi|_{L^2(0,T;H)}^{2}:\n            \\varphi \\in L^2(0,T;H),\n            z_{0,x}^{\\varphi} = z\\},\n            \\quad z \\in C([0,T];H),\n      \\end{equation}\n      where $z_{0,x}^{\\varphi}$ is the mild solution of the skeleton equation\n      \\begin{equation}\\label{eq:skeletoneqofSPDE\n            \\frac{\\dif{z_{0,x}^{\\varphi}(t)}}{\\dif{t}}\n            =\n            Az_{0,x}^{\\varphi}(t)\n            + F(z_{0,x}^{\\varphi}(t))\n            + Q^{\\frac12}\\varphi(t),\n            \\quad t \\in (0,T],\n            \\quad z_{0,x}^{\\varphi}(0) = x.\n      \\end{equation}\n\\end{Theorem}\n\n\n\n\nTo ensure the existence of invariant measures for \\eqref{eq:SPDE}, we need the following assumption on dissipativity.\n\n\n\\begin{Assumption\n\\label{ass:LFleqlambda1\n      Assume $F(0) = 0$ and $L_{F} < \\lambda_{1}$, where $L_{F}$ is the Lipschitz constant of $F$ given by \\eqref{eq:globalLF} and $\\lambda_{1}$ is the smallest eigenvalue of $-A$.\n\\end{Assumption}\n\n\n\n\n\n\n\n\n\n\nThis assumption together with\n$-Ae_{i} = \\lambda_{i}e_{i}, i \\in \\mathbb{N}^{+}$\nshows\n\\begin{equation}\\label{eq:dissipativecond\n      \\langle Au + F(u), u \\rangle\n      \\leq -c|u|^{2},\n      \\quad u \\in \\dot{H}^{2}\n\\end{equation}\nwith $c:= \\lambda_{1}-L_{F} > 0$.\nIt follows from \\cite{cerrai2005largeinvariant} that there exists $\\{t_{i}\\}_{i \\in \\mathbb{N}^{+}} \\uparrow +\\infty$ (possibly depending on $\\varepsilon$) such that the sequence of probability measures $\\{\\mu_{t_{i}}^{\\varepsilon}\\}_{i \\in \\mathbb{N}^{+}}$, defined by\n      $$\n            \\mu_{t_{i}}^{\\varepsilon}(B)\n            :=\n            \\frac{1}{t_{i}} \\int_{0}^{t_{i}}\n            \\P\\big(X_{0}^{\\varepsilon}(s) \\in B\\big)\n            \\diff{s},\n            \\quad B \\in \\mathcal{B}(H),\n      $$\nconverges weakly to some probability measure $\\mu^{\\varepsilon}$ on $(H,\\mathcal{B}(H))$, which is invariant for \\eqref{eq:SPDE}. Moreover, the following theorem shows that $\\{\\mu^{\\varepsilon}\\}_{\\varepsilon > 0}$ obeys an LDP on $H$; see, e.g., \\cite{cerrai2005largeinvariant}.  \n\n\n\n\n\\begin{Theorem}\n      Suppose that Assumptions \\ref{ass:AQ}, \\ref{ass:F} and \\ref{ass:LFleqlambda1} hold. Then  $\\{\\mu^{\\varepsilon}\\}_{\\varepsilon > 0}$ satisfies an LDP on $H$ with the rate $\\frac{1}{\\varepsilon^{2}}$ and the good rate function $V$ defined by\n      \\begin{equation}\\label{eq:invaLdpratefun\n            V(u)\n            =\n            \\inf\\{I_{0,T}^{0}(z):\n            T>0, z \\in C([0,T];H), z(0) = 0, z(T) = u\\},\n            \\quad u \\in H.\n      \\end{equation}\n\\end{Theorem}\n\n\n\n\n\n\n\n\\section{Spatial discretization and its LDPs}\n\\label{sec:spatial\n\n\nHere we discretize \\eqref{eq:SPDE} in space by the spectral Galerkin method and present its uniform LDP of sample paths in Subsection \\ref{sec:SGM}.\nSubsection \\ref{subsec:semiweakasympre} obtains the weakly asymptotical preservation for the LDP of sample paths of the original equation. After establishing the LDP of invariant measures of the spatial discretization in Subsection \\ref{subsec:semiinvameaLDPs}, we prove its weakly asymptotical preservation for the LDP of invariant measures of the original equation in Subsection \\ref{sebsec:semiinvameaLDPasy}.\n\n\n\\subsection{Spectral Galerkin method}\\label{sec:SGM\n\nFor each $n \\in \\mathbb{N}^{+}$, we define $H_{n} := \\text{span}\\{e_{1},\\ldots,e_{n}\\} \\subset H$,  the projection operator $P_{n} \\colon H \\to H_{n}$ by $P_{n}u = \\sum_{i=1}^{n}\\langle e_{i},u \\rangle e_{i}$ for every $u \\in H$, and\n$$\n      y :=  P_{n}x,\n      \\quad F_{n} :=  P_{n}F,\n      \\quad Q_{n}^{\\frac12} :=  P_{n}Q^{\\frac12},\n      \\quad W_{n} :=  P_{n}W.\n$$\nThe spectral Galerkin method applied to \\eqref{eq:SPDE} is given by\n\\begin{equation}\\label{eq:SPDEsemi\n      \\diff{X_{y}^{\\varepsilon,n}(t)}\n      =\n      \\big(A_{n}X_{y}^{\\varepsilon,n}(t)\n      +\n      F_{n}(X_{y}^{\\varepsilon,n}(t))\\big)\\diff{t}\n      +\n      \\varepsilon Q_{n}^{\\frac12}\\diff{W_{n}(t)},\n      \\quad t > 0,\n      \\quad X_{y}^{\\varepsilon,n}(0) = y,\n\\end{equation}\nwhere $A_{n} \\colon H_{n} \\to H_{n}$ is defined by $A_{n} :=  P_{n}A$ and generates a $C_{0}$-semigroup $\\{E_{n}(t) := e^{tA_{n}}\\}_{t \\geq 0}$ on $H_{n}$.\nUnder Assumptions \\ref{ass:AQ} and \\ref{ass:F}, Theorem 4.5.3 in \\cite{kloeden1992numerical} ensures that \\eqref{eq:SPDEsemi} admits a unique mild solution\n\\begin{equation}\\label{eq:SPDEsemisolu\n      X_{y}^{\\varepsilon,n}(t)\n      =\n      E_{n}(t)y\n      +\n      \\int_{0}^{t} E_{n}(t-s) F_{n}(X_{y}^{\\varepsilon,n}(s)) \\diff{s}\n      +\n      \\varepsilon\n      \\int_{0}^{t} E_{n}(t-s)Q_{n}^{\\frac12} \\diff{W_{n}(s)},\n      \\quad t \\geq 0.\n\\end{equation}\nMoreover, the solution process $\\{X_{y}^{\\varepsilon, n}(t)\\}_{t \\in [0,T]}$ belongs to $L^{p}(\\Omega;C([0,T];H_{n}))$ for any $p \\geq 1$ and $T > 0$. To show $\\{X_{y}^{\\varepsilon,n}\\}_{\\varepsilon > 0}$ satisfying an LDP on $C([0,T];H_{n})$, we first note that the skeleton equation corresponding to \\eqref{eq:SPDEsemi} is given by\n\\begin{equation}\\label{eq:SPDEsemiskeletoneq\n      \\frac{\\dif{z_{0,y}^{n,\\psi}(t)}}{\\dif{t}}\n      =\n      A_{n}z_{0,y}^{n,\\psi}(t) + F_{n}(z_{0,y}^{n,\\psi}(t)) + Q_{n}^{\\frac12}\\psi(t),\n      \\quad t \\in (0,T],\n      \\quad z_{0,y}^{n,\\psi}(0) = y\n\\end{equation}\nwith $\\psi \\in L^{2}(0,T;H_{n})$. According to\n\\cite[Theorem 1.1]{peszat1994large}, we know that \\eqref{eq:SPDEsemiskeletoneq} admits a unique mild solution $z_{0,y}^{n,\\psi} \\in C([0,T];H_{n})$, given by\n\\begin{equation}\\label{eq:skeletoneqspatial\n      z_{0,y}^{n,\\psi}(t)\n      =\n      E_{n}(t)y\n      +\n      \\int_{0}^{t} E_{n}(t-s)F_{n}(z_{0,y}^{n,\\psi}(s)) \\diff{s}\n      +\n      \\int_{0}^{t} E_{n}(t-s)Q_{n}^{\\frac12}\\psi(s) \\diff{s},\n      \\quad t \\in [0,T].\n\\end{equation}\nWith this, we define $I_{0,T}^{n,y} \\colon C([0,T];H_{n}) \\to [0,+\\infty]$ by\n\\begin{equation}\\label{eq:SPDEsemiratefun\n      I_{0,T}^{n,y}(z)\n      =\n      \\frac{1}{2}\\inf\\{|\\psi|_{L^2(0,T;H)}^{2}:\n      \\psi \\in L^2(0,T;H_{n}), z_{0,y}^{n,\\psi} = z\\},\n      \\quad z \\in C([0,T];H_{n}).\n\\end{equation}\n\n\nFollowing the approach in \\cite{peszat1994large}, one can prove the uniform LDP of sample paths of $\\{X_{y}^{\\varepsilon,n}\\}_{\\varepsilon > 0}$.\n\n\n\n\\begin{Theorem}\n\\label{thm:semisolutionLDP\n      Suppose that Assumptions \\ref{ass:AQ} and \\ref{ass:F} hold. Then $\\{X_{y}^{\\varepsilon,n}\\}_{\\varepsilon > 0}$ satisfies a uniform LDP on $C([0,T];H_{n})$ with the rate $\\frac{1}{\\varepsilon^{2}}$ and the good rate function $I_{0,T}^{n,y}$ given by \\eqref{eq:SPDEsemiratefun}, i.e.,\n\n\n      \\begin{enumerate}\n        \\item [(i)] compact level set: for any $T > 0$, $n \\in \\mathbb{N}^{+}$ and $y \\in H_{n}$, the level set $K_{0,T}^{n,y}(\\alpha) := \\{z \\in C([0,T];H_{n}): I_{0,T}^{n,y}(z) \\leq \\alpha\\}$ is compact for every $\\alpha \\geq 0$;\n        \\item [(ii)] uniform lower bound: for any $T > 0$, $n \\in \\mathbb{N}^{+}$, $\\alpha \\geq 0$, $\\delta > 0$, $\\gamma > 0$ and $K > 0$, there exists $\\varepsilon_0 > 0$ such that for any $y \\in H_{n}$ with $|y| \\leq K$ and $z \\in K_{0,T}^{n,y}(\\alpha)$,\n      \\begin{equation}\\label{eq:semisoluldplower\n            \\P(|X_{y}^{\\varepsilon,n}-z|_{C([0,T];H)} < \\delta)\n            \\geq\n            \\exp\\Big(-\\frac{I_{0,T}^{n,y}(z)+\\gamma}{\\varepsilon^2}\\Big),\n            \\quad \\varepsilon \\leq \\varepsilon_{0};\n      \\end{equation}\n        \\item [(iii)] uniform upper bound: for any $T > 0$, $n \\in \\mathbb{N}^{+}$, $\\alpha \\geq 0$, $\\delta > 0$, $\\gamma > 0$ and $K > 0$, there exists $\\varepsilon_0 > 0$ such that for any $y \\in H_{n}$ with $|y| \\leq K$,\n      \\begin{equation}\\label{eq:semisoluldpupper\n\t        \\P(\\rho^{C([0,T];H)}\n            (X_{y}^{\\varepsilon,n},K_{0,T}^{n,y}(\\alpha))\n            \\geq \\delta)\n\t        \\leq\n\t        \\exp\\Big(-\\frac{\\alpha-\\gamma}{\\varepsilon^2}\\Big),\n\t        \\quad \\varepsilon \\leq \\varepsilon_{0},\n      \\end{equation}\n      where $\\rho^{C([0,T];H)}(z,U) := \\inf\\limits_{z' \\in U}|z-z'|_{C([0,T];H)}, z \\in C([0,T];H), U \\subset C([0,T];H)$.\n      \\end{enumerate}\n\\end{Theorem}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Weakly asymptotical preservation for LDP of $\\{X_{x}^{\\varepsilon}\\}_{\\varepsilon > 0}$}\n\\label{subsec:semiweakasympre\n\n\n\nThis part aims to estimate the error between the rate functions $I_{0,T}^{n,y}$ and $I_{0,T}^{x}$. To this end, we first simplify the expression of $I_{0,T}^{x}$, which relies on the spatial regularity of the solution of the skeleton equation \\eqref{eq:skeletoneqofSPDE}.\n\n\n\\begin{Lemma}\n\\label{lem:spatialregularity\n      Suppose that Assumptions \\ref{ass:AQ} and \\ref{ass:F} hold and let $\\{z_{0,x}^{\\varphi}(t)\\}_{t \\in [0,T]}$ with $\\varphi \\in L^{2}(0,T;H)$ be given by \\eqref{eq:skeletoneqofSPDE}. If\n      $x \\in \\dot{H}^{2}$, then $z_{0,x}^{\\varphi}(t) \\in \\dot{H}^{2},t \\in [0,T]$ and there exists $C > 0$ such that\n      \\begin{equation}\\label{eq:38\n            |z_{0,x}^{\\varphi}(t)|_{\\dot{H}^{2}}\n            \\leq\n            C(1+|x|_{\\dot{H}^{2}}),\n            \\quad t \\in [0,T].\n      \\end{equation}\n\\end{Lemma}\n\n\\begin{proof}\n      Using $|E(t)|_{\\mathcal{L}(H)} \\leq e^{-\\lambda_{1}t} \\leq 1,t \\geq 0$, \\eqref{eq:globalLF} and the H\\\"{o}lder inequality\n      leads to\n      \\begin{equation*}\n      \\begin{split}\n            &~|z_{0,x}^{\\varphi}(t)|\n            \\leq\n            |E(t)x|\n            +\n            \\int_{0}^{t} |E(t-s)F(z_{0,x}^{\\varphi}(s))| \\diff{s}\n            +\n            \\int_{0}^{t} |E(t-s)Q^{\\frac12}\\varphi(s)| \\diff{s}\n            \\\\\\leq&~\n            |x|\n            +\n            C\\int_{0}^{t} (1+|z_{0,x}^{\\varphi}(s)|) \\diff{s}\n            +\n            |Q^{\\frac12}|_{\\mathcal{L}(H)} \\Big(\\int_{0}^{t} 1^{2} \\diff{s}\\Big)^{\\frac{1}{2}}\n            \\Big(\\int_{0}^{t} |\\varphi(s)|^{2} \\diff{s}\\Big)^{\\frac{1}{2}}\n            \\\\\\leq&~\n            |x| + CT\n            +\n            \\sqrt{T}|Q^{\\frac12}|_{\\mathcal{L}(H)}\n            |\\varphi|_{L^{2}(0,T;H)}\n            +\n            C\\int_{0}^{t} |z_{0,x}^{\\varphi}(s)| \\diff{s}.\n      \\end{split}\n      \\end{equation*}\n      The Gronwall inequality implies\n      \\begin{equation}\\label{eq:39393939\n           |z_{0,x}^{\\varphi}(t)|\n            \\leq\n            C(1+|x|), \\quad t \\in [0,T].\n      \\end{equation}\n      Similarly, we use $|(-A)^{\\gamma}E(t)|_{\\mathcal{L}(H)}\n\t  \\leq Ct^{-\\gamma}, t > 0,\\gamma \\geq 0$  to get\n      \\begin{align*}\n            &|z_{0,x}^{\\varphi}(t)|_{\\dot{H}^{1}}\n            \\leq\n            |E(t)x|_{\\dot{H}^{1}}\n            +\n            \\int_{0}^{t} |E(t-s)F(z_{0,x}^{\\varphi}(s))|_{\\dot{H}^{1}} \\diff{s}\n            +\n            \\int_{0}^{t} |E(t-s)Q^{\\frac12}\\varphi(s)|_{\\dot{H}^{1}} \\diff{s}\n            \\\\\\leq&\n            |x|_{\\dot{H}^{1}}\n            +\n            C\\int_{0}^{t} |(-A)^{\\frac{1}{2}}E(t-s)|_{\\mathcal{L}(H)}\n            |F(z_{0,x}^{\\varphi}(s))| \\diff{s}\n            +\n            |(-A)^{\\frac{1}{2}}Q^{\\frac12}|_{\\mathcal{L}(H)}\n            \\int_{0}^{t} |\\varphi(s)| \\diff{s}\n            \\\\\\leq&\n            |x|_{\\dot{H}^{1}} + C(1+|x|)\n            +\n            \\sqrt{T}\n            |(-A)^{\\frac{1}{2}}Q^{\\frac12}|_{\\mathcal{L}_{2}(H)}\n            |\\varphi|_{L^{2}(0,T;H)}\n            \\\\\\leq&\n            C(1+|x|_{\\dot{H}^{1}}).\n      \\end{align*}\n      In the same way,\n      \\begin{align*}\n            |z_{0,x}^{\\varphi}(t)|_{\\dot{H}^{2}}\n            \\leq&~\n            |E(t)x|_{\\dot{H}^{2}}\n            +\n            \\int_{0}^{t} |E(t-s)F(z_{0,x}^{\\varphi}(s))|_{\\dot{H}^{2}} \\diff{s}\n            +\n            \\int_{0}^{t} |E(t-s)Q^{\\frac12}\\varphi(s)|_{\\dot{H}^{2}} \\diff{s}\n            \\\\\\leq&~\n            |x|_{\\dot{H}^{2}}\n            +\n            \\int_{0}^{t} |(-A)^{\\frac{1}{2}}E(t-s)|_{\\mathcal{L}(H)}\n            |F(z_{0,x}^{\\varphi}(s))|_{\\dot{H}^{1}} \\diff{s}\n            \\\\&~+\n            \\int_{0}^{t} |(-A)E(t-s)Q^{\\frac12}|_{\\mathcal{L}(H)}|\\varphi(s)| \\diff{s}\n            \\\\\\leq&~\n            |x|_{\\dot{H}^{2}}\n            +\n            C(1+|x|_{\\dot{H}^{1}})\n            +\n            |\\varphi|_{L^{2}(0,T;H)}\n            \\Big(\\int_{0}^{t} |(-A)E(t-s)Q^{\\frac12}|_{\\mathcal{L}_{2}(H)}^{2} \\diff{s}\\Big)^{\\frac{1}{2}}\n            \\\\\\leq&~\n            |x|_{\\dot{H}^{2}} + C(1+|x|_{\\dot{H}^{2}})\n            +\n            C|\\varphi|_{L^{2}(0,T;H)} |(-A)^{\\frac{1}{2}}Q^{\\frac12}|_{\\mathcal{L}_{2}(H)}\n            \\\\\\leq&~\n            C(1+|x|_{\\dot{H}^{2}}),\n      \\end{align*}\n      where we have used \\eqref{eq:auxiliaryassumptionFuH1} and \\cite[Lemma 2.3]{wang2015note}.\n      Thus we complete the proof.\n\\end{proof}\n\n\n\n\nLemma \\ref{lem:spatialregularity} means that\nthe mild solution $\\{z_{0,x}^{\\varphi}(t)\\}_{t \\in [0,T]}$ is also a strong solution. To proceed, we denote by $W_{2}^{1,2}(T)$ the closure of $C^{\\infty}([0,T] \\times [0,1];\\mathbb{R})$ in the norm\n\\begin{equation*}\n\\begin{split}\n      |z|_{W_{2}^{1,2}(T)}\n      :=&\n      \\Big(\n      \\int_{0}^{T}\\int_{0}^{1}\n          |z(t,\\xi)|^{2}\n          +\n          \\Big|\\frac{\\partial z(t,\\xi)}{\\partial t}\\Big|^{2}\n          +\n          \\Big|\\frac{\\partial z(t,\\xi)}{\\partial\\xi}\\Big|^{2}\n          +\n          \\Big|\\frac{\\partial^{2} z(t,\\xi)}{\\partial\\xi^{2}}\\Big|^{2}\n        \\diff{\\xi}\\diff{t}\n      \\Big)^{\\frac{1}{2}}.\n\\end{split}\n\\end{equation*}\nAs $Q$ is one-to-one, we have\n\\begin{equation}\\label{eq:I0Txz\n      I_{0,T}^{x}(z)\n      =\n      \\begin{cases}\n        \\frac{1}{2}\n        \\int_{0}^{T}\n            \\Big|Q^{-\\frac{1}{2}}\n            \\Big(\\frac{\\diff{z(t)}}{\\diff{t}}\n                 -\n                 Az(t) - F(z(t))\\Big)\n            \\Big|^{2}\n        \\diff{t}\n        , & \\mbox{if } z \\in W_{2,Q}^{1,2}(T) \\text{~with~} z(0) = x, \\\\\n        +\\infty, & \\text{otherwise},\n      \\end{cases}\n\\end{equation}\nwhere\n\\begin{align*}\n      W_{2,Q}^{1,2}(T)\n      :=\n      \\Big\\{&z \\in W_{2}^{1,2}(T):\n      \\frac{\\diff{z(t)}}{\\diff{t}} - Az(t) - F(z(t)) \\in Q^{\\frac{1}{2}}(H) \\text{~for almost all~} t \\in [0,T]\n      \\\\&\\int_{0}^{T}\n      \\Big|Q^{-\\frac{1}{2}}\\Big(\\frac{\\diff{z(t)}}{\\diff{t}} - Az(t) - F(z(t))\\Big)\\Big|^{2}\\diff{t} < +\\infty\\Big\\}.\n\\end{align*}\n\n\n\n\nFor the rate function $I_{0,T}^{n,y}$, its effective domain is given by\n$$\n      \\mathcal{H}_{1}^{n,y}(T)\n      :=\n      \\Big\\{z \\in C([0,T];H_{n}):\n      z(\\cdot) = y + \\int_{0}^{\\cdot} \\upsilon(s) \\diff{s},\n      \\upsilon \\in L^{2}(0,T;H_{n})\\Big\\}.\n$$\nIt follows that\n\\begin{equation}\\label{eq:I0Tnyz\n      I_{0,T}^{n,y}(z)\n      =\n      \\begin{cases}\n        \\frac{1}{2}\n        \\int_{0}^{T}\n            \\Big|Q_{n}^{-\\frac12}\\Big(\\frac{\\diff{z(t)}}{\\diff{t}}\n                 -\n                 A_{n}z(t) - F_{n}(z(t))\\Big)\n            \\Big|^{2}\n        \\diff{t}\n        , & \\mbox{if~} z \\in \\mathcal{H}_{1}^{n,y}(T), \\\\\n        +\\infty, & \\mbox{otherwise},\n      \\end{cases}\n\\end{equation}\nwhere $Q_{n}^{-\\frac{1}{2}} \\colon H_{n} \\to H_{n}$ is the inverse operator of the bijective operator $Q_{n}^{\\frac{1}{2}} \\colon H_{n} \\to H_{n}$.\n\n\n\nSimilarly to \\cite[Definition 4.2]{chen2020large}, we introduce the definition of weakly asymptotical preservation for the LDP of sample paths by a numerical method.\n\\begin{Definition\n\\label{def:semiweakasympres\n     \n      We say that\n      the semi-discrete numerical method \\eqref{eq:SPDEsemi}\n      weakly asymptotically preserves the LDP of $\\{X_{x}^{\\varepsilon}\\}_{\\varepsilon > 0}$ if for any $\\kappa > 0$ and $z \\in W_{2,Q}^{1,2}(T)$ with $z(0) = x$, there exist $n \\in \\mathbb{N}^{+}$ and $z_{n} \\in \\mathcal{H}_{1}^{n,y}(T)$ such that\n      $$\n            |z-z_{n}|_{C([0,T];H)} < \\kappa,\n            \\quad\\quad\n            |I_{0,T}^{x}(z)-I_{0,T}^{n,y}(z_{n})| < \\kappa.\n      $$\n\\end{Definition}\n\n\n\nAfter these preparations, we have the following result.\n\n\\begin{Theorem}\n\\label{thm:spatialsolutionwap\n      Suppose that Assumptions \\ref{ass:AQ} and \\ref{ass:F} hold. If $x \\in \\dot{H}^{2}$, then\n     \n      the semi-discrete numerical method \\eqref{eq:SPDEsemi}\n      weakly asymptotically preserves the LDP of $\\{X_{x}^{\\varepsilon}\\}_{\\varepsilon > 0}$.\n\\end{Theorem}\n\n\\begin{proof}\n      For any $z \\in W_{2,Q}^{1,2}(T)$ with $z(0) = x$,\n     \n      we define\n      $$\\varphi(t)\n      :=\n      Q^{-\\frac{1}{2}}\n            \\Big(\\frac{\\diff{z(t)}}{\\diff{t}}\n                 -\n                 Az(t) - F(z(t))\\Big)$$\n      for almost all $t \\in [0,T]$. Then $\\varphi \\in L^2(0,T;H)$,  $z_{0,x}^{\\varphi} = z$ and\n      \\begin{equation}\\label{eq:IoTxz311311\n            \\frac{1}{2}|\\varphi|_{L^2(0,T;H)}^{2} = I_{0,T}^{x}(z) < +\\infty,\n      \\end{equation}\n      which together with \\eqref{eq:39393939} in Lemma \\ref{lem:spatialregularity} yields\n      \\begin{equation}\\label{eq:ztdotH2z0x313\n            |z(t)|\n            =\n            |z_{0,x}^{\\varphi}(t)|\n            \\leq\n            C(1+|x|),\\quad t \\in [0,T].\n      \\end{equation}\n      Now for any $n \\in \\mathbb{N}^{+}$, we define $\\{z_{n}(t)\\}_{t \\in [0,T]}$ by\n      \\begin{equation}\\label{eq:dzntdt312\n            \\frac{\\diff{z_{n}(t)}}{\\diff{t}}\n            =\n            A_{n}z_{n}(t) + F_{n}(z_{n}(t))\n            + Q_{n}^{\\frac{1}{2}}P_{n}\\varphi(t),\n            \\quad t \\in (0,T],\n            \\quad z_{n}(0) = P_{n}x.\n      \\end{equation}\n      It follows from $\\langle Au,u \\rangle \\leq -\\lambda_{1}|u|^{2}, u \\in H$ and \\eqref{eq:globalLF} that\n      \\begin{align*}\n            &\\frac{1}{2}\\frac{\\diff{|z(t)-z_{n}(t)|^{2}}}{\\diff{t}}\n            =\n            \\Big\\langle \\frac{\\diff{(z(t)-z_{n}(t))}}{\\diff{t}},\n            z(t)-z_{n}(t) \\Big\\rangle\n            \\\\=&\n            \\langle A(z(t)-z_{n}(t)),z(t)-z_{n}(t) \\rangle\n            +\n            \\langle (I-P_{n})F(z(t)),z(t)-z_{n}(t) \\rangle\n            \\\\&+\n            \\langle F_{n}(z(t))-F_{n}(z_{n}(t)),z(t)-z_{n}(t) \\rangle\n            +\n            \\langle (I-P_{n})Q^{\\frac{1}{2}}\\varphi(t),\n            z(t)-z_{n}(t) \\rangle\n            \\\\\\leq&\n            -\\lambda_{1}|z(t)-z_{n}(t)|^{2}\n            +\n            |(I-P_{n})F(z(t))||z(t)-z_{n}(t)|\n            \\\\&+\n            L_{F}|z(t)-z_{n}(t)|^{2}\n            +\n            |(I-P_{n})Q^{\\frac{1}{2}}\\varphi(t)|\n            |z(t)-z_{n}(t)|\n            \\\\\\leq&\n            L_{F}|z(t)-z_{n}(t)|^{2}\n            +\n            \\frac{1}{2\\lambda_{1}}|(I-P_{n})F(z(t))|^{2}\n            +\n            \\frac{1}{2\\lambda_{1}}\n            |(I-P_{n})Q^{\\frac{1}{2}}\\varphi(t)|^{2},\n      \\end{align*}\n      where the weighted Young inequality is used in the last step. Integrating yields\n      \\begin{align*}\n            |z(t)&-z_{n}(t)|^{2}\n            \\leq\n            |x-P_{n}x|^{2}\n            +\n            2L_{F}\\int_{0}^{t}|z(s)-z_{n}(s)|^{2}\\diff{s}\n            \\\\&+\n            \\frac{1}{\\lambda_{1}}\\int_{0}^{t}\n            |(I-P_{n})F(z(s))|^{2}\\diff{s}\n            +\n            \\frac{1}{\\lambda_{1}}\\int_{0}^{t}\n            |(I-P_{n})Q^{\\frac{1}{2}}\\varphi(s)|^{2}\\diff{s}\n      \\end{align*}\n      and consequently\n      \\begin{align*}\n            |z&-z_{n}|_{C([0,t];H)}^{2}\n            \\leq\n            \\Pi_{n}\n            +\n            2L_{F}\\int_{0}^{t}|z-z_{n}|_{C([0,s];H)}^{2}\\diff{s}\n      \\end{align*}\n      with\n      \\begin{equation}\n            \\Pi_{n}\n            :=\n            |(I-P_{n})x|^{2}\n            +\n            \\frac{1}{\\lambda_{1}}\\int_{0}^{T}\n            |(I-P_{n})F(z(s))|^{2}\\diff{s}\n            +\n            \\frac{1}{\\lambda_{1}}\\int_{0}^{T}\n            |(I-P_{n})Q^{\\frac{1}{2}}\\varphi(s)|^{2}\\diff{s}.\n      \\end{equation}\n      The Gronwall inequality shows\n      \\begin{equation}\\label{eq:zzn314\n            |z-z_{n}|_{C([0,t];H)}^{2} \\leq \\Pi_{n}e^{2L_{F}t} \\leq \\Pi_{n}e^{2L_{F}T}, \\quad t \\in [0,T].\n      \\end{equation}\n      Applying \\eqref{eq:globalLF} and \\eqref{eq:ztdotH2z0x313} yields $$|(I-P_{n})F(z(t))|^{2} \\leq |F(z(t))|^{2} = |F(z_{0,x}^{\\varphi}(t))|^{2} \\leq C(1+|x|^{2}),\\quad t \\in [0,T].$$\n      We use $\\lim\\limits_{n \\to \\infty}|(I-P_{n})F(z(t))|^{2} = 0, t \\in [0,T]$ and the bounded convergence theorem to get\n      \\begin{equation*}\n            \\lim_{n \\to \\infty}\n            \\frac{1}{\\lambda_{1}}\\int_{0}^{T}\n            |(I-P_{n})F(z(s))|^{2}\\diff{s}\n            =\n            0.\n      \\end{equation*}\n      In the same way, one can validate $\\lim\\limits_{n \\to \\infty}\\Pi_{n} = 0$ and thus $\\lim\\limits_{n \\to \\infty}|z-z_{n}|_{C([0,T];H)} = 0$ by \\eqref{eq:zzn314}. It follows that for any $\\kappa > 0$, there exists $n_{1} \\in \\mathbb{N}^{+}$ such that\n      \\begin{equation}\\label{eq:z-znC0TH\n            |z-z_{n}|_{C([0,T];H)} < \\kappa,\\quad n > n_{1}.\n      \\end{equation}\n      By \\eqref{eq:dzntdt312} and \\eqref{eq:I0Tnyz}, we have $z_{n} \\in \\mathcal{H}_{1}^{n,y}(T)$ and hence\n      \\begin{equation*}\n            I_{0,T}^{n,y}(z_{n})\n            =\n            \\frac{1}{2}\n            \\int_{0}^{T}\n            \\Big|Q_{n}^{-\\frac12}\n            \\Big(\\frac{\\diff{z_{n}(t)}}{\\diff{t}}\n                 -\n                 A_{n}z_{n}(t) - F_{n}(z_{n}(t))\\Big)\n            \\Big|^{2}\n            \\diff{t}.\n      \\end{equation*}\n      It immediately follows from \\eqref{eq:dzntdt312} that\n      \\begin{equation}\\label{eq:I0Tny317317\n            I_{0,T}^{n,y}(z_{n})\n            =\n            \\frac{1}{2}|P_{n}\\varphi|_{L^2(0,T;H)}^{2}.\n      \\end{equation}\n      Applying \\eqref{eq:IoTxz311311}, \\eqref{eq:I0Tny317317} and the H\\\"{o}lder inequality leads to\n      \\begin{align*}\n             |I_{0,T}^{x}(z)-I_{0,T}^{n,y}(z_{n})|\n            =&\n            \\frac{1}{2}\\Big|\\int_{0}^{T}\n            \\big\\langle (I+P_{n})\\varphi(t),\n            (I-P_{n})\\varphi(t)\\big\\rangle \\diff{t}\\Big|\n            \\\\\\leq&\n            |\\varphi|_{L^2(0,T;H)}\n            \\Big(\\int_{0}^{T}\n            |(I-P_{n})\\varphi(t)|^{2} \\diff{t}\\Big)^{\\frac{1}{2}}.\n      \\end{align*}\n      By the Lebesgue dominated convergence theorem, we conclude\n      \\begin{equation}\\label{eq:313313313\n            \\lim\\limits_{n \\to \\infty}\n            |I_{0,T}^{x}(z)-I_{0,T}^{n,y}(z_{n})| = 0,\n      \\end{equation}\n      which yields that there exists $n_{2} \\in \\mathbb{N}^{+}$ such that\n      \\begin{equation}\\label{eq:I0Txz-I0Tnyzn\n            |I_{0,T}^{x}(z)-I_{0,T}^{n,y}(z_{n})|\n            <\n            \\kappa, \\quad n > n_{2}.\n      \\end{equation}\n      By choosing $n > \\max\\{n_{1},n_{2}\\}$, we complete the proof.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\\subsection{LDP for invariant measures of spatial discretization}\n\\label{subsec:semiinvameaLDPs\nAs is well known, the uniform LDP of sample paths not only is of importance in itself but also can be used to derive the LDP of invariant measures; see, e.g., \\cite{sowers1992largeinvariant,cerrai2005largeinvariant}. With this purpose, one must first guarantee the existence of invariant measures of spatial discretization \\eqref{eq:SPDEsemi}.\nAccording to \\cite[Theorem 3.1]{CGW2020ErgodicityANM} and \\cite[Remark 7.2 and Proposition 7.10]{da2006introduction},\nthe family of probability measures $\\{\\mu_{t}^{\\varepsilon,n}\\}_{t > 0}$, defined by\n      $$\n            \\mu_{t}^{\\varepsilon,n}(B)\n            :=\n            \\frac{1}{t} \\int_{0}^{t}\n            \\P\\big(X_{0}^{\\varepsilon,n}(s) \\in B\\big)\n            \\diff{s},\n            \\quad B \\in \\mathcal{B}(H_{n}), t > 0\n      $$\nis tight. It follows from the Krylov--Bogoliubov theorem (\\cite[Theorem 7.1]{da2006introduction}) that there exists $\\{t_{i}\\}_{i \\in \\mathbb{N}^{+}} \\uparrow +\\infty$ (possibly depending on $\\varepsilon$) such that the sequence $\\{\\mu_{t_{i}}^{\\varepsilon,n}\\}_{i \\in \\mathbb{N}^{+}}$ converges weakly to some probability measure $\\mu^{\\varepsilon,n}$ on $(H_{n},\\mathcal{B}(H_{n}))$, which is invariant for \\eqref{eq:SPDEsemi}.\nTo study the LDP of invariant measures $\\{\\mu^{\\varepsilon,n}\\}_{\\varepsilon > 0}$, we define $V^{n} \\colon H_{n} \\to [0,+\\infty]$ by\n\\begin{equation}\\label{eq:Vny\n      V^{n}(v)\n      =\n      \\inf\\{I_{0,T}^{n,0}(z) : T > 0, z \\in C([0,T];H_{n}), z(0) = 0, z(T) = v \\},\n\\end{equation}\nand its level set\n$K^{n}(\\alpha) = \\{v \\in H_{n} : V^{n}(v) \\leq \\alpha\\}$ for each $\\alpha \\geq 0$.\n\n\n\n\n\n\n\n\n\n\\subsubsection{$V^{n}$ being a good rate function}\n\n\n\n\n\n\nIn this part, we will use the boundedness of $K^{n}$ to show that $V^{n}$ is a good rate function. To this end, we need the following lemma.\n\n\n\\begin{Lemma}\\label{lem:4.9\n      Suppose that Assumptions \\ref{ass:AQ}, \\ref{ass:F} and \\ref{ass:LFleqlambda1} hold and let $\\{z(t)\\}_{t \\in [0,T]}$ be the mild solution of the following equation\n      \\begin{equation}\\label{eq:diffztdifft\n            \\frac{\\diff{z(t)}}{\\diff{t}}\n            =\n            A_{n}z(t) + F_{n}(z(t)) + Q_{n}^{\\frac12}\\psi(t),\n            \\quad t \\in (0,T],\n            \\quad z(0) = y \\in H_{n}\n      \\end{equation}\n      with $\\psi \\in L^{2}(0,T;H_{n})$. Then we have\n      \\begin{equation}\\label{eq:ztHn2\n            |z(t)|^{2}\n            \\leq\n            C\\big(1+|y|^{2}\n            +\n            |Q^{\\frac12}|_{\\mathcal{L}(H)}^{2}\n            |\\psi|_{L^{2}(0,T;H)}^{2}\\big),\n            \\quad t \\in [0,T].\n      \\end{equation}\n\\end{Lemma}\n\n\n\n\n\\begin{proof}\n      Setting $O(t) :=  \\int_{0}^{t} E_{n}(t-s) Q_{n}^{\\frac12}\\psi(s) \\diff{s}, t \\in [0,T]$, we use $|E_{n}(t)|_{\\mathcal{L}(H_{n})} \\leq e^{-\\lambda_{1}t},t \\in [0,T]$ and the H\\\"{o}lder inequality to get\n      \\begin{equation}\\label{eq:OtHn\n      \\begin{split}\n            |O(t)|\n            \\leq&\n            |Q_{n}^{\\frac12}|_{\\mathcal{L}(H)}\n            \\int_{0}^{t} e^{-\\lambda_{1}(t-s)}|\\psi(s)| \\diff{s}\n            \\leq\n            |Q^{\\frac12}|_{\\mathcal{L}(H)}\n            |\\psi|_{L^{2}(0,T;H)},\n            \\quad t \\in [0,T].\n      \\end{split}\n      \\end{equation}\n      Let $\\bar{z}(t) :=  z(t)-O(t), t \\in [0,T]$, then $\\{\\bar{z}(t)\\}_{t \\in [0,T]}$ satisfies\n      $$\n            \\frac{\\dif{\\bar{z}(t)}}{\\dif{t}}\n            =\n            A_{n}\\bar{z}(t) + F_{n}(\\bar{z}(t)+O(t)),\n            \\quad t \\in (0,T],\n            \\quad \\bar{z}(0) = y.\n      $$\n      Using \\eqref{eq:dissipativecond}\n     \n     \n      and \\eqref{eq:globalLF}\n     \n      yields\n      \\begin{align*}\n            &~\\frac{\\diff{e^{ct}|\\bar{z}(t)|^{2}}}\n                 {\\diff{t}}\n            =\n            ce^{ct}|\\bar{z}(t)|^{2}\n            +\n            2e^{ct}\\big\\langle\n                 A_{n}\\bar{z}(t) + F_{n}(\\bar{z}(t)+O(t)),\\bar{z}(t)\n            \\big\\rangle\n            \\\\=&~\n            ce^{ct}|\\bar{z}(t)|^{2}\n            +\n            2e^{ct}\\big\\langle\n                 A_{n}\\bar{z}(t) + F_{n}(\\bar{z}(t)),\\bar{z}(t)\n            \\big\\rangle\n            +\n            2e^{ct}\\big\\langle\n                 F_{n}(\\bar{z}(t)+O(t))-F_{n}(\\bar{z}(t)),\\bar{z}(t)\n            \\big\\rangle\n            \\\\\\leq&~\n            -\n            ce^{ct}|\\bar{z}(t)|^{2}\n            +\n            2L_{F}e^{ct}|O(t)||\\bar{z}(t)|\n            \\leq\n            \\frac{L_{F}^{2}}{c}e^{ct}|O(t)|^{2}\n      \\end{align*}\n      due to the weighted Young inequality $ab \\leq \\kappa a^{2} +\\frac{b^{2}}{4\\kappa}$ for all $a,b \\in \\mathbb{R}$ with $\\kappa = c > 0$ in the last step. Applying \\eqref{eq:OtHn} leads to\n      \\begin{equation*}\n            |\\bar{z}(t)|^{2}\n            \\leq\n            |y|^{2}\n            +\n            \\frac{L_{F}^{2}}{c^{2}}\n            |Q^{\\frac12}|_{\\mathcal{L}(H)}^{2}\n            |\\psi|_{L^{2}(0,T;H)}^{2},\n            \\quad t \\in [0,T].\n      \\end{equation*}\n      Together with $|z(t)|^{2} \\leq 2|\\bar{z}(t)|^{2} + 2|O(t)|^{2}$ and \\eqref{eq:OtHn}, we complete the proof.\n\\end{proof}\n\n\n\n\n\n\\begin{Theorem\n\\label{th:semiinvariantgood\n      Suppose that Assumptions \\ref{ass:AQ}, \\ref{ass:F} and \\ref{ass:LFleqlambda1} hold. Then the mapping $V^{n}$ given by \\eqref{eq:Vny} is a good rate function.\n     \n\\end{Theorem}\n\n\n\n\n\\begin{proof}\n      Notice that $V^{n}$ is lower semi-continuous and $H_{n}$ is a finite dimensional space. In order to verify $V^{n}$ being a good rate function, it suffices to show that $K^{n}(\\alpha)$ is bounded. For any $y \\in K^{n}(\\alpha)$, the definition of $V^{n}(y)$ implies that for any $\\kappa > 0$ there exists $T_{\\kappa} > 0, z_{\\kappa} \\in C([0,T_{\\kappa}];H_{n}), z_{\\kappa}(0) = 0, z_{\\kappa}(T_{\\kappa}) = y$ such that\n      \\begin{equation*}\n            I_{0,T_{\\kappa}}^{n,0}(z_{\\kappa})\n            \\leq\n            V^{n}(y) + \\kappa\n            \\leq\n            \\alpha + \\kappa.\n      \\end{equation*}\n      The definition of $I_{0,T_{\\kappa}}^{n,0}(z_{\\kappa})$ means that there exists $\\psi_{\\kappa} \\in L^{2}(0,T_{\\kappa};H_{n})$ with $z_{0,0}^{n,\\psi_{\\kappa}} = z_{\\kappa}$ such that\n      \\begin{equation*}\n            \\frac{1}{2}|\\psi_{\\kappa}|_{L^{2}(0,T_{\\kappa};H)}^{2}\n            \\leq\n            I_{0,T_{\\kappa}}^{n,0}(z_{\\kappa}) + \\kappa\n            \\leq\n            \\alpha + 2\\kappa.\n      \\end{equation*}\n      Because of $z_{0,0}^{n,\\psi_{\\kappa}} = z_{\\kappa}$ satisfying \\eqref{eq:diffztdifft}, we use Lemma \\ref{lem:4.9} to get\n      \\begin{equation*}\n            |y|^{2}\n            =\n            |z_{\\kappa}(T_{\\kappa})|^{2}\n            \\leq\n            C\\big(1\n            +\n            |Q^{\\frac12}|_{\\mathcal{L}(H)}^{2}\n            |\\psi_{\\kappa}|_{L^{2}(0,T_{\\kappa};H)}^{2}\\big)\n            \\leq\n            C\\big(1\n            +\n            2(\\alpha+2\\kappa)\n            |Q^{\\frac12}|_{\\mathcal{L}(H)}^{2}\\big).\n      \\end{equation*}\n      Thus we complete the proof.\n\\end{proof}\n\n\n\n\n\n\\subsubsection{Lower bound estimate for the LDP of $\\{\\mu^{\\varepsilon,n}\\}_{\\varepsilon > 0}$}\n\n\n\n\n\n\n\\begin{Theorem\n\\label{th:semiinvariantmeasurelowerbounded}\n      Suppose that Assumptions \\ref{ass:AQ}, \\ref{ass:F} and \\ref{ass:LFleqlambda1} hold. Then for any $n \\in \\mathbb{N}^{+}$, $\\bar{y} \\in H_{n}$, $\\delta > 0$ and $\\gamma > 0$, there exists $\\varepsilon_{0} > 0$ such that\n      \\begin{equation}\\label{eq:muvnyHndelta\n            \\mu^{\\varepsilon,n}(\\{y \\in H_{n} : |y-\\bar{y}| < \\delta\\})\n            \\geq\n            \\exp\\Big(-\\frac{V^{n}(\\bar{y})+\\gamma}{\\varepsilon^{2}}\\Big),\n            \\quad \\varepsilon \\leq \\varepsilon_{0}.\n      \\end{equation}\n\\end{Theorem}\n\n\\begin{proof\n      Without loss of generality, we assume  $V^{n}(\\bar{y}) < +\\infty$, otherwise \\eqref{eq:muvnyHndelta} obviously holds. For any $\\bar{y} \\in H_{n}$ and $\\gamma > 0$, the definition of $V^{n}(\\bar{y})$ implies that there exists $\\bar{T} > 0$, $\\bar{z} \\in C([0,\\bar{T}];H_{n})$ with $\\bar{z}(0) = 0$ and $\\bar{z}(\\bar{T}) = \\bar{y}$ such that\n      $\n            I_{0,\\bar{T}}^{n,0}(\\bar{z}) \\leq V^{n}(\\bar{y}) + \\frac{\\gamma}{4}.\n      $\n      The definition of $I_{0,\\bar{T}}^{n,0}(\\bar{z})$\n      further implies that there exists $\\bar{\\varphi} \\in L^{2}(0,\\bar{T};H_{n})$ with $z_{0,0}^{n,\\bar{\\varphi}} = \\bar{z}$ such that\n      $$\n            \\frac{1}{2}|\\bar{\\varphi}|_{L^{2}(0,\\bar{T};H)}^{2}\n            \\leq\n            I_{0,\\bar{T}}^{n,0}(\\bar{z}) + \\frac{\\gamma}{4}\n            \\leq\n            V^{n}(\\bar{y}) + \\frac{\\gamma}{2}.\n      $$\n      For any $T > \\bar{T}$, $K > 0$ and $y \\in H_{n}$ with $|y| \\leq K$, we define\n      $$\n            \\tilde{\\varphi}(t)\n            :=\n            \\left\\{\n            \\begin{array}{ll}\n              0, & t \\in [0,T-\\bar{T}], \\\\\n              \\bar{\\varphi}(t-(T-\\bar{T})), & t \\in [T-\\bar{T},T]\n            \\end{array}\n            \\right.\n      $$\n      and\n      $$\n            \\tilde{z}(t)\n            :=\n            z_{0,y}^{n,\\tilde{\\varphi}}(t)\n            =\n            \\left\\{\n            \\begin{array}{ll}\n              z_{0,y}^{n,0}(t), &  t \\in [0,T-\\bar{T}], \\\\\n              z_{T-\\bar{T},z_{0,y}^{n,0}(T-\\bar{T})}^{n,\\bar{\\varphi}(\\cdot-(T-\\bar{T}))}(t),\n              & t \\in [T-\\bar{T},T].\n            \\end{array}\n            \\right.\n      $$\n      Then\n      we have\n      $|\\tilde{\\varphi}|_{L^{2}(0,T;H)}^{2}\n       =\n       |\\bar{\\varphi}|_{L^{2}(0,\\bar{T};H)}^{2}$\n      ,\n      $\\tilde{z} \\in C([0,T];H_{n})$ and\n      \\begin{equation}\\label{eq:I0Tnytildez\n      \\begin{split}\n            I_{0,T}^{n,y}(\\tilde{z})\n            =&\n            I_{0,T}^{n,y}(z_{0,y}^{n,\\tilde{\\varphi}})\n            =\n            \\frac{1}{2}\\inf\\{|\\varphi|_{L^{2}(0,T;H)}^{2}:\n            \\varphi \\in L^{2}(0,T;H_{n}), z_{0,y}^{n,\\varphi} = \\tilde{z}\\}\n            \\\\\\leq&\n            \\frac{1}{2}|\\tilde{\\varphi}|_{L^{2}(0,T;H)}^{2}\n            =\n            \\frac{1}{2}|\\bar{\\varphi}|_{L^{2}(0,\\bar{T};H)}^{2}\n            \\leq\n            V^{n}(\\bar{y}) + \\frac{\\gamma}{2}.\n      \\end{split}\n      \\end{equation}\n      In view of\n      \\begin{equation*}\n      \\begin{split}\n            \\tilde{z}(t)\n            =&\n            E_{n}(t-(T-\\bar{T}))z_{0,y}^{n,0}(T-\\bar{T})\n            +\n            \\int_{0}^{t-(T-\\bar{T})} E_{n}(t-(T-\\bar{T})-s)\n            F_{n}(\\tilde{z}(s+T-\\bar{T})) \\diff{s}\n            \\\\&+\n            \\int_{0}^{t-(T-\\bar{T})} E_{n}(t-(T-\\bar{T})-s)\n            Q_{n}^{\\frac12} \\bar{\\varphi}(s) \\diff{s},\n            \\quad t \\in [T-\\bar{T},T],\n      \\end{split}\n      \\end{equation*}\n      we set $\\hat{z}(t) :=  \\tilde{z}(t+(T-\\bar{T})), t \\in [0,\\bar{T}]$ to get\n      \\begin{equation*}\n      \\begin{split}\n            \\hat{z}(t)\n            =&\n            E_{n}(t)z_{0,y}^{n,0}(T-\\bar{T})\n            +\n            \\int_{0}^{t} E_{n}(t-s) F_{n}(\\hat{z}(s)) \\diff{s}\n            +\n            \\int_{0}^{t} E_{n}(t-s)Q_{n}^{\\frac12}\\bar{\\varphi}(s) \\diff{s},\n            \\quad t \\in [0,\\bar{T}]\n      \\end{split}\n      \\end{equation*}\n      and consequently\n      \\begin{equation*}\n            \\hat{z}(t)-z_{0,0}^{n,\\bar{\\varphi}}(t)\n            =\n            E_{n}(t)z_{0,y}^{n,0}(T-\\bar{T})\n            +\n            \\int_{0}^{t} E_{n}(t-s)\n                \\big( F_{n}(\\hat{z}(s)) - F_{n}(z_{0,0}^{n,\\bar{\\varphi}}(s)) \\big)\n            \\diff{s},\n            \\quad t \\in [0,\\bar{T}].\n      \\end{equation*}\n      Using $|E_{n}(t)|_{\\mathcal{L}(H_{n})} \\leq e^{-\\lambda_{1}t}$ for all $t \\geq 0$, \\eqref{eq:globalLF} and the Gronwall inequality yields\n      \\begin{equation}\\label{eq:hatztz00nbar\n      \\begin{split}\n            |\\hat{z}(t)-z_{0,0}^{n,\\bar{\\varphi}}(t)|\n            \\leq&\n            |z_{0,y}^{n,0}(T-\\bar{T})| e^{L\\bar{T}},\n            \\quad t \\in [0,\\bar{T}].\n      \\end{split}\n      \\end{equation}\n      As $\\{z_{0,y}^{n,0}(t)\\}_{t \\in [0,T-\\bar{T}]}$ is the solution of\n      \\begin{equation*}\n            \\frac{\\diff{z_{0,y}^{n,0}(t)}}{\\diff{t}}\n            =\n            A_{n}z_{0,y}^{n,0}(t)\n            +\n            F_{n}(z_{0,y}^{n,0}(t)),\n            \\quad t \\in (0,T-\\bar{T}],\n            \\quad z_{0,y}^{n,0}(0) = y,\n      \\end{equation*}\n      we use \\eqref{eq:dissipativecond}\n     \n      to get\n      \\begin{equation*}\n      \\begin{split}\n            \\frac{\\diff{e^{2ct}|z_{0,y}^{n,0}(t)|^{2}}}{\\diff{t}}\n            =&\n            2ce^{2ct}|z_{0,y}^{n,0}(t)|^{2}\n            +\n            2e^{2ct}\\big\\langle\n            Az_{0,y}^{n,0}(t) + F(z_{0,y}^{n,0}(t)),\n            z_{0,y}^{n,0}(t)\\big\\rangle\n            \\leq\n            0,\n      \\end{split}\n      \\end{equation*}\n      which yields $|z_{0,y}^{n,0}(t)| \\leq e^{-ct}|y|$ for all $t \\in [0,T-\\bar{T}]$ and thus\n      $\\lim\\limits_{T \\to \\infty\n      \\sup\\limits_{|y|_{H_{n}} \\leq K} |z_{0,y}^{n,0}(T-\\bar{T})| = 0$.\n      Noting \\eqref{eq:hatztz00nbar}, $\\hat{z}(\\bar{T}) = \\tilde{z}(T) = z_{0,y}^{n,\\tilde{\\varphi}}(T)$ and $z_{0,0}^{n,\\bar{\\varphi}}(\\bar{T}) = \\bar{z}(\\bar{T}) = \\bar{y}$, there exists $\\tilde{T} > \\bar{T}$ such that\n      $$\n            \\sup_{|y| \\leq K}\n            |z_{0,y}^{n,\\tilde{\\varphi}}(\\tilde{T})-\\bar{y}|\n            \\leq\n            \\frac{\\delta}{2}.\n      $$\n       This together with the invariance of $\\mu^{\\varepsilon,n}$ implies\n      \\begin{equation*}\n      \\begin{split}\n            \\mu^{\\varepsilon,n}(\\{y \\in H_{n} : |y-&\\bar{y}| < \\delta\\})\n            =\n            \\int_{H_{n}} \\P(|X_{y}^{\\varepsilon,n}(\\tilde{T})-\\bar{y}| < \\delta)\n            \\,\\mu^{\\varepsilon,n}(\\dif{y})\n            \\\\\\geq&\n            \\int_{{|y| \\leq K}} \\P(|X_{y}^{\\varepsilon,n}(\\tilde{T})-\\bar{y}| < \\delta)\n            \\,\\mu^{\\varepsilon,n}(\\dif{y})\n            \\\\\\geq&\n            \\int_{{|y| \\leq K}} \\P(|X_{y}^{\\varepsilon,n}(\\tilde{T})\n            - z_{0,y}^{n,\\tilde{\\varphi}}(\\tilde{T})| < \\delta\/2)\n            \\,\\mu^{\\varepsilon,n}(\\dif{y})\n            \\\\\\geq&\n            \\int_{{|y| \\leq K}}\n                \\P(|X_{y}^{\\varepsilon,n}\n                -z_{0,y}^{n,\\tilde{\\varphi}}|_{C([0,\\tilde{T}];H)}\n                < \\delta\/2)\n            \\,\\mu^{\\varepsilon,n}(\\dif{y}).\n      \\end{split}\n      \\end{equation*}\n      By \\eqref{eq:semisoluldplower} and \\eqref{eq:I0Tnytildez}, there exists $\\varepsilon_{1} > 0$ such that\n      \\begin{equation}\\label{eq:muvnyHnybary\n      \\begin{split}\n            \\mu^{\\varepsilon,n}(\\{y \\in H_{n} : |y-\\bar{y}| < \\delta\\})\n            \\geq&~\n            \\int_{{|y| \\leq K}}\n                \\exp\\Big(-\\frac{I_{0,\\tilde{T}}^{n,y}(z_{0,y}^{n,\\tilde{\\varphi}})+\\gamma\/2}\n                    {\\varepsilon^{2}}\\Big)\n            \\,\\mu^{\\varepsilon,n}(\\dif{y})\n            \\\\\\geq&~\n            \\mu^{\\varepsilon,n}({|y| \\leq K})\n            \\exp\\Big(-\\frac{V^{n}(\\bar{y})+\\gamma}{\\varepsilon^{2}}\\Big),\n            \\quad \\varepsilon \\leq \\varepsilon_{1}.\n      \\end{split}\n      \\end{equation}\n      It remains to estimate $\\mu^{\\varepsilon,n}({|y| \\leq K})$. Using the definition of $\\mu^{\\varepsilon,n}$ yields\n      \\begin{equation}\\label{eq:muvarepsilonnyHnK\n            \\mu^{\\varepsilon,n}(|y| > K)\n            =\n            \\lim_{i \\to \\infty}\n            \\frac{1}{t_{i}} \\int_{0}^{t_{i}}\n            \\P(|X_{0}^{\\varepsilon,n}(t)| > K) \\diff{t}.\n      \\end{equation}\n      Similarly to the proof of Lemma \\ref{lem:4.9}, we use \\eqref{eq:SPDEsemi} to obtain\n      \\begin{equation*}\n            |X_{0}^{\\varepsilon,n}(t)|\n            \\leq\n            \\varepsilon\\frac{L+c}{c}\n            |\\Gamma^{n}|_{C([0,t];H)}\n      \\end{equation*}\n      with\n      \\begin{equation}\\label{eq:Gammant\n            \\Gamma^{n}(t)\n            :=\n            \\int_{0}^{t}\n                E_{n}(t-s)Q_{n}^{\\frac12}\n            \\diff{W_{n}(s)},\n            \\quad t \\geq 0.\n      \\end{equation}\n      This together with the Chebyshev inequality shows that for any $t \\geq 0$ and $K > 0$,\n      \\begin{equation}\\label{eq:PX0vntHnK\n      \\begin{split}\n            &\\P(|X_{0}^{\\varepsilon,n}(t)| > K)\n            \\leq\n            \\P\\Big(\n            |\\Gamma^{n}|_{C([0,t];H)}\n            >\n            \\frac{cK}{\\varepsilon(L+c)}\\Big)\n            \\leq\n            \\varepsilon^{2}\\Big(\\frac{L+c}{cK}\\Big)^{2}C\n      \\end{split}\n      \\end{equation}\n      with $C$ being independent of $t$, which is due to\n      \\begin{equation}\\label{eq:GammanC0tHn2\n      \\begin{split}\n            \\mathbb{E}\\big[|\\Gamma^{n}|_{C([0,t];H)}^{2}\\big]\n            =&~\n            \\mathbb{E}\\Big[\\sup_{s \\in [0,t]}\n            \\Big|\\int_{0}^{s}\n                     E_{n}(s-r)Q_{n}^{\\frac12}\n                 \\diff{W_{n}(r)}\n            \\Big|^{2}\\Big]\n            \\leq\n            C\\int_{0}^{t}\n                 |E_{n}(t-r)Q_{n}^{\\frac12}\\big|_{\\mathcal{L}_{2}(H_{n})}^{2}\n            \\diff{r}\n            \\\\\\leq&~\n            C|(-A_{n})^{-\\frac{1}{2}}|_{\\mathcal{L}(H_{n})}^{2}\n            |(-A_{n})^{\\frac{1}{2}}Q_{n}^{\\frac12}|_{\\mathcal{L}_{2}(H_{n})}^{2}\n            \\int_{0}^{t}\n                |E_{n}(t-r)|_{\\mathcal{L}(H_{n})}^{2}\n            \\diff{r}\n            \\\\\\leq&~\n            C\\int_{0}^{t}\n                e^{-2\\lambda_{1}(t-r)}\n            \\diff{r}\n            \\leq\n            C,\n            \\quad t \\geq 0.\n      \\end{split}\n      \\end{equation}\n      By setting $\\bar{K} > 0$, \\eqref{eq:muvarepsilonnyHnK} and \\eqref{eq:PX0vntHnK} lead to\n      \\begin{equation*}\n            \\lim_{\\varepsilon \\to 0}\n            \\mu^{\\varepsilon,n}(|y| > \\bar{K})\n            \\leq\n            \\lim_{\\varepsilon \\to 0}\n            \\sup_{t \\geq 0}\\P(|X_{0}^{\\varepsilon,n}(t)| > \\bar{K})\n            \\leq\n            0\n      \\end{equation*}\n      and consequently\n      $\n            \\lim\\limits_{\\varepsilon \\to 0}\n            \\mu^{\\varepsilon,n}({|y| \\leq \\bar{K}})\n            =\n            1\n      $.\n      This in combination with \\eqref{eq:muvnyHnybary} shows that there exists sufficiently small $\\varepsilon_{0} \\leq \\varepsilon_{1}$  such that \\eqref{eq:muvnyHndelta} holds. Thus we complete the proof.\n\\end{proof}\n\n\n\n\n\n\\subsubsection{Upper bound estimate for the LDP of $\\{\\mu^{\\varepsilon,n}\\}_{\\varepsilon > 0}$}\n\nThe following lemma gives the exponential tail estimate for invariant measure $\\mu^{\\varepsilon,n}$ based on the Fernique theorem.\n\n\n\n\n\\begin{Lemma}\\label{lem:stochasticevolutiontailestimate}\n      Suppose that Assumptions \\ref{ass:AQ}, \\ref{ass:F} and \\ref{ass:LFleqlambda1} hold. Then for any $\\alpha \\geq 0$, there exists $\n      \\bar{K} > 0$ such that\n      \\begin{equation}\\label{eq:327327\n            \\mu^{\\varepsilon,n}(\\{u \\in H_{n} : |u| > \\bar{K}\\})\n            \\leq\n            \\exp\\Big(-\\frac{\\alpha}{\\varepsilon^{2}}\\Big),\n            \\quad \\varepsilon \\leq 1.\n      \\end{equation}\n\\end{Lemma}\n\n\n\n\n\\begin{proof}\n      For any $K > 0$, it follows from the definition of $\\mu^{\\varepsilon,n}$ that\n      \\begin{equation}\\label{eq:mvnHnyHnK\n            \\mu^{\\varepsilon,n}(\\{u \\in H_{n} : |u| > K\\})\n            =\n            \\lim_{i \\to \\infty}\\frac{1}{t_{i}}\n                \\int_{0}^{t_{i}} \\P(|X_{0}^{\\varepsilon,n}(t)| > K) \\diff{t}.\n      \\end{equation}\n      Similarly to the proof of Lemma \\ref{lem:4.9}, one can\n      show that there exists $C > 0$ independent of $t$ such that\n      $\n            |X_{0}^{\\varepsilon,n}(t)|\n            \\leq\n            \\varepsilon C |\\Gamma^{n}(t)|,\n      $\n      where $\\Gamma^{n}(t)$ is defined by \\eqref{eq:Gammant}.\n      Thus for any $K > 0$ and $\\kappa > 0$, we use the Markov inequality to get\n      \\begin{equation}\\label{eq:PBigsup0leqsleqt\n      \\begin{split}\n            \\P(|X_{0}^{\\varepsilon,n}(t)| > K)\n            \\leq&\n            \\P\\Big(|\\Gamma^{n}(t)|\n            > \\frac{K}{\\varepsilon C}\\Big)\n            \\\\=&\n            \\P\\Big(\n            \\exp\\big(\\kappa|\\Gamma^{n}(t)|^{2}\\big)\n            >\n            \\exp\\Big(\\kappa\\Big(\\frac{K}{\\varepsilon C}\\Big)^{2}\\Big)\\Big)\n            \\\\\\leq&\n            \\exp\\Big(-\\kappa\\Big(\\frac{K}{\\varepsilon C}\\Big)^{2}\\Big)\n            \\mathbb{E}\\big[\\exp\\big(\\kappa|\\Gamma^{n}(t)|^{2}\\big)\\big],\n            \\quad t \\geq 0.\n      \\end{split}\n      \\end{equation}\n      By \\cite[Proposition 4.28]{da2014stochastic},\n      \\begin{equation*}\n           \n            \\Gamma^{n}(t)\n            \\sim\n            \\mathcal{N}\\Big(0,\\int_{0}^{t} E_{n}(t-r)Q_{n}E_{n}(t-r)\\diff{r}\\Big)\n      \\end{equation*}\n      and for any $i = 1,\\ldots,n$,\n      \\begin{equation*}\n            \\Big(\\int_{0}^{t} E_{n}(t-r)Q_{n}E_{n}(t-r)\\diff{r}\\Big)e_{i}\n            =\n            \\Big(\\int_{0}^{t} e^{-2\\lambda_{i}(t-r)}q_{i}\\diff{r}\\Big)e_{i}\n            =\n            \\frac{q_{i}}{2\\lambda_{i}}\n            \\big(1 - e^{-2\\lambda_{i}t}\\big)e_{i}.\n      \\end{equation*}\n      Applying Fernique's theorem (\\cite[Proposition 1.13]{da2006introduction}) yields that for any $\\kappa < \\min\\big\\{\\frac{\\lambda_{i}}{q_{i}}:i =1, \\ldots,n\\big\\}$,\n     \n      \\begin{equation}\\label{eq:EBigexpBigkappa\n      \\begin{split}\n            &\n            \\mathbb{E}\\Big[\n            \\exp\\big(\\kappa|\\Gamma^{n}(t)|^{2}\\big)\n            \\Big]\n            =\n            \\Big(\\prod_{i=1}^{n}\n            \\Big(1 - \\kappa \\frac{q_{i}}{\\lambda_{i}}\n            \\big(1 - e^{-2\\lambda_{i}t}\\big) \\Big)\\Big)^{-\\frac{1}{2}}\n            \\leq\n            \\Big(\\prod_{i=1}^{n}\n            \\Big(1 - \\kappa \\frac{q_{i}}{\\lambda_{i}}\\Big)\\Big)^{-\\frac{1}{2}},\n            \\quad t \\geq 0.\n      \\end{split}\n      \\end{equation}\n      Combining \\eqref{eq:mvnHnyHnK}, \\eqref{eq:PBigsup0leqsleqt} and \\eqref{eq:EBigexpBigkappa} implies that one can choose\n      \\begin{equation*}\n            \\bar{K}\n            \\geq\n            C\\sqrt{\\frac{1}{\\kappa}\\Big(\\alpha\n            + \\varepsilon^{2}\\ln \\Big(\\prod_{i=1}^{n}\n            \\Big(1 - \\kappa \\frac{q_{i}}{\\lambda_{i}}\\Big)\\Big)^{-\\frac{1}{2}}\\Big)}\n      \\end{equation*}\n      such that \\eqref{eq:327327} holds. Thus we complete the proof.\n\\end{proof}\n\n\n\nFor any $T, L > 0$, we define\n\\begin{equation}\\label{eq:K0TnLalpha\n      K_{0,T}^{n,L}(\\alpha)\n      :=\n      \\{z \\in C([0,T];H_{n}) :\n      |z(0)| \\leq L,I_{0,T}^{n,z(0)}(z) \\leq \\alpha\\}.\n\\end{equation}\nSimilarly to the proof of \\cite[Lemma 7.1]{cerrai2005largeinvariant}, we have the following lemma.\n\n\n\n\\begin{Lemma}\\label{lem:zTzK0TnLalpha\n      Suppose that Assumptions \\ref{ass:AQ}, \\ref{ass:F} and \\ref{ass:LFleqlambda1} hold. Then for any $\\alpha \\geq 0$ and $\\delta > 0$, there exist $\\bar{L} > 0$ and $\\bar{T} > 0$ such that\n      $$\n            \\big\\{z(t): z \\in K_{0,t}^{n,\\bar{L}}(\\alpha)\\big\\}\n            \\subset\n            \\Big\\{y \\in H_{n}:\\rho^{H}(y,K^{n}(\\alpha)) < \\frac{\\delta}{2}\\Big\\},\n            \\quad t \\geq \\bar{T},\n      $$\n      where $\\rho^{H}(u,U) := \\inf\\limits_{u' \\in U}|u-u'|, u \\in H, U \\subset H$.\n\\end{Lemma}\n\n\n\n\n\n\n\nFor any $K,L > 0$, $J \\in \\mathbb{N}^{+}$, we define\n\\begin{equation}\n      H_{K,L,J\n      :=\n      \\{z \\in C([0,J];H_{n}) : |z(0)| \\leq K,\n      |z(j)| > L, j = 1,2,\\ldots,J \\}.\n\\end{equation}\nThe proof of the following lemma is similar to that of Lemma 7.2 in \\cite{cerrai2005largeinvariant}.\n\n\\begin{Lemma}\\label{lem:infI0barJnyz310\n      Suppose that Assumptions \\ref{ass:AQ}, \\ref{ass:F} and \\ref{ass:LFleqlambda1} hold. Then for any $\\alpha,\\delta > 0$ and $K > 0$, there exists $\\bar{J} \\in \\mathbb{N}^{+}$ such that\n      $$\n            \\inf\\{I_{0,\\bar{J}}^{n,z(0)}(z) : z \\in H_{K,\\bar{L},\\bar{J}}\\}\n            >\n            \\alpha,\n      $$\n      where $\\bar{L}$ is the constant introduced in Lemma \\ref{lem:zTzK0TnLalpha} corresponding to $\\alpha$ and $\\delta$.\n\\end{Lemma}\n\n\n\n\n\n\n\nBased on the above lemmas, we are in a position to show the LDP upper bound estimate.\n\\begin{Theorem}\n\\label{th:semiinvariantmeasureupperbounded}\n      Suppose that Assumptions \\ref{ass:AQ}, \\ref{ass:F} and \\ref{ass:LFleqlambda1} hold. Then for any $\\alpha \\geq 0$, $\\delta > 0$ and $\\gamma > 0$, there exists $\\varepsilon_{0} > 0$ such that\n      $$\n            \\mu^{\\varepsilon,n}(\\{y \\in H_{n} : \\rho^{H}(y,K^{n}(\\alpha)) \\geq \\delta\\})\n            \\leq\n            \\exp\\Big(-\\frac{\\alpha+\\gamma}{\\varepsilon^{2}}\\Big),\n            \\quad \\varepsilon \\leq \\varepsilon_{0}.\n      $$\n\\end{Theorem}\n\n\\begin{proof\n      By the invariance of $\\mu^{\\varepsilon,n}$ and \\eqref{eq:K0TnLalpha}, we have that for any $t \\geq 0$, $K,\\, L > 0$, $J \\in \\mathbb{N}^{+}$,\n      \\begin{equation}\\label{eq:muvarepsilonnyHn\n      \\begin{split}\n            &\\mu^{\\varepsilon,n}(\\{y \\in H_{n} :\n            \\rho^{H}(y,K^{n}(\\alpha)) \\geq \\delta\\})\n            =\n            \\int_{H_{n}} \\P(\\rho^{H}(X_{y}^{\\varepsilon,n}(t),K^{n}(\\alpha))\n             \\geq \\delta)\n            \\,\\mu^{\\varepsilon,n}(\\dif{y})\n            \\\\\\leq&\n            \\mu^{\\varepsilon,n}(\\{y \\in H_{n} : |y| > K\\})\n            +\n            \\int_{|y| \\leq K}\n                \\P(\\rho^{H}(X_{y}^{\\varepsilon,n}(t),K^{n}(\\alpha)) \\geq \\delta)\n            \\,\\mu^{\\varepsilon,n}(\\dif{y})\n            \\\\\\leq&\n            \\mu^{\\varepsilon,n}(\\{y \\in H_{n} : |y| > K\\})\n            +\n            \\sup_{|y| \\leq K}\n            \\P(X_{y}^{\\varepsilon,n} \\in H_{K,L,J})\n            \\\\&+\n            \\int_{|y| \\leq K}\n                \\P(\\rho^{H}(X_{y}^{\\varepsilon,n}(t),K^{n}(\\alpha)) \\geq \\delta,\n                X_{y}^{\\varepsilon,n} \\notin H_{K,L,J})\n            \\,\\mu^{\\varepsilon,n}(\\dif{y}).\n      \\end{split}\n      \\end{equation}\n       For the last term in \\eqref{eq:muvarepsilonnyHn}, applying\n      $\n            \\{X_{y}^{\\varepsilon,n} \\in H_{K,L,J}\\}\n            =\n            \\bigcap\\limits_{j=1}^{J}\n            \\{|X_{y}^{\\varepsilon,n}(j)| > L\\}\n      $\n      and the Markov property of $\\{X_{y}^{\\varepsilon,n}(t)\\}_{t \\geq 0}$ yields\n      \\begin{align*}\\label{eq:intyHnleqK}\n            &\\int_{|y| \\leq K}\n                \\P(\\rho^{H}(X_{y}^{\\varepsilon,n}(t),K^{n}(\\alpha)) \\geq \\delta,\n                X_{y}^{\\varepsilon,n} \\notin H_{K,L,J})\n            \\,\\mu^{\\varepsilon,n}(\\dif{y})\n            \\\\\\leq&\n            \\sum\\limits_{j=1}^{J}\n            \\int_{|y| \\leq K}\n                \\P\\big(\\rho^{H}(X_{y}^{\\varepsilon,n}(t),K^{n}(\\alpha)) \\geq \\delta,\n                |X_{y}^{\\varepsilon,n}(j)| \\leq L\\big)\n            \\,\\mu^{\\varepsilon,n}(\\dif{y})\n            \\\\=&\n            \\sum\\limits_{j=1}^{J}\n            \\int_{|y| \\leq K}\n                \\P\\big(\\rho^{H}(X_{y}^{\\varepsilon,n}(t-j),K^{n}(\\alpha)) \\geq \\delta,\n                |X_{y}^{\\varepsilon,n}(j-j)| \\leq L\\big)\n            \\,\\mu^{\\varepsilon,n}(\\dif{y})\n            \\\\\\leq&\n            \\sum\\limits_{j=1}^{J}\n            \\sup_{|y| \\leq L}\n            \\P\\big(\\rho^{H}(X_{y}^{\\varepsilon,n}(t-j),K^{n}(\\alpha)) \\geq \\delta\\big),\n            \\quad t > J.\n      \\end{align*}\n      Substituting the above estimate into \\eqref{eq:muvarepsilonnyHn} leads to\n      \\begin{equation}\n      \\begin{split}\n            \\mu^{\\varepsilon,n}&(\\{y \\in H_{n} :\n            \\rho^{H}(y,K^{n}(\\alpha)) \\geq \\delta\\})\n            \\leq\n            \\mu^{\\varepsilon,n}(\\{y \\in H_{n} : |y| > K\\})\n            \\\\&+\n            \\sum\\limits_{j=1}^{J}\n            \\sup_{|y| \\leq L}\n            \\P\\big(\\rho^{H}(X_{y}^{\\varepsilon,n}(t-j),K^{n}(\\alpha)) \\geq \\delta\\big)\n            +\n            \\sup_{|y| \\leq K}\n            \\P(X_{y}^{\\varepsilon,n} \\in H_{K,L,J}),\n            \\quad t > J.\n      \\end{split}\n      \\end{equation}\n      It follows from Lemma \\ref{lem:stochasticevolutiontailestimate} that for any $\\alpha \\geq 0$, there exists $\\bar{K} > 0$ such that\n      \\begin{equation}\\label{eq:muvnyHnK\n            \\mu^{\\varepsilon,n}(\\{y \\in H_{n} : |y| > \\bar{K}\\})\n            \\leq\n            \\exp\\Big(-\\frac{\\alpha}{\\varepsilon^{2}}\\Big),\n            \\quad \\varepsilon \\leq 1.\n      \\end{equation}\n      Lemma \\ref{lem:zTzK0TnLalpha} shows that for any $\\alpha > 0$ and $\\delta > 0$, there exists $\\bar{L} > 0$ and $\\bar{T} > 0$ such that for any $t \\geq \\bar{T}$ and $y \\in H_{n}$ with $|y| \\leq \\bar{L}$,\n      \\begin{equation*}\n            \\rho^{H}(z(t),K^{n}(\\alpha)) < \\frac{\\delta}{2},\n            \\quad z \\in K_{0,t}^{n,\\bar{L}}(\\alpha).\n      \\end{equation*}\n      Then\n      the triangle inequality\n      $\n            \\rho^{H}(X_{y}^{\\varepsilon,n}(t),K^{n}(\\alpha))\n            \\leq\n            \\rho^{H}(X_{y}^{\\varepsilon,n}(t),z(t))\n            +\n            \\rho^{H}(z(t),K^{n}(\\alpha))\n      $\n      implies $$\\{\\rho^{H}(X_{y}^{\\varepsilon,n}(t),K^{n}(\\alpha)) \\geq \\delta\\} \\subset\n      \\{\\rho^{C([0,t];H)}(X_{y}^{\\varepsilon,n},z)\n      \\geq \\frac{\\delta}{2}\\},\n      \\quad z \\in K_{0,t}^{n,\\bar{L}}(\\alpha).$$\n      By the arbitrariness of $z \\in K_{0,t}^{n,\\bar{L}}(\\alpha)$, we get\n      $$\\{\\rho^{H}(X_{y}^{\\varepsilon,n}(t),K^{n}(\\alpha)) \\geq \\delta\\} \\subset\n      \\{\\rho^{C([0,t];H)}\n            (X_{y}^{\\varepsilon,n},K_{0,t}^{n,y}(\\alpha))\n      \\geq \\frac{\\delta}{2}\\},\n      \\quad y \\in H_{n}, |y| \\leq \\bar{L}.$$\n      This together with \\eqref{eq:semisoluldpupper} implies that for any $t \\geq \\bar{T}$ there exists $\\varepsilon(t) > 0$ such that for any $y \\in H_{n}$ with $|y| \\leq \\bar{L}$,\n      \\begin{align*}\n            \\P\\big(\\rho^{H}(X_{y}^{\\varepsilon,n}(t),K^{n}(\\alpha)) \\geq \\delta\\big)\n            \\leq\n            \\P\\big(\\rho^{C([0,t];H)}(X_{y}^{\\varepsilon,n},\n                   K_{0,t}^{n,y}(\\alpha)) \\geq \\delta\/2\\big)\n            \\leq\n            \\exp\\Big(-\\frac{\\alpha-\\gamma\/2}{\\varepsilon^{2}}\\Big),\n            \\quad \\varepsilon \\leq \\varepsilon(t).\n      \\end{align*}\n      Taking $t = \\bar{T} + J$ and $\\varepsilon_{1} := \\min\\{\\varepsilon(t-j), j = 1,\\ldots,J\\}$, we have\n      $$\n            \\sum\\limits_{j=1}^{J}\n            \\sup_{|y| \\leq \\bar{L}}\n            \\P\\big(\\rho^{H}(X_{y}^{\\varepsilon,n}(t-j),K^{n}(\\alpha)) \\geq \\delta\\big)\n            \\leq\n            J \\exp\\Big(-\\frac{\\alpha-\\gamma\/2}{\\varepsilon^{2}}\\Big),\n            \\quad \\varepsilon \\leq \\varepsilon_{1}.\n      $$\n      Moreover, Lemma \\ref{lem:infI0barJnyz310} implies that there exists $\\bar{J} \\in \\mathbb{N}^{+}$ such that $\\rho^{C([0,\\bar{J}];H)}(H_{\\bar{K},\\bar{L},\\bar{J}},\n            K_{0,\\bar{J}}^{n,y}(\\alpha)) > 0$. It follows from \\eqref{eq:semisoluldpupper} that there exists $\\varepsilon_{2} > 0$ such that\n      \\begin{equation*}\n      \\begin{split}\n            \\sup_{|y| \\leq \\bar{K}}\n            \\P(X_{y}^{\\varepsilon,n} \\in H_{\\bar{K},\\bar{L},\\bar{J}})\n            \\leq&\n            \\sup_{|y| \\leq \\bar{K}}\n            \\P\\big(\\rho^{C([0,\\bar{J}];H)}\n            (X_{y}^{\\varepsilon,n},K_{0,\\bar{J}}^{n,y}(\\alpha)) \\geq\n            \\rho^{C([0,\\bar{J}];H)}(H_{\\bar{K},\\bar{L},\\bar{J}},\n            K_{0,\\bar{J}}^{n,y}(\\alpha))\\big)\n            \\\\\\leq&\n            \\exp\\Big(-\\frac{\\alpha-\\gamma\/2}{\\varepsilon^{2}}\\Big),\n            \\quad \\varepsilon \\leq \\varepsilon_{2}.\n      \\end{split}\n      \\end{equation*}\n      Combining the above estimates shows that for any $\\varepsilon \\leq \\varepsilon_{3} := \\min\\{1,\\varepsilon_{1},\\varepsilon_{2}\\}$,\n      $$\n            \\mu^{\\varepsilon,n}(\\{y \\in H_{n} :\n            \\rho^{H}(y,K^{n}(\\alpha)) \\geq \\delta\\})\n            \\leq\n            \\exp\\Big(-\\frac{\\alpha}{\\varepsilon^{2}}\\Big)\n            +\n            (1+\\bar{J})\\exp\\Big(-\\frac{\\alpha-\\gamma\/2}{\\varepsilon^{2}}\\Big).\n      $$\n      By taking sufficiently small $\\varepsilon_{0} \\leq \\varepsilon_{3}$, we complete the proof.\n\\end{proof}\n\n\n\n\n\n\\subsection{Weakly asymptotical preservation for the LDP of $\\{\\mu^{\\varepsilon,n}\\}_{\\varepsilon > 0}$}\n\\label{sebsec:semiinvameaLDPasy\n\n\nTo estimate the error between the rate functions $V^{n}$ and $V$, we denote the effective domain of $V$ by\n\\begin{equation}\n      \\mathcal{D}_{V}\n      :=\n      \\{u \\in H : V(u) < \\infty\\}.\n\\end{equation}\nNow we give the following definition of weakly asymptotical preservation  for the LDP of invariant measures by a numerical method.\n\n\\begin{Definition\n\\label{def:semiinvaweakasympres\n\n     \n      We say that \n     \n      the semi-discrete numerical method \\eqref{eq:SPDEsemi}\n      weakly asymptotically preserves the LDP of $\\{\\mu^{\\varepsilon}\\}_{\\varepsilon > 0}$ if for any $\\kappa > 0$ and $u \\in \\mathcal{D}_{V}$, there exist $n \\in \\mathbb{N}^{+}$ and $u_{n} \\in H_{n}$ such that\n      $$\n            |u-u_{n}| < \\kappa,\n            \\qquad\n            |V(u)-V^{n}(u_{n})| < \\kappa.\n      $$\n\\end{Definition}\n\n\n\n\\begin{Theorem}\\label{th:semiinvameaLDPasympre}\n      Suppose that Assumptions \\ref{ass:AQ}, \\ref{ass:F} and \\ref{ass:LFleqlambda1} hold. If\n     \n     \n      $F \\equiv \\textbf{0}$,\n      then \n     \n      the semi-discrete numerical method \\eqref{eq:SPDEsemi}\n      weakly asymptotically preserves the LDP of $\\{\\mu^{\\varepsilon}\\}_{\\varepsilon > 0}$.\n\\end{Theorem}\n\n\n\\begin{proof}\n      For any $u \\in \\mathcal{D}_{V}$, we set $u_{n} := P_{n}u, n \\in \\mathbb{N}^{+}$ and obtain $\\lim\\limits_{n \\to \\infty}|u-u_{n}| = 0$,\n      i.e., for each $\\kappa > 0$ there exists $n_{1} \\in \\mathbb{N}^{+}$ such that\n      \\begin{equation}\\label{eq:xynHkappa\n            |u-u_{n}| < \\kappa,\\quad n \\geq n_{1}.\n      \\end{equation}\n      The definition of $V(u)$ implies that for the above given $\\kappa > 0$ there exists $T_{\\kappa} > 0$ and $z_{\\kappa} \\in C([0,T_{\\kappa}];H)$ with $z_{\\kappa}(0) = 0, z_{\\kappa}(T_{\\kappa}) = u$ such that\n      $$\n            I_{0,T_{\\kappa}}^{0}(z_{\\kappa})\n            \\leq\n            V(u) + \\frac{\\kappa}{2} < +\\infty,\n      $$\n      which in combination with \\eqref{eq:I0Txz} yields $z_{\\kappa} \\in W_{2,Q}^{1,2}(T_{\\kappa})$. Define\n      \\begin{equation*}\n            \\varphi_{\\kappa}(t)\n            =\n            Q^{-\\frac{1}{2}}\n            \\Big(\\frac{\\diff{z_{\\kappa}(t)}}{\\diff{t}}\n            - Az_{\\kappa}(t) - F(z_{\\kappa}(t))\\Big)\n      \\end{equation*}\n      for almost all $t \\in [0,T_{\\kappa}]$, then $\\varphi_{\\kappa} \\in L^{2}(0,T_{\\kappa};H), z_{0,0}^{\\varphi_{\\kappa}} = z_{\\kappa}$ and $\\frac{1}{2}|\\varphi_{\\kappa}|_{L^{2}(0,T_{\\kappa};H)}^{2} = I_{0,T_{\\kappa}}^{0}(z_{\\kappa}) < +\\infty$, which together with Lemma \\ref{lem:spatialregularity} gives\n      \\begin{equation}\\label{eq:zkappatdotH2\n            |z_{\\kappa}(t)|_{\\dot{H}^{2}}\n            =\n            |z_{0,0}^{\\varphi_{\\kappa}}(t)|_{\\dot{H}^{2}}\n            \\leq\n            C,\n            \\quad t \\in [0,T_{\\kappa}].\n      \\end{equation}\n      As $P_{n}z_{\\kappa} \\in \\mathcal{H}_{1}^{n,0}(T_{\\kappa}),n \\in \\mathbb{N}^{+}$, we assert that\n      \\begin{equation}\\label{eq:I0Tkappan0Pnzkappa342}\n     \n            \\lim\\limits_{n \\to \\infty}\n            I_{0,T_{\\kappa}}^{n,0}(P_{n}z_{\\kappa})\n            =\n            I_{0,T_{\\kappa}}^{0}(z_{\\kappa}),\n      \\end{equation}\n      which shows that there exists $n_{2} \\in \\mathbb{N}^{+}$ such that\n      \\begin{equation}\\label{eq:I0Tkappan0Pnzkappa343}\n     \n            |I_{0,T_{\\kappa}}^{0}(z_{\\kappa})\n            -I_{0,T_{\\kappa}}^{n,0}(P_{n}z_{\\kappa})|\n            \\leq\n            \\frac{\\kappa}{2},\n            \\quad n \\geq n_{2}.\n      \\end{equation}\n      In fact, applying the identity\n      $|u|^{2} - |v|^{2} + |u-v|^{2} = 2\\langle u,u-v \\rangle$ for all $u,v \\in H$, \\eqref{eq:I0Txz}, \\eqref{eq:I0Tnyz}\n      and the H\\\"{o}lder inequality leads to\n      \\begin{align*}\n            &~|I_{0,T_{\\kappa}}^{0}(z_{\\kappa})\n            -I_{0,T_{\\kappa}}^{n,0}(P_{n}z_{\\kappa})|\n            \\leq\n            \\frac{1}{2}\n            \\int_{0}^{T_{\\kappa}}\n            \\Bigg|\n                \\Big|\n                Q^{-\\frac{1}{2}}\n                \\Big(\\frac{\\diff{z_{\\kappa}(t)}}{\\diff{t}}\n                - Az_{\\kappa}(t) - F(z_{\\kappa}(t))\\Big)\n                \\Big|^{2}\n                \\\\&~-\n                \\Big|\n                Q_{n}^{-\\frac12}\n                \\Big(\\frac{\\diff{P_{n}z_{\\kappa}(t)}}{\\diff{t}}\n                - A_{n}P_{n}z_{\\kappa}(t) - F_{n}(P_{n}z_{\\kappa}(t))\\Big)\n                \\Big|^{2}\n            \\Bigg|\n            \\diff{t}\n            \\\\=&~\n            \\frac{1}{2}\n            \\int_{0}^{T_{\\kappa}}\n            \\Bigg|\n            -\\Big|Q^{-\\frac{1}{2}}\n            \\Big(\\frac{\\diff{z_{\\kappa}(t)}}{\\diff{t}}\n            - Az_{\\kappa}(t) - F(z_{\\kappa}(t))\\Big)\n            -\n            Q_{n}^{-\\frac12}\n            \\Big(\\frac{\\diff{P_{n}z_{\\kappa}(t)}}{\\diff{t}}\n            - AP_{n}z_{\\kappa}(t) - F_{n}(P_{n}z_{\\kappa}(t))\\Big)\n            \\Big|^{2}\n            \\\\&~+\n            2\\Big\\langle\n            Q^{-\\frac{1}{2}}\n            \\Big(\\frac{\\diff{z_{\\kappa}(t)}}{\\diff{t}}\n            - Az_{\\kappa}(t) - F(z_{\\kappa}(t))\\Big)\n            -\n            Q_{n}^{-\\frac12}\n            \\Big(\\frac{\\diff{P_{n}z_{\\kappa}(t)}}{\\diff{t}}\n            - AP_{n}z_{\\kappa}(t) - F_{n}(P_{n}z_{\\kappa}(t))\\Big),\n            \\\\&~~~~~~~~~~Q^{-\\frac{1}{2}}\n            \\Big(\\frac{\\diff{z_{\\kappa}(t)}}{\\diff{t}}\n            - Az_{\\kappa}(t) - F(z_{\\kappa}(t))\\Big)\n            \\Big\\rangle\n            \\Bigg|\\diff{t}\n            \\\\\\leq&~\n            \\frac{1}{2}\\int_{0}^{T_{\\kappa}} \\Theta_{n}(t) \\diff{t}\n            +\n            \\Big(2I_{0,T_{\\kappa}}^{0}(z_{\\kappa})\n            \\int_{0}^{T_{\\kappa}}\\Theta_{n}(t)\\diff{t}\\Big)^{\\frac{1}{2}},\n      \\end{align*}\n      where\n      \\begin{equation*}\n            \\Theta_{n}(t)\n            :=\n            \\Big|Q^{-\\frac{1}{2}}\n            \\Big(\\frac{\\diff{z_{\\kappa}(t)}}{\\diff{t}}\n            - Az_{\\kappa}(t) - F(z_{\\kappa}(t))\\Big)\n            -\n            Q_{n}^{-\\frac12}\n            \\Big(\\frac{\\diff{P_{n}z_{\\kappa}(t)}}{\\diff{t}}\n            - AP_{n}z_{\\kappa}(t) - F_{n}(P_{n}z_{\\kappa}(t))\\Big)\n            \\Big|^{2}.\n      \\end{equation*}\n     \n      Observing $Q^{-\\frac12}u = Q_{n}^{-\\frac12}u,u \\in H_{n}$, we apply the mean value formula to get\n     \n      \\begin{align*}\n            \\Theta_{n}(t)\n            =&~\n            \\Big|(I-P_{n})Q^{-\\frac{1}{2}}\n            \\Big(\\frac{\\diff{z_{\\kappa}(t)}}{\\diff{t}}\n            - Az_{\\kappa}(t) - F(z_{\\kappa}(t))\\Big)\n            -\n            Q^{-\\frac{1}{2}}\n            \\big(F_{n}(z_{\\kappa}(t)) - F_{n}(P_{n}z_{\\kappa}(t))\\big)\n            \\Big|^{2}\n            \\\\=&~\n            \\Big|(I-P_{n})Q^{-\\frac{1}{2}}\n            \\Big(\\frac{\\diff{z_{\\kappa}(t)}}{\\diff{t}}\n            - Az_{\\kappa}(t) - F(z_{\\kappa}(t))\\Big)\n            \\\\&~-\n            \\int_{0}^{1}Q^{-\\frac{1}{2}}\n            F_{n}^{'}\\big(P_{n}z_{\\kappa}(t)\n            +r(z_{\\kappa}(t)-P_{n}z_{\\kappa}(t))\\big)\n            \\big(z_{\\kappa}(t)-P_{n}z_{\\kappa}(t)\\big)\\diff{r}\\Big|^{2}.\n      \\end{align*}\n      On one hand, we use \\eqref{eq:hybridAQ}, \\eqref{eq:auxiliaryassumption}, $F_{n}^{'} = P_{n}F'$ and $|I-P_{n}|_{\\mathcal{L}(H)} = 1, n \\in \\mathbb{N}^{+}$ to show\n      \\begin{equation*}\n      \\begin{split}\n            \\Theta_{n}(t)\n            \\leq\n            2\\Big|Q^{-\\frac{1}{2}}\n            \\Big(\\frac{\\diff{z_{\\kappa}(t)}}{\\diff{t}}\n            - Az_{\\kappa}(t) - F(z_{\\kappa}(t))\\Big)\\Big|^{2}\n            +\n            2L^{2}\\big(1+4|z_{\\kappa}(t)|_{\\dot{H}^{2}}^{2}\\big)^{2}\n            |z_{\\kappa}(t)|_{\\dot{H}^{2}}^{2}\n      \\end{split}\n      \\end{equation*}\n      with\n      \\begin{equation*}\n            \\int_{0}^{T_{\\kappa}} \\Big|Q^{-\\frac{1}{2}}\n            \\Big(\\frac{\\diff{z_{\\kappa}(t)}}{\\diff{t}} - Az_{\\kappa}(t)\n            - F(z_{\\kappa}(t))\\Big)\\Big|^{2}  \\diff{t}\n            =\n            2I_{0,T_{\\kappa}}^{0}(z_{\\kappa})\n            <\n            \\infty,\n      \\end{equation*}\n      \\begin{equation*}\n            \\int_{0}^{T_{\\kappa}}\n            \\big(1+4|z_{\\kappa}(t)|_{\\dot{H}^{2}}^{2}\\big)^{2}\n            |z_{\\kappa}(t)|_{\\dot{H}^{2}}^{2}\n            \\diff{t}\n            \\leq\n            T_{\\kappa}C^{2}\\big(1+4C^{2}\\big)^{2}\n            <\n            \\infty\n      \\end{equation*}\n      due to \\eqref{eq:zkappatdotH2}. On the other hand, using $\\lim\\limits_{n \\to \\infty} P_{n}u = u, u \\in H$ and\n      \\eqref{eq:auxiliaryassumption} yields\n      \\begin{equation*}\n      \\begin{split}\n            \\lim\\limits_{n \\to \\infty} \\Theta_{n}(t)\n            \\leq&\n            2\\lim\\limits_{n \\to \\infty}\n            \\Big|(I-P_{n})Q^{-\\frac{1}{2}}\n            \\Big(\\frac{\\diff{z_{\\kappa}(t)}}{\\diff{t}}\n            - Az_{\\kappa}(t) - F(z_{\\kappa}(t))\\Big)\\Big|^{2}\n            \\\\&+\n            2L^{2}\\big(1+4|(-A)z_{\\kappa}(t)|^{2}\\big)^{2}\n            \\lim\\limits_{n \\to \\infty}\n            |(I-P_{n})(-A)z_{\\kappa}(t)|^{2} = 0.\n      \\end{split}\n      \\end{equation*}\n      Combining the above results and using the dominated convergence theorem ensure \\eqref{eq:I0Tkappan0Pnzkappa342}.\n      As $P_{n}z_{\\kappa} \\in \\mathcal{H}_{1}^{n,0}(T_{\\kappa}) \\subset C([0,T_{\\kappa}];H_{n})$ satisfying $P_{n}z_{\\kappa}(0) = 0$ and $P_{n}z_{\\kappa}(T_{\\kappa}) = P_{n}u = u_{n}$ for each $n \\in \\mathbb{N}^{+}$, the definition of $V^{n}(u_{n})$ and \\eqref{eq:I0Tkappan0Pnzkappa343} lead to\n      \\begin{equation}\\label{eq:345\n            V^{n}(u_{n})\n            \\leq\n            I_{0,T_{\\kappa}}^{n,0}(P_{n}z_{\\kappa})\n            \\leq\n            I_{0,T_{\\kappa}}^{0}(z_{\\kappa}) + \\frac{\\kappa}{2}\n            \\leq\n            V(u) + \\kappa,\n            \\quad n \\geq \\max\\{n_{1},n_{2}\\}.\n      \\end{equation}\n      It remains to show $V(u) \\leq V^{n}(u_{n}) + \\kappa$ for sufficiently large $n \\in \\mathbb{N}^{+}$.\n      Now for each $n \\geq \\max\\{n_{1},n_{2}\\}$, the definition of $V^{n}(u_{n})$ implies that for the above given $\\kappa > 0$ there exists $T_{n,\\kappa} > 0$, $z_{n,\\kappa} \\in C([0,T_{n,\\kappa}];H_{n})$, $z_{n,\\kappa}(0) = 0$, $z_{n,\\kappa}(T_{n,\\kappa}) = u_{n}$ such that\n      \\begin{equation}\\label{eq:346\n            I_{0,T_{n,\\kappa}}^{n,0}(z_{n,\\kappa})\n            \\leq\n            V^{n}(u_{n}) + \\frac{\\kappa}{2}\n            \\leq\n            V(u) + \\frac{3}{2}\\kappa\n            <\n            + \\infty,\n      \\end{equation}\n      which in combination with \\eqref{eq:I0Tnyz} yields $z_{n,\\kappa} \\in \\mathcal{H}_{1}^{n,0}(T_{n,\\kappa})$.\n      Since\n      $z_{n,\\kappa} \\in W_{2,Q}^{1,2}(T_{n,\\kappa})$, the definition of $V(u_{n})$ in \\eqref{eq:Vny} shows\n      \\begin{equation}\\label{eq:347\n            V(u_{n})\n            \\leq\n            I_{0,T_{n,\\kappa}}^{0}(z_{n,\\kappa}).\n      \\end{equation}\n      Due to\n      $F \\equiv \\textbf{0}$,\n     \n      we have\n      \\begin{equation}\\label{eq:348\n      \\begin{split}\n            I_{0,T_{n,\\kappa}}^{n,0}(z_{n,\\kappa})\n            =&\n              \\frac{1}{2}\n              \\int_{0}^{T_{n,\\kappa}}\n                  \\Big|Q_{n}^{-\\frac12}\n                  \\Big(\\frac{\\diff{z_{n,\\kappa}(t)}}{\\diff{t}}\n                       -\n                       A_{n}z_{n,\\kappa}(t)\\Big)\n                  \\Big|^{2}\n              \\diff{t}\n            \\\\=&\n              \\frac{1}{2}\n              \\int_{0}^{T_{n,\\kappa}}\n                  \\Big|Q^{-\\frac12}\n                  \\Big(\\frac{\\diff{z_{n,\\kappa}(t)}}{\\diff{t}}\n                       -\n                       Az_{n,\\kappa}(t)\\Big)\n                  \\Big|^{2}\n              \\diff{t}\n              =\n              I_{0,T_{n,\\kappa}}^{0}(z_{n,\\kappa}).\n      \\end{split}\n      \\end{equation}\n      As $\\lim\\limits_{n \\to \\infty} u_{n} = u$, the semi-lower continuity of $V$ implies $V(u) \\leq \\liminf\\limits_{n \\to \\infty} V(u_{n})$, which means that for the above given $\\kappa > 0$ there exists $n_{3} \\in \\mathbb{N}^{+}$ such that\n      \\begin{equation}\\label{eq:349\n            V(u) \\leq \\liminf_{n \\to \\infty} V(u_{n})\n            \\leq\n            V(u_{n}) + \\frac{\\kappa}{2},\n            \\quad n \\geq n_{3}.\n      \\end{equation}\n      Combining \\eqref{eq:346}--\\eqref{eq:349} leads to\n      \\begin{equation}\\label{eq:350\n            V(u)\n            \\leq\n            V^{n}(u_{n}) + \\kappa,\n            \\quad n \\geq \\max\\{n_{1},n_{2},n_{3}\\}.\n      \\end{equation}\n      Finally, \\eqref{eq:xynHkappa}, \\eqref{eq:345} and \\eqref{eq:350} finish the proof.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Spatio-temporal discretization and its LDPs}\n\\label{sec:temporal\nIn Subsection \\ref{eq:AEEM}, we obtain a full-discrete numerical approximation $\\{Y_{y,m}^{\\varepsilon,n}\\}_{m \\in \\mathbb{N}}$ by applying the accelerated exponential Euler method to \\eqref{eq:SPDEsemi} and give the uniform LDP of sample paths of $\\{Y_{y}^{\\varepsilon, n}\\}_{\\varepsilon > 0}$, which is a continuous version of $\\{Y_{y,m}^{\\varepsilon,n}\\}_{m \\in \\mathbb{N}}$.\nThen the weakly asymptotical preservation for the LDP of $\\{X_{x}^{\\varepsilon}\\}_{\\varepsilon > 0}$ is given in Theorem \\ref{th:fullLDPasympre}.\nSubsection \\ref{subsec:fullinvameaLDP} shows the LDP of invariant measures $\\{\\mu^{\\varepsilon,n,\\tau}\\}_{\\varepsilon > 0}$ of  $\\{Y_{y}^{\\varepsilon, n}\\}_{\\varepsilon > 0}$ and establishes its weakly asymptotical preservation for the LDP of $\\{\\mu^{\\varepsilon}\\}_{\\varepsilon > 0}$.\n\\subsection{Accelerated exponential Euler method}\n\\label{eq:AEEM\n\n\n\n\nLet $\\tau > 0$ be a uniform time stepsize and $t_{m} := m\\tau, m \\in \\mathbb{N}$ the grid points, the accelerated exponential Euler method (see \\cite{jentzen2009overcoming}) applied to \\eqref{eq:SPDEsemi} is given\nby setting $Y_{y,0}^{\\varepsilon,n} = y$ and\n\\begin{equation}\\label{eq:EE\n      Y_{y,m+1}^{\\varepsilon,n}\n      =\n      E_{n}(\\tau)Y_{y,m}^{\\varepsilon,n}\n      +\n      \\int_{t_{m}}^{t_{m+1}}\n      E_{n}(t_{m+1}-s)F_{n}(Y_{y,m}^{\\varepsilon,n}) \\diff{s}\n      +\n      \\varepsilon \\int_{t_{m}}^{t_{m+1}}  E_{n}(t_{m+1}-s) Q_{n}^{\\frac12}\\diff{W_{n}(s)}\n\\end{equation}\nfor each $m \\in \\mathbb{N}$. To give a continuous time approximation of $\\{Y_{y,m}^{\\varepsilon,n}\\}_{m \\in \\mathbb{N}}$, we define $\\{Y_{y}^{\\varepsilon,n}(t)\\}_{t \\geq 0}$ by\n\\begin{equation}\\label{eq:4242\n      Y_{y}^{\\varepsilon,n}(t)\n      =\n      E_{n}(t-t_{m}) Y_{y,m}^{\\varepsilon,n}\n      +\n      \\int_{t_{m}}^{t} E_{n}(t-s)\n      F_{n}(Y_{y,m}^{\\varepsilon,n}) \\diff{s}\n      +\n      \\varepsilon \\int_{t_{m}}^{t} E_{n}(t-s) Q_{n}^{\\frac12}\\diff{W_{n}(s)}\n\\end{equation}\nfor all $t \\in [t_{m},t_{m+1}), m \\in \\mathbb{N}$.\nLet $\\lfloor \\cdot \\rfloor$ represent the largest integer no larger than the corresponding constant.\nAs $Y_{y}^{\\varepsilon,n}(t_{m}) = Y_{y,m}^{\\varepsilon,n},m \\in \\mathbb{N}$, i.e., $Y_{y}^{\\varepsilon,n}(\\tau \\lfloor t\/\\tau \\rfloor) = Y_{y,\\lfloor t\/\\tau \\rfloor}^{\\varepsilon,n}, t \\geq 0$, we have\n\\begin{equation}\\label{eq:saptiotemporalconteq\n\\begin{split}\n      &\\diff{Y_{y}^{\\varepsilon,n}(t)}\n      =\n      \\big(A_{n}Y_{y}^{\\varepsilon,n}(t)\n      +\n      F_{n}(Y_{y}^{\\varepsilon,n}(\\tau \\lfloor t\/\\tau \\rfloor)) \\big)\\diff{t}\n      +\n      \\varepsilon Q_{n}^{\\frac12}\\diff{W_{n}(t)},\n      \\quad t > 0,\n      \\quad Y_{y}^{\\varepsilon,n}(0) = y.\n\\end{split}\n\\end{equation}\nFor any $\\phi \\in L^{2}(0,T;H_{n})$, it follows from \\cite[Theorem 1.1]{peszat1994large} that the skeleton equation\n\\begin{equation*\n      \\frac{\\diff{z_{0,y}^{n,\\tau,\\phi}(t)}}{\\diff{t}}\n      =\n      A_{n}z_{0,y}^{n,\\tau,\\phi}(t)\n      +\n      F_{n}(z_{0,y}^{n,\\tau,\\phi}(\\tau \\lfloor t\/\\tau \\rfloor))\n      +\n      Q_{n}^{\\frac12}\\phi(t),\n      \\quad t > 0,\n      \\quad z_{0,y}^{n,\\tau,\\phi}(0) = y\n\\end{equation*}\nadmits a unique mild solution $\\{z_{0,y}^{n,\\tau,\\phi}(t)\\}_{t \\geq 0}$.\nWe will prove that $\\{Y_{y}^{\\varepsilon, n}\\}_{\\varepsilon > 0}$ satisfies a  uniform LDP on $C([0,T];H_{n})$ with the rate function\n$I_{0,T}^{n,\\tau,y} \\colon C([0,T];H_{n}) \\to [0,+\\infty]$\ndefined by\n\\begin{equation}\\label{eq:SPDEfullratefun\n      I_{0,T}^{n,\\tau,y}(z)\n      =\n      \\frac{1}{2} \\inf\\{|\\phi|_{L^{2}(0,T;H)}^{2}:\n      \\phi \\in L^{2}(0,T;H_{n}), z_{0,y}^{n,\\tau,\\phi}= z\\},\n      \\quad z \\in C([0,T];H_{n}).\n\\end{equation}\nBy the arguments for \\eqref{eq:I0Tnyz}, we have\n\\begin{equation}\\label{eq:I0TnMyz\n      I_{0,T}^{n,\\tau,y}(z)\n      =\n      \\begin{cases}\n        \\frac{1}{2}\n        \\int_{0}^{T}\n            \\Big|Q_{n}^{-\\frac12}\\Big(\\frac{\\diff{z(t)}}{\\diff{t}}\n                 -\n                 A_{n}z(t) - F_{n}(z(\\tau \\lfloor t\/\\tau \\rfloor))\\Big)\n            \\Big|^{2}\n        \\diff{t}\n        , & \\mbox{if } z \\in \\mathcal{H}_{1}^{n,y}(T), \\\\\n        +\\infty, & \\mbox{otherwise}.\n      \\end{cases}\n\\end{equation}\n\n\nSimilarly to the proof of Theorem \\ref{thm:semisolutionLDP}, we establish the uniform LDP of sample paths of\n$\\{Y_{y}^{\\varepsilon,n}\\}_{\\varepsilon > 0}$.\n\n\\begin{Theorem}\n\\label{th:fullLDPsamplepath\n      Suppose that Assumptions \\ref{ass:AQ} and \\ref{ass:F} hold. Then $\\{Y_{y}^{\\varepsilon,n}\\}_{\\varepsilon > 0}$ satisfies a uniform LDP on $C([0,T];H_{n})$ with the rate $\\frac{1}{\\varepsilon^{2}}$ and the good rate function $I_{0,T}^{n,\\tau,y}$ given by \\eqref{eq:SPDEfullratefun}, i.e.,\n      \\begin{enumerate}\n        \\item [(i)] compact level set: for any $T > 0$, $n \\in \\mathbb{N}^{+}$ and $y \\in H_{n}$, the level set $K_{0,T}^{n,\\tau,y}(\\alpha) :=\n            \\{z \\in C([0,T];H_{n}) : I_{0,T}^{n,\\tau,y}(z) \\leq \\alpha\\}$\n            is compact for every $\\alpha \\geq 0$;\n\n        \\item [(ii)] uniform lower bound: for any $T > 0$, $n \\in \\mathbb{N}^{+}$, $\\alpha \\geq 0$, $\\delta > 0$, $\\gamma > 0$ and $K > 0$, there exists $\\varepsilon_{0} > 0$ such that for any $y \\in H_{n}$ with $|y| \\leq K$ and $z \\in K_{0,T}^{n,\\tau,y}(\\alpha)$,\n            \\begin{equation}\\label{eq:fullsoluldplower\n            \\P\\big(|Y_{y}^{\\varepsilon,n}-z|_{C([0,T];H)} < \\delta\\big)\n            \\geq\n            \\exp\\Big(-\\frac{I_{0,T}^{n,\\tau,y}(z)+\\gamma}{\\varepsilon^2}\\Big),\n            \\quad \\varepsilon \\leq \\varepsilon_{0};\n            \\end{equation}\n        \\item [(iii)] uniform upper bound: for any $T > 0$, $n \\in \\mathbb{N}^{+}$, $\\alpha \\geq 0$, $\\delta > 0$, $\\gamma > 0$ and $K > 0$, there exists $\\varepsilon_0 > 0$ such that for any $y \\in H_{n}$ with $|y| \\leq K$,\n            \\begin{equation}\n            \\P\\big(\\rho^{C([0,T];H)}(Y_{y}^{\\varepsilon,n},\n            K_{0,T}^{n,\\tau,y}(\\alpha)) \\geq \\delta\\big)\n            \\leq\n            \\exp\\Big(-\\frac{\\alpha-\\gamma}{\\varepsilon^2}\\Big),\n            \\quad \\varepsilon \\leq \\varepsilon_{0}.\n            \\end{equation}\n      \\end{enumerate}\n\\end{Theorem}\n\n\n\n\n\n\n\n\n\n\n\n\n\nTo characterize the error between the rate functions $I_{0,T}^{n,\\tau,y}$ and $I_{0,T}^{x}$, we give the definition of weakly asymptotical preservation for the LDP of sample paths by a numerical method.\n\\begin{Definition}\n     \n      We say that\n      the fully-discrete numerical method \\eqref{eq:saptiotemporalconteq}\n      weakly asymptotically preserves the LDP of $\\{X_{x}^{\\varepsilon}\\}_{\\varepsilon > 0}$ if for any $\\kappa > 0$ and $z \\in W_{2,Q}^{1,2}(T)$ with $z(0) = x$, there exist $n \\in \\mathbb{N}^{+}$, $\\tau > 0$ and $z_{n,\\tau} \\in \\mathcal{H}_{1}^{n,y}(T)$ such that\n      $$\n            |z-z_{n,\\tau}|_{C([0,T];H)} < \\kappa,\n            \\quad\\quad\n            |I_{0,T}^{x}(z)-I_{0,T}^{n,\\tau,y}(z_{n,\\tau})| < \\kappa.\n      $$\n\\end{Definition}\n\n\nNow we show that $\\{Y_{y}^{\\varepsilon,n}\\}_{\\varepsilon > 0}$ weakly asymptotically preserves the LDP of $\\{X_{x}^{\\varepsilon}\\}_{\\varepsilon > 0}$.\n\\begin{Theorem}\n\\label{th:fullLDPasympre\n      Suppose that Assumptions \\ref{ass:AQ} and \\ref{ass:F} hold. Then\n      the fully-discrete numerical method \\eqref{eq:saptiotemporalconteq}\n      weakly asymptotically preserves the LDP of $\\{X_{x}^{\\varepsilon}\\}_{\\varepsilon > 0}$.\n\\end{Theorem}\n\n\n\\begin{proof}\n      For any $\\kappa > 0$ and $z \\in W_{2,Q}^{1,2}(T)$ with $z(0) = x$, it follows from Theorem \\ref{thm:spatialsolutionwap} that there exist $n \\in \\mathbb{N}^{+}$ and $z_{n} \\in \\mathcal{H}_{1}^{n,y}(T)$ such that\n      \\begin{equation}\\label{eq:part111\n            |z-z_{n}|_{C([0,T];H)} < \\frac{\\kappa}{2} < \\kappa,\n            \\quad\\quad\n            |I_{0,T}^{x}(z)-I_{0,T}^{n,y}(z_{n})| < \\frac{\\kappa}{2}.\n      \\end{equation}\n      Then we use \\eqref{eq:I0Tnyz}, \\eqref{eq:I0TnMyz}, the identity\n      $|u|^{2} - |v|^{2} + |u-v|^{2} = 2\\langle u,u-v \\rangle, u,v \\in H$ and the H\\\"{o}lder inequality to get\n      \\begin{align*\n            |I_{0,T}^{n,y}(z_{n})\n            -I_{0,T}^{n,\\tau,y}(z_{n})|\n            \\leq&~\n            \\frac{1}{2}\n                \\int_{0}^{T}\n                \\Bigg|\n                \\Big|Q_{n}^{-\\frac12}\n                \\Big(\\frac{\\diff{z_{n}(t)}}{\\diff{t}}\n                    -\n                    A_{n}z_{n}(t)\n                    -\n                    F_{n}(z_{n}(t))\\Big)\\Big|^{2}\n                \\\\&~-\n                \\Big|Q_{n}^{-\\frac12}\n                \\Big(\\frac{\\diff{z_{n}(t)}}{\\diff{t}}\n                    -\n                    A_{n}z_{n}(t)\n                    -\n                    F_{n}(z_{n}(\\tau \\lfloor t\/\\tau \\rfloor))\\Big)\n                    \\Big|^{2}\n                \\Bigg|\n                \\diff{t}\n            \\\\=&~\n            \\frac{1}{2}\n                \\int_{0}^{T}\n                \\Big|\n                2\\Big\\langle\n                Q_{n}^{-\\frac12}\n                \\big(F_{n}(z_{n}(\\tau \\lfloor t\/\\tau \\rfloor))\n                - F_{n}(z_{n}(t))\\big),\n                \\\\&~Q_{n}^{-\\frac12}\\Big(\\frac{\\diff{z_{n}(t)}}{\\diff{t}}\n                -\n                A_{n}z_{n}(t) - F_{n}(z_{n}(t))\\Big)\\Big\\rangle\n                \\\\&~-\n                \\big|Q_{n}^{-\\frac12}\n                \\big(F_{n}(z_{n}(\\tau \\lfloor t\/\\tau \\rfloor))\n                -F_{n}(z_{n}(t))\\big)\\big|^{2}\n                \\Big|\n            \\diff{t}\n            \\\\\\leq&~\n            \\Big(2I_{0,T}^{n,y}(z_{n})\\int_{0}^{T}\n                \\big|Q_{n}^{-\\frac12}\n                \\big(F_{n}(z_{n}(\\tau \\lfloor t\/\\tau \\rfloor))\n                  -\n                F_{n}(z_{n}(t))\\big)\n                \\big|^{2}\\diff{t}\\Big)^{\\frac12}\n                \\\\&~+\n                \\frac{1}{2}\\int_{0}^{T}\n                \\big|Q_{n}^{-\\frac12}\n                \\big(F_{n}(z_{n}(\\tau \\lfloor t\/\\tau \\rfloor))\n                -\n                F_{n}(z_{n}(t))\\big)\\big|^{2}\\diff{t}.\n      \\end{align*}\n      Applying \\eqref{eq:hybridAQ}, \\eqref{eq:auxiliaryassumption} and the mean value formula,\n     \n      we have\n      \\begin{align*}\n            &~\\big|Q_{n}^{-\\frac12}\n            \\big(F_{n}(z_{n}(\\tau \\lfloor t\/\\tau \\rfloor))\n            -\n            F_{n}(z_{n}(t))\\big)\\big|^{2}\n            \\\\\\leq&~\n            \\big|Q_{n}^{-\\frac12}(-A_{n})^{-1}\\big|_{\\mathcal{L}(H_{n})}\n            \\int_{0}^{1}\\big|(-A_{n})F_{n}^{'}\n            \\big(z_{n}(t)\n            +r(z_{n}(\\tau \\lfloor t\/\\tau \\rfloor)\n            -z_{n}(t))\\big)\n            \\big(z_{n}(\\tau \\lfloor t\/\\tau \\rfloor)\n            -z_{n}(t)\\big)\n            \\big|^{2}\\diff{r}\n            \\\\\\leq&~\n            C\\big(1+|z_{n}(t)|_{\\dot{H}^{2}}^{2}\n            +|z_{n}(\\tau \\lfloor t\/\\tau \\rfloor)|_{\\dot{H}^{2}}^{2}\\big)^{2}\n            \\big|(-A_{n})\\big(z_{n}(\\tau \\lfloor t\/\\tau \\rfloor)\n            -z_{n}(t)\\big)\\big|^{2},\n      \\end{align*}\n      where $F_{n}^{'} = P_{n}F'$. As $z_{n} \\in \\mathcal{H}_{1}^{n,y}(T)$, we define\n      $$\n          \\psi(t)\n          :=\n          Q_{n}^{-\\frac12}\n          \\Big(\\frac{\\diff{z_{n}(t)}}{\\diff{t}}\n          -\n          A_{n}z_{n}(t)\n          -\n          F_{n}(z_{n}(t))\\Big)\n      $$\n      for almost all $t \\in [0,T]$. Then $\\psi \\in L^{2}(0,T;H_{n})$, $z_{0,y}^{n,\\psi} = z_{n}$ and $\\frac{1}{2}|\\psi|_{L^{2}(0,T;H)}^{2} = I_{0,T}^{n,y}(z_{n})$.\n      Similarly to the proof of \\eqref{eq:38}, we obtain\n      $|z_{n}(t)|_{\\dot{H}^{2}} =\n      |z_{0,y}^{n,\\psi}(t)|_{\\dot{H}^{2}}\n      \\leq C(1+|y|_{\\dot{H}^{2}}),t \\in [0,T]$, and thus\n      \\begin{align*}\n            &\\big|Q_{n}^{-\\frac12}\n            \\big(F_{n}(z_{n}(\\tau \\lfloor t\/\\tau \\rfloor))\n            -F_{n}(z_{n}(t))\\big)\\big|^{2}\n            \\leq\n            C\\big|(-A_{n})\\big|_{\\mathcal{L}(H_{n})}^{2}\n            \\big(1+|y|_{\\dot{H}^{2}}^{2}\\big)^{2}|y|_{\\dot{H}^{2}}^{2}\n            <\n            +\\infty.\n      \\end{align*}\n      By $\\lim\\limits_{\\tau \\to 0}\n      \\big|z_{n}(\\tau \\lfloor t\/\\tau \\rfloor)\n      -z_{n}(t) \\big| = 0, t \\in [0,T]$, we have\n      \\begin{align*}\n            \\lim\\limits_{\\tau \\to 0}\n            \\big|Q_{n}^{-\\frac12}\n            \\big(F_{n}(z_{n}(\\tau \\lfloor t\/\\tau \\rfloor))\n            -F_{n}(z_{n}(t))\\big)\\big|^{2}\n            \\leq\n            C\\big|(-A_{n})\\big|_{\\mathcal{L}(H_{n})}^{2}\n            \\lim\\limits_{\\tau \\to 0}\n            \\big|z_{n}(\\tau \\lfloor t\/\\tau \\rfloor) - z_{n}(t)\\big|^{2}\n            =\n            0.\n      \\end{align*}\n      It follows from the bounded convergence theorem that\n      \\begin{equation}\\label{eq:410410410\n      \\lim\\limits_{\\tau \\to 0}|I_{0,T}^{n,y}(z_{n})\n            -I_{0,T}^{n,\\tau,y}(z_{n})| = 0,\n      \\end{equation}\n      and consequently that there exists $\\tau > 0$ such that\n      \\begin{equation}\\label{eq:part222\n            \\big|I_{0,T}^{n,y}(z_{n})\n            -I_{0,T}^{n,\\tau,y}(z_{n})\\big|\n            \\leq\n            \\frac{\\kappa}{2}.\n      \\end{equation}\n      For $n,\\tau$ given by \\eqref{eq:part111} and \\eqref{eq:part222}, we set $z_{n,\\tau} := z_{n}$ and use the triangle inequality to complete the proof.\n\\end{proof}\n\n\n\n\n\n\\subsection{LDP of invariant measures of spatio-temporal discretization}\n\\label{subsec:fullinvameaLDP\n\n\n\nIn this subsection, we study the LDP of invariant measures of the spatio-temporal discretization. The following lemma gives the uniform boundedness of numerical solutions, which ensures the existence of the invariant measures of the full discretization. We refer the interested readers to \\cite{chen2017approximation,hong2017numerical,\nhong2019invariant,cui2021weak} for the investigation on the invariant measures of the full discretzations for SPDEs.\n\n\n\n\n\n\\begin{Lemma}\\label{lem:uniformbound2ordermoment}\n      Suppose that Assumptions \\ref{ass:AQ}, \\ref{ass:F} and \\ref{ass:LFleqlambda1} hold. If $\\tau \\leq \\tau_{0}$ with $\\tau_{0}$ satisfying\n      $\\frac{e^{\\lambda_{1}\\tau_{0}}-1}{\\lambda_{1}\\tau_{0}} = \\frac{\\lambda_{1}+L_{F}}{2L_{F}}$, then there exists $C > 0$ independent of $t$ such that\n      \\begin{equation}\\label{eq:Y0vartHn\n            \\sup\\limits_{t \\geq 0}\n            \\mathbb{E}\\big[|Y_{y}^{\\varepsilon,n}(t)|^{2}\\big]\n            \\leq\n            C.\n      \\end{equation}\n\\end{Lemma}\n\n\n\n\\begin{proof}\n      We rewrite \\eqref{eq:EE} as\n      \\begin{align*}\n            Y_{y,m}^{\\varepsilon,n}\n            =&\n            E_{n}^{m}(\\tau)Y_{y,0}^{\\varepsilon,n}\n            +\n            \\int_{0}^{t_{m}}E_{n}(t_{m}-s)\n            F_{n}(Y_{y,\\lfloor s\/\\tau \\rfloor}^{\\varepsilon,n})\n            \\diff{s}\n            +\n            \\varepsilon \\int_{0}^{t_{m}}  E_{n}(t_{m}-s) Q_{n}^{\\frac12}\\diff{W_{n}(s)}.\n      \\end{align*}\n      Noting $\\{\\Gamma^{n}(t)\\}_{t \\geq 0}$ given by \\eqref{eq:Gammant} and setting $\\bar{Y}_{y,m}^{\\varepsilon,n} := Y_{y,m}^{\\varepsilon,n} - \\varepsilon\\Gamma^{n}(t_{m})$, it follows that\n      \\begin{align*}\n            \\bar{Y}_{y,m}^{\\varepsilon,n}\n            =&\n            E_{n}^{m}(\\tau)\\bar{Y}_{y,0}^{\\varepsilon,n}\n            +\n            \\int_{0}^{t_{m}}E_{n}(t_{m}-s)\n            F_{n}(\\bar{Y}_{y,\\lfloor s\/\\tau \\rfloor}^{\\varepsilon,n}\n            +\\varepsilon\\Gamma^{n}(\\tau \\lfloor s\/\\tau \\rfloor))\n            \\diff{s},\n            \\quad\n            \\bar{Y}_{y,0}^{\\varepsilon,n} = y\n      \\end{align*}\n      and thus\n      \\begin{align*}\n            \\bar{Y}_{y,m}^{\\varepsilon,n}\n            =&\n            E_{n}(\\tau)\\bar{Y}_{y,m-1}^{\\varepsilon,n}\n            +\n            (-A_{n})^{-1}(I-E_{n}(\\tau))\n            F_{n}(\\bar{Y}_{y,m-1}^{\\varepsilon,n}\n            +\\varepsilon\\Gamma^{n}(t_{m-1})).\n      \\end{align*}\n      Applying $|E_{n}(\\tau)|_{\\mathcal{L}(H_{n})} \\leq e^{-\\lambda_{1}\\tau}$ and\n      $|(-A_{n})^{-1}(I-E_{n}(\\tau))|_{\\mathcal{L}(H_{n})} \\leq (1-e^{-\\lambda_{1}\\tau})\/\\lambda_{1} \\leq \\tau$ yields\n      \\begin{align*}\n            |\\bar{Y}_{y,m}^{\\varepsilon,n}|_{L^{2}(\\Omega;H)}\n      \\leq&\n            e^{-\\lambda_{1}\\tau}\n            |\\bar{Y}_{y,m-1}^{\\varepsilon,n}|_{L^{2}(\\Omega;H)}\n            +\n            \\tau \\frac{1-e^{-\\lambda_{1}\\tau}}{\\lambda_{1}\\tau}\n            |F_{n}(\\bar{Y}_{y,m-1}^{\\varepsilon,n}\n            +\\varepsilon\\Gamma^{n}(t_{m-1}))|_{L^{2}(\\Omega;H)}\n      \\\\\\leq&\n            \\Big(e^{-\\lambda_{1}\\tau}\n            +\n            \\tau L_{F}\\frac{1-e^{-\\lambda_{1}\\tau}}{\\lambda_{1}\\tau}\\Big)\n            |\\bar{Y}_{y,m-1}^{\\varepsilon,n}|_{L^{2}(\\Omega;H)}\n            +\n            \\tau L_{F}\\frac{1-e^{-\\lambda_{1}\\tau}}{\\lambda_{1}\\tau}\n            |\\varepsilon\\Gamma^{n}(t_{m-1})|_{L^{2}(\\Omega;H)}.\n      \\end{align*}\n      Recalling $\\sup\\limits_{t \\geq 0}\n      |\\Gamma^{n}(t)|_{L^{2}(\\Omega;H)} \\leq C$ in \\eqref{eq:GammanC0tHn2} and\n      $\\tau \\leq \\tau_{0}$ with $\\tau_{0}$ satisfying\n      $\\frac{e^{\\lambda_{1}\\tau_{0}}-1}{\\lambda_{1}\\tau_{0}} = \\frac{\\lambda_{1}+L_{F}}{2L_{F}}$,\n      we get\n      \\begin{align*}\n            |\\bar{Y}_{y,m}^{\\varepsilon,n}|_{L^{2}(\\Omega;H)}\n      \\leq&\n            \\Big(e^{-\\lambda_{1}\\tau}\n            +\n            \\tau L_{F}e^{-\\lambda_{1}\\tau}\n            +\n            \\tau\\frac{\\lambda_{1}-L_{F}}{2}e^{-\\lambda_{1}\\tau}\\Big)\n            |\\bar{Y}_{y,m-1}^{\\varepsilon,n}|_{L^{2}(\\Omega;H)}\n            +\n            C\\tau\n      \\\\=&\n            \\Big(1 + \\tau\\frac{\\lambda_{1}+L_{F}}{2}\\Big)\n            e^{-\\lambda_{1}\\tau}\n            |\\bar{Y}_{y,m-1}^{\\varepsilon,n}|_{L^{2}(\\Omega;H)}\n            +\n            C\\tau\n      \\\\\\leq&\n            e^{-\\frac{\\lambda_{1}-L_{F}}{2}\\tau}\n            |\\bar{Y}_{y,m-1}^{\\varepsilon,n}|_{L^{2}(\\Omega;H)}\n            +\n            C\\tau\n      \\\\\\leq&\n            e^{-\\frac{\\lambda_{1}-L_{F}}{2}m\\tau}|y|\n            +\n            C\\tau\\frac{e^{\\frac{\\lambda_{1}-L_{F}}{2}\\tau}}\n                  {e^{\\frac{\\lambda_{1}-L_{F}}{2}\\tau}-1}\n      \\\\\\leq&\n            |y|\n            +\n            2C\\frac{e^{\\frac{\\lambda_{1}-L_{F}}{2}\\tau_{0}}}\n                  {\\lambda_{1}-L_{F}}.\n      \\end{align*}\n      Thus we assert that there exists $C > 0$ independent of $m \\in \\mathbb{N}$ such that $\\sup\\limits_{m \\in \\mathbb{N}}|Y_{y,m}^{\\varepsilon,n}|_{L^{2}(\\Omega;H)} \\leq C$.\n     \n      It follows from \\eqref{eq:4242} that\n      \\begin{align*}\n            |Y_{y}^{\\varepsilon,n}(t)|_{L^{2}(\\Omega;H)}\n      \\leq&\n            |Y_{y,m-1}^{\\varepsilon,n}|_{L^{2}(\\Omega;H)}\n            +\n            |F_{n}(Y_{y,m-1}^{\\varepsilon,n})|_{L^{2}(\\Omega;H)}\n            +\n            |\\varepsilon \\Gamma^{n}(t)|_{L^{2}(\\Omega;H)}\n            \\leq\n            C\n      \\end{align*}\n      with $C$ being independent of $t$. Thus we complete the proof.\n\\end{proof}\n\n\n\n\n\nBy Lemma \\ref{lem:uniformbound2ordermoment} and \\cite[Proposition 7.10]{da2006introduction}, the family of probability measures $\\{\\mu_{t}^{\\varepsilon,n,\\tau}\\}_{t > 0}$, defined by\n      $$\n            \\mu_{t}^{\\varepsilon,n,\\tau}(B)\n            :=\n            \\frac{1}{t} \\int_{0}^{t}\n            \\P\\big(Y_{0}^{\\varepsilon,n}(s) \\in B\\big)\n            \\diff{s},\n            \\quad B \\in \\mathcal{B}(H_{n}), t > 0,\n      $$\nis tight. It follows from the Krylov--Bogoliubov theorem (\\cite[Theorem 7.1]{da2006introduction}) that there exists $\\{t_{i}\\}_{i \\in \\mathbb{N}^{+}} \\uparrow +\\infty$ (possibly depending on $\\varepsilon$) such that the sequence $\\{\\mu_{t_{i}}^{\\varepsilon,n,\\tau}\\}_{i \\in \\mathbb{N}^{+}}$ converges weakly to some probability measure $\\mu^{\\varepsilon,n,\\tau}$ on $(H_{n},\\mathcal{B}(H_{n}))$, which is invariant for \\eqref{eq:saptiotemporalconteq}. Following the procedures\nin Subsection \\ref{subsec:semiinvameaLDPs}, we establish the LDP for $\\{\\mu^{\\varepsilon,n,\\tau}\\}_{\\varepsilon > 0}$ on $H_{n}$ with the rate function $V^{n,\\tau}$ defined by\n\\begin{equation}\\label{eq:Vntauy\n      V^{n,\\tau}(u)\n      =\n      \\inf\\{I_{0,T}^{n,\\tau,0}(z) : T > 0, z \\in C([0,T];H_{n}), z(0) = 0, z(T) = u \\},\n      \\quad u \\in H_{n}.\n\\end{equation}\n\n\n\\begin{Theorem}\n     \n      Under assumptions in Lemma \\ref{lem:uniformbound2ordermoment}, $\\{\\mu^{\\varepsilon,n,\\tau}\\}_{\\varepsilon > 0}$ satisfies an LDP on $H_{n}$ with the rate $\\frac{1}{\\varepsilon^{2}}$ and the good rate function $V^{n,\\tau}$ given by \\eqref{eq:Vntauy}.\n     \n\\end{Theorem}\n\n\nWe give the definition of weakly asymptotical preservation for the LDP of invariant measures by a numerical method to characterize the error between $V^{n,\\tau}$ and $V$.\n\n\n\n\n\n\\begin{Definition}\n      We say that\n      the fully-discrete numerical method \\eqref{eq:saptiotemporalconteq}\n      weakly asymptotically preserves the LDP of $\\{\\mu^{\\varepsilon}\\}_{\\varepsilon > 0}$ if for any $\\kappa > 0$ and $u \\in \\mathcal{D}_{V}$, there exist $n \\in \\mathbb{N}^{+}$, $\\tau > 0$ and $u_{n,\\tau} \\in H_{n}$ such that\n      $$\n            |u-u_{n,\\tau}| < \\kappa,\n            \\qquad\n            |V(u)-V^{n,\\tau}(u_{n,\\tau})| < \\kappa.\n      $$\n\\end{Definition}\n\n\n\n\n\n\\begin{Theorem}\\label{fullinvameaLDPasypre\n      Suppose that assumptions in Lemma \\ref{lem:uniformbound2ordermoment} hold.\n     \n      If\n      $F \\equiv \\textbf{0}$,\n      then\n      the fully-discrete numerical method \\eqref{eq:saptiotemporalconteq}\n      weakly asymptotically preserves the LDP of $\\{\\mu^{\\varepsilon}\\}_{\\varepsilon > 0}$.\n\\end{Theorem}\n\n\n\\begin{proof}\n      For any $\\kappa > 0$ and $u \\in \\mathcal{D}_{V}$, Theorem \\ref{th:semiinvameaLDPasympre} shows that there exist\n      $n \\in \\mathbb{N}^{+}$ and $u_{n} \\in H_{n}$ such that\n      \\begin{equation}\\label{eq:invapart111\n            |u-u_{n}| < \\frac{\\kappa}{2} < \\kappa,\n            \\qquad\n            |V(u)-V^{n}(u_{n})| < \\frac{\\kappa}{2}.\n      \\end{equation}\n      Then the definition of $V^{n}(u_{n})$ in \\eqref{eq:Vny} ensures that for any $\\kappa > 0$ there exist $T_{n,\\kappa} > 0, z_{n,\\kappa} \\in C([0,T_{n,\\kappa}];H_{n}), z_{n,\\kappa}(0) = 0$ and  $z_{n,\\kappa}(T_{n,\\kappa}) = u_{n}$ such that\n      \\begin{equation}\\label{eq:518\n            I_{0,T_{n,\\kappa}}^{n,0}(z_{n,\\kappa})\n            \\leq\n            V^{n}(u_{n}) + \\frac{\\kappa}{4}\n            <\n            +\\infty,\n      \\end{equation}\n      which implies $z_{n,\\kappa} \\in \\mathcal{H}_{1}^{n,0}(T_{n,\\kappa})$ by \\eqref{eq:I0Tnyz}. According to \\eqref{eq:Vntauy}, we have\n      \\begin{equation}\\label{eq:519\n            V^{n,\\tau}(u_{n})\n            \\leq\n            I_{0,T_{n,\\kappa}}^{n,\\tau,0}(z_{n,\\kappa}).\n      \\end{equation}\n      As \\eqref{eq:410410410} in Theorem \\ref{th:fullLDPasympre} ensures\n      $\n            \\lim\\limits_{\\tau \\to 0}I_{0,T_{n,\\kappa}}^{n,\\tau,0}(z_{n,\\kappa})\n            =\n            I_{0,T_{n,\\kappa}}^{n,0}(z_{n,\\kappa})$, i.e.,\n      there exists $\\tau > 0$ such that\n      \\begin{equation}\\label{eq:520\n            \\big|I_{0,T_{n,\\kappa}}^{n,\\tau,0}(z_{n,\\kappa})\n            -\n            I_{0,T_{n,\\kappa}}^{n,0}(z_{n,\\kappa})\\big|\n            \\leq\n            \\frac{\\kappa}{4},\n      \\end{equation}\n      it follows from \\eqref{eq:518}, \\eqref{eq:519} and \\eqref{eq:520} that\n      \\begin{equation}\\label{eq:521\n            V^{n,\\tau}(u_{n})\n            \\leq\n            I_{0,T_{n,\\kappa}}^{n,\\tau,0}(z_{n,\\kappa})\n            \\leq\n            I_{0,T_{n,\\kappa}}^{n,0}(z_{n,\\kappa})\n            +\n            \\frac{\\kappa}{4}\n            \\leq\n            V^{n}(u_{n}) + \\frac{\\kappa}{2}.\n      \\end{equation}\n      It remains to show $V^{n}(u_{n}) \\leq V^{n,\\tau}(u_{n}) + \\frac{\\kappa}{2}$ for the above given $\\tau > 0$. The definition of $V^{n,\\tau}(u_{n})$ in \\eqref{eq:Vntauy} implies that for $\\kappa > 0$ there exists $T_{n,\\tau,\\kappa} > 0, z_{n,\\tau,\\kappa} \\in C([0,T_{n,\\tau,\\kappa}];H_{n}), z_{n,\\tau,\\kappa}(0) = 0$ and $z_{n,\\tau,\\kappa}(T_{n,\\tau,\\kappa}) = u_{n}$ such that\n      \\begin{equation}\\label{eq:522\n            I_{0,T_{n,\\tau,\\kappa}}^{n,\\tau,0}(z_{n,\\tau,\\kappa})\n            \\leq\n            V^{n,\\tau}(u_{n}) + \\frac{\\kappa}{2}\n            \\leq\n            V^{n}(u_{n}) + \\kappa\n            <\n            +\\infty,\n      \\end{equation}\n      which together with \\eqref{eq:I0TnMyz} implies $z_{n,\\tau,\\kappa} \\in \\mathcal{H}_{1}^{n,0}(T_{n,\\tau,\\kappa})$.\n      The definition of $V^{n}(u_{n})$ in \\eqref{eq:Vny} yields\n      \\begin{equation}\\label{eq:523\n            V^{n}(u_{n})\n            \\leq\n            I_{0,T_{n,\\tau,\\kappa}}^{n,0}(z_{n,\\tau,\\kappa}).\n      \\end{equation}\n      Taking advantaging of $F \\equiv \\textbf{0}$, we have\n      \\begin{equation}\\label{eq:524\n            I_{0,T_{n,\\tau,\\kappa}}^{n,0}(z_{n,\\tau,\\kappa})\n            =\n            \\frac{1}{2}\n            \\int_{0}^{T_{n,\\tau,\\kappa}}\n            \\Big|Q_{n}^{-\\frac12}\\Big(\\frac{\\diff{z_{n,\\tau,\\kappa}(t)}}{\\diff{t}}\n                 -\n                 A_{n}z_{n,\\tau,\\kappa}(t)\n                \n                 \\Big)\n            \\Big|^{2}\n            \\diff{t}\n            =\n            I_{0,T_{n,\\tau,\\kappa}}^{n,\\tau,0}(z_{n,\\tau,\\kappa}).\n      \\end{equation}\n      This together with \\eqref{eq:522}, \\eqref{eq:523} and \\eqref{eq:524} leads to\n      \\begin{equation}\\label{eq:525\n            V^{n}(u_{n})\n            \\leq\n            I_{0,T_{n,\\tau,\\kappa}}^{n,0}(z_{n,\\tau,\\kappa})\n            =\n            I_{0,T_{n,\\tau,\\kappa}}^{n,\\tau,0}(z_{n,\\tau,\\kappa})\n            \\leq\n            V^{n,\\tau}(u_{n}) + \\frac{\\kappa}{2}.\n      \\end{equation}\n      Then \\eqref{eq:521} and \\eqref{eq:525} show\n      \\begin{equation}\\label{eq:invapart222\n            |V^{n}(u_{n})-V^{n,\\tau}(u_{n})| \\leq \\frac{\\kappa}{2}.\n      \\end{equation}\n      For $n,\\tau$ given by \\eqref{eq:invapart111} and \\eqref{eq:520}, we set $u_{n,\\tau} := u_{n}$. By \\eqref{eq:invapart111}, \\eqref{eq:invapart222} and the triangle inequality, we complete the proof.\n\\end{proof}\n\n\n\\begin{Remark}\n       When $F \\colon H \\to H$ is a linear mapping, i.e., $F(u) = Bu$ for some $B \\in \\mathcal{L}(H)$ with $|B|_{\\mathcal{L}(H)} < \\lambda_{1}$, we can redefine the unbounded linear operator $A$ by $A+B$ to vanish $F$,\n       as well as the numerical discretizations. Repeating procedure in Theorems \\ref{th:semiinvameaLDPasympre} and \\ref{fullinvameaLDPasypre} still leads to the weakly asymptotical preservation for the corresponding LDPs by the numerical approximations.\n\\end{Remark}\n\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":" \n_THE CROCHETER'S  \nTREASURE CHEST_\n\n_80 Classic Patterns for Tablecloths, Bedspreads, Doilies and Edgings_\n\nEDITED BY\n\nMary Carolyn Waldrep\n\nDOVER PUBLICATIONS, INC., NEW YORK\nCopyright \u00a9 1988 by Dover Publications, Inc.  \nAll rights reserved.\n\n**Library of Congress Cataloging in Publication Data**\n\nThe Crocheter's treasure chest.\n\n1. Crocheting\u2014Patterns. I. Waldrep, Mary Carolyn.  \nTT820.C933 1988 746.43\u20324041 88-23702\n\neISBN-13: 978-0-486-15076-5.\n\nThis Dover edition, first published in 1988, is a new selection of patterns from _Home Decoration, Book 76_ , published by The Spool Cotton Company, New York, 1936; _Modern Table Settings, Book 88_ , published by The Spool Cotton Company, 1937; _100 Useful Edgings, Book No. 129_ , published by The Spool Cotton Company, 1939; _Bedspreads, Book No. 136_ , published by The Spool Cotton Company, 1939; _Bedspreads, Book No. 151_ , published by The Spool Cotton Company, 1940; _Bedspreads to Knit and Crochet, Book No. 166_ , published by The Spool Cotton Company, 1941; _Edgings, Book No. 182_ , published by The Spool Cotton Company, 1942; _Bedspreads to Knit and Crochet, Book No. 186_ , published by The Spool Cotton Company, 1942; _Doilies, Book No. 201_ , published by The Spool Cotton Company, 1943; _Bedspreads, Book No. 244_ , published by The Spool Cotton Company, 1948; _Priscilla Centerpieces, Book No. 276_ , published by The Spool Cotton Company, 1951; _New Ideas in Doilies, Book No. 283_ , published by The Spool Cotton Company, 1952; _Edgings, Book No. 305_ , published by Coats & Clark Inc., New York, 1954; _Ruffled Doilies, Book No. 306_ , published by Coats & Clark Inc., 1954; _Edgings: Crocheted, Knitted, Tatted, Book 7_ , published by the American Thread Company, New York, n.d.; _Star Book of Doilies, Book 22_ , published by the American Thread Company, n.d.; _Conserve with Crochet_ . . . _For the Home, Star Book No. 25_ , published by the American Thread Company, n.d.; _Doilies: Crocheted and Tatted, Star Book No. 44_ , published by the American Thread Company, n.d.; _Emblems and Church Laces, Star Book No. 50_ , published by the American Thread Company, n.d.; _New Tablecloths, Book No. 57_ , published by the American Thread Company, 1948; _Doilies, Star Doily Book No. 124_ , published by the American Thread Company, 1955; _Doilies, Star Doily Book No. 151_ , published by the American Thread Company, n.d.; _Crochet \"County Fair,\" Design Book No. 51_ , published by Lily Mills Company, Shelby, North Carolina, 1950; _Tablecloths for the Seasons, Crochet Design Book No. 57_ , published by Lily Mills Company, n.d.; _Doilies to Treasure, Book 1600_ , published by Lily Mills Company, n.d.; _Crochet for Today, Tomorrow and Always, Direction Book 1700_ , published by Lily Mills Company, 1947; _Laces and Doilies, Book No. 3_ , published by Royal Society, Inc., 1943; _Crisp New Doilies, Book No. 9_ , published by Royal Society, Inc., 1948; _Doilies, Book No. 12_ , published by Royal Society, Inc., 1951. A new Introduction has been written specially for this edition.\n\nManufactured in the United States by Courier Corporation  \n25833513 2013  \nwww.doverpublications.com\n_Table of Contents_\n\n_Introduction_\n\n_Tablecloths and Placemats_\n\n_Bedspreads_\n\n_Doilies_\n\n_Edgings_\n\n_Simple Crochet Stitches_\n\n_Metric Conversion Chart_\nCROCHET ABBREVIATIONS\n\nbal. . . . . . | balance\n\n---|---\n\nbl OR blk. . . . . . | block\n\nch. . . . . . | chain\n\ndc. . . . . . | double crochet\n\ndec. . . . . . | decrease\n\nd tr. . . . . . | double treble\n\nh dc. . . . . . | half double crochet\n\ninc. . . . . . | increase\n\nincl. . . . . . | inclusive\n\nlp. . . . . . | loop\n\np. . . . . . | picot\n\npc st. . . . . . | popcorn stitch\n\nrnd. . . . . . | round\n\nrpt. . . . . . | repeat\n\nsc. . . . . . | single crochet\n\ns dc. . . . . . | short double crochet\n\nsk. . . . . . | skip\n\nsl st. . . . . . | slip stitch\n\nsp. . . . . . | space\n\nst(s). . . . . . | stitch(es)\n\ntog. . . . . . | together\n\ntr OR trc. . . . . . | treble crochet\n\ntr tr. . . . . . | triple treble (yarn over hook 4 times)\n\n* (asterisk) or \u2020 (dagger) . . . Repeat the instructions following the asterisk or dagger as many times as specified.\n\n** or \u2020 \u2020 . . . Used for a second set of repeats within one set of instructions.\n\nRepeat instructions in parentheses as many times as specified. For example: **\"(Ch 5, sc in next sc) 5 times\"** means to work all that is in parentheses 5 times.\n\nSTITCH CONVERSION CHART\n\n_U.S. Name_ | _Equivalent_\n\n---|---\n\nChain | Chain\n\nSlip | Single crochet\n\nSingle crochet | Double crochet\n\nHalf-double or short-double crochet | Half-treble crochet\n\nDouble crochet | Treble crochet\n\nTreble crochet | Double-treble crochet\n\nDouble-treble crochet | Treble-treble crochet\n\nTreble-treble or long-treble crochet | Quadruple-treble crochet\n\nAfghan stitch | Tricot crochet\n\nSTEEL CROCHET HOOK CONVERSION CHART\n\n_Introduction_\n\nCrochet has long been one of the most popular forms of needlework. Instructions for crochet have been a staple of women's magazines for well over a century, and early books on the subject were eagerly sought by practitioners of the art.\n\nDuring the first sixty years of the twentieth century, America's thread companies produced thousands of inexpensive instructional leaflets designed to promote their products. These leaflets featured beautiful crocheted tablecloths, bedspreads, doilies, edgings and other household items. By the 1960s, however, tastes in needlework had changed, and such crocheted accessories were no longer in favor.\n\nToday, there is a new interest in crochet, and these now rare instruction leaflets, and the designs featured in them, have become collector's items. Here we offer directions for 80 of the finest designs from the 30s, 40s and 50s. Modern technology permits us to present them to you exactly as they originally appeared. For your convenience, we have arranged the designs into four categories\u2014Tablecloths and Placemats, Bedspreads, Doilies and Edgings.\n\nA number of the threads called for in the directions are still available; if not, other, similar threads can easily be found. Be careful when buying threads, however, because some product names used in the past are now being reused on completely different threads. If using colored threads, be sure to buy enough at one time to complete your project, since dye lots can vary considerably.\n\nMany of the patterns in the book list a gauge\u2014the number of stitches per inch or the size of the individual motifs or blocks. However, not all patterns give this information. Doilies often list only a finished size, while a few patterns do not even do this. In these cases, a small variation in the size of the design will make little difference to the appearance of the finished piece. Whether there is a gauge listed or not, the number of stitches and rows should be the same as indicated in the directions. Work a sample of the pattern using the suggested thread and hook and compare it to the gauge if one is listed. If your piece is too big, use a smaller hook; if too small, use a larger hook. If no gauge is stated, check the appearance of your work\u2014if the stitches are loose and untidy, use a smaller hook; if they are crowded, use a larger hook. Edgings are a special case and can be made using a variety of threads, depending on the desired effect. Just remember, the finer the thread, the smaller the hook required.\n\nYour finished piece will be improved by careful washing and blocking. For large projects that are made up of many units sewn together, you may find it easier to block the individual pieces before joining them. Use a neutral soap and cool water. Gently squeeze the suds through the crochet; do not rub. Rinse thoroughly. Pad a flat surface with several layers of thick terry toweling. Using rust-proof pins, pin the piece, right side down, on the surface, pinning each picot and loop in place. When the crochet is almost dry, press it through a damp cloth with a moderately hot iron. Do not allow the iron to rest on the stitches, particularly the raised stitches.\n\nTo give a crisper look to doilies, starch them after washing. Mix the starch solution following the manufacturer's directions and immerse the piece in the solution, squeezing it through the stitches. Squeeze out the excess and pin the piece in place as described above. For a ruffled doily, use a very heavy starch solution and pin the piece _right side up_ , leaving the ruffle free. Shape the ruffle with your fingers as the piece dries.\n\nThe terminology and hooks listed in this book are those used in the United States. The charts opposite give the U.S. names of crochet stitches and their equivalents in other countries and the approximate equivalents to U.S. crochet hook sizes. Crocheters should become thoroughly familiar with the differences in both crochet terms and hook sizes before starting any project.\n\nThe stitches used in the projects in this book are explained on page 95. A metric conversion chart is located on page 96.\n_Tableclothes and Placemats_\n\n_Mayfair Dinner Set_\n\n**Materials:** Clark's O.N.T. Mercerized Crochet, Size 50, White, 13 balls; or J. & P. Coats 8 balls. 1-1\/3 yards of 36 or 39 inch pastel linen. Milward's steel crochet hook No. 11.\n\nFan Medallion measures about 4\u00bd inches on each straight edge.\n\n**Fan Medallion:** * Ch 5, d c in 1st st of ch, working off only 2 loops, d c in same st, working off only 2 loops, thread over and draw through remaining loops on hook. Repeat from * making 40 scallops. Break thread but do not fasten off. **2nd row:** Attach thread between 18th and 19th scallops, counting from start of row, ch 8, tr tr between 2nd and 3rd scallops to left, ch 8, s c between next 2nd and 3rd scallops to left. Fasten off. **3rd row:** Attach thread between next 3rd and 4th scallops from start of last row. * Ch 6, Clones knot in 4th ch from hook. (To make Clones knot, thread over, insert hook in 4th ch from hook and draw up a loop, bring it forward and up and thread over again as for a d c. Continue to draw up loops from over and under the ch for 8 times (16 loops over hook). Draw thread through all loops on hook at once, thread over and draw through loop on hook, and make an s c around ch at base of knot, drawing tight which completes the knot.) Then ch 2, skip 2 sts of ch of last row, make long tr tr (thread over 5 times) in next st. Repeat from * 4 more times. Ch 6, knot, ch 2, s c between 3rd and 4th scallops. Fasten off. **4th row:** Attach thread between next 3rd and 4th scallops from start of last row. * Ch 7, knot, ch 3, long tr tr over long tr tr of row below. Repeat from * 4 more times. Ch 7, knot, ch 3, s c between 3rd and 4th scallops. Fasten off. **5th, 6th, 7th rows:** Continue working same way, with longer chs between knots. For 5th row, ch 8, knot, ch 4; 6th row, ch 9, knot, ch 5; 7th row, ch 11, knot, ch 7. **8th row:** Attach thread close to last scallop made, * long tr tr over next long tr tr of previous row, 5 scallops (same as for 1st row), repeat 5 more times, joining last scallop with sl st to 1st scallop of 1st row. **9th row:** Ch 11, d c in 1st st of ch, * ch 4, d c between next 2 scallops. Repeat from * along both sides of fan, making ch 7, d c in same st at point. Around curve of fan, ch 5, d c between next 2 scallops, joining last ch 5 to 3rd st of 1st ch 11. Fasten off.\n\nMake 4 fan medallions for each plate doily and center runner. Pin medallions into shape and press with a damp cloth. Cut colored linen 12 by 18 inches for plate doilies and 12 by 30 inches for runner. Sew a fan medallion in place in each corner, making allowance that after linen is hemmed between medallions, the beading row of ch 5 and d c around curve of fan will extend out beyond edge of hemmed linen. Cut out corners of linen allowing \u215b inch to turn back. Slash at inside corners. Work over edge of medallion and linen closely with Six Strand Floss, using 2 strands doubled. Then hem edges of linen between medallions and work a beading row across linen edges, joining the beading rows around curve of fans. Make beading rows ch 4, d c. When complete, work around doily with 4 s c over each ch and s c over s c. **Edge.** Ch 4, 2 d c in top of last s c, working off only 1 loops of each d c, thread over and draw thread through remaining loops, * ch 11, knot, ch 7, 3 d c in 10th s c on edge, working off d c's as before. Repeat from * around.\n\n**Circular Medallion for Runner.** Make a row of 40 scallops as for 1st row of fan. Fasten off. Make another row of 20 scallops, sl st between 20th and 21st scallops of 1st row, then make 20 more scallops. Fasten off. Attach thread between 2nd and 3rd scallop of 1 row, counting from center crossing, * ch 8, d tr in center crossing, ch 8, 1 sl st between 2nd and 3rd scallops of next row. Repeat from * around and join with sl st to start of row and fasten off. **3rd row:** Attach thread between next 3rd and 4th scallops and make same as 3rd row of fan, crossing over each of the 4 rows of scallops working around complete circle. **4th to 7th rows incl:** Same as corresponding rows of fan. **8th row:** Attach thread to end of 1 row of scallops, make * 5 scallops, long tr tr over next long tr tr of previous row and repeat from * 5 more times, joining to end of next row of scallops with 1 s c. Continue around. **9th row:** Ch 9, 1 d c between next 2 scallops, * ch 5, 1 d c between next 2 scallops and repeat from * around. Fasten off. Set circular medallion into linen runner and finish with an edging the same as for the fan medallions.\n\n_Poinsettia Tablecloth_\n\n**Materials Required\u2014**\n\n**AMERICAN THREAD COMPANY  \n\"STAR\" MERCERIZED CROCHET COTTON.  \nARTICLE 20. SIZE 20**\n\n38\u2014250 yd. Balls Cream, Ecru or White or \"Gem\" Mercerized Crochet Cotton, Article 35, Size 20.\n\n32\u2014300 yd. Balls.\n\nSteel Crochet Hook #11.\n\nEach motif measures about 4\u00bc inches. 221 motifs 13\u00d717 are required for cloth measuring about 55\u00d772 inches without edge.\n\n**MOTIF** \u2014Ch 8, join to form a ring, ch 1 and work 16 s c in ring, join in 1st s c.\n\n**2nd Row** \u2014Ch 7, sl st in 5th st from hook for picot, ch 2, skip 1 s c, s c in next s c, repeat from beginning all around.\n\n**3rd Row** \u2014Sl st to center of picot, * ch 8, s c in next picot, repeat from * 6 times, ch 4, d c in 1st picot (this brings thread in position for next row).\n\n**4th Row** \u2014Ch 4, 4 d c over same loop, work 4 d c, ch 1, 4 d c over each remaining loop working 3 d c over remainder of 1st loop, join in 3rd st of ch.\n\n**5th Row** \u2014Sl st in next st of ch, * ch 7, 7 tr c with ch 1 between each tr c in next ch 1 loop, ch 7, s c in next ch 1 loop, repeat from * 3 times.\n\n**6th Row** \u2014Sl st to center of loop, s c over same loop, ** ch 5, s c in next ch 1 loop, * ch 3, s c in next ch 1 loop, repeat from * 4 times, ch 5, s c in next loop, ch 6, s c in next loop, repeat from ** twice, ch 5, s c in next ch 1 loop, * ch 3, s c in next ch 1 loop, repeat from * 4 times, ch 5, s c in next loop, ch 2, d c in 1st s c.\n\n**7th Row** \u2014Ch 3, 2 d c over same loop, ** ch 3, s c in next loop, ch 5, s c in next loop, * ch 3, s c in next loop, repeat from * 3 times, ch 5, s c in next loop, ch 3, 3 d c in next loop, repeat from ** twice, ch 3, s c in next loop, ch 5, s c in next loop, * ch 3, s c in next loop, repeat from * 3 times, ch 5, s c in next loop, ch 3, join in 3rd st of ch.\n\n**8th Row** \u2014Ch 4, 2 tr c in same space, ** ch 7, skip 1 d c, 3 tr c in next d c, ch 3, skip next 3 ch loop, s c in next 5 ch loop, ch 5, s c in next loop, * ch 3, s c in next loop, repeat from * twice, ch 5, s c in next loop, ch 3, 3 tr c in next d c, repeat from ** twice, ch 7, skip 1 d c, 3 tr c in next d c, ch 3, skip next 3 ch loop, s c in next 5 ch loop, ch 5, s c in next loop, * ch 3, s c in next loop, repeat from * twice, ch 5, s c in next loop, ch 3, join in 4th st of ch.\n\n**9th Row** \u2014Ch 4, 1 tr c in each of the next 2 tr c, ** ch 7, s c in next loop, ch 7, 1 tr c in each of the next 3 tr c, ch 3, skip next 3 ch loop, s c in next 5 ch loop, ch 5, s c in next loop, * ch 3, s c in next loop, repeat from * once, ch 5, s c in next loop, ch 3, 1 tr c in each of the next 3 tr c, repeat from ** twice, ch 7, s c in next loop, ch 7, 1 tr c in each of the next 3 tr c, ch 3, skip next 3 ch loop, s c in next 5 ch loop, ch 5, s c in next loop, * ch 3, s c in next loop, repeat from * once, ch 5, s c in next loop, ch 3, join in 4th st of ch.\n\n**10th Row** \u2014Ch 4, 1 tr c in each of the next 2 tr c, * ch 9, s c in next loop, ch 9, s c in next loop, ch 9, 1 tr c in each of the next 3 tr c, ch 3, skip next 3 ch loop, s c in next 5 ch loop, ch 5, s c in next loop, ch 3, s c in next loop, ch 5, s c in next loop, ch 3, 1 tr c in each of the next 3 tr c, repeat from * twice, ch 9, s c in next loop, ch 9, s c in next loop, ch 9, 1 tr c in each of the next 3 tr c, ch 3, skip next 3 ch loop, s c in next 5 ch loop, ch 5, s c in next loop, ch 3, s c in next loop, ch 5, s c in next loop, ch 3, join in 4th st of ch.\n\n**11th Row** \u2014Ch 4, 1 tr c in each of the next 2 tr c, ** ch 9, s c in next loop, ch 5, work 4 cluster sts with ch 5 between each cluster st in next loop (cluster st: * thread over twice, insert in loop, pull through and work off 2 loops, twice, repeat from * twice, thread over and work off all loops at one time), ch 5, s c in next loop, ch 9, 1 tr c in each of the next 3 tr c, ch 3, skip next 3 ch loop, s c in next 5 ch loop, * ch 5, s c in next loop, repeat from * once, ch 3, 1 tr c in each of the next 3 tr c, repeat from ** twice, ch 9, s c in next loop, ch 5, work 4 cluster sts with ch 5 between each cluster st in next loop, ch 5, s c in next loop, ch 9, 1 tr c in each of the next 3 tr c, ch 3, skip next 3 ch loop, s c in next 5 ch loop, * ch 5, s c in next loop, repeat from * once, ch 3, join in 4th st of ch.\n\n**12th Row** \u2014Ch 4, 1 tr c in each of the next 2 tr c, ** ch 9, s c in next loop, * ch 5, cluster st in next loop, repeat from * 4 times, ch 5, s c in next loop, ch 9, 1 tr c in each of the next 3 tr c, ch 3, skip next 3 ch loop, 1 s c in each of the next 2 loops, ch 3, 1 tr c in each of the next 3 tr c, repeat from ** twice, ch 9, s c in next loop, * ch 5, cluster st in next loop, repeat from * 4 times, ch 5, s c in next loop, ch 9, 1 tr c in each of the next 3 tr c, ch 3, skip next 3 ch loop, 1 s c in each of the next 2 loops, ch 3, join in 4th st of ch.\n\n**13th** Row\u2014Sl st across each tr c and to center st of next loop, s c over same loop, * ch 5, cluster st in next loop, repeat from * twice, ch 5, cluster st in next cluster st, ch 11, cluster st in same space, * ch 5, cluster st in next loop, repeat from * twice, ch 5, s c in next loop, ch 9, * thread over twice, insert in next tr c and work off 2 loops twice, repeat from * 5 times, thread over and work off all loops at one time, ch 9, s c in next loop, continue all around in same manner ending row with ch 9, sl st in 1st s c, break thread.\n\nWork a 2nd motif in same manner joining to 1st motif in last row as follows: sl st to center of 1st loop, s c in same loop, * ch 5, cluster st in next loop, repeat from * twice, ch 5, cluster st in next cluster st, ch 6, sl st in corresponding loop of 1st motif, ch 6, cluster st in same cluster st of 2nd motif, * ch 3, sl st in corresponding loop of 1st motif. ch 3, cluster st in next loop of 2nd motif, repeat from * twice, ch 3, sl st in corresponding loop of 1st motif, ch 3, s c in next loop of 2nd motif, ch 5, sl st in corresponding loop of 1st motif, ch 5, * thread over twice, insert in next tr c of 2nd motif, pull through and work off 2 loops twice, repeat from * 5 times, thread over and work off all loops at one time, ch 5, sl st in corresponding loop of 1st motif, ch 5, s c in next loop of 2nd motif, * ch 3, sl st in corresponding loop of 1st motif, ch 3, cluster st in next loop of 2nd motif, repeat from * twice, ch 3, sl st in corresponding loop of 1st motif, ch 3, cluster st in next cluster st of 2nd motif, ch 6, sl st in corresponding loop of 1st motif, ch 6, cluster st in same cluster st of 2nd motif and complete motif same as 1st motif.\n\nJoin 3rd motif to 2nd motif and 4th motif to 3rd and 1st motifs in same manner.\n\n**EDGE:** Attach thread at joining at right hand side before corner, ch 4, cluster st in same space, ch 4, sl st in top of cluster st for picot, ch 5, s c in next loop, ch 4, sl st in top of s c for picot, * ch 5, cluster st in next loop, ch 4, sl st in top of cluster st for picot, ch 5, s c in next loop, ch 4, sl st in top of s c for picot, repeat from * once, ch 5, cluster st in next loop, picot, ch 5, s c in top of tr c group, ch 4, sl st in top of s c for picot, * ch 5, cluster st in next loop, picot, ch 5, s c in next loop, picot, repeat from * twice, ch 5, cluster st in same space, picot, ch 6, sl st in 4th st from hook for picot, ch 2, cluster st in same space, picot, ch 5, s c in same space, picot, (corner) * ch 5, cluster st in next loop, picot, ch 5, s c in next loop, picot, repeat from * once, ch 5, cluster st in next loop, picot, ch 5, s c in top of tr c group, picot, * ch 5, cluster st in next loop, picot, ch 5, s c in next loop, picot, repeat from * twice, ** ch 5, cluster st in joining, picot, * ch 5, s c in next loop, picot, ch 5, cluster st in next loop, picot, repeat from * twice, ch 5, s c in top of tr c group, picot, * ch 5, cluster st in next loop, picot, ch 5, s c in next loop, picot, repeat from * twice, repeat from ** all around working all corner motifs same as 1st corner motif.\n\nIf napkins are desired, cut linen the required size. Roll a narrow hem and work a row of s c over hem. Finish with edge same as on cloth.\n\n_Sundial Tray Mat_\n\n**Materials:** Choose one of the following threads in White or Ecru:\n\nClark's O.N.T. Mercerized Crochet, size 10, 6 balls.\n\nJ. & P. Coats Mercerized Crochet, size 10, 6 balls.\n\nJ. & P. Coats Big Ball, size 10, 3 balls.\n\nClark's Big Ball, size 10, 3 balls.\n\nMilward's steel crochet hook No. 6.\n\nLarge motif measures about 4\u00be inches. When completed, mat measures about 14 \u00d7 19 inches.\n\n**Large Motif.** To begin, ch 10, join with sl st to form ring. **1st rnd:** Ch 3, 23 d c in ring. Join with sl st to 3rd ch of ch-3 first made. **2nd rnd:** Ch 3, d c in each of next 2 d c, ch 2, * d c in each of next 3 d c, ch 2. Repeat from * around. Join with sl st to 3rd ch of ch-3 first made. **3rd rnd:** Ch 3, d c in st from which ch-3 was started, d c in next d c, 2 d c in last d c, ch 2, * 2 d c in 1st d c of next group of d c, d c in next d c, 2 d c in last d c of group, ch 2, repeat from * around. Join with sl st. **4th rnd:** Ch 3, d c in st from which ch-3 was started, d c in each of next 3 d c, 2 d c in last d c, ch 2, * 2 d c in 1st d c of next group of d c, d c in each of next 3 d c, 2 d c in last d c of group, ch 2. Repeat from * around. Join with sl st. **5th rnd:** Ch 3, d c in st from which ch-3 was started, d c in each of next 5 d c, 2 d c in last d c, ch 3, * 2 d c in 1st d c of next group of d c, d c in each of next 5 d c, 2 d c in last d c of group, ch 3, repeat from * around. Join with sl st. **6th rnd:** Ch 3, d c in st from which ch-3 was started, d c in each of next 3 d c, ch 3, skip next d c, d c in each of next 3 d c, 2 d c in last d c, ch 3, * 2 d c in 1st d c of next group of d c, d c in each of next 3 d c, ch 3, skip next d c, d c in each of next 3 d c, 2 d c in last d c, ch 3. Repeat from * around. Join with sl st. **7th rnd:** Sl st in each d c to next ch-3, ch 6, d c in same ch-3 sp, ch 3, d c in same sp, ch 3, d c in same sp, ch 4, s c in next ch-3 sp, ch 4, * d c in next ch-3 sp, ch 3, d c in same sp, ch 3, d c in same sp, ch 3, d c in same sp, ch 4, s c in next ch-3 sp, ch 4. Repeat from * around. Join with sl st in 3rd ch of ch-6 first made. **8th rnd:** Ch 8, tr in next sp, ch 2, tr in next ch-3 sp, ch 2, tr in same sp, ch 2, tr in next ch-3 sp, ch 2, tr in same sp, ch 4, s c in next, s c, ch 4, * tr in next ch-3 sp, ch 2, tr in same sp, ch 2, tr in next ch-3 sp, ch 2, tr in same sp, ch 2, tr in next ch-3 sp, ch 2, tr in same sp, ch 4, s c in next s c, ch 4, repeat from * around. Join with sl st in 6th ch of ch-8 first made. **9th rnd:** Ch 5, s c in next ch-2 sp, * ch 5, s c in next ch-2 sp, ch 5, s c in next ch-2 sp, ch 5, s c in next ch-2 sp, ch 5, s c in next ch-2 sp, ch 6, s c in next ch-2 sp, ch 5, s c in same ch-2 sp. Repeat from * around. Join with sl st in 1st ch of ch-5 first made. **10th rnd:** Ch 5, s c in next ch-5 loop, * ch 5, s c in next ch-5 loop, ch 5, s c in next ch-5 loop, ch 5, s c in next ch-5 loop, ch 5, s c in next ch-5 loop, ch 2, s c in ch-6 loop, ch 2, s c in next ch-5 loop, ch 5, s c in same ch-2 sp. Repeat from * around. Join with sl st. Break off.\n\nMake 12 of these large motifs.\n\n**Small Motif.** To begin, ch 10, join. **1st and 2nd rnds:** Same as 1st and 2nd rnds of Large Motif. **3rd rnd:** * Ch 5, s c in 2nd d c of 3-dc group, ch 5, s c in next ch-2 sp. Repeat from * 7 more times. **4th rnd:** * Ch 5, s c in next sp. Repeat from * 16 more times. Break off.\n\nMake 6 of these small motifs.\n\n**Joining Motifs.** Place motifs in position with the small motifs to fill in spaces left open by large motifs. With a sewing needle and crochet cotton, sew motifs together with 2 or 3 over and over stitches. Do not break off thread, but make a running stitch through the ch-loop on one motif to as far as center of next ch-loop. Be careful not to draw thread too tight. Take 2 or 3 over and over stitches in the center of this loop and the one on the other motif to join the 2 motifs together. Thus continue joining around blocks.\n\n_Pond Lily Tablecloth_\n\nMaterials Required\u2014\n\nAMERICAN THREAD COMPANY\n\n\"STAR\" MERCERIZED CROCHET COTTON.\n\nARTICLE 20. SIZE 20\n\n36\u2014250 yd. Balls Cream, Ecru or White or \"Gem\" Mercerized Crochet Cotton, Articie 35, Size 20.\n\n30\u2014300 yd. Balls.\n\nSteel Crochet Hook #11.\n\nEach motif measures about 4\u00bd inches. 192 Motifs 12\u00d716 are required for cloth measuring about 56\u00d774 inches with edge.\n\n**MOTIF** \u2014Ch 8, join to form a ring, ch 3 and work 23 d c into ring, join in 3rd st of ch.\n\n**2nd Row** \u2014Ch 12, sl st in 7th st from hook, ch 5, skip 2 d c, s c in next d c, repeat from beginning all around.\n\n**3rd Row** \u20145 s c over next loop, sl st in next loop, ch 4, * thread over twice, insert in same loop, pull through and work off 2 loops twice, repeat from * once, thread over and work off all loops at one time, ** ch 5, * thread over twice, insert in same loop, pull through and work off 2 loops twice, repeat from * twice, thread over and work off all loops at one time, repeat from ** twice, ch 5, * thread over twice, insert in same loop, pull through and work off 2 loops twice, repeat from * once, thread over and work off all loops at one time, ch 4, sl st in same loop, 5 s c in next loop, repeat from beginning all around.\n\n**4th Row** \u2014Sl st across next 5 s c. the ch 4 and to the center of next loop, s c in same loop, * ch 6, s c in next loop, repeat from * twice, s c in next loop, repeat from 1st * all around, join in 1st s c.\n\n**5th Row** \u2014Sl st in each of the next 2 chs, 3 s c in same loop, * ch 6, 3 s c in next loop, repeat from * all around ending row with ch 3, d c in 1st s c (this brings thread in position for next row).\n\n**6th Row** \u2014Ch 4 (ch 4 at beginning of row counts as 1 tr c), 2 tr c in same space, * ch 5, 3 s c in next loop, repeat from * once, ch 5, then repeat from beginning all around, join in 4th st of ch.\n\n**7th Row** \u2014Ch 4, 2 tr c in same space, * ch 7, skip next tr c, 3 tr c in next tr c, ch 6, skip next loop, 5 s c in next loop, ch 6, 3 tr c in next tr c, repeat from * all around, ending row with ch 7, skip next tr c, 3 tr c in next tr c, ch 6, skip next loop, 5 s c in next loop, ch 6, join in 4th st of ch.\n\n**8th Row** \u2014Ch 4, 1 tr c in each of the next 2 tr c, * ch 7, 3 s c in next loop, ch 7, 1 tr c in each of the next 3 tr c, ch 6, skip 1 s c, 1 s c in each of the next 3 s c, ch 6, 1 tr c in each of the next 3 tr c, repeat from * all around ending row with ch 7, 3 s c in next loop, ch 7, 1 tr c in each of the next 3 tr c, ch 6, skip 1 s c, 1 s c in each of the next 2 s c, ch 6, join in 4th st of ch.\n\n**9th Row** \u2014Ch 4, 1 tr c in each of the next 2 tr c, * ch 7, 3 s c in next loop, ch 7, 3 s c in next loop, ch 7, 1 tr c in each of the next 3 tr c, ch 6, s c in center s c of s c group, ch 6, 1 tr c in each of the next 3 tr c, repeat from * all around in same manner, join in 4th st of ch.\n\n**10th Row** \u2014Ch 4 (counts as 1 tr c), 1 tr c in each of the next 2 tr c, * ch 7, 3 s c in next loop, repeat from * twice, ch 7, 1 tr c in each of the next 3 tr c, repeat from beginning all around, join, break thread.\n\nWork a 2nd motif in same manner joining to 1st motif in last row as follows: ch 4, 1 tr c in each of the next 2 tr c, * ch 7, 3 s c in next loop, repeat from * once, ch 3, sl st in corresponding loop of 1st motif, ch 3, 3 s c in next loop of 2nd motif, ch 3, sl st in next loop of 1st motif, ch 3, 1 tr c in each of the next 6 tr c of 2nd motif, * ch 3, sl st in corresponding loop of 1st motif, ch 3, 3 s c in next loop of 2nd motif, repeat from * once and complete motif same as 1st motif, break thread.\n\nJoin 3rd motif to 2nd motif and 4th motif to 3rd and 1st motifs in same manner.\n\n**JOINING MOTIF** \u2014Ch 8, join to form a ring, ** ch 4, * thread over twice, insert in ring, pull through and work off 2 loops twice, repeat from *, thread over and work off all loops at one time, ch 4, 3 s c in ring, repeat from ** 3 times.\n\n**2nd Row** \u2014Sl st to top of ch 4, ch 3, 2 d c in same space, * ch 6, skip next st, 3 d c in next st, ch 5, 3 d c in top of next ch 4, repeat from * twice, ch 6, skip next st, 3 d c in next st, ch 5, join in 3rd st of ch.\n\n**3rd Row** \u2014Ch 3 (ch 3 at beginning of row counts as 1 d c), 1 d. c in each of the next 2 d c, ch 6, 3 s c in next loop, ch 6, 1 d c in each of the next 3 d c, ch 3, s c in next loop, ch 3, repeat from beginning all around, join in 3rd st of ch.\n\n**4th Row** \u2014Ch 4, 1 tr c in each of the next 2 d c, ch 3, sl st. in 3rd free loop of any large motif, ch 3, 3 s c in next loop of joining motif, ch 3, sl st in next free loop of same large motif, ch 3, sl st in next free loop of next large motif, ch 3, 3 s c in next loop of joining motif, ch 3, sl st in next loop of same large motif, ch 3, tr c in each of the next 6 d c of joining motif and continue in same manner until joining is completed.\n\n**EDGE:** Attach thread in 3rd loop on right hand side before joining of corner motif, ch 1 and work 3 s c in same space, ch 7, 3 s c in next loop, ch 7, 3 s c in next loop, ch 3, thread over needle, insert in same loop with joining, pull through and work off 2 loops, thread over needle, insert in same loop with joining of next motif, pull through and work off all loops 2 at a time, ch 3, 3 s c in next loop, * ch 7, 3 s c in next loop, repeat from * 18 times, ** thread over needle, insert in next loop (same as joining) pull through and work off 2 loops, thread over needle, insert in same loop with joining of next motif, pull through and work off all loops 2 at a time, ch 3, 3 s c in next loop, * ch 7, 3 s c in next loop, repeat from * 10 times, repeat from ** all around working all corner motifs same as 1st corner motif.\n\n**2nd Row** \u2014Sl st into loop, ch 4, cluster st in same space, * ch 5, sl st in 4th st from hook for picot, ch 1, cluster st in same space, repeat from * twice, ch 5, 2 s c in next loop, ch 4, sl st in top of last s c for picot, 2 s c in next ch 7 loop of next motif, ** ch 5, 4 cluster sts with ch 1, picot, ch 1 between each cluster st in next loop, * ch 7, 2 s c in next loop, ch 4, sl st in top of last s c for picot, s c in same space, repeat from * twice, repeat from ** 3 times, ch 5, 4 cluster sts with ch 1, picot, ch 1 between each cluster st in next loop, ch 5, 2 s c in next loop, ch 4, sl st in top of last s c for picot, 2 s c in next ch 7 loop of next motif and continue all around in same manner.\n\nIf napkins are desired, cut linen the size required. Work a row of s c over a rolled hem.\n\n**2nd Row** \u2014Ch 7, skip 5 s c, 1 s c in each of the next 3 s c, repeat from beginning all around.\n\n**3rd Row** \u2014Same as last row of edging on cloth working the 4 cluster sts at corners only.\n\n_Empire Medallion Tea Cloth_\n\n**Materials:** Choose one of the following threads in White or Ecru:\n\nClark's O.N.T. Mercerized Crochet, size 30, 15 balls.\n\nJ. & P. Coats Mercerized Crochet, size 30, 12 balls.\n\nJ. &. P. Coats Big Ball, size 30, 6 balls.\n\nMilward's steel crochet hook No. 10, and 1\u2153 yards linen, 36 inches wide, for cloth about 50 inches square.\n\nMedallion measures about 6 inches square.\n\n**Medallions:** Ch 7, join with sl st to form ring. **1st rnd:** 12 s c in ring. Join with sl st to s c first made. **2nd rnd:** Ch 6, * d c in next s c, ch 3, repeat from * around, join with sl st to 3rd st of ch-6 (12 d c counting ch). **3rd rnd:** * Over next ch loop, 1 half d c, 3 d c, 1 half d c, repeat from * around. **4th rnd:** Ch 9, d c between next 2 scallops, * ch 6, d c between next 2 scallops and repeat from * around, ch 6, join with sl st to 3rd st of ch-9. **5th rnd:** Ch 3, * 6 d c over next ch, d c over d c, repeat from * around, join with sl st to 3rd st of ch-3. Sl st over next d c. **6th rnd:** Ch 3, d c in each of next 5 d c, * ch 5, 1 s c in 1st st to make p, ch 1, skip next d c. D c in each of next 6 d c and repeat from * around, joining last ch-1 with sl st to 3rd st of ch-3. **7th rnd:** Ch 3, d c over d c then ch 1, ch-5 p, ch 2 and d c over d c as in previous rnd.\n\n**8th, 9th and 10th rnds:** Same as 7th except for p loops between. (8th) Ch 2, ch-5 p, ch 3. (9th) Ch 3, ch-5 p, ch 4. (10th) Ch 4, ch-5 p, ch 5. **11th rnd:** Ch 3, * d c in next st, d c in next, but work off only 2 loops, then d c in next and work off 2 loops, thread over and draw through the 3 loops remaining on hook, d c in each of next 2 sts, ch 11, d c in next d c and repeat from * around, joining last ch-11 to 3rd st of ch-3. **12th rnd:** * Ch 3, d c in each of next 3 sts, working off only 2 loops of each, thread over and draw through 2 loops, thread over and draw through remaining loops on hook. Make a 5 ch p in top of d c, ch 4, sl st in next st, ch 1. Over ch 11 make 8 s c, a ch-5 p, 8 s c over remainder of ch, 1 sl st in 1st st of next group. Repeat from * around. Sl st to tip of 1st point.\n\n**13th rnd:** Ch 8, * a ch-5 p, ch 5, tr in next p, ch 5, repeat from * around, ch 5, and join to 4th st of ch-8. * Ch 8, make a ch-5 p, ch 3, d tr in next tr, ch 3, a ch-5 p, ch 3, d tr in same st, ch 3, a ch-5 p, ch 8, s c in next tr. Fasten off. Skip 3 tr, join to next tr, and repeat from * 3 more times, to form the 4 corners.\n\n**Heading rnd:** Join to the p of 1 corner loop, ch 15, d tr in next d tr, ** ch 8, skip next p, tr in 4th st of ch beyond, ch 8, skip next p loop, tr in next tr, * ch 8, tr in next tr, repeat from * once, ch 8, skip next p, 1 tr in 4th st of next ch, ch 8, 1 d tr in next d tr, ch 8, 1 d tr in next corner p, ch 8, 1 d tr in next d tr. Repeat from ** around, fastening ch-8 at end of row to 8th st of ch-15.\n\n**Beading rnd:** Ch 8, d c in same st, * ch 2, skip next 2 sts, d c in next and repeat from * around, making 1 d c, ch 5, 1 d c in same st in each corner. End rnd with sl st in 3rd st of 1st ch 8. Make 24 more medallions. Stretch and pin in perfect squares and press with a damp cloth.\n\nCut squares of linen slightly larger than medallions (to allow for hem), and hem to measure the same as the beading row along one edge of medallion.\n\nOn 12 of the linen squares sew a beading row on one side. To make beading, ch 125, d c in 8th ch from hook, * ch 2, skip 2 sts, d c in next, and repeat from * along ch. Whip medallions to linen squares in checkerboard style, using the 12 squares with beading for the outside edge. For the first row use 4 medallions and 3 linen squares, so that the completed cloth has a medallion in each corner.\n\n**Edge:** Make a solid row of d c around, with 2 d c over each ch 2 and d c over d c, and 5 d c in each corner. **2nd row:** Starting at 1 corner, ch 11, * a ch-5 p, ch 6, skip 13 sts, d tr in next, ch 6, and repeat from * around. At each corner d c make 1 d tr, ch 13 and 1 d tr. End row with sl st in 4th st of 1st loop. **3rd, 4th and 5th rows:** Make the same p loops as in 2nd row, making the d tr's over the d tr's of previous row. At each corner make a d tr in d tr, ch 6, d tr in center st of ch-13, ch 13, d tr in same st, ch 6 and continue around. **6th row:** Ch 21, d tr in next d tr, ch 13, d tr in next d tr around. At each corner make 1 d tr, ch 13, 1 d tr in same st and 1 extra ch 6 before and after corner sts. Finish row with sl st in 9th st of 1st loop. **7th row:** 13 s c over each ch-13 loop and 6 s c over each ch-6 loop. Fasten off. **8th row:** Join to 5th s c of a 13-s c loop, ch 3, 1 d c in each of next 4 s c, * ch 2, a ch-5 p, ch 4, skip 4 s c of next loop, d c in each of next 5 s c, and repeat from * around. At corner loops, skip 4 s c, 1 d c in each of next 5 s c, ch 2, a ch-5 p, ch 3, skip 4 s c, 1 d c in each of next 5 s c. Finish row with sl st in ch 3 at start of row. **9th row:** Ch 3, 1 d c in next d c, 1 d c in each of next 2 d c, working off only 2 loops of each, thread over and draw through remaining loops of hook, d c in next d c, ch 10, 5 d c in next group (working off the 3rd and 4th together), repeat around. At corners make ch 15 between groups of d c. End row with sl st in ch 3 at start of row.\n\n**10th row:** * Ch 3, d c in each of next 2 d c, working off only 2 loops of each, thread over and draw through all loops on hook. A ch-5 p, ch 3, sl st in next d c, ch 1, 5 s c, a ch 5 p, 5 s c over next ch 10, sl st in 1st d c of next group. Repeat from * around Break thread and fasten off.\n\n**Napkins:** Join to a corner of hemmed edge, ch 8, 1 d c in same place, ch 2, 1 d c in edge about \u215b inch away. Repeat around. Make 2 d c with ch 5 between at each corner. End row with sl st in 3rd st of 1st ch 8. **2nd row:** Make a solid row of d c around, making 2 d c over each ch 2 and d c in each d c, with 5 d c in the 3rd st of each ch 5 at corners. Fasten off. **3rd row:** Join to 1 corner, ch 29, 1 d tr in same place, ch 14, 1 d tr in about 14th or 15th d c along edge. Repeat around, adjusting spacing of d tr so that ch 14's are stretched nearly straight and taut. At each corner make ch 20 between d tr. End row with sl st in 9th st of 1st loop. **4th row:** 14 s c over each ch 14, and 25 s c over each corner loop. Fasten off. **5th row:** Follow directions for 8th row of edge on cloth, except that in corner loops, leave 3 s c between 2 groups of d c. Complete by following directions for 9th and 10th rows of the edging for the cloth\n\n_Chrysanthemum Tablecloth_\n\nMaterials Required\u2014\n\nAMERICAN THREAD COMPANY  \n\"STAR\" MERCERIZED CROCHET COTTON.\n\nARTICLE 20. SIZE 30\n\n40\u2014325 yd. Balls White, Ecru or Cream or \"Gem\" Mercerized Crochet Cotton, Article 35, Size 30.\n\n33\u2014400 yd. Balls.\n\nSteel Crochet Hook #12.\n\nEach motif measures about 3 inches. 567 motifs, 21\u00d727 are required for a cloth measuring about 63\u00d781 inches.\n\n**MOTIF** \u2014Ch 12, join to form a ring, ch 3, * thread over, insert in ring, pull through and work off 2 loops, repeat from *, then work off remaining loops at one time, ** ch 4, * thread over needle, insert in ring, pull through and work off 2 loops, repeat from * twice, thread over and work off remaining loops at one time (this is a cluster stitch), repeat from ** until there are 8 cluster sts in ring, ch 4, join.\n\n**2nd Row** \u2014Ch 6, sl st in 4th st from hook for picot, ch 7, sl st in 4th st from hook for picot, ch 2, s c in next cluster st, repeat from beginning all around, join.\n\n**3rd Row** \u2014Sl st to center of picot loop, * ch 9, s c in center of next picot loop, repeat from * all around, join.\n\n**4th Row** \u2014Sl st into loop, ch 3 and work 3 cluster sts with ch 3 between each cluster st in same loop, * ch 6, 3 cluster sts with ch 3 between each cluster st in next loop, repeat from * all around, ch 6, join in 1st cluster st.\n\n**5th Row** \u2014Sl st in ch 3 loop, ch 3, work 2 cluster sts with ch 3 between in same loop, ch 3, 2 cluster sts with ch 3 between in next loop, * ch 6, skip 1 loop, 2 cluster sts with ch 3 between in next loop, ch 3, 2 cluster sts with ch 3 between in next loop, repeat from * all around, ch 6, join.\n\n**6th Row** \u2014Sl st into loop, ch 3 and work 2 cluster sts with ch 3 between in same loop, * ch 3, 2 cluster sts with ch 3 between in next loop, repeat from * once, ** ch 4, s c over the ch 6 loops of 2 previous rows, ch 4, 2 cluster sts with ch 3 between in next ch 3 loop, * ch 3, 2 cluster sts with ch 3 between in next loop, repeat from * once, then repeat from ** all around, join in 1st cluster st.\n\n**7th Row** \u2014Ch 1, * 2 s c, 4 ch picot, 1 s c in each 3 ch loop, ch 6, sl st in 4th st from hook for picot, ch 2, skip the 2-ch 4 loops, repeat from * all around, join, break thread.\n\nWork another motif in same manner joining it to 1st motif in last row as follows: work 2 s c in 1st ch 3 loop, ch 2, sl st in corresponding picot of 1st motif, ch 2, complete picot, 1 s c in same loop of 2nd motif, join the next 4 picots of 2nd motif in same manner and complete motif same as 1st motif. Join 3rd motif to 2nd motif and join 4th motif to 3rd and 1st motifs in same manner leaving 7 picots free between joinings.\n\n**JOINING MOTIF** \u2014Work 1st row of large motif, * ch 5, join to 3rd free picot of large motif, ch 2, sl st in 3rd st of ch to complete picot, ch 5, skip 1 picot, sl st in next picot of same large motif, ch 2, sl st in 3rd st of ch to complete picot, ch 2, s c in next cluster st of small motif, ch 6, sl st in 4th st from hook for picot, ch 2, s c in next cluster st of small motif, repeat from * all around joining all motifs in same manner, break thread.\n\n_Filet Luncheon Set_\n\n**Materials Required-AMERICAN THREAD COMPANY  \n\"STAR\" OR \"GEM\" MERCERIZED  \nCROCHET COTTON  \nSize 20**\n\n4 400-yd. Balls or 9 150-yd. Balls.\n\nSteel Crochet Hook No. 11 or 12.\n\nEach doily measures about 11\u00d715 inches.\n\nStarting at arrow marked A, ch 51 and work 1 d c in the 9th st from hook, 1 d c in each of the next 3 chs, ch 2, skip 2 chs, 1 d c in the next 4 chs, ch 2, skip 2 chs, 1 d c in the next ch, ch 2, skip 2 chs, 1 d c in next ch, then work 27 d c on remainder of ch, ch 3, turn.\n\n**2nd Row** \u2014Work 1 d c in each of the next 3 d c, ch 2, skip 2 d c, 1 d c in next d c, continue until you have 7 o m (open meshes), Ism (solid mesh), 3 o m, 1 s m, 1 o m, 1 s m, ch 2, tr c in the same st as the last d c, ch 5, turn.\n\n**3rd Row** \u20141 d c in tr c, 2 d c in ch, 1 d c in next d c, 1 o m, 1 s m, 4 o m, 1 s m, 3 o m, 1 s m, ch 3, turn.\n\n**4th Row** \u20141 s m, 2 o m, 1 s m, 1 o m, 1 s m, 2 o m, 1 s m, 5 o m, 1 s m, 1 o m, 1 s m, ch 2, tr c in the same st as last d c, ch 5, turn.\n\n**5th Row** \u20141 s m, 1 o m, 1 s m, 3 o m, 1 s m, 2 o m, 1 s m, 1 o m, 1 s m, 3 o m, 1 s m, 1 o m, 1 s m, ch 3, turn.\n\nContinue working back and forth according to diagram to arrow marked B.\n\n**Next Row** \u2014Omit the last 3 d c in the solid row of insertion, ch 5 for turning, continue to arrow C and continue work on scallop only, complete the scallop and break thread. Join thread in the 4th solid mesh from point of scallop, ch 5, working into side of edging work 1 s m, 1 o m, 1 s m, d c into the next d c on the 11 open mesh row, ch 2, 1 d c into the next d c and work back on the scallop again, you are working on the second side of edging. When you reach the solid row where one solid mesh was omitted, work a solid mesh into side of work to complete row.\n\nContinue work until there are 8 squares, work corner, then work 4 squares and work other half to correspond joining the work in the last 2 rows by working 1st into the last row then into the 1st row until entire side is joined.\n\n**Edging.** Join at any point and work * 2 s c, ch 3, 2 s c, ch 3, 2 s c, ch 3, 2 s c in ch 5 loop, 2 s c, ch 3, 2 s c in ch 3 loop, 2 s c, ch 3, 2 s c in next ch 3 loop, 2 s c, ch 3, 2 s c in next 3 ch loop, 2 s c between scallops, 2 s c, ch 3, 2 s c in each of the next 3 loops, repeat from * all around.\n\nFit a piece of material into the square, allowing for a narrow hem. Over a narrow hem work a row of s c and attach edging using the same crochet thread.\n\n_Daisy Ring Tablecloth_\n\n_MATERIALS_ \u2014 Lily MERCROCHET Cotton size 20:\u201452-balls White, Cream or Ecru. DAISY Mercerized Crochet Cotton may be substituted if preferred. Crochet hook No. 12. Size\u201463 \u00d7 84 inches.\n\nBlock measures 3\u00bd inches point to point.\n\n_BLOCK_ \u2014Ch 8, join with sl st to form ring. _ROW 1_ \u2014Ch 6, dc in ring, (ch 3, dc in ring) 6 times, working over starting end to cover it up, ch 3. join with sl st in 3d ch of ch-6. _ROW 2_ \u2014Ch 1, sc in same st, (3 sc in next sp, sc in dc) repeated around. Join with sl st in back lp of 1st sc. _ROW 3_ \u2014Ch 5, 3 tr in same st, (ch 7, sk 3 sc, 4 tr in back lp of next se) 7 times, ch 7, sl st in top of ch-5. _ROW_ 4\u2014Ch 5, holding back the last lp of each tr on hook. make tr in back 1 ps on next 3 tr, thread over and draw thru all lps on hook (Cluster made), * ch 6. dc in 4th (center) ch of next ch-7 lp, ch 6, (tr in back lps of next 4 tr) made into a Cluster. Repeat from * around. Join to 1st Cluster. _ROW_ 5 \u2014Ch 3, * 6 dc in next sp, dc in dc, ch 6, sl st in last dc for a p, 6 dc in next sp. dc in Cluster, ch 6, sl st in one lp on 4th ch from hook for a p, ch 5, p in same way, ** ch 11, p, ch 5, p, ch 3, sl st in last dc (p-lp made). 6 dc in next sp, dc in dc, ch 6, sl st in last dc for a p, 6 dc in next sp, dc in Cluster, a ch-4 p. Repeat from * around. End with sl st in top of ch-3, ch 4, sl st in same st. Cut 6\" sl long, thread to a needle and fasten off on back.\n\n_2d BLOCK_ \u2014Repeat to ** in Row 5. Ch 3, sl st in one lp of center ch at tip of a p-lp on 1st Block, ch 7, p, ch 5, p, ch 3, sl st back in last dc on 2d Block. * 6 dc in next sp, dc in dc, ch 3, sl st in next ch-6 p on 1st Block, ch 3, sl st back in last dc on 2d Block, 6 dc in next sp, dc in Cluster, * ch 2, sl st in next ch-4 p on 1st Block, ch 2, st back in last dc. Repeat from * to *. Ch 6, p, ch 5, p, ch 3, sl st in one lp of center ch of next p-lp on 1st Block, ch 7, p, ch 5, p, ch 3, sl st back in last dc. Complete as for 1st Block. Join 3rd Block to 2d Block and 4th Block to 1st and 3d Blocks in same way.\n\nFollowing illustration make 823 Blocks and join 18 \u00d7 24 around outside. Stretch and pin cloth right-side-down on quilting or curtain frames. Lay frames over an ironing board or padded table a section at a time, steaming and pressing dry each section thru a cloth until completely blocked.\n\n_Tulip Tablecloth_\n\nMaterials Required\u2014\n\nAMERICAN THREAD COMPANY  \n\"STAR\" MERCERIZED CROCHET COTTON.\n\nARTICLE 20. SIZE 20\n\n48\u2014250 yd. Balls White, Ecru or Cream or \"Gem\" Mercerized Crochet Cotton, Article 35, Size 20.\n\n40\u2014300 yd. Balls.\n\nSteel Crochet Hook #11.\n\nEach motif measures about 6 inches. 176 motifs 11\u00d716 are required for cloth measuring 66\u00d796 inches.\n\n**LARGE MOTIF** \u2014Ch 8, join to form a ring, ch 4, * thread over twice, insert in ring, pull through and work off 2 loops twice, repeat from *, thread over and work off all loops at one time, ** ch 4, * thread over twice, insert in ring, pull through and work off 2 loops twice, repeat from * twice, thread over and work off all loops at one time (a cluster st), repeat from ** 6 times, ch 4, join in 1st cluster st.\n\n**2nd Row** \u2014Ch 4, tr c in same space, ch 7, 2 tr c in same space, * ch 3, 2 tr c, ch 7, 2 tr c in next cluster st, repeat from * all around, ch 3, join in 4th st of ch.\n\n**3rd Row** \u2014Sl st to center of loop, ** ch 5, 2 tr c in 1st st of ch, * ch 4, 2 tr c in top of last tr c, (rice st) repeat from * 7 times, s c in top of 4th rice st, ch 4, 2 tr c in same space, ch 4, 2 tr c in top of last tr c, s c in top of 2nd rice st, ch 4, 2 tr c in same space, ch 4, 2 tr c in top of last tr c, skip one loop, s c in next loop, repeat from ** all around, break thread.\n\n**4th Row** \u2014Join thread between 5th and 6th rice sts, ch 12, d c between next 2 rice sts, * ch 9, d c between next 2 rice sts, repeat from *, ** ch 9, s c between 10th and 11th rice sts. ch 7, tr c between 12th and 13th rice sts, tr c between 1st and 2nd rice sts of next group, ch 7, s c between 3rd and 4th rice sts, ch 9, d c between 5th and 6th rice sts, * ch 9, d c between next 2 rice sts, repeat from * twice, repeat from ** 6 times, ch 9, s c between 10th and 11 rice sts, ch 7, tr c between 12th and 13th rice sts, tr c between 1st and 2nd rice sts of 1st group, ch 7, s c between 3rd and 4th rice sts, ch 9, join in 3rd st of ch.\n\n**5th Row** \u2014Sl st to loop, ch 3, 2 d c in same space, ch 3, 3 d c, ch 3, 3 d c in same space, ** ch 3, 3 d c, ch 3, 3 d c, ch 3, 3 d c, ch 3, 3 d c in next loop, * ch 3, 3 d c, ch 3, 3 d c, ch 3, 3 d c in next loop, repeat from *, s c in next loop, ch 3, s c in next 7 ch loop, 3 d c, ch 3, 3 d c, ch 3, 3 d c in next loop, ch 3, 3 d c, ch 3, 3 d c, ch 3, 3 d c in next loop, repeat from ** 6 times, ch 3, 3 d c, ch 3, 3 d c, ch 3, 3 d c, ch 3, 3 d c in next loop, * ch 3, 3 d c, ch 3, 3 d c, ch 3, 3 d c in next loop, repeat from *, s c in next loop, ch 3, s c in next 7 ch loop, 3 d c, ch 3, 3 d c, ch 3, 3 d c in next loop, ch 3, join in 3rd st of ch.\n\n**6th Row** \u2014Sl st to center of next loop, * ch 5, s c in next loop, repeat from * 10 times, ** skip next 3 ch loop, s c in next 3 ch loop, ch 2, sl st in last 5 ch loop of previous scallop, ch 2, s c in next 3 ch loop of 2nd scallop, ch 2, sl st in next 5 ch loop of previous scallop, ch 2, s c in next 3 ch loop of 2nd scallop, * ch 5, s c in next loop, repeat from * 11 times, repeat from ** 6 times, skip next 3 ch loop, s c in next 3 ch loop, ch 2, sl st in last 5 ch loop of previous scallop, ch 2, s c in next 3 ch loop of 1st scallop, ch 2, sl st in next 5 ch loop of previous scallop, ch 2, s c in next 3 ch loop of 1st scallop, ch 3, d c in sl st of 1st loop, (this brings thread in position for next row).\n\n**7th Row** \u2014Ch 3, s c in next loop, * ch 7, sl st in 5th st from hook for picot, ch 2, 2 d c in next loop, repeat from * 5 times, ch 7, sl st in 5th st from hook for picot, ch 2, s c in next loop, ch 3, s c in next loop, ch 6, sl st in 5th st from hook for picot, ch 1, s c in next loop of next scallop, repeat from beginning all around, break thread.\n\nWork a 2nd motif in same manner joining to 1st motif in last row as follows: ch 3, s c in next loop, * ch 7, sl st in 5th st from hook for picot, ch 2, 2 d c in next loop, repeat from * twice, * ch 5, sl st in corresponding picot of 1st motif, ch 2, complete picot, ch 2, 2 d c in next loop of 2nd motif, repeat from * twice, ch 7, sl st in 5th st from hook for picot, ch 2, s c in next loop, ch 3, s c in next loop, ch 6, sl st in 5th st from hook for picot, ch 1, s c in next loop, ch 3, s c in next loop, ch 7, sl st in 5th st from hook for picot, ch 2, 2 d c in next loop, * ch 5, sl st in corresponding picot of 1st motif, ch 2, complete picot, ch 2, 2 d c in next loop of 2nd motif, repeat from * twice and complete motif same as 1st motif, break thread. Join 3rd motif to 2nd motif and 4th motif to 3rd and 1st motifs in same manner.\n\n**JOINING MOTIF** \u2014Work 1st 2 rows same as large motif.\n\n**3rd Row** \u2014Work 2 rice sts, ch 2, sl st in center free picot of 1st scallop, ch 2, complete picot, work 2 rice sts, skip next loop of joining motif, s c in next loop, work 2 rice sts, ch 2, sl st in center free picot of next scallop of same motif, ch 2, complete picot, work 2 rice sts, skip next loop of joining motif, s c in next loop, work 2 rice sts, ch 2, sl st in center free picot of next scallop of next motif, ch 2, complete picot and continue in same manner until joining is completed, break thread.\n\n_Sunset_\n\n**Materials:** Clark's O.N.T. Mercerized Crochet, size 30, 16 balls of Ecru and 7 balls of color Dk. Yellow; or J. & P. Coats Mercerized Crochet, 12 balls of Ecru and 5 balls of Dk. Yellow; or if Clark's Big Ball Mercerized Crochet is being used, buy 6 balls of Ecru only. Milward's steel crochet hook No. 8. \u00bd yd. 54 inch width natural linen for 4 napkins. Set consists of 4 place mats each about 11\u00d721 inches, a center mat about 13 \u00d7 28 inches, and 4 napkins each 13 inches square. The center mesh of mat is worked first, then sunset patterns are worked and sewed to each end.\n\n**Place Mat. Center Mesh:** With Ecru, ch 162, turn. **1st row:** D c in 8th ch from hook, d c in each ch across (155 d c). Ch 8, turn. **2nd row:** D c in each d c across. Ch 8, turn. **3rd row:** D c in each of first 5 d c (thus making a loop on edge), * ch 1, skip 2 d c, d c in next d c, ch 1, skip 2 d c, d c in next d c, ch 2, d c in next d c, repeat from * across, ending row with ch 1, skip 2 d c, d c in each of next 5 d c. Ch 8, turn. **4th row:** D c in each first 5 d c (thus making a loop on edge), d c in next single d c, ch 2, d c in same single d c, * ch 1, d c in next ch-2, ch 1, d c in next single d c, ch 2, d c in same single d c, repeat from * across, ending row with d c in last single d c, ch 2, d c in same single d c, d c in each of last 5 d c. Ch 8, turn. **5th row:** D c in each of first 5 d c, * ch 1, d c in next ch-2 sp, ch 1, d c in next single d c, ch 2, d c in same single d c, repeat from * across, ending row with d c in last ch-2 sp, ch 1, d c in each of last 5 d c. Ch 8, turn. Repeat 4th and 5th rows alternately, until work measures 11 inches. Then work 2 rows of d c to correspond with the beginning. Do not break off thread but continue for edging along long side as follows:\n\n**Edging. 1st row:** * Ch 3, 3 d c with ch 3 between each d c in next loop, ch 3, 2 s c in next loop, repeat from * across, ending row with 3 d c with ch 3 between in last loop, ch 3, sl st to base of last d c of 1st row. Ch 3, turn. **2nd row:** * S c in ch-3 loop, s c in d c, ch 5, s c in next loop, s c in next loop, ch 5, s c in next d c, s c in next loop, ch 2 (this ch-2 should come directly over 2-s c of previous row), repeat from * across, ending row with s c in last d c, sl st along last ch-3 loop. Turn and break off. **3rd row:** Attach Dk. Yellow, make 4 s c in ch-3 loop, * 9 s c in ch-5 loop, 4 s c in next ch-5 loop, ch 5, remove hook, insert hook in 5th s c of ch-5 loop just completed, draw loop through, 9 s c in ch-5 loop, 5 s c in next incompleted loop, 3 s c in ch-2 loop, repeat from * across, ending row to correspond with beginning. Break off. Work edging on opposite side in same way.\n\n**Sunset Pattern:** With Ecru, ch 6, join with sl st to form ring. **1st row:** Ch 6, d c in ring, * ch 3, d c in ring, repeat from * until 4 loops are made. Ch 8, turn. **2nd row:** D c in 1st loop, * ch 5, d c in next loop, repeat from * until 4 loops are made, then ch 5, d c in 4th st of turning ch. Ch 8, turn. **3rd row:** D c in first loop, * ch 8, d c in next loop, repeat from * until 5 loops are made. Ch 7, turn. **4th row:** D c in 4th st of first ch-8 loop, * ch 6, d c in 3rd st of next ch-8 loop, ch 4, skip 2 ch, d c in next ch of same ch-8 loop, repeat from * until 9 loops are made. Ch 6, turn. **5th row:** * D c in next d c, ch 6, repeat from * until 8 loops are made, then ch 3, d c in 3rd st of turning ch. Ch 3, turn. **6th row:** D c in each ch st and d c in each d c of previous row (57 d c, counting turning ch-3 as 1 d c). Ch 3, turn. **7th row:** D c in each of next 3 d c, * ch 2, d c in each of next 7 d c, repeat from * across, ending with ch 2, d c in each of last 4 d c (7 groups of 7-d c and 2 groups of 4-d c). Ch 3, turn. **8th row:** D c in each d c and ch 3 (instead of ch 2) between d c-groups. Ch 3, turn. **9th to 15th rows incl:** Work d c in each d c, making 1 additional ch st in the ch-loops between the d c-groups. (There will be ch 10 between d c-groups on 15th row.) At end of 15th row, turn and break off. **16th row:** Attach Dk. Yellow and make s c in each d c of 4-d c group, ch 18, ** sl st in 2nd d c of next 7-d c group, turn, s c in 1st st of ch-18, 1 half d c in next st, d c in each of next 2 sts, tr in each of next 3 sts, d c in each of next 2 sts, 1 half d c in next st, s c in next st (a petal made), * ch 17, turn, sl st in next d c of same 7-d c group, turn and make another petal in same way, repeat from * until 5 petals are made. Then ch 7, s c in each d c of next 7-d c group, ch 18, repeat from ** across, ending row with ch 7, s c in each of last 4 d c. Turn and break off. **17th row:** Attach Ecru, s c in each of first 3 s c, * 10 s c in next ch-loop. In each of next 4 loops (between petals) make 1 s c, 1 half d c, 2 d c, 5 tr, 2 d c, 1 half d c and 1 s c (1 shell made). Then 10 s c in next ch-loop, skip 1st s c of 7-s c group, s c in each of next 5 s c. Repeat from _*_ across, ending row with s c in each of last 3 s c. Ch 7, turn.\n\n**18th row:** Skip 4 s c of first loop, s c in next s c, ch 3, skip 3 sts of next shell, d c in next st, * ch 5, skip 1 st, d c in next st, repeat from * twice more, holding back on hook the last 2 loops of the last d c; then skip 3 sts of next shell, d c in next st, working off 2 loops, then thread over hook and draw thread through all loops on hook. Repeat from first * across the next 3 shells of this group completing the last d c on 4th shell, then ch 5, s c in center s c of next s c-loop, ch 7, s c in center s c of next s c-loop, ch 5, skip 3 sts of next shell, and continue thus across, ending row with ch 3, tr in last s c. Ch 7, turn. **19th row:** Skip first 2 loops, 2 s c in next ch-5 loop, * ch 3, 3 d c with ch 3 between d c's in next loop, ch 3, 2 s c in next loop, ch 1, 2 s c in next loop, repeat from * across the next 3 shells, after the last 2 s c are made in the 3rd loop of 4th shell, ch 3, skip next loop, d c in 3rd st of ch-7, ch 3, skip next loop, 2 s c in next loop, and continue thus across, ending row with 2 s c in 3rd loop from end, then ch 3, tr in 3rd st of turning ch. Ch 5, turn. **20th row:** 2 s c in first loop, * ch 5, 2 s c in next loop, repeat from * twice more, then ch 2, skip ch-1 sp, 2 s c in next loop, ch 5, 2 s c in next loop, repeat from first _*_ across the next 3 shells, after the last 2 s c are made in the 4th loop of 4th shell, make 4 s c in next loop, s c in next d c, 4 s c in next loop, 2 s c in next loop, ch 5, and continue thus across, ending row with 3 ch-5 loops after the last ch-2 sp. Turn and break off.\n\n**21st row:** Attach Dk. Yellow, and * work 9 s c in each of next 2 loops, 5 s c in next loop, ch 5, remove hook, then insert hook in 5th s c of s c-loop just completed and draw loop through, ch 5, remove hook, then insert hook in 5th s c of next s c-loop and draw loop through. Work 9 s c in this loop, 5 s c in next loop, ch 5, remove hook, insert hook in 5th s c of s c-loop to the right and draw loop through, work 9 s c in this loop, 4 s c in next incompleted loop, 3 s c in next incompleted loop, s c in ch-2 sp. Repeat from * across, making 7 s c over group of 9-s c between groups. Break off. This completes sunset pattern for one end. Make another one same as this and sew one to each end of place mat with over and over stitches. Make 3 more place mats.\n\n**Center Mat.** Start center mesh with ch 190 and work as for place mat until piece measures 14\u00bd inches. Then work 2 rows of d c to correspond with beginning. To make sunset pattern, starting with Ecru ch 6, join with sl st to form ring. **1st row:** Ch 6, d c in ring, * ch 3, d c in ring, repeat from * until 5 loops are made. Ch 8, turn. **2nd to 6th rows incl:** Work as for 2nd to 6th rows inch of mesh pattern on place mat, excepting that there will be 6 loops in 2nd row; 6 loops in 3rd row; 11 loops in 4th row; 11 loops in 5th row; 71 d c in 6th row. Continue as for mesh pattern on place mat. (There will be 9 groups of 7-d c, and 2 groups of 4-d c.) Make another one same as this and sew one to each end of center mat.\n\n**Napkins.** Cut linen 13 inches square. Make a \u00bc inch hem around edges. **1st row:** Attach Yellow to one corner of napkin and make ch-5 loops all around, spacing loops evenly and making an even number of loops on each side. Join and break off. **2nd row:** Attach thread to 2nd loop before next corner and make 9 s c in same loop, 9 s c in each of next 2 loops, 5 s c in next loop, * ch 5, remove hook, insert hook in 5th s c of next s c-loop (to the right) and draw loop through, repeat from * 2 more times (3 loops made), make 9 s c in loop just made, 9 s c in next loop, 9 s c in next loop, 4 s c in incompleted loop (this completes corner), ** 9 s c in next loop, 5 s c in next loop, ch 5, remove hook, insert hook in 5th s c of s c-loop (to the right) and draw loop through, 9 s c in this loop, 4 s c in next incompleted loop. Repeat from ** across, making each corner as corner just made.\n\n_Gazelle Luncheon Set_\n\n**Materials:** Choose one of the following threads in White or Ecru:\n\nClark's O.N.T. Mercerized Crochet, size 80, 5 balls.\n\nJ. & P. Coats Mercerized Crochet, size 80, 3 balls.\n\nMilward's steel crochet hook No. 9 or 10.\n\n1\u00be yards of 39 inch linen.\n\nWhen completed, luncheon set measures about 38\u00d738 inches, and each napkin about 12 inches square. Beginning at point directly below gazelle, ch 8. **1st row:** D c in 5th ch from hook, ch 2, skip 2 ch, d c in last ch, ch 7, turn. **2nd row:** D c in d c from which ch-7 started, 2 d c in sp, d c in next d c, 2 d c in next sp, d c in 3rd st of turning ch, ch 2, tr in same st with last d c, ch 7, turn. **3rd row:** D c in tr, 2 d c in sp, d c in 1st d c of 7-d c, ch 5, d c in last d c of 7-d c, 2 d c in next sp, d c in 3rd st of turning ch, ch 2, tr in same st with last d c, ch 7, turn.\n\n**4th row:** D c in tr, 2 d c in sp, d c in next d c, ch 2, skip 2 d c, d c in next d c, ch 3, skip 2 sts of ch-5, sl st in next st, ch 3, skip next 2 sts of same ch-5, d c in next d c, ch 2, skip 2 d c, d c in next d c, 2 d c in next sp, d c in 3rd st, ch 2, tr in same st with last d c, ch 7, turn. Work in this manner, following chart, increasing sps at beginning and end of rows. Make 4 corners.\n\n**Edging:** Cut corners of cloth so that crocheted pieces can be put on as in illustration. Have edges of cloth hemstitched. Sew on crocheted corners and make edging as follows: **1st row:** Attach thread to a hemstitched sp and make 2 s c in each sp around cloth and 4 s c in each sp of crocheted corner. Ch 7, turn.\n\n**2nd row:** * Skip 3 s c, d c in next s c, ch 3, repeat from * around, ch 7, turn. **3rd row:** Work sp over sp, ch 2, turn. **4th row:** 3 s c in each sp and 1 s c in each d c, making a ch-5 p in every 3rd d c.\n\n**Napkins:** Start same as for corners of cloth, and then follow chart. Make edging same as for cloth.\n\n**CHART FOR NAPKIN**\n\n**CHART FOR CLOTH**\n\n_Lily of the Valley Tablecloth_\n\n**Materials Required\u2014**\n\n**AMERICAN THREAD COMPANY**\n\n**\"STAR\" MERCERIZED CROCHET COTTON.**\n\n**ARTICLE 20. SIZE 30**\n\n36\u2014325 yd. Balls White, Ecru or Cream or \"Gem\" Mercerized Crochet Cotton, Article 35, Size 30.\n\n30\u2014400 yd. Balls.\n\nSteel Crochet Hook #12.\n\nEach motif measures about 3 inches. 567 motifs 21\u00d727 are required for cloth measuring about 63\u00d781 inches without edge.\n\n**MOTIF** \u2014Ch 8, join to form a ring, ch 4 and work 23 tr c into ring, join in 4th st of ch.\n\n**2nd Row** \u2014Ch 3, d c in same space, * ch 6, sl st in 5th st from hook for picot, ch 6, sl st in 5th st from hook for picot, ch 1, skip 2 tr c, 2 d c in next tr c, repeat from * 6 times, * ch 6, sl st in 5th st from hook for picot, repeat from *, ch 1, join in 3rd st of ch.\n\n**3rd Row** \u2014Ch 4, tr c in d c, * ch 6, sl st in 5th st from hook for picot, repeat from * 3 times, ch 1, 1 tr c in each of the next 2 d c, repeat from 1st * all around in same manner, join in 4th st of ch.\n\n**4th Row** \u2014Ch 10, * tr c in center of 4 picot loop, ch 6, tr c between next 2 tr c, ch 6, tr c in same space, ch 6, tr c in center of next 4 picot loop, ch 6, tr c between next 2 tr c, ch 6, repeat from * twice, tr c in center of next 4 picot loop, ch 6, tr c between next 2 tr c, ch 6, tr c in same space, ch 6, tr c in center of next 4 picot loop, ch 6, join in 4th st of ch.\n\n**5th Row** \u2014Ch 1, 6 s c over loop, ch 5, s c in next loop, ch-5, 3 d c in next loop, ch 3, 3 d c in same loop, * ch 5, s c in next loop, ch 5, 6 s c over each of the next 2 loops, ch 5, s c in next loop, ch 5, 3 d c in next loop, ch 3, 3 d c in same loop, repeat from * twice, ch 5, s c in next loop, ch 5, 6 s c over next loop, join in 1st s c.\n\n**6th Row** \u2014Ch 1, 1 s c in each of the next 4 s c, * ch 5, s c in next loop, repeat from * once, ** ch 5, 1 d c in each d c, 3 tr c in corner loop, ch 3, 3 tr c in same space, 1 d c in each d c, * ch 5, s c in next loop, repeat from *, ch 5, skip 2 s c, 1 s c in each of the next 8 s c, * ch 5, s c in next loop, repeat from * once, then repeat from ** all around in same manner ending row with skip 2 s c, 1 s c in each of the next 4 s c, join in 1st s c.\n\n**7th Row** \u2014Ch 1, 1 s c in each of the next 2 s c, * ch 5, s c in next loop, repeat from * twice, ** ch 5, 1 d c in each of the next 6 sts, 3 tr c in corner loop, ch 5, 3 tr c in same loop, 1 d c in each of the next 6 sts, * ch 5, s c in next loop, repeat from * twice, ch 5, skip 2 s c, 1 s c in each of the next 4 s c, * ch 5, s c in next loop, repeat from * twice, then repeat from ** all around in same manner, join in 1st s c, break thread.\n\nWork a 2nd motif in same manner joining to 1st motif in last row as follows: ch 1, 1 s c in each of the next 2 s c, * ch 5, s c in next loop, repeat from * twice, ch 5, 1 d c in each of the next 6 sts, 3 tr c in corner loop, ch 2, sl st in corresponding loop of 1st motif, 3 tr c in same loop of 2nd motif, 1 d c in each of the next 6 sts, * ch 2, sl st in corresponding loop of 1st motif, ch 2, s c in next loop of 2nd motif, repeat from * twice, ch 2, sl st in corresponding loop of 1st motif, ch 2, skip 2 s c of 2nd motif, 1 s c in each of the next 4 s c, * ch 2, sl st in corresponding loop of 1st motif, ch 2, s c in next loop of 2nd motif, repeat from * twice, ch 2, sl st in corresponding loop of 1st motif, ch 2, 1 d c in each of the next 6 sts of 2nd motif, 3 tr c in corner loop, ch 2, sl st in corresponding loop of 1st motif, ch 2, 3 tr c in same loop of 2nd motif and complete motif same as 1st motif. Join 3rd motif to 2nd motif and 4th motif to 1st and 3rd motifs in same manner.\n\n**EDGE:** With right side of work toward you and starting at corner motif, join thread in center of s c group to the right of corner, * ch 5, s c in next loop, repeat from * 3 times, ch 5, skip 4 d c, s c in next d c, ch 5, 3 tr c in corner loop, ch 5, 3 tr c in same loop, ** ch 5, skip 3 tr c and 1 d c, s c in next d c, * ch 5, s c in next loop, repeat from * 3 times, ch 5, s c in center of s c group, * ch 5, s c in next loop, repeat from * 3 times, ch 5, skip 4 d c, s c in next d c, * ch 5, s c in next loop, repeat from *, then repeat from ** all around working all corners in same manner, join.\n\n**2nd Row** \u2014Sl st to center of loop, ch 8, sl st in 5th st from hook for picot, d c in next loop, * ch 5, sl st in 5th st from hook for picot, d c in next loop, repeat from * twice, * ch 5, sl st in 5th st from hook for picot, ch 6, sl st in 5th st from hook for picot, d c in next loop, repeat from * once, ch 5, sl st in 5th st from hook for picot, * ch 6, sl st in 5th st from hook for picot, repeat from * once, d c in same loop (corner), * ch 5, sl st in 5th st from hook for picot, ch 6, sl st in 5th st from hook for picot, d c in next loop, repeat from * once, * ch 5, sl st in 5th st from hook for picot, d c in next loop, repeat from * 10 times, ch 5, sl st in 5th st from hook for picot, d c in same loop, ** ch 5, sl st in 5th st from hook for picot,. d c in next loop, * ch 5, sl st in 5th st from hook for picot, d c in next loop, repeat from * 11 times, ch 5, sl st in 5th st from hook for picot, d c in same loop, repeat from ** all around working all corners same as 1st corner.\n\nIf napkins are desired, cut linen the size required. Turn under a small hem. Work a row of s c all around.\n\n**2nd Row** \u2014Ch 7, sl st in 5th st from hook for picot, ch 2, skip 3 s c, s c in next s c, repeat from beginning all around.\n\n_Rose Filet Cloth_\n\n**Materials Required\u2014  \nAMERICAN THREAD COMPANY  \n\"STAR\" MERCERIZED CROCHET COTTON.  \nARTICLE 20. SIZE 30**\n\n30\u2014325 yd. Balls White, Cream or Ecru.\n\n3 yds of 36 inch Linen will be required for a cloth measuring about 68 \u00d7 103 inches.\n\nSteel Crochet Hook No. 12.\n\n**Gauge:** 6 mesh = 1 inch\n\nCh 200, d c in 8th st from hook, 1 d c in each of the next 3 sts of ch, * ch 2, skip 2 sts of ch, d c in next st, repeat from * 14 times, 1 d c in each of the next 3 sts of ch, * ch 2, skip 2 sts of ch, d c in next st, repeat from * 24 times, 1 d c in each of the next 3 sts of ch, * ch 2, skip 2 sts of ch, d c in next st, repeat from * 18 times, 1 d c in each of the next 3 sts of ch, ch 2, skip 2 sts of ch, d c in last st, ch 5 to turn each row.\n\n**2nd Row** \u2014D c in next d c, 1 d c in each of the next 3 d c, * ch 2, d c in next d c, repeat from * 18 times, 1 d c in each of the next 3 d c (a solid mesh), * 2 d c in mesh, 1 d c in next d c, repeat from * once, * ch 2, d c in next d c (an open mesh), repeat from * 22 times, 1 d c in each of the next 3 d c, * ch 2, d c in next d c, repeat from * 14 times, 1 d c in each of the next 3 d c, ch 2, d c in 3rd st of ch.\n\n**3rd Row** \u20141 open mesh, 1 solid mesh, 15 open meshes, 1 solid mesh, 22 open meshes, 2 solid meshes, 21 open meshes, 1 solid mesh, 1 open mesh. Continue working back and forth according to diagram to arrow, then repeat from beginning to arrow 6 times.\n\nWork 2 more lengths of insertion in same manner and finish all edges with a row of s c, working 2 s c in 1 mesh and 3 s c in next mesh and 5 s c in each corner mesh.\n\nCut 2 strips of linen 4 inches wide and required length. Cut 2 strips of linen 14 inches wide and required length. Work a row of s c over a narrow rolled hem around linen sections. Sew linen and insertion together as illustrated.\n\n**EDGE:** Join thread in corner, * 1 s c in each of the next 12 s c, ch 5, turn, s c in 6th s c from hook, ch 5, s c in 1st s c made, ch 1, turn and work 9 s c over first ch 5 loop and 5 s c over 2nd loop, ch 5, turn, s c in center st of 1st scallop, ch 1, turn and work 5 s c over loop, ch 3, slip st in top of s c for picot, 4 s c over same loop, sl st in top of 2nd scallop and finish same scallop with 4 more s c, slip st into s c of previous row, repeat from * all around, break thread. If napkins to match are desired, cut squares of linen the size required and work a row of s c over a narrow rolled hem.\n\n**EDGE:** Starting at corner, * 1 s c in each of the next 6 s c, ch 5, turn, s c in 1st s c, ch 1, turn, work 3 s c over loop, ch 3, slip st in top of s c for picot, 3 s c over same loop, repeat from * all around, break thread.\n\nFILET CROCHET INSTRUCTIONS\n\nOPEN MESH\n\nWhen worked on a chain, work the first d c in 8th ch from hook * ch 2, skip 2 sts, 1 d c in next st, repeat from *. Succeeding rows, ch 5 to turn and d c in d c, * ch 2, d c in next d c, repeat from *.\n\nSOLID MESH\n\nFour double crochets form 1 solid mesh and 3 d c are required for each additional solid mesh.\n\nTo increase a solid mesh at end of row, work a tr c in same space with last d c, work 2 more tr c working each tr c into lower loop of previous st.\n\nTo increase 1 open mesh at end of row, ch 2, tr c in same space with last d c.\n\nTo increase 1 open mesh at beginning of row, ch 8, d c in 1st d c.\n\nTo increase a solid mesh at beginning of row, ch 5, 1 d c in 4th st from hook, 1 d c in next st of ch, d c in next d c.\n\nTo decrease a mesh at beginning of row, sl st across 1st mesh.\n\n_Lace Medallion Luncheon Set_\n\n**MATERIALS** \u2014 DAISY Mercerized Crochet Cotton size 30:\u20145-balls or 4 skeins White, Cream or Ecru (sufficient for Centerpiece, 4 Place Mats and 4 Coasters). Crochet hook size 13.\n\n**BLOCK** \u2014(Size\u20143\u00bd\u2033 square) \u2014 Ch 7, sl st in 1st st. Ch 9, tr in ring, holding starting end around ring and working over it to cover it up, (ch 4, tr in ring) 6 times, ch 4, sl st in 5th ch of 1st ch-9. **ROW 2** \u2014Ch 1, sc in next ch-4 sp, * (ch 4, sc) twice in same sp, ch 4, sc in next sp. Repeat from * around. End with ch 2, dc in 1st se drawn down to make lp the same size as others (24 lps). **ROW 3** \u2014(Ch 4, sc in next lp) 23 times, ch 2, dc in next lp. **ROW 4** \u2014Repeat Row 3. **ROW 5** \u2014* Ch 12, sk 1 lp, se in next lp, (ch 4, sc in next lp) 4 times. Repeat from * around, ending with 3 ch-4 lps, ch 2, dc in next lp. **ROW 6** \u2014* Ch 2, sk 2 ch of next ch-12 lp, (3 sc, ch 4, sl st in last sc for a p) 3 times and 3 se,\u2014all in lp, leaving final 2 ch of lp uncovered too. Ch 2, sc in next ch-4, lp, (ch 4, sc in next lp) 3 times. Repeat from * around, with ch 2 and dc for final lp. **ROW 7** \u2014* (Ch 12, dc between next 2 ps) twice, ch 12, sc in next ch-4 lp (ch 4, sc in next lp) twice. Repeat from * around, with ch 2 and dc for final lp. **ROW 8** \u2014* Ch 2, sk 1st 2 ch of next ch-12 lp, (4 sc, p) twice and 4 sc all on bal. of lp, (4 sc, p, 6 sc, p, and 4 sc) all in next lp, (4 sc, p) twice and 4 sc all in next lp leaving final 2 ch of lp uncovered, ch 2, sc in next ch-4 lp, ch 4, sc in next lp. Repeat from * around with ch 2 and dc for final lp. **ROW 9** \u2014Ch 18, * dc between next 2 ps, ch 15, dc in 3d sc between ps on next lp, ** ch 21, dc in next sc, ch 15, dc between ps in next lp, ch 13, tr in ch-4 lp at tip of point, ch 13 and repeat from * around. Join with sl st in 5th ch of 1st ch-18. Cut 6\u2033 long, thread to a needle and fasten off on back.\n\n**2d BLOCK** \u2014Repeat to ** in Row 9. Ch 10, sl st in one lp of center (11th) st in a corner lp on 1st Block, ch 10, dc back in next sc on 2d Block, ch 7, sl st in next lp on 1st Block, ch 7, dc back between 2 ps on next lp on 2d Block, ch 6, sl st in next lp on 1st Block, ch 6, tr back in next ch-4 lp on 2d Block, ch 6, sl st in next lp on 1st Block, ch 6, dc back between ps on next lp on 2d Block, ch 7, sl st in next lp on 1st Block, ch 7, dc back in 3d sc between ps on next lp on 2d Block, ch 10, sl st in center st on comer lp of 1st Block, ch 10, dc back in next sc on 2d Block. Complete row as for 1st Block.\n\n**PLACE MAT** \u2014(Size\u201411\u00bd\u2033xl6\u2033) \u2014 Make and join 12 Blocks 3\u00d74 (or other desired size). **Edge** \u2014Se in one corner, ch 15, sc in same lp, * ch 15, sc in next lp, (ch 13, sc in next lp) 3 times, ch 15, sc in joining of Blocks. Repeat from * around, with an extra ch-15 lp at corners. **ROW 2** \u2014Ch 1, sc in next (corner) lp, (ch 4, sc in same lp) 5 times, * ch 2, sc in next lp, (ch 4, sc) 4 times in same lp. Repeat from * around with corner lps like 1st one. End with ch 2, sl st in 1st se. **ROW 3** \u2014Sl st to center of next lp, (ch 4, sc in next lp) twice, ch 5, sc in same last lp, (ch 4, sc in next lp) twice, * ch 1, sc in next ch-4 lp on next scallop, (ch 4, sc in next lp) 3 times. Repeat from * around, with corners like 1st one. Join and fasten off. Repeat for desired number of mats.\n\n**CENTERPIECE** \u2014(Size \u2014 11\u00bd\u2033\u00d723\u2033)\u2014Make and join 18 Blocks 3\u00d76. Repeat Edge.\n\n**COASTER** \u2014Make one Block and repeat Edge around it.\n\nStretch and pin doilies right-side-down on a padded board in true shape. Steam and press dry thru a cloth.\n_Bedspreads_\n\n_Starbright_\n\n**MATERIALS:**\n\nCLARK'S O.N.T. MERCERIZED BEDSPREAD COTTON: **Single Size Spread** \u2014 _72 \u00d7 107 inches (including fringe)_ \u2014 _54 balls of White, Ecru or Cream, or 78 balls of Bedspread Peach_. **Double Size Spread** \u2014 _88 \u00d7 107 inches (including fringe)_ \u2014 _76 balls of White, Ecru or Cream, or 108 balls of Bedspread Peach_.\n\nSTEEL CROCHET HOOK _No. 7_.\n\n**GAUGE:** Block measures 4\u00bd inches from side to side.\n\n**FIRST BLOCK** . . . Starting at center, ch 8. Join with sl St. **1st rnd:** Ch 1, 18 sc in ring. Join with sl st in first sc. **2nd rnd:** Ch 3, dc in next 2 sc, (ch 5, dc in next 3 sc) 5 times; ch 5, sl st in top st of ch-3. **3rd rnd:** Ch 3, dc in next 2 dc, (ch 6, dc in next 3 dc) 5 times; ch 6. Join. **4th rnd:** Ch 3, dc in next 2 dc, (ch 5, se under next two chains, ch 5, dc in next 3 dc) 5 times; ch 5, sc under next two chains, ch 5. Join. **5th rnd:** Ch 3, dc in next 2 dc, * 3 dc in next loop, ch 6, 3 dc in next loop, dc in next 3 dc. Repeat from * around, ending with 3 dc in last loop. Join. **6th rnd:** Ch 3, dc in next 5 dc, * ch 7, dc in next 9 dc. Repeat from * around. Join. **7th rnd:** Ch 3, dc in next 5 dc, * ch 6, sc under next two chains, ch 6, dc in next 9 dc. Repeat from * around. Join. **8th rnd:** Ch 3, dc in next 2 dc, * (ch 6, sc in next loop) twice; ch 6, skip 3 dc, dc in next 3 dc. Repeat from * around. Join. **9th rnd:** Ch 3, dc in next 2 dc, * (ch 6, sc in next loop) 3 times; ch 6, dc in next 3 dc. Repeat from * around. Join. **10th rnd:** Ch 3, dc in next 2 dc, * (ch 7, sc in next loop) 4 times; ch 7, dc in next 3 dc. Repeat from * around. Join and break off.\n\n**SECOND BLOCK** . . . Work as for First Block until the 9th rnd is completed. **10th rnd:** Ch 3, dc in next 2 dc, ch 3, sl st in corresponding loop on First Block, ch 3, sc in next loop on Second Block and complete rnd as for First Block, joining next 4 loops as previous loop was joined.\n\nFor Single Size Spread, make 10 rows of 23 blocks and 9 rows of 24 blocks. For Double Size Spread, make 12 rows of 23 blocks and 11 rows of 24 blocks. Join blocks, alternating rows of 23 and 24 blocks (see diagram, page 14).\n\n**EDGING** . . . Attach thread to any loop on outer edge, sc in same loop, * ch 7, sc in next loop. Repeat from * around. Join and break off.\n\n**FRINGE** . . . See above, having 15 strands, each 10 inches long.\n\n_Golden Wedding_\n\n**MATERIALS:** J. & P. COATS BEDSPREAD COTTON, _18 balls of White or Ecru for single size spread; 21 balls for double size spread_. MILWARD'S _steel crochet hook No. 8_.\n\nGAUGE: 4 sps make 1 inch; 4 rows make 1 inch. Completed motif measures 7\u00bd inches square. For a single size spread, about 75 \u00d7 106 inches including fringe, make 9 \u00d7 13 motifs. For a double size spread, about 91 \u00d7 106 inches including fringe, make 11 \u00d7 13 motifs.\n\n**MOTIF** . . . Starting at center section, ch 15. **1st row:** D c in 3rd ch from hook and in next 2 ch (these 4 d c make 1 bl); ch 2, skip 2 ch, d c in next ch (sp); make another sp, end with 4 d c (bl). Ch 5, turn. **2nd row:** D c in last d c of bl; make d c in next 2 ch, d c in d c, d c in next 2 ch and in d c. Ch 2, d c in top st of turning ch-3. Ch 5, turn. **3rd row:** Sp over sp and bl over bl. Ch 3, turn. **4th row:** D c in next 2 ch and in d c, 2 sps over next 2 bls, bl over next sp (center section completed). Hereafter work in rnds. **1st rnd:** Ch 5, d c at base of last d c made, ch 2, d c at base of d c (turning ch) of previous row, ch 2, d c at base of next end d c, ch 2, d c at base of next ch-3, ch 5 (corner), d c in same place, sp over next bl. Continue to make sps thus around and ch-5 at corners, ending with ch 5, join with sl st to 3rd ch of ch-5 first made. **2nd rnd:** Ch 5, d c in next d c. Now make sp over each sp around, making an extra ch-5 at each corner. Join. **3rd rnd:** Ch 5 (to count as d c of last bl and ch-2 of 1st sp), d c in next d c; 3 more sps, 7 d c (2 bls); ch 5, another d c in corner ch, 6 more d c, 4 sps, 7 d c. Continue thus around. Join. **4th rnd:** Ch 3, 4 bls over 4 sps, 2 bls over 2 bls, ch 2, d c in 3rd ch of corner ch-5, ch 5, d c in same place, ch 2, d c in next d c, 8 bls, sp, ch 5 for corner sp, sp. Continue thus around, ending with 2 bls, join. Follow chart for remainder of pattern, always making an extra ch-5 at corners and working into center st of this ch on following rnd.\n\nMake necessary number of motifs and sew together on wrong side with neat over-and-over stitches.\n\n**FRINGE . . .** Make fringe in every other sp around. Cut 8 strands of thread, each 12 inches long. Double these strands, forming a loop. Pull loop through sp and draw loose ends through. Pull tight. When fringe has been worked all around edges, trim evenly to 4 inches.\n\n**MATERIALS:**\n\n**CLARK'S O.N.T. MERCERIZED BEDSPREAD COTTON**\n\n**SINGLE SIZE** | **DOUBLE SIZE**\n\n---|---\n\n42 balls of White or Ecru. | 56 balls of White or Ecru.\n\nOR\n\n**J. & P. COATS KNIT-CRO-SHEEN**\n\n**SINGLE SIZE** | **DOUBLE SIZE**\n\n---|---\n\n42 balls of White or Ecru, or 68 balls of any color. | 56 balls of White or Ecru, or 81 balls of any color.\n\nSteel crochet hook No. 6.\n\n**GAUGE:** Each motif measures about 7 inches from edge to edge, and 7\u00be inches from point to point, before blocking. For a single size spread about 70 \u00d7 102 inches, make 161 motifs. For a double size spread about 91 \u00d7 102 inches, make 212 motifs.\n\n_Chevy Chase_\n\n**FIRST MOTIF** . . . Starting at center, ch 8. Join with sl st. **1st rnd:** Ch 3, 23 dc in ring. Join with sl st in top of 1st ch-3. **2nd rnd:** Ch 3, 4 dc in same place as sl st, drop loop from hook, insert hook in top st of ch-3 and pull loop through (a starting pc st), * ch 2, skip 1 dc, dc in next dc, ch 2, skip 1 dc, in next dc make 5 dc, drop loop from hook, insert hook in top of the 1st dc of this group and pull loop through (pc st). Repeat from * around. Join in top of 1st pc st. **3rd rnd:** Ch 3, starting pc st in top of 1st pc st, * ch 3, dc in next dc, ch 3, pc st in top of next pc st. Repeat from * around. Join. **4th rnd:** Ch 3, 4 dc in top of pc st, * ch 2, dc in next dc, ch 2, 5 dc in top of next pc st. Repeat from * around. Join. **5th rnd:** Ch 5, (dc in next dc, ch 2) 4 times; * in next dc make dc, ch 3 and dc; (ch 2, dc in next dc) 5 times; ch 2. Repeat from * around. Join with sl st in 3rd st of 1st ch-5. **6th rnd:** Sl st in next sp, ch 3, starting pc st in same sp, (ch 2, pc st in next sp) 3 times; ch 2, * in next ch-3 sp make dc, ch 3 and dc; ch 2, skip 1 sp, (pc st in. next sp, ch 2) 4 times. Repeat from * around. Join.\n\n**7th rnd:** Ch 3, starting pc st in top of 1st pc st, * (ch 2, pc st in next sp) 3 times; ch 2, pc st in top of next pc st, ch 2, in next ch-3 sp make dc, ch 3 and dc; ch 2, pc st in top of next pc st. Repeat from * around. Join. **8th rnd:** Ch 3, starting pc st in top of 1st pc st, * (ch 2, pc st in next sp) 4 times; ch 2, pc st in top of next pc st, ch 2, in next ch-3 sp make dc, ch 3 and dc; ch 2, pc st in top of next pc st. Repeat from * around. Join. **9th rnd:** Ch 3, starting pc st in top of 1st pc st, * (ch 2, pc st in next sp) 5 times; ch 2, pc st in top of next pc st, ch 2, in next ch-3 sp make dc, ch 3 and dc; ch 2, pc st in top of next pc st. Repeat from * around. Join. **10th rnd:** Sl st to next sp, ch 3, starting pc st in same sp, * (ch 4, sc in next sp, ch 4, pc st in next sp, ch 2, pc st in next sp) twice; ch 2, in next ch-3 sp make dc, ch 3 and dc; (ch 2, pc st in next sp) twice. Repeat from * around, joining last ch-2 to top of 1st pc st. **11th rnd:** Ch 6, * (sc in next loop, ch 4) twice; pc st in next sp, (ch 4, sc in next loop) twice; ch 4, (pc st in-next sp, ch 2) twice; in next ch-3 sp make dc, ch 3 and dc; (ch 2, pc st in next sp) twice; ch 4. Repeat from * around. Join to 3rd st of 1st ch-6. **12th rnd:** Sl st in next loop, ch 1, sc in same loop, (ch 4, sc in next loop) 6 times; * ch 4, pc st in next sp, ch 4, sc in next ch-3 sp, ch 4, pc st in next sp, (ch 4, sc in next loop) 8 times. Repeat from * around. Join with sl st to 1st se made. **13th rnd:** Sl st to center of next loop, ch 1, sc in same loop, * ch 5, sc in next loop. Repeat from * around. Join and fasten off.\n\n**SECOND MOTIF** . . . Work as for 1st motif until 12th rnd is completed. **13th rnd:** Sl st to center of next loop, ch 1, sc in same loop, (ch 5, sc in next loop) 9 times; (ch 2, sl st in corresponding loop on 1st motif, ch 2, sc in next loop on 2nd motif) 8 times; ch 5, sc in next loop on 2nd motif. Finish 2nd motif with no more joinings. Fasten off.\n\nMake necessary number of motifs and join them as in diagram for joining hexagon motifs on page 31. For single size bedspread make 17 motifs from A to B, and 9 motifs from A to C; for double size spread, 17 motifs from A to B, and 12 motifs from A to C.\n\n**FRINGE** . . . Make fringe in every other loop between scallops on both long sides as follows: Cut 20 strands, each 9 inches long. Double these strands forming a loop. Pull loop through 1st space and draw loose ends through loop. Pull tight (there should be 7 groups of fringe between scallops). Trim evenly.\n\n_Marguerite_\n\n**MATERIALS:**\n\nCLARK'S O.N.T. MERCERIZED BEDSPREAD COTTON: **Single Size Spread** \u2014 _68 \u00d7 104 inches\u201487 balls of White, Ecru or Cream, or 124 balls of Bedspread Yellow_. **Double Size Spread** \u2014 _90 \u00d7 104 inches\u2014108 balls of White, Ecru or Cream, or 155 balls of Bedspread Yellow_.\n\nSTEEL CROCHET HOOK _No_. 7.\n\n**GAUGE:** Block measures 12\u00bd inches from side to side.\n\n**BLOCK** . . . Starting at center, ch 9. Join with sl st. **1st rnd:** Ch 1, 18 sc in ring. Sl st in first sc. **2nd rnd:** Ch 3, 4 dc in back loop of next st, drop loop from hook, insert hook in top st of ch-3 and draw dropped loop through (pc st made), * ch 2, skip 1 sc, 5 dc in back loop of next sc, drop loop from hook, insert hook in first dc of 5-dc group and draw dropped loop through (another pc st made). Repeat from * around, ending with ch 2, sl st in top of first pc st. **3rd rnd:** Ch 5, * dc in next sp, ch 2, dc in top of next pc st, ch 2. Repeat from * around. Sl st in 3rd ch of ch-5. **4th rnd:** Ch 1, 3 sc in same place as sl st, * sc in next sp, 3 sc in next dc. Repeat from * around. Join with sl st to first sc.\n\n**Hereafter pick up only the back loop of each sc.**\n\n**5th to 16th rnds incl:** * 3 sc in next sc (center sc of 3-sc group), sc in each sc to center sc of next 3-sc group. Repeat from * around. Join (each 3-sc-group is the beginning of a petal\u201418 petals in rnd). **17th rnd:** Mark the center stitch between each petal with a colored thread. Then sl st in each sc to within 5 sc of first marker. * Holding the marker in left hand, fold the next petal back to 5 sts from marker, having right sides together. Then, working through both thicknesses, make sc in next 5 sts, ch 4. Repeat from * around, ending with ch 4. Join. **18th rnd:** Working in back loop only, make * sc in next 2 sc, 3 sc in next sc, sc in next 2 sc, sc in next 4 ch, (sc in next 5 sc, sc in next 4 ch) twice. Repeat from * around. Join. 19th to 23rd rnds incl: Se in each sc around, making 3 sc in center sc of each 3-sc group and ending with sl st in center sc of first 3-sc group. **24th rnd:** Ch 4, dc in same place as sl st, * (ch 1, skip 1 sc, dc in next sc) 18 times; ch 1, in center sc of next 3-sc group make dc, ch 1 and dc. Repeat from * around, ending with ch 1, sl st in 3rd st of ch-4 (120 sps in rnd). **25th rnd:** Sl st in sp, ch 4, dc in same sp, * (ch 1, dc in next sp) 19 times; ch 1, in next sp make dc, ch 1 and dc. Repeat from * around. Join (126 sps in rnd). **26th rnd:** * 3 sc in next st, sc in next 41 sts. Repeat from * around. Join (264 sc in rnd). **27th rnd:** Sl st in each st to center sc of first 3-sc group, * 3 sc in center sc, sc in next 3 sc, (pc st in next sc, sc in next 5 sc) 6 times; pc st in next sc, sc in next 3 sc. Repeat from * around. Join. **28th rnd:** 3 sc in next sc, sc in each st around, making 3 sc in center sc of each 3-sc group. Join. **29th rnd:** * 3 sc in center sc, sc in next 2 sc, (pc st in next sc, sc in next 5 sc) 7 times; pc st in next sc, sc in next 2 sc. Repeat from * around. Join.\n\n**30th rnd:** Repeat 28th rnd. **31st rnd:** Sl st in center sc, ch 4, dc in same place as sl st, * (ch 1, skip 1 sc, dc in next sc) 25 times; ch 1, in center sc of next 3-sc group make dc, ch 1 and dc. Repeat from * around. Join. **32nd rnd:** Sl st in next sp, ch 4, dc in same sp, * (ch 1, dc in next sp) 26 times; ch 1, in next sp make dc, ch 1 and dc. Repeat from * around. Join. **33rd rnd:** * 3 sc in next st, sc in next 55 sts. Repeat from * around. Join. **34th rnd:** * 3 sc in center sc of 3-sc group, sc in next 4 sc, (pc st in next sc, sc in next 5 sc) 8 times; pc st in next sc, sc in next 4 sc. Repeat from * around. Join. **35th rnd:** * 3 sc in center sc, sc in each st to within center sc of next 3-sc group. Repeat from * around. Join. **36th rnd:** * 3 sc in next sc, sc in next 3 sc, (pc st in next sc, sc in next 5 sc) 9 times; pc st in next sc, sc in next 3 sc. Repeat from * around. Join. **37th rnd:** Repeat 35th rnd. **38th rnd:** * 3 sc in center sc, sc in next 2 sc, (pc st in next sc, sc in next 5 sc) 10 times; pc st in next sc, sc in next 2 sc. Repeat from * around. Join. **39th rnd:** Repeat 35th rnd. **40th rnd:** Sl st in center sc, ch 4, dc in same place, * (ch 1, skip 1 sc, dc in next sc) 34 times; ch 1, in center sc of next 3-sc group make dc, ch 1 and dc. Repeat from * around. Join. **41st rnd:** Sl st in sp, ch 4, dc in same sp, * (ch 1, dc in next sp) 35 times; ch 1, in next sp make dc, ch 1 and dc. Repeat from * around. Join and break off.\n\n**HALF BLOCK** . . . Starting at center, ch 9. Join with sl st. **1st row:** 9 sc in ring, sl st in last 3 ch of ring, sl st in first sc. **2nd row:** Ch 3, 4 dc in same place as sl st and complete pc st as before, (ch 2, skip 1 sc, make a 5-dc pc st in back loop of next sc) 4 times. Break off.\n\n**Note: Hereafter, break off at end of each row and attach thread to beginning of previous row.**\n\n**3rd row:** Attach thread to tip of first pc st, ch 5, * dc in next sp, ch 2, dc in next pc st, ch 2. Repeat from * across, ending with ch 2, dc in tip of last pc st (8 sps). Break off. **4th row:** Attach thread to 3rd ch of ch-5, * sc in sp, 3 sc in next dc. Repeat from * across, ending with 3 sc in last dc. Break off. **Hereafter pick up only the back loop of each st. 5th to 16th rows incl:** Sc in each sc across, making 3 sc in center sc of each 3-sc group. Break off. **17th row:** Mark the center st between each petal with a colored thread. Place a pin in 9th sc from beginning of previous row. With wrong side facing attach thread to sc at pin, ch 5, skip 3 sc, sc in next 5 sc. Turn work so that right side is facing, * ch 5, holding the marker in left hand, fold the next petal back to 5 sts from marker, then, working through both thicknesses, make sc in next 5 sts. Repeat from * 6 more times; then with wrong side of last petal facing make sc in last 5 sc (85 sts on row). Break off. **18th row:** Attach thread to first ch and make sc in each st across (85 sc). Break off. 19th row: Attach thread to first sc, 2 sc in same place, (sc in next 27 sc, 3 sc in next sc) twice; sc in next 27 sc, 2 sc in last sc. Break off. **20th to 23 rd rows incl:** Sc in each sc across, making 2 sc in first and last sc and 3 sc in center sc of each 3-sc group (115 sc on 23rd row). **24th row:** Attach thread to first sc, ch 4, dc in same place, * (ch 1, skip 1 sc, dc in next sc) 18 times; ch 1, in center dc of next 3-sc group make dc, ch 1 and dc. Repeat from * across, ending with dc, ch 1 and dc in last sc. Break off.\n\nHereafter work as for Block to within last 2 rows, always making an extra sp at beginning and end of each row of sps and 2 sc at beginning and end of each sc-row (214 sc on last row). Now work 2 rnds all around as follows: **1st rnd:** Attach thread to first sc, ch 4, dc in same sp, * (ch 1, skip 1 sc, dc in next sc) 34 times; ch 1, in center st of next 3-sc group make dc, ch 1 and dc. Repeat from * across 3 sides, ending with dc, ch 1 and dc in last sc. Continue making ch-1 sps along remaining side, ending with ch 1, sl st in 3rd ch of ch-4. **2nd rnd:** Sl st in sp, ch 4, dc in same sp, * ch 1, dc in next sp. Repeat from * around, making dc, ch 1 and dc in each corner sp. Join and break off.\n\nFor Single Size Spread, make 3 rows of 8 blocks and 2 rows of 7 blocks. For Double Size Spread, make 4 rows of 8 blocks and 3 rows of 7 blocks. Sew blocks neatly together on wrong side. Fill in spaces at top and bottom of spread with half blocks.\n\n**FRINGE** . . . Cut 10 strands of thread, each 12 inches long. Double these strands to form a loop. Insert hook in space on edge of bedspread and draw loop through. Draw loose ends through loop and pull up tightly to form a knot. Make a fringe in every other space (or every half inch) around spread. When fringe is completed, trim ends evenly.\n\n_Popcorn Pinwheel_\n\n**MATERIALS:** J. & P. COATS BEDSPREAD COTTON, _24 balls of White or Ecru for single size spread; 30 balls for double size spread_. MILWARD'S _steel crochet hook No. 8 or 9_.\n\nGAUGE: Each motif measures about 6 inches across, from side to side. For a single size spread, about 72 \u00d7 110 inches including fringe, make 214 motifs. For a double size spread, about 92 \u00d7 110 inches including fringe, make 280 motifs.\n\n**MOTIF** . . . Starting at center, ch 12, join with sl st. **1st rnd:** Ch 3, 23 d c in ring. Join to 3rd st of ch-3. **2nd rnd:** Ch 3, 4 d c in same place as sl st; drop st from hook, insert hook back in 3rd st of ch-3 first made and draw loop through the one on hook (a pc st). * Ch 5, skip 2 d c, d c in next d c, ch 3, 5 d c in next d c; drop st from hook, insert hook back in last ch made and draw loop through the one on hook (a pc st). Repeat from * around, ending with ch 2, sl st at tip of first pc st made. **3rd rnd:** Sl st in ch-5 sp, ch 3 and complete a pc st; * ch 2, a pc st, ch 5, d c in next sp, ch 2, a pc st. Repeat from * around, ending with ch 2, sl st at tip of 1st pc st made. **4th rnd:** Sl st in next sp (between pc sts), ch 3 and complete a pc st; * ch 2, in next ch-5 sp make 2 pc sts with ch-2 between; ch 5, d c in next sp, ch 2, pc st in sp between 2 pc sts. Repeat from * around, ending as before. **5th and 6th rnds:** Same as 4th rnd, making pc sts between pc sts of previous rnd and 2 pc sts in ch-5 sp, always making ch-2 between pc sts (4 pc sts in each group on 5th rnd; 5 pc sts in each group on 6th rnd). **7th rnd:** Sl st in next sp (between 1st 2 pc sts), ch 3 and complete a pc st; * (ch 2, pc st between next 2 pc sts) 3 times; ch 3, d c at tip of next pc st, ch 3, d c in ch-5 sp, ch 3, d c in next sp, ch 2, pc st between next 2 pc sts. Repeat from * around. Join. **8th rnd:** * 3 pc sts between pc sts of previous rnd, with ch-2 between; ch 3, d c at tip of next pc st; (ch 3, d c in next sp) 4 times; ch 2 and repeat from * around. Join. **9th rnd:** * 2 pc sts between pc sts of previous rnd, with ch-2 between; ch 3, d c at tip of next pc st; (ch 3, d c in next sp) 6 times, ch 2. Repeat from * around. Join. **10th rnd:** Sl st in sp (between pc sts), ch 3 and complete a pc st; * (ch 3, d c in next sp) 3 times; ch 3, skip next d c and sp; in next d c make 3 tr, ch 3, 3 tr; ch 3, skip next sp and d c; d c in next sp; (ch 3, d c in next sp) twice, ch 2, pc st between pc sts of previous rnd. Repeat from * around. Join and fasten off.\n\nMake necessary number of motifs and sew together on wrong side with neat over-and-over stitches, as in diagram on page 29 (for single size spread, disregard motifs to left of heavy line).\n\n**FRINGE** . . . Make fringe in each sp around four sides as follows: Cut 8 strands, each 12 inches long. Double these strands, forming a loop. Pull loop through sp and draw loose ends through loop. Pull tight. When fringe has been worked around all edges, trim evenly.\n\n_Filet Medallion Bedspread_\n\n**MATERIALS:**\n\n**J. & P. COATS KNIT-CRO-SHEEN**\n\n**SINGLE SIZE** | **DOUBLE SIZE**\n\n---|---\n\n26 balls of White or Ecru,  \nor 42 balls of any color. | 37 balls of White or Ecru,  \nor 59 balls of any color.\n\nOR\n\n**CLARK'S O.N.T. MERCERIZED BEDSPREAD COTTON**\n\n**SINGLE SIZE** | **DOUBLE SIZE**\n\n---|---\n\n26 balls of White or Ecru. | 37 balls of White or Ecru.\n\nSteel crochet hook No. 7 or 8.\n\n**GAUGE:** 3 sps make 1\u00bc inches; 3 rnds make 1\u00bc inches. Each block measures about 17\u00bd inches after blocking. For a single size spread about 72 \u00d7 108 inches including edging, make 4 \u00d7 6 blocks. For a double size spread about 90 \u00d7 108 inches including edging, make 5 \u00d7 6 blocks.\n\n**BLOCK** . . . Starting at center, ch 11, join with sl st to form ring. **1st rnd:** Ch 4 (to count as tr), 4 tr in ring, (ch 7, 5 tr in ring) 3 times; ch 7. Join with sl st to top of ch-4. **2nd rnd:** Ch 4, tr in next 4 tr, * 4 tr in next sp, ch 7, 4 tr in same sp, tr in next 5 tr. Repeat from * around. Join. **3rd rnd:** Ch 4, tr in 8 tr, * 4 tr in next sp, ch 7, 4 tr in same sp, tr in next 13 tr. Repeat from * around. Join. **4th rnd:** Ch 4, tr in 4 tr, * (ch 3, skip 3 tr, tr in next tr) twice; in corner sp make 4 tr, ch 7 and 4 tr; tr in next tr, (ch 3, skip 3 tr, tr in next tr) twice; tr in next 4 tr. Repeat from * around. Join. **5th rnd:** Ch 4, tr in 4 tr, * ch 3, tr in next tr, 3 tr in next sp, tr in next 5 tr, in corner sp make 4 tr, ch 7 and 4 tr, tr in 5 tr, 3 tr in next sp, tr in next tr, ch 3, tr in next tr, tr in next 4 tr. Repeat from * around. Join. **6th to 20th rnds incl:** Work in this manner following chart, making ch-3 for sps and ch-7 for corner sps, and joining each rnd with a sl st. The heavy line on chart shows where each rnd begins. At end of 20th rnd, join and fasten off. Make necessary number of blocks and sew together on wrong side with neat over-and-over sts.\n\n**EDGING** . . . Attach thread, ch 4 and work 3 rnds of tr making corners as on blocks, being careful edging does not ruffle. Fasten off. Block to measurements given.\n\n**There are 10 spaces between heavy lines**\n\n**This lovely spellbinder with its central garland of roses, its border and pillow motif, its clusters of popcorns, makes a charming picture. Clear-cut as a cameo and just as precious.**\n\n_Cameo_\n\n**_For Double Size Bed Only_**\n\n**MATERIALS:**\n\n**J. & P. COATS BEDSPREAD COTTON**\n\n_24 balls of White or Ecru_\n\nMILWARD'S STEEL CROCHET HOOK _No. 7 or 8_.\n\n**GAUGE:**\n\n4 sps or bls make 1 inch; 4 rows make 1 inch. Finished bedspread measures about 90 \u00d7 108 inches after blocking.\n\nStarting at bottom of Chart One, make a chain (12 ch sts to 1 inch) about 3 yards long. **1st row:** D c in 8th ch from hook, * ch 2, skip 2 ch, d c in next ch. Repeat from * until there are 359 sps made. Cut off remaining chain (mark the 180th sp with a colored thread\u2014this is center). Ch 5, turn. **2nd row:** (This is right side) D c in next d c (sp over sp), 11 more sps, * make a pc st in next sp\u2014 _to make a pc st, ch 1, 5 d c in sp, drop loop from hook, insert hook in ch-1, and draw dropped loop through;_ d c in next d c. Repeat from * 8 more times; 11 sps, (5 pc sts, 7 sps) 11 times; (5 pc sts, 8 sps) twice, (5 pc sts, 7 sps) 11 times; 5 pc sts, 11 sps, 9 pc sts, 12 sps. Ch 5, turn. **3rd row:** Make 10 sps, (make a reverse pc st in next sp\u2014 _to make a reverse pc st, work as for pc st, only inserting hook in ch-1 from back of work, thus raising pc st to right side;_ d c in next d c) twice; 9 sps, 3 reverse pc sts, 7 sps, (1 reverse pc st, 5 sps) 24 times; 2 reverse pc sts, 5 sps, 2 reverse pc sts, (5 sps, 1 reverse pc st) 24 times; 7 sps, 3 reverse pc sts, 9 sps, 2 reverse pc sts, 10 sps. Ch 5, turn.\n\nNow follow chart, starting at 4th row, always working to center of row as on chart (chart shows only the first half of each row); then omitting the center sp or pc st-bl, as the case may be, follow chart back to the beginning of row. When 24th row has been completed, work as follows: Ch 5, turn. **25th row:** 2 sps, 1 pc st, 2 sps, 5 pc sts, 9 sps, 1 pc st, 28 sps, make a bl over next sp\u2014 _to make a bl, make 2 a c in sp, d c in next d c;_ 11 sps, 1 bl, 25 sps, 1 bl, 16 sps, 3 bls, 21 sps, 1 bl, 20 sps, 1 bl, 19 sps, 6 bls, 11 sps, 6 bls, 19 sps, 1 bl, 20 sps, 1 bl, 21 sps, 3 bls, 16 sps, 1 bl, 25 sps, 1 bl, 11 sps, 1 bl, 28 sps, 1 pc st, 9 sps, 5 pc sts, 2 sps, 1 pc st, 2 sps. Ch 5, turn.\n\nStarting with 26th row, follow chart to top. Reverse chart and omitting last row, work back to \"A\" (this last row completes center design). _Floral Borders_ are now worked from \"B\" toward top of chart; at the same time, keep continuity of popcorn all-over pattern and popcorn scallop design along edges until the 324th row is complete. **Next row:** Mark off the center 119 sps. Continue bedspread as before, following Chart Two when working across the center 119 sps. Continue thus, until top of Chart Two is reached. Work scallop design, floral borders and all-over pattern as before, until 408 rows are complete. Do not fasten off but work s c all around, keeping work flat. Fasten off. Block to measure 90 \u00d7 108 inches.\n\n**THERE ARE 10 SPACES BETWEEN HEAVY LINE**\n\n_Textured Beauty_\n\n**MATERIALS** \u2014Lily FROST-TONE Mercerized Crochet Cotton:\u2014Single Bed Size\u201411\u00d718 Blocks\u201474\u00d7107 inches, including fringe\u201432 cones of White, Cream or Ecru. Double Bed Size\u201414\u00d718 Blocks\u201490\u00d7107 inches, including fringe\u201441 cones.\n\nCrochet hook size 10.\n\n**BLOCK** \u2014(Size\u20145\u00bd inches when blocked)\u2014Ch 9, sl st in starting st. Ch 1, 2 sc in ring. Now make a puff\u2014Ch 7, 7 dtr in 7th ch st from hook, holding back the last lp of each dtr on hook, thread over and pull thru all 8 lps on hook at once (a Cluster), ch 4, sl st in lp at top of Cluster for a p, ch 7, sl st at base of Cluster in same st where dtr were worked (1 puff made). This will not be mentioned again. (3 sc in ring, a puff) 3 times, 1 sc in ring, sl st in 1st sc. **ROW 2** \u2014Ch 3 and holding puffs down in front, make * dc in next sc, sk puff, dc in next sc, ** (dc, ch 5, dc) in next sc. Repeat from * twice and from * to Dc in next st, ch 2, dc in top of 1st 3-ch. **ROW 3** \u2014Ch 3, turn, 2 dc in next 2-ch sp, * dc in next 4 dc, ** (3 dc, ch 5, 3 dc) in corner sp. Repeat from * twice and from * to **. 3 dc in next sp, ch 2, dc in top of 1st 3-ch. **ROW 4** \u2014Ch 3, turn, 2 dc in next 2-ch, * dc in next 4 dc, sl st in p at tip of puff, ch 1, sk 2 dc behind puff, dc in next 4 dc, ** (3 dc, ch 5, 3 dc) in corner sp. Repeat from * twice and from * to **. 3 dc in next sp, ch 2, dc in top of 1st 3-ch. **ROW 5** \u2014Ch 5, turn, dc in next dc, * (ch 2, dc in next 3d st) 5 times, ** ch 2, (dc, ch 5, dc) in 3d st of corner 5-ch, ch 2, dc in next dc. Repeat from * twice and from * to **. (Ch 2, dc in next 3d st) twice. **ROW 6** \u2014Ch 1, turn, (2 sc, a puff and 1 sc) in next sp, * sc in next dc, 2 sc in next sp, sc in dc, (sc, a puff, sc) in next sp. Repeat from * twice. Sc in dc, 2 sc in next sp, sc in dc, (1 sc, a puff, 3 sc, a puff and 1 sc) in corner sp. Repeat from * around. In final sp, make 1 sc, a puff and 1 sc, sl st in 1st sc. **ROW 7** \u2014Ch 5, * (sk 1 sc on each side of puff, dc in next 4 sc, ch 2) 4 times, (dc, ch 5, dc) in corner sc, ch 2. Repeat from * around. Sk 1 sc on each side of final puff, dc in next st, ch 2, dc in 3d st of 1st 5-ch. **ROW 8** \u2014Ch 3, turn, 2 dc in 2-ch sp, * dc in next dc, (ch 2, dc in next 4 dc) 4 times, ch 2, dc in next dc, ** (3 dc, ch 5, 3 dc) in corner sp. Repeat from * twice and from * to **. 3 dc in next sp, ch 2, dc in top of 1st 3-ch. **ROW 9** \u2014Ch 3, turn, 2 dc in 2-ch sp, * dc in next 4 dc, (sl st in p at top of next Cluster, ch 1, dc in next 4 dc) 5 times, ** (3 dc, ch 4, 3 dc) in corner sp. Repeat from * twice and from * to **. 3 dc in next sp, dc in top of 1st 3-ch. **Corner** \u2014Ch 4, turn, sk last 4 dc, * dc in next dc, (ch 2, dc in next 3d st) twice, ch 9, sk 9 sts, sc in next 2 sts, ch 9, sk 9 sts, dc in next dc, (ch 2, dc in next 3d st) twice, dc in next 3d dc. **ROW 2** \u2014Ch 4, turn, sk last 2 dc, dc in next dc, 2 dc in next sp, dc in dc, ch 2, dc in next 3d ch st, ch 7, sc in next 2 sc, ch 7, dc in next 7th ch st, ch 2, dc in next dc, 2 dc in next sp, dc in next 2 dc. **ROW 3** \u2014Ch 4, turn, sk last 4 dc, dc in next dc, 2 dc in next sp, dc in dc, 3 dc in next sp, ch 6, sc in 2 center sc, ch 6, 3 dc on left end of next 7-ch, dc in next dc, 2 dc in next sp, dc in next dc, dc in next 3d dc. **ROW 4** \u2014Ch 4, turn, sk last 4 dc, dc in next 4 dc, ch 3, dtr in 2 center sc, ch 3, dc in next 4 dc, dc in next 3d dc. **ROW 5** \u2014Ch 4, turn, sk last 4 dc, dc in next dc, 3 dc in next sp, ch 2, 3 dc in next sp, dc in next dc, dc in next 3d dc. **ROW 6** \u2014Ch 4, turn, sk last 4 dc, dc in next dc, 2 dc in next sp, dc in next dc, dc in next 3d dc. Ch 5, turn, sk last 4 dc, sl st in next dc. Cut 2 inches long and pull thru lp tightly. Sk next 7 sps down side, join to next dc, ch 4, sk 2 dc and repeat corner from *. Continue until 4 corners are completed. **Edge** \u2014Joining to one corner of Block, * make 3 sc in corner sp, ch 4, sl st in last sc for a p, 3 sc in same sp, (2 sc, a p and 2 sc in next sp) 13 times, working over end left from corner. Repeat from * around and join. Cut 6 inches long, thread to a needle and fasten off on back.\n\nMake and join Blocks to desired size by the p at corners and by the 13 ps on each side. To join, in place of a p, make 2-ch, sl st in corresponding p on 1st Block, ch 2, sl st back in last sc to complete p. Repeat with each p-joining.\n\n**EDGE** \u2014Join to one corner, ch 4, sc in same p, * ch 4, sc in next 3d sc, (ch 4, sc in next p) 13 times, ch 4, sc in next 3d sc, ch 4, sc in joining of Blocks. Repeat from * around, making an extra lp at each corner. Fasten off.\n\n**FRINGE** \u2014Cut a stiff cardboard 8 inches long. Wind thread 8 times around card and cut at one end. Double these 16-inch strands to form a lp. Insert hook up from underneath thru a 4-ch sp on Edge, catch lp and pull thru, pass ends thru lp and pull tight. Repeat in each sp around both sides and bottom of Spread. Then knot tog. 8 strands of 2 adjacent knots, \u00bd inch down from 1st row of knots. Continue thus along entire fringe. Comb out fringe and trim evenly.\n\nStretch and pin Spread right-side-down in true shape on a padded board or table, or on curtain or quilting frames. Steam and press dry thru a cloth. If stretched on frames, lay over an ironing board, and steam and press dry in sections until completed.\n\n_Berkeley Square_\n\n**The classic square motif with its placid dignity has a stately air that is at home in both modern and traditional settings.**\n\n**MATERIALS:**\n\n**CLARK'S O.N.T. OR J. 4 P. COATS  \nBIG BALL BEST SIX CORD MERCERIZED CROCHET, in size 20**\n\n**SINGLE SIZE** | **DOUBLE SIZE**\n\n---|---\n\n65 balls of White or Ecru. | 81 balls of White or Ecru.\n\nMILWARD'S STEEL CROCHET HOOK NO. 8 or 9.\n\n**GAUGE:**\n\nEach block measures about 6 inches square before blocking. For a single size spread, about 74 \u00d7 110 inches, make 12 \u00d7 18 blocks. For a double size spread, about 92 \u00d7 110 inches, make 15 \u00d7 18 blocks.\n\n**BLOCK** . . . Starting at center, ch 16, join with sl st to form ring. **1st rnd:** Ch 3, 31 d c in ring. Join to 3rd st of ch-3. **2nd rnd:** Ch 6, * skip 1 d c, d c in next d c, ch 3. Repeat from * around. Join to 3rd st of ch-6 (16 sps). **3rd rnd:** Into each ch-3 sp make s c, half d c, 3 d c, half d c and s c (a petal). **4th rnd:** Sl st in each st to 2nd d c of 1st petal incl., ch 11, tr tr in same place as sl st; * (ch 4, d c in 2nd d c of next petal) 3 times; ch 4, into 2nd d c of next petal make tr tr, ch 5 and tr tr. Repeat from * around. Join last ch-4 with sl st to 6th ch of ch-11 first made. **5th rnd:** * Into corner loop make 4 s c, ch 1 and 4 s c; 5 s c in each of next 4 loops. Repeat from * around; join. **6th and 7th rnds:** S c in each s c around, making s c, ch 1 and s c in each corner; join. **8th rnd:** Sl st to ch-1 incl. at corner, ch 8, d c in same place as sl st; * (ch 3, skip 2 s c, d c in next s c) 10 times; ch 3; into corner ch-1 make d c, ch 5 and d c. Repeat from * around, ending with ch 3, sl st in 3rd st of ch-8.\n\n**9th rnd:** Sl st in loop, ch 3; in corner loop make d c, 2 tr, ch 3, 2 tr and 2 d c; * s c in next sp; (into next sp make 2 d c, tr, ch 3, tr and 2 d c; s c in next sp) 5 times; in corner loop make 2 d c, 2 tr, ch 3, 2 tr and 2 d c. Repeat from * around; join. **10th rnd:** Sl st in each st to within corner ch-3 loop, sl st in 1st st of ch-3, s c in loop, * ch 7, s c in next loop (between tr); then (ch 6, s c in next loop) 4 times; ch 7, s c in corner loop. Repeat from * around; join. **11th rnd:** Ch 2 (to count as s c and ch-1), s c in same place as sl st, * 6 s c in next loop, s c in next s c; (7 s c in next loop, s c in next s c) 4 times; 7 s c in next loop, in corner s c make s c, ch 1 and s c. Repeat from * around; join. **12th, 13th and 14th rnds:** S c in each s c around, making s c, ch 1 and s c in each corner. **15th rnd:** Sl st in each st to ch-1 at corner incl., ch 3, d c in same place as sl st, ch 3, 2 d c in same place. * (Ch 2, skip 2 s c, d c in next 2 d c) 13 times; ch 2, into corner ch-1 make 2 d c, ch 3 and 2 d c. Repeat from * around; join. **16th rnd:** Sl st in next d c and in corner sp, ch 3, d c in same place as last sl st, ch 3, 2 d c in same place; * (ch 2, 2 d c in next sp) 14 times; ch 2, into corner ch-3 sp make 2 d c, ch 3 and 2 d c. Repeat from * around; join.\n\n**17th rnd:** Sl st in next d c and in corner sp; ch 3, make d c, ch 3 and 2 d c in same place as last sl st. * (Ch 2, 2 d c in next sp) 4 times; (d c in next 2 d c, 2 d c in next sp) 8 times; (ch 2, 2 d c in next sp) 3 times; ch 2, into corner ch-3 make 2 d c, ch 3 and 2 d c. Repeat from * around; join. **18th rnd:** Sl st in next d c and in corner sp, ch 3; make d c, ch 3 and 2 d c in same place as last sl st. * (Ch 2, 2 d c in next sp) 4 times; ch 2, skip 2 d c, d c in next 30 d c; (ch 2, 2 d c in next sp) 4 times; ch 2, into corner ch-3 make 2 d c, ch 3 and 2 d c. Repeat from * around; join. **19th rnd:** Sl st in next d c and in corner sp, ch 3; make d c, ch 3 and 2 d c in same place as last sl st. * (Ch 2, 2 d c in next sp) 5 times; ch 2, skip 2 d c, d c in next 26 d c; (ch 2, 2 d c in next sp) 5 times; ch 2, into corner ch-3 make 2 d c, ch 3 and 2 d c. Repeat from * around. Fasten off. This completes one block.\n\nMake the necessary number of blocks and sew them together on wrong side with neat over-and-over stitches. Block to measurements given.\n\n**MATERIALS:**\n\n**CLARK'S O.N.T. or J. & P. COATS BEST SIX CORD MERCERIZED CROCHET, size 20:**\n\n**SINGLE SIZE** | **DOUBLE SIZE**\n\n---|---\n\nSMALL BALL: | SMALL BALL:\n\nCLARK'S O.N.T.\u201499 balls, | CLARK'S O.N.T.\u2014135 balls,\n\nOR | OR\n\nJ. & P. COATS\u201457 balls. | J. & P. COATS\u201478 balls.\n\nBIG BALL: | BIG BALL:\n\nJ. & P. COATS\u201433 balls. | J. & P. COATS\u201445 balls.\n\nSteel crochet hook No. 8 or 9.\n\n**GAUGE:** Each motif measures 4\u00bc inches square before blocking. For single size, spread about 72 \u00d7 108 inches including edging, make 15 \u00d7 24 motifs; for double size spread about 90 \u00d7 108 inches including edging, make 20 \u00d7 24 motifs.\n\n_Nocturne_\n\n**FIRST MOTIF** . . . Ch 7, join with sl st. **1st rnd:** Ch 4 (to count as 1 tr), 23 tr in ring. Join with sl st in top st of 1st ch-4. **2nd rnd:** Ch 4, tr in next 2 tr, * ch 5, tr in next 3 tr. Repeat from * around, joining last ch-5 to top st of 1st ch-4. **3rd rnd:** Sl st in next st, ch 4, tr in next tr, 3 tr in next loop, * ch 6, skip next tr, tr in next 2 tr, 3 tr in next loop. Repeat from * around. Join. **4th rnd:** Sl st in next st, ch 4, tr in next 3 tr, 3 tr in next loop, * ch 7, skip 1 tr, tr in next 4 tr, 3 tr in next loop. Repeat from * around. Join. **5th rnd:** Sl st in next st, ch 4, tr in next 5 tr, 3 tr in next loop, * ch 11, skip next tr, tr in next 6 tr, 3 tr in next loop. Repeat from * around. Join. **6th rnd:** Sl st in next st, ch 4, tr in next 4 tr, * ch 8, in center st of loop below make tr, ch 2, tr, ch 5, tr, ch 2 and tr; ch 8, skip 1 tr, tr in next 5 tr, ch 8, sc in next loop, ch 8, skip 1 tr, tr in next 5 tr. Repeat from * around. Join. **7th rnd:** Sl st in next st, ch 4, tr in next 2 tr, * ch 8, sc in next loop, ch 8, in next ch-5 sp (corner sp) make tr, ch 2, tr, ch 6, tr, ch 2 and tr; ch 8, sc in next loop, ch 8, skip next tr, tr in next 3 tr, ch 8, sc in next loop, ch 8, sc in next loop, ch 8, skip next tr, tr in next 3 tr. Repeat from * around. Join and fasten off.\n\n**SECOND MOTIF** . . . Work 1st 6 rnds as for 1st motif. **7th rnd:** Sl st in next st, ch 4, tr in next 2 tr, ch 8, sc in next loop, ch 8, in next ch-5 sp make tr, ch 2 and tr; ch 3, sl st in corner sp on 1st motif, ch 3, in same sp on 2nd motif where last tr was made make tr, ch 2 and tr; ch 4, sl st in next loop on 1st motif, ch 4, sc in next loop on 2nd motif, ch 4, sl st in next loop on 1st motif, ch 4, skip next tr, tr in next 3 tr, (ch 4, sl st in next loop on 1st motif, ch 4, sc in next loop on 2nd motif) twice; ch 4, sl st in next loop on 1st motif, ch 4, skip next tr on 2nd motif, tr in next 3 tr, ch 4, sl st in next loop on 1st motif, ch 4, sc in next loop on 2nd motif, ch 4, sl st in next loop on 1st motif, ch 4, in ch-5 sp on 2nd motif make tr, ch 2 and tr; ch 3, sl st to corner sp of 1st motif, ch 3, in same sp on 2nd motif where last tr was made, make tr, ch 2 and tr. Complete rnd as for 1st motif. Make necessary number of motifs, joining adjacent sides as 2nd motif was joined to 1st (wherever 4 corners meet, join 3rd and 4th corners to joining of other corners).\n\n**EDGING** . . . Attach thread to 1st joining preceding a corner. Ch 1, sc in same place, (ch 10, sc in next loop) 7 times; ch 10, in corner sp make tr, ch 3, tr, ch 5, tr, ch 3 and tr; * (ch 10, sc in next loop) 7 times; ch 10, sc in joining of motifs. Repeat from * around, turning corners as 1st corner was turned and joining last ch-10 to 1st sc made. **2nd to 6th rnds incl:** Sl st to center of next loop, ch 1, sc in same loop, * ch 10, sc in next loop. Repeat from * around and continue turning corners as before. **7th rnd:** Sl st to center of next loop, ch 4, 2 tr in same loop, * ch 8, sc in next loop, ch 8, 3 tr in next loop. Repeat from * around, making 3 tr in corner sps and joining last ch-8 to 4th st of 1st ch-4. **8th rnd:** Sl st in next 2 sts, ch 4, 2 tr in next loop, ch 8, sc in same loop, * sc in next loop, ch 8, 2 tr in same loop, tr in next tr, ch 5, sc in 5th ch from hook (a p made), skip 1 tr, tr in next tr, 2 tr in next loop, ch 8, sc in same loop. Repeat from * 2 more times. For corner make sc in next loop, ch 8, 2 tr in next loop, in next tr make tr, p and tr; tr in next tr, in next tr make tr, p and tr; 2 tr in next loop, ch 8, sc in same loop; ** sc in next loop, ch 8, 2 tr in loop, tr in next tr, p, skip 1 tr, tr in next tr, 2 tr in next loop, ch 8, sc in same loop. Repeat from ** around, turning corners as 1st corner was turned. Join and fasten off.\n\n_Gracie Square_\n\n**MATERIALS:**\n\n**CLARK'S O.N.T. MERCERIZED BEDSPREAD COTTON**\n\n**SINGLE SIZE** | **DOUBLE SIZE**\n\n---|---\n\n32 balls. | 39 balls.\n\nSteel crochet hook No. 5.\n\n**GAUGE:** Each block measures 4 inches square before blocking. For a single spread 72 \u00d7 105 inches, make 18 \u00d7 26 blocks; for a double spread 90 \u00d7 105 inches, make 22 \u00d7 26 blocks.\n\n**BLOCK** . . . Starting at center ch 10. Join. **1st rnd:** Ch 1, * sc in ring, ch 5. Repeat from * 3 more times. **2nd rnd:** * Sc in next sc, 2 sc in next sp, ch 5. Repeat from * 3 more times. **3rd, 4th and 5th rnds:** * Sc in each sc around, 2 sc in sp, ch 5. Repeat from * 3 more times. **6th rnd:** Same as 5th rnd making ch 6 (instead of ch 5). **7th rnd:** * Skip next sc, sc in each sc around to within last sc of group, ch 6, sc in next loop, ch 6. Repeat from * 3 more times. **8th rnd:** * Skip next sc, sc in each sc around to within last sc of group, (ch 7, sc in next loop) twice; ch 7. Repeat from * 3 more times. **9th rnd:** * Skip next sc, sc in each sc around to within last sc of group, (ch 7, sc in next loop) 3 times; ch 7. Repeat from * 3 more times. **10th rnd:** * Skip next sc, sc in each sc around to within last sc of group, (ch 7, sc in next loop) 4 times; ch 7. Repeat from * around, ending with sc in last loop. **11th rnd:** Ch 8, dc in next loop, ch 5, dc in next loop, * ch 5, in center st of next loop make dc, ch 9 and dc; (ch 5, dc in next loop) 4 times. Repeat from * around. Join last ch-5 with sl st to 3rd st of ch-8. **12th rnd:** Ch 3, dc in each st around, making 3 dc in center st of each corner loop. Join and fasten off.\n\nMake necessary number of blocks and sew them together on wrong side with neat over-and-over stitches. Block to measurements given.\n\n_Prophecy_\n\n**MATERIALS:**\n\n**J. & P. COATS KNIT-CRO-SHEEN**\n\n**SINGLE SIZE** | **DOUBLE SIZE**\n\n---|---\n\n50 balls of White or Ecru,  \nor 80 balls of any color. | 62 balls of White or Ecru,  \nor 98 balls of any color.\n\nSteel crochet hook No. 7 or 8.\n\n**GAUGE:** Each motif measures about 5\u00be inches from point to opposite point, and about 5 inches from side to opposite side before blocking. For single size spread about 72 \u00d7 105 inches, make 348 motifs; for double size spread about 90 \u00d7 105 inches, make 430 motifs.\n\n**MOTIF** . . . Starting at center, ch 10. Join with sl st. **1st rnd:** Ch 3 (to count as dc), 23 dc in ring. Join with sl st in top st of 1st ch-3. **2nd rnd:** Ch 4 (to count as dc and ch 1), * dc in next dc, ch 1. Repeat from * around. Join with sl st to 3rd st of ch-4. **3rd rnd:** Sl st in next sp, ch 3, in same sp make 2 dc, ch 3 and 3 dc (starting shell made); * ch 2, skip 1 sp, dc in next sp, ch 2, skip 1 sp, in next sp make 3 dc, ch 3 and 3 dc (another shell made). Repeat from * around. Join. **4th rnd:** Sl st in next 2 sts, sl st in next sp, ch 3, make starting shell in same sp; * (ch 2, dc in next sp) twice; ch 2, make a shell in sp of shell below. Repeat from * around. Join. **5th rnd:** Sl st in next 2 sts, sl st in next sp, ch 3, make starting shell in same sp; * ch 2, dc in next sp, ch 2, 5 dc in next sp, drop loop from hook, insert hook in top of 1st dc of this group and pull dropped loop through (pc st made); ch 2, dc in next sp, ch 2, make shell in sp of shell below. Repeat from * around. Join.\n\n**6th rnd:** Sl st in next 2 sts, sl st in next sp, ch 3, make starting shell in same sp; * ch 2, dc in next sp, (ch 2, pc st in next sp) twice; ch 2, dc in next sp, ch 2, shell in sp of shell below. Repeat from * around. Join. **7th rnd:** Sl st in next dc, ch 3, pc st in top of next dc, in next sp make dc, ch 3 and dc; pc st in top of next dc, dc in next dc, * ch 2, dc in next sp, ch 2, (pc st in next sp, ch 2) 3 times; dc in next sp, ch 2, skip 1 dc, dc in next dc, pc st in top of next dc, in next sp make dc, ch 3 and dc; pc st in top of next dc, dc in next dc. Repeat from * around. Join. **8th rnd:** Ch 3, pc st in same place as sl st, dc in top of pc st below, dc in next dc, in next sp make dc, ch 3 and dc; dc in next dc, dc in top of pc st below, pc st in top of next dc, * (dc in next sp, ch 2) twice; (pc in next sp, ch 2) twice; dc in next sp, ch 2, dc in next sp, pc st in top of next dc, dc in top of pc st below, dc in next dc, in next sp make dc, ch 3 and dc; dc in next dc, dc in top of next pc st, pc st in top of next dc. Repeat from * around, ending with dc in last sp. Join. **9th rnd:** Ch 3, dc in top of pc st, dc in next 2 dc, in next sp make dc, ch 3 and dc; dc in next 3 dc, dc in top of pc st, pc st in next dc, * (dc in next sp, ch 2) twice; pc st in next sp, (ch 2, dc in next sp) twice; pc st in next dc, dc in top of next pc st, dc in next 3 dc, in next sp make dc, ch 3 and dc; dc in next 3 dc, dc in top of next pc st, pc st in next dc. Repeat from * around, ending with dc in last sp, pc st in last dc. Join and fasten off.\n\nMake necessary number of motifs and sew together on wrong side with neat over-and-over stitches, joining them as in diagram for joining hexagon motifs. For single size spread make 17 motifs from A to B and 20 motifs from A to C; for double size spread make 21 motifs from A to B and 20 motifs from A to C.\n\n**DIAGRAM FOR JOINING HEXAGON MOTIFS**\n\n_Doilies_\n\n_Sea Shells_\n\n_The shell motif echoes the beauty of the sea_\n\n. . . _against a background of blue linen_.\n\n**MATERIALS:**\n\n**J. & P. Coats Best Six Cord Mercerized Crochet,** Art. A. 104, Size 50: 1 ball of White . . . **Milwards Steel Crochet Hook** No. 12 . . . A piece of blue linen, 9\u00bd inches in diameter.\n\n**Centerpiece measures 18 inches in diameter.**\n\nMake a narrow hem all around linen. **1st rnd:** Attach thread to edge of linen and sc closely around, having 390 sc on rnd (about 14 sc to 1 inch). Join with sl st to first sc. **2nd rnd:** Sc in same place as sl st, sc in next 3 sc, * ch 4, skip 2 sc, sc in next 4 sc. Repeat from * around. Join (65 ch-4 sps on rnd). **3rd rnd:** Ch 6, * sc in next 2 sc, ch 3, 4 dc in next loop, ch 3, skip next sc. Repeat from * around, ending with 3 dc in last loop, sl st in 3rd ch of ch-6. **4th rnd:** * Ch 4, sc in next 4 dc. Repeat from * around. Join. **5th rnd:** Sl st in next sp, ch 3, 3 dc in same sp, * ch 3, skip next sc, sc in next 2 sc, ch 3, 4 dc in next sp. Repeat from * around. Join to top of starting ch-3. **6th rnd:** Sc in same place as sl st, sc in next 3 dc, * ch 4, sc in next 4 dc. Repeat from * around. Join. **7th rnd:** Repeat 3rd rnd. **8th rnd:** Repeat 4th rnd. **9th rnd:** Ch 3, make 3 dc in each sp and dc in each sc around. Join to top of ch-3 (456 dc on rnd, counting starting ch-3 as 1 dc). **10th rnd:** * Sc in next 13 dc, ch 8, skip next 6 dc, dc in next 13 dc, ch 8, skip next 6 dc. Repeat from * around. Join. **11th rnd:** Sl st in next sc, sc in same sc and in next 11 sc (1 sc decreased on this sc-section), * ch 8, (dc in next dc, ch 1) 12 times; dc in next dc, ch 8, skip next sc, sc in next 12 sc. Repeat from * around. Join. **12th rnd:** Sl st in next sc, sc in same place as sl st, * sc in each remaining sc on this section, ch 8, (dc in next dc, ch 2) 12 times; dc in next dc, ch 8, skip 1 sc. Repeat from * around. Join. **13th to 22nd rnds incl:** Work as for 12th rnd, having 1 sc less on each sc-section on every rnd and having 1 more ch between dc's of each de-section on every other rnd (ch 7 on 22nd rnd). **23rd rnd:** Sl st to center of next sp, sc in same sp, * (ch 7, sl st in 4th ch from hook\u2014picot made\u2014ch 3, sc in next sp) 13 times; sc in next sp, ch 1, sc in first sp of next scallop. Repeat from * around. Join and break off.\n\nStarch lightly and press.\n\n_Shining Star_\n\n**ROYAL SOCIETY SIX CORD CORDICHET,** Large Ball,\n\n_Size 20, 8 balls of White or Ecru_.\n\nSteel Crochet Hook No. 9.\n\n**Doily measures 30 inches in diameter.**\n\nStarting at center, ch 10. Join with sl st to form ring. **1st rnd:** Ch 7, (tr in ring, ch 3) 7 times. Join to 4th ch of ch-7. **2nd rnd:** Ch 4, * 4 tr in next sp, tr in next tr. Repeat from * around. Join. **3rd rnd:** Sc in same place as sl st, * ch 5, sc in next tr. Repeat from * around. Join. **4th, 5th and 6th rnds:** Sl st to center of next loop, sc in same loop, * ch 7, sc in next loop. Repeat from * around. Join. **7th rnd:** Sl st to center of next loop, ch 4, 2 tr in same loop, * (ch 7, sc in next loop) 4 times; ch 7, 3 tr in next loop. Repeat from * around. Join. **8th rnd:** Sl st in next tr, ch 4, in same tr make 2 tr, ch 3 and 3 tr (shell made); * ch 2, 3 tr in next loop, (ch 7, sc in next loop) 3 times; ch 7, 3 tr in next loop, ch 2, skip 1 tr, in next tr make 3 tr, ch 3 and 3 tr (another shell made). Repeat from * around. Join. **9th rnd:** Sl st in next 2 tr, sl st in next sp, ch 4, in same sp make 2 tr, ch 3 and 3 tr (shell made over shell); * ch 2, skip next sp, tr in next 3 tr, 3 tr in next loop, (ch 7, sc in next loop) twice; ch 7, 3 tr in next loop, tr in next 3 tr, ch 2, shell in sp of next shell. Repeat from * around. Join. **10th rnd:** * Shell over shell, ch 2, skip next sp, tr in next 6 tr, 3 tr in next loop, ch 7, sc in next loop, ch 7, 3 tr in next loop, tr in next 6 tr, ch 2. Repeat from * around. Join. **11th rnd:** * Shell over shell, ch 2, skip next sp, tr in next 9 tr, 3 tr in next loop, ch 2, 3 tr in next loop, tr in the next 9 tr, ch 2. Repeat from * around. Join.\n\n**12th rnd:** * Shell over shell, ch 2, skip next sp, tr in next 12 tr, 3 tr in next sp, tr in next 12 tr, ch 2. Repeat from * around. Join. **13th rnd:** * Shell over shell, ch 2, skip next sp, tr in next 13 tr, in next tr make tr, ch 5 and tr; tr in next 13 tr, ch 2. Repeat from * around. Join. **14th rnd:** * Shell over shell, ch 2, skip next sp, tr in next 12 tr, ch 2, 9 tr in next sp, ch 2, tr in next 12 tr, ch 2. Repeat from * around. Join. **15th rnd:** * Shell over shell, ch 2, skip next sp, tr in next 10 tr, ch 5, skip next 2 tr, (sc in next tr, ch 5) 9 times; skip 2 tr, tr in next 10 tr, ch 2. Repeat from * around. Join. **16th rnd:** * Shell over shell, ch 2, skip next sp, tr in next 8 tr, ch 7, skip next sp, (sc in next loop, ch 7) 8 times; skip 2 tr, tr in next 8 tr, ch 2. Repeat from * around. Join. **17th rnd:** * Shell over shell, ch 2, skip next sp, tr in next 8 tr, 3 tr in next loop, (ch 7, sc in next loop) 7 times; ch 7, 3 tr in next loop, tr in next 8 tr, ch 2. Repeat from * around. Join. **18th rnd:** * Shell over shell, ch 2, skip next sp, tr in next 11 tr, 3 tr in next loop, (ch 7, sc in next loop) 6 times; ch 7, 3 tr in next loop, tr in next 11 tr, ch 2. Repeat from * around. Join. **19th rnd:** * Shell over shell, ch 2, skip next sp, tr in next 14 tr, 3 tr in next loop, (ch 7, sc in next loop) 5 times; ch 7, 3 tr in next loop, tr in next 14 tr, ch 2. Repeat from * around. Join. **20th rnd:** * Shell over shell, ch 2, skip next sp, tr in next 17 tr, 3 tr in next loop, (ch 7, sc in next loop) 4 times; ch 7, 3 tr in next loop, tr in next 17 tr, ch 2. Repeat from * around. Join. **21st rnd:** * Shell over shell, ch 2, skip next sp, tr in next 20 tr, 3 tr in next loop, (ch 7, sc in next loop) 3 times; ch 7, 3 tr in next loop, tr in next 20 tr, ch 2. Repeat from * around. Join. **22nd rnd:** * Shell over shell, ch 2, skip next sp, tr in next 23 tr, 3 tr in next loop, (ch 7, sc in next loop) twice; ch 7, 3 tr in next loop, tr in next 23 tr, ch 2. Repeat from * around. Join. **23rd rnd:** * Shell over shell, ch 2, skip next sp, tr in next 26 tr, 3 tr in next loop, ch 7, sc in next loop, ch 7, 3 tr in next loop, tr in next 26 tr, ch 2. Repeat from * around. Join. **24th rnd:** * Shell over shell, ch 2, skip next sp, tr in next 29 tr, 3 tr in each of next 2 sps, tr in next 29 tr, ch 2. Repeat from * around. Join.\n\n**25th rnd:** * Shell over shell, ch 2, skip next sp, tr in next 3 tr, (ch 2, skip 2 tr, tr in next tr) 9 times; tr in next 2 tr, ch 5, tr in next 3 tr, (ch 2, skip next 2 tr, tr in next tr) 9 times; tr in next 2 tr, ch 2. Repeat from * around. Join. **26th rnd:** * Shell over shell, ch 2, skip next sp, 2 tr in next tr, tr in next 2 tr, (ch 2, tr in next tr) 7 times; ch 2, skip next sp, 2 tr in next sp, 9 tr in next sp, ch 2, 3 tr in next sp, ch 2, skip next tr, (tr in next tr, ch 2) 7 times; tr in next 2 tr, 2 tr in next tr, ch 2. Repeat from * around. Join. **27th rnd:** * Shell over shell, ch 2, skip next sp, 2 tr in next tr, tr in next 3 tr, (ch 2, tr in next tr) 6 times; ch 2, skip next sp, 3 tr in next sp, ch 5, skip next sp, (sc in next tr, ch 5) 9 times; skip next sp, 3 tr in next sp, ch 2, skip next tr, (tr in next tr, ch 2) 6 times; tr in next 3 tr, 2 tr in next tr, ch 2. Repeat from * around. Join. **28th rnd:** * Shell over shell, ch 2, skip next sp, 2 tr in next tr, tr in next 4 tr, (ch 2, tr in next tr) 5 times; ch 2, skip next sp, 3 tr in next sp, ch 7, skip next sp, (sc in next loop, ch 7) 8 times; skip next sp, 3 tr in next sp, ch 2, skip next tr, (tr in next tr, ch 2) 5 times; tr in next 4 tr, 2 tr in next tr, ch 2. Repeat from * around. Join. **29th rnd:** * Shell over shell, ch 2, skip next sp, 2 tr in next tr, tr in next 5 tr, (ch 2, tr in next tr) 6 times; ch 2, 3 tr in next sp, (ch 7, sc in next loop) 7 times; ch 7, 3 tr in next sp, ch 2, skip 2 tr, (tr in next tr, ch 2) 6 times; tr in next 5 tr, 2 tr in next tr, ch 2. Repeat from * around. Join. **30th rnd:** Sl st to sp of shell, ch 4, in same sp make 2 tr, (ch 3, 3 tr) twice; * ch 2, skip next sp, 2 tr in next tr, tr in next 6 tr, (ch 2, tr in next tr) 7 times; ch 2, 3 tr in next loop, (ch 7, sc in next loop) 6 times; ch 7, 3 tr in next loop, ch 2, skip 2 tr, (tr in next tr, ch 2) 8 times; tr in next 6 tr, 2 tr in next tr, ch 2, in sp of next shell make (3 tr, ch 3) twice and 3 tr. Repeat from * around. Join.\n\nNow work in rows as follows: **1st row:** Sl st in next 2 tr, across next sp and in next 3 tr, sl st in next sp, ch 4, in same sp make 2 tr, ch 3 and 3 tr; ch 1, skip next sp and 2 tr, tr in next 6 tr, (2 tr in next sp, tr in next tr) 8 times; tr in next 2 tr, 3 tr in next loop, (ch 7, sc in next loop) 5 times; ch 7, 3 tr in next loop, tr in next 3 tr, (2 tr in next sp, tr in next tr) 8 times; tr in next 5 tr, ch 1, skip next sp, shell in next sp. Turn. **2nd row:** Sl st in next 3 tr, sl st in sp, shell over shell, ch 1, skip 3 tr, tr in each tr across, 3 tr in next loop, (ch 7, sc in next loop) 4 times; ch 7, 3 tr in next loop, tr in each tr across to within last 3 tr, ch 1, shell over shell. Turn. **3rd and 4th rows:** Repeat last row having 1 loop less on pineapple on each row. **5th row:** Shell over shell, ch 1, skip 3 tr, tr in each tr across, 3 tr in next loop, ch 7, sc in next loop, ch 7, 3 tr in next loop, tr in each tr across to within last 3 tr, ch 1, shell over shell. Turn. **6th row:** Shell over shell, ch 1, skip 3 tr, tr in each tr across,. 3 tr in next loop, ch 1, 3 tr in next loop, tr in each tr across to within last 3 tr, ch 1, shell over shell. Turn. **7th row:** Shell over shell, ch 1, skip 3 tr, tr in each tr across, 3 tr in ch-1 sp, tr in each tr across to within last 3 tr, ch 1, shell over shell. Turn. **8th to 22nd rows incl:** Shell over shell, ch 1, skip 3 tr, tr in each tr across to center tr of 3-tr group, 3 tr in center tr, tr in each remaining tr across to within last 3 tr, ch 1, shell over shell. Turn. **23rd row:** Shell over shell, skip 3 tr, 3 tr in next tr, shell over shell. Turn. **24th row:** Shell over next 2 shells. Break off. Attach thread to next sp on 30th rnd and work remaining points to correspond. Starch lightly and press.\n\n_Floral Splendor_\n\n**MATERIALS:**\n\n**Clark's O.N.T. Best Six Cord Mercerized Crochet,** Art. B.4, Size 30: 8 balls of White . . . **Milwards Steel Crochet Hook** No. 10 . . . A piece of linen, 6 inches in diameter.\n\n**Centerpiece measures 16 inches in diameter.**\n\n**FIRST MOTIF . . .** Starting at center, ch 8. Join with sl st to form ring. **1st rnd:** Ch 9, (tr in ring, ch 5) 7 times. Join with sl st to 4th ch of ch-9. **2nd rnd:** 8 sc in each ch-5 sp around. Join. **3rd rnd:** Ch 3, dc in next sc and in each sc around. Join. **4th rnd:** Ch 4, holding back on hook the last loop of each tr make tr in next 2 dc, thread over and draw through all loops on hook (a 2-tr cluster made); * ch 6, skip next dc, 3-tr cluster over next 3 dc. Repeat from * around, ending with ch 6, sl st in top of ch-4 (16 clusters on rnd). **5th rnd:** 8 sc in each sp around. Join. **6th rnd:** Sl st in next sc, ch 3, dc in next 11 sc, * ch 5, skip 4 sc, dc in next 12 sc. Repeat from * around, ending with ch 5. Join to top of ch-3. **7th rnd:** Ch 3, dc in next 11 dc, * ch 5, sc in next sp, ch 5, dc in next 12 dc. Repeat from * around. Join. **8th rnd:** Ch 3, * dc in each dc of this dc section, (ch 5, sc in next loop) twice; ch 5. Repeat from * around. Join. **9th rnd:** Sl st in next 2 dc, ch 3, dc in next 7 dc, * (ch 5, sc in next loop) 3 times; ch 5, skip next 2 dc, dc in next 8 dc. Repeat from * around. Join. **10th rnd:** Sl st in next 2 dc, ch 3, dc in next 3 dc, * (ch 5, sc in next loop) 4 times; ch 5, skip next 2 dc, dc in next 4 dc. Repeat from * around (8 points on rnd). Join and break off.\n\n**SECOND MOTIF . . .** Work as for First Motif until 9 rnds have been completed. **10th rnd:** Work as for 10th rnd of First Motif until 7 points have been completed, ch 2, sl st in last ch-5 loop on any ch-5 loop section on First Motif, (ch 2, sc in next loop on Second Motif, ch 2, sl st in next loop on First Motif) 4 times; ch 2, skip next 2 dc, dc in next 4 dc and complete rnd as for First Motif (no more joinings).\n\nMake 5 more Motifs, joining adjacent sides as Second Motif was joined to First Motif and leaving 20 loops free on outer edge and 10 loops free on inner edge between joinings. Join first and last motifs.\n\n**OUTER EDGE . . . 1st rnd:** Attach thread to ch-2 sp preceding joining on any motif, sc in same place, * sc in corresponding ch-2 sp following joining on next motif, (ch 5, sc in next loop) 20 times; ch 5, sc in ch-2 sp preceding joining. Repeat from * around. Join. **2nd rnd:** Make 7 sc in each ch-5 loop around. Join. **3rd rnd:** Sl st in next 2 sc, ch 4, 2-tr cluster over next 2 sc, * ch 6, skip last 2 sc of this 7-sc group and next 2 sc of next 7-sc group, make a 3-tr cluster over next 3 sc. Repeat from * across motif, ending with a 3-tr cluster over last 7-sc group on motif, make a 3-tr cluster over first 7-sc group on next motif (thus omitting the ch-6 loop) and complete rnd to correspond. Join to tip of first cluster. **4th rnd:** In each ch-6 loop around make 4 sc, ch 5, sl st in 5th ch from hook (picot made) and 4 sc. Join and break off.\n\n**INNER EDGE . . . 1st rnd:** Attach thread to any joining between motifs, * (ch 5, sc in next loop) 10 times; ch 5, sc in next joining. Repeat from * around. Join. **2nd rnd:** 5 sc in each loop around. Join with sl st to first sc. **3rd rnd:** Sl st in next 12 sc, ch 9, * (tr in center sc of next loop, ch 5) twice; dc in center sc of next loop, ch 5, (tr in center sc of next loop, ch 5) twice; holding back on hook the last loop of each d tr, make a d tr in center sc of next loop, skip 2 loops on next motif, d tr in center sc of next loop, thread over and draw through all loops on hook (joint d tr made), ch 5. Repeat from * around. Join last d tr to 5th ch of ch-9. Break off.\n\nPin in place on linen and trace outline of center. Cut linen leaving \u00bc inch for hem. Roll hem and sew neatly in place. Sew inner edge to linen. Starch lightly and press.\n\n_Trio Tricks_\n\n**MATERIALS** \u2014DAISY Mercerized Crochet Cotton size 30:\u20141 skein White, Cream or Ecru. Crochet hook size 12. Size\u20149 \u00bd\u2033.\n\nCh 10, sl st in 1st st. Ch 1, 12 sc in ring. In back lps, sl st in 1st sc, ch 17, sc in 5th st from hook for a p, ch 1, a long tr (thread over 5 times and work off in twos) in same sc, (ch 5, p, ch 1, a long tr in next sc, ch 5, p, ch 1, a long tr in same sc) repeated around, joining final p-lp to 11th st of 1st lp (24 ps). **ROW 3** \u2014Ch 7. dc in next long tr, (ch 4, dc in next long tr) repeated around. Join to 3d st of 1st 7-ch. **ROW 4** \u2014Ch 1, (5 sc in next sp, 1 sc in dc) repeated around. Sl st in 1st 1-ch. **ROW 5** \u2014Ch 17, p, ch 1, a long tr in next 3d sc, (ch 5, p, ch 1, a long tr in next 3d sc) repeated around and join to 11th st of 1st lp. Repeat Row 3. **ROW 7** \u2014Ch 1, (4 sc in next sp, 1 sc in dc) repeated around. Sl st in 1st 1-ch. **ROW 8** \u2014** (Ch 4, sc in next 2d sc) 12 times, ch 2, dc in next 2 dc, sc. * Turn, (ch 4, sc in next lp) repeated across to 2d from end, ch 2, dc in end lp. Repeat from *11 times (1 lp in final row). Cut 2\u2033 long. Join to next 4th sc on center and repeat from ** around. **ROW 9** \u2014* 7 sc in tip sp of one point, (3 sc in next sp) 12 times, working over end left from point, sl st between points, (3 sc in next sp) 4 times, ch 8, turn, sl st in 12th sc up side of last point, ch 1, turn, (5 sc, ch 5, sl st in last sc for a p, and 5 sc) in lp, (3 sc in next sp) 3 times, ch 22, turn, sl st in next 9th sc up side of last point, ch 1, turn, 29 sc in lp, (3 sc in next sp) 5 times. Repeat from * around. **ROW 10** \u2014Sl st in 1st 7 sc, * ch 5, p, ch 1, a long tr (thread over 6 times and work off in twos) in 1st sc on lp between points, (ch 5, p, ch 1, a long tr in next 2d sc) 14 times, ch 5 p, ch 1, sl st in 7th sc in tip of next point. Repeat from * around. **ROW 11** \u2014Sl st to tip of point, * (ch 4, dc in next long tr) 15 times, ch 4, sc in tip of point. Repeat from * around. **ROW 12** \u2014Ch 1, (4 sc in next sp, 1 sc in next st) repeated around, sl st in 1st 1-ch. **ROW 13** \u2014Ch 13, a 2-tr-Cluster in 6th st from hook, ch 1, sc in next 10th sc (directly above a dc), * ch 6, a 2-tr-Cluster in 6th st from hook, tr in next 5th sc, ch 5, sl st in tr for a p, ch 6, a Cluster, ch 1, sc in next 5th sc. Repeat from * 5 times. Ch 6, a Cluster, dtr in sc between shells, a p, ch 6, a Cluster, ch 1, sc in next 10th sc. Repeat from * around. Join final Cluster to 7th st of 1st lp, a p and fasten off. Stretch and pin right-side-down in a true circle. Steam and press dry thru a cloth.\n\n_Cluster Stitch Doily_\n\n**Materials Required: AMERICAN THREAD COMPANY The Famous \"PURITAN\" MERCERIZED CROCHET COTTON, Article 40**\n\n3 balls White\n\nSteel crochet hook No. 7\n\n**Approximate size:** 22 inches in diameter or\n\n**\"GEM\" MERCERIZED CROCHET COTTON, Article 35,**\n\n**size 30**\n\n2 balls White\n\nSteel crochet hook No. 12\n\n**Approximate size:** 16 inches in diameter\n\n**General Notations:** A cluster st at beginning of each round is made as follows: ch 3 (to count as 1 d c), 2 more d c in same space, keeping last loop of each d c on hook, thread over and work off all loops at one time (this completes a cluster st at beginning of the round).\n\nThe regular cluster sts through 1st 12 rounds of doily are made as follows: 3 d c in same space keeping last loop of each d c on hook, thread over and work off all loops at one time.\n\nCh 9, join to form a ring, work 8 cluster sts with ch 4 between each cluster st in ring, ch 4, join in top of 1st cluster st.\n\n**2nd Round:** Sl st into loop, work 2 cluster sts with ch 3 between in same space, * ch 3, 2 cluster sts with ch 3 between in next loop, repeat from * all around, ch 3, join.\n\n**3rd Round:** Cluster st in 1st cluster st, * ch 3, cluster st in next loop, ch 3, cluster st in top of next cluster st, ch 3, skip next ch 3 loop, cluster st in top of next cluster st, repeat from * all around ending with ch 3, cluster st in next loop, ch 3, cluster st in top of next cluster st, ch 3, join.\n\n**4th Round:** Cluster st in 1st cluster st, * ch 3, 2 cluster sts with ch 3 between in next cluster st, ch 3, cluster st in next cluster st, ch 3, cluster st in next cluster st, repeat from * all around ending to correspond, ch 3, join.\n\n**5th Round:** Cluster st in 1st cluster st, ch 3, cluster st in next cluster st, * ch 3, cluster st in next loop, ch 3, 1 cluster st in each of the next 4 cluster sts with ch 3 between each cluster st, repeat from * all around ending to correspond, ch 3, join.\n\n**6th Round:** * 1 cluster st in each of the next 5 cluster sts with ch 3 between each cluster st, ch 5, repeat from * all around, join.\n\n**7th Round:** * 1 cluster st in each of the next 5 cluster sts with ch 3 between each cluster st, ch 9, repeat from * all around, join.\n\n**8th Round:** Cluster st in 1st cluster st, * ch 3, cluster st in next cluster st, ch 1, cluster st in next cluster st, ch 1, cluster st in next cluster st, ch 3, cluster st in next cluster st, ch 6, d c in center st of next loop, ch 6, cluster st in next cluster st, repeat from * all around ending to correspond, join.\n\n**9th Round:** Cluster st in 1st cluster st, * ch 3, cluster st in next cluster st, ch 1, skip 1 cluster st, cluster st in next cluster st, ch 3, cluster st in next cluster st, ch 6, d c in next d c, ch 7, d c in same space, ch 6, cluster st in next cluster st, repeat from * all around ending to correspond, join.\n\n**10th Round:** Cluster st in 1st cluster st, * ch 1, skip 1 loop, cluster st in next ch 1 space, ch 1, skip 1 loop, cluster st in next cluster st, ch 7, d c in next d c, ch 7, d c in center st of next loop, ch 7, d c in next d c, ch 7, cluster st in next cluster st, repeat from * all around ending to correspond, join.\n\n**11th Round:** Cluster st in 1st cluster, * ch 1, skip 1 cluster st, cluster st in next cluster st, ch 7, d c in next d c, ch 7, d c in next d c, ch 5, d c in same space, ch 7, d c in next d c, ch 7, cluster st in next cluster st, repeat from * all around ending to correspond, join.\n\n**12th Round:** Sl st into ch 1 space, cluster st in same space, * ch 12, s c in next d c, ch 12, skip 1 loop, s c in next ch 5 loop, ch 12, skip 1 loop, s c in next d c, ch 12, cluster st in next ch 1 space, repeat from * all around ending to correspond, join.\n\n**13th Round:** Sl st to center of next loop, ch 4 (counts as part of 1st tr c cluster st), 2 tr c in same space keeping last loop of each tr c on hook, thread over and work off all loops at one time, * ch 11, 2 tr c in next loop, ch 11, 3 tr c cluster st in next loop (tr c cluster st: 3 tr c in same space keeping last loop of each tr c on hook, thread over and work off all loops at one time), repeat from * all around ending with ch 11, 2 tr c in next loop, ch 11, join.\n\n**14th Round:** Ch 4 (always counts as part of 1st cluster st), tr c cluster st in same space, * ch 5, 5 tr c in next tr c, ch 2, tr c in same space, tr c in next tr c, ch 2, 5 tr c in same space, ch 5, 3 tr c cluster st in next cluster st, repeat from * all around ending to correspond, ch 5, join.\n\n**15th Round:** Ch 4, tr c cluster st in same space, * ch 1, tr c cluster st in same space, ch 4, 1 tr c in each of the next 5 tr c keeping last loop of each tr c on hook, thread over and work off all loops at one time, ch 5, 3 tr c in next tr c, 2 tr c in next tr c, ch 5, 1 tr c in each of the next 5 tr c keeping last loop of each tr c on hook, thread over and work off all loops at one time, ch 4, tr c cluster st in next cluster st, repeat from * all around ending to correspond, ch 4, join in 1st cluster st.\n\n**16th Round:** Ch 4, tr c cluster st in same space, * ch 4, tr c in next ch 1 space, ch 4, tr c cluster st in next tr c cluster st, ch 7, skip 2 loops, 1 s c in each of the next 5 tr c, ch 7, skip 2 loops, tr c cluster st in next tr c cluster st, repeat from * all around ending to correspond, ch 7, join.\n\n**17th Round:** Ch 4, tr c cluster st in same space, * ch 6, tr c in next tr c, ch 6, tr c cluster st in next tr c cluster st, ch 7, 1 tr c in each of the next 5 s c keeping last loop of each tr c on hook, thread over and work off all loops at one time (5 tr c cluster), ch 7, skip 1 loop, tr c cluster st in next tr c cluster st, repeat from * all around ending to correspond, ch 7, join.\n\n**18th Round:** Ch 4, tr c cluster st in same space, * ch 8, 1 tr c, ch 3, 1 tr c in next tr c, ch 8, tr c cluster st in next tr c cluster st, ch 6, s c in next 5 tr c cluster, ch 6, tr c cluster st in next tr c cluster st, repeat from * all around ending to correspond, ch 6, join.\n\nAll cluster sts are tr c cluster sts for remainder of doily and will be referred to as cluster sts only.\n\n**19th Round:** Ch 4, cluster st in same space, * ch 8, 5 tr c in next tr c, ch 2, 2 tr c in next ch 3 loop, ch 2, 5 tr c in next tr c, ch 8, cluster st in next cluster st, ch 6, s c in next s c, ch 6, cluster st in next cluster st, repeat from * all around ending to correspond, join.\n\n**20th Round:** Ch 4, cluster st in same space, ch 8, sl st in top of cluster st for picot, * ch 8, 5 tr c cluster over next 5 tr c, ch 5, 5 tr c in next tr c, ch 2, tr c in same space, tr c in next tr c, ch 2, 5 tr c in same space, ch 5, 5 tr c cluster over next 5 tr c, ch 8, cluster st in next cluster st, ch 8, sl st in top of cluster st for picot, cluster st in next cluster st (this closes the point), repeat from * all around ending with ch 8, 5 tr c cluster over next 5 tr c, ch 5, 5 tr c in next tr c, ch 2, tr c in same space, tr c in next tr c, ch 2, 5 tr c in same space, ch 5, 5 tr c cluster over next 5 tr c, ch 8, cluster st in next cluster st, join in next cluster st.\n\n**21st Round:** Sl st into picot, ch 4, cluster st in same space, ch 7, cluster st in same space, * ch 8, skip 2 loops, 5 tr c cluster over next 5 tr c, ch 5, 5 tr c in next tr c, ch 2, tr c in same space, tr c in next tr c, ch 2, 5 tr c in same space, ch 5, 5 tr c cluster over next 5 tr c, ch 8, skip 2 loops, 2 cluster sts with ch 7 between in next picot, repeat from * all around ending to correspond, join.\n\n**22nd Round:** Ch 4, cluster st in same space, * ch 5, cluster st in next loop, ch 5, cluster st in next cluster st, ch 7, skip 2 loops, 5 tr c cluster over next 5 tr c, ch 7, 3 tr c in next tr c, 2 tr c in next tr c, ch 7, 5 tr c cluster over next 5 tr c, ch 7, skip 2 loops, cluster st in next cluster st, repeat from * all around ending to correspond, join.\n\n**23rd Round:** Ch 4, cluster st in same space, ch 3, cluster st in same space, * ch 4, 2 cluster sts with ch 3 between in next cluster st, ch 4, 2 cluster sts with ch 3 between in next cluster st, ch 10, skip 2 loops, 1 s c in each of the next 5 tr c, ch 10, skip 2 loops, 2 cluster sts with ch 3 between in next cluster st, repeat from * all around ending to correspond, join.\n\n**24th Round:** Ch 4, cluster st in same space, * ch 3, cluster st in next loop, ch 3, cluster st in next cluster st, ch 4, cluster st in next cluster st, repeat from * once, ch 3, cluster st in next loop, ch 3, cluster st in next cluster st, ch 9, 5 tr c cluster over next 5 s c, ch 9, skip 1 loop, cluster st in next cluster st, repeat from first * all around ending to correspond, join.\n\n**25th Round:** Ch 4, cluster st in same space, ** ch 3, cluster st in next cluster st, ch 3, cluster st in next cluster st, * ch 4, 1 cluster st in each of the next 3 cluster sts with ch 3 between each cluster st, repeat from * once, ch 8, s c in next 5 tr c cluster, ch 8, cluster st in next cluster st, repeat from ** all around ending to correspond, join.\n\n**26th Round:** Ch 4, cluster st in same space, ch 2, cluster st in next cluster st, ch 2, cluster st in next cluster st, * ch 4, 1 tr c, ch 3, 1 tr c in next loop, ch 4, cluster st in next cluster st, ch 2, cluster st in next cluster st, ch 2, cluster st in next cluster st, repeat from * once, ch 8, s c in next s c, ch 8, cluster st in next cluster st, repeat from all around ending to correspond, join.\n\n**27th Round:** Ch 4, cluster st in same space, * skip 1 cluster st, cluster st in next cluster st, ch 4, tr c in next tr c, ch 4, tr c in next loop, ch 4, tr c in next tr c, ch 4, cluster st in next cluster st, repeat from * once, skip 1 cluster st, cluster st in next cluster st, ch 4, tr c in same space, tr c in 1st cluster st of next cluster st group, ch 4, cluster st in same space, repeat from 1st * all around ending to correspond, join in 1st cluster st.\n\n**28th Round:** Sl st between 1st 2 cluster sts, ch 11, sl st in 6th st from hook for picot, ch 7, sl st in same space, ch 6, sl st in same space, tr c between same 2 cluster sts, * ch 6, s c in next tr c, ch 6, tr c in next tr c, ch 6, sl st in top of tr c for picot, ch 7, sl st in same space, ch 6, sl st in same space (3 picot group), tr c in same space with tr c, ch 6, s c in next tr c, ch 6, tr c between next 2 cluster sts, work a 3 picot group, tr c in same space with last tr c, ch 6, s c in next tr c, ch 6, tr c in next tr c, work a 3 picot group, tr c in same space with last tr c, ch 6, s c in next tr c, ch 6, tr c between next 2 cluster sts, work a 3 picot group, tr c in same space with last tr c, ch 6, s c between next 2 tr c, ch 6, tr c between next 2 cluster sts, work a 3 picot group, tr c in same space with last tr c, repeat from * all around ending to correspond, join, cut thread.\n\n_Rock Pool_\n\n**A scalloped shell pattern that is so simple to make,**\n\n**yet has an air of fragile grace . . . . . . . .**\n\n**J. & P. COATS BIG BALL BEST SIX CORD MERCERIZED CROCHET, Art. A. 104,** Size 30: 3 balls of Ecru; or\n\n**CLARK'S BIG BALL MERCERIZED CROCHET, Art. B.34,** Size 30: 3 balls of No. 61 Ecru.\n\n**Milwards Steel Crochet Hook** No. 10.\n\n**Doily measures 15 inches in diameter, including ruffle.**\n\nStarting at center, ch 10. Join with sl st to form ring. **1st rnd:** 24 sc in ring. Join. **2nd rnd:** Ch 5, * tr in next sc, ch 1. Repeat from * around. Join with sl st to 4th ch of ch-5. **3rd rnd:** Sc in same place as sl st, * sc in next sp, sc in next tr. Repeat from * around. Join (48 sc). **4th and 5th rnds:** Repeat 2nd and 3rd rnds (96 sc on 5th rnd). **6th rnd:** Ch 4, tr in each sc around. Join. **7th rnd:** Sc in same place as sl st, sc in each tr around. Join. **8th rnd:** Sc in same place as sl st, * (ch 5, skip 2 sc, sc in next sc) twice; ch 5, skip 1 sc, sc in next sc. Repeat from * around, ending with ch 2, dc in first sc (36 loops). **9th rnd:** In loop just formed make sc, ch 3 and sc (picot made): * ch 5, in next loop make sc, ch 3 and sc (another picot). Repeat from * around, ending with ch 2, dc in first sc. **10th to 17th rnds incl:** Make a picot in loop just formed, * ch 5, make a picot in next ch-5 loop. Repeat from * around. Join as before. At end of 17th rnd, join last ch-5 with sl st to first sc. **18th rnd:** Sl st in picot. ch 3, 2 dc in same place (shell made); * picot in next ch-5 loop, 3-dc shell in next picot. Repeat from * around. Join.\n\n**19th rnd:** Sl st in next dc, picot in same place, * shell in next picot, picot in center dc of next shell. Repeat from * around. Join. **20th rnd:** Sl st in next picot, ch 3, 4 dc in same place, * picot in center dc of next shell, 5-dc shell in next picot. Repeat from * around. Join. **21st rnd:** Work as for 19th rnd, making 5 dc in each shell instead of 3 dc. **22nd rnd:** Repeat 20th rnd. **23rd rnd:** Work as for 19th rnd, making 7 dc in each shell. **24th rnd:** Sl st in next picot, ch 3, 6 dc in same place, (picot in center dc of next shell, 7-dc shell in next picot) twice; * picot in next shell, draw loop on hook out to measure \u00bc inch, thread over and draw loop through, insert hook between single and double loops and draw loop through, thread over and draw through all loops on hook (knot st made), dc in next picot, make a knot st, (picot in next shell, shell in next picot) 5 times. Repeat from * around. Join. **25th rnd:** Sl st to center dc of shell, picot in same place, (shell in next picot, picot in next shell) twice; * make a knot st, dc in next picot, make a knot st, sc under double loop of next knot st, sc in next dc, sc in next knot st, make a knot st, dc in next loop, make a knot st, (picot in next shell, shell in next picot) 4 times; picot in next shell. Repeat from * around. Join.\n\n**26th rnd:** (Shell in next picot, picot in next shell) twice; * make a knot st, dc in next loop, make a knot st, sc in next knot st, in next dc and in next knot st, make a knot st, dc in center sc of next group, make a knot st, sc in next knot st, next dc and next knot st, make a knot st, dc in next picot, make a knot st, (picot in next shell, shell in next picot) 3 times; picot in next shell. Repeat from * around. Join. **27th rnd:** Sl st to center dc of next shell, picot in same place, shell in next picot, * picot in next shell, make a knot st, dc in next picot, make a knot st, sc in next knot st, next dc and next knot st, make a knot st, dc in center sc of next group) twice; make a knot st, sc in next knot st, next dc and next knot st, make a knot st, dc in next picot, make a knot st, (picot in next shell, shell in next picot) twice. Repeat from * around. Join. **28th rnd:** Sl st in next picot, shell in same place, * picot in next shell, make a knot st, dc in next picot, (make a knot st, sc in next knot st, in next dc and next knot st, make a knot st, dc in center sc of next 3-sc group) 3 times; make a knot st, sc in next knot st, in next dc and in next knot st, make a knot st, dc in next picot, make a knot st, picot in next shell, shell in next picot. Repeat from * around. Join.\n\n**29th rnd:** Sl st to center of next shell, picot in same place, * make a knot st, dc in next picot, (make a knot st, sc in next knot st, next dc and in next knot st, make a knot st, dc in center sc of next sc group) 4 times; make a knot st, sc in next knot st, fn next dc and next knot st, make a knot st, dc in next picot, make a knot st, picot in next shell. Repeat from * around. Join. **30th rnd:** Sl st in next picot, ch 3, * (make a knot st, sc in next knot st, next dc and next knot st, make a knot st, dc in center sc of next sc group) 5 times; make a knot st, sc in next knot st, in next dc and in next knot st, make a knot st, dc in next picot. Repeat from * around. Join to top of ch-3. **31st rnd:** Sc in same place as sl st, sc in next knot st, * make a knot st, sc in next knot st, in next dc and in next knot st, make a knot st, dc in center sc of next sc group. Repeat from * around, ending with sc in last knot st. Join. **32nd rnd:** Ch 3, * make a knot st, sc in next knot st, in next dc and in next knot st, make a knot st, dc in center sc of next sc group. Repeat from * around. Join to top of ch-3. **33rd rnd:** Sc in same place as sl st, * ch 7, sc in next dc, ch 7, dc in center sc of next sc group. Repeat from * around. Join.\n\n**RUFFLE . . . 1st rnd:** 8 sc in each sp around. Join. **2nd rnd:** Sc in same place as sl st, * ch 5, skip next sc, sc in next sc. Repeat from * around, ending with ch 2, dc in first sc. **3rd to 7th rnds inch:** * Ch 5, sc in next loop. Repeat from * around, ending with ch 2, dc in dc. **8th rnd:** * Ch 6, sc in next loop. Repeat from * around, ending with ch 3, dc in dc. **9th rnd:** * Ch 7, sc in next loop. Repeat from * around, ending with ch 3, tr in dc. **10th rnd:** * Ch 8, sc in next loop. Repeat from * around, ending with ch 4, tr in tr. **11th to 14th rnds incl:** * Ch 9, sc in next loop. Repeat from * around, ending with ch 5, tr in tr. At end of 14th rnd, ch 9, sl st in tr. Break off. Starch lightly and press.\n\n_Sea Scallop_\n\nMaterials Required:\n\nAMERICAN THREAD COMPANY\n\nThe Famous \"PURITAN\" MERCERIZED STAR SPANGLED CROCHET COTTON, Article 40\n\n4 balls Silver Spangle or\n\nThe Famous \"PURITAN\" MERCERIZED CROCHET COTTON, Article 40\n\n2 balls White or color of your choice\n\nSteel Crochet hook No. 7\n\nApp. size: 19\u00bd inches in diameter for \"PURITAN\" STAR SPANGLED; 19\u00bd inches in diameter for \"PURITAN\"\n\nChain (ch) 5, slip stitch (sl st) in 1st stitch (st) of ch for picot, ch 4, 3 double crochet (dc) in 1st st of ch, repeat from beginning 3 times, join. 2nd ROUND: Ch 7, double treble crochet (d trc: 3 times over hook) in same space, * ch 2, d trc in same space, repeat from * 7 times, * 9 d trc with ch 2 between each d trc in next picot, repeat from * twice, join. 3rd ROUND: 2 single crochet (sc) in same loop, * ch 1, 2 sc in next loop, repeat from * 6 times, ch 1, * 2 sc in next loop, ch 1, repeat from * 7 times, complete round to correspond, join in 1st sc. 4th ROUND: Sl st to 1st ch 1, * ch 5, sc in next ch 1, repeat from * 5 times, ch 5, skip next ch 1, sc in next ch 1, complete round to correspond ending with ch 2, dc in 1st sc (this brings thread in position for next round). 5th to 9th ROUND: Ch 3, sc in same space, * ch 5, sc in next loop, ch 3, sc in same loop, repeat from * all around ending with dc in dc. 10th ROUND: Ch 3, dc in same space, * ch 4, 2 dc in next ch 5 loop, repeat from * all around, join in 3rd st of ch. 11th ROUND: Ch 3, dc in next dc, * 4 dc in next loop, 1 dc in each of the next 2 dc, repeat from * all around ending with 4 dc in last loop, join in 3rd st of ch. 12th ROUND: Ch 4, treble crochet (trc) in same space, 2 trc in next dc, * ch 5, skip 4 dc, sc in next dc, ch 5, sc in next dc, ch 5, skip 5 trc, 2 trc in each of the next 2 dc, repeat from beginning all around ending to correspond, join in 4th st of ch. 13th ROUND: Ch 4, trc in same space, 1 trc in each of the next 2 trc, 2 trc in next trc, * ch 5, skip next loop, sc in next loop, ch 5, sc in same loop, ch 5, 2 trc in next trc, 1 trc in each of the next 2 trc, 2 trc in last trc, repeat from * all around ending to correspond, join. 14th and 15th ROUNDS: Same as last round working 2 trc in 1st and last trc, and 1 trc in each trc between in each of the 14 trc groups. 16th ROUND: Ch 4, 3 trc in same space. 1 trc in each of the next 8 trc, 4 trc in last trc, * ch 1, skip 1 loop, sc in next loop, ch 1, 4 trc in next trc, 1 trc in each of the next 8 trc, 4 trc in last trc, repeat from * all around ending to correspond, join. 17th ROUND: Ch 5, skip 1 trc, sc in next trc, repeat from beginning twice, ch 5, skip 2 trc, sc in next trc, * ch 5, skip 1 trc, sc in next trc, repeat from * twice, sc in next trc, repeat from beginning all around. 18th ROUND: Sl st to center of loop, * ch 5, sc in next loop, ch 3, sc in same loop, repeat from * 4 times, sc in next loop, ch 4, sl st in 1st st of ch for picot, sc in next loop, repeat from 1st * all around. 19th ROUND: Sl st to center of loop, sc in same loop, * ch 3, sc in same loop, ch 5, sc in next loop, repeat from * all around ending with ch 2, dc in 1st sc. 20th to 30th ROUND: Ch 3, sc in same space, ch 5, sc in next loop, repeat from beginning all around ending with ch 2, dc in dc. 31st ROUND: Ch 4, 2 trc in same space, * ch 2, 3 trc in next ch 5 loop, repeat from * all around, ch 2, join. 32nd ROUND: Sl st to next loop, ch 8, treble treble crochet (tr trc: 4 times over hook) in same space and * ch 2, tr trc in same space, repeat from * 4 times, * ch 5, skip 1 loop, sc in next loop, ch 5, skip 1 loop, 7 tr trc with ch 2 between each tr trc in next loop, repeat from * all around ending to correspond. 33rd ROUND: Ch 4, 2 dc in 1st st of ch, sc in next tr trc, repeat from beginning 5 times, 3 sc in each of the next 2 loops, sc in next tr trc, then repeat from beginning all around ending to correspond, join, cut thread.\n\n_Three's Company_\n\nDOILY No. 2203\n\n**Materials Required\u2014AMERICAN THREAD COMPANY \"STAR\" or \"GEM\" MERCERIZED CROCHET COTTON, Size 20 or 30**\n\n1\u2014300 Yd. Ball White or Colors.\n\nSteel Crochet Hook No. 11 or 12.\n\nEach Motif measures about 4 inches. 7 motifs are required for doily illustrated. Doily measures about 10 inches through the center.\n\n**Motif.** Ch 6, join to form a ring, ch 6, d c into ring, * ch 3, d c into ring, repeat from * 3 times, ch 3, sl st into 3rd st of 6 ch loop to join.\n\n**2nd Row.** Ch 3, 2 d c in same space, * ch 5, sl st into 4th st from hook for picot, ch 1, 3 d c in next d c, repeat from * 4 times, picot loop and sl st into ch 3 to join.\n\n**3rd Row.** Ch 3, 1 d c in each of the next 2 d c, * ch 5, sl st in 4th st from hook for picot, ch 5, sl st in 4th st from hook for picot, ch 1, 1 d c in each d c, repeat from * 4 times, double picot loop and sl st in ch 3 to join.\n\n**4th Row.** Ch 3, 3 d c in center d c, 1 d c in last d c, * double picot loop, 1 d c in 1st d c, 3 d c in center d c, 1 d c in last d c, repeat from * 4 times, double picot loop and sl st in 3 ch loop to join.\n\n**5th Row.** Ch 3, 1 d c in next d c, 3 d c in center d c, 1 d c in each of the next 2 d c, * ch 7, 1 d c in each of the next 2 d c, 3 d c in center d c, 1 d c in each of the next 2 d c, repeat from * 4 times, ch 7, join in 3 ch loop.\n\n**6th Row.** Ch 1, 1 s c in each d c and 5 s c, ch 4, 5 s c over each loop.\n\n**7th Row.** Ch 10, skip 5 s c, s c in next s c, ch 10, skip 10 s c of scallop, s c in next s c, repeat from beginning all around.\n\n**8th Row.** Sl st to loop, ch 1, * 1 s c and 5 d c over loop, ch 5, 5 d c and 1 s c over same loop, s c in next s c, ch 6, tr c over next loop, ch 6, s c in next s c and repeat from * all around.\n\n**9th Row.** Sl st to 3rd d c of petal, * ch 9, s c in 3rd d c on other side of petal, ch 6, d c in tr c, ch 3, tr c in same space, ch 3, d c in same space, ch 6, s c in 3rd st of next petal and repeat from * all around.\n\n**10th Row.** 1 s c and 5 d c over next loop, ch 5, 5 d c and 1 s c over same loop, ch 5, d c in next d c, ch 5, tr c in tr c, ch 5, tr c in same, space, ch 5, d c in next d c, ch 5 and repeat from beginning all around joining row in s c of 1st petal.\n\nJoin motifs in the last row as follows, work 1 petal, ch 5, d c in d c, ch 5, tr c in tr c, ch 2, slip loop off hook, insert in center of corresponding loop of 1st motif and pull loop through (all joinings are made in same manner), ch 2, tr c in tr c of second motif, ch 5, d c in d c, ch 5, 1 s c and 5 d c over loop, ch 2, join to corresponding picot of 1st motif, ch 2, 5 d c and 1 s c over remainder of loop, ch 5, d c in next d c, ch 5, tr c in tr c, ch 2, join to corresponding loop of 1st motif, ch 2 and complete row. Join all motifs in same manner.\n\nDOILY No. 2204\n\n**Materials Required\u2014AMERICAN THREAD COMPANY \"STAR\" or \"GEM\" MERCERIZED CROCHET COTTON, Size 20 or 30**\n\n1\u2014300 Yd. Ball White, Ecru or Colors.\n\nSteel Crochet Hook No. 11 or 12.\n\nEach Motif measures about 3 inches. Doily 3 \u00d7 5 Motifs measured about 9 \u00d7 15 inches.\n\n**Motif.** Ch 8, join, ch 5, tr c into ring, * ch 2, tr c into ring, repeat from * 9 times, ch 2, sl st into 3rd st of ch to join (12 meshes).\n\n**2nd Row.** Ch 1, 3 s c in each mesh, join.\n\n**3rd Row.** Ch 1, s c in each s c taking up back loop of st only, join.\n\n**4th Row.** Ch 4, d c in next s c taking up back loop of st only, * ch 1, d c in next s c taking up back loop of st only, repeat from * all around, join in 3rd st of ch.\n\n**5th Row.** Ch 1, 2 s c in each mesh (72 s c), join.\n\n**6th Row.** Ch 4, 1 tr c in each of the next 4 s c, * ch 7, skip 1 s c, 1 tr c in each of the next 5 s c, repeat from * 10 times, ch 7, join.\n\n**7th Row.** Ch 1, 1 s c in each tr c and 4 s c, ch 4, 4 s c in each loop, join.\n\n**8th Row.** Sl st to center s c, ch 1, s c in same space, * ch 7, d c in next picot, ch 3, d c in same picot, ch 7, s c in center s c, repeat from * all around.\n\nMotifs are joined together in the last row. Work to within the last 2 points, * d c into picot, ch 1, sl st into shell of 1st motif, ch 1, d c into same picot of 2nd motif, ch 7, s c into center s c, ch 7, repeat from *. Join 3rd motif to 2nd motif and 4th motif to 1st and 3rd motif.\n\n**Joining Motif.** Fasten thread in 3 ch loop, ch 6, d c in same space, ch 5, tr c between motifs, ch 5, d c in next 3 ch space, ch 3, d c in same space and continue around joining in 4th st of ch 6.\n\n**2nd Row.** Ch 4, 1 d c in same space, ch 3, 2 d c in same space, 2 d c, ch 3, 2 d c in next 3 ch space, repeat twice. Join and break thread.\n\nDOILY No. 2205\n\n**Materials Required\u2014AMERICAN THREAD COMPANY \"STAR\" or \"GEM\" MERCERIZED CROCHET COTTON, Size 30**\n\n1\u2014300 Yd. Ball White or Colors.\n\nSteel Crochet Hook No. 11 or 12.\n\nEach Motif measures about 3\u00bd inches. 7 Motifs are required for doily illustrated which measures about 10 inches through the center.\n\n**Motif.** Ch 11, join to form a ring and work 24 s c in ring, join.\n\n**2nd Row.** Ch 4, cluster st in same st. (To make cluster st, counting the ch 4 as 1 tr c, thread over needle twice, insert in st, pull loop through, thread over and pull through 2 loops, thread over and pull through next 2 loops, thread over needle twice, insert in same st, pull through, thread over and pull through 2 loops, thread over and pull through next 2 loops, thread over and pull through all loops on needle.) * Ch 7, skip 1 s c, 3 tr c cluster st in next s c and repeat from * all around, ch 7, join to top of 1st cluster st (12 cluster sts).\n\n**3rd Row.** Sl st to center of ch 7, * ch 11, s c in center st of next loop and repeat from * all around.\n\n**4th Row.** Sl st to center st of ch 11, 6 s c over same loop, 6 s c over next loop, * ch 11, 6 s c over next loop, 6 s c over next loop and repeat from * all around, ch 11, join to 1st s c of row.\n\n**5th Row.** Ch 1, * 1 s c in each of the 1st 5 s c, skip 1 s c, 1 s c in each of the next 6 s c, ch 6, s c in center of ch 11 of previous row, ch 6 and repeat from *, join in 1st s c of row.\n\n**6th Row.** * 1 s c in each s c omitting the 1st and last s c (9 s c), ch 6, s c in next loop, s c in s c, s c in next loop, ch 6 and repeat from * all around, join.\n\n**7th Row.** Same as row 6 having 7 s c over the 9 s c of previous row and increasing 1 s c on each side of the 3 s c (5 s c).\n\n**8th Row.** Same as row 7, work 5 s c over the 7 s c of previous row and increasing 1 s c on each side of the 6 s c (7 s c).\n\n**9th Row.** Same as row 8, working 3 s c over the 5 s c of previous row and increasing 1 s c on each side of the 7 s c (9 s c).\n\n**10th Row.** * 1 s c in center of 3 s c of previous row, ch 7, 1 s c over loop, 1 s c in each of the next 4 s c, ch 7, skip 1 s c, 1 s c in each of the next 4 s c, 1 s c in loop, ch 7 and repeat from * all around, break thread.\n\nTo join motifs in last row of 2nd motif, work as follows: 1 s c in center of 3 s c of previous row, ch 7, 1 s c in loop, 1 s c in each of the next 4 s c, ch 3, join to corresponding loop of 1st motif, ch 3, skip 1 s c, 1 s c in each of the next 4 s c of 2nd motif, 1 s c in loop, ch 3, join to next loop of 1st motif, ch 3, 1 s c in center of 3 s c of 2nd motif, ch 3, join to next loop of 1st motif, ch 3, s c in next loop of 2nd motif, 1 s c in each of the next 4 s c, ch 3, join to corresponding loop of 1st motif, ch 3, skip 1 s c, 1 s c in each of the next 4 s c of 2nd motif, 1 s c in loop and finish row same as 1st motif. Join all motifs in same manner.\n\n**Edge.** Attach thread between 2 motifs, ch 5, skip 2 s c, d c in next s c, ch 5, sl st in top of d c just made for picot, ch 5, s c in next loop, ch 5, d c in s c at point, ch 5, sl st in top of d c for picot, ch 5, sl st in next loop, ch 5, skip 2 s c, d c in next s c, picot, ch 5, sl st in next loop and continue all around.\n\n_Picot Picot_\n\nMaterials Required:\n\nAMERICAN THREAD COMPANY\n\nThe Famous \"PURITAN\" MERCERIZED CROCHET COTTON, Article 40\n\n1 ball Yellow or\n\nThe Famous \"PURITAN\" STAR SPANGLED MERCERIZED CROCHET COTTON, Article 40\n\n2 balls Yellow Spangle or color of your choice\n\nSteel Crochet Hook No. 7\n\nApp size: 11\u00bd inches in diameter for \"PURITAN\"; 12\u00bd inches in diameter for \"PURITAN\" STAR SPANGLED\n\nChain (ch) 7, slip stitch (sl st) in 5th st from hook for picot, ch 7, sl st in 7th st from hook for picot, ch 5, sl st in 5th st from hook for picot, sl st in 1st picot (3 picot group), repeat from beginning 5 times, join in 1st st of ch. 2nd ROUND: Ch 15, treble treble crochet (tr trc: 4 times over hook) in same space, * ch 10, 2 tr trc with ch 10 between each tr trc in space between next 3 picot groups, repeat from * 4 times, ch 10, join in 6th st of ch. 3rd ROUND: 5 single crochet (sc), ch 5, 5 sc in each loop, join in 1st sc. 4th ROUND: Sl st to next ch 5 loop, ch 9, * double treble crochet (d trc: 3 times over hook) in same space, ch 5, repeat from * once, tr trc in same space, * ch 5, d trc in same space, repeat from * twice, ch 2, 3 sc in next loop, ch 2, d trc in next loop, ch 5, repeat from 1st * all around ending to correspond, join in 5th st of ch. 5th ROUND: Ch 1, sc in same space, * 4 sc in next loop, sc in next st, repeat from * 5 times, working over sc and into ch 5 loop of 3rd round, 3 double crochet (dc) in loop, sc in next st, repeat from 1st * all around ending to correspond, join in 1st sc. 6th ROUND: Ch 3, dc in same space, * ch 4, skip 4 sc, 2 d trc in next sc, repeat from * once, ch 4, skip 4 sc, 2 d trc, ch 4, 2 d trc, ch 4, 2 d trc in next sc, * ch 4, skip 4 sc, 2 d trc in next sc, repeat from * once, ch 4, skip 4 sc, 2 dc in next sc, 2 dc in next sc, repeat from 1st * all around ending to correspond, join in 3rd st of ch. 7th ROUND: Sl st to next loop, ch 1, 3 sc in same loop, sc in each of the next 2 d trc, * 3 sc in the next loop, 1 sc in each of the next 2 d trc, repeat from * 5 times, 3 sc in next loop, skip next dc, * thread over, insert hook in next dc, pull thread through, repeat from * once, thread over and work off all loops at one time, repeat from 1st * all around ending to correspond, join in 1st sc. 8th ROUND: Sl st over next 3 sc, ch 5, d trc in next sc, * ch 2, skip 3 sc, 2 d trc in each of the next 2 sc, repeat from * once, ch 2, skip 3 sc, * 4 d trc in next sc, ch 3, 4 d trc in next sc, ch 2, * skip 3 sc, 2 d trc in each of the next 2 sc, ch 2, repeat from * once, skip 3 sc, 1 d trc in each of the next 2 sc, skip 7 sts, 1 d trc in each of the next 2 sc, repeat from 1st * all around ending to correspond, join in 5th st of ch. 9th ROUND: Ch 1, sc in same space, work 1 sc in each d trc, 1 sc in each ch 2 loop and 3 sc in each ch 3 loop, join in 1st sc. 10th ROUND: Sl st over next 2 sc, ** work a double picot loop (double picot loop: * ch 7, sl st in 5th st from hook for picot, repeat from * once, ch 2), skip 4 sc, sc in next sc, repeat from ** twice, work a double picot loop, skip 1 sc, sc in next sc, * double picot loop, skip 4 sc, sc in next sc, repeat from * 6 times and continue all around ending to correspond, join. 11th ROUND: Sl st to center of next double picot loop, ch 3, dc in same space, * double picot loop, 2 dc in center of next double picot loop, repeat from * once, double picot loop, 2 trc, double picot loop, 2 trc in center of next double picot loop, * double picot loop, 2 dc in center of next double picot loop, repeat from * twice, work a single picot loop (single picot loop: ch 7, sl st in 5th st from hook for picot, ch 2), sc in center of next double picot loop, single picot loop, 2 dc in center of next double picot loop, repeat from 1st * all around ending to correspond, join. 12th ROUND: Sl st to center of double picot loop, ch 3, dc in same space, * double picot loop, 2 dc in center of next double picot loop, repeat from * once, double picot loop, 2 trc, double picot loop, 2 trc in center of next double picot loop, * double picot loop, 2 dc in center of next double picot loop, repeat from * twice, single picot loop, 2 dc before next single picot loop, 2 dc after next single picot loop, single picot loop, 2 dc in center of next double picot loop, repeat from 1st * all around ending to correspond, join. 13th ROUND: Sl st to center of next double picot loop, ch 3, 1 dc in same space, * double picot loop, 2 dc in center of next double picot loop, repeat from * once, double picot loop, 2 trc, double picot loop, 2 trc in center of next double picot loop, * double picot loop, 2 dc in center of next double picot loop, repeat from * twice, single picot loop, 2 dc after next single picot, 2 dc before next single picot, single picot loop, 2 dc in center of next double picot loop, repeat from 1st * all around ending to correspond, join in 3rd st of ch, cut thread.\n\n_Sea Spray_\n\n**MATERIALS:**\n\n**J. & P. COATS or CLARK'S O.N.T. BEST SIX CORD MERCERIZED CROCHET,** Size 30:\n\nSMALL BALL:\n\nJ. & P. COATS\u20142 balls of White or Ecru, or 3 balls of any color, or\n\nCLARK'S O.N.T.\u20144 balls of White, Ecru or any color.\n\nSteel Crochet Hook No. 10 or 11.\n\nThis amount is sufficient for a set consisting of 1 large doily about 10 \u00d7 15 inches and 2 small doilies each about 10 inches in diameter.\n\n**Small Doily (Make 2) . . .** Starting at center ch 10. Join with sl st to form ring. **1st rnd:** Ch 1, 16 sc in ring. Sl st in 1st sc made. **2nd rnd:** Ch 4 (to count as 1 tr), tr in same place as sl st, 2 tr in each sc around (32 tr). Sl st in top st of starting chain. **3rd rnd:** Ch 4, holding back the last loop of each tr on hook, make tr in next 2 sts, thread over and draw through all loops on hook (3-tr cluster made); * ch 9, holding back the last loop of each tr on hook make tr in same place as last tr, tr in each of next 2 sts, complete a 3-tr cluster as before. Repeat from * around. Join last ch-9 with sl st in tip of 1st cluster. **4th rnd:** Sl st to center of loop, ch 13, * in center st of next loop make tr, ch 5, sc in 4th ch from hook (p made), ch 1 and tr. ch 9. Repeat from * around, ending with ch 1. p, ch 1, sl st in 4th st of ch-13. **5th rnd:** Sl st to center of next loop, ch 15, * in center st of next ch-9 make tr, ch 1, p, ch 1 and tr, ch 11. Repeat from * around. Join. **6th rnd:** Sl st to center of next loop, ch 17, * in center st of next ch-11 make tr, ch 1, p, ch 1 and tr, ch 13. Repeat from * around. Join.\n\n**7th rnd:** Sl st to center of next loop, ch 1, sc in same loop, * ch 15, sc in center st of next loop. Repeat from * around, ending with sl st in 1st sc made. **8th rnd:** Ch 1, 9 sc in loop, * ch 8, tr tr in 8th ch from hook, ch 5, sc in tr tr, ch 7, sc in same place where tr tr was made, 9 sc in same loop, 9 sc in next loop. Repeat from * around. Sl st in 1st sc made. **9th rnd:** Sl st in next 3 sc, ch 9, * in ch-5 ring make (a 3-d tr cluster, ch 5) 5 times; skip 5 sc on 2nd half of loop, holding back the last loop of each tr on hook make tr in next sc, skip 3 sc of next loop, tr in next sc, thread over and draw through all loops on hook, ch 5. Repeat from * around. Join last tr with sl st in 4th st of ch-9. **10th rnd:** Ch 6, * skip next cluster, sc in center st of next loop, ch 5, in tip of next cluster make a 3-d tr cluster, ch 5 and 3-d tr cluster; in tip of next cluster make three 3-d tr clusters with ch-5 between, in tip of next cluster make 3-d tr cluster, ch 5 and 3-d tr cluster; ch 5, sc in center st of next loop, ch 3, dc in tip of the joined tr, ch 3. Repeat from * around. Join last ch-3 with sl st to 3rd st of ch-6. **11th rnd:** Sl st in next 3 ch, the sc and in the following 2 ch, 3 sc in same loop, * in each of next 4 loops make 5 sc, ch 3 and 5 sc; 3 sc in next loop, skip next 2 ch-3 sps, 3 sc in next loop. Repeat from * around. Join with sl st to 1st sc made. Fasten off.\n\n**Large Doily . . .** Starting at center, ch 61 (to measure about 4\u00be inches). **1st rnd:** 3 sc in 2nd ch from hook, sc in each ch across, 5 sc in last ch. Now, working along opposite side of foundation chain, make sc in each st across, ending with 2 sc in same place where first 3 sc were made. Join with sl st in 1st sc. **2nd rnd:** Ch 4, 2 tr in same place as sl st, 2 tr in each of next 3 sc, (ch 2, skip 2 sc, tr in next sc) 18 times; ch 2, skip next 2 sc, 2 tr in each of next 3 sc, 3 tr in next sc, 2 tr in each of next 3 sc, (ch 2, skip 2 sc, tr in next sc) 18 times; ch 2, skip next 2 sc, 2 tr in each of last 3 sc. Join with sl st in top st of ch-4. **3rd rnd:** Ch 4, holding back the last loop of each tr on hook make tr in next 2 sts, complete a 3-tr cluster, (ch 9, holding back the last loop of each tr on hook make tr in same place as last tr, tr in each of next 2 sts, complete a 3-tr cluster) 3 times; ch 9, cluster in next sp, ch 9, (cluster in next sp, ch 5, skip 1 sp, sc in next sp, ch 5, skip next sp, cluster in next sp, ch 9, skip 1 sp) twice; cluster in next sp, ch 5, skip 1 sp, sc in next sp, ch 5, skip 1 sp, cluster in next sp, ch 9, cluster in next space, ch 9, make a 3-tr cluster over next 3 tr, make 6 more clusters with ch-9 between, having 1st tr of each cluster in same place as last tr of previous cluster, ch 9, cluster in next sp and work to correspond with other side, joining last ch-9 with sl st in tip of 1st cluster made. **4th to 8th rnds incl:** Work exactly as for Small Doily. **9th rnd:** Sl st in next 3 sc, ch 9, * in ch-5 ring make five 3-tr clusters with ch 5 between, ch 5, skip 5 sc on 2nd half of loop, holding back the last loop of each tr on hook make tr in next sc, skip 3 sc of next loop, tr in next sc, thread over and draw through all loops on hook, ch 5. Repeat from * 2 more times. Hold the next two ch-5 rings together and work the five 3-tr clusters over both of them. Continue thus around, working over two ch-5 rings on opposite side of doily. Join last ch-9 with sl st in 4th st of ch-9. **10th and 11th rnds:** Work exactly as for Small Doily. Fasten off.\n\nStarch lightly and block to measurements given.\n\n_Scroll Doily_\n\n**Materials Required \u2013 AMERICAN THREAD COMPANY \"STAR\" MERCERIZED CROCHET COTTON, Article 30, Size 50.**\n\n1\u2013150 yd. Ball White.\n\nSteel Crochet Hook No. 13.\n\nDoily measures about 9\u00bc inches.\n\nCh 6, join to form a ring, ch 5, d c in ring, * ch 2, d c in ring, repeat from * 5 times, ch 2, join in 3rd st of ch.\n\n**2nd Row.** Ch 1 and work 2 s c over each 2 ch mesh.\n\n**3rd Row.** Ch 5, d c in next s c, * ch 2, d c in next s c, repeat from * all around, ch 2, join in 3rd st of ch (16 d c).\n\n**4th Row.** Ch 1 and work 3 s c over each 2 ch mesh.\n\n**5th Row.** Ch 18, s c in 2nd st from hook and work 19 s c over balance of ch, sl st in next 2 s c of center, ** ch 1, turn, work in back loop of st only for entire scroll, 1 s c in each of the next 19 s c on scroll, ch 3, turn, skip 2 sts of ch, 1 s c in next st of ch, 1 s c in each of the next 5 s c, * ch 1, turn, 1 s c in each of the next 4 s c, ch 3, turn, skip 2 sts of ch, 1 s c in next st of ch, 1 s c in each of the next 4 s c, 1 s c in each of the next 2 s c on base of scroll, repeat from * 6 times, sl st in next 4 s c of center, ch 17, turn, s c in 3rd picot from bottom of scroll just completed, ch 1, turn, and work 20 s c over ch, sl st in each of the next 2 s c of center, repeat from ** until 7 scrolls are completed. Work another scroll joining it to 1st scroll in the 6th picot, complete scroll, then sl st in each of the next 4 s c of center, break thread.\n\nAttach thread in 2nd free picot of any scroll, s c in same space, ** ch 5, s c in next free picot, * ch 5, s c in next free picot, repeat from *, ch 5, skip 1st picot of next scroll, s c in next free picot, repeat from ** 6 times ending row with * ch 5, s c in next picot, repeat from * twice, ch 3, d c in 1st s c, this brings thread in position for next row.\n\n**2nd Row.** S c in same space, ch 3, s c in same space, * ch 6, s c in next loop, ch 3, s c in same space, repeat from * all around ending row with ch 3, d c in first s c.\n\n**3rd, 4th & 5th Rows.** S c in same space, ch 3, s c in same space, * ch 6, s c in next 6 ch loop, ch 3, s c in same space, repeat from * all around ending each row same as last row.\n\n**6th Row.** S c in same space, ch 3, s c in same space, ** ch 6, cluster st in next loop, (cluster st: thread over needle twice, insert in loop and work off 2 loops twice, * thread over needle twice, insert in same space and work off 2 loops twice, repeat from *, thread over and work off remaining loops at one time) ch 4, sl st in top of cluster st for picot, ch 3, cluster st in same loop, picot, ch 3, cluster st in same loop, picot, ch 6, s c in next loop, ch 3, s c in same loop, repeat from ** all around ending row with ch 6, cluster st in next loop, picot, * ch 3, cluster st in same loop, picot, repeat from *, ch 3, d c in 1st s c.\n\n**7th Row.** S c in same space, ch 3, s c in same space, * ch 6, s c in next loop, ch 3, s c in same space, ch 6, s c between next 2 cluster sts, ch 6, s c between next 2 cluster sts, ch 6, s c in next loop, ch 3, s c in same space, repeat from * all around ending row with ch 3, d c in 1st s c.\n\n**8th, 9th, 10th & 11th Rows.** Same as 3rd row.\n\n**12th Row.** S c in same space, ** ch 3, s c in same space, * ch 6, s c in next 6 ch loop, ch 3, s c in same space, repeat from *, ch 5, cluster st in next ch 6 loop, picot, ch 3, cluster st in same space, picot, ch 3, cluster st in same space, picot, (cluster st group), ch 5, s c in next ch 6 loop, repeat from ** all around in same manner.\n\n**13th Row.** Sl st to center of next 6 ch loop, ch 3, s c in same space, * ch 6, s c in next loop, ch 3, s c in same space, ch 7, skip 1 loop, s c between next 2 cluster sts, ch 7, s c between next 2 cluster sts, ch 7, skip 1 loop, s c in next 6 ch loop, ch 3, s c in same space, repeat from * all around in same manner ending row with ch 3, d c in 1st s c.\n\n**14th Row.** S c in same space, ch 3, s c in same space, ** ch 6, s c in next 6 ch loop, ch 3, s c in same space, * ch 6, s c in next 7 ch loop, ch 3, s c in same space, repeat from * twice, repeat from ** all around in same manner, ending row same as last row.\n\n**15th, 16th & 17th Rows.** Same as 3rd row.\n\n**18th Row.** Same as 12th row.\n\n**19th Row.** Sl st to next 6 ch loop, ch 3, s c in same space, ch 6, s c in next loop, ch 3, s c in same space, * ch 6, skip 1 loop, cluster st group between next 2 cluster sts, cluster st group between next 2 cluster sts, ch 6, skip 1 loop, s c, ch 3, s c in next loop, ch 6, s c, ch 3, s c in next loop, repeat from * all around in same manner.\n\n**20th Row.** Sl st to next 6 ch loop, ch 3, s c in same space, * ch 6, cluster st between 1st 2 cluster sts, picot, ch 3, cluster st in same space, picot, ch 3, cluster st between next 2 cluster sts, picot, ch 3, cluster st in same space, picot, ch 1, skip 2 center cluster sts, cluster st in next 3 ch loop, picot, ch 3, cluster st in same space, picot, ch 3. cluster st in next loop, picot, ch 3, cluster st in same space, picot, ch 6, skip 1 loop, s c, ch 3, s c in next loop, repeat from * all around, break thread.\n\n_Floral Garland Doily_\n\n**ROYAL SOCIETY SIX CORD CORDICHET,**\n\n_Size 30, 2 balls of White or Ecru_.\n\nSteel Crochet Hook No. 10.\n\nPiece of linen 6 inches in diameter.\n\n**Doily measures 13\u00bd inches in diameter.**\n\nRoll edge of linen and baste. **1st rnd:** Make 272 sc around rolled edge. **2nd rnd:** Ch 3, dc in next 15 sc, * (ch 5, skip 2 sc, sc in next sc) 5 times; ch 5, skip 3 sc, dc in next 16 sc. Repeat from * around. Join with sl st. **3rd rnd:** Sl st in next 2 dc, ch 3, dc in next 11 dc, * (ch 5, sc in next loop) 6 times; ch 5, skip 2 dc, dc in next 12 dc. Repeat from * around. Join. **4th rnd:** Sl st in next 2 dc, ch 4, tr in next 7 dc, * (ch 5, sc in next loop) 7 times; ch 5, skip 2 dc, tr in next 8 dc. Repeat from * around. Join. **5th rnd:** Ch 4, holding back on hook the last loop of each tr make tr in next 3 tr, thread over and draw through all loops on hook (cluster made), * ch 3, 4-tr cluster over next 4 tr, (ch 5, sc in next loop) 8 times; ch 5, cluster over next 4 tr. Repeat from * around. Join. **6th rnd:** Sl st in next loop, ch 6, dc in next loop, * (ch 5, tr in next loop) 7 times; ch 5, dc in next loop, (ch 3, dc in next loop) twice. Repeat from * around. Join to 3rd ch of ch-6. **7th rnd:** Ch 6, dc in next dc, * (ch 5, tr in tr) twice; ch 5, sc in next tr, ch 5, 5-d tr cluster in same tr, skip next tr, 5-d tr cluster in next tr, ch 5, sl st at base of cluster, (ch 5, tr in next tr) twice; ch 5, dc in next dc, (ch 3, dc in next dc) twice. Repeat from * around. Join. **8th rnd:** Ch 6, * dc in next dc, ch 5, tr in next tr, ch 7, d tr in next tr, 6-d tr cluster in center of previous 2 clusters, (ch 9, 6-d tr cluster in same place) twice; d tr in next tr, ch 7, tr in next tr, ch 5, dc in next dc, ch 3, dc in next dc, ch 3. Repeat from * around. Join. **9th rnd:** * Ch 7, skip ch-3 sp, tr in next sp, ch 5, d tr in next sp, (ch 5, in next sp make tr, ch 5 and tr) twice; ch 5, d tr in next sp, ch 5, tr in next sp, ch 7, skip 1 dc, sc in next dc. Repeat from * around. Join. **10th rnd:** * 9 sc in next loop, 7 sc in next 7 loops, 9 sc in next loop. Repeat from * around. Join. **11th rnd:** Sl st in next 2 sc, sc in next sc, * (ch 3, skip 1 sc, sc in next sc) 31 times; ch 3, skip 4 sc, sc in next sc. Repeat from * around, ending with ch 3. Join. **12th rnd:** Sl st across first loop, * (sc in next loop, ch 3) 28 times; sc in next loop, ch 5, skip 3 loops. Repeat from * around, ending with ch 2, dc in first sc. **13th rnd:** * Ch 5, skip 1 loop, (sc in next loop, ch 3) 25 times; sc in next loop, ch 5, skip 1 loop, sc in next loop. Repeat from * around, ending with ch 2, dc in dc. **14th rnd:** Ch 7, sc in next loop, ch 7, skip 1 loop, sc in next loop, * (ch 3, sc in next loop) 22 times; ch 7, skip 1 loop, sc in next loop, ch 7, sc in next loop, ch 7, skip 1 loop, sc in next loop. Repeat from * around, ending with ch 7. Join and break off.\n\nNow complete scallops individually as follows: **1st row:** Skip first ch-3 loop, attach thread to next loop, (ch S, sc in next loop) 18 times; dc in next loop, turn. **2nd row:** Sc in next loop, (ch 3, sc in next loop) 15 times; dc in next loop, turn. **3rd row:** Sc in next loop, (ch 3, sc in next loop) 12 times; dc in next loop, turn. 4th row: Sc in next loop, (ch 3, sc in next loop) 10 times. Break off. Work each scallop in this manner.\n\n**EDGING . . .** Attach thread to first ch-7 loop between any two scallops, ch 5, holding back on hook the last loop of each d tr make d tr in same loop, (d tr in next loop) twice; thread over and draw through all loops on hook (cluster made); (ch 4, sc in tip of cluster) 3 times (triple picot made); ch 5, sc in same loop as last d tr of cluster was made, * ch 5, holding back on hook the last loop of each d tr make d tr in same loop as last sc was made, (d tr in next free ch-3 loop) twice and complete cluster as before, ch 4, sc in tip of cluster (picot made), ch 5, sc in same loop as last d tr of cluster was made. Repeat from * once more; ** ch 5, holding back on hook the last loop of each d tr make d tr in same loop as last sc was made, skip next loop, d tr in next sc, skip next loop, d tr in next loop, and complete cluster as before, picot, ch 5, sc in same loop as last d tr of cluster was made. Repeat from ** 3 more times; make 2 more picot clusters to correspond with 2nd and 3rd picot clusters made, making last d tr of last picot cluster in next ch-7 loop, ch 5, make a cluster and triple picot as before, and continue thus around. Join and break off.\n\n_Entertaining Ideas_\n\n**MATERIALS** \u2014DAISY Mercerized Crochet Cotton, size 30 in White, Cream or Ecru. 1 skein or ball is sufficient for 4 doilies. Crochet hook size 13. Sizes\u20146\u00bd to 7 inches.\n\n**A\u2014DOILY\u2014\"WHIRL AROUND\"** (Center of photograph) Ch 8, sl st in 1st st. Ch 7, tr in ring, (ch 1, tr) 16 times in ring, ch 1, sl st in 6th st of 1st 7-ch. **ROW 2** \u2014Ch 3, (2 dc in next 1-ch sp, 1 dc in tr) repeated around, sl st in 1st 3-ch. **ROW 3** \u2014(Ch 25, sc in next 3d dc) 18 times. Cut 6 inches long, thread to a needle and fasten off on back. **ROW 4** \u2014Sl st in 13th st of one lp, ch 7, tr in same st, (ch 2, tr) twice in same st, * tr in 13th st of next lp, (ch 2, tr) 3 times in same st. Repeat from * around, sl st in 5th st of 1st 7-ch. **ROW 5** \u2014Ch 1, * (1 sc, 1 hdc and 1 dc) in next 2-ch sp, dc in tr, 3 dc in next sp, dc in tr, (1 dc, 1 hdc and 1 sc) in next sp, sc between shells. Repeat from * around. Sl st up to center of 1st shell. **ROW 6** \u2014Ch 11, * (dc, ch 3, dc) in center st of next shell, ch 8. Repeat from * around. Join final 3-ch to 3d st of 1st 11-ch. **ROW 7** \u2014Ch 4, turn, dc in 3-ch sp, (ch 1, dc) twice in same sp, * ch 7, dc in next shell, (ch 1, dc) 3 times in same shell. Repeat from * around. Join final 7-ch to 3d st of 1st 4-ch. **ROW 8** \u2014Ch 9, turn, * dc in next dc, (ch 2, dc in next dc) 3 times, ch 6. Repeat from * around. Join final 2-ch to 3d st of 1st 9-ch. **ROW 9** \u2014Ch 3, turn, dc in same st, * (ch 2, 2 dc in next dc) 3 times, ch 4, 2 dc in next dc. Repeat from * around and join to 1st 3-ch. **ROW 10** \u2014Ch 6, turn, * dc in next 2 dc, (ch 2, dc in next 2 dc) 3 times, ch 3. Repeat from * around. End with 1 dc, sl st in 3d st of 1st 6-ch. **ROW 11** \u2014Ch 3, turn, * 2 dc in next dc, (ch 2, dc in next dc, 2 dc in next dc) 3 times, dc in next dc. Repeat from * around and join to 1st 3-ch. **ROW 12** \u2014Ch 1, turn, sc in next dc, * hdc in next, (ch 2, dc in next dc, 2 dc in next, 1 dc in next) twice, ch 2, hdc in next dc, sc in next 4 dc. Repeat from * around. Cut 6 inches long, thread to a needle and fasten off on back.\n\n**B\u2014DOILY\u2014\"LIGHT AND AIRY\"** (Upper right)\n\nCh 8, sl st in 1st st. (Ch 35, sc in ring) 8 times. Cut 6 inches long, thread to a needle and fasten off on back. **ROW 2** \u201410 sc across center of 1 lp, (ch 3, 10 sc in next lp) 7 times, ch 3, sl st in 1st sc. **ROW 3** \u2014* Ch 7, sk last ch st, 6 sc on ch, sl st in next sc. Ch 1, turn, sk sl st and in back lps only, sc in last 5 sc, ch 4, turn, sk last ch st, 3 sc on ch, sc in 5 sc, sl st in next sc on center. Ch I, turn, sk sl st, sc in next 7 sc, ch 4, turn, 3 sc on ch, 7 sc, sl st in next 2d sc. Ch 1, turn, 9 sc, ch 4, turn, 3 sc on ch, 9 sc, sl st in next 2d sc. Ch 1, turn, 9 sc, ch 2, turn, 1 sc on ch, 9 sc, sl st in next sc. Ch 1, turn, 7 sc, ch 2, turn, 1 sc on ch, 7 sc, sl st in next se. Ch 1, turn, 5 sc, ch 2, turn, 1 sc on ch, 5 sc, sl st in end sc, 3 sc in next 3-ch sp, sl st in next sc. Repeat from * around. Fasten off on back. **ROW 4** \u2014Join to center of 1 point, * ch 12, dtr in end of next 2d rib, dtr in 2d rib of next point, ch 12, sc in center of same point. Repeat from * around. **ROW 5** \u2014* (Ch 8, tr tr) twice in next dtr. Repeat from * once. Ch 8, sc in next sc. Repeat from * around. **ROW 6** \u2014(10 sc in next 8-ch sp, sc in tr tr) twice, * (5 sc, ch 5, sl st in last sc for a p, and 5 sc) in next sp, sc in tr tr, 10 sc in next sp, sc in tr tr, 10 sc in next 2 sps, sc in tr tr, ch 7, turn, sl st back in 11th sc up side of previous shell, ch 1, turn, (1 sc, 1 hdc, 7 dc, 1 hdc and 1 sc) all in 6-ch lp, 5 sc in half of next sp, ch 2, turn, tr in 3d st on added lp, (ch 2, tr in next st) 6 times, ch 2, sl st in next 5th sc on shell, ch 1, turn, 3 sc in next 2-ch sp, (2 sc, a p and 2 sc in next sp) 6 times, 3 sc in next sp, 5 sc in next sp, 1 sc in tr tr. Repeat from * around. Sl st in 1st 11 sc and make added shell. Fasten off on back.\n\n**C\u2014DOILY\u2014\"LATTICED AND LOVELY\"** (Lower right) Ch 8, sl st in 1st st. Ch 3, 19 dc in ring, sl st in 1st 3-ch. **ROW 2** \u2014(Ch I and in back lps only, 3 dc in next dc, ch 1, sc in next dc) 10 times. Sl st to center dc of next scallop. **ROW 3** \u2014Ch 4, (dc, ch 1, dc) in same st, (ch 1, dc) 3 times in center dc of each scallop around. Ch 1, sl st in 3d st of 1st 4-ch. **ROW 4** \u2014Ch 6, turn, * dc in next dc, (ch 1, dc in next dc) twice, ch 3 and repeat from * around. Join final 1-ch to 3d st of 1st 6-ch. **ROW 5** \u2014Ch 4, turn, dc in next dc, ch 1, dc in next dc, * ch 5, dc in next dc, (ch 1, dc in next dc) twice. Repeat from * around. Join final 5-ch to 3d st of 1st 4-ch. **ROW 6** \u2014Ch 10, turn, * dc in next dc, (ch 1, dc in next dc) twice, ch 7. Repeat from * around and join final 1-ch to 3d st of 1st 10-ch. **ROW 7** \u2014Ch 3, turn, (dc in next 2 dc, holding back the last lp of each dc, thread over and pull thru all lps at once (a Cluster made), * ch 13, (dc in next 3 dc) made into a Cluster. Repeat from * around and join to 1st Cluster. **ROW 8** \u2014Ch 3, turn, (14 dc in next sp, 1 dc in Cluster) repeated around and join to 3-ch, sl st in next dc. **ROW 9** \u2014Ch 4, dc in next 2d dc, ch 1, dc in next 2d dc, * ch 9, dc in next 5th dc, (ch 1, dc in next 2d dc) 5 times. Repeat from * around. Join final 1-ch to 3d st of 1st 4-ch. **ROW 10** \u2014Ch 4, turn, dc in next dc, (ch 1, dc in next dc) twice, * ch 9, dc in next dc, (ch 1, dc in next dc) 5 times. Repeat from * around. Join final l-ch to 3d st of 1st 4-ch. **ROW 11** \u2014Ch 4, turn, dc in next dc, ch 1, dc in next dc, * ch 6, sc under center of next 2 lps, ch 6, dc in next dc, (ch 1, dc in next dc) 5 times. Repeat from * around and join. **ROW 12** \u2014Repeat Row 10 to *. ** Ch 1, dc in next 6-ch lp, ch 10, dc in next lp, (ch 1, dc in next dc) 6 times. Repeat from ** around and join. **ROW 13** \u2014Ch 4, turn, dc in next dc, (ch 1, dc in next dc) twice, * ch 10, dc in next dc, (ch 1, dc in next dc) 7 times. Repeat from * around and join. **ROW 14** \u2014Ch 4, turn, dc in next dc, (ch 1, dc in next dc) 3 times, * ch 6, sc under center of next 2 lps, ch 6, dc in next dc, (ch 1, dc in next dc) 7 times. Repeat from * around and join. **ROW 15** \u2014Ch 11, turn, dtr in next dc, ch 3, dtr in next dc, * ch 3, tr in next dc, ch 6, sc in next sc, ch 6, tr in next dc, (ch 3, dtr in next dc) twice, (ch 3, tr tr in next dc) twice, (ch 3, dtr in next dc) twice. Repeat from * around. Join final 3-ch to 8th st of 1st 11-ch. **ROW 16** \u2014(Sc, ch 1, 3 dc, ch 1, sc) in each 3-ch sp around, 6 sc in each 6-ch sp. Cut 6 inches long, thread to a needle and fasten off on back.\n\n**D\u2014DOILY\u2014\"DANCING FANS\"** (Lower left)\n\nCh 10, tr in 1st st, (ch 4, tr in same st) 6 times, ch 4, sl st in next 5th ch st. **ROW 2** \u2014Ch 1, (5 sc in next sp, sc in tr) repeated around, sl st in 1st 1-ch. **ROW 3** \u2014* Ch 4, (dc, ch 4, dc) in next 3d sc, ch 4, sc in next 3d sc. Repeat from * 7 times. **ROW 4** \u2014(4 sc in next sp, 5 sc in next, 4 sc in next) 8 times. Sl st up to center sc of next shell. **ROW 5** \u2014(Ch 9, sc in 2 lps of 5th st from hook for a p, ch 12, p, ch 5, sc in next point) 8 times. Cut 6 inches long, thread to a needle and fasten off on back. **ROW 6** \u2014Join to center of 1 lp, (ch 10, p, ch 14, p, ch 6, sc in next lp) 8 times. Fasten off as before. **ROW 7** \u2014Join to center st of 1 lp, * ch 3, 7 dc in same st. Ch 5, turn, sk last dc, dc in next dc, (ch 2, dc in next st) 6 times. Ch 6, turn, dc in next dc, (ch 3, dc in next dc) 5 times, ch 3, dc in 3d st of end 5-ch. Ch 8, turn, dc in next dc, (ch 5, dc in next dc) 5 times, ch 5, dc in 4th st of end 6-ch. * Cut 3 inches long. ** Turn, join to center st of next lp to left of last Fan and repeat from * to *. Ch 2, sl st in corner of last Fan and cut 3 inches long. Repeat from ** around. Join 1st and last Fans. **Edge** \u2014Make 7 sc in center sp of one Fan, (7 sc in next sp) 3 times, * 1 sc between Fans, 7 sc in next sp, 4 sc in half of next sp, ch 16, turn, sl st back in center of 2d sp up side of last Fan, ch 1, turn, 1 sc in lp, (1 hdc, 5 dc, 1 hdc and 1 sc) 3 times in lp, 3 sc in bal. of sp on Fan, 4 sc in next sp, turn, (ch 10, sc in next scallop) 3 times, ch 10, sl st in center of next sp on Fan. Ch 1, turn, (6 sc, ch 5, sl st in last sc for a p, and 6 sc) in each 10-ch lp, 3 sc in bal. of sp on Fan, (7 sc in next sp) 4 times. Repeat from * around, working over ends left from Fans. Fasten off as before.\n\nStretch and pin each Doily right-side-down in a true circle. Steam and press dry thru a cloth.\n\n_Frost Flower Refreshment Set_\n\n**MATERIALS** \u2014Lily Sil-Tone Mercerized Crochet Cotton:\u20144 balls White (100-yd. balls)\u2014sufficient for Doily and 4 Coasters. Crochet hook size 12.\n\n**MOTIF** \u2014Ch 6, sl st in 1st st. Ch 5, dc in ring, (ch 2, dc in ring) 6 times, ch 2, sl st in 3d st of 1st 5-ch. **ROW 2** \u2014(Sc, ch 1, 3 dc, ch 1 and sc) in each 2-ch sp. **ROW 3** \u2014Sl st to center dc of 1st petal, ch 3 for a dc, (ch 8, sl st in 1 lp of 5th ch st from hook for a p, ch 4, dc in next petal) 7 times, ch 8, p, ch 4, sl st in 3d st of 1st lp. **ROW 4** \u2014Ch 3, turn, 2 dc in same st with sl st, (ch 8, p, ch 4, 3 dc in next dc) 7 times, ch 8, p, ch 4, sl st in 1st 3-ch. **ROW 5** \u2014Ch 3 for a dc, turn, (ch 7, p, ch 3, 2 dc in each dc) repeated around. End with 5 dc, sl st in 3d st of 1st lp. **ROW 6** \u2014Ch 3, turn, dc in sl st, 2 dc in next dc, * dc in next 2 dc, (2 dc in next dc) twice, ch 7, (2 dc in next dc) twice. Repeat from * around and join to 1st 3-ch. **ROW 7** \u2014Ch 3, turn, sc in center of last 7-ch lp, * (2 dc in next dc) twice, (1 dc in next dc, 2 dc in next) 4 times, sc in center of next lp. Repeat from * around. End with 1 less dc, sl st in 1st dc, sl st in next sc. **ROW 8** \u2014Turn, * ch 2, dc in next 2d dc, ch 2, tr in next 2d dc, (ch 2, dtr in next 2d dc) twice, ch 2, dtr in next dc, ch 2, dtr in next 2d dc, ch 2, tr in next 2d dc, ch 2, dc in next 2d st, ch 2, sc in sc between fans. Repeat from * 7 times. **ROW 9** \u2014Ch 1, turn, * 2 sc in next sp, 3 sc in next, (2 sc, ch 5, sl st in last sc for a p, and 2 sc \u2014all in next sp) 5 times, 3 sc in next sp, 2 sc in next. Repeat from * 7 times. Cut 6 inches long, thread to a needle and fasten off on back.\n\nMake a Motif for each Coaster.\n\n**DOILY** \u2014Make 8 Motifs and join into a circle by 2 shells on each side of each, joining by the last 3 ps on one shell and the 1st 3 ps on next. Leave one shell free between joinings on inside of circle. To join ps, instead of a 5-ch p, make 2-ch, sl st in corresponding p on adjoining Motif, ch 2 back, sl st in last sc to complete p. Repeat with each joining.\n\n**Center** \u2014Make another Motif. Join to center p on a shell, * ch 9, p, ch 10, p, ch 5, tr in next 2d p, tr in 1st p on next shell, ch 9, p, ch 10, p, ch 5, sc in next 2d p. Repeat from * around and fasten off. **ROW 2** \u2014Join to center of last p-lp, * ch 10, p, ch 3, sl st in center p of free shell on one Motif in circle, ch 3, tr at base of last p, ch 5, sl st in tr for a p, ch 6, sc in next lp, (ch 10, p) twice, ch 5, sl st in 1st p on next shell of same Motif, ch 5, dc back at base of last p, ch 6, p, ch 7, sl st in 2d p on next shell on next Motif, ch 5, dc back in 6th st of 7-ch, a p, ch 2, tr in center st between 1st 2 ps at start of this p-lp, ch 7, p, ch 6, sc in next p-lp of last row. Repeat from * 7 times to complete circle. Fasten off.\n\nStretch and pin right-side-down in true circles. Steam and press dry thru a cloth.\n\n_Victoriana_\n\n**MATERIALS:**\n\n**CLARK'S BIG BALL MERCERIZED CROCHET,** Art. B.34, Size 30: 1 ball each of No. 181 Shaded Lt. Yellows and No. 76 Robinette; or\n\n**J. & P. Coats Best Six Cord Mercerized Crochet,** Art. A. 104, Size 30: 1 ball each of No. 181 Shaded Lt. Yellows and a color of your own choice.\n\n**Milwards Steel Crochet Hook** No. 10.\n\nA piece of aqua linen about 9 inches square.\n\n**Doily measures 17\u00bd inches in diameter.**\n\n**BRAIDED SECTION . . .** Starting at inner edge with Shaded Lt. Yellows, ch 6, in 6th ch from hook make dc, ch 2 and dc (shell made); ch 5, turn; (in sp of shell just made, make dc, ch 2 and dc\u2014shell made over shell\u2014ch 5, turn) 10 times; shell over shell, * ch 2, insert hook through previous 3 loops on same side, thread over and complete as a sl st, thus forming a scallop, ch 2, turn; (shell over shell, ch 5, turn, shell over shell, ch 2, sl st in adjacent free loop opposite, ch 2, turn) 3 times; (shell over shell, ch 5, turn) 6 times; shell over shell, ch 2, sl st through previous 3 loops to form scallop as before, ch 2, turn. Continue making braid, joining 4 loops on one side to adjacent loops opposite. Continue making braid until there are 4 free ch-5 loops on same side as last joining, ending with shell over shell, ch 2, sl st through previous 3 loops to form scallop, ch 2, turn and continue making braid, joining 6 loops with loops opposite, ending with shell over shell; make braid until there are 2 free loops on same side as last joining, ending with shell over shell, ch 2, form scallop as before, ch 2, turn. Make braid, joining 4 loops to loops opposite, ending with shell over shell. Work braid until 2 free loops have been completed on same side as last joining, ending with shell over shell. Repeat from * around, until there are 34 scallops on both inner and outer edges, ending with ch 2, turn; dc in sp of last shell made, ch 1, sl st at base of first shell made, ch 1, dc in same sp as last dc was made. Break off.\n\n**CENTER . . . 1st rnd:** Attach Robinette to first joined loop of any scallop, ch 10, * holding back on hook the last loop of each tr, make 3 tr in the free loop of same scallop, thread over and draw through all loops on hook (cluster made), ch 5, holding back on hook the last loop of each d tr, make d tr in next joined-loop of same scallop and in first joined loop of next scallop, thread over and draw through all loops on hook (joint d tr made), ch 5. Repeat from * around, ending with d tr in last joined loop. Join with sl st to 5th ch of ch-10. **2nd rnd:** Ch 7, d tr in same place as sl st, * ch 2, cluster in tip of next cluster, ch 2, in tip of next joint d tr make d tr, ch 2 and d tr. Repeat from * around. Join to 5th ch of ch-7. **3rd rnd:** Ch 4, dc in next d tr, * ch 1, dc in tip of next cluster, (ch 1, dc in next d tr) twice. Repeat from * around. Join to 3rd ch of ch-4. Break off.\n\n**EDGING . . . 1st rnd:** Attach Robinette to free loop of any small scallop, sc in same loop, * ch 5, cluster in first free loop on next scallop, ch 5, cluster in next loop, ch 5, in next loop make (cluster, ch 5) 5 times; (cluster in next loop, ch 5) twice; sc in free loop on next small scallop. Repeat from * around. Join. **2nd rnd:** * Sc in next sp, (ch 7, sc in next sp) 9 times. Repeat from * around. Join. **3rd rnd:** Sl st in first 2 ch of next loop, sc in same loop, * (ch 9, sc in next loop) 8 times; sc in next loop. Repeat from * around. Join. **4th rnd:** Sl st in first 3 ch of next loop, sc in same loop, * (ch 9, cluster in next loop) 7 times; sc in next loop. Repeat from * around. Join and break off. **5th rnd:** Attach Shaded Lt. Yellows to any loop, in each loop around make (3 sc, ch 3) 3 times and 3 sc. Join and break off.\n\nPlace doily on linen. Cut out material in back of doily, leaving \u215b inch for hem. Sew hem neatly in place. Sew linen to doily. Starch lightly and press.\n\n_Chariot Wheels_\n\nDOILY No. 2212\n\n**Materials Required\u2014AMERICAN THREAD COMPANY \"STAR\" or \"GEM\" MERCERIZED CROCHET COTTON, Size 20 or 30**\n\n2\u2014300 Yd. Balls White or Colors.\n\nSteel Croche' Hook No. 11 or 12.\n\nDoily measures 15 inches at the widest point.\n\nCh 5, join to form a ring, ch 7, 1 d c in ring, * ch 4, 1 d c in ring, repeat from * 3 times, ch 4, join to 3rd st of ch.\n\n**2nd Row.** Ch 5, 6 d c in 1st space, * ch 2, 6 d c in next space, repeat from * all around working 5 d c in last space, join to 3rd st of ch.\n\n**3rd Row.** Ch 4, 5 tr c in 1st space, * ch 6, 6 tr c in next space, repeat from * all around, ch 6 and join.\n\n**4th Row.** Ch 4, 1 tr c in each tr c, * ch 5, 1 s c in center of ch 6, ch 5, tr c in each tr c, repeat from * all around. Work 36 more motifs and sew them together as follows: 4 in the 1st row, 5 in the 2nd row, 6 in the 3rd row and 7 in the 4th row, which is the center, continue, having one less motif in each row until there are 4 motifs in row.\n\n**Edge:** Join thread to ch 5 at point, ch 9, * s c in 1st st of next ch 5, ch 7, skip 4 sts of next loop, s c in next st, ch 7, s c in 1st st of next loop, ch 7, skip 2 loops and 4 sts of next loop, s c in next st, ch 7, repeat from * all around.\n\n**2nd Row.** Sl st in loop, ch 3, 11 d c in same loop, (ch 3 counts as 1 d c) ch 5, s c in next loop, ch 5, s c in next loop, ch 5, 12 d c in large loop, ch 5, s c in next loop, ch 5, s c in next loop, ch 5, s c in next loop, ch 5, 12 d c in next loop and continue all around.\n\n**3rd Row.** Ch 3, 1 d c with 1 ch between in every d c, ch 2, s c in next loop, ch 5, s c in next loop, ch 5, s c in next loop, ch 2, d c with 1 ch between in every d c and continue all around.\n\n**4th Row.** Ch 3, d c in d c, * ch 4, sl st in top of d c just made for picot, ch 1, d c in top of d c, repeat from * 10 times, d c in d c, ch 2, s c in 5 ch loop, s c in next loop, ch 2, * d c in top of d c, ch 4, sl st in top of d c just made for picot, repeat from * 10 times, d c in d c, ch 2, s c in next 5 ch loop, s c in next loop, s c in next loop, ch 2 and continue all around.\n_Edgings_\n\n_Edgings That are Different_\n\n**No. 8022** Starting at center of 1 fan, ch 8, join with sl st. **1st row:** 13 s c in ring. Ch 13, turn. **2nd row:** ** Make a Clones knot \u2014 _to make a Clones knot_ , * _thread over, pass hook under ch, thread over and draw loop forward. Repeat from_ * 7 _more times, thread over and draw through all loops on hook, ch 1 to fasten, s c in 4th ch from hook_. Ch 3, skip 1 s c, tr tr in next s c, ch 7. Repeat from ** across. Ch 16, turn. **3rd row:** * A Clones knot, ch 4, tr tr in next tr tr, ch 8. Repeat from * across, ending with Clones knot, ch 4, tr tr in 6th st of turning ch. Ch 20, turn. **4th row:** * Tr tr in next tr tr, ch 14. Repeat from * across, ending with tr tr in 6th st of turning ch. Ch 4, turn. **5th row:** * Skip 1 st, d c in next st, ch 1. Repeat from * across. Fasten off. Make another fan; do not break off, but ch 2, join with sl st to 3rd st of turning ch on last row of previous fan. Make necessary number of fans, joining in same way.\n\nEDGING . . . Attach thread to 1st sp of 1st fan, * ch 5, s c in next sp. Repeat from * to within last sp of 1st fan, ch 2, skip 1 sp of 2nd fan, s c in next sp, ch 5 and continue thus across. Do not break off but work along straight edge as follows: **1st row:** 3 s c in 1st sp, * 11 s c in each of next 3 sps, 7 s c in next sp, 11 s c in each of next 3 sps, 3 s c in each of next 3 sps. Repeat from * across. Ch 5, turn. **2nd row:** * Skip 2 s c, d c in next s c, ch 2. Repeat from * across. Ch 5, turn. **3rd row:** * D c in next d c, ch 2. Repeat from * across. Fasten off.\n\n**No. 8357** EDGING No. 8357-A . . . **1st row:** Ch 10, d c in 10th ch from hook. Ch 10, turn. **2nd row:** D c in d c of previous row. Ch 10, turn. Repeat 2nd row until 12 loops are made. Ch 5, s c in next loop and in each of next 5 loops. Ch 2, turn and make s c in 1st s c made. Ch 5, d c in next d c. * Ch 10, turn and continue as before until 10 loops are made. Ch 5, s c in each of next 5 loops. Ch 2, turn and make s c in 1st s c, ch 5, d c in next d c. Repeat from * for desired length. Fasten off. Attach thread to 3rd last loop, s c in same loop, * ch 5, s c in next loop, ch 5, tr in each of next 2 loops, ch 5, s c in next loop. Repeat from * across. Fasten off.\n\nINSERTION No. 8357-B . . . Work exactly as for No. 8357-A, but make a chain on both sides instead of one side.\n\n_Rose Filet Edge_\n\n(Measure about 6 inches at the widest point)\n\n**Materials Required\u2014  \nAMERICAN THREAD COMPANY  \n\"STAR\" MERCERIZED CROCHET COTTON, Article 30**\n\n1-125 yd. Ball White, size 50 will make about 7 inches of edge.\n\nSteel Crochet Hook #13.\n\nCh 113, d c in 8th st from hook, 1 d c in each of the next 6 sts of ch, * ch 2, skip 2 sts of ch, d c in next st, repeat from * 16 times, 1 d c in each of the next 3 sts of ch, * ch 2, skip 2 sts of ch, d c in next st, repeat from *, 1 d c in each of the next 6 sts of ch, ch 2, skip 2 sts of ch, 1 d c in each of the next 4 sts, * ch 2, skip 2 sts of ch, d c in next st, repeat from * 6 times, 1 d c in each of the next 3 sts of ch, ch 2, skip 2 sts of ch, d c in next st, ch 5, turn.\n\n**2nd Row** \u2014D c in d c, 1 d c in each of the next 3 d c, (solid mesh) * ch 2, d c in next d c, (open mesh), repeat from * 3 times, then work 3 solid meshes, I open mesh, 2 solid meshes, 4 open meshes, 1 solid mesh, 16 open meshes, 2 solid meshes, 1 open mesh, ch 5, turn.\n\n**3rd Row** \u2014D c in 1st d c, (an increase) 1 open mesh, 2 solid meshes, 13 open meshes, 3 solid meshes, 8 open meshes, 1 solid mesh, 2 open meshes, 1 solid mesh, 3 open meshes, 1 solid mesh, 1 open mesh, ch 5, turn.\n\nContinue working up and down according to diagram to arrow. Repeat from beginning to arrow for desired length, break thread.\n\nAttach thread in 1st mesh of lower edge at right hand corner, 2 s c in same mesh, ** 2 s c in next mesh, 5 s c in next mesh, * 5 s c in next mesh, 2 s c in next mesh, repeat from *, 3 s c in next mesh, ch 4, sl st in last s c for picot, 2 s c in same mesh, * 2 s c in each of the next 2 meshes, picot, repeat from *, 2 s c in each of the next 3 meshes, picot, 2 s c in each of the next 2 meshes, picot, 3 s c in each of the next 2 meshes, 2 s c, picot, 3 s c in next mesh, * 2 s c in next mesh, 5 s c in next mesh, repeat from * 5 s c in next mesh, 2 s c in next mesh, picot, repeat from ** across lower edge. Work a row of s c across top edge alternating 2 s c in one open mesh and 3 s c in next mesh.\n\n_Dress Up Your Bed Linens_\n\nWE SUGGEST ROYAL PERL\u00c9\n\n**No. 3-1** . . . Starting at narrow end, ch 19. **1st row:** Tr in 11th ch from hook, (ch 3, sk 3 ch, tr in next ch) twice; ch 3, d tr in same place as last tr. Ch 5, turn. **2nd row:** Dc in d tr, (ch 1, sk 1 ch, dc in next ch, ch 1, dc in next tr) 3 times; (ch 1, sk 1 ch, dc in next ch) twice. Ch 1, turn. **3rd row:** (Sc in dc, sc in sp) 9 times; sc in 4th st of turning ch. Ch 5, turn. **4th row:** Dc in 1st sc, (ch 1, sk 1 sc, dc in next sc) 5 times. Ch 7, turn. **5th row:** (Sk next dc, tr in next dc, ch 3) twice; sk next dc, tr in 4th st of turning ch, ch 3, d tr in same place as last tr. Ch 5, turn. Repeat 2nd to 5th rows inclusive for length desired, ending with the 4th row. Fasten off.\n\n**SCALLOPS . . . 1st row:** Attach thread in center ch of 1st tr corner. Ch 4, * holding back the last loop of each tr on hook make 3 tr in next dc corner, thread over and draw through all loops on hook (3-tr cluster made), (ch 5, cluster in same corner) twice; tr in next dc corner. Repeat from * across. Ch 1, turn. **2nd row:** * Sc in tr, in next ch-5 loop make 2 sc, (ch 3, 1 sc) twice; ch 3 and 2 sc. Repeat from * across. Fasten off.\n\n**HEADING . . . 1st row:** Attach thread to 1st sp, ch 1, sc where thread was attached, * 2 sc in next sp, sc in next sc, sc in next ch, 2 sc in next sp, sc in top of next d tr, 4 sc in next sp, sc in next ch. Repeat from * across. Ch 4, turn. **2nd row:** * Sk next sc, dc in next sc, ch 1. Repeat from * across. Ch 1, turn. **3rd row:** Sc in each sp and each dc across. Fasten off.\n\n**No. 3-2** . . . Make a chain slightly longer than desired length. **1st row:** Sc in 2nd ch from hook, ch 3, sc in next ch, * ch 7, sk 4 ch, sc in next ch, ch 3, sc in next ch. Repeat from * until 1st row is desired length, having an uneven number of ch-7 loops. Cut off remaining chain. Ch 10, turn. **2nd row:** * In next loop make sc, ch 3 and sc, ch 7. Repeat from * across, ending with ch 5, d tr in last sc. Ch 1, turn. **3rd row:** In 1st loop make sc, ch 3 and sc, * ch 7, in next loop make sc, ch 3 and sc. Repeat from * across. Ch 10, turn. **4th, 5th and 6th rows:** Repeat 2nd, 3rd and 2nd rows. Ch 8, turn. **7th row:** Sc in 3rd ch from hook (p made), in 1st loop make (d tr, ch 3, sc in 3rd ch from hook) 5 times and d tr; * in next loop make sc, ch 3 and sc, in next loop make (d tr, p) 6 times and d tr. Repeat from * across. Fasten off.\n\n_Wide Edgings for Smartness_\n\n**No. 8782 . . .** Starting at narrow end of heading ch 8. **1st row:** In 8th ch from hook make 3 dc, ch 2 and 3 dc. Ch 5, turn. **2nd row:** In ch-2 sp (between dc groups) make 3 dc, ch 2 and 3 dc. Ch 5, turn. Repeat 2nd row for desired length being sure there is a multiple of seven ch-5 sps along both edges. Fasten off. **Next row:** Attach thread in 1st ch-5 sp, ch 1, sc in same place where thread was attached, * ch 5, sc in next ch-5 sp. Repeat from * across. Ch 4, turn. **Following row:** * (Skip 1 ch, dc in next ch, ch l) twice; dc in next sc, ch 1. Repeat from * across. Fasten off. Attach thread in 1st ch-5 sp on opposite side and work to correspond but do not fasten off. Continue for scallop as follows: Ch 1, turn. **1st row:** Sc in 1st dc, (ch 6, skip 2 dc, sc in next dc) 5 times; ch 3, skip 2 dc, dc in last dc. Ch 1, turn. **2nd row:** In 1st and last loops make 4 sc, in each loop between make 7 sc. Ch 6, turn. **3rd row:** * Sc in center sc of loop below, ch 6. Repeat from * across ending with ch 3, dc in last sc of last loop. Ch 1, turn. **4th to 10th rows:** Repeat 2nd and 3rd rows alternately 3 times, then the 2nd row once more. At the end of the 10th row ch 3, turn. **11th row:** Dc in last sc of last loop. Fasten off.\n\nNEXT SCALLOP\u2014Attach thread in next dc of heading and work as for 1st scallop. Continue thus across. Fasten off.\n\nEDGING\u2014With right side facing, attach thread in 1st loop of 1st scallop; working along outside edge of scallops make 2 sc, ch 3 and 2 sc in the unfinished half of each outer loop, 2 sc in space between scallops. Continue thus across. Fasten off.\n\n**No. 8780 . . .** Starting at narrow end, ch 43. **1st row:** Sc in 9th ch from hook, * ch 5, skip 3 ch, sc in next 5 ch, ch 5, skip 3 ch, sc in next ch. Repeat from * once more, ch 5, skip 3 ch, sc in next 6 ch. Ch 8, turn. **2nd row:** Dc at base of ch-8, * ch 5, skip 1 sc, sc in next 2 sc, (ch 5, sc in next loop) twice. Repeat from * across. Ch 3, turn. **3rd row:** 8 dc in loop, * sc in next loop, ch 5, sc in next loop, 9 dc in next loop. Repeat from * across. Ch 7, turn. **4th row:** Skip 2 dc, sc in next 5 dc, * ch 5, sc in next loop, ch 5, skip 2 dc, sc in next 5 dc. Repeat from * across. Ch 5, turn. **5th row:** * Skip 1 sc, sc in next 2 sc, (ch 5, sc in next loop) twice; ch 5. Repeat from * 2 more times, skip 1 sc, sc in next 2 sc, ch 5, sc in next loop, ch 5, tr in same loop. Ch 5, turn. **6th row:** 9 dc in loop, * sc in next loop, ch 5, sc in next loop, 9 dc in next loop. Repeat from * across ending with sc in next loop, ch 5, sc in next loop. Ch 7, turn. **7th row:** Sc in loop, * ch 5, skip 2 dc, sc in next 5 dc, ch 5, sc in next loop. Repeat from * 2 more times; ch 5, skip 2 dc, sc in next 5 dc. Turn. **8th row:** Sl st in next st, ch 1, sc in same place, sc in next st, * (ch 5, sc in next loop) twice; ch 5, skip 1 sc, sc in next 2 sc. Repeat from * 2 more times; (ch 5, sc in next loop) twice. Ch 3, turn. **9th row:** 8 dc in loop, * sc in next loop, ch 5, sc in next loop, 9 dc in next loop. Repeat from * across ending with sc in last loop. Turn.\n\n**10th row:** Sl st in next 2 sts, sc in next 5 sts, * ch 5, sc in next loop, ch 5, skip 2 dc, sc in next 5 dc. Repeat from * across. Ch 5, turn. **11th row:** Skip 1 sc, sc in next 2 sc, * (ch 5, sc in next loop) twice; ch 5, skip 1 sc, sc in next 2 sc. Repeat from * 2 more times. Ch 3, turn. **12th row:** Sc in next loop, * 9 dc in next loop, sc in next loop, ch 5, sc in next loop. Repeat from * across. Ch 7, turn. **13th row:** Sc in loop, * ch 5, skip 2 dc, sc in next 5 dc, ch 5, sc in next loop. Repeat from * once more, ch 5, skip 2 dc, sc in next 6 dc. Ch 8, turn. Repeat the 2nd to 13th rows incl. for length desired. Fasten off. Work along scalloped edge as follows: Attach thread to last sc at end of 1st row worked. **1st row:** (Ch 5, sc in next loop) 4 times; ch 5, skip 2 dc, sc in next st (point), then work 4 loops along other side of scallop to correspond. Continue in this manner to end of row. Turn. 2nd row: 7 sc in each of 1st 4 loops, * 3 sc in next loop, ch 3, sc in 3rd ch from hook, 3 sc in same loop (point), 7 sc in each of next 8 loops. Repeat from * across. Fasten off.\n\n**No. 8340 . . . 1st row:** Ch 57. Dc in 4th ch from hook and in next 10 ch, ch 11, skip 5 ch, dc in next ch, (ch 2, skip 2 ch, dc in next ch) twice; ch 11, skip 5 ch, dc in next 18 ch, then ch 3, skip 2 ch, sc in next ch, ch 3, skip 2 ch, dc in next ch (lacet made); dc in next 2 ch. Ch 3, turn. _Hereafter work only through back loops of stitches throughout, except when making ch-2 sps_. **2nd row:** Dc in next 2 dc, ch 5, dc in 17 dc, sl st in 3rd st of ch and in next 6 ch, * ch 2, skip 1 sp, dc in next dc, ch 2, skip 1 sp, sl st in 3rd st of next ch and in next 6 ch, skip 1 dc, dc in next 10 dc and in top of turning ch. * Ch 8, turn. **3rd row:** * Dc in 4th ch from hook and in next 4 ch, dc in 6 dc, ch 11, skip 5 dc, dc in next st, (ch 2, skip 2 sts, dc in next st) twice; * ch 24, skip 2 sps, dc in next sl st, (ch 2, skip 2 sts, dc in next st) twice; ch 11, skip 5 dc, dc in next 12 dc, make a lacet, then dc in next dc and in top of turning ch. Ch 3, turn. **4th row:** Dc in next 2 dc, ch 5, dc in 11 dc, sl st in 3rd st of ch and in next 6 ch, ch 2, skip 1 sp, dc in next dc, ch 2, skip 1 sp, sl st in 3rd st of next ch. Working into each st of ch make sc, h dc, dc, 2 tr, dc, h dc, sc, 2 sl sts, sc, h dc, dc, 2 tr, dc, h dc, sc and sl st (2 petals made). Repeat between *'s of 2nd row, ch 8, turn. **5th row:** Repeat between *'s of 3rd row, ch 24, skip 2 sps, 2 petals and 2 sps; dc in next sl st, (ch 2, skip 2 sts, dc in next st) twice; ch 11, skip 5 dc, dc in next 6 dc, make a lacet and finish as for 3rd row. Ch 3, turn. **6th row:** Dc in next 2 dc, ch 5, dc in 5 dc, sl st in 3rd st of ch and in next 6 ch, ch 2, skip 1 sp, dc in next dc, ch 2, sl st in 3rd st of next ch. Make 2 petals as before, then repeat between *'s of 2nd row. Ch 8, turn. **7th row:** Repeat between *'s of 3rd row, ch 36, skip 2 sps, 2 petals and 2 sps; dc in next sl st, (ch 2, skip 2 sts, dc in next st) twice; ch 10, skip 4 dc, dc in next dc, and finish row as before. Ch 3, turn. **8th row:** Dc in next 2 dc, ch 5, skip lacet, sl st in 3rd st of next ch and in next 5 ch, ch 2, skip 1 sp, dc in next dc, ch 2, skip 1 sp, sl st in 3rd st of next ch and in next 6 ch. Make 2 petals, then sl st in next 6 ch. Repeat between *'s of 2nd row, ch 1, turn. **9th row:** * Skip 1st dc, sl st in next 6 dc (6 dc decreased); ch 3, dc in next 4 dc and in 7 sl sts, ch 11, skip 2 sps, dc in next sl st, (ch 2, skip 2 sts, dc in next st) twice *; ch 37, skip 2 petals, dc in next sl st, (ch 2, skip 2 sts, dc in next st) twice; ch 11, skip 2 sps, dc in 6 sl sts, and finish row as before. Ch 3, turn. **10th row:** Dc in next 2 dc, ch 5, dc in 5 dc, sl st in 3rd st of ch and in next 6 ch, ch 2, skip 1 sp, dc in next dc, ch 2, skip 1 sp, sl st in 3rd st of next ch and in next 6 ch. Then working into each ch make sc, h dc, dc, 2 tr, dc, h dc, sc and sl st, then make sc under all petals at center, skip 1 ch, then continue along ch as before, making sl st, sc, h dc, dc, 2 tr, dc, h dc, sc and 7 sl sts. Repeat between *'s of 2nd row, ch 1, turn. **11th row:** Repeat between *'s of 9th row, ch 11, skip 2 petals, dc in next sl st, (ch 2, skip 2 sts, dc in next st) twice; ch 11, skip 2 sps, dc in 7 sl sts and in 5 dc, and finish row. Ch 3, turn. **12th row:** Dc in next 2 dc, ch 5, dc in 11 dc, sl st in 3rd st of ch and in next 6 ch, ch 2, skip 1 sp, dc in next dc, ch 2, skip 1 sp, sl st in 3rd st of next ch and in next 6 ch. Repeat between *'s of 2nd row, ch 1, turn. **13th row:** Repeat between *'s of 9th row, ch 11, skip 2 sps, dc in 7 sl sts and 11 dc, and finish row. Ch 3, turn. **14th row:** Same as 2nd row. Repeat 3rd to 14th rows incl. for length desired. Fasten off.\n\nEDGING: Attach thread to last dc at scalloped edge of last row, * ch 3, dc in corner between 2 points, ch 4, sl st in top of dc (p made), work 2 more p's, ch 3, sc in last dc of next point. Repeat from * 2 more times, ch 3, dc between next 2 rows, 3 p's, ch 3, sc at next point. Make 2 more \"p\" groups across other half of scallop to correspond with first half, ch 3, dc between next 2 rows (between scallops), 3 p's, ch 3, sc in next point, and continue thus across. Fasten off.\n\n_Enchanting Edgings_\n\n**No. 8404** MOTIFS . . . Starting at center, ch 22. Join with sl st to form a ring. **1st rnd:** 44 s c in ring. **2nd rnd:** * Ch 5, s c in next s c, ch 5, s c in same s c, ch 5, s c in each of next 10 s c. Repeat from * around. **3rd rnd:** Sl st in each of next 3 ch of ch-5 loop, * ch 3, in next loop make s c, ch 5 and s c. Ch 3, s c in next loop, ch 3, s c in next loop. Repeat from * around. Fasten off. Make another motif same as this to within 3rd rnd.\n\n**3rd rnd:** Sl st in each of next 3 ch of ch-5 loop, * ch 3, s c in next loop, ch 2, sl st in ch-5 loop of 1st motif, ch 2, s c back in same ch-5 of 2nd motif and complete as for 3rd rnd of previous motif. Fasten off. Make the necessary number of motifs for desired length, joining to previous motif as 2nd was joined to 1st, leaving 1 point free between joinings.\n\nHEADING . . . **1st row:** Attach thread to free loop next to corner loop and in line with joining. Ch 10, d c in next loop, ch 2, s c in next center loop, * ch 2, d c in next loop, ch 4, tr tr in next ch-3 loop, tr tr in corresponding loop of next motif, ch 4, d c in next loop, ch 2, s c in next loop. Repeat from * across, ending row with tr tr in loop next to corner loop. Ch 1, turn. **2nd row:** 6 s c in 1st sp, s c in d c, 3 s c in next sp, s c in next s c, * 3 s c in next sp, s c in next d c, 6 s c in next sp, s c in each of next 2 tr tr, 6 s c in next sp, s c in next d c, 3 s c in next sp, s c in next s c. Repeat from * across. Ch 3, turn. **3rd row** : * Skip 3 s c, d c in next s c, ch 3, d c in same place as last d c. Repeat from * across, ending row with skip 3 s c, d c in last s c. Ch 1, turn. **4th row:** S c in d c, * 2 s c in sp, s c in each of 2 d c. Repeat from * across. Fasten off.\n\n**No. 8383 1st row:** Ch 7, 4 d c in 7th ch from hook. Ch 7, turn. **2nd row:** 5 d c in last d c made. Ch 7, turn. Repeat the 2nd row for desired length. Ch 6, do not turn but work along long side as follows: **1st row:** * 4 d c in next ch-7 loop, ch 3, d c in d c at base of ch-7 loop, d c at base of same d c, ch 3. Repeat from * across, ending row with 4 d c in last loop. Ch 3, turn. **2nd row:** D c in each of 4 d c, d c in next ch, * ch 2, skip 2 ch of next ch-3, d c in next ch, d c in each of next 4 d c, d c in next ch. Repeat from * across. Ch 5, turn. **3rd row:** * Skip 2 sts, d c in next st, ch 2. Repeat from * across. Fasten off.\n\n**No. 8346** Make a chain slightly longer than desired length. **1st row:** D c in 8th ch from hook, * ch 2, skip 2 ch, d c in next ch. Repeat from * across. Ch 3, turn. **2nd row:** * 2 d c in ch-2 sp, d c in next d c, ch 4, skip 1 sp, s c in next sp., ch 4, skip 1 sp, d c in next d c. Repeat from * across, ending row with 4 d c. Ch 1, turn. **3rd row:** * S c in each of 4 d c, ch 9. Repeat from * across, ending row with 4 s c. Ch 1, turn. **4th row:** S c in each of 3 s c, * ch 4, skip 4 ch of ch-9, in next ch make d c, ch 2 and d c. Ch 4, skip 1 s c, s c in each of next 3 s c. Repeat from * across. Ch 1, turn. **5th row:** Skip 1 s c, s c in next s c, ** ch 4, in ch-2 sp make: d c, * ch 5, sl st in 4th ch from hook (1 p made), ch 1, d c in same ch-2. Repeat from * 2 more times, ch 4, skip l s c, s c in next s c. Repeat from ** across.\n\n_Remember, Tatting Cotton for cobwebby fine touches\u2014heavy Pearl Cotton for sturdy peasant effects_.\n\n_Dainty Edgings for Handkerchiefs and Doilies_\n\nFor dainty Edging use American Thread Company \"Silkine\", Mercerized Crochet Cotton sizes 50 to 100 or \"Silkine\" Tatting Cotton, White or Colors.\n\n**No. 769**\n\n**1st Row.** Ch 10, thread over hook 5 times, insert in 10th st from hook work off loops two at a time leaving the last 2 on hook, tr c in same st, work off last 3 loops together, ch 15 and repeat from beginning for desired length.\n\n**2nd Row.** 14 s c over 10 chain loop, 1 s c in each of the next 3 chs, picot, 1 s c in each of the next 2 chs and repeat for entire row.\n\n**No. 770**\n\nMake a ch the desired length and work one row of open meshes.\n\n**2nd Row.** Ch 5, d c in mesh, * ch 5, sl st in 2nd st for picot, ch 1, d c in next mesh, ch 2, d c in next mesh, repeat from * across row.\n\n**No. 771**\n\nCh 4, d c in 4th st from hook, * ch 3, turn, d c in d c, ch 6, sl st in base of 1st row, turn, 4 s c, p, 4 s c over loop, sl st in d c, ch 3, d c in d c, repeat from * for desired length and work a row of 6 ch loops on other side of edging.\n\n**No. 772**\n\n**1st Row.** Ch 9, turn, s c in third st from hook, 1 s c in each ch, ch 5, turn.\n\n**2nd, 4th and 6th Rows.** d c in third s c, ch 2, skip 2 sts, d c in next st, ch 2 and turn.\n\n**3rd and 5th Rows.** 2 s c in first mesh, s c in d c, 3 s c in next mesh, ch 5 and turn.\n\n**7th Row.** 2 s c in first mesh, s c in d c, 3 s c in next mesh, ch 12, sl st in base of 4th row, ch 2, sl st in base of 2nd row and turn.\n\n**8th Row.** 15 d c over ch, ch 6, turn, skip 4 d c, sl st in next st, ch 8, skip 4 d c, sl st in next st, ch 6, skip 4 d c, sl st in next st, ch 2, turn. 10 s c over first ch, 14 s c over next ch, 10 s c over last ch, d c in 1st s c of 7th row, * ch 2, skip 2 s c d c in next s c repeat from * and repeat pattern for desired length.\n\n**No. 773**\n\n**Work a row of s c.**\n\n**2nd Row.** * Ch 5, skip 3 sts, s c in next st, repeat from *.\n\n**3rd Row.** 2 d c in loop, * ch 5, 2 d c in next loop, repeat from *.\n\n**4th Row.** Ch 3, * 2 d c in loop, repeat from * twice, ch 2, 1 s c in next loop, repeat from beginning.\n\n**No. 774**\n\nWork 1 row of s c and 1 row of open meshes.\n\n**3rd Row.** Ch 4, * 1 d c in next d c, ch 2, 1 d c in same d c, repeat from *.\n\n**4th Row.** Ch 4, **tr c in loop, * picot, tr c in same loop, repeat from * twice, ch 3, 1 s c in next loop, ch 5, picot, ch 1, s c in next loop, ch 3, repeat from **.\n\n**No. 775**\n\nWork a chain for desired length or a row of s c over material and work 1 row of open meshes.\n\n**3rd Row.** ** work s c over 3 meshes. Ch 7, turn, sl st over 2nd d c, ch 1, turn, 15 s c over loop, s c over next open mesh, ch 2, turn, * skip 1 s c, 1 d c in next s c, ch 2, repeat from * 6 times, ch 2, sl st above next d c, ch 1, turn, * 2 s c in loop, picot, repeat from * 6 times, 2 s c over last loop, 8 s c over corner loop, repeat from **.\n\n**No. 776**\n\nWork a row of s c.\n\n**2nd Row.** * 3 s c, ch 4, 1 d c in first s c, ch 4, 1 d c in top of last d c made, skip 4 s c and repeat from *.\n\n**No. 777**\n\nWork 12 s c over material, ** ch 4, turn, skip 3 sts, 1 d c in next st, ch 2, skip 2 sts, 1 d c in next st, ch 4, skip 3 sts, sl st in next st, turn, 5 s c over loop, 1 s c in d c, * ch 4, sl st in first st for picot, repeat from * twice, 1 s c in d c, 5 s c over loop, work 20 s c over material and repeat from **.\n\n**No. 778**\n\nWork a row of s c.\n\n**2nd Row.** Ch 4, skip 2 sts, 1 d c in next st, ch 2, * skip 2 sts, 1 d c in next st, picot, 1 d c in same st, ch 2, skip 2 sts, 1 d c in next st, ch 2, repeat from *.\n\n**3rd Row.** * Ch 5, 1 s c in single d c, repeat from *.\n\n**4th Row.** * 3 s c over loop, picot, 2 s c over loop, picot, 3 s c over loop, repeat from *.\n\n**No. 779**\n\n**Work a row of s c.**\n\n**2nd Row.** Ch 5, * skip 2 sts, 1 d c in next st, ch 2, skip 2 sts, 1 d c in next st, ch 2, skip 3 sts, 2 d c in next st, ch 5, 2 d c in last d c made, ch 2, 2 d c in same space, ch 2, 1 d c in same space, ch 2, sl st in same space, ch 2, skip 3 sts, 1 d c in next st, ch 5, turn work, 1 s c in 2 ch loop, ch 4, 1 s c in next loop, ch 5, sl st in next d c, ch 5, turn work, 8 s c over loop, 3 s c, picot 3 s c over next loop, 8 s c over next loop, ch 2 and repeat from *.\n\n**No. 780**\n\nWork a row of s c.\n\n**2nd Row.** * Ch 6, skip 5 sts, 1 s c in next st, repeat from *.\n\n**3rd Row.** S c in loop, * ch 6, 1 s c in next loop, ch 3, 1 s c in same loop, ch 6, 1 s c in next loop, repeat from *.\n\n**4th Row.** * Ch 4, 4 d c in loop, ch 5, skip picot, 4 d c in next loop, ch 3, 1 s c in next loop, ch 4, 1 s c in next loop, repeat from *.\n\n**5th Row.** * 6 s c in loop, 1 s c in each d c, 4 d c in next loop, picot, 4 d c in same loop, 1 s c in each d c, 6 s c in next loop, 8 s c over next loop, repeat from *.\n\n**No. 781**\n\nCh for desired length or work a row of s c.\n\n**2nd Row.** * Ch 3, skip 2 sts 1 d c in each of the next 2 sts, repeat from *.\n\n**3rd Row.** * Ch 3, 2 d c in each of the next 2 d c, ch 3, 1 s c in each of the next 2 d c, repeat from *.\n\n**4th Row.** * Ch 3, 2 d c in each d c, ch 8, turn, sl st in 1st d c, ch 1, turn, 12 s c over loop, ch 3, 1 s c in each s c, repeat from *.\n\n**No. 782**\n\nCh 17, 2 d c in 8th st from hook, ch 2, 2 d c in same st, ch 4, skip 3 sts, 1 s c in next st, ch 4, skip 3 sts, 1 d c in each of the last 2 sts, ch 3, turn.\n\n**2nd Row.** 1 d c in 2nd d c, ch 4, 1 s c in loop, ch 4, 1 s c in next loop, ch 4, 2 d c in center of shell, ch 2, 2 d c in same space, ch 5, turn.\n\n**3rd Row.** 2 d c in shell, ch 2, 2 d c in same space, ch 4, 1 s c in center loop, ch 4, 1 d c in each d c, repeat the last 2 rows for desired length.\n\n**No. 783**\n\nWork a chain the desired length and work 1 row of d c, 1 row of 1 ch meshes and 1 row of d c.\n\n**4th Row.** Ch 3, ** skip 5 sts, 1 d c in next st, * ch 1, 1 d c in same st, repeat from * twice, ch 3 and repeat from ** to end of row.\n\n**5th Row.** * 1 s c in ch, ch 2, 1 d c in first ch of shell, ch 4, d c in next ch, ch 4, d c in next ch, ch 3, repeat from *.\n\n_Simplicity in Easily Worked Edgings_\n\nWE SUGGEST ROYAL SOCIETY CORDICHET, SIZE 20 OR ROYAL PERL\u00c9, SIZE 5\n\n**No. 3-17 . . .** Starting at one narrow end, ch 38. **1st row:** Dc in 8th ch from hook, (ch 2, sk 2 ch, dc in next ch) 10 times. Ch 5, turn. **2nd row:** Dc in next dc, (ch 2, dc in next dc) 9 times; ch 2, sk 2 ch, dc in next ch (11 sps made); then ch 8, sk 2 ch, sc in next ch (at base of 1st row). Turn. **3rd row:** 5 sc in ch-8 loop, ch 5, sl st in 5th ch from hook (p made), 5 sc in same loop, sc in dc, * in next sp work dc and tr, in next dc work 2 d tr with ch 1 between, in next sp work tr and dc, then sc in next dc. Repeat from * 3 more times; ch 3, dc in next dc, ch 2, dc in next dc, ch 2, sk 2 ch, dc in next ch. Ch 5, turn. **4th row:** Dc in next dc, ch 2, dc in next dc, ch 2, sk 2 ch, dc in next ch, ch 2, sc in ch-1 sp, (ch 5, sc in next ch-1 sp) 3 times; ch 2. tr in next sc. Ch 5, turn. **5th row:** Dc in next sc, (ch 2, sk 2 ch, dc in next ch, ch 2, dc in next sc) 3 times; (ch 2, dc in next dc) 3 times; ch 2, sk 2 ch, dc in next ch. Ch 5, turn. **6th row:** Dc in next dc, (ch 2, dc in next dc) 9 times; ch 2, sk 2 ch, dc in next ch, ch 8, sc in tr at end of 4th row. Turn. Repeat the 3rd to 6th rows inclusive for length desired, ending with the 5 sc, p and 5 sc at beginning of 3rd row.\n\n**No. 3-18 . . .** Starting at bottom of chart, ch 26. **1st row:** Dc in 8th ch from hook, (ch 2, sk 2 ch, dc in next ch) 3 times; dc in next 3 ch, (ch 2, sk 2 ch, dc in next ch) twice. Ch 5, turn. **2nd row:** Dc in next dc, 2 dc in sp, dc in next dc (bl over sp made), dc in next 3 dc (bl over bl made), (ch 2, dc in next dc) 3 times (3 sps over 3 sps made). Ch 7, turn. **3rd row:** Dc in 1st dc (1 sp increased), 3 sps, 1 bl, ch 2, sk 2 dc, dc in next dc (sp over bl made), make 1 more sp. Ch 5, turn. Starting with 4th row, follow chart to top. Repeat entire chart for length desired. Work a row of sc evenly along both straight and scalloped edges. Fasten off.\n\n**CHART No. 3-18**\n\n**No. 3-19 . . .** Make a chain slightly longer than length desired. **1st row:** Sc in 2nd ch from hook and in next 3 ch, * ch 5, sk 4 ch, dc in next 5 ch, ch 5, sk 4 ch, sc in next 7 ch. Repeat from * across for length desired, ending with 4 sc. Ch 1, turn. Cut off remaining chain. **2nd row:** Sc in 3 sc, * ch 5, sk 4 ch, dc in next ch, dc in next 2 dc, ch 3, sk 1 dc, dc in next 2 dc and in next ch, ch 5, sk 1 sc, sc in next 5 sc. Repeat from * across, ending with 3 sc. Ch 1, turn. **3rd row:** Sc in 2 sc, * ch 5, sk 3 ch, dc in next 2 ch, dc in next dc, ch 5, sc in next sp, ch 5, sk 2 dc, dc in next dc and in next 2 ch, ch 5, sk 1 sc, sc in 3 sc. Repeat from * across, ending with 2 sc. Ch 1, turn. 4th row: Sc in sc, * ch 5, sk 3 ch, dc in next 2 ch, dc in next dc, ch 5, sc in loop, ch 7, sc in next loop, ch 5, sk 2 dc, dc in next dc and in next 2 ch, ch 5, sk 1 sc, sc in next sc. Repeat from * across, ending with sc. Turn. 5th row: Sl st in next 4 ch, ch 3, dc in next ch and in next dc, ch 5, then holding back on hook the last loop of each tr, work 3 tr in ch-7 loop, thread over and draw through all loops on hook (cluster made); work * (ch 5, dc in 5th ch from hook, cluster in same loop) 4 times; ch 5, sk 2 dc, dc in next dc and in next 2 ch, sk next sc and next 3 ch, work dc in next 2 ch and in next dc, ch 5, cluster in next loop. Repeat from * across, ending with 3 dc. Fasten off.\n\n_Infinite Beauty in Small Design_\n\nWE SUGGEST ROYAL SOCIETY CORDICHET, SIZES 30-50\n\n**No. 3-22** . . . Starting at narrow end, ch 11. **1st row:** Dc in 4th ch from hook and in next 7 ch. Ch 10, turn. **2nd row:** Make a cross st as follows: Thread over twice, insert hook in next dc and draw loop through, thread over and draw through 2 loops on hook, thread over, sk 2 dc, insert hook in next dc and draw loop through, thread over and draw through 2 loops 4 times; ch 2, insert hook in center of cross and draw loop through, then complete as for a dc (cross st made). Ch 1, sk 1 dc, make another cross st (making 2nd leg in top of turning ch), ch 15, sc in same place as 2nd leg of cross st. Turn. **3rd row:** In ch-15 loop make sc, h dc and 20 dc, dc in next st, 2 dc in next sp, dc in next st, dc in next sp, dc in next st, 2 dc in next sp. dc in next st. Ch 10, turn. **4th row:** Dc in next dc, (ch 1, sk 1 dc, dc in next dc) 4 times. Ch 3, turn. **5th row:** (Dc in next sp, dc in next dc) 4 times. Ch 10, turn. Repeat the 2nd to 5th rows inclusive for length desired, ending with 3rd row. Do not fasten off but ch 9 to turn and work heading along top edge as follows: * Tr in next loop, ch 2, tr in same loop, ch 2. Repeat from * across, ending with ch 9, sl st at base of last dc of 1st row of edging. Fasten off.\n\n**No. 3-23** . . . Starting at narrow end, ch 9. **1st row:** Thread over twice, insert hook in 5th ch from hook and pull loop through; thread over and draw through 2 loops on hook, thread over, sk 2 ch, insert hook in next ch and pull loop through, (thread over and draw through 2 loops) 4 times; ch 2, dc in center point of cross (cross st made), tr in last ch. Ch 4, turn. **2nd row:** Tr in 1st leg of cross st, 2 tr in ch-2 sp, tr in last leg of cross st, tr in lop st of turning ch. Ch 4, turn. **3rd row:** Thread over twice, insert hook in next tr and pull loop through; thread over and draw through 2 loops on hook, thread over, sk 2 tr, insert hook in next tr and pull loop through, complete cross st as before, tr in top st of turning ch. Ch 4, turn. Repeat 2nd and 3rd rows alternately for length desired, ending with a cross st. Ch 1 and work scallops along long edge as follows: **1st row:** Sc in top of last tr, * ch 6, sc in base of same tr, ch 6, sc in top of next tr. Repeat from * across. Ch 5, turn. **2nd row:** * Make 9 tr in next loop, dc in next loop. Repeat from * across, ending with 9 tr in last loop. Ch 5, turn. **3rd row:** Sk next tr, * tr in center 5 tr of this group, ch 5, sc in next dc, ch 5, sk 2 tr of next group. Repeat from * across, ending with tr in center 5 tr of last group. Ch 4, turn. **4th row:** * Holding back the last loop of each tr on hook make tr in next 3 tr, thread over and draw through all loops on hook (cluster made), in tip of cluster just made make ch 3, sc, ch 5, sc, ch 3 and sc, ch 5, holding back the last loop of each dc on hook make dc in each of next 2 loops, thread over and draw through all loops on hook, ch 5, sk 1 tr. Repeat from * across. Fasten off.\n\n_Edgings for the Bath_\n\n_Washcloth_. . . S-522\n\n**J. & P. COATS \"KNIT-CRO-SHEEN,\" Art. A.64:** 1 ball each of No. 1 White, No. 10-A Canary Yellow and No. 12 Black.\n\n**Milwards Steel Crochet Hook** No. 7.\n\nA washcloth.\n\nStarting with Yellow, ch 2. **1st row:** Sc in 2nd ch from hook. Ch 5, turn. **2nd row:** In sc make dc, ch 1 and dc. Ch 1, turn. **3rd row:** Sc in ch-1 sp. Ch 5, turn. Repeat 2nd and 3rd rows alternately until piece reaches around washcloth, allowing for corners. Break off.\n\n**HEADING** . . . Now working along straight side of edging, attach White to first st, sc in same place, * ch 3, skip dc row, sc in next sp. Repeat from * across. Break off.\n\n**SCALLOPED EDGE** . . . Attach Black to opposite side, sc in sc, * ch 3, sc in ch-5 loop, ch 3, sc in next sc. Repeat from * across. Break off. Attach White to first Black sc, sc in same place, ch 3, * holding back on hook the last loop of each sc, insert hook in first ch-3 sp, thread over and draw loop through, insert hook in next ch-3 sp, thread over and draw loop through, thread over and draw through all loops on hook (joint sc), ch 3, sc in next sc, ch 3. Repeat from * across. Break off. Sew edging to washcloth.\n\n_Guest Towel_. . . S-523\n\n**COATS & CLARK'S O.N.T. TATTING-CROCHET, Art. C.21,** Size 70: 2 balls of No. 1 White and 1 ball of No. 9 Yellow . . . a few yards of White \"Knit-Cro-Sheen.\"\n\n**Milwards Steel Crochet Hook** No. 14.\n\nA grey guest towel.\n\n**Each motif measures 1\u00bc inches in diameter**\n\n**RING MOTIF\u2014First Ring** . . . Starting at center with White, ch 8. Join with sl st to form ring. **1st rnd:** Working over 3 strands of \"Knit-Cro-Sheen\" make 16 sc in ring. Join with sl st to first sc. Break off.\n\n**SECOND RING** . . . Work as for First Ring and join to First Ring by making a sl st in any sc. Make 3 more rings, joining the same way and leaving 3 sc free on inner edge and 11 sc free on outer edge. Join last ring to first ring.\n\nNow work around rings as follows: **1st rnd:** Attach White to 4th free sc on any ring, sc in same place, * (ch 4, sl st in 3rd ch from hook\u2014picot made) twice; ch 1, skip 3 sc, sc in next sc, (ch 1, picot) twice; ch 1, skip first 3 sc on next ring, sc in next sc. Repeat from * around. Join. **2nd rnd:** Sl st to center of next loop between picots, sc in same place, * (ch 1, picot) twice; ch 1, sc in center of next loop. Repeat from * around. Join and break off.\n\n**FLOWER MOTIF** . . . Starting at center with Yellow, ch 10. Join with sl st to form ring. Working over 3 strands of \"Knit-Cro-Sheen,\" make sc in ring, then make 9 sc over \"Knit-Cro-Sheen\" only, * ch 1, turn, still working over \"Knit-Cro-Sheen,\" make sc in each of 9 sc just made, 2 sc in ring, turn; skip first 2 sc, sc in first 5 sc on petal, make 4 sc over \"Knit-Cro-Sheen\" only. Repeat from * until 10 petals have been completed, ending with 1 sc in ring. Join to first sc made. Break off. Sew last petal to first petal.\n\nNow work around flower as follows: Attach White to tip of any petal, sc in same place, ch 1, picot, sl st in any loop on Ring Motif, picot, ch 1, sc in tip of next petal, ch 1, picot, sl st in next loop on Ring Motif, picot, ch 1, sc in tip of next petal, * (ch 1, picot) twice; ch 1, sc in tip of next petal. Repeat from * around. Join and break off.\n\nMake another Ring Motif, join to Flower Motif as before leaving 3 loops free on each side of joining. Continue in this manner, alternating Flower and Ring Motifs until piece is long enough to reach across towel, ending with a Ring Motif.\n\n**HEADING . . . 1st row:** Attach White to 3rd free loop preceding joining on First Motif, ch 9, dc in same loop, * ch 5, sc in next loop, ch 5, dc in next loop, ch 5, holding back on hook the last loop of each tr make tr in same loop and in first free loop on next motif, thread over and draw through all loops on hook (joint tr made), ch 5, dc in same loop. Repeat from * across, ending with dc, ch 5 and tr in 3rd free loop on last motif. Ch 8, turn. **2nd row:** * Skip next 5 ch, dc in next st. Repeat from * across. Break off.\n\nMake another piece the same way. Sew to towel.\n_Simple Crochet Stitches_\n\n**No. 1\u2014Chain Stitch (CH)** Form a loop on thread insert hook on loop and pull thread through tightening threads. Thread over hook and pull through last chain made. Continue chains for length desired.\n\n**No. 2\u2014Slip Stitch (SL ST)** Make a chain the desired length. Skip one chain, * insert hook in next chain, thread over hook and pull through stitch and loop on hook. Repeat from *. This stitch is used in joining and whenever an invisible stitch is required.\n\n**No. 3\u2014Single Crochet (S C)** Chain for desired length, skip 1 ch, * insert hook in next ch, thread over hook and pull through ch. There are now 2 loops on hook, thread over hook and pull through both loops, repeat from *. For succeeding rows of s c, ch 1, turn insert hook in top of next st taking up both threads and continue same as first row.\n\n**No. 4\u2014Short Double Crochet (S D C)** Ch for desired length thread over hook, insert hook in 3rd st from hook, draw thread through (3 loops on hook), thread over and draw through all three loops on hook. For succeeding rows, ch 2, turn.\n\n**No. 5\u2014Double Crochet (D C)** Ch for desired length, thread over hook, insert hook in 4th st from hook, draw thread through (3 loops on hook) thread over hook and pull through 2 loops thread over hook and pull through 2 loops. Succeeding rows, ch 3, turn and work next d c in 2nd d c of previous row. The ch 3 counts as 1 d c.\n\n**No. 6\u2014Treble Crochet (TR C)** Ch for desired length, thread over hook twice insert hook in 5th ch from hook draw thread through (4 loops on hook) thread over hook pull through 2 loops thread over, pull through 2 loops, thread over, pull through 2 loops. For succeeding rows ch 4, turn and work next tr c in 2nd tr c of previous row. The ch 4 counts as 1 tr c.\n\n**No. 7\u2014Double Treble Crochet (D TR C)** Ch for desired length thread over hook 3 times insert in 6th ch from hook (5 loops on hook) and work off 2 loops at a time same as tr c. For succeeding rows ch 5 turn and work next d tr c in 2nd d tr c of previous row. The ch 5 counts as 1 d tr c.\n\n**No. 8\u2014Rib Stitch.** Work this same as single crochet but insert hook in back loop of stitch only. This is sometimes called the slipper stitch.\n\n**No. 9\u2014Picot (P)** There are two methods of working the picot. (A) Work a single crochet in the foundation, ch 3 or 4 sts depending on the length of picot desired, sl st in top of s c made. (B) Work an s c, ch 3 or 4 for picot and s c in same space. Work as many single crochets between picots as desired.\n\n**No. 10\u2014Open or Filet Mesh (O M.)** When worked on a chain work the first d c in 8th ch from hook * ch 2, skip 2 sts, 1 d c in next st, repeat from *. Succeeding rows ch 5 to turn, d c in d c, ch 2, d c in next d c, repeat from *.\n\n**No. 11\u2014Block or Solid Mesh (S M)** Four double crochets form 1 solid mesh and 3 d c are required for each additional solid mesh. Open mesh and solid mesh are used in Filet Crochet.\n\n**No. 12** \u2014 **Slanting Shell St.** Ch for desired length, work 2 d c in 4th st from hook, skip 3 sts, sl st in next st, * ch 3, 2 d c in same st with sl st, skip 3 sts, sl st in next st. Repeat from *. **2nd Row.** Ch 3, turn 2 d c in sl st, sl st in 3 ch loop of shell in previous row, * ch 3, 2 d c in same space, sl st in next shell, repeat from *.\n\n**No. 13\u2014Bean or Pop Corn Stitch.** Work 3 d c in same 4 space, drop loop from hook insert hook in first d c 5 made and draw loop through, ch 1 to tighten st.\n\n**No. 14\u2014Cross Treble Crochet.** Ch for desired length, thread over twice, insert in 5th st from hook, * work off two loops, thread over, skip 2 sts, insert in next st and work off all loops on needle 2 at a time, ch 2, d c in center to complete cross. Thread over twice, insert in next st and repeat from *.\n\n**No. 15\u2014Cluster Stitch.** Work 3 or 4 tr c in same st always retaining the last loop of each tr c on needle, thread over and pull through all loops on needle.\n\n**No. 16\u2014Lacet St.** Ch for desired length, work 1 s c in 10th st from hook, ch 3 skip 2 sts, 1 d c in next st, * ch 3, skip 2 sts, 1 s c in next st, ch 3, skip 2 sts 1 d c in next st, repeat from * to end of row, 2nd row, d c in d c, ch 5 d c in next d c.\n\n**No. 17\u2014Knot Stitch (Sometimes Called Lovers Knot St.)** Ch for desired length, * draw a \u00bc inch loop on hook, thread over and pull through ch, s c in single loop of st, draw another \u00bc inch loop, s c into loop, skip 4 sts, s c in next st, repeat from *. To turn make \u215c\u2033 knots, * s c in loop at right of s c and s c in loop at left of s c of previous row, 2 knot sts and repeat from *.\n_Metric Conversion Chart_\n\nCONVERTING INCHES TO CENTIMETERS AND YARDS TO METERS\n\nmm \u2014 millimeters cm \u2014 centimeters m \u2014 meters\n\nINCHES INTO MILLIMETERS AND CENTIMETERS  \n _(Slightly rounded off for convenience)_\n\nYARDS TO METERS\n\n_(Slightly rounded off for convenience)_\n\nwww.doverpublications.com\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nROUTLEDGE LIBRARY EDITIONS: WORDSWORTH AND COLERIDGE\n\n* * *\n\nVolume 10\n\nTHE DESIGN OF  \n _BIOGRAPHIA LITERARIA_\n\n* * *\n**THE DESIGN OF _BIOGRAPHIA LITERARIA_**\n\nCATHERINE MILES WALLACE\n\nFirst published in 1983 by George Allen & Unwin (Publishers) Ltd\n\nThis edition first published in 2016  \nby Routledge  \n2 Park Square, Milton Park, Abingdon, Oxon OX14 4RN\n\nand by Routledge  \n711 Third Avenue, New York, NY 10017\n\n_Routledge is an imprint of the Taylor & Francis Group, an informa business_\n\n\u00a9 1983 Catherine Miles Wallace\n\nAll rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers.\n\n_Trademark notice_ : Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe.\n\n_British Library Cataloguing in Publication Data_  \nA catalogue record for this book is available from the British Library\n\nISBN: 978-1-138-67344-1 (Set)  \nISBN: 978-1-315-56191-2 (Set) (ebk)  \nISBN: 978-1-138-66997-0 (Volume 10) (hbk)  \nISBN: 978-1-138-67001-3 (Volume 10) (pbk)  \nISBN: 978-1-315-61786-2 (Volume 10) (ebk)\n\n**Publisher's Note**  \nThe publisher has gone to great lengths to ensure the quality of this reprint but points out that some imperfections in the original copies may be apparent.\n\n**Disclaimer**  \nThe publisher has made every effort to trace copyright holders and would welcome correspondence from those they have been unable to trace.\n**The Design of _Biographia Literaria_**\n\nCATHERINE MILES WALLACE\n\n_Department of English, Northwestern University_\n\nLondon\n\nGEORGE ALLEN & UNWIN\n\nBoston Sydney\n\u00a9 Catherine Miles Wallace, 1983  \nThis book is copyright under the Berne Convention. No reproduction without permission. All rights reserved.\n\n**George Allen & Unwin (Publishers) Ltd,  \n40 Museum Street, London WC1A 1LU, UK**\n\nGeorge Allen & Unwin (Publishers) Ltd,  \nPark Lane, Hemel Hempstead, Herts HP2 4TE, UK\n\nAllen & Unwin, Inc.,  \n9 Winchester Terrace, Winchester, Mass. 01890, USA\n\nGeorge Allen & Unwin Australia Pty Ltd,  \n8 Napier Street, North Sydney, NSW 2060, Australia\n\nFirst published in 1983\n\n* * *\n\n**British Library Cataloguing in Publication Data**\n\nWallace, Catherine Miles   \nThe design of Biographia Literaria.   \n1. Coleridge, Samuel Taylor, _Biographia literaria_   \nI. Title.   \n808.1 PR4476   \nISBN 0-04-800016-7\n\n* * *\n\n**Library of Congress Cataloging in Publication Data**\n\nWallace, Catherine Miles.   \nThe design of Biographia Literaria.   \nIncludes bibliographical references and index.   \n1. Coleridge, Samuel Taylor, 1772\u20131834. Biographia   \nliteraria. I. Title.   \nPR4476.W34 1983 821\u2032.7 [B] 82-20791   \nISBN 0-04-800016-7\n\n* * *\n\nSet in 10 on 11 point Plantin by Typesetters (Birmingham) Limited  \nand printed in Great Britain  \nby Mackays of Chatham\n\n# **Contents**\n\nPreface\n\nAbbreviations\n\nAcknowledgments\n\n**1** The Chamois Hunter\n\n**2** Starting-Points\n\n**3** The Associative Fancy\n\n**4** Imagination's Synthesis of Being and Knowing\n\n**5** Imagination, Philosophic Consciousness and the 'True and Original Realism'\n\n**6** Poetry\n\n**7** Wordsworth and Poetic Diction\n\n**8** Wordsworth and the Imaginative Particular\n\n**9** Conclusion\n\nNotes\n\nIndex\n\n# **Preface**\n\nArthur Symons said it best: ' _Biographia Literaria_ is the greatest book of criticism in English, and one of the most annoying in any language.' Many have agreed that Coleridge's brilliance comes shrouded in an obscure, infuriating intricacy. In order to analyze that intricacy, and yet not to write a book as labyrinthine as the _Biographia_ itself, I have 'at all times endeavoured to look steadily at my subject': the order and relation of parts that I call 'design'. I occasionally discuss Coleridge's plagiarisms and his misquotations, but only when they create or solve interesting rhetorical problems. I discuss Coleridge's personal history only when such details reveal particularly well how he creates a literary life'. I discuss Wordsworth in somewhat greater detail, but I am not centrally concerned with illuminating that friendship and its theoretical disputes. Wordsworth appears in _Biographia Literaria_ partly as an individual, but more centrally as the ideal philosophic poet \u2014 and in all instances as Coleridge's idea of him. I step outside that conception only when recourse to Wordsworth's criticism proves requisite to the full understanding of how Coleridge designs his counter-argument. None the less, the Wordsworth portrayed in my pages remains the _Biographia_ 's image of him. Major work needs to be done on each of these issues.\n\nIn evaluating _Biographia Literaria_ as a discourse, I do not commonly engage philosophic 'first questions' in my own right: I am not a philosopher. I do point out how Coleridge's conclusions depend upon assumptions usually drawn from Christianity. I also describe how the coherence of his theory depends not only on the validity of these assumptions, but also upon the ways in which they modify and restrict each other. But in my own analyses I ordinarily grant the validity of these starting-points.\n\nAnd it should be admitted from the start that I have thoroughly enjoyed the puzzles Coleridge's discourse presents. One day a colleague commented ruefully to me that one must be a true 'Coleridgean' to love the _Biographia_. Perhaps so. But it is also true that one who becomes fond of bitter tastes never calls them sweet. I regularly distinguish between the intricacy that derives from the sophistication of Coleridge's thinking, and the obscurity that derives from the manner of his composing. Yet both his errors and his intricacies reflect consistent habits: comprehending the design as the work of both a genius and an exhausted, troubled man can render more easily accessible what is permanently valuable in _Biographia Literaria_.\n\nThis book is intended for three general kinds of readers. First, of course, sworn 'Coleridgeans'. For them, the crucial issue will be my argument about how Coleridge composed \u2014 an argument tested against the _Biographia_ because it is so central among his works. Secondly, those interested in the _Biographia_ only in parts. _Biographia Literaria_ is, alas, usually read only in parts except by the most determined; and so I allowed a certain pragmatism to prevail in my own design. This book is written so that one can discover how the 'interesting' parts fit into the whole structurally, conceptually, and rhetorically. Biographers, philosophers, Wordsworthians, literary theorists or historians, and students of poetry can each select the chapters that pertain to the parts of the _Biographia_ that matter to them. My third set are those whom Coleridge himself addressed \u2014 not literary professionals of any ilk, but that wider group of people who care about literature, who value its contributions to human culture, who seek reasonable principles of critical judgment. Accommodating the needs of these readers has at times required that I verge upon an old but useful format, the so-called 'commentary'. Coleridgeans will recognize, I trust, a certain familiar irony in this: just a few comments, if the right comments, can illuminate much.\n\nAfter my manuscript had been sent to Allen & Unwin, Cambridge University Press published Kathleen Wheeler's _Sources, Processes and Methods in Coleridge's 'Biographia Literaria'_. In a few places I have noted major points of disagreement, but these comments are necessarily limited in scope. Our agreements, however, are many. We both argue, in her words, that 'The reader of the _Biographia_ must at all times be aware that his intellectual activity is a primary subject of the text' (p. 107). The complementarity of our works may be more than coincidental: at slightly different times, and unknown to each other, we both studied with John Wright at the University of Michigan in Ann Arbor. Mr Wright directed my dissertation, which underlies the present study; I am delighted to acknowledge how much I owe to the rigor and detail of his criticisms at that stage. More recently, this study or parts of it were evaluated by Don H. Bialostosky, Charles O. Hartman, J. R. de J. Jackson, Lawrence Lipking, and Philip C. Rule, SJ. To each I am grateful for good advice. Errors and infelicities that remain are my responsibility. Both early and late, my work with Coleridge has been shared by Warren H. Wallace, MD, to whom my debts and my gratitude are deepest of all.\n\n# **Abbreviations**\n\n_AP_ | _Anima Poetae_ , ed. Ernest Hartley Coleridge (London: William Heinemann, 1895).  \n---|---  \nAR | _Aids to Reflection_ , ed. Henry Nelson Coleridge (1840; reprinted London: Kennikat Press, 1971).  \nBL | _Biographia Literaria_ , ed. J. Shawcross, 2 vols (London: Oxford University Press, 1907).  \nCCS | _On the Constitution of Church and State_ , ed. John Colmer, _The Collected Works of Samuel Taylor Coleridge_ , Vol. 10 (Princeton, NJ: Princeton University Press, 1976).  \nCL | _Collected Letters of Samuel Taylor Coleridge_ , ed. Earl Leslie Griggs, 6 vols (Oxford: Clarendon Press, 1956\u20139).  \nCN | _The Notebooks of Samuel Taylor Coleridge_ , ed. Kathleen Coburn, 3 vols in 6 pts (Princeton, NJ: Princeton University Press, 1957\u201373).  \nCOS | _Coleridge on Shakespeare: The Text of the Lectures of 1811\u20131812_ , ed. R. A. Foakes (London: Routledge & Kegan Paul, 1971).  \nF | _The Friend_ , ed. Barbara E. Rooke, 1 vol in 2 pts, _The Collected Works of Samuel Taylor Coleridge_ , Vol. 4 (Princeton, NJ: Princeton University Press, 1969).  \nLPR | _Lectures 1795 on Politics and Religion_ , ed. Lewis Patton and Peter Mann, _The Collected Works of Samuel Taylor Coleridge_ , Vol. 1 (Princeton, NJ: Princeton University Press, 1971).  \nLS | _Lay Sermons_ , ed. R. J. White, _The Collected Works of Samuel Taylor Coleridge_ , Vol. 6 (Princeton, NJ: Princeton University Press, 1969).  \nPhL | _The Philosophical Lectures_ , ed. Kathleen Coburn (London: The Pilot Press, 1949).  \nShedd III | _Biographia Literaria, The Complete Works of Samuel Taylor Coleridge_ , ed. W. G. T. Shedd, Vol. III (New York: Harper & Bros, 1884).  \nShedd V | _Literary Remains, The Complete Works of Samuel Taylor Coleridge_ , ed. W. G. T. Shedd, Vol. V (New York: Harper & Bros, 1884).  \nTT | _Specimens of the Table Talk of the Late Samuel Taylor Coleridge, The Complete Works of Samuel Taylor Coleridge_ , ed. W. G. T. Shedd, Vol. VI (New York: Harper & Bros, 1884).\n\n# **Acknowledgments**\n\nI would like to thank Marilyn Gaull, editor of _The Wordsworth Circle_ , for permission to reprint The function of autobiography in Coleridge's _Biographia Literaria_ ', an essay that appears here as chapter one; and William Kupersmith for permission to reprint portions of an essay first published in _Philological Quarterly_ as 'Coleridge's theory of language'.\n\n# **1 The Chamois Hunter**\n\n> The musician may tune his instrument in private, ere his audience have yet assembled: the architect conceals the foundation of his building beneath the superstructure. But an author's harp must be tuned in the hearing of those, who are to understand its after harmonies; the foundation stones of his edifice must lie open to common view, or his friends will hesitate to trust themselves beneath the roof.\n> \n> _The Friend_ , I, 14\n\n> A man's principles... are the life of his life.\n> \n> _The Friend_ , I, 97\n\nIn this half of the century, the familiar portrait has been redrawn: we no longer see Samuel Taylor Coleridge as a genius helplessly mired in addictions, neuroses and morbid Christianity. Growing numbers of scholars have demonstrated the depth and subtlety of Coleridge's thought. His originality has been re-evaluated; his reputation has grown. _Biographia Literaria_ deserves a new look as well. Like Coleridge himself, it is not a rambling, foggy, inconsistent traveler toward isolated, cryptic insights. It is a formal and rhetorical whole designed to engage the reader not simply as a philosopher-critic, but as a person, as a union of intellectual, emotional and moral powers. Its design reveals a coherent and genuinely imaginative vision of the necessary character of modern discourse.\n\nYet it is a difficult book. At first glance, there is ample evidence that _Biographia Literaria_ is a fragmented disaster whose difficulties can neither be resolved nor understood: the 'missing' transcendental construction in chapter XIII; the incorporations from Schelling; its origin as a preface to _Sibylline Leaves_. Such evidence appears to justify Carlyle's malefic description:\n\n... instead of answering [a question], or decidedly setting out towards an answer of it, he would accumulate formidable apparatus, logical swim-bladders, transcendental life-preservers and other precautionary and vehiculatory gear, for setting out; perhaps did at last get under way, \u2013 but was swiftly solicited, and turned aside by the glance of some radiant new game on this hand or that, into new courses; and ever into new; and before long into all the Universe.\n\n'Solicited' is the central word here: seduced, enticed, led astray. The Victorians said Coleridge lacked 'will power'; the moderns that he was hopelessly 'neurotic'. Yet the principle remains the same: the formal disorder of the works reflects the personal disorder of the man.\n\nBut the evidence for _Biographia Literaria'_ s formal disorder shrivels abruptly if examined in strong light. The infamous chapter XIII, for example, culminates an extensive pattern of reference to the intellectual powers of Englishmen; it is an elaborately planned appeal to the philosophic reader. Philosophic readers have been appropriately disappointed: the strategy succeeds at least in part. But such readers err in judging this chapter a neurotic break, or a failure of nerve, or a change in plan literally unexpected by its author. There is substantial evidence to the contrary. In a parallel way, the _Biographia_ 's origin as a preface reflects no reprehensible spontaneity on Coleridge's part: one finds plans for the _Biographia_ in letters and notebook entries dating back at least to the autumn of 1803. The plagiarisms from Schelling have been evaluated in general by Thomas McFarland, and in painstaking detail by Elinor Stoneman Shaffer, both of whom conclude that the literal appearance of plagiarism is misleading. In the light of recent scholarship, this traditionally _prima facie_ evidence loses its impressive appearance.\n\nBut the reputation of _Biographia Literaria_ has also rested on the experience of many attentive and informed readers. For a text so often described as unreadable, it has been read more often and valued more highly than quite makes sense. The paradox is revealing: there is no frustration quite the equal of intuiting a coherence that refuses to emerge into the full daylight of objectifying comprehension. There are, in fact, many brief descriptions of the _Biographia_ 's general thematic unity. Shawcross is exemplary. He attributes 'the miscellaneous character of the book' to the poor state of Coleridge's 'health and spirits', but goes on to say:\n\nIt is with this end in view ['the desire... to state clearly, and defend adequately, his own poetic creed'] that, in the autobiographical portion of the book, he describes the growth of his own literary convictions; that, in the philosophical, he seeks to refer them to first principles; and that, in the criticism of Wordsworth's poetry and poetic theory, he emphasizes the differences which, as he imagines, exist between Wordsworth and himself. ( _BL_ , I, xcii)\n\nColeridge's complaint seems to the point:\n\nNow surely a [work] the contents and purposes of which are capable of being faithfully and compleatly ennumerated in a sentence of 7 or 8 lines, and where all the points treated of tend to a common result, cannot justly be regarded as a motley Patch-work, or a Farrago of heterogenous Effusions! ( _LS_ , 114 n)\n\nThis general thematic unity is seldom denied. Yet it is one thing for an attentive reader to perceive the relations among the topics Coleridge discusses; it is something altogether different to perceive that Coleridge so manages his discourse as to define exactly how 'all points treated of tend to a common result'. A sympathetic reader can link the parts to a postulated common result, but this does not prove that _Biographia Literaria_ , as a discourse, generally succeeds in establishing these relations through various formal and rhetorical strategies. Such strategies comprise what I call 'design': the blueprint of the whole, regarded both as a governing structure, and as the intent or governing idea implicit in and enacted by this structure.\n\nThe generations of scholars and critics succeeding Shawcross have explicated Coleridge's literary convictions' and his 'first principles' to reveal a thinker known to very few in 1907: a Coleridge who is erudite, philosophically acute, and humanly profound. As we have more and more fully seen how his intricacy of mind engages complex human questions, it becomes less possible to believe that so great a thinker could have been so fundamentally incompetent a writer. Common sense questions whether such extraordinary powers of concentration and synthesis could have been so entirely suspended when he sat down to compose. He himself often insisted that genius and command of language vary together. His books are not perfectly written \u2013 whose are? \u2013 but their difficulty resides primarily in the complexity of his ideas, and in the complexity of design these ideas require for their intelligible presentation. Coleridge knew he would be charged with obscurity; he often tried both to defend himself, and to preclude the charge through elaborate explanations of his principles in writing. Understanding these principles alleviates much of the confusion and frustration that so many readers experience.\n\nColeridge's desire to make his readers think, to make them engage genuine ideas, underlies the design _of Biographia Literaria_. It also accounts in part for the book's reputation as excessively difficult reading. In what he regarded as 'the finest passage in the _Friend_ ', Coleridge acknowledges the problems he creates for his readers.\n\nAlas! legitimate reasoning is impossible without severe thinking, and thinking is neither an easy nor an amusing employment. The reader, who would follow a close reasoner to the summit and absolute principle of any one important subject, has chosen a Chamois-hunter for his guide. Our guide [Coleridge himself, as The Friend] will, indeed, take us the shortest way, will save us many a wearisome and perilous wandering, and warn us of many a mock road that had formerly led himself to the brink of chasms and precipices, or at best in an idle circle to the spot from whence he started. But he cannot carry us on his shoulders: we must strain our own sinews, as he has strained his; and make firm footing on the smooth rock for ourselves, by the blood of toil from our own feet. ( _F_ , I, 55, and n. 2)\n\nFeet that are sticky with blood \u2013 it is a lurid image, but all too often it has struck me as entirely apt. Owen Barfield grants that the relation between subject and object is no doubt the _pons asinorum_ of Coleridge's philosophic endeavor, but then argues that 'the concept, and perhaps the _experience_ , of thinking as an _act_ , or as an \"act and energy,\" are the toll-gate in the middle of the bridge, the barrier that has to be opened before we can get across'. Barfield demonstrates how thoroughly this central concept influences all of Coleridge's thought; my major concern is the way in which it shapes his composition.\n\nColeridge's compositions are designed to make the reader engage a particular problem or issue in ways that will lead him to the conclusions that Coleridge has already reached. He wants to make us think things through for ourselves, under his tutelage. To the extent that his particular interests are all versions of a predominant interest in thinking itself, one can say that he writes in ways designed to help us think about thinking. That is no small ambition. Further challenges to the reader emerge from the fact that Coleridge seldom accommodates to lesser energies or lesser talents than his own. He may lead us away from precipices, but he does not hesitate to lead us up rough and steep paths if these take us where he wants to go.\n\nSome of the reader's troubles derive from Coleridge's lack of skill in the fine art of enticing tired minds to further and further effort. But often the reader stumbles because the way is genuinely and irreducibly difficult. As Sara Coleridge notes, the original thinker cannot rely on 'known ways and smooth well-beaten tracks': he forces his readers into unaccustomed mental exercise. Both contemporary and modern readers have echoed this image of a difficult path to describe the effort Coleridge's discourse requires. The image is particularly apt: as Coleridge often observed, the Greek word we translate as 'method' translates literally as 'way' or 'path' or 'mode of transit'. The 'Essays on Method' in _The Friend_ claim that valid method demands powerful imagination. So does reading the _Biographia_. As Bishop C. Hunt, Jr, observes, 'what we might call the \"dramatic\" element in philosophy, the process of search and its written reenactment, assumes a larger significance. Much of Coleridge's best writing can be read as a kind of dramatic monologue in prose, a mimetic representation of the mind in the act of thinking something through.' Thinking along with Coleridge as he thinks something through demands more than energy or learning. It requires that we read his prose with the same use of the same skills we engage when we read his poetry.\n\nWhen Coleridge asserts that he wants his readers to _think_ , he means that we must develop and exercise the ability to observe our own minds. We must acquire the self-consciousness that is secondary imagination's most basic act. His clearest statement of this requirement is the distinction between _thinking_ and _attending_ in _Aids to Reflection_ :\n\nIt is a matter of great difficulty, and requires no ordinary skill and address, to fix the attention of men on the world within them, to induce them to study the processes and superintend the works which they are themselves carrying on in their own minds; in short, to awaken in them both the faculty of thought* and the inclination to exercise it. For alas! the largest part of mankind are nowhere greater strangers than at home.\n\nHe who would explicate creativity itself requires a thinking audience no less than he who would explicate 'moral or religious truth'. Whether the critic's 'inward experiences' be those of reading or of writing, his reader needs the imaginative power of self-consciousness as a prior condition to understanding imagination itself in rigorous theoretical terms. The primary facts essential to the intelligibility of my principles I can prove to others only as far as I can prevail on them to retire _into themselves_ and make their own minds the objects of their stedfast attention' ( _F_ , I, 21).\n\nAlone and unbalanced, this requirement provides nothing less than a license for solipcism. But, despite what has been asserted in his name, Coleridge was no advocate of an 'autonomous' imagination. His many explanations of the relation between ideas (the content or object of thinking) and experience reveal how his method excludes the irresponsible and the eccentric. Ideas and laws are correlative terms, strictly defined. An idea is a subjectively originating set of relations exactly echoed by relations observable in experience. An idea whose counterpart is a law of the natural world can be 'objectively' or physically demonstrated. One who understands will be compelled to assent. An idea whose counterpart is a law of the mind cannot be physically demonstrated, because it has a psychic not physical referent. Furthermore, assent cannot be logically compelled because the spirit itself is essentially free: it cannot be coerced into generating the confirming mental experience. None the less, 'At the annunciation of _principles_ , of _ideas_ , the soul of man awakes, and starts up, as an exile in a far distant land at the unexpected sounds of his native language, when after long years of absence, and almost of oblivion, he is suddenly addressed in his own mother-tongue' ( _LS_ , 24). The imaginative person intuitively and immediately grasps the truth of a valid idea.\n\nExperience confirms \u2013 although it cannot demonstrate \u2013 the validity of ideas about non-physical realities. For instance, experience illustrates but does not prove the existence of God:\n\nAssume the existence of God, \u2013 and then the harmony and fitness of the physical creation may be shown to correspond with and support such an assumption; \u2013 but to set about _proving_ the existence of a God by such means is a mere circle, a delusion. It can be no proof to a good reasoner.... ( _TT_ , 22 February 1834)\n\nHistorians face the same situation as theologians:\n\nIf you ask me how I can know that this idea \u2013 my own invention \u2013 is the truth, by which the phenomena of history are to be explained, I answer, in the same way exactly that you know that your eyes were made to see with; and that is, because you _do_ see with them... this idea, not only like a kaleidoscope, shall reduce all the miscellaneous fragments into order, but shall also minister strength, and knowledge, and light... . ( _TT_ , 14 April 1833)\n\nAs _Biographia Literaria_ explains in detail, the same is true of literary critics. They must illustrate their ideas by particular reference to texts, while yet remembering that criticism is not an empirical science.\n\nColeridge's intent to make his readers think, to make them engage ideas, accounts for the oft-lamented 'digressive' texture of his works, especially the _Biographia_. In _The Friend_ , he cites Plato approvingly: '\"It is difficult, excellent friend! to make any comprehensive truth compleatly intelligible, unless we avail ourselves of an example. Otherwise we may as in a dream, seem to know all, and then as it were, awakening find that we know nothing\"' ( _F_ , I, 148). His 'examples' seem digressive in part because they are lengthy, and in part because he obviously enjoys them for their own sake; but primarily they seem so because they form no part of an induction. They relate not to each other, but to the ideas they illustrate. They are not particulars from which he generalizes and concludes according to the logical principles governing the use of evidence. The rational and imaginative reader who tries (consciously or unconsciously) to synthesize the anecdotes into a proper induction will be driven to distraction.\n\nThis lack of 'system' creates hazards for the unwary, but it is easy enough to discern Coleridge's motives. According to Coleridge, strictly objective observation, abstraction and generalization are a waste of time, because the number of potentially relevant particulars is nearly infinite. Without a prior idea or, as Jackson so aptly paraphrases, a good _hunch_ , 'the life of an ante-diluvian patriarch would be expended... in merely polling the votes, and long before he could [have] commence[d] the process of simplification, or have arrived in sight of the law' ( _F_ , I, 485). Hence, tor one who possesses an idea, the role of particulars in a discourse can be largely rhetorical. And yet for Coleridge this role is not rhetorical in the degraded sense. Because empirical confirmation stands prominently among the tests for a valid idea, Coleridge's particulars are 'illuminations' somewhat as Blake uses that word. They comprise one of the major strategies whereby he tries to prevail upon and help his readers to 'retire into themselves'. As a consequence, one who reads the _Biographia_ ignoring its 'digressions' wastes his time no less surely than the empirical patriarch: these illuminations are crucial to the full intelligibility of Coleridge's principles.\n\nTo the problems inherent in the strategy one must add the fact that, as Coleridge himself admitted, he tended to offer too many of these illuminations. '[M]y illustrations swallow up my thesis \u2013 I feel too intensely the omnipresence of all in each, platonically speaking...' ( _CN_ , II, 2372). As De Quincey so shrewdly observed, that excess can confuse Coleridge's audience, and obscure the idea he is trying to illuminate:\n\nColeridge, to many people, and often I have heard the complaint, seemed to wander; and he seemed then to wander the most, when, in fact, his resistance to the wandering instinct was greatest \u2013 viz., when the compass and huge circuit by which his illustrations moved travelled farthest into remote regions before they began to revolve. Long before this coming round had commenced most people had lost him, and naturally enough supposed that he had lost himself.... However, I can assert, upon my long and intimate knowledge of Coleridge's mind, that logic the most severe was as inalienable from his modes of thinking as grammar from his language.\n\nIn the most practical of ways, then, the 'digressive' texture coherently reflects Coleridge's desire to help his readers think about ideas. Even for the most astute reader, the strategy probably fails at times, partly because it so easily breaks down into the excessive indirectness characteristic of Coleridge, and partly because any reader habituated to sustained abstract argument may be annoyed or unduly distracted by the change of pace. But it will fail less often for a reader who understands in advance what Coleridge is doing, and why.\n\nAn example of my own will illustrate the character of Coleridge's illuminations. Shawcross dismisses most of chapter II: 'the irritability of men of genius' is 'a quite irrelevant topic' ( _BL_ , I, 212). Irrelevant to what? To poetic diction? To autobiography? One who accepts Shawcross's judgment ignores Coleridge's first major account of the psychological consequences of imagination. The 'irritability' issue forges the major link between the controversy over _Lyrical Ballads_ and the theory of imagination. Without a firm grasp of this link \u2013 and the principal metaphors that subsequently call it into play \u2013 one is apt to miss nearly half of what the _Biographia_ says about imagination. Shawcross's error is provoked by the conclusion of chapter I: a profusion of anecdotes superficially concerned with Coleridge's delight in literary parody. Yet, on a closer look, the 'Higgenbottom' sonnets illustrate that the simple style he advocates can easily descend into the trivial. This story, and the related stories centered around The Ancient Mariner', reveal Coleridge as a man blessed with a good-humored, common-sense balance both in his theories and in his personality. Chapter II focusses sharply on this question of balance in personality and in literary style as evident in great poets and anonymous critics; it smoothly and coherently develops the argument about diction introduced in the preceding chapter by tracing qualities of diction to an origin in qualities of mind. The discontinuity that misled Shawcross resides only in the illustrations Coleridge uses: the first chapter draws primarily on memories of his youth and early manhood; the second on his knowledge of literary history and his encounters with contemporary criticism.\n\nColeridge's desire to make his readers think explains more than the book's superficially digressive texture. It explains the principal feature of the design: its fusion of personal history and philosophic argument. To understand this design, one must first recognize that the _Biographia_ 's autobiography is art \u2013 not chronicle. One must distinguish between Coleridge the author and 'Coleridge' the created narrator, the hero of a story that has a coherent development and formal unity that only art can impose upon a real life. Coleridge the author adapts his own history to perfect it as the vehicle for a philosophic inquiry into the distinction between fancy and imagination. The diversity of his own experience supplies the illustrations that progressively reveal the idea of imagination: the stages of the reader's intuition of this idea correspond to the stages of 'Coleridge's' intellectual life. As George Watson observes, _Biographia Literaria_ follows 'a plan neither narrative nor logical but a disconcerting combination of the two'.\n\nSara and H. N. Coleridge, and J. Shawcross, have pointed out the differences between this created self and the real man; but the necessarily delicate analysis of these differences should be done only by a psychologically astute biographer. For present purposes what matters most is that Coleridge turns the primary autobiographical act of self-creation into a vehicle for philosophic and literary discourse. This central feature of the _Biographia_ 's design in part reflects his commitment to the ideal unity of poetry and philosophy. But in part it also solves major practical problems. The philosophic analysis is not complete, yet it presents itself as capable of formal closure. The strictly literary unity of the autobiography seeks to bridge this gap by eliciting the reader's confidence that closure can be attained. The autobiographical narrative, in turn, is often and severely interrupted by abstract logical arguments. But the created self is a metaphysician from boyhood; he incorporates these arguments \u2013 not without strain, at times \u2013 as accounts of his scholarly endeavors.\n\nThe synthesis of autobiography and philosophy serves another, more crucial purpose as well. It helps Coleridge convince his readers of something he cannot prove logically: that the theory of imagination does not lead to pantheism. The created self is deeply religious and repeatedly concerned with the moral consequences of materialist philosophies. We may judge that his theory is wrong, or formulated badly; but we cannot doubt that it is orthodox in intent. For this person, the pursuit of philosophic questions and the discovery of identity have been a single process. For us, the development of a philosophic inquiry and the growth of the critic's mind are a single story. Although for practical reasons it is sometimes convenient to distinguish the text as autobiography from the text as philosophic discourse, ultimately these two aspects are inseparable. They are polar opposites informing the _Biographia_ 's design; their unifying or originating power is Coleridge's idea of imagination.\n\nEarly in chapter IX, one dense footnote focusses many scattered hints that the design of _Biographia Literaria_ is a conscious innovation on the received form of philosophic writing:\n\n... [the word] _discourse_ [as used in _Paradise Lost_ ]... does not mean what we _now_ call discoursing; but the _discursion_ of the _mind_ , the processes of generalization and subsumption, of deduction and conclusion. Thus, Philosophy has _hitherto_ been DISCURSIVE; while Geometry is _always_ and _essentially_ INTUITIVE. ( _BL_ , I, 109 n)\n\nColeridge saw himself as devising a philosophic work that would be intuitive at least in part, because it is impossible to define 'imagination' by discursive methods alone. Geometry is always intuitive, in Coleridge's terms, because the geometer seeks to communicate an idea that finds perfect echo in nature's laws ( _PhL_ , 107\u20138). That is, both the idea of a triangle and a triangle drawn in sand will have angles whose sum is one hundred eighty degrees. The triangle in sand is not the law, but the symbol of the idea (whose corresponding law is mathematical). Geometry is unique among fields of inquiry because mathematical statements fully describe both the idea in the mind and the law in nature. In other fields, the discursively formulated law is not so perfect. 'An Idea, in the _highest_ sense of that word, cannot be conveyed but by a _symbol_ ; and, except in geometry, all symbols of necessity involve an apparent contradiction' ( _BL_ , I, 100). Except in geometry, symbols can be explicated discursively only as contradictions in terms: the idea can be conveyed symbolically (through some equivalent of the triangle in sand), but the symbol itself is paradoxical. That is why 'a great idea can be taught gradually, that is by considering it as a germ which cannot appear at any one moment in all of its force' ( _PhL_ , 173). In most fields, the 'great idea' _must_ be taught gradually if it is to emerge from a discursive text.\n\nExcept in geometry, then, symbols communicate ideas more adequately than discursive logical formulations, because a symbol holds together the contradictions that logic can only break apart. A symbol, Coleridge explains, 'is characterized by a translucence of the Eternal through and in the Temporal. It always partakes of the Reality which it renders intelligible; and while it annunciates the whole, abides itself as a living part in that Unity, of which it is the representative' ( _LS_ , 30). The symbolic relation between Eternal and Temporal, or between God and Man, cannot be defined through logic; and every symbol participates in this relation.\n\nMcFarland's work suggests that the systematic understanding, once fully engaged by extended, close, logical analysis, will always object to a discourse that tries to present a symbolic resolution to logically defined problems. Yet, as he also shows, no closed logical system could accommodate Coleridge's complex view of human reality. I suggest that Coleridge himself knew this full well: the understanding cannot fully grasp the symbolic reality known through reason. A closed logical system excludes the divine:\n\nThe inevitable result of all consequent Reasoning, in which the intellect refuses to acknowledge a higher or deeper ground than it can itself supply, and weens to possess within itself the center of its own System, is \u2013 and from Zeno the Eleatic to Spinoza and from Spinoza to the Schellings, Okens, and their adherents, of the present day, ever has been \u2013 Pantheism under one or other of its modes, the least repulsive of which differs from the rest, not in its _consequences_ , which are one and the same in all, and in all alike are practically atheistic, but only as it may express the striving of the philosopher himself to hide these consequences from his own mind. ( _F_ , I, 523 n)\n\nWhen he claimed that he would one day render his system acceptable to the understanding, he probably intended to fuse a Kantean demonstration of understanding's inadequacies with an essentially imaginative and symbolic portrait of the realities known directly only by reason. Such a discourse would not be a closed logical system, but a work of art \u2013 some variation, perhaps, on 'the first genuine philosophic poem' that he wanted Wordsworth to write. It would not have persuaded the understanding _per se_ but, rather, it would have kept the understanding within its proper limits. Assent would not have been a conclusion by the powers of analytical intellect alone, but a moral and imaginative act of all our powers in concert.\n\nIn _Biographia Literaria_ , 'imagination' is the title of an idea whose symbolic forms include poetry (see also _LS_ , 29). If we are to grasp this idea, so as to assent or to refuse assent, then the _Biographia_ cannot limit itself to 'deduction and conclusion'. These cannot render its idea fully accessible to the inquiring reader. These cannot bring the reader to what J. Robert Barth, SJ, calls a symbolic 'encounter' with the idea of imagination \u2013 not the theory, but the idea that the theory can only describe paradoxically. And thus the created self who is not a pantheist, whose history is not quite Coleridge's own, cannot be regarded as merely a device. Like all good metaphors, the created self is neither a lie nor an ornament. It is Coleridge's response to Hume's demonstration that closed logical systems cannot adequately describe the character of the mind, or the nature of reality. Any orderly explanation \u2013 any human act creating order \u2013 is grounded in an act of faith. This is not directly nor necessarily a faith in God but, rather, a faith in the possibility of order and knowledge that for many individuals has culminated in religious experience. The modern philosopher, by whatever particular title he or she is known, must offer more than just the logical proofs we have come to distrust as partial at best. The philosopher must also evoke our faith in the order he has discovered. He or she must appeal not only to logic, but also to imagination, to our intuitive (rather than sensory) relation to the world around and within us. If something in our souls does not 'awake and start up', we distrust the theory even before we have detected its error.\n\nColeridge responds to Hume's challenge by creating 'Coleridge' as a gradually emerging symbol of the fusion of actual human experience and rigorous philosophic inquiry: questioner and question are subjective and objective poles of the unity that is the examined life. As Barth explains, 'the making or perceiving of a symbol is for Coleridge a kind of act of faith. It is meant to evoke a response of the whole person, in faith, hope, and love. It is a commitment of self to someone other than one's self.' 'Coleridge's' history is designed to elicit our faith in the idea 'imagination' even before rigorous proof has been established. We are first to trust the man, and then his theory. The coherence of the man's life and the validity of the theory are two forms or manifestations of a single integrity. As Richard Mallette argues, 'the narrator's personality... steps forth at every junction to shape and mould our experience.... The total effect... lends to the _Biographia_ an imaginative unity that would otherwise be lost in a work that attempts to survey the horizonless landscapes of literary theory, practical criticism, philosophical disquisition, and personal autobiography.'\n\nBut put questions of consent aside for now: the theory remains inaccessible until one understands this character 'Coleridge'. The _Biographia_ 's speaker is in part an ironic self-parody by a man who knew himself all too well. The speaker is garrulous, he is eclectic and distractable, he is learned in the 'obscure' and the 'useless'. His manner of telling reflects these qualities: his story turns freely, abruptly, from one thing to another. Only after several readings does one perceive the inexorable, orderly momentum of the whole: the impromptu is rarely random. The speaker's freely associational leaps are exquisitely calculated to insist, over and over again, that the thinker finds unity within apparent diversity. Minor issues reflect major questions because all significant problems are ultimately interrelated. Gradually one discovers the central concerns that unite and direct these associational shifts.\n\nWith that recognition comes another: Coleridge is an exceptionally witty, playful writer. As he develops the contrast between his speaker and anonymous critics, he transvalues the elements of his self-parody: the speaker is eclectic and antiquated only to those enamored of a facile, inconsistent, popularized Lockeanism. His diffident double negatives point emphasis to issues too profound for this degraded group to recognize. His unexpected leaps land him always in the heart of the matter. One who reflects carefully on the abrupt shifts will always find his way to basic questions about fancy and imagination. The strategy is at once arrogant and entertaining: it is an intriguing style occasionally carried too far. Even the best reader may be occasionally confused, not by the subtlety of the relations _per se_ , but by the unpredictability that the strategy generates. As Jackson explains, 'the reader is not permitted to know where he is being taken, or why he is being taken there; by obscuring the end in view, the intricacies of the way are allowed to become confusing and even irritating'. I suspect that Coleridge's judgment may have been overwhelmed at times by the potent combination of his anger at anonymous criticism, and his delight with parody. And perhaps, as in any autobiography, fiction merges with fact: both the author and the speaker are 'responsible' for the reader's periodic bewilderment.\n\nAlthough this 'speaker' clearly lacks the full artifice of Browning's speakers, he is far more deliberately a construct than the speakers of most autobiographies. Coleridge's created image may quite aptly resemble himself, but 'he' also serves such definite ends so well that analysis requires some means of distinguishing creator from creation. Coleridge's celebrated introspective powers should lead us to expect such delicate complexity in his intellectual autobiography. Richard Haven suggests that Coleridge was 'a born psychologist trying to write as a metaphysician', that 'Coleridge's philosophy should be seen not as an unsuccessful attempt at a logical analytical system, but, like his poetry, as a projection, an \"elaborated transformed symbol\", of his own psychological experience'. Much recent scholarship has stressed Coleridge's extraordinary fidelity to his own moral and intellectual experience, fidelity sustained despite the enormous difficulty of reconciling the diverse elements into an intelligible whole. Autobiography is the natural vehicle for such a mind, because it allows him to present experiences that illuminate major speculative insights, but it does not require formal logical unity. The autobiography of a richly imagined 'self' can make arguments Coleridge wants to make, and yet bear up under the strain of the gaps that emerge when these arguments are separated from their autobiographical enactment.\n\nAlthough Coleridge's later works are generally clearer, and more skillfully persuasive, _Biographia Literaria_ remains the central text for the study of Coleridge's prose. As James Olney explains in _Metaphors of Self: The Meaning of Autobiography:_\n\n... a man's lifework is his fullest autobiography [.]... When, moreover, a man writes, in addition to his other works, something that is confessedly autobiographical... then we may expect to be able to trace therein that creative impulse that was uniquely his: it will be unavoidably there in manner and style and, since autobiography is precisely an attempt to describe a lifework, in matter and content as well. A man's autobiography is thus like a magnifying lens, focusing and intensifying that same peculiar creative vitality that informs all the volumes of his collected works; it is the symptomatic key to all else that he did and, naturally, to all that he was.\n\nColeridge achieves what Olney describes even more fully than conventional autobiographers: his intellectual autobiography portrays not the _man_ , but the man _thinking_. In _Biographia Literaria_ one meets face to face the thinker whose methods inform all the other works: he who believes his intuitions and then seeks confirmations; he who attributes to others his own ideas, and uses others' texts without attribution; he whose penetrating syntheses revise the habitual patterns of understanding. Both Coleridge's vitality and his peculiarity are most evident in _Biographia Literaria_ : one who grasps the design of this most rewarding, most frustrating text will more easily follow Coleridge's track anywhere else.\n\nColeridge's desire to make his readers think about genuine ideas underlies the _Biographia_ 's digressive texture; his fusion of autobiography and philosophy creates the necessary supply of anecdotal illuminations. These features of the design are a coherent, intelligible response to problems arising from the unity of ideas and laws. But this does not explain \u2013 nor explain away \u2013 all the difficulties impeding the _Biographia_ 's readers. The design is also characterized by peculiar or inadequate transitions for which there is often no defense. Prudent authors of difficult arguments take particular care with transitions. Their transitions integrate parts, subordinate parts to whole, and repeatedly orient the reader within sequences of patterns large and small. Such transitions separate form from content by commenting on the form as such; this helps the reader to monitor his comprehension of major structures. Coleridge's argument is exceedingly difficult, but his transitions are hardly prudent. Often they are cryptic, just a sentence or two catapulting us into strange new terrain. More often, they seem to establish only a whimsical, undisciplined, associational flow. The speaker seems to ramble unbearably, just as Carlyle described.\n\nIn part, I grant, Coleridge's transitions are terrible. They suggest an incredible insensitivity to audience. Perhaps he so quickly penetrated to fundamental issues in his own reading that he failed to perceive the needs of blunter minds. A person as well read and as astute as Coleridge himself might have little trouble recognizing the unifying issues, but the rest of us are too apt to discover the central points only by wandering in circles. In other works, Coleridge seems to realize the useful (albeit mechanical) function of the conventional transition: major divisions commonly begin by commenting on the form as such. These transitions are often clumsy and intrusive because the underlying duality of form precludes tidy logical summary. But they help, if only in assuring the reader that the author is executing a deliberate plan. This is no small matter: without confidence in the ultimate coherence of the whole, one would not expend the effort Coleridge's prose requires.\n\nYet the famous Coleridgean carelessness does not explain everything. It is also clear that the autobiographical fiction of _Biographia Literaria_ makes it difficult for Coleridge to write transitions that maintain our sense of order, unity, and progression. Had Coleridge continually specified the relations between philosophic and autobiographical movements, and between the parts of these and the definition of imagination, then the autobiographical fiction would have become palpably fake. No real life is so direct, so fortuitously ordered. Coleridge's solution is to orient the reader primarily to the life and times of 'Coleridge' rather than to the evolving definition of imagination: as he explains, 'I have used the narration chiefly for the purpose of giving a continuity to the work' ( _BL_ , I, 1). But this solution has its awkward moments: in the opening lines of chapters IX and XIV, for instance, the abrupt return to autobiography may disorient even the most careful reader.\n\nHis fusion of narration and philosophy also creates problems. The definition of imagination does not unfold in a direct, clear, logical fashion, so the reader's progress and his locale within this definition cannot be sketched briefly. Furthermore, Coleridge cannot easily separate form and content by commenting on the form as such: his dual design _enacts_ its principal idea to a far greater extent than one commonly finds. The form is to make us think; the content is to tell us what thinking is \u2013 which is to say, to render us conscious of what we are doing as we ponder the issues _Biographia Literaria_ raises. Kathleen Wheeler explores this self-reflexivity by analyzing passages that refer both to the theory of imagination and to the experience of reading the text itself. She argues that German Romanticism provides both a motive and a rationale for such writing. So does the tradition of spiritual autobiography. For these passages to function as they must, the work as a whole must systematically lead the reader to reflect on them both objectively, for what they say, and subjectively, for why or how they emerge at particular places. The work does so, I contend, by engaging our sympathies with both the madcap adventures and the scholarly endeavors of its most colorful speaker. Transitions cannot locate the reader within this process, because there is little reason to assume that all readers of the same chapter are at the same 'point' in understanding the idea of imagination. Like the youthful speaker himself, the new reader may not fully appreciate the implications of certain experiences or discoveries. When Coleridge's transitions fail, it is often because he presumes that more is understood at a given point than an actual reader is likely to understand, even after several attentive readings. And many subtle but quite adequate transitions will fail, of course, for the reader who assumes that there can be no unity within the diversity of _Biographia Literaria_.\n\nAlthough Coleridge's design impedes conventional transitions, it encourages \u2013 and perhaps requires \u2013 the unity that metaphors can provide. The cinque-spotted water-insect for instance, belongs to an elaborate pattern of imagistic reference to imaginative synthesis. Light and water recurrently signal the rhetorical power of imaginative works. These metaphors, and others, both signal and generate the complex unity of _Biographia Literaria_ 's many discrete issues. They establish its character as an essentially imaginative \u2013 not logical \u2013 discourse because they simultaneously develop the speaker and elucidate his ideas. Some have condemned Coleridge's densely imagistic passages as pious or poetic excrescences in what ought to be sober philosophy. But to ignore these lines is to mutilate the text, and its ideas.\n\nHow does one read a text of such complexity? What methods are adequate? Arthur Symons makes the basic point: 'one who is ceaselessly attentive will be ceaselessly rewarded'. But even reasonably ceaseless attention reaches a point of diminishing return: by creating a form nearly as difficult as his ideas, Coleridge risks losing our interest in either. But recognizing the _Biographia_ 's duality, recognizing its mediation between philosophy and narrative, enormously simplifies the reader's task. From the character of the design it follows that each chapter _must_ be read both as a stage in the speaker's intellectual development, and as a stage in the effort 'to effect... a settlement of the long continued controversy concerning the true nature of poetic diction; and... to define... the real _poetic_ character of the poet, by whose writings this controversy was first kindled' by demonstrating that fancy and imagination are 'two distinct and widely different faculties' ( _BL_ , I, 1\u20132, 60). Only if each part is read in both ways will all the relations among parts become evident. Without such bifocal vision, the whole remains fragmentary and, at times, rudely incoherent. In the chapters that follow, I will often distinguish between the two aspects both to demonstrate this method, and to reveal how the two strands of development are sustained and interwoven. But distinction is not separation: philosophic and autobiographical aspects provide objective and subjective accounts of a single idea that cannot be fully defined from either perspective alone.\n\nYes, _Biographia Literaria_ is difficult \u2013 but it is not impossible. Its real flaws do not amount to so much when one considers the clarity, the acuity, and the grace that are generally evident. The reader's greatest burden is imposed not by these flaws, but by the complexity of Coleridge's thought. Coleridge wants his readers to think, and so he writes in ways designed to encourage and to require that effort.\n\n### Notes\n\n*Distinction between thought and attention. \u2013 By thought is here meant the voluntary reproduction in our minds of those states of consciousness, or... of those inward experiences, to which, as to his best and most authentic documents, the teacher of moral or religious truth refers us. In attention we keep the mind passive: in thought, we rouse it into activity. In the former, we submit to an impression \u2013 we keep the mind steady, in order to receive the stamp. In the latter, we seek to imitate the artist, while we ourselves make a copy or duplicate of his work. We may learn arithmetic, or the elements of geometry, by continued attention alone; but self-knowledge, or an insight into the laws and constitution of the human mind and the grounds of religion and true morality, in addition to the effort of attention requires the energy of thought. ( _AR_ , 69 and n)\n\n# **2 Starting-Points**\n\n> As to the Persecution of Bigots, I have all my life been exposed to them \u2013 and in for a penny, in for a pound. Nay, the latter is the better policy; for it is the Nature of these Cattle to hate in an inverse ratio to the magnitude of the Difference.\n> \n> Letter to Hyman Hurwitz, 10 March 1820, _Collected Letters_ , V, 21\n\nThe first four chapters most clearly reveal how Coleridge weaves autobiography and inquiry into a single fabric. The speaker describes his youthful encounters with both genius and its anti-type, the fanatic. From his position in the autobiographer's 'present', he offers some apparently extraneous speculation about the psychological differences between geniuses and fanatics. Take these speculations seriously, examine them closely, and you find a major account of the psychological consequences of imaginative power. The speaker attributes his youthful ability to distinguish genius from fanatic not to any precocious insight into such matters, but to the rigor of his education. He repeats his teacher's principles of writing, and derives from them the literary principles that shaped his youthful taste in poetry. His explanation of these principles allows Coleridge to define complex connections between imagination and language. He further develops the relations among personality, imagination, and language by characterizing the speaker as a genius who responds in imaginative ways to the imaginative deficiencies of anonymous critics. Coleridge also maps out the _Biographia_ 's future development when he presents the youthful speaker as seeking to substantiate these relationships. And note that Coleridge does all this without extensively committing himself to any single explanation of the complex he is sketching. This enables him to discuss parts _as parts_ later on \u2013 a flexibility that proves crucial as he maneuvers past pantheism.\n\nPhilosophy and autobiography are so closely woven in these first four chapters that in most of what follows I shall be separating them: this is the least confusing way of revealing their complementarity. But, before getting into that, let us look for a moment at the _Biographia_ 's opening paragraph. It is an extensive and entirely accurate account of what is to come; it directs the reader along rational and fruitful paths through the complexity of these opening chapters. But it is paradoxically unlikely to operate in this way for the new reader, who undoubtedly needs such assistance most keenly. This first paragraph proclaims the book's intent to resolve the controversy about poetic diction, and to define 'the real _poetic_ character' of Wordsworth. To resolve the controversy about diction requires two things. First, it demands a proper theory of diction itself. Secondly, it demands an explanation of the controversy as such: an analysis of the anonymous critic's response to Wordsworth's genius. To define Wordsworth's 'real _poetic_ character' is to define his particular character: he is the first genuine philosophic poet; he is the first to combine Shakespeare's objectivity with Milton's subjectivity so as to write about both subject and object simultaneously \u2013 so as to write about perception itself. But to define Wordsworth's particularity requires first that one define the type of which he is an instance: what is 'real _poetic_ character'? For Coleridge, poetic diction and the psychology of genius are aspects of the more encompassing question, 'what is imagination?' This introductory statement of purpose orients the reader both to major issues and to principal relations among major issues, if one reads very closely. He relies on this statement \u2013 no doubt unduly \u2013 to project patterns of emphasis into the first four chapters. The statement functions so poorly in directing such emphasis, however, because it competes so awkwardly with what appears to be the major statement of purpose.\n\nIt will be found, that the least of what I have written concerns myself personally. I have used the narration chiefly for the purpose of giving a continuity to the work, in part for the sake of the miscellaneous reflections suggested to me by particular events, but still more as introductory to the statement of my principles in Politics, Religion, and Philosophy, and an application of the rules, deduced from philosophical principles, to poetry and criticism. (I, 1)\n\nThis sounds like an immethodical miscellany. Coleridge seems to set out three domains of inquiry \u2013 politics, religion and philosophy \u2013 and to note that from the philosophy will be deduced the means of settling the Wordsworth controversy. But Wordsworth is no mere _exemplum_ , no simple application of one of three inquiries: he and the controversy swirling about him are Coleridge's central illuminations of his theory of imagination. The 'statement of my principles' anticipated here can only be one high unified set \u2013 a theory of the mind's activity \u2013 that will have implications in all directions.\n\nHis 'not the least important' intentions concerning Wordsworth are an accurate but off-handed prediction that the implications most centrally explored will be those having to do with genius and poetry. To make sense of what follows, one must simply remain alert for the enunciation of principles. A substantial number of these appear promptly in chapter II, which presents itself as an exploration of the speaker's puzzling experiences as recounted in chapter I. This mode of development also gives a particular content to the first paragraph's sketch of the relation between autobiographical narration and principles.\n\nRegarded as autobiography, these four chapters define the sequence of influences from the days at Christ's Hospital until approximately 1796. (This record of influences will continue to the end of chapter IX.) These influences introduce four of Coleridge's principal working assumptions, although without as yet offering much justification for them. First, the accounts of Boyer and Bowles provide an initial statement of the criteria for good poetry. From the Reverend Bowyer (more properly, _Boyer_ ) of Christ's Hospital he learned that all good writing is precise, that poetry is the most precise form of writing, and that the logic that a poem follows is difficult because subtle, complex, and dependent upon 'fugitive causes' (I, 4\u20135). His attribution to a teacher forms part of a pattern of statements about the importance of education, a pattern evident everywhere in Coleridge's work (I, 7\u20138; I, 27 n; II, 116\u201317). The suggestion that a Boyer-like 'index expurgatorius' should be hung in courts of law uses humor to enforce a serious suggestion: imprecise thinking and writing threaten public welfare. This public dimension will later grow: the necessary features of precise language generate the truth-claims advanced on behalf of poetry, philosophic criticism, and philosophically grounded inquiry in all fields.\n\nBowles's sonnets encourage Coleridge to seek a detailed and philosophically grounded understanding of Boyer's 'more fugitive causes' of the logic and language of poems. Such philosophic grounding is necessary to refute the judgments of those whose taste has been formed by 'Mr. Pope and his followers' (I, 11\u201314). The speaker's first attempt to organize his ideas about poetry results in two aphorisms. The first reformulates Boyer's teachings: 'whatever lines can be translated into other words of the same language, without diminution of their significance, either in sense, or association, or in any worthy feeling, are so far vicious in their diction' (I, 14). The speaker consistently describes bad poetry as either translatable, or as a translation (I, 13). The broad definition of 'significance' gradually develops into a theory of language that defines the kind of precision poetry achieves. Only as that theory is delineated (in chapters XIV to XXII) do we see the necessary connection between precision and the second aphorism, 'the poem... to which we _return_ , with the greatest pleasure, possesses the genuine power, and claims the name of _essential poetry_ ' (I, 14). Coleridge later argues that such precise control over the whole domain of significance initiates in the reader an imaginative activity closely akin to the poet's own. Such activity generates a delight \u2013 the aesthetic response \u2013 that sustains repeated reading (II, 10\u201311).\n\nSecondly, Bowles's influence leads the speaker to develop a criterion for philosophy: the philosopher must maintain a proper balance between 'metaphysic depths' and 'history, and particular _facts_ '. Neither can philosophy exclude 'the love of nature, and the sense of beauty' (I, 10). The 'preposterous pursuit' to which the speaker refers is not metaphysics itself, but metaphysical inquiry that excludes both the particular and the promptings of the heart. The lines from _Paradise Lost_ that the speaker cites to describe his youthful excursion into lifeless philosophy are Milton's description of a pastime in Hell. As Satan fights his way up through Chaos, the angels remaining in Hell amuse themselves with what Milton calls 'Vain wisdom all, and false Philosophy'. Philosophy without faith, Coleridge obliquely suggests, is not just preposterous \u2013 it is demonic. The speaker later asserts that those who deny our immediate knowledge of a common physical world are but schoolmen who 'live and move in a crowd of phrases and notions from which human nature has long ago vanished' (I, 179). The faint echo of St Paul suggests that such systems are not just foolish but immoral; the sharp accusation that such a philosophy is built on the abuse of words is an equally serious charge. Bowles's sonnets (and the 'amiable family' of Mary Evans) draw the speaker back from the abyss of false philosophy. He turns to literature, and seeks a philosophically rigorous understanding of literary merit. His first attempt results in the two aphorisms we have already examined.\n\nColeridge uses this broadly inclusive definition of philosophy to distinguish between his system and pantheism. This crucial distinction rests in large measure on his claim that epistemology must begin by assuming immediate knowledge of both an independent world and a personal God; it must begin here because in his heart he finds it impossible to doubt that he possesses such knowledge. Without these two principles, or within a more conventional definition of philosophy, what Coleridge proposes in _Biographia Literaria_ is clearly pantheist.\n\nThe lines from Milton imbedded in the account of youthful philosophizing do more than support an inclusive definition of philosophy; they also define theology as one of the speaker's longstanding interests:\n\nOf providence, fore-knowledge, will, and fate,\n\nFixd fate, free will, fore-knowledge absolute,\n\nAnd found no end in wandering mazes lost.\n\n( _Paradise Lost_ , II, 559\u201361)\n\nThe speaker's inquiry into these matters proves extensive. The character of will occupies most of chapters V to IX. Foreordination is not an explicit issue, but human freedom and moral responsibility certainly are. The relation between human will and divine knowledge occupies chapter IX, parts of chapter X, and all of chapters XII and XIII. Yet this inquiry is neither demonic nor sterile, because it is animated and in part motivated by the speaker's faith not only in God, but also in a particular world immediately known.\n\nYet note that there is very little in chapter I to alert us to the density and precision of these few apparently casual statements about the speaker's youthful interest in 'metaphysic depths'. Any good reader will remember what has been said about precision, or permanent literary value, or the anger of fanatics, because these receive the conventional emphasis of repetition. But these few statements about philosophy, although theoretically part of the 'influences' pattern that ought to convey adequate emphasis, are quite apt to be forgotten. Only the experienced 'Coleridgean' will realize that Coleridge's introspective habits and his knowledge of _The Prelude_ would have made him an extraordinarily self-conscious autobiographer. Without such confidence, a self-disciplined reader is unlikely to examine these remarks as closely as they require. This instance is further complicated by the extent to which many of Coleridge's contemporaries would regard any excursion into 'metaphysic depths' as a 'preposterous pursuit'. The speaker's ironically self-critical remarks eventually accumulate into a distinction between the good or philosophic reader, and the Lockean blockhead. But at this very early stage the fine distinctions underlying the word 'preposterous' are all too apt to be missed.\n\nThe third influential figure is Southey. The praise of Southey is abundant, yet qualified. His poems are full of pathos and just reflections, but they are never described with terms signaling imagination (I, 40; cf. I, 57\u201360; II, 13\u201320). He merits attention as a moral influence on the imprudent and impulsive speaker, and as a member of the so-called Lake School. Coleridge denies that he, Southey and Wordsworth form any school but 'that of good sense confirmed by the long-established models of the best times of Greece, Rome, Italy, and England' (I, 36 n). Reviews of Southey's work demonstrate the principles of anonymous criticism (which is thus implicitly opposed to 'good sense' and the whole Western tradition). The defense of Southey proceeds side by side with a listing of these failures: mistaking poetic genre or intent; exaggerated attention to isolated flaws; the lack of principles of judgment; and vicious personal attack (I, 43\u20134; I, 47). In short, both their concrete particulars and their abstract theories are deficient: by the criteria for philosophy just proposed, such criticism is entirely fraudulent. This account of the defamation of Southey is designed to reinforce the reader's faith in the public need for such principles as the speaker learned through Boyer and Bowles. The speaker asserts that critics must 'support their decisions by reference to fixed canons of criticism, previously established _and deduced from the nature of man_ ' (I, 44, my italics). Chapters V to XVI seek to fix such canons in exactly this way. The need for this kind of foundation, here asserted, will only later be justified by the relation between imaginative power and particular features of good poems.\n\nWordsworth himself is the final influence celebrated in these chapters. Through Wordsworth and criticism of _Lyrical Ballads_ , the speaker recognizes the differences in kind between the synthetic imagination and the associative fancy. As a result, his quest for a firm foundation in 'the nature of man' takes a specific direction: to define these two terms as an aid both to the poet and to the 'philosophical critic' (I, 62). The form of this inquiry is described as complementary to Wordsworth's inquiry \u2013 a fiction later discarded because it would obviate the need for any practical discussion of poetry.\n\nBut it was Mr. Wordsworth's purpose to consider the influences of fancy and imagination as they are manifested in poetry, and from the different effects to conclude their diversity in kind; while it is my object to investigate the seminal principle, and then from the kind to deduce the degree. (I, 64)\n\nThe paradoxical form of Coleridge's intent is easily resolved. Coleridge often insisted that when the mind acts, the whole mind acts, not some single faculty. One may distinguish among the mind's characteristic activities, although one may never regard these acts as entirely independent of each other, as if they were distinct computing programs that could be separately engaged. Coleridge's distinguishing among characteristic activities _per se_ yields difference in kind, and his 'seminal principle' is an idea about the character of all mental activity. Wordsworth's distinguishing among _products_ of mental acts, however, can go no farther than differences in degree. And the clearest understanding of these differences in degree requires a prior understanding of the difference in kind, because differences in degree are relative and contingent. Coleridge sharply disagrees with Wordsworth whenever it appears that Wordsworth's relative distinctions are contingent upon a seminal principle derived from Lockean materialism.\n\nJudgments of degree dominate these early chapters, as part of the motive to the inquiry into kind. They also dominate most of the second volume, as Coleridge puts the abstract inquiry to practical use. The speaker's judgments often assert relative degrees of imaginative power. The later Wordsworth is more imaginative than the early Wordsworth; the same is true of Southey (I, 58; I, 40). Pope's translation of the _Iliad_ is pseudo-poetry, but his original compositions are the work of genius (I, 26 n). The 'faulty elder poets' are better than the 'moderns', but they are still primarily fanciful (I, 15). Only anonymous critics seem entirely to lack imaginative power. In them we see what Barfield calls degraded fancy. But this is only an appearance: they lie to make money (I, 28\u20139; II, 129).\n\nIn the fourth chapter, Coleridge defines 'true poetic genius' by describing Wordsworth (I, 56\u201360). The influences traced in these chapters, and the chronology-breaking speculations to which memory gives rise (e.g. chapter II) have inexorably led to the recognition that the imaginative qualities of genius are responsible for both the precision and the truth of genial production. The influence and the poetic genius of Wordsworth culminate and summarize the first four chapters, so as to provide a major autobiographical link between the philosophic inquiry and the practical criticism. I will deal with the figure of Wordsworth in chapter IV more fully below.\n\nThe first four chapters also define the personality of the speaker in ways designed both to establish the validity of Coleridge's ideas, and to prepare for their orderly exposition within an autobiographical context. The crediting of influences, for instance, achieves more than an initial statement of working assumptions. In its continuity through the first nine chapters, this pattern also defines the speaker as a humble man, as one quick to give others their due because 'the obligations of intellect [are] among the most sacred of the claims of gratitude' (I, 9). The very controversial acknowledgment to Schelling (I, 102\u20135) forms part of this extensive movement. Yet this humility is no simple thing. By crediting others with ideas basic to his argument, Coleridge can offer unequivocal assertions of the truth and importance of the ideas at issue without making the speaker unbearably arrogant. Ideas credited to others are often presented as if _a priori_ truths. Yet, since egotism is always inversely proportional to genius, the speaker's grateful attributions indirectly suggest that he is a genius himself. This pattern of attributions may also be a distant echo of what Geoffrey Yarlott calls Coleridge's need for 'sheet anchor' relationships to help him sustain a modicum of genuine self-confidence and its attendant productivity.\n\nWhatever its psychological resonance, the pattern also reflects Coleridge's profoundly conservative character: the genius does not proclaim revolutionary new truths, but instead rediscovers and revitalizes the wisdom of the ages (I, 60; I, 66). Such conservatism underlies Coleridge's view of history as well, a view evident in the _Biographia_ and elsewhere. By contrast, those lacking imagination are 'ingenious' and endlessly seek novelty (I, 11; I, 27\u20138; I, 27\u20138 n). The speaker's character and personality are most substantially and colorfully defined by this comparison to anonymous critics. The ongoing contrast begins with the account of the education of an anonymous critic that immediately follows the description of the speaker's education. The speaker's humility and deference are highlighted by the 'self-conceit, shallowness, arrogance, and infidelity' taught to future anonymous critics. The portrait of such reviewers and the comparison between them and the speaker often reflect eighteenth-century satires and satirical methods; Hazlitt responds brilliantly in kind. The 'anonymous critic' figure plays an important role in the exposition of the theory of imagination, but it is also a lively and utterly polemical response to the abuse Wordsworth and his circle had suffered.\n\nThe speaker's generous good humor (which also sharply distinguishes him from anonymous critics) enforces the need for philosophically grounded critical principles. At the end of the first chapter, two anecdotes (one in the text, one in a footnote) recount the speaker's parodies of his own poetic diction and his epigram critical of The Ancient Mariner'. In each instance, someone expects that he will be outraged by the anonymous verse. The first paragraph of chapter II extends the comedy into serious issues: why should we credit the ideas he (or any other critic) advocates? Why do we suppose that a poet seriously defending himself against a critic only proves himself 'irritable'? The current situation of both criticism and the arts reveals that many poets and most critics are engaged in no more than a frivolous game of politics and personalities, a game that threatens all genuine writers (see I, 27 n; I, 25\u20139; II, 87). Lacking principles 'derived from the nature of man', poets and critics alike lack the absolute standards necessary to give intellectual substance to their arguments.\n\nIn chapter X such advocacy of principles and principled argument will be cited as evidence of philosophic genius (I, 125; II, 39). This strikingly complements the speaker's serenity, objectivity, humor and lack of egotism. Coleridge portrays the speaker as a genius so as to establish both a theoretical and a rhetorical ground for the speaker's authority. The speaker himself never advances the claim: he seems 'unconscious' of the implications of what he reveals about himself. Yet the distinction between 'speaker' and Coleridge himself is very delicate at this point. I believe that Coleridge was essentially conscious of the pattern, because its extraordinary consistency is managed with great care to avoid direct claims, and because it serves an important rhetorical aim: we must be convinced that the speaker is capable of solving the enormous philosophic problem postponed in chapter XIII.\n\nAt the end of chapter II the speaker explains that poets have a reputation for irritability simply because their expression, on any topic, will be more lively than that of the less gifted writer (I, 30). Such an explanation indirectly justifies his most lively condemnations of anonymous critics, seeking to resolve the apparent contradiction between the genially diminished sense of self and such eruptions. As I will show in more detail later, these descriptions provide a set of images whereby Coleridge can briefly characterize a man or his work as inadequately imaginative. The speaker's occasional tempestuous excess supplies the imagery so characteristic of Coleridge's dense prose; it solves a major problem of first-person narration.\n\nIn the light of several indications that the speaker is a genius, many of his self-critical remarks cannot be taken literally. The most frequently misunderstood of these concludes chapter IV. Because it is so often quoted in misleading fragments, I will quote the entire paragraph:\n\nYet even in this attempt ['to investigate the seminal principle' of the difference between fancy and imagination] I am aware, that I shall be obliged to draw more largely on the reader's attention, than so immethodical a miscellany can authorize; when in such a work (the _Ecclesiastical Polity_ ) of such a mind as Hooker's, the judicious author, though no less admirable for the perspicuity than for the port and dignity of his language; and though he wrote for men of learning in a learned age; saw nevertheless occasion to anticipate and guard against 'complaints of obscurity,' as often as he was about to trace his subject 'to the highest well-spring and fountain.' Which, (continues he) 'because men are not accustomed to, the pains we take are more needful a great deal, than acceptable; and the matters we handle, seem by reason of newness (till the mind grow better acquainted with them) dark and intricate.' I would gladly therefore spare both myself and others this labor, if I knew how without it to present an intelligible statement of my poetic creed; not as my _opinions_ , which weigh for nothing, but as deductions from established premises conveyed in such a form, as is calculated either to effect a fundamental conviction, or to receive a fundamental confutation. If I may dare once more adopt the words of Hooker, 'they, unto whom we shall seem tedious, are in no wise injured by us, because it is in their own hands to spare that labor, which they are not willing to endure.' Those at least, let me be permitted to add, who have taken so much pains to render me ridiculous for a perversion of taste, and have supported the charge by attributing strange notions to me on no other authority than their own conjectures, owe it to themselves as well as to me not to refuse their attention to my own statement of the theory, which I _do_ acknowledge; or shrink from the trouble of examining the grounds on which I rest it, or the arguments which I offer in its justification. (I, 64\u20135)\n\nNo literally 'unmethodical miscellany' provides 'deductions from established premises' or a grounded theory with supporting arguments. For Coleridge, 'immethodical' would be a term of deepest condemnation; and yet the headnote printed opposite page 1 distinctly echoes _The Friend:_ 'He wishes to spare the young those circuitous paths, on which he himself had lost his way' (I, civ; compare _F_ , II, 48\u20139 and _F_ , I, 55 and n). The geographical metaphor is repeated exactly in the extract from Hooker; and, in Coleridge's terms, no genuine reasoner will be immethodical. Furthermore, the balance between particular and general or theoretical, characteristic of the young Coleridge's philosophical work, also characterizes correct method ( _F_ , I, 448\u2013524). The delicately allusive play leading to substantial ideas, so common in Coleridge's prose, entirely undercuts the self-description 'immethodical miscellany'.\n\nSo why the irony? It acknowledges how the work will appear to the unimaginative, while encouraging the sympathetic reader to question his own \u2013 not Coleridge's \u2013 abilities (cf. I, 160\u20131). Hooker addressed 'learned men in a learned age'; the speaker, however, must address 'the public' (I, 41\u20132). Yet this public reads carelessly and inattentively (I, 26\u20137 and n). Beginning here, the speaker explains and laments that the necessary difficulty of his topic will limit his audience (I, 73\u20134; I, 105\u20137; I, 149); all of chapter XII, from the perspective of autobiography, is an elaborate explanation of why this public will be confused, and what they will fail to understand. As the pattern develops from this point to chapter XII, responsibility gradually shifts from the speaker to the reader.\n\nThe irony, then, is directed not against himself, but against the dim and inattentive reader for whom he will seem obscure every time he deals with first principles. The unequivocal undermining of 'unmethodical miscellany' as a description of the _Biographia_ presumes at least a rudimentary knowledge of method; that is, that method involves such things as deductions, fundamental proofs and confutations, grounds, arguments, and established premises. This maneuver constitutes a second appeal to the reader whom the speaker assumes will share his own response to the excesses of anonymous criticism (cf. I, 50). As responsibility for following the argument progressively shifts on to this reader, he is more and more clearly portrayed as sharing the speaker's judgment of 'the public' and its anonymous magazine-writers. When the speaker ceases transcribing chapter XIII, we are to see that as a helpless surrender to the practical power of the deficient multitude. The lines of judgment are so strictly drawn that an actual reader is to perceive every obscurity in the text as evidence of his own imaginative poverty \u2013 as long, of course, as he remains within the conceptual and rhetorical universe of _Biographia Literaria_.\n\nThe reputation of the book, then and now, suggests that this dubious strategy failed. To the informed and sympathetic reader who finds the _Biographia_ obscure, the pattern may constitute a high-handed shirking of authorial responsibility. The problem, I suspect, is not primarily that Coleridge is this arrogant; but that he is rather inept at the 'interpersonal' skills involved in an author's relation to his readers. He does not often enough and clearly enough distinguish between the genuine troubles of an attentive reader, and the perversely deliberate noncomprehension of magazine-reading blockheads. His sardonic remarks about those who cannot follow his arguments may provoke a defensive anger that subverts his rhetorical intentions. Later I will point out other instances of Coleridge's clumsiness in identifying and relating to a consistently imagined audience.\n\nThe gap in chapter XIII concludes another aspect of self-portraiture that first appears in these introductory chapters. The speaker describes himself as impractical, and constitutionally oblivious to the public consequences of his acts (I, 31\u20132). He is, as a result, quick to depend on the influence of such practical, foresightful, and dutiful men as Southey (I, 49 n). Yet in reading _Lyrical Ballads_ , friends admired principally for judgment prove themselves deficient (I, 53\u20134). Since the judgment concerns literary not practical matters, the speaker is not deceived. But when one of these 'judicious' friends advises against publication, the speaker immediately stops transcribing. This is clearly worse imprudence. Coleridge seems here to be operating ironically within the eighteenth-century opposition between judgment and genius. The portrait of the speaker as an imprudent genius subject both to his own frailties and to the miscomprehension of Englishmen serves primarily to validate the claim that Coleridge could prove that his theory of imagination is compatible with Christianity. It is intended to imply that the sympathetic reader might come to Highgate to read the withheld manuscript, or at least anticipate its rapid publication. Needless to say, these implications were false.\n\nTo summarize the autobiographical aspect: the chronological sequence and self-portraiture in these first chapters provide the starting-points of Coleridge's argument, and the conceptual and rhetorical foundations on which it will be built. Regarded as philosophy, these chapters begin to build that argument, to define imagination by contrasting fanatics (or pseudo-poets) with geniuses. The structure of this definition reveals what R. H. Fogle calls the classically Coleridgean method. A prior unity is dissolved into subjectively oriented and objectively oriented aspects, and then reconstituted as a whole. Here, Coleridge defines imagination through describing both the psychology and the works of the two contrasting types of mind. The fourth chapter ends approximately where the first began \u2013 with the desire to resolve the controversy by explaining fancy and imagination. As one always finds with Coleridge's circular progression, the second formulation substantiates or illuminates the first by drawing all that has intervened into coherent relation.\n\nThe psychology of the fanatic differs from that of the genius because the fanatic, who suffers 'a debility and dimness of the imaginative power', must consequently rely on the 'immediate impressions of the senses'. The genius, on the other hand, 'is affected by thoughts, rather than by things; and only then feels the requisite interest even for the most important events and accidents, when by means of meditation they have passed into _thoughts_ ' (I, 20). This same domination of the physical generates the pseudo-poetic diction of modern poetry, which sacrifices both passion and intellect to an elaborate, incoherent imagery (I, 15,1, 26 n). (The pseudo-poet, of course, may be a temporally prior form of the anonymous critic, who is the _Biographia_ 's foremost example of the fanatic (I, 25\u20139).)\n\nThe same 'dimness of the imaginative power' accounts for temperamental differences between fanatics and geniuses. Most of chapter II describes the notable serenity of Chaucer, Spenser, Shakespeare and Milton, contrasting it with the virulent hostility of anonymous critics (I, 19\u201320). The explanation for this state of affairs presents a major idea: The passion being in an inverse proportion to the insight, _that_ the more vivid, as _this_ the less distinct; anger is the inevitable consequence' (I, 19). The controversy concerning _Lyrical Ballads_ demonstrates this clearly. Although two-thirds of the poems in this collection were quite good, anonymous critics focussed their attention on the few flawed pieces. These flawed pieces 'provoked direct hostility when announced as intentional, as the result of choice after full deliberation' \u2013 as if the flawed poems rather than the good ones reflected the intent of the Preface (I, 51\u20134). The inconsistency of such critical judgment is self-evident, and the anonymous critics dimly sense this contradiction:\n\nIn all perplexity there is a portion of fear, which predisposes the mind to anger. Not able to deny that the author possessed both genius and a powerful intellect, they felt _very positive_ , but were not _quite certain_ , that he might not be in the right,... [and sought] alleviation by quarrelling... and by wondering at the perverseness of the man, who had written a long and argumentative essay to persuade them... that they had been all their lives admiring without judgment, and were now about to censure without reason. (I, 52)\n\nThe controversy that the speaker wants to resolve arises, _as a controversy_ , from the characteristic response of the deficient mind to the powerful genius of Wordsworth. Such genial power, in context with the flaws in the Preface and in a few poems, inevitably ignited these volatile fanatics.\n\nThroughout the _Biographia_ , excessive passion of any sort, but especially exorbitant anger, signals the lack of imaginative and intellectual power \u2013 often through images of swarming bees, swirling storms, or other naturally violent events (e.g. I, 19, 32; II, 7). 'Genial', by contrast, means both 'characteristic of genius' and 'cordial or kindly'. References to hostility or serenity, and to stupidity or insight, help sustain the development of the theory of imagination in and through an autobiographical narrative principally recounting the speaker's encounters with other people or their writings.\n\nThe works of fanatics are distinguished from the works of geniuses through several metaphors. At issue is the fact that, despite their reliance on the senses, fanatics observe poorly and thus write incoherently. In Pope's translation of the _Iliad_ , cited as the source for decadent modern imagery, it is difficult at times to tell whether 'the sense or the diction be the more absurd' (I, 27 n). Moderns sacrifice both intellect and passion to 'the glare and glitter of a perpetual, yet broken and heterogeneous imagery, or rather to an amphibious something, made up, half of image, and half of abstract meaning' (I, 15). In such poetry, we see nothing clearly, nor are we brought to understand anything distinctly. The false light of 'glare and glitter' contrasts sharply with the '\"frequent bursts of overpowering light'\" characteristic of Wordsworth's _Descriptive Sketches_ , a volume which announced 'the emergence of an original poetic genius above the literary horizon' (I, 56\u20137). The traditional association of light and enlightenment makes distinctions among qualities or sources of light a natural image for the imagination; this image is largely responsible for the first link between the terms 'imagination' and 'self-consciousness' (I, 165\u20137).\n\nAs one would expect, the works of frauds are also fraudulent: 'The difference indeed between these and the works of genius is not less than between an egg and an egg-shell; yet at a distance they both look alike' (I, 26). The fanatic's anger reveals his intellectual emptiness, 'even as the flowery sod, which covers a hollow, may be often detected by its shaking and trembling' (I, 25). In contrast to such indictments, Coleridge advances major truth-claims on behalf of genuinely imaginative works, works which arise from 'the union of deep feeling with profound thought; the fine balance of truth in observing, with the imaginative faculty in modifying the objects observed' (I, 59). The subjective synthesis of passion, observation and thought results in a poem that synthesizes power, beauty and truth. Coleridge particularly stresses the union of truth with the power to elicit response that characterizes the great poem:\n\nIn poems, equally as in philosophic disquisitions, genius produces the strongest impressions of novelty, while it rescues the most admitted truths from the impotence caused by the very circumstance of their universal admission. Truths of all others the most awful and mysterious, yet being at the same time of universal interest, are too often considered as _so_ true, that they lose all the life and efficiency of truth, and lie bedridden in the dormitory of the soul, side by side with the most despised and exploded errors. (I, 60)\n\nThese ideas reappear in different form as the 'plan' for _Lyrical Ballads_ (II, 5\u20136). This blend of truth and power is consistently represented by images of health, dew and refreshing rain (I, 59; II, 121). By contrast, the fraudulent anonymous critic has 'intellectual claims to the guardianship of the muses [which] seem, for the greater part, analogous to the physical qualifications which adapt their oriental brethren for the superintendence of the Harem' (I, 42): the image deftly combines debility and perversion.\n\nThe contrast of fanatics and geniuses serves generally to enforce this link between truth and aesthetic power. The philosophic inquiry in chapters V to XIII substantiates the truth-claim by defining the cognitive function of imagination; the further discussion of poetic diction in chapters XIV to XX substantiates the link between truth and aesthetic power by defining the imaginative origins of language itself- origins briefly signaled here by the link between verbal precision and poetic genius. These two definitions are at once the most central and the most vulnerable of Coleridge's arguments: they are philosophically, although not rhetorically, dependent on the metaphysics that chapter XIII does not provide.\n\nI want to clarify one possible misunderstanding concerning the contrast between the genius and the fanatic. The fanatic does not suffer an excess of fancy. He suffers an excessive reliance on the senses, or a deformed, exaggerated sensibility. Poets 'from Donne to Cowley' are witty or fanciful. They do not achieve the balance and unity of true poetic genius, but true poetic genius is extremely rare. Even those who are more imaginative than these 'elder poets' (Bowles, for instance, or Cowper) are never described as geniuses. They are far less imaginative than Wordsworth.\n\nA footnote at the end of chapter II clarifies this issue. The true poet is not 'irritable' because his 'profound sensibility' is counter-balanced by the associative power (fancy) and the modifying power (imagination) (I, 30 n). The fanatic's anger reveals the deficiency of this balance. 'GENIUS itself consists' in imagination; but fancy is 'a component equally essential', as the criticism of Wordsworth's poetry will later show (I, 30 n; II, 104\u20135). These intellectual powers are the polar opposites of the physical powers of sense and passion; like centripetal and centrifugal forces, they remain necessarily in balance. Fanatics and pseudo-poets can neither observe well, nor control their passions, because their fancy and imagination are feeble.\n\nThe image from physics deserves close attention. Centrifugal and centripetal forces are two manifestations of a single power (attraction) exhibited in or by the relations of bodies having mass. The outward-directed centrifugal force (sensibility) is not truly a force at all, but a way of naming a particular perspective on or experience of the inward-directed centripetal force. In the familiar physics-class example, the man sitting on the ball being twirled on a string only _feels as if_ he and the ball are being flung outward. Actually, his movement is possible only through the inward pull exerted along the string. This is, I submit, a quite precise image for what Barfield calls Coleridge's theory of '\"separative projection\"'. We inexorably _feel as if_ there is a real, independent, physical world. Any science (of things or minds) that is concerned with knowledge must accept as axiomatic that we can know an external physical world that actually does exist. But metaphysics in its strict sense \u2013 as ontology \u2013 must recognize the _logical_ priority of (inward-directed) imaginative power: that is, the priority of self-consciousness. The image reappears later to illumine the polarity of being and knowing (I, 188).\n\nThe image in this passage implicitly links Coleridge with Newton, just as Kant had compared himself to Copernicus. In chapter IX the speaker describes such a 'completion' of the master's work as the task he shares with Schelling (I, 103\u20134). A later footnote explaining the meanings and relationship of words T and 'me' further identifies Coleridge's position for the philosophically experienced reader (I, 52\u20133 n).\n\nI will have much more to say about imagination and self-consciousness later on. For now, one need only note that genius requires both fancy and imagination for its effective manifestation. The opposite of genial imagination is not fancy, but a literally mindless empiricism. The passage also reveals a most fundamental trait of Coleridge's prose: his metaphors are highly cognitive. They often (perhaps always) define relationships and ideas that will be explained 'literally' only later.\n\nOne final issue deserves attention: in what ways do these philosophical and autobiographical aspects interrelate? Coleridge says that he uses the chronological narrative 'chiefly for the purpose of giving a continuity to the work' (I, 1). Any substantial continuity has its roots in the structure of ideas and arguments presented; the sequence of influences reveals such origins. Like the landscape of Wordsworth's boyhood, the figures of Boyer and Bowles loom high because they foster the speaker's intellectual and moral character. They impress upon him and elicit from him the values and habits of mind that subsequent influences (Southey, Wordsworth, the anonymous critics) refine or draw farther into distinct consciousness. These values and habits prepare him to recognize both genius and fanaticism for what they are: the operations of imagination account for both, and for their productions as well. The speaker's forthcoming inquiry reflects his native philosophic bent, and arises \u2013 as do all genial productions \u2013 from the balanced power of his ideas (about language), his observations (of writers), and his passion (for true understanding and fair judgment).\n\n# **3 The Associative Fancy**\n\n> 24th March, 1808. In how kind and quiet a manner the _Conscience_ talks to us, in general, & at first \u2013 how _long-suffering_ it is, how delicate, & full of pity \u2013 and with what pains when the Dictates of Reason made impulsive by its own Whispers have been obstinately pushed aside, does it utter the sad, judicial, tremendous Sentence after which nothing is left to the Soul but supernatural aid. O what an aweful Being is Conscience! and how infra-bestial the Locks [sic], Priestleys, Humes, Condilliacs [sic] and the dehumanizing race of fashionable Metaphysicians. _Metapothecaries_ , said one _sportively_ , but I _seriously_ , should say Gzraphysicians (i.e. _Contra_ naturalists) when I spoke of them as _Agents_ ; but when I regard them merely in _themselves_ & _passive_ , I should call them _Hypo_ physicians, i.e. _below Nature. Zoophytes?_ \u2013 Nay, there is no contradiction in anything but degraded man. ( _CN_ , III, 3281)\n\nThe first four chapters demonstrate well the so-called 'circularity' of Coleridge's discourse. He presents a series of interrelated terms ('precision', 'beauty', 'truth', 'serenity', 'anger', 'insight', 'intellectual dimness'), each of which he links to 'fanatic' or to 'genius'. These terms appear in each chapter; progressive development unfolds the _relations_ of the terms, not just their meanings in the usual sense. One idea controls all these relations: the center of the circle is _will_. Chapter V begins an extended inquiry into will, and particularly into the relation between will and imagination.\n\nChapter V begins this inquiry by distinguishing three levels of will, and by asserting that the nature of will is epistemology's central problem:\n\nThere have been men in all ages, who have been impelled as by an instinct to propose their own nature as a problem, and who devote their attempts to its solution. The first step was to construct a table of distinctions, which they seem to have formed on the principle of the absence or presence of the WILL. Our various sensations, perceptions, and movements were classed as active or passive, or as media partaking of both.... [C]onjectures, however, concerning the mode in which our perceptions originated, could not alter the natural difference of _things_ and _thoughts_.... Our inward experiences were thus arranged in three separate classes, the passive sense, or what the school-men call the merely receptive quality of the mind; the voluntary; and the spontaneous, which holds the middle place between both. (I, 65\u20136)\n\nColeridge's attempt to base his criticism on principles 'established and deduced from the nature of man' (I, 44) formally begins with this account of the 'traditional' analysis of human nature. Later we learn that secondary imagination operates at the spontaneous level of will. It is 'an intermediate faculty, which is at once both active and passive' (I, 86). As chapter V proceeds, Coleridge asks whether associationism can account for the spontaneous activity of mind; and he concludes that it cannot. The laws of association describe the operations and the materials of fancy, but imagination itself operates in other ways. Imagination does not associate; it synthesizes. What _that_ distinction entails will be stated in chapters XII and XIII, and explained coherently in the chapters on Wordsworth.\n\nThis table of distinctions provides the epistemological and structural foundation of the _Biographia_ 's inquiry into imagination. It functions here as a substitute, in some ways, for Coleridge's more usual foundation: the distinction between reason and understanding. As the reader winds his way through chapters V to XIII the table slowly grows. The completed table looks like this:\n\n| I | II | III  \n---|---|---|---  \n(levels of will) | receptive | spontaneous | voluntary  \n(forms of cognition) | sensuous | intuitive | discursive  \n(forms of | common | self- |   \nconsciousness) |  consciousness |  consciousness | memory  \n(acts of powers) | perception | synthesis | association  \n(names of powers) | primary imagination | secondary imagination | fancy\n\nThe table has three major features. First, columns I and II are very closely related. Both common consciousness and self-consciousness are self-knowledge, although knowledge of different aspects of the self. Common consciousness is or knows the self as agent. Self-consciousness is or knows the self as an act. Because will is never actually suspended (I, 77), 'receptive' (the 'relatively passive') and 'spontaneous' name differences in degree, not kind, of mental activity (cf. I, 202). Because all knowledge is the concurrence of subject and object, 'perception' itself is 'synthetic', albeit not consciously so. Yet the terms 'sensuous' and 'intuitive' are far less closely related. One is mediated, the other immediate; one is physical, the other spiritual.\n\nThis reveals the second feature of the table. All the elements in column I describe the mind's encounter with the sensuous domain; all the elements in column II describe the mind's encounter with itself. The table thus reflects the differences, and the similarities, between our knowledge of things and our knowledge of thoughts \u2013 a distinction with which epistemology begins (I, 65). The opposition between things and thoughts has already been observed at length in the differences between sensation-dependent fanatics, and geniuses, whose minds are 'affected by thoughts, rather than by things' (I, 20). In a highly characteristic way, Coleridge moves forward by delving beneath prior topics; he makes the implicit explicit; he moves background issues into the foreground. Yet notice as well that he does not tell us that this is what he is doing. He writes as if he presumes we will anticipate these epistemological issues as the next logical step. It is the logical progression only for one who makes much of the fact that fanatics' minds are passive, while geniuses' minds are active.\n\nThe table of distinctions at first accommodates the fanatic-genius, passive-active distinction by appearing to grant that minds _are_ passive in their relation to the physical world. But as the table and the epistemology underlying it are developed, this appearance is gradually discarded. This development takes place in chapters V to IX, which are centrally concerned with the realities named in column III. The third feature of the table is the extent to which column III names the central issues in Coleridge's critical commentary on his own times \u2013 and on Wordsworth. Throughout the analysis of associationism, the speaker argues that Lockean philosophies do not explain the phenomena named in column II, those concerned with the mind's experience of itself. Materialist associationisms collapse the intuitive into the sensuous, the spontaneous into the receptive, and so on. The network of relations among these fifteen terms underlies Coleridge's contention that Wordsworth's attitude toward the landscape verges on an empiricism that is ultimately pantheist. Wordsworth, Coleridge contends, attributes to _things_ what he ought to attribute to his own imaginative _thinking_. In his theory and in his poems, he sometimes exaggerates or mismanages all that column III represents, because he has been unduly influenced by the Lockeanism of the day. He, too, tends to represent the intuitive as the receptive ('the light reflected, as a light bestowed'), although \u2013 Coleridge insists \u2013 he knows better. Coleridge's own account of fancy explains that the poet's associative processes should be controlled by imagination's spontaneous impulses, not by the spacio-temporal forms of the landscape. He offers an imaginative, not material, account of associating.\n\nThe concept of will is the _Biographia_ 's structural foundation because here Coleridge is centrally interested in discrediting the Lockean contention that the mind is a _tabula rasa_. The theory of imagination arises by contrast to this common but perniciously mistaken notion that the mind is passive. By asserting that the mind is active, he can account for the controversy over _Lyrical Ballads_ by analyzing the intellectual passivity of fanatical critics. From the same standpoint, he can undermine portions of Wordsworth's theory by contending that Wordsworth's careless writing suggests that the mind is passive in its relation to the landscape. By analyzing the merits of Wordsworth's best poems, Coleridge can champion the moral and cultural value of Wordsworth's exemplary genius. He can restore the elements in column III to their proper place in our knowledge of human nature \u2013 to the dignity and value Wordsworth's best poetry so brilliantly portrays.\n\nDespite \u2013 or perhaps because of \u2013 this structural centrality, the _Biographia_ seems to explain the idea of will everywhere in general, but very few places in particular. It is a genuinely elusive idea, but one that Coleridge regarded as crucial for his philosophy. In most rudimentary terms, the idea of will comes down to this: the mind is active. Most precisely, the mind _is_ an act. It is not a blank slate on which the material world writes by means of the sense organs. The pith of my system', he is quoted as saying, 'is, to make the senses out of the mind \u2013 not the mind out of the senses, as Locke did' ( _TT_ , 25 July 1832). The pith _of Biographia Literaria_ 's design is Coleridge's insight into the relation between literary value and moral value. He risks the abyss of pantheism because he must have ways, even interim ways, to link imaginative power as a literary fact to imaginative power as an epistemological fact. He must have this link because it is crucial to the proper evaluation of Wordsworth's poetry. The origin of great art, and the origin of moral insight, are one and the same: Wordsworth at his best is a great poet and a great thinker for one reason, not two reasons. Coleridge cannot prove the relation he sees between these two functions of imagination's power, but he endeavors to reveal to the self-consciously thinking reader that a poem's beauty and its truth elicit or require the same intellectual activity from its reader. By urging and helping us to think about our activities as readers, Coleridge elicits our consent to his new version of the old ideal, 'dulce et utile'.\n\nMany critics have puzzled over the fact that both before and after _Biographia Literaria_ Coleridge attributes to reason many of the epistemological functions that the _Biographia_ seems to attribute to imagination. This is not quite the case, despite appearances: the _Biographia_ 's account of will offers a very subtle and difficult distinction between secondary imagination and pure reason, which is the sheerly active pole that is one part of imagination's synthesis. Because this very subtle distinction supplies the crucial link between literary value and moral value, let me stop here to preview Coleridge's argument. Without such an outlook over the difficult terrain ahead, my own analysis will unduly recapitulate the baffling intricacies of Coleridge's own way.\n\nOne preliminary note: generations of scholars have described the _Biographia_ as almost a pivot in Coleridge's career: before this, poetry; after this, religion. Before this, imagination; after this, reason. Beginning in a substantial way with McFarland's and Barfield's work in the late 1960s, this dichotomy has been increasingly challenged. From his early days as a Hartleyan to his last hours in Highgate, Coleridge was very centrally concerned with morals. His 'orthodoxy' \u2013 that usually scornful term \u2013 was no coward's shelter. His Christianity was subtle, imaginative, and intellectually daring. His literary theories cannot be understood properly in a radical isolation from his religious speculation, because they rest on an analysis of consciousness that cannot be formulated without recourse to religious concepts. Similarly, his religious inquiries and allegiances cannot be appreciated by one who fails to recognize the bold imaginative powers at work in his reaffirmation of classically Western moral values. In short, let me begin by positing the integrity of Coleridge's intellectual career, just as I began by positing the unity of his intellectual autobiography, because it permits me to gain a vantage-point from which this integrity can be more clearly seen and more objectively evaluated.\n\nThe scope and significance of this assumption becomes most fully evident if one contrasts this study with Kathleen Wheeler's _Sources, Processes and Methods in Coleridge's 'Biographia Literaria_ '. Wheeler contends that There are passages in the _Biographia_ which have an imaginative, \"paganistic\" originality and individuality which no formulations explicitly Christian could possibly allow' (p. 146). Later in his career, she explains, 'uncertainties... led Coleridge to have to suppress imagination and irony as uncontrollable elements foreign to his Christian philosophy' (p. 133). 'Immediately upon finishing the _Biographia_ he retired into the safety of the mainland of ordinary conventional consciousness, and occupied much of his intellectual energy with untangling the issues within the Christian theological tradition' (p. 147). In representing Coleridge's Christianity as a retreat from full self-consciousness, from the burden of individual identity, and from the responsibilities coinherent in creativity, Wheeler portrays him as attempting to return to the dimness of sensation and enfeebled intellect characteristic of the deficient multitude. Because such a retreat is not psychologically possible, the act of denying one's own self-consciousness is profoundly self-deceptive. That is why Wheeler and those who share her view describe Coleridge's later career as signaling a radical personal failure \u2013 however scrupulously they avoid condemning him. The quality and influence of his theology offer, apparently, no amends; nor do the ways in which his inquiries challenge the orthodox _status quo_. Different ways of understanding Coleridge's career are important for the formal evaluation _of Biographia Literaria_ as a discourse, because the _Biographia_ includes many specific references to Christian beliefs. Those who regard such statements as irrelevant or contradictory change the _Biographia_ no less radically than those who oppose Milton's God change _Paradise Lost_.\n\nIn the _Biographia_ , Coleridge's arguments about Wordsworth's theories and poems presume the theory of imagination, which presumes the idea of will, which cannot be defined apart from the ideas of reason and faith that are its complements in his analysis of consciousness. Understanding the relations among reason, faith and will, and understanding the relation between this complex and the idea of imagination, are prerequisite to following the path of his argument about the relation between moral value and literary value. So let us look first at the relations among reason, faith and will; and then at the relation between will and imagination; and finally at the ways in which this relationship influences what imagination produces. This preliminary summary can do scant justice to the richness and philosophic complexity of these issues. But let complexity wait for later, as the _Biographia_ itself gradually engages these problems. What we need now is no more than a very plain, very general sense of direction.\n\nWill, faith and reason together name the single, complex, highest power of the human mind; and the ideal union of literary value and moral value derives from the _singleness_ of this power. When discussing only one of these three, Coleridge usually attributes to it the whole complex power. He does so consistently in the _Biographia_ , which focusses our attention almost exclusively on will. But when he defines all three together, he attributes to each a specific aspect of the single power: reason is cognitive, will is motive, and faith is the relation between the fullest operation of each. The most elegant and lucid definition of this psychological trinity concludes the 'Essay on Faith'.\n\nFaith subsists in the _synthesis_ of the reason and the individual will. By virtue of the latter therefore it must be an energy, and inasmuch as it relates to the whole moral man, it must be exerted in each and all of his constituents or incidents, faculties and tendencies; \u2013 it must be a total, not a partial; a continuous, not a desultory or occasional energy. And by virtue of the former, that is, reason, faith must be a light, a form of knowing, a beholding of truth. In the incomparable words of the Evangelist, therefore \u2013 _faith must be a light originating in the Logos, or substantial reason, which is co-eternal and one with the Holy Will, and which light is at the same time the life of men_. Now as life is here the sum or collective of all moral and spiritual acts, in suffering, doing, and being, so is faith the source and the sum, the energy and the principle of the fidelity of man to God, by the subordination of his human will, in all provinces of his nature to his reason, as the sum of spiritual truth, representing and manifesting the will Divine. (Shedd, V, 565)\n\nIn describing faith as both motive (the 'energy' of will) and cognitive (reason, a 'form of knowing'), Coleridge places himself approximately intermediate between Roman Catholic and Anglican Protestant doctrines. The essential act of faith is fidelity to God. Yet this is simultaneously fidelity to the self, to one's own reason as the origin of individual identity and personality. Such simultaneity is possible because reason knows the difference between right and wrong. Reason knows the difference because it knows God's will, or God as Absolute Will. In knowing the difference between right and wrong, reason itself participates in the motive aspect of faith: the entirely rational person will by definition not desire that which he knows to be wrong. Yet sin is precisely such a desire: it is the failure to subordinate desire (will) to knowledge (reason). By defining sin in this way, Coleridge portrays it as simultaneously inauthentic, psychologically diseased, and contrary to God's will. The well-formed conscience, in turn, testifies both to mental health and to morals: it reflects the unity or the disharmony of faith, will and reason.\n\nThe 'Essay on Faith' emphasizes faith's role in this psychological trinity; _Biographia Literaria_ emphasizes will's role. It describes _will_ as the mind's power to _act_ , and argues that the mind's essential act is _knowing_. Coleridge usually calls this essential cognitive act _reason_ , and then distinguishes (in basically Kantean ways) between reason's cognition and the cognition of understanding. But the distinction between reason and understanding very seldom appears in the _Biographia_. For clarity's sake, I will follow the _Biographia_ 's usage, using _will_ to name the mind's power to act, a power primarily evident in the power to know. In the _Biographia_ , then, will is both cognitive and motive, and its highest form of cognition is the act of faith or the knowledge of God.\n\nLet us turn now to the character of will's interactivity with imagination. Will is involved in 'each and all' human acts, from the most profound to the most trivial. Knowing is not a part of will, as an arm is part of a dancer. Rather, knowing is to human being as dancing is to the dancer _per se_. As chapter XII explains, the epistemologist examines the fundamental form of this human act, which is self-knowing or self-consciousness. By virtue of self-consciousness, all our other acts are different from the acts of unconscious nonhumans, just as by virtue of dancing all the movements of a dancer are (potentially, at least) more coherent, deliberate and beautiful than those of a nondancer. This turning inward of will, the self-consciousness, is the act of secondary imagination.\n\nIt is or requires the act of imagination because Coleridge here distinguishes knowing as such from the ability to dissolve and simultaneously to reconstitute the unity of consciousness. When Coleridge later in his career stops discussing imagination, it is at least in part because this distinction is not absolutely necessary to his principal arguments about reason itself. And later on he is more interested in determining how reason accounts for the richest development of human consciousness, rather than how imagination acts as an essentially necessary 'catalyst' for these supreme human achievements. The _Biographia_ , however, does emphasize imagination's role in attaining self-consciousness because ( _a_ ) imagination's access to the primal unity of will, faith and reason provides the link between moral value and the most perfectly achieved imaginative works, and ( _b_ ) because self-consciousness is only one instance of imagination's power to dissolve and recreate. The process of composing poetry is another.\n\nWhen will and imagination interact as self-consciousness (when will through imagination strives to know itself), will discovers the pure spirit or soul. It discovers its own being-which-is-knowing that Coleridge usually calls 'reason'. As we can and cannot tell the dancer from the dance, so we can and cannot distinguish our being from our knowing. The unity of being and knowing is an act, the act of self-knowing, which can be represented by the words 'I am'. The human power of knowing is an echo in the finite human mind of the infinite and absolute knowing that is God, 'the infinite I am'. One's ability through will to know one's own consciousness, to know the human being-which-is-knowing, is thus a knowledge of or an encounter with the absolute in a pure but relative form. It is consciousness of the human in the divine, or the divine in the human, or Christ within us, 'begotten not made, one in being with the Father'. In short, it comes to this: will as cognition (the reason aspect of faith) knows God; it also knows itself as pure spirit, as pure knower, and thus as both free and immortal by virtue of its essential identity with or participation in the absolute knowing that is God. It attains such knowledge only when or as it exerts the fullness of its cognitive power on the mind itself, and such self-consciousness requires an immensely powerful secondary imagination.\n\nWill thus involves two paradoxes. The first is the possibility of an absolute in relative form. This derives from the mystery of the Incarnation, or the Trinity generally, about which the _Biographia_ says almost nothing. The second is the paradoxical relation between the mind and the world: the world is both physically real and independent, and created for us through the act of knowing. Since knowing is such an important access to the divine, the contradiction may be capable of symbolic resolution. But, again, the _Biographia_ says almost nothing, repeatedly deferring solution to the Logosophia. These two paradoxes underlie Coleridge's central contention that the greatest works of literature intimately synthesize moral insights with vividly particular portraits of all that we call 'reality'. Because the paradoxes are genuinely irresolvable, the argument about literature is an essentially rhetorical construct. Coleridge seeks to persuade us by appealing to 'inner experience' generally, and by specific appeal to the 'inner experience' of the meditative reading of poems. And this mode of argument is neither outrageous nor sly. Recognizing that 'proofs' of God can no longer be valid or meaningful, Coleridge sees further that 'proofs' of the cultural and moral value of literature are equally futile. We need, he recognizes, a new defense of poetry.\n\nThis account of the interactivity of will and imagination has emphasized the role of will so as to sketch the connection between imagination and the moral realm. Explaining the same interaction with an emphasis now on imagination can sketch the connection between imagination and literature. The imagination mediates between the mind's active pole (will) and its passive pole (sensation). Imagination's own activity is synthesizing these two poles. This synthesis generates two sets of 'products'. The first set, produced through primary imagination, is common consciousness and perception. Common consciousness is the union of self-as-subject (will as act or as knowing) with self-as-object (will as agent or as being); it arises coinstantaneously and spontaneously with the act of perception. It can be represented by the phrase 'I am', whereby the T is regarded or discovered as an entity distinct from its surroundings.\n\nThe second set, produced through secondary imagination, is philosophic or imaginative self-consciousness and poetry (in the highest sense in which poetry and philosophy are equatable). These products arise when the synthesis comprising common consciousness becomes itself an object of knowledge. Common consciousness can itself be known only by one who can simultaneously dissolve and reconstitute its union of being and knowing. Such self-consciousness or philosophic consciousness is the knowing of common consciousness both as a unity and as its parts. Secondary imagination distinguishes the dancer from the dance, but in doing so realizes that the two _cannot be known in isolation from one another_. Imaginative self-consciousness discovers or generates suprasensuous knowledge because it directly intuits the will itself (the purely active pole) when it regards the unity of common consciousness as its parts. It 'dissolves, diffuses, [and] dissipates' not the object itself, but the relation between mind and object. I will explain more about imagination's direct intuition of will when we come to chapter XII.\n\nThe 'Essay on Faith' explains that faith as a product is knowledge, and faith as a process is will. The _Biographia_ explains that poetry as a _product_ is the union of universal (via will's self-scrutiny) and particular (via the senses) that Coleridge calls a symbol. Poetry's moral and human value derives from these two links, first to God and secondly to the richness of human experience or reality as humanly known. Poetry as a _process_ 'brings the whole soul of man into activity, with the subordination of its faculties to each other, according to their relative worth and dignity' (II, 12). By virtue of this whole involvement, poetry is both an intellectual and linguistic craft with its own rules, and an impassioned self-expression bounded only by the expressing self. Poetry as a process also achieves symbolic richness: it is both universal in its fidelity to rules derived from fundamental patterns of human consciousness, and particular in its presentation of both specific situations and the individuality of its author. We will see much more of Coleridge's ideas about the nature of poetry when we come to the _Biographia_ 's second volume.\n\nOne final note about imagination. It is not directly cognitive. We have only two sources of knowledge: the relatively passive sense organs, and the will (or reason). The imagination \u2013 at either of its levels \u2013 can be called cognitive only because it synthesizes cognitive powers. It does not, itself, have access to objects as the senses do, nor to the suprasensuous as the will does. If it did, then the distinction between sensuous and suprasensuous would become very badly blurred; and this is likely to land us in pantheism. And yet we have no conscious knowledge of any sort without the imagination. Because it is not necessary in every context to specify that imagination 'knows' only in this qualified and indirect sense, Coleridge in the _Biographia_ talks about imaginative knowledge or self-conscious knowledge when he wishes to contrast what may be known through sensation with what may be known through meditation. Such passages are highly condensed, or strictly adapted to local argument; but they are not contradictory. (They do, however, make it somewhat difficult to specify the relation between Coleridge's idea of imagination and Kant's.) Coleridge's point, inevitably, is that imaginative knowledge arises from the synthesis or harmony of all human powers, both spiritual and natural. Its opposite is strictly empirical or sensory observation, which does not, finally, deserve the name 'knowledge'.\n\nThe _Biographia_ says little about faith, and less about reason and understanding, because it is primarily concerned with the products of imagination, not with the full intricacy of the activity of imagination in concert with other powers. Coleridge defines only those parts of his full philosophy that he needs to establish the unity of truth, verbal precision and aesthetic power in great poems. As he says in chapter IX, he intends to offer an aid to judgment not \u2013 as later \u2013 an aid to reflection. Formal arguments concerning the exact activity of will, and its interactivity with imagination, are confined to one chapter \u2013 to the extraordinarily compressed summary and revision of Schelling's analysis of consciousness. But the practical or experiental aspects of will and imagination are worked out in a lucid and sometimes leisurely way in chapters V to XI, as the speaker gradually and simultaneously refutes materialism and recognizes the unity of being and knowing.\n\nThis refutation formally begins in chapter V, which begins with the 'table of distinctions' that we have just examined. Somehow it ought to be obvious that the three levels of will provide a basis for the philosophic inquiry projected in chapter IV. But the remarkable absence of will as a term and an explicit issue in most commentaries on the _Biographia_ demonstrates that these valuable distinctions, especially as later elaborated, largely escape even careful readers. This happens, I suspect, because Coleridge moves so rapidly to the summary of Mackintosh's lectures that he invites us to see the chapter's opening paragraph as no more than an introductory historical flourish. Furthermore, one who expects Coleridge to be always digressive and disjointed will be particularly willing to dismiss this paragraph because it does not immediately and intimately link chapter V to chapter IV. That expectation becomes self-fulfilling: the analysis of associationism appears largely irrelevant, and this misperception itself generates much subsequent confusion. The unity of the _Biographia_ begins rapidly to unravel, especially for a reader who also discarded chapter II because of its equally off-handed opening. As a composition, _Biographia Literaria_ is by far too fragile to resist and correct such readers.\n\nI doubt that it is possible to apportion 'responsibility' or 'blame' for the reader's probable misdirection at chapter V: Coleridge's genuine clumsiness here is too complexly interwoven with the traditions influencing how we approach his works. I would point out only that autobiography offered Coleridge ample resources for the devising of a more helpful transition, resources that one must assume he ignores for conscious and unconscious reasons of his own. Regarded as autobiography, chapters V to VII refute the claims advanced in lectures by Sir James Mackintosh, who according to Coleridge asserted that Hobbes had advanced a new explanation of spontaneous mental activity, an explanation fully worked out by Hartley. The speaker sharply disagrees with Mackintosh's high praise for these two philosophers. Mackintosh's lectures are not linked to the sequence of influences recorded in chapters I to IV, although since Coleridge attended them in 1800 they might have been fitted in. Since Hartley was actually a quite major influence (a fact noted in chapter X), he, too, offers a means of linking these chapters to the first four. But no autobiographical continuity links chapter V to what has gone before, until chapter IX describes the philosophers analyzed in these chapters as a sequence of studies the speaker had undertaken. This 'revision' establishes Mackintosh as a student and historian of philosophy opposite and equivalent to the speaker. It reflects helpful emphasis back to the table of distinctions, and to the question whether ancient or modern philosophers more accurately understood the spontaneous (i.e. imaginative) activity of mind. It is possible that the lack of immediate autobiographical linkage is intended to signal that _Biographia Literaria_ is not simply or exclusively an intellectual autobiography, but also an inquiry into will and poetry. If so, then the principal chronological tie becomes the speaker's _present_ intent to present his 'poetic creed' not as opinion, but as 'deductions from established premises' (I, 65): the autobiography takes the first of several leaps 'forward' into the speaker's present time.\n\nRegarded as philosophy, chapters V to VII progressively develop the issues of earlier chapters: they define fancy, and distinguish it from imagination. In defining fancy, Coleridge asserts that association is governed not by things but by thinking. The spacio-temporal relations of objects matter less than the power of will to focus attention and thereby to control the clarity with which items are present or accessible within one's memory. The spontaneous operation of will (the secondary imagination) thus can govern the activity of fancy in associating elements linked by any notion or feature at all. Coleridge sketches this interaction of fancy and imagination in the densely metaphoric and analogic 'water-insect' passage late in chapter VII. These images and analogies reappear in the criticism of Wordsworth when Coleridge contends that Wordsworth attributes to the rural landscape a power to govern association that properly belongs to secondary imagination. This error, Coleridge claims, is evident both in Wordsworth's statements about rustics' language, and in his unsuccessful poems.\n\nLet us begin with the definition of fancy, turning afterwards to the critique of materialist theories of association. Chapter V describes fancy objectively, from the perspective of things associated; chapter VI describes fancy subjectively, from the perspective of mental controls over this process. Chapter VII explores the consequences of mental control over fancy, and formulates the 'true practical general law of association' in a way that emphasizes fancy's subordination to will and imagination. Chapter VIII explores the metaphysical issues raised by this subordination; further inquiry into such issues will generate the formal definition of imagination in chapter XIII.\n\nThe objective account of what perceptions are associable begins from the same fundamental conservatism we have already noted: Tor many, very many centuries, it has been difficult to advance a new truth, or even a new error, in the philosophy of the intellect or morals' (I, 66). On this basis, the speaker prefers Aristotle to the moderns: 'In as much as later writers have either deviated from, or added to his doctrines, they appear to me to have introduced either error or groundless supposition' (I, 71). Yet, as Shawcross points out, the speaker emends Aristotle, who does not list the 'five agents or occasioning causes' of association (I, 72). Even this position is subsequently 'clarified' into something quite different. The five causes are summarized as variations on the _contemporaneity_ of original impressions; the speaker later substitutes _continuity_ to eliminate altogether any necessary temporal link (I, 87; cf. I, 70). Coleridge calls down the authority of Aristotle against the authority of the associationist tradition in English materialism, arranging and adding to Aristotle's rather sketchy remarks on the topic for his own purposes.\n\nAristotle's sketchiness very nicely deflates association from the complete theory of mind that it is in Hartley to no more than the name of one class of actions that the mind can perform. The speaker approvingly notes that Aristotle provides only 'a comprehensive survey of the different facts, and of their relations to each other without _supposition_ , i.e. a fact _placed under_ a number of facts, as their common support and explanation' (I, 72). Because Aristotle consistently distinguishes mental movement or activity from movement in space, the speaker claims that Aristotle supports his own major contention about fancy: association is controlled not by the physical relations of the objects whose impressions are associated by the mind but, rather, by strictly mental processes. Association is not a law governed by things, but a law governed by thinking \u2013 a process that the mind itself controls.\n\nChapter VI turns from this concern with the relations among associable objects to consider the mental processes by which fancy is directed. The speaker asserts that 'the will, the reason, the judgement, and the understanding' are the 'determining causes' of association (I, 76). If these higher powers failed to exercise such authority, then 'our whole life would be divided between the despotism of outward impressions, and that of senseless and passive memory' (I, 77).\n\nThere follows an example designed to support this very central contention. It begins with a verb in the imperative mood: 'Consider, how immense must be the sphere of a total impression from the top of St. Paul's church...' (I, 77). The reader who resists has demonstrated his control over association: he refuses to summon up the view associated with these words. The reader who cooperates would be swept into delirium if will did not control and limit the associative movement of mind. Either way, the speaker makes his point by eliciting the control that is at issue, rather than by constructing formal philosophic counter-arguments.\n\nBecause this strategy would probably not persuade a professional philosopher (especially a sophisticated empiricist), we have here fairly clear evidence concerning the kinds of reader Coleridge expected. This passage, like the footnote defining 'idea' (I, 69 n), or the explanation that both materialism and idealism can be traced back to the ancients (I, 66), or the footnote distinguishing objective and subjective forms of consciousness (I, 53 n), seems to suggest that Coleridge expected little formal sophistication. The argument is designed for one who shares the speaker's essential values \u2013 both his Christian ideas about the mind and about moral responsibility, and his attitudes toward inquiry. The long experiential arguments identify the speaker as a person more deeply concerned about practical consequences or moral issues than about formal proofs. We are to realize that, for this man, proofs without such practical bases and moral conviction are merely academic word-games \u2013 an abuse of language closely akin to that of anonymous critics (see I, 95\u20137; I, 178\u20139; cf. _CL_ , II, 961). In effect, these experiential arguments define the powers of will as axioms, a status that never changed (see _AR_ , 153\u20139). Personal agency is an unquestioned but well-examined fact from which Coleridge begins. The reader who is more interested in poetry and Wordsworth than in philosophy and Schelling might accept such a complex axiom. But such a reader is apt to be confused and annoyed when Coleridge suddenly shifts to a dense discourse that presumes a thorough knowledge of the implications of various philosophic formulae, as he does in chapters VIII and XII.\n\nThe professional philosopher who can read such chapters with relative ease will most likely object to the breadth of Coleridge's initial assumptions. The professional is more likely than the amateur to demand that Coleridge prove his idea of will, or argue formally on behalf of his concept of God. In the absence of such proofs and arguments, the professional might think that these illuminations are to be taken as proofs; and thence conclude that Coleridge is a bungler with no sense of formal argument, nor of the complexity of his issues. This misperception derives in part from the sharp wit that guides less experienced readers: Coleridge writes as if he has demolished his opponents' positions. His repeated statements of intent to supply formal proofs in another book should signal his recognition that such proof is necessary. Yet the confident tone, the unfinished Logosophia, and the irresolvable paradoxes leave Coleridge quite vulnerable to sharp professional attack.\n\nAnd in all this it must be remembered that in the _Biographia_ Coleridge explicitly refers to a class of readers to whom he expects to be almost entirely opaque: anonymous critics and their kind. Rather than leave his argument vulnerable to the confused apprehension of this powerful group, Coleridge's speaker periodically breaks off his discourse with a reference to the unspoken. Usually the unspoken is represented as the Logosophia or some other unwritten or unfinished work; occasionally, as in chapter VI, it is represented as a mystery that should not be revealed indiscriminately. As these references become more explicit, it gradually becomes evident that all refer to the relation between the creative powers of human cognition, and the power or activity that is God. The most famous reference to 'mysteries' is the chapter's second 'anecdote', describing the delirious serving girl. The story attributes to memory the passive recording falsely attributed to fancy itself, and asserts that _conscious_ memory represents only a fraction of the total power. The girl's ravings reveal association without the conscious control of will. The story claims that even relatively passive mental processes rest not on material but on spiritual causes: 'all thoughts are in themselves imperishable', and their record is the 'dread book of judgement' (I, 79\u201380). The concluding lines attribute the unity and permanence revealed by association operating as memory to 'the free-will, our only absolute _self_ ' (I, 80).\n\nThe passage from Plotinus links will to the aesthetic issues defined earlier. Before defining imagination, the speaker must solve several problems in epistemology and metaphysics. Because he solves them through religious meditation, moral issues often color chapters V to XIII. As a result, thematic unity demands coherent relations among morals, aesthetics, imagination and will. These relations are first defined here, when the speaker 'breaks off' a discussion of the moral implications of will to quote the following. It is 'profanation' to speak of such things, he explains,\n\n'To those to whose imagination it has never been presented, how beautiful is the countenance of justice and wisdom; and that neither the morning nor the evening star are so fair. For in order to direct the view aright, it behoves that the beholder should have made himself congenerous and similar to the object beheld. Never could the eye have beheld the sun, had not its own essence been soliform,' ( _i.e. pre-configured to light by a similarity of essence with that of light_ ) 'neither can a soul not beautiful attain to an intuition of beauty.' (I, 80 n)\n\nColeridge adds 'imagination' to the first sentence to adapt the passage to his own ends \u2013 a characteristic strategy. The speaker literally asserts that he cannot continue this line of moral inquiry because not everyone will understand (another instance of the 'deficient multitude' theme). In Coleridge's context, the passage from Plotinus implicitly equates imagination, aesthetic sensitivity, and consciousness of will. It also asserts a necessary connection between this psychological unity and the metaphysical unity of the Good, the True and the Beautiful. The speaker later substantiates this necessary connection through the analysis of will and consciousness (i.e. through the definition of imagination). The theory of imagination explains how the Good, the True and the Beautiful are evident in the most perfect instances of poetry.\n\nIn short, if the last paragraph of chapter VI is read as a transition \u2013 certainly what the reader expects in a last paragraph \u2013 one discovers both the unity of prior issues and a basis for understanding how the diversity of earlier issues will lead to a higher unity. Will interrelates imagination, beauty and the fanatic-genius opposition; the equation of imagination and the consciousness of will maps new ground. The 'breaking off' is merely stylistic, an odd and probably unsuccessful device to draw attention to the highly condensed and important integration of several central issues.\n\nChapter VII defines the ontological and metaphysical consequences of mental control over association. Its concluding definition of fancy both complements and supplants the 'Aristotelian' five causes by locating the real cause of association in the controlling power of will. The chapter is dense, quick, and tightly woven with the critique of Hartley; but every idea follows clearly from one central point: failure to recognize the agency of will destroys the ideas of consciousness, moral responsibility or personal agency, God, faith and causality.\n\nAs a philosopher (and a Christian), the speaker values final causes, as he explains in chapter IX: 'We learn all things indeed by _occasion_ of experience; but the very facts so learnt force us inward on the antecedents, that must be pre-supposed in order to render experience itself possible' (I, 94). That is, we _ought_ to be forced inward to final causes, by our own ontology. 'The facts' will only 'force' us if we establish certain standards for explanations. The speaker argues for the _necessary_ existence of suprasensible realities. His argument centers on a complex of terms ('consciousness', 'will', 'self', 'soul') that are not entirely synonymous but, rather, aspects of basic human identity. This complex is closely linked to God, in part through the traditional associations of religious language, and in part for reasons only later explained (but discussed by me at the beginning of the present chapter).\n\nThe speaker first defends consciousness itself. According to Hartley's followers (including, once, Coleridge himself), 'consciousness [is] considered as a _result_ , as a _tune_ , the common product of [material] breeze and [bodily] harp[.]... [But] what is harmony but a mode of relation, the very _esse_ of which is _percipi?_ ' (I, 81) To hear and to analyze the music made by the harp requires a third position, neither harp nor breeze but auditor. If consciousness were merely a product of the physical world and the physical body, then we could by definition never achieve the knowledge of consciousness that the image defines. 'Listening' requires the power to observe one's own mind. It requires self-consciousness. The power of self-observation is the single most central fact for Coleridge's philosophy generally; it is defined here for the first time through a metaphor. Coleridge's equation of poetry and philosophy must be taken very seriously by any student of his prose \u2013 or his poetry.\n\nThe second suprasensible is will itself, considered as moral responsibility or personal agency. If final causes are discarded, then 'the disquisition, to which I am at present soliciting the reader's attention, may be as truly said to be written by Saint Paul's church, as by _me'_ (I, 82). The reference to St Paul's echoes the earlier reference to introduce another witty portrait of common-sense experience. The passage appeals to the experience of personal agency, rather than proving its 'objective' possibility. None the less, the paragraph strongly and successfully urges the irrationality of denying personal agency.\n\nThe denial or the defense of self-consciousness and personal agency leads directly to the denial or the defense of God. Our knowledge of our own will (i.e. self-consciousness) intimately involves a knowledge of God as 'an intelligent and holy will' (I, 83). Limiting all explanations to efficient causality necessarily leads to the 'degredation of every fundamental idea in ethics or theology' because these ideas depend on entities that are not knowable through the sense organs. Spinoza's denial of both personal agency and the personality of God stands silently behind the critique of associationism. The speaker refutes not the intent of Locke or Hartley, but the consequences of object-oriented philosophy as made evident by Spinoza. By suggesting that the similarity of God and man rests on the power of will, Coleridge firmly anchors the relation between his moral and his aesthetic concerns: the activity of mind that generates art also leads to God.\n\nFinal causes must be distinguished not only from efficient causes, but also from necessary conditions (I, 85). The contemporaneity of objects (as stressed by Hartley's mechanism) is a law of matter, not of mind. The development of this idea introduces the first substantial explanation of the interaction of fancy and imagination: the famous 'water-insect' passage, followed by the formal definition of the law of association. This definition stresses the agency of will that the water-insect passage illuminates in greater detail.\n\n... the true practical general law of association is this; that whatever makes certain parts of a total impression more vivid or distinct than the rest, will determine the mind to recall these in preference to others equally linked together by the common condition of contemporaneity, or (what I deem a more appropriate and philosophical term) of _continuity_. But the will itself by confining and intensifying the attention may arbitrarily give vividness or distinctness to any object whatsoever. (I, 87)\n\nThe substitution of 'continuity' eliminates the last traces of materialism from the law of association: continuity can link past and present, or objects here with objects there, on the basis of subjectively originating similarity. Most important, the recalled feature that triggers the associative leap depends not on physical conditions, but on the 'arbitrary' or selective focussing of attention on to particular features of a given perception. Thus, the operations of fancy are determined by habits of observation (what aspects are noted?), by the power of attention or concentration (how clearly are the aspects noted?), and by the subtlety with which resemblances are noted (are the relations crude or delicate?). The important qualities of fancy are determined not by the acuity of sense organs, but by the entire character of a person's relation to the environment \u2013 a relation established through primary and secondary imagination.\n\nThis definition follows the water-insect passage; it partially explains how imagination, or the spontaneous level of will, can control the associative activity of fancy. Having looked at the definition first, we are better prepared to recognize how much more the metaphor suggests than the consecutive first reader may immediately recognize.\n\n... contemporaneity (Leibnitz's _Lex Continui_ ) is the _limit and condition_ of the laws of mind, itself being rather a law of matter, at least of phaenomena considered as material. At the utmost, it is to _thought_ the same, as the law of gravitation is to loco-motion.... Let us consider what we do when we leap. We first resist the gravitating power by an act purely voluntary, and then by another act, voluntary in part, we yield to it in order to light on the spot, which we had previously proposed to ourselves. Now let a man watch his mind while he is composing; or, to take a still more common case, while he is trying to recollect a name; and he will find the process completely analogous. Most of my readers will have observed a small water-insect on the surface or rivulets, which throws a cinque-spotted shadow fringed with prismatic colours on the sunny bottom of the brook; and will have noticed, how the little animal _wins_ its way up against the stream, by alternate pulses of active and passive motion, now resisting the current, and now yielding to it in order to gather strength and a momentary _fulcrum_ for a further propulsion. This is no unapt emblem of the mind's self-experience in the act of thinking. There are evidently two powers at work, which relatively to each other are active and passive; and this is not possible without an intermediate faculty, which is at once both active and passive. (In philosophical language, we must denominate this intermediate faculty in all its degrees and determinations, the IMAGINATION....) (I, 85\u20136)\n\nThe passage opens with a reference to gravity which, like the earlier reference (I, 30 n), asserts the _relative_ opposition of two forces between which imagination mediates. The passive aspect is sensation, which obeys laws of matter such as contemporaneity. The purely active pole is pure will (or pure reason). Despite the vivid sensual detail of the insect and its shadow, the description emphasizes not the creature itself, but its activity. It is easy to enjoy Coleridge's metaphors as rich ornament, but one dare not stop there.\n\nAs the swimming insect or the leaping person demonstrates, genuine synthesis requires that imagination share essential features with the active will and the relatively passive sense receptors. That is, the leaping body has both mass _and_ the capacity to resist gravity by virtue of the same essential property: muscle, skeleton, and nervous organization. In the same way, imagination's mediation cannot be a merely mechanical switching back and forth from activity to passivity, because then imagination would be no more than a title for the alternative dominance of one or the other over consciousness. Nothing so crude could possibly account for the delicately balanced unity that characterizes both geniuses' psyches and their works.\n\nHow does the imagination achieve its mediation between the active and passive aspects of mind? That is the question with which the consecutive first reader is left \u2013 that, and the image of progress via synthesis. This strategy provides a framework or a niche for the definition of imagination. This definition develops quite slowly until the image of synthesis reappears in the relative definitions of 'subject' and 'object' in chapter XII.\n\nLet me once again pause to preview Coleridge's answer to the question poised at this point. As I have already explained, imaginative synthesis consists partly in the relative passivity of primary imagination as perception, and partly in the activity of secondary imagination as it dissolves and reconstitutes perceptual unity and common consciousness. The whole issue arises in chapter VI because the imagination's synthesis _per se_ depends on the fancy's power to collect perceptions that have features in common, to assemble 'material' capable of synthesis into a whole by virtue of its fundamental interrelations. But, as the law of association points out, fancy cannot collect coherently except under specific direction. The other two analogies in this passage make the process quite clear. In trying to recollect a name, one summons to preconsciousness many scattered memories all of which have some bearing on the sought-for name. The secondary imagination seeks or creates the pattern in the memories that will reveal the name. Yet this creating is only theoretically distinguishable from the imagination's control of the recollecting or assembling performed by fancy, because the synthesis gets under way as soon as there is more than one 'datum' present. The 'Essays on Method' explain this more fully ( _F_ , I, 448\u2013524).\n\nThe process of composing demonstrates the same inseparable cooperation. Assembling materials can be mechanical and dull, yet clearly it is crucial. In any substantial investigation, knowing what to gather and where to look depends partly on logic and experience, but more centrally on intuition. The act of composing requires that one discover the pattern one has created, the pattern implicit and buried in the reams of notes. For this process to succeed, the writer's genius (that is, both imagination and, in the old sense, guiding spirit) must both fuse with and stand critically aside from what has been gathered. As many scholars have testified, one delight of the work is in discovering how central one's whimsical quests can prove to be. Such experiences would be far from delightful were it not for confidence that such whimsy is not blind chance but preconscious purpose. Whether poetry or prose, literature or literary criticism, imaginative works of any kind both resist and imitate the material world by means of the extraordinary care exercised by imagination through fancy in the selection of appropriate materials.\n\nMy point is that when Coleridge suggests, 'let a man watch...,' he expects that we will. Although fancy and imagination are distinct, imagination cannot operate without fancy any more than fancy can operate coherently without the focus or motive that will provides. This focus need not be imaginative, but it certainly can be. As Coleridge is quoted as saying, 'Genius must have talent as its complement and implement, just as, in like manner, imagination must have fancy. In short, the higher intellectual powers can only act through a corresponding energy of the lower' (7T, 20 August 1833). By interposing the activity of the water-insect and the other analogies in the definition of fancy, Coleridge sketches the interaction of fancy and imagination through images for it. The images function analogously to a statement of intent, yet only for the reader who knows to take them this seriously, and who is capable of carrying forward the cognitive range of a metaphor.\n\nThe definition of fancy as partial and dependent provokes curiosity concerning the powers that control fancy. This curiosity would have been even more keen for the sympathetic contemporary reader. By attributing all association to fancy, and then defining fancy as he does, Coleridge prepares ground for his own ideas about the mind. These begin to come to the foreground in chapter VIII.\n\nBut, before proceeding to chapter VIII, I should summarize one aspect of the issues in question. When Coleridge demotes 'association' from the title of an adequate philosophy to the name of one class of mental acts, he reopens the questions this popular philosophy had answered: What is perception? and What is thinking? One option is to reverse Lockean mental passivity, but this generates other, equally severe problems. The mind is neither absolutely active nor absolutely passive: as chapter VII concludes, there is the 'intermediate' or synthetic power of imagination \u2013 the spontaneous movement of mind that Mackintosh claims Hartley had explained. Chapter VIII's emphasis on dualism may seem tangential, because this word has not previously appeared. Yet Coleridge does not depart from his topic. He again delves beneath it, to the fundamental issue that has been shaping his counter-arguments all along: What is the relation between mind and external reality? He must answer this metaphysical question before answering the psychological ones; his fundamental disagreement with associationism is as much metaphysical as psychological.\n\nBut Coleridge himself had not yet answered the metaphysical question in strictly philosophical terms. (As McFarland has explained, a rigorous solution is not possible.) In the _Biographia_ , Coleridge does not attempt to close his system or to complete his argument formally. Completion is repeatedly deferred to the Logosophia, as we see for the first time in chapter VIII. As we shall see, in chapter X he offers instead a pair of assertions about faith and cognition: through the activity of faith, we have immediate knowledge of both a personal God and a real world of physical objects immediately known. Beyond these assertions, one finds only hints toward a philosophical construction. As these suggestions become more explicit, claims about the relative completeness of the Logosophia become more explicit as well.\n\nIn chapter VIII these hints are not very specific. The first two depend on Coleridge's prior critique of anonymous critics and the reading public. The speaker describes the possibility that body and spirit ' _may_ without any _absurdity_ be supposed to be different modes, or degrees in perfection, of a common substratum', but explains that this possibility was not 'the fashion' (I, 88). Yet fashion is hardly a reliable guide to philosophic truth. As Coleridge remarks elsewhere, 'From a popular philosophy and a philosophic populace, Good Sense deliver us!' A second, rhetorically similar hint emerges later: 'Leibnitz's doctrine of a pre-established harmony... was in its _common_ interpretation too strange to survive the inventor \u2013 too repugnant to our _common sense_ ; (which is not indeed entitled to a judicial voice in the courts of scientific philosophy, but whose whispers still exert a strong secret influence)' (I, 89). As he elsewhere explains, philosophic arguments 'can be of no permanent utility, while the authors themselves join in the vulgar appeal to common sense as the one infallible judge in matters, which become subjects of philosophy only, because they involve a contradiction between this common sense and our _moral_ instincts, and require therefore an arbiter, which containing both ( _eminenter_ ) must be higher than either' ( _LS_ , 110). Coleridge uses the accumulated pressure of his critique of 'popular' thinking to identify two philosophic ideas as viable: Leibnitzian pre-established harmony, and the common substratum of being and knowing.\n\nThe third hint is somewhat more difficult to explain. On the narrative level, as we shall see shortly, Coleridge in these chapters examines philosophy since Descartes and Hobbes as described by Mackintosh. Chapter VIII returns to Descartes. It briefly describes attempts to correct Descartes' erroneous supposition that mind and matter are absolutely heterogeneous, dividing these attempts into the unsuccessful and the unfashionable. This strategy delivers the speaker to an apparent dead end. No prior philosopher seems able to explain the relation between mind and physical reality. Yet he perseveres.\n\nBut it is not either the nature of man, or the duty of the philosopher to despair concerning any important problem until, as in the squaring of the circle, the impossibility of a solution has been demonstrated. (I, 89)\n\nThis statement shifts our attention from particular philosophies to the logical form of basic philosophic questions. It allows the speaker to formulate the question about mind and matter that 'the philosopher' seeks to answer, and thus to determine the necessary form of all possible answers. His formulation changes the question in crucial ways.\n\nHow the _esse_ , assumed as originally distinct from the _scire_ , can ever unite itself with it: how _being_ can transform itself into a _knowing_ , becomes conceivable on one only condition; namely, if it can be shown that the _vis representativa_ , or the Sentient, is itself a species of being; i.e. either as a property or attribute, or as an hypostasis or self subsistence. The former is, indeed, the assumption of materialism; a system which could not but be patronized by the philosopher, if only it actually performed what it promises. (I, 89\u201390)\n\nBy shifting the inquiry from mind and matter to knowing and being, Coleridge tries to sidestep the insolvable problems that radical dualisms pose. His answers represent one position \u2013 the 'hypostasis or self subsistence' of being and knowing \u2013 as the only remaining viable alternative.\n\nHe also reaches back to gather in the 'unfashionable' possibilities: these three hints tend in a single direction. Hypostasis or self-subsistence or common substratum (etymologically and historically related terms, generally speaking) all point to a pre-existent harmony between being and knowing. Chapter IX consolidates and clarifies the basic point Coleridge has asserted:\n\nThe term, Philosophy, defines itself as an affectionate seeking after the truth; but Truth is the correlative of Being. This again is [in] no way conceivable, but by assuming as a postulate, that both are _ab initio_ , identical and coinherent; _that intelligence and being are reciprocally each other's substrate_. (I, 94; my italics)\n\nAs the reference to hypostasis suggests, this is potentially a Neoplatonic and pantheist answer to the metaphysical question. But only potentially. 'Hypostasis' also translates into Latin and then English as 'person', and figures in theories of the Trinity. The subsequent reference to the Logosophia leaves no doubt about Coleridge's intention, although it provides no explanation of how this intent will be realized.\n\nThe strategy is designed to lead us to accept the substantial unity of being and knowing as both an orthodox and a necessary solution. He defers to the Logosophia only the proof for that which we should see as logically necessary. However irking for those who share Coleridge's interest in the philosophical grounds of literary theory, such a tactic is not formally unreasonable in a book with the _Biographia_ 's declared intent. Not every author is bound in every circumstance to trace the grounds of his argument back to first principles in metaphysics. Yet, because the speaker has rigorously examined Mackintosh's claims, we expect and demand equivalent rigor when he advances his own, alternative philosophy. We might tolerate the missing metaphysics had the speaker not already engaged metaphysical issues with such passion and in such detail. That earlier detail supplies the necessary grounds for the speaker's claim that his ideas are the only remaining viable ones, but the imbalance _per se_ threatens his credibility.\n\nThe reader is also apt to be disappointed by chapter VIII's brevity. Coleridge's prose is often highly condensed, because he richly utilizes the semantic powers of sentence structure and etymology. Such condensation merges with the genuine difficulty of his ideas and arguments to generate a stylistic surface that has been judged cryptic, or obscure, or impenetrable. Recent scholarship has sufficiently undercut those accusations: the need for generalized defense is past. What needs to be noted here is how Coleridge's _departure_ from his usual standards both provokes and justifies charges of obscurity. In chapter VIII undue brevity replaces condensation. The so-called hints tend to accumulate rather than develop; one finds redundance, not richness.\n\nOne easy explanation would suggest that the unusual brevity immediately reflects anxiety generated by pantheism and unresolved metaphysical questions. In reply, I would point to later places \u2013 to be examined in detail \u2013 that are far more pantheistic and yet masterfully managed. I prefer another explanation, if explanations be necessary. Coleridge appears to have lost for a time his concentered attention to a particular audience, because basic comprehension of the chapter requires recognizing the metaphysical and theological issues signaled by 'substratum', 'subsistence' and 'hypostasis'. Such is not the case generally in _Biographia Literaria_. Coleridge writes for an astute and attentive reader, but not one particularly learned in the history of philosophy and theology.\n\nThis lapse also reveals what I take to be a fundamental uncertainty underlying Coleridge's idea of his most probable audience. I have said that Coleridge writes for someone who firmly but unreflectively accepts the dominant Lockean psychology and metaphysics of his day. This expectation makes a sort of 'demographic' common sense, I grant: surely the majority of his contemporaries did accept this world-view; Wordsworth's poetry can strongly appeal to thoughtful empiricists (witness John Stuart Mill). But I wonder how likely it is that one who is _na\u00efvely_ empiricist will prove capable of thinking with the energy and rigor that the _Biographia_ demands. If one's own experience has never prompted dissatisfaction with Locke, will one possess the imaginative energy such thinking requires? And if one's experience has prompted dissatisfaction, then it is plausible that one will have read some other philosophy somewhere along the line, or that one will not be quite so persuadable as Coleridge seems to desire. Perhaps Coleridge's lapses into philosophically dense argument reveal a wavering doubt that there can exist a readership of capable, philosophically na\u00efve minds. It is as if he is just not sure how much philosophic sophistication to expect from his 'good readers', which may in turn reflect a reasonable fear that the right audience does not in fact exist. Perhaps his audience is as much an artful construct as his 'speaker'. Haven remarks that 'In the development of Coleridge's thought, the problem posed by the confrontation of Mariner and Wedding Guest is as important as the Mariner's experience itself'. This may prove true not only for Coleridge's thought, but also for his works. The ideal reader \u2013 the perfect Friend of the Friend \u2013 will both understand the technicalities and tolerate the postponed proof because he understands that symbolic realities cannot be demonstrated.\n\nColeridge's problem in imagining the right audience is no doubt a thorny one, given his times, and given the formally incomplete argument he has chosen to write. But abrupt changes such as chapter VIII must still be accounted serious flaws; he has entirely lost control of the useful contrast between good readers and Lockean blockheads. Some good readers will be perplexed by the chapter; others will find that it raises confusing questions about what level of philosophic rigor they should expect of this book generally. In his art as in his life, perhaps, Coleridge understood very little about how to elicit and to satisfy consistent, reasonable expectations from those around him. Or, conversely, disappointing expectations served some subterranean purpose of its own.\n\nThe major flaws of chapter VIII are recouped in the second paragraph of the next chapter, where Coleridge more solidly forges the link between Christian orthodoxy and the unity of being and knowing. Whatever its flaws, chapter VIII could not have been simply deleted: it summarizes materialism's failure, so as to close the preceding three chapters and to clarify those points on which future development depends. Preserving this numbered summary would have meant substantial rewriting of chapter VIII. Rather than rewrite, Coleridge repeats before proceeding, addressing once again the reader to whom earlier chapters have been addressed. Such repetitions are not at all characteristic, and may signal his recognition that chapter VIII is genuinely obscure.\n\nAs we have seen, the long discussion of fancy serves in part as a critique of contemporary associationist psychologies. This aspect of chapters V to VIII comes more fully into view if we examine them from the perspective of narrative continuity. Mackintosh's four claims reflect the speaker's perception of the philosophical _status quo_ in England. Given his early allegiance to Hartley, and the opening paragraph of chapter IX, these chapters also recount the first stages of his own philosophical development. Let me list Mackintosh's claims, for easier reference later.\n\n1 the 'law of association as established in the contemporaneity of the original impressions, formed the basis of all true psychology'\n\n2 'any ontological or metaphysical science, not contained in such (i.e. empirical) psychology, was but a web of abstractions and generalizations'\n\n3 Hobbes is 'the original _discoverer_ of this great law', while Hartley offers 'full application to the whole intellectual system'\n\n4 association is to the mind what gravitation is to matter (I, 67).\n\nThese objections, listed here in the order presented, are refuted as a group rather than individually. The only Mackintosh claim contextually emphasized concerns Hobbe's originality. One does not necessarily expect any further refutation of Mackintosh. Nor, more importantly, is it particularly evident in chapter V that the speaker fundamentally disagrees with associationism. His principal methodological criticisms could just as easily have led to a new and better associationist psychology. Although one might charge that Coleridge inadequately introduces the scope of things to come, what he introduces and how he begins suggest a reasonable respect for the power accruing to the ideas he would discredit.\n\nHis intent is not primarily to discredit Hobbes or Hartley or Mackintosh, but to discredit the domination of the material world over the mind by means of the sense organs, as advocated by the English empiricist tradition generally. His critique is not primarily ontological, not the argument that it is na\u00efve or inadequate to see reality as comprised of discrete independent objects impinging on passive human consciousness to generate our knowledge of things. On the contrary, this real world of independent objects immediately (but not passively) known remains a very English cornerstone in his system. The critique instead depends on an analysis of ordinary mental experience. As we have seen already, he consistently appeals to our knowledge of our minds rather than to our knowledge of things.\n\nHobbes is fitted into the table of distinctions through Mackintosh's claim that psychological association accounts for the _spontaneous_ activity of mind. But, as the speaker shows, Hobbesian theories deny genuine spontaneity by defining the mind as passive. When the speaker describes the imagination as spontaneous, he integrates himself into the table, and further clarifies his rivalry with Hobbes _et al_. Hobbes surrenders originality to Descartes and accuracy to Aristotle; the speaker credits him only with inventing a physiological mechanism that has no basis in fact and no power to account for the complexity of association _per se_. As Mackintosh, Shawcross and the family editors point out, this is hardly fair to Hobbes; but Coleridge may have quite complex motives here.\n\nColeridge refutes Hobbes so as to reshape the contemporary view of psychology. He wants to shift attention away from the popular (and persistent) misconception that psychology will achieve the spectacular practical results of the physical sciences by equating the real with the quantifiable or the visible, by making itself into a subordinate branch of mechanics or astronomy. By insisting that association has long been studied by philosophers, Coleridge focusses attention far more sharply on the specific recent innovations: physiological mechanisms. By discrediting the mechanisms, he can discredit associationism as a total psychology without denying its power to account for substantial areas of mental experience.\n\nHartley's mechanism is analyzed in a way that discredits any strict physiological mechanism: if they are strict or consistent, they deny will. By refuting Hartley's theory logically, experientially and morally, the speaker again asserts his criteria for philosophy. The logical refutation demonstrates well the Coleridgean difference between the conceivable and the visible (I, 74\u20136). If you visualize Hartley's mechanism as the balls on a billiard table, everything works beautifully. But if you realize that two different sensations must be two different vibrations, then it becomes impossible to understand how one vibration could ever directly propagate another vibration different from itself. Thinking of _acts_ (vibrating) as _things_ (billiard balls) leads only to mayhem. Coleridge refutes the mechanism by appeal to the distinction between visual and conceivable so as to prepare the reader for his own philosophic method: analyzing the _act_ of perceiving rather than the _things_ perceived. The difference proves crucial.\n\nI have already discussed the experimental refutations: the speaker denies the mental passivity inherent in Hartley's philosophy by appeal to the experience of psychological self-control, and the experience of personal agency. The refutation on moral grounds consists partly in the experiential appeal to personal agency, and partly in the argument that Hartley's apparently strict empiricism disallows any nonmaterial God. Moral values lose any ground but observable utility: doing good for its own sake loses meaning. Hartley's evident piety allows the speaker to renew the contrast between himself and anonymous critics: he distinguishes between the system's morality and the man's (I, 83\u20135). The distinction serves another purpose as well. It enforces the speaker's claim that bad ideas mislead good men. It underscores England's need for correct philosophy \u2013 a need argued at length in chapter X.\n\nWe have seen how the speaker responds to Mackintosh's first and third claims, concerning the validity of associationist psychology and the importance of these two figures. The speaker refutes Mackintosh's second claim by experiential arguments that turn the same charge against Hartley: his metaphysics is but 'a web of abstractions and generalizations' that both morals and concrete experience disprove. The fourth claim (association: mind::gravity:matter) restates the second claim within a different metaphor, one that emphasizes again Coleridge's disagreement with a falsely grounded imitation of physical science in the sciences of mind. A law of the mind will be found in the mind, not in the relations of material bodies: philosophers must remember the difference between things and thoughts.\n\nA law of mind will also not be found in the relations between the central nervous system and its environment. Chapter VIII highlights this underlying disagreement with the English tradition, which to the speaker represents one set of attempts to amend Descartes' erroneously absolute distinction between mind and matter. As we have seen, the chapter's first two paragraphs divide emendations into the unsuccessful and the unpopular. The speaker proposes instead that the question be located within the mind, as the relation of being and knowing, rather than outside the mind, as the relation of mind and matter.\n\nMaterialism generally \u2013 and mechanist associationism in particular \u2013 may at first appear to locate the question properly, because it regards knowing as an attribute of being. But the speaker argues that this is not the case. Consistent materialism replaces dualism with monism, but it is an object-dominated monism. Everything is absorbed in the material; the mind is reduced to passive neural mechanisms. This amounts to a denial of real individual existence, a denial logically prior to the denials of moral responsibility and suprasensible realities that we examined earlier. As I commented earlier, Spinoza stands in the shadows. It was he who demonstrated the necessity of these consequences from materialism; 'hylozoism' was Cudworth's veiled term for the argument of Spinoza's _Ethics_. As McFarland's _Coleridge and the Pantheist Tradition_ so persuasively argues, Spinoza's importance to Coleridge cannot be underestimated. Spinoza informs and enlivens Coleridge's opposition to the English materialist tradition \u2013 an opposition everywhere present in _Biographia Literaria_.\n\nThe chapter concludes with a numbered list of objections to any associationist psychology that is dependent upon a materialist epistemology (I, 92). These objections reflect and counter-balance Mackintosh's claims. They also concenter reader attention on the problem of perception. What is it? How is it possible? What may be known? These are the questions taken up in chapter IX as Coleridge begins to define the most fundamental synthesis achieved through imagination: the synthesis of being and knowing in perception and common consciousness. The complexity and apparent indirectness of chapters IX to XI derive from his attempt to describe this synthesis within the context of issues we have seen to this point: the contrast of geniuses and fanatics, with all that involves; the various activities or 'levels' of will, and how they are coordinate; the moral and ontological consequences of suprasensible will; the cultural conflict between true philosophy and its 'popular' counterparts. These issues remain so much in the foreground because the formal definition of the unity of being and knowing, and in particular _how_ that definition unifies his positions on these issues, will be repeatedly deferred to the Logosophia. It is only by interweaving these issues that the speaker can explain the unity of being and knowing without verging into formal metaphysics. We see more of the practical consequences than of the theoretical bases. _Biographia Literaria_ is an unrelentingly practical book.\n\n# **4 Imagination's Synthesis of Being and Knowing**\n\n> Truth is a good dog; but beware of barking too close to the heels of an error, lest you get your brains kicked out.\n> \n> _Table Talk_ , 7 June 1830\n\nColeridge often relies on such symmetries as the contrasts of genius and fanatic, or objective and subjective aspects of an issue. As one recognizes his favorite symmetries, the prose works become easier to follow. Chapters IX, X and XI demonstrate how such symmetries are multiplied as the _Biographia_ progresses. Yet many readers find these chapters inconsequent. Such judgments arise, I suggest, not from a lack of unity and relation, but from an excess.\n\nIn chapters V to VIII the speaker criticizes philosophies he does not accept; in chapter IX he praises those he does. The earlier chapters contrast the methods of ancient and modern philosophers; the later chapters renew the contrast to define the true modern heirs of ancient wisdom. Chapters I to IV contrast genial poetry and pseudo-poetry, with passing reference to the qualities shared by genial poetry and genial philosophy (see, e.g., I, 59\u201360). Chapters V to XI contrast genial and pseudo-philosophy (whether political philosophy, philosophy of mind, or philosophy of religion), with passing reference to qualities shared by genial poets and genial philosophers (see, e.g., I, 140). Chapters X and XI also develop the contrast between geniuses and fanatics. The earlier chapters principally contrasted their personalities, with secondary emphasis on their works. These later chapters reverse the emphasis. The genius imaginatively synthesizes the relatively active and relatively passive aspects of human nature to arrive at permanent principles, the foundation of true national harmony and well-being. The fanatic exists always at an extreme, provoking repression and revolution, because his dependence on sensation generates political notions as inadequate as the philosophic notions criticized in chapters V to VIII.\n\nThis abundance of relations can be confusing because the issues are not arranged in a distinct system. Instead, particular issues relate to each other primarily through a common relation to Coleridge's central contention that the mind is active. Although that contention is closely related to his argument about imagination, and imaginative power is crucial for insight of any sort, Coleridge provides relatively few direct links between imagination and these autobiographical anecdotes. He seems to expect that his readers will share something of his own extraordinary gift for seizing immediately the generative idea underlying any argument or any illustration, however elaborate. Yet the length and associative flow of these chapters can make that a difficult task. The readers most seriously interested in Coleridge's ideas may grow impatient with the loquacious speaker, however entertaining he may be at times. But, once having grasped the idea of will in its relation to imagination, the _Biographia_ 's reader principally needs a memory tenacious enough (or a notebook large enough) to keep track of all the issues that this idea shapes. Let me begin by collecting and interrelating the major ideas of these chapters, so that I can, later on, show how the funny stories and widely ranging illustrations are designed to explain, to justify and to emphasize these ideas.\n\nThe first of these is Coleridge's idea of philosophy. The opening paragraph of chapter IX restates and develops the speaker's encounter with an apparent dead end so as to extend his redefinition of the philosopher's central question into a redefinition of philosophy itself. Limiting inquiry to empirical procedures \u2013 observe, collect, classify \u2013 would be a 'wilful resignation of intellect' that results in the destruction of all ideas of cause and effect (I, 93\u20134). Causality has already been featured as a principal moral concern: without valid causality, there is no valid moral responsibility (I, 82\u20133). If we are to understand experience, and intelligibly defend or ground our belief in personal agency, we must (like Kant) discern and study 'the antecedents, that must be presupposed in order to render experience itself possible' (I, 94).\n\nBut philosophy is not just the science of antecedents to experience. It is also 'an affectionate seeking after the truth' (I, 94). It engages both intellect and passions. Philosophy, like poetry, is the work of the whole person. This fact requires or generates another postulate, that 'Truth', or knowing, and 'Being... are _ab initio_ , identical and coinherent; that intelligence and being are reciprocally each other's substrate' (I, 94). As I noted in the last chapter, this postulate resolves the major problems posed in chapter VIII; being can transform itself into knowing because the _esse_ and the _scire_ are not originally distinct but, rather, originally identical and coinherent.\n\nColeridge's justification for this postulate assimilates scholastic theology, neoplatonism, Descartes and post-Kantean analyses of consciousness. All of these point to the truth, _'Cogito quia sum, et sum quia cogito_ '. That is, in consciousness itself, or as the basic act of consciousness, we find affirmation that knowing and being are one. (Here, and more emphatically later, Coleridge differs from Schelling in closely linking the psychological or human union with the simultaneity of knowing and being in God.) The remainder of chapter IX and all of chapter X explain how the unity of being and knowing in consciousness is evident in various human activities: philosophy itself, politics, journalism, history and religion. Only when these practical consequences have been presented \u2013 and copiously illustrated \u2013 does Coleridge attempt a formal derivation of the unity. As he does in chapter VIII, Coleridge presents this derivation as a partial and incomplete proof of something already known to be true.\n\nColeridge most forcefully urges the necessity of this psychic or moral unity in philosophy itself. Unless philosophy engage the whole person, unless it be true to the union of being and knowing that it would explore, philosophy reduces itself to academic word-games. (And, unless this integrity be noted and remembered in Coleridge's own writing, he appears to be a pantheist, a submerged materialist and atheist \u2013 as he undoubtedly realized.) The speaker praises ill-educated but imaginative mystics in part to make this point: despite their inadequate explanations, these men possessed truths that academicians did not. The writings of mystics 'acted in no slight degree to prevent [his] mind from being imprisoned within the outline of any single dogmatic system. They contributed to keep alive the _heart_ in the _head_ ' and to lead him to the intuition that 'all the products of the mere _reflective_ faculty [i.e., understanding] partook of DEATH' (I, 98). Because the existence of will cannot be formally or logically proven, no strictly logical or 'closed' or 'dogmatic' system can include it. Yet a philosophy that denies or neglects will also denies or neglects our spiritual nature. All logically self-contained philosophic systems are pantheist:\n\nThe inevitable result of all consequent Reasoning, in which the Intellect refuses to acknowledge a higher or deeper ground than it can itself supply, and weens to possess within itself the center of its own System, is \u2013 and from Zeno the Eleatic to Spinoza and from Spinoza to the Schellings, Okens, and their adherents, of the present day, ever has been \u2013 PANTHEISM under one or another of its modes.... ( _F_ , I, 523 n)\n\nThe most a philosophic system can do is demonstrate, more or less as Kant did with the categorical forms, the necessity of postulating such a power. None the less, the actual necessity of postulating will derives not from abstract logical argument, but from the fabric of mental and physical experience. The Logosophia was to supply a very important proof, but not the crucial one. The logically _and morally_ prior demonstration of the need to postulate will derives from an examination of ordinary experience. Coleridge provides this partly in _Biographia Literaria_ , and more formally in _Aids to Reflection_.\n\nTo summarize: philosophy engages the whole person, and postulates both the twelve categorical forms and the unity of being and knowing. The revision of the _cogito_ and (in reference to Fichte) the assertion that philosophy must begin 'with an _act_ , instead of a _thing_ or _substance'_ (I, 101) suggest that the act in question is 'the mind's self-experience in the act of thinking' \u2013 the act of imagination (I, 86). In self-consciousness, the act of knowing and the state of being are simultaneous and identical. Coleridge has already urged that we take our consciousness of personal agency as evidence that causality is no chimera: the categorical forms are evident in consciousness or thought, not in things.\n\nThe second major idea defined in these chapters concerns the value, and the imaginative origins, of principles. Coleridge begins by accommodating his sketchy definition of imagination both to the definition of truth and to the union of being and knowing that were outlined in chapter VIII and the first part of IX. By defining imagination as ' _esemplastic_ ', Coleridge frees the term from its usual place in aesthetic theories derived from Locke. The subsequent discussion of pedantic expressions provides some vocabulary for distinguishing the relatively active from the relatively passive, or our knowledge of thoughts from our knowledge of things.\n\n_Knowing_ | _Being_  \n---|---  \nthe thought, or act of thinking | a thought, or the object of reflection  \nintuitive | sensuous  \nsubjective | objective  \nperceiver | perceived  \nReason | Understanding\n\nThese are the relatively opposite powers and aspects of experience that imagination fuses into the unity of common consciousness, and the unity of perception.\n\nThis table underlies chapter X's discussion of principles because it amplifies the defiinition of truth as the union of knowing and being, or \u2013 more concretely \u2013 as the union of perceiving mind and perceived thing or situation. Truth, like consciousness, is a mode of relation whose _esse_ is _percipi_ : to speak of the true and the false (as distinct from the actual and the nonexistent) is to define a relation between perceiver and perceived. Principles as well as poems thus depend upon the synthetic activity of imagination. Genial inquiry derives from principles not opinion, or it seeks to establish principles where opinion had held sway. Anonymous criticism is unprincipled in both senses of the word: it derives from opinion, politics and personality; and it seeks pernicious and sometimes immoral ends. The ongoing contrast between philosophic inquirers and anonymous critics rises again into the foreground in these chapters: autobiography proceeds side by side with a sustained account of imagination's role in various aspects of philosophic quests of several kinds.\n\nFinally, in these chapters the speaker defines the major idea of his own epistemology: under certain circumstances, the fact that something is _perceived_ must be taken as evidence that it _is_. Under certain circumstances, the unity of being and knowing in consciousness extends to an analogous unity between human knowing and independently objective being. The circumstances? That the realities perceived are not objects of the senses, and that there is 'the absence of all motive to doubt' (or that 'the law or conscience peremptorily commands' belief in) the reality thus asserted (I, 133, 135).\n\nThe speaker defines only two elements of this set: things in themselves, and the personality of God (I, 132\u20136). Coleridge here assimilates the epistemological and metaphysical problem of things in themselves to the theological problem of God's personality, so as to draw on religious traditions in support of his position. As a result, however, conscience bears an enormous philosophic burden: the theoretical ground for material reality becomes the judgment by conscience that the doubt is contra-natural ('wrong' in both senses). In short, the imaginative synthesis of being and knowing provides direct access both to God and to the real world. It establishes the twin poles of Coleridge's philosophy, the 'I am' and 'it is'. Yet, since logic cannot prove the reality of these two poles, imagination assumes another crucial task. It is also the power whereby the philosophic mind recognizes this state of affairs. That is, primary imagination knows God and the real world: secondary imagination knows that such knowledge is the work of imagination (not of the senses alone), and knows further that it is valid although not subject to logical proof.\n\nLet us step back for a moment, to see where we have come. Six chapters ago, at the end of chapter IV, the speaker resolved to understand and to explain imagination as Wordsworth had revealed it. True to his character, he begins by investigating answers others had found. These progressively sharpen his sense (and ours) of the answer he is seeking. Chapters VII, VIII and IX begin to formulate this answer: knowing and being are originally identical and coinherent; relatively to each other they are active and passive; this coinherence requires a faculty both active and passive that synthesizes the two into the unities of thought and thinking, or perception and consciousness. This is conceivable as a psychology, but will it do as a metaphysics? As we have just seen, the speaker discovers that genuine philosophy itself requires imagination. It takes imagination to understand imagination. The scheme works as a metaphysics only if we can give equal weight to sensuous and suprasensuous knowledge, if we can both allow _and allow for_ the differences between the two kinds of knowledge.\n\nColeridge's philosophic and rhetorical strategy here needs to be examined closely. In accordance with the autobiographical character of _Biographia Literaria_ , the speaker's experiences and recognitions are given. Intuitive certainty concerning God and things in themselves is explicitly such a personal recognition, and no more: the speaker realizes that 'this must be the case', but offers no supporting argument. At most, he draws on the Bible and Christian tradition to suggest that such a resolution has long been available within Christianity, or that he has just recognized the need for a truth long taught by the Christian churches. But he presents this as no more than a brief demonstration of consistency with received doctrine, not as a proof. When the Christian God and things-in-themselves reappear in chapter XII as facts, they are presented as conclusions from a proof to be presented in the Logosophia. This strategy \u2013 or sleight of hand \u2013 clearly reveals the artifice of the _Biographia_ 's autobiography. Coleridge can be intimately personal in his self-revelations when these serve his argument; less 'useful' personal facts from his complex experience are not included. Only an autobiographical form could have allowed two major assertions to be so distinctly limited without the limiting itself suggesting either authorial uncertainty or a mindlessly passive acceptance of religious doctrine.\n\nOn philosophic grounds, it is true that observation cannot, by definition, prove the existence of a material reality correspondent to sensation. An epistemology that begins with a rigorous distinction between the thing as we know it and the thing itself has generated a philosophical Humpty Dumpty. And theology has discarded formal proofs of God as incompatible with genuine faith. As McFarland has argued at length, these two great poles of human experience \u2013 the real conscious spirit and the real natural world \u2013 are always initial assumptions. Whether or not one accepts a philosophy so explicitly grounded in such complex forms of these assumptions, one cannot dismiss Coleridge's position as facile or amateur. Coleridge argues that faith nourishes the rational understanding, and that spiritual knowledge follows from natural knowledge, but this does not mean that all humanly important truths are capable of logical proof.\n\nThe philosophic argument of chapters IX and X is not difficult when its major ideas are separated like this from the wealth of anecdote and history. Yet this raises a question more or less submerged until now: how does the narrative aspect of _Biographia Literaria_ complement the philosophic aspect? More specifically, how does narrative chronology direct the reader's attention to these scattered philosophic points? The narrative successfully modulates emphasis through the consistency of the speaker's character, as this informs his relations with other thinkers and with his contemporaries.\n\nThe clearest instances are in chapter IX, where the speaker once again gratefully attributes to others the ideas that ground his theory of imagination. Once again, such attributes are paired with strong claims about the truth or value of the acquired insight. The elder speaker is more critical of his 'teachers' than the younger man had been: his evaluative comments about mystics and German philosophers interrelate the ideas he learns from each of them.\n\nAs we have seen, the first two paragraphs make explicit the philosophic basis of the speaker's position in chapters V to VIII. The inadequacy of strict empiricism demonstrates the need to postulate suprasensible realities such as causality, and thereby leads the speaker to accept Kant's explanation of the categorical forms and the function of understanding. The host of major philosophers who 'all contributed to prepare [his] mind for the reception and the welcoming' of the substantial unity of being and knowing (both in consciousness and in God) reinforce the traditional authority and thus, for the speaker, the validity of this idea. Earlier he had recognized it only hypothetically, but now further study confirms his belief.\n\nThe remainder of the chapter explores the proper relation of knowing (reason, or self as act) to being (understanding, or self as agent) through commentary on the degree of balance achieved by the mystics, Kant, Fichte and Schelling. Both Fichte and the mystics, unlike either fanatics or professional academicians (always a suspect group, especially in _Biographia Literaria_ ) penetrate to important truths about reason; directly or indirectly, both are described as imaginative. Yet both neglect understanding and the beautiful physical world that it knows \u2013 the mystics through lack of education, Fichte through an inability to synthesize his knowledge of intuitive reason with an equally sophisticated knowledge of perception. Attention to the terms of praise reveals both the priority of reason (or will) and the philosophic importance of recognizing its priority. The intuitive reason is 'the inmost centre, from which all the lines of knowledge diverge to their ever distant circumference' (I, 96). It is the foundation on which all certain knowledge ultimately rests. Furthermore, only a philosophy recognizing this fact will be 'truly systematic: (i.e. having its spring and principle within itself)'; only such a system will be 'truly metaphysical' rather than heartless and 'dogmatic' (I, 101, 98), progressively unfolding the richness of its central idea or symbol. By praising Fichte's beginning with an act, not a thing or substance, the speaker contradistinguishes him from materialists who reduce everything to substance and deny the possibility of genuine mental activity (or will). This, too, identifies Fichte's position with the speaker's. The commentary on Fichte and the mystics directs the reader to essential features of Coleridge's philosophy, provided one has learned how (and why) the speaker judges and expresses his judgments.\n\nThis is more emphatically true for the commentary on Kant. Kant 'at once invigorated and disciplined [his] understanding'; the speaker was awed, among other things, by 'the adamantine chain of the logic' (I, 99). Such praise elevates Kant's analysis of the constitutive powers of understanding to a point beyond question: this is true; this is the defense of causality so necessary for a coherent and moral world-view. The speaker also argues that Kant's work on morals 'assume[s] a higher ground (the autonomy of the will) as a POSTULATE deducible from... the conscience' (I, 99), thereby claiming that Kant recognized (the speaker's views concerning) the cognitive and epistemological significance of will and conscience. Such a position is not clearer in Kant's works, the speaker explains, because a justified fear of persecution by fanatics led him to write ' _mythically_ or equivocally' (I, 100). This strategy allows the speaker to draw on the authority of Kant in support of those ideas which distinguish his own beliefs from a Schellingean pantheism. Kant's role is analogous to Aristotle's: he is a revered authority whose ideas the speaker 'clarifies', praises and assimilates.\n\nThe praise of Schelling is as unqualified as the praise of Kant or Aristotle, and once again based on major reinterpretation or adaptation. Like Schelling, the speaker proposes to complete Kant's work, to explain what Kant knew but dared not say. Yet this completing belongs to Logosophia, not _Biographia Literaria_ (see I, 102): the unity of being and knowing as a metaphysical issue deserves separate study. The speaker's intent is more practical: To me it will be happiness and honor enough, should I succeed in rendering the system itself intelligible to my countrymen, and in the application of it to the most awful of subjects for the most important of purposes' (I, 104). To this end, we need only know that a correct analysis of being discovers the need to postulate spiritual knowledge, and that a correct analysis of knowing discovers the need to postulate a real world immediately known, to avoid a 'hyperstoic hostility to NATURE' (I, 102). Such studies have shown the speaker that imagination's synthesis must be faithful both to God and to nature \u2013 and thus to man. The Voluntary' at chapter's end reminds us that such genuinely imaginative inquiry must expect a small audience (I, 105\u20136).\n\nChapter X divides the years from 1796 to roughly 1804 into five periods demarked by both geographical moves and changing endeavors: _The Watchman_ ; Nether Stowey and poetry; 'a cottage in Somersetshire' and morals; Germany and German literature and philosophy; and finally London and the _Morning Post_ , In the first period, the tales of his campaign for subscribers enforce again the speaker's diminished sense of self and self-importance, his imprudence, and his devotion to duty. The 'exemplary old clergyman' story renews the contrast between the serene, learned philosopher and the irascible ignorant reviewer. The speaker fails at magazine writing himself, because he refuses to swear absolute allegiance to either political extreme. He watches his efforts literally go up in smoke, and wryly concludes that he is not a popular writer. Although _The Watchman_ is portrayed as a madcap adventure doomed from the start, its fate demonstrates the social power of fanatical party spirit.\n\nAs the stories of the second period illustrate, fanaticism does more than squelch small moderate periodicals: it also threatens the liberties of Englishmen. Both English and Continental history further demonstrate the violence elicited by fanatics. Yet the innkeeper and, ultimately, even Spy Nozy himself reveal the fundamental decency of ordinary people. Fanaticism, however dangerous, is an aberration of the powerful, not a character of the English people. Despite the delightful comedy of the speaker's stories, England's present relative tranquility is a fragile thing for which substantial thanks are due to Edmund Burke.\n\nIn such a context, the fact that the speaker's intellectual maturing culminates in his own advocacy of principles (in the fifth period) should be taken as clear evidence that he, too, has some measure of genial power. The evidence from personality in the earlier chapters is matched here by more substantial proof. His persecution, like that of the genial figures in chapter IX only further demonstrates the dangerous ascendancy of the least capable minds (see especially I, 31\u20132; I, 148\u201351). The speaker's grim recognition of this ascendancy triggers a crisis in the third period, 'and it was long ere [his] ark touched on an Ararat, and rested' (I, 133). This crisis \u2013 doubting not only God, but also any substantial reality \u2013 leads him to recognize that fundamental truths in morals (and in epistemology) cannot be 'wholly independent of the will' (I, 135). Absolute proof is not possible. The frustrations, failure and political violence recorded in the chapter's anecdotes and histories sharply focus our attention on this crisis and its resolution because the hapless, good-hearted speaker naturally wins our sympathy.\n\nIn chapter XII the speaker neatly summarizes his conclusions from such experiences:\n\nBut it is time to tell the truth; though it requires some courage to avow it in an age and country, in which disquisitions on all subjects, not privileged to adopt technical terms or scientific symbols, must be addressed to the PUBLIC. I say, then, that it is neither possible or necessary for all men, or for many, to be PHILOSOPHERS. (I, 163\u20134)\n\nThe speaker pleads that we defer judgment until we understand what he is trying to explain; yet paradoxically the _Biographia_ 's most difficult philosophy is his subsequent analysis of self-consciousness to explain why it is rare, and why important. His descriptions in chapter XII of the unconscious multitude who will not comprehend his philosophy summarizes all the accusations previously made about their materialism, and their generalized hostility to abstract and rigorous thought (I, 160\u20134; I, 188\u201394). The speaker thus strives to protect himself from the kinds of criticism and miscomprehension that were his lot during the years described in chapter X: he gives up on one set of readers to appeal exclusively to the 'good reader' who has been postulated all along as the sympathetic audience to his complaints about the times.\n\nChapter XI is addressed to young male members of this properly attentive and thoughtful audience. The chapter focusses the two related contrasts, speaker-populace and speaker-fanatic. The supposed audience the chapter advises are the kind of men who supported and understood the speaker's fruitless endeavors as recounted in chapter X. They are the ones who should read chapter XIII, had the overwhelming numbers of thoughtless readers not effectively 'prevented' its publication. Such astute readers, like the speaker himself, form a culturally invaluable bulwark against fanaticism: the speaker advises that young men in this group avoid his imprudent ventures into journalism, and accept positions in the Church of England. Chapter XI's description of the good they may accomplish reappears later in _On the Constitution of Church and State_ (pp. 42\u201376); the need to protect the people at large from fanaticism's influence remains a major concern of Coleridge. Even in the _Biographia_ , the contrast between the genius and the fanatic is developed in far more detail than the theory of imagination alone requires. The conflict over _Lyrical Ballads_ is but one sign of a most serious national _malaise_.\n\nThe chapter's advice to serious writers \u2013 'drawn from my own dear-bought experience' \u2013 draws imagination's _spontaneity_ into the foreground again ( _CL_ , IV, 633). The writer who tries to force the Muse, and the writer who reads and thinks only to publish, will both benumb their creative powers, because the end of imagination is comprised in the means. Imagination is an act, not a tool. Chapters VIII to X have emphasized imagination's synthesis more than its spontaneity, but this aspect of imagination comes again into the light in chapter XII.\n\n# **5 Imagination, Philosophic Consciousness and the 'True and Original Realism'**\n\n> But what are my metaphysics? merely the referring of the mind to its own consciousness for truths indispensable to its own happiness! To what purposes do I, or am I about to employ them? To perplex our clearest notions and living moral instincts? To deaden the feelings of will and free power, to extinguish the light of love and of conscience, to make myself and others worthless, soul-less, God-less? No! to expose the folly and the legerdemain of those who have thus abused the blessed machine of language; to support all old and venerable truths; and by them support, to kindle, to project the spirit; to make the reason spread light over our feelings, to make our feelings, with their vital warmth, actualize our reason: \u2013 these are my objects, these are my subjects, and are these the metaphysics which the bad spirits in hell delight in?\n> \n> _The Friend_ , I, 108\n\nChapters XII and XIII are the most difficult in the _Biographia_. They are the most dense philosophically, the least conventional rhetorically, and the most controversial. Chapter XIII's 'letter from a friend' and its 'missing' philosophical construction have angered or frustrated even the most sympathetic readers. They seem incontrovertible evidence that the _Biographia_ is a fragmentary failure. Although the chapter is undeniably peculiar, and in many ways unsuccessful in achieving its rhetorical goals, the 'missing' construction does not directly and absolutely destroy the integrity of the work as a whole. And even more mystifying than this 'gap' is Coleridge's rewriting of extensive passages from Schelling, as he bends Schelling's arguments and methods to his own, different ends. Elinor Stoneman Shaffer has studied these changes in great detail, and I am indebted to her lucid and graceful explications of the complex technical issues involved. As Kathleen Coburn has observed in her notes to the notebook drafts and translations that also underlie the chapter, much work remains to be done on Coleridge's debt to and divergence from Schelling in these pages. My premise in explicating the design of these two chapters has been that the study of their contexts in the _Biographia_ itself is reciprocally necessary to comparative studies, and yet \u2013 for practical reasons of coherence and readability \u2013 a distinct endeavor at this stage.\n\nYet a few remarks are in order. Many discussions of Coleridge's debt to Schelling in these chapters refer in passing to changes and deletions, as if these were insignificant overall, or as if they represented little more than his misunderstanding of the German philosopher. But such is not the case. Philosophical argument is a delicate thing: small changes can have large consequences. And some of Coleridge's changes are far from minor. As Shaffer demonstrates in detail:\n\nThe fact that [Coleridge] often adopted whole phrases and passages literally is deceptive; the spirit, if not the letter, has been altered. So in this instance: while quoting, compressing, and summarizing Schelling, Coleridge criticizes him in essential respects. The scandal of these pages in the _Biographia_ is not that they are based on Schelling, but that to a reader unacquainted with Schelling's treatise they must remain obscure, if not totally unintelligible.... Read against his texts in Schelling, Coleridge's Chapters XII and XIII appear as a cunning reduction of Schelling's elaborate _System_ , and a sturdy dissent from it. ('Coleridge's aesthetics', p. 4)\n\nColeridge himself later judged these chapters 'unformed and immature'. They contain, he said, 'the fragments of the truth, but it is not fully thought out' ( _TT_ , 28 June 1834). Substantial portions are revised and incorporated in later works; the changes inevitably clarify, and sometimes develop, but never change the principal idea.\n\nThe general tendency of Coleridge's dissent from Schelling can be explained briefly (see 'Coleridge's aesthetics', pp. 50\u20131). Schelling's _System des transzendentalen Idealismus_ traces the stages whereby the Ich creates the entire world of objects through its knowledge of itself. Schelling justifies our certainty that there exists a real world by trying to prove that the proposition 'things exist' is ultimately identical with the proposition 'I am'. But, according to Schelling, we know only sensations, not things themselves \u2013 a position with which Coleridge sharply disagrees (I, 92\u20133, 178\u20139, 100). He also disagrees with Schelling's collapse of the world into the self: for Coleridge our certainty that things exist is independent of, not grounded in, our certainty of self. Shaffer summarizes Coleridge's disagreement well: 'The belief in the external world... is an assumption required by the nature of consciousness or the \"I am\" itself. It is, equally with the self, groundless because it is fundamental' ('Coleridge's aesthetics', p. 26). In short, although Coleridge incorporates parts of Schelling's explanation of the mind-world relation, he defines this relation in a substantially different way. He also avoids the dialectical logic whereby Schelling demonstrates the identity of the two propositions, an argument Shaffer judges 'highly dubious' ('Coleridge's aesthetics', p. 33). Schelling invents a special faculty capable of knowing the identity of these two propositions; Coleridge rejects this, too, asserting that the philosophic or secondary imagination differs only in degree from powers possessed by all people, and that conscience demonstrates the capacity for such imagination in everyone. Finally, Coleridge transforms Schelling's 'self-consciousness in general' to God. The consequences of that change are evident less in this local argument about things themselves than in the _Biographia's_ larger concern with the nature and value of poetry.\n\nGiven these substantial differences, what attracted Coleridge to the _System_? Why bother struggling to rewrite a philosophy so profoundly uncongenial? One aspect of Schelling's view of art deeply attracted Coleridge. Schelling's work implies that 'a successful piece of philosophy must itself be a work of art, or carry no conviction: its intuitions will be doubted. A piece of art must in turn illustrate intuitions worthy of philosophy. \"Philosophic criticism\" had indeed come into its own, and art achieved a staggering dignity' ('Coleridge's aesthetics', p. 16). For Schelling, a work of art most translucently reveals the mind's range of powers: artistry is the highest human act, and the greatest human achievement. It is not pleasant escape from sterner labors, but the most profoundly important of our endeavors. Furthermore, Schelling's insistence on the 'act of freedom' underlying the identity of the two propositions easily merges with Coleridge's idea that the conscience testifies to our moral freedom \u2013 thus linking art not only with philosophy, but also with religion. Thus, Schelling's philosophy reaches a point Coleridge sought, but by means Coleridge rejected. The incorporations and revisions in Chapters XII and XIII seek the same ends by less radical, less pantheistic means, in part by introducing the ancient mystery of the one and the many in a Christianized form.\n\nIn design, chapter XII is a long parenthetical remark. The speaker explains why the multitude whom popular authors address will find chapter XIII incomprehensible, but in doing so he defines the core of his own epistemology in an unrelentingly technical and abstruse argument. Yet, despite the chapter's difficulty, it offers very few new ideas. Principally it summarizes the philosophical arguments of the first eleven chapters, defining or making explicit the relations among ideas presented earlier. Understanding the chapter requires no unravelling of peculiar structures or ineffective transitions but, rather, continually monitoring its allusions and relations to the earlier chapters \u2013 connections without which its minimal comprehensibility slides back into utter obscurity.\n\nBefore examining the chapter in detail, let me again preview Coleridge's argument. The chapter can be divided into four parts. The first (I, 164\u201374) explains that only some individuals can attain self-consciousness, although the capacity is at least latent in all. Self-consciousness is identified as intuitive, imaginative and the basis for the certainty of all knowledge; it is the proper domain of pure philosophy. Complex geometrical metaphors, delicately rewritten from Schelling, reveal that self-consciousness results from the 'spontaneous' level of will's operation.\n\nThe second section (I, 174\u201380) asks again what chapter VIII asked: how is perception possible? It establishes a context for this question by defining 'subjective' as essentially active and conscious, and 'objective' as essentially passive and without consciousness, and by defining a correlative distinction between pure philosophy and natural philosophy. In a cryptic and stylistically tangled paragraph, the speaker asserts the central assumption of his transcendental philosophy: perception is possible because our knowledge of external reality is 'unconsciously involved' in our knowledge of ourselves. This is but another formulation of chapter VIII's hypothesis that knowing and being are originally a unity; this new formulation emphasizes that perception must be the work of imagination, just as self-consciousness is. Not until chapter XIII, however, does Coleridge distinguish these two functions as 'primary' and 'secondary' operations of imagination. The speaker explains that the Logosophia will provide 'the demonstrations and constructions' of this philosophy. For his present purposes \u2013 defining 'the principles of production and of genial criticism in the fine arts' \u2013 explanation not demonstration will suffice (I, 180).\n\nIn the third section (I, 180\u20131), the ten theses attempt to offer this explanation. Most of them do little more than collect and summarize points made in earlier chapters, fitting these ideas into the general framework provided in the first two parts of chapter XII. We reach the heart of the matter in Thesis VII. If it could be proved that the certainty of all knowledge must derive from the certainty of the logical identity 'I know me' \u2013 i.e. from the imaginative act of self-consciousness \u2013 then we could be assured of the certainty of all genuinely immediate, intuitive, supra-sensuous knowledge. We could be thus assured because all intuitive knowledge is knowledge of the fundamental unity of knowing and being. This does not, of course, _solve_ the problem of perception: it does not proceed beyond chapter X's point that the 'constitution of the mind' renders it morally impossible to doubt the existence of a real world correspondent to sense-impressions (see I, 132\u20136). Only the Logosophia is to attempt to prove what the _Biographia_ awkwardly and provisionally sketches.\n\nThesis VII is as cryptic and tangled as the central paragraph of the second section: Coleridge is trying to revise Schelling's account of 'unconscious involvement' into something sharply different. Schelling's argument depends upon an esoteric faculty that none the less provides us \u2013 the incapacitated, ordinary intellects \u2013 with our only accurate information about the fundamental character of reality. Coleridge's imagination, on the other hand, is a power that all possess at least potentially. Thesis VII merely validates or provides the means to validate the 'true and original realism' that is 'common to mankind' by grounding it in an equally traditional Western theism. It is tempting to sidestep the tortuous intricacy of Coleridge's prose by assimilating it to the cool clarity of its sources in Schelling but, as Shaffer demonstrates, this seriously falsifies Coleridge's position. As Ren\u00e9 Wellek has argued in so many places, it is dishonest to credit Coleridge with particular statements or specific detailed arguments that he has copied from someone else; in this instance, confusing Coleridge with Schelling places Coleridge among the originators of an isolationist and radically formalist tradition in poetics to which he does not belong.\n\nThe fourth section (I, 188\u201394) is a Coleridgean 'miscellany' that ties the chapter's many loose ends to various major themes in the _Biographia_ generally. Viewed retrospectively, the chapter's main contention looks back to the opening paragraphs of chapter IX: philosophy is possible as a science because the certainty of all knowledge derives from the imaginative act that is self-consciousness. In self-consciousness, as in God himself, being and knowing are ' _ab initio_ , identical and coinherent' (I, 94). The Logosophia will provide this science, but meanwhile we have the major practical result of literary interest: imagination is the faculty whereby we know who we are as moral beings, and what the world's order really is. From this follow two ideas that permeate the chapters on poetry and on Wordsworth. Because imagination is also the origin of great art, it is reasonable to expect the very best art to offer profound insight into the human condition. And, secondly, the origin and thus to some extent the particular character of poems will be intimately involved with the poet's self-consciousness.\n\n## **I**\n\nIn the chapter's first part, a complex geographic image for the relation between common consciousness and philosophic consciousness develops the contention that not all men are philosophers into an explanation of the philosophic role of imagination. Most men believe that empirical knowledge is the only knowledge: they cannot distinguish _knowing_ from knowing _things_ , and will not give credence to those who can (cf. I, 64\u20135, 73\u20134, 105\u20136, 149). Such Valley-dwellers' can never attain absolute and spontaneous affirmation of immediate (suprasensuous) knowledge. The 'rivers' that carved the Valley' define the relation between sensuous and suprasensuous knowledge:\n\n... in all ages there have been a few, who measuring and sounding the rivers of the vale at the feet of their furthest inaccessible falls have learned, that the sources [of the rivers] must be far higher and far inward; a few, who even in the level streams have detected elements, which neither the vale itself or the surrounding mountains contained or could supply. How and whence to these thoughts, these strong probabilities, the ascertaining vision, the intuitive knowledge may finally supervene, can be learnt only by the fact. (I, 166)\n\nSensuous and suprasensuous interpenetrate. The stream of (common) consciousness reveals its origins in the pure spirit. They only are qualified as philosophers who can intuit spiritual (suprasensuous) realities, as the speaker does in chapter X. In both instances, Coleridge insists that the intuition is spontaneous: ideas about spiritual realities cannot be demonstrated from without (like 'the truths of abstract science', I, 135); nor can they be elicited strictly by choice (I, 167, i\u2013viii). References to the 'direction' of the streams of consciousness sustain the geographic metaphor throughout the chapter: 'the common consciousness itself will furnish proofs by its own direction, that it is connected with master-currents below the surface' (I, 167; cf. I, 76\u20137). It will prove crucial to remember that the 'direction' of consciousness is both its contents and its activity regarded as a unity originating in the suprasensuous. It is quite characteristic of Coleridge to provide an image for so complex an idea.\n\nHaving explained that the subjective basis of transcendental philosophy is the philosopher's imagination, the speaker explains that its objective basis or its character as a system is constructions from postulates. The 'direction' of consciousness is compared to the direction of the geometer's point moving in space. The geometer's point can be self-directed (indeterminate motion), directed from without (a straight line), or both, that is undirected externally but directly internally (a circle). These options exactly recapitulate the three levels of will: the voluntary or directed from within; the receptive or directed from without; and the spontaneous or intermediate. As J. B. Beer points out, the circle is one of Coleridge's favorite images for imaginative activity; it appears elsewhere in the _Biographia_ as well.\n\nThe geometer can literally construct or draw his point in motion, so as to demonstrate his idea in a physical form. The philosopher cannot. How, then, can he distinguish among the directions consciousness may take? The distinction rests on a moral (not sensuous) embodiment of the ideational or spiritual reality:\n\n... the inner sense has its direction determined for the greater part only by an act of freedom.... This more or less betrays already, that philosophy in its first principles must have a practical or moral, as well as a theoretical or speculative side. (I, 172\u20133)\n\nAs the geometer progresses by means of his constructions to a clearer intuition of space, so the philosopher progresses in his knowledge of consciousness by attending to the manifestations of moral freedom \u2013 the imperatives of conscience. Thus it is, the speaker concludes (pointing again to problems posed by his contemporaries), that those unable to intuit consciousness directly will condemn discussions of its 'directions' as metaphysical moonshine. For such readers, a text in transcendental philosophy 'is groundless and hollow, unsustained by living contact, unaccompanied with any realizing intuition which exists by and in the act that affirms its existence, which is known, because it is, and is, because it is known' (I, 173). Such a self-confirming truth, or absolute, is necessary not only for any science of knowing, but also to guarantee certainty in general (see I, 168, xiv\u2013xix). Those who cannot or \u2013 worse \u2013 who will not engage any issue involving abstract or purely intellectual truths stand on treacherous and uncertain ground even in their knowledge of objects. That is why anonymous critics and their kind are unreliable guides on _any_ issue.\n\n## **II**\n\nThis transcendental philosophy that so few will understand is the analysis of philosophic consciousness as the ground of all knowledge. The chapter's central section (I, 174\u201380) defines the major question answered by the ten theses: how is it possible that we have immediate intuitive certainty concerning our knowledge of objects external to ourselves? Or, to use the less technical form of the same question from chapter X, 'what proof... of the outward _existence_ of anything? Of this sheet of paper, for instance, as a thing in itself, separate from the phaenomena or image in... perception?' chapter X answers, 'the existence is _assumed_ by a logical necessity arising from the constitution of the mind itself...' (I, 133). The ten theses analyze that constitution to define the power and importance of secondary imagination.\n\nThis second section defines the metaphysical problem of mind and world from two perspectives. Natural philosophy assumes the priority of the objective (or being): inquiry moves from observation to theory. Ultimately, 'nature' refers both to the system of laws comprising the theory, and to the profusion of particulars that the theory encompasses. In the hypothetical 'completion' of science, when theories account for all observable particulars, 'the heavens and the earth shall declare not only the power of their maker, but the glory and the presence of their God' (I, 176). Nature will be revealed as truly 'the language of God', as his text but not as God himself. Scientific inquiry, in its progress from being to the 'equatorial' unity of being and knowing, goes from creation to God. Only a _self-conscious_ intelligence can perceive the unity of idea and law that underlies theory: to transform observations into theories is to transform 'nature' from particulars to relations \u2013 to intelligence or knowing. Science at its best is as imaginative an activity as poetry.\n\nTranscendental philosophy attains the same equatorial point from the other side: 'We begin with the I KNOW MYSELF, in order to end with the absolute I AM. We proceed from the SELF, in order to lose and find all self in GOD' (I, 186). But the transcendental philosopher's immediate concern is the movement from self (knowing) to things (being), just as the scientist's immediate concern is developing and testing theories about material reality. Coleridge sketches the movement of transcendental philosophy hypothetically (I, 178) before deriving it formally (Thesis VII). This development from hypothesis to formal assertion exactly repeats the strategy we saw at the end of chapter VIII and the beginning of chapter IX: the speaker proposes no more than a formal explanation of what we already intuitively recognize must be the case.\n\nThe transcendental philosopher's scientific scepticism uncovers two ineradicable beliefs, two products of imagination: that I exist (the _I am_ ) and that the world exists (the _it is_ ). The _I am_ 'is groundless indeed; but then in the very idea it precludes all ground, and separated from the immediate consciousness loses its whole sense and import' (I, 178). But the _it is_ retains its meaning even when separated from the philosopher's consciousness of its certainty: if _it_ really _is_ then _it is_ regardless of whether I am thinking about it. This state of affairs engenders the philosopher's hypothesis concerning the relation between self-consciousness and things. He intuits\n\nthat the former ['the existence of things'] is unconsciously involved in the latter ['the existence of our own being']; that it ['the existence of things'] is not only coherent but identical, and one and the same thing with our own immediate self-consciousness. To demonstrate this identity is the office and object of his philosophy. (I, 178)\n\nThe first clause asserts that the basis of our knowledge of our own existence encloses or includes the basis of our knowledge of the existence of things. The second clause asserts that self-consciousness is the same as the _existence_ of things without us (not the same as these _things_ , but the same as the _existence_ of these things, the same as their _being_ ). Yet self-consciousness is an act. It follows that the being of things without us is ultimately an act, and, further, that the being of all realities \u2013 conscious or not conscious, subjective spirit or objective nature \u2013 is knowing.\n\nThe hypothesis suggests that there are acts that we know _as acts_ , as the various manifestations of our power to know. And there are acts that we know _as things_ , as _beings_. Yet things are objects \u2013 passive, material, without consciousness. If they are acts, they are not human acts, for that would contradict our absolute certainty of their independent existence. They must therefore be acts of God, as no third possibility exists. God must, then, be the ground of their being \u2013 a familiar and orthodox idea. Coleridge's point is that our self-grounded knowledge of being (our own, or objects') exists in some important and intimate relation to God. Human knowing must somehow reflect the Supreme Being. Theses VI and VII later develop and substantiate what this earlier paragraph merely proposes.\n\nThe speaker strongly asserts that this hypothesis accounts for\n\nthe realism common to all mankind[.]... It is the table itself, which the man of common sense believes himself to see, not the phantom of a table, from which he may argumentatively deduce the reality of a table, which he does not see.... It is to the true and original realism, that I would direct the attention. This believes and requires neither more nor less, than [that] the object which it beholds or presents to itself, is the real and very object. In this sense, however much we may strive against it, we are all collectively born idealists, and therefore and only therefore are we at the same time realists. (I, 179)\n\nHe explains that this hypothesis will be formally established in the forthcoming Logosophia; but, as we have seen before, the Logosophia can do no more than argue formally and rigorously for that which we already realize must be true. Because the hypothesis involves knowledge of spiritual realities, no strict logical proof will be possible: logic can do no more than justify the necessity and consistency of this belief. Shaffer says it well:\n\nColeridge's objections [to Schelling] are not those of mere 'common sense.' There is simply no alternative to beginning where our consciousness begins: with 'sense certainty' of both the self and the external world, of subject and object alike. Such certainty does not constitute proof; of these fundamental assumptions there can be no proof. It is in this sense that our certainty is not prejudice, but 'faith.' All our knowing is indeed transcendental, in that it revolves on itself and depends on the system of our fundamental, necessary assumptions. We cannot escape from the mode of operation of our minds. But just for this reason we are bound to consider its natural results as real. There can be no other reality. ('Coleridge's aesthetics', p. 28)\n\nSuch fundamental assumptions cannot be transformed into conclusions without substituting some other idea as the fundamental starting-point. Coleridge insists on both beliefs, and on the primacy of both.\n\n## **III**\n\nIn the chapter's third section, the ten theses recapitulate major conclusions from the analysis of spontaneous knowing that began in chapter V but without the accompanying illustrations and history (personal or philosophic). Coleridge is trying to be exclusively abstract, keeping illustrations to a bare minimum and even then relegating them to scholia and to footnotes. I suspect that the attempt ran contrary to his particular genius: the theses have an odd objective clarity without conveying his meaning as effectively as the earlier formulations of the same ideas.\n\nThe first six theses primarily assemble earlier points into the argument that self-consciousness is the single absolute first principle of transcendental philosophy. For knowledge to be a system, there must be some one principle that is true because it is known, and known because it is true; Thesis VI identifies this absolute as the _I am_ , or self-consciousness. Yet the Scholium to VI adds an important qualification that the appended note on Descartes explains in more detail: to affirm of the _I am_ that it is true or self-evident does not necessarily imply that it exists. (It is self-evident that a circle is equi-radial, but that does not imply that there exists \u2013 physically exists \u2013 an absolutely genuine circle.) Yet, unless absolutely genuine self-consciousness exists, knowledge is not possible as a system, and therefore certain knowledge of any kind is not possible. Although the ground of being is not properly part of the science of knowing, the scholium points out that the ground of the _existence_ of self-consciousness is God, 'the absolute self, the great eternal I AM, [in whom] the principle of being, and of knowledge, of idea, and of reality; the ground of existence, and the ground of the knowledge of existence, are absolutely identical, Sum quia sum' (I, 183).\n\nThesis VI asserts that the absolute principle of transcendental philosophy is the unity of being and knowing in self-consciousness. Thesis VII asserts that from the identity 'self-knowing knows the self' we can conclude the 'immediate certainty of all intuitive knowledge' (I, 184). Yet how does the certainty of self-knowledge accrue to the intuitive (not sensuous) knowledge of objects themselves? Thesis II points out that if A is certain, and B is the same as A, then B is certain. So let me rephrase the question: in what way is our knowledge of our own being the same as our knowledge of external being? Both are known by the pure spirit, not the senses; yet the spirit knows only itself (I, 184, ix); so the _knowledge_ of the being of the self and of the being of the world must ultimately be the same knowledge. For this to be so, human being, and the being of the world, must have a single basis: the Supreme Being of God, in which human being _consciously_ participates via the triunity of will, faith and reason.\n\nAs I said before: Coleridge's argument here possesses a certain odd logical clarity despite its essential opacity. The principal obscurity in the ten theses derives from the slippery meaning of 'spirit'. This confusion is deepest when, after defining 'spirit' as 'self-consciousness' (I, 183, iii), Coleridge speaks of 'the self-consciousness of a spirit' (I, 184, iii). In doing so, he slides from a specific limited use of the word to his more usual, broader, traditional use. I suggest that by 'self-consciousness of a spirit' he means the self-knowledge of one who possesses reason, because reason is the cognitive faculty whereby we possess spiritual or suprasensuous knowledge, and thus our own identities as essentially suprasensuous entities (souls or spirits who are both free and immortal). And, as I explained in chapter 3, in a fully developed, imaginatively activated self-consciousness, the reason \u2013 pure knowing \u2013 discovers its immediate, intuitive, suprasensuous knowledge of God's being, and human being (free and immortal) and the world's being (real objects correspondent to sense data).\n\nWith at least this much clarified, it becomes possible to see that Thesis VI and its Scholium, and Thesis VII together say that pure knowing (the divine absolute in a pure but relative form) encounters pure being (God) in which all other being (including its own) participates, or by which all other being is sustained. Ergo the certainty accruing to the identity 'I know me' also accrues to the statement 'I know the table itself' because both statements are purely spiritual acts. And pure knowing or philosophical self-consciousness \u2013 the highest spiritual act \u2013 is one act not several. The certainty of all human knowing about realities other than itself depends upon God; in _Biographia Literaria_ this spiritual act is attributed to will, although elsewhere Coleridge attributes it either to faith or to reason as his particular argument warrants.\n\nOne's faith \u2013 the only correct name for it \u2013 one's faith in the existence of a real table is not to be confused with one's physical perception 'table over there'. The faith is in the reality of an object _correspondent to_ perception; it is not, very strictly speaking, a faith in the complete reliability of fragile sense organs, nor in the adequacy of sensation as the sole basis of knowledge about physical realities. But when my hands and eyes and brain are operating 'normally', then my confidence that the table _is_ there is just as valid as my confidence that I _am_ here: in both cases, the confidence derives not from logical proof, but from the way normal healthy minds work. Let me cite again Coleridge's clearest statement on this matter:\n\nFor wherein does the realism of mankind properly consist? In the assertion that there exists a something without them, what, or how, or where they know not, which occasions the objects of their perception? Oh no! This is neither connatural nor universal. It is what a few have taught and learned in the schools, and which the many repeat without asking themselves concerning their own meaning.... It is the table itself, which the man of common sense _believes himself to see_ , not the phantom of a table, from which he may argumentatively deduce the reality of a table, which he does not see. (I, 178\u20139; italics mine)\n\nNeither argumentative deductions nor, again more recently, skeptical critiques of such arguments can discredit this 'true and original realism' because it is an act of faith not a logical conclusion. As Thesis VII concludes, The self-conscious spirit therefore is a will; and freedom must be assumed as a _ground_ of philosophy, and can never be deduced from it' (I, 185). Coleridge's emphasis here on free will rather than reason reveals again the _Biographia's_ central interest in the moral implications of the idea 'imagination'. We believe some things because we choose to, because the alternatives all lead to solipcism \u2013 the dead end Coleridge describes in chapter VIII. These issues arise in the _Biographia_ because the genuinely great poet thus possesses intuitive knowledge of the moral realities sketched in Theses VI and VII. Thematic ties between chapter XII and other, more directly literary chapters appear much more clearly when one realizes that full self-consciousness so intimately requires (secondary) imagination that Coleridge legitimately uses the terms interchangeably.\n\nNote that, throughout the argument to this point, Coleridge has been talking only about the certainty of beliefs \u2013 not their objective validity. His strictly limiting the domain of transcendental philosophy is to prevent us from translating his conclusions about the certainty of our knowledge of being into statements about the nature of being itself. Such extensions \u2013 any study of the full unity of knowing and being \u2013 belong only to the total philosophy in which religion subsumes both transcendental and natural philosophy. The transcendental philosopher accepts as axiomatic that we have real knowledge of external realities, and simply asks _how_ this is so \u2013 not whether it is valid.\n\nAlthough transcendental philosophy does not ask about being, we can \u2013 and probably must \u2013 because the real independent existence of things is one of Coleridge's major practical divergences from his sources in Schelling. Does thesis VII contradict Coleridge's earlier insistence on the 'true and original realism'? Does the identical ground of _knowledge_ of being imply that spirit and objective nature are ultimately indistinguishable? No: to say that two things are known in the same way is not to say that they are the same. The chapter's several references to the Christian God suggest an orthodox interpretation instead: God creates, sustains and relates the being of both human minds and the natural world, but yet generates and maintains the necessary distinction between the two by creating the human as essentially subjective or active, and nature as objective _relative to man_ (see _PhL_ , 523\u20134). As Coleridge often insists, the essential difference between persons and things is the foundation of morals.\n\nThesis VII prompts another question as well. What about other instances of intuitive knowledge? What are they? How do we recognize them? The self's existence and the existence of things are the only intuitive _and immediately certain_ knowledge discovered by the transcendental philosopher's scientific skepticism. But there are, potentially, other instances which are not indubitable \u2013 God, for instance. Chapters XII and XIII define the objective features that characterize knowledge that is intuitive rather than sensuous in origin. Principal among these, of course, is the relation of part to whole variously called 'polar unity' or 'organic unity'. For the literary critic, an important index of the intuitive (or imaginative) origin of a work will be the unanimously favorable judgment of qualified readers over time, and the consistent effect on any sensitive reader. The critic's intuitive knowledge of artistic greatness is not indubitable or fundamental to all human minds, but it is real or genuine knowledge. For the _Biographia_ as a whole, the importance of Thesis VII is this grounding \u2013 or sketch for a grounding \u2013 of the consistent judgment of good critics on which Coleridge relies so heavily as a literary theorist.\n\nOne question about Thesis VII still remains: how can reason or the spirit maintain the unity of knowing and being _as a unity_ while yet knowing being _per se_? It can do so because it is an act, not a thing.\n\n... a spirit is that, which is its own object, yet not originally an object, but an absolute subject for which all, itself included, may become an object. It must therefore be an ACT... the spirit (originally the identity of object and subject) must in some sense dissolve this identity, in order to be conscious of it: fit alter et idem. But this implies an act, and it follows therefore that intelligence or self-consciousness is impossible, except by and in a will. The self-conscious spirit therefore is a will.... (I, 184\u20135)\n\nA notebook draft for chapter XII supplies a helpful explanation:\n\n... to be known, this Identity must be dissolved \u2013 and yet it cannot be dissolved. For its Essence consists in this Identity. This Contradiction can be solved no otherwise, than by an Act. ( _CN_ , III, 4265 f. 11)\n\nIn 'the act of constructing itself objectively to itself', the spirit knows being directly. The spirit is the continual simultaneous dissolving and recreating of the unity of being and knowing: it 'dissolves, diffuses, dissipates, in order to recreate' (I, 202). As the organ of philosophy, spirit (or self-consciousness via the philosophic [secondary] imagination) operates spontaneously. It cannot be compelled. It is the free act of the will that testifies to and grounds the essential difference between persons and things.\n\nTheses VIII, IX and X draw conclusions from the principles stated in VI and VII. Thesis VIII, on immortality, in part reveals chapter XII's origins as part of a draft for the Logosophia ( _CN_ , III, 4265 n). It also justifies the speaker's earlier claim that 'all the organs of sense are framed for a corresponding world of sense; and we have it. All the organs of spirit are framed for a correspondent world of spirit: though the latter organs are not developed in all alike. But they exist in all...' (I, 167). Thesis IX, clarifying VII, asserts that the unity _per se_ of being and knowing is will. Knowing as such is reason; _spirit_ is the more comprehensive term for knowing and being as an identity. And this unity or identity is sustained through imagination. chapter XII defines the will relative to knowing as an act, the act of imagination, just as chapter 10 defines will relative to being as faith. Thesis IX also distinguishes among natural, transcendental and total philosophy to insist that the 'equatorial point' or absolute unity of being and knowing is God. Such a division of inquiry is crucial for Coleridge's endeavors in chapter XII, because it allows transcendental philosophy to be a complete system _sui generis_ , a complete part of the whole of human knowledge.\n\nThesis X addresses misunderstandings that would reduce transcendental philosophy's first principle into a relative or mediate one. First, it repeats the distinctions among inquiries defined in Thesis IX: transcendental philosophy does not seek the ultimate ground of knowing in the absolute unity of knowing and being. Self-consciousness is not itself absolute, but only absolute for our knowing. Secondly, it argues that the knowing of knowing does not generate an infinite regress: even one who could know the knowing of knowing still comprehends nothing beyond knowing itself. Self-consciousness is not a higher form of being, antecedent to a still higher form. It is knowledge of the operations of our own minds.\n\nThe concluding paragraph of Thesis X sketches the 'interrupted' intent of chapter XIII: to abstract from self-consciousness a single power generating two polar forces, and from their interactions to evolve 'the fulness of _human_ intelligence' (I, 188). We may rightly ask what relation this one force has to the God of the Scholium to VI. The answer would be that transcendental philosophy treats God as absolute knowing, not as a person, and not as a deity; natural philosophy, correspondingly, treats God as absolute being. Only the total philosophy (in which philosophy passes into religion) deals with God as a person, as a 'moral creator, and governor' (I, 133).\n\nThat is a good and adequate answer, strictly speaking. But _Biographia Literaria_ as a whole is not strictly transcendental philosophy. Coleridge veers repeatedly into the total philosophy, particularly in his arguments about moral responsibility in chapters V to IX, and in his discovery of the common ground of the personal God and things in themselves in chapter X. The actual gap in this book is between Thesis VI and its Scholium, between the one force or single ground and the personal triune God. Philosophical resolution of that gap is repeatedly deferred to the Logosophia. Autobiographical resolution takes its place 'temporarily': the speaker knows that the orthodox Christian God underlies his coherent experience of both an independent material reality and his own moral freedom. He urges orthodox values and perspectives at every turn. That strategy sharply limits \u2013 or attempts to limit \u2013 the range of implication and inference from transcendental philosophy that the reader will perceive as valid. Despite the logical appearance of pantheism in chapter XII, the dramatic autobiographical form of the whole work is to preclude our taking that appearance literally. The problem, as I noted with regard to chapter VIII, is that the reader's logical powers have been fully engaged by chapter XII's arguments. The speaker's orthodoxy lacks the rhetorical potence necessary to preclude the pantheist interpretations. Coleridge's deficient sense of audience becomes startlingly apparent \u2013 or, perhaps, his vulnerable openness in asking us to trust the orthodoxy of his morals and the powers of his mind. On such issues as these, the man and the speaker can be only theoretically distinct.\n\n## **IV**\n\nChapter XII's concluding transition is unusually complete. The speaker presents some new terms and repeats chapter X's defense of technical language as if to suggest that we have arrived at a major internal division in his analysis of being and knowing. His repeated lament about 'popular' philosophy echoes the chapter's opening, reinforces the impression of major internal division, and ironically prepares us both for more technical argument and for the 'deletion' of that argument in the next chapter.\n\nThe chapter also concludes with a glance toward chapter XIV. The speaker explains that in rereading Wordsworth's 1815 preface he has discovered that he disagrees with Wordsworth's account of the 'poetic fruitage' of the interaction between fancy and imagination (see I, 64). Had Wordsworth's account been acceptable, of course, then much of what Coleridge argues in Chapters XIV to XXII would have been precluded. At most, Coleridge's commentary on the controversy about diction could have been formulated as a deduction from Wordsworth's _accurate_ definition of the poetic aspects of the two powers. But, rather than undertake such a reconciliation of the transcendental root with the Wordsworthian 'poetic fruitage', the speaker attributes most of the controversy to inaccuracies and imprecisions in the 1800 Preface. In short, Wordsworth assumes a role analogous to that assigned Aristotle or Kant or Schelling: he formulates ideas that Coleridge finds it convenient to adapt and to argue against as a way of structuring his own presentation. As we shall see later, Coleridge misrepresents parts of the Preface to his own ends.\n\nThe infamous chapter XIII offers little more than a selection of previously defined ideas as the basis for a rigorous 'construction' in transcendental philosophy. The lines from Milton affirm God's creativity and creation's order, yet in Coleridge's context, and given Coleridge's syncretism, no precise philosophic significance can be assigned. As Lovejoy argues, the great chain of being created by emanation from One Almighty is an ancient idea rich with its own complexities: the range of interpretations here must stretch far beyond _Paradise Lost_. Yet this much is clear: Milton's lines begin from mind and arrive at matter. Leibnitz's direction, in the succeeding passage, is just the opposite: 'all the truth about corporeal things cannot be collected from logistic and geometric axioms along'. In examining matter, one discovers mind: 'even as natural philosophers we must arrive at the same principle from which as transcendental philosophers we set out; that is, in a self-consciousness' (I, 187). The final citation, from Synesius, affirms again that the 'equatorial' unity of being and knowing is an act:\n\n> I worship the hidden order of intellectual things;\n> \n> The Mean dances and is not still.\n\nThe portion preceding the 'letter' repeats and expands Chapter XII's explanation of polarity (cf. I, 188). The transcendental philosopher contends that the universe can be rendered 'intelligible' by postulating a single power that generates two forces, 'one of which tends to expand infinitely, while the other strives to apprehend or _find_ itself in this infinity' (I, 196). The equipoise of these forces is not stasis, but 'finite generation': a real physical world, growing, reproducing, renewing \u2013 the 'process and mystery of production and life' (I, 185).\n\nAt this point the speaker 'interrupts' himself to cite a 'letter from a friend' that, as he said later, he composed without interruption as a part of the chapter. From one perspective at least, the letter is 'a philosophical route, a mad dash from an untenable position'. There are two major problems with the philosophic position delineated in chapters IX to XII. For the argument to succeed, being and knowing must be identical. Yet they must also be distinct \u2013 actually distinct, not just verbally so. Thesis VII reveals this paradox most fully, and hints at its potential resolution with the Trinitarian's formula, 'fit alter et idem'. Secondly, if being and knowing are identical, and we have immediate knowledge of independently real objects, then objective reality as portrayed by transcendental philosophy will be identical with reality as portrayed by natural philosophy, and these together will reveal God: the total philosophy will be implicit in the conclusion of either of its subsidiary branches. Coleridge cannot rigorously pursue his transcendental construction of the 'intelligible' world without engaging the whole range of unsolved metaphysical problems \u2013 an error soon recognized ( _CL_ , IV, 874).\n\nThe two parts of chapter XIII \u2013 the initial philosophy, and the letter \u2013 try to avoid the consequences of leaving us philosophically in mid-air by assimilating the missing construction to an ancient philosophic tradition, and by asserting that the metaphysical problems have been solved but withheld from publication. The lines from Milton and Synesius and the explanation of the productive union of polar opposites assert that the deferred philosophic argument will be nothing new and ingenius but, rather, the 'purification' of what the speaker calls the 'Dynamic System... begun by Bruno' but boasting such ancestors as scholastic theologians, Plato, Plotinus, Ficino, Proclus and Gemistius Pletho (see I, 94; 98, xxxiii; 103\u20134). Either this system is pantheism, or it rests on the mystery of the one and the many \u2013 a mystery that literally or logically breaks down into pantheism. Both the unity of being and knowing and the mystery of the one and the many contradict the fundamental principle of discursive logic: _A_ cannot be _not-A_. Or, as the speaker explains in chapter IX, 'An IDEA, in the _highest_ sense of that word, cannot be conveyed but by a _symbol_ ; and, except in geometry, all symbols of necessity involve an apparent contradiction' (I, 100). All other symbols depend for their intelligibility upon the complex cognitive power of faith and conscience. As the many references to the Logosophia indicate, for Coleridge the Logos is the central or organizing symbol for all inquiry. By locating his philosophy in the historical past, Coleridge seeks to bolster his claim that it is sound and valid: it is a truth neglected by those entranced by popular philosophy. Coleridge firmly and literally believed this was so: the rhetorical strategy here is no mere convenience of the moment.\n\nThe letter seeks in a different way to support the validity of Coleridge's philosophy. The 'friend' is a sympathetic dunderhead, a kindly member of the Lockean 'populace' whose miscomprehensions have been predicted all along. His letter justifies these predictions; the strategy seeks to persuade us that Coleridge should not demonstrate the transcendental construction of the intelligible world in this generally nontechnical book. In reading chapter XIII, the friend feels as if he is standing on his head. That is how readers of the Preface felt: it signals an inability to reconcile thoughts with feelings, and thus a feeble imagination (I, 51\u20132). Lest we fail to recognize this parallel, Coleridge includes a page-number reference. In a more subtle echo, the images of light and darkness in the friend's analogy match those in the speaker's earlier explanations that those who lack philosophic imagination will find transcendental philosophy baffling and disorienting. The friend reports finding himself '\" _Now in glimmer, and now in gloom;\" often in palpable darkness not without a chilly sensation of terror; then suddenly emerging into broad yet visionary lights with coloured shadows of fantastic shapes_ ' (I, 199). As the speaker had explained in chapter XII:\n\nThe first range of hills, that encircles the scanty vale of human life, is the horizon for the majority of its inhabitants.... By the many, even this range... is but imperfectly known. Its higher ascents are too often hidden by mists and clouds from uncultivated swamps.... To the multitude below these vapours appear, now as the dark haunts of terrific agents, on which none may intrude with impunity; and now all _a-glow_ , with colours not their own.... (I, 164\u20136)\n\nOr, slightly later:\n\nA system, the first principle of which it is to render the mind intuitive of the _spiritual_ in man (i.e. of that which lies _on the other side_ of our natural consciousness) must needs have a greater obscurity for those, who have never disciplined and strengthened this ulterior consciousness. It must in truth be a land of darkness, a perfect _Anti-Goshen_ , for men to whom the noblest treasures of their own being are reported only through the imperfect translation of lifeless and sightless _notions_. (I, 168)\n\nAll three passages describe the sensation of confusion as frightening darkness broken only by an equally disconcerting light. Those who achieve only partial intuition of the spiritual will be as frightened as the boy Wordsworth by the apparently external source of the power that 'suddenly shines upon us' (I, 167). To those even less disposed toward philosophy than the letter's 'author', of course, Coleridge remains entirely 'obscure'.\n\nThe friend's account of the cathedral's statuary reflects the speaker's judgment of poets and philosophers. Such upending of the popular hierarchy follows necessarily, the speaker has warned, from a true understanding of imagination (I, 11\u201316, 52 and n). The contrast to Mackintosh has further emphasized the speaker's 'unusual' view of history. The friend's complaint about substances and shadows mirrors the speaker's inversion of Mackintosh's charges about nonempirical metaphysics: Hartley's 'oscillations' are proved shadows; suprasensuous knowledge is proved solid, not vaporous.\n\nThe friend argues against publication by claiming that chapter XIII is not suited to a literary life and opinions, ignoring the speaker's claims that he wishes to offer deductions from established principles (I, 64\u20135), and that some readers will have trouble understanding him, but some will not (I, 65, 74, 105\u20137, 149, 160\u20137, 188\u201392). The friend's suggestion that there may be too much metaphysics already formulates a complaint of all but one contemporary review, and of a good many readers since then as well. The friend's derogatory comparison of _Biographia Literaria_ to Berkeley's _Siris_ provides a useful clue to the _Biographia's_ form and its tradition.\n\nThe friend's final paragraph refers to familiar features in the speaker's character: his chronic inability to consider his reputation and financial needs (I, 31\u20132, 110, 119, 145), and his diligent study (I, 14, 60\u20135, 93\u20135, 104\u20135, 112, 137\u201341, 148\u201351). When the speaker defers to the judgment of his friend, it is only worse imprudence. Yet the reader is to be convinced by the letter, and by the consistency or predictability of the speaker's response, that at least part of the Logosophia has been completed. The letter thus completes a pattern of references whereby the Logosophia's status as a work-in-progress has been gradually enhanced. What is at first 'a work, which I have many years been preparing', becomes a manuscript whose transcription for the press has been abruptly curtailed (I, 91\u20132, 102, 179\u201380). It has become more real as the relation between being and knowing has been more closely specified.\n\nFrom another perspective, then, this 'mad dash' is no dash at all, no spur-of-the-moment tactic. However severely one might judge Coleridge's strategies or his motives in chapter XIII, it is not accurate to say that he has suddenly broken off his disquisition in horrified recognition of the spectre of pantheism, or in frustration at the intractability of Schelling's text, or in confusion about his own argument. The letter is not the abrupt thing it appears, but the completion of a pattern painstakingly established. Yet to say as much solves little. If we cannot write this off as error, panic or exhaustion, where does that leave us? There are two primary sources of explanation: the text itself, and the man.\n\nLet us look first to the text. Although the speaker asserts that a sense of self is present in inverse proportion to the imaginative power, and although his stories show him unconcerned with the niceties of social status, he is outraged at what has been said about him in print, and claims that such defamation has caused him 'serious injury' (I, 150). Because truly imaginative works derive from 'our nobler being', one gifted with such power exists under some moral obligation to defend his works from scurrilous attacks (I, 32), such as those by critics 'who have taken so much pains to render me ridiculous for a perversion of taste, and have supported the charge by attributing strange notions to me on no other authority than their own conjectures' (I, 65). Such comments are scattered throughout the first four chapters' commentary on anonymous criticism. The climax of such short moments of self-defense is chapter X, which records years of study and writing coming to little because of the influence of fanatics, who have then accused him 'of having dreamed away [his] life to no purpose' (I, 150). One cannot but sympathize with the wish that\n\nthe criterion of a scholar's utility were the number and moral value of the truths, which he has been the means of throwing into the general circulation; or the number and value of the minds, whom by his conversation or letters he has excited into activity, and supplied with the germs of their after-growth! A distinguished rank might not indeed, even then, be awarded to my exertions; but I should dare look forward with confidence to an honorable acquittal.... By what I _have_ effected, am I to be judged by my fellow men; what I _could_ have done, is a question for my own conscience. (I, 149\u201351)\n\nSuch a person, with this reputation and this sensitivity to it, has in an autobiographical work asserted that a major undertaking has been accomplished.\n\nThe assertion will not help his reputation \u2013 it is past redemption, as the friend's miscomprehensions indicate \u2013 but a strong if counter-productive claim has been made: I have not wasted my life. The speaker has been portrayed all along as one who says what must be said, regardless of consequences. His (in effect) refusal here to mask his motive for deleting the construction, to rewrite so as to avoid insulting the noncomprehending populace, is a coherent and integral act. Imprudent, certainly \u2013 but he is imprudent and noted for bad judgment in such situations. However frustrated we feel, we are to recognize him as more frustrated yet \u2013 as once again stymied by the very people who damn him as a dreamer. Coleridge counts on that sympathetic understanding to sustain his authority with us, the 'good' readers. Our anger at him is to be deflected into anger at the very dangerous influence of popular philosophy.\n\nI suspect that Coleridge underestimated how vividly interested readers would become in his analysis of consciousness. Granting that he was in some ways seriously depressed, that he often felt abandoned by friends and tormented by an inability to share more widely the fruits of his monumental study, granting all this, I think it possible that he could not, in 1815, strongly and consistently enough imagine the good reader he hopefully addresses now and then. In the works written after 1815, we find these same thorny metaphysical problems handled with grace and clarity. _Aids to Reflection_ , the _Biographia's_ closest of kin, perhaps, after 'Appendix C' of the _Statesman's Manual_ , offers a bolder and far more difficult form of the same basic design found in _Biographia Literaria_ ; yet it has always enjoyed a better reputation as a coherent and integral piece of writing.\n\nBut 1825 was not 1815. In 1815, Coleridge was emerging from a long period of study, illness, addiction, family problems, problems with friends and, above all, of limited _public_ productivity. The need to assert his intellectual worth may have been strong. The ability to imagine his proper readers may also have demanded a higher level of psychic energy and self-confidence than he could sustain. Thus the _Biographia_ emerged as written for sympathetic members of the populace, for what Haven calls 'Wedding Guests', with copious illustration of relatively simple points. He brackets the abstract and difficult parts, saying (in effect) 'you may not follow this; but maybe someone else will be interested'. The fiction of a contemporary English audience for transcendental philosophy complements the fiction of the completed Logosophia. The poignant moments of self-defense suggest that, were the first real, the second might have been. Chapters XII and XIII might have proceeded with Milton and Synesius and an English 'true and original realism' rather than a pseudo-Schellingeanism. Coleridge's later writings on such topics are both lucid and graceful because he never again ventured into numbered propositions and exclusively abstract arguments.\n\nFrom the letter's clear relation to what has preceded, and its comprehensible (albeit peculiar) rhetorical purposes, one may conclude that \u2013 objectively regarded \u2013 it does no damage to the integrity or completeness of the work as a whole. Regarded subjectively, however, the letter poses a more difficult challenge, because it marks and symbolizes a highly problematic aspect of Coleridge's theory of the mind's functions. The theory is formally incomplete. But from this fact one cannot directly conclude that _Biographia Literaria_ itself is incomplete, because the character of this problematic part must be taken into account. Most of the first fourteen pages of chapter XII explain that the major principles of transcendental philosophy cannot be demonstrated. The mysteries of being and knowing, or the one and the many, would not have been converted to logically demonstrable propositions by the missing construction. Does this comprise an intolerable break in the book's thematic or ideational unity? Critical or scholarly consensus here principally exerts a rhetorical and moral force: the mystery is not made less mysterious by winning broad support.\n\nThis then is the distinction of moral philosophy \u2013 not that I begin with one or more assumptions; for this is common to all science; but \u2013 that I assume a something, the proof of which no man can give to another, yet every man may find himself.... _Omnia exeunt in mysterium_ , says a schoolman: that is, There is nothing, the absolute ground of which is not a mystery. The contrary were indeed a contradiction in terms: for how can that, which is to explain all things, be susceptible of an explanation? It would be to suppose the same thing first and second at the same time. ( _AR_ , 154\u20136)\n\nIn its most fundamental terms, the issue here is whether or not one agrees that ' _Omnia exeunt in mysterium_ '. Those who absolutely agree live in a world fundamentally different from those who absolutely disagree, separated by a gulf across which arguments may carry, but to no useful purpose. But most of us, I suspect, float about somewhat eclectically, uneasy with this mysterious assumption that we know and can discourse about a real and coherent world, yet repulsed and unconvinced by the alternatives. Ultimately, one cannot absolutely and logically prove whether or not the abstract formal integrity of _Biographia Literaria_ is matched by a theoretical or ideational unity. But we conduct our lives in accord with many beliefs that we cannot absolutely establish. In reading and writing as in living, most of us rely on persuasive probabilities, and common sense, and on those graces of explanation and argument that make more easily accessible what 'every man may find for _himself_ '. One who would reach beyond the currently fashionable nihilisms can only rely on some version of the _Biographia's_ idea that discourse is an essentially creative act designed for a free agent who consciously chooses to accept some version of the true and original realism.\n\nThe famous definitions at the end of the chapter add very little that is new, but they draw together parts of a complex and difficult argument. We knew in chapter VII that imagination exists in levels, some common and others more rare. These are now given titles: the primary imagination is the ordinary agent of perception and consciousness; the secondary \u2013 the knowing of knowing \u2013 is the direct or self-conscious knowledge of the operations of the primary. It is the 'organ' of philosophy, the poetic Vision and the faculty divine', that arises spontaneously, or 'suddenly shines upon us' (I, 173, 166, 167).\n\nA summary of this argument may make the significance of the definitions more easily apparent. In common consciousness, for the ordinary person ordinarily, the identity of being and knowing is unknown. This unity is discovered by the scientific skepticism of the transcendental philosopher. Because the unity is unknown, its two 'products' \u2013 common consciousness and the perceived world \u2013 usually appear as givens, _and as unrelated_. In the cosmos known by primary imagination, T and 'it' have no fundamental and necessary connection. 'It' is simply there \u2013 impassive, inanimate, morally remote. Material association remains on this level, but two flaws reveal its inadequacies. The first is the failure of the mechanism to explain the psychic phenomena at hand. The second is that even ordinary people sometimes experience the world not as inanimate and remote, but as beautiful, alive, and translucent with meaning and value. And everyone in basic psychic health experiences both personal agency and moral responsibility \u2013 further evidence of the higher intellectual powers.\n\nSome people can penetrate to the origin of such experiences: our fundamental unity with the world around us, the common spiritual ground of human being and the world's being. They do so by or through philosophic self-consciousness: they know not simply things, and the self as a thing among things, but the activities that are the self _per se_. This ability is a heightened power to know; it is a further development of the primary imagination, although identical with it in kind. Secondary imagination 'dissolves, diffuses, dissipates' the unity of being and knowing 'so as to recreate' this unity consciously, in full possession of both the common moral ground of being, and the uniquely human power of self-knowing (with its origins and consequences). Even when such resynthesis is impossible, the secondary imagination 'struggles to idealize and to unify' \u2013 to heighten the experience of beauty, animation, and translucent meaning by focussing attention on those aspects that embody or reflect these qualities most clearly. When secondary imagination cannot literally transform, it reorganizes so as to make the implicit spiritual form more evident even if not fully realized. The productions of imagination \u2013 whether philosophy or art or science \u2013 help confirm and encourage the occasional insights of the less self-conscious ordinary person.\n\nThe difficult question about imagination is not what the power is, or how it works, but rather how these operations account for the qualities we find in imaginative works. How does rendering the primary unity conscious generate the explicit form of spiritual values and realities through the physical world? _Biographia Literaria_ offers Coleridge's fullest statement on this point, through its poetics and practical criticism. As we shall see, this literary criticism depends upon a theory of language as intimately involved with the Logos as the analysis of consciousness, although somewhat less explicitly so. For now, however, let us turn back to Theses VI and VII to see what light they shed.\n\nAnd let us grant the _fit alter et idem_ identity of being and knowing, for without such assent (at least conditionally) most of the _Biographia_ remains incomprehensible. Since the self is not a thing but the dynamic union of knowing and being, self-knowing (the spirit) knows the dynamic union both as its parts and as a whole. _In doing so, the spirit directly intuits knowing as such, and being as such_. Knowing as such is God, the creative One Almighty. Being as such (not beings) is the spiritual essence of all reality \u2013 God, man, world, interrelated and thus knowable by man, yet distinct. As a result, secondary imagination presents the spiritual in and as the natural. It bridges the gap yet retains the distinction between the human spirit and the physical world by creating symbols. Because language (a system of symbols) is the medium of least resistance to the spirit, and a poem is the most exact use of words, in a poem we are entitled to expect not just physical detail and profound insight, but an integral unity of these two:\n\nIt was not however [Wordsworth's] freedom from false taste... which made so unusual an impression on my feelings immediately, and subsequently on my judgement. It was the union of deep feeling with profound thought; the fine balance of truth in observing, with the imaginative faculty in modifying the objects observed; and above all the original gift of spreading the tone, the _atmosphere_ , and with it the depth and height of the ideal world around forms, incidents, and situations, of which, for the common view, custom had bedimmed all the lustre, had dried up the sparkle and the dew drops.... In poems, equally as in philosophic disquisitions, genius produces the strongest impressions of novelty, while it rescues the most admitted truths from the impotence caused by the very circumstance of their universal admission. (I, 59\u201360)\n\nThe genial unity of thought and feeling, or universal and particular, both depend on secondary imagination's synthesis of the powers that penetrate to pure knowing and pure being, and to their absolute union in the triune God.\n\nThe cognitive range achieved through imagination reflects the fact that 'the self-conscious spirit... is a will' (I, 185) and 'the evidence of [religion's] doctrines could not, like the truths of abstract science, be wholly independent of the will' (I, 135). Imagination is the spontaneous level of will, intermediate between the will's absolute activity in knowing, and its relative passivity in being (cf. I, 65\u20136; I, 85\u20136; I, 174; I, 182\u20133). Imagination, or self-consciousness, draws the full range of our spiritual knowledge into relation with the full range of our sensory knowledge. Because it synthesizes the animated world, imagination is truly a 'repetition in the finite mind of the eternal act of creation in the infinite I AM' (I, 202). But this fact would remain beyond our ken if we could not know imagination's operation directly, that is, know the parts as parts (and thus purely), and yet as the whole, as the rich, endlessly fascinating panorama acknowledged by 'the child's sense of wonder and novelty' but too often 'bedimmed' by custom and familiarity (I, 59).\n\nThe integrative function of imagination is analogous to that of conscience \u2013 imagination for our being in the world, and conscience for our being in God. Because our knowledge of these two aspects of our humanity is one act, the same knowledge, Coleridge appeals to conscience to illuminate evidence of imagination. His first argument for the true and original realism appeals to conscience (I, 133\u20136), and the second to imagination (Thesis VII), and yet they are the same argument. In _Biographia Literaria_ Coleridge argues for the moral depth of art; in _Aids to Reflection_ for the imaginative power of religion. It is easy enough to argue that he advocates pantheism with a veneer of Christian piety \u2013 the most likely collapse of mystery into mechanism. It is more difficult, but more accurate, to recognize how precisely he defines _and limits_ the supralogical so as to emerge with a solid base for contending that poetry is not lies, nor mere entertainment nor simply egoist self-expression, but yet not 'the truths of abstract science' either. The character of Coleridge's philosophy is not determined by the fact 'that [he] begin[s] with one or more assumptions; for this is common to all science' ( _AR_ , 154), but by the fact that his assumptions are essentially religious.\n\nYet poetry is not religion (although great poetry is moral). One need not accept Christianity to make good use of his theory of imagination. The true and original realism makes possible a definition of imagination's poetic workings that is strongly empirical in its appeal to observation, and to critical and poetic tradition. The ultimate basis of this poetics is philosophical and thus religious; but the immediate basis is textual. Coleridge's criticism is rooted in the immediate reactions of the reader to particular linguistic features. One would never reach Coleridge's poetics through the empiricist's modes of inquiry; but, having otherwise attained them, one can explicate and defend on empirical grounds. The true and original realism guarantees and insists on no less.\n\nOne point remains. What does chapter XIII add to our understanding of fancy? Very little. By distinguishing the levels of imagination as he does, Coleridge clarifies what chapters V to VIII suggest: fancy is controlled both from above and from below. It gathers and sorts (according to the law of association) the percepts generated by primary imagination; the quality of the work performed by these two powers inevitably influences the level of synthesis or transformation that secondary imagination can achieve. Yet the influence is reciprocal: one who consciously intuits the spiritual realities animating experience will more adeptly perceive their physical signs or correlatives. Because 'the will itself by confining and intensifying the attention may arbitrarily give vividness or distinctness to any object whatsoever', fancy can collect animated perceptions with the same mechanical efficiency with which it collects any other kind of perception. It can provide associations that are trivially witty, or it can provide associations that place the object into a net of relations that illumine a profound significance. In the first, a flea is still just an insect, albeit a more interesting one; in the second, the flea assumes symbolic potence. The difference between the two depends partly on the poet's original insight into the symbolic potential of the flea, and partly on the reader's appropriate synthesis of the associative details into a whole more significant than these parts. For both reader and poet, then, the role of fancy depends on the power of imagination.\n\n# **6 Poetry**\n\n> As a fruit-tree is more valuable than any one of its fruits singly, or even than all its fruits of a single season, so the noblest object of reflection is the mind itself, by which we reflect:\n> \n> And as the blossoms, the green, and the ripe, fruit of an orange-tree are more beautiful to behold when on the tree and seen as one with it, than the same growth detached and seen successively, after their importation into another country and different clime; so it is with the manifold objects of reflection, when they are considered principally in reference to the reflective power, and as part and parcel of the same. No object, of whatever value our passions may represent it, but becomes foreign to us as soon as it is altogether unconnected with our intellectual, moral, and spiritual life. To be ours, it must be referred to the mind, either as motive or consequence, or symptom.\n> \n> _Aids to Reflection_ , Introductory Aphorism V\n\n'The office of philosophical _disquisition_ ', Coleridge explains, 'consists in just _distinction_ ; while it is the privilege of the philosopher to preserve himself constantly aware, that distinction is not division. In order to obtain adequate notions of any truth, we must intellectually separate its distinguishable parts; and this is the technical _process_ of philosophy. But having so done, we must then restore them in our conceptions to the unity, in which they actually co-exist; and this is the _result_ of philosophy' (II, 8). Chapters XIV to XVI restore the distinctions between fancy and imagination, thought and feeling, being and knowing, subject and object to their original unity: the integrity of the human mind as evident in literary language.\n\nA poem reveals the complex unity of the mind because a word names not a physical thing, nor an idea in the mind of the speaker, but the relation between speaker and reality. Words exist _within_ the unity of being and knowing, world and mind; and thus a precise use of words (by philosopher or by poet) renders that unity distinctly accessible to the conscious mind. And a poem is our most precise use of language: 'it would be scarcely more difficult to push a stone out from the pyramids with the bare hand, than to alter a word, or the position of a word, in Milton or Shakespeare, (in their most important works at least,) without making the author say something else, or something worse, than he does say' (I, 15); or 'Poetry... ha[s] a logic of its own, as severe as that of science; and more difficult, because more subtle, more complex, and dependent on more, and more fugitive causes. In the truly great poets... there is a reason assignable, not only for every word, but for the position of every word' (I, 4). A poem reveals both the mind of the poet, and the reality he knows; in examining our reaction to the text, we discover our relation both to the powers of mind and to the aspects of reality. Granting Coleridge's original realism, we see that words precisely used make true statements. The cultural and moral value of poetry depend on this link between precision and truth.\n\nR. H. Fogle said chapter XIV offers 'a microcosm of Coleridge's entire critical system', and so it does \u2013 both in what it says, and in how it delineates the complexities that a poem represents. Nowhere does he more frequently shift from objective to subjective perspectives and back again. Objectively, the poem is a use of language, a 'species of composition' (II, 10). As such, it exists in relation to the universe (it makes true or false statements). Yet it also exists in two sets of subjective relations, to the reader and to the poet. To the reader, a poem is both a thing (a text, a 'species of composition') and a psychic experience, an act of reading. In parallel ways, to the poet a poem both expresses what he _knows_ , and who he _is_ , through the medium of words. The definitions of 'poem', 'legitimate poem', 'poetry' and 'poet' explain these relations. Chapter XV translates this theoretical complex into something useful for the practical critic by explaining how a poem's features both arise from the poet's skills, and elicit the reader's reaction. The subjective orientation of Chapter XV is balanced by the objective orientation of chapter XVI which argues that language does more than reveal skills or engender response. It shapes our relation to reality. Coleridge alternates orientations so often because he is trying to define how language incarnates the essentially spiritual relation between being and knowing.\n\nAttending closely to Coleridge's statements about language reveals that his poetics has deep and intricate roots in the first volume, although the bulk of these are not in Chapters XII and XIII. Throughout the first volume, Coleridge asserts a connection between precise language and truth. The original realism is denied only by 'philosophers of the schools... who live and move in a crowd of phrases and notions from which human nature has long ago vanished' (I, 179). Edmund Burke's power as a ' _scientific_ statesman; and therefore a _seer_ ' arises from his habitual precision of expression (I, 125). The vagueness and inconsequence of anonymous critics and decadent modern poets reflect their confused understanding (I, 25\u20139 and n; I, 15). By contrast, the speaker was rigorously trained in verbal precision even at Christ's Hospital. As he is more fully portrayed as a genius, his care with words is more clearly stressed, as part of the larger argument that any apparent obscurity in a genius's work reveals the difficulty of the topic, or the deficiency of the reader, rather than authorial failing (I, 4\u20135, 64, 108\u201310, 146). The link between good poetic diction and truth is often asserted, but especially in the first major paean to Wordsworth, which shifts effortlessly from the praise of his pure diction to a celebration of the truth that his genius properly animates (I, 58\u201360). In Chapters XIV to XVI, Coleridge begins to justify this link by exploring the relations between poet and text, and between text and reader. In doing so, he develops the definitions of imagination and fancy; he does not simply apply them to literary questions.\n\nCritics and scholars have long argued about whether or not the theory of literature presented in these chapters grows coherently from the metaphysics and epistemology in the first volume. Those who would describe Coleridge as the father of the New Criticism tend to disregard these metaphysical roots, and to focus on such famous doctrines as 'poetic faith' or the opposition between poems and science. Those who take these roots seriously encounter a variety of complex and perplexing problems. One who emphasizes imagination's synthesis of being and knowing within a metaphysics of polarity can account quite well for much that is both valuable and influential in Coleridge's literary criticism. But, taken in isolation from the rest of Coleridge's philosophic thinking, these doctrines lead directly, abruptly into pantheism. That makes it difficult to justify the centrality of Coleridge's belief in the moral value of art: the belief appears either extraneous, or inconsistent, or both. Those who try to resolve the conflicts are more or less stymied by the fact that after _Biographia Literaria_ Coleridge said and wrote very little about imagination, fancy and poetry.\n\nBut he did continue to write about language itself, and the relation between metaphysics and poetics is to be found primarily in what the _Biographia_ says about language. This idea of language, like the idea of will, permeates Coleridge's works; but it is fully explicated in no one place. Pausing now to define this idea will establish a locus from which the continuity and balance of first and second volumes can be more easily described. Language integrates subjects with objects, or knowing with being, because words reveal both the real world and the person who uses them. 'I include in the _meaning_ of a _word_ ', Coleridge explains, 'not only its correspondent object, but likewise all the associations which it recalls. For language is framed to convey not the object alone, but likewise the character, mood and intentions of the person who is representing it' (II, 115\u201316). In the _Biographia_ he does not explain how, referring instead to 'some future occasion' on which he will 'attempt to prove the close connection between veracity and habits of mental accuracy; the beneficial after-effects of verbal precision in the preclusion of fanaticism... and to display the advantages which language... presents to the instructor' (II, 116\u201317).\n\n_Aids to Reflection_ explains what _Biographia Literaria_ somewhat cryptically asserts. Words signify neither things, nor ideas, but rather the mind's activity in perception, reflection and meditation. Words reveal the creative, active mind as it distinguishes between real things and mere artifacts of sensory receptors:\n\nNow when a person speaking to us of any particular object or appearance refers it by means of some common character to a known class (which he does in giving it a name), we say, that we understand him; that is, we understand his words. The name of a thing, in the original sense of the word name ( _nomen, \u03bd\u03bf\u03cd\u03bc\u03b5\u03bd\u03bf\u03bd, \u03c4\u1f78 intelligible, id quod intelligitur_ ) expresses that which is _understood_ in an appearance, that which we place (or make to _stand) under_ it, as the condition of its real existence, and in proof that it is not an accident of the senses, or _affection_ of the individual, not a phantom or apparition, that is, an appearance which is _only_ an appearance. ( _AR_ , 220\u20131)\n\nThe act of naming asserts that the thing named exists in a reality shared by speaker and auditor. Coleridge calls naming 'the condition of real existence' because the speaker effectively distinguishes between real and chimerical when he gives something a name: language does not supply names for entities beyond our imagination. Language expresses and records our knowledge through words, which signify not ideas, nor things, but the relation between an idea and its object.\n\nI can sharpen our focus on Coleridge's definition of 'word' by sketching its context. His assertion of the mind's active role in establishing the meanings of words contrasts sharply with the dominant Lockean tradition in linguistics. Throughout the third book, 'Of words', in his _An Essay concerning Human Understanding_ (first edition, 1690), Locke maintains that words signify ideas. Since ideas arise from experience of things (or reflections on such experience), words refer to objects only indirectly. In this contention, Locke himself differs sharply from the dominant linguistic tradition in his own day. The contrast can be simplified as follows: for Locke, words signify ideas (of objects); for John Wilkins and other seventeenth-century linguists, words signify (ideas of) objects. In his _An Essay Toward a Real Character, and a Philosophical Language_ (1668), Wilkins undertakes a monumental task: 'the great foundation of the thing here designed [is] a regular _enumeration_ and _description_ of all those things and notions, to which marks or names ought to be assigned according to their respective natures' (p. 1). Wilkins hopes to remedy the confusion of tongues and advance the cause of the Royal Society by inventing a language (his 'real character') in which there is a completely unequivocal one-to-one correspondence between signs and signifieds. (Despite losing his manuscript in the Great Fire, Wilkins completed this enumeration, and invented a language based on Hebrew.) Coleridge's definition of 'word' as the relation between idea and object places him intermediate between the positions represented by Locke and Wilkins, and in a line that Ernst Cassirer traces through Heraclitus, Plato, Berkeley, Herder and von Humboldt.\n\nColeridge's definition of 'word' represents language as participating intimately in the complex relation between mind and world: the process of naming and the process of knowing are represented as a single process. As the paragraph cited above continues, Coleridge develops the definition in two directions. First, as 'evidence', he cites the frequent identity in the Bible of _nomen_ (name) and _numen_ ('invisible power and presence'). His claim here has obvious affinities with such 'Platonic' theories as that of James Harris. Yet Harris contends that words refer not to 'external particulars, nor yet... particular ideas' but, rather, to 'general ideas'. These are 'a kind of superior objects; a new race of perceptions... each one of which may be found entire and whole in the separate individuals of an infinite and fleeting multitude, without departing from the unity and permanence of its own nature'. For Coleridge, the significance of language depends not on how words name objects (or universals) but, rather, on how words reveal the mind's activity.\n\nSecondly, he supports the definition of 'word' by explaining that the objects of understanding are strictly linguistic:\n\nThus, in all instances, it is words, names, or, if images, yet images used as words or names, that are the only and exclusive subjects of understanding. In no instance do we understand a thing in itself; but only the name to which it is referred.... No one would say he understands red or blue. He _sees_ the colour, and had seen it before in a vast number and variety of objects; and he understands the _word_ red, as referring his fancy or memory to this his collective experience. ( _AR_ , 222)\n\nSensation is predominantly a physical act; understanding is predominantly an intellectual one. We do not _know_ things in themselves: we feel or smell or touch them. In a characteristically adept use of Kant, Coleridge presents language as the principal vehicle for the interaction of knowing mind and concrete being:\n\nIf this be so, and so it most assuredly is \u2013 if the proper functions of the understanding be that of generalizing the notices received from the senses in order to the construction of names: of referring particular notices (that is, impressions or sensations) to their proper names; and, _vice versa_ , names to their correspondent class or kind of notices \u2013 then it follows of necessity, that the understanding is truly and accurately defined in the words of Leighton and Kant, a faculty judging according to sense. ( _AR_ , 222)\n\nIn studying language, we see how the free spirit or will can engage an intractably stable physical reality without losing its own freedom. When imagination dissolves and recreates the mind's relation to this world, words record the change. Language is framed to reveal the insights that philosophic self-consciousness attains, to express an intuitive penetration to the relation between God and man.\n\nLanguage is not only the medium of least resistance to the spirit, but also a powerful objective force in its own right. As political language so clearly reveals, those who manipulate the words that name our relations to events can very effectively shape our moral responses to the events themselves. For Coleridge, linguistic accuracy is at heart a moral issue, because precisely written texts are a moral resource. And that is the principal issue uniting the _Biographica's_ transcendental philosophy with its literary criticism.\n\nLet me explain the relation between language and truth in more detail, because it provides a major key to the philosophic origins of Coleridge's criticism. Because words reveal the mind's activity, one can say that language expresses the contents and activity of consciousness. In the manuscript 'Logic' Coleridge explains more fully an idea of grammar and logic that he barely sketches in the _Biographia_ (I, 14; II, 63\u20138, 116\u201317). Grammar, he explains, 'reflects the forms of the human Mind, and gradually familiarizes the half-conscious boy with the frame and constitution of his own Intellect'. This 'frame and constitution' is logic: 'it is plain that Logic[,] in as much as it presents the universal and necessary rules of the Understanding, must in these rules present likewise the criterion of truth, that is, of formal truth, or truth relative to the constitution or constituent forms, laws and rules of the thinking faculty'. In the 'Opus Maximum' manuscript, Coleridge clarifies his special interpretation of the common eighteenth-century equation between logic and grammar: 'In fact the science of grammar is but logic in its first exemplification or rather in its first product... i.e., _thoughts in connexion or connected language_ ' (my italics). Logic defines both the rules of correct understanding and the rules of correct word-use; as formalized in grammar, logic is the power of language to express relations. Although language itself does not constitute reality, it expresses the constituent forms of the understanding: the power to make relations.\n\nAs I explained in the first chapter, the most fundamental relation is that between ideas and the laws governing both physical and spiritual or psychological realities. Language can incarnate the relation between ideas and physical laws because it distinguishes perceptions from illusions. Language can incarnate the relation between ideas and spiritual laws because the logical, grammatical, relational features of language itself reveal the psychological laws of thinking for which the correlative idea is the Logos, the Word of God, whom we know through the fullest concurrent action of will, reason and faith. Because of these complementary capacities, language in its total complexity reveals and enacts the laws of the universe. Language embodies a dynamic and vital metaphysics sharply opposed to the fixity of materialism. In the _Philosophical Lectures_ Coleridge explores this idea of language at length, attributing it to Pythagoras and his heirs. Pythagoras was the first to recognize, Coleridge explains, that 'the very powers which in men reflect and contemplate, are in their essence the same as those powers which in nature produce the objects contemplated' ( _PhL_ , 114). These powers were 'by the Pythagoreans and Anaxagoras called the _Nous_ , (the _Logos_ or the _Word_ of Philo and St. John)' ( _PhL_ , 175). The _Philosophical Lectures_ develop this insight through analyses of such figures as Aristotle and Bacon, who study these powers as manifest in the natural world, and those such as Plato, who study them as manifest in the mind.\n\nLanguage is one of these manifestations in the mind. Plato 'direct[s] his inquiries chiefly to those objective truths that exist in and for the intellect alone, the images and representatives of which we construct for ourselves by figure, number, and word' ( _F_ , I, 492). Mathematics had long been taken as a model for absolute certitude; the apparently dramatic correspondence between the pure mathematical system and the observable world had always had metaphysical significance. Coleridge echoes the claim that words, like figures and numbers, are part of a perfect system which none the less reflects the form of an imperfect and changeable world. It is in its capacity to represent a (logical, grammatical) system of relations that language is analogous to mathematics.\n\nWords are to consciousness what geometrical figures are to space, or numbers to time ( _F_ , I, 440 n). Language symbolically represents both the relations of mind to world (the correlation of ideas and laws), and the relation of human knowledge to divine knowledge (logic and the _Logos_ ). Language differs from geometry in part because language changes and develops, while geometry does not (or would not have seemed to, for Coleridge). But this difference is minor. Although language is always changing, it is at any given point a perfect system, because it perfectly accords with consciousness itself, which is always changing. Had Coleridge recognized that the properties of space are still developing (in his sense), then he would have seen that geometry must develop apace.\n\nBecause accurate language reliably guides us to truth, and because unrecognized truths are apt to require astute linguistic distinctions for their intelligibility, many of Coleridge's arguments appeal to etymology, and to grammar and usage. Throughout his career, he urges desynonymizing as a route to clear understanding. It is, he says, 'the progress of language' ( _PhL_ , 368\u20139; see also 200\u20131). He bases arguments on the exact meanings of words, and cites or invents etymologies to make his point. He frequently structures arguments as inquiries into the precise meaning of a word, or the correct distinction between related words. These pervasive strategies are more than convenient rhetorical habits: they reflect deeply held convictions about the nature of language itself. Read against German transcendentalism, or certain strains in modern literary theory, Coleridge's view of the interactivity of language and consciousness appears to incorporate a high level of indeterminacy; but this is only an appearance. As the _Aids to Reflection_ unequivocally demonstrates, Coleridge firmly believed in the stability and universality of what used to be called 'human nature'. Eliminate this stability, or the theism underlying it, and the ideas which remain change character so radically that they should not be attributed to Coleridge.\n\nColeridge's idea of language underlies both the textual specificity of his best criticism, and its moral concerns. Poetry guides us to truth not by what it says \u2013 not by sermonizing \u2013 but by how its language reveals the fundamental relations animating consciousness and the world as humanly experienced. This idea also underlies his willingness to make value-judgments, because the distinction between intellectual knowing and physical sensing guarantees that we can judge the adequacy of verbal formulations by comparing actual experience and linguistic representation. Coleridge would take quick issue with the notion that words refer only to other words, and not to realities. He would say, I suspect, that dissolving either the connection or the distinction between word and signified is the same school-error as dissolving the connection or the distinction between sensation and object sensed: it breeds chimerical problems that only the original realism can escape.\n\nIn chapter XIV, Coleridge begins to present a theory of poetry that ties aesthetic pleasure to linguistic qualities that guarantee the moral value of art, yet without subordinating the poet's craftsmanship to a moral censor. Such a theory allows him to explain how philosophic imagination can generate both the pure diction and the moral insight that he recognized in Wordsworth's early poems (I, 56\u201360). It also allows close and sustained attention to diction _per se_ , and to the crucial role fancy plays in the effective presentation of imaginative insight. As the theory unfolds, one begins to see that the complementary functions of fancy and imagination in many ways reflect the complementarity of being and knowing, or faithful representation and self-conscious transformation. Poetics and metaphysics come into their fullest, richest balance in the critique of Wordsworth's theory and practice: Coleridge argues that Wordsworth's mistaken idea of language reflects a mistaken epistemology, and generates defective poems. Analyzing his errors, as Coleridge earlier analyzes the errors of 'the excellent and pious Hartley', reveals how the cooperative energies of fancy and imagination create a poetic cosmos in which mind and world are perfectly balanced and fully integrated.\n\nChapter XIV's account of the 'plan' for _Lyrical Ballads_ introduces an objectively oriented definition of poetry as a 'species of composition'. But as the chapter develops, Coleridge shifts the objective orientation into its opposite, primarily by changing from poem as text, to poem as reading, to poem as expression. He maintains the unity of this progression partly by anchoring himself in the poem as language, and partly by incorporating images that signal imagination's activity into his objective descriptions. The most striking such image is the reference to light in the account of _Lyrical Ballads_. The image is repeated from chapter XII and, more directly, from the description of Wordsworth's genius in chapter IV (I, 164\u20137; I, 59\u201360).\n\n... our conversations turned frequently on the two cardinal points of poetry, the power of exciting the sympathy of the reader by a faithful adherence to the truth of nature, and the power of giving the interest of novelty by the modifying colours of imagination. The sudden charm which accidents of light and shade, which moon-light or sun-set diffused over a known and familiar landscape, appeared to represent the practicability of combining both. These are the poetry of nature. (II, 5)\n\nLight affects not only how well we see, but also what we see. It both creates and reveals. The poet's rhetorical task is to modulate imagination's light so as to engage the reader's sympathy, and to avoid the disorientation reported by the 'friend' in chapter XIII. One who succeeds in this task opens new worlds to the reader, whether the purely fictional world of The Ancient Mariner', or the realistic but too often unknown world to which Wordsworth restores us (cf. I, 60; II, 6).\n\nIn defining _poem_ as 'a species of composition', Coleridge explains how the poet achieves these ends through poetic form. A poem is distinct from other uses of language by its use of rhyme, or metre, or both (II, 8), and by its immediate purpose or intent: pleasure for the reader (II, 9). This pleasure principle \u2013 the opposition between poems and 'works of science' \u2013 indirectly but no less certainly ensures that the great poem will fully embody an imaginative access into profound and fundamental truths. Because 'nothing can permanently please, which does not contain in itself the reason why it is so, and not otherwise', the features of poetic form must justify and require each other. These justifications and requirements can be for the reader's sake, or the poet's, or both; and so the argument begins to shift toward the subjective.\n\nBefore following, let us look more closely at the quality of pleasure poems afford, and its relation to truth. We delight primarily in what one may generally call the 'aptness' of a poem's expression, the reflexivity of 'form' and 'content', rather than the 'content' _per se_. But the nature of words and the nature of grammar guarantee that the utterly apt expression will truly reveal some aspect of the relations among God, man and a shared, immediately accessible reality. Lear was not a real king, perhaps; but to discredit or to ignore the play's truth on that ground is akin to discrediting Mendel's conclusions because his peas are not all peas. It is to mistake, profoundly, the character of the human activity in question. One attains truth about the laws governing physical reality by directly testing one's ideas. One attains truth about the laws governing intellectual or spiritual realities indirectly, by aiming at precise expression, by relying on the Logos. Only through primary, immediate attention to the delights of accurate and beautiful language does the poet reach poetry's 'proper ultimate end'. The ultimate truth of _King Lear_ is not Lear's fate, nor even the sum of propositions extractable from major speeches, but, rather, what the play reveals about human minds. The critic's task is to discover and to represent that truth in relation to its medium \u2013 as Coleridge himself does, for instance, in his famous study of _Hamlet's_ opening scene. This highly psychological (or epistemological) approach to literature has considerable value for much of the literature written since 1800, but it is not necessarily the most fruitful approach to Wordsworth's experiments with descriptive poetry. Coleridge sharply criticizes his attempts at a rigorous but enlightened objectivity.\n\nWhether or not truth is compatible with aesthetic pleasure from the utterly apt or self-justifying poetic form will depend upon the 'state of society' and the 'character of the author' \u2013 not the genre of the work (II, 9). Given what we have seen about contemporary society, and the authors who pander to it, this distinction between immediate and ultimate purpose resonates deeply with the controversy over _Lyrical Ballads_. Coleridge's account of the controversy often involves this distinction. When Wordsworth confuses the two ends, he sermonizes or bogs down in excessive detail. When he distinguishes and manages them properly, his diction improves; but he ensures the hostility of anonymous critics who either cannot or will not respond to genuinely imaginative works. The flaws of Wordsworth's theory make the flaccid diction of the immediately moral poems seem deliberate, and provide anonymous critics with excuses for ignoring the ultimate truth and immediate beauty of the imaginative poems.\n\nColeridge links proper aesthetic pleasure to the reader's imagination when he defines a ' _legitimate_ ' or excellent poem. Any poem must provide 'such delight from the _whole_ , as is compatible with a distinct gratification from each component _part_ ' (II, 10). But the great poem has more than such balance: it has unity in diversity. '... the parts... mutually support and explain each other; all in their proportion harmonizing with, and supporting the purpose and known influences of metrical arrangement', i.e. 'perpetual and distinct attention to each part' (II, 10). Metre enforces attention to parts, which reciprocally enforce attention to the whole. It leads the sensitive reader to the imaginative synthesis of an organic unity. Poetic language and poetic form together arouse the reader's imagination, and thereby generate aesthetic pleasure. Coleridge's account of this pleasure echoes both the first critical aphorism (I, 14) and the water-insect passage (I, 85\u20136):\n\nThe reader should be carried forward, not merely or chiefly by the mechanical impulse of curiosity, or by a restless desire to arrive at the final solution; but by the pleasureable activity of mind excited by the attractions of the journey itself. Like the motion of a serpent, which the Egyptians made the emblem of intellectual power; or like the path of sound through the air; at every step he pauses and half recedes, and from the retrogressive movement collects the force which again carries him onward. (II, 11)\n\nBecause the quintessential aesthetic pleasure arises from the act of reading, not \u2013 strictly speaking \u2013 from the content of what is read, one can reread, and reread again, with no diminution of pleasure. Novelty is not the point, although it is the principal attraction of the works written and praised by anonymous critics (I, 27 n). The identity of parts yet their unity in the whole reflects the identity of being and knowing yet their unity in consciousness. A poem can so intimately reveal or enact the nature of consciousness because language symbolizes the unity of being and knowing. Thus, the imaginative synthesis responsible for self-consciousness, and for truth, is also responsible for aesthetic pleasure.\n\nFrom this it follows that aesthetic pleasure can be generated by a work whose immediate purpose is truth, not pleasure \u2013 if the author is most powerfully imaginative (II, 11). In theory even metre is not crucial; it is principally an aid to modulating attention and thus increasing the pleasure most fully appropriate to poems. A poem differs from such poetry as Plato offers by using metre to accommodate the whole most completely to the intensely imaginative parts; it is crucial to the highest degree of such accommodation, whether from the poet's or the reader's standpoint. By using the word 'poetry' to refer to that which generates aesthetic pleasure generally, Coleridge affirms that the objective features of a poem (such as metre or poetic diction generally) exist to enhance, facilitate and sustain the reader's imaginative response. It is obvious that truth \u2013 the unity of being and knowing \u2013 cannot be the logical opposite of aesthetic pleasure. Although pleasure and truth will not be compatible for all readers, or for all poets, at all times, the highest aesthetic pleasure and the greatest truths will most probably and most often be found together.\n\nIn describing a poem's imaginative qualities, Coleridge has worked his way back to the imaginative qualities of great poets: 'What is poetry? is so nearly the same question with, what is a poet? that the answer to the one is involved in the solution of the other' (II, 12). The synthetic power of great poems can only arise from the synthetic power of great poets; the linguistically embodied relation between knowing mind and real world presumes an author for whom that relation is lucid and self-conscious. The famous portrait of the 'poet, described in _ideal_ perfection', translates the familiar features of imaginative activity generally into terms useful for describing the literary products of that activity. Imagination, we are told once again, functions within a complex of powers that must be balanced and coordinated properly. It is spontaneous, not relatively passive, nor chosen in complete independence, but retained under the 'gentle and unnoticed control' of the will and understanding (the relatively active and passive poles that it synthesizes). These familiar ideas are repeated here because they will play a major role in Coleridge's theory of metre, and his use of that theory as a basis for criticizing Wordsworth's theory and his poems. The description ends with a long catalogue of the relatively active and relatively passive qualities that imagination ' _fuses_ , each into each'. Coleridge breaks this catalogue into subordinate groups by changing whether the initial member of each group is objective or subjective:\n\nsame | different  \n---|---  \ngeneral | concrete  \nidea | image  \nrepresentative | individual  \nold and familiar objects | novelty and freshness  \norder | emotion  \njudgment | enthusiasm and feeling  \nartificial | natural  \nart | nature  \nmanner | matter  \nadmiration for the poet | sympathy with the poem\n\nThis catalogue has obvious affinities to the table of distinctions concerning will, and to the distinctions between relatively active and relatively passive in chapter IX (I, 66, 108\u20139). It also accords with chapter XVI's distinctions 'between the Poets of the present age and those of the 15th and 16th centuries' (II, 20). It represents again 'the two cardinal points of poetry': sympathy with the familiar, concrete, particular world; and interests in the novelty supplied by intellectual and artistic transformations. Because Wordsworth is quite nearly this ideal poet, these oppositions appear time and again in the commentary on his work. Coleridge's successful emphasis on this description helps make evident that Wordsworth's poetry \u2013 like all great poetry \u2013 synthesizes the powers of spirit with the animated world the spirit knows.\n\nThe catalogue serves thematic unity as well. Beginning here, the opposition between knowing and being shifts back into its earlier, more vivid and more useful forms: head and heart, judgment and passion, the centrifugal and centripetal forces of exactness and liveliness. So, too, the lines from Davies (as amended by Coleridge) translate the metaphysics of original realism into something less abstract and more useful for criticism: the insistence that imagination's powers must '\"Steal access through our senses to our mind\"' (II, 13), that imagination's powers can and must be concretely manifest in the experience of living and the experience of literature. This insistence strikes me as the quintessentially English side of Coleridge's theory, and simultaneously as the most fundamental appeal his theory makes to that soul within us who must 'awake and start up' at the annunciation of a genuine idea. A poem synthesizes intellectual, moral and emotional powers, not _by_ its words or their paraphrases, but _through_ its words, through the words' simultaneous relations to consciousness and to reality in all the complexities of both. A poem symbolizes the integrity of the human spirit in the fullness of its relation to all that is 'other'.\n\nAnd yet this unity is fully susceptible of analysis, because words are not the only medium of relation to the other. We have hands and eyes and ears to tell us whether a line successfully arouses an imaginative revision of the world we have known. In Chapter XV, on The specific symptoms of poetic power', Coleridge examines the roles played by four major components of a poem \u2013 versification, topic, imagery and ideas \u2013 to anchor the activity of imagination in particular features of the text. The chapter's design sets a pattern that later chapters often follow. On the one hand, the chapter offers numbered judgments on a single issue: the objective signs of poetic genius. On the other, the justifications for these judgments maintain an independent progression that gradually illuminates a major principle: the role of imagination in reading and writing. The numbered sequence engenders a sense of orderly momentum that is somewhat rare for Coleridge's prose, while the justifications hover in more characteristic fashion over a single but very complex idea. The strategy may be a carry-over from the conveniences of Schelling's numbered theses. Although helpful to the reader in a mechanical way, its artificiality \u2013 so rare a flaw in Coleridge \u2013 tends to obscure the center about which he circles.\n\nThe first section, on versification, defines how the poem's language sustains the poet's intimate relation with his reader. The poet's musical delight moves through the text to a reciprocal delight _in the reader_. The poet reduces the plenitude of his materials to unity of effect _on the reader_. One legitimate measure of a poem's quality, then, is the reader's response \u2013 a measure that underlies several crucial rhetorical patterns later on. These ideas come into much sharper focus when Coleridge examines Wordsworth's versification: his feeble prosody fails to move the reader; and the prosody is feeble because he himself lacks the passion or the materials that arouse musical prowess.\n\nThe second section, on the choice of subjects, explains that the poem's beauty (its imaginative fusion of universal and particular) derives from contemplation not observation \u2013 from secondary imagination, not primary. The comic tale of the indifferent statue with the divine legs enforces what is by now a familiar point in Coleridge's favorite way. The sexual passion of Venus or Adonis is even more easily distinguished from Shakespeare's creative passion (although, as with the goddess, distinction is not division: opposites meet). Once again, the poet's imaginative powers shape the reader's experience through the medium of the poem.\n\n['Venus and Adonis'] is throughout as if a superior spirit more intuitive, more intimately conscious... of the flux and reflux of the mind in all its subtlest thoughts and feelings, were placing the whole before our view; himself meanwhile unparticipating in the passions, and actuated only by that pleasurable excitement, which had resulted from the energetic fervor of his own spirit in so vividly exhibiting what it had so accurately and profoundly contemplated.... You seem to be told nothing, but to see and hear everything. Hence... the perpetual activity of attention required on the part of the reader; from the rapid flow, the quick change, and the playful nature of the thoughts and images; and above all from the alienation, and... the utter _aloofness_ of the poet's own feelings, from those of which is he at once the painter and the analyst. (II, 15\u201316)\n\nThe reader repeats the poet's dissolving and recreating through his distinct attention both to the parts as such and to the whole. The reader engages the artwork, not the sexual passion; he responds to contemplation's product, not its immediate subject. Coleridge further develops this point when he links Wordsworth's emphasis on observation and literal experience to the poems he regards as inexcusably dull.\n\nThe third section explains that the poem's images enact the transforming power of imagination. In a legitimate poem, images are shaped by the poet's creative or contemplative passion. Regarded objectively, images transcend space, time, and fragmented particularity by 'reducing multitude to unity, or succession to an instant' (II, 16). When engaged in the text, poet and reader dwell together and apart, not in 'that inanimate cold world allowed\/To the poor loveless ever-anxious crowd' but in 'A new Earth and new Heaven,\/Undreamt of by the sensual and the proud'. How? Images define relations, not just things. Pines like women with their backs to the wind are not just trees, but trees as seen by a human observer. The poet's power to observe himself observing, and his access to the spirit with all that involves, is most evident in the poem's imagistic language. By attending to the imagery, the reader sees the world as the poet sees it: the reader knows the poet's knowing. Coleridge later argues that Wordsworth's dramatic poems often fail because they delete this mediating consciousness: one sees _what_ the poet sees, not _how_.\n\nThe fourth section is the second of Coleridge's three general descriptions of poetic genius (II, 12, 19\u201320, 122\u20134). The first defines the polar oppositions that the poet reconciles in his works. This second one defines the oppositions reconciled in the consciousness of the poet _qua_ poet: the unity and identity in imagination of creative originality and fidelity to nature. Familiar images reappear to stress that this specifically poetic union develops gradually from the primary psychic unity of self-consciousness.\n\nIn Shakespeare's _poems_ the creative power and the intellectual energy wrestle as in a war embrace. Each in its excess of strength seems to threaten the extinction of the other. At length in the DRAMA they were reconciled, and fought each with its shield before the breast of the other. Or like two rapid streams, that, at their first meeting within narrow and rocky banks, mutually strive to repel each other and intermix reluctantly and in tumult; but soon finding a wider channel and more yielding shores blend, and dilate, and flow on in one current and with one voice. (II, 19)\n\nThe tumult that permanently characterizes the fanatic is also a stage of development for the genius; hence the speaker does not mind 'a certain degree of disputatiousness in a young man... provided I find him always arguing on one side of the question' (I, 13). For the genius, the will or true self soon fuses or reconciles opposing tendencies into a single, stronger power. Even in Shakespeare's early poems, however, we find his imagination clearly at work, and his 'dominion, often domination, over the whole world of language' (II, 19).\n\nAnd it is to the nature of language, particularly the poetic language of rhythm and image, that Coleridge turns in chapter XVI. Wordsworth's centrality in this poetics reflects his place in Coleridge's pantheon of great English poets. Shakespeare 'passes into all the forms of human character and passion'; Milton 'attracts all forms and things to himself, into the unity of his own IDEAL' (II, 20). There must be an intermediate kind of poetic genius, neither absorbed in nor absorbing the material world \u2013 just as a word names neither idea nor thing, but their relation. And this is the genius of Wordsworth, who is capable of writing the 'FIRST GENUINE PHILOSOPHIC POEM' (II, 129).\n\nLike chapter II, on 'the supposed irritability of men of Genius', chapter XVI at first glance seems digressive. Italian poets of the fifteenth and sixteenth centuries do not seem to have much relevance either to Wordsworth, or to the analysis of poetry and poetic genius. That is because the predominant transition or relation between chapter XVI and the two preceding rests not on poetic theory, but on the chronological narrative of the speaker's 'literary life and opinions'. The continuity of that narrative relies primarily on renewing patterns and themes established in the first four chapters. Looking back at how chapter XIV echoes these early themes will show not only how chapter XVI fits in, but also how the genius-fanatic contrast provides a crucial context for the analysis of Wordsworth.\n\nThe most important contrast renewed in chapter XIV is that between anonymous and philosophical criticism. The speaker claims that Wordsworth and he have been victims of a literary inquisition in which acrimony replaces analysis.\n\nFrom the conjunction of perceived power with supposed heresy I explain the inveteracy and in some instances, I grieve to say, the acrimonious passions, with which the controversy has been conducted by the assailants[.]... the intellectual energy of the author, which was more or less consciously felt, where it was outwardly and even boisterously denied, meeting with sentiments of aversion to his opinions, and of alarm at their consequences, produced an eddy of criticism, which would of itself have borne up the poems by the violence, with which it whirled them round and round. (II, 7)\n\nThese reviewers are not critics, but 'assailants' who 'boisterously' deny their own intuitive apprehensions. The tornado-like 'eddy' that would of itself have borne the poems to prominence repeats the rising circular image of swarming bees from the first definition of the fanatic (I, 19). Whatever the equivocations of the Preface, such irresponsible and frantic criticism reveals more about the critic than it does about the poet.\n\nThe speaker wishes to extricate himself from this situation by declaring his differences from Wordsworth, and to extricate Wordsworth by explaining what he 'really meant'. The painstaking definitions of poem, legitimate poem, poetry and poet reinforce the speaker's status as a precise thinker. The visual, aural and semantic similarities of these terms may unnecessarily confuse and delay the reader, but they enhance the appearance of extreme precision. In much the same way, Chapter XV reminds us that the speaker is a learned man who knows relevant history thoroughly. Its numbering enforces the same sense of clarity and precision: nothing 'boisterous' here.\n\nIn this context, chapter XVI's account of Italian poets further enforces the speaker's scholarly responsibility. These poets may lack novelty or particularity, but the speaker certainly prefers their work to the utter confusion reigning in the diction of his contemporaries, whether poets or prose writers: '... the composition of our novels, magazines, public harangues, &c, is commonly as trivial in thought, and yet enigmatic in expression, as if ECHO and SPHINX had laid their heads together to construct it' \u2013 in pointed contrast to his own style in the preceding two chapters (II, 21\u20132).\n\nReading the chapter this way leads us easily enough to its role in the _Biographia_ generally: the qualities of fanatics always reverse those of geniuses; the discussion of fanaticism often introduces an explanation of the ground of the difference. The genius-fanatic contrast here points in two directions at once. First, it asserts that the poet must 'fuse, each into each', the interesting surprises of novelty and the utter predictability of a mannerist style. An excess of the first generates a work that gratifies curiosity alone; an excess of the second mutes attention to particular parts, and thus the pleasure of reading as an activity. Neither work will be reread as often as truly genial works are.\n\nSecondly, the denunciation of decadent modern style introduces Coleridge's major statement of the relation between truth and precision, or knowing mind and knowledgeable expression. The Italian side of a lost balance between the novel and the predictable is better than the fanatic side, because the Italians retain precise diction. They may be dull at times, but they are exact. In a manner characteristic of the _Biographia_ , this fullest statement of a crucial unproven point is a quotation \u2013 acknowledged properly, but not translated. The passage defending the Italians' style summarizes and reemphasizes the links among imagination, truth and pure diction that first appear in chapters I to IV.\n\nNay, even of those who have most rescued themselves from this contagion [by the style of magazines and harangues], I should plead inwardly guilty to the charge of duplicity or cowardice if I withheld my conviction that few have guarded the purity of their native tongue with that jealous care which the sublime Dante, in his tract 'De la nobile volgare eloquenza,' declares to be the first duty of a poet. For language is the armoury of the human mind; and at once contains the trophies of its past, and the weapons of its future conquests. 'See how easily men fall from the wrong use of words into errors about things themselves!' [Hobbes]. There are certainly plenty of things in this short life and dark world which are worth time to study, so that we need not spend time in trying to understand confused and many-sided discussions. Alas, cloudy words are so many failures, they say so much that they say nothing \u2013 clouds, rather, from which hurricanes burst, both in church and state! What Plato has said in the Gorgias is indeed true: \"Anyone who knows words will know things too\"; and as Epictetus says, \"the study of words is the beginning of education\"; and Galen wrote most wisely, \"Confusion in our knowledge of words makes confusion in our knowledge of things.\" J. C. Scaliger has indeed said excellently, in book I of his _Plants_ : \"A wise man's first duty (he says) is to think well so that he can live for himself; the next is to speak well so that he can live for his country.\"' (II, 22)\n\nThe references to church and state, to public duties, and to hurricanes focus the basic threat posed by anonymous critics: they corrupt the language. Bad judgment follows from bad writing, and bad writing from bad thinking; the popularity of their bad judgments impedes the healing and proper influence of genius \u2013 as both the speaker's and Wordsworth's careers demonstrate. Only the precise use of words will lead us to the truth about 'things themselves'. The original realism and the nature of language guarantee that any method which leads to the astute and exact knowledge of the physical world will simultaneously lead to knowledge of the spirit, because the transcendental analysis of consciousness shows that these proceed apace.\n\nChapter XVI ends with a transparent reference to Wordsworth: 'A lasting and enviable reputation awaits that man of genius, who should attempt and realize a union' (II, 24). One who achieves the syntheses characteristic of the ideal poet will be hailed as a genius \u2013 or ought to be. When Coleridge describes Wordsworth's poetry as achieving these qualities, we are to see Wordsworth as wrongly deprived of the laurels that are his. As the portrait of genial Wordsworth gains detail, so does the condemnation of his critics (chapter XXI), and the speaker's (chapter XXIV). We are left with the idea of a choice between genial imaginative Christianity and fanatical materialist atheism, a choice that we are to see as influencing every human activity and every aspect of our own experience. Wordsworth, and to a lesser extent the speaker himself, are to stand as symbols for the great traditional values both in literature and in life.\n\n# **7 Wordsworth and Poetic Diction**\n\n> Poets (especially if philosophizers too) are apt to represent the effect made on themselves as general \u2013 the Geese of Phoebus are all Swans, & Wordsworth's Shepherds & Estatesmen all Wordsworths, even (as in old Michael) in the _un_ poetic traits of character. Whether mountains have any particular effect on the native inhabitants, by virtue of being mountains exclusively, & what that effect is, would be a difficult problem.... \u2013 But this subject I have discussed, & (if I do not flatter myself) satisfactorily in the Literary Life, & I will not conceal from _you_ , that this inferred dependency of the human soul on accidents of Birthplace & Abode together with the vague misty, rather than mystic, Confusion of God with the World & the accompanying Nature-worship, of which the asserted dependence form a part, is the Trait in Wordsworth's poetic Works that I most dislike, as unhealthful, & denounce as contagious....\n> \n> Letter to Thomas Allsop, 8 August 1820 ( _Collected Letters_ , V, 94\u20135)\n\nIn his arguments about diction, Coleridge builds upon the classical rhetoricians' belief that language can both express and evoke emotion. According to Coleridge, however, poetic language expresses and evokes not feelings but imaginative activity itself. This dual capacity arises from and requires the essential interdependence of imagery, metre, subject-matter and thought. As Abrams has so lucidly explained, Coleridge's subsequent influence on literary criticism has followed in large measure from his successfully accommodating rhetorical tools of analysis with a philosophically grounded idea of the creative imagination. In chapters XVII and XVIII, Coleridge explains at length how effective poetic diction simultaneously derives from and arouses imagination, an explanation that proceeds step by step with his critical commentary on Wordsworth's Preface to _Lyrical Ballads_.\n\nTo meet the needs of his own exposition, Coleridge 'adapts' the Preface, just as he adapts the works and the theories of Kant, Aristotle and Schelling. But in each case, and most clearly with Wordsworth, the adaptation is probably also intended to illuminate more clearly a fundamental truth \u2013 or a fundamental error \u2013 that the author himself represents only partially. Coleridge follows a principle of interpretation that he enunciates most clearly in the _Philosophical Lectures_ : 'the individual is... subordinated to the history of philosophy' ( _PhL_ , 205). In Wordsworth's case, the fundamental error is trying to describe imaginative works in associationist terms that ultimately attribute to the landscape a spirit and moral purpose that belong only to God and to man. According to Coleridge, Wordsworth's befuddled theory and its consequences combine the worst errors of both English mechanism and Continental pantheism, because he does not sufficiently emphasize the mind's priority over matter. In reading the Preface and the poems, Coleridge seizes the points where Wordsworth does not assert this logical and moral priority so as to develop at length the aesthetic inadequacies of an associationist theory of poetry and poetic diction.\n\nColeridge's criticism of Wordsworth's theory depends in part on his conflating Preface and Appendix, and in part on his ignoring how the Appendix qualifies and clarifies several major ideas. Wordsworth has been ably defended on several fronts; only a few points need to be repeated here. The most important of these concerns what Wordsworth means by contrasting the 'real language of men' with 'poetic diction'. In the opening paragraphs of the Appendix, Wordsworth contrasts 'real language' with the 'distorted language' known as 'poetic diction' \u2013 the special stock of words and images regarded as poetic because they are remote from ordinary speech and utilitarian prose. This poetic diction, Wordsworth explains, is characterized by 'various degrees of wanton deviation from good sense and nature', whereas real language maintains a 'natural connection' with 'feelings and thoughts'. The real language of men has two subclasses: 'ordinary language' and the 'genuine language of passion', which is 'daring, and figurative'. So-called poetic diction falsely imitates 'the original figurative language of passion' through the 'mechanical adoption of these figures of speech... applied... to feelings and thoughts with which they ha[ve] no natural connection whatsoever'. In time, 'by the influence of books upon men', the distortions and inanities of poetic diction become to some extent the conventional diction of all writers of verse. In proposing 'to adopt the very language of men', consequently, Wordsworth asserts his fidelity to 'good sense and nature', and to the 'daring, and figurative' language of genuine passion.\n\nHe is not proposing to wander about with a notebook, as Coleridge suggests; but he is judging poetic diction primarily by the quality of the words' relations to realities they name. Wordsworth's account of poetic diction probably reflects Hartley, Don Bialostosky explains, for whom a 'real language' is one in which words refer to sensations and feelings rather than to other words, as in a 'nominal language'. To Coleridge, this entire view of language is profoundly mistaken: words name the complex activity of the mind in knowing and reacting, not just the things known or the consequent emotions. Coleridge ignores the qualifying contexts of Wordsworth's statements about 'the real language of men' so as to focus more sharply on this mistakenly empiricist idea of language.\n\nSimilar motives underlie Coleridge's reading of the statement, ' _There neither is or can be any essential difference between the language of prose and metrical composition_ ' (II, 45). Wordsworth makes this equation to deduce the applicability of 'good sense and natural feeling' as criteria for poetic imagery. If these criteria apply, then the distortions of poetic diction lose validity. Furthermore, the prose works he has in mind are not scientific treatises, but 'works of _imagination and sentiment_ '. Coleridge ignores Wordsworth's context so as to make this statement into an appropriate opposition for his own argument that metre, imagery and passion are essentially interdependent. Once again, however, a genuine disagreement underlies Coleridge's questionable reading. As I will explain in more detail later, Wordsworth attributes a balancing power to metre that Coleridge attributes to imagination itself.\n\nColeridge seldom demonstrates much scholarly responsibility in his use of others' texts, and his handling of the Preface is no exception to this fault. But in this case, at least, some of his motives remain quite near to the surface, especially if one compares the 1800 and 1802 editions of the Preface. In 1802 and subsequent editions, Wordsworth's expanded explanation of 'real language' discredits Coleridge's account of an unduly literal observation and selection, but Wordsworth does insist that the poet cannot hope to replicate the 'liveliness and truth' of the language of actually impassioned men. Nature remains greater than art. Coleridge disagrees: 'In poetry, in which every line, every phrase, may pass the ordeal of deliberation and deliberate choice, it is possible, and barely possible, to attain that ultimatum which I have ventured to propose as the infallible test of a blameless style; its _untranslatableness_ in words of the same language without injury to the meaning' (II, 115). Deliberation and revision, not immediate passion, underlie the most lively, exact and potent expression. As Wimsatt and Brooks explain, reformers of poetic diction can usually be classified in two ways: those who appeal to 'the directly passionate, the naturally spoken word', and those who appeal to the 'educated spoken word', as models of clarity and vitality. This division is, perhaps, the poetic equivalent of the division into Aristotelians and Platonists; Wordsworth is in one group, and Coleridge in the other. Each argues that meditation and observation must be finely balanced, but each portrays the balance from the perspectives supplied by his own experience and character.\n\nThe design of chapters XVII and XVIII is at once pellucid and extraordinarily difficult to describe. As I said before, Coleridge argues the necessary interdependence of subject-matter, metre, imagery and thought in producing the highest aesthetic effects. But this whole argument is closely interwoven with his evaluation of Wordsworth, and its central point unfolds as slowly as a flower's opening. Because all of Coleridge's transitions orient us to Wordsworth, the integrity and coherence of his counter-argument appear only retrospectively; yet this counter-argument supplies the single perspective that unifies Coleridge's analysis of Wordsworth's theories and his poems. As a result, the narrative account of Wordsworth and the philosophical account of poetry and imagination are so closely interwoven that neither makes much sense without the other.\n\nLet me sketch this main point as briefly and plainly as I can. The interdependence of parts, or the 'organic' unity of a poem, derives from the poet's imagination. Poetic skill depends on contemplation, on imaginative self-consciousness, not on observation of things or imitations of traditions, because it is a verbal art. Words exist within the relation between mind and all that it knows; words embody consciousness itself. It follows that one who is most fully self-conscious will use words most precisely, and thereby reveal most reliably both the exquisite physical sensibility and the moral insight that comprise imaginative genius. Words can carry such a burden only when organized into metre (which embodies, evokes and sustains passion) and images (which characterize realities _as known by_ a human consciousness). The reality that poems 'imitate', then, is not the objective world as such but, rather, the consciousness of the poet himself _in his encounters with_ the objective world. Poets must rely not on primary imagination's rendering of the objective physical world, but on secondary imagination's rendering of man's intimate relation to that world. Only through the secondary power of 'wedding Nature' do we find in poems pine-trees like women with their backs to the wind. Only thereby do we find in poems not a world of fixed and morally remote objects but, rather, a world animated by value and meaning. It follows from this that the poet's only genuine subject-matter is himself, and the only ideas he presents will be ideas about the activity of consciousness in the world around it. Wordsworth's theories err and his poems fail when he attributes to objects \u2013 to metres, to landscapes, to garrulous seamen \u2013 imaginative qualities and aesthetic powers that properly belong only to himself. This is truly a fault of which none but the genius is capable, and so Coleridge's criticisms are a backhanded praise slowly verging into full celebration of the great philosophic poet.\n\nColeridge's adaptation of Wordsworth's theory begins with the speaker's opening summary of their agreements. The speaker praises Wordsworth's call for 'a reformation in our poetic diction', but without acknowledging Wordsworth's predominant concern with how convention and tradition can shape the reader's expectations and thus his judgment (II, 28). Wordsworth's interest in what he calls the 'contract' between reader and poet locates the question about diction in pragmatic contexts sometimes remote from Coleridge's philosophical and theoretical concerns. The speaker's opening criticism of Wordsworth's rule that proper diction for poetry 'consists altogether in a language taken, with due exceptions, from the mouths of men in real life' also fails to supply the context that makes the rule intelligible (II, 29).\n\nThese deficiencies permit Coleridge to open his own account of poetic diction by shifting Wordsworth's issue from qualities of imagery to qualities of style generally: the less declamatory kinds of poetry will use sentence structures and vocabulary not profoundly different from those found in ordinary prose or in educated speech:\n\nMy objection is, first, that in _any_ sense this rule is applicable only to _certain_ classes of poetry; secondly, that even to these classes it is not applicable, except in such a sense, as hath never by any one (as far as I know or have read) been denied or doubted; and lastly, that as far as, and in that degree in which it is _practicable_ , yet as a _rule_ it is useless, if not injurious, and therefore either need not, or ought not to be practised. (II, 29\u201330)\n\nColeridge was certainly familiar with the criticism that both denied and doubted the propriety of the less stylized diction Wordsworth advocates, and his readers would have been just as familiar. The speaker's paradoxical statement directs emphasis to what will prove to be Coleridge's major point. If by imitating the real language of men Wordsworth means no more than following the conventions of pure, distinct English, why is the rule 'as a rule... injurious'?\n\nAny rule that directs the poet toward observation at the expense of meditation threatens or weakens imaginative self-consciousness. The poet does not imitate; he contemplates himself observing. Even grammar is but 'the laws of universal logic, applied to psychological materials' (II, 38). The _Philosophical Lectures_ provide a useful gloss here: the wise man realizes, Coleridge explains, that both observation and meditation are crucial. But which is primary? For Coleridge, as we have seen, meditation is primary. The hazard posed by the Preface is in its according primacy to observation \u2013 whether of real men or of beautiful landscapes.\n\nThe speaker's initial evaluation of rustic diction turns in part on a demographic quibble: those Wordsworth imitates \u2013 shepherds like Michael \u2013 are not ' _low_ and rustic'. Wordsworth mistakenly attributes to rural residence what he ought to attribute to the economic status and educational level of the self-employed working class. The speaker's argument seems to suggest that Coleridge would have been satisfied if Wordsworth had imitated educated and prosperous rustics because they are usually free from the affectations and ambitions of the rising urban middle class. But the speaker's further critique of rural life itself begins to reveal the basic issue: Wordsworth confuses the essentially rural character with the essentially self-conscious character. Setting is ultimately irrelevant to the self-conscious mind: Wordsworth's idea of low and rustic life attributes to environment a power it does not have.\n\nThe speaker's defense of his position unfolds the aesthetic principles that ought to guide the poet's choice of characters. Properly imagined characters must synthesize an idea with its image in the natural world:\n\nI adopt with full faith the principle of Aristotle, that poetry as poetry is essentially ideal... the _persons_ of poetry must be clothed with _generic_ attributes, with the _common_ attributes of the class: not with such as one gifted individual might possibly possess, but such as from his situation it is most _probable_ before-hand that he _would_ possess. (II, 33\u20134)\n\nWordsworth's emphasis on the rural particularity of his essentially self-conscious characters is inappropriate because these particulars are not _necessarily_ related to the universal he is portraying. A long footnote describes this error as both an aesthetic flaw and a failure to meet art's ultimate end:\n\nOne of the essential properties of Geometry is not less essential to dramatic excellence; and Aristotle has accordingly required of the poet an involution of the universal in the individual. The chief differences are, that in Geometry it is the universal truth, which is uppermost in the consciousness; in poetry the individual form, in which the truth is clothed. (II, 33 n)\n\nThe poem's involution of ideal and individual transforms the reader's consciousness in the same way that the unity and identity of parts does. The symbolic union of particular and universal leads to the 'temporary oblivion' of the literal self because it is that change in the form (i.e. the activity) of the reader's consciousness called 'aesthetic pleasure'. Coleridge's argument shifts, from a critique of 'low and rustic' characters to a critique of any character particularized by what is not essential. Only through an essential and necessary relation between particular and universal does a character achieve the symbolic potence and moral value expected in literature. Only through such characters can a poet 'transport the mind to a sense of its possible greatness... suspending our individual recollections and lulling them to sleep amid the music of nobler thoughts' (II, 33 n).\n\nThe speaker's analysis of Wordsworth's poems about rustics suggests that the essentially self-conscious character is the poet himself. Wordsworth's willingness to de-emphasize the observing eye \u2013 to look steadily at the object _qua_ object \u2013 leads too often to a mere objectivity in which the reader encounters nothing he might not have seen for himself. The reader's vision is neither clarified nor purified by such poems, which fail to achieve the synthesis that first drew Coleridge to Wordsworth's verse (see I, 59\u201360). In the quoted passage from 'Michael' we watch Wordsworth describing, judging and explaining. We observe his observing. But in 'The Thorn' we see _what_ the poet observes, not _how_. Garrulous old seamen and their tales can be charming, if one is in the proper frame of mind to be charmed \u2013 but the charm is absent from the poem itself because the observing and transforming mind is relatively absent. Wordsworth in effect demands that his readers possess imaginative powers equal to his own, as if he need only supply such materials as proved congenial to his own genius. In Coleridge's eyes, such methods reveal a failure to understand the origins of aesthetic pleasure. In The Thorn' 'the parts... which might as well or still better have proceeded from the poet's own imagination, and have been spoken in his own character, are those which have given, and which will continue to give, universal delight' (II, 36). In 'The Thorn', and even more clearly in 'The Mad Mother', Wordsworth's excessively objective portraits have limited aesthetic power because the muting of imagination allows the reader's fancy to have its own way, to supply associations that do not belong to the created whole (II, 35).\n\nWordsworth's defense of rustic language _per se_ , the speaker suggests, reflects the same mistaken and excessive objectivity as his choice of rustic characters. The 'best part of language' derives not from the best objects ('the beautiful and permanent forms of nature') but from the most potent imaginations. The best part of language is relational and philosophic; it distinctly refers to the activities of the mind, not to the material world _per se_ :\n\n... the educated man chiefly seeks to discover and express those _connections_ of things, or those relative _bearings_ of fact to fact, from which some more or less general law is deducible. For _facts_ are valuable to a wise man, chiefly as they lead to the discovery of the indwelling _law_ , which is the true _being_ of things, the sole solution of their modes of existence, and in the knowledge of which consists our dignity and our power.... The best part of human language, properly so called, is derived from reflection on the acts of the mind itself. It is formed by a voluntary appropriation of fixed symbols to internal acts, to processes and results of imagination, the greater part of which have no place in the consciousness of uneducated man.... (II, 39\u201340)\n\nThe 'internal acts' that are the 'processes and results of imagination' are, of course, perception and self-consciousness. Because a word refers simultaneously to the consciousness of its speaker and to the external world, language is ideally suited for the discovery and communication of relations \u2013 and hence principles (cf. I, 124\u20135, 146\u20137). In a poet's hands, language can generate the identity and unity of pleasure and truth, particular and universal, part and whole that underlie aesthetic pleasure, because the dual reference of words (a duality most evident in a good image) reveals self-consciousness directly. The usage Wordsworth advocates, the speaker explains, is not essentially rustic but essentially imaginative, and therefore the possession of no one social or economic class. The diction ideally suited for poetry is the poet's own.\n\nIt is worth pausing here to recall I. A. Richards's often-criticized argument that the difference between imaginative and fanciful verse derives from the number and quality of relations among 'parts or units of meaning'. Grant the importance of _relation_ in Coleridge's thought and in his idea of imaginative aesthetic response, and Richards's argument does not necessarily reduce the difference between fancy and imagination to a difference merely of degree. The superior intimacy of relation of imaginative verse, as Richards so brilliantly describes it, can only derive from the fact that each 'unit' relates to the others not as object to object, but in and through the ways in which each 'unit' relates to the expressive human consciousness. The more fully the self-conscious imagination synthesizes the moral spirit with the world it knows, the more coherently and beautifully will that world be represented in words. The quality of relation that the best part of language expresses underlies both poetry's aesthetic power and its moral value.\n\nChapter XVII's concluding remarks on 'in a state of excitement' draws a necessary distinction. Not any language used by the imaginative poet is thereby a poem. However deep Coleridge's interest in 'expressive' theories of poetry, his original realism in its literary bearings locates aesthetic value in textual features. These features are defined in chapter XVIII, but first, here in the concluding lines of chapter XVII, the speaker establishes the necessary link between textual features and the poet's consciousness. Passion, he argues, translates ability into performance, arousing the fullest possible exertion of faculties subordinated to each other:\n\nFor the nature of a man's words, where he is strongly affected by joy, grief, or anger, must necessarily depend on the number and quality of the general truths, conceptions and images, and of the words expressing them, with which his mind had been previously stored. For the property of passion is not to _create_ ; but to set in increased activity. (II, 42)\n\nBut neither is any impassioned speech thereby poetry. If it were, of course, then Wordsworth would be right: it is probable that actually passionate men would use a more vivid and poetic language than poets writing silently at their desks. chapter XVIII, on metre, takes up this problem.\n\nThe transition from chapter XVII to chapter XVIII, over which I suspect most readers barely pause, deserves a moment of consideration. The transition is designed to be hurried past: chapter XVIII begins 'I conclude' and its opening paragraph closes the analysis of Wordsworth's idea of rustic language by arguing that he who can select and purify must possess principles that obviate the need to imitate at all. This style of connecting suggests that we are to see the chapters as distinct, but hardly separate. Why? Why not assemble chapters XVII and XVIII into one whole the size of chapter X or XII? The symmetries thereby established would have been elegant indeed, and utterly Coleridgean in their complexity.\n\nBut other Coleridgean subtleties are at work here. Coleridge often closes off one chapter or essay or section at the point where an underlying problem impedes progress, so that the next part can address that problem directly. One not alert to the growing impediment may feel that Coleridge has veered off into a tangent, when in fact he has delved under his main argument. These subterranean pursuits are most evident in the shifts into chapter II, on the irritability of men of genius, or chapter VIII, on idealism and materialism, or chapter IX, on mystics and German philosophers. Coleridge's habits in this regard suggest that the prudent reader must reflect carefully on chapters that end at unexpected points, and on chapters that engage unexpected issues. But the question remains \u2013 why the failure to orient us with conventional transitions? Why not tell us where he is going, and why? Coleridgean carelessness again? More inconsiderate haste? In part, perhaps; but in part also for reasons near to the heart of Coleridge's thought.\n\nThese impeding problems are commonly metaphysical ones and, as such, usually ask questions for which, Coleridge argued, no irrefutable logical answer is possible. To write a highly visible, sharply focussed transition, he would have to engage these difficult technical problems in all their original complexity. But on that technical level his own solutions are themselves lengthy and elaborate. He chooses instead \u2013 and most prudently \u2013 to engage only the practical consequences of the metaphysical problem, and furthermore to engage only those consequences immediately crucial for the problem at hand. (Given the sophistication of Coleridge's thinking, that sometimes excludes only a little of the technicalities; but in general I have observed notable restraint at various points throughout his prose works.) Readers are apt to lose themselves none the less because Coleridge's prudent decisions require transitions that are exquisitely sensitive to the needs and perceptions of the audience. Coleridge seldom demonstrates such tact. The reader perplexed by chapter divisions often has few options but rereading, searching diligently for the relations between one chapter and the next. It is an annoying task, because Coleridge has bungled a rudimentary aspect of his craft; but one who persists will always be rewarded.\n\nThe unformulated technical problem here, as I have suggested, concerns the relation between passion and language, or the distinction between the genial poet's ordinary use of language and his poetry. One might argue that any impassioned statements by an imaginative person ought to be imaginative, and thereby to possess interior relations that stir the attentive reader or auditor to imaginative aesthetic response. Common sense objects that the notion is nonsense, but chapter XVII does not preclude the error. So far as I know, chapter XVIII's definition of the relation proceeds in the only style Coleridge ever uses on the problem: practical discussions of poems and of reading. These discussions appeal to the ear and to the mind of an experienced literary critic. They argue not about linguistics, but about prosody and figures.\n\nBefore considering this argument, let me summarize the position Coleridge has advanced in opposition to Wordsworth's. A poem's best speaker or its best character must be ideal, or a fully representative instance of the universal that the poem reveals. The best language for a poem is relational or philosophic. It follows from all this that the best model for the poet, or the best diction for poetry, is the language of the impassioned philosopher \u2013 the poet himself: 'No man was ever yet a great poet, without being at the same time a profound _philosopher_ ' (II, 19). This idea underlies Coleridge's disapproval of Wordsworth's dramatic poems (II, 36, 53, 55, 109), and his suggestion that Wordsworth's theory is not what it appears to be (II, 29, 69, 77, 95).\n\nChapter XVIII begins and ends by asserting that the interrelation of parts in poetry depends upon imagination. The genuinely educated (i.e. imaginative) man can 'subordinate and arrange the different parts according to their relative importance, [so] as to convey [his meaning] at once and as an organized whole' (II, 44), whereas in 'pseudo-poesy' one finds only 'compulsory juxtaposition' (II, 65, 68). The highest degree of this accommodating of parts to the whole, we were told in chapter XIV, requires metre (II, 11). In chapter XVIII, Coleridge argues that metre is essential (II, 49\u201357), that the quality of metre and the quality of diction are interdependent (II, 57\u201363), and that both depend on the potence and facility of the poet's imagination. Accommodating Wordsworth to this argument requires interpreting his 'essential difference' as a matter of style \u2013 what Coleridge calls 'the formal construction, or architecture, of the words and phrases' (II, 48). Wordsworth's account of the relation between prose and poetry is not entirely clear, I grant; but Coleridge's interpretation of his meaning seems beyond defense. Coleridge's painstaking analysis of the word 'essential' serves primarily to rule out Wordsworth's most probable meaning: the neoclassic (or pseudoneoclassic) stock of images and figures, now reduced to blurred clich\u00e9s, are not absolutely requisite for poetry. Coleridge's motive is again out of sight, but not entirely out of reach. The error signaled by the equation, although not embedded in it, is the idea that expressive authenticity is an adequate measure of poetic diction if the expression reflects a pure and proper relation to the landscape. But the poet must look steadily at both his subject and his audience. Wordsworth's ultimate error, in Coleridge's eyes, is his somewhat na\u00efve faith that the unbiased reader will respond to the poet's 'subject' in the ways that the poet did.\n\nChapter XVIII is quintessentially Coleridgean in the style of its argument. Coleridge's theory of poetic diction, like his theory of mind, depends on intuition. Such a theory can be both accurate and rigorously conceived because all knowledge ultimately depends upon 'the IMMEDIATE, which dwells in every man, and on the original intuition or absolute affirmation of it' (I, 168). The critic, like the transcendental philosopher, cannot achieve 'objective' verification because the phenomenon he studies is not an object, but a self-conscious experience. The quality of a poem does not reside in the text as a thing in itself, but in the poem as read, in the poem's power to evoke a particular kind of response from the sensitive and experienced reader called the 'philosophic critic'. And yet the 'original realism' guarantees that the text as read and the text as a thing in itself are the same, so that it is possible to define and to argue about how specific features of the text _per se_ affect the reader's response. The experience of reading is the foundation of all critical analysis. In many ways, the whole logical structure of chapter XVIII is designed to focus our detailed and self-conscious attention on the fact that reading poetry _feels_ differently from reading prose. When this feeling is not evoked by a versified text, we are disappointed, we feel cheated of a pleasure that is our due \u2013 even if the bad lines offer interesting ideas or true statements.\n\nThe arguments on the origin and effects of metre define metre's significance both for the poet and for the reader. For the poet, metre expresses the balance of passion with order: the imagination spontaneously seeks to reassert the symmetry of passion and thought that an excess of passion threatens. This high-energy synthesis can be deliberately maintained, and its expression will be 'organized into _metre_... by a supervening act of the will and judgement, consciously and for the foreseen purpose of pleasure' (II, 50). The psychic and linguistic processes that require metre require imagery as well:\n\nthis union [of ' _spontaneous_ impulse and of _voluntary_ purpose'] can be manifested only in a frequency of forms and figures of speech (originally the offspring of passion, but now the adopted children of power) greater than would be desired or endured, where the emotion is not voluntarily encouraged and kept up for the sake of that pleasure, which such emotion, so tempered and mastered by the will, is found capable of communicating. (II, 50)\n\nThe poet's fancy and his imagination must co-operate both in the deliberate sustaining of the imagination's synthesis, and in the deliberately metrical and figurative expression of this synthesis. Voluntary and spontaneous powers of mind must interpenetrate.\n\nThe decision to write in metre is, after all, a deliberate choice. Metre _per se_ , the assembling of iambic pentameter lines, is a mechanical (albeit skilled) task. The poem will succeed, however, only to the extent that the mechanical regularity of metre can be transformed into rhythm \u2013 only to the extent that metrical regularity is enlightened by variations and substitutions that reveal the passion in all its vitality. The poem's success also depends on the propriety of the forms and figures the poet chooses: it is 'the prerogative of poetic genius to distinguish by parental instinct its proper offspring from the changelings, which the gnomes of vanity or the fairies of fashion may have laid in its cradle' (II, 64\u20135). When imagination fails to vitalize the composing process itself, the poem's images degenerate into 'mere creatures of an arbitrary purpose, cold technical artifices of ornament or connection' (II, 64). In short, the cooperative agency of fancy and imagination in poetic diction ultimately depends upon the continued governing power of imagination itself.\n\nMetre not only expresses the poet's sustained, modulated passion, but also arouses the reader's faculties. For the reader, metre 'tends to increase the vivacity and susceptibility both of the general feelings and of the attention'; it functions 'as a medicated atmosphere, or as wine during animated conversation' (II, 51). The speaker offers no proof of this point, just as he presumes the equally traditional link between the poet's passion and metrical expression. And once again he uses the connection as grounds for insisting on highly figurative diction:\n\nWhere, therefore, correspondent food and appropriate matter are not provided for the attention and feelings thus aroused, there must needs be a disappointment felt; like that of leaping in the dark from the last step of a stair-case, when we had prepared our muscles for a leap of three or four. (II, 51)\n\nThe intimate relations among the poet's passion, prosody and diction and the reader's pleasure supply the grounds for Coleridge's particular judgments about the diction of individual poems: undistinguished or prosaic diction and flaccid prosody will vary together (see II, 53\u20134).\n\nThe speaker pauses here to refute the Preface in more detail, because Coleridge's argument to this point resembles Wordsworth's in several regards. Both poets assert the commonplace connection between metre and pleasurable excitement in both poet and reader. But for Wordsworth metre modulates the poet's impassioned expression, elevating or damping its effects as necessary to maximize the reader's pleasure. For Wordsworth, metre and diction are complementary opposites; for Coleridge, they are the coinherent expressions of a prior balance within the poet's consciousness. Coleridge makes so much of this point because Wordsworth's error apparently underlies his inferior poems (II, 53). Metres do not modulate passion to create an integral aesthetic effect: the synthetic imagination does. The poems to which the speaker refers resemble each other principally in the disproportionate bulk of physical details that are not immediately requisite to the poet's impassioned response to the event depicted in the poem. Once again, Coleridge complains that Wordsworth attributes to things themselves \u2013 to metres _per se_ , or to the edematous ankles of old huntsmen \u2013 a power that resides not in the thing but in imagination.\n\n'Notwithstanding the beauties which are to be found in each of them where the poet interposes the music of his own thoughts', the speaker remarks, these poems would have been 'more delightful to me in prose... in a moral essay or pedestrian tour' (II, 53). Where Wordsworth's diction fails to provide appropriate matter for the reader's awakened feelings and attention, 'the metre itself must often become feeble' (II, 54). The speaker paradoxically attributes this excessive objectivity and prosodical flatness not to a lack of passion, as his theory would predict, but, rather, to 'the susceptibility of [Wordsworth's] own genius' (II, 54). As we shall see in much more detail later on, Coleridge deflects this potentially crucial failure into a criticism not of Wordsworth's impassioned imagination but, rather, of his fancy. Wordsworth must attend more carefully to the general state of association if he is to communicate his imaginative vision intelligibly (II, 105). When he fails to accommodate his own exquisite 'susceptibility' to the needs of blunter minds, his prosody and diction are apt not to communicate his passion. They are apt to be prosaic. And such prosaicism, like his theoretical errors, is the fault of none but a genius. Coleridge himself makes a characteristic mistake at this point: the paradox presented here gives rise to most of chapter XXII, but there is little that communicates this link to even the most sympathetic reader.\n\nThe speaker's third argument for metre defines the relation between passion and metre in more generally applicable terms. Lyrics are not the only poetic kind, after all; and even lyrics are not unrelentingly emotional. The speaker begins with references back to chapter XIV: metre is essential to poetry because it excites the distinct attention to parts that underlies aesthetic pleasure. But not all parts of a poem are essentially poetic \u2013 not all parts are the immediate expression of imaginative vision (II, 10\u201311, 55\u20136). There must be a mordaunt \u2013 a chemical fixative \u2013 to integrate these relatively unpoetic parts within the whole. This mordaunt is the poet's excitement in the act of composing. Such excitement is not an emotion _per se_ , but it is none the less a passion in the philosophic sense. As a passion, it, too, will require and involve metrical, figurative expression. The speaker's fourth argument restates this point more simply: poetry requires metre because metre underlies both the integral relation of distinguishable parts and the aesthetic pleasure of imaginative reading. The fifth argument appeals to literary tradition \u2013 a weighty argument for a conservative like Coleridge, but, as time has shown, his least reliable. Literary history no longer so unequivocally supports the necessity of metre.\n\nIn the last part of chapter XVIII the speaker apparently moves from abstract theorizing to practical examination of lines from Gray, Spenser and Daniel. But Coleridge is very seldom 'redundant' in this way. Although never losing hold of metre as the contested issue, Coleridge shifts his predominant argument to qualities of imagery and then, more generally, to diction in the comprehensive sense. He can shift emphasis gracefully because animated prosody and adeptly figurative expression are the coinherent manifestations of imaginatively sustained passion. From the proper perspective he seems not to shift at all, but to make the implicit explicit.\n\nWordsworth's use of Gray, and Coleridge's use of Wordsworth's use, can be read as subsuming many of the complex delicate shifts in literary sensibility in the late eighteenth century. Ransom has argued that Gray himself in this sonnet criticizes the inadequacies of the inherited 'poetic diction'. Coleridge uses the lines to present the guiding principles of his own idea of diction. First, the essentially poetic diction may have qualities in common with prose, but it will also possess qualities unique to itself and not suitable for prose (II, 57\u20138; see also II, 50). This argument allows the speaker to accede to Wordsworth's account of poetry's kinship with prose without 'agreeing' that they are identical twins.\n\nSecondly, the speaker asserts 'GOOD SENSE' \u2013 not equality with prose \u2013 as the only proper measure of poetic imagery. Gray's use of 'Phoebus' differs from Spenser's because Gray's line 'conveys incongruous images, [and] because it confounds the cause and the effect, the real _thing_ with the personified _representative_ of the thing' (II, 58). As the speaker has already argued at length, this pseudo-poetic modern imagery reflects both blunt sensibility and enfeebled imagination. One need not equate poetry and prose to demonstrate its fraudulence. As the contrasting lines from Spenser demonstrate, quality depends not on the image _per se_ , but on its place and function in the whole. Neither rustics' diction nor pseudo-classical diction can supply the lack of imagination.\n\nBut 'GOOD SENSE' alone is not a sufficient measure. Daniel's lines are full of good sense. Daniel's diction fails not because it is incoherent, but because it lacks the 'more frequent employment of picturesque and vivifying language' that ought to distinguish poetry from prose (II, 50). One can obliterate Daniel's metre by transcribing his verse without its proper lineation. Given Coleridge's idea of the essential interdependence of diction and metre, the failure of one testifies to the other's failure. Daniel's commonplace images \u2013 disorder as disease, society as a fabric \u2013 do not catalyze the reader's imagination into activity. There is here no 'union of deep feeling with profound thought', nor do the lines awaken any 'freshness of sensation' or 'new feeling' concerning English history (I, 59\u201360). We remain firmly planted on the Cis-Alpine side of consciousness, in the day-to-day world of inanimate objects. Yet notice that the speaker does not develop this indictment as fully as I have developed it. For Coleridge, enough has been said when he demonstrates (or invites the reader to discover) the limpness of Daniel's prosody. He relies on the reader's concurrent judgment, as he has done before when invoking the original realism. In his criticism, no less than in his metaphysics, Coleridge requires an extraordinarily active, thoughtful reader.\n\nHaving finished with Gray, Coleridge draws another statement from the Preface so as to argue that good sense and aesthetic potence apply to metre as fully as they apply to imagery. Wordsworth errs, the speaker contends, in claiming that 'the distinction of rhyme and metre is voluntary and uniform': prosody varies in quality just as imagery does. In neither case is the reader 'at the mercy of the poet', because the reader has \u2013 or ought to have \u2013 relevant criteria for judgment.\n\nAnd this point unfolds into chapter XVIII's concluding issue: literary value-judgments. The end of criticism, the speaker argues, is to establish the principles of writing, not the principles of judgment. And the first principle of writing is that the poet must be guided by a meditative and fully self-conscious imagination:\n\nBut if it be asked, by what principles the poet is to regulate his own style... I reply; by principles, the ignorance or neglect of which would convict him of being no _poet_ , but a silly or presumptuous usurper of the name! By the principles of grammar, logic, psychology!... by such a knowledge of the facts, material and spiritual, that most appertain to his art, as, if it have been governed and applied by _good sense_ , and rendered instinctive by habit, becomes the representative and reward of our past conscious reasonings, insights, and conclusions, and acquires the name of Taste[.]... by the power of imagination proceeding upon the _all in each_ of human nature[.] By _meditation_ , rather than _observation_ [.] And by the latter in consequence only of the former[.]... Could a rule be given from _without_ , poetry would cease to be poetry, and sink into a mechanical art. It would be [a fashioning] not [a creation]. The _rules_ of the IMAGINATION are themselves the very powers of growth and production. (II, 63\u20135; phrases in brackets translated from the Greek)\n\nThe speaker immediately insists that such criticism none the less supplies an adequate basis for value-judgments. Such a critic has no difficulty distinguishing Donne's apostrophes from Cowley's Pindaric Odes, or 'poetic fervor self-impassioned' from 'the startling _hysteric_ of weakness over-exerting itself' (II, 65\u20137).\n\nAdequate judgments flow from distinguishing the origins of such verse in the powers and quality of the poet's mind \u2013 provided, of course, that one has a fully developed theory of such origins, and their relations to aesthetic power and to literary form. (A theory, of course, that Wordsworth had no intention of elaborating in his Preface.) Pseudo-poesy derives neither from fancy nor from imagination, but from the mere wit (or degraded fancy). Its juxtapositions of incompatible things are not even unified by a coherent associational flow, much less by imagination's unifying vision. In asserting his grounds for value-judgment, Coleridge alludes again to the coherent development of the speaker's idea of literature, because the speaker in his youth had judged the merits of a poem by faculties to which it appealed (I, 14). Wordsworth's poems had appealed quite powerfully to the speaker's imagination, yet Wordsworth's theory (as interpreted here) neatly reverses almost every detail of an imaginative theory of composition. 'There is not', the speaker none the less affirms, '... a man now living, who has, from his own inward experience, a clearer intuition, than Mr. Wordsworth himself... [of] the true sources of _genial_ discrimination' (II, 64).\n\n# **8 Wordsworth and the Imaginative Particular**\n\n>... a man's principles, on which he grounds his Hope and his Faith, are the life of his life. We live by Faith, says the philosophic Apostle; and faith without principles is but a flattering phrase for wilful positiveness, or fanatical bodily sensation. Well, and of good right therefore, do we maintain with more zeal, than we should defend body or estate, a deep and inward conviction, which is as the moon to us; and like the moon with all its massy shadows and deceptive gleams, it yet lights us on our way, poor travellers as we are, and benighted pilgrims.\n> \n> _The Friend_ , I, 97\n\nHaving argued against Wordsworth's views on rustic language and poetic diction, the speaker claims that Wordsworth does not in fact believe the theory that the speaker has been systematically demolishing for forty pages. Fortunately so: by definition the self-conscious genius knows what he is about. Despite his own 'inward experience', Wordsworth 'suffered himself to express, in terms at once too large and too exclusive, his predilection for a style the most remote possible from the false and showy splendor which he wished to explode' (II, 64, 70). Chapter XIX defines the neutral style Wordsworth 'meant' to defend, and illustrates it with excerpts from Chaucer and Herbert. Yet even this is not what Wordsworth _really_ meant, because the style of his own poems is both specifically poetic and highly individual. Chapter XX illustrates this imaginative individuality. Chapter XXI analyzes the critical methods of those who have condemned Wordsworth; they have misunderstood Wordsworth's individuality, and mismanaged their own. Chapter XXII provides what such critics do not: a 'fair and philosophical' examination of Wordsworth's poems on their own merits.\n\nOnce again the analysis of Wordsworth also serves as an exogenous skeleton for Coleridge's arguments about other things. Chapters XX and XXI assert the need for particularity in poetry and in criticism. Chapter XXI explains the quality of particular which is essential to philosophic criticism. Chapter XXII does the same for poetry: Wordsworth's defects illustrate the proper role of fancy in providing appropriate particulars; his beauties define the synthesis of universal and particular that imagination and fancy together achieve. Through self-conscious imagination, the poet knows the interpenetration of particular and universal. The symbolic character of language expresses this interpenetration; the associative power of fancy renders the expression effective. It follows that one can work backward from objective features of texts to their origins in the relative activity of fancy and imagination. Chapter XXII thus complements chapters VII and XII, demonstrating the concrete manifestations of powers defined abstractly. The concluding chapter not only laments the disproportion between public abuse and actual deserving that has been the fate of both men; it also draws us back to the foundation of 'original realism': the unity of being and knowing in imaginative self-consciousness.\n\nThroughout these chapters Coleridge explains how imagination governs fancy. This is a major question, as Lowes demonstrates in _The Road to Xanadu_. Lowes's understatement has lost none of its force in the half-century and more since he wrote: 'imagination must have materials on which to work'. And, as both his book and Baker's _The Sacred River_ fully demonstrate, association supplies imagination with the material it needs. Yet not all materials are suitable: the thoroughness of an energetic fancy virtually ensures that much of what wells up will be inferior, or at least unworkable. The sources Lowes cites are remarkable more for their dross than for their gold.\n\nFancy's role may be compared to an architect's shipments of bricks and beams. Everything depends upon what he does with his materials, but without them he can do nothing at all. He cannot embody his prior vision. The architect's creative process stretches out over time what probably happens almost instantaneously for a poet. Fancy is a mode of memory emancipated from time and space; its associative power collects from the artist's past those words and images and rhythms generally suitable to his present purpose. Yet these remain disparate heaps of things until imagination begins to work with and within what it has 'sent' fancy to gather. In imagination's final product, the diverse materials are fragments no longer, but parts of a whole which places each within a network of relations. These relations are so many and so intimate that each part is rendered integral both to the other parts and to the whole as such. Or, to put this matter another way, fancy's function is merely instrumental, but it is none the less a crucial instrument.\n\nColeridge's analysis of Wordsworth's faults proceeds apace with his explanation of the principles by which one selects from the plethora fancy provides. In the most general of terms, Coleridge praises Wordsworth's vivid particularity and his highly individual voice, but criticizes the welter of meaningless detail and foggy moralizing that intrudes when Wordsworth loses control over his vision and his voice. These judgments strike me as apt, although in my judgment Coleridge exaggerates Wordsworth's faults to facilitate his own account of fancy. His acknowledging how seldom Wordsworth errs does not adequately counter-balance the detail to which these faults are examined, nor the tones in which they are condemned. Coleridge's tone and manner, especially concerning Wordsworth's apparently pantheist empiricism, perhaps make more sense if one keeps the 1805 _Prelude_ in mind, and the still-vital plans for the great philosophic poem. These chapters are probably relevant to much more than the poems they particularly examine. Wordsworth's faults are explicated first because fancy's role is clearest when most inappropriate. When genially empowered judgment properly manages fancy, its contribution is dissolved, diffused, transmogrified. '\"As we our food into our nature change\"' (II, 12). Celebrating Wordsworth's beauties last allows Coleridge to end his critical essay both with resounding praise for the philosophic poet, and with his last, most resonant account of imagination's power.\n\nColeridge's account of Wordsworth's genial powers is matched by renewed emphasis on the speaker's own philosophic genius. Chapter XIX's references to German literature and to the history of English pronunciation enforce again the speaker's scholarship; the passage from Chaucer, like the earlier one from Spenser (II, 59), pinpoints the issue in question most precisely. By its brevity and simplicity, the chapter qualifies as what Coleridge elsewhere calls a 'landing-place': a pause in a sustained and difficult argument, often an illustration of a practical or amusing aspect of the matter at hand. Herbert's poems illustrate both the neutral style, and the tendency for poets of his day to convey 'in the most fantastic language... the most trivial thoughts' (II, 73) or to sacrifice 'the passion and passionate flow of poetry, to the subtleties of intellect, and to the starts of wit' (I, 15). The chapter offers two short 'burlesque' passages from Drayton and from Herbert, and then three reasonably characteristic seventeenth-century religious poems. The first two of these poems represent disorder of intellect; the last two, moral disorder; the central one, the relation between physical and moral loss. Yet the poems do more than illustrate the range and power of the neutral style. On the one hand, it is clear that the three religious poems display the translucency of the spiritual within the natural. 'The Bosom Sin' reappears in _Aids to Reflection_ as an illustrative commentary on the reluctance of many to know themselves \u2013 a reluctance evident in anonymous critics ( _AR_ , 75 n-76). But, on the other hand, this whole set of poems seems inescapably comic. The 'ludicrous tone of feeling' remains. What is going on here?\n\nThe speaker playfully and indirectly suggests an analogy between his own 'obscurity' and that of metaphysical poets. Drayton's reader wonders why he must 'wrest invention'; Coleridge has abundantly explained the arresting new vision that the genius supplies. The poet's or philosopher's 'lunacy' is literal only for the unperceptive. Both 'unravel' ordinary perceptions in order to reweave, restoring our consciousness of the transcendental in part by heightening our sensitivity to words. For both, clarity and unity depend upon the reader's ability to discern the relation between verbal play and substantial idea. The speaker laughs at his reputation, and recognizes its necessity: the neutral style includes both Herbert's poems and his own prose. The earlier bitterness concerning his reputation is softened here, and the speaker as a distinct personality appears more clearly at this point than he has for several chapters. Yet Coleridge's failure to close chapter XIX leaves us up in the air. Chapter XX \u2013 ' _The former subject continued_ ' \u2013 picks up the main argument without reference to these poems. I suggest that Coleridge's fancy was caught by the analogy I have sketched, but that he could not see how to subordinate it to his account of Wordsworth's style. Rather than revise, he picks up where he left off, with a somewhat apologetic chapter headnote.\n\nChapter XX explains that the neutral style is not Wordsworth's, nor is the rustic style (II, 77, 83). It is, the speaker wryly notes, 'a singular and noticeable fact' that Wordsworth's theory advocates a diction quite different from his own (II, 77). Wordsworth's imaginative individuality is stylistically evident in 'modes of connections... [and] breaks and transitions' as well as in 'grammatical connections' \u2013 in those means whereby an author depicts not just things, but relations (II, 83\u20134). In these qualities his style is as remote as possible from actual rustics' styles, and near instead to Milton and Shakespeare. At his best, which the speaker amply illustrates, Wordsworth brings physical objects 'before the mind, as the actions of a living and acting power' (II, 84). He awakens in others a 'meditative mood' (II, 85). His poetry is both aesthetically powerful and capable of showing that the being of all things, both windswept trees and human minds, is ultimately an act. This account of Wordsworth's successful fusion of individual expression and universal truth stands between Coleridge's balanced accounts of the excessive concreteness and philosophical errors marring both Wordsworth's theory and some of his poems. It prepares us for chapter XXII's more detailed account of how his very best poetry combines vivid particularity with stunning imaginative insight.\n\nAnonymous criticism has failed to acknowledge the character of Wordsworth's individuality, condemning him for trivial characters and ludicrous or exaggerated responses, because it cannot (or will not) properly manage its own particulars. Chapter XXI analyzes the misplaced concreteness that vitiates anonymous criticism. 'Fair and philosophic' criticism requires a balance and blending of universal principles and particulars from the text to be judged. Arguments conducted in this way result in conclusions that the reader accepts 'in the light of judgement and in the independence of free-agency' (II, 85). They appeal to the activity of will, not the passivity of sensation; they engage the judgment, not the emotions. Such philosophical debate underlies popular government, just as demagoguery underlies tyranny (I, 121\u201332). Thus, the speaker would respect as nobility any group of critics following 'fair and philosophical' procedures: they are natural leaders (see I, 124\u20135, 155\u20136). Unlike the violent hurricanes of fanaticism and anonymous criticism, philosophic criticism is a windmill grinding grain: 'All the two and thirty winds alike are its friends. Of the whole wide atmosphere it does not desire a single fingerbreadth more than what is necessary for its sails to turn round in' (II, 88).\n\nThe announced plan for the _Edinburgh Review_ promised such philosophic criticism, but the magazine has not remained faithful to its plan. The speaker condemns not the 'asperity of the damnatory style' (which he does not hesitate to use himself), but the inappropriate particularity (II, 86). The particulars on which anonymous critics dwell reveal that their principles derive from politics and personality, not 'the essence of human nature' (II, 58; cf. I, 44). When they do not draw particulars from irrelevant details of the poet's private life, these critics offer either no particulars at all, or particulars that fail to illustrate the fault they would condemn. The speaker cites one such condemned passage as evidence that behind such criticism one finds a diseased moral feeling (II, 92\u20134): what the critic condemned other readers have found consonant with 'an intuitive certainty in their own inward experience' (II, 91). Minds crippled by illogic and a degraded fancy cannot respond properly to poetry. In a quick sketch of what will preoccupy chapter XXII the speaker explains that a poem's 'sense' must be 'consonant with all the best convictions of my understanding' and its 'imagery and diction [must collect] round these convictions my noblest as well as my most delightful feelings' (II, 91). Only imagination can so completely fuse both the understanding's knowledge of the true and the false with all a man's passions, and in doing so imagination relies on the drawing power of an enlightened fancy.\n\nLike Wordsworth's 'Essay Supplementary to the Preface' of 1815, this chapter's denunciation in effect gives up any hope of winning the approval of these powerful critics and their readers. It is the high point of the condemnation to which Coleridge had been urged by friends. _Biographia Literaria_ naturally provoked vitriolic reviews. However unfortunate for the subsequent reputation of the book, this reaction could not have been a surprise. In some ways, Coleridge traps the reviewers quite handily: when they denounce him for diseased egotism, rather than engage the complex philosophic problems he engages, they cast themselves as fanatics.\n\nBefore considering chapter XXII, let us look back a moment over the preceding three chapters, and then forward to chapter XXIV. Chapters XIX, XX and XXI are swifter afoot, and far less dense, than any passage of comparable length since chapter X. And, like chapter X, they serve as a prelude to a long, slow, technical argument. Just as chapter X contrasts the speaker's genial quest for principles with the political and philosophic fanaticism of his day, so these chapters contrast Wordsworth's genial reform of poetic diction with the literary fanaticism of his day. They do so in part by locating his style within the English tradition, and in part by evaluating the kinds of reviews his work has had. The speaker assimilates the apparent plainness of Wordsworth's style \u2013 its lack of high rhetorical ornament \u2013 to the 'neutrality' of poets from Chaucer to Herbert. The assimilation suggests that Wordsworth's plainness is ancient and pure, not primitivist or 'democratic' or empiricist. Although the speaker hastens to add that this poetry is not Wordsworth's best, nor his most characteristic, the chapter none the less suggests that Coleridge recognized and understood Wordsworth's aesthetic purposes rather more fully than chapters XVII and XVIII alone might suggest.\n\nAnd it is significant that this recognition comes so closely interwoven with the familiar indictments of anonymous critics. Because their imaginations are feeble, their sensibility is blunt. Their relation to reality \u2013 textual or physical \u2013 is vague and foggy. These are the very last readers from whom one ought to expect the acuity, the sensitivity or the subtlety that Wordsworth's plain and 'objective' descriptions can demand. But in _Biographia Literaria_ the phrase 'anonymous criticism' stands more for the taste and sensibility of the age than it does for the works of men like Hazlitt and Jeffrey. From this perspective, then, Coleridge censures Wordsworth not only for writing bad poetry, but also for failing to adapt his style to his actual readers. The complaint applies to Coleridge's own work, I have argued; and this elaborate, somewhat indirect reflexivity often characterized Coleridge's advice to his friends. In chapter XXII, when Coleridge contends that Wordsworth's lack of rhetorical finesse does result in bad poetry, he seeks more to explain fancy's role in aesthetically effective poetry than to account for Wordsworth's contemporary status.\n\nChapter XXI's account of anonymous criticism also connects chapters II and X with chapter XXIV, so as to integrate the 'anonymous criticism' theme with the seminal philosophic issue of causality. As we saw, in chapters V to IX the speaker complains that materialist associationism cannot account for causality, and thereby denies both personal agency and moral responsibility. He builds his alternative account of cognition upon the testimony of conscience that we are indeed responsible for our acts and our motives: the properly formed conscience requires imagination's self-consciousness, or the distinct knowledge of the acts and forms of one's own will. And, as we have seen in chapters I to IV and XII, imagination also provides distinct knowledge of the particular physical world: the genius is blessed with exquisite sensibility. The philosophic critic, no less than the poet, needs imagination, because argument no less than poetry requires not only the distinct knowledge of both causality and particulars, but also a delicate, astute apprehension of how the two are related. Anonymous critics cannot support their argument with particulars because their dislike is caused by politics not by the poems, and because their response to particulars is warped by anger and envy. The quality of their criticism reveals not just aesthetic failure, or bad taste, but a deeply grounded moral failure as well. As the speaker later explains, they write and act in bad conscience (II, 129\u201330).\n\nThe _Biographia's_ enormously complex arguments about causality and particularity lie submerged beneath the lucidity of chapter XXII. As I have suggested, from one perspective Wordsworth's misplaced causality and his excessive particularity suggest the undue influence of Locke and Hartley. But there are other perspectives; the issue is not as simple as this. Coleridge's original realism guarantees that the animated nature Wordsworth describes is 'really there'. Is the poet, then, required \u2013 as the philosopher surely is \u2013 to maintain, distinctly and explicitly, and at all times, that the animation is 'reflected' not 'bestowed'? Must good poetry first of all be good epistemology? No. Obviously not: 'a poem is that species of composition, which is opposed to works of science, by proposing for its _immediate_ object pleasure, not truth' (II, 10). Even though the ultimate object of poetry is truth, and even though the greatest works of art will be found to achieve that ultimate end, the critic can demand neither doctrinal nor philosophic orthodoxy as a measure of artistic success. All he can rightly demand is good sense and fidelity to nature, and the proper interdependence of parts. These alone directly affect the reader's aesthetic, imaginative response.\n\nBut these are sufficient measures to discredit what Coleridge calls Wordsworth's 'Vague, misty... Confusion of God with the World', because his excessive particularity and empiricism reflect a school philosophy that _does_ contradict the common sense of every man (I, 77, 82\u20133, 178\u20139). But these measures require some subtlety: too literally applied, they would devastate Wordsworth's attempts to explore the psyche, attempts surely beyond the ken of the actual common sense of most of his contemporaries. All of which is perhaps only to say that no one's critical principles \u2013 and especially not those of one like Coleridge \u2013 can be applied without regard for judgment and taste. Coleridge's application, although profoundly rhetorical, also attends closely to the interior relations of the work itself, because these relations most firmly govern the reader's response. His balancing subjective response with objective textual features presupposes his whole idea of language. Poems can most fully express _and most fully communicate_ what imagination reveals because words \u2013 regarded broadly, as language itself \u2013 express the consciousness common to their users. But words also express the particular consciousness of their individual speaker. The consciousness common to the group, and the special imaginative consciousness of the poet, are intelligibly related through the poem by means of the real world to which words also refer. In seeing not just _what_ but also _how_ the poet sees, as well as his impassioned response, we gain access to the moral unity of what it is to be a human being, and what it is to be a conscious thinker. The greatest poems render concretely accessible Coleridge's whole transcendental analysis of consciousness.\n\nAnd that is why Coleridge so sharply condemns Wordsworth's pantheism, and his faith in the aesthetic and moral power of a keenly described scene or situation. These must be flaws, _demonstrable_ flaws in his art, or what is to be the clear, concrete enactment of Coleridge's orthodox Christian transcendentalism turns instead into the ghost of Spinoza. Coleridge wants Schelling's view of art, but St John's idea of Man, God and Logos. Wordsworth's best poetry does enact this synthesis, the speaker has been saying all along; but Wordsworth's theory and at least some of his poems show all too clearly the weak side, the dark side of Coleridge's own vision. Thus it is that chapter XXII's account of the defects and the beauties of Wordsworth's verse recapitulates many of the distinctions between empiricist associationism and Coleridge's original realism. In the _Biographia's_ first volume, this distinction is focused subjectively: the two philosophies are evaluated as rival accounts of mental experience and cognition. The focus here is objective: literally faithful observation _versus_ the imaginative association of properly managed fancy as rival origins of vividly particular, aesthetically potent verse. Wordsworth's most authentic, most characteristic poems, like the speaker's most authentic, most characteristic philosophy, accommodate fully the rival claims of imagination's transforming power and the green world's serene stability.\n\nThe duality of Coleridge's design attains a culminating richness in chapter XXII, where the critique of Wordsworth's poems proceeds simultaneously with a detailed explanation of the correlative operation of fancy and imagination. Wordsworth's defects and excellences are closely related to one another (II, 115), as one would expect in the writings of genius (I, 43). His usual balance between thought and sensibility sometimes deviates into prolixity and bombast. His proper fidelity to nature sometimes degenerates into excessive detail. His pure and individual diction sometimes lapses into undistinguished, unimpassioned prosaicism. His usual balance of thought and image with subject-matter allows him to write truly philosophic and imaginative poems, but the loss of this proportion mires him in a confused variety of materialism and pantheism. Both his flaws and his beauties demonstrate that the fullest aesthetic pleasure requires appropriate particulars (i.e. properly managed fancy), because the poet reaches the reader's imagination only through his detailed representations of the reality known by both. Yet, to have their wonted effect, these representations must form parts of an aesthetic whole, one in which parts are interdependent. Wordsworth's defects are consistently described as comprising the organic unity of his poems, and thus their aesthetic power.\n\nThe first defect, occasional prosaicism, draws on the necessary relation between passion and imagery to explain that full aesthetic power requires _sustained_ imagery. The reader does not maintain the pleasurable excitement through his own imaginative power, but through the images:\n\nEven in real life, the difference is great and evident between words used as the _arbitrary marks_ of thought, our smooth market-coin of intercourse, with the image and superscription worn out by currency; and those which convey pictures either borrowed from _one_ outward object to enliven and particularize some _other_ ; or used allegorically to body forth the inward state of the person speaking[.]... excitement aris[es] from concentered attention[.]... in the perusal of works of literary _art_ , we _prepare_ ourselves for such language; and the business of the writer... is so to raise the lower and neutral tints... to produce the effect of a _whole_. Where this is not achieved in a poem, the metre merely reminds the reader of his claims in order to disappoint them. (II, 98)\n\nImages concenter attention by conveying relations which transcend space and time (II, 16) and obliterate the sense of self (II, 33 n) by changing the form of activity of consciousness. But for the poet to sustain imagery effectively, for the images to contribute to unity and identity of parts in the whole, he must adapt each image not simply to the relation it is to convey, but also to the other images, to the whole system of relations to which it belongs.\n\nBut not any images, and not any concrete descriptive details, will suffice. As the account of the second defect explains, particulars should reveal universals: the historian records particulars _per se_ , but the poet offers ' _truth operative, and by effects continually alive_ ' (II, 102). The poem's visual descriptions should generate simultaneous wholes, not geometrical constructions nor topographical surveys (II, 102\u20133; cf. II, 16\u201318). The imaginative whole is intuited, not assembled from parts; it consists in relations conveyed by images. The most effective image, consequently, is a symbol that 'partakes of the Reality which it renders intelligible' ( _LS_ , 30). The poem's particulars reveal a living truth only when they succeed in creating an imaginative whole.\n\nBut what interrelation of particulars generates an imaginative whole? The poet must forge unity from the reader's conventional associations with the poem's particulars. This associative activity unfolds to define an imaginative center that only the poet sees beforehand, and this is the only way in which rare imaginative vision becomes intelligible generally. For the poet to shape the reader's associations, the reader in turn must accept all the poem's parts and interrelations of parts. He must not discard or ignore some. Poet and reader thus co-operate: the reader must be willing to 'believe for his own sake', to 'permit the images presented to work by their own force, without either denial or affirmation of their real existence by the judgement' (II, 101, 107). The poet in turn must not provoke probability-judgments, or contrast poetic faith with real faith by too closely juxtaposing poetic fiction and real fact (II, 107). The poet must work from _within_ his best estimates of the reader's literary habits and his conventional associations.\n\nYet the Preface suggests Wordsworth's distrust of such expectations. And he refuses at times to appeal to the general state of association because he wants to 'attack and subdue' its pernicious associations with economic and social status (II, 104). The speaker responds that truth and aesthetic pleasure may differ only because of the fallen state of men, yet this is the reality with which the poet must cope.\n\nNow till the blessed time shall come, when truth itself shall be pleasure, and both shall be so united, as to be distinguishable in words only, not in feeling, it will remain the poet's office to proceed upon that state of association, which actually exists as _general_ ; instead of attempting first to _make_ it what it ought to be, and then to let the pleasure follow.... For the communication of pleasure is the introductory means by which alone the poet must expect to moralize his readers. (II, 104\u20135)\n\nBy aiming directly at truth, not pleasure, Wordsworth arouses neither aesthetic response nor moral conviction. Such works are neither good poems nor good sermons, because the particulars embody neither universals nor principles (II, 104\u20135). The speaker echoes his earlier appeal to Aristotle, and calls down the authority of Horace: the poetic particular must not perplex or contradict the reader's feeling if the poem is to achieve the ideal unity and universality proper for art. Decorum remains a useful principle because the poet cannot hope to control the reader's associations. He can only try to elicit those associations most useful for his own ends.\n\nDecorum also helps supply the quality of unity essential for aesthetic imaginative activity in the reader, and its role reveals more fully what an imaginative fancy contributes to aesthetic power. For Coleridge, the universal and permanent exist within the particular and transitory if one has grasped the relation between the being of the particular and the being of the universal. One must, then, scrutinize a single tulip, or the fleeting impression of a beggar or a bed of daffodils, to clarify not the sensations of things, but the consciousness of the selfs perceiving. But, for an image to lead the reader to an imaginative apprehension of the underlying relations (whether in reality itself, or in the poem), the image must seem an appropriate object for the poet's meditation. Images will not 'work by their own force' on a reader who judges the poet's meditative energy misspent. An imaginative fancy seeks images that will both provide an adequate vehicle for the poet's meditative passion, and appeal to the ordinary reader. Without the poet's imagination, there is no unity; without the reader's correlative response, the unity is not perceived.\n\nConcern for the reader's expectations, feelings and associations is equally evident in the speaker's account of the third defect: Wordsworth's 'undue predilection for the _dramatic_ form' (II, 109; cf. II, 36, 53, 55). Such poems are either good plays, and thus disconcerting for the reader of poems; or they are bad plays, and thus not convincing. Coleridge's enforcing generic distinctions, like his enforcing decorum, provides grounds in post-Kantean psychology for ancient and valuable critical principles. Wordsworth's errors matter not because he fails to imitate properly a given form, which in turn imitates a given aspect of reality, but because he confuses his readers. Conventional associations with dramatic dialogue generate stylistic expectations that Wordsworth disappoints.\n\nIn citing the last two defects, the speaker explains that effective images and particulars \u2013 an effective style \u2013 must be suitable not only for the audience, but also for the poem's essential balance between thought and feeling (II, 109; cf. II, 19, 49\u201351). When the feelings are disproportionate to the objects described, the thought cannot progress. The relations among parts are befuddled. When thoughts and images are disproportionate to the originating situation, the poet's impassioned response seems ridiculous. Several illustrations reveal that this error arises when Wordsworth fails to distinguish between associations appropriate to his ideas, and those accidentally linked with it in his mind from the situation in which the idea originated (II, 109\u201310). These inappropriate associations are worse than merely ineffective associations, because they are illogical uses of words:\n\nIf the words are taken in the common sense, they convey an absurdity; and if, in contempt of dictionaries and custom, they are so interpreted as to avoid the absurdity, the meaning dwindles into some bald truism. Thus you must at once understand the words _contrary_ to their common import, in order to arrive at any _sense_ ; and _according_ to their common import, if you are to receive from them any feeling of _sublimity_ or _admiration_. (II, 114)\n\nWordsworth's lapses into this pseudo-sublime lead him into an incoherent pantheism that the speaker criticizes most sharply.\n\nTo set this famous critique of the 'Intimations' ode in its full context, however, we must pause for a summary that will transform the speaker's proscriptions into prescriptions. Properly literary language, he has said, conveys pictures either to enliven and particularize objects or to embody the inward state of the speaker. These properly literary descriptions and characterizations must convey '\" _truth operative, and by effects continually alive_ \"' through the pleasure they arouse (II, 102). They must, therefore, remain generally conventional or representative or decorous, lest the reader be confused or distracted, and thus not pleased, and thus not instructed. Yet they must also remain consistent and congruous with their dramatic speaker, whose emotional and intellectual responses to the objects and characters he encounters must therefore be both authentic and believable. In sum, a genially directed fancy maintains properly literary (i.e. aesthetically potent) relations between the poetic voice and the reader by gathering vivid particulars whose associative range covers the intersection between fully imaginative sensibility and the common sensibility. Only by manipulating the ordinary reader's ordinary responses can the poet transform the reader's consciousness into an imaginative echo of his own. That is why Coleridge so consistently describes critical taste as itself an imaginative power: the best reader, the truly philosophic critic, will most fully share the poet's imaginative sensibility. When Wordsworth misgoverns his fancy, when he presumes that his readers will \u2013 or ought to \u2013 associate exactly as he associates _even when his associations are random or arbitrary_ , then he becomes dull, prosaic, confusing and fragmentary.\n\nSo would any poet. Little to this point in chapter XXII specifically characterizes Wordsworth himself, rather than Wordsworth the _Biographia's_ ideal poet. Little, that is, except the critique of 'Intimations', which defines the error sometimes visible in Wordsworth's own associations with objects. When Wordsworth attributes to things what he ought to attribute to imaginative relations among perceiver and things, then he recapitulates the error of pious Hartley: he attributes moral animation to the unconscious object _per se_. He derives from objects themselves what he ought to derive from self-conscious activity of mind. In the pantheist tendencies of Wordsworth's associations with objects, he fails to meet not only art's ultimate end, but also its immediate purpose, because he misuses words and thereby disorients his reader.\n\nThe speaker's account of this false tendency begins with 'I wandered lonely' and 'Gipsies', where Wordsworth makes no more serious error than exaggeration. But the third instance, 'Intimations', demonstrates the moral danger and the abuse of language inherent in Wordsworth's associations with the material world.\n\nIn what sense can the magnificent attributes... be appropriated to a _child_ , which would not make them equally suitable to a _bee_ , or a _dog_ , or a _field of corn_ : or even to a ship, or to the wind and waves that propel it? The omnipresent Spirit works equally in them, as in the child; and the child is equally unconscious of it as they. (II, 113)\n\nColeridge himself verges into polemical exaggeration here, and Wordsworth has not lacked defenders. Coleridge objects to Wordsworth's attributing such a rich philosophic wisdom to a yet-unconscious child. The unconscious child cannot discover moral values by meditating with full philosophic rigor on the activity of his mind. Whatever he may know for himself of those 'truths...\/Which we are toiling all our lives to find', he can only know through observation. If such a child is the 'best philosopher', then the best philosophy must be pantheism. In short, Wordsworth's vision of the child reveals the dangerous moral and philosophic tendencies of his own na\u00efve faith in beautiful landscapes and unsophisticated characters.\n\nWordsworth seldom makes this mistake, the speaker acknowledges; but defends his own tone by asserting that Wordsworth sets a dangerous example for others. That seems to me a lame excuse, another attempt to gloss over the places where the real Wordsworth's ideas and values differ from those predicated of the utterly self-conscious, fully imaginative ideal poet. The account of Wordsworth's defects as a poet is replete with qualifications: his style lapses in less than one hundred lines; his pseudo-sublimity is 'a fault of which none but a man of genius is capable' (II, 109). He mismanages fancy only by subordinating it directly instead of mediately to his moral sense. Such flaws, although significant at times, dwindle even further when contrasted to his imaginative achievement. Wordsworth's beauties reveal all that a truly philosophic imagination and an imaginative fancy together can achieve. The account of Wordsworth's virtues as a poet provides Coleridge's most complete and by far his most accessible definition of imagination's powers.\n\nPre-eminent among these powers, of course, is dominion over language. Wordsworth's 'perfect appropriateness of the words to the meaning' guarantees more than truth about things (see II, 22), because the speaker 'include[s] in the _meaning_ of a word not only its correspondent object, but likewise all the associations which it recalls. For language is framed to convey not only the object alone, but likewise the character, mood and intentions of the person who is representing it' (II, 115\u201316). An imaginatively directed fancy makes possible a portrait of the active mind itself, because it is through imagination's modulating of connotations that the poem reveals the knowing of knowing. Such precision is 'good for the understanding', because self-consciousness provides our most exact knowledge of the world around us; it is also an 'antidote' for an age of 'corrupt eloquence' because a precise use of words fosters the development of self-consciousness in the reader. The speaker's appeal to 'the beneficial after-effects of verbal precision in the preclusion of fanaticism' (II, 116\u201317) echoes his account of Christ's Hospital, and the very different 'improved pedagogy' that produces anonymous critics (I, 4\u20138). This idea of language so exactly reflects his original realism that the speaker pauses here to promise a full-scale treatise on the topic. One begins to recognize that all of his 'incomplete' or 'promised' works are one work, a discourse on the relations among mind, world and God \u2013 with applications to a multitude of practical matters.\n\nThe explanation of Wordsworth's second beauty develops the same point further. At his best, he exactly balances intellectual and emotional responses, both of which originate in his 'meditative observation' \u2013 i.e. in his imaginative response to the world around him. Although such poems have all the clarity and good sense of Samuel Daniel at his best, his most imaginative ones so directly reveal imagination itself that they are accessible only to the philosophic reader. Despite his earlier criticism of parts of 'Intimations', the speaker lavishes praise on the ode's account of the _poet's_ consciousness. Wordsworth's meditative observations of his own mind are\n\ndrawn up from depths which few in any age are privileged to visit, into which few in any age have courage or inclination to descend.... A poem is not necessarily obscure, because it does not aim to be popular.... the ['Intimations'] ode was intended for such readers only as had been accustomed to watch the flux and reflux of their inmost nature, to venture at times into the twilight realms of consciousness, and to feel a deep interest in modes of inmost being, to which they know that the attributes of time and space are inapplicable and alien, but which yet can not be conveyed save in symbols of time and space. (II, 120; cf. II, 15)\n\nIn such poems, Wordsworth so coherently relates the being of the particular with the being of the universal that his figurative, impassioned style forges symbols of imagination's activity (cf. _AR_ , 110, 236). It does not just vividly describe objects he has known, and his reactions to them.\n\nAnonymous critics' complaints about the 'Platonism' of such poems merely display their own inadequacies. The lines from Pindar that the speaker cites in response echo both the lines from Plotinus cited earlier in reference to his own discourse (I, 80), and chapter II's account of the genesis of an anonymous critic (I, 27\u20139):\n\n'I have many a swift arrow in the quiver beneath my arm to sing for those with understanding. But the masses need interpreters. The true poet is he who knows much by the light of nature, but those who have only learnt verse as a craft and are violent and immoderate in their speech waste their breath in screeching like crows against the divine bird of Zeus.'(II, 121)\n\nColeridge brings the whole force of his philosophic inquiry to bear here not only to illuminate Wordsworth's accomplishment, but also to transvalue the reviews of _Poems in Two Volumes_. The tones and judgments of such reviews are to become direct evidence for Wordsworth's genius and the incompetence of his magazine critics.\n\nThe third beauty, the 'sinewy strength and originality of single lines and paragraphs', was illustrated at length in chapter XX (II, 121). Like Shakespeare and Milton, Wordsworth achieves art's greatest universality in a highly individual way. His precision and objective truth arise from within his own deeply imaginative personality, and remain true to their origin. That fidelity guarantees universality because it proceeds from the ' _all in each_ of human nature' (II, 64): the common ground of human being in the absolute, infinite 'I Am'. The quality of his poetic diction, and his 'real poetic character' come at last to the same thing: imagination's power in and through language.\n\nThe account of the fourth and fifth beauties extends the domain of Wordsworth's accuracy. He is precise not only in his use of words and in his portrait of imaginative activity, but also in his descriptions of the natural world. These reveal 'a long and genial intimacy with the very spirit which gives the physiognomic expression to all the works of nature' (II, 121). As the speaker recognized even in Wordsworth's early poems, his works rescue truths otherwise neglected on the 'dusty high road of custom' (II, 121; cf. I, 59\u201360). His poems also reveal the richness of his sympathetic knowledge of human nature in general, 'the sympathy indeed of a contemplator, rather than a fellow-sufferer or co-mate, (spectator, haud particeps) but of a contemplator, from whose view no difference of rank conceals the sameness of the nature' (II, 122\u20133). Imagination and imaginative observation \u2013 not observation alone, nor mere obedience to 'rules' for poetry \u2013 provide both the pleasure and the truth that the very best art attains.\n\nWhen the speaker says at last that 'in imaginative power, [Wordsworth] stands nearest of all modern writers to Shakespeare and Milton', he repeats a claim and a conclusion that have been variously evident all along. Wordsworth's power to '\"add the gleam,\/The light that never was\"' confirms his mastery of 'the cardinal points of poetry': exciting the reader's sympathy with worlds beyond his usual ken (II, 124\u20135). Although his fancy is 'not always graceful, and sometimes _recondite_ ', sometimes subordinated to powers other than imagination, at least it never operates randomly. There is nothing in Wordsworth of _mere_ wit. The landscape analogy for Wordsworth's genius makes the same point more vividly: his mind is no flowery sod over a hollow, nor an egg-shell, but a towering forest rich with granite outcroppings (II, 128\u20139; cf. I, 25, 26, 42). By standing midway between Shakespeare's absorption into the object, and Milton's absorption of all objects into himself, Wordsworth is most capable of portraying imagination's mediation directly in 'the FIRST GENUINE PHILOSOPHIC POEM' (II, 129; cf. II, 20).\n\nThis analysis of Wordsworth's poetry ends abruptly with a single-sentence paragraph reorienting us to the anonymous criticism of both men:\n\nThe preceding criticism will not, I am aware, avail to overcome the prejudices of those, who have made it a business to attack and ridicule Mr. Wordsworth's compositions. (II, 129)\n\nNo rational argument will dissuade 'a gossip, backbiter and pasquillant', because his motives are personal, not rational (II, 87). The speaker claims that these reviewers are not so insensitive to imagination as their reviews suggest: one must not confuse the critic and the man. They are not insensitive louts, but liars and opportunists whose degraded 'personal' style helps to sell copies of their magazines (II, 129\u201330). They are not, as Wordsworth says, 'too petulant to be passive to a true poet, and too feeble to grapple with him' (II, 129). They are really much worse: they encourage fanaticism in the hope of financial return (see I, 130\u20131). The speaker then turns to Wordsworth's sentimental admirers. They will not be persuaded, either. Yet their indiscriminate and mawkish admiration is even more disgusting than the critics' malice. At least the critics perceive Wordsworth's genius \u2013 in private. The speaker thus finds himself once again intermediate between two equally erroneous extremes because he bases his inquiry on principles.\n\nBecause his Bristol printer mistakenly paginated for two volumes, Coleridge padded out the second volume by inserting here three of his letters from Germany (previously published in _The Friend_ ) and a long critique of the play _Bertram_. In spirit and in content, these belong in chapter X. The lively and amusing _Letters_ complement the _Watchman_ anecdotes in displaying the speaker's delight in observation, and his self-ironic humor. The critique of _Bertram_ , like the earlier commentary on Jacobinism, asserts the dangers and the depravity of political fanaticism. But, in their present place, these interpolations mask chapter XXIV's origins in the conclusion of chapter XXII. The critics who 'have made it a business to ridicule Mr. Wordsworth's compositions' have done the same to the speaker.\n\nChapter XXII's analysis of Wordsworth's poetry reveals how fully he achieves both art's immediate end and its ultimate purpose. His poetic portraits of the mind's activity can exert a profoundly moral influence because the delights of his diction successfully carry his truths alive into the hearts of his readers. In chapter XXIV the speaker explains that his rigorous philosophic inquiries into imagination's activity have generated the same abusive reviews as Wordsworth's poems. Even 'Christabel' was 'assailed with a malignity and a spirit of personal hatred that ought to have injured only the work in which such a tirade appeared' (II, 211). _The Statesman's Manual_ was reviewed 'with a malignity so avowedly and exclusively personal, as is, I believe, unprecedented even in the present contempt of all common humanity that disgraces and endangers the liberty of the press' (II, 214). As a consequence, 'the innuendo of my \"potential infidelity\"... has been received and propagated with a degree of _credence_ ' entirely disproportionate to the intellectual merits of the review (II, 214).\n\nSuch a lack of proportion confuses and disorients, and thus sends one seeking after some ground of order and coherence. As we have seen, the speaker's philosophy is the quest for such ground; here, he explains his theory of faith by comparing the intuition of God to the intuition of causality:\n\nThe sense of Before and After becomes both intelligible and intellectual when, and _only_ when, we contemplate the succession in the relations of Cause and Effect, which, like the two poles of the magnet manifest the being and unity of the one power by relative opposites, and give, as it were, a substratum of permanence, of identity, and therefore of reality, to the shadowy flux of Time. It is Eternity revealing itself in the phenomena of Time: and the perception and acknowledgement of the proportionality and appropriateness of the Present to the Past, prove to the afflicted Soul, that it has not yet been deprived of the sight of God, that it can still recognise the effective presence of a Father, though through a darkened glass and a turbid atmosphere, though of a Father that is chastising it. And for this cause, doubtless, are we so framed in mind, and even so organized in brain and nerve, that all confusion is painful. (II, 207\u20138)\n\nCausality \u2013 and, presumably, all of the Kantean categories \u2013 is one mode of God's presence to us. Although the intelligibility of all experience rests immediately on the synthetic power of imagination, it rests ultimately on the knowledge of God, on faith and reason. The speaker refutes the charge of infidelity by asserting that only his faith in God comforts him in the face of such unmerited and unreasonable abuse.\n\nHume had argued that causality cannot be empirically demonstrated; _The Statesman's Manual_ argues that miracles are not empirical proof of God. All suprasensuous knowledge, whether of God or the Kantean categories, must be self-grounded: 'REASON AND RELIGION ARE THEIR OWN EVIDENCE' (II, 215). The three branches of inquiry \u2013 natural philosophy, transcendental philosophy, and religion \u2013 come together at this point: all certainty is grounded in self-consciousness. We should not attempt 'to master by the reflex acts of the Understanding what we can only _know_ by the act of _becoming_.... [I believe, because I understand,] appears to me the dictate equally of Philosophy and Religion' (II, 216; sentence in brackets translated from the Latin).\n\nAn analysis of causality structures the speaker's very first argument that the mind is active, and that the will is a morally free and responsible agent. This argument develops into the theory of imagination, and here Coleridge works his way back again to the fundamental moral issue. He explains the true nature of faith by repeating the chain image from the scholium to thesis II. Let us look at the scholium before going on.\n\nA chain without a staple, from which all the links derive their stability, or a series without a first, has been not inaptly allegorized, as a string of blind men, each holding the skirt of the man before him, reaching far out of sight, but all moving without the least deviation in one strait line. (I, 180)\n\nAs the following theses explain, the staple or guide of this series is the activity of will. Will relative to knowledge is faith, which supplies a certainty one can never attain through empirical methods alone:\n\n... the Scheme of Christianity... though not discoverable by human Reason, is yet in accordance with it... link follows link by necessary consequence... Religion passes out of the ken of Reason only where the eye of Reason has reached its own Horizon; and... Faith is then but its continuation; even as the Day softens away into the sweet Twilight, and Twilight, hushed and breathless, steals into the Darkness. (II, 218; cf. I, 134\u20136)\n\nThis definition of faith anchors both the autobiographical self-defense that reaches its zenith in this chapter and the philosophic inquiry that has been the speaker's lifework. His 'settlement of the long-continued controversy concerning... poetic diction... [and] the real _poetic_ character of the poet' depends upon our accepting this definition of faith as the valid but intuitive ground of all knowledge beyond immediate sense data.\n\nBy concluding _Biographia Literaria_ as he does, Coleridge asserts again the unity of being and knowing. Who Wordsworth _is_ \u2013 the genuine poet \u2013 cannot be separated from what he _knows_ about words, about nature and about humanity itself. The attacks on Wordsworth's poems are then scarcely distinguishable from attacks on the man himself, and they are rendered even less distinguishable by the critics' degraded personal style. Similarly, what the speaker knows and who he is are so intimately united that one book can tell both tales. The unity of being and knowing cannot be physically demonstrated, as one might demonstrate a steam engine; it must be intuited and therefore its ultimate validity rests on an act of faith. But Coleridge trusts that the coherence of his being and knowing, and Wordsworth's being and knowing, will persuade those who apprehend it, because such unity elegantly and truly resolves a painful confusion. All critics, finally, can do little more than appeal to the persuasive force of a reasonable order.\n\n# **9 Conclusion**\n\n> In reading Dykes Campbell's book on Coleridge... I was infinitely struck with the suggestiveness of S.T.C.'s figure \u2013 wonderful, admirable figure \u2013 for pictorial treatment. What a subject some particular cluster of its relations would make for a little story, a small vivid picture.... Would not such a drama necessarily be the question of the acceptance by someone... of the general _responsibility_ of rising to the height of accepting him for what he is, recognizing his rare, anomalous, magnificent, interesting, curious, tremendously suggestive character, vices and all, with all its imperfections on its head, and _not_ being guilty of the pedantry, the stupidity, the want of imagination, of fighting him, deploring him in the details \u2013 failing to recognize that one _must_ pay for him and that on the whole he is magnificently worth it.\n> \n> Henry James, _The Notebooks of Henry James_ , ed. F. O. Matthiessen and Kenneth B. Murdock (New York: Oxford University Press, 1947), p. 152\n\nThe _Biographia's_ dual design solved problems peculiar to this text: the logical incompleteness and apparent pantheism of the theory of imagination. The design solved a more general problem as well: it makes the reader think about fundamental philosophic issues in part by relating them to a contemporary controversy. As I explained in the first chapter, making readers think was one of Coleridge's most abiding aims. Duality characterizes Coleridge's design in part for this reason, and in part for a more mundane one: the need to publish. There would not, after all, have been much of an English audience for Coleridge's technical philosophy. But he could not apply himself to hackwork. Writing about contemporary problems, not as a hack but as a philosopher, made it possible to publish, and yet remain true to his own aims and his own values.\n\nColeridge's numerous plans for works of this sort suggest the ease with which he could forge such links between the popular and the perennial. As McFarland observes, Coleridge had 'an organic, coherent, and fully worked-out viewpoint that could be quickly focused on a given topic because all the larger problems of structure had already been solved'. A look at just one of these projected works will reveal how Coleridge's habits as a writer differ from those of most serious thinkers who engage popular or local problems.\n\nAn 1810 notebook entry plans a study of Socinianism. His purpose, Coleridge states, is to undertake\n\na full and free Examination of the Credibility of Socinian Christianity, or rather _Jesuism_ \u2013 Is the historical Evidence fairly weighed sufficient to verify the facts of the O. and N. Testament, and to ascertain their _miraculous_ nature \u2013 & 2. are these miraculous Facts fit and adequate evidences of the Dogmata of these Books, supposing them to be such and only such as are held by the Unitarians \u2013 namely, the renewal of Life after Death by a resurrection of each Man's Body, to happiness or misery according to the preponderance of his good or bad actions in this life. I would proceed in this manner.... ( _CN_ , III, 3817)\n\nThere is a fairly straightforward procedure explicit in the statement of purpose. Coleridge might have divided the work in two parts, and set about answering the two questions he asks by applying criteria already available. But Coleridge's procedure runs a full page and a half in Coburn's edition of the notebooks: it describes a full-scale inquiry into the history and the psychology of faith. Coleridge divides this inquiry into three parts: (1) a study of Priestley and Socinian divines to demonstrate ( _a_ ) that materialism and necessity are inevitable deductions from Socinianism and ( _b_ ) that Socinianism has no basis in the Gospels, and ( _c_ ) that its doctrines are neither certain nor comprehensible; (2) a history of Christianity backwards from the Council of Nice to establish criteria for miracles and to provide 'a compleat Canon of Belief, deduced from History & Psychology, in all ages & Religions & Sects of Religion, Philosophy, Medicine, Superstition, &c &c'; and (3) 'an attempt to frame certain criteria of the Authenticity of Books' so as to establish the canon of the Bible.\n\nThrough this procedure, Coleridge transforms the local topic of Socinianism into an occasion to advance a major religious and philosophic principle:\n\nAbove all, I would lose no opportunity of enforcing the prodigious difference between the auxiliary evidence required for a religion whose Truth & Necessity is proved a priori, & merely a _condition_ of its being received as a Revelation, i.e. Legislator leges suas conscientiae datas nunc solemniter per intuitum sensitivum reconfirmans, and the absolute evidence which is to support a weight solely on itself.\n\nSocinianism seems to fall from view, just as poetic diction disappears as an issue for nine chapters, while Coleridge explores the underlying issues. Unlike more prudent authors, Coleridge refuses to choose between writing a sophisticated but practical analysis of Priestley, and writing a wholly philosophical and theoretical treatise on faith. One might say that Coleridge's books are too ambitious, that they try to do too much. But this same unswerving allegiance to the larger, more difficult issues is the heart of his permanence as a thinker.\n\nThe same duality characterizes the designs of such serial works as _The Friend_ and the _Philosophical Lectures_. The first volume of _The Friend_ moves inexorably to the distinction between reason and understanding in the final two essays, yet progressive unity is sustained by reference to the author's responsibility in publishing such a periodical. An author's primary duty, of course, is to communicate truth: the two strands of argument can be interwoven because one neither knows the truth, nor communicates it properly without distinguishing correctly between reason and understanding.\n\nThe second volume continues and develops the reflexivity evident in the first, until the philosophic inquiry into reason and understanding and the discourse on writing intersect in the famous 'Essays on Method' comprising most of the third volume. These derive principles of both inquiry and discourse from the nature of reason and understanding. Coleridge integrates the local political issues of volume two at the beginning of 'Section the second' in volume three. There he renders explicit the moral concerns that have been clear all along, and relates morals, discourse and politics by discussing the Sophists. Throughout all three volumes, the Friend stands between mirrors: he explains how one ought to write and to think about problems in part by writing about his thinking about his own writing.\n\nColeridge's dual design works well in serial essays: we tolerate a higher level of discontinuity, so he can move from local problem to philosophic issue and back again without disorienting us too badly. General integrity and progressive unity are sustained by long, somewhat clumsy transitions usually at the beginnings of essays. These commonly refer to the Friend's digressiveness, obscurity and prolixity, or to his excessive interest in metaphysics. Yet Coleridge's richly evident irony turn these 'confessions' into a mode of self-defense that orients the reader to his most fundamental moral and metaphysical concerns.\n\nDuality is even more evident in the design of _Philosophical Lectures_. These lectures are organized both chronologically, as a history of philosophy, and conceptually, as a history of the gradual, direct evolution of a proper idea of method. They are the 'Essays on Method' writ large. Teleological histories are common enough, of course. Coleridge differs in the centrality of his desire to explain correct method _per se_ : the lineaments of correct method provide the work's major structural elements. Historical chronology, like autobiographical narrative, is used chiefly for unifying. The _Philosophical Lectures_ are not at all polished: their 'digressive' texture is quite pronounced; transition and emphasis are idiosyncratic. But their basic design is none the less reasonably clear and accessible, probably because Coleridge need merely 'exaggerate' the conventions of teleological history.\n\nThe aphoristic style of _Aids to Reflection_ meets Coleridge's needs even better than the serial essay, by cueing the reader even more clearly to seek interrelations on his own: it is Coleridge's most successful, most sophisticated use of a dual design. James D. Boulger has astutely described both the dual design and the difficulty of this work:\n\nUnfortunately for _Aids to Reflection_ , its two structural principles of organization are intricate without being clear. Coleridge's division into Introductory, Prudential, Moral and Religious, and Spiritual aphorisms, unintentionally belies the importance and number of the religious issues considered; nor does it coordinate the numerous allusions to contemporary religious figures and problems.... On the surface this entire structure seems quite arbitrary, external, and rambling.... Is there, then, any organization in the baffling arrangement which resulted from the merger of the 'Beauties of Leighton' with the 'Moral Prudential, and Spiritual aphorisms'? If there is, it is certainly not a schematic but an ideational principle, and only corresponds occasionally with the putative structures.\n\nWe have seen this 'ideational principle' at work in _Biographia Literaria_ : Coleridge's desire to make his readers think about issues in ways that will lead them to accept the necessity of ideas whose corresponding laws cannot be directly demonstrated. _Aids to Reflection_ is more difficult reading than _Biographia Literaria_ because its issues are more subtle and more complex, but it does not squander the reader's energy as the _Biographia_ occasionally does.\n\n_Aids to Reflection_ provides three elaborate sets of spiritual exercises, designed to conduct the meditating reader from the most basic question, 'How shall I behave?' through some of the most intricate and fundamental problems in Christian doctrine. Coleridge argues that self-interested prudence leads the thinker to morality, which in turn leads him to revealed religion, and then to Christianity as the most perfect of religions. But it does so, of course, only for those who can examine the issues rigorously and correctly. Conducting this rigorous examination \u2013 with Leighton's help, and others' \u2013 allows Coleridge both to explicate his own system of thought and to engage contemporary religious controversies. He manages both tasks so well by writing for a distinct and specialized audience: students of the ministry, who presumably know both the classical and the contemporary arguments about doctrine.\n\nColeridge had tried defining a limited audience in _The Statesman's Manual_ ; but philosophical politicians have always been, I suspect, in rather short supply. Furthermore, 'political skill and foresight' usually depend at least as much on pragmatic skills as on philosophical acuity. Consequently, the contemporary, local half of the argument is both strained and feeble. Placing great chunks of the philosophical argument into appendices \u2013 an attempt, perhaps, to disguise the radical disproportion of parts \u2013 only complicates the reader's attempt to make consecutive sense of the whole. The second of the _Lay Sermons_ avoids appendices, but does not improve upon the vaguely defined local argument. The two _Lay Sermons_ are brilliant, satisfying books for readers who plunder them boldly, seeking out the intriguing parts; but their designs are clumsy.\n\nColeridge's other early dual design is not so awkward, nor does it address an utterly fictitious audience: the lectures on Shakespeare speak to one who expects literature both to entertain and to teach. One can read them both as literary criticism and as an explanation of what Peter Hoheisel calls 'the basic moral principles at work in human life and... the consequences of those principles'. For Coleridge, Hoheisel explains, it was Shakespeare's genius to have grasped and embodied these principles in his plays; the lectures are organized as simultaneous commentaries on the plays and on the moral principles.\n\nColeridge's last work, _On the Constitution of the Church and State_ , is his most conventional. The contemporary problem of Catholic emancipation, and the philosophical problem of 'church' and 'state', stand so closely together that the plaguing problems of transition and emphasis nearly solve themselves. Coleridge has at last found a contemporary issue well suited to the philosophical problems he wants to address. _Church and State_ depends, no less than his other works, on ideas and distinctions that cannot be fully demonstrated; but the difficulty of making us think is not compounded to the same extent with mundane problems of unity and progression. Yet _Church and State_ plods a bit, at least to my sensibilities. It is less playful, less exuberant, more labored than his earlier works.\n\nYet had the _Biographia_ such sobriety, it might have infuriated fewer readers. But something vital would be lost if Coleridge returned to edit away the Baroque complexities of _Biographia Literaria_. To wish them away is somewhat like wishing away Milton's endlessly involuted sentences, or Wordsworth's timelessly undulating pentameters. And perhaps this is my last but most important point: one must read Coleridge's prose as if it were poetry. His prose requires that same patiently acquired knowledge of the text, that same minute attention to meanings and etymologies and repetitions of words, that same sensitivity to images. But, in the end, the sad, stubborn fact remains: _Biographia Literaria_ is not a poem after all. Coleridge criticized Wordsworth for the obscurities introduced by the intrusion of deliberate philosophy into his poetry; I suspect that the poet's judgment usurps the philosopher's at times in Coleridge's prose. The 'speaker' is an image in whom we are to believe for our own sakes, for the imaginative pleasure and insight we can derive from contemplating an image in full poetic faith. But the accounts of 'his' scholarship are not strongly enough integrated into the autobiography to preclude their provoking the wrong kinds of expectations. And these mistaken or at best confused expectations place inordinate stress on all other unifying devices, most of which are far too delicate to support the extra burden. Like Wordsworth's excessively objective poems, _Biographia Literaria_ requires a level of sympathetic imagination that it does not itself elicit and sustain. If one willingly expends the energy Coleridge's poetic prose requires, it is because, in James's words, 'on the whole he is magnificently worth it'.\n\nAnd because Coleridge is worth it, one tolerates his failings. Yes, the _Biographia_ is flawed, but it is also brilliant. Both judgments are accurate, but one must begin by acknowledging what generations of thoughtful readers have affirmed: the book is obscure beyond what its imaginative purposes can justify. I have worked too hard for too many years untangling the snarled threads of Coleridge's discourse to credit any claim to the contrary. And yet, and yet, with equal equanimity do I assert that the book deserves its status as 'masterpiece' no less fully than it deserves various other epithets. When its design is understood, when Coleridge's imaginative purposes are distinguished from his inept execution, then his mistakes become manageable. Nothing can make the _Biographia_ graceful, but such knowledge renders it moderately accessible.\n\n'Vices and all, with all its imperfections on its head', _Biographia Literaria_ none the less succeeds in illuminating the imaginative powers animating art. Generations of thoughtful, weary readers have also affirmed this. Proposing as his immediate object truth, not pleasure, Coleridge offers a vision of how the greatest poems can reconcile lively delight and moral value. While demanding his reader's unflagging energy, Coleridge illuminates his difficult path with imaginative appeals that assert, over and over again, the intimate relation between literary criticism and life itself. Presenting no rigorous proof, he relies instead on our intuitive assent, on our 'ascertaining vision'. He relies on our imagination.\n\n# **Notes**\n\n## **Chapter 1**\n\n _Life of John Sterling_ , in _The Works of Thomas Carlyle_ (New York: Charles Scribner's Sons, 1897), Vol. XI, p. 56.\n\n Among the Victorians, see, for instance: Matthew Arnold, _The Complete Prose Works of Matthew Arnold_ , ed. R. H. Super (Ann Arbor: University of Michigan Press, 1960\u201377), Vol. III, p. 189, and Vol. IX, p. 237; Walter Pater, _Appreciations_ , 3rd edn (London: Macmillan, 1901), pp. 65\u2013104; John Ruskin, _Ruskin as Literary Critic_ , ed. A. H. R. Ball (London: Cambridge University Press, 1928), pp. 267\u20138; Leslie Stephen, _Hours in a Library_ (1909; reprinted London: John Murray, 1919), Vol. III, pp. 324, 331, and Stephen's entry on C in _The Dictionary of National Biography_ , Vol. IV. C's influence on his century was substantial. See Graham Hough, 'Coleridge and the Victorians' in _The English Mind: Studies in the English Moralists presented to Basil Willey_ , ed. D. H. Sykes and G. Watson (London: Cambridge University Press, 1964), pp. 175\u201392; and Philip C. Rule, SJ, 'C's reputation as a religious thinker: 1816\u20131972', _Harvard Theological Review_ , vol. 67 (1974), pp. 289\u2013320. Thomas McFarland analyzes the history of C's reputation in the introduction and first chapter of _C and the Pantheist Tradition_ (Oxford: Clarendon Press, 1969).\n\n Discussed in Chapters 2 and 5.\n\n These plans have been discussed by George Whalley, 'The integrity of _Biographia Literaria', Essays and Studies_ , n.s., vol. 6 (1953), pp. 87\u2013101; and by Earl Leslie Griggs, _CL_ , III, xlvii-lii; and most recently by Kathleen Wheeler, _Sources, Processes and Methods in C's 'Biographia Literaria'_ (Cambridge: Cambridge University Press, 1981), pp. 8\u201328. See also Daniel Mark Fogle, 'A compositional history of the _Biographia Literaria', Studies in Bibliography: Papers of the Bibliographical Society of the University of Virginia_ , vol. 30 (1977), pp. 219\u201334. On the original intention to write a preface, see _CL_ , IV, 584\u20135, 578\u20139.\n\n McFarland, _C and the Pantheist Tradition;_ Elinor Stoneman Shaffer, 'Studies in C's aesthetics', _Dissertation Abstracts_ , 28 (1967), p. 1409, col. A (Columbia University), esp. ch. 2 (pp. 18\u201370), which reappears in very condensed form as 'The \"Postulates in Philosophy\" in _Biographia Literaria', Comparative Literature Studies_ , vol. 7 (1970), pp. 297\u2013313. The issue is a complex one, as McFarland's history shows. Other views on the issue are discussed below, Ch. 5, n. 2.\n\n The work referred to is _Lay Sermons_. C continues: 'Before a just tribunal of Criticism I could apply still more triumphantly the same test... to two distinct Treatises in the Literary Life, besides the Essay on Authorship as a Trade [Ch. XI].' C would not have written 'in' if he meant 'comprising' or 'constituting'; these two essays are probably chs I to X, and XIV to XXII.\n\n This thematic unity has been defined most fully by George Whalley, 'The integrity of _Biographia Literaria_. George Watson extends one aspect of Whalley's argument in his introduction to _Biographia Literaria_ , ed. George Watson, Everyman's Library (London: Dent, New York: Dutton, 1965). Very short summaries, less comprehensive than Shawcross's, are numerous. J. R. de J. Jackson defines another aspect of the _Biographia_ 's unity: it methodical basis and consistency. See his _Method and Imagination in C's Criticism_ (Cambridge, Mass: Harvard University Press, 1969), esp. pp. 63\u201372. Most recently, Kathleen Wheeler ( _Sources, Processes and Methods_ ) has argued that C uses a higher or Socratic irony to devise a work whose unity depends upon the reader's imaginative response to certain densely metaphoric passages that ultimately render the _Biographia_ almost entirely self-reflexive.\n\n Owen Barfield, _What C Thought_ (Middletown, Conn.: Wesleyan University Press, 1971), p. 13.\n\n Introduction to _Biographia Literaria_ , ed. Henry Nelson Coleridge and Sara Coleridge, in Shedd III, p. xxii.\n\n See Sara Coleridge (n. 9, above); Henry Nelson Coleridge, preface to _TT_ , p. 233; Thomas De Quincey, _De Quincey's Collected Writings_ , ed. by David Masson, 2nd edn (Edinburgh: Adam & Charles Black, 1889\u201390), Vol. II, pp. 152\u20133; William Wordsworth as reported by Christopher Wordsworth, _Memoirs of William Wordsworth_ (London: Edward Moxon, 1851), Vol. II, p. 443; and, more recently, Jackson, _Method and Imagination in C's Criticism_ p. 72; or Arthur Symons, introduction to _Biographia Literaria_ , ed. Ernest Rhys, Everyman's Library (London: Dent, 1906), pp. ix\u2013x. Traveling provides many conventional images for the process of reading; these uses of the 'travel' metaphor are notable for the clarity and consistency with which the authors use the image to distinguish between the clarity or value of the ideas, and the difficulties of the style or manner.\n\n Bishop C. Hunt, Jr, 'C and the endeavor of philosophy', _PMLA_ , vol. 91 (1976), pp. 829\u201339.\n\n C's marginalia on Schelling document his response to arguments for rare and radically independent intellectual powers. These marginalia are compiled and annotated by Henri Nidecker, 'Notes marginales de S. T. Coleridge en marge de Schelling', _Revue de litt\u00e9rature compar\u00e9e_ , Vol. 7 (1927), pp. 736\u201346. For a discussion of the issue, see Shaffer, 'C's aesthetics', pp. 38\u201340, and below, Ch. 3, n. 12. See also Wheeler's account of the ways in which C avoids solipsistic idealism: _Sources, Processes and Methods_ , pp. 34\u201341.\n\n For definitions of idea, see _CCS_ , 12; _LS_ , 113\u201314. For the relations between idea and law, see _PhL_ , 107\u20138; _CCS_ , 13, 19\u201320.\n\n See _BL_ , II, 85\u201394.\n\n Jerome C. Christensen ingeniously seizes this fact to argue that _BL_ is radically self-deconstructed: it reveals C's understanding that statements are prisons, or that freedom is possible only within indeterminacy. A deconstructionist no doubt serves major strategic ends by arguing that even _BL_ actively undermines its own traditional humanism, but C _never_ valued the kind of indeterminacy Christensen describes. None the less, Christensen's description of C's 'marginal' method of composition holds great promise as the basis for a systematic theory of C's use of his own notebooks and others' texts. See 'The genius in the _Biographia Literaria', Studies in Romanticism_ , vol. 17 (1978), pp. 215\u201331; and 'C's marginal method in the _Biographia Literaria_ ', _PMLA_ , vol. 92 (1977), pp. 928\u201340. Lawrence Buell takes much the same approach, contending that C is systematically 'deformalizing' his text: 'The question of form in C's _Biographia Literaria', ELH_ , vol. 46 (1979), pp. 399\u2013417. On the whole, Wheeler's account of C's facility with the higher or Socratic irony explains far more convincingly why it is that he so persistently represents his works as fragments or fragmentary: _Sources, Processes and Methods_ , pp. 59\u201380.\n\n Jackson, _Method and Imagination_ , p. 40.\n\n 'Coleridge', _Tait's Magazine_ , n.s., vol. 1 (Aug 1834), p. 514; reprinted in _De Quincey's Collected Writings_ , ed. Masson, Vol. II, pp. 152\u20133.\n\n J. A. Appleyard, SJ, _C's Philosophy of Literature: The Development of a Concept of Poetry, 1791\u20131819_ (Cambridge, Mass: Harvard University Press, 1965), reveals even more fully the hazards of such dismissal. He discounts chs II and III (p. 174, n. 10) and chs X and XI (p. 188), because he has chosen to study the _Biographia_ as 'Coleridge's own immediate deliberations on his theories' of poetry and imagination, a perspective sharply distinguished from studying it as 'an exposition of a philosophy or even as a biographical sketch of literary opinions' (p. 170, see also p. 176). Such an approach foreordains his inability to see how the theory of secondary imagination can explain both the spontaneous activity of the mind requisite for poetry, and the unity of being and knowing in perception: 'Coleridge sets up requirements for the imagination that it cannot possibly fulfill in terms of its original [i.e. temporally prior] conception' (p. 183). Appleyard's willingness to discount not only major sections of the text, but also C's explicit intention to provide the philosophy underlying his literary theories, leads necessarily to his conclusion that the _Biographia_ is a 'remarkable failure' (p. 169). Appleyard's methods also lead directly to his reading the epistemological function of imagination in unduly Schellingean terms: imagination is not the source and guarantor of truth. As the neglected ch. X explains, God is this source and guarantor.\n\n Introduction to _Biographia Literaria_ , ed. Watson, p. xv.\n\n Documenting and analyzing C's grasp of this equation are among the principal endeavors of McFarland, _C and the Pantheist Tradition_ , and Wheeler, _Sources, Processes and Methods_.\n\n For a fuller explanation of this view of C's idea of symbol, see J. Robert Barth, SJ, _The Symbolic Imagination: C and the Romantic Tradition_ (Princeton, NJ: Princeton University Press, 1977), pp. 3\u201321. A submerged issue here is quite complex: how can human knowledge generally attain the precision and predictive value characteristic of mathematics? This is one of the major issues Kant addresses in _Prolegomena to Any Future Metaphysics_ and, at greater length, in _The Critique of Pure Reason_.\n\n Barth, _The Symbolic Imagination_ , pp. 14\u201317.\n\n ibid., p. 14.\n\n Richard Mallette, 'Narrative technique in the _Biographia Literaria_ ', _Modern Language Review_ , vol. 70 (1975), pp. 32\u201340.\n\n Jackson, _Method and Imagination_ , p. 72.\n\n Richard Haven, _Patterns of Consciousness: An Essay on Coleridge_ (Amherst: University of Massachusetts Press, 1969), p. 12.\n\n James Olney, _Metaphors of Self : The Meaning of Autobiography_ (Princeton NJ: Princeton University Press, 1972), pp. 3\u20134. Olney argues for the propriety of autobiography as a vehicle for philosophy. Coleridge's adaptation of his literal life and his enigmatic promise in the last chapter to write a _personal_ autobiography suggest a highly developed understanding of the philosophic potential of the genre as Olney has described it.\n\n See, for instance, _F_ , I, 107\u201326, 149\u201353; _LS_ , 43\u201352; _AR_ , 167\u20138, 199\u2013200, 235\u20138, 276\u201381. The lengthy summarizing transitions in _The Friend_ and the _Lay Sermons_ are not literal summaries, but attempts to provide a single perspective from which the reader might integrate what has gone before, and preview terrain ahead. These do not appear in _Biographia Literaria_ , perhaps because Coleridge relies instead on the perspectival function of chronology: one who continually tries to relate each part of the _Biographia_ to the autobiography does in fact begin to integrate the whole. Transitions become relatively shorter and more effective in _Aids to Reflection_ , and quite graceful in _On the Constitution of Church and State_. He also seems to have objected to the 'orienting' transition itself: 'Hotheaded men confuse, your cool-headed Gentry jumble, the man of warm feelings only produces order and true connection \u2014 in what a jumble M. & H. write \u2014 every third paragraph beginning with \u2014 \"Let us now return\" or \"We come now to the consideration of such a thing\" \u2014 i.e. what I _said_ I _would_ come to in the Contents prefixed to the Chapter' ( _CN_ , I, 868). When Coleridge deviates from his own ideal, it is obviously toward hotheadedness.\n\n Wheeler, _Sources, Processes and Methods_.\n\n Symons, introduction to _Biographia Literaria_ , ed. Rhys, pp. ix\u2013x.\n\n## **Chapter 2**\n\n Beginning with Ch. 2, citations to the _Biographia_ will be identified by volume and page number alone; _'BL'_ will not appear.\n\n In the 'Essays on Method' in _The Friend_ , C calls education 'the most weighty and concerning of all sciences... the nisus formativus of social man' _(F_ , I, 493\u20134). His abiding interest in the process of thinking underlies his complementary arguments about education and about method. For further discussion, see Jackson, _Method and Imagination_ , ch. 2; and Barfield, _What C Thought_ , chs 1 and 2.\n\n C's accounts of the intellectual and emotional hazards of heartless philosophy are often misread as indictments of philosophizing generally, especially in commentaries on 'Dejection: An Ode'. Geoffrey Yarlott offers a statement that is exemplary in its failure to distinguish false from genuine philosophy: 'Since experience showed that thought and feeling must go together it followed that to differentiate and isolate them was merely inviting trouble. It led to what he called \"the thinking disease\", on which \"no moral being ever becomes healthy\". Though metaphysics provided incidental compensation (it being a salve for his wounded self-esteem that here was a sphere in which he outshone all rivals), because it was basically a neurotic activity it could afford no final anodyne' ( _C and the Abyssinian Maid_ (London: Methuen, 1967), p. 222). Major portions of C's achievements are waved aside here as 'neurotic', and radically isolated from what Yarlott rightly depicts as the central issue of C's value-system: reconciling ideas and feelings. Yet, in the midst of the years to which Yarlott refers, C writes to Southey, 'Believe me, Southey! a metaphysical Solution, that does not instantly _tell_ for something in the Heart, is grievously to be suspected as apocry[p]hal' ( _CL_ , II, 961). The 'blessed interval' to which C refers at I, 10, is the period before his complex emotional difficulties rendered it distressing to think about his feelings; 'abstruse researches', although technically useful, become morally and intellectually dangerous if they are mistaken for the enterprise of philosophy itself (see I, 98, _F_ , I, 523 n, and Haven, _Patterns of Consciousness)_. As McFarland argues, C was always a poet and always a philosopher; and he was a philosopher of such erudition that his work poses major methodological problems for its students ( _C and the Pantheist Tradition_ , pp. 112\u201316, xxiii\u2013xi).\n\n Unless epistemology anchor itself to a real world immediately known, C contends, it sinks into extreme forms of materialism and idealism \u2014 extremes that are inimical not only to religion, but also to the most rudimentary forms of moral responsibility. For a further discussion of C's rejection of these extremes, see McFarland, _C and the Pantheist Tradition_. I. A. Richards's analysis of the two extremes remains a model of lucidity and economy ( _C on Imagination_ (1935; 2nd edn, New York: W. W. Norton & Co., 1950), ch. 3). C's definition of faith as both volitional and cognitive is the key to the solution he sought; for an introduction to the doctrinal controversy, see J. Robert Barth, SJ, _C and Christian Doctrine_ (Cambridge, Mass.: Harvard University Press, 1969), ch. II.\n\n There are two thorny problems here: Did C complete a system? and, Is it pantheist? McFarland contends that no formally closed philosophic system could encompass all that C wanted to include ( _C and the Pantheist Tradition_ , p. 192, and ch. 3 passim). Barfield says that one can judge C pantheist only by ignoring his statements about the Trinity ( _What C Thought_ , pp. 149\u201350). Although Barfield is not as concerned with formal philosophic criteria as McFarland (ibid., pp. 149\u201350), he none the less explicates in great detail the consistency of C's thought (see pp. 5, 13). In a review, McFarland contends that the polar metaphysics Barfield explicates is ultimately pantheist, and was later abandoned ( _The Wordsworth Circle_ , Vol. 4 (1973), pp. 165\u201370). But McFarland also acknowledges that Barfield's book remains a most valuable guide to C's thinking on many issues, particularly at the stage represented by _Biographia Literaria_. At one point, Barfield describes his whole book as basically a commentary on ch. 12 (p. 63). The major theoretical issues engaged by Barfield and McFarland cannot be addressed thoroughly within the scope of the present study, although I will later note both the ways in which C diverges from Schelling to link his psychology, his poetics and his metaphysics to the orthodox Christian God, and the ways in which this philosophically frail link is strengthened rhetorically.\n\n In _'Quisque sui faber:_ Coleridge in the _Biographia Literaria_ ', _Philological Quarterly_ , Vol. 50 (1971), pp. 208\u201329, M. G. Cooke attributes the 'letter' in ch. XIII in part to C's loss of confidence after literally stopping his composition to reread Wordsworth's 1815 preface, but this judgment does not take adequately into account either this distinct reference to the central disagreement with Wordsworth, or the elaborate pattern culminating in the 'letter' (discussed below, Ch. 5). Neither does it fully enough consider the body of evidence testifying to C's much earlier recognition that he and Wordsworth disagree in significant ways (cited and discussed by Whalley, 'The integrity of _Biographia Literaria_ '). There is none the less great merit in Cooke's describing the autobiographical self as a self-creative psychological act; Cooke very sensitively evaluates the stylistic and philosophic consequences of C's own recognition that he had not fully and adequately defined his theory of imagination. The rhetorical functions I describe add to the _Biographia_ 's complexity as a psychological document: much work remains to be done.\n\n Barfield, _What C Thought_ , p. 87.\n\n Shawcross points out that C exaggerates his revision of his poems in response to early reviews (I, 3; I, 204\u20135), and shortens his list of publications by omitting _The Watchman, The Friend_ and _Conciones ad Populum_ (I, 38; I, 218). Yet since the speaker later describes his efforts with _The Watchman_ in great detail, and several times quotes or refers to _The Friend_ , it is difficult to put a precise interpretation on the misrepresentation (I, 114\u201321; I, 60; I, 110; I, 119). In general, the character or achievements of the speaker are exaggerated or deflated from those of the real man so as to sharpen the contrast between philosophic and anonymous criticism.\n\n Yarlott, _C and the Abyssinian Maid_ , ch. I et passim.\n\n For a discussion of C's philosophy of history, and the influence of his views, see Robert Preyer, _Bentham, C, and the Science of History_ , Beitrage zur englischen Phil., 41. Heft. (Bochum-Langendreer: H. Popinghaus, 1958).\n\n C's portrait of anonymous critics at I, 25\u20139, should be compared to Pope's portrait of the evolution of the inept critic in 'An Essay on Criticism', which of course has origins of its own, as the lines from Pindar at II, 121, suggest. In this context, a footnote in ch. III accuses Jeffrey of maliciously misrepresenting his opinion of C (I, 36 n). Later, C accuses him of lying in the same way about Wordsworth (II, 129). Jeffrey responds in a long footnote to Hazlitt's review in the _Edinburgh Review_ ( _C: The Critical Heritage_ , ed. J. R. de J. Jackson (London: Routledge, 1970), pp. 314 n-318); Hazlitt in turn echoes Dryden's 'MacFlecknoe' (ibid., p. 298). The controversy is picked up in later reviews in _Blackwood's Edinburgh Magazine_ (ibid., pp. 344\u20135) and in _British Critic_ (ibid., pp. 366\u20139). There are still later notes in a letter to the editor in _Blackwood's_ (ibid., p. 353), and in Crabb Robinson's diary, published as _Books and Their Writers_ , ed. Edith J. Morley (London: Dent, 1938), Vol. I, p. 209. This exchange of views is but a brief indication of the extent to which C's remarks about reviewers provoked lively reaction. For further discussion, see David Erdman and Paul Zall, 'C and Jeffrey in controversy', _Studies in Romanticism_ , vol. 14 (1975), pp. 75\u201383; and Nathaniel Teich, 'C's _Biographia Literaria_ and the contemporary controversy about style', _The Wordsworth Circle_ , vol. 3 (1972), pp. 61\u201370. For an account of C's own work as a reviewer, see David Erdmann, 'Coleridge and the review business', _The Wordsworth Circle_ , vol. 6 (1975), pp. 3\u201350. For discussion of C's use of eighteenth-century satirical modes, see Mallette, 'Narrative technique in the _Biographia Literaria'_.\n\n For discussion, see David Erdman, 'Coleridge as \"Nehemiah Higgenbottom'\", _MLN_ , vol. 73 (1958), pp. 569\u201380.\n\n The paragraph from _Ecclesiastical Polity_ uses metaphors of depth four times to describe first principles. The metaphor of height that C quotes is the only one. See Vol. I, bk 1, sect. 2.\n\n The continual insistence that criticism be personally unbiased and properly derived from first principles once found a sympathetic audience among professional academic critics, as did the insistence that poetry has a logic of its own, as severe as that of science but more subtle and more complex. Yet the _British Critic_ review disputed the first claim, asserting that criticism is best served by a literary equivalent of the legal adversary system; Wilson was outraged by the second (Jackson (ed.), _Critical Heritage_ , pp. 361\u20133, 336\u20137).\n\n In _Sources, Processes and Methods in Cs 'Biographia Literaria'_ , Kathleen Wheeler takes a different position on this issue. She suggests that those who find _BL_ obscure are themselves unimaginative, passive or lazyminded (see, e.g., pp. 6, 109, 110, 112, 128). I grant that lazy and unimaginative readers will understand very little of C's argument \u2013 and little of Wheeler's, either. But such are not usually taken as representative instances of the 'competent' reader. If one grants Iser's contention that texts do not 'contain' meanings but, rather, provide 'instructions' for the reader's active endeavor to assemble meanings, then some of _BL_ 's instructions must be judged inadequate, and many more criticized as excessively idiosyncratic. Wheeler's sweeping indictment fails to take adequately into account the close attention paid to the book by many outstanding scholars, critics and poets both in this century and the last.\n\n Southey's role in C's marriage to Sara Fricker casts a shadow of irony over C's statements. In subtle ways, judgment is associated with fancy throughout the _Biographia;_ judgment itself is the work of understanding. This link becomes most evident when Wordsworth's disproportions \u2014 clearly failures of judgment \u2014 are represented as failures to direct fancy appropriately (II, 104\u20135). One who has exceptionally good judgment, particularly in matters of rhetoric, is one whose fancy is most strongly attuned to 'that state of association, which actually exists as _general'_ (I, 105). In coordination with imagination, such attunement is crucially important for poetry; but in isolation from imagination it generates faint praise from C.\n\n R. H. Fogle, _The Idea of C's Criticism_ (Berkeley: University of California Press, 1962), p. 40.\n\n See Barfield, _What C Thought_ , pp. 64\u20137 et passim. McFarland explores at length the consequences of Coleridge's dual allegiance to a creative mind and a real physical world immediately known _(C and the Pantheist Tradition)_. In places Wheeler tends to assimilate Coleridge to Kant on this issue (e.g. _Sources, Processes and Methods_ , pp. 40\u20131).\n\n Preface to Second Edition, _Critique of Pure Reason_ , trans. Norman Kemp Smith (New York: St Martin's Press, Toronto: Macmillan, 1965), pp. 22, 25 n. On Newton, see _TT_ , 8 Oct 1830.\n\n## **Chapter 3**\n\n For C's explanation of will's centrality, see _AR_ , 153\u201360, 108 n, 246 n. 'I assume a something, the proof of which no man can give to another, yet every man may find for himself ( _AR_ , 154).\n\n Those seeking a more detailed explication of the relations among will, faith and reason should consult: Barth, _C and Christian Doctrine_ , esp. chs II, IV, V; Barfield, _What C Thought_ , esp. ch. 12; and James D. Boulger, _Cas Religious Thinker_ (New Haven, Conn.: Yale University Press, 1961), chs II, III, V. See also Laurence S. Lockridge, _Coleridge the Moralist_ (Ithaca, NY: Cornell University Press, 1977).\n\n 'Your know that every intellectual act, however you may distinguish it by name, in respect to the originating faculties, is truly the act of the entire man' ( _TT_ , 29 July 1830).\n\n On the psychological Trinity of reason, will and faith, see _LS_ , 62; _AR_ , 243 n. On the relation between faith and reason, see _AR_ , 72\u20134, 301\u20133; _LS_ , 47\u20138, 175\u20136. On reason and will (sometimes called speculative reason and practical reason), see _AR_ , 181, 234; _LS_ , 60, n. 2.\n\n On C and the history of doctrines of faith, see: Barth, C _and Christian Doctrine_ , pp. 31\u20133; Boulger, _C as Religious Thinker_ , pp. 37\u201342.\n\n Cs idea of self is related in complex ways to his theory of the Trinity, on which his major statement is the yet-unpublished _Opus Maximum._ Barth discusses this material in detail ( _C and Christian Doctrine_ , ch. IV; cf. _AR_ , 183, n. 2 ff.). See also McFarland, _C and_ _the Pantheist Tradition_ , ch. 4, esp. pp. 235\u201344. On personality and reason, see _F_ , I, 97\u20138. On personality and will, see _AR_ , 108.\n\n On conscience, see _LS_ , 66\u20137; _F_ , I, 150\u20131, 159; _AR_ , 145\u20136. On sin, see _AR_ , 248\u201350, 259\u201362, 273\u20134.\n\n The distinction between reason and understanding takes two forms, explained at length in the _Statesman's Manual_ (pp. 59\u201360 and nn) and the _Aids to Reflection_ (pp. 211\u201325, and 'Appendix', pp. 353\u20134). Regarded _theoretically_ , '... the Understanding... concerns itself exclusively with the quantities, qualities, and relations _of particulars_ in time and space. The UNDERSTANDING, therefore, is the science of phaenomena, and their subsumption under distinct kinds and sorts, ( _genus_ and _species)._ Its functions supply the rules and constitute the possibility of EXPERIENCE; but remain mere logical _forms_ , except as far as _materials_ are given by the senses or sensations. The REASON, on the other hand, is the science of the _universal_ [.]... The Reason first manifests itself in man by the _tendency_ to the comprehension of all as one. We can neither rest in an infinite that is not at the same time a whole, nor in a whole that is not infinite' ( _LS_ , 59\u201360). In its theoretical function, Reason discovers or invents theories, explanations in which all particulars are treated as a unified whole, as _one_ not a conglomerate: 'from individual (or particular) and contingent facts and forms [Reason] concludes universal, necessary, and permanent Truth' ( _LS_ , 19, n. 1). The _practical_ version of the distinction centers on morals, not inquiry. The practical reason is 'the power of determining the Will by Ideas, as _Ultimate_ ends'; it is often equated with will or with conscience ( _LS_ , 61 n). This power distinguishes man as a moral being from the animals. The practical Understanding is 'the faculty of selecting and adapting means to _proximate_ ends'; it is a more highly developed form of instinct ( _LS_ , 61 n). The two sets of distinctions are so closely related that C often discusses theoretical and practical aspects simultaneously. He calls the distinction between Reason and Understanding the _'Gradus ad Philosophiam' (TT_ , 14 May 1830); it is the first point that a student of his prose must master \u2013 although the distinction is an ancient one. C himself directs beginners first to _F_ , I 154\u201361, and then to _LS_ , 59\u201393 ( _CL_ , IV, 851). One should also consult _AR_ , 221\u20135, which richly illustrates the distinction, and relates it to the nature of language.\n\n On self-knowledge and the knowledge of God, see _AR_ , 79, n 2; _LS_ , 68, n. 3. On the Incarnation, see above, n. 6. The passage in quotation marks is from the Nicene Creed.\n\n For a statement of the progression from self-knowledge to knowledge of God, freedom and immortality, see _CCS_ , 47 n; _F_ , I, 112.\n\n Another, more common way of describing this is to say that imagination mediates between reason and understanding and again between understanding and sensation. Commentary on this issue has usually centered on a marginal note in Tenneman's _Geschichte der Philosophie_ , printed for the first time in _C on the Seventeenth Century_ , ed. Roberta Brinkley (Durham, NC: Duke University Press, 1955), pp. 693\u20134. J. A. Appleyard, SJ, writes that before _Biographia Literaria_ there is no discussion of imagination that gives the faculty a general epistemological function ( _C's Philosophy of Literature_ , p. 207). See below, Ch. 5, n. 27.\n\n This philosophic consciousness (secondary imagination) is not to be confused with Schelling's _intellektuelle Anschauung_ , which is a rare faculty whereby the mind catches itself in the act of creating the external world (see Shaffer, 'C's aesthetics', pp. 32\u201340; and Gian Orsini, _C and German Idealism_ (Carbondale: Southern Illinois University Press, 1969), p. 202). There are two principal differences. First, for C the external world is independently real and existent, not a creation of consciousness. Secondly, for C philosophic consciousness is not the knowledge of anything fundamentally unconscious but rather an awareness of one's activity as a thinker, an awareness of one's _activity_ in sorting, judging, attending or ignoring, believing, concluding. It is the ability to watch oneself think, and to reflect upon such observations.\n\n For relevant discussions of symbol, see _LS_ , 28\u201330, 69\u201370.\n\n As McFarland points out, the relations among imagination, understanding and reason are not entirely clear even in Kant's own works; further complexities derive from the fact that both Kant and C derive theories of imagination from Tetens (The origin and significance of C's theory of secondary imagination', in _New Perspectives on C and Wordsworth_ , ed. Geoffrey Hartman (New York: Columbia University Press, 1972), pp. 195\u2013246). George Whalley sorts through the technical philosophic issues underlying the relation between C and Kant on imagination in _Poetic Process_ (London: Routledge & Kegan Paul, 1953), pp. 46\u201363. See also A. O. Lovejoy, 'C and Kant's two worlds', _ELH_ , vol. 7 (1940), pp. 341\u201362.\n\n Among the exceptions is Michael G. Cooke, _The Romantic Will_ (New Haven, Conn., and London: Yale University Press, 1976): 'In this story of an individual which modulated into a story of being... the presence of the will, though it has gone largely unnoticed, is radical and pervasive' (p. 28). And Geoffrey Yarlott is no doubt correct when he describes C's life as dominated by his quest to unite thought and feeling or power and strength, and when he notes that C 'tended to identify this want of strength with some atrophy or deficiency of the _Will'_ ( _C and the Abyssinian Maid_ , p. 15). Yarlott's analysis of C's psychological difficulties suggests that the centrality of will in _BL_ must have had a personal significance that might have given the work a tighter unity for C than it ordinarily possesses for us.\n\n Mackintosh replies to C's charge of historical errors in a note to his _Dissertation on the Progress of Ethical Philosophy_ (Edinburgh: A. & C. Black, 1836), note T-U. Mackintosh's commentary on Hobbes and Hartley in this work suggests that he would not have praised them so richly as C here claims. C's notes on the first five lectures of the second (1800) series attribute to Mackintosh certain materialist suppositions refuted in chs V-VIII: e.g. 'Makes Idea (of course) mean Image' (see _CN_ , I, 634 and n). The story of Hartley's influence has often been told. Richard Haven argues persuasively that Hartley's linking associationist psychology to Christianity explains C's early interest in him. See 'C, Hartley, and the mystics', _Journal of the History of Ideas_ , vol. 20(1959), pp. 477\u201394; for a further development of the same general perspective on C's career, see also his _Patterns of Consciousness._ For a full and detailed history of C's associationism, see James V. Baker, _The Sacred River: C's Theory of the Imagination_ (Baton Rouge: Louisiana State University Press, 1957), esp. ch. 2.\n\n In commenting on the story of the serving girl, a review signed 'Oriel College, Oxford', complains that 'on such a vague and indefinite statement, no true philosopher could, we think, venture to found any serious speculation' ('David Hume charged by Mr. Coleridge with plagiarism from St. Thomas Aquinas', _Blackwood's Edinburgh Magazine_ , vol. 3 (Sept 1818), pp. 653\u20137). Yet prior claims about will show that the story is merely to illustrate or at most confirm something already known. The facts about will that it illuminates are among those that experience confirms but cannot teach. This short, interesting review stands apart from its fellows in praising C's 'singularly acute metaphysics', especially in chs V-VIII. The review is not listed or reprinted in William S. Ward (ed.), _Literary Reviews in British Periodicals, 1798\u20131820: A Bibliography_ , 2 vols (New York: Garland, 1972), or Donald Reiman (ed.), _The Romantics Reviewed: Contemporary Reviews of British Romantic Writers_ , 3 vols in 9 pts (New York: Garland, 1973).\n\n See _Enneads_ , I, 6 [4], [9], trans. Stephen Mackenna, 4th edn, rev. B. S. Page (New York: Pantheon Books of Random House, 1969), pp. 59, 63\u20134; and see also trans. Thomas Taylor, in _Thomas Taylor the Platonist_ , ed. with intro. Kathleen Raine and George Mills Harper, Bollingen Series LXXXVIII (Princeton, NJ: Princeton University Press, 1969), pp. 150, 158. C cites sun imagery form the _Enneads_ again in ch. XII (I, 167).\n\n For C on an author's obligation to remember his audience as part of his obligation to communicate truth, see _F_ , I, 34\u201369. On this reference to mysteries as part of a pattern in _BL_ , see Hunt, 'C and the endeavor of philosophy.\n\n For an explanation of the consequences of object-oriented philosophies such as Locke's or Spinoza's, see McFarland, _C and Pantheist Tradition_ , pp. 53\u20134, 69\u201370.\n\n On the relation between fancy and imagination, see also Ch. 8, below, esp. pp. 133\u201340.\n\n For an explanation of the necessary systematic or logical openness of C's philosophy, see McFarland, _C and the Pantheist Tradition_ , pp. 191\u20134, 53\u20137.\n\n _LS_ , 38; on popular philosophy, see also _LS_ , 35\u20138 and nn; _F_ , I, 34\u201367; _CN_ , III, 3281. On the quality of popular writing, see _EL_ , I, 25\u20137, 26 n.\n\n See in particular the 'letter' in ch. XIII, discussed above, Ch. 2, and below, Ch. 5.\n\n Haven, _Patterns of Consciousness_ , pp. 36\u20137.\n\n J. A. Appleyard, SJ, argues that C's theory of imagination did evolve from associationism. See 'C and criticism: I. critical theory', in _Writers and Their Background: STC_ , ed. R. L. Brett (Athens, Ohio: Ohio University Press, 1972), p. 128. See also Baker, _The Sacred River._\n\n On Hobbes's originality, see Shedd III, 209 n; Shawcross, I, 229; Mackintosh, _Dissertation on the Progress of Ethical Philosophy_ , p. 197. Shawcross's reference to Mackintosh's note 'S' is mistaken. The key to C's use of Hobbes may lie in Hobbes's commentary on wit, judgment and fancy, rather than in Mackintosh or mechanisms for association. C's use of both Hobbes and Descartes deserves further investigation.\n\n 'Schemes of conduct, grounded on calculations of self-interest, or on the average consequences of actions, supposed to be general, form a branch of political economy, to which let all due honor be given. Their utility is not here questioned. But however estimable within their own sphere such schemes, or any one of them in particular, may be, they do not belong to moral science, to which, both in kind and purpose, they are in all cases foreign, and, when substituted for it, hostile' ( _AR_ , 268). See also John Stuart Mill's famous distinction of the thinkers of his day into Coleridgeans and Benthamites ('Coleridge', in _Autobiography and Other Writings_ , ed. Jack Stillinger (Boston, Mass.: Houghton-Mifflin, 1969), pp. 259\u2013309, see esp. pp. 259\u201369).\n\n The 'Essays on Method' in _The Friend_ define the proper relation between methods of inquiry in the physical sciences and in the humanities ( _F_ , I, 448\u2013524). On C's resolution of the competing ontologies of science and religion, see _AR_ , 277, and C. Miles Wallace, 'C's _Biographia Literaria_ and the evidence for Christianity', in _Interspace and the Inward Sphere_ , ed. Norman A. Anderson and Margene E. Weiss (Macomb, Ill.: _Essays in Literature_ Books, 1978), pp. 19\u201332.\n\n John Arthur Passmore, _Ralph Cudworth_ (London: Cambridge University Press, 1951), pp. 5\u20136. The reference may be submerged because C in _BL_ defends Spinoza's philosophy as not necessarily incompatible with religion. In this regard, note the link between Spinoza and Leibnitz in ch. VIII's headnote. Cudworth's _The True Intellectual System of the Universe_ may have been one of C's resources in writing _BL._ According to Passmore, Cudworth's book offers a 'sustained polemic' against Hobbes, and against 'aetheistic materialism' (pp. 3, 9). Cudworth, like Coleridge, insists on the continuity of tradition in philosophy, and protests the modern emphasis on originality and innovation (pp. 13\u201314). Cudworth also insists that Cartesian 'clear and distinct' ideas need no evidence (p. 9), an argument C adapts in ch. X. C's emphasis on the errors of Cartesian dualism, rather than the errors of dualism _per se_ , may partly reflect Cudworth's admiration for the doctrine (Passmore, p. 8). Elsewhere, C describes Descartes' 'Mechanic or Corpuscular scheme' as leading to the philosophies of Berkeley and Spinoza \u2013 opposite forms of the same error ( _AR_ , 344 and n). Writing in the _Encyclopedia of Philosophy_ (New York: Macmillan, 1967), Passmore also notes that opposition to fanaticism was a principal concern of Benjamin Whichcote, influential teacher of Cudworth and the other Cambridge Platonists (Vol. II, p. 9). See also W. Schrickx, 'C and the Cambridge Platonists', _Review of English Literature_ , vol. 7 (1966), pp. 71\u201389.\n\n## **Chapter 4**\n\n Walter Jackson Bate explains that C 'wanted to cut off the word \"imagination\" from any associations it still had with a mere \"image-making\" faculty which produces, separates, or joins together images derived from sensation' ( _Coleridge_ (New York: Collier Books, 1973), p. 162). He does so not by wrenching 'imagination' into an entirely new meaning, but by more closely examining the relation of mind to world underlying the old definition. See also Thomas McFarland, The origin and significance of C's theory of secondary imagination', in _New Perspectives on C and Wordsworth_ , ed. Geoffrey Hartman (New York and London: Columbia University Press, 1972), pp. 195\u2013246; George Watson, '\"Imagination\" and \"Fancy,\"' _Essays in Criticism_ , vol. 3 (1953), pp. 201\u201314; W. J. Bate and J. Bullitt, 'Distinctions between fancy and imagination in eighteenth century criticism', _MLN_ , vol. 60 (1945), pp. 8\u201315; E. R. Wasserman, 'Another eighteenth-century distinction between fancy and imagination', _MLN_ , vol. 64 (1949), pp. 23\u20135; Wilma L. Kennedy, _The English Heritage of C of Bristol, 1798: The Basis in Eighteenth Century English Thought for His Distinction between Imagination and Fancy_ (New Haven, Conn.: Yale University Press, 1947).\n\n For more on C's interest in principles, see Jackson, _Method and Imagination_ , pp. 21\u201347.\n\n See Shaffer, The \"Postulates in Philosophy\" in the _Biographia Literaria', Comparative Literature Studies_ , vol. 7 (1970), pp. 297\u2013313.\n\n On faith and human intelligence, see _AR_ , 61\u20132. On the grounds of our knowledge of will, see _AR_ , 122. On the union of religion and morals re logic and language, see _AR_ , 106\u20138, cf. _AR_ , 129 n). On proofs of God, see _TT_ , 22 Feb 1834; _AR_ , 187\u20138.\n\n 'Nether Stowey' and 'a cottage in Somersetshire' refer to the same place.\n\n Compare self-descriptions at I, 14, 62\u20135, 115, 137, 145\u201351.\n\n## **Chapter 5**\n\n Shaffer, 'C's aesthetics'. Shaffer translates the relevant sections from Schelling's _System of Transcendental Idealism_ in an appendix. The principal arguments of the dissertation reappear in three journal articles, but the discussion of Schelling and the _Biographia_ is so severely condensed that the thesis remains preferable. See also 'The \"Postulates in Philosophy\" in the _Biographia Literaria_ '; 'C's theory of aesthetic interest', _Journal of Aesthetics and Art Criticism_ , vol. 27 (1969), pp. 399\u2013408; 'C's revolution in the standard of taste', ibid., vol. 28 (1969), pp. 213\u201321.\n\n _CN_ , III, 4265 n. McFarland both analyzes and extends the work that has already been done in _C and the Pantheist Tradition_ , ch. 1. The most detailed and reliable analysis that judges C derivative is Orsini, _C and German Idealism_. According to Orsini, C's 'rightful place in the history of philosophical ideas' is as a translator of Schelling (p. 221). One should also consult Ren\u00e9 Wellek's discussions of Coleridge in _The Romantic Age_ , in _A History of Modern Criticism_ , Vol. II (New Haven, Conn.: Yale University Press, 1955), pp. 151\u201387; and in _The English Romantic Poets: A Review of Research and Criticism_ , ed. Frank Jordan, Jr, 3rd edn (New York: MLA, 1972), pp. 209\u201358. One interested in C's comments on the matter might begin with _CN_ , II, 2375 and 2546; CN, I, 1695.\n\n For instance, _CCS_ , 44\u20135 echoes the Cis- and Trans-Alpine passage (I, 164\u20137); _LS_ , 137, repeats the first half of the scholium to Thesis II (I, 174); _AR_ , 108 n, repeats the definitions of nature, subject and object (I, 174); _AR_ , 122 and 224 n, clarifies the comparison between the transcendental philosopher and the geometer (I, 171\u20133). Given C's habitual incorporating and rewriting of others' texts, his rewriting of his own work cannot by itself constitute evidence of dissatisfaction with the original. But one does find direct evidence of such dissatisfaction. As early as July 1817, C refers to 'a few opinions which better information and more reflection would now annul. But even these will, I trust, be found only in the lesser branches, as knotts [sic] & scars that may exist without implying either canker at the root, or malignant quality in the general sap of the tree' ( _CL_ , IV, 758). And in September of 1818 he criticizes Schelling's 'making all knowledge bi-polar, Transcendental Idealism as one Pole and Nature as the other', and laments his use of the idea in _BL_ ( _CL_ , IV, 874). As I describe below, C's sharp division between transcendental idealism and the total philosophy does not successfully distinguish ch. XII's philosophy from pantheism (below, pp. 81\u20134).\n\n On C's views of the relation between art and religion, see _LS_ , 62; _COS_ , 88\u20139.\n\n See McFarland, _C and the Pantheist Tradition_ , pp. 116\u201323.\n\n See Kathleen Coburn, 'The interpenetration of Man and Nature', _Proceedings of the British Academy_, vol. 49 (1963), pp. 95\u2013113, for an explanation of this theme in C's thought.\n\n This belief about common consciousness probably underlies C's early interest in Hartley and associationism. See above, Ch. 3, n. 16.\n\n Roman numerals following a page number indicate line numbers, which I will provide whenever such exact reference may be helpful.\n\n See Shaffer, 'C's aesthetics', pp. 1\u201330, for an explanation of the technical issue with which Schelling wrestles. C translates his specialized references to geometry into an explanatory analogy.\n\n J. B. Beer, _C the Visionary_ (London: Chatto & Windus, 1959), pp. 70\u20131.\n\n See above, Ch. 3, pp. 30\u201342, and Ch. 4, pp 61\u20133.\n\n This traditional reference is a favorite of C's. See _LPR_ , 94, n. 3, for a listing of other instances, and of his sources. For discussion, see C. Miles Wallace, 'C's theory of language', _Philological Quarterly_ , vol. 59 (1980), pp. 338\u201352.\n\n See McFarland, _C and the Pantheist Tradition_ , pp. 53\u20138, 107\u201312. C's 'original realism' is not, as Appleyard claims, a na\u00efve position to which he falls back because he lacks a theory ( _C's Philosophy of Literature_ , p. 196).\n\n For thesis I, see I, 94, 174; for thesis II, see I, 168; for thesis III, see I, 168, 172\u20134, 178; for thesis V, see I, 66, 175\u20138; for thesis VI, see I, 94\u20135.\n\n 'Organic' is a more complicated and specialized term in C's thought than it later became in critical theory generally. See, e.g., _TT_ , 18 Dec 1831, where he proposes 'a sheaf of corn' as an example of the 'inorganic'. The issue is explicated at length in the _Theory of Life_. On organicism generally, see, e.g., Barfield, _What C Thought_ , pp. 41\u201362 and 210 n. 3; R. H. Fogle, _The Idea of C's Criticism_ , pp. 18\u201368; M. H. Abrams, _The Mirror and the Lamp_ (London: Oxford University Press, 1953), esp. pp. 167\u201377, 218\u201355; and James Benziger, 'Organic unity: Leibnitz to Coleridge', _PMLA_ , vol. 46, no. 2 (1951), pp. 24\u201348.\n\n Not all agree that these are equivalent terms, but the alternative is to conclude that there must be as many kinds of imagination as there are kinds of human endeavor. S. V. Pradhan analyzes the patterns of C's usage, and concludes this is the case ('C's \"Philocrisy\" and his theory of fancy and secondary imagination', _Studies in Romanticism_ , vol. 13 (1974), pp. 235\u201354). Parsimony is better served by agreeing with Barfield that C shifts vocabulary to suit his context ( _What C Thought_ , p. 76). As an example, compare I, 167, xvii\u2013xxvii, and I, 173, x\u2013xxxi. I take it that in the second passage C refers to the effective development of the faculty, and in the first to its 'genotypic' presence.\n\n In the _Philosophical Lectures_ , C traces how philosophers and scientists, working from 'opposite' ends, discover the need to postulate a single force, which Christ reveals as the personal triune God.\n\n My position depends in large measure on taking seriously C's division of all inquiry into three domains. Grosvenor Powell reads this thesis (in effect) as part of the total philosophy, and argues well that C does establish an infinite regress that badly confuses the distinction between God and man. See 'C's \"imagination\" and the infinite regress of consciousness', _ELH_ , vol. 39 (1972), pp. 266\u201378. On the validity of this division as a bulwark against pantheism, see below, pp. 81\u20134.\n\n For C's response to the 1815 preface as part of the genesis of _BL_ , see above, Ch. 1, n. 4.\n\n Arthur Lovejoy, _The Great Chain of Being_ (Cambridge, Mass.: Harvard University Press, 1936, 1964), esp. ch. III.\n\n Cited in translation supplied by Watson in the Everyman's Library edition (1965), p. 161, n. 2.\n\n Cited in translation supplied by Watson, ibid., p. 162, n. 1. Compare a different translation of the same lines in a notebook entry, _CN_ , III, 4189 n. The _dance_ metaphor is present in the Greek.\n\n This idea is explored at length in the _Theory of Life_. See also above, Ch. 5, nn. 6 and 15.\n\n See _CL_ , IV, 728.\n\n McFarland, _C and the Pantheist Tradition_ , p. 156.\n\n See above, Ch. 3, n. 17.\n\n These definitions are the _locus classicus_ for arguments about C's theory of imagination. These arguments can be very roughly divided into two sorts: those principally concerned with the theory as part of C's philosophy, and those predominantly concerned with its role in his criticism. On the second of these groups, see below, Ch. 6, nn. 4, 5, 7. Arguments about the philosophical significance of imagination have more or less centered on the problematic relation between reason and imagination. To what extent do the qualities C attributes to imagination and all its works essentially derive from reason itself? What are the proper qualities of imagination, as distinguishable from the powers and elements it synthesizes? The questions are complicated enormously by C's intricate relations to Kant, to Schelling, to Tetens, and to others (see above, Ch. 3, n. 14). Principal studies include: Barfield, _What C Thought_ ; James D . Boulger, 'C on imagination revisited', _The Wordsworth Circle_ , vol. 4 (1973), pp. 13\u201324; Jackson, _Method and Imagination_ ; Roy Park, 'C and Kant: poetic imagination and practical reason', _British Journal of Aesthetics_ , vol. 8 (1968), pp. 335\u201346, 'Coleridge's two voices as a critic of Wordsworth', _ELH_ , vol. 36 (1969), pp. 361\u201381, and 'Coleridge: philosopher and theologian as literary critic', _University of Toronto Quarterly_ , vol. 38 (1969), pp. 17\u201333; Pradhan, 'C's \"Philocrisy\" and his theory of fancy and secondary imagination'. The questions remain vexing and, in the last analysis, probably without adequate answers. The powers of the mind are not independent programs which can each run the same computer. When one studies the mind in strict abstraction, it is easy enough to define reason, to define imagination, and to distinguish the definitions. In a parallel way, anatomy is distinct from physiology, from biochemistry, from pathology, and from genetics. But when one studies the mind not as an abstraction but as an act, then the tidy distinctions begin to blur. As C insists, when the mind acts, the whole mind acts; and poetry requires the richest, fullest mental activity. When a gleeful child runs pell-mell down the street, who can say what is due to anatomy, to physiology, to genetics? The more closely one studies the interactivity of reason and imagination, the farther one moves from abstraction to life.\n\n Sara Coleridge observes that her father later lined out this clause in a copy of the book (Shedd III, p. 363 n). Shawcross's repetition of her note refers somewhat ambiguously to the 'sentence', leading some readers to believe that C had deleted _all_ reference to primary imagination. _BL_ provides C's only statements concerning primary imagination.\n\n## **Chapter 6**\n\n My views on C's idea of language appeared first as 'C's theory of language'.\n\n R. H. Fogle, _The Idea of C's Criticism_ , p. 8.\n\n Compare Wordsworth: 'If words be not... an incarnation of the thought but only a clothing for it, then surely will they prove an ill gift' ('Essays upon Epitaphs, III', _The Prose Works of William Wordsworth_ , ed. W. J. B. Owen and Jane Worthington Smyser (Oxford: Clarendon Press, 1974), Vol. II, pp. 84\u20135; cited afterwards as _Pr. Wk W._ ). On Wordsworth's idea of language in relation to his idea of poetry, see M. H. Abrams, 'Wordsworth and C on diction and figures', _English Institute Essays, 1952_ , ed. Alan S. Downer (New York: Columbia University Press, 1954) pp. 171\u2013201; Frances Ferguson, _Wordsworth: Language as Counter-Spirit_ (New Haven, Conn.: Yale University Press, 1977); and W. J. B. Owen, _Wordsworth as Critic_ (Toronto: University of Toronto Press, 1969). See also below, Ch. 7, nn. 5, 12.\n\n R. H. Fogle, for instance, in his introduction to Baker's _The Sacred River_ , describes C as offering 'a rallying-cry' and a 'trumpet call' to 'our famous New Critics' (p. xii). But C has a wonderfully complex relation to these critics, who agreed with many of his statements about the character of an artwork, but who understood the mode of relation between art and values in different terms. One alternative, then, is to argue that C's criticism and his philosophy are incompatible (and, often, that his philosophy is a shabby, incoherent, inconsequent quilt of plagiarisms). The strongest of such arguments are by Ren\u00e9 Wellek (see above, Ch. 5, n. 2); for specific application to _BL_ , see Appleyard, _C's Philosophy of Literature_ , esp. p. 252. New Critics and Neo-Humanists alike none the less criticized C's work. Clarence D. Thorpe summarizes such criticism, and defends C in 'C as aesthetician and critic', _Journal of the History of Ideas_ , vol. 5 (1944), pp. 387\u2013414. For a lengthy and quite polemical account of C's relation to New Criticism, see Manfred Wojcik, 'The mimetic orientation of C's aesthetic thought', _Zeitschrift f\u00fcr Anglistik und Amerikanistic_ , vol. 17 (1969), pp. 344\u201391. Wojcik develops his point in three subsequent numbers of the same journal, writing what ought to have been a book on the topic. See also 'C and the problem of transcendentalism', ibid., vol. 18 (1970), pp. 30\u201358; 'C: symbol, organic unity, and modern aesthetic subjectivism', ibid., vol. 18 (1970), pp. 355\u201390; and 'C: symbolization, expression, and artistic creativity', ibid., vol. 19 (1971), pp. 117\u201354. On the difference between C's concept of organic form and the New Critical account, see R. S. Crane, The critical monism of Cleanth Brooks', _Critics and Criticism_ , ed. R. S. Crane (Chicago, Ill.: University of Chicago Press, 1952), pp. 82\u2013107.\n\n See, for instance, Walter Jackson Bate, 'C on the function of art', _Perspectives of Criticism_ , ed. Harry Levin (Cambridge, Mass.: Harvard University Press, 1950), pp. 129\u201359, and his _Coleridge_ , pp. 146\u201357; R. H. Fogle, _The Idea of C's Criticism_. M. H. Abrams deftly summarizes this approach in _The Mirror and the Lamp_ , pp. 118\u201319.\n\n See above, Ch. 6, n. 4. The error here, I suspect, is excessively assimilating C's view of art to that of the German transcendental idealists, esp. Schelling. Roy Park sharply and properly distinguishes C from his German forebears and contemporaries, and concludes that the secondary imagination is not free but subordinated to the moral vision of reason. See above, Ch. 5, n. 27. See also below, pp. 100\u2013102, on C's distinction between the immediate and the ultimate purposes of art.\n\n There is a considerable body of critical commentary attempting to reconcile C's philosophy with his criticism, much of it focused on the relation between imagination's role in cognition and its role in art. One error is particularly common: some claim that primary imagination provides access not to the day-to-day world but, rather, to a reality already translucent with value. Those who define primary imagination in this way then define secondary imagination as the specifically artistic power of making poems from the materials thus supplied, through a sort of transcendentalized mimeticism. This is the wrong route to a crucial distinction between imaginative vision and artistic production. C locates the distinction not between primary and secondary imagination, but between imaginative power and linguistic control over ideas and poetic forms. The best studies include Abrams, _The Mirror and the Lamp_ ; Baker, _The Sacred River_ , J. Robert Barth, SJ, _The Symbolic Imagination: C and the Romantic Tradition_ (Princeton, NJ: Princeton University Press, 1977); Stephen Prickett, _Coleridge and Wordsworth: The Poetry of Growth_ (London: Cambridge University Press, 1970); Whalley, _Poetic Process_ , pp. 46\u201363; and see also Ch. 5, n. 27. For an overview of the questions from the perspective of English literary theory, see R. L. Brett, 'Coleridge's theory of imagination', _Essays and Studies_ , n.s., vol. 2 (1949), pp. 75\u201390. For the same overview from the Continental perspective, see Herbert Read, 'C as critic', _Sewanee Review_ , vol. 56 (1948), pp. 597\u2013624; and of course McFarland, _C and the Pantheist Tradition_.\n\n The volume division was the printer's choice, not C's. See below, Chapter 8, n. 11. In this approach I am of course preceded by I. A. Richards. There needs to be much further study of these issues by one well versed in present-day philosophies of language.\n\n On the Lockean tradition in linguistics, see Hans Aarsleff, _The Study of Language in England, 1780\u20131860_ (Princeton, NJ: Princeton University Press, 1967), pp. 13\u201333; and Stephen K. Land, _From Signs to Propositions: The Concept of Form in Eighteenth Century Semantic Theory_ , Longman Linguistics Library no. 16 (London: Longman, 1974), pp. 1\u201320. On the contrast represented between Locke and Wilkins, see Murray Cohen, _Sensible Words: Linguistic Practice in England, 1640\u20131785_ (Baltimore, Md.: The Johns Hopkins University Press, 1977), pp. 1\u201342. On the tradition in which C participates, see Ernst Cassirer, _The Philosophy of Symbolic Forms_ , trans. Ralph Mannheim (New Haven, Conn.: Yale University Press, 1953), Vol. I, _Language_ , pp. 117\u201376, esp. re Heraclitus (pp. 119\u201322), Plato (pp. 124\u20136), Berkeley (pp. 138\u20139), Herder (pp. 151\u20133) and von Humboldt (pp. 155\u201363). For Cassirer's delineation of the position underlying this history, see pp. 85\u201393. Cassirer would call C's work an attempt to define a 'metaphysical-speculative solution' that seeks 'to understand how the concrete totality of particular forms develops from a single original principle' (p. 95). C would respond that a 'critical solution' like Cassirer's is 'a cycle of equal truths without a common and central principle, which prescribes to each its proper sphere' (I, 181). It is no more likely, he would say, than the string of blind men walking a straight line without a sighted leader.\n\n _Hermes; or a Philosophical Inquiry concerning Universal Grammar_ (1751; rev. ed. London: W. Simprin & R. Marshall, 1816), pp. 113, 119\u201321; see also pp. 123\u201331.\n\n Cited by Alice D. Snyder, _C on Logic and Learning_ (New Haven, Conn.: Yale University Press, 1929), p. 79, 86.\n\n ibid., p. 128.\n\n See above, Ch. 1, pp. 9\u201311.\n\n See above, Ch. 3, pp. 63\u201374.\n\n See also the 'Essays on Method' in _The Friend_. Method and language are both essentially relational; method itself is concerned both with inquiry and with discourse. See also above, Ch. 6, n. 1.\n\n The relation between language and mathematics had been variously explored both by philosophers and by those eager to make language a more precise tool. See, for instance, David Hartley, _Observations on Man: His Frame, His Duty and His Expectations_ (1749; facsimile reprint in 1 vol., Gainesville, Fla: Scholar's Facsimiles and Reprints, 1966), I, 280\u20131. See also Land, _From Signs to Propositions_ , pp. 128\u201354.\n\n On C's etymologies and his relation to scientific linguistics, see James Holly Hanford, 'C as a philologian', _Modern Philology_ , vol. 16 (1918\u201319), pp. 615\u201336; Joshua H. Neumann, 'C on the English language', _PMLA_ , vol. 63 (1948), pp. 642\u201361; L. A. Willoughby, 'C as a philologist', _Modern Language Review_ , vol. 31 (1936), pp. 176\u2013201.\n\n On C's blending of moral and literary concerns, see Peter Hoheisel, 'C on Shakespeare: method amid the rhetoric', _Studies in Romanticism_ , vol. 13 (1974), pp. 15\u201323.\n\n On C's account of their aims, see Mark L. Reed, 'Wordsworth, C, and the 'plan' of the _Lyrical Ballads_ ', _University of Toronto Quarterly_ vol. 34 (1965), pp. 238\u201353.\n\n On light as a central symbol of imagination for both poets, see Stephen Prickett, _Coleridge and Wordsworth: The Poetry of Growth_.\n\n On how C's idea of this reflexivity differs from more recent versions, see R. S. Crane, 'The critical monism of Cleanth Brooks', in _Critics and Criticism_ , ed. R. S. Crane, pp. 82\u2013107. See also James Benziger, 'Organic unity: Leibnitz to C', _PMLA_ , vol. 46, no. 2 (1951), pp. 24\u201348.\n\n See also his study of _Romeo and Juliet, COS_ , 75\u201397.\n\n For defenses of Wordsworth, see below, Ch. 7, nn. 5, 12, 13, 20; Ch. 8, nn. 7, 8.\n\n Cited in translation supplied by Watson in the Everyman's Library edition, pp. 182\u20133, nn. 1\u20133.\n\n## **Chapter 7**\n\n Abrams, 'Wordsworth and C on diction and figures', pp. 171\u2013201. See also above, Ch. 6, n. 3.\n\n On C's role in the Preface, see _CL_ , II, 811\u201312, 829\u201330, and Max F. Schulz, 'C, Wordsworth, and the 1800 \"Preface\" to _Lyrical Ballads_ ', _Studies in English Literature 1500\u20131900_ , vol. 5 (1965), pp. 619\u201339. On the relation between Wordsworth and C generally, see Earl Leslie Griggs, 'Wordsworth through C's eyes', in _Wordsworth: Centenary Studies Presented at Cornell and Princeton Universities_, ed. Gilbert T. Dunklin (Princeton, NJ: Princeton University Press, 1951), pp. 45\u201390; Thomas McFarland, 'The symbiosis of C and Wordsworth', _Studies in Romanticism_ vol. 11 (1972), pp. 263\u2013303; Prickett, _Coleridge and Wordsworth: The Poetry of Growth_. On the relation between C's criticism of the Preface and that in the magazines, see John O. Hayden, 'C, the reviewers, and Wordsworth', _Studies in Philology_ , vol. 68 (1971), pp. 105\u201319. Those who defend Wordsworth from C's criticism are cited in notes below (5, 12, 13, 20); for a defense of C's position, see T. M. Raysor, 'C's criticism of Wordsworth, _PMLA_ , vol. 54 (1939), pp. 496\u2013510.\n\n W. K. Wimsatt and Cleanth Brooks, _Literary Criticism: A Short History_ (New York: Alfred A. Knopf, 1966), pp. 339\u201343.\n\n _Pr. Wk W._ , I, 160\u20131.\n\n Don H. Bialostosky, 'C's interpretation of Wordsworth's Preface to _Lyrical Ballads_ ', _PMLA_ , vol. 93 (1978), pp. 912\u201324. As Wellek observes, Wordsworth's later qualifications of the intent to imitate the real language of men 'surely leaves all the leeway anybody could demand' ( _A History of Modern Criticism_ , Vol. II, p. 134).\n\n _Pr. Wk W._ , I, 134. The italicized statement in the _Biographia_ is a compound of two of Wordsworth's sentences.\n\n ibid., I, 164.\n\n ibid., I, 137\u201343, esp. I, 138.\n\n Wimsatt and Brooks, _Literary Criticism_ , p. 343.\n\n See Nathaniel Teich, 'C's _Biographia_ and the contemporary controversy about style', _The Wordsworth Circle_ , vol. 3 (1972), pp. 61\u201370.\n\n On grammar and logic, see my Ch. 6, n. 1.\n\n For a defense of Wordsworth's objectivity, see Gene W. Ruoff, 'Wordsworth on language: toward a radical poetics for English Romanticism', _The Wordsworth Circle_ , vol. 3 (1972), pp. 204\u201311; for a defense of Wordsworth's essential subjectivity, see Frederick A. Pottle, 'The eye and the object in the poetry of Wordsworth', in Dunklin (ed.), _Wordsworth: Centenary Studies_ , pp. 23\u201343; for an argument that the real issue is the propriety of dramatic monologues, see Stephen Maxfield Parrish, 'The Wordsworth-C controversy', _PMLA_ , vol. 73 (1958), pp. 367\u201374.\n\n For a defense of 'The Thorn', see ibid.\n\n _Pr. WK W._ , I, 124. See also Ch. 7, n. 5, and Ch. 6, pp. 95\u20136.\n\n I have argued elsewhere that poems most fully meet the criteria for methodical discourse. See Ch. 6, n. 1.\n\n Richards, _C on Imagination_ , ch. 4. On the poetic function of fancy, see Baker, _The Sacred River_.\n\n This passage echoes the opening paragraphs of the 'Essays on Method' ( _F_ , I, 448\u20139). The 'Essays on Method' argue even more clearly than the opening chapters of _BL_ that genuine education does and ought to foster the development of imagination. On method, see Paul Alkon, 'Critical and logical concepts of method from Addison to C', _Eighteenth Century Studies_ , vol. 5 (1971), pp. 97\u2013121.\n\n See Abrams, _The Mirror and the Lamp_ , pp. 100\u201324; and above, Ch. 6, n. 3.\n\n _Pr. Wk W._ , I, 144\u20138. For discussion of this same disagreement from Wordsworth's perspective, see Stephen Maxfield Parrish, 'Wordsworth and C on metre', _Journal of English and Germanic Philology_ , vol. 59 (1960), pp. 41\u20139.\n\n John Crowe Ransom, 'William Wordsworth: notes toward an understanding of his poetry', in Dunklin (ed.), _Wordsworth: Centenary Studies_ , pp. 91\u2013113.\n\n Wordsworth, of course, did not contend that they were twins. See _Pr. Wk W._ , I, 134.\n\n## **Chapter 8**\n\n John Livingston Lowes, _The Road to Xanadu: A Study in the Ways of the Imagination_ (Boston, Mass.: Houghton-Mifflin, 1927), p. 49. See also above, Ch. 7, n. 16.\n\n For an account of this urging, see David Erdman and Paul Zall, 'C and Jeffrey in controversy', _Studies in Romanticism_ , vol. 14 (1975), pp. 75\u201383.\n\n _CL_ , V, 95. See also Park, 'C's two voices as a critic of Wordsworth', _ELH_ , vol. 36 (1969), pp. 361\u201381.\n\n See I, 87: by concentering the attention, will can render any objects 'associable'. Because this function is not bound by the literal contemporaneity of the original impressions, the genius's immediate impressions _and his memories_ can be accommodated to imagination's ends. See also above, Ch. 3, pp. 46\u20139.\n\n II, 106\u20137; see also II, 33\u20134 and n; II, 101\u20132; and George Whalley, 'The Aristotle-C axis', _University of Toronto Quarterly_ , vol. 42 (1973), pp. 93\u2013109.\n\n On dramatic monologues and dialogues, see Parrish, 'The Wordsworth-C controversy.\n\n Among the most astute defenses of the 'Intimations' ode are those by Richards and Ransom. See _C on Imagination_ , pp. 130\u20137; and above, Ch. 7, n. 20.\n\n At least parts of this treatise are present in _Aids to Reflection_ (see Ch. 6, n. 1). I suspect that C's interests in language substantially merged with his interests in Biblical hermaneutics and the Higher Criticism. See below, ch. 9, n. 2.\n\n Cited in translation supplied by Watson in the Everyman's Library edition, p. 268 n.\n\n For an account of C's problems with the printer, see Grigg's introduction, _CL_ , III, xlvii\u2013lii, and his notes to some of the relevant letters, _CL_ , IV, 657\u2013660 nn.\n\n See above, Ch. 3, and _LS_ , 137; _AR_ , 246 and n.\n\n## **Chapter 9**\n\n McFarland, _C and the Pantheist Tradition_ , p. xxvi.\n\n For an account of C's place in Biblical hermaneutics and the Higher Criticism, see Elinor S. Shaffer, _'Kubla Khan' and the Fall of Jerusalem: The Mythological School in Biblical Criticism and Secular Literature, 1770\u20131880_ (London: Cambridge University Press, 1975).\n\n Boulger, _C as Religious Thinker_ , pp. 5\u20138.\n\n Note that arguments to and about political leaders are barely evident in the principal summary of his argument ( _LS_ , 43\u20139).\n\n Peter Hoheisel, 'C on Shakespeare: method amid the rhetoric'.\n\n# **Index**\n\nAbrams, M. H: on Coleridge's influence on literary criticism\n\n_Aids to Reflection_ , , \u2013, , ; on human nature ; 'The Bosom Sin' ; duality\n\n'Ancient Mariner' stories ,\n\nAnecdotes \u2013; autobiographical , , ; delirious serving girl\n\nAristotle , , ,\n\nArt: Schelling's view of ; great art ; moral depths of\n\nAssociationism ; laws of , , ; tradition of ; mental control over ; and Wordsworth\n\nAutobiography ; used as illustration , ; symbolic of Coleridge's experience ; intellectual nature ; provides continuity , , ; artifice of\n\nBacon, Francis\n\nBaker, James V: _The Sacred River_\n\nBarfield, Owen ; on understanding _Biographia Literaria_ ; on anonymous critics ; on separative projection\n\nBarth, J. Robert: on imagination ; on faith\n\nBeer, J. B: on circle\n\nBerkeley, George: _Siris_\n\n_Bertram_ : critique of\n\nBialostosky, Don H: on poetic diction\n\nBoulger, James D: on _Aids to Reflection_\n\nBowles, William , ,\n\nBowyer, Rev. James ,\n\nBrevity of style\n\nBurke, Edmund ,\n\nCarlyle, Thomas \u2013,\n\nCassirer, Ernst: on linguistics\n\nCausality , ,\n\nChaucer, Geoffrey ,\n\n'Christabel'\n\nChristianity ; orthodox interpretation of God\n\n_Church and State, seeOn the Constitution of the Church and State_\n\nCicularity of argument\n\nCoburn, Kathleen: on Coleridge and Schelling\n\nColleridge, H. N: on autobiographical elements\n\nColeridge, Samuel Taylor: changing reputation ; defence of _Biographia Literaria_ \u2013; educational influences ; personality defined ; on Chapters XII and XIII ; recovery from problems ; victim of literary inquisition ; and publication \u2013\n\nColeridge, Sara: on difficulty of _Biographia Literaria_ ; on autobiographical elements\n\n_Coleridge and the Pantheist Tradition, see_ McFarland, T.\n\nComposition, process of \u2013\n\nCondensation of style\n\nConsciousness: defended ; common and philosophic \u2013\n\nConservatism\n\nContinuity of design: supplied by autobiography\n\nCriticism: not empirical science ; and defamation of Southey ; New Criticism ; aim of\n\nCritics, anonymous: lack of imagination ; contrasted with Coleridge ; engaged in literary game ; corruption of language by ; misunderstanding of Wordsworth , ; Coleridge's attack on ; attacks on Coleridge\n\nCudworth, Ralph: on Spinoza\n\nDaniel, Samuel ,\n\nDante Alighieri\n\nDe Quincey, Thomas: on illustrations\n\nDescartes, Ren\u00e9 , ,\n\n_Descriptive Sketches, see_ Wordsworth, William\n\nDesign: definition of\n\nDiction, poetic , \u2013; and the imagination ; Wordsworth on , \u2013; best ; verbal precision\n\nDifficulty of _Biographia Literaria_ ; acknowledged in _The Friend_ \u2013; recognized by Coleridge \u2013\n\nDigression \u2013; inherent unity \u2013; in _Philosophical Lectures_\n\nDisorder\n\nDrayton, Michael\n\nDualism\n\nDuality of design , ,\n\n_Ecclesiastical Polity, see_ Hooker, R.\n\n_Edinburgh Review_\n\nEmpiricism: opposite of genius\n\nEpictetus\n\nEpistemology ; Coleridge's defined \u2013\n\n_Essay concerning Human Understanding, see_ Locke, J.\n\n'Essay on faith'\n\n_Essay toward a Real Character, see_ Wilkins, J.\n\n'Essays on Method' ( _The Friend_ ) , ,\n\n_Ethics, see_ Spinoza, B.\n\nEtymology, _see_ Words\n\nEvans, Mary\n\nExaggeration\n\nExperience: illustrates ideas\n\nFaith , ,\n\nFanatic: v. genius , , ; psychology of ; works of\n\nFancy: essential feature of poetry ; defined ; interaction with imagination ; Wordsworth's view of ; dependent upon imagination ; governed by imagination ; supplies material of poet\n\nFichte, Johann Gottlieb ; and mystics\n\nFogle, R. H: on Coleridgean method ; on Coleridge's critical system\n\nFormal disorder\n\n_Friend, The_ ; acknowledges difficulty of _Biographia Literaria_ \u2013; stresses reader's necessary powers of thought ; quotes Plato ; on pantheism ; duality of design\n\nGalen\n\nGenius: v. fanatic ; philosophic ; psychology of ; lies in imagination ; poetic ; Wordsworth as \u2013\n\nGerman Romanticism\n\n'Gipsies', _see_ Wordsworth, William\n\nGod: orthodox interpretation of\n\nGray, Thomas\n\nHarris, James: on words\n\nHartley, David , , , ; and real language ; influence on Wordsworth ,\n\nHaven, Richard: on Coleridge as psychologist ; on 'Ancient Mariner' ; on readership\n\nHazlitt, William\n\nHerbert, George , ,\n\n'Higginbottom' sonnets\n\nHobbes, Thomas , ,\n\nHoheisel, Peter\n\nHooker, Richard: _Ecclesiastical Polity_\n\nHorace\n\nHume, David: on closed logical systems ; on causality\n\nHunt, Bishop C, Jr: on Coleridge's prose\n\nHypostasis\n\n'I wandered lonely', _see_ Wordsworth, William\n\nIdea: defined\n\n_Iliad, see_ Pope, A.\n\nIllustration \u2013; autobiographical , ,\n\nImagery: use of by fanatic ; in poetry ,\n\nImagination: not pantheistic ; Wordsworth as illustration of ; defined ; sign of genius ; table of distinctions \u2013; cognitive ; imaginative synthesis ; interaction with fancy ; act not tool ; philosophical role of ; moral implications of ; governs fancy ; logical incompleteness\n\nImagination, primary\n\nImagination, secondary: and will ; and Ten theses ; poetic vision \u2013\n\nImmortality\n\n'Intimations' ode, _see_ Wordsworth, W.\n\nItalian poets\n\nJackson, J. R. de J: on intricacies of construction\n\nJames, Henry ,\n\nKant, Immanuel \u2013,\n\nKnowing ; and self-consciousness\n\nLanguage: theory of ; poetic , , ; and truth ; analogous to mathematics ; corruption by anonymous critics ; rustic \u2013\n\n_Lay Sermons_ : defence of _Biographia Literaria_ \u2013; on ideas \u2013; on symbols ; clumsy design\n\nLeibnitz, Gottfried ,\n\n'Letter from a friend' \u2013\n\nLiterature, theory of\n\nLocke, John: materialism ; philosophy of ; linguistics ; _Essay concerning Human Understanding_ ; causality\n\n'Logic'\n\nLogosophia , , , , , , , , ,\n\nLovejoy, Arthur\n\nLowes, John L: _The Road to Xanadu_\n\n_Lyrical Ballads_ ; controversy , , ; plan for , ; 1800 Preface ; 1802 Preface ; 1805 Preface ; 1815 Preface , ; Appendix\n\nMcFarland, Thomas , ; on plagiarism from Schelling ; on symbols ; _Coleridge and the Pantheist Tradition_ ; on Coleridge's philosophy ; on Coleridge's writing technique \u2013\n\nMackintosh, Sir James ; lectures , ; claims\n\n'Mad Mother, The', _see_ Wordsworth, W.\n\nMallette, Richard: on autobiography\n\nMaterialism: Lockean ; refuted by Coleridge ; English ; and monism\n\nMechanism ,\n\nMetaphors: provide unity \u2013; cognitive ; water-insect ,\n\n_Metaphors of Self, see_ Olney, J.\n\nMetaphysical poets\n\n'Method, Essays on' ( _The Friend_ ),\n\nMetre , \u2013\n\n'Michael', _see_ Wordsworth, William\n\nMilton, John ; _Paradise Lost_ ,\n\nMonism\n\nNarrative: link with philosophy\n\nNature: language of God\n\nNeoplatonism\n\nNew Criticism\n\nNewton, Sir Isaac\n\n_Notebooks_ : on illustrations ; reason as an act ; on projected work\n\nNozy, Spy\n\nObscurity , ; of Ten Theses ; set out in 'Letter from a friend'\n\nOlney, James: _Metaphors of Self_ \u2013\n\n_On the Constitution of the Church and State_ ,\n\n'Opus Maximum'\n\nPantheism , , , , , , , ; in Wordsworth , , ,\n\n_Paradise Lost, see_ Milton, John\n\nParody, literary ,\n\nParticularity \u2013,\n\nPerception ,\n\nPhilosophic writing: received form\n\n_Philosophical Lectures, The_ : on intuitive thought ; on God ; and language ; principle of interpretation ; on observation and meditation ; duality of design \u2013; digressional style\n\nPhilosophy: Coleridge's concept of \u2013, \u2013; ancient and modern ; natural ; transcendental \u2013; , , \u2013; total\n\nPindar\n\nPlagiarism: from Schelling , \u2013\n\nPlato , ; _Gorgias_\n\nPlatonism\n\nPlotinus \u2013,\n\n_Poems in Two Volumes, see_ Wordsworth, William\n\nPoet: irritability of , ; portrait of ; Italian poets \u2013; impassioned philosopher\n\nPoetry: Coleridge's concept of , , \u2013; defense of ; language in ; theory of ; pleasure principle ; legitimate ; metre , \u2013; versification ; topic ; imagery , , \u2013; decadent modern style ; choice of characters ; reading of ; particularity ; metaphysical ; unity ; decorum\n\nPolarity \u2013,\n\nPope, Alexander: _Iliad_ ,\n\n_Prelude, The, see_ Wordsworth, W.\n\nPrinciples: defined\n\nPsychology ; contemporary view of\n\nPythagoras\n\nRansom, John Crowe: on Gray\n\nReader: author's relation with ; non-philosophical ; ideal , ; poorly imagined by Coleridge\n\nReason \u2013\n\nRepetition\n\nReviewers, _see_ Critics, anonymous\n\nRichards, I. A: on imaginative and fanciful verse\n\nRomanticism, German\n\n_Sacred River, The, see_ Baker, J. V.\n\nScaliger, J. C: _Plants_\n\nSchelling, Friedrich Wilhelm Joseph von , ; plagiarism from ; acknowledgement to ; shares Coleridge's task ; analysis of consciousness ; Coleridge differs from ; rewritten \u2013; _System des transzendentalen Idealismus_ \u2013; view of art ; on unconscious involvement\n\nSelf-consciousness , , , ,\n\nSelf-parody\n\nShaffer, Elinor Stoneman: on plagiarism from Schelling , \u2013, ; on knowing\n\nShakespeare, William ; _King Lear_ \u2013; _Hamlet_ ; poems \u2013; lectures on\n\nShawcross, J: on thematic unity ; on irrelevancy of Chapter II \u2013; on autobiographic elements ; on Coleridge and Aristotle\n\n_Sibylline Leaves_\n\nSin: defined\n\n_Siris, see_ Berkeley, G.\n\nSocinianism: study of\n\nSolipcism\n\n_Sources, Processes and Methods in Coleridge's Biographia Literaria, see_ Wheeler, K.\n\nSouthey, Robert: defense of ; and imagination ; practicality\n\nSpeaker: psychology of ; genius of ; self-criticism\n\nSpenser, Edmund ,\n\nSpinoza, Benedict ; _Ethics_\n\n_Statesman's Manual_ ; criticism of ; on miracles , \u2013\n\nStyle: digression \u2013; condensation ; repetition ; exaggeration\n\nSymbol \u2013, ,\n\nSymmetrics\n\nSymons, Arthur ix,\n\nSynesius\n\n_System des transzendentalen Idealismus, see_ Schelling, F.\n\n_Table Talk_ : on experience  Theses, Ten , ,\n\n'Thorn, The', _see_ Wordsworth, W.\n\nThought: Coleridge's definition of\n\nTransitions \u2013, \u2013, \u2013\n\nTruth: union of knowing and being ; purpose of poetry\n\nUnity, thematic \u2013,\n\nVersification\n\n_Watchman, The_ ,\n\nWater-insect metaphor ,\n\nWatson, George: on plan of _Biographia Literaria_\n\nWellek, Ren\u00e9: on plagiarism\n\nWheeler, Kathleen: _Sources, Processes and methods in Coleridge's Biographia Literaria_ x; on influence of German Romanticism ; on Coleridge's Christianity\n\nWilkins, John: _Essay toward a Real Character_\n\nWill: character of ; relation to imagination ; structural foundation of work , ; defined ,\n\nWimsatt, William K. and Brooks, Cleanth\n\nWit\n\nWord: definition of \u2013; etymology ; verbal precision\n\nWordsworth, William: real poetic character to be defined ; _The Prelude_ ; _Lyrical Ballads_ , , ; _Descriptive Sketches_ ; pantheism ; versification ; genius of , ; victim of literary inquisition ; poetic theory adapted by Coleridge ; 'Michael' \u2013; 'The Thorn' ; The Mad Mother' ; poetic style , ; 'Essay Supplementary to the Preface' (1815) ; defects \u2013, ; dramatic form ; 'Intimations' ode \u2013; ; 'I wandered lonely' ; 'Gipsies' ; virtues \u2013; _Poems in Two Volumes_ ; universality\n\nYarlott, Geoffrey: on Coleridge's character \n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n# Ace Books by Ilona Andrews\n\nThe Kate Daniels Novels\n\nMagic Bites\n\nMagic Burns\n\nMagic Strikes\n\nMagic Bleeds\n\nMagic Slays\n\nMagic Rises\n\nMagic Breaks\n\nMagic Shifts\n\nThe World of Kate Daniels\n\nGunmetal Magic\n\nThe Edge Novels\n\nOn the Edge\n\nBayou Moon\n\nFate's Edge\n\nSteel's Edge\n\nSpecials\n\nMagic Mourns\n\nMagic Dreams\nAlphas: Origins\n\nIlona Andrews\n\nInterMix Books, New York\n\nAN IMPRINT OF PENGUIN RANDOM HOUSE LLC\n\n375 HUDSON STREET, NEW YORK, NEW YORK 10014\n\nMAGIC GIFTS\n\nAn InterMix Book \/ published by arrangement with the author\n\n\"Alphas: Origins\" previously appeared in _Angels of Darkness_ , published by Berkley.\n\nCopyright \u00a9 2011 by Andrew Gordon and Ilona Gordon.\n\nExcerpt from _Magic Shifts_ copyright \u00a9 2015 by Andrew Gordon and Ilona Gordon.\n\nExcerpt from _On the Edge_ copyright \u00a9 2009 by Andrew Gordon and Ilona Gordon.\n\nPenguin supports copyright. Copyright fuels creativity, encourages diverse voices, promotes free speech, and creates a vibrant culture. Thank you for buying an authorized edition of this book and for complying with copyright laws by not reproducing, scanning, or distributing any part of it in any form without permission. You are supporting writers and allowing Penguin to continue to publish books for every reader.\n\nINTERMIX and the \"IM\" design are trademarks of Penguin Random House LLC.\n\nFor more information, visit penguin.com.\n\neBook ISBN: 978-0-451-48790-2\n\nPUBLISHING HISTORY\n\nBerkley trade edition \/ October 2011\n\nInterMix eBook edition \/ April 2016\n\nThis is a work of fiction. Names, characters, places, and incidents either are the product of the author's imagination or are used fictitiously, and any resemblance to actual persons, living or dead, business establishments, events, or locales is entirely coincidental.\n\nVersion_1\n\n# Contents\n\n_Ace Books by Ilona Andrews_\n\n_Title Page_\n\n_Copyright_\n\nChapter 1\n\nChapter 2\n\nChapter 3\n\nChapter 4\n\nChapter 5\n\nChapter 6\n\nChapter 7\n\nChapter 8\n\n_Excerpt from_ Magic Shifts\n\n_Excerpt from_ On the Edge\n\n_About the Author_\n\n# **CHAPTER 1**\n\nKarina Tucker took a deep breath. \"Jacob, do _not_ hit Emily again. Emily, let go of his hair. Don't make me stop this car!\"\n\nHer daughter's face swung into the rearview mirror, outraged as only a six-year-old could be. \"Mom, he started it!\"\n\n\"I don't care who started it. If you don't be quiet right now, things will happen!\"\n\n\"What things?\" Melissa whined. Megan, her twin, stuck her tongue out.\n\nKarina furrowed her eyebrows, trying to look mean in the rearview mirror. \"Horrible things.\"\n\nThe four children quieted in the back of the van, trying to figure out what \"horrible things\" meant. The quiet wouldn't last. Karina drove on. The next time Jill called to ask her if she would chaperone a gaggle of first graders for a school field trip, she would claim to have the bubonic plague instead.\n\nThe trip itself wasn't that awful. The sun shone bright, and the drive down to the old-timey village, forty-five minutes from Chikasha, was downright pleasant. Nothing but clear sky and flat Oklahoma fields with an occasional thin line of forest between them to break the wind. But now, after a day of hayrides and watching butter being churned and iron nails being hammered, the kids were tired and cranky. They'd been on the road for twenty minutes and the lot of them had already engaged in a World War III\u2013scale conflict three times. She imagined the other parents hadn't fared any better. As the six cars made their way up the rural road, Karina could almost hear the whining emanating from the vehicles ahead of her.\n\nThey should have just gotten a school bus. But Jill had panicked half of the parents over the bus not having seat belts. In retrospect, the whole thing seemed silly. Thousands of children rode school buses every day with no problems, seat belts or not. Unfortunately, creating panic was one of her best friend's talents. Jill meant well, but her life was a string of self-created emergencies, which she then cheerfully overcame. Usually Karina pulled her off the edge of the cliff, but with Emily involved, it was hard to maintain perspective.\n\nThis pointless worry really had to stop. Emily wasn't made of glass. Eventually Karina would have to let her go on a trip or to a sleepover without her mommy. The thought made Karina squirm. After Jonathan died, she'd taken Emily to a grief counselor, who offered to work with her as well. Karina had turned it down. She'd already been through it, when her parents passed away, and it hadn't made things any easier.\n\nHer cell beeped. Karina pushed the button on her hands-free set. \"Yes?\"\n\n\"How are you holding up?\" Jill's voice chirped.\n\n\"Fantastic.\" Would be even better if she didn't have to talk on the phone while driving. \"You?\"\n\n\"I need to go potty!\" Jacob announced from the back.\n\n\"Robert called Savannah a B word. Other than that we're good,\" Jill reported.\n\n\"I really need to go. Or I'll poop in my pants. And then there'll be a big stain . . .\"\n\n\"Listen, Jacob needs to go potty.\" She caught sight of a dark blue sign rising above the trees. \"I'm going to pull over at the motel ahead of you.\"\n\n\"What motel?\"\n\n\"The one on the right. With the big blue sign, says Motel Sunrise?\"\n\n\"Where?\" Jill's voice came through tinted with static. \"I don't see it.\"\n\n\"I don't see a motel,\" Megan reported.\n\n\"Look at the blue sign.\" Emily pointed at the window.\n\n\"Well, I don't see it,\" Jacob declared.\n\n\"That's because you're a doofus,\" Emily said.\n\n\"You suck!\"\n\n\"Quiet!\" Karina barked.\n\nThe exit rolled up on her right. Karina angled the car into it. \"I'm taking this exit,\" she said to the cell phone. \"I'll catch up with you in a minute.\"\n\n\"What exit? Karina, where are you? You were right there and now you're gone. I don't see you in my rearview mirror . . .\"\n\n\"That's because I took the exit.\"\n\n\"What exit?\"\n\nOh, for the love of God. \"I'll talk to you later.\"\n\nThe paved road brought them to a two-story building covered with dark gray stucco. Only one car, an old Jeep, sat in the parking lot.\n\nKarina pulled up before the entrance and hesitated. The building, a crude box with small narrow windows, looked like some sort of institutional structure, an office, or even a prison. It certainly didn't look inviting.\n\n\"Now I see it,\" Megan said.\n\nKarina shook her head. You'd think if you owned a motel, you'd want to make it seem hospitable. Plant some flowers, maybe choose a nice color for the walls, something other than battleship gray. It only made good business sense. As it was, the place radiated a grim, almost menacing air. She had a strong urge to just keep on driving.\n\n\"I have to go!\" Jacob announced and farted.\n\nKarina jumped out of the van and slid the door open. \"Out.\"\n\nFifteen seconds later, she herded them inside a small lobby. The lone woman standing behind the counter turned her head at their approach. She was skeletally thin, with long red hair dripping down past her shoulders. Karina glanced at her face and almost marched back out. The woman had eyes like a rattlesnake, no compassion, no kindness, no anger. Nothing at all.\n\n\"I'm sorry,\" Karina said. \"Could we please use your facilities? The little boy needs to go to the bathroom.\"\n\nThe woman nodded to the archway on Karina's right. Charming. That's okay. They just needed to get in and get out. \"Thank you! Come on, kids.\"\n\nThe archway opened into a long hallway. On the left, several doors punctuated the wall, one marked \"Bathroom\" and another, at the very end, marked \"Stairs.\" On the right an older man stood in the middle of the hallway. Heavily muscled, with a face like a bulldog, he'd planted himself as if he were about to be overrun by rioters. His eyes watched her with open malice. The kids sensed it, too, and clustered around her. Karina didn't blame them.\n\n\"Hi!\"\n\nThe man said nothing.\n\nOkay. She marched to the bathroom and swung the door open. A single-person bathroom, relatively clean. No scary strangers hiding anywhere. \"In you go.\" She ushered Jacob inside and stood guard by the door.\n\nMinutes ticked off, long and viscous. The man hadn't moved. The children kept quiet under his scrutiny, like tiny rabbits sensing a predator.\n\nKarina knocked on the door gently. \"Come on, Jacob. Let the other kids have a turn.\"\n\n\"Almost done.\"\n\nKarina waited. The man kept staring at her. Gradually his face took on a new expression. Instead of staring her down, he was now studying her as if she were some bizarre alien life-form. That was even more disturbing. Karina fought a shiver.\n\n\"Jacob, we need to go.\"\n\nShe heard the toilet flush. Finally.\n\nJacob emerged from the bathroom. \"I washed my hands with soap,\" he informed her. \"Do you want to smell them?\"\n\n\"No. Does anybody else need to go?\"\n\nThey shook their heads. Emily hugged her leg. \"I want to go home, Mom.\"\n\n\"Excellent idea.\" Karina led them down the hallway.\n\nThe man moved to block their way. \"Thank you for letting us use the bathroom,\" Karina said. \"We'll be on our way now.\"\n\nThe man leaned forward. His nostrils fluttered. He sucked in the air through his nose and his face split in a grin. He didn't smile; he showed her his teeth: abnormally large and sharp, triangular like shark teeth, and definitely not human.\n\nIce skittered down Karina's spine.\n\nThe man took a step forward. \"You ssshmell like a donor.\" His teeth took up so much space in his mouth, he slurred the words.\n\nKarina backed away, holding her hands out to shield the kids behind her. She wished she had a can of mace or a gun\u2014some weapon in her purse other than Kleenex, her pocketbook, and a cell phone with a dead battery.\n\n\"Let us out!\"\n\nThe man advanced. \"Rishe! The woman ishh a donor.\"\n\n\"We'll be leaving now!\" Karina put some steel into her voice. Sometimes if you looked like you were ready to fight, people backed down and looked for an easier target.\n\nThe man bared his teeth again and she glimpsed what looked like a second row of fangs behind the first in his mouth. \"No, you won't,\" he said.\n\nTime for emergency measures. \"Help!\" Karina screamed at the top of her lungs. \"Help!\"\n\n\"No help,\" he assured her.\n\nThe kids began to cry.\n\n_Maybe this is a nightmare,_ flashed in her head. Maybe she was dreaming.\n\n\"Mom?\" Emily clutched at her jeans.\n\nDream or not, Karina couldn't let him get a hold of her or the kids. She kept backing away to the door behind her, the one labeled \"Stairs.\"\n\n\"Let us go!\"\n\nHe kept coming. \"Rishe! Where are you?\"\n\nThe wall on their right exploded.\n\nSplintered pieces of wood peppered the hallway, knocking the shark-toothed man back and missing Karina by mere inches. Stunned, she glanced into the gap in the wall. The redheaded woman\u2014Rishe?\u2014jumped over the counter and ran directly at Karina and the children, her face twisted into a grotesque mask. The skin on Rishe's neck bulged, rolling up, as if a tennis ball slid up her throat into her mouth.\n\n_This is just crazy . . ._\n\nThe woman spat.\n\nSomething dark flickered through the air. Pain stung Karina's left side. A long thin needle, like the quill of a porcupine, sprouted from her stomach, just under the ribs. She yanked it out on pure instinct. She should've been terrified, but there was no time . . .\n\nSomething hit the red-haired woman from behind, arresting her in midstep. Rishe's mouth gaped in a terrified silent scream. Huge claws grasped her face, jerked, and her head twisted completely around.\n\n_Oh, my God._\n\nRishe's body fell, and beyond it Karina glimpsed a _thing_. Huge, dark, inhuman, it stared back at her with malevolent eyes. Its very existence was so at odds with everything Karina knew, that her mind simply refused to believe it was real.\n\nAn odd odor saturated the air, dry and slightly metallic, like copper warmed by the sun. The thing stepped over the woman, its gaze fixed on her.\n\n\"Run!\" Karina turned on her heel and dashed down the hallway, herding the children before her.\n\nThe man with shark teeth rose slowly, pulled a wooden splinter out of his eye, tossed it aside, and, with a deep bellow, charged into the lobby through the hole in the wall.\n\nA snarl answered him, a promise of pain and death. It whipped Karina into a frenzy. She swiped Jacob off the floor\u2014he was the smallest\u2014and ran faster to the heavy door barring the stairs. She jerked it open. \"Up the stairs, go, go!\"\n\nThey ran up, whimpering and sobbing. The same fear that drove her propelled them up the stairs better than anything she could've screamed.\n\nKarina slammed the door closed, balancing Jacob on one arm, and looked for something to bar it, but the stairway was empty. Her stomach burned, the pain from the needle puncture spreading up and down her body as if her skin had caught on fire. She ran after the kids. The boy in her arms was stone heavy. They reached the top of the staircase and crowded on the landing.\n\nBelow something clanged. There it was again, the scent of hot metal burning her lungs.\n\nKarina set Jacob down and wrenched the door open. They burst into the upstairs hallway. She scanned the rows of doors and tried to shove the nearest one open, but it was locked.\n\nAnother\u2014locked, too.\n\nThird\u2014locked.\n\n_This is a nightmare. It has to be a nightmare._\n\nA vicious snarl chased them. Emily screamed, a high-pitched shriek that could've broken glass. Karina grabbed her daughter by the hand and dragged her down the hall, to the single window. \"Follow me!\"\n\nBeyond the window a fire escape waited.\n\nKarina grasped the window latch and jerked it up. Stuck.\n\nHer head swam. The air around her had grown scalding hot. Every breath burned her lungs from the inside out. She stumbled, caught herself on the windowsill, and pulled the sash upward with all her strength. The wood groaned and suddenly the frame slid up.\n\nA door thumped. Kids screamed. The terrible dark beast had made it into the hallway.\n\nShe grabbed the nearest child and hurled her onto the fire escape, then the next, and the next. Little feet thudded, running down the metal stairs. Emily was last. Karina clutched her daughter to her and climbed out on the fire escape.\n\nA black van waited below. Several men stood by the van. They had the children. They stood there silently, watching her, so calm while the kids screamed, and suddenly she realized that they and the beast inside were allies. They were trapped.\n\nA growl washed over her.\n\nThe world gained crystal clarity, everything becoming painfully vivid and sharp. Slowly Karina turned. Her daughter hugged her, her breath a tiny warm cloud on her neck. The metal rail of the fire escape dug into Karina's back. The thudding of her heart sounded so loud, each beat shook her rib cage like a blow from a sledgehammer. Every breath was a gift.\n\nShe saw the thing emerge from the darkness. Slowly, it solidified out of the gloom, one gargantuan paw on the windowsill, then another. Enormous claws scratched the wood. It climbed onto the windowsill and perched there, a mere foot from her. Karina stared into its eyes, inhaled its scent, and knew with absolute certainty that she was going to die.\n\nThe thing opened its maw, revealing huge fangs. Its deep voice issued forth in a single mangled word. \"Donor.\"\n\n\"Are you sure?\" asked a male voice from below.\n\nThe beast snarled. Karina jerked back, shielding Emily with her hands. Her legs gave out and she fell to her knees.\n\n\"My lady?\" said the voice from below, closer now.\n\nShe barely turned her head, not daring to take her gaze from the monster in the window. A dark-haired man climbed the fire escape toward her. His face was preternaturally beautiful, his eyes a dark, intense blue. \"I have a proposition for you, my lady . . .\"\n\nHis voice faded, replaced by darkness and the feel of cotton against her body.\n\n_I agree._\n\nKarina sat up . . . She was in her bed. The room lay dark about her. A nightmare. That was all.\n\nHer heart thudded in her chest. She rubbed her face and her hands came away slick with cold moisture.\n\n_I agree. \"I agree\" to what?_ What did she agree to in her dream?\n\nIt didn't matter. It was a nightmare. In the morning, she'd call the grief counselor.\n\nKarina frowned and pushed free of the blankets. She felt a strange sense of wrongness, as if there was something very important she was missing. Something vital. A small lamp waited on the table next to the bed. She flicked it on and a cone of soft electric light illuminated the room.\n\nThe bedroom wasn't hers.\n\n* * *\n\nFor a moment Karina froze, and then fear caught her in its fist and squeezed. \"Emily?\" she whispered. \"Emily?\"\n\nNo answer.\n\nShe was alone in a strange bedroom.\n\nThere could be a rational explanation for this. There had to be. She just didn't know what it was.\n\n_I agree._ An echo of her voice from the dream. She had a terrible suspicion the unfamiliar bedroom and those two words were connected.\n\nHer clothes were gone. She wore only underwear and a giant T-shirt, three sizes too big.\n\nA pair of carefully folded jeans lay on a chair next to the bed. Her jeans, the ones she had worn on the field trip. Karina pulled them on. She had to find Emily.\n\nThe door swung open with ease and she found herself in a hallway. To the left, the hallway ended in a stairway leading up. To her right, a pool of electric light brightened the wooden floor and the rust-colored rug. Quiet voices carried on a soft discussion.\n\nShe followed the voices and stepped into a kitchen, blinking against the light. Three men sat at the round table in the center. They turned to look at her. The one sitting farthest from her wore the unearthly face of the man from her dream. His name surfaced from the depths of her memory. Arthur.\n\n_I agree._\n\nArthur nodded to her. \"Ah. You're up. Why don't you sit down with us? Henry, please get a chair for Lady Karina.\" His soft, intimate voice caressed her almost like a touch. It should've been soothing. Instead her insides clenched into a tight knot.\n\nA tall man with a shy smile rose and held a chair out for her. So oddly domestic, all three drinking tea. Nobody was startled by her appearance. Clearly she was expected.\n\nKarina sat. \"Thank you.\" The automatic response rolled from her lips before she even realized it.\n\n\"You're welcome,\" Arthur said. He leaned back with a quiet elegance, artfully posed without putting any effort into it. His hair was soft, black, and brushed back from his perfectly sculpted face. His eyebrows were equally black and so were his eyelashes, long and soft like velvet. They framed big eyes, crystalline blue, distant, and cold. Angelic, she thought. He looked like an angel, not a plump cherub, but an angel who roamed freely in the sky, possessed of heart-wrenching beauty and terrible power, an angel who had stared into the bottomless blue for so long that his eyes had absorbed its color.\n\n\"Would you like some tea?\" Arthur asked.\n\n\"Children . . . ?\"\n\n\"Safe,\" he said and she believed the sincerity of his words even though she had no reason to do so.\n\nArthur rose, took a small blue mug from the shelf behind him, and poured steaming tea into it from a large kettle on the stove. He set the cup in front of her. \"Please drink. It will steady your nerves.\"\n\nKarina looked at the cup.\n\nHe drank from his own cup and smiled in encouragement.\n\nShe picked up the cup and took a sip. Green tea. Odd taste, slightly sour.\n\nMaybe she was still dreaming. The whole scene had that slightly absurd wrongness found only in dreams.\n\nKarina looked about the table. The man who had offered her the chair, Henry, sat to her right. He was tall and whipcord lean. His face, serious with somber intelligence, lacked Arthur's magnetism, but its sharp angles drew her all the same. His tawny hair was cut close to the scalp, but still showed a trace of a curl. His green eyes regarded her and she read pity in their depths.\n\nThe man on her left was model pretty. Strong masculine jaw, deep, dark blue eyes, high cheekbones, a mane of golden wavy hair dripping down to below his waist, hiding half of his face . . . His eyes flashed with wild humor. He gave her a wink, grinned, exposing even white teeth, and tossed his hair back. An ugly scar ripped his left cheek, almost as if something had taken a bite out of him and his flesh hadn't healed right. She fought an urge to look away. He reached for her hand . . .\n\n\"Daniel.\" Arthur's voice gained a slight edge. \"That's extremely unwise.\"\n\nDaniel sat back.\n\n\"Just because she didn't scream when she saw your face doesn't mean you get to touch.\" Henry refilled his cup.\n\n\"Please forgive Daniel. He doesn't mean to be rude. He's just forbidden to speak for the time being. Your tea is getting cold,\" Arthur said.\n\n\"He tends to cause problems when he speaks,\" Henry said.\n\nDaniel gave her a smoldering smile.\n\nShe faced Arthur. \"What did I agree to?\"\n\nArthur sighed. \"I see.\"\n\nHenry leaned forward. \"Perhaps we should mend this.\"\n\n\"Yes. The sooner, the better. Lucas might return and that would make things considerably more complicated.\"\n\nDaniel laughed softly. If wolves could laugh, they would sound just like him.\n\nHenry held out his hand. \"It's easier if you hold on to me.\"\n\nKarina hesitated.\n\n\"You do want to remember, don't you?\" Arthur asked.\n\nShe put her hand into Henry's. His long warm fingers closed about hers. The world tore in two and she was back on the landing of the fire escape at the not-motel, cradling Emily. Her whole body burned with a terrible ache.\n\nArthur leaned his head to the side, looked at them for a moment, and plucked Emily from her arms.\n\n\"No!\" Karina struggled to hold on, but her hands had lost all strength.\n\nEmily didn't kick. Didn't scream. Her face was completely blank, as if she had turned into a doll. Arthur turned and handed her to someone behind him on the stairs.\n\n\"Emily!\" Karina tried to crawl after her but her body refused to obey.\n\nArthur touched the hem of her black top and edged it upward. His fingers touched her stomach. Pain pierced her and she cried out.\n\n\"Ah. Now see, this isn't good.\" Arthur shook his head mournfully. \"All of this must seem terribly confusing to you and our time is short, so I will keep the explanations simple. This is the house where monsters live. We are the killers of monsters. I suppose that also makes us monsters simply by necessity. I don't know why you're here. It's probably a pure coincidence. An unlucky roll of the dice. You and your children were caught in the cross fire. One of the monsters poisoned you with her throat dart. The wound is fatal. You're dying.\"\n\nFear shot down Karina's spine in an icy rush. She didn't think she could have gotten more scared, but his tone, that patient, pleasant, even tone, as if he were discussing lunch, terrified her. _It's not a dream,_ she realized. _It's happening. It's happening to me right now. God, please let Emily be okay. Please. I'll do anything._\n\n\"I can smell your fear,\" Arthur said. \"It rolls off your skin. A better man would feel discomfort at your pain. But I'm not a good man. I feel nothing for you. We rarely have to deal with innocent bystanders and when we do, we strive to send them back unharmed, not out of some altruistic impulse, but because we dislike attention. If you hadn't been injured, Henry here would wipe your memory and the five of you would go merrily on your way. As it is, however, you will be dead in the next thirty minutes.\"\n\nThe words refused to leave her mouth. Karina strained and forced them out. \"Why are you telling me this?\"\n\nHis ice-cold smile made her heart jump. \"I'm talking to you because I'm about to offer you a deal. You have something we want, my lady. Your body has a genetic predisposition toward producing certain hormones one of us desperately needs. Your subspecies isn't unique, but it's rare enough to make you valuable. I suspect that's also how you were able to find this place, and that's why the yadovita, the redheaded woman, took the time to poison you instead of defending herself from us. Listen carefully, my lady, because I won't repeat myself.\"\n\nShe stared at him, committing each word to memory.\n\n\"The creature behind you requires your blood. He will feed on you. His venom will counteract the poison that's killing your body. In return, he will consume the chemicals your body will produce. You will give yourself to the House of Daryon. You will let the beast feed on you. You will live in quarters of our choosing. You can never leave. You can have no contact with the outside world. For your agreement to this, we will spare your life and the lives of the children.\"\n\nThe thing on the windowsill let out a low whine of anticipation. That . . . that beast would feed on her. _Forever_. _Oh, dear God. I can't do it . . . I can't . . ._\n\nArthur leaned forward, his face showing no emotion beyond the pleasant, calm composure. \"Consider carefully before you answer. I don't offer this deal to you because I like you or because I'm moved by some noble emotion. I do it because we need you. What I propose won't be pleasant for you. You won't enjoy it. In fact, many would say you're better off dying now.\"\n\nFog gathered on the edge of her mind, threatening to smother her. Karina clawed at reality, trying to remain conscious.\n\n\"My daughter . . .\"\n\nThe beast growled on the windowsill.\n\n\"He guarantees her safety,\" Arthur said.\n\n\"The children . . . will be returned to their families?\"\n\n\"Yes.\"\n\n\"I agree.\"\n\nA gentle hand seized her mind and pulled her back through time and space to the reality of the round table and the hot tea mug in her hand. She looked at Arthur.\n\n\"My daughter, Emily?\"\n\nHe didn't answer.\n\n\"You promised me the children would be returned to their families. Her father is dead. I'm the only family Emily has. Where is she?\"\n\nHe smiled, a flat curving of lips without any emotion. \"She's at the main house for the time being.\"\n\nNo, she wasn't, Karina realized. He was lying. \"I want my daughter. We made a deal. Bring me my daughter, or I am leaving.\"\n\nDaniel rocked back on his chair and laughed. A door slammed. Footsteps echoed through the house. \"Cooperate with Lucas and your daughter will be brought to you,\" Arthur said.\n\nA man walked into the kitchen. Tall, corded with muscle that bulged his T-shirt, he dwarfed the doorway. He wasn't just large, he was massive and wrapped in menace, as if he were a whirlwind of violence, barely contained in the shell of his body. Black hair fell on his hard, aggressive face in long strands. He glanced at her, his eyes green and merciless. She met his gaze and gulped. It was like looking into the eyes of a tiger. His stare promised death.\n\nRecognition sparked in his green irises and flared into rage.\n\nHe lunged forward, inhumanly quick, and hit the table with his palm. She jerked back.\n\n\"Get your hands off of her!\" His voice rippled with a snarl.\n\nHenry raised his hands in the air. The man grasped the chair, Henry still on it, and tossed it aside. Steely fingers grabbed her elbow and pulled her up. He swiped her off the floor with ridiculous ease, locking her in the crook of his arm, and snapped like a rabid dog, \"Mine!\"\n\n\"We have no intention of taking her from you.\" Arthur sipped his tea.\n\n\"Don't any of you fuckers touch her!\"\n\nShe flailed in his arms, trying to break free, but it was like trying to push back a semi.\n\n\"You must forgive Lucas,\" Arthur told her. \"He tends to be overprotective of his food.\"\n\nA familiar scent of heated metal invaded her nostrils. Panic squirmed through her. She fought harder, but her feet kicked only air. He carried her away out of the kitchen back to the bedroom where she had awakened.\n\n# **CHAPTER 2**\n\nLucas dropped her on the bed and went to lock the door. \"Stay away from Arthur. He's a sick fuck.\"\n\nHe turned and strode toward her, enormous, overwhelming in his sheer size. Karina shrank back until her spine hit the wall.\n\nHe looked her over, a long, lingering stare that made her want to cover herself, frowned and ducked into a doorway on the left. Water gushed. Lucas reappeared with a tall glass of water and handed it to her. \"Drink this. It will help.\"\n\nShe drank.\n\nHe sat on a chair across from her and pulled off his socks. Only now she noticed that he wasn't wearing any shoes. He balled the socks into a clump and tossed them into the room where he'd gotten the water, then shrugged off his T-shirt. Karina's breath caught in her throat. Faded ragged scars crisscrossed his massive back. His legs were long, his waist narrow in comparison to his vast shoulders. His lines were almost perfect. As he squared his shoulders, muscles rolled under his skin, forming hard ridges. He didn't move\u2014he stalked and prowled, like a huge predatory animal, menace cascading from him in waves along with his hot metallic scent.\n\nHer memory thrust Jonathan before her. Her husband had been handsome and well built, an average-sized man. Lucas could've snapped him in half and wouldn't have given it a second thought. He'd just toss the broken body aside and continue on his way. She had no chance. In a physical fight, Lucas would destroy her.\n\n\"Drink,\" he said.\n\nKarina forced some more water down. Her throat had gone dry and she drank again. Suddenly Lucas gathered himself. His gaze fixed on the door. His body tensed, his expression alert. His feet gripped the bare floorboards, his legs bent lightly, as he readied to launch himself into a leap. Muscles bunched and knotted across his shoulders and back. His arms lifted slightly, spread wide, the fingers of his big hands like talons, ready to grasp and crush. His eyes ignited with a hot, hungry fire. Poised like this, he was barely human.\n\nSomeone's knuckles rapped on the door.\n\n\"What?\" Lucas growled.\n\n\"Do you want the sedative?\" Henry's voice asked.\n\nLucas glanced at her and asked quietly, \"Do you want to be drugged?\"\n\n\"No.\"\n\n\"She said no,\" he snarled.\n\nThe footsteps retreated. Lucas eased, relaxing slowly, muscle by muscle. He glanced at her with his light green eyes and she shrank from his gaze.\n\n\"How much did they tell you?\" he asked.\n\n\"I know what I agreed to.\" She hesitated. \"Are you . . . ?\"\n\n\"I am.\"\n\nShe tried to reconcile the beast and the man, and couldn't. That dark, grotesque creature was huge, twice as big as Lucas. A horrible meld of ape, dog, bear\u2014Karina struggled for a comparison, a point of reference, and could find none. Her memory was fuzzy. She remembered fangs and baleful eyes, and massive shoulders sheathed in dark fur. How was it possible? Her mind refused to admit that thing existed. But her body felt Lucas near and knew the beast was real.\n\nShe had to have an explanation. Anything at all. \"Are you a vampire?\" she asked.\n\n\"No.\"\n\n\"What are you?\"\n\nHe sighed. \"There's no myth or legend or cute explanation. People here call those like me Demons. It's just a name, nothing religious attached to it. You might also hear people call me Subspecies 30. The rest is complicated.\" He took her half-empty glass and went to top it off. \"I don't actually need your blood to sustain me. I require the endocrine hormones your body will secrete in response to my bite.\"\n\n\"For what?\"\n\n\"To counteract the effects of my venom. It hurts me.\"\n\nHe handed her the full glass, rubbed his hand across the back of his neck, and held the palm to her face. The odor of hot metal hit her nostrils and she drew back.\n\n\"That smell means I'm hungry for you.\"\n\nHe was too close. The cup trembled in Karina's fingers. God, she was scared. It took all of her will not to scream and run. \"Will it hurt?\"\n\n\"Yes. It's not like vampire movies, where the vampire bites the woman and she moans softly and comes all over herself. There's no rapture involved. No climax. Just me chewing on you.\"\n\nHe took her by the chin, lifting her face, and peered into her eyes. Karina pulled back. He leaned closer. She tried to scramble away, but he grasped her shoulder, keeping her still. His lips touched her forehead. \"Fever.\" Lucas grimaced. \"Your eyes are still bloodshot.\"\n\nHis presence pressed on her like a physical burden. Karina closed her eyes. She sat there, world shut out, and pretended that everything would be okay even if every instinct assured her it wouldn't. She had to survive and adapt. She had to do whatever was necessary to get her daughter back.\n\nWhen she opened her eyelids, he waited for her with a synthetic cord in his hands. She hadn't heard him move.\n\n\"To keep you still.\" He moved toward her, uncoiling the cord.\n\nNo. Lying there tied up and completely helpless while he drank her blood would be too much. \"That's okay,\" she said quickly. \"I won't change my mind.\"\n\nLucas kept coming.\n\n\"I won't change my mind.\" Desperation put steel into her voice. \"I've agreed to this to save my daughter. They'll let me see her after you feed. I won't run or fight.\"\n\nHe halted.\n\n\"Arthur said I would stay here for as long as I live. That means you have to feed frequently. Might as well start it right.\"\n\nLucas gripped the rope. His biceps bulged. He snapped the rope apart. Karina winced. \"If you're trying to intimidate me, it's too late. I'm already as scared as I'm going to get.\"\n\n\"I'm not trying to scare you.\" He rolled the section of the rope into a tight wad, wrapped the end about it several times, tied it, and dropped it in her lap. \"To bite down. In case it gets too rough.\"\n\nShe picked it up.\n\nLucas sat next to her. \"Arthur isn't in charge of your daughter. I am. I guaranteed her safety. Both of you belong to me.\"\n\nLucas leaned to look into her face. She expected rage, hunger, some violent emotion, but instead she saw only steady calm.\n\n\"I promise you that no matter what happens between you and me, your daughter will be safe. I will never use her against you. Everyone is afraid of me, and she will never be bullied or mistreated.\"\n\nKarina stared at him in surprise.\n\n\"You wanted to start this right,\" he said. \"We can do that. Let's be honest. The bitch in the hotel poisoned you. Technically she infected you with a virus that secretes a toxin into your bloodstream. To counteract the virus, you need my venom. I've already bitten you once but it will take several feedings before you're in the clear.\"\n\n\"You've bitten me?\"\n\n\"Left thigh,\" he said. \"I was in the attack variant at the time, and biting you anywhere else would've caused too much damage.\"\n\nShe grabbed at her leg, trying to feel the wound through the fabric of the jeans.\n\n\"It was a very quick bite,\" he said. \"To keep you from dying. This will be worse.\"\n\nHe was serious. The thought of him feeding on her, chewing on her, was almost too much to contemplate. \"Can we do a blood transfusion instead?\"\n\n\"No. We've tried in the past and failed. There is some sort of relationship between your blood, my venom, and my saliva that we don't understand. I have to feed on you. You need me to survive and I need you to . . .\" He paused. \"To counteract my venom.\"\n\nHe was holding something back, she could feel it.\n\nLucas's eyes held no mercy. \"I'm a predator and my body knows that you're my prey. Your fear is exciting. Try not to be so scared. Don't struggle. The more you flail about and whimper, the more excited I'll get. If you get me excited enough, I'll chew up your veins and end up fucking you in a puddle of blood. I take it you don't want that.\"\n\n\"No.\"\n\n\"Then stay calm.\" He nodded at the cord in her lap. \"You sure you don't want to be tied?\"\n\n\"Yes.\"\n\nLucas stretched out on the bed, took her by the waist, and pulled her down, flush against him. They lay together, her butt pressed against his groin, her back tight against his chest. Like two lovers. Jonathan and she used to lie like this after sex. The perversity of it made her shiver.\n\n\"Lie still.\" His arms pulled her tighter to him. The hard shaft of his erection dug into her butt. She tried to edge away from it.\n\n\"Don't worry. I can't help it, but I won't molest you. Unless you start moaning and rubbing your ass against me.\"\n\nShe stopped moving. The odor of hot copper was overpowering now. Karina cleared her throat. \"I feel light-headed.\"\n\n\"You're breathing in my scent. Your body's reacting. It will speed things up.\"\n\nThat explained the shirt coming off. He wanted no fabric barriers between her and that smell, so it could roll off his skin and take her under. \"Do I need to do anything?\"\n\n\"Just lie there and endure. Your body needs my venom. As I said, I've bitten you already to kill the poison, but you got just enough to keep you alive. This will take some time.\"\n\nShe brushed her hair from her neck, exposing skin. No point in drawing this out.\n\nA low laugh answered her. He spoke into her ear, his breath a warm touch on her skin. \"You ever watch hockey?\"\n\n\"No.\"\n\n\"The Buffalo Sabres had a goalie\u2014Clint Malarchuk. Steve Tuttle, a guy on another team, was trying to score a goal, and as he charged at the crease, a defenseman grabbed him from behind and swung him up. Tuttle's skate caught Malarchuk's neck. A shallow cut, only severed the exterior jugular. Blood sprayed like water from a hose. Covered the whole crease in seconds.\"\n\nFor some reason she couldn't understand, his quiet voice steadied her nerves. \"Did he survive?\"\n\n\"He did. Had the skate cut a bit deeper, he would've been dead in about two minutes.\" He gathered her even tighter against himself. \"The neck nuzzling is fun, but the pressure within the jugular would expel your blood so quickly, it would kill you.\" His finger traced an outline on the vein on her neck, sending electric shivers along her skin. She wished he hadn't done that.\n\n\"If not the neck, then where?\"\n\n\"The arm works well.\"\n\n\"Can you . . . get on with it?\"\n\n\"Not yet. The longer we wait, the less painful it will be for you.\"\n\nHis body was hot against hers, his heat seeping into her. His scent enveloped her completely now. Her head spun.\n\n\"That's it,\" he prompted. \"Go limp. Don't strain.\"\n\n\"I'm scared,\" she told him.\n\n\"I'm sorry.\" The undercurrent of violence that permeated everything he said muted slightly.\n\n\"What will happen after you feed?\"\n\n\"You'll pass out. It's like giving blood except messier. Your body will go into shock from my venom. If you survive, you'll get used to the feedings.\"\n\n\"I might die?\"\n\n\"Yes.\"\n\n\"This just gets better and better.\"\n\n\"Life's a bitch.\"\n\nThe room crawled. \"I'm not dreaming, am I?\"\n\n\"If this is your dream, you're seriously fucked up.\"\n\n\"Who are you . . . all of you?\"\n\n\"You ask too many questions.\"\n\nHe pulled away from her, turned her arm to him, and bit into the soft flesh just above the elbow. Pain lanced through her. Her body tensed in response, but his arms clamped her down and she could barely breathe.\n\nIt hurt. It hurt and hurt, but worse than the pain was the awful sensation of his gnawing teeth and the prickly heat squirming its way up her arm. It spread into her shoulder and fanned out, claiming her body. She wanted to break free, to get away, but Lucas held her tight.\n\n\"Promise me you will make sure my daughter is safe if I die.\"\n\nHe didn't answer.\n\n\"Promise me.\"\n\n\"I promise,\" he said.\n\nKarina let herself sink into the pain. Gradually it eased into a steady ache. Her limbs relaxed. She tried to think of something else, anything else, of Emily, of their safe little apartment, of being far away in a different place. But the reality refused to recede. And so she lay there and waited it out, her entire body humming with a distinct unusual pain, until her dizziness blotted out the world and she slipped under.\n\n* * *\n\nLucas nuzzled her thin neck. Feverish. Not too bad. She was healthy. And clean. The blood work from the main house had shown no abnormalities aside from the poison. That was what donors were. Resilient; resistant to most disease.\n\nAnd grounded. She didn't seem like she would snap, but he'd seen enough people break under the weight of the transition to let his guard slip. And then there was her daughter. Children complicated things.\n\nShe just lay there and let him feed.\n\nHis first donor, Robert Milder, had to be sedated for the feedings. After him, there was Galatea. He had to tie her up. Every time. She had resented her role, loathed being restrained, despised him, and yet pulled him into her bed; and when they fucked, she drained him so completely, he felt blissfully empty, as if he had poured not only his seed, but his pain into her. She took it all and reveled in it, enjoying the power she wielded over him. He wasn't a fool. He knew she was driven by revenge, but he came back to her again and again, an idiot thirsty for a poisoned spring.\n\nAnd now he had Karina.\n\nA soothing cold spread through his veins, melting the needles of pain that always prickled him in the aftermath of his transformation from the attack variant. Funny. He had survived for six years on injections, shooting himself up every couple of days, but the synthetic hormones failed to soothe the ache. They managed to dull the pain, yet it had still gnawed at him, until he became convinced it would grind him down to nothing. Karina's body had barely had a chance to respond to his poison, yet even this tiny dose of the hormones brought relief to him. He had forgotten what it was like not to hurt.\n\nLucas breathed in her scent. The memory of the chase through the motel danced through his mind. He wanted to chase her again. He felt drunk.\n\nHe slipped the narrow strap of the tank top off Karina's shoulder, baring her left breast. Bigger, fuller, softer than he had expected. He imagined sliding his palm over the mound, brushing the nipple with his thumb. He pictured how her body would tighten in response, how the nipple would feel erect against his fingers.\n\nHe slid his fingers under the waistband of her jeans, pulled it up, and looked at the triangle of her white underwear. His cock ached. He wanted to mount her and thrust it inside her.\n\nSo what was stopping him?\n\nLucas slid his hand up, to her slightly rounded stomach, holding her gently, trying to puzzle it out. Had he tied her up before feeding from her, he would've fucked her by now, of that he was certain.\n\nTrust, he realized. She'd held up her part of the deal. It had cost her. She'd cried toward the end, once her grip on consciousness slipped\u2014silent tears that left wet tracks on her cheeks. Her arm would be sore as hell tomorrow. Provided the fever didn't rise, the poison didn't kill her, and there was a tomorrow in her future. He wanted her to live, but he had done all he could to help her.\n\nThe feeding had cost her, but she lay there and let him do his thing, as she had promised, and she expected him to hold up his end of the bargain. And the bargain didn't include fucking rights. She'd made that crystal clear.\n\nHe tugged her tank top back into place, covering her up, and pulled her to him, sliding his arm over her. She was his. She would take away his pain and he would guard her in return. That was the agreement.\n\n# **CHAPTER 3**\n\nKarina awoke to an empty room. Bright morning light flooded through the open window, drawing a yellow rectangle on the wooden floor. A draft brought an acrid stink of burning bacon.\n\nEmily.\n\nShe pushed free of the sheets and almost fell. Her head swam. Slowly, very slowly she slid off the bed and stood upright. Her throat was so dry, it hurt. A full glass of water sat on the bedside table beside a pair of binoculars and a yellow sticky that read \"Drink it.\" She could practically hear Lucas's growl.\n\nThe memory of his gnawing teeth squirmed through her, dragging nausea in its wake. Karina bent over, gripped the night table to steady herself, and saw a square bandage on her arm. She tugged at it, sending a jolt of pain through her limb. The bandage remained stuck. Karina pulled harder, trying to rip it away as if she could shed the memory of Lucas with it. She struggled with it for a few seconds, pain pounding up her biceps in hot prickly bursts, and finally tore it free.\n\nA big bruise stained the bend of her arm. Dark purple, it sat there like a brand. Lucas's proof of ownership. Dried blood was caked in the center, where his teeth had mangled her veins.\n\nThe price she paid for Emily's life. And her own. The ache in her arm pushed her to scream at the sheer mind-boggling unfairness of it: at being attacked, kidnapped, hurt, held down by brute force, robbed of her daughter, stripped of her freedom . . . At being plucked from her life. Only a day ago, she felt reasonably safe, secure in the knowledge that she could dial 911 at any moment and bring a police cruiser to her door. She had rights. She had protections. She was a person.\n\nShe felt the hot wet tears well in her eyes and clenched her teeth. She had to get a grip. Thinking like a victim would get her nowhere. Yes, it was terrifying. Yes, it hurt. But it didn't kill her. She was still alive and as long as she breathed, she had to fight for herself and her child. She had to obey and be sweet. She had to ingratiate herself. That was her only chance at survival and escape. Karina dropped the bandage on the night table and drained the glass. It was time to find her daughter.\n\nA harsh screech made her turn to the window. She walked to it, picking up the binoculars off the night table on the way. A wide green expanse spread before her, a wooded slope gently rolling away and down, toward mountains, brown and rust, fading to blue and eventually gray in the distance. A scrub forest hugged the roots of the mountains, dotting the grassy prairie in clumps of green. The wind fanned her face, bringing moisture and the tart fragrance of some unknown flower.\n\nIt was the middle of summer in southern Oklahoma and the prairie she'd seen through her windshield the day before had been a brown sea of dried grass. This, this looked like spring after weeks of rains somewhere in the foothills of rugged mountains.\n\nWhere the hell was she? Looked like complete wilderness, probably miles from any road, any people. Any help. If she escaped, crossing across rugged country with a six-year-old would be very difficult. She would have to plan well and bring a lot of water.\n\nThe brush quaked. A small brown animal burst from the growth. It resembled a dog, or maybe a coyote. It dashed across the grass, zigzagging in sheer panic. It didn't run like a coyote.\n\nWhat in the world?\n\nKarina raised the binoculars to her eyes.\n\nThe creature wasn't a dog. If anything it looked like a tiny horse, no more than two feet tall.\n\nThe brush shivered and spat three gray shapes onto the grass, one large and two others smaller. They ran upright on a pair of massively muscled legs, their bodies sheathed with gray feathers speckled with spots of black. Long, powerful necks supported heads armed with enormous beaks. The binoculars picked up every detail, from the crests of long feathers on their heads to the tiny vicious eyes.\n\nThe horse galloped for its life, veering left. The bird closest to it slid and swung toward the house to right itself. A flash of pale red shot through the empty air, as if the bird had run into an invisible net stretched tight, and the pressure of its body caused the threads to glow. The bird screeched and fell, catapulted back. For a moment it lay on the grass stunned, and then it rolled back to its feet and rejoined the chase.\n\nThe small horse was getting tired. It slowed. Foam dripped from its mouth.\n\nThe largest bird sprinted. The monstrous beak rose, then came down like an ax, chopping at the horse and knocking it off its feet. The horse rolled in the grass and staggered upright. The three birds danced about it, jabbing and pecking. The horse cried out and fell. Bloody beaks rose again and again . . .\n\nKarina lowered the binoculars.\n\nShe didn't know much about zoology, but she knew enough. They weren't emus; they weren't ostriches; no, these were something vicious, something ancient, something that should not exist in Texas or the Ozarks. Or the twenty-first century.\n\nSuddenly she was cold, freezing from head to toe.\n\nA triumphant screech rolled up from the plain.\n\nKarina dropped the binoculars on the side table and slammed the window shut.\n\n* * *\n\nA cloud of oily smoke greeted Karina in the kitchen. By the stove, Henry cursed, slid several charred pieces of bacon out of a pan with a spatula, and deposited them onto a plate. He saw her and waved the spatula around, flinging hot drops of grease onto the table. \"Good morning.\"\n\n\"Good morning,\" she answered on autopilot. \"I saw . . . birds.\"\n\n\"Terror birds.\" Henry nodded. \"Nasty creatures. Don't worry, there is a large fence around the entire hill. We call it the net\u2014it's thin wire with a powerful current running through it. You're completely safe within the vicinity of the house. They won't come close. Besides, they are mostly cowards. An adult human has nothing to worry about.\"\n\nOne of those things would kill a child. A vision of a bloody beak coming down like a hatchet flashed before Karina's eyes. She swallowed. \"My daughter?\"\n\nThe spatula pointed to Henry's right. \"Through that doorway.\"\n\nKarina forced herself not to run. She skirted the table and walked through the doorway into a living room. Her heart pounded.\n\nA small shape was curled on the couch, hidden by a green blanket. Karina pulled back the covers. Emily lay on the pillow. Her mouth was open slightly, her eyes closed, her hair a tangled mess.\n\nKarina knelt and hugged her gently. Emily stirred and she put her face to her daughter's cheek and clenched herself, trying not to cry.\n\n\"Daniel brought her early in the morning. Arthur told him he would let him speak in return,\" Henry said softly from the doorway. \"I've wiped her memory of the assault in the motel\u2014it was too traumatic\u2014so she won't recall anything about the place, and that entire day will be dim for her. There are no long-term effects to memory wipes, but there are some short-term consequences: she will sleep a lot more, she will seem confused, and she might have some anxiety. It should last for about a week. Lucas already called the main house. They have a nice room set up for her.\"\n\nKarina turned. \"I want her to stay with me.\"\n\nHenry looked uncomfortable. \"There is a reason why the three of us are separated from the main house.\"\n\n\"Three? I thought Arthur lived here.\"\n\nHenry shook his head. \"Arthur stays at the main compound. Of our entire group, Lucas is the most feared, Daniel is the most despised, and I'm the least trusted.\" He paused. \"This house isn't the best place for a child.\"\n\nShe paused. \"Henry, why in the world would anyone not trust you?\" Of the four men she'd met so far, Henry seemed the least insane.\n\nHe smiled, apologetic, almost vulnerable, and leaned closer. \"I can make you forget we ever had this conversation. I can make you forget about Lucas, about the motel, and, if I strain a little, you won't remember you ever had a daughter.\"\n\nShe paused. It seemed insane, but no less insane than the idea of a man who turned into a nightmarish beast. \"Can you read thoughts?\"\n\n\"Nobody can read thoughts.\" Henry shook his head. \"Not even combat-grade operatives like me.\"\n\nCombat with whom? Why? He was wording his replies very carefully, thinking about them for a moment before answering. If she pushed him too hard, he would stop talking. \"I'm not sure I understand. Do you wipe your enemies' memories?\"\n\nHenry took his glasses off and cleaned the lenses with the corner of Emily's blanket. Without his glasses, he seemed younger. \"The mind doesn't just store memories. It also governs many functions of the body. I can mentally scout the enemy and tell you their numbers. Obviously the more of them there are, the higher the margin of error is, but typically I'm not off by a significant number. I can find your mind in a crowd of people and attack it, so you'll think you're drowning. I can disconnect your brain from the rest of you and starve it of oxygen until you become a vegetable. My subspecies isn't called Memory Wiper. It's called Mind Bender.\"\n\nFor a moment she was more terrified of him than she was of Lucas, and thinking that he might somehow crack her skull open and peer into her brain scared her even more.\n\nHenry glanced at Emily on the couch. \"Do you trust me now? Do you want your daughter near me?\"\n\nNo. She didn't trust any of them. But the main house, whatever it was, would be full of strangers. The thought of someone full of violent rage, like Lucas, or cold like Arthur, being in charge of Emily without her to shield her daughter made her wince.\n\nKarina clenched her hands. Screaming and hysterics would do her no good. She had to reason with them. She had to be smart. Use logic. \"Henry, I'd rather take you and Lucas over a house of people I don't know. Emily woke up alone, without me. She must've been frightened. She's my daughter, Henry. She's safest with me, because I'm her mother and I would give my life to keep her from harm.\"\n\n\"Speak to Lucas,\" Henry suggested. \"I'm sure he will permit some sort of visitation.\"\n\nLucas. Lucas had said he owned both of them. She had to make him understand. Karina fixed Emily's blanket and rose. \"Can I make her breakfast? Or should I ask Lucas's permission?\"\n\nHenry stepped aside. \"You're welcome to any food we have.\" He cleared his throat.\n\nThe fridge contained eggs, several pounds of bacon, some slimy cold cuts, a hunk of mozzarella cheese\u2014dried, yellow, and brittle\u2014and a pack of green-looking hot dogs. Karina pulled out eggs and bacon. \"Flour?\"\n\nHenry dug in one of the cabinets, looking lost, frowned, and opened a door, revealing a huge supply room. \"I think in here somewhere.\"\n\nShe stepped into the room. Rows and rows of wooden shelves, filled with cans and jars, a huge spice rack, fifty-pound bags of sugar, flour, rice . . . three large freezers filled with meat. Enough food to feed these men for years. \"Are you expecting a long siege?\"\n\n\"You never know,\" Henry said with a thin smile. \"We've had a few.\"\n\n\"You, Daniel, Lucas, me, Emily. . . is anybody else coming?\"\n\n\"No. Does this mean we're invited to the meal?\"\n\n\"I'm using your food.\"\n\nHenry exhaled, picked up the plate of black bacon strips, and dumped them into the trash. \"Thank God.\"\n\nKarina opened the window first, so the kitchen would air out, and set about making breakfast. Henry parked himself by the refrigerator and watched her. There was something disquieting about Henry. When she looked at him, she got an impression of length: long limbs, long frame, long face. Even though she vaguely recalled that he was slightly shorter than Lucas, he appeared taller. He seemed lean, almost thin, but that notion was deceiving\u2014his sweatshirt sleeves were pulled up to his elbows, revealing forearms sculpted with hard muscle. He smiled often, but the curving of his lips lacked emotion. His smile was paper-thin, an automatic, knee-jerk reaction like blinking.\n\nA Mind Bender. If what he said was true, he could kill Emily in front of her, wipe Karina's mind clean, and she would never remember it.\n\nKarina found Granny Smith apples in the bottom of the fridge and checked the drawers. On the third try she hit what looked like a utility drawer: knives, screwdrivers, bottle openers, and wooden spoons. She fished a medium-sized knife from the drawer, peeled the apples, cored and chopped them, and set them to fry slowly, sprinkling them with brown sugar.\n\n\"It smells divine,\" Henry murmured.\n\n\"Is there cinnamon?\"\n\n\"I am sure there is. It's brown powder, right?\" Henry stepped into the pantry.\n\n\"Yes.\" She grabbed the knife, pulled the fabric of her jeans away from her hip, and slid the knife into her pocket. The point of the blade cut the lining and she jammed the knife all the way down to the hilt. The blade scraped against her skin. She glanced down. No blood. Karina exhaled. Cutting herself was a calculated risk\u2014she had no other place to hide the knife. Anywhere else it would make a bulge. She pulled her T-shirt down over it.\n\nHenry came out of the pantry. She held her breath. Maybe he could read thoughts. Maybe he would pluck the image of the knife out of her head. She had to stop thinking about it, but she couldn't. The shape of the knife was probably glowing in her brain.\n\nHenry shook a plastic container of cinnamon. \"Found it.\"\n\nShe had to say something or he would realize things were wrong. Karina willed her mouth to move. \"Thank you.\" She took the cinnamon and sprinkled it on the apples.\n\nThe bacon rack was missing in action, or perhaps they didn't have one. She layered a plate with paper towels, placed the strips on top, and popped it into the microwave.\n\n\"You don't cook often?\" she asked.\n\n\"On the contrary. I cook quite frequently, out of sheer necessity. Unfortunately, most of what I produce is inedible. Daniel's cooking is even worse than mine, if such a thing is possible. Lucas can grill quite well when pushed to it, but in the kitchen his idea of a meal involves a raw piece of meat, burned on the outside. Adrino was our cook.\"\n\n\"Where is he now?\"\n\n\"Dead. About nine months ago.\"\n\nShe paused to look at him. \"I'm sorry.\"\n\nHenry nodded. \"Thank you.\"\n\nKarina resumed stirring the pancake batter. \"How did he die?\"\n\n\"Lucas bit him in half.\"\n\nShe stopped. \"Was he a member of your family?\"\n\n\"He was. He was Lucas's cousin on his mother's side, and my stepbrother.\"\n\nKarina found the griddle and set it on the burners to heat up. She stirred the apples with a wooden spoon, then pulled the bacon out of the microwave and peeled it from the paper towels.\n\n\"I can do that,\" Henry offered.\n\n\"Thank you.\" She poured the pancake batter on the griddle in quick drips and watched the first pancake puff and bubble at the edges. \"Why did Lucas kill him?\"\n\n\"Adrino tried to murder Arthur.\"\n\n\"Why?\"\n\nHenry smiled, a quick baring of teeth, meaningless and flat like a mask. \"Adrino had raped a woman on base. As a punishment, Arthur had him chained for two months.\"\n\n\"Chained?\"\n\n\"In the courtyard. Eventually Adrino was let off the chain and everything went quite well, until he attempted to solidify Arthur's blood during the last Christmas dinner. In retrospect, we should have expected it. His subspecies is prone to rashness.\" Henry smiled again. \"You will find that we're a violent, vicious lot, Lady Karina. All of us hate Arthur, hate each other, hate who we are, what we are, why we are. This hate is so deep within us, it's in our bones. Lucas hates stronger than most of us for his own reasons. But Lucas is also far more controlled in his rages than he lets on. He recognizes the simple truth: Arthur is the glue that holds us together. Arthur makes mistakes, and he's brutal, but he's also fair. Every tribe must have a leader. Without the leader there is chaos. May I just mention that your pancakes smell delicious? I don't suppose there is any way I could steal one right now, is there?\"\n\n* * *\n\nThirty minutes later, the pancakes were done, the bacon was cooked, and Karina crossed the room to her daughter.\n\n\"Emily? Wake up . . .\"\n\n\"Mommy!\" Emily clutched Karina around the neck and hung on with surprisingly fierce strength.\n\nKarina scooped her off the couch and held her close, afraid to hug the tiny body too hard. \"I'm here, baby. I love you.\" Emily never said \"mommy.\" It was always \"mom.\"\n\n\"You won't leave?\"\n\nA hard knot formed in Karina's throat. \"Leaving\" was Emily's euphemism for dying. Her daughter thought she had died.\n\n\"I will try very hard not to,\" she promised.\n\nEmily hung on, and Karina gently carried her into the kitchen. \"I made your favorite apples.\"\n\nSlowly Emily's hold on her neck eased. A few seconds later she allowed herself to be put into a chair at the table.\n\nDaniel marched into the kitchen. \"Food.\"\n\nHenry nodded. \"Yes.\"\n\nDaniel pulled out a chair, sat, and reached for the pancakes.\n\n\"Let's wait for Lucas,\" Henry said.\n\n\"Fuck Lucas.\"\n\nKarina looked at Daniel. Henry sighed. Daniel looked back at them, glanced at Emily, and shrugged. \"They don't like it that I swear. Do you mind if I swear?\"\n\nEmily shook her head.\n\n\"See, she doesn't mind.\"\n\nLucas loomed in the doorway. One moment it was empty and the next he was just there, green eyes watching her every move with a hungry light. Karina took her chair, trying to ignore it, but his gaze clasped her like an invisible chain. She looked back at him. _Yes, I belong to you. You don't have to ram it down my throat._\n\nEmily's eyes had grown big. She shied a little when Lucas stepped to the table, aware of his movements. Karina read fear in her daughter's face and reached over to hold her hand. He'd given Emily no reason to fear him, yet she was clearly scared, almost as if she sensed on some primal level that he was a threat.\n\nLucas sat next to Karina, opposite of Daniel, and reached for the pancakes. She watched him load his plate: four pancakes, four links of sausage, six strips of bacon . . . The plate would hold no more. He pondered it, frustrated, then piled the apples atop the pancakes and drenched the whole thing in maple syrup.\n\nIt was good that she had made enough for ten people.\n\nLucas sliced pancakes with his fork, pierced a slice of the apple, and maneuvered the whole thing into his mouth. Karina sat on the edge of her seat, listening to the elevated tempo of her own heartbeat, watching him chew, and waited for him to throw the plate across the table. She wanted them to like the food; no, she desperately needed the three of them to like the food. Her survival depended on it.\n\nLucas swallowed. \"Good,\" he said and reached for more.\n\nKarina slumped a little in her chair, unable to hide her relief.\n\n\"Good? It's fucking divine,\" Daniel said. \"It's the first decent meal we've had in weeks.\"\n\nLucas leveled a heavy stare at him but said nothing.\n\n\"Mom,\" Emily said.\n\n\"What, baby?\"\n\n\"I left my backpack at Jill's house. It has my school stuff in it.\"\n\nThe three men ate, watching her.\n\n\"That will be okay, baby,\" Karina said. \"You have to change schools anyway.\"\n\n\"Why?\"\n\n\"Because we live here now and you'll go to a special school.\" The words came out painfully.\n\n\"Do I have to ride the bus?\"\n\nKarina swallowed a lump that had formed in her throat. Acknowledging where they were was hard, as if she were driving nails into her own coffin. \"No.\"\n\n\"Why do we have to stay here?\"\n\n\"This is where I work now.\"\n\n\"Your mother is a slave,\" Daniel said. \"Lucas owns her.\"\n\nIf only she could have reached across the table, she would have hit him with a closed fist so it would hurt. Karina forced neutrality into her face, pulling it on like a mask. Show nothing. Betray no weakness.\n\n\"Is a slave better than a payroll supervisor?\" Emily asked.\n\n\"They're not that different,\" Karina lied. So many times before she had thought she worked like a slave, pulling in long hours, picking up project after project, perpetually behind, trying to get to the bottom of her to-do stack. She thought she had experienced the worst life could throw at her. All of it seemed so pointless now. Her memories belonged to someone else, a happier, flightier, younger person. She had a new life now and new priorities, chief of which was the welfare of her daughter. She had to keep Emily safe.\n\nEmily poked her pancake with a fork. \"What about the house? All our stuff is there . . . my Hello Kitty blanket . . .\"\n\n\"We'll get new things.\" She cast a quick glance around the table but none of the three men said anything to break down her fragile promises.\n\n\"Will I get my own room?\"\n\nKarina looked to Lucas. _Please. Don't separate me from my daughter._\n\nHe wiped his mouth with a napkin, his movements unhurried. \"You have to stay at the main house. You can come to visit your mother on weekends. We'll set up a room.\"\n\n\"I want to stay with Mom.\" Emily's voice was tiny.\n\n\"You can't,\" Lucas said.\n\nEmily bit her lip.\n\n\"You'll have a good place at the main house. A room you'll share with a nice girl. Toys. Clothes. Everything you need. If anybody tries to be mean to you, tell them you belong to Lucas. Everyone is afraid of me. Nobody will harm you.\"\n\n\"No,\" Emily said.\n\nLucas stopped eating. Karina tensed.\n\n\"Are you telling me no?\" Lucas asked. His voice was calm.\n\nEmily raised her chin with all of the defiance a six-year-old could muster. \"I'm tired and I'm scared, and I'm not going. I'm staying with my mom. Are you going to yell at me?\"\n\n\"No,\" Lucas said. \"I don't need to.\"\n\n\"You're not my dad. My dad left.\"\n\nLucas glanced at Karina.\n\n\"I'm a widow,\" she said quietly.\n\n\"I'm not your father, but I'm in charge,\" Lucas said. \"You will obey me anyway.\"\n\n\"Why?\" Emily asked.\n\nLucas leaned forward and stared at Emily. \"Because I am big, strong, and scary. And you are very small.\"\n\n\"You're not nice.\" Emily held his gaze, but Karina could tell it wasn't out of courage. Emily had simply frozen like a baby rabbit looking into the eyes of a wolf.\n\n\"It's not a nice world and I can't always be nice,\" Lucas said. \"But I will try and I won't be mean to you without a reason.\"\n\nKarina put her hand on his forearm, trying to tear his attention away from Emily. It worked; he looked at her.\n\n\"Please.\" It took all of her will to keep the tremble out of her voice. \"Please let her stay.\"\n\n\"I want to stay,\" Emily said. \"I'll be good. I'll do all my chores.\"\n\n\"I'll think about it,\" Lucas said.\n\n# **CHAPTER 4**\n\nA half hour later, breakfast was finished. The men rose one by one, rinsed their plates, and loaded the dishes and silverware into the dishwasher with surprising efficiency. Karina put the last of the food away. Henry had stepped out, but Daniel remained in the kitchen, leaning against the counter, watching her. Lucas loomed by the door, watching Daniel.\n\n\"Can I go outside?\" Emily asked.\n\nKarina paused. \"I don't think that is a good idea.\"\n\n\"Why not?\" Daniel arched an eyebrow.\n\n\"Because there are scary birds out there.\"\n\n\"There are scary birds? What kind of scary birds?\"\n\n\"It's safe,\" Lucas said. \"The net keeps everything out.\"\n\nKarina remembered the bird's body hitting the invisible fence. \"What if she walks into this net?\"\n\n\"She'd have to walk a mile and a half down the hill before she reached it,\" Lucas said.\n\n\"I want to see the birds,\" Emily said. \"Please?\"\n\nIt would get them out of the house, away from the men and out into the open. She could get a better look around. Maybe she would see a road, or a house, some avenue of escape. Karina wiped her hands with a towel and hung it on the back of the chair. \"Okay. But we're going to stay by the house.\"\n\n\"I'll come with you,\" Lucas said.\n\nAll she wanted was the illusion of being alone with her daughter. He wouldn't let her have it. Karina clenched her teeth.\n\n\"That's right,\" Daniel said. \"Bite your tongue. It will come in handy.\"\n\nLucas gave him a flat stare. For a moment they stood still, then Daniel rolled his eyes and casually wandered out of the kitchen into the side hallway. Lucas moved in the opposite direction, through the doorway. Karina took Emily by the hand. \"Come on, baby.\"\n\nThe hallway cut through the house, straight to the door. They passed rooms: a library filled with books from floor to ceiling on the right, a large room with a giant flat-screen TV on the left, and then Lucas opened the door and they stepped on the porch into the sunlight. The yard was grass, small scrawny oaks and brush flanking it on both sides. A path led down the hill into the distance. To the left a huge oak out of sync with the rest of the scrub forest and probably planted, spread its branches.\n\nA shaggy brown dog stepped out from behind the oak. As tall as a Great Dane, it trotted forward on massive legs, its long tail held straight behind it. There was something odd in the way it walked, waddling slightly, more like a bear than a dog.\n\nKarina stepped between Emily and the beast.\n\nThe animal stopped. Large brown eyes stared at them from a massive head crowned with round ears.\n\n\"Don't worry, he's tame,\" Lucas said behind her.\n\nThe meld of dog and bear peered at Lucas and let out a short snort.\n\n\"He doesn't like it when I phase into my attack variant,\" Lucas said. \"It weirds him out for a couple of days. Cedric, don't be a dick. Let the kid pet you.\"\n\nAnother snort. She couldn't really blame the dog. Considering how Lucas looked in his \"attack variant\" it was a wonder the dog stuck around at all.\n\nCedric pondered them for a long moment and waddled over. Emily stretched out her hand. Karina's insides clenched into a tight knot.\n\nCedric nudged Emily's hand with his nose, snorted again, and bumped the bulge in the front pocket of her hoodie.\n\n\"What do you have in your pocket?\" Karina asked.\n\nEmily dug into her pocket and pulled out a half-eaten apple.\n\nNot again. Karina kept her voice gentle. \"Emily, you know you're not supposed to have that . . .\"\n\nCedric sniffed at the apple. His mouth gaped open, revealing huge teeth.\n\n\"He won't hurt her,\" Lucas said with absolute certainty in his voice.\n\nEmily held the apple out. Very carefully, almost gently, Cedric swiped it off her hand, sat on his behind, and raised the fruit to his mouth, holding it with long, dark claws. The black nose sniffed the apple, the jaws opened and closed, and the beast bit a small chunk from the fruit and chewed in obvious pleasure.\n\n\"He likes it!\" Emily announced and jumped down off the steps into the yard. \"Come on, Cedric!\"\n\n\"Where are you going?\" Karina took a step to follow.\n\n\"Just to the tree.\"\n\nThe oak was barely fifty feet away. Karina bit her lip. Her instincts told her to clutch her child and not let go, but Emily needed to feel normal. She needed to play. Her daughter didn't understand how precarious their situation was, she had no idea how vulnerable they were, and Karina had to keep it that way.\n\nEmily was looking at her. \"Can I go?\"\n\n\"Yes. You can go.\"\n\nEmily headed toward the tree. Cedric finished his apple in a hurried gulp, rolled to his paws, and followed her to the tree.\n\nLucas leaned on a porch post next to Karina. She had expected him to somehow shrink in daylight, as if he were some sort of evil creature of the night whose power faded with the sun, but he remained just as big and menacing. If anything, the sun made it worse\u2014she could see every detail of his severe face. Everything about him, the way he leaned against the rail, the way muscle bulged on his arms and chest, the way he surveyed the yard, inspecting his territory, communicated predator.\n\nLucas raised his face to the sun, closed his eyes, and smiled. The smile lasted only a moment, gone like a leaf blown by the breeze, but she had seen it. He was handsome and the danger he emanated sharpened that beauty to a lethal edge. He was beautiful in the same deadly way a tiger was beautiful, and now she was locked in a cage with him.\n\nIf he was on her side, nobody would ever bother them.\n\nAt the tree, Emily picked up a twig and tossed it. Cedric looked at the twig and back at her, slightly puzzled.\n\n\"What is he?\" Karina asked.\n\n\"A bear-dog. We played with their genetics a few generations back. He's gentle like a collie with the kids and he's a lot smarter than an average dog. What's the problem with her having an apple?\"\n\nKarina sat on the stairs. \"She hoards food.\"\n\n\"Why?\"\n\nShe didn't want to tell him. The less he knew about them, the less information he could use against her. \"It makes her feel safe.\"\n\nBy the tree Emily clapped her hands and explained something to Cedric. He sat on his butt again, listening to her.\n\n\"Was she adopted?\" Lucas asked quietly.\n\n\"No.\" She wouldn't have expected him to know that adoptive children sometimes exhibited food hoarding. Now she had to explain more just to keep him from getting the wrong idea. \"It happened after her father's death. It's not an eating disorder. She doesn't want extra food; she's just trying to control her environment. We handled it, but with everything that's happened she might have relapsed. Please don't berate her or yell at her for it. It . . .\"\n\n\"It makes things worse,\" he finished for her. \"I know.\"\n\n\"Let me have her,\" she said, suddenly desperate. \"Let me have her here with me. I thought I lost her on that fire escape. You have everything else\u2014my freedom, my body, everything\u2014and I just want one thing. Just let me keep my baby.\"\n\nLucas frowned. He didn't seem vicious now. \"I'm not doing this to be an asshole.\"\n\n\"I'll keep her out of the way . . .\"\n\nCedric snarled at the bushes, baring his teeth, and lunged into the thicket.\n\nKarina jumped to her feet. Before her knees had straightened, Lucas had leaped over the railing and was sprinting to the tree.\n\nEmily stumbled back. Her mouth gaped in a surprised O.\n\nKarina ran but she was so agonizingly slow.\n\nLucas reached Emily, pushed her back, out of the way, and crashed through the underbrush.\n\nKarina lunged forward. Her hand closed about Emily's shoulder. She grabbed her daughter and backed away, keeping her hand on her pocket, feeling the hard knife handle through the fabric of her jeans.\n\nLucas jerked something out of the bushes. Long and green and brown, it writhed in his hand, flailing, the elongated olive tail brushing the ground. He roared, a deep growl that nearly made her jump. \"Henry!\"\n\nThe thing jerked, its throat caught in Lucas's hand. He turned and Karina finally saw it: it resembled a freakishly large bearded dragon lizard bristling with inch-long spikes on its cheeks and sides. As the creature twisted, a crest snapped open along its back, the spikes standing up like the razor-sharp fins of some deepwater fish. The lizard creature clawed at Lucas's arm with long black claws. Blood welled in the scratches.\n\n\"A monster!\" Emily squeaked.\n\n\"No, just a big lizard.\" Karina kept a death grip on Emily.\n\nBehind her the door burst open. Daniel charged onto the porch. His face contorted. Something brushed past Karina like a sudden draft. The beast jerked and hung motionless, its legs abruptly gone limp.\n\nLucas carried the lizard to the porch. \"Henry!\"\n\nHenry burst onto the porch.\n\nLucas slammed the lizard down onto the porch boards. The creature blinked but lay completely still. Henry knelt by the lizard. His hands touched the back of the creature's skull. He closed his eyes, focused for a long moment, and glanced up. \"Its mind is inert. It didn't transmit.\"\n\nLucas looked at him. \"Sure?\"\n\nHenry pushed his glasses back up his nose. \"Yes. If it transmitted, there would be evidence of a spike in neural activity.\"\n\nLucas raised his fist and brought it down like a hammer. She barely had enough warning to spin Emily around before his fist crushed the lizard's skull, flattening it like an empty Coke can.\n\n\"Daniel, call the main house.\" Lucas turned to her. \"Take Emily and go to our room. Don't come out until I get you.\"\n\nKarina didn't ask what was going on. She just picked Emily up, ran inside the house, and didn't stop until the door of Lucas's room closed behind her.\n\n* * *\n\nThe day burned down to the afternoon. Emily investigated the room, then she whined about being bored, and finally she fell asleep in the overstuffed chair in the corner. At first Karina listened for every noise and creak outside the door. Her nerves were wound so tight, she could barely sit still.\n\nIf the creature in the bushes had been just an ordinary lizard, Lucas would've killed it right away. She had no doubt of it. No, this beast had created an emergency. She had no idea why and that somehow made everything so much worse. Eventually her own anxiety wore her out and she sank into a light sleep, a kind of wakeful drowsiness, where every stray noise made her raise her head.\n\nThe room was so quiet. Karina closed her eyes for a moment, opened them, and then Lucas was there, walking across the room. She hadn't heard the door open.\n\nLucas scooped Emily out of the chair. Karina surged to her feet. \"Where are you taking her?\"\n\n\"To a different room,\" he said quietly and went out. She followed him down the hallway to a small bedroom. A bed with a red comforter stood against one wall, next to a bookcase filled with children's books. A desk offered a small computer with a flat-screen monitor.\n\nHe'd made her a room. He'd changed his mind.\n\nLucas deposited Emily on the bed and stepped out. Karina pulled the blanket over Emily's shoulders. She was so tiny on the bed. Karina's mind replayed Lucas clenching the lizard's throat. One squeeze and Emily would be dead.\n\nHe waited for her now, in the hallway. Karina made herself step away from the bed and walked out. Lucas closed the door, locked it, and handed her the key. \"This is for her protection. Our room doesn't have a lock. Daniel is pissed off tonight, and I'm feeling surly, which makes the house a dangerous place to be, so it's best she stays in this room. This is for tonight only. Tomorrow she will go to the main house.\"\n\nBut the room\u2014it was a child's room, made for a little girl. The blankets and the pillowcases looked brand-new and the rug still had the price sticker on it.\n\nSo he hadn't changed his mind. She had from now until morning to convince him to let her keep her daughter. Karina opened her mouth and said the only thing she could think of. \"Are you hungry?\"\n\nLucas nodded. \"I could eat.\"\n\n\"Any preference?\"\n\n\"Meat would be nice.\" He turned away.\n\n\"Lucas?\"\n\nHe glanced at her over his shoulder. \"Yes?\"\n\n\"What's going on?\" Karina asked him softly. \"What was that thing?\"\n\nLucas grimaced. \"It's a long explanation.\"\n\n\"Please. I want to know.\" Whatever he would tell her had to be better than not knowing.\n\nLucas sighed. \"The woman who poisoned you has friends. Her people are looking for our base, so they are sending scouts out. The lizard was one of them. It's basically a walking camera\u2014it records what it sees and then transmits the information to its owners in short bursts. Luckily we caught this one before any transmissions had gone out.\"\n\n\"And if it had sent this transmission?\"\n\n\"We'd be evacuating,\" Lucas said. \"We still may. We'll know more in the morning.\"\n\nKarina hugged her shoulders. \"Lucas, where are we?\"\n\nHe was looking directly at her. \"We're on base.\"\n\n\"Where is this base? I've seen those birds. There are no birds like that in North America.\"\n\nLucas examined her face for a long breath. \"You want the truth?\"\n\n\"Yes.\"\n\nHe grimaced. \"You asked for it. As the planet rotates, fluctuations between the forces of gravity and nuclear reactions on the subleptron and subquark level cause a ripple effect in reality, where time and space are not constant but dynamic. Parts of space-time become incompatible with the current reality and are discarded. In essence, Earth continuously sheds chunks of itself. They linger for a time and dissipate, some slower, some faster. We're in one such chunk\u2014we call them fragments. It was shed sometime during the late Pliocene, approximately two and a half million years ago in what is now Texas. This pocket is stable and shouldn't begin to dissipate for another couple thousand years. Can you make cubed steak?\"\n\n\"What?\" Karina stared at him, sure she had misheard.\n\n\"I asked if you can cook cubed steak. I just realized I'd really like some.\"\n\n\"Yes, I can. You're not joking?\"\n\n\"About the steak?\"\n\n\"About the fragments.\"\n\nLucas shook his head.\n\nThis was just insane. \"So we're in an alternate reality? Like in a parallel dimension? Like in _Star Trek_?\"\n\n\"No. A mirror dimension is a self-contained, complete reality. We're in a dimensional fragment.\" Lucas leaned back against the wall. \"Okay, think of an onion. The inner layers are white, and the outer layer is brown. Suppose the outer layer rots. The onion makes a replacement layer, identical to this outer one, and sheds the rotten layer in bits and pieces, some big, some tiny. We are in a piece of that rotten layer.\"\n\nShe stared at him. If he wasn't lying, they weren't anywhere near Oklahoma. They weren't even on the same planet. Escape was impossible.\n\n\"Don't think about it too much,\" Lucas said. \"Subquantum mechanics will drive you insane.\"\n\n\"Can we get back? To normal Earth?\"\n\n\"It depends on how close the layer is to its reality. The motel where you were attacked was in a layer that had barely begun to separate, so we could cross in and out easily. But this pocket has peeled much too far away for you and I to exit on our own. We need someone to rip it. To open a gateway.\" Lucas pushed off from the wall.\n\n\"But we _can_ go back?\" Surely they had to go back occasionally. Their clothes had tags; their plates had Corelle stamped on the back. Microwaves and refrigerators didn't sprout on prehistoric trees, which meant the people of Daryon had to pop back and forth from the normal Earth to here and back on a whim.\n\nLucas leaned toward her. His gaze fixed on her. Suddenly he was occupying too much space. She took a step back, her spine pressing against the wall.\n\nA slow smile curved Lucas's lips. \"Yes. You can go back. But never without me. If you ever try, I will find you and bring you back.\" His smile grew wider. \"And then all bets are off.\"\n\nHe was looking at her with an open sexual hunger, so intense, for a second she didn't think it could be sincere. She froze, terrified. And then a small part of her responded to it. For a second, Karina wondered what it would be like to cross the distance between them, laugh right into that stare, and walk away, leaving him standing there like an idiot. But as long as he controlled Emily, she could do nothing.\n\nHe leaned forward a quarter inch, like a predatory cat about to pounce.\n\nIn her mind, Karina gulped and fled down the hallway, her heart hammering too fast and too loud. But showing weakness wasn't an option. Lucas had told her before that he was a predator. If she ran, the predator would chase.\n\nShe raised her face toward him. \"If I do go back without you, don't find me.\"\n\nHe turned his head to the side, like a dog, studying her. \"Or?\"\n\n\"Or I will kill you.\"\n\nHe laughed, a low rich sound that sent shivers of alarm down her spine. \"How?\"\n\n\"I'll think of something.\"\n\nShe turned her back to him and forced herself to walk slowly toward the kitchen.\n\n* * *\n\nLucas tilted his head and watched Karina retreat down the hallway. The look in her eyes, the angle of her face, the way she stood, everything communicated defiance. She challenged him. She had no idea how exciting this made her. He wanted to pin her against the wall, until she acknowledged that he was strong enough and powerful enough for her. He wanted to kiss and taste and grind and own. Different standards, he reminded himself. For him it would be flirting. For her, it would be a prelude to rape.\n\nLucas looked at the ceiling. He knew exactly where this violent impulse was coming from. It was an evolutionary echo, the same echo that told him to murder every other male in the house and then hunt her until she gave in. He made a choice to reject it daily. Strangely, it wasn't getting any easier.\n\nHenry's light steps approached him. \"Physical assault is probably not the best way to go,\" Henry murmured.\n\nSometimes Lucas could swear the man could read thoughts, even though every Mind Bender Lucas had ever met maintained it was impossible. \"Playing in my head?\"\n\n\"Of course not.\" Henry smiled at him. \"Your fists are clenched and it's written all over your face.\"\n\nHe'd figured as much. \"She's beginning to ask questions.\"\n\n\"That's a little faster than I expected.\" Henry frowned. \"I wiped almost twelve hours of severe pain from her. Usually a wipe of that extent leaves people inert longer. You're pacing the explanations?\"\n\nLucas nodded. \"Not my first time.\"\n\nHe'd helped bring people over a few times before. A human mind could only accept so much. If he flooded her with the information contradicting her view of reality, the impact of it, combined with her physical trauma, would cause her to snap under the pressure. Her body was at its limits already, fighting the poison and coping with his venom and its consequences, which would soon follow.\n\nLucas started down the hallway. He needed a shower and some time away from everyone to soothe the excitement rushing through his veins.\n\n\"Lucas?\" Henry called.\n\nLucas turned.\n\nHis cousin looked at him for a long moment. \"Be kind.\"\n\n* * *\n\nAn hour later Karina put the dinner on the table. The encounter in the hallway kept replaying in her head and she couldn't decide if she'd botched it or handled it well. Emily still slept. Henry had said the fatigue was normal, but she worried all the same.\n\n\"Cubed steak.\" Henry slid into his seat. \" 'Beef. It's what's for dinner.' \"\n\nKarina took her seat. Lucas sat to the right of her. Too close. She should have served the dinner in the dining room instead of the kitchen. The bigger table would've given her more space.\n\nLucas crowded her, drinking in her anxiety. Karina swallowed, unable to help herself. He was simply too large and he watched her constantly. Even when she couldn't see him, she couldn't get rid of the pressure his gaze brought. He leaned toward her, emanating menace, and she shrank from him out of sheer self-defense.\n\nHis lips stretched and Lucas showed her his teeth, large and sharp. \"Am I scary?\"\n\nShe met his stare. \"Yes,\" she said. \"But you know that already. Making me admit it makes you cruel. Corn or beans?\"\n\nHe drew back. His eyes widened and for a moment the burden of his presence eased. \"Corn.\"\n\nShe passed the dish of corn to him.\n\nDaniel sauntered into the room. While Henry migrated from place to place and Lucas stalked, his steps soundless and full of fierce grace, Daniel strode as if his feet did the ground a great favor. He didn't walk but floated, devastating in his beauty and perfectly aware of it.\n\nDaniel took a seat directly opposite her. He speared a steak and dropped it on his plate. \"Are you going to do this every day? Cook the dinner, be the dinner?\"\n\n\"Yes,\" Karina said with a calm she didn't feel.\n\n\"Why? Are you totally spineless? What do you think sucking up will earn you? Look at him.\" Daniel pointed at Lucas. \"He doesn't care.\"\n\n\"I'm not doing it for him.\"\n\n\"Then why?\"\n\n\"Here we go.\" Henry rolled his eyes.\n\nDaniel pushed off from the table, balancing his chair on its back legs, and crossed his arms. \"No, I want her to enlighten me. How deeply has Stockholm syndrome set in?\"\n\nKarina put down her fork. Her instinct told her that whatever she said next would define her place in this house. The idea of some flattering subterfuge crossed her mind and died. She wondered if she should say nothing at all. In the end, she decided on honesty.\n\n\"I understand that I can die at any moment. Lucas's cousin died at the last Christmas dinner. For all I know, Lucas might die tomorrow, killed by your enemies or by your family members. Without Lucas I have no worth. My daughter is here because of me. If I'm no longer needed, I expect that neither will she be. I've seen enough of your family to realize we won't be allowed to leave. You will dispose of us as if we never existed. I have to find some way to make myself valued beyond Lucas. Then, if he dies, both my daughter and I might survive.\"\n\n\"And you do this by becoming our housekeeper?\" Daniel grinned. \"Cooking, cleaning up after us? Tell me, how low will you stoop? If I leave some shit in the bathroom for you, will you clean it up?\"\n\n\"No,\" Karina said. \"You'll clean your own shit. Unless you're sitting in a pile of it right now, you must know how to aim for the toilet and wipe your own ass.\"\n\nThe amusement in Daniel's eyes crystallized into anger. \"If you want to ingratiate yourself, there's a much easier way of doing it. You can come over here right now and suck my cock. That will put you into my good graces much faster than scrubbing the sink.\"\n\nKarina glanced at Lucas. He cut a piece of steak, chewed with obvious pleasure, and threw her a look that said, _Sit tight._\n\n\"She isn't a fool, Daniel.\" Henry snagged another roll from the bread basket. \"These are delicious. She knows that servicing you would put you and Lucas at each other's throats. You're playing this game for your personal gratification, but Lucas depends on her for his survival. She'd have to be mentally deficient to choose you over him.\"\n\nDaniel shifted to Lucas. \"So what does his lordship think of all this? Your snack has you buried already. Are you flattered?\"\n\nLucas cut into his third steak.\n\n\"What would you do in her place? Would you mop the floors, O mighty one?\"\n\nLucas thought about it. \"In her place I would've killed the two of you already. But I'm not in her place. And I'm not her. I'm not smaller and weaker than everyone around me, nor do I have a child's life in my hands. She's being prudent, given her situation.\"\n\nDaniel smirked. \"Never thought you'd be so agreeable at the idea of your own death.\"\n\n\"We all must come to terms with it one way or another,\" Lucas said.\n\n\"Maybe I'll help you on your way, then, since you're all prepared. Seems a shame to waste the opportunity.\"\n\n\"Think you can?\" Lucas asked with genuine interest.\n\n\"Careful, Daniel,\" Henry said. \"That kind of talk will end with you breaking a nail or messing up your hair.\"\n\nDaniel ignored him and glared at Lucas. \"Bring it.\"\n\nLucas put down his fork, smiled, and shoved the table aside like it weighed nothing. Karina scrambled out of the way. Lucas's huge hand clamped Daniel's throat. Daniel clawed at Lucas's forearm. The bigger man jerked him off his feet, shook him the way a dog shakes a rat, and slammed him down onto the table. Dishes flew. Trapped in a corner between the counter and the stove, Karina threw her hands in front of her face. A ceramic dish shattered next to her, spraying green beans over the counter.\n\n\"No,\" Henry screamed. \"Not inside! Not inside!\"\n\nRed marks sliced Lucas's forearms. His skin bulged as if his bones were trying to break free.\n\n\"Yeah!\" he snarled. \"Hurt me more. Is that all you got?\" His hand still locked on Daniel's throat, he pulled him up and smashed him onto the table again. \"Need some more?\" Daniel's face had grown bright red. Lucas jerked him up. \"Not done yet?\" He drove Daniel back down.\n\nWith a thunderous snap, the table broke in two. The two halves fell apart and Daniel crashed onto the floor, Lucas atop him, still crushing his windpipe. Daniel's feet drummed the ground. Veins bulged on his face, his skin turning magenta. His eyes rolled back into his skull.\n\n\"Here we go.\" Henry sighed. \"We lose all the good dishes this way.\" He showed Karina the bread basket. \"At least I saved the rolls. And don't worry, I'm keeping Emily asleep.\"\n\nLucas released Daniel. The blond man lay unmoving. Lucas stepped over him, his eyes blazing with fury. His gaze locked on her. \"Bedtime,\" Lucas growled and lunged at her. An unstoppable force swept Karina off her feet and she found herself slung over Lucas's back.\n\n\"Let me go!\" She struggled to pull free.\n\nHe swung around to face Henry. \"Leave the mess for when he wakes up.\"\n\n\"Will do.\" Henry saluted him with a roll.\n\nLucas headed out of the kitchen. Karina tried to grab onto the door frame, but her fingers slipped and she was carried through the darkness of the hallway to the bedroom.\n\n# **CHAPTER 5**\n\nThe room swung as Lucas slapped the door closed. Karina expected him to hurl her on the bed but he lowered her to the floor. She stumbled, dizzy from being spun back and forth, and scrambled to get away. Steely fingers caught her arm. He held on to her and sniffed at the sleeve of her sweatshirt. \"Green beans. You want a shower?\"\n\nHis tone was calm. She glanced at his face. All of the rage had gone out of him. He looked worn out, his fury muted to mere smoldering coals.\n\n\"Yes.\" She hesitated. \"I don't have any clean clothes.\"\n\n\"That's a problem,\" Lucas agreed. \"I'm sorry about the dinner.\"\n\n\"That's okay.\" His sudden calm threw her off balance. She stood still, expecting him to swing at her or maybe roar into her face.\n\nLucas reached into the dresser and pulled out a white T-shirt. \"That's the best I can do for now. I'll have something sent up from the main house in the morning.\"\n\nShe took the T-shirt. He didn't offer her any underwear. She would be naked under it.\n\n\"Come on.\" Lucas pulled off his shirt and dropped it on the floor. Carved muscle bunched on his back. Nude, clothed\u2014he could rape her at any point. Clothes wouldn't provide much of a defense.\n\nHe paused, his hand on the door of the bathroom. \"Are you coming?\"\n\n_Not if I can help it._ \"I'll wait until you're done.\"\n\n\"I'll be in here for hours,\" he said. \"The shower stall is enclosed. You can take your clothes off and I'll see nothing.\"\n\nFor hours . . . Why would he be in the bathroom for hours? \"I thought you needed to feed.\"\n\n\"I do, but I won't be feeding for a while.\"\n\nShe followed him, despite knowing better, eager for any crumb of information. \"How long is a while?\"\n\n\"Couple of weeks. Maybe longer. Depends on how quickly you deal with my venom.\"\n\n\"Why?\"\n\n\"Because too much of my toxin at once will kill you.\"\n\nShe remembered his explanation from the night before. \"You said your venom hurts you. Does it hurt now?\"\n\nHe nodded.\n\n\"Always?\"\n\nLucas looked at her. \"Always. Worse after I am injured and much worse after I phase out of the attack variant. Sometimes I have seizures after phasing out.\"\n\nIf he hurt always, he would have to feed always . . . \"How often do you . . .\"\n\nAs if reading her thoughts, he shrugged. \"Once the optimal ratio of my venom to your hormones is reached in my blood, I'll need to feed every three weeks to maintain it. I won't be drinking as much as the last time. Come on. You need a shower and I need to sit down.\"\n\nHe stepped out of her way. During the day she had used the bathroom in the hallway, near the kitchen. She had assumed this one would be the same.\n\nA room almost as big as the bedroom itself greeted her. A dark green hot tub was sunk into the sealed wooden floor. Beyond it a shower stall stretched the entire length of the wall. Its frame matched the hot tub, but the stall itself consisted of wide, dark green panels, either glass or plastic, thick and frosted from the inside. Lucas hadn't lied\u2014he might be able to discern her shadow, but that was about it. To the right was another stall, which she assumed hid the toilet, next to a large sink.\n\nLucas flipped a switch on the wall and the hot tub jets started, whipping the water into froth.\n\nThe shower called to Karina. To go on and disrobe while he was in the tub was insane, but she was covered in food and his scent from the previous night still stained her skin. She could wash him off.\n\nKarina bit her lip and slipped past Lucas to the shower. She closed the door and saw a latch. Relief flooded her. She could lock herself in and for a few minutes pretend she was safe. She slid the latch closed and almost cried.\n\nThe shower stall was divided into a dressing area and the shower itself, separated by a curtain. Karina dug into the pocket of her jeans and fished out the knife. The blade seemed so small compared to Lucas. If she stuck it into his back, he might not even notice. She put it on the small metal shelf next to the soap and, pulled off her clothes, dropping them into a rumpled pile on the bench. An array of shampoo bottles and soaps waited her selection. She took the bottle with the picture of a green apple on the side, picked up a bar of soap at random, and stepped into the shower. Jets surrounded her on three sides. She turned the big wheel of the faucet and a wide sheet of water spilled on her from above in a warm, soothing waterfall. She dropped the shampoo and the soap. All around her water sprayed and cascaded, drenching her, washing away the scent of warm copper. She stepped into the deluge, closed her eyes, and swayed.\n\n* * *\n\nLucas slid into the hot water. He liked it near scalding. It wasn't quite hot enough, but it was getting there. The currents pummeled his body. He switched the two nearest jets off. The sharp claws of pain that scraped his ribs dulled to a low ache as he healed. His right arm still throbbed. Daniel was getting stronger.\n\nOne day one of them would get careless and they might finish each other off. Lucas closed his eyes and submerged. There were worse ways to go than being killed by your brother.\n\nThe rage that had driven him these past few days was gone, burned out in an adrenaline rush of violence.\n\nHe came up for air and settled with his head on the ledge, positioned in the dip of the shelf, the only place he could sit with the water lapping at his neck.\n\nSo tired . . .\n\nThe healing was draining his inner resources and he felt thin and weak, as if all of his muscles were a threadbare shirt hanging off his bones. From here he could see the door and the shower stall. She was in there. Naked. Wet. A fruity synthetic scent teased him\u2014she was washing her hair. He pictured her body under the water, her hands sliding over her breasts and down . . .\n\nA dull thud made him lift his head. In the shower, a dark shadow slumped, pressed against the glass.\n\nIt had hit her finally. He'd waited the whole day for it.\n\nLucas climbed out of the hot tub. The shower-stall door was locked. He hit it with his palm and the lock popped open. Karina lay curled in a corner of the shower, a small wet clump. Her legs shivered. Her skin had gained a pale, almost gray tint. He scooped her off the floor.\n\n\"No,\" she stuttered. Her lips had turned blue. Not a good sign.\n\nHe bent down. She lashed out. He caught a glint of metal and pulled back, letting the knife blade miss him. Where had she even gotten one? Ah, yes. The kitchen. He plucked the knife out of her fingers and picked her up off the floor.\n\n\"No.\" She pushed against his chest.\n\n\"Shhh,\" he told her. \"I'm not going to hurt you.\"\n\nHe carried her out. Her wet skin was ice-cold against his.\n\nShe fought him even as he climbed into the tub and lowered her onto the shelf, sinking her up to her chin in the hot water. \"Let me go . . .\"\n\nAfraid to agitate her any further, he put the full width of the tub between them, giving her room. No need to strain her. If she passed out, the chances of her survival would drop to almost nothing.\n\nIt took a full three minutes before her teeth stopped chattering. She looked at him. \"Everything hurts.\"\n\n\"Your body is reacting to the venom,\" he said. \"Hot water will help. It soothes the muscles. It's normal.\" Technically everything he said was true. He just didn't go into the rest of the details. Not yet.\n\nA short bitter laugh slipped from her lips. \"Normal? Nothing about this is normal.\"\n\nTrue. Not for her anyway. For him, it was business as usual. \"Thirsty?\"\n\n\"Yes.\"\n\nHe waded through the tub, reached for the small fridge beside it, and extracted a bottle of water.\n\nShe took the bottle, clamped the plastic cap in her teeth, twisted it off, and drank, draining nearly a third in a single long draft. _That's it . . . Drink, Karina._\n\nHe recalled Galatea's first time. She'd known exactly what would happen. She had been raised for precisely this purpose: to support him. And she loathed him for it. Hate would've been too personal of a word; he didn't rank that high in her mental roster. Galatea hated the family; she hated Arthur because he was in charge; but Lucas she merely despised, disgusted by his touch. The older he got, the more he realized that sex with him was her way of revenge. In feeding he dominated her and she had no choice but to submit. In bed, for a few fleeting moments Galatea dominated him. That first time, when she cried and screamed as her body struggled with its initial dose of his venom, he had tried to hold her. She was so pretty, so fragile . . . He didn't want to break her. She had sensed that small spark of compassion in him, clutched on to it, and twisted it, used it against him again and again, until finally he could stand it no longer. Living with Galatea meant fighting a constant war. Living with Karina so far was like sparring with an honest fighter. She defied him, but she would never stick a knife in his back. She would try to stab him in plain view.\n\nLucas sank down into the water and closed his eyes. Thinking about Galatea left a foul taste in his mind. His ribs ached again. Drowsiness came, threatening to smother his mind like a heavy blanket.\n\nKarina's voice tugged on him before he passed out. \"Why are you being nice to me?\"\n\n\"'Nice' isn't in my vocabulary. I'm just tired.\"\n\n\"Your ribs are bruised.\"\n\n\"Daniel.\"\n\n\"I didn't see him hit you.\"\n\n\"He doesn't have to. I'm a Demon, and he's an Acoustic. He can mimic voices and wrench the bones from my body with a focused sound wave.\" He raised his arms and stood up, showing her the long angry welts outlining his ribs. \"If he really pushed, you'd see bone shards puncturing the skin.\"\n\nShe stared at him in horrified silence. He sank back down and closed his eyes.\n\n\"Why do you fight like that?\" she asked.\n\n\"There's no single reason. Sometimes he doesn't like something I've done. Sometimes I do it because he annoys me.\"\n\n\"What about today?\"\n\nLucas sighed. She wouldn't let him be. \"Today we fought because Daniel argued with Arthur. Daniel wants to evacuate. Arthur doesn't. Daniel insisted and Arthur bruised his pride. I took Arthur's side. Evacuating the base is costly. One scout isn't reason enough to do it. It's a bad sign\u2014we had seen scouts before in the neighboring fragments, but never this close. But we can't just run at the first hint of trouble.\"\n\nShe frowned. \"So twisting bones out of your sockets is the way he demonstrates his displeasure at being pushed around?\"\n\n\"Pretty much. Daniel wants to be taken seriously. So I treated him as a serious threat and made a big production of it. I was a substitute fight. What he really wanted was a shot at Arthur, which I can't let him take, because Arthur will kill him.\" Lucas thought of leaving it at that, but something nagged him to explain. \"It's complicated. We live by different rules. In your other life, people undergo strict social conditioning that evolved over hundreds of years. They grow up in relative safety and under constant supervision. Parents, schools, peers\u2014all of their interactions fine-tune their behavior until they are . . .\"\n\n\"Safe?\" she suggested.\n\n\"Socialized. But Daniel and I grew up as outcasts, with only the extremes of our behavior corrected\u2014so we don't murder someone whenever the urge strikes us. Our interactions are simpler than yours, less layered and closer to . . .\" Lucas grappled for the right word. When it came to him, he didn't like it. \"Animals. Both of us reached sexual maturity a while ago. We have a strong urge to mate and have our own territory, our own families, and separate lives. Instead we're stuck with each other, in this house, with an illusion of privacy and an excess of aggression. And now there is you. Daniel doesn't really want you for your own sake. He wants you because he views me as competition and now I have something he doesn't. I am the only consequence he fears. He's hostile and defensive, and Arthur made him sit down and shut up today. Daniel had to vent and I'm the only one who would put up with it.\"\n\n\"Why?\" she asked softly.\n\n\"Because he is my brother.\"\n\nThere was a tiny pause. \"But he is not a Demon like you.\"\n\n\"Different fathers,\" he told her. \"All of us within the House of Daryon carry genes from many different subspecies. Our mother was a Demon. My father was a normal human. Daniel's father was a powerful Acoustic. We both played the genetic lottery and got different prizes.\"\n\nHe left out rape, imprisonment, and murder. It sounded much better this way.\n\n\"Did Daniel hoard food as a child?\"\n\nShe was perceptive. He would have to remember that. \"Yes.\"\n\n\"And you took care of him?\"\n\n\"Yes.\" Because nobody else would.\n\n\"Why doesn't he just leave?\" she asked. \"Why don't you? You don't seem to like living here.\"\n\n\"Because we have a job to do. We guard you from genocide.\" The mission overrode everything. A logical part of him assured Lucas that life outside of the original mandate existed. He just couldn't picture himself living it. \"As long as we exist, you survive.\"\n\n\"I don't understand.\"\n\nHe sighed. This was another long explanation and he had no energy for it today. Nor did he want to shock her again. She'd been through enough. \"Monsters exist. They call themselves Ordinators. They want to kill people like you. Normal ordinary people. We exist to keep them from succeeding. That's all there is to it.\"\n\n\"But what do they want?\"\n\n\"They want you to die.\"\n\n\"Why do they hate us so much?\"\n\nHe sighed. \"They don't hate you. They simply want you not to be. It's a genetic cleansing, a mass extermination. They view the current situation as a mistake, which they're trying to correct. They feel that they are ordained to take your place. Subspecies 61, the 'normal' human, has no value to them, except maybe as an occasional food source in a pinch.\"\n\n\"They're cannibals?\" Her voice spiked a little.\n\n\"Only some of them. I meant a food resource for their war animals. Do you know what a daeodon is?\"\n\n\"No.\"\n\n\"It's a nasty breed of entelodon, a prehistoric boar. Picture a predatory pig, twelve feet long, seven feet tall at the shoulder, jaws like a crocodile. It eats anything, and once you mess with its genetics, it gets smart and breeds fast. They need a lot of meat.\"\n\nWhen he opened his eyes, he found her looking at him. Karina sat submerged so deeply, only her face floated above the water. Warm color had returned to her cheeks. Her hair, slicked by the shower, swirled in the roiling water.\n\n_Mmmmm. Mine._\n\nLucas could reach out and pull her to him and run his hands up and down her body, to feel the heavy fullness of her breasts, the curve of her ass . . . If it wasn't for fatigue, and the fact that she trusted him, anchoring him to the spot, he might have done it.\n\nHis thoughts must've reflected on his face, because she pulled as far from him as the tub would allow. A haunted look claimed her face, sharpening her features. Like a stray dog, he thought, shivering, scared, and ready to bite. He held the key to her: turn it one way and break her; turn it the other and the pressure would ease. He'd been just like that a few years ago. The memory of being scared of everyone was still fresh.\n\n\"You know I can't stop you. What consequences do you fear?\" Karina asked.\n\n\"Right now I just don't want to fight with you,\" Lucas said. \"I fight with Arthur, with Daniel, with Henry. I'm tired.\" And he wanted her to stop jerking back every time he looked at her. It made him feel like he was a monster and he had enough help with that already.\n\n\"If you want peace, let me have Emily.\"\n\n\"No.\"\n\nShe clenched her teeth.\n\n\"Maybe later. Down the road.\"\n\n\"Why not now?\"\n\nIrritation flared in him. \"Because I can't watch the two of you every moment of every day and you are stealing knives.\"\n\n\"The knife was for protection. I won't take another one. I won't try to stab you again . . .\"\n\n\"It's not me I'm worried about.\"\n\nShe became utterly still. \"Oh, my God.\" Her eyes widened. \"You think I would hurt my own daughter?\"\n\n\"You wouldn't be the first one.\" Not by a long shot. \"Shock is a bitch. Especially when mixed with venom fucking with your hormones.\"\n\n\"She is everything I have.\"\n\nShe looked on the verge of tears. He forced himself to sound calmer. \"And that's why you could slit her throat the second I gave her to you. You're both my responsibility. I said I would keep you safe. I don't want you to hurt her or yourself.\"\n\n\"I had the knife since breakfast,\" she told him. \"You sent me into the room with Emily. I didn't kill her. If I'd tried, you couldn't have stopped me . . .\"\n\n\"Henry was monitoring your mind. Had your stress level spiked, he would've shut you down.\"\n\n\"Then ask him if I tried to kill her or myself. I had the opportunity. I got the knife so I could hurt you. Not myself.\"\n\nLucas rose and crossed the tub, pinning her between his body and the tub wall. The feel of her body against his shoved him right to the edge. In his mind all the leashes he put on himself were snapping one by one. Karina turned to the side, trying to hide from him.\n\n\"Look at me.\"\n\nKarina looked at him. Lucas peered into her eyes, looking for some sort of indicator of sanity. \"If you had a loaded gun in your hand, would you shoot me?\"\n\n\"No. If I killed you, I would be next. Either Daniel, Henry, or Arthur would murder me, and Emily would have nobody.\"\n\nAn honest rational answer. \"Do you want to die?\" He wanted her. He wanted to crush her in his arms and see her want him.\n\n\"No.\" She shook her head.\n\n\"What do you want?\" He knew what he wanted. She was right there, caught against his chest. His heart was beating too fast.\n\n\"I want to escape,\" she told him. \"I want to go back to my life.\"\n\nShe was sane and stable, or as sane as he could expect. Lucas released her and Karina scrambled away from him.\n\n\"What would you do if I let you have your daughter, Karina?\"\n\nShe stopped. He read the answer on her face. _Anything._ She would do anything. She would let _him_ do anything, and if he demanded, she would pretend to like it.\n\nIt was the answer his mother would've given.\n\n\"What do you want?\" she asked hoarsely. He felt the tension hidden in her words, as if she stood on the edge of a chasm, waiting for him to push her in.\n\n\"Can you bake a chocolate cake?\"\n\nThere was a tiny pause before she answered. \"Yes.\"\n\n\"Make one. For Daniel. It's his favorite.\"\n\nShe waited. When he didn't say anything, she finally asked, \"That's it?\"\n\n\"Yes.\"\n\nLucas waited for relief on her face, but she just sat there, clenched up. Still looking for the catch, he realized.\n\n\"You'll really let me have her?\" He barely heard her voice. \"No conditions?\"\n\n\"Yes.\" And the more fool he for it. Nothing good would come of it, not with the way they fought. Henry would think him insane. But Lucas felt weary. He didn't have the strength to fight yet another war. And he didn't want her to be miserable. \"Make a list of what you both will need, and I'll send it to the main house tomorrow. Last time I checked, you could buy Hello Kitty blankets in any department store . . .\"\n\nKarina covered her face and cried.\n\nHe sat there and watched her shudder and sob, not knowing what to do with himself. Uncomfortable, as if he were intruding on something private. Guilt rose in him and he wasn't sure where it came from.\n\n\"Stop,\" Lucas growled finally.\n\n\"I can't.\"\n\nHer sobs died gradually. She splashed some water on her face. \"Can I stay with her in her room?\"\n\n\"No. You'll stay with me.\"\n\n\"Can I sleep on the floor?\"\n\n\"No. You'll sleep in my bed, just like last night.\"\n\n\"Why?\"\n\n_Because you're mine._ And because he would know if she got up in the middle of the night. \"Because I want it that way.\"\n\n\"I could\u2014\"\n\nHe closed his eyes and leaned his head back. \"Quiet. No more talking.\"\n\n\"Thank you,\" she said softly.\n\n\"You're welcome.\"\n\n# **CHAPTER 6**\n\nKarina awoke alone. She dimly recalled seeing Lucas get out of the water, his huge muscled body wet, and feeling a sharp inner clench, the same clench that gripped her when he'd caught her in the tub. She would've liked to pretend it was fear or anxiety, but that would mean lying to herself. When he rose to show her the bruises Daniel had made, she stared at him for a moment too long and it wasn't to study his injured ribs.\n\nLucas had brought her a towel and when he turned away, giving her a fragile illusion of privacy, she'd draped it around herself and escaped into the bedroom. He didn't follow her. She toweled off, slipped on the giant T-shirt he'd given her, and slid into bed, curling under the blanket into a worried ball. Her nervousness should've kept her awake, but her body simply gave out. Lucas took his time getting to bed and by the time he lay down on the other side, she was half asleep. He asked her something, but her feverish haze mugged her and dragged her under into a dreamless sleep.\n\nKarina struggled to sit up. She felt the steady heat of her slowly burning, low fever. At least she was alive. She forced herself all the way up. Her head swam and the dizziness nearly took her back down.\n\n_Up. Up, come on, you can do it._\n\nAnd now she was talking to herself. Outstanding.\n\nKarina walked to the shower, swaying on wobbling feet. She'd rinsed her underwear last night, and it still hung on the towel hook where she'd left it. Karina touched it. Dry. She slipped the panties on and went to use the bathroom.\n\nA couple of minutes later she made it to the sink. A new toothbrush, still in its case, waited for her. Karina stared at it.\n\nLucas hadn't kidnapped her. He hadn't forced her into human slavery at gunpoint. She'd been attacked by Rishe and the shark-toothed man, and she'd been given a choice: to die or to live on Lucas's terms. She was a victim of circumstance. That didn't change the fact that Lucas owned her now.\n\nThe House of Daryon had stripped every shred of independence from her. She depended on Lucas for everything: her food, her safety, her clothes, the safety and survival of her daughter. He had the power to tell her when to go to bed, where to sleep, when to shower . . . He was protecting her and Emily from some sort of terrible enemy she couldn't understand and he could kill them both at a moment's notice. Any relaxation of the rules became a kindness on his part. A small thing, like a toothbrush, seemed like some great favor. But it wasn't, she told herself. It wasn't. It was a basic necessity for any human being.\n\nThen again, she could've been a slave without any freedom at all. She could've lost her daughter. She could've been raped. All he had to do was say, \"I'll give you your daughter,\" and she would've done anything. The very fact that he thought to leave her a toothbrush was a small miracle.\n\nHer own drive to survive was interfering with her sense of reality. Her instincts drove her to forge an emotional bond. The more Lucas liked her, the less likely he was to murder her or Emily. The more she liked him . . .\n\nKarina took a deep breath. Lucas was physically overwhelming. The memory of his arms around her flashed before her. Lucas was . . . He was . . .\n\nShe stared at herself in the bathroom mirror. _Just say it. Say it, acknowledge it, and walk away from it._\n\nSeductive. Desirable. Shocking. He was masculine in the way women fantasized men to be: powerful, strong, dangerous. If she had met him at a party or in a professional setting, when he wore a suit and she wore something other than his T-shirt and a pair of underwear she'd washed in the shower, she would've sought him out. If he had spoken to her, she would've been flattered.\n\nFor a while, after Jonathan's death, she was so wrapped up in guilt, and in Emily's well-being, she forgot men existed. It took almost a year before she became aware of them again: a man with a nice smile in the checkout line, a random stranger in good shape stepping out of the car in the parking spot next to her. A small part of her wanted to be noticed again and checked to see if she was. She was vulnerable and the way Lucas looked at her left her no doubt that if she gave him the tiniest indication that she wanted him, he would rush to oblige and mow down whatever stood in his way.\n\nThere was an odd desperation in Lucas under all that violence. Karina sensed a deep overpowering need to be . . . not accepted exactly, but to be liked. If she were ruthless, she would seduce him to make sure he would become dependent on her, but that kind of manipulation was beyond her. She couldn't bring herself to do it.\n\nKarina looked at her reflection. She could practically see him in the mirror next to her. She could recall him with crystal clarity: every powerful line of his body; the promise of raw violence in the way he moved; the precise curve of his mouth, almost sardonic; the look in his eyes, the wild, unfiltered look of pure male lust. No, more than lust. Need.\n\nThinking of him was like playing with fire.\n\nShe had been married; she knew very well that a healthy relationship hinged on respect and constant compromise. With Lucas there could be no respect and no compromise, because they were not equals. He owned her. She was his property and once she opened the door to a relationship, he wouldn't let her close it.\n\nKarina shut her eyes. She could picture herself wrapped in those powerful arms. It would feel safe, so safe. Her life was broken like a mirror and the shards kept cutting her fingers. She was desperate to forget that she was little more than a slave. She craved that illusion of safety as if it were a drug and she had to score a hit. She wanted to feel the heat of his strong body warming her skin. And she wanted to see him bend, to find out what it would be like to see the vulnerability of intimacy in those hard eyes. She was completely powerless and she needed to feel powerful, as a woman does who is wanted so badly by a man, he would do anything for her.\n\nThere it was. All of it, out in the open.\n\n_You're sick_ , she told her reflection.\n\nWell, now it was out. She owned all of it.\n\nShe had to keep things in perspective. He was strong and she was weak and vulnerable and not in her right mind. She would take it one day at a time, wait until the last of the poison cleared out of her system, and when a chance to escape presented itself, she would take it\u2014and they would never find her and Emily again. And if she let herself buy into her own lies, she would never wonder what it would have been like to feel him inside her . . . She cut off that thought. The less she imagined it, the better.\n\nKarina opened the toothbrush. She would brush her teeth, locate her jeans, and check on her daughter. And then she would go out there and make a chocolate cake.\n\n* * *\n\nEmily seemed to have no memory of Lucas and Daniel's fight the previous night. She slept well and when Karina had come to get her, she got a hug. The violent episode had passed her daughter by completely. Karina held her for a long time, breathing in the scent of her hair. They were both alive. She would get to keep Emily with her. It would be okay. It would be hard and painful, but it would be okay.\n\nKarina took Emily to the kitchen. Sunlight poured in through the open window. Nobody waited for her. Nobody demanded breakfast. The house was quiet and serene. Karina exhaled her tension, pulled the ingredients from the pantry, and started mixing the cake batter.\n\nHenry walked into the kitchen, looking a bit lost. \"Good morning!\"\n\n\"Good morning!\" Emily chirped.\n\n\"I have something for you.\" Henry put a drawing pad and a set of watercolor pencils on the table.\n\n\"For me?\"\n\n\"For you.\"\n\nEmily pried at the pencil case.\n\n\"What do you say?\" Karina murmured on autopilot.\n\n\"Thank you!\"\n\n\"You're welcome.\" Henry offered her a small smile.\n\n\"Where is everyone?\"\n\n\"They've gone to check the perimeter net. What is it you're making?\"\n\nKarina glanced at him. \"A chocolate cake. Did they go to check for signs of those people who sent the lizards to spy on us?\"\n\nHenry nodded.\n\n\"Lucas called them Ordinators. Henry, who are they? Who are you?\"\n\nHenry smiled again and slid his glasses up his nose. \"It's a long and complicated explanation. It's better to wait a couple of days. Too much new information too fast will only make things worse.\"\n\n\"I'd like to know.\"\n\nHe shook his head. \"You've been through a great deal of violence in the past two days and you've been exposed to things that conflict with your worldview. I don't want to be the one to add to it.\"\n\n\"Henry, not knowing is worse. All I'm asking is that you don't treat me like a slave who is told where to be and what to do and isn't owed any explanation.\"\n\n\"No,\" he said quietly.\n\nThey looked at each other over the table. Karina held his gaze. It might not have been wise, but she wouldn't back down now.\n\n\"Look, Mom, I drew Cedric!\"\n\nKarina looked down at the ball of brown fluff that looked like a sheep with a sabertooth's fangs. \"That's looks very nice, Emily.\"\n\nWhen she looked up, the kitchen was empty. Henry had escaped.\n\n* * *\n\nThe cake smelled of chocolate and vanilla. When Karina took the two round pans out of the oven and set them out to cool, the familiar scents floated through the kitchen, so reminiscent of home and happy times, she almost cried.\n\nA door banged. She looked up just in time to see Lucas loom in the doorway. His face was grim. He glanced at the cake, then at her. She stared back, suddenly terrified that all her thoughts would pour out through her eyes.\n\nHe didn't seem to notice. \"Would you like new clothes?\"\n\n\"Yes.\" Oh, God, yes.\n\nHe jerked his head toward the door. \"They have some things prepared for you at the main house. I didn't know what size, so you have to come and try them on. Come on, I'll walk with you.\"\n\n\"Can I come?\" Emily slid off the chair.\n\n\"Yes,\" Lucas said. \"They have clothes for you, too.\"\n\n\"And Cedric?\"\n\n\"Cedric doesn't need clothes,\" Lucas said.\n\n\"Can he come with us?\" Karina asked.\n\n\"Sure.\"\n\nKarina washed her hands, wiped them on a towel, and followed Lucas out. The sun shone bright. Cedric already waited for them at the foot of the stairs. Emily stepped down and the bear-dog rolled to his feet and trotted next to her, nearly as tall as she was.\n\nLucas led them out of the yard and down a dirt path. It wound around the hill, flanked on the left by stunted oaks and shrubs climbing up the slope and rolling off to the prairie on the right. Cedric and Emily pulled ahead a couple dozen yards. Karina watched them, aware of Lucas striding next to her, like some tiger who had learned to walk upright. The air was dry, and the heat beat down on them from the pale, burned-out sky, painting the path in stripes of bright yellow sunshine.\n\n\"We're in a fragment of reality,\" Karina said.\n\n\"Yes,\" Lucas said.\n\n\"Why is the sun shining? Why is there air?\"\n\n\"Because the fluctuation occurs on the universal level,\" Lucas said.\n\n\"So it's a duplicate sun?\"\n\n\"No, it's the same sun the Earth has. We just get access to it on a different level. Think of a house with many rooms. We walked out of the main room into a smaller side bedroom, but we're still under the same roof.\"\n\nKarina sighed. \"It makes my head hurt.\"\n\n\"Don't talk about dimensions to any Rippers, then,\" Lucas said.\n\n\"Rippers?\"\n\n\"They make inter-dimensional rents that let people like you and me travel back and forth. You get one of them started on the subject and the insanity pours out until you want to stick your head in a bucket of water just to wash it out of your mind. When a man has to continuously cut himself, because pain helps him punch through dimensions, you can't expect him to be lucid anyway.\"\n\nKarina glanced at him. \"You seem irritated.\"\n\nLucas's thick black eyebrows knitted together. \"We found out how the lizard got through the net. It tunneled under it. A long, deep tunnel, almost twenty-five meters.\"\n\n\"And?\"\n\n\"There was more than one tunnel,\" Lucas said.\n\nMore than one tunnel meant other lizards. \"Did you track them down?\"\n\nLucas nodded.\n\n\"Did they transmit what they saw?\"\n\nAnother nod.\n\n\"So the enemy knows where we are?\"\n\nLucas grimaced. \"Difficult to say. The Rippers are saying there was too much inter-dimensional interference for the transmission to have gone through fully. But it's possible.\" He clenched his teeth, pondering something, and said, \"We had perimeter alarms, infrared, microwave, and frequency sensors. The sensors are very specific: if you look on Cedric's collar, you'll see a transmitter. The transmitter broadcasts a code. The sensors check this code against the database and if the code is active, the sensors don't register an alarm. For some reason someone loaded an old set of codes into the system. The lizards came through fitted with transmitters of their own and when they broadcast the outdated set of codes, the system didn't flag them.\"\n\n\"How did they know which codes to load?\"\n\nLucas's eyes turned darker. \"There was a woman. Galatea. She was a donor like you.\"\n\nHe said her name like she was a plague. \"Was she your donor?\"\n\n\"Yes. She defected.\"\n\nHe'd clenched his teeth again. There was more to this story. \"Were you lovers?\"\n\nLucas stopped and for a moment she thought she might have pushed him too far. \"We fucked,\" he said.\n\nAha. She kept pushing. \"For how long?\"\n\nThere was a short pause before he answered. \"For four years.\"\n\n\"That's some long fucking,\" Karina said. He'd loved Galatea. He was in love, and she betrayed him, and now he wanted to kill her. Any woman past the age of fifteen would've connected these dots. He must've been young\u2014it had obviously left a deep scar. \"What was she like?\"\n\nLucas took a step toward her. A wild thing looked back at her from his eyes, the thing full of lust and aggression. She realized that in his mind he was peeling off her clothes and thinking of what it would be like, and suddenly she was back in the tub, naked, sitting two feet away from him and afraid he would cross the distance.\n\nHe stared at her. \"Would you like me to tell you about it?\"\n\nShe squared her shoulders. \"No.\"\n\n\"Are you sure?\"\n\n\"Yes.\"\n\n\"Okay, then.\"\n\nHe turned and they sped up to narrow the gap between themselves and Emily. Karina kept the pace, exhaling quietly. He had no brakes, at least not the ones she was used to as a woman. Ordinary men didn't end dinners by breaking the table with their brother's spine, they didn't kill lizards by caving their heads in, they didn't turn into monsters, and they didn't feed on women. Ordinary men didn't behave like this outside of movie screens and when they did it on the screen, other men ridiculed them for it. This was a game she couldn't afford to play, because he held the best cards. She had to survive this.\n\nKarina chanced a glance at him. The wild, hungry thing in his eyes was still there. \"Since someone had to have uploaded this old code, someone on the inside is helping Galatea,\" she said, trying to steer him away from whatever he was thinking.\n\n\"Looks that way. And when I find them, they'll wish they were never born.\" His voice contained so much malice, the hairs on the back of her neck stood up.\n\nIf this enemy was coming, Emily would be in danger. \"Should we evacuate?\"\n\n\"That's up to Arthur.\"\n\n\"Do you think we should?\"\n\nLucas glanced at her. \"It depends on how many people they bring to the fight. This is an old base, and we are actively mining this fragment for aluminum and beryllium. If the Ordinators are coming, they're coming fast. So even if we begin full base evacuation now, we'll take a hit in equipment. The base is run by means of a fiber network. It's a sophisticated computer system that coordinates mining operations, bio-support, communications, and so on. It also has the locations of the nearest bases. If the Ordinators gain access to it, a lot of us will die, which is why the network must be destroyed before the evacuation is complete. Detonating it will make this base uninhabitable. Fragments like this, with a stable climate and ecosystem, are rare. Most fragments we find are dead: no plants, no animals, often no atmosphere. You have to wear a suit and live in a hermetically sealed bunker. And popping back and forth through dimensions leaves a trail. If the Ordinators don't know where we are, they will once we start ripping.\"\n\nThe path ended, joining a larger road that rolled down the hill toward the prairie. In the distance a group of small horses galloped across the grass, ducking in and out of the brush. The vast prairie rolled to the towering mountain ridge, savage and ancient and somehow so much bigger than the modern landscape, that for a moment Karina stopped and simply stared, caught by the natural majesty of it.\n\n\"This is paradise compared to some of the fragments I've seen,\" Lucas said. \"If we have a chance, we'll fight for it. Come on.\"\n\nHe turned and strode up the hill. She sped up to keep pace, Emily and Cedric in tow.\n\nThey rounded a bend and suddenly before them stood two tall white columns marking an entrance. Thrusting twenty feet up, they curved like the ribs of some prehistoric giant. An intricate network of designs covered the columns, etched into their surface. It drew the eye, hypnotic in its complexity. Once you looked, your gaze just kept sliding and sliding, up along the grooves and curved lines . . .\n\nA hand rested on her shoulder. Karina turned, saw Lucas's fingers on her shoulder, and jerked away. He held his hand in empty air for a second and lowered it.\n\nKarina turned to Emily. Her daughter stood next to her, staring at the column, her expression blank.\n\n\"Come,\" Lucas said.\n\nKarina bent down and took Emily's hand. \"Come on, baby.\"\n\nEmily blinked, as if waking up from a deep sleep, and walked with her. They passed through the arches and Karina stopped again.\n\nPale buildings with curved roofs spread before her. On second thought, the complex was all one huge building in the shape of a horseshoe, rising three stories high. A beautiful garden lay in the crook of the horseshoe, crisscrossed by covered passageways, stonelined paths, and lush flowerbeds, artfully bordering artificial ponds. Picturesque shrubs spread their branches. Flowers bloomed, blue, orange, yellow . . . The wind brought the by-now-familiar tart flower scent.\n\nA large white sign stood next to the wide path leading into the garden, its smooth surface marked with an odd script. It had to be writing of some sort\u2014groups of symbols separated by spaces\u2014but it wasn't any language Karina was familiar with.\n\n\"What does it say?\"\n\nA string of odd words spilled from Lucas's lips, lyrical and surprisingly familiar. She waited for the meaning.\n\n\"It says 'The Mandate is everything.' \"\n\n\"What is the Mandate?\"\n\n\"The Original Mandate. It's hard to explain in English. There is a word in the primary language, _ile_. It means 'we,' 'us,' but it also means civilization, the best of us, the best of our kind. The mandate is ' _Ile_ must survive.'\"\n\nThat explained nothing. \"That's it?\"\n\n\"That's it. On this world, under this set of circumstances, the people among whom you lived are _ile_. We exist to make sure they survive. When we're no longer needed, we'll die out like many other subspecies before us.\"\n\nThe more he explained things, the more confused she became. For now she had to just gather the crumbs of information and hope all would make sense sooner or later.\n\nLucas walked on, down the wide path of smooth stones. Karina scrambled to follow. They walked side by side along the path and over a bridge. The gardens burrowed into nooks in the buildings here and there, forming small sitting areas. To the left two women sat on a bench, discussing something. They looked so normal. Both wore jeans; the older of the pair had on a flowered top, white on blue; the younger woman wore a familiar yellow blouse\u2014Karina had looked at it in J. C. Penney last week.\n\nLast week. A lifetime ago.\n\nThe women saw Lucas. Their faces took on a certain tightness, as if they were straining to keep calm. They looked her over next. Karina met their gaze and saw pity in their eyes. Suddenly it made her furious. If Lucas grabbed her throat right now, they wouldn't lift a finger to help her. They would just sit there and watch him choke her to death and feel sorry for her. She raised her chin and stared at Lucas's back. No, thank you. She didn't need anyone's pity.\n\nHenry's words came back to her. _Lucas is the most feared._ \"They're afraid of you,\" she said.\n\n\"I'm the security specialist here; I have the right of judgment,\" he said. \"I can kill anyone on base at any point without any retribution.\"\n\n\"You protect them, and all you get in return is fear. Why do you keep doing this?\"\n\nLucas kept walking. \"Because everyone must have a purpose. The Mandate tells me what I am doing is right and must be done and because I'm the biggest and the strongest it's my duty to put myself between my people and danger. I would do it for you.\"\n\nHe would. She believed him. \"Lucas . . .\"\n\n\"Yes?\"\n\nShe wanted to tell him that if he ever shielded her or Emily, she wouldn't be afraid of him. She wanted to tell him that he didn't have to put up with people shrinking away from him, but inside a cold rational voice warned her that she was losing her grip on reality. The plan had to be to escape. The plan couldn't be to fall for Lucas and be that one sole person who comforted him.\n\nHe was looking at her.\n\n\"I'm really confused right now,\" she told him. \"So this actually doesn't mean anything.\"\n\nHe nodded. \"Okay.\"\n\n\"Bend your arm at the elbow.\"\n\nHe did. Karina reached out. _What am I doing?_ She put her hand on his forearm and raised her chin. The two women on the bench stared at them, openmouthed.\n\n\"Now we walk,\" she murmured, avoiding looking at him.\n\n\"We can do that,\" he agreed. They started down the walkway. His arm was rock-steady under her fingers. A few moments, and the dense greenery of rhododendron shrubs hid the women from their view.\n\n\"Why?\" he asked.\n\nBecause she lost it, that's why. \"Would you hurt those two women?\"\n\n\"Not unless they tried to hurt someone else first.\"\n\n\"Then they're in no danger and they know it, but they still make a big production out of you walking by, minding your own business.\"\n\n\"That still doesn't answer my question,\" he said.\n\n\"Can we stop talking about this?\"\n\nHe didn't say anything. They simply kept walking. It was surreal, Karina reflected. Beautiful flowers, Emily and a tame bear-dog, and she and Lucas striding side by side.\n\n\"I'm tired,\" Emily said.\n\nKarina bent down and picked her up. The effort nearly made her lose her balance. Apparently she was weaker than she thought.\n\nCedric sniffed at her feet.\n\n\"Let her ride him,\" Lucas offered.\n\n\"What?\"\n\n\"Let her ride him. He doesn't mind.\"\n\n\"I want to ride!\" Emily squirmed in her arms.\n\nKarina surveyed the bear-dog. He was almost as big as a pony. Gingerly she lowered Emily on his back.\n\n\"Hold on to his fur,\" Lucas said. Emily dug her fingers into Cedric's brown mane and they were off again.\n\nThey emerged from the stand of rhododendrons. Lucas stepped aside, revealing a round plaza paved with dark red stone. A bronze statue rose in the center, a nude man, muscled with crisp precision. Enormous wings thrust from his shoulders. An angel, but not a garden cupid or some mournful cemetery statue. The angel leaned forward, one arm stretched out, his muscles knotted on his frame. The wings thrust up and out, featherless, as if made of sharp bone. The angel's perfect face stared into the distance, its gaze focused. Everything about it communicated fury and power. This was a predatory being about to kill its victim. Metal letters beveled on the side of the statue read \"A. Rodin.\"\n\nKarina glanced at Lucas. \"A. Rodin? The sculptor who created _The Thinker_?\"\n\nLucas shrugged. \"He says so, but I wouldn't put it past him to have the name slapped on there over the actual sculptor's signature. He is vain enough.\"\n\nWhat? He who? She scrutinized the statue.\n\nOh, God.\n\nThe angel wore Arthur's face. It had to be figurative\u2014she hadn't seen any wings on Arthur's back when he offered her tea.\n\n\"But Rodin died in the beginning of the last century.\"\n\nLucas circled the statue and kept walking.\n\n\"Lucas!\"\n\nHe turned and looked at her over his shoulder, light eyes under black eyebrows like two chunks of ice. \"Arthur is a Wither. Subspecies 21. They live a long time.\"\n\n\"How long?\"\n\n\"Long enough to have met Rodin. Come.\"\n\nShe wanted to freak out. She wanted to scream and kick her feet in panic, because right here, in cold bronze, was the final proof that this was not a nightmare. Instead Karina waved Cedric ahead of her and they kept going deeper into the garden.\n\nLucas turned left, down a path leading to a section of the building structured with an almost Japanese flair. Except for the white roof, it could've been part of a teahouse. An older woman waited on the covered porch, a stack of clothes neatly folded next to her.\n\nThey were twenty feet away from the porch when the siren ripped the quiet into shreds.\n\n# **CHAPTER 7**\n\nKarina pulled Emily off the bear-dog and into her arms.\n\n\"Stay close,\" Lucas barked as he turned and ran back up the path. She followed him, trying not to stumble. They pounded over the bridge they'd crossed on the way in.\n\n\"What's happening, Mommy?\"\n\n\"I don't know, baby. Hold on tight.\"\n\nEmily was so heavy. Karina never remembered her being that heavy. It was like all of the strength had somehow gone out of her arms.\n\nThey cleared the garden and burst into the open space between the two spires, Lucas ahead and she, out of breath, a few dozen yards behind. A group of people stood by the spires, where the road out of the settlement rolled down the hill. A familiar face looked at her with merciless sky eyes. Arthur. Daniel's golden mane swung into view. He grinned at her, a deranged wild grin that had too much mirth. On the periphery a few yards away, Henry stood with his eyes closed, tense, his face raised to the sky. A young girl, barely a teenager, stood next to him in an identical pose. To the right an older, dark-skinned woman and another man, tall and gaunt, imitated them.\n\n\"Good of you to join us,\" Arthur said.\n\nLucas walked up to stand next to him.\n\nA huge sound came from the distance, deep, booming, as if someone was playing a foghorn like a trumpet.\n\nThe girl at Henry's side inhaled sharply and dropped to her knees, breathing in ragged, painful gasps. Henry's eyes snapped open. He thrust his hand out and clenched it into a fist. \"Oh no, you don't.\"\n\nA desperate scream of pure pain came from the distance.\n\nHenry smiled. His face glowed with vicious joy, so shocking that Karina took a step back. He stared into the distance. \"Not as fun to pick on someone your own size?\"\n\nThe scream kept ringing higher and higher, pausing for the mere fraction of a second that it took the agonized being that was making it to gulp some air.\n\nBehind Henry the fallen girl opened her eyes and rose to her feet. The older couple awakened from their trance.\n\nHenry twisted his fist and jerked it, as if ripping something in half.\n\nThe scream died.\n\n\"Thank you,\" the girl said.\n\n\"It's all right. Next time remember to cloak.\" Henry turned to Arthur. \"They have two hundred civs, fifty pigs, two heavy field artillery batteries, six squads of twenty-five men each, and seven Mind Benders. Minus one.\"\n\nHe'd killed an enemy Mind Bender, Karina realized. Kind, shy Henry crushed him, but not before he made him suffer.\n\n\"Too many,\" someone muttered.\n\n\"It's overkill,\" Daniel said.\n\n\"There is at least one Demon, too,\" Henry said.\n\nLucas laughed, a bitter, self-assured chuckle.\n\nThey had a Demon like Lucas. Lucas would fight it. She saw it in his face. She didn't want him to die.\n\nSomething climbed over the crest of the distant hill, spilling onto the prairie. Karina squinted. What in the world . . .\n\nArthur's face remained serene. \"Begin immediate full base evacuation.\"\n\nA dark-haired woman on Karina's left held out binoculars to her. \"Here. Looks like I won't need them.\"\n\n\"Thank you.\" Karina lowered Emily to the ground and took the binoculars. \"Stay with me, baby.\"\n\nThe woman turned and ran, back toward the garden. A moment later the alarm sounded again, but this time in two short bursts.\n\nPeople peeled off from the group and headed back, deeper into the base. Now was her chance. If she could slip away and go through the gate, she could get away. Nobody would find her in the confusion . . .\n\n\"Lady Karina,\" Arthur's voice rang out.\n\nShe snapped back to look at him.\n\nThe gaze of his blue eyes bore into her. \"Stay close. We must hold until the evacuation is complete. Lucas may have need of your services.\"\n\nHis voice was soft but his eyes left her no doubt\u2014he knew what she was thinking and escape was futile.\n\nArthur turned and looked out to the plain. She looked, too, raising the binoculars to her eyes. The mountains swung into view, suddenly clear. She tilted the binoculars lower . . .\n\nPeople came walking over the hill. To the right a middle-aged man in filthy khakis and a ripped shirt with thin blue stripes climbed over a rock. Next to him two dark-skinned men in jeans helped a third limp forward. On the left a woman in business clothes walked on, stumbling. The binoculars captured her face. Her features, caked with grime and dust, twisted into an expression of abject terror.\n\nKarina inhaled sharply. A red-haired teenage girl followed the woman. Her ruffled black skirt hung limply around her skinny legs in torn stockings. She shuddered as she walked and Karina realized she was sobbing.\n\nKarina jerked the binoculars down. \"There are people out there!\"\n\n\"They are captives,\" Lucas said. \"People the Ordinators snatched up here and there, the missing. The pigs are running them at the net. It's designed to stop high-impact projectiles, but if enough body mass hits it at once, it will overload and collapse.\"\n\nThe memory of the bird shocked by that red glow flashed before her. \"They will die!\"\n\n\"That's the idea,\" Daniel said. \"They're trying to break through before we have a chance to detonate the network.\"\n\n\"Can't they just use a tank or a vehicle?\"\n\n\"The net would fry it,\" Lucas said grimly. \"Biomass is the best way to go.\"\n\nThe people on the right broke into a run. Karina raised the binoculars.\n\nA creature bounded over the hill. Huge and brown, it looked like a seven-foot-tall boar moving too fast on surprisingly long and skinny legs. The pig paused. Its long crocodilian jaws gaped open, flashing fangs as large as her fingers, wider, wider, until the pig's entire head seemed to split in half. A hoarse roar burst forth. The daeodon.\n\nThe people in front of the creature scattered like minnows, sprinting across the rough ground toward the net in a ragged herd, a blond man in a once white tank top leading the run. The daeodon roared again and gave chase.\n\nOn the left, a second pig crested the hill, sending another group of prisoners into flight. An older man in a torn flannel shirt stumbled and fell, splaying in the dirt. The pig bore down on him. The long jaws dipped down. A shriek rang out, vibrating with the sheer terror of a man who knew his life was ending, and vanished, cut off in midnote.\n\nOn the right, the blond man ran headfirst into the net and jerked, caught by a deep carmine glow. His body convulsed, his legs and arms flailing, as if he were being shocked by a live wire. The man directly behind him tried to slow down, but his momentum carried him right into the red glow and he shook, caught in a similar seizure.\n\nKarina whipped to Lucas. \"Can't you do something? Anything? They're dying!\"\n\n\"We can give them a quick death once they break through,\" Lucas said.\n\n\"But . . .\"\n\n\"Lucas is correct,\" Arthur said. \"We will spare them the pain.\"\n\nThe air around Arthur shimmered. People backed away. He bowed his head and stood very still.\n\nOn the prairie, the prisoners tried to swerve away from the red glow, but the pigs drove them forward. One by one the bodies crashed into the net. Karina turned Emily around. \"Don't look, baby.\"\n\n\"What are they doing?\"\n\n_Lie_ , she told herself. _Lie._ But the words spilled out on their own. \"They are dying, Emily.\"\n\n\"Why?\"\n\n\"Because the bad guys are killing them.\"\n\n\"Are the bad guys going to get us?\"\n\n\"No, little one,\" Henry said. \"Arthur and Lucas will kill them.\"\n\nThe red glow bent forward under the weight of many bodies, and still more people were coming across the prairie, herded by the daeodons like sheep. Arthur didn't move. His eyes stared into the distance, somewhere far away.\n\n\"How long till the detonation?\" Lucas asked.\n\nHenry closed his eyes and opened them. \"Three minutes.\"\n\nLucas rolled his head right, then left, cracking his neck.\n\nWith a bright flash the net collapsed under the weight of the bodies. People fell into the gap, tumbling over each other, convulsing on the ground. The four huge pigs who'd herded them to the net galloped into the gap, trampling the bodies beneath their hooves. The daeodons charged up the slope.\n\nLucas grunted. His skin seemed to peel off his bones in thick slabs. Bloody mist filled the air. Karina stared, unable to look away. Bones bent, ligaments twisted, and the beast burst forth. It was bigger than she remembered. In her memory, he had morphed into a dark, featureless shadow, but here, in the light of day, she saw every bulge of terrifying muscle, every fang, every sickle claw, every hair in the black crest of his mane.\n\nFear washed over her, setting every nerve on fire.\n\nThe beast turned his head. Lucas's green eyes looked at her from a horrid face.\n\n_Don't flinch,_ she told herself. He was about to fight for them. He could die in the next few moments. She didn't want him to go into it thinking she was disgusted by what he was. Whatever Lucas's faults were, he was about to put himself between the pigs and her daughter. He deserved better than the blind fear the two women in the garden showed him.\n\nShe met his gaze. They looked at each other.\n\n\"Good luck,\" she said.\n\nThe daeodons roared, pounding up the slope.\n\nThe beast who was Lucas nodded to her, leaped down, and smashed into the first pig. His claws sliced across the daeodon's neck and it went down. Lucas swerved away from the gaping jaws, leaped onto the second daeodon, and thrust his claws through the brown hide and wrenched a bloody shard of its spine out.\n\nThe third pig halted, unsure. The fourth veered left, around the carnage, and charged up the hill, digging into the hard dirt with its hooves.\n\nKarina clenched Emily closer. Her instinct told her to run, but around her nobody moved.\n\nTwenty yards. Fifteen. Ten.\n\nDaniel stepped forward and clenched his fist. With a dry crunch, the bones of the pigs' front legs snapped. White bone sliced through the muscles and skin. The pig squealed, crashed on its side, and rolled down the hill. Lucas rose from the body of the third pig, leaped over the fallen daeodon as it tumbled down, and smashed its skull with one brutal punch.\n\n\"Are we in a story, Mommy?\"\n\nKarina looked down into Emily's big brown eyes. _I wish we were. I wish we were dreaming._ She reached deep inside herself, through the fear and anxiety and disbelief, and when she spoke, her voice was calm and confident. \"It will be okay, baby. We will be just fine.\"\n\nMore daeodons spilled from the prairie, dashing toward the base; so many, she couldn't even count. A huge beast led the charge. He looked just like Lucas, except for the reddish fur. The red beast sprinted, widening the distance between himself and the mass of daeodons, moving in powerful leaps that devoured the prairie.\n\nLucas backed two steps up the slope and planted his giant feet.\n\nThe beast thundered at them, hurtling like a cannonball. It jumped and sailed over the mass of writhing human bodies.\n\nLucas leaped. The two monsters collided in midair and Karina realized that Lucas was visibly smaller. They rolled down the hill, snarling and tearing at each other like two massive feral cats.\n\nThe larger beast raked Lucas's side. Blood wet the dirt in a hot spray.\n\nKarina spun to Daniel. \"Help him!\"\n\n\"I can't,\" he growled. \"I need a clear target.\"\n\nThe beasts brawled and snapped, biting and ripping in a tornado of claws and teeth.\n\nThe alarm blared again, this time a single long note followed by a short beep. Daniel whirled to an older woman standing next to him. She was short and plump, with an elaborate knot of tiny braids on her head. Her gray pantsuit was pristine, her makeup flawless. She looked like a secretary or a receptionist for an upscale business firm.\n\n\"Rip it,\" Daniel said. \"Now.\"\n\nThe woman pulled a knife out of her pantsuit, jerked the sleeve back, and slashed a gash across her skin. Blood welled. The pain must've been excruciating, because she bent nearly double, cradling her arm.\n\nAt the bottom of the hill, the larger beast hurled Lucas aside. He flew, flipped in the air, and landed on all fours. Blood streamed from his flanks. The two creatures squared off and collided again.\n\nThe woman straightened. A pale green glow burst from her stomach, twisting into thin strands of light. The strands snapped out, flared, and split the empty air in half. A seven-foot circle appeared, filled with darkness.\n\n_So that's what the dimensional rip looks like._\n\nArthur raised his head.\n\nThe ground shook under his feet. Tiny rocks bounced up and down. The vibration pounded the bottoms of Karina's shoes.\n\n\"Lucas! End it!\" Daniel screamed. \"End it now!\"\n\nThe reddish beast leaped, striking with an enormous paw, claws out like daggers. Lucas spun, rolling to the side, inhumanely fast. The large beast landed in the dirt. The moment his paws touched the ground, Lucas vaulted onto his back. Huge teeth flashed and he clamped onto the rival beast's neck. The creature screamed, kicking and trying to roll. Two beasts plunged down.\n\nKarina held her breath.\n\nThe black beast rose, slowly.\n\nShe exhaled.\n\nLucas pondered the body of his fallen opponent as if he wasn't sure where he was or what he was doing there. Behind him, the captives, caught between him and the sea of pigs, scrambled to their feet.\n\nThe vibration below the surface increased, hitting Karina's feet like the blow of an underground hammer. Tiny red sparks flickered around Arthur.\n\n\"Hurry,\" Henry whispered next to her. His gaze was fixed on Lucas, his voice an insistent low whisper, almost a command. \"Hurry.\"\n\nLucas jerked. His head snapped up. He saw them and bounded up the hill.\n\nThe sparks around Arthur danced faster. Arthur's feet left the ground. He rose three feet into the air, his body tense, looking down at the prairie stretching before him.\n\nOh, God.\n\nThe beast reached the apex of the hill, crashed down in a sickening revolt of flesh, and rose again, as Lucas, bloody and shaking. He shuddered on his feet, careened, and Karina caught him. For a moment his entire weight rested on her. She looked into his eyes and saw pain. And then Daniel pulled him off her and dragged him forward to the rip.\n\nIn the distance the foghorn blared frantically. The daeodons closed in. Karina swept Emily into her arms.\n\nHenry wrapped his arm around her. \"We must go. You don't want to see this.\"\n\nThey hurried to the rent. She looked back over her shoulder, as if pulled by some invisible force. The sparks darting around Arthur's shoulders paused. For a fraction of a breath they hung motionless, then blinked, then sparked into brilliant light. Red radiance burst from Arthur's shoulders in twin streams, boiling with flashes of white and orange, unfurling into two enormous wings knitted of lightning.\n\n\"Come on.\" Henry pulled her toward the rip. It loomed before them, lightless and frightening, a hole in reality itself.\n\nThe red lightning flashed. The front row of captives fell to their knees. Fire spilled from their eyes and mouths, as if they were being incinerated from the inside out. Their faces turned to ash. The second row followed and on and on and on . . . Jets of flames spurted from the ground. The whole hill quaked as if caught in the grip of a powerful earthquake.\n\n_Oh, dear God. So that's what a Wither does . . ._\n\n\"Now!\" Henry barked.\n\nKarina took a deep breath, cradled Emily, and stepped into the darkness.\n\n* * *\n\nIt was like being underwater. As if she were walking through a flooded tunnel of crystal-clear liquid filled with sunlight. Her body was very light, almost weightless. It lasted a lifetime or a single moment\u2014Karina couldn't tell\u2014and then she stepped onto beige carpet.\n\nFor a second she was afraid to move, afraid to do anything, and then she remembered to breathe. The air tasted sweet.\n\nEmily looked at her, blinking.\n\n\"Are you okay?\" Karina whispered, her voice strained.\n\nEmily stirred. \"I know!\"\n\n\"Know what, Emily?\"\n\n\"Mom, I know, I know! I am the Courageous Princess. Like in the comic book.\"\n\nKarina exhaled and hugged her. For some reason, she wanted to cry.\n\nThey stood in a foyer. There were people around her, both men and women. In front of her a glass wall guarded a conference room, a long black table with matching chairs; and beyond that a floor-to-ceiling window offered a view of an evening city from above, lit up with electric lights. They had to be on the twentieth floor.\n\nThey had gotten away.\n\nIn her mind the bodies still burned, vomiting fire and ashes. What the hell was Arthur? What were all of them?\n\n\"We shouldn't be here,\" Henry said next to her, his voice vibrating with alarm. \"This is wrong.\"\n\nA woman behind her snarled. \"The fucking Ripper dropped us into the wrong base.\"\n\nA soft thud made her turn. Lucas crashed onto the carpet and Daniel tried to pick him up. Lucas's eyes were closed. He looked so pale, his skin had gained an almost greenish tint.\n\nShe set Emily down and knelt by him, sliding her hand on his forehead. His skin was cold, almost clammy. Blood clung to his rib cage and a big purple bruise stained the right side of his stomach. He looked like he was dying. The heavy metallic scent rolled off him, so thick she almost choked. He wasn't just hungry for her blood. He was starving for it and he hurt.\n\n\"What's wrong?\"\n\n\"Too much venom,\" Daniel spat out. \"He shouldn't have phased into the attack variant so soon after the last fight.\"\n\nArthur stepped onto the carpet out of thin air. \"He will be fine.\"\n\nA grimace skewed Daniel's face, stretching his scar. He looked like a rabid dog. \"We should've evacuated yesterday. You overwork him. You know he needs at least two weeks between phasings, but you counted on him to save your ass anyway, because you knew he would do it. Look at him. Look at him, Arthur. He's dying from the venom.\"\n\nArthur glanced at the skyline. \"Not now, Daniel. Where is the Ripper?\"\n\n\"You are a fucking asshole!\"\n\nHenry closed his eyes and opened them. \"She isn't in the building.\"\n\n\"Daniel, stop your hysterics and search the building . . .\"\n\n\"Fuck you!\"\n\n\"Will the two of you shut up?\" Lucas said. His eyes were still closed. A shudder gripped him. He arched his back, his heels digging into the carpet, his arms rigid, his massive body straining against the pain.\n\nIdiots. Karina wrapped her arms around Lucas, trying to hold him down, but it was like trying to hold down a bull. \"We need something for his mouth. He's grinding his teeth.\"\n\n\"Vault, now,\" Arthur snapped. \"Pick him up.\"\n\nPeople swarmed Lucas, brushing her away. He lashed out, convulsing, throwing a man aside like a rag doll. They pulled Lucas up and dragged him down the hall.\n\nArthur bent down, grasped her by the elbow, and pulled her to her feet. \"Come with us.\"\n\n\"My daughter . . .\"\n\nArthur's fingers clenched her arm like a vise. He pulled her down the hallway, after the clump of people trying to move the convulsing Lucas forward.\n\nEmily ran after her. \"Mommy!\"\n\nKarina jerked. \"Let go of me! You're scaring her!\"\n\n\"Do you want your daughter to live?\" Arthur asked.\n\n\"Yes!\" Bastard.\n\n\"Then do as you're told.\"\n\nThey were almost to the end of the tunnel. Something swung open with a heavy metallic sound. Karina caught a glimpse of a huge vault door standing ajar. The people carrying Lucas ducked into the round opening and parted, and Karina saw a room beyond the door. It lay empty and the light of the white fluorescent lamps reflected off the metal floor and walls.\n\nThey would put her into the vault with him. Lucas hurt so badly, he was convulsing. He required her blood and he'd rip her to pieces to get it. If she crossed that threshold, she would die.\n\n\"Mommy!\"\n\nShe dug her heels in. \"Emily!\"\n\nHenry picked Emily up. \"It's okay, little one.\"\n\n\"You agreed to the contract,\" Arthur said. \"Time to honor it. Get in there and do whatever you have to do to keep him alive.\"\n\nIf she didn't go in, they would throw her in. She heard it in Arthur's voice.\n\nKarina jerked her arm out of his hand. \"Take care of my baby, Henry.\"\n\n\"I will,\" he promised.\n\nKarina took a deep breath and walked inside.\n\n\"No sudden movements,\" Henry called out.\n\nThe door behind her clanged shut.\n\n# **CHAPTER 8**\n\nLucas curled into a ball on the floor. The pain scoured the inside of his spine as if someone were scraping his vertebrae with steel wool. It stretched in tight strings through his ligaments; it pooled in his joints, in his fingertips, under his tongue. He felt it in his teeth. It ground him like a grain of wheat between two millstones.\n\nHis ears caught the sound of approaching steps.\n\nHe forced his eyes open.\n\nKarina knelt by him. He inhaled her scent and felt it spark a deep, angry hunger inside him. She pulled him like a magnet. His body screamed for her blood and the end of the pain. Tearing into her would be bliss.\n\nShe was rolling up her sleeve. Her lips were pinched together.\n\nHe had to speak now. It hurt and he was tired, but he managed. \"Don't.\"\n\n\"Arthur said you had to feed.\"\n\n\"Arthur is a sick fuck. I told you that.\"\n\n\"I can smell you,\" she said. \"You need to feed.\"\n\n\"If I feed now, you'll die.\"\n\n\"If you don't, you will, and then they'll kill Emily.\"\n\nAh. For a second he thought she had felt sorry for him, but no. \"Nobody will touch Emily. And I'm not dying. Just hurting.\"\n\n\"You look awful.\" He heard a soft note in her voice. In spite of everything, she cared a little bit. He would take that. That was more than he usually got from anyone.\n\nShe hadn't shied back when he phased. Her knees had trembled but she didn't flinch. For that he was grateful.\n\nKarina brushed the grime off his face, her eyes kind, her voice gentle. \"Lucas, don't be an idiot. Feed. It will make you feel better.\"\n\n\"The pain isn't fatal. It will pass. You'll need all of your blood before long.\"\n\nShe pulled back. \"What does that mean?\"\n\n\"Do you have a fever?\"\n\n\"Yes.\"\n\n\"Tired?\"\n\n\"Yes. Lucas, what is happening to me?\"\n\nHe almost told her the truth. \"I told you before, you're reacting to the venom.\"\n\nThe ache had burrowed deep into the base of his spine. Lucas forced himself to turn, trying to shift his weight, and it exploded into a blinding white, mind-numbing haze, twisting his limbs. Like being punched in the mouth by a star. He passed out.\n\nWhen he awoke, her scent was everywhere. The hunger stirred inside him, demanding. Lucas clenched his teeth and felt a light touch on his cheek. His eyes snapped open. She was sitting next to him, her back resting against the wall.\n\n\"How long was I out?\"\n\n\"Maybe a minute or two.\"\n\n\"Try to time the next one. I need to know if they're getting shorter.\"\n\n\"Is there anything else I can do?\"\n\nThe ache rolled back at him. \"Talk to me.\"\n\n\"About what?\"\n\n\"You never did tell me exactly why Emily hoards food.\"\n\nShe sighed and brushed the brown lock of hair from her face. \"It happened after Jonathan died.\"\n\n\"Your husband?\"\n\n\"Yes. I don't want to talk about it.\"\n\n\"Why?\"\n\nShe met his gaze. \"Because then you will know things about me.\"\n\n\"And that would be bad?\" Lucas asked.\n\n\"Yes.\"\n\nNow he wanted to know more.\n\n\"Does it hurt to be the beast?\" she asked.\n\n\"No. Phasing is like being a superhero. I'm faster, stronger. Everything is sharper. There are no consequences. I can let myself off the leash. But my attack variant's venom is toxic to my human phase variant. Turning back into a man is a bitch.\"\n\nA small tremor shook his legs. Lucas grunted and closed his eyes, trying to will the pain away.\n\n\"How long will we be locked in here?\"\n\n\"Until I pull through. Hours. Arthur is trying to keep me safe. I'm an asset and I'm rare and difficult to replace. We shouldn't have come here, to this building.\" The words came slowly. \"This base is not secure. We rent five floors here. We don't own the buildings and don't control access to it.\"\n\nKarina bent down, looking closer into his eyes. Tiny red rosettes marked the skin on her cheeks and forehead. Her own transformation was closing in. Shit. He hoped she would have another day. He didn't want her to phase here, in the vault, without medical help, without Henry to keep her calm. She could die and he wanted her to live. He had to heal fast.\n\n_Heal,_ Lucas willed in his mind. _Heal._\n\nThe pain exploded in a white burst and dragged him under.\n\nWhen the light faded he heard her voice, soothing, calm, warm. Like sitting back in the hot tub, soaking his exhausted body while she floated nearby. \". . . met in college. Jonathan was handsome. Funny. His father was the CFO for Drivers Company. It's a big insurance company in the Southwest. Brian's very driven, very conscious about his appearance. Brooks Brothers suits, expensive watch, a new BMW every couple of years. He and Lynda had Jonathan when they were much older, in their forties. Jonathan could do no wrong. He was their golden child. Good at sports, good at academics. He was easygoing and charming. The perfect son.\"\n\nShe leaned her head against the wall. He moved closer to her and rested the back of his head on her ankle. She let him do it. From here he could see her face. He could touch her hand. Lucas closed his eyes and let himself sink into her voice.\n\n\"Things always went Jonathan's way. I used to watch a cartoon when I was younger. Two mice were living in a lab, and one was very smart and the other one was a knucklehead. So every night the knucklehead mouse, Pinky, would ask the smart mouse, 'And what are we going to do today, Brain?' And Brain would say, 'Try to take over the world!' And Pinky would get all excited. See, Brain was serious. He was trying to take over the world. But to Pinky it was all a big game. That's kind of how Jonathan was. The world was his huge playground and every day he'd play at taking it over. Some days he was an athlete; other days he was a student. When we met, he was finishing his MBA and I was getting my bachelor's in accounting. My parents had died in a car accident when I was a senior in high school. I had just turned eighteen when they passed.\"\n\n\"I'm sorry,\" Lucas said and meant it.\n\n\"Thank you. They left me just enough money to get me through college and I had to work to feed myself. Before they died, I wanted to go into art history.\" She laughed a little, a bitter, quiet sound. \"I wanted to be an art appraiser. You know, the person who examines art for auctions and museums to determine if it's authentic. I always thought it would be so neat. But I was on my own then, so I went into accounting instead. It seemed . . . sensible. I was trying to be sensible. To have some structure. And then Jonathan shot into my world like a comet. He could make anything seem exciting. He made things fun. His parents were always very formal with me. I don't think they ever understood why he liked me, but Jonathan picked me and he could do no wrong.\"\n\nHe very badly wanted to murder Jonathan.\n\n\"It was great at first. Jonathan's father's connections got him a position in a private equity firm. During the day he got to play a businessman and during the night he got to play a husband. And then Emily was born. Well, you've seen her.\"\n\n\"She is pretty,\" Lucas said.\n\n\"She is. Jonathan loved her. It was yet another new game: being a dad. He used to show her off like a cute purebred puppy.\" She sighed again. \"I should've seen it then. Anyway, everything was great for a few years and then the bottom fell out of the economy. Suddenly it wasn't fun anymore.\"\n\n\"The party was over,\" Lucas guessed.\n\n\"Yes. Jonathan had to start working for his living and buckle down, or the firm would cut him loose. I worked, too, and we were doing okay, but we had to mind our p's and q's and Jonathan didn't want to be bogged down with details. We used to have the stupidest conversations. He couldn't understand why he couldn't drop thirty grand on a membership at a country club. It's like his brain couldn't digest the concept of a budget. I mean, the man had a master's degree in business management, for crying out loud.\" Her voice rose too high and Karina fell silent.\n\n\"What happened?\" he prompted.\n\n\"Finally he decided he was tired of playing with us. He started sending me these long rambling e-mails about how he felt constrained and unhappy and about the need to find himself. He wanted to live fully, he said. To find the zest in life. At first I was concerned, then I thought he was cheating, but he wasn't. It's not like we were ever on the verge of bankruptcy. We just couldn't do exciting things anymore, like ordering champagne for the entire bar. I offered to move; he didn't want to do it. No solution I suggested was good enough. He tortured me like that for about four months. In the end I didn't even care anymore. I should've fought harder maybe, but I remember one of my friends calling and telling me she saw Jonathan at her office party without me, and you know what I thought?\"\n\nShe paused. Her dark eyes were huge on her pretty face. \"I thought, 'Good. Maybe he'll meet someone and I can divorce him.' That's an awful thing to think about your husband. That's when I knew the marriage was over. We were heading downhill, except there was Emily. How do you explain to a four-year-old that Daddy decided he doesn't want her anymore because he needs to go find himself? So I spoke to his parents. I thought maybe they would talk some sense into him.\"\n\nLucas grimaced. \"You said he could do no wrong.\"\n\n\"Yes, it was stupid, but I was desperate. They called him over to have a heart-to-heart. Jonathan took me out to dinner at the end of the week. I knew something was up; I could just tell. It wasn't a date. He told me he had filed for divorce. He had no problem paying me alimony, and I could retain all my parental rights.\"\n\nA shadow passed over her face. She seemed small all of a sudden.\n\n\"We were in the car, going to pick up Emily from the sitter's. We were fighting about his generosity in regard to my 'parental rights.' \" Her voice dripped with bitterness. \"He wanted to leave and stay gone. I insisted that Emily needed a father and he couldn't just take off. He was mad. He told me that everyone had a right to be happy. He wanted to be free of me and Emily but he didn't want to be judged for it. And then, all of a sudden, he lost consciousness. It was like someone had flipped a switch. We shot into the opposing lane. I remember headlights. I woke up in the hospital.\"\n\nShe fell silent. \"He had a stroke,\" Karina said finally in a flat voice. \"He had fibromuscular dysplasia. Nobody knew. He was healthy as a horse, played racquetball, and then he just died. It was touch and go for me for a little while but I bounced back. I was in the hospital for two weeks. Emily had to stay with his parents. They didn't feed her.\"\n\n\"What?\"\n\n\"Brian, Jonathan's father, always eats out. When Jonathan died, he spent all his time at a country club. He said it was his way to cope. Lynda is in her seventies. She has a touch of dementia. All she did was eat candy all day, but she wouldn't give Emily any\u2014it would ruin her teeth. She would forget to give Emily lunch, and when she did remember to feed her, she would either try to cook and burn it or she'd give Emily food that had been in the fridge for so long, it wasn't just moldy, it was blooming.\"\n\nShe was crying, not from pity but from anger. There were no tears, but he heard it in Karina's voice, hidden behind the flat tone.\n\n\"They had a bowl of nuts set out and Emily told me she would pretend to fall asleep and then sneak out and steal them. When I got out of the hospital, she was six pounds lighter. She barely weighs anything as it is. So now you know why she hoards food. She was terrified, her father had just died, her mother was in the hospital, and her own grandparents wouldn't feed her. I told Arthur she doesn't have anyone except me. I meant it. We are not welcome at that house. They blame me and Emily for Jonathan's stroke. We made his life so difficult, he died to escape.\"\n\nThe red rosettes on her face were turning darker. Karina touched her hand to her forehead and looked at it. Her eyes widened. She rubbed his forearm.\n\n\"This is another reaction to the venom?\"\n\n\"Mmhh-hhm,\" Lucas said.\n\n\"I told you my story. Tell me yours now. It's fair.\"\n\n\"What do you want to know?\" he asked, wondering what she would think if she looked inside his mind and saw him strangling her husband.\n\n\"Who are you? All of you. Who are you really? I need to know what's happening to me.\"\n\nLucas sighed.\n\n* * *\n\nShe had told him too much, Karina decided. As much as she wanted for it to be a bribe, a down payment for the information he held, at least in part she told him what she did because he was lying beside her, bruised, beat up, bloody, and hurting. He needed a distraction and she had enough compassion to give him one. But she hadn't meant to pour her heart out. It just happened. He was in pain, and although she had the means to ease his suffering, he refused to feed, because he didn't want to hurt her. He wasn't willing to trade his pain for hers. The least she could do was talk and try to distract him.\n\nKarina reached over and touched his hand. His fingers closed on hers. Lucas glanced at her, surprised. They had that in common now\u2014both of them treated any act of kindness with suspicion. She didn't expect kindness anymore, except from him. But she was an outsider. He wasn't.\n\n\"There are no scared women here to watch us,\" he told her.\n\n\"It was never for them. It was for you.\"\n\nShe almost cried and couldn't even understand why. It was the stress, Karina told herself. The trauma of watching hundreds of people die at once. And the fever, which kept rising and rising. Her breath felt hot when she exhaled. Her skin was dry and too tight. And now there were rings of red dots all over her arms.\n\nShe had never told the entire story of her marriage to anyone. _It's the fever. Of course it is._\n\nLucas was looking at her. Sprawled like that, even battered, he looked enormous. If a week ago someone had told her she would be locked in a vault with a nude, bloody man who was trying his best not to devour her to stop his pain, she would've dialed 911 to report a lunatic running amok.\n\n\"I'm going to tell you a story,\" Lucas said. His voice was laced with fatigue. \"You can choose to believe it or not. It can be the truth or just a story. It's your choice.\"\n\n\"Okay.\"\n\nLucas closed his eyes. \"Suppose there is a civilization. A powerful country. It has taken over all of its available territory, but it knows that it must expand. It must continue to grow outward, or it will rot and collapse. This civilization sends colonists out to explore new territories. They find fertile lands and colonize them. When they succeed, they let the knowledge of the large civilization fade. The small colonies grow and prosper on their own, and when they develop enough, they rediscover their mother civilization and rejuvenate it with their unique achievements.\"\n\nHe glanced at her.\n\n\"Okay,\" Karina said. \"I can see how that would happen.\"\n\n\"Suppose a new island was found for colonization. An island with an abundant ecosphere and great resources. The civilization had done this many times before and they had developed a protocol. The colony ships arrived and the colonists created thirteen small settlements, Houses, one for each colony ship.\n\n\"Genetically, all the colonists belonged to the Base Strain. It's a very stable breed of human, long-lived, resistant to diseases, armed with superior DNA repair mechanisms to counteract mutation. To successfully colonize a new environment, a species must adapt to it. To facilitate this adaptation, most of the colonists were exposed to an agent inhibiting their cellular and DNA repair and vulnerability to native viruses.\"\n\n\"They deliberately made their people weaker? How does that make sense?\"\n\n\"They didn't just want a colony,\" Lucas said. \"They wanted a unique colony, perfectly in tune with this new island. That's how the civilization kept itself from stagnation. The colonists wanted an explosion of mutations in the future generations, and they needed a shorter life span and faster sexual maturity to pass the new changes on to their offspring. That's why scientists experiment on mice: they breed quickly and don't live very long. The shorter life span goes hand in hand with faster sexual maturity. But it also brings negative anthropological consequences: immaturity, inability to pass on knowledge, loss of ethics and culture, and so on. These consequences were considered acceptable. The colony had to develop on its own without the knowledge of its origin anyway. The sooner people forgot where they came from, the better. A small group of the colonists remained as Base Strain for control purposes. They lived in the settlements, the Houses, and monitored the whole thing. With me?\"\n\nSort of. \"Go on.\"\n\n\"Mutations bloomed. A succession of several dozen subspecies of human followed. Some subspecies developed variations, people with similar powers or physiology. Subspecies 29 showed all of the adaptations necessary for survival, but all eight of its types were plagued by sensitivity to heat and alarmingly low fertility. Subspecies 44, type 3, produced exceptional Mind Benders, who were prone to insanity.\"\n\n\"Is that what Henry is?\"\n\nLucas nodded.\n\n\"We're not talking about islands, are we?\"\n\n\"Some say islands,\" Lucas said. \"Some say planets. It's just a story.\"\n\nA story, right. \"Aliens.\" She stared at him. \"Are you trying to tell me that all of us are aliens?\"\n\nLucas sighed. \"You could say that. You could also say that once the planet shaped us and twisted our DNA, we are now just as native as anybody else.\"\n\n\"What about Subspecies 30?\" _What about you?_\n\nLucas's eyes fixed on her. \"Subspecies 30, types 1 through 5, otherwise known as Demons. A venomous, carnivorous, predatory variant of human with the ability to drastically alter its morphology. They were powerful, aggressive, territorial, and they dominated their point of origin for a few hundred years, hunting in small packs, but this subspecies was not viable long term. They were crippled because their bodies couldn't produce a set of small molecules necessary for their survival, so they had to cannibalize other humans to get it.\"\n\n\"Cannibalize?\"\n\n\"At that point the various subspecies of human had only a rudimentary language and no memory of where they came from,\" Lucas said. \"No ethics, no morals, nothing. They were forming fledgling societies and 'might is right' was the law. If I need your blood, and there is nothing in my upbringing or experience that tells me I shouldn't, why wouldn't I kill you and eat your flesh? Being a nice guy is a modern concept.\"\n\nHe was serious. He was actually serious.\n\n\"Should I keep going?\" he asked.\n\n\"Yes.\"\n\n\"This went on for hundreds of years. The small remaining pockets of Base Strain, the original colonists, kept as a control group, meticulously documented all of it from their Houses. They didn't interfere. They just cataloged what occurred.\n\n\"Then suddenly Subspecies 48 popped on the scene. The Rippers had a fatal vulnerability to cancers but also the ability to rupture holes in reality, accessing dimensional fragments. This was a new development, unknown to the colonists, and nobody knew what to do about it. Some Houses took Ripper children and raised them within the settlements to study them.\n\n\"The mutations bloomed and bloomed, until one subspecies emerged as best adapted. It did well in almost every climate. It reproduced quickly, showed mental agility, and demonstrated decent DNA repair. At approximately six thousand planetary cycles, Subspecies 61 was declared viable. The colonists had done their job: they had created the type of human with the best ability to survive and prosper. Now nature needed to take over. All support for other strains ceased, as dictated by the Original Mandate. _Ile_ must survive. Subspecies 61 became _ile_. Everyone else needed to die to make room.\"\n\n\"Subspecies 61. Humans,\" Karina guessed. \"Us.\"\n\n\"No,\" Lucas said. \"Them. Your neighbors, your friends. But not you.\"\n\nHer fever was now so high, she was freezing and melting at the same time. \"You said them, not me. What do you mean, not me?\"\n\n\"I'm getting to that. Other subspecies were dying out, while Subspecies 61 went on to multiply and claim the island.\"\n\n\"The planet.\" Karina didn't need him to keep babying her.\n\n\"The planet,\" he agreed. \"The colony cities began to gradually phase out their technology. They were letting themselves disappear. But there was a protocol breach at one of the cities, as a result of which Subspecies 29, the one that had trouble with heat, discovered where they came from.\"\n\n\"What do you mean?\"\n\nLucas sighed. \"I mean that the scientists at the Mare House fucked up. Subspecies 29 produced several unusually smart children. A sudden explosion of kids with genius-level intelligence was rare and odd, so the idiots thought it would be a good idea to study them further. They extracted these children and raised them within Mare with the full knowledge of their history. Well, the kids grew up and decided they didn't want to go gently into that good night while some other breed of human took over.\n\n\"There was a quiet coup. By the time it was discovered, Strain 29 and their captive personnel had genetically corrected their shortcomings. Now they had no trouble with heat and they bred like rabbits. They decided that they were more viable than Strain 61. They, not humans, were _ile._ A mistake was made and they decided it had to be corrected. They were ordained to take over the Earth.\"\n\nNow it made sense. \"They became the Ordinators?\"\n\n\"Yes.\"\n\n\"So this is it? They've been trying to kill us off for thousands of years?\"\n\n\"More or less. They went to war, using the colonists' original technology. The other cities opposed them, but they were weak by that point and in the process of dissolving themselves, so they plucked people from different strains with combat potential. The Ordinators were broken and would've been wiped out, except they acquired Rippers and began hopping through dimensional fragments. Eventually, so did we.\n\n\"Strain 61, the _ile_ humans, was reproducing too quickly, and their numbers grew too numerous. They saw us and started forming religions and folklore. We had to disappear.\"\n\n\"So this is how it is,\" Karina said.\n\nHe nodded. \"People like me have been keeping the Ordinators at bay for over thirty thousand years. Occasionally they break through with a new weapon. Sometimes it's a virus that kills the food supply. Sometimes it's bubonic plague. Sometimes they find a way to fiddle with the climate. The problem is that the Ordinators breed faster than us, they're better organized, and their job is easier: it's much simpler to destroy something than to protect it.\n\n\"There were thirteen Houses, one for each landing site. They have one House, the House of Mare. There are probably between one and two hundred thousand of them. We are the soldiers of the remaining twelve Houses. There are maybe fifty thousand of us. We crossbreed and have children with weird powers instead of dying out the way we should. This is the planet where everything went wrong. As humanity moves closer and closer to interstellar space flight, the Ordinators are getting desperate, because once we reconnect with the root civilization, it's all over for them. They abandoned the original mandate and they will be exterminated. They're attacking with everything they've got and we're losing the fight.\"\n\nShe stared at him. \"And where do I fit in?\"\n\nHe took her hand and squeezed it gently. \"You know why my people died out?\"\n\n\"Because their own venom poisoned them?\" Karina said.\n\n\"That. But also because the colonists had done some projections. It was decided that if we were allowed to exist, we would destroy the other subspecies and then die out before reaching the level of medical sophistication necessary to fix our defect. They poisoned us, wiped out the entire species almost completely. They were right\u2014even now the synthetic substitutes are just a Band-Aid. See, if we could've overcome this handicap, they would've let us murder everyone else, but the problem is that only one very specific subspecies produces the hormones we need. The Base Strain. The donors. The ones who gave rise to all of us.\"\n\nShe jerked her hand back. \"You mean I am a descendant of the original colonists?\"\n\n\"Yes.\"\n\n\"That's not possible.\"\n\n\"It is. Your type has a remarkably stable genome.\"\n\n\"But my parents were normal people!\"\n\n\"They may not have known who they were. Maybe only one of them was a donor. A donor and Subspecies 61 will produce donor offspring.\"\n\n\"But what about this?\" She held out her arms, speckled with brilliant red. \"Explain this!\"\n\nLucas sat up. \"When I fed on you, the mutation agent entered your bloodstream. In normal humans the mutation agent has grown weak over the generations. But I am carrying a near-full dose and I gave it to you during the feeding. You are changing.\"\n\n\"Into what?!\"\n\n\"I don't know. I don't know what's in your DNA besides the donor genes. The mutation agent is an inhibitor. It will release the brakes within your body, short-circuiting your DNA repair, and let you develop into something that's already there in your genotype, acquired over the centuries of crossbreeding with different human subspecies but suppressed. You could become Subspecies 61, but I doubt it. Chances are, it will be one of our subspecies instead.\"\n\nThey had taken her freedom, her home, and her dignity, and now they were taking away her body. \"No! No, I am not doing it! I won't! You hear me?\" Karina surged to her feet. She managed two steps. Pain shot up through her bones. She cried out. The world went red and she crashed onto the floor.\n\n* * *\n\nIt hurt. It hurt more than any pain she could remember. At first she begged, then she prayed, then she screamed and whimpered, squeezing her eyes shut, opening them again, glimpsing Lucas's face against the harsh light of the vault, and then sinking into more pain. If only she could pass out completely and be done with it, but no, every time she tried, he shook her back, into the place of hurt.\n\n\"Come on, stay with me. Stay awake. Snap out of it.\"\n\n\"Let me be,\" she snarled.\n\n\"You pass out, you die. Come on. Stay with me.\"\n\n\"I hate you! You did this to me!\"\n\n\"That's right,\" Lucas snarled right back. \"Hate me. Fight with me. Stay awake. You die, Emily will be alone. You don't want to leave your daughter with an asshole like me.\"\n\nShe just wanted the torture to stop.\n\nAnother bout of agony rocked her. When it was over, she was so tired, she could barely breathe.\n\n\"The other woman . . .\" Karina whispered. Forcing the words out felt like trying to swallow glass. \"Did she have to do this?\"\n\n\"Yes.\"\n\n\"Did you kidnap her, too?\"\n\n\"No.\" Lucas gathered her closer, holding her against him. \"She was one of us. Her family were donors of Daryon.\"\n\n\"Did she hurt, too?\"\n\n\"Yes.\"\n\nLucas's eyes were so dark, they seemed almost brown.\n\n\"Tell me about her.\" She wasn't sure why she wanted to know, but she did.\n\n\"She was very smart. And she looked beautiful. Very graceful, fragile, elegant.\"\n\n\"Not like me, then?\" Nobody would call her fragile. Or elegant, for that matter.\n\n\"Nothing like you,\" he told her quietly.\n\nThe agony burned through her in a crippling spasm. \"Why does it sound like a compliment?\"\n\n\"Because she only looked beautiful. In our world nobody has the luxury of doing nothing,\" he said. \"Everyone has a function. I protect. Someone else oversees mining. Someone else oversees stocks and finances. Galatea's family had only one function: to provide Base Strain to the House. For that they were sheltered, fed, and protected. Galatea never worked a day in her life.\"\n\n\"Must be nice,\" Karina whispered.\n\n\"She didn't think so. She wanted the mutating agent.\"\n\n\"She wanted this? Why?\"\n\n\"Power,\" Lucas said. \"She thought she would become something much more prized than a donor and she would be free of me. Her father was my first donor. She wasn't supposed to become one, but he died, and she had to take his place. She thought I was an animal. She was convinced that once I fed, she would become a Ripper and could use it as leverage to be free of me.\"\n\n\"What did she become?\"\n\n\"An Electric. She senses electric currents. It's not an uncommon subspecies. A lot of technicians come from it.\"\n\n\"Uh-oh,\" Karina managed. Her lips were so dry, but there was no water. \"Let me guess: it was your fault, right?\"\n\nHe nodded. \"It was everyone's fault. She used to scream and throw fits, and then she wanted to fuck and she wanted me to beg for it. I was young and stupid. She was older, smarter, and beautiful.\"\n\nKarina raised her hand and touched his haggard face. \"You loved her.\"\n\n\"Yes. And I was so dumb, I thought it was enough. That's why I let it go that far. She once told me that we, the House, had stolen her life. She wanted to stroll the streets of London, visit the Tate Modern, go to concerts in Royal Albert Hall. I offered to take her. She told me that it wouldn't be the same. My presence would poison London for her.\"\n\n\"She sounds charming,\" Karina managed.\n\n\"I am what I am,\" Lucas said. \"No illusions. Life with me is hard, but she made a personal hell for me and her. I wasn't the one who started sex, but I finished it. I dealt with it for four years and when I turned twenty-two, I decided I was done. I went on synthetics and told Arthur to find her a different place. He transferred her to a technical work crew. She tried to stab me with a knife when she found out. Galatea was never fond of getting her hands dirty. Three months later, during an attack, she disappeared. The next time Henry sensed her presence, we ran into the Ordinators.\"\n\n\"She betrayed you.\"\n\n\"Yes, she did.\" Lucas shifted her carefully. \"And now you know the whole story.\"\n\n\"Do you miss her?\" she asked.\n\nHe peered at her face. \"How did you know?\"\n\n\"I miss my husband,\" she whispered. \"I don't blame you, you know.\"\n\n\"For what?\"\n\n\"For any of it. For the motel, for the feeding, for this.\" Karina tried to swallow the pain away, but it remained. She wouldn't make it. She could feel death crouching just a few feet away. \"Lucas, you're not a bad person. You have no idea how scary you are, but you're kind and patient. If things were different . . . It has to start right . . . And we just can't, because I would never be more than a slave and you would always own me. Please take care of Emily for me. Don't let anyone hurt her. She's a great kid.\"\n\nHe didn't answer. He just held her.\n\n* * *\n\nKarina awoke slowly. Within her body, the pain subsided, gradually, like a receding tide, fighting for every step of its retreat.\n\nShe opened her eyes and saw Lucas's neck. Her face was buried in it.\n\nHe was kneeling on the floor, looking up. She was wrapped in his arms.\n\nHer voice shook. \"Why are you holding me?\"\n\nLucas turned to look at her. His face was too close to hers. \"I didn't want you to die alone on the floor.\"\n\nShe said things. Stupid, stupid things. Maybe it was a dream. His eyes assured her that it wasn't.\n\n\"Please put me down.\"\n\nHe let her go slowly. Karina slid down onto her knees and sat clumsily on the floor. Her legs shook a little. She felt light, so light and cold. \"Is my change over?\"\n\n\"Yes,\" he said.\n\nShe had survived. \"I don't feel any different.\"\n\n\"The change isn't always obvious. Something will trigger it sooner or later.\" He was looking up again. She glanced up, too, and saw a monitor in the ceiling. It showed an empty hallway.\n\nA man in dark clothes darted across the hallway, brandishing a machine gun, and hid behind the wall.\n\n\"We're being attacked,\" Lucas said. His voice was calm, almost casual.\n\n\"How is that possible?\" Emily. Henry had her. If they were being attacked, her daughter would be in danger.\n\nMore people flickered past on the screen.\n\n\"The Ripper must have been an Ordinator mole,\" Lucas said. \"We should've gone to a ranch in Montana\u2014that's our evacuation route from that base. Instead we're in Detroit. This building is nearly abandoned; only the bottom three floors and the top five\u2014those are ours\u2014are operational. The blocks in a one-mile radius around it are basically deserted. We're sitting ducks here.\"\n\n\"Why didn't Arthur evacuate us?\"\n\n\"I don't know,\" Lucas said. \"The Ordinators likely blocked the exits. We landed into a trap.\" His face was dark. \"Our best chance is to stay here.\"\n\nNo. No, she had to go and find Emily. \"Why?\"\n\n\"I'm at my limit. Normally I would be drugged and sleeping this off for the next two or three days until my body came to terms with my venom. I could barely hold you. Most likely Arthur has sent for reinforcements. The vault is solid and must be opened from the inside. It will take them several hours to get through the door, so it's likely they won't bother with us right away. By the time they get around to it, we might be reinforced. Our best bet is to stay here and wait it out. We probably die either way, but here we have more of a chance. Especially if we're quiet.\"\n\n\"You have to let me out.\"\n\nHe looked at her, obviously trying to decide if she was crazy. She had to convince him she wasn't.\n\n\"Henry has Emily,\" she said. \"She's out there somewhere.\" Out in an abandoned building full of people with guns and God knows what sort of weird powers.\n\nLucas looked at her for a long moment.\n\n\"I have to find her, Lucas. You don't have to come with me. All I ask is that you help me open the door, because I don't know how. I'll find her myself.\"\n\n* * *\n\nLucas looked at the door. If they opened the vault, he would walk out of it a dead man. She stood before him, her eyes huge and brimming with worry. She just wanted her little girl back and she didn't understand how far gone he was or how many enemies they would face.\n\nEveryone dies, Lucas reflected. He'd been a selfish bastard all of his life. If he walked out of that door and died helping her find her child, at least he'd die doing something worthwhile, not cowering like a dog in the vault, waiting to be gunned down.\n\nAnd she couldn't go out there alone. She would be dead in minutes.\n\nHe sighed, rose, and stepped to the wall. Karina clenched her hands. She couldn't read his face. He touched it and a section of it slid open, revealing a number keypad and a small speaker. His fingers played with the keys. \"Cousin?\" Lucas said softly.\n\nA faint hiss of static issued from the wall, then Henry's faint voice came through. \"Lucas. Red, gray, seven, pinned.\"\n\nLucas grimaced. \"Is the little girl with you?\"\n\n\"Yes. Black.\"\n\n\"How bad?\"\n\n\"I'll live.\"\n\n\"Don't move. I'm coming to get you.\"\n\n\"That's unwise,\" Henry said.\n\nLucas slid the panel back in place. \"He is two floors below us. He's been shot. Emily is okay; he is keeping her under. He can't move because it's too dangerous and he is cloaking, which makes him harder to find, but they will locate him eventually. The moment we leave this vault, you and I must fight to survive. Remember how you tried to cut me with your knife?\"\n\n\"Yes.\"\n\n\"Find that woman and be her.\"\n\nHe had no idea how hard she had worked on hiding that woman and how ready she was to let her out.\n\n\"Don't move.\" Lucas walked over to the vault door, punched in a combination in the small number pad, and turned the wheel in the door's center. Something clanged inside the door. Lucas moved to stand on the side. With a soft hiss, the door swung open and Karina stared straight at a man with a gun.\n\n\"Hands up!\"\n\nShe didn't move.\n\nThe barrel of the machine gun glared at her, black and huge, like the mouth of a cannon.\n\n\"I said hands up!\"\n\nLucas nodded at her. She raised her hands.\n\n\"Subspecies?\" the man demanded.\n\n\"I'm a donor,\" she said.\n\nThe man's eyes widened. \"Get up and walk to me.\"\n\nLucas shook his head.\n\n\"I can't,\" Karina said, keeping her voice monotone. \"I'm sick. I can't walk.\"\n\nThe man moved into the vault, one step at a time, careful, the gun pointing at her. He took three steps in. Lucas lunged, so quick she barely saw it. His hands closed about the man's neck. Bones crunched, and the man sagged down on the floor, limp.\n\nA week earlier, she would've screamed. Now she just got up and ran over to the body.\n\nLucas staggered, leaned against the wall, and pushed himself upright. He wasn't joking. He really was at his limit.\n\nShe crouched by the body and began going through the man's pockets. \"I can do this alone.\"\n\n\"Yeah, yeah.\" He picked up the man's machine gun and handed it to her. \"Safety here.\" He flipped a small switch. \"Point and pull the trigger. Your instinct will tell you to keep clenching it. Don't. Count to three in your head and let go of the trigger. Short bursts.\"\n\nKarina took the gun and raised it. It was heavy like a cement block. \"You do realize that I can kill you with this.\" She didn't mean to say it. It just came out.\n\n\"Yes.\" He turned his back to her and went out of the vault. A pair of jeans and a sweatshirt lay by the door. Lucas pulled on the clothes and started down the hallway. She followed him. He moved like a cat, soundless on bare feet.\n\nThey came to the end of the hallway. Lucas leaned against the wall, glanced around the corner, and looked at her. \"Point and pull the trigger,\" he whispered.\n\n\"Count to three,\" she whispered back.\n\nHe nodded.\n\nThere were people at the end of that hallway. People she would have to kill. _It's them or us._ Kill or be killed.\n\nShe took a deep breath, stepped into the hallway, and pulled the trigger. The gun spat thunder. Bullets ripped into four distant shadows. She thought there would be blood, but no. They just jerked and went down, screaming. She pounded the bullets into the bodies for another long breath and let go. Lucas moved next to her.\n\nIt was a test, she realized. He had to know if he could rely on her. Well, he could. She'd kill every one of them to get to Emily.\n\n\"What happened to letting go on three?\"\n\n\"There were four of them,\" she said. Movies and books told her she should be throwing up now, but she didn't feel queasy. Her mouth was dry. It would probably hit her later, but now only Emily mattered. \"I decided to take two extra seconds.\"\n\n* * *\n\nKarina followed Lucas through the dark passageways as fast as she could. She was squeezing everything she had out of her exhausted body. Now that the first flush of adrenaline had worn off, fatigue set in. She didn't walk, she dragged herself forward, shot when Lucas shot, stopped when he stopped. Only the next step mattered and she gritted her teeth and managed it again and again.\n\nThey made it to a small door. Lucas punched a code into the lock, the door snapped open, and they went through onto a concrete landing. Lucas punched the lock and the small square light in its corner turned red.\n\n\"We rest,\" he said. \"Two minutes.\"\n\nKarina sank down to the concrete and he sprawled next to her. The grimy floor was like heaven.\n\n\"Why are you helping me?\"\n\nHis voice was a quiet growl. \"Because I like you. And your little girl.\"\n\nShe closed her eyes, feeling the cold concrete under her cheek. That wasn't it. Lucas was making up for his past sins, but that wasn't all of it, either. She knew the true answer. She could read it in his worn-out face. He wanted to save her, because he wanted her to stop flinching when she looked at him.\n\n\"Thank you,\" she told him. \"Thank you for helping me.\"\n\n\"Time to get up.\" He rose.\n\nShe cried out as he pulled her off the floor and followed him down the stairs. An odd sensation clenched her, almost like some internal spring had compressed inside her and now begged to be released. She stumbled, and it vanished.\n\nOne floor. The landing. They were midway down the next flight of stairs when the door below swung open.\n\nAn icy presence clenched her mind in a hard grip. It shut her off, trapping her. She couldn't move; she couldn't speak. Time slowed to a crawl.\n\nThe door kept opening, wider and wider. She saw inside it; she saw armed people pour out onto the landing. She knew she had to fire. Instead she just stood there, disconnected from her body.\n\nAnd then Lucas shoved her down and sprayed the landing with bullets.\n\nThe presence gripped her mind and squeezed. She couldn't even scream.\n\nOrange sparks flared on Lucas's gun. It died.\n\nMore people spilled into the landing over the bodies. Lucas leaped into the attackers. He smashed one out of the way, cracking the man's skull against concrete like a walnut. The man slid down, leaving a bright red stain on the wall. Lucas ripped a woman's throat out with his hand, backhanded another man down the stairs, and shuddered as a handgun barked. Red spray shot out of Lucas's side. He lunged forward and broke the gunman like a twig and dived into the doorway.\n\nThe sound faded. She was completely disconnected from her body now. Only her vision worked.\n\nLucas emerged from the door, bloody, his eyes furious. He must've jerked her up, because her view changed and suddenly he was directly above her. He barked something, angry. The world shook. He dived down. His lips closed on hers. She felt nothing. He jerked back up and rocked back and forth, screaming again.\n\n_Henry,_ she read his lips calling. _Henry._\n\nHe kissed her again and rocked, his face jerking up and down. His hands pushed on her chest. She saw the muscles on his arms flex, but felt nothing. The red stain on his sweatshirt spread wider. Was he doing CPR? Was she dying?\n\n_Henry._\n\nThe ice cracked. She heard a distant female scream somewhere impossibly far. Warmth flooded into her. Something popped inside her mind and she saw a radiant light, bright and glorious.\n\n_She's gone now,_ Henry's voice said in her mind. _She won't bother you again. You're free. Breathe, Karina. Breathe._\n\nThe world snapped back to its normal speed, jerking her back into her body. She felt everything at once: pain, the hardness of the stair under her back, and the rhythmic push of Lucas's hands on her chest. She gasped. He pulled her up, into his arms.\n\n\"Mind Bender attack,\" he told her. \"Up. Keep moving.\"\n\nThe scent of heated metal rising from Lucas was so thick, she almost choked. He wasn't just hurt. He had to be close to dying. If he died, she would be free, but in this moment she didn't care. She just wanted him to survive. \"You've been shot.\"\n\n\"We must move,\" he told her and pulled her up to her feet. \"Faster!\"\n\nHe drove her down the stairs, through the door, and along the narrow hallway. They dashed past a row of offices. Lucas rammed a door head-on and they burst into a small conference room. Henry lay slumped in the corner, his back pressed against a wall that was mirrored floor to ceiling. His cracked glasses sat slightly askew on his blood-smeared face. Emily was curled in the crook of his arm.\n\nKarina cleared the room in a desperate sprint and dropped to her knees. \"Is she okay?\"\n\n\"She's fine,\" Henry said softly. \"She woke up a little when I had to help you, but now she's sleeping again.\"\n\nKarina hugged her, cradling Emily's small body. Finally.\n\nLucas shoved the table against the door and landed next to them.\n\n\"I see you're bleeding, too, cousin.\" Henry smiled. \"Nice of you to join me.\"\n\n\"Where are the others?\" Lucas growled.\n\n\"I don't know. We were hit two minutes after you went into the vault. It was a concentrated assault. They came prepared. The seventeenth floor fell within ten minutes. We were retreating, when I got cut off. I went into cloak almost immediately. Our people may have evacuated.\"\n\n\"Without us?\" Karina stared at them.\n\n\"Arthur probably thought I fed,\" Lucas said. \"Your blood would give me enough of a boost to either get Henry and me clear or to hide.\"\n\n\"They are surrounding us,\" Henry said. \"What's the plan?\"\n\n\"You and I go. They stay,\" Lucas said.\n\n\"Ah.\" Henry nodded. \"I thought it might be something like that.\"\n\n\"What are you talking about?\" Karina gathered Emily closer.\n\n\"We're going to open that door,\" Lucas said. \"Henry and I will take off. Henry will make sure they concentrate on us and I will make sure to keep them busy. They will follow us. You will wait here for three minutes, then you will take Emily, go out into the hallway, and turn right. You will come to an intersection. Turn right again. That will get you to the stairs. Shoot anyone you see. Then you get the hell out. If you make it out of the building, Arthur won't look for you right away, since I'll be dead and he won't need a donor immediately. Don't use credit cards, don't stay twice in the same\u2014\"\n\n\"They will kill you!\" No, that was not how this would go. The spring of tension inside her shivered, compressing.\n\n\"It was never about me surviving,\" Lucas said. \"I died when we opened the vault.\"\n\n\"He's right,\" Henry said.\n\nGod, he pissed her off. \"No.\" She shook her head, trying to keep a lid on her anger. \"We go to the stairs together and fight our way down. Together.\"\n\nLucas grabbed her, jerking her close. \"You will do as you're told.\"\n\n\"No,\" she said into his snarl. \"I won't. We go together.\"\n\nThe pressure inside her built.\n\n\"This isn't a democracy!\"\n\n\"Lucas, I can't carry Emily and shoot at the same time. I can barely hold this stupid gun with two hands. Do you think I'm Rambo? It's suicide for me, Emily, _and_ you.\"\n\n\"She has a point,\" Henry said.\n\n\"See? They will kill me and your grand sacrifice will be wasted. I don't want you to die for nothing. I don't want you to die at all.\"\n\n\"Why the hell not?\"\n\n\"Because I care if you live or die! My God, you are a moron! We fight our way to the stairs together. We have a better chance that way.\"\n\nHe shook her. \"I'm trying to save your daughter, you idiot! I've been doing this a long time and I am telling you, if we go out there, we'll all die.\"\n\n\"He also has a point,\" Henry said.\n\nKarina exhaled. Emily's life was all that mattered. \"Then drink my blood and get her out of here.\"\n\n\"I would have to drain you dry. I'm barely conscious!\"\n\n\"Do it.\" Karina told him, furious. \"You have the best chance of getting out of here with Emily alive. Drain me.\"\n\n\"No!\" he snarled.\n\n\"Do it, Lucas!\"\n\n\"That's nice,\" Henry said. \"But the Ordinators are coming.\"\n\n\"Drain me or we go to the stairs,\" Karina said.\n\n\"No, we'll do this my way.\"\n\n\"Your way, I die, you die, Emily dies!\"\n\n\"There is no time,\" Henry said calmly. \"You missed your opportunity. We are all about to die. Don't let them take you alive. You will regret it.\"\n\nThe back wall of the conference room shuddered. Cracks crisscrossed the wood. It shattered and rained down in a waterfall of tiny splinters. People stood behind it, people with automatic weapons and dark helmets shielding their faces. In front of them a tall man with pale hair down to his waist slowly lowered his hand, smiling. She looked into his face and saw her own death there.\n\nIt hit her like a punch. Emily, she, Lucas, and Henry\u2014the four of them really were about to die.\n\nFor nothing. They would die for nothing.\n\nLucas surged to his feet, trying to shield her.\n\nNo. No, this was not happening. She was tired and scared and pissed off and she was done with this shit.\n\nFuck them all.\n\nThe coiled spring inside her snapped free. Fiery power surged through her in a glorious cascade. It was time to set things right.\n\nThe smile slid off the blond Ordinator's face. He opened his mouth.\n\nThe power surged from her, up and over her shoulders in twin streams.\n\nShe looked right into his eyes and said, \"Die!\"\n\nHis face turned green, as if dusted with emerald powder. He crumpled and fell to the floor. She stared at the men behind him and they collapsed like rag dolls.\n\nTwo others burst into her view from the left. She turned and _looked_ at them and watched them die in midstep.\n\n\"Anybody else?\" she called out. Her voice rang through the building. \"Does anybody else want some? Because I've got plenty!\"\n\nNobody answered. She marched out into the hallway, turned the corner, and saw a hallway full of people.\n\n_Die._\n\nThey collapsed as one.\n\nThey wanted to exterminate humanity. They had declared a war. Fine. If the Ordinators wanted a war, she would introduce them to one.\n\nKarina turned. Lucas was staring at her, his mouth hanging open. Next to him Henry stood, blinking as if he hoped that one of the times when he reopened his eyes he would see something different.\n\nKarina looked above them and saw her own reflection in the mirror wall. Twin streams of green lightning spread out from her shoulders in two radiant green wings. Like Arthur's red ones.\n\n\"A Wither,\" Henry said in a small voice, still blinking. \"She's a Wither.\"\n\nThe memory of burning faces flashed before her and she brushed it aside. Fine. She was a Wither and nobody would ever push her around again.\n\nLucas closed his mouth. His gaze met hers and she saw pride and defiance in his eyes. \"Do it quick,\" he said.\n\nHe expected her to kill him.\n\nAfter everything she'd said to him, he expected her to kill him.\n\nKarina stepped to him. Her lightning wings burned around them. \"Don't worry,\" she told him. \"I'm the biggest and the strongest and I'll protect you. We are walking out of here.\"\n\nHenry stopped blinking.\n\n* * *\n\nIt took them forty-five minutes to get down the stairs. Karina inhaled the night air. It smelled of acrid smoke and rotting garbage, but she didn't care.\n\nBehind her the building rose like a grim tower. It now belonged to the dead. She had walked through every hallway and checked every room, while Henry and Lucas sat waiting and bleeding on the stairs. She had no idea how many people she killed, but it had to be dozens. She checked their faces to make sure they were dead. They all looked the same: features sunken in, emerald green tint painting their skin.\n\nAnd now, finally, she was done.\n\nHer lightning wings had vanished, her power exhausted. Reality returned slowly, in bits and pieces.\n\nNext to her Lucas stirred. \"If you want to disappear, now is the time. You killed them because they were caught unaware. The House of Daryon won't be. I don't know what your plan is but I know that once Arthur realizes what you are, he'll do everything he can to keep you within the House. You are too powerful to cut loose. He'll kill you if you refuse, and I don't know if I can stop him.\"\n\n\"He's right,\" Henry said. \"It's alarming how often I keep repeating that. Withers, Subspecies 21, have several types. You're type 4. Arthur is type 7. He is more powerful and he has a lot more experience. At your best you can't take him, and it will take you a long time to build your reserves back up to do anything on a massive scale again. Sometimes it takes years. Not to mention that we will have to fight you if you try to kill Arthur.\"\n\nKarina looked at Lucas. \"If I leave, how will you feed?\"\n\n\"Synthetics,\" he said. \"They take the edge off.\"\n\nHis entire body was tense, like a string pulled too tight. He didn't want her to go. \"Why?\" she asked.\n\n\"That's what you want,\" he said. \"Freedom. One more day or maybe many. It's yours. Take it.\"\n\nHenry cleared his throat. \"The Ordinators . . .\"\n\nLucas looked at him. Henry closed his mouth with a click.\n\nKarina peered at Lucas's face. \"Didn't you promise me you would find me if I escaped?\"\n\n\"I did. I promise you it will take me a really long time to find you. Go now.\"\n\nShe hesitated. Emily stirred in Lucas's arms, waking up.\n\nLucas could find her\u2014she saw the certainty of it in his eyes. If he could find her, the Ordinators could find her as well, and they would be much more motivated. And even if she did escape, she would always be living on the run, hiding from everyone and afraid of every shadow. She had no doubt that Emily was a donor. She had a responsibility to her child\u2014she had to teach Emily how to protect herself or when they would be found, Emily would be caught unaware, just like she was.\n\nKarina looked out into the city. That way lay freedom. Even twelve hours before, Karina Tucker would've taken it in a blink. But she was no longer that Karina Tucker. Nothing would ever be the same. There was a chasm between her old self and her new self, and it was filled with Ordinator bodies. Too much had happened. It changed her and there was no going back.\n\nThe woman who only days before had driven four children on a school trip was dead. She had been a nice girl, kind and a little naive, because she thought she knew what tragedy was. That woman had a small, secure, cozy life. Karina missed her and she took a moment to mourn her. It hurt to let go of that life. She shed it anyway, but not like a butterfly breaking free of the cocoon. More like a snake leaving its old skin. And this new Karina took risks. She was stronger, harder, and more powerful. There was a war going on and she would take part in it.\n\nAnd even if she chickened out and tried to walk away, the memory of Lucas would keep her from going too far. She had more in common with a man who turned into a monster than she did with Jill and her endless worry over seat belts. She couldn't leave him behind now, back in the place where everyone was scared of him, where Arthur used him with no regard for Lucas's life, where his brother continuously bickered and fought with him. She had Emily. Lucas had no one and he wanted her so badly. And she wanted him. Right or wrong, she no longer cared. It was her decision and she made it.\n\n\"Decide,\" Lucas told her. \"We can't stay out in the open.\"\n\nOnly one question remained. Karina took a deep breath and closed the distance between her and Lucas. She lifted her face and looked into his green eyes and kissed him.\n\nFor a moment he stood still and then he kissed her back, his mouth eager and hungry for her. When they broke apart, Henry was staring at them.\n\n\"I am confused,\" Henry said.\n\n\"Well, I can't let you go back on your own,\" Karina said. \"All beat up and sad. Arthur might kill you somehow, or Daniel will bring the house down, or Henry, you might poison everyone with your cooking.\"\n\nEmily opened her eyes. \"Mommy!\"\n\n\"Hi, baby.\"\n\n\"Where are we?\"\n\n\"In Detroit. We had to make a stop here for a little while, but Lucas and Henry are taking us home with them now.\"\n\nThere had to be words to describe the look on Lucas's face, but she didn't know them. He probably didn't know them, either. He looked like he wasn't sure if he were surprised, relieved, happy, or mad.\n\n\"I believe there is a fast-food place three blocks north,\" Henry said. \"We could go there, use their phone, and drink coffee while we wait to get picked up. I could use some coffee.\"\n\n\"Can you make it?\" Lucas asked.\n\n\"If I faint, just leave me in the street.\"\n\nLucas slid his shoulder under Henry's arm.\n\n\"Thank you.\"\n\nThey started down the street.\n\n\"You don't own me anymore,\" Karina said quietly.\n\n\"Fine,\" Lucas said.\n\n\"And I will have my own room.\"\n\n\"Fine.\"\n\n\"And if you need to feed, you will ask me. Nicely.\"\n\nHe stopped and glared at her.\n\n\"Nicely,\" she told him.\n\n\"Fine.\"\n\n\"But all kidding aside, you will still cook, right?\" Henry asked. \"You said\u2014\"\n\n\"Yes, I will definitely cook.\"\n\n\"Oh, good,\" Henry said. \"I was afraid you would quit and we would have to eat Lucas's cooking.\"\n\n\"My cooking is fine,\" Lucas said.\n\nAhead, the familiar yellow-on-red sign rose on the corner.\n\n\"Are we going there, Mommy?\" Emily pointed at the sign.\n\n\"Yes.\"\n\n\"Do we have money to get ice cream?\"\n\n\"I have twenty dollars,\" Henry said. \"It's a little bloody, but they will take it.\"\n\n\"They'll take it,\" Lucas said grimly.\n\nKarina pictured Lucas, a little bloody and a little pissed off, breaking the McDonald's counter in half. Hopefully it wouldn't come to that.\n\n\"Don't worry, baby. We'll get you all the ice cream you want.\" Karina glanced back at the husk of the skyscraper. For a second she thought she saw her own self waving good-bye. Her new self smiled back. People who knew the old Karina would judge her, if they knew, but that didn't matter. She made her own choices now.\n\nShe put her hand on Lucas's arm. He bent it at his elbow, letting her fingers rest on his muscled forearm, and they walked side by side into the night.\nRead on for an exciting excerpt from the latest Kate Daniels novel\n\nMAGIC SHIFTS\n\n_Available now from Ace Books_\n\n\"Ilona Andrews's books are guaranteed good reads.\"\n\n\u2014Patricia Briggs, #1 _New York Times_ bestselling author\n\n\" _Magic Shifts_ is a perfectly balanced mixture of action, adventure, mystery, humor, and a delicious romance. Ilona Andrews only continues to excel in this wonderfully complex and magical series that revolves around a woman, her sword, and her battle to save her small piece of the world.\"\n\n\u2014Heroes and Heartbreakers\n\n#\n\nI rode through the night-drenched streets of Atlanta on a mammoth donkey. The donkey's name was Cuddles. She was ten feet tall, including the ears, and her black-and-white hide suggested she might have held up a Holstein cow in some dark alley and was now wearing her clothes. My own blood-spattered outfit suggested I'd had an interesting night. Most horses would've been nervous about letting a woman covered with that much blood on their back, but Cuddles didn't seem to mind. Either it didn't bother her or she was a pragmatist who knew where her carrots were coming from.\n\nThe city lay in front of me, deserted, quiet, and steeped in magic, unfurling its streets to the starlight like a moonlight flower. Magic ran deep through Atlanta tonight, like a current of some phantom river, slipping into the shadowy places and waking hungry things with needle-long teeth and glowing eyes. Anyone with a drop of common sense hid behind reinforced doors and barred windows after dark. Unfortunately for me, common sense was never among my virtues. As Cuddles quietly clopped her way down the streets, the sounds of her hoofbeats unnaturally loud, the night shadows watched us and I watched them back. _Let's play who can be a better killer. My sword and I love this game._\n\nNone of the monsters took the bait. It might have been because of me, but most likely it was because one of them was moving parallel to my route. They smelled him, and they hid and hoped he would pass them by.\n\nIt was almost midnight. I'd had a long day. My back ached, my clothes smelled of fetid blood, and a hot shower sounded heavenly. I had made two apple pies last night, and I was pretty sure that at least one piece would be left for me. I could have it tonight with my tea before I went to bed . . .\n\nAn annoying spark of magic ignited in my mind. A vampire. Oh goody.\n\nThe spark \"buzzed\" in my brain like an angry mosquito and moved closer. The Immortuus pathogen, the disease responsible for vampirism, killed the minds of its victims, leaving behind an empty shell driven by an all-consuming bloodlust. Left to its own devices, a vampire would hunt and slaughter, and when it ran out of things to kill, it would starve to death. This particular bloodsucker wasn't free to rampage, because its blank mind was held in a telepathic grip by a necromancer. The necromancer, or navigator as they were called, sat in a room far away, directing the vampire with his will as if it were a remote-controlled car. The navigator heard what the vampire heard, saw what the vampire saw, and if the vampire opened its mouth, the navigator's words would come out of it.\n\nMeeting a bloodsucker this far south meant it belonged to the People, an odd hybrid of a corporation and a research facility, whose personnel dedicated themselves to the study of the undead and making money on the side. The People avoided me like the plague. Two months ago they had figured out that the man behind their organization, the nearly immortal wizard with godlike powers and legendary magic, happened to be my father. They had some difficulty with that development. So the vampire wasn't for me.\n\nStill . . . I knew most of the People's patrol routes and this undead was definitely off-course. Where the hell was it going?\n\nNo. Not my circus, not my undead monkeys.\n\nI felt the vampire make a ninety-degree turn, heading straight for me.\n\nHome, shower, apple pie. Maybe if I said it like a prayer, it would work.\n\nThe distance between us shrank. _Home, shower . . ._\n\nAn undead leaped off the roof of the nearest two-story house and landed on the road next to me, gaunt, each shallow muscle visible under the thick hide, as if someone had crafted a human anatomy model out of steel wire and poured a paper-thin layer of rubber over it.\n\nDamn it.\n\nThe undead unhinged its mouth and Ghastek's dry voice came out. \"You're difficult to find, Kate.\"\n\nWell, well. The new head of the People's Atlanta office had come to see me personally. I'd curtsy but I was too tired to get off my donkey and the sword on my back would get in the way. \"I live in the suburbs and come home almost every night. My business phone number is in the book.\"\n\nThe vampire tilted its head, mimicking Ghastek's movements. \"You're still riding that monstrosity?\"\n\n\"Feel free to stomp him,\" I told Cuddles. \"I'll back you up.\"\n\nCuddles ignored me and the vampire, defiantly clopping past it. The bloodsucker turned smoothly and fell into step next to me. \"Where is your . . . significant other?\"\n\n\"He's around.\" He was never too far. \"Why, are you worried he'll find out about this romantic rendezvous?\"\n\nThe vampire froze for a second. \"What?\"\n\n\"You're meeting me in secret on a lonely street in the middle of the night . . .\"\n\nGhastek's voice was so sharp, if it were a knife, I would've been sliced to ribbons. \"I find your attempts at humor greatly distressing.\"\n\nHee-hee.\n\n\"I assure you, this is strictly business.\"\n\n\"Sure it is, sweet cheeks.\"\n\nThe vampire's eyes went wide. In an armored room deep in the bowels of the People's Casino, Ghastek was probably having a heart attack from the outrage.\n\n\"What are you doing out in my neck of the woods?\"\n\n\"Technically, the entire city is your neck of the woods,\" Ghastek said.\n\n\"True.\"\n\nTwo months ago my father had decided to dramatically claim Atlanta as his own domain. I tried to stop him in an equally dramatic fashion. He knew what he was doing, I didn't, and I ended up accidentally claiming the city in his stead. I was still fuzzy on how exactly the claiming worked, but apparently it meant that I had assumed guardianship of the city and the safety of Atlanta was now my responsibility. In theory, the magic of the city was supposed to nourish me and make my job easier, but I had no idea how exactly that worked. So far I didn't feel any different.\n\n\"But still, I heard you were promoted. Don't you have flunkies to do your bidding?\"\n\nThe vampire twisted his face into a hair-raising leer. Ghastek must've grimaced.\n\n\"I thought you would be happy,\" I said. \"You wanted to be the head honcho.\"\n\n\"Yes, but now I have to deal with you. _He_ spoke to me, personally.\"\n\nHe said \"he\" with the kind of reverence that could only mean Roland, my father.\n\n\"He believes that you may hesitate to kill me because of our shared experiences,\" Ghastek continued. \"Which makes me uniquely qualified to lead the People in your territory.\"\n\nShowing how freaked out I was about having a territory would severely tarnish my City Guardian cred. \"Aha.\"\n\n\"I'm supposed to cooperate with you. So, in the spirit of cooperation, I'm informing you that our patrols have sighted a large group of ghouls moving toward the city.\"\n\nGhouls were bad news. They followed the same general pattern of infection, incubation, and transformation as vampires and shapeshifters, but so far nobody had managed to figure out what actually turned them into ghouls. They were smart, supernaturally fast, and vicious, and they fed on human carrion. Unlike vampires, whom they somewhat resembled, ghouls retained some of their former personality and ability to reason, and they quickly figured out that the best way to get human carrion was to butcher a few people and leave the corpses to rot until they decomposed enough to be consumed. They traveled around in packs of three to five members and attacked isolated small settlements.\n\n\"How large is the group?\"\n\n\"Thirty plus,\" Ghastek said.\n\nThat wasn't a group. That was a damn horde. I had never heard of a ghoul pack that large.\n\n\"Which way are they coming?\"\n\n\"The old Lawrenceville Highway. You have about half an hour before they enter Northlake. Best of luck.\"\n\nThe vampire took off into the night.\n\nA few decades ago, Northlake would have been only a few minutes away. Now a labyrinth of ruins lay between me and that part of the city. Our world suffered from magic waves. They began without warning a few decades ago in a magic-induced apocalypse called the Shift. When magic flooded our world, it took no prisoners. It smothered electricity, dropped planes out of the sky, and toppled tall buildings. It eroded asphalt off the roads and birthed monsters. Then, without warning, the magic would vanish again and all of our gadgets and guns once again worked.\n\nThe city had shrunk post-Shift, after the first magic wave caused catastrophic destruction. People sought safety in numbers, and most of the suburbs along the old Lawrenceville Highway stood abandoned. There were some isolated communities in Tucker, but people settling there knew what to expect from the magic-fueled wilderness and it would be difficult for a pack of ghouls to take them down. Why bother, when less than five miles down the road Northlake marked the outer edge of the city? It was a densely populated area, filled with suburban houses and bordered by a few watchtowers along a ten-foot fence topped with razor wire. The guards could handle a few ghouls, but with thirty coming in fast, they would be overrun. The ghouls would scale the fence in seconds, slaughter the tower guards, and turn the place into a bloodbath.\n\nThere would be no assistance from the authorities. By the time I found a working phone and convinced the Paranormal Activity Division that a pack of ghouls six times the typical size was moving toward Atlanta, Northlake would be an all-you-can-devour ghoul buffet.\n\nAbove me a huge dark shape dashed along the rooftops and leaped, clearing the gap between two buildings. The starlight caught it for a heart-stopping second, illuminating the powerfully muscled torso, four massive legs, and the dark gray mane. The hair on the back of my neck stood up. It was as if the night itself had opened its jaws and spat out a prehistoric creature, something born of human fear and hungry animal growls echoing in the dark. I only saw him for a moment, but the image imprinted itself in my mind as if chiseled in stone. My body instantly recognized that he was predator and I was prey. I'd known him for three years now, and the instinctual response still hit every single time.\n\nThe beast landed, turned north, and vanished into the night, heading toward Northlake.\n\nInstead of running away as fast as I could like any sane person would do, I nudged Cuddles, hurrying her until she broke into a gallop. One doesn't let her fianc\u00e9 fight a horde of ghouls by himself. Some things were just not done.\n\n* * *\n\nThe empty expanse of the Lawrenceville Highway spread before me. The road cut through a shallow hill here, and stone walls held back the slope on both sides. I parked myself at the mouth of the hill, just before it melted into a vast, completely flat field. As good a place as any to make a stand.\n\nI stretched my neck slowly, one side, then the other. I'd left Cuddles tethered to a tree half a mile back. Ghouls normally would have no interest in her, but she smelled like me and one of them might try to rip her neck open just out of spite.\n\nThe moon rolled out of the clouds, illuminating the fields. The night sky was impossibly high, the stars like diamonds in its icy depth. A cold breeze came, tugging at my clothes and my braid. It was the beginning of March, and the onset of spring was sudden and warm, but at night winter still bared its fangs.\n\nThe last time I was this far from the city, I had been the Consort of the Pack, the largest shapeshifter organization in the South. That was behind me now. Thirty ghouls would be rough without backup. Lucky for me, I had the best backup in the city.\n\nWhen I had claimed Atlanta, the claiming had created a boundary. I felt it fifty feet in front of me, an invisible line of demarcation. I should've gone to inspect the boundary sooner, but I'd been busy trying to separate myself from the Pack and setting up the new house and working my ass off, because eventually our savings would run out . . . But pretending that the claiming hadn't happened did me no favors.\n\nSomething moved in the distance. I focused on it. The movement continued, the horizon rippling slightly. A few breaths and the shiver broke into individual shapes running in an odd loping gait, leaning on their arms like gorillas but never fully shifting into a quadrupedal run.\n\nWow, that's a lot of ghouls.\n\nShowtime. I reached for the sword on my back and pulled Sarrat out of its sheath. The opaque, almost white blade caught the weak moonlight. Single-handed, with a razor-sharp edge, the blade was a cross between a straight sword and a traditional saber, with a slight curve that made it excellent for both slashing and thrusting. Sarrat was fast, light, flexible, and razor-sharp, and it was about to get a hell of a workout.\n\nThe distorted shapes kept coming. Knowing there were thirty ghouls was one thing. Seeing them gallop toward you was completely different. A spark of instinctual fear shot through me, turning the world sharper, and melted into calm awareness.\n\nThin tendrils of vapor rose from Sarrat's surface in response. I turned the saber, warming up my wrist.\n\nThe ghoul horde drew closer. How the hell did I get myself into these things?\n\nI walked toward them, sword in my hand, point down. I had few social skills, but intimidation I did well.\n\nThe ghouls saw me. The front ranks slowed, but the back rows were still running at full speed. The mass of ghouls compacted like a wave breaking against a rock and finally screeched to a halt just before the boundary. We stopped, them on one side of the invisible magic divide, me on the other.\n\nThey were lean and muscular, with disproportionately powerful arms and long, spadelike hands, each finger tipped by a short curved claw. Bony protrusions, like short knobby horns, thrust through their skin at random spots on their back and shoulders. The horns were a defensive mechanism. If someone tried to pull the ghoul out of its burrow, the horns would wedge against dirt. A werewolf armed with superhuman strength would have a difficult time plucking a ghoul out of the ground. I'd seen the horns grow as long as four inches, but most of the ones decorating this crowd barely reached half an inch. Their skin was dark gray on the chest, neck, and faces, the kind of gray that was most often found on military urban camouflage. Small splotches of muddy brown dotted their backs and their shoulders. If not for the watery yellow glow of their irises, they would've blended into the road completely.\n\nNone of them were lame, starved, or weak. The odds weren't in my favor. I had to think of a strategy and fast.\n\nThe ghouls peered at me with oddly slanted eyes, the inner corners dipping much lower than the outer ones.\n\nI waited. The moment you start speaking, you become less scary, and I had no intention of being less scary. The ghouls were sentient, which meant they could feel fear, and I needed every bit of advantage I could scrounge up.\n\nA large ghoul shouldered its way to the front of the pack. Well-fed, with a defined powerful body, he crouched in front of me. If he stood upright, he would be close to seven feet tall. At least two hundred pounds, all of it hard muscle and sharp claws. The brown pattern on his back was almost nonexistent. Instead, long alternating stripes of paler and darker gray slid down his flanks.\n\nThe ghoul rocked forward. His face touched the boundary and he pulled back and stared at me. He wasn't sure what he was sensing, but he knew that the boundary and I were somehow connected.\n\nSome ghouls were scavengers. They were harmless and sometimes even gainfully employed. We lived in an unsafe world. Too often bodies couldn't be recovered because they were under debris or the scene was too grisly for the next of kin to identify the remains. Putting the bodies into a mass grave was a recipe for disaster. Human bodies emanated magic even after death and there was no telling what the next magic wave would do to that mass grave. Most often the remains were cremated, but occasionally the authorities would bring in ghouls to clean the site. It was cheaper and faster.\n\nI'd bet my arm these ghouls weren't licensed scavenge workers, but I had to be absolutely sure.\n\nThe ghoul stared at me. I gave him my best psychotic smile.\n\nThe ghoul blinked his yellowish eyes, tensed like a dog about to charge, and opened his mouth, stretching his lips in a slow deliberate grin. _That's right, show me your big teeth, pretty boy._\n\nA row of thick sharp teeth decorated the front of his jaw. Toward the back, the teeth thinned out, becoming more bladelike, with serrated edges. _Got you._\n\nThe ghoul unhinged his jaw. A rough raspy voice came out. \"Who are you?\"\n\n\"Turn around now and you'll live.\"\n\nHe clamped his mouth shut. Apparently this wasn't the answer he'd expected. Kate Daniels, master of surprises. _Don't worry, I'm just getting started._\n\n\"We're a licensed cleanup crew,\" the leader ghoul said.\n\n\"No.\"\n\nHalf a mile behind the ghouls, a dark shape moved through the field, so silent, for a second I thought I was seeing things. My mind refused to accept that a creature that large could be so quiet. _Hi, honey._\n\nThe ghouls didn't notice him. They were conditioned to pay attention to human flesh and I was standing right in front of them, providing a nice convenient target.\n\nThe leader ghoul turned, displaying a tattoo on his left shoulder.\n\nColumbia, SC\n\n014\n\nLocation of license and license number. He thought I was born yesterday.\n\n\"We're a peaceful group,\" the ghoul continued.\n\n\"Sure you are. You're just running into the city to borrow a cup of sugar and invite people to your church.\"\n\n\"You're interfering with official municipal business. This is discrimination.\"\n\nThe dark shadow emerged onto the road and started toward us. I'd need to buy him some time to get within striking range.\n\nI looked at the ghoul. \"Do you know what is so special about ghouls? You have an unrivaled adaptability. Your bodies change to match their environment faster than ninety-nine percent of anything we've seen in nature.\"\n\nMy favorite monster crept closer on huge paws.\n\nI raised my saber and rested the opaque blade on my shoulder. Faint tendrils of vapor escaped from Sarrat's surface. The sword sensed trouble and was eager for it.\n\n\"Let me tell you what I see. Your color has changed from brown to gray, because you no longer have to blend in with the dirt. Your stripes tell me you spend a lot of time moving through the forest. Your horns are short, because you no longer hide in your burrows.\"\n\nThe ghouls shifted closer. Their eyes glowed brighter. They didn't like where this was going.\n\n\"Your claws aren't long and straight to help you dig. They are curved and sharp to rend flesh.\"\n\nThe ghouls bared their teeth at me. They were a hair away from violence. I had to keep talking.\n\n\"Your pretty teeth have changed, too. They're no longer narrow and serrated. They are thick, strong, and sharp. The kind of teeth you get when you need to hold struggling prey in your mouth. And your fancy tattoo is two years out of date. All ghouls' licenses in Columbia now have the year tattooed under the license number.\"\n\nThe ghouls had gone completely silent, their eyes like dozens of tiny shiny moons all focused on me. Just a few more seconds . . .\n\n\"Kill her,\" another ghoul chimed in. \"We have to hurry.\"\n\n\"Kill her. He's waiting,\" a third voice chimed in.\n\n\"Kill her. Kill her.\"\n\nThey seemed awfully desperate. Something weird was going on.\n\n\"Who is waiting?\" I asked.\n\n\"Shut up!\" the leading ghoul snarled.\n\nI leaned forward and gave the leader ghoul my hard stare. \"You look plump. You've been raiding the countryside and growing fat from gorging yourself on the people you've murdered. I gave you a chance to leave. Now it's too late. Pay attention to this moment. Look at the stars. Breathe in the cold air. This is your last night. These are the last breaths you take. I will kill every one of you.\"\n\nThe leader ghoul snarled, dropping all pretense. \"You and what army?\"\n\nI began pulling magic to me. This would hurt. This always hurt. \"That's the great thing about werelions. You don't need an army. You just need one.\"\n\nThe ghoul twisted his face. \"You're not a werelion, meat.\"\n\n\"I'm not.\" I nodded behind them. \"He is.\"\n\nThe leader ghoul spun around.\n\nTwo gold eyes stared at him from the darkness. The enormous lionlike beast opened his mouth and roared. Until I met him, I had never heard an actual lion roar. It sounded like thunder. Deafening, ravenous heart-dropping thunder that severed some vital link between logic and control of your body deep inside your brain. It was a blast of sound so powerful, I had seen hundreds of shapeshifters cringe when they heard it. A wolf howl heard in the middle of the night raised the hair on the back of your neck, but a lion's roar punched through all of your training and reason straight to the secret place hidden deep inside that screamed at you to freeze.\n\nThe ghouls stopped, motionless.\n\nI opened my mouth and spat a power word. _\"Osanda.\"_ Kneel.\n\nPower words came from a long-forgotten age, so ancient that they commanded raw magic. Few people knew about them and even fewer could use them, because to learn a power word, you had to own it. You made it yours or it killed you. I knew a handful of power words, far more than anyone else I'd met, but using even one came with a heavy price tag. For my father, the power words were a language, one he spoke fluidly and without repercussions. They didn't hurt him, but I always paid a price.\n\nThe magic ripped out of me. I braced for the familiar twist of agony. The backlash bit at me, tearing through my insides, but this time something must've blunted its teeth, because it didn't hurt nearly as much as I remembered.\n\nThe magic smashed into the petrified ghouls. Their knees and elbows crunched in unison and they crashed to the asphalt. It would buy me at least ten seconds. If the magic wave had been stronger, I would've broken their bones.\n\nI swung my sword. Sarrat met a ghoul's bony neck and sliced through cartilage and thick hide like butter. Before its dead body fell to the ground, I thrust my blade into the chest of the second ghoul and felt Sarrat's tip pierce the tight ball of its heart.\n\nThe lion's body boiled, snapping upright. Bones thrust upward; powerful muscle spiraled up the new skeleton. A blink and a new monster lunged forward, a nightmarish mix of man and lion, seven and a half feet tall, with steel-hard muscle sheathed in gray fur and curved, terrible claws. A ghoul leaped at him. He grabbed the creature by its throat and shook it, as if he were snapping a wet towel. A sickening snap echoed through the night and the ghoul went limp.\n\nI carved the third ghoul into two separate pieces and sliced the fourth one's throat.\n\nThe ghouls woke up. They swarmed us. The leonine beast swung his claws and disemboweled a ghoul with a precise swipe. Intestines rained onto the road. The bitter stench of ghoul blood mixed with the unmistakable sour reek of a gut wound singed my nostrils.\n\nClaws ripped through my clothes, drawing agonizing scalding-hot lines across my back. _You want to play? Fine._ I needed a workout anyway.\n\nMy saber became a razor-sharp wall. It cut, sliced, and pierced, ripping flesh and hissing as the ghoul blood that washed it boiled from its magic. I moved fast, sidestepping claws and blocking teeth. Another fiery gash stung my back. A ghoul clamped onto my boot and I ripped my leg free and stomped his skull into the pavement. A welcome heat spread through me, turning my muscles flexible and pliant. The world turned crystal clear. Time stretched, helping me. The ghouls lunged, but I was faster. They raked at me with their claws, but my blade found them first. I savored it all, every second of the fight, every drop of blood flying past me, every moment of resistance when Sarrat caught my target on its edge.\n\nThis was what I was raised and trained for. For better or worse, I was a killer. This was my calling, and I made no excuses for it.\n\nA ghoul loomed before me. I sliced it down in a classic overhand stroke. It fell. Nobody took its place. I pivoted on my toes, looking for a fight. To the left the werelion tossed a broken body to the ground and turned to me. A single ghoul hugged the ground, caught between us.\n\n\"Alive,\" the werelion snarled.\n\nWay ahead of you. Let's find out who the mysterious \"he\" is. I started toward the ghoul, sword in hand.\n\nIt shivered, looked right, then left, looked at the werelion, then at me. _That's right. You're trapped and not going anywhere._ If it ran, we would chase it down.\n\nThe ghoul reared, jerked its clawed hands to its throat, and sliced it open. Blood gushed. The ghoul gurgled and collapsed on the ground. The light went out of its eyes.\n\nWell, that was a hell of a thing.\n\nThe lion monster opened his mouth and a human voice came out, his diction perfect. \"Hey, baby.\"\n\n\"Hey, honey.\" I pulled a piece of cloth out of my pocket and carefully wiped down Sarrat's blade.\n\nCurran stepped over to me and put his arm around my shoulders, pulling me close. I leaned against him, feeling the hard muscle of his torso against my side. We surveyed the road strewn with broken bodies.\n\nThe adrenaline faded slowly. The colors turned less vivid. One by one the cuts and gashes made themselves known: my back burned, my left hip hurt too, and my left shoulder ached. I'd probably wake up with a spectacular bruise tomorrow.\n\nWe'd survived another one. We'd get to go home and keep on living.\n\n\"What the hell was all this about?\" Curran asked me.\n\n\"I have no idea. They don't typically gather into large packs. The biggest marauder pack ever sighted had seven ghouls, and that was considered a fluke. They are solitary and territorial. They only band together for protection, but clearly someone was waiting for them. Do you think Ghastek is connected to this?\"\n\nCurran grimaced. \"It's not like him. Ghastek only moves when he has something to gain. Having us kill ghouls doesn't help him in any way. He knows what we can do. He had to realize we'd go through them.\"\n\nCurran was right. Ghastek had to know we'd dispatch the ghouls. He wouldn't have used us to do his dirty work either. For all of his faults, Ghastek was a premier navigator, a Master of the Dead, and he loved his job. If he wanted the ghouls dead, he would've sliced this group to pieces with a couple of vampires or he would've used this opportunity as a training exercise for his journeymen.\n\n\"This isn't making any sense to me,\" I said, pulling traces of my blood toward me. It slid and rolled in tiny drops, forming a small puddle on the pavement. I pushed it to the side, solidified it, and stomped on it. It shattered under my foot into inert powder. Blood retained its magic even when separated from the body. For as long as I could remember, I had to guard my blood because if it were examined, it would point to my father like an arrow. There was a time where I had to set any trace of my blood on fire, but now it obeyed me. I couldn't decide if it made me a better fighter or just a worse abomination. \"They seemed desperate. Driven, almost, as if they had some sort of goal to get to.\"\n\n\"We'll figure it out,\" Curran told me. \"It's almost midnight. I say we go home, get cleaned up, and climb into bed.\"\n\n\"Sound like a plan.\"\n\n\"Hey, is there any of that apple pie left?\" Curran asked.\n\n\"I think so.\"\n\n\"Oh good. Let's go home, baby.\"\n\nOur home. It still hit me like a punch, even after months of us being together\u2014he was right there, waiting for me. If something attacked me, he'd kill it. If I needed help, he would help me. He loved me and I loved him back. I was no longer alone.\n\nWe were walking to my donkey when he said, \"Sweet cheeks?\"\n\n\"I couldn't help it. Ghastek's got a stick up his ass the size of a railroad track. Did you see the look on the vampire's face? He looked constipated.\"\n\nCurran laughed. We found Cuddles and went home.\nKeep reading for an excerpt from the first in the romantic urban fantasy series of the Edge\n\nON THE EDGE\n\nAvailable now from Ace Books\n\n\"A fascinating world combined with pulse-pounding action and white-hot romance makes On the Edge a winner!\n\n\u2014Jeaniene Frost, New York Times bestselling author\n\n\"A great, fun romance, an offbeat mix of old-fashioned rural magics, contemporary life (complete with Wal-Mart and comic book shops), and magic sword-wielding warriors.\"\n\n\u2014Locus\n\n#\n\n\"ROSIE!\" Grandpa's bellow shook the foundation of the house.\n\n\"Why me?\" Rose wiped the dish-soap suds from her hands with a kitchen towel, swiped the crossbow from the hook, and stomped onto the porch.\n\n\"Roooosie!\"\n\nShe kicked the screen door open. He towered in the yard, a huge, shaggy bear of a man, deranged eyes opened wide, tangled beard caked with blood and quivering grayish shreds. She leveled the crossbow at him. Drunk as hell again.\n\n\"What is it?\"\n\n\"I want to go to the pub. I want a pint.\" His voice slipped into a whine. \"Gimme some money!\"\n\n\"No.\"\n\nHe hissed at her, swaying unsteadily on his feet. \"Rosie! This is your last chance to give me a dollar!\"\n\nShe sighed and shot him. The bolt bit between the eyes, and Grandpa toppled onto his back like a log. His legs drummed the ground.\n\nRose rested the butt of her crossbow on her hip. \"All right, come out.\"\n\nThe two boys slipped from behind the huge oak spreading its branches over the yard. Both were filthy with reddish mud, sap, and the other unidentifiable substances an eight-and a ten-year-old could find in the Wood. A jagged scratch decorated Georgie's neck, and brown pine straw stuck out of his blond hair. Red welts marked the skin between Jack's knuckles. He saw her looking at his hands. His eyes got big, amber irises flaring yellow, and he hid his fists behind his back.\n\n\"How many times do I have to say it: don't touch the ward stones. Look at Grandpa Cletus! He's been eating dog brains again, and now he's drunk. It will take me half an hour to hose him off.\"\n\n\"We miss him,\" Georgie said.\n\nShe sighed. \"I miss him, too. But he's no good to anybody drunk. Come on, you two, let's take him back to his shed. Help me get the legs.\"\n\nTogether they dragged Grandpa's inert form back to the shed at the edge of the clearing and dumped him on his sawdust. Rose uncoiled the metal chain from the corner, pulled it across the shed, locked the collar on Grandpa's neck, and peeled back his left eyelid to check the pupil. No red yet. Good shot\u2014he would be out for hours. Rose put her foot on his chest, grasped the bolt, and pulled it out with a sharp tug. She still remembered Grandpa Cletus as he was, a tall, dapper man, uncanny with his rapier, his voice flavored with a light Scottish brogue. Even as old as he was, he would still win against Dad one out of three times in a sword fight. Now he was this . . . this thing. She sighed. It hurt to look at him, but there was nothing to be done about it. As long as Georgie lived, so did Grandpa Cletus.\n\nThe boys brought the hose. She turned it on, set the sprayer on jet, and leveled the stream at Grandpa until all the blood and dog meat were gone. She had never quite figured out how \"going down to the pub\" equaled chasing stray dogs and eating their brains, but when Grandpa got out of his ward circle, no mutt was safe. By the time she was done washing him, the hole in his forehead had closed. When Georgie raised things from the dead, he didn't just give them life. He made them almost indestructible.\n\nRose stepped out of the shed, locked the door behind her, and dragged the hose back to the porch. Her skin prickled as she crossed the invisible boundary: the kids must've put the ward stones back. She squinted at the grass. There they were, a line of small, seemingly ordinary rocks, spaced three, four feet from each other. Each rock held a small magic charge. Together they created an enchanted barrier, strong enough to keep Grandpa in the shed if he broke the chain again.\n\nRose waved the boys to the side and raised the hose. \"Your turn.\"\n\nThey flinched at the cold water. She washed them off methodically, from top to bottom. As the mud melted from Jack's feet, she saw a two-inch rip in his Skechers. Rose dropped the hose.\n\n\"Jack!\"\n\nHe cringed.\n\n\"Those are forty-five-dollar shoes!\"\n\n\"I'm sorry,\" he whispered.\n\n\"Tomorrow is the first school day! What were you doing?\"\n\n\"He was climbing up the pines to get at the leech birds,\" Georgie said.\n\nShe glared. \"Georgie! Thirty-minute time-out tonight for snitching.\"\n\nGeorgie bit his lip.\n\nRose stared at Jack. \"Is that true? You were chasing the leech birds?\"\n\n\"I can't help it. Their tails are so flittery . . .\"\n\nShe wanted to smack him. It was true, he couldn't help it\u2014it wasn't his fault he was born as a cat\u2014but those were brand-new shoes she had bought him for school. Shoes for which she had painstakingly tweaked their budget, scrimping every penny, so he wouldn't have to wear Georgie's old beat-upuncanny with his rapier, his voice flavored with a light Scottish brogue. Even as old as he was, he would still win against Dad one out of three times in a sword fight. Now he was this . . . this thing. She sighed. It hurt to look at him, but there was nothing to be done about it. As long as Georgie lived, so did Grandpa Cletus.\n\nThe boys brought the hose. She turned it on, set the sprayer on jet, and leveled the stream at Grandpa until all the blood and dog meat were gone. She had never quite figured out how \"going down to the pub\" equaled chasing stray dogs and eating their brains, but when Grandpa got out of his ward circle, no mutt was safe. By the time she was done washing him, the hole in his forehead had closed. When Georgie raised things from the dead, he didn't just give them life. He made them almost indestructible.\n\nRose stepped out of the shed, locked the door behind her, and dragged the hose back to the porch. Her skin prickled as she crossed the invisible boundary: the kids must've put the ward stones back. She squinted at the grass. There they were, a line of small, seemingly ordinary rocks, spaced three, four feet from each other. Each rock held a small magic charge. Together they created an enchanted barrier, strong enough to keep Grandpa in the shed if he broke the chain again.\n\nRose waved the boys to the side and raised the hose. \"Your turn.\"\n\nThey flinched at the cold water. She washed them off methodically, from top to bottom. As the mud melted from Jack's feet, she saw a two-inch rip in his Skechers. Rose dropped the hose.\n\n\"Jack!\"\n\nHe cringed.\n\n\"Those are forty-five-dollar shoes!\"\n\n\"I'm sorry,\" he whispered.\n\n\"Tomorrow is the first school day! What were you doing?\"\n\n\"He was climbing up the pines to get at the leech birds,\" Georgie said.\n\nShe glared. \"Georgie! Thirty-minute time-out tonight for snitching.\"\n\nGeorgie bit his lip.\n\nRose stared at Jack. \"Is that true? You were chasing the leech birds?\"\n\n\"I can't help it. Their tails are so flittery . . .\"\n\nShe wanted to smack him. It was true, he couldn't help it\u2014it wasn't his fault he was born as a cat\u2014but those were brand-new shoes she had bought him for school. Shoes for which she had painstakingly tweaked their budget, scrimping every penny, so he wouldn't have to wear Georgie's old beat-up sneakers, so he could look just as nice as all the other second graders. It just hurt.\n\nJack's face pinched into a rigid white mask\u2014he was about to cry.\n\nA small spark of power tugged on her. \"Georgie, stop trying to resurrect the shoes. They were never alive in the first place.\"\n\nThe spark died.\n\nAn odd desperation claimed her, her pain shifting into a sort of numbness. Pressure built in her chest. She was so sick of it, sick of counting every dollar, sick of rationing everything, sick to death of it all. She had to go and get Jack a new pair of shoes. Not for Jack's sake, but for the sake of her own sanity. Rose had no clue how she would make up the money, but she knew she had to buy him a new pair of shoes right now, or she would explode.\n\n\"Jack, do you remember what will happen if a leech bird bites you?\"\n\n\"I'll turn into one?\"\n\n\"Yes. You have to stop chasing the birds.\"\n\nHe hung his head. \"Am I punished?\"\n\n\"Yes. I'm too mad to punish you right now. We'll talk about it when we get home. Go brush your teeth, comb your hair, put on dry clothes, and get the guns. We're going to Wal-Mart.\"\n\n* * *\n\nTHE old Ford truck bounced on the bumps in the dirt road. The rifles clanged on the floor. Georgie put his feet down to steady them without being asked.\n\nRose sighed. Here, in the Edge, she could protect them well enough. But they were about to pass from the Edge into another world, and their magic would die in the crossing. The two hunting rifles on the floor would be their only defense. Rose felt a pang of guilt. If it wasn't for her, they wouldn't need the rifles. God, she didn't want to be jumped again. Not with her brothers in the car.\n\nThey lived between worlds: on one side lay the Weird and the other the Broken. Two dimensions, existing side by side, like mirror images of each other. In the place where the dimensions \"touched,\" they intersected slightly, forming a narrow ribbon of land that belonged to both of them\u2014the Edge. In the Weird, magic pooled deeply; in the Edge it was a shallow trickle. But in the Broken, no magic shielded them at all.\n\nRose eyed the Wood hugging the road, its massive trees crowding the narrow ribbon of packed dirt. She drove this way every day to her job in the Broken, but today the shadows between the gnarled trunks filled her with anxiety. \"Let's play the 'You Can't' game,\" she said to stave off the rising dread. \"Georgie, you go first.\"\n\n\"He went first the last time!\" Jack's eyes shone with amber.\n\n\"Nyaha!\"\n\n\"Yaha!\"\n\n\"Georgie goes first,\" she repeated.\n\n\"Past the boundary, you can't raise dead things,\" Georgie said.\n\n\"Past the boundary, you can't grow fur and claws,\" Jack said.\n\nThey always played the game when driving through to the Broken. It was a good reminder to the boys of what they could and could not do, and it worked much better than any lecture. Very few people in the Broken knew of the Edge or the Weird, and it was safer for everyone involved to keep it that way. Experience had taught her that trying to explain the existence of magic to a person in the Broken would do no good. It wouldn't get you committed into a mental institution, but it did land you into the kooky idiot category and made people give you a wide berth during lunch hour.\n\nFor most people of the Broken, there was no Broken, no Edge, and no Weird. They lived in the United States of America, on the continent of North America, on the planet Earth\u2014and that was that. For their part, most people in the Weird couldn't see the boundary either. It took a special kind of person to find it, and the kids needed to remember that.\n\nGeorgie touched her hand. It was her turn. \"Past the boundary, you can't hide behind a ward stone.\" She glanced at them, but they kept going, oblivious to her fears.\n\nThe road lay deserted. Few Edgers drove up this way this time of the evening. Rose accelerated, eager to get the trip over with and be back to the safety of the house.\n\n\"Past the boundary, you can't find lost things,\" Georgie said.\n\n\"Past the boundary, you can't see in the dark.\" Jack grinned.\n\n\"Past the boundary, you can't flash,\" Rose said.\n\nThe flash was her greatest weapon. Most Edgers had their own specific talents: some prophesied, some cured tooth-aches, some raised the dead like Georgie. Some cursed like Rose and her grandmother. But flashing could be learned by anyone with a drop of magic. It wasn't a matter of talent but of practice. You took a hold of the magic inside you and channeled it from your body in a controlled burst that looked like a whip or a ribbon of lightning. If you had magic and patience, you could learn to flash, and the lighter the color of your flash, the hotter and more potent it was. A powerful bright flash was a terrible weapon. It could slice through a body like a hot knife through butter. \"Georgie, you go first.\"\n\n\"He went first the last time!\" Jack's eyes shone with amber.\n\n\"Nyaha!\"\n\n\"Yaha!\"\n\n\"Georgie goes first,\" she repeated.\n\n\"Past the boundary, you can't raise dead things,\" Georgie said.\n\n\"Past the boundary, you can't grow fur and claws,\" Jack said.\n\nThey always played the game when driving through to the Broken. It was a good reminder to the boys of what they could and could not do, and it worked much better than any lecture. Very few people in the Broken knew of the Edge or the Weird, and it was safer for everyone involved to keep it that way. Experience had taught her that trying to explain the existence of magic to a person in the Broken would do no good. It wouldn't get you committed into a mental institution, but it did land you into the kooky idiot category and made people give you a wide berth during lunch hour.\n\nFor most people of the Broken, there was no Broken, no Edge, and no Weird. They lived in the United States of America, on the continent of North America, on the planet Earth\u2014and that was that. For their part, most people in the Weird couldn't see the boundary either. It took a special kind of person to find it, and the kids needed to remember that.\n\nGeorgie touched her hand. It was her turn. \"Past the boundary, you can't hide behind a ward stone.\" She glanced at them, but they kept going, oblivious to her fears.\n\nThe road lay deserted. Few Edgers drove up this way this time of the evening. Rose accelerated, eager to get the trip over with and be back to the safety of the house.\n\n\"Past the boundary, you can't find lost things,\" Georgie said.\n\n\"Past the boundary, you can't see in the dark.\" Jack grinned.\n\n\"Past the boundary, you can't flash,\" Rose said.\n\nThe flash was her greatest weapon. Most Edgers had their own specific talents: some prophesied, some cured tooth-aches, some raised the dead like Georgie. Some cursed like Rose and her grandmother. But flashing could be learned by anyone with a drop of magic. It wasn't a matter of talent but of practice. You took a hold of the magic inside you and channeled it from your body in a controlled burst that looked like a whip or a ribbon of lightning. If you had magic and patience, you could learn to flash, and the lighter the color of your flash, the hotter and more potent it was. A powerful bright flash was a terrible weapon. It could slice through a body like a hot knife through butter. Most Edgers never could get their flash bright enough to kill or injure anything with it. They were mongrels, living in a place of diluted magic, and most flashed red and dark orange. Some lucky few managed green or blue.\n\nIt was her flash that had started all of their trouble.\n\nNo, Rose reflected, they'd had plenty of trouble before her. Draytons were always unlucky. Too smart and too twisted for their own good. Grandpa was a pirate and a rover. Dad was a gold digger. Grandma was stubborn like a goat and always thought she knew better than anyone else. Mom was a tramp. But all those problems didn't affect anyone but the individual Draytons. When Rose flashed white at the Graduation Fair, she focused the attention of countless Edge families squarely on their little clan. Even now, even with the rifles on the floor, she didn't regret it. She felt guilty about it, she wished things hadn't gone the way they did, but given a chance, she would do it again.\n\nAhead the road curved. Rose took the turn a bit too fast. The truck's springs creaked.\n\nA man stood in the road, like a gray smudge against the encroaching twilight.\n\nShe slammed on the brakes. The Ford skidded in a screech on the hard, dry dirt of the road. She caught a glimpse of long pale hair and piercing green eyes staring straight at her.\n\nThe truck hurtled at him. She couldn't stop it.\n\nThe man leapt straight up. Feet in dark gray boots landed on the hood of the truck with a thud and vanished. The man vaulted over the roof to the side and disappeared into the trees.\n\nThe truck slid to a stop. Rose gulped the air. Her heart fluttered in her chest. Her fingertips tingled, and she tasted bitterness on her tongue.\n\nShe stabbed the seat belt release button, threw the door open, and jumped out onto the road. \"Are you hurt?\"\n\nThe Wood lay quiet.\n\n\"Hello?\"\n\nNo answer. The man was gone.\n\n\"Rose, who was that?\" Georgie's eyes were the size of small saucers.\n\n\"I don't know.\" Relief flooded her. She hadn't hit him. She got scared out of her wits, but she hadn't hit him. Everybody was fine. Nobody was hurt. Everybody was fine . . .\n\n\"Did you see the swords?\" Jack asked.\n\n\"What swords?\" All she'd seen were the blond hair, green eyes, and some kind of cloak. She couldn't even recall his face\u2014just a pale smudge.\n\n\"He had a sword,\" Georgie said. \"On his back.\" \"Two swords,\" Jack corrected. \"One on the back and one on his belt.\"\n\nSome of the older locals liked to play with swords, but none of them had long blond hair. And none of them had eyes like that. Most people facing a truck head-on would be scared. He stared her down as if she had insulted him by nearly running him over. Like he was some sort of king of the road.\n\nStrangers were never good in the Edge. It wasn't wise to linger.\n\nJack sniffed the air, wrinkling his nose the way he did when he looked for a scent trail. \"Let's find him.\"\n\n\"Let's not.\"\n\n\"Rose . . .\"\n\n\"You're on thin ice already.\" She climbed into the truck and shut the door. \"We're not chasing after some knucklehead who thinks he's too important to walk on the shoulder.\" She snorted, trying to get her heart rate under control.\n\nGeorgie opened his mouth.\n\n\"Not another word.\"\n\nA couple of minutes later, they reached the boundary, the point where the Edge ended and the Broken began. Rose always recognized the precise moment when she passed into the Broken. First, anxiety stabbed right through her chest, followed by an instant of intense vertigo, and then pain. It was as if the shiver of magic, the warm spark that existed somewhere inside her, died during the crossing. The pain lasted only a blink, but she always dreaded it. It left her feeling incomplete. Broken. That's how the name for the magic-less dimension had come about.\n\nThere was an identical boundary on the opposite end of the Edge, the one that guarded the passage to the Weird. She never tried to cross it. She wasn't sure her magic would be strong enough for her to survive.\n\nThey entered the Broken without any trouble. The Wood ended with the Edge. Mundane Georgia oaks and pines replaced the ancient dark trees. The dirt became pavement.\n\nThe narrow two-lane road brought them past the twin gas stations to the parkway. Rose checked the parkway for oncoming traffic, took a right, and headed toward the town of Pine Barren.\n\nAbove them an airplane thundered, fixing to land at the Savannah airport only a couple of miles away. The woods gave way to half-finished shopping plazas and construction equipment, scattered among heaps of red Georgia mud. Ponds and streams interrupted the landscape\u2014with the coast only forty minutes away, every hole in the ground sooner or later filled up with water. They passed hotels, Comfort Inn, Knights Inn, Marriott, Embassy Suites,\n\nSome of the older locals liked to play with swords, but none of them had long blond hair. And none of them had eyes like that. Most people facing a truck head-on would be scared. He stared her down as if she had insulted him by nearly running him over. Like he was some sort of king of the road.\n\nStrangers were never good in the Edge. It wasn't wise to linger.\n\nJack sniffed the air, wrinkling his nose the way he did when he looked for a scent trail. \"Let's find him.\"\n\n\"Let's not.\"\n\n\"Rose . . .\"\n\n\"You're on thin ice already.\" She climbed into the truck and shut the door. \"We're not chasing after some knucklehead who thinks he's too important to walk on the shoulder.\" She snorted, trying to get her heart rate under control.\n\nGeorgie opened his mouth.\n\n\"Not another word.\"\n\nA couple of minutes later, they reached the boundary, the point where the Edge ended and the Broken began. Rose always recognized the precise moment when she passed into the Broken. First, anxiety stabbed right through her chest, followed by an instant of intense vertigo, and then pain. It was as if the shiver of magic, the warm spark that existed somewhere inside her, died during the crossing. The pain lasted only a blink, but she always dreaded it. It left her feeling incomplete. Broken. That's how the name for the magic-less dimension had come about.\n\nThere was an identical boundary on the opposite end of the Edge, the one that guarded the passage to the Weird. She never tried to cross it. She wasn't sure her magic would be strong enough for her to survive.\n\nThey entered the Broken without any trouble. The Wood ended with the Edge. Mundane Georgia oaks and pines replaced the ancient dark trees. The dirt became pavement.\n\nThe narrow two-lane road brought them past the twin gas stations to the parkway. Rose checked the parkway for oncoming traffic, took a right, and headed toward the town of Pine Barren.\n\nAbove them an airplane thundered, fixing to land at the Savannah airport only a couple of miles away. The woods gave way to half-finished shopping plazas and construction equipment, scattered among heaps of red Georgia mud. Ponds and streams interrupted the landscape\u2014with the coast only forty minutes away, every hole in the ground sooner or later filled up with water. They passed hotels, Comfort Inn, Knights Inn, Marriott, Embassy Suites, stopped at a light, crossed the overpass, and finally turned into a busy Wal-Mart parking lot.\n\nRose parked on the side and held the door open, letting the boys out. Jack's eyes had lost their amber sheen. Now they were plain dark hazel. She locked the truck, checked the door just in case\u2014locked up tight\u2014and headed to the brightly lit doors.\n\n\"Now remember,\" she said as they joined the herd of evening shoppers. \"Shoes and that's it. I mean it.\"\n**Ilona Andrews** is the pseudonym for a husband-and-wife writing team. Together they are the coauthors of the #1 _New York Times_ bestselling Kate Daniels urban-fantasy series and the romantic urban-fantasy novels of the Edge. They currently reside in Texas with their two children and numerous pets.\n**Looking for more?**\n\nVisit Penguin.com for more about this author and a complete list of their books.\n\n**Discover your next great read!**\n\n  1. Cover\n  2. Ace Books by Ilona Andrews\n  3. Title Page\n  4. Copyright\n  5. Contents\n  6. Chapter 1\n  7. Chapter 2\n  8. Chapter 3\n  9. Chapter 4\n  10. Chapter 5\n  11. Chapter 6\n  12. Chapter 7\n  13. Chapter 8\n  14. Excerpt from MAGIC SHIFTS\n  15. Excerpt from ON THE EDGE\n  16. About the Author\n\n  1. Contents\n  2. Cover\n  3. Start\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nAlso by MICHAEL R. PITTS  \nAND FROM MCFARLAND\n\n* * *\n\n_Allied Artists Horror, Science Fiction and Fantasy Films_ (2011)\n\n___Columbia Pictures Horror, Science Fiction and Fantasy Films, 1928\u20131982_ (2010)\n\n_Western Film Series of the Sound Era_ (2009)\n\n_Poverty Row Studios, 1929\u20131940: An Illustrated History of 55 Independent Film Companies, with a Filmography for Each_ (1997; paperback 2005)\n\n_Charles Bronson: The 95 Films and the 156 Television Appearances_ (1999; paperback 2003)\n\n_Horror Film Stars,_ 3d ed. (2002)\n\n_Western Movies: A TV and Video Guide to 4200 Genre Films_ (1986; paperback 1997)\n\n_Horror Film Stars, 2d ed._ (1991)\n\n_Hollywood and American History: A Filmography of Over 250 Motion Pictures Depicting U.S. History_ (1984)\n\n_Horror Film Stars_ (1981)\n\n# WESTERN MOVIES\n\n#### _A Guide to 5,105 Feature Films_\n\n* * *\n\nSECOND EDITION\n\n* * *\n\n#### Michael R. Pitts\n\nMcFarland & Company, Inc., Publishers  \n_Jefferson, North Carolina, and London_\n\nLIBRARY OF CONGRESS CATALOGUING DATA ARE AVAILABLE\n\nBRITISH LIBRARY CATALOGUING DATA ARE AVAILABLE\n\n**e-ISBN: 978-1-4766-0090-1**\n\n\u00a9 2013 Michael R. Pitts. All rights reserved\n\n_No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying or recording, or by any information storage and retrieval system, without permission in writing from the publisher._\n\nOn the cover: Poster art from the 1960 film _The Magnificent Seven_ (United Artists\/Photofest)\n\n_McFarland & Company, Inc., Publishers  \n_ _Box 611, Jefferson, North Carolina 28640  \n_ _www.mcfarlandpub.com_\n\nFor Carolyn, Angela, Juan Carlos and Jacob\n\n### Table of Contents\n\n_Preface_\n\n**THE FEATURE FILMS AND SERIALS**\n\n_Appendix 1: The Cowboys (and Cowgirls) and Their Horses_\n\n_Appendix 2: Screen Names_\n\n_Selected Bibliography_\n\n_Index to Entry Numbers_\n\n### Preface\n\nA quarter of a century has passed since the first edition of _Western Movies_ was published, and this new edition updates and expands the initial volume. More than 5,100 feature films are included, with more than 900 newly added. Many of the original entries have been fine-tuned and expanded. The criterion for inclusion is simple: an extant film that is, or has been, available for public viewing in some format. This includes theatrical, television, 16mm, 8mm and Super 8mm film and various video formats, along with prints housed in archives. The initial volume had purchase sources for some titles but with the ever changing video market this has been dispensed with since a check of the internet can usually determine availability.\n\nEach entry includes film title, release company and year, running time; if available in color (otherwise the movie is in black and white); a thorough cast listing, a plot synopsis and a brief critical review.\n\nThe following two abbreviations are used in the entries: D for Director and SC for Script.\n\nOnly feature films (running nearly four reels or approximately 40 minutes or more) are included in the text; there are no X-rated movies unless an R-rated version has been released. Running times may vary according to source. When films have been edited (mainly for television) they have a shorter running time; the original running times are included in the text. A number of films, especially from Republic Pictures, were edited to 54 minutes for television showings.\n\nRegarding cast listings, sometimes spellings vary and a Screen Names appendix is provided to help sort these out as well as alternate names. Actors' sons often drop the \"Jr.\" in the later years of their careers, such cases including Noah Beery, Jr., Lon Chaney, Jr., and Alan Hale, Jr.\n\nThe book includes all aspects of the Western film genre, not just shoot-'em-ups, with the bottom line having the plot take place on the frontier. As America developed frontier boundaries extended westward, thus _Drums Along the Mohawk_ is a frontier drama of the 1760s while _Stagecoach_ is part of the post\u2013Civil War frontier of the West. In these pages will be found north woods dramas, south of the border action films, outdoor adventures and foreign titles that either deal with the American frontier or have plots indigenous to the Western.\n\nAs noted in the first volume, opinions on movies are purely subjective and should be taken for that and nothing more. What appeals to one viewer may not appeal to another. My reviews of the movies are my own opinion and are no more than suggestions for the reader and hardly the final word. It is my hope the readers of this volume will find it enjoyable as well as useful.\n\nAny additions, corrections or comments regarding this volume are welcome, sent in care of the publisher.\n\nThanks goes to John Hellstrom and Douglas Deegan for additional title suggestions and the Moving Image Section, Library of Congress (Rosemary Hanes), for reference assistance.\n* * *\n\n### THE FEATURE FILMS AND SERIALS\n\n* * *\n\n**  \n**\n\n**1** _ **Abilene Town**_ **** United Artists, 1946. 89 min. D: Edwin L. Marin. SC: Harold Shumate. With Randolph Scott, Ann Dvorak, Edgar Buchanan, Rhonda Fleming, Lloyd Bridges, Helen Boyce, Howard Freeman, Richard Hale, Jack Lambert, Hank Patterson, Eddy Waller, Dick Curtis, Earl Schenck, Guy Wilkerson, Walter Baldwin, Buddy Roosevelt, Paul Brinegar, Dick Elliott, Harry Tenbrook, Chubby Johnson, Morgan Flowers, Bob Perry, Chief Tarachee, Chick Hannon, Victor Cox, Polly Bond, Maryellen Sennett. A sheriff tries to stop range fights between settlers and cattlemen in Kansas after the Civil War. Good Randolph Scott vehicle from the novel _Trail Town_ by Ernest Haycox.\n\n**2** _ **Abilene Trail**_ **** Monogram, 1951. 54 min. D: Lewis D. Collins. SC: Harry Fraser. With Whip Wilson, Andy Clyde, Noel Neill, Tommy Farrell, Steve Clark, Dennis Moore, Marshall Reed, Lee Roberts, Milburn Morante, Ted Adams, Bill Kennedy, Stanley Price, Lyle Talbot, Al Haskell, Clarke Stevens, Jack Low. Two suspected horse thieves come to the aid of a young rancher who is having trouble driving his herd to market. Solid Whip Wilson outing.\n\n**3** _ **Ace High**_ **** Paramount, 1969. 120 min. Color. D-SC: Giuseppi Colizzi. With Terence Hill, Eli Wallach, Bud Spencer, Brock Peters, Kevin McCarthy, Steffan Zacharias, Livio Lorenzon, Tiffany Hoyveld, Remo Capitani, Armando Bandini, Isa Foster, Rick Boyd. Sentenced to hang, an outlaw gets out of prison and teams with two enemies to fleece a crooked casino owner. Well made but overlong and hard to follow. Filmed in Italy as _**Quattro Dell'Ave Maria**_ (Ave Maria Four) and released in Great Britain as _**Revenge at El Paso**_.\n\n**4** _ **Ace of Cactus Range**_ **** Aywon, 1924. 43 min. D: Denver Dixon (Victor Adamson). SC: Irving Goldstein, Nellie Whitefield and Al Martin. With Art Mix (George Kesterson), Virginia Warwick, Clifford Davidson, Harvey Stafford, Dorothy Chase, Charles Colby, H. Paul Walsh, A.W. Dearie, Charles Mears. Diamond thieves threaten a young woman and her father who are helped by a lawman. Interesting silent curio from producer-director Victor Adamson; one of the few extant silent films starring George Kesterson as Art Mix.\n\n**5** _ **Ace of Clubs**_ **** Rayart, 1926. 60 min. D: J.P. McGowan. SC: G.A. Durlam. With Al Hoxie, Peggy Montgomery, Jules Cowles, Minna Redman, Andrew Waldron, Charles \"Slim\" Whittaker, Frank Ellis, Mutt (dog). A girl comes to live with her uncle and his son, not knowing they are rustling a widow's cattle. Slapped together, rock-bottom Al Hoxie vehicle that survives in a 37-minute version.\n\n**6** _ **Aces and Eights**_ **** Puritan, 1936. 62 min. D: Sam Newfield. SC: George A. Durlam. With Tim McCoy, Luana Walters, Wheeler Oakman, Rex Lease, John Merton, Charles Stevens, Joe Girard, Jimmie Aubrey, Earle Hodgins, J. Frank Glendon, Frank Ellis, Karl Hackett, Milburn Morante, Jack Kirk, Oscar Gahan, Robert Walker, Artie Ortego, Clyde McClary, Fred Parker. A gambling man plans to fleece the populace of a small town. Slow-paced Tim McCoy outing saved by the star's fine performance as a gambler.\n\n**7** _ **Aces 'N' Eights**_ RHI Entertainment, 2008. 80 min. Color. D: Craig R. Baxley. SC: Ronald M. Cohen and Dennis Shryack. With Casper Van Dien, Bruce Boxleitner, Ernest Borgnine, Jeff Kober, Jack Noseworthy, William Atherton, Jake Thomas, Rodney Scott, Deirdre Quinn, Victoria Chalaya, Alan Fudge, George LePorte, Emily Warfield, Ron Rogge, Michael H. Barnett, Kanin Howell, Deborah Ann Woll, Jared Ward, Jayden Lund, Melissa Bickerton, Gregory J. Barnett, Jeffrey G. Barnett. A railroad executive teams with a cowboy to stop two gunmen out to get rid of ranchers whose spreads are wanted for rights-of-way. By the numbers drama with nice action sequences.\n\n**8** _ **Aces Wild**_ **** Commodore, 1936. 57 min. D: Harry Fraser. SC: Monroe Talbot. With Harry Carey, Gertrude Messinger, Phil Dunham, Roger Williams, Fred \"Snowflake\" Toones, Ed Cassidy, Chuck Morrison, Theodore Lorch, William McCall, Bill Patton, Francis Walker, Jack Evans, Ray Henderson. A newspaper editor is threatened by outlaws when he tries to stop their activities. Low budget but nicely done Harry Carey film.\n\n**9** _ **Across the Badlands**_ **** Columbia, 1950. 55 min. D: Fred F. Sears. SC: Barry Shipman. With Charles Starrett, Smiley Burnette, Helen Mowery, Stanley Andrews, Robert Wilke, Harmonica Bill (William Russell), Dick Elliott, Hugh Prosser, Robert W. Cavendish, Charles Evans, Paul Campbell, Richard Alexander, Bob Woodward. The general manager of a railroad calls in a lawman to investigate constant attacks on the line's surveying crews. Well made \"Durango Kid\" entry. Released in Great Britain as _**The Challenge**_.\n\n_**Across the Border**_ see _**The Lone Rider Crosses the Border**_\n\n**10** _ **Across the Great Divide**_ **** Pacific International, 1977. 101 min. Color. D-SC: Stewart Raffill. With Robert Logan, Heather Rattray, Mark Edward Hall, George \"Buck\" Flower, Hal Bokar, Frank F. Salsedo, Fernando Celis, Loren Ewing, Tiny Brooks, James Elk, Stanley Cowley. In 1876 two orphans and a vagabond brave the harsh elements of the West to trek to Oregon so the youngsters can claim inherited land. Too long and a bit slow but nice scenery.\n\n**11** _ **Across the Line**_ **** High Water Films, 2000. 99 min. Color. D: Martin Spotti. SC: Sigal Erez and Michael Spotti. With Brad Johnson, Sigal Erez, Adrienne Barbeau, Brian Bloom, J.C. Quinn, Marshall Teague, Justin Urich, Julie Dolce Vita, Carlos Carrasco, Mark Adair-Rios, Stephen Spacek, Roger Velasco, Julia Vera, Steve Vinovich, Courtney Gebhart, John Vargas, Dave Silva, Timothy Dale Agee, Joshua Gallegos, Ernesto Garcia, Amit Knust, Don William Owen, Tony Perez, Luis Contreras, Elise Robbins, Bea Silvern, Tom Rosales. A beautiful illegal Mexican immigrant observes a murder and the lawman investigating the crime falls in love with her. Pretty fair contemporary Western drama focusing on the problem of undocumented workers.\n\n**12** _ **Across the Plains**_ **** Associated Independent Producers, 1928. 55 min. D-SC: Robert J. Horner. With Pawnee Bill, Jr. (Ted Wells), Ione Reed, Martha Barclay, Jack Richardson, Boris Bullock, Cliff Lyons. A crooked lawman tries to hang a ranch foreman for shooting a gambler in a dishonest poker game but a saloon girl comes to the rescue. Bottom rung entry in the low grade \"Pawnee Bill Jr.\" series. The film survives in a 37-minute version.\n\n**13** _ **Across the Plains**_ **** Monogram, 1939. 57 min. D: Spencer Gordon Bennet. SC: Robert Emmett (Tansey). With Jack Randall, Joyce Bryant, Frank Yaconelli, Hal Price, Dennis Moore, Glenn Strange, Robert Card, Bud Osborne, Dean Spencer, Wylie Grant. As youngsters two brothers are separated and years later they meet again but on the opposite sides of the law. The hackneyed plot does not help this average Jack Randall vehicle.\n\n**14** _ **Across the Rio Grande**_ **** Monogram, 1949. 55 min. D: Oliver Drake. SC: Ronald Davidson. With Jimmy Wakely, Dub Taylor, Reno Browne, Riley Hill, Dennis Moore, Kenne Duncan, Ted Adams, Myron Healey, Bud Osborne, Polly (Bergen) Burgin, Bob Curtis, Carol Henry, Boyd Stockman, Frank Ellis, Ben Corbett. A young lawyer becomes involved with border ore smuggling. Typical Jimmy Wakely musical western enhanced by sidekick Dub Taylor.\n\n**15** _ **Across the Sierras**_ **** Columbia, 1941. 58 min. D: D. Ross Lederman. SC: Paul Franklin. With Bill Elliott, Richard Fiske, Luana Walters, Dub Taylor, Dick Curtis, LeRoy Mason, Ruth Robinson, Art Mix, John Dilson, Milton Kibbee, Ralph Peters, Tex Cooper, Eddie Laughton, Edmund Cobb, Tom London, James Pierce, Carl Knowles, Ed Coxen, Lew Meehan, Blackjack Ward, Curly Dresden, Rube Dalroy, Jack Tornek. Wild Bill Hickok plans to settle down in the Oklahoma Territory but gets involved with an old friend on the wrong side of the law and a bad man he once sent to jail. Surprisingly austere \"Wild Bill Hickok\" series entry with Dick Curtis stealing the show as villainous Mitch Carew; a very good programmer. British title: _**Welcome Stranger**_.\n\n**16** _ **Across the Wide Missouri**_ **** Metro-Goldwyn-Mayer, 1951. 79 min. Color. D: William A. Wellman. SC: Talbot Jennings. With Clark Gable, Ricardo Montalban, John Hodiak, Adolphe Menjou, J. Carrol Naish, Jack Holt, Alan Napier, George Chandler, Richard Anderson, Henri Letondal, Douglas Fowley, Maria Elena Marques, Louis Niccolletti Whitmore, Russell Simpson, John Hartman, Frankie Darro, James Whitmore, Henry Wills, Frank McGrath, Fred Graham, Chuck Roberson, Evelyn Finley. Story of the opening of the trail west from St. Louis in the nineteenth century. Well done and action filled; good entertainment.\n\n_**Adios**_ see _**The Lash**_\n\n**17** _ **Adios, Amigo**_ **** Atlas, 1976. 87 min. Color. D-SC: Fred Williamson. With Fred Williamson, Richard Pryor, James Brown, Robert Phillips, Mike Henry, Victoria Jee, Lynne Jackson, Shuaila Farhat, Thalmus Rasulala, Liz Treadwell. A con man and his fall guy pal ply their trade in the Old West. Nice comedy.\n\n**18** _ **Adios Gringo**_ **** Explorer Film\/Fono Roma\/Trebal Film C.C.-Les Films Corona, 1965. 97 min. Color. D: George Finlay (Giorgio Stegani). SC: Giorgio Stegani, Jose Jerez and Michele Villerot. With Giuliano Gemma (Montgomery Wood), Evelyn Stewart (Ida Galli), Roberto Camardiel, Peter Cross, Jesus Puente, Grant Laramy, Jean Martin, Max Dean, Monique Saint Clare, Ted Carter, Frank Brana. A rancher, cheated out of his cattle and forced to kill a man in self defense, sets out to clear his name and becomes involved with a woman raped by three men. Very popular in Europe and not bad for this type of fare; remade as _**Wanted**_ (q.v.), also headlining Giuliano Gemma.\n\n**19** _ **Adios Hombre**_ **** Germania-Film, 1966. 85 min. Color. D: Mario Caiano. SC: Eduardo Manzanos (Brochero). With Craig Hill, Eduardo Fajardo, Piero Lulli, Giulia Rubini, Nello Passerine, Eleanora Vargas, Spartaco Conversi, Roberto Camardiel, Jacques Herlin, Elio Angelucci, Caterina Trentini, Pino Polidori, Massimo Carocci, Tomas Pico, Nazzareno Natale, Nazzareno Zamperla. An escaped convict tries to stop a madman and his gang who have taken over a small border town while waiting for a gold shipment. Typically violent Spaghetti Western. Also called _**Seven Pistols for a Massacre**_.\n\n**20** _ **Adios, Sabata**_ **** United Artists, 1971. 104 min. Color. D: Frank Kramer (Gianfranco Parolini). SC: Renato Izzo and Gianfranco Parolini. With Yul Brynner, Dean Reed, Pedro Sanchez, Susan Scott (Nieves Navarro), Gerard Herter, Sal Borgese, Franco Fantasia, Joseph Persuad, Gianni Rizzo, Salvatore Billa, Massino Carocci, Antonio Gardoli, Andea Scotti, Rick Boyd. A gunslinger is induced to aid Mexican rebels but he is also lured by a shipment of buried gold. Fair Italian production. Originally made as _**Indio Black**_ , the title character was changed to Sabata; Lee Van Cleef starred in its predecessor, _**Sabata**_ (q.v.), and its successor, _**The Return of Sabata**_ (q.v.).\n\n**21** _ **Adrenaline Cowboys**_ **** Ardusty Home Entertainment, 2004. 84 min. Color. D-SC: Steven Dieveney. With Kelly Armstrong, Bo Derek, Mike Lee, Adriano Moraes, Ty Murray. Video documentary on the lives of cowboys seeking a bull riding championship. For fans of this sport. Also called _**Adrenaline Cowboys: 8 Seconds to Glory**_.\n\n**22** _ **Advance to the Rear**_ **** Metro-Goldwyn-Mayer, 1964. 97 min. Color. D: George Marshall. SC: Samuel A. Peeples. With Glenn Ford, Stella Stevens, Melvyn Douglas, Jim Backus, Joan Blondell, Andrew Prine, Jesse Pearson, Alan Hale, James Griffith, Whit Bissell, Michael Pate, Yvonne Craig, Chuck Roberson, Bill Troy, Frank Mitchell, Harlan Warde, Paul Langton, Charles Horvath, Eddie Quillan, Paul Smith, Harvey Stephens, Gregg Palmer. During the Civil War a group of misfit raw recruits are mistakenly ordered by the Union army to guard a shipment of gold. A comedy that is only average but the cast is good despite the material.\n\n**23** _ **An Adventure of the Texas Kid: Border Ambush**_ **** United International, 1954. 60 min. Color. D: Robert Tansey. SC: Robert Emmet (Tansey). With Hugh Hooker, John Laurenz, Pamela Blake, Monte Blue, Terry Frost, James Kirkwood, Frank Scannell, John Carpenter, Noble \"Kid\" Chissell, Frank Marlowe. Two undercover operatives run into trouble with outlaws led by a corrupt lawyer after oil land. Tacky telefeature made up of two unsold pilots and also called _**Adventures of the Texas Kid**_.\n\n**24** _ **The Adventurer of Tortuga**_ **** Liber Film, 1965. 100 min. Color. D: Luigi Capuano. SC: De Riso and Poggi. With Guy Madison, Nadia Gray, Rik Battaglia, Inge Schoener, Mino Doro, Aldo Bufi Landi, Andrea Aureli, Guilio Marchetti, Linda Sini. A pirate is at odds with a Spanish governor for the hand of a beautiful Indian princess, the niece and heiress of a wealthy man. Mediocre dubbed costume melodrama from Italy originally issued as _**L'Avventuriero della Tortgua**_ (The Adventurer from Tortuga). Video title: _**Cold Steel for Tortuga**_.\n\n**25** _ **Adventures in Silverado**_ **** Columbia, 1948. 75 min. D: Phil Karlson. SC: Kenneth Gamet, Tom Kilpatrick and Joe Pagano. With William Bishop, Gloria Henry, Edgar Buchanan, Forrest Taylor, Edgar Barrier, Irving Bacon, Joseph Crehan, Paul E. Burns, Patti Brady, Fred F. Sears, Joe Wong, Charles Kane, Eddy Waller, Netta Parker, Trevor Bardette, George Chesebro, Bud Osborne. Traveling west, author Robert Louis Stevenson is on a stage robbed by a masked highwayman called \"The Monk\" and when the driver is accused of being in cahoots with the bad man, the writer plans to capture the outlaw. Well done action drama based on Stevenson's story \"Silvarado Squatters.\"\n\n_**The Adventures of Bear Tooth**_ see _**Beartooth**_\n\n**26** _ **The Adventures of Bullwhip Griffin**_ **** Buena Vista, 1967. 110 min. Color. D. James Nielson. SC: Lowell S. Hawley. With Roddy McDowall, Suzanne Pleshette, Karl Malden, Harry Guardino, Richard Haydn, Hermoine Baddeley, Bryan Russell, Liam Redmond, Cecil Kellaway, Joby Baker, Mike Mazurki, Alan Carney, Parley Baer, Arthur Hunnicutt, Dub Taylor, Pedro Gonzalez-Gonzalez, Gil Lamb, Burt Mustin, Dave Willock, John Qualen. A Boston boy and his stuffy butler travel to California to hunt for gold. Amusing genre spoof.\n\n**27** _ **The Adventures of Don Coyote**_ **** United Artists, 1947. 65 min. D: Reginald LeBorg. SC: Bob Williams and Ralph Cohn. With Frances Rafferty, Richard Martin, Marc Cramer, Bennie Bartlett, Frank Fenton, Byron Foulger, Eddie Parker, Pierce Lyden, Frank McCarroll, Val Carlo. Two caballeros try to help a young woman whose ranch is being attacked by outlaws. Nice little \"B\" outing.\n\n**28** _ **The Adventures of Frank and Jesse James**_ Republic, 1948. 13 chapters. D: Fred C. Brannon and Yakima Canutt. SC: Frankly Adreon, Sol Shor and Basil Dickey. With Clayton Moore, Noel Neill, Steve Darrell, George J. Lewis, Stanley Andrews, John Crawford, Sam Flint, House Peters, Jr., Dale Van Sickel, Tom Steele, James Dale, I. Stanford Jolley, Gene (Roth) Stutenroth, Lane Bradford, George Chesebro, Jack Kirk, Steve Clark, Dub Taylor, Carey Loftin, Frank Ellis, Art Dillard, Fred Graham, Guy Teague, Joe Yrigoyen, Eddie Parker, Bud Osborne, Rosa Turich, David Sharpe, Bob Reeves, Kenneth Terrell, Bud Wolfe. A crooked mine foreman tries to prevent Frank and Jesse James from repaying their robbery debts with proceeds from a gold mine they operate with a man and his daughter. Action-filled cliffhanger.\n\n**29** _ **The Adventures of Frontier Fremont**_ **** Sunn Classic, 1976. 85 min. Color. D: Richard Friedenberg. SC: David O'Malley. With Dan Haggerty, Denver Pyle, Norman Goodman, Tony Mirrati. In 1835 a man decides to live in the wilderness and has to overcome many hardships before finding contentment in the wild. Low budget but entertaining family-oriented adventure feature.\n\n**30** _ **Adventures of Gallant Bess**_ **** Eagle Lion, 1948. 71 min. D: Lew Landers. SC: Matthew Rapf. With Cameron Mitchell, Audrey Long, Fuzzy Knight, James Millican, John Harmon, Edward Gargan, Cliff Clark, Harry V. Cheshire, Evelynn Eaton, Herman Hack, Jack Tornek, Phil Arnold, Gallant Bess the Wonder Horse. A cowboy captures a wild horse and turns her into a trick rodeo attraction. Pleasant low budget affair.\n\n**31** _ **The Adventures of Grizzly Adams at Beaver Dam**_ **** NBC-TV, 1981. 60 min. Color. With Dan Haggerty, Denver Pyle, Don Shanks, Bozo the Bear. A mountain man tries to stop a family of beavers from building a dam which he fears will flood his valley home. Genial segment of the TV series \"The Life and Times of Grizzly Adams\" (NBC-TV, 1977\u201378) issued as a video feature. Filmed in the high Uinta Mountain Range of Utah.\n\n**32** _ **The Adventures of Grizzly Adams: Blood Brothers**_ **** NBC-TV, 1980. 59 min. Color. With Dan Haggerty, Denver Pyle, Don Shanks, Bozo the Bear. Trying to live in the wild after being falsely accused of a crime, a man meets an Indian brave who becomes his blood brother. Okay segment from \"The Life and Times of Grizzly Adams\" (NBC-TV, 1977\u201378) released on video.\n\n**33** _ **The Adventures of Grizzly Adams: The Renewal**_ **** NBC-TV, 1980. 73 min. D: Jack B. Hively. With Dan Haggerty, Denver Pyle, Patrick Wayne, Don Shanks, Ned Romero, Rudy Ramos, John Bishop, Brian Erickson, Bozo the Bear. After aiding a settler and his son, whose wagon is struck by lightning, a mountain man helps an Indian tribe find their sacred bird of spring. Colorful video feature derived from \"The Life and Times of Grizzly Adams\" (NBC-TV, 1977\u201378).\n\n**34** _ **The Adventures of Ned Blessing: Dead Man's Revenge**_ **** Trinity Home Entertainment, 2004. 89 min. Color. D: Dan Lerner. SC: William D. Wittliff. With Brad Johnson, Luis Avalos, Brenda Bakke, Rob Campbell. As Ned Blessing battles to keep his evil enemies from taking back the town of Plumb Creek, the ghost of the town's murdered sheriff aids him. Offbeat video feature derived from \"Ned Blessing: The Story of My Life and Times\" (CBS-TV, 1993).\n\n**35** _ **The Adventures of Ned Blessing: Return to Plumb Creek**_ **** Trinity Home Entertainment, 2004. 95 min. Color. D: Jack Bender. SC: William D. Wittliff. With Brad Johnson, Luis Avalos, Brenda Bakke, Rob Campbell, Bill McKinney, Richard Riedle, Wes Studi, Gregory Scott Cummins. Ned Blessing returns to his small town Texas home and finds it has been taken over by two tyrants, a man and his wife. The first of a trio of video features lifted from \"Ned Blessing: The Story of My Life and Times\" (CBS-TV, 1993); fair entertainment.\n\n**36** _ **The Adventures of Ned Blessing: The Return of the Hooded Man**_ **** Trinity Home Entertainment, 2004. 90 min. Color. D: Dan Lerner. SC: William D. Wittliff. With Brad Johnson, Luis Avalos, Brenda Bakke, Rob Campbell, Tim Scott, Wes Studi. Ned Blessing rescues his adopted father from being hung by the Texas Rangers and becomes a fugitive. Fans of the title character will like this video segment of \"Ned Blessing: The Story of My Life and Times\" (CBS-TV, 1993).\n\n_**The Adventures of Neeka**_ see _**Neeka**_\n\n**37** _ **Adventures of Red Ryder**_ **** Republic, 1940. 12 Chapters. D: William Witney and John English. SC: Franklyn Adreon, Ronald Davidson, Norman S. Hall, Barney A. Sarecky and Sol Shor. With Don \"Red\" Barry, Noah Beery, Tommy Cook, Maude Pierce Allen, Vivian Coe, Harry Worth, Hal Taliaferro, William Farnum, Robert Kortman, Carleton Young, Ray Teal, Gene Alsace, Gayne Whitman, Hooper Atchley, John Dilson, Lloyd Ingraham, Charles Hutchison, Gardner James, Wheaton Chambers, Lynton Brent, Edward Hearn, Dickie Jones, Matty Roubert, Roy Brent, Ed Cassidy, William Benedict, Curley Dresden, Joe De La Cruz, Bud Geary, Jack Rockwell, Post Park, Fred Burns, Dan White, Kenneth Terrell, Reed Howes, Budd Buster, Ed Brady, Augie Gomez, Al Taylor, Frank Conklin, Walter James, Ernest Sarracino, Bob Burns, Jack Kirk, James Fawcett, Duke Green, Art Dillard, Art Mix, David Sharpe, Joe Yrigoyen, Bill Yrigoyen, William Nestel, James Carlisle, Max Waizman, Chester Conklin, Jack O'Shea, Robert Wilke, Chick Hannon, Rose Plummer. A crooked banker murders several citizens in his efforts to seize land to be used by the railroad and after he kills Red Ryder's father, Red and his pal Little Beaver vow revenge. Top notch cliffhanger which made Don Barry a western star; well worth watching.\n\n**38** _ **The Adventures of the Masked Phantom**_ **** Equity, 1939. 59 min. Color. D: Charles Abbott. SC: Joseph O'Donnell and Clifford Sanforth. With Monte Rawlins, Betty Burgess, Larry Mason (Art Davis), Sonny La Mont, Merrill McCormick, Matty Kemp, Jack Ingram, Curley Dresden, Boots (Dog), Thunder (horse). A law officer dawns a mask in order to hunt outlaws smuggling stolen gold plates from a mine with low grade ore. Tattered, campy musical western curio.\n\n_**The Adventures of the Texas Kid**_ see _**The Texas Kid**_ (1943)\n\n_**Adventures of the Texas Kid**_ (1954) see _**An Adventure of the Texas Kid: Border Ambush**_\n\n**39** _ **Adventures of the Wilderness Family**_ **** Pacific International, 1975. 100 min. Color. D-SC: Stewart Raffill. With Robert Logan, Susan Damante Shaw, Hollye Holmes, Ham Larsen, George \"Buck\" Flower, William Cornford. A modern-day family rejects civilization and moves to the rugged Rocky Mountains. Big moneymaking family film, slight on plot but visually satisfying. Followed by _**The Further Adventures of the Wilderness Family**_ and _**Mountain Family Robinson**_ (q.v.).\n\n**40** _ **Africa**_ **\u2014** _ **Texas Style!**_ **** Paramount, 1967. 109 min. Color. D: Andrew Marton. SC: Andy White. With Hugh O'Brian, John Mills, Nigel Green, Tom Nardini, Adrienne Corri, Ronald Howard, Charles Hayes, Haley Mills, Charles Malinda, Honey Wamala, Stephen Kikumu, Ali Twaha. A rancher in Kenya hires two American cowpokes to try to save African wildlife by herding and domesticating the animals, but a rival tries to ruin the plan. Although this Africa-set Western has a lot of promise it fails to deliver much in the way of entertainment although it was the basis for the television series \"Cowboy in Africa\" (ABC-TV, 1967\u201368) starring Chuck Connors with Tom Nardini and Ronald Howard repeating their screen characters.\n\n**41** _ **Against a Crooked Sky**_ **** Cinema Shares, 1976. 100 min. Color. D: Earl Bellamy. SC: Eleanor Lamb and Douglas Stewart. With Richard Boone, Stewart Peterson, Henry Wilcoxon, Clint Ritchie, Shannon Farnon, Geoffrey Land, Vincent St. Cyr. After his teenage sister is kidnapped by Indians and his parents give her up for dead, a young boy joins an old trapper in trying to find her. Appealing, picturesque family feature.\n\n**42** _ **Al Jennings of Oklahoma**_ **** Colum-bia, 1950. 77 min. Color. D: Ray Nazarro. SC: George Bricker. With Dan Duryea, Gale Storm, Dick Foran, Gloria Henry, Guinn Williams, Raymond Greenleaf, Stanley Andrews, John Ridgely, James Millican, Harry Shannon, Robert Bice, Helen Brown, George J. Lewis, Jimmie Dodd, Edwin (Eddie) Parker, James Griffith, William Phillips, John Dehner, Charles Meredith, William Norton Bailey, Louis Jean Heydt, Harry Cording, Myron Healey, George Lloyd, Hank Patterson, George Chesebro, Earle Hodgins, John R. Hamilton, Harry Tyler, Guy Beach, Boyd Stockman, Tommy Ivo, Frank Matts. Al Jennings is forced to give up his law practice and soon becomes a famous outlaw. Highly fabricated, but entertaining version of Jennings' life, based on his book, with fine work by Dan Duryea in the title role.\n\n**43** _ **The Alamo**_ **** United Artists, 1960. 192 min. Color. D: John Wayne. SC: James Edward Grant. With John Wayne, Richard Widmark, Laurence Harvey, Richard Boone, Frankie Avalon, Carlos Arruza, Patrick Wayne, Linda Cristal, Joan O'Brien, Chill Wills, Joseph Calleia, Ken Curtis, Hank Worden, Denver Pyle, Aissa Wayne, Julian Trevino, Jester Hairston, Veda Ann Borg, Olive Carey, Wesley Lau, Tom Hennessey, Bill Henry, John Dierkes, Guinn Williams, Jack Pennick, Fred Graham, Chuck Roberson, Boyd \"Red' Morgan, Ruben Padilla. The story leading up to the heroic sacrifice of the men at the Alamo, which led to Texas independence from Mexico. Too long and detailed but still a magnificent effort with excellent work by its stars and well staged battle sequences.\n\n**44** _ **The Alamo**_ **** Buena Vista, 2004. 137 min. Color. D: John Lee Hancock. SC: Leslie Bohem, Stephen Gaghan and John Lee Hancock. With Dennis Quaid, Billy Bob Thornton, Jason Patric, Patrick Wilson, Emilio Echevarria, Jordi Molia, Leon Rippy, Tom Davidson, Marc Blucas, Robert Prentiss, Kevin Page, Joe Stevens, Stephen Bruton, Laura Clifton, Ricardo S. Chavira, Steven Chester Price, Craig Erickson, Nick Kokich, Richard Nance, Jeff Garner, Estephania Lebaron, Alerno Omilami, Edwin Hodge, Emily Deschanel, Blue Deckert, Turk Pipkin, Brandon Smith, Tommy G. Kendrick, W. Earl Brown, Tom Everett, Rance Howard, Stewart Finlay-McLennan, Matt O'Leary, John S. Davies, Kit Gwin, Castulo Guerra, Francisco Philbert, Mauricio Zatarain, Flavio Hinoiosa, Hugo Perez, Jesus Mayorga, Hector Garcia, Roland Uribe, Ruben G. Rojas, Lanell Pena, Michael Crabtree, Anna Reyesn, Sonia Montoya, Elena Hurst, Lynn Mathis, Charles Sanders, Rutherford Cravens, Dameon Clarke, Tim Mateer, Nathan Price, Don Javier Castillo, Lonnie Rodriguez, Julio Cesar Cedillo, Buck Taylor, Oscar D. Silva, Marc Menchaca, Safia Gray, Eric Montoya, Michael Clossin, Robert Bassetti, Nathan Walker, Frank Matthews, Krystal Morton, Aidan Black, Daniel Zubiate, Bert Beatson, Tony Wolford, Wendy Bonn, Clint Tidwell, Crystal Marie Dudley, Frank Thompson, Charles E. Gray, Ann Taylor, Alyssa Peterson, Amanda Peterson, Celina Hernandez, Robert C. Pemelton, Richard Jones. Another elaborate re-telling of the battle of the Alamo; revisionist history and a box office bust.\n\n**45** _ **The Alamo: Thirteen Days to Glory**_ **** NBC-TV, 1987. 140 min. Color. D: Burt Kennedy. SC: Clyde Ware and Norman McLeod Morrill. With James Arness, Brian Keith, Alec Baldwin, Raul Julia, David Ogden Stiers, Lorne Greene, Gene Evans, Buck Taylor, Ethan Wayne, Stan Ivar, Jim Metzler, Tom Schanley, Fernando Allende, Kathleen York, Isela Vega, Michael Wren, Jon Lindstrom, Hinton Battle, David Sheiner, Noble Willingham, Eloy Casados, Tony Becker, Thomas Callaway, Jerry Potter, Grainger Hines, Tom Everett, Jan Triska, Gary Kasper, John Furlong, Jay Baker, Dale Swann, Laura Fabian, Loyda Ramos, Bel Sandre, Laura Martinez Harring, Nick Blair, Red West. Very good television movie about the epic 1836 confrontation at the Alamo. The battle scenes are first rate as are the performances, especially James Arness as Jim Bowie and Brian Keith as Davy Crockett.\n\n**46** _ **Alaska**_ **** Monogram, 1944. 76 min. D: George Archainbaud. SC: George Wallace Sayre, Harrison Orkow and Malcolm Stuart Boyland. With Kent Taylor, Margaret Lindsay, Dean Jagger, John Carradine, Nils Asther, Iris Adrian, George Cleveland, Dewey Robinson, Lee \"Lasses\" White, John Rogers, John Maxwell, Warren Jackson. A prospector trying to hold off claim jumpers is unjustly accused of murder and is given three days to find the killer. Nicely done version of Jack London's short story \"Flush of Gold.\"\n\n**47** _ **Alaska Highway**_ **** Paramount, 1943. 66 min. D: Frank McDonald. SC: Lewis R. Foster and Maxwell Shane. With Richard Arlen, Jean Parker, Ralph Sanford, Bill Henry, Joseph Sawyer, Eddie Quillan, Jack Wegman, Harry Shannon, Edward Earle, Keith Richards, Lane Chandler, Kit Guard, Gary Gray, Charles Sullivan. Two hard headed brothers battle over the same girl as their father tries to head a crew building a roadway to Alaska to help the war effort. Rugged World War II program feature from the Pine-Thomas unit.\n\n**48** _ **Alaska Safari**_ **** American National Enterprises, 1968. 120 min. Color. D-SC: Arthur R. Dubs. With Arthur R. Dubs (narrator). Documentary on Alaska including its animal life, mountains and giant ice packs. Entertaining nature film followed by _**White Fury**_ (q.v.).\n\n**49** _ **Albuquerque**_ **** Paramount, 1948. 90 min. Color. D: Ray Enright. SC: Clarence Upson Young and Gene Lewis. With Randolph Scott, Barbara Britton, George \"Gabby\" Hayes, Lon Chaney, Russell Hayden, Catherine Craig, George Cleveland, Irving Bacon, Bernard Nedell, Karolyn Grimes, Russell Simpson, Jody Gilbert, John Halloran, Dan White, Walter Baldwin, Lane Chandler, Cliff Clark, Forrest Taylor, Dick Elliott, Sam Flint, Lorin Raker, Lee \"Lasses\" White, Buddy Roosevelt, Frank Ellis, Gregg Barton, Sailor Vincent, Lee Bennett, Chuck Roberson, Artie Ortego, Iron Eyes Cody, Leander De Cordova, Warren Jackson, Jack Low, Tex Cooper, George Morrell, Chick Hannon, Foxy Callahan, George Hazel, Joe Murphy, Cap Somers, Tom Monroe, Sam Lufkin, Ray Hyke, Augie Gomez, Ralph Bucko, Roy Bucko. A man tries to stop his crooked uncle from bankrupting his freight line competitors and ends up falling in love with a rival owner's pretty sister. Worth watching, especially for the brutal fight between Randolph Scott and Lon Chaney and Gabby Hayes' antics.\n\n**50** _ **Albur de Amor**_ (Chance of Love) Studio Latino, 1947. 90 min. D-SC: Alfonso Patino Gomez. With Pedro Armendariz, Susanna Cora, Gilberto Gonzalez, Emma Roldan, Alfonso Bedoya, Alfredo Varela, Julio Ahuet, Jose Muno, Pascual Garcia Pena, Ignacio Pedon. A cowboy is asked to look after a ranch owner's young daughter and he falls in love with her but when she rejects him because of his class he leaves the area, vowing to win her. Interesting Mexican Western drama.\n\n**51** _ **Alias Billy the Kid**_ **** Republic, 1946. 54 min. D: Thomas Carr. SC: Earle Snell and Betty Burbridge. With Sunset Carson, Peggy Stewart, Roy Barcroft, Tom London, Russ Whiteman, Tom Chatterton, Tex Terry, Pierce Lyden, Stanley Price, Ed Cassidy, Jack Rockwell, Jack Kirk, Jack O'Shea. A ranger lets a convicted murderer escape from jail so he can trial him to his gang but the man gets away and the lawman becomes involved with a female gang leader. Lots of action in this average Sunset Carson programmer.\n\n**52** _ **Alias El Alacran**_ **** (Alias the Scorpion) **** Radaent Films, 1961. 90 min. D: Arturo Martinez. With Rodolfo de Anda, Jaime Fernandez, Gina Romand, Dagoberto Rodriguez, Sonia Infante, Carlos Lopez Moctezuma, Oscar Pulido. After escaping to Mexico, Billy the Kid changes his name and works on a ranch where the owner is to wed a beautiful woman threatened by killers. Okay pseudo-historical Mexican Western, a sequel to _**El Muchacho de Durango**_ (The Boy of Durango) [q.v.].\n\n**53** _ **Alias Jesse James**_ **** United Artists, 1959. 92 min. Color. D: Norman McLeod. SC: William Bowers and Daniel B. Beauchamp. With Bob Hope, Rhonda Fleming, Wendell Corey, Jim Davis, Hugh O'Brian, Ward Bond, James Arness, Roy Rogers, Fess Parker, Gail Davis, James Garner, Gene Autry, Jay Silverheels, Bing Crosby, Gary Cooper, Gloria Talbott, Will Wright, Mary Young, Sid Melton, George E. Stone, James Burke, Joseph Vitale, Lyle Latell, Harry Tyler, Mike Mazurki, Mickey Finn, Nestor Paiva, Emory Parnell, I. Stanford Jolley, Michael Whalen, Richard Alexander, Oliver Blake, Jack Lambert, Ethan Laidlaw, Glenn Strange, J. Anthony Hughes, Iron Eyes Cody, Bob Gunderson, Fred Kohler, Jr. An insurance mn is sent to protect Jesse James after the company discovers it has a policy on the outlaw and the agent ends up being mistaken for the gunman. Highly amusing Bob Hope vehicle filled with genre stars and old-timers.\n\n**54** _ **Alias John Law**_ **** Supreme, 1935. 54 min. D: Robert North Bradbury. SC: Forbes Parkhill. With Bob Steele, Roberta Gale, Earl Dwire, Jack Rockwell, Buck Connors, Bob McKenzie, Roger Williams, Steve Clark, Horace Murphy. A young man returns home to claim land where oil has been discovered while an outlaw gang leader masquerades as the heir. Bob Steele fans will like this speedy effort.\n\n**55** _ **Alias Smith and Jones**_ **** Universal\/ABC-TV, 1971. 90 min. Color. D: Gene Levitt. SC: Matthew Howard and Glen A. Larson. With Peter Duel, Ben Murphy, Forrest Tucker, Susan Saint James, James Drury, Jeanette Nolan, Earl Holliman, John Russell, Bill Fletcher, Bill McKinney, Peter Brocco, Sid Haig, Jon Shank. Two outlaws are given amnesty if they agree to bring in a vicious gang. Adequate TV comedy Western feature, the pilot for the \"Alias Smith and Jones\" (ABC-TV, 1971\u201373) series.\n\n**56** _ **Alias the Badman**_ **** Tiffany, 1931. 66 min. D: Phil Rosen. SC: Earle Snell. With Ken Maynard, Virginia Brown Faire, Frank Mayo, Charles King, Lafe McKee, Robert Homans, Irving Bacon, Ethan Allen, Earl Dwire, Jack Rockwell, Jim Corey. A cowboy pretending to be an outlaw in order to catch the man who murdered his father falls in love with a girl whose father is a suspected cattle rustler. Good Ken Maynard vehicle.\n\n**57** _ **The Alien Encounters**_ **** Gold Key, 1976. 90 min. Color. D-SC: James T. Flocker. With Augie Tribuck, Matt Boston, Phil Catalli, Bonnie Henry, Patricia Hunt, Lukas Jackson, Chris Lee Jackson, Amy Dalton. In the desert country, a man tries to locate persons who have had encounters with alien beings. Slow moving and poorly made, the film manages to make its intriguing plot premise dull; desert scenery is the only interest.\n\n_**Alien Thunder**_ see _**Dan Candy's Law**_\n\n_**Alleluja and Sartana Are Sons**_ **...** _ **Sons of God**_ see _**Halleluja and Sartana Strike Again**_\n\n_**All Faces West**_ see _**Call of the Rockies**_\n\n**58** _ **All Hat**_ **** Odeon Films, 2007. 89 min. Color. D: Leonard Farlinger. SC: Brad Smith. With Luke Kirby, Keith Carradine, Noam Jenkins, Lisa Ray, Rachael Leigh Cook, David Alpay, Ernie Hudson, Joel Keller, Graham Greene, Gary Farmer, Stephen McHattie, Michelle Nolden, Lorne Brass, Michael Mahonen, Elley-Ray Snow, Charlotte Laurier, Tony Rannelli, Brooke Johnson, Nigel Hamer, Doug Murray, Christopher Bolton, Trent McMullen, Shawn Orr, John Robinson, Deanna Dezmari, David Gardener, Tracy Wright, Sandy Hawley, Chad Beckon, Brian Bochinski, Tyrone Harding, Eldridge Lindsay, Ray Raganauth, Kris Robb, Augustus Siddo, Andrew Foster, Jessica Barrow, Sima Fisher, Diego Fuentes, Dan Loiselle. A former baseball player gets out of jail and goes back to his rural Ontario home only to get mixed up in romance and a land scam. Brad Smith adapted his novel for this pleasant modern day Canadian oater.\n\n**59** _ **All Hell Broke Loose**_ **** North American Motion Pictures, 2009. 90 min. Color. D: Christopher Forbes. SC: Christopher Forbes and Jim Hilton. With David Carradine, Dave Long, Allison Tysinger, Stan Fink, Jim Hilton, Scotty Sparks, Jerry Chesser, Dianne All, Harry All, Ronald Bumgardner, Tripp Courtney, Alex Daniel, Michael Hilton, Richard Kinsey, Dan Beck. A bumbling outlaw is hired to carry out a hit only to be hounded by a United States marshal. Sub-standard direct-to-video affair with a fuzzy plot.\n\n**60** _ **All Mine to Give**_ **** Universal-International\/RKO Radio, 1957. 102 min. Color. D: Allen Reisner. SC: Dale Eunson and Katherine Eunson. With Glynis Johns, Cameron Mitchell, Rex Thompson, Patty McCormack, Ernest Truex, Hope Emerson, Alan Hale, Sylvia Field, Ralph Sanford, Steven Wooten, Butch Bernard, Yolanda White, Terry Ann Ross, Roy Engel, Ellen Corby, Reta Shaw, Royal Dano, Rita Johnson, Margaret Brayton. In frontier Wisconsin a pioneer family struggles to survive. Heart warming and very well done family movie; recommended.\n\n**61** _ **All Out**_ **** Stellar IV, 1968. 88 min. Color. D: Umberto Lenzi. SC: Eduardo Manzanos Brochero and Nino Stresa. With Mark Damon, John Ireland, Monica Randall, Fernando Sancho, Raf Baldassarre, Spartaco Conversi, Eduardo Fajardo, Armando Calvo, Jose Torres, Calisto Calisti, Miguel Del Castillo, Lisa Halverson, Tito Garcia, Joaquin Parra, Franco Guia, Frank Brana, Luis Induni, Ivan Scratuglia, Luis Barboo, Emilio Rodriguez, Rafael Albaicin, Fabian Conde, Claudio Scarchilli. A bounty hunter is hired to locate an outlaw suspected of hiding a cache of gold from a bank robbery. A cast full of Spaghetti Western favorites helps to cover the empty plot in this Italian-Spanish co-production made as _**Tutto per Tutto**_ (All for All) and also called _**Go for Broke**_.\n\n**62** _ **All the Pretty Horses**_ **** Miramax Films _ **,**_ 2000. 116 min. Color. D: Billy Bob Thornton. SC: Ted Tally. With Matt Damon, Henry Thomas, Penelope Cruz, J.D. Young, Laura Poe, Sam Shepard, Robert Patrick, Lucas Black, Yvette Diaz, Imelda Colindres, Agustin Solis, Ruben Blades, Elizabeth Ibarra, Bruce Dern, Miriam Colon, Lonnie Rodriguez, Raul Malo, Frederick Lopez, Ferron Lucero, Jr., Manuel Sanchez, Katie Harro, Denes Lujan, Leeann Lyons, Julio Oscar Mechoso, Edwin Figuera, Matthew E. Montoya, Julian Prada, Robert Enrique Pineda, Vincente Ramos, George R. Lopez, J.D. Garfield, Jo Harvey Allen, Julio C. Cedillo, Marc Miles, Brian Orr, Daniel Lanois, Jesse Plemons, Angelina Torres, Clark Sanchez, Theodore Grivas, Chris Talley, Richard Barela, James Roach, Dennis Chavez, Anthony Dilio, Philip Olivas, J. Nathan Simmons, Carlos Taboada, Rene Mungula, David Miguel Estrada, Dennis E. Garber, Patricia Miller, Jason Page. Two young men head to Mexico after World War II hoping to become cowboys but find the world is changing around them. Average coming of age drama.\n\n**63** _ **Allegheny Uprising**_ **** RKO Radio, 1939. 81 minutes. D: William A. Seiter. SC: P.J. Wolfson. With Claire Trevor, John Wayne, George Sanders, Brian Donlevy, Wilfred Lawson, Robert Barrat, John F. Hamilton, Moroni Olsen, Eddie Quillan, Chill Wills, Ian Wolfe, Wallis Clark, Monte Montague, Eddy Waller, Clay Clement, Olaf Hytten, Charles Middleton, Douglas Spencer, Bud Osborne, Stanley Blystone, Tom London. A frontiersman goes against his colony's commanding officer and seeks out a crooked trader whose selling weapons to the Indians is threatening peace. Set in the pre\u2013Revolutionary War period and based on fact, which makes it all the more interesting. British title: _**The First Rebel**_.\n\n**64** _ **Along Came Jones**_ **** RKO Radio, 1945. 90 min. D: Stuart Heisler. SC: Nunnally Johnson. With Gary Cooper, Loretta Young, William Demarest, Dan Duryea, Frank Sully, Russell Simpson, Arthur Loft, Willard Robertson, Don Costello, Ray Teal, Walter Sande, Lane Chandler, Frank Cordell, Tommy Coats, Tony Roux, Erville Alderson, Paul Sutton, Ernie Adams, Paul E. Burns, Chris-Pin Martin, Ralph Dunn, John Merton, Lee Phelps, Robert Kortman, Frank McCarroll, Hank Bell, Lou Davis, Ed Randolph, Herbert Heywood, Frank Hagney, Ralph Littlefield, Lane Watson, Doug Morrow, Jack Baxley, Geoffrey Ingram, Tom Herbert, Charles Morton, Lee Phelps, Billy Engle, Chalky Williams. A cowboy is mistaken for a hunted outlaw and becomes the target of both the posse looking for the fugitive and the bad man himself. Gary Cooper produced and starred in this International Pictures' feature with tepid results.\n\n**65** _ **Along the Great Divide**_ **** Warner Bros., 1951. 88 min. D: Raoul Walsh. SC: Walter Doniger and Lewis Meltzer. With Kirk Douglas, Virginia Mayo, John Agar, Walter Brennan, Ray Teal, Hugh Sanders, Morris Ankrum, James Anderson, Charles Meredith, Lane Chandler, Kenneth MacDonald, Steve Clark, Carl Harbaugh, Zon Murray, Sam Ash, Steve Darrell, Al Ferguson, Guy Wilkerson. A sheriff who feels responsible for his father's death tries to bring a prisoner in for trial and along the way faces opposition from the man's daughter as well as a posse that wants him to kill the captive. Well made, suspenseful and well acted, especially by Virginia Mayo as the prisoner's pretty daughter.\n\n**66** _ **Along the Mohawk Trail**_ **** International Television Corporation (ITC), 1964. 89 min. D: Sam Newfield. SC: Nat Tanchuck. With John Hart, Lon Chaney, Bill Walsh, Stan Francis, John Vernon A frontiersman and his Indian friend try to help the people of a small community stop a man who has set himself up as dictator. Pleasant paste-up of three episodes of the Canadian-filmed syndicated television series \"Hawkeye and the Last of the Mohicans\" (1957).\n\n**67** _ **Along the Navajo Trail**_ **** Republic, 1945. 66 min. D: Frank McDonald. SC: Gerald Geraghty. With Roy Rogers, George \"Gabby\" Hayes, Dale Evans, Estelita Rodriguez, Douglas Fowley, Nestor Paiva, Emmett Vogan, Roy Barcroft, Sam Flint, Bob Nolan and The Sons of the Pioneers (Doye O'Dell, Tim Spencer, Ken Carson, Shug Fisher, Hugh Farr, Karl Farr), Tex Terry, Budd Buster, Eddie Kane, Marin Sais, Kit Guard, George Morrell, Hank Bell, Frank O'Connor, Bert Moorhouse, Frank Stephens, Marlene Ames, David Cota, Ed Cassidy. Cowboy Roy Rogers aids pioneers and gypsies against land grabbers. Dull going except for the exciting climax.\n\n**68** _ **Along the Oregon Trail**_ **** Republic, 1947. 64 min. Color. D: R.G. Springsteen. SC: Earle Snell. With Monte Hale, Adrian Booth, Max Terhune, Clayton Moore, Roy Barcroft, Foy Willing and The Riders of the Purple Sage, Will Wright, LeRoy Mason, Tom London, Forrest Taylor, Kermit Maynard, Wade Crosby, Frank Ellis. A cowboy finds himself at odds with a madman who wants to build an empire for himself in the West. Average Monte Hale outing helped by color and sidekick Max Terhune.\n\n**69** _ **Along the Rio Grande**_ **** RKO Radio, 1941. 61 min. D: Edward Killy. SC: Arthur V. Jones and Morton Grant. With Tim Holt, Ray Whitley, Emmett Lynn, Robert Fiske, Betty Jane Rhodes, Hal Taliaferro, Carl Stockdale, Slim Whitaker, Monte Montague, Ruth Clifford, Ernie Adams, Bob Baker, Frankie Marvin, Ben Corbett, David Sharpe. In order to avenge the murder of their ex-boss, three men assume the guise of outlaws and head to Mexico and join the gang responsible for the killing. Typically good Tim Holt vehicle with Ray Whitley making his first appearance in the role of Smokey. A remake of _**Whistlin' Dan**_ (q.v.).\n\n**70** _ **Along the Sundown Trail**_ **** Producers Releasing Corporation, 1942. 59 min. D: Peter Stewart (Sam Newfield). SC: Arthur St. Clair. With Bill \"Cowboy Rambler\" Boyd, Art Davis, Lee Powell, Julie Duncan, Charles King, Karl Hackett, Howard Masters, John Merton, Jack Ingram, Kermit Maynard, Herman Hack, Frank Ellis, Ted Adams, Al St. John, Reed Howes, Art Dillard, Tex Palmer, Curley Dresden, Steve Clark, Hal Price, Jimmy Aubrey. A trio of lawmen hunt for outlaws out to rob a tungsten mine. Low grade entry in PRC's \"Frontier Marshal\" series.\n\n**71** _ **Alvarez Kelly**_ **** Columbia, 1966. 114 min. Color. D: Edward Dmytryk. SC: Franklin Coen and Elliott Arnold. With William Holden, Richard Widmark, Janice Rule, Patrick O'Neal, Victoria Shaw, Roger C. Carmel, Richard Rust, Arthur Franz, Donald Barry, Harry Carey, Jr., Mauritz Hugo, Robert Morgan, Stephanie Hill, Paul Lukather, Clint Ritchie. An adventurer leads a herd of cattle East to sell to the Union army during the Civil War but he is kidnapped by Confederates who want the beef for themselves. Fairly entertaining; could have been better.\n\n**72** _ **The Amazing Zorro**_ **** Nickelodeon Network, 2002. 72 min. Color. D: Scott Henning. SC: Bob Forward. With Cusse Mankuma, Nancy Cortes, Mark Acheson, Carmen Aquirre, Kathleen Barr, Eli Gabay, Santo Lombardo, John Novak, Sylvia Maldonado, Dale Wilson (voices). The foppish Don Diego becomes the dashing masked Zorro to save his homeland from a cruel dictator. Fun animated rehash of Johnston McCulley's story \"The Curse of Capistrano.\"\n\n**73** _ **Ambush**_ **** Metro-Goldwyn-Mayer, 1949. 89 min. D: Sam Wood. SC: Marguerite Roberts. With Robert Taylor, John Hodiak, Arlene Dahl, Don Taylor, Jean Hagen, Bruce Cowling, Leon Ames, John McIntire, Pat Moriarity, Charles Stevens, Chief Thundercloud, Ray Teal, Robin Short, Richard Bailey, Lane Chandler, Reed Howes, Cliff Clark, William Haade, Florence Lake, Hank Mann, Heinie Conklin, Bill Hale, Ray Bennett, Charles Cane, Tom Forman, Robert Hoy, Carol Henry, James Van Horn, Walt LaRue, Pat O'Malley, Fred Somers, Archie Butler. A scout finds himself at odds with a cavalry captain when he is assigned to bring back a woman captured by renegade Indians. The MGM gloss somewhat helps this average \"A\" outing.\n\n**74** _ **Ambush at Cimarron Pass**_ **** 20th Century\u2013Fox, 1958. 70 min. D: Jodie Copelan. SC: Richard G. Taylor and John K. Butler. With Scott Brady, Margia Dean, Clint Eastwood, Irving Bacon, Frank Gerstle, Dirk London, Baynes Barron, Keith Richards, John Merrick. A rancher, once a Confederate, teams with soldiers to thwart an Indian attack. Arid feature of interest for giving Clint Eastwood his first major Western role.\n\n**75** _ **Ambush at Tomahawk Gap**_ **** Columbia, 1953. 73 min. Color. D: Fred F. Sears. SC: David Lang. With John Hodiak, David Brian, Maria Elena Marques, John Derrell, Ray Teal, John Qualen, Otto Hullett, Percy Helton, Trevor Bardette, John Doucette. Four convicts escape from jail to prove their innocence and become involved in an Indian attack. Well done and suspenseful.\n\n**76** _ **Ambush Trail**_ **** Producers Releasing Corporation, 1946. 59 min. D: Harry Fraser. SC: Elmer Clifton. With Bob Steele, Syd Saylor, Lorraine Miller, I. Stanford Jolley, Charles King, Bob (John) Cason, Budd Buster, Kermit Maynard, Frank Ellis, Ed Cassidy, Al Ferguson, Henry Hall, Wally West, Ray Jones, Roy Brent, Lew Morphy. A rancher comes to the aid of friends as they oppose a desperado after their spreads. Cheaply made PRC oater saved by star Bob Steele.\n\n**Kermit Maynard, Charles King, John Cason, unidentified player, Bob Steele and Syd Saylor in** _**Ambush Trail**_ **(PRC, 1946).**\n\n**77** _ **Ambush Valley**_ **** Reliable, 1936. 57 min. D: Raymond Samuels (Bernard B. Ray). SC: Bennett Cohen. With Bob Custer, Victoria Vinton, Vane Calvert, Eddie Phillips, Wally Wales, Oscar Gahan, Ed Cassidy, Denver Dixon, Wally West, Roger Williams, John Elliott. A lawman finds himself opposing the views of his future father-in-law who threatens to kill nesters who settle on his land. Low grade Bob Custer outing.\n\n**78** _ **American Bandits: Frank and Jesse James**_ **** EI Entertainment, 2010. 88 min. Color. D-SC: Fred Olen Ray. With Peter Fonda, Tim Abbell, George Stults, Jeffrey Combs, Michael Gaglio, Anthony Tyler Quinn, Siri Baruc, Ian Patrick Williams, Ted Monte, Christopher Weir, Franc Ross, Peter Sherayko, Lauren Eckstrom, Jake Thornton, Patrick Gorman, Randy Mulkey. A U.S. marshal is on the trail of the James gang with a showdown set in a ghost town. Poor.\n\n**79** _ **American Empire**_ **** United Artists, 1942. 82 min. D: William McGann. SC: J. Robert Bren, Gladys Atwater and Ben Grauman Kohn. With Richard Dix, Preston Foster, Frances Gifford, Leo Carrillo, Guinn Williams, Robert Barrat, Jack LaRue, Cliff Edwards, Chris-Pin Martin, Richard Webb, William Farnum, Hal Taliaferro, Tom London, Guy Rodin, Etta McDaniel. Two partners built a cattle empire in Texas despite personal problems between them over a woman and their battles with a Mexican bandit leader. Highly entertaining and well made with especially good work by Richard Dix and Preston Foster as the partners and Leo Carrillo as the bandit.\n\n**80** _ **American Outlaws**_ **** Warner Bros., 2001. 94 min. Color. D: Les Mayfield. SC: Roderick Taylor and John Rogers. With Colin Farrell, Scott Caan, Ali Larter, Gabriel Macht, Gregory Smith, Harris Yulin, Kathy Bates, Timothy Dalton, Will McCormack, Ronny Cox, Terry O'Quinn, Nathaniel Arcand, Ty O'Neal, Joe Stevens, Barry Tubb, Jack Watkins, Tom Schuster, Lee Ritchey, Robin Christian, Ed Goldart, Brad Leland, Craig Erickson, Mark Walters, Michael Costello, Jack Gould, Morgana Shaw, Brady Coleman, Richard Jones, Steven \"Dooky\" Bland, Jerry Cotton, Muse Watson, Lane Thomas Wilson, Ron Hayden, Darryl Cox, Riley Flynn, Joe Brown, Shawn Patrick Nash, Jessica M-E Nitsch, Big Skinny Brown, Troy Dillinger, Johnny Bartee, Pei-San Brown, Susan E. Denison, Frank Matthews, Paul Wright, Steve Crawford, Chris Warner, Frank G. Curcio, Marvin Schroeder, Lisa Del Dotto, Kirk Hunter, Philip Olivas, Jeremy Denzlinger, Charles E. Gray, David Jachin Kelley, Rana Morrison. The James and Younger brothers team to oppose a railroad baron out to fleece homesteaders of their lands. Well worth watching.\n\n**81** _ **An American Tail: Fievel Goes West**_ **** Universal, 1991. 75 min. Color. D: Phil Nibbling and Simon Wells. SC: Flint Dille. With Voices of James Stewart, Phillip Glasser, Erica Yohn, Cathy Cavadini, Nehemiah Persoff, Dom DeLuise, Amy Irving, John Cleese, Jon Lovitz, Larry Moss, Robert Watts, Mickie McGowan, Jennifer Darling, Sherry Lynn, Fausto Bara, Jack Angel, Vanna Bonta, Nigel Pegram, Philip Clarke, Lev Mailer, David Tate, Lisa Raggio, Patrick Penney, Annie Holiday, Lawrence Steffan. A mouse family is finagled by a sleazy cat into moving West while one of their friends follows along after his girlfriend. Pleasant animated feature, a sequel to _**An American Tail**_ (1986) and James Stewart's last film.\n\n**82** _ **The Americano**_ **** RKO Radio, 1955. 85 min. D: William Castle. SC: Guy Trosper. With Glenn Ford, Frank Lovejoy, Cesar Romero, Ursula Theiss, Abbe Lane, Rodolfo Hoyos, Tom Powers, Dan White, Frank Marlowe. A Texas cowboy is sent to Brazil to deliver prize Brahma bulls and runs into outlaws as well as a pretty girl. Typical mid\u20131950s \"Western\" set in Brazil instead of the Old West.\n\n**83** _ **El Ametralladora**_ **** (The Machine Gunner) Jalisco Films, S.A., 1943. 98 min. D-SC: Aurelio Robles Castillo. With Pedro Infante, Margarita Mora, Angel Garasa, Victor Manuel Mendoza, Arturo Soto Rangel, Alfredo Varela, Antonio Bravo, Manuel Arvide, Eugenia Galindo, Neomi Beltran, Manuel Noriega, Jose Torvay, Francisco Pando, Robert Canedo, Mariachi Vargas, Las Tres Morenas. A man called \"The Machine Gunner\" loses his girlfriend to another after being falsely accused of crimes committed by his rival. Okay Mexican Western, a sequel to _**Ay, Jalisco, no te Rajes!**_ (Oh, Jalisco, Don't Break Down!) (1941).\n\n_**Amigo, Stay Away**_ see _**Ben and Charlie**_\n\n**84** _ **Among Vultures**_ **** Rialto-Film\/Jadran-Film\/Atlantis-Film, 1964. 98 min. Color. D: Alfred Vohrer. SC: Eberhard Sendoff and Johanna Sibelius. With Stewart Granger, Pierre Brice, Elke Sommer, Gotz George, Walter Barnes, Mario Girotti, Renato Baldini, Sieghardt Rupp, Louis Velle, Mila Blach. Frontier scout Old Surehand and his Indian comrade Winnetou head a wagon train carrying the daughter of a diamond dealer; she is kidnapped by outlaws masquerading as Indians. Flavorful screen adaptation of Karl May's novel with Stewart Granger adding underplayed comedy as Old Surehand. Made in West Germany as _**Unter Geirn**_ (Among Vultures) and issued in the U.S. in 1968 by Columbia as _**Frontier Hellcat**_.\n\n**85** _ **And God Said to Cain**_ **** ZIV International, 1970. 93 min. Color. D: Anthony Dawson (Antonio Margheriti). SC: Giovanni Addessi and Antonio Margheriti. With Klaus Kinski, Peter Carsten, Marcella Michelangeli, Lee Burton, Antonio Cantafora, Giuliano Raffaelli, Alan Collins, Lucio De Santis, Maria Luisa Sala, Joaquin Blanco, Giacomo Furia, Furio Meniconi, Gigi Bonos, Marco Morelli, Franco Gula, Amerigo Santarelli, Osiride Pevarello, Ettore Arena, Renzo Pevarello, Pedro Mendiconi. After a decade in prison, a man gets out and vows revenge on those who framed him. Improbable Spaghetti Western filmed in Italy as _**E Dio Disse a Caino**_ **...** in 1969.\n\n**86** _ **And Now Miguel**_ **** Universal, 1966. 95 min. Color. D: James B. Clark. SC: Ted Sherdeman and Jane Klove. With Pat Cardi, Guy Stockwell, Clu Gulagher, Michael Ansara, Joe De Santis, Pilar Del Rey, Peter Robbins, Buck Taylor, James Hall, Emma Tyson. A young boy longs to go with his father into the mountains for the summer grazing of their sheep herd but it takes an artist to teach him the meaning of patience in growing up. Nicely done period piece set in the Southwest.\n\n**87** _ **And Starring Pancho Villa as Himself**_ **** HBO Films, 112 min. Color. D: Bruce Beresford. SC: Larry Gelbart. With Antonio Banderas, Eion Bailey, Alan Arkin, Jim Broadbent, Matt Daly, Michael McKean, Colm Feore, Alexa Davalos, Anthony Stewart Head, Kyle Chandler, Saul Rubinek, Cosme Alberto, Pedro Armendariz (Jr.), Michael F. Boyle, Peter Gregory, Jay Kimball, Jose Concepcion Macias, Darrell Pritchett, Diego Sandoval, John Wharton, Maria Guillermo Ramirez, Benjamin Long, Damian Alcazar, Rita Lopez Carrasco, Jorge Jimenez, Jose Manuel Lambarri, Carl Dillard, Marcelo Garcia, Fernando Becerril, Madeline Lee, Julian Sedgwick, Sacha Oberlander Tasletikye, Lilia Zeninna, Patricia Schweers, Gabriela Reynosa, Barbara May, Lynnanne Zager, Steve Calcote, Jake Koenig, Adrian Hernandez, Isalas Jimenez, Maurico Magana, Lorena Minguez, Cesar Di Parra. Needing money to fight Mexican government forces, Pancho Villa makes a deal with Hollywood to film his revolutionary activities. Interesting historical fiction made for television; the plot includes noted filmmakers D.W. Griffith (Colm Feore), Christy Cabanne (Michael McKean) and Raoul Walsh (Kyle Chandler).\n\n_**And Then a Time for Killing**_ see _**Tequila Joe**_\n\n**88** _ **And They Smelled the Strange, Exciting, Dangerous Scent of Dollars**_ **** Samy Film, 1973. 92 min. Color. D: Italo Alfaro (Piero Regnoli). SC: Piero Regnoli. With Robert Malcolm, Piero Vida, Rosalba Neri, Luigi Meccia, Salvatore Puntillo, Peter Landers, Spartaco Conversi, Claudio Ruffini, Franco Ukmar, Dante Maggio, Rocco Lerro, Amerigo Castrighella, Ottorino Polentini. The agent in charge of delivering a government railroad payroll finds himself beset by Mexican outlaws, a preformed priest and a saloon keeper, all of them after the money. Light, pleasant Italian Spaghetti Western released there as _**Sentivano**_ **...** _ **uno Stano, Eccitante, Pericoloso Puzzo di Dollari**_ (They Felt ... the Strange, Exciting, Dangerous Stench of Dollars).\n\n**89** _ **Angel and the Badman**_ **** Republic, 1947. 100 min. D-SC: James Edward Grant. With John Wayne, Gail Russell, Harry Carey, Bruce Cabot, Irene Rich, Lee Dixon, Stephen Grant, Tom Powers, Paul Hurst, Olin Howlin, John Halloran, Joan Barton, Craig Woods, Marshall Reed, Hank Worden, Pat Flaherty, Jack Kirk. An outlaw on the run is reformed by the love of a devout Quaker girl. Fine John Wayne vehicle; the type of fare William S. Hart and Harry Carey did in the silent era.\n\n**90** _ **Angel and the Badman**_ **** Hallmark Channel\/Barnholtz Entertainment, 2009. 92 min. Color. D: Terry Ingram. SC: Jack Nasser. With Lou Diamond Phillips, Deborah Kara Unger, Luke Perry, Terrance Kelly, Merrilyn Gann, Gig Morton, Michael Teigan, John Tench, Scott McNeil, Don Thompson, Brendan Wayne, Winston Rekert, Garry Chalk, Jennifer Copping, Melanie Papalia, Matthew Robert Kelly, Noah Beggs, Charles Andre, Jim Shield, Luis Javier, Stefan Arngim, Stephen Dimopoulos, Teach Grant. A wounded gunfighter is taken in by a widow who is raising a young son. Tepid TV remake of the John Wayne classsic.\n\n**91** _ **Angel in Exile**_ **** Republic, 1948. 90 min. D: Allan Dwan and Philip Ford. SC: Charles Larson. With John Carroll, Adele Mara, Thomas Gomez, Barton MacLane, Alfonso Bedoya, Grant Withers, Paul Fix, Art Smith, Tom Powers, Ian Wolfe, Howard Chamberlain, Elsa Lorraine Zepeda, Mary Currier, Don Haggerty, Mickey Simpson, Gloria Varela, Al Haskell, Soledad Jiminez, Charles Marsh, Julia Montoya, Elias Gamboa, Joe Dominguez, Rose Mary Lopez. In rural Mexico an ex-convict plans to use a played out mine to smuggle gold stolen in a robbery but begins to have a change of heart when the locals think he is sent by God. Pleasant inspirational drama.\n\n**92** _ **Animal Called Man**_ **** Trans-World Entertainment, 1972. 83 min. Color. D-SC: Roberto Mauri. With Vassili Karis, Gillian Bray, Craig Hill, Gilberto Galimberti, Amero Capanna, Carla Mancini, Paolo Magalotti, Roberto Dell'Acqua, Sergio Serafini. After a slick bandit wins his girl in a sharp shooting contest, a town despot vows revenge. Low grade, eccentric takeoff of the Spaghetti Western genre. Filmed as _**Animale Chiamato Uomo**_ (Animal Called Man).\n\n**93** _ **The Animals**_ **** Levitt-Pickman, 1971. 86 min. Color. D: Ron Joy. SC: Hy Mizrahi. With Henry Silva, Keenan Wynn, Michele Carey, John Anderson, Joseph Turkel, Pepper Martin, Bobby Hall, Peter Hellmann, William Bryant, Peggy Stewart. A young school teacher is raped by five thugs, after they hold up a stagecoach she is riding, and swears revenge against them. Brutal oater which had limited theatrical release. British title: _**Five Savage Men**_.\n\n**94** _ **Annie Get Your Gun**_ **** Metro-Goldwyn-Mayer, 1950. 107 min. Color. D: George Sidney. SC: Sidney Shelton. With Betty Hutton, Howard Keel, Louis Calhern, J. Carrol Naish, Edward Arnold, Keenan Wynn, Benay Venuta, Clinton Sundberg, James H. Harrison, Chief Yowlachie, Lee Tung Foo, William Tannen, Anne O'Neal, John Hamilton, Edward Earle, Marjorie Wood, Frank Wilcox, John Mylong, Carl Sepulveda, Carol Henry, Fred Gilman, Eleanor Brown, John War Eagle, Edith Mille, Dorinda Clifton, Ed Kilroy, Nolan Leary, Al Rhein, Budd Fine. Annie Oakley rises from backwoods target practice to the stop sharpshooter with Buffalo Bill's Wild West Show as she tries to win the man of her dreams. Lively screen adaptation of the Irving Berlin musical which starred Ethel Merman and Ray Middleton on Broadway.\n\n**95** _ **Annie Oakley**_ **** RKO Radio, 1935. 90 min. D: George Stevens. SC: Joel Sayre and John Twist. With Barbara Stanwyck, Preston Foster, Melvyn Douglas, Pert Kelton, Moroni Olsen, Andy Clyde, Chief Thundercloud, Delmer Watson, Margaret Armstrong, Adeline Craig, Willie Best. A tomboy crack sharpshooter falls in love with the world's top marksman and becomes the featured attraction of Buffalo Bill Cody's show. Fictional account of the life of Annie Oakley but still pleasant entertainment.\n\n**96** _ **Another Man, Another Chance**_ **** United Artists, 1977. 132 min. Color. D: Claude Lelouch. SC: Jacques Lefrancois (Claude Lelouch). With James Caan, Genevieve Bujold, Francie Huster, Jennifer Warren, Susan Tyrrell, Rossie Harris. A young woman arrives in the West in 1880 with a photographer who dies suddenly and she meets and marries a widowed veterinarian. Well acted and visually interesting retelling of director Claude Lelouch's _**A Man and a Woman**_ (Allied Artists, 1966), this time set in the Old West. Made in France as _**Un Autre Homme, Un Autre Chance**_ (Another Man, Another Chance). Alternate title: _**Another Man, Another Woman**_.\n\n**97** _ **Another Pair of Aces: Three of a Kind**_ **** CBS-TV, 1991. 100 min. Color. D: Bill Bixby. SC: Rob Gilmer. With Willie Nelson, Kris Kristofferson, Joan Severance, Dan Kamin, Ken Farmer, Rip Torn, Sammy Allfred, Bud Shrake, Marvin Shapiro, Billy Streater, Bill Ballis, Harvey Christiansen, Bernard Engel, Gabriel Folse, James Harrell, Charles Gunning, Turk Pipkin, Julius Tennon, Larry Hovis, Richard Jones, Christine Poole, Tonie Perensky, Tracy Kristofferson, Michael Griffith, Paula Nelson, Shelby Lynne, Terry Mross. An ex-safecracker teams with a Texas Ranger to prove the innocence of another lawman accused of murdering a convict. Average follow-up to _**Pair of Aces**_ (q.v.).\n\n**98** _ **Any Gun Can Play**_ **** Golden Eagle, 1968. 105 min. Color. D: Enzo G. Castellari. SC: Enzo G. Castellari, Romolo Guerrieri, George Simmonelli and Fabio Carpo. With George Hilton, Edd Byrnes, Gilbert Roland, Kareen O'Hara, Pedro Sanchez, Gerard Herter, Jose Torres. Three men, a bandit, a stranger and a banker, join forces so they can divide a fortune in stolen gold. Better than average Italian Western made so by the strong macho performance of Gilbert Roland as one of the lead players. Released in Italy in 1967 as _**Vado**_ **...** _ **l'Ammazzo e Torno**_ by FIDA Cinematografica and also called _**Blood River**_ and _**Go Kill and Come Back**_.\n\n**Gilbert Roland (center) in** _**Any Gun Can Play**_ **(Golden Eagle, 1968).**\n\n**99** _ **Anything for a Friend**_ **** Tarquinia Film, 1973. 90 min. Color. D: Miles Deem (Demofilo Fidani). SC: Demofilo Fidani, Mila Vitally and Filippo Perrone. With Red Carter (Ettore Manni), Bud Randall (Paolo Rosani), Rick Boyd (Federico Boldo), Gordon Mitchell, Simone Blondell (Simonetta Vitelli), Carla Mancini, Dennis Colt (Benito Pacifico), Sleepy Warren, Angela Portaluri, Raimondo Toscano, Custer Gail (Amerigo Castrighella), Paul Crain (Enzo Pulcrano), Michele (Branca) Francia, Antonio Basile, Luciano Conti, Vinicio Raimondi. Two conmen join a convoy heading to the gold fields and decide to take revenge after being robbed by outlaws. Pathetic Italian production played more for comedy than action; made as _**Amico Mio, Frega Tu**_ **...** _ **Che Frego Io!**_\n\n**100** _ **Apache**_ **** United Artists, 1954. 91 min. Color. D: Robert Aldrich. SC: James R. Webb. With Burt Lancaster, Jean Peters, John McIntire, Charles (Bronson) Buchinsky, John Dehner, Paul Guifoyle, Ian MacDonald, Walter Sande, Morris Ankrum, Monte Blue, Philip Van Zandt, Rory Mallinson, Paul E. Burns, Dick Rich, John George, Lonnie Burr. An Indian gives up his pacifistic ways to combat the U.S. cavalry when the rights of his people are threatened. Not a very remarkable film with supporting players Charles Bronson and Monte Blue more impressive than star Burt Lancaster.\n\n**101** _ **Apache Ambush**_ **** Columbia, 1955. 68 min. D: Fred F. Sears. SC: David Lang. With Bill Williams, Richard Jaeckel, Alex Montoya, Movita, Adele August, Tex Ritter, Ray Corrigan, Ray Teal, Don C. Harvey, James Griffith, James Flavin, George Chandler, Forrest Lewis, George Keymas, Victor Milan, Harry Lauter, Bill Hale, Robert Foulk, Edmund Cobb, Clayton Moore, Kermit Maynard, Lane Chandler, Iron Eyes Cody. While leading a cattle drive to Texas after the Civil War, an ex\u2013Union soldier is faced with trouble from marauding Indians and renegade Confederates. Average outing of interest because of Tex Ritter and Ray Corrigan in supporting roles.\n\n**102** _ **Apache Blood**_ **** Key International, 1975. 86 min. Color. D: Tom Quillen. SC: Jack Lee and Dewitt Lee. With Ray Danton, Dewitt Lee, Troy Nabors, Diane Taylor, Eva Kovacs, Jason Clark, David Robart, William Chadwick, Carl Mancini, Earl Baldwin, Wilford \"Whizzer\" White, Carl Nelson, Jack Lee. An Indian plans to get revenge for the Army massacring his tribe. Tatty low budget effort filmed in Arizona in 1971 as _**Pursuit**_.\n\n**103** _ **Apache Chief**_ **** Lippert, 1949. 60 min. D: Frank McDonald. SC: Gerald Green and Leonard Picker. With Alan Curtis, Tom Neal, Russell Hayden, Carol Thurston, Fuzzy Knight, Francis MacDonald, Trevor Bardette, Roy Gordon, Charles Soldani, Rodd Redwing. Two Indian brothers, once peaceful and one warlike, oppose each other to see who will lead their tribe. Low budget Lippert outing of interest because of its three stars.\n\n**104** _ **Apache Country**_ **** Columbia, 1952. 62 min. D: George Archainbaud. SC: Norman S. Hall. With Gene Autry, Pat Buttram, Carolina Cotton, Harry Lauter, Francis X. Bushman, Mary Scott, Sidney Mason, Gregg Barton, Tom London, Byron Foulger, Frank Matts, Mickey Simpson, Iron Eyes Cody, Tony Whitecloud, Jemez Indians, Cass County Boys (Jerry Scoggins, Bert Dodson, Fred Martin), Frankie Marvin, Frank Ellis, Bob Woodward. A government agents tries to thwart an outlaw gang masquerading as Indians as cover for their illegal activities. Pretty fair Gene Autry opus filmed in Sepiatone.\n\n**105** _ **Apache Drums**_ **** Universal-Internatinonal, 1951. 75 min. Color. D: Hugo Fregonese. SC: David Chandler. With Stephen McNally, Coleen Gray, Willard Parker, Arthur Shields, James Griffith, Armando Silvestre, Georgia Backus, Clarence Muse, James Best, Ray Bennett. After being run out of a small town by a corrupt official, a gambler returns to help the citizens fight off an Indian attack. Despite its medium budget trappings, this one is not of much interest.\n\n_**Apache Fury**_ see _**Fury of the Apaches**_\n\n**106** _ **Apache Gold**_ **** Columbia, 1963. 91 min. Color. D: Harald Reinl. SC: Harald G. Petersson. With Lex Barker, Pierre Brice, Mario Adorf, Marie Versini, Ralf Wolter, Walter Barnes, Milivoje Popovic-Mavid, Dunja Rajter, Chris Howland, Niksa Stefanini, Branko Spoljar, Husein Cokic, Demeter Bitenc, Gojko Mitic. A frontiersman and his Indian friend try to protect a local tribe from marauding whites who are after their gold. Action filled outing in the Karl May series from West Germany based on the author's 1893 novel _Winnetou, Der Rote Gentleman_. Issued in Europe as _**Winnetou I**_ and _**Winnetou the Warrior**_ at 111 minutes.\n\n**107** _ **The Apache Kid**_ **** Republic, 1941. 56 min. D: George Sherman. SC: Eliot Gibbons and Richard Murphy. With Don \"Red\" Barry, Lynn Merrick, LeRoy Mason, Robert Fiske, John Elliott, Forbes Murray, Monte Montague, Al St. John, Fred \"Snowflake\" Toones, Charles King, Frank Brownlee, John Cason, Cactus Mack, Kenne Duncan, Hal Price, Buddy Roosevelt. A young adventurer leads a wagon train of friends and neighbors westward to Oregon but finds out the trip was instigated by his crooked uncle who has a government contract to build a road. Nicely done Don Barry star vehicle with pretty Lynn Merrick adding to the scenery.\n\n**108** _ **The Apache Kid's Escape**_ **** Horner Productions, 1930. 50 min. D-SC: Robert J. Horner. With Jack Perrin, Josephine Hill, Fred Church, Virginia Ashcroft, Henry Roquemore, Bud Osborne, Fred Burns, Buzz Barton, Horace B. Carpenter, Charles Le Moyne, Starlight (horse). An outlaw masquerades as a cowpoke and helps a friend who is in trouble, even giving up the girl he loves. Tattered early talkie mainly of curio value.\n\n_**Apache Massacre**_ see _**Face to the Wind**_\n\n**109** _ **Apache Rifles**_ **** 20th Century\u2013Fox, 1964. 92 min. Color. D: William Witney. SC: Kenneth Gamet and Richard Schayer. With Audie Murphy, Michael Dante, Linda Lawson, L.Q. Jones, Ken Lynch, John Archer, Charles Watts, Hugh Sanders. A Cavalry captain in Arizona in 1879 is sent to capture marauding Indians who have been slaughtering miners and settlers. Speedy Audie Murphy vehicle somewhat hurt by stock footage and a mediocre plot.\n\n**110** _ **Apache Rose**_ **** Republic, 1947. 75 min. Color. D: William Witney. SC: Gerald Geraghty. With Roy Rogers, Dale Evans, Bob Nolan and The Sons of the Pioneers (Tim Spencer, Lloyd Perryman, Pat Brady, Hugh Farr, Karl Farr), Olin Howlin, George Meeker, Russ Vincent, Minerva Urecal, LeRoy Mason, Terry Frost, Tex Terry, John Laurenz, Donna (Martell) DeMario, James Linn. Oil wildcatter Roy Rogers discovers a rich deposit on a Mexican ranch but the owner is heavily in debt to a man who tries to kill a cousin who has half-interest in the land. Average Roy Rogers entry enhanced by Trucolor.\n\n**111** _ **Apache Territory**_ **** Columbia, 1958. 75 min. D: Ray Nazarro. SC: Charles R. Marion and George W. George. With Rory Calhoun, Barbara Bates, John Dehner, Carolyn Craig, Leo Gordon, Myron Healey, Frank De Kova, Reg Parton, Bob Woodward, Thomas Pittman. While crossing the desert, a drifter comes across a girl who is the sole survivor of a wagon train attack and the two unite to fight attacking Indians. Average but fairly interesting Rory Calhoun vehicle.\n\n**112** _ **Apache Trail**_ **** Metro-Goldwyn-Mayer, 1943. 66 min. D: Richard Thorpe. SC: Maurice Geraghty. With Lloyd Nolan, Donna Reed, William Lundigan, Ann Ayars, Connie Gilchrist, Chill Wills, Miles Mander, Gloria Holden, Ray Teal, Grant Withers, Fuzzy Knight, Trevor Bardette, Frank M. Thomas, George Watts. After their ceremonial grounds are desecrated by whites, Indians go on the warpath and innocent settlers face the consequences. Okay M-G-M oater from the World War II era with a fine cast and fast direction.\n\n**113** _ **Apache Uprising**_ **** 20th Century\u2013Fox\/CBS-TV, 1956. 45 min. With Ricardo Montalban, John Lupton, Rita Moreno, John Conte (host). A scout is assigned to convince an Apache chief to stop his attacks on U.S. mail carriers. Originally an episode of \"The 20th Century\u2013Fox Hour\" on CBS-TV, this interesting telefeature was based on _**Broken Arrow**_ (q.v.) and served as the pilot for the \"Broken Arrow\" (ABC-TV, 1956\u201360) series.\n\n**114** _ **Apache Uprising**_ **** Paramount, 1966. 90 min. Color. D: R.G. Springsteen. SC: Harry Sanford. With Rory Calhoun, Corinne Calvet, John Russell, Lon Chaney, Gene Evans, Richard Arlen, Robert H. Harris, Arthur Hunnicutt, DeForrest Kelley, George Chandler, Johnny Mack Brown, Jean Parker, Abel Fernandez, Don \"Red\" Barry, Jim Mitchum, Dan White, Reg Parton, Roy Jenson, Rodd Redwing, Ben Stanton. A diverse group of stage passengers head for a way station where a robbery is set to occur and an Indian attack looms. Better-than-average A.C. Lyles Western with a good cast, highlighted by Lon Chaney as a happy-go-lucky stage driver.\n\n**115** _ **Apache War Smoke**_ **** Metro-Goldwyn-Mayer, 1952. 67 min. D: Harold Kress. SC: Jerry Davis. With Gilbert Roland, Glenda Farrell, Robert Horton, Barbara Ruick, Gene Lockhart, Henry (Harry) Morgan, Patricia Tierani, Hank Worden, Myron Healey, Emmett Lynn, Argentina Brunetti, Bobby Blake, Douglass Dumbrille, Chubby Johnson, Iron Eyes Cody. Bandits head for a stop after robbing a stagecoach only to find it is about to be attacked by Indians. MGM Western with a strong performance by Gilbert Roland in a good, bad man role.\n\n**116** _ **Apache Warrior**_ **** 20th Century\u2013Fox, 1957. 73 min. D: Elmo Willams. SC: Kurt Neumann and Eric Norden. With Keith Larsen, Jim Davis, Rodolfo Acosta, John Miljan, Eddie Little Sky, Michael Carr, George Keymas, Lane Bradford, Eugenia Paul, Damian O'Flynn, Ray Kellogg, Allan Nixon, Karl Davis, Boyd Stockman. An Indian scout for the Army turns renegade when his brother is killed and his former white brother is forced to hunt him down. Fairly interesting outing with good work by Keith Larsen as the renegade, Jim Davis as the hunter and Rodolfo Acosta as the Indian responsible for the trouble.\n\n**117** _ **Apache Woman**_ **** American Releasing Corporation, 1955. 82 min. D: Roger Corman. SC: Lou Russoff. With Lloyd Bridges, Joan Taylor, Lance Fuller, Morgan Jones, Paul Birch, Paul Dubov, Jonathan Haze, Dick Miller, Chester Conklin, Lou Place, Gene Marlowe, Jean Howell. A government agent investigates several mysterious deaths blamed on reservation Indians. Roger Corman's second Western is a low grade affair that moves well and should satisfy his followers.\n\n_**Apache's Last Battle**_ see _**Old Shatterhand**_\n\n**118** _ **Apocalypse Joe**_ **** Columbia Film-Verleih, 1970. 90 min. Color. D: Leopoldo Savona. SC: Eduardo Brochero and Leopoldo Savona. With Anthony Steffen, Eduardo Fajardo, Mary Paz Pondal, Ferando Cerulli, Vernoica Korosec, Giulio Baraghini, Fernando Bilboa, Flora Carosello, Virginia Garcia, Ugo Adinolfi, Sergio Sagnotti, Renato Lupi, Miguel Del Castello, Brunio Arie, Angelo Susani, Gilberto Gailimberti, Riccardo Pizzuli, Silvano Spadaccino, Stelio Candelli, Omero Capanna, Artemio Antonini. An actor-gunfighter finds out a town boss murdered his uncle and took over his mine which rightfully belongs to him and he plans to get even. Fast paced Italian action thriller made as _**L'Uomo Chiamoto Apocalisse Joe**_ (A Man Called Apocalypse Joe).\n\n**119** _ **The Appaloosa**_ **** Universal, 1966. 98 min. Color. D: Sidney J. Furie. SC: James Bridges and Roland Kibbee. With Marlon Brando, Anjanette Comer, John Saxon, Emilio Fernandez, Alex Montoya, Miriam Colon, Rafael Campos, Frank Silvera, Argentina Brunetti, Larry Mann. A prize horse is stolen from an cowboy and he heads into the Mexican wilderness at the turn of the century to retrieve it. Slow moving and not very interesting, but highlighted by good photography.\n\n**120** _ **Appaloosa**_ **** Warner Bros., 2008. 115 min. Color. D: Ed Harris. SC: Robert Knott and Ed Harris. With Bobby Jauregui, Jeremy Irons, Timothy V. Murphy, Lucie Rains, Jim Tarwater, Boyd Kestner, Gabriel Marantz, Ed Harris, Viggo Mortensen, Lance Henriksen, Adam Nelson, Corby Griesenbeck, Benjamin Rosenshein, Cerris Morgan-Moyer, James Gammon, Timothy Spall, Tom Bower, Erik J. Bockemier, Fred Hice, Neil Summers, Tim Carroll, Rene Zellweger, Bounthanh Xaynhachack, Ariana Gil, Art Usher, Clark Sanchez, Cliff Gravel, Mike Watson, Rex Linn, Bob Harris, Daniel T. Parker, Martin Connelly, Ed Pennybacker, Alvin William \"Dutch\" Lunak. Two cowpokes are hired to rid a small town of a murderous rancher and his gang with one of them falling for a newly arrived young widow. Pleasant and well made Western with the title referring to the New Mexico town where the action takes place.\n\n**121** _ **The Apple Dumpling Gang**_ **** Buena Vista, 1975. 100 min. Color. D: Norman Tokar. SC: Don Tait. With Bill Bixby, Susan Clark, Don Knotts, Tim Conway, David Payne, Slim Pickens, Harry Morgan, Clay O'Brien, Brad Savage, Iris Adrian. A gambler becomes the guardian of three homeless children and tries to get rich by pulling off a fantastic bank robbery. Well done Disney comedy highlighted by the antics of Don Knotts and Tim Conway.\n\n**122** _ **The Apple Dumpling Gang Rides Again**_ **** Buena Vista, 1979. 89 min. Color. D: Bernard McEveety. SC: Don Tait. With Tim Conway, Don Knotts, Tim Matheson, Kenneth Mars, Elyssa Davalos, Jack Elam, Robert Pine, Harry Morgan, Ruth Buzzi, Audrey Totter, Richard X. Slattery, John Crawford, Cliff Osmond, Ted Gehring. Two blunderers are hunted throughout the West as outlaws. Mediocre sequel to _**The Apple Dumpling Gang**_ (q.v.).\n\n**123** _ **Los Apuros de dos Gallos**_ (The Tight Spot of Two Roosters). Oro Films, 1963. 93 min. Color. D-SC: Emilio Gomez Muriel. With Miguel Aceves Meija, Marco Antonio Muniz, Lilian de Celis, Lucha Villa, Angel Garaza, Fernando Soto \"Mantequilla,\" Miguel Angel Ferriz, Guillermo Alvarez Blanchi, Arthur \"Bigoton\" Castro, Julien de Meriche, Mario Orea, Emilio Garibay, Manuel Arvide, Jose Luis Fernandez, Martha Arlette, Pepe Hernandez, Jose Jasso. Four singing cowboys serenade the denizens of a rancho. Average Mexican Western musical comedy.\n\n**124** _ **Aqui esta Juan Colorado**_ **** Clasa-Mohme, 1946. 105 min. D: Rolando Aguilar. SC: Raul de Anda, Rolando Aguilar and Carlos Gaytan. With Luis Aguilar, Raul de Anda, Aurora Cortes, Iram Torres, Jose Torvay, Jose L. Murillo, Yadira Jiminez, Lidia Franco, Pepe Vava, Jose Paradve, Rafael Icardo. After helping to defeat government troops, a rebel leaders tries to settle down with his new love. Good historical Western from Mexico based on a folk hero.\n\n**125** _ **Arctic Flight**_ **** Monogram, 1952. 78 min. D: Lew Landers. SC: Robert Hill and George Bricker. With Wayne Morris, Lola Albright, Alan Hale, Jr., Carol Thurston, Phil Tead, Tom Richards, Anthony Garson, Kenneth MacDonald, Paul Bryar, Dale Van Sickel. An Alaskan bush pilot is hired by a big game hunter to fly him to an area supposedly to find polar bears but actually he is a Soviet spy. Well done little action melodrama with good location footage.\n\n**126** _ **Arctic Fury**_ **** RKO Radio, 1949. 61 min. D-SC: Norman Dawn. With Del Cambre, Eve Miller, Gloria Petroff, Merrill McCormick, Fred Smith. A plane carrying a doctor to a plague ridden village crashes in the Arctic and he fights the elements to survive. Slapped together programmer also called _**Tundra**_.\n\n**127** _ **Arctic Manhunt**_ **** Universal-International, 1949. 69 min. D: Ewing Scott. SC: Oscar Brodney and Joel Malone. With Mikel Conrad, Carol Thurston, Wally Cassell, Helen Brown, Harry Harvey, Chet Huntley, Paul E. Burns, Quianna. Insurance agents are on the trail of an ex-convict who has fled to Alaska with money taken from a robbery. Low budget action outing based on director Ewing Scott's book _Narana of the North_.\n\n**128** _ **Arena**_ **** Metro-Goldwyn-Mayer, 1953. 83 min. Color. D: Richard Fleischer. SC: Harold Jack Bloom. With Gig Young, Jean Hagen, Polly Bergen, Henry (Harry) Morgan, Barbara Lawrence, Robert Horton, Lee Aaker, Lee Van Cleef, Marilee Phelps, Jim Hayward, George Wallace, Stuart Randall, Morris Ankrum. A rodeo star lets success go to his head and this causes the near failure of his marriage. Mediocre modern day oater highlighted by rodeo footage from the annual Fiesta de los Vaqueros in Tucson, Arizona.\n\n**129** _ **Arizona**_ **** Columbia, 1940. 127 min. D: Wesley Ruggles. SC: Claude Binyon. With Jean Arthur, William Holden, Warren William, Regis Toomey, Paul Harvey, George Chandler, Byron Foulger, Porter Hall, Colin Tapley, Edgar Buchanan, Griff Barnett, Paul Lopez, Frank Darien, Syd Saylor, Addison Richards, Carleton Young, Jack Ingram, Emmett Lynn, I. Stanford Jolley, Uvaldo Varela, Earl Crawford, Ludwig Hart, Patrick Moriarty, Wade Crosby, Frank Hill, Nina Campana, Ralph Peters, Emmett Lynn, Walter Baldwin, William Harrigan, Walter Sande, John Arledge, Frank Richards, Kermit Maynard, Frank Brownlee, Lou Fulton, Stanley Brown, Richard Fiske, Fred Parker, Merrill McCormick, Julia Montoya. A young Arizona woman, with the aid of a Missouri man, sets out to battle corrupt elements in running a successful freight business and a large ranch. Entertaining \"A\" budget affair that is much too long, with a nice villainous portrayal by Warren William.\n\n_**Arizona**_ (1970) see _**Arizona Colt Returns**_\n\n**130** _ **Arizona Badman**_ **** Willis Kent, 1935. D: S. Roy Luby. SC: Willis Kent. With Reb Russell, Lois January, Charles (Slim) Whitaker, Edmund Cobb, Richard Botiller, Tommy Bupp, Ann Howard, Walter James, Ben Corbett, Tracy Layne, Lionel Backus, Ray Henderson, Silver Harr. An outlaw attempts to use his pretty stepdaughter to lure a cattlemen's association agent away from the area while he pulls off a rustling job. Tacky Reb Russell series affair with good work by leading lady Lois January and Slim Whitaker, Edmund Cobb and Richard Botiller as rustlers.\n\n_**Arizona Bill**_ see _**The Road to Fort Alamo**_\n\n**131** _ **Arizona Bound**_ **** Monogram, 1941. 57 min. D: Spencer Gordon Bennet. SC: Jess Bowers (Adele Buffington). With Buck Jones, Tim McCoy, Raymond Hatton, Luana Walters, Dennis Moore, Tristram Coffin, Kathryn Sheldon, Gene Alsace, Slim Whitaker, Artie Ortego, I. Stanford Jolley, Horace Murphy, Hal Price, Jack Daly. Three retired U.S. marshals on special assignment each take on different guises as they come to a town to find out who is the leader of a gang of stagecoach robbers. The first of eight films in the \"Rough Riders\" series and a good kickoff for the popular teaming of Buck Jones, Tim McCoy and Raymond Hatton.\n\n**132** _ **Arizona Bushwackers**_ **** Paramount, 1968. 86 min. Color. D: Lesley Selander. SC: Steve Fisher. With Howard Keel, Yvonne De Carlo, John Ireland, Marilyn Maxwell, Scott Brady, Brian Donlevy, Barton MacLane, James Craig, Reg Parton, Montie Montana, Eric Cody, Roy Rogers, Jr. During the Civil War a Confederate officer becomes the sheriff of an Arizona town and uncovers a gun runner dealing with local Indians. Well-done A.C. Lyles production highlighted by a top notch veteran cast with narration by James Cagney.\n\n_**Arizona Colt**_ see _**The Man from Nowhere**_\n\n**133** _ **Arizona Colt Returns**_ **** Izaro Films, 1970. 90 min. Color. D: Sergio Martino. SC: Ernesto Gastaldi and Joaquin Romero Hernandez. With Anthony Steffen, Rosalba Neri, Marcella Michelangeli, Aldo Sambrell, Roberto Camardiel, Raffaele [Raf] Baldassarre, Emilio Delle Piane, Gildo Di Marco, Jose Manuel Martin, Florentino Alonso, Silvio Bagolini, Luis Barboo, Enrico Marciani, Brizio Montinaro. An ex-convict and former bounty hunter takes revenge on the gang leader who murdered his sweetheart and his best friend. Average Spaghetti Western with the usual modicum of violence; also called _**Arizona**_.\n\n**134** _ **The Arizona Cowboy**_ **** Republic, 1950. 57 min. D: R.G. Springsteen. SC: Bradford Ropes. With Rex Allen, Teala Loring, Gordon Jones, Minerva Urecal, James Cardwell, Roy Barcroft, Stanley Andrews, Harry Cheshire, Edmund Cobb, Joseph Crehan, Steve Darrell, Douglas Evans, John Elliott, Chris-Pin Martin, Frank Reicher, George Lloyd, Lane Bradford. After service in World War II, a cowboy becomes the top attraction in a rodeo but bad guys falsely accuse him of being involved in a robbery. Rex Allen's starring debut is a good one and he became know by the moniker of the film's title.\n\n**135** _ **Arizona Cyclone**_ **** Universal, 1941. 57 min. D: Ray Taylor. SC: Sherman Lowe. With Johnny Mack Brown, Fuzzy Knight, Nell O'Day, Kathryn Adams, Herbert Rawlinson, Dick Curtis, Buck Moulton, Glenn Strange, Jack Clifford, Kermit Maynard, Frank Ellis, Carl Sepulveda, Chuck Morrison, Robert Strange, The Notables. The driver for a freight line tries to find out who murdered his boss over a telegraph hauling contract. Well done Johnny Mack Brown vehicle shackled by a trio of mediocre songs.\n\n_**Top:**_ **Yvonne De Carlo and Howard Keel in** _**Arizona Bushwhackers**_ **(Paramount, 1968).** _**Bottom:**_ **Fuzzy Knight, Nell O'Day and Johnny Mack Brown in** _**Arizona Cyclone**_ **(Universal, 1941).**\n\n**136** _ **Arizona Days**_ **** Syndicate, 1928. 50 min. D: J.P. McGowan. SC: Mack V. Wright. With Bob Custer, Peggy Montgomery, J.P. McGowan, John Lowell Russell, Mack V. Wright, Jack Ponder. A cowboy working for the cattlemen's association pretends to be an outlaw in order to join a rustling gang and finds a local rancher is doing the same thing. Fairly action filled Bob Custer silent vehicle.\n\n**137** _ **Arizona Days**_ **** Grand National, 1937. 56 min. D: John English. SC: Lindsley Parsons. With Tex Ritter, Eleanor Stewart, Snub Pollard, Ed Cassidy, William Faversham, Forrest Taylor, Glenn Strange, Horace Murphy, Earl Dwire, Budd Buster, Salty Holmes, William Desmond, Al Taylor. A drifter joins a traveling minstrel show which is burned by outlaws and he becomes a tax collector in order to replace the attraction's equipment and has a showdown with the villain responsible for the fire. Entertaining Tex Ritter outing.\n\n**138** _ **Arizona Frontier**_ **** Monogram, 1940. 55 min. D: Al Herman. SC: Robert Emmett (Tansey). With Tex Ritter, Arkansas Slim Andrews, Evelyn Finley, Frank LaRue, Tristram Coffin, Gene Alsace, Richard Cramer, James Pierce, Jim Thorpe, Hal Price, Sherry Tansey, Chick Hannon, Art Wilcox and His Arizona Rangers. A government agent is sent to investigate a series of Indian raids and becomes convinced the commander of a local Army post is behind the lawlessness. Filmed in Arizona, this Tex Ritter affair is pretty good with Tex singing \"Red River Valley.\"\n\n**139** _ **Arizona Gang Busters**_ **** Producers Releasing Corporation (PRC), 1940. 57 min. D: Peter Stewart (Sam Newfield). SC: Joseph O'Donnell and William Lively. With Tim McCoy, Pauline Haddon, Lou Fulton, Ted Adams, Forrest Taylor, Otto Reichow, Julian Rivero, Arno Frey, Kenne Duncan, Carl Mathews, Ben Corbett, Frank Ellis, Curley Dresden. A cowboy in Arizona uncovers a Nazi fifth column and sets out to expose the spies. Nicely entertaining, topical Tim McCoy vehicle.\n\n**140** _ **Arizona Gunfighter**_ **** Republic, 1937. 58 min. D: Sam Newfield. SC: George Plympton. With Bob Steele, Jean Carmen, Ted Adams, Ernie Adams, Lew Meehan, Steve Clark, John Merton, Karl Hackett, Frank Ball, Sherry Tansey, Jack Kirk, Hal Price, Budd Buster, Horace B. Carpenter, Tex Palmer, Allen Greer, Oscar Gahan. A cowboy is on the trail of the man who murdered his father. Entertaining and well-done with the usual Bob Steele plot motif.\n\n**141** _ **The Arizona Kid**_ **** Davis Distributing, 1928. 50 min. D-SC: Horace B. Carpenter. With Art Acord, Carol Lane, Cliff Lyons, Lynn Sanderson, Bill Conant, George Hollister, Horace B. Carpenter, James Tromp, Al Hoxie, Star (horse), Rex (dog). A U.S. marshal pretends to be a foppish bandit in order to round up a gang that robbed an express shipment and took the guard and his daughter hostage. Fast paced but low grade Art Acord vehicle, one of his last but not one of his best. Re-titled: _**Pursued**_.\n\n**142** _ **The Arizona Kid**_ **** Fox, 1930. 90 min. D: Alfred Santell. SC: Ralph Black. With Warner Baxter, Mona Maris, Carol(e) Lombard, Theodore Von Eltz, Arthur Stone, Solidad Jiminez, Walter P. Lewis, Jack Herrick, Wilfred Lucas, Hank Mann, James Gibson, Larry McGrath, De Sacia Mooers. Posing as a romantic Mexican miner, a bandit carries out his illegal activities while pursuing many girls until he falls for a married Eastern woman. Slow moving early talkie in which Warner Baxter carries on his Cisco Kid-like tradition.\n\n**143** _ **The Arizona Kid**_ **** Republic, 1939. 61 min. D: Joseph Kane. SC: Luci Ward and Gerald Geraghty. With Roy Rogers, George \"Gabby\" Hayes, Dorothy Sebastian, Stuart Hamblen, Sally March, David Kerwin, Earl Dwire, Peter Fargo, Fred Burns, Ed Cassidy, Jack Ingram, Ted Mapes, Frank McCarroll, Robert Middlemass, Georgia Simmons, Ben Corbett, Herman Hack, Tom Smith. During the Civil War, Roy and Gabby fight for the South and oppose a guerrilla who is allied with Roy's pal. Good action sequences and minor songs in this well done film with an effective performance by Stuart Hamblen as a pseudo\u2013Quantrill.\n\n**144** _ **The Arizona Legion**_ **** RKO Radio, 1939. 58 min. D: David Howard. SC: Oliver Drake. With George O'Brien, Lorraine Johnson (Laraine Day), Chill Wills, Carlyle Moore, Jr., Edward Le Saint, Harry Cording, Tom Chatteron, William Royle, Glenn Strange, Monte Montague, Bob Burns, John Dilson, Lafe McKee, Guy Usher, Robert Kortman, Wilfred Lucas, Jim Mason, Art Mix. An undercover agent, at a cavalry post commanded by a former friend who no longer trusts him, is assigned to expose a corrupt official. Highly competent and entertaining George O'Brien opus; remade as _**Fighting Frontier**_ (q.v.).\n\n**145** _ **Arizona Mahoney**_ **** Paramount, 1936. 61 min. D: James Hogan. SC: Robert Yost and Stuart Anthony. With Joe Cook, Robert Cummings, June Martel, Larry \"Buster\" Crabbe, Marjorie Gateson, Fred Kohler, John Miljan, Dave Chasen, Irving Bacon, Richard Carle, Billy Lee, Fuzzy Knight, Si Jenks, James P. Burtis, Frank Mayo, Jim Mason, Jack Perrin, Charles Williams, Frank McGlynn, Jr., Tiny Newland, Victor Potel, Anna Demetrio, Charlotte Wyatt, Dot Farley, Jimmy Conlin, James C. Morton, John \"Skins\" Miller, Chester Gan, Harry Tyler, Spike Spackman, Frank Cordell, Cecil Kellogg, Al Burk, Bill Hurley, Johnny Eckert. An Eastern tenderfoot is falsely accused of being an outlaw but the real crooks proves his innocence. Standard \"B\" outing from Zane Grey's _Stairs of Sand_ with an emphasis on comedy, which was first filmed under that title in 1929 by Paramount with Wallace Beery, Jean Arthur, Phillips Holmes and Fred Kohler. Reissue title: _**Arizona Thunderbolt**_.\n\n**146** _ **Arizona Manhunt**_ **** Republic, 1951. 60 min. D: Fred C. Brannon. SC: William Lively. With Michael Chapin, Eilene Janssen, James Bell, Lucille Barkley, Roy Barcroft, John Baer, Harry Harvey, Stuart Randall, Ted Cooper, Hazel Shaw, Herman Hack, Foxy Callahan. Two youngsters aid an old sheriff and his deputy in defeating a gang of outlaws. Mediocre entry in the \"Rough Ridin' Kids\" series not helped much by the Republic sheen.\n\n_**Arizona Mission**_ see _**Gun the Man Down**_\n\n**147** _ **Arizona Raiders**_ **** Paramount, 1936. 54 min. D: James Hogan. SC: Robert Yost and John Krafft. With Larry \"Buster\" Crabbe, Marsha Hunt, Raymond Hatton, Jane Rhodes, Grant Withers, Johnny Downs, Don Rowan, Arthur Aylesworth, Richard Carle, Herbert Heywood, Petra Silva, Augie Gomez, Ken Cooper, James P. Burtis, Spike Spackman. A gunman sides with settlers who are being terrorized by a gang of outlaws in early Arizona. Fair \"B\" action film based on Zane Grey's _Raiders of Spanish Peaks_. Reissued as _**Bad Men of Arizona**_.\n\n**148** _ **Arizona Raiders**_ **** Columbia, 1965. 88 min. Color. D: William Witney. SC: Alex Gottlieb, Mary Willingham and Willard Willingham. With Audie Murphy, Michael Dante, Ben Cooper, Buster Crabbe, Gloria Talbott, Ray Stricklyn, Read Morgan, George Keymas, Willard Willingham, Fred Graham. After the Civil War a former member of Quantrill's Raiders joins the newly formed Arizona Rangers in hunting down his former cohorts who have been raiding area settlements. Better-than-average Audie Murphy vehicle not hurt by William Witney's direction or a good supporting cast, including Buster Crabbe.\n\n**149** _ **The Arizona Ranger**_ **** RKO Radio, 1948. 63 min. D: John Rawlins. SC: Norman Houston. With Tim Holt, Jack Holt, Nan Leslie, Richard Martin, Steve Brodie, Paul Hurst, Robert Bray, Jim Nolan, Richard Benedict, William Phipps, Harry Harvey. Two new rangers join forces with an old-time lawman in taking on an outlaw gang. Typically good Tim Holt outing enhanced by his father, Jack Holt, as the veteran ranger.\n\n**150** _ **Arizona Roundup**_ **** Monogram, 1942. 56 min. D: Robert Emmett Tansey. SC: Robert Emmett (Tansey) and Frances Kavanaugh. With Tom Keene, Hope Blackwood, Frank Yaconelli, Sugar Dawn, Jack Ingram, Steve Clark, Nick Moro, Tom Seidel, Hal Price, I. Stanford Jolley, Ed Cassidy, Tex Palmer, Gene Alsace, Fred Hoose, Horace B. Carpenter, James Sheridan (Sherry Tansey. A federal agent goes to work for a rancher selling wild horses to the government and tries to break up a combine formed by two crooks. Okay Tom Keene vehicle.\n\n**151** _ **Arizona Stage Coach**_ **** Monogram, 1942. 58 min. D: S. Roy Luby. SC: Arthur Hoerl. With Ray Corrigan, John King, Max Terhune, Nell O'Day, Kermit Maynard, Charles King, Carl Mathews, Slim Whitaker, Steve Clark, Frank Ellis, Roy Harris, Jack Ingram, Stanley Price, Forrest Taylor, Richard Cramer, Eddie Dean, Slim Harkey, Jimmy Aubrey, Milburn Morante, Denver Dixon, Herman Hack. The Range Busters try to help a man wrongly accused of murder by hunting for the real killer. Action filled entry in the \"Range Busters\" series (the last with the trio of Corrigan, King and Terhune) but it is hurt by too much stock footage and forced comedy.\n\n**152** _ **Arizona Territory**_ **** Monogram, 1950. 56 min. D: Wallace Fox. SC: Adele Buffington. With Whip Wilson, Andy Clyde, Nancy Saunders, Dennis Moore, John Merton, Carl Mathews, Ted Adams, Carol Henry, Bud Osborne, Frank Austin. A lawman is after counterfeiters who are transferring fake currency to merchants in the East. Average Whip Wilson vehicle.\n\n**153** _ **The Arizona Terror**_ **** Tiffany, 1931. 64 min. D: Phil Rosen. SC: John Francis (Jack) Natteford. With Ken Maynard, Lina Basquette, Hooper Atchley, Nena Quartaro, Michael Visaroff, Murdock MacQuarrie, Charles King, Tom London, Edmund Cobb, Fred Burns, Jack Rockwell, Jim Corey. Falsely accused of murder, a man is saved by a rancher and his daughter and he soon becomes aware of a plot to take over his benefactor's land. Nicely done early sound feature, fast paced with scenic locations and Michael Visaroff is effective as a good-badman character.\n\n**154** _ **Arizona Terrors**_ **** Republic, 1942. 56 min. D: George Sherman. SC: Doris Schroeder and Taylor Cavan. With Don \"Red\" Barry, Lynn Merrick, Al St. John, Reed Hadley, Rex Lease, John Maxwell, Frank Brownlee, Lee Shumway, Tom London, John Merton, Fred \"Snowflake\" Toones, Curley Dresden, Herman Hack, Kermit Maynard, Bud Osborne, Jack Kirk. Two cowboys come to the aid of ranchers who are being terrorized and heavily taxed by a tyrant who claims the land they have settled belongs to him via a Spanish land grant. The old saw about an ancient land grant is given fresh air here and the end result is lots of fast action. Remake of _**The Night Riders**_ (1939) [q.v.].\n\n_**Arizona Thunderbolt**_ see _**Arizona Mahoney**_\n\n**155** _ **Arizona Trail**_ **** Universal, 1943. 57 min. D: Vernon Keays. SC: William Lively. With Tex Ritter, Fuzzy Knight, Dennis Moore, Janet Shaw, Johnny Bond and His Red River Valley Boys, Jack Ingram, Erville Alderson, Joseph Greene, Glenn Strange, Dan White, Art Fowler, Roy Brent, George Gray, William Yip, Ray Jones, Bill Wolfe. A young man returns home to find his dad battling land-grabbers and he joins in the fight. Fairly well done Tex Ritter vehicle.\n\n**156** _ **Arizona Trails**_ **** Superior Talking Pictures, 1935. 56 min. D: Al James (Victor Adamson\/Denver Dixon). SC: Tom Camden. With Bill Patton, Edna Aslin, Ed Carey, Tom Camden, Wallace Pindell, Delmar Costello, Herman Hack, Fred Parker, Ernest Scott, Frank Ball, Denver Dixon. A cowboy comes to the aid of a young man accused of murdering a gambler to whom he owed debts. Sub-par Victor Adamson production; a useless attempt to bring Bill Patton back to stardom.\n\n**157** _ **Arizona Whirlwind**_ **** Monogram, 1944. 59 min. D: Robert Tansey. SC: Frances Kavanaugh. With Ken Maynard, Hoot Gibson, Bob Steele, Myrna Dell, Ian Keith, Donald Stewart, Charles King, Karl Hackett, George Chesebro, Dan White, Frank Ellis, Charles Soldani, Charley Murray, Jr. A trio of U.S. marshals come to a small town to combat a crooked banker who his behind a diamond smuggling ring. Nicely done \"Trail Blazers\" feature with lots of action and good humor; Ken Maynard's last film in the series.\n\n**158** _ **The Arizona Wildcat**_ **** 20th Century\u2013Fox, 1939. D: Herbert I. Leeds. SC: Barry Trivers and Jerry Cady. With Jane Withers, Leo Carrillo, Pauline Moore, Henry Wilcoxon, William Henry, Douglas Fowley, Ethienne Girardot, Harry Woods, Russell Simpson, Lew Kelly, Rosita Harlan, Bruce Mitchell, Arthur Loft, Chris-Pin Martin, Julian Rivero, Fred Malatesta, Charles Stevens, Bob McKenzie, Art Dupuis, Buck Moulton, Donald Haines, Jack Baxley, Ed Brady, Neal Hart, Russ Powell, Anne Schaefer, Stub Musselman, George Ovey. After he is falsely accused of a crime, a former bandit is helped by his adopted daughter in exposing a crooked sheriff and an outlaw gang. Another venture out West with Jane Withers and strictly for her fans.\n\n**159** _ **The Arizonian**_ **** RKO Radio, 1935. 75 min. D: Charles Vidor. SC: Dudley Nichols. With Richard Dix, Margot Grahame, Preston Foster, Louis Calhern, James Bush, Ray Mayer, Francis Ford, J. Farrell MacDonald, Joseph Sawyer, Edward Van Sloan, Robert Kortman, Ted Oliver, Willie Best, Etta McDaniel, Jim Thorpe, Hank Bell. An honest lawman tries to protect his brother and his girl friend by cleaning up a small town run by a crook. Very entertaining Richard Dix vehicle and one of the best \"A\" budget Westerns of the 1930s.\n\n**160** _ **Arkansas Judge**_ **** Republic, 1941. 72 min. D: Frank McDonald. SC: Dorrell McGowan and Stuart McGowan. With The Weaver Brothers and Elviry, Roy Rogers, Pauline Moore, Spring Byington, Frank M. Thomas, Veda Ann Borg, Monte Blue, Eily Malyon, Loretta Weaver, Minerva Urecal, Harrison Greene, Frank Darien, Russell Hicks, Edwin Stanley. A working woman is accused of a crime actually committed by the daughter of a judge, the man who wrongly blamed her. Vehicle for the corn comedy and music of The Weaver Brothers and Elviry, with Roy Rogers along for bait; humorous.\n\n**161** _ **Arkansas Swing**_ **** Columbia, 1948. 63 min. D: Ray Nazarro. SC: Barry Shipman. With Gloria Henry, The Hoosier Hotshots (Charles \"Gabe\" Ward, Ken Trietsch, Paul \"Hezzie' Trietsch, Gil Taylor), Stuart Hart, Mary Eleanor (Elinor) Donahue, June Vincent, Dorothy Porter, Douglas Fowley, Syd Saylor, Eddy Waller, Pierre Watkin, Dick Elliott, Cottonseed Clark, The Texas Rangers (Robert \"Captain Bob\" Crawford, Edward \"Tookie\" Cronenbold, Francis \"Irish\" Mahaney, Roderick \"Dave\" May). A small girl enlists the aid of The Hoosier Hotshots in helping her conceal a prize race horse from a grasping feed store owner and a rich socialite who owns a rival steed. Average hokum Columbia \"B\" musical. Also called _**Texas Sandman**_.\n\n**162** _ **Armed and Dangerous: Time and Heroes of Bret Harte**_ **** Gorky Film, 1977. 99 min. Color. D: Vladimir Vaynshtok. SC: Vladimir (Vaynshtok) Vladimirov and Pavel Finn. With Donalas Banionis, Mircea Verolu, Lyudmila Senchina, Leonid Bronevoy, Lev Durov, Algimantas Masiulis, Sergei Martinson, Maria Ploae, Oleg Zhakov, Vesvolod Abdulov, Talgat Nigmatulin, Dimtrie Craciun, Ferenc Bencze, Grigori Lyampe, Jan Schanilec. In the late 1880s a homesteader discovers oil on his land with the local banker blackmailing his wife and former mistress in an attempt to get the property. Posh Soviet production, based on Bret Harte's novel _Gabriel Conroy_ , that is deluged with left wing propaganda. Filmed as _**Vooruzhyon I Ochen Opasen**_.\n\n**163** _ **El Arracadas**_ (The Pendants). Cima Films, 1978. 110 min. Color. D: Alberto Mariscal. SC: Adolfo Torres Portillo. With Vicente Fernandez, Fernando Almada, Roberto Canedo, Patricia Rivera, Mario Almada, Raquel Olmedo, Maria Teresa Alvarez, Umberto Elizondo, Wanda Seux, Alfredo Gutierrez, Pedro Munoz, Patricia Maldonado, Luis de Alba, Alejandro Fernandez, Arturo Martinez, Jr. A gunman sets out to find the man who murdered his father. Violent Mexican revenge oater.\n\n**164** _ **Arriba las Manos Texano**_ (Hands Up, Texan) Estudios America, 1969. 85 min. Color. D: Alfredo B. Crevenna. SC: Alfredo Ruanova and Alfonso Morones A. With Rodolfo de Anda, Ofelia Montesco, Eric del Castillo, David Silva, Rosa Maria Gallardo, Cynthia Mandan, Juan Gallardo, Quintin Bulnes, Alfonso Mejia, Bruno Rey, Ricardo Munoz, Jose Luis Caro, Juan Garza, Carlos Suarez, Jesus Gomez, Victor Jordan, Raul Montoya. A cowboy teams with a gunman to stop a ruthless man and his gang from forcing ranchers to sell out cheap so they can get control of the area. Nicely done Mexican Western.\n\n**165** _ **Arrow in the Dust**_ **** Allied Artists, 1954. 80 min. Color. D: Lesley Selander. SC: Don Martin. With Sterling Hayden, Coleen Gray, Keith Larsen, Tom Tully, Lee Van Cleef, Tudor Owen, Jimmy Wakely, John Pickard, Carleton Young, Iron Eyes Cody. A cavalry soldier deserts his unit and comes across a dying officer and assumes his identity which he later uses to take command of a wagon train under siege by Indians. Rather well done action drama with good direction by Lesley Selander; Jimmy Wakely sings \"The Weary Stranger.\"\n\n**166** _ **Arrowhead**_ **** Paramount, 1953. 105 min. Color. D-SC: Charles Marquis Warren. With Charlton Heston, Jack Palance, Katy Jurado, Brian Keith, Mary Sinclair, Milburn Stone, Richard Shannon, Lewis Martin, Frank De Kova, Robert Wilke, Peter Coe, John Pickard, Pat Hogan, Mike Ragan, Chick Hannon, James Burke. In the Southwest a cavalry officer and the chief of the Tonto Apaches are at odds when the Indians refuse to sign a peace treaty and the soldier is ordered to consummate such a signing. Fairly entertaining feature greatly aided by the work of Charlton Heston and Jack Palance in the lead roles.\n\n_**As I Rode Down to Laredo**_ see _**Return of the Gunfighter**_\n\n**167** _ **El Asesino Enmascarado**_ (The Masked Assassin). Producciones Sotomayer, 1961. 86 min. D: Manuel Munoz. SC: Manuel Munoz and Jose Maria Fernandez Usain. With Miguel Aceves Mejia, Ana Bertha Lepe, Joaquin Cordero, Lilia Pardo, Eduardo Noriega, Arturo Martinez, Luis Aragon, Ramon Bugarini, Armando Velazco, El Enano Santanon, Jose Chavez, Lupe Carriles, Chel Lopez. In a lawless area where the locals are being victimized, a masked man emerges to defend the innocent. Average Mexican Western with the usual masked hero; sequel to _**Asesinos de la Lucha Libre**_ (Assassins of the Wrestlers), also released in 1961 and starring Miguel Aceves, Joaquin Cordero and Lilia Pardo.\n\n**168** _ **Los Asesinos**_ (The Assassins). Filmica Vergara, 1968. 85 min. Color. D: Jaime Salvador. SC: Federico Curiel and Ramon Obon. With Nick Adams, Regina Torne, Pedro Armendariz, Jr., Elsa Cardenas, Amadee Chabot, Carlos East, Chuck Anderson, Pancho Cordova, Juan Gallardo, Manuel Donde, Jose Eduardo Perez, Raul Perez, Prieto, Alfonso Mungula, Tito Novaro, Guillermo Ayala, Ali Junco, Roberto Iglesias, Queta Carrasco, Juan Garza, Rene Barrera, Jose Dupeyron, Ignacio Ballalvan, Ernesto Juarez, Carlos Ortigoza. Two bandits with a deep distrust of each other vie for control of a small border town. Mexican Western produced by Luis Enrique Vergara and mainly of interest because it was Nick Adams' final film.\n\n**169** _ **The Assassination of Jesse James by the Coward Robert Ford**_ **** Warner Bros., 2007. 160 min. Color. D-SC: Andrew Dominik. With Brad Pitt, Mary-Louise Parker, Brooklynn Prouix, Dustin Bollinger, Casey Affleck, Sam Rockwell, Jeremy Renner, Sam Shepard, Michael Parks, Garret Dillahunt, Paul Schneider, Joel McNichol, James DeFelice, J.C. Roberts, Darrell Orydzuk, Jonathan Erich Drachenberg, Torben S. Hansen, Alison Elliott, Lauren Calvert, Kailin See, Tom Aldredge, Jesse Frechette, Pat Healy, Ted Levine, Joel Duncan, James Carville, Stephanie Wahlstrom, Adam Arlukiewicz, Ian Ferrier, Michael Rogers, Calvin Blid, Sarah Lind, Nick Cave, Matthew Walker, Zooey Deschanel, Michael Copeman, Laryssa Yanchak, Hugh Ross, Myrna Vallance, Doug Christian, Sarah Murphy-Dyson, Barb Mitchell. Although he idolizes Jesse James, young Bob Ford plans to kill him for a reward and public recognition. Fair screen adaptation of Ron Hansen's novel.\n\n**170** _ **At Gunpoint**_ **** Allied Artists, 1955. 81 min. Color. D: Alfred Werker. SC: Daniel B. Ullman. With Fred MacMurray, Dorothy Malone, Walter Brennan, Tommy Rettig, Skip Homeier, John Qualen, Harry Shannon, Whit Bissell, Irving Bacon, Jack Lambert, Frank Ferguson, James Anderson, John Pickard, Charles Morton, Anabel Shaw, Rick Vallin, Kim Charney, Mimi Gibson, James Griffith, Harry Lauter, Byron Foulger, Keith Richards, Lyle Latell, Barbara Woodell, Gertrude Astor, Harry Strang, Stephen Wootten, James Lilburn. A businessman is forced to kill a robber and finds himself deserted by his friends and aided only by his pretty wife when the dead man's brother vows revenge against him. Tame take-off of _**Nigh Noon**_ (q.v.) with a good cast.\n\n**171** _ **Audaz y Bravero**_ (Bold and Wild). Cinematografica Jalisco, S.A., 1965. 85 min. D: Alfonso Corona Blake. SC: Eduardo Gazon, Alfredo Corona Blake and Jesus Murcielago Velazquez. With Luis Aguilar, Lilia Pardo, Ofelia Montesco, Dagoberto Rodriguez, Noe Murayama, Arturo Martinez, Mario Chavez, Agustin Isunza, Guillero Merrara, Armando Gutierrez, Emilio Garibay, Sergio Barrios, Jose Alfredo Jiminez. A wealthy rancher loses his fiancee when another man spirits her away on their wedding day. Solid Mexican Western.\n\n**172** _ **The Aurora Encounter**_ **** New World, 1986. 90 min. Color. D: Jim McCullough, Sr. SC: Jim McCullough, Jr. With Jack Elam, Peter Brown, Carol Bagdararian, Dottie West, Spanky McFarland, Charles B. Pierce, Mickey Hays, Will Mitchell, Mindy Smith, Carly McCullough, Tracy Kuehert, Paula Barrett, Foster Litton, Kaye Winters, Lois Lane, Ann Hazlett, Don Pirl, Cyrus Thiebeault, Rick Phiffer, Big Charles Gibbons. Evil aliens take over a Western town and a newspaperwoman tries to write an expose of the invasion. Cheap Texas made sci-fi Western.\n\n**173** _ **The Avenger**_ **** Awyon, 1924. 50 min. D: Charles R. Seeling. With Guinn Williams, Kathleen Collins, Fred Maletesta. A cowboy tries to prove to his girl that her other suitor, a real estate agent, is a crook. Low grade poverty row affair for fans of Guinn \"Big Boy\" Williams.\n\n**174** _ **The Avenger**_ **** Columbia, 1931. 65 min. D: Roy William Neill. SC: George Morgan and Jack Townley. With Buck Jones, Dorothy Revier, Ed Peil, Sr., Otto Hoffman, Sidney Bracey, Edward Hearn, Walter Percival, Paul Fix, Frank Ellis, Al Taylor, Slim Whitaker, Blackjack Ward. Taking on the guise of a Mexican bandit, a man plans to take revenge on the gang who murdered his family. Buck Jones' sixth sound film is a fine one with a south of the border setting and the star is very good in his Mexican disguise. Remade as _**Vengeance of the West**_ (q.v.).\n\n**175** _ **The Avenger**_ **** B.R.C.\/Estela Films, 1966. 92 min. Color. D: Ferdinando Baldi. SC: Franco Rossetti and Ferdinando Baldi. With Franco Nero, Cole Kitosch, Elisa Montes, Hugo Blanco, Alberto Dell'Acqua (Robert Widmark), Livio Lorenzon, Jose Suarez, Luigi Pistilli, Antonella Murgia, Jose Guardiola. Trying to avenge the murder of his father, a man finds out the killer is the father of his younger half-brother. Fair Italian horse opera, a bit on the slow side. Released in Italy as _**Texas, Addio**_ (Goodbye Texas).\n\n**176** _ **The Avengers**_ **** Republic, 1950. 90 min. D: John Auer. SC: Lawrence Kimble and Aeneas MacKenzie. With John Carroll, Adele Mara, Mona Maris, Roberto Airaldi, Jorge Villoldo, Vivian Bay, Fernando Lamas, Vincent Padula, Cecile Lezard, Juan Olaguivel, Andre LeBlanc. When settlers are attacked by bandits in South America, the heroic Don Careless rides to their rescue and avenges the murder of his father. Okay South American \"Western\" based on the novel _Don Careless_ by Rex Beach.\n\n**177** _ **The Avenging**_ **** Comworld Pictures, 1981. 91 min. Color. D-SC: Lyman Dayton. With Michael Horse, Efrem Zimbalist, Jr., Matt Stetson, Sherry Hursey, Taylor Lacher, Joseph Running Fox, Cam Clarke, Brenda Venus, Dan August, Dorothy Romero. A well-educated half-breed finds himself the victim of racial persecution and accused of horse stealing. Passable melodrama with good work by Efrem Zimbalist, Jr.\n\n**178** _ **Avenging Angel**_ **** Hallmark Channel, 2007. 81 min. Color. D: David S. Cass, Sr. SC: William Sims Myers. With Kevin Sorbo, Nick Chinlund, Cynthia Watros, Richard Lee Jackson, Lorin McCraley, Wings Hauser, Joey King, Jim Haynie, Jack Riley, Sam Sorbo, Willow Geer, Dennis Fitzgerald, Tom O'Keefe, Brandon Parrish, Van Epperson, David Wells, Earl H. Bullock, Bobbi Stamm, Fred Cross, David Atkinson, Tom Carey, Tim Trobec, Brad Carter, Todd Royal, Dave Rowden, Rachel Abendroth. After losing his family to a land baron, a preacher becomes a gunfighter and eventually comes to the aid of a religious group being evicted from their land by a crooked lawman. Average TV Western.\n\n**179** _ **The Avenging Rider**_ **** RKO Radio, 1943. 55 min. D: Sam Nelson. SC: Harry O. Hoyt. With Tim Holt, Cliff Edwards, Ann Summers, Davison Clark, Norman Willis, Karl Hackett, Earle Hodgins, Ed Cassidy, Kenne Duncan, Bud Osborne, Robert Kortman, Guy Usher, Lloyd Ingraham, David Sharpe. A man wants to clear himself and his buddy of the murder of his partner in a gold mine. Good Tim Holt vehicle enhanced by sidekick support from Cliff \"Ukulele Ike\" Edwards.\n\n**180** _ **Avenging Waters**_ **** Columbia, 1936. 56 min. D: Spencer Gordon Bennet. SC: Nate Gatzert. With Ken Maynard, Beth Marion, John Elliott, Zella Russell, Ward Bond, Wally Wales, Tom London, Edmund Cobb, Buffalo Bill, Jr., Glenn Strange, Edward Hearn, Buck Moulton, Cactus Mack. A cowboy leads a cattle herd to be sold to a rancher and runs into a range feud over fencing rights. Pretty good Ken Maynard action entry, slightly marred by mediocre processing shots involving the climactic flood.\n\n**181** _ **Aventuras de las Hermanas X**_ (Adventures of the Sisters X) Estudios America, 1963. 87 min. D: Federico Curiel. SC: Federico Curiel and Alfredo Ruanova. With Kitty de Hoyos, Dacia Gonzalez, Dagoberto Rodriguez, Rene Cardona, Jose Chavez, Rafael Bertrand, Pancho Cordoba, Santanon, Roberto Ramirez Garza, Mario Alberto Rodriguez, Celia Viveros, Antonio Raxel, Agustin Fernandez. A dozen years after their parents are murdered, a traveling entertainer and her sister seek the killer. The first of a quartet of exciting female avenger features starring Kitty de Hoyos and Dacia Gonzalez, followed by _**Las Vengadoras Enmascaradas**_ , _**Las Hijas del Zorro**_ and _**Las Invencibles**_ (qq.v.).\n\n**182** _ **The Awakening Land**_ **** NBC-TV, 1978. 111 min. Color. D: Boris Sagal. SC: James Lee Barrett and Liam O'Brien. With Elizabeth Montgomery, Hal Holbrook, Jane Seymour, Steven Keats, Louise Latham, Tony Mockus, Derin Atay, Michelle Stacy, Barney McFadden, W.H. Macy, Jeanette Nolan, James D. O'Reilly, Charles Tyner, Dorrie Kavanaugh, Bert Remsen, Sandra Wheeler, Art Kassul, Louis Plante, George Holcomb, Charles Gowan, Sean Frye, Johnny Timko, Pia Romans, Theresa Landreth, Dennis Dimster, Tracy Kieronomos, Katy Kurtzman, Bryne Piven, Devon Ericson, Martin Scanlan, Jane Alderman, Julie Briggs, Bernie Kuby, Bill Neal, Paul Swanson, Joan Tompkins, George Womack, Robert Padilla, Ann Eggert, John Kirk, Allen Hamilton, Oseley Cole. The life and times of a frontier woman in the Ohio Territory in the late 1700s and early 1800s. Sprawling three-part TV drama from the novels by Conrad Richter.\n\n_**Awkward Hands**_ see _**Manos Torpes**_\n\n**183** _ **Back in the Saddle**_ Republic, 1941. 73 min. D: Lew Landers. SC: Richard Murphy and Jesse Lasky, Jr. With Gene Autry, Smiley Burnette, Mary Lee, Edward Norris, Jacqueline Wells (Julie Bishop), Addison Richards, Arthur Loft, Edmund Cobb, Reed Howes, Stanley Blystone, Curley Dresden, Fred \"Snowflake\" Toones, Frank Ellis, Jack O'Shea, Herman Hack, Bob Burns, Frankie Marvin, Bill Nestell, Art Dillard, Jess Cavin, John Indrisano, Bob Woodward, Bob Card, Jack C. Smith, Roy Bucko, Jack Montgomery. A cowpoke inherits a ranch and finds it is rich in copper, causing a local boom, but crooks are soon after the property. Fair Gene Autry outing with a heavy emphasis on songs.\n\n**184** _ **Back to God's Country**_ Canadian Photoplays\/First National, 1919. 60 min. D: David M. Hartford. SC: James Oliver Curwood. With Nell Shipman, Wheeler Oakman, Wellington Palyter, Ralph Laidlaw, Charles Arling. A husky dog comes to the aid of a young woman chased by lecherous crooks. Interesting silent adaptation of James Oliver Curwood's story \"Wapi the Walrus\" with especially good photography and scenery; filmed in northern Canada.\n\n**185** _ **Back to God's Country**_ Universal-International, 1953. 78 min. Color. D: Joseph Pevney. SC: Tom Reed. With Rock Hudson, Steve Cochran, Marcia Henderson, Hugh O'Brian, Chubby Johnson, Tudor Owen, John Cliff, Bill Radovich, Arthur Space, Pat Hogan. Carrying a cargo of valuable furs, a sea captain and his wife land in a remote Canadian harbor where a trader plots to steal the pelts, murder the seaman and take the woman. Color and the villainy of Steve Cochran greatly help this adaptation of James Oliver Curwood's story.\n\n**186** _ **The Back Trail**_ Universal, 1924. 47 min. D: Clifford Smith. SC: Isadore Bernstein. With Jack Hoxie, Eugenia Gilbert, Claude Payton, Alton Stone (Al Hoxie), William Lester, William McCall, Buck Connors, Pat Harmon. After losing his memory in the war, a man is told he has a criminal past and is blackmailed into breaking his father's will, harming the foster sister he loves, but he is aided by a ubiquitous tramp. Average Jack Hoxie silent feature from a story by Walt Coburn.\n\n**187** _ **Back Trail**_ Monogram, 1948. 54 min. D: Christy Cabanne. SC: J. Benton Cheney. With Johnny Mack Brown, Raymond Hatton, Mildred Coles, Ted Adams, Pierce Lyden, Jimmy Horne, Jr., Snub Pollard, Marshall Reed, Bob Woodward, Carol Henry, George Morrell. A saloon owner is blackmailing a banker for a crime he did not commit and the victim asks for help from a State Protective League special investigator. Average.\n\n**188** _ **Backfire**_ Aywon, 1922. 50 min. D-SC: Alvin J. Neitz (Alan James). With Jack Hoxie, Florence Gilbert, George Sowards, Lew Meehan, William Lester, William Gould, Bert Rollins, Nellie Anderson, Poke Williams. A cowboy and his pal are framed for the murder of a Wells Fargo agent during a holdup, the crime being committed by a gang headed by a corrupt ranch foreman. Jack Hoxie action silent feature that will please his fans.\n\n**189** _ **Backlash**_ Universal-International, 1956. 84 min. Color. D: John Sturges. SC: Borden Chase. With Richard Widmark, Donna Reed, William Campbell, John McIntire, Barton MacLane, Edward Platt, Harry Morgan, Robert Wilke, Reg Parton, Robert Foulk, Roy Roberts, Rex Lease, Glenn Strange, I. Stanford Jolley, Kermit Maynard, Jack Lambert, Gregg Barton, Fred Graham, Phil Chambers, Frank Chase. After five men die in an Indian attack, a survivor is hunted by a posse because they think he escaped with a fortune in gold. Entertaining affair done in the typically good John Sturges fashion.\n\n**190** _ **Backtrack**_ Universal, 1969. 97 min. Color. D: Earl Bellamy. SC: Borden Chase. With James Drury, Neville Brand, Doug McClure, Rhonda Fleming, Ida Lupino, Fernando Lamas, Peter Brown, William Smith, Philip Carey, Royal Dano, Gary Clarke, Randy Boone, L.Q. Jones, Carol Byron, Ross Elliott, Hal Baylor, George Savalas, Alberto Morin, Ruben Moreno, Teresa Terry, Pricilla Garcia. On the way to Mexico to get a prize bull four Texas Rangers get involved with bandits. Tacky compilation of episodes of \"The Virginian\" and \"Laredo\" television series and issued theatrically.\n\n**191** _ **Bad Bascomb**_ **** Metro-Goldwyn-Mayer, 1946. 112 min. D: S. Sylvan Simon. SC: William Lipman and Grant Garrett. With Wallace Beery, Margaret O'Brien, Marjorie Main, J. Carrol Naish, Frances Rafferty, Marshall Thompson, Russell Simpson, Warner Anderson, Donald Curtis, Connie Gilchrist, Sara Haden, Renie Riano, Henry O'Neill, Frank Darien, Jane Green, Stanley Andrews, Joseph Crehan, Arthur Space, Eddie Acuff, John Gallaudet, Wally Cassell, Clyde Fillmore. Two bandits take refuge with a group of Mormons and one pays them back by stealing their money while the other remains to aid them during an Indian raid. Fans of Wallace Beery, Margaret O'Brien and Marjorie Main will go for this one.\n\n**192** _ **Bad Company**_ Paramount, 1972. 93 min. Color. D: Robert Benton. SC: David Newman and Robert Benton. With Jeff Bridges, Barry Brown, Jim Davis, David Huddleston, John Savage, Jerry Houser, Damon Cofer, Geoffrey Lewis, Ed Lauter, John Quade, Jean Allison, Charles Tyner, Claudia Bryar, Todd Martin. Two young draft dodgers head West on a robbery spree and are hounded by a relentless lawman during the Civil War. Underrated Western greatly helped by a fine cast, especially Jim Davis as the sheriff.\n\n**193** _ **Bad Day at Black Rock**_ Metro-Goldwyn-Mayer, 1955. 81 min. Color. D: John Sturges. SC: Millard Kaufman and Don McGuire. With Spencer Tracy, Robert Ryan, Anne Francis, Dean Jagger, Walter Brennan, John Ericson, Ernest Borgnine, Lee Marvin, Russell Collins, Walter Sande. A one-armed man arrives in a small Western town and uncovers a secret that upsets the locals. Excellent modern-day Western, well made, acted and directed.\n\n**194** _ **Bad Girls**_ 20th Century\u2013Fox, 1994. 99 min. Color. D: Jonathan Kaplan. SC: Ken Friedman and Yolande Finch. With Madeleine Stowe, Mary Stuart Masterson, Andie MacDowell, Drew Barrymore, James Russo, James LeGros, Robert Loggia, Dermot Mulroney, Jim Beaver, Nick Chinlund, Neil Summers, Daniel O'Haco, Richard Reeves, Alex Kubik, Will MacMillen, Harry Northrup, Don Hood, Donald L. Montoya, Zoaunne LeRoy, Jimmy Lewis, Jr., Millie Weddles, Vince Davis, Blue Deckert, Robert Boyce, Nick Hagler, Mark Feitch, Max Bode, Morgan Blanchard, Cooper Huckabee, Richard Robbins, Beulah Quo, Rick Lundin, Mark Carlton, Amber Leigh, Chuck Bennett, R.C. Bates. Four hookers head for Colorado after a shooting to start new lives but one is hounded by a lawman from her past. Not very interesting.\n\n**195** _ **Bad Jim**_ 21st Century Film Corporation, 1990. 90 min. Color. D-SC: Clyde Ware. With James Brolin, Richard Roundtree, John Clark Gable, Harry Carey, Jr., Rory Calhoun, Ty Hardin, Pepe Serna, Bruce Kirby, Joe George, Suzanne Wouk, Pierette Grace, Scotty Wrght, Teresa Van der Woude, Tonya Townsend, Humberto Ortiz, William J. Ware. Three none-too-bright cowpokes decide to become outlaws and after pulling off a bank heist they find they are wanted men. Only fair, with its main interest being it co-starred Clark Gable's son, John Clark Gable.\n\n**196** _ **Bad Lands**_ RKO Radio, 1939. 70 min. D: Lew Landers. SC: Clarence Upson Young. With Robert Barrat, Noah Beery, Jr., Guinn Williams, Robert Coote, Douglas Walton, Andy Clyde, Addison Richards, Paul Hurst, Francis Ford, Francis McDonald, Jack (John) Payne. An Army officer leads his men into the desert in pursuit of renegade Indians and the soldiers get picked off, one by one. Modest but entertaining \"B\" drama, showing Lew Landers was not a hack director. Western remake of _**The Lost Patrol**_ (RKO Radio, 1934) with Douglas Walton in both versions but in different roles.\n\n**197** _ **The Bad Man**_ First National, 1930. 90 min. D: Clarence Badger. SC: Howard Estabrook. With Walter Huston, Dorothy Revier, James Rennie, O.P. Heggie, Sidney Blackmer, Marion Byron, Guinn Williams, Arthur Stone, Edward Lynch, Harry Semels, Erville Alderson, Myrna Loy. Because he once saved his life, a Mexican bandit comes to the aid of a man about to lose his ranch. Early talkie mainly of interest to Walter Huston fans. First filmed in 1923 by Associated First National with Jack Mulhall, Holbrook Blinn and Enid Bennett.\n\n**198** _ **The Bad Man**_ Metro-Goldwyn-Mayer, 1941. 70 min. D: Richard Thorpe. SC: Wells Root. With Wallace Beery, Lionel Barrymore, Laraine Day, Ronald Reagan, Henry Travers, Chris-Pin Martin, Tom Conway, Chill Wills, Nydia Westman, Charles Stevens, Artie Ortego, Joe Dominguez, Daniel Rea. An elderly rancher is forced to depend on an old friend, a bandit with a price on his head, to help him save his land from crooks. Third screen version of Porter Emerson Browne's play is basically a vehicle for the delightful hamming of Wallace Beery and Lionel Barrymore.\n\n_**Bad Man from Big Bend**_ see _**Swing, Cowboy, Swing**_\n\n**199** _ **Bad Man from Red Butte**_ **** Universal, 1940. 58 min. D: Ray Taylor. SC: Sam Robins. With Johnny Mack Brown, Fuzzy Knight, Bob Baker, Anne Gwynne, Lloyd Ingraham, Lafe McKee, Bill Cody, Jr., Roy Barcroft, Norman Willis, Earle Hodgins, Myrna McKinney, Art Mix, Texas Jim Lewis and His Lone Star Cowboys. Arriving in a small Western town, a cowboy is mistaken for his killer twin. Average Johnny Mack Brown oater.\n\n**200** _ **The Bad Man of Brimstone**_ **** Metro-Goldwn-Mayer, 1937. 89 min. D. J. Walter Reuben. SC: Maurice Rapf and J. Walter Reuben. With Wallace Beery, Virginia Bruce, Dennis O'Keefe, Joseph Calleia, Lewis Stone, Guy Kibbee, Bruce Cabot, Guinn Williams, Cliff Edwards, Noah Beery, Charley Grapewin, Arthur Hohl, John Qualen, Robert Barrat, Raymond Hatton, Art Mix, John Wray, Virginia Brissac, Stanley Andrews, Eddy Waller, Robert Glecker, Jules Cowles, John T. Murray, George Regas, Olin Howland, Frank Hagney, E. Allyn Warren, Spencer Charters, Mitchell Lewis, Robert Middlemass, Harry Wilson, Larry McGrath, Bob Perry, Sidney Jarvis. When an old-time outlaw learns a young man is his son, it completely changes his life. Sentimental fare for Wallace Beery fans.\n\n**201** _ **Bad Man of Deadwood**_ **** Republic, 1941. 61 min. D: Joseph Kane. SC: Joseph R. Webb. With Roy Rogers, George \"Gabby\" Hayes, Carol Adams, Sally Payne, Henry Brandon, Herbert Rawlinson, Hal Taliaferro, Jay Novello, Monte Blue, Horace Murphy, Ralf Harolde, Jack Kirk, Yakima Canutt, Curley Dresden, Fred Burns, Lynton Brent, Lloyd Ingraham, George Lloyd, Robert Frazer, Archie Twitchell, Karl Hackett, Harry Harvey, Eddie Acuff, Tom London, Jack Rockwell, Ernie Adams, Jack O'Shea, George Morrell, Wally West, Bob Woodward, Horace B. Carpenter, Harrison Greene. In an attempt to turn his back on a lawless past, a young man joins a medicine show as a sharpshooter and becomes allied with citizens opposed to businessmen who are terrorizing them in order to thwart competition. Very good Roy Rogers vehicle with a fine supporting cast of familiar faces.\n\n_**Bad Man of Harlem**_ see _**Harlem on the Prairie**_\n\n**202** _ **Bad Man's River**_ **** Scotia International, 1972. 89 min. Color. D: Eugenio (Gene) Martin. SC: Eugenio Martin and Philip Yordan. With Lee Van Cleef, Gina Lollobrigida, James Mason, Simon Andrev, Eduardo Fajardo, Diana Lorys, Gianni Garko, Jose Manuel Martin, Aldo Sambrell, Daniel Martin, Barta Barri. In 1905 an outlaw gang his hired by a Mexican revolutionary leader to blow up a safe but the crooks get involved with a beautiful woman and a double-cross. This Spanish-Italian-French co-production does not know whether to be a Western or a comedy and it fails miserably at both. Filmed in Spain as _**El Hombre del Rio Malo**_ (The Man of the Bad River). Alternate video title: _**Hunt the Man Down**_.\n\n_**Bad Men of Arizona**_ see _**Arizona Raiders**_ (1938)\n\n**203** _ **Bad Men of Missouri**_ **** Warner Bros., 1941. 71 min. D: Ray Enright. SC: Charles Grayson. With Dennis Morgan, Jane Wyman, Wayne Morris, Arthur Kennedy, Victor Jory, Alan Baxter, Walter Catlett, Howard Da Silva, Faye Emerson, Russell Simpson, Virginia Brissac, Erville Alderson, Hugh Sothern, Sam McDaniel, Dorothy Vaughn, William Gould, Ann Todd, Roscoe Ates, Robert Winkler, Duncan Renaldo, Tom Tyler, Creighton Hale, Charless Middleton, Frank Mayo, Arthur Loft, Paul Panzer, Wade Boteler, Trevor Bardette, Stuart Holmes, Bud Osborne, Frank Wilcox, Spencer Charters, Jack Mower, Eddie Acuff, Eddy Waller, Herbert Heywood, Dix Davis, Sonny Bupp, Art Miles, Dutch Hendrian, Howard Mitchell, Leah Baird, Arthur Aylesworth, Joel Friedkin, Glen Cavender, Ray Teal, Bob Perry, Milton Kibbee, Vera Lewis, Ed Stanley, Henry Blair, Jack Carr, Tom Wilson. The story of the Younger Brothers and how they were pushed into a life of crime by carpetbaggers in Missouri after the Civil War. Entertaining but complete fiction.\n\n**204** _ **Bad Men of the Border**_ **** Universal, 1945. 56 min. D: Wallace Fox. SC: Adele Buffington. With Kirby Grant, Fuzzy Knight, Armida, John Eldredge, Barbara Sears, Francis McDonald, Soledad Jiminez, Edward Howard, Edmund Cobb, Pierce Lyden, Gene (Roth) Stutenroth, Roy Brent, Glenn Strange, Ethan Laidlaw, Charles Stevens. A U.S. marshal masquerading as an outlaw and a female Mexican agent investigating a counterfeit ring join forces to bring in the crooks. If one can overlook the implausible plot, Kirby Grant's initial series vehicle for Universal is acceptable entertainment.\n**205** _ **Bad Men of the Hills**_ **** Columbia, 1942. 58 min. D: William Berke. SC: Luci Ward. With Charles Starrett, Russell Hayden, Cliff Edwards, Luana Walters, Alan Bridge, Guy Usher, Joel Friedkin, Norman Jean Wooters, John Shay, Richard Botiller, Art Mix, Jack Ingram, Ben Corbett, Carl Sepulveda, Frank Ellis, John Cason, Steve Clark, Budd Buster, Ray Jones. Crooks try to murder a man investigating the killing of a marshal but he is befriended by the citizens of a lawless town who are really ranchers opposing the killers. A complicated plot and lots of action make this Charles Starrett-Russell Hayden vehicle pleasant viewing.\n\n**206** _ **Bad Men of Thunder Gap**_ **** Producers Releasing Corporation, 1943. 60 min. D: Al Herman. SC: Elmer Clifton. With Dave O'Brien, Jim Newill, Guy Wilkerson, Janet Shaw, Jack Ingram, Charles King, Tom London, Michael Vallon, Lucille Vance, I. Stanford Jolley, Bud Osborne, Jimmie Aubrey, Kermit Maynard, Hank Bell, Carl Mathews, Artie Ortego, Cal Shrum's Rhythm Rangers. When outlaws plague a small town, members of the Texas Rangers try to stop them. Dull entry in the \"Texas Rangers\" series. Reissued in 1947 by Eagle Lion in a 39-minute version entitled _**Thundergap Outlaws**_.\n\n**207** _ **Bad Men of Tombstone**_ **** Allied Artists, 1949. 74 min. D: Kurt Neumann. SC: Jay Monaghan. With Barry Sullivan, Marjorie Reynolds, Broderick Crawford, Fortunio Bonanova, Guinn Williams, John Kellogg, Mary Newton, Louis Jean Heydt, Virginia Carroll, Dick Wessell, Claire Carleton, Ted Hecht, Harry Cording, Lucien Littlefield, Harry Hayden, Olin Howlin, Robert Barrat, Julie Gibson, Joseph Crehan, Ted Mapes, Rory Mallinson, Ted French, Douglas Fowley, Dennis Hoey, Morris Ankrum, George Chesebro, Tom Fadden, Billy Gray, William Yip, Dick Foote, Gerald Courtemarche, Bonnie Lou Donaldson. During the Gold Rush era a young man tries to make a fortune but instead turns to a life of crime as a gunfighter. Average plot with good work by Barry Sullivan and Broderick Crawford as ruthless gunmen.\n\n**208** _ **The Badge of Marshal Brennan**_ **** Allied Artists, 1959. 74 min. D: Albert C. Gannaway. SC: Thomas G. Hubbard. With Jim Davis, Arleen Whelan, Carl Smith, Lee Van Cleef, Louis Jean Heydt, Marty Robbins, Harry Lauter, Douglas Fowley, Lawrence Dobkin, Rick Vallin, Eddie Crandall, Darryl Guy, Edward Colemans. A outlaw on the run is mistaken for a dead marshal and goes up against an evil land baron. Low budget but pleasing affair sporting country music stars Carl Smith and Marty Robbins.\n\n**209** _ **The Badlanders**_ **** Metro-Goldwyn-Mayer, 1958. 85 min. Color. D: Delmer Daves. SC: Richard Collins. With Alan Ladd, Ernest Borgnine, Katy Jurado, Claire Kelly, Kent Smith, Nehemiah Persoff, Anthony Caruso, Robert Emhardt, Adams Williams, Ford Rainey, John Day, Richard Devon, Gregg Barton, Henry Wills. At the turn of the last century, two men plan to rob gold from an Arizona mine while each plans to double cross the other. Fairly good Western remake of _**The Asphalt Jungle**_ (Metro-Goldwyn-Mayer, 1950).\n\n_**Badlands Drifter**_ see _**Challenge of the MacKennas**_\n\n**210** _ **Badlands of Dakota**_ **** Universal, 1941. 74 min. D: Alfred E. Green. SC: Gerald Geraghty. With Robert Stack, Ann Rutherford, Broderick Crawford, Frances Farmer, Richard Dix, Hugh Herbert, Lon Chaney, Jr., Fuzzy Knight, Andy Devine, Addison Richards, Samuel S. Hinds, Eddie Dew, Kermit Maynard, Hank Bell, Charles King, Bradley Page, Carleton Young, Glenn Strange, Don Barclay, Emmett Vogan, Willie Fung, Edward Fielding, The Jesters, Richard Alexander, Jeanne Kelly, Harry Cording, Alan Bridge, Robert Barron, Chuck Morrison, William Ruhl, Jane Farley, Carl Sepulveda, Joe King. In Deadwood, a crooked saloon owner sends his younger brother to bring home his fiancee and the two end up falling in love. Solid Western with Frances Farmer (as Calamity Jane) and Richard Dix (as Wild Bill Hickok) stealing the show.\n\n**211** _ **Badlands of Montana**_ **** 20th Century\u2013Fox, 1957. 75 min. D-SC: Daniel B. Ullman. With Rex Reason, Margia Dean, Beverly Garland, Keith Richards, Emile Meyer, William Phipps, Stanley Farrar, John Pickard, Ralph Peters, Paul Newlan, Russ Bender, Jack Kruschen, William Tannen. Two friends, a marshal and an outlaw, are forced into a showdown. Arid, mediocre affair.\n\n**212** _ **Badman's Country**_ **** Warner Bros., 1958. 68 min. D: Fred F. Sears. SC: Orville Hampton. With George Montgomery, Neville Brand, Buster Crabbe, Karin Booth, Gregory Walcott, Malcolm Atterbury, Russell Johnson, Richard Devon, Morris Ankrum, Dan Riss, Lewis Martin, Fred Graham, John Harmon, William Bryant. Sheriff Pat Garrett joins forces with Wyatt Earp, Bat Masterson and Buffalo Bill Cody to go against Butch Cassidy and his gang. Familiar faces add zest to his otherwise average oater. Title song sung by The Mellowmen.\n\n**213** _ **Badman's Gold**_ **** Eagle Lion Classics, 1951. 58 min. D-SC: Robert Tansey. SC: Robert Emmett (Tansey) and Alyn Lockwood. With John Carpenter, Alyn Lockwood, Kenne Duncan, Emmett Lynn, Jack Daly, Daisy (dog). When a series of robberies take place on a stage line carrying gold shipments, a marshal is called in to investigate. Made on a shoestring and it looks it.\n\n**214** _ **Badman's Territory**_ **** RKO Radio, 1946. 98 min. D: Tim Whelan. SC: Jack Natteford and Luci Ward. With Randolph Scott, Ann Richards, George \"Gabby\" Hayes, Lawrence Tierney, Tom Tyler, Steve Brodie, John Halloran, Phil Warren, William Moss, James Warren, Isabel Jewell, Morgan Conway, Nestor Paiva, Chief Thundercloud, Ray Collins, Virginia Sale, Andrew Tombes, Harry Holman, Richard Hale, Emory Parnell, George Chesebro, Ethan Laidlaw, Kermit Maynard, Bud Osborne, Ben Johnson, Carl Faulkner, Jason Robards, Buddy Roosevelt, Jack Clifford, Monte Montague, Elmo Lincoln, Boyd Stockman, Budd Buster, Robert Wilke, Phillip Morris, John Hamilton, John Elliott, Robert Homans, Frank LaRue, Harry Harvey, Wilbur Mack. A marshal is at a loss on how to stop a group of notorious outlaws who have taken refuge in a town outside government jurisdiction. Great Randolph Scott vehicle with a good supporting cast.\n\n**215** _ **Bajo el Cielo de Sonora**_ (The Faint Sky of Sonora) **** Clasa-Mohme, 1948. 90 min. D: Rolando Aguilar. SC: Raul de Anda. With Raul de Anda, Leonora Amar, Carlos Lopez Moctezuma, Jose L. Murillo, Domingo Soler, Roberto Meyer, Jose Elias Moreno, Silviano Sanchez, Pepe Nava. For his final military assignment, a Mexican Army officer is assigned to make peace with the Yaqui tribe only to discover his government is persecuting the Indians. Well done south of the border drama.\n\n**216** _ **Baker's Hawk**_ **** Doty-Dayton, 1976. 105 min. Color. D: Lyman D. Dayton. SC: Dan Greer and Hal Harrison, Jr. With Clint Walker, Burl Ives, Diane Baker, Alan Young, Lee H. Montgomery, Taylor Lacher, Bruce M. Fisher, Cam Clarke, Danny Bouaduce, Phil Hoover, Brian Williams. A boy learns about growing up when he aids an injured hawk and helps his father fight crooks. Overlong but fairly interesting outdoor drama.\n\n**217** _ **Una Bala es Mi Testigo**_ (A Bullet Is My Witness) **** Alameda Films, 1960. 90 min. D: Chano Urueta. SC: Ramon Obon. With Gaston Santos, Pedro de Aguillon, Jaime Fernandez, Mauricio Garces, Irma Castillon, Jose Castro, Emilio Garibay, Rita Macedo, Jose Luis Fernandez, Hortensia Lantovena. A stranger comes to a village where his friend stole 200,000 pesos and saves a rancher and his wife from an outlaw gang. Fast paced Mexican Western.\n\n**218** _ **Ballad of a Bounty Hunter**_ **** United Pictures\/Trebol Film, 1970. 83 min. Color. D: Joaquin L. Romero Merchant. SC: Joaquin L. Romero Merchant, Giovanni Simonelli and Victor Aux. With James Philbrook, Norma Bengell, Simon Andreu, Luis Induni, Emilio Caba, Alfonso Rojas, Maria Silva, Alvaro de Luna, Angel Ortiz. A bounty hunter falls in love with a young girl and then is forced to trail her brother. Low budget Spanish-Italian co-production made as _**Lo Non Perdono...Uccido!**_ (I Do Not Forgive...I Kill!) in 1968.\n\n**219** _ **Ballad of a Gunfighter**_ **** Parade, 1964. 84 min. D-SC: Bill Ward. With Marty Robbins, Joyce Redd, Bob Barron, Nestor Paiva, Michael Davis, Laurette Luez, Gene Davis, Traveler (horse). An outlaw's plan to rob a stagecoach is foiled by another man who not only takes the bandit's gold but also his woman. Marty Robbins' first starring feature is a low budget affair but his fans will enjoy his singing \"El Paso\" and \"San Angelo.\"\n\n**220** _ **The Ballad of Ben and Charlie**_ **** Variety International, 1972. 107 min. Color. D: Michele Lupo. SC: Sergio Donati and Luigi Montefiori (George Eastman). With Guiliano Gemma, George Eastman, Marisa Mell, Vittorio Congia, Giacomo Rossi Stuart, Luciano Lorcas, Remo Capitani, Giovani Pazzafini, Aldo Sambrell, Franco Fantasia, Jose Manuel Martin, Cris Huerta, George Rigaud, Luis Induni, Roberto Camardiel, Claudio Ruffini, Jesus Guzman, Tom Felleghy, Francisco Sanz, Carla Mancini. Two petty crooks who detest each other team for a robbery spree that turns them into famous outlaws. Somewhat tongue-in-cheek Spaghetti Western is fun to watch. Issued in Italy as _**Amico, Stammi Lontano Alemno un Palmo**_ and also called _**Amigo, Stay Away**_ and _**Ben and Charlie**_.\n\n**221** _ **The Ballad of Cable Hogue**_ **** Warner Bros., 1970. 120 min. Color. D: Sam Peckinpah. SC: John Crawford and Edmund Penny. With Jason Robards, Stella Stevens, David Warner, Strother Martin, L.Q. Jones, Peter Whitney, R.G. Armstrong, Gene Evans, William Mims, Kathleen Freeman, Vaughn Taylor, James Anderson, Victor Izay, Mary Munday, William Faralla, Matthew Peckinpah. A prospector is left in the desert to die by his partners but he manages to survive and build a depot station around a water hole and become prosperous. Rambling, overrated but somewhat ingratiating Sam Peckinpah feature highlighted by Stella Stevens as a gorgeous, good hearted whore.\n\n_**Ballad of Death Valley**_ (1965) see _**The Return of Ringo**_\n\n_**Ballad of Death Valley**_ (1970) see _**Sartana in the Valley of Vultures**_\n\n**222** _ **The Ballad of Gregorio Cortez**_ **** PBS-TV, 1982. 99 min. Color. D: Robert M. Young. SC: Victor Villasenor. With Edward James Olmos, Tom Bower, James Gammon, Bruce McGill, Brion James, Pepe Serna, Alan Vint, Tim Scott, Michael McGuire, Jack Kehoe, Barry Corbin, Rosana DeSoto, Victoria Plata, William Sanderson. In 1901 a Mexican cowboy who shot a lawman who killed his brother tries to elude a 600-man posse across the Texas desert. Grim and somewhat slow made-for-Public TV movie given theatrical release in 1983 by Embassy Pictures.\n\n**223** _ **The Ballad of Josie**_ **** Universal, 1968. D: Andrew V. McLaglen. SC: Harold Swanton. With Doris Day, Peter Graves, George Kennedy, Andy Devine, William Talman, David Hartman, Guy Raymond, Karen Jensen, Robert Lowery, Paul Fix, Audrey Christie, Elisabeth Fraser, Linda Meiklejohn, Shirley O'Hara, Timothy Scott, Don Stroud, Harry Carey, Jr., John Fiedler, Teddy Quinn, George Ives, Bill Quinn, J. Edward McKinley, Harry Swoger, Edward Faulkner, Alexander Lockwood, James Seay, Mike Lally, Ollie O'Toole, Jonathan Wynne. A widow in 1890 tries to build up a rundown ranch to raise sheep and starts a feud with cattlemen. Dud Doris Day Western that is supposed to be funny but is not.\n\n**224** _ **The Ballad of Little Jo**_ **** Fine Line Features, 1993. 121 min. Color. D-SC: Maggie Greenwald. With Suzy Arne, Bo Hopkins, Ian McKellen, David Chung, Heather Graham, Rene Auberjonois, Carrie Snodgrass, Anthony Heald, Melissa Leo, Sam Robards, Olinda Turturro, Ruth Maleczech, Jeffrey Andrews, Cathy Haase, Peadair S. Addie, Sr., Irina Pasmur, Michael Rudd, Sasha Pasmur, David Ruben Plowman, Rusty Pegar, Troy Smith, Keith Kamppinen, Jenny Lynch, Vince O'Neil, Dennis McNiven, Barbara Jean Marsh, Robert Erickson, Tom Bower, Sean Murphy, Renee Tafoya, Richard Osterman, Karen Johnson, Jaime Crabtree, Tracy Mayfield, Julianne Kirst, Deborah J. Richard, Netha Goodrich, Becca Busch, Jim Dunkin, Homer Simon, Eryn L. Bent, Peter Plowman, Joe Freed, Anne Plowman, Melissa Ladvala, Yeugeuly Yasyriu, Duane Ebel. After being disowned by her parents for having an illegitimate child, a young woman poses as a man in order to survive in a mining community. Suzy Arne's performance carries this well modulated frontier saga, allegedly based on a true story.\n\n**225** _ **Bandera Bandits**_ **** K-Tel International, 1972. 92 min. Color. D: Sergio Corbucci. SC: Sergio Corbucci and Mario Amendola. With Susan George, Tomas Milan, Telly Savalas, Eduardo Fajardo, Rosanna Yanni, Franco Giacobini, Herbert Fux, Werner Pochath, Victor Israel, Laura Betti, Alvaro de Luna, Gene Collins, Rafael Albaicin, Jose Canalejas, Simon Arriaga, Lorenzo Robledo. Two robbers, an ex-convict and a young girl, are relentlessly pursued by a lawman. Action filled, low key Italian-Spanish-West German Spaghetti Western serio-comedy; sort of \"Bonnie and Clyde\" out West. Also called **Sonny and Jed**.\n\n**226** _ **Bandido**_ **** United Artists, 1956. 92 min. Color. D: Richard Fleischer. SC: Earl Fenton. With Robert Mitchum, Ursula Thiess, Gilbert Roland, Zachary Scott, Rodolfo Acosta, Henry Brandon, Douglas Fowley, Victor Junco, Jose I. Torvay. Arriving in Mexico in 1916 to sell weapons during the revolution, an American adventurer finds himself up against a treacherous rival. Robert Mitchum and Zachary Scott as the foes and Gilbert Roland as the rebel leader make this a fast moving and pleasant feature.\n\n**227** _ **Bandidos**_ **** Epic Film\/Hesperia, 1967. 94 min. Color. D: Max Dillman (Massimo Dallamano). SC: Romano Migliorini, Giambattista Mussetto and Juan Cobos. With Enrico Maria Salerno, Terry Jenkins, Marco Guglielmi, Luigi Pistilli, Chris Huerta, Antonio Pica, Maria Martin, Venantino Venantini, Victor Israel, Roberto Messina, Valentino Morchi, Maria Martin, Arthur Chase, Giancarlo Bastianoni, Jesus Puente, Giancarlo Sisti, Franco Morici, Juan Jose Milian. After his hands are mutilated by a rival, a gunfighter trains an escaped convict to get even for him. Well modulated revenge themed Spaghetti Western; nicely directed and scored; also called **Banditos**.\n\n**228** _ **Bandit King of Texas**_ **** Republic, 1949. 60 min. D: Fred C. Brannon. SC: Olive Cooper. With Allan \"Rocky\" Lane, Eddy Waller, Helen Stanley, Harry Lauter, Jim Nolan, Robert Bice, John Hamilton, Lane Bradford, George Lloyd, Steve Clark, I. Stanford Jolley, Richard Emory, Danni Nolan. A government investigator is after racketeers who sell bogus land to settlers and then murder them for their money. Fast moving Allan Lane vehicle.\n\n**229** _ **Bandit Queen**_ **** Lippert, 1950. 71 min. D: William Berke. SC: Victor West and Budd Lester. With Barbara Britton, Willard Parker, Philip Reed, Barton MacLane, Martin Garralaga, John Merton, Jack Ingram, Victor Kilian, Thurston Hall, Jack Perrin, Chuck Roberson, Margia Dean, Angie (Angelo Rossitto), Paul Martin, Pepe Hern, Lalo Rios, Mike Conrad, Carl Pitti, Hugh Hooker. When vicious land grabbers murder her family in Old California, a young woman takes on the guise of a masked crusader and leads a vigilante group against the killers. Passable Lippert feature with plenty of action and pretty Barbara Britton in the title role.\n\n**230** _ **Bandit Ranger**_ **** RKO Radio, 1942. 64 min. D: Lesley Selander. SC: Bennett Cohen and Morton Grant. With Tim Holt, Cliff Edwards, Joan Barclay, Kenneth Harlan, LeRoy Mason, Glenn Strange, Jack Rockwell, Frank Ellis, Robert Kortman, Bud Geary, Dennis Moore, Ernie Adams, Russell Wade, Tom London, Lloyd Ingraham, Cliff Parkinson, Jess Cavin, Charles Murphy, Ray Johnson, Bob Clark, Wayne McCoy. Finding a dying Ranger, a man takes on the guise of the lawman to save his daughter from a murder-kidnap plot. Good drama interpolated with action and a workman-like cast.\n\n**231** _ **The Bandit Trail**_ **** RKO Radio, 1941. 60 min. D: Edward Killy. SC: Norton S. Parker. With Tim Holt, Ray Whitley, Janet Waldo, Lee \"Lasses\" White, Morris Ankrum, Roy Barcroft, J. Merrill Holmes, Eddy Waller, Glenn Strange, Frank Ellis, Guy Usher, Jack Clifford, Joseph Eggenton, Bud Osborne, John Merton, Bud Geary, Lew Meehan, Terry Frost, Carl Stockdale, James Farley, Al Ferguson, Bob Burns. A man and his two pals become outlaws after the murder of his father but when he falls in love with a pretty girl the trio go to the right side of the law and round up a gang. Typically solid Tim Holt vehicle.\n\n_**Banditos**_ see _**Bandidos**_\n\n**232** _ **The Bandits**_ **** Lone Star, 1979. 89 min. Color. D: Robert Conrad and Alfredo Zacharias. SC: Robert Conrad, Edward Di Lorenzo and Alfredo Zacharias. With Robert Conrad, Antonio Aguilar, Jan-Michael Vincent, Pilar Pellicar, Maria Duval, Roy Jenson, Pedro Armendariz, Jr., Manuel Lopez Ochoa, Jose Chavez, Elizabeth Dupeyron, Pascual Garcia Pena, Fanny Schuller, Leighton, Gould, Enrique Larcero,. A Mexican saves three cowpokes from being hanged and they accompany him to his homeland where they have a series of adventures. Average action feature filmed in Mexico in 1966 as _**Los Bandidos**_ (The Bandits). Issued on video as **Bandits**.\n\n**233** _ **Bandits of Dark Canyon**_ **** Republic, 1947. 59 min. D: Philip Ford. SC: Bob Williams. With Allan \"Rocky\" Lane, Eddy Waller, Bob Steele, Roy Barcroft, Linda Johnson, John Hamilton, Francis Ford, Eddie Acuff, LeRoy Mason, Gregory Marshall, Norman Willis. When a mine owner is falsely accused of killing his foreman, he escapes with the help of friends and tries to find the murderer. Better than average Allan Lane entry, enhanced by co-star Bob Steele.\n\n**234** _ **Bandits of El Dorado**_ **** Columbia, 1949. 56 min. D: Ray Nazarro. SC: Barry Shipman. With Charles Starrett, Smiley Burnette, George J. Lewis, Fred F. Sears, Clayton Moore, Mustard and Gravy, John Dehner, Jock (Mahoney) O'Mahoney, John Doucette, Max Wagner, Henry Kulky, Edmund Cobb, John Merton, Ray Bennett, Kermit Maynard, Jack Evans, Carl Mathews, Monte Montague, Blackie Whiteford, Merrill McCormick, Jack Tornek, Victor Cox, Herman Hack, Al Haskell, Ted Mapes, Ray Jones. A government investigator fakes the murder of a Texas Ranger in order to become wanted so he can learn how criminals are escaping across the Mexican border. Fair \"Durango Kid\" outing with footage from another series entry, _**Galloping Thunder**_ (q.v.). British title: _**Tricked**_.\n\n**235** _ **Bandits of the Badlands**_ **** Republic, 1945. 55 min. D: Thomas Carr. SC: Doris Schroeder. With Sunset Carson, Peggy Stewart, Si Jenks, Monte Hale, John Merton, Forrest Taylor, Jack Ingram, Fred Graham, Robert Wilke, Tex Terry, Jack O'Shea, Jack Kirk, Horace B. Carpenter, Charles Stevens, Marshall Reed, Bob Reeves, Post Park, Foxy Callahan, Bert LeBaron. When his brother is murdered by outlaws in retaliation for his having killed one of their members during a holdup, a ranger resigns from the force and pretends to be an escaped convict to bring in the culprits. Fast paced and entertaining Sunset Carson vehicle.\n\n_**Bandits of the Natchez Trace**_ see _**Natchez Trace**_\n\n**236** _ **Bandits of the West**_ **** Republic, 1953. 54 min. D: Harry Keller. SC: Gerald Geraghty. With Allan \"Rocky\" Lane, Eddy Waller, Cathy Downs, Roy Barcroft, Trevor Bardette, Ray Montgomery, Byron Foulger, Harry Harvey, Robert Bice. Outlaws try to sabotage a gas company and a U.S. marshal is sent to stop them. Despite declining production values in \"B\" Westerns in the early 1950s this Allan Lane entry is well made and fast paced.\n\n**237** _ **Bandolero!**_ **** 20th Century\u2013Fox, 1968. 106 min. Color. D: Andrew V. McLaglen. SC: James Lee Barrett. With James Stewart, Dean Martin, Raquel Welch, George Kennedy, Andrew Prine, Will Geer, Denver Pyle, Tom Heaton, Rudy Diaz, Sean McClory, Harry Carey, Jr., Donald Barry, Guy Raymond, Perry Lopez, Jock Mahoney, Big John Hamilton, Dub Taylor, John Mitchum, Joseph Patrick Cranshaw, Roy Barcroft, Bob Adler. Masquerading as a hangman, a man saves his brother and their outlaw gang and then flee south of the border with a pretty hostage. Okay entertainment with good production values and a likable cast.\n\n**Dean Martin and James Stewart in** _**Bandolero!**_ **(20th Century-Fox, 1968).**\n\n**238** _ **The Bang Bang Kid**_ **** Ajay Films, 1967. 90 min. Color. D: Stanley Prager. SC: Howard Beck. With Guy Madison, Tom Bosley, Sandra Milo, Riccardo Garrone, Jose Maria Caffarel, Dianik Zurakowska, Giustino Durano, Jose Caffarel. A bungler finds himself in a mining town lorded over by a notorious gunman but he is saved by a gun-toting robot. Amusing Old West parody co-produced by the U.S., Italy and Spain.\n\n**239** _ **Banjo Hackett**_ **** Columbia\/NBC-TV, 1976. 100 min. Color. D: Andrew V. McLaglen. SC: Ken Trevey. With Don Meredith, Ike Eisenmann, Chuck Connors, Jennifer Warren, Dan O'Herlihy, Jeff Corey, Gloria De Haven, L.Q. Jones, Jan Murray, Anne Francis, Slim Pickens, David Young, Richard Yung, Stan Haze. In 1880 a horse trader and his orphaned nephew wander the frontier looking for the boy's prize horse that was stolen by a bounty hunter. Run-of-the-mill television feature, the pilot for a series that did not sell. Originally called _**Banjo Hackett: Roamin' Free**_.\n\n**240** _ **Bar L Ranch**_ **** Big 4, 1930. 60 min. D: Harry S. Webb. SC: Bennett Cohen. With Buffalo Bill, Jr., Yakima Canutt, Wally Wales, Betty Baker, Ben Corbett, Fern Emmett, Robert Walker, Tom London, Hank Bell, Bud Pope. After being fired from their jobs, a cowboy and his pal learn the ranch foreman is rustling the cattle of their ex-boss, a pretty young woman. Tattered early talkie.\n\n**241** _ **Bar 20**_ **** United Artists, 1943. 55 min. D: Lesley Selander. SC: Morton Grant, Norman Houston and Michael Wilson. With William Boyd, Andy Clyde, Dustine Farnum, George Reeves, Victor Jory, Douglas Fowely, Betty Blythe, Robert Mitchum, Francis McDonald, Earle Hodges, Henry Wills, Roy Bucko. Hopalong Cassidy, California Carlson and Lin Bradley try to help a young woman and her mother after they lose their valuables in a holdup with the chief suspects being the girl's fiance and his best man. Fairly good \"Hopalong Cassidy\" entry, draggy in spots with Robert Mitchum having a large role as a ranch owner.\n\n**242** _ **Bar 20 Justice**_ **** Paramount, 1938. 70 min. D: Lesley Selander. SC: Arnold Belgard and Harrison Jacobs. With William Boyd, George Hayes, Russell Hayden, Paul Sutton, Gwen Gaze, Pat J. O'Brien, Joseph De Stefani, William Duncan, Walter Long, Bruce Mitchell, John Beach, Dick Dickinson, Wen Wright. Hopalong Cassidy tries to help a young widow in re-opening her mine while crooks try to keep it closed so they can get all its ore. Entertaining \"Hopalong Cassidy\" series feature.\n\n**243** _ **Bar 20 Rides Again**_ **** Paramount, 1935. 61 min. D: Howard Bretherton. SC: Doris Schroeder and Gerald Geraghty. With William Boyd, James Ellison, Jean Rouverol, George Hayes, Harry Worth, Frank McGlynn, Jr., Howard Lang, Ethel Wales, Paul Fix, J.P. McGowan, Joe Rickson, Al St. John, John Merton, Frank Layton, Chill Wills and His Avalon Boys, Jim Mason, Jack Kirk, Chuck Baldra, Tracy Layne, Sid Jordan. The cowpokes from the Bar 20 find themselves up against a land baron who fancies himself another Napoleon. The third feature in the \"Hopalong Cassidy\" series and one of the very best; highly recommended.\n\n**244** _ **Bar Z Badmen**_ **** Republic, 1937. 57 min. D: Sam Newfield. SC: George Plympton. With Johnny Mack Brown, Lois January, Tom London, Ernie Adams, Dick Curtis, Jack Rockwell, Milburn Morante, Horace Murphy, Budd Buster, Frank Ellis, George Morrell, Tex Palmer, Horace B. Carpenter, Art Dillard, Oscar Gahan. When he falsely accused of rustling a fellow rancher's cattle, a man seeks the real culprits. Typically good Johnny Mack Brown-Republic production with a cast of familiar faces.\n\n**245** _ **Barbarosa**_ **** Universal, 1982. 90 min. Color. D: Fred Schepisi. SC: William D. Wittliff. With Gary Busey, Willie Nelson, Gilbert Roland, Isela Vega, Danny De La Paz, George Voskovec, Alma Martinez, Howard Chamberlain, Wolf Muser, Kai Wuf, Harry Caesar, Sharon Compton, Roberto Contreros, Luis Contreros, Sonia De Leon, Joanelle Romera. A young farm boy, on the run for the accidental killing of his brother-in-law, is befriended by a notorious outlaw who teaches him to survive. Offbeat oater which got little theatrical release is fair although Willie Nelson is likable in the title role and Gilbert Roland is powerful as a vengeful patron.\n\n**246** _ **Barbary Coast**_ **** United Artists, 1935. 91 min. D: Howard Hawks. SC: Ben Hecht and Charles MacArthur. With Miriam Hopkins, Edward G. Robinson, Joel McCrea, Walter Brennan, Frank Craven, Brian Donlevy, Otto Hoffman, Rollo Lloyd, Donald Meek, Harry Carey, Robert Gray, Clyde Cook, J.M. Kerrigan, Matt McHugh, Wong Chung, Russ Powell, Frederik Vogeding, David Niven, Edward Gargan, Herman Bing, Tom London, Heinie Conklin, Art Miles, Charles West. In San Francisco in 1849, a dance hall queen throws over a corrupt gambling house operator for an honest, but broke, young man. Fast paced and entertaining period piece with a bevy of fine performances.\n\n**247** _ **The Barbary Coast**_ **** Paramount\/ABC-TV, 1975. 100 min. Color. D: Bill Bixby. SC: Douglas Heyes. With William Shatner, Dennis Cole, Charles Aidman, Michael Ansara, Neville Brand, Bobbi Jordan, Richard Kiel, John Vernon, Lynda Day George, Leo Gordon, Bob Hoy, Terry Lester, Simon Scott, Todd Martin, Charles Picerni, Michael Carr, Bill Bixby. When a Confederate officer sets off an extortion plot he finds himself opposed by a casino owner and a government agent in San Francisco in the 1860s. Moderately sustaining telefeature that served as the pilot for the \"Barbary Coast\" (ABC-TV, 1975\u201376) series starring William Shatner and Doug McClure.\n\n**248** _ **Barbary Coast Gent**_ **** Metro-Goldwyn-Mayer, 1944. 87 min. D: Roy Del Ruth. SC: William R. Lipman, Grant Garrett and Harry Ruskin. With Wallace Beery, Binnie Barnes, John Carradine, Bruce Kellogg, Frances Rafferty, Chill Wills, Noah Beery, Henry O'Neill, Ray Collins, Morris Anrkum, Donald Meek, Addison Richards, Harry Hayden, Paul E. Burns, Paul Hurst, Victor Kilian, Cliff Clark, Louise Beavers, Robert Emmett O'Connor, Ray Teal, Earle Hodgins, Jack Norton, Harry Shannon, Fred \"Snowflake\" Toones, Byron Foulger, Lee Phelps, Anne O'Neal, James Farley, Edgar Dearing. In the 1890s a likable crook makes a quick getaway from San Francisco and goes to Nevada where he plans to sell phony mining stock but to his surprise he reforms. Fun Wallace Beery vehicle.\n\n**249** _ **The Bargain**_ **** Paramount, 1914. 50 min. D: Reginald Barker. SC: William H. Clifford and Thomas H. Ince. With Clara Williams, J. Barney Sherry, William S. Hart, J. Frank Burke, James Dowling. A notorious bandit is injured while trying to rob a stage and is rescued by a rancher and nursed back to health by the man's pretty daughter, with whom he falls in love. William S. Hart's first feature, in which he is third billed as the outlaw, is a good silent effort.\n\n**250** _ **Barbed Wire**_ **** Columbia, 1952. 61 min. D: George Archainbaud. SC: Gerald Geraghty. With Gene Autry, Pat Buttram, Anne James, William Fawcett, Leonard Penn, Michael Vallon, Terry Frost, Clayton Moore, Edwin Parker, Sandy Sanders. When he can no longer find fresh herds, a cattle buyer uncovers a plot by a rancher to get homesteaders' lands so he can construct a railroad. Fast moving and entertaining Gene Autry affair.\n\n**251** _ **The Baron of Arizona**_ **** Lippert, 1950. 97 min. D-SC: Samuel Fuller. With Vincent Price, Ellen Drew, Beulah Bondi, Reed Hadley, Vladimir Sokoloff, Robert Barrat, Robin Short, Barbara Woodell, Tina Rome, Margia Dean, Edward Keene, Gene Roth, Karen Kester, Joseph Green, Fred Kohler, Jr., Tristram Coffin, I. Stanford Jolley, Terry Frost, Angelo Rossitto, Zachary Yaconelli, Wheaton Chambers, Stuart Holmes, Jonathan Hale, Stanley Price, Sam Flint, Richard Cramer, Adolfo Ornelas, Robert O'Neil, Stephen Harrison, George Meader, Ed East, Al Haskell. A clerk in the Arizona land office spends years falsifying documents in order to prove he is the legal heir to thousands of acres in Arizona. Top notch work by Vincent Price in the title role, leading lady Ellen Drew and director-writer Samuel Fuller make this historical based film worth watching.\n\n**252** _ **Barricade**_ **** Warner Bros., 1950. 75 min. Color. D: Peter Godfrey. SC: William Sackheim. With Dane Clark, Raymond Massey, Ruth Roman, Robert Douglas, Morgan Farley, Walter Loy, George Stern, Robert Griffin, Frank Marlowe, Tony Martinez. Two outlaws come to the rescue of the denizens of a mining camp who are under the ruthless control of a tyrant. Fairly good Western reworking of _**The Sea Wolf**_ (Warner Bros., 1941).\n\n**253** _ **Barricade on the Big Black**_ **** NBC-TV, 1957. 54 min. D-SC: Anthony Spinner. With Richard Crenna, Mary LaRoche, Andrew Duggan, George Galbreth. An Indian-hating Army lieutenant finds himself falling in love with the white wife of a hostile warrior. Originally telecast March 27, 1957, as a segment of \"Matinee Theatre,\" this drama was issued to TV as a feature film.\n\n**254** _ **The Barrier**_ **** Paramount, 1937. 90 min. D: Lesley Selander. SC: Bernard Schubert, Harrison Jacobs and Mordaunt Shairp. With Leo Carrillo, Jean Parker, James Ellison, Otto Kruger, J.M. Kerrigan, Robert Barrat, Andy Clyde, Sally Martin, Sara Haden, Addison Richards, Allen Davies. After her true background is revealed to her lover, a young woman decides to join her father, an Alaskan sea captain. Filmed at Washington's Mount Baker National Forest, this drama from Rex Beach's novel is an above average programmer. First filmed in 1926 by Metro-Goldwyn-Mayer with Norman Kerry, Henry B. Walthall, Lionel Barrymore and Marceline Day.\n\n**255** _ **Barquero**_ **** United Artists, 1970. 115 min. Color. D: Gordon Douglas. SC: George Schenck and William Marks. With Lee Van Cleef, Warren Oates, Kerwin Mathews, Forrest Tucker, Mariette Hartley, Augie Gomez, Armando Silvestre, John Davis Chandler, Harry Lauter, Brad Weston, Craig Littler, Ed Bakey, Richard Lapp. A ferryman has to protect his job against bandits and when the gang wipes out a small Mexican village he hunts them down. Action filled but extremely violent and bloody oater, in the style Lee Van Cleef did in Europe.\n\n**256** _ **The Bat People**_ **** American International, 1974. 95 min. Color. D: Jerry Jameson. SC: Lou Shaw. With Stewart Moss, Marianne McAndrew, Michael Pataki, Paul Carr, Arthur Space, Robert Berk, Pat Delaney, George Paulsin, Bonnie Van Dyke, Jeni Kulik, Laura Brooks Jefferson, Herb Pierce. While on his honeymoon, a doctor is bitten by a bat in New Mexico's Carlsbad Caverns and becomes a monster. Average modern-day horror Western, also known as _**It Lives by Night**_.\n\n**257** _ **The Battle of Apache Pass**_ **** Universal-International, 1951. 85 min. Color. D: George Sherman. SC: Gerald Drayson Adams. With John Lund, Jeff Chandler, Beverly Tyler, Susan Cabot, Bruce Cowling, John Hudson, James Best, Regis Toomey, Richard Egan, Hugh O'Brian, William Reynolds, Jay Silverheels, Tommy Cook, Jack Elam, Richard Garland, Jack Ingram, John Baer. An Army major and the Indian chief Cochise try to work together to keep the renegade Geronimo from slaughtering settlers. Jeff Chandler repeats the role of Cochise from _**Broken Arrow**_ (q.v.) but this outing is no competition for that classic.\n\n**258** _ **Battle of Greed**_ **** Crescent, 1937. 59 min. D: Howard Higgin. SC: John T. Neville. With Tom Keene, Gwynne Shipman, James Bush, Jimmy Butler, Budd Buster, Lloyd Ingraham, Bob Callahan, Henry Roquemore, Rafael (Ray) Bennett, Robert Fiske, Carl Stockdale, William Worthington. A young man becomes involved in treachery following the discovery of the Comstock Lode in Virginia City. One of the pseudo-historical features Tom Keene made for Crescent, it even includes Mark Twain in the adventure; average for the series.\n\n**259** _ **Battle of Rogue River**_ **** Columbia, 1954. 71 min. Color. D: William Castle. SC: Douglas Heyes. With George Montgomery, Martha Hyer, Richard Denning, John Crawford, Emory Parnell, Michael Granger, Bill Bryant, Charles Evans, Lee Roberts, Steven Ritch, Frank Sully, Bill Hale, Jimmy Lloyd, Willis Bouchey. Settlers in Oregon in 1850 want statehood but this cannot be accomplished without the signing of a peace treaty with local Indians. Producer Sam Katzman must have spent the budget on Technicolor because the title battle is not much although the film entertains adequately.\n\n**260** _ **The Battles of Chief Pontiac**_ **** Realart, 1952. 71 min. D: Felix Feist. SC: Jack DeWitt. With Lex Barker, Lon Chaney, Helen Westcott, Berry Kroeger, Roy Roberts, Larry Chance, Katherine Warren, Ramsey Hill, Guy Teague, James Fairfax, Abner George. In the early 1760s a ranger tries to negotiate a peace treaty between the British and Chief Pontiac only to find that a Hessian renegade is selling guns to the Indians. Cheaply made and historically inaccurate but not bad entertainment.\n\n**261** _ **Battlin' Buckaroo**_ **** Anchor, 1924. 60 min. D-SC: Alvin J. Neitz (Alan James). With Bill Patton, Peggy O'Day, Andrew Waldron, Lew Meehan, Anthony Freendenthall, Fred Hank. A nester loves a rancher's daughter but gets the blame when her father is cheated by his crooked foreman. Low grade Bill Patton (whose screen persona was like that of a silent Jimmy Wakely) vehicle, action filled with nice photography (Marvin Hughes) and desert locales.\n\n**262** _ **Battling Buckaroo**_ **** Willis Kent, 1932. 60 min. D: Armand Schaefer. SC: Oliver Drake. With Lane Chandler, Doris Hill, Yakima Canutt, Lafe McKee, Bill Patton, Ted Adams, Olin Francis, Bart Carre, Herman Hack, Pat Harmon, Cliff Lyons, Fred Parker. When crooks try to steal their gold, a Mexican rancher and his daughter are helped by an outlaw. Not much to recommend this one except for star Lane Chandler.\n\n**263** _ **Battling Marshal**_ **** Astor, 1950. 55 min. D: Oliver Drake. SC: Rose Kreves. With Sunset Carson, Pat Starling, Lee Roberts, Forrest Matthews, Al Terry, A.J. Baxley, Richard Bartell, Bob Curtis, Pat Gleason, Stephen Keyes, Don Gray, Dale Carson, William Val, Buck Buckley, Joe Hiser. Two federal marshals arrive in a small town to investigate attempts made on the life of a rancher. Very low grade Sunset Carson vehicle.\n\n_**Battling Outlaw**_ see _**Billy the Kid in Texas**_\n\n**264** _ **Battling with Buffalo Bill**_ **** Universal, 1931. 12 Chapters. D: Ray Taylor. SC: George Plympton and Ella O'Neal. With Tom Tyler, Lucille Brown, Rex Bell, William Desmond, Francis Ford, Yakima Canutt, Chief Thundercloud, Franklyn Farnum, Joe Bonomo, Art Mix, Bud Osborne, John Beck, George Regas, Jim Thorpe, Bobby Nelson, Edmund Cobb, Fred Humes. Scout Buffalo Bill Cody helps a young woman whose father's mine is sought by a crook and his gang who have incited Indian attacks. Not much history here but there is plenty of action to make up for it in this cliffhanger.\n\n**265** _ **The Bear**_ **** Tri Star Pictures, 1988. 96 min. Color. D: Jean-Jacques Annaud. SC: Gerard Brach. With Bart (bear), Youk (bear cub), Tcheky Karyo, Jack Wallace, Andre Lacombe. An orphan bear cub is protected by a huge Kodiak Bear who is being pursued by hunters. Picturesque and loving adaptation of _The Grizzly King_ by James Oliver Curwood.\n\n_**Bearheart of the Great Northwest**_ see _**Legend of Bearheart**_\n\n**266** _ **Beartooth**_ **** Filmark Entertainment Group, 1978. 86 min. Color. D: Zach Belcher. SC: Mary H. Belcher. With Dub Taylor, Buck Taylor, Johnny Bush, Judy Nugent. A man has to coexist with a huge bear so both of them can survive a terrible blizzard in the savage wilderness. Tacky production whose only asset is fine photography by James A. Sullivan; also called _**The Adventures of Bear Tooth**_.\n\n**267** _ **The Bears and I**_ **** Buena Vista, 1974. 89 min. Color. D: Bernard McEveety. SC: John Whedon. With Patrick Wayne, Chief Dan George, Andrew Duggan, Michael Ansara, Robert Pine, Val De Vargas, Hal Baylor. A Vietnam War veteran moves to the north woods where he becomes embroiled in a dispute between Indians and settlers. This Walt Disney production is only average but the scenery is nice.\n\n**268** _ **The Beast**_ **** West Devon\/Nadir Cinematographica, 1970. 90 min. Color. D: Mario Costa. SC: Franco Calabrese and Mario Costa. With Klaus Kinski, Gabriella Girogelli, Luisa Rivelli, Steven Tedd, Lee Burton, Gianni Pallavicino, Andrea Aureli, Remo Capitani, Giuliano Raffaelli, Paul Sullivan, Grazia De Marze, Fioni Florence, Giora Garson, Cristina Iosani, Vittorio Mangano, Ivana Novak. A madman and his gang travels through the West leaving a path of rape and murder, planning to kidnap an heiress about to come into a huge inheritance. For Klaus Kinski followers, but well done.\n\n**269** _ **The Beast of Hollow Mountain**_ **** United Artists, 1956. 80 min. Color. D: Edward Nassour and Ismael Rodriquez. SC: Robert Hill. With Guy Madison, Patricia Medina, Eduardo Noriega, Carlos Rivas, Marjo Navarro, Pascual Garcia Pena, Margarito Luna. When cattle begin disappearing from his ranch, a man tries to learn the cause and discovers a huge dinosaur. Taken from a story by Willis O'Brien, this film combines the horror and Western genres, but not very well, although the prehistoric beast looks good. Filmed in Mexico as _**El Monstruo de la Montana Hueca**_ (The Monster of the Hollow Mountain).\n\n**270** _ **Beau Bandit**_ **** RKO Productions, 1930. 68 min. D: Lambert Hillyer. SC: Wallace Smith. With Rod La Rocque, Doris Kenyon, Mitchell Lewis, Walter Long, Charles Middleton, George Duryea (Tom Keene\/Richard Powers), James Donlan, Charles Brinley, Barney Furey, Bill Patton, Kenneth Cooper, Bob Erickson, Gordon Jones, Walt Robbins, Ben Corbett. A notorious bandit plans a bank heist but becomes enamored with a pretty girl only to find she is being sought after by the institution he is planning to rob. This early sound oater, definitely a curio today, was considered old-fashioned when it was first released.\n\n**271** _ **The Beautiful Blonde from Bashful Bend**_ **** 20th Century\u2013Fox, 1949. 77 min. Color. D-SC: Preston Sturges. With Betty Grable, Cesar Romero, Rudy Vallee, Olga San Juan, Sterling Holloway, Hugh Herbert, El Brendel, Porter Hall, Danny Jackson, Emory Parnell, Alan Bridge, Chris-Pin Martin, Patti Behrs, Margaret Hamilton, J. Farrell MacDonald, Richard Hale, Georgia Caine, Esther Howard, Harry Hayden, Chester Conkin, Torben Meyer, Dewey Robinson, Richard Kean, Harry Tyler, Dudley Dickerson, Russell Simpson, Marie Windsor, Marie Monica McDonald, Snub Pollard, Frank Moran, Joseph Turkel, George Lynn, James Joseph O'Neill, Philo McCullough, George Melford, Tom McGuire, Eddie Gribbon, Emil Sitka, Gertrude Astor, George Magrill, Hank Mann, Frank Mills, Blackie Whiteford, Max Wagner, Ray Spiker. After a run-in with the law, a gun-toting young woman goes to a town where she is mistaken for the new school marm, but soon runs afoul of crooks. Bland Preston Sturges \"comedy\" that was a box office bust when it was issued, although a fine cast (Rudy Vallee and Emory Parnell are especially good) does its best.\n\n**Rudy Vallee, Betty Grable, Danny Jackson and Sterling Holloway in _ **The Beautiful Blonde from Bashful Bend**_ (20th Century\u2013Fox, 1949).**\n\n**272** _ **Beauty and the Bandit**_ **** Monogram, 1946. 77 min. D: William Nigh. SC: Charles Belden. With Gilbert Roland, Ramsay Ames, Martin Garralaga, Frank Yaconelli, Vida Aldana, George J. Lewis, William Gould, Dimas Sotello, Felipe Turich, Glenn Strange, Alex Montoya, Artie Ortego. The Cisco Kid joins forces with a beautiful female bandit who is fighting corrupt officials. Nicely done \"Cisco Kid\" series adventure for which star Gilbert Roland wrote additional dialogue; the film is greatly helped by the presence of the luscious Ramsay Ames as a female Robin Hood.\n\n**273** _ **Before the White Man Came**_ **** FC, 1921. 50 min. D: John Maple. SC: William E. Wing. The story of the American Indian and the tribes who lived in the West before the arrival of white settlers. Rather interesting, if somewhat crude, semi-documentary made in the B Horn Mountains of Montana and Wyoming with the cooperation of the Crow people (the film has an all Native American cast) who had adopted director John Maple into their tribe. Production was made in association with the Department of Indian Affairs and endorsed by the Department of the Interior.\n\n**274** _ **Behind Southern Lines**_ **** Monogram, 1952. 51 min. D: Thomas Carr. SC: Melvin Levy and Maurice Tombragel. With Guy Madison, Andy Devine, Rand Brooks, Milburn Stone, Gloria Saunders, Robert Shayne, Murray Alper, Jonathan Hale, William Ruhl, Parke MacGregor, Bill McKenzie, Duke Green, Lee Phelps, Bill Meader, Orley Lindgren. During the Civil War, Union officer Wild Bill Hickok and his pal Jingle P. Jones go on a spy mission into the Confederacy and after the war, as U.S. marshals, they try to stop a gunman forcing a protection racket on miners. Okay initial theatrical release of episodes (\"Behind Southern Lines\" and \"The Silver Mine Protection Story\") of \"The Adventures of Wild Bill Hickok\" (1951\u201358).\n\n**275** _ **Behind the Mask of Zorro**_ **** Duca\/Hispamer\/Promidex Film, 1965. 97 min. Color. D: Richard Blasco. SC: Richard Blasco, Mario Amendola, Jose Gallardo, Luis Lucas Ojeda and Daniel Ribera. With Tony Russel, Rosita Yarza, Jesus Puente, Maria Jose Alfonso, Jose Maria Seoane, Agustin Gonzalez, Mireya Meravigilia, Jose Rubio, Felix Garcia Sancho, Roberto Paoleti, Naria Seoane, Angel Soler, Maria Luisa Arias, Aldo Decconi, Angel Soler, Riccardo Lillo. The California governor's butler is really Zorro who is trying to capture a bandit fomenting a revolution. Another in the long line of European \"Zorro\" sagas, this one being fairly good. An Italian-Spanish co-production filmed as _**El Zorro Cabalga Otra Vez**_ (Zorro Rides Another Time) and also called _**Oath of Zorro**_.\n\n**276** _ **Behind Two Guns**_ **** Aywon, 1924. 58 min. D-SC: Robert North Bradbury. With J.B. Warner, Hazel Newman, Jim Welch, Otto Lederer, William Calles, Marin Sais, Jay Morley, Jack Waltemeyer, Emile Gerdes, Bart Carre, Robert North Bradbury. A physician and his Indian cohort try to solve mysterious strongbox robberies. This fair silent Anthony J. Xydias production provides a chance to see a J.B. Warner vehicle.\n\n**277** _ **Belle Le Grand**_ **** Republic, 1951. 90 min. D: Allan Dwan. SC: D.D. Beauchamp. With Vera Ralston, John Carroll, William Ching, Muriel Lawrence, Hope Emerson, Grant Withers, Stephen Chase, John Qualen, Henry (Harry) Morgan, Charles Cane, Thurston Hall, Marietta Canty, Glenn Vernon, Don Beddoe, Isabel Randolph, John Holland, Frank Wilcox, Paul Maxey, Pierre Watkin, John Hart, Edward Keane, Russell Hicks, Sam Flint, Ed Cassidy, John Hamilton, Perry Ivins, William Schallert, Maude Eburne, Carl \"Alfalfa\" Switzer, Queenie Smith, Peter Brocco, Hal Price, Dick Elliott, Andrew Tombes, Eddie Parks, Fred Hoose, James Kirkwood, John Wengraft, Howard Negley, Ruth Robinson, Gino Corrado, Thomas Browne Henry, James Arness, Eddie Dunn, Emory Parnell, Chester Clute, Sam Flint, Rodney Bell, John Close, John Vosper, Howard Mitchell, Jimmy Ogg, Don C. Harvey, Jerry Miley, Joseph Granby, Sam Sebby. In Virginia City a lady gambler marries a no-account she loves despite the fact the man has a yen for her younger sister. Big production, good supporting cast, empty script\u2014mediocre film.\n\n**278** _ **Belle of the Yukon**_ **** RKO Radio, 1944. 84 min. D: William A. Seiter. SC: James Edward Grant. With Randolph Scott, Gypsy Rose Lee, Dinah Shore, Bob Burns, Charles Winninger, William Marshall, Guinn Williams, Robert Armstrong, Florence Bates, Victor Kilian, Wanda McKay, Edward Fielding, Charles Soldani, Jane Hale, The Yukon Belles. A crooked gambling hall proprietor in the Yukon during the gold rush finally becomes an honest man to make his sweetheart happy. Fair adventure with emphasis on music and comedy.\n\n**279** _ **Belle Starr**_ **** 20th Century\u2013Fox, 1941. 87 min. D: Irving Cummings. SC: Lamar Trotti. With Randolph Scott, Gene Tierney, Dana Andrews, John Shepperd (Shepperd Strudwick), Elizabeth Patterson, Chill Wills, Louise Beavers, Olin Howland, Paul Burns, Joseph Sawyer, Joseph Downing, Charles Trowbridge, Howard Hickman, James Flavin, Charles Middleton, Matthew \"Stymie\" Beard, Mae Marsh, Kermit Maynard, Franklyn Farnum, Cecil Weston, Hugh Chapman, Davison Clark, George Reed, Norman Willis, Clarence Muse, Clinton Rosemond, Michael Morris, Dolores Hurlic, George Melford, Herbert Ashley, Billy Wayne, Dick Rich. After the Civil War, a Confederate guerilla leader and his bandit-wife wage a private war against local carpetbaggers. Not much fact in this screen biography of Belle Starr but who cares with Gene Tierney to look at?\n\n**280** _ **Belle Starr**_ **** CBS-TV, 1980. 97 min. Color. D: John A. Alonzo. SC: James Lee Barrett. With Elizabeth Montgomery, Cliff Potts, Michael Cavanaugh, Fred Ward, Jesse Vint, Allan Vint, Geoffrey Lewis, Gary Combs, Sandy McPeak, David Knell, Geno Silva, Michelle Stacy, Peter Hobbs, Morgan Paull, Sarah Cunningham, Stony Bower, Burt Edwards, James Burke, Dee Cooper, Gilbert Combs, Kate Williams, John Edwards. Another retelling of the life of Belle Starr, this time showing her as a rebellious female who deserts her lover to ride with the likes of the James, Dalton and Younger brothers. Like its theatrical predecessor, this TV film has little foundation in fact but Elizabeth Montgomery is good in the title role.\n\n**281** _ **Belle Starr Story**_ **** Eureka Films\/Mercurfilm, 1968. 103 min. Color. D-SC: Nathan Wich (Piero Cristofani and Lina Wertmuller). With Elsa Martinelli, Robert Woods, George Eastman, Dan Harrison, Francesca Righini, Bruno Corazzari, Vladmer Nedar, Eugene Walter, Remo De Angelis. The two men she loves leads lovely Belle Starr into a life of crime in the Old West and when one of the is captured during a robbery she comes to his rescue. Not much in the way of historical fact in this Italian production, made as _**Il Mio Corpo per un Poker**_ (My Body for a Poker), but Elsa Martinelli is easy to look at in the title role.\n\n**282** _ **Belle Starr's Daughter**_ **** 20th Century\u2013Fox, 1947. 85 min. D: Lesley Selander. SC: W.R. Burnett. With George Montgomery, Rod Cameron, Ruth Roman, Wallace Ford, Charles Kemper, William Phipps, Edith King, Chris-Pin Martin, Jack Lambert, J. Farrell MacDonald, Fred Libby, Larry Johns, Isabel Jewell, Kenneth MacDonald, Christine Larson, Frank Darien, Charles Stevens, William Perrott, Mary Foran, Paul E. Burns, Lane Chandler, Alvin Hammer, Harry Harvey, Bill Kennedy, Herbert Heywood, Henry Hall, John \"Skins\" Miller, William Ruhl, Hank Patterson, Carol Henry, Rudy Bowman, Johnny Reese. A marshal is blamed for the murder of Belle Starr and her daughter comes to town to avenge her mother's death. Better than average outlaw yarn aided by its trio of stars, steady direction and a good script.\n\n**283** _ **Bells of Capistrano**_ **** Republic, 1942. 73 min. D: William Morgan. SC: Lawrence Kimble. With Gene Autry, Smiley Burnette, Virginia Grey, Lucien Littlefield, Morgan Conway, Claire DuBrey, Charles Kane, Joe Strauch, Jr., Marla Shelton, Tristram Coffin, Jay Novello, Alan Bridge, Eddie Acuff, Jack O'Shea, Julian Rivero, William Forrest, Ken Christy, Dick Wessell, Guy Usher, Ralph Peters, Joe McGuinn, Terrisita Osta, Howard Hickman, William Kellogg, Peggy Satterlee, Frankie Marvin, Ray Jones. Singer Gene Autry and his crew join a rodeo targeted for destruction by a rival outfit. Gene Autry's last pre\u2013World War II starrer is on the sluggish side.\n\n**284** _ **Bells of Coronado**_ **** Republic, 1950. 67 min. Color. D: William Witney. SC: Sloan Nibley. With Roy Rogers, Dale Evans, Grant Withers, Foy Willing and The Riders of the Purple Sage, Pat Brady, Clifton Young, Robert Bice, Stuart Randall, Leo Cleary, John Hamilton, Edmund Cobb, Rex Lease, Lane Bradford, Eddie Lee, Henry Rowland, Loren Riebe, Duke Green. Insurance investigator Roy Rogers tries to find missing uranium ore thought to be sought after by a foreign power. Good action and an entertaining storyline make this an above average later Roy Rogers feature.\n\n**285** _ **Bells of Innocence**_ **** Good Times, 2003. 110 min. Color. D: Alin Bijan. SC: Chris Bessey. With Chuck Norris, Mike Norris, Carey Scott, David A.R. White, Marshall R. Teague, Scarlett McAllister, Grant James, Gabby Di Ciolli, Scotty Veale, Matthew Grear, Cindy Michelle, Dennis O'Neil, Julie Arebalo, Marcus Mozier, Tara Di Leva, Alysse Cook. After the death of his daughter, a man and two friends embark on a journey and end up in a remote Texas town where they have to confront a evil mystery man who controls the area. Tepid modern-day horror Western based on a story by co-star Mike Norris, star Chuck Norris' son.\n\n**286** _ **Bells of Rosarita**_ **** Republic, 1945. 68 min. D: Frank McDonald. SC: Jack Townley. With Roy Rogers, George \"Gabby\" Hayes, Dale Evans, Adele Mara, Grant Withers, Bob Nolan and The Sons of the Pioneers (Tim Spencer, Ken Carson, Shug Fisher, Hugh Farr, Karl Farr), Roy Barcroft, Earle Hodgins, Addison Richards, Janet Martin, Syd Saylor, Ed Cassidy, Kenne Duncan, Rex Lease, Robert Wilke, Ted Adams, Wally West, The Robert Mitchell Boychoir, Helen Talbot, Poddles Hanneford, Hank Bell, Eddie Kane, Tom London, Marin Sais, Sam Ash, Barbara Elliott, Mary McCarty, Bill Elliott, Allan Lane, Robert Livingston, Don \"Red\" Barry, Sunset Carson, Duke Taylor, Harvey Parry, Cactus Mack, Duke Green, Roger Creed, Frank McDonald, Frank McCarroll, Jack Richardson, Gil Perkins, Buster Brodie, Marian Kerrigan, Billy Cummings, Larry Williams. Movie star Roy Rogers tries to keep a circus from being taken over by crooks and he enlists the help of other screen cowboy heroes (Don \"Red\" Barry, Sunset Carson, Bill Elliott, Allan Lane, Bob Livingston). Interesting Roy Rogers outing with too much music and pretty Adele Mara stealing the acting honors.\n\n**287** _ **Bells of San Angelo**_ **** Republic, 1947. 71 min. Color. D: William Witney. SC: Sloan Nibley. With Roy Rogers, Dale Evans, Andy Devine, John McGuire, Olaf Hytten, David Sharpe, Fritz Leiber, Hank Patterson, Fred \"Snowflake\" Toones, Eddie Acuff, Bob Nolan and The Sons of the Pioneers (Tim Spencer, Lloyd Perryman, Pat Brady, Hugh Farr, Karl Farr), Dale Van Sickel, Buck Bucko, James Linn, Eddie Parker, Fred Graham, Luana Walters, Keefe Brasselle, Rex Rossi, Charles Sullivan, Ray Turner. Government investigator Roy Rogers is after a gang smuggling silver across the Mexican border. Fairly entertaining Roy Rogers entry.\n\n**288** _ **Bells of San Fernando**_ **** Screen Guild, 1947. 74 min. D: Terry Morse. SC: Jack DeWitt and Renault Duncan (Duncan Renaldo). With Donald Woods, Gloria Warren, Monte Blue, Shirley O'Hara, Byron Foulger, Paul Newlan, Anthony Warde, Claire Du Brey, Gordon Clark, Angelo Rossitto, David Leonard, Frank Cody, Felipe Turich, Drew Allen, Luisa Triana. An evil man takes control of a small town in Southern California and rules like a tyrant until he is opposed by an Irish immigrant. Cheaply made but entertaining \"B\" feature.\n\n**289** _ **Below the Border**_ **** Monogram, 1942. 57 min. D: Howard Bretherton. SC: Jess Bowers (Adele Buffington). With Buck Jones, Tim McCoy, Raymond Hatton, Linda Brent, Charles King, Dennis Moore, Roy Barcroft, Ted Mapes, Bud Osborne, Eva Puig, Merrill McCormick, Jack Rockwell, Howard Masters, Walter McGrail, Reed Howes, Tex Palmer, Kermit Maynard, Frank Ellis, Kansas Moehring, Bill Nestell, Wally West, Jack Daly, Denver Dixon, Foxy Callahan. Three lawmen assume various guises as they head to the Mexican border to put a stop to a gang of cattle rustlers who have stolen jewels from a young woman whose fiance is involved with the crooks. Fine entry in \"The Rough Riders\" series.\n\n**Advertisement for** _**Below the Border**_ **(Monogram, 1942).**\n\n**  \n**\n\n_**Ben and Charlie**_ see _**The Ballad of Ben and Charlie**_ (1972)\n\n**290** _ **Bend of the River**_ **** Universal-International, 1952. 91 min. Color. D: Anthony Mann. SC: Borden Chase. With James Stewart, Julia (Julie) Adams, Arthur Kennedy, Rock Hudson, Lori Nelson, Jay C. Flippen, Henry (Harry) Morgan, Chubby Johnson, Royal Dano, Stepin Fetchit, Howard Petrie, Jack Lambert, Frank Ferguson, Frances Bavier, Cliff Lyons, Lillian Randolph, Britt Wood, Gregg Barton, Philo McCullough, Donald Kerr, Jennings Miles, Frank Chase, Hugh Prosser, Harry Arnie, Denver Dixon. A former border raider saves a man from hanging and the two lead a wagon train of fruit farmers to Oregon and along the way they fight hostile Indians. Big, brawling oater which is nicely produced; good entertainment.\n\n**291** _ **Beneath Western Skies**_ **** Republic, 1944. 56 min. D: Spencer Gordon Bennet. SC: Albert De Mond and Bob Williams. With Robert Livingston, Smiley Burnette, Effie Laird, Joe Strauch, Jr., LeRoy Mason, Kenne Duncan, Bud Geary, Jack Kirk, Tom London, Frank Jacquet, Jack Ingram, Budd Buster, Robert Wilke, Tom Steele, Herman Hack, Carl Sepulveda. A sheriff tries to stop an outlaw gang but a blow to the head causes him to have amnesia and becoming a pawn of the crooks. One of a trio of \"John Paul Revere\" films starring Robert Livingston; enjoyable.\n\n**292** _ **The Best Man Wins**_ **** Columbia, 1948. 73 min. D: John Sturges. SC: Edward Huebsch. With Edgar Buchanan, Anna Lee, Robert Shayne, Gary Gray, George Lynn, Hobart Cavanaugh, Stanley Andrews, Bill Sheffield, Paul E. Burns, Marietta Canty. A man who deserted his family returns home and tries to win back his wife with the help of their son. Okay adaptation of Mark Twain's story \"The Celebrated Jumping Frog of Calaveras County.\"\n\n**293** _ **Best of the Badmen**_ **** RKO Radio, 1951. 84 min. Color. D: William D. Russell. SC: Robert Hardy Andrews and John Twist. With Robert Ryan, Claire Trevor, Jack Buetel, Robert Preston, Walter Brennan, Bruce Cabot, John Archer, Lawrence Tierney, Barton MacLane, Tom Tyler, Robert Wilke, John Cliff, Lee MacGregor, Emmett Lynn, Carleton Young, Byron Foulger, Larry Johns, Harry Woods, William Tannen, Ed Max, David McMahon, Everett Glass, Robert Kortman. A Union officer takes Jesse James, the Ringo Kid and the Younger Brothers into custody only to be framed on a murder charge by a crooked detective. All-star galloper is quite good and will warm the hearts of genre fans.\n\n**294** _ **The Better Man Wins**_ **** Sanford Productions, 1922. 64 min. D-SC: Frank S. Mattison and Marcel Perez. With Pete Morrison, Dorothy Wood, E.L. Van Sickle, Jack Walters, Gene Crosby, Tom Bay. A cowboy comes to the assistance of a ranch owner's daughter when her father becomes ill, but is soon enticed by a pretty house guest, a cabaret performer. Fun Pete Morrison silent feature.\n\n**295** _ **Between Fighting Men**_ **** World Wide, 1932. 60 min. D: Forrest Sheldon. SC: Betty Burbridge and Forrest Sheldon. With Ken Maynard, Ruth Hall, Wallace MacDonald, Josephine Dunn, Albert J. Smith, Walter Law, James Bradbury, Jr., John Pratt, Charles King, Edmund Cobb, Robert Kortman, Jack Rockwell, Jack Kirk, Bud McClure, Roy Bucko, Blackjack Ward. Two men try to halt a range war by stopping sheep men from moving their herds through town and at the same time they become rivals for the same girl. Very pleasant Ken Maynard vehicle with good use of comedy and surprisingly little action.\n\n**296** _ **Between God, the Devil and a Winchester**_ **** Hispamex, 1968. 98 min. Color D: Dario Silvestri (Marino Girolami). SC: Marino Girolami, Manuel Martinez Remis and Tito Carpi. With Gilbert Roland, Richard Harrison, Ennio Girolami, Folco Lulli, Raf Baldassare, Dominique Boschero, Roberto Camardiel, Luis Barboo, Humberto Sempere, Gonzalo Esquiroz, Rocco Lerro, Jose Luis Lluch, Mirella Pamphili, Jose Sanchez, Enzo G. Castellari, Xan das Bolas, Rafael de la Rosa, Jose Maria Ecenarro, Jose Sacristan. A gang leader and a priest pretending to be a gunman try to enlist the aid of a scout in finding stolen treasure. Pretty fair Spaghetti Western adventure dominated by good performances by Gilbert Roland and Richard Harrison in this reworking of Robert Louis Stevenson's _Treasure Island_. Filmed as _**Anche nel West C'era una Volta Dio**_ (God Was Also in the West at One Time).\n\n**297** _ **Between Men**_ **** Supreme, 1935. 59 min. D: Robert North Bradbury. SC: Charles Francis Royal. With Johnny Mack Brown, Beth Marion, William Farnum, Earl Dwire, Lloyd Ingraham, Wally Wales, Frank Ball, Harry Downing, Horace B. Carpenter, Forrest Taylor, Bud Osborne, Sherry Tansey, Milburn Morante, Artie Ortego, Horace Murphy, Budd Buster, Francis Walker, Jack Kirk, George Morrell, Jim Corey, Tex Phelps, Archie Ricks, Clyde McClary, Silver Tip Baker. A young man goes West to find the rejected daughter of the man who raised him and he not only locates her but also his real father who thought he had been killed. Complicated plotline, good performances and competent production help this one add up to pleasing entertainment; the fist fight between Brown and Farnum is a corker with the latter more than holding his own.\n\n**298** _ **Beyond the Border**_ **** Producers Distributing Corporation, 1925. 55 min. D: Scott R. Dunlap. SC: Harvey Gates. With Harry Carey, Mildred Harris, Tom Santschi, Jack Richardson, William Scott. A lawman brings in a girl's brother who has been falsely accused of a crime only to find a local crook has taken his job. Entertaining Harry Carey silent feature.\n\n_**Beyond the Frontier of Hate**_ see _**Four Came to Kill Sartana**_\n\n**299** _ **Beyond the Last Frontier**_ **** Republic, 1943. 57 min. D: Howard Bretherton. SC: John K. Butler and Morton Grant. With Eddie Dew, Smiley Burnette, Lorraine Miller, Robert Mitchum, Harry Woods, Kermit Maynard, Ernie Adams, Richard Cramer, Jack Kirk, Wheaton Chambers, Jack Rockwell, Cactus Mack, Art Dillard, Tom Steele, Henry Wills, Curley Dresden. In an effort to stop border gun runners, the Texas Rangers have one of their own infiltrate the gang. The first of two films which starred Eddie Dew in the \"John Paul Revere\" series, this one fails to do much other than have the hero take a backseat to villain Bob Mitchum.\n\n**300** _ **Beyond the Law**_ **** Syndicate, 1930. 60 min. D: J.P. McGowan. SC: G.A. Durlam. With Robert Frazer, Louise Lorraine, Lane Chandler, Charles King, Jimmy Kane, William Walling, Franklyn Farnum, Harry Holden, George Hackathorne, Ed Lynch, Robert Graves, Al St. John, Bob Reeves, Blackie Whiteford, Tex Phelps. Two cowpokes ride into a region where the law is in cahoots with a hoodlum who hires a famous road agent to run ranchers off their spreads. Slow moving early talkie that is poorly recorded, includes too much stock footage and has two inane musical numbers by a group of singing soldiers.\n\n**301** _ **Beyond the Law**_ **** Columbia, 1934. 60 min. D: D. Ross Lederman. SC: Harold Shumate. With Tim McCoy, Shirley Grey, Lane Chandler, Addison Richards, Dick Rush, Harry C. Bradley, Morton Laverre (John Merton). When a girl's father is sent to prison for a murder committed during a holdup, a railroad detective, who believes him innocent, tries to find the real culprits. Good Tim McCoy programmer made as a part of his non-Western series for Columbia but basically a genre entry.\n\n**302** _ **Beyond the Law**_ **** Sancrosiap\/Roxy Film, 1968. 110 min. Color. D: Giorgio Stegani. SC: Giorgio Stegani and Fernando Di Leo. With Lee Van Cleef, Antonio Sabato, Lionel Stander, Graziella Granata, Bud Spencer, Ann Smyrner, Herbert Fux, Carlo Gaddi, Enzo Fiermonte, Gordon Mitchell, Hans Elsenspoek, Gunther Stoll, Carlo Pedersoli. A notorious bandit and his two cohorts rob a stage and he later befriends a man who saves his life and in a small town, is made sheriff and must protect a money shipment from a gang of vicious holdup men. Violent but well done Lee Van Cleef vehicle made in Italy as _**Al Di La Della Legge**_ (Beyond the Law). Issued in the U.S. in 1973 by Cinema Shares running 78 minutes.\n\n**303** _ **Beyond the Pecos**_ **** Universal, 1945. 59 min. D: Lambert Hillyer. SC: Bennett Cohen. With Rod Cameron, Eddie Dew, Fuzzy Knight, Jennifer Holt, Ray Whitley and His Bar-Six-Cowboys, Gene Roth, Robert Homans, Jack Ingram, Frank Jaquet, Henry Wills, Jack Rockwell, Jim Thorpe, Dan White, Al Ferguson, Forrest Taylor, William Desmond, Herman Hack, Artie Ortego, Merle Travis, Jerome Sheldon, Buster Brodie. Two men from feuding families fight over rich oil land rights and the love of a pretty girl. Stout Rod Cameron vehicle nicely helmed by veteran Lambert Hillyer.\n\n**304** _ **Beyond the Purple Hills**_ **** Columbia, 1950. 70 min. D: John English. SC: Norman S. Hall. With Gene Autry, Pat Buttram, Jo Dennison, Don Beddoe, James Millican, Don Kay Reynolds, Hugh O'Brian, Robert Wilke, Roy Gordon, Harry Harvey, Gregg Barton, Ralph Peters, Frank Ellis, John Cliff, Sandy Sanders, Merrill McCormick, Tex Terry, Maudie Prickett, Pat O'Malley, Herman Hack, Cliff Barnett, Frank O'Connor, Frankie Marvin, Bobby Clark, Boyd Stockman, Lynton Brent. An acting sheriff is forced to arrest his pal when the latter's father is murdered although he believes his friend is innocent. Adequate Gene Autry musical western, a remake of _**Sheriff of Las Vegas**_ (q.v.).\n\n**305** _ **Beyond the Rio Grande**_ **** Big 4, 1930. 60 min. D: Harry S. Webb. SC: Carl Krusada. With Jack Perrin, Franklyn Farnum, Charline Burt, Emma Tansey, Buffalo Bill, Jr., Pete Morrison, Edmund Cobb, Henry Roquemore, Henry Taylor. When his partner robs a bank, a man is falsely blamed for the crime and is forced to head south of the border. Pretty fair early talkie that will appeal to fans of Jack Perrin and Franklyn Farnum.\n\n**306** _ **Beyond the Rockies**_ **** RKO Radio, 1932. 60 min. D: Fred Allen. SC: Oliver Drake. With Tom Keene, Rochelle Hudson, Julian Rivero, Hank Bell, Ernie Adams, William Welsh, Ted Adams, Tom London, Marie Wells, Slim Whitaker, Frank Ellis, Jim Corey, Al Taylor, Rosa Turich. The governments sends a undercover agent to Texas to look into cattle rustling. David O. Selznick was the executive producer on his well made and fast moving production, starring a bit too gung-ho Tom Keene.\n\n**307** _ **Beyond the Sacramento**_ **** Columbia, 1940. 58 min. D: Lambert Hillyer. SC: Luci Ward. With Bill Elliott, Evelyn Keyes, Dub Taylor, John Dilson, Bradley Page, Frank LaRue, Norman Willis, Steve Clark, Jack Clifford, Don Beddoe, Bud Osborne, George McKay, Olin Francis, Tex Cooper, Ned Glass, Harry Bailey, Blackjack Ward, Jack Low, Clem Horton, Chick Hannon, Eddie Laughton, George Morrell, Tex Phelps, Jack Tornek, Tom Moray. Settlers in California are plagued by lawlessness and Wild Bill Hickok tries to put a stop to the terrorism. Faced paced Bill Elliott affair with nice work by Evelyn Keyes as the heroine. British title: _**Power of Justice**_.\n\n**308** _ **Beyond the Trail**_ **** Chesterfield, 1926. 50 min. D: Al Herman. With Bill Patton, Janet Dawn, Eric Mayne, Sheldon Lewis, Stuart Holmes, Clara Horton, James F. Fulton. After he accidentally causes two robbers to shoot each other, a bumbling cowpoke is assigned to bring in a thieving ranch foreman and ends up defeating him in a fight for the girl he loves. Tepid silent effort headlining klutzy hero (!) Bill Patton.\n\n**309** _ **The Big Bonanza**_ **** Republic, 1945. 68 min. D: George Archainbaud. SC: Dorrell McGowan, Stuart McGowan and Paul Gengelin. With Richard Arlen, Robert Livingston, Jane Frazee, George \"Gabby\" Hayes, Lynne Roberts, Bobby Driscoll, J.M. Kerrigan, Russell Simpson, Frank Reicher, Cordell Hickman, Roy Barcroft, Fred Kohler, Jr., Charles King, Jack Rockwell, Henry Wills, Fred Graham, Dan White, Robert Wilke, Monte Hale, Tom Steele. During the Civil War a Union soldier is wrongly accused of cowardice and goes West where he meets up with a old friend, the gambling house proprietor who framed him. Republic B plus production benefited by the performances Richard Arlen and Bob Livingston as the good and bad guys.\n\n**310** _ **Big Boy Rides Again**_ **** Beacon, 1935. 55 min. D: Al Herman. SC: William Nolte. With Guinn Williams, Connie Bergen, Charles K. French, Lafe McKee, Victor Potel, Bud Osborne, William Gould, Augie Gomez. A cowboy returns home to aid his father in protecting a hidden treasure only to have his dad killed and himself stalked by the murderer. Atmospheric, spooky dark house thriller set in the Old West. Remade as _**Saddle Mountain Roundup**_ (q.v.).\n\n**311** _ **Big Calibre**_ **** Supreme, 1935. 58 min. D: Robert North Bradbury. SC: Perry Murdock. With Bob Steele, Peggy Campbell, Georgia O'Dell, Earl Dwire, Bill Quinn, John Elliott, Forrest Taylor, Perry Murdock, Si Jenks, Frank Ball, Frank McCarroll, Blackie Whiteford. When he is falsely accused of killing his father, a rancher is nearly lynched before he escapes and proves his innocence. Bob Steele is on the run again in this series oater, which is not typical in that a physically grotesque chemist uses poison gas to kill his victims.\n\n**312** _ **The Big Cat**_ **** Eagle-Lion, 1949. 75 min. Color. D: Phil Karlson. SC: Morton Grant. With Preston Foster, Lon McCallister, Forrest Tucker, Peggy Ann Garner, Skip Homeier, Sara Holden, Irving Bacon, Gene Reynolds. The long time hatred of two men over a girl is triggered by the arrival of her son in a Utah valley plagued by a rampaging cougar and a deadly drought. Better-than-average outdoor melodrama helped by nice location scenery and a good cast.\n\n_**Big Chuck, Little Chuck**_ see _**Hollywood, It's a Dog's Life**_\n\n**313** _ **The Big Country**_ **** United Artists, 1958. 156 min. Color. D: William Wyler. SC: James R. Webb, Sy Bartlett and Robert Wilder. With Gregory Peck, Jean Simmons, Charlton Heston, Burl Ives, Carroll Baker, Charles Bickford, Alfonso Bedoya, Chuck Connors, Chuck Hayward, Buff Brady, Jim Burk, Dorothy Adams, Chuck Roberson, Bob Morgan, John McKee, Jay Slim Talbot, John Morgan. A one-time sea captain arrives in the West to marry a rancher's daughter and finds himself in the middle of a feud over water rights. Sprawling Western which is splendidly made but overblown and verbose.\n\n_**A Big Deal at Dodge City**_ see _**A Big Hand for the Little Lady**_\n\n**314** _ **The Big Diamond Robbery**_ **** Film Booking Offices (FB0), 1929. D: Eugene Forde. SC: John Stuart Twist. With Tom Mix, Kathryn McGuire, Frank Beal, Martha Mattox, Ernest Hilliard, Barney Furey, Ethan Laidlaw, Tony (horse). A ranch foreman exposes a robbery gang led by the friend of his employer's daughter, the crooks being after her priceless diamond ring. Modern-day Tom Mix adventure that survives today only in a 22-minute version, approximately the middle third of the feature.\n\n**315** _ **Big Foot**_ **** Ellman Enterprises\/Gemini-American\/Western-International, 1971. 94 min. Color. D: Robert Slatzer. SC: Robert Slatzer and James Gordon White. With Chris Mitchum, Joi Lansing, John Carradine, Ken Maynard, Lindsay Crosby, James Craig, Judy Jordan, John Mitchum, Joy Wilkerson, Doodles Weaver, Dorothy Keller, Noble \"Kid\" Chissel, Nick Raymond, James Stellar, Lois Red Elk, Lonesome Fawn. When her plane crashes in Northern California, a woman is captured by an ape-like creature and after a rescue party saves her a peddler makes plans to exhibit the monster. Combination of horror and Western genres for the drive-in trade; interesting because of special billed Ken Maynard's return to the screen as a storekeeper. Also known as _**Bigfoot**_.\n\n**316** _ **The Big Gundown**_ **** Columbia, 1968. 90 min. Color. D: Sergio Sollima. SC: Sergio Donati and Sergio Sollima. With Lee Van Cleef, Tomas Milian, Fernando Sancho, Luisa Rivelli, Nieves Navarro, Benito Stefanelli, Walter Barnes, Angel Del Pozo, Maria Granada, Lanfranco Ceccarelli, Roberto Camardiel, Tom Felleghy, Gerard Herter. A famous Texas man hunter gets on the trail of a Mexican accused of raping and murdering a small girl but as the search continues he begins to realize the chase is being used to cover up another crime. Handsomely made feature starring Lee Van Cleef that was popular on both sides of the Atlantic; tremendous Ennio Morricone music score. Released in 1967 in Italy, where it was filmed, as _**La Resa Del Conti**_ (Account Rendered) by P.E.A.\/Tulio De Micheli.\n\n_**Big Gundown 2**_ see _**Run, Man, Run**_\n\n**317** _ **A Big Hand for the Little Lady**_ **** Warner Bros., 1966. 95 min. Color. D: Fielder Cook. SC: Sidney Carroll. With Henry Fonda, Joanne Woodward, Jason Robards, Paul Ford, Charles Bickford, Burgess Meredith, Kevin McCarthy, Robert Middleton, John Qualen, James Kenny, Allen Collins, Jim Boles, Gerald Michenaud, Virginia Gregg, Chester Conklin, Ned Glass, Mae Clarke, James Griffith, Noah Keene, Milton Seltzer, Louise Glenn, William Cort. Trying to rid her husband of his compulsive gambling addiction, and recoup his losses, a woman enters a five-card poker showdown in Laredo in 1896. Somewhat of a genre takeoff, this production is amusing and well done. Also called _**A Big Deal at Dodge City**_.\n\n**318** _ **Big Jack**_ **** Metro-Goldwyn-Mayer, 1949. 85 min. D: Richard Thorpe. SC: Gene Fowler, Marin Borowsky, Otto Van Eyss and Robert Thoere. With Wallace Beery, Richard Conte, Marjorie Main, Edward Arnold, Vanessa Brown, Clinton Sundberg, Charles Dingle, Clem Bevans, Jack Lambert, Will Wright, William \"Bill\" Phillips, Syd Saylor, Andy Clyde, Ann Doran, Vince Barnett, Trevor Bardette, Francis McDonald, Edith Evanson, Tom Fadden, Robert B. Williams, Eddie Dunn, Minerva Urecal, Milton Parsons, Hank Bell, Richard Alexander, Del Henderson, Lane Bradford, Fred Gilman, Holly Bane, Frank McCarroll, James Pierce, E. Mason Hopper, Florence Auer, Henry Sylvester, William Norton Bailey, Casey MacGregor, Cactus Mack, Carl Sepulveda, Billy Dix, Robert Filmer, Frank McGrath, Carol Henry, Joan Blair, Eddie Parks, John Phipps, Jimmy Martin, Lynn Farr. In Colonial times a man and a woman make a living as road agents until they are reformed by a righteous doctor. Fairly good Wallace Beery vehicle with emphasis on humor; Beery's last film and he looks in poor health.\n\n**319** _ **Big Jake**_ **** National General, 1971. 110 min. Color. D: George Sherman. SC: Harry Julian Fink and R.M. Fink. With John Wayne, Maureen O'Hara, Richard Boone, Patrick Wayne, Chris Mitchum, Bobby Vinton, Bruce Cabot, Glenn Corbett, Harry Carey, Jr., John Doucette, Jim Davis, John Agar, Gregg Palmer, Robert Warner, Jim Burke, John Ethan Wayne, Virginia Capers, William Walker, Jerry Gatlin, Tom Hennessy, Don Epperson, Everett Creach, Jeff Wingfield, Hank Worden, Jerry Summers, Chuck Roberson, Bernard Fox, Roy Jenson. When his grandson is kidnapped by outlaws, a rich land owner sets out to rescue him. Typically good John Wayne vehicle with plenty of action and a good story line.\n\n**John Wayne and Maureen O'Hara in _ **Big Jake**_ (National General, 1971).**\n\n**320** _ **The Big Land**_ **** Warner Bros., 1957. 98 min. Color. D: Gordon Douglas. SC: David Dortort and Martin Rackin. With Alan Ladd, Virginia Mayo, Edmond O'Brien, Anthony Caruso, Julie Bishop, John Qualen, Don Castle, David Ladd, Jack Wrather, Jr., George J. Lewis, James Anderson, Don Kelly, Charles Watts, John Doucette, Henry Rowland. Businessmen try to cheat cattlemen and farmers in the post\u2013Civil War era and the latter two unite in an attempt to build a railroad spur that will connect them to better markets. Good Alan Ladd vehicle with fine production values.\n\n**321** _ **Big Money Rustlas**_ **** Psychopathic Records, 2010. 95 min. Color. D: Paul Andresen. SC: Paul Andresen, Studebaker Ducham and Joe Bruce (Violent J). With Violent J, Shaggy 2 Dope, 2 Tuff Tony, Boondox, Blaze, Mark Jury, Jason Mewes, Ron Jeremy, Bridget Powerz, Jody Sadler, Jamie Madrox, Monoxide, Jumpsteady, Brian David, Cindie Haynie, Corporal Robinson, Clay, Otis from Axe Murder Boys, Mike E. Clark, Erwin Shepansky, Billy Bill, Scott Hall, Jimmy Hart. A gambling tycoon is forced into a showdown with a lawman trying to save the town of Mudbug. Western comedy for Joggolo fans.\n\n_**The Big North**_ see _**The Wild North**_\n\n**322** _ **Big Red**_ **** Buena Vista, 1962. 89 min. Color. D: Norman Tokar. SC: Louis Pelletier. With Walter Pidgeon, Gilles Payant, Emile Genest, Janette Bertrand, Doris Lussier, Rolland Bedard, George Bouvier, Teddy Burns Goulet. An orphan boy comes to live at the ranch of a wealthy dog fancier and he develops a rapport with a previously incorrigible canine. Enjoyable Disney family film made in Canada.\n\n**323** _ **The Big Show**_ **** Republic, 1936. 59 min. D: Mack V. Wright. SC: Dorrell McGowan and Stuart McGowan. With Gene Autry, Smiley Burnette, Kay Hughes, Max Terhune, Sally Payne, William Newell, Charles Judels, Rex King, Harry Worth, Mary Russell, Christine Maple, Jerry Larkin, Jack O'Shea, Slim Whitaker, George Chesebro, Edward Hearn, Cliff Lyons, The Sons of the Pioneers (Bob Nolan, Len Slye [Roy Rogers], Tim Spencer, Hugh Farr, Karl Farr), Tracy Layne, Jack Rockwell, Frankie Marvin, Cornelius Keefe, Horace B. Carpenter, Frances Morris, Richard Beach, Art Mix, I. Stanford Jolley, Sally Rand, The SMU 50 (Southern Methodist University Marching Band), The Light Crust Doughboys, The Beverly Hill Billies (Elton Britt, Aleth Hansen, Rudy Sooter), The Jones Boys. When a stuck-up cowboy film hero refuses to appear at a rodeo his stunt double takes his place, becomes mixed up with gangsters and eventually wins screen stardom. Very entertaining Gene Autry feature filmed at the Texas Centennial Exposition with a plethora of country and western acts; Max Terhune's screen debut.\n\n_**Big Showdown**_ see _**The Grand Duel**_\n\n**324** _ **The Big Sky**_ **** RKO Radio, 1952. 140 min. D: Howard Hawks. SC: Dudley Nichols. With Kirk Douglas, Dewey Martin, Elizabeth Threatt, Arthur Hunnicutt, Buddy Baer, Steven Geray, Hank Worden, Jim Davis, Henri Letondal, Robert Hunter, Booth Colman, Paul Frees, Frank De Kova, Guy Wilkerson, Cliff Clark, Fred Graham, George Wallace, Max Wagner, Charles Regan, Sam Ash, Don Beddoe, Jim Hayward, Anthony Jochim, Nolan Leary, Frank Lackteen, Ray Hyke, Eugene Borden, Veola Vonn, Cactus Mack, Crane Whitley. Two pals heading west join a fur trapping expedition and along the way both all in love with an Indian girl. Sprawling adaptation of A.B. Guthrie, Jr.'s novel; cuts for TV at 122 minutes make it more watch able; also available in a computerized color version.\n\n**325** _ **The Big Sombrero**_ **** Columbia, 1949. 82 min. Color. D: Frank McDonald. SC: Olive Cooper. With Gene Autry, Elena Verdugo, Stephen Denne, George J. Lewis, Vera Marshe, William Edmunds, Martin Garralaga, Gene (Roth) Stutenroth, Neyle Morrow, Bob (John) Cason, Pierce Lyden, Jose Alvarado, Alex Montoya, Joe Kirk, Artie Ortego, Joe Dominguez. A cowboy tries to stop a crook from marrying a young woman for her ranch, which he plans to sell. Finely done and entertaining Gene Autry opus.\n\n**326** _ **Big Stakes**_ **** East Coast Productions, 1922. 61 min. D: Clifford S. Elfelt. SC: Frank Howard Clark. With J.B. Warner, Elinor Fair, Les Bates, Willie May Carson, H.S. Karr, Robert Grey, Ethelbert Knott, Louis Emmons. An adventurer romances the betrothed of a rich Mexican rancher but switches his affections to a woman kidnapped by night riders. Action filled, light hearted, silent J.B. Warner vehicle.\n\n**327** _ **The Big Stampede**_ **** Warner Bros., 1932. 53 min. D: Tenny Wright. SC: Kurt Kempler. With John Wayne, Noah Beery, Mae Madison, Luis Alberni, Berton Churchill, Paul Hurst, Sherwood Bailey, Frank Ellis, Hank Bell, Lafe McKee, Slim Whitaker, Ed Phillips, Bud Osborne, Joe Girard, Jim Corey, Al Taylor, Blackjack Ward, Chuck Baldra, G. Raymond Nye, Fred Burns, Iron Eyes Cody, Bob Fleming, Tom Bay, Al Haskell, Tex Phelps, John Ince, Leonard Trainor, Rose Plummer, S.S. Simon, Edward Burns. A deputy sheriff enlists the help of a bandit in opposing a ranch owner who kills lawmen so his gang can rustle cattle. Pleasing early John Wayne vehicle with the usual larger-than-life villainy of Noah Beery. Remake of Ken Maynard's (who is clearly visible with his horse Tarzan in some of the footage) _**The Land Beyond the Law**_ (First National, 1927) and remade under that title (q.v.) by Warner Bros. in 1936 with Dick Foran. The film's theme song is \"Under a Texas Moon.\"\n\n**328** _ **The Big Trail**_ **** United Artists, 1930. 158 min. D: Raoul Walsh. SC: Jack Peabody, Marie Boyle and Florence Postal. With John Wayne, Marguerite Churchill, Tyrone Power (Sr.), El Brendel, Tully Marshall, David Rollins, Ian Keith, Frederick Burton, Russ Powell, Charles Stevens, Helen Parrish, Louise Carver, William V. Mong, Dodo Newton, Jack Peabody, Ward Bond, Marcia Harris, Marjorie Leet, Emslie Emerson, Frank Rainboth, Andy Shuford, Gertrude Van Lent, Lucille Van Lent, DeWitt Jennings, Alphonse Ethier, Chief Big Tree, Don Coleman, Pete Morrison, Iron Eyes Cody. While leading settlers west a young wagon master plans to avenge the murder of a trapper friend by one of the travelers. Originally shot in both 35 mm and 70 mm, this expansive production is a delight to the eye as well as providing John Wayne with a fine first starring role; it is hard to understand why this film was not more popular when it was released. Recommended. Later cut to 125 and 110-minute versions.\n\n**329** _ **The Big Trees**_ **** Warner Bros., 1952. 89 min. Color. D: Felix Feist. SC: John Twist and James R. Webb. With Kirk Douglas, Eve Miller, Patrice Wymore, Edgar Buchanan, John Archer, Alan Hale, Jr., Roy Roberts, Charles Meredith, Harry Cording, Ellen Corby, William Challee, Lester Sharp, Mel Archer, Duke Watson, Lillian Bond, Vicki Raaf, Kay Marlow, Sue Casey, Ann Stuart, Art Millan, Iris Adrian, William Vedder. An unscrupulous timber man gets into the confidence of a valley of settlers so he can cut their redwood trees and sell them. Color adds a nice touch to his lumbering yarn although \"hero\" Kirk Douglas is none-too-likable in the lead.\n\n_**Bigfoot**_ see _**Big Foot**_\n\n**330** _ **Billy Jack**_ **** Warner Bros, 1971. 112 min. Color. D: T.C. Frank (Delores Taylor). SC: Frank Christina and Teresa Christina. With Tom Laughlin, Delores Taylor, Clark Howard, Bert Freed, Julie Webb, Kenneth Tobey, Victor Izay, Debbie Schock, Stan Rice, Teresa Kelly, John McClune, Katy Moffatt. When the existence of a freedom school for runaway teenagers on an Indian reservation in Arizona is threatened by the citizens of a small town, a half-breed Vietnam veteran comes to its aid. Fans of the \"Billy Jack\" series should like this feature but otherwise beware of its peace-preaching.\n\n**331** _ **Billy the Kid**_ **** Metro-Goldwyn-Mayer, 1930. 90 min. D: King Vidor. SC: Wanda Tuchok and Lawrence Stallings. With John (Johnny) Mack Brown, Wallace Beery, Kay Johnson, Wyndham Standing, Karl Dane, Russell Simpson, Blanche Frederici, Roscoe Ates, Warner Richmond, James Marcus, Nelson McDowell, Jack Carlyle, John Beck, Christopher Martin, Soledad Jiminez, Don Coleman, Jack Rockwell, Frank Hagney, Blackjack Ward, Hank Bell. Billy the Kid murders a cattle baron in revenge for the death of a pal, then gets married, but he and his bride are trailed by Sheriff Pat Garrett, and a posse. Early talkie shot in wide screen is something of a novelty, but definitely worth a look. TV title: _**The Highwayman Rides**_.\n\n**332** _ **Billy the Kid**_ **** Metro-Goldwyn-Mayer, 1941. 95 min. D: David Miller. SC: Gene Fowler. With Robert Taylor, Brian Donlevy, Ian Hunter, Mary Howard, Gene Lockhart, Henry O'Neill, Frank Puglia, Cy Kendall, Connie Gilchrist, Ethel Griffies, Chill Wills, Guinn Williams, Olive Blakeney, Lon Chaney, Jr., Frank Conlan, Mitchell Lewis, Dick Curtis, Ted Adams, Earl Gunn, Frank Dunn, Grant Withers, Joe Yule, Carl Pitti, Arthur Housman, Lew Harvey, Priscilla Lawson, Kermit Maynard, Slim Whitaker, Ray Teal, George Chesebro, Frank Hagney, Edwin Brady, Tom London, Buck Mack, Ben Petti. A long-time lawman friend of Billy the Kid is forced to hunt him down when the outlaw kills the corrupt rancher who murdered his pal. A fair remake of the 1930 _**Billy the Kid**_ (q.v.) with Robert Taylor miscast in the lead although Brian Donlevy is very good as Pat Garrett, here called Jim Sherwood.\n\n**333** _ **Billy the Kid**_ **** Turner Home Entertainment, 1989. 96 min. Color. D: William A. Graham. SC: Gore Vidal. With Val Kilmer, Wilford Brimley, Rene Auberjonois, Michael Parks, Albert Salmi, Duncan Regehr, Julie Carmen, Tom Everett, Will Hannah, Clark Ray, Billy Joe Patton, Andrew Bicknell, Ned Vaughn, Mike Casper, Nate Esformes, Jack Dunlap, Richard Glover, John O'Hurley, Ric San Nicholas, Richard Blake, Bing Blenman, Patrick Massett, Kirk Nelson, Burr Steers, Ed Adams, Roberto Guajardo, Tiny Wells, Henry Max Kedrick, Sam Smiley, Red West, Rich Wheeler, Sidney Laen, Gore Vidal. Young Billy Bonney finds himself in the middle of the Lincoln County Cattle War, leading to his becoming a wanted outlaw. Passable entertainment; revisionist history. Also called _**Gore Vidal's Billy the Kid**_.\n\n**Billy the Kid in Blazing Frontier** see _**Blazing Frontier**_\n\n_**Billy the Kid in Cattle Stampede**_ see _**Cattle Stampede**_\n\n_**Billy the Kid in Fugitive of the Plains**_ see _**Fugitive of the Plains**_\n\n**334** _ **Billy the Kid in Santa Fe**_ **** Producers Releasing Corporation, 1941. 64 min. D: Peter Stewart (Sam Newfield). SC: Joseph O'Donnell. With Bob Steele, Al St. John, Rex Lease, Dennis Moore, Marin Sais, Karl Hackett, Steve Clark, Hal Price, Charles King, Frank Ellis, Dave O'Brien, Kenne Duncan, Curley Dresden, John Elliott, Reed Howes, Milburn Morante, Henry Wills, Art Dillard, Wally West, George Morrell, Ray Henderson, Chick Hannon, Artie Ortego, Foxy Callahan, Oscar Gahan, Jack Evans, Denver Dixon, Herman Hack, Barney Beasley, George Hazel, Jack Tornek, Herman Howlin. A crook falsely accuses Billy the Kid of a killing so he escapes to Santa Fe where he makes an alliance with a man whose brother was hanged by the accuser. Run-of-the-mill PRC oater, fast and furious but not overly good.\n\n**Steve Clark, Bob Steele, Dennis Moore and Marin Sais in _ **Billy the Kid in Santa Fe**_ (PRC, 1941).**\n\n**335** _ **Billy the Kid in Texas**_ **** Producers Releasing Corporation, 1940. 63 min. D: Peter Stewart (Sam Newfield). SC: Joseph O'Donnell. With Bob Steele, Terry Walker, Al St. John, Carleton Young, Charles King, John Merton, Lew Meehan, Frank LaRue, Slim Whitaker, Curley Dresden, Tex Palmer, Merrill McCormick, George Morrell, Denver Dixon, Bob Woodward, Sherry Tansey, Herman Hack, Oscar Gahan, Wally West, Ray Henderson, Augie Gomez, Pascale Perry, Art Dillard, Chick Hannon, Jack Evans, Ben Corbett, Al Haskell. Escaping from New Mexico, Billy the Kid thwarts a robbery and becomes a town marshal only to discover his brother heads an outlaw gang. Low grade but action filled; also called **Battling Outlaw**.\n\n**336** _ **Billy the Kid Outlawed**_ **** Producers Releasing Corporation, 1940. 52 min. D: Peter Stewart (Sam Newfield). SC: Oliver Drake. With Bob Steele, Louise Currie, Al St. John, Carleton Young, John Merton, Ted Adams, Reed Howes, Kenne Duncan, Walter McGrail, Hal Price, George Chesebro, Jack Perrin, Steve Clark, Joe McGuinn, Carl Mathews, Budd Buster, Sherry Tansey. Billy the Kid and his pals are sought as outlaws by a corrupt sheriff who is in cahoots with local crooks planning a big swindle. The first of a half dozen \"Billy the Kid\" features starring Bob Steele, this one is fairly good considering its origins.\n\n**337** _ **Billy the Kid Returns**_ **** Republic, 1938. 56 min. D: Joe (Joseph) Kane. SC: Jack Natteford. With Roy Rogers, Smiley Burnette, Mary Hart (Lynne Roberts), Morgan Wallace, Fred Kohler, Wade Boteler, Edwin Stanley, Horace Murphy, Joseph Crehan, Robert Emmett Keane, Al Taylor, Ray Nichols, Chris-Pin Martin, Dorothy Vaughn, Betty Jean Hainey, Patsy Parsons, Tom Smith, Jim Corey, Bud McClure, Lloyd Ingraham, Rudy Sooter, Oscar Gahan, Bruce MacFarlane, Ralph Dunn, Jack Kirk, Al Haskell, Bob McKenzie, George Morrell, Bob Burns, George (Montgomery) Letz, Fred Burns, Fern Emmett, Betty Roadman, Tex Phelps, Fred Parker, Jack Evans, Silver Tip Baker, Bob Card, Frank O'Connor. After killing Billy the Kid, Sheriff Pat Garrett asks the outlaw's look-a-like to pretend to be the outlaw in order to stop a range war. Top notch Roy Rogers starring vehicle.\n\n_**Billy the Kid Rides Again**_ see _**The Kid Rides Again**_\n\n**338** _ **Billy the Kid Trapped**_ **** Producers Releasing Corporation, 1942. 59 min. D: Sherman Scott (Sam Newfield). SC: Oliver Drake and Joseph O'Donnell. With Buster Crabbe, Al St. John, Anne Jeffreys, Bud McTaggart, Glenn Strange, Walter McGrail, Ted Adams, Jack Ingram, Milton Kibbee, Eddie Phillips, Budd Buster, Jack Kenney, Jimmy Aubrey, Wally West, Art Dillard, Kenne Duncan, George Chesebro, Carl Mathews, Richard Cramer, Curley Dresden, Horace B. Carpenter, Jim Mason, Hank Bell, Oscar Gahan, Herman Hack, Ralph Bucko, Roy Bucko, Ray Henderson, Augie Gomez, Jack Evans, Jack Tornek, Pascale Perry. Three outlaws who claim to be Billy the Kid and his cohorts attack a sheriff but the real Billy rescues the lawman and formulates plans to round up the culprits. Average series entry.\n\n**339** _ **Billy the Kid vs. Dracula**_ **** Embassy, 1966. 89 min. Color. D: William Beaudine. SC: Carl K. Hittleman. With John Carradine, Chuck Courtney, Melinda Plowman, Virginia Christine, Walter Janovitz, Bing Russell, Lenni Geer, Roy Barcroft, Olive Carey, Mannie Landman, Marjorie Bennett, George Cisar, Charlita, William Forrest, Richard Reeves, Harry Carey, Jr., Max Kelvin, Jack Williams, William Challee. Billy the Kid is the foreman of a ranch whose nubile young owner is the lecherous object of Count Dracula, who poses as her uncle. As bad as its sounds with Dracula in bat form flying around in the daylight!\n\n**340** _ **Billy the Kid Wanted**_ **** Producers Releasing Corporation, 1941. 64 min. D: Sherman Scott (Sam Newfield). SC: Fred Myton. With Buster Crabbe, Al St. John, Dave O'Brien, Choti Sherwood, Glenn Strange, Charles King, Slim Whitaker, Howard Masters, Joe Newfield, Budd Buster, Frank Ellis, Curley Dresden, Wally West, Steve Clark, Reed Howes, Ray Henderson, Art Dillard, Al Taylor, Pascale Perry, Kenne Duncan, Archie Hall, Augie Gomez, George Morrell, Chick Hannon. Billy the Kid hides at the ranch of a friend to elude a posse that framed him for a crime he did not commit. Cheaply made but entertaining.\n\n**341** _ **Billy the Kid's Fighting Pals**_ **** Producers Releasing Corporation, 1941. 62 min. D: Sherman Scott (Sam Newfield). SC: George Plympton. With Bob Steele, Al St. John, Phyllis Adair, Hal Price, Carleton Young, George Chesebro, Forrest Taylor, Budd Buster, Julian Rivero, Wally West, Ray Henderson, Curley Dresden, Ed Piel, Sr., Art Dillard, Stanley Price, Sherry Tansey, Frank Ellis, Al Taylor, Milburn Morante, George Morrell, Jack Evans, Jack Tornek. When a sheriff is murdered, Billy the Kid gets the lawman's look-a-like to pose as him so he can capture the killer. Low grade but okay PRC sagebrush yarn. TV title: _**Trigger Men**_.\n\n**342** _ **Billy the Kid's Gun Justice**_ **** Producers Releasing Corporation, 1940. 59 min. D: Peter Stewart (Sam Newfield). SC: Joseph O'Donnell. With Bob Steele, Louise Currie, Al St. John, Carleton Young, Charles King, Rex Lease, Kenne Duncan, Forrest Taylor, Ted Adams, Al Ferguson, Karl Hackett, Ed Peil, Sr., Julian Rivero, Joe McGuinn, George Morrell, Blanca Vischer, Oscar Gahan, Richard Cramer, Curley Dresden, Wally West, Carl Mathews, Augie Gomez. Billy the Kid and his pals come to the aid of ranchers who bought properties only to learn a crook has diverted the local water supply. Another \"Billy the Kid\" effort, this one on the tacky side.\n\n**343** _ **Billy the Kid's Range War**_ **** Producers Releasing Corporation, 1941. 58 min. D: Peter Stewart (Sam Newfield). SC: William Lively. With Bob Steele, Al St. John, Joan Barclay, Rex Lease, Carleton Young, Milton Kibbee, Karl Hackett, Ted Adams, Julian Rivero, John Ince, Buddy Roosevelt, Ralph Peters, Alden Chase, Howard Masters, George Chesebro, Charles King, Steve Clark, Tex Palmer, Blanca Vischer, Wally West, Sherry Tansey, Carl Mathews, Curley Dresden, Jack Tornek. Billy the Kid is accused of several killings by a steamboat operator who is trying to stop construction of a stage line road. Better-than-average \"Billy the Kid\" series entry later released on 16mm as **Texas Trouble**.\n\n**344** _ **Billy the Kid's Roundup**_ **** Producers Releasing Corporation, 1941. 58 min. D: Sherman Scott (Sam Newfield). SC: Fred Myton. With Buster Crabbe, Al St. John, Carleton Young, Joan Barclay, Glenn Strange, Dick Curtis, Slim Whitaker, John (Elliott) Webster, Charles King, Dennis Moore, Kenne Duncan, Curley Dresden, Richard Cramer, Wally West, Tex Palmer, Tex Cooper, Horace B. Carpenter, Jim Mason, Oscar Gahan, Herman Hack, Horace B. Carpenter, George Morrell, Denver Dixon, Tex Phelps, Barney Beasley, Lew Morphy, Art Dillard, Augie Gomez, Jack Evans, Morgan Flowers, Tom Smith. A small town sheriff is murdered and Billy the Kid urges a female newspaper operator to use her business to combat the culprits. Another okay \"Billy the Kid\" series outing.\n\n**345** _ **Billy the Kid's Smoking Guns**_ Producers Releasing Corporation, 1942. 63 min. D: Sherman Scott (Sam Newfield). SC: George Milton. With Buster Crabbe, Al St. John, Joan Barclay, Dave O'Brien, John Merton, Milton Kibbee, Ted Adams, Frank Ellis, Karl Hackett, Budd Buster, Joe Newfield, Slim Whitaker, Bert Dillard, Curley Dresden, Steve Clark, George Morrell, Art Dillard, Chick Hannon, Foxy Callahan, Lou Fulton. Billy the Kid and his pal Fuzzy help ranchers harassed by a gang of hoodlums. Plenty and action and shooting in this PRC feature. British title: _**Smoking Guns**_.\n\n**346** _ **Billy Two Hats**_ **** United Artists, 1974. 97 min. Color. D: Ted Kotcheff. SC: Alan Sharp. With Gregory Peck, Desi Arnaz, Jr., Jack Warden, David Huddleston, Sian Barbara Allen, John Pearce, Dawn Little Sky, W. Vincent St. Cyr, Henry Medicine Hat, Zev Berlinsky, Anthony Scott. An old Irishman befriends a half-breed Indian boy and the two are chased by a relentless lawman after robbing a bank. Western filmed Israel with Gregory Peck making it palatable in his offbeat characterization of the Irish rogue.\n\n**347** _ **Birth of a Legend**_ **** Gold Key, 1973. 96 min. Color. An orphaned coyote, who has learned to herd sheep instead of hunt them, is mistaken by an Navajo Indian for the reincarnation of his herdsman grandfather. Entertaining docudrama. Also called **Navajo Coyote**.\n\n**348** _ **Bite the Bullet**_ **** Columbia, 1975. 131 min. Color. D-SC: Richard Brooks. With Gene Hackman, Candice Bergen, James Coburn, Jan-Michael Vincent, Ian Bannen, Ben Johnson, John McLiam, Jerry Gatlin, Robert Donner, Robert Hoy, Dabney Coleman, Paul Stewart, Jean Willes, Sally Kirkland, Buddy Van Horn. Numerous cowboys and adventurers enter an endurance horse race over 600 miles of badlands in 1908 for a $2,000 prize. Well produced but overlong Western which Ben Johnson steals with his poignant performance that should have won him a second Oscar.\n\n**349** _ **Bitter Creek**_ **** Allied Artists, 1954. 74 min. D: Thomas Carr. SC: George Waggner. With Bill Elliott, Beverly Garland, Veda Ann Borg, Carleton Young, Claude Akins, John Harmon, John Pickard, Jim Hayward, Forrest Taylor, Mike Ragan, Zon Murray, John Larch, Florence Lake, Earle Hodgins, Jane Easton, Joe Devlin, Dabbs Greer, Stanley Price. When his rancher brother is shot in the back a man plans to avenge the murder. Later Bill Elliott film is compact, well acted and nicely scripted.\n\n**350** _ **Bitter Springs**_ **** British Empire Films, 1950. 86 min. D: Ralph Smart. SC: W.P. Lyescomb. With Tommy Trinder, Chips Rafferty, Gordon Jackson, Jean Blue, Charles Tingwell, Nonnie Piper, Nicky Yardley, Michael Pate, Henry Murdock. In frontier Australia, a sheepherder and his family face danger from attacking aborigines. Well made and exciting Australian drama also called _**Savage Justice**_.\n\n**351** _ **Black Aces**_ **** Universal, 1937. 59 min. D: Buck Jones. SC: Frances Guihan. With Buck Jones, Kay Linaker, Robert Frazer, Charles King, Fred Mackaye, W.E. Lawrence, Raymond Brown, Robert Kortman, Frank Campeau, Charles LeMoyne, Arthur Van Slyke, Bob McKenzie, Lee Shumway, Bernard Phillips, Herman Hack, Ben Corbett, Jim Corey, Archie Ricks. Outlaws rustle a rancher's cattle herd and he sets out to round up the gang. Buck Jones produced and directed this entry in his Universal series and the results are good.\n\n**352** _ **Black Arrow**_ **** Columbia, 1944. 15 Chapters. D: Lew Landers. SC: Sherman Lowe, Jack Stanley, Leighton Brill and Royal K. Cole. With Robert Scott, Adele Jergens, Kenneth MacDonald, Robert Williams, Charles Middleton, Martin Garralaga, George J. Lewis, Chief Thundercloud, Nick Thompson, George Navarro, I. Stanford Jolley, Harry Harvey, John Laurenz, Dan White, Eddie Parker, Stanley Price, Ted Mapes, Iron Eyes Cody, Charles King, Forrest Taylor, Davison Clark, Elmo Lincoln, Kit Guard, Brooks Benedict. An Indian brave, who is really white, is run off the reservation for refusing to take revenge for the murder of his supposed father and he tries to prevent warfare when crooks want to steal gold from tribal land. Competent cliffhanger.\n\n**353** _ **Black Bandit**_ **** Universal, 1938. 60 min. D: George Waggner. SC: Joseph West (George Waggner). With Bob Baker, Marjorie Reynolds, Hal Taliaferro, Jack Rockwell, Forrest Taylor, Glenn Strange, Arthur Van Slyke, Carleton Young, Dick Dickinson, Rex Downing, Schuyler Standish. Twin boys are separated when they are young and later meet as grown men, one being a sheriff and the other an outlaw wanted for murder. Good production values add greatly to this Bob Baker series entry.\n\n**354** _ **Black Bart**_ **** Universal-International, 1948. 80 min. Color. D: George Sherman. SC: Luci Ward, Jack Natteford and William Bowers. With Yvonne De Carlo, Dan Duryea, Jeffrey Lynn, Percy Kilbride, Lloyd Gough, Frank Lovejoy, Don Beddoe, John McIntire, Ray Walker, Soledad Jiminez, Eddy Waller, Anne O'Neal, Chief Many Treaties, Douglas Fowley, Paul Maxey, Milton Kibbee, Ray Harper, Eddie Acuff, Ray Teal, Russ Conway, Ray Bennett, George Douglas, Reed Howes, Everett Shields, Wayne Treadway, Earl Audet, William Norton Bailey, Bert Davidson, Nina Campana, Bill O'Leary. On a tour of the West, dancer Lola Montez becomes involved with the famous highwayman Black Bart, who masquerades as a rancher, and she tries to reform him. Dan Duryea in the title role, Yvonne De Carlo as Lola Montez, and color, add zest to this fast-moving Western fantasy.\n\n**355** _ **The Black Cyclone**_ **** Path\u00e9, 1925. 70 min. D: Fred Jackman. SC: H.M. Walker and Malcolm Stuart Boylan. With Rex (horse), Guinn Williams, Kathleen Williams, Christian Frank; Killer, Pest, Lady (horses). A wild stallion, rescued from quicksand by a cowboy, aids his new master in fighting a crook as well as saving his mate from a herd led by a killer stallion. Good action filled silent adventure based on Hal Roach's story.\n\n**356** _ **The Black Dakotas**_ **** Columbia, 1954. 68 min. Color. D: Ray Nazarro. SC: Roy Buffum and DeVallon Scott. With Gary Merrill, Wanda Hendrix, John Bromfield, Noah Beery, Jr., Fay Roope, Howard Wendell, Robert Simon, James Griffith, Richard Webb, Peter Whitney, Clayton Moore, Jay Silverheels, George Keymas, Robert Griffin, Frank Wilcox. In order to steal money from the Sioux Indians and start an uprising to cover their escape, two crooks murder an emissary. A good cast and color can do little to save this mundane bow-and-arrows \"B\" outing.\n\n**357** _ **Black Eagle**_ **** Columbia, 1948. 76 min. D: Robert Gordon. SC: Edward Huebsch and Hal Smith. With William Bishop, Virginia Patton, Gordon Jones, James Bell, Trevor Bardette, Will Wright, Edmund MacDonald, Paul E. Burns, Harry V. Cheshire, Al Eben, Ted Mapes, Richard Talmadge, Ray Teal, Chuck Hamilton, Johnny Luther, Kernan Cripps, Glenn Thompson. A young man who tries to avoid trouble finds himself involved with a crooked livestock agent. Average action programmer.\n\n**358** _ **Black Eagle of Santa Fe**_ **** International Television Corporation (ITC), 1966. 86 min. Color. D: Ernest Hofbauer. SC: Jack Lewis. With Brad Harris, Tony Kendall, Joachim Hansen, Horst Frank, Helga Sommerfeld, Werner Peters, Thomas Moore, Pinkas Braun, Serge Marquand. A power hungry rancher tries to goad the Comanches into war so he can steal their lands but a government agent and a newsman attempt to thwart him. Typically violent West German-French-Italian co-production filmed in West Germany by Rapid-Film as _**Die Schwarzen Adler von Santa Fe**_ (The Black Eagle of Santa Fe) and originally running 93 minutes.\n\n**359** _ **Black Fox: Black Horse**_ **** CBS-TV, 1995. 92 min. Color. D: Steven H. Stern. SC: John Binder. With Christopher Reeve, Raoul Trujillo, Tony Todd, Janet Bailey, Nancy Sorel, Chris Wiggins, Chris Benson, Lawrence Dane, Cyndy Preston, Dale Wilson, Leon Goodstriker, Morningstar Mecredi, Jole Phage-Wright, Byron Chief-Moon, Buffalo Child, Denis Lacroix, David Lereaney, Pat Johnston, Lorette Clow. A former slave tries to retrieve hostages from Indians in Texas during the Civil War. Sturdy TV movie based on Matt Braun's book.\n\n**360** _ **Black Fox: Good Men and Bad**_ **** CBS-TV, 1995. 90 min. Color. D: Steven H. Stern. SC: Michael Michaelian. With Christopher Reeve, Kim Coates, Tony Todd, Janet Bailey, Nancy Sorel, Kelly Rowan, David Fox, Lawrence Dane, Alan Shearman, Graham McPherson, Beverly Elliott, Tom McBeath, Rainbow Francks, Alan Van Sprang, Billy Morton, Ronald Carothers, John Dodds. A one-time slave becomes a deputy marshal in order to find his former owner and friend along with the man who murdered his pal's wife. Another good telefeature from characters created by Matt Braun.\n\n**361** _ **Black Fox: The Prince of Peace**_ **** CBS-TV, 1995. 96 min. Color. D: Steven H. Stern. SC: John Binder, Joe Byrne and Jed Rosebrook. With Christopher Reeve, Raoul Trujillo, Tony Todd, Janet Bailey, Nancy Sorel, Chris Wiggins, Dale Wilson, John Blackwood, Cyndy Preston, Rainbow Francis, Don S. Davis, Michael Rhoades, Luc Corbeil. Two friends, a former plantation owner and the slave he freed, try to help a Kiowa tribe from being raided by a rancher whose wife left him for a warrior. Equally good entry in the trilogy of TV movies from the novel _Black Fox_ by Matt Braun.\n\n_**The Black Ghost**_ see _**The Last Frontier**_ (1932)\n\n**362** _ **Black Gold**_ **** United Artists, 1947. 90 min. D: Phil Karlson. SC: Agnes Christine Johnson. With Anthony Quinn, Katherine De Mille, Elyse Knox, Kane Richmond, Moroni Olsen, Ducky Louie, Darryl Hickman, Raymond Hatton, Thurston Hall, Charles Trowbridge, Jonathan Hale, Alan Bridge, Jack Norman, H.T. Tsiang. An Indian couple discover oil on their land, become millionaires and begin breeding horses, one of which wins the Kentucky Derby. Well-intentioned feature badly hurt by poor production values.\n\n**363** _ **Black Gold**_ **** Warner Bros., 1963. 76 min. D: Leslie H. Martinson. SC: Bob Duncan and Wanda Duncan. With Philip Carey, Diane McBain, Claude Akins, James Best, Fay Spain, William Phipps, Dub Taylor, Ken Mayer, Iron Eyes Cody, Vincent Barbi, Rusty Wescoatt. When a ruthless tycoon plans to cheat a young woman out of her oil-rich lands, his foreman works to stop him. Typical modern-day drilling melodrama.\n\n**364** _ **Black Hills**_ **** Eagle-Lion, 1947. 60 min. D: Ray Taylor. SC: Joseph Poland. With Eddie Dean, Roscoe Ates, Shirley Patterson, Terry Frost, Andy Parker and The Plainsmen, Steve Clark, William Fawcett, Nina Bara, Lane Bradford, Lee Morgan, George Chesebro, Bud Osborne, Steve Crane, Carl Mathews, Eddie Parker. A man kills a rancher who has discovered a rich gold vein but is thwarted by a lawman and his pal. Poor Eddie Dean vehicle, except for some good songs.\n\n**365** _ **Black Hills Ambush**_ **** Republic, 1952. 54 min. D: Harry Keller. SC: M. Coates Webster and Ronald Davidson. With Allan \"Rocky\" Lane, Eddy Waller, Lesley Banning, Roy Barcroft, Michael Hall, John Vosper, Ed Cassidy, John Cason, Michael Barton. A raider gang terrorizes a frontier area and a U.S. marshal is called in to stop them. Typical but good Allan Lane series entry.\n\n**366** _ **The Black Hills Express**_ Republic, 1943. 56 min. D: John English. SC: Norman Hall and Fred Myton. With Don \"Red\" Barry, Wally Vernon, Ariel Heath, George Lewis, William Halligan, Hooper Atchley, Charles Miller, Pierce Lyden, Jack Rockwell, Robert Kortman, Al Taylor, LeRoy Mason, Milton Kibbee, Wheaton Chambers, Marshall Reed, Curley Dresden, Frank Ellis, Carl Sepulveda, Ray Jones. A famous outlaw is given a month's immunity when the manager of the Black Hills division of Wells Fargo wants him to round up the gang robbing the company's express lines. Very good Don Barry vehicle heaped with action and fine emoting.\n\n**367** _ **Black Horse Canyon**_ **** Universal-International, 1954. 82 min. Color. D: Jesse Hibbs. SC: Geoffrey Homes. With Joel McCrea, Mari Blanchard, Murvyn Vye, Irving Bacon, Ewing Mitchell, John Pickard, Henry Wills. A veteran cowpoke and the niece of a cattle breeder team to capture a rebellious black stallion but are opposed by a neighboring rancher. Easy going oater for Joel McCrea fans.\n\n_**Black Killer**_ see _**Uccisore Nero**_\n\n**368** _ **The Black Lash**_ **** Western Adventure, 1952. 54 min. D: Ron Ormond. SC: Kathy McKeel. With Lash LaRue, Al St. John, Peggy Stewart, Ray Bennett, Kermit Maynard, Byron Keith, Jimmie Martin, John Cason, Clarke Stevens, Bud Osborne, Roy Butler, Larry Barton, Jim Bannon, George Chesebro, Sarah Padden, Lee Morgan, Sandy Sanders, Forrest Matthews. U.S. Marshal Lash LaRue and his deputy Fuzzy Q. Jones try to put outlaw Deuce Rago back in prison. Produced by Ron Ormond, this follow-up to _**Frontier Revenge**_ (q.v.) contains too much stock footage from that feature.\n\n**369** _ **Black Market Rustlers**_ **** Monogram, 1943. 58 min. D: S. Roy Luby. SC: Patricia Harper. With Ray Corrigan, Dennis Moore, Max Terhune, Evelyn Finley, Steve Clark, Glenn Strange, Carl Sepulveda, George Chesebro, Frank Ellis, Hank Worden, John Merton, Hal Price, Stanley Price, Wally West, Carl Mathews, Tex Cooper, Claire McDowell, Frosty Royce, James Austin, Little Jean Austin, Ingrid Austin, Art Fowler, Foxy Callahan, Tex Palmer, Bert Dillard, George Morrell, Dick Rush, Tom Smith, Rube Dalroy, Barney Beasley. The Range Busters are sent by the government to stop a gang rustling cattle and killing ranchers while supplying beef for the black market. Pretty good \"Range Busters\" entry helped by trick-riding heroine Evelyn Finley.\n\n_**Black Mountain Stage**_ see _**Riders of Black Mountain**_\n\n**370** _ **Black Noon**_ **** CBS-TV\/Columbia, 1971. 73 min. Color. D: Bernard Kowalski. SC: Andrew J. Fenady. With Roy Thinnes, Yvette Mimieux, Ray Milland, Lyn Loring, Henry Silva, Gloria Grahame, Willliam Bryant, Buddy Foster, Hank Worden, Stan Barrett. Joshua Bryant, Jennifer Bryant, Charles McCready, Leif Garrett, Dave Cass, Suzan Sheppard. A circuit riding preacher and his wife arrive in a remote Western town infested by a weird religious cult and an evil gunfighter. Combination of the Western and horror genres is pretty well handled in this TV movie.\n\n**371** _ **Black Patch**_ **** Warner Bros., 1957. 83 min. D: Allen H. Miner. SC: Leo Gordon. With George Montgomery, Diane Brewster, Leo Gordon, Tom Pittman, House Peters, Jr., Lynn Cartwright, Sebastian Cabot, Peter Brocco, Strother Martin, George Trevino, Dan Blocker. After the Civil War a sheriff is accused of killing a bank robber, the husband of the woman he once loved, and taking the money and hiding it. Mundane.\n\n**372** _ **Black Rodeo**_ **** Cinerama, 1972. 87 min. Color. D-SC: Jeff Kanew. With Woody Strode, Muhammad Ali, Bud Bramwell, Cleo Hern, Skeets Henderson, Rocky Watson, Lisa Bramwell. Various noted black rodeo performers appear in an event staged in New York City. Documentary including comments by black personalities as well as some background on black history; for those interested in the subject.\n\n**373** _ **Black Spurs**_ **** Paramount, 1965. 81 min. Color. D: R.G. Springsteen. SC: Steve Fisher. With Rory Calhoun, Linda Darnell, Scott Brady, Lon Chaney, Terry Moore, Richard Arlen, Bruce Cabot, Patricia Owens, Jerome Courtland, James Best, DeForest Kelley, James Brown, Joseph Hoover, Manuel Padilla, Robert Carricart, Joe Forte, Lorraine Bendix, Jeanne Baird, Guy Wilkerson, Read Morgan, Chuck Roberson, Reg Parton, Roy Jenson, Patricia King, Rusty Allen, Sally Nichols, William Bickmore, Max Powers, William Meador, Mike Mahoney. A cowboy gains the alliance of several important citizens in a small town in a scheme to make a nearby community so wild the railroad will bypass it and build locally. Not one of producer A.C. Lyles' best but still nice to see all the veteran players.\n\n**Rory Calhoun and Lon Chaney in _ **Black Spurs**_ (Paramount, 1965).**\n\n**374** _ **Black Star**_ **** Ambrosiana Cinematografica, 1966. 93 min. Color. D-SC: Giovanni Grimaldi. With Robert Woods, Elga Andersen, Renato Rossini, Franco Lantieri, Jane Tilden, Andrea Scotti, Harald Wolff. A gambler-banker rides roughshod over a town in Mexico but his authority is threatened by the arrival of a mysterious man who begins defending the locals. Robert Woods is a Robin Hood of the Old West, Italian-style, in this violent opus originally called _**Johnny Colt**_ and _**Starblack**_.\n\n**375** _ **The Black Whip**_ **** 20th Century\u2013Fox, 1956. 77 min. D: Charles Marquis Warren. SC: Orville Hampton. With Hugh Marlowe, Coleen Gray, Adele Mara, Angie Dickinson, Richard Gilden, Strother Martin, Paul Richards, Charles Gray, Patrick O'Moore, Sheb Wooley, John Pickard, Harry Landers, Howard Culver. Whey they rescue a quartet of dance hall girls in a Western town, two brothers find themselves up against a whip-yielding bad man. Mundane and rather pointless oater.\n\n**376** _ **Blackie the Pirate**_ **** Hispamex, 1971. 99 min. Color. D: Vincent Thomas (Lorenzo Gicca Palli). SC: Enzo Gicca (Lorenzo Gicca Palli). With Terence Hill, Silvia Monti, George Martin, Diana Lorys, Edmund Purdom, Monica Randall, Salvatore Borghese, Pat Basil, Ferdinando Bilboa, Bud Spencer, Aldo Cecconi, Paolol Magalotti, Gustavo Re, Alan Collins, Carlo Reali, Giuliano Dower, Luciano Catenacci. A buccaneer buys the captured wife of the Spanish viceroy in hopes of obtaining a treasure in gold. Italian-Spanish swashbuckler set in the early days of New World colonization.\n\n**377** _ **Blackjack Ketchum, Desperado**_ **** Columbia, 1956. 76 min. D: Earl Bellamy. SC: Luci Ward and Jack Natteford. With Howard Duff, Victor Jory, Maggie Mahoney, Angela Stevens, David Orrick, William Tannen, Ken Christy, Martin Garalaga, Don C. Harvey, Pat O'Malley, Ralph Sanford, Charles Wagenheim, Holly Bane, Kermit Maynard, Bob Woodward. A famous outlaw wants to live a peaceful life but in order to do so he is forced to fight a gang of cattle thieves. Howard Duff is fine in the title role in this otherwise average outing.\n\n**378** _ **Blade Rider: Revenge of the Indian Nations**_ **** Reel World, 1991. 102 min. Color. D: Vincent McEveety, Allen Reisner and Harry Harris. SC: Larry Cohen, Frederick Louis Fox, Ken Pettus, John Wilder and Jerry Ziegman. With Chuck Connors, Burt Reynolds, Lee Van Cleef, Greg Morris, David Brian, Noah Beery, Jr., Kathie Browne, Robert Lansing, Michael Pate, H.M. Wynant, Vaughn Taylor, Michael Keep, Felix Locher, Richard Tatro. After the Civil War, a man falsely accused of being a coward tries to start a new life for himself in the West. Cobbled together video feature made up of three episodes of the \"Branded\" (NBC-TV, 1965\u201366) series: \"Call to Glory,\" \"Fill No Glass for Me\" and \"Now Join the Human Race.\"\n\n_**Blake's Marauders**_ see _**Payment in Blood**_\n\n**379** _ **Blazing Across the Pecos**_ **** Columbia, 1948. 55 min. D: Ray Nazarro. SC: Norman S. Hall. With Charles Starrett, Smiley Burnette, Patricia (Barry) White, Chief Thundercloud, Paul Campbell, Charles Wilson, Thomas Jackson, Pat O'Malley, Jock Mahoney, Frank McCarroll, Pierce Lyden, Paul Conrad, Jack Ingram, Red Arnall and The Western Aces, Post Park, Jack Evans, Blackie Whiteford, Jack Tornek, Ralph Bucko. When outlaws attempt to goad an Indian uprising against local settlers, the Durango Kid tries to stop them. Anemic entry in the \"Durango Kid\" series. British title: _**Under Arrest**_.\n\n_**Blazing Arrows**_ see _**Fighting Caravans**_\n\n**380** _ **Blazing Bullets**_ **** Monogram, 1951. 51 min. D: Wallace Fox. SC: George Daniels. With Johnny Mack Brown, Lois Hall, House Peters, Jr., Stanley Price, Dennis Moore, Edmund Cobb, Milburn Morante, Forrest Taylor, Ed Cassidy, Carl Mathews. A U.S. marshal tries to find a man who has been kidnapped along with gold bullion, since the victim's daughter is suspected of the crime. Mild action show.\n\n**381** _ **The Blazing Forest**_ **** Paramount, 1952. 90 min. Color. D: Edward Ludwig. SC: Lewis R. Foster and Winston Miller. With John Payne, Susan Morrow, Richard Arlen, Agnes Moorehead, William Demarest, Roscoe Ates, Lynne Roberts, Ewing Mitchell, Walter Reed, Jim Davis, Joey Ray, Joe Garcia, Brett Houston, Max Wagner. A logger is contracted by a woman to cut timber on her north woods lands and he falls in love with her pretty niece but has troubles with his no-account brother. Handsome production with especially good work by Susan Morrow as the city yearning young woman and Richard Arlen as the recalcitrant sibling.\n\n**382** _ **Blazing Frontier**_ **** Producers Releasing Corporation, 1943. 61 min. D: Sam Newfield. SC: Patricia Harper. With Buster Crabbe, Al St. John, Marjorie Manners, Milton Kibbee, I. Stanford Jolley, Kermit Maynard, Frank Hagney, George Chesebro, Frank Ellis, Hank Bell, Jimmy Aubrey, Kenne Duncan, Robert Hill, Slim Whitaker, Pascale Perry, Morgan Flowers, Budd Buster, Charles King, Cactus Mack, Chick Hannon, Jack Evans, Augie Gomez, Rube Dalroy, Bert Dillard, Curley Dresden, Frank McCarroll, Tex Palmer, Ray Jones, Herman Hack, Bill Wolfe, John Elliott, Barney Beasley. When settlers and railroad officials fight over rights-of-way, Billy the Kid and Fuzzy Jones discover land agents are the cause of the trouble. Low grade but fast paced, the last \"Billy the Kid\" PRC series film; also called _**Billy the Kid in Blazing Frontier**_.\n\n**383** _ **Blazing Guns**_ **** Monogram, 1943. 55 min. D: Robert Tansey. SC: Frances Kavanaugh. With Ken Maynard, Hoot Gibson, Kay Forrester, LeRoy Mason, Emmett Lynn, Kenne Duncan, Frank Ellis, Lloyd Ingraham, Roy Brent, Charles King, Weldon Heyburn, Eddie Gribbon, George Kamel, Robbie Kavanaugh, Charles Murray, Jr., Robert Allen (Lee Roberts), John Bridges, Wally West, Victor Cox, Dan White. Two U.S. marshals are called to a locale where a gang of thugs is stealing land in order to put together a cattle empire. Slickly done and well-directed entry in \"The Trail Blazers\" series.\n\n_**Blazing Guns**_ (1950) see _**Marshal of Heldorado**_\n\n**384** _ **Blazing Justice**_ **** Spectrum, 1936. 60 min. D: Albert Herman. SC: Zara Tazil. With Billy Cody, Gertrude Messinger, Gordon Griffith, Mil Moranti (Milburn Morante), Budd Buster, Frank Yaconelli, Charles Tannen, Buck Morgan, Curley Baldwin. After bringing in two wanted criminals, a cowboy is falsely accused of stealing money belonging to a rancher. Awkward, dull Bill Cody vehicle with about enough plot to fill a slow two reeler.\n\n**385** _ **Blazing Saddles**_ **** Warner Bros., 1974. 94 min. Color. D: Mel Brooks. SC: Mel Brooks, Norman Steinberg, Andrew Bergman, Richard Pryor and Alan Unger. With Cleavon Little, Gene Wilder, Slim Pickens, David Huddleston, Liam Dunn, Alex Karras, John Hillerman, George Furth, Madeline Kahn, Harvey Korman, Mel Brooks, Carol Arthur, Dom DeLuise, Don Megowan, Burton Gilliam, Count Basie, Harvey Parry, Tom Steele. A black man, recently on a chain gang, becomes the sheriff of a Western town and must face local prejudice as well as dishonest state officials out to take over the area. Every clich\u00e9 imaginable is kidded in this comedy oater which varies in quality from very funny to distasteful to boring. Best at the beginning when Frankie Laine sings the title song.\n\n**386** _ **Blazing Six Shooters**_ **** Columbia, 1940. 61 min. D: Joseph H. Lewis. SC: Paul Franklin. With Charles Starrett, Iris Meredith, The Sons of the Pioneers (Bob Nolan, Tim Spencer, Pat Brady, Lloyd Perryman, Hugh Farr, Karl Farr), Dick Curtis, Alan Bridge, George Cleveland, Henry Hall, Stanley Brown, John Tyrrell, Eddie Laughton, Francis Walker, Edmund Cobb, Bruce Bennett. A cowboy tries to stop a crook from cheating an old man out his ranch that contains a silver deposit. Fair Charles Starrett vehicle.\n\n**387** _ **Blazing Sixes**_ **** Warner Bros., 1937. 55 min. D: Noel Smith. SC: John T. Neville. With Dick Foran, Helen Valkis, John Merton, Myra McKinney, Kenneth Harlan, Glenn Strange, Wilfred Lucas, Henry Otho, Milton Kibbee, Gordon Hart, Bud Osborne, Artie Ortego, Tom Forman, Ben Corbett, Malcolm Waite, Tom Burns, Tom Wilson, Jack Mower, Gene Alsace, Frank Ellis, Cactus Mack. A government agent is assigned to capture outlaws robbing gold shipments and to accomplish his mission he masquerades as a bandit. Pleasing Dick Foran entry with the star handling the action in addition to singing a few ditties.\n\n**388** _ **Blazing Stewardesses**_ **** Independent-International, 1975. 85 min. Color. D: Al Adamson. SC: Samuel M. Sherman and John R. D'Amato. With Yvonne De Carlo, Robert Livingston, Don \"Red\" Barry, The Ritz Brothers (Harry and Jimmy Ritz), Geoffrey Land, Regina Carrol, Connie Hoffman, T.A. King, Lon Bradshaw, Don Shanks, David Sharpe. Three sexy stewardesses come to the rescue of a rancher friend being robbed by his crooked foreman who is in cahoots with the local madam romancing the cattleman. Forgetting the brief opening sex scenes, this is a mild, amusing tribute to Westerns of yore, complete with a masked hero and a score made up of Gordon Zahler's music; lots of fun, especially from Harry and Jimmy Ritz, last minute replacements for The Three Stooges (Moe Howard, Joe Da Rita, Emil Sitka). Re-titled _**Cathouse Girls**_ , _**The Great Truck Robbery**_ , _**Texas Layover**_ and _**Up Like a Shot!**_\n\n**389** _ **The Blazing Sun**_ **** Columbia, 1950. 70 min. D: John English. SC: Jack Townley. With Gene Autry, Pat Buttram, Lynne Roberts, Anne Gwynne, Edward Norris, Kenne Duncan, Alan Hale, Jr., Gregg Barton, Steve Darrell, Tom London, Sandy Sanders, Frankie Marvin, Bob Woodward, Boyd Stockman, Lewis Martin, Virginia Carroll, Sam Flint, Charles Coleman, Pat O'Malley, Amira Sessions, Nolan Leary, Chris Allen. Lawman Gene Autry is on the trail of two bank robbers. Okay modern-day oater.\n\n**390** _ **Blazing the Overland Trail**_ **** Columbia, 1956. 15 Chapters. D: Spencer Gordon Bennet. SC: George Plympton. With Lee Roberts, Dennis Moore, Norma Brooks, Gregg Barton, Don C. Harvey, Lee Morgan, Pierce Lyden, Ed Coch, Reed Howes, Kermit Maynard, Al Ferguson, Bud Osborne, Jack O'Shea, Ray Jones. A crooked rancher organizes a gang to raid the overland trail but he is opposed by an Army scout and a Pony Express agent. Tacky cliffhanger, the last made in the U.S. and a sad finale to a grand genre. It includes footage from _**Overland with Kit Carson**_ and _**White Eagle**_ (1941) [qq.v.].\n\n**391** _ **Blazing the Western Trail**_ **** Columbia, 1945. 60 min. D: Vernon Keays. SC: J. Benton Cheney. With Charles Starrett, Tex Harding, Dub Taylor, Carole Mathews, Bob Wills and His Texas Playboys, Alan Bridge, Nolan Leary, Virginia Sale, Steve Clark, Mauritz Hugo, Ethan Laidlaw, Edmund Cobb, Frank LaRue, Forrest Taylor, Francis Walker, James T. \"Bud\" Nelson, Budd Buster, Ted Mapes, John Tyrrell, Robert Williams, Chick Hannon, Edward Howard. The Durango Kid comes to the aid of a stagecoach operator who is being forced out of business by a rival. Pretty good \"Durango Kid\" episode. British title: _**Who Killed Waring?**_\n\n_**  \n**_\n\n**Edmund Cobb in** _**Blazing the Western Trail**_ **(Columbia, 1945).**\n\n**  \n**\n\n_**Blazing Trail**_ (1932) see _**Guns for Hire**_\n\n**392** _ **The Blazing Trail**_ **** Columbia, 1949. 56 min. D: Ray Nazarro. SC: Barry Shipman. With Charles Starrett, Smiley Burnette, Marjorie Stapp, Hank Penny and Slim Duncan, Jack O'Mahoney (Jock Mahoney), Steve Darrell, Fred F. Sears, Steve Pendleton, Robert Malcolm, Trevor Bardette, John Cason, Frank McCarroll, John Merton, Merrill McCormick, Herman Hack, Frank O'Connor, George Morrell, Rube Dalroy, Jack Evans, Blackie Whiteford. A sheriff and a newspaper editor believe fraud exists when a rancher is killed and his will leaves only a worthless mine to one brother while the other gets the rest his property. Fair mystery plot highlights this \"Durango Kid\" affair. British title: _**The Forged Will**_.\n\n**393** _ **Blind Justice**_ **** Home Box Office (HBO), 1994. 85 min. Color. D: Richard Spence. SC: Daniel Knauf. With Armand Assante, Elisabeth Shue, Robert Davi, Adam Baldwin, Ian McElhinney, Danny Nucci, M.C. Gainey, Titus Welliver, Jack Black, Michael O'Neill, Douglas Roberts, Gary Cervantes, Jesse Dabson, Stanton Davis, Jimmy Herman, Clayton Landey, James Oscar Lee, Daniel O'Haco, Jeff O'Haco, Joe Pennell, Jason Rodriguez, Ric San Nicholas, Forrie J. Smith, Michael A. Goorjian, Tom Hodges. A nearly blind gunman, who is protecting a baby, joins forces with the cavalry to help deliver a gold shipment. Pretty good TV movie.\n\n**394** _ **Blindman**_ **** 20th Century\u2013Fox, 1972. 105 min. Color. D: Ferdinando Baldi. SC: Vincenzo Cerami, Piero Anchisi and Tony Anthony. With Tony Anthony, Ringo Starr, Agneta Eckemyr, Lloyd Battista, Magda Konopka, Raf Baldassarie, David Dreyer, Ken Wood, Lucretia Love, Isabella Savona. A blind gunfighter tries to stop a ruthless Mexican bandit who has kidnapped fifty mail order brides. Typically violent, and entertaining, Spaghetti Western, co-produced and co-written by star Tony Anthony, with more nudity than usual for this fare.\n\n**395** _ **The Blocked Trail**_ **** Republic, 1943. 56 min. D: Elmer Clifton. SC: John K. Butler and Jacquin Frank. With Bob Steele, Tom Tyler, Jimmie Dodd, Helen Deverall, George J. Lewis, Walter Soderling, Kermit Maynard, Pierce Lyden, Carl Mathews, Hal Price, Budd Buster, Earle Hodgins, Bud Osborne, Al Taylor, Art Dillard, Bud Geary, Ellen Lowe, Matty Roubert, Nolan Leary, Cliff Parkinson, Roy Bucko, Jess Cavin, Rose Plummer, Bill Wolfe, Kelly Flint. The Three Mesquiteers are suspected of killing an eccentric man and they try to expose the murderer as well as his motive. A mystery motif with the killing of the miner only witnessed by his dwarf horse makes this \"Three Mesquiteers\" a bit different.\n\n**396** _ **Blood and Guns**_ **** Filmamerica, 1968 90 min. Color. D: Guilio Petroni. SC: Guilio Petroni and Franco Solinas. With Tomas Milian, Orson Welles, John Steiner, Jose Torres, Luciano Casamonica, Anna Maria Lanciaprima, Giancarlo Badessi, Angel Ortiz, George Wang, Paco Sanz. During the Mexican Revolution in 1917, government soldiers and a young English doctor both want revenge on an illiterate peon who is leading a small band of rebels. Hard to follow and violent Italian oater, originally called **Tepea** and **Viva la Revolucion**.\n\n**397** _ **Blood and Honor**_ **** DML, 2000. 273 min. D-SC: Donald Farmer. With Maria Ortiz, Miles O'Keefe, Rena Watts, Stancy Clements, Andy Hamrick, Joseph Casterline, Autumn Vena, Mary Tretchell, Michelle Bauer, Jeffrey Alfiero, Philip Newman, Shelley Holmes, Tavia Upshaw, Andre Buckner, Rick Martin, Earl Clark, Jennifer Huss, Dr. Maurice J. Fagan, Jr., Larissa LaRenne, Kathy Bell, Mark Johnson, Greg Perkins. At the close of the Civil War, a renegade Union officer takes control of a Southern plantation and its two beautiful sister inhabitants. Clunky, overlong melodrama.\n\n**398** _ **Blood and Steel**_ **** Independent Pictures, 1925. 60 min. D: J.P. McGowan. SC: George Plympton. With Helen Holmes, William Desmond, Robert Edeson, Ruth Stonehouse, Mack V. Wright, Albert J. Smith, C.L. Sherwood, Paul Walters, Walter Fitzroy. An engineer is hired to help in the completion of a railroad and he learns a rival company plans to sabotage the project. This silent teaming of action stars Helen Holmes and William Desmond provides its quota of thrills.\n\n**399** _ **Blood Arrow**_ **** 20th Century\u2013Fox, 1958. 78 min. D: Charles Marquis Warren. SC: Fred Freiberger. With Scott Brady, Phyllis Coates, Paul Richards, Don Haggerty, Rocky Shahan, Patrick O'Moore, Jeanne Bates, John Dierkes. When her people need a serum, a young Mormon woman treks through hostile country to get it and is exposed to Indian attacks. Phyllis Coates does well as the serum seeking Mormon in this interesting drama.\n\n_**Blood at Sundown**_ (1969) see _**Why Kill Again?**_\n\n_**Blood City**_ see _**Welcome to Blood City**_\n\n**400** _ **Blood for a Silver Dollar**_ **** Teleworld, 1965. 92 min. Color. D: Kelvin Jackson Paget (Giorgio Ferroni). SC: George Finlay and Kelvin Jackson Paget (Giorgio Ferroni). With Montgomery Wood (Giuliano Gemma), Evelyn Stewart (Ida Galli), Peter Cross, John MacDouglas, Frank Farrel, Tor Altmayer, Max Dean, Andrew Scott (Andrea Scotti), Nicholas St. John, Benny Reeves, Frank Liston, Pedro Sanchez, Nello Pazzafini, Jean Martin, Peter Surtess, Benny Farber. After the Civil War, a man goes West in order to get the money to send for his wife but ends up in the middle of a double cross that nearly costs him his life and when his wife shows up thinking him dead she takes up with the man who betrayed him. Very violent Italian oater made by Fono Roma\/Dorica\/Explorer\/Les Films Corona as _**Un Dollaro Bucato**_ (A Dollar with a Hole in It). Also called _**One Silver Dollar**_.\n\n_**Blood Money: Stranger and the Gunfighter**_ see _**The Stranger and the Gunfighter**_\n\n**401** _ **Blood on the Arrow**_ **** Allied Artists, 1964. 91 min. Color. D: Sidney Salkow. SC: Robert E. Kent. With Dale Robertson, Martha Hyer, Wendell Corey, Dandy Curran, Paul Mantee, Ted de Corsia, Elisha Cook, Jr., Tom Reese, John Matthews, Bloyce Wright, Michael Hammond, Leland Wainscott, Robert Carricart. An Army prisoner is the only survivor of an Indian attack and he takes refuge with a couple whose son has been kidnapped by the tribe and ransomed for rifles. None-too-interesting Indians-on-the-warpath affair; looks like it should have been made a decade before in black and white.\n\n**402** _ **Blood on the Moon**_ **** RKO Radio, 1948. 88 min. D: Robert Wise. SC: Lillie Hayward. With Robert Mitchum, Barbara Bel Geddes, Robert Preston, Walter Brennan, Phyllis Thaxter, Frank Faylen, Tom Tully, Charles McGraw, Clifton Young, Tom Tyler, George Cooper, Richard Powers (Tom Keene), Bud Osborne, Zon Murray, Robert Bray, Al Ferguson, Chris-Pin Martin, Robert Malcolm, Ruth Brennan, Harry Carey, Jr., Hal Taliaferro, Iron Eyes Cody, Al Murphy. When his rustler pal hires a gunman to run a young woman and her father off their ranch either by persuasion or cattle theft, the gunfighter finds himself falling for the girl. Studio bound Western should have been more interesting but is greatly helped by its trio of stars, especially Robert Mitchum.\n\n**403 Blood River** Constantin, 1966. 93 min. Color. D: Piero Pierotti. SC: Piero Pierotti and Arpad de Riso. With Alan Steel, Toni Sailer, Mario Petri, Brigit Heiberg, Wolfgang Lukschy, Dada Galltti, Elisabetta Fanti, Anna Maria Polani, Pierre Cressoy. A gambler and his pal find an Inca treasure after suspecting a card sharp murdered a woman's father and placed the blame on another man. Fair West German oater released there as _**Samson und der Schatz der Inkas**_ (Samson and the Treasure of the Incas) and also called _**Lost Treasure of the Aztecs**_ and _**Lost Treasure of the Incas**_.\n\n_**Blood River**_ (1968) see _**Any Gun Can Play**_\n\n**404** _ **Blood Shack**_ **** Program Releasing Corporation, 1971. 55 min. Color. D: Wolfgang Schmidt (Ray Dennis Steckler). SC: Christopher Edwards. With Carolyn Brandt, Ron Haydock, Jason Wayne, Laurel Spring, John Bates, Steve Edwards, Linda Steckler, Laura Steckler. A woman inherits a ranch said to be plagued by a mysterious phantom. All this modern-day Western offers is a slim, slim budget, a few murders and some tepid rodeo footage. Alternate title: _**The Chopper**_. Some prints run 60 and 70 minutes.\n\n**405** _ **Bloody Trail**_ **** Academy Entertainment, 1972. 91 min. Color. D: Richard Robinson. SC: Gale Robinson and David Allen Russell (Hagen Smith). With Paul Harper, Rance Howard, John Mitchum, Rickey Richardson, Hagen Smith, Eve York, Richard Benedict. During Reconstruction after the Civil War in the South, an ex\u2013Union soldier runs afoul of both former Confederates and slaves. Pretty poor affair with nudity, violence and a supernatural subplot. Video title: _**White Justice**_.\n\n**406** _ **Blowing Wild**_ **** Warner Bros., 1953. 90 min. Color. D: Hugo Fregonese. SC: Philip Yordan. With Gary Cooper, Barbara Stanwyck, Ruth Roman, Anthony Quinn, Ward Bond, Ian MacDonald, Richard Karlan, Juan Garcia. A wildcatter puts up all his money in hopes of striking it rich with a gusher while his ex-love, now the wife of a rich oil tycoon, want to renew their relationship. Filmed in Mexico, this steamy oil fields drama promises more than it delivers; Frankie Laine sings the title song.\n\n**407** _ **Blue**_ **** Paramount, 1968. 113 min. Color. D: Silvio Narizzano. SC: Meade Roberts and Ronald M. Cohen. With Terence Stamp, Joanna Pettet, Karl Malden, Ricardo Montalban, Anthony Costello, Joe De Santis, James Westerfield, Stathis Giallelis, Carlos East, Robert Lipton, Kevin Corcoran, Wes Bishop, Sally Kirkland, William Shannon, Michael Nader, Marian Mason, Jerry Gatlin. Raised by a Mexican bandit, a young man finds himself resented by the outlaw's three sons and distrusted by Americans as well as Mexicans. Big, expensive Western that proves foreign genre directors cannot improve their craft in Hollywood; not much of a film.\n\n**408** _ **Blue Blazes Rawden**_ **** Paramount-Artcraft, 1918. 55 min. D: William S. Hart. SC: J.G. Hawks. With William S. Hart, Maude George, Gertrude Claire, Robert McKim, Robert Gordon, Hart (Jack) Hoxie. A lumber camp worker wins a saloon in a bet but is forced to shoot and kill the owner only to be reformed by the dead man's mother. Another frontier morality play from William S. Hart; good entertainment.\n\n**409** _ **Blue Canadian Rockies**_ **** Columbia, 1952. 58 min. D: George Archainbaud. SC: Gerald Geraghty. With Gene Autry, Pat Buttram, Gail Davis, Carolina Cotton, Russ Ford, Tom London, Mauritz Hugo, Don Beddoe, Gene Roth, John Merton, David Garcia, Bob Woodward, Billy Wilkerson, The Cass County Boys (Jerry Scoggins, Bert Dodson, Fred Martin). Sent to Canada by his employer to stop the man's daughter from marrying a no-good, Gene Autry find she has turned her home into a dude ranch and game preserve with the place plagued by a series of murders. Compact and interesting Gene Autry film.\n\n_**Blue Eyes**_ see _**Retribution Road**_\n\n**410** _ **Blue Montana Skies**_ **** Republic, 1939. 56 min. D: B. Reeves Eason. SC: Gerald Geraghty. With Gene Autry, Smiley Burnette, June Storey, Harry Woods, Tully Marshall, Alan Bridge, Glenn Strange, Dorothy Granger, Edmund Cobb, Jack Ingram, John Beach, Elmo Lincoln, Walt Shrum and His Colorado Hillbillies, Allan Cavan, Buffalo Bill, Jr., Augie Gomez, Robert Winkler, Frankie Marvin, Wally West, Curley Dresden, Ted Mapes, Ray Henderson, Buck Moulton. When fur thieves begin smuggling pelts into the United States, the government sends an agent to trace the origins of the activities and round up the gang. Fairly good Gene Autry vehicle with a well-written script and ingratiating music interludes.\n\n**411** _ **Blue Steel**_ **** Monogram, 1934. 54 min. D-SC: Robert North Bradbury. With John Wayne, Eleanor Hunt, George Hayes, Ed Peil, Sr., Yakima Canutt, George Cleveland, George Nash, Lafe McKee, Hank Bell, Earl Dwire, Artie Ortego, Theodore Lorch, Horace B. Carpenter, Silver Tip Baker, Fern Emmett, Perry Murdock, Herman Hack, Bud McClure, Barney Beasley, Jack Evans, Herman Nowlin, Tex Phelps, Ralph Bucko. A U.S. marshal is after the notorious Polka Dot bandit and he and a fellow lawman come to the aid of a young woman whose father was murdered by the outlaw who is working for a rancher out to take over the area because of a rich gold deposit. Good John Wayne-Lone Star Western with an atmospheric opening sequence at a half-way house during a thunderstorm where a safe robbery takes place. Reworking of _**Son of the Plains**_ (q.v.) and colorized as _**Stolen Goods**_.\n\n**Yakima Canutt, John Wayne and George Hayes in a posed scene from** _**Blue Steel**_ **(Monogram, 1934).**\n\n**  \n**\n\n**412** _ **Bobbie Jo and the Outlaw**_ **** American International, 1976. 88 min. Color. D: Mark Lester. SC: Vernon Zimmerman. With Marjore Gortner, Lynda Carter, Jesse Vint, Peggy Stewart, Merrie Lynn Ross, Gerrit Graham. Believing he is the reincarnation of Billy the Kid, a punk tries to live up to the outlaw's reputation. Really bad modern-day Western.\n\n**413** _ **The Boiling Point**_ **** Allied, 1932. 70 min. D: George Melford. SC: Daniel W. Lee, Harry Neumann and Tom Gallaghan. With Hoot Gibson, Helen Foster, Skeeter Bill Robbins, Lafe McKee, Tom London, George Hayes, Wheeler Oakman, William Nye, Charles Bailey, Billy Bletcher, Frank Ellis, Lew Meehan, Hattie McDaniel, Bob Burns, Art Mix, Merrill McCormick, Artie Ortego. A cowboy is sent to a neighbor's ranch and told to hold his temper for a month or lose an inheritance but once there he gets involved in a fight over a girl. Overlong, slow moving Hoot Gibson vehicle.\n\n**414** _ **The Bold Caballero**_ **** Republic, 1936. 69 min. Color. D-SC: Wells Root. With Robert Livingston, Heather Angel, Sig Rumann, Robert Warwick, Ian Wolfe, Emily Fitzroy, Charles Stevens, Walter Long, Ferdinand Munier, King (Chris-Pin) Martin, John Merton, Jack Kirk, Slim Whitaker, George Plues, Chief Thundercloud, Carlos De Valdez, Soledad Jiminez, Jack Rockwell, Henry Hall, Steve Clark, Harrison Greene, Si Jenks, Pascale Perry, Herman Hack, Rube Dalroy, Jimmy Aubrey, Artie Ortego, Dick Botiller, Wally West, Ben Corbett, Sherry Tansey, Bud McClure, Ed Phillips, Vinegar Roan, Yakima Canutt, Henry Morris, Bill Wolfe. In Spanish California a man becomes the masked avenger Zorro to stop the tyranny of crooked officials. The initial sound \"Zorro\" feature is a good one, enhanced by Magna Color, fast action and an ingratiating performance by Bob Livingston as Zorro. Some prints are in black and white.\n\n**415** _ **The Bold Frontiersman**_ **** Republic, 1948. 60 min. D: Philip Ford. SC: Bob Williams. With Allan \"Rocky\" Lane, Eddy Waller, Roy Barcroft, Fred Graham, John Alvin, Francis McDonald, Ed Cassidy, Edmund Cobb, Harold Goodwin, Jack Kirk, Kenneth Terrell, Marshal Reed, Al Murphy. A government investigator resorts to trickery to capture an outlaw and his gang. Standard, but entertaining, Allan Lane vehicle.\n\n**416** _ **The Boldest Job in the West**_ **** Promofilm, 1971. 101 min. Color. D-SC: Jose Antonio de la Loma. With Mark Edwards, Carmen Sevilla, Fernando Sancho, Charley Bravo, Piero Lulli, Yvan Verella, Frank Brana. A gang plans to carry off the robbery of a small town bank but everything goes wrong. Slight Italian Western shows how badly the genre was slipping at the time. Made as _**El Mas Fabulosi Golpe del Far West**_ (The Most Fabulous Coup of the Far West) and also known as _**Nevada**_.\n\n**417** _ **Bonanza: The Next Generation**_ **** LBS Communications, 1988. 93 min. Color. D: William F. Claxton. SC: Paul Savage. With Robert Fuller, John Ireland, Mark Peter Richman, John Amos, Michael Landon, Jr., Barbara Anderson, Brian A. Smith, Richard Bergman, Gillian Greene, Kevin Hagen, William Benedict, Dabbs Greer, Gary Reed, Les McLaughlin, Robert Hoy, Rex Linn, Jack Lilley, Jerry Gatlin, Jeffrey Meyer, Robert Jaurequi, Jeffrey Boudov, Laurie Rude, Clayton Staggs, David Q. Combs, Michael Delta Femina, Patrick Joseph O'Neill, Joyce Anderson, John D. Ward, M.C. Christopher, Jack O'Leary, Robert J. Fuller, Jeannette Knight, William James Anderson, Mike Silvera. Descendants of the Cartwright family run the Ponderosa ranch but are at odds with each other and a mining company out to exploit the spread. Mediocre TV movie follow up to \"Bonanza\" (NBC-TV, 1959\u201373).\n\n**418** _ **Bonanza Town**_ **** Columbia, 1951. 56 min. D: Fred F. Sears. SC: Barry Shipman and Bart Forswell. With Charles Starrett, Smiley Burnette, Fred F. Sears, Luther Crockett, Slim Duncan, Myron Healey, Charles Horvath, Ted Jordan, Al Wyatt, Marshall Reed, Vernon Dent, Paul McGuire, Nancy Saunders, Glenn Stuart, I. Stanford Jolley, George Chesebro, Robert J. Wilke, Nolan Leary, Steve Clark, Zon Murray, Bud Osborne, Guy Teague, George Magrill. The Durango Kid is after an outlaw thought to be dead but is actually in cahoots with a corrupt town boss. Average \"Durango Kid\" offering that is a sequel to _**West of Dodge City**_ (q.v.). British title: _**Two-Fisted Agent**_.\n\n**419** _ **The Boogens**_ **** Jensen Farley Pictures\/Taft International, 1982. 95 min. Color. D: James L. Conway. SC: David O'Malley and Bob Hunt. With Rebecca Balding, Fred McCarren, Anne-Marie Martin, Jeff Harland, John Crawford, Med Flory, Jon Lormer, Peg(gy) Stewart, Scott Wilkinson, Marcia Reider. The reopening of a silver mine in a remote Utah town results in letting lose prehistoric monsters buried there since a 1912 disaster. Eerie modern-day horror Western provides shivers for fans.\n\n**420** _ **Boom Town**_ **** Metro-Goldwyn-Mayer, 1940. 116 min. D: Jack Conway. SC: John Lee Mahin. With Clark Gable, Claudette Colbert, Spencer Tracy, Hedy Lamarr, Frank Morgan, Lionel Atwill, Chill Wills, Marion Martin, Minna Gombell, Joe Yule, Horace Murphy, Roy Gordon, Richard Lane, Casey Johnson, George Lessey, Sara Haden, Frank Orth, Frank McGlynn, Sr., Curt Bois, Dick Curtis, Baby Quintanilla, Nestor Paiva, Parker Barnett. Two partners strike it rich in the oil fields but soon part over money and a woman. Brawling big budget feature voted one of the top ten films of 1940 by _Film Daily_ does not hold up well, but who cares with Hedy Lamarr to look at?\n\n**421** _ **Boot Hill**_ **** Film Ventures, 1971. 87 min. Color. D-SC: Giuseppe Colizzi. With Terence Hill, Bud Spencer, Eduardo Ciannelli, Woody Strode, Victor Buono, Lionel Stander, Alberto Dell'Acqua (Robert Widmark). Under the cover of a circus a man escapes from jail and is joined by two others in seeing revenge against an outlaw gang. Complicated and somewhat hard to follow Spaghetti Western issued in Italy in 1969 as _**La Collina Degli Stivali**_ (The Hill of Boots).\n\n**422** _ **Boot Hill Bandits**_ **** Monogram, 1942. D: S. Roy Luby. SC: Arthur Durlam. With Ray Corrigan, John King, Max Terhune, Jean Brooks, John Merton, Glenn Strange, I. Sanford Jolley, Steve Clark, Richard Cramer, George Chesebro, Budd Buster, Milburn Morante, Jimmy Aubrey, Carl Mathews, Merrill McCormick, Hank Bell, Horace B. Carpenter, Snub Pollard, Archie Ricks, Harry Willingham, Ray Henderson, James Sheridan, Wally West, Jack Tornek, Tom Smith, Jack Evans, Denver Dixon, Tex Palmer, Herman Hack, Bert Dillard. The Range Busters arrive in a town plagued by a series of Wells Fargo gold shipment holdups and try to find out who is behind the lawlessness. Muddled entry in \"The Range Busters\" series although Glenn Strange is grand as a murderous prospector.\n\n**423** _ **Boothill Brigade**_ **** Republic, 1937. 58 min. D: Sam Newfield. SC: George Plympton. With Johnny Mack Brown, Claire Rochelle, Dick Curtis, Horace Murphy, Frank La Rue, Ed Cassidy, Bobby Nelson, Frank Ball, Steve Clark, Frank Ellis, Lew Meehan, Jim Corey, Tex Palmer, Sherry Tansey. A crook holds the mortgage on a rancher's land and forces him to do his bidding which upsets the man's daughter and her fiance, his foreman. Final film in Johnny Mack Brown's series for producer A.W. Hackel; above average and fast moving.\n\n**424** _ **Boots and Saddles**_ **** Republic, 1937. 59 min. D: Joseph Kane. SC: Oliver Drake. With Gene Autry, Smiley Burnette, Judith Allen, Ra Hould, Guy Usher, Gordon (Bill) Elliott, Max Terhune, John Ward, Frankie Marvin, Chris-Pin Martin, Stanley Blystone, Bud Osborne, Merrill McCormick, Al Taylor, Nelson McDowell, Bob Reeves, Jerry Frank. Gene Autry becomes involved with the pretty daughter of an Army colonel when he sells horses to the service but finds a rival is out to close his business. Although action filled, this Gene Autry opus is somewhat hampered by a complicated plot.\n\n**Judith Allen and Gene Autry in** _**Boots and Saddles**_ **(Republic, 1937).**\n\n**  \n**\n\n**425** _ **Boots of Destiny**_ **** Grand National, 1937. 56 min. D: Arthur Rosson. SC: Philip White. With Ken Maynard, Claudia Dell, Vince Barnett, Walter Patterson, Martin Garralaga, George Morrell, Fred Cordova, Ed Cassidy, Sid D'Albrook, Carl Mathews, Wally West. A cowboy is put in jail when the local sheriff thinks he is a famous bandit after buried treasure on a young woman's ranch. Low budget effort but beefy Ken Maynard could still carry a film and this one is more than passable. The title refers to traces of clay on the villain's boots, which causes his capture.\n\n**426** _ **The Border**_ **** Universal\/RKO, 1982. 107 min. Color. D: Tony Richardson. SC: Deric Washsburn, Walon Green and David Freeman. With Jack Nicholson, Valerie Perrine, Harvey Keitel, Warren Oates, Jeff Morris, Dirk Blocker, Lonny Chapman, Elpidia Carillo, Shannon Wilcox. A U.S.-Mexican border guard finds himself caught in the middle with the smuggling of aliens. Mediocre modern-day melodrama.\n\n**427** _ **Border Badmen**_ **** Producers Releasing Corporation, 1945. 59 min. D: Sam Newfield. SC: George Milton. With Buster Crabbe, Al St. John, Lorraine Miller, Marilyn Gladstone, Charles King, Marin Sais, Budd Buster, Bud Osborne, John Cason, Ray Bennett, Archie Hall, Robert Kortman, Slim Whitaker, Wally West, Steve Clark, Henry Hall, Frank Ellis, Victor Cox, Ray Jones, Ray Henderson, Roy Bucko. Fuzzy thinks he is the heir to an estate and on the way to claim it he and Billy Carson are arrested by a gang led by a man who wants the property for himself. Seedy but fast \"Billy Carson\" outing made entertaining by Al St. John's many pratfalls.\n\n**428 Border Bandits** Monogram, 1946. 58 min. D: Lambert Hillyer. SC: Frank Young. With Johnny Mack Brown, Raymond Hatton, Rosa del Rosario, Riley Hill, John Merton, Tom Quinn, Frank LaRue, Steve Clark, Charles Stevens, Bud Osborne, Terry Frost, I. Stanford Jolley, Ray Jones, Lucio Villegas, Pat R. McGee, Rube Dalroy, Julia Villirea. Two U.S. marshals come to the aid of a young woman whose father has been robbed of valuable jewels. Mundane penultimate entry in the \"Nevada Jack McKenzie\" series.\n\n**429** _ **Border Brigands**_ **** Universal, 1935. 58 min. D: Nick Grinde. SC: Stuart Anthony. With Buck Jones, Lona Andre, Fred Kohler, Frank Rice, Gertrude Astor, Edward Keane, J.P. McGowan, Hank Bell, Alan Bridge, Lew Meehan. When his brother is murdered by a gang leader who escapes across the Canadian border into the U.S., a Mountie quits the force and heads south for revenge. Another good outing for Buck Jones in his Universal series.\n\n**430** _ **Border Buckaroos**_ **** Producers Releasing Corporation, 1943. 61 min. D-SC: Oliver Drake. With Dave O'Brien, Jim Newill, Guy Wilkerson, Christine McIntyre, Eleanor Counts, Charles King, Jack Ingram, Ethan Laidlaw, Michael Vallon, Kenne Duncan, Reed Howes, Kermit Maynard, Bud Osborne, Slim Whitaker, Roy Brent. Mistaken for outlaws in a town where a murder has been committed, three lawmen join a gang to find out its leader. Typically paltry fourth entry in \"The Texas Rangers\" series.\n\n**431** _ **Border Caballero**_ **** Puritan, 1936. 54 min. D: Sam Newfield. SC: Joseph O'Donnell. With Tim McCoy, Lois January, Ralph Byrd, Ted Adams, J. Frank Glendon, Earle Hodgins, John Merton, Bob McKenzie, Oscar Gahan, Harrison Greene, Si Jenks, Richard Botiller, Frank McCarroll, Tex Phelps, George Morrell, Jack Evans, Ray Henderson, Wally West, Bill Patton, Steve Clark, Herman Hack, Jimmy Aubrey, Artie Ortego, Slim Whitaker, Henry Hall, Jack Rockwell, Ben Corbett, Sherry Tansey, Bud McClure, Ed Phillips, Rube Dalroy, Bill Wolfe. A medicine show sharpshooter, a former federal agent, rejoins the service following the murder of his pal by an outlaw gang. Well done and fast paced Tim McCoy vehicle in which the star first used a Mexican disguise.\n\n**432** _ **Border Caf\u00e9**_ **** RKO Radio, 1937. 67 min. D: Lew Landers. SC: Lionel Houser. With Harry Carey, John Beal, Armida, Walter Miller, Marjorie Lord, J. Carrol Naish, Lee Patrick, Paul Fix, George Irving, Leona Roberts, Max Wagner, Alec Craig, Hooper Atchley, Dudley Clements. In order to rehabilitate himself, a young man goes West and ends up fighting outlaws to save a community. Pretty good \"B+\" effort with a sturdy performance by Harry Carey.\n\n**433** _ **Border City Rustlers**_ **** Allied Artists, 1953. 54 min. D: Frank McDonald and Wesley Barry. SC: William Raynor. With Guy Madison, Andy Devine, Gloria Talbott, George J. Lewis, Steve Pendleton, Murray Alper, Jerome Sheldon, George Eldredge, Don Turner, Isabel Randolph, Robert Bice, John Carpenter, House Peters, Jr., Gregg Barton, Pierce Lyden, Bruce Edwards, Jo Carroll Dennison. Wild Bill Hickok and Jingles pretend to be murder victims in order to catch outlaws and they help a cowpoke imprisoned for crimes he did not commit. Pleasant theatrical release made up of \"Border City\" and \"Ex-Convict Story,\" two episodes of the \"Wild Bill Hickok\" (1951\u201358) TV series.\n\n**434** _ **Border Devils**_ **** Artclass, 1932. 60 min. D: William Nigh. SC: Harry P. (Fraser) Crist. With Harry Carey, Kathleen Collins, Niles Welch, Ray Gallagher, Olive Fuller Golden, Murdock MacQuarrie, George Hayes, Albert Smith, Maston Williams, Art Mix, Merrill McCormick, Tetsu Komai, Frank Ellis, Jack Gallagher. Falsely accused of a crime, a man breaks jail to prove his innocence. Good Harry Carey vehicle.\n\n**435** _ **Border Fence**_ **** Astor, 1951. 89 min. D: H.W. Kier and Norman Sheldon. SC: Norman Sheldon. With Walt Wayne, Lee Morgan, Mary Nord, Steve Raines, Henry Garcia, LeRoy Fisher, Frank Savage, Charles Clark, Frank Miller, Alvin France, Chester Scott, Jr., Ray Young, Jerry O'Dell. Out of prison after taking the blame for his rustler pal, a rancher again comes under suspicion when his friend steals from another cattleman. Bottom rung oater filmed in San Antonio, Texas; also called _**Cactus Barrier**_.\n\n**436** _ **Border Feud**_ **** Producers Releasing Corporation, 1947. 54 min. D: Ray Taylor. SC: Joseph O'Donnell. With Lash LaRue, Al St. John, Ian Keith, Gloria Marlen, Kenneth Ferrell, Ed Cassidy, Bob Duncan, Brad Slaven, Mikel Conrad, Bud Osborne, Frank Ellis, Richard Cramer, Henry Wills, Bob Woodward, Casey MacGregor. Sheriffs Cheyenne Davis and Fuzzy Q. Jones are after a mysterious outlaw, The Tiger, and Cheyenne masquerades as the bad man in order to get the goods on him and his gang. Fast moving, but of little interest.\n\n**437** _ **Border G-Man**_ **** RKO Radio, 1938. 60 min. D: David Howard. SC: Oliver Drake and Bernard McConville. With George O'Brien, Laraine (Day) Johnson, Ray Whitley, John Miljan, Rita LaRoy, Edgar Dearing, William Stelling, Edward Keane, Bob Burns, Ethan Laidlaw, Hugh Sothern, Ken Card, The Phelps Brothers, Herman Hack, Hank Bell. Posing as a ranch foreman, an FBI agent tries to find out who is heading a smuggling ring along the West Coast. Action packed George O'Brien Western, one of his best. Ray Whitley sings \"Back in the Saddle Again,\" which he co-wrote with Gene Autry, who used it as his theme song.\n\n**438** _ **Border Guns**_ **** Awyon, 1935. 55 min. D-SC: Robert J. Horner. SC: Ollie Milliken. With Bill Cody, Franklyn Farnum, Janet Morgan (Blanche Mehaffey), George Chesebro, Fred Church, William Desmond, Jimmy Aubrey, Wally Wales, Doris Brook, Oscar Gahan, Nelson McDowell, Buck Morgan, Fred Parker, Frank Clark, Archie Ricks, Bud Pope, Barney Beasley. A cowpoke is aided by a notorious gunman in stopping outlaws from taking over range land. Crude and vapid, helped only by Franklyn Farnum's bravura performance as the good-bad man.\n\n**439** _ **Border Incident**_ **** Metro-Goldwyn-Mayer, 1949. 95 min. D: Anthony Mann. SC: John C. Higgins and George Zuckerman. With George Murphy, Ricardo Montalban, Howard Da Silva, James Mitchell, Arnold Moss, Alfonso Bedoya, Teresa Celli, Charles McGraw, Jose Torvay, John Ridgely, Arthur Hunnicutt, Sig Rumann, Otto Waldis, John McGuire, Jack Lambert, Nedrick Young, Fred Graham, Jose Dominguez, Al Haskell, Mitchell Lewis, Elias Gamboa, Martin Garralaga, Paul Marion, Manuel Lopez, William \"Bill\" Phillips, Lita Baron, Frank Conlan, Lynn Whitney, Harry Antrim, Tony Barr, Rozene Jones. U.S. Immigration agents try to put a stop to the smuggling of Mexicans into in the country across the Texas-Mexican border. Good expose, if somewhat violent, of modern-day slave trade.\n\n_**Border Land**_ see _**Borderland**_\n\n**440** _ **Border Law**_ **** Columbia, 1931. 63 min. D: Louis King. SC: Stuart Anthony. With Buck Jones, Lupita Tovar, Frank Rice, Jim Mason, Don Chapman, Louis Hicks, F.R. Smith, John Wallace, Bob Burns, Glenn Strange, Fred Burns, Art Mix, Jack Evans, Herman Hack. When his brother is killed in a fight, a cowboy vows revenge. Excellent Buck Jones vehicle, which he remade as _**The Fighting Ranger**_ (q.v.); also redone as _**Rio Grande Ranger**_ (q.v.).\n\n**441** _ **The Border Legion**_ **** Paramount, 1930. 80 min. D: Otto Brower and Edwin F. Knoff. SC: Percy Heath and Edward E. Paramore, Jr. With Richard Arlen, Jack Holt, Fay Wray, Eugene Pallette, Stanley Fields, E.H. Calvert, Ethan Allen, Syd Saylor. In Idaho an outlaw gang rescues a young cowboy about to be lynched for a crime committed by one of them, and the cowpoke agrees to join the band. Well done early sound version of the 1916 Zane Grey novel; first filmed in 1919 starring Hobart Bosworth, with Paramount remaking it in 1924 headlining Antonio Moreno.\n\n**442** _ **The Border Legion**_ **** Republic, 1940. 58 min. D: Joseph Kane. SC: Olive Cooper and Louis Stevens. With Roy Rogers, George \"Gabby\" Hayes, Carol Hughes, Joseph Sawyer, Maude Eburne, Jay Novello, Hal Taliaferro, Dick Wessell, Paul Porcasi, Robert Emmett Keane, Ted Mapes, Fred Burns, Post Park, Art Dillard, Chick Hannon, Chuck Baldra, Art Mix, Eddie Acuff, Ed Peil, Sr., Lew Kelly, Monte Montague, Jack Kirk, Ed Brady, Curley Dresden, Cactus Mack, Bob Woodward, Pascale Perry, Jack Montgomery, Bob Card. A New York doctor heads to Idaho a wanted fugitive after taking the blame for a crime committed by his girl's brother, and ends up joining an outlaw gang in order to bring them to justice. Pretty good Roy Rogers film although it bears little resemblance to the Zane Grey work. TV title: _**West of the Badlands**_.\n\n_**Border Lust**_ see _**Lust to Kill**_\n\n**443** _ **Border Menace**_ **** Aywon, 1936. 55 min. D: Jack Nelson. SC: Robert J. Horner. With Bill Cody, Miriam Rice, George Chesebro, Jimmy Aubrey, Ben Corbett, Frank Clark, Jim Donnelly, Fred Parker, Robert Walker, Herman Hack, Oscar Gahan, Jack Evans, Tex Palmer, Barney Beasley. A Secret Service agent (called \"The Shadow') tries to prevent a crooked banker and his henchman from cheating a man and his daughter out of their oil lands. Rock bottom cinema and one of the very worst \"B\" Westerns.\n\n**444** _ **Border Outlaws**_ **** United International\/Eagle-Lion, 1950. 59 min. D: Richard Talmadge. SC: Arthur Hoerl. With Space Cooley, Maria Hart, Bill Edwards, Bill Kennedy, George Slocum, John Laurenz, Douglas Wood, Bud Osborne, John Carpenter, The Metzetti Brothers. Authorities post a big reward for the capture of the \"Phantom Rider\" who is wanted for smuggling drugs. Another sorry attempt at sagebrush stardom by country swing bandleader Spade Cooley, which mainly belongs to Bill Edwards. Richard Talmadge co-produced, directed and appears as one of the Metzetti Brothers.\n\n**445** _ **Border Patrol**_ **** Path\u00e9, 1928. 50 minutes. D: James P. Hogan. SC: Finis Fox. With Harry Carey, Kathleen Collins, Richard Tucker, Phillips Smalley, James Neil, James Marcus. A Texas Ranger falls in love with a young woman not knowing she is being used by a band of counterfeiters that includes her father. Modern-day Western with much footage in El Paso, Texas, containing lots of comedy and well photographed chase sequences.\n\n**446** _ **Border Patrol**_ **** United Artists, 1943. 65 min. D: Lesley Selander. SC: Michael Wilson. With William Boyd, Andy Clyde, Jay Kirby, Russell Simpson, Claudia Drake, Cliff Parkinson, George Reeves, Pierce Lyden, Duncan Renaldo, Robert Mitchum, Earle Hodgins, Charles Stevens, Merrill McCormick, Herman Hack, Robert Kortman, Dan White, Bill Nestell, Hugh Prosser, Henry Wills, Leo J. McMahon, Charles Murphy, Denver Dixon, Roy Bucko. Texas Rangers Hopalong Cassidy, California Carlson and Johnny Travers try to help a woman find out who killed a mine operator and uncover a crooked judge, the leader of a murderous gang. Good, exciting \"Hopalong Cassidy\" entry with Russell Simpson excellent as a tyrant.\n\n**447** _ **The Border Patrolman**_ **** 20th Century\u2013Fox, 1936. 60 min. D: David Howard. SC: Dan Jarrett and Bennett Cohen. With George O'Brien, Polly Ann Young, Roy (LeRoy) Mason, Mary Doran, Smiley Burnette, Tom London, Al Hill, Murdock MacQuarrie, John St. Polis, Cyril Ring, William P. Carlton, Martin Garralaga, Chris-Pin Martin. A wealthy family hires a border patrolman to keep their spoiled daughter in line but she goes to Mexico and gets involved with an international jewel theft ring and he has to come to her rescue. Pleasant George O'Brien vehicle interpolating comedy and action.\n\n**448** _ **Border Phantom**_ **** Republic, 1937. 58 min. D: S. Roy Luby. SC: Fred Myton. With Bob Steele, Harley Wood, Don Barclay, Karl Hackett, Horace Murphy, Miki Morita, John Peters, Perry Murdock, Frank Ball, Hans Joby, Budd Buster, Horace B. Carpenter, Clyde McClary. A cowpoke and his tenderfoot pal stumble onto a young woman whose professor father has been murdered. Mystery element provides zest to his exciting Bob Steele feature.\n\n**449** _ **Border Rangers**_ **** Lippert, 1950. 57 min. D: William Berke. SC: Victor West. With Don Barry, Robert Lowery, Wally Vernon, Pamela Blake, Lyle Talbot, Bill Kennedy, John Merton, George Keymas, Tom Kennedy, Eric Norden, Bud Osborne. An outlaw gang robs a bank and a Texas Ranger takes on the guise of a bandit to arrange their capture. Fast moving but cheaply produced.\n\n**450** _ **Border River**_ **** Universal-International, 1954. 81 min. Color. D: George Sherman. SC: William Sackheim and Louis Stevens. With Joel McCrea, Yvonne De Carlo, Pedro Armendariz, Howard Petrie, Erika Nordin, Alfonso Bedoya, George J. Lewis, Nacho Galindo, Ivan Triesault, George Wallace, Martin Garralaga, Lane Chandler, Louis Horvath, Britt Wood, Fred Beir, Monte Montague, Pilar Del Rey, Felipe Turich, Jack Del Rio, John Vernon, Robert Hoy, Joe Bassett. Near the end of the Civil War, a Confederate officer crosses the Rio Grande River into Zona Libra, a territory separated from Mexico, to buy weapons from a self-serving general. Color outing provides good entertainment.\n\n**451** _ **Border Romance**_ **** Tiffany, 1930. 60 min. D: Richard Thorpe. SC: John Francis (Jack) Natteford. With Armida, Don Terry, Marjorie \"Babe\" Kane, Victor Potel, Wesley Barry, Nita Martan, Frank Glendon, Harry Von Meter, William Costello. Three cowpokes have their cattle stolen by a notorious Mexican bandit and one of them romances a girl to find the whereabouts of the thief, although he is in love with a pretty senorita. Dated curio.\n\n**452 Border Roundup** Producers Releasing Corporation, 1942. 57 min. D: Sam Newfield. SC: Stephen Worth (Joseph O'Donnell). With George Houston, Al St. John, Smoky (Dennis) Moore, Patricia Knox, Charles King, I. Stanford Jolley, Ed Peil, Sr., Jimmy Aubrey, John Elliott, Nick Thompson, Frank Ellis, Curley Dresden, Lynton Brent, Dale (Gale) Sherwood, Jack Kirk, Herman Hack, Ray Henderson, Dan White. The Lone Rider assists a friend who has been framed for murder by crooks wanting a gold mine. Good entry in \"The Lone Rider\" series. TV title: _**The**_ _**Lone Rider in Border Roundup**_.\n\n**453** _ **Border Saddlemates**_ **** Republic, 1952. 57 min. D: William Witney. SC: Albert DeMond. With Rex Allen, Mary Ellen Kay, Slim Pickens, Forrest Taylor, Roy Barcroft, Jimmy Moss, Zon Murray, Keith McConnell, Bud Osborne, The Republic Rhythm Riders, Pat O'Malley, James Magrill, Post Park, Joe Yrigoyen, Billy Dix. A government agent is sent to a community on the U.S.-Canadian border to doctor silver foxes and uncovers a crooked scheme. The \"B\" Western was on the way out by the time this Rex Allen effort came along and the decline shows.\n\n**454** _ **Border Sheriff**_ **** Universal, 1926. D-SC: Robert North Bradbury. With Jack Hoxie, Olive Hasbrouck, S.E. Jennings, Gilbert \"Pee Wee\" Holmes, Buck Moulton, Tom Lingham, Bert De Marc, Frank Rice, Floyd Criswell, Leonard Trainer. Working incognito, a lawman becomes the ally of a wealthy rancher he suspects is the head of a narcotics smuggling operation. Colorful Jack Hoxie film that shows why he was so popular in the 1920s.\n\n**455** _ **Border Shootout**_ **** Turner Pictures, 1990. 100 min. Color. D-SC: C.T. McIntyre. With Glenn Ford, Cody Glenn, Jeff Kaake, Lizabeth Rohovit, Michael Horse, Russell Todd, Michael Ansara, Michael Forest, Kim Kelley, Charlene Tilton, George Salazar, Danny Nelson, Sam Smiley, Don Starr, Ed Gable, Josef Ranier, Gary Matansky, Rudy Martinez, Fred Jay Nelson, Connie McKenzie, Stanley Grover, Gale Wingfield, Bruce Paul Barbour, Sergio Calderon, Frank Plenchner, Emily Blanton, Steve Blanchard. As a long time sheriff trails cattle thieves, townspeople appoint a young rancher his deputy but he finds himself up against one of the area's founders in trying to clean up lawlessness. Glenn Ford's final Western is a pretty good TV movie.\n\n**456** _ **Border Treasure**_ **** RKO Radio, 1950. 60 min. D: George Archainbaud. SC: Norman Houston. With Tim Holt, Jane Nigh, Richard Martin, John Doucette, House Peters, Jr., Inez Cooper, Julian Rivero, Kenneth MacDonald, Vince Barnett, David Leonard. Money collected for charity by a young woman is taken from her by outlaws and a cowboy plans to retrieve it. Quality Tim Holt series entry.\n\n**457** _ **Border Vengeance**_ **** Awyon, 1925. 60 min. D: Harry S. Webb. SC: Forrest Sheldon. With Jack Perrin, Josephine Hill, Minna Redman, Vondell Darr, Jack Richardson, Bud Osborne, Leonard Clapham (Tom London), Hugh Saxon. A rancher is at odds with a gambler and his assayer cohort over a woman and a mine. Low grade silent offering.\n\n**458** _ **Border Vengeance**_ **** Willis Kent, 1935. 57 min. D: Ray Heinz. With Reb Russell, Mary Jane Carey, Clarence Geldert, Kenneth MacDonald, Jane Bupp, Ed Phillips, Norman Feusier, Ben Corbett, Narty Joyce, Slim Whitaker, Fred Burns, Pat Harmon, Glenn Strange, Eddie Parker, Bart Carre, Silver Tip Baker, Bud Pope, Bill Gillis, Hank Bell, Rex Bell, Montie Montana, Mabel Strickland. A rodeo performer gets involved in a revenge plot set up by a crook who forced his family off their land by trumping up a fake murder charge against them. Very poor Reb Russell vehicle with lots of stock rodeo footage; Rex Bell and Montie Montana make brief appearances.\n\n**459** _ **Border Vigilantes**_ **** Paramount, 1941. 63 min. D: Derwin Abrahams. SC: J. Benton Cheney. With William Boyd, Russell Hayden, Andy Clyde, Victor Jory, Frances Gifford, Morris Ankrum, Ethel Wales, Tom Tyler, Hal Taliaferro, Jack Rockwell, Britt Wood, Hank Worden, Hank Bell, Edward Earle, Al Haskell, Curley Dresden, Chuck Morrison, Ted Wells, Wen Wright, John Beach, Johnny Luther, Lem Sowards, Herman Howlin, Joe Garcia, Foxy Callahan, Tex Cooper, Henry Wills, Arthur Thalasso, Jess Cavin, Charles Murphy, George Sowards. Hopalong Cassidy and pals Lucky and Califonria try to help a rancher friend stop raids by an outlaw gang. Formula, but fast paced \"Hopalong Cassidy\" entry.\n\n**460** _ **Border Wolves**_ **** Universal, 1938. 57 min. D: Joseph H. Lewis. SC: Norton S. Parker. With Bob Baker, Constance Moore, Fuzzy Knight, Dickie Jones, Frank Campeau, Glenn Strange, Ed Cassidy, Oscar O'Shea, Jack Montgomery, Willie Fung, Dick Dorrell, Frank Ellis, Hank Bell, Jack Kirk, Ed Brady, Jack Evans, Eva McKenzie. During the California Gold Rush, a cowboy tries to clear himself after being falsely accused of criminal activities. Some fancy camerawork from director Joseph H. Lewis and star Bob Baker's pleasant personality and songs make this add up to more than a passable outing.\n\n**461** _ **Borderland**_ **** Paramount, 1937. 82 min. D: Nate Watt. SC: Harrison Jacobs. With William Boyd, James Ellison, George Hayes, Stephen Morris (Morris Ankrum), Charlene Wyatt, John Beach, Nora Lane, George Chesebro, Trevor Bardette, Earle Hodgins, Alan Bridge, John St. Polis, Slim Whitaker, Cliff Parkinson, Karl Hackett, Robert Walker, Frank Ellis, Ed Cassidy, J.P. McGowan, Jack Evans, Harry Bernard, Francis Walker, Roy Bucko, Ralph Bucko, Leo J. McMahon, Buffalo Bill, Jr., Herman Hack, Jim Corey, Frosty Royce, Joe Dominguez. Hopalong Cassidy pretends to take the side of lawlessness in order to aid the Texas Rangers and the Mexican Secret Service in capturing a notorious border bad man called \"The Fox.\" Overlong but entertaining \"Hopalong Cassidy\" entry.\n\n_**Borderland Rangers**_ see _**The Man from God's Country**_ (1924)\n\n**462** _ **Borderline**_ **** Universal-International, 1950. 88 min. D: William A. Seiter. SC: Devery Freeman. With Fred MacMurray, Claire Trevor, Raymond Burr, Roy Roberts, Morris Ankrum, Jose Torvay, Charles Lane, Don Diamond, Nacho Galindo, Pepe Hern, Grazia Narciso. A narcotics agent and a newspaperwoman team to track down dope smugglers on the U.S.-Mexican border. Ho-hum feature that unsuccessfully treads a thin line between comedy and drama.\n\n**463** _ **Borderline**_ **** Associated Film Distributors, 1980. 105 min. Color. D: Jerrold Freeman. SC: Steve Kline and Jerrold Freeman. With Charles Bronson, Bruno Kirby, Bert Remsen, Michael Lerner, Kenneth McMillan, Ed Harris, Karmin Murcelo, Enrique Castillo, A. Wilford Brimley, Norman Alden, James Victor, John Ashton, Lawrence Casey, Charles Cyphers, Panchito Gomez, John Roselius, Murray MacLeod, Jerry De Wilde, Katherine Pass, Virgil Frye, Luis Contreras, Eduardo Ricard, John O'Banion, Rodger La Rue, Tanya Russell, Virginia Bingham, Antony Munoz, Ray Ochoam, Ross Reynolds. When a fellow border patrolman is murdered while investigating the smuggling of aliens into the country, an officer sets out to avenge the crime and stop the illegal traffic in humans. Taut and well executed Charles Bronson thriller.\n\n**464** _ **Bordertown Gun Fighters**_ **** Republic, 1943. 56 min. D: Howard Bretherton. SC: Norman S. Hall. With Wild Bill Elliott, George \"Gabby\" Hayes, Anne Jeffreys, Ian Keith, Harry Woods, Edward Earle, Roy Barcroft, Bud Geary, Karl Hackett, Charles King, Carl Sepulveda, Edward Keane, Frank McCarroll, Wheaton Chambers, Kenneth Terrell, Neal Hart, Frosty Royce, Marshal Reed, Charles Sullivan, Jack Kenney, Nino Bellini, Ben Johnson, Budd Buster, Post Park, Jack Rockwell, Herman Willingham, Al Haskell, Foxy Callahan, Rose Plummer, James Mitchell, Jim Massey, Ralph Bucko, Bill Woolf. Wild Bill Elliott is assigned to break up a small town lottery racket, but also finds romance. Typically fast moving \"Wild Bill Elliott\" entry.\n\n**465** _ **Bordertown Trail**_ **** Republic, 1944. 55 min. D: Lesley Selander. SC: Bob Williams and Jesse Duffy. With Smiley Burnette, Sunset Carson, Ellen Lowe, Weldon Heyburn, Jack Luden, Addison Richards, Francis McDonald, John James, Jack Kirk, Harry Willis, Jack O'Shea, Neal Hart, Chick Hannon, Robert Wilke, Ted Wells, Roy Darmour, Earl Dobbins. The U.S. Border Patrol is up against a smuggling gang in cahoots with a self-serving politician. Early action filled Sunset Carson entry with as many plot twists as fights; one-time genre star Jack Luden appears as Sunset's brother.\n\n**466** _ **Born Reckless**_ **** Warner Bros., 1959. 79 min. D: Howard Koch. SC: Richard Landau and Aubrey Schenck. With Mamie Van Doren, Jeff Richards, Arthur Hunnicutt, Carol Ohmart, Tom Duggan, Tex Williams, Donald Barry, Nacho Galindo, Orlando Rodrigues, Johnny Olenn and Group. Rodeo drama about a champion performer and his beautiful blonde girlfriend. Title pretty much tells it all.\n\n**467** _ **Born to Battle**_ **** Path\u00e9, 1927. 46 min. D: Allan J. Neitz (Alan James). SC: L.V. Jefferson. With Bill Cody, Barbara Luddy, Sheldon Lewis, Frank McGlynn, Jr., Olin Francis, Ralph Yearsley, Nora Cecil, J.P. Lockney, Lew Meehan, Sailor Sharkey. A half-crazed woman wants revenge on the man who killed her husband while his daughter is in love with a cowpoke whose uncle wants the trouble to continue so he can get both feuding parties' ranches. Silent Bill Cody feature with a complicated plot, quite a bit of slapstick comedy and the star doing some nice stunt riding as well as using a bullwhip.\n\n**468** _ **Born to Battle**_ **** Reliable, 1935. 53 min. D: Harry S. Webb. SC: Rose Gordon and Carl (Krusada) Hartman. With Tom Tyler, Jean Carmen, Earl Dwire, Julian Rivero, Nelson McDowell, William Desmond, Richard Alexander, Charles King, Ralph Lewis, Ben Corbett, Jimmy Aubrey, Roger Williams, Robert Walker, Bob McKenzie, George Morrell, Blackie Whiteford. A wild living cowboy is bailed out of jail by a representative of the cattlemen's association and assigned the task of locating a notorious rustler and his gang. Action filled, but shoddy, Tom Tyler vehicle.\n\n**469** _ **Born to Buck**_ **** A.N.E., 1968. 93 min. Color. With Casey Tibbs; Henry Fonda, Rex Allen (narrators). Documentary about rodeo champion Casey Tibbs breeding his own bucking broncos on the Teton Sioux Indian Reservation in South Dakota, then driving the herd of some 400 wild horses halfway across the state and back to his ranch. A different kind of film with fine scenic values.\n\n**470** _ **Born to the Saddle**_ **** Astor, 1953. 73 min. Color. D: William Beaudine. SC: Adele Buffington. With Donald Woods, Leif Erickson, Karen Morley, Rand Brooks, Chuck Courtney, Glenn Strange, Dolores Priest, Fred Kohler, Jr., Dan White, Milton Kibbee, Boyd Davis. A young boy is befriended by a man who hires him to train a horse for an important race although he is really a gambler and the event has been fixed. Astor's low production values do little to enhance this Western racing drama.\n\n**471** _ **Born to the West**_ **** Paramount, 1937. 59 min. D: Charles Barton. SC: Stuart Anthony and Robert Yost. With John Wayne, Johnny Mack Brown, Marsha Hunt, John Patterson, Syd Saylor, Monte Blue, Lucien Littlefield, Nick Lukats, James Craig, Jack Kennedy, Vester Pegg, Earl Dwire, Jim Thorpe, Jennie Boyle, Alan Ladd, Jack Daly, Lee Prather. Two cowboys wander into a small town where one of them falls for the boss' girl and is framed by crooks. Fairly interesting Zane Grey outing with nice locales. Reissued as _**Hell Town**_.\n\n**472** _ **Borrowed Trouble**_ **** United Artists, 1948. 59 min. D: George Archainbaud. SC: Charles Belden. With William Boyd, Andy Clyde, Rand Brooks, Anne O'Neal, John Kellogg, Earle Hodgins, Cliff Clark, Helen Chapman, John Parrish, Herbert Rawlinson, Don Haggerty, James Harrison, Clarke Stevens, George Sowards, Eilene Janssen, Nancy Stowe, Jimmy Crane, Bill O'Leary, Norman Ollestad, Jr., Byron Foulger, Herman Hack, Al Thompson. Hoppy and his pals get in the middle of a feud between a spinster teacher and a saloon owner who has his business adjacent to her school. Amusing \"Hopalong Cassidy\" entry enhanced by good performances.\n\n**473** _ **The Boss Cowboy**_ **** Superior, 1934. 51 min. D: Denver Dixon (Victor Adamson). SC: B. Burdoge (Betty Burbridge). With Buddy Roosevelt, Frances Morris, Sam Pierce, Fay McKenzie, Bud Osborne, George Chesebro, Lafe McKee, William (Merrill) McCormick, Allen Holbrook, Clyde McClary, Denver Dixon. A ranch foreman falls for two women, one of whom is robbed by her own foreman, a wanted killer. Pretty poor stuff.\n\n**474** _ **Boss Nigger**_ **** Dimension, 1974. 87 min. Color. D: Jack Arnold. SC: Fred Williamson. With Fred Williamson, D'Urville Martin, William Smith, Barbara Leigh, R.G. Armstrong, Don \"Red\" Barry, Carmen Hayworth, Ben Zeller. Two black bounty hunters take over a small town as lawmen and capture an outlaw gang. Violent oater directed by cult favorite Jack Arnold. Re-titled **Boss**.\n\n**475** _ **Boss of Boomtown**_ **** Universal, 1944. 56 min. D: Ray Taylor. SC: William Lively. With Rod Cameron, Tom Tyler Fuzzy Knight, Vivian Austin, Ray Whitley, Jack Ingram, Robert Barron, Marie Austin, Max Wagner, Sam Flint, Richard Alexander, Forrest Taylor, Beverlee Mitchell. Two buddies ride into a brawling Western town where they come up against the city boss as well as finding romance. Rod Cameron's first starring Western is an entertaining one and he and co-star Tom Tyler make a good duo.\n\n**476** _ **Boss of Bullion City**_ **** Universal, 1941. 61 min. D: Ray Taylor. SC: Arthur St. Claire and Victor McLeod. With Johnny Mack Brown, Fuzzy Knight, Nell O'Day, Maria Montez, Harry Woods, Melvin Lang, Richard Alexander, Earl Hodgins, Karl Hackett, Frank Ellis, Tex Terry, Kermit Maynard, Bill Nestell. A ruthless tyrant rules an area with an iron fist and a lawman is called in to put a stop to his reign. Well produced Johnny Mack Brown vehicle; of interest to Maria Montez fans because she plays the second female lead, a girl who hero worships the marshal.\n\n**477** _ **The Boss of Hangtown Mesa**_ Universal, 1942. 59 min. D: Joseph H. Lewis. SC: Oliver Drake. With Johnny Mack Brown, Fuzzy Knight, William Farnum, Helen Deverell, Hugh Prosser, Jimmy Wakely, The Pals of the Golden West, Nora Lou Martin, Robert Barron, Michael Vallon, Fred Kohler, Jr., Henry Hall. Following the robbery of a telegraph company employee, a lawman is assigned to clean up a lawless town. Typically plotted entry in the Johnny Mack Brown Universal series, but this one has a plethora of music.\n\n**478** _ **Boss of Lonely Valley**_ **** Universal, 1937. 60 min. D: Ray Taylor. SC: Frances Guihan. With Buck Jones, Muriel Evans, Harvey Clark, Walter Miller, Lee Phelps, Ted Adams, Matty Fain, Ezra Pallette, Dickie Howard. A crooked town boss obtains land through fraud and a cowboy arrives in the area and tries to put a stop to his unlawful activities. Buck Jones produced this outing in his Universal series, a fairly good effort based on Forrest Brown's novel.\n\n**479** _ **Boss of Rawhide**_ **** Producers Releasing Corporation, 1943. 57 min. D-SC: Elmer Clifton. With Dave O'Brien, Jim Newill, Guy Wilkerson, Nell O'Day, Ed Cassidy, Jack Ingram, Charles King, Billy Bletcher, George Chesebro, Robert Hill, Dan White, Lucille Vance, Robert Kortman, Karl Hackett, Frank Ellis, Bud Osborne, Slim Whitaker, Jimmy Aubrey, Curley Dresden, Wally West, Budd Buster, Herman Hack, Fred Burns, Tex Cooper, Rose Plummer. The Texas Rangers are sent to an area where mysterious killings have been taking place and they uncover a gang who has unlawfully set up toll gates across range land. Another fast but cheap entry in PRC's \"The Texas Rangers\" series.\n\n**480** _ **The Boss Rider of Gun Creek**_ **** Universal, 1936. 60 min. D: Lesley Selander. SC: Frances Guihan. With Buck Jones, Muriel Evans, Harvey Clark, Tom Chatterton, Josef Swickard, Lee Phelps, Ernest Hilliard, Mahlon Hamilton, Alphonse Ethier, Edward Hearn. A man convicted of a murder he did not commit escapes from prison and takes on the guise of his look-a-like in order to clear himself. Buck Jones plays a dual role in this entertaining series vehicle.\n\n**481** _ **Both Barrels Blazing**_ **** Columbia, 1945. 57 min. D: Derwin Abrahams. SC: William Lively. With Charles Starrett, Tex Harding, Dub Taylor, Pat Parrish, Emmett Lynn, Alan Bridge, The Jesters, Dan White, Edward Howard, Jack Rockwell, Charles King, Robert Barron, Mauritz Hugo, James T. \"Bud\" Nelson, John Cason, Bert Dillard, Tex Palmer, Hansel Warner, Wally West, Rube Dalroy. A crook uses an old panhandler as a front for shipping stolen gold and a Texas Ranger, alias the Durango Kid, is on his trail. Fair entry in the \"Durango Kid\" series. British title: _**The Yellow Streak**_.\n\n**482** _ **The Bottom of the Bottle**_ **** 20th Century\u2013Fox, 1956. 88 min. Color. D: Henry Hathaway. SC: Sydney Boehm. With Van Johnson, Joseph Cotten, Ruth Roman, Jack Carson, Margaret Hayes, Bruce Bennett, Brad Dexter, Peggy Knudsen, Jim Davis, Margaret Lindsay, Nancy Gates, Gonzales-Gonzales (Pedro Gonzalez Gonzalez), John Lee, Ted (Tod) Griffin, Ernestine Barrier, Walter Woolf King, Sandy Descher, Henry (Harry) Morgan, Mimi Gibson, Kim Charney, Frances Dominguez, Maria M. Valerani, George Trevino, Joanne Jordan, George Anderson, Oscar Humberto Stevens, Martin F. Gerrish, Leo Gonzalez, Shirley Patterson (Shawn Smith), Carleton Young, John Doucette, Rosa Rey, Robert Adler, Alma Beltran. An alcoholic escaped convict takes refuge at his rich rancher brother's spread on the U.S.-Mexican border and asks his help in escaping with his family. Sturdy modern-day Western melodrama.\n\n**483** _ **Bounty**_ **** Barholtz Entertainment, 2009. 90 min. Color. D-SC: Jared Isham. With Jarret LeMaster, Michelle Acuna, Austin O'Brien, Bruce Isham, Rorick Lee Goins, Steve Savage, Jon Wyatt Davis, Peter Sherayko, Joe Pepper, Johnnie Oberg, Rebecca Oda, Patrick McCoy, Rafael Rio, Ben Barber. Needing to collect a bounty to pay off a debt, a reformed outlaw tries to break a pretty female prisoner out of jail. Low budge effort with productions gaffs but over all is not too bad.\n\n**484** _ **The Bounty Hunter**_ **** Warner Bros., 1954. 79 min. Color. D: Andre De Toth. SC: Winston Miller. With Randolph Scott, Dolores Dorn, Marie Windsor, Howard Petrie, Harry Antrim, Robert Keys, Ernest Borgnine, Dub Taylor, Tyler MacDuff, Archie Twitchell, Paul Picerni, Phil Chambers, Mary Lou Holloway, Charles Delaney, Fess Parker. A bounty hunter is on the trail of three killers who pretend to be average citizens. Randolph Scott fans will love this hard, relentless chase film that also features a fight sequence between heroine Dolores Dorn and saloon gal Marie Windsor.\n\n**485** _ **The Bounty Hunter**_ **** Action International, 1989. 90 min. Color. D: Robert Ginty. SC: Thomas Baldwin and Robert Ginty. With Robert Ginty, Bo Hopkins, Leota Waterdown, Melvin Holt, John White, Robert Knott, Jay Bullbear, Rocky Smith, Lisa Kious, Randy Whalen, Lance Lansford, Harvey Snell, Mark Brrager, Ted Vansickle, Dann Daigle, Seth Pollack, Steve Rosich, Michael Nauman, Shari Shanahar, Kenny Sullivan, Suzanne Sanders, Paul Vica, Barry Friedman, David McCally, Rex Linn. A veteran is at odds with a dishonest small town sheriff supposedly investigating the murder of his Vietnam War pal while in cahoots with an oil company in forcing a local Indian tribe to sell their land cheap. Low budget but nicely done independent modern-day Western; music by Rita Coolidge.\n\n**Dan Duryea and Audrey Dalton in** _**The Bounty Killer**_ **(Embassy, 1965).**\n\n**  \n**\n\n**486** _ **The Bounty Killer**_ **** Embassy, 1965. 92 min. Color. D: Spencer Gordon Bennet. SC: R. Alexander and Leo Gordon. With Dan Duryea, Rod Cameron, Audrey Dalton, Richard Arlen, Buster Crabbe, Fuzzy Knight, Johnny Mack Brown, Bob Steele, G.M. \"Bronco Billy\" Anderson, Peter Duryea, Eddie Quillan, Norman Willis, Edmund Cobb, I. Stanford Jolley, Frank Lackteen, Dan White, Grady Sutton, Emory Parnell, Red Morgan, Tom Kennedy, Michael Hinn. A dude from the East is forced to defend himself against an outlaw gang and when he kills the lot of them it turns him into a vicious bounty hunter. Production values and script are none-too-great but who cares with all the veteran genre stars and character players populating this Alex Gordon production? Buster Crabbe is especially good as a vicious outlaw.\n\n**487** _ **The Bounty Man**_ **** ABC-TV\/ABC Circle Films, 1972. 74 min. Color. D: John Llewellyn Moxey. SC: Jim Byrnes. With Clint Walker, Richard Basehart, John Ericson, Margot Kidder, Gene Evans, Arthur Hunnicutt, Rex Holman, Wayne Sutherlin, Paul Harper, Dennis Cross. Two rival bounty hunters track a young outlaw to an isolated valley and find themselves being attacked by his vicious gang. This telefeature shows just how good TV movies can be when care is taken with them.\n\n**488** _ **Bowery Buckaroos**_ **** Monogram, 1947. 66 min. D: William Beaudine. SC: Tim Ryan and Edmond Seward. With Leo Gorcey, Huntz Hall, Bobby Jordan, Gabriel Dell, Billy Benedict, David Gorcey, Julie Gibson, Bernard Gorcey, Minerva Urecal, Jack Norman (Norman Willis), Russell Simpson, Chief Yowlachie, Iron Eyes Cody, Rosa Turich, Sherman Sanders, Billy Wilkerson, Jack O'Shea, Bud Osborne, Cathy Carter. When their drugstore owner pal is accused of murder the Bowery Boys head West to track down the real killer. Average \"Bowery Boys\" series entry that will please their fans.\n\n**489** _ **The Boy from Oklahoma**_ **** Warner Bros., 1954. 88 min. Color. D: Michael Curtiz. SC: Frank Davis and Winston Miller. With Will Rogers, Jr., Nancy Olson, Lon Chaney, Anthony Caruso, Sheb Wooley, Merv Griffin, Clem Bevans, Louis Jean Heydt, Wallace Ford, Slim Pickens, Harry Lauter, James Griffith, Charles Watts, John Cason, Guy Teague, Tom Monroe, George Chesebro, George Lloyd, Joan Weldon, Forrest Taylor, Jack Daly, Guy Wilkerson, Britt Wood, Frank Marlowe, Emile Avery, Bud Osborne, Charles Waggenheim, Denver Pyle, Tyler MacDuff, Ted Mapes. An easy going law student becomes the sheriff of a rough town which he manages to clean up with the urging of a pretty girl, the daughter of his murdered predecessor. Pleasant semi-funny Western which is a good vehicle for Will Rogers, Jr.\n\n**490** _ **The Boy Who Talked to Badgers**_ **** Buena Vista, 1975. 100 min. Color. D: Gary Nelson. With Christian Juttner, Carl Betz, Salome Jens, Denver Pyle (narrator). A youngster has the ability to communicate with animals and he runs away from home to the wilds of Canada where his life is endangered. Well made Walt Disney family film originally telecast as a two-part segment of the series on NBC-TV.\n\n**491** _ **Boy's Ranch**_ **** Metro-Goldwyn-Mayer, 1946. 97 min. Color. D: Roy Rowland. SC: William Ludwig. With James Craig, Butch Jenkins, Skippy (Skip) Homeier, Dorothy Patrick, Ray Collins, Darryl Hickman, Sharon McManus, Minor Watson, Arthur Space, Robert Emmett O'Connor, Moroni Olsen, Geraldine Wall. An ex\u2013baseball player, trying to raise two orphans on his cattle ranch, makes it possible for delinquent boys to rehabilitate themselves by working there. Dated juvenile oriented film.\n\n**492** _ **Brand of Fear**_ **** Monogram, 1949. 56 min. D: Oliver Drake. SC: Basil Dickey. With Jimmy Wakely, Dub Taylor, Gail Davis, Tom London, Ray Whitley, Marshall Reed, William Ruhl, William Norton Bailey, Boyd Stockman, Dee Cooper, Frank McCarroll, Holly Bane, Bob Curtis, Myron Healey, Bob Woodward, Denver Dixon, Ray Jones, Bill Potter. A cowpoke falls in love with a beautiful girl only to discover she is the daughter of an ex-convict. Jimmy Wakely's singing and a good supporting cast make this one passable.\n\n**493** _ **Brand of Hate**_ **** Supreme, 1934. 63 min. D: Lewis D. Collins. SC: John F. (Jack) Natteford. With Bob Steele, Lucille Brown, William Farnum, George Hayes, Archie Ricks, James Flavin, Charles K. French, Jack Rockwell, Mickey Rentschler, Blackie Whiteford, Bill Patton, Bob Burns, Fred Burns, Al Haskell, Roy Bucko, Bob Card, Rose Plummer, Lionel Backus. A rancher is forced to harbor his cattle stealing half-brother and gang while the man's daughter is in love with the son of a neighbor, the enforcer investigating local cattle thefts. Average Bob Steele vehicle with an action filled finale and a fine cast.\n\n**494** _ **Brand of the Devil**_ **** Producers Releasing Corporation, 1944. 62 min. D: Harry Fraser. SC: Elmer Clifton. With Dave O'Brien, Jim Newill, Guy Wilkerson, Ellen Hall, Charles King, I. Stanford Jolley, Reed Howes, Budd Buster, Karl Hackett, Kermit Maynard, Ed Cassidy, Wally West, John Cason, Hank Bell, Rose Plummer, Jess Cavin, Jack Evans, Jack Tornek. An outlaw gang called \"Brand of the Devil\" is plaguing area ranchers and the Texas Rangers trio try to stop them. Another cheap entry in PRC's \"The Texas Rangers\" series and the penultimate one for James (Jim) Newill.\n\n**495** _ **Brand of the Outlaws**_ **** Supreme, 1936. 60 min. D-SC: Robert North Bradbury. With Bob Steele, Margaret Marquis, Jack Rockwell, Charles King, Virginia True Boardman, Ed Cassidy, Frank Ball, Bud Osborne, Horace Murphy, Bob Reeves, Budd Buster, Clyde McClary. After saving the life of a lawman, a cowboy joins a gang not knowing they are cattle thieves and the sheriff captures him and brands him an outlaw, making him try to prove his innocence. Well done and fast paced Bob Steele vehicle.\n\n**496** _ **Branded**_ **** 1931. 61 min. D: D. Ross Lederman. SC: Randall Faye. With Buck Jones, Ethel Kenyon, Wallace MacDonald, Al Smith, Fred Burns, Philo McCullough, John Oscar, Robert Kortman, Clark Burroughs, Lafe McKee, Archie Ricks, Sam McDaniel, Harry Todd, Ben Corbett, Blackjack Ward. Inheriting a ranch, a man becomes involved with a pretty neighbor whose crooked foreman plans to rustle his cattle. Good Buck Jones early talkie.\n\n**497** _ **Branded**_ **** Paramount, 1951. 95 min. Color. D: Rudolph Mate. SC: Sidney Boehm and Cyril Hume. With Alan Ladd, Mona Freeman, Charles Bickford, Robert Keith, Joseph Calleia, Peter Hansen, Tom Tully, Milburn Stone, Martin Garralaga, Edward Clark, John Butler, John Berkes, Selena Royle, Olan Soule, Robert Kortman, George J. Lewis, Ed Peil, Sr., Salvador Baguez. Outlaws find a man in the wilderness and come up with a scheme to use him to bilk a rancher by making him think he is his long lost son. There is enough action and romance in this Alan Ladd outing to satisfy his fans.\n\n**498** _ **Branded a Bandit**_ **** Arrow, 1924. 58 min. D-SC: Paul Hurst. With Yakima Canutt, Alys Murrell, Wilbur McGaugh, Judge Hamilton, Cliff Lyons. A rancher is in love with a prospector's daughter but is framed for killing the man by a bandit. Lots of action in this exciting Yakima Canutt silent affair.\n\n**499** _ **Branded a Coward**_ **** Supreme, 1935. 57 min. D: Sam Newfield. SC: Richard Martinsen. With Johnny Mack Brown, Billie Seward, Roger Williams, Syd Saylor, Lloyd Ingraham, Yakima Canutt, Lee Shumway, Frank McCarroll, Rex Downing, Robert Kortman, Ed Peil, Sr., Joe Girard, Wally West, Artie Ortego, Sherry Tansey. After seeing his parents killed when he was a boy, a man overcomes his fear of gunmen, becomes a marshal and tries to find a criminal who turns out to be his long lost brother. Johnny Mack Brown's first series Western is a good one despite low production values.\n\n**500** _ **Branded Men**_ **** Tiffany, 1931. 70 min. D: Phil Rosen. SC: Earle Snell. With Ken Maynard, June Clyde, Charles King, Irving Bacon, Donald Keith, Jack Rockwell, Hooper Atchley, Edmund Cobb, Slim Whitaker, Billy Bletcher, Al Taylor, Bud McClure. A cowboy and his pals join the side of the law in order to round up a bad man and his outlaw gang. Standard Ken Maynard vehicle without the fast pace of some of his other features.\n\n_**Brandy Sheriff**_ see _**Ride and Kill**_\n\n**501** _ **The Brass Legend**_ **** United Artists, 1956. 79 min. D: Gerd Oswald. SC: Don Martin. SC: Hugh O'Brian, Nancy Gates, Raymond Burr, Raba Tassell, Donald MacDonald, Robert Burton, Eddie Firestone, Stacy Harris, Norman Leavitt, Russell Simpson. After a young boy assists a sheriff in the capture of a vicious killer the lawman tries to save the lad when the bad man seeks revenge. Average oater with future TV stars Hugh O'Brian as the sheriff and Raymond Burr as the villain to recommend it.\n\n**502** _ **The Bravados**_ **** 20th Century\u2013Fox, 1958. 98 min. Color. D: Henry King. SC: Philip Yordan. With Gregory Peck, Joan Collins, Stephen Boyd, Albert Salmi, Henry Silva, Kathleen Gallant, Barry Coe, George Voskovec, Herbert Rudley, Lee Van Cleef, Andrew Duggan, Ken Scott, Gene Evans, Joe Da Rita, Robert Adler, Robert Griffin. Wanting vengeance for the rape and murder of his wife, a man plans to kill those who committed the crimes but after a time he comes to realize he has become no better than those he is hunting. Austere but very good film with Gregory Peck excellent as the hunter and Joe Da Rita (later one of The Three Stooges) giving a surprisingly harrowing performance as the \"hangman.\"\n\n**503** _ **Brave Warrior**_ **** Columbia, 1952. 73 min. Color. D: Spencer Gordon Bennet. SC: Robert E. Kent. With Jon Hall, Christine Larson, Jay Silverheels, Michael Ansara, Harry Cording, James Seay, George Eldredge, Leslie Denison, Rory Mallinson, Rusty Wescoatt. Peace between settlers and Native Americans in Indiana is threatened in 1811 by the interference of the British. Not even Technicolor can help this dull Sam Katzman historical production.\n\n**504** _ **Braveheart**_ **** Producers Distributing Corporation, 1925. 60 min. D: Alan Hale. SC: Mary O'Hara. With Rod La Rocque, Lillian Rich, Robert Edeson, Arthur Housman, Frank Hagney, Jean Acker, Tyrone Power (Sr.), Sally Rand, Henry Victor. An Indian brave goes to college to study law in order to defend his tribe's fishing rights but after becoming a top athlete he tries to save a friend and ends up in disgrace. Well done silent feature presented by Cecil B. DeMille.\n\n**505** _ **BraveStarr: The Legend**_ **** Taurus Entertainment, 1988. 91 min. Color. D: Tom Tataranowicz. SC: Bob Forward and Steve Hayes. With Pat Fraley, Susan Blu, Charles Adler, Ed Gilbert, Alan Oppenheimer (voices). When robots invade the planet of New Texas, a super powered cowboy and his pals come to the rescue. Fair sci-fi animated Western culled from the 1987\u201388 TV series \"BraveStarr.\"\n\n**506** _ **The Bravos**_ **** ABC-TV\/Universal, 1972. 100 min. Color. D: Ted Post. SC: Christopher Knopf and Ted Post. With George Peppard, Pernell Roberts, Belinda Montgomery, L.Q. Jones, George Murdock, Barry Brown, Dana Elcar, John Kellogg, Bo Svenson, Vincent Van Patten, Clint Ritchie, Randolph Mantooth, Joaquin Martinez. After the Civil War an officer is assigned the command at a small Western post but trouble with Indians arises and his young son is kidnapped. There is nothing special about this TV Western despite a good story premise.\n\n_**Brawlers**_ see _**El Buscabullas**_\n\n**507** _ **The Brazen Bell**_ **** Universal, 1963. 74 min. Color. D: James Sheldon. SC: Roland Kibbee and Charles Marquis Warren. With James Drury, Lee J. Cobb, Doug McClure, Gary Clarke, Pippa Scott, Roberta Shore, Anne Meacham, Royal Dano, John Davis Chandler, Robert J. Stevenson, Ross Elliott, Kay Stewart, Justin Smith, Walter Matthews, Lester Maxwell, Rick Murray. A frightened school teacher attempts to escape the harsh realities of the West but in a forced showdown proves he can stand up and fight. Pretty good drama originally telecast October 17, 1962, as an episode of \"The Virginian\" (NBC-TV, 1962\u201370) and issued theatrically abroad.\n\n**508** _ **Breakheart Pass**_ **** United Artists, 1976. 95 min. Color. D: Tom Gries. SC: Alistair MacLean. With Charles Bronson, Ben Johnson, Richard Crenna, Jill Ireland, Charles Durning, Ed Lauter, David Huddleston, Roy Jenson, Casey Tibbs, Archie Moore, Joe Knapp, Read Morgan, Robert Rothwell, Rayford Barnes, Scott Newman, Bill McKinney, Eddie Little Sky, Robert Tessier, Eldon Burke, John Mitchum, Keith McConnell, Doug Atkins, Sally Kirkland, Sally Kemp, Irv Falling, Bill Klem. Masquerading as a cowardly prisoner, an undercover agent is put on a train in an effort to expose gun runners. Alistair MacLean adapted his novel to the screen for this taut Western mystery.\n\n**509** _ **Breakout**_ **** NBC-TV\/Universal, 1970. 91 min. Color. D: Richard Irving. SC: Sy Gomberg. With James Drury, Red Buttons, Kathryn Hays, Woody Strode, Sean Garrison, Victor Meyerlink, Bert Freed, Mort Mills, William Mims, Harold J. Stone, Don Wilbanks, Kenneth Tobey, Ric Roman. A criminal works out a plan to escape from a mountain prison to be near his wife and the half-million dollars he has hidden but the scheme is endangered by a small boy lost in the snowy area. Well done TV movie.\n\n**510** _ **Breed of the Border**_ **** Monogram, 1933. 58 min. D: Robert North Bradbury. SC: Harry O. Jones. With Bob Steele, Marion Byron, George Hayes, Ernie Adams, Wilfred Lucas, Henry Roquemore, Fred Cavens, Robert Cord, Perry Murdock, John Elliott, Hal Price, Horace B. Carpenter, Blackie Whiteford, Ray Jones. A race car drive and his pal join forces with a female undercover agent to capture smugglers working along the U.S.-Mexican border. Complicated but fast moving Bob Steele vehicle with some nice fencing scenes between the star and Fred Cavens.\n\n**Poster for** _**Breed of the Border**_ **(Monogram, 1933).**\n\n**  \n**\n\n**511** _ **Breed of the West**_ **** Big 4, 1930. 55 min. D-SC: Alvin J. Neitz (Alan James). With Wally Wales, Virginia Brown Faire, Buzz Barton, Robert Walker, Lafe McKee, Bobby Dunn, George Gerwin, Hank (Bell) Cole, Edwin (Edmund) Cobb, Art Mix, Frank Ellis, Slim Andrews, Bud Osborne, Ben Corbett, Slim Whitaker, Bob Burns, Fred Burns. A cowboy is in love with his boss' daughter but has a rival in the ranch foreman who plans to rob the old man. More romance than action in this fair Wally Wales film.\n\n_**The Bride Comes to Yellow Sky**_ see _**Face to Face**_\n\n**512** _ **Bridger**_ **** ABC-TV\/Universal, 1976. 100 min. Color. D: David Lowell Rich. SC: Merwin Gerard. With James Wainwright, Ben Murphy, Dirk Blocker, John Anderson, William Windom, Sally Field, Margarita Cordova, Tom Middleton, X Brands. Jim Bridger is commissioned by President Andrew Jackson to open a trail from the Rocky Mountain to the West Coast in forty days in order to obtain land for the government. Fair historical drama with John Anderson as Andrew Jackson being its best moments.\n\n**513** _ **Brigham**_ **** Sunset Films, 1977. 96 min. Color. D: Tom McGowan. SC: Philip Yordan. With Maurice Grandmaison, Charles (Richard) Moll, John Mason, Howard Culver, Alan Richardson, Michael L. Goodman, Faith Clift, Robin Russell, Francis L. Urry, Terrence Gehr, Larry Ruup, James Arrington. The story of the Mormon Church from the time of Joseph Smith through the Indian Wars of the 1850s, using footage from _**Brigham Young, Frontiersman**_ (q.v.). This overblown, low budget effort was altered and reissued as _**Savage Journey**_.\n\n_**Brigham Young**_ see _**Brigham Young, Frontiersman**_\n\n**514** _ **Brigham Young, Frontiersman**_ **** 20th Century\u2013Fox, 1940. 114 min. D: Henry Hathaway. SC: Lamarr Trotti. With Tyrone Power, Linda Darnell, Dean Jagger, Jane Darwell, Brian Donlevy, John Carradine, Mary Astor, Vincent Price, Jean Rogers, Ann Todd, Willard Robertson, Moroni Olsen, Marc Lawrence, Stanley Andrews, Frank Thomas, Fuzzy Knight, Dickie Jones, Selmer Jackson, Russell Simpson, Arthur Aylesworth, Chief Big Tree, Claire Du Brey, Tully Marshall, Davison Clark, Dick Rich, Edwin Maxwell, Edmund MacDonald, Charles Halton, Lee Shumway, Charles Middleton, Frank LaRue, Cecil Watson, Ruth Robinson, Murdock MacQuarrie, Frederick Burton, Ralph Dunn, George Melford, David Kirkland, Phillip Morris, Paul E. Burns, Frank Shannon, William Haade, Herbert Heywood, Eddy Waller, Harry Tyler, Edmund Elton. The trek of the Mormons to Salt Lake, from the death of Joseph Smith in Illinois to the establishment of their colony in Utah, is retold in this historical drama. Very well done with a great performance by Dean Jagger in the title role; recommended.\n\n**515** _ **Brighty of Grand Canyon**_ **** Feature Film Corporation of America, 1967. 92 min. Color. D-SC: Norman Foster. With Joseph Cotten, Pat Conway, Dick Foran, Karl Swenson, Dandy Curran, Jiggs (burro). When his master is murdered a little burro meets a famed hunter, a boy and Theodore Roosevelt as he brings the killer to justice. Location filming and a good story make this pleasant entertainment.\n\n**516** _ **Brimstone**_ **** Republic, 1949. 90 min. Color. D: Joseph Kane. SC: Thames Williams. With Rod Cameron, Adrian Booth, Forrest Tucker, Walter Brennan, Jack Holt, Jim Davis, James Brown, Guinn Williams, Jack Lambert, Will Wright, David Williams, Harry V. Cheshire, Hal Taliaferro, Herbert Rawlinson, Stanley Andrews, Charlita, Jack Perrin, George Chesebro, Emmett Lynn, Jack O'Shea, Hank Bell, David Williams, Chester Conklin, Jody Gilbert, Tex Terry, Sam Flint, Helen Brown, Augie Gomez, Charles Cane, Leo Cleary. A lawman is sent to a territory where cattle rustling is rampant and he learns his friend, now a crooked sheriff, is working with a rancher and his two sons in the thefts. Action filled, brutal Western that is above average.\n\n**517** _ **Bring Me the Head of Alfredo Garcia**_ **** United Artists, 1974. 112 min. Color. D: Sam Peckinpah. SC: Sam Peckinpah and Gordon Dawson. With Warren Oates, Isela Vega, Robert Webber, Gig Young, Helmut Dantine, Emilio Fernandez, Kris Kristofferson, Chano Urueta, Donny Fritts, Jorge Russek, Chalo Gonzalez, Don Levy, Enrique Lucero, Janine Maldonado, Tamara Garina, Farnesio de Bernal, Ahui Camacho, Monica Miguel, Paco Pharres, Juan Manuel Diaz, Rene Dupeyron, Yolanda Ponce, Juan Jose Palacios, Manolo, Nery Ruiz, Roberto Dumont, Richard Bright, Conrad Hool, Whitey Hughes, Sharon Peckinpah, Garner Simmons. A bounty hunter duo teams with a piano player as they try to collect the reward for a man's head offered by a rich Mexican rancher. Another violent, bloody effort from writer-director Sam Peckinpah; not one of his better efforts although Warren Oates is fine as the pianist. Co-star Helmut Dantine was the film's executive producer.\n\n_**Broadway to Cheyenne**_ see _**From Broadway to Cheyenne**_\n\n**518** _ **Brokeback Mountain**_ **** Cinemac Films, 2005. 134 min. Color. D: Ang Lee. SC: Larry McMurtry and Diana Ossana. With Heath Ledger, Jake Gyllenhaal, Randy Quaid, Valerie Planche, David Trimble, Victor Reyes, Lachlan Mackintosh, Michelle Williams, Larry Reese, Marty Antonini, Tom Carey, Dan McDougall, Don Bland, Steven Cree Molison, Anne Hathaway, Duval Lang, Dean Barrrett, Scott Michael Campbell, Mary Liboiron, Graham Beckel, Kade Philips, Steffen Cole Moser, Brooklyn Prouix, Keanna Dube, James Baker, Pete Seadon, Sarah Hyslop Jacey Kenny, Jerry Callagan, Cayla Wolever, Cheyenne Hill, Jake Church, Ken Zilka, John Tench, Linda Cardellini, Anna Faris, David Harbour, Kae Mara, Will Martin, Gary Lauder, Christian Fraser, Cam Sutherland, Roberta Maxwell, Peter McRobbie, Mary McBride, Willie Nelson (voice). Two cowboys form a romantic, but uneven, life long relationship. Controversial gay-themed money maker from the story by Annie Prouix.\n\n**519** _ **Broken Arrow**_ **** 20th Century\u2013Fox, 1950. 93 min. Color. D: Delmer Daves. SC: Michael Blankfort. With James Stewart, Jeff Chandler, Debra Paget, Basil Ruysdael, Arthur Hunnicutt, Will Geer, Joyce MacKenzie, Raymond Bramley, Jay Silverheels, Argentina Brunetti, Jack Lee, Robert Adler, Harry Carter, Robert Griffin, Billy Wilkerson, Mickey Kuhn, Charles Soldani, Iron Eyes Cody, John Doucette, Trevor Bardette. Following the Civil War, an agent marries an Indian maiden and tries to bring peace between the government and the Chiricahua Apaches led by Cochise. Very colorful and entertaining Western.\n\n**520** _ **Broken Lance**_ **** 20th Century\u2013Fox, 1954. 96 min. Color. D: Edward Dmytryk. SC: Richard Murphy. With Spencer Tracy, Jean Peters, Robert Wagner, Richard Widmark, Katy Jurado, Hugh O'Brian, Carl Benton Reid, Eduard Franz, Earl Holliman, E.G. Marshall, Philip Ober, Robert Burton, Robert Adler, Robert Grandlin, Harry Carter, Nacho Galindo, Julian Rivero, Edmund Cobb, Russell Simpson, King Donovan, George E. Stone, Paul Kruger, Arthur Bryan. Following his second marriage, an aging Western land baron begins to feel his empire is crumbling due to conflicts with his sons. Western remake of _**House of Strangers**_ (20th Century\u2013Fox, 1949); a powerful and well acted film.\n\n**521** _ **The Broken Land**_ **** 20th Century\u2013Fox, 1962. 60 min. D: John Bushelman. SC: Edward Lasko. With Kent Taylor, Jody McCrea, Dianna Darrin, Robert Sampson, Gary Sneed, Don Orlando, Jack Nicholson, Helen Joseph, H. Tom Cain, Bob Pollard. A town is ruled by a sadistic sheriff who is at odds with a trio of young people who enlist the aid of the lawman's deputy in bringing about his downfall. Compact oater with an impressive performance by Kent Taylor as the sheriff.\n\n**522** _ **The Broken Law**_ **** Goodwill, 1924. 65 min. D: Paul Hurst. SC: Daniel F. Whitcomb. With Jack Mower, Alma Rayford, Vester Pegg, Frank Abbott, Carl Silvera, Bob Burns, Chief Tachachee. A cowboy is falsely accused of killing an Indian for his treasure map but he is defended by his female boss, with whom he has fallen in love, against her ranch foreman, the real culprit. Star Jack Mower produced this fair poverty row silent Western.\n\n**523** _ **Broken Sabre**_ **** Columbia, 1966. 89 min. Color. D: Bernard McEveety. SC: Jameson Brewer. With Chuck Connors, Kamala Devi, Peter Breck, Macdonald Carey, John Carradine, Wendell Corey, Rochelle Hudson, Robert Q. Lewis, Cesar Romero, Patrick Wayne, William Bryant, Steve Malo, H.M. Wynant, John Lormer, Jay Jostyn, Montie Plyler. A man is convicted of being a coward during the Battle of Bitter Creek and he tries to prove himself in the Arizona frontier after being dismissed from the military. Feature made from several episodes of \"Branded\" (NBC-TV, 1965\u201366) and issued theatrically in Great Britain.\n\n**524** _ **The Broken Star**_ **** United Artists, 1956. 82 min. D: Lesley Selander. SC: John C. Higgins. With Howard Duff, Lita Baron, Bill Williams, Henry Calvin, Douglas Fowley, Addison Richards, Joel Ashby, John Pickard, William Phillips, Dorothy Adams. A deputy marshal claims he killed a man in self defense when he actually murdered him for his gold. Interesting plot does not help this average oater.\n\n**525** _ **Broken Trail**_ **** American Movie Classics, 2006. 184 min. Color. D: Walter Hill. SC: Alan Geoffrion. With Robert Duvall, Thomas Haden Church, Greta Scacchi, Gwendoline Yeo, Chris Mulkey, Rusty Schwimmer, Scott Cooper, Valerie Tian, Caroline Chan, Olivia Cheng, Jadyn Wong, Donald Fong, James Russo, Tod Allan, Bill Baksa, Dusty Bews, Morris Birdyellowhead, Duncan Fraser, Philip Granger, Shaun Johnston, Sandy Kellerman, Greg Lawson, Scarlet Li, William Marquez, Stephen E. Miller, Donnelly Rhodes, Pat Richards, Peter Skagen, Patricia Stutz, Xuefeng Tan. Two cowboys, a man and his nephew, care for five abused Chinese girls while trying to save them from outlaws and also carry out a horse herd drive. Well staged TV mini-series.\n\n**526** _ **Bronco Billy**_ **** Warner Bros., 1980. 119 min. Color. D: Clint Eastwood. SC: Dennis Hacklin. With Clint Eastwood, Sondra Locke, Geoffrey Lewis, Scatman Crothers, Bill McKinney, Sam Bottoms, Dan Vadis, Sierra Pecheur, Walter Barnes, Hank Worden, Woodrow Parfrey, William Prince, Tanya Russell, Douglas McGrath, Beverlee McKinsey, Pam Abbas, Edye Byrde. An heiress with a bad temper, deserted by her husband, reluctantly joins a rag-tag Wild West show as the head showman's assistant. Beautifully done production that nicely captures the dream image of the cowboy hero.\n\n**527** _ **Bronco Buster**_ **** Universal-International, 1952. 80 min. Color. D: Budd Boetticher. SC: Horace McCoy and Lillie Hayward. With John Lund, Scott Brady, Joyce Holden, Chill Wills, Don Haggerty, Casey Tibbs, Dan Poore. A veteran rodeo performer befriends a young man and teaches him the trade only to become his rival in both work and romance. Not bad but the rodeo sequences are superior to the plot.\n\n_**Bronco Busters**_ see _**Little Moon and Jud McGraw**_\n\n**528** _ **Bronco: Death of an Outlaw**_ **** ABC-TV\/Warner Bros., 1960. 48 min. D: Herbert L. Strock. SC: Gerald Drayson Adams. With Ty Hardin, Allan Lane, Stephen Joyce, Rhodes Reason, Jean Allison, Barry Atwater, Miriam Colon, Forrest Lewis, Morris Ankrum, Alan Caillou, Dick Gering, Howard McLeod, James Beck, Bartlett Robinson, Harry Swoger, John Verros. A cowboy is aided by Billy the Kid during a range war although the outlaw is hunted by Sheriff Pat Garrett. An episode of the \"Bronco\" (ABC-TV, 1958\u201362) series issued on tape as a feature film.\n\n_**Bronson's Revenge**_ see _**Cut-Throats Nine**_\n\n**529** _ **The Bronze Buckaroo**_ **** Sack Amusement, 1938. 60 min. D-SC: Richard C. Kahn. With Herb (Jeffries) Jeffrey, Spencer Williams, Rellie Hardin, Artie Young, Clarence Brooks, F.E. Miller, Lucius Brooks, Lee Calmes, Earl J. Morris, The Four Tunes. A cowboy and his pal go to a ranch to help a girl whose father has been bushwhacked and they try to find the culprit. Interesting curio with an all-black cast starring singing hero Herb Jeffries. Filmed at N.B. Murray's black dude ranch near Victorville, California. The same year Herb Jeffries was also a cowboy crooner in the featurette _**Rhythm Rodeo**_.\n\n**530** _ **Brother of the Wind**_ **** Sun International, 1973. 91 min. Color. D: Dick Robinson. SC: John Mahon and John Champion. With Dick Robinson, Leon Ames (narrator). A mountain man finds his life of solitude changing after he adopts four motherless wolf cubs. Filmed in the forests of the Canadian Rockies, this heartwarming feature is a treat for the eyes.\n\n**531** _ **Brother Outlaw**_ **** Trans World Films, 1971. 84 min. Color. D: Edward G. Muller (Edoardo Mulargia). SC: Alessandro Schiro and Edoardo Mulargia. With Tony Kendall, James Rogers, Sophia Kammar, Dean Stratford, Omero Gargano, Sergio Sagnotti, Mimmo Maggio, Celso Faria, Fortunato Arena, Attilio Dottesio, Luciano Conti, Michele Branca, Nio Musco, Franco Marletta, Bruno Boschetti, Enzo Pulcrano, Omero Capanna. After his brother breaks him out of jail following conviction on a false charge, the siblings track down the gang behind the robbery that sent him behind bars. Standard Spaghetti Western filmed in Italy as _**Rimase uno Solo e Fu la Morte per Tutti**_ (It Remained Only One was Death for All).\n\n**532** _ **Brothers Blue**_ **** Warner Bros., 1973. 81 min. Color. D: Marc Meyer (Luigi Bazzoni). SC: Augusto Caminito. With Jack Palance, Tina Aumont, Antonio Falsi, Guido Mannari, Maurizio Bonuglia, Lee Burton. In order to stop crooks from taking over the West, a gang stages a series of holdups but they are tracked by a hired gunman. Jack Palance's fans should go for this Italian action feature issued there in 1971 by Felix Cinematografica as _**Blu Gang Vissero per Sempre Felici e Ammazzati**_ (Blue Gang Always Lived Happy and Murdered) and also called _**A Few Happy Days of the Brothers Ken**_ and _**The Short and Happy Life of the Brothers Blue**_.\n\n**533** _ **Brothers in Arms**_ **** Sony Pictures Entertainment, 2005. 96 min. Color. D: Jean-Claude La Marre. SC: Jean-Claude La Marre and Tyger Torrez. With David Carradine, Ed Lauter, Gabriel Casseus, Raymond Cruz, Jared Day, Idalis DeLeon, Nancy De Mayo, David Gianopoulos, Peter Greene, Garry Guerrier, Joel Harkham, Jerri Harris, William B. Jackson, Kurupt, Jean-Claude La Marre, Cameron Monagan, Kenya Moore, Lenee Pedersen, Michelle Penick, Glenn Plummer, Clifton Powell, Barry Ratcliffe, Karen Salkin, Peter Sherayko, Antwon Tanner, Tasha Dixon, Vakisha Coleman. Two outlaw brothers plan to get revenge on the man who murdered their relatives. Pretty poor revisionist Western.\n\n**534** _ **Brothers in the Saddle**_ **** RKO Radio, 1949. 60 min. D: Lesley Selander. SC: Norman Houston. With Tim Holt, Richard Martin, Steve Brodie, Virginia Cox, Carol Forman, Richard Powers (Tom Keene), Stanley Andrews, Robert Bray, Francis McDonald, Emmett Vogan, Monte Montague, Ted Adams. Two brothers go to opposite sides of the law, with one becoming a gambler who sinks deeper into crime despite the help given him by his sibling. A fast moving and exciting Tim Holt vehicle.\n\n**535** _ **Brothers of the West**_ **** Victory, 1937. 58 min. D: Sam Katzman. SC: Basil Dickey. With Tom Tyler, Lois Wilde, Dorothy Short, Lafe McKee, Bob Terry, Dave O'Brien, Roger Williams, Jim Corey, James C. Morton, Tiny Lipson, George Morrell. A range detective tries to prove his brother did not rob a bank and kill its president. Paltry entry in Tom Tyler's Victory series directed by producer Sam Katzman.\n\n**536** _ **The Brothers O'Toole**_ **** CVD\/American National Enterprises, 1973. 94 min. Color. D: Richard Erdman. SC: Tim Kelly and Marion Hargrove. With John Astin, Pat Carroll, Hans Conreid, Lee Meriwether, Allyn Joslyn, Jesse White, Richard Jury, Steve Carlson, Richard Erdman, Miranda Barry, Jacques Hampton. Two shiftless brothers come to a Colorado town and one is mistaken for a notorious highwayman and sentenced to be hanged. Dull, terrible Western-comedy.\n\n**537** _ **The Brute and the Beast**_ **** American International, 1968. 87 min. Color. D: Lucio Fulci. SC: Fernando Di Leo. With Franco Nero, George Hilton, Nino Castelnuovo, Lyn Shane, John MacDouglas (Giuseppe Addobatti), Rita Franchetti, Aysanoa Runachagua, Tom Felleghy, Franco Morici, Rina Franchetti, Tschang Yu, Aysanoa Runachaqua, John Bartha, Sal Borghese, Franco Guia, Mario Dionisi. A man returns home and eventually gets the aid of his half-brother in fighting a crook and his gun crazy son who have taken over their ranch. Brutal and violent Italian oater issued there in 1966 as _**Tempo di Massacro**_ (Time of Massacre) and also called _**Massacre Time**_.\n\n**538** _ **Brute Corps**_ **** General Film Corporation, 1972. 90 min. Color. D: Jerry Jameson. SC: Mike Kars and Abe Polsky. With Paul Carr, Jennifer Billingsley, Joseph Kaufmann, Alex Rocco, Michael Pataki, Charles Macaulay, Roy Jenson, Felton Perry, Joseph Bernard, Parker West. While on a camping trip in rural Mexico, a young American couple meet a group of mercenaries with tragic results. Violent modern-day Western.\n\n**539** _ **Buchanan Rides Alone**_ **** Columbia, 1958. 78 min. Color. D: Budd Boetticher. SC: Charles Lang, Jr. With Randolph Scott, Craig Stevens, Barry Kelley, Jennifer Holden, Tol Avery, Peter Whitney, Manuel Rojab, William Leslie, Don C. Harvey, L.Q. Jones, Robert Anderson, Joe De Santis, Nacho Galindo, Roy Jenson, Frank Scannell, Terry Frost, Riley Hill, Al Wyatt, Barbara James. A Texan rides into a border town and befriends a Mexican who opposes the tyrant running the area. One of the well regarded features star Randolph Scott, producer Harry Joe Brown and director Budd Boetticher made in the late 1950s and one that deserves its reputation.\n\n**540** _ **Buck and the Preacher**_ **** Columbia, 1972. 102 min. Color. D: Sidney Poitier. SC: Ernest Kinoy. With Sidney Poitier, Harry Belafonte, Cameron Mitchell, Ruby Dee, Denny Miller, Rita Talbot, John Kelly, Tony Brubaker, James McEachin, Clarence Muse, Ken Menard, Julie Robinson. A trail guide taking ex-slaves West so they can homestead is forced to ally himself with a con-man preacher when their journey is threatened by bounty hunters who want to return the travelers back to the South as cheap labor. Interesting premise does not unfold well in this mostly black cast oater that has far more talk than action.\n\n**541** _ **Buck Benny Rides Again**_ **** Paramount, 1940. 82 min. D: Mark Sandrich. SC: William Morrow and Edmund Belion. With Jack Benny, Ellen Drew, Andy Devine, Phil Harris, Dennis Day, Eddie \"Rochester\" Anderson, Don Wilson, Virginia Dale, Lillian Cornell, Theresa Harris, Kay Linaker, Ward Bond, Morris Ankrum, Charles Lane, James Burke, Merriel Abbott Dancers, Edward Gargan, Eddie Acuff, George Melford, Dick Rich, George Barrows, Billy Bletcher, Leyland Hodgson, Eddy Chandler, George Hickman, George Guhl, Monte Collins, Edgar Dearing, Willie Fung, Harry Baldwin, Allen Wood, Buddy Roosevelt, Ernest Whitman, Max Wagner, Archie Twitchell, Arthur Stuart Hull, Martin Faust, Roger Gray, George Hickman, John Laird; Fred Allen, Portland Hoffa, Mary Livingstone (voices). Radio comedian Jack Benny tries to win a girl's affections by showing her he is an all-American cowboy. Frequently very funny comedy with lots of help from Benny's radio crew; probably his best film.\n\n**542** _ **Buckaroo from Powder River**_ **** Columbia, 1947. 55 min. D: Ray Nazarro. SC: Norman S. Hall. With Charles Starrett, Smiley Burnette, Eve Miller, Forrest Taylor, The Cass County Boys (Jerry Scoggins, Fred Martin, Bert Dodson), Paul Campbell, Douglas D. Coppin, Philip Morris, Casey MacGregor, Ted Adams, Ethan Laidlaw, Edmund Cobb, Frank McCarroll, Kermit Maynard, Roy Butler, Phil Arnold, Buster Brodie, Tex Palmer. An outlaw gang leader plans to counterfeit government bonds stolen by his gang in a bank holdup but his nephew, who is in love with the sheriff's daughter, will not go along with the scheme and becomes a murder target. Entertaining \"Durango Kid\" series segment.\n\n**543** _ **The Buckaroo Kid**_ **** Universal, 1926. 64 min. D-SC: Lynn Reynolds. With Hoot Gibson, Ethel Shannon, Burr McIntosh, Harry Todd, James Gordon, Newton House, Joe Rickson, Arthur Thalasso, Clark Comstock, Arthur Millett. While managing a man's dilapidated ranch, a cowboy falls for the owner's daughter. Light hearted and charming silent Hoot Gibson vehicle.\n\n**544** _ **Buckaroo Sheriff from Texas**_ **** Republic, 1951. 60 min. D: Philip Ford. SC: Arthur Orloff. With Michael Chapin, Eilene Janssen, James Bell, Hugh O'Brian, Steve Pendleton, Tristram Coffin, William Haade, Selmer Jackson, Ed Cassidy, Eddie Dunn, Alice Kelley, Bob Reeves, George Taylor, Steve Dunhill, Tommy Coats, Cactus Mack. Two siblings help to bring in a notorious outlaw. First of four films in the \"Rough Ridin' Kids\" series showing the business of the law in the Old West is best left to grownups.\n\n**545** _ **Buckeye and Blue**_ **** Academy Entertainment, 1988. 94 min. Color. D-SC: J.C. Compton. With Robyn Lively, Jeff Osterhage, Rick Gibbs, Will Hannah, Michael Horse, Kenneth Jensen, Stuart Rogers, Patrick Johnson, James Gooden, Howard Allen, Anthony Auriemma, Daniel Frank Webster, Dan Gunther, Gina Genova. In order to find her desperado husband, a young woman joins an outlaw gang claiming to know his whereabouts. Standard low budget effort.\n\n**546** _ **Bucking Broadway**_ **** Universal, 1917. 53 min. D: Jack (John) Ford. SC: George Hively. With Harry Carey, Molly Malone, L.M. Wells, Vester Pegg, William (Steele) Gettinger, Gertrude Astor, Martha Mattox. A ranch foreman falls in love with his boss' daughter but she is lured to the big city by a crooked stockbroker and the cowboy goes after her. Average effort in Harry Carey's \"Cheyenne Harry\" silent series, thought lost but found and restored.\n\n**547** _ **Buckshot John**_ **** Bosworth-Paramount, 1915. 55 min. D: Hobart Bosworth. SC: Hetty Grey. With Hobart Bosworth, Helen Wolcott, Courtenay Foote, Carl Von Schiller, Herbert Standing, Marshall Stedman, Frank Lanning, Oscar Linkenhelt, Art Acord, Rhea Haines, Hoot Gibson. After fifteen years in prison an outlaw is supposedly helped by a medicine show quack and reveals the location of hidden loot, only to find out he has been deceived. Only two reels of this silent effort from director-star Hobart Bosworth exist.\n\n**548** _ **Buckskin**_ **** Paramount, 1968. 97 min. Color. D: Michael Moore. SC: Steve Fisher. With Barry Sullivan, Joan Caulfield, Wendell Corey, Lon Chaney, John Russell, Barbara Hale, Bill Williams, Richard Arlen, Gerald Micheaud, Barton MacLane, Aki Aleong, Michael Larrain, Leo Gordon, George Chandler, Emile Meyer, Robert Riordan, Manuela Thiess, LeRoy Johnson. A territorial marshal opposes a Montana land baron who is trying to drive off remaining settlers around a small town by diverting their water supply. Except for the veteran players there is little to recommend A.C. Lyles' final Paramount Western.\n\n**549** _ **Buckskin Frontier**_ **** United Artists, 1943. 74 min. D: Lesley Selander. SC: Norman Houston. With Richard Dix, Jane Wyatt, Lee J. Cobb, Victor Jory, Albert Dekker, Lola Lane, Max Baer, Joseph Sawyer, George Reeves, Francis McDonald, Harry Allen, Bill Nestell, George Plues. A man fights corruption in a Western town in the 1860s when businessmen try to top the building of the railroad to further their cattle empire. Big, action-filled Harry Sherman production with a well conceived shoot-out.\n\n**550** _ **Buckskin Lady**_ **** United Artists, 1957. 66 min. D: Carl K. Hittleman. SC: David Lang and Carl K. Hittleman. With Patricia Medina, Richard Denning, Gerald Mohr, Henry Hull, Robin Short, Richard Reeves, Dorothy Adams, Hank Worden, Frank Sully, George Cisar, Louis Lettieri, Byron Foulger, John Dierkes. A lady gambler, who supports her drunken father, falls for the new medico in town but her gunman boyfriend objects. Okay programmer.\n\n**551** _ **Buddy Goes West**_ **** Alex Cinematografica, 1981. 90 min. Color. D: Michele Lupo. SC: Sergio Donati and Gene Luotto. With Bud Spencer, Amidou, Joe Bugner, Renato Scarpa, Piero Trombetta, Sara Franchetti, Andrea Heuer, Marilda Dona, Pino Patti, Riccardo Pizzuli. When gold is found under a village the locals are joined by two outlaws in fighting the town's greedy sheriff and a gang of crooks. Pleasant Spaghetti Western comedy released in Italy as _**Occhio Alla Penna**_.\n\n**552** _ **Buffalo Bill**_ **** 20th Century\u2013Fox, 1944. 90 min. Color. D: William A. Wellman. SC: Aeneas MacKenzie, Clements Ripley and Cecile Kramer. With Joel McCrea, Maureen O'Hara, Linda Darnell, Thomas Mitchell, Edgar Buchanan, Anthony Quinn, Moroni Olsen, Frank Fenton, Matt Briggs, George Lessey, Frank Orth, George Chandler, Chief Many Treaties, Chief Thundercloud, Sidney Blackmer, Evelyn Beresford, Cecil Watson, Fred Graham, Harry Tyler, Arthur Loft, Syd Saylor, Robert Homans, John Dilson, Edwin Stanley, Kermit Maynard, Ben Corbett, Henry Wills, George Bronson, Margaret Martin, Cordell Hickman, John Reese, Eddie Nichols, Gerald Mackey, Vincent Graeff. The story of Buffalo Bill Cody, from his days as a cavalry scout to the time he became a famous showman and owner of the most authentic wild west show in America. Entertaining but basically glossy picture of a legend.\n\n**553** _ **Buffalo Bill**_ **** Gloria Film, 1964. 95 min. Color. D: J.W. Fordson (Mario Costa). SC: Nino Stresa and Luciano Martino. With Gordon Scott, Richard Stuyvesant (Mario Brega), Catherine Ribeiro, Jan Hendriks, Peter Lull (Piero Lulli), Rolando Lupi, Hans von Borsody, Mirko Ellis, Ingeborg Schoener, Feodor Chaliapin (Jr.), Hugo Arden (Ugo Sasso), Andrew Scott (Andrea Scotti), Jacques Herlin, Frank Farrell (Franco Fantasia), Ronald Parish, Luigi Tosi, Rinaldo Zamperla. To stop attacks led by Yellow Hand on settlers, President Grant sends Buffalo Bill Cody to a fort where the scout learns a trader has been responsible for supplying the Indians with firearms. Handsomely mounted European treatment of an American historical subject, although none-too-accurate. Made in Italy by Filmes\/Corona\/Gloria Film as _**Buffalo Bill, l'roe Del Far West**_ (Buffalo Bill, Hero of the Far West).\n\n**554** _ **Buffalo Bill and the Indians, or Sitting Bull's History Lesson**_ **** United Artists, 1976. 123 min. Color. D: Robert Altman. SC: Alan Rudolph and Robert Altman. With Paul Newman, Burt Lancaster, Joel Grey, Kevin McCarthy, Harvey Keitel, Allan Nicholls, Geraldine Chaplin, John Considine, Robert Doqui, Mike Kaplan, Bert Remsen, Bonnie Leaders, Denver Pyle, Will Sampson, Pat McCormick, Shelley Duvall. Buffalo Bill Cody uses fraud and treachery to build a reputation for himself as an Indian fighter and frontiersman. Filmed in Canada and supposedly a Bicentennial presentation, this Robert Altman production is dull and pointless.\n\n**555** _ **Buffalo Bill in Tomahawk Territory**_ **** United Artists, 1952. 66 min. D: Bernard B. Ray. SC: Sam Neuman and Nat Tanchuck. With Clayton Moore, Arkansas Slim Andrews, Sharon Dexter, Chief Yowlachie, Chief Thundercloud, Rodd Redwing, Charles Hughes, Eddie Phillips, Tom Hubbard, Helena Dare, Charles Harvey, Merrill McCormick, Al Haskell, Chuck Hayward, Bill Coontz. When outlaws try to steal Indian lands, Buffalo Bill Cody and his sidekick come to the rescue. Low grade affair although Clayton Moore is fine in the title role.\n\n**556** _ **Buffalo Bill on the U.P. Trail**_ **** Sunset, 1926. 60 min. D: Frank S. Mattison. With Roy Stewart, Kathryn McGuire, Cullen Landis, Sheldon Lewis, Earl Metcalfe, Milburn Morante, Hazel Howell, Fred De Silva, Felix Whitefeather, Jay Morley, Eddie Harris, Dick LaReno, Harry Fenwick. Buffalo Bill Cody and his pal plan to build a town along a railroad route but run into trouble when they prevent a locator from buying into their plans. Standard, but fast paced, silent \"historical\" feature. Also called _**With Buffalo Bill on the U.P. Trail**_.\n\n**557** _ **Buffalo Bill Rides Again**_ **** Screen Guild, 1947. 70 min. D: Bernard B. Ray. SC: Barney Sarecky and Frank Gilbert. With Richard Arlen, Jennifer Holt, Lee Shumway, Gil Patrick, Edmund Cobb, Ed Cassidy, Ted Adams, Charles Stevens, Chief Many Treaties, Holly Bane, Frank McCarroll, Carl Mathews, George Sherwood, Fred Graham, Frank O'Connor, Dorothy Curtis, Shooting Star. Buffalo Bill Cody comes to the aid of a girl and her father whose ranch is sought by fur thieves. Stars Richard Arlen and Jennifer Holt help to make this more than passable entertainment.\n\n**558** _ **Buffalo Girls**_ **** CBS-TV, 1995. 180 min. Color. D: Ron Hardy. SC: Cynthia Whitcomb. With Anjelica Huston, Melanie Griffith, Jack Palance, Sam Elliott, Gabriel Byrne, Peter Coyote, Tracey Walter, Floyd \"Red Crow\" Westerman, Charalyne Woodard, John Diehl, Live Schreiber, Andrew Bicknell, Paul Lazar, Russell Means, Reba McIntire, Jane E. Goold, Michael Eiland, Jerry King, Rob Nicholas, Jeanine O'Connell, Dennis Robbins, Boots Sutherland, Geoffrey Bateman, Julie Bevan, Peter Birch, Graham Gadd, David Garver, Hannah Taylor-Gordon, Robert Harnsberger, Brian Knight, Russell Milton, J. Michael Oliva, Richard Simpson, Hanley Smith, Jenny Saxon, Gilley Grey, Tisha Frazier, Robyn Reede, Dennis E. Garber. Calamity Jane tries to find her long lost daughter and also recapture the West of her glorious past. Average TV Western from Larry McMurty's book.\n\n**559** _ **Buffalo Gun**_ **** Globe, 1962. 72 min. D: Albert C. Gannaway. SC: A.L. Milton. With Wayne Morris, Webb Pierce, Marty Robbins, Carl Smith, Mary Ellen Kay, Donald Barry, Douglas Fowley, Harry Lauter, Ed Crandall, Bill Coontz, Chris Little, Charles Saldoni, The Jordanaires. Three singing government agents are sent West to investigate the thefts of shipments to the Indians. Cheaply made oater that capitalizes on its trio of country-western stars as cowboys; Wayne Morris' final film.\n\n**560** _ **Buffalo Rider**_ **** Starfire Films, 1978. 90 min. Color. D: George Lauris. With John Freeman, Rick Guinn, Priscilla Lauris, George Sager, Rich Scheeland, Lane Caudell (voice). The story of C.J. \"Buffalo\" Jones, who tries to stop the slaughter of the Plains bison. Co-produced by Dick Robinson, this sorry affair was also called _**The Life and Legend of Buffalo Jones**_.\n\n_**Buffalo Soldiers**_ (1970) see _**Soul Soldier**_\n\n**561** _ **Buffalo Soldiers**_ **** Turner Network Television (TNT), 1997. 100 min. Color. D: Charles Haid. SC: Frank Military and Susan Rhinehart. With Lamont Bentley, Tom Bower, Timothy Busfield, Gabriel Casseus, Danny Glover, Bob Gunton, Keith Jefferson, Robert Knott, Carl Lumbly, Clifton Powell, Matt Ross, Glynn Thurman, Michael Warren, Mykelti Williamson, David Jean Thomas, Chesley Wilson, Jeri Brunoe-Samson, Dutch Lunak, Harrision Lowe, Mark Bustamante, Chris Gatewood, Barrie Tompkins, Matthew T. Wilson, Mike Lutz, Tony Brubaker. A black cavalry troop tries to capture an Apache warrior killing settlers in New Mexico. Good made-for-TV Western.\n\n_**Buffalo Stampede**_ see _**The Thundering Herd**_\n\n**562** _ **Bugles in the Afternoon**_ **** Warner Bros., 1952. 85 min. Color. D: Roy Rowland. SC: Geoffrey Homes and Harry Brown. With Ray Milland, Helena Carter, Hugh Marlowe, Forrest Tucker, Barton MacLane, George Reeves, James Millican, Gertrude Michael, Stuart Randall, William Phillips, Hugh Beaumont, Dick Rich, John Pickard, John War Eagle, Sheb Wooley, Charles Evans, Nelson Leigh, Ray Montgomery, Virginia Brissac, John Doucette, Bud Osborne, Harry Lauter, Bob Steele, Mary Adams, Lucille Shamburger. Branded a coward during the Civil War, a cavalry sergeant meets an old rival in the Dakota Territory and plans to settle a score on the eve of the Little Big Horn battle. Average outing with big budget trappings.\n\n**563** _ **Bull of the West**_ **** Universal, 1971. 78 min. Color. D: Paul Stanley and Jerry Hopper. SC: Richard Fielder and Don Ingalls. With James Drury, Charles Bronson, Lee J. Cobb, Brian Keith, Lois Nettleton, Bob Random, George Kennedy, Ben Johnson, Geraldine Brooks, De Forrest Kelley, Vito Scotti, Diane Roter, Doug McClure, Randy Boone, Gary Clarke, Paul Fix (narrator). An embittered man tries to make his ranch a success without the help of others. Two episodes of \"The Virginian\" (NBC-TV, 1962\u201370) issued as a feature in Europe to cash in on Charles Bronson's international popularity. An unofficial remake of _**Man Without a Star**_ and _**A Man Called Gannon**_ (qq.v.); also called _**Hot Lead**_.\n\n**564** _ **Bulldog Courage**_ **** Puritan, 1935. 60 min. D: Sam Newfield. SC: Joseph O'Donnell and Frances Guihan. With Tim McCoy, Joan Woodbury, Karl Hackett, John Elliott, Ed Cassidy, Edmund Cobb, George Morrell, Paul Fix, Jack Rockwell, Bud Osborne, Art Mix, Slim Whitaker, Frank Ellis, Jack Mower, Edward Hearn, Roy Bucko. A man returns home after twenty years to even the score with the crook who cheated his father out of a gold mine and caused his death. Exciting Tim McCoy vehicle with good direction by Sam Newfield.\n\n**565 Bullet and the Flesh** Ultima\/Hesperia\/Cineurope, 1964. 85 min. Color. D: Fred Wilson (Marino Girolami). SC: Marino Girolami and Gino De Santis. With Rod Cameron, Patricia Viterbo, Thomas Moore, Dan Harrison, Carol Brown, Manolo Zarzo, Alfred Mayo, Marie Versini, Manuelo Lupo, Julio Pena, Piero Lulli, Marco Mariani, George Lynn, Franco Latni, Enzo Girolami. A lumber king plans to plunder Cherokee lands while his daughter falls in love with an Indian chief. Pretty good Italian made oater with a fine villainous performance by Rod Cameron. Made as _**I Sentieri dell'Odio**_ (Paths of Hate); also called _**Bullet in the Flesh**_.\n\n**566** _ **Bullet Code**_ **** RKO Radio, 1940. 58 min. D: David Howard. SC: Doris Schroeder. With George O'Brien, Virginia Vale, Harry Woods, Slim Whitaker, Robert Stanton (Kirby Grant), Walter Miller, William Haade, Bob Burns, Howard Hickman, Lew Meehan, Bob McKenzie, Jack C. Smith, Cactus Mack. A cowboy, who mistakenly thinks he killed his pal, goes to work for the dead man's father and sister who are having their cattle rustled. Handsome George O'Brien vehicle; remake of _**Gun Law**_ (1933) [q.v.].\n\n**567** _ **Bullet for a Badman**_ **** Universal, 1964. 80 min. Color. D: R.G. Springsteen. SC: Mary Willingham and Willard Willingham. With Audie Murphy, Darren McGavin, Ruta Lee, Beverly Owens, Skip Homeier, George Tobias, Alan Hale, Bob Steele, Edward C. Platt, Mort Mills, Kevin Tate, Buff Brady. The happiness of a man and his wife is threatened when her ex-husband, who deserted her for the outlaw life, swears revenge on them. Competently done Audie Murphy feature.\n\n**568** _ **Bullet for a Stranger**_ **** Flora Film\/National Cinematografica, 1971. 94 min. Color. D: Anthony Ascott (Guiliano Carmineo). SC: E.B. Clucher (Enzo Barboni). With Gianni Garko, William Berger, Christopher Chittell, John Fordyce, Ugo Fangareggi, Raimondo Penne, Franco Ressel, Ivano Staccioli, Nello Pazzafini, Gianni De Benedetto, Ugo Adinolfi, Aldo Barberito, Gildo di Marco, Bill Vanders, Pinuccio Ardia, Amerigo Santarelli, Frank Brana, Mino Loy, Furio Meniconi, Goffredo Unger, Frank Ukmer, Claudio Ruffini, Aldo Cicconi, Roberto Messina. A mysterious stranger helps two greenhorn brothers after they get into trouble for throttling an extortion gang member. Pleasant Spaghetti Western part-comedy, one of a string of features with Gianni Garko as the Stranger. Issued in Italy as _**Gli Fumavano le Colt...Lo Chiamavano Camposanto**_ (His Pistols Smoked...They Call Him Cemetery).\n\n**569** _ **A Bullet for Billy the Kid**_ **** Associated Distributors Producers (ADP), 1963. 61 min. Color. D: Rafael Baledon. SC: Raymond Obon. With Gaston (Santos) Sands, Steve Brodie, Lloyd Nelson, Maria Blaine, Richard McIntyre, Rita (Macedo) Mace, Gilbert Cramer, Peter Gillon, Jaime Fernandez, Maurico Garces, Jose Cortez. When he goes up against corrupt forces, Billy the Kid soon finds himself the target of an assassin. Hacked up U.S. release of a Mexican film, _**Una Bala es Mi Testigo**_ (A Bullet Is My Witness) [Alameda Films, 1959], with new footage directed by the notorious Jerry Warren, known for such scissors-and-paste efforts as _**Attack of the Mayan Mummy**_ (1964), _**Face of the Screaming Werewolf**_ and _**Invasion of the Animal People**_ (both 1965).\n\n**570** _ **A Bullet for Sandoval**_ **** UMC Pictures, 1970. 96 min. Color. D: Julio Buchs. SC: Ugo Guerro, Jose Luis Martinez Molla, Frederic De Urrutia and Julio Buchs. With George Hilton, Ernest Borgnine, Gustavo Rojo, Alberto De Mendoza, Leo Anchoriz, Annabella Incontrera, Antonio Pica, Jose Manuel Martin, Manuel De Blas, Manuel Miranda. During the Civil War, a Confederate soldier returns home to get even with the land baron he blames for the starvation deaths of his wife and small son. Violent Spaghetti Western issued in Italy in 1969 as _**Quei Disperati Che Puzzano di Sudore e di Morte**_ (Those Desperate Men, Smelling of Sweat and Death) and also called _**Vengeance Is Mine**_.\n\n**571** _ **A Bullet for the General**_ **** Avco-Embassy, 1966. 95 min. Color. D: Damiano Damiani. SC: Salvatore Laurani and Franco Solinas. With Gian Maria Volonte, Klaus Kinski, Martine Beswick, Lou Castel, Bianca Manini, Jaimie Fernandez, Andrea Checci, Jose Manuel Martin, Spartaco Conversi, Joaquin Parra, Aldo Sambrell. During the Mexican Revolution a government agent is hired to kill a rebel general and he must gain the loyalty of a guerilla leader. Made in Italy as _**Quien Sabe?**_ , this is an above average European Western.\n\n_**Bullet in the Flesh**_ see _**Bullet and the Flesh**_\n\n**572** _ **A Bullet Is Waiting**_ **** Columbia, 1954. 82 min. Color. D: John Farrow. SC: Thames Williamson and Casey Robinson. With Rory Calhoun, Jean Simmons, Stephen McNally, Brian Aherne. A resolute lawman treks through the desert with a prisoner and they become stranded with a man and his pretty daughter. Intriguing psychological Western, heavy on characterization by its compact cast.\n\n**573** _ **Bullets and Saddles**_ **** Monogram, 1943. 56 min. D: Anthony Marshall. SC: Elizabeth Beecher. With Ray Corrigan, Dennis Moore, Max Terhune, Julie Duncan, Budd Buster, Rose Plummer, Forrest Taylor, Glenn Strange, Steve Clark, John Merton, Ed Cassidy, Joe Garcia, Silver Harr, Carl Mathews, Robert Kortman, Tom London, Denver Dixon, Wally West, Victor Cox, Frank McCarroll, Jack Evans, George Morrell, Hal Price. The Range Busters are called in to stop a crooked businessman who is trying to get control of an area with his gang. Final \"Range Busters\" series film is on the anemic side, using footage from an earlier entry, _**Fugitive Valley**_ (q.v.). Also called _**Vengeance in the Saddle**_.\n\n**574** _ **Bullets Don't Argue**_ **** Walter Manley Productions, 1965. 93 min. Color. D: Mike Perkins (Manfred Rieger). SC: Manuel Waller and Donald Mooch. With Rod Cameron, Dick Palmer, Vivi Bach, Kai Fisher, Angel Aranda, Horst Frank, Hans Nielsen, Ludwig Duran, Jose Manuel Martin. On his wedding day, Sheriff Pat Garrett is forced to go after two outlaw brothers who robbed the town bank and killed area citizens. One of the earliest, and best, European Westerns; full of lively action and helped by good work by Rod Cameron as Pat Garrett. It also includes one of Ennio Morricone's best, and sadly underrated, scores. Made by Jolly\/Trio\/Constantin as _**Die Letzen Zwei Vom Rio Bravo**_ (The Last Two from Rio Bravo) and also known as _**Guns Don't Argue**_ and _**The Two from Rio Bravo**_.\n\n**575** _ **Bullets for Bandits**_ **** Columbia, 1942. 55 min. D: Wallace Fox. SC: Robert Lee Johnson. With Bill Elliott, Tex Ritter, Dorothy Short, Frank Mitchell, Forrest Taylor, Ralph Theodore, Edythe Elliott, Eddie Laughton, Joe McGuinn, Tom Moray, Art Mix, Harry Harvey, Hal Taliaferro, John Tyrrell, Bud Osborne. Wild Bill Hickok comes to the aid of a woman rancher whose property is sought by a crook. Uneven entry in the \"Wild Bill Hickok\" series.\n\n**576** _ **Bullets for Rustlers**_ **** Columbia, 1940. 58 min. D: Sam Nelson. SC: John Rathmell. With Charles Starrett, Lorna Gray, The Sons of the Pioneers (Bob Nolan, Tim Spencer, Lloyd Perryman, Pat Brady, Hugh Farr, Karl Farr), Kenneth MacDonald, Jack Rockwell, Edward Le Saint, Francis Walker, Eddie Laughton, Lee Prather, Hal Taliaferro, Herman Hack, Jack Evans. A cattlemen's association undercover agent pretends to be a rustler so he can join the gang he is trying to arrest. Action filled Charles Starrett affair.\n\n**577** _ **Bullwhip!**_ **** Allied Artists, 1956. 80 min. Color. D: Harmon Jones. SC: Adele Buffington. With Guy Madison, Rhonda Fleming, James Griffith, Don Beddoe, Peter Adams, Dan Sheridan, Burt Nelson, Al Terry, Hank Worden, Barbara Woodell, Rhys Williams, Jay Reynolds, Tim Graham, Rick Vallin. In order to get control of a trading firm, a crook forces a cowboy to either marry the young woman who is to inherit the business or face hanging over a false murder charge. Average Guy Madison vehicle; Frankie Laine sings the title song with more conviction than there is in the movie itself.\n\n**578** _ **Burning Daylight**_ **** First National, 1928. 72 min. D: Charles Brabin. SC: Louis Stevens, Rufus McCosh and Dwinelle Benthall. With Milton Sills, Doris Kenyon, Arthur Stone, Big Boy (Guinn) Williams, Lawford Davidson, Jane Winton, Stuart Holmes, Edmund Breese, Howard Truesdale, Frank Hagney, Harry Northrup. A prospector makes and loses two fortunes, in the Klondike and San Francisco, before finding love and success. Sturdy melodrama with a chance to see popular silent film star Milton Sills.\n\n**579** _ **Burning Gold**_ **** Republic, 1936. 58 min. D: Sam Newfield. SC: Earle Snell. With Bill (William) Boyd, Judith Allen, Lloyd Ingraham, Fern Emmett, Frank Mayo, Bud Flanagan (Dennis O'Keefe). An oil driller gets rich from a gusher, finds romance and loses his fortune in a fire. Poor stuff from producer Nat Levine.\n\n**580** _ **The Burning Hills**_ **** Warner Bros., 1956. 94 min. Color. D: Stuart Heisler. SC: Irving Wallace. With Tab Hunter, Natalie Wood, Skip Homeier, Eduard Franz, Earl Holliman, Claude Akins, Ray Teal, Frank Puglia, Hal Baylor, Tyler MacDuff, Rayford Barnes. On the run from cattle thieves, a young man finds shelter with a half-breed girl and they fall in love. Average outing which its stars can do little to help.\n\n**581** _ **The Burrowers**_ **** Lions Gate, 2008. 96 min. Color. D-SC: J.T. Petty. With Clancy Brown, Stephanie Delgado, David Busse, William Mapother, Jocelin Donahue, Alexandra Edmo, Brighid Fleming, Karl Geary, Christopher Hagen, Doug Hutchison, Galen Hutchison, Laura Leighton, Harley Coriz, Suzi McLaughlin, Tatanka Means, David Midthunder, John Kristian Moore, Anthony Parker, Cole Resch, R.J. Rice, Sean Patrick Thomas, Chris Grabner. When a frontier family disappears, a rescue party thinks Indians are to blame but they are soon faced with an unknown evil. Fairly scary horror Western.\n\n**582** _ **Bury Me Not on the Lone Prairie**_ Universal, 1941. 61 min. D: Ray Taylor. SC: Sherman Lowe and Victor McLeod. With Johnny Mack Brown, Fuzzy Knight, Nell O'Day, Kathryn Adams, Harry Cording, Jack Rockwell, Ernie Adams, Edward Cassidy, Don House, Lee Shumway, Pat J. O'Brien, Frank O'Connor, William Desmond, Bud Osborne, Slim Whitaker, Kermit Maynard, Robert Kortman, Jim Corey, Charles King, Ethan Laidlaw, Frank Ellis, Jimmy Wakely and His Rough Riders (Johnny Bond, Dick Reinhart). After his brother is murdered, along with the brother of a young woman, a mining engineer gets on the trail of the killer. Good Johnny Mack Brown vehicle with both action and music.\n\n**583** _ **El Buscabullas**_ (The Troublemakers) **** Cine Vision, 1976. 88 min. Color. D: Raul de Anda, Jr. SC: Raul de Anda and Ramon Obon. With Rodolfo de Anda, Hector Suarez, Jorge Russek, Bruno Rey, Silvia Manriquez, Yolanda Lievana, Carlos Lopez Moctezuma, Rebecca Iturbide, Jose L. Murillo, Ricardo Carrion, Gerardo Zepeda. A cowboy and his pal attempt to rescue a boy who has been kidnapped by his prospector father's partner. Satisfactory Mexican Western. Video title: _**Brawlers**_.\n\n**584** _ **The Bushwackers**_ **** Realart, 1952. 70 min. D-SC: Rod Amateau and Thomas Gries. With John Ireland, Wayne Morris, Lawrence Tierney, Dorothy Malone, Lon Chaney, Myrna Dell, Frank Marlowe, Bill Holmes, Jack Elam, Bob Wood, Charles Trowbridge, Stuart Randall, George Lynn, Norman Leavitt, Eddie Parks, Ted Jordan, Kit Guard. At the close of the Civil War a soldier returns home to Missouri to find a ruthless man and his daughter have gained control of the area. Cheaply done Jack Broder production helped by its fine cast.\n\n**585** _ **Butch and Sundance: The Early Years**_ **** 20th Century\u2013Fox, 1979. 111 min. Color. D: Richard Lester. SC: Allan Burns. With Tom Berenger, William Katt, Jeff Corey, John Schuck, Michael C. Gwynne, Peter Weller, Brian Dennehy, Jill Eikenberry, Chris (Christopher) Lloyd, Joel Fluellen, Regina Baff, Peter Brocco, Vincent Schiavelli, Hugh Gillin, Sherril Lynn Katzman, Jack Riley, Charles Knapp, John Megna, Frrank Doubleday, John Mark Robinson, Shay Duffin, Noble Willingham, Elya Baskin, Carol Ann Williams, Paul Price, Paul Michael Plunkett, Patrick Stewart, Reg Parton, Ben Zeller, Arthur Hill. The story of how Butch Cassidy and the Sundance Kid met and teamed up as outlaws. Average.\n\n**586** _ **Butch Cassidy and the Sundance Kid**_ **** 20th Century\u2013Fox, 1969. 110 min. Color. D: George Roy Hill. SC: William Goldman. With Paul Newman, Robert Redford, Katharine Ross, Strother Martin, Henry Jones, Jeff Corey, George Furth, Cloris Leachman, Ted Cassidy, Kenneth Mars, Donnelly Rhodes, Jody Gilbert, Timothy Scott, Don Keefer, Nelson Olmstead, Paul Bryar, Charles Akins, Eric Sinclair, Percy Helton. The saga of outlaws Butch Cassidy and Harry \"The Sundance Kid\" Lonbaugh, along with the Kid's girlfriend Etta Place, including their various robberies, a spree in New York City and a trip to Bolivia, where they are chased by law enforcers. Glamorized account of the two law breakers that was very popular when released and still holds up as good entertainment.\n\n**587** _ **Butterfly**_ **** Analysis Film Releasing, 1981. 108 min. Color. D: Matt Cimber. SC: John Goff and Matt Cimber. With Pia Zadora, Stacy Keach, James Franciscus, Edward Albert, Orson Welles, Lois Nettleton, Stuart Whitman, June Lockhart, Ed McMahon, Paul Hampton, George \"Buck\" Flower, Ann Dane, Guy Gault, John O'Connor White, Peter Jason, John Goff. In 1937 the caretaker of a closed Nevada silver mine lusts for a teenage girl he believes is his daughter. Mediocre adaptation of James M. Cain's sizzling novel; Orson Welles' self-indulgent performance as a local judge has to be seen to be believed. About the only interest for genre fans is the use of Johnny Bond's recording of \"Silver on the Sage.\n\n**588** _ **Buzzy and the Phantom Pinto**_ **** Ellkay, 1941. 55 min. D: Richard C. Kahn. SC: E.C. Robertson. With Buzzy Henry, Dave O'Brien, Dorothy Short, George Morrell, Sven Hugo Borg, Milburn Morante, Frank Marlo, Harry Norman, Don Kelly, Philip Arnold. A young cowpoke and a ranch foreman try to capture an elusive horse. Second and last film in the Buzzy Henry starring series; a mediocre affair. Also called _**Phantom Pinto**_ and reissued in 1948 by Astor as _**Western Terror**_.\n\n**589** _ **Buzzy Rides the Range**_ **** Ellkay, 1940. 60 min. D: Richard C. Kahn. SC: E.C. Robertson. With Buzzy Henry, Dave O'Brien, Claire Rochelle, George Morrell, George Eldredge, Frank Marlo, Don Kelly. A young boy and a range detective team to track down outlaws. So-so independent feature intended as the first of a series to star juvenile Robert \"Buzzy\" (later Buzz) Henry but followed only by _**Buzzy and the Phantom Pinto**_ (q.v.).\n\n**590** _ **By Dawn's Early Light**_ **** Showtime, 2000. 100 min. Color. D: Arthur Alan Seidelman. SC: Jacqueline Feather and David Seidler. With Richard Crenna, David Carradine, Chris Olivero, Patrick David, Gary Bisiq, Ben Cardinal, Tim Henry, Stella Stevens, Blair Slater, Lachian Murdoch, Sandra Nelson, Greg Kean, Don MacKay, Peter Raffan, Lisa Marie Caruk, Lulie Patzwald, Tyler Labine, David Coles, Caley Wilson, Frank C. Turner, Colin Foo, Anthony Harrison, Mark Holden, Joanna Piros. When a teenager wants to go home after being sent to spend the summer with his grandfather in Colorado, the old cowboy has them make the trip on horseback. Refreshing modern-day TV movie.\n\n_**By Whose Hand?**_ see _**Rustlers of the Bandlands**_\n\n**591** _ **La Cabeza de Pancho Villa**_ (The Head of Pancho Villa) **** Clasa-Mohme, 1957. 94 min. D: Chano Urueta. SC: Ramon Obon. With Luis Aguilar, Flor Silvestre, Jaime Fernandez, Fernando Oses, Carlos Suarez, Pascual Garcia Pena, Guillermo Cramer, Salvador Godinez, Francisco Reiguera, Alberto Pedret, Eduardo Bonada, Elvira Lodi, Ennedina Diaz de Leon, Antonio Sandoval. A singing cowboy and his sidekick find themselves at odds with a mysterious cult that worships the head of Pancho Villa. Atmospheric, diverting Mexican horror Western.\n\n_**Cactus Barrier**_ see _**Border Fence**_\n\n**592** _ **The Cactus Kid**_ **** Reliable, 1935. 56 min. D: Harry S. Webb. SC: Carl Krusada. With Jack Perrin, Jayne Regan, Slim Whitaker, Tom London, Fred Humes, Wally Wales, Philo McCullough, Joe De La Cruz, Tina Menard, Kit Guard, Lew Meehan, George Chesebro, Gordon DeMain, George Morrell. When his partner is murdered a cowboy plots revenge. Poorly done later Jack Perrin vehicle.\n\n**593** _ **Cactus Trails**_ **** Aywon, 1925. 55 min. D: Harry S. Webb. With Jack Perrin, Alma Rayford, Nelson McDowell, Wilbur McGaugh, Barney Furey, Martin Turner, Floyd Ames, Bob McFarland, Chris-Pin Martin. Returning home from World War I, a cowpoke rescues a girl during a runaway and later is broken out of jail for a crime he did not commit by her father before saving her from kidnappers. Average Jack Perrin silent film highlighting his beautiful steed Starlight.\n\n**594** _ **Cahill, United States Marshal**_ Warner Bros., 1973. 103 min. Color. D: Andrew V. McLaglen. SC: Harry Julian Fink and Rita M. Fink. With John Wayne, George Kennedy, Gary Grimes, Neville Brand, Clay O'Brien, Marie Windsor, Morgan Paull, Dan Vadis, Royal Dano, Scott Walker, Denver Pyle, Jackie Coogan, Rayford Barnes, Dan Kemp, Harry Carey, Jr., Walter Barnes, Paul Fix, Pepper Martin, Vance Davis, Chuck Roberson, Ken Wolger, Hank Worden, James Nusser, Murray MacLeod, Hunter Von Leer. A dedicated U.S. marshal neglects his two sons in deference to duty and in going after a robbery gang he learns his boys are mixed up in the crime. Not one of John Wayne's better outings but still good entertainment.\n\n_**Cain's Cutthroats**_ see _**Cain's Way**_\n\n**John Wayne in** _**Cahill, United States Marshal**_ **(Warner Bros., 1973).**\n\n**  \n**\n\n**595** _ **Cain's Way**_ **** M.D.A. Associates, 1970. 95 min. Color. D: Kent Osborne. SC: Wilton Denmark. With Scott Brady, John Carradine, Robert Dix, Don Epperson, Adair Jamison, Darwin Jaston, Bruce Kimball, Teresa Shaw, Willis Martin. Seven bikers in a modern Western town are transported back to the frontier of the 1870s. Overly violent and cheaply made production combining the Western and fantasy genres, none-too-successfully.\n\n**596** _ **Calaboose**_ **** United Artists, 1943. 45 min. D: Hal Roach, Jr. SC: Arnold Belgard. With Jimmy Rogers, Noah Beery, Jr., Mary Brian, Marc Lawrence, Bill Henry, Paul Hurst, William B. Davidson, Jean Porter, Iris Adrian, Sarah Edwards. Two cowboys come to the aid of a sheriff and his daughter by opposing a big city gangster. One of a brief series of short features starring Jimmy Rogers (Will's son) and Noah Beery, Jr. Average.\n\n**597** _ **Calamity Jane**_ **** Warner Bros., 1953. 101 min. Color. D: David Butler. SC: James O'Hanlon. With Doris Day, Howard Keel, Philip Carey, Allyn Ann McLerie, Dick Wesson, Paul Harvey, Chubby Johnson, Gale Robbins, Francis McDonald, Monte Montague, Forrest Taylor, Zon Murray, Kenne Duncan, Lane Chandler, Edmund Cobb, Jack Perrin, Rex Lease, Buddy Roosevelt, Robert Fuller, Terry Frost, Reed Howes, I. Stanford Jolley, Franklyn Farnum, Donald Kerr, Emmett Lynn, Gene Roth, Glenn Strange, Stanley Blystone, Budd Buster, Billy Bletcher, Jack Mower, Herman Hack, Clem Fuller, Pierce Lyden, Frank Mills, Jack Kinney, Ray Jones, Kermit Maynard, Denver Dixon, Bill Hale, Ethan Laidlaw, Lee Shumway, Burt Mustin, Sailor Vincent, Harry Wilson, Tom Smith, Major Sam Harris, Lee Morgan, Augie Gomez, Kansas Moehring, Jack Montgomery, Tom Monroe, Bess Flowers. The wildest sharp shooting gal in the West decides to tame the heart of Wild Bill Hickok. Musical hokum makes for fun viewing and there's the song \"Secret Love\" too.\n\n**598** _ **Calamity Jane**_ **** CBS-TV, 1984. 104 min. Color. D: James Goldstone. SC: Suzanne Clauser. With Jane Alexander, Frederic Forrest, Ken Kercheval, Walter Olkewicz, Talia Balsam, Walter Scott, David Hemmings, Isabell Monk, Jack Murdock, Larry Cedar, Doug Toby, Laurie O'Brien, Sara Abeles, Gillian Eaton, Don Hepner, Jessica Nelson, Henry M. Kenrick, Gloria Henry, Mavis Neal Palmer, Theresa DePaolo. In the 1870s wild west gal Calamity Jane gets involved with Wild Bill Hickok and Buffalo Bill Cody as she tries to make an independent life for herself. A mediocre TV movie about the Western legend from a feminist viewpoint.\n\n**599** _ **Calamity Jane and Sam Bass**_ **** Universal-International, 1949. 85 min. Color. D: George Sherman. SC: Maurice Geraghty. With Yvonne De Carlo, Howard Duff, Dorothy Hart, Willard Parker, Norman Lloyd, Marc Lawrence, Houseley Stevenson, Milburn Stone, Clifton Young, John Rodney, Roy Roberts, Ann Doran, Charles Cane, Walter Baldwin, Paul Maxey, George Carleton, Harry Harvey, Jack Ingram, Francis McDonald, Douglas Walton, Nedrick Young, Russ Conway, Pierce Lyden, I. Stanford Jolley, Stanley Blystone, Roy Butler, Frank McCarroll, Bob Perry. When his prize horse is killed by crooks, Sam Bass takes to crime and meets Calamity Jane, but he prefers a sheriff's sister. Surprisingly good film, although mostly fiction, with Yvonne De Carlo and Howard Duff in top form as the leads.\n\n**600** _ **Calgary Stampede**_ **** Universal, 1925. 60 min. D: Herbert Blanche. SC: Raymond L. Schrock, Donald W. Lee and E. Richard Schayer. With Hoot Gibson, Virginia Brown Faire, Clark Comstock, Ynez Seabury, Jim Corey, Philo McCullough, W.T. McCulley, Ena Gregory, Charles Sellon, Tex Young, Bob Gillis. A rodeo rider goes to Canada and falls in love with a girl but has to take it on the lam when her father, who objected to their romance, is murdered by a poacher who puts the blame on the cowboy. Calgary Stampede rodeo scenes are the highlight of this otherwise standard Hoot Gibson film.\n\n**601** _ **Calibre 44**_ (Caliber 44) **** Producciones Sotomayer, 1960. 95 min. Color. D: Julian Soler. SC: Jose Maria Fernandez Unsain. With Pedro Armendariz, Rosita Quintana, Jaime Fernandez, Rodolfo Landa, Lalo Gonzalez Piporro, Amando Soto La Marina \"Chicote,\" Carlos Muzquiz, Jose Eduardo Perez, Guillermo Cramer, Caroline Barrett, Manuel Donde, Guillermo Hernandez. Two rival gunman each adopt a victim's twin boys who grow up to try and stop the feud between them. Interesting Mexican Western.\n\n_**The Calico Queen**_ see _**The Hanging of Jake Ellis**_\n\n**602** _ **California**_ **** Paramount, 1946. 97 min. Color. D: John Farrow. SC: Frank Butler and Theodore Strauss. With Ray Milland, Barbara Stanwyck, Barry Fitzgerald, George Coulouris, Albert Dekker, Anthony Quinn, Frank Faylen, Gavin Muir, James Burke, Eduardo Ciannelli, Roman Bohnen, Argentina Brunetti, Howard Freeman, Julia Faye, Crane Whitely, Lane Chandler, Will Wright, Francis Ford, Stanley Andrews, Don Beddoe, Si Jenks, Jeff Corey, Stanley Blystone, Martin Garralaga, Tom Chatterton, Ralph Dunn, Russ Clark, William Hall, Tommy Tucker, Pedro Regas, John Sheehan, Eddy Chandler, Frances Morris, Virginia Farmer, Minerva Urecal, Sam Flint, Harry Hayden, Ian Wolfe, Kathryn Sheldon, Ethan Laidlaw, Gertrude W. Hoffman, Alan Bridge, Bud Geary, Dick Wessell, Tom Fadden, Rex Lease, Guy Wilkerson, Frank Hagney, George Magrill, Lester Dorr, Al Ferguson, Phil Dunham, Philip Van Zandt, Harry Cording, George Lloyd, Jack Clifford, Perc Launders, Lee Phelps, Kernan Cripps, Clancy Cooper, Frank Ferguson, Darby Jones, LeRoy Edwards, Betty Farrington, Joey Ray. During the Gold Rush a wagon master with a past and a shady lady get involved in a scheme by crooks to keep California from attaining statehood. Fans of the two stars will like this one, others beware; the color helps.\n\n**603** _ **California**_ **** American International, 1963. 86 min. D: Hamil Petroff. SC: James West. With Jock Mahoney, Faith Domergue, Michael Pate, Susan Seaforth, Rodolfo Hoyos, Penny Santon, Nestor Paiva, Felix Locher, Charles Horvath. In 1841 the people of California revolt against Mexican oppression and ask the U.S. for statehood. Fast paced, but cheaply made, vehicle for Jock Mahoney and Faith Domergue.\n\n**604** _ **California Conquest**_ **** Columbia, 1952. 79 min. Color. D: Lew Landers. SC: Robert E. Kent. With Cornel Wilde, Teresa Wright, Alfonso Bedoya, Lisa Ferraday, Eugene Iglesias, John Dehner, Ivan Lebedeff, Tito Renaldo, Renzo Cesana, Baynes Barron, Rico Alaniz, Alex Montoya, Hank Patterson, George Eldredge. A young nobleman aids the Spanish government when Russia tries to lay claim to California. Producer Sam Katzman's mild account of a little known aspect of California history; originally conceived as a \"Zorro\" movie.\n\n**605** _ **California Firebrand**_ **** Republic, 1948. 63 min. Color. D: Philip Ford. SC: J. Benton Cheney and John K. Butler. With Monte Hale, Adrian Booth, Paul Hurst, Tristram Coffin, Foy Willing and The Riders of the Purple Sage, Alice Tyrell, Douglas Evans, LeRoy Mason, Sarah Edwards, Dan Sheridan, Duke York, Lanny Rees, Glenn Strange. Disguised as a notorious outlaw, a cowboy investigates a series of mining claim thefts. Adventuresome Monte Hale opus, enhanced by Trucolor.\n\n**606** _ **California Frontier**_ **** Columbia, 1938. 55 min. D: Elmer Clifton. SC: Monroe Schaff and Arthur Hoerl. With Buck Jones, Carmen Bailey, Milburn Stone, Jose Perez, Soledad Jiminez, Stanley Blystone, Carlos Villarias, Glenn Strange, Paul Ellis, Ernie Adams, Forrest Taylor, Tom London, Frank Ellis, Herman Hack, Bob Terry, Carl Mathews, Chick Hannon, Tex Phelps, Ray Jones, James Morton, Billy Bletcher, Tom Smith. When Mexican ranchers are forced off their land by crooks an Army captain is sent to California to stop the injustice. Buck Jones' final Columbia series Western is an okay effort but not up to the standards of some of his earlier features for the company.\n\n**607** _ **California Gold Rush**_ **** Republic, 1946. 56 min. D: R.G. Springsteen. SC: Bob Williams. With Wild Bill Elliott, Bobby Blake, Alice Fleming, Peggy Stewart, Russell Simpson, Dick Curtis, Kenne Duncan, Monte Hale, Tom London, Joel Friedkin, Wen Wright, Jack Kirk, Budd Buster, Bud Osborne, Neal Hart, Frank Ellis, Herman Hack, Dickie Dillon, Mary Arden, Nolan Leary, Freddie Chapman, Post Park, Pascale Perry, Henry Wills, Frances Gladwin, Dorothy Stevens, Marian Kerrigan, Beverly Reedy, Roy Bucko, Jess Cavin, Kansas Moehring, James Mitchell. In order to save a stage line from outlaws, Red Ryder pretends to be a killer called the Idaho Kid. Pretty good \"Red Ryder\" series film.\n\n**608** _ **California Gold Rush**_ **** NBC-TV, 1981. 100 min. Color. D: Jack B. Hively. SC: Tom Chapman and Roy London. With Robert Hays, John Dehner, Henry Jones, Ken Curtis, Gene Evans, Victor Mohica, Coleman Creel, Cliff Osmond. Writer Bret Harte comes West in 1849 and becomes involved with Captain John Sutter and the gold hunt frenzy. Interesting \"Classics Illustrated\" TV movie based on Harte's \"The Luck of Roaring Camp\" and \"The Outcasts of Poker Flats.\"\n\n_**California in 1878**_ see _**Fighting Thru, or California in 1878**_\n\n**609 California in '49** Arrow, 1925. 60 min. D: Jacques Jaccard. SC: Karl Coolidge. With Edmund Cobb, Neva Gerber, Charles Brinley, Ruth Royce, Wilbur McGaugh, Yakima Canutt, Clark Coffey. Captain John Sutter and his friends plan to build an empire in California and after aiding the snowbound Donner party he helps settlers revolting against the Mexican government. Fast moving silent historical drama, taken from the 1924 Arrow serial _**Days of '49**_.\n\n**610** _ **California Joe**_ **** Republic, 1943. 55 min. D: Spencer Gordon Bennet. SC: Norman S. Hall. With Don \"Red\" Barry, Helen Talbot, Wally Vernon, Terry Frost, Twinkle Watts, Edward Earle, LeRoy Mason, Charles King, Pierce Lyden, Edmund Cobb, Karl Hackett, Robert Kortman, Edward Keane, Tom London, Jack O'Shea, Robert Wilke, Jack Kirk, Ernest Hillard, Foxy Callahan, Bob Burns, Lee Morgan, Larry Steers. During the Civil War corrupt politicians plan to make California a separate empire as a Union solider tries to thwart them. Typically speedy Don \"Red\" Barry feature.\n\n**611** _ **California Mail**_ **** Warner Bros., 1936. 56 min. D: Noel Smith. SC: Harold Buckley and Roy Chanslor. With Dick Foran, Linda Perry, Edmund Cobb, Glenn Strange, Bob Woodward, Wilfred Lucas, Jack Kirk, Lew Meehan, Tex Palmer, The Sons of the Pioneers (Bob Nolan, Len Slye [Roy Rogers], Tim Spencer, Hugh Farr, Karl Farr), Milton Kibbee, Tom Brower, James Farley, Edward Keane, Ben Hendricks, Cliff Saum, Gene Alsace, Fred Burns, Smoke the Wonder Horse. Crooks try to obtain a mail contract and when three stagecoach lines have the same bids a race is staged to decide the winner. Well done Dick Foran vehicle with songs by the star and The Sons of the Pioneers, including Roy Rogers.\n\n_**California Outpost**_ see _**Old Los Angeles**_\n\n**612** _ **California Passage**_ **** Republic, 1950. 90 min. D: Joseph Kane. SC: James Edward Grant. With Forrest Tucker, Adele Mara, Estelita Rodriguez, Jim Davis, Bill Williams, Paul Fix, Rhys Williams, Francis McDonald, Eddy Waller, Peter Miles, Charles Kemper, Charles Stevens, Iron Eyes Cody, Alan Bridge, Ruth Brennan, Hal Taliaferro, Marshall Reed, I. Stanford Jolley, Rory Mallinson, Frank Richards. A woman falls for the saloon owner who accidentally killed her brother, despite his being accused of a stagecoach robbery actually committed by his dishonest partner. Interesting \"A\" budget affair with Forrest Tucker as a good guy for a change.\n\n**613** _ **The California Trail**_ **** Columbia, 1933. 67 min. D: Lambert Hillyer. SC: Jack Natteford. With Buck Jones, Helen Mack, Emile Chautard, George Humbert, Charles Stevens, Evelyn Sherman, Chris-Pin Martin, Carmen LaRoux, Carlos Villarias, Augie Gomez, John Paul Jones, Allan Garcia, Robert Steele, Juan DuVal. An American scout comes to the aid of a village in Old Mexico that is ruled by two ruthless brothers out to take the locals' land. Well made Buck Jones vehicle highlighted by a good story and direction.\n\n**614** _ **The Californian**_ **** 20th Century\u2013Fox, 1937. 61 min. D: Gus Meins. SC: Gilbert Wright. With Ricardo Cortez, Marjorie Weaver, Katherine De Mille, Maurice Black, Morgan Wallace, Nigel de Brulier, Ann Gillis, Helen Holmes, James Farley, George Regas, Pierre Watkin, Edward Keane, Gene Reynolds, Richard Botiller, Tom Forman, Bud Osborne, Monte Montague, William Fletcher. Sent to Spain by his father to become a gentleman, a young man returns home to find the area plagued by crooks. Fair programmer adaptation of the Zane Grey work. TV title: _**Gentleman from California**_.\n**615** _ **Call of the Canyon**_ **** Republic, 1942. 71 min. D: Joseph Santley. SC: Olive Cooper. With Gene Autry, Smiley Burnette, Ruth Terry, Joe Strauch, Jr., Thurston Hall, Cliff Nazarro, Dorothea Kent, Bob Nolan and The Sons of the Pioneers (Tim Spencer, Lloyd Perryman, Pat Brady, Hugh Farr, Karl Farr), Edmund MacDonald, Marc Lawrence, John Holland, Eddy Waller, Budd Buster, Frank Jaquet, Lorin Baker, Johnny Duncan, Ray Bennett, Anthony Marsh, Fred Santley, Frank Ward, Earle Hodgins, John Harmon, Al Taylor, Frankie Marvin, Bob Burns, Charles Williams, Joy Barton. While in the big city trying to get fair prices for ranchers from a meat packer, Gene Autry gets involved with a pretty radio singer who ends up renting his ranch from his pal Frog. Very well done Gene Autry musical oater.\n\n**616** _ **Call of the Coyote**_ **** Imperial, 1934. 50 min. D: Patrick (Pat) Carlyle. SC: Robert Emmett (Tansey). With Ken Thompson, Pat Carlyle, Sally (Darling) Dolling, Merrill McCormick, Baby Marie Bracco, Charles Stevens, Barthlett (Bartlett\/Bart) Carre, Morgan Galloway, Wallace Sheperd, Jack Pollard, Howard Fossett, Jack Evans. A gold mine owner is murdered by his partner leaving an orphaned little girl and a map which is discovered by a caballero who vows revenge for the killing. Rock bottom cinema from a story by director-star Pat Carlyle with Merrill McCormick playing both the villain and his victim and top billed Ken Thompson apparently invisible; some video prints run 44 minutes but are no less painful.\n\n**617** _ **Call of the Desert**_ **** Syndicate, 1930. 55 min. D: J.P. McGowan. SC: Sally Winters and Barney Williams. With Tom Tyler, Sheila (Bromley) LeGay, Cliff Lyons, Bud Osborne, Bobby Dunn. A cowboy rides into the desert in search of a gold claim left by his father with his partner leaving him for dead and stealing the map. Fairly entertaining and fast moving Tom Tyler vehicle with a scenic opening of snow in the desert. Issued with a music score but no dialogue.\n\n**618** _ **Call of the Forest**_ **** Lippert, 1949. 74 min. D: John F. Link. SC: Craig Burns. With Robert Lowery, Ken Curtis, Chief Thundercloud, Martha Sherrill, Charles Hughes, Tom Hanley, Fred Gildart, Eula Guy, Black Diamond (horse), Jimmy (crow), Ready (raccoon), Ripple (deer), Fuzzy (bear). The adventures of a young boy and the animals he befriends in the great north woods. Pleasant youth oriented outing.\n\n**619** _ **The Call of the Klondike**_ **** Rayart, 1926. 45 min. D: Oscar Apfel. SC: John F. (Jack) Natteford. With Gaston Glass, Dorothy Dwan, Earl Metcalfe, Sam Allen, William Lowery, Olin Francis, Harold Holland, Jimmy Aubrey, Lightning Girl (dog). A young woman and her father rescue a mining engineer in the Klondike and later in Alaska the man, now a miner accused of murdering his partner, escapes from jail to save her from the advances of a crook. Okay silent melodrama that originally ran one hour but only survives in a truncated version.\n\n**620** _ **Call of the Klondike**_ **** Monogram, 1950. 67 min. D: Frank McDonald. SC: Charles Lang. With Kirby Grant, Anne Gwynne, Lynne Roberts, Tom Neal, Russell Simpson, Paul Bryar, Duke York, Pat Gleason, Marc Krah, Chinook (dog). A Mountie and a woman search for the latter's missing father and when they find his gold mine they are attacked by crooks. One of the better efforts in the Kirby Grant series based on the stories of James Oliver Curwood.\n\n**621** _ **Call of the Prairie**_ **** Paramount, 1936. 63 min. D: Howard Bretherton. SC: Doris Schroeder and Vernon Smith. With William Boyd, James Ellison, Muriel Evans, George Hayes, Chester Conklin, Alan Bridge, Hank Mann, Willie Fung, Howard Lang, Al Hill, John Merton, Jim Mason, Chill Wills and His Avalon Boys, John St. Polis, Bob McKenzie, Tom London, Pascale Perry, Denver Dixon. An outlaw gang frames Johnny Nelson for a series of crimes but Hopalong Cassidy comes to his rescue. Fine entry in the \"Hopalong Cassidy\" series.\n\n**622** _ **Call of the Rockies**_ **** Syndicate, 1931. 62 min. D: Raymond K. Johnson. With Ben Lyon, Marie Prevost, Gladys Johnston, Anders Randolf, Russell Simpson, James (Jim) Mason, Tex Driscoll, The Four Hawks. Three crooks, including a beautiful girl, try to swindle pioneers who plan to settle a secluded valley but the woman falls in love with one of the homesteaders. Financed by the Mormon church and filmed as a silent in Utah in 1928, this mundane feature did not see theatrical release until 1931 when, with a talking prologue relating the story added on, it was previewed as _**All Faces West**_ and also called _**West of the Rockies**_.\n\n**623** _ **Call of the Rockies**_ **** Columbia, 1938. 54 min. D: Alan James. SC: Ed Earl Repp. With Charles Starrett, Iris Meredith, Donald Grayson, The Sons of the Pioneers (Bob Nolan, Tim Spencer, Lloyd Perryman, Hugh Farr, Karl Farr), Dick Curtis, Edward Le Saint, Edmund Cobb, Art Mix, John Tyrrell, George Chesebro, Alan Bridge, Glenn Strange, Jack Rockwell, Franklyn Farnum, Fred Burns, Hank Bell. A cowboy helps a young woman who is in debt and about to lose her ranch to a dishonest land dealer. Entertaining Charles Starrett series Western.\n\n**624** _ **Call of the Rockies**_ **** Republic, 1944. 58 min. D: Lesley Selander. SC: Bob Williams. With Smiley Burnette, Sunset Carson, Ellen Hall, Kirk Alyn, Harry Woods, Frank Jaquet, Charles Williams, Jack Kirk, Tom London, Robert Kortman, Edmund Cobb, Jack O'Shea, Rex Lease, Frank McCarroll, Bud Geary, Robert Wilke, Kit Guard, Carl Sepulveda, Horace B. Carpenter. Two freight haulers lose their cargo and learn a mine owner and a doctor are in cahoots in trying to control all the area lodes. Sunset Carson's first starring film (he is second billed behind Smiley Burnette) is a fast paced rip-snorter and what the star lacks in the thespian department he more than makes up for in his ability to fight and ride.\n\n**625** _ **Call of the West**_ **** Columbia, 1930. 70 min. D: Albert Ray. SC: Colin Clements. With Dorothy Revier, Matt Moore, Catherine Clark Ward, Tom O'Brien, Alan Roscoe, Victor Potel, Nick De Ruiz, Joe De La Cruz, Blanche Rose, Bud Osborne. While in Texas a cabaret entertainer marries a rancher but when he joins a posse to track down rustlers she returns to New York City and is romanced by a former suitor. Trite early talkie.\n\n**626** _ **Call of the Wild**_ **** United Artists, 1935. 91 min. D: William A. Wellman. SC: Gene Fowler and Leonard Praskins. With Clark Gable, Loretta Young, Jack Oakie, Reginald Owen, Frank Conroy, Sidney Toler, Charles Stevens, Katherine De Mille, James Burke, John T. Murray, Bob Perry, Sid Grauman, Herman Bing, Wade Boteler, John Ince, Syd Saylor, Joan Woodbury, Arthur Aylesworth, Lalo Encinas, Tommy Jackson, Russ Powell, George MacQuarrie, Frank Whitson, Tyler Brooke, Arthur Housman, Marie Wells, LeRoy Mason, Frank Campeau, Perry Ivins, Walter McGrail, Frank Moran, John Ince, Pat Flaherty, Larry McGrath, Jack Stoney, Helene Chadwick, Mary MacLaren, Ted Lorch, Bud Osborne, Frank Mills, Harry Wood, Buck (dog). Two prospectors search for gold in the frozen Klondike with one finding love and both being threatened by a vicious claim jumper. Likable version of the Jack London story with more romance than London; Reginald Owen is a delight as the bad guy.\n\n**627** _ **Call of the Wild**_ **** Intercontinental Releasing Corporation, 1973. 102 min. Color. D: Ken Annakin. SC: Hubert Frank and Tibor Reves. With Charlton Heston, Michele Mercier, Maria Rohm, George Eastman, Raymond Harmstorf, Friedhelm Lehmann, Horst Heuck, Sancho Garcia. An adventurer and his dog travel over the north most parts of the Pacific Ocean, from the waters of Alaska to the gold fields of the Yukon, where they see men driven by greed for wealth. West German production of Jack London's novel, filmed in Finland and somewhat weakened by its handling of the material. Issued in West Germany in 1973 as _**Ruf Der Wildnis**_ (Call of the Wild) by CCC Filmkunst.\n\n**628** _ **Call of the Wild**_ **** NBC-TV, 1976. 100 min. Color. D: Jerry Jameson. SC: James Dickey. With John Beck, Bernard Fresson, John McLiam, Michael Pataki, Penelope Windust, Billy Green Bush, Johnny Tillotson, Ray Guth, Dennis Burkley. In 1903 a young prospector and a veteran trapper face the wilds of the Yukon in search of gold. Pretty good telefilm of the Jack London work.\n\n**629** _ **Call of the Wilderness**_ **** Associated Exhibitors, 1926. 55 min. D: Jack Nelson. SC: Van Pelt Brothers and Lon Young. With Sandow (dog), Lewis Sargent, Edna Marion, Sydney DeGrey, Albert J. Smith, Max Asher, Tom Connelly, George Harvey. A free spirited man and his dog settle in a small Western town where he is at odds with a prospector after buying land from an agent with a pretty daughter. Lethargic modern-day Western headlining long forgotten Rin Tin Tin rival Sandow.\n\n_**Call of the Wilderness**_ (1932) see _**Trailing the Killer**_\n\n**630** _ **Call of the Yukon**_ **** Republic, 1938. 70 min. D: B. Reeves Eason. SC: Gertrude Orr and William Bartlett. With Richard Arlen, Beverly Roberts, Lyle Talbot, Ray Mala, Garry Owen, Ivan Miller, James Lono, Emory Parnell, Al St. John, Anthony Hughes, Nina Campana, Buck (dog). A trapper and a woman writer, along with two canines, search for gold and a story in the wilds of the Yukon. Fair adventure yarn with many animals and scenic avalanche footage.\n\n**631** _ **Call the Mesquiteers**_ **** Republic, 1938. 55 min. D: John English. SC: Luci Ward. With Robert Livingston, Ray Corrigan, Max Terhune, Lynne Roberts, Sammy McKim, Earle Hodgins, Eddy Waller, Maston Williams, Eddie Hart, Pat Gleason, Roger Williams, Warren Jackson, Hal Price, Frank Ellis, Curley Dresden, Jack Ingram, Ralph Peters, Ethan Laidlaw, Tom Steele, Al Taylor, Jim Corey, Bob Burns, Bob Card, Francis Walker, Loren Riebe, Flash (dog). Mistaken for train robbers, the Three Mesquiteers get involved with a medicine show operator and his two offspring in a ghost town as they try to clear themselves and expose silk smugglers. Slick, speedy entry in \"The Three Mesquiteers\" series.\n\n_**Call to Glory**_ see _**Ride to Glory**_\n\n**632** _ **Callaway Went Thataway**_ **** Metro-Goldwyn-Mayer, 1951. 91 min. D-SC: Norman Panama and Melvin Frank. With Fred MacMurray, Dorothy McGuire, Howard Keel, Jesse White, Fay Roope, Natalie Schaefer, Douglas Kennedy, Elizabeth Fraser, Johnny Indrisano, Stan Freberg, Don Haggerty, Dorothy Andre, Glenn Strange, Mae Clarke, Hugh Beaumont, Earle Hodgins, Douglas Fowley, Ethan Laidlaw, Emmett Lynn, Rocky Camron, Ned Glass, Paul Bryar, Dorothy Andre, Billy Dix, John Banner, Carl Sepulveda, Clark Gable, Elizabeth Taylor, Esther Williams. Two advertising agents resurrect the old films of a forgotten cowboy star, making him popular again on television but when he cannot be found a double is used to make more movies. A delightful spoof of the \"Hopalong Cassidy\" craze with Howard Keel as both a drunken, woman chasing cowboy star and his real life double. Well worth seeing.\n\n**633** _ **The Calling of Dan Matthews**_ **** Columbia, 1936. 63 min. D: Phil Rosen. SC: Dan Jarrett, Don Swift and Karl Brown. With Richard Arlen, Charlotte Wynters, Douglass Dumbrille, Mary Kornman, Donald Cook, Carlyle Blackwell, Jr., Frederick Burton, Lee Moran, Tommy Dugan, Edward McWade. A militant clergyman fights corruption and gangsters in the modern-day West. Fairly good programmer from Harold Bell Wright's 1909 novel.\n\n**634** _ **Calling Wild Bill Elliott**_ **** Republic, 1943. 55 min. D: Spencer Gordon Bennet. SC: Anthony Coldeway. With Wild Bill Elliott, George \"Gabby\" Hayes, Anne Jeffreys, Buzz Henry, Fred Kohler, Jr., Roy Barcroft, Herbert Heyes, Eve March, Charles King, Frank Hagney, Bud Geary, Lyndon Brent, Frank McCarroll, Burr Caruth, Forbes Murray, Ted Mapes, Herman Hack, Yakima Canutt, Al Taylor, Budd Buster, George Hazel, Hank Bell, Forbes Murray, Bill Nestell, Fred Burns, Horace B. Carpenter, Rose Plummer, Foxy Callahan, Lew Morphy, Jack Evans, Roy Bucko. In order to catch a corrupt a cattle baron, Wild Bill Elliott poses as the governor of a new territory. First film in Bill Elliott's Republic series, this is a fast moving adventure, short on plot but heavy on fights, chases, etc.\n\n**635** _ **Caminos de Sangre**_ (Path of Blood) **** Clasa-Mohme, 1945. 90 min. D-SC: Rolando Aguilar. With Luis Aguilar, Amanda del Llano, Carlos Lopez Moctezuma, El Chicote (Armando Soto La Marina), Miguel Inclan, Salvador Quiroz, Lauro Benitz, Maria Gentil Arcos, Jose L. Murillo, Alicia Rodriguez. When his little sister is wounded by an outlaw gang, a cowboy sets out to bring the marauders to justice. Good Mexican adventure from prolific producer Raul de Anda, who wrote the story.\n\n**636** _ **Campbell's Kingdom**_ **** Lopert, 1958. 102 min. Color. D: Ralph Thomas. SC: Robin Estridge and Hammond Innes. With Dirk Bogarde, Stanley Baker, Michael Craig, Barbara Murray, James Robertson Justice, Athene Seyler, Robert Brown, John Laurie, Sidney James, Mary Merrall, George Murcell, Ronald Brand, Finlay Currie, Peter Illing, Stanley Maxted, Gordon Tanner, Richard McNamara. In the Canadian Rockies, a land owner is at odds with a man who wants to build a huge dam. British-made adventure, produced in Canada, which is very entertaining. Released in Great Britain in 1957 by the Rank Organization.\n\n**637** _ **Canadian Mounties vs. Atomic Invaders**_ **** Republic, 1953. 12 Chapters. D: Franklin Adreon. SC: Ronald Davidson. With Bill Henry, Susan Morrow, Arthur Space, Dale Van Sickel, Pierre Watkin, Mike Ragan, Stanley Andrews, Harry Lauter, Hank Patterson, Edmund Cobb, Gayle Kellogg, Tom Steele, Jean Wright, Bob Reeves, Fred Graham, George DeNormand, William Fawcett, Jane Wood, Joe Yrigoyen, Carey Loftin, Drew Cahill, Kenner Kemp, Duke Taylor, Duane Thorsen, Gordon Armitage, Paul Palmer, Bob Jamison, Earl Bunn, Jimmy Fawcett, David Sharpe. The Canadian Mounted Police are on the trail of a gang of foreign agents mysteriously working in the upper reaches of the country. Slow moving cliffhanger, re-edited into a 100 minute TV feature called _**Missile Base at Taniak**_.\n\n**Lobby card for** _**Canadian Mounties vs. Atomic Invaders**_ **(Republic, 1953).**\n\n**  \n**\n\n**638** _ **Canadian Pacific**_ **** 20th Century\u2013Fox, 1949. 95 min. Color. D: Edwin L. Marin. SC: Jack De Witt and Kenneth Gamet. With Randolph Scott, Jane Wyatt, Nancy Olson, J. Carrol Naish, Victor Jory, Robert Barrat, Walter Sande, Don Haggerty, Grandon Rhodes, Mary Kent, John Parrish, John Hamilton, Richard Wessel, Howard Negley, Richard Alexander. A railroad advance man becomes romantically involved with both a female doctor and a frontier gal while fighting Indians and trappers out to halt his mission. Handsome Randolph Scott epic in Cinecolor.\n\n**639** _ **The Canadians**_ **** 20th Century\u2013Fox, 1961. 85 min. Color. D-SC: Burt Kennedy. With Robert Ryan, John Dehner, Torin Thatcher, Teresa Stratas, Burt Metcalfe, John Sutton, Jack Creley, Scott Peters, Richard Alden, Michael Pate. Following the Custer massacre, the Sioux Indians move to Canada and there a trio of Mounties convince them to remain peaceful or be driven back south of the border. Fairly colorful yarn that wastes opera singer Teresa Stratas as an Indian squaw.\n\n**640** _ **Cannon**_ **** CBS-TV, 1971. 100 min. Color. D: George McGowan. SC: Ed Hume. With William Conrad, Vera Miles, J.D. Cannon, Lynda Day (George), Barry Sullivan, Keenan Wynn, Murray Hamilton, Earl Holliman, John Fiedler, Lawrence Pressman, Ross Hagen. In a modern Western town a private detective uncovers local corruption while trying to help an ex-girlfriend accused of murdering her husband. Entertaining TV movie, the pilot for the popular \"Cannon\" (CBS-TV, 1971\u201376) series.\n\n**641** _ **Canon for Cordoba**_ **** United Artists, 1970. 104 min. Color. D: Paul Wendkos. SC: Stephen Kandell. With George Peppard, Giovanna Ralli, Raf Vallone, Peter Duel, Don Gordon, Nico Minardos, John Russell, Francine York, John Larch, Charles Stainaker, John Clark, Gabrielle Tinti, Hans Meyer. An Army intelligence officer and a small group of men attempt to recover canons stolen from General Pershing's army by Mexican revolutionaries. Average action outing that attempts to imitate European Westerns of the period.\n\n**642** _ **Can't Help Singing**_ **** Universal, 1944. 90 min. Color. D: Frank Ryan. SC: Lewis R. Foster and Frank Ryan. With Deanna Durbin, Robert Paige, Akim Tamiroff, David Bruce, June Vincent, Ray Collins, Olin Howlin, Leonid Kinsky, Clara Blandick, Thomas Gomez, Andrew Tombes, George Cleveland, Olin Howlin, Edward Earle, Almira Sessions, Chester Conklin, George Eldredge, Roscoe Ates, Barbara Pepper, Ruby Dandridge, Harry Woods, Glenn Strange, Frank Hagney, Forrest Taylor, Bob McKenzie, Dennis Moore, George J. Lewis, James Bush, Max Wagner, Eddie Hart, Renie Riano, Herbert Heywood, Frank Lackteen, Nina Campana, Jay Novello, Jody Gilbert, Heinie Conklin, Jimmy Aubrey, Irving Bacon, Gertrude Astor, Virginia Sale, Jack Clifford, Art Miles, Robert Homans, Kernan Cripps, Joseph E. Bernard, Frank Darien, Fred Steele, George Lloyd, John James, Bob Perry, Frank Melton, Eddie Acuff, Phil Warren, Victor Potel, Harry Semels, Nana Bryant, Fern Emmett, William Desmond, Theodore Rand, Geneva Holt, Gladys Blake, Manuel Paris, Jim Thorpe, Iron Eyes Cody. In 1849 a young girl, over her senator father's objections, heads to California to marry an Army lieutenant but finds romance along the trail. Sprightly musical-comedy-Western sure to delight Deanna Durbin fans.\n\n**643** _ **Canyon Ambush**_ **** Monogram, 1952. 53 min. D: Lewis D. Collins. SC: Joseph Poland. With Johnny Mack Brown, Phyllis Coates, Lee Roberts, Dennis Moore, Hugh Prosser, Marshall Reed, Denver Pyle, Pierce Lyden, Carol Henry, Stanley Price, Frank Ellis, Russ Whiteman. A government agent arrives in a community to help the local sheriff and concerned citizens in combating a gang led by a masked rider. There is fair entertainment in Johnny Mack Brown's final series outing.\n\n**644** _ **Canyon City**_ **** Republic, 1943. 56 min. D: Spencer Gordon Bennet. SC: Robert Yost. With Don \"Red\" Barry, Helen Talbot, Wally Vernon, Twinkle Watts, LeRoy Mason, Pierce Lyden, Forbes Murray, Ed Peil, Sr., Eddie Gribbon, Tom London, Morgan Conway, Emmett Vogan, Stanley Andrews, Roy Barcroft, Jack Kirk, Kenne Duncan, Bud Geary, Bud Osborne, Hank Worden. The Nevada Kid helps locals in their fight with an Eastern gangster out to steal their land. Not the best of Don Barry's Westerns, especially with little Twinkle Watts along as some kind of sidekick.\n\n**645** _ **Canyon Crossroads**_ **** United Artists, 1955. 83 min. D: Alfred Werker. SC: Emmett Murphy and Leonard Heideman. With Richard Basehart, Phyllis Kirk, Stephen Elliott, Russell Collins, Charles Wagenheim, Richard Hale, Tommy Cook. A man searches for uranium in Utah while crooks plan to get the claim before he can record its location. Taut modern-day Western using helicopters more than horses.\n\n**646** _ **Canyon Hawks**_ **** Big 4, 1930. 55 min. D-SC: Alvin J. Neitz (Alan James). With Yakima Canutt, Buzz Barton, Rene Borden, Wally Wales, Robert Walker, Bob Reeves, Cliff Lyons, Bobby Dunn. A cowboy befriends a young woman and her brother, selling them land for their sheep herd, and later comes to her rescue when she is abducted by a bad man. Low grade effort that gives fans a chance to see Yakima Canutt in a talkie starring role.\n\n**647** _ **Canyon of Missing Men**_ **** Syndicate, 1930. 55 min. D: J.P. McGowan. SC: George H. Williams. With Tom Tyler, Shelia (Bromley) LeGay, Bud Osborne, Tom Forman, J.P. McGowan, Cliff Lyons, Bobby Dunn, Arden Ellis, Perry Murdock, Bill Nestell, Rube Dalroy, Jack Low. An outlaw falls for a rancher's daughter and betrays his gang when they kidnap her for ransom. Fair silent (with sound effects) Tom Tyler item with mostly outdoor shots and a slight story.\n\n**648** _ **Canyon Passage**_ **** Universal, 1946. 99 min. Color. D: Jacques Tourneur. SC: Ernest Pascal. With Dana Andrews, Susan Hayward, Brian Donlevy, Patricia Roc, Hoagy Carmichael, Ward Bond, Andy Devine, Stanley Ridges, Lloyd Bridges, Fay Holden, Victor Cutler, Tad Devine, Denny Devine, Onslow Stevens, Rose Hobart, Dorothy Peterson, Halliwell Hobbes, James Cardwell, Ray Teal, Virginia Patton, Francis McDonald, Erville Alderson, Ralph Peters, Jack Rockwell, Gene Roth, Karl Hackett, Jack Clifford, Richard Alexander, Chief Yowlachie, Wallace Scott, Peter Whitney, Harry Shannon, Chester Clute, Frank Ferguson, Eddie Dunn, Harlan Briggs, Rex Lease, Jack Ingram, Ann Burr. In 1856 a store owner-mule freight hauler and a crooked gambling banker both love the same woman in the rugged Oregon country. Beautifully produced feature showing both the glory and harshness of frontier life; a very good motion picture.\n\n**649** _ **Canyon Raiders**_ **** Monogram, 1951. 54 min. D: Lewis D. Collins. SC: Jay Gilgore. With Whip Wilson, Fuzzy Knight, Phyllis Coates, Jim Bannon, Bill Kennedy, Barbara Woodell, I. Stanford Jolley, Marshal Reed, Riley Hill, William Fawcett, Bob Woodward, Ray Jones. Two ranchers try to stop a gang that rustled 500 horses and plans to sell them to the Army with a forged bill of sale. There is enough action to carry this Whip Wilson vehicle.\n\n**650** _ **Canyon River**_ **** Allied Artists, 1956. 80 min. Color. D: Harmon Jones. SC: Daniel B. Ullman. With George Montgomery, Marcia Henderson, Peter Graves, Richard Eyer, Walter Sande, Robert Wilke, Alan Hale (Jr.), John Harmon, Jack Lambert, William Fawcett, Bud Osborne, Lee Roberts. The foreman of a cattle drive from Oregon to Wyoming has his life saved by an outlaw gang leader but when the bandits attack he is forced to fight them. Average oater helped by George Montgomery and color; a remake of _**The Longhorn**_ (q.v.).\n\n**651** _ **Captain Apache**_ **** Scotia International, 1971. 95 min. Color. D: Alexander Singer. SC: Philip Yordan and Milton Sperling. With Lee Van Cleef, Carroll Baker, Stuart Whitman, Percy Herbert, Elisa Montes, Tony Vogel, Hugh McDermott, Charles Stalnaker, Charley Bravo, Faith Clift, Dan Van Husen, D. Pollock, George Margo, Jose Bodalo. When the Indian commissioner is murdered an Apache warrior is assigned by the Army to find out to committed the crime. Tepid British produced oater with all the violence of Continental Westerns and Lee Van Cleef singing the title song. Alternate title: _**Deathwork**_.\n\n**652** _ **Captain John Smith and Pocahontas**_ **** United Artists, 1953. 75 min. Color. D: Lew Landers. SC: Aubrey Wisberg and Jack Pollexfen. With Anthony Dexter, Jody Lawrence, Alan Hale (Jr.), Robert Clarke, Stuart Randall, James Seay, Philip Van Zandt, Shepard Menken, Douglass Dumbrille, Anthony Eustral, Henry Rowland, Franchesca di Scaffa, Joan Nixon. Captain John Smith tries to establish a colony in Virginia despite being opposed by those who want to hunt for gold or use it as a base for privateers; he tries to make peace with the Indians only to be captured. Colorful, but average, romantic history.\n\n**653** _ **Captain Thunder**_ **** Warner Bros., 1930. 66 min. D: Alan Crosland. SC: Gordon Rigby and William K. Wells. With Victor Varconi, Fay Wray, Charles Judels, Don Alvarado, Robert Elliott, Natalie Moorhead, Bert Roach, Frank Campeau, Robert Emmett Keane, John St. Polis. A dashing Mexican bandit helps a young man who tries to capture him when the latter's pretty fiancee is forced to marry a rival. Static early talkie.\n\n**654** _ **The Capture**_ **** RKO Radio, 1950. 81 min. D: John Sturges. SC: Niven Busch. With Lew Ayres, Teresa Wright, Victor Jory, Duncan Renaldo, Jacqueline White, Jimmy Hunt, Barry Kelley, William Bakewell, Milton Parsons, Edwin Rand, Frank Matts, Felipe Turich, Rosa Turich, Paul Marion, Manuel Paris, Rodolfo Hoyos, Rico DeMontez, Paul Fierro, Vito Soctti, Tina Menard, Charles Morton, Pepe Hern, Rico Alaniz, Chuck Roberson, Alberto Morin, Manuel Lopez, Alex Gerry, Paul Regas, Harry Vejar, Gil Herman, Tommy Lee, Francisco Villalobos. A detective feels he may have killed the wrong man in a robbery attempt and tries to reinvestigate the case. Modern-day drama with much of its footage in rural Mexico; good viewing.\n\n**655** _ **The Capture of Bigfoot**_ **** Studio Film Corporation, 1979. 93 min. Color. D: Bill Rebane. SC: Ingrid Neumayer and Bill Rebane. With Stafford Morgan, Katherine Hopkins, Richard Kennedy, George \"Buck\" Flower, John Goff, Otis Young, John Eimerman, Randy Scott, Durwood McDonald, Greg Gault, Wally Flaherty, Nelson C. Sheppo, William Dexter, Harry Youstos, Doug Ibold, Verkina Flower, Mitzi Kress, Woody Jarvis, Bill Cannon, Mitch Irish, Jeana Tomasino, Patty Holzmann. A diverse group, including a lawman, two hunters, a forest ranger and a trapper, try to snare two Bigfoot creatures lurking around a snowy, remote community. Almost as mind numbing as the cold climate in which it was filmed.\n\n**656** _ **Capture of Billy the Kid**_ **** Republic, 1952. 54 min. D: Fred C. Brannon. SC: M. Coates Webster and Richard Wormser. With Allan \"Rocky\" Lane, Penny Edwards, Grant Withers, Clem Bevans, Roy Barcroft, Mauritz Hugo, Frank McCarroll. Outlaws after a treasure hidden by Billy the Kid are hunted by a marshal out to round them up and uncover the loot. Nicely paced Allan Lane series effort; his fans will like it.\n\n**657** _ **The Capture of Grizzly Adams**_ **** NBC-TV, 1982. 100 min. Color. D-SC: Arthur Heinemann. With Dan Haggerty, Kim Darby, Chuck Connors, Noah Beery (Jr.), Keenan Wynn, June Lockhart, Peggy Stewart, Sidney Penny, G.W. Bailey. A mountain man is framed on a fake murder charge by those who want him out of the way. Average TV movie based on the feature _**The Life and Times of Grizzly Adams**_ (q.v.) and the 1977\u201378 NBC-TV series of the same title.\n\n**658** _ **The Caravan Trail**_ **** Producers Releasing Corporation, 1946. 57 min. Color. D: Robert Emmett (Tansey). SC: Frances Kavanaugh. With Eddie Dean, Emmett Lynn, Al \"Lash\" LaRue, Jean Carlin, Robert Malcolm, Charles King, Robert Barron, Forrest Taylor, Bob Duncan, Jack O'Shea, Terry Frost, George Chesebro, Bud Osborne, Lee Roberts, Wylie Grant, Lee Bennett, Lloyd Ingraham, Herman Hack, George Morrell, Ray Jones, Cliff Parkinson. The leader of a wagon train enlists the assistance of an outlaw in stopping land grabbers who have stolen the pioneer's homesteads. Passable Eddie vehicle that helped launch Lash LaRue's series; best when Eddie Dean sings \"Wagon Wheels.\"\n\n_**El Carcel de Cananea**_ see _**Pursuit Across the Desert**_\n\n**659** _ **The Cariboo Trail**_ **** 20th Century\u2013Fox, 1950. 81 min. Color. D: Edwin L. Marin. SC: Frank Gruber. With Randolph Scott, Karin Booth, George \"Gabby\" Hayes, Bill Williams, Douglas Kennedy, Jim Davis, Dale Robertson, Mary Stuart, James Griffith, Lee Tung Foo, Anthony Hughes, Mary Kent, Ray Hyke, Jerry Root, Cliff Clark, Fred Libby, Dorothy Adams, Michael Barret, Smith Ballew, Kermit Maynard, Tom Monroe. While searching for the location of a ranch in Canada, a cattleman discovers gold. Typically good Randolph Scott feature with beautiful locations.\n\n**660** _ **Carnival Boat**_ **** RKO Path\u00e9, 1932. 62 min. D: Alfred S. Rogell. SC: James Seymour. With Bill (William) Boyd, Ginger Rogers, Hobart Bosworth, Fred Kohler, Marie Prevost, Edgar Kennedy, Harry Sweet, Charles Sellon, Eddy Chandler, Walter Percival, Jack Carlyle, Joe Smith Marba, Jim Mason, Sam Harris, Bob Perry, Larry McGrath, Hal Price, Charles Sullivan. The head of a logging camp is at odds with his son when the young man marries a carnival boat entertainer while a rival logger plots to take over his position. Engaging north woods drama enhanced by the teaming of William Boyd and Ginger Rogers, the latter being especially good as entertainer Honey.\n\n**661** _ **Carolina Cannonball**_ **** Republic, 1955. 74 min. D: Charles Lamont. SC: Barry Shipman. With Judy Canova, Andy Clyde, Ross Elliott, Sig Rumann, Leon Askin, Jack Kruschen, Frank Wilcox, Roy Barcroft, Emil Sitka. A group of bumbling foreign agents capture an atomic controlled missile but end up causing it to land on a woman's ranch. Later, but still typical, Judy Canova comedy corn-fare that will appeal to her fans.\n\n**662** _ **Carolina Moon**_ **** Republic, 1940. 65 min. D: Frank McDonald. SC: Winston Miller. With Gene Autry, Smiley Burnette, June Storey, Mary Lee, Eddy Waller, Hardie Albright, Texas Jim Lewis and His Texas Cowboys, Frank Dale, Terry Nibert, Robert Fiske, Etta McDaniel, Paul White, Fred Ritter, Ralph Sanford, Jack Kirk. Rodeo performers Gene Autry and Frog Millhouse try to help a man and his daughter who are being cheated out of their prize horse by crooks but the girl believes Gene is in league with the hoodlums. Fairly vapid Gene Autry vehicle.\n\n**663** _ **Carry on Cowboy**_ **** Anglo-Amalgamated\/Filmways, 1966. 95 min. Color. D: Gerald Thomas. SC: Talbot Rothwell. With Sidney James, Kenneth Williams, Joan Sims, Jim Dale, Percy Herbert, Angela Douglas, Davy Kaye, Bernard Bresslaw, Charles Hawtrey, Peter Butterworth, Sydney Bromley, Sally Douglas, Jon Pertwee, Edina Romay, Peter Gilmore, Garry Colleano. The evil Rumpo Kid kills the sheriff and takes over the town of Stodge City before being faced by a marshal from the British Sanitary Engineers. Typically loony segment in the British \"Carry On...\" series. Also called _**The Rumpo Kid**_.\n\n**664** _ **Carson City**_ **** Warner Bros., 1953. 87 min. Color. D: Andre De Toth. SC: Sloan Nibley and Winston Miller. With Randolph Scott, Lucille Norman, Raymond Massey, Richard Webb, James Millican, Larry Keating, George Cleveland, William Haade, Thurston Hall, Vince Barnett, Don Beddoe, Jack Woody, James Smith, Guy Tongue, Billy Vincent, Ida Moore, Sarah Edwards, Edgar Dearing, Russ Clark, Iris Adrian, Nick Thompson, Frank McCarroll, Post Park, Jack Daly, Mickey Simpson, Edmund Cobb, John Halloran, Mikel Conrad, Zon Murray, House Peters, Jr., Rory Mallinson, Ray Bennett, Karen Hale, Stanley Blystone, Stanley Andrews, Richard Reeves, George Eldredge, Charles Evans, Kenneth MacDonald, George Sherwood, Pierce Lyden, Lee O'Pace. A railroad boss is at odds with a miner and a young girl who do not want him to complete his construction job. Rugged Randolph Scott oater; well worth seeing.\n\n**665** _ **Carson City Cyclone**_ **** Republic 1943. 55 min. D: Howard Bretherton. SC: Norman S. Hall. With Don \"Red\" Barry, Lynn Merrick, Noah Beery, Emmett Lynn, Bryant Washburn, Stuart Hamblen, Roy Barcroft, Bud Osborne, Jack Kirk, Bud Geary, Curley Dresden, Reed Howes, Tom London, Frank Ellis, Horace B. Carpenter, Ed Cassidy, Tom Steele, Jack O'Shea, Frank McCarroll, Roy Brent. During a court trial a novice lawyer is accused of bribing a witness and he tries to find the real culprit. More than passable Don Barry entry with good villainy by Noah Beery.\n\n**666** _ **Carson City Kid**_ **** Republic, 1940. 57 min. D: Joseph Kane. SC: Robert Yost and Gerald Geraghty. With Roy Rogers, George \"Gabby\" Hayes, Bob Steele, Noah Berry, Jr., Pauline Moore, Francis McDonald, Hal Taliaferro, Arthur Loft, Chester Gan, Paul Hurst, George Rosener, Hank Bell, Ted Mapes, Jack Ingram, Jack Kirk, Jack Rockwell, Art Dillard, Hal Price, Yakima Canutt, Kit Guard, Curley Dresden, Oscar Gahan. In 1849 Sonora gambling house owner Jessop is sought by a bandit, The Carson City Kid, who believes he is responsible for the murder of his younger brother. Top notch Roy Rogers film dominated by Bob Steele, given special billing as the villain; it also includes the song \"Sonora Moon.\"\n\n**667** _ **Carson City Raiders**_ **** Republic, 1948. 60 min. D: Yakima Canutt. SC: Earle Snell. With Allan \"Rocky\" Lane, Eddy Waller, Beverly Jons, Frank Reicher, Hal Landon, Steve Darrell, Harold Goodwin, Dale Van Sickel, Edmund Cobb, Holly Bane, Robert Wilke, Herman Hack. A U.S. marshal tries to capture a murderer but the victim's son wants to avenge the killing himself. Another fine entry in the Allan Lane \"Famous Westerns\" series, with lots story movement from director Yakima Canutt.\n\n**668** _ **Caryl of the Mountains**_ **** Reliable, 1936. 60 min. D: Bernard B. Ray. SC: Tom Gibson. With Rin Tin Tin, Jr., Francis X. Bushman, Jr., Lois Wild(e), Joseph Swickard, Earl Dwire, Robert Walker, George Chesebro, Steve Clark, Jack Hendricks. A Mountie and his faithful German shepherd dog are on the trail of an outlaw. Low budget quickie for fans of the Royal Mounted; supposedly based on a James Oliver Curwood work.\n\n**669** _ **La Casa Colorada**_ (The Colorada House) **** Ventura, 1947. 110 min. D: Miguel Morayta. With Pedro Armendariz, Dolores Camarillo, Lidia Franco, Jorge Arriaga, Roberto Cobo, Gilberto Gonzalez, Rita Macedo, Jose Eduardo Perez, Joaquin Roche, Amanda del Llano, Ramon G. Larrea, Jose Romero, Armando Silvestre. A revolutionary fighting government forces because his parents were killed by them finds his life stabilized by a woman. Interesting Mexican Western melodrama.\n\n**670** _ **Cassidy of Bar 20**_ **** Paramount, 1938. 59 min. D: Lesley Selander. SC: Norman Houston. With William Boyd, Russell Hayden, Frank Darien, Nora Lane, Robert Fiske, John Elliott, Margaret Marquis, Carleton Young, Gertrude W. Hoffman, Gordon Hart, Ed Cassidy, John Beach, Wen Wright, Jim Toney, Charles Murphy. The Bar 20 boys go to the aid of Hoppy's ex-sweetheart and come up against a crooked landowner. Pretty fair outing in the \"Hopalong Cassidy\" series.\n\n**671** _ **Cast a Long Shadow**_ **** United Artists, 1959. 82 min. D: Thomas Carr. SC: Martin G. Goldsmith. With Audie Murphy, Terry Moore, John Dehner, James Best, Denver Pyle, Ann Doran, Robert Foulk, Rita Flynn, Wright King, Stacy Harris, Terry Frost, Rusty Wescoatt, Kermit Maynard, Ray Jones, Dale Van Sickel. Troubled by being illegitimate, a young man turns to the bottle but given the responsibility of running a ranch he begins to make something of his life. None-too-interesting Audie Murphy vehicle.\n\n**672** _ **The Castaway Cowboy**_ **** Buena Vista, 1974. 91 min. Color. D: Bernard McEveety. SC: Don Tait. With James Garner, Vera Miles, Robert Culp, Eric Shea, Shug Fisher, Elizabeth Smith, Gregory Sierra, Manu Tupou. A cowboy shipwrecked in Hawaii meets a young widow and her son whose land is sought by a crook and he proceeds to turn it into a cattle ranch. Pleasant Walt Disney affair with the Old West transferred to Hawaii.\n\n**673** _ **The Cat**_ **** Embassy, 1966. 87 min. Color. D: Ellis Kadison. SC: William Redlin and Laird Koenig. With Roger Perry, Peggy Ann Garner, Barry Coe, Dwayne Redlin, George \"Shug\" Fisher, Ted Darby, John Todd Roberts, Richard Webb, Les Bradley. A young boy looking for a wildcat witnesses a rustler murder a rancher. Appealing family film.\n\n**674** _ **Cat Ballou**_ **** Columbia, 1965. 96 min. Color. D: Elliott Silverstein. SC: Walter Newman. With Lee Marvin, Jane Fonda, Michael Callan, Dwayne Hickman, Nat \"King\" Cole, Stubby Kaye, Tom Nardini, John Marley, Reginald Denny, Jay C. Flippen, Arthur Hunnicutt, Bruce Cabot, Burt Mustin, Paul Gilbert, Harvey Clark, Oscar Blank, Ted White, Carol Veazie, Erik Sorenson. A timid schoolmarm comes West and soon becomes a wanted outlaw who teams with a drunken gunman to take on his notorious gunfighter brother. Overrated comedy does not hold up well although Lee Marvin's Oscar winning performance is worth watching.\n\n_**Cathouse Girls**_ see _**Blazing Stewardesses**_\n\n**675** _ **Catlow**_ **** Metro-Goldwyn-Mayer, 1971. 101 min. Color. D: Sam Wanamaker. SC: Scott Finch and J.J. Griffith. With Yul Brynner, Richard Crenna, Daliah Lavi, Leonard Nimoy, Jo Ann Pflug, Jeff Corey, David Ladd, Bessie Love, Michael Delano, Julian Mateos. While trying to steal two million dollars in gold from a pack train, an outlaw is forced to avoid his marshal friend and a bounty hunter, both of whom are after him. Average Western filmed in Spain.\n\n**676** _ **Cattle Annie and Little Britches**_ Universal, 1981. 95 min. Color. D: Lamont Johnson. SC: David Eyre and Robert Ward. With Burt Lancaster, Rod Steiger, John Savage, Diane Lane, Amanda Plummer, Scott Glenn, Michael Conrad, Steven Ford, Roger Cudney, Jr., John Quade, Jerry Gatlin, Yvette Sweetman, Tom Delaney, Matthew Taylor, Redmond Gleesoni, Buck Taylor, William Russ, Ken Call, Chad Hastings, Perry Lang, John Sterlini, Mike Moroff, John Hock, Russ Hoverson. Two feisty young girls track down the remnants of the once notorious Doolin-Dalton gang and urge them to return to lawlessness. Pleasant tongue-in-cheek affair filmed in Mexico.\n\n**677** _ **Cattle Drive**_ **** Universal-International, 1951. 77 min. Color. D: Kurt Neumann. SC: Jack Natteford and Lillie Hayward. With Joel McCrea, Dean Stockwell, Chill Wills, Leon Ames, Henry Brandon, Bob Steele, Howard Petrie, Griff Barnett, Chuck Roberson, Harry Carey, Jr., Carol Henry. The sheltered young son of a railroad tycoon learns life's values as he goes with a veteran cowboy on a cattle drive across the desert. A different kind of Western and quite entertaining.\n\n**678** _ **Cattle Empire**_ **** 20th Century\u2013Fox, 1958. 83 min. Color. D: Charles Marquis Warren. SC: Andre Boehm and Eric Norden. With Joel McCrea, Gloria Talbott, Phyllis Coates, Don Haggerty, Bing Russell, Paul Brinegar, Hal K. Dawson, Richard Shannon, Charles Gray, Patrick O'Moore, Steve Raines, Nesdon Booth, Bill Hale, Howard Culver, Bill McGraw. When it becomes imperative they get their cattle to market, a group of citizens ask a trail boss, a man they once sent to jail, to lead their herd and he agrees but plans to double cross them. Entertaining Joel McCrea oater with several neat plot twists.\n\n**679** _ **Cattle King**_ **** Metro-Goldwyn-Mayer, 1963. 88 min. Color. D: Tay Garnett. SC: Thomas Thompson. With Robert Taylor, Joan Caulfield, Robert Loggia, Robert Middleton, Larry Gates, Malcolm Atterbury, William Windom, Virginia Christine, Ray Teal, Richard Devon, Robert Ivers, Maggie Pierce, John Mitchum. In 1883 Wyoming Territory a rich rancher wants to have fenced in ranges but other cattlemen oppose him and the situation becomes so tense President Chester A. Arthur is forced to intervene. Fairly interesting Western with a novel plot, but Larry Gates looks nothing like Chester A. Arthur.\n\n**680** _ **Cattle Queen**_ **** United International, 1951. 72 min. D-SC: Robert Emmett Tansey. With Maria Hart, Drake Smith, William Fawcett, Robert Gardette, John Carpenter, Edward Clark, Emile Meyer, Jim (James) Pierce, Joe Bailey, Douglas Wood, Alvin Lockwood, I. Stanford Jolley, Lane Chandler, William Norton Bailey, Frank Marlowe, Roger Anderson, Vern Teters, Steve Conte, Robert H. Robinson, Whitey Hughes. A woman ranch owner battles for her rights as she assists a town in cleaning up lawlessness with the aid of paroled criminals. Sparse affair, starring whip carrying Maria Hart, from producer-co-star John Carpenter.\n\n**681** _ **Cattle Queen of Montana**_ **** RKO Radio, 1954. 88 min. Color. D: Allan Dwan. SC: Robert Blees and Howard Estabrook. With Barbara Stanwyck, Ronald Reagan, Gene Evans, Lance Fuller, Anthony Caruso, Jack Elam, Yvette Dugay, Morris Ankrum, Chubby Johnson, Myron Healey, Rodd Redwing, Paul Birch, Byron Foulger, Burt Mustin, Roy Gordon. An Army undercover agent helps a woman rancher whose father was murdered by a renegade white and his Indian accomplice. Filmed in SuperScope near the Glacier National park, this one has little interest for anyone outside fans of its two stars.\n\n**682** _ **The Cattle Raiders**_ **** Columbia, 1938. 61 min. D: Sam Nelson. SC: Joseph Poland and Ed Earl Repp. With Charles Starrett, Donald Grayson, Iris Meredith, The Sons of the Pioneers (Bob Nolan, Lloyd Perryman, Hugh Farr, Karl Farr), Dick Curtis, Allen Brook, Ed LeSaint, Edmund Cobb, George Chesebro, Art Mix, Ed Coxen, Steve Clark, Alan Sears, Ed Peil, Sr., Jim Thorpe, Hank Bell, Blackie Whiteford, Jack Clifford, Frank Ellis, Curley Dresden, Merrill McCormick, Forrest Taylor, Horace B. Carpenter, George Morrell, Bob Burns, Wally West, Jim Mason, Clem Horton. A man is falsely accused of murder by a pal who is deeply in debt to a dishonest cattle dealer. Action filled Charles Starrett vehicle.\n\n**683** _ **Cattle Stampede**_ **** Producers Releasing Corporation, 1943. 59 min. D: Sam Newfield. SC: Joseph O'Donnell. With Buster Crabbe, Al St. John, Frances Gladwin, Glenn Strange, Charles King, Ed Cassidy, Hansel Warner, Ray Bennett, Frank Ellis, Steve Clark, Roy Brent, Reed Howes, John Elliott, Budd Buster, Hank Bell, Tex Cooper, Ted Adams, Frank McCarroll, Ray Jones, George Morrell, Carl Mathews, Art Dillard, Curley Dresden, Roy Bucko, Hal Price, Rose Plummer, Wally West, Ed Peil, Sr. Billy the Kid and Fuzzy Jones assist a cattleman who is caught in the middle of a range war. Crude, but fans of the PRC series will like this one. Also called _**Billy the Kid in Cattle Stampede**_.\n\n**684** _ **The Cattle Thief**_ **** Columbia, 1936. 57 min. D: Spencer Gordon Bennet. SC: Nate Gatzert. With Ken Maynard, Geneva Mitchell, Ward Bond, Roger Williams, Jim Mason, Sheldon Lewis, Edward Cecil, Jack Kirk, Edward Hearn, Glenn Strange, Al Taylor, Dick Rush, Bud McClure, Jack King. A cattlemen's association agent masquerades as a dimwit peddler to get the goods on an outlaw gang trying to cheat ranchers out of their land. Fine Ken Maynard vehicle with the star showing just how good he could be in a character role. Inside joke: the name of the owner of the Bottleneck Ranch is Carl Pierson, a well known film editor and sometime director.\n\n**685** _ **Cattle Town**_ **** Warner Bros., 1952. 71 min. D: Noel Smith. SC: Tom Blackburn. With Dennis Morgan, Rita Moreno, Philip Carey, Paul Picerni, Amanda Blake, George O'Hanlon, Ray Teal, Jay Novello, Robert Wilke, Sheb Wooley, Charles Meredith, Merv Griffin, Boyd Morgan, Carol Henry. When trouble erupts between cattlemen and a land baron, a gunfighter is brought in to restore peace. Throwaway Dennis Morgan vehicle made on the cheap.\n\n**686** _ **Caught**_ **** Paramount, 1931. 71 min. D: Edward Sloman. SC: Agnes Brand Leahy and Kenne Thompson. With Richard Arlen, Frances Dee, Louise Dresser, Syd Saylor, Edward J. LeSaint, Tom Kennedy, Martin Burton, Marcia Manners, Guy Oliver, Charles K. French, Jim Mason, Jack Clifford. An Army lieutenant is on the trail of saloon owner Calamity Jane who is wanted for a series of crimes. Dresser gives an interesting true-to-life unglamorous portrayal of Calamity Jane in this appealing drama.\n\n**687** _ **Cavalcade of the West**_ **** Diversion, 1936. 59 min. D: Harry Fraser. SC: Norman Houston. With Hoot Gibson, Rex Lease, Marion Shilling, Adam Goodwin, Nina Guilbert, Steve Clark, Earl Dwire, Phil Dunham, Bob McKenzie, Jerry Tucker, Barry Downing, Budd Buster, Blackie Whiteford, Milburn Morante, Oscar Gahan, Francis Walker, Herman Hack, Jack Evans, Art Dillard, William McCall, Dick Morehead. On the trail of an outlaw, a Pony Express rider learns the man he seeks is his long lost brother. A sparse budget hurts this otherwise acceptable Hoot Gibson feature from producer Walter Futter.\n\n**688** _ **Cavalier of the West**_ **** Artclass, 1931. 65 min. D-SC: J.P. McCarthy. With Harry Carey, Kane Richmond, Carmen LaRoux, Paul Panzer, Ted Adams, George Hayes, Lew Meehan, Ben Corbett, Maston Williams, Carlotta Monti, Elena Verdugo. With war between Indians and settlers imminent, an Army captain tries to restore peace. Harry Carey fans should like this low grade effort.\n\n**689** _ **Cavalry**_ **** Republic, 1936. 63 min. D: Robert North Bradbury. SC: George Plympton. With Bob Steele, Frances Grant, Karl Hackett, William Welch, Earl Ross, Hal Price, Ed Cassidy, Perry Murdock, Budd Buster, Earl Dwire, William Desmond, Horace B. Carpenter. After the Civil War subversives attempt to set up a separate state in the West and the Army sends a cavalry officer to stop them. Good Bob Steele effort, well paced with a literate script.\n\n_**Cavalry Charge**_ see _**The Last Outpost**_\n\n**690** _ **Cavalry Command**_ **** Parade, 1963. 88 min. Color. D-SC: Eddie Romero. With John Agar, Richard Arlen, Myron Healey, Alica Verge, Pancho Magalona, William Phipps, Eddie Infante. During the American occupation of the Philippines in 1902, cavalry troops who befriend the people are opposed by a guerilla leader. So-So Philippine made oater, interesting because of its trio of stars.\n\n**Pancho Magalona, Richard Arlen, William Phipps and John Agar in** _**Cavalry Command**_ **(Parade, 1963).**\n\n**  \n**\n\n**691** _ **Cavalry Scout**_ **** Monogram, 1951. 78 min. Color. D: Lesley Selander. SC: Dan Ullman and Thomas Blackburn. With Rod Cameron, Audrey Long, Jim Davis, James Millican, James Arness, John Doucette, William Phillips, Stephen Chase, Rory Mallinson, Paul Bryar, Bud Osborne, Chief Yowlachie. When two Gatling guns and other weapons are stolen from an Army arsenal, a civilian scout is assigned to find them. Fast moving oater that will more than pass muster for Rod Cameron fans.\n\n**692** _ **Cave of the Outlaws**_ **** Universal-International, 1951. 75 min. Color. D: William Castle. SC: Elizabeth Wilson. With Macdonald Carey, Alexis Smith, Edgar Buchanan, Victor Jory, Hugh O'Brian, Houseley Stevenson, Charles Horvath, Kenneth MacDonald, Russ Tamblyn, John Carpenter, Jack Ingram. An ex-convict searches for gold hidden after a Wells Fargo holdup and on his trail and looking for the loot are an investigator and a crooked miner. Considering the cast and story, this film should have been a lot better.\n\n_**The Century Turns**_ see _**Hec Ramsey**_\n\n_**The Challenge**_ see _**Across the Badlands**_\n\n**693** _ **The Challenge of Rin Tin Tin**_ **** Burt Leonard Productions, 1957. 90 min. D: Robert G. Walker. SC: Lee Berg, Jennings Cobb and John O'Dea. With Rin Tin Tin V (dog), James Brown, Lee Aaker, Joseph Sawyer, Rand Brooks, Pierre Watkin. An orphan boy and his dog, both adopted as honorary troopers by the cavalry soldiers of Fort Apache, aid the military and locals against lawlessness. Enjoyable telefeature culled from episodes of \"The Adventures of Rin Tin Tin\" (ABC-TV, 1954\u201359); released on video as _**The Courage of Rin Tin Tin**_.\n\n**694** _ **Challenge of the MacKennas**_ Picturemedia, 1969. 101 min. Color. D: Leon Klimovsky. SC: Jose L. Navarro. With John Ireland, Robert Woods, Annabella Incontrera, Roberto Camardiel, Daniela Giordano, Vidal Molina, Ken Wood (Giovanni Cianfriglia). A rancher rules the land with an iron hand but begins to be opposed by a gunman and his own family. Typically violent Spaghetti Western made in Italy by Filmar\/Atlantida Films. Also called **Badlands Drifter**.\n\n**695** _ **Challenge of the Range**_ **** Columbia, 1949. 56 min. D: Ray Nazarro. SC: Ed Earl Repp. With Charles Starrett, Smiley Burnette, Paula Raymond, William (Billy) Halop, Steve Darrell, Henry Hall, Robert Filmer, George Chesebro, John McKee, Frank McCarroll, John Cason, Kermit Maynard, Edmund Cobb, Ray Bennett, Cactus Mack, The Sunshine Boys (Eddie Wallace, J.D. Sumner, M.H. Richman, Freddie Daniel), Jock Mahoney, Leroy Johnson, Emile Avery, Matty Roubert, Pat O'Malley, Milton Kibbee, Marshall Reed, Frank O'Connor, Rose Plummer. The Durango Kid gets involved when ranchers accuse each other of acts of lawlessness, resulting in a range war. Good entry in \"The Durango Kid\" series that utilizes footage from a previous entry, _**Galloping Thunder**_ (q.v.). British title: _**Moonlight Raid**_.\n\n**696** _ **Challenge to Be Free**_ **** Pacific International, 1976. 88 min. Color. D: Tay Garnett and Ford Beebe. SC: Chuck Keen, Anne Bosworth and Tay Garnett. With Mike Mazurki, Jimmy Kane, Fritz Ford, Vic Christy, Tay Garnett, John McIntire (narrator). A trapper who loves animals is hunted by the law through the Yukon after he accidentally shoots a trooper. Filmed in the Yukon in 1972 as _**The Mad Trapper**_ , this somewhat crude production makes for good entertainment, especially for the scenery and Mike Mazurki in the title role. The story was also used for **Death Hunt** (q.v.).\n\n_**Challenge to Survive**_ see _**Land of No Return**_\n\n**697** _ **Challenge to White Fang**_ **** Premiere Releasing, 1974. 98 min. Color. D: Lucio Fulci. SC: Alberto Silvestri, Roberto Gianviti and Lucio Fulci. With Franco Nero, Virna Lisi, John Steiner, Harry Carey, Jr., Raimund Harmstorf, Yanti Somer, Werner Pochath, Hannelore Elsner, Renato De Carmine, Renato Cestie, Donald O'Brien, Rolf Hartmann, John Bartha, Paolo Magalotti, Sergio Smacchi, Ezio Marano, Stanislaus Gunawan, Vittorio Fanfoni, Carla Mancini, Goffredo Unger, Pietro Torrisi, Riccardo Petrazzi, Missaele. In northern Canada a young boy befriends a wolf dog and they help a writer and his friend stop a ruthless despot from taking over a small town. So-so, overly violent Italian feature with sub-par production values; made as _**Il Ritorno di Zanna Bianca**_ (The Return of White Fang) and a sequel to _**White Fang**_ (1972) [q.v.].\n\n**698** _ **Champions of Justice**_ **** Wrather Corporation, 1956. 75 min. Color. D: Earl Bellamy and Oscar Randolph. SC: Thomas Seller, Doane Hoag, Robert E. Schaefer, Eric Friewald and Robert Leslie Bellem. With Clayton Moore, Jay Silverheels, Allen Pinson, Wayne Burson, Myron Healey, Dennis Moore, David Sharpe, Harry Strang, Don C. Harvey, Steve Raines, George Barrows, Robert Homans, Sydney Mason, Watson Downs, Dan Barton, Walt LaRue, Brad Jackson, Florence Lake, Carlos Vera, Linda Wrather, Tom Noel, Byron Foulger, Kathryn Riehl, Nolan Leary, William Fawcett, Zon Murray. The Lone Ranger and Tonto are almost hung for a murder committed by an outlaw gang; they try to keep a boy from going to the wrong side of the law; and they unravel the mystery of a man's murder. Well done telefilm from \"The Lone Ranger\" (ABC-TV, 1949\u201357) episodes \"The Angel and the Outlaw,\" \"Blind Witness\" and \"Clover in the Dust.\"\n\n**699** _ **La Chamuscada**_ (The Scorched) **** Producionnes Matonk, 1971. 90 min. Color. D: Alberto Mariscal. SC: Luis Alcoriza and Juan de la Cabada. With Luis Aguilar, Rodolfo de Anda, Emilio Fernandez, Irma Serrano, Adriana Roel, Enrique Rocha, Jose Alfredo Jiminez, Guillermo A. Bianchi, Robert Guinart, Consuelo Frank, Victor Alcocer, Carlos Leon, Tito Novaro, Eva Calvo, Diana Ochoa, Colo Cora, Ricardo Fuentes, Raymon Rey, Rodolfo Rey. During the Mexican Revolution a peon joins the rebels in opposing government forces. Entertaining Mexican historical drama.\n\n_**Charge!**_ see _**Those Dirty Dogs**_\n\n**700** _ **The Charge at Feather River**_ **** Warner Bros., 1953. 96 min. Color. D: Gordon Douglas. SC: James R. Webb. With Guy Madison, Frank Lovejoy, Vera Miles, Helen Wescott, Dick Wesson, Onslow Stevens, Steve Brodie, Ron Hagerthy, Fay Roope, Neville Brand, Henry Kulky, Lane Chandler, Fred Carson, James Brown, Adele Jergens, Rand Brooks, Ben Corbett, Ralph Brooke, Carl Andre, Fred Kennedy, Dub Taylor, John Pickard, Vivian Mason, Dennis Dengate. Two women are kidnapped by Indians and the cavalry rescues them only to start an uprising. Nothing special in this oater, originally issued in 3-D.\n\n**701** _ **Charley-One-Eye**_ **** Paramount, 1973. 96 min. Color. D: Don Chaffey. SC: Keith Leonard. With Richard Roundtree, Roy Thinnes, Nigel Davenport, Jill Peason, Aldo Sambrell, Rafael Albaicin, Alex Davion, Johnny Sekka, Madeline Hinde, Patrick Mower, Imogene Hassall, Edward Woodward, William Mervyn, David Lodge, Luis Aller. A black army deserter and a wounded Indian join forces to survive in the desert only to find themselves trailed by a bounty hunter. More drama than action in this British production.\n\n**702** _ **Charlie Cobb: Nice Night for a Hanging**_ **** NBC-TV\/Universal, 1977. 100 min. Color. D: Richard Michaels. SC: Peter S. Fischer. With Clu Gulager, Ralph Bellamy, Blair Brown, Christopher Connelly, Pernell Roberts, Stella Stevens, Carmen Matthews, George Furth, Tricia O'Neil. A private detective is hired by a rancher to return a girl he believes is his long lost daughter, while the man's wife and cohorts work to stop the sleuth. Mediocre TV Western, a pilot for a series that did not sell.\n\n**703** _ **Charlie, the Lonesome Cougar**_ **** Buena Vista, 1970. 75 min. Color. D: Winston Hibler. SC: Jack Speirs. With Rex Allen (narrator), Ron Brown, Brian Russell, Linda Wallace, Jim Wilson, Clifford Peterson, Lewis Sample, Edward C. Moller. Members of a north woods logging camp befriend a small cougar. Very pleasant Disney family film.\n\n**704** _ **Charro!**_ **** National General, 1969. 98 min. Color. D-SC: Charles Marquis Warren. With Elvis Presley, Ina Balin, Victor French, Barbara Werle, Solomon Sturges, Lynn Kellogg, Paul Brinegar, James Sikking, Harry Landers, Tony Young, James Almanazar, Charles H. Gray, Rodd Redwing, Gary Walbert, Duane Grey, John Pickard, J. Edward McKinley, Robert Luster, Chrisa Lang, Robert Karnes. In a border town, a former outlaw must face members of his gang while romancing a pretty saloon owner. Elvis Presley tries his best (he sings only the title song) but is defeated by a mediocre production.\n\n**705** _ **El Charro de las Calaveras**_ (The Rider of the Skulls) **** Columbia, 1965. D-SC: Alfredo Salazar. With Dagoberto Rodriguez, David Silva, Alicia Caro, Pascual Garcia Pena, Laura Martinez, Rosario Montes, Carlos del Muro, Jose Luis Cabrera, Gabriel Agrasanchez, Alfonso Ortiz, Alfredo Salazar. A masked cowboy avenger finds himself up against a headless swordsman, a vampire and a werewolf, plus a witch and a talking corpse. Fun Mexican three part horror Western hampered by tacky looking monsters.\n\n**706** _ **El Charro Negro**_ (The Black Cowboy) **** Cinexport Distributing, 1940. 70 min. D-SC: Raul de Anda. With Pedro Armendariz, Carolina Barrett, Emilio Fernandez, Raul de Anda, Alfonso Bedoya, Miguel Inclan, Roberto Canedo, Manuel Noriega, Max Largla, Agustin Isunza, Consuelo Segane, Luis G. Barreiro, Miguel Angel Ferriz. An outlaw who robs the men to whom he sold land is pursed by a cowboy, with both outfitted in black. Nicely done Mexican Western with fine heroics by Pedro Armendariz and villainy by Emilio Fernandez.\n\n**707** _ **El Charro Negro Contra la Banda de los Cuervos**_ (The Black Cowboy Against the Band of the Ravens) **** Radeant Films, 1962. 87 min. D: Arturo Martinez. SC: Fernando Faliana, Carlos E. Taboada and Raul de Anda. With Rodolfo de Anda, Fernando Soto \"Mantequilla,\" Laila Buentello, Alejandro Parodi, Lucha Villa, Miguel Manzano, Pascual Garcia Pena, Miguel Arenas, David Reynoso, Mario Chavez, Roberto Moreno, Armando Arreola, Jose Hernandez, Emilio Garibay, Agustin Fernandez. A masked rider and his two pals attempt to rid a village of an outlaw and his gang. Typical Mexican Western adventure in the \"Charro Negro\" series.\n\n**708** _ **El Charro Negro en el Norte**_ (The Black Cowboy in the North) **** Peliculas Mexicanas, S.A., 1949. 73 min. D: Raul de Anda. SC: Gabriel Ramirez Osante. With Raul de Anda, Carmen Gonzalez, Dagoberto Rodriguez, Raul de Anda, Jr., El Chicote (Armando Soto La Marina), Jorge Arriaga, Agustin Isunza, Jose L. Murillo, Emilio Garibay, Jose Munoz, Gregorio Acosta. After their father is killed by gunmen, a young woman and her little brother try to keep the title to a gold mine and are helped by a cowboy dressed in black. Action filled outing in Mexican producer-writer-star Raul de Anda's \"Charro Negro\" films.\n\n**709** _ **Chato's Land**_ **** United Artists, 1972. 92 min. Color. D: Michael Winner. SC: Gerald Wilson. With Charles Bronson, Jack Palance, Richard Basehart, James Whitmore, Simon Oakland, Ralph Waite, Richard Jordan, Victor French, William Watson, Roddy McMillan, Paul Young, Lee Patterson, Rudy Ugland, Raul Castro, Sonia Rangan, Clive Endersby, Peter Dyneley, Hugh McDermott, Verna Harvey, Sally Adez, Rebecca Wilson, Florencio Amarilla, Clestino Gonzalez, Roland Grant, Louis Amarilla. When an Indian is forced to kill a lawman, a posse murders his family and follows him into the desert only to become the hunted. Well-made Western reworking of _**The Lost Patrol**_ (RKO Radio, 1935); filmed in Spain and originally running 100 minutes.\n\n**710** _ **Chatterbox**_ **** Republic, 1943. 77 min. D: Joseph Santley. SC: George Carleton Brown and Frank Gill, Jr. With Joe E. Brown, Judy Canova, Rosemary Lane, John Hubbard, Gus Schilling, Chester Clute, Anne Jeffreys, Emmett Vogan, George Byron, Billy Bletcher, The Mills Brothers, Spade Cooley, Roy Barcroft, Marie Windsor, Earle Hodgins, Pierce Lyden, Tex Williams, Robert \"Buzz\" Henry, Matty Kemp, Frank Melton, Ben Taggart, Charles Williams, Tom Quinn, Nora Lane, Herbert Heyes, Sam Flint, Gary Bruce, Judy Clark, Dorothy Andre, Maxine Doyle, Mary Kenyon, Jane Weeks, Edward Earle, Mary Armstrong, Gordon DeMain, Ray Parsons, Ruth Robinson, Art Whitney, Joe Phillips, Lloyd Perryman, Pat Starling, Bill Yrigoyen, Joe Yrigoyen. A fading Hollywood cowboy star gets involved with a yokel girl while on location and she not only saves his reputation but becomes his co-star and fiancee. Fun genre spoof nicely pairing Joe E. Brown and Judy Canova.\n\n_**The Cheat**_ see _**The Lone Hand Texan**_\n\n_**Cheatin' Hearts**_ see _**Paper Hearts**_\n\n_**The Cheat's Last Throw**_ see _**Heading West**_\n\n**711** _ **Check Your Guns**_ **** Eagle Lion, 1948. 55 min. D: Ray Taylor. SC: Joseph O'Donnell. With Eddie Dean, Roscoe Ates, Nancy Gates, George Chesebro, Andy Parker and The Plainsmen, I. Stanford Jolley, Mikel Conrad, Lane Bradford, Terry Frost, Mason Wynn, Dee Cooper, William Fawcett, Ted Adams, Marshall Reed, Steve Clark, Frank Ellis, Budd Buster, Wally West. After wandering into a remote town, a cowboy is made its sheriff and tries to stop an outlaw gang. Passable Eddie Dean musical vehicle.\n\n_**Checkmate**_ see _**The Desert Horseman**_\n\n**712** _ **Cherokee Flash**_ **** Republic, 1945. 58 min. D: Thomas Carr. SC: Betty Burbridge. With Sunset Carson, Linda Stirling, Roy Barcroft, Bud Geary, George Chesebro, Fred Graham, Tom London, John Merton, Frank Jaquet, Joe McGuinn, Pierce Lyden, James Lynn, Edmund Cobb, Bud Osborne, Bill Woolf, Hank Bell, Chick Hannon, Tommy Coats, Cactus Mack, George Sowards, Duke Green. A once famous outlaw is falsely blamed for a robbery committed by his old gang and his stepson tries to clear his name. Sturdy Sunset Carson fare with Roy Barcroft as a good guy for a change.\n\n**713** _ **The Cherokee Kid**_ **** Home Box Office (HBO), 1996. 91 min. Color. D: Paris Barclay. SC: Tim Kazurinsky and Denise DeClue. With Sinbad, James Coburn, Gregory Hines, A Martinez, Ernie Hudson, Mark Pellegrino, Vanessa Bell Calloway, Hal Williams, Burt Reynolds, Obba Babatunde, Reginald T. Dorsey, Dawnn Lewis, Lorraine Toussaint, Paris Barclay, W. Earl Brown, Herb Jeffries, Michael Kagan, Carlos Cervantes, Roy Fegan, Troy Garity, Walton Goggins, Kareem R. Woods, Hattie Winston, James Lashly, Jim Lewis, Jr., Ivory Ocean, Jack Rubens, Spice Williams, Stuart Proud Eagle Grant, Martin Grey, DeJuan Guy, Alaina Reed-Hall, Edward B. Hicks, Charles Hyman, Jim Cody Williams, Tim Kazurinsky, Robert L. Minor, Angela Means, June Kyoto Lu, Tim Sampson, Nancy Legehan, Arnetia Walker, Roxanne Reese. A black man plans to get revenge on the crook who murdered his family and he learns the ways of survival from various characters before a final showdown with a hired gun who may have killed his brother. Vapid TV Western feature wasting a good cast.\n\n**714** _ **Cherokee Strip**_ **** Warner Bros., 1937. 55 min. D: Noel Smith. SC: Joseph K. Watson and Luci Ward. With Dick Foran, Jane Bryan, David Carlyle (Robert Paige), Helen Valkis, Edmund Cobb, Gordon Hart, Joseph Crehan, Frank Faylen, Milton Kibbee, Jack Mower, Tom Brower, Walter Soderling, Tommy Bupp, Glenn Strange, Bud Osborne, Ben Corbett, Artie Ortego, Jack Kirk. Settlers out for free homesteads in the Oklahoma Territory take part in land rush as a cowboy has his horse lamed by a man who wants the parcel he plans to claim. Better than average Dick Foran vehicle, thanks to the plot and the song \"My Little Buckaroo.\"\n\n**715** _ **Cherokee Strip**_ **** Paramount, 1940. 86 min. D: Lesley Selander. SC: Norman Houston and Bernard McConville. With Richard Dix, Florence Rice, Victor Jory, Andy Clyde, William Henry, Tom Tyler, George E. Stone, Morris Ankrum, Charles Trowbridge, Douglas Fowley, Addison Richards, Hal Taliaferro, William Haade, Ray Teal, Jack Rockwell, Tex Cooper, Robert Winkler. A newly appointed lawman tries to bring order to the town of Goliath in the Cherokee Strip. Entertaining Richard Dix feature, well produced and action filled.\n\n**716** _ **Cherokee Uprising**_ **** Monogram, 1950. 60 min. D: Lewis D. Collins. SC: Dan Ullman. With Whip Wilson, Andy Clyde, Lois Hall, Iron Eyes Cody, Sam Flint, Forrest Taylor, Marshall Reed, Chief Yowlachie, Lee Roberts, Stanley Price, Lyle Talbot, Edith Mills. A government agent tries to find out what is behind a threatened Indian revolt. Standard, but more than passable, Whip Wilson entry.\n\n**717** _ **Chetan, Indian Boy**_ **** Autoren, 1972. 94 min. Color. D-SC: Mark Bohm. With Marquard Bohm, Deschingis Bowakow, Willi Schultes, Horst Schram. In the northwest wilderness a shepherd frees an Indian lad held captive by a farmer who comes searching for him. Nicely done West German production issued in that country as _**Tschetan der Indianer Junge**_ (Chetan the Indian Boy).\n\n**718** _ **Cheyenne**_ **** Warner Bros., 1947. 100 min. D: Raoul Walsh. SC: Alan LeMay and Thomas Williamson. With Dennis Morgan, Jane Wyman, Janis Paige, Bruce Bennett, Alan Hale, Arthur Kennedy, John Ridgely, Barton MacLane, Tom Tyler, Bob Steele, John Compton, John Alvin, Monte Blue, Ann O'Neal, Tom Fadden, Britt Wood, Lee \"Lasses\" White, Snub Pollard, Ethan Laidlaw, Robert Filmer, Ray Teal, Kenneth MacDonald, Ben Corbett, Philo McCullough, Houseley Stevenson, Clancy Cooper, Artie Ortego, Jack Norman, Bob Alderette, Carl Harbaugh, Jasper Palmer, Jameson Slade, Hubert Brill. While trying to capture an outlaw, a gambler falls in love with the man's wife. A good cast and steady direction greatly aid this \"A\" effort. TV title: _**The Wyoming Kid**_.\n\n**719** _ **Cheyenne**_ **** 1996. Bruder Releasing. 90 min. Color. D: Dimitri Logothetis. SC: Frederick Bailey. With Bobbie Phillips, Gary Hudson, Bo Svenson, M.C. Hammer, Bobby Bell, Tobin Bell, Ritchie Montgomery, Cole McKay, Dimitri Logothetis, Jay Nethercott, Dennis Burkley, John Diab, Shaun Monson, Dave Marshall, Juan Fernandez, Jared Chandler, Joe Hutchins, George Purcells, Charlie Hutchins, Verle Green, Aaron Heck. After shooting her brutal, two-timing spouse and taking his money, a feisty young woman is trailed by a bounty hunter hired by the husband and another reward claimer with his evil Indian dwarf associate. R-rated Western filmed in Utah makes for fair viewing.\n\n**720** _ **Cheyenne Autumn**_ **** Warner Bros., 1964. 145 min. Color. D: John Ford. SC: James R. Webb. With Richard Widmark, Carroll Baker, James Stewart, Karl Malden, Edward G. Robinson, Sal Mineo, Dolores Del Rio, Ricardo Montalban, Gilbert Roland, Arthur Kennedy, Elizabeth Allen, John Carradine, Patrick Wayne, Victor Jory, Mike Mazurki, George O'Brien, Sean McClory, Judson Pratt, Carmen D'Antonio, Ken Curtis, Walter Baldwin, Shug Fisher, Nancy Hsueh, Chuck Roberson, Harry Carey, Jr., Ben Johnson, Jimmy O'Hara, Chuck Hayward, Lee Bradley, Walter Reed, Willis Bouchey, Carleton Young, Denver Pyle, John Qualen, Dan Borgaze, Dean Smith, Bing Russell. Nearly 300 Cheyenne Indians try to return to their homes in the Dakotas from an Oklahoma reservation but are pursed by the cavalry. Although not a totally successful film, John Ford's final Western is more hit than miss; it is nice to see George O'Brien again and Mike Mazurki excels in a part Victor McLaglen would have done two decades before.\n\n**721** _ **Cheyenne Cyclone**_ **** Willis Kent, 1932. 57 min. D: Armand L. Schaefer. SC: Oliver Drake. With Lane Chandler, Connie LaMont, Frankie Darro, Edward Hearn, J. Frank Glendon, Josephine Hill, Henry Roquemore, Yakima Canutt, Marie Quillan, Jay Hunt, Charles \"Slim\" Whitaker, Jack Kirk, Helen Gibson, Hank Bell, Bart Carre. Stranded in a small town with an acting troupe, a cowboy goes to work for a rancher who is about to lose his cattle to a crook. Rawboned Saturday matinee fodder, although Lane Chandler is a likable Western hero.\n\n**722** _ **The Cheyenne Kid**_ **** West Coast Pictures, 1930. 55 min. D: Jacques Jaccard. SC: Jacques Jaccard and Yakima Canutt. With Buffalo Bill, Jr., Joan Jaccard, Yakima Canutt, Jack Mower, Frank Ellis, Fred Burns, Violet McKay, Tom Forman, Lafe McKee, Cliff Lyons. Falsely accused of a payroll car holdup, a cowboy accidentally shoots a young woman and ends up in jail but a U.S. marshal believes he is innocent. Slow moving, complicated poverty row item co-authored by Yakima Canutt.\n\n**723** _ **The Cheyenne Kid**_ **** RKO Radio, 1933. 54 min. D: Robert Hill. SC: Kenne Thompson. With Tom Keene, Mary Mason, Roscoe Ates, Alan Bridge, Otto Hoffman, Alan Roscoe, Anderson Lawlor, Ken Cooper, Gordon James, Bill Hurley, Buff Jones. An easy going cowboy is blamed for a murder and to prove his innocence tries to find the real killer. Likable Tom Keene feature.\n\n**724** _ **The Cheyenne Kid**_ **** Monogram, 1940. 50 min. D: Raymond K. Johnson. SC: Tom Gibson. With Jack Randall, Louise Stanley, Kenneth (Kenne) Duncan, Frank Yaconelli, Reed Howes, Charles King, George Chesebro, Forrest Taylor, Tex Palmer. A notorious outlaw wants to reform and goes to work for a cattleman but crooks try to frame him on murder and rustling charges. Fairly entertaining Jack Randall vehicle.\n\n**725** _ **Cheyenne Rides Again**_ **** Victory, 1937. 56 min. D: Robert Hill. SC: Basil Dickey. With Tom Tyler, Lucille Brown, Jimmie Fox, Lon Chaney, Jr., Roger Williams, Ed Cassidy, Theodore Lorch, Bud Pope, Francis Walker, Carmen LaRoux, Jed Martin, Slim Whitaker, Bob Hill, Merrill McCormick, Oscar Gahan, Jack C. Smith, Wilbur McCauley. A detective for the cattlemen's association tries to pass himself off as an outlaw in order to infiltrate a gang. Action filled entry in Tom Tyler's Victor series, heavy on outdoor scenes.\n\n**726** _ **Cheyenne Roundup**_ **** Universal, 1942. 59 min. D: Ray Taylor. SC: Elmer Clifton and Bernard McConville. With Johnny Mack Brown, Tex Ritter, Fuzzy Knight, Jennifer Holt, Harry Woods, Roy Barcroft, Robert Barron, Gil Patrick, The Jimmy Wakely Trio (Jimmy Wakely, Johnny Bond, Scotty Harrell), William Desmond, Kenne Duncan, Kermit Maynard, Budd Buster, Carl Mathews. An outlaw attempts to murder a lawman who is after him but is himself killed as his twin brother pretends to be him in order to help clean up the territory. Satisfactory remake of _**Bad Man from Red Butte**_ (q.v.) with a good song, \"On the Rainbow Trail.\"\n\n**727** _ **The Cheyenne Social Club**_ **** National General, 1970. 103 min. Color. D: Gene Kelly. SC: James Lee Barrett. With James Stewart, Henry Fonda, Shirley Jones, Sue Anne Langdon, Elaine Devry, Robert Middleton, Arch Johnson, Dabbs Greer, Jackie Russell, Jackie Joseph, Sharon De Bord, Richard Collier, Charles Tyner, Jean Willes, Robert Wilke, Carl Reindel, J. Pat O'Malley, Jason Wingreen, John Dehner, Hal Baylor, Charlotte Stewart, Albert Morin, Myron Healey, Warren Kemmerling, Dick Johnstone, Red Morgan, Bill Davis, Richard Alexander. Two lowbrow cowboys find themselves the owners of a brothel and have to defend the honor of their \"girls.\" Fairly pleasant comedy; the final screen teaming of James Stewart and Henry Fonda.\n\n**728** _ **Cheyenne Takes Over**_ **** Eagle Lion, 1947. 56 min. D: Ray Taylor. SC: Arthur E. Orloff. With Lash LaRue, Al St. John, Nancy Gates, George Chesebro, Lee Morgan, John Merton, Steve Clark, Bob Woodward, Marshall Reed, Budd Buster, Carl Mathews, Dee Cooper, Brad Slaven, Hank Bell. U.S. marshals Cheyenne Davis and Fuzzy Q. Jones help a young woman who witnessed the murder of a man over a ranch. A slow drama with mediocre production values, dominated by George Chesebro as the bad guy.\n\n**729** _ **The Cheyenne Tornado**_ **** Willis Kent, 1935. 55 min. D: William A. O'Connor. SC: Oliver Drake. With Reb Russell, Victoria Vinton, Roger Williams, Edmund Cobb, Tina Menard, Perry Winton, Richard Botiller, Ed Porter, Lafe McKee, Bart Carre, Tracy Layne, Jack King, Hank Bell, Art Dillard, Bert Dillard, Jack Evans, Oscar Gahan, Clyde McClary, Bud Pope, Lew Morphy, Arthur Thalasso, Jack Jones, Bill Hickey. The Cheyenne Kid tries to help a female cattle rancher whose father was supposedly murdered by sheep owners but he comes to believe some one else is behind the trouble. Rock bottom Willis Kent production with Smiley Burnette singing the title song.\n\n**730** _ **Cheyenne Warrior**_ **** Libra Pictures, 1994. 90 min. Color. D: Mark Griffiths. SC: Michael B. Druxman. With Kelly Preston, Bo Hopkins, Dan Haggerty, Clint Howard, Rick Dean, Charles Powell, Noah Colton, Louise Baker, Pato Hoffman, Nik Winterhawk, Patricia Van Ingen, Frankie Avina, Joseph Wolves Kill, Terrance Fredericks, Mark S. Brien, Mark Cota, Jules Desjarlais, Dan Clark, Jesse Flores, Ezra Gabey, Lewis Ninham. Just before the Civil War, a young woman pioneer and a Cheyenne brave survive an Indian attack and begin to feel a mutual attraction. Well made TV Western.\n\n**731** _ **Cheyenne Wildcat**_ **** Republic, 1944. 56 min. D: Lesley Selander. SC: Randall Faye. With Wild Bill Elliott, Bobby Blake, Alice Fleming, Peggy Stewart, Francis McDonald, Roy Barcroft, Tom London, Tom Chatterton, Kenne Duncan, Bud Geary, Steve Clark, Jack Kirk, Bud Osborne, Robert Wilke, Rex Lease, Tom Steele, Forrest Taylor, Franklyn Farnum, Horace B. Carpenter, Frank Ellis, Bob Burns, Jack O'Shea, Frederick Howard, Dickie Dillon, Charles King, Lucille Ward, Rudy Bowman, Merlyn Nelson, Jack Tornek, Tom Smith, Willie Keefer, Wade Crosby, Charles Morton. Red Ryder and Little Beaver assist the citizens of Cheyenne whose bank is being sought by a crook from the East. Nicely done entry in the \"Red Ryder\" series.\n\n**732** _ **Un Chico Valiente**_ (A Valiant Boy) **** Cinematografica Grovas, 1958. 90 min. D: Mauricio de la Serna. SC: Mauricio de la Serna and Josefina Vicens. With Joaquin Cordero, Irma Castillon, Celia Tejeda, Marina Camacho, Jose Dupeyron, Ramon Sanchez, Raul Guerrero, Jose Munoz, Salvador Terroba, Gabriel Gracida, Jose Rojo de la Vega, Carlos Leon, Jose Eduarod Perez, Emilio Garibay, Chela Najera, Jose Eduardo Perez. With his horse and dog, a boy runs away from home when his sister plans to marry their ranch foreman and joins a gang, unaware they are rustlers. Okay juvenile Mexican Western.\n\n**733** _ **Chief Crazy Horse**_ **** Universal-International, 1955. 86 min. Color. D: George Sherman. SC: Gerald Drayson Adams and Franklin Coen. With Victor Mature, Suzan Ball, John Lund, Ray Danton, Keith Larsen, Paul Guifoyle, David Janssen, Robert Warwick, James Millican, Morris Ankrum, Donald Randolph, Robert F. Simon, Stuart Randall, Pat Hogan, Dennis Weaver, John Peters, Henry Wills, Charles Horvath, David Miller. Young brave Crazy Horse believes in the old prophecy that a warrior will destroy the whites and he proves it to be true by defeating General Custer. Predictable oater focusing on the famous Indian chief with Victor Mature stalwart in the title role. British title: _**Valley of Fury**_.\n\n**734** _ **Child Bride of Short Creek**_ **** NBC-TV, 1981. 104 min. Color. D: Robert Michael Lewis. SC: Joyce Eliason. With Christopher Atkins, Diane Lane, Conrad Bain, Kiel Martin, Helen Hunt, Warren Vanders, Joan Shawlee, Babetta Dick, Melinda Almquist, Julianne Slocum, C. Duane Tuft, Trisha Lynn Tibbs, X.V. Kelly. A Korean War veteran attempts to stop his polygamist father from marrying an underage girl. Nicely done drama based on a true story and set in Arizona in the 1950s.\n\n**735** _ **A Child of the Prairie**_ **** Aywon, 1924. 45 min. D-SC: Tom Mix. With Tom Mix, Louella Maxam, Baby Norma, Edward J. Brady, Leo Maloney, Fay Robinson, Frank Campeau. A gambler steals the wife and child of a rancher and the little girl grows up to be reunited with her father who wants revenge on the bad man. Fairly interesting silent made up of two Tom Mix Selig shorts.\n\n_**The Children's West**_ see _**A Stranger in Town**_ (1969)\n\n**736** _ **China 9, Liberty 37**_ **** Lorimar\/Titanus\/Compagnia Europea Cinematographica, 1978. 102 min. Color. D: Monte Hellman. SC: Jerry Harvey and Douglas Verturelli. With Warren Oates, Fabio Testi, Jenny Agutter, Sam Peckinpaugh, Isabel Mestres, Gianrico Tondivelli, Franco Interlenghi, Carlos (Charley) Bravo, Helga Line, Paco Benlloch, Richard C. Adams, Sydney Lassick, Natalie Kim, Luis Prendes, Yvonne Sentis, Frank Clement, Matthieu Ettori, David Thompson, Ramano Puppo, Tony Brandt, Piero Fondi, Luciano Sapdoni, Daniel Panes, Jose Murillo, Raphael Albaicon, Luis Barboo. Saved from execution by corrupt railroad tycoons, a man is sent to murder a former gunman for his land and finds the intended victim's young wife is willing to help him. Well acted but rather tiresome U.S.-Italian-Spanish co-production. Ronee Blakley sings the theme song; the film's title is a trail sign.\n\n_**Chinchero**_ see _**The Last Movie**_\n\n**737** _ **Chingachgook, the Great Snake**_ **** VEB Progress Film\u2014Vertrieb, 1967. 92 min. Color. D: Richard Groschopp. SC: Wolfgang Ebeling and Richard Groschopp. With Gojko Mitic, Rolf Romer, Helmut Schreiber, Jurgen Frohriep, Lilo Grahn, Andrea Drahota, Johannes Knitell, A.P. Hoffman, Heinz Klevenow, Jr., Milian Jablonsky, Horst Preusker, Rudolf Ulrich, Karl Zugowski, Gunter Schaumburg, Hans-Peter Pieper. The story of the struggle between the British and the French for the New World in the area of Lake Ontario. Colorful, but stilted, East German screen adaptation of James Fenimore Cooper's _The Deerslayer_ filmed as _**Chingachgook, die Grosse Schlange**_ (Chingachgook, the Great Snake).\n\n**738** _ **Chino**_ **** Intercontinental Releasing Corporation, 1976. 98 min. Color. D: John Sturges. SC: Dino Maiuri, Massimo De Rita and Clair Huffaker. With Charles Bronson, Jill Ireland, Vincent Van Patten, Marcel Buzzuffi, Melissa Chimenti, Fausto Tozzi, Ettore Manni, Adolfo Thous, Florencia Amarilla, Corrado Gaida, Diana Lorys, Melissa Chimenti. In 1880 a half-breed horse raiser befriends a teenage boy who helps him on his ranch but their life is interrupted when the man falls in love with the sister of a powerful neighbor who vows to destroy him. Entertaining, colorful Charles Bronson vehicle, made in Spain and issued abroad in 1973 by CIC as _**The Valdez Horses**_.\n\n**739** _ **Chip of the Flying U**_ **** Universal, 1939. D: Ralph Staub. SC: Larry Rhine and Andrew Bennison. With Johnny Mack Brown, Bob Baker, Fuzzy Knight, Doris Weston, Forrest Taylor, Anthony Warde, Karl Hackett, Henry Hall, Claire Whitney, Ferris Taylor, Kermit Maynard, Cecil Kellogg, The Texas Rangers, Hank Bell, Harry Tenbrook, Chester Conklin, Victor Potel, Hank Worden, Charles K. French, Budd Buster, Frank Ellis. Foreign agents rob a bank and shoot its president with the foreman of a nearby ranch being blamed for the crime. Entertaining, but unfaithful, adaptation of Bertha Muzzy Porter's chestnut which was first filmed by Selig in 1914 as a Tom Mix vehicle. In 1920 Bud Osborne starred as Chip in _**The Galloping Dude**_ and six years later Hoot Gibson had the title role when it was done again under its original title.\n\n**740** _ **The Chisholms**_ **** CBS-TV, 1979. 270 min. Color. D: Mel Stuart. SC: Evan Hunter. With Robert Preston, Rosemary Harris, Ben Murphy, Brian Kerwin, Jimmy Van Patten, Stacey Nelkin, Susan Swift, Charles Frank, Glynnis O'Connor, Sandra Griego, David Hayward, Anthony Zerbe, Brian Keith, Doug Kershaw, Tom Taylor, Gavin Troster, Dean Hill, David Allen, Don Shanks. The story of a pioneer family's journey from Virginia to Wyoming and their eventual settlement in California. Originally telecast as a mini-series, this well done drama was issued theatrically in a shorter version in 1979 by New Line International Releasing.\n\n**741** _ **Chisum**_ **** Warner Bros., 1970. 110 min. Color. D: Andrew V. McLaglen. SC: Andrew J. Fenaday. With John Wayne, Forrest Tucker, Christopher George, Pamela McMyler, Geoffrey Deuel, Ben Johnson, Glenn Corbett, Bruce Cabot, Andrew Prine, Patric Knowles, Richard Jaeckel, Lynda Day (George), John Agar, Lloyd Battista, Robert Donner, Ray Teal, Edward Faulkner, Ron Soble, John Mitchum, Glenn Langan, Alan Baxter, Alberto Morin, William Bryant, Pedro Armendariz, Jr., Christopher Mitchum, Abraham Sofaer, Gregg Palmer, Chuck Roberson, Hank Worden, Ralph Volkie, Pedro Gonzalez Gonzalez, John Pickard. New Mexico cattle baron John Chisum opposes crooks who are trying to steal his lands, resulting in the famous Lincoln County Cattle Wars. Big, brawling John Wayne action filled production; quite entertaining and one his last really exciting vehicles.\n\n_**The Chopper**_ see _**Blood Shack**_\n\n**742** _ **The Christmas Kid**_ **** Producers Releasing Organization, 1968. 87 min. Color. D: Sidney Pink. SC: Jim Henaghan and Rodrigo Rivero. With Jeffrey Hunter, Louis Hayward, Gustavo Rojo, Perla Cristal, Luis Prendes, Reginald Gilliam, Fernando Hilbeck, Jack Taylor, Eric Chapman, Carl Rapp. A loner rebelling against society takes a stand to stop a corrupt gambler from taking over a town. Cheaply made but engaging European Western.\n\n_**Christmas Miracle at Sage Creek**_ see _**Miracle at Sage Creek**_\n\n**743** _ **Christmas Mountain**_ **** Gold Coast, 1980. 90 min. Color. D: Pierre De Moro. SC: Mark Miller. With Slim Pickens, Mark Miller, Barbara Stanger, Tina Minard, Fran Ryan, John Hart, Brian Poelman. Caught in the mountains in a blizzard, an aging cowboy learns the true meaning of Christmas when he finds shelter with a widow and her children. Obscure but rewarding holiday fare re-titled _**The Cowboy Angels**_ for video.\n\n_**Chuck Moll**_ see _**The Unholy Four**_\n\n**744** _ **Chuka**_ **** Paramount, 1967. 105 min. D: Gordon Douglas. SC: Richard Jessup. With Ernest Borgnine, John Mills, Luciana Paluzzi, James Whitmore, Angela Dorian (Victoria Vetri), Louis Hayward, Michael Cole, Hugh Reilly, Barry O'Hara, Joseph Sirola, Marco Antonio, Gerald York, Lucky Carson. A gunman arrives at a fort to tell the Indian hating soldiers that unless the tribes are given food there will be warfare. Interesting concept that is not totally successful, resulting in a mediocre movie.\n\n**745** _ **El Ciclon**_ (The Cyclone) **** Madera, 1957. 83 min. D: Gilberto Martinez Solares. SC: Felipe Mier Miranda and Jose Pichel. With Miguel Aceves Mejia, Flor Silvestre, Sonia Furio, Raul Ramirez, Dagoberto Rodriguez, Juan Garcia, Ceclia Vieros, Jose Luis Fernandez, Miguel Angel Ferriz, Emilio Garibay, Agustin Isunza. A lawman and his pal go to a rural town to get revenge for the death of the peacemaker's brother. Fair Mexican Western drama with music; from a story by director Gilberto Martinez Solares.\n\n**746** _ **Cielito Lindo**_ (Beautiful Sky) **** Peliculas Nacionales, 1957. 90 min. D: Miguel M. Delgado. SC: Eduardo Galindo and Ramon Perez. With Rosita Quintana, Luis Aguilar, Carlos Lopez Moctezuma, Pedro Galindo, Alfredo Varela, Jr., Miguel Manzano, Genaro de Alba, Luis Aragon, Amgraso Arzamena, Alfonso Torres, Roberto Meyer, Eldonia Hernandez, Jose Munoz, Emilio Garibay, Carlos Guameros. A revolutionary in love with the daughter of a Federalist family is captured and sentenced to die but his lady love tries to save him. Pleasant Mexican musical drama.\n\n**747** _ **Cimarron**_ **** RKO, 1931. 131 min. D: Wesley Ruggles. SC: Howard Estabrook. With Richard Dix, Irene Dunne, Estelle Taylor, Nance O'Neil, William Collier, Jr., Roscoe Ates, George E. Stone, Stanley Fields, Robert McWade, Edna May Oliver, Frank Darien, Eugene Jackson, Dolores Brown, Gloria Vonic, Otto Hoffman, William Orlamond, Frank Dillaway, Junior Johnson, Douglas Scott, Lillian Lane, Henry Roquemore, Nell Craig, Bob McKenzie, Robert Kortman, Bud Flanagan (Dennis O'Keefe), William Janney, Frank Beal, Nancy Dover, Clara Hunt. A young woman marries a drifter-gunfighter and they get in the Oklahoma land rush but go their separate ways, she becoming a newspaper editor and later a member of Congress, while dies as an oil worker. This dated adaptation of Edna Ferber's novel won three Oscars, including best film and script, and is still worth viewing.\n\n**748** _ **Cimarron**_ **** Metro-Goldwyn-Mayer, 1960. 140 min. Color. D: Anthony Mann. SC: Arnold Schulman. With Glenn Ford, Maria Schell, Anne Baxter, Arthur O'Connell, Russ Tamblyn, Mercedes McCambridge, Vic Morrow, Robert Keith, Charles McGraw, Harry Morgan, David Opatoshu, Aline MacMahon, Lili Darvas, Edgar Buchanan, Mary Wickes, Royal Dano, L.Q. Jones, George Brelin, Vladimir Sokoloff, Helen Westcott, Ivan Triesault, Eddie Little Sky, Dawn Little Sky. A man with wanderlust marries a pretty girl and moves to the Oklahoma Territory where they split up, she to become a success while he drifts into obscurity. Indifferent remake of the Edna Ferber work, relying too much on color and modern film techniques and not enough on the book.\n\n**749** _ **The Cimarron Kid**_ **** Universal-International, 1951. 84 min. Color. D: Ted Richmond. SC: Louis Stevens. With Audie Murphy, Beverly Tyler, James Best, Yvette Dugay, John Hudson, Leif Erickson, Noah Berry, Jr., John Hubbard, Hugh O'Brian, Palmer Lee (Gregg Palmer), Rand Brooks, William Reynolds, Roy Roberts, David Wolfe, John Bromfield, Frank Silvera, Richard Garland, Eugene Baxter, Frank Ferguson, Tristram Coffin, Rory Mallinson, Jack Ingram, Harry Harvey, David Sharpe. A gunman, who leads a gang of bank robbers, falls in love with a woman who tries to get him to give up his life of crime. Average Audie Murphy vehicle, helped by Technicolor and a fine cast.\n\n**750** _ **Cinco Asesinos Esperan**_ (Five Waiting Killers). Alameda Films, 1964. 70 min. D: Chano Urueta. SC: Ramon Obon. With Carlos Cortes, Jorge Martinez de Hoyos, Noe Murayama, Dagoberto Rodriguez, Arturo Martinez, Sonia Infante, Alicia Caro, Jose Dupeyron, Humberto Dupeyron, Alfred Wally Barron, Luis Lomeli, Carlos Rotzinger. A sheriff finds his family is being threatened by an outlaw gang. More than passable Mexican Western.\n\n**751** _ **Cipolla Colt**_ **** Worldwide Entertainment Group, 1976. 96 min. Color. D: Enzo Girolami (Enzo G. Castellari). SC: Sergio Donati and Luciano Vincenzoni. With Franco Nero, Sterling Hayden, Martin Balsam, Emma Cohen, Leo Anchoriz, Ramano Puppo, Neno Zamperla, Massimo Vanni, Helmut Brasch, Duilio Curciani, Fernando Castro, Wal Davis, Dan van Husen, Dick Butkus, Daniel Martin, George (Jorge) Rigaud, Charly (Carlos) Bravo, David Warbeck, Antonio Pica, Xan das Bolas, Juan Antonio Rubio, Enzo G. Castellari, Manuel Zarzo. An onion farmer wants to settle on land he has purchased but the orphans of the former owner will not leave and an oil baron covets the property. Sloppy Italian genre comedy with too much slapstick; also called _**Spaghetti Western**_. Reissued by Joseph Green Pictures in 1980.\n\n**752** _ **Circle Canyon**_ **** Superior, 1934. 48 min. D: Victor Adamson (Denver Dixon). SC: B.R. (Burl) Tuttle. With Buddy Roosevelt, June Mathews, Clarise Woods, Bob Williamson (John Tyke), Allen Holbrook, Clyde McClary, Harry Leland, Bud Osborne, Mark Harrison, Ernest Scott, William McCall, Sherry Tansey, Barney Beasley, Tex Miller. A cowboy tries to protect his adopted daughter from outlaws who want her oil land. Bottom rung Buddy Roosevelt oater, also filmed the same year as _**'Neath the Arizona Skies**_ (q.v.).\n\n**753** _ **Circle of Death**_ **** Willis Kent, 1935. 60 min. D: J. Frank Glendon. SC: Roy Claire (S. Roy Luby) and Willis Kent. With Montie Montana, Tove Linden, Henry Hall, Yakima Canutt, Ben Corbett, J. Frank Glendon, Jack Carson, John Ince, Princess Ah-Tee-Ha, Richard Botiller, Chief Standing Bear, Slim Whitaker, Hank Bell, Budd Buster, Bart Carre, George Morrell, Olin Francis, Marin Sais, Bob Burns. The son of an Indian chief, actually a white man rescued by braves years before following a massacre, helps a rancher who is being blackmailed by those who believe there is gold on his land. Near the bottom of the barrel but still worth a look to see the great Montie Montana in his only starring Western.\n\n**754** _ **The Circus Cyclone**_ **** Universal, 1925. 55 min. D-SC: Albert S. Rogell. With Art Acord, Nancy Deaver, Albert J. Smith, Jim Corey, Cesare Gravina, Hilliard Karr, George F. Austin, Moe McCrea, Ben Corbett, Raven (horse), Rex (dog). A cowboy helps a pretty circus bareback rider lusted after by her boss who tries to frame her clown father for a bank robbery. Only part of this action filled Art Acord Universal Blue Steak Western is known to exist.\n\n_**Cisco**_ see _**El Cisco**_\n\n**755** _ **The Cisco Kid**_ **** Fox, 1931. 60 min. D: Irving Cummings. SC: Alfred A. Cohn. With Warner Baxter, Edmund Lowe, Conchita Montenegro, Nora Lane, Frederick Burt, Willard Robertson, James Bradbury, Jr., Jack (John Webb) Dillon, Charles Stevens, Chris-Pin Martin, Douglas Haig, Marilyn Knowlden, Rita Flynn, Consuelo Castillo de Bonzo. The Cisco Kid steals $5,000 to help a young woman pay off her ranch and that amount is placed on his head as reward money. Pleasing follow up to Warner Baxter's Academy Award winning performance as the Cisco Kid in **In Old Arizona** (q.v.).\n\n_**The Cisco Kid**_ (1966) see _**El Cisco**_\n\n**756** _ **The Cisco Kid**_ **** Turner Pictures, 1994. 91 min. Color. D: Luis Valdez. SC: Luis Valdez and Michael Kane. With Jimmy Smits, Cheech Marin, Sadie Frost, Bruce Payne, Ron Pearlman, Tony Amendola, Tim Thompson, Pedro Armendariz, Jr., Phil Esparza, Clayton Landey, Charles McGaugh, Tony Pandolfo, Roger Cudney, Joaquin Garrido, Guillermo Rios, Miguel Sandoval, Tomas Goros, Rufino Echegoyen, Teresa Lagunes, Honorato Magaloni, Luis Valdez, Yareli Arizmendi, Marisol Valdez, Julius Jansland, Mario Ecati Zapata, Mario Alberto, Boris Peguero, Maya Zapaa, Gerardo Zepeda, Lorena Victoria, Valentina Ponzanneli, Pedro Altamirano, Geraldo Martinez, Rojo Grau, Guido Bolanos, Roberto Olivo, Roberto Antunez, Pablo Zuack, Lakin Valdez, Mario Valdez, Luisa Coronel, Amelia Zapata, Alexandra Vicencio, Moctesuma Esparza, Susan Benedict, Corinna Duran, Patricia Brown, Claire Lewin, Carolyn Caldera, Herendia Silva. The Cisco Kid joins forces with Pancho, who he meets in jail, to sell weapons purchased from two former Confederates to revolutionaries trying to overthrow Emperor Maximilian in 1867 Mexico. Revisionist look at the O. Henry character in a none too good TV movie.\n\n**757** _ **The Cisco Kid and the Lady**_ **** 20th Century\u2013Fox, 1940. 74 min. D: Herbert I. Leeds. SC: Frances Hyland. With Cesar Romero, Marjorie Weaver, Chris-Pin Martin, George Montgomery, Virginia Field, Robert Barrat, Harry Green, John Beach, Ward Bond, J. Anthony Hughes, James Burke, Harry Hayden, James Flavin, Ruth Warren, Gloria Ann White, Eddy Waller, Adrian Morris, Ivan Miller, Virginia Brissac, Eddie Dunn, Arthur Rankin, Harry Strang, Lester Door, Paul Sutton, Paul E. Burns. The Cisco Kid gets involved with a crook trying to steal a gold mine, an orphaned baby and a woman who loves another man. First of a half-dozen \"Cisco Kid\" adventures headlining Cesar Romero; too long on romance and running time and too short on action.\n\n_**The Cisco Kid in In Old New Mexico**_ see _**In Old New Mexico**_\n\n_**The Cisco Kid in South of the Rio Grande**_ see _**South of the Rio Grande**_\n\n**758** _ **The Cisco Kid Returns**_ **** Monogram, 1945. 64 min. D: John P. McCarthy. SC: Betty Burbridge. With Duncan Renaldo, Martin Garralaga, Roger Pryor, Cecilia Callejo, Anthony Warde, Fritz Leiber, Vicky Lane, Jan Wiley, Sharon Smith, Cy Kendall, Eva Puig, Bud Osborne, Bob Duncan, Carl Mathews, Emmett Lynn, Elmer Napier, Jerry Shields, Walter Clinton, Neyle Morrow. The Cisco Kid and Pancho suspect a respected businessman of being behind a series of crimes. Duncan Renaldo's first appearance as \"The Cisco Kid\" is a standard affair that will appeal to his fans. TV title: **The Daring Adventurer**.\n\n**759** _ **City of Badmen**_ **** 20th Century\u2013Fox, 1953. 82 min. Color. D: Harmon Jones. SC: George W. George and George F. Slavin. With Dale Robertson, Jeanne Crain, Richard Boone, Lloyd Bridges, Carole Mathews, Carl Betz, Whitfield Connor, Hugh Sanders, Rodolfo Acosta, Pasquel Garcia Pena, Harry Carter, Robert Adler, John Doucette, Alan Dexter, Don Haggerty, Leo Gordon, Gil Perkins, John Day, James Best, Richard Cutting, Douglas Evans, Kit Carson, Barbara Fuller, Anthony Jochim, George Melford, George Selk, Charles Tannen, Tristram Coffin, Reed Howes, I. Stanford Jolley, Percy Helton, Frank Ferguson, Earle Hodgins. When the heavyweight championship boxing bout between James J. Corbett and Bob Fitzsimmons is staged in Carson City, Nevada, in 1897, outlaws plan to steal the box office receipts. Pretty good Western crime melodrama, highlighted by the restaging of the famous fight which was also the subject of _**Vigilantes of Boomtown**_ (q.v.).\n\n**760** _ **City Slickers**_ **** Columbia, 1991. 112 min. Color. D: Ron Underwood. SC: Lowell Ganz and Babaloo Mandel. With Billy Crystal, Daniel Stern, Bruno Kirby, Patricia Wettig, Helen Slater, Jack Palance, Noble Willingham, Tracey Walter, Josh Mostel, David Paymer, Bill Henderson, Jeffrey Tambor, Phil Lewis, Kyle Secor, Dean Hallo, Karla Tamburelli, Yeardley Smith, Roberto Costanzo, Walker Brandt, Molly McClure, Jane Alden, Lindsay Crystal, Jake Gyllenhaal, Danielle Harris, Eddie Palmer, Howard Honig, Fred Malo, Jayne Meadows, Alan Charof, Anne Lockhart, Lana Underwood, Robert Mickelson. A trio of middle aged businessmen decide to get away from it all and take part in a two week cattle drive where they come under the wing of hardened trail boss. Big box office comedy with Jack Palance winning a best supporting actor Oscar for his role of Curly Washburn.\n\n**761** _ **City Slickers II: The Legend of Curly's Gold**_ **** Columbia, 1994. 116 min. Color. D: Paul Weiland. SC: Billy Crystal, Lowell Ganz and Babaloo Mandel. With Billy Crystal, Daniel Stern, Jon Lovitz, Jack Palance, Patricia Wettig, Pruitt Taylor Vince, Bill McKinney, Lindsay Crystal, Beth Grant, Noble Willingham, David Paymer, Josh Mostel, Jayne Meadows, Alan Charof, Kenneth S. Allan, Jennifer Crystal, Molly McClure, Helen Siff, Irmise Brown, Bill McIntosh, Mario Roberts, Bob Balaban, James Boyd III, Kent Kasper, Lesley Boone. A radio advertising salesman teams with a friend and freeloader brother in trying to find gold hidden in the Arizona desert as revealed on a map left by an old cowboy. Fair follow up to _**City Slickers**_ that was not nearly as successful financially; wonderful Utah locations.\n\n**762** _ **The Civilized Men**_ **** NBC-TV\/Universal, 1969. 74 min. Color. With Robert Stack, Jack Kelly, Rod Cameron, Jill St. John, Kaz Garas, Susan Saint James, Phil Philbin. A former F.B.I. Agent, how the senior editor of a news magazine, travels to Florida to investigate modern-day cattle rustling on ranches there. Very good telefeature, originally an shown November 28, 1969, as a segment of \"The Name of the Game\" (NBC-TV, 1968\u201372).\n\n**763** _ **The Claim**_ **** United Artists, 2000. 120 min. Color. D: Michael Winterbottom. SC: Frank Cottrell Boyce. With Peter Mullan, Milia Jovoich, Wes Bently, Nastassia Kinski, Sarah Polley, Shirley Henderson, Julian Richings, Sean McGinley, Randy Birch, Tom McCamus, Frank Zotter, Artur Ciastkowski, Barry Ward, Karolina Muller, David Leareney, Valene Plache, Marie Brassard, Phillips Peak, Kate Hennig, Fernando Savalos, Marc Hollogne, Ron Anderson, Marty Antoni, Lydia Lau, Royal Sproule, Duncan Fraser, Karen Minish. Two decades after he sold his wife and daughter for a gold claim, a wealthy town boss is beset by a young woman claiming to be his offspring and surveyors working for an incoming railroad. Very fine adaptation of Thomas Hardy's novel _The Mayor of Casterbridge_.\n\n_**Claim Jumpers**_ see _**Lucky Texan**_\n\n**764** _ **Clancy of the Mounted**_ **** Universal, 1932. 12 Chapters. D: Ray Taylor. SC: Basil Dickey, Harry O. Hoyt and Ella O'Neill. With Tom Tyler, Jacqueline Wells (Julie Bishop), Earl McCarthy, William Desmond, Rosalie Roy, W.L. Thomas, Leon Duval, Francis Ford, Tom London, Edmund Cobb, William Thorne, Al Ferguson, Fred Humes, Frank Lackteen, Monte Montague, Steve Clemente. Crooks after a dead man's gold mine frame a Mountie's brother on a murder charge and the lawman is assigned to bring him to trial. Tom Tyler fans will love his action packed cliffhanger.\n\n**765** _ **Clash of the Wolves**_ **** Warner Bros., 1925. 60 min. D: Noel Mason Smith. SC: Charles A. Logue. With Rin Tin Tin, June Marlowe, Charles Farrell, Heinie Conklin, Will Walling, Pat Hartigan. A half-breed dog, the leader of a wolf pack, has a price on his head but his befriended by a borax prospector who he aids when a rival tries to kill him. Action filled Rin Tin Tin vehicle; good entertainment.\n\n_**The Claw Strikes**_ see _**Landrush**_\n\n**766** _ **Claws**_ **** Alaska Pictures, 1977. 90 min. Color. D: Richard Bansbach and R.E. Pierson. SC: Chuck D. Keen and Brian Russell. With Myron Healey, Leon Ames, Jason Evers, Anthony Caruso, Carla Layton, Glenn Sipes, Buck Young, Fred Otah. A killer grizzly terrorizes the north woods and hunters plan to destroy the beast. Beautifully photographed adventure tale made in Alaska and similar to _**Grizzly**_ (q.v.); also called _**The Devil Bear**_.\n\n**767** _ **Clearing the Range**_ **** Allied, 1931. 65 min. D: Otto Brower. SC: Jack Natteford. With Hoot Gibson, Sally Eilers, Hooper Atchley, Robert Homans, Ed Peil, Sr., George Mendoza, Edward Hearn, Maston Williams, Ben Corbett, Jim Corey, Eva Grippon. In trying to find out to killed his brother, a cowpoke pretends to be both a pacifist and the bandit El Capitan. Slow moving, nicely photographed (by Ernest Miller) with exciting fight sequences.\n\n**768** _ **Climb an Angry Mountain**_ **** NBC-TV\/Warner Bros., 1972. 100 min. Color. D: Leonard Horn. SC: Joseph Cavelli and Sam Rolfe. With Fess Parker, Marj Dusay, Arthur Hunnicutt, Barry Nelson, Stella Stevens, Joe Kapp, Clay O'Brien, Jewel Branch, Richard Brian Harris, Casey Tibbs, Kenneth Washington, J.C. McElroy. An Indian running from the law kidnaps a sheriff's son and heads up California's Mount Shasta, with the lawman and a New York City policeman, at odds over police procedure, in pursuit. Better than average telefeature with pleasant scenic values.\n\n**769** _ **Clint the Nevada Loner**_ **** Balcazar\/International Germania Film\/Lux Film, 1967. 92 min. Color. D: Alfonso Balcazar. SC: Alfonso Balcazar and Jose Antonio de la Loma. With George Martin, Marianne Koch, Gerhard Riedmann, Pinkas Braun, Xan das Bolas, Osvaldo Genazzani, Beni Deus, Francisco Jose Huetos, Remo De Angelis, Fernando Sancho, Renato Baldini, Walter Barnes, Gustavo Re, Luis Barboo, Paolo Gozlino. A gunman returns home and is taken back by his wife after he promises to give up his guns but a confrontation with a cattle baron and his gang causes him to lose the respect of his son. Well done Spaghetti Western followed by _**There's a Noose Waiting for You Trinity**_ (q.v.); also called _**Clint the Stranger**_.\n\n_**Clint the Stranger**_ see _**Clint the Nevada Loner**_\n\n_**The Clue**_ see _**Outcasts of Mesa Flats**_\n\n**770** _ **Cocaine Cowboys**_ **** International Harmony, 1979. 87 min. Color. D: Ulli Lommel. SC: Ulli Lommel, Spencer Compton, Tom. Sullivan and Victor Bockris. With Jack Palance, Tom Sullivan, Andy Warhol, Suzanna Love, Esther Bedham-Faran, Winnie Hollman, Richard Young, Tony Manafo, Richard Bassett, Pete Huckabee, The Cowboy Island Band. A rock group working on an album and planning a tour also makes a living smuggling dope. Strictly amateur night at Andy Warhol's house, where this \"Eastern Western\" dud was filmed; the nadir of Jack Palance's film career.\n\n**771** _ **The Cockeyed Cowboys of Calico County**_ **** Universal, 1970. 97 min. Color. D: Tony Leader. SC: Ranald MacDougall. With Dan Blocker, Nanette Nabray, Jim Backus, Wally Cox, Jack Elam, Jack Cassidy, Henry Jones, Stubby Kaye, Mickey Rooney, Noah Beery, Jr., Marge Champion, Donald Barry, Hamilton Camp, Tom Basham, Iron Eyes Cody, James McCallion, Byron Foulger, Ray Ballard. Fearing they will lose their blacksmith, whose mail order bride failed to arrive, the citizens of a Western town try to find him a mate. Dull comedy made for TV but first issued to theatres.\n\n**772** _ **Code of Honor**_ **** Syndicate, 1930. 55 min. D: J.P. McGowan. SC: G.A. Durlam. With Mahlon Hamilton, Doris Hill, Lafe McKee, Robert Graves, Jr., Stanley Taylor, Jimmy Aubrey, Harry Holden, William Dyer. A gambler falls for a young woman and goes up against a crook who has used her brother to obtain the land grant to her father's ranch. Obscure, cheaply made, early talkie.\n\n**773** _ **Code of the Cactus**_ **** Victory, 1939. 57 min. D: Sam Newfield. SC: Edward Halperin. With Tim McCoy, Dorothy Short, Dave O'Brien, Ben Corbett, Ted Adams, Alden Chase, Forrest Taylor, Bob Terry, Slim Whitaker, Frank Wayne, Kermit Maynard, Art Davis, Carl Mathews, Carl Sepulveda, Jimmy Aubrey, Clyde McClary, Jack King, Robert Walker, Lew Porter, George Morrell, Milburn Morante, Tex Palmer, James Sheridan, Jim Corey, Bob Card, Lee Burns, Reed Howes. Ranchers enlist the help of lawman Lightning Bill Carson to help stop a rang of rustlers using trucks to steal their cattle. Low budget but fast moving Tim McCoy \"Lightning Bill Carson\" entry.\n\n**774** _ **Code of the Fearless**_ **** Spectrum, 1939. 56 min. D: Raymond K. Johnson. SC: Fred Myton. With Fred Scott, Claire Rochelle, John Merton, Walter McGrail, George Sherwood, Harry Harvey, William Woods, Don Gallaher, Roger Williams, Carl Mathews, Frank LaRue, Gene Howard, James \"Buddy\" Kelly, Art Mix, George Morrell, Phil Dunham, Denver Dixon. A Texas Ranger pretends to be drummed out of the service to infiltrate an outlaw gang. The same old plot ploy does nothing for this average Fred Scott vehicle, nor do a trio of mediocre songs.\n\n**775** _ **Code of the Lawless**_ **** Universal, 1945. 60 min. D: Wallace Fox. SC: Patricia Harper. With Kirby Grant, Fuzzy Knight, Poni (Jane) Adams, Hugh Prosser, Barbara Sears, Edward M. Howard, Stanley Andrews, Rune H. Hultman, Rex Lease, Budd Buster, Edmund Cobb, Roy Brent, Pierce Lyden, Bob McKenzie, Pietro Sosso, Carey Harrison, Blackie Whiteford, Fred Graham, Brick Sullivan. A government agent poses as the son of the boss of a holding company illegally taxing local ranchers. Kirby Grant's second Universal film is an acceptable affair. Also called _**The Mysterious Stranger**_.\n\n**776** _ **Code of the Mounted**_ **** Ambassador, 1935. 60 min. D: Sam Newfield. SC: George W. Sayre. With Kermit Maynard, Robert Warwick, Lillian Miles, Syd Saylor, Wheeler Oakman, Eddie Phillips, Dick Curtis, Stanley Blystone, Roger Williams, Jim Thorpe, Jack Perrin, George Morrell, Artie Ortego, Frank McCarroll, Dick Botiller, Art Dillard, Carl Mathews, Jack Casey, Lester Dorr, Pascale Perry, Ben Hendricks, Jr. Two constables are assigned to bring in the gang responsible for the murders of fur trappers. Scenic action filled James Oliver Curwood adaptation later reworked as _**Dawn on the Great Divide**_ (q.v.).\n\n**777** _ **Code of the Outlaw**_ **** Republic, 1942. 57 min. D: John English. SC: Barry Shipman. With Bob Steele, Tom Tyler, Rufe Davis, Melinda Leighton, Weldon Heyburn, Bennie Bartlett, Don (Donald) Curtis, John Ince, Kenne Duncan, Phil Dunham, Max Waizmann, Chuck Morrison, Carleton Young, Al Taylor, Robert Frazer, Richard Alexander, Forrest Taylor, Jack Ingram, Wally West, Ed Peil, Sr., Bud Osborne, Hank Worden, Cactus Mack, Pascale Perry, Chuck Baldra, Sonny Bupp, Harry McKim, Jack Kirk, Bob Burns, Adele Smith, George Billings, Merlyn Nelson. An outlaw responsible for a mine payroll theft is hunted by the Three Mesquiteers. Typically fast entry in the Republic series with its likable trio stars, but not as good as those of yore.\n\n_**Code of the Plains**_ see _**The Renegade**_\n\n**778** _ **Code of the Pony Express**_ **** Columbia, 1950. 15 Chapters. D: Spencer Gordon Bennet. SC: David Matthews, Lewis Clay and Charles Condon. With Jock O'Mahoney (Mahoney), Dickie Moore, Peggy Stewart, William Fawcett, Tom London, Helena Dare, George J. Lewis, Pierce Lyden, Jack Ingram, Rick Vallin, Frank Ellis, Ross Elliott, Ben Corbett, Rusty Wescoatt. The Army assigns an undercover agent to find out who is behind a series of stagecoach raids, the work of a shady lawyer and his gang who work for an eastern syndicate out to corral transportation routes. Passable Columbia cliffhanger.\n\n**779** _ **Code of the Prairie**_ **** Republic, 1944. 56 min. D: Spencer Gordon Bennet. SC: Albert DeMond and Anthony Coldeway. With Smiley Burnette, Sunset Carson, Peggy Stewart, Weldon Heyburn, Tom Chatterton, Roy Barcroft, Bud Geary, Tom London, Jack Kirk, Tom Steele, Robert Wilke, Frank Ellis, Rex Lease, Henry Wills, Kenneth Terrell, Charles King, Nolan Leary, Hank Bell, Karl Hackett, Jack O'Shea, Horace B. Carpenter. A cowboy and his photographer pal help a woman and her father who plan to start a newspaper but are opposed by an outlaw gang secretly led by the town barber. Nifty Sunset Carson vehicle with Smiley Burnette along for some fun as the picture taker.\n\n**780** _ **Code of the Range**_ **** Columbia, 1936. 55 min. D: C.C. Coleman, Jr. SC: Ford Beebe. With Charles Starrett, Mary Blake, Ed Coxen, Allan Cavan, Ed Peil, Sr., Edmund Cobb, Ed LeSaint, Ralph McCullough, George Chesebro, Art Mix, Albert J. Smith. Cattlemen are at odds with each other over allowing sheep men to use range land for grazing their herds and a saloon owner attempts to inflame the situation for his own gain. Quite good Charles Starrett film which is very well written.\n\n**781** _ **Code of the Rangers**_ **** Monogram, 1938. 56 min. D: Sam Newfield. SC: Stanley Roberts. With Tim McCoy, Judith Ford, Rex Lease, Wheeler Oakman, Frank LaRue, Roger Williams, Zeke Clements, Kit Guard, Frank McCarroll, Jack Ingram, Budd Buster, Ed Peil, Sr., Hal Price, Herman Hack. Two brothers are Texas Rangers but one joins with outlaws and it is up to the other one to bring him to justice. Well paced Tim McCoy entry with good-bad guy work by Wheeler Oakman.\n\n**782** _ **Code of the Rangers**_ **** Sundown Productions, 1972. 58 min. D: Frank James. SC: Travis Cole. With Tex Hill, Linda Leon, Tony Harris, Billy Young, Ben Traywick. A Texas Ranger and his pals team with a saloon girl to thwart a dishonest politician trying to control a frontier town. Totally inept production not released until 2005 by Film Baby on DVD in a mercifully short 30-minute version.\n\n_**Code of the Redmen**_ see _**King of the Stallions**_\n\n**783** _ **Code of the Saddle**_ **** Monogram, 1947. 53 min. D: Thomas Carr. SC: Eliot Biggons. With Johnny Mack Brown, Raymond Hatton, Kay Morley, Riley Hill, William Norton Bailey, Zon Murray, Gary Garrett, Ken Duncan, Jr., Ted Adams, Bud Osborne, Boyd Stockman, Ray Jones, Chick Hannon. A rancher is murdered and two cowboys visiting him and his daughter attempt to find the killer, although a neighbor has been accused of the crime by the sheriff. A good story enhances this Johnny Mack Brown vehicle.\n\n**784** _ **Code of the Silver Sage**_ **** Republic, 1950. 60 min. D: Fred C. Brannon. SC: Arthur Orloff. With Allan \"Rocky\" Lane, Eddy Waller, Kay Christopher, Roy Barcroft, Rex Lease, Lane Bradford, William Ruhl, Richard Emory, Forrest Taylor, Kenne Duncan, Hank Patterson, John Butler. A madman plans to become dictator of the Arizona Territory and a U.S. cavalry lieutenant is sent to stop him. Another good one for Allan Lane; full of action with an interesting story line.\n\n**785** _ **Code of the West**_ **** Syndicate, 1929. 53 min. D: J.P. McGowan. SC: G.A. Durlam and Sally Winters. With Bob Custer, Vivian Ray, Bobby Dunn, Martin Cichy, Cliff Lyons, Bud Osborne, Tom Bay, Buck Bucko, Merrill McCormick, Dick Dickinson, Alfred Hewston. A railroad agent teams with a sheriff to capture a gang stealing express mail and then collecting on the insurance. Average silent Bob Custer entry.\n\n**786** _ **Code of the West**_ **** RKO Radio, 1947. 57 min. D: William Berke. SC: Norman Houston. With James Warren, Debra Allen, John Laurenz, Steve Brodie, Robert Clarke, Carol Foreman, Rita Lynn, Harry Woods, Raymond Burr, Harry Harvey, Phil Warren, Emmett Lynn, Forrest Taylor, William Desmond. Two cowboys try to help a man and his daughter who want to open an honest bank but are opposed by a corrupt town boss. Okay adaptation of the Zane Grey story.\n\n_**Coffin for the Sheriff**_ see _**Lone and Angry Man**_\n\n**787** _ **A Cold Day in Hell**_ **** Lions Gate, 2001. 95 min. Color. D: Christopher Forbes. SC: Christopher Forbes and Jim Hilton. With Michael Madsen, Ronald Bumgardner, Kimberly Campbell, Stan Fink, Ed Janostak, Michael Hilton, Jim Hilton, Tomme Hilton, Debra Carlsen, Heather Clark, Braxton Williams, David Topp, Dave Long, Tripp Courtney, Richard Kinsey, W. Clay Lee, Angela Lewis, Sara Lewis, Dave Long, Allison Tysinger, Christopher Forbes. Having lost his wife and home, a Civil War veteran heads to the high country hoping to lose himself in the wilderness. Mediocre low budget affair.\n\n**788** _ **Cold River**_ **** Shapiro, 1979. 94 min. Color. D-SC: Fred G. Sullivan. With Suzanne Weber, Pat Petersen, Richard Jaeckel, Robert Earl Jones, Brad Sullivan, Elizabeth Hubbard, Augusta Dabney. Modern-day outdoor adventure film about a man and an woman and their attempts to tame a raging river. Nice locations make this a pleasant adaptation of William Judson's best selling book.\n\n_**Cold Vengeance**_ see _**The Dawn Rider**_\n\n**789** _ **Cole Younger, Gunfighter**_ **** Allied Artists, 1958. 78 min. Color. D: R.G. Springsteen. SC: Daniel Manwaring. With Frank Lovejoy, Abby Dalton, James Best, Jan Merlin, Douglas Spencer, Frank Ferguson, Myron Healey, George Keymas, Dan Sheridan, John Mitchum, Ainslie Pryor. In early 1870s Texas, Cole Younger gets a reputation as a gunfighter for his opposition to corrupt lawmen. Frank Lovejoy handles the title role well and the film moves along at a good clip. Remake of _**The Desperado**_ (q.v.).\n\n**790** _ **Colorado**_ **** Republic, 1940. 57 min. D: Joseph Kane. SC: Louis Stevens and Harrison Jacobs. With Roy Rogers, George \"Gabby\" Hayes, Pauline Moore, Milburn Stone, Maude Eburne, Hal Taliaferro, Vestor Pegg, Fred Burns, Lloyd Ingraham, Jay Novello, Tex Palmer, Joseph Crehan, Ed Cassidy, Robert Fiske, Stanley Andrews. During the Civil War a Union lieutenant and his sidekick are sent to Denver to find out who is causing trouble with the Indians. Good Roy Rogers drama, fast paced and well acted with a fine desert finale.\n\n**791** _ **Colorado Ambush**_ **** Monogram, 1951. 51 min. D: Lewis D. Collins. SC: Myron Healey. With Johnny Mack Brown, Lois Hall, Myron Healey, Tommy Farrell, Christine McIntyre, Lyle Talbot, Lee Roberts, Marshall Bradford, John Hart. A ranger investigates the murders of three Wells Fargo agents and learns a man supplying horses to the freight hauler is also giving information to a hotel hostess. Myron Healey wrote this one and he also plays the dastardly villain, the highlight of this more than passable Johnny Mack Brown entry.\n\n**792** _ **Colorado Kid**_ **** Republic, 1937. 60 min. D: Sam Newfield. SC: Charles Francis Royal. With Bob Steele, Marion Weldon, Karl Hackett, Ted Adams, Ernie Adams, Frank LaRue, Horace Murphy, Kenne Duncan, Budd Buster, Frank Ball, John Merton, Horace B. Carpenter, Wally West. When he is unjustly accused of murder a cowboy escapes from jail to prove his innocence. Pretty action filled Bob Steele affair, slickly produced.\n\n**793** _ **Colorado Pioneers**_ **** Republic, 1945. 57 min. D: R.G. Springsteen. SC: Earle Snell. With Wild Bill Elliott, Bobby Blake, Alice Fleming, Roy Barcroft, Bud Geary, Billy Cummings, Freddie Chapman, Frank Jaquet, Tom London, Monte Hale, Buckwheat Thomas, George Chesebro, Emmett Vogan, Tom Chatterton, Ed Cassidy, Fred Graham, Horace B. Carpenter, Bill Woolf, Jack Rockwell, George Morrell, Jack Kirk, Roger Williams, Richard Lydon, Howard M. Mitchell, Frank O'Connor, Cliff Parkinson, Gary Armstrong, Bobby Anderson, Robert Goldschmidt, Romey Foley, Wally West, Rose Plummer, Chick Hannon, Jess Cavin. A gang of tough city kids sent West to be reformed help Red Ryder in stopping a rancher after the Duchess' land. An out-of-the-ordinary plot adds some zest to this \"Red Ryder\" segment.\n\n**794** _ **Colorado Ranger**_ **** Lippert, 1950. 57 min. D: Thomas Carr. SC: Ron Ormond and Maurice Tombragel. With James Ellison, Russell Hayden, Fuzzy Knight, Raymond Hatton, Betty (Julie) Adams, Tom Tyler, George J. Lewis, John Cason, Stanley Price, Dennis Moore, George Chesebro, Bud Osborne, Gene Roth, I. Stanford Jolley, Stephen Carr, Jimmie Martin, Joseph Richards. Shamrock and Lucky arrive at the former's family ranch to get his mother's inheritance and find his stepfather has been kidnapped. Arid entry in \"The Irish Cowboys\" series, with little to recommend it except its cast. TV title: _**Guns of Justice**_.\n\n**795** _ **Colorado Serenade**_ **** Producers Releasing Corporation, 1946. 68 min. Color. D: Robert Emmett Tansey. SC: Frances Kavanaugh. With Eddie Dean, Roscoe Ates, May Kenyon, David Sharpe, Forrest Taylor, Dennis Moore, Abigail Adams, Warner Richmond, Lee Bennett, Bob McKenzie, Bob Duncan, Charles King, Herman Hack, John Carpenter, George DeNormand, John Bridge. Two cowboys save a judge from being ambushed and learn one of the attackers is the man's son, who refuses to believe the jurist is his father. Action filled Eddie Dean vehicle, one of his better starring efforts.\n\n**796** _ **Colorado Sundown**_ **** Republic, 1952. 67 min. D: William Witney. SC: Eric Taylor and William Lively. With Rex Allen, Mary Ellen Kay, Slim Pickens, June Vincent, Fred Graham, John Daheim, Louise Beavers, Chester Clute, Clarence Straight, The Republic Rhythm Riders, Rex Lease, Tex Terry, Harry Harvey, Bud Osborne, Hal Price. While trying got help a friend keep a spread he inherited, a fellow rancher is accused of murder. Nicely done Rex Allen film, with emphasis on lots of movement.\n\n**797** _ **Colorado Sunset**_ **** Republic, 1939. 61 min. D: George Sherman. SC: Betty Bur bridge and Stanley Roberts. With Gene Autry, Smiley Burnette, June Storey, Barbara Pepper, Larry \"Buster\" Crabbe, Robert Barrat, William Farnum, Patsy Montana, Frankie Marvin, Purnell B. Pratt, Kermit Maynard, Jack Ingram, Elmo Lincoln, Ethan Laidlaw, Fred Burns, Jack Kirk, Budd Buster, Ed Cassidy, Slim Whitaker, Murdock MacQuarrie, Ralph Peters, The CBS-KMBC Texas Rangers, Francis Ford, Herman Hack, Chuck Baldra. A musical troupe buys a cattle ranch but the herd turns out to be milk cows and they find they are being pressured by crooks to join a combine. Pretty fair Gene Autry vehicle featuring the great Patsy Montana.\n\n**798** _ **Colorado Territory**_ **** Warner Bros., 1949. 94 min. D: Raoul Walsh. SC: John Twist and Edmund H. North. With Joel McCrea, Virginia Mayo, Dorothy Malone, Henry Hull, John Archer, James Mitchell, Morris Ankrum, Basil Ruysdael, Frank Puglia, Ian Wolfe, Harry Woods, Houseley Stevenson, Victor Kilian, Oliver Blake, Monte Blue, Charles Horvath, Hallene Hill, Fred Kelsey, Maudie Prickett, Artie Ortego, Jack Daly, Irene Elinor, Jack Montgomery, Bert Dillard, Ben Corbett, Frosty Royce, Glenn Thompson, Charles Miller, Steve Stephens, Merlyn Nelson, Carl Harbough, Paul Kruger, George Bell, Robert Filmer, Carl Andre, Harry Strang, Grey Eyes. An outlaw escapes from jail and joined by his girlfriend tries to hide but is trapped by a posse in a mountain area. Raoul Walsh's Western remake of his gangster classic _**High Sierra**_ (Warner Bros., 1941); a very good motion picture re-done by Warners in 1955 as the crime drama _**I Died a Thousand Times**_.\n\n**799** _ **Colorado Trail**_ **** Columbia, 1938. 55 min. D: Sam Nelson. SC: Charles Francis Royal. With Charles Starrett, Iris Meredith, The Sons of the Pioneers (Bob Nolan, Lloyd Perryman, Pat Brady, Hugh Farr, Karl Farr), Ed LeSaint, Alan Bridge, Robert Fiske, Dick Curtis, Hank Bell, Ed Peil, Sr., Edmund Cobb, Jack Clifford, Richard Botiller. A young man joins cattlemen in a range war with his father on the opposite side. Pretty fair Charles Starrett film.\n\n**800** _ **Colt Comrades**_ **** United Artists, 1943. 67 min. D: Lesley Selander. SC: Michael Wilson. With William Boyd, Andy Clyde, Jay Kirby, George Reeves, Lois Sherman, Earle Hodgins, Victor Jory, Douglas Fowley, Herbert Rawlinson, Robert Mitchum, Jack Mulhall, Russell Simpson, Dewey Robinson, Art Dillard, William Gould, Jack Shannon, Cliff Lyons, Bill Wolfe, Fred Kohler, Jr., Henry Wills, Blackjack Ward, Jim Corey, Roy Bucko, Ralph Bucko, Tex Phelps, George Sowards, Tex Cooper, George Plues. Crooks frame Hopalong Cassidy for cattle rustling and he has to prove his innocence. Standard entry in the \"Hopalong Cassidy\" series.\n\n**801** _ **Colt .45**_ **** Warner Bros., 1950. 74 min. Color. D: Edwin L. Marin. SC: Thomas Blackburn. With Randolph Scott, Ruth Roman, Zachary Scott, Lloyd Bridges, Alan Hale, Ian MacDonald, Chief Thundercloud, Luther Crockett, Walter Coy, Charles Evans, Buddy Roosevelt, Hal Taliaferro, Art Miles, Barry Reagan, Howard Negley, Paul Newlan, Aurora Navarro, Franklyn Fanrum, Ed Peil, Sr., Jack Watt, Carl Andre, Ben Corbett, Artie Ortego, Bob Burrows, William Steele. When his gun samples are stolen by an outlaw, a salesman is accused of being a member of the gang and attempts to capture the thief and prove his innocence. Pretty good Randolph Scott opus. TV title: _**Thunder Cloud**_.\n\n**802** _ **The Colt Is My Law**_ **** Proscensa, 1966. 93 min. Color. D: Al Bradley (Alfonso Brescia). SC: Al Bradley (Alfonso Brescia), Franco Cobianchi, Ramon Comas and Mario Musy. With Anthony Clark (Angel del Pozo), Lucy Gilly (Luciana Gilli\/Lucia Gil Fernandez), Michael Martin (Miguel de la Riva\/Michael Rivers), Peter White (Pietro Tordi), Jim Clay (Aldo Cecconi), Grant Laramie (Germano Longo), Henry Colt (Enrico Glori), Rafael Alcantara, Livio Lorenzon, Nino Nini, Franco Cobianchi, Milo Quesada, Stella Finney, Charles Johnson, Dan Silver. A railroad payroll disappears from a small town and two federal marshals try to get to the bottom of the trouble. Tepid, but violent, Spaghetti Western filmed as _**La Colt e la Mia Legge**_ (The Colt is My Law) and released in the U.S. as _**My Gun Is the Law**_ and in England as _**The Colt Is My Law**_. Issued on video as _**La Rey del Revolver**_ (The Law of the Revolver).\n\n**803** _ **Column South**_ **** Universal-International, 1953. 84 min. Color. D: Frederick De Cordova. SC: William Sackheim. With Audie Murphy, Joan Evans, Robert Sterling, Ray Collins, Palmer Lee (Gregg Palmer), Ralph Moody, Dennis Weaver, Johnny Downs, Russell Johnson, Bob Steele, Jack Kelly, Ray Montgomery, Richard Garland, James Best, Ed Rand, Rico Alaniz. In order to prevent fighting between Indians and Army troops, agitated by an intolerant captain, a young lieutenant tries to help the Navajos before they are forced into war. Fairly interesting Audie Murphy cavalry affair.\n\n**804** _ **Comanche**_ **** United Artists, 1956. D: George Sherman. SC: Carl Krueger. With Dana Andrews, Kent Smith, Linda Cristal, John Litel, Henry Brandon, Nestor Paiva, Mike Mazurki, Stacy Harris, Lowell Gilmore, Reed Sherman. Trying to halt skirmishes along the U.S.-Mexican border and to bring last peace with the Indians, two scouts are assigned to task of locating the Comanche chief and offering him peace. Director George Sherman infuses quite a bit of action into this oater to cover up a mundane story.\n\n**805** _ **Comanche Moon**_ **** CBS-TV, 2008. 360 min. Color. D: Simon Wincer. SC: Larry McMurtry and Diana Ossana. With Karl Urban, Steve Zahn, Ryan Merriman, Val Kilmer, Rachel Griffith, Linda Cardellini, Troy Baker, Melanie Lynskey, James Rebhorn, Arron Shiver, Wes Studi, Wally Welch, Elizabeth Banks, Adam Beach, Toby Metcalf, David Midthunder, Steve Reevis, Keith Robinson, Ray McKinnon, Rod Rondeaux, Kristine Sutherland, Frederick Lopez, Josh Berry, Scott Augare, Bill Flynn, Geraldine Keams, Jack Burning, Jake Busey, Brady Coleman, Grover Coulson, Jonathan Joss, Bill Flynn, Sal Lopez, Tatanka Means, Savion Rose, Nathon S. Lewis, Jonathan Scorza, T.A. Taylor, Christy Summer Lopez, Sarah Majors, Byran Lane, Jack Caffrey, Barbara Bartleson. Two Texas Rangers are on the trail of outlaws and also get involved with fighting Comanches. Big budget TV miniseries that is a confusing pre-sequel to _**Lonesome Dove**_ (q.v.).\n\n**806** _ **Comanche Station**_ **** Columbia, 1960. 74 min. Color. D: Budd Boetticher. SC: Burt Kennedy. With Randolph Scott, Nancy Gates, Skip Homeier, Richard Rust, Rand Brooks, Dyke Johnson, Foster Hood, Joe Molina, Vince St. Cyr, John Patrick Noland. A man enlists the aid of three outlaws in helping him find his wife who has been captured by Indians. Entertaining Randolph Scott feature, very well made and paced.\n\n**807** _ **Comanche Territory**_ **** Universal-International, 1950. 76 min. Color. D: George Sherman. SC: Oscar Brodney and Lewis Meltzer. With Maureen O'Hara, Macdonald Carey, Will Geer, Charles Drake, Pedro de Cordoba, Ian MacDonald, Rick Vallin, Parley Baer, James Best, Edmund Cobb, Glenn Strange, Iron Eyes Cody, Terry Frost, John Carpenter, I. Stanford Jolley, John Cason. When outlaws try to steal Indian lands because of rich silver deposits, frontiersman Jim Bowie comes to the rescue. Historical fiction turned into romantic pap.\n\n**808** _ **The Comancheros**_ **** 20th Century\u2013Fox, 1961. 107 min. Color. D: Michael Curtiz (and uncredited John Wayne). SC: James Edward Grant and Clair Huffaker. With John Wayne, Stuart Whitman, Ina Balin, Nehemiah Persoff, Lee Marvin, Michael Ansara, Patrick Wayne, Bruce Cabot, Joan O'Brien, Jack Elam, Edgar Buchanan, Guinn Williams, Bob Steele, Henry Daniell, Richard Devon, Steve Baylor, John Dierkes, Roger Mobley, Luisa Triana, Iphigenie Castiglioni, Aissa Wayne, George J. Lewis, Gregg Palmer, Don Brodie, Jon Lormer, Phil Arnold, Alan Carney, Ralph Volkie, Dennis Cole. A captain in the Texas Rangers teams with a gambler to thwart run runners and then carry a consignment of weapons to the stronghold of the Comancheros, white men allied with the Indians. Top notch John Wayne vehicle with lots of action and good humor; Guinn Williams has a great cameo as a dense gun runner.\n\n**809** _ **Come On, Cowboy!**_ **** Toddy Pictures, 1948. 72 min. With Mantan Moreland, Maurytne Brent, Johnny Lee, F.E. Miller, Fred Wilson. A man is sent West to prepare his boss' ranch for his arrival with a new bride not knowing outlaws are using it as a hideout, claiming the place is haunted. Obscure black cast comedy with songs, highlighted by the repartee between Mantan Moreland and Johnny Lee.\n\n**810** _ **Come on Cowboys!**_ **** Republic, 1937. 59 min. D: Joseph Kane. SC: Betty Burbridge. With Robert Livingston, Ray Corrigan, Max Terhune, Maxine Doyle, Ed Peil, Sr., Horace Murphy, Ann Bennett, Ed Cassidy, Roger Williams, Willie Fung, Fern Emmett, Yakima Canutt, Merrill McCormick, Al Taylor, George Plues, Milburn Morante, Carleton Young, George Morrell, Ernie Adams, Jim Corey, Jack Kirk, George Burton, Loren Riebe, Victor Allen, Jack O'Shea, Ernie Adams, Henry Hall, Tom Smith, Rose Plummer, James A. Marcus, Oscar Gahan. When an old pal from the circus gets involved with crooks the Three Mesquiteers come to his rescue. Action filled entry in \"The Three Mesquiteers' series with some big-top excitement thrown in for good measure.\n\n**811** _ **Come on Danger**_ **** RKO Radio, 1932. 60 min. D: Robert Hill. SC: Bennett Cohen. With Tom Keene, Julie Haydon, Roscoe Ates, Robert Ellis, Wade Boteler, William Scott, Harry Tenbrook, Bud Osborne, Roy Stewart, Frank Lackteen, Nell Craig, Monte Montague, Hank Potts, Puff Jones. A ranger and his pal set out to capture the lawman's brother's killer only to find the gang leader they are after is a young woman framed for the crime. Mature and well executed Tom Keene production remade as _**Renegade Ranger**_ (q.v.) and again under its original title (q.v.) in 1942.\n\n**812** _ **Come On, Danger!**_ RKO Radio, 1942. 58 min. D: Edward Killy. SC: Norton S. Parker. With Tim Holt, Frances Neal, Ray Whitley, Lee \"Lasses\" White, Karl Hackett, Bud McTaggart, Glenn Strange, Davison Clark, John Elliott, Slim Whitaker, Henry Roquemore, Evelyn Dickson, Kate Harrington. A Texas Ranger is assigned to bring in the female leader of a gang of outlaws and after she is wounded he discovers a crooked tax collector is behind the trouble. Passable third version of the 1932 Tom Keene film (q.v.), remade earlier as _**Renegade Ranger**_ (q.v.).\n\n**813** _ **Come On, Rangers**_ **** Republic, 1938. 57 min. D: Joseph Kane. SC: Gerald Geraghty and Jack Natteford. With Roy Rogers, Mary Hart, Raymond Hatton, J. Farrell MacDonald, Purnell B. Pratt, Harry Woods, Bruce MacFarlane, Lane Chandler, Chester Gunnels, Lee Powell, Robert Kortman, George (Montgomery) Letz, Frank McCarroll, Chick Hannon, Jack Kirk, Al Taylor, Horace B. Carpenter, Robert Wilke, Al Ferguson, Allan Cavan, Ben Corbett, Burr Caruth. Due to the lack of funds the Texas Rangers are disbanded and crooks pour into the state under the control of a dishonest senator behind a protection racket using a gang of raiders. Very entertaining and well made early Roy Rogers vehicle.\n\n**814** _ **Come On, Tarzan**_ **** World Wide, 1932. 61 min. D-SC: Alan James. With Ken Maynard, Merna Kennedy, Kate Campbell, Niles Welch, Roy Stewart, Ben Corbett, Bob McKenzie, Jack Rockwell, Nelson McDowell, Jack Mower, Edmund Cobb, Robert Walker, Hank Bell, Slim Whitaker, Jim Corey, Blackjack Ward, Al Taylor, Bud McClure. A ranch foreman, at odds with his pretty female boss, fights outlaws who are killing horses to be used as dog food. A bit different for Ken Maynard, but still a good film.\n\n**815** _ **The Comeback Trail**_ **** Dynamic Entertainment, 1982. 76 min. Color. D-SC: Harry Hurwitz. With Buster Crabbe, Chuck McCann, Ina Balin, Robert Staats, Jara Kahout, Henny Youngman, Professor Irwin Corey, Monti Rock III, Joe Franklin, Lenny Schultz, Hugh Hefner, Mike Gentry. Two dishonest movie producers hire a faded cowboy star to appear in their film, only they plan to kill him and collect the insurance. Western comedy made in 1970 that got some release in Canada in 1979 as _**Crazy Movie**_ ; except for Buster Crabbe, who is very good as one-time star Duke Montana, the film is a real bust.\n\n**816** _ **Comes a Horseman**_ **** United Artists, 1978. 118 min. Color. D: Alan J. Pakula. SC: Dennis Lynton Clark. With Jane Fonda, James Caan, Jason Robards, George Grizzard, Richard Farnsworth, Jim Davis, Mark Harmon, Macon McCalman, Basil Hoffman, James Kline, James Keach, Clifford A. Pellon. Small ranchers in 1940s Colorado are being squeezed out by a land hungry tyrant. Standard, but well made, production filmed in Colorado's Wet Mountain Valley and sporting a good performance by Richard Farnsworth as Dodger.\n\n**817** _ **Comin' at Ya!**_ Filmways, 1981. 91 min. Color. D: Ferdinando Baldi. SC: Lloyd Battista, Wolf Lowenthal and Gene Quintana. With Tony Anthony, Gene Quintana, Victoria Abil, Ricardo Palacios, Lewis Gordon. Two evil brothers working the white slave trade kidnap a cowboy's girlfriend and later leave her to die in the desert. Lumbering 3-D Spaghetti Western produced by star Tony Anthony.\n\n**818** _ **Comin' Round the Mountain**_ **** Republic, 1936. 55 min. D: Mack V. Wright. SC: Oliver Drake, Dorrell McGowan and Stuart McGowan. With Gene Autry, Smiley Burnette, Ann Rutherford, LeRoy Mason, Raymond Brown, Ken Cooper, Tracy Layne, Bob McKenzie, John Ince, Frank Lackteen, Frankie Marvin, Jim Corey, Al Taylor, Steve Clark, Frank Ellis, Hank Bell, Richard Botiller. Gene Autry comes to the aid of a young woman ranch owner who has had money stolen from her. Well made Gene Autry film.\n\n**819** _ **The Command**_ **** Warner Bros., 1953. 88 min. Color. D: David Butler. SC: Russell Hughes and Samuel Fuller. With Guy Madison, Joan Weldon, James Whitmore, Carl Benton Reid, Harvey Lembeck, Ray Teal, Bob Nichols, Don Shelton, Gregg Barton, Red Morgan, Jim Bannon, Reed Howes, Kermit Maynard, Iron Eyes Cody, Chubby Johnson. An Army captain leads his troops and civilians through Wyoming Territory, battling Indians and smallpox, in order to take possession of the area. Fairly good adaptation of James Warner Bellah's novel.\n**820** _ **Companeros!**_ **** 20th Century\u2013Fox, 1971. 118 min. Color. D: Sergio Corbucci. SC: Dino Maiuri, Massimo De Rita, Fritz Ebert and Sergio Corbucci. With Franco Nero, Jack Palance, Tomas Milian, Fernando Rey, Iris Berben, Francisco Bodalo, Eduardo Fajardo, Karin Schubert, Luizi Pernice, Alvarado De Luna, Jesus Fernandez, Claudio Scarchilli, Lorenzo Robelod, Gioanni Petti, Gerard Tichy, Giovanni Pulone, Simon Arriaga, Victor Israel. A Swedish mercenary works as a gun runner in revolution torn Mexico at the turn of the century. Fans of Franco Nero and Jack Palance may find some interest in this overlong, bloody Spaghetti Western.\n\n**821** _ **Con el Diablo en el Cuero**_ (With the Devil in the Body) **** Cinematografica Intercontinental, 1954. 90 min. D: Raul de Anda. SC: Raul de Anda and Gilberto Gazcon. With Luis Aguilar, Linda Cristal, Dagoberto Rodriguez, Jose L. Murillo, Domingo Soler, Emilio Garibay, Arturo Martinez, Jose Munoz, Enedina Diaz de Leon, Humberto Rodriguez, Juan Jose Hurato, Jose Eduardo Perez, Cecilia Leger, Ignacio Peon, Antonio Maciel, Jose Pardave, America Martin. A rancher falls in love with a beautiful woman but his past interferes with their happiness. Nicely done Mexican Western drama from producer-director-writer Raul de Anda.\n\n**822** _ **Conagher**_ **** Turner Network Television (TNT), 1991. 94 min. Color. D: Reynaldo Villalobos. SC: Jeffrey M. Meyer, Sam Elliott and Katharine Ross. With Sam Elliott, Katharine Ross, Barry Corbin, Billy Green Bush, Ken Curtis, Paul Koslo, Gavan O'Herlihy, James Parks, Daniel Quinn, Pepe Serna, Buck Taylor, Dub Taylor, Cody Braun, Anndi McAfee, James Gammon, Richard Jury, Jeffrey M. Meyer, Peter P. Oliver, Craig Pinkard, Archie Smith, Adam Taylor, R.L. Tolbert, Ben Quinters, John Furlough, Kate Hall, Angelique L'Amour, Ted White. An aging cowboy with wanderlust becomes involved with a pretty widow who is raising two children while managing a way station. First rate TV movie based on Louis L'Amour's book; co-adapted by stars Sam Elliott and Katharine Ross.\n\n**823** _ **The Concentratin' Kid**_ **** Universal, 1930. 60 min. D: Arthur Rosson. SC: Harold Tarshis. With Hoot Gibson, Kathryn Crawford, Duke R. Lee, Jim Mason, Robert E. Homans. A cowboy in love with a radio singer he has never met bets his pals he can win her or he will give them a radio. Fun early talkie from Hoot Gibson who also was its producer.\n\n_**Condemned in Error**_ see _**Quick on the Trigger**_\n\n**824** _ **Conflict**_ **** Universal, 1936. 60 min. D: David Howard. SC: Charles Logue and Walter Weems. With John Wayne, Jean Rogers, Ward Bond, Tommy Bupp, Bryant Washburn, Frank Sheridan, Harry Woods, Margaret Mann, Eddie Borden, Frank Hagney, Lloyd Ingraham, Glenn Strange, Bruce Mitchell, Harry Bowen, Ed Peil, Sr., Fred Parker, Richard Perry, Leonard Kibrick, Billie Morris, Walter Weems. A boxer works as a shill to cheat loggers in fixed boxing bouts until he turns honest after adopting an orphan boy and falling for a pretty reporter. Standard action program feature from producers Trem Carr and Paul Malvern, based on Jack London's novel _The Abysmal Brute_.\n\n**825** _ **The Conquering Horde**_ **** Paramount, 1931. 76 min. D: Edward Sloman. SC: Grover Jones and William McNutt. With Richard Arlen, Fay Wray, George Mendoza, Ian MacLaren, Claude Gilllingwater, James Durkin, Claire Ward, Charles Stevens, Arthur Stone, Frank Rice, Ed Brady, Robert Kortman, Harry Cording, John Elliott. After the Civil War a Texan returns home to help rebuild the state which is plagued by carpetbaggers. Old fashioned oater, a bit slow moving, but Richard Arlen and Fay Wray's fans will want to view it. A remake of _**North of '36**_ **** (Paramount, 1924).\n\n**826** _ **The Conquerors**_ **** RKO Radio, 1932. 88 min. D: William A. Wellman. SC: Robert Lord. With Richard Dix, Ann Harding, Edna May Oliver, Julie Haydon, Guy Kibbee, Donald Cook, Harry Holman, Skeets Gallagher, Walter Walker, Wally Albright, Jr., Marilyn Knowlden, Jason Robards, Jed Prouty, E.H. Calvert, J. Carrol Naish, Robert Greig, Elizabeth Patterson. A young couple marry and go West where they start a bank that proliferates into a financial empire which survives three panics. One of director William A. Wellman's most underrated features, the film spans the half-century between the 1870s and 1932 with Richard Dix particularly good as the financier. TV title: _**Pioneer Builders**_.\n\n**827** _ **Conquest of Cheyenne**_ **** Republic, 1946. 56 min. D: R.G. Springsteen. SC: Earle Snell. With Wild Bill Elliott, Bobby Blake, Alice Fleming, Peggy Stewart, Jay Kirby, Milton Kibbee, Tom London, Emmett Lynn, Kenne Duncan, George Sherwood, Frank McCarroll, Jack Kirk, Tom Chatterton, Ted Mapes, Jack Rockwell, Bob Burns, Jack O'Shea, Bert Dillard, LeRoy Mason (voice). When a corrupt banker tries to steal a pretty girl's oil lands, Red Ryder comes to the rescue. Another fine \"Red Ryder\" film, well handled by director R.G. Springsteen.\n\n**828** _ **Conquest of Cochise**_ **** Columbia, 1953. 70 min. Color. D: William Castle. SC: Arthur Lewis and DeVallon Scott. With Robert Stack, John Hodiak, Joy Page, Rico Alaniz, Fortunio Bonanova, Edward Colmans, Alex Montoya, Steven Ritch, Carol Thurston, John Crawford, Rodd Redwing, Robert E. Griffith, Joseph Waring. In the 1850s cavalry officers are sent to New Mexico to keep the peace and stop raids by Cochise and his braves. Nothing special about this color effort.\n\n**829** _ **Convict Cowboy**_ **** Metro-Goldwyn-Mayer, 1995. 98 min. Color. D: Rod Holcomb. SC: Rick Way and Jim Lindsay. With Jon Voight, Kyle Chandler, Marcia Gay Harden, Ben Gazzara, Glenn Plummer, Stephen McHattie, Dean Wray, Tom Heaton, Jeremy Ratchford, Bill Crook, Zook Matthews, Fred Perron, Tyron Beskin, Nathaniel DeVeaux, Dave Houlsen, Matt Huson, Mark Acheson, Deejay Jackson, Truman Hoszouski, Chris Nannarone, Sefan Stasiuk. An older cowboy is sent to prison where he develops a bond with a younger prisoner as they tend horses, join in a rodeo and try to avoid getting involved in the drug trade. Okay modern-day prison\u2013Western made for TV.\n\n**830** _ **Convict Stage**_ **** 20th Century\u2013Fox, 1965. 71 min. D: Lesley Selander. SC: Daniel Mainwaring. With Harry Lauter, Donald Barry, Jodi Mitchell, Hanna Landy, Joe Patridge, Eric Matthews, Walter Reed, Michael Carr, Fred Krone, George Sawaya, Karl MacDonald, Fred Beir. A man vows revenge on the two brothers who murdered his sister and plans to get them as they are being taken to prison by stagecoach. Fair programmer, thanks mainly to director Lesley Selander and its two stars.\n\n**831** _ **Coogan's Bluff**_ **** Universal, 1968. 100 min. Color. D: Don Siegel. SC: Herman Miller, Dean Riesner and Howard Rodman. With Clint Eastwood, Susan Clark, Lee J. Cobb, Tisha Sterling, Don Stroud, Betty Field, Tom Tully, Melodie Johnson, James Edwards, Rudy Diaz, David Doyle, Marjorie Bennett. An Arizona deputy sheriff comes to New York City to track down and extradite a killer. Sturdy, action filled Clint Eastwood melodrama which his fans will love.\n\n**832** _ **Copper Canyon**_ **** Paramount, 1950. 83 min. Color. D: John Farrow. SC: Jonathan Latimer. With Ray Milland, Hedy Lamarr, Macdonald Carey, Mona Freeman, Harry Carey, Jr., Frank Faylen, Hope Emerson, Taylor Holmes, Peggy Knudsen, James Burke, Percy Helton, Philip Van Zandt, Francis Pierlot, Erno Verebes, Paul Lees, Robert Watson, Georgia Backus, Ian Wolfe, Robert Kortman, Nina Mae McKinney, Len Hendry, Earle Hodgins, Robert Stephenson, Buddy Roosevelt, Julia Faye, Joe Whitehead, Hank Bell, Ethan Laidlaw, Russell Kaplan, Alan Dinehart III, Rex Lease, Stanley Andrews, Kit Guard, Stuart Holmes, Trevor Bardette, Erville Alderson. In the post\u2013Civil War West a former soldier joins a carnival as a sharpshooter and gets involved with a beautiful woman. Glossy affair without much interest except to look at Hedy Lamarr.\n\n**833** _ **Copper Sky**_ **** 20th Century\u2013Fox, 1957. 79 min. D: Charles Marquis Warren. SC: Eric Norden. With Jeff Morrow, Coleen Gray, Strother Martin, Paul Brinegar, John Pickard, Patrick O'Moore, Rocky Shahan, Rush Williams, Rodd Redwing. A drunken ex-soldier and a school teacher survive an Indian attack on a remote town and then trek across the desert to the nearest outpost. Stars Jeff Morrow and Coleen Gray try hard but the arid script defeats them.\n\n**834** _ **Cornered**_ **** Columbia, 1932. 60 min. D: B. Reeves Eason. SC: Wallace MacDonald. With Tim McCoy, Raymond Hatton, Noah Beery, Shirley Grey, Niles Welch, Claire McDowell, Walter Long, Walter Brennan, Wheeler Oakman, Robert Kortman, Edmund Cobb, Tom London, Lloyd Ingraham, Charles King, John Elliott, Art Mix, Merrill McCormick, Artie Ortego, Jim Corey, Ed Peil, Sr., Ray Jones, Jack Evans, Blackie Whiteford, Jack Kirk. A sheriff and a ranch foreman both like the same girl but when her father is murdered the latter is blamed, escapes from jail and joins an outlaw gang. Top grade Tim McCoy vehicle dominated by madman villain Noah Beery who says there are two things worth living for, \"to kill or be killed\" and \"to get revenge.\"\n\n**835** _ **Coroner Creek**_ **** Columbia, 1948. 90 min. Color. D: Ray Enright. SC: Kenneth Gamet. With Randolph Scott, Marguerite Chapman, George Macready, Sally Eilers, Edgar Buchanan, Barbara Reed, Wallace Ford, William Bishop, Forrest Tucker, Joseph Sawyer, Russell Simpson, Douglas Fowley, Lee Bennett, Forrest Taylor, Phil Schumacher, Warren Jackson. A cowpoke, with the help of a pretty hotel owner, plans revenge on the man responsible for the death of his girl friend. High standard Randolph Scott color opus.\n\n**Advertisement for** _**Coroner Creek**_ **(Columbia, 1948).**\n\n**  \n**\n\n**836** _ **Corpus Christi Bandits**_ **** Republic, 1945. 55 min. D: Wallace Grissell. SC: Norman S. Hall. With Allan \"Rocky\" Lane, Helen Talbot, Twinkle Watts, Tom London, Francis McDonald, Jack Kirk, Roy Barcroft, Kenne Duncan, Robert Wilke, Ed Cassidy, Emmett Vogan, Neal Hart, Horace B. Carpenter, Hal Price, Frank Ellis, Frank McCarroll, Henry Wills, Cliff Parkinson, Eva Novak, George Bell, Carl Faulkner. A pilot learns the story of how his grandfather became an outlaw because of carpetbaggers after the Civil War. A different kind of plot adds zest to this above average Allan Lane vehicle.\n\n**837** _ **El Correo del Norte**_ (The Northern Courier) **** Universal, S.A., 1960. 65 min. D: Zacarias Gomez Urquiza. With Luis Aguilar, Jaime Fernandez, Fernando Alamada, Rosa de Castilla, Arturo Martinez, Jose Chavez, Rosario Galvez, Fernando Fernandez. A secret society is involved in trading weapons between rebels and government troops during the Mexican Revolution. Standard \"B\" effort from south of the border.\n\n_**Cost of Dying**_ see _**Taste of Death**_\n\n**838** _ **Cotter**_ **** Gold Key, 1973. 94 min. Color. D: Paul Stanley. SC: William D. Gordon. With Don Murray, Carol Lynley, Rip Torn, Sherry Jackson, R.G. Armstrong, Lonny Chapman, James McCallion, Michael Forest, Ford Rainey, Larry D. Mann, Mark Allen, Carolyn Fleming, Walter Scott, Christopher Knight. After losing his job in a rodeo due to drink, an Indian returns home only to be blamed for the murder of a wealthy rancher and chased by a lynch mob. Nicely done, although somewhat obscure, modern-day oater.\n\n**839** _ **Cougar**_ **** Sidney A. Snow Productions, 1933. 70 min. With Jay Bruce, Edwin C. Hill, Ranger (dog), Sidney A. Snow (narrator). An expedition heads into California's Caly Hills in search of mountain lions and other game. Good vintage documentary also called _**Cougar, the King Killer**_.\n\n**840** _ **Cougar Country**_ **** Gold Key, 1970. 91 min. Color. With Ernest Wilkinson, Whiskers (cougar), Michael Rye (narrator). A cougar, over a two year span, grows from a cub to a powerful hunter. Filmed in southern Colorado, this outdoor adventure is ideal family fare.\n\n_**Cougar, the Killer**_ see _**Cougar**_\n\n**841** _ **Count the Clues**_ **** Wrather Corporation, 1956. 75 min. Color. D: Earl Bellamy and Oscar Rudolph. SC: Doane Hoag, Wells Root, Robert E. Schaefer and Eric Friewald. With Clayton Moore, Jay Silverheels, Allen Pinson, Wayne Burson, Richard Crane, Claire Carleton, Bud Osborne, William Challee, Rand Brooks, Slim Pickens, Mickey Simpson, Steven Ritch, House Peters, Jr., Jason Johnson, Frank Scanner, Gordon Mills, Roy Engle, Barbara Knudsen, Sydney Mason, Walt LaRue, Ron Hagerthy, Lee Roberts, John Berardino, Tudor Owen, Carlos Vera, Brad Morrow, Baynes Barron. The Lone Ranger and Tonto fight blackmailers, abet a man against outlaws and chase a robbery gang into the badlands. Entertaining \"Lone Ranger\" telefeature made up of three 1956\u201357 episodes of the popular ABC-TV series: \"Wooden Rifle,\" \"Sheriff of Smoketree\" and \"Ghost Town Fury.\"\n\n**842** _ **Count the Hours**_ **** RKO Radio, 1953. 76 min. D: Don Siegel. SC: Doane R. Hoag. With Teresa Wright, Macdonald Carey, Dolores Moran, Adele Mara, Edgar Barrier, John Craven, Jack Elam, Ralph Sanford, Ralph Dumke, John Harmon, Richard Kipling, Norman Rice, Kay Riehl, Lee Phelps, I. Stanford Jolley, William E. Green, Edward Hearn, Sam Flint, George Pembroke, Roy Engel, Michael McHale, Gene Roth, Charles Victor, Brick Sullivan, Gayle Kellogg, Jess Kirkpatrick, Lee Morgan, Marshall Bradford, Richard Emory, Dolores Fuller, Harlan Howe, Ralph Brooks, Al Hill, Jack Carr, Paul Hoffman, Benny Burt, Robert Carson, Vernon Rich, Lorin Raker, Herbert Lytton, Richard Norris, Kathleen O'Malley, Michael Vallon, Lanny Rees, Joey Ray, Dick Scott, Allan Ray, Carl Sklover. A ranch hand is falsely accused of murdering the couple he worked for and his wife and a district attorney try to prove his innocence. Taut modern-day oater shot in only nine days; worth viewing.\n\n**843** _ **Count Three and Pray**_ **** Columbia, 1955. 102 min. Color. D: George Sherman. SC: Herb Meadows. With Van Heflin, Joanne Woodward, Philip Carey, Raymond Burr, Allison Hayes, Myron Healey, Nancy Kulp, James Griffith, Richard Webb, Kathryn Givney, Robert Burton, Vince Townsend, John Carson, Jean Willes, Adrienne Marden, Steve Raines, Jimmy Hawkins, Juney Ellis. After the Civil War a man with a past becomes a pastor in a small town and is enamored with an orphaned girl. Okay melodrama with good dramatics from its stars.\n\n_**Count Your Blessings**_ see _**Face to the Wind**_\n\n**844** _ **The Country Beyond**_ **** 20th Century\u2013Fox, 1936. 69 min. D: Eugene Forde. SC: Lamar Trotti and Adele Commandini. With Rochelle Hudson, Paul Kelly, Robert Kent, Alan Hale, Alan Dinehart, Matt McHugh, Andrew Tombes, Paul McVey, Claudia Coleman, Holmes Herbert, Jack Mulhall, Creighton Hale, Harry Strang, Pat O'Malley, Chester Gan, Chief Thundercloud, Charles Stevens, Niles Welch, Landers Stevens, Lew Harvey, Fred Walton, George H. Reed, Buck (dog), Prince (wolf). A young woman and her dog aid two Mounties in capturing a murderous fur thief. More than satisfactory north country follow-up to _**Call of the Wild**_ (1935) [q.v.].\n\n**845** _ **The Courage of Kavil, the Wolf Dog**_ **** NBC-TV, 1980. 100 min. Color. D: Peter Carter. SC: George Malko. With Ronny Cox, John Ireland, Linda Sorenson, Andrew Ian McMillan, Chris Wiggins. Taken from his family, a champion sled dog undergoes the arduous trek of 2,000 miles through the Alaskan wilderness to return to them. Average TV fare with nice scenery.\n\n_**The Courage of Rin Tin Tin**_ see _**The Challenge of Rin Tin Tin**_\n\n**846** _ **Courage of the North**_ **** Stage and Screen, 1935. 55 min. D-SC: Robert Emmett (Tansey). With John Preston, June Love, William Desmond, Tom London, Jimmy Aubrey, Charles King, James Sheridan (Sherry Tansey), Jim Thorpe, Montie Montana, Dynamite (horse), Captain (dog). A gang of fur thieves working in the north country is tracked by a Canadian Mounted Policeman. Low budget north woods affair with a wooden leading man and good photography by Brydon Baker.\n\n**847** _ **Courage of the West**_ **** Universal, 1937. 56 min D: Joseph H. Lewis. SC: Norton S. Parker. With Bob Baker, Lois January, J. Farrell MacDonald, Fuzzy Knight, Carl Stockdale, Harry Woods, Albert Russell, Charles K. French, Oscar Gahan, Richard Cramer, Jack Montgomery, Thomas Monk, Buddy Cox. After outlaws rob Wells Fargo messengers and express offices, rangers are assigned to stop them. Bob Baker's initial series outing is a fairly fast affair, helped by a good script and direction.\n\n**848** _ **The Courageous Avenger**_ **** Supreme, 1935. 58 min. D: Robert North Bradbury. SC: Charles Francis Royal. With Johnny Mack Brown, Helen Erickson, Warner Richmond, Ed Cassidy, Frank Ball, Eddie Parker, Forrest Taylor, Bob Burns, Earl Dwire, George Morrell, Wally West, Herman Hack, Art Dillard, Francis Walker, Fred Parker. A wagon driver is murdered and a sheriff investigates, learning outlaws are tapping a silver vein and using prisoners to mine it. Average Johnny Mack Brown film with a rather novel plot.\n\n**849** _ **The Court-Martial of General George Armstrong Custer**_ **** NBC-TV\/Warner Bros., 1977. 100 min. Color. D: Glenn Jordan. SC: John Gay. With Brian Keith, James Olson, Ken Howard, Blythe Danner, Stephen Elliott, Richard Dysart, Nicholas Coster, J.D. Cannon, William Daniels, James Ray. Teledrama about what might have occurred had General Custer survived the Battle of the Little Big Horn. Based on Douglas C. Jones' book, it was originally shown on \"The Hallmark Hall of Fame\" on NBC-TV on December 1, 1977, and is of interest to history buffs.\n\n**850** _ **Courtin' Trouble**_ **** Monogram, 1948. 58 min. D: Ford Beebe. SC: Ronald Davidson. With Jimmy Wakely, Dub Taylor, Virginia Belmont, Leonard Penn, Steve Clark, Marshall Reed, House Peters, Jr., Frank LaRue, Bob Woodward, Bud Osborne, Boyd Stockman, Bill Bailey, Bill Potter, Bill Hale, Carol Henry, Don Weston, Louis Armstrong, Arthur Smith, Frank Ellis, Ray Jones. A singing cowboy returns home to find warfare between businessmen and cattle ranchers. Fair Jimmy Wakely musical opus with an action filled second half.\n\n**851** _ **Courtin' Wildcats**_ **** Universal, 1929. 56 min. D: Jerome Storm. SC: Dudley McKenna. With Hoot Gibson, Eugenia Gilbert, Monte Montague, Joseph Girard, James Farley, Harry Todd, John Oscar, Lon Poff, Pete Morrison, Joe Bonomo, Fred Gilman, Arthur Millett, Ben Corbett, Frank Ellis, Jim Corey, Blackie Whiteford, Iron Eyes Cody. A free spirited college man is put into a wild west show where he wins the heart of a pretty bronco rider after he helps her when she shoots a crook. Hoot Gibson's second talkie is a fair adaptation of William Dudley Pelly's novel _Courtin' Calamity_.\n\n**852** _ **The Covered Wagon**_ **** Paramount, 1923. 98 min. D: James Cruze. SC: Jack Cunningham. With J. Warren Kerrigan, Lois Wilson, Ernest Torrence, Charles Ogle, Ethel Wales, Alan Hale, Tully Marshall, Guy Oliver, Johnny Fox, Tim McCoy. Two wagon trains leave Kansas City for Oregon but one of them cuts off from the main convoy and heads for the California gold rush. One of the pioneer epic Westerns highlighted by its semi-documentary style and Karl Brown's photography; slow by today's standards but still a must see for all genre followers.\n\n**853** _ **Covered Wagon Days**_ **** Republic, 1940. 56 min. D: George Sherman. SC: Earle Snell. With Robert Livingston, Raymond Hatton, Duncan Renaldo, Kay Griffith, George Douglas, Ruth Robinson, Paul Marion, John Merton, Tom Chatterton, Guy D'Ennery, Tom London, Reed Howes, David Newell, Jack Kirk, Al Taylor, Lee Shumway, Edward Earle, Elias Gamboa, Richard Alexander, Edward Hearn, Art Mix, Frank McCarroll, Herman Hack, Kenneth Terrell, Tex Palmer, Jack Montgomery, Bob Card, Arthur Loft, Rosa Turich, Barry Hays, Pascale Perry, Herman Howlin, Roy Bucko, Chick Hannon, Henry Wills, Bud McClure. The Three Mesquiteers get mixed up with silver smugglers when a dishonest businessman, the head of the gang, tries to force Rico's uncle to sell his mine. Pretty fair south-of-the-border \"Three Mesquiteers\" segment.\n\n**854** _ **Covered Wagon Raid**_ **** Republic, 1950. 60 min. D: R.G. Springsteen. SC: M. Coates Webster. With Allan \"Rocky\" Lane, Eddy Waller, Lyn Thomas, Alex Gerry, Byron Barr, Dick Curtis, Marshall Reed, Pierce Lyden, Sherry Jackson, Rex Lease, Lester Dorr, Lee Roberts, Edmund Cobb, Wee Willie Keeler. A cowboy is on the trail of a vicious outlaw gang terrorizing a small community. Nicely done Allan Lane action oater.\n\n**855** _ **Covered Wagon Trails**_ **** Syndicate, 1930. 50 min. D: J.P. McGowan. SC: Sally Winters. With Bob Custer, Phyllis Bainbridge, Perry Murdock, Charles Brinley, Martin Cichy, J.P. McGowan, Bud Osborne, Cliff Lyons. Lawman Smoke Sanderson is after a gang of crooks working along the Mexican border and falls for the sister of one of the outlaws. Without being hampered by dialogue, Bob Custer comes across fairly well in this silent effort with a music score.\n\n**856** _ **Covered Wagon Trails**_ **** Monogram, 1940. 52 min. D: Raymond K. Johnson. SC: Tom Gibson. With Jack Randall, Sally Cairns, Lafe McKee, David Sharpe, Budd Buster, Glenn Strange, Hank Bell, Kenne Duncan, Frank Ellis, George Chesebro, Carl Mathews, Edward Hearn, Art Mix, Jack Montgomery, Frank McCarroll, John Elliott, Tex Terry, Jimmy Aubrey. A cowboy opposes corrupt cattlemen who are trying to stop settlers from farming the range. So-so Jack Randall vehicle.\n\n**857** _ **Cow Country**_ **** Allied Artists, 1953. 82 min. D: Lesley Selander. SC: Tom W. Blackburn and Adele Buffington. With Edmond O'Brien, Helen Westcott, Robert Lowery, Barton MacLane, Peggie Castle, Robert Barrat, James Millican, Don Beddoe, Robert Wilke, Raymond Hatton, Chuck Courtney, Steve Clark, Rory Mallinson, Marshall Reed, Tom Tyler, Sam Flint, Jack Ingram, George J. Lewis, Brett Houston, Lane Chandler, Lee Roberts, Chuck Roberson, Ray Jones. In the Texas Panhandle of the 1880s ranchers struggle to keep their spreads despite drought and depression, along with the machinations of a dishonest banker. Downbeat oater that is well made and worth watching; from the novel _Shadow Range_ by Curtis Bishop.\n\n**858** _ **Cow Town**_ **** Columbia, 1950. 70 min. D: John English. SC: Gerald Geraghty. With Gene Autry, Gail Davis, Harry Shannon, Jock (Mahoney) O'Mahoney, Clark \"Buddy\" Burrroughs, Harry Harvey, Steve Darrell, Sandy Sanders, Ralph Sanford, Bud Osborne, Robert Hilton, Ted Mapes, Charles (Chuck) Roberson, House Peters, Jr., Blackie Whiteford, Pat O'Malley, Victor Cox, Frankie Marvin, Herman Hack, Frank McCarroll, Felice Raymond, Holly Bane, Ray Jones. When he supports the use of barbed wire to stop rustling, Gene Autry finds himself disliked by a female rancher and in the middle of a range war. Action filled Gene Autry vehicle with several good songs, including \"Down in the Valley\" and \"Powder Your Face with Sunshine.\"\n\n**859** _ **Cowboy**_ **** Columbia, 1958. 92 min. Color. D: Delmer Daves. SC: Edmund H. North. With Glenn Ford, Jack Lemmon, Anna Kashfi, Brian Donlevy, Dick York, Victor Manuel Mendoza, Richard Jaeckel, King Donovan, Vaughn Taylor, Donald Randolph, James Westerfield, Frank De Kova, Eugene Iglesias, Buzz Henry, William Leslie, Guy Wilkerson. A young hotel clerk in the 1870s joins a cattle drive and is toughened into a man with the help of the trail boss. Delightful drama, realistic and entertaining.\n\n**860** _ **Cowboy**_ **** CBS-TV, 1983. 100 min. Color. D: Jerry Jameson. SC: Stanley Cherry, Carole Cherry and Dennis Capps. With James Brolin, Annie Potts, Randy Quaid, Ted Danson, George DiCenzo, Edward Holmes, Robert Keith, Jerry Gatlin, Dan Doucette, Ben Scott. A former teacher returns home to find crooks are after his ranch. Made-for-television modern Western that holds up pretty well.\n\n**861** _ **The Cowboy**_ **** Lippert, 1954. 69 min. Color. D: Elmo Williams. SC: Lorraine Williams. With Tex Ritter, William Conrad, John Dehner, Lawrence Dobkin (narrators). The history and present day existence of the American cowboy, shown on the trail, at roundups, rodeos, festivals, etc. A very good documentary and one worth viewing; besides partially narrating Tex Ritter sings \"Dodge City Trail\" on the soundtrack; issued on DVD as _**The True Story of...The Cowboy**_.\n\n**862** _ **The Cowboy and the Bandit**_ **** Superior, 1935. 57 min. D: Al Herman. SC: Jack Jeyne. With Rex Lease, Janet Morgan (Blanche Mehaffey), Bobby Nelson, Richard Alexander, Wally Wales, William Desmond, Bill Patton, Franklyn Farnum, Art Mix, Lafe McKee, Ben Corbett, George Chesebro, Victor Potel, Jack Kirk, Herman Hack, George Morrell, Bud Pope. When an outlaw gang tries to take her ranch a young widow is helped by a fun-loving cowboy. Cheaply produced Rex Lease vehicle.\n\n**863** _ **The Cowboy and the Blonde**_ **** 20th Century\u2013Fox, 1941. 68 min. D: Ray McCarey. SC: Walter Bullock. With Mary Beth Hughes, George Montgomery, Alan Mowbray, Robert Conway, John Miljan, Richard Lane, Robert Emmett Keane, Minerva Urecal, Fuzzy Knight, George O'Hara, Mae Marsh, Trevor Bardette, Robert Homans, Tom London, Monica Bannister, William Halligan, Robert Cornell, Charles Tannen, Pauline Garon, Ralph Dunn, Hugh Beaumont, Pat West, Harry Strang, Robert Homans, Jack Chefe, Bettye Avery, Lillian Porter, Barbara Pepper, Frank Fanning, Albert Conti, Harold Goodwin, Donald Kerr, Kitty McHugh, Jill Dennett, Ernie Alexander, Addie McPhail. A real-life cowboy attempts to become a Western star but fails a screen test and ends up romancing a beautiful blonde actress. For fans of the two stars only.\n\n**864** _ **The Cowboy and the Indians**_ **** Columbia, 1949. 68 min. D: John English. SC: Dwight Cummins and Dorothy Yost. With Gene Autry, Sheila Ryan, Frank Richards, Hank Patterson, Clayton Moore, Jay Silverheels, Claudia Drake, George Nokes, Charles Stevens, Alex Frazer, Frank Lackteen, Chief Yowlachie, Lee Roberts, Nolan Leary, Maudie Prickett, Harry Macklin, Charles Quigley, Gilberto Alonzo, Roy Gordon, Jose Alvarado, Ray Beltram, Felipe Gomez, Iron Eyes Cody, Shooting Star, Romere Darlaing, Evelyn Finley. A young brave is blamed when the chief of the Navajo tribe is murdered by a trader and his men but Gene Autry and a female doctor try to prove his innocence. A good script highlights this Gene Autry outing in which he sings four songs, including \"Here Comes Santa Claus.\"\n\n**865** _ **The Cowboy and the Kid**_ **** Universal, 1936. 58 min. D: Ray Taylor. SC: Frances Guihan. With Buck Jones, Dorothy Revier, Billy Burrud, Harry Worth, Charles LeMoyne, Dick Rush, Lafe McKee, Bob McKenzie, Burr Caruth, Eddie Lee, Kernan Cripps, Oliver Eckhardt, Mary Mersch, Mildred Gover. A happy-go-lucky cowpoke blames himself for the death of a rancher and decides to raise the man's orphaned son. Good Buck Jones vehicle with a fine mixture of drama, comedy and pathos.\n\n**866** _ **The Cowboy and the Lady**_ **** United Artists, 1938. 91 min. D: H.C. Potter. SC: Sonya Levien. With Gary Cooper, Merle Oberon, Patsy Kelly, Walter Brennan, Fuzzy Knight, Mabel Todd, Henry Kolker, Harry Davenport, Emma Dunn, Walter Walker, Berton Churchill, Charles Richman, Frederick Vogeding, Arthur Hoyt, Ernie Adams, Russ Powell, Irving Bacon, George Chandler, Jack Baxley, Johnny Judd, Billy Wayne, Mabel Colcord. The snobbish daughter of a presidential candidate meets and falls in love with a lanky rodeo cowboy. Producer Samuel Goldwyn's teaming of Gary Cooper and Merle Oberon in this Western-comedy is now a dated bore.\n\n**867** _ **Cowboy and the Prizefighter**_ **** Eagle-Lion, 1949. 59 min. Color. D: Lewis D. Collins. SC: Jerry Thomas. With Jim Bannon, Little Brown Jug (Don Kay Reynolds), Emmett Lynn, Marin Sais, Don Haggerty, Karen Randle, Lou Nova, John Hart, Lane Bradford, Marshall Reed, Forrest Taylor, Frank Ellis, Bud Osborne, Steve Clark, Frank O'Connor, Herman Hack, Ray Jones, Jack Low. To help a pal prove his father did not commit suicide, Red Ryder agrees to a prize fight with a giant boxer managed by the crooked promoter who is the killer. Only a fair \"Red Ryder\" entry and the last of a quartet of Cinecolor efforts produced by Equity Pictures starring Jim Bannon as the Fred Harman character.\n\n**868** _ **The Cowboy and the Senorita**_ **** Republic, 1944. 78 min. D: Joseph Kane. SC: Gordon Kahn. With Roy Rogers, Mary Lee, Dale Evans, Guinn Williams, Bob Nolan and The Sons of the Pioneers (Tim Spencer, Ken Carson, Hugh Farr, Karl Farr), John Hubbard, Hal Taliaferro, Jack Kirk, Fuzzy Knight, Dorothy Christy, Lucien Littlefield, Jack O'Shea, Rex Lease, Lynton Brent, Julian Rivero, Robert Wilke, Wally West, Jane Beebe, Ben Rochelle, Spanky McFarland, Kirk Alyn, Cappella and Patricia, Tito Valdes, Corinne Valdes. Two cowboys are falsely accused of kidnapping a young girl and after she gives them a job on her ranch they find out that crooks are after high grade ore in her late father's mine. Passable Roy Rogers entry, with Guinn Williams as a good comedy sidekick, hampered by mediocre songs and production numbers; the initial teaming of Roy Rogers and Dale Evans.\n\n_**The Cowboy Angel**_ see _**Christmas Mountain**_\n\n**869** _ **Cowboy Blues**_ **** Columbia, 1946. 65 min. D: Ray Nazarro. SC: J. Benton Cheney. With Ken Curtis, Jeff Donnell, Guy Kibbee, Guinn Williams, Mrs. Uppington (Isabel Randolph), Robert Scott, Peg La Centra, The Town Criers, Deuce Spriggins' Band, Carolina Cotton, The Plainsmen (Andy Parker, George Barnby, Paul Birch, Charles Morgan), The Hoosier Hotshots (Paul \"Hezzie\" Trietsch, Ken Trietsch, Gil Taylor, Charles \"Gabe\" Ward), Alan Bridge, Vernon Dent, Jack Rockwell, Forbes Murray, Dick Elliott, Henry Vridon, Coulter Irwin. Two cowboys try to help a fellow ranch employee whose snobbish daughter, her fiance and his mother, think he is the owner. Hodgepodge of comedy and music featuring bucolic guest stars and eleven songs.\n\n**870** _ **Cowboy Canteen**_ **** Columbia, 1944. 72 min. D: Lew Landers. SC: Paul Gangelin and Felix Adler. With Charles Starrett, Jane Frazee, Vera Vague, Guinn Williams, Dub Taylor, Max Terhune, Emmett Lynn, Edythe Elliott, Bill Hughes, John Tyrrell, The Mills Brothers, Jimmy Wakely and His Saddle Pals, Chickie and Buck, Roy Acuff and His Smokey Mountain Boys and Girls, The Tailor Maids, Ted French, Ben Taggart, Herbert Heyes, Eleanor Counts, Joanne Frank, Vivian Mason, Craig Woods,. A ranch owner joins the Army and finds his newly hired hands are all female with the service sending him back home to establish a canteen. Fun musical Western with a thin plot and plenty of guest stars.\n\n**871** _ **Cowboy Cavalier**_ **** Monogram, 1948. 57 min. D: Derwin Abrahams. SC: Ronald Davidson and J. Benton Cheney. With Jimmy Wakely, Dub Taylor, Jan Bryant, Douglas Evans, Claire Whitney, William Ruhl, Steve Clark, Milburn Morante, Bud Osborne, Carol Henry, Bob Woodward. A stage-freight line operated by a young woman is being harassed by bandits with a singing cowboy and his pal coming to her rescue. Typically low grade and not very entertaining Jimmy Wakely vehicle.\n\n**872** _ **Cowboy Commandos**_ **** Monogram, 1943. 55 min. D: S. Roy Luby. SC: Elizabeth Beecher. With Ray Corrigan, Dennis Moore, Max Terhune, Evelyn Finley, Johnny Bond, Budd Buster, John Merton, Edna Bennett, Steve Clark, Bud Osborne, Frank Ellis, Hank Bell, Denver Dixon, Artie Ortego, George Chesebro, Ray Jones, Pascale Perry, Augie Gomez, Jack Evans, Herman Hack, Kansas Moehring, Archie Ricks, Jack Tornek, Foxy Callahan, Carl Sepulveda. The Range Busters uncover a nest of Nazis trying to sabotage a magnesium mine. Delightful \"Range Busters\" series entry; Johnny Bond sings \"I'll Shoot the Fuehrer, Sure as Shootin'.\"\n\n**873** _ **The Cowboy Counselor**_ **** Allied, 1932. 62 min. D: George Melford. SC: Jack Natteford. With Hoot Gibson, Sheila Mannors, Skeeter Bill Robbins, Bobby Nelson, Fred Gilman, Jack Rutherford, William Humphreys, Gordon DeMain, Merrill McCormick, Alan Bridge, Frank Ellis, Slim Whitaker. A frontier lawyer finds himself at odds with a gang of crooks. Leisurely paced and somewhat humorous Hoot Gibson film lacking the budget necessary to make it really good.\n\n**874** _ **The Cowboy from Brooklyn**_ **** Warner Bros., 1938. 80 min. D: Lloyd Bacon. SC: Earl Baldwin. With Dick Powell, Pat O'Brien, Priscilla Lane, Dick Foran, Ann Sheridan, Ronald Reagan, Johnnie David, Emma Dunn, Granville Bates, James Stephenson, Hobart Cavanaugh, Elisabeth Risdon, Dennis Moore, Rosella Towne, May Boley, Harry Barris, Candy Candido, Donald Briggs, Jeffrey Lynn, John Ridgely, William Davidson, Mary Field, Emmett Vogan, Eddy Chandler, Ben Hendricks, Dorothy Vaughn, Monty Vandergrift, Sam Hayes, Jack Moore, Eddie Graham, Jack Wise, Cliff Saum. A crooner with a fear of animals gets a job at a dude ranch in Wyoming and a slick promoter tries to make him as a singing cowboy. Fun Hal B. Wallis genre spoof redone as _**Two Guys from Texas**_ (q.v.).\n\n**875** _ **Cowboy from Lonesome River**_ **** Columbia, 1944. 55 min. D: Benjamin Kline. SC: Luci Ward. With Charles Starrett, Vi Athens, Dub Taylor, Kenneth MacDonald, Ian Keith, John Tyrrell, Bud Geary, Steve Clark, Jack Rockwell, Ozie Waters, Jimmy Wakely and His Saddle Pals (Arthur A. Wenzel, Shelby D. Atchison, Foy [Willing] Willingham, Al Sloey), Craig Woods, Frank O'Connor, Frank LaRue, Davison Clark, Eleanor Counts, Vivian Mason, Judy Malcolm, Eddie Bruce. A dishonest water company executive has a senator opposing him killed and replaced with his look-a-like brother while the head of a ranchers' group tries to stop him from over charging them. Okay modern-day oater giving dual roles to Kenneth MacDonald.\n\n**876** _ **The Cowboy from Sundown**_ **** Monogram, 1940. 58 min. D: Spencer Gordon Bennet. SC: Roland Lynch and Robert Emmett (Tansey). With Tex Ritter, Pauline Haddon, Roscoe Ates, Carleton Young, George Pembroke, Dave O'Brien, Patsy Moran, Tristram Coffin, Chick Hannon, Arkansas Slim Andrews, Bud Osborne, Glenn Strange, Wally West, Sherry Tansey. A sheriff is forced to quarantine cattle due to hoof and mouth disease and this angers ranchers who need to get herds to market to stop a banker from foreclosing their mortgages. A fairly interesting plot helps this Tex Ritter outing.\n\n**877** _ **Cowboy Holiday**_ **** Beacon, 1934. 57 min. D: Robert Hill. SC: Rock Hawkey (Robert Hill). With Guinn Williams, Janet Chandler, Julian Rivero, Richard Alexander, John Elliott, Alma Chester, Frank Ellis, William Gould, Julia Bejarano. A man disguised as a Mexican bandit causes havoc on the border and a cowboy plans to bring him to justice. Cheaply made but rugged, and often amusing, Guinn \"Big Boy\" Williams film.\n\n**878** _ **Cowboy in the Clouds**_ **** Columbia, 1943. 55 min. D: Benjamin Kline. SC: Elizabeth Beecher. With Charles Starrett, Dub Taylor, Julie Duncan, Jimmy Wakely, Hal Taliaferro, Charles King, Davison Clark, Dick Curtis, Ed Cassidy, Ted Mapes, John Tyrrell, Paul Zarema, The Jesters, Foy Willing, Bryant Washburn, Vernon Dent, Henry Hall, Shelby Atkinson, Patti Sheldon, Walter Carlson, Guy Bonham, Gwen Seager, Dwight Latham. A cowpoke fights for his country by joining the Civil Air Patrol and combating enemy agents. Topical and well done.\n\n**879** _ **The Cowboy Millionaire**_ **** Fox, 1935. 74 min. D: Edward Cline. SC: George Waggner and Dan Jarrett. With George O'Brien, Evelyn Bostock, Edgar Kennedy, Alden Chase, Maude Allen, Dan Jarrett, Lloyd Ingraham, Thomas Curran. During her vacation at a dude ranch, a titled English woman falls for a cowboy and after many misunderstandings they eventually find happiness. As much of a romantic comedy as a Western, this outing should appeal to George O'Brien fans.\n\n_**Cowboy Reckoning**_ see _**Enemy of the Law**_\n\n_**Cowboy Roundup**_ see _**Ride 'Em Cowboy (1936)**_\n\n**880** _ **Cowboy Serenade**_ **** Republic, 1942. 66 min. D: William Morgan. SC: Olive Cooper With Gene Autry, Smiley Burnette, Fay McKenzie, Cecil Cunningham, Rand Brooks, Addison Richards, Tristram Coffin, Arkansas Slim Andrews, Melinda Leighton, Johnny Berkes, Forrest Taylor, Hank Worden, Si Jenks, Ethan Laidlaw, Hal Price, Bud Wolfe, Forbes Murray, Bud Geary, Frankie Marvin, Tom London, Kenneth Terrell, Ken Cooper. When professional gamblers get control of a cattle herd, Gene Autry plans to retrieve the beef. Fair Gene Autry vehicle with some good music.\n\n**881** _ **The Cowboy Star**_ **** Columbia, 1936. 56 min. D: David Selman. SC: Frances Guihan. With Charles Starrett, Iris Meredith, Si Jenks, Marc Lawrence, Ed Peil, Sr., Wally Albright, Ralph McCullough, Landers Stevens, Winifred Hari, Nick Copeland, Lew Meehan, Richard Terry, Frank Melton, Robert Fiske, Ann Merill, Richard Powell, Lucille Lund, Eric Alden, James McDonald, Eleanor Huntley, Gale Goodson, Jane Weir. Wanting a rest, a cowboy film star goes to a small town incognito and there proves himself a real-life hero. Breezy and entertaining tongue-in-cheek jab at \"B\" Westerns; lots of fun for fans.\n\n**882** _ **Cowboy Up**_ **** Destination Films, 2001. 105 min. Color. D: Xavier Koller. SC: James Redford. With Kiefer Sutherland, Marcus Thomas, Daryl Hannah, Melinda Dillon, Molly Ringwald, Russell Means, Anthony Lucero, Bo Hopkins, Peter Postlethwaite, Timothy Daly, Julian Daly, Al Corley, Georginia Lightning, Nataanil Nez Means, Robert G. Miranda, Steven Barr, Robyn Peterson, James Lurie, Brian Connell, Judd Leffew, Kieu Chinh, Kerstin Caujolle, Tiffany Beard, Bret Leffew, Dave \"Rooster\" Kuden, Ernie Garrett, Helena Quintanar, Luanne Robinson, Eddie Kutz, Bill Dunn, Karina Logue, Donnie Gray, Pam Minick, Michael Hollon, Brian Moore, Patrick Cunningham, Paul F. Foster, Marty O'Brien, Blain Street, Billy Potoroff, Elizabeth Fields, Walter Ludwig. Two brothers working the rodeo circuit, one a bull rider and the other a clown, have a falling out when one of them falls for a beautiful performer. Not much to brag about in this Las Vegas filmed drama, made in 1998 and given some release as _**Ring of Fire**_.\n\n**883** _ **The Cowboys**_ **** Warner Bros., 1972. 121 min. Color. D: Mark Rydell. SC: Irving Ravetch, Harriet Frank, Jr. and William Dale Jennings. With John Wayne, Roscoe Lee Browne, Bruce Dern, Colleen Dewhurst, Slim Pickens, A. Martinez, Alfred Barker, Jr., Nicholas Beauvy, Steve Benedict, Robert Carradine, Norman Howells, Jr., Stephen Hudis, Sean Kelly, Clay O'Brien, Sam O'Brien, Mike Pyeat, Lonny Chapman, Sarah Cunningham, Charles Tyner, Allyn Ann McLerie, Matt Clark, Jerry Gatlin, Tap Canutt, Chuck Courtney, Henry Wills, Joe Yrigoyen, Casey Tibbs, Chuck Roberson, Kent Hays, Gary Epper, J.R. Randall. When his drovers quit, a cattleman rounds up a group of boys and trains them to drive his herd to market. Handsomely made and very good, although violent, John Wayne vehicle; recommended. Frank De Kova as Chief Joseph was cut from the final release print. The film served as the source for \"The Cowboys\" (ABC-TV, 1974) starring Jim Davis.\n\n**884** _ **Cowboys and Aliens**_ **** Universal, 2011. 118 min. Color. D: Jon Favreau. SC: Robert Orci, Alex Kurtzman, Damon Lindelof, Mark Fergus and Haw Ostby. With Daniel Craig, Harrison Ford, Abigail Spencer, Buck Taylor, Olivia Wilder, Sam Rockwell, Matthew Taylor, Cooper Taylor, Clancy Brown, Paul Dano, Chris Browning, Adam Beach, Ana de la Reguera, Noah Ringer, Brian Duffy, Keith Carradine, Brendan Wayne, Gavin Grazer, Toby Husa, Wyatt Russell, Jimmy Jatho, Kenny Call, Walton Goodins, Julio Cesar Cedillo, Garnett James Noel, David O'Hara, Troy Gilbert, Chad Randal, Scout Hendrickson, Raoul Trujillo. A man suffering from amnesia in 1870s Arizona comes to realize he has seen the vanguard of an space alien invasion and attempts to get the locals to stop them. Somewhat anemic sci-fi Western.\n\n_**Cowboys and Zombies**_ see _**The Dead and the Damned**_\n\n**885** _ **Cowboys Don't Cry**_ **** Atlantis Films, 1988. 96 min. Color. D-SC: Anne Wheeler. With Ron White, Zachary Ansley, Janet-Laine Green, Val Pearson, Candace Ratcliffe, Thomas Hauff, Rebecca Jenkins, Michael Hogan, Thomas Peacock, Joshua Ansley, Michael Hogan, Janet Wright, Barney O'Sullivan, Wendell Smith, Graham McPherson, Jason Wolff, Georgie Collins, William Korbut, Ruby Swekla, Ryan Byrne, Ernie Marshall, Bill Kehler, Ivan Daines. Following the death of his wife in an auto accident, a heavy drinking cowboy tries to make a new life with his teenage son on a ranch they inherited. Okay modern-day drama; somewhat confusing.\n\n**886** _ **Cowboys from Texas**_ **** Republic, 1939. 57 min. D: George Sherman. SC: Oliver Drake. With Robert Livingston, Raymond Hatton, Duncan Renaldo, Carole Landis, Betty Compson, Charles Middleton, Ethan Laidlaw, Yakima Canutt, Walter Wills, Ed Cassidy, Bud Osborne, Charles King, Forbes Murray, Horace Murphy, Henry Strang, Jack Kirk, David Sharpe, Lew Meehan, Jack O'Shea, Charles Miller, Ivan Miller, Harry McKim, Murdock MacQuarrie, William Nestell, Al Haskell. When cattle ranchers and homesteaders declare war over open range, the Three Mesquiteers try to bring the matter to a peaceful solution. Another slick, speedy entry in the long running series based on William Colt MacDonald's literary characters.\n\n**887** _ **Cowboy's Run**_ **** American World Pictures, 2003. 83 min. Color. D: Alan Smithee (Philip Spink). SC: Annie Frazier Henry. With David Hasselhoff, Gordon Tootoosis, Michael Moriarty, Kimberly Hawthorne, Steven Cree Molison, Barb Mitchell, Michelle Thrush, Vincent Gale. A former rodeo rider and an Indian lawyer, who dislike each other, are accused of a bingo robbery and head into the wilderness followed by an incompetent law officer. Hard to follow, light hearted Canadian production; also called _**Fugitives Run**_.\n\n**888** _ **El Coyote**_ (The Coyote) **** Centauro Films\/Oro Films, 1955. 75 min. D: Joaquin Luis Romero Marchent. SC: J. Chamor (Pedro Chamorro) and Jesus (Jess) Franco. With Abel Salazar, Gloria Marin, Manuel Monrov, Rafael Bardem, Santiago Rivero, Antonio Garcia Quijada, Mario Moreno, Jose Calvo, Xan das Bolas, Alfred Munoz, Angel Alvarez. In the mid\u20131800s a mild mannered young man returns to California and becomes a masked avenger fighting corrupt officials. Standard Mexico-Spain \"Zorro\" imitation co-production, followed by _**La Justicia del Coyote**_ (q.v.).\n\n**889** _ **Coyote Trails**_ **** Reliable, 1935. 60 min. D: Bernard B. Ray. SC: Rose Gordon. With Tom Tyler, Alice Dahl, Ben Corbett, Lafe McKee, Richard Alexander, Slim Whitaker, George Chesebro, Lew Meehan, Jack Evans, Art Dillard, Jimmy Aubrey, Bud McClure, Tex Palmer, Phantom (horse). Two cowboys try to capture a stallion who they believe has been falsely accused of rustling a rancher's horses. The story has been done both before and since and usually much better than this shoddy Tom Tyler effort.\n\n**890** _ **Crashin' Through**_ **** Anchor, 1924. 50 min. D: Robert J. Horner. SC: Alvin J. Neitz (Alan James). With Jack Perrin, Aline Goodwin, Jack Richardson, Steve Clement, Dick La Reno, Jena Riley, Taylor Graves. Wanting to sell his ranch to a man who seeks real Western wildness, a cowboy plans such a masquerade only to have a real outlaw gang show up. Fun tongue-in-cheek silent Jack Perrin film.\n\n**891** _ **Crashing Broadway**_ **** Monogram, 1933. 61 min. D: John P. McCarthy. SC: Wellyn Totman. With Rex Bell, Doris Hill, Harry Bowen, Charles King, George Hayes, Ann Howard, Blackie Whiteford, Perry Murdock, Henry Roquemore, Gordon DeMain, Tex Palmer, George Morrell. Heading East for the first time, a cowpoke runs into trouble with hoodlums in the big city. Breezy Rex Bell vehicle.\n\n**892** _ **Crashing Thru**_ **** Monogram, 1939. 60 min. D: Elmer Clifton. SC: Sherman Lowe. With James Newill, Jean Carmen, Warren Hull, Dave O'Brien, Milburn Stone, Robert Frazer, Walter Byron, Stanley Blystone, Joseph Girard, Earl Douglas, Ted Adams, Roy Barcroft, Iron Eyes Cody, Horace Murphy, Wally West. A brother and sister are accused of hijacking a gold shipment by a pair of Mounties, although they claim a mining company owner is the culprit. Second and final entry in the \"Renfrew of the Mounted\" series for Grand National Pictures (although the studio folded and it was issued by Monogram) and a pleasing one.\n\n**893** _ **Crashing Thru**_ **** Monogram, 1949. 58 min. D: Ray Taylor. SC: Adele Buffington. With Whip Wilson, Andy Clyde, Christine Larson, Kenne Duncan, Tristram Coffin, George J. Lewis, Jan Bryant, Virginia Carroll, Steve Darrell, Jack Richardson, Tom Quinn, Dee Cooper, Boyd Stockman, Bob Woodward, Merrill McCormick, Wally West. An undercover insurance agent poses as a murdered ranger to trap the gang responsible for the killing. Whip Wilson's first starring vehicle is well written and produced plus greatly helped by a fine supporting cast.\n\n_**Crazy Horse and Custer**_ see _**The Legend of Custer**_\n\n_**Crazy Horse and Custer\u2014The Untold Story**_ see _**The Legend of Custer**_\n\n_**Crazy Movie**_ see _**The Comeback Trail**_\n\n_**Crazy Westerners**_ see _**Rita of the West**_\n\n**894** _ **The Crimson Trail**_ **** Universal, 1935. 58 min. D: Al Rabock. SC: Jack Natteford. With Buck Jones, Polly Ann Young, Carl Stockdale, Charles K. French, Ward Bond, Robert Kortman, Bud Osborne, Paul Fix, Robert Walker. Two rival ranchers oppose each other in an election but when one of them is shot the man's nephew tries to find the culprit and falls in love with the other man's daughter. Somewhat complicated, but appealing Buck Jones fare.\n\n**895** _ **Cripple Creek**_ **** Columbia, 1952. 78 min. Color. D: Ray Nazarro. SC: Richard Shayer. With George Montgomery, Karin Booth, Jerome Courtland, William Bishop, Richard Egan, Don Porter, John Dehner, Robert Armstrong, Roy Roberts, George Cleveland, Byron Foulger, Cliff Clark, Harry Cording, Chris Alcaide, Robert Bice, Grandon Rhodes, Peter Brocco, John Hamilton, Emmett Lynn. When outlaws steal shipments from gold mines, two government agents try to reveal their identities by pretending to be bandits. Nothing special about this George Montgomery feature from producer Edward Small.\n\n**896** _ **Crooked River**_ **** Lippert, 1950. 58 min D: Thomas Carr. SC: Ron Ormond and Maurice Tombragel. With James Ellison, Russell Hayden, Betty (Julie) Adams, Fuzzy Knight, Raymond Hatton, Tom Tyler, George J. Lewis, John Cason, Stanley Price, Stephen Carr, Dennis Moore, George Chesebro, Bud Osborne, Jimmie Martin, Cliff Taylor, Helen Gibson, Carl Mathews, George Sowards, Scoop Martin, Joe Phillips. A cowboy learns his folks have been brutally murdered and he sets out to catch the outlaws. A fine cast can do nothing to save this entry in \"The Irish Cowboys\" series containing lots of stock footage from 1930s Bob Steele Westerns for Supreme Pictures and the end from _**The Star Packer**_ (q.v.). TV title: _**The Last Bullet**_.\n\n**897** _ **The Crooked Trail**_ **** Supreme, 1936. 60 min. D: S. Roy Luby. SC: George Plympton. With Johnny Mack Brown, Lucille Brown, John Merton, Charles King, Ted Adams, Dick Curtis, John Van Pelt, Ed Cassidy, Horace Murphy, Earl Dwire, Artie Ortego, Hal Price. A cowboy saves two men from dying of thirst in the desert and when he later becomes a sheriff he refuses to believe one of them is a thief. Johnny Mack Brown is the quick-on-the-draw lawman in this satisfying effort.\n\n**898** _ **Cross Fire**_ **** RKO Radio, 1933. 55 min. D: Otto Brower. SC: Harold Shumate and Tom McNamara. With Tom Keene, Betty Furness, Edgar Kennedy, Lafe McKee, Charles K. French, Edward (Eddie) Phillips, Murdock MacQuarrie, Stanley Blystone, Jules Cowles, Thomas (Tom) Brower, Nick Cogley, Kid Wagner, Tom Kennedy, Lew Meehan, Jim Corey, Jack Perry. A soldier returns home from the World War to find gangsters have invaded the range. Tom Keene's final RKO starring vehicle is okay but its plot is nothing new.\n\n**899** _ **Crossed Trails**_ **** Monogram, 1948. 60 min. D: Lambert Hillyer. SC: Colt Remington. With Johnny Mack Brown, Raymond Hatton, Kathy Frye, Lynne Carver, Douglas Evans, Steve Clark, Ted Adams, Zon Murray, Pierce Lyden, Milburn Morante, Frank LaRue, Mary MacLaren, Henry Hall, Bud Osborne, Artie Ortego. A young woman is the heir to a ranch with valuable mineral rights and her guardian, who refuses to sell the land, is jailed on a false murder charge. Entertaining Johnny Mack Brown-Raymond Hatton series entry.\n\n**900** _ **Crossfire Trail**_ **** Turner Network Television (TNT), 2001. 92 min. Color. D: Simon Wincer. SC: Charles Robert Carner. With Tom Selleck, Virginia Madsen, Wilford Brimley, David O'Hara, Christian Kane, Barry Corbin, Joanna Miles, Ken Pogue, Patrick Kilpatrick, Rex Linn, William Sanderson, Daniel T. Parker, Marshall Teague, Brad Johnson, Mark Harmon, Kyla Anderson, Michael O'Shea, Carmen Moore, James Nicholas, Mark Acheson. Keeping a promise to a friend he saw die at sea, a man goes to Wyoming to look after his ranch and widow and learns a land baron has been courting the woman to get her property. Top notch, faithful small screen adaptation of Louis L'Amour's book with star Tom Selleck serving as executive producer; filmed in Canada.\n\n**901** _ **Cry Blood Apache**_ **** Golden Eagle, 1970. 82 min. Color. D: Jack Starrett. SC: Sean MacGregory. With Jody McCrea, Dan Kemp, Jack Starrett, Don Henley, Robert Tessier, Carolyn Stellar, Joel McCrea. An old-timer recalls events from his youth involving a feud between whites and Indians after the Mexican War. Anemic oater enhanced only by Joel McCrea's brief cameo.\n\n_**Cry for Me, Billy**_ see _**Face to the Wind**_\n\n**902** _ **A Cry in the Wild**_ **** Concorde, 1990. 82 min. Color. D: Mark Griffiths. SC: Gary Paulsen. With Jared Rushton, Ned Beatty, Pamela Sue Martin, Stephen Meadows, Terence H. Winkless, Louise Baker, Deke Anderson, John Jakes, Lois Mallory, Ollie Mann. After surviving a Yukon plane crash, a young teen must fend for himself to survive in the wild. Well made outdoor drama from producer Julie Corman, which Gary Paulsen adapted to the screen from his novel _Hatchet_ ; followed by _**White Wolves: A Cry in the Wild II**_ , _**White Wolves II: Legend of the Wild**_ and _**White Wolves III: Cry of the White Wolf**_ (q.q.v.).\n\n**903** _ **A Cry in the Wilderness**_ **** ABC-TV\/Universal, 1974. 74 min. Color. D: Gordon Hessler. SC: Stephen Knarpf and Elinor Knarpf. With George Kennedy, Joanna Pettet, Lee H. Montgomery, Collin Wilcox-Horne, Liam Dunn, Roy Poole, Bing Russell, Irene Tedrow, Robert Brubaker, Anne Seymour, Paul Sorenson. After being bitten by a rabid skunk, a farmer tries to protect his family by chaining himself inside a barn only to learn a flood is coming. Fairly suspenseful outing made for television.\n\n**904** _ **Cry of the Black Wolves**_ **** Cinema Shares, 1972. 85 min. Color. D: Harald Reinl. SC: Kurt Nachmann. With Ron Ely, Gila von Weitershausen, Raimund Harmstorf, Arthur Brauss, Catharina Conti, Jean-Claude Hoffman, Angelica Ott, Hans Terofal, Carl Lange, Alexander Grill, Dan van Husen, Heinrich Schweiger, Kurt Bulau, Tony Berger, Gunter Clemens, Karin Lorson, Untine Frohlich, Sigfrit Steiner, Ernst H. Hilbich, Jan Groth. In 1903 Alaska, a corrupt prospector steals a fur trapper's dog sled but when he is found murdered the trapper is blamed. Rugged West German melodrama based on a Jack London story and filmed as _**Der Schrei der Schwarzen Wolfe**_ (The Cry of the Black Wolves) by Lisa-Film.\n\n**905** _ **Cry of the Wild**_ **** American National Enterprises, 1973. 91 min. Color. D-SC: Bill Mason. Documentary on wolves, both at large and in captivity, telling of their habits and exposing many myths about these supposedly savage beasts. Director-writer Bill Mason also did the camera work for this film, which is quite entertaining.\n\n**906** _ **Cry to the Wind**_ **** Sebastian International, 1979. 90 min. Color. D: Robert W. Davison. SC: David James Nielsen. With Sheldon Woods, Cameron Garrick, Aaron Card, Bonnie Card, Lamont Topaum. A young man attempts to conquer the wilderness and learns how to survive and respect his surroundings. Capable adventure yarn with lots of scenic value.\n\n**907** _ **Cuando Canto la Ley**_ (When on the Side of the Law) **** Dario Productions, 1939. 77 min. D: Richard Harlan and Gabriel Navarro. SC: Richard Harlan and Jack Natteford. With Tito Guizar, Tana, Martin Garralaga, Paul Ellis, Pilar Arcos, Jose Luis Tortosa, Carlos Ruffino, Carlos Montalban, Raul Lechuga, Jose Pena. On the trail of an embezzler, a Mexican Secret Service agent pretends to be a cowboy and goes to work for a female rancher. Zesty Mexican musical Western in which star Tito Guizar performs five songs he co-wrote; released in the U.S. by Paramount cut to 67 minutes.\n\n**908** _ **Los Cuatro Juanes**_ (The Four Juans) **** Producciones Zacarias, S.A., 1966. 95 min. D: Miguel Zacarias. SC: Alfredo Zacarias. With Luis Aguilar, Antonio Aguilar, Javier Solis, Narciso Busquets, Alma Delia Fuentes, Ofelia Monesco, Rosario Galvez, Conrado Cortes, Antonio Raxel, Jorge Russek, Jose Torvay, Emilio Garibay, Antonio Haro Oliva, Stim Segar, Adolfo Aguilar, Manuel Arvide. Four men named Juan, including folk heroes Juan Colorado and Juan Charrasqueado, join forces to oppose lawlessness in Old Mexico. Fast paced adventure from south of the border.\n\n**909** _ **The Culpepper Cattle Company**_ **** 20th Century\u2013Fox, 1972. 92 min. D: Dick Richards. SC: Eric Berovici and Gregory Prentiss. With Gary Grimes, Billy Green Bush, Luke Askew, Bo Hopkins, Geoffrey Lewis, Wayne Sutherlin, John McLiam, Matt Clark, Raymond Guth, Anthony James, Charles Martin Smith, Larry Finley, Bob Morgan, Jan Burrell, Gregory Sierra, Royal Dano, Hal Needham, Jerry Galtin. A teenager becomes part of a trail drive and the hardships along the way teach him to be a man. Fairly good dramatic offering although a bit on the violent side.\n\n**910** _ **The Curse of Bigfoot**_ **** Gold Key, 1971. 87 min. Color. D: Don Fields. SC: J.T. Fields. With William Simonsen, Robert Clymire, Jan Swihart, Ken Kloepfer, Dennis Kottmier, Ruth Ann Mannella, Mary Browles, Augie Tribach. A group of archaeological students uncover an ancient beast interred in an Indian burial ground and it goes on a killing spree. Pitiful production with new footage augmenting scenes shot years before.\n\n_**Curse of the Demon Mountain**_ see _**The Shadow of Chikara**_\n\n**911** _ **The Curse of the Headless Horseman**_ **** DLM, 1972. 80 min. Color. D: John Kirkland. SC: Kenn Riche. With Ultra Violet, Marland Proctor, Don Carrara, Claudia Ream, B.G. Fisher, Margo Dean, Lee Byers, Joe Cody. A doctor inherits a ranch where a headless phantom is said to take revenge on the eight gunmen who murdered him. Pathetic horror film in a Western setting; torturous viewing.\n\n**912** _ **Curse of the Lost Gold Mine**_ **** Yaletown Entertainment Group, 1994. 50 min. Color. D: Michael Collier. SC: P.J. Reece. With Donnelly Rhodes, Norman Natrall, Rolf Cutts, Mike Billy, Mark Antone, Dave Ponsart, Peter McIlvaney, Donald E. White, John F.N. Thompson. Semi-documentary about the ancient Indian legend of Slumach and those who sought his treasure. Okay entertainment made for video release.\n\n**913** _ **Curse of the Undead**_ **** Universal-International, 1959. 79 min. D-SC: Edward Dein. With Eric Fleming, Michael Pate, Kathleen Crowley, John Hoyt, Bruce Gordon, Jimmy Murphy, Helen Kleeb, Jay Adler, Edwin (Eddie) Parker, John Truax, Frankie Van, Rush Williams, Edward Binns, Edward Colmans, Nancy Kilgas, Alan Reynolds, Margaret Bert, Jeanna Cross, Charles Keane, Forrest Stanley, Don Sullivan, Amzie Strickland. A mysterious gunman dressed in black is hired to expedite a range war but he is really a vampire after a beautiful woman rancher. Different, atmospheric and a good horror Western.\n\n**The Curse of the Viking Grave** see _**Lost in the Barrens II: The Curse of the Viking Grave**_\n\n**914** _ **Curtain Call at Cactus Creek**_ **** Universal-International, 1950. 86 min. Color. D: Charles Lamont. SC: Howard Dimsdale. With Donald O'Connor, Gale Storm, Walter Brennan, Vincent Price, Eve Arden, Chick Chandler, Joseph Sawyer, Harry Shannon, Rex Lease, I. Stanford Jolley, Eddy Waller, Hank Worden, Edmund Cobb, Lane Bradford, Paul Maxey, Terry Frost, John Carpenter, Ferris Taylor, Al Haskell. An acting struck stagehand with a touring troupe in the Old West accidentally captures a local bank robber. Amusing satire helped by an eager cast.\n\n**Donald O'Connor and Vincent Price in** _**Curtain Call at Cactus Creek**_ **(Universal-International, 1950).**\n\n**  \n**\n\n**915** _ **Custer of the West**_ **** Cinerama, 1967. 146 min. Color. D: Robert Siodmak and Irving Lerner. SC: Bernard Gordon and Julian Halvey. With Robert Shaw, Mary Ure, Jeffrey Hunter, Ty Hardin, Robert Ryan, Charles Stalnaker, Robert Hall, Lawrence Tierney, Kieron Moore, Marc Lawrence, Jack Taylor, Fred Kohler, Jr., John Clarke, Bud Strait, Robert Reynolds, Barta Barri, Clemence Bettany, Jack Gaskins, Bill Christmas, Joe Zboran, Carl Rapp, Jack Cooper, Luis Rivera, John Dillon. After being a Civil War hero, George Armstrong Custer is given a command in the West where he must deal with warring Indians and Army rivals. Fairly accurate rendering of the Custer saga, filmed in Spain. Also called _**A Good Day for Fighting**_.\n\n**916** _ **Custer's Last Fight**_ **** Quality Amusements, 1925. 55 min. D: Thomas H. Ince. SC: Richard V. Spencer. With Francis Ford, Grace Cunard, J. Barney Sherry, William Eagle Shirt, Ann Little, Charles K. French, Lillian Christy, Art Acord, Clayton Monroe Teters, Snowball (horse). The story of the showdown between General Custer and Sitting Bull at the Little Big Horn River. An expanded version of the 1912 Ince three reel film; one of the first really good Westerns and well worth viewing. In 1912 Francis Ford was Custer in _**The Invaders**_ (q.v.), also for Thomas H. Ince.\n\n**917** _ **Custer's Last Stand**_ **** Stage and Screen, 1936. 15 Chapters. D: Elmer Clifton. SC: George A. Durlam, Eddy Graneman and Bob Lively. With Rex Lease, Jack Mulhall, Ruth Mix, Dorothy Gulliver, William Farnum, Lona Andre, Reed Howes, Bobby Nelson, Frank McGlynn, Jr., William Desmond, Helen Gibson, Nancy Casell, Chief Thundercloud, Josef Swickard, Creighton Hale, George Chesebro, Milburn Morante, Ted Adams, George Morrell, Robert Walker, Walter James, Cactus Mack, Budd Buster, Carl Mathews, Artie Ortego, Franklyn Farnum, Lafe McKee, Allen Greer, James Sheridan (Sherry Tansey), Ken Cooper, Chief Big Tree, Iron Eyes Cody, Bill Thompson, Walter Gable, White Feather, Buddy Fisher, Whiten Sovern, Charles Hunter, William Hunt, William Bartlett. A scout for General Custer tries to help settlers attacked by Indians led by a renegade after a medicine arrow, the clue to a hidden treasure. Lumbering, dull, poorly made and badly paced cliffhanger culminating an exciting re-enactment of the Battle of the Little Big Horn; helped only by a large veteran cast with George Chesebro a standout as a dishonest soldier turned good. Also issued in an equally inert 65 minute feature version.\n\n**918** _ **Cut-Throats Nine**_ **** United International, 1973. 90 min. Color. D: Joaquin L. Romero Merchant. SC: Santiago Moncada and Joaquin L. Romero Merchant. With Robert Hundar (Claudio Undari), Emma Cohen, Alberto Dalbes, Manuel Tejada, Ricardo Diaz, Carlos Romero Merchant, Antonio Iranzo, Jose Manuel Martin, Rafael Hernandez, Eduardo Calvo, Lorenzo Robeldo, Emilio Rodriguez, Tomas Ares, Francisco Nieto, Antonio Padilla, Simon Arriaga, Juan Antonio Elices, Mabel Karr, Dan van Husen. A Union Army sergeant and his pretty daughter lead criminals on a 400 mile journey to a government gold mine so the convicts can work it for the North. One of the more brutal, gore filled Spaghetti Westerns, produced by Films Triunfo in Spain as _**Condenados a Vivir**_ (Condemned to Live).\n\n**919** _ **Cutter's Trail**_ **** CBS-TV\/CBS Studio Center, 1970. 100 min. Color. D: Vincent McEveety. SC: Paul Savage. With John Gavin, Marisa Pavan, Beverly Garland, Joseph Cotten, J. Carrol Naish, Nehemiah Persoff, Manuel Padilla, Jr., Shug Fisher, Ken Swofford, Victor French, Bob Random, Robert Totten, Tom Brown. A marshal returns to Santa Fe to find the town pillaged by an outlaw gang and only a young Mexican mother and her small son will help him track the marauders. Fairly acceptable television oater that originally ran 75 minutes and was expanded for subsequent showings.\n\n**920** _ **Cutting Horse**_ **** Image Entertainment, 2002. 124 min. Color. D: Larry Clark. SC: Larry Clark and David Heintz. With Albert Harris, Cesar Flores, Robert Earl Crudup, Rufus Norris, Mellisa Cellura, Susan Martino, Roberto Bethel, Christopher Upham, Joy Garner, Sigi Lobas, Sherry Al-Mufti, Fred Barson, Robert J. Ramsey III, H. Lee Burton, Ian Davidson, Scott Campbell, Michael Orlando, Artis Fountain, Peter Carlstrom, Larry Roszkowiak, Coy Sanders, Jeanne Sapieza, Nadia Tarzi, Paula Martin, Jamie Lujian, Lisa Cortez Walden, Bud Sisson, Carol Wilkinson, Stu Richel, Joe Lewis, Gordon Wong. After a decade a cowboy returns to his family's horse breeding ranch to find them being pressured to sell out to a greedy conglomerate. Fair low budget, R-rated, modern-day Western with attractive locales.\n\n**921** _ **Cyclone Cavalier**_ **** Rayart, 1925. 55 min. D: Albert S. Rogell. SC: Krag Johnson and Burke Jenkins. With Reed Howes, Carmelita Geraghty, Wilfred Lucas, Jack Mower, Eric Mayne, Johnny Sinclair, Ervin Renard. Sent to a Central American republic where he falls in love with the president's daughter, an adventurer tries to thwart a palace revolution. Low budget affair that gives viewers a chance to see Reed Howes (the Arrow Collar man) in a starring role.\n\n**922** _ **Cyclone Fury**_ **** Columbia, 1951. 54 min. D: Ray Nazarro. SC: Barry Shipman and Ed Earl Repp. With Charles Starrett, Smiley Burnette, Fred F. Sears, Clayton Moore, Robert Wilke, Louis Lettieri, George Chesebro, Frank O'Connor, Merle Travis and His Bronco Busters, Jay Silverheels, Edmund Cobb, Kermit Maynard, Ray Bennett, Matty Roubert, Slim Duncan, John Merton, Lane Bradford, Frank Moran, Robert E. Scott, Richard Alexander, Lew Morphy. An agent assigned to insure the delivery of horses to the government gets suspicious after a rancher is murdered. Fair \"Durango Kid\" film interpolating footage from the earlier series outings _**Galloping Thunder**_ , _**Landrush**_ and _**Prairie Raiders**_ (qq.v.).\n\n**923** _ **Cyclone Jones**_ **** Aywon, 1923. 50 min. D: Charles R. Seeling. SC: John F. (Jack) Natteford. With (Guinn) Big Boy Williams, Kathleen Collins, J.P. (Lafe) McKee, Bill Patton, Fred Burns, Fatty Alexander. A cowboy falls for a sheep herder's daughter who spurns him until he saves her life when a cattleman hires a no good to run off her family. Only fair Guinn Williams silent vehicle interspersed with humor.\n\n**924** _ **The Cyclone Kid**_ **** Big 4, 1931. 60 min. D: J.P. McGowan. SC: George Morgan. With Caryl Lincoln, Buzz Barton, Francis X. Bushman, Jr., Ted Adams, Lafe McKee, Blackie Whiteford, Nadja, Silver Harr. A ranch foreman, in love with the boss' daughter, is helped by a young boy in opposing outlaws. Poor, low grade production.\n\n**925** _ **The Cyclone Kid**_ **** Republic, 1942. 56 min. D: George Sherman. SC: Richard Murphy. With Don \"Red\" Barry, Lynn Merrick, John James, Alex Callam, Joel Friedkin, Slim Andrews, Rex Lease, Joe McGuinn, Monte Montague, Frank LaRue, Edmund Cobb, Budd Buster, Hal Price, Jack Rockwell, Jack O'Shea, Curley Dresden, Bob Woodward. When his lawyer brother comes West and learns of his true activities, a gunman turns on his crooked cattle baron boss. Typically good Don Barry series film with a plot that is a bit hard to take, but the action compensates.\n\n**926** _ **Cyclone of the Saddle**_ **** Superior, 1935. 53 min. D: Elmer Clifton. SC: Elmer Clifton and George Merrick. With Rex Lease, Janet Chandler, Bobby Nelson, William Desmond, Yakima Canutt, Art Mix, Chief Thundercloud, Helen Gibson, Milburn Morante, George Chesebro, Glenn Strange, George Morrell, The Range Ranglers, Chief Standing Bear, Black Fox (horse). After an outlaw gang causes trouble with settlers and Indians, the Army assigns an officer to stop them. Tacky production values detract from this Rex Lease vehicle.\n\n**927** _ **Cyclone on Horseback**_ **** RKO Radio, 1941. 60 min. D: Edward Killy. SC: Norton S. Parker. With Tim Holt, Marjorie Reynolds, Ray Whitley, Lee \"Lasses\" White, Dennis Moore, Harry Worth, Monte Montague, John Dilson, Lew Kelly, Terry Frost, Slim Whitaker, Eddie Dew, John Ince, Walter Shumway, Jack Kirk, Lloyd Ingraham, Cactus Mack, Rube Schaefer, John Daheim, Cliff Lyons, Tom Steele, Marty Faust, Jane Patton, Art Dupuis. Three cowpokes come to the aid of a pretty girl and her brother whose attempt to string a telegraph wire, in order to win a contract, is being thwarted by hoodlums. Well paced and action filled Tim Holt film, with an especially exciting finale.\n\n**928** _ **Cyclone Prairie Rustlers**_ **** Columbia, 1944. 55 min. D: Benjamin Kline. SC: Elizabeth Beecher. With Charles Starrett, Dub Taylor, Jimmie Davis, Constance Worth, Jimmy Wakely and His Saddle Pals, Robert Fiske, Clancy Cooper, Ray Bennett, I. Stanford Jolley, Edward M. Phillips, Edmund Cobb, Forrest Taylor, Paul Zaremba, Ted Mapes, Steve Clark, Edna Harris. A cowboy and his pals try to stop Nazis from sabotaging cattle, crops and equipment in the West. Topical and fast paced Charles Starrett vehicle.\n\n**929** _ **The Cyclone Ranger**_ **** Spectrum, 1935. 60 min. D: Robert Hill. SC: Oliver Drake. With Bill Cody, Nena Quartero, Eddie Gribbon, Soledad Jiminez, Earle Hodgins, Zara Tazil, Donald Reed, Colin Chase, Budd Buster, Herman Hack, Buck Morgan. An outlaw is befriended by a blind woman and pretends to be her son who was killed by a posse. Sentimental, but mediocre, Bill Cody outing; remake of _**Gun Law**_ (1933) [q.v.].\n\n**930** _ **Dakota**_ **** Republic, 1945. 82 min. D: Joseph Kane. SC: Lawrence Hazard. With John Wayne, Vera Hruba Ralston, Walter Brennan, Ward Bond, Mike Mazurki, Ona Munson, Hugo Haas, Olive Blakeney, Nicodemus Stewart, Paul Fix, Grant Withers, Robert Livingston, Olin Howlin, Pierre Watkin, Robert Barrat, Jonathan Hale, Bobby Blake, Paul Hurst, Eddy Waller, Sarah Padden, Jack LaRue, George Cleveland, Selmer Jackson, Claire DuBrey, Roy Barcroft, Victor Varconi, Cliff Lyons, Fred Graham, Linda Stirling, Kenne Duncan, Yakima Canutt, Lorna Gray (Adrian Booth), Rex Lease, Tom London, Houseley Stevenson, Paul E. Burns, Cay Forester, William Haade, Larry Thompson, Michael Visaroff, Dorothy Christy, Bob Burns, Dick Wessel, Jack O'Shea, Art Miles, Jack Roper, Hector Sarno, Eugene Borden, Noble \"Kid\" Chissel, Tom Smith, Al Murphy, Larry Thompson, Betty Shaw, Martha Carroll, Frances Gladwin, Harriette Haddon, Virginia Wave, Rosemonde James, Marian Kerrigan, Melva Anstead, Beverly Reedy, Dorothy Stevens. An ex-soldier and his heiress wife go to the Dakotas to invest in land on which a railroad is to be built and find themselves at odds with two crooks trying to force farmers off their homesteads. Big, brawling drama, not as good as it should have been, but it will appeal to Duke's fans.\n\n**931** _ **Dakota**_ **** Miramax Films, 1988. 97 min. Color. D: Fred Holmes. SC: Lynn Kuntz and Darryl Kuntz. With Lou Diamond Phillips, Eli Cummins, Dee Dee Norton, Jordan Burton, Steven Ruge, John Hawkes, Tom Campitelli, Herta Ware, Lawrence Montaigne, Leslie Mullen, Connie Colt, Susan Crippin, Rodger Boyce, Robert Lemus, Ben Jones, Cecilia Flores, Robert Ahola, Helena Humann, John Glenn, Tom McRae, Kendall Thomas, H. Laverne Smith, Audeen Casey, Mark Schulte, Abel O. Zapata. After taking a job on a Texas ranch a young man with a troubled past comes to grips with his life as he learns to help others. Minor modern-day drama.\n\n**932** _ **Dakota Incident**_ **** Republic, 1956. 88 min. Color. D: Lewis R. Foster. SC: Frederick Louis Fox. With Linda Darnell, Dale Robertson, John Lund, Ward Bond, Skip Homeier, Regis Toomey, Irving Bacon, John Doucette, Whit Bissell, William Fawcett, Malcolm Atterbury, Charles Horvath, Diane Du Blis, Eva Novak, Boyd \"Red\" Morgan, Fred Coby, Rankin Mansfield. Passengers on a stagecoach are attacked by Indians and must defend themselves as well as settle their differences. Interesting premise but a none-too-good production.\n\n**933** _ **The Dakota Kid**_ **** Republic, 1951. 60 min. D: Philip Ford. SC: William Lively. With Michael Chapin, Eilene Janssen, James Bell, Margaret Field, Robert Shayne, Roy Barcroft, Danny Morton, Mauritz Hugo, House Peters, Jr., Lee Bennett, Michael Ragan (Holly Bane), Art Dillard. Two youngsters assist the law in rounding up an outlaw gang. One of the quartet of features in the \"Rough Ridin' Kids\" series and just as mediocre as the others.\n\n**934** _ **Dakota Lil**_ **** 20th Century\u2013Fox, 1950. 88 min. Color. D: Lesley Selander. SC: Maurice Geraghty. With George Montgomery, Rod Cameron, Marie Windsor, John Emery, Wallace Ford, Jack Lambert, Larry Johns, Marion Martin, James Flavin, J. Farrell MacDonald, Walter Sande, Joel Friedkin, Jack Perrin, Soledad Jiminez, Nacho Galindo, Felipe Turich, Lillian Bronson, Clancy Cooper, Rosa Turich, Frank Lackteen, Saul Gorss, Albert Morin, Ken MacDonald, Bill Perrott, Tom Greenway, John Dako, Ben Harris, Bryan Hightower. A Treasury agent is on the trail of a team of counterfeiters in the Old West. Pleasing drama with nice performances in the leading roles.\n\n**935** _ **Dallas**_ **** Warner Bros., 1950. 94 min. Color. D: Stuart Heisler. SC: John Twist. With Gary Cooper, Ruth Roman, Steve Cochran, Raymond Massey, Barbara Payton, Leif Erickson, Antonio Moreno, Jerome Cowan, Reed Hadley, Will Wright, Monte Blue, Byron (Brian) Keith, Gil Donaldson, Zon Murray, Al Ferguson, Gene Evans, Fred Kelsey, Buddy Roosevelt, Ben Corbett, Charles Horvath, Carl Andre, O.Z. Whitehead, Frank McCarroll, Larry McGrath, Dewey Robinson, Slim Talbot, Tom Fadden, Hal K. Dawson, Alex Montoya, Fred Graham, Roy Bucko, Frank Kreig. In post\u2013Civil War Texas, a former Confederate plans go get revenge on the carpetbaggers who murdered his family. Big budget Gary Cooper opus that hits the entertainment mark; Barbara Payton is good as heroine Ruth Roman's pal.\n\n**936** _ **The Dalton Gang**_ **** Lippert, 1949. 59 min. D-SC: Ford Beebe. With Don Barry, Robert Lowery, Betty (Julie) Adams, James Millican, Byron Foulger, J. Farrell MacDonald, Greg McClure, George J. Lewis, Marshall Reed, Ray Bennett, Lee Roberts, Dick Curtis, Stanley Price, Cactus Mack, Cliff Taylor. Two lawmen are assigned to round up the infamous Dalton brothers. Outside of its stars there is not much to recommend this pedestrian effort. TV title: _**The Outlaw Gang**_.\n\n**937** _ **The Dalton Girls**_ **** United Artists, 1957. 71 min. D: Reginald LeBorg. SC: Maurice Tombragel. With Merry Anders, Penny Edwards, John Russell, Lisa Davis, Sue George, Johnny Western, Malcom Atterbury, Douglas Henderson, Red Morgan, Ed Hinton. After the Dalton Gang is arrested, several young female relatives band together and form their own outlaw coterie. Director Reginald LeBorg and the cast tries hard but they are defeated by cheap production values.\n\n**938** _ **The Dalton That Got Away**_ **** Dalton Film Company, 1960. 69 min. D: Jimmy (Jaime) Salvador. SC: E.L. Erwin. With Mike Connors, Elsie (Elsa) Cardenas, Carlos Rivas, Felix Moreno, Zachary Milton, Stillman Segar, George Russell, Reed Howes, Francisco Reynolds, Quintin Buines, Sam Murphy, Arlene King. Two of the notorious outlaw Dalton brothers have a falling out over the affections of an Indian princess. Obscure, flimsy low budget affair filmed in Mexico in 1957.\n\n**939** _ **The Daltons Ride Again**_ **** Universal, 1945. 70 min. D: Ray Taylor. SC: Roy Chanslor and Paul Gangelin. With Alan Curtis, Lon Chaney, Kent Taylor, Noah Berry, Jr., Martha O'Driscoll, Jess Barker, Thomas Gomez, John Litel, Walter Sande, Douglass Dumbrille, Virginia Brissac, Milburn Stone, Stanley Andrews, Fern Emmett, Cyril Delevanti, Wheaton Chambers, Davison Clark, Jack Rockwell, Robert Wilke, Dick Dickinson, George Chesebro, Paul Birch, Ed Cassidy, Ethan Laidlaw, Henry Hall, Richard Alexander. The exploits of the Dalton brothers outlaw gang is told by the only survivor of the Coffeyville, Kansas, shootout. Compact and very entertaining \"B plus\" Western, with the four stars doing excellent work as the notorious siblings.\n\n**940** _ **The Daltons' Women**_ **** Western Adventure, 1951. 80 min. D: Thomas Carr. SC: Ron Ormond and Maurice Tombragel. With Tom Neal, Pamela Blake, Jack Holt, Lash LaRue, Al St. John, Jacqueline Fontaine, Raymond Hatton, Lyle Talbot, Tom Tyler, J. Farrell MacDonald, Terry Frost, Stanley Price, Bud Osborne, Lee Roberts, June Benbow, Cliff Taylor, Clarke Stevens, Archie Twitchell, Duke Johnson, Jimmie Martin, Buff Brady. A saloon owner is in cahoots with an outlaw gang terrorizing a town, but U.S. marshals are assigned to stop them. Notorious stitched together production is best taken as a curio, otherwise it is pretty poor.\n\n**941** _ **Dan Candy's Law**_ **** Cinerama Releasing Corporation\/American International, 1973. 95 min. Color. D: Claude Fournier. SC: George Malko. With Donald Sutherland, Chief Dan George, Kevin McCarthy, Jean Duceppe, Francine Rocette, Jack Creely. After his partner is killed, a Mountie tracks the Indian accused of the crime but eventually realizes he is the hunted instead of the hunter. Nice scenery highlights this otherwise tiresome Canadian feature. Also called _**Alien Thunder**_.\n\n**942** _ **Dances with Wolves**_ **** Orion, 1990. 181 min. Color. D: Kevin Costner. SC: Michael Blake. With Kevin Costner, Mary McDonnell, Graham Greene, Rodney A. Grant, Floyd Red Crow Westerman, Tantoo Cardinal, Robert Pastorelli, Charles Rocket, Maury Chaykin, Jimmy Herman, Nathan Lee Chasing His Horse, Michael Spears, Jason R. Lone Hill, Tony Pierce, Doris Leader Charge, Tom Everett, Larry Joshua, Kirk Baltz, Wayne Grace, Donald Hotton, Annie Costner, Conor Duffy, Elisa Daniel, Percy White Plume, John Tail, Steve Reevis, Sheldon Wolfchild, Wes Studi, Buffalo Child, Clayton Big Eagle, Richard Leader Charge, Redwing Ted Nez, Marvin Holy, Raymond Newholy, David J. Fuller, Ryan White Bull, Otakuye Conroy, Maretta Big Crow, Steve Chambers, William H. Burton, Bill W. Curry, Nick Thompson, Carter Hanner, Kent Hays, Robert Goldman, Frank P. Costanza, James A. Mitchell, R.L. Curtin, Jim Wilson, Michael Horton, Teddy and Buck (wolves). A Civil War hero is assigned to a remote deserted Western post where he makes friends with a wolf, an Indian tribe and the young white girl they raised. Big box office winner co-produced by director-star Kevin Costner; extended version runs 224 minutes and the director's cut is 236 minutes.\n\n**943** _ **Danger Ahead**_ **** Monogram, 1940. 60 min. D: Ralph Staub. SC: Edward Halperin. With James Newill, Dorothea Kent, Dave O'Brien, Guy Usher, Maude Allen, Harry Depp, John Dilson, Earl Douglas, Bob Terry, Lester Dorr, David Sharpe. Officer Renfrew and his Mountie pals are at odds with a stubborn young woman who refuses to help them in capturing an outlaw gang wanted for murder. Okay entry in the \"Renfrew of the Royal Mounted\" series based on Laurie York Erskine's _Renfrew's Long Trail_.\n\n**944** _ **Danger Patrol**_ **** RKO Radio, 1937. 60 min. D: Lew Landers. SC: Helen Vreeland and Hilda Vincent. With Sally Eilers, John Beal, Harry Carey, Frank M. Thomas, Crawford Weaver, Lee Patrick, Edward Gargan, Paul Guilfoyle, Solly Ward. Working as a nitro shooter in the oil fields, a medical student falls in love with the daughter of the man who is training him. Sturdy oil drilling drama highlighted by Harry Carey as the mentor-father.\n\n**945** _ **Danger Trails**_ **** Beacon\/First Division, 1935. 62 min. D: Robert Hill. SC: Rock Hawkey (Robert Hill). With Guinn Williams, Marjorie Gordon, Wally Wales, Edmund Cobb, John Elliott, George Chesebro, Steve Clark, Ace Cain, Francis Walker, Wally West, George Morrell, Bob Hill, Buck Morgan, Ray Henderson. A man, educated in the East, plans to take revenge on the outlaw gang that murdered his family. Cheaply made but entertaining Guinn \"Big Boy\" Williams vehicle for which the star wrote the original story.\n\n**946** _ **Danger Valley**_ **** Monogram, 1938. 58 min. D: Robert North Bradbury. SC: Robert Emmett (Tansey). With Jack Randall, Lois Wilde, Charles King, Hal Price, Frank LaRue, Chick Hannon, Earl Dwire, Ernie Adams, Tex Palmer, Merrill McCormick, Oscar Gahan, Denver Dixon, Sherry Tansey, Jimmy Aubrey, Glenn Strange, Bud Osborne. Two cowpokes arrive in a ghost town where an old prospector has discovered gold but is being harassed by an outlaw gang. Passable Jack Randall singing Western.\n\n**947** _ **The Dangerous Days of Kiowa Jones**_ **** ABC-TV\/Metro-Goldwyn-Mayer, 1966. 100 min. Color. D: Alex March. SC: Frank Fenton and Robert W. Thompson. With Robert Horton, Diane Baker, Sal Mineo, Nehemiah Persoff, Gary Merrill, Robert H. Harris, Lonny Chapman, Royal Dano, Zalman King, Harry Dean Stanton, Val Avery. A dying marshal asks a cowboy to transport two criminals to jail and along the way the trio must elude bounty hunters. Early network telefeature is on the mediocre side and failed to sell as a series.\n\n**948** _ **Dangerous Nan McGrew**_ **** Paramount, 1930. 62 min. D: Malcolm St. Clair. SC: Paul Gerard Smith. With Helen Kane, Victor Moore, James Hall, Stuart Erwin, Frank Morgan, Roberta Robinson, Louise Closser Hale, Allan Forrest, John Hamilton, Robert Milash. A young woman working in a medicine show is stranded in the Canadian northwest and ends up capturing a bank robber. Vintage Helen Kane starring vehicle with songs will satisfy the curious, but otherwise beware.\n\n**949** _ **Dangerous Odds**_ **** Independent Pictures, 1925. 50 min. D: William J. Craft. With Bill Cody, Eileen Sedgwick, Milton Fahrney, Claude Payton, Monte Collins, Al Hallett. When a bank manager, who has withdrawn funds for his ranch, is murdered, a cowboy is accused of the crime and escapes a lynching party to prove his innocence. Okay Bill Cody silent effort, but nothing special.\n\n**950** _ **Dangerous Venture**_ **** United Artists, 1947. 59 min D: George Archainbaud. SC: Doris Schroeder. With William Boyd, Andy Clyde, Rand Brooks, Fritz Leiber, Douglas Evans, Harry Cording, Betty Alexander, Francis McDonald, Neyle Morrow, Ken (Kenneth) Tobey, Patricia Tate, Bob Faust, Jack Quinn, Bill Nestell. Hopalong Cassidy and his pals are in the middle of Indian warfare caused by archaeologists trying to locate a sacred burial ground treasure. Mystery elements highlight this later \"Hopalong Cassidy\" series offering.\n\n**951** _ **Dangers of the Canadian Mounted**_ **** Republic, 1938. 12 Chapters. D: Fred C. Brannon and Yakima Canutt. SC: Franklyn Adreon, Basil Dickey, Sol Shor and Robert G. Walker. With Jim Bannon, Virginia Belmont, Anthony Warde, Dorothy Granger, Dale Van Sickel, Tom Steele, I. Stanford Jolley, Phil Warren, Lee Morgan, James Dale, Ted Adams, John Crawford, Jack Clifford, Eddie Parker, Frank O'Connor, Kenneth Terrell, Robert Wilke, Marshall Reed, House Peters, Jr., Holly Bane, Ted Mapes, Jack Kirk, Al Taylor, Harry Cording, Bud Wolfe, Roy Bucko, David Sharpe. Canadian Mounties are after a gang of crooks trying to locate a hidden Chinese treasure in the north country. Mediocre cliffhanger. TV feature title: _**R.C.M.P**_ **.** _**and the Treasure of Genghis Khan**_ (100 minutes).\n\n**952** _ **Daniel Boone**_ **** RKO Radio, 1936. 77 min. D: David Howard. SC: Daniel Jarrett. With George O'Brien, Heather Angel, John Carradine, Ralph Forbes, Clarence Muse, George Regas, Dickie Jones, Huntley Gordon, Harry Cording, Aggie Herring, Crauford Kent, Keith Kenneth, Dick Curtis, John Merton, Chief Big Tree, James Lichter, Ed Peil, Sr., Tom Ricketts. Daniel Boone leads settlers across the Cumberland Mountains in 1775 to settle in Kentucky and encounters hostile Indians led by the evil Simon Girty. Fine historical drama with George O'Brien making an excellent Daniel Boone and John Carradine equally as good as the villainous Girty; well worth seeing.\n\n**953** _ **Daniel Boone: Frontier Trail Rider**_ **** 20th Century\u2013Fox, 1966. 91 min. Color. D: George Sherman. SC: D.D. Beauchamp and Jack Guss. With Fess Parker, Ed Ames, Patricia Blair, Armando Silvestre, Dallas (Dal) McKennon, Jacqueline Evans, Roy Jenson, Barbara Turner DeHubp, Charles Horvath, Robert (Bob) Terhune, Jack Williams, Chuck Roberson, Ted White, Felix Gonzalez. While leading settlers into Kentucky to start a new settlement, Daniel Boone falls in love with a pretty servant girl who is also wanted by a gambler. Okay feature made up of two segments of \"Daniel Boone\" (NBC-TV, 1964\u201370); a remake of _**Bend of the River**_ (q.v.).\n\n**954** _ **Daniel Boone Thru the Wilderness**_ **** Sunset, 1926. 62 min. D-SC: Robert North Bradbury. With Roy Stewart, Kathleen Collins, Edward Hearn, Jay Morley, Bob Bradbury, Jr. (Bob Steele), Thomas Lingham, Frank Rice, James O'Neil, Emile Gertes. Daniel Boone leads settlers into the wilderness to establish a new settlement and meets opposition from Indians. Compact, nicely done feature from producer Anthony J. Xydias; also called _**With Daniel Boone Thru the Wilderness**_.\n\n**955** _ **Daniel Boone, Trail Blazer**_ **** Republic, 1956. 76 min. Color D: Albert C. Gannaway and Ismael Rodriguez. SC: Tom Hubbard and Jack Patrick. With Bruce Bennett, Lon Chaney, Faron Young, Damion O'Flynn, Ken Dibbs, Jacqueline Evans, Freddy Fernandez, Nancy Rodman, Fred Kohler, Jr., Lee Morgan. Daniel Boone leads settlers from North Carolina to build a fort at Boonesborough in Kentucky where they are attacked by Indians, renegade French and Tories. Partially filmed in Mexico, this historical drama is more than passable with Bruce Bennett making a stalwart Daniel Boone and Lon Chaney an excellent Chief Blackfish, although country singer Faron Young is miscast as scout Callaway.\n\n**956** _ **Daredevils of the West**_ **** Republic, 1943. 12 Chapters. D: John English. SC: Ronald Davidson, Basil Dickey, Joseph O'Donnell, Joseph Poland and William Lively. With Allan Lane, Kay Aldridge, Eddie Acuff, William Haade, Robert Frazer, Ted Adams, George J. Lewis, Stanley Andrews, Jack Rockwell, Charles Miller, John Hamilton, Budd Buster, Kenneth Harlan, Kenne Duncan, Rex Lease, Chief Thundercloud, Eddie Parker, Ray Jones, Chief Many Treaties, Tom Steele, Jack O'Shea, George Magrill, Pierce Lyden, George Plues, Edmund Cobb, Al Taylor, Frank McCarroll, Tom London, George Pembroke, Ed Cassidy, Herbert Rawlinson, Tex Cooper, Charles Soldani, Crane Whitley, Augie Gomez. A cowboy assists a young woman whose stage line is being threatened by mysterious attacks. Exciting Republic cliffhanger.\n\n_**The Daring Adventurer**_ see _**The Cisco Kid Returns**_\n\n**957** _ **The Daring Caballero**_ **** United Artists, 1950. 60 min. D: Wallace Fox. SC: Betty Burbridge. With Duncan Renaldo, Leo Carrillo, Kippee Valez, Charles Halton, Pedro de Cordoba, Stephen Chase, Edmund Cobb, David Leonard, Frank Jaquet, Mickey Little. The Cisco Kid and his sidekick Pancho help a banker falsely convicted of robbery and murder. More than adequate \"Cisco Kid\" series programmer re-titled _**Guns of Fury**_ for television.\n\n**958** _ **Daring Danger**_ **** Columbia, 1932. 60 min. D: D. Ross Lederman. SC: Michael Trevelyan. With Tim McCoy, Alberta Vaughn, Wallace MacDonald, Robert Ellis, Ed LeSaint, Bobby Nelson, Max Davidson, Richard Alexander, Vernon Dent, Murdock MacQuarrie, Edmund Cobb, Art Mix, Bud Osborne, Artie Ortego, Jim Corey, Arthur Millett, Ben Corbett, Charles Brinley. A crook tries to starve an old man and his daughter off their ranch but a cowboy teams with a cattlemen's agent to help them. While a bit on the slow side, this Tim McCoy effort entertains and has a good finale.\n\n_**The Daring Rogue**_ see _**The Gay Amigo**_\n\n**959** _ **Dark Before Dawn**_ **** PSM Entertainment, 1988. 95 min. D: Robert Totten. SC: Reparatta Mazzola. With Sonny Gibson, Doug McClure, Ben Johnson, Reparatta Mazzola, Morgan Woodward, Billy Drago, Rance Howard, Buck Henry, Paul Newson, Jeffrey Osterhage, Red Steagall, John Martin. When they are about to lose their properties due to government corruption and crooked businessmen, farmers and Vietnam veterans team to protect what is theirs. Violence laced modern-day melodrama.\n\n**960** _ **Dark Command**_ **** Republic, 1940. 94 min. D: Raoul Walsh. SC: Lionel Hosier and F. Hugh Herbert. With Claire Trevor, John Wayne, Walter Pidgeon, Roy Rogers, George \"Gabby\" Hayes, Porter Hall, Marjorie Main, Raymond Walburn, Joseph Sawyer, Helen MacKellar, J. Farrell MacDonald, Trevor Bardette, Harry Woods, Glenn Strange, Alan Bridge, Jack Rockwell, Ernie Adams, Edward Hearn, Edmund Cobb, Hal Taliaferro, Yakima Canutt, Ben Alexander, Tom London, Cliff Lyons, Al Haskell, Tex Cooper, Bob Woodward, Hank Bell, Tom Smith. During the Civil War, a Kansas school teacher becomes the leader of a notorious band of guerillas as he and a sheriff vie for the woman they both love. Thrilling Republic production that is probably the best screen version of Quantrill's Raiders, although the character is called Will Cantrell in the film; worth viewing. Remade as _**Law of the Golden West**_ (q.v.). A colorized version is available.\n\n**961** _ **Dark Mountain**_ **** Paramount, 1944. 66 min. D: William Berke. SC: Maxwell Shane. With Robert Lowery, Ellen Drew, Regis Toomey, Eddie Quillan, Elisha Cook, Jr., Byron Foulger, Walter Baldwin, Ralph Dunn, Virginia Sale, Eddie Kane, Alex Callam, Rose Plummer, John Fisher, Angelo Desfis. A young woman marries a gangster instead of the forest ranger who truly loves her. Very well made and action filled Pine-Thomas production.\n\n**962** _ **The Darkening Trail**_ **** Mutual, 1915. 62 min. D: William S. Hart. SC: C. Gardner Sullivan. With William S. Hart, Enid Markey, Louise Glaum, George Fisher, Nona Thomas, Milton Ross, Roy Laidlaw. In the Yukon a cad marries a young woman but soon loses interest in her but she is still loved by his chief rival. Interesting and very somber early William S. Hart silent film, directed by the star.\n\n**963** _ **A Daughter of the Sioux**_ **** Davis Distributing, 1925. 55 min. D: Ben Wilson. SC: George W. Pyper. With Ben Wilson, Neva Gerber, Robert Walker, Fay Adams, William Lowery, Rhody Hathaway. A government surveyor suspects a young Indian girl is giving information about fortifications to renegade braves. Low budget silent effort from the popular team of Ben Wilson and Neva Gerber.\n\n**964** _ **Daughter of the West**_ **** Film Classics, 1949. 77 min. Color. D: Harold Daniels. SC: Irving R. Franklyn and Raymond L. Schrock. With Martha Vickers, Philip Reed, Donald Woods, James J. Griffith, Tommy Cook, Pedro de Cordoba, William Farnum, Milton Kibbee, Marion Carney, Anthony Barr. A woman working on an Indian reservation attempts to help the Navajos when a corrupt agent tries to steal their copper lands. Low grade follow-up to _**Ramona**_ (1936) (q.v.).\n\n**965** _ **The Daughters of Joshua Cabe**_ **** ABC-TV, 1972. 74 min. Color. D: Philip Leacock. SC: Paul Savage. With Buddy Ebsen, Karen Valentine, Lesley Ann Warren, Sandra Dee, Don Stroud, Henry Jones, Jack Elam, Leif Erickson, Michael Anderson, Jr., Paul Koslo, Ron Soble. When a new homestead law requires a man to have children in order to keep his land, a veteran trapper hires three young women with tainted pasts to pretend to be his daughters. Fairly amusing telefilm. Sequel: _**The Daughters of Joshua Cabe Return**_ (q.v.).\n\n**966** _ **The Daughters of Joshua Cabe Return**_ **** ABC-TV, 1975. 74 min. Color. D: David Lowell Rich. SC: Kathleen Hite. With Dan Dailey, Dub Taylor, Ronnie Troup, Christina Hart, Brooke Adams, Kathleen Freeman, Carl Betz, Arthur Hunnicutt, Terry Wilson, Robert Burton. One of the three women hired by an old trapper to pose as his daughter is kidnapped by her father, who holds her for ransom. Sequel to _**The Daughters of Joshua Cabe**_ (q.v.), this TV movie amounts to little more than a poor time killer.\n\n**967** _ **Davy Crockett and the River Pirates**_ **** Buena Vista, 1956. 81 min. Color. D: Norman Foster. SC: Tom Blackburn and Norman Foster. With Fess Parker, Buddy Ebsen, Jeff York, Kenneth Tobey, Clem Bevans, Irvin Ashkenazy, Mort Mills, Paul Newlan, Frank Richards, Walter Catlett, Douglass Dumbrille, Mike Mazurki, William Bakewell, George J. Lewis, William Fawcett. In 1810 Davy Crockett and his sidekick George Russell agree to a flatboat race with Big Mike Fink, the self proclaimed \"King of the (Ohio) River.\" Enjoyable fiction follow-up to _**Davy Crockett, King of the Wild Frontier**_ (q.v.), and like its predecessor was first shown on Walt Disney's TV program (as a two part episode) before successful theatrical release; Duke York is grand as Big Mike Fink.\n\n**968** _ **Davy Crockett at the Fall of the Alamo**_ **** Sunset, 1926. 60 minutes. D: Robert North Bradbury. SC: Ben Allan Newman. With Cullen Landis, Kathryn McGuire, Edward Hearn, Bob Fleming, Joe Rickson, Jay Morley, Frank Rice, Bob Bradbury, Jr. (Bob Steele), Ralph McCullough, Fletcher Norton, Anne Berryman, Thomas Lingham, Betty Brown, Steve Clemente. Texans led by Davy Crockett, Jim Bowie and Colonel Travis make an heroic stand against the tyrant Santa Ana and his army at the Alamo. Somewhat creaky silent version of the 1836 massacre from producer Anthony J. Xydias, who used the well staged climactic battle scenes in his _**Heroes of the Alamo**_ (q.v.) a decade later; only a 34-minute version of the movie survives.\n\n**969** _ **Davy Crockett, Indian Scout**_ **** United Artists, 1950. 71 min. D: Lew Landers. SC: Richard Shayer. With George Montgomery, Ellen Drew, Philip Reed, Noah Beery, Jr., Paul Wilkerson, John Hamilton, Chief Thundercloud, Kenne Duncan, Ray Teal, Jimmy Moss, Vera Marshe. Davy Crockett's nephew leads a wagon train that is attacked by Indians with the chief's daughter working as a spy. Tepid pseudo-historical drama with good work by George Montgomery in the title role. TV title: _**Indian Scout**_.\n\n**970** _ **Davy Crockett, King of the Wild Frontier**_ **** Buena Vista, 1955. 93 min. Color. D: Norman Foster. SC: Tom Blackburn. With Fess Parker, Buddy Ebsen, Basil Ruysdael, Hans Conreid, William Bakewell, Kenneth Tobey, Pat Hogan, Helene Stanley, Nick Cravat, Don Megowan, Mike Mazurki, Jeff Thompson, Henry Joyner, Benjamin Hornbuckle, Hal Youngblood, Jim Maddux, Robert Booth, Eugene Brindel, Ray Whitetree, Campbell Brown. The story of frontiersman Davy Crockett, from his days as an Indian fighter with Andrew Jackson, through serving in Congress and his heroic stand at the Alamo. Although historically glossy, this feature made up of three segments of Walt Disney's TV series, is dandy entertainment and was the cause of the 1950s' Davy Crockett phenomena. Sequel: _**Davy Crockett and the River Pirates**_ (q.v.).\n\n**971** _ **Davy Crockett: Rainbow in the Thunder**_ **** NBC-TV, 1988. 94 min. Color. D: David Hemmings. SC: William Blinn. With Tim Dunigan, Johnny Cash, Cheryl L. Arutt, Samantha Eggar, David Hemmings, Matt Salinger, Gary Grubbs, Richard Tyson, Jill Gamley, Brenda Grichlow, Jeff Irvine, Blu Mankuma, Fred Perry, Matt Walker. Davy Crockett and Andrew Jackson relive their part in putting down an Indian uprising a quarter of a century earlier. Pleasant historical fiction, originally an episode of \"Walt Disney's Wonderful World of Color.\"\n\n**972** _ **Dawn at Socorro**_ **** Universal-International, 1954. 80 min. Color. D: George Sherman. SC: George Zuckerman. With Rory Calhoun, Piper Laurie, David Brian, Kathleen Hughes, Alex Nicol, Edgar Buchanan, Mara Corday, Skip Homeier, Roy Roberts, James Millican, Lee Van Cleef, Stanley Andrews, Richard Garland, Paul Brinegar, Philo McCullough, Forrest Taylor, Tristram Coffin, Terry Frost, Dick Curtis, Ray Bennett, William Fawcett. A reformed gunman, waiting in a small town for a train, is forced into one last gunfight. Predictable but entertaining.\n\n**973** _ **Dawn on the Great Divide**_ **** Monogram, 1942. 70 min. D: Howard Bretherton. SC: Jess Bowers (Adele Buffington). With Buck Jones, Raymond Hatton, Mona Barrie, Rex Bell, Robert Lowery, Harry Woods, Maude Eburne, Christine McIntyre, Betty Blythe, Robert Frazer, Tristram Coffin, Jan Wiley, Roy Barcroft, Dennis Moore, Steve Clark, Reed Howes, Bud Osborne, I. Stanford Jolley, Artie Ortego, George Morrell, Milburn Morante, Ray Jones, Lee Shumway, Warren Jackson, Ben Corbett, Spade Cooley, Al Haskell, Art Mix, Jack Daly, Horace B. Carpenter, George Sowards, Kansas Moehring, Rube Dalroy, Herman Hack, Merrill McCormick, Chief Yowlachie, Iron Eyes Cody, Charles Soldani, Denver Dixon. Three buddies lead a wagon train with munitions for the railroad but two brothers plan to hijack the explosives and hire an outlaw gang to dress as Indians in order to put the blame on a local tribe. Buck Jones' final film is short on action but has good production values, an interesting plot and a very fine cast. Based on the James Oliver Curwood story \"Wheels of Fate.\"\n\n**974** _ **The Dawn Rider**_ **** Monogram, 1935. 51 min. D-SC: Robert North Bradbury. With John Wayne, Marion Burns, Reed Howes, Denny Meadows (Dennis Moore), Joe DeGrasse, Yakima Canutt, Earl Dwire, Nelson McDowell, Bert Dillard, Jack Jones, James Sheridan (Sherry Tansey), Herman Hack, Jack Evans, Chuck Baldra, Fred Parker, Tex Palmer, George Morrell, Tex Phelps, Archie Ricks, Bob Morrison. A cowboy tries to capture the robber who murdered his father and becomes involved with the man's pretty sister, who is also loved by his pal. Better than average John Wayne-Lone Star vehicle, next to the last in the series before Republic took over the Paul Malvern production releases. Remade as _**Western Trails**_ (q.v.) and colorized as _**Cold Vengeance**_.\n\n**975** _ **The Dawn Trail**_ **** Columbia, 1930. 66 min. D: Christy Cabanne. SC: John T. Neville. With Buck Jones, Miriam Seegar, Charles Morton, Charles King, Hank Mann, Erville Alderson, Ed LeSaint, Inez Gomez, Vester Pegg, Slim Whitaker, Bob Burns, Buck Connors, Art Mix, Bob Fleming, Jack Curtis, Bill Patton, William McCall, Charles West, Jack King, Charles Brinley, Jack Low, Violet Axzelle, Roy Bucko. In an area plagued by a cattlemen-sheep herders war, a sheriff must hold his girl's brother for murder. Excellent Buck Jones early talkie, remade as _**Texas Stampede**_ (q.v.).\n\n**976** _ **Day of Anger**_ **** National General, 1969. 109 min. Color. D: Tonino Valerii. SC: Ernesto Gastaldi, Tonino Valerii and Renzo Genta. With Lee Van Cleef, Giuliano Gemma, Walter Rilla, Christa Linder, Piero Lulli, Ennio Balboa, Lukas Ammann, Andrea Bosic, Pepe Calvo, Giorgio Gargiullo, Anna Orso, Benito Stefanelli, Yvonne Sanson. A gunslinger befriends a young man and the two take over a town to get money owed to the gunman but eventually his partner comes to dislike his mentor's ways. Not one of Lee Van Cleef's better efforts but it has enough action and violence to please his fans. Issued in Italy in 1967 by Sancrosiap\/Corona\/ KG Divina Films as _**I Giorni dell'Ira**_ (The Days of Wrath). Some video prints run 78 minutes.\n\n**977** _ **A Day of Fury**_ **** Universal-International, 1956. 78 min. Color. D: Harmon Jones. SC: James Edmiston and Oscar Brodney. With Dale Robertson, Jock Mahoney, Mara Corday, Carl Benton Reid, Jan Merlin, John Dehner, Dayton Lummis, Sheila Bromley, Terry Frost, Howard Wendell, Henry Wills, James Bell, Dani Crayne, Charles Cane, Phil Chambers. Seeing the decline of lawlessness in the Old West, a young rebel tries to terrorize a small town. A bit different plot adds some interest to this oater.\n\n**978** _ **Day of the Animals**_ **** Film Ventures International, 1977. 95 min. Color. D: William Girdler. SC: William Norton. With Christopher George, Leslie Nielsen, Lynda Day George, Richard Jaeckel, Michael Ansara, Ruth Roman, Andrew Stevens, Gil Lamb, Jon Cedar, Paul Mantee. A group of hikers are attacked by animals in the wilderness after the beasts have gone mad from damage done to the Earth's ozone layer. Fairly competent Western-sci-fi venture with a pleasingly adept score by Lao Shifrin. Also called _**Something Is Out There**_.\n\n**979** _ **Day of the Bad Man**_ **** Universal-International, 1958. 82 min. Color. D: Harry Kellar. SC: Lawrence Roman. With Fred MacMurray, Joan Weldon, John Ericson, Robert Middleton, Marie Windsor, Edgar Buchanan, Skip Homeier, Eduard Franz, Peggy Converse, Robert Foulk, Ann Doran, Lee Van Cleef, Eddy Waller, Christopher Dark, Don Haggerty, Chris Alcaide, Kenneth MacDonald, William Henry, I. Stanford Jolley, Tom London, Steve Darrell, Ralph Littlefield, Jess Kirkpatrick, Hank Patterson, Harry Tyler, Frank O'Connor, Chuck Hamilton, Eddie Parker, Paul Petersen. A circuit judge sentences an outlaw to be hanged but in order to carry out the verdict he must hold off the man's brothers, who plan to rescue him. Passable programmer.\n\n**980** _ **Day of the Evil Gun**_ **** Metro-Goldwyn-Mayer, 1968. 93 min. Color. D: Jerry Thorpe. SC: Charles Marquis Warren and Eric Bercovici. With Glenn Ford, Arthur Kennedy, Dean Jagger, Pilar Pellicer, John Anderson, Paul Fix, Nico Minardos, (Harry) Dean Stanton, Parley Baer, Barbara Babcock, James J. Griffith, Royal Dano, Ross Elliott, Peter Mark Richman, Lee J. Cobb, Olan Soule, Jose Chavez, Jaime Fernandez, Peter Ford, Jane Geffrey, Jorge Martinez de Hoyos. Two enemies join forces to rescue the wife and children of one of them after they were abducted by Indians. Fairly exciting and entertaining outing.\n\n**981** _ **Day of the Outlaw**_ **** United Artists, 1959. 91 min. D: Andre De Toth. SC: Philip Yordan. With Robert Ryan, Burl Ives, Tina Louise, Alan Marshal, Nehemiah Persoff, David Nelson, Venetia Stevenson, Donald Elson, Helen Westcott, Robert Cornthwaite, Jack Lambert, Lance Fuller, Frank De Kova, Paul Wexler, William Schallert, Arthur Space, Betsy Jones Moreland, Elisha Cook (Jr.), George Ross, Dabbs Greer. An outlaw gang on the run after a robbery, with their injured leader, rides into a small town and is detained there by a blizzard and opposed by a strong willed cattleman. Better than average melodrama; well done with nice winter location filming.\n\n**982** _ **The Day of the Wolves**_ **** Gold Key, 1973. 91 min. Color. D-SC: Ferd(e) Grofe, Jr. With Richard Egan, Martha Hyer, Rick Jason, Jan Murray, Frankie Randall, Andre Marquis, Henry Capps, Smokey Roberds, Zaldy Zshomak, John Lupton. Sean McClory, Jack Bailey, Biff Elliott, Percy Helton, Herb Vigran, John Dennis, John Gunn, Danny Rees, Len Travis, Steve Manone, Ben Summers, Doc Richards, Elizabeth Thomas. Seven criminals plan to loot the Western town of Wellterton but a recently fired sheriff stands up to the invaders. Modern-day drama with a fairly good plot, a fine performance by Richard Egan as the lawman but sparse production values.\n\n_**Day of Vengeance**_ see _**Long Days of Vengeance**_\n\n**983** _ **Days of Adventure, Dreams of Gold**_ **** William Bronson, 1975. 60 min. Color. D: William Bronson and Denver Sutton. With Hal Holbrook (narrator). Documentary on the last gold rush in the Yukon Territory in 1897. History buffs will like this one.\n\n**984** _ **Days of Buffalo Bill**_ **** Republic, 1946. 56 min. D: Thomas Carr. SC: William Lively and Doris Schroeder. With Sunset Carson, Peggy Stewart, Tom London, James Craven, Rex Lease, Edmund Cobb, Eddie Parker, Michael Sloan, Jay Kirby, George Chesebro, Ed Cassidy, Tex Cooper, Kit Guard, Tommy Coats, Pascale Perry, Roy Bucko. A cowpoke and his buddy are framed for murder and escape a posse to prove their innocence. Buffalo Bill Cody is nowhere to be seen, but fans of Sunset Carson will like it anyway.\n\n_**Days of '40**_ see _**California in '49**_\n\n**985** _ **Days of Heaven**_ **** Paramount, 1978. 95 min. Color. D-SC: Terrence Malick. With Richard Gere, Brooke Adams, Sam Sheppard, Linda Manz, Robert Wilke, Jackie Shultis, Stuart Margolin, Tim Scott, Gene Bell, Doug Kershaw, Richard Libertini, Frenchie Lemond, Sahbra Markus, Bob Wilson, Murile Joliffe, John Wilkinson, King Cole. In rural Texas farm workers strive to bring in a harvest while being at odds with their rancher boss as two of them love the same woman. Beautifully photographed drama set in the pre\u2013World War I period, filmed in Canada.\n\n**986** _ **Days of Jesse James**_ **** Republic, 1939. 63 min. D: Joseph Kane. SC: Jack Natteford. With Roy Rogers, George \"Gabby\" Hayes, Pauline Moore, Donald Barry, Harry Woods, Arthur Loft, Wade Boteler, Ethel Wales, Scotty Beckett, Harry Worth, Glenn Strange, Olin Howlin, Monte Blue, Jack Rockwell, Fred Burns, Bud Osborne, Jack Ingram, Carl Sepulveda, Lynton Brent, Pasquel Perry, Eddie Acuff, Horace B. Carpenter. A railroad detective after Jesse James runs afoul of an opportunistic sheriff and a crooked banker who commit a series of robberies and placed the blame on the famous outlaw. Excellent Roy Rogers feature with Don Barry stealing the show as Jesse James.\n\n**987** _ **Days of Old Cheyenne**_ **** Republic, 1943. 55 min. D: Elmer Clifton. SC: Norman S. Hall. With Don \"Red\" Barry, Lynn Merrick, Herbert Rawlinson, William Haade, Emmett Lynn, Robert Kortman, William Ruhl, Nolan Leary, Kenne Duncan, Eddie Parker, Bob Reeves, Art Dillard. A cowboy helps citizens in fighting a corrupt political leader in the Wyoming Territory. Typically action filled Don Barry vehicle.\n\n**988** _ **Dead Aim**_ **** Producciones Jaguar, 1975. 97 min. Color. D: Jose Antonio Balanos. SC: Jose Antonio Balanos and Pedro F. Mirt. With Glen Lee, Venetia Vianello, James Westerfield, Virgil Frye, Evaristo Marquez, Granville van Deusen, Barbara Angely, Carlos East, George (Jorge) Russek, Tony Monaco, Billy Joe Roucke, Eduardo Bonada. After being saved from a rattle snake bite by a mortician, a gun loving youth becomes his savior's bodyguard and later runs afoul of a man over his wife. Confusing Italian-Mexican co-production lensed in Mexico. Also called _**Lucky Johnny**_.\n\n**989** _ **The Dead and the Damned**_ **** Inception Media Group, 2010. 82 min. Color. D-SC: Rene Perez. With David A. Lockhart, Camille Montgomery, Rick Mora, Robert Amstler, Pat McIntire, Randall Marshall Dillon, Autumn J.D. Harrison, Heather Montanez, Mandy Pauline, Nathan J. Yeisley, Colin Hussey, Harry Bruce, Lauren C. Kelly, George Anderson. During the 1849 California gold rush a meteor crashes near a mining settlement releasing spoors that turn the locals into zombies. Poor horror Western released in Great Britain as _**Cowboys and Zombies**_.\n\n_**Dead Are Countless see Garringo**_\n\n**990** _ **The Dead Don't Dream**_ **** United Artists, 1948. 62 min. D: George Archainbaud. SC: Francis Rosenwald. With William Boyd, Andy Clyde, Rand Brooks, John Parrish, Leonard Penn, Mary Tucker, Francis McDonald, Richard Alexander, Bob Gabriel, Stanley Andrews, Forbes Murray, Don Haggerty. Hoppy, California and Lucky visit a remote ranch where several of the owner's relatives have been murdered. Good atmospheric mystery angle makes this one of the better later \"Hopalong Cassidy\" films.\n\n**991** _ **Dead for a Dollar**_ **** Denwer Films, 1968. 106 min. Color. D: Osvaldo Civirani. SC: Tito Carpi, Osvaldo Civirani and Luciano Gregoretti. With George Hilton, Sandra Milo, John Ireland, Gordon Mitchell, Don Palmer (Mimmo Palmara), Andrew Scott (Andrea Scotti), Piero Vida, Franco Ressel, Monica Pardo, Franco Guia, Carla Brait, Rossella Bergamonti, Renato Chiantoni, Giovanni Scratuglia, Enzo Andronico, Roberto Messina, Mario De Vico. Three men vie for stolen money they lost to a now deceased crook. Spanish made Western that is overlong without much action; filmed as _**T'ammazzo!...Raccoomandati a Dio**_ (She Tortured You...Sent to God) and also called _**Trusting Is Good...Shooting Is Better**_.\n\n**992** _ **Dead Man**_ **** Miramex Films, 1995. 121 min. Color. D-SC: Jim Jarmusch. With Johnny Depp, Gary Farmer, Crispin Glover, Lance Henriksen, Michael Wincott, Eugene Byrd, John Hurt, Robert Mitchum, Iggy Pop, Gabriel Byrne, Jared Harris, Miili Avital, Jimmie Ray Weeks, Mark Bringelson, John North, Peter Schrum, Mike Dawsonk Billy Bob Thornton, Michelle Thrush, Gibby Haines, Richard Boes, George Duckworth, Thomas Bettles, Alfred Molina, Daniel Chas Stacy, Todd Pfeiffer, Leonard Bowechop, Cecil Cheeka, Michael McCarthy. An accountant goes West, is mistaken for a killer, pursued by bounty hunters and after being wounded is helped by an Indian who thinks he is the reincarnation of poet William Blake, his namesake. Strange combination of the Western and fantasy genres with probably unintended humor.\n\n**993** _ **Dead Man's Bounty**_ **** Barnholtz Entertainment, 2006. 94 min. Color. D-SC: Piotr Uklanski. With Boquslaw Linda, Karel Roden, Katarzyna Figura, Val Kilmer, Marek Barbasiewicz, Anna Baniwoska, Romuald Andrzej Klos, Rafal Mohr. A man from another country arrives in a small Western town with a wanted outlaw. Disappointing R-rated production filmed in Poland.\n\n**994** _ **Dead Man's Gold**_ **** Screen Guild, 1948. 60 min. D: Ray Taylor. SC: Moree Herring and Gloria Welsch. With Lash LaRue, Al St. John, Peggy Stewart, John Cason, Terry Frost, Lane Bradford, Pierce Lyden, Steve Keys, Cliff Taylor, Britt Wood, Marshall Reed, Bob Woodward. Two men ride into Gold Valley to help a buddy, run into outlaws and learn their friend has been murdered. Fans of Lash LaRue and Al St. John will enjoy this fun film.\n\n**995** _ **Dead Man's Gulch**_ **** Republic, 1943. 56 min. D: John English. SC: Norman S. Hall and Robert Williams. With Don \"Red\" Barry, Lynn Merrick, Rex Lease, Emmett Lynn, Clancy Cooper, Bud McTaggart, Jack Rockwell, Pierce Lyden, Lee Shumway, Robert Frazer, Robert Fiske. A one-time Pony Express rider learns he is being used by crooks to cheat ranchers on freight rates. Another good entry in Don Barry's Republic series.\n\n**996** _ **Dead Man's Revenge**_ **** Universal, 1994. 100 min. Color. D: Alan J. Levi. SC: Jim Byrnes and David Chisholm. With Bruce Dern, Michael Ironside, Vondie Curtis-Hall, Keith Coulouris, Daphne Ashbrook, Tobin Bell, John M. Jackson, Melora Walters, Jack Rader, Doug McClure, Randy Travis, Ping Wu, Robert Cornthwaite, Eric Boles, Larry Cedar, David Dunard, Robert Mason Ward, William Newman, Ritch Brinkley, Jeffrey Roth, Bradley Pierce, Luis Contreras, Heath Kizzier, Mark Nearing, Kenny Call, Anthony Reynolds, Steve Kelso, Aliza Washabaugh, Ken Parham. Framed for a crime he did not commit by a railroad tycoon who wanted his land, a homesteader breaks out of jail and a bounty hunter is hired to bring him back. Fair Western made for television.\n\n**997** _ **Dead Man's Trail**_ **** Monogram, 1952. 59 min. D: Lewis D. Collins. SC: Joseph Poland. With Johnny Mack Brown, James Ellison, Barbara Allen, I. Stanford Jolley, Terry Frost, Lane Bradford, Gregg Barton, Richard Avonde, Dale Van Sickel, Stanley Price, John Hart. The brother of an outlaw, who has been murdered by his own gang, helps a sheriff in tracking down the bad men and recovering stolen money. Okay Johnny Mack Brown outing and his last film with James Ellison.\n\n**998** _ **Dead Man's Walk**_ **** ABC-TV, 1996. 272 min. Color. D: Yves Simoneau. SC: Larry McMurtry and Diana Ossana. With David Arquette, Jonny Lee Miller, F. Murray Abraham, Brian Dennehy, Keith Carradine, Harry Dean Stanton, Patricia Childress, Edward James Olmos, Eric Schweig, Jennifer Garner, Ray McKinnon, Tim Blake Nelson, Alastair Duncan, Brad Greenquist, Kieran Mulroney, Jared Rushton, Joaquim de Almeida, Haviland Morris, Akosua Busia, Rodger Boyce, Ed Cantrell, Rutherford Cravens, Matt Davison, Eulra Doonkeen, Grant James, Jonathan Joss, Steve Larson, Gretchen Mol, Bert Roberts, Marc Miles, Adam Lamberg, Chris Penn, Manuel Calderon, Toby Metcalf, Robert Norsworthy, Jimmie F. Skaggs, Marvin \"Skeeter\" Roubison, Victor Aaron, Hugo Urrutia, Julius Tennon, Booth Southerland. Two young westerners fight to survive on the Texas frontier. Entertaining pre-sequel to _**Lonesome Dove**_ (q.v.), followed by _**Comanche Moon**_ (q.v.).\n\n**999** _ **Dead Men Don't Make Shadows**_ **** A.B. Films, 1970. 98 min. Color. D: Miles Deem (Demofilo Fidani). SC: Francesco Munich and Demofilo Fidani. With Hunt Powers, Chet Davis, Gordon Mitchell, Dennis Colt, Simone Blondell, Ettore Manni, Pietro Fumelli, Custer Gail, Dean Reese, Arizona Masochist (Joe D'Amato). A bounty hunter who is being stalked shows up in a small town to take out the ruthless mine owner who controls the area. Hazy, slow moving Italian Western photographed by Joe D'Amato; made as _**Inginocchiati Straniero...I Cadaveri non Fanno Ombra!**_ and also called _**Stranger That Kneels Beside the Shadow of a Corpse**_.\n\n**1000** _ **Dead Noon**_ **** Barnholtz Entertainment, 2007. 85 min. Color. D: Andrew Wiest. SC: Andrew Wiest, Matthew Taggart and Keith Suta. With Kane Hodder, Robert Milo Andrus, Robert Baer, Jen Kelsey, Nick Martin, Tye Nelson, James Teague, Scott Phillips, Nick Quintilliani, Charles Stoll, Jordan Jansen-Mecca, Ed Bosco, Andrew Wiest, Elizabeth Mouton, Kelsey McCann, M.J. Somer. An outlaw comes back from the dead to take revenge on a town and he enlists the forces of Hell to aid him. Cheap, terrible, video horror Western.\n\n**1001** _ **Dead or Alive**_ **** Producers Releasing Corporation, 1944. 56 min. D: Elmer Clifton. SC: Harry Fraser. With Tex Ritter, Dave O'Brien, Guy Wilkerson, Marjorie Clements, Charles King, Rebel Randall, Ray Bennett, Reed Howes, Bud Osborne, Henry Hall, Ted Mapes, Frank Ellis, Ed Cassidy, Jimmy Aubrey, Wen Wright, Ray Henderson. Three lawmen, using various guises, come to the aid of a judge when an outlaw gang tries to take a young woman's ranch. Lots of action in this \"Texas Rangers\" entry but crudely made with cheap sets and a mundane plot. Among the songs sung by Tex Ritter are \"I'm Gonna Leave You Like I Found You\" and \"Don't Care Since You Said Goodbye.\" British title: _**Wanted by the Law**_.\n\n**1002** _ **Deadline**_ **** Columbia, 1931. 60 min. D-SC: Lambert Hillyer. With Buck Jones, Loretta Sayers, Robert Ellis, Raymond Nye, Ed Brady, Knute Erickson, George Ernest, Harry Todd, Jack Curtis, James Farley, Robert Kortman, Ed LeSaint. A quick tempered cowboy is paroled from jail but soon finds himself in trouble with outlaws. Exceedingly fine Buck Jones vehicle thanks to a literate script and good production values.\n\n**1003** _ **Deadline**_ **** Astor, 1948. 57 min. D: Oliver Drake. SC: O.C. (Oliver) Drake. With Sunset Carson, Pat Starling, Al Terry, Pat Gleason, Lee Roberts, Steven Keyes, Frank Ellis, Forrest Matthews, Robert Curtis, Philip Arnold, Jose Hisar, Don Grey, Buck Monroe, Al Wyatt. While making his final run before the use of the telegraph, a Pony Express rider uncovers a plot by a rancher to force a company out of business and steal its land for his own profit. Low grade and very boring Sunset Carson film.\n\n**1004** _ **Deadlock**_ **** Cinerama, 1970. 94 min. Color. D-SC: Roland Klick. With Mario Adorf, Anthony Dawson, Marquard Bohm, Mascha Elm-Rabben, Sigurd Fitzek, Betty Segal. After pulling off a robbery, two bandits meet in a deserted mining town to divide their loot but a vagabond tries to steal it from them. Interesting West German modern drama filmed in Israel.\n\n**1005** _ **The Deadly Companions**_ **** Path\u00e9-America, 1961. 90 min. Color. D: Sam Peckinpaugh. SC: A.S. Fleischman. With Maureen O'Hara, Steve Cochran, Brian Keith, Chill Wills, Strother Martin, Will Wright, John Hamilton, Jim O'Hara. When a former soldier accidentally shoots a boy he agrees to lead the lad's funeral procession across the desert in Apache Territory so his mother can prove her son's legitimacy. Austere melodrama; slow moving but not without interest.\n\n**1006** _ **Deadly Reactor**_ **** Action International, 1989. 88 min. Color. D-SC: David Heavener. With Stuart Whitman, David Heavener, Darwyn Swalve, Allyson Davis, Kimberly Cassey, Arvid Holmberg, Barbara Kerek, Ingrid Vold. A gunman-preacher becomes a sheriff and does battle with a biker gang. Cheap, violent futuristic sci-fi Western.\n\n_**Deadly Shooter**_ see _**The Shooter**_\n\n**1007** _ **The Deadly Trackers**_ **** Warner Bros., 1973. 104 min. Color. D: Barry Shear. SC: Lukas Heller. With Rod Taylor, Richard Harris, Al Lettieri, Neville Brand, William Smith, Paul Benjamin, Pedro Armendariz, Jr., Kelly Jean Peters, Sean Marshall, Red Morgan, William Bryant. After outlaws pull off a robbery, murdering a banker's wife and son, a peaceful man sets out on a quest of avenging the crimes. Despite a good plot, this Western is a misfire and will not likely please genre fans. Also called _**Killbrand**_ and _**Riata**_.\n\n**1008** _ **Deadwood Dick**_ **** Columbia, 1940. 15 Chapters. D: James W. Horne. SC: Wyndham Gittens, Morgan B. Cox, George Morgan and John Cutting. With Don Douglas, Lorna Gray (Adrian Booth), Harry Harvey, Marin Sais, Lane Chandler, Jack Ingram, Charles King, Ed Cassidy, Robert Fiske, Lee Shumway, Edmund Cobb, Ed Peil, Sr., Edward Hearn, Karl Hackett, Roy Barcroft, Bud Osborne, Joe Girard, Tom London, Kenne Duncan, Yakima Canutt, Fred Kelsey, Edward Cecil, Kit Guard, Al Ferguson, Franklyn Farnum, Jim Corey, Eddie Featherston, Charles Hamilton, Constantine Romanoff. Deadwood Dick, a mysterious figure of the plains, tries to thwart the nefarious activities of \"The Skull\" and his gang which has been terrorizing the citizens of South Dakota. Quick paced affair that will please serial lovers.\n\n_**Top:**_ **Sean Marshall (left) and Richard Harris in** _**The Deadly Trackers**_ **(Warner Bros., 1973).** _**Bottom:**_ **Poster for** _**Deadwood Dick**_ **(Columbia, 1940).**\n\n**  \n**\n\n**1009** _ **Deadwood Pass**_ **** Monarch\/Freuler, 1933. 61 min. D: J.P. McGowan. SC: John Wesley Patterson. With Tom Tyler, Wally Wales, Alice Dahl, Lafe McKee, Edmund Cobb, Slim Whitaker, Merrill McCormick, Carlotta Monti, Buffalo Bill, Jr., Duke Lee, Blackie Whiteford, Bill Nestell, Bud Osborne, J.P. McGowan, Ben Corbett, Jack Kirk, Bud McClure, Chuck Baldra. A government agent poses as the notorious outlaw \"The Hawk\" so he can find out where his gang hid stolen loot. Fast moving and action filled Tom Tyler film; one of his better sound outings.\n\n**Poster for** _**Deadwood Pass**_ **(Monarch\/Freuler, 1933).**\n\n**  \n**\n\n**1010** _ **Deadwood '76**_ **** Fairway-International, 1965. 94 min. Color. D: James Landis. SC: Arch Hall, Jr. and William Watters (Arch Hall). With Arch Hall, Jr., Melissa Morgan, Jack Lester, William Watters (Arch Hall), Robert Dix, Rex Marlow, John Bryant, Barbara Moore, Red Morgan, John Cardos, Little Jack Little, Ray Vegas, Harold Bizzy. Heading to the Dakotas to take part in a gold rush, an ex-soldier is mistaken for Billy the Kid. Wild Bill Hickok, Calamity Jane, Sam Bass and Chief Crazy Horse are just a few of the historical characters who show up in this inane production that has to be seen to be believed.\n\n**1011** _ **Deaf Smith and Johnny Ears**_ **** Metro-Goldwyn-Mayer, 1973. 91 min. Color. D: Paolo Cavara. SC: Harry Essex and Oscar Saul. With Anthony Quinn, Franco Nero, Pamela Tiffin, Ira Furstenberg, Franco Graziosi, Renato Romano, Adolfo Lastretti, Tom Felleghy. Two pals try to stop a would be dictator from taking over the newly formed Republic of Texas in 1836. Mediocre Italian Western with good interplay between the title characters portrayed by Anthony Quinn and Franco Nero.\n\n**1012** _ **Dean Teaster's Ghost Town**_ **** Barnholtz Entertainment, 2007. 115 min. Color. D: Dean West (Dean Teaster) and Jeff Kennedy. SC: D.J. Perry and Dean West (Dean Teaster). With Herbert \"Cowboy\" Coward, Bill McKinney, D.J. Perry, Princess Lucai, Rance Howard, Renee O'Connor, Tony Becker, Stella Parton, Terence Knox, Sammy Kershaw, Charles Edwin Powell, Terry Jerrigan, Charles Matthau, Dean West (Dean Teaster), Anthony Hornus, Robert Bradley, Russ Stine, Jordan Engle, Bill Steele, Paul Projos, Fred Griffith, Greg Mason, Tommy Dippel, Tom Chaudoin, Austin Two Feathers, Mark Jones, Patrick Walker, John McElrath, Bill Whitworth, Mary Beth Hampton, Harry Valentine, Tammy Stephens Teaster, Phoebe Bond, Ralphene Rathbone, Bobby Teaster, Ed Mantell. Western filmed in North Carolina and centered around the Teaster family's theme park with staged gunfights. Of local interest only; also called _**Ghost Town: The Movie**_.\n\n**1013** _ **Death Goes North**_ **** Warwick Films, 1939. 56 min. D: Frank McDonald. SC: Edward R. Austin. With Rin Tin Tin, Jr., Edgar Edwards, Sheila Bromley, Dorothy Bradshaw, Jameson Thomas, Walter Byron, Arthur Kerr, James McGrath, Vivian Combe, Reginald Hincks. A Mountie and his dog attempt to bring in the killer of lumberman in the north woods. Well made and pleasing Canadian made programmer issued in that country by Columbia in 1938.\n\n**1014** _ **Death Hunt**_ **** 20th Century\u2013Fox, 1981. 97 min. Color. D: Peter Hunt. SC: Michael Craig and Mark Victor. With Charles Bronson, Lee Marvin, Andrew Stevens, Carl Weathers, Ed Lauter, Angie Dickinson, Scott Hylands, Henry Beckman, William Sanderson, Jon Cedar, James McConnell, Len Lesser, Dick Davalos, Maury Chaykin, James McIntire, Rayford Barnes, August Schellenberg, Dennis Wallace, Maurice Kowalski, Sean McCann, Steve O.Z. Finkel, Denis Lacroix, Tantoo Martin, Amy Marie George. In the Yukon in 1931 a reclusive trapper, forced to commit murder, is tracked over the frozen wastes by a resolute Mounted Policeman and a vicious posse. Nice pictorial fact based film, similar to _**Challenge to Be Free**_ (q.v.); colorful and entertaining.\n\n**1015** _ **Death of a Gunfighter**_ **** Universal, 1969. 100 min. Color. D: Allen Smithee (Robert Totten and Don Siegel). SC: Joseph Calvelli. With Richard Widmark, Lena Horne, Carroll O'Connor, John Saxon, Kent Smith, David Opatoshu, Jacqueline Scott, Morgan Woodward, Larry Gates, Dub Taylor, Victor French, Michael McGreevey, Darleen Carr, Mercer Harris, James O'Hara, Harry Carey, Jr., Jimmy Lydon, Kathleen Freeman, Royal Dano, Walter Sande, Robert Sorrells, Amy Thomson, Charles Kuenstle, Sara Taft. An old time marshal tries to prevent the citizens of his town from taking away his job. Fairly interesting production made for TV but given theatrical release.\n\n**1016** _ **Death Rides a Horse**_ **** United Artists, 1969. 114 min. Color. D: Guilio Petroni. SC: Luciano Vincenzoni. With Lee Van Cleef, John Philip Law, Luigi Pistilli, Anthony Dawson, Jose Torres, Mario Brega, Carla Cassola, Archie Savage, William Bogart, Bruno Corazzaari. Fifteen years after the brutal murder of his parents, a young man tries to find the killers and joins forces with an ex-convict who may know their whereabouts. Over long, but action filled, Italian Western; mainly for Lee Van Cleef fans. Issued in its homeland in 1967 by P.E.C. as _**Da Uomo a Uomo**_ (As Man to Man).\n\n**Poster for** _**Death Rides a Horse**_ **(United Artists, 1969).**\n\n**  \n**\n\n**1017** _ **Death Rides the Plains**_ **** Producers Releasing Corporation, 1943. 56 min. D: Sam Newfield. SC: Joseph O'Donnell. With Robert Livingston, Al St. John, Nica Doret, Ray Bennett, I. Stanford Jolley, George Chesebro, John Elliott, Kermit Maynard, Slim Whitaker, Karl Hackett, Frank Ellis, Ted Mapes, Jimmy Aubrey, Dan White, Curley Dresden, Wally West, Kansas Moehring, Hank Bell, Lane Bradford, Milburn Morante, Oscar Gahan, Tex Cooper, Art Dillard, George Morrell, Tex Palmer, Jack Evans, Art Fowler, Rube Dalroy, Ralph Bucko, Roy Bucko, Jack Tornek, Lew Morphy. A rancher offers to sell his land and then kills the buyers for their money with the Lone Rider stumbling onto the scheme and trying to stop him. Good entry in PRC's \"Lone Rider\" series.\n\n**1018** _ **Death Rides the Range**_ **** Colony, 1940. 58 min. D: Sam Newfield. SC: William Lively. With Ken Maynard, Fay McKenzie, Ralph Peters, Julian Rivero, Charles King, John Elliott, William Costello, Sven Hugo Borg, Michael Vallon, Richard Alexander, Bud Osborne, Murdock MacQuarrie, Wally West. A cowboy and his pal find themselves at odds with foreign agents after a helium gas deposit. An interesting story and plenty of action make this later Ken Maynard vehicle a must-see for his fans.\n\n**1019** _ **Death Sentence**_ **** B.L. Vision, 1967. 90 min. Color. D-SC: Mario Lanfranchi. With Richard Conte, Robin Clarke, Adolfo Celi, Tomas Milian, Enrico Maria Salerno, Lilli Lembo. A gunman is out for revenge on the gang who killed his brother in a robbery some years before. Typically violent Spaghetti Western helped by good acting by Richard Conte and Adolfo Celi; made in Italy as _**Sentenza di Morte**_ (Sentence of Death).\n\n**1020** _ **Death Valley**_ **** Screen Guild, 1946. 70 min. Color. D: Lew Landers. SC: Doris Schroeder. With Nat Pendleton, Helen Gilbert, Robert Lowery, Sterling Holloway, Barbara Reed, Russell Simpson, Paul Hurst, Dick Scott, Stan(ley) Price, Bob Benton. A man buys a fake claim map in Death Valley and there the lure of gold drives him mad. Well done adventure melodrama with the added attraction of location filming in Cinecolor.\n\n**1021** _ **Death Valley**_ **** Universal, 1982. 87 min. Color. D: Dick Richards. SC: Richard Rothstein. With Paul Le Mat, Catherine Hicks, Stephen McHattie, A. Wilford Brimley, Peter Billingsley, Edward Herrmann, Jack O'Leary. A woman, her young son and boyfriend travel through the desert on a vacation and run into a murderous psychopath. Modern-day Western is only fairly suspenseful but genre fans will like seeing a good display of Ken Maynard film clips.\n\n**1022** _ **Death Valley Gunfighter**_ **** Republic, 1949. 60 min. D: R.G. Springsteen. SC: Bob Williams. With Allan \"Rocky\" Lane, Gail Davis, Eddy Waller, Jim Nolan, William Henry, Harry Harvey, Mauritz Hugo, George Chesebro, Forrest Taylor, Lane Bradford. While looking into a payroll robbery, a peace officer is attacked by outlaws. Typically good Allan Lane \"Famous Westerns\" series entry.\n\n**1023** _ **Death Valley Manhunt**_ **** Republic, 1943. 55 min. D: John English. SC: Norman S. Hall and Anthony Coldeway. With Wild Bill Elliott, George \"Gabby\" Hayes, Anne Jeffreys, Weldon Heyburn, Herbert Heyes, Davison Clark, Pierce Lyden, Jack Kirk, Bud Geary, Marshall Reed, Charles Murray, Jr., Edward Keane, Curley Dresden, Eddie Phillips, Al Taylor, Jesse Graves, Charles Sullivan, Walter McGrail, Neal Hart, Franklyn Farnum, Frank Ellis, Art Dillard, Kansas Moehring, Silver Harr. Brought out of retirement by an oil company to look into in the sabotage of their wells in Death Valley, Wild Bill Elliott tries to track down the responsible party. Highly competent \"Wild Bill Elliott\" series vehicle with lots of action and a good script.\n\n**1024** _ **Death Valley Outlaws**_ **** Republic, 1941. 54 min. D: George Sherman. SC: Don Ryan and Jack Lait, Jr. With Don \"Red\" Barry, Lynn Merrick, Rex Lease, Bob McKenzie, Milburn Stone, Karl Hackett, Jack Kirk, Fred \"Snowflake\" Toones, Robert Kortman, Curley Dresden, John Cason, Griff Barnett, Lee Shumway, Reed Howes, George J. Lewis, Harry Strang, Michael Owen, Wally West, Tex Palmer, Sam Lufkin. Wanting to find his missing brother, a cowboy gets involved with a lawless gang. Nice going for Don Barry in one of his earlier starring efforts.\n**1025** _ **Death Valley Rangers**_ **** Monogram, 1943. 59 min. D: Robert Tansey. SC: Elizabeth Beecher. With Ken Maynard, Hoot Gibson, Bob Steele, Linda Brent, Weldon Heyburn, Bryant Washburn, Glenn Strange, Forrest Taylor, Karl Hackett, Charles King, George Chesebro, John Bridges, Al Ferguson, Steve Clark, Wally West, Lee Roberts, Frank Ellis. The Trail Blazer try to help a town fight gold shipment robbers by having Bob masquerade as an outlaw and infiltrate the gang. Fine \"Trail Blazers\" affair that moves very quickly with the three heroes in good form.\n\n_**Deathworks**_ see _**Captain Apache**_\n\n**1026** _ **Decision at Sundown**_ **** Columbia, 1957. 95 min. Color. D: Budd Boetticher. SC: Charles Lang, Jr. With Randolph Scott, John Carroll, Karen Steele, Valerie French, Noah Beery, Jr., John Archer, Andrew Duggan, James Westerfield, John Litel, Ray Teal, Vaughn Taylor, Richard Deacon, H.M. Wynant, Guy Wilkerson, Bob Steele, Abel Fernandez, Reed Howes, Jim Hayward. A cowboy searches for three years for the man who stole his wife and finds him in a town where he is about to marry a local girl. Highly competent, brooding Western; one of Randolph Scott's best.\n\n**1027** _ **The Decoy**_ **** Echo Bridge Home Entertainment, 2006. 109 min. Color. D: Justin Kreinbrink. SC: Justin Kreinbrink and Tara Kreinbrink. With Justin Kreinbrink, Susan Arnold, Marchelle Scarnier, William Killian, John Michael Bartish, Santiago Craig, Clint James, Brian Mulligan, Jeffrey Scott Holland, Gregory Sweet, Howard Allen, Janee Page, Rachel Owens, Inna Rohr, Leonard Batson, Tom Bushee, Amos Carver, Jake McDaniel, Kevin Market, Sandy Cooper, John Lushbaugh, Keith Lushbaugh, Wendi Evans, Linda Kay Gross, Jay Gammons, Joanne Gammons, Chad Grimes. A deputy sheriff takes his best friend to be hung for killing the lawman's in-laws. Low budget but enjoyable affair, produced, directed and co-written by star Justin Kreinbrink.\n\n**1028** _ **Deep in the Heart of Texas**_ **** Universal, 1942. 62 min. D: Elmer Clifton. SC: Grace Norton. With Johnny Mack Brown, Tex Ritter, Jennifer Holt, Fuzzy Knight, William Farnum, Harry Woods, Kenneth Harlan, Pat O'Malley, Eddie Polo, Earle Hodgins, Roy Brent, Edmund Cobb, Rod Cameron, Jimmy Wakely Trio (Jimmy Wakely, Johnny Bond, Scotty Harrell), Budd Buster, Frank Ellis. Appointed commissioner of public affairs, a man returns home to find himself at odds with his father, the leader of a guerilla band. Good drama, action and music make this fine viewing.\n\n**1029** _ **Deep Valley**_ **** Warner Bros., 1947. 104 min. D: Jean Negulesco. SC: Salka Vietrel and Stephen Morehouse. With Ida Lupino, Dane Clark, Wayne Morris, Fay Bainter, Henry Hull, Willard Robertson, Jack Mower, Ian MacDonald, Rory Mallinson, Harry Strang, Eddie Dunn, William Haade, Clancy Cooper, Ralph Dunn, Ray Teal, John Alvin, Bob Lowell, Lennie Bremen, Ross Ford. The life of a bitter and lonely farm woman changes when she meets an escapee from a chain gang. Downbeat and brooding drama makes good entertainment.\n\n**1030** _ **Deep West**_ **** Cambist Films, 1971. 94 min. Color. D: Anthony Ascott (Giuliano Carnimeo). SC: Tito Carpi. With George Hilton, Charles Southwood, Agata Flori, Roberto Camardiel, Rick Boyd (Federico Boido), Paolo Gozlino, Andrea Bosic, Aldo Barberito, Franco Pesce, Ugo Adinolfi, Fortunato Arena, Luciano Rossi, Goffredo Unger, Lina Sini, Aldo Berti, Furio Meniconi, Paolo Magalotti, Sergio Smacchi, Lina Franchi, Lino Coletta, John Bartha, Rocco Lerro, Claudio Ruffini, Gaetano Scala. A gunman is hired by a Mexican general to rob jewels from the Army, but he is also after a gun runner, a Russian and a nun who is really as spy. Convoluted, but fun, fast moving Italian Western made as _**Testa t'Ammazzo, Croce...Sei Morto...Mi Chiamano Alleluia**_ ; also called _**Heads You Die, Tails I Kill You**_ and _**They Call Me Hallelujah**_. Video title: _**Guns for Dollars**_.\n\n**1031** _ **The Deerslayer**_ **** Cameo Distributing, 1923. 60 min. D: Arthur Wellin. SC: Robert Heymann. With Emil Mamelok, Bela Lugosi, Herta Heden, Gottfried Kraus, Edward Eyseneck, Margot Sikolowska. Hawkeye and his blood brother Chingachgook help British settlers harassed by the French and Indians in upper New York state. Picturesque, but jumbled German silent version of the James Fenimore Cooper novel, originally issued in Europe in 1920 by Luna Film as _**Lederstrumpf**_ (Leatherstocking); heavily cut but worth seeing for Bela Lugosi as Chingachgook.\n\n**1032** _ **The Deerslayer**_ **** Republic, 1943. 67 min. D: Lew Landers. SC: P.S. Harrison and E.B. Derr. With Bruce Kellogg, Jean Parker, Larry Parks, Warren Ashe, Wanda McKay, Yvonne De Carlo, Addison Richards, Robert Warwick, Johnny Michaels, Philip Van Zandt, Trevor Bardette, Chief Many Treaties, Clancy Cooper, Princess Whynemah, William Edmunds. Natty Bumpo, the Deerslayer, comes to the aid of a tribe whose pretty princess is coveted by a rival Huron brave who burns their village and kidnaps the maiden. Tacky presentation of the James Fenimore Cooper work; this independent production from script writers P.S. Harrison and E.D. Derr was issued theatrically by Republic.\n\n**1033** _ **The Deerslayer**_ **** 20th Century\u2013Fox, 1957. 78 min. Color. D-SC: Kurt Neumann. With Lex Barker, Forrest Tucker, Cathy O'Donnell, Rita Moreno, Jay C. Flippen, Carlos Rivas, John Halloran, Joseph Vitale, Rocky Shahan, Carol Henry. Hawkeye and his Mohican blood brother try to avert an Indian war when they learn a white man, living on an isolated island with his two daughters, is a scalp hunter. Colorful adaptation of the James Fenimore Cooper book.\n\n**1034** _ **The Deerslayer**_ **** NBC-TV\/Schick Sunn Classics, 1974. 74 min. Color. D: Dick Friedenbert. SC: S.S. Schweitzer. With Steve Forrest, Ned Romero, John Anderson, Victor Mohica, Joan Prather, Charles Dierkop, Brian Davies, Ted Hamilton, Madeline Stowe, Ruben Moreno, Alma Bettran. When an Indian chief's daughter is abducted by a rival tribe, Hawkeye and Chingachgook come to their assistance. \"Classics Illustrated\" TV version of the James Fenimore Cooper work; pretty good viewing. Follow-up to the previous year's _**The Last of the Mohicans**_ (q.v.).\n\n**Lex Barker and Rita Moreno in** _**The Deerslayer**_ **(20th Century** **\u2013** **Fox, 1957).**\n\n**  \n**\n\n**1035** _ **Defiance**_ **** Lion's Gate, 2002. 72 min. Color. D: Doveed Linder. With Brandon Bollig, Walker Deibel, Jim Freivogel, Stephanie Vogt, Dave Wassilak, John Gerbin, Anastasia Roark, Robert Nolan Clark, Brenda Sue Fowler, Craig Thrasher, Charles Heuvelman, Kenn Drescher, Alicia Skirball. A young man grows up bent on getting revenge on the town boss who murdered his father. Poor, cheaply made R-rated production.\n\n_**Deliver Us from Evil**_ (1973) see _**Running Wild**_\n\n**1036** _ **Deliver Us from Evil**_ **** ABC-TV, 1973. 78 min. Color. D: Boris Sagal. SC: Jack B. Sowards. With George Kennedy, Jan-Michael Vincent, Bradford Dillman, Jim Davis, Charles Aidman, Jack Weston, Allen Pinson. Five men on a hiking trip in the mountains come across an injured skyjacker with $600,000; they kill him and then fall out among themselves. Fairly entertaining TV feature.\n\n**1037** _ **A Demon for Trouble**_ **** Supreme, 1934. 58 min. D: Robert Hill. SC: Jack Natteford. With Bob Steele, Gloria Shea, Walter McGrail, Don Alvarado, Lafe McKee, Nick Stuart, Carmen LaRoux, Perry Murdock, Blackie Whiteford, Jimmy Aubrey. A cowboy uncovers a plot in which land buyers are murdered and their money stolen after they have purchased range property. Very good Bob Steele vehicle.\n\n**1038** _ **The Demon Rider**_ **** Davis Distributing, 1925. 50 min. D: Paul Hurst. SC: Jay Inman Kane. With Ken Maynard, Alma Rayford, Fred Burns, Tom London, James Low, Hollywood Beauty Sextette. A ranch foreman single handedly captures an outlaw gang but when he attempts to return the gold they stole he is mistaken by the sheriff for \"Black Hawk,\" the leader of the desperados. Ken Maynard's second starring feature for J. Charles Davis is an action filled modern-day Western.\n\n**1039** _ **Denver and the Rio Grande**_ **** Paramount, 1952. 89 min. Color. D: Byron Haskin. SC: Frank Gruber. With Edmond O'Brien, Sterling Hayden, Dean Jagger, Laura Elliott, Lyle Bettger, J. Carrol Naish, ZaSu Pitts, Tom Powers, Robert Barrat, Paul Fix, Don Haggerty, James Burke. Two rival companies compete in the building of the Denver and Rio Grande railroad in the 1870s. Competent oater with a good script and cast.\n\n**1040** _ **The Denver Kid**_ **** Republic, 1948. 60 min. D: Philip Ford. SC: Bob Williams. With Allan \"Rocky\" Lane, Eddy Waller, Carole Gallagher, William Henry, Douglas Fowley, Rory Mallinson, George Lloyd, George Meeker, Emmett Vogan, Marshall Reed, Hank Patterson, Tom Steele. A border patrol agent is after a notorious murderer. Nicely staged Allan Lane vehicle.\n\n_**The Deputies**_ see _**Law of the Land**_\n\n**1041** _ **Deputy Marshal**_ **** Lippert, 1949. 75 min. D-SC: William Berke. With Frances Langford, Jon Hall, Dick Foran, Julie Bishop, Russell Hayden, Joseph Sawyer, Clem Bevans, Vince Barnett, Mary Gordon, Kenne Duncan, Stanley Blystone, Wheaton Chambers, Forrest Taylor, Ted Adams. A lawman is on the trail of gunmen brothers and a map belonging to the railroad. Outside of the cast, there is not much to recommend this pedestrian production.\n\n**1042** _ **Desert Bandit**_ **** Republic, 1941. 54 min. D: George Sherman. SC: Eliot Gibbons and Bennett Cohen. With Don \"Red\" Barry, Lynn Merrick, James Gillette, William Haade, Dick Wessell, Tom Chatterton, Robert Strange, Curley Dresden, Jim Corey, Merrill McCormick, Charles King, Jack Montgomery, Jack O'Shea, Tom Ewell. Texas Rangers are after a band of gun smugglers. Action filled outing in Don Barry's Republic series; remade as _**Riders of the Deadline**_ (q.v.).\n\n**1043** _ **Desert Gold**_ **** Paramount, 1936. 58 min. D: James Hogan. SC: Stuart Anthony and Robert Yost. With Larry \"Buster\" Crabbe, Robert Cummings, Marsha Hunt, Tom Keene, Monte Blue, Raymond Hatton, Glenn (Leif) Erickson, Walter Miller, Frank Mayo, Phillip Morris, Si Jenks, Art Mix, Bob McKenzie, Willis Marks, Billy Bletcher, James P. Burtis, Ed Thorpe, Anders Van Haden, John Merkyl, Gertrude Simpson. An outlaw gang leader attempts to kidnap a young woman loved by both a soldier and his Eastern friend. Sturdy programmer adaptation of the Zane Grey novel, first filmed by Paramount in 1926 with Neil Hamilton, Shirley Mason, Robert Frazer and William Powell.\n\n**1044** _ **Desert Greed**_ **** Goodwill, 1926. 50 min. D: Jacques Jaccard. With Yakima Canutt, Rose Blossom, Henry Hebert, Dick LaReno. After a Texas Ranger helps a young woman get the wages due her he accompanies her back home only to discover her sadistic stepfather plans to sell her to a crooked lawyer who has found out she is an heiress. Pedestrian Yakima Canutt (he was the producer) silent affair with a speedy climax.\n\n**1045** _ **Desert Guns**_ **** Beaumont, 1936. 60 min. D: Charles Hutchison. SC: Jacques Jaccard. With Conway Tearle, Margaret Morris, Charles K. French, Budd Buster, William Gould, Marie Werner, Kate Brinker, Duke Lee, Art Felix, Slim Whitaker, Bull Montana. A lawman pretends to be a young woman's long lost brother in order to save her inheritance from crooks Stilted poverty row Conway Tearle series vehicle, although the star makes a good genre hero even in his fifties.\n\n**1046** _ **The Desert Horseman**_ **** Columbia, 1946. 57 min. D: Ray Nazarro. SC: Sherman Lowe. With Charles Starrett, Smiley Burnette, Adelle Roberts, Richard Bailey, John Merton, Walt Shrum and His Colorado Hillbillies, George Morgan, Tommy Coats, Jack Kirk, Bud Osborne, Riley Hill, Tex Williams, Herman Hack, Bert Dillard, Tex Cooper. Falsely accused of robbing an Army payroll, a captain takes on the alias of the Durango Kid to clear himself and find the real culprit. Fair \"Durango Kid\" entry. British title: _**Checkmate**_.\n\n**1047** _ **Desert Justice**_ **** Atlantic, 1936. 60 min. D: Lester Williams (William Berke). SC: Gordon Phillips and Lewis Kingdom. With Jack Perrin, Maryan Downing, Warren Hymer, David Sharpe, Dennis (Moore) Meadows, Roger Williams, Budd Buster, William Gould, Fred \"Snowflake\" Toones, Earl Dwire, Starlight (horse), Braveheart (dog). Border patrolman Casey is on the trail of a gang of smugglers. Cheaply made, but Jack Perrin is a pleasing player.\n\n**Poster for** _**Desert Justice**_ **(Atlantic, 1936).**\n\n**  \n**\n\n**1048** _ **Desert of Lost Men**_ **** Republic, 1951. 54 min. D: Harry Keller. SC: M. Coates Webster. With Allan \"Rocky\" Lane, Mary Ellen Kay, Irving Bacon, Roy Barcroft, Ross Elliott, Cliff Clark, Red Morgan, Kenneth MacDonald. A deputy marshal tries to track down and capture an outlaw gang made up of notorious bad men from all over the West. Interesting plot adds zest to this \"Famous Westerns\" opus.\n\n**1049** _ **Desert of the Lost**_ **** Path\u00e9, 1927. 58 min. D: Richard Thorpe. SC: Frank L. Inghram. With Wally Wales, Peggy Montgomery, William J. (Bill) Dyer, Edward (Ed) Cecil, Richard Neill, Kally Cafford, Ray Murro, George Magrill, Charles (Slim) Whitaker, Lafe McKee. Hunted by a detective for shooting a man in self-defense, a cowboy goes to Mexico where he falls in love with a young woman whose father wants to marry her off to a half-breed outlaw with a secret gold mine. Somewhat austere but enjoyable silent Wally Wales production.\n\n**1050** _ **Desert Passage**_ **** RKO Radio, 1952. 61 min. D: Lesley Selander. SC: Norman Houston. With Tim Holt, Richard Martin, Joan Dixon, Walter Reed, Clayton Moore, Dorothy Patrick, John Dehner, Lane Bradford, Denver Pyle, Francis McDonald. After he is paroled from prison, a man tries to find hidden bank robbery money but is trailed by a crooked lawyer and a gunman. Tim Holt's final \"B\" Western is a good one with a literate script and fine acting.\n\n**1051** _ **Desert Patrol**_ **** Republic, 1938. 60 min. D: Sam Newfield. SC: Fred Myton. With Bob Steele, Marion Weldon, Rex Lease, Ted Adams, Forrest Taylor, Budd Buster, Steve Clark, Jack Ingram, Tex Palmer. After a Texas Ranger is murdered, his comrade sets out to get the smuggling gang that killed him. The revenge plot is typical for a Bob Steele oater and this one is a good entry in his series for producer A.W. Hackel.\n\n**1052** _ **Desert Phantom**_ **** Supreme, 1936. 64 min. D: S. Roy Luby. SC: Earle Snell. With Johnny Mack Brown, Sheila Mannors, Ted Adams, Karl Hackett, Hal Price, Nelson McDowell, Charles King, Forrest Taylor, Roger Williams, George Morrell, Art Dillard, Fred Parker. A cowboy is hired by a young woman to run her ranch which is being raided by an outlaw gang. Cheaply made but more than passable Johnny Mack Brown fare.\n\n**1053** _ **Desert Pursuit**_ **** Monogram, 1952. 71 min. D: George Blair. SC: W. Scott Darling. With Wayne Morris, Virginia Grey, Anthony Caruso, George Tobias, Gloria Talbott, Emmett Lynn, Frank Lackteen, John Doucette, Robert Bice. A prospector and a fortune hunting woman are trailed by outlaws as they search for hidden gold in the California desert. Low budget but entertaining.\n\n**1054** _ **The Desert Rider**_ **** Sunset, 1923. 54 min. D: Robert North Bradbury. With Jack Hoxie, Evelyn Nelson, Frank Rice, Claude Peyton, Tom Lingham, Walter Wilkinson. A cowboy finds a dying miner and plans to take care of his young son as he hunts the man's claim and brings in his killer. Somewhat drawn out, but still pleasing, Jack Hoxie silent feature, although not one of his better efforts.\n\n**1055** _ **The Desert Secret**_ **** Madoc, 1924. 55 min. D-SC: Frederick Reel, Jr. With Bill Patton, Pauline Curley, Fred Burns, Lew Meehan, James Truax. A prospector becomes despondent when the claim he and his partner worked is taken over by a gambler, but he does not know the girl he loves has filed on it for him. Tacky modern-day silent Western gets little help from laconic star Bill Patton.\n\n**1056** _ **Desert Snow**_ **** Anchor Bay, 1989. 90 min. Color. D: Paul De Gruccio. With Frank Capizzi, Flint Carney, Shelly Hinkle, Sam Incorvia, Carolyn Jacobs, Frank McGill, Cynthia Miles, Paul Beauvais, Simo Maceo, Peter De Falco, Steve Labatt, Ray Gamboa, Richard Trujillo. Both drug pushers and cowboys want to take over a tiny Western town. Okay modern-day affair.\n\n**1057** _ **The Desert Trail**_ **** Monogram, 1935. 53 min. D: Cullen Lewis (Lewis D. Collins). SC: Lindsley Parsons. With John Wayne, Mary Kornman, Paul Fix, Eddy Chandler, Carmen LaRoux, Al Ferguson, Lafe McKee, Henry Hall, Theodore Lorch, Tommy Coats, Fred Parker, Jack Evans, Tex Palmer, Ray Henderson, Silvertip Baker, Frank Brownlee, Frank Ellis, Lew Meehan, Wally West, Archie Ricks, Frank Ball, Herman Hack, Olin Francis, Artie Ortego, Tex Palmer, Dick Dickinson, Jack Hendricks. A rodeo star and his gambler pal are falsely accused of a hold-up and set out to clear their names. A bit stilted, but pleasing \"Lone Star\" feature from producer Paul Malvern.\n\n**1058** _ **Desert Vengeance**_ **** Columbia, 1931. 55 min. D: Louis King. SC: Stuart Anthony. With Buck Jones, Barbara Bedford, Douglas Gilmore, Al Smith, Ed Brady, Buck Connors, Gilbert \"Pee Wee\" Holmes, Slim Whitaker, Robert Ellis, Bob Fleming, Joe Girard, Barney Beardsley. A bandit running a remote town stronghold falls for a young woman who deceives him but he later saves her when she is left to die in the desert by her partner and the two face an attack by a rival gang. The plot is a bit different for a Buck Jones film but it is quite good.\n\n**1059** _ **Desert Vigilante**_ **** Columbia, 1949. 56 min. D: Fred F. Sears. SC: Earle Snell. With Charles Starrett, Smiley Burnette, Peggy Stewart, Tristram Coffin, The Georgia Crackers, Mary Newton, George Chesebro, Jack Ingram, Paul Campbell, Tex Harding, I. Stanford Jolley, Ted Mapes, George Morrell, Blackie Whiteford, Sandy Sanders, Lew Morphy, Roy Bucko, Jerry Hunter. A government agent is after silver smugglers working along the Mexican border and he meets a young woman whose father was murdered by the band. Mediocre \"Durango Kid\" production.\n\n**1060** _ **A Desert Wooing**_ **** Paramount, 1918. 55 min. D: Jerome Storm. With Enid Bennett, Jack Holt, Donald MacDonald, John P. Lockney, Charles Spere, Elinor Hancock. A society woman in need of money sells her pretty daughter in marriage to a rugged rancher with the spoiled girl, who is also romanced by a cad, eventually learning to love her husband. Thomas H. Ince supervised this silent curio that will be of interest to Jack Holt fans.\n\n**1061** _ **The Deserter**_ **** Triangle, 1916. 59 min. D: Walter Edwards. SC: R.V. Spencer. With Charles Ray, Rita Stanwood, Wedgwood Nowell, Hazel Belford, Joseph Dowling. Spurned by a colonel's daughter and after a fight with another soldier, a young man deserts his post but eventually proves he is a hero in a fight with Indians. Fair, action filled silent drama.\n\n**1062** _ **The Deserter**_ **** Paramount, 1971. 99 min. Color. D: Burt Kennedy. SC: Clair Huffaker. With Bekim Fehmiu, Richard Crenna, Chuck Connors, Ricardo Montalban, Brandon De Wilde, Slim Pickens, Albert Salmi, Woody Strode, Patrick Wayne, Ian Bannen, John Houston, John Alderson, Mimmo Palmara. A cavalryman deserts from the Army to carry on a one man war against the Apaches for the mutilation of his wife. Passable Europe filmed drama but nothing more; also known as _**Ride to Glory**_.\n\n_**Desperado**_ see _**Now They Call Him Sacramento**_\n\n**1063** _ **The Desperado**_ **** Allied Artists, 1954. 82 min. D: Thomas Carr. SC: Geoffrey Homes. With Wayne Morris, Beverly Garland, James Lydon, Dabbs Greer, Rayford Barnes, Lee Van Cleef, Nestor Paiva, Roy Barcroft, John Dierkes, I. Stanford Jolley, Florence Lake, Richard Shackelton, Charles Garland, Reed Howes, Stanley Price, Lyle Talbot, Robert Shayne, William Fawcett. A young lawman teams with an outlaw to oppose the carpetbagger government in Texas in 1870. Nicely done \"B\" outing and one of the last series Westerns. Remade as _**Cole Younger, Gunfighter**_ (q.v.).\n\n**1064** _ **Desperado**_ **** Universal Television, 1987. 104 min. D: Virgil Vogel. SC: Elmore Leonard. With Alex McArthur, Lise Cutter, David Warner, Yaphet Kotto, Pernell Roberts, Robert Vaughn, Gladys Knight, Donald Moffat, Dirk Blocker, Sydney Walsh, Stephen Davis, Richard Marcus, Ed Adams, Linda Almond, Bruce Barber, William P. Brown, Cathy Browning, Tony Brubaker, Townsend Canyon, Dave Cass, Roydon Clark, Rick Currens, Dan Delligati, Steve Gray. When land grabbers try to take a woman's ranch she is helped by a mysterious stranger. Average TV Western that spawned a trio of sequels: _**Desperado: Avalanche at Devil's Ridge**_ , _**Desperado: Badland's Justice**_ and _**Desperado: The Outlaw Wars**_ (qq.v.).\n\n**1065** _ **Desperado**_ **** Columbia, 1995. 103 min. Color. D-SC: Robert Rodriguez. With Antonio Banderas, Salma Hayek, Joaquim de Almeida, Steve Buscemi, Richard \"Cheech\" Marin, Carlos Gomez, Angel Aviles, Danny Trejo, Tito Larriva, Quentin Tarantino, Carlos Gallardo. A guitar playing gunfighter arrives in a Mexican village to end the reign of a local tyrant. Violent affair that makes little sense.\n\n**1066** _ **Desperado: Avalanche at Devil's Ridge**_ **** Universal Television, 1988. 96 min. Color. D: Richard Compton. SC: Larry Cohen. With Alex McArthur, Rod Steiger, Lise Cutter, Hoyt Axton, Alice Adair, Lee Paul, Dwier Brown, Arch Archamboult, Katherine Engel, Tim Scott, Leslie Schwartz, Ben Zeller, Suzanne Lederer, Laura Martinez Herring, John David Garfield, Blake Conway, Ben Connors, John Barks, Jack Caffey. A wealthy businessman saves an outlaw from the gallows and hires him to find his abducted daughter. Passable TV Western.\n\n**1067** _ **Desperado: Badland's Justice**_ **** Universal Television, 1989. 96 min. D: E.W. Swackhamer. SC: Leslie Boehm. With Alex McArthur, John Rhys-Davies, James B. Sikking, Patricia Charbonneau, Gregory Sierra, Robert O'Reilly, Deborah Slaboda, Edward Wiley, Joel Colodner, Leslie Neale, Anne Curry, Geoffrey Rivas. Seeking to clear his name, a wanted outlaw finds himself opposed by crooks in a mining community. The tired finale in the TV movies' \"Desperado\" saga.\n\n**1068** _ **Desperado: The Outlaw Wars**_ **** Universal Television, 1989. 100 min. Color. D: E.W. Swackhamer. SC: William Wister. With Alex McArthur, Richard Farnsworth, James Remar, Brad Douriff, Tom Bower, Whip Hubley, Brion James, Debra Feuer, Buck Taylor, Geoffrey Lewis, Lise Cutter, Dron Richmond. To stay with his former girlfriend and their child, a wanted man agrees to bring in an outlaw in order to get a pardon. Shoddy production values coupled with a mediocre script make for a poor \"Desperado\" outing.\n\n**1069** _ **The Desperado Trail**_ **** Columbia, 1965. 93 min. Color. D: Harald Reinl. SC: Harald G. Petersson and J. Joachim Bartsch. With Lex Barker, Pierre Brice, Rik Battaglia, Ralf Wolter, Carl Lange, Sophie Hardy, Mihail Baloh, Dusan Antonijevic, Aleksandar Gavric, Illija Ivezic, Veljiko Maricic, Slobodan Dimitrijevic, Gojko Mitic, Milan Micic, Dusan Vujisic, Sime Jagarinac, Dragomir Felba, Ivo Kristof, Miroslav Buhin, Joachim Nottke (narrator). Land speculators attempt to get Indians on the warpath to steal their lands but frontiersman Shatterhand and his friend, Apache chief Winnetou, try to stop them. Familiar but colorful West German oater in the Karl May series; released in Europe by Rialto\/Jadran-Film as _**Winnetou III**_.\n\n**Lobby card for** _**The Desperado Trail**_ **(Columbia, 1965).**\n\n**  \n**\n\n**1070** _ **The Desperadoes**_ **** Columbia, 1943. 85 min. Color. D: Charles Vidor. SC: Robert Carson. With Randolph Scott, Claire Trevor, Glenn Ford, Evelyn Keyes, Edgar Buchanan, Raymond Walburn, Guinn Williams, Irving Bacon, Porter Hall, Joan Woodbury, Glenn Strange, Bernard Nedell, Ethan Laidlaw, Slim Whitaker, Edward Pawley, Chester Clute, Charles King, Jack Kinney, Francis Ford, Bud Osborne, Bill Wolfe, Tom Smith. A lawman reforms a young hellion and the two team to round up an outlaw gang. Well made and quite entertaining, based on a Max Brand story.\n\n**1071** _ **The Desperadoes Are in Town**_ **** 20th Century\u2013Fox, 1956. 78 min. D: Kurt Neumann. SC: Earle Snell and Kurt Neumann. With Robert Arthur, Kathy Nolan, Rhys Williams, Rhodes Reason, Dave O'Brien, Kelly Thordsen, Mae Clarke, Robert Osterloh, Morris Ankrum, Frank Sully, William Challee, Byron Foulger, Richard Wessel, Carl Mathews, Dorothy Granger, Bill (Willliam) Phipps, Todd Griffin, Nancy Evans, Carol Kelly. After a former outlaw befriends him and then is killed by two ex-partners, a young man plots revenge. Uninteresting and dull.\n\n**1072** _ **Desperadoes of Dodge City**_ **** Republic, 1948. 60 min. D: Philip Ford. SC: Bob Williams. With Allan \"Rocky\" Lane, Eddy Waller, Mildred Coles, Tristram Coffin, Roy Barcroft, William Phipps, James Craven, John Hamilton, Ed Cassidy, House Peters, Jr., Dale Van Sickel, Ted Mapes, Robert Wilke. An outlaw gang is out to stop a wagon train of settlers. A mystery element helps this Allan Lane vehicle.\n\n**1073** _ **Desperadoes of the West**_ **** Republic, 1950. 12 Chapters. D: Fred C. Brannon. SC: Ronald Davidson. With Richard Powers (Tom Keene\/George Duryea), Judy Clark, Roy Barcroft, I. Stanford Jolley, Lee Phelps, Lee Roberts, Cliff Clark, Edmund Cobb, Dale Van Sickel, Tom Steele, Sandy Sanders, John Cason, Guy Teague, Bud Osborne, Stanley Blystone, Chuck Hayward, Frank O'Connor, George Chesebro, Art Dillard, Holly Bane, Duke Taylor, Cactus Mack, Ken Cooper, Dennis Moore, Steve Clark, Chick Hannon, Mauritz Hugo, Al Taylor, Bob Reeves, Eddie Parker, Fred Kohler, Jr., Harold Goodwin, Jack Ingram, Augie Gomez, Merrill McCormick. A crook and his outlaw gang try to prevent ranchers from drilling for oil so he can get the leases on their properties for his Eastern syndicate bosses. Action filled cliffhanger starring Tom Keene.\n\n**1074** _ **Desperadoes' Outpost**_ **** Republic, 1952. 54 min. D: Philip Ford. SC: Albert De Mond and Arthur Orloff. With Allan \"Rocky\" Lane, Eddy Waller, Roy Barcroft, Myron Healey, Lyle Talbot, Claudia Barrett, Lee Roberts, Lane Bradford, Ed Cassidy. When a number of stagecoaches are mysteriously sabotaged, a government agent is assigned to find the culprits. Another good entry in the \"Famous Westerns\" series starring Allan Lane.\n\n**1075** _ **The Desperados**_ **** Columbia, 1969. 90 min. Color. D: Henry Levin. SC: Walter Brough. With Vincent Edwards, Jack Palance, George Maharis, Neville Brand, Sylvia Syms, Christian Roberts, Kate O'Mara, Kenneth Cope, John Paul, Patrick Holt, Christopher Malcolm, John Clarke, Benjamin Edney. After the Civil War a man and his three sons go West and lead an outlaw band but one of the boys deserts, marries and settles down only to have his land invaded by his family. Fairly interesting feature.\n\n**1076** _ **Desperate Chance**_ **** Rayart, 1926. 55 min. D: J.P. McGowan. SC: Charles Saxton. With Bob Reeves, Ione Reed, Leon De La Motte, Charles \"Slim\" Whitaker, Gypsy Clarke, Harry Hurley. Two men try to get even with a crook who harmed them but when he is murdered one is accused of the crime. Low grade silent Bob Reeves vehicle.\n\n_**Desperate Men**_ see _**El Dorado Pass**_\n\n**1077** _ **Desperate Mission**_ **** ABC-TV\/20th Century\u2013Fox, 1971. 98 min. Color. D: Earl Bellamy. SC: Jack Guss and Richard Collins. With Ricardo Montalban, Slim Pickens, Rosey (Roosevelt) Grier, Ina Balin, Earl Holliman, Miriam Colon, Jim McMullan, Armando Silvestre, Robert Wilke, Anthony Caruso, Charles Horvath, Barbara Turner. A bandit helps the locals in Spanish California fight outlaws and dishonest government officials. Another re-telling of the Joaquin Murieta saga, with Ricardo Montalban good in the title role, but overall a mediocre film. Initially released abroad in 1970 as _**Joaquin Murieta**_ and _**Murieta**_.\n\n**1078** _ **Desperate Search**_ **** Metro-Goldwyn-Mayer, 1953. 73 min. D: Joseph H. Lewis. SC: Walter Doniger. With Howard Keel, Jane Greer, Keenan Wynn, Robert Burton, Lee Aaker, Linda Lowell, Michael Dugan, Elaine Stewart, Jonathan Cott, Jeff Richards, Dick (Richard) Simmons. After his children are stranded in the Canadian north woods following a plane crash, a man attempts to find them. Low budget but well done and exciting drama.\n\n_**Desperate Siege**_ see _**Rawhide**_ (1950)\n\n**1079** _ **The Desperate Trail**_ **** Motion Picture Corporation of America, 1995. 93 min. Color. D: P.J. Pesce. SC: P.J. Pesce and Tom Abrams. With Sam Elliott, Craig Sheffer, Linda Florentino, Frank Whaley, John Furlong, Robin Westphal, Boots Southerland, Joey Hamlin, Danny O'Haco, Bradley Whitford, Jim Scott Momaday, R.L. Tolbert, P.J. Pesce, Rockne Tarkington, Michael Huddleston, Peter Gregory, George Cook, Elliot \"Bub\" Tolbert, Andrea Camarena Lindsay, Jerry Gardner, Sam Gauny, Wally Welsch, Tom Berto, Ramon Frank, Jeff O'Haco, Tom Abrams, Jon Maldonado, Malissa Feruzzi, Cecile Krevoy, Gretechen Becker, Cliff Gravel, Joe Bernier. Sentenced to be hanged by her father-in-law marshal for the killing of his brutal son, a woman escapes during a stagecoach robbery and teams with an outlaw to avoid the lawman. Obscure but fairly good drama.\n\n**1080** _ **Desperate Trails**_ **** Universal, 1939. 58 min. D: Albert Ray. SC: Andrew Bennison. With Johnny Mack Brown, Bob Baker, Fuzzy Knight, Frances Robinson, Russell Simpson, Clarence Wilson, Bill Cody, Jr., Ralph Dunn, Charles Stevens, Ed Cassidy, Horace Murphy, Fern Emmett, Frank Ellis, Frank McCarroll, Cliff Lyons, Eddie Parker. A crooked sheriff and banker are behind a group of night riders trying to rustle a young woman's cattle. Johnny Mack Brown's first Universal series Western and a good one.\n\n**1081** _ **Desperate Women**_ **** NBC-TV, 1978. 100 min. Color. D: Earl Bellamy. SC: Jack B. Sowards. With Susan Saint James, Dan Haggerty, Ronee Blakley, Ann Dusenberry, Susan Mayers, Randy Powell, Max Gail, Michael Delano, Taylor Lacher, Tiger Williams, Bob Hoy, James Griffith, Rudy Diaz, John Crawford, Clint Ritchie, William Vaughan, Ed Fury. Three women prisoners and two orphaned children are left in the desert with an Army deserter and they team with a gunman to oppose an outlaw gang. Passable TV movie, nothing more.\n\n**1082** _ **Destry**_ **** Universal-International, 1955. 95 min. Color. D: George Marshall. SC: Felix Jackson, Edmund North and D.D. Beauchamp. With Audie Murphy, Mari Blanchard, Lyle Bettger, Lori Nelson, Thomas Mitchell, Edgar Buchanan, Wallace Ford, Mary Wickes, Alan Hale, Jr., Lee Aaker, Trevor Bardette, Walter Baldwin, Rex Lease, George Wallace, John Doucette, Richard Reeves, Ralph Peters, Frank Richards, Henry Wills, Mitchell Lawrence. A shy young man becomes the sheriff of a rough town and falls for a saloon singer. This retelling of the Max Brand story is not bad but not up to the two earlier versions of _**Destry Rides Again**_ (q.q.v.) with George Marshall returning to direct after having also done the 1939 outing.\n\n**1083** _ **Destry Rides Again**_ **** Universal, 1932. 64 min. D: Ben Stoloff. SC: Richard Shayer and Robert Keith. With Tom Mix, Claudia Dell, Stanley Fields, ZaSu Pitts, Earle Fox, Ed Peil, Sr., Francis Ford, Frederick Howard, George Ernst, John Ince, Ed LeSaint, Charles K. French. A cowboy intends to clean up a corrupt town by running for sheriff but crooks frame him on a murder charge. Tom Mix's first sound feature is a good one, proving why he is one of the all-time great Western stars. TV title: _**Justice Rides Again**_.\n\n**1084** _ **Destry Rides Again**_ **** Universal, 1939. 94 min. D: George Marshall. SC: Felix Jackson, Gertrude Purcell and Harry Myers. With James Stewart, Marlene Dietrich, Mischa Auer, Charles Winninger, Brian Donlevy, Allen Jenkins, Warren Hymer, Irene Hervey, Una Merkel, Tom Fadden, Samuel S. Hinds, Lillian Yarbo, Edmund MacDonald, Billy Gilbert, Virginia Brissac, Ann Todd, Dickie Jones, Jack Carson, Joe King, Harry Cording, Richard Alexander, Bill (Steele) Gettinger, Minera Urecal, Bob McKenzie, Billy Bletcher, Lloyd Ingraham, Bill Cody, Jr., Harry Tenbrook, Chief Big Tree, Philo McCullough, Robert Keith, Loren Brown, Harold De Carro. A tenderfoot is drafted into becoming the lawman of a wild town and falls under the spell of a seductive saloon gal. Second version of the Max Brand novel still holds up well, mainly for James Stewart in the title role and Marlene Dietrich's singing \"See What the Boys in the Backroom Will Have.\"\n\n**1085** _ **The Devil and Leroy Bassett**_ **** American National Enterprises, 1973. 85 min. Color. D-SC: Robert E. Pearson. With Cody Bearpaw, John F. Goff, George \"Buck\" Flower, James Ward, Dick Winslow, Elliott Lindsey, Bobbi Shaw, Hal Bokar, Lillian MacBride, Siegfried Anton, Jim Beach, Don Epperson, James Howard, Bob Padilla, Gordon James, Jerry Mills, Paul Kalin, Zeno Russell, George Engelson, Cliff McDonald, Joe Herrera, John Banks, Imagene Goodshot, Richard Breeding, Janice Hallums, Aly Yoder, Woody Lee. After accidentally shooting a deputy sheriff, a man is taken into custody but is saved by his two brothers and the trio take refuge in a family's mountain cabin before trying to make an escape. Typically violent 1970's modern Western.\n\n**1086** _ **The Devil and Miss Sarah**_ **** ABC-TV\/Universal, 1971. 73 min. Color. D: Michael Caffey. SC: Calvin Clements. With Gene Barry, James Drury, Janice Rule, Charles McGraw, Slim Pickens, Logan Ramsey, Donald Moffat. A young couple capture a gunman and attempt to take him to the nearest law but along the way he uses his hypnotic powers to take possession of the wife. There is not much to recommend this TV made horror Western.\n\n_**The Devil Bear**_ see _**Claws**_\n\n**1087** _ **The Devil Horse**_ **** Path\u00e9, 1926. 62 min. D: Fred Jackman. SC: Hal Roach. With Rex (horse), Yakima Canutt, Gladys Morrow, Robert Kortman, Roy Clements, Fred Jackson, Killer (horse). A cowboy, the sole survivor of an Indian massacre as a boy, is aided by a wild stallion in rescuing a major's daughter who has been kidnapped by a renegade brave. Action filled silent feature from producer Hal Roach, who also wrote the script.\n\n**1088** _ **The Devil Horse**_ **** Mascot, 1932. 12 Chapters. D: Otto Brower. SC: George Morgan, Barney A. Sarecky, George Plympton and Wyndham Gittens. With Harry Carey, Noah Beery, Frankie Darro, Greta Granstedt, Barrie O'Daniels, Ed Peil, Sr., Jack Mower, Alan Bridge, Jack Byron, J. Paul Jones, Carli Russell, Lou Kelly, Dick Dickinson, Lane Chandler, Fred Burns, Yakima Canutt, Ken Cooper, Wes Warner, Al Taylor, Apache (horse). Trying to capture a wild horse a man kills a forest ranger whose brother swears revenge and enlists the help of a boy who has run with the stallion since he was a child. There is plenty of movement in this Nat Levine production but genre purists may be put off by excessive \"cheat\" footage; also released in a feature version.\n\n**1089** _ **Devil on Horseback**_ **** Grand National, 1936. 71 min. Color. D-SC: Crane Wilbur. With Lili Damita, Fred Keating, Del Campo, Tiffany Thayer, Jean Chatburn, Renee Torres, Juan Torena, Blancha Visher. A visiting radio star and her troupe are kidnapped and held for ransom by a Central American ranch owner. Dull comedy-drama filmed in Hirlicolor by producer George A. Hirliman.\n\n**1090** _ **Devil Riders**_ **** Producers Releasing Corporation, 1943. 56 min. D: Sam Newfield. SC: Joe (Joseph) O'Donnell. With Buster Crabbe, Al St. John, Patti McCarty, Charles King, John Merton, Kermit Maynard, Frank LaRue, Jack Ingram, George Chesebro, Ed Cassidy, Al Ferguson, Frank Ellis, Bert Dillard, Bud Osborne, Artie Ortego, Herman Hack, Hank Bell, Steve Clark, Rose Plummer, Jimmy Aubrey, Ralph Bucko, Roy Bucko, Art Dillard, Bert Dillard, Kansas Moehring, Curley Dresden, Big Slicker Quartet (Tex Williams, Deuce Spriggins, Smokey Rogers, Don Weston). A dishonest lawyer and his cohort are after land the government has designated for a stage route with Billy Carson and his pal Fuzzy helping the local franchise owner combat the crooks. Okay first entry in PRC's \"Billy Carson\" series.\n\n**1091** _ **The Devil's Bedroom**_ **** Manson Distributing, 1964. 78 min. D: L.Q. Jones. SC: Claude Hall and Morgan Woodward. With John Lupton, Valerie Allen, Dick Jones, Alvy Moore, Morgan Woodward, Justice McQueen, Mrs. Arch Pearson, Claude Hall, Bill Buckner, Thomas Commack, Merv Dawson, E.B. Jolly, Lawrence Mooney, W.H. Handles, Ken Ariola, Ralph G. Edwards. When they learn land is located on a valuable oil deposit, a couple tries to drive the owner insane in order to get his ranch. Obscure, violent modern-day Western melodrama from the production team of Alvy Moore and L.Q. Jones; filmed in color but released in black and white.\n\n**1092** _ **Devil's Canyon**_ **** RKO Radio, 1953. 92 min. Color. D: Alfred Werker. SC: Frederick Hazlitt Brennan and Harry Essex. With Virginia Mayo, Dale Robertson, Stephen McNally, Arthur Hunnicutt, Robert Keith, Jay C. Flippen, George J. Lewis, Whit Bissell, Morris Ankrum, James Bell, William Phillips, Earl Holliman, Irving Bacon, Paul Fix, Glenn Strange, Larry Blake, Mickey Simpson, Fred Coby, Jim Hayward, Gregg Martell, Harvey Parry, Murray Alper, John Cliff, Harold Kruger. After killing two men in self-defense, a marshal is railroaded into prison where he becomes involved in a riot. Offbeat oater; not without interest.\n\n**1093** _ **Devil's Doorway**_ **** Metro-Goldwyn-Mayer, 1950. 84 min. D: Anthony Mann. SC: Guy Trosper. With Robert Taylor, Paula Raymond, Louis Calhern, Marshall Thompson, James Mitchell, Edgar Buchanan, Rhys Williams, Spring Byington, James Millican, Fritz Leiber, Chief Big Tree, Kermit Maynard, Bruce Cowling, Harry Antrim, Tom Fadden, Frank Conlan, William \"Bill\" Phillips, William Norton Bailey, Philo McCullough, Roy Butler, Lee Phelps, John Maxwell, George Sky Eagle, Henry Marco, Dabbs Greer, Dan Foster. A Shoshone Indian, honored for bravery fighting for the North during the Civil War, returns home to find he has to save his people's lands. Robert Taylor is very good as the Indian brave and this outing is well worth viewing.\n\n**1094** _ **The Devil's Mistress**_ **** Emerson, 1968. 66 min. Color. D-SC: Orville Wanzer. With Joan Stapleton, Robert Gregory, Forest Westmoreland, Douglas Warren, Oren Williams, Arthur Resley. Four cowboys murder a man and take his mistress as their servant as she plans a terrible revenge on them. Low budget horror-Western filmed in New Mexico.\n\n**1095** _ **The Devil's Partner**_ **** Mutual\/Truart, 1926. 57 min. D-SC: Frederick Becker. With Edward Hearn, Nancy Deaver, Philo McCullough, Carl Stockdale, Florence Lee, Will Walling, Harvey Clark, Billie Lattimer, Fred Becker, Hayden Stevenson. The leader of a rustling gang has a romantic rival framed for a crime and then kidnaps the girl. Poverty row silent oater that should please genre fans.\n\n**1096** _ **The Devil's Partner**_ **** American International\/Filmgroup, 1961. 61 min. D: Charles Rondeau. SC: Stanley Clements and Laura Mathews. With Ed Nelson, Richard Crane, Edgar Buchanan, Jean Allison, Spencer Carlisle, Byron Foulger, Claire Carleton, Brian O'Hara, Harry Fleer, Joe Hooker, Riley Hill, Hugh Hooker. When his uncle dies a man arrives in a small desert town and strange things begin to happen. Low budget, and low grade, horror Western filmed in 1958. Alternate DVD title: _**Enter the Devil**_.\n\n**1097** _ **The Devil's Playground**_ **** United Artists, 1946. 65 min. D: George Archainbaud. SC: Ted Wilson. With William Boyd, Andy Clyde, Rand Brooks, Elaine Riley, Robert Elliott, Joseph J. Greene, Francis McDonald, Ned (Nedrick) Young, George Eldredge, Earle Hodgins, Everett Shields, John George, Glenn Strange, Dewey Robinson, Herman Hack, Jack Evans, Blackie Whiteford, Henry Wills, Merrill McCormick, Hank Bell, Tex Cooper. Hoppy, California and Lucky try to help a young woman who is hunted by a crooked judge after her \"friend's\" gold. Lots of action, a good story and nice locations in this later \"Hopalong Cassidy\" offering.\n\n**The Devil's Price** see _**The Lone Star Vigilante**_\n\n**1098** _ **The Devil's Rain**_ **** Bryanston, 1975. 86 min. Color. D: Robert Fuest. SC: Gabe Essoe, James Ashton and Gerald Hopman. With Ernest Borgnine, Eddie Albert, William Shatner, Ida Lupino, Tom Skerritt, Joan Prather, Keenan Wynn, Woodrow Chambliss, George Sawaya, Lisa Todd, Claudio Brook, Anton LaVey, John Travolta, Robert Wallace, Erika Carlson, Tony Cortez. A man finds himself the victim of a cult of devil worshippers in a small town. Awful horror film set in the modern-day West.\n\n**1099** _ **Devil's Saddle Legion**_ **** Warner Bros., 1937. 52 min. D: Bobby Connolly. SC: Ed Earl Repp. With Dick Foran, Anne Nagel, Willard Parker, Granville Owen, Carlyle Moore, Jr., Glenn Strange, Frank Orth, Jack Mower, Milton Kibbee, George Chesebro, Ray Bennett, Dick Botiller, Bud Osborne, Art Mix, Artie Ortego, Ben Corbett. A cowboy is falsely accused of being an outlaw gang leader and is sent to work on building a dam designed to divert water needed by ranchers. More than passable entry in Dick Foran's Warner Bros. series.\n\n_**The Devil's Spawn**_ see _**The Last Gunfighter**_\n\n**1100** _ **The Devil's Trail**_ **** Columbia, 1942. 61 min. D: Lambert Hillyer. SC: Robert Lee Johnson. With Bill Elliott, Tex Ritter, Eileen O'Hearn, Noah Beery, Frank Mitchell, Ruth Ford, Art Mix, Joel Friedkin, Joe McGuinn, Edmund Cobb, Tristram Coffin, Paul Newlan, Steve Clark, Sarah Padden, Bud Osborne, Stanley Brown, Buck Moulton, Art Mix, Art Dillard. During the Kansas slavery question struggle a federal marshal tries to help his pal Wild Bill Hickok, who has been falsely accused of murder. Top notch effort in the Bill Elliott-Tex Ritter series with writer Robert Lee Johnson adapting his story \"The Town in Hell's Backyard\"; a grand performance by Noah Beery as villain Bull McQuade.\n\n**1101** _ **El Diablo de la Frontera**_ (The Devil of the Frontier) **** World Magic Films, 2005. 115 min. Color. D: Carlos Valdemar. With Salvador Salinas, Armando Infante, Carmen del Valle, Gerardo Albarran, Karla Barahona, Alfredo Gutierrez, Rojo Grau. After a child is killed, several cowboys try to steal a notorious champion racing horse from its wealthy female owner and her son. Overlong Mexican video Western.\n\n**1102** _ **The Diamond Trail**_ **** Monogram, 1932. 60 min. D: Harry Fraser. SC: Harry Fraser and Sherman Lowe. With Rex Bell, Frances Rich, Lloyd Whitlock, Bud Osborne, Norman Feusier, Jerry Storm, John Webb Dillon, Billy West, Harry LaMont. A New York City reporter infiltrates a band of jewel thieves who head West to murder a cattleman, a go-between for the gang. Only fair Rex Bell vehicle.\n\n**Advertisement for** _**The Diamond Trail**_ **(Monogram, 1932).**\n\n**  \n**\n\n**1103** _ **Dig That Uranium**_ **** Allied Artists, 1956. 61 min. D: Edward Bernds. SC: Elwood Ullman and Bob Lawrence. With Leo Gorcey, Huntz Hall, Bernard Gorcey, Mary Beth Hughes, Raymond Hatton, Myron Healey, Richard Powers (Tom Keene), Harry Lauter, Francis McDonald, David (Gorcey) Condon, Bennie Bartlett, Paul Fierro, Frank Jenks, Don C. Harvey, Carl \"Alfalfa\" Switzer. The Bowery Boys buy a mine in Nevada but when they arrive to claim it they find themselves at odds with crooks. Typically amusing \"Bowery Boys\" affair enhanced by the participation of genre favorites Raymond Hatton, Myron Healey, Tom Keene, Harry Lauter and Francis McDonald.\n\n**1104** _ **La Diligencia de la Muerte**_ (The Diligence of the Dead) **** Filmadora Chapultepec, 1961. 70 min. D-SC: Rogelio A. Gonzalez, Jr. With Luis Aguilar, Armando Silvestre, Luz Maria Aguilar, Raul Ramirez, Jose Chavez, Agustin Isunza, Nora Veryan, Alfredo Vergara, Jose Munoz, Gregorio Acosta. Outlaws change road signs and rob travelers with the local authorities unable to stop them but one of the thieves has a falling out with the leader for the allocation of spoils. Fun south of the border Western teaming favorites Luis Aguilar and Armando Silvestre.\n\n**1105** _ **Ding Dong Williams**_ **** RKO Radio, 1946. 61 min. D: William Berke. SC: Brenda Weinberg and M. Coates Webster. With Glenn Vernon, Marcy McGuire, Felix Bressart, Anne Jeffreys, James Warren, William B. Davidson, Tom(my) Noonan, Cliff Nazarro, Ruth Lee, Jason Robards (Sr.), Bob Nolan and The Sons of the Pioneers (Tim Spencer, Shug Fisher, Ken Carson, Hugh Farr, Karl Farr), Richard Korbel, Robert Clarke, Tanis Chandler, Harry Harvey, Virginia Belmont, Myrna Dell, Helen Eby-Rock, Bruce Edwards, C. Bakaleinikoff, Jack Gargan, Billy Vernon, Betty Gillette, Edmund Glover, Tom Quinn, Rodney Hildebrand, James Pilcher, Jimmy Jordan, Larry McGrath, Foster H. Phinney, Lee Phelps, William J. O'Brien, Bob Thorn, Bob Alden, Aina Constant. A jazz clarinetist, who cannot read or write music, agrees to compose the score for a film after meeting his hero, a singing cowboy. Pleasant programmer poking fun at Hollywood musicals and featuring James Warren as a cowboy crooner.\n\n**1106** _ **Dirty Dingus Magee**_ **** Metro-Goldwyn-Mayer, 1970. 91 min. Color. D: Burt Kennedy. SC: Tom Waldman, Frank Waldman and Joseph Heller. With Frank Sinatra, George Kennedy, Anne Jackson, Lois Nettleton, Jack Elam, Michele Carey, John Dehner, Henry Jones, Donald Barry, Harry Carey, Jr., Paul Fix, Mike Wagner, Terry Wilson, Tom Fadden, Lisa Todd, Carol Anderson, Grady Sutton. A small time crook and saddle tramp has troubles in a Western town with a dumb sheriff, a woman mayor-madam, Indians and the Army. Unfunny Western spoof, badly cut for TV release at 79 minutes.\n\n**1107** _ **Dirty Little Billy**_ **** Columbia, 1972. 100 min. Color. D: Stan Dragoti. SC: Charles Moss and Stan Dragoti. With Michael J. Pollard, Lee Purcell, Richard Evans, Charles Aidman, Dran Hamilton, Willard Sage, Josip Elic, Mills Watson, Alex Wilson, Ronnie Graham, Dick Stahl, Gary Busey, Doug Dirksen, Cherie Franklin, Dick Van Patten, Rosary Nix, Frank Welker. The story of the early years of Billy the Kid and how he got into a life of crime. Low class biopic not likely to appeal to genre fans.\n\n**1108** _ **The Dirty Outlaws**_ **** Transvue, 1971. 103 min. Color. D-SC: Franco Rossetti. With Chip Gorman (Andrea Giordana), Rosmarie Dexter, Franco Giornelli, Dana Ghia, Aldo Berti, Giovanni Petrucci, Piero Lilli, John Bartha. A bandit tries to pass himself off as a blind man's son in order to get gold buried in a deserted town but he finds an outlaw gang is also after the riches. Another in the long line of violent Spaghetti Westerns; issued in Italy in 1967 by Daiano\/Leone Film as _**El Desperado**_ (The Desperado).\n\n**1109** _ **Disappearances**_ **** Truly Indie, 2006. 103 min. Color. D-SC: Jay Craven. With Kris Kristofferson, Charlie McDermott, Gary Farmer, William Sanderson, Genevieve Bujold, Lothaire Bluteau, Heather Rae, Bill Raymond, Luis Guzman, John Griesemer, Christy Scott Cashman, Rusty Dewees, Steve Small, Josh Pellerin, Munson Hicks, Ken Winter, Tessa Klein, William Rough, Bow Thayer, Marc Gregoire. To save his cattle, a Vermont farmer hopes to get money by reverting to his family's heritage of smuggling whiskey across the Canadian border. Eye pleasing scenery enhances a jumbled plot set during Prohibition.\n\n**1110** _ **The Disciple**_ **** Ince\/Triangle, 1915. 60 min. D: William S. Hart and Clifford Smith. SC: S. Barret McCormick and Thomas H. Ince. With William S. Hart, Dorothy Dalton, Robert McKim, Charles French, Thelma Salter, Robert Kortman, Jean Hersholt. In a Western town a new \"sky pilot\" loses his wife to a gambler and denounces God only to return to the faith when his small daughter becomes ill. Well done silent William S. Hart morality film with none of the slickness associated with later genre movies.\n\n_**Disciples of Death**_ see _**Enter the Devil**_ (1971)\n\n**1111** _ **Distant Drums**_ **** Warner Bros., 1951. 101 min. Color. D: Raoul Walsh. SC: Niven Busch and Martin Rackin. With Gary Cooper, Mari Aldon, Richard Webb, Ray Teal, Arthur Hunnicutt, Robert Barrat, Clancy Cooper, Dan White, Lee Roberts, Gregg Barton, Sheb Wooley, Kenneth MacDonald, Warren MacGregor, Angelita McCall, Beverly Brandon, Mel Archer, Larry Carper, Sidney Capo. An Indian fighter leads troops into the Florida Everglades to put down a Seminole uprising. Filmed in Florida, this is not one of Gary Cooper's better genre efforts.\n\n**1112** _ **A Distant Trumpet**_ **** Warner Bros., 1964. 116 min. Color. D: Raoul Walsh. SC: John Twist. With Troy Donahue, Suzanne Pleshette, James Gregory, Diane McBain, William Reynolds, Claude Akins, Kent Smith, Judson Pratt, Bartlett Robinson, Bobby Bare, Richard X. Slattery, Guy Eltsosis, Larry Ward, Mary Patton, Russell Johnson, Lane Bradford. At a frontier post, a cavalry officer falls in love with a lieutenant's wife and when the man is killed the officer's fiancee arrives just as an Indian attack is imminent. Director Raoul Walsh's final film his not a worthy swan song.\n\n**1113** _ **Django**_ **** B.R.C.\/Tecisa, 1965. 90 min. Color. D: Sergio Corbucci. SC: Franco Rossetti and Jose G. Maesso. With Franco Nero, Loredana Nusciak, Jose Bodalo, Angel Alvarez, Eduardo Fajardo, Jimmy Douglas, Simone Arrag, Ivan Scratuglia. A mysterious stranger arrives in a border town during a battle between Mexican and American soldiers and flees with gold belonging to the Mexican army. Good first film in the long running \"Django\" series; very violent.\n\n**1114** _ **Django Defies Sartana**_ **** P.A.C., 1971. 89 min. Color. D-SC: William Redford (Pasquale Squitieri). With Tony Kendall, George Ardisson, Jose Torres, Bernard Faber, Maria (Malisa) Longo, Adler Gray, Jose (Rivas) Jaspe, Salvatore Billa, Fulvio Mingozzi, Augusto Pesarini, Mirella Pompili, Rick Boyd (Federico Boido), Claudio Trionfi, Teodoro (Doro) Corra, John Alvar, Fortunato Arena, Anna Maria Perego, Goffredo Ungar, Pasquale Squitieri, Tania Alvarado. Sartana and Django team to find the bank robbers who framed the former's brother, causing him to be lynched. Okay Spaghetti Western with a novel plot twist, filmed as _**Django Sfida Sartana**_ (Django Challenges Sartanta); song \"They Call Him Django\" sung by John Balfour.\n\n**1115** _ **Django, Kill!**_ Golden Era, 1967. 117 min. Color. D: Giulio Questi. SC: Franco Arcalli, Maria del Carmen Martinez Roman and Guilio Questi. With Tomas Milian, Raymond (Ray) Lovelock, Piero Lulli, Milo Quesada, Roberto Camardiel, Miguel Serrano, Angel Silva, Felix Sancho Garcia, Marilu Tolo, Mirelli Panfili, Panco Sanz, Patrizia Vaiturri, Frank Brana, Daniel Martin, Gene Collins, Fernando Villena, Calogero Azzaretto, Herman Reynoso. After helping a gang rob a stagecoach, Django is shot and left for dead but he claws out of his grave craving revenge. Beware of this overlong, stomach turning Spaghetti Western filmed as _**Se Sei Vivo Spara**_ (If You Live Shoot) and also called _**Django, Kill...If You Live, Shoot!**_.\n\n**1116** _ **Django Meets Sartana**_ **** Cinepatrizia, 1970. 90 min. Color. D: Miles Deem (Demofilio Fidani). SC: Maria Rosa Valenza and Demofilio Fidani. With Hunt Powers, Fabio Testi, Dean Stratford, Dennis Colt, Lucky McMurray, Simone Blondell, Dan Reesy, Celso Faria, Robert Dannish, Antonio Basile, Michael Brank, Franco Pasquetto, Pietro Torrisi, Nuria Torray, Calogero Caruana, Franco Graziosi, Luciano Pallotta, Mariella Palmich. Bounty hunter Django helps a sheriff in stopping an outlaw gang out to take over Black City and use it to smuggle guns. Quickly made, low grade Italian production released there as _**Quel Maledetto Giorno d'Inverno...Django e Sartana all'Ultimo Sangue**_ (One Damned Day at Dawn...Django Meets Sartana).\n\n_**Django Rides Again**_ see _**Keoma**_\n\n**1117** _ **Django Shoots First**_ **** FIDA Cinematografica, 1966. 95 min. Color. D: Alberto De Marino. SC: Sandro Continenza, Massimo Capriccioli, Alberto Fiorenzo Capri and Vincenzo Flamini. With Glenn Saxon, Fernando Sancho, Evelyn Stewart (Ida Galli), Erika Blanc, Jose Manuel Martin, Lee Burton (Guido Lollobrigida), Alberto Lupo, Nando Gazzolo, Diana Lorys, Marcello Tusco, Antonio Piretti, Valentino Macchi, George Eastman, Osiride Pevarello, Bruno Arie, Fortunato Arena, Riccardo Pizzuti, Attilio Severini. A crooked banker frames his partner for a crime he did not commit and when he is killed his son promises revenge. Another violent outing in the \"Django\" series released in Italy as _**Django Spara per Primo**_ (Django Shoots First).\n\n_**Django Strikes Back**_ see _**Return of Django**_\n\n**1118** _ **Django the Avenger**_ **** New Line Cinema, 1969. 107 min. Color. D: Sergio Garrone. SC: Antonio De Teffe and Sergio Garrone. With Anthony Steffan (Antonio De Teffe), Palo Gozlino, Lu Kamante, Teodoro Corra, Jean Louis, Carlo Gaddi, Tomas Rudi, Lucia Bomez, Emy Rossi Scott, Rada Rassimov, Osiride Pevarello, Furio Meniconi, Ennio Balbo, Celso Faria. A Yankee soldier returns from the dead to get revenge on the three comrades who betrayed him. Spooky, entertaining Spaghetti Western co-produced by Herman Cohen and star Anthony Steffen; also called _**Django the Bastard**_ and _**Stranger's Gundown**_.\n\n_**Django the Bastard**_ see _**Django the Avenger**_\n\n**1119** _ **Django the Killer**_ **** Constantin Film, 1967. 88 min. Color. D: Joseph Warren (Giuseppe Vari). SC: Augusto Caminito. With George Eastman, Anthony Ghidra, Dana Ghia, Daniele Vargas, John Hamilton (Gianni Medici), Mirko Ellis, John MacDouglas (Giuseppe Addobatti), Frank Fargas, Fred Coplan, John Mathews, Giuseppe Castellano, Anton de Cortes, Paolo Figlia, Valentino Macchi, Paolo Reale. Gunman Django is hired by a land baron to shoot an informer but he is bushwhacked and saved by a young man whose family was wiped out on his employer's orders. Well made but ponderous Italian production issued there as _**L'Ultimo Killer**_ (The Last Killer).\n\n**1120** _ **Djurado**_ **** Studio T\/Compagnia Cinematografica Astro, 1966. 90 min. Color. D-SC: John Farrell (Gianni Narzisi). With Montgomery Clark, Scilla Gabel, Margaret Lee, Mary Jordan, Isacro Ravaioli, Luis Induni, Goyo Lebrero. A gambler arrives in a border town and wins half interest in a saloon before opposing a murderous tyrant. Sub-par Italian oater.\n\n**1121** _ **Doc**_ **** United Artists, 1971. 96 min. Color. D: Frank Perry. SC: Pete Hamill. With Stacy Keach, Faye Dunaway, Harris Yulin, Mike Witney, Denver John Collins, Dan Greenberg, Penelope Allen, Hedy Sontag, Bruce M. Fisher, James Green, Richard MacKenzie, John Scanlon, Antonia Rey, John Bottoms, Philip Shafer, Marshall Efron, Fred Dennis, Mart Hulswit, Gene Collins. Hard-drinking, tubercular Doc Holliday joins forces with prostitute Kate Elder and they end up helping Wyatt Earp in his battle with the Clanton clan. Murky and basically boring re-telling of the Wyatt Earp-Doc Holliday story.\n\n**1122** _ **Doc Hooker's Bunch**_ **** Empire Pictures, 1976. 88 min. Color. D: Zack Belcher. SC: Mary H. Belcher. With Dub Taylor, Buck Taylor, Otis Sistrunk, Gaetana Campbell, Danielle Hibbard, Linda Mann, John Davis Chandler, John Furlong, Tac Tharp, David MacKay, R. Peter Dracup, Bob Turnbull, Mike Williams, Phyllis Julian, T.J. Raccio, Doyce Hutson, Dirty Will Landis, Bruce Cameron, Space, Cheyenne Rivera, Tom Paxton, Kyle T. Melick. In the Old West a theatrical troupe entertains in small towns but also stages bank robberies as its female performers fleece the local yokels. Fun comedy Western.\n\n**1123** _ **Doc West**_ **** Grindstone Entertainment Group, 2009. 97 min. Color. D: Terence Hill and Giulio Base. SC: Jess Hill, Marcello Olivieri, Luca Biglione and Marco Barboni. With Terence Hill, Paul Sorvino, Ornella Muti, Boots Southerland, Adam Taylor, Clare Carey Alessio Di Clemente, Kisha Sierra, Micah Alberti, Linus Huffman, Maria Bethke, Gianni Biasetti, Darrian Chavez, Benjamin Petry, Darren Gibson, Fabrizio Bucci, Dylan Kenin, Christina Judy Kim, Mercedes Leggett, Gisella Margeno, Jimmy Ning, Mary Petruolo, Mark Siversten, Sheila Ivy Traister. While seeking the bandit who stole his winnings, a gambler gunman finds himself in the middle of a town feud between the forces of good and evil. Okay Italian TV movie originally shown in two parts, running 180 minutes; followed by a sequel, _**Triggerman**_ (q.v.) [2009].\n\n**1124** _ **Doctor Quinn, Medicine Woman**_ **** CBS-TV, 1993. 90 min. Color. D: Jeremy Paul Kagan. SC: Beth Sullivan. With Jane Seymour, Diane Ladd, Joe Lando, Guy Boyd, Chad Allen, Erika Flores, Shawn Toovey, Colm Meaney, Geoffrey Lower, Verna Bloom, Helene Udy, Nick Ramus, Frank Collison, Adrian Sparks, Ivory Ocean, Larry Sellers, William Shockley, Heidi Kozak, Mary Gregory. When Colorado Springs advertises for a doctor, a woman from the East shows up for the job and must prove her worth to the town's citizens. Very well done pilot for the TV series of the same name that ran on CBS-TV from 1993 to 1998.\n\n**1125** _ **Doctor Quinn, Medicine Woman: The Heart Within**_ **** CBS-TV, 2001. 120 min. Color. D: Jerry London. SC: Beth Sullivan. With Jane Seymour, Joe Lando, Jessica Bowman, Elinor Donahue, Sara McRae, Shawn Toovey, Vlasta Vrana, Victoria Barkoff, Stephen Spreekmeester, Bibi Burton, Micihael Rudder, Gary Plaxton, Brandon Douglas, Francis Xavier McCarthy, Georgann Johnson, Una Kay, Daniel Libman, Jeffrey Aarles, Frank Fontaine, Mark Walker, John Walsh, Brian Wrench, Jude Beny, Maria Bircher, James Bradford, Richard Jutras, Joel Miller. Dr. Quinn, her husband and three children leave the frontier and return to Boston. Okay telefilm based on the characters created by Beth Sullivan.\n\n**1126** _ **Doctor Quinn, Medicine Woman: The Movie**_ **** CBS-TV, 1999. 86 min. Color. D: James Keach. SC: Beth Sullivan. With Jane Seymour, Joe Lando, Jim Knobloch, Frank Collison, Henry G. Sanders, Shawn Toovey, Jonelle Allen, Geoffrey Lower, Larry Sellers, Orson Bean, Barbara Babcock, Kalie Zaretsky, Chad Carr, Les Lannom, Jacqueline Torres, P.B. Hutton, Mark Collie, Rudy Ramos, Stephen Meadows, Luis Contreras, William Marquez, Leonardo Guerra, Mimi Lesseos, Rudy Ugland, Geoff Erwin, Eduardo Yenez, Makenzie Vega, Craig Carter. A female doctor, with three orphaned children to look after, is aided by a loner cowboy as she tries to establish a medical practice in frontier Colorado Springs. Competently done telefilm follow-up to the popular series \"Doctor Quinn, Medicine Woman\" (CBS-TV, 1993\u201398); also called _**Doctor Quinn: Revolutions**_.\n\n_**Doctor Quinn: Revolutions**_ see _**Doctor Quinn, Medicine Woman: The Movie**_\n\n_**The Doctor's Alibi**_ see _**The Medico of Painted Springs**_\n\n**1127** _ **Dodge City**_ **** Warner Bros., 1939. 104 min. Color. D: Michael Curtiz. SC: Robert Buckner. With Errol Flynn, Olivia de Havilland, Ann Sheridan, Bruce Cabot, Frank McHugh, Alan Hale, John Litel, Henry Travers, Henry O'Neill, Victor Jory, Guinn Williams, Bobs Watson, William Lundigan, Gloria Holden, Douglas Fowley, Georgia Caine, Charles Halton, Ward Bond, Cora Witherspoon, Russell Simpson, Monte Blue, Nat Carr, Clem Bevans, Joseph Crehan, Thurston Hall, Chester Clute, Ralph Sanford, Milton Kibbee, James Burke, George Chesebro, Robert Homans, George Guhl, Spencer Charters, Wilfred Lucas, Earl Dwire, Richard Cramer, Steve Clark, Francis Sayles, Merrill McCormick, Pat O'Malley, Vera Lewis, Bud Osborne, Horace B. Carpenter, Earle Hodgins, Jack Mower, Ed Peil, Sr., Tom Chatterton, Fred Graham, Guy Wilkerson, Bruce Mitchell, Frank Mayo, Pat Flaherty, Henry Otho, James Farley, Frank Pharr. An Irish soldier of fortune becomes the sheriff of Dodge City and is determined to make the area safe for homesteaders. Colorful Errol Flynn feature is short on story but big on action and color.\n\n**1128** _ **Dodge City Trail**_ **** Columbia, 1936. 56 min. D: C.C. Coleman, Jr. SC: Harold Shumate. With Charles Starrett, Marion Weldon, Donald Grayson, Russell Hicks, Si Jenks, Alan Bridge, Art Mix, Ernie Adams, Lew Meehan, Hank Bell, George Chesebro, Jack Rockwell, John Elliott, Blackie Whiteford, Richard Botiller, Charles E. Brinley, Fred \"Snowflake\" Toones, Buck Moulton, Bill Patton, Blackjack Ward, Tommy Coats, Ed Warren, Dick Bodkin, Al Haskell, George Burton, Tom Sutton, Roy Bucko, Buck Bucko, Slim Hazel. While on a cattle drive, a ranch foreman helps rescue a kidnapped young woman whose father is the brains behind a gang of bandits. Pedestrian Charles Starrett series effort.\n\n_**Dollars for a Fast Gun**_ see _**$100,000 for Lassiter**_\n\n**1129** _ **Domino Kid**_ **** Columbia, 1957. 74 min. D: Ray Nazarro. SC: Kenneth Gamet. With Rory Calhoun, Kristine Miller, Andrew Duggan, Yvette Dugay, Peter Whitney, Robert Burton, James J. Griffith, Roy Barcroft, Denver Pyle, Ray Corrigan, Eugene Iglesias, William Christensen, Don Orlando, Bart Bradley, Dennis Moore, Don C. Harvey, Tom London, Frank Sully. Returning home to Texas after the Civil War, a man finds his father and brother have been murdered and he tries to find the killers. Star Rory Calhoun wrote the story for this fairly entertaining \"B\" outing.\n\n_**Don Amigo**_ see _**The Girl from San Lorenzo**_\n\n**1130** _ **Don Daredevil Rides Again**_ **** Republic, 1951. 12 Chapters. D: Fred C. Brannon. SC: Ronald Davidson. With Ken Curtis, Aline Towne, Roy Barcroft, Lane Bradford, Robert Einer, John Cason, I. Stanford Jolley, Hank Patterson, Lee Phelps, Sandy Sanders, Guy Teague, Tom Steele, Michael Ragan, Cactus Mack, Art Dillard, Bud Osborne, Saul Gorss, Gene (Roth) Stutenroth, James Magill, David Sharpe, Charles Horvath, Dale Van Sickel, Jack Ingram, George Lloyd, Carey Loftin, Forrest Taylor, Don C. Harvey, Tex Terry, Bob Reeves, Chick Hannon, Herman Hack, Joe Phillips, Roy Bucko, Frank McCarroll, Jack Harden, Carlie Taylor, Bert LeBaron, James Linn, Gene Christopher, Tony DeMario, Frank Meredith. A homesteader takes on the guise of the masked Don Daredevil to stop a political boss who is trying to run settlers off their spreads by claiming an old land grant is a fake. This pseudo-Zorro cliffhanger has a good plot but too much footage from earlier Republic efforts.\n\n**1131** _ **Don Q, Son of Zorro**_ **** United Artists, 1925. 113 min. D: Donald Crisp. SC: Lotta Woods. With Douglas Fairbanks, Mary Astor, Jack MacDonald, Donald Crisp, Stella De Lanit, Warner Oland, Jean Hersholt, Albert MacQuarrie, Lottie Pickford Forrest, Charles Stevens, Tote Du Crow, Martha Franklin, Juliette Belanger, Roy Coulson, Enrique Acosta. Sent to Spain by his father Don Diego, a young Californian falls in love with a pretty girl and gets involved in court intrigue. Action filled Douglas Fairbanks diversion, a sequel to his _**The Mark of Zorro**_ (1920) [q.v.].\n\n**1132** _ **Don Ricardo Rides Again**_ **** Producers Releasing Corporation, 1946. 63 min. D: Terry Morse. SC: Jack De Witt and Renault Duncan (Duncan Renaldo). With Fred Coby, Isabelita, Martin Garralaga, Paul Newton, Claire DuBrey, David Leonard, Anthony Warde, Michael Visaroff. Learning he has been declared dead and his cousin has taken his place, a Spanish don masquerades as a peon and seeks the aid of a mission priest in proving his inheritance. Pleasing outing from PRC, co-written by Duncan Renaldo.\n\n**1133** _ **Donner Pass: The Road to Survival**_ **** NBC-TV\/Schick Sunn Classics, 1978. 100 min. Color. D: James L. Conway. SC: S.S. Schweitzer. With Robert Fuller, Andrew Prine, Michael Callan, Diane McBain, John Anderson, John Doucette, Cynthia Eibacher, Royal Dano, Gregory Walcott, Lance LeGault, Whit Bissell, Peg(gy) Stewart, Reid Cruichshanks, Robert Carricart, Rudy Diaz, John Hansen, George Barrows. A wagon train is stranded in the mountains during a blizzard and the travelers are eventually forced into cannibalism. Despite its subject matter, this \"Classics Illustrated\" TV movie is pretty good.\n\n**1134** _ **Don't Fence Me In**_ **** Republic, 1945. 71 min. D: John English. SC: Dorrell McGowan, Stuart McGowan and John K. Butler. With Roy Rogers, George \"Gabby\" Hayes, Dale Evans, Robert Livingston, Moroni Olson, Arthur Space, Bob Nolan and The Sons of the Pioneers (Tim Spencer, Shug Fisher, Ken Carson, Hugh Farr, Karl Farr), Marc Lawrence, Lucille Gleason, Andrew Tombes, Paul Harvey, Douglas Fowley, Stephen Barclay, Edgar Dearing, Helen Talbot, Tom London, Sam Ash, Michael Branden, Ray Teal, Eddie Fetherston, John Ince, Lee Phelps, Kenner G. Kemp, Charles Teske, Arleen Claire, Phil Dennis, Sherry Hall, James Linn, Diane Quilland, Brick Sullivan, Frank Fanning. A female magazine writer reveals an old man was once a famous outlaw and this gets him into trouble until he finds a killer and collects a reward. Better than average Roy Rogers vehicle that mainly belongs to Gabby Hayes. One interesting sequence has Gabby in a funeral parlor pretending to be dead while the Sons of the Pioneers sing \"Headin' for the Last Roundup,\" while another has him using a bullwhip, pre\u2013Lash LaRue.\n\n**1135** _ **Don't Turn the Other Cheek**_ **** International Amusements, 1974. 103 min. Color. D: Duccio Tessari. SC: Dino Maiuri, Massimo De Rita, Gunter Ebert and Juan De Orduna y Fernandez. With Franco Nero, Eli Wallach, Lynn Redgrave, Horst Janson, Eduardo Fajardo, Jose Moreno, Victor Israel, Gisela Hahn, Jose Jaspe, Enrique Espinosa, Gunda Hiller, Furio Meniconi (Men Fury), Dale Van Husen, Rudy Gaebell, Carla Mancini, Mirko Ellis, Marilu Tolo, Lorenzo Robelod, Luigi Antonio Guerra. Trying to start a revolt by peons in Mexico at the time of the Revolution, an Irish woman writer teams with a bandit and a supposed Russian prince but the two men are really after hidden gold. Complicated but fun Italian Western released in that country as _**Viva la Muerte...Tua!**_ by Titanus in 1971.\n\n**1136** _ **The Doolins of Oklahoma**_ **** Columbia, 1949. 90 min. D: Gordon Douglas. SC: Kenneth Gamet. With Randolph Scott, Louise Allbritton, George Macready, John Ireland, Virginia Huston, Charles Kemper, Noah Beery, Jr., Dona Drake, Robert Barrat, Lee Patrick, Griff Barnett, Frank Fenton, Jock (Mahoney) O'Mahoney, James Kirkwood, Robert Osterloh, Virginia Brissac, John Sheehan, George Chesebro, Stanley Andrews, Trevor Bardette, Reed Howes, Stanley Blystone, Alan Bridge, Harry Hayden, Eddie Dunn, Vernon Dent, John Sheehan, Al Hill, William Haade, Ethan Laidlaw, George DeNormand, Paul Scanlon, Anne O'Neal, Frank O'Connor, Hank Patterson, Paul E. Burns, Chuck Hamilton, Harry Tyler, Mira McKinney, Joe Palma, Aleth Hansen, Claire Meade, David Clark, Jack Parker, George Bell, Paul Scardon, Rose Higgins. An ex-bandit attempts to go straight but his brothers continue their lawless ways and the entire family ends up being hunted by a sheriff and his posse. Well staged, and quite good, Randolph Scott vehicle.\n\n**1137** _ **Doomed at Sundown**_ **** Republic, 1937. 60 min. D: Sam Newfield. SC: George Plympton. With Bob Steele, Lorraine Hayes (Laraine Day), Warner Richmond, David Sharpe, Earl Dwire, Horace B. Carpenter, Sherry Tansey, Harold Daniels, Budd Buster, Jack Kirk, Horace Murphy, Charles King, Lew Meehan, Jack Ingram. When his father is murdered by outlaws, a cowpoke pretends to be a bad man in order to infiltrate the gang. Bob Steele is again on the trail of his father's murderer and this one provides lots of thrills for his legion of fans.\n\n**1138** _ **Doomed Caravan**_ **** Paramount, 1941. 62 min. D: Lesley Selander. SC: Johnston McCulley and J. Benton Cheney. With William Boyd, Russell Hayden, Andy Clyde, Minna Gombell, Morris Ankrum, Georgia Hawkins, Trevor Bardette, Pat J. O'Brien, Raphael (Ray) Bennett, Jose Luis Tortosa, Ed Cassidy, Martin Garralaga, Wen Wright, Fred Burns, Charles Murphy, Art Dillard. Hoppy and the Bar 20 boys agree to help a woman whose freight wagon has been raided but he becomes suspicious when she calls in the cavalry, actually a band of outlaws. Good \"Hopalong Cassidy\" segment, very well done and exciting.\n\n_**The Doomed Ranch**_ see _**Fury of the Apaches**_\n\n_**Doomsday**_ see _**Drummer of Vengeance**_\n\n**1139** _ **Las Dos Hectareas**_ (The Two Acres) **** Mexcinema Video Corporation, 1999. 100 min. Color. D: Manuel Ramirez. SC: Manolo Cardenas. With Manuel Ramirez, Alicia Encinas, Manolo Cardenas, Irene Arcila, Valentin Trujillo, Roberto Ballesteros. A gunman returns home to win back his family and his rancho. Average Mexican video Western.\n\n_**Double Alibi**_ see _**Law and Order**_ (1942)\n\n**1140** _ **Double Deal**_ **** RKO Radio, 1950. 64 min. D: Abby Berlin. SC: Charles S. Belden and Lee Berman. With Marie Windsor, Richard Denning, Taylor Holmes, Fay Baker, James Griffith, Carleton Young, Tom Browne Henry, Frank Fenton, Walter Burke, Richard Reeves, Gil Perkins, Edgar Dearing, Jim Hayward, Ned Roberts, Charles Wagenheim, Art Dupuis, Sid Gorss, Paul E. Burns. A woman tries to get back oil land her brother left to his girlfriend who has hired an engineer to drill a well. More than passable \"B\" effort.\n\n_**Double Identity**_ (1940) see _**River's End**_ (1940)\n\n_**Double Identity**_ (1941) see _**Hurricane Smith**_\n\n**1141** _ **Down Dakota Way**_ **** Republic, 1949. 67 min. Color. D: William Witney. SC: John K. Butler and Sloan Nibley. With Roy Rogers, Dale Evans, Pat Brady, Montie Montana, Foy Willing and The Riders of the Purple Sage, Elisabeth Risdon, Byron Barr, James Cardwell, Roy Barcroft, Emmett Vogan, Victor Cutler, George Lamond. When his cattle are found to have hoof-and-mouth disease, a rancher hires a gunman to kill the veterinarian who made the diagnosis before he is forced to destroy his herd. Nicely plotted and action filled Roy Rogers outing.\n\n**1142** _ **Down Laredo Way**_ **** Republic, 1953. 54 min. D: William Witney. SC: Gerald Geraghty. With Rex Allen, Slim Pickens, Dona Drake, Marjorie Lord, Roy Barcroft, Clayton Moore, Judy Nugent, Percy Helton, Zon Murray. A gang of diamond smugglers get away with a series of robberies until they are hunted by a rodeo star. Good direction and a fast paced script add up to fine entertainment in this Rex Allen opus.\n\n**1143** _ **Down Mexico Way**_ **** Republic, 1941. 77 min. D: Joseph Santley. SC: Olive Cooper and Albert Duffy. With Gene Autry, Smiley Burnette, Fay McKenzie, Harold Huber, Sidney Blackmer, Joseph Sawyer, Andrew Tombes, Murray Alper, Arthur Loft, Duncan Renaldo, Paul Fix, Julian Rivero, Ruth Robinson, Thornton Edwards, Eddie Dean, The Herrera Sisters, Frankie Marvin, Esther Estrella, Sam Appel, Helen MacKellar, Elias Gamboa, Rico de Montez, Charles Rivero, Paquito del Rey, Jose Manero, Carmela Cansino, Reed Howes, Hank Bell, Fred Burns, Al Haskell, Jack O'Shea. Three cowpokes are after a gang who bilked townspeople out of money on the pretext of producing a movie in their community. Pleasing Gene Autry affair with a good story and nice songs (i.e., \"South of the Border\" and \"Maria Elena\").\n\n**1144** _ **Down Missouri Way**_ **** Producers Releasing Corporation, 1946. 73 min. D: Josef Berne. SC: Sam Neuman. With Martha O'Driscoll, John Carradine, Eddie Dean, William Wright, Roscoe Ates, Renee Godfrey, Mabel Todd, Eddie Craven, Chester Clute, Will Wright, Paul Scardon, Earle Hodgins, TheTailor-Maids, The Notables, Shirley (mule). A wind-bag producer wants to make a film about an intelligent mule and finds one at a Missouri college. Not much to recommend this so-called comedy except for Eddie Dean singing a half-dozen tunes.\n\n**1145** _ **Down Rio Grande Way**_ **** Columbia, 1942. 57 min. D: William Berke. SC: Paul Franklin. With Charles Starrett, Russell Hayden, Britt Wood, Rose Anne Stevens, Norman Willis, Davison Clark, Edmund Cobb, Budd Buster, Paul Newlan, William Desmond, Jim Corey, Steve Clark, Forrest Taylor, Ed Piel, Sr., John Cason, Art Mix, Kermit Maynard, Frank McCarroll, Joseph Eggenton, Betty Roadman, Tom Smith. Two cowboys get involved with the movement for independence in Texas. Slim production values hurt the overall effectiveness of this pseudo-historical Charles Starrett vehicle.\n\n**1146** _ **Down Texas Way**_ **** Monogram, 1942. 57 min. D: Howard Bretherton. SC: Jess Bowers (Adele Buffington). With Buck Jones, Tim McCoy, Raymond Hatton, Luana Walters, Dave O'Brien, Lois Austin, Harry Woods, Glenn Strange, John Merton, Tom London, Jack Daley, Kansas Moehring, Frank Ellis, Bill Nestell, Jack Holmes, Blackie Whiteford, Wally West, George Morrell, Foxy Callahan, Chick Hannon, Ben Corbett, Artie Ortego, Charles Murphy, Jack Tornek, Milburn Morante. An outlaw gang has a young woman pose as the widow of a murdered man in order to claim his ranch, which rightfully belongs to the victim's son. Good one in the popular \"Rough Riders\" series with a complicated, but interesting, plot. Remade as _**Western Renegades**_ (q.v.).\n\n**1147** _ **Down the Long Hills**_ **** Disney Television, 1986. 95 min. Color. D: Burt Kennedy. SC: Jon Povare and Ruth Povare. With Bruce Boxleitner, Bo Hopkins, Michael Wren, Don Shanks, Ed Bruce, Buck Taylor, Jack Elam, Lisa MacFarlane, Thomas Wilson Brown, Richard J. Martin. After escaping from a wagon train massacre two children find they are chased by outlaw, an Indian and a marauding bear. Average TV family fare.\n\n**1148** _ **Down the Wyoming Trail**_ **** Monogram, 1939. 62 min. D: Al Herman. SC: Peter Dixon and Roger Merton. With Tex Ritter, Mary Brodel, Horace Murphy, Bobby Lawson, Charles King, Bob Terry, Jack Ingram, Earl Douglas, Karl Hackett, Frank LaRue, Ernie Adams, Ed Coxen, Jean Southern, Charles Sargent, The Northwesterners. A cowpoke is after an outlaw gang planning to drive their stolen cattle herd through a mountainous area. Fair Tex Ritter vehicle benefiting from scenic locales and nice songs.\n\n**1149** _ **Drag Harlan**_ **** Fox, 1920. 64 min. D: J. Gordon Edwards. SC: H.P. Keeler. With William Farnum, Jackie Saunders, G. Raymond Nye, Herschel Mayall, Frank Thurwald, Kewpie Morgan, Al Fremont, Earl Crain. A gunman finds a dying miner who gives him the map to his claim and asks him to take it to his daughter, who is being threatened by the gang who shot her father and is after his gold. Stout silent melodrama greatly enhanced by the work of William Farnum in the title role.\n\n**1150** _ **Dragoon Wells Massacre**_ **** Allied Artists, 1957. 88 min. Color. D: Harold Schuster. SC: Oliver Drake and Warren Douglas. With Barry Sullivan, Dennis O'Keefe, Mona Freeman, Katy Jurado, Sebastian Cabot, Jack Elam, Trevor Bardette, Hank Worden, Warren Douglas, Casey Adams (Max Showalter), Jon Shepodd, Judy Stranges, Alma Betran, John War Eagle. In 1860 a group of people, including outlaws and lawmen, are cornered in a fort about to be attacked by Indians. Well staged and highly entertaining oater.\n\n**1151** _ **Drango**_ **** United Artists, 1957. 92 min. D: Hall Bartlett and Jules Bricken. SC: Hall Bartlett. With Jeff Chandler, Joanne Dru, Julie London, Ronald Howard, Donald Crisp, John Lupton, Morris Ankrum, Helen Wallace, Walter Sande, Parley Baer, Charles Horvath, Mimi Gibson, Paul Lukather, Damian O'Flynn, Milburn Stone, Edith Evanson. After the Civil War a Union officer is assigned to govern a town he was once forced to plunder. Fairly enjoyable yarn.\n\n**1152** _ **Draw!**_ Home Box Office Premiere Films, 1984. 100 min. Color. D: Steven H. Stern. SC: Stanley Mann. With Kirk Douglas, James Coburn, Alexandra Bastedo, Graham Jarvis, Derek McGrath, Jason Michas, Len Birman, Maurice Brand, Graham McPherson, Vladimir Valenta, Linda Sorenson, Gerard Parkes, Richard Donat, Frank Adamson, Stuart Gillard, Miles Vasey, James DeFelice, James Forsythe, Sherrill De Marco, Larry Musser, Bonar Bain, Wilf Rowe, Brian Fustukian, Frank C. Turner, Brian George, Victor Bain, Joan Hurley, Alan Stebbings, Vincent Gale. A once famous outlaw is forced to shoot a sheriff and take a girl hostage while citizens demand his capture by a respected, but alcoholic, lawman. Made-for-pay-TV feature is only average despite good work by its two stars and fine production trappings.\n\n_**Dream of Zorro**_ see _**Grandsons of Zorro**_\n\n**1153** _ **Drift Fence**_ **** Paramount, 1936. 56 min. D: Otho Lovering. SC: Robert Yost and Stuart Anthony. With Larry \"Buster\" Crabbe, Katherine DeMille, Tom Keene, Benny Baker, Glenn (Leif) Erickson, Stanley Andrews, Richard Carle, Irving Bacon, Effie Ellser, Jan Duggan, Walter Long, Chester Gan, Richard Alexander, Bud Fine, Jack Pennick, Henry Roquemore, Frank O'Connor, Jack Clifford, Don Roberts. A dude gets a wrangler take over his identity in order to take control of a ranch he has inherited. Very well done \"B\" adaptation of the Zane Grey novel; Benny Baker is very good as comical tenderfoot Jim Traft. Reissued as _**Texas Desperadoes**_.\n\n**1154** _ **The Drifter**_ **** Willis Kent, 1932. 60 min. D: William O'Connor. SC: Oliver Drake. With William Farnum, Noah Beery, Phyllis Barrington, Charles Sellon, Bruce Warren, Russell Hopton, Ann Brody, Ynez Seabury. A man gets involved in a lumber feud not knowing one of the leaders is his brother. Pretty good low budget melodrama highlighted by stars William Farnum and Noah Beery.\n\n**1155** _ **The Drifter**_ **** Producers Releasing Corporation, 1944. 61 min. D: Sam Newfield. SC: Patricia Harper. With Buster Crabbe, Al St. John, Carol Parker, Kermit Maynard, Jack Ingram, Roy Brent, George Chesebro, Ray Bennett, Jimmy Aubrey, Slim Whitaker, Wally West, Robert Hill, Herman Hack, Foxy Callahan. Billy Carson impersonates a sharpshooter, the leader of an outlaw gang, to find out who is behind a series of robberies. Interesting plot, mundane execution.\n\n**1156** _ **The Driftin' Kid**_ **** Monogram, 1941. 55 min. D: Robert Emmett Tansey. SC: Robert Emmett (Tansey) and Frances Kavanaugh. With Tom Keene, Betty Miles, Frank Yaconelli, Arkansas Slim Andrews, Stanley Price, Gene Alsace, Glenn Strange, Steve Clark, Sherry Tansey, Fred Hoose, Frank McCarroll, Wally West. Outlaws plan to kill a rancher for his spread and government contract and when a federal agent is sent to investigate the theft of his cattle it turns out the two men are look-alikes. Okay entry in Tom Keene's second Monogram series; remade as _**Stars Over Texas**_ (q.v.).\n\n**1157** _ **Driftin' River**_ **** Producers Releasing Corporation, 1946. 57 min. D: Robert Emmett Tansey. SC: Frances Kavanaugh. With Eddie Dean, Shirley Patterson, Roscoe Ates, Lee Bennett, William Fawcett, Dennis Moore, Lottie Harrison, Forrest Taylor, Robert Callahan, Lee Roberts, Don Murphy, The Sunshine Boys (Eddie Wallace, J.D. Sumner, M.H. Richman, Freddie Daniel). Two cowpokes join a gang of cattle rustlers to capture those responsible for the massacre of an Army platoon carrying money to buy a woman's horse herd. Slow drama aided by pretty Shirley Patterson and Eddie Dean's singing of the title tune and \"Way Back in Oklahoma\"; refashioned into _**The Tioga Kid**_ (q.v.).\n\n**1158** _ **Drifting Along**_ **** Monogram, 1946. 60 min. D: Derwin Abrahams. SC: Adele Buffington. With Johnny Mack Brown, Raymond Hatton, Lynne Carver, Douglas Fowley, Smith Ballew, Milburn Morante, Steve Clark, Marshall Reed, Jack Rockwell, Terry Frost, Lynton Brent, Curt Barrett and The Trailsmen, Harry V. Cheshire, Ted Mapes, Ted French, Hollywood Exhibition Square Dancers, Thornton Edwards. A ranch foreman learns his pretty boss' fiance is a cattle rustler. Fairly good outing in the Johnny Mack Brown-Raymond Hatton series, with a quartet of tunes including Johnny's singing a number.\n\n**1159** _ **Drifting Westward**_ **** Monogram, 1939. 58 min. D: Robert Hill. SC: Robert Emmett (Tansey). With Jack Randall, Frank Yaconelli, Edna Duran, Stanley Blystone, Carmen Bailey, Julian Rivero, Dave O'Brien, James Sheridan (Sherry Tansey), Tom London, Dean Spencer, Octavio Giraud. A cowboy tries to help when outlaws harass a Spanish family in possession of a map to a rich mine claim. Fair Jack Randall vehicle.\n\n**1160** _ **Drop Them or I'll Shoot**_ **** Films Marceau, 1969. 90 min. Color. D-SC: Sergio Corbucci. With Johnny Hallyday, Francoise Fabian, Silvie Fennec, Serge Marquand, Mario Adorf, Gastone Moschin, Gino Pernkice. An outsider goes up against outlaws plaguing the people of a frontier town. French-Italian-West German co-production first issued as _**Gli Specialisti**_ (The Specialist) offers little other than lots of violence.\n\n_**Dr. Quinn, Medicine Woman**_ see _**Doctor Quinn, Medicine Woman**_\n\n_**Dr. Quinn, Medicine Woman**_ see _**Doctor Quinn, Medicine Woman: The Heart Within**_\n\n_**Dr. Quinn, Medicine Woman: The Movie**_ see _**Doctor Quinn, Medicine Woman: The Movie**_\n\n_**Dr. Quinn: Revolutions**_ see _**Doctor Quinn, Medicine Woman: The Movie**_\n\n**1161** _ **Drum Beat**_ **** Warner Bros., 1954. 111 min. Color. D-SC: Delmer Daves. With Alan Ladd, Audrey Dalton, Marisa Pavan, Charles Bronson, Robert Keith, Rodolfo Acosta, Warner Anderson, Elisha Cook, Jr., Anthony Caruso, Richard Gaines, Edgar Stehli, Hayden Rorke, Frank De Kova, Isabel Jewell, Perry Lopez, Willis Bouchey, George J. Lewis, Frank Ferguson, Peter Hansen, Peggy Converse, Pat Lawless, Paul Wexler, Richard Cutting, Strother Martin, Rico Alaniz, John Veitch, George Ross, Victor Millan, Ken Smith, Maurice Jara, Jonas Applegarth, Felix Noriego, James Griffith, Frank Gerstle, Carol Nugent, Michael Daves, Leonard Penn, Oliver Blake, Dan Borgaze, George Lloyd, Ron Hargrave. In 1869 President Grant appoints a peace commissioner to negotiate a treaty with renegade Indians. Good action film with Charles Bronson giving a fine performance as the Indian leader.\n\n**1162** _ **Drum Taps**_ **** World Wide, 1932. 61 min. D: J.P. McGowan. SC: Alan James. With Ken Maynard, Dorothy Dix, Hooper Atchley, Alan Bridge, Charles Stevens, Junior Coghlan, Harry Semels, Jim Mason, Slim Whitaker, Neal Hart, Art Mix, Kermit Maynard, Leo Willis, Lloyd Ingraham, Merrill McCormick, Tex Palmer, Jack Rockwell, Bud McClure, Fred Burns, Pascale Perry, Blackjack Ward, Boy Scout Troup 107. A cowboy helps the Boy Scouts in trying to thwart land grabbers. Entertaining Ken Maynard vehicle with an exciting climax.\n\n**1163** _ **Drummer of Vengeance**_ **** Times Films, 1974. 90 min. Color. D-SC: Robert Paget (Mario Gariazzo). With Ty Hardin, Rossano Brazzi, Craig Hill, Gordon Mitchell, Rosalba Neri, Ida de Benedetto, Lee Burton, Ralf Baldassarre. After an outlaw gang murders his wife and child a man seeks revenge but learns their leader is a sheriff. Fair British-Italian co-production, better for its performances than plot. Made in 1972 as _**Doomsday**_.\n\n**1164** _ **Drums Across the River**_ **** Universal-International, 1954. 78 min. Color. D: Nathan Juran. SC: John K. Butler. With Audie Murphy, Lisa Gaye, Lyle Bettger, Walter Brennan, Mara Corday, Hugh O'Brian, Jay Silverheels, Regis Toomey, Morris Ankrum, James Anderson, George Wallace, Bob Steele, Lane Bradford, Emile Meyer, Gregg Barton, Howard McNear, Kenneth Terrell, Edmund Cobb, Robert Bray, Rusty Wescoatt, Chief Yowlachie. A young man mistakenly joins a group of gold hunters who go into Indian Territory but he soon sees his mistake and joins with his father in trying to restore peace. Fair Audie Murphy vehicle.\n\n**1165** _ **Drums Along the Mohawk**_ **** 20th Century\u2013Fox, 1939. 103 min. Color. D: John Ford. SC: Lamar Trotti and Sonya Levien. With Claudette Colbert, Henry Fonda, Edna May Oliver, Eddie Collins, John Carradine, Doris Bowden, Jessie Ralph, Arthur Shields, Robert Lowery, Roger Imhoff, Francis Ford, Ward Bond, Kay Linaker, Russell Simpson, Chief Big Tree, Spencer Charters, Arthur Aylesworth, Si Jenks, Jack Pennick, Charles Tannen, Tom Tyler, Paul McVey, Clarence Wilson, Edwin Maxwell, Clara Blandick, Mae Marsh, Noble Johnson, Elizabeth Jones, Robert Greig, Lionel Pape, Beulah Hall Jones. Newlyweds struggle against adversity, Tories and their Indian allies, in the Mohawk Valley at the start of the Revolutionary War. A John Ford classic, a must see film.\n\n**Edna May Oliver, Claudette Colbert and Henry Fonda in** _**Drums Along the Mohawk**_ **(20th Century** **\u2013** **Fox, 1939).**\n\n**  \n**\n\n**1166** _ **Drums in the Deep South**_ **** RKO Radio, 1951. 87 min. Color. D: William Cameron Menzies. SC: Philip Yordan and Sidney Harmon. With James Craig, Barbara Payton, Guy Madison, Barton MacLane, Craig Stevens, Tom Fadden, Robert Osterloh, Taylor Holmes, Lewis Martin, Peter Brocco, Dan White, Robert Easton, Louis Jean Heydt, Myron Healey, Kenne Duncan, James Griffith, Guy Wilkerson, Mickey Simpson, Tom Monroe. Two friends who love the same woman find themselves on opposite sides as General Sherman marches through Georgia. Mediocre Civil War yarn greatly enhanced by Lionel Lindon's photography.\n\n**1167** _ **Drums of Destiny**_ **** Crescent, 1937. 64 min. D: Ray Taylor. SC: Roger Whatley and John T. Neville. With Tom Keene, Edna Lawrence, Budd Buster, Rafael (Ray) Bennett, Robert Fiske, David Sharpe, John Merton, Carlos De Valdez, Chief Flying Cloud. In 1815 a cavalry captain plans to go into Spanish Florida to put down Creek Indian attacks, an action contrary to government policy. Well done entry in Crescent's historical series starring Tom Keene.\n\n**1168** _ **The Drylanders**_ **** Columbia, 1963. 70 min. D: Donald Haldane. SC: M. Charles Cohen. With Frances Hyland, James Douglas, Lester Nixon, Mary Savage, William Fruete, Don Francks, Irena Mayeska, William Weintraub. A man takes his wife and sons west to homestead in Canada where thy fight the elements to survive and build a new life. Reasonably good entertainment; National Film Board of Canada's first feature, originally done for television.\n\n_**Dual**_ see _**Dual: The Lone Drifter**_\n\n**1169** _ **Dual: The Lone Drifter**_ **** Cinema Epoch, 2009. 90 min. Color. D: Steven R. Monroe. SC: Michael Worth. With Tim Thomerson, Karen Kim, Michael Worth, Margot Farley, Warren Neff, Karen Hustus, Sandy Cooper, Don Hearn, Jr., Hank Hustus, David Barr. A stranger arrives in a small town to find the citizens have been savagely killed and he decides to find out why. Award winning, violent independent Western.\n\n**1170** _ **The Duchess and the Dirtwater Fox**_ **** 20th Century\u2013Fox, 1976. 104 min. Color. D: Melvin Frank. SC: Barry Sandler, Jack Rose and Melvin Frank. With George Segal, Goldie Hawn, Conrad Janis, Thayer David, Jennifer Lee, Roy Jenson, Pat Ast, Si Gould, Bob Hoy, E.J. Andre, Richard Farnsworth, John Alderson, Prentiss Rowe, Jerry Gatlin. A crooked gambler is forced to team with a dancehall girl when they head for the desert with stolen loot. Mediocre Western \"comedy\" for fans of its two stars.\n\n**1171** _ **Duck, You Sucker**_ **** United Artists, 1972. 139 min. Color. D: Sergio Leone. SC: Sergio Leone, Sergio Donati and Luciano Vincenzoni. With Rod Steiger, James Coburn, Romolo Valli, Maria Monti, Rik Battaglia, Franco Graziosi, Domingo Antoine, Goffredo Pistoni, Roy Bosier, John Frederick. During the 1913\u201314 Mexican Revolution a foreigner helps a local rebel while planning a bank robbery. Big Italian-made violent oater, mainly for fans of Sergio Leone. Issued in Italy in 1971 as _**Giu la Testa**_ (Down with Your Head) by Rafran\/San Marco\/Miura Film, running 158 minutes. TV title: _**A Fistful of Dynamite**_.\n\n**1172** _ **Dude Bandit**_ **** Allied, 1933. 62 min. D: George Melford. SC: Jack Natteford. With Hoot Gibson, Gloria Shea, Hooper Atchley, Skeeter Bill Robbins, Neal Hart, Lafe McKee, Gordon DeMain, Fred Burns, Art Mix, Fred Gilman, George Morrell, Merrill McCormick, Charles King, Slim Whitaker, Pete Morrison, Frank Ellis, Horace B. Carpenter, Charles Brinley, Blackie Whiteford, Bill Gillis. Pretending to be a dimwit, a cowboy investigates the murder of a friend and learns a banker is responsible and takes on the guise of a bandit to stop the crook. Rambling production with Hoot Gibson as a character similar to the one in _**Spirit of the West**_ (q.v.); Skeeter Bill Robbins is a most annoying sidekick.\n\n**1173** _ **The Dude Cowboy**_ **** Film Booking Offices (FBO), 1926. 55 min. D: Jack Nelson. SC: Paul M. Bryan and James Ormont. With Bob Custer, Flora Bramley, Billy Bletcher, Howard Truesdell, Bruce Gordon, Amber Norman, Sabel Johnson, Edward Gordon. A cowboy and his pal rescue a tourist and his daughter from outlaws and join them at a dude ranch where the foreman plans to rob the guests. Fairly entertaining silent Bob Custer affair. TV title: _**Secret Rancher**_.\n\n**1174** _ **Dude Cowboy**_ **** RKO Radio, 1941. 59 min. D: David Howard. SC: Morton Grant. With Tim Holt, Marjorie Reynolds, Louise Currie, Ray Whitely, Lee \"Lasses\" White, Helen Holmes, Eddie Kane, Eddie Dew, Byron Foulger, Glenn Strange, Tom London, Lloyd Ingraham. A Treasury Department Secret Service agent goes slumming on a dude ranch in order to break up a counterfeiting operation. Highly entertaining Tim Holt film.\n\n**1175** _ **The Dude Goes West**_ **** Allied Artists, 1948. 86 min. D: Kurt Neumann. SC: Richard Sale and Mary Loos. With Eddie Albert, Gale Storm, Gilbert Roland, James Gleason, Binnie Barnes, Barton MacLane, Douglas Fowley, Tom Tyler, Harry Hayden, Chief Yowlachie, Sarah Padden, Catherine Doucet, Edward Gargan, Frank Yaconelli, Olin Howlin, Dick Elliott, Lee \"Lasses\" White, Si Jenks, George Meeker, Ben Weldon, Charles Williams, Francis Pierlot, Tom Fadden. A Bowery shopkeeper becomes a sharpshooter, heads West and goes up against an outlaw gang. Very pleasant Western comedy with a good cast.\n\n**1176** _ **Dude Ranch**_ **** Paramount, 1931. 72 min. D: Frank Tuttle. SC: Percy Heath, Grover Jones and Lloyd Corrigan. With Jack Oakie, Stuart Erwin, Mitzi Green, June Collyer, Eugene Pallette, Charles Sellon, Guy Oliver, George Webb, James Crane, Cecil Weston. In order to impress a girl, an actor goes to a dude ranch where he poses as a cowboy. Early talkie is still fun.\n\n**1177** _ **The Dude Ranger**_ **** Fox, 1934. 68 min. D: Edward F. Cline. SC: Barry Barringer. With George O'Brien, Irene Hervey, Syd Saylor, LeRoy Mason, Henry Hall, Jim Mason, Lloyd Ingraham, Earl Dwire, Si Jenks, Lafe McKee, Hank Bell, Jack Kirk. After inheriting a ranch in Arizona from his uncle, a man finds out it is being plagued by rustlers. Entertaining adaptation of Zane Grey's story.\n\n**1178** _ **The Dude Wrangler**_ **** Sono Art-World Wide, 1930. 60 min. D: Richard Thorpe. SC: Robert N. Lee. With Lina Basquette, George Duryea (Tom Keene\/Richard Powers), Francis X. Bushman, Clyde Cook, Sojin, Margaret Seddon, Ethel Wales, Wilfred North, Alice Davenport, Virginia Sale, Julia Swayne Gordon, Louis Payne, Fred Parker, Aileen Carlyle, Jack Richardson. A young man borrows money to buy a dude ranch but one of the guests plots to sabotage it so he can impress the woman they both want. Interesting early talkie for Tom Keene fans.\n\n**1179** _ **Dudes Are Pretty People**_ **** United Artists, 1942. 46 min. D: Hal Roach, Jr. SC: Louis Kaye. With Jimmy Rogers, Noah Beery, Jr., Marjorie Woodworth, Paul Hurst, Marjorie Gateson, Russell Gleason, Grady Sutton, Bob Gregory, Frank Moran. Two cowboys work at a dude ranch where one of them falls for a guest who already has a boyfriend. Silly opener for the brief featurette series starring Jimmy Rogers and Noah Beery, Jr.\n\n**1180** _ **Due Mafiosi Nel Far West**_ (Two Mafia Men in the Far West) **** FIDA\/Epoca Film, 1964. 102 min. Color. D-SC: Giorgio Simonelli. With Franco Franchi, Ciccio Ingrassi, Aroldo Tieri, Helene Chanel, Fernando Sancho, Anna Casares, Aldo Giuffre, Felix De Fauce, Alfredo Rizzo. Two loony Italians inherit a Texas goldmine and get mixed up with outlaws. Lame \"comedy\" from the team of Franco Franchi and Ciccio Ingrassi.\n\n**1181** _ **Due Sergenti Del Generale Custer**_ (Two Sergeants of General Custer) **** FIDA\/Balcazar, 1965. 97 min. Color. D-SC: Giorgio Simonelli. With Franco Franchi, Ciccio Ingrassi, Fernando Sancho, Margaret Lee, Moira Orfei, Arolo Tieri, Riccardo Garrone, Ernesto Calindri. Two men are mourned as having fallen at the Alamo when in reality they are deserters who later redeem themselves by spying on Confederates. Inane, silly Italian Western.\n\n**1182** _ **Duel at Apache Wells**_ **** Republic, 1957. 70 min. D: Joseph Kane. SC: Bob Williams. With Anna Maria Alberghetti, Ben Cooper, Jim Davis, Harry Shannon, Francis McDonald, Bob Steele, Frank Puglia, Argentina Brunetti, Ian MacDonald, John Dierkes, Ric Roman, Dick Elliott. A man returns home to face the crook who murdered his father and stole his land. Fairly satisfying oater with Jim Davis as the dastardly villain and Bob Steele as his vicious henchman.\n\n**1183** _ **Duel at Diablo**_ **** United Artists, 1966. 103 min. Color. D: Ralph Nelson. SC: Marvin H. Albert and Michael M. Grilikhes. With James Garner, Sidney Poitier, Bibi Andersson, Dennis Weaver, Bill Travers, William Redfield, John Hoyt, John Crawford, John Hubbard, Kevin Coughlin, Jay Ripley, Jeff Cooper, Ralph Bahnsen, Bobby Crawford, Richard Lapp, Dawn Little Sky, Eddie Little Sky, Al Wyatt, Phil Schumacher, Richard Farnsworth, Joe Finnegan, Bill Hart. A diverse group of people travel through the desert with a convoy of munitions as they face the threat of Indian attack. Surprisingly good account of the old plot ploy.\n\n**1184** _ **Duel at Silver Creek**_ **** Universal-International, 1952. 77 min. Color. D: Donald Siegel. SC: Gerald Dryson Adams and Joseph Hoffman. With Audie Murphy, Faith Domergue, Stephen McNally, Susan Cabot, Gerald Mohr, Lee Marvin, Eugene Iglesias, James Anderson, Walter Sande, George Eldredge, Tex Terry, John Carpenter, Harry Harvey, Lee Morgan. Coming to town to gamble, the Silver Kid finds himself teaming with a sheriff to stop a gang of murderous claim jumpers. Fast paced and lots of fun.\n\n**1185** _ **Duel at the Rio Grande**_ **** Teleworld, 1964. 93 min. D: Mario Gaiano. SC: Guido Malatesta, Andre Tabet and Arturo Rigal. With Sean Flynn, Danielle de Metz, Folco Lulli, Armando Calvo, Gaby Andre, Helga Line, Enrique Diosdado, Carlo Tamberlani, Walter Barnes, Mino Doro, Mario Petri, Alfredo Rizzo, Ugo Sasso, Gigi Bonos, Manrico Melchiorre, Elena Barrios. After returning home to Mexico and finding his father murdered, a man leads a band of revolutionaries in stopping a dictator. Mediocre re-working of the Zorro theme with Sean Flynn only average in an attempt to recreate the swashbuckling image of his father, Errol Flynn. Released in Europe as _**Il Segno di Zorro**_ (The Sign of Zorro) and also called _**The Sign of Zorro**_.\n\n_**Duel in Durango**_ see _**Gun Duel in Durango**_\n\n**1186** _ **Duel in Eclipse**_ **** Hispamer\/Prodimex, 1968. 98 min. Color. D: Eugenio Martin and Jose Luis Merino. SC: Enrico Colombo and Giuliana Garavaglia. With Lang Jeffries, Fernando Sancho, Femy Benussi, Carlo Gaddi, Ruben Rojo, Aldo Sambrell, Carlo Simoni, Carlo Gaddi, Giuly Garr, Angel Alvarez, Marisa Paredes. An gunman-astronomer seeks revenge against a gold hungry outlaw and his gang for the vicious murder of his brother. Well made, out of the ordinary, Spanish-Italian Spaghetti Western, released theatrically in Europe as _**Requiem para el Gringo**_ (Requiem for a Gringo).\n\n**1187** _ **Duel in the Sun**_ **** Selznick Releasing, 1946. 138 min. Color. D: King Vidor (and uncredited Otto Brower, William Dieterle, Sidney Franklin, William Cameron Menzies, David O. Selznick and Josef von Sternberg). SC: David O. Selznick and Oliver H.P. Garrett. With Jennifer Jones, Gregory Peck, Joseph Cotten, Lionel Barrymore, Lillian Gish, Walter Huston, Herbert Marshall, Charles Bickford, Joan Tetzel, Harry Carey, Otto Kruger, Sidney Blackmer, Tilly Losch, Scott McKay, Butterfly McQueen, Francis McDonald, Victor Milian, Griff Barnett, Frank Cordell, Dan White, Steve Dunhill, Lane Chandler, Johnny Bond, Lloyd Shaw, Bert Roach, Si Jenks, Hank Worden, Rose Plummer, Guy Wilkerson, Lee Phelps, Al Taylor, Bob McKenzie, Charles Dingle, Thomas Dillon, Orson Welles (narrator). As their senator-land baron father battles the railroad over building track on his land, two brothers fight it out over a half-breed girl. Very bad \"epic\" Western from David O. Selznick, that is still worth a look just to see how big budget producers can bungle a film.\n\n**1188** _ **Duel on the Mississippi**_ **** Columbia, 1955. 72 min. Color. D: William Castle. SC: Gerald Drayson Adams. With Lex Barker, Patricia Medina, Warren Stevens, Craig Stevens, John Dehner, Ian Keith, Chris Alcaide, John Mansfield, Celia Lovsky, Lou Merrill, Mel Welles, Jean Del Val, Baynes Barron, Vince M. Townsend, Jr. In 1820s New Orleans a man becomes a bond servant so his planter father will not go to jail and eventually stops raids on sugar plantations by bayou renegades. Standard costume drama that moves along at a good clip.\n\n_**Duelo a Muerte**_ see _**La Venganza del Lobo Negro**_\n\n**1189** _ **Duelo en el Desierto**_ (Duel in the Desert) **** Radaent Films, 1964. 85 min. D: Arturo Martinez. SC: Raul de Anda. With Rodolfo de Anda, Fanny Cano, Dabogerto Rodriguez, Miguel Arenas, Carlos Lopez Moctezuma, Jose Chavez, Armando Arriola, Sergio Barrios, Emilio Garibay, Hortensia Santovena. When a woman marries an ex-gunman her brother has him put in jail on a trumped up charge and then makes plans to eliminate him. Standard Mexican Western produced by Raul de Anda, Jr.\n\n**1190** _ **Duelo en El Dorado**_ (Duel in El Dorado) **** Cinematografica Calderon S.A., 1969. 87 min. Color. D: Rene Cardona. SC: Ramon Obon and Roberto G. Rivera. With Luis Aguilar, Lola Beltran, Lilia Predo, Emilio Fernandez, German Valdes (Tin Tan), Crox Alvarado, Raul Martinez, Jose Torvay, Roberto Canedo, Lupita Ferrer, Rogelio Gaona, Eleazar Garcia \"Cheleo,\" Ramon Valdes, Yolanda Ponce, Lila Brado, Cuco Sanchez, Roberto G. Rivera. An orphan is fought over by two men, an ex-gunman who adopted him and a rich man who claims to be his father. Average Mexican Western, sequel to _**La Conquista de El Dorado**_ (The Conquest of El Dorado) (1965).\n\n**1191** _ **Dugan of the Badlands**_ **** Monogram, 1931. 66 min. D-SC: Robert North Bradbury. With Bill Cody, Andy Shuford, Blanche Mehaffey, Earl Dwire, Ethan Laidlaw, Julian Rivero, John Elliott. A cowboy adopts a young boy whose father has been killed and together they help a lawman in tracking a crooked deputy. Fair first entry in the \"Bill and Andy\" series.\n\n**1192** _ **The Durango Kid**_ **** Columbia, 1940. 61 min. D: Lambert Hillyer. SC: Paul Franklin. With Charles Starrett, Luana Walters, Kenneth MacDonald, Francis Walker, Forrest Taylor, Melvin Lang, The Sons of the Pioneers (Bob Nolan, Tim Spencer, Lloyd Perryman, Pat Brady, Hugh Farr, Karl Farr), Frank LaRue, Ralph Peters, Jack Rockwell, Marin Sais, Roger Gray, Jack Kirk, Steve Clark, George Russell, John Tyrrell, Silver Tip Baker. A rancher tries to find out who murdered his father, with the killer, a rival cattleman, putting the blame on nesters. Satisfying production that eventually spawned the long running \"Durango Kid\" series headlining Charles Starrett.\n\n**1193** _ **Durango Valley Raiders**_ **** Republic, 1938. 60 min. D: Sam Newfield. SC: George Plympton. With Bob Steele, Louise Stanley, Karl Hackett, Forrest Taylor, Ted Adams, Steve Clark, Horace Murphy, Jack Ingram, Ernie Adams, Budd Buster, Frank Ball. A cowpoke tries to help citizens fighting an outlaw gang and he learns the local sheriff is behind the terrorism. Plenty of action in this nicely done Bob Steele movie.\n\n**1194** _ **Dust**_ **** Lion's Gate, 2001. 127 min. Color. D-SC: Milcho Manchevski. With Joseph Fiennes, David Wehnam, Adrian Lester, Anne Brochet, Nikolina Kujaca, Rosemary Murphy, Vlado Jovanovski, Salaetin Bilal, Vera Farmiga, Matt Ross, Meg Gibson, Tamer Ibrahim, Vladimir Jacey, Vladimir Gjorgiloski, Zora Georgieva, Jordan Simonov, Josif Josifovski, Joe Mosso, Saunra McClain, Nick Andow, Bruce MacVittie, Tom Strauss, Milica Stajanova, Stanko Stoilkov, Petar Mircevski, Mladen Krstevski, Stoja Arev, Pavie Dameski, Randy Duke. A gunman loses the girl he loves to his brother, becomes a mercenary in Greece and a century later his story his old by an old woman to the young thug who tries to rob her. Entertaining but somewhat confusing drama.\n\n**1195** _ **Dust to Dust**_ **** DML, 1994. 91 min. Color. D-SC: Gerald Cain. With Robert Vaughn, Willie Nelson, Shaw Jones, Lisa Cangelosi, Gary Carter, Sascha Biesi, Alfredo Huereca, Russ Marker, Cindy Curtis, Stuart Wasser, George Eads, Michael Roland Williams, Grant James, Dustin Sautter, Angel Arroyo, Omari Miller, Scott Ward, Jeffery Mills, Tricia M. Chaney, Tom Davidson, Bob Peterson, J.P. Schwan, Ryan Wickerham, Chris Morris, Rex Owens, Jon Lacey, Vincent Gaskins, Bill Coldwell, Victoria Parello, Carrie Cain, Jim Henry, Montgomery A. Atterbury, Joshua Inge, Shelley Durham, David Lee, Mike Shanks. In the remote town of Bramble, a former Union officer tries to get an insane woman to sign over her land for a railroad right-of-way and is opposed by a gang of youths, the leader falling in love with his daughter. Mixed up morality play filmed near Austin, Texas, by The Water Hole Gang; Willie Nelson makes only a fleeting appearance as a lawyer.\n\n_**Dynamite and Gold**_ see _**Where the Hell's the Gold**_\n\n**1196** _ **Dynamite Canyon**_ **** Monogram, 1941. 58 min. D: Robert Emmett Tansey. SC: Robert Emmett (Tansey) and Frances Kavanaugh. With Tom Keene, Evelyn Finley, Arkansas Slim Andrews, Sugar Dawn, Stanley Price, Kenne Duncan, Jack Perrin, Gene Alsace, Fred Hoose, Tom London. After an outlaw leader murders two men over a copper deposit, a ranger is assigned to investigate and masquerades as the wanted Trigger Jones in order to infiltrate the gang. Pretty fair Tom Keene entry.\n\n**1197** _ **Dynamite Jim**_ **** Balcazar\/Lux, 1966. 86 min. Color. D: Alfonso Balcazar. SC: Alfonso Balcazar and Jose A. De La Loma. With Luis Davila, Fernando Sancho, Rosalba Neri, Maria Pia Conte, Aldo Sambrell, Charles Sola. During the Civil War an Army colonel tries to bring a gold shipment from Mexico to aid the Union cause but some of his men get greedy. Fairly interesting Spanish made oater.\n\n**1198** _ **Dynamite Joe**_ **** Seven Film\/Hispamer, 1966. 94 min. Color. D: Anthony Dawson (Antonio Margheriti). SC: Maria Del Carmin Martinez. With Rick Van Nutter, Halina Zalewska, Mercedes Caracuel, Renato Baldini, Barta Barry, Aldo Cecconi, Alfonso Rocas, Mario De Grassi, Santiago Rivero. Special agent Dynamite Joe Ford is assigned by a senator to protect government gold shipments from attacking Comancheros. Better-than-average Spaghetti Western.\n\n**1199** _ **Dynamite Pass**_ **** RKO Radio, 1950. 61 min. D: Lew Landers. SC: Norman Houston. With Tim Holt, Richard Martin, Lynne Roberts, Regis Toomey, Robert Shayne, Don C. Harvey, Cleo Moore, John Dehner, Dan Haggerty, Ross Elliott, Denver Pyle, Stuart Randall. Fearing competition, a toll-road owner tries to stop the construction of a new highway. Well done Tim Holt vehicle.\n\n**1200** _ **Dynamite Ranch**_ **** World Wide, 1932. 60 min. D: Forrest Sheldon. SC: Barry Barringer and Forrest Sheldon. With Ken Maynard, Ruth Hiatt, Alan Roscoe, Jack Perrin, Arthur Hoyt, Al Smith, John Beck, George Pierce, Lafe McKee, Martha Mattox, Edmund Cobb, Charles LeMoyne, Cliff Lyons, Kermit Maynard. When a train is robbed during a fake stop a playful cowboy is blamed for the crime and escapes from jail to prove his innocence. Okay Ken Maynard feature.\n\n_**Each Man for Himself**_ see _**The Ruthless Four**_\n\n**1201** _ **The Eagle and the Hawk**_ **** Paramount, 1950. 104 min. Color. D: Lewis R. Foster. SC: Geoffrey Homes and Lewis R. Foster. With John Payne, Rhonda Fleming, Dennis O'Keefe, Thomas Gomez, Fred Clark, Frank Faylen, Eduardo Noriega, Grandon Rhodes, Walter Reed, Margaret Martin. The U.S. government sends two law enforcers to Mexico in the 1860s in an attempt to stop the plot to make Maximilian emperor. Mildly entertaining historical fiction.\n\n**1202** _ **The Eagle's Brood**_ **** Paramount, 1935. 61 min. D: Howard Bretherton. SC: Doris Schroeder and Harrison Jacobs. With William Boyd, James Ellison, William Farnum, Addison Richards, George Hayes, Nana Martinez (Joan Woodbury), Frank Shannon, Dorothy Revier, Paul Fix, John Merton, Al Lydell, George Mari, Juan Torena, Henry Sylvester, Cliff Lyons, Jim Corey, Rube Dalroy. When outlaws kill a young couple and steal their child, Hopalong Cassidy intervenes and goes up against a murderous saloon owner. Top notch second film in the \"Hopalong Cassidy\" series and a portent of more good things to come.\n\n**1203** _ **The Eagle's Claw**_ **** Aywon, 1924. 50 min. D: Charles R. Seeling. SC: Donald L. Buchanan. With Guinn Williams, Kathleen Collins, Lew Meehan, Lafe McKee, William Gunn. A cowboy inherits a gold mine and becomes the target of attacks from an old enemy. Cheap silent effort that will interest Guinn \"Big Boy\" Williams fans.\n\n**1204** _ **Eagle's Wing**_ **** International Pictures, 1979. 111 min. Color. D: Anthony Harvey. SC: John Briley. With Martin Sheen, Sam Waterston, Harvey Keitel, Stephane Audran, John Castle, Caroline Langrishe, Jorge Russo, Manuel Ojeda, Jorge Luke, Pedro Damian, Claudio Brook, Jose Carlos Ruiz, Farnesio de Bernal, Cecilia Camacho, Enrique Lucero, Julio Lucena. After murdering his partner, a cowboy steals a Comanche horse and is chased by a warrior determined to retrieve it as his braves attack a stagecoach, carry off a beautiful young woman and are tracked by a posse of Mexicans. Obscure but fast paced, violent and action filled feature.\n\n**1205** _ **Edge of Eternity**_ **** Columbia, 1959. 81 min. D: Don Siegel. S: Knut Swenson and Richard Collins. With Cornel Wilde, Victoria Shaw, Mickey Shaughnessy, Edgar Buchanan, Rian Garrick, Jack Elam, Dabbs Greer, Alexander Lockwood, Tom Fadden, John Roy, Wendell Holmes. A criminal uses a resort area as a hideout after murdering a mining executive and he is pursed by a deputy sheriff. Well done modern-day Western with an exciting climax in the Grand Canyon.\n\n**1206** _ **80**_ _**Steps to Jonah**_ **** Warner Bros., 1969. 107 min. Color. D: Gerd Oswald. SC: Frederic Louis Fox. With Wayne Newton, Diana Ewing, Jo Van Fleet, Keenan Wynn, R.G. Armstrong, Slim Pickens, Mickey Rooney, Sal Mineo, Brandon Cruz, Teddy Quinn, Susan Mathews, Dennis Cross, James Bacon, Erin Moore, Butch Patrick. On the run from the law, a young man stumbles on a ranch for blind children and it changes his life. Okay drama although singer Wayne Newton is somewhat miscast in the lead role.\n\n**1207** _ **El Cisco**_ **** Filmepoca, 1966. 90 min. Color. D: Sergio Bergonzelli. SC: Sergio Bergonzelli, Aldo Greci and Paolo Lombardo. With William Berger, George Wang, Antonella Murgia, Tom Felleghy, Nino Vingelli, Cristina Gajoni, Renato Chiantoni, Lucye Bomez. El Cisco, a man unjustly accused of a crime with a price on his head, foils a bank robbery attempt by a deputy sheriff in league with Mexican bandits but ends up being blamed for the hold-up. Somewhat involved, but fair Italian oater with a good music score by Bruno Nicolai. Also called _**Cisco**_ and _**The Cisco Kid.**_\n\n**1208** _ **El Condor**_ **** National General, 1970. 98 min. Color. D: John Guillermin. SC: Larry Cohen and Steven Carabatsos. With Lee Van Cleef, Jim Brown, Mariana Hill, Patrick O'Neal, Elisha Cook, Iron Eyes Cody, Imogen Hassall, Gustavo Rojo, Florencio Amarilla, Julio Pena, John Clark. Two men trek to Mexico with plans to steal the gold of Maximilian, which is hidden in the fort at El Condor. Veteran director Andre DeToth produced this foreign filmed oater, which is fair entertainment.\n\n**1209** _ **El Diablo**_ **** Home Box Office Pictures, 1990. 115 min. Color. D: Peter Markle. SC: Tommy Lee Wallace, John Carpenter and Bill Phillips. With Anthony Edwards, Louis Gossett, Jr., John Glover, Joe Pantoliano, Robert Beltran, M.C. Gainey, Miguel Sandoval, Sarah Trigger, Branscombe Richmond, Jim Beaver, Geno Silva, David Dunard, Craig Reay, Nick Young, Don Collier, Luis Contreras, Jesse Doran, Kathleen Erickson, Don Pendergrass, Frank Koppala, Michael Francis Kelly, Wilfredo Hernandez, Todd Fitzpatrick. A tenderfoot teacher teams with a black gunfighter to rescue a young girl, his student, kidnapped by the notorious outlaw El Diablo. Amusing Western comedy made for television.\n\n**1210** _ **El Diablo, El Sainto y El Tonto**_ (The Devil, The Saint and the Fool) **** Cumbre Films, 1987. 92 min. Color. D: Rafael Villasenor Kuri. SC: Adolfo Torres Portillo. With Vicente Fernandez, Sasha Montenegro, Pedro Weber, Carmelita Gonzales, Frank Tostado, Felipe Arriaga, Eulalio Gonzales, Jorge Noble, Martha Ortiz, Luc Maria Rico. Three unalike half-brothers meet to divide their late rancher father's estate with astonishing results. Associate Producer\u2013star Vicente Fernandez plays all three title roles in this zany Mexican Western comedy as well as performing five ranchero songs.\n\n**1211** _ **El Diablo Rides**_ **** Metropolitan, 1939. 55 min. D: Ira Webb. SC: Carl Krusada. With Bob Steele, Claire Rochelle, Carleton Young, Ted Adams, Kit Guard, Robert Walker, Rob Robinson, Hal Carey. A cowboy finds himself in the middle of a range war between cattlemen and sheep herders. Low grade but action filled Bob Steele film.\n\n**1212** _ **El Dorado**_ **** Paramount, 1967. 127 min. Color. D: Howard Hawks. SC: Leigh Brackett. With John Wayne, Robert Mitchum, James Caan, Charlene Holt, Michele Carey, Arthur Hunnicutt, R.G. Armstrong, Edward Asner, Paul Fix, Christopher George, Robert Donner, John Gabriel, Jim Davis, Marina Chane, Anne Newman, Johnny Crawford, Robert Rothwell, Adam Roarke, Chuck Courtney, Bill Henry, Nacho Galindo, Victoria George, John Mitchum. An aging gunfighter helps his sheriff pal in opposing a corrupt land baron. Entertaining reworking of _**Rio Bravo**_ (q.v.), although not up to the previous effort by director Howard Hawks, writer Leigh Brackett and star John Wayne.\n\n**Advertisement for** _**El Dorado**_ **(Paramount, 1967).**\n\n**  \n**\n\n**1213** _ **El Dorado Pass**_ **** Columbia, 1948. 56 min. D: Ray Nazarro. SC: Earle Snell. With Charles Starrett, Smiley Burnette, Elena Verdugo, Steve Darrell, Rory Mallinson, Shorty Thompson and His Saddle Rockin' Rhythm, Ted Mapes, Blackie Whiteford, Stanley Blystone, Harry Vejar, Russell Meeker, Gertrude Chorre. Framed for a crime he did not commit, a cowboy breaks out of jail and becomes the masked Durango Kid, aiding a Mexican rancher and his daughter who have been robbed of the money they planned to use to buy cattle. Passable \"Durango Kid\" series segment. British title: _**Desperate Men**_.\n\n**1214** _ **El Paso**_ **** Paramount, 1948. 101 min. Color. D-SC: Lewis R. Foster. With John Payne, Gail Russell, Sterling Hayden, George \"Gabby\" Hayes, Dick Foran, Henry Hull, Mary Beth Hughes, Eduardo Noriega, H.B. Warner, Catherine Craig, Arthur Space, Bobby Ellis, Peggy McIntyre, Chief Yowlachie, Steven Geray, Lawrence Tibbett, Jr., Pierre Watkin, Gloria Winters, Reed Howes, Lane Chandler, Nacho Galindo, John Hart, Herbert Heywood, Don Haggerty, Jesse Graves, Chief Yowlachie, Dewey Robinson, Lorin Raker, John Merton, Peggy McIntyre, Jack Perrin, Lee \"Lasses\" White, Denver Pyle, Max Wagner, Renata Vanni, Argentina Brunetti, Dan White, Lee Roberts, Irving Bacon, Joe Devlin, Eddie Parks, Ray Hyke, Jack Hendricks, Keith Richards, Tom Smith, Harry Tenbrook. In Texas after the Civil War a lawyer learns the gun is the only way to rid the area of lawlessness. Standard but entertaining big budget action Western.\n\n**1215** _ **The El Paso Kid**_ **** Republic, 1946. 54 min. D: Thomas Carr. SC: Norman Sheldon. With Sunset Carson, Marie Harmon, Robert Filmer, Wheaton Chambers, Zon Murray, John Carpenter, Hank Patterson, Edmund Cobb, Tex Terry, Robert Wilke, Ed Cassidy. After quitting an outlaw gang over a killing, a cowboy is made a sheriff and eventually redeems himself for his past. Although a good rider and fighter, Sunset Carson is a mediocre actor at best and this does not help the proceedings.\n\n**1216** _ **El Paso Stampede**_ **** Republic, 1953. 53 min. D: Harry Keller. SC: Arthur E. Orloff. With Allan \"Rocky\" Lane, Phyllis Coates, Eddy Waller, Stephen Chase, Roy Barcroft, Edward Clark, Tom Monroe, Stanley Andrews, William Tannen, John Hamilton. During the Spanish-American War rustlers hijack cattle intended for the Army and the government sends special agent Rocky Lane to investigate. None too interesting final outing in the \"Famous Westerns\" series with lots of stock footage and re-use of the old chestnut of having cattle hidden behind a waterfall; a picture of Grant Withers is used to portray unseen rustler Jose Delgado.\n\n**1217** _ **El Puro**_ **** Filmar Cinematografica, 1972. 89 min. Color. D: Fabrizio Gianni. SC: Ignacio Iquino, Eduardo Mulargia and Fabrizio Gianni. With Robert Woods, Rosalba Neri, Maurizio Bonuglia, Mario Brega, Mariangela Giordano, Aldo Berti, Attilo Cottesio, Fabrizio Gianni, Gustavo Re. After accepting a reward to bring in an outlaw alive, a cowboy is forced to shoot the wanted man. Another violent Spaghetti Western issued in 1970 in Italy as _**La Taglia e Tua, I'Umo, I'Anamazzo Io**_ and in France as _**El Puro, La Rancon Est Pour Toi**_ (El Puro, the Ransom Is for You); also called _**The Reward's Yours, the Man's Mine**_.\n\n_**El Rancho Grande**_ see _**Rancho Grande**_\n\n**1218** _ **El Topo**_ (The Mole) **** ABKCP\/Producciones Panicas, 1971. 123 min. Color. D-SC: Alexandro Jodorowsky. With Alexandor Jodorowsky, Brontis Jodorowsky, Maria Lorenzio, David Silva, Paula Romo, Jacqueline Luis, Robert John. A mysterious figure rides through the desert and meets a woman who urges him to seek out and kill four sharp-shooting masters. Strange, ambiguous and extremely violent Mexican production that has developed a cult following but will not appeal to the average filmgoer.\n\n**1219** _ **The Electric Horseman**_ **** Universal\/Columbia, 1979. 120 min. Color. D: Sydney Pollack. SC: Robert Garland and Paul Gaer. With Robert Redford, Jane Fonda, Willie Nelson, Valerie Perrine, John Saxon, Nicholas Coster, Allan Arbus, Wilford Brimley, Will Hare, Basil Hoffman, Timothy Scott, James B. Sikking, James Kline, Frank Speiser, Quinn Redeker, Lois Areno, Sarah Harris, Tasha Zemrus, James Novak, Debra L. Maxwell, Michele Heyden, Robin Timm, Patricia Blair, Gary M. Fox, Richard Perlmutter, Carol Eileen Montgomery, Theresa Ann Dent, Perry Sheehan Adair, Sarge Allen, Sylvie Strauss, Richard Knoll, Angelo Giouzelis, Mark Jamison, Brendan Kelly, Sheila B. Wakely, X.V. Kelly, Gary Shermaine, Gary Liddiard. A faded rodeo star heads to his desert hideout after one of his \"deals\" fails to work, is sought out by a TV reporter and the two fall in love. Lackluster feature that is too long to be interesting although Willie Nelson is just fine as the \"hero's\" pal.\n\n_**Emperor of the North**_ see _**Emperor of the North Pole**_\n\n**1220** _ **Emperor of the North Pole**_ **** 20th Century\u2013Fox, 1973. 118 min. Color. D: Robert Aldrich. SC: Christopher Knopf. With Lee Marvin, Ernest Borgnine, Keith Carradine, Charles Tyner, Malcolm Atterbury, Simon Oakland, Harry Caesar, Hal Baylor, Matt Clark, Elisha Cook, Joe Di Reda, Liam Dunn, Diane Dye, Robert Foulk, James Goodwin, Ray Guth, Sid Haig, Karl Lukas, Edward McNally, John Steadman, Vic Tayback, Dave Willock. A hobo carries out a personal vendetta against a conductor who brutally murders men trying to get free rides on his train. Very violent melodrama; Marty Robbins sings the title song, \"A Man and a Train.\" Also called _**Emperor of the North**_.\n\n**1221** _ **Empty Holsters**_ **** Warner Bros., 1937. 62 min. D: B. Reeves Eason. SC: John T. Neville. With Dick Foran, Pat Wathall, Edmund Cobb, Glenn Strange, George Chesebro, J.P. McGowan, Milton Kibbee, Emmett Vogan, Art Mix, Artie Ortego, Earl Dwire, Jack Mower, Ben Corbett, Merrill McCormick, Wilfred Lucas, Neal Hart, Henry Otho, Charles LeMoyne, Tom Brower, Anderson Lawlor. After being released from prison, having been framed on robbery and murder charges, a cowboy plans to clear his name by finding the real culprit. Old story is given plenty of fast action in this retelling.\n\n**1222** _ **Empty Saddles**_ **** Universal, 1936. 62 min. D: Lesley Selander. SC: Frances Guihan. With Buck Jones, Louise Brooks, Harvey Clark, Niles Welch, Gertrude Astor, Frank Campeau, Charles Middleton, Lloyd Ingraham, Claire Rochelle, Robert Adair, Ben Corbett, Earl Askam. Crooks spark a war between cattle ranchers and sheep men and a cowboy tries to stop the feuding. Well mounted Buck Jones vehicle of double interest because silent film siren Louise Brooks is the leading lady.\n\n**1223** _ **En Peligro de Muerte**_ (On Risk of Death) **** Producciones Zacarias S.A., 1962. 90 min. Color. D: Rene Cardona. SC: Alfredo Zacarias and Roberto Gomez Bolanos \"Chespirito.\" With Viruta (Marco Antonio Campos), Capulina (Gaspar Henaine), Tin Tan (German Valdes), Lorena Velazquez, Jorge Russek, Rene Cardona, Jr., Tere Velazquez, Jorge Zamora, Mayte Carol, Manuel Donde, Guillermo Bravo Sosa, Sara Gabriela, Jesus Gomez, Victor Velazquez, Eduardo Lugo, Ramon Valdes. Two bungling prospectors are captured by Indians but use their metal detector and flash camera to make them think they have magical powers before rescuing two beautiful women and taking them to their uncle, a sheriff whose town is besieged by outlaws. Silly Mexican comedy Western.\n\n**1224** _ **The Enchanted Valley**_ **** Eagle Lion, 1948. 72 min. D: Robert Emmett Tansey. SC: Frances Kavanaugh. With Alan Curtis, Anne Gwynne, Charley Grapewin, Donn Gift, Joseph Crehan, Joseph Devlin, Al (Lash) LaRue, John Bleifer, Rocky Camron, Jerry Riggio. A young man's happy existence is interrupted by the arrival of two bandits and their woman companion. Average programmer.\n\n**1225** _ **End of a Gun**_ **** CBS-TV\/20th Century\u2013Fox, 1957. 45 min. D: Lewis Allen. SC: Sam Peckinpah. With Richard Conte, John Barrymore, Jr., Marilyn Erskine, Lyle Bettger, Richard Crane, Alix Talton, Frank Ferguson, Carl Benton Reid, Frank Sully, Michael Landon, Maudie Prickett, Percy Helton, Peter Brocco, John Cliff, Jamie Farr, Mort Mills, Jim Hayward, Richard Collier, John Conte (host). A famous gunman, who wants to lead a peaceful life, is forced into one last showdown. TV version of _**The Gunfighter**_ (q.v.), originally telecast as a segment of \"The 20th Century\u2013Fox Hour\" on CBS-TV on January 9, 1957, and later syndicated as a feature film; Sam Peckinpah adapted the teleplay from William Bowers' screenplay.\n\n**1226** _ **End of the Rope**_ **** NBC-TV, 1957. 54 min. Color. D: Albert McCleery. SC: Sheldon Stark. With John Barrymore, Jr., Susan Oliver, George Peppard, Norma Moore, Parley Baer, John Conte (host). A man finds himself in a Arizona town where the populace wants to lynch him. Telefeature first shown as an episode of \"Matinee Theatre\" (NBC-TV, 1955\u201358) on April 1, 1957.\n\n**1227** _ **End of the Trail**_ **** Columbia, 1932. 60 min. D: D. Ross Lederman. SC: Stuart Anthony. With Tim McCoy, Luana Walters, Wheeler Oakman, Wally Albright, Lafe McKee, Wade Boteler, Chief White Eagle, Henry Hall. A soldier, forced out of the Army after being falsely accused of giving guns to Indians, goes to live with the Arapahos when his adopted son is killed, and eventually thwarts a massacre. Exceedingly well done film not hurt by a tacked-on happy ending; perhaps Tim McCoy's best.\n\n**1228** _ **End of the Trail**_ **** Columbia, 1936. 70 min. D: Erle C. Kenton. SC: Harold Shumate. With Jack Holt, Louise Henry, Guinn Williams, Douglass Dumbrille, George McKay, Gene Morgan, John McGuire, Ed Le Saint, Frank Shannon, Erle C. Kenton, Hank Bell, Art Mix, Blackie Whiteford, Blackjack Ward, Edgar Dearing, Albert J. Smith, Paul Guifoyle, Pat Flaherty, Carl Stockdale, Bob McKenzie, Richard Cramer, Stanley Blystone, Bud Osborne, Frank Moran, James B. Kenton, Frank Ellis, Chuck Hamilton, Olin Francis, Dick Bodkin, John Tyrrell, Hal Price, Eddie Fetherston, Allan Cavan, Ted Mapes, Fred Parker, Charles Brinley, Lee Prather, Cecil Kellogg, Ed Warren, Curley Gibson, Earle Bunn, Walter Merrill, Herbert White. Two friends, who grew up together but are on opposite sides of the law, both love the same girl. Top notch Jack Holt feature with an adult theme; director Erle C. Kenton portrays Theodore Roosevelt.\n\n**1229** _ **Enemy of the Law**_ **** Producers Releasing Corporation, 1945. 59 min. D-SC: Harry Fraser. With Tex Ritter, Dave O'Brien, Guy Wilkerson, Kay Hughes, Jack Ingram, Charles King, Frank Ellis, Kermit Maynard, Henry Hall, Karl Hackett, Ed Cassidy, Ben Corbett, Jack Evans. Three Texas Rangers track outlaws who years before robbed a safe and hid the loot. Dull \"Texas Rangers\" series affair somewhat saved by Tex Ritter singing \"Teach Me to Forget\" and \"You Will Have to Pay.\" Reissued in 16mm as _**Cowboy Reckoning**_.\n\n_**Enter the Devil**_ (1962) see _**The Devil's Partner**_\n**1230** _ **Enter the Devil**_ **** Artists International, 1971. 86 min. Color. D-SC: Frank Q. Dobbs and David Cass. With Josh Bryant, Irene Kelly, David Cass, Carle Benson, Linda Rascoe, John Martin, Nodris Dominque, Wanda Wilson, Ed Geldert, Happy Shahan. An anthropologist looking for cult sub-cultures in the Texas desert comes across a group of devil worshippers. Mediocre horror Western originally called _**Disciples of Death**_.\n\n**1231** _ **Entre las Patas de los Caballos**_ (Between the Legs of a Horse) **** Lagunas Productions, 2000. 90 min. Color. D: Arturo Martinez. SC: Oswaldo Vizcarra. With Rodolfo de Anda, Jose Vargas, Luis Gatica, Fernando Casanova, Maricarmen Resendiz, Julio Aldama, Elizabeth Aburto, Giannina Ruiz, Gustavo Diaz, Marcial \"El Jalisco.\" Two Mexican families have a generations' old feud until the son and daughter from each fall in love. Okay adaptation of \"Romeo and Juliet\" in the Old West of Mexico.\n\n**1232** _ **Epitaph for a Fast Gun**_ **** Jack H. Harris, 1967. 82 min. Color. D: Nick Nostro. SC: Astrain Bada and Ignacio Iquino. With Michael Riva, Diana Garson, Albert Farley, Indio Gonzales, Jack Rocks, Mario Maranzana, Diana Sorel. An aging sheriff teams with a cowboy, who loves the man's daughter, in cleaning up a tough town. Average Italian-Spanish concoction first released by Cineproduzioni Associate\/I.F.I.S.A. as _**Un Dollaro di Fuoco**_ (A Dollar of Fire).\n\n**1233** _ **The Erotic Adventures of Zorro**_ **** Entertainment Ventures, 1972. 102 minutes Color. D: Colonel Robert Freedman. SC: David F. Friedman. With Douglas Frey, Roby Whiting, Penny Boran, Jude Farese, Robert W. Creese, Michelle Simon, Bruce Gibson, Sebastian Gregory, Mike Perratta, Ernie Dominy, Allen Bloomfield, Becky Pearlman, Kathy Hilton, Gerald Broulard, Cory Brandon, David Villa, Fermin Castillo del Muro, Jesus Valez, David F. Friedman. While pretending to be a gay cavalier, Don Diego Vega is really a lusty Zorro who finds a local tyrant and beds all the town's beautiful women. Silly adult \"Zorro\" comedy.\n\n**1234** _ **Escape from Fort Bravo**_ **** Metro-Goldwyn-Mayer, 1953. 98 min. Color. D: John Sturges. SC: Frank Fenton. With William Holden, Eleanor Parker, John Forsythe, William Demarest, William Campbell, John Lupton, Richard Anderson, Polly Bergen, Carl Benton Reid, John Lupton, Glenn Strange, Forrest Lewis, Harry V. Cheshire, Charles Stevens, Howard McNear, Alex Montoya, Fred Graham, Michael Dugan, Richard P. Beedle, William Newell, Phil Rich, Frank Matts, Eloise Hardt, Valerie Vernon. During the Civil War a woman manages to help her Confederate fiance and follow prisoners escape from the Yankees only to be attacked by hostile Indians. A good story and a compact cast make this interesting viewing.\n\n**1235** _ **Escape from Red Rock**_ **** 20th Century\u2013Fox, 1958. 79 min. D-SC: Edward Bernds. With Brian Donlevy, Eilene Janssen, Gary Murray, Jay C. Flippen, William Phipps, Michael (Myron) Healey, Nesdon Booth, Rick Vallin, Dan White, Andre Adoree, Courtland Shepard, Tina Menard, Zon Murray, Ed Hinton, Frosty Royce, Frank Richards, Hank Patterson, Eileen Stevens, Frank Marlowe, Dick Crockett, Sailor Vincent. To save his brother's life a young rancher takes part in a robbery and later he and his girlfriend flee a posse although Indians are on the warpath. Entertaining and action filled oater with good work by Brian Donlevy as a gang leader, Myron Healy as a vicious gang member and Eilene Janssen as the girl.\n\n**1236** _ **Escape in the Desert**_ **** Warner Bros., 1945. 81 min. D: Edward A. Blatt. SC: Thomas Job. With Philip Dorn, Helmut Dantine, Alan Hale, Jean Sullivan, Irene Manning, Samuel S. Hinds, Bill Kennedy, Kurt Kreuger, Rudolph Anders, Hans Schumm, Monte Blue, Alan Bridge, Trevor Bardette, Cliff Clark, George Sherwood, Selmer Jackson, Angela Greene, Victor Kilian, Blaney Lewis, Tom Fadden, Charles Cane, Jack Mower, James Notaro, Oliver Prickett. A flier tries to capture an escaped Nazi he spots in an Arizona desert caf\u00e9. Mediocre war time reworking of _**The Petrified Forest**_ (q.v.).\n\n**1237** _ **Escape to Grizzly Mountain**_ **** Miracle Entertainment\/Emmett\/Furla Film, 2000. 95 min. Color. D: Anthony Delesandro. SC: Boon Collins. With Dan Haggerty, Jan-Michael Vincent, Miko Hughes, Cody McMains, Ellina McCormick, Nik Winterhawk, Cynthia Palmer, John J. Dalesandro, Charlotte Dodds, Miles O'Keefe, William Stallings, Sam Scarber, Gloria Iglesias, Bobbie Thomas, Steven Erdek, Jay Tavare, Dennis Fimple, Lora Lyn Peterson, Colin Malone, Shannon Welles, Jason De Hoyos, Tracie Amico, Charity Nicole James, Alexandra Shraub, Anat Schraub, Larry Layton, Michael Fino, Haley Jackson Sawyers. Trying to help an abused circus bear cub, a young boy locates a cave where he transported back a century in time and meets a mountain man who agrees to assist him. Poor attempt to produce a \"Grizzly Adams\" look-a-like, even having Dan Haggerty as a mountain man, with a fantasy theme.\n\n_**Escondido**_ see _**A Minute to Pray, a Second to Die**_\n\n**1238** _ **Escort West**_ **** United Artists, 1959. 75 min. D: Francis D. Lyon. SC: Leo Gordon and Fred Hartsook. With Victor Mature, Elaine Stewart, Faith Domergue, Reba Waters, Noah Beery, Jr., Leo Gordon, Rex Ingram, John Hubbard, Harry Carey, Jr., Slim Pickens, Roy Barcroft, William Ching, Ken Curtis, X Brands, Chuck Hayward, Charles Soldani, Claire DuBrey, Syd Saylor. An ex\u2013Confederate and his small daughter head West at the end of the Civil War, are snubbed by a Union wagon train but later find its two female survivors after an Indian attack. Competent but only average, sagebrush yarn.\n\n**1239** _ **Eureka Stockade**_ **** British-Path\u00e9, 1949. 103 min. D-SC: Harry Watt. With Chips Rafferty, Gordon Jackson, Peter Illing, Peter Finch, Jack Lambert, Ralph Truman, Sydney Loder, John Fernside, Grant Taylor, Jane Barrett, Mary Ward, Betty Ross, Leigh O'Malley, Nick Yardley, Marshal Crosby, Jean Blue, Nigel Lovell, Paul Delmar, Charles Tasman, John Wiltshire, Alexander Cann, Rex Dawe, John Fegas, Al Thomas, Ron Whelan, John Cazabon, John Clark, Clement Maloney. In 1853 Australia a miner fights for the right to hunt for gold., the events leading to the designing of the nation's first flag. Exciting and well produced frontier melodrama from producer J. Arthur Rank; also called _**Massacre Hill**_.\n\n**1240** _ **Everyboy's Dancin'**_ Lippert, 1950. 67 min. D: William Berke. SC: Bob Nunes and Spade Cooley. With Spade Cooley, Dick (Richard) Lane, Hal Derwin, James Millican, Lyle Talbot, Michael Whalen, Sid Melton, The Sons of the Pioneers (Lloyd Perryman, Ken Curtis, Shug Fisher, Hugh Farr, Karl Farr, Tommy Doss), Roddy McDowall, Adele Jergens, James Ellison, Russell Hayden, Barbara Woodell, Ginny Jackson, Tex Cromer, Bobby Hyatt, Chuy Reyes Orchestra, Les Anderson, Fred Kelsey, Dorothy Lloyd, Bobby Hyatt, George Meader, Dan Rense, The Flying Taylors, The Great Velardi, The Medians, Virginia MacPherson. Crooks are out to get a ballroom owner's business and several acts come to his rescue. Cheaply made but flavorful Western musical co-scripted by the star, western swing band leader Spade Cooley.\n\n**1241** _ **Everyman's Law**_ **** Supreme, 1936. 61 min. D: Albert Ray. SC: Earle Snell. With Johnny Mack Brown, Beth Marion, Frank Campeau, Roger Gray, Lloyd Ingraham, John Beck, Horace Murphy, Richard Alexander, Slim Whitaker, Ed Cassidy, Jim Corey, George Morrell, Herman Hack, Francis Walker, Art Dillard, Tex Palmer, Jack Evans. A trio of lawmen pretend to be hired guns to get the goods on a rancher harassing homesteaders. A good entry in Johnny Mack Brown's Supreme series for producer A.W. Hackel.\n\n**1242** _ **Everything Happens to Me**_ **** Tobis Filmkunst, 1980. 99 min. Color. D: Michele Lupo. SC: Marcello Fondato and Francesco Scardamaglia. With Bud Spencer, Cary Guffey, Robert Hundar (Claudio Undari), Ferruccio Amendola, John Bartha, Carlo Reali, Giancarlo Biastianoni, Giovanni Cianfrigilia, Ottaviano Dell'Acqua, Paola Figlia, Lorenzo Fineschi, Clayton Landey, Amedeo Lerurini, Vincenzo Maggio, Lawrence Montague, Riccardo Pizzuti, Larry Quackenbush, Sergio Smacchi, Marco Stefanelli. A Western sheriff and his young alien pal are helped by a crook when the military kidnaps the boy. Fun, fast paced sci-fi Western comedy sequel to _**The Sheriff and the Satellite Kid**_ (q.v.), produced in West Germany as _**Chissa Perche...Capitano Tutte a Me**_.\n\n**1243** _ **Evil Roy Slade**_ **** NBC-TV\/Universal, 1972. 100 min. Color. D: Jerry Paris. SC: Garry Marshall and Jerry Belson. With John Astin, Edie Adams, Dick Shawn, Milton Berle, Pamela Austin, Mickey Rooney, Dom DeLuise, Henry Gibson, Arthur Batanides, Larry Hankin, Milton Frome, Luana Anders, Robert Liberman, Connie Sawyer, Pat Morita, Leonard Barr. A notorious outlaw tries to mend his ways after falling in love with an innocent school teacher but he is trailed by a sheriff out to capture him. Poorly conceived and executed Western comedy.\n\n**1244** _ **Extreme Prejudice**_ **** Tri Star Pictures, 1987. 104 min. Color. D: Walter Hill. SC: Deric Washburn and Harry Kleiner. With Nick Nolte, Powers Boothe, Michael Ironside, Maria Conchita Alonso, Rip Torn, Clancy Brown, William Forsythe, Matt Mulhern, Larry B. Scott, Dan Tullis, Jr., John Dennis Johnston, Luis Contreras, Carlos Cervantes, Tommy \"Tiny\" Lister, Marco Rodriguez, James Lashly, Tony Frank, Mickey Jones, Kent Lipham, Sam Gauny, Gil Reyes, Rick Garcia, Richard Duran, Larry Duran, Christina Garcia, Charles Lewis, Humberto De La Torre, Erin Bowen, Thomas Rosales, Jr. Once childhood friends, a Texas Ranger and an drug pusher are at odds when the former's girlfriend tries to protect the latter's business. Fairly good modern-day Western.\n\n**1245** _ **An Eye for an Eye**_ **** Embassy, 1966. 106 min. Color. D: Michael Moore. SC: Bing Russell and Sumner Williams. With Robert Lansing, Pat(rick) Wayne, Slim Pickens, Gloria Talbott, Paul Fix, Strother Martin, Henry Wills, Jerry Gatlin, Rance Howard, Clint Howard, Herman Hack, George Sowards, William Dooley, John Dillon, Walt Ryerson, Dean Spencer, Marshall George, Kathleen King, Art Sasser, Tiny Wells, James Mizell, Mary Mizell, Jack White, Joe Miller. An ex\u2013bounty hunter who is out to get the hoodlums who murdered his wife and son enlists the aid of a man to help him although both are physically handicapped. An interesting plot and good photography add some life to this feature. Also called _**Talion**_.\n\n**1246** _ **Eyes of Texas**_ **** Republic, 1948. 70 min. Color. D: William Witney. SC: Sloan Nibley. With Roy Rogers, Lynne Roberts, Andy Devine, Bob Nolan and The Sons of the Pioneers (Tim Spencer, Lloyd Perryman, Pat Brady, Hugh Farr, Karl Farr), Nana Bryant, Roy Barcroft, Danny Morton, Francis Ford, Stanley Blystone, Pasquale Perry, Bob Reeves. A rancher turns his spread into a camp for war orphans but outlaws are after the land. Okay Roy Rogers feature; fast on action and nice use of color.\n\n**1247** _ **The Fabulous Texan**_ **** Republic, 1947. 96 min. D: Edward Ludwig. SC: Lawrence Hazard and Horace McCoy. With William Elliott, John Carroll, Catherine McLeod, Andy Devine, Albert Dekker, Jim Davis, Ruth Donnelly, Russell Simpson, James Brown, George Beban, Tommy Kelly, Johnny Sands, Harry Davenport, John Miles, Robert Coleman, Robert Barrat, Douglass Dumbrille, Reed Hadley, Roy Barcroft, Frank Ferguson, Glenn Strange, Selmer Jackson, Harry V. Cheshire, Harry Woods, Karl Hackett, John Hamilton, Pierre Watkin, Ed Cassidy, Tristram Coffin, Stanley Andrews, Olin Howlin, Kenneth MacDonald, Jack Ingram, Ted Mapes, Pierce Lyden, Al Ferguson, Ethan Laidlaw, Ray Teal, Franklyn Farnum, Ivan Parry, Craig Reynolds, Richard Foote, William Forrest, George Lloyd, George Eldredge, Crane Whitley, Helen Brown, Regina Wallace, Douglas Wood, Russell Hicks, Wade Crosby, Eddie Acuff, Dick Elliott, Ralph Dunn, Mickey Simpson, Edythe Elliott, Tom Chatterton, Nolan Leary, Howard Mitchell, Harland Tucker, Pat Flaherty, Jerry Jerome, Charles Morton, Sarah Selby, Paul Scardon, Frank Austin. When carpetbaggers take over Texas after the Civil War, a man is forced to become a bandit in order to protect himself and his rights. Very well done William Elliott feature with a top notch cast of character actors.\n\n**1248** _ **Face of a Fugitive**_ **** Columbia, 1959. 81 min. Color. D: Paul Wendkos. SC: David T. Chantler and Daniel B. Ullman. With Fred MacMurray, Lin McCarthy, Dorothy Green, Alan Baxter, Myrna Fahey, James Coburn, Francis De Sales, Gina Gillespie, Paul E. Burns, Robert \"Buzz\" Henry, James Gavin, Hal K. Dawson, Harrison Lewis, Ron Hayes, John Milford, Rankin Mansfield, Stanley Farrar. Falsely accused of murder, a man takes a new identity in another town but finds he cannot shake his past. Fairly entertaining oater with Fred MacMurray good in the lead.\n\n**1249** _ **Face to Face**_ **** RKO Radio, 1952. 92 min. D: Bretaigne Windust. SC: James Agee. With Robert Preston, Marjorie Steele, Minor Watson, Dan Seymour, Olive Carey, James Agee. A man brings his new bride to a small Western town where they plan to settle down. Dull adaptation of the Stephen Crane story \"The Bride Comes to Yellow Sky\" makes up one-half of this feature film, the other portion being James Mason in Joseph Conrad's \"The Secret Sharer.\"\n\n**1250** _ **Face to the Wind**_ **** Warner Bros., 1974. 93 min. Color. D: William A. Graham. SC: David Markson. With Cliff Potts, Xochitl, Harry Dean Stanton, Don Wilbanks, Woodrow Chambliss, James Gammon, Roy Jenson, William Carstens, Richard Breeding. When a young drifter falls in love with an Indian maiden the two find themselves the object of hate and violence. Obscure and violent melodrama first issued in 1972 as _**Cry for Me, Billy**_. Alternate titles: _**Apache Massacre**_ , _**Count Your Bullets**_ and _**The Long Tomorrow**_.\n\n**1251** _ **Fade-In**_ **** Paramount, 1968. 93 min. Color. D: Allen Smithee (Jud Taylor). SC: Jerry Ludwig and Mart Crowley. With Burt Reynolds, Barbara Loden, Patricia Casey, Noam Pitlik, James Hampton, Joseph Perry, Lawrence Heller, Wage Tucker, Sally Kirkland, George Savalas, Jason Heller, Jud Taylor. During the filming of the movie _**Blue**_ (q.v.) a cowboy working as an extra falls in love with an attractive film editor. This production received no official release and was made simultaneously with _**Blue**_ by co-producer Silvio Marizzano, who directed the former film; nothing special but passable.\n\n**1252** _ **Fair Warning**_ **** Fox, 1931. 74 min. D: Alfred Werker. SC: Ernest Pascal. With George O'Brien, Louise Huntington, Mitchell Harris, George Brent, Nat Pendleton, Willard Robertson, Ernie Adams, John Sheehan, Erwin Connelly, Alphonse Ethier. A cowboy tries to prove two men were responsible for the robbery of a saloon. Entertaining George O'Brien \"B plus\" opus.\n\n**1253** _ **The Falcon Out West**_ **** RKO Radio, 1944. 64 min. D: William Clemens. SC: Billy Jones and Morton Grant. With Tom Conway, Carole Gallagher, Barbara Hale, Joan Barclay, Cliff Clark, Minor Watson, Don Douglas, Edward Gargan, Lyle Talbot, Lee Trent, Perc Launders, Wheaton Chambers, Chief Thundercloud, Robert Anderson, Edmund Glover, Rosemary LaPlanche, Elaine Riley, Shirley O'Hara, Patti Brill, Bert Roach, Norman Willis, Kernan Cripps, Slim Whitaker, Bill Nestell, Tom Burton, Steve Winston, Mary Halsey, Daun Kennedy, Chef Milani, Michael St. Angel, Eddie Clark, Joe Cody, Zedra Conde, Norman Mayes. After a wealthy Texas rancher is murdered in New York City, detective Tom Lawrence, alias the Falcon, travels to the victim's ranch in order to catch the killer. Nicely done entry in the \"Falcon\" series with a good Western flavor.\n\n**1254** _ **False Colors**_ **** United Artists, 1943. 65 min. D: George Archainbaud. SC: Bennett Cohen. With William Boyd, Andy Clyde, Jimmy Rogers, Claudia Drake, Douglass Dumbrille, Robert Mitchum, Tom Seidel, Glenn Strange, Pierce Lyden, Roy Barcroft, Sam Flint, Earle Hodgins, Elmer Jerome, Tom London, Dan White, George Morrell, Bob Burns, Glen Walters, Franklyn Farnum, Denver Dixon, Jack Montgomery, Frank O'Connor. A Bar 20 wrangler is killed soon after inheriting a ranch and Hoppy agrees to look after the place, along with the dead man's sister, but a crooked banker behind the killing is out to get the spread for himself. Excellent photography (by Russell Harlan), a good plot and plenty of action make this a \"Hopalong Cassidy\" winner.\n\n_**False Hero**_ see _**Roaring Rangers**_\n\n**1255** _ **False Paradise**_ **** United Artists, 1948. 59 min. D: George Archainbaud. SC: Harrison Jacobs and Doris Schroeder. With William Boyd, Andy Clyde, Rand Brooks, Elaine Riley, Joel Friedkin, Cliff Clark, Kenneth MacDonald, Don Haggerty, Richard Alexander, William Norton Bailey, Zon Murray, George Eldredge. Hopalong Cassidy and his pals California Carlson and Lucky Jenkins try to help ranchers whose silver rich lands are being sought by a crooked banker who holds their mortgages. Penultimate film in the \"Hopalong Cassidy\" series and a long way from the best.\n\n**1256** _ **Family Honeymoon**_ **** Universal-International, 1948. 90 min. D: Claude Binyon. SC: Dan Lussier. With Claudette Colbert, Fred MacMurray, Rita Johnson, William Daniels, Gigi Perreau, Jimmy Hunt, Peter Miles, Lillian Bronson, Hattie McDaniel, Chill Wills, Catherine Doucet, Paul Harvey, Irving Bacon, Chick Chandler, Frank Jenks, Wally Brown, Anne Nagel, Fay Baker, Lois Austin, Sarah Edwards, Beatrice Roberts, Minerva Urecal, Nancy Evans, John Gallaudet, Wilton Graff, Edmund Cobb, Snub Pollard, Frank Orth, Holmes Herbert, Syd Saylor, Frank MacGregor, Constance Purdy, O.Z. Whitehead, William Bailey, Heinie Conklin, Tom Chatterton, Joel Fluellen, Harold Goodwin, Smoki Whitfield, Nick Thompson, Lois Hall, Harry Hayden, Lorin Raker, John O'Connor, Denise Gray, Herbert Heywood, Almira Sessions, Edward C. Short, Jay Silverheels, Bess Flowers. Newlyweds find their Grand Canyon honeymoon trip sabotaged by the widowed bride's pesky children and the professor husband's vengeful ex-girlfriend. Erose outing fails to follow-up the successful teaming of Claudette Colbert and Fred MacMurray in _**The Egg and I**_ (1947).\n\n**1257** _ **Fancy Pants**_ **** Paramount, 1950. 92 min. Color. D: George Marshall. SC: Edmund Hartman and Robert O'Brien. With Bob Hope, Lucille Ball, Bruce Cabot, Lea Penman, Hugh French, Eric Blore, Joseph Vitale, John Alexander, Norma Varden, Virginia Kelley, Colin Keith-Johnston, Joe Wong, Chester Conklin, Robert Kortman, Ray Bennett, Almira Sessions, Percy Helton, Ida Moore, Oliver Blake, Ethel Wales, Major Sam Harris, Hank Bell, Olaf Hytten, Edgar Dearing, Jimmie Dundee, Howard Petrie, Robin Hughes, Hope Sansberry, Charles Cooley, Mira McKinney, Howard Mitchell. A high class British butler is hired by a newly rich Western woman to bring culture to her community. Okay remake of _**Ruggles of Red Gap**_ (q.v.) with the star billed as \"Mr. Robert Hope.\"\n\n**1258** _ **Fangs of Fate**_ **** Chesterfiled, 1925. 60 min. D-SC: Horace B. Carpenter. With Bill Patton, Dorothy Donald, Ivor McFadden, Beatrice Allen, William Bertram, Merrill McCormick, Tex Starr, Carl Silvera. An outlaw goes straight for the sake of a girl but when his old gang refuses to quite he becomes a deputy sheriff to capture them. Fairly good silent horse opera, although Bill Patton is a pallid hero.\n\n**1259** _ **Fangs of the Arctic**_ **** Monogram, 1953. 62 min. D: Rex Bailey. SC: Bill Raynor and Warren Douglas. With Kirby Grant, Lorna Hansen, Warren Douglas, Leonard Penn, Richard Avonde, Robert Sherman, John Close, Roy Gordon, Phil Tead, Kit Carson, Chinook (dog). A Royal Canadian Mountie and his loyal husky dog are on the trail of crooks engaged in illegal trapping. Passable production of the James Oliver Curwood story, made on the cheap.\n\n**1260** _ **Fangs of the Wild**_ **** Astor\/Metropolitan, 1941. 55 min. D: Raymond K. Johnson. SC: R.D. Persall. With Rin Tin Tin, Jr., Dennis Moore, Luana Walters, Tom London, Mae Busch, Theodore (Ted) Adams, George Chesebro, James (Jimmy) Aubrey, Bud Osborne, George Morrell, Martin Spellman. A federal investigator and his trusty dog try to learn who is behind the thefts of silver foxes from breeding ranches. Faced paced and somewhat scenic poverty row action film.\n\n**1261** _ **Fangs of the Wild**_ **** Lippert, 1954. 71 min. D: William F. Claxton. SC: Orville Hampton and William F. Claxton. With Charles Chaplin, Jr., Onslow Stevens, Margia Dean, Freddie Ridgeway, Phil Tead, Robert Stevenson, Buck (dog). A boy witnesses a murder in the north woods but cannot convince his father of what he saw and becomes the target of the killer. Cheaply made but visually attractive and quite entertaining \"B\" drama. TV title: _**Follow the Hunter**_.\n\n**1262** _ **Far and Far Away**_ **** Universal, 1992. 140 min. Color. D: Ron Howard. SC: Bob Dolman. With Tom Cruise, Nicole Kidman, Thomas Gibson, Robert Prosky, Barbara Babcock, Cyril Cusack, Eileen Pollock, Colm Meaney, Douglas Gillison, Michelle Johnson, Wayne Grace, Niall Toibin, Barry McGovern, Gary Lee Davis, Jared Harris, Steven O'Donnell, Wesley Murphy, Derry Power, Noel O'Donovan, Macdara O'Fatharta, Brendan Ellis, Clint Howard, Jeffrey Andrews, Judith McIntyre, Rynagh O'Grady, Frank Coughlan, Hoke Howell, William Preston, Rance Howard, Ian Elliot, Bob Dolman, Phillip V. Caruso, Mark Wheeler, Brendan Cauldwell, Jimmy Keogh. In the early 1890s a young couple flee Ireland and migrate to Oklahoma where they try to start a new life together only to face hardships and the threat of the girl's parents taking her home. High class Western drama, a box office winner.\n\n**1263** _ **The Far Country**_ **** Universal-International, 1955. 96 min. Color. D: Anthony Mann. SC: Borden Chase. With James Stewart, Ruth Roman, Corinne Calvet, Walter Brennan, John McIntire, Jay C. Flippen, Harry Morgan, Steve Brodie, Connie Gilchrist, Robert Wilke, Chubby Johnson, Royal Dano, Jack Elam, Kathleen Freeman, Guy Wilkerson, John Doucette, Eddy Waller, Eugene Borden, Robert Foulk, Paul Bryar, Edwin (Eddie) Parker, Stuart Randall, Terry Frost, Robert Bice, Marjorie Stapp, Don C. Harvey, Ted Mapes, Dick Dickinson, Gene Holland, Damian O'Flynn, Dick Taylor, John Macklin, Carl Harbaugh. A loner and his pal take their cattle herd by boat to Alaska and find lots of trouble in the mining camps. Big budget, very enjoyable oater.\n\n**1264** _ **The Far Frontier**_ **** Republic, 1948. 67 min. Color. D: William Witney. SC: Sloan Nibley. With Roy Rogers, Gail Davis, Andy Devine, Francis Ford, Roy Barcroft, Clayton Moore, Foy Willing and The Riders of the Purple Sage, Lane Bradford, Edmund Cobb, Holly Bane, Clarence Straight, Tom London, Anthony Caruso, Robert Strange, John Bagni, Emmett Lynn, Stanley Blystone, Keith Richards, Jack O'Shea, Robert Wood. Two men smuggle gangsters who have been deported back into the country and plan to buy out a rancher but are opposed by Roy Rogers and sidekick Cookie Bullfincher. Pretty good action feature in which the villains and supporting cast dominate the proceedings in deference to the star.\n\n**1265** _ **The Far Horizons**_ **** Paramount, 1955. 108 min. Color. D: Rudolph Mate. SC: Winston Miller and Edmund H. North. With Fred MacMurray, Charlton Heston, Donna Reed, Barbara Hale, William Demarest, Alan Reed, Eduardo Noriega, Larry Pennell, Argentina Brunetti, Ralph Moody, Herbert Heyes, Lester Matthews, Helen Wallace, Walter Reed, Voltaire Perkins, Joe Canutt. The story of the Meriwether Lewis-William Clark expedition into the recently purchased Louisiana Territory. Colorful but highly fictional account of the famous trek with Donna Reed badly miscast as Sacajawea.\n\n**1266** _ **The Far Out West**_ **** Universal, 1967. 87 min. Color. D: Joe Connelly. SC: George Tibbles. With Ann Sheridan, Ruth McDevitt, Douglas V. Fowley, Gary Vinson, Carole Wells, Robert Lowery, Morgan Woodward, Lon Chaney, Jay Silverheels, Alex Henteloff, Stanley Adams, Lee Patrick, Charles Meredith, Gil Lamb, Quinn O'Hara, Fred Williams, George Murdock, Bill Oberlin, Willis Bouchey. A gun-toting frontier family is at odds with a greedy saloon owner and his hired gunman. Amusing telefeature sewn together from episodes of the series \"Pistols 'n Petticoats\" (CBS-T, 1967\u201368).\n\n**1267** _ **The Far Side of Jericho**_ **** First Look International, 2006. 99 min. Color. D: Tim Hunter. SC: Ron Sullivan and James Crumley. With Suzanne Andrews, Judith Burnett, Lissa Negrin, Patrick Bergin, Lawrence Pressman, James Gammon, John Diehl, C. Thomas Howell, Jason Connery, Bill Doyle, Zachary Ray Sherman, Tim DeKay, Costance Forslund, Vanessa Zima, John Erison, Christian Aubert, Carlos Cervantes, Steve Cromier, Jack Burning, Boots Southerland, Matt Langseth, Oliver Page, James Tarwater, Debrianna Mansini, John David Garfield, Rio Alexander, Dale Malley, Peter Sherayko, Wendi Andres, Ricardo Andres, Brent Lambert, James Blackburn. Vigilantes, bad men and strange Indians chase the widows of hanged outlaw brothers because they may know the whereabouts of hidden loot. Nicely photographed but otherwise poor horror Western.\n\n**1268** _ **Fargo**_ **** Monogram, 1952. 69 min. D: Lewis D. Collins. SC: Joseph Poland and Jack DeWitt. With Bill Elliott, Phyllis Coates, Myron Healey, Fuzzy Knight, Arthur Space, Robert Wilke, Jack Ingram, Terry Frost, Robert Bray, Tim Ryan, Florence Lake, Stanley Andrews, Richard Reeves, Gene Roth, House Peters, Jr., Bud Osborne, Denver Pyle, Stanley Price. After his brother is murdered by a cattle baron, a man returns to North Dakota and helps small ranchers fence off the range, starting a conflict. Nicely done tale of barbed wire being introduced in the West.\n\n**1269** _ **Fargo Express**_ **** World Wide\/Fox, 1933. 61 min. D: Alan James. SC: Alan James and Earle Snell. With Ken Maynard, Helen Mack, Paul Fix, William Desmond, Roy Stewart, Jack Rockwell, Claude Payton, Joe Rickson, Hank Bell, Bud McClure, Charles King, Ben Corbett, Pat Harmon, Blackjack Ward, Buck Bucko. A cowboy tries to help a young man who robbed an express shipment because he loves the bandit's sister. Pretty good Ken Maynard vehicle.\n\n**1270** _ **The Fargo Kid**_ **** RKO Radio, 1940. 63 min. D: Edward Killy. SC: W.C. Tuttle. With Tim Holt, Jane Drummond, Ray Whitley, Emmett Lynn, Cy Kendall, Ernie Adams, Paul Fix, Paul Scardon, Glenn Strange, Mary MacLaren, Dick Hogan, Carl Stockdale, Harry Harvey, Lee Phelps, Betty McLaghlin (Sheila Ryan), Ezra Paulette, Ken Card, Charlie Quirk. A cowboy is mistaken for an outlaw and two crooked businessmen try to hire him to kill a man so they can get ore rich land from the intended victim's widow. Well done remake of _**The Cheyenne Kid**_ (q.v.).\n\n**1271** _ **Fast Bullets**_ **** Reliable, 1936. 59 min. D: Henri Samuels (Harry S. Webb). SC: Carl Krusada and Rose Gordon. With Tom Tyler, Rex Lease, Margaret Nearing, Alan Bridge, William Gould, Robert Walker, Slim Whitaker, Jimmy Aubrey, Nelson McDowell, Lew Meehan, George Chesebro, Charles King, Frank Ellis. A government ranger is after smugglers and enlists the help of a gang member whose pretty sister is the object of the head man's affections. Pretty low grade Tom Tyler effort.\n\n**1272** _ **Fast on the Draw**_ **** Lippert, 1950. 57 min. D: Thomas Carr. SC: Maurice Tombragel and Ron Ormond. With James Ellison, Russell Hayden, Raymond Hatton, Fuzzy Knight, Betty (Julie) Adams, Tom Tyler, George J. Lewis, John Cason, Dennis Moore, Judith Webster, Bud Osborne, Helen Gibson, Stanley Price, Ray Jones, I. Stanford Jolley, Cliff Taylor, Jimmie Martin, Carl Mathews, Scoop Martin, Bud Hooker, Joe Phillips, Roy Butler, George Plues. Texas Rangers are after a dishonest landowner and one of them poses as an outlaw to get the goods on him. Tacky production wasting a good cast; last release in the \"Irish Cowboys\" series and a remake of _**Branded a Coward**_ (q.v.). TV title: _**Sudden Death**_.\n\n**1273** _ **The Fastest Guitar Alive**_ **** Metro-Goldwyn-Mayer, 1967. 88 min. Color. D: Michael Moore. SC: Robert E. Kent. With Roy Orbison, Sammy Jackson, Maggie Pierce, Joan Freeman, Lyle Bettger, John Doucette, Patricia Donahue, Ben Cooper, Douglas Kennedy, Len Hendry, Iron Eyes Cody, Victoria Carroll, Maria Korda. At the end of the Civil War, Confederate spies steal gold from the U.S. mint in San Francisco and when they learn the conflict is over they have to replace it without being detected. Sam Katzman cheapie made to exploit the popularity of singer Roy Orbison who sings plenty of songs to please his fans.\n\n**1274** _ **The Fastest Gun Alive**_ **** Metro-Goldwyn-Mayer, 1956. 89 min. D: Russell Rouse. SC: Frank D. Gilroy and Russell Rouse. With Glenn Ford, Jeanne Crain, Broderick Crawford, Russ Tamblyn, Allyn Joslyn, Leif Erickson, John Dehner, Noah Beery, Jr., J.M. Kerrigan, Rhys Williams, Virginia Gregg, Chubby Johnson, John Doucette, William Phillips, Paul Birch, Dub Taylor, Addison Richards, Earle Hodgins, Glenn Strange, Kenneth MacDonald, Walter Baldwin, John Dierkes, Louis Jean Heydt, Kermit Maynard, Bud Osborne, Michael Dugan, Ray Jones. A peaceful storekeeper, once a famous gunfighter, is forced into a showdown with a bad man who threatens to destroy the town. The old saw about the ex-gunman trying to live down his past is nicely retold in this engaging feature.\n\n_**Fasthand**_ see _**Fasthand Is Still My Name**_\n\n**1275** _ **Fasthand Is Still My Name**_ **** France-Inter Cinema\/Jugendfilm-Verleih, 1973. 85 min. Color. D: Frank Bronston (Mario Bianchi). SC: Vittorio Salerno, Eduardo Manazanos Brochero and Alberto Cardone. With Alan Steel, William Berger, Frank Brana, Fernando Bilboa, Gill Rolland, Cecline Bessy, Francisco Sanz, Welma Truccolo, Ettore Ribotta, Sergio Dolfin, Stefano Oppedisano, Francesco D'Adda. Two years after Indians save him following torture by a band of Confederate renegades, an Army captain returns for revenge. Better than average Spaghetti Western dominated by William Berger as the sadist rebel leader. Released in Italy as _**Lo Chiamavano Requiscat Fasthand**_ (They Call Him Rest in Peace Fasthand) and in Italy as _**Mano Rapida**_ (Fast Hand).\n\n_**Father Kino, Padre on Horseback**_ see _**Mission to Glory: A True Story**_\n\n_**The Father Kino Story**_ see _**Mission to Glory: A True Story**_\n\n**1276** _ **The Fearless Rider**_ **** Universal, 1928. 51 min. D: Edgar Lewis. SC: Basil Dickey and Gardner Bradford. With Fred Humes, Barbara Worth, Ben Corbett, Pee Wee Holmes, Buck Connors, William Steele, Al Taylor. A cowboy and his foreman become suspicious after a doctor suggests to a young woman that her miner father go to a hospital although he was not hurt in a cave-in. Average silent Western that gives viewers a chance to see Fred Humes in a starring role.\n\n**1277** _ **Female Artillery**_ **** ABC-TV\/Universal, 1973. 73 min. Color. D: Marvin Chomsky. SC: Bud Freeman. With Dennis Weaver, Ida Lupino, Sally Ann Howes, Linda Evans, Lee Harcourt Montgomery, Albert Salmi, Nina Foch, Anna Navarro, Charles Dierkop, Robert Sorrells, Bobby Eilbacher. An outlaw steals gold from another gang and hides it in a wagon train consisting of women who find the loot and blackmail him into taking them to a fort. None too amusing TV Western comedy wasting a good cast.\n\n**1278** _ **The Female Bunch**_ **** Dalia, 1971. 86 min. D: Al Adamson and John Cardos. SC: Jale Lockwood and Brent Nimrod. With Russ Tamblyn, Jenifer Bishop, Lon Chaney, Nesa Renet, Geoffrey Land, Regina Carrol, Don Epperson, John Cardos, Albert Cole, A'Lesha Lee, Jackie Taylor, Leslie MacRae, William Bonner, Bobby Clark. A gang of hell raising young women work with a former movie stuntman in smuggling drugs over the U.S.-Mexican border. Violent, tacky modern-day oater of interest because it was Lon Chaney's final film and he gives a touching performance; filmed in 1969 as _**A Time to Run**_.\n\n**1279** _ **Fence Riders**_ **** Monogram, 1950. 57 min. D: Wallace Fox. SC: Eliot Gibbons. With Whip Wilson, Andy Clyde, Reno Browne, Riley Hill, Myron Healey, Ed Cassidy, Terry Frost, Frank McCarroll, George DeNormand, Holly Bane, John Merton, Buck Bailey. A cowboy and his pal help a pretty rancher whose cattle are being rustled. Okay Whip Wilson vehicle.\n\n**1280** _ **The Ferocious Pal**_ **** Principal, 1934. 55 min. D: Spencer Gordon Bennet. SC: Joe Roach. With Kazan (dog), Ruth Sullivan, Gene Toler, Robert Manning, Tom London, Grace Wood, Edward Cecil, Henry Roquemore, Nelson McDowell, Prince (dog). A boy and his dog team with a man and a young woman to fight a vicious sheep thief and his killer cur. Exceedingly low budget but likable juvenile programmer.\n\n**1281** _ **The Feud Maker**_ **** Republic, 1938. 60 min. D: Sam Newfield. SC: George Plympton. With Bob Steele, Marion Weldon, Karl Hackett, Frank Ball, Budd Buster, Lew Meehan, Roger Williams, Forrest Taylor, Steve Clark, Lloyd Ingraham, Sherry Tansey, Wally West, Jack C. Smith, Tex Palmer. A cowboy attempts to stop a crook who has instigated a feud between ranches and homesteaders. Typically action filled Bob Steele oater for producer A.W. Hackel.\n\n**1282** _ **Feud of the Range**_ **** Metropolitan, 1939. 55 min. D: Harry S. Webb. SC: Carl Krusada. With Bob Steele, Gertrude Messinger, Jean Cranford, Richard Cramer, Frank LaRue, Bob Burns, Budd Buster, Jack Ingram, Charles King, Denver Dixon, Carl Mathews. A bad man instigates a range war to obtain land while a cowboy tries to stop him. Cheap Bob Steele outing for Harry S. Webb, loaded with stock footage. Also called _**Feud on the Range**_.\n\n**1283** _ **Feud of the Trail**_ **** Victory, 1937. 56 min. D: Robert Hill. SC: Basil Dickey. With Tom Tyler, Harley Wood, Milburn Morante, Roger Williams, Lafe McKee, Richard Alexander, Slim Whitaker, Jim Corey, Eddie Gribbon, Francis Walker, Colin Chase. A family belonging to the grange has its gold stolen by crooks and a cowboy tries to recover it. Very cut rate Sam Katzman production.\n\n**1284** _ **Feud of the West**_ **** Diversion, 1936. 62 min. D: Harry Fraser. SC: Phil Dunham. With Hoot Gibson, Joan Barclay, Buzz Barton, Reed Howes, Robert Kortman, Ed Cassidy, Nelson McDowell, Roger Williams, Lew Meehan, Allen Greer, Richard Cramer. A rodeo rider is hired by a rancher to get evidence on the gang that killed his son and nephew but the cowpoke is accused of murder and is forced to head for the hills. The lack of good production values hurt this Hoot Gibson vehicle for producer Walter Futter.\n\n_**Feud on the Range**_ see _**Feud of the Range**_\n\n**1285** _ **Feudin' Rhythm**_ **** Columbia, 1949. 65 min. D: Edward Bernds. SC: Barry Shipman. With Eddy Arnold, Gloria Henry, Kirby Grant, Isabel Randolph, Tommy Ivo, Fuzzy Knight, Carolina Cotton, Mustard and Gravy, The Oklahoma Wranglers (The Willis Brothers), John Dehner, Edward Gargan, Maxine Gates, Snub Pollard, Gene Roth, Dick Elliott, George Lloyd, Emil Sitka, John Cason. Singer Eddy Arnold helps a radio star rancher get financing for his new television show despite problems with a cultured female sponsor. Limp Western musical enhanced by Eddy Arnold singing his hit record \"Cattle Call.\"\n\n**1286** _ **A Few Dollars for Django**_ **** Filmar Cinematografica, 1966. 87 min. Color. D: Leon Klimovsky. SC: Tito Carpi and Manuel Sebares. With Anthony Steffen, Gloria Osuna, Frank Wolff, Enzo Girolami, Thomas Moore, Alfonso Rojas, Angel Ter, Joe Kamel. A bounty hunters tracks a robber into Old Montana and ends up teaming with him to wipe out a dictator and his band. Brutal but bland Italian production; for genre fans only. Original title: _**Per Pochi Dollari per Django**_ (A Few Dollars for Django). TV title: _**A Few Dollars for Gypsy**_.\n\n_**A Few Dollars for Gypsy**_ see _**A Few Dollars for Django**_\n\n**1287** _ **A Few Dollars More**_ **** RAF Industries, 1969. 90 min. Color. D-SC: Julio Buchs. With Peter Lee Lawrence, Gloria Milland, Fausto Tozzi, Dianik Zuratowska, Antonio Pica, Luis Prendes, Paco Sanz, Tomas Blanco, Luis Induni. Teenager Billy Bonney becomes a fugitive when he kills the man who raped his mother and seeks refuge with Pat Garrett, a friend of his late father, but eventually turns to a life of crime. Violent, inaccurate Italian retelling of the Billy the Kid saga. Issued in Italy in 1967 by Kinesis\/Altor Film as _**...E Divenne il Piu Spietato Bandito del Sud**_ (...And He Became the Most Ruthless Bandit in the South) and also known as _**For a Few Bullets More**_.\n\n**1288** _ **The Fiddlin' Buckaroo**_ **** Universal, 1933. 63 min. D: Ken Maynard. SC: Nate Gatzert. With Ken Maynard, Gloria Shea, Fred Kohler, Fred Rice, Jack Rockwell, Jack Mower, Bob McKenzie, Joe Girard, Slim Whitaker, Pascale Perry, Frank Ellis, Roy Bucko, Buck Bucko, Bud McClure, Hank Bell, Jack Kirk, Robert Walker, Clem Horton. Arrested for supposedly helping a gang during a robbery, an undercover government man breaks jail when the outlaws kidnap a rancher's pretty daughter. A bit on the slow side with too much music, this Ken Maynard production was also directed by the star.\n\n**1289** _ **The Fiend Who Walked the West**_ **** 20th Century\u2013Fox, 1958. 101 min. D: Gordon Douglas. SC: Harry Brown and Philip Yordan. With Hugh O'Brian, Robert Evans, Dolores Michaels, Linda Cristal, Stephen McNally, Edward Andrews, Ron Ely, Ken Scott, Emile Meyer, Gregory Morton, Georgia Simmons. A madman escapes from prison and goes on a killing spree while his cellmate is let loose in order to stop him. Savage melodrama, a reworking of _**Kiss of Death**_ (20th Century\u2013Fox, 1947).\n\n**1290** _ **Fiesta**_ **** Metro-Goldwyn-Mayer, 1947. 104 min. Color. D: Richard Thorpe. SC: George Bruce and Lester Cole. With Esther Williams, Akim Tamiroff, Ricardo Montalban, John Carroll, Mary Astor, Cyd Charisse, Fortunio Bonanova, Hugo Haas, Jean Van, Joey Preston, Frank Puglia, Los Bocheros, Alan Napier, Ben Welden, John Maxwell, Dewey Robinson, Jane Ross. When a matador gives up his career for music his twin sister takes over in the bull ring. Fairly lavish but weak Western musical.\n\n_**Fiery Spur**_ see _**Love Desperados**_\n\n**1291** _ **$50,000 Reward**_ **** Davis Distributing, 1924. 49 min. D: Clifford S. Smith. SC: Frank Howard Clark. With Ken Maynard, Esther Ralston, Bert Lindley, Ed Peil, Sr., Lillian Leighton, Charles Newton, Frank Whitson. A cowboy inherits a ranch where a dam is to be built and a crooked banker and a lawyer want the land for themselves. Ken Maynard's first series film is an exciting affair, successfully launching his genre career.\n\n**1292** _ **Fight for Gold**_ **** Dimension Pictures, 1973. 97 min. Color. D: Harald Reinl. SC: Johannes Weiss. With Doug McClure, Harald Leipnitz, Heinz Reichke, Klaus Lowitsch, Kristina Nel, Angelica Ott, Roberto Blanco, Kurt Bluau, Ivan Stimac, Miha Baloh, Fabro Konjhodzic, Branko Spoljar, Vladimir Krstulovic, Illija Ivozic, Mirko Roman, Vladimir Medar, Vojo Govedarica. A trapper, after being robbed of his gold by bandits, saves the life of a kid but is later accused of being in cahoots with the killer of the boy's prospector father. Pretty poor West German production released there as _**Die Blutigen Geier von Alaska**_ (The Bloody Vulture from Alaska) and also called _**The Hellhounds of Alaska**_.\n\n**1293** _ **The Fighter**_ **** United Artists, 1952. 78 min. D: Herbert Kline. SC: Aben Kendall and Herbert Kline. With Richard Conte, Vanessa Brown, Lee J. Cobb, Frank Silvera, Roberta Haynes, Hugh Sanders, Claire Carleton, Martin Garralaga, Argentina Brunetti, Rodolfo Hoyos, Jr., Margaret Padilla. After his family is murdered in the 1910 Mexican Revolution, a young man takes up boxing in order to get money to buy weapons for guerrillas. Good screen version of Jack London's story \"The Mexican.\"\n\n**1294** _ **Fighters in the Saddle**_ **** Davis Productions, 1929. 50 min. D-SC: Robert J. Horner. With Art Acord, Peggy Montgomery, John Lowell, Tom Bay, Betty Carter, Lynn Sanderson, Cliff Lyons, Jack Ponder. The owner of a land company wants a ranch for county road expansion and frames the brother and sister who have leased the property. Cheaply made Art Acord vehicle (one of his last films) but it should please his many fans. Alternate title: _**Fighters of the Saddle.**_\n\n_**Fighters of the Saddle**_ see _**Fighters in the Saddle**_\n\n**1295** _ **Fightin' Jack**_ Goodwill, 1926. 52 min. D: Louis Chaudet. SC: Peggene Olcott. With Bill Bailey, Hazel Deane, Frona Hale, John Byron, Sailor Sharkey, Herma Cordova. A cowboy rescues a girl after she falls from a cliff and learns a crook and his gang are after her ranch. Fast action and nice scenery make up for a mediocre story in this silent Bill Bailey vehicle.\n\n**1296** _ **Fighting Bill Carson**_ **** Producers Releasing Corporation, 1945. 55 min. D: Sam Newfield. SC: Louise Rousseau. With Buster Crabbe, Al St. John, Lorraine Miller, Kay Hughes, I. Stanford Jolley, Kermit Maynard, Bob (John) Cason, Budd Buster, Bud Osborne, Charles King, Wally West, Lynton Brent, Augie Gomez, Jimmy Aubrey, Roy Bucko, Jack Tornek, Rube Dalroy, Foxy Callahan, George Morrell, Ray Jones, Rose Plummer. Billy Carson and pal Fuzzy Q. Jones rescue a young woman from a stagecoach holdup only to later discover she is part of an outlaw gang. Average \"Billy Carson\" entry.\n\n**1297** _ **Fighting Bill Fargo**_ **** Universal, 1942. 58 min. D: Ray Taylor. SC: Paul Franklin, Dorcas Cochran and Arthur V. Jones. With Johnny Mack Brown, Fuzzy Knight, Jeanne Kelly, Kenneth Harlan, Nell O'Day, Ted Adams, James Blaine, Alan Bridge, The Eddie Dean Trio, Robert Kortman, Earle Hodgins, Tex Palmer, Harry Tenbrook, Kermit Maynard, Blackie Whiteford, Merrill McCormick, Bud Osborne. A man returns home to help his sister run their late father's newspaper and becomes involved with crooks trying to rig the election of the town's sheriff. Good Johnny Mack Brown affair with songs by Eddie Dean and his trio.\n\n**1298** _ **The Fighting Buckaroo**_ **** Columbia, 1943. 58 min. D: William Berke. SC: Luci Ward. With Charles Starrett, Kay Harris, Arthur Hunnicutt, Ernest Tubb, Johnny Luther's Ranch Boys, Wheeler Oakman, Forrest Taylor, Stanley Brown, Robert Kellard, John Tyrrell, Norma Jean Wooters, Roy Butler, Jessie Arnold, Ray Jones, Lane Bradford, Chuck Baldra, Bob Burns, Stephen Keyes, Eddie Laughton, Rose Plummer, John Hoffman. A cowboy and his pal attempt to help an old friend accused of assisting a gang of cattle rustlers. Pretty good Charles Starrett vehicle in which sidekick Arthur Hunnicutt is billed as Arthur \"Arkansas\" Hunnicutt; country singer Ernest Tubb made his screen debut singing two of his biggest records, \"Walking the Floor Over You\" and \"Blue Eyed Elaine.\"\n\n**1299** _ **Fighting Caballero**_ **** Superior\/First Division, 1935. 59 min. D: Elmer Clifton. SC: Elmer Clifton and George Merrick. With Rex Lease, Dorothy Gulliver, George Chesebro, Robert Walker, Wally Wales, Earl Douglas, Milburn Morante, George Morrell, Carl Mathews, Franklyn Farnum, Paul Ellis, Artie Ortego, Frank Yaconelli, Barney Furey, Marty Joyce, Pinky Barnes. A cowboy helps a female silver mine owner being harassed by criminals. Cheaply produced Rex Lease vehicle.\n\n**1300** _ **Fighting Caravans**_ **** Paramount, 1931. 91 min. D: Otto Brower and David Burton. SC: Edward G. Paramore, Jr., Kenne Thompson and Agnes Brand Leahy. With Gary Cooper, Lily Damita, Ernest Torrance, Fred Kohler, Tully Marshall, Eugene Pallette, Roy Stewart, May Boley, James Farley, James Marcus, Eve Southern, Donald MacKenzie, Syd Saylor, E. Alyn Warren, Frank Campeau, Charles Winninger, Frank Hagney, Jane Darwell, Irving Bacon, Merrill McCormick, Tiny Sanford, Chief Big Tree. A wagon master and his two sidekicks lead pioneers across the plains, fighting outlaws and Indians. Better-than-average action film, some of its footage was later used in _**Wagon Wheels**_ (q.v.). TV title: _**Blazing Arrows**_.\n\n**1301** _ **The Fighting Champ**_ **** Monogram, 1932. 56 min. D: J.P. McCarthy. SC: Wellyn Totman. With Bob Steele, Arletta Duncan, Kit Guard, George Chesebro, George Hayes, Charles King, Henry Roquemore, Lafe McKee, Frank Ball, Si Jenks, Hank Bell, Perry Murdock, Denny Meadows (Dennis Moore), Gilbert \"Pee Wee\" Holmes, Buzz Barton, Dorothy Vernon, Ed Peil, Sr., Ed Coxen, George Morrell, Al Haskell, Archie Ricks, Jack Tornek, Jack Evans, Bud Pope, Clyde McClary, Barney Beasley, Tex Palmer, Jack Jones, Fred Parker, Harry Leroy. After foiling a stagecoach holdup a cowboy is drafted into fighting a traveling boxer. A well staged boxing match highlights this Bob Steele film that includes a good performance by George Chesebro as an oily fight manager.\n\n**1302** _ **The Fighting Code**_ **** Columbia, 1934. 65 min. D-SC: Lambert Hillyer. With Buck Jones, Diane Sinclair, Niles Welch, Ward Bond, Richard Alexander, Louis Natheux, Alf James, Erville Alderson, Gertrude Howard, Robert Kortman, Charles Brinley, Buck Moulton, Frank Ellis, Jim Corey. A cowboy tries to find out who murdered a young woman's father. Excellent Buck Jones vehicle enhanced by a mystery motif.\n\n**1303** _ **The Fighting Cowboy**_ **** Superior, 1933. 58 min. D: Denver Dixon (Victor Adamson). SC: L.V. Jefferson. With Buffalo Bill, Jr., Genee Boutell, Allen Holbrook, William Ryno, Marin Sais, Tom Palky, Bart Carre, Jack Evans, Boris Bullock, Ken Brocker, Betty Butler, Clyde McClary. An investigator tries to stop a crook from stealing an old man's tungsten claim, as well as his pretty daughter. About as shoddy as a Western can get.\n\n**1304** _ **The Fighting Deputy**_ **** Spectrum, 1937. 60 min. D: Sam Newfield. SC: William Lively. With Fred Scott, Al St. John, Phoebe Logan, Marjorie Beebe, Charles King, Lafe McKee, Frank LaRue, Eddie Holden, Sherry Tansey, Jack C. Smith, Chick Hannon, Jack Evans. A cowboy takes his dad's lawman job to hunt for the man who ambushed him only to find the culprit is his fiancee's long lost brother. Although the plot is not new, this outing is well staged and Fred Scott makes a grand singing cowboy cavalier.\n\n**1305** _ **Fighting Fists of Shanghai Joe**_ **** Beacon Film, 1974. 98 min. Color. D: Mario Caiano. SC: Mario Caiano and T.F. Karter (Fabrizio Trifone Trecca). With Chen Lee, Klaus Kinski, Gordon Mitchell, Robert Hundar (Claudio Undari), Katsutoshi Mikuriya, Carla Romanelli, Giacomo Rossi Stuart, Carla Mancini, Paco Sanz, George Wang, Rick Boyd (Federico Boido), Piero Lulli, Andrea Aureli, Tito Garcia, Dante Maggio, Alfonso de la Vega, Alberto Dell'Acqua (Robert Widmark), Osiride Pevarello, Angelo Susani, Francisco Sanz, Claudio Ruffini, Giovanni Sabbatini, Dante Cleri, Umberto D'Orsi, Veriano Ginesi. A Chinese martial arts expert is hired by a rancher to work as a cowboy but he soon realizes his boss is running a cattle theft operation. Fast moving, violent Kung Fu-Spaghetti Western filmed in Italy; also called _**The Dragon Strikes Back**_ and _**Shanghai Joe**_. Sequel: _**The Return of Shanghai Joe**_ (q.v.).\n\n**1306** _ **The Fighting Fool**_ **** Columbia, 1932. 58 min. D: Lambert Hillyer. SC: Frank Clark. With Tim McCoy, Marceline Day, Mary Carr, Robert Ellis, Ethel Wales, Dorothy Granger, Robert Kortman, Arthur Rankin, Harry Todd, William V. Mong, Lew Meehan, Herman Hack, Jack Kirk, Dick Dickinson, Al Taylor, Ray Henderson. A cowboy is on the trail of an outlaw gang led by a masked phantom. Rather slow going in this Tim McCoy film.\n\n**1307** _ **Fighting for Justice**_ **** Columbia, 1932. 60 min. D: Otto Brower. SC: Robert Quigley. With Tim McCoy, Joyce Compton, Robert Frazer, Hooper Atchley, William Norton Bailey, Lafe McKee, Walter Brennan, Harry Todd, Harry Cording, Murdock MacQuarrie, William V. Mong, Charles King, Henry Sedley, Fuzzy Knight, Glenn Strange, Mickey Condon, Johnny Luther, Milburn Morante, Bob Burns, Horace B. Carpenter, Fred Burns, Hank Bell, Art Dillard, Merrill McCormick, Ray Jones, Charles Brinley, Jack Kirk, Al Taylor, Tiny Sanford, Rose Plummer. Crooks kill a man who has bought a ranch and the former owner must clear himself of the murder charge. A good script and effective action make this a good Tim McCoy vehicle.\n\n**1308** _ **Fighting Frontier**_ **** RKO Radio, 1943. 57 min. D: Lambert Hillyer. SC: J. Benton Cheney and Norton S. Parker. With Tim Holt, Cliff Edwards, Ann Summers, Eddie Dew, William Gould, Davison Clark, Slim Whitaker, Tom London, Monte Montague, Jack Rockwell, Bud Osborne, Russell Wade, Fern Emmett, Steve Clark, Hank Bell, Ray Jones. An undercover agent is assigned by the governor to join an outlaw gang and get the goods on its leader. Good entry in Tim Holt's popular RKO series.\n\n**1309** _ **The Fighting Frontiersman**_ **** Columbia, 1946. 62 min. D: Derwin Abrahams. SC: Ed Earl Repp. With Charles Starrett, Smiley Burnette, Helen Mowery, Hank Newman and The Georgia Crackers, Robert Filmer, George Chesebro, Emmett Lynn, Zon Murray, Jim Diehl, Maudie Prickett, Jacques O'Mahoney (Jock Mahoney), Frank Ellis, Frank Larue, Herman Hack, Russell Meeker, Jack Evans, Jack Tornek, Foxy Callahan, Victor Cox, Kit Guard, Ben Corbett, George Plues, Blackie Whiteford, Ray Jones. A ranger and his pal arrive at the behest of a saloon gal to investigate the disappearance of an old prospector who found gold left in the area by Santa Ana. Poor \"Durango Kid\" effort, part of which was later recycled into the final series outing, _**The Kid from Broken Gun**_ (q.v.). British title: _**Golden Lady**_.\n\n**1310** _ **Fighting Fury**_ **** J.D. Trop, 1934. 61 min. D: Robert Hill. SC: Myron Dattlebaum. With Kazan (dog), John King, Bonita Baker, Tom London, Lafe McKee, Philo McCullough, Bart Carre, Del Morgan, Jack Donovan, Cactus (horse). Kazan the wonder dog and Cactus, a beautiful white stallion, help a cowboy dubbed the Lone Ranger as he opposes an outlaw gang. Low grade action programmer produced by male lead John King (not the later John \"Dusty\" King of \"Range Busters\" fame), the owner-trainer of Kazan. Also called _**Outlaw's Highway**_.\n\n**1311** _ **The Fighting Gringo**_ **** RKO Radio, 1939. 59 min. D: David Howard. SC: Oliver Drake. With George O'Brien, Lupita Tovar, Lucio Villegas, William Royle, Glenn Strange, Slim Whitaker, LeRoy Mason, Mary Field, Martin Garralaga, Richard Botiller, Bill Cody, Cactus Mack, Chris-Pin Martin, Ben Corbett, Forrest Taylor, Hank Bell. The leader of a group of hired guns helps in the rescue of a gold shipment from bandits and saving a man's ranch. Lots of fast and furious action in this George O'Brien film.\n\n**1312** _ **Fighting Hero**_ **** Reliable, 1934. 55 min. D: Harry S. Webb. SC: Carl Krusada and Rose Gordon. With Tom Tyler, Renee Borden, Edward Hearn, Richard Botiller, Ralph Lewis, Murdock MacQuarrie, Nelson McDowell, Tom London, George Chesebro, Rosa Rosanova, J.P. McGowan, Lew Meehan, Chuck Baldra, Jimmy Aubrey. An undercover agent pretends to be a wanted man so he can round up an outlaw gang. Low grade, tacky, hard to follow Tom Tyler vehicle.\n\n**1313** _ **The Fighting Kentuckian**_ **** Republic, 1949. 100 min. D-SC: George Waggner. With John Wayne, Vera Ralston, Philip Dorn, Oliver Hardy, Marie Windsor, John Howard, Hugo Haas, Odette Myrtil, Grant Withers, Paul Fix, Mae Marsh, Jack Pennick, Mickey Simpson, Fred Graham, Mabelle Koenig, Shy Waggner, Crystal White, Hank Worden, Charles Cane, Cliff Lyons, Chuck Roberson, Steve Darrell, Alberto Morin, Tony Travers, Charles Andre, Al Murphy, Ralph Dunn, Michael Ross, Dave Anderson, Billy Green, William Hawes, Fred Libby. In 1818 Alabama a Kentucky rifleman falls in love with a French woman and uncovers a plot to deprive her people of their land. Not John Wayne's best by any means but still entertaining with Oliver Hardy quite good as Duke's bumbling pal; also released in a colorized version.\n\n**1314** _ **The Fighting Lawman**_ **** Allied Artists, 1953. 71 min. D: Thomas Carr. SC: Dan Ullman. With Wayne Morris, Virginia Grey, John Kellogg, Harry Lauter, Myron Healey, John Pickard, Rick Vallin, Dick Rich, Stanley Price, Denver Pyle, Herman Hack. When a quartet of outlaws rob a bank a deputy marshal tries to capture them and becomes involved with a woman who wants the stolen loot. Well written and acted drama which belies its small budget.\n\n**1315** _ **The Fighting Legion**_ **** Universal, 1930. 75 min. D: Harry J. Brown. SC: Bennett Cohen. With Ken Maynard, Dorothy Dwan, Ernie Adams, Stanley Blystone, Frank Rice, Harry Todd, Robert Walker, Jack Fowler, Lee Bates, Bill Nestell, Slim Whitaker. A crooked cattleman, in cahoots with a dishonest banker, murders a Texas Ranger and tries to pin the crime on a cowboy who romances the girl he wants. This Ken Maynard part-talkie is a bit overlong and somewhat stagy at times but overall provides good entertainment.\n\n**1316** _ **Fighting Luck**_ **** Rayart, 1926. 50 min. D: J.P. McGowan. With Bob Reeves, Ione Reed, William Ryno, Lew Meehan, J.P. McGowan, Eddie Beery. A hired gunman is wounded by an outlaw and his gang and he later falls in love with a rancher's daughter who is abducted by the bad man. This silent poverty row offering is fun to view.\n\n**1317** _ **Fighting Mad**_ **** Monogram, 1939. 60 min. D: Sam Newfield. SC: George Rosener and John Rathmell. With James Newill, Sally Blane, Dave O'Brien, Benny Rubin, Milburn Stone, Walter Long, Warner Richmond, Ted Adams, Chief Thundercloud, Horace Murphy, Ole Olsen, Iron Eyes Cody. Bandits capture a young woman who has witnessed a robbery and the Mounties try to rescue her. Okay entry in the \"Renfrew of Royal the Mounted\" series with good singing by James Newill and nice locales; also called _**Renfrew of the Royal Mounted in Fighting Mad**_.\n\n**1318** _ **Fighting Mad**_ **** New Realm, 1956. 53 min. D: Denis Kavanagh. SC: Jennifer Wyatt. With Joe Robinson, Adrienne Scott, Jack Taylor, Beckett Bould, Colin Cleminson. After accidentally killing two opponents, a boxer gives up the ring, moves to Canada and helps his uncle fight timber thieves and oil claim jumpers. Stilted British dual bill production.\n\n**1319** _ **Fighting Man of the Plains**_ **** 20th Century\u2013Fox, 1949. 94 min. Color. D: Edwin L. Marin. SC: Frank Gruber. With Randolph Scott, Jane Nigh, Bill Williams, Victor Jory, Douglas Kennedy, Joan Taylor, Barry Kroeger, Rhys Williams, Barry Kelley, James Todd, Paul Fix, James Millican, Burt Symon, Dale Robertson, Herbert Rawlinson, J. Farrell MacDonald, Harry V. Cheshire, James Griffith, Tony Hughes, John Hamilton, John Halloran, Cliff Clark, Anthony Jochim, James Harrison, Matt Willis, Kermit Maynard. A notorious gunman wants to find the man who killed his brother and ends up as the sheriff of a lawless town. Another fine action feature starring Randolph Scott.\n\n**1320** _ **The Fighting Marshal**_ **** Columbia, 1931. 60 min. D: D. Ross Lederman. SC: Frank Howard Clark. With Tim McCoy, Dorothy Gulliver, Matthew Betz, Mary Carr, Pat O'Malley, Ed Le Saint, Lafe McKee, W.A. Howell, Dick Dickinson, Bob Perry, Harry Todd, Ethan Laidlaw, Lee Shumway, Blackie Whiteford, Art Mix, Glenn Strange, Artie Ortego, Frank Ellis, Blackjack Ward, Arthur Millett, Frank Lanning, Al Taylor, Milton Brown, Bob Card, Barney Beasley, Bob Roper. Escaping from prison before he receives a reprieve, a man takes on the identity of a lawman and helps round up an outlaw gang, later learning he has been exonerated. Nice going in this action filled Tim McCoy outing.\n\n**1321** _ **Fighting Mustang**_ **** Astor, 1948. 60 min. D: Oliver Drake. SC: Rita Ross. With Sunset Carson, Pat Starling, Al Terry, Polly McKay, William Val, Forrest Matthews, Joe Hiser, Lee Roberts, Felice Raymond, Bob Curtis, Stephen Keyes, Tex Wilson, Al Ferguson, Hugh Hooker, Dale Harrison, Little Joe's Wranglers. Two rangers stationed near the badlands try to combat an outlaw gang stealing wild horses. Tattered Sunset Carson vehicle.\n\n**1322** _ **The Fighting Parson**_ **** Allied, 1933. 70 min. D-SC: Harry Fraser. With Hoot Gibson, Marceline Day, Robert Frazer, J. Farrell MacDonald, Stanley Blystone, Skeeter Bill Robbins, Charles King, Jules Cowles, Phil Dunham, Ethel Wales, Frank Nelson, Frank Ellis, Merrill McCormick, Horace B. Carpenter, Blackie Whiteford, Tex Palmer. Two cowboys kicked off a ranch when falsely accused of dishonesty arrive in a town run by crooks and one of them takes on the guise of a minister to route the gang. Overlong and slow moving Hoot Gibson affair.\n\n_**The Fighting Phantom**_ see _**The Mysterious Rider**_ (1933)\n\n**1323** _ **Fighting Pioneers**_ **** Resolute, 1935. 54 min. D: Harry Fraser. SC: Harry Fraser and Chuck Roberts. With Rex Bell, Ruth Mix, Buzz Barton, Stanley Blystone, Earl Dwire, John Elliott, Chief Thundercloud, Roger Williams, Guate Mozin, Chuck Morrison, Chief Standing Bear, Francis Walker, Bob Burns, Blackjack Ward, Barney Beasley. Gun runners are stirring up trouble between Indians and whites and a cavalry officer enlists the aid of the chief's daughter in stopping the trouble. Low grade but probably the best of the quartet of Rex Bell-Ruth Mix-Buzz Barton vehicles for Resolute.\n\n**1324** _ **The Fighting Ranger**_ **** Columbia, 1934. 60 min. D: George B. Seitz. SC: Harry O. Hoyt. With Buck Jones, Dorothy Revier, Frank Rice, Ward Bond, Bradley Page, Paddy O'Flynn, Art (Mix) Smith, Frank LaRue, Jack Wallace, Bud Osborne, Lew Meehan, Denver Dixon, Jim Corey, Steve Clemente, Frank Ellis, Mozelle Britton. A ranger quits the service and heads to Mexico to round up a murderous outlaw gang. Action filled remake of star Buck Jones' _**Border Law**_ (q.v.).\n\n**1325** _ **The Fighting Ranger**_ **** Monogram, 1948. 57 min. D: Lambert Hillyer. SC: Ronald Davidson. With Johnny Mack Brown, Raymond Hatton, Christine Larson, Marshall Reed, Steve Clark, I. Stanford Jolley, Bob Woodward, Eddie Parker, Milburn Morante, Charlie Hughes, Peter Perkins. A ranger learns a man framed his cousin for murder so he could inherit his ranch and gets a job on the spread to capture the culprit. Another passable entry in Johnny Mack Brown's Monogram series.\n\n**1326** _ **The Fighting Redhead**_ **** Eagle Lion, 1949. 55 min. Color. D: Lewis D. Collins. SC: Paul Franklin and Jerry Thomas. With Jim Bannon, Little Brown Jug (Don Kay Reynolds), Emmett Lynn, Marin Sais, Peggy Stewart, John Hart, Lane Bradford, Forrest Taylor, Lee Roberts, Bob Duncan, Sandy Sanders, Billy Hammond, Ray Jones, Spooky Reynolds, Fess Reynolds. Red Ryder helps a young woman capture cattle rustlers who murdered her homesteader father. Average entry in the revived \"Red Ryder\" series.\n\n**1327** _ **Fighting Renegade**_ **** Victory, 1939. 60 min. D: Sam Newfield. SC: William Lively. With Tim McCoy, Joyce Bryant, Dave O'Brien, Ben Corbett, Budd Buster, Forrest Taylor, Ted Adams, Reed Howes, John Elliott, Carl Mathews, Artie Ortego, Frank Wayne, Tom Smith, Wally West, Herman Hack, Dan White, Chick Hannon, Tex Palmer. Using a Mexican disguise, a man sets out to clear himself of a six-year-old murder charge. Average outing, fast but with low production values. Tim McCoy, as Lightning Bill Carson, wears a Mexican disguise throughout the film.\n\n_**The Fighting 7th**_ see _**Little Big Horn**_\n\n**1328** _ **Fighting Shadows**_ **** Columbia, 1935. 60 min. D: David Selman. SC: Ford Beebe. With Tim McCoy, Robert (Bob) Allen, Geneva Mitchell, Ward Bond, Si Jenks, Otto Hoffman, Ed Le Saint, Bud Osborne, Ethan Laidlaw, Richard Alexander, Jim Mason, Charles Brinley, Howard C. Hickman, George C. Pearce, Allan Sears, Walter Shumway, Jess Cavin, Fred Malatesta, Steve Clark, Blackjack Ward, Jack Evans, Jack Mower, Tex Phelps, Iron Eyes Cody, Monte Carter, Robert Wilber, Rube Dalroy, Rhody Hathaway. A Northwest Mounted Police constable is assigned to find out who is behind a gang of fur thieves. Well done Tim McCoy film, aided by good scenery, a tight script and nice photography (by George Meehan).\n\n**1329** _ **The Fighting Sheriff**_ **** Columbia, 1931. 67 min. D: Louis King. SC: Stuart Anthony. With Buck Jones, Loretta Sayers, Robert Ellis, Harlan Knight, Paul Fix, Lillian Worth, Nena Quartero, Clarence Muse, Lillian Leighton, Lew Meehan, Lafe McKee, Slim Whitaker, Ernie Adams, Bill Patton, Merrill McCormick, Hank Bell, Fred Burns, Blackjack Ward. A sheriff falls for a young woman but she turns against him when a rival informs her the lawman shot her brother who was riding with the gang led by the crook. One of Buck Jones' best early talkies; a good film.\n\n**1330** _ **The Fighting Stallion**_ **** Eagle Lion, 1950. 62 min. Color. D: Robert Tansey. SC: Frances Kavanaugh. With Bill Edwards, Doris Merrick, Forrest Taylor, Rocky Camron, John Carpenter, Maria Hart, Don C. Harvey, Bob (John) Cason, Merrill McCormick, Concha Ybarra. A war veteran, who is going blind, captures and trains a wild stallion with the steed later saving his life during a forest fire. Cheaply made but action filled melodrama for the juvenile trade.\n\n**1331** _ **The Fighting Strain**_ **** William Steiner, 1923. 51 min. D-SC: Neal Hart. With Neal Hart, Beth Mitchell, William Quinn, Bert Wilson, Gladys Gilland, James McLaughlin. Returning home, a soldier goes after the crook who abducted his sister and fleeced his girlfriend's father, the trail leading him to Canada. Complicated Western drama written, produced and directed by star Neal Hart.\n\n**1332** _ **The Fighting Texan**_ **** Ambassador, 1937. 59 min. D: Charles Abbott. SC: Joseph O'Donnell. With Kermit Maynard, Elaine Shepard, Frank LaRue, Budd Buster, Ed Cassidy, Bruce Mitchell, Murdock MacQuarrie, Art Miles, Merrill McCormick, Blackie Whiteford, Wally West, John Merton, Bob Woodward, Dick Morehead, Glenn Strange, Art Dillard, Bert Dillard, Curley Dresden, Rube Dalroy, Jack Evans, Clem Horton, Cherokee Alcorn. When his new partner is killed, a rancher accuses a rival and his daughter of being involved in the crime. Somewhat involved but still entertaining Kermit Maynard vehicle.\n\n**1333** _ **The Fighting Texans**_ **** Monogram, 1933. 60 min. D: Armand L. Schaefer. SC: Wellyn Totman and Charles Roberts. With Rex Bell, Luana Walters, Betty Mack, Wally Wales, George Hayes, Lafe McKee, Yakima Canutt, Alan Bridge, Frank LaRue, Ann Howard, Gordon DeMain, George Nash, Slim Whitaker, Frank Ellis, Si Jenks, Duke R. Lee, George Morrell, Gilbert \"Pee Wee\" Holmes, Fred Parker, Jack Evans, Tommy Coats, Dick Dickinson, Tex Palmer. After falling in love with the sheriff's daughter, an oil stock salesman tangles with crooked promoters and almost gets lynched because of their bogus well. Fair Rex Bell feature from producers Trem Carr and Paul Malvern.\n\n**1334** _ **Fighting Through**_ **** Willis Kent\/Cristo, 1934. 55 min. D-SC: Harry Fraser. With Reb Russell, Lucille Lund, Yakima Canutt, Edward Hearn, Wally Wales, Chester Gan, Steve Clemento, Bill Patton, Frank McCarroll, Ben Corbett, Hank Bell, Slim Whitaker, Nelson McDowell, Lew Meehan, Jack Kirk, Jack Jones, Chuck Baldra, Herman Hack, Bart Carre, Jack Evans, Ray Jones, Buck Morgan, Ed Rowland. Two cowpokes become friends when one saves the other's life after a crooked card game and the duo get jobs on a girl's ranch and save her from kidnappers. Low grade but action filled Reb Russell opus.\n\n**1335** _ **Fighting Thru or California in 1878**_ **** Tiffany, 1930. 61 min. D: William Nigh. SC: Jack (John Francis) Natteford. With Ken Maynard, Jeannette Loff, Wallace MacDonald, Carmelita Geraghty, William L. Thorne, Charles King, Fred Burns, William Nestell, Art Mix, Chuck Baldra, Jack Kirk, Bud McClure, Jim Corey, Tommy Bay, Jack Fowler. A gold miner is accused of killing his partner but deduces a gambler and a saloon girl are the culprits. Fairly standard Ken Maynard early talkie with a somewhat tangled plot.\n\n**1336** _ **Fighting to Live**_ **** Principal, 1934. 60 min. D: Eddie (Edward F.) Cline. SC: Robert Ives. With Marion Shilling, Gaylord (Steve) Pendleton, Reb Russell, Eddie Phillips, Lloyd Ingraham, Henry Hall, John Strohback, Bruce Mitchell, Captain and Lady (dogs). Two dogs, muzzled and left to die in the desert, are hunted by a posse for stealing chickens but are defended by a young lawyer. Crude juvenile-oriented programmer, poorly photographed and recorded.\n\n**1337** _ **The Fighting Trooper**_ **** Ambassador, 1934. 57 min. D: Ray Taylor. SC: Forrest Sheldon. With Kermit Maynard, Barbara Worth, Walter Miller, Robert Frazer, LeRoy Mason, George Regas, Charles Delaney, Joe Girard, George Chesebro, Charles King, Artie Ortego, Lafe McKee, Milburn Morante, Gordon DeMain, Nelson McDowell, George Morrell, Merrill McCormick. A rookie Mountie is assigned to bring in a trapper suspected of killing a veteran trooper. Well done Northwest melodrama based on James Oliver Curwood's story \"Footprints\"; and Kermit Maynard's first starring sound film. British title: _**The Trooper**_.\n\n**1338** _ **Fighting Valley**_ **** Producers Releasing Corporation, 1943. 62 min. D-SC: Oliver Drake. With Dave O'Brien, Jim Newill, Guy Wilkerson, Patti McCarty, John Merton, Robert Bice, Stanley Price, Mary MacLaren, John Elliott, Charles King, Dan White, Carl Mathews, Curley Dresden, Jimmy Aubrey, Hal Price, Jess Cavin, Tex Williams, Don Weston. The Texas Rangers step in to help a man who is having ore from his smelting mine stolen by hijackers. Typical low grade entry in PRC's \"Texas Rangers\" series.\n\n**1339** _ **The Fighting Vigilantes**_ **** Eagle Lion, 1947. 61 min D: Ray Taylor. SC: Robert Churchill. With Lash LaRue, Al St. John, Jennifer Holt, George Chesebro, Lee Morgan, Marshall Reed, Steve Clark, Carl Mathews, Russell Arms, John Elliott, Felice Richmond. Posing as vigilantes, two marshals arrive in a town plagued by murder and violence in an attempt to get to the bottom of the crimes. Mostly dull going except for an exciting climax.\n\n_**The Fighting Westerner**_ see _**Rocky Mountain Mystery**_\n\n**1340** _ **Fighting with Kit Carson**_ **** Mascot, 1933. 12 Chapters. D: Armand L. Schaefer and Colbert Clark. SC: Jack Natteford, Barney A. Sarecky, Colbert Clark and Wyndham Gittens. With Johnny Mack Brown, Noah Beery, Betsy King Ross, Tully Marshall, Robert Warwick, William Farnum, Lane Chandler, Noah Beery, Jr., Edward Hearn, Edmund Breese, Lafe McKee, Ernie Adams, Alan Bridge, Reed Howes, Jack Mower, Maston Williams, Iron Eyes Cody, Frank Ellis, Slim Whitaker, DeWitt Jennings, Yakima Canutt. A dishonest trader, the leader of a gang called the Mystery Riders, is after government gold stolen in a pack train massacre and is opposed by scout Kit Carson. Slow moving and none-too-entertaining cliffhanger; Johnny Mack Brown's first serial.\n\n**1341** _ **The Final Hour**_ **** Universal, 1963. 74 min. Color. D: Robert Douglas. SC: Bernard Girard, Ward Hawkins and Harry Kleiner. With Lee J. Cobb, James Drury, Doug McClure, Ulla Jacobsson, Gary Clarke, Roberta Shore, Jacques Aubuchon, Bert Freed, Don Galloway, Dean Fredericks, Myron Healey, Sheldon Allman, Ross Elliott, Whit Bissell, Richard Garland, Peter Mamakos, Ted Knight, Anthony Jochim. Trouble erupts between ranchers and imported Polish miners over the affections of a young woman. Entertaining telefilm, first shown as an episode of \"The Virginian\" (NBC-TV, 1962\u201370) on May 5, 1963.\n\n**1342** _ **Find a Place to Die**_ **** Gadabout Gaddis Productions, 1971. 100 min. Color. D: Anthony Ascott (Giuliano Carmineo). SC: Hugo Fregonese and Ralph Grave. With Jeffrey Hunter, Pascale Petit, Piero Lulli, Daniela Giordano, Gianni Pallavicini, Aldo Lastretti, Rez Fahzeli, Ted Carter (Nello Pazzafini). Outlaws attack a gold mine, wounding a man as his sister goes for help, bringing back an assorted group of men, some of whom plan to steal the ore. Very violent Spaghetti Western, issued in Italy in 1968 by Aico Film as _**Joe...Cercati un Posto per Morire**_ (Joe...Look for a Place to Die).\n\n**1343** _ **The Firebrand**_ **** 20th Century\u2013Fox, 1962. 63 min. D: Maury Dexter. SC: Harry Spalding. With Kent Taylor, Valentin De Vargas, Lisa Montell, Joe Raciti, Chubby Johnson, Barbara Mansell, Troy Melton, Fred Krone, Sid Haig, Felix Locher, Jerry Summers, Allen Jaffe. The leader of a gang of Mexican bandits goes on a killing spree when he learns a former cohort caused the murders of some of his men. Supposedly based on the exploits of Joaqin Murieta, this compact oater entertains.\n\n**1344** _ **Firebrand Jordan**_ **** Big 4, 1930. 60 min. D: Alvin J. Neitz (Alan James). SC: Carl Krusada. With Lane Chandler, Aline Goodwin, Yakima Canutt, Sheldon Lewis, Marguerite Ainslee, Tom London, Lew Meehan, Frank Yaconelli, Alfred Hewston, Fred Harvey, Cliff Lyons. A cowboy after counterfeiters meets a woman whose father is missing as she is being compromised by a man the cowpoke suspects of being a gang member. Low class early talkie.\n\n**1345** _ **Firebrands of Arizona**_ **** Republic, 1944. 55 min. D: Lesley Selander. SC: Randall Faye. With Smiley Burnette, Sunset Carson, Peggy Stewart, Earle Hodgins, Roy Barcroft, LeRoy Mason, Tom London, Jack Kirk, Rex Lease, Charles Morton, Bud Geary, Robert Wilke, Fred \"Snowflake\" Toones, Pierce Lyden, Budd Buster, Bob Burns, Jack O'Shea, Hank Bell, William Desmond, Maxine Doyle, Grace Cunard, Horace B. Carpenter, Bob Woodward, Tom Steele, Victor Cox, Roy Butler, Phil Dunham, Pascale Perry, Bill Nestell, Dickie Dillon. Sunset Carson and his pal Frog Millhouse, a hypochondriac, are going to see a doctor when the law mistakes Frog for his look-a-like, outlaw Beefsteak Disco. Thanks to Smiley Burnette in a dual role and lots of action plus good direction from Lesley Selander, this Sunset Carson film is a cut above average.\n\n**1346** _ **Firecreek**_ **** Warner Bros.\/Seven Arts, 1968. 104 min. Color. D: Vincent McEveety. SC: Calvin Clements. With James Stewart, Henry Fonda, Inger Stevens, Gary Lockwood, Dean Jagger, Ed Begley, Jay C. Flippen, Jack Elam, James Best, Barbara Luna, Jacqueline Scott, Brooke Bundy, J. Robert Porter, Morgan Woodward, John Qualen, Louise Latham, Kevin Tate, Christopher Shea. A small town sheriff realizes he must defend his citizens against a gang of rowdies and their cold blooded leader. This James Stewart-Henry Fonda film promises a lot more entertainment than it delivers; average at best.\n\n_**The First Rebel**_ see _**Allegheny Uprising**_\n\n**1347** _ **The First Texan**_ **** Allied Artists, 1956. 82 min. Color. D: Byron Haskin. SC: Daniel S. Ullman. With Joel McCrea, Felicia Farr, Jeff Morrow, Wallace Ford, Abraham Sofaer, Jody McCrea, Chubby Johnson, Dayton Lummis, Rodolfo Hoyos, William Hopper, Roy Roberts, Frank Puglia, James Griffith, Nelson Leigh, David Silva, Carl Benton Reid, William Phipps, Scott Douglas, Lane Chandler, Myron Healey, William Tannen, Salvador Baguez. The story of Sam Houston, who is urged by President Andrew Jackson to take the lead in making Texas independent of Mexican rule. Fine film study of Sam Houston with Joel McCrea very good in the part.\n\n**1348** _ **The First Traveling Saleslady**_ **** RKO Radio, 1956. 92 min. Color. D: Arthur Lubin. SC: Stephen Longstreet and Devery Freeman. With Ginger Rogers, Carol Channing, Barry Nelson, James Arness, Robert F. Simon, Frank Wilcox, Dan White, Harry V. Cheshire, John Eldredge, Clint Eastwood, Ed Cassidy, Fred Essler, Lane Chandler, Lester Dorr, Bill Hale, Mauritz Hugo, Kathy Marlowe, Nora Bush, Hans Herbert. After a Broadway show closes because some of her corsets were used in a number, a designer and her secretary head West to sell barbed wire and run into all kinds of trouble. Ginger Rogers and Carol Channing try hard but nothing can help this dull Western comedy originally written for Mae West.\n\n**1349** _ **Fish Hawk**_ **** JAD Films International, 1980. 95 min. Color. D: Donald Shebib. SC: Blanche Hanalis. With Will Sampson, Charlie Fields, Geoffrey Bowes, Mary Pirie, Don Francks, Chris Wiggins. A young boy befriends a drunken Indian and helps him to give up the bottle and enjoy life. G-rated fare that is only average; filmed in Canada.\n\n**1350** _ **A Fistful of Dollars**_ **** United Artists, 1964. 96 min. Color. D: Bob Robertson (Sergio Leone). SC: Sergio Leone and Duccio Tessari. With Clint Eastwood, Marianne Koch, Gian Maria Volonte, Wolfgang Lukschy, Sieghart Rupp, Antonio Prieto, Jose Calvo, Benny Reeves, Daniel Martin. A stranger rides into a border town, finds two factions at war and decides to make money by heating up the rivalry. Filmed in Italy as _**Per un Pugno di Dollari**_ (For a Fistful of Dollars), this feature started the Spaghetti Western craze of the 1960s and launched Clint Eastwood to international stardom; a refashioning of the Japanese feature _**Yojimbo**_ (1961), it is not as good as its follow-up, _**For a Few Dollars More**_ (q.v.).\n\n_**A Fistful of Dynamite**_ see _**Duck, You Sucker**_\n\n_**Fistful of Knuckles**_ see _**Per un Pugno nell'Occhio**_ (For a Fist in the Eye)\n\n**1351** _ **Fistful of Lead**_ **** Colt Produziones Cinematografica, 1970. 93 min. Color. D: Anthony Ascott (Giuliano Carmineo). SC: Tito Carpi. With George Hilton, Charles Southwood, Erika Blanc, Peter Carter (Piero Lulli), Linda Sini, Carlo Giordana, Nello Pazzafini, Carlo Gaddi, Aldo Barberito, Marco Zuanelli, Lou Kamante (Luciano Rossi), Armando Calvo, Spartaco Conversi, Rick Boyd, Gigi Bonos, John Bartha, Fanco Fantasia, Ettore Arena, Furio Meniconi, Umberto Di Grazia. A mysterious gunman sees a holdup and is hired by a town boss to protect his gold shipments and gets forced into a showdown with another gunfighter. Somewhat muddled but fun Italian Western released in that country as _**C'e Sartana...Vendi la Pistola e Comprati la Barba**_ (I am Sartana...Trade Your Guns for a Coffin).\n\n**1352** _ **Five Bloody Graves**_ **** Independent-International, 1970. 98 min. Color. D: Al Adamson. SC: Robert Dix. With Robert Dix, Scott Brady, Jim Davis, John Carradine, Paula Raymond, John Cardos, Tara Ashton, Kent Osborne, Vicki Volante, Denver Dixon, Ray Young, Julie Edwards, Fred Meyers, Maria Polo, Keith Murphy, Ray Goldrup, Tom Goldrup, Gene Raymond (narrator). A notorious gunman after the savage Indian who murdered his wife gets involved with gun runners and wagon train outcasts stranded in hostile country. Low grade violent oater, mainly of interest for its cast. TV title: _**Gun Riders**_ (81 minutes).\n\n**1353** _ **Five Bold Women**_ **** Citation, 1960. 82 min Color. D: Jorge Lopez Portillo. SC: Mortimer Braus and Jack Pollexfen. With Jeff Morrow, Merry Anders, Irish McCalla, Guinn Williams, Kathy Marlowe, Jim Ross, Dee Carroll, Lucita Blain. The husband of one of five women being taken to prison attacks the lawman transporting them and they escape, only to be threatened by Indians. Passable low budget programmer; title song sung by Dean West.\n\n**1354** _ **Five Card Stud**_ **** Paramount, 1968. 103 min. Color. D: Henry Hathaway. SC: Marguerite Roberts. With Dean Martin, Robert Mitchum, Inger Stevens, Roddy McDowall, Katherine Justice, John Anderson, Ruth Springford, Yaphet Kotto, Denver Pyle, Bill Fletcher, Whit Bissell, Ted De Corsia, Don Collier, Roy Jenson, Boyd \"Red\" Morgan, Jerry Gatlin, Chuck Hayward, Louise Lorimer, Hope Summers. A sheriff teams with a gun-toting preacher to locate a man who plans to murder five gamblers who hanged their sixth player. Mystery element added to the two leads make this a good time for genre buffs.\n\n**1355** _ **Five Giants from Texas**_ **** Miro Cinematografica\/Balcazar, 1966. 90 min. Color. D: Aldo Florio. SC: Aldo Florio, Alfonso Balcazar and Jose Luis De La Loma. With Guy Madison, Monica Randall, Vidal Molina, Antonio Molino Rojo, Vassili Karamesinis, Giovanni Cianfrigilia, Jose Manuel Martin, Gianni Solaro. Several years after their friend is murdered by bandits hired by his wife's jealous cousins, five men show up to take revenge. Violent oater mainly for Guy Madison fans, this Italian-Spanish co-production was issued in Italy as _**I Cinque Della Vendetta**_ (Five for Revenge) and in Spain as _**Los Cunco de la Venganza**_ (Five for Vengeance).\n\n**1356** _ **Five Guns to Tombstone**_ **** United Artists, 1962. 71 min. D: Edward L. Cahn. SC: Richard Shayer. With James Brown, John Wilder, Walter Coy, Robert Karnes, Joe Haworth, Quent Sondergaard, Boyd \"Red\" Morgan, Jon Locke, Della Sharman, Gregg Palmer, Willis Bouchey, John Eldredge, Jeff DeBenning, Boyd Stockman, Al Wyatt, Bob Woodward. A reformed gunfighter learns his brother is trying to lure him back to lawlessness by framing him for a crime he did not commit. Standard programmer with good work by James Brown in the lead; a remake of _**Gun Belt**_ (q.v.).\n\n**1357** _ **Five Guns West**_ **** American Releasing, 1955. 79 min D: Roger Corman. SC: R. Wright Campbell. With John Lund, Dorothy Malone, Touch (Michael) Connors, Jonathan Haze, Paul Birch, Jack Ingram, Larry Thor. Five murderers are freed from prison to join the Confederate Army but after stealing gold from a Union stagecoach they decide to keep the proceeds. Roger Corman's directorial debut is a passable affair.\n\n**1358** _ **The Five Man Army**_ **** Metro-Goldwyn-Mayer, 1970. 105 min. Color. D: Don Taylor. SC: Dario Argento and Marc Richards. With Peter Graves, James Daly, Bud Spencer, Tetsuro Tamba, Nino Castelnuovo, Daniela Giordano, Marc Lawrence, Piero Lulli, Claudio Gora, Annabella Andreoli Carlo Alighiero, Jack Stuart, Jose Torres, Marino Mase. In 1914 five men team to rob a half-million dollars in gold from a train with four of them seeking the loot for themselves while the fifth wants it for the Mexican Revolution. Made in Italy as _**Un Esercito di 5 Uomini**_ (An Army of 5 Men), this feature provides some good excitement.\n\n_**Five Savage Men**_ see _**The Animals**_\n\n**1359** _ **$5,000 on One Ace**_ **** International Germania\/Balcazar\/FIDA, 1966. 91 min. Color. D: Alfonso Balcazar. SC: Alessandro Continenza and Helmut Harun. With Robert Wood, Fernando Sancho, Maria Sebalt, Norman Preston, Hans Nielsen, Helmut Schmidt, Antonio Molina Rojo, Giacomo Rossi Stuart, Paco Sanz, Fernando Rubio. A gambler wins part of a ranch from a man he is forced to kill and with his new partners, a brother and sister, he has to fight a land grabbing blackmailer to keep it. Typical European Western from the mid\u20131960s, short on plot but long on violence. German title: _**Die Gejagten der Sierra Nevada**_ (The Hunted of the Sierra Nevada).\n\n**1360** _ **Flame of Barbary Coast**_ **** Republic, 1945. 91 min. D: Joseph Kane. SC: Borden Chase. With John Wayne, Ann Dvorak, Joseph Schildkraut, William Frawley, Virginia Grey, Russell Hicks, Jack Norton, Paul Fix, Manart Kippen, Eve Lynne, Marc Lawrence, Bufferfly McQueen, Rex Lease, Jack Mulhall, Kenne Duncan, Stuart Hamblen, Edmund Cobb, Si Jenks, Frank Jaquet, Frankie Marvin, Eddie Acuff, William Halligan, Hugh Prosser, Adele Mara, Patricia Knox, Tom London, Dorothy Christy, Larry Steers, Emmett Vogan, Victor Potel, Bud Geary, Lee Shumway, Frank McCarroll, Roy Butler, Joe Rickson, Jan Ullrich, Al Murphy, Hank Bell, Charles Sullivan, Carl Wood, George Boyce, Joe Evans. A Montana cowboy is fleeced by a San Francisco gambling house proprietor and returns to set up a rival saloon and take the man's girl, only to have his plans interrupted by an earthquake. Okay John Wayne feature, but not up to his usual \"A\" efforts; colorized for video release.\n\n**1361** _ **The Flame of New Orleans**_ **** Universal, 1941. 79 min. D: Rene Clair. SC: Norman Krasna. With Marlene Dietrich, Bruce Cabot, Roland Young, Mischa Auer, Andy Devine, Frank Jenks, Eddie Quillan, Laura Hope Crews, Franklin Pangborn, Theresa Harris, Clarence Muse, Melville Cooper, Anne Revere, Bob Evans, Emily Fitzroy, Virginia Sale, Dorothy Adams, Herbert Rawlinson, Anthony Marlowe, Gitta Alpar, Reed Hadley, Gus Schilling, Shemp Howard, Frank Sully, Tony Paton, Joe Devlin, Frank Moran, Jack Raymond, Rex Evans, James Guifoyle, Mary Treen, Roy Harris, Lowell Drew, Bess Flowers. In 1841 a woman comes to New Orleans seeking a rich husband and falls in love with a riverboat captain after a fling with the town's richest man. Marlene Dietrich fans will go for this feature which provides solid entertainment but not much action.\n\n**1362** _ **Flame of the West**_ **** Monogram, 1945. 71 min. D: Lambert Hillyer. SC: Adele Buffington. With Johnny Mack Brown, Raymond Hatton, Joan Woodbury, Douglass Dumbrille, Lynne Carver, Harry Woods, John Merton, Riley Hill, Steve Clark, Bud Osborne, Jack Rockwell, Ray Bennett, Tom Quinn, Jack Ingram, Eddie Parker, John Cason, Frank McCarroll, Hal Price, Ted Mapes, Kermit Maynard, Pee Wee King and His Golden West Cowboys, Horace B. Carpenter, Henry Wills, Hank Bell, Dick Dickinson. A pacifistic doctor, thought to be a coward by his girlfriend, takes up his guns when a sheriff is murdered by gamblers. One of Johnny Mack Brown's best Westerns; highly recommended.\n\n**1363** _ **Flaming Bullets**_ **** Producers Releasing Corporation, 1945. 59 min. D-SC: Harry Fraser. With Tex Ritter, Dave O'Brien, Guy Wilkerson, Patricia Knox, Charles King, I. Stanford Jolley, Bud Osborne, Kermit Maynard, Richard Alexander, Robert Hill, John Cason, Bob Duncan, Dan White. A lawman pretends to be a wanted outlaw so he can capture a gang that murders criminals to collect reward money. The last effort in PRC's \"Texas Rangers\" series is okay, with Tex Ritter singing a couple of tunes.\n\n**1364** _ **Flaming Feather**_ **** Paramount, 1952. 77 min. Color. D: Ray Enright. SC: Gerald Drayson Adams and Frank Gruber. With Sterling Hayden, Arleen Whelan, Forrest Tucker, Barbara Rush, Richard Arlen, Victor Jory, Edgar Buchanan, Carol Thurston, Ian MacDonald, George Cleveland, Robert Kortman, Ethan Laidlaw, Paul E. Burns, Ray Teal, Nacho Galindo, Frank Lackteen, Donald Kerr, Forrest Taylor, Hank Mann, Heinie Conklin, Perc Launders, Kermit Maynard, Larry McGrath, Boyd \"Red\" Morgan, Sailor Vincent, Chuck Hamilton, Herman Nowlin, Max Wagner, Slim Hightower, Don Dunning. Vigilantes try to rescue a young woman captured by renegade Indians and held prisoner in Montezuma Castle. Well written, action laced feature.\n\n**1365** _ **Flaming Frontier**_ **** 20th Century\u2013Fox, 1958. 70 min. D: Sam Newfield. SC: Louis Stevens. With Bruce Bennett, Jim Davis, Don Garrard, Paisley Maxwell, Cecil Linder, Bill Walsh, Larry Mann, Peter Humphreys, Ben Lennick. After trouble develops between whites and Sioux Indians, a half-breed cavalry officer tries to intervene but runs into prejudice. Fair effort, made on the cheap in Canada.\n\n**Hermina Pipinic and Stewart Granger in** _**Flaming Frontier**_ **(Warner Bros.** **\u2013** **Seven Arts, 1968).**\n\n**  \n**\n\n**1366** _ **Flaming Frontier**_ **** Warner Bros.\u2013Seven Arts, 1968. 93 min. Color. D: Alfred Vohrer. SC: Eberhard Keindorff and Fred Denger. With Stewart Granger, Pierre Brice, Letitia Roman, Larry Pennell, Mario Girotti (Terence Hill), Wolfgang Lukschy, Erik Schumann, Paddy Fox (Milan Srdoc), Bata Zivonjinovic, Dusan Antonijevic, Aleksandar Gavric, Vladimir Medar, Jelina Zigon, Voja Miric, Dusco Janicijevic, Hermina Pipinic, Jelena Jovanovic. When an Indian chief's son is murdered by a gang of outlaws, frontiersman Old Surehand enlists the aid of his Apache blood brother Winnetou in averting war. Sturdy West German oater with Stewart Granger and Pierre Brice good in the leads; issued in Europe in 1965 by Constantin as _**Old Surehand**_.\n\n**1367** _ **Flaming Frontiers**_ **** Universal, 1938. 15 Chapters. D: Ray Taylor and Alan James. SC: Wyndham Gittens, Paul Perez, Basil Dickey, George Plympton and Ella O'Neill. With Johnny Mack Brown, Ralph Bowman (John Archer), Eleanor Hansen, Charles Middleton, James Blaine, George Stevens, William Royle, Horace Murphy, Michael Slade, John Rutherford, Chief Thundercloud, Roy Barcroft, Eddy Waller, Ed Cassidy, Karl Hackett, Iron Eyes Cody, Pat J. O'Brien, Earle Hodgins, J.P. McGowan, Jim Corey, Frank Ellis, Hank Bell, Horace B. Carpenter, Tom Steele, Slim Whitaker, Frank LaRue, Alan Bridge, Blackjack Ward, Ferris Taylor, Bob Woodward, Helen Gibson, George Plues. An Indian scout comes to the aid of a young woman courted by a crook who wants to marry her for her father's gold mine. There is not much logic in this serial but it is made up for by endless action.\n\n**1368** _ **Flaming Gold**_ **** RKO Radio, 1934. 54 min. D: Ralph Ince. SC: Malcolm S. Boylan and John Goodrich. With Bill (William) Boyd, Pat O'Brien, Mae Clarke, Rollo Lloyd, Helen Ware, Robert McQuade. A man is sent to rural Mexico by an oil company to put two rivals out of business but ends up stopping a field fire. Confusing melodrama.\n\n**1369** _ **Flaming Guns**_ **** Universal, 1932. D: Arthur Rosson. SC: Jack Cunningham. With Tom Mix, Ruth Hall, William Farnum, George Hackathorne, Clarence Wilson, Bud Osborne, Duke Lee, Gilbert \"Pee Wee\" Holmes, Bill Steele, Fred Burns, Slim Whitaker, Jimmy Shannon. Assigned to run a ranch, a cowboy gets opposition from the crusty owner as well as an outlaw gang. The weakest of Tom Mix's Universal series, a remake of the Hoot Gibson silent feature _**The Buckaroo Kid**_ (Universal, 1926).\n\n**1370** _ **Flaming Lead**_ **** Colony, 1939. 57 min. D: Sam Newfield. SC: Joseph O'Donnell. With Ken Maynard, Eleanor Stewart, Dave O'Brien, Ralph Peters, Walter Long, Tom London, Carleton Young, Reed Howes, Kenne Duncan, John Merton, Carl Mathews, Bob Terry, Lew Meehan, Ed Peil, Sr., Ernie Adams, Budd Buster, Foxy Callahan, Chick Hannon, Tex Palmer. A cowboy assists a rancher about to lose an Army contract for horses due to constant rustling. One of the best of Ken Maynard's later films with a good story and lots of movement.\n\n**1371** _ **Flaming Star**_ **** 20th Century\u2013Fox, 1960. 101 min. Color. D: Don Siegel. SC: Claire Huffaker and Nunnally Johnson. With Elvis Presley, Barbara Eden, Steve Forrest, Dolores Del Rio, John McIntire, Rodolfo Acosta, Karl Swenson, Ford Rainey, Richard Jaeckel, Anne Benton, L.Q. Jones, Douglas Dick, Tom Reese, Roy Jenson, Virginia Christine, Rodd Redwing, Perry Lopez, Tom Fadden, The Jordanaires, Ted Jacques, Marian Goldina, Sharon Bercutt, Monte Burkhardt. A family gets caught in the middle of an Indian war with their half-breed son forced to choose between whites and Indians. Elvis Presley is good in this well done melodrama many consider his finest film.\n\n**1372** _ **Flap**_ **** Warner Bros., 1970. 105 min. Color. D: Carol Reed. SC: Clair Huffaker. With Anthony Quinn, Shelley Winters, Claude Akins, Victor Jory, Victor French, Tony Bill, Rodolfo Acosta, Anthony Caruso, Susan Miranda, William Mims, John War Eagle, Rudy Diaz, Pedro Regas. A renegade Indian claims the city of Phoenix actually belongs to him. Films tries to show the plight of Native Americans but despite Anthony Quinn's performance it cannot decide to be a comedy or tragedy. British title: _**The Last Warrior**_.\n\n**1373** _ **Flashing Guns**_ **** Monogram, 1947. 59 min. D: Lambert Hillyer. SC: Frank H. Young. With Johnny Mack Brown, Raymond Hatton, Jan Bryant, Riley Hill, James Logan, Douglas Evans, Ted Adams, Gary Garrett, Edmund Cobb, Ray Jones, Jack O'Shea, Steve Clark, Frank LaRue, Jack Rockwell, Bob Woodward. A sheriff helps a rancher and his daughter when a banker, who is after the spread's silver ore, has the man's loan payment stolen before it can be deposited. Routine but entertaining Johnny Mack Brown film.\n\n**1374** _ **Flashing Steeds**_ **** Chesterfield, 1925. 60 min. D: Horace B. Carpenter. With Bill Patton, Dorothy Donald, Merrill McCormick, Ethel Childers, Alfred Hewston, Dick La Reno, Harry O'Connor. A government agent pretending to be a ranch hand tries to stop two swindlers masquerading as British nobility from stealing a retired sea captain's valuable pearls. Cheap but enjoyable Bill Patton silent vehicle.\n\n**1375** _ **Flashpoint**_ **** TriStar, 1984. 94 min. Color. D: William Tannen. SC: Dennis Shryack and Michael Butler. With Kris Kristofferson, Treat Williams, Rip Torn, Kevin Conway, Kurtwood Smith, Miguel Ferrer, Jean Smart, Guy Boyd, Mark Slade, Roberts Blossom, Tess Harper, Terry Alexander, Ana Marie Auther, Barry Davis, Sam Edelman, Robert Elliott, William Frankfeather, Robin Wayne Fugett, Nora Heflin, Henry Max Kendrick, Justin Lord, Joaquin Martinez, Will Morton, Grant Wheeler, Dick O'Neill. After finding a buried jeep containing a corpse, $800,000 and a rifle, two Texas border patrol guards begin to realize they may be involved in the Kennedy assassination cover-up. Pretentious modern-day melodrama.\n\n**1376** _ **La Flecha Envenenada**_ (The Poison Arrow) **** Alameda Films, 1957. 75 min. D: Rafael Baledon. SC: Ramon Obon. With Gaston Santos, Pedro de Aguillon, Otilia Larranaga, Emma Roldan, Leonor Gomez, Carlos Suarez, Guillermo Hernandez, Guillermo Cramer, Armando Arriola, Vicente Larra, Chel Lopez, Bruno Marquez, Salvador Lozano Mena. A cowboy sees an old man murdered and follows the killers to a town where he faces them in a cantina. Good pastime Western from Mexico.\n\n**1377** _ **Flesh and the Spur**_ **** American International, 1956. 77 min. D: Edward L. Cahn. SC: Charles B. Griffith and Mark Hanna. With John Agar, Marla English, Touch (Michael) Connors, Raymond Hatton, Joyce Meadows, Kenne Duncan, Maria Monay, Frank Lackteen, Richard Alexander, Kermit Maynard, Bud Osborne, Buddy Roosevelt, Michael Harris, Mel Gaines, Eddie Kafafian. While searching for the killer of his twin brother, a cowpoke meets a young woman and a gunman who lead him into outlaw territory. Well modulated Alex Gordon production, with a cast of familiar faces and fine performances (especially Raymond Hatton), makes for good entertainment; Mike Connors was one of the producers.\n\n_**Flight from Adventure**_ see _**Tales of Adventure**_\n\n**1378** _ **Flowing Gold**_ **** Warner Bros., 1940. 82 min. D: Alfred E. Green. SC: Kenneth Gamet. With John Garfield, Frances Farmer, Pat O'Brien, Raymond Walburn, Cliff Edwards, Tom Kennedy, Granville Bates, Jody Gilbert, Edward Pawley, Frank Mayo, William Marshall, Virginia Sale, John Alexander, Sol Gorss, E.E. Clive, Eily Malyon, Robert Elliott, Eddie Acuff, Erville Alderson, Alan Bridge, Heinie Conklin, Cliff Clark, Eddy Chandler, Stuart Holmes, G. Pat Collins, Glen Cavender, William Gould, Russell Wade, George Haywood, Dutch Hendrian, Dick Wessel, Jack Mower, Sailor Vincent, Harry Strang, Lee Phelps, Frank O'Connor, George Haywood, William Haade, Walter Soderling, Charles Sullivan, Phillip Morris, Cliff Saum, Charles Sherlock, Monica Bannister, Phyllis Godfrey. A drifter goes to work on an oil drilling operation and becomes the foreman's rival for the boss' daughter. Action filled melodrama, based on Rex Beach's 1922 novel, with nice performances from the three leads and fine comedy support from Raymond Walburn, Cliff Edwards, Tom Kennedy and Jody Gilbert. First filmed by First National in 1924 starring Milton Sills and Anna Q. Nilsson.\n\n**1379** _ **Flying Lariats**_ **** Big 4, 1931. 53 min. D-SC: David Kirkland. With Wally Wales, Buzz Barton, Bonnie Jean Gray, Fred Church, Sam J. Garrett, Joe Lawliss, Etta Dalmas, Tete Brady, Lorraine Laval, Don Wilson, Gus Anderson. Two brothers fall for the same girl and try to raise money for her to ride in a rodeo as a crook schemes with a banker to steal the proceeds from the event. Tiresome, lumbering affair that is poorly photographed and recorded.\n\n_**Follow the Hunter**_ see _**Fangs of the Wild**_ (1954)\n\n**1380** _ **Fool's Gold**_ **** United Artists, 1946. 64 min. D: George Archainbaud. SC: Doris Schroeder. With William Boyd, Andy Clyde, Rand Brooks, Jane Randolph, Robert Emmett Keane, Stephen Barclay, Forbes Murray, Harry Cording, Earle Hodgins, Wee Willie Davis, Ben Corbett, Fred \"Snowflake\" Toones, Glen Gallagher, Johnny Luther, George Sowards, Bob Bentley. An army lieutenant, who deserted when summoned for court martial, is captured by outlaws and Hopalong Cassidy agrees to help his colonel father rescue him. Entertaining later \"Hopalong Cassidy\" series feature.\n\n_**For a Few Bullets More**_ see _**A Few Dollars More**_\n\n**1381** _ **For a Few Dollars More**_ **** United Artists, 1965. 125 min. Color. D: Sergio Leone. SC: Luciano Vincenzoni and Sergio Leone. With Clint Eastwood, Lee Van Cleef, Gian Maria Volonte, Jose Egger, Rosemarie Dexter, Mara Krup, Klaus Kinski, Mario Brega, Aldo Sambrell, Luigi Pistilli, Antonio Molina Rojo, Peter Lee Lawrence, Tomas Blanco, Roberto Camardiel, Benito Stefanelli, Luis Rodriguez, Panos Papadopulos. Two bounty hunters form an uneasy alliance as they track the leader of a vicious outlaw gang. Sergio Leone's follow-up to _**A Fistful of Dollars**_ (q.v.) is a better film, mainly because co-star Lee Van Cleef adds life to the proceedings; filmed in Italy as _**Per Qualche Dollaro in Piu**_ (For a Few Dollars More).\n\n**1382** _ **For Some Dollars Less**_ **** Panda Cinematografica, 1966. 90 min. Color. D: Mario Mattoli. SC: Sergio Corbucci, Bruno Corbucci and Mario Guerra. With Lando Buzzaca, Raimondo Vianello, Gloria Paul, Alberto Giraldi, Valeria Ciangottini, Elio Pandolfi, Lucia Modungo, Angela Luce, Tony Renis. Trying to replace missing bank funds, a cashier teams with his gambler cousin in a scheme that backfires and puts him in jail while the relative gambles away the reward money he collected on him. Silly Italian Western comedy, kidding Sergio Leone's movies, issued there as _**Per Qualche Dollaro in Meno**_ (For a Few Dollars Less).\n\n**1383** _ **For the Love of Mike**_ **** 20th Century\u2013Fox, 1960. 84 min. Color. D: George Sherman. SC: D.D. Beauchamp. With Richard Basehart, Stuart Erwin, Danny Bravo, Arthur Shields, Armando Silvestre, Elsa Cardenas, Rex Allen, Michael Steckler. An Indian boy finds an injured colt, nurses him back to health and trains him for a race so he can use his winnings for a new church. Minor but satisfying family fare.\n\n**1384** _ **For the Service**_ **** Universal, 1936. 65 min. D: Buck Jones. SC: Isadore Bernstein. With Buck Jones, Beth Marion, Fred Kohler, Clifford Jones, Edward Keane, Frank McGlynn, Ben Corbett, Chief Thundercloud, Lafe McKee, Richard Cramer, Slim Whitaker, Jack Ingram, Francis Walker, Alan Sears, Phillip Trent. In Indian territory a government agent tries to outwit an outlaw gang. Action filled Buck Jones film produced and directed by the star.\n\n**1385** _ **For the Taste of Killing**_ **** Hercules Cinematografica\/Montana Films, 1966. 89 min. Color. D: Tonino Valerii. SC: Victor Auz. With Craig Hill, George Martin, Fernando Sancho, Peter Carter (Piero Lulli), Diana Martin, Frank Ressel, Rada Rassimov, Graham Sotty, George Wang, Jose Marco. A bounty hunter, who only trails convoys of stolen gold, is encouraged to bet his own money that a shipment will not be robbed. Craig Hill plays a character called \"Lanky Fellow\" in this fair Italian made oater issued there as _**Per Il Gusto di Uccidere**_ (For the Taste of Killing) and shown in Great Britain as _**A Taste for Killing**_.\n\n**1386** _ **El Forastero Vengador**_ (The Avenging Stranger) **** Artistas Nacionales Asociados, 1966. 95 min. Color. D: Fernando Fernandez. SC: Jose Delfos and Fernando Fernandez. With Jaime Fernandez, Eleazar Garcia Chelelo, Ofelia Guilmain, Dagoberto Rodriguez, Crox Alvarado, Fernando Soto \"Mantequilla,\" Agustin Fernandez, Mario Garcia \"Harapos,\" Glenda Castro, Oscar Alatorre, Juan Garza, Ignacio Villalbazo, Bruno Rey, Jesus Gomez, Roberto Porter, Felix Gonzalez, Cesar Jimenez. A female rancher is faced with a challenge from a newly arrived stranger when he refuses her request to kill her daughter's boyfriend. Okay Western drama from Mexico.\n\n**1387** _ **The Forbidden Trail**_ **** Sunset, 1923. 62 min. D-SC: Robert North Bradbury. With Jack Hoxie, Evelyn Nelson, Frank Rice, William Lester, Joe McDermott, Tom Lingham, Steve Clemento. Hunting his father's killer, a cowboy falls in love with a girl he believes is the murderer's daughter. Fast moving and entertaining Jack Hoxie silent feature.\n\n**1388** _ **Forbidden Trail**_ **** Columbia, 1932. 71 min. D: Lambert Hillyer. SC: Milton Krims. With Buck Jones, Barbara Weeks, Mary Carr, George Cooper, Ed Brady, Frank Rice, Al Smith, Frank LaRue, Wallis Clark, Tom Forman, Dick Rush. A cowboy arrives in town and helps a young woman newspaper editor fight a rustler-land grabber. Too much comedy makes this Buck Jones film a bit slow although it does have an exciting finale.\n\n**1389** _ **Forbidden Trails**_ **** Monogram, 1941. 60 min. D: Robert North Bradbury. SC: Jess Bowers (Adele Buffington). With Buck Jones, Tim McCoy, Raymond Hatton, Dave O'Brien, Tristram Coffin, Christine McIntyre, Charles King, Glenn Strange, Lynton Brent, Hal Price, Richard Alexander, Bud Osborne, Jerry Sheldon, Frank Yaconelli, Marin Sais, Tom London, Ed Peil, Sr., Eddie Phillips, Milburn Morante, Bill Nestell, Lee Shumway, Kansas Moehring, Herman Hack, Silver Tip Baker, Dan White, Lew Morphy, Tex Palmer, Jack Kirk, Jess Cavin, Jack Evans, Rube Dalroy. The Rough Riders help a mine owner who is being forced to sign a hauling contract. Fast action entry in the fine \"Rough Riders\" series.\n\n**1390** _ **Forbidden Valley**_ **** Universal, 1938. 67 min. D-SC: Wyndham Gittens. With Noah Beery, Jr., Frances Robinson, Robert Barrat, Fred Kohler, Henry Hunter, Samuel S. Hinds, Stanley Andrews, Spencer Charters, Charles Stevens, Margaret McWade, John Ridgely, Ray Jones, Alonzo Price, Soledad Jiminez, Ferris Taylor, Glenn Strange, Robert Kortman, Sarah Padden, James Foran, Robert Rosson. After growing up in a secret canyon, a young man attempts to round up a herd of wild horses and take them to market but rustlers steal them. Nicely done adventure program feature; remade as _**Sierra**_ (q.v.).\n\n**1391** _ **The Force on Thunder Mountain**_ **** American National Enterprises, 1978. 93 min. Color. D-SC: Peter B. Good. With Christopher Cain, Todd Dutson, Borge West, David Fogg, James Lyle Strong. In 1888 two prospectors die looking for gold and ninety years later a father and son go camping on the same mountain and experience strange phenomena. Cheaply made but fun family fantasy Western.\n\n**1392** _ **The Forest Rangers**_ **** Paramount, 1942. 87 min. Color. D: George Marshall. SC: Harold Shumate. With Fred MacMurray, Paulette Goddard, Susan Hayward, Lynne Overman, Albert Dekker, Eugene Pallette, Regis Toomey, Rod Cameron, Clem Bevans, James Brown, Kenneth Griffith, Monte Blue, Keith Richards, William Cabanne, George Chandler, Tim Ryan, Lee Phelps, Chester Clute, Pat West, Sarah Edwards, Jimmy Conlin, Robert Kent, Jack Mulhall, George Turner, Robert Kortman, Perc Launders, Ethan Laidlaw, Louise LaPlanche, Edwin Brady, Harry Woods, Al Thompson, Arthur Loft, Robert Homans, Katharine Booth, Bert Stevens, Pat West, Wade Boteler, Byron Foulger, Frank Coleman, Howard Mitchell, James Millican, Carl Saxe. A forest ranger marries a wealthy heiress and his ex-love tries to prove to him that he made a mistake. Technicolor, lots of action and a forest fire help to make up for a mundane plot.\n\n_**The Forged Will**_ see _**The Blazing Trail**_\n\n**1393** _ **Forgotten Pistolero**_ **** Constantin Film\/Izaro Film, 1969. 88 min. Color. D: Ferninando Baldi. SC: Ferninando Baldi, Vincenzo Cerami, Federico De Urrutia and Mario di Nardo. With Leonard Mann, Luciana Paluzzi, Peter Martell, Alberto de Mendoza, Pilar Velazquez, Piero Lulli, Luciano Rossi, Jose Suarez, Barbara Nelly, Franco Pesce, Mirella Pompili, Enzo Fiermonte, Silvana Bacci, Francesco Gula, Jose Manuel Martin, Jose Riesco, Nicola Solari, Eugenio Galadini, Jose Terron, Renzo Pevarello, Omero Capanna, Osiride Pevarello. A gunman seeks the murderer of his father and along the way falls in love with a beautiful woman with a mysterious past. Well made, action filled Spaghetti Western reworking of Euripides' \"Orestes\"; released in Spain as _**Tierra de Gigantes**_ (Land of Giants); also called _**Gunman of Ave Maria**_.\n\n**1394** _ **Forlorn River**_ **** Paramount, 1937. 56 min. D: Charles Barton. SC: Stuart Anthony and Robert Yost. With Larry \"Buster\" Crabbe, June Martel, Harvey Stephens, John Patterson, Chester Conklin, Lew Kelly, Syd Saylor, William Duncan, Rafael (Ray) Bennett, Lee Powell, Robert Homans, Purnell B. Pratt, Merrill McCormick, Vester Pegg, Buffalo Bill, Jr., Hank Bell. A cowboy is on the trail of a wily outlaw who framed him for a crime he did not commit. Nicely done version of the Zane Grey story, filmed as a silent in 1926 with Jack Holt. TV title: _**River of Destiny**_.\n\n_**The Fort**_ see _**Renegade of the Sage**_\n\n**1395** _ **Fort Apache**_ **** RKO Radio, 1948. 127 min. D: John Ford. SC: Frank S. Nugent. With John Wayne, Henry Fonda, Shirley Temple, Pedro Armendariz, John Agar, Ward Bond, Irene Rich, George O'Brien, Anna Lee, Victor McLaglen, Dick Foran, Jack Pennick, Guy Kibbee, Grant Withers, Miguel Inclan, Mae Marsh, Movita Castenada, Francis Ford, Frank Ferguson, Mickey Simpson, Ray Hyke, Mary Gordon, Hank Worden, Archie Twitchell, William Forrest, Cliff Clark, Fred Graham, Philip Keiffer, Ben Johnson, Harry Tenbrook, Dan Borgaze, Gil Perkins, Frank McGrath. An arrogant lieutenant is at odds with a captain at a remote Army post threatened by an Indian attack. Classic John Ford cavalry drama; well worth seeing.\n\n**1396** _ **Fort Bowie**_ **** United Artists, 1958. 80 min. D: Howard W. Koch. SC: Maurice Tombragel. With Ben Johnson, Kent Taylor, Jan Harrison, Jana Davi, Larry Chance, Ian Douglas, Peter Mamakos, Jerry Frank, Johnny Western, Ed Hinton, Barbara Parry. A fort commander believes an officer is romancing his wife while the two men face the danger of an Indian attack. Ben Johnson and Kent Taylor bring some life to this programmer.\n\n**1397** _ **Fort Courageous**_ **** 20th Century\u2013Fox, 1965. 72 min. D: Lesley Selander. SC: Richard Landau. With Fred Beir, Donald Barry, Hanna Landy, Harry Lauter, Walter Reed, Michael Carr, Fred Krone, George Sawaya, Joseph Patridge, Cheryl MacDonald. A court-martialed sergeant takes over command of a fort beleaguered by Indian raids. Low budget affair of interest to Don Barry and Harry Lauter fans.\n\n**Harry Lauter and George Sawaya in** _**Fort Courageous**_ **(20th Century** **\u2013** **Fox, 1965).**\n\n**  \n**\n\n**1398** _ **Fort Defiance**_ **** United Artists, 1951. 81 min. Color. D: John Rawlins. SC: Louis Lantz. With Dane Clark, Ben Johnson, Peter Travey, Tracey Roberts, Dennis Moore, George Cleveland, Ralph Sanford, Iron Eyes Cody, Craig Woods, Dick Elliott. Relationships build between several people at an outpost about to be attacked by Navajo Indians. Interesting \"B\" melodrama with more emphasis on characterization than action.\n\n**1399** _ **Fort Dobbs**_ **** Warner Bros., 1958. 90 min. D: Gordon Douglas. SC: Burt Kennedy and George W. George. With Clint Walker, Virginia Mayo, Brian Keith, Richard Eyer, Russ Conway, Michael Dante, John Day. A widow and her young son are escorted through Indian country by the man she believes killed her husband. Well done oater with fine performances by Clint Walker and Virginia Mayo.\n\n**1400** _ **Fort Dodge Stampede**_ **** Republic, 1951. 60 min. D: Harry Keller. SC: Richard Wormser. With Allan \"Rocky\" Lane, Mary Ellen Kay, Roy Barcroft, Chubby Johnson, Trevor Bardette, Bruce Edwards, Wesley Hudman, William Forrest, Chuck Roberson, Rory Mallinson, Jack Ingram, Kermit Maynard, Wally West. A deputy sheriff has to give up his badge when he goes after an outlaw gang looking for hidden loot. Another action filled and well written entry in Allan Lane's \"Famous Westerns\" series.\n\n**1401** _ **Fort Doom**_ **** Silver Nitrate, 2004. 93 min. Color. D: J. Christian Ingvordsen. SC: Matthew M. Howe. With Debbie Rochon, Billy Drago, Rick Washburn, Joshua Park, J. Christian Ingvordsen, Jennifer Lauren Grant, Mya Sagara, Aaron Roman Weiner, Meyer DeLeeuw, Melissa Paladino, Jesse Steccato, Matthew M. Howe. A group of hookers decides to take up residence in an abandoned fort despite the warnings from locals about a mad killer in the area. Cheap horror-Western-comedy video production.\n\n**1402** _ **Fort Massacre**_ **** United Artists, 1958. 80 min. Color. D: Joseph M. Newman. SC: Martin N. Goldsmith. With Joel McCrea, Forrest Tucker, Susan Cabot, John Russell, Anthony Caruso, Robert Osterloh, Denver Pyle, Rayford Barnes, Guy Prescott, Irving Bacon, Claire Carleton, Francis McDonald, George N. Neise. A cavalry sergeant leads a band of soldiers constantly being attacked by warring Indians. A nice combination of action and characterization; well done.\n\n**1403** _ **Fort Osage**_ **** Monogram, 1952. 72 min. Color. D: Lesley Selander. SC: Dan Ullman. With Rod Cameron, Jane Nigh, Douglas Kennedy, Morris Ankrum, John Ridgely, William Phipps, I. Stanford Jolley, Dorothy Adams, Francis McDonald, Myron Healey, Lane Bradford, Iron Eyes Cody, Barbara Woodell, Russ Conway, Marshall Reed, Carol Henry, Lee Roberts, Fred Graham, Ray Jones. As he leads a wagon train West a scout learns the people who hired him are behind an Indian uprising. Considering those involved, this is a bit of a disappointment.\n\n**1404** _ **Fort Savage Raiders**_ **** Columbia, 1951. D: Ray Nazarro. SC: Barry Shipman. With Charles Starrett, Smiley Burnette, John Dehner, Trevor Bardette, Peter Thompson, Fred F. Sears, John Cason, Frank Griffin, Sam Flint, Dusty Walker. The Durango Kid and three pals, along with a West Point man, track an Army deserter and his gang that has been raiding the countryside. Well made \"Durango Kid\" episode with a sympathetic villain well played by John Dehner.\n\n**1405** _ **Fort Ti**_ **** Columbia, 1953. 73 min. Color. D: William Castle. SC: Robert E. Kent. With George Montgomery, Joan Vohs, Irving Bacon, James Seay, Ben Astar, Phyllis Fowley, Howard Petrie, Lester Matthews, Louis Merrill, Cicely Browne, George Lee. In 1759 Rogers' Rangers join the English in fighting the French and Indians at Fort Ticonderoga. This low budget Sam Katzman production covers the same ground as _**Northwest Passage**_ (q.v.) but not nearly as well; shown in 3-D.\n\n**1406** _ **Fort Utah**_ **** Paramount, 1967. 83 min. Color. D: Lesley Selander. SC: Steve Fisher and Andrew Craddock. With John Ireland, Virginia Mayo, Scott Brady, John Russell, Robert Strauss, James Craig, Richard Arlen, Jim Davis, Donald Barry, Harry Lauter, Read Morgan, Reg Barton, Eric Cody. An Indian agent and a cowboy are forced to defend a wagon train from attacking Native Americans. Mediocre A.C. Lyles production, although the cast is worth watching.\n\n**Virginia Mayo and John Ireland in** _**Fort Utah**_ **(Paramount, 1967).**\n\n**  \n**\n\n**1407** _ **Fort Vengeance**_ **** Allied Artists, 1953. 75 min. Color. D: Lesley Selander. SC: Dan Ullman. With James Craig, Rita Moreno, Keith Larsen, Reginald Denny, Morris Ankrum, Guy Kingsford, Paul Marion, Emory Parnell, Charles Irwin, Michael Granger, Patrick Whyte, Jack Ingram, William Forrest. Two men, one of whom is wanted by the law, go to Canada, join the Mounties and become involved with fur thieves, an Indian uprising and romance. Colorful action melodrama.\n\n**1408** _ **Fort Worth**_ **** Warner Bros., 1951. 80 min. Color. D: Edwin L. Marin. SC: John Twist. With Randolph Scott, David Brian, Phyllis Thaxter, Helena Carter, Dick Jones, Ray Teal, Paul Picerni, Emerson Treacy, Bob Steele, Lawrence Tolan, Walter Sande, Chubby Johnson, Don C. Harvey, Lee Roberts, Bud Osborne, Kermit Maynard, Zon Murray. Arriving in Fort Worth via wagon train, a man starts a newspaper and accuses the trail boss of committing murder on the orders of the town's leading citizen. Okay \"A\" Western.\n\n**1409** _ **Fort Yuma**_ **** United Artists, 1955. 78 min. Color. D: Lesley Selander. SC: Danny Arnold. With Peter Graves, Joan Vohs, Joan Taylor, Abel Fernandez, Stanley Clements, John Pickard, Addison Richards, John Hudson, William \"Bill\" Phillips, Lee Roberts, Edmund Penry. When a homesteader kills an Apache chief, the Indians go on the warpath. Average action drama that moves at a good clip.\n\n**1410** _ **Fort Yuma Gold**_ **** Gala, 1969. 100 min. Color. D: Calvin Jackson Paget (Giorgio Ferroni). SC: Augusto Finocchi and Massimiliano Capriccioli. With Montgomery Wood (Giuliano Gemma), Dan Vadis, Jacques Sernas, Jose Calvo, Sophie Daumier, Angel Del Pozo, Nello Pazzafini, Alfonso Rojas, Jacques Herlin, Andrea Bosic, Antonio Molina Rojo. At the end of the Civil War, a Southern major plans to attack a Western fort to get its gold. Lethargic but violent Spaghetti Western issued in Italy in 1966 as _**Per Pochi Dollari Ancora**_ (For a Few Extra Dollars) by FIDA\/Epoca Film.\n\n**1411** _ **40 Graves for 40 Guns**_ **** Boxoffice International, 1971. 95 min. Color. D: Paul Hunt. SC: Steve Fisher. With Robert Padilla, Stanley Adams, Richard Rust, Mahita Saint Duvall, Rita Rogers, Steven Oliver, David Eastman, Rockne Tarkington, Michael Christine, Owen Orr, Michael Green. An outlaw gang heads south of the border, raids a small village and carries off a priceless gold cross only to be relentlessly pursued by the Mexican army. Brutal action filled melodrama filmed in Arizona as _**El Salvejo**_ (The Savage) and reissued in a toned down version in 1977 by Sun Productions as _**The Great Gundown**_ ; also called _**Maschimo\u201440 Graves for 40 Guns**_ , _**Savage Red\u2014Outlaw White**_ and _**The Revenge of the Wild Bunch**_. ****\n\n**1412** _ **Forty Guns**_ **** 20th Century\u2013Fox, 1957. 80 min. D-SC: Samuel Fuller. With Barbara Stanwyck, Barry Sullivan, Dean Jagger, John Ericson, Gene Barry, Robert Dix, Jidge Carroll, Paul Dubov, Gerald Milton, Ziva Rodann, Hank Worden, Neyle Morrow, Chuck Roberson, Chuck Hayward, Eve Brent, Eddie Parks. A tough woman appoints herself the ruler of Tombstone, Arizona, and finds opposition from an ex-gunfighter, now working for the U.S. attorney general, and his brothers. Film is interesting for Barbara Stanwyck's work in the leading role and it will also appeal to Sam Fuller followers.\n\n**1413** _ **40 Guns to Apache Pass**_ **** Columbia, 1967. 95 min. Color. D: William Witney. SC: Willard Willingham and Mary Willingham. With Audie Murphy, Kenneth Tobey, Michael Burns, Laraine Stephens, Robert Brubaker, Kay Stewart, Kenneth MacDonald, Byron Morrow, Michael Blodgett, Michael Keep, Willard Willingham, Ted Gehring, Jackson Beck. When Cochise and his braves declare war, a cavalry captain leads settlers to safety and goes after the trader who sold the Indians stolen rifles. Average Audie Murphy vehicle.\n\n**1414** _ **The Forty-Niners**_ **** Monarch\/Freuler, 1932. 59 min. D: J.P. McCarthy. SC: F. McGrew Willis. With Tom Tyler, Betty Mack, Alan Bridge, Gordon (DeMain) Wood, Fern Emmett, Mildred Rogers, Fred Ritter, Frank Ball, Florence Wells. Trying to help settlers on a wagon train going westward, a cowboy faces trouble from outlaws along with a buffalo stampede. Crudely made Tom Tyler movie.\n\n**1415** _ **The Forty-Niners**_ **** Allied Artists, 1954. 71 min. D: Thomas Carr. SC: Dan Ullman. With Bill Elliott, Virginia Grey, Henry (Harry) Morgan, John Doucette, Lane Bradford, I. Stanford Jolley, Ralph Sanford, Gregg Barton, Harry Lauter, Earle Hodgins, Dean Cromer, Stanley Price, Jack O'Shea. To find out the identities of three men involved in a killing, a marshal takes on the guise of a murderer. Well done Bill Elliott feature, his last Western.\n\n**1416** _ **Forty Thieves**_ **** United Artists, 1944. 61 min. D: Lesley Selander. SC: Michael Wilson and Bernie Kamins. With William Boyd, Andy Clyde, Jimmy Rogers, Louise Currie, Douglass Dumbrille, Kirk Alyn, Herbert Rawlinson, Robert Frazer, Glenn Strange, Jack Rockwell, Robert Kortman, Hal Taliaferro, Earle Hodgins, Bill Nestell, Herman Hack, Richard Botiller, Tex Harper, Lew Morphy, George Sowards, Hank Worden, Denver Dixon. A fixed election causes Hopalong Cassidy to lose his job as sheriff and he sets out to track down those responsible for stuffing the ballot boxes. Cheaply made but action filled \"Hopalong Cassidy\" oater, the final one produced by Harry Sherman.\n\n**1417** _ **Four Bullets for Joe**_ **** J.J. Films, S.A., 1964. 82 min. Color. D: Agustin Navarro. SC: Fernando Galiana and Julio Porter. With Paul Piaget, Fred Canow (Fernando Casanova), Liz (Poitel) Poiter, Barbara (Nelli) Nelly, Angela Cavo, Frank (Paco) Moran, Tullio Altamura, Rafael Bardem, Juan Cazalilla, Juan Cortes, Miguel Del Castillo, Jose Angel Espinosa \"Ferrusquilla,\" Tito Garcia, John (Jose) Marco, Fernando Montes, Jose Riesgo, Brunco Scipioni. A vengeful gunman arrives in a Kansas town to avenge the death of his sister who was falsely accused of killing her boyfriend. Violent French-Italian-Spanish co-production made as _**Cuatro Balazos**_ (Four Shots) and also called _**Shots Ring Out!**_\n\n**1418** _ **Four Came to Kill Sartana**_ **** Tarquinia Film, 1969. 97 min. Color. D: Miles Deem (Demofilo Fidani). SC: Demofilo Fidani and Mila Vitelli. With Jeff Cameron (Geoffredo Scarciofolo), Anthony G. Stanton (Franco Ricci), Celso Faria, Dennis Colt (Benito Pacifico), Peter Torres (Pietro Torrisi), Simone Blondell (Simonetta Vitelli), Robert Danish (Roberto Danesi), Umberto Raho, Grazia Giuvi, Frank Fargas, Paul Carter, Gualtiero Rispoli, Custer Gail, Mariella Palmich, Luciano Conti. Sartana tries to stop a mysterious bad man known as The Mormon from using his gunmen to kidnap citizens for ransom. Threadbare \"Sartana\" series entry; filmed in Italy as _**E Vennero in Quattro...per Uccidere Sartana**_ (Four Came to Kill Sartana) and re-released in 1972 as _**Beyond the Frontiers of Hate**_.\n\n**1419** _ **Four Dollars for Revenge**_ **** GAR Film, 1966. 88 min. Color. D: J. Warren (Alfonso Balcazar). SC: Bruno Corbucci and Giovanni Grimaldi. With Robert Woods, Ghia Arlen, Jack Stuart, Dan Vadis, Jose Torres, Rosy Zichel, John MacDouglas (Giuseppe Addobbati), Dick Reagan (Riccardo Garrone), Angelo Infanti, Antonio Casas, Jose Manuel Martin, Gerard Tichy, Tomas Torres, Antonio Molino, Rojo, Giulio Maculani, Osvaldo Genazzani, Gardenia Polito, Lucio Rosato, Gustavo Re, Sergio Dore, Carlos Ronda, Gianluigi Crescenzi. At the close of the Civil War a Union officer, falsely accused of being complicit in a massacre that resulted in the deaths of his men while escorting Confederate gold, escapes from prison to find the real bandits. Well made and photographed (by Victor Monreal) Spaghetti Western issued in Italy as _**Quattro Dollari di Vendetta**_ (Four Dollars for Vengeance) and in Spain as _**Cuatro Dolares de Venganza**_ (Four Dollars of Vengeance).\n\n**1420** _ **Four Faces West**_ **** United Artists, 1948. 90 min. D: Alfred E. Green. SC: Graham Baker. With Joel McCrea, Frances Dee, Charles Bickford, Joseph Calleia, William Conrad, Martin Garralaga, Raymond Largay, John Parrish, Dan White, Davison Clark, Eva Novak, Houseley Stevenson, Sam Flint, Forrest Taylor, George McDonald. A man robs a bank to get money to save his father's ranch and is pursued by a sheriff but helped by a railroad nurse and a saloon keeper. Entertaining but fairly non-violent oater with fine performances. British title: _**They Passed This Way**_.\n\n**1421** _ **Four Fast Guns**_ **** Universal-International, 1960. 74 min. D: William Hole, Jr. SC: James Edmiston and Dallas Gaultois. With James Craig, Martha Vickers, Edgar Buchanan, Brett Halsey, Paul Richards, Richard Martin, Blu Wright, John Swift, Paul Raymond, Jim Hurley, Grizzly Green, Roger Anderson. When a gunman is hired to rid a town of its lawless element, he is forced into a showdown with his own brother. There is nothing special about this average oater.\n\n**1422** _ **4 for Texas**_ **** Warner Bros., 1963. 124 min. Color. D: Robert Aldrich. SC: Allan Weiss. With Frank Sinatra, Dean Martin, Anita Ekberg, Ursula Andress, Victor Buono, Charles Bronson, Richard Jaeckel, Eric Connor, Nick Dennis, Mike Mazurki, Wesley Addy, Marjorie Bennett, Jack Elam, Fritz Feld, Percy Helton, Jonathan Hale, Jack Lambert, Paul Langton, Bob Steele, Virginia Christine, Ellen Corby, Ralph Volkie, The Three Stooges (Moe Howard, Larry Fine, Joe Da Rita), Teddy Buckner and His All Stars, Arthur Godfrey, Jessalyn Fax, Allyson Ames. Two feuding conmen become involved with a crooked banker and join forces to thwart his nefarious activities. Poorly conceived Western comedy with only villains Victor Buono and Charles Bronson plus a lot of fine character actors to recommend it.\n\n**1423** _ **Four Guns to the Border**_ **** Universal-International, 1954. 83 min. Color. D: Richard Carlson. SC: George Van Marter and Franklin Coen. With Rory Calhoun, Colleen Miller, George Nader, Walter Brennan, Nina Foch, John McIntire, Charles Drake, Jay Silverheels, Nestor Paiva, Mary Field, Reg Parton, Paul Brinegar, Henry Wills. After holding up a bank, an outlaw gang helps an ex-gunman and his daughter who are being attacked by Indians. A different story line makes this oater acceptable entertainment; directed by actor Richard Carlson.\n\n**1424** _ **Four of the Apocalypse**_ **** Coralta Cinematografica, 1975. 87 min. Color. D: Lucio Fulci. SC: Ennio De Concini. With Fabio Testi, Lynne Frederick, Michael J. Pollard, Tomas Milian, Harry Baird, Adolfo Lastretti, Bruno Corazzari, Giorgio Trestini, Donald O'Brien, Claudio Ruffini, Goffredo Unger, Charles Borromel, Salvatore Puntillo, Lorenzo Robeldo, Edward Mannix (narrator). After escaping a massacre, four criminals, including a prostitute, try to survive in frontier Utah but are harassed by a sadistic Mexican outlaw. A Spaghetti Western filled with symbolism from director Lucio Fulci, best known for his gory horror films; released in Italy as _**I Quattro dell'Apocalisse**_ (The Four of the Apocalypse).\n\n**1425** _ **Four Rode Out**_ **** ADA Films\/Sagittarius Productions, 1969. 99 min. Color. D: John Peyser. SC: Paul Harrison and Don Balluck. With Sue Lyon, Pernell Roberts, Leslie Nielsen, Julian Mateos, Maria Martin, John Clark, Bob Hall, Leonard Bell, Charles Drace, Neil Wright, Janis Ian, Albert Salmi. When accused of robbing a bank and committing murder, a Mexican heads into the desert followed by a sheriff, his girlfriend and a Pinkerton man. The acting is the best thing about this U.S.-Spanish co-production filmed in Spain; the story is by actor Dick Miller and the music by Janis Ian.\n\n**1426** _ **The Fourth Horseman**_ **** Universal, 1932. 63 min. D: Hamilton MacFadden. SC: Jack Cunningham. With Tom Mix, Margaret Lindsay, Fred Kohler, Raymond Hatton, Rosita Marstini, Edmund Cobb, Richard Cramer, Herman Nolan, Paul Shawhan, Donald Kirke, Harry Allan, Duke Lee, C.E. Anderson, Helene Millard, Martha Mattox, Buddy Roosevelt, Frederick Howard, Grace Cunard, Walter Brennan, Pat Harmon, Hank Mann, Jim Corey, Delmar Watson, Fred Burns, Bud Osborne, Harry Tenbrook, Charles Sullivan, Augie Gomez. A cowboy wants to help a young woman save her ghost town property since irrigation will revive the area but he learns outlaws are using it as a hideout. Entertaining and well made Tom Mix vehicle.\n\n**1427** _ **The Foxes of Harrow**_ **** 20th Century\u2013Fox, 1947. 117 min. D: John M. Stahl. SC: Wanda Tuchock. With Rex Harrison, Maureen O'Hara, Richard Haydn, Victor McLaglen, Vanessa Brown, Patricia Medina, Gene Lockhart, Charles Irwin, Hugo Haas, Roy Roberts, Dennis Hoey, Marcel Journet, Helen Crozier, Sam McDaniel, Libby Taylor, Renee Beard, Suzette Marbin, Percy William Ward, Clear Nelson, Jr., James Lagano, Dorothy Adams, Celia Lovsky, Eugene Borden, Gordon Clark, James Kirkwood, Robert Emmett Keane, Bernard DeRoux, Frederick Burton, Wee Willie Davis, Randy Stuart, William Norton Bailey, William Walker, Mary Currier, William Schallert, Paul Maxey, Andre Charlot, Georges Renavent, Joseph Crehan, Maynard Holmes, Russ Conklin, John Doucette, Cy Schindel, Jim Toney, John Hamilton, Alberto Morin, Perry Ivins, John Bagni, A.C. Bilbrew. In 1820 New Orleans a gambler woos and weds a society belle only to leave her. Colorful frontier soap opera based on the Frank Yerby novel.\n\n**1428** _ **Foxfire**_ **** Universal-International, 1955. 93 min. Color. D: Joseph Pevney. SC: Ketti Frings. With Jane Russell, Jeff Chandler, Dan Duryea, Mara Corday, Robert F. Simon, Frieda Inescort, Barton MacLane, Charlotte Wynters, Eddy Waller, Celia Lovsky, Arthur Space, Phil Chambers, Robert Bice, Vici Raaf, Grace Lenard, Guy Wilkerson, Lillian Bronson, Dabbs Greer, Hal K. Dawson, Billy Wilkerson, Charles Soldani. A pretty socialite weds a Western mining engineer and his quest for gold almost destroys their marriage. Murky melodrama with star Jeff Chandler doing a good job singing the title song.\n\n**1429** _ **Frank and Jesse**_ **** Trimark Pictures, 1995. 105 min. Color. D-SC: Robert Boris. With Rob Lowe, Bill Paxton, Randy Travis, Dana Wheeler-Nicholson, Maria Pitillo, Luke Askew, Sean Patrick Flanery, Alexis Arquette, Todd Field, John Pyper-Ferguson, Nick Sadler, William Atherton, Tom Chick, Mary Neff, Richard Maynard, Jim Flowers, Mari Askew, William Michael Evans, Lyle Armstrong, Cole McKay, Dennis Letts, John Stiritz, Micah Dyer, Jackie Stewart, Chad Linley, Rhed Khilling, Jerry Saunders, D.C. \"Dash\" Foff, Robert Moniot, Norman Hawley, Jeffrey Paul Johnson, Bryce Anthony Thomason, John Paxton, Elizabeth Hatcher-Travis, Sudie Henson, Ron Licardi. When railroad tycoons cheat them out of their land, Frank and Jesse James team with the Ford and Younger brothers and other outlaws to carry out a series of successful robberies, causing them to be pursued by Pinkerton agents. Fair retelling of the James boys' saga.\n\n**1430** _ **Frankie and Johnnie**_ **** Republic, 1936. 66 min. D: Chester Erskine. SC: Lou Goldberg and Moss Hart. With Helen Morgan, Chester Morris, Lilyan Tashman, Florence Reed, Walter Kingsford, William Harrigan, John Larkin, Cora Witherspoon, Montagu Love, Jean Brooks. In 1870 a Mississippi riverboat huckster romances a dance hall singer but later two times her over a saloon girl. Tepid cinematic version of the famous folk song set for release in 1934 by RKO Radio but sold to Republic.\n\n**1431** _ **Freckles**_ **** 20th Century\u2013Fox, 1960. 83 min. Color. D: Andrew V. McLaglen. SC: Harry Spalding. With Martin West, Carol Christensen, Jack Lambert, Steven Peck, Roy Barcroft, Lorna Thayer, Ken Curtis, John Eldredge. An orphaned young man comes to Oregon's Limberlost country where he is befriended by a girl and a pretty school teacher and becomes a guard against timber thieves. Shot on location, this is a fair cinematic retelling of Gene Stratton Porter's 1904 novel, previously filmed by Paramount in 1917 with Jack Pickford, Hobart Bosworth and Louise Huff and directed by Marshall Neilan; in 1928 by Film Booking Offices (FBO) with Johnny Fox, the work's author, Gene Stratton Porter, and Hobart Bosworth repeating the role of McLean from the 1917 production; and in 1935 by RKO Radio starring Tom Brown, Carol Stone and Virginia Weidler.\n\n_**Freddie Goes West**_ see _**Vacation Days**_\n\n**1432** _ **Freighters of Destiny**_ **** RKO Radio, 1931. 60 min. D: Fred Allen. SC: Adele Buffington. With Tom Keene, Barbara Kent, Frank Rice, Mitchell Harris, Fred Burns, Slim Whitaker, Billy Franey, Frederick Burton, William Welsh, Fred Burns, Art Mix, George Hayes, Bill Nestell, Jim Corey, Hank Bell, Jack Kirk, Bud McClure, Chuck Baldra, Edward Burns, Charles Brinley, Tom Bay, Bob Roper. A cowboy helps lead a wagon train carrying pioneers westward. Well produced entry in Tom Keene's RKO series.\n\n**1433** _ **Frenchie**_ **** Universal-International, 1951. 80 min. Color. D: Louis King. SC: Oscar Brodney. With Joel McCrea, Shelley Winters, Paul Kelly, Elsa Lanchester, John Russell, Marie Windsor, John Emery, George Cleveland, Regis Toomey, Paul E. Burns, Frank Ferguson, Larry (Lawrence) Dobkin, Vincent Renno, Lucille Barkley, Tudor Owen, George Eldredge, Jack Ingram, Jack Perrin, Al Ferguson, Chuck Hamilton, Chilli Williams, Chubby Johnson, Billy Wayne, Perc Launders, Max Wagner, Frank McCarroll, Harry Tenbrook, Brick Sullivan, Forbes Murray, Frank Mills, John Pickard, Monte Montague, Jerry Paris, Jack Stoney, Steve Clark, Kit Guard, Sam Flint, Art Dupuis, Helen Dickson, John Cliff, Roy Butler, Sherry Hall, Eileen Howe, Mike Lally, Frank Malet, George Ryland, Paul Palmer, William J. O'Brien, Mary Ellen Gleason, Shirley Ballard, Marie Allison, Sam Finn. After her father is murdered by a gunman, a young woman returns to a Western town, opens a saloon and plans to avenge his death. Mediocre re-filming of Max Brand's \"Destry Rides Again.\"\n\n**1434** _ **The Friendly Persuasion**_ **** Allied Artists, 1956. 140 min. Color. D: William Wyler. SC: (uncredited) Michael Wilson. With Gary Cooper, Dorothy McGuire, Marjorie Main, Anthony Perkins, Richard Eyer, Phyllis Love, Robert Middleton, Mark Richman, Walter Catlett, Richard Hale, Joel Fluellen, Theodore Newton, John Smith, Mary Carr, Edna Skinner, Russell Simpson, Charles Halton, Everett Glass, Richard Garland, James Dobson, John Compton, James Seay, Diane Jergens, Ralph Sanford, Nelson Leigh, William Schallert, John Craven, Frank Jenks, Frank Hagney, Marjorie Durant, Frances Farwell, Jean Inness, Helen Kleeb, Marty Jackson. In Indiana during the Civil War, a Quaker must choose between his religious beliefs and taking revenge on the man who murdered his friend. Very good screen version of Jessamyn West's novel.\n**1435** _ **The Friendly Persuasion**_ **** ABC-TV\/International Television Productions\/Allied Artists, 1975. 100 min. D: Joseph Sargent. SC: William P. Wood. With Richard Kiley, Shirley Knight, Clifton James, Michael O'Keefe, Kevin O'Keefe, Tracie Savage, Sparky Marcus, Paul Benjamin, Erik Holland, Maria Grimm, Bob Minor. During the Civil War a Hoosier Quaker couple jeopardize themselves and their family when they harbor two runaway slaves. Pretty fair TV remake of the Jessamyn West book.\n\n**1436** _ **Frisco Kid**_ **** Warner Bros., 1935. 77 min. D: Lloyd Bacon. SC: Warren Duff and Seton I. Miller. With James Cagney, Margaret Lindsay, Ricardo Cortez, Lily Damita, Donald Woods, Barton MacLane, George E. Stone, Addison Richards, Joseph King, Robert McWade, Joseph Crehan, Robert Strange, Joseph Sawyer, Fred Kohler, Edward McWade, Claudia Coleman, John Wray, Lee Phelps, Don Barclay, Jack Curtis, Milton Kibbee, Karl Hackett, Wilfred Lucas, James Farley, Charles Middleton, Landers Stevens, Frank Sheridan, Edward Keane, Ed LeSaint, Dick Rush, William Desmond, Helene Chadwick. When he opposes gambling on the San Francisco waterfront, a sailor emerges as a kingpin only to be threatened by vigilantes. Warner Bros.' answer to Samuel Goldwyn's _**Barbary Coast**_ (q.v.) is nothing more than an imitation.\n\n**1437** _ **The Frisco Kid**_ **** Warner Bros., 1979. 122 min. Color. D: Robert Aldrich. SC: Michael Ellis and Frank Shaun. With Gene Wilder, Harrison Ford, Ramon Bieri, Val Bisoglio, George Ralph DiCenzo, Leo Fuchs, Penny Peyser, William Smith, Jack Tomack, Cliff Pellow, Alan Rich. En route to head his new congregation in 1850 San Francisco, a penniless Polish Orthodox Rabbi teams with a good-hearted outlaw. Lame, overlong genre comedy.\n\n**1438** _ **Frisco Sal**_ **** Universal, 1945. 63 min. D: George Waggner. SC: Curt Siodmak and Gerald Geraghty. With Susanna Foster, Alan Curtis, Turhan Bey, Andy Devine, Thomas Gomez, Colette Lyons, Samuel S. Hinds, Fuzzy Knight, Ernie Adams, George Lloyd, Reed Howes, Beatrice Roberts, Ethan Laidlaw, Harry Hayden, Jack O'Shea, Billy Wayne, Billy Green, Bert Fiske, Syd Saylor, Earle Hodgins, Billy Wilkerson, Carlyle Blackwell, Dick Dickinson, Cyril Ring, Kit Guard, Lois Austin, James Carlisle, Isabelle La Mal. After her brother is murdered, a singer from New England comes to California to avenge his death. Vehicle for beautiful singer-actress Susanna Foster; basically for her fans.\n\n**1439** _ **Frisco Tornado**_ **** Republic, 1950. 61 min. D: R.G. Springsteen. SC: M. Coates Webster. With Allan \"Rocky\" Lane, Eddy Waller, Martha Hyer, Stephen Chase, Ross Ford, Mauritz Hugo, Lane Bradford, Hal Price, Rex Lease, George Chesebro, Edmund Cobb, Ted Adams, Bud Geary. When outlaws force ranchers to submit to a protection racket, a U.S. marshal plans to break up their illegal activities. Another fast moving segment in Allan Lane's \"Famous Westerns\" series.\n\n**1440** _ **From Broadway to Cheyenne**_ **** Monogram, 1932. 62 min. D: Harry Fraser. SC: Wellyn Totman. With Rex Bell, Marceline Day, Robert Ellis, Roy D'Arcy, Gwen Lee, George Hayes, Huntley Gordon, Matthew Betz, John Elliott, Alan Bridge, Theodore Lorch, Gordon (DeMain) D. Wood, Ernie Adams, Earl Dwire, Si Jenks, Hank Bell, Dick Dickinson, Harry Semels, Rae Daggett, Silvertip Baker. Two cowpokes head East to the big city and run into trouble with hoodlums and romance. Title tells all in this average Rex Bell vehicle; also called _**Broadway to Cheyenne**_.\n\n**1441** _ **From Hell to Texas**_ **** 20th Century\u2013Fox, 1958. 100 min. Color. D: Henry Hathaway. SC: Robert Buckner and Wendell Mayes. With Don Murray, Diane Varsi, Chills Wills, Dennis Hopper, R.G. Armstrong, Jay C. Flippen, Margo, John Larch, Ken Scott, Rodolfo Acosta, Harry Carey, Jr., Jose Torvay, Malcolm Atterbury, Salvador Baquez, Jerry Oddo, Dayton Lummis, James Philbrook, Tom Greenway, Rush Williams, Silvia Pineiro, Adelina Pedroza, Anna Navarro, Julia Montoya, Jon Lormer, Harry Fleer. After accidentally killing a rancher's son, a young man flees into the desert pursed by the dead man's father and two brothers and is helped by a cattle man and his tomboy daughter. Colorful, entertaining feature strong on characterization.\n\n**1442** _ **From Noon Till Three**_ **** United Artists, 1976. 99 min. Color. D-SC: Frank D. Gilroy. With Charles Bronson, Jill Ireland, Douglas V. Fowley, Stan Haze, Damon Douglas, Betty Cole, Don \"Red\" Barry, Sonny Jones, Hector Morales, Howard Brunner. When she thinks her third-rate outlaw lover has been killed, a pretty widow writes a best selling book about their three hour romance and he becomes a legend. Excellent Western satire; probably Charles Bronson's most underrated film.\n\n**1443** _ **Frontier Agent**_ **** Monogram, 1945. 56 min. D: Vernon Keays. SC: Norman S. Hall. With Johnny Mack Brown, Raymond Hatton, Reno Blair, Kenneth MacDonald, Dennis Moore, Riley Hill, Frank LaRue, Ted Adams, William Ruhl, Lane Bradford, Bob Woodward, Boyd Stockman. A land promoter tries to sabotage the completion of a telegraph line while a trouble-shooter for the company comes to the aid of a rancher who is using his own money to complete the project. Action filled Johnny Mack Brown entry, with a good script.\n\n**1444** _ **Frontier Badmen**_ **** Universal, 1943. 80 min. D: William McGann and Ford Beebe. SC: Gerald Geraghty and Morgan B. Cox. With Robert Paige, Anne Gwynne, Noah Beery, Jr., Diana Barrymore, Leo Carrillo, Lon Chaney, Andy Devine, Thomas Gomez, Tex Ritter, William Farnum, Frank Lackteen, Robert Homans, Tom Fadden, Norman Willis, Arthur Loft, Jack Rockwell, Stanley Price, Carl Sepulveda, William Desmond, Gil Patric, Eddy Waller, Charles Wagenheim, Frank Austin, William Ruhl, Fern Emmett, George Eldredge, Earle Hodgins, Bob Reeves, Kermit Maynard, Michael Miller, Jack C. Smith, Beverlee Mitchell. In 1869 a cattleman organizes an exchange for the sale of various herds after a syndicate takes over the Chisholm Trail. Authentic looking oater with a good script, excellent cast and plenty of action; well above average.\n\n**1445** _ **Frontier Crusader**_ **** Producers Releasing Corporation, 1940. 63 min. D: Peter Stewart (Sam Newfield). SC: William Lively. With Tim McCoy, Dorothy Short, Forrest Taylor, Ted Adams, John Merton, Lou Fulton, Karl Hackett, Hal Price, Kenne Duncan, Frank LaRue, George Chesebro, Frank Ellis, Carl Mathews, Reed Howes, Lane Bradford, Sherry Tansey. A mysterious riders shows up as outlaws plan to rob a payroll in order to get control of a mine. Well scripted and effective Tim McCoy feature.\n\n**1446** _ **Frontier Days**_ **** Spectrum, 1934. 61 min. D: Robert Hill. SC: James Shawkey. With Bill Cody, Ada Ince, Wheeler Oakman, Franklyn Farnum, Lafe McKee, William Desmond, Bill Cody, Jr., Victor Potel, Bob McKenzie, Harrison Martel. A town's leading citizen (banker, justice of the peace, lawyer) has a man killed for his ranch with the Pinto Kid blamed for the crime. Slow moving Bill Cody affair hampered by sub-par production values.\n\n**1447** _ **Frontier Feud**_ **** Monogram, 1945. 54 min. D: Lambert Hillyer. SC: Jess Bowers (Adele Buffington). With Johnny Mack Brown, Raymond Hatton, Christine McIntyre, Dennis Moore, Jack Ingram, Lloyd Ingraham, Mary MacLaren, Steve Clark, Jack Rockwell, Edwin Parker, Terry Frost, Frank LaRue, Ted Mapes, Charles King, Edmund Cobb, Stanley Price, Dan White, Lynton Brent, Wally West, Pierce Lyden, Frank McCarroll, Horace B. Carpenter, Ray Jones, Ray Henderson, Rube Dalroy. Two cowpokes arrive in an Arizona town to find a rancher about to be lynched for the murder of a rival. Okay entry in the \"Nevada Jack McKenzie\" series.\n\n_**Frontier Fighters**_ see _**Western Cyclone**_\n\n**1448** _ **Frontier Fugitives**_ **** Producers Releasing Corporation, 1946. 57 min. D: Harry Fraser. SC: Elmer Clifton. With Tex Ritter, Dave O'Brien, Guy Wilkerson, Lorraine Miller, I. Stanford Jolley, Jack Ingram, Frank Ellis, Jack Hendricks, Charles King, Karl Hackett, Budd Buster, Robert Kortman, Carl Mathews, George Morrell. The Texas Rangers get mixed up with a crook who kills a trapper for his hidden furs only to learn he is associated with a dishonest Indian agent. Rambling, stilted affair saved only by Tex Ritter singing \"Too Late to Worry, Too Blue to Cry,\" \"I'll Wait for You, Dear\" and \"Long Time Gone.\" British title: _**Fugitives of the Frontier**_.\n\n_**Frontier Fury**_ (1941) see _**The Lone Rider in Frontier Fury**_\n\n**1449** _ **Frontier Fury**_ **** Columbia, 1943. 55 min. D: William Berke. SC: Betty Burbridge. With Charles Starrett, Roma Aldrich, Arthur Hunnicutt, Jimmie Davis and His Singing Buckaroos, Johnny Bond, Clancy Cooper, I. Stanford Jolley, Edmund Cobb, Bruce Bennett, Ted Mapes, Billy Wilkerson, Stanley Brown, Joel Friedkin, Frank LaRue, Chief Yowlachie, Elmo Lincoln, Franklyn Farnum, Jack Rockwell, Eddie Borden, Jack Kirk, George Russell, Jessie Arnold. When funds belonging to Indians are stolen, a government agent is fired and he attempts to find the robbers. Pleasant Charles Starrett action drama.\n\n**1450** _ **Frontier Gal**_ **** Universal, 1945. 84 min. Color. D: Charles Lamont. SC: Michael Fessler and Ernest Pagano. With Yvonne De Carlo, Rod Cameron, Andy Devine, Fuzzy Knight, Sheldon Leonard, Andrew Tombes, Beverly Sue Simmons, Clara Blandick, Frank Lackteen, Claire Carleton, Eddie Dunn, Harold Goodwin, Jan Wiley, Rex Lease, George Eldredge, Jack Ingram, Jack Overman, Edward Howard, Joseph Haworth, Lloyd Ingraham, Joan Shawlee, Jack O'Shea, Billy Engle, Cliff Lyons, Eddie Borden, William Desmond, Kit Guard, Jack Rutherford, Lou Wood, Karen Randle, Joseph E. Bernard, Eddie Lee. After a one night honeymoon with a fiery French woman, a man returns home from prison to find his wife owns a saloon and he is the father of a little girl. Colorful, brawling oater that will more than satisfy fans of its two stars. British title: _**The Bride Wasn't Willing**_.\n\n**1451** _ **Frontier Gambler**_ **** Associated Releasing, 1956. 75 min. D: Sam Newfield. SC: Orville Hampton. With John Bromfield, Coleen Gray, Jim Davis, Kent Taylor, Margia Dean, Veda Ann Borg, Tracey Roberts, Stanley Andrews, Roy Engel, Frank Sully, Pierce Lyden, Rick Vallin, John Merton. When the female ruler of a small town is murdered and her ex-lover accused of the crime, a deputy marshal is sent to investigate. Good script and a quartet of fine stars help this low budget entry.\n\n**1452** _ **Frontier Gun**_ **** 20th Century\u2013Fox, 1959. 70 min. D: Paul Landres. SC: Stephen Kandel. With John Agar, Joyce Meadows, Barton MacLane, Robert Strauss, Morris Ankrum, James Griffith, Lyn Thomas, Leslie Bradley, Doodles Weaver, Mike Ragan (Holly Bane), Claire DuBrey. Riding into a remote town, a man becomes its unwilling sheriff and has to stand up to the local bosses, a gambler and saloon owner. Average oater.\n\n**1453** _ **Frontier Gun Law**_ **** Columbia, 1946. 60 min. D: Derwin Abrahams. SC: Bennett Cohen. With Charles Starrett, Tex Harding, Dub Taylor, Jean Stevens, Al Trace and His Silly Symphonies, Jack Guthrie, Weldon Heyburn, Jack Rockwell, Frank LaRue, John Elliott, Robert Kortman, Stanley Price, Bill Nestell, Hank Worden, John Tyrrell. The Durango Kid chases an outlaw band called \"The Phantoms\" that have been raiding area ranchers. Pretty fair \"Durango Kid\" entry; British title: _**Menacing Shadows**_.\n\n_**Frontier Hellcat**_ see _**Among Vultures**_\n\n_**Frontier Horizon**_ see _**The New Frontier**_ (1939)\n\n**1454** _ **Frontier Investigator**_ **** Republic, 1949. 60 min. D: Fred C. Brannon. SC: Robert Williams. With Allan \"Rocky\" Lane, Eddy Waller, Clayton Moore, Gail Davis, Roy Barcroft, Robert Emmett Keane, Marshall Reed, Francis Ford, Claire Whitney, Harry Lauter, Tom London, George Lloyd, Hank Bell, Tom Steele. A lawman is on the trail of a killer who murders victims with a special telescopic device mounted on his rifle. There is plenty of action in this Allan Lane outing; also called _**Frontier Marshal**_.\n\n**1455** _ **Frontier Justice**_ **** First Division\/Grand National, 1935. 58 min. D: Robert McGowan. SC: W. Scott Darling. With Hoot Gibson, Jane Barnes, Richard Cramer, Franklyn Farnum, Lloyd Ingraham, Joseph Girard, Fred \"Snowflake\" Toones, Roger Williams, George Yoeman, John Elliott, Lafe McKee, Henry Hall, Jack Hendricks, The Beverly Hill Billies (Rudy Sooter, Aleth Hansen, Harley Luse), Bill Patton, Sherry Tansey, Steve Clark, Pat Harmon, William McCall, Olin Francis, Jack Evans, Fred Parker, Barney Beasley, Clyde McClary, Silvertip Baker. Returning home, a man finds his father has been committed to an asylum and their heavily mortgaged ranch suffering from rustling raids. Complicated but entertaining Hoot Gibson film hurt by a low budget; based on the novel by Col. George B. Rodney.\n\n**1456** _ **Frontier Law**_ **** Universal, 1943. 59 min. D-SC: Elmer Clifton. With Russell Hayden, Fuzzy Knight, Jennifer Holt, Dennis Moore, Johnny Bond and His Red River Valley Boys, Jack Ingram, Hal Taliaferro, George Eldredge, I. Stanford Jolley, Frank LaRue, James Farley, Michael Vallon, Tex Cooper, Neal Hart, Earle Hodgins, Bob Reeves, Harry Tenbrook, Art Fowler, Pascale Perry, Frosty Royce, Hank Bell, Victor Cox, Roy Butler. Two cowboys ride into a locale plagued by cattle rustlers and learn their pal is working for the gang leader. Fair Universal programmer with Russell Hayden (doubled by Rod Cameron) replacing ailing Tex Ritter.\n\n**1457** _ **Frontier Marshal**_ **** Fox, 1934. 66 min. D: Lewis Seiler. SC: William Conselman and Stuart Anthony. With George O'Brien, Irene Bentley, George E. Stone, Alan Edwards, Ruth Gillette, Berton Churchill, Frank Conroy, Ward Bond, Ed LeSaint, Russell Simpson, Jerry Foster. A lawman arrives in Tombstone, Arizona, where the crooked mayor controls all the dishonest elements after killing his banking partner. First screen version of Stuart N. Lake's novel _Wyatt Earp, Frontier Marshal_ (although Earp is called Michael Wyatt here) and it is a good one.\n\n**1458** _ **Frontier Marshal**_ **** 20th Century\u2013Fox, 1939. 71 min. D: Allan Dwan. SC: Sam Hellman. With Randolph Scott, Nancy Kelly, Cesar Romero, Binnie Barnes, John Carradine, Edward Norris, Eddie Foy, Jr., Ward Bond, Lon Chaney, Jr., Tom Tyler, Chris-Pin Martin, Joseph Sawyer, Del Henderson, Harry Hayden, Ventura Ybarra, Si Jenks, Gloria Roy, Pat O'Malley, Charles Stevens, Harry Woods, Richard Alexander, Hank Mann, Ed LeSaint, Heinie Conklin, George Melford, Fern Emmett, Kathryn Sheldon, Ferris Taylor, Arthur Aylesworth, Eddie Dunn, Philo McCullough, Ethan Laidlaw, Margaret Brayton, John Butler, John Bleifer, Hank Bell, Harlan Briggs, Dick Elliott, Jimmy Aubrey, Post Park, Henry Clive. Sheriff Wyatt Earp, with the help of Doc Holliday, tries to bring law and order to the town of Tombstone, Arizona. Second filming of Stuart N. Lake's book contains an excellent recreation of the shootout at the O.K. Corral.\n\n_**Frontier Marshal**_ (1949) see _**Frontier Investigator**_\n\n**1459** _ **Frontier Outlaws**_ **** Producers Releasing Corporation, 1944. 58 min. D: Sam Newfield. SC: Joseph O'Donnell. With Buster Crabbe, Al St. John, Frances Gladwin, Marin Sais, Charles King, Jack Ingram, Kermit Maynard, Ed Cassidy, Emmett Lynn, Budd Buster, Frank Ellis, Bert Dillard, Ray Henderson, Wally West, Dan White, Silver Harr, John Cason, Tex Cooper, Jimmy Aubrey, George Morrell, Horace B. Carpenter, Jack Tornek, Silver Tip Baker, Herman Hack, Carl Mathews, Artie Ortego, Tex Williams. Billy Carson is framed for murder after opposing an outlaw gang trying to take over a valley. Low grade, but entertaining second \"Billy Carson\" series entry.\n\n**1460** _ **Frontier Outpost**_ **** Columbia, 1950. 55 min. D: Ray Nazarro. SC: Barry Shipman. With Charles Starrett, Smiley Burnette, Lois Hall, Steve Darrell, Fred F. Sears, Hank Penny and Slim Duncan, Robert Wilke, Paul Campbell, Jock (Mahoney) O'Mahoney, Bud Osborne, Chuck Roberson, Pierre Watkin, Dick Wessel, Everett Glass. The Durango Kid robs a stage carrying a government gold shipment so the money cannot be stolen by outlaws. Rather jumbled \"Durango Kid\" episode.\n\n**1461** _ **The Frontier Phantom**_ **** Western Adventure, 1952. 56 min. D: Ron Ormond. SC: Maurice Tombragel and June Carr. With Lash LaRue, Al St. John, Virginia Herrick, Archie Twitchell, Clarke Stevens, Bud Osborne, Cliff Taylor, Kenne Duncan, George Chesebro, Sandy Sanders, Buck Garrett, Jack O'Shea, Frank Ellis, Roy Butler, Larry Barton, Dee Cooper, Dan White, Ted Adams, Lee Roberts, Nancy Saunders, John Merton, Steve Dunhill, Dee Cooper, Artie Ortego, Al Haskell. Two U.S. marshals try to find out who is the ringleader of a counterfeiting outfit and one of them takes on the guise of his outlaw brother. Lash LaRue's final starring series Western is exciting and action filled; contains footage from _**Outlaw Country**_ (q.v.).\n\n**Lash LaRue, Dan White and Jack O'Shea in** _**The Frontier Phantom**_ **(Western Adventure, 1952) in a scene originally from** _**Outlaw Country**_ **(Western Adventure, 1949).**\n\n**  \n**\n\n**1462** _ **Frontier Pony Express**_ **** Republic, 1939. 58 min. D: Joseph Kane. SC: Norman S. Hall. With Roy Rogers, Mary Hart, Raymond Hatton, Edward Keane, Noble Johnson, Monte Blue, Donald Dillaway, William Royle, Ethel Wales, Bud Osborne, George (Montgomery) Letz, Charles King, Fred Burns, Jack Kirk, Ernie Adams, Hank Bell, Jack O'Shea, Chris-Pin Martin, House Peters, Jr., Art Dillard. In 1861 a crooked senator plans to set up a republic in California by pretending to aid the Confederacy, using the Pony Express in his scheme. A very pleasant Roy Rogers film; includes the songs \"Rusty Spurs\" and \"My Old Kentucky Home.\"\n\n**1463** _ **Frontier Revenge**_ **** Screen Guild\/Western Adventure, 1948. 58 min. D-SC: Ray Taylor. With Lash LaRue, Al St. John, Peggy Stewart, Jim Bannon, Ray Bennett, Sarah Padden, Jimmie Martin, Jack Hendricks, Lee Morgan, Sandy Sanders, Billy Dix, Cliff Taylor, Steve Raines, Bud Osborne, Charles Chesebro, Kermit Maynard, Jack Evans. In order to unmask the leader of an outlaw gang terrorizing a small town, Lash and Fuzzy pose as two famous outlaws and join the marauders. Okay Lash LaRue vehicle; a remake of _**Panamint's Bad Man**_ (q.v.), also directed by Ray Taylor.\n\n**1464** _ **Frontier Scout**_ **** Grand National, 1938. 62 min. D: Sam Newfield. SC: Frances Guihan. With George Houston, Beth Marion, Al St. John, Dave O'Brien, Guy Chase, Jack Ingram, Jack C. Smith, Dorothy Fay, Slim Whitaker, Kenne Duncan, Carl Mathews, Kit Guard, Bob Woodward, Walter Byron, Budd Buster, Frank LaRue, Minerva Urecal, Mantan Moreland, Roger Williams, Joe Girard, Jim Thorpe. Wild Bill Hickok helps ranchers plagued by cattle rustlers and Indian raids. George Houston's first Western is a sturdy affair, sure to please his fans.\n\n**1465** _ **Frontier Town**_ **** Grand National, 1938. 60 min. D: Ray Taylor. SC: Lindsley Parsons. With Tex Ritter, Ann Evers, Snub Pollard, Horace Murphy, Charles King, Forrest Taylor, Jack C. Smith, Ed Cassidy, Karl Hackett, Lynton Brent, Don Marion, Hank Worden, John Elliott, Jimmie LeFieur's Saddle Pals. Despite the events being fixed by a gang of crooks, a singing cowboy tries to win the big prize money at a rodeo. Cheaply made but fairly exciting Tex Ritter vehicle, helped by the star singing a few good songs.\n\n**1466** _ **Frontier Uprising**_ **** United Artists, 1961. 68 min. D: Edward L. Cahn. SC: Owen Harris. With Jim Davis, Nancy Hadley, Ken Mayer, Nestor Paiva, Don O'Kelly, Stuart Randall, David Renard, Tudor Owen, Addison Richards, Jan Arvan, Sid Kane, Barbara Mansell. A wagon train heads West to California with its passengers not knowing the U.S. and Mexico are at war with the latter making an alliance with local Indians. More than passable small budget affair, with Jim Davis doing a good job as a frontier scout.\n\n**1467** _ **Frontier Vengeance**_ **** Republic, 1940. 54 min. D: Nate Watt. SC: Bennett Cohen and Barry Shipman. With Don \"Red\" Barry, Betty Moran, George Offerman, Jr., Ivan Miller, Yakima Canutt, Kenneth MacDonald, Cindy Walker, Jack Rockwell, Griff Barnett, Jack Lawrence, Fred \"Snowflake\" Toones, Obed Packard. When a crooked stage line operator tries to run a rival company out of business, a driver steps in to help the female owner. Typically breezy Don Barry film.\n\n**1468** _ **Frontiers of '49**_ **** Columbia, 1939. 54 min. D: Joseph Lovering. SC: Nate Gatzert. With Bill Elliott, Luana de Alcaniz, Hal Taliaferro, Charles King, Slim Whitaker, Al Ferguson, Jack Walters, Octavio Girard, Carlos Villarias, Jose De La Cruz, Kit Guard, Bud Osborne, Jack Ingram, Lee Shumway, Ed Cassidy, Tex Palmer, Buzz Barton, Chick Hannon, Fred Parker. Two government men are sent to California to stop the dictatorial activities of a crook forcing many Spanish ranchers off their ranchos. Compact and action filled Bill Elliott vehicle.\n\n**1469** _ **The Frontiersman**_ **** Paramount, 1938. 74 min. D: Lesley Selander. SC: Norman Houston and Harrison Jacobs. With William Boyd, Russell Hayden, George Hayes, Evelyn Venable, William Duncan, Clara Kimball Young, Charles A. Hughes, Dickie Jones, Roy Barcroft, Emily Fitzroy, John Beach, George Morrell, Jim Corey, Robert Mitchell and His St. Brendan's Boys Choir, Dorothy Vernon, Jack Evans, Rube Dalroy, Charles Brinley, Jess Cavin, Blackjack Ward. A crook, who is in love with a school teacher, rustles Bar 20 cattle and then murders his partner. Mediocre entry in the \"Hopalong Cassidy\" series, interesting only for silent film stars William Duncan (as Buck Peters) and Clara Kimball Young; contains useless filler of the St. Brendan's Boys Choir in a school sequence.\n\n_**Fuerte Perdido**_ see _**Massacre at Fort Perdition**_\n\n**1470** _ **The Fugitive**_ **** Monogram, 1933. 61 min. D: Harry Fraser. SC: Harry O. Jones (Harry Fraser). With Rex Bell, Cecilia Parker, George Hayes, Robert Kortman, Tom London, Gordon DeMain, Theodore Lorch, Dick Dickinson, Earl Dwire, George Nash, Lloyd Whitlock, Phil Dunham, Tommy Coats, Arthur Thalasso. A cowboy, falsely accused of a crime, must run from the law until he can prove his innocence. Low budget Rex Bell outing.\n\n**1471** _ **The Fugitive**_ **** RKO Radio, 1947. 104 min. D: John Ford. SC: Dudley Nichols. With Henry Fonda, Dolores Del Rio, Pedro Armendariz, Ward Bond, Leo Carrillo, J. Carrol Naish, Robert Armstrong, John Qualen, Fortunio Bonanova, Chris-Pin Martin, Miguel Inclan, Fernando Fernandez, Jose Torvay. A priest, who supports the revolutionary cause in Mexico, is hunted by the police and befriended by a man who later turns him in for money. Low key John Ford film that will please Henry Fonda fans.\n\n_**The Fugitive**_ (1966) see _**El Fugitivo**_\n\n**1472** _ **Fugitive from Sonora**_ **** Republic, 1943. 55 min. D: Howard Bretherton. SC: Norman S. Hall. With Don \"Red\" Barry, Lynn Merrick, Wally Vernon, Harry Cording, Ethan Laidlaw, Frank McCarroll, Pierce Lyden, Kenne Duncan, Karl Hackett, Slim Whitaker, Art Dillard, Augie Gomez, Kansas Moehring. A one-time outlaw comes to a town and tries to stop a range war between homesteaders and cattlemen. Another good one in Don Barry's Republic series; it introduced Barry's long time comedy sidekick Wally Vernon.\n\n**1473** _ **Fugitive of the Plains**_ **** Producers Releasing Corporation, 1943. 56 min. D: Sam Newfield. SC: George W. Sayre. With Buster Crabbe, Al St. John, Maxine Leslie, Jack Ingram, Kermit Maynard, Karl Hackett, Hal Price, George Chesebro, Frank Ellis, John Merton, Budd Buster, Artie Ortego, Carl Sepulveda, Curley Dresden, Art Dillard, Jimmy Aubrey, Tex Harper, Hank Bell, Kansas Moehring. Billy the Kid and Fuzzy Jones help a young woman forced into lawlessness by crooks. Standard \"Billy the Kid\" series entry; reissued in 1947 by Eagle Lion as _**Raiders of Red Rock**_ (38 min.). Also called _**Billy the Kid in Fugitive of the Plains**_.\n\n**1474** _ **The Fugitive Sheriff**_ **** Columbia, 1936. 58 min. D: Spencer Gordon Bennet. SC: Nate Gatzert. With Ken Maynard, Beth Marion, Walter Miller, Hal Price, John Elliott, Arthur Millett, Virginia True Boardman, Frank Ball, Edwin (Edmund) Cobb, William Gould, Art Mix, Vernon Dent, Fred Burns, Frank Ellis, Slim Whitaker, Horace Murphy, Theodore Lorch, Bob Burns, Horace B. Carpenter, Oscar Gahan, William McCall, Lafe McKee, Glenn Strange, Bud McClure, Bud Osborne, Rudy Sooter, Lew Meehan, Bob Reeves, Fred Parker, Tex Cooper, Jack King, Bud Jamison, Bob Card, Herman Hack, Art Dillard, Tex Palmer, Al Taylor, Blackjack Ward, Silvertip Baker, Ralph Bucko, Roy Bucko. After being elected town sheriff, a cowboy is framed for a train robbery and has to prove his innocence. Average Ken Maynard vehicle, his last for Columbia.\n\n**1475** _ **Fugitive Valley**_ **** Monogram, 1941. 60 min. D: S. Roy Luby. SC: John Vlahos and Robert Finkle. With Ray Corrigan, John King, Max Terhune, Julie Duncan, Glenn Strange, Robert Kortman, Tom London, Reed Howes, Ed Brady, Carl Mathews, Ed Peil, Sr., Doye O'Dell, Frank McCarroll, Ray Jones. In Arizona, outlaws led by \"The Whip\" terrorize the countryside and the Range Busters get into the gang to stop them. Okay \"Range Buster\" series affair with a bit too much humor although Glenn Strange and Robert Kortman are great in villainous roles; some footage later used in _**Bullets and Saddles**_ (q.v.).\n\n_**Fugitives of the Frontier**_ see _**Frontier Fugitives**_\n\n_**Fugitive's Run**_ see _**Cowboy's Run**_\n\n**1476** _ **El Fugitivo**_ (The Fugitive) **** Productora Filmica Mexico, 1966. 95 min. Color. D: Emilio Gomez Muriel. SC: Emilio Gomez Muriel and Alfred Ruanova. With Luis Aguilar, Lucha Villa, Amparo Rivelles, Alma Delia Fuentes, Jorge Russek, Jose Chavez, Rita Macedo, Victor Alcocer, Ramon Bugarini, Raul Ramirez, Arturo Castro \"Bigoton,\" Roberto Canedo, Arturo Correa, Rafael del Rio. Falsely convicted of a crime, a man plots revenge against the former friend whose testimony put him in prison. Okay Mexican Western featuring the masked hero Black Rider.\n\n**1477** _ **Full House for the Devil**_ **** Devon Film\/Flora Film, 1968. 87 min. Color. D: Giovanni Fago. SC: Ernesto Gastaldi. With George Hilton, Paul Stevens (Paolo Gozlino), Claudie Lange, Gerard Herter, Krista Nell, Carlo Gaddi, Aldo Cecconi, Paul Muller, Ferruccio Viotti, Rex Purdom, Gill Rolland, Angela Ellison, Adriana Giuffre, Pietro Tordi, Ugo Adinolfi, Silvio Bagolini, Robert Anthony (Espartaco Santoni), Aldolfo Belletti, Renato Pinciroli, Mirko Valentin, Rodolfo Valadier, Franco Aloisi, Pino Sciacqua, Freddy Unger. An effete gunman and a bumbling bandit form an alliance to take revenge on the land speculator who murdered a minister. Standard Spaghetti Western with a violent shootout; filmed in Italy and issued there as _**Uno di Piu all'Inferno**_ (One More to Hell) and also called _**To Hell and Back**_.\n\n**1478** _ **The Furies**_ **** Paramount, 1950. 109 min. D: Anthony Mann. SC: Charles Schnee. With Barbara Stanwyck, Walter Huston, Wendell Corey, Gilbert Roland, Judith Anderson, Thomas Gomez, Beulah Bondi, Albert Dekker, John Bromfield, Wallace Ford, Blanche Yurka, Louis Jean Heydt, Frank Ferguson, Movita, Myrna Dell, Charles Evans, Craig Kelly, Eddy Waller, Arthur Hunnicutt, Nolan Leary, Jane Novak, Pepe Hern, Lou Steele, Rosemary Petit, James Davies, Douglas Grange, Joe Dominguez, Sam Finn. A stubborn, self-made cattle rancher clashes with the strong willed daughter he cannot control. None-too-interesting psychological Western with a lot of hidden undertones for those with a symbolic bent.\n\n**1479** _ **The Further Adventures of the Wilderness Family**_ **** Pacific International, 1978. 105 min. Color. D: Frank Zuniga. SC: Arthur R. Dubs. With Robert Logan, Susan Damante Shaw, Hollye Holmes, Ham Larsen, George \"Buck\" Flowers, Brian Cutler. A modern-day family, having deserted the big city for the pioneer life in the Rocky Mountains, further experiences the joys and tribulations of going through a harsh winter. The second of a three part series and just as good as the others; preceded by _**The Adventures of the Wilderness Family**_ (q.v.) and followed by _**Mountain Family Robinson**_ (q.v.). Also called _**Wilderness Family, Part Two**_.\n\n**1480** _ **Fury at Furnace Creek**_ **** 20th Century\u2013Fox, 1948. 88 min. D: H. Bruce Humberstone. SC: Charles G. Booth. With Victor Mature, Coleen Gray, Glenn Langan, Reginald Gardiner, Albert Dekker, Fred Clark, Charles Kemper, Robert Warwick, George Cleveland, Roy Roberts, Willard Robertson, Griff Barnett, Frank Orth, J. Farrell MacDonald, Jay Silverheels, Robert Adler, Mauritz Hugo, Howard Negley, Harry Carter, Harlan Briggs, Si Jenks, Guy Wilkerson, Edmund Cobb, Kermit Maynard, Paul Newlan, Ted Mapes, George Chesebro, Al Hill, Minerva Urecal, Ray Teal, Alan Bridge, Oscar O'Shea, Jerry Miley. A man tries to prove his father did not cause a massacre and uncovers evidence that three no-accounts were the real culprits. Routine oater with Victor Mature trying hard as the avenger.\n\n**1481** _ **Fury at Gunsight Pass**_ **** Columbia, 1956. 68 min. D: Fred F. Sears. SC: David Lang. With David Brian, Neville Brand, Richard Long, Lisa Davis, Kathleen Warren, Percy Helton, Morris Ankrum, Addison Richards, Joe Forte, Wally Vernon, Paul E. Burns, Frank Fenton, James Anderson, George Keymas, Robert Anderson, I. Stanford Jolley, Harry Harvey. When a wedding halts their attempt to rob a bank, an outlaw gang decides to take over the town. An out-of-the ordinary plot adds some spice to this low budget affair from producer Wallace MacDonald.\n\n**1482** _ **Fury at Showdown**_ **** United Artists, 1957. 75 min. D: Gerd Oswald. SC: Jason Thomas. With John Derek, John Smith, Carolyn Craig, Nick Adams, Gage Clarke, Robert Griffin, Malcolm Atterbury, Rusty Lane, Frances Morris, Tyler MacDuff, Robert Adler, Norman Leavitt, Ken Christy. A one-time outlaw is branded a coward for refusing to use a gun but when a bad man takes his girl hostage he comes to her defense. Brooding Western with fine performances.\n\n**1483** _ **Fury in Paradise**_ **** Filmmakers\/Alfonso Sanchez-Tello, 1956. 77 min. Color. D-SC: George Bruce. With Peter Thompson, Carlos Rivas, Rea Iturbi, Eduardo Noriega, Felipe Nolan, Claud Brooks. An American tourist in Mexico nearly ends up before a firing squad after getting involved with a man and his pretty daughter, who turn out to be revolutionaries. Low budget melodrama filmed in Mexico.\n\n**1484** _ **Fury of the Apaches**_ **** Castilla, 1966. 84 min. Color. D: Joe Lacy (Jose M. Elorrieta). SC: Jose M. Elorrieta and Jose Luis Navarro. With Frank Latimore, Yvonne Bastion, Georges Gordon, Liza Moreno, George Martin, Angel Ortiz, Nuria Torray, Jesus Puente. A cavalry unit rescues settlers attacked by Indians and they are taken to a nearby fort to await another assault. Adequate Spanish-made oater also called _**Apache Fury**_ ; a remake of _**Massacre at Fort Perdition**_ (q.v.).\n\n**1485** _ **Fury River**_ **** Metro-Goldwyn-Mayer, 1962. 74 min. D: Jacques Tourneur, Alan Crosland, Jr., Joe Waggner and Otto Lang. SC: Gerald Drayson Adams, Anthony Ellis and Sloan Nibley. With Keith Larsen, Buddy Ebsen, Don Burnett, Philip Tonge, Lisa Davis, Larry Chance, Jim Hayward, Pat Hogan, Lisa Gaye, Harry Lauter, Luis Van Rooten, Denny Miller, Paul Picerni, Rayford Barnes. Rogers' Rangers search for a waterway to the ocean while they battle the French and Indians in frontier Canada. Average telefeature from episodes of \"Northwest Passage\" (NBC-TV, 1958\u201359) and issued abroad theatrically.\n\n**1486** _ **Fuzzy Settles Down**_ **** Producers Releasing Corporation, 1944. 60 min. D: Sam Newfield. SC: Louise Rousseau. With Buster Crabbe, Al St. John, Patti McCarthy, Charles King, John Merton, Frank McCarroll, Robert Hill, Ted Mapes, Tex Palmer, Ed Peil, Sr., John Elliott, Hal Price, Horace B. Carpenter, Ray Jones, Artie Ortego, Wally West, Steve Clark, Ben Corbett, John Cason, Holly Bane, Herman Hack, Dan White, Jack Tornek, Morgan Flowers, George Morrell, Ray Henderson, Chick Hannon, Jimmy Aubrey, Hank Bell, Silver Tip Baker. Billy Carson and Fuzzy Q. Jones capture two notorious outlaws and Fuzzy uses his portion of the reward money to buy a newspaper in a town where the citizens want a telegraph line in order to break up a gang of rustlers. Low grade but entertaining.\n\n**1487** _ **The Gal Who Took the West**_ **** Universal-International, 1949. 84 min. Color. D: Frederick De Cordova. SC: William Bowers and Oscar Brodney. With Yvonne De Carlo, Scott Brady, Charles Coburn, John Russell, Myrna Dell, James Millican, Clem Bevans, Bob Stevenson, Houseley Stevenson, Robin Short, Russell Simpson, John Litel, James Todd, Edward Earle, Jack Ingram, Francis McDonald, Glenn Strange, William Tannen, Steve Darrell, Pierce Lyden, Ross Elliott, John James, Howard Negley, Charles Cane, William Haade, Louise Lorimer, Forrest Taylor, Paul Brinegar, House Peters, Jr., Russ Whiteman, Roger Moore, Forbes Murray, Gary Teague, Richard Farmer, Martin Cichy, Audrey Young, Ann Pierce, Jane Fulton, Patricial Hall, Charles Jordan, George Stern, William Donnelly, Steve Crandall, Jon Riffel, Fraser McWinn, Peggy Leon, Ella Ethridge, Vera Kirman, William Norton Bailey, David Alison, Mildred Sellers, Louise Bates, Philip Ahn, Helen Dickson, Harlan Hoagland, Chalky Williams, Paul Palmer, Patrick Griffin. An opera singer comes to Arizona in the 1890s and two feuding brothers fall in love with her. Seriocomic Western is not good on either count; mediocre.\n\n**1488** _ **The Gallant Defender**_ **** Columbia, 1935. 60 min. D: David Selman. SC: Ford Beebe. With Charles Starrett, Joan Perry, Harry Woods, Ed LeSaint, Jack Clifford, Alan Bridge, George Chesebro, Edmund Cobb, Frank Ellis, Jack Rockwell, Tom London, Stanley Blystone, The Sons of the Pioneers (Bob Nolan, Len Slye [Roy Rogers], Tim Spencer, Hugh Farr, Karl Farr), Lew Meehan, Merrill McCormick, Glenn Strange, Al Ferguson, Slim Whitaker, Bud Osborne, George Billings, Buck Connors, Oscar Gahan, Jack Kirk, Richard Botiller, Bud McClure, Al Haskell, Chuck Baldra, Ray Jones, Tom Smith, Pascale Perry, Bob Card. A cowboy helps homesteaders harassed by cattlemen who do not want them to settle on their range land. Charles Starrett's initial series film is a sturdy affair enhanced by Ford Beebe's fine script.\n\n**1489** _ **The Gallant Fool**_ **** Monogram, 1933. 61 min. D: Robert North Bradbury. SC: Robert North Bradbury and Harry O. (Fraser) Jones. With Bob Steele, Arletta Duncan, John Elliott, Theodore Lorch, Perry Murdock, George Hayes, Si Jenks, Art Mix, George Nash, Pascale Perry, Vane Calvert, Anne Howard, Herman Hack, Blackie Whiteford, Bob Burns, Billy Franey, Dick Dickinson, Steve Clemente, Silvertip Baker. After being falsely accused of murder, a man takes refuge in a circus with his small son. Nicely done and action filled Bob Steele vehicle.\n\n**1490** _ **The Gallant Legion**_ **** Republic, 1946. 88 min. D: Joseph Kane. SC: Gerald Adams. With William Elliott, Adrian Booth, Joseph Schildkraut, Bruce Cabot, Andy Devine, Jack Holt, Adele Mara, Grant Withers, James Brown, Hal Taliaferro, Russell Hicks, Herbert Rawlinson, Marshall Reed, Harry Woods, Roy Barcroft, Bud Osborne, Hank Bell, Jack Ingram, George Chesebro, Jack Perrin, Noble Johnson, Rex Lease, John Hamilton, Emmett Vogan, Trevor Bardette, Gene Roth, Ferris Taylor, Iron Eyes Cody, Kermit Maynard, Jack Kirk, Merrill McCormick, Fred Kohler, Glenn Strange, Tex Terry, Joseph Crehan, Lester Sharpe. When a crooked politician tries to split Texas in half by disbanding the Texas Rangers, a lawman attempts to stop him and is helped by a female reporter. Very fine William Elliott film, strong in story, action and cast.\n\n**1491** _ **Galloping Dynamite**_ **** Ambassador, 1937. 58 min D: Harry Fraser. SC: Sherman Lowe and Charles Condon. With Kermit Maynard, Ariane Allen, John Merton, John Ward, Stanley Blystone, David Sharpe, Earl Dwire, Francis Walker, Tracy Layne, Bob Burns, Allen Greer, Budd Buster, Bruce Mitchell, Oscar Gahan. A Texas Ranger finds out three men have murdered his prospector brother to get a ranch containing a valuable vein of gold. Kermit Maynard joins the legion of singing cowboys in this average outing.\n\n**1492** _ **Galloping Gallagher**_ **** Film Booking Offices (FBO), 1924. 50 min. D: Albert S. Rogell. SC: Marion Jackson. With Fred Thomson, Hazel Keener, Frank Hagney, Nelson McDowell, Shorty Hendrix, Andy Morris, Lew Meehan, Bob Reeves, George F. Marion, Silver King (horse). A newly elected sheriff, with the help of his horse, rids a town of outlaws and saves a lady preacher from the clutches of their leader, a crooked banker. Fast paced Fred Thomson film that only survives in a 29-minute version.\n\n**1493** _ **Galloping On**_ **** Action\/Weiss Brothers\/Artclass, 1925. 53 min. D: Richard Thorpe. SC: Frank L. Ingraham and Betty Burbridge. With Wally Wales, Jessie Cruzon, Louise Lester, Charles \"Slim\" Whitaker, Richard Belfield, Gretchen Waterman, Art Phillips, Lawrence Underwood. Returning home after being falsely sent to prison, a man learns the crook who framed him, now the town banker, wants to send him back and he is helped by a young girl in getting evidence against the bad man. A good silent \"B\" item with a top notch performance by Wally Wales in the lead role.\n\n**1494** _ **Galloping Romeo**_ **** Monogram, 1933. 60 min. D: Robert North Bradbury. SC: Harry O. (Fraser) Jones. With Bob Steele, Doris Hill, George Hayes, Frank Ball, Ernie Adams, Lafe McKee, Ed Brady, George Nash, Earl Dwire, Hal Price, Dick Dickinson, Tex Palmer, Silvertip Baker. A cowboy teams with an old timer to prove his innocence when he is unjustly accused of a crime. Okay Bob Steele outing that has too much footage from his previous films.\n\n**1495** _ **Galloping Through**_ **** Sunset, 1923. 50 min. D-SC: Robert North Bradbury. With Jack Hoxie, Priscilla Banner, William Lester, Lorraine Lorimer, William McCall, Tom Lingham, Janet Ford, Scout (horse). A cowpoke helps a family of homesteaders when the husband is accused of a crime he did not commit. Jack Hoxie vehicle his fans will like.\n\n**1496** _ **Galloping Thru**_ **** Monogram, 1932. 58 min. D: Lloyd Nosler. SC: Wellyn Totman. With Tom Tyler, Betty Mack, Alan Bridge, Si Jenks, Stanley Blystone, G.D. Woods (Gordon DeMain), John Elliott, Artie Ortego, Art Mix. A cowboy returns home to see his father murdered and tries to find the assailant. Low grade, but action filled, Tom Tyler outing.\n\n**1497** _ **Galloping Thunder**_ **** Columbia, 1946. 54 min. D: Ray Nazarro. SC: Ed Earl Repp. With Charles Starrett, Smiley Burnette, Adelle Roberts, Merle Travis and His Bronco Busters, Richard Bailey, Edmund Cobb, Kermit Maynard, Ray Bennett, Curt Barrett, John Merton, Nolan Leary, Budd Buster, Forrest Taylor, Merrill McCormick, Roy Butler, Bob Reeves, Gordon Harrison. Outlaws are preventing ranchers from shipping mustang herds to the government for Army use and an agent, the Durango Kid, is sent to investigate. Only a passable effort in the \"Durango Kid\" series.\n\n**1498** _ **Gallowwalker**_ **** Intandem Films, 2010. 90 min. Color. D: Andrew Goth. SC: Andrew Goth and Joanne Reay. With Wesley Snipes, Tanit Phoenix, Riley Smith, Patrick Bergin, Dallas Page, Jenny Gago, Simona Brhikova, Alvssa Pridham, Kevin Howarth, Steven Elder, Alex Avant, Hector Hank, Jack Bowyer, Arthur Berenzin, Jonathan Garcia, Joe Zmztsky, Vito Vilonel, Sean Naude, Pierre Roos, William Venter, Shani Maritz, Derek Soutwork, Vicky Moller-Forbes, Frederick Haraseb, Roberto Husselmann, Derek Griffith, Martin Strauss, Wotan Swiegers. A cursed gunman, hunted by a band of outlaws he shot and killed, is aided by a young warrior. So-so horror Western.\n\n_**The Gambler**_ see _**Kenny Rogers as the Gambler**_\n\n_**The Gambler\u2014The Adventure Continues**_ see _**Kenny Rogers as the Gambler\u2014The Adventure Continues**_\n\n_**The Gambler, Part III: The Legend Continues**_ see _**Kenny Rogers as the Gambler, Part III: The Legend Continues**_\n\n**1499** _ **The Gambler from Natchez**_ **** 20th Century\u2013Fox, 1954. 88 min. Color. D: Henry Levin. SC: Gerald Drayson Adams and Irving Wallace. With Dale Robertson, Debra Paget, Kevin McCarthy, Thomas Gomez, Lisa Davis, Douglas Dick, John Wengraf, Jay Novello, Woody Strode, Peter Mamakos, Donald Randolph. When a man is falsely accused of cheating at cards and gunned down by three men, his son plans to avenge his murder. Entertaining frontier drama set in the 1840s.\n\n**1500** _ **The Gambler Returns:**_ _**The Luck of the Draw**_ **** NBC-TV, 1991. 240 min. Color. D: Dick Lowry. SC: Jeb Rosebrook and Joe Byrne. With Kenny Rogers, Rick Rossovich, Reba McEntire, Claude Akins, Dion Anderson, Gene Barry, Paul Brinegar, Jere Burns, David Carradine, Chuck Connors, Johnny Crawford, Juli Donald, James Drury, Linda Evans, Brian Keith, Jack Kelly, Patrick Macnee, Doug McClure, Hugh O'Brian, Park Overall, Christopher Rich, Mickey Rooney, Brad Sullivan, Dub Taylor, Clint Walker, Lisa Rieffel, Sheryl Lee Ralph, Zelda Rubinstein, Ray McKinnon, Alma Martinez, Teri Copley, Kent Broadhurst, Mary Cadorette, Melissa Hurley, Tammy Anderson, Marianne Gordon, Christopher Cody Rogers, Sean Staek, Dell Young, Jorge Cervera, Jr., Sam Whippie, Tim Choate, Kelly Junkerman, Ann Gillespie, Debra Christofferson, Norman Large, Pepper Sweeney, Mike Pniewski, Kevin Furlong, Dean Cochran, Don S. Davis, Pete Antico, Max Grodenchik, Rex Linn, John Fleck, Kelley Menighan Hensley, Jack Lilley, Doug McDonald. After winning a big game, a gambler and his four madam financial backers head to San Francisco for one final high stakes play but they are pursued by outlaws who want their money. Enjoyable, nostalgic TV Western with several series actors (Gene Barry, Paul Brinegar, David Carradine, Chuck Connors, Johnny Crawford, James Drury, Brian Keith, Jack Kelly, Doug McClure, Hugh O'Brian, Clint Walker) reprising their noted small screen characters.\n\n**1501** _ **The Gambler, the Girl and the Gunslinger**_ **** Hallmark Channel, 2009. 95 min. Color. D: Anne Wheeler. SC: Bob Barbash and Larry Cohen. With Dean Cain, James Tupper, Allison Hossack, Keith Mackechnie, Michael Eklund, John Desantis, Teach Grant, Serge Houde, Alejandro Abellan, Garwin Sanford, Sheldon Yamkovy, Quentin Schneider, Eli Zaquodakis, Jonathan Field, Kyle Thomson, Mike Mitchell, Dean Wray, John Shield, Raugi Yu, Chad Krowchuk. Two rival gamblers win a ranch and both want the same woman but end up uniting to fend off bandits. Pleasant TV Western comedy.\n\n**1502** _ **The Gambler Wore a Gun**_ **** United Artists, 1961. 66 min. D: Edward L. Cahn. SC: Owen Harris. With Jim Davis, Merry Anders, Mark Allen, Addison Richards, Don Dorrell, Robert Anderson, Keith Richards, John Craig, Charles Cane, Joe McGuinn, Boyd \"Red\" Morgan, Boyd Stockman, Jack Kenney, Brad Trumbull. An honest gambler buys a ranch but cannot take possession because the owner died without signing the final papers, and in trying to help the dead man's children he learns the place is being used by outlaws for hiding stolen cattle. Competent programmer remake of _**The Lone Gun**_ (q.v.).\n\n**1503** _ **The Gambling Terror**_ **** Republic, 1937. 60 min. D: Sam Newfield. SC: George Plympton and Fred Myton. With Johnny Mack Brown, Iris Meredith, Charles King, Ted Adams, Earl Dwire, Dick Curtis, Horace Murphy, Bobby Nelson, Frank Ellis, Frank Ball, Budd Buster, Lloyd Ingraham, Sherry Tansey, Steve Clark, George Morrell, Art Dillard, Tex Palmer, Jack Montgomery, Herman Hack, Oscar Gahan, Buck Morgan, Clyde McClary, Ray Henderson, Roy Bucko. A man pretends to be a gambler to stop a crook running a cattle protection racket. Okay Johnny Mack Brown entry in his series for producer A.W. Hackel.\n\n**1504** _ **Gangs of Sonora**_ **** Republic, 1941. 56 min. D: John English. SC: Albert De Mond and Doris Schroeder. With Robert Livingston, Bob Steele, Rufe Davis, June Johnson, Bud McTaggart, Helen MacKellar, Robert Frazer, William Farnum, Budd Buster, Hal Price, Wally West, Bud Osborne, Bud Geary, Jack Kirk, Griff Barnett, Curley Dresden, Burr Caruth, Max Waizmann, Jack O'Shea, Al Taylor, Buddy Roosevelt, Herman Hack, Jack Lawrence. Three cowboys come to the aid of newspaperwoman Kansas Kate after a dishonest rival tries to take over her business. Pleasing outing for \"The Three Mesquiteers.\"\n\n**1505** _ **Gangster's Den**_ **** Producers Releasing Corporation, 1945. 55 min. D: Sam Newfield. SC: George Plympton. With Buster Crabbe, Al St. John, Sidney Logan, Charles King, I. Stanford Jolley, Emmett Lynn, Kermit Maynard, Ed Cassidy, George Chesebro, Karl Hackett, Bob (John) Cason, Michael Owen, Wally West, Herman Hack, Steve Clark, Jimmy Aubrey, Artie Ortego, Frank McCarroll, Art Fowler, Jack Montgomery, Morgan Flowers, Art Mix, Matty Roubert, Jack Evans, Horace B. Carpenter, Foxy Callahan, Victor Cox, Rube Dalroy. Billy Carson and Fuzzy Q. Jones help a young woman whose ranch is coveted by a crook. Good \"Billy Carson\" series entry with Charles King not playing a villain for a change; here he is a lovable drunk in a brief, but hilarious, barroom sequence.\n\n**1506** _ **Gangsters of the Frontier**_ **** Producers Releasing Corporation, 1944. 58 min. D-SC: Elmer Clifton. With Tex Ritter, Dave O'Brien, Guy Wilkerson, Patti McCarty, Harry Harvey, Betty Miles, I. Stanford Jolley, Marshall Reed, Clarke Stevens, Charles King, Ted Mapes, Henry Hall, Wally West, Robert Barron, Herman Hack, Victor Cox, Ray Henderson, Lew Morphy, George Morrell, Jack Evans. The Texas Rangers come to a small town taken over by two brothers, prison escapees who are forcing the townspeople to work in the local mines. Dreary entry in \"The Texas Rangers\" series although Tex Ritter does well by the songs, including \"Please Remember Me\" and \"Ride, Ranger, Ride.\" British title: _**Raiders of the Frontier**_.\n\n**1507** _ **Garden of Evil**_ **** 20th Century\u2013Fox, 1954. 100 min. Color. D: Henry Hathaway. SC: Frank Fenton. With Clark Gable, Susan Hayward, Richard Widmark, Hugh Marlowe, Cameron Mitchell, Rita Moreno, Victor Manuel Mendoza, Fernando Wagner, Arturo Soto Bangel, Manuel Donde, Antonio Bribiesca, Salvado Terroba. A woman hires three soldiers of fortune to find her husband who disappeared in the Mexican gold fields. Fans of the stars will have a good time with this steamy melodrama.\n\n**1508** _ **Garringo**_ **** Interpeninsular, 1969. 84 min. Color. D: Rafael Romero Merchant. SC: Joaquin Romero Merchant and Vittorio Salerno. With Anthony Steffen, Peter Lee Lawrence, Solvi Stubing, Jose Bodalo, Raf Baldassarre, Luis Barboo, Frank Brana, Luis Marin, Luis Induni, Barta Barri, Alfonso Rojas, Tito Garcia, Rossana Rovere, Marta Monterrey, Lorenzo Robelod, Antonio Molino Rojo, Guillermo Mendez, Xan das Bolas, Carlos Romero Merchant, Mario Morales. A merciless Army lieutenant is assigned to hunt down the gunman who killed a fellow officer. Somewhat different Spaghetti Western in that the madman hates all officers because some of them executed his father years before; also called _**Dead Are Countless**_.\n\n**1509** _ **The Gas House Kids Go West**_ **** Producers Releasing Corporation, 1947. 62 min. D: William Beaudine. SC: Robert E. Kent, Robert A. McGowan and Eugene Conrad. With Chili Williams, John Sheldon, Carl \"Alfalfa\" Switzer, Vince Barnett, Bennie Bartlett, Tommy Bond, Emory Parnell, William Wright, Lela Bliss, Ronn Martin, Ray Dalcianne, Rudy Wissler. Youngsters win a trip to California so they deliver a car to a dealer but they find out the auto has been stolen. Typical low grade segment in PRC's \"Gas House Kids\" series.\n\n**1510** _ **The Gatling Gun**_ **** Ellman Enterprises, 1971. 93 min. Color. D: Robert Gordon. SC: Joseph Van Winkle and Mark Hanna. With Guy Stockwell, Robert Fuller, Barbara Luna, Woody Strode, Patrick Wayne, Pat Buttram, John Carradine, Phil Harris, Judy Jordan, Carlos Rivas, Tommy Cooke, Steve Conte. A cavalry officer and his men must protect a gatling gun and a westward bound family from marauding Indians. Action filled tale with well staged battle scenes.\n\n**1511** _ **The Gaucho**_ **** United Artists, 1928. 95 min. D: F. Richard Jones. SC: Lotta Woods. With Douglas Fairbanks, Lupe Velez, Geraine Greer, Eve Southern, Gustav von Seyffertitz, Michael Vavitch, Charles Stevens, Nigel de Brulier, Albert MacQuarrie, Mary Pickford. A gaucho is turned over to the law by the one who loves him because she is jealous of his interest in a \"miracle girl.\" Fun, fast moving Douglas Fairbanks romp.\n\n**1512** _ **Gaucho Serenade**_ **** Republic, 1940. 66 min. D: Frank McDonald. SC: Betty Burbridge and Bradford Ropes. With Gene Autry, Smiley Burnette, June Storey, Mary Lee, Duncan Renaldo, Cliff Severn, Jr., Lester Matthews, Smith Ballew, Joseph Crehan, William Ruhl, Wade Boteler, Ted Adams, Fred Burns, Jean Porter, Julian Rivero, George Lloyd, Ed Cassidy, Olaf Hytten, Fred \"Snowflake\" Toones, Jack Kirk, Harry Strang, Hank Worden, Jim Corey, Tom London, Walter Miller, Frankie Marvin, Gene Morgan, Al Taylor. Gene Autry and his pals get mixed up with a group of show girls and a pompous singing cowboy. Limp and action less Gene Autry film that did produce the title song and \"The Singing Hills,\" both hit records for the star and Dick Todd. Reissued as _**Keep Rollin'.**_\n\n**1513** _ **Gauchos of El Dorado**_ **** Republic, 1941. 56 min. D: Lester Orleback. SC: Albert DeMond. With Bob Steele, Tom Tyler, Rufe Davis, Lois Collier, Duncan Renaldo, Yakima Canutt, Norman Willis, Rosina Galli, William Ruhl, Edmund Cobb, Eddie Dean, Terry Frost, John Merton, Si Jenks, Ted Mapes, Bob Woodward, Horace B. Carpenter, Tony Roux, Ray Bennett, Virginia Farmer, Jack Holmes, Al Taylor, Bud Geary, Matt Roubert, Roy Bucko, Ray Jones, Lynton Brent, Bob Burns. A dishonest banker and his cohorts try to cheat a woman out of her ranch in order to get its rich bauxite deposits but they are opposed by the Three Mesquiteers. Another fast episode in the long running Republic series; one of four remakes of _**Gun Law**_ (1933) [q.v.].\n\n**1514** _ **El Gavilan Pollero**_ (The Chicken Hawk) **** Mier and Brooks, 1950. 107 min. D-SC: Rogelio A. Gonzalez. With Pedro Infante, Lilia Prado, Antonio Badu, Ana Maria Villasenor, Armando Arriola, Jose Munoz, Heckor Mateos, Victor Alcocer. A singing adventurer is pitted against his friend by the woman he loves. Okay Mexican musical comedy Western.\n\n**1515** _ **Los Gavilanes Negros**_ (The Black Sparrowhawks) **** Filmadora Chapultepec, 1966. 85 min. D: Chano Urueta. SC: Pedro Galindo, Jr. and Jose Maria Fernandez Unsain. With Luis Aguilar, Irma Serrano, Fernando Casanova, Pedro Armendariz, Jr., Ramon Bugarini, Guillermo Rivas, Carlos Leon, Notahel (Nathanael) Leon, Armando Acosta, Jose Luis Fernandez, Felipe del Castillo, Carlos Guarneros \"Don Cuco.\" A woman falls in love with a man who has been hurt by love in the past and refuses to become involved in romance. Routine Mexican romantic Western comedy.\n\n**1516** _ **The Gay Amigo**_ **** United Artists, 1949. 62 min. D: Wallace Fox. SC: Doris Schroeder. With Duncan Renaldo, Leo Carrillo, Armida, Joseph Sawyer, Fred Kohler, Jr., Walter Baldwin, Kenneth MacDonald, George DeNormand, Clayton Moore, Fred Crane, Helen Servis, Bud Osborne, Sam Flint, Beverly Jons, Al Ferguson, David Sharpe, Lee Tong Foo, Dick Elliott, Billy Wayne. The Cisco Kid and Pancho are blamed by the cavalry for a series of robberies committed by a gang disguised as Mexicans and masterminded by two corrupt businessmen. Fast moving \"Cisco Kid\" dual bill item. TV title: _**The Daring Rogue**_.\n\n**1517** _ **The Gay Buckaroo**_ **** Allied, 1932. 61 min. D: Phil Rosen. SC: Philip Graham White. With Hoot Gibson, Merna Kennedy, Roy D'Arcy, Ed Peil, Sr. Charles King, Lafe McKee, Sidney DeGrey, Skeeter Bill Robbins, The Hoot Gibson Cowboys, Glenn Strange, Maston Williams, Kit Guard, Ben Corbett, George Sowards, Lem Sowards, Milton Brown, William Gillis, Goober Glenn. A rancher and a gambler are rivals for the love of a pretty girl. Hoot Gibson opus that is on the slow side.\n\n**1518** _ **The Gay Caballero**_ **** Fox, 1932. 60 min. D: Alfred Werker. SC: Barry Conners and Philip Klein. With George O'Brien, Victor McLaglen, Conchita Montenegro, Linda Watkins, C. Henry Gordon, Weldon Heyburn, Willard Robertson, Martin Garralaga, Juan Torena, Al Garcia, Cecilia Parker, Lew Meehan, Charles Stevens, Wesley Giraud, Harry Semels, George Reed. A college football hero returns to his Western ranch home to find a crooked Mexican cattle baron has taken control of his family and their money. Well made and entertaining George O'Brien vehicle.\n\n**1519** _ **The Gay Caballero**_ **** 20th Century\u2013Fox, 1940. 58 min. D: Otto Brower. SC: Albert Duffy and John Larkin. With Cesar Romero, Sheila Ryan, Chris-Pin Martin, Robert Sterling, Janet Beecher, Edmund MacDonald, Jacqueline Dalya, Hooper Atchley, C. Montague Shaw, Ethan Laidlaw, George Magrill, LeRoy Mason, Jim Pierce, John Byron, Tom London, Dave Morris, Jack Stoney, Lee Shumway, Frank Lackteen. The Cisco Kid comes to the rescue of a young woman being swindled out of her ranch by two crooks. Delightful \"Cisco Kid\" feature that moves along at a fast clip.\n\n**1520** _ **The Gay Cavalier**_ **** Monogram, 1946. 65 min. D: William Nigh. SC: Charles Belden. With Gilbert Roland, Ramsay Ames, Martin Garralaga, Nacho Galindo, Helen Gerald, Drew Allen, Tristram Coffin, Iris Flores, John Merton, Frank LaRue, Ray Bennett, Artie Ortego, Pierre Andre, Joseph Burlando, Iris Bocignon, Terry Frost, Pierce Lyden, Dusty Rhodes, Delmar Costello, Ralph Johns, Alex Montoya, Jack La Tour, Gabriel Peralta, Bob Butt, Mike J. Rodriguez, Clem Fuller, Lynton Brent, Elvira Aldana, George J. Lewis, Wally West, Dorothy Michaels, Don Driggers, Ernie Adams, Jack Cheatham, Larry Steers, Dee Cooper, Eddie Majors, Ted Mapes. When a rancher is plagued by outlaw attacks the Cisco Kid comes to his defense. Gilbert Roland is dashing as Cisco, Ramsay Ames is nice to look at and the Roland-Tristram Coffin sword fight is exciting, but overall this \"Cisco Kid\" entry is only passable entertainment.\n\n**1521** _ **The Gay Desperado**_ **** United Artists, 1936. 88 min. D: Rouben Mamoulian. SC: Wallace Smith. With Nino Martini, Ida Lupino, Leo Carrillo, Harold Huber, Mischa Auer, Stanley Fields, James Blakeley, Paul Hurst, Adrian Rosley, Allan Garcia, Frank Puglia, Michael Visaroff, Chris-Pin Martin, Harry Semels, George Du Count, Alphonso Pedroza, Len Brixton, Travolores Chinacos, M. Alvarez Maciste. A Mexican bandit, influenced by American gangster movies, kidnaps a singing caballero along with a feisty heiress and her fiance. Picturesque musical-comedy spoof of Westerns and gangster films; pleasant entertainment highlighted by Nino Martini's singing.\n\n**1522** _ **The Gay Ranchero**_ **** Republic, 1948. 72 min. Color. D: William Witney. SC: Sloan Nibley. With Roy Rogers, Tito Guizar, Jane Frazee, Andy Devine, Estelita Rodriguez, George Meeker, LeRoy Mason, Dennis Moore, Keith Richards, Betty Gagnon, Robert Rose, Ken Terrell, Bob Nolan and The Sons of the Pioneers (Doye O'Dell, Tim Spencer, Pat Brady, Hugh and Karl Farr), David Sharpe. A lawman and a bullfighter team to thwart hijackers after gold shipments from taking over an airport. Pleasant Roy Rogers musical Western.\n\n**1523** _ **Gene Autry and the Mounties**_ **** Columbia, 1951. 70 min. D: John English. SC: Norman S. Hall. With Gene Autry, Pat Buttram, Elena Verdugo, Carleton Young, Herbert Rawlinson, Richard Emory, Trevor Bardette, Francis McDonald, Jim Frasher, Gregg Barton, House Peters, Jr., Jody Gilbert, Nolan Leary, Boyd Stockman, Teddy Infuhr, Billy Gray, Roy Butler, Chris Allen. Two Montana marshals joins forces with a Mounted Policeman to bring in a bank robber. Average, but scenic, Gene Autry vehicle.\n\n**1524** _ **General Custer at Little Big Horn**_ **** Sunset, 1926. 60 min. D: Harry Fraser. SC: Carrie E. Rawles and L.V. O'Connor. With Roy Stewart, Helen Lynch, Edmund Cobb, John Beck, Arthur Morrison, Nora Lindley, Andre Farneur. A scout and an evil Army captain both romance a pioneer girl with the military man causing an Indian uprising that leads to the Battle of Little Big Horn. Considering its limited budget, this silent historical romance from producer Anthony J. Xydias is pretty good; also called _**With General Custer at Little Big Horn**_ and _**With Custer at Little Big Horn**_.\n\n_**Genius**_ see _**A Genius, Two Friends and an Idiot**_\n\n**1525** _ **A Genius, Two Friends and an Idiot**_ **** Tobis Filmkunst, 1975. 126 min. Color. D: Damiano Damiani. SC: Damiano Damiani, Ernesto Gastaldi and Fulvio Morsella. With Terence Hill, Patrick McGoohan, Miou Miou, Robert Charlebois, Raimund Harmstorf, Piero Vida, Rik Battaglia, Mario Valgoi, Mario Brega, Frederick von Ledebur, Jean Martin, Klaus Kinski, Clara Colosimo, Ferdinando Cerulli, Benito Stefanelli, Renato Baldini, Roy Bosier, Gerard Boucaron, Miriam Mahler, Carla Cassola, Vittorio Fanfoni, Armando Bottin, Valerio Ruggeri, Lina Franchi, Pietro Torrisi, Karl Braun. A crook teams with a dim-witted half-breed and a none too bright young woman to cheat an Army major out of $300,000 but end up caught in an Indian attack. Light hearted French-Italian-West German Spaghetti Western filmed as _**Un Genio, due Compari, un Pollo**_ (A Genius, Two Friends, a Dupe) and also known as _**Genius**_ and _**Trinity Is Back Again**_.\n\n**1526** _ **Gentle Annie**_ **** Metro-Goldwyn-Mayer, 1944. 80 min. D: Andrew Martin. SC: Lawrence Hazard. With James Craig, Donna Reed, Marjorie Main, Barton MacLane, Morris Ankrum, Henry (Harry) Morgan, Paul Langton, John Philliber, Noah Beery, Frank Darien, Robert Emmett O'Connor, John Merton, Lee Phelps, Arthur Space, Norman Willis, Lee Shumway, Art Miles, Jack Clifford, Wade Crosby, Charles Williams, Jim Farley. A woman and her two sons commit a series of robberies and are tracked by a marshal disguised as a bum. Low key and amusing production with Marjorie Main stealing the show as the outlaw gang leader; based on the novel by MacKinlay Kantor.\n\n**1527** _ **Gentle Savage**_ **** Cinemation Industries, 1973. 85 min. Color. D-SC: Sean MacGregor. With William Smith, Gene Evans, Joe Flynn, Barbara Luna, R.G. Armstrong, Ned Romero, Henry Brandon, Robert Tessier, Arch Johnson, Kevin Hagen, Betty Ann Carr, Cody Bearpaw, Richard Schuyler, Larry Watson, Robert Reynolds, Darlene Conley, C.J. Hincks, Owen Orr, Robert Reynolds. In order to cover up the rape and murder of his stepdaughter, a man blames the crimes on an Indian who is hunted by two lawmen. Pretty good drama produced by star William Smith.\n\n**1528** _ **The Gentleman from Arizona**_ **** Monogram, 1939. 71 min. Color. D: Earl Haley. SC: Earl Haley and Jack O'Donnell. With John King, Joan Barclay, J. Farrell MacDonald, Craig Reynolds, Ruthie Reece, Nora Lane, Johnny Morris, Doc Pardee. A wandering cowboy comes to a ranch looking for a job and ends up entering a big horse race. Innocuous little programmer shot in an early Cinecolor process.\n\n_**Gentleman from California**_ see _**The Californian**_\n\n**1529** _ **The Gentleman from Texas**_ **** Monogram, 1946. 55 min. D: Lambert Hillyer. SC: J. Benton Cheney. With Johnny Mack Brown, Raymond Hatton, Claudia Drake, Reno Blair, Christine McIntyre, Tristram Coffin, Marshall Reed, Ted Adams, Frank LaRue, Steve Clark, Terry Frost, Tom Carter, Jack Rockwell, Lynton Brent, Pierce Lyden, Curt Barrett and The Trailsmen, George Morrell, Artie Ortego, Wally West, Chick Hannon, Rube Dalroy. Two lawmen arrive in a town where a crook has taken over by bullying the citizens. Nicely done Johnny Mack Brown entry with Tristram Coffin especially good as the bad man.\n\n**1530** _ **Gentleman Killer**_ **** Castilla Films, 1967. 97 min. Color. D: George Finley (Giorgio Stegani). SC: Jaime J. Balcazar. With Anthony Steffen, Eduardo Fajardo, Silvia Solar, Vidal Molina, Benito Stefaneli, Angel Lombardo, Gaspar Gonzalez, Antonio Iranzo, Anna Orso, Frank Oliveras, Juan Torres, Luis Barboo, Joaquin Blanco, Raul Aparici, Tomas Torres, Carlos Frigola, Isidro Martin, Jose Halufi, Valentino Macchi. A stranger looking for his brother's killer rides into a disputed border town ruled by marauding outlaws. Lots of fast action and a good music score by Bruno Nicolai highlight this Italian-Spanish co-production released in Europe as _**Gentleman Jo...Uccidi**_ (Gentleman Jo...Killer).\n\n**1531** _ **Gentlemen with Guns**_ **** Producers Releasing Corporation, 1946. 53 min. D: Sam Newfield. SC: Fred Myton. With Buster Crabbe, Al St. John, Patricia Knox, Steve Darrell, George Chesebro, Karl Hackett, Budd Buster, Frank Ellis, George Morrell, Herman Hack, Jack Evans, Art Dillard, Bert Dillard. When Fuzzy refuses to sell the water rights on his land, a crook has him framed for murder and Billy Carson rides to the rescue. Not one of the better \"Billy Carson\" episodes.\n\n**1532** _ **Geronimo**_ **** Paramount, 1939. 89 min. D-SC: Paul H. Sloane. With Preston Foster, Ellen Drew, Andy Devine, William Henry, Ralph Morgan, Gene Lockhart, Marjorie Gateson, Kitty Kelly, Monte Blue, Addison Richards, Pierre Watkin, Joseph Crehan, Chief Thundercloud, Joe Dominguez, William Haade, Ivan Miller, Frank M. Thomas, Syd Saylor, Richard Denning, Steve Gaylord, Francis Ford, Russell Simpson, Archie Twitchell, Pat West, Cecil Kellogg, Harry Templeton, Guy Wilkerson. An Army captain tries to stop a war between whites and Indians. Trite melodrama made up of lots of stock footage and despite its title Geronimo (Chief Thundercloud) is barely in evidence.\n\n**1533** _ **Geronimo**_ **** United Artists, 1962. 101 min. Color. D: Arnold Laven. SC: Pat Fielder. With Chuck Connors, Kamala Devi, Ross Martin, Pat Conway, Adam West, Enid Jaynes, Lawrence Dobkin, Denver Pyle, Armando Silvestre, John Anderson, Joe Higgins, Robert Hughes, Mario Navarro, Bill Hughes, James Burk. After bad treatment from a dishonest Indian agent, Geronimo puts together a small band of braves and plans to attack the U.S. cavalry. Anemic retelling of the Geronimo saga.\n\n**1534** _ **Geronimo:**_ ****_**An American Legend**_ **** Columbia, 1993. 115 min. Color. D: Walter Hill. SC: John Milius and Larry Gross. With Jason Patric, Gene Hackman, Robert Duvall, Wes Studi, Matt Damon, Rodney A. Grant, Kevin Tighe, Steve Reevis, Carlos Palomino, Victor Aaron, Stuart Proud Eagle Grant, Stephen McHattie, John Finn, Lee de Broux, Rino Thunder, Hoke Howell, Richard Martin, Jr., J. Young, Raleigh Wilson, Jackie Old Coyote, Monty Bass, Pato Hoffman, Scott Crabbe, Patricia Pretzinger, Roger Callard, Juddson Keith Linn, Mark Boone Junior, M.C. Gainey, Michael Ruud, Michael Minjarez, Burnette Bennett, Davina Smith, Jonathan Ward, Luis Contreras, Jacquelin Lee, Jim Manygoats, Scott Wilson, Eva Larson, Jim Beaver. Geronimo and his band of braves break away from the Apache reservation and fight government troops in order to stay free. Fairly well done historical Western; big budget but a box office bust.\n\n**1535** _ **Geronimo's Revenge**_ **** Buena Vista, 1965. 61 min. Color. D: James Neilson and Harry Keller. SC: D.P. Harmon. With Tom Tryon, Darryl Hickman, Betty Lynn, Brian Corcoran, Onslow Stevens, Harry Carey, Jr., Allan Lane, Pat Hogan, Charles Maxwell, James Edwards, Annette Gorman, Jay Silverheels. A Texas Ranger tries to help the chief of the Natchez Indians when Geronimo disobeys orders and goes on the warpath. Standard action film originally telecast as a segment of the \"Texas John Slaughter\" series on Walt Disney's ABC-TV program on March 4, 1960, and issued abroad theatrically.\n\n**1536** _ **Get Mean**_ **** Strange Films, 1976. 90 min. Color. D: Ferdinando Baldi. SC: Lloyd Battista and Wolfe Lowenthal. With Tony Anthony, Lloyd Battista, Diana Lorys, Raf Baldassare, David Dreyer, Mirta Miller. A stranger single handedly attacks a fortress, the stronghold of the man who tried to kill him, and finds a treasure. Outlandish, violent Spaghetti Western produced by star Tony Anthony and taking place in Medieval times.\n\n**1537** _ **Ghost City**_ **** Monogram, 1932. 60 min. D: Harry Fraser. SC: Wellyn Totman. With Bill Cody, Andy Shuford, Helen Forrest, Walter Miller, Charles King, Walter Shumway, Si Jenks, Al Taylor, Kate Campbell, Jack Carlisle, Thomas Curran, Hank Bell. A cowboy helps a young woman who is trying to obtain her rightful gold filed inheritance from a crook. Fair Billy Cody-Andy Shuford series vehicle.\n\n**1538** _ **Ghost Dance**_ **** Trans World Entertainment, 1980. 96 min. Color. D: Peter F. Buffa. SC: Peter F. Buffa and Robert M. Sutton. With Julie Amato, Victor Mohica, Henry Bal, Frank Sotonoma Salsedo, James Andronica, Patricia Alice Albrecht, Deloris Maaske, J. Christopher Senter, Henry Max Kendrick, Felicia Leon, Ramon Chavez, Frank A. Soto, Jim Brockett, Kirk Irving Koskella, Susan Carol Stymore, Donald L. Shanks, Quentin Sondergaard, Peter Garcia, Patrick Garcia, Inez Perez, Joe Faust, Gil Escandon, Don Zapian, Laurie Ball. An Indian shaman becomes a mad killer when the spirit of a dead warrior takes over his body. Low grade horror Western.\n\n**1539** _ **Ghost Dancing**_ **** ABC-TV, 1983. 100 min. Color. D: David Greene. SC: Phil Penningroth. With Dorothy McGuire, Bruce Davison, Bill Erwin, Richard Farnsworth, Wings Hauser, Bo Hopkins, Victoria Racimo, Rod Colbin, Richard Lineback, Karen Machon, Fran Ryan, Sierra Pecheur, John Bellah, Scotch Byerly, Robert Clotworthy, Henry Hamilton, Eleanor Zee. When a city uses its authority to drain the water from a fertile valley, a poor widow rancher destroys the pipeline in hopes of bringing the area's plight to public attention. Nicely done TV movie enhanced by Dorothy McGuire's strong performance as the protagonist.\n\n**1540** _ **Ghost Guns**_ **** Monogram, 1944. 60 min. D: Lambert Hillyer. SC: Frank H. Young. With Johnny Mack Brown, Raymond Hatton, Evelyn Finley, John Merton, Tom Quinn, Sarah Padden, Marshall Reed, Ernie Adams, Jack Ingram, Frank LaRue, Steve Clark, Bob (John) Cason, George Morrell, Riley Hill, Ray Jones, Dick Rush, Jack Evans, Chick Hannon, Dee Cooper, Dick Dickinson. Two U.S. marshals try to get to the bottom of the murders of ranchers and the rustling of their cattle by crooks out to take over a valley because a railroad spur will be built there. Good segment in Johnny Mack Brown's \"Nevada Jack McKenzie\" programmers.\n\n**1541** _ **Ghost of Crossbones Canyon**_ **** Monogram, 1952. 56 min. D: Frank McDonald. SC: Maurice Tombragel. With Guy Madison, Andy Devine, Russell Simpson, John Doucette, Bart Davidson, Gordon Jones, Mike Ragan, Ray Bennett, Marjorie Bennett, Sam Flint, Joe Greene, James Guifoyle, Billy Bletcher. On the trail of robbers, Wild Bill Hickok and his deputy Jingles arrive in a ghost town allegedly haunted by the spirit of a famous outlaw. Average paste-up feature from two episodes of the \"Wild Bill Hickok\" (1951\u201358) TV series, \"The Tax Collecting Story\" and \"Ghost Town Story.\" Issued theatrically in some areas under the Allied Artists banner.\n\n**1542** _ **Ghost of Hidden Valley**_ **** Producers Releasing Corporation, 1946. 56 min. D: Sam Newfield. SC: Ellen Coyle. With Buster Crabbe, Al St. John, Jean Carlin, John Meredith, John Cason, Charles King, Jimmy Aubrey, George Morrell, Bert Dillard, Karl Hackett, Silver Harr, Zon Murray, Bob Burns, Milburn Morante, Wally West, Ray Henderson, Herman Hack, Denver Dixon, Jack Evans. Billy Carson and Fuzzy Q. Jones are on the trail of cattle rustlers who use a range belonging to a young married couple to hide stolen herds. Passable \"Billy Carson\" series entry.\n\n**1543** _ **Ghost of Zorro**_ **** Republic, 1949. 12 Chapters. D: Fred C. Brannon. SC: Royal Cole, William Lively and Sol Shor. With Clayton Moore, Pamela Blake, Roy Barcroft, George J. Lewis, Gene Roth, John Crawford, I. Stanford Jolley, Steve Clark, Steve Darrell, Dale Van Sickel, Tom Steele, Marshall Reed, Jack O'Shea, Holly Bane, Bob Reeves, Eddie Parker, Stanley Blystone, Joe Yrigoyen, George Chesebro, Charles King, Kenneth Terrell, Robert Wilke, Art Dillard, Frank Ellis, Chuck Roberson. When crooks try to halt the construction of a telegraph line, a descendant of Don Diego takes on the guise of Zorro to stop them. Fast moving cliffhanger released in a feature version in 1959.\n\n**1544** _ **Ghost Patrol**_ **** Puritan, 1938. 56 min. D: Sam Newfield. SC: Joseph O'Donnell. With Tim McCoy, Claudia Dell, Walter Miller, Wheeler Oakman, Lloyd Ingraham, Dick Curtis, Slim Whitaker, Artie Ortego, Art Dillard, Frank Ellis, Bruce Mitchell, Jack Cheatham, Blackie Whiteford. A scientist invents a ray machine that outlaws use to bring down mail shipment planes in order to rob them. The sci-fi element adds some zest to this fast moving Tim McCoy vehicle.\n\n**1545** _ **The Ghost Rider**_ **** Superior\/First Division, 1935. 56 min. D: Jack (Jevne) Levine. SC: John West (Jack Jevne). With Rex Lease, Bobby Nelson, Ann Carol, William Desmond, Franklyn Farnum, Art Mix, Bill Patton, Lloyd (Ingraham) Ingram, Blackie (Whiteford) Whitcomb, Roger Williams, Ed (Eddie) Parker, Lafe McKee, Jack (Blackjack) Ward, John Alexander, Ed Coxen, Ernie Adams, Jack Kirk, Denver Dixon. A deputy sheriff after outlaws finds himself helped by a ghostly masked phantom. Cheap Louis Weiss production enhanced by its mystery plot.\n\n**1546** _ **The Ghost Rider**_ **** Monogram, 1943. 54 min. D: Wallace Fox. SC: Jess Bowers (Adele Buffington). With Johnny Mack Brown, Raymond Hatton, Beverly Boyd, Harry Woods, Charles King, Edmund Cobb, Bud Osborne, Milburn Morante, George Morrell, Tom Seidel, Artie Ortego, George DeNormand, Bill Hunter, Wally West, Bill Nestell, Jack Daley, Horace B. Carpenter, Ray Miller, Art Fowler, Jess Cavin, Herman Hack, Foxy Callahan, Kansas Moehring, Ralph Bucko, Roy Bucko. A man known as \"The Ghost Rider\" tries to find his father's killer and teams with a U.S. marshal to catch him. Good start to the \"Nevada Jack McKenzie\" series starring Johnny Mack Brown and Raymond Hatton.\n\n**1547** _ **Ghost Riders**_ **** New World, 1987. 85 min. Color. D: Alan L. Stewart. SC: James Desmarais and Clay McBride. With Bill Shaw, Jim Peters, Ricky Long, Cari Powell, Mike Sammns, Arland Bishop, Beverly Cleveland, James Desmarais, Cari Young, Bill Moses, Wade \"Jesse\" Mason, Steve Fincher, Doc Lipsey, Gerald Stewart, David Miller. Hanged members of an outlaw gang return from the dead to take vengeance on the descendants of those who sent them to the gallows. Low grade horror Western made in Texas.\n\n**1548** _ **Ghost Riders**_ **** K and K Motion Pictures, 1993. 75 min. Color. D-SC: Ronald Koontz. With Ronald Koontz, Doss Bryant, Vernon Koontz, Barbara Manns, Genia Richardson, Doyle Thomason, Ted Thomason, Barry Rentz, Joel Koontz, Mike Koontz, Danny Branson, Jerry Earles, Steve Thomason, Ronnie Sanders, Roger Leonard, Bart Leonard, Scott Foster, Kermite Omen, Jr., Mark Styers, David Soloman, Todd Carver. Gunfights, bank holdups and cattle drives are retained in the memories of a remote Western town. Obscure production made in North Carolina by writer-director-editor-photographer and star Ronald Koontz.\n\n_**Ghost Riders of the West**_ see _**The Phantom Rider**_ (1946)\n\n**1549** _ **Ghost Rock**_ **** Lion's Gate, 2004. 101 min. Color. D: Dustin Rikert. SC: Michael Worth. With Gary Busey, Michael Worth, Jeff Fahey, Adrienne Barbeau, Craig Wasson, Jenya Lano, James Hong, John Laughlin, Rance Howard, David Jean Thomas, Christa Sauls, April Hong, Peter Kwong, Michiko Nishiwaki, Daniel Southworth, Shane Lacey, Pauline Hemmer, Mike Vaughn, Conroy Kanter, Renee Roland, Yan Birch, Peggy Seagren, Joyce Westergaard, Shauna Sand-Lamas, Art Monte, Peter Sherayko, Lindy Teague, Dustin Rikert, Wade Rikert, Kyle Rikert, Jonathan DePaz, Sal Cardile, Jennifer Walden, Tom Ford, Greg Debeneditti, Preston Gamblin, Rob Jensen, Bobby Havens, Kathy Messick, Todd Swindell, Matt B. Davis, Jennifer Masisaac, Philip Zabriskie, Holly Baron, Jerusha Rubi, Jamie Vaughan. Two decades after he survived a massacre, a man returns to where it happened and joins forces with a female gunfighter to kill the town boss responsible. Not bad action filled, revenge motivated drama.\n\n**1550** _ **Ghost Town**_ **** Commodore, 1936. 60 min. D: Harry Fraser. SC: Monroe Talbot. With Harry Carey, Ruth Findlay, Jane Novak, David Sharpe, Lee Shumway, Ed Cassidy, Roger Williams, Earl Dwire, Phil Dunham, Chuck Morrison, Sonny (horse). When claim jumpers go after a mine, a cowboy tries to help its owner. Low budget but quite satisfying Harry Carey film from producer William Berke.\n\n_**Ghost Town**_ (1941) see _**The Lone Rider in Ghost Town**_\n\n**1551** _ **Ghost Town**_ **** United Artists, 1956. 75 min. D: Allen Miner. SC: Jameson Brewer. With Kent Taylor, John Smith, Marian Carr, John Doucette, William Phillips, Serena Sande, Gary Murray. After a stagecoach arrives at a way station recently raided by Indians, the passengers decide to head for a ghost town in order to avoid the hostilities. More than passable programmer.\n\n**1552** _ **Ghost Town**_ **** New World, 1988. 85 min. Color. D: Richard Governor. SC: Duke Sandefur. With Franc Luz, Catherine Hickland, Jimmie F. Skaggs, Penelope Windust, Bruce Glover, Zitto Kazann, Blake Conway, Laura Schaefer, Michael Alldredge, Ken Kolb, Will Hannah, Henry Max Kendrick, James Oscar Lee, Charles Robert Harden, Edward Gabel, Jackson Fisher, Julie Kausier. A deputy sheriff arrives in a ghost town looking for missing girls and finds it haunted by an outlaw gang. Early effort from executive producer Charles Band is a better than average horror Western.\n\n_**Ghost Town\u2014The Movie**_ see _**Dean Tester's Ghost Town**_\n\n**1553** _ **Ghost Town Gold**_ **** Republic, 1936. 57 min. D: Joseph Kane. SC: John Rathmell and Oliver Drake. With Robert Livingston, Ray Corrigan, Max Terhune, Kay Hughes, Yakima Canutt, Frank Hagney, LeRoy Mason, Burr Caruth, Robert Kortman, Milburn Morante, Horace Murphy, Earle Hodgins, Ed Peil, Sr., Harry Harvey, Hank Worden, Bud Osborne, Bob Burns, I. Stanford Jolley, Wally West, Don Roberts, F. Herrick, Robert C. Thomas, Harry Tenbrook, Budd Buster, Charles Sullivan, Billy Franey, Horace B. Carpenter, Jess Cavin, Bill Hickey, Art Dillard, Rube Dalroy. The Three Mesquiteers are after an outlaw gang that hides its stolen loot in a ghost town. Second entry in \"The Three Mesquiteers\" series based on the William Colt MacDonald characters is solid entertainment; it is the first one with the great Max Terhune as Lullaby Joslin (and, of course, Elmer).\n\n**1554** _ **Ghost Town Law**_ **** Monogram, 1942. 62 min. D: Howard Bretherton. SC: Jess Bowers (Adele Buffington). With Buck Jones, Tim McCoy, Raymond Hatton, Virginia Carpenter, Murdock MacQuarrie, Charles King, Howard Masters, Ben Corbett, Tom London, Milburn Morante, Robert Walker, Jack Baxley, Eddie Phillips, Jack Ingram, Frank Lackteen, Artie Ortego. While investigating the murders of two fellow lawmen, the Rough Riders help a woman whose brother has disappeared. A mystery element and good staging help to makes this an entertaining \"Rough Riders\" romp.\n\n**1555** _ **Ghost Town Renegades**_ **** Producers Releasing Corporation, 1947. 58 min. D: Ray Taylor. SC: Patricia Harper. With Lash LaRue, Al St. John, Jennifer Holt, Jack Ingram, Terry Frost, Steve Clark, Lane Bradford, Lee Roberts, William Fawcett, Henry Hall, Dee Cooper, Mason Wynn. The Cheyenne Kid and his pal Fuzzy Q. Jones try to stop crooks from taking over a property for its gold, causing the Kid to be framed for murder. Action filled Lash LaRue vehicle.\n\n**1556** _ **Ghost Town Riders**_ **** Universal, 1938. 54 min. D: George Waggner. SC: Joseph West (George Waggner). With Bob Baker, Fay (McKenzie) Shannon, George Cleveland, Hank Worden, Forrest Taylor, Glenn Strange, Jack Kirk, Martin Turner, Reed Howes, Murdock MacQuarrie, Merrill McCormick, George Morrell, Frank Ellis, Oscar Gahan, Tex Phelps. While leading a horse herd, two cowboys arrive in a ghost town where a gang plans to start a fake gold rush. Pretty good Bob Baker feature predating writer-director George Waggner's Universal horror efforts.\n\n**1557** _ **Ghost Valley**_ **** RKO Path\u00e9, 1932. 54 min. D: Fred Allen. SC: Adele Buffington. With Tom Keene, Merna Kennedy, Buck Moulton, Kate Campbell, Harry Brown, Mitchell Harris, Harry Semels, Ted Adams, Al Taylor, Slim Whitaker, George Hayes, Tom London, Jack Kirk, Yakima Canutt. Mysterious happenings take place after a young woman and her brother inherit a gold mine and a cowboy tries to help them. Eerie and atmospheric Tom Keene vehicle; solid entertainment.\n\n**1558** _ **Ghost Valley Raiders**_ **** Republic, 1940. 54 min. D: George Sherman. SC: Bennett Cohen. With Don \"Red\" Barry, Lona Andre, LeRoy Mason, Tom London, Jack Ingram, Horace Murphy, Ralph Peters, Curley Dresden, Yakima Canutt, John Beach, Bud Osborne, Al Taylor, Jack Montgomery, Fred Burns. A cowboy assumes another identity in trying to capture a notorious stagecoach robber. Don Barry's initial series feature shows why he quickly established himself as one of the genre's most popular and durable players.\n\n**1559** _ **Ghosts That Still Walk**_ **** Gold Key, 1977. 96 min. Color. D-SC: James T. Flocker. With Matt Boston, Ann Douglas, Caroline Howe, Jerry Jensen, Rita Crafts, Phil Catalli, Lee James, Janice Renney, David Kane. The spirit of an Native American mummy takes over a boy's body and murders his family. Cheap Western ghost tale.\n\n**1560** _ **Giant**_ **** Warner Bros., 1956. 201 min. Color. D: George Stevens. SC: Fred Guiol and Ivan Moffat. With Elizabeth Taylor, Rock Hudson, James Dean, Carroll Baker, Chill Wills, Mercedes McCambridge, Jane Withers, Sal Mineo, Robert Nichols, Dennis Hopper, Elsa Cardenas, Fran Bennett, Earl Holliman, Paul Fix, Judith Evelyn, Carolyn Craig, Rodney (Rod) Taylor, Alexander Scourby, Monte Hale, Mary Ann Edwards, Charles Watts, Maurice Jara, Victor Millan, Sheb Wooley, Ray Whitley, Tina Menard, Mickey Simpson, Noreen Nash, Guy Teague, Max Terhune, Ray Bennett, Barbara Barrie, George Dunne, Slim Talbot, Tex Driscoll. A wealthy Texas rancher marries a strong willed woman as they face problems with worker discontent and an ambitious foreman who becomes a rich oilman. Over long but good adaptation of Edna Ferber's novel; the film has developed a cult following.\n\n**Elizabeth Taylor and James Dean in** _**Giant**_ **(Warner Bros., 1956).**\n\n**  \n**\n\n**1561** _ **The Giant Gila Monster**_ **** Hollywood Pictures, 1959. 74 min. D: Ray Kellogg. SC: Jay Simms. With Don Sullivan, Lisa Simone, Pat Reaves, Shug Fisher, Jerry Cortwright, Beverly Thurman, Clarke Browne, Pat Simmons, Fred Graham, Grady Vaughn, Howard Ware, Don Flourney, Bob Thompson. The denizens of a New Mexico desert town are terrorized by a giant lizard. El cheapo production from producer Ken Curtis, with a silly (and non-scary) blow-up monster; also available in a colorized edition.\n\n**1562** _ **The Girl and the Gambler**_ **** RKO Radio, 1939. 63 min. D: Lew Landers. SC: Joseph A. Fields and Clarence Upson Young. With Leo Carrillo, Steffi Duna, Tim Holt, Donald MacBride, Chris-Pin Martin, Paul Fix, Julian Rivero, Frank Puglia, Esther Muir, Paul Sutton, Charles Stevens, Frank Lackteen, Edward Raquello, Henry Roquemore. An American falls in love with a Mexican girl who is also being wooed by a revolutionary leader. Okay remake of _**The Dove**_ (United Artists, 1927) and also done previously by RKO as _**Girl of the Rio**_ (q.v.) in which Leo Carrillo also played the Pancho Villa-type rebel.\n\n**1563** _ **Girl Crazy**_ **** RKO Radio, 1932. 75 min. D: William A. Seiter. SC: Tim Whelan. With Bert Wheeler, Robert Woolsey, Dorothy Lee, Eddie Quillan, Mitzi Green, Brooks Benedict, Kitty Kelly, Arline Judge, Stanley Fields, Lita Chevret, Chris-Pin Martin, Monte Collins, Nat Pendleton, Rochelle Hudson, Dick Curtis, Frank Ellis, Bob Reeves, Ethan Laidlaw, Jim Mason, Artie Ortego, Jerry Mandy, High Eagle, Al Cooke, Esther Garcia, Josefina Ramos, Max Steiner. Sent to an Arizona town by his father to get him away from women, a man is helped by his zany pal and a cab driver in fighting local bad men. Only average screen version of the Broadway production with music by George and Ira Gershwin; re-takes were directed by Norman Taurog who helmed the 1943 remake (q.v.).\n\n**1564** _ **Girl Crazy**_ **** Metro-Goldwyn-Mayer, 1943. 99 min. D: Norman Taurog. SC: Fred F. Finklehoffe. With Mickey Rooney, Judy Garland, Gil Stratton, Robert E. Strickland, Rags Ragland, June Allyson, Nancy Walker, Guy Kibbee, Tommy Dorsey and His Orchestra, Frances Rafferty, Howard Freeman, Henry O'Neill, Sidney Miller, Eve Whitney, Carole Gallagher, Kay Williams, Jess Lee Brooks, Roger Moore, Charles Coleman, Harry Depp, Richard Kipling, Henry Roquemore, Alphonse Martel, Barbara Bedford, Victor Potel, William Beaudine, Jr., Irving Bacon, George Offerman, Jr., Georgia Carroll, Noreen Nash, Inez Cooper, Hazel Brooks, Don Taylor, James Warren, Helen Dickson, Julia Griffith, Lillian West, Bess Flowers, Harry C. Bradley, Bill Hazlett, Spec O'Donnell, Frank Jaquet, Peter Lawford, Jimmy Butler, Bob Lowell, John Eaton, Rose Higgins, Kathleen Williams, Leo Diamond Harmonica Band. A youthful playboy is sent to a Western college by his publisher father to avoid gold diggers and upon arriving he flirts with the postmistress. While better than the 1932 (q.v.) outing, this second filming of the Gershwins' musical is not top notch although the finale staged by Busby Berkeley (the film's initial director) is worth seeing. Filmed a third time in 1958 as _**When the Boys Meet the Girls**_ (q.v.).\n\n**1565** _ **The Girl from Alaska**_ **** Republic, 1942. 75 min. D: Nick Grinde. SC: Edward T. Lowe and Robert Ormond Case. With Ray Middleton, Jean Parker, Jerome Cowan, Robert Barrat, Ray Mala, Francis McDonald, Raymond Hatton, Milton Parsons, Nestor Paiva, Edmund Cobb, Jack O'Shea, Frank Lackteen, Johnnhy Kascier, Matty Roubert, Bob Jameson, Iron Eyes Cody, Bill Wilkerson, Augie Gomez, Art Dupuis. A prospector wanted by the law is used by a gang in Alaska to cheat a young woman out of her gold claim. Robert Ormond Case co-adapted his story \"The Golden Porage\" for this pretty good melodrama.\n\n**1566** _ **The Girl from Calgary**_ **** Monogram, 1932. 66 min. D: Phil Whitman. SC: Leon D'Usseau. With Fifi D'Orsay, Paul Kelly, Astrid Allwyn, Robert Warwick, Eddie Fetherston, Edwin Maxwell, Adrienne Dore, Geneva Mitchell, Rolfe Sedan, Tiny Sanford, Harry Bowen. A pretty Canadian rodeo champion sets her cap for the man she plans to marry. Standard light hearted action programmer.\n\n**1567** _ **Girl from God's Country**_ **** Republic, 1940. 54 min. D: Sidney Salkow. SC: Elizabeth Meehan, Robert Lee Johnson and Malcolm Stuart Boylan. With Chester Morris, Jane Wyatt, Charles Bickford, (Ray) Mala, Kate Lawson, John Bleifer, Mamo Clark, Ferike Boros, Don Zelaya, Clem Bevans, Edward Gargan, Spencer Charters, Thomas Jackson, Victor Potel, Si Jenks, Gene Morgan, Ace (dog). Hunted by the law for the mercy killing of his father, a doctor is helped by a pretty nurse in Alaska. Fairly pleasing drama.\n\n**1568** _ **The Girl from San Lorenzo**_ **** United Artists, 1950. 59 min. D: Derwin Abrahams. SC: Ford Beebe. With Duncan Renaldo, Leo Carrillo, Jane Adams, Leonard Penn, Edmund Cobb, David Sharpe, Lee Phelps, Bill Lester, Don C. Harvey, Byron Foulger, Wes Hudman, Henry Wills. Outlaws carry out a series of stage robberies and place the blame on the Cisco Kid and Pancho. The final theatrical release in \"The Cisco Kid\" series is action filled. Also called _**Don Amigo**_.\n\n**1569** _ **Girl in the Woods**_ **** Republic, 1957. 71 min. D: Tom Gries. SC: Oliver Crawford and Marcel Klauber. With Forrest Tucker, Maggie Hayes, Barton MacLane, Diana Francis, Murvyn Vye, Paul Langton, Joyce Compton, Kim Charney, Mickey Finn, Bartlett Robinson, George Lynn. A veteran lumberman comes to work for a new outfit and trouble erupts with a rival over a young woman. Cheaply made but passable north woods melodrama.\n\n**1570** _ **The Girl of the Golden West**_ **** First National, 1930. 100 min. D: John Francis Dillon. SC: Waldemar Young. With Ann Harding, James Rennie, Harry Bannister, Ben Hendricks, Jr., J. Farrell MacDonald, George Cooper, Johnny Walker, Richard Carlyle, Arthur Stone, Arthur Houseman, Norman McNeil, Fred Warren, Joe Girard, Newton House, Princess Noola, Chief Yowlachie, Francis Ford, Russell Simpson, Cy Kendall, Richard Tucker, Frank McGlynn, E. Alyn Warren, Chief Big Tree, Victor Potel, Pedro Regas, Virginia Howell, Hank Bell, Nick Thompson, Alberto Morin, Joe Dominguez, Tom Mahoney. A pretty saloon owner falls in love with a notorious bandit and wins his freedom in a poker game with a lawman. Fair first sound version of the David Belasco play, filmed in 1915 by Cecil B. DeMille for Paramount and again in 1923 by Associated First National with J. Warren Kerrigan, Sylvia Breamer, Russell Simpson and Rosemary Theby.\n\n**1571** _ **The Girl of the Golden West**_ **** Metro-Goldwyn-Mayer, 1938. 120 min. D: Robert Z. Leonard. SC: Isabel Dawn and Boyce McGrew. With Jeanette MacDonald, Nelson Eddy, Walter Pidgeon, Leo Carrillo, Buddy Ebsen, Cliff Edwards, Leonard Penn, Priscilla Lawson, Bob Murphy, Olin Howland, Billy Bevan, Brandon Tynan, H.B. Warner, Monty Woolley, Charles Grapewin, Noah Beery, Bill Cody, Jr., Ynez Seabury, Victor Potel, Nick Thompson, Chief Big Tree, Russell Simpson, Curley Wright, Pedro Regas, Alberto Morin, Joe Dominguez, Frank McGlynn, Cy Kendall, E. Alyn Warren, Hank Bell, Francis Ford, Richard Tucker. A young woman competes in a battle of wits with a sheriff who is after her bandit lover. David Belasco's 1905 evergreen is brought to the screen for the fourth time (Cecil B. DeMille filmed the first version in 1914 and Associated First National did a remake in 1923 followed by the 1930 [q.v.] feature) but not even a score by Gus Kahn and Sigmund Romberg can save the aged story; for fans of Jeannette MacDonald and Nelson Eddy.\n\n**1572** _ **The Girl of the Rio**_ **** RKO Radio, 1932. 69 min. D: Herbert Brenon. SC: Elizabeth Meehan. With Dolores Del Rio, Leo Carrillo, Norman Foster, Ralph Ince, Lucille Gleason, Edna Murphy, Stanley Fields, Frank Campeau. A beautiful Mexican cabaret entertainer falls in love with a cardsharp and gambles for his life with an outlaw. Fairly charming Dolores Del Rio movie from Willard Mack's play _The Dove_ and first made under that title by United Artists in 1928 with Norma Talmadge, Gilbert Roland and Noah Beery, and done for a third time by RKO in 1939 as _**The Girl and the Gambler**_ (q.v.) with Leo Carrillo repeating the role of the bandit.\n\n**1573** _ **Girl Rush**_ **** RKO Radio, 1944. 65 min. D: Gordon Douglas. SC: Robert E. Kent. With Alan Carney, Wally Brown, Frances Langford, Robert Mitchum, Vera Vague (Barbara Jo Allen), Paul Hurst, Patti Brill, Sarah Padden, Cy Kendall, John Merton, Diana King, Rita Corday, Elaine Riley, Rosemary La Planche, Daun Kennedy, Virginia Belmont, Michael Vallon, Sherry Hall, Kernan Cripps, Wheaton Chambers, Chilli Williams, Ernie Adams, Dale Van Sickel, Kenneth Terrell, Bud Osborne, Byron Foulger. When all their patrons head for the gold fields in 1849, a show troupe follows them to entertain the miners. Average Western musical comedy.\n\n**1574** _ **Git Along Little Dogies**_ **** Republic, 1937. 60 min. D: Joseph Kane. SC: Dorrell McGowan and Stuart McGowan. With Gene Autry, Smiley Burnette, Judith Allen, William Farnum, Weldon Heyburn, The Maple City Four, Carleton Young, Will and Gladys Ahern, Willie Fung, The Cabin Kids, G. Raymond Nye, Frankie Marvin, George Morrell, Horace B. Carpenter, Earl Dwire, Lynton Brent, Jack Kirk, Al Taylor, Frank Ellis, Jack C. Smith, Murdock MacQuarrie, Oscar Gahan, Monte Montague, Sam McDaniel, Eddie Parker, Bob Burns. Oil drillers vie with cattlemen over range land while Gene Autry tries to clear up the conflict and romance a banker's daughter. Pleasant Gene Autry production with its nice blend of songs and action.\n\n_**Glory Glory**_ see _**Hooded Angels**_\n\n**1575** _ **The Glory Guys**_ **** United Artists, 1965. 112 min. Color. D: Arnold Laven. SC: Sam Peckinpah. With Tom Tryon, Harve Presnell, Michael Anderson, Jr., Senta Berger, James Caan, Andrew Duggan, Slim Pickens, Peter Breck, Jeanne Cooper, Laurel Goodwin, Adam Williams, Erik Holland, Wayne Rogers, Alice Backus. A soldier, with a troop of untrained recruits, is ordered by his superiors to do battle with rampaging Sioux Indians. Filmed in Mexico, this oater is a pedestrian drama with little to recommend it.\n\n**1576** _ **The Glory Trail**_ **** Crescent, 1937. 64 min. D: Lynn Shores. SC: John T. Neville. With Tom Keene, Joan Barclay, E.H. Calvert, Frank Melton, William Royle, Walter Long, Allen Greer, William Crowell, Harve Foster, Ann Hovey, John Lester Johnson, Etta McDaniel, James Bush, Jack Ingram, Oscar Gahan, Carl Mathews, Fred Parker, Denver Dixon, Tom Steele. After the Civil War a cowboy takes part in the settlement of the West and the events leading to the Bozeman Massacre. Colorful historical drama in the series produced by E.B. Derr starring Tom Keene.\n\n_**Go for Broke**_ see _**All Out**_\n\n_**Go Kill and Come Back**_ see _**Any Gun Can Play**_\n\n**1577** _ **Go West**_ **** Metro-Goldwyn-Mayer, 1940. 80 min. D: Edward Buzzell. SC: Irving Brecher. With The Marx Brothers (Groucho, Harpo, Chico), John Carroll, Diana Lewis, Robert Barrat, Walter Woolf King, June MacCloy, George Lessey, Mitchell Lewis, Tully Marshall, Clem Bevans, Joe Yule, Arthur Houseman, Joan Woodbury, Iris Adrian, Edgar Dearing, Edward Gargan, Billy Wayne, Barbara Bedford, Frederick Burton, Harry Tyler, Lew Harvey, Slim Lucas, Fred Warren, Henry Sylvester. Three zanies head West to reclaim a land deed stolen by a crook. None-too-good Marx Brothers vehicle, only a fair satire and certainly not much of a Western; for diehard Marx Brothers fans.\n\n**1578** _ **Go West, Young Girl**_ **** ABC-TV\/Columbia, 1978. 74 min. Color. D: Alan J. Levi. SC: George Yanok. With Karen Valentine, Sandra Will, Stuart Whitman, Richard Jaeckel, Michael Bell, Carl Bellini, David Dukes, Charles Frank, Richard Kelton, William Larsen, John Quade, Gregg Palmer, Pepe Callahan. Two young women, a New England writer and a cavalry officer's widow, team to hunt for Billy the Kid. Fairly amusing Western comedy made for TV.\n\n**1579** _ **Go West, Young Lady**_ **** Columbia, 1941. 70 min. D: Frank R. Strayer. SC: Richard Flournoy and Karen De Wolf. With Penny Singleton, Glenn Ford, Ann Miller, Charles Ruggles, Allen Jenkins, Onslow Stevens, Bob Wills and His Texas Playboys, Edith Meiser, Bill Hazlett, Jed Prouty, Kermit Maynard, The Foursome, Kenneth MacDonald, Stanley Brown, George Chesebro, Al Ferguson, Edmund Cobb, Bud Osborne, Ned Glass, Hank Bell, Fern Emmett, Bill Wilkerson, Art Miles, Waffles, Dorothy Vaughn. A saloon keeper sends for his nephew who turns out to be a pretty young lady plagued by a series of misadventures. Tepid Western musical comedy that is buoyed by Penny Singleton's singing and Ann Miller's dancing.\n\n**1580** _ **God Forgives:**_ _**His Life Is Mine**_ **** Italian International, 1968. 90 min. Color. D: Paolo Bianchi. SC: Fernando Di Leo. With Dean Reed, Peter Martell, Piero Lulli, Agnes Spaak, Linda Veras, Ivano Staccioli, Fidel Gonzales, Ivan Scratuglia, Piero Mazzinghi, Rossella Bergamonti, Bruno Arie, Giuseppe Alizeri, Appio Cartei. A bounty hunter takes the job of stopping a banker who is the head of a gold robbery gang operating on the Mexican border. Star Dean Reed sings the title song in this better than average Italian production filmed as _**Dio li Crea...Io li Ammazzo!**_ (God Made Them...I Kill Them).\n\n**1581** _ **God Forgives, I Don't**_ **** American International, 1969. 101 min. Color. D-SC: Giuseppe Colizzi. With Terence Hill, Bud Spencer, Frank Wolff, Gina Rovere, Jose Manuel Martin, Frank Brana, Tito Garcia, Paco Sanz, Giovanni Lenzi, Luis Barboo. A gunman and an insurance salesman team to find loot hidden by a brutal gunman who they think is dead but is really on their trail. Slow moving Italian oater issued there in 1966 by Crono Cinematografica\/PEFSA as _**Dio Perdona...lo No!**_ (God Forgives...I Don't!).\n\n**1582** _ **God Holds the Bullet**_ **** Danny Film, 1967. 90 min. Color. D: Amerigo Anton. SC: Mario Amendola. With Robert Mark (Rod Dana), Larry Ward, Gordon Mitchell, Elina De Witt, Fabrizio Moroni, Andrea Bosic, Albert Farley (Alberto Farnese), Benjamin May (Beniamino Maggio), Tony Rogers, Mary Land, Men Fury (Furio Meniconi), Remo Capitani, Ivan Giovanni Scratuglia, Renato Terra, Walter Conroy. A mysterious fiddler, actually a wanted man, shows up in a small town and falls in love with a woman whose family is in the middle of a deadly feud. Lots of slaughter and other violence are the main ingredients in this Spaghetti Western released in Italy in 1966 by Regalfilm as _**Uccidi o Muori**_ (Kill or Die); also known as _**Kill or Be Killed**_ and _**Ringo Against Johnny Colt**_.\n\n**1583** _ **The Godchild**_ **** ABC-TV\/Metro-Goldwyn-Mayer, 1974. 78 min. Color. D: John Badham. SC: Ron Bishop. With Jack Palance, Jack Warden, Keith Carradine, Ed Lauter, Jose Perez, Bill McKinney, Jesse Vint, Fionnuala Flanagan, John Quade, Simon Deckard, Ed Bakey, Kermit Murdock. Three Civil War deserters rob a bank and head into the desert pursued by cavalry and Indians, find a dying woman and agree to take her newborn baby to safety. Not bad television adaptation of the oft-filmed Peter B. Kyne's story \"Three Godfathers.\"\n\n**1584** _ **Godmonster of Indian Flats**_ **** Ellman Film Enterprises, 1973. 89 min. Color. D-SC: Fredric Hobbs. With Christopher Brooks, Stuart Lancaster, E. Kerrigan Prescott, Peggy Browne, Richard Marion, Karen Ingenthron, Robert Hirschfeld, Steven Kent Browne, Erica Gavin, Terry Wills, Evalyn Stanley, Carolyn Beaupre, Andre Brummer, Marianne Browne, Ann Wagner, Gordon Lane, Ann Lane, P.S. Kreiger, Frank Ford, Walter Daniels, Richard Walton, George Costello. A scientist believes a mutant sheep exposed to a mysterious chemical may hold the key to creation. Obscure, pitiful horror Western.\n\n**1585** _ **God's Country**_ **** Action Pictures\/Screen Guild, 1946. 62 min. Color. D: Robert Tansey. SC: Frances Kavanaugh. With Robert Lowery, Helen Gilbert, William Farnum, Buster Keaton, Si Jenks, Stanley Andrews, Trevor Bardette, Estelle Zarco, Juan Reyes, Al Ferguson, Sandy McTavish, Howard King, Turk Monroe, Old Tarr, White Cloud, Lee Roberts, Timber Cross, River Starr, Little Eagle, Whip Wilson. On the lam from the law, a man and his pal come to the north country and help a young woman and her father save their forest from a dishonest lumber company boss. Nice color and a good James Oliver Curwood story make this very pleasant viewing.\n\n**1586** _ **God's Country and the Man**_ **** Syndicate, 1931. 59 min. D: John P. McCarthy. SC: Wellyn Totman. With Tom Tyler, Lillian Bond, Alan Bridge, Andy Shuford, Jack Perrin, Ernie Adams, Gordon De Main, Slim Whitaker, Fern Emmett, Carmen LaRoux, Henry Roquemore, Merrill McCormick, William Bertram, Al Taylor, Al Haskell, Tom Smith. A Texas ranger and his ex-outlaw sidekick try to bring to justice a murderous gang leader. Fairly good Tom Tyler early talkie with Alan Bridge as a fiddle playing killer. Also called _**Man's Country**_ and _**Rose of the Rio Grande**_.\n\n**1587** _ **God's Country and the Man**_ **** Monogram, 1937. 56 min. D: Robert North Bradbury. SC: Robert Emmett (Tansey). With Tom Keene, Betty Compson, Charlotte Henry, Charles King, Billy Bletcher, Eddie Parker, Bob McKenzie, Merrill McCormick, Sherry Tansey, Glenn Strange, Lafe McKee, Milburn Morante, Henry Hall, Chick Hannon, Tex Palmer, Jack Evans, Tex Cooper, Bud Pope. A cowboy is after the gang leader who murdered his father. Tom Keene's first Monogram series vehicle is a pretty sturdy outing with stock footage from _**The Trail Beyond**_ (q.v.).\n\n**1588** _ **God's Country and the Woman**_ **** Warner Bros., 1937. 71 min. Color. D: William Keighley. SC: Norman Reilly Raine. With George Brent, Beverly Roberts, Barton MacLane, Robert Barrat, Alan Hale, Joseph King, El Brendel, Joseph Crehan, Addison Richards, Roscoe Ates, Billy Bevan, Bert Roach, Victor Potel, Mary Treen, Herbert Rawlinson, Harry Hayden, Pat Moriarty, Max Wagner, Susan Fleming, Eily Malyon. A playboy arrives in the north woods to manage a lumber company and gets involved in a business dispute with the female owner of a rival outfit. Okay action drama originally assigned to Bette Davis, who refused it.\n\n**1589** _ **God's Gun**_ **** Cannon Films, 1977. 90 min. Color. D-SC: Frank Kramer (Gianfranco Parolini). With Lee Van Cleef, Jack Palance, Richard Boone, Leif Garrett, Sybil Danning, Robert Lipton, Cody Palance, Ian Sadler, Pnina Golan, Zila Carni, Heinz Bernard, Didi Lukov, Ricardo David, Chin Chin, Rafi Ben Ami, Franco Pesce, Carolyn Stellar. A gunman and his five gang members terrorize a town until a former bounty hunter, whose priest brother was murdered by the outlaws, stands up to them. Fair action oater filmed in Israel.\n\n**1590** _ **Goin' South**_ **** Paramount, 1978. 101 min. Color. D: Jack Nicholson. SC: Al Ramus, Charles Shyer and Alan Mendel. With Jack Nicholson, Mary Steenburgen, Christopher Lloyd, Veronica Cartwright, John Belushi, Richard Bradford, Lucy Lee Flippen, Jeff Morris, Danny De Vito, Tracey Walter, Gerald H. Reynolds, Luana Anders, George W. Smith, Ed Begley, Jr., Britt Leack, R.L. Armstrong, Dennis Fimple. In order to save himself from the gallows a crook agrees to marry a pretty spinster but she turns out to be a stern taskmaster. Dull going.\n\n**1591** _ **Goin' to Town**_ **** Paramount, 1935. 74 min. D: Alexander Hall. SC: Mae West. With Mae West, Paul Cavanaugh, Gilbert Emery, Marjorie Gateson, Ivan Lebedeff, Fred Kohler, Monroe Owsley, Grant Withers, Luis Alberni, Tito Coral, Lucio Villegas, Mona Rico, Wade Boteler, Paul Harvey, Joe Frye, Adrienne D'Ambricourt, Bert Roach, Tom London, Syd Saylor, Irving Bacon, Francis Ford, Dewey Robinson, Julian Rivero, Stanley Price, Morgan Wallace, Tom Ricketts, J.P. McGowan, Jack Pennick, James Pierce, Leonid Kinsky, Lew Kelley, Jules Cowles, George Guhl, Virginia Hammond, Nell Craig, Cyril Ring, Frank Mundin. A dance hall singer marries a rich cattle baron and when he dies she inherits his money and tries to become socially prominent. Typically bawdy and funny Mae West feature.\n\n**1592** _ **Gold**_ **** Majestic, 1932. 58 min. D: Otto Brower. SC: W. Scott Darling. With Jack Hoxie, Alice Day, Hooper Atchley, Tom London, Robert Kortman, Lafe McKee, Matthew Betz, Jack Clifford, Jack Byron, Jack Kirk, Hank Bell, Dynamite (horse). When his partner in a gold claim is murdered, the victim's daughter blames a cowpoke for the crime and he tries to find the killer. Dull and slow moving, the film contains a most austere finale with villain Hooper Atchley tied to a wagon disguised as an intended victim and gunned down by his own men.\n\n**1593** _ **Gold Fever**_ **** Monogram, 1952. 63 min. D: Leslie Goodwins. SC: Edgar B. Anderson, Jr. and Cliff Lancaster. With John Calvert, Ann Cornell, Ralph Morgan, Gene Roth, Tom Kennedy, Judd Holdren, George Morrell, Danny Reese, Robert Graham. A man puts up money for an old prospector to work a hidden claim but crooks get wind of the operation and try to take over. Star John Calvert produced this cheaply made melodrama.\n\n**1594** _ **Gold Is Where You Find It**_ **** First National, 1938. 90 min. Color. D: Michael Curtiz. SC: Warren Duff and Robert Buckner. With George Brent, Olivia de Havilland, Claude Rains, Margaret Lindsay, John Litel, Marcia Ralston, Barton MacLane, Tim Holt, Sidney Toler, Henry O'Neill, Willie Best, Robert McWade, George Hayes, Harry Davenport, Russell Simpson, Clarence Kolb, Moroni Olsen, Granville Bates, Robert Homans, Eddy Chandler, Wilfred Lucas, Edmund Cobb, Douglas Wood, James Farley, Charles Halton, Erville Alderson, Cy Kendall, Guy Wilkerson, Karl Hackett, Milton Kibbee, Spec O'Donnell, Richard Botiller, Jack Mower, Arthur Aylesworth, Chester Gan, Jack Rutherford, John Harron, Sarah Edwards, Alan Davis, Cliff Saum, Al Herman, Frank Pharr, Walter Rogers. When gold is discovered on California farmland a terrible feud erupts between ranchers and miners. Elaborate and action filled feature.\n\n**1595** _ **Gold Mine in the Sky**_ **** Republic, 1938. 60 min. D: Joseph Kane. SC: Betty Bur- bridge and Jack Natteford. With Gene Autry, Smiley Burnette, Carol Hughes, Craig Reynolds, Cupid Ainsworth, LeRoy Mason, Frankie Marvin, Robert Homans, Eddie Cherkose, Ben Corbett, Milburn Morante, Jim Corey, George Guhl, Jack Kirk, Fred \"Snowflake\" Toones, The Stafford Sisters, J.L. Franks' Golden West Cowboys, George (Montgomery) Letz, Charles King, Lew Kelly, Joe Whitehead, Earl Dwire, Maudie Prickett, Al Taylor, Art Dillard, Herman Hack, George Plues. Gene Autry is made the administrator of a property owned by a wild spending young woman and when he refuses to turn it into a dude ranch her boyfriend hires Chicago gangsters to eliminate him. Interesting Gene Autry movie with enough action and music, plus a good plot, to entertain his fans.\n\n**1596** _ **Gold of the Seven Saints**_ **** Warner Bros., 1961. 88 min. D: Gordon Douglas. SC: Leigh Brackett. With Clint Walker, Roger Moore, Leticia Roman, Robert Middleton, Chill Wills, Gene Evans, Roberto Contreras, Jack C. Williams, Art Stewart. Two trappers make a gold strike but end up being chased across the desert by marauders. Utah locations are the film's main interest.\n\n**1597** _ **Gold Raiders**_ **** United Artists, 1951. 56 min. D: Edward Bernds. SC: Daniel Ullman and William Lively. With George O'Brien, The Three Stooges (Moe Howard, Larry Fine, Shemp Howard), Sheila Ryan, Clem Bevans, Monte Blue, Lyle Talbot, John Merton, Al Baffert, Hugh Hooker, Bill Ward, Fuzzy Knight, Dick Crockett, Roy Canada. An ex-marshal sells miners insurance to protect their gold shipments while three zanies with a traveling store end up chasing the crooks. Fans of The Three Stooges will like this one.\n\n**1598** _ **The Gold Rush**_ **** United Artists, 1925. 85 min. D-SC: Charles Chaplin. With Charles Chaplin, Georgia Hale, Mack Swain, Tom Murray, Betty Morrissey, Malcolm Waite, Henry Bergman. The Lone Prospector finds adventure in the Klondike as he falls for a saloon girl and helps a friend reclaim a stolen gold mine. Chaplin's classic comedy is as fresh today as when originally released; recommended.\n\n**1599** _ **Gold Rush Maisie**_ **** Metro-Goldwyn-Mayer, 1940. 82 min. D: Edwin L. Marin. SC: Betty Reinhardt and Mary C. McCall, Jr. With Ann Sothern, Lee Bowman, Virginia Weidler, John F. Hamilton, Mary Nash, Slim Summerville, Scotty Beckett, Irving Bacon, Louis Mason, Victor Kilian, Wallace Reed, Jr., Clem Bevans, John Sheehan, Charles Judels, Virginia Sale, Frank Orth, Kathryn Sheldon, Eddy Waller, Charles Middleton, Eddie Gribbon, Ray Teal, Cy Kendall, Barbara Bedford, Mitchell Lewis, Robert Middlemass, Henry Roquemore, Eddy Chandler, Ivan \"Dusty\" Miller, Wesley Giraud, Anna Chandler, Lee Phelps, Lew Harvey, Jessie Arnold, Dorothy Appleby, Margaret Bert, Naomi Childers, Ed O'Neill, Albert Russell, Henry Sylvester, Martie Faust. An out of work entertainer ends up at a mining camp and joins a poor family in searching for gold but soon softens the heart of a rancher and takes up farming. Definitely one of the lesser entries in the long running \"Maisie\" series.\n\n_**Gold Strike River**_ see _**The Lucky Texan**_\n\n_**Gold Train**_ see _**30 Winchesters for El Diablo**_\n\n**1600** _ **The Golden Eye**_ **** Monogram, 1948. 69 min. D: William Beaudine. SC: Scott Darling. With Roland Winters, Wanda McKay, Mantan Moreland, Victor Sen Yung, Bruce Kellogg, Tim Ryan, Evelyn Brent, Ralph Dunn, Lois Austin, Forrest Taylor, Lee \"Lasses\" White, Edmund Cobb, John Merton, Tom Tyler, George L. Spaulding, Jean Fong, Richard Loo, Lee Tung Foo, Michael Gaddis, Sam Flint, Geraldine Cobb. Charlie Chan is called to an Arizona mine to help a man who has been injured and he uncovers a smuggling ring. Atmospheric Chan mystery in a Western setting; also called _**The Mystery of the Golden Eye**_.\n\n**1601** _ **Golden Girl**_ **** 20th Century\u2013Fox, 1951. 108 min. Color. D: Lloyd Bacon. SC: Walter Bullock, Charles O'Neal and Gladys Lehman. With Mitzi Gaynor, Dale Robertson, Dennis Day, James Barton, Una Merkel, Raymond Walburn, Gene Sheldon, Carmen D'Antonio, Michael Ross, Harry Carter, Lovyss Bradley, Emory Parnell, Luther Crockett, Harris Brown, Kermit Maynard, Robert Nash, Jessie Arnold, Jimmie Dodd, Harry Seymour, Emmett Lynn, Duke York, Roger Moore, Nolan Leary, Chuck Hamilton, Kit Guard, George Magrill, Joe Dominguez, George Navarro, George Regas, Rico Alaniz, Alex Montoya, J. Farrell MacDonald, Kathryn Sheldon, Jean Moorehead, Claire Howard, Jack Davidson, Sherry Hall, Ferris Taylor, Frank Mills. In California during the Civil War, actress Lotta Crabtree falls in love with a man who is a Confederate spy. Pleasant entertainment more for its musical numbers than plot.\n\n_**Golden Lady**_ see _**The Fighting Frontiersman**_\n\n**1602** _ **The Golden Stallion**_ **** Mascot, 1927. 10 Chapters. D: Harry S. Webb. SC: Carl Krusada and William Lester. With Maurice \"Lefty\" Flynn, Molly Malone, Joe Bonomo, Josef Swickard, Burr McIntosh, Billy Franey, Tom London, White Fury (horse). A Mounted Policeman tries to stop an outlaw gang from capturing a horse that has the clue to a fabulous treasure branded on its neck. Fast paced silent serial also issued in an enjoyable feature version.\n\n**1603** _ **The Golden Stallion**_ **** Republic, 1949. 67 min. Color. D: William Witney. SC: Sloan Nibley. With Roy Rogers, Dale Evans, Estelita Rodriguez, Pat Brady, Foy Willing and The Riders of the Purple Sage, Chester Conklin, Douglas Evans, Greg McClure, Frank Fenton, Dale Van Sickel, Clarence Straight, Karl Hackett, Mauritz Hugo, Buff Brady, Jack Sparks. A horse trader uncovers a band of diamond smugglers who use a herd to smuggle gems across the U.S.-Mexican border. Action film with good photography but marred by inane Pat Brady and \"Nellybelle.\"\n\n**1604** _ **The Golden Trail**_ **** Monogram, 1940. 52 min. D: Al Herman. SC: Rolland Lynch, Robert Emmett (Tansey) and Roger Merton. With Tex Ritter, Arkansas Slim Andrews, Ina Guest, Patsy Moran, Gene Alsace, Stanley Price, Warner Richmond, Eddie Dean, Forrest Taylor, Frank LaRue, Chuck Morrison, Chick Hannon, Tex Palmer, Denver Dixon, Sherry Tansey, James Pierce, Hal Price, Ernie Adams, Richard Cramer, Bill Wells. A mining town is controlled by murderous crooks who want two workers out of the way and frame them with false evidence. Pretty fair Tex Ritter film with the star singing a song he co-wrote, \"Gold is Where You Find It.\"\n\n**1605** _ **The Golden West**_ **** Fox, 1932. 74 min. D: David Howard. SC: Gordon Rigby. With George O'Brien, Janet Chandler, Marion Burns, Onslow Stevens, Julia Swayne Gordon, Everett Corrigan, Edmund Breese, Sam West, Arthur Pierson, Bert Hanlon, Hattie McDaniel, Charles Stevens, Stanley Blystone, George Regas, Dorothy Ward, Sam Adams, Ed Dillon, Chief Big Tree, John War Eagle. After his father is killed by Indians, a young man grows up with the tribe, learns to hate whites and leads an attack on settlers. Fine adaptation of the Zane Grey novel, a remake of _**The Last Trail**_ (1927).\n\n_**The Golden Yukon**_ see _**The Grub Stake**_\n\n**1606** _ **Goldtown Ghost Raiders**_ **** Columbia, 1953. 59 min. D: George Archainbaud. SC: Gerald Geraghty. With Gene Autry, Smiley Burnette, Gail Davis, Kirk Rile, Carleton Young, Neyle Morrow, Denver Pyle, John Doucette, Steve Conte. A frontier circuit judge must decide if a man must go back to prison for killing his ex-partner a second time, since the victim survived the first attempt. Okay Gene Autry film enhanced by a good script.\n\n**1607** _ **Gone to Texas**_ **** CBS-TV, 1986. 144 min. Color. D: Peter Levin. SC: John Binder. With Sam Elliott, Claudia Christian, Devon Ericson, Michael C. Gwynne, Donald Moffat, John Quade, Ned Romero, William Russ, John P. Ryan, James Stephens, Richard Yriguez, Michael Beck, Bo Hopkins, G.D. Spradlin, Ritch Brinkley, John de Lancie, Peter Gonzales Falcon, Javier Grajeda, Cynthia Cuprill, Blue Deckert, Abrossio Guerra, Jerry Heynes, Robert F. Hoy, Brad Leland, Dennis Letts, Ivy Pryce, Andrew Stahl, Dave Tanner, Joe Morales, David Perfia, Luis Munoz, John B. Wells, John Nixon, Jimmy Ray Pickens, Kevin R. Young, Katharine Ross, William Schallert (narrator). From being governor of Tennessee, Sam Houston becomes the leader of the Texas independence movement against Mexico. Fair TV biopic.\n\n_**Gone with the West**_ see _**Little Moon and Jud McGraw**_\n\n_**A Good Day for Fighting**_ see _**Custer of the West**_\n\n**1608** _ **Good Day for a Hanging**_ **** Columbia, 1958. 85 min. Color. D: Nathan Juran. SC: Daniel B. Ullman and Maurice Zinn. With Fred MacMurray, Maggie Hayes, Robert Vaughn, Joan Blackman, James Drury, Edmond Ryan, Wendell Holmes, Stacy Harris, Kathryn Card, Emile Meyer, Bing Russell, Russell Thorson, Denver Pyle, Phil Chambers, Howard McNear, Rusty Swope, Harry Lauter, Gregg Barton, Tom London, William Fawcett, Bob Bice. A lawman brings in a wanted killer only to discover the citizens do not care if the man stands trial since they believe him innocent. Well done, but neglected, oater.\n\n**1609** _ **A Good Day to Die**_ **** CBS-TV, 1995. 175 min. Color. D: David Greene. SC: Joyce Eliason. With Sidney Poitier, Michael Moriarty, Joanna Going, Hart Bochner, Regina Taylor, Bill Wirth, Shirley Knight, Grace Zabriskie, Basil Wallace, James Caviezel, Robert Guillaume, Farrah Fawcett, John Pyper Ferguson, Byron Chief-Moon, Kevin McNutty, Katherine Isobel, Michael LaPlante, Zachary Savard, Jesse Lipscombe, Wilma Pelly, Lindsey Campbell, Brent Stait, Charles Andre, Eric Keenleyside, Jack Ackroyd, Tom Schanley, Brian Jensen, Dale Wilson, Michael Elias, Crystal Verge, Donna Belleville, Michelle Thrush, Edward C.K. Richardson III, Joshua Myers. A black gunfighter joins the Army in massacring a Cheyenne village and years later while helping his people build a settlement he is confronted by a now grown survivor of the carnage. Vapid attempt to elucidate racism in the Old West; also called _**Children of the Dust**_.\n\n_**A Good Day for Fighting**_ see _**Custer of the West**_\n\n**1610** _ **The Good Guys and the Bad Guys**_ **** Warner Bros., 1969. 90 min. Color. D: Burt Kennedy. SC: Ronald M. Cohan and Dennis Shryack. With Robert Mitchum, George Kennedy, David Carradine, Tina Louise, Douglas V. Fowley, Lois Nettleton, Martin Balsam, John David Chandler, John Carradine, Marie Windsor, Dick Peabody, Kathleen Freeman, Jimmy Murphy, Garrett Lewis, Nick Dennis. A has-been lawman and his long time outlaw foe, who has been discarded by his gang, join forces to thwart a train robbery. Fanciful Western comedy with fine work by Robert Mitchum and George Kennedy in the lead roles.\n\n**1611** _ **The Good Old Boys**_ **** Turner Network Television (TNT), 1995. 130 min. Color. D: Tommy Lee Jones. SC: Tommy Lee Jones and J.T. Allen. With Tommy Lee Jones, Terry Kinney, Frances McDormand, Sam Shepard, Sissy Spacek, Wilford Brimley, Walter Oikewicz, Matt Damon, Blayne Weaver, Bruce McGill, Larry Mahan, Richard Jones, Karen Jones, Park Overall, Laura Poe, Joaquin Jackson, Jeff Gore, Norberto Navarette, Margaret Bowman, James N. Harrell, Bernard Engel, Larry Lynch, Rodger Boyce, Joe Sears, Tony Epper, Jimmy Don Cox, Cliff Teinert, Patrick Scott, Ted J. Crum, Tom Hadley, Clay M. Lindley. A cowboy must choose between his family, a new love and a desire to roam free. Adequate TV movie directed and co-written by star Tommy Lee Jones.\n\n**1612** _ **The Good, the Bad and the Ugly**_ **** United Artists, 1968. 155 min. Color. D: Sergio Leone. SC: Sergio Leone and Luciano Vincenzoni. With Clint Eastwood, Lee Van Cleef, Eli Wallach, Aldo Giuffre, Chelo Alonso, Rada Rassimov, Silvana Bacci, Mario Brega, Lugi Pistilli, Livio Lorenzon, Enzo Petito, Al Muloch. Three Civil War veterans form an uneasy alliance as they search for a cash box full of gold hidden in an unmarked grave. Overrated and overlong, this is still probably the most famous Spaghetti Western and it is not without interest, especially for Lee Van Cleef's portrayal of a sadist killer. Released in Italy in 1966 as _**Il Buono, il Bruto, il Cattivo**_ (The Good, the Bad, the Wicked).\n\n_**Above:**_ **Advertisement for** _**The Good Guys and the Bad Guys**_ **(Warner Bros., 1969).** _**Left:**_ **Lee Van Cleef in** _**The Good, the Bad and the Ugly**_ **(United Artists, 1968).**\n\n**  \n**\n\n**1613** _ **Goodnight for Justice**_ **** Hallmark Channel, 2011. 88 min. Color. D: Jason Priestley. SC: Tippi Dobrofsky and Neal Dobrofosky. With Luke Perry, Lara Gilchrist, Winston Rekert, Daryl Shuttleworth, Darren Moore, Sean Wei Mah, Darla Fay, Sam Duke, Adom Osei, Melanie Papalia, John Shaw, Michael Teigen, Brett Dier, Hal Myshrall, John Tench, Reg Tupper, Maya Massar, Hayden Davies, Luis Javier. After witnessing the murder of his family as a boy, a man becomes a circuit judge, dispensing justice on the frontier until he gets a clue to the killings. Average made-for-TV flick.\n\n_**Goodbye Texas**_ see _**Adios, Texas**_\n\n**1614** _ **Gordon of Ghost City**_ **** Universal, 1933. 12 Chapters. D: Ray Taylor. SC: Ella O'Neill, Basil Dickey, George Plympton, Harry O. Hoyt and Het Mannheim. With Buck Jones, Madge Bellamy, Walter Miller, William Desmond, Francis Ford, Edmund Cobb, Tom Ricketts, Hugh Enfield, Bud Osborne, Dick Rich, Ethan Laidlaw, Jim Corey, Bill Steele, Artie Ortego, Tom London, Cliff Lyons. When a mysterious figure and his outlaw gang attempt to gain control of a gold strike, a cowboy is hired to bring peace to the area. Fast paced and very entertaining cliffhanger; Buck Jones' initial serial and his fans will love it.\n\n_**Gore Vidal's Billy the Kid**_ see _**Billy the Kid**_ (1989)\n\n**1615** _ **La Gran Aventura del Zorro**_ (The Great Adventure of Zorro) **** Cine Vision\/Estudios America, S.A., 1976. 90 min. Color. D: Raul de Anda. SC: Raul de Anda, Raul de Anda, Jr. and Rodolfo de Anda. With Rodolfo de Anda, Pedro Armendariz, Jr., Ricardo Carrion, Helena Rojo, Jorge Russek, Carlos Lopez Montezuma, Jorge Arvizu, Carlos Leon, Jorge Marcos, Jose L. Murillo. Zorro is blamed for the murder of a ranch owner, the culprit being the man who bought the spread and then took back his money after committing the crime. Well done Mexican \"Zorro\" feature.\n\n**1616** _ **Grand Canyon**_ **** Screen Guild, 1949. 69 min. D: Paul Landres. SC: Jack Harvey and Milton Luban. With Richard Arlen, Mary Beth Hughes, Reed Hadley, James Millican, Olin Howlin, Grady Sutton, Joyce Compton, Charlie Williams, Margia Dean, Stanley Price, Holly Bane, Anna May Slaughter, Zon Murray, Frank Hagney, Noble \"Kid\" Chissell. A movie director who connives to make a film on location in the Grand Canyon discovers a local prospector and elevates him to stardom. Pretty bad Robert Lippert production (which mentions the company name at every opportunity) and not even star Richard Arlen can save it.\n\n**1617** _ **Grand Canyon Massacre**_ **** Titanus, 1964. 89 min. Color. D: Stanley Corbett (Sergio Corbucci and Albert Band). SC: Albert Band, Sergio Corbucci and Fede Arnaud. With James Mitchum, Jill Powers, George Ardisson, Giacomo Rossi Stuart, Burt Nelson, Eduardo Ciannelli, Andrea Girodana, Nando Poggi, Benito Stefanelli, Milla Sannoner, Renato Terra Caizzi, Medar Vladimir, Gavrik Vlastimir, Attilio Severini. While hunting his father's killer, a young man becomes embroiled in a range feud over pasture land. Early Spaghetti Western with James Mitchum as a lackluster hero and a script short on action; released in Italy as _**Massacro al Grande Canyon**_ (Massacre at Grand Canyon). Rod Dana sings the title song.\n\n**1618** _ **Grand Canyon Trail**_ **** Republic, 1948. 67 min. D: William Witney. SC: Gerald Geraghty. With Roy Rogers, Andy Devine, Jane Frazee, Robert Livingston, Roy Barcroft, James Finlayson, Emmett Lynn, Foy Willing and The Riders of the Purple Sage, Charles Coleman, Kenneth Terrell, Zon Murray, Tommy Coats. Roy Rogers and pal Cookie Bullfincher try to combat a crook looking for the location of a silver vein in a ghost town. Good blend of action, comedy and suspense with Robert Livingston a fine villain and Jimmy Finlayson providing slapstick comedy relief as the local sheriff.\n\n**1619** _ **The Grand Duel**_ **** Cinema Shares, 1974. 92 min. Color. D: Giancarlo Santi. SC: Ernesto Gastaldi. With Lee Van Cleef, Peter O'Brien (Alberto Dentice), Marc Mazza, Klaus Grunberg, Horst Frank, Jess Hahn, Anthony Vernon (Antonio Casale), Dominique Darel, Sandra Cardini, Gastone Pescussi, Elvira Cortese, Anna Maria Gherardi, Hans Terofal, Salvatore Baccaro, Ray O'Connor (Remo Capitani), Franco Balducci, Giovanni Filidoro, Angelo Susani, Giovanni Cianfrigilia, Giorgio Trestini, Furio Meniconi, Franco Fantasia, Ottorino Polentini, Bob Clark. A veteran gunfighter becomes the self-appointed protector of a young man sought by an outlaw gang for the murder of their leader. Another typically violent European Western starring Lee Van Cleef, although this one uses the mystery element to good effect. An Italian-West German co-production it was released in Italy in 1972 as _**Il Grande Duello**_ (The Grand Duel); also called _**Big Gundown**_ and _**Storm Rider**_.\n\n**1620** _ **Grandpa Goes to Town**_ **** Republic, 1940. 54 min. D: Gus Meins. SC: Jack Townley. With James Gleason, Lucille Gleason, Russell Gleason, Harry Davenport, Lois Ranson, Maxie Rosenbloom, Tommy Ryan, Ledda Godoy, Noah Beery, Douglas Meins, Gary Owen, Ray Turner, Lee \"Lasses\" White, Walter Miller, Emmett Lynn, Joe Caits, Arturo Godoy. When a family inherits a bankrupt hotel in a Nevada ghost town the false rumor of a gold strike causes a boom. Adequate \"Higgins Family\" series entry.\n\n**1621** _ **Grandsons of Zorro**_ **** Dania Film\/Medusa, 1975. 100 min. Color. D: Mariano Laurenti. SC: Mario Mariani, Mariano Laurent and Luci Tortelli. With Franco Franchi, Ciccio Ingrassia, Mario Colli, Gianni Musi, Pedro Sanchez, Paola Tedesco, Maurizio Arena, Vito Pecory, Ugo Bonardi, Rod Licari, Mario Carotenuto, Grazia di Marza, Renato Malavasi, Vittorio Daverio. After dreaming of becoming Zorro, a meek man is given a special potion that turns him into a master swordsman so he can save his girl from a lecher. Unbearable Franco and Ciccio Italian spoof of \"Zorro,\" made there as _**Il Sogno di Zorro**_ (The Grandsons of Zorro) and also called _**Dream of Zorro**_.\n\n**1622** _ **Granny Get Your Gun**_ **** Warner Bros., 1940. 56 min. D: George Amy. SC: Kenneth Gamet. With May Robson, Margot Stevenson, Harry Davenport, Hardie Albright, Clem Bevans, William B. Davidson, Clay Clement, Arthur Aylesworth, Granville Bates, Ann Todd, Vera Lewis, Max Hoffman, Archie Twitchell, Walter Wilson, Nat Carr. When her granddaughter is falsely accused of murder, an elderly woman returns to Nevada, where she made a fortune as a gold miner, to find the killer. Slim adaptation of Erle Stanley Gardner's _Case of the Dangerous Dowager_ although May Robson and Harry Davenport are delightful in the lead roles.\n\n_**Grass Lands**_ see _**Hex**_\n\n**1623** _ **Grayeagle**_ **** American International, 1977. 104 min. Color. D-SC: Charles B. Pierce. With Ben Johnson, Alex Cord, Lana Wood, Iron Eyes Cody, Jack Elam, Paul Fix, Jacob Daniels, Charles B. Pierce. A rancher goes in search of his daughter who has been kidnapped by an Indian brave and eventually learns a startling truth. Murky, over long drama helped by a good cast.\n\n**1624** _ **Greaser's Palace**_ **** Cinema 5, 1972. 91 min. Color. D-SC: Robert Downey. With Allan Arbus, Albert Henderson, Elsie Downey, Luana Anders, Woodrow Chambliss, Michael Sullivan, James Antonio, George Morgan, Ron Nealy, Larry Moyer, John Paul Hudson, Jackson Haynes, Lawrence Wolf, Alex Hitchcock, Pablo Ferro, Toni Basil, Stan Gottlieb, Herve Vellechaize, Rex King, Joe Madden, Don Smolen, Donald Calfe. In a sleazy Western town a gunman comes to realize he is really Jesus Christ and begins to perform miracles. Low jinks production full of \"inside humor.\"\n\n**1625** _ **The Great Adventure**_ **** Pacific International, 1976. 87 min. D: Paul Elliotts (Gianfranco Baldanello). SC: Jay Anson and Elliot Geisinger. With Jack Palance, Joan Collins, Fred Romero (Fernando Romero), Elisabetta Virgili, Manuel de Blas, Remo de Angelis. Stranded in the Alaskan Rockies, a young boy finds companionship and protection with a faithful dog. Average outdoor drama loosely based on a Jack London story and made in Europe.\n\n**1626** _ **The Great Adventures of Wild Bill Hickok**_ **** Columbia, 1938. 15 Chapters. D: Mack V. Wright and Sam Nelson. SC: George Rosener, Charles Arthur Powell and George Arthur Durlam. With Gordon (Bill) Elliott, Monte Blue, Carole Wayne, Frankie Darro, Dickie Jones, Sammy McKim, Kermit Maynard, Roscoe Ates, Monty Collins, Reed Hadley, Chief Thundercloud, Mala, Walter Willis, J.P. McGowan, Eddy Waller, George Chesebro, Alan Bridge, Jack Perrin, Slim Whitaker, Walter Miller, Lee Phelps, Robert Fiske, Earle Hodgins, Earl Dwire, Ed Brady, Ray Jones, Edmund Cobb, Art Mix, Hal Taliaferro, Blackie Whiteford, Kenne Duncan, Budd Buster, Richard Cramer, Jack Rockwell, Francis Walker, Herman Hack, Bill Patton, Frank Lackteen, William Gould, Ethan Laidlaw, Tom London, Lew Meehan, Frank Ellis, Steve Cark, Buck Connors, Hank Bell, Edward Hearn, George Morrell, Robert Walker, Horace B. Carpenter, Richard Botiller, Carl Mathews, Ray Henderson, Al Haskell, Iron Eyes Cody, Jack Evans, Gene Alsace, Ted Mapes, Charles Brinley, Bob Burns, Al Thompson, Chuck Hamilton, Blackjack Ward, Bud McClure, Jack Montgomery, Artie Ortego, Curley Dresden, Allan Cavan, Ernie Adams, David McKim, Jesse Graves, Bruce Lane. Abilene marshal Wild Bill Hickok organizes a group of youngsters called the \"Flaming Arrows\" to help him combat renegade phantom raiders trying to stop a cattle drive from Texas over the Chisholm Trail. Well made and exciting cliffhanger that launched Bill Elliott to genre stardom; a grand supporting cast.\n\n**1627** _ **The Great Alaskan Mystery**_ **** Universal, 1944. 13 Chapters. D: Ray Taylor and Lewis D. Collins. SC: Maurice Tombragel and George H. Plympton. With Milburn Stone, Marjorie Weaver, Edgar Kennedy, Samuel S. Hinds, Martin Kosleck, Ralph Morgan, Joseph Crehan, Fuzzy Knight, Harry Cording, Anthony Warde, Richard Powers (Tom Keene), Jay Novello, Edward Gargan, Jack Clifford, Perc Launders, George Chesebro, Edmund Cobb, Reed Howes, Ray Bennett, Jack Ingram, Jack Rockwell, Stanley Price, Ed Peil, Sr., Artie Ortego, Bill Hunter, Dick Rush, Clarence Straight, Ben Taggart, Jean Trent, Charles Sullivan, Kernan Cripps, Curley Dresden, Roy Bucko, Kenneth Terrell, Carl Vernell, Joel Allen, James Dime, Carey Harrison, Bill Healy. An expedition heads to Alaska hoping to locate an old mine containing ore for a new weapon but one of them is really an enemy agent. Action filled wartime chapter play combining the spy and sci-fi genres.\n\n**1628** _ **The Great American Cowboy**_ **** American National Enterprises\/Sun International, 1974. 90 min. Color. D: Keith Merrill. SC: Douglas Kent Hall. With Joel McCrea (narrator), Larry Mahan, Phil Lyne, Elias Arriola. The story of the year long quest for the world's rodeo championship between veteran star Larry Mahan and young competitor Phil Lyne. Academy Award winning documentary that contains exciting rodeo footage and pleasing narration by Joel McCrea.\n\n**1629** _ **The Great American Indian**_ **** Doty-Dayton, 1974. 90 min. Color. D: Keith Merrill. The lives of Native Americans, how they live today, their history and heritage. Well done documentary from the people that made _**The Great American Cowboy**_ (q.v.).\n\n**1630** _ **The Great American Wilderness**_ **** Bill Burrud Productions, 1977. 95 min. Color. D: Barry Clark. With Marvin Miller (narrator). A travelogue of the wilds of North America, including the Great Plains, Rocky Mountains, deserts of the Southwest and the Arctic. Just the type of fare to please nature lovers.\n\n**1631** _ **The Great Bank Robbery**_ **** Warner Bros.\u2013Seven Arts, 1969. 98 min. Color. D: Hy Averback. SC: William Peter Blatty. With Kim Novak, Zero Mostel, Clint Walker, Claude Akins, Akim Tamiroff, Larry Storch, John Anderson, Sam Jaffe, Mako, Elisha Cook, Jr., Ruth Warrick, John Fiedler, John Larch, Peter Whitney, Norman Alden, Grady Sutton, Bob Steele, Mickey Simpson, Guy Wilkerson, Burt Mustin, Philo McCullough, William Zuckert, Jerry Summers, Byron Keith, Ben Aliza, Roy Agata, Bob Mitchell Boys' Choir. A woman and her friends pose as church leaders in order to rob a bank built by the notorious James, Dalton and Younger brothers. Poor Western comedy.\n\n_**The Great Bar 20**_ see _**Gunfighter (1999)**_\n\n_**The Great Barrier**_ see _**Silent Barriers**_\n\n**1632** _ **The Great Call of the Wilderness**_ **** American National Enterprises, 1976. 95 min. Color. With Larry Jones. In the American Northwest, a man fights to build a large natural preserve for the local animal life. Another okay drama for outdoors fans.\n\n**1633** _ **Great Day in the Morning**_ **** RKO Radio, 1956. 92 min. Color. D: Jacques Tourneur. SC: Lesser Samuels. With Virginia Mayo, Robert Stack, Ruth Roman, Alex Nicol, Raymond Burr, Leo Gordon, Regis Toomey, Peter Whitney, Dan White, Donald McDonald, Lane Chandler, Dennis Moore, Kermit Maynard, Ben Corbett. During the Colorado gold rush, trouble develops between prospectors along with secession activities. Colorful adaptation of Robert Hardy Andrews' novel.\n\n**1634** _ **The Great Divide**_ **** First National, 1929. 73 min. D: Reginald Barker. SC: Fred Myton and Paul Perez. With Dorothy Makaill, Ian Keith, Myrna Loy, Lucien Littlefield, Creighton Hale, George Fawcett, Claude Gillingwater, Roy Stewart, Ben Hendricks, Jr., Jean Laverty, Marjorie Kane. An arrogant Eastern debutante vacations in the West with her fiance and friends but finds herself courted by a mine owner. Picturesque early talkie from William Vaughn Moody's 1909 play; first filmed in 1925 by Metro-Goldwyn-Mayer with Alice Terry, Conway Tearle and Wallace Beery.\n\n_**The Great Gundown**_ see _**40 Graves for 40 Guns**_\n\n**1635** _ **The Great Jesse James Raid**_ **** Lippert, 1953. 73 min. Color. D: Reginald LeBorg. SC: Richard Landau. With Willard Parker, Barbara Payton, Tom Neal, Wallace Ford, James Anderson, Jim Bannon, Richard Cutting, Barbara Woodell, Marin Sais, Earle Hodgins, Joan Arnold, Steve Pendleton, Rory Mallinson, Frank Ellis, Ted Mapes, Fred Graham, Ray Jones. A \"retired\" Jesse James agrees to join Bob Ford in stealing gold hidden in a closed mine tunnel. Tawdry melodrama made to get box office mileage from the tabloid romance between plump Barbara Payton and Tom Neal, although neither has much to do in the proceedings.\n\n**1636** _ **The Great K and A Train Robbery**_ **** Fox, 1926. 53 min D: Lewis Seiler. SC: John Stone. With Tom Mix, Dorothy Dwan, William Walling, Harry Grippe, Carl Miller, Ed Peil, Sr., Curtis McHenry, Sammy Cohen. A railroad detective disguises himself as a bandit to board a train plagued by robberies and expose the culprits. Grand Tom Mix and Tony silent Western; very entertaining. John Wayne appeared as an extra.\n\n**1637** _ **The Great Locomotive Chase**_ **** Buena Vista, 1956. 85 min. Color. D: Francis D. Lyon. SC: Lawrence S. Watkin. With Fess Parker, Jeffrey Hunter, Jeff York, John Lupton, Eddie Firestone, Kenneth Tobey, Don Megowan, Claude Jarman, Jr., Harry Carey, Jr., Lennie Geer, Stan Jones, Slim Pickens, Morgan Woodward, Harvey Hester. A Union spy leads a group of volunteers into the South disguised as Confederates with a plan to steal a train and take it North. Fine Disney action drama.\n\n**1638** _ **The Great Man's Lady**_ **** Paramount, 1942. 90 min. D: William A. Wellman. SC: W.L. River. With Barbara Stanwyck, Joel McCrea, Brian Donlevy, Katherine Stevens, Thurston Hall, Lloyd Corrigan, Lillian Yarbo, Damian O'Flynn, Charles Lane, George Chandler, Anna Q. Nilsson, George P. Huntley, Milton Parsons, Etta McDaniel, Mary Treen, Helen Lynd, Lucien Littlefield, Frank M. Thomas, William B. Davidson, Fred \"Snowflake\" Toones, John Hamilton, Monte Blue, Eleanor Stewart, Theodore Von Eltz, George Irving, Fern Emmett, Pat O'Malley, Irving Bacon, Lee Phelps, Lee Moore, Charles Williams, Hank Bell, Ottola Nesmith, David Clyde, Bob Perry, Buck Mack. A young woman falls in love with a man who wants to build an oil empire but eventually he loses her to a gambler. Interesting performances aid this rather stilted drama.\n\n**1639** _ **The Great Meadow**_ **** Metro-Goldwyn-Mayer, 1931. 75 min. D: Charles Brabin. SC: Elizabeth Roberts. With Eleanor Boardman, Johnny Mack Brown, Lucille LaVerne, Anita Louise, Gavin Gordon, Guinn Williams, Russell Simpson, Sarah Padden, Helen Jerome Eddy, William Bakewell, Andy Shuford, James Marcus, Gardner James, Chief Whitespear. Pioneers join a wagon train and travel from Virginia to settle new land in Kentucky. Dated historical melodrama with some good action sequences and an early genre role for Johnny Mack Brown as the wagon master.\n**1640** _ **The Great Missouri Raid**_ **** Paramount, 1951. 83 min. Color. D: Gordon Douglas. SC: Frank Gruber. With Macdonald Carey, Ellen Drew, Wendell Corey, Ward Bond, Bruce Bennett, Bill Williams, Anne Revere, Edgar Buchanan, Louis Chastland, Louis Jean Heydt, Barry Kelly, James Millican Guy Wilkerson, Ethan Laidlaw, Tom Tyler, Paul Fix, James Griffith, Steve Pendleton, Paul Lees, Robert Bray, Frank Ferguson, King Donovan, Ray Teal, John Pickard. The James and Younger brothers take to the wrong side of the law after they are treated badly by Union soldiers after the Civil War. Color is a big help in this otherwise pedestrian effort.\n\n**1641** _ **The Great Northfield, Minnesota Raid**_ **** Universal, 1972. 91 min. Color. D-SC: Philip Kaufman. With Cliff Robertson, Robert Duvall, Luke Askew, R.G. Armstrong, Dana Elcar, Donald Moffatt, John Pearce, Matt Clark, Barry Brown, Wayne Sutherlin, Robert H. Harris, Jack Manning, Elisha Cook, Royal Dano, Mary Robin Redd, Bill Calloway, Craig Curtis, Nolan Leary, Henry Hunter, Valda Hansen, William Challee, Liam Dunn, Erik Holland, Madeline Taylor Holmes, Inger Stratton, Herbert Nelson, Marjorie Durant, Robert Gravage. When they fail to get amnesty from the law, the James and Younger brothers plan to pull off a big robbery in Northfield, Minnesota. Interesting account of the famous outlaws' last big bungle with good performances from the leads.\n\n**1642** _ **The Great Scout and Cathouse Thursday**_ **** American International, 1976. 102 min. Color. D: Don Taylor. SC: Richard Shapiro. With Lee Marvin, Oliver Reed, Robert Culp, Elizabeth Ashley, Kay Lenz, Strother Martin, Sylvia Miles, Howard Platt, Leticia Robles, Erika Carlson, Ana Verdugo. Two prospectors strike gold with one stealing the proceeds and becoming a wealthy, influential citizen while the other tries to ruin him. Western comedy is bawdy and has its moments but is mostly dull.\n\n**Lee Marvin and Oliver Reed in** _**The Great Scout and Cathouse Thursday**_ **(American International, 1976).**\n\n**  \n**\n\n**1643** _ **The Great Silence**_ **** Les Films Corona, 1968. 105 min. Color. D: Sergio Corbucci. SC: Sergio Corbucci, Bruno Corbucci, Mario Amendola and Vittoriano Petrilli. With Jean-Louis Trintignant, Klaus Kinski, Frank Wolff, Luigi Pistilli, Vonetta McGee, Mario Brega, Carlo D'Angelo, Marisa Merlini, Maria Mizar, Marisa Sally, Raf Baldassarre, Spartaco Conversi, Remo De Angelis, Mirella Pamphill, Claudio Ruffini, Fortunato Arena, Mimmo Poli, Bruno Corazzari, Benito Pacifico. A mute gunman helps outlaws hunted by a sadistic sheriff and his deputies. The bad guys are the good guys in this interesting French-Italian co-production set in a mountainous area; issued in Italy as _**Il Grande Silenzio**_ (The Grand Silence).\n\n**1644** _ **The Great Sioux Masssacre**_ **** Columbia, 1965. 91 min. Color. D: Sidney Salkow. SC: Fred C. Dobbs (Marvin Gluck). With Joseph Cotten, Darren McGavin, Philip Carey, Julie Sommars, Nancy Kovack, John Matthews, Michael Pate, Don Haggerty, Frank Ferguson, Stacy Harris, Iron Eyes Cody, House Peters, Jr., John Napier, William Tannen. The events leading up to General Custer's last stand at the Little Big Horn are recounted. Average oater adding nothing new to the historical event.\n\n**1645** _ **The Great Sioux Uprising**_ **** Universal-International, 1953. 80 min. Color. D: Lloyd Bacon. SC: Melvin Levy, J. Robert Bren and Gladys Atwater. With Jeff Chandler, Faith Domergue, Lyle Bettger, Stacy Harris, Walter Sande, Clem Fuller, Glenn Strange, Ray Bennett, Charles Arnt, Peter Whitney, John War Eagle, Stephen Chase, Rosa Rey. A former Army officer befriends Chief Red Cloud and stops rustlers from causing the Sioux nation to go on the warpath. Fair bow-and-arrows oater.\n\n**1646** _ **Great Stagecoach Robbery**_ **** Republic, 1945. 56 min. D: Lesley Selander. SC: Randall Faye. With Bill Elliott, Bobby Blake, Alice Fleming, Francis McDonald, Don Costello, Sylvia Arslan, Bud Geary, Leon Tyler, Henry Wills, Hank Bell, Robert Wilke, John James, Tom London, Horace B. Carpenter, Grace Cunard, Freddie Chapman, Dickie Dillon, Bobby Wilson, Raymond ZeBrack, Patsy May, Chris Wren, Ginny Wren, Frederick Howard, Fred Graham, Lucille Byron, Dorothy Stevens. A young man plans to follow in the footsteps of his famous outlaw father but find his actions are opposed by Red Ryder. Only average.\n\n**1647** _ **The Great Train Robbery**_ **** Republic, 1941. 61 min. D: Joseph Kane. SC: Olive Cooper, Garnett Weston and Robert T. Shannon. With Bob Steele, Claire Carleton, Milburn Stone, Helen MacKeller, Si Jenks, Monte Blue, Hal Taliaferro, George Guhl, Jay Novello, Dick Wessel, Yakima Canutt, Lew Kelly, Guy Usher, Franklyn Farnum, Jack Ingram, Horace Murphy, Jack O'Shea, Eddie Acuff, Henry Hall, Philip Trent, Cactus Mack, Charles Williams. A railroad detective is assigned to locate a train carrying a gold shipment that disappeared en route. Although it bears no resemblance to the 1903 classic of the same title, this programmer is a clever blend of the action, mystery and Western genres; remade as _**The Last Bandit**_ (q.v.).\n\n**1648** _ **The Great Treasure Hunt**_ **** Continental, 1972. 95 min. Color. D: Tonino Ricci. SC: Fabrizio Tallevi and Tonino Ricci. With Mark Damon, Stan Cooper, Stelvio Rosi, Luis Marin, Rosalba Neri, Alfredo Mayo, Giancarlo Badessi, Jose Luis Chinchilla, Adolfo Thous, Francisco Sanz, Bruno Are. After he rescues his brother from jail, the two siblings join a munitions expert and his woman in helping a blind man raid a Mexican warlord's fort for its gold. Not one of the better Spaghetti Westerns, filmed in Italy and released there as _**Monta in Sella, Figlio Di...**_ (Mounted in the Saddle, the Son of...) in 1967.\n\n_**The Great Truck Robbery**_ see _**Blazing Stewardesses**_\n\n**1649** _ **Green Grass of Wyoming**_ **** 20th Century\u2013Fox, 1948. 89 min. Color. D: Louis King. SC: Martin Berkeley. With Peggy Cummins, Charles Coburn, Robert Arthur, Lloyd Nolan, Burl Ives, Geraldine Wall, Robert Adler, Will Wright, Richard Garrick, Charles Tannen. Two rival families breed and raise trotting horses with the boy and girl from each falling in love. Fairly entertaining juvenile outing.\n\n**1650** _ **The Grey Fox**_ **** United Artists, 1983. 90 min. Color. D: Philip Boros SC: John Hunter. With Richard Farnsworth, Jackie Burroughs, Wayne Robson, Ken Pogue, Timothy Webber, Gary Reineke, David Petersen, Don MacKay, Samantha Langevin, Tom Heaton, James McLarty, George Dawson, Ray Michal, Stephen E. Miller, David L. Crowley, David McCulley, Gary Chalk, Jack Leaf, Isaac Bishop, Sean Sullivan, Bill Murdoch, Jack Ackroyd, Nicolas Rice, Frank C. Turner, Bill Melen, David Raines, Paul Coeur, Mel Tuck, Peter John, Anthony Holland, Jon York, John Owen, Lisa Westman, Tom Glass, Paul Whitney, David Ackridge, Murray Ord. After thirty years in prison an old-time outlaw decides to become a train robber. Leisurely paced Western filmed in Canada with a grand performance by Richard Farnsworth as the outlaw.\n\n**1651** _ **The Grey Vulture**_ **** Davis Distributing, 1926. 58 min. D: Forrest Sheldon. SC: George Hively. With Ken Maynard, Hazel Deane, Sailor Sharkey, Boris Bullock, Fred Burns, Nancy Zann, Whitehorse, Olive Trevor, Marie Woods, Flora Maitland, Dorothy Dodd, Fern Lorraine. A cowboy, who falsely believes he murdered a man, goes to work for a rancher, falls in love with his daughter and saves the boss' stolen cattle. Amusing, pleasant Ken Maynard silent feature.\n\n**1652** _ **Grim Prairie Tales:**_ _**Hit the Trail...to Terror**_ **** Academy Entertainment, 1990. 86 min. Color. D-SC: Wayne Coe. With James Earl Jones, Brad Dourif, Will Hare, Marc McClure, Michelle Joyner, Willliam Atherton, Lisa Eichhorn, Wendy J. Cooke, Scott Paulin, Jennifer Barlow, Dan Leegant, William M. Brennan, Tom Simcox, Bruce Discher, James Glick, Hannah Fixco, Joan Lemmo, Jole Shoptesse, Oscar Fragosa, Mony Bass, Darice Sampson, Jessica Vega Vasquez, Elena Lopez, Erica Vega Vasquez, Robert Kent Ball, Bob Terhune, Justin Lundin, Ray Saniger. A clerk and a black bounty hunter spend the night on the prairie and each relates stories of the supernatural. Weird horror Western.\n\n_**Gringo**_ (1965) see _**Gunfight at Red Sands**_\n\n**1653** _ **Gringo**_ **** Cemofilm, 1968. 89 min. Color. D: Frank Corlish (Bruno Corbucci). SC: Dean Whitcomb (Bruno Corbucci and Mario Amendola). With Brian Kelly, Fabrizio Moroni, Keenan Wynn, Folco Lulli, Erika Blanc, Rik Battaglia, Furio Meniconi, Gigi Bonos, Gianni Pallavicino. A wealthy Mexican land baron hires a gunman to bring back his son but the shootist discovers the boy is really the man's wife's illegitimate child and he plans to torture him. Very violent, but well done, Italian oater made as _**Spara Gringo, Spara**_ (Shoot Gringo, Shoot) and also called _**The Longest Hunt**_ and _**Rainbow**_.\n\n**1654** _ **Grizzly**_ **** Film Ventures International, 1976. 91 min. Color. D: William Girdler. SC: Harvey Flaxman and David Sheldon. With Christopher George, Andrew Prine, Richard Jaeckel, Joan McCall, Joe Dorsey, Victoria Johnson, Charles Kissinger. A giant grizzly bear murders two teenage girls at an amusement park and three men hunt him before he attacks again. Fairly suspenseful horror Western with the bad bear being the main interest. TV title: _**Killer Grizzly**_.\n\n**1655** _ **Grizzly Adams and the Legend of Dark Mountain**_ **** Joda Productions, 1999. 98 min. Color. D: John Huneck and David Sheldon. SC: Larry Bischof and William Brian Lowry. With Tom Tayback, Lindsay Bloom, Jennifer Waldman, May Nutter, Joseph Campanella, Joel Rooks, Eva Andrews, Billy Davis, Jr. Bill Gonta, Chase Montgomery Nutter, Joey D. Vieira, Link Wyler, Joshua Lee Patton, Marilyn McCoo, Mickey Jones, Frances Peach, Jenny Regli, Frank Sontonoma Salsedo, Tom Schuster, Billy Daydoge, Lexey Dennison, Billy Day Dodge, Jim Elk, Luis Gomes, Mark S. Brien, Janetta Walker. When he thinks two crooked prospectors have kidnapped three orphans, mountain man Grizzly Adams and his bear Samson get on their trail. One of the lesser \"Grizzly Adams\" adventures.\n\n**1656** _ **Grizzly and the Treasure**_ **** Gold Key, 1974. 98 min. Color D: James T. Flocker. With Scott Beach (narrator), Andrew Gordon, Robert Shelbe, Susan Bucklinie, Terry Bough, Mark Ostrander. A man takes his wife and young son to Alaska to search for gold and they suffer many hardships as a result, including a blizzard. Satisfying G-rated family fare.\n\n**1657** _ **Grizzly Mountain**_ **** Legacy Releasing, 1997. 96 min. Color. D: Jeremy Haft. SC: Jeremy Haft and Peter White. With Dan Haggerty, Dylan Haggerty, Nicole Lund, Kim Morgan Greene, Perry Stephens, E.E. Bell, Robert Patteri, Andrew Craig, Robert Dubaska, Martin Kove, Marguerite Hickey, Don Borza, Mark Abbott, Gilbert Revilla, Bill Stalling, John Nolan, Megan Haggerty, Jacqueline Anderson, Charlie Sammut, Sam Ferrer, Dwane Christiansen. Going camping with their parents, two siblings find a mountain cave where they are taken back to 1870 and encounter a mountain man. Another mediocre attempt to capitalize on Dan Haggerty playing a character similar to Grizzly Adams.\n\n**1658** _ **The Groom Wore Spurs**_ **** Universal-International, 1951. 80 min. D: Richard Whorf. SC: Robert Libott and Frank Burt. With Ginger Rogers, Jack Carson, Joan Davis, Stanley Ridges, James Brown, John Litel, Victor Sen Young, Mira McKinney, Gordon Nelson, George Meader, Kemp Niver, Robert B. Williams, George Chesebro, John Eldredge, Don Brodie, George Pembroke, Ross Hunter, Franklyn Farnum, Allen K. Wood, Benny Burt, William \"Bill\" Phillips, Jess Kirkpatrick, Douglas Evans, Robert Carson, Kate Drain Lawson, Ralph Roberts, Richard Whorf. When her cowboy film star husband is falsely accused of murder a lawyer helps prove his innocence. Funny comedy with some behind-the-scene looks at the fantasy world of Western moviemaking.\n\n**1659** _ **The Grub Stake**_ **** American Releasing, 1923. 80 min. D: Bert Van Tuyle and Nell Shipman. SC: Nell Shipman. With Nell Shipman, Alfred Allen, Lillian Leighton, George Berrell, Hugh Thompson, C.K. Van Auker, Ah Wing, Marjorie Warfield, Lloyd Peters. After finding out a Yukon gambler who has proposed marriage plans to sell her to a dance hall proprietor, a young woman, her sick father and a miner search for a remote gold claim. Enjoyable outdoor melodrama from Nell Shipman Productions, filmed in Idaho and Washington; released in Great Britain as _**The Romance of Lost Valley**_ and re-issued by Aywon as _**The Golden Yukon**_.\n\n**1660** _ **Guardian of the Wilderness**_ **** Sunn Classics Pictures, 1977. 112 min. Color. D: David O'Malley. SC: Casey Conlon and Charles E. Sellier, Jr. With Denver Pyle, John Dehner, Ken Berry, Cheryl Miller, Don Shanks, Cliff Osmond, Jack Kruschen, Ford Rainey, Norman Fell, Prentiss Rowe, Brett Palmer, Melissa Jones, Yosemite (bear), Hardtack (Rhodesian Ridgeback). After recovering his health in the wilderness, a man carries out a fight to save the great sequoia trees in Yosemite Valley from timber jacks. A biopic of Galen Clark, who was responsible for making Yosemite a national refuge; quite entertaining. Alternate TV title: _**Mountain Man**_.\n\n**1661** _ **La Guerrillera de Villa**_ (Villa's Warrior Women). Oro Films, 1969. 97 min. Color. D: Miguel Morayta. SC: Fernando Galiana and Miguel Morayta. With Carmen Sevilla, Vicente Parra, Julio Aleman, Jose Elias Moreno, Jaime Fernandez, Carlos (Charly) Bravo, Sergio Virel, Jose Baviera, Roberto Porter, Jose Angel Espinosa \"Ferrusquilla,\" Oscar Morelli, Enrique Garcia Alvarez, Carlos Lopez Figuerosa. During the Mexican Revolution a government soldier, after falling in love with a pretty singer, joins Pancho Villa's rebel forces. More than adequate historical adventure drama from Mexico.\n\n**1662** _ **Guilty Trails**_ **** Universal, 1939. 57 min. D: George Waggner. SC: Joseph West (George Waggner). With Bob Baker, Marjorie Reynolds, Hal Taliaferro, Georgia O'Dell, Jack Rockwell, Carleton Young, Glenn Strange, Murdock MacQuarrie, Jack Kirk, Tom London, Tex Palmer. A dishonest banker stages a fake robbery to steal proof of a young woman's ranch inheritance and a lawman tries to help her. Pretty fair Bob Baker vehicle highlighted by the song \"Ring Around the Moon Tonight.\" Remake of _**Texas Terror**_ (q.v.).\n\n**1663** _ **The Gun and the Pulpit**_ **** ABC-TV, 1974. 74 min. Color. D: Daniel Petrie. SC: William Bowers. With Marjoe Gortner, Slim Pickens, David Huddleston, Geoffrey Lewis, Estelle Parsons, Pamela Sue Martin, Jeff Corey, Karl Swenson, Jon Lormer, Robert Phillips, Larry Ward, Joan Goodfellow, Walter Barnes, Melanie Fullerton, Steve Tackett, Jason Clark, Ron Nix. A gunman on the run takes the guise of a minister and in a small town stands up to a tyrant. Standard TV movie enhanced by a good cast.\n\n**1664** _ **Gun Battle at Monterey**_ **** Allied Artists, 1957. 74 min. D: Carl K. Hittleman and Sidney A. Franklin, Jr. SC: Jack Leonard and Lawrence Resner. With Sterling Hayden, Pamela Duncan, Ted De Corsia, Mary Beth Hughes, Lee Van Cleef, Charles Cane, Byron Foulger, Mauritz Hugo, I. Stanford Jolley, Michael Vallon, Pat Comiskey, Fred Sherman, George Baxter, John Damler. After he is bushwhacked by a supposed friend and left for dead, an outlaw recovers and seeks revenge. Cheaply made but with good scenic locations. Alternate title: _**Gun Battle of Monterey**_.\n\n_**Gun Battle of Monterey**_ see _**Gun Battle at Monterey.**_\n\n**1665** _ **Gun Belt**_ **** United Artists, 1953. 72 min. Color. D: Ray Nazarro. SC: Richard Shayer and Jack De Witt. With George Montgomery, Tab Hunter, Helen Westcott, John Dehner, William Bishop, Jack Elam, Hugh Sanders, Willis Bouchey, James Millican, Bruce Cowling, Boyd Stockman, Douglas Kennedy, Boyd \"Red\" Morgan, William Phillips, Rex Lease, Joe Hayworth, Chuck Roberson. A once famous gunfighter wants to get married and settle down but his former gang implicates him in a crime. More than passable melodrama with strong work by George Montgomery in the lead role; remade as _**Five Guns to Tombstone**_ (q.v.).\n\n**1666** _ **Gun Brothers**_ **** United Artists, 1956. 79 min. D: Sidney Salkow. SC: Gerald Drayson Adams. With Buster Crabbe, Ann Robinson, Neville Brand, Michael Ansara, Walter Sande, Lita Milan, James Seay, Roy Barcroft, Slim Pickens, Dorothy Ford. A man sets up a homestead and is joined by his reformed ex-outlaw brother but the latter's old gang attacks them. Fairly entertaining oater that will satisfy Buster Crabbe fans.\n\n**1667** _ **Gun Code**_ **** Producers Releasing Corporation, 1940. 57 min. D: Peter Stewart (Sam Newfield). SC: Joseph O'Donnell. With Tim McCoy, Inna Gest, Lou Fulton, Dave O'Brien, Alden Chase, Carleton Young, Ted Adams, Robert Winkler, George Chesebro, Jack Richardson, John Elliott, Carl Mathews. A federal agent arrives in a town to break up a protection racket. Low budget, but Tim McCoy gives a good account as the stern G-man.\n\n**1668** _ **Gun Duel in Durango**_ **** United Artists, 1957. 73 min. D: Sidney Salkow. SC: Louis Stevens. With George Montgomery, Ann Robinson, Steve Brodie, Bobby Clark, Frank Ferguson, Donald Barry, Henry Rowland, Denver Pyle, Mary Treen, Roy Barcroft, Pierce Lyden, Red Morgan, Al Wyatt, Joe Yrigoyen. Trying to go straight, an ex-outlaw is forced to shoot it out with his old gang in order to reform. Mediocre oater enhanced by its star. TV title: _**Duel in Durango**_.\n\n**1669** _ **Gun Fever**_ **** United Artists, 1958. 81 min. D: Mark Stevens. SC: Stanley H. Silverman. With Mark Stevens, John Lupton, Larry Storch, Jana Davi, Aaron Saxon, Jerry Barclay, Norman Frederic, Clegg Hoyt, Jean Inness, Russell Thorson, Michael Hinn, Iron Eyes Cody, Cyril Delevanti, George Selk. When is father is murdered, a young boy tries to find the killer, not realizing he is a close family friend. Okay drama for Mark Stevens (who also directed) fans.\n\n**1670** _ **Gun Fight**_ **** United Artists, 1961. 68 min. D: Edward L. Cahn. SC: Gerald Drayson Adams and Richard Shayer. With Jim Brown, Joan Staley, Gregg Palmer, Ron Soble, Ken Mayer, Charles Cooper, Walter Coy, James Parnell, Andy Albin, Jon Locke, John Damler, Robert Nash, Jack Kenney, Frank Eldredge, Gene Coogan, Bill Koontz, Boyd Stockman, Bob Woodward. An ex-soldier heads West to join his brother in ranching only to find he is an outlaw. Fair dual bill item.\n\n**1671** _ **Gun for a Coward**_ **** Universal-International, 1957. 88 min. Color. D: Abner Biberman. SC: R. Wright Campbell. With Fred MacMurray, Jeffrey Hunter, Janice Rule, Chill Wills, Dean Stockwell, Josephine Hutchinson, Betty Lynn, Iron Eyes Cody, Robert Hoy, Jane Howard, John Larch, Paul Birch, Bob Steele, Frances Morris, Marjorie Stapp. A successful rancher has trouble with his two sons, one being a hothead and the other branded a coward. Nothing new in this psychological approach to the genre.\n\n_**A Gun for Ringo**_ see _**A Pistol for Ringo**_\n\n**1672** _ **Gun Fury**_ **** Columbia, 1953. 83 min. Color. D: Raoul Walsh. SC: Irving Wallace and Roy Huggins. With Rock Hudson, Donna Reed, Philip Carey, Roberta Haynes, Lee Marvin, Neville Brand, Leo Gordon, Ray Thomas, Forrest Lewis, John Cason, Pat Hogan, Mel Welles, Post Park, Robert Herron, Don Carlos, Frank Fenton, Henry Rowland, Dan White. When his new bride is kidnapped by an outlaw, his girlfriend and an Indian, a man goes after them. Originally issued in 3-D, this Arizona filmed oater is a colorful outing.\n\n**1673** _ **Gun Glory**_ **** Metro-Goldwyn-Mayer, 1957. 89 min. Color. D: Roy Rowland. SC: William Ludwig. With Stewart Granger, Rhonda Fleming, Chill Wills, Steve Rowland, James Gregory, Jacques Aubuchon, Arch Johnson, William Fawcett, Carl Pitti, Lane Bradford, Rayford Barnes, Ed Mundy, Gene Coogan, Michael Dugan, Jack Montgomery, Bud Osborne, May McAvoy, Charles Herbert, Steve Widders. An ex-gunman returns to his hometown and is shunned by the locals until they are threatened by another gunfighter. Stewart Granger is good in the lead but the story, based on Philip Yordan's novel _Man of the West_ , is only passable.\n\n**1674** _ **Gun Grit**_ **** Atlantic, 1936. 60 min. D: Lester Williams (William Berke). SC: Gordon Phillips. With Jack Perrin, Ethel Beck, David Sharpe, Jimmy Aubrey, Ed Cassidy, Earl Dwire, Horace Murphy, Roger Williams, Ralph Peters, Frank Hagney, Oscar Gahan, Budd Buster, Starlight (horse), Braveheart (dog). Big city racketeers go West to sell protection to cattlemen and an FBI agent is assigned to stop them. Low grade film with a lot of Hollywood locales; Jack Perrin is a pleasant player.\n\n**1675** _ **The Gun Hawk**_ **** Allied Artists, 1963. 92 min. Color. D: Edward Ludwig. SC: Jo Heims. With Rory Calhoun, Rod Cameron, Ruta Lee, Rod Lauren, Morgan Woodward, Robert Wilke, John Litel, Rodolfo Hoyos, Lane Bradford, Lee Bradley, Glenn Stensel, Joan Conners, Ron Whelan, Gregg Barton, Jody Daniels, Frank Gardner, Harry Fleer, Natividad Vacio. A notorious gunman tries to prevent a young gunfighter from continuing his life of crime. Stars Rory Calhoun and Rod Cameron make this one interesting.\n\n**1676** _ **Gun in His Hand**_ **** CBS-TV\/20th Century\u2013Fox, 1956. 45 min. D: Lewis Allen. SC: Steve Fisher. With Robert Wagner, Debra Paget, Charles Drake, Ray Collins, Royal Dano, Richard Crane, Trevor Bardette, Charles Meredith, George E. Stone, Paul McGuire, Stuart Randall, Herbert C. Lytton, Charles Cane, Brick Sullivan, John Cliff, Paul Stader, Mark Hanna, Mort Mills, Norman Leavitt, Paul Baxley. When he takes part in a bank holdup with his father, who is killed, a young man tries to redeem himself in the eyes of the law by hunting the other robbers. Telefeature originally shown as a segment of \"The 20th Century\u2013Fox Hour\" (CBS-TV, 1955\u201357) on April 4, 1956, and issued abroad theatrically.\n\n**1677** _ **Gun Justice**_ **** Universal, 1933. 62 min. D: Alan James. SC: Robert Quigley. With Ken Maynard, Cecilia Parker, Hooper Atchley, Walter Miller, Jack Rockwell, Francis Ford, Fred McKaye, William Dyer, Jack Richardson, Ed Coxen, William Gould, Sheldon Lewis, Lafe McKee, Ben Corbett, Robert McKenzie, Horace B. Carpenter, Frank Ellis, Hank Bell, Bud McClure, Roy Bucko, Buck Bucko, Pascale Perry, Cliff Lyons, Blackjack Ward. Crooks murder a man for his ranch and then hire a look-a-like to impersonate his nephew, who has inherited one-half of the property. Star Ken Maynard also produced this fair action drama.\n\n**1678** _ **Gun Law**_ **** Majestic, 1933. 59 min. D: Lewis D. Collins. SC: Lewis D. Collins and Oliver Drake. With Jack Hoxie, Betty Boyd, J. Frank Glendon, Mary Carr, Harry Todd, Edmund Cobb, Ben Corbett, Paul Fix, Richard Botiller, Bob Burns, Horace B. Carpenter, Jack Kirk, William T. Burt, Archie Ricks, Otto Lederer. Lawmen are after the notorious Sonora Kid, who has been terrorizing the Arizona countryside. A good chance to see Jack Hoxie in one of his half-dozen sound films; passable entertainment. Remade as _**Bullet Code**_ , _**Cyclone Ranger**_ , _**Gauchos of El Dorado**_ and _**Melody of the Plains**_ (qq.v.).\n\n**1679** _ **Gun Law**_ **** RKO Radio, 1938. 60 min. D: David Howard. SC: Oliver Drake. With George O'Brien, Rita Oehman, Ray Whitley, Paul Everton, Ward Bond, Francis McDonald, Edward Pawley, Robert Glecker, Frank O'Connor, Hank Bell, Paul Fix, Ethan Laidlaw, Lloyd Ingraham, Bob Burns, Jim Mason, Neal Burns, Ken Card, The Phelps Brothers. As a series of stagecoach robberies take place, a U.S. marshal pretends to be an outlaw to capture the hold up men. Highly exciting and action filled; a remake of _**West of the Law**_ (Film Booking Offices, 1928) starring Tom Tyler, and _**The Reckless Rider**_ (Willis Kent, 1932) with Lane Chandler.\n\n**1680** _ **Gun Law Justice**_ **** Monogram, 1949. 54 min. D: Lambert Hillyer. SC: Basil Dickey. With Jimmy Wakely, Dub Taylor, Jane Adams, Ray Whitley, John James, Myron Healey, I. Stanford Jolley, Lee Phelps, Edmund Cobb, Bud Osborne, Carol Henry, Tom Chatterton, Bob Curtis, Zon Murray, Eddie Majors, Herman Hack, Merrill McCormick, George Morrell, Ray Jones. A singing cowboy and his pal try to help an ex-outlaw gang leader who is trying to abide by the law. Okay Jimmy Wakely singing sagebrush yarn with a good supporting cast.\n\n**1681** _ **Gun Lords of Stirrup Basin**_ **** Republic, 1937. 60 min. D: Sam Newfield. SC: George Plympton and Fred Myton. With Bob Steele, Louise Stanley, Karl Hackett, Ernie Adams, Frank LaRue, Frank Ball, Steve Clark, Lew Meehan, Frank Ellis, Jim Corey, Budd Buster, Lloyd Ingraham, Jack Kirk, Horace Murphy, Milburn Morante, Bobby Nelson, Tex Palmer, Horace B. Carpenter, Jack Evans, Sherry Tansey, Chuck Baldra, Rose Plummer. Outlaws ignite a feud between two families but the plan is thwarted when a boy and girl from each line fall in love. Action filled Bob Steele vehicle for producer A.W. Hackel. TV title: _**Gunlords of Stirrup Basin**_.\n\n**1682** _ **Gun Packer**_ **** Monogram, 1938. 51 min. D: Wallace Fox. SC: Robert Emmett (Tansey). With Jack Randall, Louise Stanley, Charles King, Barlowe Borland, Raymond Turner, Lloyd Ingraham, Lowell Drew, Ernie Adams, Glenn Strange, Forrest Taylor, Curley Dresden, Sherry Tansey. A lawman investigates a series of stage robberies and learns the bandits are using gold to salt a mine in a planned swindle. One of the better Jack Randall series films with a sci-fi subplot of a gold transforming formula; remade as _**Range Land**_ (q.v.).\n\n**1683** _ **The Gun Ranger**_ **** Republic, 1937. 60 min. D: Robert North Bradbury. SC: George Plympton. With Bob Steele, Eleanor Stewart, Hal Taliaferro, John Merton, Ernie Adams, Earl Dwire, Budd Buster, Frank Ball, Horace Murphy, Lew Meehan, Horace B. Carpenter, Jack Kirk, George Morrell, Tex Palmer, Oscar Gahan, Archie Ricks, Clyde McClary. When a young woman's father is killed, a ranger tries to find the murderer. Exciting and well done Bob Steele feature.\n\n**1684** _ **Gun Runner**_ **** Monogram, 1949. 56 min. D: Lambert Hillyer. SC: J. Benton Cheney. With Jimmy Wakely, Dub Taylor, Noel Neill, Mae Clarke, Kenne Duncan, Steve Clark, Marshall Reed, Ted Adams, Bud Osborne, Carol Henry, Bob Woodward, Pascale Perry, Ray Jones, Eddie Majors. A woman is illegally smuggling guns to local Indians and a cowboy tries to stop her. A pretty fair movie in need of a star.\n\n**1685** _ **Gun Smoke**_ **** Paramount, 1931. 66 min. D: Edward Sloman. SC: Grover Jones and William McNutt. With Richard Arlen, Mary Brian, William Boyd, Eugene Pallette, Louise Fazenda, Charles Winninger, James \"Junior\" Durkin, J. Carrol Naish, Dawn O'Day (Anne Shirley), Guy Oliver, Brooks Benedict, William V. Mong, Willie Fung. Gangsters take over a Western town but are opposed by a cowboy and his pals. Interesting interpolation of the Western and gangster themes with Richard Arlen as the hero and William Boyd as the lead villain.\n\n**1686** _ **Gun Smoke**_ **** Monogram, 1945. 57 min. D: Howard Bretherton. SC: Frank H. Young. With Johnny Mack Brown, Raymond Hatton, Jennifer Holt, Riley Hill, Wen Wright, Ray Bennett, Steve Clark, Bob (John) Cason, Roy Butler, Frank Ellis, Marshall Reed, Chick Hannon, Louis Hart, Kansas Moehring, Dimas Sotello, Elmer Napier, Jack Baxley, Horace B. Carpenter, George Morrell. Two marshals find a stagecoach with all the passengers murdered and discover an outlaw gang is after Indian relics. Interesting plot highlights this \"Nevada Jack McKenzie\" series entry.\n\n**1687** _ **Gun Smoke**_ **** Willis Kent, 1935. 57 min. D: Barlett (Bart) Carre. SC: Oliver Drake. With Buck Coburn (Gene Alsace), Marion Shilling, Bud Osborne, Benny (Ben) Corbett, Henry Hall, Roger Williams, Dick Botiller, Nelson McDowell, Lloyd Ingraham, Tracy Layne, Philo McCullough, Lafe McKee, Lew Meehan, Steve Clark, Herman Hack, Bill Patton, Chief Thundercloud, Francis Walker, Fred Parker, Charles Murphy, Bob Burns, Allen Greer, Frank Austin, Bart Carre, Clyde McClary, Bud McClure, Silvertip Baker, Ed Carey, Barney Beasley, Ralph Bucko, Roy Bucko. A cowboy goes to work for a rancher trying to stop the influx of sheep on the range, the trouble being stirred by a gunman hired by a dishonest lawyer who wants the cattleman's daughter. Although billed as a Montie Montana Production, this threadbare fare was made by Willis Kent; also called _**Gunsmoke on the Guadalupe**_.\n\n**1688** _ **Gun Smugglers**_ **** RKO Radio, 1948. 62 min. D: Frank McDonald. SC: Norman Houston. With Tim Holt, Martha Hyer, Richard Martin, Gary Gray, Paul Hurst, Douglas Fowley, Robert Warwick, Don Haggerty, Frank Sully, Robert Bray, Harry Harvey, Al Ferguson, Monte Montague, Steve Savage. A small boy, who is in the custody of an honest man, is used by crooks in a plot to fleece his guardian and steal Army guns for enemy agents. Tim Holt outing that moves at a fast clip.\n\n**1689** _ **Gun Street**_ **** United Artists, 1961. 67 min. D: Edward L. Cahn. SC: Sam C. Freddle. With James Brown, Jean Willes, Mel Florey, John Clarke, John Pickard, Peggy Stewart, Sandra Stone, Warren Kemmerling, Neston Booth, Herb Armstrong. A sheriff tries to stop a convict from murdering the man who sent him to prison and then married his wife. Pretty fair programmer.\n\n**James Brown in** _**Gun Street**_ **(United Artists, 1961).**\n\n**  \n**\n\n**1690** _ **Gun Talk**_ **** Monogram, 1947. 57 min. D: Lambert Hillyer. SC: J. Benton Cheney. With Johnny Mack Brown, Raymond Hatton, Christine McIntyre, Geneva Gray, Douglas Evans, Wheaton Chambers, Frank LaRue, Ted Adams, Carl Mathews, Zon Murray, Carol Henry, Bill Hale, Boyd Stockman. While searching for his missing cousin, a man thwarts a stage robbery and gets involved in capturing the holdup men. A somewhat complicated plot hampers this otherwise routine oater.\n\n**1691** _ **The Gun That Won the West**_ **** Columbia, 1955. 71 min. Color. D: William Castle. SC: James R. Gordon. With Dennis Morgan, Paula Raymond, Richard Denning, Chris O'Brien, Robert Bice, Michael Morgan, Roy Gordon, Howard Wright, Richard Cutting, Kenneth MacDonald, Howard Negley, Dennis Moore, Don C. Harvey. In Wyoming, the cavalry and its scouts use a new weapon to restore peace with the Indians. Title refers to the Springfield Rifle in this competent program feature.\n\n**1692** _ **Gun the Man Down**_ **** United Artists, 1956. 78 min. D: Andrew V. McLaglen. SC: Burt Kennedy. With James Arness, Angie Dickinson, Robert Wilke, Emile Meyer, Don Megowan, Michael Emmet, Harry Carey, Jr., Pedro Gonzalez Gonzalez, Robert Hinkle, Frank Fenton. After being wounded in a robbery, an outlaw is abandoned by his cohorts and he swears revenge. Average oater with a good plot and cast, co-produced by John Wayne. TV title: _**Arizona Mission**_.\n\n**1693** _ **Gun Town**_ **** Universal, 1946. 53 min. D: Wallace Fox. SC: William Lively. With Kirby Grant, Fuzzy Knight, Lyle Talbot, Louise Currie, Claire Carleton, Dan White, Gene Garrick, Ray Bennett, Earle Hodgins, George Morrell, Tex Cooper, Merrill McCormick, Bill Sundholm. Two cowpokes help a female stage line owner who is unaware her fiancee is leading the outlaws harassing her. Compact Kirby Grant vehicle with fine villainous work by Lyle Talbot.\n\n**1694** _ **A Gunfight.**_ Paramount, 1971. 90 min. Color. D: Lamont Johnson. SC: Harold Jack Bloom. With Kirk Douglas, Johnny Cash, Jane Alexander, Raf Vallone, Karen Black, Eric Douglas, Dana Elcar, Robert Wilke, Keith Carradine, Paul Lambert, Philip L. Mead, John Wallwork, David Burleson, Dick O'Shea, Paul Lambert, Neil David, Douglas Doran, Paula Dillenschneider. Two aging gunmen are forced into a showdown in a small town so they decide to charge admission to the event. Offbeat production hampered by Johnny Cash as one of the gunslingers.\n\n**1695** _ **Gunfight at Black Horse Canyon**_ **** Revue, 1962. 100 min. Color. D: R.G. Springsteen. SC: Anthony Lawrence and Jack Turley. With Dale Robertson, Rod Cameron, William Demarest, Jack Ging, Claude Akins, Philip Carey, Patricia Owens, Jon Lormer, Mary Jayne Saunders, Steve Darrell, George Kennedy, Gene Roth, Stafford Repp, Lenny Geer, Lory Patrick, Coleman Francis, Paul McGuire, Phil Barselow, William Hunter, Richard Warren, C.W. Rankin, Kermit Maynard. A Wells Fargo agent has to contend with a female writer who has a prejudice against the West and a recently released outlaw who vows vengeance for his having sent him to prison. Acceptable TV movie made up of \"Assignment in Gloribee\" and \"The Dodger\" episodes of \"Tales of Wells Fargo\" (NBC-TV, 1957\u201362).\n\n**1696** _ **Gunfight at Comache Creek**_ **** Allied Artists, 1963. 90 min. Color. D: Frank McDonald. SC: Edward Bernds. With Audie Murphy, Ben Cooper, Coleen Miller, DeForrest Kelley, Jan Merlin, John Hubbard, Damian O'Flynn, Susan Seaforth, Adam Williams, Mort Mills, Douglas Kennedy, Thomas Browne Henry, William Wellman, Jr., Eddie Quillan, Laurie Mitchell, Tim Graham, Michael Mikler. Hired to stop an outlaw gang, a detective ingratiates himself into the band hoping to uncover its mysterious leader. Mundane Audie Murphy vehicle; remake of _**Star of Texas**_ and _**Last of the Badmen**_ (qq.v.).\n\n**1697** _ **The Gunfight at Dodge City**_ **** United Artists, 1959. 81 min. D: Joseph M. Newman. SC: Daniel B. Ullman and Martin M. Goldsmith. With Joel McCrea, Julie Adams, John McIntire, Nancy Gates, Richard Anderson, James Westerfield, Walter Coy, Wright King, Don Haggerty, Harry Lauter, Myron Healey, Mauritz Hugo, Henry Kulky, Timothy Carey, Bill Henry, Don C. Harvey, Earle Hodgins. Bat Masterson is recruited to be the sheriff of a town whose citizens do not approve of his trying to clean out the lawless element. Nicely done melodrama with Joel McCrea making a grand Bat Masterson.\n\n**1698** _ **Gunfight at La Mesa**_ **** Grindstone Entertainment Group, 2010. 88 min. Color. D: Chris Fickley. SC: Chris Fickley and Walter Haynes. With Walter Haynes, Dan Braun, Bruce Ladd, Kristine Proctor, Ronnie Sheadman, Matt Lott, Jolane Lentz, Chris Fickley, Terance Berry, Dick Rundell, Francesca Pearce, Brennan W. Patrick, Robb Osaba, Josh Deshlongchamps, Jessie Coliver, Marc W. Havener, Luke Schelhaas. A man returns to the town where his parents were murdered and in trying to find the killer gets help from a boyhood friend, now the sheriff. Poor video Western.\n\n**1699** _ **Gunfight at Nigh Noon**_ **** Centauro Film\/PEA, 1963. 97 min. Color. D: J.R. (Joaquin Luis Romero) Marchent. SC: J.R. (Joaquin Luis Romero) Marchent, Rafael Romero Marchent, Jesus Navarro Carrion and Marcello Fondato. With Richard Harrison, Robert Hundar (Claudio Undari), Gloria Milland, Billy Hyden (Miguel Palenzuela), Fernando Sancho, Evelyn Merrill (Gloria Osuna), Andrew Scott (Andrea Scotti), Raf Baldassare, Pablito Alonso, Luis Miguel Arranz, Enrique Hernandez, Luis Induni, Jose Riesgo, Emilio Rodriguez, Alfonso Rojas, Carlos Romero Merchant, Francisco Sanz, Jose Truchado, Rafael Vaquero, Gaspar \"Indio\" Gonzalez, Rufino Ingles, Ricardo G. Lillo, Dina Loy, Jose Manuel Martin, Aldo Sambrell, Miguel Merino, Freddie Tormil. A man seeks vengeance for the murder of his father and eventually tracks down the last culprit only to find he is the father of his marshal brother's fiancee. Early Italian-Spanish co-production with a mixed up plot highlighted by good photography and direction; originally called _**El Sabor de la Venganza**_ (The Taste of Vengeance) in Spain, _**I Tre Spietati**_ (The Three Ruthless Ones) in Italy and _**Sons of Vengeance**_ on U.S. TV.\n\n**1700** _ **Gunfight at Red Sands**_ **** Tecisa\/Jolly Film, 1965. 97 min. Color. D: Ricardo Blasco. SC: Alfredo Antonini (Albert Band) and Ricardo Blasco. With Richard Harrison, Mikaela Wood, Giacomo Rossi Stuart, Daniel Martin, Aldo Sambrell, Barta Barri, Sara Lezana, Sam Field. When his adopted brother is wounded and their gold stolen, a cowboy seeks revenge on the perpetrators. Pretty good Spaghetti Western with a fine music score by Ennio Morricone (as Dan Savio); made in 1963 as _**Gringo**_.\n\n**1701** _ **Gunfight at Sandoval**_ **** Buena Vista, 1963. 74 min. D: Harry Keller. SC: Frank D. Gilroy and Maurice Tombragel. With Tom Tryon, Dan Duryea, Lyle Bettger, Beverly Garland, Norma Moore, Harry Carey, Jr., Judson Pratt. A Texas Ranger hunts an outlaw gang who murdered his pal when he tried to stop them from robbing a bank. Well done drama issued theatrically in Europe although in this country it was shown on Walt Disney's ABC-TV program as \"Showdown at Sandoval\" on January 23, 1959, a part of the \"Texas John Slaughter\" mini-series.\n\n**1702** _ **Gunfight at the O.K. Corral**_ **** Paramount, 1957. 122 min. Color. D: John Sturges. SC: Leon Uris. With Burt Lancaster, Kirk Douglas, Rhonda Fleming, Jo Van Fleet, John Ireland, Lyle Bettger, Frank Faylen, Earl Holliman, Ted De Corsia, Dennis Hopper, Whit Bissell, George Mathews, John Hudson, DeForrest Kelley, Martin Milner, Kenneth Tobey, Lee Van Cleef, Joan Camden, Olive Carey, Brian Hutton, Nelson Leigh, Jack Elam, Don Castle, Dennis Moore, Ethan Laidlaw, William Norton Bailey, Joe Forte, Charles Herbert, Mickey Simpson, Henry Wills, Lee Roberts, Richard Reeves, Frank Hagney, Bing Russell, Tony Merrill. Wyatt Earp teams with Doc Holliday to oppose the notorious Ike Clanton and his outlaw sons. Another retelling of the famous showdown, colorful but historically empty; Frankie Laine sings the haunting title song throughout the proceedings.\n\n**1703** _ **Gunfight in Abilene**_ **** Universal, 1967. 86 min. Color. D: William Hale. SC: Bernie Giler and John D.F. Black. With Bobby Darin, Emily Banks, Leslie Nielsen, Donnelly Rhodes, Don Galloway, Frank McGrath, Michael Sarrazin, Barbara Werle, Johnny Seven, William Phipps, William Mims, Don Dubbins. During the Civil War, the ex-sheriff of Abilene develops a fear of guns and when he returns home the citizens want him to take over his old job. Competent melodrama with pop singer Bobby Darin in a dramatic role.\n\n**1704** _ **The Gunfighter**_ **** 20th Century\u2013Fox, 1950. 84 min. D: Henry King. SC: William Bowers and William Sellers. With Gregory Peck, Helen Wescott, Millard Mitchell, Jean Parker, Karl Malden, Skip Homeier, Anthony Ross, Verna Felton, Ellen Corby, Richard Jaeckel, Alan Hale, Jr., John Pickard, Angela Clarke, Cliff Clark, Alberto Morin, Kenneth Tobey, Michael Brandon, Ferris Taylor, Hank Patterson, Mae Marsh, Kim Spaulding, Harry Shannon, Houseley Stevenson, James Millican, Edmund Cobb, Dick Curtis, Dan White, Ted Mapes. A famous gunman, pursued by the brothers of his latest victim, returns to the town where his ex-wife and son live and tries to start a new life. Top notch affair with fine writing and performances, a near genre classic.\n\n_**The Gunfighter**_ (1983) see _**The Kid and the Gunfighter**_\n\n**1705** _ **Gunfighter**_ **** Amazing Movies, 1999. 94 min. Color. D-SC: Christopher Coppola. With Robert Carradine, Martin Sheen, Chris Lybbert, Pat Rourke, Adrienne Stout-Coppola, Clu Gulager, Will Hutchins, Dale Groves, Font Camps, Bud Clark, Tong Dingman, Dale Groves, William J. Lindsey, Marty Glaser, Tom McDermott, Claude Sheehan, Peter Ridet, Tom Gulager, Cliff Davis, Dick Jones, Charlie Mariluch, Lou Schweibert, Rick Haugh, Kenny Mills, John Gulager, Marcus Pinkney, John Gourly. A gunman out to avenge the massacre of his town's citizens learns an outlaw has kidnapped his lady love. Sorry, patchwork attempt to revive Clarence H. Mulford's Bar 20 characters that began filming a decade before its release; Johnny Rivers sings the closing tune.\n\n**1706** _ **The Gunfighters**_ **** Columbia, 1947. 87 min. Color. D: George Waggner. SC: Alan LeMay. With Randolph Scott, Barbara Britton, Dorothy Hart, Bruce Cabot, Charles Grapewin, Steven Geray, Forrest Tucker, Charles Kemper, Grant Withers, John Miles, Griff Barnett. A former gunman becomes a wrangler on a ranch where the owner's daughter loves a killer. Fine screen adaptation of Zane Grey's _Twin Sombreros_ with Randolph Scott returning to the author whose material gave him screen stardom a decade before _ **.**_\n\n**1707** _ **The Gunfighters**_ **** Columbia TriStar, 1987. 96 min. D: Clay Borris. SC: Jim Byrnes. With Art Hindle, Reiner Schoene, Tony Addabbo, George Kennedy, Michael Kane, Lori Hallier, Howard Kruschke, Francis Damberger, Beverly Hendry, Wendell Smith, Moira Wally, Dale Wilson, Bill Mellen, Bryan Fustukian, Eric Kramer, Blair Haynes, Alex Green, Paul Whitney, Raoul Tome, Mike Evans, Jay Smith, Dennis Robinson, Glenn Beck, Steve Atkinson, James DeFelice, Tom Glass, Kent Gallie, Paul Wood, Kevin Smith, Wilf Rowe, Damien Keene, Larry Yachimec, Calvin Cairnes, Lisa Skinner. When a dishonest rancher frames a man for murder, his brother and cousin help him to get free and the trio make plans to rob their antagonist. Only average oater helped by George Kennedy as the villain.\n\n_**Gunfighters Die Harder**_ see _**If You Meet Sartana, Pray For Your Death**_\n\n**1708** _ **Gunfighter's Moon**_ **** Rysher Entertainment, 1995. 91 min. Color. D-SC: Larry Ferguson. With Lance Henriksen, Kay Lenz, David McIlwraith, Nikki Deloach, Ivan Sergei, James Victor, Brent Stait, Yareli Arizmendi, Matthew Walker, Walter Marsh, Kevin McNulty, Alex Diakun, Ken Camroux, Barney O'Sullivan, John Payne, Dave \"Squatch\" Ward, Thell Reed, Byron Chief-Moon, Balinder Johal, Reese McBeth, Jed Dixon, Fabricio Santin, Mina E. Mina. A hardened gunman learns from a former lover, now the wife of a businessman, that he is the father of their daughter. Very well done melodrama.\n\n**1709** _ **Gunfighters of Abilene**_ **** United Artists, 1960. 66 min. D: Edward L. Cahn. SC: Orville H. Hampton. With Buster Crabbe, Barton MacLane, Rachel Ames, Arthur Space, Eugenia Paul, Russell Thorson, Kenneth MacDonald, Richard H. Cutting, Richard Devon, Lee Farr, Jan Arvan, Hank Patterson, Reed Howes, Boyd \"Red\" Morgan. A veteran gunslinger rides into a town looking for the killers of his brother. More than adequate program feature with Buster Crabbe dominating the proceedings as the gunman.\n\n**1710** _ **Gunfighters of Casa Grande**_ **** Metro-Goldwyn-Mayer, 1965. 92 min. Color. D: Roy Rowland. SC: Borden Chase and Clarke Reynolds. With Alex Nicol, Jorge Mistral, Dick Bentley, Steve Rowland, Phil Posner, Maria Granada, Diana Lorys, Mercedes Alonso, Aldo Sambrell. A notorious outlaw enlists the aid of other crooks in pulling off a big cattle theft and then tries to double cross them. Average Spanish made oater that got good distribution in the U.S.; produced in 1964 as _**Los Pistoleros de Casa Grande**_ (The Gunfighters of Casa Grande).\n\n**1711** _ **Gunfighters of the Northwest**_ **** Columbia, 1954. 15 Chapters. D: Spencer Gordon Bennet. SC: Arthur Hoerl, Royal R. Cole and George H. Plympton. With Jack (Jock) Mahoney, Clayton Moore, Phyllis Coates, Don C. Harvey, Marshall Reed, Rodd Redwing, Lyle Talbot, Tommy Farrell, Terry Frost, Lee Roberts, Joseph Allen, Jr., Gregg Barton, Chief Yowlachie, Pierce Lyden, John Hart, Gene Roth, Bud Osborne, Kermit Maynard, Zon Murray, William Fawcett. A Canadian Mounted Policeman is faced with marauding Indians and an avalanche in the great northwest. Lame cliffhanger although its trio of stars do their best with the tired material.\n\n_**A Gunfighter's Pledge**_ see _**The Pledge**_\n\n**1712** _ **Gunfire**_ **** Resolute, 1934. 56 min. D: Harry Fraser. SC: Harry C. (Fraser) Crist. With Rex Bell, Ruth Mix, Buzz Barton, Milburn Morante, Theodore Lorch, Philo McCullough, Ted Adams, Lew Meehan, Willie Fung, Mary Jane Irving, Jack Baston, Fern Emmett, Howard Hickey, Chuck Morrison, Mary Jo Ellis, William Demarest, Slim Whitaker. Rivals frame a rancher for murder but a cowboy and an Eastern woman come to his rescue. Low grade Rex Bell affair.\n\n**1713** _ **Gunfire**_ **** Lippert, 1950. 60 min. D: William Berke. SC: William Berke and Victor West. With Don Barry, Robert Lowery, Wally Vernon, Pamela Blake, Gaylord (Steve) Pendleton, Tommy Farrell, Leonard Penn, Dean Reisner, Claude Stroud, Steve Conte, Robert Anderson, William Norton Bailey, Charles King, Lee Roberts, Barbara Woodell, Carol Henry, Dale Van Sickel. A Frank James look-a-like uses the former outlaw's name in a series of holdups and the real Frank James comes out of seclusion to stop him. Low budget programmer enhanced by Don Barry's excellent work in dual roles.\n\n**1714** _ **Gunfire at Indian Gap**_ **** Republic, 1957. 70 min. D: Joe (Joseph) Kane. SC: Barry Shipman. With Vera Ralston, Anthony George, George Macready, Barry Kelley, John Doucette, George Keymas, Chubby Johnson, Glenn Strange, Dan White, Steve Warren, Chuck Hicks, Sarah Selby. At a remote shipment station three outlaws are after gold and a half-breed girl. Cheap Vera Ralston vehicle with the star too old for the part.\n\n_**Gunlords of Stirrup Basin**_ see _**Gun Lords of Stirrup Basin**_\n\n**1715** _ **The Gunman**_ **** Monogram, 1952. 52 min. D: Lewis D. Collins. SC: Fred Myton. With Whip Wilson, Fuzzy Knight, Phyllis Coates, Rand Brooks, Terry Frost, I. Stanford Jolley, Lane Bradford, Gregg Barton, Russ Whiteman, Richard Avonde. Citizens of an outlaw plagued area send to Texas Territory for a marshal and his deputy to help them. Anemic Whip Wilson film.\n\n**1716** _ **The Gunman from Bodie**_ **** Monogram, 1941. 62 min. D: Spencer Gordon Bennet. SC: Jess Bowers (Adele Buffington). With Buck Jones, Tim McCoy, Raymond Hatton, Christine McIntyre, Dave O'Brien, Robert Frazer, Charles King, Lynton Brent, Max Waizman, Jerry Sheldon, Jack King, Earle Douglas, Warren Jackson, Billy Carro, Frederick Gee, John Merton, Frank LaRue, Gene Alsace, Kernan Cripps, Wilbur Mack, Billy Carr. A man masquerades as a gunfighter to learn who is committing murders near a small town and he is helped by a U.S. marshal and a ranch cook. Action filled \"Rough Riders\" feature that does not reveal the hero trio until the finale; reworked as _**Riders of the Dawn**_ (1945) (q.v.).\n\n**1717** _ **Gunman in Town**_ **** Devon Film\/Copercines, 1970. 99 min. Color. D: Anthony Ascott (Giuliano Carmineo). SC: Tito Carpi. With Gianni Garko, Susan Scott, Piero Lulli, Nieves Navarro, Massimo Serato, Jose Jaspe, Bruno Corazzari, Frank Brana. A gunslinger breaks a convicted killer out of prison and returns him to the scene of the crime so he can prove his innocence. Fairly interesting segment in the \"Sartana\" series with a good music score by Bruno Nicolai. Italian title: _**Una Nuvola de Porvere...Un Grido di Morte...Ariva Sartana**_ (A Cloud of Dust...A Cry of Death...Sartana Is Coming).\n\n_**Gunman of Ave Maria**_ see _**Forgotten Pistolero**_\n\n**1718** _ **Gunman's Code**_ **** Universal, 1946. 55 min. D: Wallace Fox. SC: William Lively. With Kirby Grant, Fuzzy Knight, Jane Adams, Danny Morton, Bernard Thomas, Karl Hackett, Charles Miller, Frank McCarroll, Dan White, Artie Ortego, Jack Montgomery, Carl Mathews. Two Wells Fargo agents arrive in a town and try to capture outlaws robbing the company's stagecoaches. Pretty fair Kirby Grant vehicle; a remake of _**Road Agent**_ (q.v.).\n\n**1719** _ **Gunman's Walk**_ **** Columbia, 1958. 97 min. Color. D: Phil Karlson. SC: Frank Nugent. With Van Heflin, Tab Hunter, Kathryn Grant, James Darren, Mickey Shaughnessy, Robert F. Simon, Edward Platt, Ray Teal, Paul Birch, Will Wright, Bert Convy, Paul E. Burns, Paul Bryar, Everett Glass, Dorothy Adams. A stern rancher raises his two sons to walk the straight and narrow but there is a personality clash and one of them kills the other's girlfriend. Okay psychological oater with Van Heflin excelling as the patriarch.\n\n**1720** _ **Gunmen from Laredo**_ **** Columbia, 1959. 67 min. D: Wallace MacDonald. SC: Clarke Reynolds. With Robert Knapp, Jana Davi, Walter Coy, Paul Birch, Don C. Harvey, Clarence Straight, Ron Hayes, Charles Horvath, Jean Moorehead, X Brands. With the aid of an Indian girl a rancher escapes from jail and hunts the men who framed him and murdered his wife. Low grade outing from producer-director Wallace MacDonald who acted in Tim McCoy's Columbia Westerns in the 1930s.\n\n**1721** _ **Gunmen of Abilene**_ **** Republic, 1950. 60 min. D: Fred C. Brannon. SC: M. Coates Webster. With Allan \"Rocky\" Lane, Eddy Waller, Donna Hamilton, Roy Barcroft, Peter Brocco, Selmer Jackson, Duncan Richardson, Don C. Harvey, Donald Dillaway, George Chesebro, Steve Clark, Arthur Walsh, Tom Steele. An outlaw gang plots to steal a gold shipment but is opposed by an undercover deputy marshal. Another good, action filled Allan Lane opus.\n\n**1722** _ **Gunmen of the Rio Grande**_ **** Allied Artists, 1965. 86 min. Color. D: Tulio Demichelli. SC: Gene Luotto. With Guy Madison, Madeleine Lebeau, Carolyn Davys, Massimo Serato, Gerard Tichy, Fernando Sancho, Olivier Hussenot, Beni Deus, Dario Michaelis, Natividad Zaro, Alvaro de Luna, Xan Das Bolas, Juan Jajan, E. Marn, H. Morrow. Taking on the guise of a drifter, Wyatt Earp arrives in a settlement to help a woman whose silver interests are being sought by a ruthless mine owner. Pretty good European Western that will please Guy Madison fans since he portrays Wyatt Earp. Issued in Great Britain as _**Duel at Rio Bravo**_ and made in Italy under the title _**Jennie Lee Ha una Nuova Pistola**_ (Jennie Lee Has a New Pistol) by West-Film\/Flora Film\/Illama Films\/Path\u00e9-Cinema.\n\n**1723** _ **Gunners and Guns**_ **** Beaumont, 1935. 57 min. D: Jerry Callahan and Robert Hoyt. SC: Ruth Runell. With Black King (horse), Edwin (Edmund) Cobb, Edna Aselin, Edward Allen Biby, Eddie Davis, Ned Norton, Lois Glaze, Felix Vallee, Jack Cheatham, Ruth Runell, Frank Walker. A foreman is falsely accused of murdering his dude ranch boss, the deed actually done by men who where once part of his outlaw gang. Bottom of the barrel oater that includes a beautiful horse and a chance to see Edmund Cobb star in a sound feature. Given brief release in 1934 by Aywon as _**Racketeer Round-Up**_ with new footage added for general release the next year.\n\n**1724** _ **Gunning for Justice**_ **** Monogram, 1948. 60 min. D: Ray Taylor. SC: J. Benton Cheney. With Johnny Mack Brown, Raymond Hatton, Max Terhune, Evelyn Finley, House Peters, Jr., Ted Adams, I. Stanford Jolley, Bud Osborne, Dan White, Bob Woodward, Carol Henry, Boyd Stockman, Dee Cooper, Artie Ortego. A man and his pals find a map showing the location of gold hijacked during the Civil War and try to find it. _**The Good, the Bad and the Ugly**_ (q.v.) it is not but it is a pleasant affair.\n\n**1725** _ **Gunning for Vengeance**_ **** Columbia, 1946. 56 min. D: Ray Nazarro. SC: Louise Rosseau and Ed Earl Repp. With Charles Starrett, Smiley Burnette, Marjean Neville, Curt Barrett and The Trailsmen, Robert Kortman, George Chesebro, Frank LaRue, Lane Chandler, Phyllis Adair, Robert Williams, Jack Kirk, John Tyrrell, Nolan Leary, Frank Fanning, Dick Rush, Herman Hack, Tommy Coats, Matty Roubert, Chick Hannon, Blackie Whiteford, Herman Howlin, Bob Reeves. The Durango Kid helps a small girl whose father was bushwhacked by a gang extorting protection money from area ranchers. Fairly good \"Durango Kid\" effort. Also called _**Jail Break**_.\n\n**1726** _ **Gunplay**_ **** RKO Radio, 1951. 61 min. D: Lesley Selander. SC: Ed Earl Repp. With Tim Holt, Richard Martin, Joan Dixon, Marshall Reed, Robert Bice, Robert Wilke, Mauritz Hugo, Harper Carter, Jack Hill. The father of a young boy is murdered and the youngster is befriended by two cowpokes who try to find the killer. There is solid entertainment in this later Tim Holt feature.\n\n**1727** _ **Gunpoint**_ **** Universal, 1966. 86 min. Color. D: Earl Bellamy. SC: Mary Willingham and Willard Willingham. With Audie Murphy, Joan Staley, Warren Stevens, Edgar Buchanan, Denver Pyle, Royal Dano, Nick Dennis, William Bramley, Kelly Thorsden, David Macklin, Morgan Woodward, Robert Pine, Mike Ragan (Holly Bane), Ford Rainey, Roy Barcroft, John Hoyt, Bill Henry, Carol Henry. An outlaw gang robs a train and kidnaps a young girl as the sheriff of a local town forms a posse and chases them into New Mexico Territory. Pretty interesting Audie Murphy vehicle.\n\n_**Guns A' Blazing**_ see _**Law and Order**_ (1932)\n\n_**Guns Along the Trail**_ see _**Paradise Canyon**_\n\n**1728** _ **Guns and Guitars**_ **** Republic, 1936. 56 min. D: Joseph Kane. SC: Dorrell McGowan and Stuart McGowan. With Gene Autry, Smiley Burnette, Dorothy Dix, Earle Hodgins, J.P. McGowan, Tom London, Charles King, Frankie Marvin, Jack Rockwell, Ken Cooper, Harrison Greene, Eugene Jackson, Pascale Perry, Bob Burns, Tracy Layne, Jack Kirk, George Morrell, Sherry Tansey, Jack Evans, Art Davis, George Plues, Denver Dixon, Wes Warner, Jim Corey. In an area plagued by rustling, cattle fever and quarantines, Gene Autry arrives with a medicine show and tries to stop the lawlessness by running for sheriff. Top grade Gene Autry vehicle with nice songs and a strong plot.\n\n**1729** _ **Guns and Guts**_ **** Azteca, 1974. 98 min. Color. D: Rene Cardona, Jr. SC: Fernando Galiana. With Jorge Rivero, Pedro Armendariz, Jr., Rogelio Guerra, Zulma Faiad, Quintin Bulnes, Rene Cardona, Chano Urueta, Jose Antonio Mena, Rebecca Silva, Leticia Robles, Diana Selga, Susana Polga, Gladys Vivas, Olivia Pasos, Enrique Ponton, Carlos Agosti, Arturo Silva, Jaime Moreno, Rene Barrera, Guillermo Hernandez, Jesus Gomez, Armando Acosta. A gunfighter, about to retire to a life of ease with his mistress, is hired for one last job of killing a sheriff who has taken refuge in an old monastery. Intriguing, well staged Mexican Western from producer-director Rene Cardona, Jr., with a violent finale showdown; original title: _**Las Viboras Cambian de Piel**_ (The Vipers Change Skin).\n\n_**Guns Don't Argue**_ see _**Bullets Don't Argue**_\n\n**1730** _ **Guns for Hire**_ **** Willis Kent, 1932. 59 min. D-SC: Oliver Drake. With Lane Chandler, Sally Darling, Neal Hart, Yakima Canutt, John Ince, Slim Whitaker, Jack Rockwell, Ben Corbett, Steve Clemente, Bill Patton, Hank Bell, John P. McGuire, Frances Morris, Nelson McDowell, John Bacon, Edward Porter, Roy Bucko, Buck Bucko, Bud McClure, Gene Alsace, Bud Pope, Jack O'Shea, Ray Jones. A gunman joins forces with a rancher fighting crooks but finds out the other side employs the man who taught him his trade. Low budget but entertaining Lane Chandler vehicle with silent star Neal Hart as the rival gunfighter. TV title: _**Blazing Trail**_. Remade as _**The Tulsa Kid**_ (q.v.).\n\n**1731** _ **Guns for San Sebastian**_ **** Metro-Goldwyn-Mayer, 1967. 111 min. Color. D: Henri Verneuil. SC: James R. Webb. With Anthony Quinn, Anjanette Comer, Charles Bronson, Sam Jaffe, Silvia Pinal, Jorge Martinez de Hoyos, Jaime Fernandez, Pedro Armendariz, Jr., Rosa Furman, Leon Askin, Ivan Desny, Jorge Russek, Jose Chavez, Fernand Gravey, Aurora Clavel, Julio Aldama, Ferrusquilla, Pancho Cordova, Enrique Lucero, Chano Urueta, Noe Murayama, Guillermo Hernandez, Francisco Reisguera, Carlos Berriochoa, Armando Acosta, Guy Fox, Rico Lopez. Mistaken for a priest, an outlaw helps the people of a Mexican village defeat raiding Yaqui Indians. Weak drama although Anthony Quinn does his best as the outlaw as does Charles Bronson as a half-breed. ****\n\n**1732** _ **Guns in the Dark**_ **** Republic, 1937. 56 min. D: Sam Newfield. SC: Charles Francis Royal. With Johnny Mack Brown, Claire Rochelle, Syd Saylor, Ted Adams, Frank Ellis, Budd Buster, Merrill McCormick, Richard Cramer, Jack C. Smith, Dick Curtis, Roger Williams, Steve Clark, Jim Corey, Julian Madison, Slim Whitaker, Lew Meehan, Tex Palmer, Oscar Gahan, Sherry Tansey. After he mistakenly believes he killed his pal in a Mexican saloon brawl, a cowboy returns to the U.S. to work for a woman who has a contract to build a dam but the project is being sabotaged by rustlers. Interesting Johnny Mack Brown vehicle with all kinds of subplots, including drug smuggling.\n\n**1733** _ **Guns of a Stranger**_ **** Universal, 1973. 91 min. Color. D: Robert Hinkle. SC: Charles W. Aldridge. With Marty Robbins, Chill Wills, Dovie Beams, Steve Tackett, Shug Fisher, Ronny Robbins, Melody Hinkle, Charles Aldridge, Neil Summers, Fred Graham, Bill (Coontz) Foster. A singing drifter rides into a Western town and has a profound effect on the lives of its citizens. Tepid oater starring country music favorite Marty Robbins, who sings several songs, including his self-penned \"Oh, Virginia\" and \"Lonely Old Bunkhouse\"; Monte Hale was scheduled to appear in this feature.\n\n**Marty Robbins in** _**Guns of a Stranger**_ **(Universal, 1973).**\n\n**  \n**\n\n**1734** _ **Guns of Diablo**_ **** Metro-Goldwyn-Mayer, 1964. 76 min. Color. D: Boris Sagal. SC: Bernie Giler. With Charles Bronson, Susan Oliver, Kurt Russell, Jan Merlin, John Fiedler, Douglas Fowley, Rayford Barnes, Morris Ankrum, Russ Conway, Robert Carricart, Ron Hagherthy, Maurice Wells, Mike De Anda, Susan Flannery, Byron Foulger, Marguerita Cordova. The leader of a wagon train stops at a settlement where he becomes involved with a former adversary and an ex-love. Telefeature derived from the episode \"The Day of Reckoning\" (telecast March 15, 1964) of \"The Travels of Jamie McPheeters\" (ABC-TV, 1963\u201364) and issued theatrically in Europe; well made and finely acted by Charles Bronson and Susan Oliver.\n\n**1735** _ **Guns of Fort Petticoat**_ **** Columbia, 1957. 82 min. Color. D: George Marshall. SC: Walter Doniger. With Audie Murphy, Kathryn Grant, Hope Emerson, Jeff Donnell, Jeannette Nolan, Sean McClory, James Griffith, Madge Meredith, Ernestine Wade, Peggy Maley, Isobel Elson, Kim Charney, Ray Teal, Nestor Paiva, Charles Horvath, Reed Howes, John Dierkes, Hugh Sanders, Francis McDonald. During the Civil War a lieutenant about to be court-martialed deserts and goes to Texas where he finds a band of women, whose husbands are away fighting in the war, and trains them to defend their settlement against marauding Indians. Entertaining Audie Murphy vehicle with nice support from Hope Emerson.\n\n_**Guns of Fury**_ see _**The Daring Caballero**_\n\n**1736** _ **Guns of Hate**_ **** RKO Radio, 1948. 62 min. D: Lesley Selander. SC: Norman Houston and Ed Earl Repp. With Tim Holt, Richard Martin, Nan Leslie, Steve Brodie, Myrna Dell, Tony Barrett, Jason Robards, Robert Bray, Jim Nolan. When crooks try to take over a gold mine two cowboys find themselves involved in the dispute. Standard Tim Holt affair enhanced by its mystery element.\n\n_**The Guns of Juana Gallo**_ see _**Juana Gallo**_\n\n_**Guns of Justice**_ see _**Colorado Ranger**_\n\n**1737** _ **Guns of Nevada**_ **** Cineproduzioni Associate\/IFISA, 1965. 93 min. Color. D-SC: Ignacio Iquino. With George Martin, Audrey Amber, Katya Loritz, John MacDouglas (Giuseppe Addobbati), Stan Bart, Miguel De La Riva. A man falls in love with two women, one a silver mine owner and the other a saloon proprietor, and he opposes a crook trying to steal the first woman's property. Passable Italian-Spanish co-production made as _**La Sfida Degli Implaccabili**_ (Challenge by the Implacable Ones); also called _**Joe Dexter**_.\n\n**1738** _ **Guns of the Law**_ **** Producers Releasing Corporation, 1944. 56 min. D-SC: Elmer Clifton. With Dave O'Brien, Jim Newill, Guy Wilkerson, Jennifer Holt, Jack Ingram, Robert Kortman, Robert Barron, Frank McCarroll, Charles King, Budd Buster, Bud Osborne, Slim Whitaker, Dan White. A crooked lawyer and his gang try to steal a valuable property as a trio of lawmen come to the owner's rescue. Low grade entry in the popular \"The Texas Rangers\" series.\n\n**1739** _ **Guns of the Magnificent Seven**_ **** United Artists, 1969. 106 min. Color. D: Paul Wendkos. SC: Herman Hoffman. With George Kennedy, Monte Markham, James Whitmore, Reni Santoni, Bernie Casey, Joe Don Baker, Scott Thomas, Michael Ansara, Frank Silvera, Tony Davis, Wende Wagner, Luis Rivera, Fernando Rey, Sancho Garcia. A gunslinger and a half-dozen hired cohorts agree to spring a Mexican revolutionary leader from prison so he can resume his cause. Action filled third feature in the \"Magnificent Seven\" outings.\n\n**1740** _ **Guns of the Pecos**_ **** Warner Bros., 1937. 56 min. D: Noel Smith. SC: Harold Buckley. With Dick Foran, Ann Nagel, Gordon (Bill) Elliott, Gordon Hart, Joseph Crehan, Eddie Acuff, Robert Middlemass, Monte Montague, Gaby Fay (Holden), Milton Kibbee, Bud Osborne, Bob Burns, Douglas Wood, Glenn Strange, Gene Alsace, Bob Woodward, Frank McCarroll, Jack Kirk, Ray Jones. Rustlers murder an Army major purchasing horses for the service and the Texas Rangers are assigned to track down the killers. Fair entry in Warner Bros.' Dick Foran series.\n\n**1741** _ **Guns of the Timberland**_ **** Warner Bros., 1960. 91 min. Color. D: Robert D. Webb. SC: Joseph Petracca. With Alan Ladd, Jeanne Crain, Gilbert Roland, Frankie Avalon, Lyle Bettger, Noah Beery, Jr., Regis Toomey, Johnny Seven, Alana Ladd, Verna Felton, George Selk, Paul E. Burns, Henry Kulky, George J. Lewis. Ranchers and townspeople oppose loggers who are clearing the land with the aid of a government grant. Colorful feature with an interesting plot centered on business interests versus ecology.\n\n_**Guns for Dollars**_ see _**Deep West**_\n\n**1742** _ **Gunsight Ridge**_ **** United Artists, 1957. 85 min. D: Francis D. Lyon. SC: Talbot Jennings and Elizabeth Jennings. With Joel McCrea, Joan Weldon, Mark Stevens, Darlene Fields, Addison Richards, Carolyn Craig, Robert Griffin, Slim Pickens, I. Stanford Jolley, George Chandler, Herbert Vigran, Jody McCrea, Martin Garralaga, Cindy Robbins. The people of Arizona Territory hire a new deputy marshal to stop a series of robberies and he learns supposedly respectable citizens are behind the holdups. Fast moving Joel McCrea feature sure to satisfy his fans.\n\n**1743** _ **Gunslinger**_ **** American Releasing, 1956. 78 min. Color. D: Roger Corman. SC: Mark Hanna and Charles B. Griffith. With John Ireland, Beverly Garland, Allison Hayes, Martin Kingsley, Jonathan Haze, Chris Alcaide, Dick Miller, Bruno Ve Sota, William Schallert, Margaret Campbell, Kermit Maynard. When her marshal husband is murdered a woman takes over his job and a crooked saloon boss hires a gunman to kill her. Early six day Roger Corman cheapie that is rather appealing.\n\n**1744** _ **Gunslingers**_ **** Monogram, 1950. 55 min. D: Wallace Fox. SC: Adele Buffington. With Whip Wilson, Andy Clyde, Reno Browne, Dennis Moore, Riley Hill, George Chesebro, Sarah Padden, Bill Kennedy, Hank Bell, Steve Clark, Carl Mathews, Frank McCarroll, Reed Howes, Carol Henry, George DeNormand, Frank Elllis, Ray Jones. A saloon keeper wants to foreclose on drought stricken spreads in order to sell them to the railroad and he concocts a scheme to have a rancher hung for rustling but the man is helped by a drifting cowboy. Well done Whip Wilson film with an involved plot.\n\n**1745** _ **Gunslinger's Revenge**_ **** Cecchi Gori Distribuzione, 1998. 87 min. Color. D: Giovanni Veronesi. SC: Leonardo Piercing and Giovanni Veronesi. With Leonardo Pieraccioni, Harvey Keitel, David Bowie, Sandrine Holt, Alessia Marcuzzi, Jim van der Woude, Yudii Mercredi, Michelle Gomez, Kwame Kwei-Armah, Stephen Jenn, Rosalind Knight, Jimmy Gardner, Jean Heywood, Jessica James, Lorenzo White, Sean Baker, Andrew Dunford, Valentina Carnelutti, Cristina Moglia, Clive Kneller, Chris John Hartz, Donald Hodson, James Weedon, Giustina Morganti, Nicholas Hunt, Danilo Mattei, Bruce Byron. A gunman retires to his son's farm only to be harassed by a crazed shootist who wants a showdown. Poor script and acting, but pretty scenery, in this Italian Western.\n\n**1746** _ **Gunsmoke**_ **** Astor, 1947. 52 min. D: Fred King. SC: Reg Browne. With Nick Stuart, Carol Foran, Robert Garden, Billy Jones, Craig Lawrence, Marie Harmon, Clark Bush, Lee \"Stormy\" Weathers, Smokey Joe LaDue, Curley Fletcher, Larraine Jensen, Dan Dowling, Lee Carling, Charles Quirk. A young man is wounded in a gunfight when he kills his father's murderer and after being helped by a girl he joins a gang not knowing the leader is the twin brother of the man he shot. Tattered Nevada filmed affair reissued as _**Gunsmoke Killers**_.\n\n**1747** _ **Gunsmoke**_ **** Universal-International, 1953. 79 min. Color. D: Nathan Juran. SC: D.D. Beauchamp. With Audie Murphy, Susan Cabot, Paul Kelly, Charles Drake, Mary Castle, Jack Kelly, Jesse White, William Reynolds, Chubby Johnson, Edmund Cobb, Clem Fuller, Holly Bane, Denver Pyle, George Eldredge, Gregg Barton, Forrest Taylor, William Fawcett, Henry Wills. An outlaw is hired to run a family off their ranch but instead he takes over, rounds up their cattle for market and falls in love with the owner's daughter. Somewhat offbeat, nicely done oater.\n\n**1748** _ **Gunsmoke in Tucson**_ **** Allied Artists, 1958. 79 min. Color. D: Thomas Carr. SC: Paul Leslie Peil and Robert Joseph. With Mark Stevens, Forrest Tucker, Gale Robbins, Vaughn Taylor, Kevin Hagen, Billy Henry, Richard Reeves, Gail Kobe, George Keymas, Terry Frost, Zon Murray, John Ward, John Cliff, I. Stanford Jolley, Paul Engle, Anthony Sydes. In the Arizona Territory turbulence between a cattle baron and settlers erupts causing a showdown between a marshal and his outlaw brother. The two stars highlight this otherwise routine effort.\n\n_**Gunsmoke Killers**_ see _**Gunsmoke**_ (1947)\n\n**1749** _ **Gunsmoke Mesa**_ **** Producers Releasing Corporation, 1944. 59 min. D: Harry Fraser. SC: Elmer Clifton. With Dave O'Brien, Jim Newill, Guy Wilkerson, Patti McCarthy, Jack Ingram, Kermit Maynard, Robert Barron, Richard Alexander, Roy Brent, Michael Vallon, Jack Rockwell, Budd Buster, Don Weston, Rose Plummer. The Texas Rangers trio witness a murder but when they report it they are arrested for the crime and have to break jail to prove their innocence. Okay outing in \"The Texas Rangers\" series, the last with Jim Newill.\n\n_**Gunsmoke on the Guadalupe**_ see _**Gun Smoke**_ (1935)\n\n**1750** _ **Gunsmoke:**_ _**One Man's Justice**_ **** CBS-TV, 1994. 91 min. Color. D: Jerry Jameson. SC: Harry Longstreet and Renee Longstreet. With James Arness, Bruce Boxleitner, Amy Stock-Poynton, Alan Scarfe, Christopher Bradley, Mickey LeBeau, Kelly Morgan, Apesanahkwat, Hallie Foote, Clarke Heathcliffe Brolly, Don Collier, Ed Adams, Wayne Anthony, Bing Bleman, Tom Brinson, Dave Adams, Sandy Gibbons, Mike Kevil, Richard Lundin, Kyle Marsh, Jonathan Mincks, Billy Joe Patton, Ric San Nicholas, Forrie J. Smith, Robin Wayne. Matt Dillon and a friend try to stop a young man from being killed by the gang he pursues for murdering his mother. The final \"Gunsmoke\" (CBS-TV, 1955\u201375) telefeature is another winner in the series.\n\n**1751** _ **Gunsmoke Ranch**_ **** Republic, 1937. 56 min. D: Joseph Kane. SC: Oliver Drake. With Robert Livingston, Ray Corrigan, Max Terhune, Julia Thayer (Jean Carmen), Kenneth Harlan, Sammy McKim, Oscar and Elmer, Yakima Canutt, Burr Caruth, Horace B. Carpenter, Robert Walker, Jack Ingram, Jack Kirk, Jack Padjan, Fred \"Snowflake\" Toones, John Merton, Robert McKenzie, Ed Peil, Sr., Fred Burns, Allen Connor, Jane Keckley, Loren Riebe, Vinegar Roan, Wes Warner, Jack O'Shea, Bud McClure, William McCall, Eva McKenzie, Bob Card, Silver Tip Baker, Bobby Burns, June Johnson, Peggy McKim, Richard Beach, Lee Ford, Al Taylor. When settlers are nearly ruined by a flood, a crooked politician tries to steal their lands but is opposed by the Three Mesquiteers. Exciting entry in the popular Republic series.\n\n**1752** _ **Gunsmoke:**_ _**The Last Apache**_ **** CBS-TV, 1990. 94 min. Color. D: Charles Correll. SC: Earl W. Wallace. With James Arness, Richard Kiley, Amy Stock-Poynton, Geoffrey Lewis, Joe Lara, Sam Vhalos, Hugh O'Brian, Michael Learned, Peter Murnik, Robert Covarrubias, Ned Bellamy, Dave Florek, Joaquin Martinez, Kevin Sifuentes, Robert Bran Wilson, Blake Boyd, James Milanesa. After his daughter is kidnapped by a renegade Apache, Matt Dillon enlists the help of an Army scout and offers Geronimo's son in return. The second \"Gunsmoke\" (CBS-TV, 1955\u201375) TV movie and a good one.\n\n**1753** _ **Gunsmoke:**_ _**The Long Ride**_ **** CBS-TV, 1993. 94 min. Color. D: Jerry Jameson. SC: Bill Stratton. With James Arness, James Brolin, Amy Stock-Poynton, Christopher Bradley, Patrick Dollaghan, Don McManus, Marco Sanchez, Ali MacGraw, Tim Choate, Michael Greene, Stewart Moss, Jim Beaver, Sharon Mahoney, Rick Dano, Ed Adams, John David Garfield, Victor Izay, Doug Katenay, Fred Lopez, Cliff Gravel. Finding a dead or alive bounty has been placed on his head, Matt Dillon tries to clear himself by tracking down the real killer. The fourth \"Gunsmoke\" (CBS-TV, 1955\u201375) telefilm is a pretty good production.\n\n**1754** _ **Gunsmoke:**_ _**The Return to Dodge**_ **** CBS-TV, 1987. 100 min. Color. D: Vincent McEveety. SC: Jim Byrnes. With James Arness, Amanda Blake, Steve Forrest, Buck Taylor, Fran Ryan, Earl Holliman, Ken Olandt, W. Morgan Sheppard, Patrice Martinez, Tantoo Cardinal, Mickey Jones, Frank M. Totino, Robert Koons, Walter Kaasa, Georgie Collins, Tony Epper, Louie Elias, Ken Kirzinger, Denny Arnold, Alex Green, Paul Daniel Wood, Larry Musser, Robert Clinton, Frank Huish, Jacob Rupp, Mary Jane Wildman, Ken Curtis, Milburn Stone, Glenn Strange, Tom Brown, Ted Jordan. Former U.S. marshal Matt Dillon and Long Branch Saloon owner Kitty Russell return to Dodge City where they are stalked by a murderer they helped send to prison a dozen years before. The initial \"Gunsmoke\" (CBS-TV, 1955\u201375) reunion TV movie is a delightful nostalgic affair that was so successful it spawned four follow-ups.\n\n**1755** _ **Gunsmoke:**_ _**To the Last Man**_ **** CBS-TV, 1992. 91 min. Color. D: Jerry Jameson. SC: Earl W. Wallace. With James Arness, Pat Hingle, Amy Stock-Poynton, Matt Mulhern, Jason Lively, Joseph Bottoms, Morgan Woodward, Mills Watson, James Booth, Amanda Wyss, Jim Beaver, Herman Poppe, Ken Swofford, Don Collier, Ed Adams, Kathleen Todd Erickson, Loy W. Burns, Andy Sherman, Clark A. Ray, Michael F. Woodson, Erol Landis, William J. Fisher, Stephen C. Foster, Ric San Nicholas, Jimmy Don Cox, Richard Glover. While on the tail of rustlers, Matt Dillon finds himself in the middle of a range war. Very entertaining third \"Gunsmoke\" (CBS-TV, 1955\u201375) telefilm.\n\n**1756** _ **Gunsmoke Trail**_ **** Monogram, 1938. 57 min. D: Sam Newfield. SC: Fred Myton. With Jack Randall, Louise Stanley, Al St. John, John Merton, Henry Roquemore, Ted Adams, Alan Bridge, Glenn Strange, Hal Price, Harry Strang, Kit Guard, Jack Ingram, Slim Whitaker, Art Dillard, Carleton Young, Sherry Tansey, George Morrell, Oscar Gahan, Blackjack Ward, Wally West, Carl Mathew. A cowboy helps a young woman whose property is wanted by a killer pretending to be her uncle. Better than average Jack Randall vehicle with a fine supporting cast.\n\n**1757** _ **Gypsy Colt**_ **** Metro-Goldwyn-Mayer, 1954. 72 min. Color. D: Andrew Marton. SC: Martin Berkeley. With Donna Corcoran, Ward Bond, Frances Dee, Larry Keating, Lee Van Cleef, Nacho Galindo, Rodolfo Hoyos, Peggy Maley, Joe Dominguez. Drought causes a family to sell their daughter's prize racing horse to a faraway stable and the loyal steed undertakes a 500 mile journey to return home. Heartwarming family film; a reworking of _**Lassie Come Home**_ (Metro-Goldwyn-Mayer, 1943).\n\n**1758** _ **Hail to the Rangers**_ **** Columbia, 1943. 57 min. D: William Berke. SC: Gerald Geraghty. With Charles Starrett, Leota Atcher, Arthur Hunnicutt, Bob Atcher, Norman Willis, Lloyd Bridges, Ted Adams, Ernie Adams, Tom London, Davidson Clark, Jack Kirk, Edmund Cobb, Budd Buster, Art Mix, Eddie Laughton, Richard Botiller, Herman Hack, Eddie Phillips, Rusty Cline, George Bamby. An ex-ranger assists a rancher pal who is about to lose his range to an influx of settlers. The plot twist of having homesteaders as the bad guys adds some zest to this Charles Starrett vehicle.\n\n**1759** _ **Hair Trigger Baxter**_ **** Film Booking Offices (FBO), 1926. 55 min. D: Jack Nelson. SC: Paul M. Bryan and James Ormont. With Bob Custer, Eugenia Gilbert, Lew Meehan, Murdock MacQuarrie, Fanny Midgley, Jim Corey, Ernie Adams, Hugh Saxon. A range detective saves the girl he loves from the clutches of a town boss and a crooked rancher. A convoluted plot does not help this Bob Custer silent effort that is only available in a 30-minute version.\n\n**1760** _ **Hair Trigger Casey**_ **** Atlantic, 1936. 59 min. D: Harry S. Fraser. SC: Monroe Talbot. With Jack Perrin, Betty Mack, Wally Wales, Fred \"Snowflake\" Toones, Ed Cassidy, Robert Walker, Phil Dunham, Denny Meadows (Dennis Moore), Victor Wong, Starlight (horse). A cowboy tries to stop a smuggling gang working along the U.S.-Mexican border. Better than average Jack Perrin vehicle for producer William Berke, with plenty of action and some good comedy.\n\n**1761** _ **Half Breed**_ **** Hampton Films, 1973. 90 min. Color. D: Harald Phillipp. SC: Fred Denger. With Lex Barker, Pierre Brice, Ralf Wolter, Gotz George, Walter Barnes, Ursula (Uschi) Glas, Ilija Dzuvalekovski, Mihail Balon, Marinko Cosic, Petar Dobric, Vladimir Leib, Nada Kasapic, Abdurrahaman Shala, Marija Crnobori, Sime Jagarinac, Zvonko Dorbin, Ivo Kristof, Branko Spoljar, Rikard Brezeska, Mile Gatara, Adam Vedernjak. After she inherits her father's gold mine, a half-breed Indian girl is kidnapped by outlaws but Old Shatterhand and his Apache blood brother Winnetou come to her rescue. Sturdy action film in the Karl May series; made in West Germany and issued there in 1966 by Rialto\/Jadran Film as _**Winnetou und Has Halbblut Apanatschi**_ (Winnetou and the Half-Blood Apanatschi).\n\n**1762** _ **The Half-Breed**_ **** RKO Radio, 1952. 81 min. Color. D: Stuart Gilmore. SC: Harold Shumate and Richard Wormser. With Robert Young, Janis Carter, Jack Buetel, Barton MacLane, Reed Hadley, Porter Hall, Connie Gilchrist, Sammy White, Damian O'Flynn, Frank Wilcox, Judy Walsh, Charles Delaney, Tom Monroe, Lee MacGregor, Caleen Calder, Marietta Elliott, Jeane Cochran, Betty Leonard, Shirley Whitney, Mary Menzies, Shelah Hackett, Stuart Randall, Chief Thundercloud, Chief Yowlachie, Jay Silverheels, Franklyn Farnum, John Merton, Perry Ivins, Al Hill, Ted Cooper, Frank O'Connor, Herman Nowlin, William J. O'Brien, Kenner G. Kemp, Phyllis Kennedy, Jeffrey Sayre, Albert Cavens, Jack Chefe, Barry Brooks, Chalky Williams. Dishonest profiteers incite a half-breed Apache into leading his tribe against white settlers in Arizona. Average film that should have turned out better.\n\n**Stuart Randall, Robert Young and Jay Silverheels in** _**The Half-Breed**_ **(RKO Radio, 1952).**\n\n**  \n**\n\n**1763** _ **Halfway to Hell**_ **** Path\u00e9-Alpha, 1962. 75 min. D: Denver Dixon (Victor Adamson). SC: Alan Greedy. With Lyle Felice, Carroll Montour, Sergio Virell, Shirley Tegge, David Lloyd, Don Carlos, Gene Sterling, Rick Adams (Al Adamson), Monte Joe Oyler, Gene Walker, Joe Lane, Al Shelly, Bob Regas. In 1902 Mexico the daughter of a wealthy family falls in love with an aide to a would be revolutionary leader. Last feature directed by the legendary Denver Dixon, this sparse production was partially filmed in Mexico as early as 1957.\n\n**1764** _ **Halleluja and Sartana Strike Again**_ **** Gloria Film, 1972. 90 min. Color. D: Mario Siciliano. SC: Adriano Belzoni and Kurt Nachmann. With Ron Ely, Robert Widmark (Alberto Dell'Acqua), Uschi Glas, Angelica Otto, Alan Abbott (Ezio Marano), Wanda Vismara, Dan van Husen, Stelio Candelli, Dan May (Dante Maggio), Enzo Andronico, Lars Bloch, Domenico Maggio, Carla Mancini, Giulio Massimini, Fury Men (Furio Meniconi), Giovanni Sabbatini, Sergio Testori, Nello Pazzafini, Osiride Pevarello, Renzo Pevarello, Roberto Dell'Acqua, Pietro Torissi. A horse thief and a bogus sky pilot team to swindle a pretty widow but end up helping citizens from being cheated by crooks. Asinine slapstick Spaghetti Western \"comedy\" filmed as _**Alleluja e Sartana Figli di...Dio**_ (Alleluja and Sartana Are Sons...Sons of God); video title: _**Halleluja and Sartana...They Are Sons of God**_ ****\n\n_**Halleluja and Sartana...They Are Sons of God**_ see _**Halleluja and Sartana Strike Again**_\n\n**1765** _ **The Hallelujah Trail**_ **** United Artists, 1965. 167 min. Color. D: John Sturges. SC: John Gay. With Burt Lancaster, Lee Remick, Jim Hutton, Pamela Tiffin, Donald Pleasence, Brian Keith, Martin Landau, John Anderson, John Dehner, Tom Stern, Robert Wilke, Jerry Gatlin, Larry Duran, Jim Burk, Dub Taylor, John McKee, Helen Kleeb, Noam Pitlik, Carl Pitti, Bill Williams, Marshall Reed, Carroll Adams, Ted Markland. In the winter of 1867 an Army officer is assigned to take a shipment of whiskey to Denver but his detail is beset by several groups, including Indians and female temperance workers. Thin, overlong Western comedy that is not very good.\n\n**1766** _ **The Halliday Brand**_ **** United Artists, 1957. 79 min. D: Joseph H. Lewis. SC: George W. George and George S. Slavin. With Joseph Cotten, Viveca Lindfors, Betsy Blair, Ward Bond, Bill Williams, Christopher Dark, Jeanette Nolan, Jay C. Flippen, John Dierkes, Glenn Strange, I. Stanford Jolley, Jay Lawrence, George Lynn, John Halloran, Michael Hinn. A wealthy rancher rides roughshod over his family but trouble erupts with his son when he allows a mob to hang his daughter's half-breed lover. Surprisingly appealing psychological Western with a strong performance by Joseph Cotten as the patriarch.\n\n**1767** _ **Hands Across the Border**_ **** Republic, 1944. 73 min. D: Joseph Kane. SC: Bradford Ropes and J. Benton Cheney. With Roy Rogers, Ruth Terry, Guinn Williams, Bob Nolan and The Sons of the Pioneers (Tim Spencer, Ken Carson, Shug Fisher, Hugh Farr, Karl Farr), Onslow Stevens, Mary Treen, Joseph Crehan, Duncan Renaldo, LeRoy Mason, Janet Martin, The Wiere Brothers, Roy Barcroft, Frederick Burton, Julian Rivero, Kenne Duncan, Jack O'Shea, Jack Kirk, Curley Dresden, Bob Reeves. Roy Rogers is forced to ride Trigger in a race to win a cavalry contract after a crook deprives an honest rival of the agreement. Okay action film with the plot subordinate to songs and a big musical finale.\n\n**1768** _ **Hands Across the Rockies**_ **** Columbia, 1941. 56 min. D: Lambert Hillyer. SC: Paul Franklin. With Bill Elliott, Mary Daily, Dub Taylor, Kenneth MacDonald, Frank LaRue, Donald Curtis, Tom Murray, Stanley Brown, Slim Whitaker, Harrison Greene, Art Mix, Eddy Waller, Hugh Prosser, Edmund Cobb, George Chesebro, John Tyrrell, George Morrell, Kathryn Bates, Eddie Laughton, Ethan Laidlaw, Buck Moulton, Tex Cooper, Curley Dresden. After his pal Cannonball's father is murdered, Will Bill Hickok helps him search for the killer and in a small town they aid a young woman, a witness to the crime, who is being forced to marry the man who did the deed. Fairly action filled \"Wild Bill Hick\" series entry.\n\n**1769** _ **Hang 'Em High**_ **** United Artists, 1968. 114 min. Color. D: Ted Post. SC: Leonard Freeman and Mel Goldberg. With Clint Eastwood, Inger Stevens, Ed Begley, Pat Hingle, Ben Johnson, Charles McGraw, Ruth White, Bruce Dern, Alan Hale, Arlene Golonka, Bob Steele, James Westerfield, Dennis Hopper, L.Q. Jones, Michael O'Sullivan, James MacArthur, Bert Freed, Russell Thorson, Rick Gates, Bruce Scott, Tod Andrews, Roy Glenn, Paul Sorenson, Jack Ging, Ned Romero, Tony Di Milo, Dennis Dengate, William Zuckert, Hal England, Robert B. Williams, John Wesley, Richard Angarola, Larry Blake, Ted Thorpe, Robert Jones, Barry Cahill. When a man is unjustly lynched by a posse for a crime he did not commit, he is saved and sets out to take revenge on those who tried to hang him. Fairly successful attempt by Hollywood to imitate the feel of the then popular European oaters with a strong performance by Bob Steele as the only repentant hangman.\n\n**Clint Eastwood and Inger Stevens in** _**Hang 'Em High**_ **(United Artists, 1968).**\n\n**  \n**\n\n**1770** _ **The Hanged Man**_ **** ABC-TV, 1974. 74 min. Color. D: Michael Caffey. SC: Ken Trevey. With Steve Forrest, Cameron Mitchell, Sharon Acker, Dean Jagger, Will Geer, Barbara Luna, Rafael Campos, Brendon Boone, Bobby Eilbacher, Ray Teal, Steve Marlo, John Mitchum, William Bryant, Hank Worden, John Pickard. A one-time gunslinger survives his own hanging and turns to the right side of the law, assisting a woman whose silver mine is sought by a grasping land baron. Better than average TV Western.\n\n_**Hanging for Django**_ see _**No Room to Die**_\n\n**771** _ **The Hanging of Jake Ellis**_ **** Hollywood Cinemart, 1969. 81 min. Color. D-SC: J. Van Hearn. With Charles Napier, Deborah Downey, James Lemp, Louis Ojena, Don Derby, Rod Wilmot, Chuy Castro, Sol Bar, Miki MacDonald, Jerry Patterson, larry Martinelli, Don Angelo. While trying to help a family save their cattle from his enemy, a cowboy is falsely accused of murdered and nearly hung. Cheap soft core Western buoyed by Charles Napier in the title role. Also called The Calico Queen.\n\n**1772** _ **The Hanging Tree**_ **** Warner Bros., 1959. 106 min. Color. D: Delmer Daves. SC: Wendell Mayes and Halsted Welles. With Gary Cooper, Maria Schell, Karl Malden, Ben Piazza, George C. Scott, Karl Swenson, Virginia Gregg, John Dierkes, King Donovan, Slim Talbot, Guy Wilkerson, Bud Osborne, Annette Claudier, Clarence Straight, Baron James Lichter, Frank Hagney, Billy Benedict, Cactus Mack, Bob Morgan, Boyd Stockman, Sailor Vincent, Don Turner, Danny Borgaze, John Hudkins, Dick Hudkins, Frank Marlow, Harold Millen, Fern Barry, Martin Eric, Dorothy Klewer, Karen Norris. In a rough mining settlement, a doctor trying to forget his past falls in love with a young woman he nurses back to health. Colorful, better than average, but not totally successful oater, best at showing the raw frontier; Marty Robbins sings the title song.\n\n**1773** _ **The Hangman**_ **** Paramount, 1959. 86 min. D: Michael Curtiz. SC: Dudley Nichols. With Robert Taylor, Tina Louise, Fess Parker, Jack Lord, Mickey Shaughnessey, Gene Evans, Shirley Harmer, James Westerfield, Mabel Albertson, Lucille Curtis, Regis Toomey, Betty Lynn, Nelson Leigh, Lorne Greene, Frank Richards, Jose Gonzales-Gonzales, Pedro Gonzalez-Gonzalez, Chuck Hamilton, Sam Wolfe, Robert Adler, Paul Salata, Nolan Leary, Sara Taft, Joseph Hamilton, Richard Collier, James Hope, Dorothy Crehan. A deputy marshal, on the trail of a wanted man, tracks his prey to a locale where he finds the citizens are shielding him. Offbeat, interesting feature with a solid performance by Robert Taylor as the lawman.\n\n**1774** _ **Hangman's Knot**_ **** Columbia, 1952. 84 min. Color. D-SC: Roy Huggins. With Randolph Scott, Donna Reed, Claude Jarman, Jr., Frank Faylen, Glenn Langan, Richard Denning, Lee Marvin, Jeanette Nolan, Clem Bevans, Ray Teal, Guinn Williams, Monte Blue, John Call, Reed Howes, Edward Earle, Post Park, Frank Hagney, Frank Yaconelli. In the closing days of the Civil War a Confederate detachment is ordered to attack a Union outfit transporting gold and after the successful effort the men learn the war is over and their commander wants the loot for himself. Highly competent Randolph Scott feature with a good cast and plot.\n\n**1775** _ **Hannah Lee**_ **** Realart, 1953. 79 min. Color. D-SC: John Ireland and Lee Garmes. With Macdonald Carey, Joanne Dru, John Ireland, Stuart Randall, Frank Ferguson, Ralph Dumke, Don Haggerty, Tom Powers, Tristram Coffin, Norman Leavitt, Peter Ireland. Cattlemen hire a notorious gunslinger to rid the range of settlers but he is opposed by a sheriff and a female caf\u00e9 operator. Cheap Jack Broder production based on MacKinlay Kantor's story; originally issued in 3-D. The Sons of the Pioneers sing the title song.\n\n**1776** _ **Hannie Caulder**_ **** Paramount, 1972. 85 min. Color. D: Burt Kennedy. SC: Z.X. Jones (Burt Kennedy and David Haft). With Raquel Welch, Robert Culp, Stephen Boyd, Ernest Borgnine, Jack Elam, Strother Martin, Christopher Lee, Diana Dors, Aldo Sambrell, Luis Barboo, Brian Lightburn, Forencio Amarilla. A woman wants to take revenge on the bank robbery gang who raped her and murdered her husband. Better than one might expect considering the plot and star.\n\n**1777** _ **Hard Bounty**_ **** Image Entertainment, 1995. 88 min. Color. D: Jim Wynorski. SC: Karen Kelly. With Matt McCoy, Kelly LeBrock, John Terlesky, Kimberly Kelley, Rochelle Swanson, Felicity Waterman, Jay Richardson, Ross Hagen, George \"Buck\" Flower, Jason Emard, Bill Alderson, Phillip Connery, Richard Gabai, Dibo Attar, Matthew Seiden, Steve Restivo, Jonathan Bierner, Robert Peters, Peter Sherayko, Antonia Dorian, Tereance O'Connor, Fred Olen Ray, Steve Barkett. When a town's saloon owner and sheriff refuse to track the killer of a prostitute, a madam and her girls seek revenge for the murder. Average low budget affair.\n\n**1778** _ **Hard Day at Blue Nose**_ **** M.P.C.\/Stonehenge, 1974. 86 min. Color. D: Herbert Kenwith. With John Astin, Patty Duke Astin, John Gavin, Philip Carey, Royal Dano. On vacation at a dude ranch in Nevada, a New York City detective gets involved in solving the murder of a female guest. Average telefeature originally shown as an episode of \"Wide World Mystery\" (ABC-TV, 1973\u201378).\n\n**1779** _ **Hard Fists**_ **** Universal, 1927. 50 min. D: William Wyler. SC: William Lester and George Plympton. With Art Acord, Louise Lorraine, Les Bates, Gilbert \"Pee Wee\" Holmes, Albert J. Smith. A rider, who takes part in fixed races because of blackmail, saves a woman's life and falls in love with her daughter. Only the first two reels of this exciting Universal Blue Streak Western starring Art Acord are extant.\n\n**1780** _ **Hard Ground**_ **** Hallmark Channel, 2003. 120 min. D: Frank Q. Dobbs. SC: Frank Q. Dobbs and David S. Cass, Sr. With Burt Reynolds, Bruce Dern, Amy Jo Johnson, Seth Peterson, David Figlioli, Martin Kove, Larry Hankin, Michael Shamus Wiles, Bill Henderson, Sergio Calderon, Randy Stripling, David Atkinson, Edward Faulkner, David Cass, Sr., Frank Sharp, Shawn Patrick Nash, Brad Heiner, William Rick Young, Steve Cobbs, Dennis Fitzgerald, Lance Lanfear, Robert A. Nowotny. A veteran lawman gets his brother paroled from prison and with a new deputy tries to stop a gang of marauders looting along the U.S.-Mexican border. Slovenly TV Western.\n\n**1781** _ **Hard Hombre**_ **** Allied, 1931. 65 min. D: Otto Brower. SC: Jack Natteford. With Hoot Gibson, Lina Basquette, Skeeter Bill Robbins, Mathilde Comont, Jessie Arnold, Raymond Nye, Christian Frank, Jack Byron, Bob Burns, Glenn Strange, Tiny Sanford, Florence Lawrence, Fred Burns, Clare Hunt. When crooks threaten to take his mother's property, a cowboy comes to her rescue. Typically fanciful Hoot Gibson movie.\n\n**1782** _ **The Hard Man**_ **** Columbia, 1957. 80 min. Color. D: George Sherman. SC: Leo Katcher. With Guy Madison, Valerie French, Lorne Greene, Barry Atwater, Robert Burton, Rudy Bond, Trevor Bardette, Rickie Sorenson, Frank Richards, Myron Healey, Renata Vanni, John Cason, Kermit Maynard. While investigating the murder of a rancher who refused to sell out to a cattle baron, a deputy marshal finds himself falling in love with the dead man's beautiful widow. Somewhat offbeat oater with a psychological tinge; entertaining.\n\n**1783** _ **Hard on the Trail**_ **** United American Films, 1971. 78 min. Color. D-SC: Greg Corarito. With Lash LaRue, Donna Bradley, Bob Romero, Bruce (Kimball) Kemp, Robert Dalton, Arne Dhean, Mary Donahue, Adam Stan, Greg Corarito, John Bloom, Monica Gayle, Richard Hoyt, Randy Starr, Phil Hoover, Mike Armstrong, Victoria Tobin, Ron Wade, Scott Wells, Jim Feazell, Mal Hutton. A vicious gang murders a rancher's wife and rapes her daughter while trying to obtain a map showing the location of a hidden mine. Cheap, violent adult Western also released in an XXX rated version called _**The Hard Trail**_.\n\n**1784** _ **A Hard Road to Vengeance**_ **** NBC-TV\/Universal, 1973. 98 min. Color. D: Alex March. SC: Harold Jack Bloom and Shimon Wincelberg. With Richard Boone, Stuart Whitman, Ruth Roman, Keenan Wynn, Rita Moreno, Harry Morgan, Rick Lenz, Sharon Acker, Dennis Rucker, Jean Allison, Harry Hickox, Fred Brookfield, James G. Richardson. A one-time lawman arrives in a Western town to clear his name in a thirteen year old murder case. Viewable telefilm originally an episode of producer Jack Webb's \"Hec Ramsey\" (NBC-TV, 1972\u201374), first telecast November 25, 1973.\n\n**1785** _ **Hard Rock Harrigan**_ **** Fox, 1935. 70 min. D: David Howard. SC: Raymond L. Schrock and Dan Jarrett. With George O'Brien, Irene Hervey, Fred Kohler, Dean Benton, Frank Rice, Victor Potel, Olin Francis, William Gould, George Humbert, Edward Keane, Lee Shumway, Glenn Strange, Jack Kirk, Lee Phelps, Curley Dresden. Two tunnel drillers love the same woman and fight over her affections. Solid entertainment based on the Zane Grey story.\n\n_**The Hard Trail**_ see _**Hard on the Trail**_\n\n**1786** _ **Hardcase**_ **** ABC-TV, 1972. 74 min. D: John Llewellyn Moxey. SC: Harold Jack Bloom and Sam Rolfe. With Clint Walker, Stefanie Powers, Pedro Armendariz, Jr., Alex Karras, Luis Mirando, Martin LaSalle, E. Lopez Rojas. A drifter returns home to find his ranch sold and his wife missing but later learns she is with a gang of Mexican revolutionaries. Well directed and not-too-bad TV oater, considering the plot.\n\n**1787** _ **Harlem on the Prairie**_ **** Associated Features, 1938. 54 min. D: Sam Newfield. SC: Fred Myton and F.E. Miller. With Herbert (Herb) Jeffries, F.E. Miller, Mantan Moreland, Spencer Williams, Jr., Connie Harris, George Randall, Nathan Curry, The Four Tones, Edward Brandon, James Davis, The Four Blackbirds. A black cowboy tries to stop a crooked Los Angeles cop from cheating club owners. Interesting curio, one of a quartet Westerns starring Herb Jeffries as a black singing cowboy; worth a look. Also called _**Bad Man of Harlem**_.\n\n**1788** _ **Harlem Rides the Range**_ **** Sack Amusement Enterprises, 1939. 58 min. D: Richard C. Kahn. SC: Spencer Williams, Jr. and F.E. Miller. With Herbert Jeffrey (Herb Jeffries), Spencer Williams, Jr., Lucius Brooks, F.E. Miller, Artie Young, Clarence Brooks, Tom Southern, The Four Tones, John Thomas, Wade Dumas, Leonard Christmas, Stardusk (horse). A cowboy attempts to stop a crook from getting control of his girl's father's radium mine. Very low budget all-black feature starring crooner Herb Jeffries.\n\n**1789** _ **The Harmony Trail**_ **** Mattox Pictures, 1944. 57 min. D: Robert Emmett (Tansey). SC: Frances Kavanaugh. With Ken Maynard, Max Terhune, Eddie Dean, Rocky Camron (Gene Alsace), Ruth Roman, Glenn Strange, Bob McKenzie, Charles King, Bud Osborne, Dan White, Hal Price, Warner Richmond, George Morrell, Bob (John) Cason, Chick Hannon, Sherry Tansey, Tex Cooper. A lawman calls on three pals to help him capture the gang that robbed a local bank. Ken Maynard's last \"B\" Western is a low budget but fairly entertaining one in which Eddie Dean croons \"On the Banks of the Sunny San Juan\" (which he wrote with Glenn Strange). Reissued in 1947 by Astor as _**White Stallion**_.\n\n**1790** _ **Harpoon**_ **** Screen Guild, 1948. 83 min. D: Ewing Scott. SC: Girard Smith and Ewing Scott. With John Bromfield, Alyce Louis, James Cardwell, Patricia Garrison, Jack George, Edgar Hinton, Frank Hagney, Holly Bane, Ruth Castle, Grant Means, Sally Davis, Alex Sharp, Lee Roberts, James Martin, Willard Jillson, Gary Garrett. In 1880s' Alaska a young man seeks revenge against his father's enemies. Okay action drama.\n\n**1791** _ **Harry Tracy\u2014Desperado**_ **** IMC\/Isram, 1982. 100 min. D: William A. Graham. SC: David Lee Henry and R. Lance Hill. With Bruce Dern, Helen Shaver, Michael C. Gwynne, Gordon Lightfoot, Jacques Hubert, Daphne Goldrick, Lynne Kolber, Alex Willows, Frank C. Turner, Fred Diehl, Charles Siegel, Jack Ackroyd, Suzie Payne, Richard MacBride, Kerry Salisbury, Jim Roberts, Tom Braidwood, Jim Defelice, Dennis Robertson, Joe Dodds, Peter Manning, Conrad Fitzgerald. Harry Tracy, known as a friend of the poor and gallant toward women, finds himself becoming a legendary outlaw and relentlessly hunted by the law. Fairly entertaining biopic filmed in Canada.\n\n**1792** _ **The Harvey Girls**_ **** Metro-Goldwyn-Mayer, 1946. 101 min. Color. D: George Sidney. SC: Edmund Beloin and Nathaniel Curtis. With Judy Garland, John Hodiak, Ray Bolger, Preston Foster, Angela Lansbury, Virginia O'Brien, Kenny Baker, Marjorie Main, Chill Wills, Cyd Charisse, Selena Royle, Jack Lambert, Ruth Brady, Edward Earle, Morris Ankrum, William \"Bill\" Phillips, Ben Carter, Norman Leavitt, Horace (Stephen) McNally, Catherine McLeod, Virginia Hunter, Mitchell Lewis, Jack Clifford, Vernon Dent, Robert Emmett O'Connor, Paul Newlan, Shirley Patterson, Joe Karnes, John Merton, Ray Teal, Lee Phelps, Robert Emmett O'Connor, Paul Newlan, Tex Cooper, Frank Austin, Sam Garrett, Dona Dax, Dorothy Tuttle, Al Rhein, Charles Regan, Tom Quinn, Al Kunde. Westward expansion brings railroad restaurants to Western communities and the waitresses help to tone down raucous citizens. Dated musical best remembered for the song \"On the Atchison, Topeka and the Santa Fe,\" popularized on record by Kate Smith.\n\n**1793** _ **Hate for Hate**_ **** Metro-Goldwyn-Mayer, 1968. 79 min. Color. D: Domencio Paolella. SC: Bruno Corbucci and Fernando Di Leo. With Antonio Sabato, John Ireland, Fernando Sancho, Gloria Milland, Mirko Ellis, Nadia Marconi, Piero Vida. In the Southwest two men join forces to escape into Mexico with gold sought by a revolutionary leader. A well produced, but violent oater from Italy, issued there in 1967 by West Film as _**Odio per Odio**_ (Hate for Hate).\n\n**1794** _ **Hate Thy Neighbor**_ **** Cinecidi, 1968. 86 min. Color. D: Fernando Baldi. SC: Fernando Baldi, Luigi Angelo and Roberto Natale. With Clyde Garner (Spiros Focas), George Eastman (Luigi Montefiori), Nicoletta Machiavelli, Horst Frank, Ivy Holzer, Robert Rice, Paolo Magalotti, Franco Fantasia, Claudio Castellani, Ivan Scratuglia, Franco Gula. Trying to locate his outlaw brother's killer, a man teams with a mortician and heads to Mexico where the land owner who hired the hit is after a map to a gold mine. Watch able but offbeat, and somewhat sadistic, Italian Western filmed there as _**Odia il Prossimo Tuo**_ (Hate Your Neighbor).\n\n**1795** _ **The Haunted**_ **** A.B. Enterprises\/International Film Industries, 1977. 85 min. Color. D-SC: Michael De Gaetano. With Aldo Ray, Virginia Mayo, Jim Negele, Ann Michelle, Brad Rearden, Fred Carroll, June Ely, Carl Belfor, Grady Daugherty, Paul Vincenzo, George Smith, Leigh Hunt Wilson, Harry Tresize, Robert Bickston, Barry Cooper, Linda Best, Michael Collins, Leo Krokos, Ron Rhode. A family near Arizona's Superstition Mountain finds themselves haunted by the spirit of a dead Indian woman who places a vengeful spirit in the body of their young daughter. Obscure horror Western worth a look for stars Virginia Mayo and Aldo Ray.\n\n**1796** _ **Haunted Gold**_ **** Warner Bros., 1932. 57 min. D: Mack V. Wright. SC: Adele Buffington. With John Wayne, Sheila Terry, Erville Alderson, Harry Woods, Otto Hoffman, Martha Mattox, Blue Washington, Slim Whitaker, Jim Corey, Ben Corbett, Bud Osborne, Blackjack Ward, John T. Prince, Bob Burns, Mack V. Wright, Charles Le Moyne, Tom Bay, Edward Burns. A cowboy and his partner go to a deserted mine that was half-owned by his late father and find a spooky situation with the partner's daughter there along with crooks after hidden gold. The \"Cat and the Canary\" of \"B\" Westerns, this outing is well done and atmospheric with footage from the Ken Maynard silent _**The Phantom City**_ (First National, 1928), of which it is a remake with Ken and Tarzan easily spotted in some of the stock shots; the statue of the Maltese Falcon appears atop a piano.\n\n**1797** _ **The Haunted Mine**_ **** Monogram, 1951. 60 min. D: Derwin Abrahams. SC: Frank Young. With Johnny Mack Brown, Raymond Hatton, Linda Johnson, Riley Hill, Claire Whitney, John Merton, Marshall Reed, Terry Frost, Lynton Brent, Ray Bennett, Frank LaRue, Ray Jones. Crooks try to steal a mine belonging to two women and a U.S. marshal is assigned to find out who has been murdering those interested in the property. The mystery motif adds some flavor to this otherwise pedestrian affair.\n\n**1798** _ **Haunted Ranch**_ **** Monogram, 1943. 57 min. D: Robert Tansey. SC: Elizabeth Beecher. With John King, David Sharpe, Max Terhune, Julie Duncan, Rex Lease, Charles King, Bud Osborne, Budd Buster, Steve Clark, Glenn Strange, Tex Palmer, Fred \"Snowflake\" Toones, Carl Mathews, Jimmy Aubrey, Hank Bell, Jim Corey, Augie Gomez. The Range Busters try to help a young woman whose land is besieged by crooks after a hidden treasure. Spooky atmosphere gives a lift to this \"Range Busters\" saga.\n\n**1799** _ **Haunted Range**_ **** Davis Distributing, 1926. 55 min. D: Paul Hurst. SC: Frank Howard Clark. With Ken Maynard, Alma Rayford, Harry Moody, Tom London, Al Hallett, Fred Burns, Bob Williamson. Given six months to solve a murder in order to keep a ranch he has inherited, a cowboy looks into the appearance of an alleged phantom as well as helping the brother, who is in the snare of crooks, of a young woman with whom he has fallen in love. Flavorful mystery Western with an involved plot; survives only in a 30-minute version.\n\n**1800** _ **Haunted Trails**_ **** Monogram, 1949. 60 min. D: Lambert Hillyer. SC: Adele Buffington. With Whip Wilson, Andy Clyde, Reno Browne, Dennis Moore, I. Stanford Jolley, Myron Healey, John Merton, Mary Gordon, William Ruhl, Steve Clark, Milburn Morante, Eddie Majors, Bud Osborne, Bill Potter, Carl Mathews, Thornton Edwards, Chuck Roberson, Carol Henry, Ben Corbett. Trailing the outlaws who killed his brother, a cowboy finds them trying to take control of a ranch using an imposter to pose as the late owner's sibling. One of the best in the Whip Wilson series; a remake of _**The Mexicali Kid**_ (q.v.).\n\n**1801** _ **Have a Good Funeral, My Friend**_ **** Flora Film, 1971. 90 min. Color. D: Anthony Ascott (Giuliano Carmineo). SC: Giovanni Simonelli and Roberto Gianviti. With John (Gianni) Garko, Antonio Vilar, Daniela Giordano, Antonio (Ivano) Staccioli, Helga Line, Luis Induni, Franco Pesce, Rick Boyd, George Wang, Franco Ressel, Roberto Dell'Acqua, Aldo Berti, Attilio Dottesio, Rocco Lerro, Jean-Pierre Clarain. A gunman arrives in a town looking for a swindler and finds the man's niece beset by two prominent townsmen who are after a vein of gold discovered by her prospector uncle. Fairly interesting Spaghetti Western in the popular \"Sartana\" series released in Italy as _**Buono Funerale, Amigos!...Paga Sartanta**_ (A Good Funeral, Friends!...Sartana Is Paying); also called _**Have a Nice Funeral**_ and _**Shanghai Gold**_.\n\n**1802** _ **Hawaiian Buckaroo**_ **** 20th Century\u2013Fox\/Principal, 1938. 60 min. D: Ray Taylor. SC: Dan Jarrett. With Smith Ballew, Evelyn Knapp, Harry Woods, Benny Burt, George Regas, Carl Stockdale, Pat J. O'Brien, Fred \"Snowflake\" Toones, Laura Treadwell. A dishonest realtor sells a cowboy worthless land, forcing him to take a job on a young woman's ranch. The Hawaiian setting (the movie was made in California) makes this seffort a bit different but the plot is still mundane; star Smith Ballew pleasantly croons \"Hawaiian Memories.\"\n\n_**The Hawk**_ (1935) see _**Trail of the Hawk**_\n\n_**The Hawk**_ (1936) see _**The Phantom of Santa Fe**_\n\n**1803** _ **The Hawk of Powder River**_ **** Eagle Lion, 1948. 54 min. D: Ray Taylor. SC: George Smith. With Eddie Dean, Roscoe Ates, Jennifer Holt, June Carlson, Eddie Parker, Terry Frost, Lane Bradford, Carl Mathews, Ted French, Steve Clark, Tex Palmer, Budd Buster, Bob Woodward, Andy Parker and The Plainsmen. A cowboy and his pal find themselves at odds with an outlaw gang led by \"The Hawk,\" who turns out to be a beautiful woman. Despite the use of footage from earlier Eddie Dean Westerns, this is one of the star's better efforts and Jennifer Holt is very good in the title role.\n\n**1804** _ **Hawk of the Hills**_ **** Path\u00e9, 1927. 10 Chapters. D: Spencer Gordon Bennet. SC: George Arthur Gray. With Allene Ray, Walter Miller, Frank Lackteen, Paul Panzer, Wally Oetell, Jack Pratt, Jack Ganzhorn, Parks Jones, Fred Dana, Evangeline Russell, George Magrill, Chief White Horse. A notorious Montana outlaw raids mining claims but one of his gang betrays him to protect a pretty girl the bad man plans to murder. Fun filled action-mystery silent cliffhanger.\n\n**1805** _ **Hawk of the Wilderness**_ **** Republic, 1938. 12 Chapters. D: William Witney and John English. SC: Barry Shipman, Rex Taylor and Norman Hall. With Herman Brix (Bruce Bennett), (Ray) Mala, Monte Blue, Jill Martin, Noble Johnson, William Royle, Tom Chatterton, George Eldredge, Patrick J. Kelly, Dick Wessel, Fred \"Snowflake\" Toones, Lane Chandler, George (Letz) Montgomery, Iron Eyes Cody, Ann Evers, Earl Askam, Jerry Sheldon, Fred Miller, William Stahl, Harry Tenbrook, Lorne Riebe, Chief Big Tree, Joe Garcia, Art Felix, Art Miles, Henry Wills, Jack O'Shea, Ted Mapes, Alex Montoya, Gertrude Chorre, Cy Shindell, Wally Rose, Sonny Chorre, Tuffie (dog). On a lost remote isle north of the Bering Strait, the son of a dead explorer tries to save the natives who raised him from murderous treasure seekers. Action filled, fun serial re-edited into a 100 TV feature entitled _**Lost Island of Kioga**_.\n\n**1806** _ **The Hawk of Wild River**_ **** Columbia, 1952. 54 min. D: Fred F. Sears. SC: Howard J. Green. With Charles Starrett, Smiley Burnette, Jack (Jock) Mahoney, Clayton Moore, Eddie Parker, Jim Diehl, Lane Chandler, Syd Saylor, John Cason, Leroy Johnson, Jack Carry, Sam Flint, Donna Hall. Two government men are sent to a town to stop lawlessness caused by a gang led by \"The Hawk,\" a desperado using a bow and arrows. Fast moving \"Durango Kid\" series outing.\n\n**1807** _ **Hawken's Breed**_ **** New World, 1987. 89 min. D-SC: Charles B. Pierce. With Peter Fonda, Jack Elam, Serene Hedin, Chuck Pierce, Jr., Sue Anne Langdon, Dennis Fimple, Royce Clark, Bill Thurman, Seamon Glass, Joe Kurtzo, Ivan Green, Bear Pierce, Brandon Lewis, Don Lewis, Robert Lewis, Charles Gibbons, Steve Lyons, Vernon Traver, Leo Aubret, Walker Flame, Andy Heden, Maggi Slivey, Charles B. Pierce (voice). A Tennessee fur trapper saves a young Indian woman from a various predators, including rogue tribesmen, a fur trapper, a posse and a corrupt land owner. Lots of plot but not much interest.\n\n**1808** _ **Hawmps!**_ **** Mulberry Square, 1976. 113 min. Color. D: Joe Camp. SC: William Bickley, Michael Warren and Joe Camp. With James Hampton, Christopher Connelly, Slim Pickens, Jack Elam, Denver Pyle, Gene Conforti, Mimi Maynard, Lee de Broux, Herbert Vigran, Jesse Davis, Frank Inn, Mike Travis, Larry Swartz, Tiny Wells, Dick Drake, Henry Kendricks, Don Starr, Cynthia Smith, Roy Gunzburg, Rex Janssen, Catherine Hearne, Larry Strawbridge, James Weir, Alvin Wright, Lee Tiplitsky, Joey Camp, Perry Martin, Richard Lundin, Charles Starkey. A remote Army post is chosen as the training ground for the use of camels as mounts in the desert. Fairly amusing genre spoof; well made.\n\n**1809** _ **He Rides Tall**_ **** Universal, 1964. 84 min. D: R.G. Springsteen. SC: Charles W. Irwin and Robert Creighton Williams. With Tony Young, Dan Duryea, Jo Morrow, Madlyn Rhue, R.G. Armstrong, Joel Fluellen, Carl Reindel, Mickey Simpson, George Murdock, Michael Carr, George Petrie, Bob Steele, Myron Healey, Roy Barcroft, William \"Bill\" Henry, Charles Irwin, George Keymas, Fred Carson, Jack Tornek. On the eve of his wedding, a marshal must tell his foster father that he was forced to gun down his son. Good dramatic Western held together by Dan Duryea's smooth villain.\n\n**1810** _ **Headin' East**_ **** Columbia, 1937. 67 min. D: Ewing Scott. SC: Ethel LaBlanche and Paul Franklin. With Buck Jones, Ruth Coleman, Donald Douglas, Elaine Arden, Shemp Howard, Earle Hodgins, John Elliott, Stanley Blystone, Frank Faylen, Dick Rich, Al Herman, Harry Lash, Leo Gorcey. When gangsters try to take advantage of lettuce growers, a rancher comes to the big city to stop them. Out-of-the-ordinary Buck Jones vehicle.\n\n**1811** _ **Headin' for God's Country**_ **** Republic, 1943. 78 min. D: William Morgan. SC: Elizabeth Meehan and Houston Branch. With William Lundigan, Virginia Dale, Harry Davenport, Addison Richards, Harry Shannon, J. Frank Hamilton, Eddie Acuff, Wade Crosby, Skelton Knaggs, John Bleifer, Eddy Waller, Charlie Lung, Ernie Adams, Eddie Lee, James B. Leong, Anna Q. Nilsson, Edmund Cobb, Frank Lackteen, Harrison Greene, Charles Miller, Jack Gardner, Howard Banks, George Lee, Ace the Wonder Dog. To get even with the people of an Alaskan village, a prospector tells them the U.S. is at war, which proves to be true. Fairly interesting programmer; topical when issued.\n\n**1812** _ **Headin' for the Rio Grande**_ **** Grand National, 1936. 60 min. D: Robert North Bradbury. SC: Robert Emmett (Tansey). With Tex Ritter, Eleanor Stewart, Warner Richmond, Syd Saylor, Snub Pollard, Charles King, Earl Dwire, Forrest Taylor, William Desmond, Charles K. French, Bud Osborne, Budd Buster, Tex Palmer, Jack C. Smith, Sherry Tansey, Jim Mason, Ed Cassidy. A cowboy brings a wounded cattleman into a town controlled by an outlaw and is jailed for murder. Tex Ritter's second feature is a good one and in it he sings the title tune and \"Night Herding Song.\"\n\n**1813** _ **Headin' for Trouble**_ **** Big 4, 1931. 60 min. D: J.P. McGowan. SC: George Morgan. With Bob Custer, Betty Mack, John Ince, Buck Connors, Andy Shuford, Robert Walker, Duke Lee, William McCall, Jack Kirk, Oscar Gahan, Ace Spriggins, Jack Jones, Jack Harvey, Jack Evans, Barney Beasley, Ray Henderson, Alfred Hewston. A cowboy tries to keep a compulsive gambler from losing his ranch to a crook. Stilted Bob Custer film.\n\n**1814** _ **Headin' North**_ **** Tiffany, 1930. 59 min. D-SC: J.P. McCarthy. With Bob Steele, Barbara Luddy, Perry Murdock, Walter Shumway, Eddie Dunn, Fred Burns, Gordon DeMain, Jim Welsh. Wrongly sent to jail, an escaped convict and his pal exchange identities with two vaudevillians as they search for the man who cheated the escapee's father out of his money. Pretty poor Bob Steele vehicle, more of a vaudeville show than a Western.\n\n**1815** _ **Heading West**_ **** Columbia, 1946. 54 min. D: Ray Nazarro. SC: Ed Earl Repp. With Charles Starrett, Smiley Burnette, Doris Houck, Norman Willis, Nolan Leary, Hank Penny and His Plantation Boys, Bud Geary, Frank McCarroll, John Merton, Tom Chatterton, Hal Taliaferro, Stanley Price, Tommy Coats, Charles Soldani, Matty Roubert, Richard Botiller, Herman Hack. A machinery company owner attacks miners to get their land and places the blame on the Durango Kid. Fair outing in the long running series. British title: _**The Cheater's Last Throw**_.\n\n_**Heads You Die, Tails I Kill You**_ see _**Deep West**_\n\n**1816** _ **Heart of Arizona**_ **** Paramount, 1938. 68 min. D: Lesley Selander. SC: Norman Houston and Harrison Jacobs. With William Boyd, George Hayes, Russell Hayden, John Elliott, Billy King, Natalie Moorhead, Dorothy Short, Stephen Alden Chase, John Beach, Lane Chandler, Leo McMahon, Lee Phelps, Bob McKenzie, Ben Corbett. Outlaws are after Buck Peters' prize breeding stock and steal them when Lucky Jenkins takes an injured girl back to her mother, with Hoppy finding out the woman's foreman is the gang leader. Entertaining \"Hopalong Cassidy\" film with good locations, nice photography and a downbeat ending.\n\n**1817** _ **The Heart of Texas Ryan**_ **** Selig, 1917. 45 min. D: E.A. Martin. SC: Gilsen Willets. With Tom Mix, Bessie Eyton, George Fawcett, Goldie Colwell, Frank Campeau, William Ryno, Leo Maloney, Charles Gerard, Sid Jordan, Hoot Gibson. A cowboy tries to impress his sweetheart by outwitting rustlers, Mexican bandits and a corrupt lawman. Fun Tom Mix silent feature, his last for Selig; re-released in 1923 as _**Single Shot Parker**_.\n\n**1818** _ **Heart of the Golden West**_ **** Republic, 1942. 65 min. D: Joseph Kane. SC: Earl Fenton. With Roy Rogers, Smiley Burnette, George \"Gabby\" Hayes, Ruth Terry, Bob Nolan and The Sons of the Pioneers (Tim Spencer, Lloyd Perryman, Pat Brady, Hugh Farr, Karl Farr), Walter Catlett, Paul Harvey, Edmund MacDonald, Leigh Whipper, William Haade, The Hall Johnson Choir, Hal Taliaferro, Cactus Mack, Hank Bell, Fred Burns, Carl Mathews, Horace B. Carpenter, Frank McCarroll, Art Dillard. When ranchers refuse to pay unjust shipping charges levied against them for hauling their cattle to market, Roy Rogers and his pals convince a steamboat owner to transport the beef. Very good Roy Rogers entry that teams sidekicks Smiley Burnette and George \"Gabby\" Hayes.\n\n**1819** _ **Heart of the High Country**_ **** ITV, 1985. 150 min. Color. D-Sam Pillsbury. SC: Elizabeth Gowans. With Kenneth Cranham, Valerie Gogan, John Howard, David Letch. A young woman leaves her English home to face a new life in 1880s frontier New Zealand, resulting in an unhappy marriage, wealth and eventual ruin. Nicely done feature from the British TV mini-series.\n\n**1820** _ **Heart of the North**_ **** Warner Bros., 1938. 85 min. Color. D: Lewis Seiler. SC: William Byron Mowery. With Dick Foran, Gloria Dickson, Patric Knowles, Allen Jenkins, Janet Chapman, James Stephenson, Arnold Averill, Joseph Sawyer, Russell Simpson, Joseph King, Garry Owen, Pedro de Cordoba, Robert Homans, Gale Page, Emmett Vogan, Harry Cording, Bill Cody, Artie Ortego, Kansas Moehring, Frank Clark, Don Turner, Sol Gorss, Buster Wicks, Lightning (dog). A Canadian Mounted Policeman is unjustly dismissed from the service but redeems himself and proves his innocence when he saves a man from being lynched. Good Dick Foran feature, not considered part of his \"B\" series for Warners.\n\n**1821** _ **Heart of the Rio Grande**_ **** Republic, 1942. 70 min. D: William Morgan. SC: Lillie Hayward and Winston Miller. With Gene Autry, Smiley Burnette, Fay McKenzie, Edith Fellows, Pierre Watkin, Joe Strauch, Jr., William Haade, Sarah Padden, Jean Porter, Milton Kibbee, Edmund Cobb, The Jimmy Wakely Trio (Jimmy Wakely, Johnny Bond and Dick Reinhart), Budd Buster, Frank Mills, Nora Lane, Mady Lawrence, Harry Deep, Frankie Marvin. A cattleman's widow plans to turn her place into a dude ranch and is helped by a singing cowboy. There is nothing special about this Gene Autry opus that is not as dull in its cut 54 minute TV version.\n\n**1822** _ **Heart of the Rockies**_ **** Republic, 1937. 57 min. D: Joseph Kane. SC: Jack Natteford and Oliver Drake. With Robert Livingston, Ray Corrigan, Max Terhune, Lynn(e) Roberts, J.P. McGowan, Sammy McKim, Yakima Canutt, Hal Taliaferro, Maston Williams, Guy Wilkerson, Ranny Weeks, Georgia Simmons, Nelson McDowell, Herman's Mountaineers, Frankie Marvin, Slim Whitaker, Blackie Whiteford, George C. Pearce. Three ranch owners discover a mountain family is behind cattle rustling and the illegal trapping of game and try out to stop them. Very fine entry in \"The Three Mesquiteers\" series.\n\n**1823** _ **Heart of the Rockies**_ **** Republic, 1951. 67 min. D: William Witney. SC: Eric Taylor. With Roy Rogers, Penny Edwards, Gordon Jones, Foy Willing and The Riders of the Purple Sage, Ralph Morgan, Fred Graham, Mira McKinney, Buzz Henry, William Gould, Pepe Hern, Rand Brooks, Ray Bennett, Ted Adams, Jack Ingram, Terry Frost, Tex Terry, George Lloyd, Julia Montoya. Roy Rogers is in charge of a highway construction project using juvenile offenders from a work camp but the boys get the blame for crimes committed by a ranch foreman. Top notch Roy Rogers film; very entertaining.\n\n**1824** _ **Heart of the West**_ **** Paramount, 1936. 60 min. D: Howard Bretherton. SC: Doris Schroeder. With William Boyd, James Ellison, George Hayes, Lyn Gabriel, Sidney Blackmer, Charles Martin, John Rutherford, Warner Richmond, Walter Miller, Ted Adams, Fred Kohler, Bob McKenzie, John Elliott, Leo J. McMahon, Roy Bucko. Hopalong Cassidy and Johnny Nelson get involved with two landowners fighting over the same pasture. Although a bit on the slow side this \"Hopalong Cassidy\" entry has an exciting climax; Al Bowlly sings the title song.\n\n**Jeff Bridges in** _**Hearts of the West**_ **(Metro-Goldwyn-Mayer, 1975).**\n\n**  \n**\n\n**1825** _ **The Heart of Wetona**_ **** Select, 1918. 69 min. D: Sidney A. Franklin. SC: Mary Murillo and George Scharborough. With Norma Talmadge, Thomas Meighan, Fred Huntley, Gladden James, Fred Turner, Princess Uwane Yea, Charles Elder. The pretty half-breed daughter of a Comanche chief is seduced by an engineer who refuses to marry her but she eventually finds love with an Indian agent. Poor silent melodrama, not one of Norma Talmadge's better vehicles.\n\n**1826** _ **Heartland**_ **** Filmhaus, 1979. 96 min. Color. D: Richard Pearce. SC: Beth Farris and Bill Kittredge. With Rip Torn, Conchata Farrell, Barry Primus, Lilia Skala, Megan Folsom, Amy Wright, Jerry Hardin, Mary Boylan, Jeff Boschee, Robert Overholzer, Bob Sirucek, Marvin Berg. In 1910 a widow brings her small daughter to a remote Wyoming town and takes the job as housekeeper for a Scottish rancher with the two marrying for convenience. Stark, rugged, down-to-earth saga of frontier life based on the diaries of a pioneer woman, Elinore Stewart, well played by Conchata Farrell.\n\n**1827** _ **Hearts in Bondage**_ **** Republic, 1936. 72 min. D: Lew Ayres. SC: Olive Cooper, Bernard Schubert and Karl Brown. With James Dunn, Mae Clarke, David Manners, Charlotte Henry, Henry B. Walthall, Fritz Leiber, George Irving, Irving Pichel, J.M. Kerrigan, Frank McGlynn, Sr., Ben Alexander, Oscar Apfel, Clay Clement, Edward Gargan, Russell Hicks, George Hayes, Douglas Wood, Bodil Rosing, Erville Alderson, John Hyams, Etta McDaniel, Lane Chandler, Hopper Atchley, Frankie Marvin, Smiley Burnette, Henry Roquemore, Robert Paige, Charles King, Warner Richmond, Sonny Bupp, Wally West, Bob Card, Harry Strang, Allan Cavan, Marc Cramer, Ethan Laidlaw, Jack Evans, Jack Ingram, Lloyd Ingraham, Herman Hack, Eugene Jackson, Cecil Watson, Pat Flaherty, Clinton Rosemond, Arthur Wanzer, Helen Seamon, Earl Eby, Maurice Brierre. A Navy man finds love during the Civil War but becomes involved in the battle between the _Monitor_ and the _Merrimac_. Howard Lydecker's special effects highlight this well done historical fiction directed by actor Lew Ayres; cut to 54 minutes for TV.\n\n**1828** _ **Hearts of the West**_ **** Metro-Goldwyn-Mayer\/United Artists, 1975. 102 min. Color. D: Howard Zieff. SC: Rob Thompson. With Jeff Bridges, Andy Griffith, Blythe Danner, Donald Pleasence, Alan Arkin, Richard B. Shull, Herb Edelman, Alex Rocco, Frank Cady, Anthony James, Burton Gilliam, Matt Clark, Candy Azzara, Thayer David, Wayne Storm, Marie Windsor, Dub Taylor, Anne Seymour, Jane Dulo, Forrest Smith. A young man arrives in Hollywood dreaming of becoming a Western novel writer, ends up getting involved in the movies and is promoted to starring in a \"B\" cowboy film. Flavorful and affectionate look at the world of the low budget Western in the 1930s; vocals by Nick Lucas add authenticity.\n\n**1829** _ **Heat Lightning**_ **** Warner Bros., 1934. 64 min. D: Mervyn LeRoy. SC: Brown Holmes and Warren Duff. With Aline MacMahon, Ann Dvorak, Preston Foster, Lyle Talbot, Glenda Farrell, Frank McHugh, Ruth Donnelly, Theodore Newton, Willard Robertson, Harry C. Bradley, James Durkin, Jane Darwell, Edgar Kennedy, Muriel Evans, Jill Dennett, Chris-Pin Martin, Eddie Schubert, Margareta Montez, Sam Hayes (voice). Two sisters run a gas station-motor camp in the desert frequented by a series of characters including two gangsters on the lam after a holdup, one of the having been the lover of the oldest sibling. Excellent screen adaptation of the play by Leon Abrams and George Abbott, one of the best \"B\" movies of all time; remade as _**Highway West**_ (q.v.).\n\n**1830** _ **Heaven Only Knows**_ **** United Artists, 1947. 98 min. D: Albert S. Rogell. SC: Ernest Haycox, Art Arthur and Rowland Leigh. With Robert Cummings, Brian Donlevy, Marjorie Reynolds, Stuart Erwin, Jorja Curtright, Bill Goodwin, John Litel, Gerald Mohr, Edgar Kennedy, Lurene Tuttle, Peter Miles, Ray Bennett, Will Orlean. An angel is assigned to come to Earth and try to reform a gambler born without a soul. Western-comedy-fantasy is a different kind of film and very pleasant. Alternate title: _**Montana Mike**_.\n\n**1831** _ **Heaven with a Barbed Wire Fence**_ **** 20th Century\u2013Fox, 1939. 62 minutes. D: Ricardo Cortez. SC: Dalton Trumbo, Leonard Hoffman and Ben Grauman Kohn. With Jean Rogers, Raymond Walburn, Marjorie Rambeau, Glenn Ford, Nicholas (Richard) Conte, Eddie Collins, Ward Bond, Irving Bacon, Kay Linaker, Fred Kelsey, Billy Wayne, Nigel De Brulier, Edward Gargan, Tom McGuire, Paul E. Burns, George Melford, Pat McKee, Nick Copeland, Paul Hurst, Dave Morris, Harry Strang, Victor Potel, Otto Hoffman, Paul Kruger, Dorothy Vernon, Jack Perry, Mae Marsh, Tiny Lipson. A New York City store clerk uses his savings to buy an Arizona ranch and teams with a hobo and a pretty Spanish refugee in getting there. Pleasant contemporary comedy drama program feature; the film debuts of Glenn Ford and Richard Conte.\n\n**1832** _ **Heaven with a Gun**_ **** Metro-Goldwyn-Mayer, 1969. 101 min. Color. D: Lee H. Katzin. SC: Richard Carr. With Glenn Ford, Carolyn Jones, Barbara Hershey, John Anderson, David Carradine, J.D. Cannon, Noah Beery (Jr.), Harry Townes, William Bryant, Virginia Gregg, James Griffith, Roger Perry, Barbara Babcock, Angelique Pettyjohn, Jessica James, Al Wyatt, Bill Catching. An ex-gunman becomes the sheriff of a town only to find himself in the middle of a range feud between cattlemen and sheep herders over water rights. Considering the plot and cast the film should have been better.\n\n**1833** _ **Heaven's Gate**_ **** United Artists, 1980. 210 minutes Color. D-SC: Michael Cimino. With Kris Kristofferson, Isabelle Huppert, Christopher Walken, John Hurt, Sam Waterston, Brad Dourif, Joseph Cotten, Jeff Bridges, Geoffrey Lewis, Paul Koslo, Ronnie Hawkins, Richard Masur, Roseanne Vela, Mary C. Wright, Nicholas Woodeson, Stefan Shcherby, Waldemar Kalinowski, Terry O'Quinn, John Conley, Margaret Benczak, James Knobeloch, Erika Petersen, Robin Bartlett, Tom Noonan, Marat Yusim, Alvars Smits, Gordana Rashovich, Jariath Conroy, Allen Keller, Caroline Kava, Mady Kaplan, Anna Levine, Pat Hodges, Mickey Rourke, Kevin McClarnon, Jerry Sullivan, David Mansfield, David Cass, Peter Ususky, Michael Christensen, Bobby Faber. A Harvard graduate migrates to nineteenth century Wyoming where he becomes involved in the struggle over land between wealthy businessmen and immigrants. Over long, often incomprehensible and basically boring, this financial bust offers excellent photography by Vilmos Szigmond and fine acting by Isabelle Huppert. Some TV prints run 149 minutes.\n\n**1834** _ **Hec Ramsey**_ **** NBC-TV\/Universal, 1972. 100 min. Color. D: Daniel Petrie. SC: Harold Jack Bloom. With Richard Boone, Rick Lenz, Sharon Acker, Harry Morgan, Ray Middleton, R.G. Armstrong, Robert Pratt, Dick Van Patten, Perry Lopez, Dennis Rucker, Bill Vint. At the turn of the 20th century an aging gunfighter agrees to work as a deputy to a young, college trained lawman. Okay pilot for \"Hec Ramsey\" (NBC-TV, 1972\u201374) released to TV as a feature film. Alternate TV title: _**The Century Turns**_.\n\n**1835** _ **Heir to Trouble**_ **** Columbia, 1935. 59 min. D: Spencer Gordon Bennet. SC: Nate Gatzert. With Ken Maynard, Joan Perry, Harry Woods, Wally Wales, Martin Faust, Harry Brown, Dorothy Wolbert, Fern Emmett, Pat O'Malley, Art Mix, Frank Yaconelli, Hal Price, Frank LaRue, Jim Corey, Lafe McKee, Jack Rockwell, Slim Whitaker, Bud McClure, Artie Ortego. After he adopts the small son of his late saddle pal, a cowboy runs into trouble with a rival who wants to steal his girl as well as his mine. Fair Ken Maynard oater, but a bit slim on coherent plot.\n\n**1836** _ **Heldorado**_ **** Republic, 1946. 70 min. D: William Witney. SC: Gerald Geraghty and Julian Zimet. With Roy Rogers, George \"Gabby\" Hayes, Dale Evans, Paul Harvey, Rex Lease, LeRoy Mason, Eddie Acuff, Bob Nolan and The Sons of the Pioneers (Tim Spencer, Lloyd Perryman, Pat Brady, Hugh Farr, Karl Farr), Clayton Moore, Steve Darrell, Doye O'Dell, Charles Williams, John Bagni, Barry Mitchell, Tex Terry, George Chandler, Eddie Kane, Victor Potel, Virginia Carroll, Keith Richards, Shug Fisher, Phil Arnold, Sam Ash, Emmett Vogan, Jr., Frank Henry, Walter Lawrence, Joaquin Bascon. Las Vegas rancher Roy Rogers joins the local sheriff and federal investigators in tracking down racketeers passing thousand dollar bills not subject to taxes. Better than average Roy Rogers effort.\n\n**1837** _ **Hell Bent for Leather**_ **** Universal-International, 1960. 82 min. Color. D: George Sherman. SC: Christopher Knopf. With Audie Murphy, Felicia Farr, Stephen McNally, Robert Middleton, Rad Fulton, Jan Merlin, Herbert Rudley, Malcolm Atterbury, Allan Lane, John Qualen, Bob Steele, Joseph Ruskin, Steve Gravers, Beau Gentry, Eddie Little Sky, Olan Soule, Holly Bane (Mike Ragan), Roy Engel, Laurie Mitchell, David Wanger. A cowboy is ambushed by a wanted murderer but ends up being arrested by a reward hungry sheriff who claims he is the real killer. Fair Audie Murphy vehicle with a fine cast.\n\n**1838** _ **Hell Canyon Outlaws**_ **** Republic, 1957. 72 min. D: Paul Landres. SC: Allan Kaufman and Max Glandbard. With Dale Robertson, Brian Keith, Rossana Rory, Dick Kallman, Don Megowan, Mike Lane, Buddy Baer, George Pembroke, Tom Hubbard, Charles Fredericks, Alexander Lockwood, James Nusser, James Maloney, William Pullen, George Ross, Vincent Padula. A former sheriff takes on an outlaw gang controlling a small town. Action filled feature from Republic's last days.\n\n**1839** _ **Hell-Fire Austin**_ **** Tiffany, 1932. 70 min. D: Forrest Sheldon. SC: Betty Burbridge. With Ken Maynard, Ivy Merton, Nat Pendleton, Jack Perrin, Charles LeMayne, Lafe McKee, Alan Roscoe, William Robyns, Fargo Bussey, Jack Rockwell, Jack Ward, Bud McClure, Lew Meehan, Ben Corbett, Slim Whitaker, Jim Corey, Jack Pennick. Two soldiers return home to Texas after World War I, receive a poor welcome and land in jail but are paroled when one of them agrees to ride a horse in a cross country race. Well written, action laden Ken Maynard outing; good entertainment.\n\n**1840** _ **Hell to Pay**_ **** Echo Bridge Home Entertainment, 2005. 99 min. Color. D-SC: Chris McIntyre. With Lee Majors, Stella Stevens, Buck Taylor, James Drury, Bo Svenson, Denny Miller, Peter Brown, William Smith, Andrew Prine, Tim Thomerson, Eden Rountree, Dale Kimsey, Rachel Kimsey, Jason Shaw, Griff Furst, William Gregory Lee, Kevin Kazakoff, Alison Vasan, Mark Warner, Katie A. Keane, Rachel Kimsey, Rico Nance, Tommy Gunn. Two brothers, a war hero and a gambler, fall in love with the same woman and one of them must oppose a vicious outlaw gang. Nicely done Western filled with genre veterans.\n\n_**Hell Town**_ see _**Born to the West**_\n\n**1841** _ **The Hellbenders**_ **** Avco-Embassy, 1967. 92 min. Color. D: Sergio Corbucci. SC: Albert Band and Ugo Liberatore. With Joseph Cotten, Norma Bengell, Julian Mateos, Aldo Sambrell, Angel Aranda, Gino Pernice, Claudio Gora, Maria Martin, Al Mulock, Julio Pena, Ennio Girolami, Rafael Vaquero. Following the Civil War a Confederate colonel refuses to accept the South's defeat and attempts to organize an army to continue the conflict. Action filled but static Italian oater with a fine work by Joseph Cotten as a madman; made in 1966 by Alba Cinematografica\/Tesica as _**I Crudeli**_ **** (The Cruel Ones).\n\n**1842** _ **Heller in Pink Tights**_ **** Paramount, 1960. 100 min. Color. D: George Cukor. SC: Dudley Nichols and Walter Bernstein. With Sophia Loren, Anthony Quinn, Margaret O'Brien, Steve Forrest, Eileen Heckart, Ramon Novarro, Edmund Lowe, George Mathews, Edward Binns, Warren Wade, Frank Silvera, Robert Palmer, Leo V. Matranga, Cal Bolder, Taggart Casey, Howard McNear, Richard Simmons, Geraldine Wall, Ken Clark, Frank Cordell, Lorraine Crawford, Harry Cheshire, Amanda Randolph, Eddie Little Sky, Iron Eyes Cody, Chief Yowlachie, Cactus Mack, Rodd Redwing, Bryn Davis, Harry Wilson, Jeffrey Sayre, Alfred Tonkel, Cathy Cox, Robert Darin, Robert Adler, David Armstrong. A theatrical troupe travels through Wyoming as the owner romances his leading lady and fights off creditors as well as Indians. A different kind of Western, not totally satisfying but worth watching; Bernard Nedell doubled in several scenes for ailing Edmund Lowe.\n\n**1843** _ **Hellfire**_ **** Republic, 1949. 90 min. Color. D: R.G. Springsteen. SC: Dorrell McGowan and Stuart McGowan. With William Elliott, Marie Windsor, Forrest Tucker, Jim Davis, H.B. Warner, Grant Withers, Paul Fix, Emory Parnell, Esther Howard, Jody Gilbert, Harry Woods, Denver Pyle, Trevor Bardette, Dewey Robinson, Harry Tyler, Roy Barcroft, Hank Worden, Kenneth MacDonald, Eva Novak, Richard Alexander, Louis Faust, Edward Keane, Olin Howlin, Stanley Price, Lillian Molieri, Crane Whitley, Fred Kohler, Jr., Paula Hill, Robert O'Neill, Elizabeth Marshall, Heenan Elliott. A gunman is redeemed by religion and plans to build a church but his opposed by a crook as he seeks to reform a woman outlaw. Very fine William Elliott feature, the type of fare William S. Hart did in the silent days, with an excellent performance by Marie Windsor as a wanted woman.\n\n**1844** _ **Hellgate**_ **** Lippert, 1952. 87 min. D-SC: Charles Marquis Warren. With Sterling Hayden, Joan Leslie, Ward Bond, James Arness, Peter Coe, John Pickard, Robert Wilke, Richard Emory, Marshal Bradford, Sheb Wooley, Rory Mallinson, Timothy Carey, Rodd Redwing, Stanley Price, Kermit Maynard. Falsely sent to prison, a man takes part in an aborted break attempt but eventually atones in the eyes of the law. Low budget, but credible, reworking of the story of Dr. Samuel Mudd, first filmed by 20th Century\u2013Fox in 1936 as _**The Prisoner of Shark Island**_ (q.v).\n\n_**The Hellhounds of Alaska**_ see _**Fight for Gold**_\n**1845** _ **Hellhounds of the Plains**_ **** Goodwill, 1927. 60 min. D-SC: Jacques Jaccard. With Yakima Canutt, Neva Gerber, Lafe McKee, Al Ferguson, Bud Osborne, Cliff Lyons, Roy Bassett, Jack Woods, Boy (horse). A cowboy is in love with the half-sister of a rancher's son who is the leader of a gang of horse thieves. Low grade but fast paced Yakima Canutt silent feature.\n\n**1846** _ **The Hellion**_ **** Columbia, 1961. 79 min. Color. D: Ken Annakin. SC: Harold Swanton, Patrick Kirwan and Harold Ruth. With Richard Todd, Anne Aubrey, Jamie Uys, Marty Wilde, Lionel Jeffries, James Booth, Al Mulock, Colin Blakely, Ronald Fraser, Zena Walker, George Moore, Bill Brewer, Jan Bruyns, Lorna Cowell. In 1860s South African Transvaal a man and his sons arrive in a remote town intent on seeking revenge against a lawman. Fairly interesting British production containing all the necessary Western plot elements.\n\n**1847** _ **Hello, Everybody!**_ **** Paramount, 1933. 70 min. D: William A. Seiter. SC: Dorothy Yost and Lawrence Hazard. With Kate Smith, Randolph Scott, Sally Blane, Charley Grapewin, George Barbier, Julia Swayne Gordon, Wade Boteler, Erville Alderson, Paul Kruger, Frank Darien, Fern Emmett, Jerry Tucker, Marguerite Campbell, Jack Pennick, William B. Davidson, Ted Collins, Russell Simpson, Frank Jenks, Edward Davis, Frank McGlynn, Sr., Hallene Hall, Lon Poff, Irving Bacon, Dennis O'Keefe, Edmund Mortimer, Hal Price, Nat Brusiloff. When the building of a dam threatens the farms near a small Western town, a local girl becomes a radio singing sensation to raise the money needed to fight the project. Kate Smith's solo starring feature is a good one, highlighted by her singing \"Twenty Million People\" and \"Moon Song.\"\n\n**1848** _ **Hello Trouble**_ **** Columbia, 1932. D-SC: Lambert Hillyer. With Buck Jones, Lina Basquette, Wallace MacDonald, Spec O'Donnell, Ruth Warren, Otto Hoffman, Ward Bond, Frank Rice, Russell Simpson, Alan Roscoe, Al Smith, King Baggott, Bert Roach, Walter Brennan, Morgan Galloway. A Texas Ranger after a trio of cattle rustlers shoots one of them only to learn it is his friend so he quits the service only to get involved in hunting the killer of a rancher. Somewhat slow moving and erratic Buck Jones vehicle, but still entertaining.\n\n**1849** _ **Hell's Crossroads**_ **** Republic, 1957. 73 min. Color. D: Franklin Adreon. SC: John K. Butler and Barry Shipman. With Stephen McNally, Peggie Castle, Robert Vaughn, Barton MacLane, Harry Shannon, Henry Brandon, Douglas Kennedy, Grant Withers, Myron Healey, Frank Wilcox, Jean Howell, Morris Anrkum, Heenan Elliott, Eddie Baker, Chip Carson, John Patrick. A member of the James gang wants to reform but his former cohorts try to foil his efforts. Average oater with a fine cast.\n\n**1850** _ **Hell's Heroes**_ **** Universal, 1930. 65 min. D: William Wyler. SC: Tom Reed. With Charles Bickford, Raymond Hatton, Fred Kohler, Fritzi Ridgeway, Maria Alba, Jose De La Cruz, Buck Connors, Walter James. Three outlaws fleeing a posse into the desert come across a dying woman and agree to take her newborn to its father. First sound version of Peter B. Kyne's _The Three Godfathers_ is a sturdy affair that holds up well.\n\n**1851** _ **Hell's Hinges**_ **** Triangle, 1915. 55 min. D: William S. Hart and Charles Swickard. SC: C. Gardner Sullivan. With William S. Hart, Clara Williams, Louise Glaum, Jack Standing, Alfred Hollingsworth, Robert McKim, J. Frank Burke, Robert Kortman, Leo Willis, Jean Hersholt, John Gilbert, Wheeler Oakman. A dishonest gambler hires a gunslinger to stop the work of a new minister but the gunman falls in love with the clergyman's sister and helps his cause. Top notch William S. Hart film, a faithful recreation of the Old West with plenty of action and violence.\n\n**1852** _ **Hell's Outpost**_ **** Republic, 1954. 90 min. D: Joseph Kane. SC: Kenneth Gamet. With Rod Cameron, Joan Leslie, John Russell, Chill Wills, Jim Davis, Kristine Miller, Ben Cooper, Taylor Holmes, Barton MacLane, Ruth Lee, Oliver Blake, Harry Woods, John Dierkes, Arthur Q. Bryan, Buzz Henry, Sue England, Almira Sessions, Don Kennedy, Paul Stader, Don Brodie, Alan Bridge, Ruth Brennan, Edward Clark, Gil Harman, James Lilburn, Elizabeth Slifer, George Dockstader. A war veteran comes to a small town determined to work his mining claim but his opposed by a crooked banker. Rugged drama with Rod Cameron as the strong hero and John Russell a bad, bad villain.\n\n**1853** _ **Henry Goes Arizona**_ **** Metro-Goldwyn-Mayer, 1939. 66 min. D: Edwin L. Marin. SC: Florence Ryerson and Milton Merlin. With Frank Morgan, Virginia Weidler, Guy Kibbee, Slim Summerville, Douglas Fowley, Owen Davis, Jr, Gordon Jones, Porter Hall, Chester Conklin, Ann Morriss, Olin Howland, Eddie Dunn, Ted Adams, Jack Kirk, Joe Whitehead, Robert Emmett Keane, Erville Alderson, Jim Thorpe, Matty Faust, Tenen Holtz, Robert Spinola, George Noisom. When a broke actor inherits his brother's ranch, a gang of outlaws try to take it for themselves. Amusing genre comedy second feature.\n\n_**Hercules and the Treasure of the Incas**_ see _**Blood River**_\n\n**1854** _ **Heritage of the Desert**_ **** Paramount, 1932. 63 min. D: Henry Hathaway. SC: Harold Shumate and Frank Partos. With Randolph Scott, Sally Blane, J. Farrell MacDonald, David Landau, Gordon Wescott, Guinn Williams, Vince Barnett, Susan Fleming, Charles Stevens, Fred Burns. A young man, raised by a desert rancher, tries to stop a claim jumper from taking his property. Adequate screen adaptation of the Zane Grey novel, first filmed by Paramount in 1924 with Bebe Daniels, Lloyd Hughes, Ernest Torrance and Noah Beery; reissued as _**When the West Was Young**_.\n\n**1855** _ **Heritage of the Desert**_ **** Paramount, 1939. 74 min. D: Lesley Selander. SC: Norman Houston and Harrison Jacobs. With Donald Woods, Evelyn Venable, Russell Hayden, Robert Barrat, Sidney Toler, C. Henry Gordon, Willard Robertson, Paul Guifoyle, Paul Fix, John \"Skins\" Miller, Reginald Barlow, Frank Ellis, Charles Brinley, Hank Bell, Tex Phelps, Charles Murphy. An Easterner comes West to claim an inheritance and gets mixed up with outlaws. Very fine third screen version of the Zane Grey work.\n\n**1856** _ **Los Hermanos Centella**_ (The Centella Brothers) **** Artistas Nacionales Asociados, 1967. 90 min. D: Fernando Fernandez. SC: Fernando Fernandez and Jose Delfos. With Dacia Gonzalez, Jaime Fernandez, Daboberto Rodriguez, Guillermo Rivas, Fernando Soto \"Mantequilla\", Crox Alvarado, Agustin Fernandez, Ana Maria Garcia, Juan Garza, Jesus Gomez, Roberto Porter, Ignacio Villalbazo, Oscar Alatorre, Gloria Berrones. Years after their families are massacred by an outlaw gang, a government agent and the woman he loves seek revenge. Okay Mexican Western.\n\n**1857** _ **Los Hemanos Diablo**_ (The Diablo Brothers) **** Alameda Films, 1959. 75 min. D: Fernando Menez and Chano Urueta. SC: Alfredo Salazar. With Maurcio Garces, Abel Salazar, Rafael Baledon, Dacia Gonzalez, Carlos Suarez, Tito Novarro, Jose Castro, Consuelo Oviedo, David Reynoso, Guillermo Rivas, Alfonso Torres, Carlos Nieto, Antonio Raxel. Three brothers fight to keep their American ranch free of Indian raiders and outlaws. Action filled oater from south of the border.\n\n**1858** _ **Heroes of Fort Worth**_ **** Fenix Film, 1964 90 min. D: Herbert Martin (Alberto De Martino). SC: Eduardo Manzanos. With Edmund Purdom, Priscilla Steele, Paul Piaget, Aurora Julia, Isarco Ravaioli, Umberto Raho, Miguel Del Castillo, Eduardo Fajardo, Tomas Blanco. During the Civil War a group of Confederates head to Mexico to enlist the aid of Emperor Maximilian in helping their cause but they meet opposition at a Western fort. Action filled Spaghetti Western with lots of historical ingredients but few facts. Issued in Italy as _**La Carica del 7 Cavalleggeri**_ (The Charge of the Seventh Cavalry).\n\n**1859** _ **Heroes of the Alamo**_ **** Columbia, 1938. 74 min. D: Harry S. Fraser. SC: Ruby Wentz. With Earle Hodgins, Lane Chandler, Rex Lease, Roger Williams, Ed Peil, Sr., Julian Rivero, Jack C. Smith, Bruce Warren, Ruth Findlay, Lee Valianos, William Costello, Steve Clark, Sherry Tansey, Denver Dixon, George Morrell, Tex Cooper, Oscar Gahan, Ben Corbett, Lafe McKee, Slim Whitaker, Budd Buster, Frank Ellis, Richard Cramer, Fred \"Snowflake\" Toones, Francis Walker, Tex Phelps, Milburn Morante, Curley Dresden, Herman Hack, Al Taylor, Hal Price, Merrill McCormick, Carl Mathews, Jim Corey, Dorothy Vernon, Tex Phelps, Tom Smith, Bob Roper. Texans strive for independence from Mexico, resulting in their gallant stand at the Alamo against Santa Ana's army. Pretty good independent production from Anthony J. Xydias and issued by his Sunset Pictures in 1937 before being distributed by Columbia the next year. Reissue title: _**Remember the Alamo**_.\n\n**1860** _ **Heroes of the Hills**_ **** Republic, 1938. 56 min. D: George Sherman. SC: Betty Burbridge and Stanley Roberts. With Robert Livingston, Ray Corrigan, Max Terhune, Priscilla Lawson, LeRoy Mason, James Eagles, Roy Barcroft, Carleton Young, Forrest Taylor, Maston Williams, John Beach, Roger Williams, Kit Guard, Jack Kirk, Curley Dresden, Robert Hays, John Wade, Jerry Frank, Gloria Rich, I. Stanford Jolley, Bob Card, Buck Morgan, Lew Meehan, Tommy Coats, Art Dillard, Chuck Baldra. The Three Mesquiteers turn their spread into a work farm for prison trustees but when they try to get other ranchers to join the program a crooked contractor tries to thwart them. Another fine and exciting \"Three Mesquiteers\" entry.\n\n**1861** _ **Heroes of the Range**_ **** Columbia, 1936. 58 min. D: Spencer Gordon Bennet. SC: Nate Gatzert. With Ken Maynard, June Gale, Harry Woods, Harry Ernest, Robert Kortman, Bud McClure, Tom London, Bud Osborne, Frank Hagney, Jack Rockwell, Lafe McKee, Wally Wales, Buffalo Bill, Jr., Bud Jamison, Bob Reeves, Marin Sais, Oscar Gahan, Jack Evans, Buck Moulton. A cowboy comes to the aid of a pretty girl whose brother is in the clutches of an outlaw gang. Delightful Ken Maynard vehicle; this is the one where Ken masquerades as an outlaw and when the crooks ask him to prove his identity he says, \"I know what ye want\" and proceeds to play the fiddle and sing.\n\n**1862** _ **Heroes of the Saddle**_ **** Republic, 1940. 59 min. D: William Witney. SC: Jack Natteford. With Robert Livingston, Raymond Hatton, Duncan Renaldo, Patsy Lee Parsons, Loretta Weaver, Byron Foulger, Vince Barnett, William Royle, Reed Howes, Al Taylor, Kermit Maynard, Tex Terry, Matt McHugh, Harrison Greene, Jack Roper, Ethel May Halls, Patsy Carmichael, Harry Strang, Tommy Coats, Douglas Deems, Darwood Kaye, Tom Hanlon, Art Dillard, Bob Card, Chief John Big Tree, Bob Burns. When money belonging to an orphanage is reported stolen, the Three Mesquiteers investigate and find the institution is run by crooks. Modern-day setting somewhat detracts from the overall effectiveness of this series outing.\n\n**1863** _ **Heroes of the West**_ **** Universal, 1932. 12 Chapters. D: Ray Taylor. SC: George Plympton, Basil Dickey, Joe Roach and Ella O'Neill. With Noah Beery, Jr., Diane Duval (Julie Bishop\/Jacqueline Wells), Onslow Stevens, William Desmond, Martha Mattox, Philo McCullough, Harry Tenbrook, Frank Lackteen, Edmund Cobb, Jules Cowles, Francis Ford, Grace Cunard, Lafe McKee, Chief Thunderbird. Moving to Wyoming with his children, a contractor tries to complete a transcontinental railroad with the help of an engineer but they are opposed by forces trying to sabotage the project. Fair serial with a more interesting cast than plot.\n\n**1864** _ **Hex**_ **** 20th Century\u2013Fox, 1973. 90 min. Color. D: Leo Garen. SC: Leo Garen and Steve Katz. With Keith Carradine, Robert Walker, Hilarie Thompson, Tina Herazo, Scott Glenn, Gary Busey, John Carradine, Mike Combs, Doria Cook, Patricia Ann Porter. Following World War I a motorcycle gang becomes involved with the occult in a Western town after a mysterious girl casts a spell on them. Okay horror Western; also called _**Grass Lands**_.\n\n**1865** _ **Hi Gaucho!**_ **** RKO Radio, 1936. 60 min. D: Thomas Atkins. SC: Adele Buffington. With John Carroll, Steffi Duna, Rod LaRoque, Montagu Love, Ann Codel, Tom Ricketts, Paul Porcasi, Julian Rivero, Sam Appel, Frank Mills, Harold Daniels, Ferike Boros, Enrique DeRosas. A South American cowboy prefers a pretty Castilian senorita to taking part in a long standing family feud. Passable comedy-action program feature.\n\n**1866** _ **Hiawatha**_ **** Allied Artists, 1952. 80 min. Color. D: Kurt Neumann. SC: Arthur Strawn and Daniel B. Ullman. With Vincent Edwards, Yvette Dugay, Keith Larsen, Morris Ankrum, Eugene Iglesias, Ian MacDonald, Stuart Randall, Katherine Emery, Stephen Chase, Armando Silvestre, Michael Tolan, Richard Bartlett, Michael Granger, Robert Bice, Henry Corden. The chief of an Indian tribe tries to bring peace with his people's long time foes. Producer Walter Mirisch's adaptation of the Henry Wadsworth Longfellow poem is on the dull side.\n\n**1867** _ **Hidalgo**_ **** Buena Vista, 2004. 136 min. Color. D: Joe Johnston. SC: John Fusco. With Viggo Mortensen, Zuleikha Robinson, Omar Sharif, Louise Lombard, Adam Alexi-Malle, Said Taghmaoul, Silas Carson, Harsh Nayyar, J.K. Simmons, Adoni Maropis, Victor Talmadge, Peter Mensah, Joshua Wolf Coleman, Frankly Mwangi, Floyd Red Crow Westerman, Elizabeth Berridge, C. Thomas Howell, Steven Rimkus, Jerry Hardin, Frank Collison, Chris Owen, Marshal Manesh, Philip Sounding Sides, George Gerdes, Todd Kimsey, Ednah New Rider Weber, Adam Ozturk, John Prosky, Michael Canavan, David Midthunder, Sam Sako, Jaek Miller, Mary Ellis, Zachary Badasci, Malcolm McDowell, Joseph J. Dawson. A Pony Express rider goes to Arabia to compete in a long distance race with his horse and becomes involved with a Sheik and his beautiful daughter. Sumptuous, entertaining production that flopped at the box office.\n\n**1868** _ **Hidden Danger**_ **** Monogram, 1948. 55 min. D: Ray Taylor. SC: J. Benton Cheney and Eliot Gibbons. With Johnny Mack Brown, Raymond Hatton, Max Terhune, Christine Larson, Myron Healey, Marshall Reed, Kenne Duncan, Edmund Cobb, Steve Clark, Milburn Morante, Carol Henry, Bill Hale, Boyd Stockman, Bill Potter, Bob Woodward. The head of a cattlemen's protective group persuades local ranchers to sell him their cattle but he pays less than the market price and is soon being investigated by a lawman. Last Johnny Mack Brown series film with Raymond Hatton is a competent affair.\n\n**1869** _ **Hidden Gold**_ **** Universal, 1932. 60 min. D: Arthur Rosson. SC: Jack Natteford and James Milhauser. With Tom Mix, Judith Barrie, Raymond Hatton, Eddie Gribbon, Donald Kirke, Wallis Clark, Roy Moore, Buffalo Bill, Jr., Bud Osborne, Gilbert \"Pee Wee\" Holmes, Ed LeSaint, Olin Francis. A cowboy goes to jail to get in good with an outlaw gang that has hidden loot from a robbery. Not one of Tom Mix's better sound films.\n\n**1870** _ **Hidden Gold**_ **** Paramount, 1940. 61 min. D: Lesley Selander. SC: Jack Merserveau and Gerald Geraghty. With William Boyd, Russell Hayden, Britt Wood, Ruth Rogers, Roy Barcroft, Minor Watson, Ethel Wales, Lee Phelps, George Anderson, Jack Rockwell, Eddie Dean, Raphael Bennett, Walter Long, Robert Kortman, Merrill McCormick, Cliff Parkinson, Art Dillard, Bruce Mitchell. After becoming a ranch foreman, Hopalong Cassidy is at odds with a crook who wants to steal a gold mine from his former outlaw partner. Pretty fair series outing.\n\n**1871** _ **Hidden Guns**_ **** Republic, 1956. 66 min. D: Albert C. Gannaway. SC: Sam Roeca and Albert C. Gannaway. With Bruce Bennett, Richard Arlen, John Carradine, Faron Young, Angie Dickinson, Lloyd Corrigan, Damian O'Flynn, Irving Bacon, Tom Hubbard, Guinn Williams, Edmund Cobb, Ben Welden, Gordon Terry, Bill Coontz, Ron Kennedy. A sheriff and his son try to save their town from a dishonest gambler, his henchman and their hired guns. Fairly good program feature with nice work by its trio of stars.\n\n**John Carradine and Bruce Bennett in** _**Hidden Guns**_ **(Republic, 1956).**\n\n**  \n**\n\n**1872** _ **Hidden Valley**_ **** Monogram, 1932. 60 min. D: Robert North Bradbury. SC: Wellyn Totman. With Bob Steele, Gertrude Messinger, Francis McDonald, Ray Haller, John Elliott, Arthur Miller, V.L. Barnes, Dick Dickinson, George Hayes, Tom London, Captain Verner L. Smith. A cowboy is after a gang seeking treasure hidden in a peaceful locale. Pretty good Bob Steele vehicle with a mystery plot plus dirigible sequences and a lost Indian tribe worshiping a skull god.\n\n**1873** _ **Hidden Valley Outlaws**_ **** Republic, 1944. 56 min. D: Howard Bretherton. SC: John K. Butler and Bob Williams. With Wild Bill Elliott, George \"Gabby\" Hayes, Anne Jeffreys, Roy Barcroft, Kenne Duncan, John James, Charles Miller, Budd Buster, Tom London, LeRoy Mason, Earle Hodgins, Yakima Canutt, Fred \"Snowflake\" Toones, Jack Kirk, Tom Steele, Bud Geary, Frank McCarroll, Ed Cassidy, Robert Wilke, Cactus Mack, Forbes Murray, Frank O'Connor, Charles Morton, Tom Smith, Harry Leroy, Kansas Moehring. When a vicious outlaw gang terrorizes a southwest town a marshal comes to the rescue. Fast moving, entertaining \"Wild Bill Elliott\" film; the last in the series.\n\n**1874** _ **The High Country**_ **** Crown-International, 1981. 101 min. D: Harvey Hart. SC: Bud Townsend. With Timothy Bottoms, Linda Purl, George Sims, Bill Berry, Jim Lawrence, Walter Mills, Paul Jolicoeur, Dick Butler, Elizabeth Alderton, Barry Graham, John Duthie, Marsha Stonehouse. An escaped convict persuades an illiterate young woman to take him into Alberta's high country in order to elude the law. Fair drama but nothing exceptional.\n\n**1875** _ **High Hell**_ **** Paramount, 1958. 87 min. D: Burt Balaban. SC: Irene Tunich. With John Derek, Elaine Stewart, Patrick Allen, Jerold Wells, Al Mulock, Rodney Burke, Colin Craft, Nicholas Stuart. A man returns to his mountain mine to find his wife there with his partner and the three are soon snowbound for the winter. Brooding, British made melodrama.\n\n**1876** _ **High Lonesome**_ **** Eagle-Lion, 1950. 81 min. Color. D-SC: Alan LeMay. With John Barrymore, Jr., Chill Wills, Lois Butler, Christine Miller, John Archer, Basil Ruysdael, Jack Elam, Dave Kashner, Frank Cordell, Clem Fuller, Hugh Aiken, Howard Joslin. A young man becomes involved with two escaped convicts planning to commit a murder. Fairly involved psychological Western.\n\n**1877** _ **High Noon**_ **** United Artists, 1952. 84 min. D: Fred Zinneman. SC: Carl Forman. With Gary Cooper, Thomas Mitchell, Lloyd Bridges, Katy Jurado, Grace Kelly, Otto Kruger, Lon Chaney, Henry (Harry) Morgan, Ian MacDonald, Eve McVeagh, Harry Shannon, Lee Van Cleef, Robert Wilke, Sheb Wooley, Tom London, Ted Stanhope, Larry Blake, William Phillips, Jeanne Blackford, William Newell, Lucien Prival, Guy Beach, Howard Chamberlin, Morgan Farley, Virginia Christine, Virginia Farmer, Jack Elam, Paul Dubov, Harry Harvey, Tim Graham, Nolan Leary, Tom Greenway, Dick Elliott, John Doucette. On the eve of his wedding, a veteran marshal must face three vengeful killers without the help of the town's citizens. Classic Western drama with several excellent performances; still it would be slow going if it were not for Tex Ritter's singing the title song throughout.\n\n**1878** _ **High Noon**_ **** Turner Broadcasting System (TBS), 2000. 88 min. Color. D: Ron Hardy. SC: Carl Foreman and T.S. Cook. With Tom Skerritt, Susanna Thompson, Reed Diamond, David Lereaney, Maria Conchita Alonso, Dennis Weaver, August Schellenberg, Michael Madsen, Matthew Walker, Frank C. Turner, Shaun Johnson, Terry M. King, Kate Newby, Brian Stollery, Noel Fisher, Joe Norman Shaw, Trevor Leigh, Colin Campbell, Jim Leyden, Stephen Eric McIntyre, Jim Shield, Royal Sproule, Tom McBeth, Andy Maton, Bob Chomyn, Jacqueline Robbins, Joyce Robbins, Brent Woolsey, Thomas Legg, Judith Buchan. A newly married lawman is challenged by a killer, who he sent to jail, and his gang. Mundane TV remake of the genre classic.\n\n**1879** _ **Nigh Noon, Part Two:**_ _**The Return of Will Kane**_ **** CBS-TV, 1980 100 min. Color. D: Jerry Jameson. SC: Elmore Leonard. With Lee Majors, David Carradine, Pernell Roberts, J.A. Preston, Michael Pataki, Katherine Cannon, Britt Leach, Frank Campanella, M. Emmet Walsh, Tracy Walter, Charles Benton, Sanford Gibbs, Stonewall Jackson, Francese Javis, Henry Max Kenrick, Kirk Koshelle, Warren Stanhope, Tiny Weller, Clint Austin. An ex-lawman, now a rancher, protects a friend wanted by a marshal and is forced to take up his guns to defend him. Average small screen follow-up to the original _**Nigh Noon**_ (q.v.).\n\n**1880** _ **High Plains Drifter**_ **** Universal, 1973. 105 min. Color. D: Clint Eastwood. SC: Ernest Tidyman. With Clint Eastwood, Verna Bloom, Marianna Hill, Mitchell Ryan, Jack Ging, Stefan Gierasch, Ted Hartley, Billy Curtis, Geoffrey Lewis, Walter Barnes, Paul Brinegar, Dan Vadis, Jack Kosslyn, Belle Mitchell, John Mitchum, Pedro Regas, Dan Vadis, Richard Bull, Jack Kasslyn, Russ McCubbin, Carl Pitti, Chuck Waters, Buddy Van Horn, Robert Donner, John Hillerman, Anthony James, William O'Cornell, John Quade, James Gosa, Jane Auld, Reid Cruickshanks. After a gunslinger is hired by the citizens of a town to protect them against a gang of killers they being to wonder if their new sheriff is really mortal. Likable film that seems to be a cross between Spaghetti Westerns and a genre spoof.\n\n**1881** _ **High Plains Invaders**_ **** RHI Entertainment, 2009. 87 min. D: K.T. Donaldson (Kristoffer Tabori). SC: Richard Beattie. With James Marsters, Cindy Sampson, Sebastian Knapp, Sanny van Heteren, Dan Bordeianu, Antony Byrne, Sorin Cristea, James Jordan, Angus MacInnes, Dugald Bruce Lockhart, Constantin Barbulesu, Adriana Butoi. About to be hung, an outlaw is saved when an alien attacks a remote town and he vows to destroy the insectoid. Far fetched, but not too bad sci-fi oater.\n\n_**High Stakes**_ see _**Two Fisted Stranger**_\n\n**1882** _ **High Voltage**_ **** Path\u00e9, 1929. 60 min. D: Howard Higgin. SC: James Gleason. With William Boyd, Owen Moore, Carol(e) Lombard, Diane Ellis, Billy Bevan, Phillips Smalley. During a snowstorm bus passengers are marooned in High Sierras church and one of them, a young woman going to jail, falls for a lineman who is wanted by the law. Although disliked by critics when first released, this early talkie holds up rather well.\n\n**1883** _ **High, Wide and Handsome**_ **** Paramount, 1937. 112 min. D: Rouben Mamoulian. SC: Oscar Hammerstein II. With Irene Dunne, Randolph Scott, Dorothy Lamour, Elizabeth Patterson, Raymond Walburn, Akim Tamiroff, Charles Bickford, Ben Blue, William Frawley, Alan Hale, Irving Pichel, Stanley Andrews, James Burke, Roger Imhof, Lucien Littlefield, Purnell B. Pratt, Edward Gargan, Helen Lowell, Russell Hopton, Frank Sully, Tommy Bupp, Claire McDowell, Edward Keane, Lew Kelly, Dell Henderson, John T. Murray, Frank Shannon, Jack Clifford, Billy Bletcher, Constance Bergen, Marjorie Cameron, Pat West, Rolfe Sedan, John Maurice Sullivan, John Marshall, Ernest Wood, Philip Morris, Paul Kruger, George MacQuarrie. A circus star marries an oil driller and they become involved with farmers in 1859 Pennsylvania who try to save their lands from the railroads. Big budget musical that is on the bland side.\n\n_**The Highwayman Rides**_ see _**Billy the Kid**_ (1930)\n\n**1884** _ **Highway West**_ **** Warner Bros., 1941. 63 min. D: William McGann. SC: Allen Rivkin, Charles Kenyon and Kenneth Gamet. With Brenda Marshall, Arthur Kennedy, Olympe Bradna, William Lundigan, Willie Best, Frank Wilcox, John Ridgely, Dorothy Tree, Noel Madison, Pat Flaherty, Victor Zimmerman, William B. Davidson, Dick Rich, James Westerfield, Nat Carr, Paul Panzer, Creighton Hale, Erville Alderson, Herbert Anderson, Leo White, Dorothy Adams, Fred Graham, Guy Usher, Wade Boteler, Harry Strang, John Dilson, Douglas Evans. After her husband is sentenced to life in prison, a woman opens a desert motel-caf\u00e9 with her younger sister and grandfather but the spouse makes a getaway and holes up with his ex-wife, who has found love with another man, and begins romancing the sister. Mediocre remake of the excellent _**Heat Lightning**_ (q.v.).\n\n**1885** _ **Las Hijas del Zorro**_ (The Daughters of Zorro) **** Pelicuas Rodriguez, S.A., 1964. 87 min. D: Federico Curiel. SC: Federico Curiel and Alfredo Ruanova. With Kitty de Hoyos, Dacia Gonzalez, Rafael Bertrand, Eduardo Fajardo, Santanon, Eric de Castillo, Alvaro Ortiz, Pancho Cordova, Luz Marquez, Rogelio Guerra, Tito Novaro. Eighteen years after Zorro is jailed by an evil military captain, his two daughters take his place to fight tyranny. Action filled Mexican adventure film followed by _**Las Invencibles**_ (q.v.).\n\n**1886** _ **El Hijo del Charro Negro**_ (The Son of the Black Cowboy) **** Radaent Films, 1961. 82 min. D: Arturo Martinez. SC: Raul de Anda. With Rodolfo de Anda, Jaime Fernandez, Crox Alvarado, Graciela Lara, Emma Roldan, Andres Soler, Domingo Soler, Cecilia Leger, Guillermo Alvarez Bianchi. The son of the Black Cowboy tries to rescue his childhood sweetheart after she is abducted by a rejected suitor. Fun Mexican Western in the \"Charro Negro\" series.\n\n**1887** _ **El Hijo del Diablo**_ (The Son of the Devil) **** Producion Filmica Mexico, 1966. 88 min. D: Zacarias Gomez Urquiza. SC: Felipe Mier and Zacarias Gomez Urquiza. With Joaquin Cordero, Alma Delia Fuentes, Jorge Russek, Irma Serrano, Jose Baviera, Roberto Meyer. A mysterious masked avenger opposes two cruel brothers and their bandit gang as they terrorize a region with one of them lusting after a rancher's daughter. Fast paced, enjoyable south of the border Western highlighted by Joaquin Cordero in the title role.\n\n**1888** _ **Hills of Oklahoma**_ **** Republic, 1950. 67 min. D: R.G. Springsteen. SC: Olive Cooper and Victor Arthur. With Rex Allen, Fuzzy Knight, Elizabeth Fraser, Elisabeth Risdon, Roscoe Ates, Rex Lease, Robert Karns, Robert Emmett Keane, Trevor Bardette, Lee Phelps, Edmund Cobb, Ted Adams, Lane Bradford, Johnny Downs, Michael Carr. Rustlers try to steal a cattle herd being driven to market by the head of the cattlemen's protective association. Average Rex Allen feature.\n\n**1889** _ **Hills of Old Wyoming**_ **** Paramount, 1937. 79 min. D: Nate Watt. SC: Maurice Geraghty. With William Boyd, George Hayes, Russell Hayden, Gail Sheridan, Stephen Morris (Morris Ankrum), Clara Kimball Young, Earle Hodgins, Steve Clemente, Chief Big Tree, John Beach, George Chesebro, Jim Mason, Paul Gustine, Leo J. McMahon, John Powers. Hoppy, Windy and Lucky oppose crooks trying to blame Indians for cattle rustling. Colorful \"Hopalong Cassidy\" feature with plenty of action and the lovely title song later associated with Eddie Dean; Russell Hayden's first film in the series as Lucky Jenkins.\n\n**1890** _ **The Hills of Utah**_ **** Columbia, 1951. 70 min. D: John English. SC: Gerald Geraghty. With Gene Autry, Pat Buttram, Elaine Riley, Onslow Stevens, Denver Pyle, Donna Martell, William Fawcett, Harry Lauter, Tom London, Kenne Duncan, Sandy Sanders, Teddy Infur, Lee Morgan, Boyd Stockman, Stanley Price, Bob Woodward, Tommy Ivo, Billy Griffith. Arriving in a town trying to find out who killed his father, a frontier doctor becomes embroiled in a feud between a mine operator and ranchers. Entertaining Gene Autry opus with more emphasis on drama than songs.\n\n**1891** _ **The Hills Run Red**_ **** United Artists, 1967. 89 min. Color. D: Lee W. Beaver (Carlo Lizzani). SC: Dean Craig (Mario Pierotti). With Thomas Hunter, Henry Silva, Dan Duryea, Nando Gazzolo, Nicolettea Machiavelli, Gianna Serra, Loris Loddi, Geoffrey Copleston, Paolo Magalotti, Tiberio Mitri, Vittorio Bonos, Mirko Valentin. A mysterious gunman helps an ex-convict out for revenge against the former partner who sent him to jail and kept the payroll money they stole. Fast moving and violent Italian oater hugely helped by Dan Duryea's work as the gunslinger; issued in Europe in 1966 as _**Un Fiume di Dollari**_ (A River of Dollars).\n\n_**Hired Gun**_ see _**The Last Gunfighter**_\n\n**1892** _ **The Hired Gun**_ **** Metro-Goldwyn-Mayer, 1957. 64 min. D: Ray Nazarro. SC: David Lang and Buckley Angell. With Rory Calhoun, Anne Francis, Vincent Edwards, John Litel, Chuck Connors, Robert Burton, Guinn Williams, Reg Parton, Salvador Baquez, Pierce Lyden, Edgar Dearing, Chuck Roberson, William Tannen, Nolan Leary, Joe Haworth, Dan Riss, Bart Braverman, Beulah Archuletta. A lawman bringing a young woman back to a settlement to hang for murder becomes convinced of her innocence and tries to find the real killer. Compact and entertaining \"B\" drama.\n\n**1893** _ **The Hired Hand**_ **** Universal, 1971. 98 min. Color. D: Peter Fonda. SC: Alan Sharp. With Peter Fonda, Warren Oates, Verna Bloom, Robert Pratt, Severn Dardern, Ted Markland, Rita Rogers, Megan Denver, Ann Doran, Michael McClure. After being away from home for seven years a man returns to work for his ex-wife and daughter, eventually trying to free a buddy held prisoner by outlaws. Uneven film that is fairly well acted.\n\n**1894** _ **His Brother's Ghost**_ **** Producers Releasing Corporation, 1945. 58 min. D: Sam Newfield. SC: George Milton (Milton Raison and George Wallace Sayre). With Buster Crabbe, Al St. John, Charles King, Karl Hackett, Archie Hall, Frank McCarroll, Bud Osborne, Bob (John) Cason, Roy Brent, George Morrell, Richard Alexander, Carl Mathews, Charles Soldani, Art Dillard, Frank Ellis, Herman Hack, Jimmy Aubrey, Ray Henderson, Rube Dalroy. Ambushed by outlaws, a rancher sends for his twin brother and after his death the bad men see the sibling and think it is the man's ghost. Typically cheap \"Billy Carson\" entry slightly distinguished by Al St. John's work in dual roles, showing how good he could be in a dramatic part.\n\n**1895** _ **His Fighting Blood**_ **** Ambassador, 1935. 60 min. D: John English. SC: Joseph O'Donnell. With Kermit Maynard, Polly Ann Young, Paul Fix, Ben Hendricks, Jr., Ted Adams, Joseph Girard, Frank LaRue, John McCarthy, Frank O'Connor, Charles King, Jack Cheatham, Ed Cecil, Theodore Lorch, The Singing Constables (Glenn Strange, Chuck Baldra, Jack Kirk). Upon his release from prison after serving a term for a crime committed by his brother, a man joins the Mounties and goes after an outlaw gang that includes his sibling. Complicated Kermit Maynard north woods melodrama.\n\n**1896** _ **His Name Was King**_ **** Foro Film, 1971. 90 min. Color. D: Don Reynolds (Renato Savino). SC: Renato Savino. With Richard Harrison, Klaus Kinski, Anne Puskin, John Silver (Goffredo Unger), Lorenzo Fineschi, Vassili Karis, Lucio Zarina, Tom Felleghy, Giuseppe Monteverdi, John Bartha, Rick Boyd (Federico Boido), Paolo Magalotti, Ada Pometti, Giorgio Dolfin, Luciano Pigozzi, Marco Zuanelli. A bounty hunter is on the trail of gun runners who murdered his brother and raped his sister-in-law, the trail leading to Mexico where the gang is headed by his best friend, a lawman. Fair Italian Western released there as _**Lo Chiamavano King**_ (They Called Him King).\n\n_**His Name Was Sam Walbash, but They Call Him Amen**_ see _**Savage Guns**_ (1971)\n\n**1897** _ **Hit and Run**_ **** Universal, 1924. 45 min. D: Edward Sedgwick. SC: Edward Sedgwick and Raymond L. Schrock. With Hoot Gibson, Marion Harland, Cyril Ring, Harold Goodwin, DeWitt Jennings, Mike Donlin, William A. Steele. A baseball player from a desert town is signed by a major team but after he romances a scout's daughter the two are kidnapped by crooks out to fix the world series. Pleasant comedy combination of sagebrush yarn and baseball.\n\n**1898** _ **Hit the Saddle**_ **** Republic, 1937. 57 min. D: Mack V. Wright. SC: Oliver Drake. With Robert Livingston, Ray Corrigan, Max Terhune, Rita (Hayworth) Cansino, Yakima Canutt, J.P. McGowan, Ed Cassidy, Sammy McKim, Harry Tenbrook, Robert Smith, Ed Boland, Bob Burns, Russ Powell, Allan Cavan, George Morrell, Budd Buster, Kernan Cripps, George Plues, Tex Palmer, Jack Kirk, Oscar Gahan, Rudy Sooter, Robert Hoag, Harley Luse, Sheila Terry, Herman Hack, Jack Tornek, Tex Phelps, Wally West. Two pals have a falling out when one of them romances a gold digging fandango dancer but reconcile to fight a gang rustling wild horses in a protected area. Average entry in the popular \"The Three Mesquiteers' series.\n\n**1899** _ **Hitched**_ **** NBC-TV\/Universal, 1973. 73 min. Color. D: Boris Sagal. SC: Richard Alan Simmons. With Sally Field, Tim Matheson, Neville Brand, Slim Pickens, John Fiedler, Denver Pyle, John McLiam, Kathleen Freeman, Don Knight, Bo Svenson, Bill Zuckert, Charles Lane. A young newlywed couple out West find themselves the victims of a crooked scheme. Sub-par TV movie, sequel to _**Lock, Stock and Barrel**_ (q.v.).\n\n**1900** _ **Hittin' the Trail**_ **** Grand National, 1937. 58 min. D: Robert North Bradbury. SC: Robert Emmett (Tansey). With Tex Ritter, Jerry Bergh, Tommy Bupp, Earl Dwire, Jack C. Smith, Heber Snow (Hank Worden), Ed Cassidy, Snub Pollard, Archie Ricks, Charles King, Ray Whitley and The Range Ramblers, Francis Walker, George Morrell, Oscar Gahan, Joe Weaver, Tex Ritter's Tornados, Smokey (dog). Arrested after accidentally becoming involved with an outlaw gang, a cowboy tries to prove his innocence by rounding up the bad men. Okay Tex Ritter yarn, although the series is beginning to show signs of excessive economic measures by this time; Tex sings a half-dozen songs, including \"Blood on the Saddle.\n\n_**Hi-Yo Silver**_ see _**The Lone Ranger**_ (1938)\n\n**1901** _ **Hoedown**_ **** Columbia, 1950. 64 min. D: Ray Nazarro. SC: Barry Shipman. With Eddy Arnold, Jeff Donnell, Jock (Mahoney) O'Mahoney, Guinn Williams, Carolina Cotton, Fred F. Sears, Don C. Harvey, Charles Sullivan, Douglas Fowley, Ray Walker, Harry Harvey, The Pied Pipers, The Oklahoma Wranglers (The Willis Brothers). A cowboy film star arrives at a dude ranch and finds bank robbers hiding there. Fun country music Western musical with a good job by Jock Mahoney as the movie hero.\n\n**1902** _ **A Hole Between the Eyes**_ **** Tigielle 33, 1968. 90 min. Color. D: Joseph Warren (Giuseppe Vari). SC: Adriano Bolzoni. With Anthony Ghidra, Robert Hundar (Claudio Undari), Rosy Zichel, Corinne Fontaine, John MacDouglas (Giuseppe Addobbati), John Bryan, Giorgio Gargiullo, Elsa Janet Waterston, Mario Darnell, Luigi Marturano, Bruno Cattaneo, Giuseppe Castellano. A bounty hunter obtains one of three cards giving the location of hidden treasure and tries to find the others, one belonging to a tyrant. Average but fairly engaging Spaghetti Western with good work by Anthony Ghidra in the lead role; released in Italy as _**Un Boco in Fronte**_ (A Hole in the Forehead).\n\n_**A Hole in the Sky**_ see _**The Ranger, the Cook and a Hole in the Sky**_\n\n**1903** _ **Hollywood Barn Dance**_ **** Screen Guild, 1947. 72 min. D: Bernard B. Ray. SC: Dorothy Knox Martin. With Ernest Tubb, Helen Royce, Earle Hodgins, Frank McGlynn, Dotti Hackett, Pat Combs, Jack Guthrie, Phil Arnold, Cyril Ring, The Texas Troubadours. When a country band accidentally burns down their hometown church while rehearsing they go on the road to earn money to rebuild it and get mixed up with a crooked promoter and his pretty daughter. Cheap musical made to cash in on Ernest Tubb's popularity; he sings nine songs.\n\n**1904** _ **Hollywood Cowboy**_ **** RKO Radio, 1937. 65 min. D: Ewing Scott. SC: Dan Jarrett and Ewing Scott. With George O'Brien, Cecilia Parker, Maude Eburne, Joe Caits, Frank Milan, Charles Middleton, Lee Shumway, Walter De Palma, Al Hill, William Royle, Al Herman, Frank Hagney, Dan Wolheim, Slim Balch, Sid Jordan, Lester Dorr, Harold Daniels, Horace B. Carpenter, Robert Walker, Donald Kerr, Hal Price, Jack Evans. While vacationing in a Western town, a cowboy film star learns local ranchers are being cheated by a protective association and decides to help them. Very good George O'Brien vehicle with a pleasant mixture of comedy and action. Reissue and TV title: _**Wings Over Wyoming**_.\n\n**1905** _ **Hollywood, It's a Dog's Life**_ **** Maricopa Films, 2004. 80 min. D-SC: Byron Quisenberry. With Mike Moroff, Anne Lockhart, Brent Davis, Robert Havice, Peter Bown, Tonjua Swann, Zoe Keller, Scott Bailey, Danielle Rayne, Peggy Stewart, Mayf Nutter, Tom Schultz, Robert Hoy, Jack Williams, Jason Newman, Hank Calia, Geno Ghiselli, Sam Maloff, Kevin Quisenberry, John Nowak, Jeff Snee, Bob Diamond, Keven Whitaker, Ryan Young, April Wade, Jeremy Lucas, Ronnie Liu, Martin Kove, Tony Brubaker, Elle Travis, Robert Weiland, Robert James Elliott, Sean Quiseberry, Lillian Byrd, Wesley Scott, Kiva Lawrence, Carlyle Taylor, Zane Taylor. A former movie stuntman, now a horse trainer, lives with his two dogs and when his granddaughter moves in she becomes attached to one of them and dreams of becoming an animal trainer. Charming modern-day Western family fare. Video title: _**Big Chuck, Little Chuck**_.\n\n**1906** _ **Hollywood Round-Up**_. Columbia, 1937. 63 min. D: Ewing Scott. SC: Joseph Hoffman and Monroe Shaff. With Buck Jones, Helen Twelvetrees, Grant Withers, Shemp Howard, Dickie Jones, Eddie Kane, Monty Collins, Warren Jackson, Lester Dorr, Lee Shumway, Edward Keane, Slim Whitaker, George Beranger. Because of a film star's jealousy, a stuntman is fired but gets a job with another movie company only to become the fall guy when they rob a bank. Pretty fair Buck Jones vehicle with some pleasant kidding of the Hollywood scene.\n\n**1907** _ **A Holy Terror**_ **** Fox, 1931. 53 min. D: Irving Cummings. SC: Ralph Brock. With George O'Brien, Sally Eilers, Rita LaRoy, James Kirkwood, Humphrey Bogart, Stanley Fields, Robert Warwick, Richard Tucker, Earl Pingree, Slim Whitaker, John Elliott, George Chandler, Fred Kohler, Jr., Buffalo Bill, Jr., Julian Rivero, Jerry Mandy, Bud Geary, Walter Hiers, Franklin Parker, Oscar Smith, Wong Chung, Otto Han, Ralph Bucko. When his father is murdered in Wyoming, a polo playing playboy heads West to find the killers. Entertaining George O'Brien vehicle based on a Max Brand novel first filmed as _**Trailin'**_ (q.v.).\n\n**1908** _ **Hombre**_ **** 20th Century\u2013Fox, 1967. 110 min. Color. D: Martin Ritt. SC: Irving Ravetch and Harriet Frank, Jr. With Paul Newman, Fredric March, Richard Boone, Diane Cilento, Cameron Mitchell, Barbara Rush, Margaret Blye, Peter Lazer, Martin Balsam, Skip Ward, Frank Silvera, Val Avery, David Canary, Linda Cordova, Pete Hernandez, Merrill Isbell. A white man raised by Apaches learns to dislike his fellow passengers aboard a stagecoach but when they are attacked by outlaws he is forced to defend them. Average psychological action melodrama.\n\n**1909** _ **El Hombre de la Furia**_ (The Man of Fury) **** Cinematografica Fermont, 1966. 95 min. D: Fernando Orozco. SC: Mario de la Pedroza, Victor Eberg and Fernando Orozco. With Javier Solis, Dacia Gonzalez, Fernando Soto \"Mantequilla,\" Raymond Belmonte, Ignacio Navarro, William Gomez, Cuco Sanchez, Miguel Angel Landa, Guillemo Galvez. After his rancher father is murdered and he is raised by a trio of rustlers, a young man plans to avenge the killing. Fairly good Mexican Western.\n\n**1910** _ **Un Homre Llamado el Diablo**_ (A Man Called the Devil) **** Produciones Matouk, 1983. 90 min. Color. D: Rafael Villasenor Kuri. D: Alfonso Torres Portillo. With Vicente Fernandez, Miguel Angel Rodriguez, Tere Alvarez, Manuel Ojeda, Enrique Lucero, Antonio de Hud, Raul Meraz, Leonor Llausas, Ignacio Reles. A stranger shows up in a remote town seeking revenge and affects the lives of the locals. Interesting Mexican oater produced by star Vicente Fernandez.\n\n**1911** _ **Un Hombre Peligroso**_ (A Dangerous Man) **** Radient Film, 1965. 85 min. D: Arturo Martinez. SC: Raul de Anda. With Rodolfo de Anda, Ofelia Monesco, Victor Junco, Andres Soler, Guillermo Herrera, Claudio Brook, Jorge Mateos, Jose Chavez, Mario Chavez, Miguel Arenas, Emma Roldan, Tito Novaro, Jose L. Murillo, Eduardo Lugo, Federico Falcon, Jose Torvay, Pepito Velazguez. After he kills a gambler in self defense a gunman adopts his son but years later the young man learns the truth, resulting in a showdown between the two men. Rodolfo de Anda plays a character called El Zurdo (The Left-Handed) in this tense south of the border oater produced and written by his father, Raul de Anda.\n\n**1912** _ **Hombres de Roca**_ (Men of Stone) **** Radient Films, 1960. 88 min. D: Raul de Anda, Jr. SC: Fernando Galiana. With Rodolfo de Anda, Jaime Fernandez, Victor Parra, Sonia Infante, Maura Monti, Arturo Martinez, Alfredo Gutierrez, Antonio de Anda, Jose L. Murillo, Carlos Hennings, Rita Macedo, Martin Plute, Ernesto Suarez, Federico Falcon, Jesus Gomez, Adolfo Aguilar. Made a sheriff, an ex-gunslinger has a falling out with his son who kidnaps a girl, forcing the lawman to track him. Well done Mexican Western drama produced by Raul de Anda.\n\n**1913** _ **Home from the Hill**_ **** Metro-Goldwyn-Mayer, 1960. 150 min. Color. D: Vincent Minnelli. SC: Harriet Frank, Jr. and Irving Ravetch. With Robert Mitchum, Eleanor Parker, George Peppard, George Hamilton, Everett Sloane, Luana Patten, Anne Seymour, Constance Ford, Ken Renard, Ray Teal, Guinn Williams, Charlie Briggs, Hilda Haynes, Denver Pyle, Dan Sheridan, Orville Sherman, Dub Taylor, Stuart Randall, Tom Gilson, the Rev. Duncan Gray, Jr., Joe Ed Russell, Burt Mustin. A wealthy Texas land baron is estranged from his wife and illegitimate son and hunted by a man who thinks he seduced his young daughter. Powerful psychological melodrama with an excellent performance by Robert Mitchum as the landowner.\n\n**1914** _ **Home in Oklahoma**_ **** Republic, 1946. 72 min D: William Witney. SC: Gerald Geraghty. With Roy Rogers, George \"Gabby\" Hayes, Dale Evans, Carol Hughes, George Meeker, Arthur Space, Frank Reicher, George Carleton, Bob Nolan and The Sons of the Pioneers (Doye O'Dell, Tim Spencer, Lloyd Perryman, Pat Brady, Hugh Farr, Karl Farr), Lanny Rees, Ruby Dandridge, George Lloyd, Johnny Walsh, The Flying \"L\" Ranch Quartette. A rancher is murdered and his fortune is left to a young boy with Roy Rogers and a newspaper woman trying to find the killer. An exciting and well directed Western mystery.\n\n**1915** _ **Home in Wyomin'**_ **** Republic, 1942. 67 min. D: William Morgan. SC: Robert Tasker and M. Coates Webster. With Gene Autry, Smiley Burnette, Fay McKenzie, Olin Howlin, Chick Chandler, Joe Strauch, Jr., Forrest Taylor, James Seay, George Douglas, Charles Lane, Hal Price, Bud Geary, Ken Cooper, Jean Porter, James McNamara, Kermit Maynard, Rex Lease, Roy Butler, Billy Benedict, Cyril Ring, Spade Cooley, Ted Mapes, Jack Kirk, William Kellogg, Betty Farrington, Tom Hanlon, Lee Shumway. Gene Autry becomes involved in helping the owner of a rodeo straighten out his wayward son. Good Autry opus enhanced by a mystery angle; based on a story by detective fiction master Stuart Palmer.\n\n**1916** _ **Home on the Prairie**_ **** Republic, 1939. 58 min. D: Jack Townley. SC: Arthur Powell and Paul Franklin. With Gene Autry, Smiley Burnette, June Storey, George Cleveland, Jack Mulhall, Walter Miller, Gordon Hart, Hal Price, Earle Hodgins, Ethan Laidlaw, John Beach, Jack Ingram, Bob Woodward, Dorothy Vernon, Olin Francis, Art Dillard, Fred Burns, Burr Caruth, Chuck Baldra, Sherven Brothers Rodeoliers. Crooks cause a young woman's ranch to be quarantined by placing sick cattle there while they try to ship the rest of the diseased herd to market before the hoof-and-mouth disease is discovered. Standard Gene Autry entry.\n\n**1917** _ **Home on the Range**_ **** Paramount, 1935. 54 min. D: Arthur Jacobson. SC: Ethel Doherty and Grant Garrett. With Jackie Coogan, Randolph Scott, Evelyn Brent, Dean Jagger, Addison Richards, Fuzzy Knight, Howard Wilson, Phillip Morris, Albert Hart, Allen Wood, Richard Carle, Ralph Remley, C.L. Sherwood, Clara Lou (Ann) Sheridan, Francis Sayles, Jack Clark, Joe Morrison, Alfred Delacambre. An outlaw gang plans to shoot a valuable racing pony belonging to two brothers. Surprisingly inferior adaptation of Zane Grey's _Code of the West_ , first filmed under that title by Paramount in 1925.\n\n**1918** _ **Home on the Range**_ **** Republic, 1946. 55 min. Color. D: R.G. Springsteen. SC: Betty Burbridge. With Monte Hale, Adrian Booth, Bob Nolan and The Sons of the Pioneers (Tim Spencer, Ken Carson, Shug Fisher, Pat Brady, Hugh Farr, Karl Farr), Bobby Blake, LeRoy Mason, Roy Barcroft, Kenne Duncan, Tom Chatterton, Budd Buster, Jack Kirk, John Hamilton, Frank O'Connor, Patsy Moran. Wanting to protect wildlife, a rancher finds his ideas put him at odds with area cattlemen. Monte Hale's first starring effort is only average despite the use of Magnacolor.\n\n**1919** _ **The Homesteaders**_ **** Allied Artists, 1953. 62 min. D: Lewis D. Collins. SC: Sol Theil and Milton Raison. With Bill Elliott, Robert Lowery, Barbara Allen, George Wallace, Emmett Lynn, Buzz Henry, Rick Vallin, Stanley Price, William Fawcett, James Seay, Tom Monroe, Ray Walker, Barbara Woodell. Outlaws are after Army dynamite on a wagon train and two Oregon homesteaders try to stop them. Well above average \"B\" picture; well done.\n\n**1920** _ **Homesteaders of Paradise Valley**_ **** Republic, 1947. 59 min. D: R.G. Springsteen. SC: Earle Snell. With Allan Lane, Bobby Blake, Martha Wentworth, Ann Todd, Gene (Roth) Stutenroth, John James, Mauritz Hugo, Emmett Vogan, Milton Kibbee, Tom London, Edythe Elliott, George Chesebro, Ed Cassidy, Jack Kirk, Herman Hack, Marshall Reed, Freddie Chapman, Pat Hennigan, Frank O'Connor, Al Ferguson, Jack Sparks, Bob Burns, Roy Bucko, Post Park, Bud Geary, Foxy Callahan, Pascale Perry, Tom Steele, Cactus Mack. When homesteaders try to build a dam in their new valley home they are opposed by two brothers. Average \"Red Ryder\" affair.\n\n**1921** _ **Hondo**_ **** Warner Bros., 1953. 83 min. Color. D: John Farrow. SC: James Edward Grant. With John Wayne, Geraldine Page, Ward Bond, Michael Pate, James Arness, Rodolfo Acosta, Leo Gordon, Tom Irish, Lee Aaker, Paul Fix, Rayford Barnes, Chuck Roberson, Frank McGrath, Morry Ogden. An ex-gunman in the Southwest comes across a widow and her small son living on a ranch about to be attacked by Indians. Fine psychological Western mixed with action; nicely adapted from the Louis L'Amour story.\n\n**1922** _ **Hondo and the Apaches**_ **** Metro-Goldwyn-Mayer, 1967. 85 min. Color. D: Lee H. Katzin. SC: Andrew J. Fenady. With Robert Taylor, Ralph Taeger, Kathie Browne, Randy Boone, Michael Rennie, Noah Beery (Jr.), Gary Clarke, Gary Merrill, John Smith, Buddy Foster, Michael Pate, Victor Lundin, Jim Davis, Steve Marlo, John Pickard, William Bryant. A loner takes an assignment from the Army to keep peace with the Indians and becomes involved with a mine owner who meets the son he has never seen. Issued abroad theatrically, this sturdy telefilm was the pilot for the TV series, \"Hondo\" (ABC-TV, 1967).\n\n**1923** _ **Honeychile**_ **** Republic, 1951. 89 min. Color. D: R.G. Springsteen. SC: Jack Townley and Charles E. Roberts. With Judy Canova, Eddie Foy, Jr., Alan Hale, Walter Catlett, Roy Barcroft, Claire Carleton, Karolyn Grimes, Brad Morrow, Leonid Kinsky, Fuzzy Knight, Gus Schilling, Irving Bacon, Roscoe Ates, Ida Moore, Sarah Edwards, Emory Parnell, Dick Elliott, Dick Wessel. A song publisher thinks a tune written by a hick girl was actually done by a famous composer. Typical Judy Canova outing for her fans; others beware.\n\n**1924** _ **The Honkers**_ **** United Artists, 1972. 101 min. Color. D: Steve Ihnat. SC: Steve Ihnat and Stephen Lodge. With James Coburn, Lois Nettleton, Slim Pickens, Anne Archer, Jim Davis, Joan Huntington, Richard Anderson, Ramon Bieri, Ted Eccles, Mitchell Ryan, Wayne McLaren, John Harmon, Richard O'Brien, Pitt Herbert, Larry Mahon, Chuck Henson, Jerry Gatlin. A once famous rodeo star tries to make a comeback to impress his son and re-win his estranged wife. Action filled but not very involving character study.\n\n**1925** _ **Honky Tonk**_ **** Metro-Goldwyn-Mayer, 1941. 105 min. D: Jack Conway. SC: Marguerite Roberts and John Sanford. With Clark Gable, Lana Turner, Claire Trevor, Frank Morgan, Marjorie Main, Albert Dekker, Chill Wills, Henry O'Neill, John Maxwell, Morgan Wallace, Douglas Wood, Betty Blythe, Hooper Atchley, Harry Worth, Veda Ann Borg, Dorothy Granger, Sheila Darcy, Cy Kendall, Erville Alderson, John Farrell, Don Barclay, Ray Teal, Esther Muir, Ralph (Francis X., Jr.) Bushman, Art Miles, Anne O'Neal, Russell Hicks, Henry Roquemore, Lew Harvey, Dick Curtis, Sheila Darcey, Syd Saylor, Hooper Atchley, Heinie Conklin, Alan Bridge, Ed Cassidy, Eddy Waller, Cy Kendall, Eddie Gribbon, Carl Stockdale, Horace Murphy, Will Wright, Ralph Peters, Harry Semels, Frank Mills, Dick Rush, Joe Devlin, Lew Kelly, Monte Montague, William Haade, Al Hill, Ed Brady, Jack Baxley, Howard Mitchell, John Sheehan, Jack C. Smith, Tom Chatterson, Gordon DeMain, Pat O'Malley, Lee Phelps, Tiny Newlan, Dorothy Ates, Elliott Sullivan, Dorothy Granger, John Carr, Art Belasco, Fay Holderness, Charles McAvoy, Earl Gunn, Ted Oliver, Malcolm Waite, Charles Sullivan, William Pagan. The nice daughter of a drunk falls in love with a crooked gambler who has taken over a Western town. Big budget vehicle for Clark Gable and Lana Turner is only average screen fare.\n\n**1926** _ **Honky Tonk**_ **** NBC-TV\/Metro-Goldwyn-Mayer, 1974. 74 min. Color. D: Don Taylor. SC: Douglas Heyes. With Richard Crenna, Stella Stevens, Will Geer, Margot Kidder, John Dehner, Geoffrey Lewis, Gregory Sierra, Robert Casper, Dub Taylor, Dennis Fimple, John Quade, Richard Evans, Richard Stahl. A con man comes to Nevada in the 1880s to take advantage of gold strikes in the boom towns. TV movie reworking of the 1941 film (q.v.) is fairly good.\n\n**1927** _ **Honor of the Mounted**_ **** Monogram, 1932. 57 min. D-SC: Harry Fraser. With Tom Tyler, Stanley Blystone, Cecilia Ryland, Francis McDonald, Charles King, Tom London, William Dyer, Arthur Millet, Gordon (DeMain) Wood, Theodore (Ted) Lorch, Earl Dwire, Dick Dickinson, Perry Murdock, Cactus Mack, Barney Beasley, Harry Fraser. A Mounted Policeman, falsely blamed for a murder, tries to prove his innocence and sets out to get the culprit. Tom Tyler is the Mountie but the results are only passable, although Stanley Blystone makes a despicable villain.\n\n**1928** _ **Honor of the Range**_ **** Universal, 1934. 61 min. D: Alan James. SC: Nate Gatzert. With Ken Maynard, Cecilia Parker, Fred Kohler, James Marcus, Frank Hagney, Eddie Barnes, Franklyn Farnum, Irving Bacon, Jack Rockwell, Albert J. Smith, Slim Whitaker, Ben Corbett, Fred McKaye, Wally Wales, Jack Kirk, Hank Bell, Art Mix, Lafe McKee, Bill Patton, Bud McClure, Nelson McDowell, Pascale Perry, Blackjack Ward, Roy Bucko, Buck Bucko, Fred Burns, Jim Corey, Cliff Lyons, Chuck Baldra. A sheriff is on the trail of an outlaw who is actually his look-alike brother. Ken Maynard, who produced this polished series outing, handles the dual roles quite well.\n\n**Poster for** _**Honor of the Range**_ **(Universal, 1934).**\n\n**  \n**\n\n**1929** _ **Honor of the West**_ **** Universal, 1939. 58 min. D: George Waggner. SC: Joseph West (George Waggner). With Bob Baker, Marjorie Bell (Marge Champion), Carleton Young, Jack Kirk, Dick Dickinson, Frank O'Connor, Reed Howes, Glenn Strange, Forrest Taylor, Murdock MacQuarrie, Walter Long, Walter Wills, Oscar Gahan, Arthur Thalasso. A sheriff after rustlers learns his girl's brother is part of the gang. Plenty of action plus some nice songs make his one of the better Bob Baker films.\n\n**1930** _ **Hooded Angels**_ **** Monarch, 2002. 102 min. Color. D-SC: Paul Matthews. With Gary Busey, Steven Bauer, Amanda Donohoe, Chantell Stander, Juliana Venter, David Dukas, Gideon Emery, Jenna Dover, Julie Hartley, Candice Argall, Jennifer Steyn, Anna Katerina, Michelle Bradshaw, Cordell McAueen, Greg Melvill Smith, Ron Smerczak, Daniel Lee, Andre Jacques van der Merwe, Nick Boraine, Lynne White, Dale Cutts, Russel Savadier, Wilson Dunster, Robin Smith, Conner Dowds, Marcel Van Heerden. Bounty hunters form a posse to track a deadly band of beautiful women pillaging the countryside in the post\u2013Civil War era. Fair feminist inclined Western, also called _**Glory Glory**_.\n\n**1931** _ **Hop-A-Long Cassidy**_ **** Paramount, 1935. 63 min. D: Howard Bretherton. SC: Doris Schroeder. With William Boyd, James Ellison, Paula Stone, Kenneth Thomson, Robert Warwick, Charles Middleton, Frank McGlynn, Jr., George Hayes, Jim Mason, Frank Campeau, Ted Adams, Willie Fung, Franklyn Farnum, John Merton, Wally West, Monte Rawlins, Pascale Perry, Sid Jordan. The foreman of the Bar 20 ranch, Hopalong Cassidy, attempts to find out who is behind a series of rustling jobs as his boss tries to keep his water rights. Initial entry in the \"Hopalong Cassidy\" series is a leisurely effort, short on action but very entertaining with a fine performance by George \"Gabby\" Hayes as Uncle Ben. Reissue and TV title: _**Hopalong Cassidy Enters**_.\n\n_**Hopalong Cassidy Enters**_ see _**Hop-A-Long Cassidy**_\n\n**1932** _ **Hopalong Cassidy Returns**_ **** Paramount, 1936. 74 min. D: Nate Watt. SC: Harrison Jacobs. With William Boyd, George Hayes, Gail Sheridan, Evelyn Brent, Stephen Morris (Morris Ankrum), William Janney, Irving Bacon, Grant Richards, John Beck, Ernie Adams, Al St. John, Ray Whitley, Joe Rickson, Claude Smith, Gwynne Shipman, William J. O'Brien, Bill Nestell, Leo J. McMahon, Frank Ellis, Bob Burns, Bud McClure, Jack Montgomery, George Plues, Jim Corey, Hank Bell. After his newspaper editor friend is ambushed and killed, Hopalong Cassidy takes over as sheriff of a town and finds himself at odds with a seductive female saloon owner who wants to control the area because of a nearby gold mine. One of the best of the Cassidy series with a good story, cast, locations and very fine camera work by Archie Stout.\n\n**1933** _ **Hopalong Rides Again**_ **** Paramount, 1937. 67 min. D: Lesley Selander. SC: Norman Houston. With William Boyd, George Hayes, Russell Hayden, William Duncan, Lois Wilde, William (Billy) King, Nora Lane, Harry Worth, John Rutherford, Ernie Adams, Frank Ellis, John Beach, Artie Ortego, Ben Corbett, William J. O'Brien, Blackjack Ward. A rustler uses the guise of a professor hunting for dinosaur bones in order to steal Bar 20 cattle. Action laden \"Hopalong Cassidy\" feature with an exciting scene of a wagon buried by an avalanche.\n\n**1934** _ **Hoppy Serves a Writ**_ **** United Artists, 1943. 69 min. D: George Archainbaud. SC: Gerald Geraghty. With William Boyd, Andy Clyde, Jay Kirby, Victor Jory, George Reeves, Jan Christy, Hal Taliaferro, Forbes Murray, Byron Foulger, Earle Hodgins, Roy Barcroft, Ben Corbett, Robert Mitchum, Art Mix, Steve Clark, Herman Hack, Bob Burns, Cliff Parkinson, Roy Bucko. Hopalong Cassidy comes to Mesa City to stop a murder and gets involved with a gang of rustlers and their boss. Pretty good entry in the Cassidy series featuring a nifty scrap between Hoppy and villain Victor Jory.\n\n**1935** _ **Hoppy's Holiday**_ **** United Artists, 1947. 60 min. D: George Archainbaud. SC: J. Benton Cheney, Bennett Cohen and Ande Lamb. With William Boyd, Andy Clyde, Rand Brooks, Andrew Tombes, Jeff Corey, Mary Ware, Leonard Penn, Donald Kirke, Holly Bane, Gil Patric, Frank Henry, Johnny Luther, Ben Corbett, Jack Evans, Jack Montgomery, Bob Burns, Rube Dalroy, Kansas Moehring, Tex Cooper, Denver Dixon, Roy Bucko, Glen Walters. While the Bar 20 boys are in town for a celebration, California Carlson is accidentally given stolen loot from a robbery and after he is arrested Hopalong Cassidy tries to locate the real criminals. Lethargic series entry containing a neat climax with the bad guys using a horseless carriage for a getaway.\n\n**1936** _ **Horizons West**_ **** Universal-International, 1952. 81 min. Color. D: Budd Boetticher. SC: Louis Stevens. With Robert Ryan, Julia (Julie) Adams, Rock Hudson, John McIntire, Judith Braun, Raymond Burr, James Arness, Frances Bavier, Dennis Weaver, Tom Powers, Rodolfo Acosta, John Hubbard, Douglas Fowley, Walter Reed, Raymond Greenleaf, Tom Monroe, Dan White, John Harmon, Robert Bice, Dan Moore, Mae Clarke, Alberto Morin, Peter Mamakos, Eddie Parker, Monte Montague, Forbes Murray, Buddy Roosevelt, Ewing Mitchell, Frank Chase. Two brothers return from the Civil War with one becoming a sheriff while the other takes to a life of crime. Well acted but so-so oater.\n\n_**A Horse Called Comanche**_ see _**Tonka**_\n\n**1937** _ **The Horse Soldiers**_ **** United Artists, 1959. 119 min. Color. D: John Ford. SC: John Lee Mahin and Martin Rackin. With John Wayne, William Holden, Constance Towers, Althea Gibson, Hoot Gibson, Anna Lee, Russell Simpson, Stan Lee, Carleton Young, Basil Ruysdael, Willis Bouchey, Ken Curtis, O.Z. Whitehead, Judson Pratt, Denver Pyle, Strother Martin, Hank Worden, Walter Reed, Jack Pennick, Fred Graham, Chuck Hayward, Charles Seel, Stuart Holmes, Major Sam Harris, Richard Cutting, Bing Russell, William Leslie, Ron Haggerty, William Forrest, Fred Kennedy, Bill Henry, Dan Borgaze, Jan Stine, William Wellman, Jr., Cliff Lyons. During the Civil War a Union outfit, led by two feuding officers, makes a daring move into the Confederacy to cut communication lines. Big budget production sure to satisfy fans of John Wayne and William Holden; Hoot Gibson has a nice supporting role.\n\n**1938** _ **Horsemen of the Sierras**_ **** Columbia, 1949. 56 min. D: Fred F. Sears. SC: Barry Shipman. With Charles Starrett, Smiley Burnette, Lois Hall, Tommy Ivo, T. Texas Tyler, John Dehner, Jason Robards, Dan Sheridan, Jock (Mahoney) O'Mahoney, George Chesebro, Emile Avery, Ethan Laidlaw, Charles Soldani, Al Wyatt. While trying to find out who murdered a government surveyor, an undercover agent gets involved in a feud between two families. Standard \"Durango Kid\" drama. British title: _**Remember Me**_.\n\n**1939** _ **Hostile Country**_ **** Lippert, 1950. 61 min. D: Thomas Carr. SC: Ron Ormand and Maurice Tombragel. With Jimmie (James) Ellison, Russell Hayden, Fuzzy Knight, Raymond Hatton, Betty (Julie) Adams, Tom Tyler, George J. Lewis, John Cason, Stanley Price, Bud Osborne, Dennis Moore, George Chesebro, Stephen Carr, Jimmie Martin, I. Stanford Jolley, J. Farrell MacDonald, Ray Jones, Cliff Taylor, Judith Webster, George Sowards, James Van Horn, Wally West, Carl Mathews. Shamrock and Lucky arrive in an area for the former to take over half-interest in his stepfather's ranch and they get involved in a range feud. Some interesting camera work by Ernest Miller highlights this better than average first \"Irish Cowboys\" series entry; still the film is not overly interesting although Tom Tyler and John Cason are good as the villainous Brady boys. TV title: _**Outlaw Fury**_.\n\n**1940** _ **Hostile Guns**_ **** Paramount, 1967. 91 min. Color. D: R.G. Springsteen. SC: Steve Fisher and Sloan Nibley. With George Montgomery, Yvonne De Carlo, Tab Hunter, Brian Donlevy, John Russell, Richard Arlen, James Craig, Leo Gordon, Robert Emhardt, Pedro Gonzalez Gonzalez, Emile Meyer, Donald Barry, Fuzzy Knight, William Fawcett, Joe Brown, Reg Parton, Read Morgan, Eric Cody. A sheriff and his deputy lead a prison wagon through hostile country and are stalked by an outlaw gang with a score to settle with the lawman. Mediocre A.C. Lyles production wasting a good cast.\n\n_**The Hot Horse**_ see _**Once Upon a Horse**_\n\n**1941** _ **Hot Lead**_ **** RKO Radio, 1951. 60 min. D: Stuart Gilmore. SC: William Lively. With Tim Holt, Richard Martin, Joan Dixon, Ross Elliott, John Dehner, Stanley Andrews, Robert Wilke, Kenneth MacDonald, Paul Marion, Lee MacGregor, Paul E. Burns. To gain information about gold shipments, an outlaw gang substitutes one of its members in place of the local telegrapher. Average Tim Holt series entry.\n\n_**Hot Lead (1965)**_ see _**Bull of the West**_\n\n**1942** _ **Hot Lead and Cold Feet**_ **** Buena Vista, 1978. 90 min. Color. D: Robert Butler. SC: Joe McEveety, Arthur Alsberg and Don Nelson. With Jim Dale, Karen Valentine, Don Knotts, Jack Elam, Darren McGavin, John Williams, Warren Vanders, Debbie Lytton, Michael Sharrett, Dave Cass, Richard Wright, Don \"Red\" Barry, Jimmy Van Patten, Gregg Palmer, Ed Bakey, John Steadman, Eric Server, Paul Lukather, Hap Lawrence, Robert Rothwell, Dallas McKennon, Stanley Clements, Don Brodie, Warde Donovan, Brad Weston, Art Burke. Three brothers try to win a town's obstacle race, opposing the dishonest activities of its mayor. Fair, but fast moving, Disney comedy oater.\n\n_**Hot Spur**_ see _**Love Desperados**_\n\n**1943** _ **Hour of the Gun**_ **** United Artists, 1967. 101 min. Color. D: John Sturges. SC: Edward Anhalt. With James Garner, Jason Robards, Robert Ryan, Albert Salmi, Charles Aidman, Steve Ihnat, Michael Tolan, Frank Converse, Sam Melville, Austin Willis, Richard Bull, Larry Gates, Karl Swenson, Bill Fletcher, Robert Phillips, Jon Voight, Willliam Schallert, Lonnie Chapman, Monte Markham, William Windom, Edward Anhalt, Walter Gregg, Dave Perna, Jim Sheppard, Jorge Russek. Wyatt Earp and Doc Holliday give chase to Ike Clanton and the other survivors of the famous Tombstone shootout. Mediocre follow-up by director John Sturges to his earlier _**Gunfight at the O.K. Corral**_ (q.v.).\n\n**1944** _ **Houston:**_ _**The Legend of Texas**_ **** Taft Entertainment Group, 1986. 156 min. Color. D: Peter Levin. SC: John Binder. With Sam Elliott, Davon Ericson, Michael Beck, Bo Hopkins, John P. Ryan, G.D. Spradlin, Richard Yniguez, James Stephens, Claudia Christian, Michael G. Gwynne, Donald Moffat, John Quade, Ned Romero, William Russ, Ritch Brinkley, John De Lancie, Peter Gonzales, Robert F. Hoy, Dennis Letts, Katharine Ross, William Schallert (narrator). Sam Houston leaves the United States senate to become the leader of the Texas independence movement. Nicely done TV biopic.\n\n**1945** _ **How the West Was Won**_ **** Metro-Goldwyn-Mayer, 1962. 162 min. Color. D: John Ford, Henry Hathaway and George Marshall. SC: James R. Webb. With John Wayne, James Stewart, Gregory Peck, Carroll Baker, George Peppard, Henry Fonda, Carolyn Jones, Karl Malden, Robert Preston, Debbie Reynolds, Richard Widmark, Eli Wallach, Walter Brennan, Raymond Massey, David Brian, Agnes Moorehead, Harry Morgan, Andy Devine, Russ Tamblyn, Ken Curtis, Lee J. Cobb, Brigid Bazlen, Mickey Shaughnessy, Lee Van Cleef, Karl Swenson, Jack Lambert, Christopher Dark, Jay C. Flippen, Joseph Sawyer, James Griffith, Claude Johnson, Walter Reed, Carleton Young, Rodolfo Acosta, Dean Stanton, Kim Charney, Bing Russell, Gene Roth, Clinton Sundberg, Walter Burke, John Larch, Edward J. McKinney, Barry Harvey, Jamie Ross, Mark Allen, Craig Duncan, Charles Briggs, Paul Bryar, Tudor Owen, Chuck Roberson, Boyd \"Red\" Morgan, Beulah Archuletta, Spencer Tracy (narrator). The saga of western migration from the late 1830s to 1889 as seen through the eyes of three generations of pioneers. Vast, sprawling story of the development of the American West that is well acted by a big cast; the best sequence is probably director Henry Hathaway's \"The Rivers\" with Walter Brennan stealing the show as a vicious river pirate.\n\n**1946** _ **How the West Was Won**_ **** ABC-TV\/Metro-Goldwyn-Mayer, 1977. 300 min. Color. D: Burt Kennedy and Daniel Mann. SC: Jim Byrnes and William Kelley. With James Arness, Eva Marie Saint, Bruce Boxleitner, Anthony Zerbe, Don Murray, Brit Lind, William Kirby Cullen, Kathryn Holcomb, Vicki Schreck, Royal Dano, John Dehner, Jack Elam, David Huddleston, Robert Padilla, Richard Angarola, Bridget Hanley, Parley Baer, Paul Fix, William Conrad (narrator). The adventures of Eastern homesteaders and their mountain man uncle as they try to settle in the West after the death of the family's father. This fine telefeature was originally shown in three parts and was the follow-up to the popular TV feature _**The Macahans**_ (q.v.).\n\n**1947** _ **The Howards of Virginia**_ **** Columbia, 1940. 117 min. D: Frank Lloyd. SC: Sidney Bochman. With Cary Grant, Martha Scott, Sir Cedric Hardwicke, Alan Marshal, Richard Carlson, Paul Kelly, Irving Bacon, Elisabeth Risdon, Anne Revere, Tom Drake, Phil Taylor, Rita Quigley, Libby Taylor, Richard Gaines, George Houston, Sam McDaniel, Virginia Sale, Ralph Byrd, Dickie Jones, Buster Phelps, Wade Boteler, Mary Field, R. Wells Gordon, Charles Francis, Olaf Hytten, Emmett Vogan, J. Anthony Hughes, Lane Chandler, Brandon Hurst, Alan Ladd, Pat Somerset, James Westerfield. A young man marries into a wealthy Virginia family and against his wife's wishes joins the colonial cause during the Revolutionary War. Overlong but authentic looking historical drama with George Houston as George Washington.\n\n**1948** _ **Hud**_ **** Paramount, 1963. 112 min. D: Martin Ritt. SC: Irving Ravetch and Harriet Frank, Jr. With Paul Newman, Melvyn Douglas, Patricia Neal, Brandon De Wilde, Whit Bissell, John Ashley, Crahan Denton, Val Avery, Sheldon Allman, Pitt Herbert, Peter Brooks, Curt Conway, Yvette Vickers, George Petrie, David Kent, Montie Montana, Carl Saxe, Sharyn Hillyer. A teenager is torn between the love of his grandfather and admiration for the man's rebellious son. Stark, well acted modern Western based on Larry McMurtry's novel _Horseman, Pass By_.\n\n**1949** _ **Hudson's Bay**_ **** 20th Century\u2013Fox, 1941. 95 min. D: Irving Pichel. SC: Lamar Trotti. With Paul Muni, Gene Tierney, Laird Cregar, John Sutton, Virginia Field, Vincent Price, Nigel Bruce, Montagu Love, Morton Lowry, Robert Greig, Chief Thundercloud, Fredric Worlock, Ian Wolfe, Chief Big Tree, Jody Gilbert, Jean Del Val, Eugene Borden, Constant Franke, John Rogers, Reginald Sheffield, Dorothy Dearing, Florence Bates, Lumsden Hare, Boyd Irwin, Denis Green, Eily Malyon, Lionel Pape. Banished to Canada, an Englishman joins forces with two French trappers to form the Hudson's Bay Company for the export of furs. Another in producer Darryl F. Zanuck's historical film series, one of the weakest and least accurate.\n\n**1950** _ **Human Targets**_ **** Big 4, 1932. 55 min. D: J.P. McGowan. SC: George Morgan. With Rin-Tin-Tin, Jr., Buzz Barton, Francis X. Bushman, Jr., Nancy Price, Tom London, Edmund Cobb, Ted Adams, Leon Kent, John Ince, Edgar Lewis, Pauline Parker, Helen Gibson, Franklyn Farnum, Fred \"Snowflake\" Toones. A young boy, a dog and a cowboy fight for a gold claim against crooks. Surprisingly well done production from poverty row.\n\n_**Hunt the Man Down**_ see _**Bad Man's River**_\n\n_**Hunt to Kill**_ see _**The White Buffalo**_\n\n**1951** _ **The Hunting Party**_ **** United Artists, 1971. 102 min. Color. D: Don Medford. SC: William Norton, Gilbert Alexander and Lou Morheim. With Oliver Reed, Candice Bergen, Gene Hackman, Simon Oakland, Ronald Howard, Mitchell Ryan, L.Q. Jones, G.D. Spradlin, Bernard Kay, William Watson, Rayford Barnes, Ralph Brown, Marian Collier, Max Slaten, Carlos Bravo, Emilio Rodrigues, Deal Selmier, Ritchie Adams, Eugenio Escudero. An outlaw and his gang kidnap a woman thinking she is a school teacher who can teach them to read, but she turns out to be the wife of a land baron who comes after them with his men. A thin story and too much violence make this poor viewing.\n\n**1952** _ **Hurricane Horseman**_ **** Willis Kent, 1931. 50 min. D: Armand L. Schaefer. SC: Oliver Drake. With Lane Chandler, Marie Quillan, Walter Miller, Yakima Canutt, Lafe McKee, Richard Alexander, Slim Whitaker, Jack Kirk, Chuck Baldra, Pascale Perry, Blackjack Ward, Hank Bell, Bill Wolfe. A gunsmith comes to the aid of a pretty senorita held for ransom by an outlaw gang. Mediocre Lane Chandler vehicle for producer Willis Kent.\n\n**1953** _ **Hurricane Smith**_ **** Republic, 1941. 68 min. D: Bernard Vorhaus. SC: Robert Presnell. With Ray Middleton, Jane Wyatt, Harry Davenport, J. Edward Bromberg, Henry Brandon, Charles Trowbridge, Frank Darien, Howard Hickman, Emmett Vogan, Casey Johnson. When he is falsely accused of theft, a cowboy escapes, assumes a new identity and marries only to find that someone from the past has recognized him. Republic's one try to make Ray Middleton a Western star was not successful due to a paltry script rather than the presence of Middleton, a fine actor and singer. TV title: _**Double Identity**_.\n\n**1954** _ **I Am Sartana, Trade Your Guns for a Coffin**_ **** Colt Produzioni Cinematografica, 1970. 93 min. Color. D: Anthony Ascott (Giuliano Carmineo). SC: Tito Carpi. With George Hilton, Charles Southwood, Erika Blanc, Peter Carter (Piero Lulli), Linda Sini, Nelio Pazzafini, Carlo Gaddi, Aldo Barberito, Marco Zuanelli, Lou Kamante (Luciano Rossi), John Bartha, Furio Meniconi, Franco Fantasia, Rick Boyd (Federico Boido), Gigi Bonos, Fortunato Arena, Armando Calvo, Umberto Di Grazia, Gaetano Imbro. Gunman Sartana his hired to protect gold shipments and ends up in a showdown with an outlaw gang leader called Sabbath. Average Italian Western issued there as _**C'e Sartana...Vendi la Pistola e Comprati la Bara**_ (Here's Sartana...Trade Your Pistol for a Coffin).\n\n**1955** _ **I Am Sartana, Your Angel of Death**_ **** Societa Ambrosiana Cinematografica, 1969. 102 min. Color. D: Anthony Ascott (Giuliano Carmineo). SC: Tito Carpi, Enzo Dell'Aquila and Ernesto Gastaldi. With John (Gianni) Garko, Frank Wolff, Ettore Manni, Klaus Kinski, Salvatore (Sal) Borghese, Renato Baldini, Jose Torres, Gordon Mitchell, Rick Boyd (Federico Boido), Tullio Altamura, Brunco Boschetti, Giovanni Petrucci, Lorenzo Piani, Samson (Sammy) Burke, John Bartha, Franco Pesce, Jean Louis, Celso Farina, Ermelinda De Felice, Giuseppe Mattei, Roberto Messina, Tchang Yu. Accused of a bank robbery, Sartana attempts to find the culprits to clear his name and is helped by a friend, but is hunted by bounty seekers. Fast moving, episodic Italian Western, the fifth in the popular \"Sartana\" series, made as _**Sono Sartana, il Vostro Becchino**_ (Sartana, Your Gravedigger); video title: _**Sartana the Gravedigger**_.\n\n**1956** _ **I Killed Geronimo**_ **** Eagle Lion, 1950. 63 min. D: John Hoffman. SC: Sam Neuman and Nat Tanchuck. With James Ellison, Virginia Herrick, Chief Thundercloud, Smith Ballew, Dennis Moore, Ted Adams, Myron Healey, Luther Crockett, Jean Andren, Forrest Taylor, Jack Kenney, Wesley Hudman, Sam Wolfe, Joseph C. Green, Harte Wayne. An Army captain tries to track outlaws supplying arms to the Indians and ends up in hand-to-hand combat with Geronimo. Cheap feature with little entertainment value.\n\n**1957** _ **I Killed Wild Bill Hickok**_ **** Associated Artists\/Wheeler Company, 1956. 63 min. D: Richard Talmadge. SC: John Carpenter. With John (Carpenter) Forbes, Helen Westcott, Tom Brown, I. Stanford Jolley, Denver Pyle, Frank Carpenter, Virginia Gibson. A horse dealer arrives in town with his daughter to do business with the Army but when a gunfight takes place the girl is killed and he seeks revenge by gunning for Wild Bill Hickok. Tiny budget affair from producer-writer-star John Carpenter that seeks to prove legendary Wild Bill Hickok was really a bad guy who deserved his fate.\n\n_**I Live for Your Death**_ see _**A Long Ride from Hell**_\n\n**1958** _ **I Married Wyatt Earp**_ **** NBC-TV, 1983. 100 min. Color. D: Michael O'Herlihy. SC: I.C. Rappaport. With Marie Osmond, Bruce Boxleitner, John Bennett Perry, Jeffrey DeMunn, Allison Arngrim, Ross Martin, Ron Manning, Joe Rainer, Dee Maaske, Earl Smith, Randy Wells, Joe Corcoran, Charles Benton, Tom Assalone, Elayne Stein, Donna Brown, Linda Jergens, Ron Chapman, Claud Hereford, Kirk Kostella, Joseph Bottoms. A young woman comes to Tombstone, wins the love of marshal Wyatt Earp and sees him through the famed O.K. Corral shootout. Based on the memoirs of Josephine Marcus Earp, this TV film is another recounting of the famed gun battle; average.\n\n**1959** _ **I Shot Billy the Kid**_ **** Lippert, 1950. 58 min. D: William Berke. SC: Ford Beebe and Orville Hampton. With Don Barry, Robert Lowery, Wally Vernon, Tom Neal, Judith Allen, Wendy Lee, Barbara Woodell, Richard Lane, Sid Nelson, Archie Twitchell, John Merton, Bill Kennedy, Jack Perrin, Frank Ellis, Carol Henry, Tom Monroe. The story of Billy the Kid, from his first crime to the final showdown with Sheriff Pat Garrett. Don Barry is very good in the title role of this low budget but entertaining outing.\n\n**1960** _ **I Shot Jesse James**_ **** Lippert\/Screen Guild, 1949. 83 min. D-SC: Samuel Fuller. With John Ireland, Preston Foster, Barbara Britton, J. Edward Bromberg, Victor Kilian, Barbara Woodell, Tom Tyler, Reed Hadley, Tommy Noonan, Byron Foulger, Eddie Dunn, Margia Dean, Chuck Roberson, Stanley Price, Gene Collins. Bob Ford gains fame for killing Jesse James but his life declines after the incident, even to losing the girl he loves. Fairly interesting \"B\" picture with a fine performance by Tom Tyler as Frank James.\n\n**1961** _ **I Take This Woman**_ **** Paramount, 1931. 76 min. D: Marion Gering. SC: Vincent Lawrence. With Gary Cooper, Carole Lombard, Helen Ware, Lester Vail, Charles Trowbridge, Clara Blandick, Gerald Fielding, Albert Hart, Guy Oliver, Syd Saylor, Mildred Van Horn, Leslie Palmer, Ara Haswell, Frank Darien, Lew Kelly, David Landau. Sent to a Wyoming ranch to avoid being involved in a divorce scandal, a young woman meets and marries a cowboy and is disinherited by her wealthy father. Fair screen adaptation of the Mary Roberts Rinehart story \"Lost Ecstasy.\"\n\n**1962** _ **I Will Fight No More Forever**_ **** ABC-TV, 1975. 74 min. Color. D: Richard T. Heffron. SC: Jed Rosebrook and Theodore Strauss. With James Whitmore, Ned Romero, Sam Elliott, John Kauffman, Emilio Delgado, Nick Ramus, Linda Redfearn, Frank Salsedo, Vincent St. Cyr, Delro White. In 1877 Chief Joseph of the Nez Perces tribe refuses to be sent to a reservation and tries to lead his people to Canada, but is opposed by the U.S. Army. Well done TV movie with appeal to history buffs.\n\n**1963** _ **Ice Palace**_ **** Warner Bros., 1960. 143 min. Color. D: Vincent Sherman. SC: Harry Kleiner. With Richard Burton, Robert Ryan, Carolyn Jones, Martha Hyer, Jim Backus, Ray Danton, Diane McBain, Karl Swenson, Shirley Knight, Barry Kelley, Sheridan Comerate, George Takei, Steve Harris, Sheila Bromley, Sam McDaniel, I. Stanford Jolley, John Bleifer, Judd Holdren, Norma French, Sol Gorss, J. John Launer, Clarence Straight, Robert Griffin, Sal Ponti, Alan Roberts, Carol Nicholson, Maurice Wells, William Yip, Robert \"Buddy\" Shaw. Two men battle over two women and the issue of Alaskan statehood during the course of several decades. There is not much to recommend this overlong soap opera except for some fairly good performances.\n\n**1964** _ **Idaho**_ **** Republic, 1943. 70 min. D: Joseph Kane. SC: Roy Chanslor and Olive Cooper. With Roy Rogers, Smiley Burnette, Bob Nolan and The Sons of the Pioneers (Tim Spencer, Lloyd Perryman, Pat Brady, Hugh Farr, Karl Farr), Virginia Grey, Harry Shannon, Ona Munson, Dick Purcell, Onslow Stevens, Arthur Hohl, Hal Taliaferro, Tristram Coffin, Roy Barcroft, Jack Mulhall, Tom London, Rex Lease, Jack Ingram, James Bush, Forrest Taylor, Jack Kirk, Fred Burns, The Robert Mitchell Boychoir. A judge, who was once an outlaw, is blackmailed by two hoodlums and a female gambling house owner when he refuses to help them rob a bank. Exciting Roy Rogers feature.\n\n**1965** _ **The Idaho Kid**_ **** Colonly, 1936. 54 min. D: Robert F. Hill. SC: George Plympton. With Rex Bell, Marion Shilling, David Sharpe, Earl Dwire, Lafe McKee, Lane Chandler, Charles King, Phil Dunham, Dorothy Woods, Herman Hack, Ed Cassidy, George Morrell, Jimmy Aubrey, Sherry Tansey, Richard Botiller, William McCall, Jack Evans, Buck Morgan. A cowboy returns home to stop a long time feud between his father and the man who raised him. Pleasant Paul Malvern production highlighted by Rex Bell's fine performance in the title role.\n\n**1966** _ **If You Meet Sartana, Pray for Your Death**_ **** Paris Etolie Films\/Parnass Film, 1968. 95 min. Color. D: Frank Kramer (Gianfranco Parolini). SC: Gianfranco Parolini, Werner Hauff and Renato Izzo. With John (Gianni) Garko, Klaus Kinski, Fernando Sancho, William Berger, Sydney Chaplin, Gianni Rizzo, Andrew Scott (Andrea Scotti), Carlo Tamberlani, Franco Pesce, Heidi Fischer, Maria Pia Conte, Sabine Sun, Gianfranco Parolino, Sergio Jossa, Patricia Carr, Arrigo Peri, Antonietta Florita, Ugo Adinolfi, Sal Borghese, Gilberto Galimberti. Sartana searches for a strong box filled with money and the vicious outlaws who massacred stagecoach passengers to get the loot. Pretty good first entry in the popular \"Sartana\" series about a frontier avenger; a French-Italian-West German co-production filmed as _**Se Incontri Sartana Prega per la Tua Morte**_ (If You Meet Sartana Pray for Death) and also called _**Sartana**_ and _**Gunfighters Die Harder**_.\n\n**1967** _ **I'm from the City**_ **** RKO Radio, 1938. 66 min. D: Ben Holmes. SC: Nicholas T. Barrows, Robert St. Clair and John Grey. With Joe Penner, Lorraine Kruger, Richard Lane, Paul Guilfoyle, Kay Sutton, Ethan Laidlaw, Lafe McKee, Edmund Cobb, Kathryn Sheldon, Willie Best, Chris-Pin Martin, Clyde Kinney. A circus performer, when hypnotized by his oily manager, becomes a famous trick rider. Typical Joe Penner comedy and one that will satisfy his fans.\n\n**1968** _ **In a Colt's Shadow**_ **** Warner Bros., 1967. 86 min. Color. D: Giovanni Grimaldi. SC: Giovanni Grimaldi and Aldo Barni. With Stephen Forsyte, Conrado San Martin, Anne Sherman, Helga Line, Andrew Scott (Andrea Scotti), Frank Ressel, Aldo Sambrell, Jose Calvo, Graham Sooty. Two gunmen, who work together, become alienated when the younger one falls in love with the other's daughter. Lots of violence and action in this Italian-Spanish co-production made in Italy in 1965 as _**All'Ombra di una Colt**_ (In the Shadow of a Colt). TV title: _**In the Shadow of a Colt**_.\n\n**1969** _ **In Early Arizona**_ **** Columbia, 1938. 53 min. D: Joseph Levering. SC: Nate Gatzert. With Bill Elliott, Dorothy Gulliver, Harry Woods, Art Davis, Jack Ingram, Franklyn Farnum, Charles King, Ed Cassidy, Slim Whitaker, Frank Ellis, Al Ferguson, Bud Osborne, Lester Dorr, Tom London, Kit Guard, Jack O'Shea, Frank Ball, Tex Palmer, Sherry Tansey, Dick Dorrell, Oscar Gahan, Buzz Barton, Jess Cavin, Symona Boniface, Chick Hannon, Bob Card, Cliff Lyons. A peaceful man takes up a gun and badge to clean up a town controlled by outlaws. Bill Elliott's first starring series oater is a corker, with plenty of action, a good script and a great roundup of genre bad guys.\n\n**1970** _ **In Line of Duty**_ **** Monogram, 1931. 60 min. D: Bert Glennon. SC: G.A. Durlam. With Sue Carol, Noah Beery, Francis McDonald, James Murray, Richard Cramer, Frank Seider, Henry Hall. A Mountie tries to bring in a man on a murder charge but learns the fugitive is his girl's brother. North woods quickie mainly of interest because of James Murray, a tragic silent star whose career tanked in the sound era.\n\n**1971** _ **In Old Amarillo**_ **** Republic, 1951. 67 min. D: William Witney. SC: Sloan Nibley. With Roy Rogers, Estelita Rodriguez, Penny Edwards, Pinky Lee, Roy Barcroft, Pierre Watkin, Ken Howell, Elizabeth Risdon, William Holmes, Alan Bridge, Kermit Maynard, The Roy Rogers Riders, Larry J. Blake, Lee Shumway, Archie Twitchell, Frank O'Connor, Ethan Laidlaw, Jack O'Shea, Angela Stevens, Brick Sullivan, Tom Steele, Bob Burns, Al Haskell, Bert Dillard, Paul Livermore, Mike Lally, Ralph Bucko, Frank Dae. When families are hit hard by drought, a cowboy wants to bring in a scientific rainmaker but his opposed by a crook who plans to buy the ranchers' cattle cheap to start a meat packing plant. One of the lesser efforts by Roy Rogers from the latter part of his Republic series.\n\n**1972** _ **In Old Arizona**_ **** Fox, 1929. 95 min. D: Raoul Walsh and Irving Cummings. SC: Tom Barry. With Warner Baxter, Edmund Lowe, Dorothy Burgess, J. Farrell MacDonald, Soledad Jiminez, Fred Warren, Henry Armetta, Tom Santschi, Frank Campeau, Pat Hartigan, Roy Stewart, James Bradbury, Jr., John Webb Dillon, Frank Nelson, Duke Martin, James Marcus, Joe Brown, Alphonse Ethier, Helen Lynch, Ed Peil, Sr., Jim Farley, Ivan Linow, Lola Salvi, Chris-Pin Martin. The Cisco Kid, a notorious bandit, loves a beautiful, but two timing, woman who plans to betray him to a lawman for reward money. Although a bit creaky today, this landmark production demonstrated that Westerns could adapt to sound and it also contains Warner Baxter's Academy Award winning performance as The Cisco Kid; it was the first sound Western to have a theme song, \"My Tonia,\" popularized on record by Nick Lucas.\n\n**1973** _ **In Old Caliente**_ **** Republic, 1939. 57 min. D: Joseph Kane. SC: Norman Houston and Gerald Geraghty. With Roy Rogers, Mary Hart, George \"Gabby\" Hayes, Jack LaRue, Katherine De Mille, Frank Puglia, Harry Woods, Merrill McCormick, Paul Marion, Ethel Wales, Bill Nestell, Al Taylor, Fred Burns, Jim Corey, Blackie Whiteford, Tom Smith. After being falsely accused of betraying his employer, a wealthy Spanish landowner, a cowboy joins a wagon train and soon discovers who is trying to rustle all his ex-boss' cattle. Better than average Roy Rogers vehicle enhanced by attractive seaside locations.\n\n**1974** _ **In Old California**_ **** Republic, 1942. 88 min. D: William McGann. SC: Gertrude Purcell and Frances Hyland. With John Wayne, Binnie Barnes, Albert Dekker, Helen Parrish, Patsy Kelly, Edgar Kennedy, Dick Purcell, Harry Shannon, Charles Halton, Emmett Lynn, Bob McKenzie, Milton Kibbee, Paul Sutton, Anne O'Neal, Frank McGlynn, Hooper Atchley, Jack O'Shea, Ruth Robinson, Frank Jaquet, Jack Kirk, Lynne Carver, Horace B. Carpenter, James Morton, Olin Howlin, Chester Conklin, Ralph Peters, Forrest Taylor, Richard Alexander, Donald Curtis, George Lloyd, Stanley Blystone, Slim Whitaker, Frank Ellis, Frank Hagney, Bud Osborne, Guy Usher, Minerva Urecal, Martin Garralaga, Rex Lease, Karl Hackett, Art Mix, Robert Homans, Merrill McCormick, Ed Brady, Bob Woodward, Harry McKim, Emily LaRue, Esther Estrella, Michael Miller, Lew Kelly, Bob Reeves, Cecil Weston, Fred Walburn, Matt Willis, Blackie Whiteford, Chick Hannon, Art Dillard, Frank Brownlee, Harry Tenbrook, Jessie Arnold, Zeke Canova, Joe McGuinn, Wade Crosby, Fern Emmett, Carl Miller, Jim Farley, Jack Carr, Max Waizmann, Harry Tyler, Dorothy Granger, Tom Quinn, Frank Mills, Jim Corey, Martin Faust, Frank O'Connor, Charles Murphy, Pearl Early, Sam Bernard, Heenan Elliott. A pharmacist arrives in Sacramento to set up practice and ends up at odds with the town boss who is jealous of his saloon singer girlfriend. Rather tame John Wayne outing, not one of his better features, but still fast paced with a grand cast.\n\n**1975** _ **In Old Cheyenne**_ **** Sono Art-World Wide, 1931. 60 min. D: Stuart Paton. SC: Betty Burbridge. With Rex Lease, Dorothy Gulliver, Jay Hunt, Harry Woods, Harry Todd, Slim Whitaker, Pete Morrison, Pee Wee Holmes, Ben Corbett, Blackie Whiteford, Hank Bell. A cowboy defends a beautiful horse accused of rustling actually carried out by a dishonest ranch foreman. Cheap Rex Lease vehicle with a worn out plot.\n\n**1976** _ **In Old Cheyenne**_ **** Republic, 1941. 58 min. D: Joseph Kane. SC: Olive Cooper. With Roy Rogers, George \"Gabby\" Hayes, Joan Woodbury, J. Farrell MacDonald, Sally Payne, William Haade, Hal Taliaferro, Billy Benedict, George Rosenor, Jack Kirk, Bob Woodward, Jim Corey, Cactus Mack, George Lloyd, Jack O'Shea, Ed Peil, Sr., Merrill McCormick, Ted Mapes, Fred Burns, Ben Corbett, Frank Ellis. The _New York Inquirer_ sends reporter Roy Rogers to Wyoming to cover a range war between cattlemen and an outlaw gang leader. A good Roy Rogers outing that has him become engaged to Spanish dancer Joan Woodbury at the finale.\n\n**Rex Lease in** _**In Old Cheyenne**_ **(Sono Art** **\u2013** **World Wide, 1931).**\n\n**  \n**\n\n**1977** _ **In Old Colorado**_ **** Paramount, 1941. 66 min. D: Howard Bretherton. SC: Norton S. Parker and J. Benton Cheney. With William Boyd, Russell Hayden, Andy Clyde, Margaret Hayes, Morris Ankrum, Sarah Padden, Cliff Nazarro, Stanley Andrews, James Seay, Morgan Wallace, Weldon Heyburn, Glenn Strange, Eddy Waller, Philip Van Zandt, Henry Wills, Curley Dresden, John Beach, Wen Wright, Ted Wells, Bill Nestell, Denver Dixon. Hopalong Cassidy is sent to buy cattle from a woman rancher who is feuding with a fellow landowner and he soon realizes a crook is after both their spreads. Good Cassidy production with a fine cast, scenic locations and an exciting story.\n\n_**In Old Los Angeles**_ see _**Old Los Angeles**_\n\n**1978** _ **In Old Mexico**_ **** Paramount, 1938. 67 min. D: Edward D. Venturini. SC: Harrison Jacobs. With William Boyd, George Hayes, Russell Hayden, Paul Sutton, Allan Garcia, Jane (Jan) Clayton, Trevor Bardette, Betty Amann, Glenn Strange, Anna Demetrio, Tony Roux, Fred Burns, Cliff Parkinson. In New Mexico, Hoppy, Windy and Lucky are after \"The Fox,\" the mysterious leader of a band of rustlers. Leisurely paced \"Hopalong Cassidy\" outing with nice scenery and good work by Paul Sutton as the villain.\n\n**1979** _ **In Old Montana**_ **** Spectrum, 1939. 60 min. D: Raymond K. Johnson. SC: Jackson Parks, Homer King Gordon, Raymond K. Johnson and Barney Hutchison. With Fred Scott, Jean Carmen, John Merton, Harry Harvey, Walter McGrail, Wheeler Oakman, Gene Howard, Frank LaRue, Allan Cavan, Jane Keckley, Richard Cramer, James Kelly, Carl Mathews. A medicine show entertainer stops to visit his dad and finds the area in a range war between sheepherders and cattlemen. Passably good Fred Scott vehicle with several nice songs to help it along.\n\n**1980** _ **In Old Monterey**_ **** Republic, 1939. 73 min. D: Joseph Kane. SC: Gerald Geraghty and Dorrell McGowan and Stuart McGowan. With Gene Autry, Smiley Burnette, George \"Gabby\" Hayes, June Storey, The Hoosier Hot Shots (Charles \"Gabe\" Ward, Paul \"Hezzie\" Triesch, Ken Triesch, Frank Kettering), Sarie and Sally, The Ranch Boys (Curley Bradley, Ken Carson, Jack Ross), Stuart Hamblen, Billy Lee, Jonathan Hole, Robert Warwick, William Hall, Eddy Conrad, Curley Dresden, Victor Cox, Robert Wilke, Hal Price, Tom Steele, Jack O'Shea, Rex Lease, Edward Earle, Jim Mason, Fred Burns, Dan White, Frank Ellis, Jim Corey, I. Stanford Jolley, Shorty Carlson. An Army sergeant is sent West to convince ranchers and townspeople to support the military's request for land to be used for bomb maneuver practice. Patriotic affair with Gene Autry in uniform; fairly good entertainment.\n\n**1981** _ **In Old New Mexico**_ **** Monogram, 1945. 62 min. D: Phil Rosen. SC: Betty Burbridge. With Duncan Renaldo, Martin Garralaga, Gwen Kenyon, Norman Willis, Lee \"Lasses\" White, Pedro de Cordoba, Frank Jaquet, Bud Osborne, Artie Ortego, Edward Earle, James Farley, Aurora Roche, Donna Dax, John (Laurenz) Lawrence, Richard Gordon, Carr-Bert Dancers, Ken Terrell, Harry Depp, The Jesters. The Cisco Kid and pal Pancho come to the aid of a nurse accused of murder. Pleasant \"Cisco Kid\" outing with Cisco and Pancho re-dubbed as Chico and Pablo for TV prints. Also called _**The Cisco Kid in In Old New Mexico**_.\n\n_**In Old Oklahoma**_ see _**War of the Wildcats**_\n\n**1982** _ **In Old Sacramento**_ **** Republic, 1946. 89 min. D: Joseph Kane. SC: Frances Hyland and Frank Gruber. With William Elliott, Constance Moore, Hank Daniels, Ruth Donnelly, Eugene Pallette, Lionel Stander, Jack LaRue, Grant Withers, Bobby Blake, Charles Judels, Paul Hurst, Victoria Horne, Dick Wessel, Hal Taliaferro, Jack O'Shea, Marshall Reed, Eddy Waller, William Haade, Boyd Irwin, Lucien Littlefield, Ethel Wales, Kenne Duncan, William B. Davidson, Ellen Corby, Fred Burns, Elaine Lange, H.T. Tsiang, Wade Crosby. A gambler hunted by vigilantes sets out to clean up the lawless element in Sacramento. Bill Elliott's first \"A\" feature is a good one; reissued as _**Flame of Sacramento**_.\n\n**1983** _ **In Old Santa Fe**_ **** Mascot, 1934. 64 min. D: David Howard and (uncredited Joseph Kane). SC: Colbert Clark. With Ken Maynard, Evelyn Knapp, H.B. Warner, Kenneth Thomson, George Hayes, Gene Autry, Lester \"Smiley\" Burnette, Frankie Marvin, Wheeler Oakman, George Chesebro, Jack Rockwell, Jim Corey, Jack Kirk, Edward Hearn, Frank Ellis, Horace B. Carpenter, George Burton, Stanley Blystone, Art Dillard, Charles Brinley, William McCall, Wally West. A racing cowboy loses his steed in a crooked dude ranch event and gangsters after the place frame him on a murder charge. Well made and exciting Ken Maynard vehicle with a good story, plenty of action and a polished production; Gene Autry is impressive in his screen debut singing two songs with Ken Maynard's vocals dubbed by Bob Nolan.\n\n**1984** _ **In the Days of the Thundering Herd**_ **** Selig, 1914. 41 min. D: Colin Campbell. SC: Gordon Willets. With Tom Mix, Bessie Eyton, Princess Red Wing, Wheeler Oakman, John Bowers, Major Gordon Lillie (Pawnee Bill), Sally Madison. When a cowboy and his sweetheart are captured by Indians they are forced to fight off the tribe in order to gain freedom. Fast paced Tom Mix silent feature, one that will please his fans although it lacks the finesse of his later Fox efforts.\n\n_**In the Shadow of a Colt**_ see _**In a Colt's Shadow**_\n\n_**In the Valley of Death**_ see _**Winnetou and Shatterhand in the Valley of Death**_\n\n**1985** _ **Incident at Phantom Hill**_ **** Universal, 1966. 88 min. Color. D: Earl Bellamy. SC: Frank Nugent. With Robert Fuller, Dan Duryea, Jocelyn Lane, Tom Simcox, Linden Chiles, Claude Akins, Noah Beery (Jr.), Paul Fix, Denver Pyle, William Phipps, Don Collier, Mickey Finn, Harlan Warde, Lila Finn, Mimi Doyle, Max Mellinger, Lia Waggner, Frank Leo. Two men and a young woman trek across the desert in search of hidden gold, battling the elements, hostile Indians and themselves. Pretty good action thriller with an impressive performance by Dan Duryea.\n\n**1986** _ **The Incredible Rocky Mountain Race**_ **** NBC-TV\/Sunn Classics, 1977. 100 min. Color. D: James L. Conway. SC: Tom Chapman and David O'Malley. With Christopher Connelly, Forrest Tucker, Larry Storch, Jack Kruschen, Mike Mazurki, Parley Baer, Whit Bissell, Bill Zuckert, Don Haggerty, Sam Edwards, Sandy Gibson, William Kazele, John Hansen, Robert Easton, David O'Malley, Allen Wood. Long time rivals Mark Twain and Mike Fink engage in a cross county, no holds barred, race from Missouri to California. Satisfying TV Western spoof.\n\n**1987** _ **Independence**_ **** Sunn Classics, 1987. 104 min. Color. D: John Patterson. SC: Gordon Dawson. With John Bennett Perry, Isabella Hofman, Anthony Zerbe, Sandy McPeak, R.G. Armstrong, Macon McCalam, Amanda Wyss, Julius J. Curry III, Davin Hollscher, Christian Clemenson, Adam Gregor, Joseph Brutsman, Joshua Julian, Ola David Verploegh, John Davis Chandler, Gisli Bjorgvinsson, Paul Brinegar, Tommy Bush, Curtis Conoway, Gordon Dawson, Doug Duran, Cory Eubanks, Ben Zellar, Stephanie Dunnam, J. Michael Flynn, Jose Garcia, Scott Jones, Adam Taylor, Bruce Watson. Several years after a gang murders his family, a man plans revenge when the marauders return to molest his second wife and children. Well done television Western.\n\n**1988** _ **Indian Agent**_ **** RKO Radio, 1948. 65 min. D: Lesley Selander. SC: Norman Houston. With Tim Holt, Noah Beery, Jr., Richard Martin, Nan Leslie, Lee \"Lasses\" White, Richard Powers (Tom Keene), Harry Woods, Claudia Drake, Robert Bray, Bud Osborne, Iron Eyes Cody. A cowboy uncovers evidence that a government agent as been selling food intended for an Indian reservation, causing the tribe to nearly starve. Good Tim Holt series entry.\n\n**1989** _ **The Indian Fighter**_ **** United Artists, 1955. 88 min. Color. D: Andre De Toth. SC: Frank Davis and Ben Hecht. With Kirk Douglas, Elsa Martinelli, Walter Matthau, Walter Abel, Diana Douglas, Eduard Franz, Alan Hale (Jr.), Lon Chaney, Elisha Cook, Michael Winkelman, Harry Landers, William Phipps, Buzz Henry, Ray Teal, Frank Cady, Hank Worden, Lane Chandler. The leader of a wagon train negotiates a treaty with an Indian chief to let his people pass peacefully through to Oregon but two bad guys soon have the tribe on the warpath. Colorful and action filled, but slight on plot.\n\n**Kirk Douglas and Elsa Martinelli in** _**The Indian Fighter**_ **(United Artists, 1955).**\n\n**  \n**\n\n_**Indian Love Call**_ see _**Rose Marie**_ (1936)\n\n**1990** _ **Indian Paint**_ **** Eagle-American Films\/Crown-International, 1965. 91 min. Color. D-SC: Norman Foster. With Johnny Crawford, Jay Silverheels, Pat Hogan, Robert Crawford, Jr., Robert Crawford, Sr., George J. Lewis, Joan Hollmark. A young Indian boys tries to tame a beautiful wild horse to keep him from joining a wild herd. Okay juvenile oriented feature.\n\nIndian Scout see Davy Crockett, Indian Scout\n\n**1991** _ **Indian Territory**_ **** Columbia, 1950. 70 min. D: John English. SC: Norman S. Hall. With Gene Autry, Pat Buttram, Gail Davis, Kirby Grant, James Griffith, Philip Van Zandt, Pat Collins, Roy Gordon, Charles Stevens, Robert Carson, Chief Thundercloud, Chief Yowlachie, Frank Lackteen, Boyd Stockman, Sandy Sanders, Frank Ellis, Frankie Marvin, Kenne Duncan, Wes Hudman, Roy Butler, Robert Hilton, John R. McKee, Bert Dodson, Chief Thundersky. A Union Army undercover agent finds out a foreign mercenary is stirring up the Indians and he tries to stop him. Entertaining Gene Autry vehicle filmed in Sepiatone.\n\n**1992** _ **Indian Uprising**_ **** Columbia, 1952. 74 min. Color. D: Ray Nazarro. SC: Kenneth Gamet and Richard Schayer. With George Montgomery, Audrey Long, Carl Benton Reid, Eugene Iglesias, Joe Baer, Joseph Sawyer, Eddy Waller, Douglas Kennedy, Robert Shayne, Miguel Inclan, Hugh Sanders, Hank Patterson, Robert Griffith, Fay Roope, Robert Dover. A cavalry captain facing court martial ends up trying to thwart an attack by Geronimo and his braves. Fair action outing enhanced by Super Cinecolor.\n\n**1993** _ **The Indians Are Coming**_ **** Universal, 1930. 12 Chapters. D: Henry MacRae. SC: Ford Beebe and George Plympton. With Tim McCoy, Allene Ray, Edmund Cobb, Francis Ford, Charles Roy, Wilbur McGaugh, Charles F. Royal, Lafe McKee, Bud Osborne, Don Francis, Bob Reeves, Jim Corey, Dick Hatton, Art Mix, Chief Thunderbird, George Plues, Les Bates, Bud McClure, Monte Montague, Frank Ellis, Jack Kirk, Bill Patton, Archie Ricks, Tex Phelps, Al Taylor, Buck Moulton, Chuck Baldra, Bob Card, Charles Le Moyne, Ben Corbett, Jack Jones, Charles Murphy, Dynamite (dog), Buck Connors (narrator). Going West with a wagon train carrying his girl and her father, a scout fights off a ruffian after the young woman and her uncle's gold claim, as well as marauding Indians. Popular but rough hewn early sound serial.\n\n**1994** _ **El Indio**_ (The Indian). Producciones Rodas, S.A., 1972. 85 min. Color. D: Rodolfo de Anda. With Pedro Armendariz, Jr., Rodolfo de Anda, Jorge Rivero, Monica Favel, Jorge Russek, Emilio Fernandez, Mario Almade, Arturo Martinez, Ray Moyer, Helena Rojo, Amparo Rivelles. When a peon refuses to tell a greedy landowner the location of priceless Indian relics he is almost hung but is saved by a cowboy who unites the locals in a rebellion against their oppressors. Taut Mexican Western drama.\n\n_**Indio Black**_ see _**Adios, Sabata**_\n\n**1995** _ **Inferno**_ **** 20th Century\u2013Fox, 1963. 87 min. Color. D: Roy Ward Baker. SC: Francis M. Cockrell. With Robert Ryan, Rhonda Fleming, William Lundigan, Larry Keating, Henry Hull, Carl Betz, Robert Burton, Harry Carter, Everett Glass, Robert Adler, Adrienne Marden, Barbara Pepper, Dan White, Charles Tannen. After breaking his leg falling off a horse, a pompous, drunken rich man is left to perish in the desert by his beautiful wife and her boyfriend. Edgy, effective 3-D melodrama remade for TV as _**Ordeal**_ (q.v.).\n\n_**An Innocent Man**_ see _**The Sagebrush Trail**_\n\n**1996** _ **Inside Straight**_ **** Metro-Goldwyn-Mayer, 1951. 89 min. D: Gerald Mayer. SC: Guy Trosper. With David Brian, Arlene Dahl, Barry Sullivan, Mercedes McCambridge, Paula Raymond, Claude Jarman, Jr., Lon Chaney, John Hoyt, Roland Winters, Barbara Billingsley, Hayden Rourke, Jerry Hartleben, Dale Hartlben, Lou Nova, Richard Hale, Percy Helton, John R. Hamilton, Marshall Bradford, Matt Moore, Cameron Grant, William Lewis, Sherry Hall, Philo McCullough, George Sherwood, Jack Shea, James Pierce, Harry Lauter, Mae Clarke, Richard Alexander, Dewey Robinson, John Bryant, Mitchell Lewis. In 1870s San Francisco a crooked gambler cheats everyone so he can get rich and later learns it is all without worth. Told mostly in flashbacks, this melodrama is an adequate affair, nothing more.\n\n**1997** _ **Into the Badlands**_ **** USA Network, 1991. 89 min. Color. D: Sam Pillsbury. SC: Dick Beebe, Marjorie David and Gordon Dawson. With Bruce Dern, Mariel Hemingway, Helen Hunt, Dylan McDermott, Lisa Pelikan, Andrew Robinson, Adan Sanchez, Jerry Gardner, Glen Burns, Steve Tyler, Oryan Walsky, Loren Haynes, Michael J. Metzger, Reynaldo Cantu, Steven Schwartz-Hartley, Royce O'Donnell, Dick Beebe. A bounty hunter is on the trail of a notorious outlaw. Poorly conceived TV Western based on stories by Marcia Muller, Bryce Walton and Heck Allen.\n\n**1998** _ **The Intruders**_ **** NBC-TV, 1970. 95 min. Color. D: William A. Graham. SC: Dean Riesner. With Don Murray, Anne Francis, Edmond O'Brien, John Saxon, Gene Evans, Edward Andrews, Shelly Novack, Dean Stanton, Stuart Margolin, Zalman King, Harrison Ford, Gavin MacLeod, Philip Alford, John Hoyt, Marlene Tracy, Ken Swofford, Robert Donner, Edward Faulkner, James Gammon, Len Wayland, Ted Gehring, Robert P. Lieb, William Phipps, Mickey Sholdar. As a former gunman tries to reform as a lawman, a half-breed Indian struggles to make a life for himself. Fair television feature.\n\n**1999** _ **The Invaders**_ **** Kay Bee, 1912. 37 minutes. D: Francis Ford and Thomas H. Ince. SC: C. Gardner Sullivan. With Francis Ford, Ethel Grandin, Ann Little, Ray Myers, Chief Eagleshirt, Art Acord. After the government breaks a treaty with the Indians, two tribes unite to wipe out a team of railroad surveyors and attack a fort with a chief's daughter trying to avert the slaughter. Well staged and acted Thomas H. Ince early silent production; well worth viewing.\n\n**2000** _ **The Invasion of Johnson County**_ **** NBC-TV, 1976. 100 min. Color. D: Jerry Jameson. SC: Nicholas E. Bachr. With Bill Bixby, Bo Hopkins, John Hillerman, Billy Green Bush, Stephen Elliott, Lee DeBroux, M. Emmet Walsh, Mills Watson, Alan Fudge, Luke Askew, Edward Winter, David Donner, Ted Gehring. When land grabbers and their hired guns attempt to steal land from small ranchers, two men, an Easterner and a cowboy, team to stop them. Loaded with action TV movie.\n\n**Bill Bixby and Bo Hopkins in** _**The Invasion of Johnson County**_ **(NBC-TV, 1976).**\n\n**  \n**\n\n**2001** _ **Las Invencibles**_ (The Invincible Women) **** Peliculas Rodriguez, S.A., 1964. 86 min. D: Federico Curiel. SC: Federico Curiel and Alfredo Ruanova. With Kitty de Hoyos, Dacia Gonzalez, Dagoberto Rodriguez, Eduardo Fajardo, Eric del Castillo, Pancho Cordova, Rafael Bertrand, Celia Viveros, Rogelio Guerra, Fernando Curiel, Federico Curiel, Noe Murayama, Fanny Schiller. Two sisters join forces with a masked avenger out to stop a mysterious figure from taking over his niece's estate. Satisfying follow-up to _**Las Hijas del Zorro**_ (q.v.).\n\n**2002** _ **Invitation to a Gunfighter**_ **** United Artists, 1964. 92 min. Color. D: Richard Wilson. SC: Elizabeth Wilson and Richard Wilson. With Yul Brynner, Janice Rule, Brad Dexter, Alfred Ryder, Mike Kellin, George Segal, Clifford David, Pat Hingle, Bert Freed, Curt Conway, Clifton James, Clarke Gordon, Strother Martin, Arthur Peterson. The citizens of a Western town hire a gunman to rid them of a local killer but the plan develops a surprise twist. Too much talk and not enough action in this oater.\n\n**Janice Rule and Brad Dexter in** _**Invitation to a Gunfighter**_ **(United Artists, 1964).**\n\n**  \n**\n\n**2003** _ **The Irish Gringo**_ **** Keith Productions, 1935. 54 min. D: William C. Thompson. SC: Patrick Petersalia (Patrick Carlyle and William C. Thompson). With Pat (Patrick) Carlyle, William Farnum, Bryant Washburn, Elena Duran, Milt (Milburn) Morante, Karlyn (Karla) May, Olin Francis, Don Orlando, Ace Cain, Rudolph Cornell, Josef Swickard, Kit Guard, Marjorie Medford, Foxy Callahan, Horace B. Carpenter, Paul Blackman, Chito Montoya, Art Felix, Herman Hack, Clyde McClary, Tex Palmer, Bud Pope. A cowboy and his pals fight an outlaw gang who murdered a rancher for the secret to the Lost Dutchman gold mine. Sparse poverty row clinker worth a look for William Farnum's hammy performance.\n\n**2004** _ **The Iron Horse**_ **** Fox, 1924. 119 min. D: John Ford. SC: Charles Kenyon. With George O'Brien, Madge Bellamy, Cyril Chadwick, Fred Kohler, Gladys Hulette, James Marcus, J. Farrell MacDonald, James Welch, Walter Rogers, George Waggner, Jack Padjan, Charles O'Malley, Charles Newton, Charles Edward Bull, Colin Chase, Delbert Mann, Chief Big Tree, Chief White Spear, Ed Peil, James Gordon, Winston Miller, Peggy Cartwright, Stanhope Wheatcroft, Frances Teague, Will Walling. Looking for this father's killer, a man ends up romancing his childhood sweetheart and the daughter of a builder of the Transcontinental Railroad. One of the truly great classic Westerns; a must see silent epic.\n\n**2005** _ **The Iron Mistress**_ **** Warner Bros., 1952. 110 min. Color. D: Gordon Douglas. SC: James Webb. With Alan Ladd, Virginia Mayo, Joseph Calleia, Phyllis Kirk, Alf Kjellin, Douglas Dick, Anthony Caruso, Ned Young, Don Beddoe, Robert Emhardt, Richard Carlyle, Jay Novello, George J. Lewis, Darla Massey, George Voskovec, Nick Dennis, Frank Ferguson, Sara Selby, Gordon Nelson, Harold Gordon. The story of Jim Bowie, his invention of the famous knife and his love for a woman who tries to take advantage of him. Mediocre historical drama, although Virginia Mayo is quite good as the ruthless seductress.\n\n**2006** _ **Iron Mountain Trail**_ **** Republic, 1953. 54 min. D: William Witney. SC: Gerald Geraghty. With Rex Allen, Slim Pickens, Nan Leslie, Grant Withers, Roy Barcroft, Alan Bridge, Forrest Taylor, George Lloyd, John Hamilton, Kenneth Terrell, Dee Cooper, Frank O'Connor, Post Park, Alex Montoya, Cactus Mack. Post Office inspector Rex Allen is assigned to find out why mail is being lost during clipper ship transportation. Nothing outstanding although henchman Roy Barcroft does have a pet monkey.\n\n**2007** _ **The Iron Rider**_ **** Goodwill, 1926. 60 min. D-SC: Jacques Jaccard. With Yakima Canutt, Vola Vale, Elsie Benham, Jim Corey, Lee Sepulveda, Alfred Hewston, Nelson McDowell, Boy (horse), Lad (dog). A cowpoke who wants to marry gets into trouble when he is cheated out of his horse by a dishonest gambler while trying to win enough money to set up housekeeping. Below average silent oater greatly helped by star Yakima Canutt, who does a lot of trick riding and stunt work in addition to giving a fine performance.\n\n**2008** _ **The Iron Sheriff**_ **** United Artists, 1957. 73 min. D: Sidney Salkow. SC: Seeleg Lester. With Sterling Hayden, Constance Ford, John Dehner, Kent Taylor, Darryl Hickman, Walter Sande, Frank Ferguson, King Donovan, Mort Mills, Peter Miller, Kathy Nolan, I. Stanford Jolley, Will Wright, Ray Walker, Bob Williams. When his son is accused of robbery and murder, the local lawman believes he is innocent and tries to prove it. Strong \"B\" Western acted by a fine cast.\n\n**2009** _ **The Iroquis Trail**_ **** United Artists, 1950. 85 min. D: Phil Karlson. SC: Richard Shayer. With George Montgomery, Brenda Marshall, Glenn Langan, Reginald Denny, Monte Blue, Sheldon Leonard, Paul Cavanaugh, Holmes Herbert, Dan O'Herlihy, John Doucette, Don Gerner, Marcel Gourmet, Arthur Little, Jr., Esther Somers. After his brother is attacked while delivering a dispatch for the British, scout Hawkeye and his Indian guide Sagamore try to stop the evil Simon Girty and his Huron chief ally from helping the French led by General Montcalm. Okay retelling of James Fennimore Cooper's 1826 novel _The Last of the Mohicans_ with fine work by George Montgomery as Hawkeye and Monte Blue as Sagamore.\n\n**2010** _ **Ishi:**_ ****_**The Last of His Tribe**_ **** NBC-TV, 1978. 100 min. Color. D: Robert Ellis Miller. SC: Dalton Trumbo and Christopher Trumbo. With Dennis Weaver, Eloy Phil Casados, Devon Erickson, Joaquin Martinez, Geno Silva, Joseph Running Fox, Lois Red Elk, Gregory Norman Cruz, Ariliene Nofchisssey Williams, Michael Medina, Peter Brandon, Patricia Ganera, Eddy Marques, Dennis Dimster, Wayne Heffey, Miss Gold, Jay W. MacIntosh, Ernest D. Paul. In 1911 in Northern California the lone survivor of a reclusive Indian tribe is found by a rancher and nursed back to health with his story eventually revealed by an anthropologist. Well done TV movie from the novel _Ishi in Two Worlds_ by Theodora Kroeber Quinn; remade as _**The Last of His Tribe**_ (q.v.) in 1992.\n\n**2011** _ **It Can Be Done Amigo**_ **** EMI\/Atlantida\/Terzafilms, 1971. 95 min. Color. D: Maurizio Lucidi. SC: Rafael Azcona. With Jack Palance, Bud Spencer, Fancisco Rabal, Renato Cestie, Dany Saval, Giovanni Pazzafini, Luciano Catenacci, Sal Borgese. A bounty hunter and his sister pursue a man who dishonored the woman who is put in charge of a small boy about to inherit a rich with oil deposits. Amusing, rambling Spanish-made oater with little bloodshed. Spanish title: _**En El Oeste Se Puede Hacer...Amigo**_ (In the West It Can Be Done...Friend).\n\n**2012** _ **It Happened Out West**_ **** 20th Century\u2013Fox, 1937. 59 min. D: Howard Bretherton. SC: Earle Snell. With Paul Kelly, Judith Allen, Johnny Arthur, LeRoy Mason, Nina Campana, Steve Clemento, Frank LaRue, Reginald Barlow, Russell Hicks, Ted Adams, Henry Otho, Ben Corbett, Lew Kelly, Edwin Brady, Evelyn Zelle, Tom Forman, Archie Ricks, Charles Treadwell, Slim Lucas, Jack Shannon. Crooks try to cheat a young woman out of her dairy ranch when a silver vein is discovered on the property. Well done Sol Lesser production, an adaptation of the Harold Bell Wright novel.\n\n_**It Lives by Night**_ see _**The Bat People**_\n\n**2013** _ **It's a Big Country**_ **** Metro-Goldwyn-Mayer, 1951. 89 min. D: Richard Thorpe, John Sturges, Charles Vidor, Don Weis, Clarence Brown, William A. Wellman and Don Hartman. SC: William Ludwig, Helen Deutsch, George Wells, Allen Rivkin, Dorothy Kingsley, Dore Schary and Isobel Lennart. With Ethel Barrymore, Keefe Brasselle, Gary Cooper, Nancy Davis, Van Johnson, Gene Kelly, Janet Leigh, Marjorie Main, Fredric March, George Murphy, William Powell, S.Z. Sakall, Lewis Stone, James Whitmore, Keenan Wynn, Leon Ames, Angela Clarke, Bobby Hyatt, Sharon McManus, Elisabeth Risdon, Bill Baldwin, Mickey Martin, William H. Welsh, Ned Glass, Sherry Hall, Fred Santley, Henry Sylvester, Roger Moore, Roger Cole, Harry Stanton, Benny Burt, June Hedin, Luana Mehlberg, Jeralyn Alton, Jacqueline Kenley, David Alpert, Tiny Francone. Stories of several Americans, showing the greatness of this country, with Gary Cooper hosting a segment entitled \"Texas\" in which he brags about the Lone Star state backed by newsreel footage. Enjoyable patriotic fare.\n\n**2014** _ **The Ivory-Handled Gun**_ **** Universal, 1935. 60 min. D: Ray Taylor. SC: Jack (John T.) Neville and Charles E. Barnes. With Buck Jones, Charlotte Wynters, Walter Miller, Carl Stockdale, Frank Rice, Joseph Girard, Robert Kortman, Stanley Blystone, Lafe McKee, Lee Shumway, Charles King, Ben Corbett, Eddie Phillips, Niles Welch, Jim Thorpe, Lew Meehan, Robert Walker, Herman Hack, Bud McClure, Blackjack Ward, Arthur Thalasso, Archie Ricks, Al Taylor, George Sowards, Jack Montgomery, Kernan Cripps, Bob Roper, Iron Eyes Cody, Charles McMurphy, Ralph Bucko, Roy Bucko. Two men involved in a long standing feud are on opposite sides when rustlers being stealing sheep. Good Buck Jones vehicle with a surprise finale.\n\n**2015** _ **The Jack Bull**_ **** Home Box Office (FBO), 1999. 116 min. Color. D: John Badham. SC: Dick Cusack. With John Cusack, John Goodman, L.Q. Jones, Miranda Otto, John C. McGinley, John Savage, Rodney A. Grant, Kurt Fuller, Rex Linn, Jay O. Sanders, Drake Bell, Nicholas E. Gillie, Duncan Fraser, Ken Pogue, Glen Morshower, Ned Bellamy, Brent Briscoe, Scott Wilson, Valerie Planche, Nathaniel DeVeaux, Bruce Flewelling, J.C. Roberts, Esther Purves-Smith, Patrick Richards, John Payne, Corry Glass, Byme Piven, Raoul Ganeev, Dick Cusack, Robert Lewis, Jimmy Herman, Chad Nobert, Gina Williams, Madeleine Lefebvre, Rick Poltaruk, Ken Hurlburt, Tom Heaton, Campbell Lane, Bill Cusack, Ron Webber. Two ranchers are at odds when one of them abuses the horse and Indian caretaker of the other and this nearly costs Wyoming its bid for statehood. Pretty fair TV movie.\n\n_**Jack London's Klondike Fever**_ see _**Klondike Fever**_\n\n_**Jack London's Tales of Adventure**_ see _**Tales of Adventure**_\n\n**2016** _ **Jack London**_ **** United Artists, 1943. 99 min. D: Alfred Santell. SC: Ernest Pascal. With Michael O'Shea, Susan Hayward, Osa Massen, Harry Davenport, Frank Craven, Virginia Mayo, Ralph Morgan, Jonathan Hale, Louise Beavers, Leonard Strong, Regis Toomey, Albert van Antwerp, Paul Hurst, Lumsden Hare, Hobart Cavanaugh, Sarah Padden, Edward Earle, Morgan Conway, Robert Homans, Arthur Loft, Wallis Clark, Ernie Adams, Sven Hugo Borg, Charles Miller, Jack Roper, Davison Clark, Dick Curtis, Ted Billings, Brooks Benedict, Pierre Watkin, Edmund Cobb, Richard Loo, Sidney D'Albrook, Olin Howland, Torben Meyer, John Kelly, Harry Semels, Jack Roper, Roy Gordon, Johnny Fisher, Rose Plummer, Evelyn Finley, Robert Katcher, John Kelly, Eddie Laughton, Harold Minjir, Frank Mills, Eddie Lee, Paul Fung, Mei Lee Foo, Charles Lung, Charlene Newman, Bruce Wong, Hank Worden, Tex Cooper. Vagabond Jack London leaves a cannery job to go on a seal hunting expedition, then prospects for gold in the Yukon and goes to college, along the way romancing several women, ending up a writer. Only a fair biopic of the famous author although Michael O'Shea is good in the title role and he is backed by a fine supporting cast.\n\n**2017** _ **Jack McCall, Desperado**_ **** Columbia, 1953. 76 min. Color. D: Sidney Salkow. SC: John O'Dea. With George Montgomery, Angela Stevens, Douglas Kennedy, James Seay, Eugene Iglesias, William Tannen, Jay Silverheels, John Hamilton, Selmer Jackson, Stanley Blystone, Gene Roth, Joe McGuinn. During the Civil War a Southerner joins the Union army, is framed on the charge of giving information to the enemy, convicted of treason and sentenced to die, but escapes to find the man who accused him. Pretty good Sam Katzman production.\n\n**2018** _ **Jack Slade**_ **** Allied Artists, 1953. 90 min. D: Harold Schuster. SC: Warren Douglas. With Mark Stevens, Dorothy Malone, Barton MacLane, John Litel, Paul Langton, Harry Shannon, John Harmon, Jim Bannon, Lee Van Cleef, Ron Hargrave, David Day, Sammy Ogg, Nelson Leigh, John Halloran, Robert Reeves, Dorothy Kennedy, Duane Thorsen, Harry Landers, Ann Navarro, Steve Darrell, Hank Patterson. A rebellious young man becomes a noted lawman but eventually turns against the woman he loves and becomes a criminal. Fairly interesting account of a good man gone bad although Mark Stevens' Jack Slade must be the grimiest leading man in movie history.\n\n**2019** _ **The Jackals**_ **** 20th Century\u2013Fox, 1967. 93 min. Color. D: Robert D. Webb. SC: Lamar Trotti and Austin Medford. With Vincent Price, Dana Ivarson, Robert Gunner, Bob Courtnet, Bill Brewer, Johnny Whitney. The 1883 gold rush in South Africa's Transvaal area brings a quartet of bank robbers who try to steal ore from an old prospector and his pretty granddaughter. Somewhat obscure South African reworking of _**Yellow Sky**_ (q.v.) with a fine performance by Vincent Price as the prospector.\n\n**2020** _ **Jackass Mail**_ **** Metro-Goldwyn-Mayer, 1942. 80 min. D: Norman McLeod. SC: Lawrence Hazard. With Wallace Beery, Marjorie Main, J. Carrol Naish, Dick Curtis, William Haade, Darryl Hickman, Hobart Cavanaugh, Joe Yule, Esther Howard, Babe London, Al Ferguson, LeRoy Mason, Harry Fleischmann, Louis Mason, George Carleton, Bobby Larson, Mary Currier, Harry Woods, Duke York, Paul Newlan, Edward Hearn, Frank Darien, Howard Mitchell, Wade Boteler, Ruth Warren, Harry Worth, Robert Emmett O'Connor, Murdock MacQuarrie, Eddie Hart, Joe Whitehead, Malcolm White, Jack Kenney, Billy Wayne, Bobby Barber, Ted Oliver, Art Belasco, Robert Perry, Charles R. Dorety. An old-time bad man is pursued by the female owner of a mail wagon team and saloon, but escaping from a hanging posse he stops a robbery and becomes a hero. Very pleasant Wallace Beery-Marjorie Main teaming sure to delight their fans.\n\n**2021** _ **Jaguar**_ **** Republic, 1956. 66 min. D: George Blair. SC: John Fenton Murray and Benedict Freedman. With Sabu, Chiquita, Barton MacLane, Jonathan Hale, Touch (Michael) Connors, Jay Novello, Fortunio Bonanova, Nacho Galindo, Rodd Redwing, Pepe Hern, Raymond Rosas. In order to keep others away from rich oil deposits, an old prospector uses the guise of a jaguar to eliminate his competitors. El cheapo program feature; the associate producer is Mickey Rooney.\n\n_**Jail Break**_ see _**Gunning for Vengeance**_\n\n**2022** _ **The James Brothers of Missouri**_ **** Republic, 1950. 12 Chapters. D: Fred C. Brannon. SC: Royal Cole, William Lively and Sol Shor. With Keith Richards, Robert Bice, Noel Neill, Roy Barcroft, Patricia Knox, Lane Bradford, Gene Roth, John Hamilton, Edmund Cobb, Hank Patterson, Dale Van Sickel, Tom Steele, Lee Roberts, Frank O'Connor, Marshall Reed, Wade Ray, Nolan Leary, David Sharpe, Art Dillard, John Crawford, Post Park, Duke Taylor, Al Ferguson, Cactus Mack, Tommy Coats, Kenneth Terrell, Robert Wilke, Forrest Burns, Herman Hack, Chick Hannon, Chuck Roberson, Bud Wolfe, Frosty Royce, Rocky Shahan. Using aliases, Jesse and Frank James join an ex\u2013gang member's freight line and when he is murdered by rivals they agree to help his sister run the business and capture the culprits. Fair pseudo-historical cliffhanger.\n\n**2023** _ **James Michener's Dynasty**_ **** NBC-TV, 1976. 100 min. Color. D: Lee Phillips. SC: Sidney Carroll. With Sarah Miles, Stacy Keach, Harris Yulin, Harrison Ford, Amy Irving, Granville Van Dusen, Charles Weldon, Gerrit Graham, Stanley Clay, Tony Swartz, John Carter, Stephanie Faulkner, Rayford Barnes, Sair Price, Norbert Schiller, Ian Wolfe, Guy Raymond, Don Eitner, James Houghton, J. Jay Saunders, William Challee, Francis De Sales, Dennis Larson. In frontier Ohio of the 1820s, a man, his wife and brother-in-law turn a family business into a financial empire. Pretty fair TV movie; well acted.\n\n**Harris Yulin, Sarah Miles and Stacy Keach in** _**James Michener's Dynasty**_ **(NBC-TV, 1976).**\n\n**  \n**\n\n**2024** _ **Jaws of Justice**_ **** Principal, 1933. 58 min. D: Spencer Gordon Bennet. SC: Joseph Anthony Roach. With Kazan (dog), Richard Terry (Jack Perrin), Ruth Sullivan, Robert Walker, Gene Tolar, Lafe McKee, Teddy (dog). In the north country a Canadian Mounted Policeman and his loyal dog team to bring in a killer. Interesting, fast moving low budget outdoor melodrama filmed at Lake Tahoe and based on Edgar Allan Poe's \"The Gold Bug.\"\n\n**2025** _ **The Jayhawkers**_ **** Paramount, 1959. 100 min. Color. D: Melvin Frank. SC: Melvin Frank, Joseph Petracca, Frank Fenton and A.I. Bezzerides. With Jeff Chandler, Fess Parker, Nicole Maurey, Henry Silva, Herbert Rudley, Leo Gordon, Don Megowan, Kenneth MacDonald, Ned Glass, Frank DeKova, Frank Wilcox, Al Wyatt, Richard Shannon, Barbara Knudson, Joe Forte, Howard Joslin, Tony Regan. An outlaw gang leader is pursued by a man determined to capture him but they both fall in love with the same woman. Tepid melodrama from the production team of Norman Panama and Melvin Frank.\n\n**2026** _ **Jeep Herders**_ **** Astor, 1949. 46 min. D-SC: Richard Talmadge and Harvey Perry. With John Day, June Carlson, Pat Michaels, Steve Clark, Ashley Cowan, Slim Gault, Paul Bradley, Dale Van Sickel, Tom Steele, Saul Gorss, Richard Fitch, Fred Kennedy, Frank McCarroll, Victor Metzetti (Richard Talmadge). Returning home from World War II, a man finds his ranch workers have all gone to nearby oil fields for higher wages so he hires his war buddies from a convalescent hospital to help him run the place. Low grade outing with plenty of action and stunts; originally released by Planet in 1946 in 16mm.\n\n**2027** _ **Jeepers Creepers**_ **** Republic, 1939. 67 min. D: Frank McDonald. SC: Dorrell McGowan and Stuart McGowan. With The Weavers and Elviry (Leon, Frank, June [Elviry]), Roy Rogers, Maris Wrixon, Billy Lee, Lucien Littlefield, Thurston Hall, Johnny Arthur, Loretta Weaver, Milton Kibbee, Ralph Sanford, Dan White, Bud Geary, Gladys Gale, Joe McGuinn, Robert Wilke, Curley Dresden, Si Jenks, Tex Phelps. A backwoods family is cheated out of their land by a rich industrialist so they try to get it back and humanize him in the process. Fun homespun feature with a pleasing title tune and Roy Rogers as a sheriff.\n\n**2028** _ **Jeremiah Johnson**_ **** Warner Bros., 1972. 108 min. Color. D: Sidney Pollack. SC: John Milius and Edward Anhalt. With Robert Redford, Will Geer, Stefan Gierasch, Allyn Ann McLerie, Charles Tyner, Paul Benedict, Matt Clark, Joaquin Martinez. A man becomes a recluse in the wilderness, learning to survive in the environment and fight an Indian curse. Somewhat interesting, but too long, adventure melodrama.\n\n**2029** _ **Jericho**_ **** Black Knight Productions, 2000. 101 min. Color. D: Merlin Miller. SC: Frank Dana Frankolino, George Leonard Briggs and Robert Avard Miller. With Mark Valley, Leon Coffee, R. Lee Ermey, Lisa Stewart, Mark Collie, Morgana Shaw, Buck Taylor, Katerie Walker, Kevin Stapleton, Renny Rozzoni, Woody P. Snow, Lashawn McIvor, Kyle Ingram, Ryon Marshall, David Crowe, David Alvarado, Tommy Worrell, Bob Brown, James Wallace, Thomas E. Blaylock, James Ham, Gil Dorland, Jim Ryan, Kelsey Bruce, Mason McWilliams, Mike Bowlin, Bob Balderson, William L. Moody IV, Charles Gamero, Richard Curilla, Jack Lewis. An amnesiac cowboy is nursed back to health by an ex-slave preacher and the two take part in a cattle drive and run a ranch before the cowpoke discovers his secret past. Average Western mystery.\n\n**2030** _ **Jesse and Lester**_ **** H.P. International, 1972. 97 min. Color. D: James London (Richard Harrison). SC: Renzo Genta and Richard Harrison. With Richard Harrison, Donald O'Brien, Gino Marturano, Anna Zinneman, George Wang, Rick Boyd (Federico Boido), Aldo Cecconi, John Bartha, Fernando Cerulli, Salvatore Baccaro, Fortunato Arena, Calogero Caruana, Emilio Messina, Osiride Pevarello, Luciano Rossi, Claudio Ruffini, Goffredo Unger, Daniela Meroni, Gianfranco Barra. Two half brothers who are opposites in personality join forces to obtain a stolen inheritance. Well made, amusing Italian Western originally called _**Due Fratelli**_ (Two Brothers) and also known as _**Jesse and Lester in a Place Called Trinity**_ , _**A Place Called Trinity**_ , _**Trinity**_ and _**Two Brothers in Trinity**_.\n\n_**Jesse and Lester in a Place Called Trinity**_ see _**Jesse and Lester**_\n\n**2031** _ **Jesse James**_ **** 20th Century\u2013Fox, 1939. 105 min. Color. D: Henry King. SC: Nunnally Johnson. With Tyrone Power, Henry Fonda, Nancy Kelly, Randolph Scott, Henry Hull, Slim Summerville, J. Edward Bromberg, Brian Donlevy, John Carradine, Donald Meek, John Russell, Jane Darwell, Charles Tannen, Claire DuBrey, Willard Robertson, Paul Sutton, Ernest Whitman, Paul Burns, Spencer Charters, Arthur Aylesworth, Lon Chaney, Jr., Charles Halton, George Chandler, Erville Alderson, Harry Tyler, George Breakston, John Elliott, Virginia Brissac, Don Douglas, Edward LeSaint, Wylie Grant, Harry Holman, Ethan Laidlaw, Charles Middleton, James Flavin, Eddy Waller, Victor Kilian, John Beck, Morgan Brown, George O'Hara. Brothers Jesse and Frank James become outlaws after their mother is murdered by carpetbaggers in post\u2013Civil War Missouri. Historically inaccurate but highly entertaining, well done and finely acted; recommended. Followed by _**The Return of Frank James**_ (q.v.). In the silent era Fred Thompson had the title role in _**Jesse James**_ (Paramount, 1927).\n\n**2032** _ **Jesse James at Bay**_ **** Republic, 1941. 56 min. D: Joseph Kane. SC: James R. Webb. With Roy Rogers, George \"Gabby\" Hayes, Sally Payne, Pierre Watkin, Hal Taliaferro, Gale Storm, Roy Barcroft, Jack Kirk, Jack O'Shea, Billy Benedict, Rex Lease, Ed Peil, Sr., Jack Rockwell, Curley Dresden, Hank Bell, Fern Emmett, Budd Buster, Lloyd Ingraham, Karl Hackett, Fred Burns, Kit Guard, Chester Conklin, Theodore Lorch, Art Mix, Al Taylor, Pascale Perry, Bob Reeves, Paul Sells, Charles Moore, Bill Wolfe, Ken Card, Luke Cosgrove, Rick Anderson. When the railroad and a crooked banker try to cheat farmers out of their land they send for Jesse James to help them. Pleasant fiction with Roy Rogers in dual roles, Jesse James and gambler Clint Burns.\n\n**Advertisement for** _**Jesse James at Bay**_ **(Republic, 1941).**\n\n**  \n**\n\n**2033** _ **Jesse James, Jr.**_ **** Republic, 1942. 56 min. D: George Sherman. SC: Richard Murphy, Taylor Cavan and Doris Schroeder. With Don \"Red\" Barry, Lynn Merrick, Al St. John, Douglas Walton, Robert Kortman, Karl Hackett, Lee Shumway, Stanley Blystone, Jack Kirk, George Chesebro, Frank Brownlee, Forbes Murray, Jim Corey, Kermit Maynard. A cowboy tries to thwart crooks who are out to destroy a telegraph headquarters. Fun, action packed Don Barry film with good comedy support from Al St. John. TV title: _**Sundown Fury**_.\n\n**2034** _ **Jesse James Meets Frankenstein's Daughter**_ **** Embassy, 1966. 82 min. Color. D: William Beaudine. SC: Carl K. Hittleman. With Estelita (Rodriguez), John Lupton, Jim Davis, Cal Bolder, Steven Geray, Narda Onyx, Felipe Turich, Rosa Turich, Rayford Barnes, William Fawcett, Nestor Paiva, Dan White, Page Slattery, Roger Creed. A female descendant of Dr. Frankenstein uses one her ancestor's artificial brains to turn Jesse James' henchman into monster. Despite its title, this is a pretty fair horror Western; issued theatrically with _**Billy the Kid vs. Dracula**_ (q.v.).\n\n**2035** _ **Jesse James Rides Again**_ **** Republic, 1947. 13 Chapters. D: Fred C. Brannon and Thomas Carr. SC: Franklin Adreon, Basil Dickey, Jesse Duffy and Sol Shor. With Clayton Moore, Linda Stirling, Roy Barcroft, John Compton, Tristram Coffin, Tom London, Holly Bane, Edmund Cobb, Gene Roth, LeRoy Mason, Ed Cassidy, Dave Anderson, Eddie Parker, Tom Steele, Dale Van Sickel, Robert Blair, Ted Mapes, Tex Terry, Gil Perkins, Tex Palmer, Emmett Lynn, Charles Morton, Duke Taylor, Monte Montague, Lee Shumway, Herman Hack, Chuck Roberson, Carl Sepulveda, Kenneth Terrell, Pascale Perry, Chester Conklin, Tommy Coats, George Chesebro, Bud Wolfe, Tom Chatterton, Charles King, Robert Riordan, Howard Mitchell, Richard Alexander, Keith Richards. Fleeing from the law for a crime he did not commit, Jesse James and a pal arrives in an area plagued by attacks from masked raiders after oil. Typical later Republic serial a great cast.\n\n**2036** _ **Jesse James Under the Black Flag**_ **** Mesco Pictures, 1921. 69 min. D-SC: Franklin B. Coates. With Jesse James, Jr., Harry Hall, Marguerite Hungerford, F.G. McCabe, Sunshine Baker, Ralph Johnson, Hortense Espey, Jack Wall, Mrs. Cart, William Baker, Frances Coffrey, Franklin B. Coates, Diana Reed, Jack Neil. After being a member of Quantrill's Raiders and then becoming an outlaw, Jesse James is pardoned and falls in love. The main interest in the obscure silent melodrama is seeing Jesse James, Jr., portraying his famous father.\n\n**2037** _ **Jesse James vs. The Daltons**_ **** Columbia, 1954. 65 min. Color. D: William Castle. SC: Robert E. Kent and Samuel Newman. With Brett King, Barbara Lawrence, James Griffith, Bill Phillips, John Cliff, Rory Mallinson, William Tannen, Richard Garland, Nelson Leigh, Raymond Largay. Believing he is Jesse James's son, a man finds himself in a showdown with the Dalton brothers. Standard 3-D production from Sam Katzman.\n\n**2038** _ **Jesse James' Women**_ **** United Artists, 1954. 83 min. Color. D: Donald Barry. SC: D.D. Beauchamp. With Don Barry, Jack Buetel, Peggie Castle, Lita Baron, Joyce Rhed, Betty Brueck, Laura Lee, Sam Keller. The James gang plans a robbery in a small town but ends up getting involved in romance. Don \"Red\" Barry directed and starred in this low budget hijacks that is more laughs than action.\n\n**2039** _ **Jessie's Girls**_ **** Mason Distributing, 1975. 84 min. Color. D: Al Adamson. SC: Budd Donnelly. With Sondra Currie, Rod Cameron, Geoffrey Land, Ben Frank, Regina Carrol, Jenifer Bishop, Ellen Stern, Joe Cortese, Jon Shank, Biff Yeager, Gavin Murrell, Rigg Kennedy, William Hammer, Hugh Warden, Joe Arrowsmith, John Durren. In 1879 newlyweds are attacked by an outlaw gang, the husband murdered, the wife raped, shot and left for dead but she is rescued by an old prospector who teaches her to survive and she sets out to get revenge on the killer. Tacky and violent; Rod Cameron as the prospector is the only interest. Alternate title: _**Wanted Woman**_.\n\n**2040** _ **Jiggs and Maggie Out West**_ **** Monogram, 1950. 66 min. D: William Beaudine. SC: Barney Gerard and Adele Buffington. With Joe Yule, Renie Riano, Tim Ryan, Jim Bannon, Riley Hill, Pat Golden, June Harrison, Henry (Kulky) Kulkovich, Terry McGinnis, Billy Griffith, George McManus, Lane Chandler, Kenne Duncan, Jimmy Aubrey. Jiggs and Maggie head West when she inherits a ranch and a goldmine but a crook wants them for himself. Pleasant screen adaptation of the popular, long running comic strip \"Bringing Up Father\" with its author, George McManus, appearing as himself; good fun.\n\n**2041** _ **El Jinete Sin Cabeza**_ (The Headless Rider) **** Clasa-Mohme, 1957. D: Chano Urueta. SC: Ramon Obon. With Luis Aguilar, Flor Silvestre, Jaime Fernandez, Pascual Garcia Pena, Crox Alvarado, Patricia Nieto, Guillermo Carmer, Alberto Pedret, Elvira Lodi, Carlos Suarez, Salvador Godinez, Fernando Oses, Salvador Lozano, Fernando Yapur. A masked rider tries to expose the machinations of a hooded cult worshipping a disembodied hand and searching for hidden treasure. Creepy, atmospheric Mexican horror Western; the title character also appeared in _**La Cabeza de Pancho Villa**_ and _**La Marca de Satana**_ (qq.v.).\n\n**2042** _ **Los Jinetes de la Bruja**_ (The Riders of the Witch) **** Almada Films, 1966. 93 min. Color. D: Vicente Orona. SC: Vicente Orona and Vicente Orona, Jr. With Blanca Sanchez, Kitty de Hoyos, Fernando Almada, Mario Almada, Dagoberto Rodriguez, Roberto Canedo, Rafael del Rio, Jose A. Espinoza \"Ferrusquilla,\" Alicia Bonet, Consuelo Frank, Antonio Ravel, Carlos Rotzinger, Jose Eduardo Perez, Jorge Mateos, Manuel Arvide, Agustin Fernandez, Carlos Suarez. When an elderly rancher is accused of killing a puppet master, his family seeks the help of a witch who avenges the murder with her spectral horse riders. Obscure Mexican horror Western filmed in Guanajuato and issued on video as _**La Herencia de la Bruja**_ (The Heritage of the Witch).\n\n_**Joan of Cattle Country**_ see _**Straight Shooting**_\n\n_**Joaquin Murieta**_ see _**Desperate Mission**_\n\n**2043** _ **Joe Dakota**_ **** Universal-International, 1957. 79 min. Color. D: Samuel Fuller. SC: Norman Jolley and William Talman. With Jock Mahoney, Luana Patten, Charles McGraw, Barbara Lawrence, Claude Akins, Lee Van Cleef, Anthony Caruso, Paul Birch, George Dunn, Steve Darrell, Rita Lynn, Gregg Barton, Jeanne Wood, Juney Ellis, Anthony Jochim, Francis McDonald. Arriving in a town where the citizens are cold and unfriendly, a cowboy tries to humanize them and instill respect for their community. A different kind of Western that succeeds more than it fails; Jock Mahoney is fine in the lead and the supporting cast is very good.\n\n_**Joe Dexter**_ see _**Guns of Nevada**_\n\n**2044** _ **Joe Kidd**_ **** Universal, 1972. 88 min. Color. D: John Sturges. SC: Elmore Leonard. With Clint Eastwood, Robert Duvall, John Saxon, Don Stroud, Stella Garcia, James Wainwright, Paul Koslo, Gregory Walcott, Dick Van Patten, Lynne Marta, John Carter, Pepe Hern, Chuck Hayward, Buddy Van Horn. Mexicans invade a remote village and a powerful land owner hires a drifter-gunman to stop them. Another in the line of features that pushed Clint Eastwood to super stardom; little better than mediocre.\n\n**Stella Garcia and Clint Eastwood in** _**Joe Kidd**_ **(Universal, 1972).**\n\n**  \n**\n\n**2045** _ **Joe Panther**_ **** Artists Creation, 1976. 110 min. Color. D-SC: Paul Krasny. With Brian Keith, Ricardo Montalban, Ray Tracey, A. Martinez, Cliff Osmond, Alan Feinstein, Lois Red Elk. Against great adversity, a young Indian brave tries to make a life for himself in the modern world and still hold onto his heritage. Okay drama.\n\n_**Johnny Colt**_ see _**Black Star**_\n\n**2046** _ **Johnny Concho**_ **** United Artists, 1956. 85 min. D: Don McGuire. SC: David Harmon and Don McGuire. With Frank Sinatra, Phyllis Kirk, Keenan Wynn, Wallace Ford, William Conrad, Dorothy Adams, Christopher Dark, Howard Petrie, Harry Bartell, Willis Bouchey, Robert Osterloh, Jean Byron, Leo Gordon, Claude Akins, John Qualen, Ben Wright, Dan Russ. A cowardly bully lives in the glory of his gunman brother until he is killed, then he must learn to be a man and face up to another gunfighter. Frank Sinatra is good in the title role of this otherwise average oater.\n\n**2047** _ **Johnny Firecloud**_ **** Entertainment Ventures, 1975. 99 min. Color. D: William A. Castleman. SC: Wilton Denmark. With Ralph Meeker, Victor Mohica, David Canary, Frank De Kova, Sachean Little Feather, Christina Hart, Jason Ledger, John F. Goff, Richard Kennedy, George \"Buck\" Flower, Wayne Storm, Elliott Lindsey, Sterling Franck, Seamon Glass, Barry Cooper, James A. Ward, Norman Sheridan, Michael Morrison. Returning home from the Army, an Indian learns his ex-girlfriend's town boss father and the local sheriff are harassing his tribe and that his sister has been raped. Low budget, violent modern-day Western.\n\n**2048** _ **Johnny Guitar**_ **** Republic, 1954. 110 min. Color. D: Nicholas Ray. SC: Philip Yordan. With Joan Crawford, Sterling Hayden, Scott Brady, Mercedes McCambridge, Ward Bond, Ben Cooper, Ernest Borgnine, John Carradine, Royal Dano, Frank Ferguson, Paul Fix, Rhys Williams, Ian MacDonald, Will Wright, John Maxwell, Robert Osterloh, Frank Marlowe, Trevor Bardette, Sumner Williams, Sheb Wooley, Denver Pyle, Clem Harvey. The ruthless female owner of a small town saloon is reunited with her gunman ex-lover who she calls on to defend her against a nearby town boss and a woman cattle raiser. Tough, symbolic Western is probably the only film Joan Crawford made that is lionized by pointed heads, but overall it is mostly on the dull side.\n\n**2049** _ **Johnny Hamlet**_ **** Transvue, 1972. 91 min. Color. D: Enzo G. Castellari. SC: Bruno Corbucci, Tito Carpi and Enzo G. Castellari. With Chip Gorman (Andrea Giordana), Gilbert Roland, Francoise Prevost, Gabriella Grimaldi, Horst Frank, Enzo Girolami, Pedro Sanchez, Stefania Careddu. A man returns home from the Civil War to find his father is dead and his mother has married his uncle. Violent affair that is a refashioning of William Shakespeare's tragedy _Hamlet_ ; it was made in 1968 in Italy as _**Quella Sporca Storia del West**_ (Dirty Story of the West) by Daiano Film\/Leone Film.\n\n_**Johnny Oro**_ see _**Ringo and His Golden Pistol**_\n**2050** _ **Johnny Reno**_ **** Paramount, 1966. 83 min. Color. D: R.G. Springsteen. SC: Steve Fisher. With Dana Andrews, Jane Russell, Lon Chaney, John Agar, Lyle Bettger, Tom Drake, Richard Arlen, Tracy Olsen, Paul Daniel, Dale Van Sickel, Robert Lowery, Reg Parton, Rodd Redwing, Charles Horvath, Chuck Hicks, Edmund Cobb. A sheriff brings the accused killer of an Indian chief's son to town for trail but finds most of the citizens support the prisoner. Average A.C. Lyles production enhanced by good performances from its veteran cast; Jerry Wallace sings the title song.\n\n**2051** _ **Johnny Tiger**_ **** Universal, 1966. 100 min. Color. D: Paul Wendkos. SC: Paul Crabtree and R. John Hough. With Robert Taylor, Geraldine Brooks, Chad Everett, Brenda Scott, Marc Lawrence, Ford Rainey, Carol Seflinger, Steven Wheeler, Pamela Melendez, Deanna Lund. In Florida a half-breed Seminole youth, in love with his teacher's pretty daughter, must decide whether to take over the leadership of his diminishing tribe or try for a new life for himself. Cheaply made but adequate modern-day drama with a fine performance by Robert Taylor as a dedicated teacher.\n\n**2052** _ **Johnny Tremain**_ **** Buena Vista, 1957. 80 min. Color. D: Robert Stevenson. SC: Tom Blackburn. With Hal Stalmaster, Luana Patten, Jeff York, Sebastian Cabot, Richard Beymer, Walter Sande, Rusty Lane, Whit Bissell, Will Wright, Virginia Christine, Walter Coy, Geoffrey Toone, Ralph Clanton, Gavin Gordon, Lumsden Hare, Anthony Ghazlo, Jr., Charles Smith. In 1773 a young silversmith apprentice loses his position and becomes involved with the Sons of Liberty, leading to the Revolutionary War. Good Walt Disney family film filled with lots of history.\n\n**2053** _ **Johnny West**_ **** C.E.A. Distribucion, 1965. 109 min. Color. D: Frank Kramer (Gianfranco Parolini). SC: Gianfranco Parolini, Gianfranco Simonelli, Robert De Nesle and Jose Luis Jerez Aloza. With Dick Palmer (Mimmo Palmara), Mike Anthony (Adriano Micantoni), Roger Delaporte, Andre Bollet, Mara Cruz, Diana Garson (Dada Gallotti), Barta Barri, Roberto Camardiel, Bob Fenton, Spanny Convery, Bruno Arie. A half-breed ends up in jail but later helps a lawman bring in an outlaw gang. Deservedly obscure and overlong Spaghetti Western released in Italy as _**Johnny West il Mancino**_ (Johnny West, the Left Handed) and also called _**Left Handed Johnny West**_.\n\n**2054** _ **Johnny Yuma**_ **** Clover Films, 1967. 99 min. Color. D: Romolo Guerrieri. SC: Fernando Di Leo. With Mark Damon, Rosalba Neri, Lawrence Dobkin, Louis Vanner, Fidel Gonzales, Gus Harper, Leslie Daniel, Dada Galotti, Gianni Solaro, Nando Poggi, Frank Liston. After inheriting his uncle's ranch, a cowboy learns the man's young wife had him murdered. Very violent and bloody oater made in Italy in 1966.\n\n**2055** _ **Johnson County Wars**_ **** Hallmark Channel, 2002. 240 min. Color. D: David S. Cass, Sr. SC: Larry McMurtry and Diana Ossana. With Tom Berenger, Luke Perry, Burt Reynolds, Rachel Ward, Michelle Forbes, Adam Storke, Christopher Cazenove, Jack Conley, Silas Weir Mitchell, Fay Masterson, Ken Pogue, Blu Makuma, William Samples, Stephen Bridgewater, P. Adrien Dorval, Stevie Mitchell, Jimmy Herman, Henry Beckman, Ron Hartman, Tim Koetting, Hal Kerbes, Paul Coeur, Tom Heaton, Lyle St. Goddard, John F. Parker, Stephen Warner, Billy Morton, Dave LeReaney, Doug Lennox, Kirk Jarrett, Steve Shayler, Joe Norman Shaw, Chris Ippolito, Tom Carey, J.C. Roberts, Joe Dodds, Peter Strand Rumpel, Shawn Orr, Jonathan Cole, Rodger Yule, Dusty Bews, Tom Eirikson, Cam Macdonald, Dan Heather, Bunk Duncan. Three feuding brothers, who love the same woman, unite to stop a coterie of cattle ranchers from taking over the range. Too long but more than passable TV mini-series from Frederick Manfred's novel _Riders of Judgement_.\n\n**2056** _ **Jory**_ **** Avco-Embassy, 1972. 97 min. Color. D: Jorge Fons. SC: Jerry Herman and Robert Irving. With John Marley, B.J. Thomas, Robby Benson, Brad Dexter, Claudio Brook, Ben (Benny) Baker, Patricia Aspillaga, Todd Martin, Linda Purl, Anne Lockhart, Betty Sheridan, Ted Markland, Quintin Buines, Carlos Cortes, John Kelly, Eduardo Lopez Rojas, Betty Sheridan, Howard Hesseman. A teenage boy sets out to get revenge on the men who murdered his father and friends. Made in Mexico, this oater was released theatrically only on an experimental basis but it is pretty good and marks Robby Benson's screen debut; Benny Baker gives a fine performance as the rancher who first shelters the boy.\n\n**2057** _ **Joshua**_ **** Lone Star, 1976. 90 min. Color. D: Larry Spangler. SC: Fred Williamson. With Fred Williamson, Calvin Bartlett, Brenda Venus, Isela Vega, Budd Stout, Henry Hendrick, Ralph Willingham, Kathryn Jackson, Neil Summers, Stonewall Jackson, Stacey Newton. Following the Civil War, a black soldier comes home to find his mother has been murdered by a vicious gang and he becomes a bounty hunter to get revenge. Fair, low key outing heavy on visuals and light on dialogue.\n\n**2058** _ **Journey Through Rosebud**_ **** Avco-Embassy, 1972. 92 min. Color. D: Tom Gries. SC: Albert Ruben. With Robert Forster, Kristoffer Tabori, Victoria Racimo, Eddie Little Sky, Roy Jenson, Wright King, Larry Pennell, Robert Cornwaithe, Steve Shemayne. A draft dodger hides from the law on an Indian reservation where he gets involved in the politics of the tribe and its troubles with the government. Tepid anti-war, pro\u2013Indian feature filmed in South Dakota.\n\n**2059** _ **Journey to Shiloh**_ **** Universal, 1968. 101 min. Color. D: William Hale. SC: Gene Coon. With James Caan, Michael Sarrazin, Brenda Scott, Paul Petersen, Don Stroud, Michael Burns, Michael Vincent, Harrison Ford, John Doucette, Noah Beery (Jr.), Tisha Sterling, James Gammon, Clark Gordon, Robert Pine, Wesley Lau, Chet Stratton, Bing Russell, Lane Bradford, Rex Ingram, Myron Healey, Eileen Wesson. A group of young men in 1862 Texas head East to join the Confederate Army without any idea as to what they are fighting for or the meaning of the war. A good premise gone awry in this rambling and none too satisfying melodrama.\n\n**2060** _ **The Journeyman**_ **** Dream Entertainment, 2001. 93 min. Color. D-SC: James Crowley. With Brad Hunt, Daniel Lapaine, Dash Mihok, Arie Verveen, Willie Nelson, Barry Corbin, Assumpta Serna, Burton Gilliam, John Beasley, Leon Singer, Chris Dahlberg, Joe Stevens, Octavia Spenser, Boots Sutherland, Daniel Grant, James Crowley, Matt Bearden, Tate Taylor, L.J. Burleson, Bela Armendariz, Alex Armendariz, Joey Hudgins, Trevor Nelson, Alex Smith, Ronnie Patilla, Big Jim Dicuffa, David Little, Marla Banda, E. Scott Perez, Ervin Laird, Juliana Sheffield, Lisa Hargus, Julianna Gilcrist. Two boys are spared by an outlaw gang who murder their rancher father and years later one of them tracks the other, who has become a ruthless morphine addict, in hopes of redeeming him. Pretty good independent Western.\n\n**2061** _ **Juan Charrasqueado**_ **** Filmadora Chapultepec, 1948. 90 min. D: Ernesto Cortazar. SC: Ramon Perez and Ignacio Villarreal. With Pedro Armendariz, Miroslava, Fernando Soto \"Mantequilla,\" Arturo Martinez, Luis Aceves Castaneda, Fernando Casanova, Angel Merino, Carlos Muquiz, Georgina Barragan, Silvia Rey. A vagabond romancer returns to his ranch home only to have a violent confrontation with his fianee's suitor. Fair Mexican Western helped by the presence of stars Pedro Armendariz and Miroslava.\n\n_**Juan Charrasqueado and Gabino Barrera**_ see _**Juan Charrasqueado y Gabino Barrera, Su Verdadera Historia**_\n\n**2062** _ **Juan Charrasqueado y Gabino Barrera, Su Verdadera Historia**_ (Juan Charrasqueado and Gabino Barrera, Their Truthful History). Cima Films, 1982. 105 min. Color. D: Rafael Villasenor. SC: Rafael Garcia Travesi. With Vicente Fernandez, Blanca Guerra, Miguel Angel Rodriguez, Jose Chavez, Jorge Fegan, Guillermo Lagunes, Albeerto Arvizu, Mario Arevalo. Two adventurers cross paths many times but eventually have a falling out over a woman. Pretty good Mexican Western released on video as _**Juan Charrasqueado and Gabino Barrera**_.\n\n**2063** _ **Juana Gallo**_ **** Producciones Zacarias, S.A., 1961. 120 min. Color. D-SC: Miguel Zacarias. With Maria Felix, Luis Aguilar, Jorge Mistral, Christiane Martel, Sonia Infante, Rene Cardona, Jose Alfredo Jiminez, Noe Murayama, Rita Macedo, Ignacio Lopez Tarso, Marina Camacho. After her father and fiance are murdered by government forces, a woman joins revolutionaries and urges her townspeople to fight. Well staged Mexican historical drama, culminating in the battle at Zacatecas; also called _**The Guns of Juana Gallo**_.\n\n**2064** _ **Juarez**_ **** Warner Bros., 1939. 132 min. D: William Dieterle. SC: John Huston, Aeneas MacKenzie and Wolfgang Reinhardt. With Paul Muni, Bette Davis, Brian Aherne, Claude Rains, John Garfield, Donald Crisp, Gale Sondergaard, Joseph Calleia, Gilbert Roland, Henry O'Neill, Pedro de Cordoba, Montagu Love, Harry Davenport, Walter Fenner, Alex Leftwich, Robert Warwick, John Miljan, Irving Pichel, Walter Kingsford, Monte Blue, Louis Calhern, Vladimir Sokoloff, Georgia Caine, Hugh Sothern, Fred Malatesta, Carlos de Valdez, Frank Lackteen, Gilbert Emery, Francis McDonald, Bill Wilkerson, Frank Reicher, Holmes Herbert, Egon Brecher, Mickey Kuhn, Noble Johnson, Martin Garralaga, Grant Mitchell, Charles Halton, Frank Mayo, Douglas Wood, Gennaro Curci, Walter O. Stahl, Manuel Diaz, Lillian Nicholson, William Edmunds. Benito Juarez rises to become the leader of the Mexican Revolution after French ruler Louis Napoleon tries to establish Maximilian as the emperor of the country. Overlong and basically boring biopic, mainly due to Paul Muni's stoic performance in the title role, although Bette Davis and Brian Aherne are great as Carlotta and Maximilian.\n\n**2065** _ **Jubal**_ **** Columbia, 1956. 101 min. Color. D: Delmer Daves. SC: Russell S. Hughes and Delmer Daves. With Glenn Ford, Ernest Borgnine, Rod Steiger, Valerie French, Felicia Farr, Basil Ruysdael, Noah Beery, Jr., Charles Bronson, John Dierkes, Jack Elam, Robert Burton, Robert Knapp, Juney Ellis, Don C. Harvey, Guy Wilkerson, Larry Hudson, Mike Lawrence, Buzz Henry, John Cason, Ann Kunde, William Rhinehart. A man is forced to shoot his best friend when falsely accused of having an affair with the dead man's wife, but finds love with a religious girl who hides him from a posse. Taut psychological drama of interest for its cast rather than the steamy plot.\n\n**2066** _ **Jubilee Trail**_ **** Republic, 1954. 103 min. Color. D: Joseph Kane. SC: Bruce Manning. With Vera Ralston, Joan Leslie, Forrest Tucker, John Russell, Ray Middleton, Pat O'Brien, Buddy Baer, Jim Davis, Barton MacLane, Richard Webb, James Millican, Nina Varela, Martin Garralaga, Charles Stevens, Nacho Galindo, Don Beddoe, John Holland, William Haade, Alan Bridge, John Halloran, Stephen Chase, Dan White, Eugene Borden, Rodolfo Hoyos, Bud Wolfe, Paul Stander, Marshall Reed, Maurice Jara, Rosa Turich, Manuel Lopez, Perry Lopez, Claire Carleton, Victor Sen Yung, Edward Colmans, George Navarro, Grant Withers, Frank Puglia, Pepe Hern, Glenn Strange, Felipe Turich, Joe Dominguez, Emil Sitka, Emmett Lynn, Tex Terry, Rocky Shahan, Chuck Hayward, Jack O'Shea, Jack Elam, Tina Menard, Buzz Henry, Pilar Del Rey, Charles Sullivan, Rico Alaniz, Ralph Brooks, Sayre Dearing, Morris Buchanan, Frances Dominguez. A woman wanted for murder travels West with a young widow and her baby, the infant being kidnapped by the mother's crooked brother-in-law. Big, brawling adaptation of Gwen Bristow's popular novel provides good screen fare.\n\n**2067** _ **Judgment Book**_ **** Beaumont, 1935. 61 min. D: Charles Hutchison. SC: E.J. Thornton. With Conway Tearle, Bernadine Hayes, Howard Lang, Richard Cramer, William Gould, Jack Pendleton, Roy Rice, Jimmy Aubrey, Ray Gallagher, Dick Rush, Blackie Whiteford, Francis Walker, Edward Clayton. When his newspaper editor uncle is murdered by ruthless cattlemen intent on dominating a town, a Easterner shows up to take over the business and oppose the lawless. Pretty fair Conway Tearle vehicle if one can accept the British actor, then in his fifties, as a young man.\n\n**2068** _ **Junction City**_ **** Columbia, 1952. 54 min. D: Ray Nazarro. SC: Barry Shipman. With Charles Starrett, Smiley Burnette, Jack (Jock) Mahoney, Kathleen Case, John Dehner, Steve Darrell, George Chesebro, Anita Castle, Mary Newton, Robert Bice, Hal Price, Hal Taliaferro, Chris Alcaide, Bob Woodward, Frank Ellis, Joel Friedkin, Harry Tyler, The Sunshine Boys (Eddie Wallace, J.D. Sumner, M.H. Richman, Freddie Daniel). The Durango Kid helps a stage driver falsely accused of kidnapping his fiancee, who is really in hiding to prevent her guardian from killing her for the rich mine she inherited. Passable \"Durango Kid\" film, the penultimate release in the series.\n\n**2069** _ **Junior Bonner**_ **** Cinerama Releasing, 1972. 100 min. D: Sam Peckinpah. SC: Jed Rosebrook. With Steve McQueen, Robert Preston, Ida Lupino, Ben Johnson, Joe Don Baker, Barbara Leigh, Mary Murphy, Bill McKinney, Sandra Deel, Donald Barry, Dub Taylor, Charles Gray, Matthew Peckinpah, Sundown Spencer, Rita Garrison, Casey Tibbs, Rod Hart. A rodeo circuit performer returns home to take part in a local show and tries to re-establish a relationship with his parents. Nicely done melodrama of rodeo life.\n\n**2070** _ **Just Pals**_ **** Fox, 1920. 50 min. D: John Ford. SC: Paul Schofield. With Buck Jones, Helen Ferguson, George E. Stone, Duke R. Lee, William Buckley, Edwin Booth Tilton, Eunice Murdock, Burt Apling, Slim Padgett, Pedro Leone, Ida Tenbrook, John J. Cooke. A town loafer is made into a he-man when he befriends a young vagabond boy. Well made and entertaining Buck Jones feature, one of his few available silent vehicles.\n\n**2071** _ **Just Tony**_ **** Fox, 1922. 58 min. D-SC: Lynn Reynolds. With Tom Mix, Claire Adams, J.P. Lockney, Duke R. Lee, Frank Campeau, Walt Robbins. A cowboy saves a wild mustang from men who want to beat it and the horse later returns the favor by rescuing his benefactor and a rancher's daughter from trouble. Tom Mix's beautiful horse Tony is spotlighted in this delightfully action packed silent oater.\n\n**2072** _ **Just Travelin'**_ **** Sierra Pictures, 1925. 54 min. D-SC: Horace B. Carpenter. With Bob Burns, Dorothy Donald, Tex (Alfred) Hewston, Lew Meehan, Harry O'Connor, Jack Radke. A cowboy and his sidekick go up against a ruthless outlaw who captures a miner as well as his pretty daughter, wanting the location of the prospector's valuable mine and wanting to marry the girl. Less than satisfying silent oater with stoic hero Bob Burns.\n\n**2073** _ **Justice of the Range**_ **** Columbia, 1935. 58 min. D: David Selman. SC: Ford Beebe. With Tim McCoy, Billie Seward, Ward Bond, Guy Usher, Ed LeSaint, Allan Sears, Jack Rockwell, Jack Rutherford, George Hayes, Bill Patton, Stanley Blystone, Earl Dwire, Dick Rush, J. Frank Glendon, Frank Ellis, Tom London, Bud Osborne, Richard Botiller, Henry Hall, Wally West, Ray Jones. A cowboy is hired to find out who is behind a cattle rustling gang but when a ranch foreman is murdered he is accused of the crime and tries to clear himself. Very fine entry in Tim McCoy's Columbia series with a good script and acting.\n\n**2074** _ **Justice of the West**_ **** Wrather Corporation, 1956. 75 min. Color. D: Earl Bellamy and Oscar Randolph. SC: Robert Schaefer, Eric Friewald, Walter A. Thompson and Robert Leslie Bellem. With Clayton Moore, Jay Silverheels, Allen Pinson, Wayne Burson, Terry Frost, Denver Pyle, Bill Henry, Joseph Crehan, House Peters, Jr., Tom Steele, Steven Ritch, Russell Sanders, Robert Burton, Henry Rowland, Ric Roman, Ron Hagherty, John Berardino, Mickey Simpson, Tudor Owen, Will Wright, James D. Parnell, Gary Lee Marshall. The Lone Ranger and Tonto track down a stolen million dollar gold shipment, hunt marauders who murdered an elderly sheriff and try to save a man from being hanged for a robbery he did not commit. Good telefeature made up of three episodes of \"The Lone Ranger\" (ABC-TV, 1949\u201357): \"No Handicap,\" \"Outlaw Masquerade\" and \"Quicksand.\"\n\n_**Justice Rides Again**_ see _**Destry Rides Again**_ (1932)\n\n**2075** _ **La Justicia del Coyote**_ (The Justice of the Coyote) **** Centauro Films\/Oro Films, 1956. 75 min. D: Joaquin Luis Romero Marchent. SC: J. Chamor (Pedro Chamorro) and Jesus (Jess) Franco. With Abel Salazar, Gloria Morin, Manuel Monroy, Rafael Bardem, Miguel Pastor Mata, Antonio Garcia Quijada, Emilio Rodriguez, Carlos Otero, Julio Gorostegui, Mario Moreno, Jose Rey, Luis Dominguez, Antonio Fornes, Angel Alvarez, Manuel San Roman, Alfred Muniz, Pepita Bravo, Jose Riesgo, Joaquin Burgos, Hector Mayro. A mysterious masked man helps peasants being evicted from their land by corrupt military men. A Mexican-Spanish co-production, this is an okay sequel to _**El Coyote**_ (q.v.).\n\n**2076** _ **Justin Morgan Had a Horse**_ **** Buena Vista, 1972. 91 min. Color. D: Hollingsworth Morse. SC: Calvin Clements, Jr. and Rod Peterson. With Don Murray, Lana Wood, R.G. Armstrong, Whit Bissell, Gary Crosby, John Smith, James Hampton, John Hubbard, E.W. Firestone, Mike Road. A cowboy raises a colt to become the sire of a famous line of horses. Well done telefilm originally shown in two parts on NBC-TV's \"Walt Disney's Wonderful World of Color.\"\n\n**2077** _ **J.W. Coop**_ **** Columbia, 1972. 112 min. Color. D-SC: Cliff Robertson. With Cliff Robertson, Geraldine Page, Christina Farrare, R.G. Armstrong, R.L. Armstrong, John Crawford, Wade Crosby, Marjorie Durant Dye, Paul Harper, Son Hooker, Richard Kennedy, Bruce Kirby, Claude Stroud. After a decade in prison, a one time rodeo performer decides to return to the circuit and become all-around cowboy but finds the times and ways of the sport have changed. Cliff Robertson does a good job in the title role and he is equally fine as the film's director and writer.\n\n**2078** _ **Der Kaiser von Kalifornien**_ (The Emperor of California) **** Rota-Film Verleik AG, 1936. 97 min. D-SC: Luis Trenker. With Luis Trenker, Viktoria von Ballasko, Werner Kunig, Karli Zwingmann, Elise Aulinger, Bernhard Minetti, Hans Zesch-Ballot, Marcella Albani, Walter Franck, Reginald Pasch, August Eichhorn, Luis Gerold, Paul Verhoeven, Melanie Horeschovsky, Berta Drews, Alexander Golling, Heinrich Marlow, Rudolf Klein-Rogge, Otto Stockel, Bruno Ziener, Josef Reithofer, Jakob Sinn, Erich Dunskus, Armin Schwiezer, Jim Diehl, Jim Simmons. Immigrant Johann Sutter, who builds an empire in California, loses everything when gold is discovered on his land and it is over run by prospectors. Outstanding German production from producer-director-star Luis Trenker, with location filming in California; issued in the U.S. in 1937 by American Tobis Company in a dubbed version.\n\n**2079** _ **Kangaroo**_ **** 20th Century\u2013Fox, 1951. 84 min. Color. D: Lewis Milestone. SC: Harry Kleiner. With Maureen O'Hara, Peter Lawford, Richard Boone, Finlay Currie, Chips Rafferty, Letty Graydon, Charles Tingwell, Ron Whelan, John Fegan, Guy Doleman, Reg Collins. Two Americans in Australia become involved with murder, a beautiful woman and a cattle drive. The Australian scenery is the main asset of this otherwise mundane Down Under oater.\n\n**2080** _ **Kangaroo Kid**_ **** Eagle-Lion, 1950. 73 min. D: Lesley Selander. SC: Anthony S. Veitch. With Jock (Mahoney) O'Mahoney, Martha Hyer, Douglass Dumbrille, Veda Ann Borg, Guy Doleman, Alec Kellaway, Alan Gifford, Grant Taylor, Frank Ransome, Haydee Seldon, Clarrie Woodland. Sent to Australia to bring back a fugitive, a frontier detective gets blamed for a gold heist after signing on as a stage driver. Slight Australian Western.\n\n**2081** _ **The Kansan**_ **** United Artists, 1943. 79 min. D: George Archainbaud. SC: Harold Shumate. With Richard Dix, Jane Wyatt, Victor Jory, Albert Dekker, Eugene Pallette, Robert Armstrong, Clem Bevans, Rod Cameron, Francis McDonald, Willie Best, Glenn Strange, Douglas Fowley, Jack Norton, Eddy Waller, Ray Bennett, Sam Flint, Merrill McCormick, Jack Mulhall, Hobart Cavanaugh, Eleanor Counts, Byron Foulger, Russell Simpson, The King's Men, Beatrice Gray. A frontiersman is hired to rid a town of the James gang but ends up opposing the corrupt officials who employed him. Action filled and entertaining Richard Dix vehicle from Harry Sherman Productions.\n\n**2082** _ **Kansas Cyclone**_ **** Republic, 1941. 58 min. D: George Sherman. SC: Oliver Drake and Doris Schroeder. With Don \"Red\" Barry, Lynn Merrick, Dorothy Sebastian, William Haade, Milton Kibbee, Harry Worth, Jack Kirk, Forrest Taylor, Charles Moore, Eddie Dean, Reed Howes, Guy Usher, Ed Peil, Sr., Yakima Canutt, Cactus Mack, Bob Woodward, Tex Terry, George J. Lewis, Buddy Roosevelt. Outlaws attacking Wells Fargo shipments are hunted by a U.S. marshal determined to stop the holdups. Another action filled, speedy Don Barry Western.\n\n**2083** _ **Kansas Pacific**_ **** United Artists, 1953. 73 min. Color. D: Ray Nazarro. SC: Daniel B. Ullman. With Sterling Hayden, Eve Miller, Barton MacLane, Harry Shannon, Reed Hadley, Tom Fadden, Douglas Fowley, Irving Bacon, Myron Healey, James Griffith, Clayton Moore, Jonathan Hale, Bob Keys, Lane Bradford, Lee Roberts, I. Stanford Jolley, Riley Hill, Carol Henry, Fred Graham. An engineer tries to build the Kansas Pacific Railroad during the Civil War but the project is plagued by Confederate guerrilla raids. Pretty fare outing with good work by Clayton Moore in a villainous role.\n\n**2084** _ **Kansas Raiders**_ **** Universal-International, 1950. 80 min. Color. D: Ray Enright. SC: Robert L. Richards. With Audie Murphy, Brian Donlevy, Marguerite Chapman, Scott Brady, Tony Curtis, Richard Arlen, Richard Long, James Best, John Kellogg, Dewey Martin, George Chandler, Charles Delaney, Richard Egan, Jack Perrin, David Wolfe, Mira McKinney, Sam Flint, Buddy Roosevelt, Larry McGrath, Ed Peil, Sr., Helen Gibson, Robert Anderson, Lee Fredericks, David Newell, Richard Farmer, Ray Grimes, Jennings Miles. The James brothers join Quantrill during the Civil War with Jesse caring for the rebel leader when he blinded and later Quantrill saves his life. Colorful and well acted, but historically empty.\n\n**2085** _ **Kansas Territory**_ **** Monogram, 1952. 65 min. D: Lewis D. Collins. SC: Daniel B. Ullman. With Bill Elliott, Peggy Stewart, Lane Bradford, Marshall Reed, I. Stanford Jolley, House Peters, Jr., Lyle Talbot, Terry Frost, John Hart, William Fawcett, Fuzzy Knight, Stanley Andrews, Lee Roberts, Ted Adams, Pierce Lyden. A man wrongly wanted on an old charge returns home to avenge the death of his brother. Very well done with a good story, cast and action; partially filmed in Sepiatone.\n\n**2086** _ **The Kansas Terrors**_ **** Republic, 1939. 57 min. D: George Sherman. SC: Jack Natteford and Betty Burbridge. With Robert Livingston, Raymond Hatton, Duncan Renaldo, Jacqueline Wells (Julie Bishop), Howard Hickman, George Douglas, Frank Lackteen, Myra Marsh, Yakima Canutt, Ruth Robinson, Artie Ortego, Richard Alexander, Merrill McCormick, Curley Dresden, Al Haskell, Ann Baldwin, Henry Wills, Rosa Turich, Richard Botiller, Joe Dominguez, Billy Bletcher. Mesquiteers Stony Brooke and Rusty Joslin take a job delivering horses for the government to a Caribbean island and there they team with Rico to defeat a tyrant. Although the plot is fairly interesting, the sudden change of locale and characters does not help this \"Three Mesquiteers\" entry.\n\n**2087** _ **Kate Bliss and the Ticker Tape Kid**_ **** ABC-TV, 1978. 100 min. Color. D: Burt Kennedy. SC: William Bowers and John Zodorow. With Suzanne Pleshette, Don Meredith, Harry Morgan, David Huddleston, Tony Randall, Burgess Meredith, Buck Taylor, Jerry Hardin, Gene Evans, Don Collier, Alice Hirson, Harry Carey, Jr., Don \"Red\" Barry, Richard Herd, James Brewer, Blair Burrows, Peggy Rea, George Dunn, Alice Backus, John Wheeler, Ned Wertimer, Alvy Moore, John Hart, John Pickard, Mike Wagner. At the turn of the century a lady detective from the East gets on the trail of a masked renegade and his band who oppose a British land baron after a ranch. Surprisingly well done tongue-in-cheek TV Western.\n\n**2088** _ **Kazan**_ **** Columbia, 1949. 65 min. D: Will Jason. SC: Arthur A. Ross. With Stephen Dunne, Lois Maxwell, Joseph Sawyer, Roman Bohnen, George Cleveland, John Dehner, Ray Teal, Loren Gage, Zorro (dog). Stolen by crooks, a huge sled dog escapes and sets out to find his master. Fair program feature adaptation of the James Oliver Curwood novel.\n\n_**Keep Rollin'**_ see _**Gaucho Serenade**_\n\n**2089** _ **Keep the Change**_ **** Turner Pictures, 1992. 95 min. Color. D: Andy Tennant. SC: John Miglis. With William L. Petersen, Rachel Ticotin, Lolita Davidovich, Buck Henry, Jeff Kober, Fred Dalton Thompson, Jack Palance, Angela Paton, Lois Smith, Frank Collison, James Ellis, William Frankfather, Ron Ray, Charlie Carpenter, Clive Rosengren, Jim Bishop, Bret Tuomi, Peter Walther, Sydney Warner. After failing as an artist, a man returns to his family's mountain ranch where he confronts his past. Fine modern-day Western made for cable TV.\n\n**2090** _ **Keeping the Promise**_ **** CBS-TV, 1997. 95 min. Color. D: Sheldon Larry. SC: Gerald Di Pego. With Keith Carradine, Annette O'Toole, Brendan Fletcher, Gordon Tootoosis, Maury Chaykin, Camilla Scott, Allegra Denton, William Lightning, Michael Stevens, Ned Geisslinger, Darrell Dennis. Prior to the Revolutionary War, a Massachusetts family moves to the Maine frontier seeking a better life. Average pioneer drama made for television; also called _**Sign of the Beaver**_.\n\n**2091** _ **Kelly**_ **** Paramount\/Famous Players Film Corporation, 1981. 93 min. Color. D: Christopher Chapman. SC: Robert Logan. With Robert Logan, Twyla-Dawn Vokins, George Clutesi, Elaine Natee, Doug Lennox, Alec Willows, Dan Granier, Jack Leaf, Mona Cozart. A young girl with a perceptual handicap goes to Canada to live with her bush pilot father. Nice scenery and a fair amount of action make this Canadian production okay viewing.\n\n**2092** _ **Kenny Rogers as the Gambler**_ **** CBS-TV, 1980. 105 min. Color. D: Dick Lowry. SC: Jim Byrnes. With Kenny Rogers, Bruce Boxleitner, Clu Gulager, Harold Gould, Christine Belford, Lee Purcell, Lance LeGault, Ronnie Scribner, Bruce M. Fischer, Noble Willingham, Borah Silver, Lew Brown, Robert Lussier, Edward Walsh, Marianne Gordon, Dave Cass, Cathy Worthington, Jerry Willis, Neil Summers, Charles Knapp, Ed Bakey. In the Southwest, a gambler returns to the town where his son and the woman he never married live, although an enemy waits there for him. Based on the Grammy Award winning song, this TV movie should please Kenny Rogers fans; also called _**The Gambler**_.\n\n**2093** _ **Kenny Rogers as the Gambler\u2014The Adventure Continues**_ **** CBS-TV, 1982. 200 min. Color. D: Dick Lowry. SC: Jim Byrnes. With Kenny Rogers, Linda Evans, Bruce Boxleitner, Mitchell Ryan, Charlie Fields, Harold Gould, Cameron Mitchell, Gregory Sierra, Ken Swofford, Paul Koslo, David Hedison, Johnny Crawford, Brion James, Robert Hoy, Macon McCalman, Lee Paul, Roy Jenson, Gary Cox, Ann Gillespie, Marianne Gordon, Bill Hart, Kelly Junkermann, Hank Kendrick, Joe Massengale, Cliff McLaughlin, Gene McLaughlin, Patrick O'Brien, John Putch, Roy Rogers, Bob Terhune, Henry Wills, Earl Smith, Lelan Rogers, Monty Simons, John Tatum, Bunky Young, Cathy Worthington, Ron Colby, Debbie Atkinson, Randy Patrick, Whitney Rybeck. A gambler whose son has been abducted by outlaws joins forces with a buddy and a female bounty hunter to track down the gang. Leisurely TV film follow-up to _**Kenny Rogers as the Gambler**_ (q.v.).\n\n**2094** _ **Kenny Rogers as the Gambler, Part III:**_ ****_**The Legend Continues**_ **** CBS-TV, 1987. 240 min. Color. D: Dick Lowry. SC: Jeb Rosebrook and Roderick Taylor. With Kenny Rogers, Bruce Boxleitner, Linda Gray, Melanie Chartoff, Matt Clark, George Kennedy, Dean Stockwell, Charles Durning, Jeffrey Jones, Marc Alaimo, Lowell D. Smith, Jimmie F. Skaggs, Brenda Strong, George American Horse, Marvin J. McIntyre, Michael Berryman, Lenora May, Richard Chaves, Sandy Martin, Terrence Evans, Rion Hunter, Ann Gillespie, Jeff Allin, Colin Meaney, Manny Twofeathers, Larry Sellers, Monty Stuart, Marco Rodriguez, Gene McLaughlin. A gambler and his friend find corruption by government agents who are supposed to be supplying food to the Sioux nation. This third chapter of the \"Gambler\" saga wears pretty thin.\n\n**2095** _ **The Kentuckian**_ **** United Artists, 1955. 104 min. Color. D: Burt Lancaster. SC: A.B. Guthrie, Jr. With Burt Lancaster, Dianne Foster, Diana Lynn, John McIntire, Una Merkel, Walter Matthau, John Carradine, Donald MacDonald, John Litel, Rhys Williams, Edward Norris, Lee Erickson, Clem Bevans, Lisa Ferraday, Douglas Spencer, Paul Wexler. In the 1820s a man and his son head West to Texas Territory but along the way they are sidetracked by two pretty women, a servant girl, a school teacher and a corrupt town boss. Okay frontier drama with adequate entertainment value.\n\n**2096** _ **Kentucky Rifle**_ **** Howco International, 1956. 80 min. Color. D: Carl K. Hittleman. SC: Carl K. Hittleman and Lee J. Hewitt. With Chill Wills, Lance Fuller, Cathy Downs, Henry Hull, Jess Barker, Jeanne Cagney, Sterling Holloway, John Pickard, John Alvin, I. Stanford Jolley, Rory Mallinson, George Keymas. Due to breakdowns, pioneers are forced to leave a wagon train in Comanche territory and later find travel impeded by Indians who want their cargo of rifles. Cheap production values hurt his otherwise adequate tale; Stanley Price was the feature's dialogue coach and Ira S. Webb its executive producer.\n\n**2097** _ **Keoma**_ **** Vadib International Films, 1976. 105 min. Color. D: Enzo G. Castellari. SC: Enzo G. Castelllari, Nico Ducci, Lugi Montefiori (George Eastman) and Mino Roli. With Franco Nero, William Berger, Olga Karlatos, Woody Strode, Orso Maria Guerrini, Gabriella Giacobbe, Antonio Marsina, John Loffredo, Donald O'Brien, Leon Lenoir, Wolfgango Soldati, Victoria Zinny, Alfio Caltabliano, Riccardo Pizzuti. A half-breed returns home from the Civil War to find a Rebel raider and his men, including his half-brother, in control of the area. One of the better Spaghetti Westerns, also called _**Django Rides Again**_.\n\n_**Kettle Creek**_ see _**Mountain Justice**_\n\n**2098** _ **The Kid and the Gunfighter**_ **** Saga Films International, 1983. 92 min. Color. D: Romy Suzara. SC: Tony Calvento. With Chuck Biller, Cole McKay, Paul Jones, Connie Angeles, Lito Lapid, Brad Fletcher, Terry Reynolds, Bret Davidson, Curt Campau, Jerry Hall, Emil Varga, Linda King. An Indian boy whose family was slaughtered by outlaws is adopted by a gunman. Low budget affair also called _**Born to Fight**_ and _**The Gunfighter**_.\n\n**2099** _ **The Kid and the Killers**_ **** Cinema Shares, 1979. 90 min. Color. D: Ralph Bluemke. SC: Ralph Bluemke and John Garces. With Jon Cypher, John Garces, Gerry Ross, Elida Alicia, Ralph Bluemke, Jamie Delgado, Joel Douglas, Susan Douglas, Gino Eqcclo, Eduardo Mosquera, Hector Rojas, Dan Ross. After two bandits take refuge with a young boy and his sister, one of them rapes and murders the girl and takes off with stolen loot as the brother and the other outlaw team to get him. Meandering drama enhanced by its rural Mexican locales.\n\n**2100** _ **Kid Blue**_ **** 20th Century\u2013Fox, 1973. 100 min. Color. D: James Frawley. SC: Edwin Shrake. With Dennis Hopper, Warren Oates, Peter Boyle, Ben Johnson, Lee Purcell, Janice Rule, Ralph Waite, Clifton James, Jose Torvay, Mary Jackson, Howard Hesseman, Jay Varela, Emmett Walsh. A hellion outlaw tries to settle down in a Texas town at the turn of the century but finds it rough going, especially after his friend's wife seduces him. None too amusing genre spoof filmed in Mexico.\n\n**2101** _ **The Kid Comes Back**_ **** Warner Bros., 1938. 61 min. D: B. Reeves Eason. SC: George Bricker. With Wayne Morris, June Travis, Barton MacLane, James Robbins, Joseph Crehan, Dickie Davis, Maxie Rosenbloom, Frank Otto, David Carlyle, Herbert Rawlinson, Robert Homans, Ken Niles. In order to raise money to save his ranch a cowboy turns to prizefighting and falls in love with a rival's sister. Fast paced action program feature with genre elements only at the beginning before turning into a boxing drama.\n\n**2102** _ **Kid Courageous**_ **** Supreme, 1935. 53 min. D-SC: Robert North Bradbury. With Bob Steele, Renee Borden, Kit Guard, Arthur Loft, Jack Powell, Lafe McKee, Vane Calvert, Perry Murdock, John Elliott, Barry Seury. After a series of thefts, a man heads West to catch the robber and saves a pretty girl from a marriage she does not want. Cheaply made but fast moving and very entertaining Bob Steele opus.\n\n**2103** _ **The Kid from Amarillo**_ **** Columbia, 1951. 56 min. D: Ray Nazarro. SC: Barry Shipman. With Charles Starrett, Smiley Burnette, Harry Lauter, Fred F. Sears, Don Megowan, The Cass County Boys (Jerry Scoggins, Bert Dodson, Fred S. Martin), Scott Lee, Guy Teague, Charles Evans, George J. Lewis, Henry Kulky, George Chesebro. Two U.S. Treasury agents ride to the Mexican border to capture a gang of clever silver smugglers. Fair \"Durango Kid\" outing; remade as an episode of \"Tales of Texas Rangers\" (ABC-TV, 1958\u201359) with Harry Lauter playing the same role in both productions. British title: _**Silver Chains**_.\n\n**2104** _ **The Kid from Arizona**_ **** Cosmos, 1931. 55 min. D: Robert J. Horner. SC: Robert Walker. With Jack Perrin, Josephine Hill, Robert Walker, George Chesebro, Henry Roquemore, Ben Corbett. A marshal is sent to the badlands to stop raids by marauding Indians. Any connection between this film and entertainment is purely coincidental.\n\n**2105** _ **The Kid from Broken Gun**_ **** Columbia, 1952. 56 min. D: Fred F. Sears. With Ed Earl Repp and Barry Shipman. With Charles Starrett, Smiley Burnette, Jack (Jock) Mahoney, Angela Stevens, Tristram Coffin, Myron Healey, Pat O'Malley, Helen Mowery, Chris Alcaide, John Cason, Mauritz Hugo, Edgar Dearing, Eddie Parker, Charles Horvath, Edward Hearn, The Sunshine Boys (Eddie Wallace, J.D. Sumner, M.H. Richman, Freddie Daniel). Two men arrive in a town to help their friend, an ex-fighter, who has been falsely accused of robbery and murder. Sewn together finale to the \"Durango Kid\" series with much footage from _**Fighting Frontiersman**_ (q.v.).\n\n**2106** _ **The Kid from Gower Gulch**_ **** Friedgen, 1950. 57 min. D: Oliver Drake. SC: Elmer (Clifton) S. Pond. With Spade Cooley, Wanda Cantlon, Bob Gilbert, Billy Dix, Jack Baxley, Little Joe Hiser, William Val, Stephen Keyes, Robert Curtis. A Hollywood sagebrush star who has no western background is roped into representing a ranch at a local rodeo and has to take a crash course in being a cowboy. One of the most incredibly cheap oaters of all time\u2014even the stock shots do not match; must be seen to be believed!\n\n**2107** _ **The Kid from Santa Fe**_ **** Monogram, 1940. 57 min. D: Raymond K. Johnson. SC: Carl Krusada. With Jack Randall, Claire Rochelle, Forrest Taylor, Clarene Curtis, Tom London, George Chesebro, Dave O'Brien, Jimmy Aubrey, Kenne Duncan, Steve Clark, Carl Mathews, Buzz Barton, Tex Palmer. The Santa Fe Kid helps a sheriff in tracking down a gang of smugglers. Fair Jack Randall vehicle.\n\n**2108** _ **The Kid from Texas**_ **** Metro-Goldwyn-Mayer, 1939. 71 min. D: S. Sylvan Simon. SC: Florence Ryerson and Albert Mannheimer. With Dennis O'Keefe, Florence Rice, Anthony Allan, Jessie Ralph, Buddy Ebsen, Virginia Dale, Robert Wilcox, Jack Carson, Helen Lynd, J.M. Kerrigan, Tully Marshall, George Meeker, Syd Saylor, Spencer Charters, George DeNormand, Artie Ortego, Howard Hickman, Gerald Oliver-Smith, Tommy Mack, Harry C. Bradley, Eddy Chandler, Jerry Frank, Allen Pomeroy, Jim Dundee, Snowy Baker. A polo playing cowboy comes East, becomes the manager of a Long Island estate and falls in love with the daughter of the proprietor of a wild west show. Humorous program feature.\n\n**2109** _ **The Kid from Texas**_ **** Universal-International, 1950. 78 min. Color. D: Kurt Neumann. SC: Robert Hardy Andrews and Karl Lamb. With Audie Murphy, Gale Storm, Albert Dekker, Shepperd Strudwick, Will Geer, William Talman, Martin Garralaga, Robert Barrat, Walter Sande, Frank Wilcox, Dennis Hoey, Ray Teal, Don Haggerty, Paul Ford, Zon Murray, Rosa Turich, Pilar Del Rey, Harold Goodwin, Edmund Cobb, John Carpenter, Terry Frost, Pierce Lyden, Rory Mallinson, William Fawcett, Jack Ingram, John Phillips, Tom Trout, Dorita Pallais. Billy the Kid goes to work for a rancher but when the man will not sell his cattle at prices a rival wants he is murdered and Billy vows revenge. Not much history but there is plenty of action.\n\n**2110** _ **The Kid Ranger**_ **** Supreme, 1936. 57 min. D-SC: Robert North Bradbury. With Bob Steele, Geraine Greer (Joan Barclay), William Farnum, Earl Dwire, Charles King, Lafe McKee, Frank Ball, Buck Moulton, Horace Murphy, Reetsy Adams, Paul and Paulina, Robert North Bradbury. On the trail of a gang of outlaws, a ranger accidentally shoots the wrong man. Another fast paced Bob Steele film from producer A.W. Hackel.\n\n**2111** _ **The Kid Rides Again**_ **** Producers Releasing Corporation, 1943. 57 min. D: Sherman Scott (Sam Newfield). SC: Fred Myton. With Buster Crabbe, Al St. John, Iris Meredith, Glenn Strange, Charles King, I. Stanford Jolley, Ed Peil, Sr., Ted Adams, Slim Whitaker, Karl Hackett, Kenne Duncan, Curley Dresden, Snub Pollard, John Merton, Jim Mason, Steve Clark, Frank McCarroll, Roy Bucko, Tex Phelps, Tex Cooper, Milburn Morante, Al Haskell, Rose Plummer. Billy the Kid is arrested for a train robbery he did not commit and breaking out of jail to find the real culprits he discovers them posing as honest ranchers. Typical fast and cheap \"Billy the Kid\" PRC series entry. Also called _**Billy the Kid Rides Again**_.\n\n**2112** _ **Kid Rodelo**_ **** Paramount, 1966. 91 min. D: Richard Carlson. SC: Jack Natteford. SC: Don Murray, Janet Leigh, Broderick Crawford, Richard Carlson, Jose Nieto, Julia Pena, Miguel Del Castillo, Emilio Rodriguez, Jose Villa Sante, Miguel Brendel. After spending a year in jail for being in the company of an outlaw, an embittered man plans to find the hidden $50,000 in gold he was accused of stealing. Based on a story by Louis L'Amour, this adequate oater was filmed in Spain.\n\n**2113** _ **Kid Vengeance**_ **** Cannon Films, 1977. 94 min. Color. D: Joe Manduke. SC: Budd Robbins and Jay Telfer. With Lee Van Cleef, Jim Brown, John Marley, Leif Garrett, Glynnis O'Connor, Matt Clark, Timothy Scott, David Loden. When outlaws murder his folks and abduct his sister, a young man seeks revenge along with rescuing the girl. Overly violent, none too entertaining feature made in Israel.\n\n**2114** _ **The Kid's Last Ride**_ **** Monogram, 1941. 55 min. D: S. Roy Luby. SC: Earle Snell. With Ray Corrigan, John King, Max Terhune, Luana Walters, Edwin Brian, Alan Bridge, Glenn Strange, Frank Ellis, John Elliott, George Havens, Tex Palmer, George Morrell, Carl Mathews. A crook blackmails a man into revealing where ranchers' cattle sale money is hidden and the Range Busters show up pretending to help the bad man. Fair series affair that drags a bit but the songs include John King's warbling of \"Call of the Wild,\" Ray Corrigan and King doing a bit of \"Home on the Range\" and the trio, plus Elmer, singing \"It's All a Part of the Game.\"\n\n_**Kill and Pray**_ see _**Requiescant**_\n\n_**Kill Johnny R**_ see _**Who Killed Johnny R?**_\n\n_**Kill or Be Killed**_ see _**God Holds the Bullet**_\n\n**2115** _ **Kill the Wicked!**_ **** R.K. Cinematografica\/Danny Film, 1967. 95 min. Color. D: Amerigo Anton (Tanio Boccia). SC: Mike Ashley. With Benny Hudson, Robert Mark (Rod Dana), Men Fury (Furio Meniconi), Max Dean (Massimo Righi), Maria Silva, Daniela Igliozzi, Vivi Gio, Benito Stefanelli, Jose Bastida, Luis Ferrin. A cowboy and a young widow ride into a ghost town where they are taken prisoners by four outlaws. A tangled plot does not help this otherwise mediocre Italian-Spanish co-production filmed in Italy as _**Dio non Paga, il Sabato**_ (God Does Not Pay on Saturday).\n\n**2116** _ **Kill Them All and Come Back Alone**_ **** Fanfare, 1970. 93 min. Color. D: Enzo G. Castellari. SC: Tito Carpi, Enzo G. Castellari and Joaquin Romero Hernandez. With Chuck Connors, Frank Wolff, Ken Wood (Giovanni Cianfriglia), Franco Citti, Leo Anchoriz, Alberto Dell'Acqua (Robert Widmark), Hercules Cortes, John Bartha, Men Fury (Furio Meniconi), Antonio Molin Rojo, Alfonso Rojas. An escaped prisoner during the Civil War takes part in the robbery of gold from an ammunition depot and when he is double crossed and left for dead by the mastermind of the heist he plans revenge. Complicated and bloody Spaghetti Western produced in Italy in 1968 as _**Ammazzali Tutti de Torna Solo**_ (Go and Kill Everybody and Come Back Alone).\n\n_**Killbrand**_ see _**The Deadly Trackers**_\n\n_**Killer Grizzly**_ see _**Grizzly**_\n\n**2117** _ **Kimberley Jim**_ **** Embassy, 1965. 81 min. Color. D-SC: Emil Nofal. With Jim Reeves, Madeleine Usher, Clive Parnell, Arthur Swemmer, Mike Holt, Tromp Terre'blanche, Vonk de Ridder, David Van Der Walt, June Neething, George Moore, The Blue Boys. Two dishonest gamblers win a diamond mine in a fixed poker game but find out it is worthless. Pleasant South African musical Western that will appeal most to Jim Reeves fans.\n\n**2118** _ **The King and Four Queens**_ **** United Artists, 1956. 86 min. Color. D: Raoul Walsh. SC: Margaret Fitts and Richard Alan Simmons. With Clark Gable, Eleanor Parker, Jo Van Fleet, Jean Willes, Barbara Nichols, Sara Shane, Roy Roberts, Arthur Shields, Jay C. Flippen, Florenz Ames, Chuck Roberson. A soldier of fortune is after gold buried by four men and in the search he finds himself in the company of their lovely wives. Hardly Clark Gable's best movie but still fairly good.\n\n**2119** _ **King of Dodge City**_ **** Columbia, 1941. 63 min. D: Lambert Hillyer. SC: Gerald Geraghty. With Bill Elliott, Tex Ritter, Dub Taylor, Judith Linden, Guy Usher, Rick Anderson, Kenneth Harlan, Pierce Lyden, Francis Walker, Harrison Greene, Jack Rockwell, Edmund Cobb, George Chesebro, Tristram Coffin, Steve Clark, Jack Ingram, Ned Glass, George Morrell, Horace B. Carpenter, Ted Mapes, Russ Powell, Frosty Royce, Tex Cooper, Ed Coxen, Lee Prather, Jay Lawrence, Theodore Lorch, Herman Hack, Jack Evans, Carl Mathews, Tom Smith. In 1861 an ex-lawman and a roving sheriff team to oppose a crook and his gang who are trying to take over a Kansas town. Steady Bill Elliott-Tex Ritter teaming but not one of their better efforts.\n\n**2120** _ **King of Texas**_. Turner Network Television (TNT), 2002. 95 min. Color. D: Uli Edel. SC: Stephen Harrigan. With Patrick Stewart, Marcia Gay Harden, Lauren Holly, Roy Scheider, David Alan Grier, Colm Meaney, Patrick Bergin, Matt Letscher, Liam Waite, Steven Bauer, Julie Cox, Richard Lineback, Lynne Goulet, Memo Escobedo, Clint Allen, Anna Doddrige, Roger Cudney, Andres Garcia, Fernando Banda, Juan Pablo Gamboa, Ehecatl Chavez. A wealthy cattle baron decides to divide his fortune among his three daughters who then turn against him. _King Lear_ out West is paltry entertainment with miscast star Patrick Stewart also serving as an executive producer.\n\n**2121** _ **King of the Arena**_ **** Universal, 1933. 62 min. D-SC: Alan James. With Ken Maynard, Lucille Brown, John St. Polis, Robert Kortman, James Marcus, Michael Visaroff, Frank Rice, Jack Rockwell, Bobby Nelson, Edgar \"Blue\" Washington, Jack Mower, Iron Eyes Cody, Ed Coxen, Lafe McKee, Fred McKaye, Willliam Walker, William Steele, Helen Gibson, Pascale Perry, Bud McClure, Horace B. Carpenter, Buck Bucko, Jack Kirk, Chief Big Tree, Artie Ortego, Merrill McCormick, Bob Burns. A one time circus performer, now a Texas Ranger hunting for a vicious outlaw dubbed the \"Black Death,\" rejoins the big top, the main locale of the bad man's operations. Ken Maynard produced this interesting feature which has an authentic circus background.\n\n**2122** _ **King of the Bandits**_ **** Monogram, 1947. 64 min. D: Christy Cabanne. SC: Bennett Cohen. With Gilbert Roland, Angela Greene, Chris-Pin Martin, Anthony Warde, Laura Treadwell, William Bakewell, Rory Mallinson, Pat Goldin, Cathy Carter, Boyd Irwin, Antonio Filauri, Jasper Palmer, Bill Cabanne, Frank Marlo, Guy Teague, James Harrison, George Douglas, Douglas Aylesworth, Gene Roth, Jack O'Shea, Bill Neff. The Cisco Kid and Pancho are falsely accused of a series of robberies really carried out by a saloon proprietor and his gang. Director Christy Cabanne wrote the story for this moderately entertaining \"Cisco Kid\" outing, the last of six with Gilbert Roland in the title role.\n\n**2123** _ **King of the Bullwhip**_ **** Western Adventure, 1951. 59 min. D: Ron Ormond. SC: Jack Lewis and Ira Webb. With Lash LaRue, Al St. John, Jack Holt, Anne Gwynne, Tom Neal, Dennis Moore, George J. Lewis, Michael Whalen, Willis Houck, Cliff Taylor, Frank Jaquet, Jimmie Martin, Roy Butler, Hugh Hooker, Tex Cooper. A bank president sends for two U.S. marshals to help his town in combating a mysterious masked bandit called El Azote. Well done Lash LaRue adventure with interesting camera work and brutal fight sequences, especially the opening and climactic ones involving whips.\n\n**Lash LaRue in** _**King of the Bullwhip**_ **(Western Adventure, 1951).**\n\n**  \n**\n\n**2124** _ **King of the Cowboys**_ **** Republic, 1943. 67 min. D: Joseph Kane. SC: Olive Cooper and J. Benton Cheney. With Roy Rogers, Smiley Burnette, Bob Nolan and The Sons of the Pioneers (Tim Spencer, Lloyd Perryman, Pat Brady, Hugh Farr, Karl Farr), Peggy Moran, Gerald Mohr, Dorothea Kent, Lloyd Corrigan, James Bush, Russell Hicks, Irving Bacon, Stuart Hamblen, Emmett Vogan, Eddie Dean, Forrest Taylor, Dick Wessel, Jack Kirk, Edward Earle, Yakima Canutt, Charles King, Jack O'Shea, Rex Lease, Herbert Rawlinson, Reed Howes, Eddie Dew, Earle Hodgins, Norman Willis, Ed Cassidy, Lynton Brent, Bud Geary, William Gould, Harrison Greene, Richard Alexander, Herbert Heyes, Dick Rich, Jack Ingram, Ralph Peters, John Dilson, Ray Bennett, Ed Peil, Sr., Charles Sullivan, Hugh Sothern, Fred Johnson, Elmer Jerome, Kate Lawson, Harry Burns, Jack Ray. Roy Rogers and his pals try to smash a sabotage ring fronted by a fake mind reader and a governor's assistant. Film does not live up to its title although it is fairly interesting viewing and topical in its day; great supporting cast.\n\n**2125** _ **King of the Forest Rangers**_ **** Republic, 1946. 12 Chapters. D: Spencer Gordon Bennet. SC: Albert DeMond, Basil Dickey, Jesse Duffy and Lynn Perkins With Larry Thompson, Helen Talbot, Stuart Hamblen, Anthony Warde, LeRoy Mason, Scott Elliott, Tom London, Walter Soderling, Bud Geary, Harry Strang, Ernie Adams, Eddie Parker, Jack Kirk, Tom Steele, Dale Van Sickel, Stanley Blystone, Marin Sais, Buddy Roosevelt, Robert Wilke, Sam Ash, Carey Loftin, Jay Kirby, Joe Yrigoyen, Kenneth Terrell, Bud Wolfe, Wheaton Chambers, Rex Lease, Charles Sullivan, David Sharpe. A Forest Ranger uncovers a plot by an amateur scientist to steal treasure buried in a national park's ancient Indian ruins. Fair cliffhanger but light on production values.\n\n**2126** _ **King of the Grizzlies**_ **** Buena Vista, 1970. 93 min. Color. D: Ron Kelly. SC: Jack Speirs. With John Yesno, Chris Wiggins, Hugh Webster, Jack Van Evers, Winston Hibler (narrator). A young Cree Indian boy raises a bear cub but when the animal grows up he is faced with the beast while alone in the wilds. Average Disney production with nice outdoor material; filmed in the Canadian Rockies.\n\n**2127** _ **King of the Lumberjacks**_ **** Warner Bros., 1940. 58 min. D: William Clemens. SC: Crane Wilbur. With John Payne, Gloria Dickson, Stanley Fields, Joseph Sawyer, Victor Kilian, Earl Dwire, Herbert Heywood, G. Pat Collins, John Sheehan, Pat West, Nat Carr, Jack Mower, John \"Skins\" Miller. Two men battle over a pretty girl and a lumber contract in the north woods. Adequate \"B\" action melodrama utilizes footage from _**Valley of the Giants**_ (Warner Bros., 1928) and _**God's Country and the Woman**_ (q.v.).\n\n**2128** _ **King of the Mounties**_ **** Republic, 1942. 12 Chapters. D: William Witney. SC: Ronald Davidson, Joseph Poland, William Lively, Joseph O'Donnell and Taylor Davan. With Allan Lane, Peggy Drake, Gilbert Emery, Russell Hicks, George Irving, Abner Biberman, William Vaughn, Nestor Paiva, Bradley Page, Douglass Dumbrille, William Bakewell, Duncan Renaldo, Francis Ford, Jay Novello, Anthony Warde, Norman Nesbitt, John Hiestand, Allen Jung, Paul Fung, Avron Dale, Kenneth Terrell, Duke Taylor, Tor Johnson, Harry Cording, Carleton Young, Tom Steele, Hal Taliaferro, Stanley Price, Tommy Coats, Bob Jamison, Jack Kenney, Forrest Taylor, Frank Wayne, Duke Green, Allen Jung, Norman Nesbitt, David Sharpe. A Mounted Policeman uncovers a plot by three enemy agents who are devising the Axis invasion of North America. Action filled serial follow-up to _**King of the Royal Mounted**_ (1940) [q.v.].\n\n**2129** _ **King of the Pecos**_ **** Republic, 1936. 54 min. D: Joseph Kane. SC: Bernard McConville, Dorrell McGowan and Stuart McGowan. With John Wayne, Muriel Evans, Cy Kendall, Jack Clifford, J. Frank Glendon, Herbert Heywood, Arthur Aylesworth, John Beck, Mary MacLaren, Bradley Metcalfe, Jr., Yakima Canutt, Edward Hearn, Earl Dwire, Tex Palmer, Jack Kirk, Horace B. Carpenter, Bud Pope, Tex Phelps, Tracy Layne, James A. Marcus, Jack Curtis. A law student seeks revenge on the crook who murdered his parents a decade earlier because they would not give up their land. Fast moving John Wayne vehicle highlighted by Cy Kendall's crafty villain.\n\n_**King of the Range**_ see _**The Marauders**_ (1947)\n\n**2130** _ **King of the Rodeo**_ **** Universal, 1929. 60 min. D: Henry MacRae. SC: B.M. Bower. With Hoot Gibson, Kathryn Crawford, Joseph W. Girard, Bodil Rosing, Charles K. French, Harry Todd, Slim Summerville, Jack Knapp, Monte Montague. The \"new Chip of the Flying U\" is thrown off the family ranch by his father for refusing to return to college so he joins a rodeo, becomes a headliner, gets involved with thieves and eventually is reunited with his folks. Lots of action and rodeo footage in this Hoot Gibson outing, one of his last silent films.\n\n**2131** _ **King of the Royal Mounted**_ **** 20th Century\u2013Fox, 1936. 61 min. D: Howard Bretherton. SC: Earle Snell. With Robert Kent, Rosalind Keith, Jack Luden, Alan Dinehart, Frank McGlynn, Jr., Grady Sutton, Arthur Loft, Artie Ortego, Frank O'Connor, Cecil Elliott, Lawrence Underwood. A Canadian Mountie attempts to stop a gang after a young woman's interest in a mine. Average program feature based on Zane Grey's comic strip; remade four years later by Republic as a serial with the same title (q.v.).\n\n**2132** _ **King of the Royal Mounted**_ **** Republic, 1940. 12 Chapters. D: William Witney and John English. SC: Franklyn Adreon, Sol Shor, Barney A. Sarecky, Norman S. Hall and Joseph Poland. With Allan Lane, Lita Conway, Robert Kellard, Robert Strange, Herbert Rawlinson, Harry Cording, Bryant Washburn, Budd Buster, Stanley Andrews, John Davidson, John Dilson, Paul McVey, Lucien Prival, Norman Willis, Tony Paton, Kenneth Terrell, Charles Thomas, Ted Mapes, Major Sam Harris, George Plues, Richard Simmons, Wallace Reid, Jr., William Justice, John Bagni, Earl Gunn, Frank Wayne, Curley Dresden, George DeNormand, Bud Geary, Tommy Coats, Dale Van Sickel, Bob Jamison, Al Taylor, David Sharpe, William Stahl, Douglas Evans, Duke Green, Robert Wayne. A Mounted Policeman discovers enemy agents are in the north country trying to locate a valuable mineral and he tries to stop them. Anti-Axis cliffhanger has plenty of action; issued as a feature in 1942 called _**The Yukon Patrol**_ by Republic.\n\n**2133** _ **King of the Sierras**_ **** Grand National, 1938. 55 min. D: Samuel Diege. SC: W. Scott Darling. With Rex, Sheik (horses), Hobart Bosworth, Harry Harvey, Jr., Frank Campeau, Harry Harvey, Jack Lindell. An old man tells a small boy the story of how a stallion tries to protect his harem of mares from a rival. Fair low budget production.\n\n**2134** _ **King of the Stallions**_ **** Monogram, 1942. 63 min. D: Edward Finney. SC: Sherman Lowe and Arthur St. Clair. With Chief Thundercloud, Dave O'Brien, Princess Bluebird (Barbara Felker), Rick Vallin, Sally Cairns, Ted Adams, G.D. Woods (Gordon DeMain), Chief Yowlachie, Forrest Taylor, Bill Wilkerson, Chief Many Treaties, Iron Eyes Cody, George Sky Eagle, Joe (J.W.) Cody, Charles Brunner, Chris Willow Bird, Fred Burns, Nakoma, Paint (horses). A ranch foreman tries to help Indians in capturing a beautiful stallion, the leader of a herd of wild horses. Competent follow-up to _**Silver Stallion**_ (q.v.) from producer-director Edward Finney, but filled with stock footage; also known as _**Code of the Redmen**_.\n\n**2135** _ **King of the Texas Rangers**_ **** Republic, 1941. 12 Chapters. D: William Witney and John English. SC: Ronald Davidson, Norman S. Hall, Joseph Poland, William Lively and Joseph O'Donnell. With Slingin' Sammy Baugh, Neil Hamilton, Pauline Moore, Duncan Renaldo, Charles Trowbridge, Kermit Maynard, Roy Barcroft, Kenne Duncan, Jack Ingram, Monte Montague, Iron Eyes Cody, Hooper Atchley, Ed Cassidy, Buddy Roosevelt, David Sharpe, Herbert Rawlinson, Frank Darien, Robert O. Davis, Monte Blue, Stanley Blystone, Joe Forte, Lucien Prival. When his father, a captain in the Texas Ranger, is murdered by enemy agents, a man joins the service and uncovers saboteurs working along the U.S.-Mexican border. Football hero Sammy Baugh is a mediocre serial hero but silent star Neil Hamilton nearly saves this one as the villainous traitor.\n\n**2136** _ **The King of the Wild Horses**_ **** Path\u00e9, 1924. 50 min. D: Fred Jackman. SC: Carl Himm. With Rex (horse), Charles Parrott (Charley Chase), Edna Murphy, Sidney DeGray, Leon Barry, Pat Hartigan, Frank Butler, Sidney D'Albrook. A cowboy in love with a ranch owner's pretty daughter captures a wild stallion who helps him stop rustlers. Producer Hal Roach wrote the story for this action filled adventure for Rex, the wild stallion, who would also star from him in outings like _**Black Cyclone**_ and _**The Devil Horse**_ (qq.v.).\n\n**2137** _ **King of the Wild Horses**_ **** Columbia, 1933. 62 min. D: Earl Haley. SC: Fred Myton. With Rex (horse), William Janney, Dorothy Appleby, Wallace MacDonald, Harry Semels, Art Mix, Ford West, King and Lady (horses). A cowboy tames a wild horse who has been mistreated by bad men. Okay juvenile fare also called _**Rex, King of the Wild Horses**_ and remade in 1947 (q.v.).\n\n**2138** _ **King of the Wild Horses**_ **** Columbia, 1947. 79 min. D: George Archainbaud. SC: Brenda Weisberg. With Preston Foster, Gail Patrick, Bill Sheffield, Guinn Williams, Buzz Henry, Charles Kemper, Patti Brady, John Kellogg, Ruth Warren, Louis Faust. A young boy befriends and tames a wild stallion. Average kid and horse drama with good scenic values; remake of the 1932 film (q.v.).\n\n**2139** _ **King of the Wild Stallions**_ **** Allied Artists, 1959. 75 min. Color. D: R.G. Springsteen. SC: Ford Beebe. With George Montgomery, Diane Brewster, Edgar Buchanan, Emile Meyer, Byron Foulger, Denver Pyle, Dan Sheridan, Rory Mallinson, Jerry Hartleben. A widow and her son fight to save their ranch from a crooked rival and are helped by a cowboy and a wild stallion. Charming little drama provides good entertainment.\n\n**2140** _ **Kingdom of the Spiders**_ **** Dimension, 1977. 95 min. Color. D: John \"Bud\" Cardos. SC: Richard Robinson and Alan Caillou. With William Shatner, Tiffany Bolling, Woody Strode, Altovise Davis, Joe Rose, Hoke Howell, Marcy Lafferty, Roy Engel, Liuex Dressler, David McLean, Natascha Ryan, Adele Malis. In the Arizona desert a veterinarian and an entomologist discover that tarantulas, with venom five times more toxic than normal, have formed an army and plan to attack a small town. Harrowing horror thriller with a nicely done shock ending.\n\n**2141** _ **Kings of the Sun**_ **** United Artists, 1963. 108 min. D: J. Lee Thompson. SC: Elliott Arnold and James R. Webb. With Yul Brynner, George Chakiris, Shirley Anne Field, Richard Basehart, Brad Dexter, Barry Morse, Armando Silvestre, Leo Gordon, Victoria Vetri, Rudy Solari, Ford Rainey, Angel Di Steffano, Chuck Hayward, Jose Moreno, James Coburn (narrator). After he and his people escape their enemies and settle in the American West, a Mayan chief battles a local Indian tribe leader for the affections of a princess. Well done clash of cultures melodrama.\n\n**2142** _ **Kino\u2014Le Legenda del Cura Negro**_ (Kino\u2014The Legend of the Black Priest) **** Cineclipse, 1993. 109 min. Color. D: Felipe Cazals. SC: Felipe Cazals, Gerardo de la Torre and Tomas Perez Turrens. With Enrique Rocha, Rodolfo de Acosta, Manuel Ojeda, Fernando Balzaretti, Carlos Cardian, Aaron Hernan, Leonardo Daniel, Ernesto Yanez, Julian Pastor, Alvaro Carcano, Max Kerlov, Jorge Fagan, Jorge Hernandez. Padre Kino is sent by Queen Isabella of Spain to explore the New World and he meets obstacles from Spanish soldiers, Native Americans and his own church. Top notch award winning Mexican drama.\n\n**2143** _ **Kiss of Fire**_ **** Universal-International, 1955. 87 min. Color. D: Joseph M. Newman. S: Franklin Coen and Richard Collins. With Jack Palance, Barbara Rush, Rex Reason, Martha Hyer, Alan Reed, Leslie Bradley, Lawrence Dobkin, Pat Hogan, Henry Rowland, Joseph Waring, Karen Kadler, Steven Geray, Bernie Gozier, Paul Marion, Robert F. Hoy, Dave Kashner, Shooting Star, Charles Horvath, David Alpert, John Mansfield. A Spanish princess travels to the New World, falls in love with a soldier and refuses to return home when she is named queen. Colorful but empty nonsense set in Spanish America.\n\n**2144** _ **The Kissing Bandit**_ **** Metro-Goldwyn-Mayer, 1948. 102 min. Color. D: Laslo Benedek. SC: Isobel Lennart and John Briard Harding. With Frank Sinatra, Kathryn Grayson, J. Carrol Naish, Mildred Natwick, Ricardo Montalban, Ann Miller, Cyd Charisse, Mikhail Rasummy, Clinton Sundberg, Carleton Young, Edna Skinner, Vicente Gomez, Henry Mirelez, Nick Thompson, Joe Dominguez, Alberto Morin, Pedro Regas, Julian Rivero, Mitchell Lewis, Byron Foulger. In Old California a man takes over his father's position as head of a group of daring highwaymen. An overlong, dull musical Western.\n\n**2145** _ **Kit Carson**_ **** United Artists, 1940. 96 min. D: George B. Seitz. SC: George Bruce. With Jon Hall, Lynn Bari, Dana Andrews, Harold Huber, Ward Bond, Renie Riano, Clayton Moore, Rowena Cook, Raymond Hatton, Harry Strang, C. Henry Gordon, Lew Merrill, Stanley Andrews, Edwin Maxwell, Peter Lynn, William Farnum, Charles Stevens, Harry Semels, Al Kikume, Blaney Harris. Frontier scout Kit Carson leads a wagon train through Indian territory as he and a cavalry officer fight over the same woman. There is nothing special or historically accurate about this so-called biopic, but it does entertain.\n\n**2146** _ **Klondike**_ **** Monogram, 1932. 68 min. D: Phil Rosen. SC: Tristram Tupper. With Lyle Talbot, Thelma Todd, Tully Marshall, Henry B. Walthall, Ethel Wales, George Hayes, Myrtle Steadman, Pat O'Malley, Jason Robards, Lafe McKee, Frank Hawks, Priscilla Dean, Earl Dwire. A doctor on the run from the law manages to turn his life around in the Klondike. Well done program feature with a fine cast including famed aviator Frank Hawks in a small part; remade as _**Klondike Fury**_ (q.v.).\n\n**2147** _ **Klondike Annie**_ **** Paramount, 1936. 77 min. D: Raoul Walsh. SC: Mae West. With Mae West, Victor McLaglen, Philip Reed, Helen Jerome Eddy, Harry Beresford, Harold Huber, Lucille Webster Gleason, Conway Tearle, Esther Howard, Soo Yong, John Rogers, Ted Oliver, Lawrence Grant, Gene Austin, Vladimir Bykoff, Tetsu Komai, James Burke, George Walsh, Chester Gan, Jack Daly, Jack Wallace, Philo McCullough, William Norton Bailey, Jim Thorpe, Guy D'Ennery, Carl Harbaugh, Otto \"Coco\" Heimel, Howard Lang, Katherine Clare Ward, D'Arcy Corrigan, Nell Craig, Nella Walker, Philip Ahn, Maidel Turner, Huntley Gordon, Paul Kruger, Edward Brady, John Lester Johnson, Laura Treadwell, George MacQuarrie. After killing a man in self defense, a woman heads to Alaska, takes on the identity of a deceased evangelist and sets up a Settlement House. Funny Mae West vehicle with some good songs composed by Gene Austin.\n\n**2148** _ **Klondike Fever**_ **** CFI, 1980. 106 min. Color. D: Peter Carter. SC: Charles Israel and Martin Lager. With Rod Steiger, Angie Dickinson, Lorne Greene, Jeff East, Barry Morse, Lisa Langlois, Robin Gammel, Michael Hogan, Gordon Pinsent, Sherry Lewis, D.D. Winters. The story of writer Jack London during his days in the Klondike gold rush. Average adventure drama with little historical value; also called _**Jack London's Klondike Fever**_.\n\n**2149** _ **Klondike Fury**_ **** Monogram, 1942. 68 min. D: William K. Howard. SC: Henry Blankfort and Tristram Tupper. With Edmund Lowe, Lucille Fairbanks, Bill Henry, Ralph Morgan, Mary Forbes, Jean Brooks, Vince Barnett, Clyde Cook, Robert Middlemass, John Roche, Monte Blue, Kenneth Harlan, Marjorie Wood, Dick Purcell, John Hamilton, Leonid Snagoff, Gil Frye, Frank Pershing. After having lost faith in himself, a physician learns a new meaning to life in the Klondike after performing a successful operation. Enjoyable remake of _**Klondike**_ (q.v.) highlighted by Edmund Lowe's fine work as the doctor.\n\n**2150** _ **Klondike Kate**_ **** Columbia, 1943. 64 min. D: William Castle. SC: Houston Branch and M. Coates Webster. With Ann Savage, Glenda Farrell, Tom Neal, Constance Worth, Sheldon Leonard, Lester Allen, George Cleveland, George McKay, Dan Seymour, Lewis Wilson, Richard Alexander, Minerva Urecal, Edward Earle, Lee \"Lasses\" White, Edward Keane, Gwen Seager, Harry Bradley, Hank Bell, Harry Cording, Tommy Kingston. During the gold rush in Alaska a hotel owner is nearly lynched for a murder he did not commit. Based on the true story of Kate Rockwell Matson, this \"B\" effort fails to deliver much entertainment.\n\n_**A Knife for the Ladies**_ see _**Silent Sentence**_\n\n**2151** _ **Knight of the Plains**_ **** Spectrum, 1938. 57 min. D: Sam Newfield. SC: Fred Myton. With Fred Scott, Al St. John, Marion Weldon, Richard Cramer, John Merton, Frank LaRue, Lafe McKee, Emma Tansey, Steve Clark, Carl Mathews, Sherry Tansey, Jimmy Aubrey, George Morrell, Cactus Mack, Tex Palmer, Olin Francis, Bob Burns, Budd Buster. A cowboy comes to the aid of settlers being harassed by rustlers and land grabbers. Good Fred Scott musical oater; produced by Stan Laurel Pictures.\n\n**2152** _ **Knights of the Range**_ **** Paramount, 1940. 68 min. D: Lesley Selander. SC: Norman Houston. With Russell Hayden, Jean Parker, Victor Jory, Britt Wood, J. Farrell MacDonald, Morris Ankrum, Ethel Wales, Rad Robinson, Raphael (Ray) Bennett, Ed Cassidy, Eddie Dean, The King's Men (Ken Darby, Bud Linn, Jon Dobson), Franklyn Farnum, John St. Polis, Ed Cassidy, Chuck Baldra, Charles Murphy, Bill Nestell, Blackjack Ward, Bob Burns. A cowboy gets involved with outlaws but switches to the right side of the law to help a young woman and her father in Oklahoma's Cimarron country. Nice program feature by producer Harry Sherman from Zane Grey's novel.\n\n**2153** _ **Konga, the Wild Stallion**_ **** Columbia, 1940. 65 min. D: Sam Nelson. SC: Harold Shumate. With Fred Stone, Rochelle Hudson, Richard Fiske, Eddy Waller, Robert Warwick, Don Beddoe, Carl Stockdale, George Cleveland, Burr Caruth, James Craig, Murdock MacQuarrie, John Tyrrell, John Dilson, Sam Ash, Herbert Heywood, Harry Bernard, Chuck Hamilton, Lee Prather, Lee Millar, Edmund Elton, Counto (horse), Boots (dog). When a man shoots his horse, a rancher kills him and is sent to prison but years later he is reunited with the animal who has been cared for by his daughter. Simple but pleasant tale enhanced by Fred Stone's performance as the rancher.\n\n**2154** _ **Kung Fu**_ **** ABC-TV\/Warner Bros., 1972. 75 min. Color. D: Jerry Thorpe. SC: Ed Spielman and Howard Friedlander. With David Carradine, Barry Sullivan, Albert Salmi, Wayne Maunder, Benson Fong, Keye Luke, Philip Ahn, Richard Loo, Victor Sen Yung, Keith Carradine, Radmaes Pera, Roy Fuller, Robert Ito, John Leoning, David Chow. Running away from murder charges in China, a Chinese-American kung fu expert comes to the United States and opposes the exploitation of railroad workers in the West. Popular TV movie that helped start the kung fu movie craze as well as serve as the pilot for the \"Kung Fu\" (ABC-TV, 1972\u201375) series which was revived two decades later as \"Kung Fu: The Legend Continues\" (Prime Time Entertainment Network, 1993\u201397).\n\n**2155** _ **Kung Fu:**_ _**The Movie**_ **** Warner Bros., 1986. 104 min. Color. D: Richard Lang. SC: Durrell Royce Crays. With David Carradine, Kerrie Keane, Mako, Brandon Lee, William Lucking, Luke Askew, Benson Fong, Martin Landau, Ellen Geer, Keye Luke, Robert Harper, Roy Jenson, Paul Rudd, John Alderman, Michael Paul Chan, Patience Cleveland, Roland Harrah III, Jim Haynie. A Kung Fu master goes back to the late nineteenth century to engage in mortal combat with a vicious martial arts killer. Fair TV movie reprise of _**Kung Fu**_ (q.v.), co-produced by star David Carradine and Skip Ward.\n\n**2156** _ **Lacy and the Mississippi Queen**_ **** NBC-TV\/Paramount, 1978. 74 min. Color. D: Robert Butler. SC: Kathy Donnell and Madeline DeMaggio-Warner. With Kathleen Lloyd, Debra Feuer, Edward Andrews, Jack Elam, Matt Clark, Les Lannom, Christopher Lloyd, James Keach, Anthony Palmer, David Byrd, Alvy Moore, Sandy Ward, Elizabeth Rogers, David Comford, Cliff Pellow, Robert Casper. Two sisters, who are direct opposites, team to hunt down the train robbers they believe killed their father. Just passable TV fare.\n\n**2157** _ **Lad:**_ _**A Dog**_ **** Warner Bros., 1962. 98 min. Color. D: Aram Avakian and Leslie Martinson. SC: Lillie Hayward and Robert O. Hodes. With Peter Breck, Peggy McCay, Carroll O'Connor, Angela Cartwright, Maurice Dallimore, Alice Pearce, Jack Daly, Charles Fredericks, Tim Graham, Lillian Buyeff, Lad (dog). A beautiful white collie brings happiness to the life of a small disabled girl. Pleasant screen adaptation of Albert Payson Terhune's novel.\n\n**2158** _ **Lady for a Night**_ **** Republic, 1942. 87 min. D: Leigh Jason. SC: Isabel Dawn and Boyce De Gaw. With Joan Blondell, John Wayne, Ray Middleton, Philip Merivale, Blanche Yurka, Edith Barrett, Leonid Kinskey, Hattie Hoel, Montagu Love, Carmel Myers, Dorothy Burgess, Guy Usher, Ivan Miller, Patricia Knox, Dewey Robinson, The Hall Johnson Choir, Lew Payton, Marilyn Hare, Dolores Gray, Gertrude Astor, Frances Gladwin, Jack Kinney, Pierre Watkin, Betty Hill, Forbes Murray, Frank Orth, Roy Gordon, Kathryn Sheldon, Minerva Urecal, Howard Hickman, Dudley Dickerson, Paul White, Charles Miller, Dick Rush, Charles McAvoy, Mickey Simpson, Neely Edwards, Lloyd Whitlock, Howard Mitchell, Leigh Whipper, Edith Evanson, Martin Turner, Margaret Armstrong, Eula Morgan. A riverboat entertainer yearns for social position so she throws over her gambler lover and marries an impoverished planter. Above average Republic production with good work by Joan Blondell in the title role although John Wayne is subordinate as the gambler.\n\n**2159** _ **The Lady from Cheyenne**_ **** Universal, 1941. 87 min. D: Frank Lloyd. SC: Kathryn Scola and Warren Duff. With Loretta Young, Robert Preston, Edward Arnold, Frank Craven, Gladys George, Jessie Ralph, Stanley Fields, Willie Best, Samuel S. Hinds, Spencer Charters, Clare Verdera, Alan Bridge, Joseph Sawyer, Ralph Dunn, Harry Cording, Marion Martin, Gladys Blake, Sally Payne, Iris Adrian, June Wilkins, Erville Alderson, Emmett Vogan, Roger Imhoff, William B. Davidson, James Kirkwood, Emory Parnell, Dorothy Granger, Richard Alexander, Griff Barnett, Esther Howard, Charles Williams, Wade Boteler, Charles T. Aldrich, Jeff Corey, Matt McHugh, Larry Lawson, John Dilson, Charles Halton, Harry Seymour, Delmar Watson, Victor Potel, Harry Stubbs, Kernan Cripps, Stanley Blystone, Frank Austin, Herbert Heywood, Kathryn Sheldon, Charles Ray, Jessie Arnold, Murdock MacQuarrie, Lloyd Ingraham, Joe Eggenton, Sue Moore, Phyllis Kennedy, Bob Larson, Loren Baker, Roger Gray. In Wyoming a pretty teacher tries to start a school in a wild town and at the same time obtain voting rights for women. Production about early women's liberation activities is fun to view.\n\n_**Lady from Frisco**_ see _**Rebellion**_\n\n**2160** _ **Lady from Louisiana**_ **** Republic, 1941. 82 min. D: Bernard Vorhaus. SC: Vera Caspary, Michael Hogan and Guy Endore. With John Wayne, Ona Munson, Ray Middleton, Henry Stephenson, Helen Westley, Jack Pennick, Dorothy Dandridge, Shimen Ruskin, Jacqueline Dalya, Paul Scardon, James C. Morton, Maurice Costello, Major James H. McNamara. In old New Orleans a lawyer tries to destroy a corrupt gang and ends up falling in love with the leader's daughter. Interesting period melodrama with John Wayne as the crusader.\n\n**2161** _ **The Lady from Texas**_ **** Universal-International, 1951. 78 min. Color. D: Joseph Pevney. SC: Gerald Drayson Adams and Connie Lee Bennett. With Howard Duff, Mona Freeman, Josephine Hull, Gene Lockhart, Craig Stevens, Ed Begley, Barbara Knudson, Lane Bradford, Chris-Pin Martin, Kenneth Patterson, Jay C. Flippen, Edmund Cobb, William Fawcett, John Maxwell, Dabbs Greer, Morgan Farley, Donald Kerr, Eddie Parks, Alice Richey, Roy Butler, Ada Adams, Buddy Roosevelt, Kathryn Sheldon, Helen Dietrich, Forbes Murray, Frank O'Connor. Crooks try to declare insane a widow whose husband was killed in the Civil War but a cowboy and a young woman come to her aid. Delightful Western comedy.\n\n**2162** _ **The Lady Is My Wife**_ **** NBC-TV\/Universal, 1967. 49 min. Color. D: Sam Peckinpah. SC: Halstead Welles and Jack Laird. With Jean Simmons, Bradford Dillman, Alex Cord, Alan Baxter, L.Q. Jones, Roberto Contreras, E.J. Andre, Jim Boles, Begona Palacios, Lillian Bronson, Billy M. Greene, Larry Watson, Bob Hope (host). In a mining town following the Civil War, a gambler and a cowboy play a game of mounted pool for each other's possessions. Originally telecast February 1, 1967, as a segment of \"Bob Hope Chrysler Theatre\" (NBC-TV, 1963\u201367), this fair outing was later issued to TV as a feature film.\n\n**2163** _ **A Lady Takes a Chance**_ **** RKO Radio, 1943. 86 min. D: William A. Seiter. SC: Robert Ardrey. With Jean Arthur, John Wayne, Charles Winninger, Phil Silvers, Mary Field, Don Costello, John Philliber, Grady Sutton, Grant Withers, Hans Conreid, Peggy Carroll, Ariel Heath, Sugar Geise, Joan Blair, Tom Fadden, Eddy Waller, Nina Quartaro, Cy Kendall, Charles D. Brown, Butch and Buddy, The Three Peppers, Alex Melesh, Paul Scott, Lane Chandler, Ralf Harolde, Hank Worden, Clarence Straight, Warren Jackson, Eddie Dew, Bud Geary, Fred Burns, Monty Collins, Horace Murphy, Syd Saylor, Eddie Borden, Bob McKenzie, Dorothy Granger, Harry Semels, George DeNormand, Bennie Bartlett, Patsy Moran, Bobby Barber, Bert Dillard, Jack Daly, Herbert Evans, J.W. Cody, Chalky Williams, Frank Melton, Joe Bernard, Robert Cherry. An Eastern woman with three unacceptable suitors heads West and is romanced by a rodeo rider. Dated, mundane genre comedy.\n\n**2164** _ **Land Beyond the Law**_ **** Warner Bros., 1937. 58 min. D: B. Reeves Eason. SC: Luci Ward and Joseph K. Watson. With Dick Foran, Linda Perry, Wayne Morris, Irene Franklin, Gordon Hart, Joseph King, Cy Kendall, Frank Orth, Glenn Strange, Harry Woods, Milton Kibbee, Edmund Cobb, Henry Otho, Tom Brower, Paul Panzer, Julian Rivero, Artie Ortego, Jim Corey, Bud Osborne, Wilfred Lucas, Gene Alsace, Frank McCarroll. A rancher finds himself in trouble with the law when, by mistake, he gets mixed up with outlaws. Pretty good Dick Foran series film, briskly directed by B. Reeves \"Breezy\" Eason; a remake of the 1927 First National Ken Maynard film of the same title and _**The Big Stampede**_ (q.v.).\n\n**2165** _ **Land of Fighting Men**_ **** Monogram, 1938. 53 min. D: Alan James. SC: Joseph O'Donnell. With Jack Randall, Herman Brix (Bruce Bennett), Louise Stanley, Dickie Jones, Walt Shrum and His Colorado Hillbillies, Wheeler Oakman, Bob Burns, John Merton, Lane Chandler, Rex Lease, Ernie Adams, Spade Cooley. Framed for killing a rancher, a cowboy must find the murderer to clear his name. Average outing in Jack Randall's Monogram series enhanced by a fine cast.\n\n_**Land of Fury**_ see _**The Seekers**_\n\n**2166** _ **Land of Hunted Men**_ **** Monogram, 1943. 58 min. D: S. Roy Luby. SC: Elizabeth Beecher. With Ray Corrigan, Dennis Moore, Max Terhune, Phyllis Adair, Charles King, John Merton, Ted Mapes, Frank McCarroll, Forrest Taylor, Steve Clark, Fred \"Snowflake\" Toones, Carl Sepulveda, Augie Gomez, Hank Bell, Tex Palmer, Jack Evans, Ray Jones, Al Haskell. The Range Busters are on the trail of an outlaw gang terrorizing the countryside. Ray Corrigan returned to the series and Dennis Moore joined it in this feature, an otherwise mediocre \"Range Busters\" affair.\n\n**2167** _ **The Land of Missing Men**_ **** Tiffany, 1930. 58 min. D: J.P. McCarthy. SC: J.P. McCarthy and Bob Quigley. With Bob Steele, Al St. John, Caryl Lincoln, Al Jennings, Edward Dunn, Fern Emmett, Emilio Fernandez, Noah Hendricks, C.R.Dufau, S.S. Simon, Frank Ellis, Jim Corey. Two cowpokes falsely accused of a stage holdup rescue a young woman from another coach and then infiltrate an outlaw gang in order to capture the real culprits. Bob Steele's fans will like this rather interesting early talkie that includes several brutal scenes.\n\n**2168** _ **Land of No Return**_ **** International Picture Show, 1981. 84 min. Color D-SC: Kent Bateman. With Mel Torme, William Shatner, Donald Moffat, Caesar (eagle), Romulus (wolf). A television animal trainer crashes his private plane in the wilds of Utah and tries to survive with only the aid of a pet eagle. No much here but the scenery; alternate titles: _**Challenge to Survive**_ and _**Snowman**_.\n\n**2169** _ **Land of the Lawless**_ **** Monogram, 1947. 60 min. D: Lambert Hillyer. SC: J. Benton Cheney. With Johnny Mack Brown, Raymond Hatton, Christine McIntyre, Tristram Coffin, June Harrison, Marshall Reed, I. Stanford Jolley, Steve Clark, Edmund Cobb, Roy Butler, Cactus Mack, Gary Garrett, Ben Corbett, Carl Sepulveda. Arriving in a remote town, a man finds his best friend murdered so he works with a prospector to rid the area of a corrupt female saloon boss and her crooked cohort. Pretty good Johnny Mack Brown-Raymond Hatton entry.\n\n**2170** _ **Land of the Open Range**_ **** RKO Radio, 1942. 60 min. D: Edward Killy. SC: Morton Grant. With Tim Holt, Ray Whitley, Janet Waldo, Lee \"Lasses\" White, Hobart Cavanaugh, Lee Bonnell, Roy Barcroft, John Elliott, Frank Ellis, Tom London, Ruth Clifford, Henry Roquemore, Bud Geary, Fern Emmett, John Ince, Pierce Lyden, Lindy Wade, J. Merrill Holmes, James Carlisle, Duke Green, Frank Mills, Charles Phipps. When a convict's will states that a large of tract of land he owned can only be homesteaded by ex-convicts, a deputy tries to stop the lawlessness they cause. A different plot adds some interest to this Tim Holt film.\n\n**2171** _ **Land of the Outlaws**_ **** Monogram, 1944. 58 min. D: Lambert Hillyer. SC: Joseph O'Donnell. With Johnny Mack Brown, Raymond Hatton, Nan Halliday, Stephen Keyes, Hugh Prosser, Charles King, John Merton, Steve Clark, Art Fowler, Tom Quinn, Ray Elder, Chick Hannon, Bob (John) Cason, Kansas Moehring, George Morrell, Ben Corbett, Bud Wolfe, John Judd, Bob Woodward, Dick Rush, Rube Dalroy, Jack Evans. A marshal is trailing a outlaw gang behind the hijacking of ore shipments. Pretty fair \"Nevada Jack McKenzie\" series outing.\n\n**2172** _ **Land of the Six Guns**_ **** Monogram, 1940. 54 min. D: Raymond K. Johnson. SC: Tom Gibson. With Jack Randall, Louise Stanley, Kenne Duncan, Glenn Strange, Bud Osborne, George Chesebro, Jack Perrin, Steve Clark, Frank LaRue, Carl Mathews, Buzz Barton, Richard Cramer, Jimmy Aubrey, Tex Palmer. A lawman buys a ranch but finds it being used by rustlers bringing in cattle from Mexico. Poor Jack Randall vehicle.\n\n**2173** _ **Land of Wanted Men**_ **** Monogram, 1932. 59 min. D-SC: Harry Fraser. With Bill Cody, Andy Shuford, Sheila Mannors, Gibson Gowland, Frank Lackteen, James A. Marcus, Jack Richardson, Jack Evans. An cowboy gets a job as a lawman in country where sheep are being brought onto cattle range land. So-so Bill Cody-Andy Shuford series effort.\n\n**2174** _ **Land Raiders**_ **** Columbia, 1970 101 min. Color. D: Nathan Juran. SC: Ken Pettus. With Telly Savalas, George Maharis, Arlene Dahl, Janet Landgard, Jocelyn Lane, George Coulouris, Guy Rolfe, Phil Brown, Peter Dane, Marcella St. Amant, Paul Picerni, Robert Carricart, Gustavo Rojo, Fernando Rey, Ben Tatar, John Clark, Charles Stahlnaker, Susan Harvey. A man and a woman survive a wagon train massacre by Indians, the attack being caused by his brother, a town boss paying for scalps. Spanish made, mediocre, violent film from producer Charles H. Schneer.\n\n**2175** _ **Landrush**_ **** Columbia, 1946. 54 min. D: Vernon Keays. SC: Michael Simmons. With Charles Starrett, Smiley Burnette, Doris Houck, Emmett Lynn, Ozie Waters and His Colorado Rangers, Bud Geary, Stephen Barclay, Robert Kortman, George Chesebro, Bud Osborne, Ted French, George Russell, George Hoey, Ethan Laidlaw, John Tyrrell, Russell Meeker, Roy Butler, Curt Barrett, Nolan Leary, Herman Hack, Scotty Harrell, John Hawks, Sam Garrett. Outlaws try to keep homesteaders off land recently opened for settlement but the Durango Kid comes to their rescue. More than passable \"Durango Kid\" series segment; footage from the film was later used in _**Cyclone Fury**_ and _**Streets of Ghost Town**_ (qq.v.). British title: _**The Claw Strikes**_.\n\n**2176** _ **Laramie**_ **** Columbia, 1949. 56 min. D: Ray Nazarro. SC: Barry Shipman. With Charles Starrett, Smiley Burnette, Elton Britt, Fred F. Sears, Tommy Ivo, Marjorie Stapp, Robert Wilke, Myron Healey, Shooting Star, Jay Silverheels, Jim Diehl, Ethan Laidlaw, Bob (John) Cason, George Lloyd, Rodd Redwing, Nolan Leary, Kermit Maynard. When an evil Army scout selling rifles to the Indians kills a chief, warfare nearly erupts with the Durango Kid trying to stop impending bloodshed. Mediocre \"Durango Kid\" outing that benefits greatly from Elton Britt singing \"Chime Bells\" and \"Mollie Darling.\"\n\n**2177** _ **The Laramie Kid**_ **** Reliable, 1935. 57 min. D: Harry S. Webb. SC: Carl Kursada and Rose Gordon. With Tom Tyler, Alberta Vaughn, Al Ferguson, Murdock MacQuarrie, George Chesebro, Snub Pollard, Steve Clark, Artie Ortego, Jimmy Aubrey, Wally Wales, Nelson McDowell, Budd Buster, Lafe McKee, Lew Meehan, Bob McKenzie, Frank Ellis, Art Dillard, Robert Walker, Herman Hack, Blackie Whiteford, Jack Hendricks. A cowboy goes to jail so his girl's father can collect reward money to pay off his ranch, keeping the young woman from marrying a banker she does not love. The production is as bad as the plot in this Tom Tyler vehicle.\n\n**2178** _ **Laramie Mountains**_ **** Columbia, 1952. 53 min. D: Ray Nazarro. SC: Barry Shipman. With Charles Starrett, Smiley Burnette, Jack (Jock) Mahoney, Fred F. Sears, Marshall Reed, Rory Mallinson, Zon Murray, John War Eagle, Robert Wilke, Chris Alcaide, Boyd \"Red\" Morgan, Frank McCarroll, Jay Silverheels. A government Indian agent tries to prevent warfare after several attacks on the cavalry and he learns the trouble is caused by dishonest scouts. Average \"Durango Kid\" affair.\n\n**2179** _ **The Laramie Trail**_ **** Republic, 1944. 54 min. D: John English. SC: J. Benton Cheney. With Robert Livingston, Smiley Burnette, Linda Brent, George J. Lewis, John James, Emmett Lynn, Leander de Cordova, Slim Whitaker, Bud Osborne, Bud Geary, Kenne Duncan, Roy Barcroft, Marshall Reed, Martin Garralaga, John Whitley. Arriving at a Spanish hacienda, two pals try to help a young man falsely accused of murder. The final teaming of Robert Livingston and Smiley Burnette, following the \"John Paul Revere\" series, is a nifty one with nicely atmospheric mystery touches.\n\n_**Lariats' End**_ see _**Mystery Brand**_\n\n**2180** _ **Lasca of the Rio Grande**_ **** Universal, 1931. 65 min. D: Edward Laemmle. SC: Randall Faye. With Johnny Mack Brown, Leo Carrillo, Dorothy Burgess, Slim Summerville, Frank Campeau, Chris-Pin Martin, Tom London, John Ince, Jim Corey. A Texas Ranger and a Mexican bandit both love a dance hall girl and when she kills a man in self defense the lawman lets her go. A bit creaky but still fun, especially for Leo Carrillo as the bandit.\n\n**2181** _ **The Lash**_ **** First National, 1930. 75 min. D: Frank Lloyd. SC: Bradley King. With Richard Barthelmess, Mary Astor, Marian Nixon, James Rennie, Fred Kohler, Robert Edeson, Barbara Bedford, Erville Alderson, Arthur Stone, Mathilde Comont, Frank Lackteen, Francis McDonald, William L. Thorne, Chris-Pin Martin, Pedro Leon. Returning home to Old California from school, a man find that crooks have taken over the locale and he sets out to stop them with daring raids, earning the nickname \"El Puma.\" Entertaining early talkie that moves at a good clip although it is dated; originally called _**Adios**_.\n\n**2182** _ **The Lash of the Law**_ **** Goodwill, 1926. 50 min. D: Paul Hurst. SC: Al Jennings and Jay Inman (Joseph) Kane. With Bill Bailey, Alma Rayford, Dick LaReno, Marcel Perez, Bud Osborne, Milton Fahrney, Roy Watson. A cowboy on the trail of an outlaw gang helps a boy and girl brutalized by their stepfather. Well made silent, worth viewing; star Bill Bailey later became William Norton Bailey.\n\n_**Lassie's Adventures in the Gold Rush**_ see _**The Painted Hills**_\n\n**2183** _ **Lassie's Great Adventure**_ **** 20th Century\u2013Fox, 1963. 103 min. D: William Beaudine. SC: Monroe Manning and Charles O'Neal. With Lassie, Jon Provost, June Lockhart, Hugh Reilly, Richard Simmons, Richard Kiel, Walter Stocker, Robert Howard, Will J. White, Patrick Waltz, Leo Needham, Patrick Westwood. Lassie and her young master are carried away in a hot air balloon that takes them to the wilds of Canada where an Indian decides to make the boy a replacement for his son. Very well done family drama taken from four episodes of \"Lassie\" (CBS-TV, 1954\u201371) and issued theatrically both here and abroad. Who says William Beaudine was not a good director?\n\n**2184** _ **The Last Bandit**_ **** Republic, 1949. 80 min. Color. D: Joseph Kane. SC: Thomas Williamson. With William Elliott, Adrian Booth, Forrest Tucker, Andy Devine, Jack Holt, Grant Withers, Minna Gombell, Virginia Brissac, Louis Faust, Stanley Andrews, Martin Garralaga, Joseph Crehan, Charles Middleton, Rex Lease, Emmett Lynn, Gene Roth, George Chesebro, Hank Bell, Jack O'Shea, Steve Clark, Tex Terry, George Eldredge, Chick Hannon, Howard Mitchell, Monte Montague, Rocky Shahan, Chuck Baldra, Rodney Bell, Vera Marshe, George Backus, Al Murphy, Len Torrey, David Williams, Cecil Combs, Steve Drake, Buster West, Frank Dae. A reformed outlaw tries to go straight as an express agent but runs afoul of a saloon singer and his own crooked brother, who is planning a big gold heist by robbing the local railroad. Pretty good William Elliott action film; quite colorful. A remake of _**The Great Train Robbery**_ (1941) [q.v.].\n\n_**The Last Bandit**_ (1979) see _**The Bandits**_\n\n_**The Last Bullet**_ see _**Crooked River**_\n\n**2185** _ **The Last Challenge**_ **** Metro-Goldwyn-Mayer, 1967. 105 min. Color. D: Richard Thorpe. SC: John Sherry and Robert Emmett Ginna. With Glenn Ford, Angie Dickinson, Chad Everett, Gary Merrill, Jack Elam, Delphi Lawrence, Royal Dano, Kevin Hagen, Florence Sundstrom, Marian Collier, Robert Sorrells, Frank McGrath, John Milford. A once famous gunman takes on a life of leisure as the sheriff of a small town until a punk arrives to gun him down. A cliched story does nothing to help this pedestrian feature; also called _**Pistolero of Red River**_.\n\n**2186** _ **The Last Chance**_ **** Chesterfield, 1926. 52 min. D-SC: Horace B. Carpenter. With Bill Patton, Dorothy Donald, Merrill McCormick, Harry O'Connor, Walter Patton, Theodore Henderson, Walter Blunt. A postal inspector infiltrates an outlaw gang and saves a stage driver and his daughter who have been taken hostages by the robbers. Labored Bill Patton silent film poorly helmed by Horace B. Carpenter.\n\n**2187** _ **The Last Command**_ **** Republic, 1955. 110 min. Color. D: Frank Lloyd. SC: Warren Duff. With Sterling Hayden, Anna Maria Alberghetti, Richard Carlson, Arthur Hunnicutt, Ernest Borgnine, J. Carrol Naish, Ben Cooper, John Russell, Jim Davis, Virginia Grey, Eduard Franz, Otto Kruger, Russell Simpson, Roy Roberts, Slim Pickens, Hugh Sanders, Morris Ankrum, Harry Woods, Kermit Maynard. Jim Bowie and his followers join the Texas fight for independence and become martyrs at the Alamo. Very fine account of the fall of the Alamo with well staged battle sequences.\n\n**2188** _ **The Last Day**_ **** NBC-TV\/Paramount, 1975. 100 min. Color. D: Vincent McEveety. SC: Jim Byrnes and Steve Fisher. With Richard Widmark, Christopher Connelly, Robert Conrad, Gene Evans, Richard Jaeckel, Tim Matheson, Barbara Rush, Tom Skerrit, Loretta Swit, Morgan Woodward, Kathleen Cody, Jon Locke, Bryan O'Byrne, Harry Morgan (narrator). When the Dalton gang threatens to rob a town bank, a retired gunman is forced to take up his guns again. Nicely entertaining telefeature from producer A.C. Lyles.\n\n**2189** _ **Last Days of Boot Hill**_ **** Columbia, 1947. 55 min. D: Ray Nazarro. SC: Norman S. Hall. With Charles Starrett, Smiley Burnette, Virginia Hunter, Paul Campbell, The Cass County Boys (Jerry Scoggins, Bert Dodson, Fred Martin), Mary Newton, Bill Free, J. Courtland Lytton, Robert Wilke, Alan Bridge, Tex Harding, Syd Saylor, John Cason, Blackie Whiteford, Victor Cox, Carole Mathews, Emmett Lynn, Charles King, Nolan Leary, Steve Clark, Robert Barron, Mauritz Hugo, John Tyrrell. A supposedly dead Durango Kid is on the trail of gold stolen by an outlaw, now a ranch foreman. Tacky \"Durango Kid\" series outing mostly made up of footage from _**Both Barrel Blazing**_ (q.v.), thus the appearance of Tex Harding who had been out of the series for almost two years. British title: _**On Boot Hill**_.\n\n**2190** _ **The Last Days of Frank and Jesse James**_ **** NBC-TV, 1986. 100 min. Color. D: William A. Graham. SC: William Stratton. With Johnny Cash, Kris Kristofferson, Ed Bruce, Willie Nelson, Gail Youngs, David Allan Coe, Andy Stahl, June Carter Cash, Marcia Cross, Darrell Wilks, Margaret Gibson, James Sinclair, Cherie Elledge Grapes, Peter Bradshaw, Earl Pooderall, Jack Barlow, Mac Bennett, John Brown, Dan Butler, Glen Clark, David Cobb, Bruce Darnaham, Ed Evans, Marshal Falwell, Buck Ford, Donnie Fritts, Lecille Harris, Mary Jane Harrill, Dan Hoffman, John Jay Hecker, Jr., Slick Lawson, William Newman, John Jackson Routh, Jimmy Tittle, Denis Tucker, Charlie Williams. Outlaw brothers Frank and Jesse James try to lead normal lives but soon re-form a gang and continue their robbery sprees. Not bad TV movie retelling of the James gang saga hampered by the casting of June Carter Cash as the outlaws' mother!\n\n**2191** _ **The Last Frontier**_ **** RKO Radio, 1932. 12 Chapters. D: Spencer Gordon Bennet. SC: George Plympton and Robert F. Hill. With Creighton (Lon, Jr.) Chaney, Dorothy Gulliver, Mary Jo Desmond, Francis X. Bushman, Jr., Joe Bonomo, Slim Cole, Judith Barrie, Richard Neil, William Desmond, LeRoy Mason, Yakima Canutt, Pete Morrison, Claude Peyton, Fritzi Fenn, Bill Nestell, Ben Corbett, Fred Burns, Frank Lackteen, Barbara Bushman, Harriet Spencer, Walt Robbins, Leo Cooper, Ray Steel. A crusading newspaper editor takes on the guise of a hooded avenger in fighting an outlaw gang after gold on settlers' lands. RKO's only cliffhanger is on the slow side although Lon Chaney's fans will like it; also released in a 65 minute feature version called _**The Black Ghost**_.\n\n**2192** _ **The Last Frontier**_ **** Columbia, 1955. 98 min. Color. D: Anthony Mann. SC: Philip Yordan and Russell S. Hughes. With Victor Mature, Guy Madison, Robert Preston, Anne Bancroft, James Whitmore, Russell Collins, Peter Whitney, Pat Hogan, Manuel Donde, Guy Williams, Mickey Kuhn, Guillermo Calles, Jack Pennick, John Cason, Terry Wilson, Reg Parton, Allen Pinson, Robert St. Angelo, William Traylor. Three frontier scouts finds themselves at odds with a know-it-all fort commander who leads his men into an Indian massacre. Fairly colorful and action laded frontier saga; TV title: _**Savage Wilderness**_.\n\n**2193** _ **Last Frontier Uprising**_ **** Republic, 1947. 67 min. Color. D: Lesley Selander. SC: Harvey Gates. With Monte Hale, Adrian Booth, Foy Willing and The Riders of the Purple Sage, James Taggert, Roy Barcroft, Edmund Cobb, Philip Van Zandt, John Ince, Frank O'Connor, Bob Blair, Doye O'Dell, Tom London. A cowboy buying horses for the government finds himself at odds with a rustling gang. Predictable but still enjoyable Monte Hale vehicle.\n\n**2194** _ **The Last Gun**_ **** Rasfilm, 1964. 88 min. Color. D: Serge Bergone (Sergio Bergonzelli). SC: Dick Fullmer, Ambrogio Molleni and James Wilde, Jr. With Cameron Mitchell, Celina Cely, Carl Mohner, Livio Lorenzon, Kitty Karver, Mary Gordon, Armand Keith, Dony Batner, Mario Munoz, John Mathews, Fanny Clair, Harris Cooper, Lina Albert, Diego Wells, Paul Solvay, Ugo Mudd, Calisto Calisti. In 1866 Arizona a town is threatened by ruthless bandits and an ex-gunfighter is forced to strap on his guns for one final shootout. Cameron Mitchell is the chief interest as the protagonist in this fair Italian made oater originally called _**Jim il Primo**_ (Jim the First) and released in England as _**Killer's Canyon**_ by British Lion; title song sung by Peter Tevis.\n\n**2195** _ **The Last Gunfighter**_ **** Joseph Brenner, 1961. 56 min. D-SC: Lindsay Shontref. With Don Borisenko, Tass Tory, Jay Shannon, Michael Zenon, Ken James, Gordon Clark, James (Hagan) Beggs, Art Janoff, James Barron, Mike Conway. A gunman his hired by ranchers to protect them against a land baron but trouble soon develops over a woman. Tattered, violent drama filmed in Canada in 1959. Also called _**The Devil's Spawn**_ and _**Hired Gun**_.\n\n**2196** _ **The Last Hard Men**_ **** 20th Century\u2013Fox, 1976. 98 min. Color. D: Andrew V. McLaglen. SC: Guerdon Trueblood. With Charlton Heston, James Coburn, Barbara Hershey, Christopher Mitchum, Michael Parks, Jorge Rivero, Larry Wilcox, Morgan Paull, Thalmus Rasulala, Robert Donner, John Quade, Sam Gilman, James Bacon, Riley Hill, Dick Alexander, Yolanda Schutz, Alberto Pina, David Herrera, Earl W. Smith, Lee Lazarow, Robert Apel. To get revenge on the sheriff who sent him to prison an outlaw and his gang kidnap the lawman's young daughter. Sturdy action feature; good entertainment.\n\n**2197** _ **The Last Horseman**_ **** Columbia, 1944. 58 min. D: William Berke. SC: Ed Earl Repp. With Russell Hayden, Dub Taylor, Bob Wills and His Texas Playboys, Ann Savage, John Maxwell, Frank LaRue, Nick Thompson, Art Mix, John Merton, Blackie Whiteford, Ted Mapes, Forrest Taylor, Curley Dresden. Crooks plan to rob a bank and use an innocent female bank teller to help them but a cowboy and his pals save the day. Lots of action and music help this Russell Hayden vehicle.\n\n**2198** _ **The Last Hunt**_ **** Metro-Goldwyn-Mayer, 1956. 108 min. Color. D-SC: Richard Brooks. With Robert Taylor, Stewart Granger, Lloyd Nolan, Debra Paget, Russ Tamblyn, Constance Ford, Joe De Santis, Ainslie Pryor, Terry Wilson, Ralph Moody, Fred Graham, Ed Lonehill, Dan White, William Phillips, Jerry Martin, Roy Barcroft, Rosemary Johnston, Joe Rickson. A rancher, whose cattle herd has been destroyed by rampaging buffalos, teams with a sadistic hunter to destroy the animals. Very good psychological Western interpolated with fast action and mostly filmed in Custer State Park in South Dakota.\n\n**2199** _ **The Last Movie**_ **** Universal, 1971. 108 min. Color. D: Dennis Hopper. SC: Stewart Stern. With Dennis Hopper, Stella Garcia, Julie Adams, Peter Fonda, Rod Cameron, Kris Kristofferson, John Philip Law, Jim Mitchum, Michelle Phillips, Dean Stockwell, Russ Tamblyn, Don Gordon, Tom Baker, Daniel Ades, Severn Dardern, Samuel Fuller, Roy Engel, Fritz Ford, George Hill, Ted Markland, Tomas Milian, Sylvia Miles, Robert Rothwell, Dennis Stock, Clint Kimbrough, Warren Finnerty. A cowboy-stuntman, who enjoys violence for its own sake, goes to Peru to work on a Western movie and stays only to end up as a sacrifice by the natives. Extremely unpleasant and disappointing feature that wastes a good cast; also called _**Chinchero**_.\n\n**2200** _ **The Last Musketeer**_ **** Republic, 1952. 67 min. D: William Witney. SC: Arthur Orloff. With Rex Allen, Mary Ellen Kay, Slim Pickens, James Anderson, Boyd \"Red\" Morgan, Monte Montague, Michael Hall, Alan Bridge, Stan Jones, The Republic Rhythm Riders. On a cattle buying trip, a cowboy stops in a town and helps the locals who have been plagued by a gang of outlaws. Typical Rex Allen vehicle with the usual amount of action.\n\n**2201** _ **The Last of His Tribe**_ **** Home Box Office (HBO), 1992. 90 min. Color. D: Harry Hook. SC: Stephen Hartman. With Jon Voight, Graham Greene, David Ogden Stiers, Jack Blessing, Anne Archer, Diana Benzali, Christianne Hauber, Charles Martinet, Carl D. Parker, Angela Paton, Benne Alder, Marie Bain, Loryn Barlese, Gilbert Bear, Lance Brady, Roy Conrad, Kevin Goetz, Neva Hutchison, John Ickes, Trudy Kinerson, Beverly La Bau, Stephen Lofaro, George Maguire, Edward Markmann, Wanda McCaddon, Sam David McClelland, C.W. Morgan, James O'Connell, John Olson, Stephen Pocock, Chris Pray, Victor Preston, Nick Scoggin, Toby Stmp, Brain VanDerWilt. The lone survivor of an Indian tribe makes his way to civilization and is befriended by an anthropologist. Okay TV movie retelling of a true story previously filmed as _**Ishi:**_ _**The Last of His Tribe**_ (q.v.).\n\n**2202** _ **Last of the Badmen**_ **** Allied Artists, 1957. 80 min. Color. D: Paul Landres. SC: Daniel B. Ullman and David Chandler. With George Montgomery, Keith Larsen, James Best, Douglas Kennedy, Robert Foulk, Tom Greenway, Meg Randall, Willis Bouchey, Michael Ansara, Addison Richards, John Doucette, Harlan Warde, Walter Reed, John Damler. When outlaws kill a detective, his agency sends two agents West to round up the gang. Okay action drama; remake of _**Star of Texas**_ (q.v.) and remade as _**Gunfight at Comanche Creek**_ (q.v.).\n\n**2203** _ **The Last of the Clintons**_ **** Ajax, 1935. 64 min. D: Harry Fraser. SC: Weston Edwards. With Harry Carey, Betty Mack, Del Gordon, Victor Potel, Earl Dwire, Ruth Findlay, Tom London, Slim Whitaker, Lafe McKee, Lew Meehan, Francis Walker, Allen Greer, William McCall, Barney Beasley, Tex Palmer. A gang has been raiding the countryside and a range detective pretends to be an outlaw so he can infiltrate them. Cheap production values do not help his oater but Harry Carey fans will still like it.\n\n**2204** _ **Last of the Comanches**_ **** Columbia, 1953. 85 min. Color. D: Andre De Toth. SC: Kenneth Gamet. With Broderick Crawford, Barbara Hale, Lloyd Bridges, Johnny Stewart, Mickey Shaughnessy, George Mathews, Hugh Sanders, Ric Roman, Chubby Johnson, Martin Milner, Milton Parsons, Jack Woody, John War Eagle, Carleton Young, George Chesebro, Harry Harvey, Jay Silverheels, Bud Osborne, Rodd Redwing. The six surviving members of a cavalry unit attacked by Indians join the passenger of a stagecoach in staving off Comanches. Tired oater, a reworking of _**Sahara**_ (Warner Bros., 1943).\n\n**2205** _ **Last of the Desperados**_ **** Associated Film Distributors, 1955. 75 min. D: Sam Newfield. SC: Orville Hampton. With James Craig, Jim Davis, Margia Dean, Barton MacLane, Myrna Dell, Bob Steele, Stanley Clements, Herbert Vigran, Thomas Browne Henry, Jack Perrin, John Hart, Mike Ragan, Brad Johnson, Frank Sully. After his final shootout with Billy the Kid, sheriff Pat Garrett finds he is more of a hunted man than his former adversary. Action filled and different \"B\" oater; Charles King shows up at the start of the film via stock footage.\n\n**2206** _ **Last of the Dogmen**_ **** Savoy Pictures, 1995. 118 min. Color. D-SC: Tad Murphy. With Tom Berenger, Barbara Hershey, Kurtwood Smith, Steve Reevis, Andrew Miller, Gregory Scott Cummins, Mark Boone Junior, Helen Calahasen, Eugene Blackbear, Dawn Lavand, Sidel Standing Elk, Hunter Bodine, Graham Jarvis, Parley Baer, Sherwood Price, Zip, Molly Parker, Antony Holland, Robert Donley, Brian Stollery, Mitchell LaPlante, Wilford Brimley (narrator). A bounty hunter is on the trail of trail of three escaped convicts in the Montana wilderness. Interesting fantasy Western.\n\n**2207** _ **Last of the Duanes**_ **** Fox, 1930. 62 min. D: Alfred L. Werker. SC: Ernest Pascal. With George O'Brien, Lucille Brown, Myrna Loy, Walter McGrail, James Bradbury, Jr., Nat Pendleton, Blanche Frederici, Roy Stewart, Frank Campeau, Jim Mason, Lloyd Ingraham, Willard Robertson, Richard Alexander, Pat Harmon, G. Raymond Nye, Ed Brady, Merrill McCormick, Carl Stockdale, Cliff Lyons, Lillian Lawrence, Harry Tenbrook. Complications arise for a cowboy because after he saves a young woman from a bad man the outlaw's wife falls for him. Slow moving adaptation of Zane Grey's novel with songs added; first filmed in 1919 by Fox with William Farnum and remade by the studio in 1924 starring Tom Mix. Filmed simultaneously with the George O'Brien vehicle was a Spanish language version called _**El Ultimo de los Vargas**_ (The Last of the Vargas), directed by David Howard and starring George J. Lewis.\n\n**2208** _ **Last of the Duanes**_ **** 20th Century\u2013Fox, 1941. 57 min. D: James Tinling. SC: Irving Cummings, Jr. and William Couselman, Jr. With George Montgomery, Lynne Roberts, Eve Arden, Francis Ford, George E. Stone, William Farnum, Joseph Sawyer, Truman Bradley, Russell Simpson, Don Costello, Harry Woods, Andrew Tombes, Tom London, Tim Ryan, Lane Chandler, Arthur Aylesworth, Ann Carter, Harry Hayden, Walter McGrail, Russ Clark, Lew Kelly, Jack Stoney, Tom Moray, Syd Saylor, LeRoy Mason, Paul Sutton, Ethan Laidlaw, Erville Alderson, J. Anthony Hughes, Paul E. Burns, William Pagan, Hank Worden, Max Wagner, Robert Winkler. Setting out to avenge the murder of his father, a cowboy gets an unjust reputation as a gunman. Fifth filming of the Zane Grey work is an okay \"B\" effort that helped George Montgomery on the road to stardom.\n\n**2209** _ **The Last of the Fast Guns**_ **** Universal-International, 1958. 82 min. Color. D: George Sherman. SC: David P. Harmon. With Jock Mahoney, Gilbert Roland, Linda Cristal, Eduard Franz, Lorne Greene, Carl Benton Reid, Edward Platt, Eduardo Noriega, Jorge Trevino, Lee Morgan, Richard Cutting. A gunfighter is hired to find a man's missing brother and heads to Mexico to carry out the assignment. Pretty good action drama with fine work by Jock Mahoney and Gilbert Roland.\n\n**2210** _ **The Last of the Knucklemen**_ **** Hexagon, 1981. 93 min. Color. D-SC: Tim Burstall. With George Kennedy, Michael Preston, Peterr Hehir, Michael Duffield, Dennis Miller, Stephen Bisley, Michael Caton, Stewart Fatchney. In frontier Australia a camp boss tries to keep order among his gang of rowdy hired hands. Entertaining Australian yarn with good character studies.\n\n**2211** _ **The Last of the Mohicans**_ **** Associated Producers, 1920. 75 min. D: Maurice Tourneur and Clarence Brown. SC: Robert Dillon. With Wallace Beery, Barbara Bedford, Albert Roscoe, Lillian Hall, Henry Woodward, Boris Karloff, Harry Lorraine, Nelson McDowell, George Hawkathorne, Jack McDonald, James Gordon. A frontiersman and his Mohican blood brother try to save the daughters of a British fort commander captured by a rival tribe loyal to the French. Well made silent version of the James Fenimore Cooper novel; some prints run 50 minutes.\n\n**2212** _ **The Last of the Mohicans**_ **** Mascot, 1932. 12 Chapters. D: Ford Beebe and B. Reeves Eason. SC: Colbert Clark, Jack (John Francis) Natteford, Ford Beebe and Wyndham Gittens. With Harry Carey, Hobart Bosworth, Junior Coghlan, Edwina Booth, Lucille Brown, Walter Miller, Robert Kortman, Walter McGrail, Nelson McDowell, Edward Hearn, Mischa Auer, Yakima Canutt, Chief Big Tree, Joan Gale, Tully Marshall, Allan Cavan, High Feather, Little Pine, White Feather, Jewel Richford. During the French and Indian War, scout Hawkeye and his Indian blood bother attempt to stop an evil chief and his tribe from helping the French against British settlers. Slow moving serial version of the James Fenimore Cooper work although Harry Carey is good as Hawkeye and Robert Kortman makes an excellent evil Magua; a feature version was called _**The Return of the Mohicans**_.\n\n**2213** _ **The Last of the Mohicans**_ **** United Artists, 1936. 91 min. D: George B. Seitz. SC: Philip Dunne. With Randolph Scott, Binnie Barnes, Henry Wilcoxon, Bruce Cabot, Heather Angel, Hugh Buckler, Robert Barrat, Philip Reed, Willard Robertson, William Stack, Frank McGlynn, Will Stanton, William V. Mong, Olaf Hytten, Claude King, Lumsden Hare, Reginald Barlow, Lionel Belmore, Ian MacLaren, Art Dupuis, John Sutton, Ethan Laidlaw, Harry Cording. A scout and his Indian friend escort two British women through the wilderness and fall in love with them as the British and French fight for control of North America. Nicely done adaptation of the James Fenimore Cooper's 1826 book.\n\n**2214** _ **The Last of the Mohicans**_ **** International German\/Balcazar\/Cineproduzione, 1965. 88 min. Color. D: Harald Reinl. SC: Joachim Bartsch. With Joachim Fuchsberger, Karin Dor, Carl Lange, Antonio De Teffe (Anthony Steffen), Dan Martin, Jose Marco, Luis Induni, Angel Ter, Stellio Candelli, Marie France. When trouble starts between warring Indian tribes, Hawkeye and Chingachgook attempt to save the lives of the kidnapped daughters of a British colonel. Despite dubbing, this West German production of the James Fenimore Cooper story is pretty good, with a fine music score by Peter Thomas; West German title: _**Der Letzte Mohikaner**_ (The Last Mohican). Alternate title: _**The Last Tomahawk**_.\n\n**2215** _ **The Last of the Mohicans**_ **** NBC-TV\/Schick Sunn Classics, 1977. 100 min. Color. D: James L. Conway. SC: Stephen Lord. With Steve Forrest, Ned Romero, Andrew Prine, Don Shanks, Robert Tessier, Jane Actman, Michele Marsh, Robert Easton, Whit Bissell, Beverly Rowland. Scout Hawkeye and his blood brother Chingachgook help a British major lead a party through hostile country during the French and Indian War. Well done \"Classics Illustrated\" production made for TV.\n\n**2216** _ **The Last of the Mohicans**_ **** 20th Century\u2013Fox, 1992. 113 min. Color. D: Michael Mann. SC: Michael Mann and Christopher Crowe. With Daniel Day-Lewis, Madeleine Stone, Russell Means, Eric Schweig, Jodhi May, Steven Waddington, Wes Studi, Maurice Roeves, Patrice Chereau, Colm Meaney, Peter Postlethwaite, Patrice Chereau, Edward Blatchford, Terry Kinney, Tracey Ellis, Justin M. Rice, Dennis Banks, Mac Andrews, Malcoln Storry, David Schofield, Eric D. Sandgren, Mike Phillips, Mark A. Blake, Dylan Baker, Tim Hopper. A frontiersman tries to avoid taking sides in the French and Indian War but eventually aids the British after rescuing and falling in love with a beautiful woman captured by Huron warrior Magua. Pictorially appealing update of the James Fenimore Cooper novel, adapted from Philip Dunne's script for the 1936 version (q.v.).\n\n**2217** _ **Last of the Pony Riders**_ **** Columbia, 1953. 59 min. D: George Archainbaud. SC: Ruth Woodman. With Gene Autry, Smiley Burnette, Kathleen Case, Dick Jones, Howard Wright, Arthur Space, Gregg Barton, Buzz Henry, Harry Mackin, Harry Hines, Kermit Maynard, John Downey, Frankie Marvin, Bob Woodward. Pony Express rider Gene Autry loses his job when he buys a stagecoach and learns that crooks are working to sabotage the operation in order to get a mail contract. Gene Autry's final theatrical starring film is a pleasant, leisurely paced affair; filmed in Sepiatone.\n\n**2218** _ **Last of the Redmen**_ **** Columbia, 1947. 77 min. Color. D: George Sherman. SC: Herbert Dalmas and George Plympton. With Jon Hall, Evelyn Ankers, Michael O'Shea, Julie Bishop, Buster Crabbe, Rick Vallin, Buzz Henry, Frederick Worlock, Emmett Vogan, Chief Many Treaties, Guy Hedlund. The lone surviving member of the Mohican tribe risks his life to save a group of white settlers led into ambush by the Iroquis. Tame adaptation of James Fenimore Cooper's _The Last of the Mohicans_ ; filmed in VitaColor.\n\n**2219** _ **Last of the Renegades**_ **** Columbia, 1966. 94 min. Color. D: Harald Reinl. SC: Harald G. Petersson. With Lex Barker, Pierre Brice, Anthony Steel, Karin Dor, Klaus Kinski, Mario Girotti (Terence Hill), Renato Baldini, Eddi Arent, Marie Noelle, Ilija Ivezic, Velemir Chytil, Stole Arandjelovic, George Heston, Mirko Boman, Rikard Brzeska, Gojko Mitic, Sime Jarainac, Jozo Kovacevic, Antun Nalis, Ivo Kristof, Curt Ackerman (narrator). Apache chief Winnetou wants to keep his people from going to war after the death of his father, but a ruthless oilman tries to start an uprising so the Indians will be slaughtered and he can get their lands. Flavorful West German Western in Constantin's popular Karl May series; issued in Europe in 1964 by Rialto\/Jadran-Film as _**Winnetou II**_ , running 104 minutes.\n\n**2220** _ **Last of the Warrens**_ **** Supreme, 1936. 56 min. D-SC: Robert North Bradbury. With Bob Steele, Margaret Marquis, Charles K. French, Lafe McKee, Charles King, Horace Murphy, Blackie Whiteford, Jim Corey, Steve Clark, Julian Madison, Herman Hack, Art Dillard, Frank Ball, Horace B. Carpenter, Tex Palmer, Chuck Baldra. After becoming an aviator hero during the war, a cowboy returns home to find a local businessman has stolen his property. Typically action packed and entertaining Bob Steele oater.\n\n**2221** _ **Last of the Wild Horses**_ **** Screen Guild, 1948. 86 min. D: Robert L. Lippert. SC: Jack Harvey. With James Ellison, Mary Beth Hughes, Jane Frazee, Douglass Dumbrille, Reed Hadley, James Millican, Olin Howlin, Grady Sutton, William Haade, Stanley Andrews, Rory Mallinson. A cowboy gets himself in the middle of a range dispute when a crooked businessman and his sheriff henchman accuse a wealthy, crippled rancher of cattle thefts. More than passable production with an extremely brutal fight between hero James Ellison and villain Reed Hadley.\n\n**2222** _ **The Last Outlaw**_ **** Paramount, 1927. 70 min. D: Arthur Rosson. SC: John Stone and J. Walter Ruben. With Gary Cooper, Betty Jewel, Jack Luden, Herbert Prior, Jim Corey, Billy Butts, Flash (horse). When a lawman is falsely accused of killing his girl's brother he tries to bring in the real murderers who are also cattle thieves. Vintage Gary Cooper silent film that will appeal to his fans.\n\n**2223** _ **The Last Outlaw**_ **** RKO Radio, 1936. 62 min. D: Christy Cabanne. SC: John Twist and Jack Townley. With Harry Carey, Hoot Gibson, Tom Tyler, Henry B. Walthall, Margaret Callahan, Ray Mayer, Harry Jans, Frank M. Thomas, Russell Hopton, Frank Jenks, Maxine Jennings, Joseph Sawyer, Fred Scott, Jack Mulhall, Harry Woods, Alfred P. James, Ethel Wales, Ralph Byrd, Bud Flanagan (Dennis O'Keefe), Alan Curtis, Stanley Blystone, George Magrill, Jack Rice, Ed Jones, Ben Hewlett, Jerry Frank, Harry Depp, James B. Burtis. A once famous outlaw is released from prison after a quarter of a century behind bars and returns home to find the West he knew is gone and opposes modern-day gangsters. John Ford co-wrote the story for this film which is a corker, combining a nostalgic look at the genre with plenty of action from its veteran stars.\n\n**2224** _ **The Last Outlaw**_ **** Home Box Office (HBO), 1993. 93 min. Color. D: Geoff Murphy. SC: Eric Red. With Mickey Rourke, Dermot Mulroney, Ted Levine, John C. McGinley, Steve Buscemi, Keith David, Daniel Quinn, Gavan O'Herlihy, Richard Fancy, Tom Connor, Sid Klinge, Phil Mead, Paul Ben-Victor, Greg Doty, John David Garfield, Jake Walker, Marvin Gilbert, Edward Proudfoot, Joey Rourke, Darrly Shay. After taking part in a foiled, bloody train robbery, an outlaw is betrayed by his gang and he joins a posse in hunting them down. Violent, well made TV movie.\n\n**2225** _ **The Last Outpost**_ **** Paramount, 1951. 88 min. Color. D: Lewis R. Foster. SC: Geoffrey Homes, George Worthington Yates and Winston Miller. With Ronald Reagan, Rhonda Fleming, Bruce Bennett, Bill Williams, Peter Hanson, Noah Beery, Jr., Hugh Beaumont, John Ridgeley, Lloyd Corrigan, Charles Evans, Chuck Roberson, Iron Eyes Cody, Chief Yowlachie, Burt Mustin. When the woman he loves is staying at a fort threatened by Indians, a Confederate cavalry officer leads his men in helping defend the Union held garrison. Good, colorful action film from the Pine-Thomas unit; alternate video title: _**Cavalry Charge**_.\n\n_**Left to right:**_ **Rhonda Fleming, Bruce Bennett and Ronald Reagan in** _**The Last Outpost**_ **(Paramount, 1951).**\n\n**  \n**\n\n**2226** _ **The Last Posse**_ **** Columbia, 1953. 73 min. D: Alfred L. Werker. SC: Seymour Bennett, Connie Lee Bennett and Kenneth Gamet. With Broderick Crawford, John Derek, Charles Bickford, Wanda Hendrix, Warner Anderson, Henry Hull, Will Wright, Tom Powers, Raymond Greenleaf, James Kirkwood, Eddy Waller, Skip Homeier, James Bell, Guy Wilkerson, Mira McKinney, Helen Wallace, Harry Hayden, Monte Blue, Paul Maxey, Reed Howes, Brick Sullivan, Stanley Blystone, Frank Ellis, Franklyn Farnum, Frank J. Scannell, Frank Hagney, Billy Wilkerson, Bob Burns. A sheriff leads a posse into the desert in pursuit of the culprits who stole a rancher's money. Very fine, compact drama from producer Harry Joe Brown.\n\n**2227** _ **The Last Rebel**_ **** Sterling World Distributors, 1961. 83 min. Color. D-SC: Miguel Contreras Torres. With Carlos Thompson, Ariadne Welter, Rodolfo Acosta, Charles Fawcett, Lee Morgan, Eduardo Noriega, Federico Curiel, John Kelly, Rebecca Iturbide, Manuel Orvide, Tony Carbajal, Carlos Muzquiz. A happy-go-lucky Mexican outlaw gang leader defeats a group of murderous gold hunters and is stalked by the Texas Rangers. Mexican filmed oater is pretty fair viewing; reissued in 1968.\n\n**2228** _ **The Last Rebel**_ **** Columbia, 1971. 89 min. Color. D: Denys McCoy. SC: Warren Kiefer (Luicano Ricci). With Joe Namath, Jack Elam, Woody Strode, Ty Hardin, Victoria George, Renato Romano, Marina Coffa, Annamaria Chio, Mike Forrest, Bruce Ewelle, Jessica Dublin, Larry Laurence, Herb Andress. In a Missouri town after the Civil War, two Confederate soldiers rescue an ex-slave from being lynched by vigilantes. Very poor Italian production.\n\n**2229** _ **The Last Ride of the Dalton Gang**_ **** NBC-TV, 1979. 150 min. Color. D: Dan Curtis. SC: Earl W. Wallace. With Cliff Potts, Larry Wilcox, Randy Quaid, Dale Robertson, Jack Palance, Bo Hopkins, Sharon Farrell, Harris Yulin, Matt Clark, Royal Dano, Julie Hill, John Karlen, Mills Watson, Bo Hopkins, Dennis Fimple, R.G. Armstrong, Don Collier, Harry Townes, Scott Brady, H.M. Wynant, Sid Conrad, Jack Collins, Elliott Street, Terry Kiser, John Fitzpatrick, Eric Lawson, James Crittenden, Jorge Moreno, Tony Palmer, Mitch Carter, Don Scarbrough, Larry Bloch, John Calvin, Dean Smith, Robert Karnes. The life and times of the legendary outlaw Dalton brothers, culminating in their final robbery attempt at Coffeyville, Kansas. Overlong and somewhat dull tongue-in-cheek TV movie.\n\n**2230** _ **The Last Ride to Santa Cruz**_ **** Casino Films, 1964. 99 min. Color. D: Rolf Olsen. SC: Otto Pischin, Herta Hareson and Leo Metzenni. With Edmund Purdom, Marion Cook (Marianne Koch), Mario Adorf, Marissa Mell, Klaus Kinski, Walter Giller, Edmund Haskins, Thomas Fritson. When a dishonest lawman unjustly sends a man to prison he escapes and returns for revenge. Fairly entertaining West German oater issued there by Magnet as _**Der Letzte Ritt nach Santa Cruz**_ (The Last Ride to Santa Cruz).\n\n**2231** _ **The Last Roundup**_ **** Syndicate, 1929. 50 min. D: J.P. McGowan. SC: Sally Winters. With Bob Custer, Hazel Mills, Bud Osborne, Cliff Lyons, Hank Bell, J.P. McGowan, Ada Bell Driver, Tom Bay. After a ranch foreman falls out with one his hands, the latter rustles his boss' cattle and kidnaps the new schoolmarm. Low grade Bob Custer silent in which his character's name is Denver Dixon!\n\n**2232** _ **The Last Round-Up**_ **** Paramount, 1934. 61 min. D: Henry Hathaway. SC: Jack Cunningham. With Randolph Scott, Barbara (Fritichie) Adams, Monte Blue, Fred Kohler, Fuzzy Knight, Richard Carle, Barton MacLane, Charles Middleton, Frank Rice, Dick Rush, Buck Connors, Bob Miles, Sam Allen, Ben Corbett, J. Merrill Holmes, Jim Corey, James (Jim) Mason, Bud Osborne, Charles Brinley, Charles Murphy. Falsely accused of murder, a cowboy is saved from being convicted of the crime by an outlaw and he joins his rescuer's gang. Fair adaptation of Zane Grey's 1916 novel _The Border Legion_ , previously filmed under that title in 1918 by Goldwyn with Hobart Bosworth, in 1924 by Paramount starring Antonio Moreno and by the same studio in 1930 (q.v.).\n\n**2233** _ **The Last Round-Up**_ **** Columbia, 1947. 77 min. D: John English. SC: Jack Townley and Earle Snell. With Gene Autry, Jean Heather, Ralph Morgan, Bobby Blake, Bud Osborne, Jay Silverheels, John Cason, Carol Thurston, Mark Daniels, Russ Vincent, Shug Fisher, Trevor Bardette, Lee Bennett, John Halloran, Roy Gordon, Dale Van Sickle, Ed Peil, Sr., George Carleton, Nolan Leary, Ted Adams, Steve Clark, Frankie Marvin, Kernan Cripps, Iron Eyes Cody, Blackie Whiteford, Robert Walker, Virginia Carroll, Rodd Redwing, Alex Montoya. A land baron causes trouble with local Indians in order to stop an aqueduct project that would interfere with his takeover of more range land. Modern-day Western is too long and Gene Autry is very stiff although it does utilize an amusing sequence involving television.\n\n**2234** _ **Last Stagecoach West**_ **** Republic, 1957. 70 min. D: Joseph Kane. SC: Barry Shipman. With Jim Davis, Mary Castle, Victor Jory, Lee Van Cleef, Grant Withers, Roy Barcroft, John Alderson, Glenn Strange, Francis McDonald, Tristram Coffin, Willis Bouchey, Lewis Martin, Kelo Henderson, Henry Wills, Percy Helton. After losing a mail contract because crooks sabotaged his stage line, a man joins with his outlaw pal to get revenge. Just passable action drama with a good cast, mediocre plot and average production values.\n\n**2235** _ **The Last Stand**_ **** Universal, 1938. 57 min. D: Joseph H. Lewis. SC: Harry O. Hoyt and Norton S. Parker. With Bob Baker, Marjorie Reynolds, Fuzzy Knight, Earle Hodgins, Forrest Tucker, Glenn Strange, Jack Kirk, Jimmy Phillips, Sam Flint, Frank Ellis, Jack Montgomery. When his father is murdered, a cowpoke pretends to be an outlaw to infiltrate the gang responsible for the crime. Fair Bob Baker vehicle in which he croons a trio of songs.\n\n**2236** _ **Last Stand at Sabre River**_ **** Turner Network Television (TNT), 1997. 96 min. Color. D: Dick Lowry. SC: Ronald M. Cohen. With Tom Selleck, Suzy Amis, Rachel Duncan, Haley Joel Oment, Keith Carradine, David Carradine, Tracey Needham, Chris Stacy, Harry Carey, Jr., Patrick Kilpatrick, Michael Osment, Denis Forest, David Dukes, Lumi Cavazos, Raymond Cruz, Frederick Lopez, Rosalie De Aragon, Ramon Frank, Paul Blott, Rex Linn, R.A. Hilder, John Howard Young, Von Engels, John David Garfield, Hector Mercado, Rudy Ugland, Tim Carroll, Billy Lockwood. An ex\u2013Confederate tries to start a new life on a homestead in Arizona but finds it taken over by squatters sympathetic to the Union. Well made TV movie adapted from Elmore Leonard's novel, worth watching.\n\n**2237** _ **The Last Sunset**_ **** Universal, 1961. 112 min. Color. D: Robert Aldrich. SC: Dalton Trumbo. With Rock Hudson, Kirk Douglas, Dorothy Malone, Joseph Cotten, Carol Lynley, Neville Brand, Regis Toomey, Rad Fulton, Adam Williams, Jack Elam, John Shay, Jose Torvay, Margarita Luna, Jose Trevino. A lawman is on the trail of the man who murdered his brother-in-law and finds him on a cattle drive with a drunken rancher and romancing the man's pretty daughter. Complicated Mexico-filmed oater with murder, alcoholism, adultery and incest.\n\n_**The Last Tomahawk**_ see _**The Last of the Mohicans**_ (1964)\n\n**2238** _ **The Last Trail**_ **** Fox, 1927. 58 min. D: Lewis Seiler. SC: John Stone. With Tom Mix, Carmelita Geraghty, William Davidson, Frank Hagney, Lee Shumway, Robert Brower, Jerry Madden, Oliver Eckhardt. A cowboy helps a man and his daughter in their contest with a crook for a government mail contract. Fast moving and entertaining silent version of the Zane Grey novel, first filmed by Fox in 1921 with Maurice Flynn, Eva Novak, Wallace Beery and Rosemary Theby.\n\n**2239** _ **The Last Trail**_ **** Fox, 1933. 59 min. D: James Tinling. SC: Stuart Anthony. With George O'Brien, Claire Trevor, J. Carrol Naish, El Brendel, Matt McHugh, Lucille LaVerne, Ed LeSaint, Ruth Warren, George Reed. A man finds out that gangsters have taken over his family's ranch and he sets out to stop them. Pleasant adaptation of the Zane Grey work with as much comedy as action.\n\n**2240** _ **Last Train from Gun Hill**_ **** Paramount, 1959. 94 min. Color. D: John Sturges. SC: James Poe. With Kirk Douglas, Anthony Quinn, Carolyn Jones, Earl Holliman, Brad Dexter, Brian Hutton, Ziva Rodann, Val Avery, Walter Sande, Lars Henderson, John P. Anderson, Lee Hendry, William Newell, Sid Tomack, Charles Stevens, Julius Tannen, Ken Becker, Courtland Shepard, Ty Hardin, Glenn Strange, Hank Mann, William Benedict. After his wife is brutally murdered by two men, a sheriff tracks them to a frontier town only to find the locals oppose him taking them to stand trial. Intense and well made action drama.\n\n**2241** _ **The Last Wagon**_ **** 20th Century\u2013Fox, 1956. 99 min. Color. D: Delmer Daves. SC: James Edward Grant, Delmer Daves and Gwen Bagni. With Richard Widmark, Felicia Farr, Susan Kohner, Tommy Rettig, Stephanie Griffin, Ray Stricklyn, Nick Adams, Carl Benton Reid, Douglas Kennedy, George Mathews, James Drury, Ken Clark, Timothy Carey, Juney Ellis, Abel Fernandez, George Ross. A man kills the two men who raped his wife and murdered their children but when he joins a wagon train he learns he is wanted by the law. Very well done psychological melodrama, finely acted, especially by Richard Widmark.\n\n_**Last Warrior**_ see _**Flap**_\n\n**2242** _ **El Latigo**_ (The Whip). Peliculas Latinoamericanas, S.A., 1978. 82 min. Color. D: Alfred B. Crevenna. SC: Ramon Obon. With Juan Miranda, Gustavo Rojo, Yolanda Ochoa, Mario Almada, Angelica Chain, Jose Luis Fernandez, Alfonso Milan, Francisco Almorza, Baltazar Ramos, Carlos Lopez Figuerosa, Armando Hernandez, Esperanza Lobos. A masked avenger fights with El Dios Tigre (The Tiger God) over a hidden treasure. Typical Mexican Western horror fantasy, followed by _**El Latigo Contra Santana**_ and _**El Latigo Contra los Momias Asesinas**_ (qq.v.).\n\n**2243** _ **El Latigo Contra Las Momias Asesinas**_ (The Whip Against the Killer Mummies) **** Peliculas Latinoamericanas, S.A., 1980. 82 min. Color. D-SC: Angel Rodriguez Vazquez. With Juan Miranda, Rosa Gloria Chagoyan, Marcko D'Carlo, Manuel Lea \"El Tinieblas,\" Guillermina Oropeza, Alfonso Millian, Leopoldo Guerrero, Cesar De Guatemala, Bernabe Palma. When four mummies try for immortality by murdering a family they are opposed by the masked El Latigo. Third and last in the Mexican \"El Latigo\" series, a bit scarier than the others; preceded by _**El Latigo**_ and _**El Latigo Contra Santanas**_ (qq.v.).\n\n**2244** _ **El Latigo Contra Santanas**_ (The Whip Against Satan) **** Peliculas Latino-americanas, 1979. 80 min. Color. D: Alfredo B. Crevanna. SC: Ramon Obon, Alfredo B. Crevanna and Roberto Rodriguez. With Juan Miranda, Noe Murayama, Ruben Rojo, Yolanda Ochoa, Victor Alcocer, Manuel Resendess, Ernesto Merida, Maria Teresa Martinez, Leonardo Moran. A masked hero saves a beautiful woman accused of witchcraft from a devil worshipping cult. Pretty fair horror Western from Mexico, preceded by _**El Latigo**_ and followed by _**El Latigo Contra las Momias Asesinas**_ (qq.v.).\n\n**2245** _ **Laughing Boy**_ **** Metro-Goldwyn-Mayer, 1934. 77 min. D: W.S. Van Dyke. SC: John Colton and John Lee Mahin. With Ramon Novarro, Lupe Velez, William B. Davidson, Chief Thunderbird, Catalina Rambula, Tall Man's Boy, F.A. Armenta, Deer Spring, Pellicana, Sidney Bracey, Anita Sheldon, Ferdinand Munier, Edward Hearn, Kathryn Sheldon, Nora Cecil, Ruth Channing, Carl Stockdale, William Steele, Grace Hayle, Carol Flores, Standing Bear, Jim Mason, Francis Gillman, Chief Meyers. A Navajo brave falls in love with a free-spirited young woman not knowing she is the mistress of a wealthy rancher. Ramon Novarro's last starring feature for MGM is only fair with the two leads superior to the maudlin material.\n\n**2246** _ **The Law and Jake Wade**_ **** Metro-Goldwyn-Mayer, 1958. 86 min. Color. D: John Sturges. SC: William Bowers. With Robert Taylor, Richard Widmark, Patricia Owens, Robert Middleton, Henry Silva, DeForest Kelley, Burt Douglas, Eddie Firestone, Henry Wills, Rory Mallinson, Al Ferguson, Roy Engel, Richard H. Cutting, Gene Coogan, Reginald Simpson. A reformed outlaw is about to be married but he and his fiancee are kidnapped by his ex-partner who wants to know the location of hidden loot. Nicely done action drama with good work by the two stars.\n\n**2247** _ **Law and Lawless**_ **** Majestic, 1932. 58 min. D: Armand Schaefer. SC: Oliver Drake. With Jack Hoxie, Hilda Moreno, Wally Wales, Yakima Canutt, Julian Rivero, Jack Mower, J. Frank Glendon, Edith Fellows, Helen Gibson, Bob Burns, Fred Burns, Al Taylor, Slim Whitaker, Ben Corbett, Hank Bell, Dixie Starr, Gracia Granadas' Orchestra. Riding into an area plagued by cattle rustlers, a cowboy sets out to round up the gang. Jack Hoxie's fans will like this one.\n\n**2248** _ **Law and Lead**_ **** Colony, 1936. 62 min. D: Robert Hill. SC: Basil Dickey. With Rex Bell, Wally Wales, Harley Wood, Earl Dwire, Soledad Jiminez, Donald Reed, Roger Williams, Lane Chandler, Lloyd Ingraham, Karl Hackett, Ed Cassidy, Lew Meehan, George Morrell. A cattleman's association agent is assigned to bring in the Juarez Kid for lawless activities on the border but the operative does not believe the Kid is guilty of the crimes. Okay low budget affair.\n\n**2249** _ **Law and Order**_ **** Universal, 1932. 80 min. D: Edward L. Cahn. SC: John Huston and Tom Reed. With Walter Huston, Harry Carey, Raymond Hatton, Russell Hopton, Ralph Ince, Harry Woods, Richard Alexander, Russell Simpson, Alphonse Ethier, Andy Devine, Lois Wilson, Dewey Robinson, Walter Brennan, Nelson McDowell, D'Arcy Corrigan, George Dixon, Arthur Wanzer, Neal Hart, Richard Cramer, Art Mix, Hank Bell, William Dyer, Eddie Gribbon, Fred Humes, Russ Powell, Lew Meehan, Frank Lanning, Charlie Hall, Stanley Blystone, Frank Brownlee, Artie Ortego, Denver Dixon, Tex Phelps, Dick Rush, Tom Smith, Edgar Lewis, Dorothy Vernon, Cliff Lyons, George Morrell, Pascale Perry, Charles Murphy, Barney Beasley, Charles Brinley, Jim Corey, Joe De La Cruz. A famous lawman and his three pals are hired to clean out the untamed element in a small town, culminating in a shootout. Well made, directed and acted, this is one of the all-time great Westerns\u2014a must see! Reissued as _**Guns A'Blazing**_.\n\n**2250** _ **Law and Order**_ **** Universal, 1940. 57 min. D: Ray Taylor. SC: Sherman Lowe and Victor McLeod. With Johnny Mack Brown, Nell O'Day, Fuzzy Knight, James Craig, Harry Cording, Earle Hodgins, Robert Fiske, Jimmie Dodd, William Worthington, Ted Adams, Ethan Laidlaw, Robert Kortman, Jim Corey, Charles King, The Notables, Harry Humphrey, George Plues, Kermit Maynard, Frank McCarroll, Frank Ellis, Lew Meehan, Eddie Polo, Herman Hack, Bob Reeves, Bill Nestell, Cliff Lyons, Cliff Parkinson, Victor Cox, Al Taylor, Roy Bucko, Jack Shannon, Scoop Martin, Wong Chung. An ex-lawman is helped by a reformed gambler in cleaning up lawlessness in a Western community. Second screen adaptation of W.R. Burnett's novel _Saint Johnson_ but no where as good as the 1932 version (q.v.); this one even contains music.\n\n**2251** _ **Law and Order**_ **** Producers Releasing Corporation, 1942. 56 min. D: Sherman Scott (Sam Newfield). SC: Sam Robins. With Buster Crabbe, Al St. John, Tex (Dave) O'Brien, Sarah Padden, Wanda McKay, Charles King, Hal Price, John Merton, Kenne Duncan, Ted Adams, Budd Buster, Kermit Maynard, George Morrell, Steve Clark, Herman Hack, Bert Dillard, Carl Mathews, Jack Kirk, Art Dillard, Jimmy Aubrey, Wally West, Hank Bell, Tex Cooper, Augie Gomez. In order to stop a woman from marrying a crook, Billy the Kid poses as her nephew, an Army officer who has been murdered. An out-of-the-ordinary plot adds a bit of zest to this otherwise mundane \"Billy the Kid\" series entry. TV title: _**Billy the Kid's Law and Order**_ ; released in Great Britain as _**Double Alibi**_.\n\n**2252** _ **Law and Order**_ **** Universal-International, 1953. 80 min. Color. D: Nathan Juran. SC: John Bagni, Owen Bagni and D.D. Beauchamp. With Ronald Reagan, Dorothy Malone, Alex Nicol, Preston Foster, Ruth Hampton, Russell Johnson, Barry Kelley, Chubby Johnson, Dennis Weaver, Jack Kelly, Valerie Jackson, Don Garner, Thomas Browne Henry, Richard Garrick, Tristram Coffin, Mike Ragan, John Carpenter, Buddy Roosevelt, Richard Cutting, Britt Wood, Martin Garralaga, Don Gordon, Gregg Barton, Wally Cassell, William O'Neal. A lawman gives up his badge to marry and become a rancher but when his brother, who has taken his job, is killed by bad men he decides to clean up the town. Still another screen version of _Saint Johnson_ by W.R. Burnett and mediocre compared to the 1932 classic (q.v.).\n\n**2253** _ **The Law and the Outlaw**_ **** Exclusive, 1920. 45 min. D: William Duncan. SC: Tom Mix and J. Edward Hungerford. With Tom Mix, Lester Cuneo, Myrtle Stedman, Florence Dye, Marshall Stedman, Rex De Rosselli, William Duncan, Old Blue (horse). A cowboy, on the run from the law for a crime committed by his brother, takes a job at a ranch where he falls in love with the owner's daughter. Originally a 1913 two reel film, this re-working added footage from another short to make it a feature, capitalizing on Tom Mix's growing popularity.\n\n**2254** _ **Law Beyond the Range**_ **** Columbia, 1935. 60 min. D-SC: Ford Beebe. With Tim McCoy, Billie Seward, Robert Allen, Guy Usher, Harry Todd, Walter Brennan, Si Jenks, Tom London, J.B. Denton, Ben Hendricks, Jr., Jack Rockwell, George Chesebro, Lew Meehan, Steve Clark, Samuel S. Hinds, Max Davidson, Slim Whitaker, Jack Kirk, Gene Alsace, Jack Evans. Allowed to escape by his pals after being falsely accused of murder and drummed out of the service, a man goes to a town where he helps a female newspaper editor oppose the mysterious crime boss, El Poder (The Power). One of Tim McCoy's better Columbia Westerns with a good story by Lambert Hillyer and an exciting shootout at the climax.\n**2255** _ **The Law Comes to Gunsight**_ **** Monogram, 1947. 56 min. D: Lambert Hillyer. SC: J. Benton Cheney. With Johnny Mack Brown, Raymond Hatton, Reno Blair, Lanny Rees, William Ruhl, Zon Murray, Frank LaRue, Ernie Adams, Kermit Maynard, Ted Adams, Lee Roberts, Artie Ortego. When a crooked mayor mistakes a lawman for a gunman he hires him to be the town's sheriff and he sets out to round up the lawless. Predictable but more than passable oater.\n\n**2256** _ **The Law Comes to Texas**_ **** Columbia, 1939. 55 min. D: Joseph Lovering. SC: Nate Gatzert. With Bill Elliott, Veda Ann Borg, Charles King, Bud Osborne, Slim Whitaker, Leon Beaumont, Edmund Cobb, Lee Shumway, Frank Ellis, Paul Everton, Jack Ingram, Frank LaRue, David Sharpe, Forrest Taylor, Lane Chandler, Budd Buster, Dan White, Ben Corbett, Francis Walker, Buzz Barton, Oscar Gahan, Fred Parker, Tex Palmer. Texas is plagued by cattle rustling and murder and a man tries to restore law and order by helping form the Texas Rangers. Okay action filled outing from producer Larry Darmour headlining Bill Elliott.\n\n**2257** _ **The Law Commands**_ **** Crescent, 1937. 58 min. D: William Nigh. SC: Bennett Cohen. With Tom Keene, Lorraine Hayes (Laraine Day), Budd Buster, Matthew Betz, Robert Fiske, John Merton, Carl Stockdale, David Sharpe, Marie Stoddard, Fred Burns, Horace B. Carpenter, Charlotte Treadway, Allan Cavan, Bill Nestell, Oscar Gahan, Olin Francis, William McCall, Herman Hack, Ray Jones, Artie Ortego, Ada Belle Driver, Bob Burns. Land grabbers try to steal farms from settlers who have come to Iowa in 1862 under the Homestead Act. Pretty good entry in Tom Keene's Crescent Pictures historical series.\n\n**2258** _ **Law for Tombstone**_ **** Universal, 1937. 59 min. D: Buck Jones. SC: Frances Guihan. With Buck Jones, Muriel Evans, Harvey Clark, Carl Stockdale, Earle Hodgins, Alexander Cross, Chuck Morrison, Mary Carney, Charles LeMoyne, Ben Corbett, Francis Walker, Robert Kortman, Slim Whitaker, Tom Forman, Bill Patton, Frank McCarroll, Chick Hannon. When a stage line is hit with gold shipment robberies a special agent is hired to uncover the culprits. Sturdy Buck Jones vehicle produced and directed by the star.\n\n**2259** _ **Law Men**_ **** Monogram, 1944. 55 min. D: Lambert Hillyer. SC: Glenn Tryon. With Johnny Mack Brown, Raymond Hatton, Jan Wiley, Kirby Grant, Robert Frazer, Edmund Cobb, Art Fowler, Hal Price, Marshall Reed, Isabel Withers, Ben Corbett, Ted Mapes, Steve Clark, Bud Osborne, Jack Rockwell, George Morrell, Ray Jones, Ted French, Bob Woodward, Denver Dixon, Artie Ortego, Jack Evans, Rube Dalroy. Two lawmen investigate a series of holdups with one joining the gang while the other sets himself up in business in order to capture the thieves. Petty good action entry in the \"Nevada Jack McKenzie\" series written by former screen star Glenn Tryon; also known as _**Lawmen**_ **.**\n\n_**Law of 45s**_ see _**The Law of the 45s**_\n\n**2260** _ **Law of the Badlands**_ **** RKO Radio, 1951. 60 min. D: Lesley Selander. SC: Ed Earl Repp. With Tim Holt, Richard Martin, Joan Dixon, Robert Livingston, Leonard Penn, Harry Woods, Larry Johns, Robert Bray, Kenneth MacDonald, John Cliff. On the trail of a counterfeiting gang, two Texas Rangers pretend to be outlaws in order to infiltrate the operation. Fairly paced Tim Holt vehicle, average for the series.\n\n**2261** _ **Law of the Barbary Coast**_ **** Columbia, 1949. 65 min. D: Lew Landers. SC: Robert Libott and Frank Burt. With Gloria Henry, Stephen Dunne, Adele Jergens, Robert Shayne, Stefan Schnabel, Edwin Max, Ross Ford, J. Farrell MacDonald, Grandon Rhodes, Peter Brocco, Ann Lawrence, Robert Williams, Jessie Arnold, Everett Glass, William Stubbs, Dewey Robinson, Myron Healey, Jo Jordan, Rube Schaffer. When her brother is murdered, a young woman takes a job in a Barbary Coast gambling house to get the proof needed to convict his killer. Okay action melodrama set in the 1880s.\n\n**2262** _ **Law of the Canyon**_ **** Columbia, 1947. 55 min. D: Ray Nazarro. SC: Eileen Gary. With Charles Starrett, Smiley Burnette, Nancy Saunders, Buzz Henry, Texas Jim Lewis and His Lone Star Cowboys, Fred F. Sears, George Chesebro, Edmund Cobb, Zon Murray, Jack Kirk, Robert Wilke, Frank Marlo, Stanley Price, Douglas D. Coppin, Art Dillard, Tommy Coats. The Durango Kid steps in when outlaws blackmail citizens into paying protection money or have their property stolen and held for ransom. Low grade \"Durango Kid\" feature. British title: _**The Price of Crime**_.\n\n**2263** _ **The Law of the 45s**_ **** Normandy Pictures\/First Division\/Grand National, 1935. 56 min. D: John McCarthy. SC: Robert (Emmett) Tansey. With Guinn Williams, Molly O'Day, Al St. John, Ted Adams, Lafe McKee, Fred Burns, Martin Garralaga, Curly Baldwin, Sherry Tansey, Glenn Strange, Bill Patton, Jack Kirk, Jack Jones, Francis Walker, Jack Evans, Tex Palmer, Merrill McCormick, George Morrell, William McCall, Broderick O'Farrell, Ace Cain, Herman Hack, Art Felix, Buck Morgan, Ralph Bucko, Budd Buster, Chuck Baldra. Tucson Smith and Stony Martin, after an outlaw gang terrorizing the area, help a rancher and his daughter save their spread. This \"Three Mesquiteers\" film, based on the books by William Colt MacDonald, is minus a member (with a surname change for Stony Brooke) but is otherwise an interesting low budget offering. Also called _**Law of 45s**_ and released in Great Britain as _**The Mysterious Mr. Sheffield**_.\n\n**2264** _ **Law of the Golden West**_ **** Republic, 1949. 59 min. D: Philip Ford. SC: Norman S. Hall. With Monte Hale, Paul Hurst, Gail Davis, Roy Barcroft, John Holland, Scott Elliott, Lane Bradford, Harold Goodwin, John Hamilton, Bill Hale, Wally West, Jack O'Shea, Chuck Roberson, Bob Reeves, Chuck Baldra, Al Haskell, Frank McCarroll, Lew Morphy. Buffalo Bill Cody tries to put a stop to outlaws attacking an area while trailing his father's killer. Okay costume entry in Monte Hale's Republic series, a remake of _**Dark Command**_ (q.v.).\n\n**2265** _ **Law of the Land**_ **** NBC-TV, 1976. 100 min. Color. D: Virgil Vogel. SC: John Wilder and Sam Rolfe. With Jim Davis, Barbara Parkins, Andrew Prine, Moses Gunn, Glenn Corbett, Charles Martin Smith, Dana Elcar, Don Johnson, Cal Bellini, Nicholas Hammond, Darleen Carr, Ward Costello, Paul Stevens, Barney Phillips. When a man goes on a rampage killing hookers, a lawman and a stranger team to pursue him. Pretty good TV fare, highlighted by Jim Davis' performance as the old time sheriff. Also called _**The Deputies**_.\n\n**2266** _ **Law of the Lash**_ **** Producers Releasing Corporation, 1947. 54 min. D: Ray Taylor. SC: William L. Nolte. With Lash LaRue, Al St. John, Lee Roberts, Mary Scott, Jack O'Shea, Charles King, Carl Mathews, Richard Cramer, Slim Whitaker, John Elliott, Ted French, Brad Slaven, Tex Palmer, Hank Bell, Ben Corbett. Outlaws run settlers out of a town and take over but two U.S. marshals arrive to restore peace. Fairly typical Lash LaRue oater with Charles King a good guy for a change.\n\n**2267** _ **Law of the Lawless**_ **** Paramount, 1964. 87 min. Color. D: William F. Claxton. SC: Steve Fisher. With Dale Robertson, Yvonne De Carlo, William Bendix, Lon Chaney, Bruce Cabot, Barton MacLane, John Agar, Richard Arlen, Kent Taylor, Jody McCrea, Bill Williams, Rod Lauren, George Chandler, Donald Barry, Romo Vincent, Lorraine Bendix, Joe Forte, Alex Sharp, Leigh Chapman, Laurel Goodwin, Fred Rapport, George Taylor, Jerry Summers, Reg Parton, Wally West, Lori Campbell, Dick Ryan. A hanging judge comes to town to try a man for murder but the defendant turns out to be the son of an old friend who tries to blackmail the jurist. The first in a series of low budget Westerns from producer A.C. Lyles with veteran casts, and it is a good one.\n\n**2268** _ **Law of the North**_ **** Monogram, 1932. 56 min. D-SC: Harry Fraser. With Bill Cody, Andy Shuford, Nadine Dore, Al St. John, William L. Thorne, Heinie Conklin, Jack Carlyle, Gill Pratt, Earl Dwire, Blackie Whiteford, Gilbert \"Pee Wee\" Holmes, Perry Murdock, Dick Dickinson, Barney Beasley, Herman Hack, Jack Evans, Tex Palmer, Al Taylor, Charles West, F.R. Smith, Clyde McClary, Jack Low. A lawman is after an elusive outlaw who has been able to escape previous arrest attempts. Slow moving, vapid entry in the Bill Cody-Andy Shuford series.\n\n**2269** _ **Law of the Northwest**_ **** Columbia, 1943. 57 min. D: William Berke. SC: Luci Ward. With Charles Starrett, Shirley Patterson, Arthur Hunnicutt, Stanley Brown, Douglas Leavitt, Donald Curtis, Douglass Drake, Davison Clark, Reginald Barlow, John Tyrrell, John Shay, Edward Cassidy, Eddie Laughton, Chuck Baldra, Wesley Tuttle, Al Boles, Eddie Tudor, Johnny Luther. The Mounties are after a crooked contractor trying to stop work on a rival's road in order to get a valuable war contract. Average, but well photographed (by Benjamin Kline), north woods opus with three pleasant songs composed by Johnny Marvin.\n\n**2270** _ **Law of the Pampas**_ **** Paramount, 1939. 72 min. D: Nate Watt. SC: Harrison Jacobs. With William Boyd, Russell Hayden, Steffi Duna, Sidney Toler, Sidney Blackmer, Pedro De Cordoba, William Duncan, Anna Demetrio, Eddie Dean, Glenn Strange, Tony Roux, Martin Garralaga, The King's Men, Jojo LaSadio, Johnny Luther Roy Brent, George Sowards, Herman Hack, George Plues, Tex Phelps, Jack Montgomery. In South America bringing cattle to a rancher, Hoppy beings to suspect the foreman of being behind the killing of the owner's two children so he can have the place for himself. Nice locations, interesting story and some good action make this a fine \"Hopalong Cassidy\" segment.\n\n**2271** _ **Law of the Panhandle**_ **** Monogram, 1950. 55 min. D: Lewis D. Collins. SC: Joseph Poland. With Johnny Mack Brown, Jane Adams, Riley Hill, Marshall Reed, Myron Healey, Ted Adams, Lee Roberts, Carol Henry, Milburn Morante, Bob Duncan, Kermit Maynard, Boyd Stockman, George DeNormand, Tex Palmer, Ray Jones, George Morrell, Denver Dixon. When a town is plagued by rustlers after range land to be used for a railroad, the sheriff calls in a U.S. marshal. Good Johnny Mack Brown outing with fine work by Myron Healey as a vicious gang leader.\n\n**2272** _ **Law of the Plains**_ **** Columbia, 1938. 56 min. D: Sam Nelson. SC: Maurice Geraghty. With Charles Starrett, Iris Meredith, Robert Warwick, The Sons of the Pioneers (Bob Nolan, Tim Spencer, Lloyd Perryman, Pat Brady, Hugh Farr, Karl Farr), Dick Curtis, Ed Le Saint, Edmund Cobb, Art Mix, Jack Rockwell, George Chesebro, Jack Long, John Tyrrell, Blackie Whiteford, Ernie Adams, Blackjack Ward. The foreman of a ranch, whose owner is being threatened by an outlaw gang, tries to stop the crooks. Well done Charles Starrett film.\n\n**2273** _ **Law of the Range**_ **** Universal, 1941. 59 min. D: Ray Taylor. SC: Sherman Lowe. With Johnny Mack Brown, Fuzzy Knight, Nell O'Day, Roy Harris (Riley Hill), Pat O'Malley, Elaine Morey, Ethan Laidlaw, The Texas Rangers, Alan Bridge, Hal Taliaferro, Lucille Walker, Charles King, Bud Osborne, Robert Kortman, Slim Whitaker, Jack Rockwell, Terry Frost, Jim Corey, Herman Hack, Frank Hagney, Chuck Morrison, Ray Henderson, Sam Garrett, Carl Sepulveda, Jack Casey, Jerome Hart. A cowboy, whose family is involved in a range feud over cattle and sheep, gets the blame for the death of a rancher whose daughter he loves. Sturdy Johnny Mack Brown feature.\n\n**2274** _ **Law of the Ranger**_ **** Columbia, 1937. 58 min. D: Spencer Gordon Bennet. SC: Nate Gatzert. With Bob Allen, Elaine Shepard, Hal Taliaferro, Lafe McKee, John Merton, Tom London, Lane Chandler, Slim Whitaker, Ernie Adams, Bud Osborne, Jimmy Aubrey, Buffalo Bill, Jr., Lee Shumway, George Morrell, Frank Ball, Herman Hack, Ray Henderson, Arthur Millett, Francis Walker, Bill Patton, Al Taylor, Tex Palmer, Jim Corey, Wally West, Buck Morgan, Ralph Bucko. Two rangers arrive incognito to investigate the intimidation of settlers by a crook and his gang, the bad man trying to get control of the area's water rights. Entertaining entry in Bob Allen's brief \"Texas Rangers\" series, this one again teaming him with Hal Taliaferro (Wally Wales).\n\n**2275** _ **Law of the Rio Grande**_ **** Syndicate, 1931. 57 min. D: Forrest Sheldon. SC: Betty Burbridge and Bennett Cohen. With Bob Custer, Betty Mack, Edmund Cobb, Nelson McDowell, Harry Todd, Lafe McKee, Fred Burns, Hank Bell, Carlton King. A ex-outlaw tries to go straight as a ranch foreman but a former cohort wants to get him back on the wrong side of the law. This outing is a bit better than most Bob Custer movies thanks to Edmund Cobb as the bad guy.\n\n**2276** _ **Law of the Saddle**_ **** Producers Releasing Corporation, 1943. 57 min. D: Melville DeLay. SC: Fred Myton. With Robert Livingston, Al St. John, Betty Miles, Lane Chandler, John Elliott, Reed Howes, Frank Ellis, Curley Dresden, Al Ferguson, Frank Hagney, Jimmy Aubrey, Bob Hill, Bert Dillard, Jack Evans, Jack Tornek, Wally West, Al Taylor, Herman Hack, Foxy Callahan, Lew Morphy, Bill Wolfe, Denver Dixon, George Morrell, Pascale Perry. The Lone Rider is after an outlaw gang that goes from town to town electing one of its members sheriff and then taking the citizen's money. Interesting plot his harmed by poor production values in this \"Lone Rider\" affair although Melville DeLay's (usually an assistant director) direction is very good. Also called _**The Lone Rider in Law of the Saddle**_.\n\n**2277** _ **Law of the Sixgun**_ **** Sunshadow Productions, 1978. 81 min. D: Al Frakes. SC: Tex Hill. With Tex Hill, Renae Richard, Kathryn Kinley, J.D. Bunges, Mike Redding, William Cooley, Tim Cooley, Steven Wright, Jack Walton, Sr., Richard Reilly, Ron Redding. Faced with a lawman out to stop his criminal activities, a gang leader sends for a gunslinger not realizing he is the peacemaker's brother. They do not come any worse than this; so bad it was not issued until 2005 on DVD by Film Baby.\n\n**2278** _ **Law of the Texan**_ **** Columbia, 1938. D: Elmer Clifton. SC: Monroe Shaff and Arthur Hoerl. With Buck Jones, Dorothy Fay, Don Douglas, Kenneth Harlan, Joe Whitehead, Matty Kemp, Forrest Taylor, Robert Kortman, Jose Torosa, Melissa Sierra, Tommy Mack, Jack Ingram, Dave O'Brien, Jack Kirk, Carl Mathews, Ray Henderson, Buck Morgan. A Texas Rangers commander learns a cattle rustling attempt was a ruse to hide the theft of an ore shipment, the plot masterminded by a mysterious figure called El Coyote. A strong script and scenic values add up to a good Buck Jones feature.\n\n**2279** _ **Law of the Timber**_ **** Producers Releasing Corporation, 1941. 65 min. D: Bernard B. Ray. SC: Jack Natteford. With Marjorie Reynolds, Monte Blue, J. Farrell MacDonald, Hal Brazeale, George Humbert, Sven Hugo Borg, Earl Eby, Milt Moranti (Milburn Morante), J. Merrill Holmes, Rudy Sooter, Zero (dog), Betty Roadman, Eddie Phillips. After the death of her father, a young woman tries to fulfill his government lumber contract despite sabotage efforts by her foreman. Nice action melodrama, from a story by James Oliver Curwood.\n\n**2280** _ **Law of the Valley**_ **** Monogram, 1944. 59 min. D: Howard Bretherton. SC: Joseph O'Donnell. With Johnny Mack Brown, Raymond Hatton, Lynne Carver, Kirk Barron, Hal Price, Edmund Cobb, Tom Quinn, Charles King, Marshall Reed, George DeNormand, Steve Clark, George Morrell, Charles McMurphy, Horace B. Carpenter, Snub Pollard, Rose Plummer, George Sowards, Dee Cooper, Bud Pope. Outlaws want range land that controls an area's water supply because a railroad plans to build a spur on it and two marshals are called in to stop them. Pleasant entry in the \"Nevada Jack McKenzie\" series. ****\n\n**2281** _ **Law of the West**_ **** World Wide, 1932. 58 min. D-SC: Robert North Bradbury. With Bob Steele, Nancy Drexel, Ed Brady, Hank Bell, Charles West, Earl Dwire, Dick Dickinson, Rose Plummer, Frank Ellis, Perry Murdock, Jack Low, F.R. Smith. Kidnapped by a gang as a baby, a young man believes his father is their leader, who plans to have him shoot his real dad, a town marshal. Well written and executed Bob Steele vehicle, with equal emphasis on drama and action.\n\n**2282** _ **Law of the West**_ **** Monogram, 1949. 54 min. D: Ray Taylor. SC: J. Benton Cheney. With Johnny Mack Brown, Raymond Hatton, Max Terhune, Bill Kennedy, Gerry Pattison, Jack Ingram, Eddie Parker, Riley Hill, Steve Clark, Jack Harrison, Bob Woodward, Marshall Reed, Kenne Duncan, Bud Osborne, Frank Ellis. A real estate agent is faking deeds to properties and then stealing them from the owners with a vacationing U.S. marshal coming to the rescue. Mediocre Johnny Mack Brown outing somewhat saved by Max Terhune's comedy and ventriloquist work.\n\n**2283** _ **The Law of the Wild**_ **** Mascot, 1934. 12 Chapters. D: B. Reeves Eason and Armand L. Schaefer. SC: Sherman Lowe and B. Reeves Eason. With Rex (horse), Rin Tin Tin, Jr. (dog), Ben Turpin, Bob Custer, Lucille Browne, Richard Cramer, Ernie Adams, Richard Alexander, Edmund Cobb, Slim Whitaker, George Chesebro, Wally Wales, Charles King, Lafe McKee, Hank Bell, Art Mix, Bud Osborne, Glenn Strange, Al Taylor, Jack Evans, Bud McClure, Herman Hack. A rancher owns a beautiful stallion that he tamed but one of his men, along with two cohorts, plan to steal animal and use him as a race horse. Fun Mascot low budget cliffhanger with hero Bob Custer getting billed behind the two animal stars and the comedy relief, silent film great Ben Turpin.\n\n**2284** _ **Law of the Wolf**_ **** Ziehm, 1941. 55 min. D: Raymond K. Johnson. SC: Joseph Murphy. With Rin Tin Tin, Jr., Dennis Moore, Luana Walters, George Chesebro, Steve Clark, Jack Ingram, Robert Frazer, Jimmy Aubrey, Martin Spellman, Bobby Gordon. A man is helped by his girl friend and a dog in battling crooks. Low grade but action packed program feature.\n\n_**Law of Vengeance**_ see _**To the Last Man**_\n\n**2285** _ **The Law Rides**_ **** Supreme, 1936. 57 min. D: Robert North Bradbury. SC: Al Martin. With Bob Steele, Harley Wood, Charles King, Buck Connors, Margaret Mann, Jack Rockwell, Barney Furey, Ted Mapes, Horace Murphy, Budd Buster, George Morrell, Art Dillard, Blackie Whiteford, Tex Palmer, Ray Henderson, George Ball. A gold strike results in outlaws robbing and killing miners with a cowboy tracking down the culprits. Fast paced Bob Steele affair, a bit shy on production values.\n\n**2286** _ **The Law Rides Again**_ **** Monogram, 1943. 58 min. D: Alan James. SC: Frances Kavanaugh. With Ken Maynard, Hoot Gibson, Betty Miles, Kenneth Harlan, Jack LaRue, Chief Thundercloud, Bryant Washburn, Emmett Lynn, Hank Bell, John Bridges, Fred Hoose, Charles Murphy, Jr., Chief Many Treaties, John Merton, Kenne Duncan, Steve Clark, Roy Brent, Budd Buster, Wally West, Chick Hannon, Foxy Callahan. Two lawmen are after a man masquerading as an Indian agent and fleecing the tribes. Second in the \"Trail Blazers\" series and sure to please its two stars' fans.\n\n**2287** _ **The Law vs. Billy the Kid**_ **** Columbia, 1954. 73 min. Color. D: William Castle. SC: John T. Williams. With Scott Brady, Betta St. John, James Griffith, Alan Hale, Jr., Paul Cavanagh, William Phillips, Benny Rubin, Steve Darrell, William Tannen, Martin Garralaga, Richard Cutting, Frank Sully, William Fawcett, Robert Griffin, George Berkeley, John Cliff, Otis Garth, Gregg Barton, John Cason, Rory Mallinson, Bud Osborne. On the run from the law, Billy the Kid is befriended by a rancher and falls in love with his pretty daughter who is also wanted by the ranch foreman. Average fiction about Billy the Kid from producer Sam Katzman.\n\n**2288** _ **The Law West of Tombstone**_ **** RKO Radio, 1938. 72 min. D: Glenn Tryon. SC: John Twist and Clarence Upson Young. With Harry Carey, Tim Holt, Evelyn Brent, Jean Rouverol, Clarence Kolb, Allan Lane, Esther Muir, Bradley Page, Paul Guifoyle, Robert Moya, Ward Bond, George Irving, Monte Montague, Robert Kortman, Eddy Waller, Don Barclay, Nina Campana, Martin Garralaga, Eddie Hart, Chief Many Treaties, Chief Thundercloud, Pat West, Charles Middleton, Horace Murphy, Frank Moran, Bud Osborne, Cactus Mack, Jack O'Shea, Henry Roquemore, Russ Powell, Syd Saylor, Harry Hayden, Selmer Jackson, Donald Kerr, Spencer Charters, Frank Ellis, Art Dillard, Robert Greig, Victor Cox, Ralph Bucko, Roy Bucko, Kermit Maynard, Gerald Oliver Smith. A judge who rules by the gun sets up court in a small town and opposes his daughter's upcoming wedding to a no-good who is soon dispatched by a young gunman. Top-notch oater that takes a look at the career of Judge Roy Bean, here called Bill Parker.\n\n**2289** _ **The Lawless**_ **** Wrather Corporation, 1956. 75 min. Color. D: Earl Bellamy and Oscar Rudolph. SC: Thomas Seller and Doane Hoag. With Clayton Moore, Jay Silverheels, Myron Healey, Allen Pinson, George J. Lewis, Trevor Bardette, Pierce Lyden, Zon Murray, William Fawcett, Tudor Owen, John Berardino, Mickey Simpson, Maria Manay, Joe Vitale, David Armstrong, Rocky Shahan, John Cason, Robert Roark, David Kashner, J. Anthony Hughes, Louise Lewis, Paul Engle, Mercedes Shirley. The Lone Ranger and Tonto are after an outlaw gang masquerading as cavalrymen; they stop vigilantes from breaking the law; and aid a retired lawman in capturing two killers. Well done telefeature compiled from the \"Return of Don Pedro O'Sullivan,\" \"Sam's Boy\" and \"The Tarnished Star\" episodes of \"The Lone Ranger\" (ABC-TV, 1949\u201357).\n\n**2290** _ **Lawless Borders**_ **** Spectrum, 1935. 58 min. D: John P. McCarthy. SC: Zara Tazil. With Bill Cody, Molly O'Day, Martin Garralaga, Ted Adams, John Elliott, Merrill McCormick, Roger Williams, Budd Buster, Wally West, Joe De La Cruz, Curley Baldwin, William McCall. After his pal is murdered, a cowboy seeks revenge on the killers. Typical low grade Bill Cody outing.\n\n**2291** _ **Lawless Breed**_ **** Universal, 1946. 58 min. D: Wallace Fox. SC: Robert Williams. With Kirby Grant, Fuzzy Knight, Jane Adams, Harry Brown, Dick Curtis, Charles King, Karl Hackett, Hank Worden, Claudia Drake, Ernie Adams, Harry Wilson, Artie Ortego. Two cowboys ride into a town where they get mixed up with an outlaw gang, are accused of murder and forced to flee a lynch mob. Fairly good penultimate entry in Kirby Grant's Universal series. TV title: _**Lawless Clan**_.\n\n**2292** _ **The Lawless Breed**_ **** Universal-International, 1952. 83 min. Color. D: Raoul Walsh. SC: Bernard Gordon. With Rock Hudson, Julia (Julie) Adams, John McIntire, Mary Castle, Hugh O'Brian, Forrest Lewis, Lee Van Cleef, Tom Fadden, William Pullen, Dennis Weaver, Glenn Strange, Richard Garland, Race Gentry, Carl Pitti, Ned Davenport, Robert Anderson, Stephen Chase, Richard Wessel, Emory Parnell, George Wallace, Edward Earle, Michael Ansara, Paul \"Tiny\" Newlan, Francis Ford, I. Stanford Jolley, Buddy Roosevelt, Ethan Laidlaw, Stanley Blystone, Wheaton Chambers, John Pickard, Bobby Hoy. After sixteen years in prison, John Wesley Hardin returns home to find his teenage son idolizes him as a gunman so he decides to take part in one last lawless act to show the boy the error of his ways. Appealing drama with slick production values and a good story.\n\n_**Lawless Clan**_ see _**Lawless Breed (1946)**_\n\n**2293** _ **Lawless Code**_ **** Monogram, 1949. 58 min. D: Oliver Drake. SC: Basil Dickey. With Jimmy Wakely, Dub Taylor, Ellen Hall, Tristram Coffin, Riley Hill, Kenne Duncan, Terry Frost, Myron Healey, Steve Clark, Bud Osborne, Bob Curtis, Frank McCarroll, Beatrice Maude. The nephew of a man murdered by outlaws is accused of the killing and a cowboy comes to his rescue. Not even a top notch bunch of screen bad guys can save this Jimmy Wakely warbler.\n\n**2294** _ **Lawless Cowboys**_ **** Monogram, 1951. 58 min. D: Lewis D. Collins. SC: Maurice Tombragel. With Whip Wilson, Fuzzy Knight, Jim Bannon, Pamela Duncan, Lee Roberts, Marshall Reed, Lyle Talbot, Steve Clark, I. Stanford Jolley, Bruce Edwards, Stanley Price, Richard Emory, Ace Malloy, Richard Avonde, Roy Butler, Pierce Lyden, Forrest Taylor, Pascale Perry. An ex\u2013Texas Ranger is hired to look into a scheme where participants are fixing rodeo events. Okay modern-day action entry in the Whip Wilson series.\n\n**2295** _ **The Lawless Eighties**_ **** Republic, 1958. 70 min. D: Joseph Kane. SC: Kenneth Gamet. With Buster Crabbe, John Smith, Marilyn Saris, Ted De Corsia, Anthony Caruso, John Doucette, Frank Ferguson, Sheila Bromley, Walter Reed, Buzz Henry, Will J. White, Bob Swan. A gunman comes to the aid of a circuit rider beaten by outlaws who he saw mistreat Indians. Pretty good drama that should please Buster Crabbe fans.\n\n**2296** _ **Lawless Empire**_ **** Columbia, 1945. 58 min. D: Vernon Keays. SC: Bennet Cohen. With Charles Starrett, Tex Harding, Dub Taylor, Mildred Law, Bob Wills and His Texas Playboys, Johnny Walsh, John Calvert, Ethan Laidlaw, Forrest Taylor, Jack Rockwell, George Chesebro, Boyd Stockman, Lloyd Ingraham, Jessie Arnold, Tom Chatterton, Ray Jones, Edward Howard, Bud Nelson, Frank LaRue, Joe Galbreath, John Tyrrell, Jack Kirk. The Durango Kid helps a minister and his wife who are trying to assist settlers harassed by a gang of raiders. Choppy, but fast moving, \"Durango Kid\" episode. British title: _**Power of Possession**_.\n\n**2297** _ **The Lawless Frontier**_ **** Monogram, 1934. 52 min. D-SC: Robert North Bradbury. With John Wayne, Sheila Terry, George Hayes, Earl Dwire, Yakima Canutt, Jack Rockwell, Buffalo Bill, Jr., Bud Wood (Gordon DeMain), Eddie Parker, Artie Ortego, Herman Hack, Tex Phelps, Arthur Millett, Tommy Coats. A cowboy whose family was murdered by a Mexican bandit leader teams with an old man and his daughter when the outlaw plans to abduct the girl. Likable, rawboned entry in the John Wayne \"Lone Star\" series; Jack Rockwell is fun as the lunkheaded sheriff.\n\n**2298** _ **Lawless Land**_ **** Republic, 1936. 55 min. D: Albert Ray. SC: Andrew Bennison. With Johnny Mack Brown, Louise Stanley, Ted Adams, Julian Rivero, Horace Murphy, Frank Ball, Ed Cassidy, Roger Williams, Frances Kellogg, Ana (Anita) Camargo, Horace B. Carpenter, Jack Kirk, Bud McClure, Cliff Parkinson, Bert Dillard, George Hazel, Al Haskell, Chuck Baldra, Ed Carey, Art Dillard, Clyde McClary, Jack Tornek, George Morrell, Chiquita Hernandez Orchestra. A Texas Rangers arrives in a town to investigate a series of murders and learns the marshal is the killer. Fairly good Johnny Mack Brown series entry.\n\n**2299** _ **The Lawless Nineties**_ **** Republic, 1936. 56 min. D: Joseph Kane. SC: Joseph Poland. With John Wayne, Ann Rutherford, Harry Woods, George Hayes, Alan Bridge, Lane Chandler, Fred \"Snowflake\" Toones, Etta McDaniel, Tom Brower, Cliff Lyons, Jack Rockwell, Al Taylor, Charles King, George Chesebro, Tom London, Sam Flint, Earl Seaman, Tracy Layne, Philo McCullough, Jimmy Harrison, Chuck Baldra, Henry Hall, Lloyd Ingraham, Bud Osborne, Edward Hearn, Lew Meehan, Jack Kirk, Blackjack Ward, George Morrell, Helen Gibson, Art Dillard, Steve Clark, Bob Burns, Jim Corey, Horace B. Carpenter, Curley Dresden, Emma Tansey, Sherry Tansey, Bert Lindley, Tex Palmer, William McCall, Pascale Perry, Bud Pope, Rose Plummer. A federal investigator is sent to Wyoming to see that elections are not rigged and finds himself opposed by an outlaw gang against statehood. Action filled, well done John Wayne feature.\n\n**2300** _ **Lawless Range**_ **** Republic, 1935. 56 min. D: Robert North Bradbury. SC: Lindsley Parsons. With John Wayne, Sheila Mannors, Frank McGlynn, Jr., Earl Dwire, Yakima Canutt, Jack Curtis, Wally Howe, Glenn Strange, Jack Kirk, Fred Burns, Slim Whitaker, Julia Griffin, Robert Kortman, George Ovey, Frank Ellis, Francis Walker, Sam Flint, Henry Hall, Herman Hack, Charles Brinley, Pascale Perry, Ray Henderson, Fred Parker, Sherry Tansey, John Ince, Bob Burns, Tex Palmer, Jack Hendricks, Denver Dixon, The Wranglers (Glenn Strange, Jack Kirk, Chuck Baldra, Charles Sargent). Upon his father's request, a cowboy tries to locate a missing friend and finds an area plagued by an outlaw gang secretly led by a crooked banker. Fast paced John Wayne movie in which he serenades (dubbed) Sheila Mannors with the Eddie Dean-Glenn Strange classic, \"On the Banks of the Sunny San Juan.\"\n\n**2301** _ **The Lawless Rider**_ **** United Artists, 1954. 72 min. D: Yakima Canutt. SC: John Carpenter. With John Carpenter, Texas Rose Bascom, Douglass Dumbrille, Frankie Darro, Frank \"Red\" Carpenter, Noel Neill, Kenne Duncan, Weldon Bascom, Bud Osborne, Bill Coontz, Tap Canutt, Hank Caldwell and His Saddle Kings, Roy Canada, Lou Roberson, Earl Bascom. When a gunman takes over a town a woman rancher appeals to a deputy marshal for help. Tacky John Carpenter outing which was Edward D. Wood, Jr.'s first released film; he was the associate producer.\n\n**2302** _ **A Lawless Street**_ **** Columbia, 1955. 78 min. Color. D: Joseph H. Lewis. SC: Kenneth Gamet. With Randolph Scott, Angela Lansbury, Warner Anderson, Jean Parker, Wallace Ford, John Emery, James Bell, Ruth Donnelly, Michael Pate, Don Megowan, Jeannette Nolan, Peter Ortiz, Frank Hagney, Frank Ferguson, Harry Tyler, Harry Antrim, Jay Lawrence, Reed Howes, Guy Teague, Hal K. Dawson, Stanley Blystone, Eddy Chandler, John Cason, Kermit Maynard, Jack Perrin, Franklyn Farnum, Wally West, Philo McCullough, G. Pat Collins, Leonard Geer, Augie Gomez, Don Carlos, Sam Harris, Frank O'Connor, Artie Ortego, Charles Williams, Frank J. Scannell, Jack Kenney, Denver Dixon, Bess Flowers. A frontier town doctor learns a businessman has marked him for murder because he loves the physician's estranged wife, a recently imported opera singer. A good plot and lots of action highlight this Randolph Scott drama.\n\n**2303** _ **Lawless Valley**_ **** Progressive\/Willis Kent, 1934. 50 min. D: J.P. McGowan. SC: Oliver Drake. With Lane Chandler, Gertrude Messinger, Dick (Richard) Cramer, J.P. McGowan, Si Jenks, Anne Howard, Art Mix, Jack Kirk, Hank Bell, Chuck Baldra. A cattlemen's association detective is assigned to bring in a notorious rustler called El Lobo who has been terrorizing a remote area. Fair Lane Chandler vehicle from producer Willis Kent.\n\n**2304** _ **Lawless Valley**_ **** RKO Radio, 1938. 58 min. D: David Howard. SC: Oliver Drake. With George O'Brien, Kay Sutton, Fred Kohler, Jr., Walter Miller, George MacQuarrie, Lew Kelly, Earle Hodgins, Chill Wills, Dot Farley, Robert Stanton (Kirby Grant), George Chesebro, Carl Stockdale, Ben Corbett, Bob McKenzie, Jack O'Shea, Landers Stevens, Frank O'Connor, Jim Mason, Carl Miller, Tommy Coats, Dick Hunter, The Four Tunes. Falsely sent to prison, a man returns home to prove his innocence, find his father's killer and claim his girl, but finds himself opposed to a self-appointed town boss and his son. Very good George O'Brien vehicle; remade as _**Thunder Town**_ (q.v.).\n\n**2305** _ **Lawman**_ **** United Artists, 1971. 99 min. Color. D: Michael Winner. SC: Gerald Wilson. With Burt Lancaster, Robert Ryan, Lee J. Cobb, Sheree North, Joseph Wiseman, Robert Duvall, Albert Salmi, J.D. Cannon, John McGiver, Richard Jordan, John Beck, Ralph Waite, William Watson, Charles Tyner, John Hillerman, Robert Emhardt, Richard Bull, Hugh McDermott, Lou Frizzell, Walter Brooke, Bill Bramley. A marshal shows up to arrest a cattle baron for the killing of an old man and finds he is opposed by the locals, including the town's weak willed sheriff. Average big budget oater helped by a fine cast.\n\n**2306** _ **A Lawman Is Born**_ **** Republic, 1937. 61 min. D: Sam Newfield. SC: George Plympton. With Johnny Mack Brown, Iris Meredith, Warner Richmond, Charles King, Dick Curtis, Mary MacLaren, Earle Hodgins, Al St. John, Frank LaRue, Steve Clark, Jack C. Smith, Sherry Tansey, Wally West, Budd Buster, Lew Meehan, Tex Palmer, Oscar Gahan. A cowboy opposed to local crooks becomes a town's sheriff and tries to stop several big ranchers from monopolizing the cattle trade. While it has a complicated plot, this one does not lack for action.\n\n_**Lawmen**_ see _**Law Men**_\n\n**2307** _ **The Law's Lash**_ **** Path\u00e9, 1928. 60 min. D: Noel Mason Smith. SC: Edward Meagher. With Klondike (dog), Robert Ellis, Mary Mayberry, Jack Marsh, Richard R. Meill, LeRoy Mason, William Walters. A Canadian Mountie assisted by a police dog searches for the killer of a fellow trooper with the chief suspect being his girl's father. Passable silent adventure film.\n\n**2308** _ **Lay That Rifle Down**_ **** Republic, 1955. 71 min. D: Charles Lamont. SC: Barry Shipman. With Judy Canova, Robert Lowery, Jil Jarmyn, Jacqueline de Witt, Richard Deacon, Robert Burton, James Bell, Leon Tyler, Tweeny Canova, Pierre Watkin, Marjorie Bennett, William Fawcett, Paul E. Burns, Edmund Cobb, Donald MacDonald, Mimi Gibson, Rudy Lee. An overworked young woman, the drudge of a small hotel, dreams of becoming rich. Another genre affair with Judy Canova and one sure to please her fans.\n\n**2309** _ **The Lazarus Man**_ **** Castle Rock Entertainment, 1996. 90 min. Color. D: Norman S. Powell. SC: Dick Beebe. With Robert Urich, Elizabeth Dennehy, David Marshall Grant, John Diehl, Brion James. Suffering from amnesia, a man rises from a shallow grave and tries to find out his identity, learning he is linked to the Lincoln assassination. Okay pilot for the 1995\u201396 TV series of the same title.\n\n**2310** _ **Leadville Gunslinger**_ **** Republic, 1952. 54 min. D: Harry Keller. SC: M. Coates Webster. With Allan \"Rocky\" Lane, Eddy Waller, Grant Withers, Elaine Riley, Roy Barcroft, Richard Crane, I. Stanford Jolley, Kenneth MacDonald, Mickey Simpson, Art Dillard, Ed Hinton, Wesley Hudman. An outlaw gang carrying out a series of robberies and killings finds itself the target of a U.S. marshal. Typically good Allan Lane outing in his \"Famous Westerns\" series.\n\n**2311** _ **The Leather Burners**_ **** United Artists, 1943. 66 min. D: Joseph Henaberry. SC: Jo Pagano. With William Boyd, Andy Clyde, Jay Kirby, Victor Jory, George Givot, Shelley Spencer, Bobby Larson, George Reeves, Hal Taliaferro, Forbes Murray, Robert Mitchum, Robert Kortman, Herman Hack, Art Mix, Christian Rub, Cal Shrum, Bill Nestell, Bob Burns, Merrill McCormick, Jack Casey, George Morrell, Kit Guard. Hoppy and California help Johnny and a local rancher in trying to rid the area of rustlers with Cassidy ingratiating himself with a man mixed up with the gang in order to find out its leader. Well done \"Hopalong Cassidy\" segment with an exciting climax in a mine; based on the novel by Bliss Lomax (Harry Sinclair Drago).\n\n**2312** _ **Leave Your Guns at the Door!**_ **** Agata Film, 1972. 83 min. Color. D: Leopoldo Savona. SC: Norbert Blake and Leopoldo Savona. With Mark Damon, Richard Melville, Veronica Korocia, Pietro Ceccarelli, Floranna Di Bernardo, Carla Mancini, Allesandro Perrella. A former Confederate turned gunman joins forces with two pretty women and their homesteader father in trying to fleece the people of a small town with several \"miracle\" schemes. Undistinguished Italian Western issued in France as _**Desposez les Colts**_ (Dispose of the Colts) and also called _**Pistol Packin' Preacher**_.\n\n**2313** _ **Left for Dead**_ **** Grindstone Entertainment Group, 2007. 87 min. Color. D: Albert Pyun. SC: Chad Leslie. With Maria Alhe, Victoria Maurette, Soledad Arocena, Andres Bagg, Mariana Seligmann, Janet Barr, Brad Krupsaw, Javier de la Vega, Oliver Kolker, Adnen Helali. A killer inhabits a ghost town and murders anyone who crosses his path until he meets a female vigilante and her posse. A different kind of supernatural low budget effort.\n\n**2314** _ **The Left-Handed Gun**_ **** Warner Bros., 1958. 102 min. D: Arthur Penn. SC: Leslie Stevens. With Paul Newman, Lita Milan, John Dehner, Hurd Hatfield, James Congdon, James Best, Colin Keith-Johnston, John Dierkes, Bob Anderson, Wally Brown, Ainslie Pryor, Martin Garralaga, Denver Pyle, Nestor Paiva, Robert Foulk, Paul Smith, Jo Summers, Anne Barton. Young Billy the Kid is befriended by a ranger who is brutally murdered and Billy seeks revenge for the killing. Psychological approach to the Billy the Kid saga should appeal to Paul Newman fans.\n\n**2315** _ **Left Handed Law**_ **** Universal, 1937. 63 min. D: Lesley Selander. SC: Frances Guihan. With Buck Jones, Robert Frazer, Noel Francis, Frank LaRue, Lee Phelps, Matty Fain, George Regas, Lee Shumway, Nena Quartero, Charles LeMoyne, Budd Buster, Frank Lackteen, Jim Toney, Bill Wolfe, Jack Evans, Jim Corey. A town terrorized by lawlessness hires an Army colonel to get rid of the outlaws. Top notch Buck Jones vehicle.\n\n**2316** _ **The Legacy of the Incas**_ **** Marischka\/PEA\/Orbita Films, 1966. 100 min. Color. D: Georg Marischka. SC: Georg Marischka, Winfried Groth and Franz Marischka. With Guy Madison, Guela Nuni, Fernando Rey, Rik Battaglia, Chris Howland, Heinz Ehrhardt, William Rothlein, Carlo Tamberlani, Francesco Rabal, Walter Giller. The president of Peru assigns the last descendant of the Incas the task of trying to stop an Indian tribe, which is allied with a bandit king and a revolutionary, from trying to drive out all whites and resurrect the Inca empire. Entertaining West German feature based on a Karl May work. West German title: _**Das Vermachtnis des Inka**_ (The Legacy of the Incas).\n\n**2317** _ **Legend of a Gunfighter**_ **** Nora Film, 1964. 95 min. Color. D: Rolf Olson. SC: Donald Sharp and Paul Clydeburn. With Thomas Fritsch, Ron Randell, Judith Dornys, Walter Giller, Heidemarie Hatheyer, Gustav Knuth, Peter Neusser, Rudolf Schundler, Ingrid van Bergen, Ilse Peternell. Three years after his parents were murdered in a stagecoach ambush, a man returns to his home town seeking revenge. Better than average early West German oater with a neat plot twist, released there as _**Heiss Weht der Wind**_ (The Wind Is Blowing Hot), running 102 minutes; also called _**Midnight Canyon**_.\n\n**2318** _ **The Legend of Alfred Packer**_ **** American National Enterprises, 1980. 95 min. Color. D: Jim Roberson. SC: Chuck Meyers and Burton Raffel. With Patrick Dray, Ron Haines Jim Dratfield, Bob Damon, Dave Ellingson, Ron Holiday, William Brooks, Cynthia Noonan, Chuck Meyers, Mark Webb, Dick Morgan, George Warrar, Stephen Franton, Jim Roberson, Sam Kiernan. A half dozen men search for gold in Colorado in 1873 but only one of them survives the bitter Rocky Mountains winter by resorting to cannibalism. Not very interesting feature supposedly based on a true story but highlighted by scenic on-location filming.\n\n**2319** _ **Legend of Bearheart**_ **** Alpha-Path\u00e9, 1978. 83 min. Color. D: Rand Brooks. SC: Jennings Cobb. With Marshall Reed, Joey Young, Dana Dillaway, Denver Pyle, William (Bill) Zuckert, Anna Lee, James Edwards, Fritz Feld, Barbara Knudson, Larry Chance, Percy Helton, Ken Hooker, Tim Stafford. When his master is murdered by a drunk trapper, a dog seeks revenge and later saves the life of a little girl only to be hunted by her father. Heartwarming north woods drama (made in 1964) produced and directed by actor Rand Brooks, who also wrote the original story; quite scenic. Also called _**Bearheart of the Great Northwest**_ and _**Legend of the Northwest**_.\n\n**2320** _ **Legend of Black Thunder Mountain**_ **** Tom Beemer, 1979. 83 min. Color. D: Tom Beemer. SC: Tom Beemer and Susan Shadburne. With Holly Beemer, Steve Beemer, Ron Brown, F.A. Milovich, Vance Cleveland, Keith Sexson, John Sexson, Tim Staab, Glen Porter, Dick Albertson (narrator). Two youngsters, unknowingly carrying a treasure map, are pursued into the wilderness by murderous gold hunters. Okay children's adventure film; vocals by Don Brown.\n\n**2321** _ **The Legend of Butch and Sundance**_ **** Barnholz Enterprises, 2006. 89 min. Color. D: Sergio Mimica-Gezzan. SC: John Fasano. With David Rogers, Ryan Browning, Rachelle Lefevre, Blake Gibbons, Jay Brazeau, Michelle Harrison, Susan Ruttan, Mark Consuelos, Michael Biehn, Marty Antonini, Hamish Boyd, Tom Carey, Mara Casey, John Escobar, John Fasano, Lucia Fasano, Greg Lawson, Jaime Alvarez, Steve Strachan, Peter Skagen. The story of the teaming of the infamous outlaws Butch Cassidy and the Sundance Kid. Lame plotted TV movie lacks historical background but does have good photography.\n\n_**Legend of Cougar Canyon**_ see _**Secret of Navajo Cave**_\n\n**2322** _ **The Legend of Custer**_ **** Filmways\/20th Century\u2013Fox, 1968. 94 min. Color. D: Norman Foster and Sam Wanamaker. SC: Samuel A. Peeples and Shimon Wincelberg. With Wayne Maunder, Slim Pickens, Robert F. Simon, Peter Palmer, Michael Dante, Mary Ann Mobley, William Mims, Rodolfo Acosta, Alex Davion, Grant Woods, Richard Schuyler, Hick Hill. After trouble with Army brass, General George Armstrong Custer is assigned to a dead end post in the Dakotas in 1870 and develops his men into a group known as the Fighting Seventh. So-so look at Custer's early years in the West, the pilot for \"The Legend of Custer\" (ABC-TV, 1969); video titles: _**Crazy Horse and Custer**_ and _**Crazy Horse and Custer\u2014The Untold Story**_.\n\n**2323** _ **Legend of Death Valley**_ **** American National Enterprises, 1977. 90 min. Color. D: Kent Durden. With Robert Dawson. A man attempts to trace his great grandfather's trips to Death Valley in search of gold. Basically a documentary on Death Valley, detailing its history as well as its flora and fauna; nicely photographed and fairly interesting.\n\n**2324** _ **The Legend of Earl Durand**_ **** Howco International, 1974. 110 min. Color. D: John D. Patterson. SC: J. Frank James. With Peter Haskell, Slim Pickens, Keenan Wynn, Martin Sheen, Anthony Caruso, Albert Salmi, Ivy Bethune, Phil Lopp, Hal Boker, Hal Wright. During the last days of the Depression in Wyoming, a young man steals to give to the poor and is hunted by a posse. Okay low budget modern-day Western.\n\n**2325** _ **The Legend of Frank Woods**_ **** Variety International, 1977. 88 min. Color. D: Deno Paoli and Hagen Smith. SC: David Allen Russell (Hagen Smith). With Brad Stewart (Hagen Smith), Troy Donahue, Kitty Vallacher, Michael Christian, Richard Hurst, Emile Meyer, Orville Sherman, Eileen Brown, Rance Howard, Timothy Scott, Ivy Jones, Tom Monroe, Hank Worden, Howard Wright, James Bacon, William Dooley, Paul Pint, Joe Miller, Jim Davis, Art Sasser. Returning to the U.S. from Mexico, a gunman is mistaken for a priest when he arrives in a border town. Offensive redemption Western re-edited, with added footage, from _**To Hell You Preach**_ (q.v.).\n\n**2326** _ **The Legend of Frenchie King**_ **** SNC\/K-Tel, 1973. 97 min. Color. D: Christian-Jacque and Guy Casaril. SC: Marie-Anges Anies, Jean Nemours, Guy Casaril, Clement Bywood and Daniel Boulanger. With Brigitte Bardot, Claudia Cardinale, Michael J. Pollard, Emma Cohen, Micheline Presle, Patty Shepard, Luis Induni, Chris Huerta, Georges Beller, Henri Czarniak, Patrick Prejan. In the 1880s a group of sisters at a French settlement in Mexico turn to lawlessness to get the things they want in life. Mediocre European co-production although Bardot, et al., are nice on the eyes. French title: _**Les Petroleuses**_ (The Bandits).\n\n**2327** _ **The Legend of God's Gun**_ **** Indican Pictures, 2007. 78 min. Color. D-SC: Mike Bruce. With Robert Bones, Kirkpatrick Thomas, Dave Koenig, Julie Patterson, Mike Bruce, Henry Evans, Scott Dyeswell, Sally Fay Dalton, Samantha Smith, Jarid Southard, Randy America, Joseph Campanella (narrator). A gun toting preacher, an evil bandit leader and a bounty hunter all converge on a wicked town for a final shootout. The bandit drinks scorpion venom which pretty much sums up the thrust of this mini-budget, low grade affair.\n\n**2328** _ **The Legend of Grizzly Adams**_ **** VCI, 1990. 76 min. Color. D-SC: Ken Kennedy. With Gene Edwards, Anthony Caruso, L.Q. Jones, Acquanetta, Neil Summers, Kirstin Dattilo, Wayne Brennan, Anita Merritt, Red West, Link Wyler, Warner McKay, Kenny Stabler, W. Randolph Galvin. A mountain man and his bear companion lead settlers not knowing there is gold on their wagon train which is pursued by inept outlaws. Another minor attempt to revive the Grizzly Adams character.\n\n**2329** _ **The Legend of Jedediah Carver**_ **** Xenon, 1976. 90 min. Color. D: DeWitt Lee. SC: DeWitt Lee and Jack Lee. With DeWitt Lee, Joshua Hoffman, Richard Montgomery, Val Chapman, Wally Broberg, Clark Graves, Odie Chapman, James Tryon, Teri Trepow, Sabra, Al Chapman, Adam Lee, David Terril. Trying to survive in the desert, a rancher is forced to fight Indians as well as the harsh elements. Low grade independent production.\n\n**2330** _ **The Legend of Lobo**_ **** Buena Vista, 1962. 67 min. Color. SC: Dwight Hauser and James Algar. With Rex Allen (narrator\/songs), The Sons of the Pioneers [Lloyd Perryman, Pat Brady, Karl Farr, Dale Warren, Tommy Doss] (songs). The story of a wolf, from birth to growing up to lead a pack and save his mate from a rustler. Another documentary winner from Walt Disney; a very good film.\n\n**2331** _ **The Legend of Nigger Charley**_ **** Paramount, 1972. 98 min. Color. D: Martin Goldman. SC: Martin Goldman and Larry G. Spangler. With Fred Williamson, D'Urville Martin, Don Pedro Cooley, Gertrude Jeannette, Marcia McBroom, Alan Gifford, Joe Ryan, Will Hussung, Mill Moor, Thomas Anderson, Jerry Gatlin, Tricia O'Neil, Doug Rowe, Keith Prentice, Tom Pemberton, Joe Santos, Fred Lerner. When a Virginia slave is forced to kill a vicious plantation overseer he finds himself a fugitive hunted by the law. Exploitation feature with heavy doses of action and comedy; followed by _**The Soul of Nigger Charley**_ (q.v.).\n\n**2332** _ **The Legend of the Boy and the Eagle**_ **** Buena Vista, 1967. 48 min. Color. D-SC: Jack Couffer. With Stanford Lomakema; Frank De Kova (narrator). A Hopi Indian boy is banished from his tribe for freeing an eagle intended for sacrifice but is later saved by the bird and becomes an expert hunter. Interesting telling of a Native American traditional tale.\n\n**2333** _ **The Legend of the Golden Gun**_ **** NBC-TV\/Columbia, 1979. 100 min. Color. D: Alan J. Levi. SC: James D. Parriott. With Jeffrey Osterhage, Carl Franklin, Hal Holbrook, Keir Dullea, Robert Davi, Michelle Carey, John McLiam, Elissa Leeds, R.G. Armstrong, R.L. Tolbert, William Bryant, J. Brian Pizer, Rex Holman, Michael Yamaha, Walt Davis. A young farmer, taught to shoot by a legendary gunman, teams with a runaway slave to bring in Quantrill and his raiders. Fairly interesting TV film.\n\n**2334** _ **The Legend of the Lone Ranger**_ **** Apex Film Corporation, 1949. 75 min. D-SC: George B. Seitz, Jr. With Clayton Moore, Jay Silverheels, Glenn Strange, Walter Sande, George Chesebro, Jack Clifford, Tristram Coffin, Guy Wilkerson, Ralph Littlefield, George J. Lewis, Frank Fenton, Horace Murphy. The lone survivor of an ambushed band of Texas Rangers is nursed back to health by an Indian and the two team to round up the Butch Cavendish gang, the outlaws responsible for the massacre. Excellent version of the origins of the Lone Ranger, made up of the first three episodes of the TV series that ran on ABC-TV from 1949 to 1957. Clayton Moore and Jay Silverheels are the epitome of the Lone Ranger and Tonto and Glenn Strange is fine as the vicious Cavendish; ten times better than the theatrical misfire of the same title issued in 1981 (q.v.).\n\n**2335** _ **The Legend of the Lone Ranger**_ **** Universal\/Associated Film Distribution, 1981. 98 min. Color. D: William A. Fraker. SC: Ivan Goff, Ben Roberts, Michael Kane and William Roberts. With Klinton Spilsbury, Michael Horse, Jason Robards, Christopher Lloyd, Matt Clark, Juanin Clay, John Bennett Perry, David Hayward, John Hart, Richard Farnsworth, Lincoln Tate, Ted Flicker, Marc Gilpin, Patrick Montoya, David Bennett, R.L.Tolbert, Ted White, Jim Burke, Henry Wills, Larry Randles, Robert F. Hoy, Ted Gehring, Buck Taylor, Chuck Hayward, Tom Laughlin, Terry Leonard, Bonita Granville, James Keach (voice). A young lawyer becomes the Lone Ranger to combat the evil Butch Cavendish gang's plans to kidnap President Ulysses S. Grant. There is not much to recommend this sad rehash of the famous story although the film does prove one thing: Clayton Moore IS the Lone Ranger.\n\n_**Legend of the Northwest**_ see _**Legend of Bearheart**_\n\n**2336** _ **Legend of the Phantom Rider**_ **** A-Mark Entertainment, 2002. 100 min. Color. D: Alex Erikiletian. SC: Robert Ray. With Denise Crosby, Robert McRay, Stefan Gierasch, Zen Gesner, Angus Scrimm, George Murdock, Rance Howard, Scott Eberlein, Jamie McShane, Robert Peters, Saginaw Grant, Irwin Keyes, Julie Erickson, John Henry Whitaker, G. Larry Butler, Mark Coliver, Al Fleming, Lee McKechnie, Bo Greigh, Ross Clay, Phil Quigley, Michael Heistand, Jason Tatum, Clark Ray, Chris Schaar, Alexis Bond, Tony Romeo, John Proudstar, Rudy Red Eagle, Maria Ortiz. Heading West after the Civil War, a family is attacked by an outlaw gang as the mother and daughter survive to get to a remote town, only to find it controlled by the raiders who face vengeance from a mysterious stranger. Bizarre, strung out horror Western.\n\n**2337** _ **Legend of the Wild**_ **** Taft International, 1981. 93 min. Color. D: James L. Conway. SC: Arthur Heineman. With Dan Haggerty, Denver Pyle, Ken Curtis, Jack Kruschen, Kristen Curry, Don Shanks, Lucky Hayes, Henry Max Kendrick. A man seeks contentment by living a rustic life in the mountains and helps find a married couple trapped by a blizzard as well as saving a bear cub. Scissor and paste re-tread made up of footage from _**The Adventures of Frontier Fremont**_ and _**Once Upon a Starry Night**_ (q.v.) as well as \"The Life and Times of Grizzly Adams\" (NBC-TV, 1977\u201378).\n\n_**Legend of the Northwest**_ see _**Bearheart of the Great Northwest**_\n\n**2338** _ **The Legend of Tillamook's Gold**_ **** Moving Pictures Film and Television, 2006. 107 min. Color. D: Jane Beaumont Hall. SC: Richard A. Doyon and Jane Beaumont Hall. With Brian McNamara, Julia Campbell, Brian Thompson, Suzanne Marie Doyon, Max Gail, Floyd \"Red Crow\" Westerman, Bradley Stryker, Janine Doyon, Richard Doyon, Mary Stein, Escher Holloway, Phillip Huber, Elizabeth Erickson, Tony Hyde, Matthew Jared, Steve Meltzer, David Welborn, Imie Lane, Karara Muhoro, Bob Doyon. A teenage girl discovers the clue to a 16th century buried Spanish treasure near her seaside Oregon home. Pleasant modern-day family feature; also called _**The Tillamook Treasure**_.\n\n**2339** _ **The Legend of Tom Dooley**_ **** Columbia, 1959. 79 min. D: Ted Post. SC: Stan Sheptner. With Michael Landon, Jo Morrow, Jack Hogan, Richard Rust, Dee Pollack, Ted Lynch, Howard Wright, Ralph Moody, John Cliff, Anthony Jochim, Jeff Morris, Bill Hale, Sandy Sanders, Boyd Santell, Boyd \"Red\" Morgan, Jason Johnson, Joe Yrigoyen, Maudie Prickett, Juney Ellis. At the end of the Civil War, a young Confederate soldier robs a Union stage unaware the conflict is over and becomes a wanted criminal. Pretty good movie based on the popular song \"Tom Dooley\" recorded by the Kingston Trio.\n\n**2340** _ **The Legend of Walks Far Woman**_ **** NBC-TV, 1982. 110 min. Color. D: Mel Damski. SC: Evan Hunter. With Raquel Welch, Bradford Dillman, George Clutesi, Nick Mancuso, Nick Ramos, Eloy Phil Casados, Frank Sotonoma Salsedo, Hortensia Colorado, Alex Hubik, Branscombe Richmond, Dehl Berti, Nocana Aranda, Henry K. Bal, Gerald Red Elk, Janice Harman, Philip Beaumont, Rudy Diaz. A Blackfoot Indian woman, who is captured by the Sioux, sees the end of the way of life for the Plains Indians as she lives to be 102 year old. Over long and rather boring TV feature, filmed in 1979.\n\n**2341** _ **The Legend of Zorro**_ **** Columbia, 2005. 129 min. Color. D: Martin Campbell. SC: Roberto Orci and Alex Kurtzman. With Antonio Banderas, Catherine Zeta-Jones, Rufus Sewell, Nick Chinlund, Pedro Armendariz, Jr., Julio Oscar Mechoso, Mary Crosby, Leo Burmester, Adrian Alonso, Alberto Reyes, Gustavo Sanchez Para, Giovanna Zacarias, Carlos Cobos, Michael Emerson, Mauricio Bonet, Fernando Beccerril, Xavier Marc, Tony Amendola, Brandon Wood, Alejandro Galan, Pedro Altamirano, Philip Meheux, Pedro Mira, Raul Mendez, Mar Carrera, Silverio Palacios, Matthew Stirling, Shuler Hensley, Pepe Olivares, Alexa Benedetti. After he is alienated from his wife, Zorro learns the man she is seeing is the head of a secret society bent on taking over the United States. Fanciful Zorro outing, full of historical inaccuracies; sequel to _**The Mask of Zorro**_ (1998) [q.v.].\n\n**2342** _ **Legion of the Lawless**_ **** RKO Radio, 1940. 59 min. D: David Howard. SC: Doris Schroeder. With George O'Brien, Virginia Vale, Herbert Heywood, Norman Willis, Hugh Sothern, Billy Benedict, Eddy Waller, Delmer Watson, Bud Osborne, Monte Montague, Slim Whitaker, Mary Field, Richard Cramer, John Dilson, Martin Garralaga, Ed Peil, Sr., Lloyd Ingraham, Wilfred Lucas, Henry Wills, Horace Murphy, Herman Nowlin, Sid Jordan, John Ince. A lawyer leads the fight to help homesteaders and ranchers in opposing a vigilante group out to steal land wanted for a railroad right-of-way. Pretty good George O'Brien action outing.\n\n**2343** _ **Lemonade Joe**_ **** Allied Artists, 1967. 90 min. Color. D: Oldrich Lipsky. SC: Jiri Brdeca and Oldrich Lipsky. With Carl (Karel) Fiala, Olga Schoberova, Kveta Fialova, Miles Kopecky, Rudy Dale (Rudolph Deyl), Joseph Nomaz (Josef Hlinomaz), Karel Effa, Waldemar Matuska, Bohus Zahorsky, Eman Fiala, Jiri Steimar, Oldrich Lukes, Viktor Ocasek. The representative of a lemonade franchise teams with a temperance father and daughter to drum out the evil of liquor in the Old West. Quite amusing Czech-made genre takeoff.\n\n**2344** _ **Let Freedom Ring**_ **** Metro-Goldwyn-Mayer, 1939. 100 min. D: Jack Conway. SC: Ben Hecht. With Nelson Eddy, Virginia Bruce, Victor McLaglen, Lionel Barrymore, Edward Arnold, Guy Kibbee, Charles Butterworth, H.B. Warner, Raymond Walburn, Dick Rush, Trevor Bardette, George F. Hayes, Louis Jean Heydt, Sarah Padden, Eddie Dunn, C.E. Anderson, Luis Alberni, Emory Parnell, Mitchell Lewis, Victor Potel, Billy Bevan, Lionel Royce, Syd Saylor, Ted Thompson, Ralph (Francis X., Jr.) Bushman, Philo McCullough, Harry Fleischmann, Tenen Holtz, Constantine Romanoff. When a Harvard educated man returns home his family wants him to lead the fight by homesteaders against a crook so he joins the man's gang as a spy. Dandy entertainment with good work by Nelson Eddy, who belts out a number of songs, including \"Dusty Road\" and \"Love's Serenade\"; very patriotic.\n\n_**Let Them Rest**_ see _**Requiescant**_\n\n**2345** _ **La Ley del Mas Rapido**_ (The Law of the Fastest) **** Filmadora Independiente, 1958. 75 min. D: Rene Cardona. SC: Jesus Cardenas. With Rene Cardona, Jr., Sofia Alvarez, Lorena Velazquez, Dagoberto Rodriguez, Juan Manuel Guerrero, Rene Cardona, Leonor Llausas, Rodolfo Landa, Wally Barron, Jorge Alzaga, Victor Velazquez, Aurora Zermeno, Armando Gutierrez, Rafael Estrada, Salvador Lozano, Dacia Gonzalez, Andres Soler, David Reynoso, Miguel Manzano. A frontier lawyer is swindled out of a gold claim by a female outlaw and her gang. Okay sequel to the Mexican film _**El Puma**_ and followed by _**A Tiro Limpio**_ (qq.v.).\n\n_**La Ley del Revolver**_ (The Law of the Revolver) see _**The Colt Is My Law**_\n\n**2346** _ **La Leyenda del Bandido**_ (The Legend of the Bandit) **** Radeant Films, 1967. 87 min. Color. D-SC: Raul de Anda. With Rene Cardona, Rodolfo de Anda, Sonia Infante, Arturo Martinez, Tito Novaro, Martha Rios, Jose Dupeyron, Manuel Donde, Alfredo Gutierrez, Guillermo Sanchez, Jose L. Murillo, Ernesto Juarez, Federico Gonzalez, Angel Garasa, Samuel Moreno, Manuel Vergara \"Manver,\" Raul Ramirez, Luis Salgado, Martin Sanchez, Martin Plata. Returning home to wed the woman he loves, a Mexican rebel learns most of his comrades have been slaughtered by the Federales and he must choose between love and revenge. Well executed Mexican Western.\n\n_**The Life and Legend of Buffalo Jones**_ see _**Buffalo Rider**_\n\n**2347** _ **The Life and Times of Grizzly Adams**_ **** Sunn Classic, 1975. 93 min. Color. D: Richard Friedenberg. SC: Larry Dobkin. With Dan Haggerty, Denver Pyle, Don Shanks, Marjorie Harper, Lisa Jones. A fur trapper is hunted by the law for a crime he did not commit and he finds peace and contentment in the wilderness. Popular road show production is only average but it spawned a television series of the same title that ran on NBC-TV from 1977\u201378, plus several movie sequels.\n\n**2348** _ **The Life and Times of Judge Roy Bean**_ **** National General, 1972. 102 min. Color. D: John Huston. SC: John Milius. With Paul Newman, Jacqueline Bisset, Ava Gardner, Stacy Keach, Anthony Perkins, Tab Hunter, John Huston, Roddy McDowall, Victoria Principal, Anthony Zerbe, Ned Beatty, Roy Jenson, LeRoy Johnson, Matt Clark, Dean Smith, Bill McKinney, Fred Krone, Jack Colvin, David Sharpe, Gary Combs, Neil Summers. Judge Roy Bean rules as the only law west of the Pecos River, carrying on a one-sided love affair from afar with actress Lily Langtry. Pretty bad tongue-in-cheek look at the Roy Bean legend.\n\n**2349** _ **Life in the Raw**_ **** Fox, 1933. 62 min. D: Louis King. SC: Stuart Anthony. With George O'Brien, Claire Trevor, Greta Nissen, Francis Ford, Warner Richmond, Gaylord (Steve) Pendleton, Alan Edwards, Nigel De Brulier, LeRoy Mason, Si Jenks, Stanley Price, Paul Panzer, Otto Hoffman, Ed Peil, Sr., Sam McDaniel, George Reed, Frank Atkinson. A cowboy falls for a pretty girl and sets out to reform her no-good brother. Claire Trevor made her film debut in this fun George O'Brien vehicle based on a Zane Grey story; film has lots of humor.\n\n**2350** _ **Life on the Mississippi**_ **** PBS-TV, 1980. 120 min. Color. D: Peter H. Hunt. SC: Philip H Reisman, Jr. With Robert Lansing, David Knell, James Keane, Donald Madden, John Pankow, Bill Holliday, Luke Reilly, Marcy Walker, Don Brady, Jack Lawrence, John Kirk, Bill Atwood, Thom Thomas, Jim Babrowski, Douglas Wells, Barbara Chaney, Collins Bell, Lyla Owen, Don Lutenbacher, Harry Gorsuch, Robert Borwick, Wayne Dickson, Norma Schwied, Thomas Kent, Stanley J. Reyes. Twenty-two year-old Samuel L. Clemens signs on a Mississippi riverboat wanting to earn a pilot's license and comes under the tutelage of a stern captain. Fine TV adaptation of the Mark Twain work set in the pre\u2013Civil War era.\n\n**2351** _ **The Light in the Forest**_ **** Buena Vista, 1958. 93 min. Color. D: Herschel Daugherty. SC: Lawrence E. Watkin. With James MacArthur, Carol Lynley, Fess Parker, Wendell Corey, Joanne Dru, Jessica Tandy, Joseph Calleia, John McIntire, Rafael Campos, Frank Ferguson, Norman Frederic, Marian Seldes, Stephen Bekassy, Sam Buffington. In 1764 a peace treaty results in a young white boy, who bas been raised by the Indians, being returned home with his finding it difficult to adjust to a new life. Pleasant Disney family film based on the novel by Conrad Richter.\n\n**2352** _ **The Light of the Western Stars**_ **** Paramount, 1932. 80 min. D: Otto Brower and Edwin H. Knopf. SC: Grover Jones and William Slavens McNutt. With Richard Arlen, Mary Brian, Harry Green, Regis Toomey, Fred Kohler, William LeMaire, George Chandler, Syd Saylor, Guy Oliver, Gus Saville. A cowboy falls in love with the sister of his murdered friend and when a lawman, in cahoots with the killer, tries to take her ranch for back taxes, the cowpoke stages a robbery and steals gold to pay off the debt. Pleasant early sound adaptation of the Zane Grey novel first filmed by Sherman\/United in 1918 with William Farnum and remade by Paramount in 1925 starring Jack Holt, Billie Dove and Noah Beery; Paramount filmed it again in 1940 (q.v.). Reissued by Favorite Films as _**Winning the West**_.\n\n**2353** _ **The Light of the Western Stars**_ **** Paramount, 1940. 63 min. D: Lesley Selander. SC: Norman Houston. With Victor Jory, Jo Ann Sayers, Russell Hayden, Noah Beery, Jr., J. Farrell MacDonald, Morris Ankrum, Ruth Rogers, Tom Tyler, Rad Robinson, Eddie Dean, Esther Estrella, Georgia Hawkins, Alan Ladd, Earl Askam, Lucio Villegas, Bob Burns, Merrill McCormick. A ranch foreman, on the verge of becoming an outlaw, is helped by a pretty girl who has faith in him as he opposes a dishonest lawman and gun runners. Although it strays from Zane Grey's book, this fourth filming is a high class \"B\" effort that provides good entertainment.\n\n**2354** _ **Light the Fuse...Sartana Is Coming**_ **** Copercines\/Devon Film, 1970. 99 min. Color. D: Anthony Ascott (Giuliano Carmineo). SC: Tito Carpi, Eduardo Manzanos and Ernesto Gastaldi. With John (Gianni) Garko, Susan Scott (Nieves Navarro), Piero Lulli, Bruno Corazzari, Frank Brana, Massimo Serato, Jose Jaspe, Dan Van Husen, Luis Induni, Fernando Bilboa, Salvatore Borghese, Francisco Sanz, Mara Krupp, Giuseppe Castellano. Gunman Sartana rides into a town seeking hidden gold and finds himself at odds with several vicious citizens. One of the better Spaghetti Westerns, this Italian-Spanish co-production was released in Europe as _**Una Nuvola di Polvere...Un Grido di Morte...Arriva Sartana**_ (Cloud of Dust...Cry of Death...Sartana is Coming).\n\n**2355** _ **Lightin' Bill Carson**_ **** Puritan, 1936. 73 min. D: Sam Newfield. SC: (George) Arthur Durham and Joseph O'Donnell. With Tim McCoy, Lois January, Rex Lease, Harry Worth, Karl Hackett, John Merton, Lafe McKee, Edmund Cobb, Roger Williams, Richard Botiller, Jack Rockwell, Joe Girard, Frank Ellis, Slim Whitaker, Jimmy Aubrey, Oscar Gahan, Artie Ortego, Herman Hack, Franklyn Farnum, George Morrell, Arthur Thalasso, Wally West, Francis Walker, Harrison Greene, Clyde McClary, Jack Evans, Barney Beasley, Tom Smith. A U.S. marshal on the trail of the outlaw brother of the girl he loves is also hunted by a notorious gunman. A bit long for a series \"B\" Western, this pretty good Tim McCoy feature introduced the character of G-Man Lightning Bill Carson, a characterization he would continue in his 1938\u201339 Victory series.\n\n**2356** _ **Lightnin' Crandall**_ **** Republic, 1937. 60 min. D: Sam Newfield. SC: Charles Francis Royal. With Bob Steele, Lois January, Dave O'Brien, Horace Murphy, Charles King, Ernie Adams, Earl Dwire, Richard Cramer, Frank LaRue, Lew Meehan, Lloyd Ingraham, Ed Carey, Art Felix. After buying a ranch in Arizona a cowboy finds himself in the middle of a range war between two feuding families. Sturdy and entertaining Bob Steele opus.\n\n**2357** _ **Lightnin' in the Forest**_ **** Republic, 1948. 58 min. D: George Blair. SC: John K. Butler. With Lynne Roberts, Don Barry, Warren Douglas, Adrian Booth, Lucien Littlefield, Claire DuBrey, Roy Barcroft, Paul Harvey, Al Eben, Jerry Jerome, George Chandler, Eddie Dunn, Dale Van Sickel, Bud Wolfe, Hank Worden. A gang of crooks, on the run from the law, kidnap a rich and spoiled young socialite and hold her hostage in a mountain cabin. Adequate program feature.\n\n**2358** _ **Lightning Bill**_ **** Superior, 1934. 46 min. D: Victor Adamson (Denver Dixon). SC: L.V. Jefferson. With Buffalo Bill, Jr., Alma Rayford, Allen Holbrook, George Hazel, Nelson McDowell, Bud Osborne, William McCall, Lafe McKee, Eva McKenzie, Blackjack Ward, Bob McKenzie, Fred Parker, Barney Beasley, Jack Jones, Denver Dixon. A cowboy is out to round up a notorious horse rustler and his gang. Tattered rock bottom Victor Adamson production notorious for its misspelled title card.\n\n**2359** _ **Lightning Bryce**_ **** National Film Corporation (Arrow), 1919. 15 Chapters. D: Paul Hurst. SC: Harvey Gates and Joe Brandt. With Jack Hoxie, Ann Little, Steve Clemente, Ben Corbett, Walter Patterson, Jill Woodward, George Champion, Slim Lucas, George Hunter, Paul Hurst, Noble Johnson, Yakima Canutt. Outlaws try to steal valuable clues to the location of a gold mine discovered by the parents of a man and woman who are also looking for the claim. Rare silent serial should please Jack Hoxie and Ann Little fans.\n\n**2360** _ **Lightning Carson Rides Again**_ **** Victory, 1938. 59 min. D: Sam Newfield. SC: Joseph O'Donnell. With Tim McCoy, Joan Barclay, Ben Corbett, Bob Terry, Jane Keckley, Ted Adams, Karl Hackett, Sherry Tansey, Frank Wayne, Forrest Taylor, Reed Howes, Frank LaRue, James Flavin, Slim Whitaker, Wally West. A lawman helps his nephew when he is accused of robbing and killing his partner, actually the work of an outlaw gang. Tim McCoy resumes the role of Lightning Bill Carson in his initial entry in the Victory series; a fairly good outing.\n\n**2361** _ **Lightning Guns**_ **** Columbia, 1950. 55 min. D: Fred F. Sears. SC: Victor Arthur. With Charles Starrett, Smiley Burnette, Gloria Henry, William Norton Bailey, Edgar Dearing, Ken Houchins, Raymond Bond, Jock (Mahoney) O'Mahoney, Chuck Roberson, Frank Griffin, Joel Friedkin, George Chesebro, Merrill McCormick, Billy Williams. The Durango Kid tries to discover who is the ringleader of a gang constantly sabotaging the construction of a new dam. Well done \"Durango Kid\" episode.\n\n**2362** _ **Lightning Jack**_ **** Anchor, 1924. With Jack Perrin, Josephine Hill, Lew Meehan, Jack Richardson, Jack Phipps, Horace B. Carpenter, Thomas Foster. A cowboy, about to enter his fast horse in a race, is framed on a murder charge. There is nothing special about this silent Jack Perrin outing, but it moves fast and is fun to view.\n\n**2363** _ **Lightning Raiders**_ **** Producers Releasing Corporation, 1945. 61 min. D: Sam Newfield. SC: Elmer Clifton. With Buster Crabbe, Al St. John, Mady Lawrence, Henry Hall, Steve Darrell, I. Stanford Jolley, Karl Hackett, Roy Brent, Marin Sais, Al Ferguson, John Cason, Budd Buster, Frank Ellis, Bert Dillard, Victor Cox, Carl Mathews, Jack Evans, Bob Burns, Rube Dalroy, Herman Hack, Tex Cooper, Rose Plummer. Billy Carson and Fuzzy Q. Jones uncover a scheme where a banker leads a gang that steals mail in order to obtain land by foreclosures. Average \"Billy Carson\" segment with a funny scene where Fuzzy accidentally eats Mexican jumping beans.\n\n**2364** _ **Lightning Range**_ **** Superior, 1935. 50 min. D: Victor Adamson (Denver Dixon). SC: L.V. Jefferson. With Buddy Roosevelt, Patsy Bellamy, Genee Boutell, Betty Butler, Anne Howard, Si Jenks, Denver Dixon, Jack Evans, Boris Bullock, Clyde McClary, Bart Carre, Olin Francis, Lafe McKee, Merrill McCormick, Ken Broeker, Jack Bronston. A cowboy tries to help a pretty girl whose money has been stolen by a gang of crooks. Typically tacky Victor Adamson film, sure to appeal to fans of low grade cinema. ****\n\n**2365** _ **Lightning Strikes West**_ **** Colony, 1940. 57 min. D: Harry Fraser. SC: Martha Chapin. With Ken Maynard, Claire Rochelle, Michael Vallon, Charles King, Bob Terry, Reed Howes, Dick Dickinson, George Chesebro, John Elliott, William Gould, Tex Palmer, Carl Mathews, Chick Hannon. A U.S. marshal goes undercover to capture an escaped convict who has re-teamed with his gang to find buried loot stolen from a government dam project. Ken Maynard's last solo starring series oater is a fast moving and entertaining affair with the star doing a good job masquerading as a vagrant.\n\n**2366** _ **Lightning Triggers**_ **** Willis Kent\/Marcy, 1935. 60 min. D: S. Roy Luby. SC: E.B. Mann. With Reb Russell, Fred Kohler, Yvonne Pelletier, Jack Rockwell, Edmund Cobb, Lillian Castle, Lew Meehan, William McCall, Richard Botiller, Olin Francis, Artie Ortego, Steve Clark, Ed Carr, Jerry Meacham, Ed Porter. Joining an outlaw gang in order to bring them to justice, a cowboy finds out their leader is his father. Fair Reb Russell vehicle, enhanced by Fred Kohler's performance as the outlaw chief.\n\n**2367** _ **The Lightning Warrior**_ **** Mascot, 1931. 12 Chapters. D: Armand L. Schaefer and Benjamin Kline. SC: Wyndham Gittens, Ford Beebe and Colbert Clark. With Rin-Tin-Tin, Frankie Darro, George Brent, Hayden Stevenson, Georgia Hale, Pat O'Malley, Theodore Lorch, Lafe McKee, Robert Kortman, Frank Lanning, Frank Brownlee, Kermit Maynard, Dick Dickinson, Helen Gibson, William Desmond, Steve Clemente, George Magrill, Yakima Canutt, Bertee Beaumont, Cliff Lyons. A young boy and a German shepherd dog try to find out the true identity of the Wolf Man, the killer responsible for the murders of the boy's father and the dog's master. Action packed cliffhanger with excellent stunt work by Yakima Canutt.\n\n**2368** _ **Lights of Old Santa Fe**_ **** Republic, 1944. 76 min. D: Frank McDonald. SC: Gordon Kahn and Bob Williams. With Roy Rogers, George \"Gabby\" Hayes, Dale Evans, Lloyd Corrigan, Richard Powers (Tom Keene), Claire DuBrey, Arthur Loft, Roy Barcroft, Lucien Littlefield, Bob Nolan and The Sons of the Pioneers (Tim Spencer, Ken Carson, Shug Fisher, Hugh Farr, Karl Farr), Sam Flint, Jack Kirk, Larry Steers, Roy Bucko, Gertrude Astor, Mary Kenyon, Arlyn Roberts. Roy Rogers and the Sons of the Pioneers work for a rodeo that is being sabotaged by a rival outfit. Only a fair film with no real dramatic climax, just a big rodeo finale.\n\n**2369** _ **Li'l Scratch**_ **** American National Enterprises, 1972. 93 min. Color. D: Larry Jones. With Larry Jones. An outdoorsman on a photographic excursion in the wilderness makes friends with an orphaned bear cub. Pleasant and amusing documentary.\n\n**2370** _ **The Lion and the Horse**_ **** Warner Bros., 1952. 83 min. Color. D: Louis King. SC: Crane Wilbur. With Steve Cochran, Sherry Jackson, Ray Teal, Bob Steele, Harry Antrim, George O'Hanlon, Ed Hinton, William Fawcett, House Peters, Jr., Lee Roberts, Lane Chandler, Tom Tyler, John Merton, Dick Curtis, Frank Nelson (voice), Wildfire (horse). In order to save his beloved stallion from an uncaring new owner, a cowboy takes the horse into the wilds, seeking sanctuary with an old rancher and his little granddaughter. Well written and action filled family fare.\n\n**2371** _ **The Lion's Den**_ **** Puritan, 1936. 59 min. D: Sam Newfield. SC: John T. Neville. With Tim McCoy, Joan Woodbury, Don Barclay, J. Frank Glendon, John Merton, Arthur Millet, Karl Hackett, Dick Curtis, Jack Evans, Art Felix, Bud McClure, Jack Rockwell, Frank Ellis. A sharpshooter who has agreed to help ranchers fight terrorism arrives in a town and is mistaken for a hired gunman by the man causing the trouble. A bit complicated but entertaining Tim McCoy vehicle.\n\n**2372** _ **Little Big Horn**_ **** Lippert, 1951. 86 min. D: Charles Marquis Warren. SC: Charles Marquis Warren and Harold Shumate. With Lloyd Bridges, Marie Windsor, John Ireland, Reed Hadley, Jim Davis, Wally Cassell, Hugh O'Brian, Sheb Wooley, King Donovan, Rodd Redwing, Richard Emory, John Pickard, Ted Avery. A group of soldiers attempt to rescue General Custer and his men at the Little Big Horn but become involved with personal differences. Cheaply made but well acted; a different kind of Western. Also called _**The Fighting 7th**_.\n\n**2373** _ **Little Big Man**_ **** National General, 1970. 150 min. Color. D: Arthur Penn. SC: Calder Willingham. With Dustin Hoffman, Faye Dunaway, Martin Balsam, Richard Mulligan, Chief Dan George, Jeff Corey, Amy Eccles, Jean Peters, Carole Androsky, Robert Little Star, Cal Bellini, Thayer David, James Anderson, Jesse Vint, Jack Bannon. A aged man recounts his life, including living with the Indians, returning to his people and taking part in the Battle of the Little Big Horn. Overlong and sometimes confusing drama, mainly for Dustin Hoffman fans.\n\n**2374** _ **Little House:**_ _**Bless All the Dear Children**_ **** NBC-TV, 1984. 100 min. Color. D: Victor French. SC: Chris Abbott-Fish. With Melissa Gilbert, Dean Butler, Victor French, Richard Bull, Kevin Hagen, Patricia Pearcy, Robin Clarke, Harvey Vernon, Allison Balson, Robert Casper, Pamela Boylance, Joel Graves, Dick Friedman, Lindsay Kennedy, Shannon Doherty, Leslie Landon, Michael Landon (narrator). While Christmas shopping in Mankato, the Wilders' small daughter is kidnapped by a woman who lost her own baby in childbirth. Shown after, but probably filmed before _**Little House: The Last Farewell**_ (q.v.), this telefilm is another segment of the long running \"Little House on the Prairie\" (NBC-TV, 1974\u201383); passable holiday fare.\n\n**2375** _ **Little House:**_ _**Look Back to Yesterday**_ **** NBC-TV, 1983. 100 min. Color. D: Victor French. SC: Vince R. Gutierrez. With Michael Landon, Melissa Gilbert, Victor French, Dean Butler, Richard Bull, Henry Brandon, Kevin Hagen, Dabbs Greer, Matthew Laborteaux, Melora Hardin, Jonathan Gilbert, Cooper Huckabee, James T. Callahan, Charles Cyphers, Allison Balson, Pamela Boylance, Robert Casper, Leslie Landon. Pa Ingalls returns to Walnut Grove to find the area in a recession and his adopted son about to die from a blood disease. Telefeature spin-off from \"Little House on the Prairie\" (NBC-TV, 1974\u201383) is a bit maudlin but fans will enjoy it, although Katherine MacGregor's character Mrs. Oleson is sorely needed to enliven the proceedings.\n\n**2376** _ **Little House on the Prairie**_ **** NBC-TV, 1974. 96 min. Color. D: Michael Landon. SC: Blanche Hanalis. With Michael Landon, Karen Grassle, Melissa Gilbert, Melissa Sue Anderson, Victor French, Lindsay and Sidney Greenbush, Vic Mohica, Cal Bellini, Sam Vlahos, Richard Alarian, Marian Breedler. A pioneer family tries to adjust to a new life on the Kansas plains. Excellent telefeature based on the popular Laura Ingalls Wilder books and the pilot for the long running series of the same name on NBC-TV from 1974 to 1982; it ran another season as \"Little House: A New Beginning\" during 1982\u201383.\n\n**2377** _ **Little House:**_ _**The Last Farewell**_ **** NBC-TV, 1984. 100 min. Color. D-SC: Michael Landon. With Michael Landon, Karen Grassle, Melissa Gilbert, Victor French, Dean Butler, Richard Bull, Kevin Hagen, Dabbs Greer, James Karen, Dennis Robertson, Roger Torrey, Rod Colbin, Alvy Moore, Bill McLennan, Jonathan Gilbert, Allison Balson, Stan Ivar, Pamela Roylance, Lindsay Kennedy, David Friedman, Leslie Landon, Robert Casper, Sherri Stoner, Shannon Doherty, Diane Kennedy, Steve Rumph, Alex Sharp, Ruth Foster, Jack Lilley. The citizens of Walnut Grove find they are going to lose their town to a ruthless land baron who has the law on his side. Well done TV movie finale to \"Little House on the Prairie\" (NBC-TV, 1974\u201383) but followed by _**Little House:**_ _**Bless All the Dear Children**_ (q.v.).\n\n**2378** _ **Little Joe the Wrangler**_ **** Universal, 1942. 64 min. D: Lewis D. Collins. SC: Sherman Lowe and Elizabeth Beecher. With Johnny Mack Brown, Tex Ritter, Fuzzy Knight, Jennifer Holt, Florine McKinney, James Craven, Hal Taliaferro, Glenn Strange, The Jimmy Wakely Trio (Jimmy Wakely, Johnny Bond, Scotty Harrell), Ethan Laidlaw, Slim Whitaker, Michael Vallon, Robert F. Hill, Evelyn Cooke, Dave Allen, Bill Patton, Carl Sepulveda A stranger is framed on a robbery and murder charge but the local lawman believes him innocent and they try to find the real culprits. Overly involved Johnny Mack Brown-Tex Ritter vehicle with too much music too boot.\n\n**2379** _ **Little Moon and Jud McGraw**_ **** International Cine Corporation, 1979. 80 min. Color. D: Bernard Girard. SC: Monroe Manning, Douglas May Stewart and Marcus Demian. With James Caan, Stefanie Powers, Aldo Ray, Sammy Davis, Jr., Barbara Werle, Robert Walker (Jr.), Peter Fonda, Mike Lane, Michael Conrad, Kenny Adams, Anne Barton, Paul Bergen, Fabian Dean, Noel Drayton, Anthony Gordon, Pepper Martin, Boyd \"Red\" Morgan, Chuck Hayward, Reed Sherman, Jay York, Dick Shane, Buck Lee, Bill Foster, Danny Redeznick, Gillian Sampson, Chris Calebrese, James McHale, Benny Dobbins, Julie Ann Johnson, Ginger Irwin, Lenore Stevens, Sherise Roland. A newspaper reporter and his girlfriend visit a ghost town and are told the story of how a cowboy and an Indian maid teamed to get revenge on a bad man and his gang. Tacky, rambling feature rounded out with filler material; absolutely awful. Filmed in 1969 by Cinema Releasing Corporation as _**Man Without Mercy**_ and issued briefly in 1975 as _**Gone With the West**_ by International Cinefilm; also called _**Bronco Busters**_.\n\n**2380** _ **The Little Patriot**_ **** Amsell Entertainment, 1995. 90 min. Color. D: J. Christian Ingvordsen. SC: J. Christian Ingvordsen and Rick Washburn. With Dan Haggerty, Ryan Washburn, Jacqueline Knox, John Christian (J. Christian Ingvordsen), John Weiner, Rick Washburn, Jeff Mazzola, Sam Bon Lorn, R. Bobby Persad, Pete Williams, Joseph P. Dandry, Timothy Oman, Eric Marshall, Maraya Chase, Jeffrey Howard, Ian Stewart, Cameron Jones, Whip Randall, Dan Leiner, Ernie Dorsett, Steve Kokinos, Andrea Sirrenberg, Kyle Gabriel, Eric Heimbold, Eli Kabilio, Steven Cea, Jim Downey. During the Revolutionary War a boy is captured by the British in their attempt to take over the frontier and he escapes to ally himself with area Indians against the invaders. Only fair family historical drama filmed in upper New York state and Ontario, Canada; issued on video as _**Sign of the Otter**_.\n\n**2381** _ **The Little Shepherd of Kingdom Come**_ **** 20th Century\u2013Fox, 1961. 108 min. Color. D: Andrew V. McLaglen. SC: Barre Lyndon. With Jimmie Rodgers, Luana Patten, Chill Wills, Linda Hutchins, Robert Dix, George Kennedy, Shirley O'Hara, Ken Miller, Neil Hamilton, Lois January, Jack Holland, Edward Faulkner, Morris Ankrum, Nelson Leigh, Lane Chandler, Diana Darrin, I. Stanford Jolley, Jerry Summers, Dan Simmons, Helen Scott, Glen Marshall. A Southerner, who fought for the Union during the Civil War, returns to his rural Kentucky home and tries to resume a normal life. Slow moving version of the old chestnut first filmed in 1920 by Goldwyn with Jack Pickford and remade in 1928 by First National headlining Richard Barthelmess.\n\n**2382** _ **The Littlest Outlaw**_ **** Buena Vista, 1955. 75 min. Color. D: Roberto Gavaldon. SC: Bill Walsh. With Andres Velasquez, Pedro Armendariz, Joseph Calleia, Rodolfo Acosta, Pepe Ortiz, Laila Maley, Gilberto Gonzales, Jose Torvay, Ferrusquilla, Enriqueta Zazueta, Margarita Luna. A Mexican boy becomes a fugitive when he runs away with a general's horse because the animal was ordered killed. Charming Walt Disney feature.\n\n**2383** _ **The Living Coffin**_ **** Alameda Films, 1959. 72 min. Color. D: Fernando Mendez. SC: Ramon Obon. With Gaston Santos, Maria Duval, Pedro D'Aguillon, Carlos Ancira, Carolina Baarret, Antonio S. Raxel, Hortensia Santovena, Quentin Buines, Jose Chavez, Eugenia Galindo, Jose Dupeyron, Hernan Vera, Guillermo Alvarez Bianchi. After finding a statue of a crying woman, a cowboy and his pal stop at a ranch where they are told it depicts a ghost seen by the locals. Atmospheric but rather bland Mexican horror Western made as _**El Grito de la Muerte**_ (The Cry of the Dead).\n\n**2384** _ **The Living Desert**_ **** Buena Vista, 1953. 73 min. Color. D: James Algar. SC: James Algar, Winston Hibler and Ted Sears. With Winston Hibler (narrator). The American desert is shown, zeroing in on its animal life. Academy Award winning documentary feature from Walt Disney; a must see for nature lovers.\n\n**2385** _ **The Llano Kid**_ **** Paramount, 1939. 70 min. D: Edward Venturini. SC: Wanda Tuchock. With Tito Guizar, Gale Sondergaard, Alan Mowbray, Jane (Jan) Clayton, Emma Dunn, Minor Watson, Chris-Pin Martin, Carlos de Valdez, Anna Demetrio, Glenn Strange, Tony Roux, Harry Worth, Eddie Dean, Bob McKenzie, Gertrude Astor. A Mexican bandit poses as the long lost heir to an old lady's fortune. Okay version of the O. Henry story \"Double-Dyed Deceiver\" which was first filmed in 1930 as _**The Texan**_ (q.v.) with Gary Cooper.\n\n**2386** _ **Loaded Pistols**_ **** Columbia, 1949. 70 min. D: John English. SC: Dwight Cummings and Dorothy Yost. With Gene Autry, Barbara Britton, Chill Wills, Jack Holt, Robert Shayne, Russell Arms, Fred Kohler, Jr., Vince Barnett, Leon Weaver, Clem Bevans, Sandy Sanders, Budd Buster, John R. McKee, Stanley Blystone, Hank Bell, Felice Raymond, Richard Alexander, Frank O'Connor, Reed Howes, Snub Pollard, Heinie Conklin, William Sundholm. Gene Autry and his friends find themselves up against a crooked rancher. There is plenty of action in this fun Gene Autry feature.\n\n**2387** _ **El Lobo Negro**_ (The Black Wolf) **** Telecines, 1981. 90 min. Color. D: Rafael Romero Marchent. SC: Joaquin Romero Hernandez and Rafael Romero Marchent. With Fernando Allende, Maria Silva, Carlos Ballesteros, Lola Forner, Julian Ugarde, Esperanza Roy, Fernando Sancho, Jose Maria Caffarei, Frank Brana, Barta Barry, Alfonso del Real, Roberto Camardiel, Alejandro De Encizo, Tomas Zori, Dum Dum Pacecho, Francisco Jones, Luis Gaspar, Paul Benson, Francisco Camoiras, Jose Luis Lespe, Fernando Sanchez Polack, Jose Yepes. In Old California a masked swordsman fights for the people against government oppressors. Standard Spanish \"Zorro\" imitation, followed by _**La Venganza del Lobo Negro**_ (q.v.).\n\n**2388** _ **The Local Bad Man**_ **** Allied, 1932. 60 min. D: Otto Brower. SC: Philip White. With Hoot Gibson, Sally Blane, Ed Peil, Sr., Hooper Atchley, Skeeter Bill Robbins, Edward Hearn, Milt Brown, Jack Clifford, Lew Meehan, Bud Osborne, Olin Francis, George Sowards, Lem Sowards. Two dishonest bankers plan to rob their own express shipment and place the blame on the driver. Pretty fair Hoot Gibson outing.\n\n**2389** _ **Lock, Stock and Barrel**_ **** NBC-TV\/Universal, 1971. 96 min. Color. D: Jerry Thorpe. SC: Richard Alan Simmons. With Tim Matheson, Belinda Montgomery, Claude Akins, Jack Albertson, Neville Brand, Burgess Meredith, Robert Emhardt, John Beck, Charles Dierkop, Joe Di Reda, Mills Watson, Timothy Scott, Dan Jenkins. When a young couple elope and head to Oregon, the girl's father gives chase as they encounter a series of adventures. Passable genre spoof made for television.\n\n**2390** _ **Lone and Angry Man**_ **** Estele Films, 1965. 95 min. Color. D: William Hawkins (Mario Caiano). SC: James Reed (Guido Malatesta) and David Moreno. With Anthony Steffen, Eduardo Fajardo, Fulvia Franco, Jorge (George) Rigaud, Armando Calvo, Arthur Kent (Arturo Dominici), Luciana Galli, Miguel Del Castillo, Jesus Fordesillas, Tomas Torres, Mario Vico, Frank Brana, Luis Barboo. A gunman infiltrates an outlaw gang, led by an ex-lawyer and a Mexican bandit, to find a murderer. Tolerable Italian-Spanish co-production made as _**Una Bara per lo Sceriffo**_ (A Coffin for the Sheriff) and also called _**Tomb for the Sheriff**_.\n\n**2391** _ **The Lone Avenger**_ **** World Wide\/Fox, 1933. 61 min. D-SC: Alan James. With Ken Maynard, Muriel Gordon, Jack Rockwell, Charles King, Alan Bridge, Jim Mason, Niles Welch, William Norton Bailey, Ed Brady, Clarence Geldert, Lew Meehan, Horace B. Carpenter, Jack Ward, Bud McClure, Fern Emmett, Jack Kirk, Robert Walker, Merrill McCormick, Olin Francis, Herman Hack, Buck Morgan. A cowboy tries to stop an outlaw gang from taking over a town by causing a bank panic. Top notch Ken Maynard movie with plenty of action to suit his legion of fans.\n\n**2392** _ **The Lone Bandit**_ **** Empire, 1935. 60 min. D: J.P. McGowan. SC: Ralph (Cushman) Consumana. With Lane Chandler, Doris Brook, Wally Wales, Slim Whitaker, Ray Gallagher, Ben Corbett, Jack Prince, Philo McCullough, Forrest Taylor, Frank Ellis, Wally West, Horace B. Carpenter. After a masked bandit steals his horse a cowboy is accused of being an outlaw but is cleared and tries to get back his mount and bring in the mystery man. A somewhat complicated plot does not detract from the overall enjoyment of this low budget affair.\n\n**2393** _ **Lone Cowboy**_ **** Paramount, 1934. 75 min. D: Paul Sloane. SC: Agnes Brand Leahy and Bobby Vernon. With Jackie Cooper, Lila Lee, Barton MacLane, Addison Richards, Charles Middleton, Gavin Gordon, Herbert Corthell, John Wray, J.M. Kerrigan, Del Henderson, Irving Bacon, Lillian Harmer, William LeMaire, George Pearce, Joe Barton, William Robyns, Leonard Kibrick, Rose Levine, Buster Guelich, Harry C. Bradley, Charles Kean, Harold Goodwin, Jerome Storm, James Adamson, Col. Starrett Ford. A delinquent from Chicago is sent West to live with his dad's pal, a cowboy, and the two become friends when faced with outlaws. A different kind of genre offering and a good film, based on Will James' novel; remade as _**Shoot Out**_ (q.v.).\n\n**2394** _ **The Lone Defender**_ **** Mascot, 1930. 12 Chapters. D: Richard Thorpe. SC: William Presley Burt, Harry Fraser and Bennett Cohen. With Rin Tin Tin, Walter Miller, June Marlowe, Buzz Barton, Josef Swickard, Lee Shumway, Frank Lanning, Robert Kortman, Arthur Morrison, Lafe McKee, Bob Irwin, Arthur Metzeth, Bill McGowan, Victor Metzetti, Julia Beharano. Crooks murder a dog's master for the map to a secret mine and then try to kidnap the canine because they believe he can lead them to the gold. Mascot's first all-talking serial is a slow affair but worth a look to see Rin Tin Tin.\n\n**2395** _ **The Lone Gun**_ **** United Artists, 1954. 78 min. Color. D: Ray Nazarro. SC: Don Martin and Richard Schayer. With George Montgomery, Dorothy Malone, Neville Brand, Frank Faylen, Skip Homeier, Douglas Kennedy, Robert Wilke, Douglas Fowley, Fay Roope, Emmett Vogan. While after a gang of cattle thieves in Texas, a lawman falls in love with a rancher's pretty daughter. Average oater remade as _**The Gambler Wore a Gun**_ (q.v.).\n\n**2396** _ **The Lone Hand**_ **** Universal-International, 1953. 79 min. Color. D: George Sherman. SC: Joseph Hoffman. With Joel McCrea, Barbara Hale, Alex Nicol, Charles Drake, Jimmy Hunt, James Arness, Wesley Morgan, Roy Roberts, Frank Ferguson, Helen Spring, Denver Pyle, Eddie Parker, George Wallace, Stanley Blystone, Eddie Dew, Frank Ellis, Tom Hubbard, Donald Kerr, Hugh Prosser, John Carpenter, Chuck Hamilton, William Kerwin, Jack Mower, Charles Regan, Brian Garfield. A rancher with a small son and a new wife risks losing their respect when he is forced to secretly work undercover to infiltrate a gang of rustlers. Pretty good Joel McCrea film.\n\n**2397** _ **The Lone Hand Texan**_ **** Columbia, 1947. 54 min. D: Ray Nazarro. SC: Ed Earl Repp. With Charles Starrett, Smiley Burnette, Mary Newton, Fred F. Sears, Mustard and Gravy (Frank Rice and Ernest L. Stokes), Maudie Prickett, George Chesebro, Robert Stevens, Bob (John) Cason, Jim Diehl, George Russell, Jasper Weldon, Post Park, Art Dillard, Matty Roubert, Herman Hack, Blackie Whiteford. Outlaws try to sabotage an oil drilling operation with the Durango Kid trying to stop them and learn the identity of their leader. Okay \"Durango Kid\" outing. British title: _**The Cheat**_.\n\n**2398** _ **The Lone Prairie**_ **** Columbia, 1942. 58 min. D: William Berke. SC: Ed Earl Repp and J. Benton Cheney. With Russell Hayden, Bob Wills and The Texas Playboys, Dub Taylor, Lucille Lambert, John Merton, John Maxwell, Jack Kirk, Edmund Cobb, Ernie Adams, Kermit Maynard, Art Mix, Steve Clark, Herman Hack, Carl Mathews, Jack Evans, Fred Burns, Ray Jones, Rube Dalroy. Crooks are after a man's ranch for a railroad right-of-way and they steal his cattle but a buyer comes to his rescue. Pretty fair Russell Hayden vehicle with nice musical interludes by Bob Wills and his group.\n\n**2399** _ **The Lone Ranger**_ **** Republic, 1938. 15 Chapters. D: William Witney and John English. SC: Barry Shipman, George Worthington Yates, Franklyn Adreon, Ronald Davidson and Lois Eby. With Chief Thundercloud, Lee Powell, Herman Brix (Bruce Bennett), Lynne Roberts, William Farnum, Stanley Andrews, George Cleveland, Hal Taliaferro (Wally Wales), Lane Chandler, George (Montgomery) Letz, John Merton, Sammy McKim, Tom London, Ray Bennett, Maston Williams, Frank McGlynn, Reed Howes, Allan Cavan, Walter James, Francis Sayles, Murdock MacQuarrie, Ted Adams, Jack Kirk, Art Dillard, Frank Ellis, Carl Stockdale, Bud Osborne, Fred Burns, Forbes Murray, Charles King, Jack Perrin, Slim Whitaker, Edmund Cobb, Jack Rockwell, Frankie Marvin, Lafe McKee, Charles Williams, Robert Kortman, Post Park, George Plues, Al Taylor, Blackie Whiteford, Griff Barnett, Jane Keckley, Bob Card, Ben Wright, Edna Lawrence, J.W. Cody, Hank Bell, Al Taylor, Curley Dresden, Ray Henderson, Forrest Burns, Art Felix, Vinegar Roan, Bert Dillard, Duke Taylor, Yakima Canutt, Duke Green, Tex Cooper, Jack Ingram, Carl Saxe, Millard McGowan, Bill Yrigoyen, Joe Yrigoyen; Billy Bletcher, Earl Graser (voices), Silver King (horse). After the Civil War five lawmen team to combat outlaws and they are aided by a masked man and his Indian friend, with one of the crusaders being the Lone Ranger. One of the all-time great sound serials; a must see for genre fans. Issued in a 69 minute feature version by Republic in 1940 as _**Hi-Yo Silver**_ with new footage featuring Raymond Hatton telling the story to Dickie Jones.\n\n**2400** _ **The Lone Ranger**_ **** Warner Bros., 1956. 86 min. Color. D: Stuart Heisler. SC: Herb Meadows. With Clayton Moore, Jay Silverheels, Lyle Bettger, Bonita Granville, Perry Lopez, Robert Wilke, John Pickard, Beverly Washburn, Michael Ansara, Frank DeKova, Charles Meredith, Mickey Simpson, Zon Murray, Lane Chandler, Lee Roberts, Malcolm Atterbury, Edward Colmans, William Schallert, Robert Williams, Hank Patterson, Elmore Vincent, Hal Taggart, Rush Williams, Kermit Maynard, Robert Filmer, Paul Power, Fred Kelsey, Robert Malcolm. The Lone Ranger and Tonto are assigned to look into unrest between whites and Indians and they learn a wealthy rancher is opposing efforts for statehood. Well done theatrical feature with Clayton Moore and Jay Silverheels successfully repeating their TV roles, with fine work by Lyle Bettger as the psychotic rancher.\n\n**2401** _ **The Lone Ranger**_ **** WB Television Network, 2003. 95min. Color. D: Jack Bender. SC: Jonathan Penner and Stacy Tile. With Chad Michael Murray, Nathaniel Arnold, Anita Brown, Fay Masterson, Sebastian Spence, Dylan Walsh, Wes Studi, Bradford Tatum, Jeffrey Nording, Lauren German, Tod Thawley, Gil Birmingham, Paul Schulze, David Franco, Martha Hackett, Mike Weinberg, Antoinette Broderick, Cassie Pappas, Laura Beth Cohen, Joel Marshall, Brian J. White, James Kyson Lee. When his ranger brother is murdered and he is badly injured and nursed back to health by an Indian, a man falls in love with his savior's sister and vows revenge on he gang who killed his sibling. Sorry TV movie retelling of the Lone Ranger saga.\n\n**2402** _ **The Lone Ranger and the Lost City of Gold**_ **** United Artists, 1958. 80 min. Color. D: Lesley Selander. SC: Robert Schaefer and Eric Friewald. With Clayton Moore, Jay Silverheels, Douglas Kennedy, Charles Watts, Noreen Nash, Lisa Montell, Ralph Moody, Norman Frederic, John Miljan, Maurice Jara, Bill Henry, Lane Bradford, Belle Mitchell, Bob Woodward, Herman Hack. After hooded riders murder members of an Indian tribe, the Lone Ranger and Tonto uncover a plot to steal five medallions that reveal the location of a sacred city of gold. While not quite up to the 1956 feature _**The Lone Ranger**_ (q.v.), this follow-up makes for fine viewing.\n\n**2403** _ **The Lone Ranger Rides Again**_ **** Republic, 1939. 15 Chapters. D: William Witney and John English. SC: Franklyn Adreon, Ronald Davidson, Sol Shor and Barry Shipman. With Robert Livingston, Chief Thundercloud, Duncan Renaldo, Jinx Falken(burg), Ralph Dunn, J. Farrell MacDonald, William Gould, Rex Lease, Ted Mapes, Henry Otho, John Beach, Glenn Strange, Stanley Blystone, Edwin (Eddie) Parker, Al Taylor, Carleton Young, Ernie Adams, Slim Whitaker, David Sharpe, Art Felix, Chick Hannon, Eddie Dean, Howard Chase, Nelson McDowell, Walter Wills, Jack Kirk, Fred Burns, Lew Meehan, Wheeler Oakman, Forrest Taylor, Frank Ellis, Herman Hack, Bud Wolfe, Duke Taylor, Forrest Burns, George DeNormand, Tommy Coats, Ted Wells, Carl Sepulveda, Roger Williams, Buddy Roosevelt, Jack Montgomery, Post Park, Art Dillard, Horace B. Carpenter, Cactus Mack, Lafe McKee, Charles Hutchison, Monte Montague, Griff Barnett, Augie Gomez, Buddy Messinger, Betty Roadman, Tom Smith, Jim Corey, Duke R. Lee, Augie Gomez, Bill Yrigoyen, Joe Yrigoyen, Francis Walker, Blackjack Ward, Billy Bletcher (voice), Silver King (horse). The Lone Ranger and Tonto come to the aid of a wagon train whose settlers are thought to be the victims of attacks by greedy cattlemen. Exciting and entertaining cliffhanger sequel to _**The Lone Ranger**_ (1938) [q.v.].\n\n**2404** _ **The Lone Rider**_ **** Columbia, 1930. 60 min. D: Louis King. SC: Forrest Sheldon. With Buck Jones, Vera Reynolds, Harry Woods, George Pearce, Lafe McKee, Blackjack Ward, Charles Le Moyne, Buck Connors, Jim Mason, Jack Kirk, George Plues, Cliff Lyons, Tex Phelps, Tom Bay, Ralph Bucko, Roy Bucko. An outlaw quits his gang, thwarts a stagecoach robbery and ends up heading the town's vigilante committee. Buck Jones' first talkie is only fair; remade as the much better _**The Man Trailer**_ , again starring Jones, and _**The Thundering West**_ (qq.v.).\n\n**2405** _ **The Lone Rider**_ **** Proton Film\/Tele Talia Films, 1960. 85 min. D: Rafael Baledon. SC: Rafael Baledon and Eva Guerrero Larranga. With Jeff Stone, Maria Rivas, Demetrio Gonzalez, Pedro de Aguillon, Carlos Suarez, Jose Dupeyron, Rafael Estrada, Lupe Andrade, Pedro Ortega, Humberto Rodriguez, Ramon Sanchez. Zorro attempts to free a mine owner captured by an outlaw gang after his gold. Obscure Mexican production first released there in 1958 as _**El Jinete Solitario en el Valle de los Desaparecidos: La Venganza del Jinete Solitario**_ (The Lone Rider in the Valley of the Desperadoes: The Vengeance of the Lone Rider) and also called _**El Valle de los Desaparecidos**_ (The Valley of the Desperadoes) and _**Zorro nella Valle dei Fantsasmi**_ (Zorro and the Valley of the Phantoms) before being reissued in 1964 as _**Vendetta de Zorro**_ (Vendetta of Zorro).\n\n**2406** _ **Lone Rider**_ **** Larry Levinson Productions, 2008. 80 min. Color. D: David S. Cass, Sr. SC: Frank Sharp. With Lou Diamond Phillips, Stacy Keach, Vincent Spano, Marta DuBois, Terry Maratos, Cynthia Preston, Angela Alvarado Rosa, Mike Starr, Robert Baker, Timothy Bottoms, Ann Walker, Tom Schanley, Daniel Trainer, Wendy Riordan, Maria Jordan. Returning home, a war hero finds a town boss is trying to take over his family's business and fights back. So-so TV Western feature.\n\n**2407** _ **The Lone Rider Ambushed**_ **** Producers Releasing Corporation, 1941. 63 min. D: Sam Newfield. SC: Oliver Drake. With George Houston, Al St. John, Maxine Leslie, Frank Hagney, Jack Ingram, Hal Price, Ted Adams, George Chesebro, Ralph Peters, Steve Clark, Charles King, Carl Mathews, Tex Palmer, Lew Porter, Ray Henderson, Wally West, Lyndon Brent, Curley Dresden, Ray Jones, Augie Gomez, Barney Beasley. The Lone Rider is the double for a wanted outlaw so he pretends to be the bad man to prove the innocence of a bank teller accused of robbing his employer. Entertaining entry in \"The Lone Rider\" series.\n\n**2408** _ **The Lone Rider and the Bandit**_ **** Producers Releasing Corporation, 1942. 55 min. D: Sam Newfield. SC: Steve Braxton. With George Houston, Al St. John, Smoky (Dennis) Moore, Vicki Lester, Glenn Strange, Jack Ingram, Milton Kibbee, Kenne Duncan, Eddie Dean, Slim Whitaker, Hal Price, Slim Andrews, Carl Sepulveda, Curley Dresden, Frank Ellis, Jack Kirk, Merrill McCormick, Oscar Gahan, Wally West, Milburn Morante, Tex Phelps, George Morrell, Pascale Perry, Steve Clark, Rube Dalroy, Augie Gomez, Jack Kinney. Pretending to be an entertainer, the Lone Rider arrives in a town to help the sheriff in catching crooks forcing local miners to sell their claims. Okay \"Lone Rider\" outing with George Houston singing four songs composed by Johnny Lange and Lew Porter.\n\n_**The Lone Rider and the Outlaws of Boulder Pass**_ see _**Outlaws of Boulder Pass**_\n\n**2409** _ **The Lone Rider Crosses the Rio**_ **** Producers Releasing Corporation, 1941. 63 min. D: Sam Newfield. SC: William Lively. With George Houston, Al St. John, Raquell Verrin, Charles King, Alden Chase, Julian Rivero, Thornton Edwards, Howard Masters, Frank Ellis, Philip [Felipe] Turich, Jay Wilsey (Buffalo Bill, Jr.), Frank Hagney, Curley Dresden, Steve Clark, Lane Bradford, Joe Dominguez, Carl Mathews, George Morrell, Wally West, James Sheridan (Sherry Tansey), Art Dillard, Ray Henderson. In Mexico the Lone Rider attempts to untangle a romantic problem between two families and ends up solving a kidnapping. Pleasant, somewhat different \"Lone Rider\" feature. Also known as _**Across the Border**_.\n\n**2410** _ **The Lone Rider Fights Back**_ **** Producers Releasing Corporation, 1941. 64 min. D: Sam Newfield. SC: Joseph O'Donnell. With George Houston, Al St. John, Dorothy Short, Dennis Moore, Frank Hagney, Charles King, Frank Ellis, Hal Price, Jack O'Shea, Merrill McCormick, Pascale Perry, Walter James, Horace B. Carpenter, Milburn Morante, George Morrell, Wally West, Art Mix. When his pal is murdered over a mine, the Lone Rider joins an outlaw gang to get the goods on the killer. Average \"Lone Rider\" affair.\n\n_**The Lone Rider in Border Roundup**_ see _**Border Roundup**_\n\n**2411** _ **The Lone Rider in Cheyenne**_ **** Producers Releasing Corporation, 1942. 59 min. D: Sam Newfield. SC: Elizabeth Beecher. With George Houston, Al St. John, Smoky (Dennis) Moore, Ella Neal, Roy Barcroft, Kenne Duncan, Lynton Brent, Milton Kibbee, Karl Hackett, Jack Ingram, George Chesebro, Jack Holmes, Curley Dresden, Lew Porter, Jack Kirk, Ray Henderson, Richard Cramer, Wally West, Al Taylor, Ed Peil, Sr., Milburn Morante, Tex Palmer, Hank Bell, Pascale Perry, Jack Evans, Augie Gomez. An innocent man is accused of murdering an outlaw who took part in a robbery and the Lone Rider tries to find the real killer. Like most in the \"Lone Rider\" series this one is cheaply made, short on plot but appealing due to star George Houston's singing and Al St. John's antics.\n\n**2412** _ **The Lone Rider in Frontier Fury**_ **** Producers Releasing Corporation, 1941. 62 min. D: Sam Newfield. SC: Fred Myton. With George Houston, Al St. John, Hillary Brooke, Karl Hackett, Ted Adams, Archie Hall, Budd Buster, Virginia Card, Ed Peil, Sr., John Elliott, Tom London, Frank Ellis, Reed Howes, Dan White, Horace B. Carpenter, Tex Cooper, Tex Palmer, Curley Dresden, Wally West, Herman Hack, Milburn Morante, Augie Gomez. The Lone Rider is falsely convicted of murdering a rancher whose niece begins to suspect he is innocent. Fair \"Lone Rider\" episode enhanced by a good cast.\n\n**2413** _ **The Lone Rider in Ghost Town**_ **** Producers Releasing Corporation, 1941. 64 min. D: Sam Newfield. SC: Joseph O'Donnell. With George Houston, Al St. John, Elaine Brandes, Budd Buster, Frank Hagney, Alden Chase, Reed Howes, Charles King, George Chesebro, Ed Peil, Sr., Archie Hall, Karl Hackett, Jay Wilsey (Buffalo Bill, Jr.), Curley Dresden, Frank Ellis, Steve Clark, Jack Ingram, Lane Bradford, Byron Vance, Don Forrest, Wally West, Herman Hack, Chick Hannon, Augie Gomez, Dan White. To keep a man from exercising an option on a mine, crooks kidnap his daughter and the Lone Rider tries to find her. A good story and a big cast of genre veterans make this \"Lone Rider\" entry a bit better than usual. Reissued on 16mm as _**Ghost Town**_.\n\n_**The Lone Rider in Law of the Saddle**_ see _**Law of the Saddle**_\n\n**2414** _ **The Lone Rider in Texas Justice**_ **** Producers Releasing Corporation, 1942. 60 min. D: Sam Newfield. SC: Steve Braxton (Sam Robins). With George Houston, Al St. John, Dennis Moore, Wanda McKay, Claire Rochelle, Karl Hackett, Curley Dresden, Steve Clark, Ray Davis, Archie Hall, Slim Whitaker, Ed Peil, Sr., Julian Rivero, Dirk Thane, Horace B. Carpenter, Frank Ellis, Merrill McCormick, Art Dillard, Jack Montgomery. The Lone Rider and his pal Fuzzy buy a ranch only to learn the former owner has been framed for rustling cattle. Above average series Western that moves fast and is fairly exciting. Also called _**Texas Justice**_.\n\n**2415** _ **The Lone Rider Rides On**_ **** Producers Releasing Corporation, 1941. 61 min. D: Sam Newfield. SC: Joseph O'Donnell. With George Houston, Al St. John, Hillary Brooke, Lee Powell, Buddy Roosevelt, Alan Bridge, Frank Hagney, Tom London, Karl Hackett, Forrest Taylor, Frank Ellis, Curley Dresden, Harry Harvey, Jr., Isabel La Mal, Don Forrest, Robert Kortman, Wally West, Steve Clark, Bobby Winkler, Richard Cramer, Wally West, Jay Wilsey (Buffalo Bill, Jr), Lew Meehan, Augie Gomez, George Morrell, Herman Hack, Ray Henderson. The Lone Rider investigates the murder of a man planning to take possession of land he had purchased and notices the killing was similar to that of his parents years before. The first in the popular \"Lone Rider\" series and a good one, enhanced by the ingratiating acting style of George Houston and his powerful singing voice. Reissued on 16mm as _**Rider of the Plains**_.\n\n**2416** _ **Lone Star**_ **** Metro-Goldwyn-Mayer, 1952. 90 min. D: Vincent Sherman. SC: Borden Chase. With Clark Gable, Ava Gardner, Broderick Crawford, Lionel Barrymore, Beulah Bondi, William Farnum, Ed Begley, James Burke, Lowell Gilmore, Moroni Olsen, Russell Simpson, William Conrad, Rex Bell, Lucius Cook, Ralph Reed, Ric Roman, Victor Sutherland, Jonathan Cott, Charles Cane, Nacho Galindo, Trevor Bardette, Harry Woods, Dudley Sadler, Emmett Lynn, Rex Lease, Stanley Andrews, Julian Rivero, Earle Hodgins, Chief Yowlachie, Roy Gordon, George Hamilton, Mitchell Lewis, Tony Roux, Harry Tenbrook, Davison Clark, Harry Wilson, Charles Sherlock, Warren MacGregor. President Andrew Jackson asks a trusted friend to persuade Sam Houston to lead the fight for Texas independence from Mexico and in doing so Sam confronts a bad man and meets a pretty newspaper editor. Good, big budget feature sure to appeal to action fans. Available in a colorized version.\n\n**2417** _ **Lone Star**_ **** Columbia TriStar, 1996. 135 min. Color. D-SC: John Sayles. With Kris Kristofferson, Chris Cooper, Elizabeth Pena, Clifton James, Matthew McConaughey, Stephen J. Lang, Tony Plana, Frances McDormand, Stephen Mendillo, Oni Faida Lampley, Eleese Lester, Joe Stevens, Gonzalo Castillo, Richard Coca, Tony Frank, Miriam Colon, Jeff Monahan, Joe Morton, LaTanya Richardson, Eddie Robinson, Ron Canada, Chandra Wilson, Damon Guy, Dee Macaluso, Luis Cobo, Marco Perella, Don Phillips, Jesse Borrego, Carina Martinez, Richard A. Jones, Beatrice Winde, Gabriel Casseus, Randy Stripling, Richard Reyes, Leo Burmester, Carmen de Lavallade, Tony Amendola, Gordon Tootosis, Lisa Suarez, Jesus Ramirez, Eduardo Martinez. While wanting to get back together with a former sweetheart, a Texas sheriff investigates the discovery of a skeleton buried for forty years in the desert. Overlong but well modulated modern-day Western.\n\n**2418** _ **Lone Star Law Men**_ **** Monogram, 1941. 58 min. D: Robert Emmett Tansey. SC: Robert Emmett (Tansey) and Frances Kavanaugh. With Tom Keene, Betty Miles, Frank Yaconelli, Sugar Dawn, Glenn Strange, Charles King, Gene Alsace, James Sheridan (Sherry Tansey), Stanley Price, Fred Hoose, Franklyn Farnum, Jack Ingram, Reed Howes, Dan White, Chick Hannon, Tex Palmer, Jack Roper. A cowboy and his pal find a peace officer who has been shot and left for dead and the cowpoke agrees to help the lawman capture his attackers by pretending to be an outlaw and joining the gang. Passable Tom Keene vehicle.\n\n_**Lone Star Lawman**_ see _**Texas Lawman**_\n\n**2419** _ **Lone Star Moonlight**_ **** Columbia, 1946. 67 min. D: Ray Nazarro. SC: Louise Rousseau. With Ken Curtis, Joan Barton, Guy Kibbee, Robert Stevens, Claudia Drake, Arthur Loft, Vernon Dent, Sam Flint, The Hoosier Hotshots (Charles \"Gabe\" Ward, Paul \"Hezzie\" Trietsch, Ken Trietsch, Gil Taylor), Merle Travis Trio, Judy Clark and Her Rhythm Cowgirls, The Smart Set, Emmett Lynn, Hank Penny, Fred F. Sears, Matty Roubert, Jerry Jarrett, Cy Malis, Vic Holbrook, Ralph Bucko, Roy Bucko. A returning G.I. learns his father has let their radio station slip in quality and a rival is after his girlfriend. Fair modern-day Western musical.\n\n**2420** _ **Lone Star Pioneers**_ **** Columbia, 1939. 56 min. D: Joseph Lovering. SC: Nate Gatzert. With Bill Elliott, Dorothy Gulliver, Lee Shumway, Slim Whitaker, Charles King, Jack Ingram, Harry Harvey, Buzz Barton, Frank LaRue, Budd Buster, David Sharpe, Frank Ellis, Kit Guard, Merrill McCormick, Jack Rockwell, Tex Palmer, Art Davis, Marin Sais, Jack C. Smith, George Plues. In Texas after the Civil War, a federal marshal disguises himself as an outlaw to infiltrate a guerrilla gang raiding supply wagons. Okay Bill Elliott vehicle that tends to be a bit slow; Jack Ingram is a good guy for a change.\n\n**2421** _ **Lone Star Raiders**_ **** Republic, 1940. 57 min. D: George Sherman. SC: Joseph March and Barry Shipman. With Robert Livingston, Bob Steele, Rufe Davis, June Johnson, George Douglas, Sarah Padden, John Elliott, John Merton, Rex Lease, Bud Osborne, Jack Kirk, Tom London, Hal Price, Tommy Coats, John Beach, Bob Card, Matty Roubert, Chick Hannon, Harrison Greene, Al Haskell, Reed Howes, Herman Hack, Bert Dillard, Cactus Mack, Art Dillard, Bud McClure, George Sowards, Foxy Callahan, Augie Gomez, Herman Newlin, Duke Taylor, Roy Bucko, Bill Wolfe. The Three Mesquiteers round up wild horses for the government and find themselves at odds with outlaws rustling the herds. Mediocre entry in the long running series with too much stock footage.\n\n**2422** _ **The Lone Star Ranger**_ **** Fox, 1930. 70 min. D: A.F. Erickson. SC: Seton I. Miller and John Hunter Booth. With George O'Brien, Sue Carol, Walter McGrail, Warren Hymer, Russell Simpson, Roy Stewart, Lee Shumway, Colin Chase, Richard Alexander, Joel Franz, Joe Rickson, Oliver Eckhardt, Caroline Rankin, Elizabeth Patterson, Billy Butts, Delmar Watson, William Steele, Bob Fleming, Ralph Le Fevre, Joe Chase, Ward Bond, Jane Keckley, Jack Perrin, Hank Bell. Falsely accused of crimes, a man is given the chance to redeem himself by capturing an outlaw gang. Strong George O'Brien vehicle from the 1915 Zane Grey novel, first filmed by Fox in 1919 starring William Farnum and again in 1923 with Tom Mix and Billie Dove.\n\n**2423** _ **The Lone Star Ranger**_ **** 20th Century\u2013Fox, 1942. 58 min. D: James Tinling. SC: William Counselman, Jr., George Kane and Irving Cummings, Jr. With John Kimbrough, Sheila Ryan, William Farnum, Truman Bradley, Jonathan Hale, George E. Stone, Russell Simpson, Dorothy Burgess, Tom Fadden, Fred Kohler, Jr., Eddy Waller, Harry Holden, George Melford, Tom London, Syd Saylor, Almira Sessions, Eva Puig, Jeff Corey, Robert Homans, Herbert Ashley, Alec Craig. A man quits being an outlaw, marries a pretty girl and is given a chance to get a pardon by bringing in his old gang. Futile attempt to make a genre star of John Kimbrough in this tepid fourth filming of the Zane Grey work.\n\n**2424** _ **The Lone Star Trail**_ **** Universal, 1943. 58 min. D: Ray Taylor. SC: Oliver Drake. With Johnny Mack Brown, Tex Ritter, Fuzzy Knight, Jennifer Holt, Earle Hodgins, Jack Ingram, Robert Mitchum, George Eldredge, Michael Vallon, Ethan Laidlaw, Harry Strang, The Jimmy Wakely Trio (Jimmy Wakely, Johnny Bond, Dick Reinhart), William Desmond, Art Mix, Henry Roquemore, Denver Dixon, Billy Engle, Carl Mathews, Bob Reeves, Eddie Parker, Fred Graham, Tom Steele. A rancher, falsely sentenced to jail for bank robbery, is paroled and with the help of a marshal tries to find out who framed him. The last of the Johnny Mack Brown\u2013Tex Ritter series, this one is fast from start to finish; one of the first films to bring notice to Robert Mitchum, cast as a saloon owner.\n\n**2425** _ **The Lone Star Vigilantes**_ **** Columbia, 1942. 58 min. D: Wallace Fox. SC: Luci Ward. With Bill Elliott, Tex Ritter, Virginia Carpenter, Frank Mitchell, Luana Walters, Budd Buster, Forrest Taylor, Gavin Gordon, Lowell Drew, Edmund Cobb, Ethan Laidlaw, Rick Anderson, Eddie Laughton, John Tyrrell, Buel Bryant, Francis Walker, Steve Clark, Charles Hamilton, Dick Botiller, Paul McVey. After the Civil War two men return to Texas to find their town under the thumb of bandits masquerading as Army troops. Not an overly distinguished Bill Elliott-Tex Ritter teaming but Tex does sing the Johnny Marvin songs \"Headin' Home to Texas\" and \"When the Moon Is Shining on the Old Corral.\" Reissued by Astor in 1950. British title: _**The Devil's Price**_.\n\n**2426** _ **The Lone Texan**_ **** 20th Century\u2013Fox, 1959. 70 min. D: Paul Landres. SC: James Landis and Jack Thomas. With Willard Parker, Grant Williams, Audrey Dalton, Douglas Kennedy, June Blair, Dabbs Greer, Barbara Heller, Rayford Barnes, Tyler McVey, Lee Farr, Jimmy Murphy, Dick Monahan, Robert Dix, I. Stanford Jolley, Gregg Barton, Sid Melton, Hank Patterson, Tom London, Frank Marlowe, Boyd Stockman, Jerry Summers, Bill Coontz, Shirley Haven. Following the war between the states, a Union cavalry officer goes back to Texas to find he is labeled a traitor with his brother a corrupt sheriff. Rather interesting drama inhibited by a low budget.\n\n**2427** _ **Lone Texas Ranger**_ **** Republic, 1945. 56 min. D: Spencer Gordon Bennet. SC: Bob Williams. With Bill Elliott, Bobby Blake, Alice Fleming, Roy Barcroft, Helen Talbot, Jack McClendon, Rex Lease, Tom Chatteron, Jack Kirk, Nelson McDowell, Dale Van Sickel, Frank O'Connor, Robert Wilke, Bud Geary, Budd Buster, Hal Price, Horace B. Carpenter, Nolan Leary, Tom Steele, Larry Olsen, Frederick Howard, Earl Dobbins, Bill Stevens, LeRoy Mason (voice). After killing a highly respected lawman, who was actually an outlaw leader, Red Ryder faces a challenge from the man's son. A strong script highlights this action packed \"Red Ryder\" episode.\n\n**2428** _ **The Lone Trail**_ **** Syndicate, 1932. 61 min. D: Forrest Sheldon and Harry S. Webb. SC: Bennett Cohen and Elizabeth (Betty) Burbridge. With Rex Lease, Virginia Brown Faire, Joe Bonomo, Billy O'Brien, Jack Mower, Robert Walker, Harry Todd, Josephine Hill, Edmund Cobb, Muro (dog). After his sister is murdered by an outlaw, a ranger learns the killer has kidnapped a young woman and plans to force her to marry him. Low grade feature supposedly edited from the 1931 serial _**Sign of the Wolf**_ (q.v.) but with a different plot emphasis that was remade as _**Skull and Crown**_ (q.v.).\n\n**2429** _ **Lone Wolf McQuade**_ **** Orion, 1983. 107 min. Color. D: Steve Carver. SC: B.J. Nelson. With Chuck Norris, David Carradine, Barbara Carrera, Leon Isaac Kennedy, Robert Beltran, L.Q. Jones, Dana Kimmell, R.G. Armstrong, Jorge Cervera, Jr., Sharon Farrell, Daniel Frishman, William Sanderson, John Anderson, Robert Jordan, Oscar Hidalgo, Anthony E. Caglia, Tommy Ballard, Gary Pike, William J. Wagner, Hector Serrano, Joe Kaufenberg, Susan Kaufenberg, Don Pike, Jeffrey Bannister. When his partner is killed in a gun fight with dope smugglers, a Texas Ranger uses martial arts to combat a drug kingpin. Colorful action filled modern-day Western.\n\n**2430** _ **Lonely Are the Brave**_ **** Universal, 1962. D: David Miller. SC: Dalton Trumbo. With Kirk Douglas, Gena Rowlands, Walter Matthau, Michael Kane, Carroll O'Connor, William Schallert, Karl Swenson, George Kennedy, Dan Sheridan, Bill Raish, William Mims, Martin Garralaga, Lalo Rios. A free spirited cowboy escapes from jail and is hunted by a posse using modern technology. This yesterday versus today's values oater is a good one, with excellent work by Kirk Douglas as the escapee.\n\n**2431** _ **The Lonely Man**_ **** Paramount, 1957. 87 min. D: Henry Levin. SC: Harry Essex. With Jack Palance, Anthony Perkins, Neville Brand, Robert Middleton, Elaine Aiken, Elisha Cook, Jr., Claude Akins, Lee Van Cleef, Harry Shannon, James Bell, Adam Williams, Denver Pyle, John Doucette, Paul Newlan. A gunman wants to lead a lawful life but is forced into one last showdown. Fairly interesting Western helped by good production values and acting.\n\n**2432** _ **The Lonely Trail**_ **** Republic, 1936. 56 min. D: Joseph Kane. SC: Bernard McConville and Jack Natteford. With John Wayne, Ann Rutherford, Cy Kendall, Robert Kortman, Fred \"Snowflake\" Toones, Etta McDaniel, Sam Flint, Denny Meadows (Dennis Moore), Jim Toney, Yakima Canutt, Lloyd Ingraham, Bob Burns, James A. Marcus, Rodney Hildebrand, Eugene Jackson, Jack Kirk, Jack Ingram, Bud Pope, Tex Phelps, Tracy Layne, Floyd Shackelford, Charles King, Horace B. Carpenter, Henry Hall, Oscar Gahan, Lafe McKee, Francis Walker, Clifton Young, Bob Card, Clyde Kenney, Leon Lord, Nina Mae McKinney. At the close of the Civil War, the governor of Texas asks a rancher, who fought for the Union, to help rid the state of carpetbaggers. Very good John Wayne \"B\" film with fine work by Cy Kendall as the bad guy.\n\n**2433** _ **Lonesome Dove**_ **** CBS-TV, 1989. 384 min. Color. D: Simon Wincer. SC: William D. Wittliff and Larry McMurtry. With Robert Duvall, Tommy Lee Jones, Diane Lane, Danny Glover, Robert Urich, Frederic Forrest, D.B. Sweeney, Rick Schroder, Angelica Huston, Chris Cooper, Timothy Scott, Glenne Headley, Barry Corbin, William Sanderson, Barry Tubb, Gavan O'Herlihy, Steve Buscemi, Frederick Coffin, Travis Swords, Kevin O'Morrison, Ron Weyand, Lanny Flaherty, David Carpenter, James McMurtry, Charlie Haynie, Terry McIlvain, Sonny Carl Davis, Jorge Martinez de Hoyos, Leon Singer, Thomas Connor, Jerry Biggs, Missy Crider, Sean Hennigan, Lauren Stanley, Julie Tennon, Jack Caffrey, Adam Faraizi, James Pickens, Jr., Bradley Gregg, Nina Siemaszko, Joel Palmer, Howard Young, Elberta Hunter, Matthew Hotspinpiller. Two retired Texas Rangers head a cattle drive to Montana with sometimes tragic results. Top notch TV miniseries based on Larry McMurtry's sprawling, ingratiating novel; followed by the small screen series of the same title that ran during the 1994\u201395 season for 21 episodes and \"Lonesome Dove: The Outlaw Years,\" a 22 episode series telecast during the 1995\u201396 season.\n\n**2434** _ **The Lonesome Trail**_ **** Syndicate, 1930. 60 min. D: Bruce Mitchell. SC: G.A. Durlam. With Charles Delaney, Virginia Brown Faire, George Berlinger, Willliam von Bricken, George Hackathorne, George Regas, Yakima Canutt, Art Mix, Ben Corbett, Lafe McKee, Jimmy Aubrey, Monte Montague, Bob Reeves, William McCall. A cowboy brings a cattle herd to a buyer who is in cahoots with a bandit and his gang. Stilted early talkie with a bunch of wheezy vaudeville gangs and hero Charles Delaney singing \"Oh, Susannah.\"\n\n**2435** _ **The Lonesome Trail**_ **** Monogram, 1945. 57 min. D: Oliver Drake. SC: Louise Rousseau. With Jimmy Wakely, Lee \"Lasses\" White, John James, Iris Clive, Horace Murphy, Lorraine Miller, Eddie Majors, The Saddle Pals, The Sunshine Girls, Colleen Summers (Mary Ford), Zon Murray, Roy Butler, Frank McCarroll, Jack Clifford, Arthur Smith, Carl Mathews, Carl Sepulveda, Jack Rivers. Much to the chagrin of his two pals, one of the three owners of a ghost town sells an interest in it to crooks who immediately start a false gold rush rumor. The music is good but the plot is weak in this Jimmy Wakely outing.\n\n**2436** _ **The Lonesome Trail**_ **** Lippert, 1955. 73 min. D: Richard Bartlett. SC: Richard Bartlett and Ian MacDonald. With Wayne Morris, John Agar, Margia Dean, Edgar Buchanan, Adele Jergens, Earle Lyon, Ian MacDonald, Douglas Fowley, Richard Bartlett, Betty Blythe. When land grabbers try to steal his place, a man retaliates with a bow and arrows instead of a six gun. Rather interesting low budget affair although top billed Wayne Morris has only a small role as a bartender.\n\n**2437** _ **The Long Chase**_ **** Universal, 1972. 89 min. Color. D: Alexander Singer. SC: John Thomas James (Roy Huggins) and Dick Nelson. With Roger Davis, Ben Murphy, Rod Cameron, Buddy Ebsen, James Drury, Frank Sinatra, Jr., Marie Windsor, J.D. Cannon, Larry Storch, Mills Watson, George Keymas, Walt Davis, Dave Garroway, Laurie Ferrone, Stephen Hardis, Monty Laird, Clarke Gordon, Jon Lormer, Renee Tetro, Tom Waters, Ralph Story (narrator). Two ex-outlaws try to go straight but find themselves at odds with a grizzled, villainous bounty hunter. Not bad for a paste-up of two episodes of \"Alias Smith and Jones\" (CBS-TV, 1970\u201373).\n\n**2438** _ **The Long Days of Vengeance**_ **** Mercurio Films, 1967. 106 min. Color. D: Stan Vance (Florestano Vancini). SC: Fernando Di Leo and Augusto Caminito. With Giuliano Gemma, Francisco Rabal, Gabriella Giorgelli, Conrado San Martin, Franco Cobi D'Este, Nieves Navarro (Susan Scott), Pajarito (Manuel Muniz), Doro Coro, Milo Quesada, Ivan Scratuglia, Pedro Basauri \"Pedrucho,\" Carlos Hurado, Juan Antonio Rubio. After serving three years at hard labor for the murder of his father, a man seeks revenge on the trio who framed him. Entertaining Spaghetti Western, a French-Italian-Spanish co-production filmed as _**I Lunghi Giorni della Vendetta**_ (The Long Days of Revenge) and issued on video as _**Day of Vengeance**_.\n\n**2439** _ **The Long Kill**_ **** CBS-TV, 1999. 96 min. Color. D: Bill Corcoran. SC: Gene Quintano. With Willie Nelson, Kris Kristofferson, Travis Tritt, Waylon Jennings, Chad Willett, Sancho Garcia, Jonathan Banks, Simon Andreu, Jorge Bosso, Aldo Sambrell, Eduardo Hererra, May Heatherly, Vincent Ginarbi, Ignacio Duran, Francis Butler, Danny Sullivan, Leonor Watling, Pablo Scola, Tony Isbert, Bill Holden, Lane Kinsey, Manuel San Martin, Marina Saura. Four veteran gunslingers join forces to get revenge for the murder of an old comrade. Fans of the four country music stars will like them in this cowboy outing.\n\n_**Long Live Your Death**_ see _**Don't Turn the Other Cheek**_\n\n**2440** _ **The Long, Long Trail**_ **** Universal, 1929. 60 min. D: Arthur Rosson. SC: Howard Green. With Hoot Gibson, Sally Eilers, Kathryn McGuire, Jim Mason, Archie Ricks, Walter Brennan, Howard Truesdale. A fun loving cowboy falls for a pretty girl as he uncovers a plot to steal rodeo proceeds. Pleasant Hoot Gibson initial talkie, a remake of his 1923 Universal outing _**The Ramblin' Kid**_.\n\n_**The Long, Long Trail**_ (1942) see _**Texas to Bataan**_\n\n**2441** _ **A Long Ride from Hell**_ **** Cinerama, 1970. 94 min. Color. D: Alex Burkes (Luigi Bazzoni). SC: Steve Reeves. With Steve Reeves, Wayde Preston, Dick Palmer (Mimmo Palmara), Silvana Venturelli, Lee Burton, Ted Carter (Nello Pazzafini), Rosalba Neri, Franco Fantasia, Aldo Sambrell, Spartaco Conversi, Mario Maranzana. A rancher, wrongly sent to prison for a robbery he did not commit, escapes to clear his name and get revenge for the murder of his family. Okay European oater for Steve Reeves fan, adapted by the star from the novel _Judas Gun_ by Gordon Shirreffs; issued in Italy in 1968 by B.R.C. as _**Vivo per la Tua Morte**_ (I Live for Your Death).\n\n_**The Long Ride Home**_ see _**A Time for Killing**_ (1967)\n\n**2442** _ **The Long Ride Home**_ **** Lions Gate Films, 2003. 87 min. D: Rob Marcarelli. SC: Vaughn Taylor. With Randy Travis, Eric Roberts, Ernest Borgnine, Vaughn Taylor, Paul Tinder, Alec Medlock, Steve Nave, Jeff McGrail, Garry Marshall, Michele Calcin, Stella Stevens, Rance Howard, Jerry Doyle, Jeff Dolan, Sam Dolan, Peter Sherayko, Sal Cardile, Larry A. Zeug, Dan Erwin, Elizabeth Tinder, Greg Stanina. In the late 1860s, a cowboy must find a killer before a relentless posse hangs him for the crime. Pretty good Western, sure to appeal to Randy Travis followers.\n\n**2443** _ **The Long Riders**_ **** United Artists, 1980. 100 min. Color. D: Walter Hill. SC: Bill Bryden, Steven Philip Smith, Stacy Keach and James Keach. With David Carradine, Keith Carradine, Robert Carradine, James Keach, Stacy Keach, Dennis Quaid, Randy Quaid, Kevin Brophy, Harry Carey, Jr., Christopher Guest, Nicholas Guest, Shelby Leverington, Felice Orlandi, Pamela Reed, James Remar, Fran Ryan, Savannah Smith, Amy Stryker, James Whitmore, Jr., John Bottoms, Wes Buchanan, Lin Shayne, Stuart Mossman, Prentiss Rowe, Allan Graf, William Taylor, Chris Mulicoy, Thomas R. Myers, Hugh McGraw, Tom Sauber, Kalen Keach. A trio of outlaw families form the James-Younger-Miller gang and carry out a series of daring robberies culminating in the ill-fated Northfield, Minnesota, raid. Fair look at the famous bandits' lives, interesting for the casting of acting brothers in the main roles.\n\n**2444** _ **The Long Rifle and the Tomahawk**_ **** International Television Corporation (ITC), 1964. 89 min. D: Sam Newfield and Sidney Salkow. SC: Richard Shayler and Charles Marion. With John Hart, Lon Chaney, John Vernon, Pegeen Ross, Ed Holmes, Michael Ansara, Stacy Harris, Casey Adams (Max Showalter), Frank De Kova, Daryl Masters. Hawkeye and his blood brother Chingachgook help English settlers in upper New York state during colonial times, including saving a British fort from the French and their Indian allies. Pretty good telefilm made up of three episodes of \"Hawkeye and the Last of the Mohicans\" (Syndicated, 1956), produced by Sigmund Neufeld and filmed in Canada.\n\n**2445** _ **The Long Rope**_ **** 20th Century\u2013Fox, 1961. 61 min. D: William Witney. SC: Robert Hamner. With Hugh Marlowe, Lisa Montell, Alan Hale (Jr.), Robert Wilke, John Alonzo, Madaleine Holmes, David Renard, Jeffrey Morris, Chris Robinson, Scott Randall, Jack Conlin, Kathryn Hart, Stephen Welles, Linda Cordova, Alex Cordellis. A circuit riding judge enlists the aid of a gunman to protect a man from townspeople during a murder trial. Compact, entertaining effort produced by Margia Dean.\n\n_**The Long Tomorrow**_ see _**Face to the Wind**_\n\n**2446** _ **Longarm**_ **** ABC-TV, 1988. 96 min. Color. D: Virgil Vogel. SC: Daniel Chisholm. With John T. Gerlesky, Whitne Kershow, John Laughlin, Malachi Throne, Deborah Dawn Slaboda, Lee de Broux, Daphne Ashbrook, Rene Auberjonois, John Quade, Shannon Tweed, John Dennis Johnson, Noble Willingham. In the late 1800s in New Mexico Territory a lady loving deputy U.S. marshal is forced to face the outlaws who raised him. Fair TV Western also called _**Showdown in Silver City**_.\n\n**2447** _ **The Longest Drive**_ **** Columbia Pictures International, 1976. 91 min. Color. D: Bernard McEveety. SC: Michael Michaelian and Katharyn Michaelian Powers. With Kurt Russell, Tim Matheson, Dan O'Herlihy, Keenan Wynn, Woody Strode, Erik Estrada, Sander Johnson, Cooper Huckabee, John Rubinstein, Gary Lockwood, Dick Davalos, Angela May, Meegan King, John Alvin, Duncan McLeod, Judith Hanson, Glenn Buttkus, Bill Smillie, Frank Salsedo, Peter Haas, Reid Rondell, Mary Angela, Jane Kellem. While searching for their lost sister, two brother try to help a friend, who is about to lose his ranch, by leading a cattle drive. Okay telefeature culled from the 13 episode series \"The Quest\" (NBC-TV, 1976); also called _**The Quest:**_ _**The Longest Drive**_.\n\n_**The Longest Hunt**_ see _**Gringo**_\n\n_**The Longest Spur**_ see _**Love Desperados**_\n\n**2448** _ **The Longhorn**_ **** Monogram, 1951. 70 min. D: Lewis D. Collins. SC: Dan Ullman. With Bill Elliott, Phyllis Coates, Myron Healey, John Hart, Marshall Reed, William Fawcett, Lee Roberts, Carol Henry, Zon Murray, Steve Clark, Lane Bradford, Herman Hack, Carl Mathews, I. Stanford Jolley, Marshall Bradford, Roy Bucko. A cowboy organizes a cattle drive so he can cross-breed stock but his co-called friend and gang plan to rustle the herd along the way. Very fine Bill Elliott vehicle, the initial film in his last Western series, with a good script, cast and fast action; recommended. Remade as _**Canyon River**_ (q.v.).\n\n**2449** _ **Look-Out Sister**_ **** Astor, 1948. 64 min. D: Bud Pollard. SC: John E. Gordon. With Louis Jordan, Suzette Harbin, Monte Hawley, Glenn Allen, Tommy Southern, Jack Glisby, Maceo Sheffield, Peggy Thomas, Louise Franklin, Bob Scott and Louis Jordan Tympany Six, Anice Clark, Dorothy Seamans, The Champion Cowboys. A bandleader on a dude ranch tries to save it from foreclosure. Music man Louis Jordan, plus eleven good tunes, add some zest to this low budget black cast oater.\n\n**2450** _ **Looped for Life**_ **** Madoc Sales, 1924. 50 min. D: Park B. Frame. SC: J. Anthony Roach. With Art Acord, Marcella Pershing, Jack Richardson, Charles Adler. Two cowboys both love the same girl with one of them committing a bank robbery and the other assigned by the sheriff to bring him in. One of the few complete extant Art Acord silent vehicles and thus worth viewing.\n\n**2451** _ **Lost Canyon**_ **** United Artists, 1943. 61 min. D: Lesley Selander. SC: Harry O. Hoyt. With William Boyd, Andy Clyde, Jay Kirby, Lola Lane, Douglas Fowley, Herbert Rawlinson, Guy Usher, Karl Hackett, Hugh Prosser, Robert Kortman, The Sportsmen Quartette, Si Jenks, John Cason, Keith Richards, Herman Hack, Murdock MacQuarrie, George Morrell, Gertrude Astor, Henry Wills, Bill Nestell, Aleth Hansen, Spade Cooley, Cliff Parkinson, Merrill McCormick, Milburn Morante, Jack Evans, Dorothy Vernon, Charles Murphy, Frank Mills. Bar 20 wrangler Johnny Travers is accused of a bank robbery but Hopalong Cassidy investigates and discovers a lawyer is behind the theft. Average Cassidy entry with stock footage from the finale of _**Rustlers' Valley**_ (q.v.) showing Lee J. Cobb, Ted Adams and Al Ferguson.\n\n**2452** _ **Lost in Alaska**_ **** Universal-International, 1952. 76 min. D: Jean Yarborough. SC: Martin A. Ragaway and Leonard Stern. With Bud Abbott, Lou Costello, Tom Ewell, Mitzi Green, Bruce Cabot, Emory Parnell, Jack Ingram, Rex Lease, Joe Kirk, Minerva Urecal, Howard Negley, Maudie Prickett, Billy Wayne, Paul Newlan, Michael Ross, Iron Eyes Cody, Donald Kerr, Bobby Barber, William Gould, Sherry Moreland, Chuck Hamilton, Fred Aldrich, Brick Sullivan, Harry Tyler, Charles Sullivan, Mike Lally, Julia Montoya, Jean Hartelle, Bert LeBaron. Three men head for Alaska to find a fortune in buried gold but meet with opposition from the locals. Tired Abbott and Costello comedy; for their fans only.\n\n**2453** _ **Lost in Silver Canyon**_ **** Total Content, 2006. 72 min. Color. D: Don Ross. SC: Patricia Oviatt. With Joe Kimpel, Vanessa Baker, Vito Aversano, Wes Baker, Patricia Baker, David Mook, De Witt Jones. Two children get stranded in a ghost town while on vacation with their family. Low budget inspirational feature filmed in Arizona.\n\n**2454** _ **Lost in the Barrens**_ **** Atlantis Films, 1990. 93 min. Color. D: Michael J.F. Scott. SC: Keith Ross Leckie. With Nicholas Shields, Evan Adams, Lee J. Campbell, Graham Greene, Marianne Jones, Victor Cowie, Paul Grau, Harry Nelken, Ken Babb, Jeff Madden, Brian Richardson, Fred Robinson, Eric Robinson, Adam Beach, Louie Camerone, Bart (bear). Two young boys, one white and one Indian, become lost in the North Canadian wilderness and work to save themselves from the harsh elements and find their way home. Pretty fair family film based on Farley Mowat's novel.\n\n**2455** _ **Lost in the Barrens II:**_ _**The Curse of the Viking Grave**_ **** Atlantis Films, 1991. 120 min. Color. D: Michael J.F. Scott. SC: Malcolm MacRury. With Nicholas Shields, Evan Adams, Michelle St. John, Gordon Tootoosis, Cedric Smith, Jay Brazeau, Wayne Robson, Lee J. Campbell, Marianne Jones, Joe Mercredi, Victor Cowie, Michael Meeches, John Bluethner, Robert Enright, Karen Barker, Jennie Tootoo. Two teenage boys help an archaeologist trek to Canada's Manitoba province in search of a Viking grave containing a valuable crucifix. Over long but okay juvenile fare based on the novel _Curse of the Viking Grave_ by Farley Mowat. Also called _**Curse of the Viking Grave**_.\n\n_**Lost Island of Kioga**_ see _**Hawk of the Wilderness**_\n\n**2456** _ **Lost Ranch**_ **** Victory, 1937. 56 min. D: Sam Katzman. SC: Basil Dickey. With Tom Tyler, Jeanne Martel, Lafe McKee, Forrest Taylor, Harry Harvey, Jr., Marjorie Beebe, Howard Bryant, Theodore Lorch, Slim Whitaker, Roger Williams, Bud Pope. A young woman and a friend come West to find her missing father and his \"secret\" and are attacked by outlaws but saved by a cowboy. Economical but entertaining Tom Tyler vehicle with mostly outdoor action and few interiors.\n\n**2457** _ **The Lost Trail**_ **** Monogram, 1945. 57 min. D: Lambert Hillyer. SC: Jess Bowers (Adele Buffington). With Johnny Mack Brown, Raymond Hatton, Jennifer Holt, Riley Hill, Kenneth MacDonald, Milburn Morante, Steve Clark, Eddie Parker, Lynton Brent, John Ince, Frank LaRue, Frank McCarroll, Dick Dickinson, George Morrell, John Bridges, Cal Shrum and His Rhythm Rangers, Henry Vroom, Denver Dixon, Tex Cooper, Cactus Mack, Chick Hannon, Victor Cox, Ray Henderson, Carl Mathews, Jack Tornek, Ralph Bucko. Two lawmen help a young woman whose stage line is being attacked by outlaws. Pretty good entry in the \"Nevada Jack McKenzie\" series, a remake of the initial \"Rough Riders\" film _**Arizona Bound**_ (q.v.).\n\n_**Lost Treasure of the Aztecs**_ see _**Blood River**_\n\n_**Lost Treasure of the Incas**_ see _**Blood River**_\n\n_**Lost Women**_ see _**Mesa of Lost Women**_\n\n**2458** _ **The Lottery Bride**_ **** United Artists, 1930. 80 min. D: Paul S. Stein. SC: Horace Jackson and Howard Emmett Rogers. With Jeanette MacDonald, John Garrick, Joe E. Brown, ZaSu Pitts, Robert Chisholm, Joseph Macaulay, Harry Gribbon, Carroll Nye, Murdock MacQuarrie, Paul Hurst, Max Davidson, Stanley Fields, Frank Brownlee, Torben Meyer, Bobby Dunn, Robert Homans, Budd Fine, Clarence Geldert. In order to pay off her brother's gambling debts, a young woman agrees to become a lottery bride in the King's Bay settlement of northern Norway. Although not a Western in the strictest sense, this early talkie musical is built around a frontier settlement, albeit Norway and not the U.S.; the film creaks a bit with age but the pleasant Rudolf Friml score holds up well.\n\n_**Louis L'Amour's Shaughnessy**_ see _**Shaughnessy**_\n\n_**Louisiana Gal**_ see _**Old Louisiana**_\n\n**2459** _ **Love Comes Softly**_ **** Hallmark Channel, 2003. 88 min. Color. D: Michael Landon, Jr. SC: Cindy Kelley and Michael Landon, Jr. With Katherine Heigi, Dale Midkiff, Skye McCole Bartusiak, Corbin Bersen, Theresa Russell, Oliver Macready, Tiffany Knight, Adam Loeffler, Nick Scoggins, Jaimz Woolvett, Rutanya Alda, Janet Rotbaltt, Christina A. Wod, David Mine, Dani Goldman. After her husband dies suddenly while they are traveling West with a wagon train, a young woman ends up living with a widower and his little daughter. Pleasant frontier TV movie drama based on the book by Janette Oke; sequel: _**Love's Enduring Promise**_ (q.v.).\n**2460** _ **Love Desperados**_ **** Olympic International Films, 1968. 91 min. Color. D-SC: R.L. Frost. With James Arena, Virginia Gordon, Joseph Mascolo, Wes Bishop, Tom McFadden, John Alderman, Paul Frank, Paul Wilmoth, John Riazzi, Bill Martin, Rod Wilmoth, Angel Carter, Laura McLaughlin, Monique Heguy. In the early 1860s bandits attack women at a frontier ranch when their husbands are away and a worker kidnaps the boss' wife to avenge the rape of his sister. Tacky soft core Western also called _**Fiery Spur**_ , _**The Hot Spur**_ , _**The Longest Spur**_ and _**Naked Spur**_.\n\n**2461** _ **Love Finds a Home**_ **** Hallmark Channel, 2009. 88 min. Color. D: David S. Cass, Sr. SC: Donald Davenport. With Sarah Jones, Haylie Duff, Jordan Bridges, Patty Duke, Courtney Halverson, Michael Trevino, Jeffrey Muller, Dahlia Salem, Thomas Kopache, Chad W. Smathers, Daniel Beer, Jeff Clarke, Jennifer Wetzel, Matthew Florida, Michelle Josette, Time Winters, Eric Shakelford. A female doctor in a Missouri town is at odds with her best friend's mother-in-law who wants to use homeopathic treatment for the young woman's pregnancy. Dull TV movie from Janette Oke's novel.\n\n**2462** _ **Love Me Tender**_ **** 20th Century\u2013Fox, 1956. 89 min. D: Robert D. Webb. SC: Robert Buckner. With Richard Egan, Debra Paget, Elvis Presley, Robert Middleton, William Campbell, Neville Brand, Mildred Dunnock, Bruce Bennett, James Drury, Russ Conway, Ken Clark, Barry Coe, L.Q. Jones, Paul Burns, Jerry Sheldon, James Stone, Ed Mundy, Joe Di Reda, Bobby Rose, Tom Greenway, Jay Jostyn, Steve Darrell. Two brothers love the same girl and when one leaves home to fight for the South during the Civil War the younger one marries her, causing a conflict when the older sibling comes back. Average adaptation of Maurice Geraghty's novel; best known as Elvis Presley's film debut.\n\n**2463** _ **Love Takes Wing**_ **** Hallmark Channel, 2009. 88 min. Color. D: Lou Diamond Phillips. SC: Rachel Stuhler. With Lou Diamond Phillips, Haylie Duff, Ernin Cottrell, Cloris Leachman, Jordan Bridges, John Bishop, Sarah Jones, Kevin Scott Richardson, Bonnie Root, Annalise Basso, Patrick Duffy, Yvonne Boismier Phillips, Jonathon Forrester, Andy Scott Harris, Mary-Jessica Pitts, June Angela. A woman becomes a physician and plans to leave her small Western town for the big city. Another average television movie from a Janette Oke work.\n\n**2464** _ **Love's Abiding Joy**_ **** Hallmark Channel, 2006. 87 min. Color. D: Michael Landon, Jr. SC: Michael Landon, Jr., Douglas Lloyd McIntosh and Bridget Terry. With Erin Cottrell, Dale Midkiff, Logan Bartholomew, Frank McRae, W. Morgan Sheppard, Drew Tyler Bell, Brett Coker, Mae Whitman, John Laughlin, Kevin Gate, Brianna Brown, James Tupper, Stephen W. Bridgewater, Blake Gibbons, Madison Leile, Thomas Stanley. Newlyweds travel West in an attempt to make a new life for themselves. Standard Hallmark Channel movie of a Janette Oke book.\n\n**2465** _ **Love's Enduring Promise**_ **** Hallmark Channel, 2004. 88 min. Color. D: Michael Landon, Jr. SC: Cindy Kelley and Michael Landon, Jr. With January Jones, Logan Bartholomew, Dale Midkiff, K'Sun Ray, Logan Arens, Mackenzie Astin, Cliff De Young, Matthew Peters, Michael Bartel, Dominic Scott Kay, Blaine Pate, Cara DeLizia, Robert F. Lyons, Douglas Fisher, E.J. Callahan, Katia Coe, Gary Sievers. When a stranger saves his son's life, a farmer gives him a job as a hired hand and the man falls in love with his daughter. Pleasant family oriented sequel to _**Love Come Softly**_ (q.v.), from a Janette Oake novel.\n\n**2466** _ **Love's Long Journey**_ **** Hallmark Channel, 2005. 88 min. Color. D: Michael Landon, Jr. SC: Douglas Lloyd McIntosh, Michael Landon, Jr. and Cindy Kelley. With Erin Cottrell, Logan Bartholomew, W. Morgan Sheppard, James Tupper, Frank McRae, Johann Urb, John Savage, Jeff Kober, Richard Lee Jackson, Graham Phillips, Irene Bedard, Gil Birmingham, Colin McCabe, Stephen Bridgewater, Diane Louise Salinger, Robert Norswrothy, Willie Arnold Davis, Jerry Louie-McGee, Tucker Louie-McGee, Dale Midiff, Angus Malcolmson, Gary Sievers. A young couple moves to a ranch where the wife has to deal with a pregnancy and missing her family. Another appealing entry in the \"Love\" TV movie series based on Janette Oke's books.\n\n**2467** _ **Love's Unending Legacy**_ **** Hallmark Channel, 2007. 84 min. Color. D: Mark Griffith. SC: Pamela Wallace. With Erin Cottrell, Dale Midkiff, Victor Browne, Samantha Smith, Holliston Coleman, Brett Coker, Hank Stratton, Braeden Lemasters, Stephanie Nash, Dave Florek, Bret Loehr, Tanner Richie, Ned Schmidtke, Dale Waddington Horowitz, Ken Magee, Trevor Gordon, Tyler Gordon, Andre Alexsen, Jeremy Shade, Lyndon Smith, Araksi Willebrand, Eva-Maria Leonardou. A young widow and her daughter return to her parents' home where she takes a teaching job, adopts an orphan and finds romance with the town sheriff. Lesser entry in the Hallmark Channel's movie series adaptations of the works of Janette Oke.\n\n**2468** _ **Lovin' Molly**_ **** Columbia, 1974. 98 min. Color. D: Sidney Lumet. SC: Stephen Friedman. With Anthony Perkins, Beau Bridges, Blythe Danner, Edward Binns, Susan Sarandon, Conrad Fowkes, Claude Transverse, John Henry Faulk. Two men love the same woman over a four decade period in rural Texas, although she also has a husband. Fair drama based on Larry McMurtry's novel _Leaving Cheyenne_.\n\n**2469** _ **The Luck of Roaring Camp**_ **** Monogram, 1937. 59 min. D: Irvin W. Willat. SC: Harvey Gates. With Owen Davis, Jr., Joan Woodbury, Charles Brokaw, Forrest Taylor, Robert Kortman, Charles King, Byron Foulger, Bob McKenzie, John Wallace. The birth of a baby boy brings luck to the inhabitants of a gold rush mining town. Expanded version of Bret Harte's short story makes for average entertainment.\n\n**2470** _ **Lucky Boots**_ **** Beacon\/Equity, 1935. 59 min. D: Al Herman. SC: William L. Nolte. With Guinn \"Big Boy\" Williams, Marion Shilling, Frank Yaconelli, Wally Wales, Charles K. French, Tom London, Roger Williams, Gordon Griffith, Barney Beasley, Si Jenks, Richard Botiller, Julian Rivero, George Morrell, Buck Morgan. Two cowpokes become involved in a treasure hunt after a Mexican bandit leader is killed and his boots contain the clue to where he hid stolen loot. Pretty good poverty row oater; Guinn Williams sings \"Home on the Range.\" Original title: _**Gun Play**_.\n\n**2471** _ **Lucky Cisco Kid**_ **** 20th Century\u2013Fox, 1940. 68 min. D: H. Bruce Humberstone. SC: Robert Ellis and Helen Logan. With Cesar Romero, Mary Beth Hughes, Dana Andrews, Evelyn Venable, Chris-Pin Martin, Joseph Sawyer, Dick Rich, Johnny Sheffield, Francis Ford, William Royle, Otto Hoffman, Bob Hoffman, Boyd \"Red\" Morgan, Harry Strang, Gloria Roy, Lillian Yarbo, Adrian Morris, Jimmie Dundee, William Pagan, Lew Kelly, Milton Kibbee, Sarah Edwards, Frank Lackteen, James Flavin, Thornton Edwards, Henry Roquemore, Syd Saylor, Blackie Whiteford, Ethan Laidlaw, Frank Ellis, Spencer Charters. The Cisco Kid romances two lovely ladies while on the trail of crooks raiding local ranchers. Slow moving and overly romantic \"Cisco Kid\" adventure.\n\n_**Lucky Johnny**_ see _**Dead Aim**_\n\n**2472** _ **Lucky Larkin**_ **** Universal, 1930. 66 min. D: Harry Joe Brown. SC: Marion Jackson. With Ken Maynard, Nora Lane, Harry Todd, Charles Clary, Paul Hurst, James Farley, Jack Rockwell, Edgar \"Blue\" Washington, Jim Corey A cowboy agrees to ride in a big race so he can get the money needed to save a ranch for the father of the girl he loves. Okay Ken Maynard silent feature, also issued with sound and music effects.\n\n**2473** _ **Lucky Larrigan**_ **** Monogram, 1932. 58 min. D: J.P. McCarthy. SC: Wellyn Totman. With Rex Bell, Helen Foster, George Chesebro, John Elliott, Stanley Blystone, Julian Rivero, G.D. Wood (Gordon DeMain), Wilfred Lucas, Herman Hack, Perry Murdock, Tex Palmer, Arthur Thalasso, Julia Bejarano. After going West, a self-centered polo player ends up helping his father, as well as his girl's father, in fighting outlaws. Poor Rex Bell vehicle.\n\n_**Lucky Luke**_ (1971) see _**Lucky Luke: Daisy Town**_\n\n**2474** _ **Lucky Luke**_ **** Tobis Filmkunst, 1991. 92 min Color. D: Terence Hill. SC: Lori Hill. With Terence Hill, Nancy Morgan, Fritz Sperberg, Dominic Barto, Bo Gray, Jack Elam, Roger Miller, John Quade, Arsenio \"Sonny\" Trinidad, Mark Hardwick, Neil Summers, Ron Carey, Buff Douhitt, Sky Fabin, Marc Mouchet, Radha Delamarter, Robin Westphal, Deborah Mansy, Kenny Dickerson, Nicholas Anthony, Dave Thomas, Kee Bahe Elsisie, Frederick Lopez, Jose Rey Toledo, Carl Allrunner Vicenti, William P. Yazzie, Douglas Eckberg. A love sick cowboys falls for pretty Lotta Leggs and battles the Dalton brothers. Poor Italian-U.S. co-production based on the popular European comic by Goscinny.\n\n**2475** _ **Lucky Luke:**_ _**La Ballade des Dalton**_ (Lucky Luke: The Ballad of the Daltons) Dargaud Films, 1978. 82 min. Color. D: Rene Goscinny, Henri Gruel, Morris and Pierre Watrin. SC: Pierre Tschernia. With Daniel Ceccaldi, Roger Carel, Jacques Balutin, Pierre Trabaud, Pierre Tronade, Gerard Hernandez, Rosy Varte, Rene Goscinny, Jacques Fabbri, Roger Lumont, Ada Lonati, Jean-Marc Thibault, Henri Virlojeux, Eric Kristy, Michel Elias, Bernard Heller, Jacques Legras, Jacques Morel, Henri Labssiere (voices). Luky Luke teams with the Dalton brothers in an attempt to get rid of the men who jailed them so they can inherit a fortune. Silly animated feature based on the Goscinny comics.\n\n**2476** _ **Lucky Luke:**_ _**Daisy Town**_ **** Dargaud Films, 1971. 76 min. Color. D: Rene Goscinny. SC: Morris, Goscinny and Pierre Tchernia. With Marcel Bozzuffi, Pierre Trabaud, Jacques Balutin, Jacaues Jouanneau, Pierre Tornade, Jean Berger, Roger Carel, Jacques Fabbri, Jacques Legras, Claude Dasset, Jacques Bodoin, Georges Atlas, Andre Legal, Jacques Hilling, Rosy Varte, Denise Bosc, Rich Little, Nicole Croisille, Gerard Dinal, Pat Woods (voices). Lucky Luke is hired to put a stop to lawlessness in a new settled Western town. French-Belgian animated feature for children from the Goscinny comic characters; also called _**Lucky Luke**_.\n\n**2477** _ **Lucky Luke 2**_ **** Paloma Films, 1991. 90 min. Color. D: Ted Nicolaou and Richard Schlesinger. SC: Carl Sautter. With Terence Hill, Nancy Morgan, Fritz Sperberg, Ron Carey, Bo Greigh. The further adventures of Lucky Luke, including his romance with Lotta Leggs and his continuing feud with the Dalton brothers. Video feature culled from the six episode 1993 \"Lucky Luke\" TV series.\n\n**2478** _ **Lucky Terror**_ **** First Division, 1936. 61 min. D-SC: Alan James. With Hoot Gibson, Lona Andre, Charles Hill, George Chesebro, Wally Wales, Bob McKenzie, Jack Rockwell, Frank Yaconelli, Charles King, Art Mix, Horace B. Carpenter, Horace Murphy, Hank Bell, Nelson McDowell, Milburn Morante, Bob Reeves, George Morrell. A drifter accidentally finds a cache of gold, hides it and joins a medicine show where he helps a young woman when crooks try to steal her mine. Highly entertaining Hoot Gibson feature, one of his best in the sound era.\n\n**2479** _ **The Lucky Texan**_ **** Monogram, 1934. 55 min. D-SC: Robert North Bradbury. With John Wayne, Barbara Sheldon, Lloyd Whitlock, Yakima Canutt, George Hayes, Gordon (DeMain) Demaine, Ed (Eddie) Parker, Earl Dwire, Jack Rockwell, Artie Ortego, Tex Palmer, Tex Phelps, George Morrell, Phil Dunham, Wally Wales, Philip Kieffer, Jack Evans, John Ince, Tommy Coats, Julie Kingdon. A cowboy plans to join his late father's partner in a mining venture but they have to fight a crooked assayer and his partner. Average entry in John Wayne's \"Lone Star\" series for producer Paul Malvern; colorized as _**Claim Jumpers**_ and _**Gold Strike River**_.\n\n**2480** _ **Lumberjack**_ **** United Artists, 1944. 65 min. D: Lesley Selander. SC: Norman Houston and Barry Shipman. With William Boyd, Andy Clyde, Jimmy Rogers, Ellen Hall, Douglass Dumbrille, Francis McDonald, Herbert Rawlinson, Ethel Wales, John Whitney, Hal Taliaferro, Henry Wills, Charles Morton, Frances Morris, Jack Rockwell, Bob Burns, Hank Worden, Earle Hodgins, Pierre Lyden, Bill Nestell. Hopalong Cassidy and his Bar 20 pals oppose a gang of outlaws in lumber country. Fast paced and action packed \"Hopalong Cassidy\" feature; one of the better films in the series' mid\u2013United Artists period.\n\n_**Lure of the Range**_ see _**Speeding Hoofs**_\n\n**2481** _ **Lure of the Wasteland**_ **** Al Lane Pictures, 1939. 55 min. Color. D: Harry Fraser. SC: Monroe Talbot. With Grant Withers, LeRoy Mason, Marion Arnold, Snub Pollard, Karl Hackett, Henry Roquemore, Tom London, Bob Terry, James Sheridan (Sherry Tansey), Budd Buster, Oscar Gahan, Norman Willis, George Morrell, Carl Mathews, Jack Evans. A federal agent works undercover to infiltrate an outlaw gang in an attempt to find out what happened to loot stolen in a robbery years before. Low grade independent outing, mainly of interest because it was filmed in Telco Color; Al Lane is a pseudonym for Robert Emmett Tansey. It contains the song \"Winds of the Wasteland\" by Glenn Strange.\n\n**Poster for** _**Lure of the Wasteland**_ **(Al Lane Pictures, 1939).**\n\n**  \n**\n\n**2482** _ **Lust for Gold**_ **** Columbia, 1949. 90 min. D: S. Sylvan Simon and (uncredited George Marshall). SC: Ted Sherdeman and Richard English. With Ida Lupino, Glenn Ford, Gig Young, William Prince, Edgar Buchanan, Will Geer, Paul Ford, Jay Silverheels, Antonio Moreno, Eddy Waller, Will Wright, Virginia Mullen, Myrna Dell, Tom Tyler, Paul E. Burns, Hayden Rorke, Elspeth Dudgeon, Si Jenks, Arthur Hunnicutt, Richard Alexander, Trevor Bardette, Edmund Cobb, George Chesebro, Hank Bell, Fred F. Sears, Matty Fain, Billy Gray, Virginia Farmer, Robert Malcolm, Baynes Barron, William Tannen, Anne O'Neal, Harry Strange, Rex Lease, Dabbs Greer, John Doucette, Percy Helton, Karolyn Grimes, Arthur Space, Dorothy Vernon, Howard Negley, Paul Bryar, Kermit Maynard, Guy Beach, Louis Mason, Maudie Prickett, George Morrell, Bill Woolf. A woman who is married to a no-good pretends to be single in order to win over a cowpoke who knows the location of the Lost Dutchman gold mine. Interesting juxtaposing of the past and present in this well made and acted grim tale of greed and murder; reworked as _**Secret of Treasure Mountain**_ (q.v.).\n\n**2483** _ **Lust in the Dust**_ **** Fox Run, 1984. 87 min. Color. D: Paul Bartel. SC: Philip Taylor. With Tab Hunter, Divine (Harris Glenn Milstead), Lainie Kazan, Geoffrey Lewis, Henry Silva, Cesar Romero, Woody Strode, Pedro Gonzalez-Gonzalez, Gina Gallego, Nedia Volz, Courtney Gains, Daniel Fushman, Erni Shinagawa. A woman attacked by an outlaw gang meets a mysterious gunman and they go to a town where the populace is after hidden treasure. Outlandish, but surprisingly funny, R-rated genre satire, filmed in New Mexico.\n\n**2484** _ **Lust to Kill**_ **** Barjul International\/Emerson, 1960. 69 min. D: Oliver Drake. SC: Sam Roeca and Tom Hubbard. With Jim Davis, Don Megowan, Allison Hayes, Gerald Milton, Toni Turner, Sandra Giles, Tom Hubbard, Claire Carleton, John Holland, James Maloney, Fred Sherman, Roger Williams, Al Terry, Gene Street. An outlaw's girlfriend helps him escape from a lawman so he can get revenge on the gang who killed his brother. Low grade but violent, and near adult, action feature made in 1957; TV title: _**Border Lust**_.\n\n**2485** _ **The Lusty Men**_ **** RKO Radio, 1952. 113 min. D: Nicholas Ray. SC: Horace McCoy and David Dortort. With Robert Mitchum, Susan Hayward, Arthur Kennedy, Arthur Hunnicutt, Frank Faylen, Walter Coy, Carol Nugent, Maria Hart, Lorna Thayer, Burt Mustin, Karen King, Jimmie Dodd, Eleanor Todd, Riley Hill, Robert Bray, Sheb Wooley, Marshall Reed, Paul E. Burns, Dennis Moore, George Wallace, Lane Bradford, Glenn Strange, George Sherwood, Lane Chandler, Ralph Volkie. A veteran rodeo star trains a younger performer while both vie for the love of a hell-raising woman. Neatly done tale of the rodeo circuit, especially well acted by its trio of stars.\n\n**2486** _ **Lynch Mob**_ **** 20th Century\u2013Fox, 1955. 45 min. D: Gerd Oswald. SC: David Dortort. With Cameron Mitchell, E.G. Marshall, Robert Wagner, Wallace Ford, Raymond Burr, Hope Emerson, Jay Brooks, Taylor Holmes, Walter Sande, Russell Simpson, Ray Teal, Robert Adler, Michael Ansara, James Westerfield, Willis Bouchey, Tyler MacDuff, Rodolfo Hoyos, Jr., Robert Griffin, Kermit Maynard, Robert Foulk, Nacho Galindo, Eddie Erwin, Eddie Firestone. When a cattleman is murdered three strangers are accused of the crime and threatened with hanging. Nice small screen adaptation of _**The Ox-Bow Incident**_ (q.v.), originally telecast November 2, 1955, on \"The 20th Century\u2013Fox Hour\" (CBS-TV, 1955\u201357); available to TV as a feature film and issued abroad theatrically.\n\n**2487** _ **The Macahans**_ **** ABC-TV\/Metro-Goldwyn-Mayer, 1976. 125 min. Color. D: Bernard McEveety. SC: Jim Byrnes. With James Arness, Eva Marie Saint, Richard Kiley, Bruce Boxleitner, Kathryn Holcomb, William Kirby Cullen, Vicki Schreck, Gene Evans, Vic Mohica, Frank Ferguson, Ann Doran, Ben Wilson, Mel Stevens, Rudy Diaz, John Crawford, William Conrad (narrator). On the eve of the Civil War, a Virginia farmer decides to move his family West and enlists the help of his brother, a mountain scout. This telefeature was a big ratings favorite as well as solid entertainment; James Arness is specially good as Zeb Macahan. It was followed by the mini-series \"How the West Was Won\" (ABC-TV, 1977\u201379).\n\n**James Arness in** _**The Macahans**_ **(ABC-TV, 1976).**\n\n**  \n**\n\n_**Machismo\u201440 Graves for 40 Guns**_ see _**40 Graves for 40 Guns**_\n\n**2488** _ **Macho Callahan**_ **** Avco-Embassy, 1970. 110 min. Color. D: Bernard Kowalski. SC: Cliff Gould. With David Janssen, Jean Seberg, Lee J. Cobb, James Booth, Pedro Armendariz, Jr., David Carradine, Anne Revere, Richard Anderson, Matt Clark, Richard Evans, Bo Hopkins, Diane Ladd, Cyril Delevanti, William Bryant, Bob Morgan, Bucklind Beery, Steve Raines, Ian Scott, David Carlile, Bill Catching, Mike Masters, Jim (James) Gammon, Curt Conway, Ron Soble, John McKee. During the Civil War, a man escapes from prison and schemes to get even with those who put him there. None-too-good oater helped by Mexican locales.\n\n**2489** _ **MacKenna's Gold**_ **** Columbia, 1969. 128 min. Color. D: J. Lee Thompson. SC: Carl Foreman. With Gregory Peck, Omar Sharif, Telly Savalas, Camilla Sparv, Keenan Wynn, Julie Newmar, Eli Wallach, Raymond Massey, Edward G. Robinson, Anthony Quayle, Burgess Meredith, Lee J. Cobb, Eduardo Ciannelli, Rudy Diaz, Ted Cassidy, Dick Peabody, Robert Phillips, J. Robert Porter, Pepe Callahan, Duke Hobbie, Trevor Bardette, Madeleine Taylor Holmes, John Garfield, Jr., Shelley Morrison, Victor Jory (narrator). A group of people try to find a lost canyon filled with gold but are not only at odds with themselves but also with Apaches and the cavalry. Surprisingly poor big budget Western hurt by pre-release cuts.\n\n**2490** _ **MacKintosh and T.J.**_ **** Penland, 1975. 96 min. Color. D: Marvin Chomsky. SC: Paul Savage. With Roy Rogers, Clay O'Brien, Billy Green Bush, Joan Hackett, Andrew Robinson, James Hampton, Walter Barnes, Dean Smith, Larry Mahan. An aging drifter takes a homeless youth under his wing, together they fight a rabies epidemic and hunt for a madman hiding on a large ranch. Roy Rogers' return to the screen is a pleasant affair sadly overlooked when first shown; music by Waylon Jennings, Willie Nelson and The Waylors. Reissued in 1984.\n\n**2491** _ **Mad at the Moon**_ **** Republic, 1992. 98 min. Color. D: Martin Donovan. SC: Martin Donovan and Richard Pelusi. With Mary Stuart Masterson, Hart Bochner, Stephen Blake, Fionnula Flanagan, Melissa Moore, Pat Atkins, Tom Mustin, Daphne Zuniga, Eleanor Baggett, Stephen Cole, Cec Verrell, Colin Firth, Cameron Frankley, Lori Ashton, Stephen Cole, Barbara Dow, Travis Jordan Marsh, Kathy Messick, Bob Shurtleff, Morgan Stuart, Jonathan Tripp, Hank Stone, Jackie Stansbury, Michael Sladek, Alix Koromzay, Reed Hollister, Raymond de Felitta. Although she loves a gunslinger, a young woman marries a sodbuster not realizing he turns into a werewolf during the cycle of the full moon. Fair horror Western with an unimpressive monster.\n\n**2492** _ **Mad Dog**_ **** Cinema Shares, 1980. 110 min. Color. D-SC: Philippe Mora. With Dennis Hopper, Jack Thomson, David Gulpilil, Frank Thring, Michael Pate, Wallace Eaton, Bill Hunter, John Hargreaves, Martin Harris, Robin Ramsay, David John, Philip Ross, Norman Kaye. The story of famous nineteenth century Australian outlaw \"Mad Dog\" Morgan. Relatively entertaining R-rated Western from Australia that won the John Ford Memorial Award as Best Western of the Year in 1976. Originally released in Australia in 1976 as _**Mad Dog Morgan**_ at 93 minutes.\n\n_**Mad Dog Morgan**_ see _**Mad Dog**_\n\n_**The Mad Trapper**_ see _**Challenge to Be Free**_\n\n**2493** _ **A Made-to-Order Hero**_ **** Universal, 1928. 50 min. D: Edgar Lewis. SC: William Lester and Gardner Bradford. With Ted Wells, Marjorie Bonner, Pearl Sindelar, Jack Pratt, Ben Corbett, Pee Wee Holmes, Scotty Mattraw, Dick L'Estrange. Wanting to marry his sweetheart, a ranch owner sets up a fake stage holdup to impress her aunt with his heroics only to find out the men he hired are actually outlaws who abduct the girl and steal her aunt's jewels. Rather fun silent Western giving viewers a chance to see Ted Wells in his heyday.\n\n**2494** _ **Madron**_ **** Four Star\/Excelsior, 1971. 92 min. Color. D: Jerry Hopper. SC: Edward Chappell and Leo McMahon. With Richard Boone, Leslie Caron, Paul Smith, Gabi Amrani, Chaim Banai, Avraham Telya, Willy Gafni, Aharon Ipale, Yaakov Banai, Sami Shmueli, Mosko Alkalay. A nun, the sole survivor of an Indian attack on a group of French Canadian sisters, is found by a hunter and the two are captured by brutal drifters. Less than average oater filmed in Israel.\n\n**2495** _ **The Magnificent Bandits**_ **** Tritone\/Medusa, 1971. 90 min. Color. D: Giovanni Fago. SC: Giovanni Fago, Antonio Troisio, Bernardino Zappoin and Jose Luis Jerez. With Tomas Milian, Ugo Pagliani, Eduardo Fajardo, Howard Ross (Renato Rossini), Alfredo Santa Cruz, Jesus Guzman, Leo Anchoriz, Claudio Scarchelli. In 1920s Brazil, farmers unite to fight the government over the destruction of their farmlands. Well made Italian feature, originally called _**O Cangaceiro**_ (Bandit), set in South America.\n\n**2496** _ **Magnificent Roughnecks**_ **** Allied Artists, 1956. 75 min. D: Sherman A. Rose. SC: Stephen Kandel. With Jack Carson, Mickey Rooney, Nancy Gates, Jeff Donnell, Myron Healey, Willis Bouchey, Eric Feldary, Alan Wells, Frank Gerstle, Larry Carr, Matty Fain, Joe Locke. Two oil wildcatters in South America try to bring in a new series of wells but meet with opposition. Very boring \"comedy\" with surprisingly poor results from the potentially great teaming of Jack Carson and Mickey Rooney.\n\n**2497** _ **The Magnificent Seven**_ **** United Artists, 1960. 126 min. Color. D: John Sturges. SC: William Roberts. With Yul Brynner, Eli Wallach, Steve McQueen, Horst Buchholz, Charles Bronson, Robert Vaughn, Brad Dexter, James Coburn, Vladimir Sokoloff, Rosenda Monteros, Jorge Martinez de Hoyos, Whit Bissell, Val Avery, Bing Russell, Rico Alaniz, Robert Wilke. The inhabitants of a remote Mexican village obtain the services of seven hired guns to protect them from the ravages of bandits. Finely done and near classic Western that is a refashioning of the famous 1954 Japanese film _**Seven Samurai**_.\n\n**Advertisement for** _**The Magnificent Seven**_ **(United Artists, 1960).**\n\n**2498** _ **The Magnificent Seven Ride!**_ **** United Artists, 1972. 100 min. Color. D: George McGowan. SC: Arthur Rowe. With Lee Van Cleef, Stefanie Powers, Mariette Hartley, Michael Callan, Luke Askew, Pedro Armendariz, Jr., William Lucking, James B. Sikking, Melissa Murphy, Darrell Larson, Ed Lauter, Carolyn Conwell, Jason Wingreen, Allyn Ann McLerie, Elizabeth Thompson, Ralph Waite, Rita Rogers, Robert Jaffe, Gary Busey, Rodolfo Acosta. When his bride is kidnapped by outlaws, an ex-gunman joins a friend in trying to find her and they end up helping five escaped convicts defend a town against the gang. Fourth feature in \"The Magnificent Seven\" series and a pretty good one.\n\n**Stefanie Powers and Lee Van Cleef in** _**The Magnificent Seven Ride!**_ **(United Artists, 1972).**\n\n**  \n**\n\n**2499** _ **The Magnificent Texan**_ **** Hispano Foxfilms, S.A., 1967. 103 min. Color. D: Lewis King (Luigi Capuano). SC: Luigi Capuano, Arpad DeRiso and Manuel Martinez Remis. With Glenn Saxson, John Barracuda (Massimo Serato), Barbara Loy, Benny Deus, Gloria Osuna, Lola Larsen (Fulvia Franco), George Greenwood (Giorgio Cerioni), Nerki Berkoff (Nerio Bernardi), Luis Induni, Richard Stark (Ricardo Pizzuti), Mary Sullivan (Mirella Pamphili), Patricia Carr (Rossella Bergamonti), Helen Wart (Anna Miserocchi), Glauco Onorato, Osiride Pevarello, Mimmo Poli, Roberto Messina, Egnazio Balsamo, Elio Angelucci. While seeking revenge for the massacre of his parents fifteen years before, a gunman helps Mexican peons oppressed by a cruel landowner and his son. Average, but overlong, Spanish Western originally released as _**Il Magnifico Texano**_ (The Magnificent Texan).\n\n**2500** _ **Mail Order Bride**_ **** Metro-Goldwyn-Mayer, 1974. 83 min. Color. D-SC: Burt Kennedy. With Buddy Ebsen, Keir Dullea, Lois Nettleton, Warren Oates, Barbara Luna, Bill Smith, Jimmy Mathers, Marie Windsor, Paul Fix, Doodles Weaver, Denver Pyle, Kathleen Freeman, Abigail Shelton, Diane Sayer, Ted Ryan. A reckless young man inherits a ranch but his guardian feels he needs to settle down so he sends for a bride for his charge. Fair genre comedy helped by a good cast.\n\n**2501** _ **Major Dundee**_ **** Columbia, 1965. 124 min. Color. D: Sam Peckinpah. SC: Harry Julian Fink, Oscar Paul and Sam Peckinpah. With Charlton Heston, Richard Harris, Jim Hutton, James Coburn, Michael Anderson, Jr., Senta Berger, Mario Adorf, Brock Peters, Warren Oates, Ben Johnson, R.G. Armstrong, L.Q. Jones, Slim Pickens, Karl Swenson, Michael Pate, John Davis Chandler, Dub Taylor, Albert Carter, Jose Carlos Ruiz. With a group of Confederate prison volunteers, a Union Army major tries to capture a rampaging Indian leader and his band in New Mexico territory. Colorful oater with a fine cast and script. Restored DVD version runs 136 minutes.\n\n**2502** _ **A Man Alone**_ **** Republic, 1955. 96 min. Color. D: Ray Milland. SC: John Tucker Battle. With Ray Milland, Mary Murphy, Ward Bond, Raymond Burr, Arthur Space, Lee Van Cleef, Alan Hale, Douglas Spencer, Thomas Browne Henry, Grandon Rhodes, Martin Garralaga, Kim Spalding, Howard Negley, Julian Rivero, Lee Roberts, Minerva Urecal, Thorpe Whiteman, Dick Rich, Frank Hagney. A loner mistakenly accused of murder is hunted by the law but sheltered by the sheriff's pretty daughter. Ray Milland made his directorial debut and did a good job helming and starring in this very fine drama which is basically a silent movie for the first third of its running time; well worth seeing.\n\n**2503** _ **Man and Boy**_ **** Levitt-Pickman, 1972. 98 min. Color. D: E.W. Swackhamer. SC: Harry Essex and Oscar Paul. With Bill Cosby, George Spell, Floria Foster, Douglas Turner Ward, Yaphet Kotto, Shelley Morrison, Leif Erickson, John Anderson, Henry Silva, Dub Taylor. A black Civil War veteran and his little son are on the trail of the thief who stole their horse. Standard drama aimed at the family trade.\n\n**2504** _ **The Man Behind the Gun**_ **** Warner Bros., 1952. 82 min. Color. D: Felix Feist. SC: John Twist. With Randolph Scott, Patrice Wymore, Philip Carey, Dick Wesson, Lina Romay, Roy Roberts, Morris Ankrum, Alan Hale (Jr.), Douglas Fowley, Anthony Caruso, Clancy Cooper, Robert Cabal, James Brown, Reed Howes, Rory Mallinson, John Logan, Vicki Raaf, Lee Morgan, Edward Hearn, Terry Frost, Charles Horvath, Art Millan, Rex Lease, James Bellah, Jack Parker, Billy Vincent, Alberto Morin, Edward Colmans, Ray Spiker, Herbert Deans. In 1850 one of the founders of Los Angeles tries to keep the territory from splitting into slave and non-slave holding areas. Fast moving historical drama sure to please Randolph Scott fans.\n\n_**Man Called Blade**_ see _**Mannaja**_\n\n**2505** _ **Man Called Django**_ **** 14 Luglio Cinematographica, 1971. 90 min. Color. D: Edward G. Muller (Edoardo Mulargia). SC: Nino Stresa. With Anthony Steffen, Stelio Candelli, Clauco Onorato, Cris Avram, Esmeralda Barros, Donato Castellaneta, Benito Stefanelli, Riccardo Pizzuti, Simone Blondell, Furio Meniconi, Alessandro Perrella, Paolo Figlia, Attilio Severini, Giovanni Cianfriglia, Gilberto Galimberti, Remo Capitani, Lorenzo Piani, Fortunato Arena. On the trail of the bandits who raped and murdered his lover, Django saves a horse thief from hanging and the two go after the killers. Another violent, but entertaining, Spaghetti Western also called _**Viva! Django**_.\n\n**2506** _ **A Man Called Gannon**_ **** Universal, 1969. 105 min. Color. D: James Goldstone. SC: Gene Kearney, Borden Chase and D.D. Beauchamp. With Tony Franciosa, Michael Sarrazin, Judi West, Susan Oliver, John Anderson, David Sheiner, James Westerfield, Gavin MacLeod, Eddie Firestone, Ed Peck, Harry Davis, Robert Sorrells, Terry Wilson, Eddra Gale, Harry Basch, James Callahan, Cliff Potter, Jason Evers, Jack Perkins. A cowboy makes a young Easterner his prot\u00e9g\u00e9 and when they takes jobs with a widowed ranch owner they find themselves at odds with neighbors over the size of their herd. Tepid remake of _**Man Without a Star**_ (q.v.).\n\n**2507** _ **Man Called Gringo**_ **** International Germania\/Procusa\/Domiziana, 1964. 90 min. Color. D: Roy Rowland. SC: Clarke Reynolds and Helmut Harun. With Gotz George, Alexandra Stewart, Helmut Schmid, Dan Martin, Sieghardt Rupp, Silvia Solar, Peter Tordy. A stranger arrives in a Western town to unravel a twenty year old mystery involving his father. Violent oater from Europe, this one an Italian-Spanish-West German co-production released in West Germany as _**Sie Nannten ihn Gringo**_ (They Call Him Gringo).\n\n**2508** _ **A Man Called Horse**_ **** National General, 1970. 114 min. D: Elliott Silverstein. SC: Jack De Witt. With Richard Harris, Judith Anderson, Jean Gascon, Manu Tupou, Dub Taylor, Corinna Tsopei, William Jordan, James Gammon, Eddie Little Sky, Manuel Padilla, Iron Eyes Cody, Lina Marin. While on a hunting expedition in the Dakotas, a British nobleman is captured by the Sioux Indians and made their slave. Interesting but brutal and gory drama that resulted in two sequels, _**Return of a Man Called Horse**_ and _**Triumphs of a Man Called Horse**_ (qq.v.).\n\n**2509** _ **The Man Called Noon**_ **** National General, 1973. 97 min. Color. D: Peter Collinson. SC: Scott Finch. With Richard Crenna, Stephen Boyd, Rosanna Schiaffino, Farley Granger, Patty Shepard, Angel Del Pozzo, Howard Ross (Renato Rossini), Aldo Sambrell, Jose Jaspe, Charley (Carlos) Bravo, Ricardo Palacios, Fernando Hilbeck, Bruce Fisher. Aided by the woman who loves him, an amnesiac gunman searches for his identity and hidden loot. Based on Louis L'Amour's novel, this Spaghetti Western is better than most such fare, helped by fine characterizations; released in Italy as _**Lo Chiamavano Mezzogiorno**_ (They Call Him Noon).\n\n**2510** _ **A Man Called Sledge**_ **** Columbia, 1971. 92 min. Color. D: Vic Morrow and (uncredited) Giorgio Gentili. SC: Vic Morrow and Frank Kowalsky. With James Garner, Dennis Weaver, Claude Akins, John Marley, Laura Antonelli, Allan Jones, Ken Clark, Tony Young, Wayde Preston, Steffan Zacharias, Paola Barbara, Mario Valgoi, Lorenzo Piani, Laura Betti, Franco Giornelli. A wanted outlaw joins three men in stealing a half-million dollars in gold from a prison but the gang then has a falling out over the loot. Okay oater mainly for James Garner fans.\n\n**2511** _ **The Man from Bitter Ridge**_ **** Universal-International, 1955. 80 min. Color. D: Jack Arnold. SC: Lawrence Roman and Teddi Sherman. With Lex Barker, Mara Corday, Stephen McNally, Trevor Bardette, Ray Teal, John Dehner, Myron Healey, Warren Stevens, Richard Garland, Jennings Miles, John Cliff, John Harmon, Frank Sully, Bob Herron, Elizabeth Slifer, Lane Chandler, Henry Rowland, Rick Vallin, Dan White, Brick Sullivan, Chuck Hamilton, Lee Morgan, Reg Parton, George DeNormand, Billy Dix, Ed Hinton, Paul McGuire, Robert Paquin, Joel Allen, Martin Cichy, Jack Gargan, Dennis Moore (voice). While working undercover to learn who is behind several stage holdups, a special agent is accused of the robberies by the outlaws. Fast moving Lex Barker vehicle with good direction by Jack Arnold.\n\n**2512** _ **The Man from Black Hills**_ **** Monogram, 1952. 57 min. D: Thomas Carr. SC: Joseph O'Donnell. With Johnny Mack Brown, James Ellison, Rand Brooks, Stanley Andrews, Florence Lake, Robert Bray, I. Stanford Jolley, Lane Bradford, Denver Pyle, Stanley Price, Ray Bennett, Joel Allen, Bud Osborne, Merrill McCormick. When a man finds his long lost father he also discovers another fellow is masquerading as him in order to inherit a mine. Average Johnny Mack Brown film with a bit different plot.\n\n**2513** _ **The Man from Button Willow**_ **** AFC Filmmakers\/United Screen Arts, 1965. 81 min. Color. D-SC: David Detiege. With Dale Robertson, Howard Keel, Edgar Buchanan, Barbara Jean Wong, Herschel Bernardi, Ross Martin, Cliff Edwards, Verna Felton, Edward Platt, Clarence Nash, Buck Buchanan, Thurl Rovenscraft, John Hiestand, Shep Menken, Pinto Colvig (voices). In 1869 the first undercover agent tries to help settlers being fleeced of their lands during the construction of the intercontinental railroad. Fairly pleasing animated feature aimed at the juvenile market.\n\n**2514** _ **Man from Cheyenne**_ **** Republic, 1942. 60 min. D: Joseph Kane. SC: Winston Miller. With Roy Rogers, George \"Gabby\" Hayes, Sally Payne, Gale Storm, Bob Nolan and The Sons of the Pioneers (Tim Spencer, Lloyd Perryman, Pat Brady, Hugh Farr, Karl Farr), Lynne Carver, William Haade, James Seay, Jack Ingram, Jack Kirk, Fred Burns, Jack Rockwell, Al Taylor, Chick Hannon, Art Dillard, Frank Brownlee, Ivan Miller, Monte Montague, Guy Usher, Ed Peil, Sr., Lynton Brent, Spade Cooley, Bob Burns, Foxy Callahan, Joe Yrigoyen, Ted Mapes, Eddie Lee, Tommy Coats. An outlaw gang terrorizes a small town and a cowboy returns home to try and stop them. Typical Roy Rogers film with enough action and songs to keep it going.\n\n**2515** _ **The Man from Colorado**_ **** Columbia, 1949. 99 min. Color. D: Henry Levin. SC: Robert D. Andrews and Ben Maddow. With Glenn Ford, William Holden, Ellen Drew, Ray Collins, Edgar Buchanan, Jerome Courtland, James Millican, Jim Bannon, Bill Phillips, Denver Pyle, James Bush, Mikel Conrad, David Clarke, Ian MacDonald, Clarence Chase, Stanley Andrews, Myron Healey, Craig Reynolds, Ray Teal, Fred Coby, Walter Baldwin, Eddie Fetherston, Pat O'Malley, Ben Corbett, Ray Hyke, Symona Boniface, Fred F. Sears, Fred Graff. A vicious Army officer is appointed a federal judge for Colorado Territory and he uses the bench to destroy his enemies. A different kind of offering with good acting and lots of violence.\n\n**2516** _ **The Man from Dakota**_ **** Metro-Goldwyn-Mayer, 1949. 74 min. D: Leslie Fenton. SC: Lawrence Stallings. With Wallace Beery, Dolores Del Rio, John Howard, Donald Meek, Robert Barrat, Addison Richards, Frederick Burton, William Haade, John Wray, Gregory Gaye, John Butler, Francis Ford, Wade Boteler, Hugh Sothern, Edward Hearn, Howard Hickman, Karl Hackett, Frank Hagney, Selmer Jackson, Frank M. Thomas, Erville Alderson, Tom Fadden, George Magrill, Buddy Roosevelt, Ted Oliver, Hal Wynants, Louise Robinson. A Union prisoner of war who has a bad past attempts to redeem himself by stealing Confederate secret plans, escapes from jail with another inmate and heads north to deliver the papers to General Grant. Seriocomic feature sure to please Wallace Beery fans.\n\n**2517** _ **The Man from Death Valley**_ **** Monogram, 1931. 64 min. D: Lloyd Nosler. SC: G.A. Durlam. With Tom Tyler, Betty Mack, Si Jenks, Gino Corrado, John Oscar, Stanley Blystone, Hank Bell, Bob Roper. When a cowboy arrives home to find his girl engaged to a lawyer he overhears a plan to rob the bank and he does it himself to keep the crooks from getting the money. Confused, low grade Tom Tyler offering.\n\n**2518** _ **The Man from Del Rio**_ **** United Artists, 1956. 82 min. D: Harry Horner. SC: Richard Carr. With Anthony Quinn, Katy Jurado, Peter Whitney, Douglas Fowley, John Larch, Whit Bissell, Douglas Spencer, Guinn Williams, Barry Atwater, Otto Waldis, Marc Hamilton, Carl Thayler, Adrienne Marden, Paul Harber, Bill Erwin, Jack Hogan, Frank Richards, Katherine DeMille. A Mexican bandit wins respect for himself when he tires to save a village from a brutal outlaw gang. Still another telling of the old story but a film that is not without merit.\n\n**2519** _ **The Man from Galveston**_ **** Warner Bros., 1964. 57 min. D: William Conrad. SC: Dean Riesner and Michael Zagor. With Jeffrey Hunter, Preston Foster, James Coburn, Joanna Moore, Edward Andrews, Kevin Hagen, Martin West, Ed Nelson, Karl Swenson, Grace Lee Whitley, Claude Stroud, Sherwood Price, Arthur Malet, Marjorie Bennett. Sam Houston's lawyer son, Temple Houston, comes to a small Texas town to defend a woman accused of murder. Entertaining pilot to \"Temple Houston\" (NBC-TV, 1963\u201364) that was issued theatrically.\n\n**2520** _ **The Man from God's Country**_ **** Phil Goldstone, 1924. 50 min. D: Alvin J. Neitz (Alan James). SC: George C. Hill. With William Fairbanks, Dorothy Revier, Lew Meehan, Milton Ross, Carl Silvera, Andrew Waldron. Two pals, an American cowboy and his Mexican vaquero buddy, fall for the same pretty girl but the former is blamed when she is brutalized by a ranch foreman. Slow melodrama from the silent days, also called _**Borderland Rangers**_.\n\n**2521** _ **Man from God's Country**_ **** Allied Artists, 1958. 72 min. Color. D: Paul Landres. SC: George Waggner. With George Montgomery, Susan Cummings, Randy Stuart, House Peters, Jr., James Griffith, Kim Charney, Frank Wilcox, Gregg Barton, Philip Terry, Al Wyatt, Kenneth MacDonald, I. Stanford Jolley, Kermit Maynard, Frank Sully, Byron Foulger, Ted Mapes. In Montana ranchers work together to obtain land needed by the railroad and are opposed by a fast shooting sheriff and his pal. Tame outing mainly for George Montgomery fans.\n\n**2522** _ **The Man from Guntown**_ **** Puritan, 1935. 60 min. D: Ford Beebe. SC: Ford Beebe and Thomas H. Ince, Jr. With Tim McCoy, Billie Seward, Rex Lease, Jack Clifford, Wheeler Oakman, Bob McKenzie, Jack Rockwell, George Chesebro, Eva McKenzie, Eddie Gribbon, Horace B. Carpenter, Hank Bell, George Pierce, Charles King, Oscar Gahan, Bud Pope, Chuck Baldra. Falsely accused of a crime, a cowboy is helped by the town marshal to break jail so he can get the goods on the real culprit. Pretty good Tim McCoy outing with a nice story and direction.\n\n**2523** _ **The Man from Hell**_ **** Willis Kent, 1934. 58 min. D: Lewis D. Collins. SC: Melville Shyer. With Reb Russell, Fred Kohler, Ann D'Arcy, George Hayes, Jack Rockwell, Yakima Canutt, Slim Whitaker, Roy D'Arcy, Tracy Layne, Mary Gordon, Tommy Bupp, Charles K. French, Murdock MacQuarrie, Ben Corbett, Jack Kirk, Bud McClure, Hank Bell, Ray Jones. After being released from Yuma Prison a man tries to get proof of his innocence by bringing in the one who framed him. Better than expected Reb Russell vehicle enhanced by a fine cast.\n\n**2524** _ **The Man from Hell's Edges**_ **** World Wide, 1932. 60 min. D-SC: Robert North Bradbury. With Bob Steele, Nancy Drexel, Julian Rivero, Robert Homans, George Hayes, Gilbert \"Pee Wee\" Holmes, Earl Dwire, Dick Dickinson, Perry Murdock, Blackie Whiteford, Bud Osborne, Blackjack Ward, Jack Evans, Ray Henderson, Duke Green, Buck Bucko, Buck Carey. Escaping from prison and ending up in a small town, a young man saves the sheriff's life, becomes his deputy and begins to suspect a caballero is the leader of an outlaw gang. Above average Bob Steele movie with a fine performance by Julian Rivero as the slick villain, Lobo. The \"Hell's Edges\" in the title refers to the Walla Walla, Washington, penitentiary.\n\n**2525** _ **The Man from Laramie**_ **** Columbia, 1955. 104 min. Color. D: Anthony Mann. SC: Philip Yordan and Frank Burt. With James Stewart, Arthur Kennedy, Donald Crisp, Cathy O'Donnell, Alex Nicol, Aline MacMahon, Wallace Ford, Jack Elam, John War Cloud, James Millican, Gregg Barton, Boyd Stockman, Frank DeKova, Frosty Royce, Eddy Waller, Frank Cordell, Bill Catching, Jack Carry. A mule team driver tries to find the men who murdered his younger brother. Exciting revenge melodrama with fine production values.\n\n**2526** _ **The Man from Montana**_ **** Universal, 1941. 57 min. D: Ray Taylor. SC: Bennett Cohen. With Johnny Mack Brown, Fuzzy Knight, Jeanne Kelly (Jean Brooks), Butch and Buddy, Nell O'Day, William Gould, James Blaine, Richard Alexander, Karl Hackett, Edmund Cobb, Kermit Maynard, Murdock MacQuarrie, The King's Men, Frank Ellis, Blackjack Ward, Tex Phelps. A crook tries to start a range war between homesteaders and ranchers because he wants both their land and cattle. Okay Johnny Mack Brown affair with a quartet of tunes, including \"Little Joe the Wrangler.\"\n\n**2527** _ **The Man from Monterey**_ **** Warner Bros., 1933. 57 min. D: Mack V. Wright. SC: Lesley Mason. With John Wayne, Ruth Hall, Luis Alberni, Francis Ford, Nina Quartero, Lafe McKee, Donald Reed, Lillian Leighton, Slim Whitaker, Jim Corey, Tom London, Chris-Pin Martin, Frank Ellis, Clarence Geldert, Joe Dominguez, Blackjack Ward, George Hazel, Bud McClure, Hank Bell, Jack Evans, Ralph Bucko, Roy Bucko, Jack Kirk, Denver Dixon. In Old California an Army captain tries to persuade a Spanish rancher to register his property which is being sought by a dishonest neighbor whose son is courting the man's daughter. Very good costume drama, the last in John Wayne's Warner Bros. Four Star Western series; a remake of Ken Maynard's _**Canyon of Adventure**_ (First National, 1927).\n\n**2528** _ **The Man from Montreal**_ **** Universal, 1940. 60 min. D: Christy Cabanne. SC: Owen Francis. With Richard Arlen, Andy Devine, Anne Gwynne, Jerry Marlowe, Kay Sutton, Reed Hadley, Addison Richards, Tom Whitten, Lane Chandler, Don Brodie, Karl Hackett, Pat Flaherty, Eddy Waller, William Royle, Eddy Conrad. A trapper falsely arrested for carrying stolen pelts tries to find the real fur thieves. Fast paced north woods melodrama, part of the Richard Arlen-Andy Devine series of action films for Universal.\n\n**2529** _ **Man from Music Mountain**_ **** Republic, 1938. 58 min. D: Joseph Kane. SC: Bernard McConville. With Gene Autry, Smiley Burnette, Carol Hughes, Sally Payne, Polly Jenkins and Her Plowboys, Ivan Miller, Al Terry, Dick Elliott, Hal Price, Cactus Mack, Ed Cassidy, Howard Chase, Lew Kelly, Frankie Marvin, Earl Dwire, Lloyd Ingraham, Gordon Hart, Joe Yrigoyen, Harry Harvey, Lillian Drew, Murdock MacQuarrie, Horace B. Carpenter, Lee Shumway, Tex Phelps, Bill Wolfe. Near the site of Boulder Dam, crooks try to swindle settlers by supposedly revamping a ghost town but a singing cowboy and his pals oppose them. Good Gene Autry vehicle with fine interpolation of story, music and action.\n\n**2530** _ **Man from Music Mountain**_ **** Republic, 1943. 71 min. D: Joseph Kane. SC: J. Benton Cheney and Bradford Ropes. With Roy Rogers, Bob Nolan and The Sons of the Pioneers (Tim Spencer, Ken Carson, Lloyd Perryman, Pat Brady, Hugh Farr, Karl Farr), Ruth Terry, Paul Kelly, Ann Gillis, George Cleveland, Hal Taliaferro, Jay Novello, Paul Harvey, Roy Barcroft, Renie Riano, Hank Bell, I. Stanford Jolley, Jack O'Shea, Slim Whitaker, Robert Kortman, Bob Burns, Fred Burns, Jane Isbell, Isabel Lamal, Roy Butler, Timmy Miller. A radio singer's homecoming is ruined by a dishonest rancher who uses a cattlemen versus sheep men feud to get control of new grazing licenses. Pretty entertaining Roy Rogers series film; re-titled _**Texas Legionaires**_ for television.\n\n**2531** _ **The Man from New Mexico**_ **** Monogram, 1932. 60 min. D: J.P. McCarthy. SC: Harry O. Hoyt. With Tom Tyler, Caryl Lincoln, Robert Walker, Lafe McKee, Jack Richardson, Frank Ball, Lewis Sargent, Blackie Whiteford, Slim Whitaker, Jack Long, William Nolte. A cattlemen's association detective works undercover to find out who is rustling area herds. Okay Tom Tyler vehicle.\n\n**2532** _ **The Man from Nowhere**_ **** Leone Film\/Orphee, 1966. 107 min. Color. D: Michele Lupo. SC: Luciano Martino and Ernesto Gastaldi. With Giuliano Gemma, Corrine Marchand, Fernando Sancho, Roberto Camardiel, Rosalba Neri, Nello Pazzafini, Gianni Solaro, Mirko Ellis, Andrea Bosic, Jose Manuel Martin. A saloon owner hires a gunman to kill the man who murdered his daughter but the price includes a night with the man's other daughter. Endlessly violent Spaghetti Western from Italy, originally called _**Arizona Colt**_.\n\n**2533** _ **The Man from Oklahoma**_ **** Rayart, 1926. 55 min. D: Harry S. Webb and Forrest Sheldon. With Jack Perrin, Josephine Hill, Lew Meehan, Lafe McKee, Martin Turner, Edmund Cobb, Molly Malone, Bud Osborne, Buzz Barton, Jim Corey. A cowboy discovers a romantic rival committed murder and he sends his dog for the law as he fights to save the girl he loves. Fairly pleasant Jack Perrin silent effort.\n\n**2534** _ **The Man from Oklahoma**_ **** Republic, 1945. 68 min. D: Frank McDonald. SC: John K. Butler. With Roy Rogers, George \"Gabby\" Hayes, Dale Evans, Roger Pryor, Bob Nolan and The Sons of the Pioneers (Tim Spencer, Ken Carson, Shug Fisher, Hugh Farr, Karl Farr), Arthur Loft, Maude Eburne, Sam Flint, Si Jenks, June Bryde, Elaine Lange, Charles Soldani, Edmund Cobb, George Sherwood, Eddie Kane, George Chandler, Wally West, Tex Terry, Robert Wilke, Tom London, Horace B. Carpenter, Cactus Mack. Roy Rogers becomes involved in a feud between rival ranchers that is secretly being instigated by a supposed friend. Typical Western fantasy of the time with an exciting wagon chase, good songs and the use of a movie camera to uncover the villain.\n\n**2535** _ **The Man from Oklahoma**_ **** International Germania\/Cineproduction Associates\/Balcazar, 1966. 85 min. Color. D: Robert M. White (Jaime J. Balcazar). SC: Helmut Harun. With Rick Horne, Sabine Bethmann, Tom Felleghi, Leontine May, John MacDouglas (Giuseppe Addobbati), Karl Otto Alberty, George Herzog. The newly appointed sheriff of a New Mexico town believes a local rancher is behind a hated outlaw gang. Slow paced European oater, this one made in West Germany as _**Oklahoma John**_.\n\n**2536** _ **The Man from Painted Post**_ **** Paramount-Artcraft, 1917. 55 min. D: Joseph Henaberry. SC: Douglas Fairbanks. With Douglas Fairbanks, Eileen Percy, Frank Campeau, Herbert Standing, Monte Blue, Charles Stevens, W.E. Lowery. A detective is assigned to investigate the disappearance of cattle near a remote Wyoming community. Fun Douglas Fairbanks silent comedy-drama.\n\n**2537** _ **Man from Rainbow Valley**_ **** Republic, 1946. 56 min. Color. D: R.G. Springsteen. SC: Betty Burbridge. With Monte Hale, Adrian Booth, Jo Ann Marlowe, Ferris Taylor, Emmett Lynn, The Sagebrush Serenaders, Bud Geary, Tom London, Kenne Duncan, Doye O'Dell, Bert Roach. A cowboy helps a comic strip writing rancher who is being swindled by a crooked rodeo owner. Monte Hale's second film is slow going despite an interesting plot.\n\n**2538** _ **The Man from Snowy River**_ **** 20th Century\u2013Fox, 1982. 104 min. Color. D: George Miller. SC: John Dixon. With Kirk Douglas, Jack Thompson, Tom Burlinson, Sigrid Thornton, Lorraine Bayly, Chris Haywood, Terence Donovan, June Jago, Tony Bonner, Bruce Kerr, John Nash. In frontier Australia a teenager grows into manhood working for a cattle baron and falls in love with the man's daughter. Fine Australian \"Western\" with good work by Kirk Douglas in dual roles as brothers.\n\n**2539** _ **Man from Sonora**_ **** Monogram, 1951. 54 min. D: Lewis D. Collins. SC: Maurice Tombragel. With Johnny Mack Brown, Phyllis Coates, House Peters, Jr., Lyle Talbot, Lee Roberts, John Merton, Stanley Price, Dennis Moore, Ray Jones, Pierce Lyden, Sam Flint, George DeNormand, George Sowards, Ray Jones. A marshal assists a local sheriff trying to find out who is behind a bullion shipment theft and the murder of another lawman working undercover as a traveling salesman. Tired Johnny Mack Brown film from the Monogram assembly line.\n\n**2540** _ **The Man from Sundown**_ **** Columbia, 1939. 58 min. D: Sam Nelson. SC: Paul Franklin. With Charles Starrett, Iris Meredith, Richard Fiske, The Sons of the Pioneers (Bob Nolan, Tim Spencer, Lloyd Perryman, Pat Brady, Hugh Farr, Karl Farr), Jack Rockwell, Alan Bridge, Richard Botiller, Robert Fiske, Ed Peil, Sr., Clem Horton, Forrest Dillon, Tex Cooper, Al Haskell, Ed Le Saint, Kit Guard, George Chesebro, Oscar Gahan, Frank Ellis. A Texas Ranger tries to find out who killed a rancher planning to testify against an outlaw gang. Average Charles Starrett vehicle.\n\n**2541** _ **The Man from Texas**_ **** Aywon, 1924. 50 min. D-SC: Tom Mix. With Tom Mix, Goldie Colwell, Sid Jordan, Leo Maloney, Roy Watson, Inez Walker, Pat Christian, Hoot Gibson. While searching for the man responsible for his sister's death, a cowboy falls in love with a rancher's daughter. Tom Mix fans will find this an interesting curio, being an expanded version of his 1915 Selig two reel short of the same title.\n\n**2542** _ **Man from Texas**_ **** Monogram, 1939. 60 min. D: Al Herman. SC: Robert Emmett (Tansey). With Tex Ritter, Ruth Rogers, Hal Price, Charles B. Wood, Kenne Duncan, Vic Demoruelle, Jr., Roy Barcroft, Frank Wayne, Tom London, Chick Hannon, Charles King, Sherry Tansey, Jim Thorpe, Chuck Baldra, Bud Pope, Walter Wilson. A lawman tries to help a rancher plagued by threatens and ends up against a gunman whose life he once saved. Pretty good Tex Ritter vehicle with more emphasis on drama than music.\n\n**2543** _ **The Man from Texas**_ **** Eagle-Lion, 1948. 71 min. D: Leigh Jason. SC: Joseph Fields and Jerome Chodorov. With James Craig, Lynn Bari, Johnny Johnston, Sara Allgood, Una Merkel, Harry Davenport, Wallace Ford, Vic Cutler, Reed Hadley, Clancy Cooper, Bert Conway, John Qualen, King Donovan, Lee Roberts, Stanley Andrews, Robert Malcolm, Eddie Dunn, Hope Landin, Charles Wagenheim, Brick Sullivan, Suzanne O'Connor, Erville Alderson, Ray Bennett, Lyle Latell, James Logan, Jim Farley, Dick Foote, Paul E. Burns, Glen Arthur. The El Paso Kid, a once notorious outlaw, weds and tries to lead a peaceful life but his past keeps following him. Okay feature with good work by James Craig in the lead; based on the play by E.B.Ginty.\n\n**2544** _ **The Man from the Alamo**_ **** Universal-International, 1953. 79 min. Color. D: Budd Boetticher. SC: Steve Fisher and D.D. Beauchamp. With Glenn Ford, Julia (Julie) Adams, Chill Wills, Victor Jory, Hugh O'Brian, Neville Brand, Jeanne Cooper, Marc Cavell, Edward Norris, Guy Williams, Dennis Weaver, John (Daheim) Day, Dan Poore, Myra Marsh, George Eldredge, Howard Negley, Kenneth MacDonald, Trevor Bardette, Stuart Randall, Arthur Space, John McKee, Evan Loew, Robert Carson, Chuck Hamilton, Brett Halsey, Guy Wilkerson, Smoki Whitfield, Walter Reed, Robert Smiley, Stuart Whitman, David Sharpe, Jack Mower, Ethan Laidlaw, Alberto Morin, Robert F. Hoy, Frank Wilcox, Monte Montague, Emile Avery, Eddie Parker, Fred Coby, Richard Cutting, Helen Gibson, Carl Andre, Polly Burson, Erik Neilson, Patsy Weil. A man leaves the Alamo to warn settlers of Santa Anna's advance and later finds himself branded a coward. Good action melodrama and interesting historical fiction.\n\n**2545** _ **The Man from the Rio Grande**_ **** Republic, 1943. 55 min. D: Howard Bretherton. SC: Norman S. Hall. With Don \"Red\" Barry, Wally Vernon, Twinkle Watts, Kirk Alyn, Nancy Gay, Roy Barcroft, Harry Cording, Paul Scardon, LeRoy Mason, Earle Hodgins, Kenneth Terrell, Robert E. Homans, Tom London, Bud Geary, Kenne Duncan, Jack Kirk, Jack O'Shea, Kansas Moehring. A man murders his brother in order to inherit a big cattle ranch but a cowboy and his pal plan to expose him. Only average Don Barry series entry.\n\n**2546** _ **The Man from Thunder River**_ **** Republic, 1943. 55 min. D: John English. SC: J. Benton Cheney. With Wild Bill Elliott, George \"Gabby\" Hayes, Anne Jeffreys, Ian Keith, John James, Georgia Cooper, Jack Ingram, Eddie Lee, Charles King, Bud Geary, Jack Rockwell, Ed Cassidy, Roy Brent, Alan Bridge, Al Taylor, Edmund Cobb, Robert Barron, Jack O'Shea, Curley Dresden, Frank McCarroll. When he tries to find out who is stealing gold ore, Wild Bill Elliott ends up saving a woman's life. Strong \"Wild Bill Elliott\" series film, well written and directed.\n\n**2547** _ **The Man from Tumbleweeds**_ **** Columbia, 1940. 59 min. D: Joseph H. Lewis. SC: Charles Francis Royal. With Bill Elliott, Iris Meredith, Dub Taylor, Raphael (Ray) Bennett, Francis Walker, Ernie Adams, Al Hill, Stanley Brown, Richard Fiske, Ed LeSaint, Don Beddoe, Eddie Laughton, John Tyrrell, Edward Cecil, Jack Low, Olin Francis, Jay Lawrence, Bruce Bennett, George Chesebro, Hank Bell, Steve Clark, Ray Jones, Buel Bryant, Frank McCarroll, Art Dillard, Jack Evans, Blackie Whiteford, Herman Howlin, Ray Jones, George Fiske, Jack King, George Hazel, Billy Wilson, Tex Cooper, Jack Tornek. Wild Bill Saunders enlists the help of parolees to help bring law and order to a town controlled by an outlaw gang. Speedy and entertaining \"Wild Bill Saunders\" series episode with fine work by Ray Bennett as the villainous Powder Kilgore.\n\n**2548** _ **The Man from Utah**_ **** Monogram, 1934. 55 min. D: Robert North Bradbury. SC: Lindsley Parsons. With John Wayne, Polly Ann Young, Ed Peil, Sr., Anita Campillo, George Hayes, Yakima Canutt, Lafe McKee, George Cleveland, Earl Dwire, Artie Ortego, Tex Phelps, Archie Ricks, Phil Dunham, Perry Murdock, Tex Palmer, Silver Tip Baker, Herman Hack, Bud McClure, Sam Garrett. A cowboy tries to get the goods on a gang of crooks committing murders on the rodeo circuit. Action filled John Wayne \"Lone Star\" Western hurt by poorly interpolated stock rodeo footage; Wayne sings (dubbed by Jack Kirk) \"Desert Breeze.\" Remade as _**Utah Trail**_ (q.v.) and colorized as _**Rodeo Racketeers**_.\n\n**2549** _ **The Man in the Saddle**_ **** Columbia, 1951. 87 min. Color. D: Andre De Toth. SC: Kenneth Gamet. With Randolph Scott, Joan Leslie, Ellen Drew, Alexander Knox, John Russell, Richard Rober, Alfonso Bedoya, Guinn Williams, Clem Bevans, Cameron Mitchell, Richard Crane, Frank Sully, George Lloyd, James Kirkwood, Frank Hagney, Don Beddoe, George Wallace, Frank Ellis, Tennessee Ernie Ford, Reed Howes, John Crawford, G. Raymond Nye, Kermit Maynard, Jim Mason, Herman Hack, Augie Gomez, Bob Burns, Frank O'Connor, Rosa Turich, Dorothy Phillips, Peter Virgo, Charles Rivero, Curley Gibson, Ada Adams, David O. McCall. A wealthy rancher swears revenge on a neighbor who has the love of the man's wife. Nicely made and entertaining Randolph Scott feature greatly enhanced by Tennessee Ernie Ford's singing of the title song throughout.\n\n**2550** _ **Man in the Shadow**_ **** Universal-International, 1957. 80 min. D: Jack Arnold. SC: Gene L. Coon. With Jeff Chandler, Orson Welles, Colleen Miller, Ben Alexander, Barbara Lawrence, John Larch, James Gleason, Royal Dano, Paul Fix, Leo Gordon, Martin Garralaga, Mario Siletti, Charles Horvath, William Schallert, Joseph J. Greene, Forrest Lewis, Harry Harvey, Sr., Joe Schneider, Mort Mills. A Mexican youth dies after being ordered beaten by the owner of a large ranch and an honest sheriff tries to find who killed the boy and why. Fairly interesting, but brooding, modern-day oater from producer Albert Zugsmith.\n\n**2551** _ **Man in the Wilderness**_ **** Warner Bros., 1971. 105 min. Color. D: Richard C. Sarafian. SC: Jack DeWitt. With Richard Harris, Henry Wilcoxon, John Huston, Prunella Ransome, John Bindon, Ben Carruthers, James Doohan, Bruce M. Fisher, Percy Herbert, Bryan Marshall, Norman Rossington, Robert Russell, Dennis Waterman, Paul Castro, Judith Furst, Manolo Landau, William Layton, Sheila Raynor, Joaquin Solis, Dean Selmier, Ines Acosta, Tony Cyrus, Tamara Sie, Martha Tuck, Rudy Althoff, Peggy (bear). In the northwest in 1820 a trapper is left for dead after being mauled by a bear and he fights to survive and get back to civilization. Rugged outdoor drama.\n\n**2552** _ **Man of Conquest**_ **** Republic, 1938. 97 min. D: George Nichols, Jr. SC: Wells Root, Jan Fortune and E.E. Paramore, Jr. With Richard Dix, Gail Patrick, Joan Fontaine, Edward Ellis, George Hayes, Victor Jory, Robert Barrat, C. Henry Gordon, Robert Armstrong, Ralph Morgan, Max Terhune, Janet Beecher, Pedro de Cordoba, George (Montgomery) Letz, Guy Wilkerson, Charles Stevens, Hal Taliaferro, Lane Chandler, Ethan Laidlaw, Edmund Cobb, Billy Benedict, Tex Cooper, Kathleen Lockhart, Leon Ames, Ferris Taylor, Francis Sayles, Arthur Aylesworth, Max Waizman, Russell Hicks, Stanley Blystone, William Desmond, Fred Kohler, Jr., George J. Lewis, Mary MacLaren, Fay McKenzie, Buddy Roosevelt, Jason Robards, Harry Strang, Slim Whitaker, Ernie Adams, Budd Buster, Richard Botiller, George Morrell, Nelson McDowell, Merrill McCormick, Tom Chatterson, Pauline Haddon, Edward Hearn, Jack Ingram, Cy Kendall, Chris-Pin Martin, Jane Keckley, Earle Hodgins, Edward Earle, Chief Yowlachie, Horace Murphy, Robert Wilke, Rosa Turich, Bill Nestell, Frank O'Connor, Sarah Padden, Chief Thundercloud, Cyril Ring, William Royle, Jim Thorpe, Helen Brown, Chief Many Treaties, Olaf Hytten, Otto Hoffman, Sam Harris, Mildred Gover, Jack Gargan, Iron Eyes Cody, Rube Dalroy, Sonny Chorre, Noble \"Kid\" Chissel, Rose Plummer, Bill Wolfe, Ethyl May Halls. The story of Sam Houston, from his days as governor of Tennessee to his leading the Texas rebellion for independence. Good historical drama enhanced by excellent production values and a great cast.\n\n**2553** _ **Man of the East**_ **** United Artists, 1974. 122 min. Color. D-SC: E.B. Clucher (Enzo Barboni). With Terence Hill, Gregory Walcott, Harry Carey, Jr., Dominic Barton, Yanti Somer, Steffen Zachariah, Tony Norton, Riccardo Pizzuti, Jean Louis, Enzo Fiermonte, Sal Borgese, Dan Sturkie. A stuffy young man from New England heads West to take over his father's ranch and runs into trouble. Overlong but somewhat amusing Italian oater issued in that country in 1972 as _**E Poi lo Chiamarono il Magnifico**_ by P.E.A.\/Artstes Associes.\n\n**2554** _ **Man of the Forest**_ **** Paramount, 1933. 62 min. D: Henry Hathaway. SC: Jack Cunningham and Harold Shumate. With Randolph Scott, Harry Carey, Verna Hillie, Noah Beery, Larry \"Buster\" Crabbe, Barton MacLane, Guinn Williams, Vince Barnett, Blanche Frederici, Tempe Piggot, Tom Kennedy, Frank McGlynn, Jr., Duke R. Lee, Lew Kelly, Merrill McCormick, Tom London, Hank Bell. A crook wants to steal an ex-convict's timber land and plans to kidnap his niece so the property cannot be signed over to her. Beautiful scenic locations and excellent photography (by Ben Reynolds) make this Zane Grey series feature a very good one; reissued as _**Challenge of the Frontier**_.\n\n_**Man of the Frontier**_ see _**Red River Valley**_ (1936)\n\n**2555** _ **Man of the Law**_ **** CBS-TV\/20th Century\u2013Fox, 1957. 45 min. D: Lewis Allen. SC: David Lang. With Wendell Corey, Ron Randell, Marsha Hunt, Constance Ford, Johnny Washbrook, Trevor Bardette, Sean McClory, William Challee, Denver Pyle, Robert Foulk, Mike Ragan (Holly Bane), Clancy Cooper, Robert Griffin, Percy Helton, John Conte (host). Three witnesses must come forth to testify after a sheriff arrests an outlaw for a murder committed during a holdup. Originally shown as an episode of \"The 20th Century\u2013Fox Hour\" (CBS-TV, 1955\u201357) on February 20, 1957, this taut telefilm was issued theatrically abroad.\n\n**2556** _ **Man of the West**_ **** United Artists, 1958. 100 min. Color. D: Anthony Mann. SC: Reginald Rose. With Gary Cooper, Julie London, Lee J. Cobb, Arthur O'Connell, Jack Lord, John Dehner, Royal Dano, Robert Wilke, Jack Williams, Guy Wilkerson, Chuck Roberson, Frank Ferguson, Emory Parnell, Tina Menard, Joe Dominguez, Dick Elliott. A reformed outlaw is a passenger on a stagecoach held up by his former gang cohorts who are now led by his uncle. Average Western made on a big scale with Gary Cooper too old for the lead.\n\n**2557** _ **Man or Gun**_ **** Republic, 1958. 79 min. D: Albert C. Gannaway. SC: Vance Skarstedt and James C. Cassity. With Macdonald Carey, James Craig, Audrey Totter, James Gleason, Warren Stevens, Harry Shannon, Jil Jarmyn, Robert Burton, Ken Lynch, Karl Davis, Larry Grant, Julian Burton, Carl York, Harry Keekas, Mel Gains, Ron McNeil. A drifter arrives in a town ruled by a ruthless family and he decides to help the locals in getting rid of them. Well acted but rather dreary oater.\n\n**2558** _ **The Man Trailer**_ **** Columbia, 1934. 59 min. D-SC: Lambert Hillyer. With Buck Jones, Cecilia Parker, Arthur Vinton, Clarence Geldert, Steve Clark, Charles West, Tom Forman, Lew Meehan, Richard Botiller, Artie Ortego, George Chesebro, Jack Rockwell, Charles Brinley, Tommy Coats, Bud McClure, Buck Bucko, Roy Bucko. On the run from the law for a murder he did not commit, a cowboy saves the money taken during a stagecoach robbery and is made sheriff only to be blackmailed by an outlaw aware of his past. One of the all time best \"B\" Westerns; a remake of _**The Lone Rider**_ (1930) and redone as _**The Thundering Trail**_ (qq.v.).\n\n**2559** _ **The Man Who Killed a Ghost**_ **** NBC-TV\/Universal, 1971. 74 min. Color. With Robert Wagner, Lex Barker, Janet Leigh, Kim Stanley, Gene Barry, Susan Saint James, David Hartman, Alfred Ryder, Donald Barry, Lurene Tuttle, William Bryant, Jack Soo, Teddy Eccles. A reporter investigates a food franchise and runs into a former Hollywood cowboy star who does not live up to his screen image. Okay episode of \"The Name of the Game\" (NBC-TV, 1968\u201371) issued to TV as a feature film.\n\n**2560** _ **The Man Who Loved Cat Dancing**_ **** Metro-Goldwyn-Mayer, 1973. 114 min. Color. D: Richard C. Sarafin. SC: Eleanor Perry. With Burt Reynolds, Sarah Miles, Lee J. Cobb, Jack Warden, George Hamilton, Bo Hopkins, Robert Donner, Sandy Kevin, Nancy Malone, Jay Silverheels, Jay Varela, Owen Bush, Larry Littlebird. An outlaw gang pulls off a robbery, takes a woman hostage and she finds herself falling in love with the leader. Fairly entertaining, but rather brutal, adaptation of Marilyn Durham's novel. Jack Warden is especially good as a vicious gang member.\n\n**2561** _ **The Man Who Shot Liberty Valance**_ **** Paramount, 1962. 122 min. D: John Ford. SC: Willis Goldbeck and James Warner Bellah. With John Wayne, James Stewart, Vera Miles, Lee Marvin, Edmond O'Brien, Andy Devine, Ken Murray, John Carradine, Jeanette Nolan, John Qualen, Willis Bouchey, Carleton Young, Woody Strode, Denver Pyle, Strother Martin, Lee Van Cleef, Robert F. Simon, O.Z. Whitehead, Paul Birch, Joseph Hoover, Jack Pennick, Anna Lee, Charles Seel, Shug Fisher, Earle Hodgins, Stuart Holmes, Dorothy Phillips, Buddy Roosevelt, Gertrude Astor, Eva Novak, Slim Talbot, Montie Montana, Bill Henry, Helen Gibson, Major Sam Harris, Ted Mapes, Jack Kenney, Sherry Jackson, John B. Whiteford. A noted politician recalls how he came to power through the guise of another man killing a town bully. Extremely well done drama with an excellent cast and script; John Ford's last great Western.\n\n**2562** _ **Man with the Golden Pistol**_ **** Balcazar, 1966. 107 min. Color. D: Alfonso Balcazar. SC: Giovanni Simonelli, Alfonso Balcazar and Jose Antonio De La Loma. With Carl Mohner, Gloria Milland, Fernando Sancho, Luis Davila, Umberto Raho, Pedro Gil, Irene Mir, Oscar Pelliceri. A wanted man finds the body of a murdered gunman, takes his identity and is hired by villagers to protect them from marauders. Violent but pretty good Italian-Spanish co-production released in Europe as _**L'Uomo dalla Pistola d'Oro**_ (The Man with the Golden Pistol).\n\n_**Man with the Golden Winchester**_ see _**Son of Zorro**_ (1974)\n\n**2563** _ **Man with the Gun**_ **** United Artists, 1955. 83 min. D: Richard Wilson. SC: N.B. Stone, Jr. and Richard Wilson. With Robert Mitchum, Jan Sterling, Karen Sharpe, Henry Hull, Emile Meyer, John Lupton, Barbara Lawrence, Ted De Corsia, Leo Gordon, James Westerfield, Florenz Ames, Robert Osterloh, Jay Adler, Amzie Strickland, Stafford Repp, Maudie Prickett, Angie Dickinson, Claude Akins, Burt Mustin, Renie Riano, Maara McAfee, Norma Calderon. A gunman, whose wife has deserted him, is hired to clean up a town lorded over by a wealthy rancher. Slow moving and brooding drama with good work by Robert Mitchum as the gunfighter.\n\n**2564** _ **Man with the Steel Whip**_ **** Republic, 1954. 12 Chapters. D: Franklyn Adreon. SC: Ronald Davidson. With Richard Simmons, Barbara Bestar, Dale Van Sickel, Mauritz Hugo, Lane Bradford, Pat Hogan, Roy Barcroft, Stuart Randall, Edmund Cobb, I. Stanford Jolley, Guy Teague, Alan Wells, Tom Steele, Art Dillard, Chuck Hayward, Charles Stevens, Jerry Brown, Harry Harvey, Bob Clark, Charles Sullivan, Gregg Barton, Tex Terry, George Eldredge, Herman Hack, Robert Henry, Tom Monroe, Chris Mitchell, Walt LaRue. Taking on the guise of the masked rider El Latigo, a rancher tries to keep Indians from being blamed for raids conducted by a saloon owner after gold rich land. Republic's final serial, a Zorro imitation, is a hodgepodge of footage from previous endeavors.\n\n**2565** _ **Man Without a Star**_ **** Universal-International, 1955. 89 min. Color. D: King Vidor. SC: Borden Chase and D.D. Beauchamp. With Kirk Douglas, Jeanne Crain, Claire Trevor, William Campbell, Richard Boone, Jay C. Flippen, Myrna Hansen, Mara Corday, Eddy Waller, Sheb Wooley, George Wallace, Roy Barcroft, James Hayward, Paul Birch, Malcolm Atterbury, William Challee, William Phipps, Ewing Mitchell, Mark Hanna, Frank Chase, Gil Patrick, Casey MacGregor, Jack Ingram, Carl Andre, Jack Elam, Myron Healey, Lee Roberts. A drifter becomes involved in helping a rancher oppose a rival woman landowner who wants all the range for herself. Fine, stout effort from director King Vidor; Frankie Laine sings the title song.\n\n**2566** _ **Manchurian Avenger**_ **** Facet Films, 1984. 87 min. Color. D: Ed Warrick. SC: Timothy Stephenson. With Bobby Kim, Bill Wallace, Leila Lee Olsen, Leila Hee, Barbara Minardi, Bruce Purcell, Jose Payo, Bob Coulson, Derek Abernathy. Years after a gang killed his father while looking for gold, a man returns to Colorado to find the one who raised him as been murdered by outlaws. There is fair entertainment in this Kung Fu Western.\n\n**2567** _ **Manhattan Cowboy**_ **** Syndicate, 1928. 54 min. D: J.P. McGowan. SC: Sally Winters and Ernest Vajda. With Bob Custer, Mary Mayberry, Lafe McKee, Charles (Slim) Whitaker, John Lowell Russell , Lynn Sanderson, Mack V. Wright, Cliff Lyons, Dorothy Vernon, J.P. McGowan. Sent West to stay on a ranch, a playboy falls in love with the owner's daughter who is lusted after by a cowpoke, who with a cohort, kidnaps her. Well done, fast paced and entertaining Bob Custer silent feature.\n\n**2568** _ **The Manhunt**_ **** Samuel Goldwyn Company, 1984. 91 min. Color. D: Larry Ludman (Fabrizio De Angelis). SC: Larry Ludman (Fabrizio De Angelis) and David Parker, Jr. (Dardano Sacchetti). With John Ethan Wayne, Henry Silva, Bo Svenson, Ernest Borgnine, Raimund Harmstorf, Terry Lynch, Don Taylor, Randy Mulkey, Farris Castleberry, Susan Wilson, Robin Fugett, Jack Dunlap, Danny O'Haco, Red Wolverton, Claude Hereford, Herny Maxkendrick, Austin Ludson, Charles Julian, Rick Schieffer, Eddie Neufang, Lawrence Niemi, Arthur Rothbard, Ed Adams. After buying two horses a cowboy is accused of theft and is sent to prison where he escapes and tries to locate the seller. Modern-day Western mainly of interest because it stars John Ethan Wayne, the Duke's youngest son.\n\n**2569** _ **Mannaja**_ **** Medusa Distribuzione, 1977. 101 min. Color. D: Sergio Martino. SC: Sergio Martino and Sauro Scavolini. With Maurizio Merli, John Steiner, Sonia Jeannine, Donald O'Brien, Salvatore Puntillo, Nino Casale, Enzo Fiermonte, Rik Battaglia, Aldo Rendine, Enzo Maggio, Sergio Tardioli, Sophia Lombardo, Philippe Leroy, Ted Carter (Nello Pazzafini), Martine Brochard, Claudio Ruffini, Alberto Dell'Acqua (Robert Widmark); Michael Forest, Nick Alexander (voices). A hatchet toting bounty hunter is hired by a town boss to find his missing daughter. Darkly violent, but pretty good Spaghetti Western made in Italy; also called _**A Man Called Blade**_ and issued on video as _**Mannaja\u2014A Man Called Blade**_.\n\n**2570** _ **Manos Torpes**_ (Awkward Hands) **** Izaro Films, 1970. 93 min. Color. D: Rafael Romero Merchent. SC: Joaquin Romero Merchent and Santiago Monicada. With Peter Lee Lawrence, Alberto de Mendoza, Pilar Velazquez, Aldo Sambrell, Luis Induni, Frank Brana, Antonio Casas, Vidal Molina, Antonio Pica, Manuel de Blas, Yelena Samarina, Dina Loy, Beni Deus, Enrique Vazquez. Beaten and forcefully separated from the woman he loved, a cowboy becomes a gunman, returning to extract revenge on the girl's father and he man she was supposed to marry. A muddled plot does not slow down this fast paced Spaghetti Western.\n\n**2571** _ **Man's Best Friend**_ **** Regal, 1935. 62 min. D: Edward A. Kull and Thomas Storey. SC: Tom Sawyer (Thomas) Storey. With Lightning the Wonder Dog, Douglas Haig, Frank Brownlee, Mary MacLaren, Patricia Chapman, Samson (bear). When a teenager brings home a litter of pups sired by his faithful dog, his cruel father refuses to let him keep them, resulting in a confrontation between the two. Rawboned, low budget poverty row double bill item.\n\n_**Man's Country**_ (1932) see _**God's Country and the Man**_ (1932)\n\n**2572** _ **Man's Country**_ **** Monogram, 1938. 55 min. D: Robert Hill. SC: Robert Emmett (Tansey). With Jack Randall, Marjorie Reynolds, Ralph Peters, Walter Long, Forrest Taylor, Bud Osborne, Dave O'Brien, Ernie Adams, David Sharpe, Charles King, Sherry Tansey, Chick Hannon, Budd Buster, Denver Dixon, Tex Palmer. An undercover agent for the rangers befriends a family by making them think he is a wanted outlaw in order to find out who committed two murders. Standard Jack Randall vehicle.\n\n**2573** _ **A Man's Land**_ **** Allied, 1932. 65 min. D: Phil Rosen. SC: Adele Buffington. With Hoot Gibson, Marion Shilling, Skeeter Bill Robbins, Alan Bridge, Charles King, Ethel Wales, Hal Burney, Robert Ellis, Bill (G. Raymond) Nye, Merrill McCormick, Slim Whitaker, Fred Gilman, Charles K. French, Glenn Strange, Bud Osborne, Frank Ellis, Hank Bell, Edgar Lewis. A young woman and a foreman each inherit one-half of a ranch that is plagued by rustlers. Okay Hoot Gibson post\u2013Universal starring film.\n\n**2574** _ **Manuel Saldivar, el Texano**_ (Manuel Saldivar, The Texan). Productora Filmica Mexico, 1972. 85 min. Color. D-SC: Rene Cardona. With Rodolfo de Anda, Pilar Pellicer, Jorge Russek, Katherine Riddle, Aaron Hernan, Rene Cordona, Herman Guida, Luis Aguilar, Victor Alcocer, Gustavo del Castillo, Sandra Boyd, Susanna Hill, Alfredo Gutierrez, Jesus Gomez, Jorge Fegan, Rene Barrera, Raul Hernandez, Miguel Suarz. A gunman hired to rid a town of bandits falls for a young woman involved in the lawlessness. Sturdy Mexican Western produced by star Rodolfo de Anda.\n\n**2575** _ **Many Rivers to Cross**_ **** Metro-Goldwyn-Mayer, 1955. 92 min. Color. D: Roy Rowland. SC: Harry Brown and Guy Trosper. With Robert Taylor, Eleanor Parker, Victor McLaglen, Rosemary De Camp, Jeff Richards, Russ Tamblyn, James Arness, Alan Hale (Jr.), John Hudson, Rhys Williams, Josephine Hutchinson, Sig Rumann, Russell Johnson, Ralph Moody, Abel Fernandez. In 1798 a wild young woman pursues a frontiersman but their romance is complicated by warring Indians. Likable frontier satire.\n\n**2576** _ **Mara of the Wilderness**_ **** Allied Artists, 1966. 90 min. Color. D: Frank McDonald. SC: Tom Blackburn. With Adam West, Linda Saunders, Theo Marcuse, Denver Pyle, Sean McClory, Eve Brent, Roberto Contreras, Ed Kemmer, Stuart Walsh, Lelia Walsh. A seven year old girl is left to live with wolves after her parents die in a plane crash and a dozen years later a forest ranger finds her but while he wants to reintroduce her to civilization a hunter plans to sell her to a sideshow. Well done human interest drama.\n\n**2577** _ **The Marauders**_ **** United Artists, 1947. 64 min. D: George Archainbaud. SC: Charles Belden. With William Boyd, Andy Clyde, Rand Brooks, Ian Wolfe, Dorinda Clifton, Mary Newton, Harry Cording, Earle Hodgins, Dick Bailey, Richard Alexander, Herman Hack. When the Bar 20 trio take shelter from a storm in an old church they find it inhabited by a woman and her daughter, who are being harassed by an outlaw gang. A mystery plot and a spooky goings on add some flavor to this otherwise mundane \"Hopalong Cassidy\" affair. TV title: _**King of the Range**_.\n\n**2578** _ **The Marauders**_ **** Metro-Goldwyn-Mayer, 1955. 81 min. Color. D: Gerald Mayer. SC: Jack Leonard and Earl Fenton. With Dan Duryea, Jarma Lewis, Keenan Wynn, Jeff Richards, John Hudson, Harry Shannon, David Kasday, James Anderson, Richard Lupino, Peter Mamakos, John Mills, Michael Dugan. A rancher fights to save his spread when a greedy land baron hires gunmen two run him off it. Pretty good program feature with nice production values.\n\n**2579** _ **La Marca de Santanas**_ (The Mark of Satan). Clasa-Mohme, 1957. 90 min. D: Chano Urueta. SC: Ramon Obon. With Luis Aguilar, Crox Alvarado, Jaime Fernandez, Flor Silvestre, Pascual Garcia Pena. A mystery man called the Masked Tiger tries to solve his brother's murder but is framed for a crime by devil worshippers. Scary Mexican horror Western.\n\n**2580** _ **Mariachis**_ **** Febus Films, 1950. 90 min. Color. D: Adolfo Fernandez Bustamante. SC: Adolfo Fernandez Bustamante and Max Aub. With Antonio Badu, Isabel del Puerto, Jose Angel Espinosa \"Ferrusquilla,\" Lupe Rivas Cacho, Beatriz Aguirre, Nelly Montiel, Carlo Pulido, Armando Arriola, Fernando Casanova, Chula Prieto, Luis Perez Meza, Rodolfo Castillo, Felix de la Fuente, Ignacio Peon, Jose Pardave, Francisco Pando, Raul Guerro, Paco Martinez, Pepe Martinez. Two mariachi band leaders each love a girl in the others' group. Pleasant Mexican Western musical comedy.\n\n_**Mark of the Apache**_ see _**Tomahawk Trail**_\n\n**2581** _ **Mark of the Gun**_ **** Emerson, 1968. 90 min. D: Ross Hagen. With Ross Hagen, Brad Thomas, Chris Carter, Gabrielle St. Claire, Katye Martine, Joan McCrea, Wallace J. Campodanio, Erick Lindberg. Jack Slade and his outlaw gang rampage through the West. Obscure, violent, low grade action thriller.\n\n**2582** _ **Mark of the Lash**_ **** Screen Guild, 1948. 58 min. D: Ray Taylor. SC: Moree Herring and Gloria Welsch. With Lash LaRue, Al St. John, Suzi Crandall, Jimmie Martin, John Cason, Marshall Reed, Tom London, Lee Roberts, Steve Dunhill, Harry Cody, Cliff Taylor, Britt Wood, Jack Hendricks. Lash and Fuzzy are out to rid the Red Rock area of an outlaw gang after local water rights. Another fast action Lash LaRue outing.\n\n**2583** _ **Mark of the Renegade**_ **** Universal-International, 1951. 81 min. Color. D: Hugo Fregonese. SC: Louis Solomon and Robert Hardy Andrews. With Ricardo Montalban, Cyd Charisse, J. Carrol Naish, Gilbert Roland, Andrea King, George Tobias, Antonio Moreno, Georgia Backus, Robert Warwick, Armando Silvestre, Bridget Carr, Alberto Morin, Renzo Cesana, Robert Cornthwaite, Edward C. Rios, David Wolfe. In 1824 California a tyrant captures a bandit and forces him to romance the pretty daughter of the area's governor. A different kind of frontier drama but none too enjoyable.\n\n**2584** _ **Mark of the Spur**_ **** Big 4, 1932. 58 min. D: J.P. McGowan. SC: Frederic Chapin and Stephen G. Hunt. With Bob Custer, Lillian Rich, George Chesebro, Lafe McKee, Anna Belle Driver, Franklyn Farnum, Blackie Whiteford, Bud Osborne, Charles Adler, Frank Ball, Jack Long, Harry Todd, Blackjack Ward. A cowboy gets on the wrong side of his female boss but later wins her affections when he helps her oppose conniving relatives. Flat Bob Custer vehicle.\n\n**2585** _ **The Mark of Zorro**_ **** United Artists, 1920. 90 min. D: Fred Niblo. SC: Elton Thomas. With Douglas Fairbanks, Marguerite de la Motte, Robert McKim, Noah Beery, Charles Hill Mailes, Claire McDowell, George Periolat, Walt Whitman, Sidney de Grey, Tote du Crow, Snitz Edwards, Gilbert Clayton, Charles Stevens, Noah Berry, Jr., Milton Berle. In Old California a young snob takes on the guise of a masked man who fights government oppression. Lively silent with Doug Fairbanks ideal as Zorro; a must see fun adventure film.\n\n**2586** _ **The Mark of Zorro**_ **** 20th Century\u2013Fox, 1940. 94 min. D: Rouben Mamoulian. SC: John Taintor Foote. With Tyrone Power, Linda Darnell, Basil Rathbone, Gale Sondergaard, Eugene Pallette, J. Edward Bromberg, Montagu Love, Janet Beecher, Robert Lowery, Chris-Pin Martin, George Regas, Belle Mitchell, John Bleifer, Frank Puglia, Pedro de Cordova, Guy D'Ennery, Eugene Borden, Fred Malatesta, Fortunio Bonanova, Harry Worth, Michael (Ted) North, Ralph Byrd, Stanley Andrews, Victor Kilian, Hector Sarno, Franco Corsaro, Paul Sutton, Charles Stevens, William Edmunds, Robert Cauterio, Rafael Corio, Frank Yaconelli, Gino Corrado, George Sorel, Lucia Villegas, Francisco Maran, Jean Del Val, Art Dupuis. A foppish nobleman tries to fight tyranny in Spanish California by wearing a mask and leading the people in their quest for freedom. Remake of the 1920 classic (q.v.) is fine entertainment with a colorful story, lots of action and some well staged dueling sequences.\n\n**2587** _ **The Mark of Zorro**_ **** ABC-TV\/20th Century\u2013Fox, 1974. 78 min. Color. D: Don McDougall. SC: Brian Taggart. With Frank Langella, Ricardo Montalban, Gilbert Roland, Yvonne De Carlo, Louise Sorel, Anne Archer, Robert Middleton, Tom Lacy, Jorge Cervera Jr., Jay Hammer, Robert Carricart, John Rose, Alfonso Tafoya, Inez Perez, Frank Soto. A man returns to his family's California hacienda to find the region under the thumb of tyranny and he becomes the masked Zorro to find the villains. Pale TV version of Johnston McCulley's story although it is greatly helped by Gilbert Roland as the elder Vega, the father of Frank Langella's lackluster Zorro.\n\n**2588** _ **Mark of Zorro**_ **** Starlight, 1976. 97 min. Color. D: Franco Lo Cascio. SC: Francisco Lara and Augusto Finocchi. With George Hilton, Lionel Stander, Charo Lopez, Rod Licari, Antonio Pica, Flora Carosello, Tito Garcia, Gino Pagnani. While a bumbling man masquerades as Zorro to overthrow a tyrant in frontier California but his heroics are orchestrated by a monk. Italian-Spanish co-production spoof that soon becomes tiresome; issued in Europe as _**Ah si? E Io Io Dico Zzzorro!**_ (Who's Afraid of Zzzorro!) and _**Nuevas Adventuras de Zorro**_ (New Adventures of Zorro).\n\n**2589** _ **Marked for Murder**_ **** Producers Releasing Corporation, 1945. 58 min. D-SC: Elmer Clifton. With Tex Ritter, Dave O'Brien, Guy Wilkerson, Marilyn McConnell, Ed Cassidy, Henry Hall, Charles King, Jack Ingram, Robert Kortman, The Milo Twins, Kermit Maynard, Wen Wright, Wally West, Frank Ellis, Ray Henderson, Art Felix, Jack Evans, Jimmy Aubrey, Herman Hack, Chick Hannon, Roy Bucko, George Sowards. In the 1880s a trio of Texas Rangers learn who is behind a range war between cattlemen and sheep herders. Entertaining and action filled \"Texas Rangers\" outing with Tex Ritter singing \"Froggie Went a Courtin',\" \"Long Time Gone\" and \"Tears of Regret.\"\n\n**2590** _ **Marked Trails**_ **** Monogram, 1944. 58 min. D: J.P. McCarthy. SC: J.P. McCarthy and Victor Hammond. With Bob Steele, Hoot Gibson, Veda Ann Borg, Ralph Lewis, Mauritz Hugo, Steve Clark, Charles Stevens, Lynton Brent, Bud Osborne, George Morrell, Allen B. Sewell, Ben Corbett, John Cason, Tex Palmer, Silver Tip Baker, Rose Plummer, Silver Harr. Two lawmen are after a notorious outlaw gang with one of them posing as a bad man to infiltrate the band. Sub-standard Bob Steele-Hoot Gibson vehicle.\n\n**2591** _ **The Marksman**_ **** Allied Artists, 1953. 62 min. D: Lewis D. Collins. SC: Dan Ullman. With Wayne Morris, Elena Verdugo, Rick Vallin, Frank Ferguson, I. Stanford Jolley, Tom Powers, Robert Bice, Stanley Price, Tim Ryan, Russ Whiteman, William Fawcett, Brad Johnson, Jack Rice. Because he is an expert with a telescopic rifle, a man is hired as a town marshal so he can track down an outlaw gang. The \"B\" Western was on its last legs as a series format and this vapid entry is a good example.\n\n**2592** _ **Marshal of Amarillo**_ **** Republic, 1948. 60 min. D: Philip Ford. SC: Bob Williams. With Allan \"Rocky\" Lane, Eddy Waller, Mildred Coles, Clayton Moore, Roy Barcroft, Trevor Bardette, Minerva Urecal, Denver Pyle, Charles Williams, Tom Chatterton, Tom London, Lynn Castile, Peter Perkins. A murder takes place at a stage line halfway house and a marshal and his pal arrive to investigate. Well staged mystery motif greatly helps this \"Famous Westerns\" feature.\n\n**2593** _ **Marshal of Cedar Rock**_ **** Republic, 1954. 54 min. D: Harry Keller. SC: M. Coates Webster. With Allan \"Rocky\" Lane, Eddy Waller, Phyllis Coates, Roy Barcroft, Bill Henry, Robert Shayne, John Crawford, John Hamilton, Kenneth MacDonald, Herbert Lytton, Art Dillard. A man is falsely accused of taking part in a bank holdup and a U.S. marshal steps in to help him prove his innocence. A \"B plus\" segment in Allan Lane's \"Famous Westerns\" series.\n\n**2594** _ **Marshal of Cripple Creek**_ **** Republic, 1947 58 min. D: R.G. Springsteen. SC: Earle Snell. With Allan Lane, Bobby Blake, Martha Wentworth, Tom London, Trevor Bardette, Roy Barcroft, Gene (Roth) Stutenroth, William Self, Helen Wallace, Budd Buster, Frank O'Connor, Art Dillard, Silver Harr, George Russell, Herman Hack, Jack Sparks, Leonard Wood. Crooks try to take advantage of the situation when a settlement is turned into a boom town following the discovery of gold. Allan Lane's last entry in the \"Red Ryder\" series and hardly one of the best.\n\n**2595** _ **Marshal of Gunsmoke**_ **** Universal, 1944. 58 min. D: Vernon Keays. SC: William Lively. With Tex Ritter, Russell Hayden, Jennifer Holt, Fuzzy Knight, Harry Woods, Herbert Rawlinson, Ethan Laidlaw, Ray Bennett, Michael Vallon, Ernie Adams, Slim Whitaker, George Chesebro, William Desmond, James Farley, Dan White, Roy Brent, Bud Osborne, Johnny Bond and His Red River Valley Boys (Wesley Tuttle, Paul Sells, Jimmie Dean). A marshal and his lawyer brother enlist the help of a saloon singer in trying to stop her dishonest boss from taking over the town. Appealing Tex Ritter-Russell Hayden film with Jennifer Holt doing a couple of songs while Tex sings \"Git Along Little Dogies.\" British title: _**Sheriff of Gunsmoke**_.\n\n**2596** _ **Marshal of Heldorado**_ **** Lippert, 1950. 63 min. D: Thomas Carr. SC: Ron Ormond and Maurice Tombragel. With James Ellison, Russell Hayden, Raymond Hatton, Fuzzy Knight, Betty (Julie) Adams, Tom Tyler, George J. Lewis, John Cason, Stanley Price, Stephen Carr, Dennis Moore, George Chesebro, Bud Osborne, Jimmie Martin, Wally West, Carl Mathews, Ray Henderson, George Sowards, Cliff Taylor, James Van Horn, Jack Geddes, Ned Roberts. A dude and a buffalo hunter team to take on a murderous clan after stolen money belonging to a colonel and his daughter. Weary entry in \"The Irish Cowboys\" series given some life by Tom Tyler, George J. Lewis, John Cason, Dennis Moore and Stephen Carr as the vicious Tulliver brothers; a remake of _**The Rider of the Law**_ (q.v.) and also called _**Blazing Guns**_.\n\n**2597** _ **Marshal of Laredo**_ **** Republic, 1945. 56 min. D: R.G. Springsteen. SC: Bob Williams. With Wild Bill Elliott, Bobby Blake, Alice Fleming, Peggy Stewart, Roy Barcroft, Tom London, George Carleton, Wheaton Chambers, Tom Chatterton, George Chesebro, Don Costello, Bud Geary, Sarah Padden, Jack O'Shea, Lane Bradford, Kenneth Terrell, Dorothy Granger, Dick Scott, Mary Arden, Jack Kirk, Rose Marie Morei, Melva Anstead. An honest lawyer, opposed to an outlaw gang, is almost hanged for his trouble before being rescued by Red Ryder. Another action filled \"Red Ryder\" segment.\n\n**2598** _ **Marshal of Madrid**_ **** CBS-TV\/20th Century\u2013Fox, 1972. 100 min. Color. D: Richard Donner. SC: Anthony Lawrence, Charles Larson and Jack Turley. With Glenn Ford, Edgar Buchanan, Linda Cristal, Bobby Darin, Victor Campos, James Gregory, Rudolfo Acosta, Taylor Lacher, David Doyle, Warren Kemerling, Peter Ford, Leif Garrett, Richard Kelton, Sandy Kevin, Linda Dangcie, Bert Santos, Richard Rust, Tim Scott, Simon Scott, Michael (Mike) Stokey, Margaret Markov, Richard Yniguez. The marshal of a rural New Mexico county finds himself involved with two cases: a smuggling operation resulting in killings and an ex-convict who is convinced he is Billy the Kid. A fairly entertaining modern-day Western, this telefilm was culled from the \"Crisscross\" and \"A Gun for Billy\" episodes of \"Cade's County\" (CBS-TV, 1971\u201372).\n\n**2599** _ **The Marshal of Mesa City**_ **** RKO Radio, 1939. 62 min. D: David Howard. SC: Jack Lait, Jr. With George O'Brien, Virginia Vale, Leon Ames, Henry Brandon, Harry Cording, Lloyd Ingraham, Slim Whitaker, Joe McGuinn, Mary Gordon, Frank Ellis, Wilfred Lucas, Carl Stockdale, Gaylord (Steve) Pendleton, Monte Montague, Harry Tenbrook, Ed Peil, Sr., Bob Burns, Ben Corbett, Jack Cheatham, Bill Patton, Ed Brady, Spade Cooley, Rube Schaefer, Speed (Aleth) Hansen, Cactus Mack. After saving a girl from the unwanted attentions of a nearby city's sheriff, a cowpoke is made marshal of a lawless town and tries to bring law and order. A bit complicated but still fairly interesting and well made George O'Brien vehicle.\n\n**2600** _ **Marshal of Reno**_ **** Republic, 1944. 54 min. D: Wallace Grissell. SC: Anthony Coldeway. With Wild Bill Elliott, Bobby Blake, Alice Fleming, Herbert Rawlinson, Jay Kirby, Tom London, Kenne Duncan, Charles King, Jack Kirk, LeRoy Mason, Robert Wilke, Fred Burns, Tom Steele, Edmund Cobb, Fred Graham, Blake Edwards, Hal Price, Bud Geary, Jack O'Shea, Al Taylor, Marshall Reed, Tom Chatterton, Carl Sepulveda, Kenneth Terrell, Horace B. Carpenter, Charles Sullivan, George Chesebro, Chick Hannon, Pascale Perry, Jim Corey, Augie Gomez, Neal Hart, Ted Wells, Roy Barcroft (voice). When two towns resort to violence over the question of which one will be the new county seat, Red Ryder arrives to restore order. Fast moving \"Red Ryder\" adventure with a top notch genre cast.\n\n**2601** _ **The Marshal's Daughter**_ **** United Artists, 1953. 71 min. D: William Berke. SC: Bob Duncan. With Hoot Gibson, Laurie Anders, Ken Murray, Harry Lauter, Robert Bray, Bob Duncan, Preston Foster, Jimmy Wakely, Johnny Mack Brown, Buddy Baer, Forrest Taylor, Tom London, Steve Clark, Cecil Elliot, Bette Lou Walters, Francis Ford, Julian Upton, Bob Gross, Lee Phelps, Ted Jordan, Harry Harvey, Danny Duncan, Tex Ritter (narrator). The daughter of a U.S. marshal becomes a masked rider in order to capture an outlaw gang. Fairly amusing genre spoof that should please fans; produced by Ken Murray.\n\n**2602** _ **Marshals in Disguise**_ **** Allied Artists, 1954. 54 min. D: Frank McDonald. SC: William Raynor and Maurice Tombragel. With Guy Madison, Andy Devine, Tristram Coffin, Norma Eberhardt, Rick Vallin, John Merton, Leonard Penn, Fred Kelsey, Bill Hale, Pat Mitchell, Bud Osborne, John Eldredge, Guy Beach, John Reynolds, Anthony Sydes, David Sharpe, James Bush, Don Turner. A bank clerk uses money he stole from gold shipments to try and buy the establishment where he works with Wild Bill Hickok and Jingles P. Jones investigating thefts and the two lawmen come to the aid of a prospector about to be fleeced of his claim by a dishonest assayer. Fair theatrical release made up of two episodes, \"Civilian Clothes Story\" and \"Lost Indian Mine,\" of the 1951\u201358 TV series \"The Adventures of Wild Bill Hickok.\"\n\n**2603** _ **Martyrs of the Alamo**_ **** Triangle, 1915. 71 min. D-SC: Christy Cabanne. With Sam De Grasse, A.D. (Allan) Sears, Walter Long, Alfred Paget, Fred Burns, John T. Dillon, Douglas Fairbanks, Juanita Hansen, Ora Carew, Tom Wilson, Augustus Carney, Monte Blue, Betty Marsh, Jack Prescott, Joseph Belmont. Texas settlers make an heroic stand at the Alamo against the army of General Santa Anna. Pretty fair re-staging of the famed historical event, supervised by D.W. Griffith, with future stars Douglas Fairbanks and Monte Blue in small roles.\n\n_**Mask of the Musketeers**_ see _**Zorro and the Three Musketeers**_\n\n**2604** _ **The Mask of Zorro**_ **** Columbia\/Tri-Star, 1998. 136 min. Color. D: Martin Campbell. SC: John Eskow, Ted Elliott and Terry Rossio. With Anthony Hopkins, Antonio Banderas, Catherine Zeta-Jones, Matt Letscher, William Marquez, Pedro Armendariz (Jr.), L.Q. Jones, Jose Perez, Tony Amendola, Julieta Rosen, Victor Rivers, Maury Chaykin, Moises Suarez, Humberto Elizondo, Erika Carlson, Vanessa Bauche, Eduardo Lopez, Manolo Pastor, Rudy Miller, Fernando Becerril, Alberto Carreras, Gonzalo Lora, Paul Ganus, Enrike Palma, Diego Sieres, Jose Maria de Tavira, Paco Morayta, Pedro Altamirano, Luisa Huertas, Tony Genaro. After being imprisoned for two decades, Zorro befriends a young man whose brother has been murdered by a cruel captain and the two plan to stop the wicked former Spanish governor from using gold mined by prisoners to purchase California from Mexico. Smooth, big theatrical moneymaker followed by _**The Legend of Zorro**_ (q.v.).\n\n_**The Masked Conqueror**_ see _**Zorro and the Three Musketeers**_\n\n**2605** _ **The Masked Raiders**_ **** RKO Radio, 1949. 60 min. D: Lesley Selander. SC: Norman Houston. With Tim Holt, Richard Martin, Marjorie Lord, Gary Gray, Frank Wilcox, Charles Arnt, Tom Tyler, Harry Woods, Clayton Moore, Houseley Stevenson, Bill George (Jay Kirby). When dishonest bankers try to take ranchers' land, the latter form a group of masked riders led by a young woman. Typically good entry in Tim Holt's RKO series with nice work by Marjorie Lord as the head of the vigilantes.\n\n**2606** _ **The Masked Rider**_ **** Arrow, 1919. 15 Chapters. D-SC: Aubrey M. Kennedy. With Ruth Stonehouse, Harry Myers, Paul Panzer, Edna Holland, Marie Treador, Blanche Gillespie, Robert Taber, Jack Chapman, Boris Karloff, George Murdock, George Cravy. The Texas Rangers try to stop a Mexican cattle rustler and his gang from taking over a region along the U.S.-Mexican border. Thirteen chapters from this violent vintage serial have survived and is well worth viewing by serial fans.\n\n**2607** _ **The Masked Rider**_ **** Universal, 1941. 58 min. D: Ford Beebe. SC: Sherman Lowe and Victor McLeod. With Johnny Mack Brown, Fuzzy Knight, Nell O'Day, Grant Withers, Roy Barcroft, Guy D'Ennery, Virginia Carroll, Richard Botiller, Fred Cordova, The Guadalajara Trio, Jose Cansino Dancers, Al Haskell, Robert O'Connor, Rico De Montez, Carmela Cansino. In South America, two U.S. cowboys get jobs in a silver mine and go after a masked man stealing ore shipments. Pleasant, somewhat tongue-in-cheek, Johnny Mack Brown vehicle.\n\n**2608** _ **Mason of the Mounted**_ **** Monogram, 1932. 58 min. D-SC: Harry Fraser. With Bill Cody, Andy Shuford, Nancy Drexel, Art Smith (Art Mix\/George Kesterson), Jack Carlisle, Blackie Whiteford, Nelson McDowell, James Marcus, Joe Dominguez, Leroy Mason, Dick Dickinson, Frank Hall Crane, Earl Dwire, Jack Long, Gordon McGee. A Canadian Mountie comes to the U.S. to bring back a notorious outlaw, the leader of a horse stealing gang. Fair Bill Cody feature.\n\n**2609** _ **Masquerade**_ **** Wrather Corporation, 1956. 75 min. Color. D: Earl Bellamy. SC: Wells Root, Charles Carson, Robert E. Schaefer, Eric Friedwald and Robert Leslie Bellem. With Clayton Moore, Jay Silverheels, Allen Pinson, Wayne Burson, Myron Healey, Helen Marshall, Margaret Stewart, Rand Brooks, Louise Lewis, Don C. Harvey, Pierce Lyden, John Cason, William Fawcett, Zon Murray, Nolan Leary, George Barrows, David Saber, Paul Ingle, William Challee, Jason Johnson, John Cliff, Sandy Sanders, John Maxwell. Pretending to be a Mexican with hearing problems, the Lone Ranger tries to thwart a gold shipment robbery, then he and Tonto assist a Mexican revolutionary leader and try to capture a gang of masked marauders. Well made TV movie from \"The Lone Ranger\" (ABC-TV, 1949\u201357) episodes \"Code of Honor,\" \"Dead Eye\" and \"The Turning Point.\"\n\n**2610** _ **Massacre**_ **** First National, 1934. 70 min. D: Alan Crosland. SC: Ralph Block and Sheridan Gibney. With Richard Barthelmess, Ann Dvorak, Dudley Digges, Claire Dodd, Henry O'Neill, Robert Barrat, Arthur Hohl, Philip Faversham, George Blackwood, Sidney Toler, Clarence Muse, Charles Middleton, Tully Marshall, Wallis Clark, William V. Mong, DeWitt Jennings, Juliet Ware, James Eagles, Frank McGlynn, Agnes Narcha. An educated Indian chief tries to remove corrupt government officials who have been cheating his people. A good look at injustice in the reservation system makes this a film worth viewing; Richard Barthelemess is fine as the caring Indian leader.\n\n**2611** _ **Massacre**_ **** 20th Century\u2013Fox, 1956. 76 min. Color. D: Louis King. SC: D.D. Beauchamp. With Dane Clark, James Craig, Martha Roth, Miguel Torruco, Jaimie Fernandez, Jose Munoz. Crooked traders sells guns to Indians resulting in the needless killing of settlers. Average affair filmed in Mexico.\n\n_**Massacre at Fort Grant**_ see _**Massacre at Fort Perdition**_\n\n_**Massacre at Fort Holman**_ see _**A Reason to Live, a Reason to Die!**_\n\n**2612** _ **Massacre at Fort Perdition**_ **** Avco-Embassy, 1965. 95 min. Color. D: J. Douglas (Jose Maria Elorrieta). SC: Jose Luis Navarro and Jose Maria Elorrieta. With Jerry Cobb (German Cobos), Marta May, Ethel Rojo, Georges Gordon, Hugh Pepper, Mariano Vidal, Aldo Sambrell, Luis Villar, Cris Huerta, Frank Brana, Luis Barboo, Angel Ortiz, Jose Sancho, Julio Perez Tabernero, Guillermo Mendez. A soldier, in civilian clothes, and his new bride are found by a rescue force at an outpost where the rest of the inhabitants have been murdered and he is branded a traitor. Another violent European oater filmed in Spain as _**Fuerte Perdido**_ (Doomed Fort) and remade as _**Fury of the Apaches**_ (q.v.); British title: _**Massacre at Fort Grant**_.\n\n**2613** _ **Massacre at Fort Phil Kearney**_ **** NBC-TV\/Universal, 1966. 49 min. Color. SC: Harold Swanton. With Richard Egan, Robert Fuller, Robert Pine, Peter Duryea, Phyllis Avery, Carroll O'Connor, Michael Sarrazin, Jeffrey Scott, Brandon Carroll, Tom Anthony. Two Army officers have different views on how to deal with Indians, one wants to pacify them while the other believes in force. Satisfying telefeature originally shown as a segment of \"The Bob Hope Chrysler Theatre\" (NBC-TV, 1963\u201367) on October 26, 1966.\n\n**2614** _ **Massacre at Grand Canyon**_ **** Columbia, 1965. 90 min. Color. D: Alfredo Antonini (Abert Band). SC: E.C. Geltman and Alfredo Antonini. With James Mitchum, Jill Powers, Eduardo Ciannelli, Giorgio (George) Ardisson, Burt Nelson, Giacomo Rossi Stuart, Andrea Giordana, Milla Sannoner, Nando Poggi. Feuding families hire gunmen to kill a sheriff while the man's brother and fiancee try to get the locals to fight back. Well done European Western with nice photography by Enzo Barboni; Rodd Dana sings the title song. Produced in Italy in 1963 as _**I Pascoli Rossi**_ (Red Pastures) by Ultra Film\/Prodi Cinematografica with some sources claiming Sergio Corbucci co-directed.\n\n**2615** _ **Massacre at Marble City**_ **** Rapid-Film, 1964. 87 min. Color. D: Paul Martin (Franz Gottlieb). SC: Alex Berg, Hans Billian and W.P. Zibaso. With Brad Harris, Mario Adorf, Dieter Borsche, Horst Frank, Marianne Hoppe (Dorothee Parker), Ralf Walter, Thomas Adler, Serge Marquand, Philippe Lemaire. When Indians bring gold for trade at a frontier settlement, greedy miners try to find the location of the ore and violence results. More bloodshed from Europe in this West German oater made as _**Die Goldsucher von Arkansas**_ (The Gold Hunter of Arkansas).\n\n**2616** _ **Massacre at Sand Creek**_ **** CBS-TV\/Columbia, 1956. 74 min. D: Arthur Hiller. SC: William Sackheim. With John Derek, Everett Sloane, Gene Evans, H.M. Wynant, William Schallert, Roy Roberts, Ken Mayer, Rick Vallin, Michael Granger, William Henry, William Bryant, Anthony Lawrence, Marshall Bradford, Robert Shield, Robert Bice, Ben Wright, Dick Joy (host). A tribe of ill-armed Cheyenne braves are attacked by an Indian-hating colonel and his troops. Fair telefilm originally shown on December 27, 1956, as a segment of \"Playhouse 90\" (CBS-TV, 1956\u201361).\n\n**2617** _ **Massacre Canyon**_ **** Columbia, 1954. 64 min. D: Fred F. Sears. SC: David Lang. With Philip Carey, Audrey Totter, Douglas Kennedy, Jeff Donnell, Guinn Williams, Charlita, Ross Elliott, Ralph Dumke, Mel Welles, Chris Alcaide, Steven Ritch, John Pickard, James Flavin, Bill Hale. A sergeant and two Army privates are assigned to guard a shipment of rifles wanted by marauders. Threadbare action film from producer Wallace MacDonald.\n\n_**Massacre Hill**_ see _**Eureka Stockade**_\n\n**2618** _ **Massacre River**_ **** Allied Artists, 1949. 75 min. D: John Rawlins. SC: Louis Stevens and Otto Englander. With Guy Madison, Rory Calhoun, Carole Mathews, Cathy Downs, Johnny Sands, Steve Brodie, Art Baker, Iron Eyes Cody, Gregg Barton, Emory Parnell, Queenie Smith, Eddy Waller, James Bush, John Holland, Douglas Fowley, Harry Brown, Kermit Maynard, Olin Howlin, J.W. Cody. A trio of cavalry officers assigned to the West after the Civil War jeopardize their friendship over a colonel's pretty daughter and the machinations of a gambling establishment owner. Routine oater with some star appeal; based on Harold Bell Wright's 1916 novel _When a Man's a Man_ and filmed under that title in 1924 and 1935 (qq.v.).\n\n_**Massacre Time**_ see _**The Brute and the Beast**_\n\n**2619** _ **The Master Gunfighter**_ **** Taylor-Laughlin Distributing Company, 1975. 120 min. Color. D: Frank Laughlin. SC: Harold Lapland. With Tom Laughlin, Ron O'Neal, Lincoln Kilpatrick, Victor Campos, Geo Anne Sosa, Barbara Carrera, Hector Elias, Michael Lane, Patti Clifton, Henry Wills, Angelo Rossitto, Alberto Morin, Franco Casaro, Robert Tafur, Edward Colmans, Robert Hoy, Burgess Meredith (narrator). A gunman hates his trade but goes after a Spanish land baron who murders Indians for their gold so he can pay his taxes. Very bad psychological Western; a reworking of the 1966 Japanese feature _**Goyokin**_.\n\n**2620** _ **Masterson of Kansas**_ **** Columbia, 1955. 73 min. Color. D: William Castle. SC: Douglas Heyes. With George Montgomery, Nancy Gates, James Griffith, Jean Willes, Benny Rubin, William Henry, David Bruce, Bruce Cowling, Gregg Barton, Donald Murphy, Sandy Sanders, Gregg Martell, Jay Silverheels, John Maxwell. Lawmen Bat Masterson, Wyatt Earp and Doc Holliday team to save a negotiator who has made a treaty giving grazing lands to Indians instead of cattlemen. There is not much to recommend this Sam Katzman production of pseudo-historical pap other than it is competently made.\n\n**2621** _ **Matalo!**_ **** Rofima\/Copercines, 1971. 83 min. Color. D: Cesare Canavari. SC: Eduardo Manazos. With Lou Castel, Corrado Pani, Antonio Salinas, Luis Davila, Claudia Gravy, Miguel Del Castillo, Ana Maria Noe, Ana Maria Mendoza, Mirella Pamphili, Diana Sorel, Bruno Boschetti, Joaquin Parra. A bounty hunter is hired by Wells Fargo to retrieve gold stolen in a holdup by four outlaws, one of whom is left for dead by his cohorts. Fast paced Italian-Spanish co-production; remake of _**Kill the Wicked**_ (q.v.).\n\n**2622** _ **The Matchmaking Marshal**_ **** Allied Artists, 1955. 54 min. D: S. Roy Luby and Frank McDonald. SC: Maurice Tombragel and William Raynor. With Guy Madison, Andy Devine, Lyle Talbot, Douglas Fowley, Rand Brooks, Karl \"Killer\" Davis, Henry Kulky, House Peters, Jr., Frank Scannell, Ann Carrol, Louise Lorimer, Paul McGuire, Robert Jordan, Nelson Leigh, Forrest Taylor, Ed Cassidy, Fred Sherman. Wild Bill Hickok and his pal Jingles investigate a murder involving two feuding families and Wild Bill has Jingles challenge a wrestler so they can stop a bank holdup. Okay theatrical feature made of up two episodes of \"The Adventures of Wild Bill Hickok\" (1951\u201358): \"Marriage Feud of Ponca City\" and \"Wrestling Story.\"\n\n**2623** _ **Maverick**_ **** Warner Bros., 1994. 127 min. Color. D: Richard Donner. SC: William Goldman. With Mel Gibson, Jodie Foster, James Garner, Graham Greene, Alfred Molina, James Coburn, Dub Taylor, Geoffrey Lewis, Paul L. Smith, Dan Hedaya, Dennis Fimple, Denver Pyle, Clint Black, Max Perlich, Art La Fleur, Leo V. Gordon, Paul Tuerpe, Jean De Baer, Paul Brinegar, Hal Ketchum, Corey Feldman, John Woodward, Jesse Eric Carroll, Toshonnie Touchin, John Meier, Steve Chambers, Read Morgan, Vilmos Zsigmond, Waylon Jennings, Kathy Mattea, Carlene Carter, Vince Gill, Janice Gill, William Smith, Chuck Hart, Doug McClure, Henry Darrow, Michael Paul Chan, Richard Blum, Bert Remsen, Robert Fuller, Donal Gibson, William Marshall, Bill Henderson, Carl Bartlett, Linda Hunt, Charles Kierkop, James Drury, Danny Glover, Will Hutchins, Margot Kidder, Reba McIntire, Don Stark. A card sharp teams with a saloon woman to get the money needed to enter a high stakes poker game. Overlong but popular revival of \"Maverick\" (ABC-TV, 1957\u201362) with original series star James Garner cast as a lawman.\n\n**2624** _ **The Maverick Queen**_ **** Republic, 1956. 90 min. Color. D: Joseph Kane. SC: Kenneth Gamet and DeVallon Scott. With Barbara Stanwyck, Barry Sullivan, Scott Brady, Mary Murphy, Wallace Ford, Jim Davis, Howard Petrie, Emile Meyer, Walter Sande, George Keymas, John Doucette, Taylor Holmes, Pierre Watkin, Tristram Coffin, Jack O'Shea, Robert Swan, William Loftos, Herbert Jones, Jack Harden, Carol Brewster, Karen Scott. A woman hotel keeper works with an outlaw gang but finds herself falling in love with a newly arrived man not knowing he is an undercover Pinkerton agent about to break up the rustlers. The stars and director rise above the mediocre material to make this entertaining.\n\n**2625** _ **McCabe and Mrs. Miller**_ **** Warner Bros., 1971. 107 min. Color. D: Robert Altman. SC: Robert Altman and Brian McVey. With Warren Beatty, Julie Christie, Rene Auberjonois, Hugh Naughton, Shelley Duvall, Michael Murphy, John Schuck, Corey Fisher, Keith Carradine, William Devane, Anthony Holland, Bert Remsen, Elizabeth Murphy. At the turn of the century, a gambler and his lady friend set up a successful brothel in a small town but hoodlums try to take it over. Overrated drama will satisfy Robert Altman followers.\n\n**2626** _ **McClintock!**_ **** United Artists, 1963. 127 min. Color. D: Andrew V. McLaglen. SC: James Edward Grant. With John Wayne, Maureen O'Hara, Yvonne De Carlo, Patrick Wayne, Stefanie Powers, Jack Kruschen, Chill Wills, Jerry Van Dyke, Edgar Buchanan, Bruce Cabot, Perry Lopez, Michael Pate, Strother Martin, Gordon Jones, Robert Lowery, H.W. Gim, Ed Faulkner, Aissa Wayne, Chuck Roberson, Mari Blanchard, John Stanley, Hal Needham, Pedro Gonzales, Jr., Hank Worden, Leo Gordon, Karl Noven, Bob Steele, Big John Hamilton, Ralph Volkie. A rich land baron wants state government to get rid of incompetent officials and he also has domestic problems with his estranged wife and daughter. Well done, somewhat tongue-in-cheek, John Wayne production that is sure to please his fans.\n\n**2627** _ **McCloud:**_ _**Who Killed Miss U.S.A.?**_ **** NBC-TV\/Universal, 1970. 100 min. Color. D: Richard A. Colla. SC: Stanford Whitmore, Richard Levinson and Willliam Link. With Dennis Weaver, Diana Mauldaur, Craig Stevens, Mark Richman, Julie Newmar, Terry Carter, Mario Alcalde, Raul Julia, Shelly Novack, Michael Bow, Nefti Millet, Kathy Stritch, Gregory Sierra, Bill Baldwin. A marshal from the Southwest arrives in New York City with a witness who is promptly kidnapped, leading the lawman into a murder case. Telefeature very similar to _**Coogan's Bluff**_ (q.v.) and the pilot for \"McCloud\" (NBC-TV, 1971\u201376); better than average for this kind of fare. Retitled _**Portrait of a Dead Girl**_.\n\n**2628** _ **McKenna of the Mounted**_ **** Columbia, 1932. 66 min. D: D. Ross Lederman. SC: Stuart Anthony. With Buck Jones, Greta Grandstedt, James Flavin, Walter McGrail, Niles Welch, Mitchell Lewis, Claude King, Glenn Strange, Bud Osborne, Edmund Cobb, Bob Reeves, Jack Kennedy, Albert J. Smith, Merrill McCormick, Maston Williams, John Lowell. A disgraced Mounted Policeman leaves the service and joins an outlaw gang but is really working to bring in the desperadoes. Not one of Buck Jones' better efforts.\n\n_**The McMaster...Tougher Than the West Itself**_ see _**The McMasters**_\n\n**2629** _ **The McMasters**_ **** Chevron, 1970. 97 min. Color. D: Alf Kjellin. SC: Harold Jacob Smith. With Burl Ives, Brock Peters, David Carradine, Nancy Kwan, Jack Palance, John Carradine, L.Q. Jones, R.G. Armstrong, Dane Clark, Frank Raiter, Alan Vint, Marian Brash, Neil Davis, William Kiernan, Richard Alden, David Strong. Following the Civil War an ex-slave who fought for the North returns home to be given half-interest in the farm he once worked but finds he is resented by the locals. Unsuccessful drama that was issued theatrically in two versions with different endings, one running 89 minutes; also called _**The McMaster...Tougher Than the West Itself**_.\n\n**2630** _ **Me Gustan Valentones!**_ (I Like Boasters) **** Producciones Sotomayeer, 1959. 92 min. D-SC: Julian Soler. With Rosita Quintana, Luis Aguilar, Eulalio Gonzalez, Piporro, Andres Soler, Carlota Solares, Tito Novaro, Augustin Fernandez, Salvador Lozano, Emilio Garibay, Virginia Manzano, Alejandro Reyna, Yoya Velazaquez, Arturo Fernandez. A ranch girl, who is pursued by all the men in the area, decides to marry a man she met by mail. Pleasant Mexican Western comedy.\n\n_**Mean Justice**_ see _**This Rugged Land**_\n\n**2631** _ **The Meanest Men in the West**_ **** Universal, 1976. 92 min. Color. D: Samuel Fuller and Charles S. Dubin. SC: Ed Waters and Samuel Fuller. With Lee J. Cobb, Charles Bronson, Lee Marvin, Miriam Colon, James Drury, Albert Salmi, Don Mitchell, Sara Lane, Brad Weston, Charles Grodin, Ross Hagen, Gary Clarke, Michael Conrad, Warren Kemmerling, Michael Mikler, Jan Stine, Lance Kerwin, Betty Baird, Regis Cordic, Bonnie Bartlett, Ron Soble, Doug McClure. Growing up hating his younger brother, an outlaw plans to use him in a scheme to rustle cattle from the rancher judge who sent him to prison. Uneven feature made up of the \"It Tolls for Thee\" (telecast November 21, 1962) and \"The Reckoning\" (telecast September 13, 1967) episodes of \"The Virginian\" (NBC-TV, 1962\u201371) and issued abroad theatrically.\n\n**2632** _ **Meanwhile, Back at the Ranch**_ **** Rancho Films, 1977. 70 min. D-SC: Richard Patterson. With John Wayne, Buck Jones, Ken Maynard, Gene Autry, Roy Rogers, Eddie Dean, Tim McCoy, William Boyd, George \"Gabby\" Hayes, Smiley Burnette, Bob Steele, George O'Brien, Buster Crabbe, Lash LaRue, Monte Hale, Hoot Gibson, Johnny Mack Brown, Rex Allen, Don \"Red\" Barry, Wild Bill Elliott, Tex Ritter, Charles Starrett, Tom Tyler, Dale Evans, Tim Holt, Tom Keene, Allan \"Rocky\" Lane, Fred Scott, The Sons of the Pioneers, Iron Eyes Cody, Bobby Blake, Eilene Janssen, Raymond Hatton, Charles King, Al St. John, Dub Taylor; Pat Buttram (narrator). All of the great cowboy stars are united for one mighty feature film but due to contract disputes it is never released. This film attempts to make a completely new movie of clips from genre features of the past, but it is only of interest due to its oddity value. Shown at the Cannes Film Festival in 1977 but not issued theatrically; Eddie Dean sings the title song.\n\n**2633** _ **The Medico of Painted Springs**_ **** Columbia, 1941. 58 min. D: Lambert Hillyer. SC: Winston Miller and Wyndham Gittens. With Charles Starrett, Terry Walker, Richard Fiske, Ray Bennett, Wheeler Oakman, The Simp-Phonies, Ben Taggert, Bud Osborne, Edmund Cobb, Edythe Elliott, Steve Clark, Lloyd Bridges, George Chesebro, Charles Hamilton, Jim Corey, Art Mix, Buck Connors, Hank Bell, Eddie Laughton, John Tyrrell, George Huggins, Carl Sepulveda. An Army doctor recruiting men for the Rough Riders finds himself in the middle of warfare between sheep men and cattle raisers. Good start to the all too brief \"Dr. Monroe\" or \"Medico\" series, based on the works of James L. Rubel; followed by _**Thunder Over the Prairie**_ and _**Prairie Stranger**_ (qq.v.). Also called _**The Doctor's Alibi**_.\n\n**2634** _ **Melody of the Plains**_ **** Spectrum, 1937. 55 min. D: Sam Newfield. SC: Bennett Cohen. With Fred Scott, Al St. John, Louise Small, Hal Price, Lew Meehan, Slim Whitaker, Lafe McKee, David Sharpe, Bud Jamison, Carl Mathews, George Fiske, George Morrell. A cowpoke, who mistakenly thinks he killed a man, goes to work for the supposed victim's rancher father whose spread is being sought by the crooks responsible for the shooting. Somewhat complicated and low grade but still worth a look to hear Fred Scott's singing since he was one of the best of the Western warblers. Remake of _**Gun Law**_ (1933) [q.v.].\n\n**2635** _ **Melody Ranch**_ **** Republic, 1940. 84 min. D: Joseph Santley. SC: Jack Moffitt and F. Hugh Herbert. With Gene Autry, Jimmy Durante, Ann Miller, Barton MacLane, Barbara Jo Allen (Vera Vague), George \"Gabby\" Hayes, Jerome Cowan, Mary Lee, Joseph Sawyer, Horace McMahon, Clarence Wilson, William Benedict, Ruth Clifford, Maxine Ardell, Veda Ann Borg, George Chandler, Jack Ingram, Horace Murphy, Lloyd Ingraham, Tom London, John Merton, Edmund Cobb, Slim Whitaker, Curley Dresden, Dick Elliott, Billy Bletcher, Art Mix, George Chesebro, Tiny Jones, Herman Hack, Jack Kirk, Merrill McCormick, Wally West, Bob Wills and His Texas Playboys, Frankie Marvin, Carl Cotner, Tex Cooper, Chick Hannon, Jim Corey, Jane Keckley, Patricia Bonner, Frank Hagney, Jack Montgomery, Buck Bucko, Joe Yrigoyen. Radio star Gene Autry returns home to become an honorary sheriff and finds the area plagued by racketeers. Entertaining Gene Autry opus butchered for TV by a half-hour.\n\n**2636** _ **Melody Trail**_ **** Republic, 1935. 60 min. D: Joseph Kane. SC: Sherman Lowe. With Gene Autry, Smiley Burnette, Ann Rutherford, Wade Boteler, Wally Costello, Alan Bridge, Marie Quillan, Gertrude Messinger, Tracy Layne, Abe Lefton, George DeNormand, Jane Barnes, Ione Reed, Marion Downing, Herman Hack, Chick Hannon, Tex Cooper, Tom Smith, Buck (dog). Cowpokes Gene Autry and Frog Millhouse get jobs as cooks on a ranch where the workers have quit and the owner uses his daughter's girl friends as hands. Pleasant Gene Autry vehicle with more emphasis on comedy and music than action.\n\n**2637** _ **Men of America**_ **** RKO Radio, 1932. 58 min. D: Ralph Ince. SC: Jack Jungmeyer and Humphrey Pearson. With Bill (William) Boyd, Charles \"Chic\" Sale, Dorothy Wilson, Ralph Ince, Henry Armetta, Alphonse Ethier, Theresa Maxwell Conover, Eugene Strong, Fatty Layman, Fred Lindstrom, Frank Mills, Inez Palange, F. Flink, Ernie Adams, Harry Strang, Harry Sullivan. A newcomer is blamed for a series of robberies and killings, committed by escaped convicts, and he teams with an old store owner to uncover the truth. Pretty fair \"B\" picture which some sources claim was co-directed by William Boyd.\n\n**2638** _ **Men of Texas**_ **** Universal, 1942. 71 min. D: Ray Enright. SC: Harold Shumate. With Robert Stack, Anne Gwynne, Broderick Crawford, Jackie Cooper, Ralph Bellamy, Jane Darwell, Leo Carrillo, John Litel, William Farnum, Janet Beecher, J. Frank Hamilton, Kay Linaker, Joseph Crehan, Addison Richards, Frank Hagney, Lane Chandler, Rex Lease, Alan Bridge, Edmund Cobb, Ethan Laidlaw, Harry Strang, Edgar Dearing, Kernan Cripps, Ben Taggart, Jack Cheatham, Dorothy Vaughan, Delos Jewkes, John Peters, Cordell Hickman, Bob Barron, David Clarke, Sherman E. Sanders, Charles Salerno. At the end of the Civil War a newspaper reporter and photographer are sent to Texas to check on reports that an uprising may take place. Typically slick drama from Universal that should please genre fans.\n\n**2639** _ **Men of the North**_ **** Metro-Goldwyn-Mayer, 1930. 61 min. D: Hal Roach. SC: Richard Schayer. With Gilbert Roland, Barbara Leonard, Arnold Korff, George Davis, Robert Elliott, Nina Quartero, Robert Graves, Jr., Frank Lackteen, Lew Meehan, Bud Osborne, Fletcher Norton, Dorothy DeBorba. A Canadian adventurer is falsely accused of stealing gold from a mine but his innocence is believed by the owner's pretty daughter. Vintage melodrama is worth a look to see a youthful Gilbert Roland who starred (billed as Luis Alonso, his real name) in a Spanish language version entitled _**Monsieur Le Fox**_ , filmed simultaneously by producer-director Hal Roach, who also made French (starring Andre Luguet) and German (starring John Reinhardt) versions as _**Monsieur Le Fox**_ , and in Italian (starring Franco Corsaro) as _**Luigi La Volpe**_.\n\n**2640** _ **Men of the Plains**_ **** Colony, 1936. 63 min. D: Robert Emmett (Tansey). SC: Robert Emmet (Tansey) and Jack Cowell. With Rex Bell, Joan Barclay, George Hall, Charles King, Forrest Taylor, Roger Williams, Ed Cassidy, Lafe McKee, Jimmy Aubrey, Jack Cowell, Budd Buster, Sherry Tansey, Denver Dixon, Bud Pope, Jack Evans, Art Felix, George Morrell, Barney Beasley. A government agent is assigned to look into gold shipment thefts and learns two of the town's leading citizens are involved. Passable action feature produced by Arthur and Max Alexander.\n\n**2641** _ **Men of the Timberland**_ **** Universal, 1941. 62 min. D: John Rawlins. SC: Maurice Tombragel and Griffin Jay. With Richard Arlen, Andy Devine, Linda Hayes, Francis McDonald, Willard Robertson, Paul E. Burns, Gaylord (Steve) Pendleton, Hardie Albright, Roy Harris (Riley Hill), John Ellis, Jack Rice, Ethan Laidlaw, Tom London, Ralph Sanford, Art Miles, Chuck Morrison, Ken Christy, Jack Voglin, Sue Moore. A lumber man uncovers a plan by a crook to cut timber over a large area, the illegal scheme having been approved by bribed government officials. Action filled \"B\" effort from the popular Richard Arlen-Andy Devine series for Universal.\n\n_**Men with Steel Faces**_ see _**The Phantom Empire**_\n\n_**Men with Whips**_ see _**Rangle River**_\n\n**2642** _ **Men Without Law**_ **** Columbia, 1930. 60 min. D: Louis King. SC: Dorothy Howell. With Buck Jones, Carmelita Geraghty, Tom Carr, Lydia Knott, Harry Woods, Fred Burns, Syd Saylor, Fred Kelsey, Lafe McKee, Ben Corbett, Art Mix, Donald Reed, Bob Burns. Returning home from World War I, a man finds his younger brother has been arrested for taking part in a bank robbery and he is soon captured by an outlaw leader. Good early Buck Jones sound film with a well lighted climactic fight sequence. Background music includes \"La Paloma\" and the gang members singing \"Bury Me Not on the Lone Prairie.\"\n\n_**Mercenaries of the Rio Grande**_ see _**The Treasure of the Aztecs**_\n\n**2643** _ **The Mercenary**_ **** United Artists, 1970. 105 min. Color. D: Sergio Corbucci. SC: Luciano Vincenzoni, Sergio Spina and Sergio Corbucci. With Jack Palance, Franco Nero, Tony Muscante, Giovanna Ralli, Eduardo Fajardo, Julio Pena, Raf Baldasssare, Franco Ressel, Bruno Corazzari, Remo de Angeles, Joe Kamel, Franco Giacobini, Vicente Jja, Jose Riesgo, Angel Ortiz, Fernando Villena, Tito Garcia, Angel Alvarez. Two bitter mercenary enemies are after a treasure but it is also sought by a revolutionary, a peasant girl and a miner. Pretty good Italian oater with the usual violence and some not-so-usual humor. Made by Produzioni Europee Associate\/Produzioni Associate Delphos S.P.A.\/Profilms as _**El Mercenario**_ (The Mercenary) and also called _**Revenge of a Gunfighter**_.\n\n**2644** _ **Mesa of Lost Women**_ **** Howco International, 1953. 69 min. D: Herbert Tevos and Ron Ormond. SC: Herbert Tevos and uncredited Orville M. Hampton. With Jackie Coogan, Richard Travis, Allan Nixon, Mary Hill, Robert Knapp, Tandra Quinn, Harmon Stevens, Samuel Wu, George Barrows, Chris-Pin Martin, Nico Lek, Dean Riesner, Fred Kelsey, Kelly Drake, Katina Vea (Katherine Victor), John Martin, Angelo Rossitto, Margia Dean, Julian Rivero, Suzanne Ridgeway, John George, Doris Hart, Dolores Fuller, Sherry Moreland, Chris Randall, Jack Low, Lyle Talbot (narrator). On a remote mesa in the Mexican desert a mad scientist tries to develop a serum that will create a super-race of spider women. Low grade and awful. TV title: _**Lost Women**_.\n\n**2645** _ **Mesquite Buckaroo**_ **** Metropolitan, 1939. 55 min. D: Harry S. Webb. SC: George Plympton. With Bob Steele, Carolyn (Clarene) Curtis, Frank LaRue, Charles King, Ted Adams, Joe Whitehead, Ed Brady, Snub Pollard, Carleton Young, John Elliott, Juanita Fletcher, Gordon Roberts, Jimmy Aubrey. A rodeo cowboy gets involved with a gang of crooks while romancing a pretty girl. Shoddy Bob Steele film with far too much rodeo stock footage.\n\n**2646** _ **A Message to Garcia**_ **** 20th Century\u2013Fox, 1936. 90 min. D: George Marshall. SC: W.P. Lipscomb and Gene Fowler. With Wallace Beery, Barbara Stanwyck, John Boles, Herbert Mundin, Martin Garralaga, Enrique Acosta, Jose Luis Tortosa, Joan Torena, Alan Hale, Mona Barrie, Warren Hymer, Frederik Vogeding, Sam Appel, Yorke Sherwood, Iris Adrian, Davison Clark, Lon Chaney, Jr., Del Henderson, Rita (Hayworth) Cansino, Philip Morris, Blanche Vischer, Pat Moriarty, Octavio Giraud, Fred Goday, Art Dupuis, Manuel Paris, Manuel Peluffo, David Clyde, George Irving, John Carradine (voice). President William McKinley sends an Army officer to Cuba to warn a guerrilla leader that the U.S. has declared war on Spain. Silly historical melodrama which not even its stars can save.\n\n_**Meteor Monster**_ see _**Teenage Monster**_\n\n**2647** _ **The Mexicali Kid**_ **** Monogram, 1938. 58 min. D: Wallace Fox. SC: Robert Emmett (Tansey). With Jack Randall, Eleanor Stewart, Wesley Barry, Ed Cassidy, Bud Osborne, George Chesebro, William von Bricken, Sherry Tansey, Ernie Adams, Frank LaRue, Buzz Barton, Archie Ricks, Denver Dixon, Hal Price, Glenn Strange, Chester Gan, Fred Parker, Billy Bletcher. A cowboy befriends an outlaw only to learn the man has been hired by a crooked foreman to help him take over a valuable ranch. Fair Jack Randall vehicle. Remade as _**Haunted Trails**_ (q.v.).\n\n**2648** _ **Mexicali Rose**_ **** Republic, 1939. 60 min. D: George Sherman. SC: Gerald Geraghty. With Gene Autry, Smiley Burnette, Noah Beery, Luana Walters, William Farnum, LeRoy Mason, William Royle, Wally Albright, Kathryn Frey, Roy Barcroft, Richard Botiller, John Beach, Merrill McCormick, Fred \"Snowflake\" Toones, Sherry Hall, Al Taylor, Josef Swickard, Tom London, Jack Ingram, Eddie Parker, Henry Otho, Joe Dominguez, Al Haskell. Bogus oil company officials hire Gene Autry, who is not aware they are crooks, to help sell their stock to the public. Good Gene Autry film with fine work by Noah Beery as a lovable bandit leader.\n\n_**Mexican Gold**_ see _**Return of the Outlaws**_\n\n**2649** _ **Mexican Spitfire Out West**_ **** RKO Radio, 1940. 76 min. D: Leslie Goodwins. SC: Charles E. Roberts and Jack Townley. With Lupe Velez, Leon Errol, Donald Woods, Elisabeth Risdon, Cecil Kellaway, Linda Hayes, Lydia Bilbrook, Grant Withers, Charles Coleman, Charles Quigley, Eddie Dunn, Tom Kennedy, Gus Schilling, Fred Kelsey, Kernan Cripps, Frank Orth, Ferris Taylor, Dick Hogan, Rafael Storm, Lester Dorr, Warren Jackson, Sammy Stein, Paul Overton, John Sheehan, Herta Margot. After an argument with her husband the Mexican Spitfire heads West for a divorce but her uncle makes plans to stop her. Average entry in the \"Mexican Spitfire\" series, mainly its for fans.\n\n**2650** _ **Mi Amigo**_ **** Azalea Film Corporation, 2002. 90 min. Color. D-SC: Milton Brown. With Josh Holloway, Tom Everett, Ed Bruce, Channon Roe, Jackie Schell, Burton Gilliam, Jack Armstrong, Freddy Fender, Alejandro De Hoyos, Jo Harvey Allen, Francisco Gonzalez. After one of them commits a crime, two friends separate and do not meet for three decades and then have to determine if they will end their estrangement. Low budget, confusing Western.\n\n**2651** _ **The Michigan Kid**_ **** Universal, 1928. 55 min. D: Irvin Willat. SC: Peter Milne, Walter Anthony, J. Grubb Alexander and uncredited J.G. Hawks, Charles Logue and Irvin Willat. With Renee Adoree, Conrad Nagel, Fred Esmelton, Virginia Grey, Maurice Murphy, Adolph Miller, Lloyd Whitlock, Donald House. Wanting to wed his girl, a man heads to Alaska to make a fortune but once there he finds his romantic rival has sent for the woman so he can marry her. The plot is not much but the acting and locales make this silent effort worth watching.\n\n**2652** _ **Michigan Kid**_ **** Universal-International, 1947. 70 min. Color. D: Ray Taylor. SC: Roy Chanslor. With Jon Hall, Rita Johnson, Victor McLaglen, Andy Devine, Byron Foulger, Stanley Andrews, Milburn Stone, William Brooks, Joan (Shawlee) Fulton, Leonard East, Ray Teal, Eddy Waller, George Chandler, Edmund Cobb, Karl Hackett, Robert Wilke, Guy Wilkerson, Art Dillard, Charles Trowbridge, Griff Barnett, Dewey Robinson, Alan Bridge, Syd Saylor, Rex Lease, Ernie Adams, Tom Quinn, Ralph Dunn, Norman Leavitt, Harry Strang, Budd Buster, George Davis, George Magrill, William Fawcett, Howard Mitchell, Spec O'Donnell, George Reed, Bert LeBaron. Four strangers come to the aid of a young woman whose ranch is coveted by corrupt town officials. Pleasant adaptation of Rex Beach's short story.\n\n_**Midnight Canyon**_ see _**Legend of a Gunfighter**_\n\n**2653** _ **The Million Dollar Dixie Deliverance**_ **** Buena Vista, 1978. 100 min. Color. D: Russell Mayberry. SC: Lawrence Montaigne. With Brock Peters, Christian Juttner, Chip Courtland, Alicia Fleer, Joe Dorsey, Christian Berrigan, Kyle Richards, Kip Niven, Kenneth Daniel, Ben Jones, Ernie Brown, Grace Zabriskie, Mike Vines, Sonny Shroyer, Stuart Culpepper, Richard Reiner, Mary Neil Santacroce, Kermit Echols, Frank Rickman, Raylon C. Ruggles, Henry Blankenship. During the Civil War a captured Union soldier tries to help five children held for a one million dollar ransom by Confederates. The Disney studios' first feature film developed especially for network TV is good family fare.\n\n**2654** _ **The Mine with the Iron Door**_ **** Columbia, 1936. 70 min. D: David Howard. SC: Howard Swift and Dan Jarrett. With Richard Arlen, Cecilia Parker, Henry B. Walthall, Stanley Fields, Horace Murphy, Spencer Charters, Charles Wilson, Barbara Bedford. A tenderfoot prospector is given the location of a valuable mine but is at odds with a bandit who has kidnapped a young woman he wants to ransom for it. Entertaining adaptation of the 1923 Harold Bell Wright novel first filmed by producer Sol Lesser, who also did this version, in 1924 for his Principal Pictures with Pat O'Malley, Dorothy Mackaill, Raymond Hatton, Charles Murray, Creighton Hale and Mary Carr.\n\n**2655** _ **Minnesota Clay**_ **** Harlequin International, 1964. 89 min. Color. D: Sergio Corbucci. SC: Adriano Bolzoni. With Cameron Mitchell, Georges Riviere, Diana Martin, Ethel Rojo, Fernando Sancho, Anthony Ross, Antonio Casas, Julia Pena, Gino Pernice, Alberto Cevenini. Going blind, a prisoner escapes and returns to the town where a man can give him an alibi to prove his innocence and there he finds rival gangs at war. Good French-Italian-Spanish co-production with fine work by Cameron Mitchell in the title role.\n\n**2656** _ **A Minute to Pray, a Second to Die**_ **** Cinerama, 1968. 97 min. Color. D: Franco Giraldi. SC: Ugo Liberatore and Louis Garfinkle. With Alex Cord, Arthur Kennedy, Robert Ryan, Nicoletta Machiavelli, Mario Brega, Renato Romano, Gianpiero Albertini, Dan Martin, Jose Manuel Martin, Enzo Fiermonte, Spean Covery, Aldo Sambrell, Alberto Dell'Acqua (Robert Widmark). A wanted outlaw takes refuge in a remote village but is soon located by his enemies, including the law, rival bad men and bounty hunters. Well made but average European oater originally called _**Escondido**_ , running 103 minutes.\n\n**2657** _ **Miracle at Sage Creek**_ **** American World Pictures, 2005. 93 min. Color. D: James Intveld. SC: Thadd Turner. With David Carradine, Wes Studi, Michael Parks, Tim Abell, Sarah Aldrich, Irene Bedard, Mark Rolston, Daniel Quinn, Darian Weiss, Masam Holden, Buck Taylor, Tracy Nelson, Rance Howard, Francine York, Brian Libby, Carey Thompson, Wyatt Turner, Marissa Baca, Fred Griffith, Myron Natwick, Thadd Turner, Jeff Prather, Tommy Dippel, Anthony Homus, DJ Perry, Peter J. Brown, Amos Carter, Tanya Turner, Terry Jacobson. In 1888 Wyoming a young boy is miraculously saved at Christmas time, causing two feuding families to become friends. Heartwarming Western also called _**Christmas Miracle at Sage Creek**_.\n\n_**Miracle in the Sand**_ see _**The Three Godfathers**_ (1936)\n\n**2658** _ **Miracle in the Wilderness**_ **** Turner Network Television (TNT), 1992. 88 min. Color. D: Kevin James Dobson. SC: Michael Michaelian and Jim Byrnes. With Kris Kristofferson, Kim Cattrall, John Dennis Johnston, Rino Thunder, David Oliver, Sheldon Peters Wolfchild, Steve Reevis, Peter Alan Morris, Joannelle Nadine Romero, Otakuye Conroy, Matthew E. Montoya, David Bull Plume, Volley Reed, Johnny Looking Cloud, Patrick N. Augare, Robby Dunn, Terry Fredericks, Loren Cuny. The story of Christ brings peace between an Indian chief and the family he captured in revenge for the killing of his son. Intriguing adaptation of the Nativity to Native American culture.\n\n**2659** _ **The Miracle of the Hills**_ **** 20th Century\u2013Fox, 1959. 73 min. D: Paul Landres. SC: Charles Hoffman. With Rex Reason, Theona Bryant, Jay North, June Vincent, Nan Leslie, Betty Lou Gerson, Gilbert Smith, Tracy Strattford, Gene Roth, I. Stanford Jolley, Gene Collins, Paul Wexler, Kenneth Mayer, Pat O'Hara, Tom Daly, Cecil Elliott, Charles Arnt, Claire Carleton. A young minister tries to bring spiritual rebirth to an 1880s mining town run by a wealthy ex\u2013dance hall hostess. Pleasing entertainment.\n\n**2660** _ **The Miracle Rider**_ **** Mascot, 1935. 15 Chapters. D: Armand L. Schaefer and B. Reeves Eason. SC: John Rathmell. With Tom Mix, Joan Gale, Charles Middleton, Jason Robards, Robert Kortman, Edward Earle, Edward Hearn, Tom London, Niles Welch, Edmund Cobb, Ernie Adams, Max Wagner, Charles King, George Chesebro, Jack Rockwell, Stanley Price, George Burton, Wally Wales, Buffalo Bill, Jr., Dick Curtis, Frank Ellis, Richard Alexander, Earl Dwire, Lafe McKee, Hank Bell, Pat O'Malley, Slim Whitaker, Robert Frazer, Art Ardigan, Chief Big Tree, Forrest Taylor, Fred Burns, Chief Standing Bear, Cliff Lyons, Black Hawk, Nick Thompson, Artie Ortego, Joe Weaver, Richard Botiller, Jim Thorpe, Henry Hall, Yakima Canutt, Tex Cooper, Jack Mower, Charles Brinley, Bud Geary, John Merton, Chief Two Hawks, Frank O'Connor. A Texas Ranger helps an Indian tribe whose lands are being secretly mined by a crook and his cohorts who after a valuable mineral needed to create a super explosive. Somewhat static but action filled cliffhanger; Tom Mix's last film.\n\n**Advertisement for** _**The Miracle Rider**_ **(Mascot, 1935).**\n\n**  \n**\n\n**2661** _ **The Misfits**_ **** United Artists, 1961. 125 min. D: John Huston. SC: Arthur Miller. With Clark Gable, Marilyn Monroe, Montgomery Clift, Eli Wallach, Thelma Ritter, James Barton, Estelle Winwood, Kevin McCarthy, Dennis Shaw, Philip Mitchell, Walter Rampage, Peggy Barton, Rex Bell, Ralph Roberts. A divorcee becomes upset at the cruelty to horses during a roundup and she appeals to new found cowboy friends to help her stop it. Overlong and overrated, this film's main appeal is it is the last for both Clark Gable and Marilyn Monroe.\n\n_**Missile Base at Taniak**_ see _**Canadian Mounties vs. Atomic Invaders**_\n\n**2662** _ **The Missing**_ **** Columbia, 2003. 137 min. Color. D: Ron Howard. SC: Ken Kaufman. With Tommy Lee Jones, Cate Blanchett, Evan Rachel Wood, Jenna Boyd, Aaron Eckhart, Val Kilmer, Sergio Calderon, Eric Schweig, Steve Reevis, Jay Tavere, Simon Baker, Ray McKinnon, Max Perlich, Ramon Frank, Deryle J. Lujan, Matthew Montoya, Jose Saenz, Gandi Shaw, Rod Rondeaux, Juddson Linn, Dutch Lunak, Elisabeth Moss, Josephine Schwan, Alexandra Elich, Yolanda Nez, Angelina C. Tores, Deborah Martinez, Clint Howard, Arron Shiver, David Midthunder, Paul Scallan, David Garver. In 1885 New Mexico, Apaches kidnap a woman doctor's daughter and she allies herself with her estranged father to rescue the girl. Fair Western, big budget box office bust; a longer version runs 154 minutes.\n\n**2663** _ **Mission to Glory: A True Story**_ **** Key International, 1977. 116 min. Color. D-SC: Ken Kennedy. With Richard Egan, Ricardo Montalban, John Ireland, Aldo Ray, Cesar Romero, Stephen McNally, Rory Calhoun, Keenan Wynn, Victor Jory, Michael Ansara, John Russell, Joseph Campanella, Anthony Caruso, Tristram Coffin, Henry Brandon, William Dozier, Danny Zapien, Joe Petrullo. After losing his mission, a Spanish priest tries to bring peace between his people and the Apaches in Old California. Choppy, convoluted waste of a fine cast. Issued to TV as _**Father Kino, Padre on Horseback**_ and on video as _**The Father Kino Story**_ and _**Savage Hunter**_.\n\n**2664** _ **The Mississippi Gambler**_ **** Universal-International, 1953. 99 min. Color. D: Rudolph Mate. SC: Seton I. Miller. With Tyrone Power, Piper Laurie, Julia (Julie) Adams, John McIntire, John Baer, Paul Cavanagh, Ron Randall, William Reynolds, Guy Williams, Robert Warwick, Ralph Dumke, Hugh Beaumont, King Donovan, Gwen Verdon, Alan Dexter, Al Wyatt, Dale Van Sickel, Michael Dale, Bert LeBaron, Dennis Weaver, Frank Wilcox, Edward Earle, Dorothy Bruce, Angela Stevens, Rolfe Sedan, Tony Hughes, Fred Cavens, David Newell, Buddy Roosevelt, Anita Ekberg, Jackie Loughery, Paul Bradley, Marcel De La Brosse, Maya Van Horn, Eduardo Cansino, Jon Shepodd, Renate Hoy, LeRoi Antienne. An honest gambler decides to set up a business in frontier New Orleans. Passable costume melodrama for Tyrone Power fans.\n**2665** _ **Mississippi Rhythm**_ **** Monogram, 1949. 69 min. D: Derwin Abrahams. SC: Louise Rousseau. With Jimmie Davis, Veda Ann Borg, Lee \"Lasses\" White, James Flavin, Lyle Talbot, Sue England, Guy Beach, Paul Maxey, Paul Bryar, Joel Marston, Duke York, Wheaton Chambers, Charlie Jordan. When dishonest gamblers try to stop settlement by homesteaders, a land agent comes to their rescue. Pleasant musical drama starring two term Louisiana governor Jimmie Davis.\n\n**2666** _ **The Missouri Breaks**_ **** United Artists, 1976. 126 min. Color. D: Arthur Penn. SC: Thomas McGuane. With Marlon Brando, Jack Nicholson, Kathleen Lloyd, Randy Quaid, Fredric Forrest, Harry Dean Stanton, John McLiam, John Ryan, Sam Gilman, Steve Franken, Richard Bradford, James Greene, Luana Anders, R.L. Armstrong, Dan Ades, Charles Wagenheim. When horse thieves steal herds belonging to Mormon ranchers they agree to hire a gunman to dispose of the menace. An overlong and surprisingly poor feature with mediocre performances from the two stars.\n\n**2667** _ **A Missouri Outlaw**_ **** Republic, 1941. 54 min. D: George Sherman. SC: Doris Schroeder and Jack Latt, Jr. With Don \"Red\" Barry, Lynn Merrick, Noah Beery, Al St. John, Paul Fix, Frank LaRue, Kenne Duncan, John Merton, Carleton Young, Frank Brownlee, Fred \"Snowflake\" Toones, Karl Hackett, Lee Shumway, Ray Bennett, Bob McKenzie, Kermit Maynard, Frank McCarroll, Curley Dresden, Herman Hack. A sheriff is forced to arrest his son, a notorious gunman. Very good entry in Don Barry's Republic series, greatly helped by the presence of pretty Lynn Merrick and the great Noah Beery.\n\n**2668** _ **The Missouri Traveler**_ **** Buena Vista, 1958. 103 min. Color. D: Jerry Hopper. SC: Norman Shannon Hall. With Brandon de Wilde, Lee Marvin, Gary Merrill, Mary Hosford, Paul Ford, Ken Curtis, Cal Tinney, Frank Cady, Mary Field, Kathleen Freeman, Will Wright, Roy Jenson, Earle Hodgins, Tony Tiner, Bill Bryant, Barry Curtis, Eddie Little Sky, Rodney Bell, Helen Brown, William Newell. In 1915 an orphaned boy tries to make a go of it in a rural Missouri town. Pleasant Walt Disney production of a bucolic nature.\n\n**2669** _ **The Missourians**_ **** Republic, 1950. 60 min. D: George Blair. SC: Arthur Orloff. With Monte Hale, Paul Hurst, Lyn Thomas, Roy Barcroft, Howard Negley, Robert Neil, Lane Bradford, John Hamilton, Sarah Padden, Charles Williams, Perry Ivins. A refugee from Poland attempts to take up ranching but is opposed by the locals although the town marshal tries to stop such prejudice. Mediocre Monte Hale feature.\n\n**2670** _ **Mistaken Orders**_ **** Anchor\/Rayart, 1926. 59 min. D: J.P. McGowan. With Helen Holmes, Jack Perrin, Henry Barrows, Hal Walters, Harry Tenbrook, Cecil Kellogg, Mack V. Wright, Arthur Millett, Alice Belcher. The daughter and playboy son of a railroad executive fight to stop sabotage and are helped by the woman's boyfriend. Fast action silent feature highlighted by several train wrecks and a runaway engine.\n\n**2671** _ **Mr. Horn**_ **** CBS-TV, 1979. 200 min. Color. D: Jack Starrett. SC: William Goldman. With David Carradine, Richard Widmark, Karen Black, Richard Masur, Clay Tanner, Pat McCormick, Jack Starrett, John Durren, Jeremy Slate, Enrique Lucero, Stafford Morgan, Don Collier, Lewis James Oliver, John Alderman, William Smith, Jr., Ian McLean, Marilyn Starr, Dan Vadis, James Steward, Tiger Williams, Romero Rameriz, Alexis Jacks. The story of Tom Horn, who captured Geronimo and became a bounty hunter. Overlong two part TV movie enhanced by good work from stars David Carradine and Richard Widmark.\n\n**2672** _ **Mohawk**_ **** 20th Century\u2013Fox, 1956. 80 min. Color. D: Kurt Neumann. SC: Maurice Geraghty and Milton Krims. With Scott Brady, Rita Gam, Neville Brand, Lori Nelson, Allison Hayes, John Hoyt, Vera Vague, Rhys Williams, Ted De Corsia, Mae Clarke, John Hudson, Tommy Cook, Michael Granger. Land owners try to top settlement in the Mohawk Valley by inciting the Indians to war but their efforts are opposed by an Easterner and his Indian girlfriend. Okay action feature enhanced by footage from _**Drums Along the Mohawk**_ (q.v.).\n\n**2673** _ **Mojave Firebrand**_ **** Republic, 1944. 55 min. D: Spencer Gordon Bennet. SC: Norman S. Hall. With Wild Bill Elliott, George \"Gabby\" Hayes, Anne Jeffreys, LeRoy Mason, Jack Ingram, Harry McKim, Karl Hackett, Forrest Taylor, Hal Price, Marshall Reed, Kenne Duncan, Bud Geary, Jack Kirk, Fred Graham, Tom London, Frank Ellis, Tom Steele, Bob Burns, Art Dillard, Bud Osborne, Larry Steers, Horace B. Carpenter, Jess Cavin, Jack Tornek, Silver Harr, Bill Nestell, Tom Smith, Victor Cox. Crooks attempt to steal a silver mine from an old prospector but a peace officer comes to his rescue. Well made and fast moving \"Wild Bill Elliott\" penultimate series feature.\n\n**2674** _ **Molly and Lawless John**_ **** Producers Distributing Corporation, 1973. 96 min. Color. D: Gary Nelson. SC: Terry Kingsley-Smith. With Vera Miles, Sam Elliott, Clu Gulagher, John Anderson, Cynthia Mayers, Charles A. Pinney, Robert Westmoreland, Melinda Chavaria, Charles LeBow, Grady Hill. An outlaw convinces a sheriff's wife to help him escape from jail and after they run away together his interest in her wanes. Fair film helped by good acting, but nothing special.\n\n**2675** _ **The Moment to Kill**_ **** Constantin Film, 1968. 92 min. Color. D: Anthony Ascott (Giuliano Camimeo). SC: Fabio Piccioni. With George Hilton, Walter Barnes, Horst Frank, Giogrio Sammartino, Loni von Friedl, Carlo Alighiero, Renato Romero, Rudolf Schundler, Remo De Angelis, Piertro Ceccarelli. Two cowboys search for a woman who knows the location of hidden Confederate gold that is also coveted by others. Darkly photographed, hard to follow Italian-West German co-production titled _**Il Momente di Uccidere**_ (The Moment to Kill).\n\n**2676** _ **Money,**_ _**Women and Guns**_ **** Universal-International, 1958. 80 min. Color. D: Richard H. Bartlett. SC: Montgomery Pittman. With Jock Mahoney, Kim Hunter, Tim Hovey, Gene Evans, William Campbell, Lon Chaney, Tom Drake, James Gleason, Jeffrey Stone, Judi Meredith, Philip Terry, Richard Devon, Ian MacDonald, Don Megowan, Nolan Leary, Kelly Thordsen. When an old prospector is murdered, a detective is hired to find his heirs as well as the killer. Pleasant Western with some good performances, albeit a bit tame.\n\n_**Monsieur Le Fox**_ see _**Men of the North**_\n\n**2677** _ **Montana**_ **** Warner Bros., 1950. 76 min. D: Ray Enright. SC: James R. Webb, Borden Chase and Charles O'Neal. With Errol Flynn, Alexis Smith, S.Z. Sakall, Douglas Kennedy, James Brown, Ian MacDonald, Charles Irwin, Paul E. Burns, Tudor Owen, Lester Mathews, Nacho Galindo, Lane Chandler, Monte Blue, Billy Vincent, Warren Jackson, Jack Perrin, Creighton Hale, Jack Mower, Gertrude Astor, Forrest Taylor, Philo McCullough, Dorothy Adams, Carl Andre, Nita Talbot, Maudie Prickett, Jessie Adams, Joe Dominguez, Almira Sessions, Joseph J. Green, David Cota, George Lee. An Australian sheep man migrates to Montana and is opposed by a woman cattle rancher. Standard genre outing.\n\n**2678** _ **Montana Belle**_ **** RKO Radio, 1952. 82 min. Color. D: Allan Dwan. SC: Horace Webster and Norman S. Hall. With Jane Russell, George Brent, Scott Brady, Forrest Tucker, Andy Devine, Jack Lambert, Ray Teal, Rory Mallinson, Roy Barcroft, John Litel, Ned Davenport, Dick Elliott, Eugene (Gene) Roth, Stanley Andrews, Holly Bane, George Chesebro, Glenn Strange, Pierce Lyden, Dennis Moore, Gregg Barton, Kenneth MacDonald, Rex Lease, Rodney Bell, Iron Eyes Cody, Charles Soldani, Hank Bell, Franklyn Farnum, Frank Ellis. The notorious Belle Starr expands her lawless activities by joining forces with the Dalton Brothers. An oater so dull that not even Jane Russell can save it; made in 1948.\n\n**2679** _ **Montana Desperado**_ **** Monogram, 1951. 51 min. D: Wallace Fox. SC: Dan Ullman. With Johnny Mack Brown, Virginia Herrick, Myron Healey, Marshall Reed, Steve Clark, Edmund Cobb, Lee Roberts, Carl Mathews, Ben Corbett. A masked rider has been killing ranchers to get their land and a cattleman tries to find out his identity. Compact affair with a good premise but cheap production values.\n\n**2680** _ **Montana Incident**_ **** Monogram, 1952. 54 min. D: Lewis D. Collins. SC: Dan Ullman. With Whip Wilson, Rand Brooks, Noel Neill, Peggy Stewart, Hugh Prosser, Russ Whiteman, William Fawcett, Terry Frost, Marshall Reed, Lyle Talbot, Bruce Edwards, Barbara Woodell, Stanley Price. Two railroad surveyors attempt to help the people of a town ruled by a ruthless woman. Better than average Whip Wilson outing, due mainly to pretty Peggy Stewart as the villain.\n\n**2681** _ **The Montana Kid**_ **** Monogram, 1931. 60 min. D: Harry Fraser. SC: G.A. Durlam. With Bill Cody, Andy Shuford, Doris Hill, W.L. Thorne, G.D. Wood (Gordon DeMain), John Elliott, Paul Panzer. A cowpoke tries to help a boy whose father was murdered after being forced to sign over his ranch to a crooked gambler. A cut above most entries in the \"Bill and Andy\" series.\n\n_**Montana Mike**_ see _**Heaven Only Knows**_\n\n**2682** _ **Montana Moon**_ **** Metro-Goldwyn-Mayer, 1930. 88 min. D: Malcolm St. Clair. SC: Sylvia Thalberg. With Joan Crawford, John(ny) Mack Brown, Ricardo Cortez, Dorothy Sebastian, Cliff Edwards, Benny Rubin, Karl Dane, Lloyd Ingraham, Pete Morrison, Bud McClure, George Reed. A spoiled heiress runs away from her father's private train and meets and later marries a cowboy but she is soon romanced by a gigolo. Dated but entertaining Joan Crawford vehicle with songs like \"The Moon Is Low\"; Johnny Mack Brown's first sound Western.\n\n**2683** _ **La Montana Sin Ley**_ (The Mountain Without Law) **** Ignacio Ferres Iquano, S.A., 1953. 95 min. D: Miguel Lluch. With Jose Suarez, Isabel de Castro, Francisco Martinez Soria, Luis Induni, Jorge Morales, Carlos Orero, Maria Zaldivar. Zorro tries to expose the leader of a pirate gang. Okay Mexican feature based on the Johnston McCulley character.\n\n**2684** _ **Montana Territory**_ **** Columbia, 1952. 64 min. Color. D: Ray Nazarro. SC: Barry Shipman. With Lon McCallister, Preston Foster, Wanda Hendrix, Hugh Sanders, Clayton Moore, Jack Elam, Robert Griffith, Myron Healey, Eddy Waller, George Russell, Ethan Laidlaw, Ruth Warren, Trevor Bardette, George Chesebro, Richard Alexander, Zon Murray, Carol Henry. After witnessing a murder, a young man is made a deputy sheriff and assigned to bring in the killers. Average production with an interesting plot.\n\n**2685** _ **Monte Walsh**_ **** National General, 1970. 98 min. Color. D: William Fraker. SC: Lukas Heller and David Zelag Goodman. With Lee Marvin, Jack Palance, Jeanne Moreau, Mitchell Ryan, Jim Davis, G.D. Spradlin, John Hudkins, Ray Guth, John McKee, Michael Conrad, Tom Heaton, Ted Gehring, Bo Hopkins, John McLiam, Allyn Ann McLerie, Matt Clark, Billy Green Bush, Charles Tyner, Jack Colvin, Guy Wilkerson, Roy Barcroft. In the 1890s two cowpokes find it difficult to get jobs so one becomes a storekeeper but his soon killed by a wrangler who is forced to become an outlaw and the victim's pal vows to avenge his death. Good account of the end of the wild life of the cowboy as civilization took over the West. Worth seeing.\n\n**2686** _ **Montezuma's Lost Gold**_ **** Bill Burrud Productions\/Gold Key Entertainment, 1978. 90 min. Color. D: John Burrud and Miles Hinshaw. SC: Jeff Fellows and John Schwartz. With Miles Hinshaw, Tom Hinshaw, Michael Carr, William Lewis, Bill Burrud (narrator) A drifter-prisoner knows the whereabouts of the legendary treasure of the Aztecs, buried in North America after the Spanish plundered Mexico. Adequate docudrama; Tod Connor sings the title song, \"Gold.\"\n\n**2687** _ **Moon Over Montana**_ **** Monogram, 1946. 56 min. D: Oliver Drake. SC: Louise Rosseau and Ande Lamb. With Jimmy Wakely, Lee \"Lasses\" White, Jennifer Holt, Jack Ingram, Terry Frost, Louise Arthur, Woody Woodell and His Riding Rangers, Stanley Blystone, Brad Slaven, Eddie Majors, Bob Duncan, Arthur Smith, John Elliott, Ray Jones, Denver Dixon. A cowboy leads a group of cattlemen in opposing a corrupt rancher trying to get control of a railroad. Poorly done and uninteresting Jimmy Wakely musical opus.\n\n**2688** _ **Moonlight and Cactus**_ **** Universal, 1944. 60 min. D: Edward Cline. SC: Eugene Conrad and Paul Gerard Smith. With The Andrews Sisters (Patty, Maxene and LaVerne), Leo Carrillo, Elyse Knox, Tom Seidel, Shemp Howard, Eddie Quillan, Murray Alper, Tom Kennedy, Frank Lackteen, Minerva Urecal, Jacqueline De Wit, Mary O'Brien, Mady Correll, Mitchell Ayres and His Orchestra, Chatita and Lolita Tovar. A Naval officer returns home to his Western ranch and finds it is being run by a trio of female singers. The Andrews Sisters are out West in this pleasant blend of music, comedy and nonsense.\n\n**2689** _ **Moonlight on the Prairie**_ **** Warner Bros., 1935. 63 min. D: D. Ross Lederman. SC: William Jacobs. With Dick Foran, Sheila Mannors, George E. Stone, Gordon (Bill) Elliott, Joseph Sawyer, Robert Barrat, Herbert Heywood, Dickie Jones, Joseph King, Milton Kibbee, Raymond Brown, Richard Carle, Bud Osborne, Ben Corbett, Gene Alsace, Glenn Strange, Victor Potel, Cactus Mack, Jack Kirk. A rodeo's cowboy singing star is led to believe he will be accused of murder at the show's next stop. Dick Foran's initial series outing is a pleasant one with a good script and cast enhanced by the star's singing.\n\n**2690** _ **Moonlight on the Range**_ **** Spectrum, 1937. 60 min. D: Sam Newfield. SC: Fred Myton. With Fred Scott, Lois January, Al St. John, Dick Curtis, Frank LaRue, Oscar Gahan, Jimmy Aubrey, Carl Mathews, Wade Walker, William McCall, Shorty Miller, Rudy Sooter, Lew Meehan, Ed Cassidy, Tex Palmer, George Morrell, Forrest Taylor, Steve Clark, Hank Worden, Herman Hack, Jack Evans, Sherry Tansey. A cowboy hits the trail planning revenge when his look-a-like cattle rustler half-brother murders his best pal. Fred Scott plays both the hero and lead villain in this fairly good action foray that is highlighted by his singing four good songs.\n\n_**Moonlight Raid**_ see _**Challenge of the Range**_\n\n**2691** _ **The Moonlighter**_ **** Warner Bros., 1953. 77 min. D: Roy Rowland. SC: Niven Busch. With Barbara Stanwyck, Fred MacMurray, Ward Bond, William Ching, John Dierkes, Morris Ankrum, Jack Elam, Charles Halton, Norman Leavitt, Sam Flint, Myra Marsh. A cattle rustler returns home for his own funeral and meets his ex-girlfriend and her hate for him soon returns to love. Originally issued in 3-D, this dull oater has little to recommend it other than its stars; the title refers to a cattle thief.\n\n**2692** _ **More Dead Than Alive**_ **** United Artists, 1969. 101 min. Color. D: Robert Sparr. SC: George Schenck. With Clint Walker, Anne Francis, Vincent Price, Paul Hampton, Mike Henry, Craig Littler, Beverly Powers, Clarke Gordon, William Woodson. After eighteen years in prison an ex-gunman is released and tries to live a peaceful life but his past keeps catching up with him. Mediocre drama highlighted by Vincent Price's performance as the showman who hires the shooter as his star attraction.\n\n**2693** _ **More Than Magic**_ **** Wrather Corporation, 1956. 75 min. Color. D: Earl Bellamy and Oscar Rudolph. SC: Thomas Seller, Robert E. Schaefer, Eric Friewald and Hilary Green Rhodes. With Clayton Moore, Jay Silverheels, Allen Pinson, Wayne Burson, Mary Ellen Kay, Harry Lauter, Rand Brooks, Don C. Harvey, Tom Brown, Ben Welden, Edmond Hashim, Robert Swan, Charles Stevens, Mike Ragan, Louis Lettieri, Sandy Sanders, John Cliff, William Challee, Barbara Ann Knudsdy, Sydney Mason, David Dwight, John Maxwell, Walt LaRue, John Pickard. The Lone Ranger and Tonto try to capture a band of disappearing road agents, get on the trail of an outlaw gang and help an Indian chief in choosing his heir. Pleasant telefeature from \"The Lone Ranger\" (ABC-TV, 1949\u201357) episodes \"Hot Spell in Panamint,\" \"Outlaws in Greasepaint\" and \"White Hawks' Decision.\"\n\n**2694** _ **More Wild Wild West**_ **** CBS-TV, 1980. 100 min. Color. D: Burt Kennedy. SC: William Bowers and Tony Hayden. With Robert Conrad, Ross Martin, Jonathan Winters, Harry Morgan, Rene Auberjonois, Liz Torres, Victor Buono, Jack La Lanne, Randi Brough, Candi Brough, Dr. Joyce Brothers. Two federal undercover agents are on the trail of a wily madman. Fans of \"Wild Wild West\" (CBS-TV, 1965\u201370) will enjoy this telefilm, a follow-up to _**The Wild Wild West Revisited**_ (q.v.).\n\n**2695** _ **Mosby's Marauders**_ **** Buena Vista, 1967. 80 min. Color. D: Michael O'Herlihy. SC: Harold Swanton. With James MacArthur, Nick Adams, Jack Ging, Kurt Russell, Peggy Lipton, Donald Harron, Jeanne Cooper, James Callahan, Robert Sorrells, E.J. Andre, Michael Forest, Steve Raines, Michael Pate, Michael Kearney, Robert Random, Amzie Strickland. A young Confederate officer joins the forces of Mosby's Marauders in a daring capture of a Union general behind enemy lines in 1863. Exciting, and fairly accurate, Walt Disney historical drama originally shown on Disney's ABC-TV series as \"Willie and the Yank: The Mosby Raiders\" on January 8, 15 and 22, 1967.\n\n**2696** _ **Mother Lode**_ **** Agamemnon Films, 1982. 101 min. D: Charlton Heston. SC: Fraser Clarke Heston. With Charlton Heston, Nick Mancuso, Kim Basinger, John Marley, Dale Wilson, Ricky Zantolas, Marie George. A bush pilot and a young woman head into the mountains of British Columbia to look for the man's gold-seeking buddy and find a Scottish miner who will stop at nothing to protect his claim. Very entertaining action feature with fine photography, plus good second unit work by Joe Canutt; filmed in British Columbia's Cassair Mountains.\n\n**2697** _ **Mountain Charlie**_ **** American National Enterprises, 1982. 96 min. Color. D: George Stapleford. SC: Karen Hoffman. With Dick Robinson, Rick Guinn, Lynne Seus, Roger Clark, John Sechser, Karl Wesson, Merlin Barlow, Wallace Bitseedy, David Sterago, Guy Faris, William Stewart, Charles Stewart, Denise Neilson, Donne Johnson. A mountain girl has her life changed when she encounters three vagabonds. Scenic family adventure fare.\n\n**2698** _ **Mountain Family Robinson**_ **** Pacific International, 1979. 98 min. Color. D: John Cotter. SC: Arthur R. Dubs. With Robert Logan, Susan Damante Shaw, Heather Rattray, Ham Larsen, George \"Buck\" Flower, William Bryant, Calvin Bartlett, Jim Davidson. The government demands the Robinson family move from their cabin home in the Rocky Mountains because the land they are on is a mining, not a homestead, claim. The final film in the \"Wilderness Family\" trilogy is a heartwarming affair with excellent photography by James Roberson; preceded by _**The Wilderness Family**_ and _**Further Adventures of the Wilderness Family**_ (qq.v.).\n\n**2699** _ **Mountain Justice**_ **** Universal, 1930. 64 min. D: Harry Joe Brown. SC: Bennett Cohen and Leslie Mason. With Ken Maynard, Kathryn Crawford, Otis Harlan, Paul Hurst, Richard Carlyle, Les Bates, Gilbert \"Pee Wee\" Holmes, Edgar \"Blue\" Washington, Fred Burns, Jim Mason, Frank Ellis, Jim Corey, Bud McClure, Blackjack Ward, Buck Bucko. An Oklahoma cowboy heads to the Kentucky hills pretending to be deaf in order to find the man who shot his father. Good Ken Maynard early talkie originally called _**Kettle Creek**_.\n\n_**Mountain Man**_ see _**Guardian of the Wilderness**_\n\n**2700** _ **The Mountain Men**_ **** Columbia, 1980. 102 min. Color. D: Richard Lang. SC: Fraser Clarke Heston. With Charlton Heston, Brian Keith, Stephen Macht, Victoria Racimo, Victor Jory, Seymour Cassell, David Ackroyd, John Glover, Cal Bellini, Bill Lucking, Ken Ruta, Danny Zapien, Tim Haldeman, Bob Terhune, Chuck Roberson, Roy Jenson, Henry Wills, Bennie Dobbins. In the 1840s an aging trapper keeps searching for untouched wilderness and finds trouble with hostile Indians. Well made and beautifully photographed frontier tale, but only average.\n\n**2701** _ **Mountain Rhythm**_ **** Republic, 1939. 61 min. D: B. Reeves Eason. SC: Gerald Geraghty. With Gene Autry, Smiley Burnette, June Storey, Maude Eburne, Ferris Taylor, Walter Fenner, Jack Pennick, Hooper Atchley, Bernard Suss, Ed Cassidy, Jack Ingram, Tom London, Frankie Marvin, Roger Williams, Slim Whitaker, Curley Dresden, Al Taylor, Herman Hack, Buck Morgan, Augie Gomez, Horace B. Carpenter, Bob Burns, Duke R. Lee, Silver Tip Baker, John Beach, Jack Baxley, Dirk Thane, Bud McClure, Lew Morphy, George Sowards, Bill Hickey. Eastern crooks want grazing land for a tourist resort and set up a government auction in order to buy it out from under the ranchers who use it. Average Gene Autry vehicle.\n\n**2702** _ **Mounted Fury**_ **** World Wide, 1931. 63 min. D: Stuart Paton. SC: Betty Burbridge. With John Bowers, Blanche Mehaffey, Lina Basquette, Frank Rice, Robert Ellis, John Ince, George Regas, Lloyd Whitlock, Jack Trent. A Canadian Mounted Policeman is assigned to bring in a man wanted for murder. Quickie production that gives viewers a chance to see silent star John Bowers in one of his few sound efforts.\n\n**2703** _ **The Mounted Stranger**_ **** Universal, 1930. 60 min. D-SC: Arthur Rosson. With Hoot Gibson, Louise Lorraine, Francis Ford, Milton Brown, Buddy Hunter, Fred Burns, James (Jim) Corey, Walter Patterson, Francelia Billington, Gilbert \"Pee Wee\" Holmes, Glenn Strange, Bud McClure, Archie Ricks. As a boy a man witnessed the murder of his father by an outlaw gang leader and years later he tries to bring the culprit to justice. Action packed Hoot Gibson early talkie, more serious than most of his features; a remake of the star's 1924 Universal film _**The Ridin' Kid from Powder River**_.\n\n**2704** _ **The Mountie**_ **** Grindstone Entertainment Group, 2011. 90 min. Color. D: Wyeth Clarkson. SC: Wyeth Clarkson, Grant Sauve and Charles Johnston. With Andrew W. Walker, Jessica Pare, George Buza, Earl Pastko, Tony Munch, Andrey Ivchenko, Matthew G. Taylor, Dean Williams, John Wildman, Kestrel Martin, Ada H. Chan. A Royal Canadian Mounted Police officer arrives in a remote Yukon village to find a killer and uncovers vast corruption. Mediocre period drama filmed in Canada; released on video as _**Way of the West**_.\n\n**2705** _ **Mrs. Mike**_ **** United Artists, 1949. 99 min. D: Louis King. SC: Alfred Lewis and DeWitt Bodeen. With Dick Powell, Evelyn Keyes, John Miljan, J.M. Kerrigan, Angela Clarke, Will Wright, Nan Boardman, Frances Morris, Joel Nester, Jean Inness, Chief Yowlachie, Clarence Straight, James Fairfax, Donald Pietro. A Mountie marries a city girl and brings her with him to a remote outpost where she learns to face the harsh realities of frontier existence. A wonderful motion picture, deftly written and finely performed by stars Dick Powell and Evelyn Keyes; a must see.\n\n**2706** _ **Mrs. Sundance**_ **** ABC-TV\/20th Century\u2013Fox, 1974. 78 min. Color. D: Marvin Chomsky. SC: Christopher Knopf. With Elizabeth Montgomery, Robert Foxworth, L.Q. Jones, Arthur Hunnicutt, Lurene Tuttle, Claudette Nevins, Robert Donner, Dean Smith, Tod Shelhorse. When school teacher Etta Place learns the Sundance kid did not die with Butch Cassidy she sets out to meet him but is aware bounty hunters are on her trail. TV movie follow-up to _**Butch Cassidy and the Sundance Kid**_ (q.v.) is acceptable thanks to Elizabeth Montgomery in the title role; followed by a sequel, _**Wanted:**_ ****_**The Sundance Woman**_ (q.v.)\n\n_**Mrs. Sundance Rides Again**_ see _**Wanted: The Sundance Woman**_\n\n**2707** _ **El Muchacho Alegre**_ (The Happy Boy). Clasa-Mohme, 1948. 90 min. D: Alejandro Galindo. SC: Raul de Anda. With Luis Aguilar, Victor Parra, Carmelita Gonzalez, Manuel Donde, Pascual Garcia Pena, Jorge Arriaga, Jose L. Murillo, Fernando Soto, Arturo Soto Rangel, Cecilia Leger, Gloria Lozano, Estela Matute, Sara Montes, Pepe Nava, Jose Munoz, Jose Pardave, Conchita Gentil Acros, Roberto Canedo, Eduardo Arozamena, Felipe Montoya. Three friends want the same woman but when one of them is murdered it causes the other two pals to become estranged as they try to find the killer. Highly regarded Mexican Western drama with songs, from producer-writer Raul de Anda.\n\n**2708** _ **El Muchacho de Durango**_ (The Kid from Durango) **** Radaent Films, 1962. 90 min. D: Arturo Martinez. SC: Raul de Anda. With Rodolfo de Anda, Gina Romand, Dagoberto Rodriguez, Jaime Fernandez, Sonia Infante, Oscar Pulido, Tito Novaro, Ernesto Cabiati. Showing up in a Sonora town, a cowboy gets a job offer but is soon at odds with a newly arrived stranger. Okay Mexican Western from producer-writer Raul de Anda, followed by _**Alias El Alacran**_ (q.v.).\n\n**2709** _ **La Mula de Cullen Baker**_ (The Mule of Cullen Baker) **** Radaent Films, 1971. 85 min. Color. D-SC: Rene Cardona. With Rodolfo de Acosta, Anel, Jorge Russek, Carlos Agosti, Juan Gallardo, Armando Acosta, Carlos Leon, Jose Dupeyron, Gloria Chavez, Carlos Cardan, Rene Cardona, Rene Cardona III, Jose L. Murillo, Jesus Gomez, Raul Valerio, Gerardo Zepeda, Alfredo Gutierrez, Regino Herrera, Ismael Larumbe, Roger Oropeza, Christa von Humboldt. A mule riding loner takes revenge on society by becoming a ruthless gunman after coming across a cache of revolvers. Star Rodolfo de Acosta co-produced this sturdy south of the border oater with his father, Raul de Anda.\n\n**2710** _ **Mule Feathers**_ **** Monarch Releasing, 1977. 79 min. Color. D-SC: Donald R. von Mizener. With Rory Calhoun, Angela Richardson, Richard Webb, Dee Cooper, Cathy Carricaburu, Frank Otterman, Doodles Weaver, Noble \"Kid\" Chissell, Arthur Roberts, Ken Smedberg, Ken Johnson, Ted Lehman, Nicholas Worth, Pat Crenshaw, Nedra Volz, Ruth Vinson, Tony Mancuso, Dorinda Carey, Robin von Mizener, Don Knotts (voice). A preacher's talking mule relates how his master forsook the gospel in search of a cache of gold and his showdown with a mean adversary. Mind numbing excuse for a Western comedy from producer Robert F. Slatzer; made in 1975 and also known as _**The West Is Still Wild**_.\n\n**2711** _ **Mule Train**_ **** Columbia, 1950. 70 min. D: John English. SC: Gerald Geraghty. With Gene Autry, Pat Buttram, Sheila Ryan, Robert Livingston, John Miljan, Frank Jaquet, Vince Barnett, Syd Saylor, Sandy Sanders, Gregg Barton, Kenne Duncan, Roy Gordon, Stanley Andrews, Robert Hilton, Robert Wilke, Robert Carson, Pat O'Malley, Eddie Parker, George Morrell, John R. McKee, George Slocum, Frank O'Connor, Norman Leavitt, Evelyn Finley, Bob Woodward. Marshal Gene Autry helps two prospectors who claim a natural cement deposit was stolen by a contractor in cahoots with a dishonest female sheriff. Another entertaining Gene Autry outing enhanced by the title song plus a trio of other country\/western favorites, \"Cool Water,\" \"The Old Chisholm Trail\" and \"Room Full of Roses.\"\n\n**2712** _ **Murder on the Yukon**_ **** Monogram, 1940. 59 min. D: Louis Gasnier. SC: Milton Raison. With James Newill, Polly Ann Young, Dave O'Brien, Al St. John, William Royle, Chief Thundercloud, Budd Buster, Karl Hackett, Kenne Duncan, Snub Pollard, Earl Douglas, Jack Clifford, Frank Campeau, Gertrude Chorre. Two Mounties on vacation investigate the homicides of Yukon prospectors who they suspect are the victims of counterfeiters. Nicely paced entry in the \"Renfrew of the Royal Mounted\" series; also called _**Renfrew of the Royal Mounted in Murder on the Yukon**_.\n\n_**Murieta**_ see _**Desperate Mission**_\n\n**2713** _ **Mustang**_ **** United Artists, 1959. 73 min. D: Peter Stephens. SC: Tom Gries. With Jack Buetel, Madelyn Trahey, Steve Keyes, Milton Swift, Robert (Bob) Gilbert, Paul Spahn, Max M. Gilford, Autumn Moon (horse). A rodeo star goes to work for a rancher and opposes a man who wants to kill a wild stallion. Poor program feature made in 1955; title song sung by Champ Butler.\n\n**2714** _ **Mustang Country**_ **** Universal, 1976. 79 min. Color. D-SC: John Champion. With Joel McCrea, Robert Fuller, Patrick Wayne, Nika Mina, Tiger (horse), Rote (dog). On the Montana-Canadian border an aging cowboy helps a young Indian boy round up a wild stallion. Joel McCrea returned to the screen in this very entertaining drama that is a good bet for the entire family; recommended.\n\n**2715** _ **Mutiny at Fort Sharp**_ **** Walter Manley Enterprises, 1966. 91 min. Color. D: Fernando Cerchio. SC: Ugo Liberatore and Fernando Cerchio. With Broderick Crawford, Elisa Montes, Mario Valdemarin, Umberto Ceriani, Hugo Arden, Julio Pena, Carlos Mendi, Tomas Pico, Nando Angelini. In 1864 French troops accidentally cross into Confederate territory in Texas from Mexico and are forced to join the rebels at a fort about to be besieged by Indians. Mediocre Spanish-made and poorly dubbed oater originally called _**Per un Dollaro di Gloria**_ (For a Dollar of Glory); Broderick Crawford is lethargic as the Confederate fort commander.\n\n**2716** _ **Mutiny in the Arctic**_ **** Universal, 1941. 64 min. D: John Rawlins. SC: Maurice Tombragel and Victor McLeod. With Richard Arlen, Andy Devine, Anne Nagel, Don Terry, Addison Richards, Oscar O'Shea, Harry Cording, Jeff Corey, John Rogers, John Bagni, Stanley Blystone, Sam Adams, Gibson Gowland, Eddie Dew, David Sharpe, Peter Potter, Charles Sullivan, Leo Abbey, Dave Wengren. Two explorers searching for a pitchblende mountain in the Arctic are betrayed by the financier of the venture and they end up on an iceberg. Fair program feature in the Richard Arlen\u2013Andy Devine action series for Universal.\n\n**2717** _ **Mutiny on the Blackhawk**_ **** Universal, 1939. 66 min. D: Christy Cabanne. SC: Michael L. Simmons. With Richard Arlen, Andy Devine, Constance Moore, Noah Beery, Guinn Williams, Mala, Thurston Hall, Sandra Kane, Paul Fix, Richard Lane, Mabel Albertson, Charles Trowbridge, Bill Moore, Byron Foulger, Francisco Maran, Eddy Waller, Mamo Clark. In 1840 a naval investigator looks into slave running between California and the Sandwich Islands and later the Mexican government tries to wipe out American settlers in California. Two stories in one film highlight this pretty good initial entry in the Richard Arlen\u2013Andy Devine series.\n\n**2718** _ **My Darling Clementine**_ **** 20th Century\u2013Fox, 1946. 97 min. D: John Ford. SC: Samuel G. Engel and Winston Miller. With Henry Fonda, Linda Darnell, Victor Mature, Walter Brennan, Tim Holt, Ward Bond, Cathy Downs, Alan Mowbray, John Ireland, Grant Withers, Roy Roberts, Jane Darwell, Russell Simpson, Francis Ford, J. Farrell MacDonald, Don Garner, Ben Hall, Arthur Walsh, Jack Pennick, Louis Mercier, Mickey Simpson, Fred Libby, Harry Woods, Charles Stevens, Mae Marsh, Hank Bell. Wyatt Earp and Doc Holliday go after the Clanton clan when they steal the Earp's cattle and murder his youngest brother. Pictorially interesting but historically inaccurate retelling of the events leading up to the gunfight at O.K. Corral; mainly for John Ford buffs.\n\n**2719** _ **My Friend Flicka**_ **** 20th Century\u2013Fox, 1943. 89 min. Color. D: Harold Shumate. SC: Lillie Hayward and Frances Edwards Faragoh. With Preston Foster, Rita Johnson, Roddy McDowall, James Bell, Jeff Corey, Diana Hale, Arthur Loft, Jimmy Aubrey. Against his father's wishes, a young boy tries to tame a wild horse he has grown to love. Fine family film, the basis for the popular series of the same title on CBS-TV from 1956 to 1957 starring Gene Evans, Anita Louise and Johnny Washbrook; followed by a sequel, _**Thunderhead, Son of Flicka**_ (q.v.).\n\n**2720** _ **My Friend Irma Goes West**_ **** Paramount, 1950. 90 min. D: Hal Walker. SC: Cy Howard and Parke Levy. With Marie Wilson, John Lund, Diana Lynn, Dean Martin, Jerry Lewis, Corinne Calvet, Lloyd Corrigan, Don Porter, Harold Huber, Joseph Vitale, Charles Evans, Kenneth Tobey, Wendell Niles, James Flavin, David Clark, Chief Yowlachie, Jimmie Dundee, George Humbert, Roy Gordon, Al Ferguson, Julia Montoya, Rose Higgins, Jasper Weldon, Gregg Palmer, Gil Herman, Link Clayton, Napoleon Whiting, Mike Mahoney, Bob Johnson, Stan Johnson, Charles Dayton, Joe Hecht, Maxie Thrower. A zany blonde goes with her roommate when the latter's singer boyfriend and his partner are signed to a Hollywood contract by a producer who, unknown to them, has escaped from an asylum. Fair follow-up to _**My Friend Irma**_ (Paramount, 1950).\n\n_**My Gun Is the Law**_ see _**The Colt Is My Law**_\n\n**2721** _ **My Heroes Have Always Been Cowboys**_ **** Samuel Goldwyn Company, 1991. 106 min. Color. D: Stuart Rosenberg. SC: Joel Don Humphreys. With Scott Glenn, Kate Capshaw, Tess Harper, Gary Busey, Ben Johnson, Balthazar Getty, Clarence Williams III, Mickey Rooney, Cynthia H. Mackey, Bill Clymer, Benjamin Rosenberg, Megan Parlen, Jim Robinett, Jennifer Johnson, Joan Hoag, Dub Taylor, Harold Suggs, Will Hussong, Dennis Fimple, Clu Gulagher, Terry McIlvain, Robert Knott, David Honeycutt Hamilton, Theresa Bell, Rex Linn, Sarah Bratton, Clem McSpadden, Don Endsley, Gary Sievers. Returning home after many years, a cowboy goes back to the rodeo circuit to raise money after finding his father in an assisted living facility and his ex-girlfriend a widow with two children. Modern-day Western, mundane at best.\n\n**2722** _ **My Little Chickadee**_ **** Universal, 1940. 83 min. D: Edward Cline. SC: Mae West and W.C. Fields. With Mae West, W.C. Fields, Joseph Calleia, Dick Foran, Margaret Hamilton, George Moran, Si Jenks, James Conlin, Gene Austin, Candy and Coco (Russell \"Candy\" Hall and Otto \"Coco\" Heimel), Fuzzy Knight, Anne Nagel, Ruth Donnelly, Donald Meek, Willard Robertson, William B. Davidson, Addison Richards, Jackie Searle, Fay Adler, Jan Duggan, Morgan Wallace, Wade Boteler, Harlan Briggs, Eddie Butler, Bing Conley, John Kelly, Walter McGrail, Otto Hoffman, Billy Benedict, Delmar Watson, Chester Gan, George Melford, Lita Chevret, Bob McKenzie, James Morton, Joe Whitehead, Lloyd Ingraham, Dick Rush, Hank Bell, Lane Chandler, Alan Bridge, Edward Hearn, Al Ferguson, Vester Pegg, Frank Ellis, Blackie Whiteford, Bob Burns, Charles McMurphy, Bob Reeves, Jack Roper, Dorothy Vernon, Slim Gault, Buddy Harris, Bill Wolfe. Thrown out of town by snobs, a beautiful woman meets a con man and pretends to marry him in order to gain social acceptance. The teaming of Mae West and W.C. Fields is hardly a classic comedy but it is a funny affair that has delighted audiences ever since its release and is well worth viewing.\n\n**Mae West and Dick Foran in** _**My Little Chickadee**_ **(Universal, 1940).**\n\n**  \n**\n\n**2723** _ **My Name Is Nobody**_ **** Universal, 1974. 115 min. Color. D: Tonino Valerii. SC: Ernesto Gastaldi. With Terence Hill, Henry Fonda, Jean Martin, Piero Lulli, Leo Gordon, R.G. Armstrong, Neil Summers, Steve Kanaly, Geoffrey Lewis, Mario Brega, Benito Stefanelli, Mark Mazza. In 1899 an aging gunman heads to retirement in New Orleans but comes across a younger shooter who soon makes him his idol. Pleasant take-off of Spaghetti Westerns made in Italy; some sources claim Sergio Leone co-directed.\n\n**2724** _ **My Name Is Pecos**_ **** Golden Era, 1968. 83 min. Color. D: Maurizio Lucidi. With Robert Woods, Peter Carsten, Lucia Modugno, Norman Karlk, Christina Josani, Max Dean, Norman Clark (Pier Paolo Capponi), Lou Castel, Guilano Raffaeli, Morris Boone, Umi Raho, George Eastman (Luigi Montefiori), Dario De Grassi, Peter Martell, Sal Borghese, Franco Gula. A man plans to take revenge on the outlaws who murdered his family, the gang now in control of a small town. Bloody Italian Western issued there in 1966 as _**Il Mio Nome e Pecos**_ (My Name is Pecos) and _**Due Once di Piombo**_ (Two Ounces of Lead).\n\n_**My Name Is Shanghai Joe**_ see _**Shanghai Joe**_\n\n**2725** _ **My Outlaw Brother**_ **** Eagle Lion, 1951. 78 min. D: Elliott Nugent. SC: Gene Fowler, Jr. and Albert L. Leavitt. With Mickey Rooney, Wanda Hendrix, Robert Preston, Robert Stack, Carlos Muzquiz, Jose Torvay, Fernando Waggner. A man finds out his brother is an outlaw so he joins the Texas Rangers to fight lawlessness. More than passable feature based on Max Brand's \"South of the Rio Grande.\"\n\n**2726** _ **My Pal the King**_ **** Universal, 1932. 63 min. D: Kurt Neumann. SC: Jack Natteford and Tom J. Crizer. With Tom Mix, Finis Barton, Stuart Holmes, Mickey Rooney, Paul Hurst, Noel Francis, James Kirkwood, Jim Thorpe, Christian Frank, Clarissa Selwynne, Ferdinand Schuman-Heink, Wallis Clark. A cowboy takes his frontier show into a kingdom where plotters are trying to take the throne from a boy king. A different kind of Tom Mix film but one that is well done and quite entertaining.\n\n**2727** _ **My Pal Trigger**_ **** Republic, 1946. 79 min. D: Frank McDonald. SC: Jack Townley and John K. Butler. With Roy Rogers, George \"Gabby\" Hayes, Dale Evans, Jack Holt, Bob Nolan and The Sons of the Pioneers (Tim Spencer, Lloyd Perryman, Pat Brady, Shug Fisher, Hugh Farr, Karl Farr), LeRoy Mason, Roy Barcroft, Sam Flint, Kenne Duncan, Ralph Sanford, Francis McDonald, Harlan Briggs, William Haade, Alan Bridge, Paul E. Burns, Frank Reicher, Fred Graham, Ted Mapes, Tom London, Earle Hodgins, George Magrill, Eddie Parker. Singing star Roy Rogers helps a rancher about to lose his spread to a crooked rival, ends up in jail on a bogus charge and almost loses Trigger. Good Roy Rogers feature although Jack Holt steals the show as the villain; remade as _**Rodeo King and the Senorita**_ (q.v.).\n\n**2728** _ **My Side of the Mountain**_ **** Paramount, 1969. 100 min. Color. D: James B. Clark. SC: Ted Sherdeman, Jane Klove and Joanna Crawford. With Tommy Eccles, Theodore Bikel, Tudi Wiggins, Frank Perry, Peggi Boder, Gina Dick, Karen Pearson. A boy decides to emulate his hero Thoreau and live close to nature after his father reneges on a promised camping trip. This drama, filmed in Canada, will be especially good for family viewing.\n\n**2729** _ **My Uncle Antoine**_ **** Gendon\/Janus Films, 1972. 110 min. Color. D: Claude Jutra. SC: Clement Perron and Claude Jutra. With Jean Duceppe, Lynde Champagne, Olivette Thibault, Claude Jutra, Jacaues Gagnon, Lionel Villeneuve, Helene Loiselle, Mario Dubuc, Lise Brunelle, Alain Legendre, Serge Evers, Robin Marcous, Monique Mercure, Georges Alexander, Rene Aslvatore Catta, Jean Dubost, Benoit Marcoux, Dominique Joly, Lisa Talbot, Michel Talbot, Simeon Dallaire, Sydney Harris, Roger Garand. A teenager learns about growing up as he works as stock boy in his uncle's store in a backwoods Canadian mining town. Leisurely paced, pleasant fare.\n\n**2730** _ **The Mysterious Avenger**_ **** Columbia, 1936. 60 min. D: David Selman. SC: Ford Beebe. With Charles Starrett, Joan Perry, Wheeler Oakman, Ed Le Saint, Lafe McKee, Hal Price, Charles Locher (Jon Hall), George Chesebro, Jack Rockwell, Edmund Cobb, Richard Botiller, The Sons of the Pioneers (Bob Nolan, Len Slye [Roy Rogers], Tim Spencer, Hugh Farr, Karl Farr), Ella McKenzie, Gertrude Astor, Edward Hearn, Tom London, Bob McKenzie, Jack Carlyle, George Plues, Dick Rush, Jack Kenney, Eva McKenzie, Blackie Whiteford, Jack Evans, Bert Dillard, Joe Schilling, Blackjack Ward, William McCall, Lillian Lawrence, Jack Walters, George Burton, Cecil Kellogg, Vic Allen, Sam Coster, Ed O'Neill, Eddie Evans, Robert Wilber. A Texas Ranger returns home to help end a feud between his father and another rancher caused by a rustlers. Very sturdy Charles Starrett vehicle.\n\n**2731** _ **The Mysterious Desperado**_ **** RKO Radio, 1949. 61 min. D: Lesley Selander. SC: Norman Houston. With Tim Holt, Richard Martin, Movita, Robert Livingston, Edward Norris, Frank Wilcox, William Tannen, Robert B. Williams, Kenneth MacDonald, Frank Lackteen, Kermit Maynard. Crooks want land recently inherited by a man and they try to frame him on a false charge so he will lose its ownership. Better than average Tim Holt feature.\n\n_**The Mysterious Mr. Sheffield**_ see _**The Law of the 45s**_\n\n**2732** _ **The Mysterious Rider**_ **** Paramount, 1933. 61 min. D: Fred Allen. SC: Harvey Gates and Robert Niles. With Kent Taylor, Lona Andre, Gail Patrick, Warren Hymer, Berton Churchill, Irving Pichel, Cora Sue Collins, E.H. Calvert, Sherwood Bailey, Niles Welch, Clarence Wilson. A cowboy takes on the guise of a hooded phantom to protect area ranchers from outlaws. Kent Taylor is good in the lead, abetted by a fine supporting cast, although it has little to do with the Zane Grey work on which it was supposedly based.; reissued as _**The Fighting Phantom**_ and remade in 1938 (q.v.).\n\n**2733** _ **The Mysterious Rider**_ **** Paramount, 1938. 78 min. D: Lesley Selander. SC: Maurice Geraghty. With Douglass Dumbrille, Sidney Toler, Russell Hayden, Charlotte Fields, Weldon Heyburn, Monte Blue, Stanley Andrews, Earl Dwire, Glenn Strange, Jack Rockwell, Leo McMahon, Ben Corbett, Ed Brady, Robert Kortman, Richard Alexander, Arch Hall, Price Mitchell. A drifter rides into an area plagued by outlaw raids and becomes a masked avenger to help ranchers. Like its predecessor this film bears little resemblance to the Zane Grey book but it is a corker of a good movie anyway.\n\n**2734** _ **The Mysterious Rider**_ **** Producers Releasing Corporation, 1942. 57 min. D: Sam Newfield. SC: Steve Braxton. With Buster Crabbe, Al St. John, Caroline Burke, John Merton, Edwin Brian, Jack Ingram, Slim Whitaker, Kermit Maynard, Ted Adams, Guy Wilkerson, Frank Ellis, Jimmy Aubrey, Bert Dillard, Joe Phillips, Augie Gomez. Two children are being cheated out of a mine they inherited with Billy the Kid coming to their rescue. Typically fast moving and cheap looking PRC \"Bill the Kid\" series entry; reissued in 1947 by Eagle Lion in a re-edited 39-minute version called _**Panhandle Trail**_.\n\n_**The Mysterious Stranger**_ see _**Code of the Lawless**_\n\n**2735** _ **The Mystery Brand**_ **** Rayart, 1927. 52 min. D: Ben Wilson. With Ben Wilson, Neva Gerber, Al Ferguson, Ted Henderson, Lafe McKee. A representative of the Cattlemen's Association gets on the trail of a gang of horse thieves. Nothing to brag about, but this silent effort lets the viewer see the poplar serial team of Ben Wilson and Neva Gerber in one of their many oaters. Alternate title: _**Lariat's End**_.\n\n**2736** _ **Mystery Man**_ **** United Artists, 1944. 58 min. D: George Archainbaud. SC: J. Benton Cheney. With William Boyd, Andy Clyde, Jimmy Rogers, Eleanor Stewart, Don Costello, Francis McDonald, Jack Rockwell, Pierce Lyden, John Merton, Bob Burns, Ozie Waters, Art Mix, George Morrell, Bob Baker, Hank Bell, Bill Hunter, Bill Nestell, Lew Meehan, Herman Hack, Henry Wills, Lew Morphy. Hopalong Cassidy, California Carlson and Jimmy Rogers find themselves at odds with a gang of robbers led by a man pretending to be a respectable citizen. Rather dull Hoppy film with an exciting climax.\n\n**2737** _ **Mystery Mountain**_ **** Mascot, 1934. 12 Chapters. D: B. Reeves Eason and Otto Brower. SC: Bennett Cohen and Armand L. Schaefer. With Ken Maynard, Verna Hillie, Edward Earle, Edmund Cobb, Lynton Brent, Syd Saylor, Carmencita Johnson, Lafe McKee, Alan Bridge, Edward Hearn, Robert Kortman, Wally Wales, Tom London, George Chesebro, Philo McCullough, Frank Ellis, Steve Clark, Gene Autry, Smiley Burnette, Jack Kirk, Jim Mason, Lew Meehan, Jack Rockwell, Art Mix, William Gould, Curley Dresden, Hooper Atchley, Cliff Lyons, Dick Dickinson, Al Haskell, Pascale Perry. A railroad detective is on the trail of a mysterious wrecker called \"The Rattler\" who is trying to stop the construction of a tunnel through a mountain containing a rich vein of gold. Fun, fast paced Mascot cliffhanger with good mystery element to heighten the suspense.\n\n**2738** _ **The Mystery of Chalk Hill**_ **** NBC-TV\/Universal, 1973. 98 min. Color. D: Harry Morgan. SC: Harold Swanton. With Richard Boone, Sharon Acker, Bruce Davison, Robert Fuller, Louise Latham, Harry Morgan, Pat Hingle, John Anderson, Henry Jones, Jeannette Nolan, Lee Paul, Bernie Hamilton, Rick Lenz, Leo Gordon, Dennis Rucker, Terry Wilson, Tony Russel. Ex-gunman turned detective Hec Ramsey wants to find the killer of a lawman's bride-to-be and her young son. More than competent mystery\u2013Western telefilm first shown as an episode of \"Hec Ramsey\" (NBC-TV, 1972\u201374).\n\n_**The Mystery of the Golden Eye**_ see _**The Golden Eye**_\n\n**2739** _ **The Mystery of the Green Feather**_ **** NBC-TV\/Universal, 1972. 98 min. Color. D: Herschel Daugherty. SC: John Mestor. With Richard Boone, Rory Calhoun, Marie Windsor, Lorraine Gary, Harry Morgan, Alan Hewitt, Morgan Woodward, Lloyd Bochner, John Fiedler, Dennis Rucker, Rick Lenz. Indians are blamed for the massacre of a white family after a sacred medicine bag is found at the scene but detective Hec Ramsey does not believe they are guilty and tries to prove it. Entertaining TV movie originally telecast as \"The Green Feather Mystery\" episode of \"Hec Ramsey\" (NBC-TV, 1972\u201374).\n\n**2740** _ **Mystery of the Hooded Horsemen**_ **** Grand National, 1937. 60 min. D: Ray Taylor. SC: Edmund Kelso. With Tex Ritter, Iris Meredith, Horace Murphy, Charles King, Forrest Taylor, Earl Dwire, Joe Girard, Lafe McKee, Heber Snow (Hank Worden), Oscar Gahan, Jack C. Smith, Chick Hannon, Tex Palmer, Lynton Brent, Ray Whitley and His Range Ramblers, Allen Greer, Sherry Tansey, Ray Henderson, Victor Cox, Rube Dalroy. A masked gang murder a man for his mine and a cowpoke and his pal try to learn who is behind the crime. Well done Tex Ritter film with a good mystery plot and songs, including \"Ride, Ride, Ride\" and \"Ridin' Old Paint.\"\n\n**2741** _ **Mystery Ranch**_ **** Fox, 1932. 65 min. D: David Howard. SC: Al Cohn. With George O'Brien, Cecilia Parker, Charles Middleton, Roy Stewart, Charles Stevens, Forrester Harvey, Virginia Herdman, Betty Francisco, Noble Johnson, Russ Powell, Frank Rice, Steve Clemente. A cowboy attempts to rescue a young woman abducted by a vicious rancher, her late father's business partner. Engaging horror Western highlighted by Charles Middleton's performance as the deranged, power hungry rancher.\n\n**2742** _ **Mystery Ranch**_ **** Reliable, 1934. 56 min. D: Ray Bernard (Bernard B. Ray). SC: Carl Krusada and Rose Gordon. With Tom Tyler, Roberta Gale, Jack (Perrin) Gable, Louise Gabo, George Chesebro, Frank Hall Crane, Charles King, Jimmy Aubrey, Tom London, Lafe McKee, Lew Meehan, Robert Walker, John Elliott, Jim Corey. A mystery writer visits a ranch where he thwarts all kinds of practical jokes but accidentally becomes involved with crooks who have stolen gold bullion. The light hearted plot helps this low grade Tom Tyler feature.\n\n**2743** _ **Mystery Range**_ **** Victory, 1937. 55 min. D: Robert Hill. SC: Basil Dickey. With Tom Tyler, Jerry Bergh, Milburn Morante, Lafe McKee, Roger Williams, Richard Alexander, Jim Corey, Slim Whitaker, Steve Clark, George Morrell, Lafe McKee, Robert Hill, Wally West, Bud Pope, Buck Morgan. A cattleman's protective association agent masquerades as an outlaw to investigate a man trying to cheat his niece out of her ranch because the land is wanted by a railroad. Surprisingly good Sam Katzman production with fine work by Lafe McKee as the villainous uncle.\n\n**2744** _ **Mystery Range**_ **** Dorado, 1947. 55 min. D-SC: Ande Lamb. With Lee \"Lasses\" White, Don Haggerty, Texas Jim Lewis, Ruth (Grace Lee) Whitney, Jack Elam, Forrest Taylor, Frank Austin, Dutch Schlickenmayer, Ed Ray, Ronald Marriott, Pat Henry, Clyde Jackman. A circuit judge and his deputy try to find the killer of a man whose brother is accused of the crime. Produced by writer-director Ande Lamb and Bart Carre for Louis Weiss, this one is near the bottom of the barrel.\n\n**2745** _ **The Mystery Trooper**_ **** Syndicate, 1931. 10 Chapters. D: Stuart Paton. SC: Carl Krusada. With Blanche Mehaffey, Buzz Barton, Al Ferguson, Charles King, William von Brincken, William Bertram, Henry Roquemore, Jack Perrin, Lafe McKee, Tom McGuire, Buffalo Bill, Jr., Al Taylor, Dick Dickinson, Bill Nestell, Robert Walker, Harry Beery, White Cloud (horse). A group of people looking for a lost gold mine in Canada are harassed by outlaws also after the bonanza, but they are protected by a mysterious Mountie. Low grade, but diverting, cliffhanger reissued as _**Trail of the Royal Mounted**_ in 1938 by Guaranteed Pictures.\n\n**2746** _ **The Mystic Warrior**_ **** ABC-TV, 1984. 200 min. Color. D: Richard T. Heffron. SC: Jeb Rosebrook. With Robert Beltran, Nick Ramus, Devon Ericson, Victoria Racimo, Roger Campo, Will Sampson, Ned Romero, Douglas Toby, David Yanez. The story of a Plains Indian tribe in the early 1800s with a boy destined to grow into a warrior and lead his people. Long, leisurely and probably accurate attempt at recreating Native American life on the plans before and during the arrival of white settlers. Well worth watching.\n\n**2747** _ **The Naked Dawn**_ **** Universal-International, 1955. 82 min. Color. D: Edgar G. Ulmer. SC: Nina Schneider and Herman Schneider. With Arthur Kennedy, Betta St. John, Eugene Iglesias, Roy Engel, Charlita, Tony Martinez, Francis McDonald. A bandit robs a train, takes refuge with a Mexican farmer and schemes to steal the man's pretty wife. Lionized as a classic by Edgar G. Ulmer followers, this film does make good use of color and contains fine performances by Arthur Kennedy and Betta St. John, but overall is basically a talky soap opera with a few artistic touches.\n\n**2748** _ **The Naked Gun**_ **** Associated Film Releasing, 1956. 73 min. D: Edward Dew. SC: Ron Ormond and Jack Lewis. With Willard Parker, Mara Corday, Barton MacLane, Tom Brown, Chick Chandler, Veda Ann Borg, Timothy Carey, Billy House, Morris Ankrum, Bill Phillips, X Brands, Steve Raines, Rick Valllin, Jim Hayward, Jody McCrea, Tony McCoy, Bill Ward, Elena Di Vinci, Ben Frommer, Gil Donaldson. Attempting to deliver jewels to the heirs of an estate, an insurance agent becomes involved with crooks who are after an Aztec treasure. Poor production values hurt this otherwise acceptable drama; director Edward Dew is the Eddie Dew who starred in Westerns for Republic and Universal in the 1940s.\n\n**2749** _ **The Naked Hills**_ **** Allied Artists, 1956. 73 min. Color. D-SC: Josef Shaftel. With David Wayne, Keenan Wynn, James Barton, Jim Backus, Marcia Henderson, Denver Pyle, Myrna Dell, Lewis Russell, Frank Fenton, Fuzzy Knight, Jim Hayward, Steve Terrell, Chris Olsen. An Indiana farmer deserts his wife and family to prospect for gold in California. More than passable melodrama about gold fever.\n\n**2750** _ **Naked in the Sun**_ **** Allied Artists, 1957. 88 min. Color. D: John Hugh. SC: John Cresswell. With James Craig, Lita Milan, Barton MacLane, Robert Wark, Jim Boles, Tony Hunter, Douglas Wilson, Bill Armstrong, Dennis Cross, Peter Dearing, Tony Morris, Mike Recco. The Osceola and Seminole Indian tribes in Florida unite to oppose an evil slave trader. Offbeat plot somewhat compensates for the obvious low budget.\n\n**2751** _ **The Naked Spur**_ **** Metro-Goldwyn-Mayer, 1953. 91 min. Color. D: Anthony Mann. SC: Sam Rolfe and Harold Jack Bloom. With James Stewart, Janet Leigh, Robert Ryan, Ralph Meeker, Millard Mitchell. A bounty hunter after a wanted man plans to use the reward money to buy back land he lost during the Civil War. Relentless chase drama with good work by its compact cast.\n\n_**Naked Spur**_ (1968) see _**Love Desperados**_\n\n**2752** _ **Nakia**_ **** ABC-TV\/Screen Gems\/Columbia, 1974. 74 min. Color. D: Leonard Horn. SC: Christopher Trumbo, Michael Butler and Sy Salkowitz. With Robert Forster, Arthur Kennedy, Linda Evans, Stephen McNally, George Nader, Robert Donner, Maria Elena Cordero, Joe Kapp, Chief George Clutesi, Taylor Lacher, Jay Varella, Barbara Sigel. An Indian deputy sheriff is caught between his tribe's desire to save an historic mission and the local citizens wanting to use the spot for a housing development. A fairly interesting television movie that was the pilot for the short lived 1974 ABC-TV series of the same title.\n\n**2753** _ **The Narrow Trail**_ **** Paramount-Artcraft, 1917. 55 min. D: Lambert Hillyer. SC: Harvey E. Thew. With William S. Hart, Sylvia Breamer, Milton Ross, Robert Kortman. An outlaw falls for the niece of a vice king and later poses as a rich rancher in order to fleece his girl's uncle but she will not go along with the scheme. Well made William S. Hart silent feature.\n\n**2754** _ **Natchez Trace**_ **** Paramount, 1960. 80 min. D: Alan Crosland, Jr. SC: D.D. Beauchamp and William R. Cox. With Zachary Scott, William Campbell, Marcia Henderson, Irene James, Ann Kelly, Jim Reppert, Annette Alexander, Al Scott, Mario Galento, Frank Cunningham, Frank White, Robert Booth, Tommy Moore, Roy Haggard, Sr., Curtis Dossett, Doug Underwood, Cecil Scaiffe, Gloria Adams, Willie Adams, Kenne Duncan, Bill Ward. In the 1850s a power hungry man plans to rule an empire in Mississippi and Tennessee but when he kills a plantation owner and his daughter his fiancee plots his downfall. Okay melodrama with good work by Zachary Scott in the lead role; also called _**Bandits of the Natchez Trace**_.\n\n**2755** _ **Nate and the Colonel**_ **** MTI, 2003. 106 min. Color. D-SC: Paul Winters. With Paul Winters, Ricco Ross, Mark S. Brien, Al Harrington, Carlos Milano, Lee Whitestar, Kansas Carradine, Victoria Ramirez, David Midthunder, Micah May, Joe Seely, Karen Genaro, Nik Winterhawk, Ken Reiching, Michael Franco, Kevin P. Kearns, Mark Irvingsen. Once boyhood friends, an ex-slave and a Confederate colonel join forces after the Civil War to help Indians harassed by the U.S. cavalry. Different but effective video Western.\n\n**2756** _ **Naughty Marietta**_ **** Metro-Goldwyn-Mayer, 1935. 106 min. D: W.S. Van Dyke. SC: John Lee Mahin, Frances Goodrich and Albert Hackett. With Jeanette MacDonald, Nelson Eddy, Frank Morgan, Elsa Lanchester, Douglass Dumbrille, Joseph Cawthorn, Cecilia Parker, Walter Kingsford, Greta Meyer, Akim Tamiroff, Harold Huber, Edward Brophy, Marjorie Main, Mary Doran, Jean Chatburn, Pat Farley, Jane Barnes, Kay English, Linda Parker, Jane Mercer, Walter Long, Olive Carey, William Desmond, Cora Sue Collins, Guy Usher, Louis Mercier, Bob McKenzie, Harry Tenbrook, Edward Keane, Edward Norris, Ralph Brooks, Richard Powell, Wilfred Lucas, Arthur Belasco, Tex Driscoll, Edward Hearn, Edmund Cobb, Charles Dunbar, Frank Hagney, Ed Brady, Dr. Edouard Lippe, Roger Gray, Henry Roquemore, William Burress, Helen Shipman, Catherine Griffith, Billy Dooley, James C. Morton, William Moore, Harry Tyler, Ben Hall. Fleeing from an unhappy romance in France, a princess arrives in the wilds of Canada where she falls in love with a captain. The first teaming of Jeanette MacDonald and Nelson Eddy is a delightful screen romance filled with good music.\n\n**2757** _ **Navajo**_ **** Lippert, 1951. 70 min. D-SC: Norman Foster. With Francis Kee Teller, John Mitchell, Billy Draper, Mrs. Teller, Sammy Ogg. A Navajo Indian boy tries to come to terms with his heritage and modern-day life on the reservation. Splendid low budget Hal Bartlett production filmed on location in northern Arizona.\n\n_**Navajo Coyote**_ see _**Birth of a Legend**_\n\n**2758** _ **Navajo Joe**_ **** United Artists, 1967. 89 min. Color. D: Sergio Corbucci. SC: Dean Craig (Mario Pierotti) and Fernando Di Leo. With Burt Reynolds, Nicoleta Machiavelli, Aldo Sambrell, Fernando Rey, Tanya Lopert, Franca Polesello, Lucie Modungo, Pierre Cressoy, Nino Imparato, Alvaro De Luna, Valeria Sabel, Mario Lanfranchi, Lucio Rosator, Simon Arraga, Chris Huerta, Angel Ortiz, Angel Alvarez, Fianni De Stolfo, Rafael Albaicin. The survivor of a massacre plans to take revenge on an outlaw gang by systematically killing them one by one. Burt Reynolds fans may be interested in this violent Italian oater but its main asset is its rousing opening music theme by Ennio Morricone; released in Europe in 1966 as _**Un Dollaro a Testa**_ (A Dollar a Head).\n\n**2759** _ **The Navajo Kid**_ **** Producers Releasing Corporation, 1945. 59 min. D-SC: Harry Fraser. With Bob Steele, Syd Saylor, Caren Marsh, Ed Cassidy, Bud Osborne, Henry Hall, Stanley Blystone, Edward Howard, Charles King (Jr.), Budd Buster, Gertrude Glorie, Rex Rossi, Bert Dillard, I. Stanford Jolley, Herman Hack, George Morrell, Carl Mathews, Ray Jones, Victor Cox, Tom Smith. When his Indian agent foster father is murdered, a cowboy sets attempts to find the killer and the identity of his natural father. Fairly good entry in Bob Steele's final starring series.\n\n**2760** _ **Navajo Run**_ **** American International, 1964. 75 min. D: Johnny Seven. SC: Jo Heims. With Johnny Seven, Warren Kemmerling, Virginia Vincent, Ron Soble. A young half-breed Navajo is nursed back to health by a frontier family only be hunted by the mute brother of the girl he loves. Passable low budget drama.\n\n**2761** _ **The Navajo Trail**_ **** Monogram, 1945. 56 min. D: Howard Bretherton. SC: Jess Bowers (Adele Buffington). With Johnny Mack Brown, Raymond Hatton, Jennifer Holt, Riley Hill, Edmund Cobb, Bud Osborne, Charles King, Ray Bennett, Ed Cassidy, Tom Quinn, Mary MacLaren, Josh (John) Carpenter, Earl Crawford, Jim Hood, Jasper L. Palmer. When a Texas Ranger is murdered, two of his comrades try to find the killer with one of them infiltrating a gang planning to steal horses from an Indian tribe. Good entry in the \"Nevada Jack McKenzie\" series.\n\n**2762** _ **Navajo Trail Raiders**_ **** Republic, 1949. 60 min. D: R.G. Springsteen. SC: M. Coates Webster. With Allan \"Rocky\" Lane, Eddy Waller, Barbara Bestar, Robert Emmett Keane, Hal Landon, Dick Curtis, Dennis Moore, Ted Adams, Forrest Taylor, Marshall Reed, Steve Clark, Chick Hannon. A government man helps some friends being harassed by a band of outlaws. Fast moving \"Famous Westerns\" series entry.\n\n**2763** _ **Near the Rainbow's End**_ **** Tiffany, 1930. 57 min. D: J.P. McGowan. SC: Sally Winters and Charles A. Post. With Bob Steele, Louise Lorraine, Al Ferguson, Lafe McKee, Alfred Hewston, Hank Bell, Perry Murdock, Merrill McCormick, Cliff Lyons, Johnny Luther, Tex Palmer, Jim Corey, Carl Comstock. A rancher and his son fence off their range to prevent cattle thefts but when a sheep man is murdered the young man is blamed. Bob Steele's first talkie is a pleasant affair.\n\n**2764** _ **Near the Trail's End**_ **** Tiffany, 1931. 55 min. D: Wallace Fox. SC: G.A. Durlam. With Bob Steele, Marion Shockley, Hooper Atchley, Si Jenks, Jay Morley, Murdock MacQuarrie, Henry Roquemore, Fred Burns, Artie Ortego, Gordon DeMain, Hank Bell, F.R. Smith, Silver Tip Baker, Perry Murdock, Herman Hack, Blackie Whiteford, Rube Dalroy, Barney Beasley, Milton Brown, Herman Willingham. When a woman sees outlaws commit two murders a cowboy helps her locate the culprits. Fair Bob Steele early sound film.\n\n_**'Neath Arizona Skies**_ see _**'Neath the Arizona Skies**_\n\n**2765** _ **'Neath Canadian Skies**_ **** Screen Guild, 1946. 41 min. D: B. Reeves Eason. SC: Arthur V. Jones. With Russell Hayden, Inez Cooper, Douglas Fowley, I. Stanford Jolley, Jack Mulhall, Cliff Nazarro, Richard Alexander, Kermit Maynard, Boyd Stockman, Jimmie Martin, Gil Patric, Pat Hurst, Joe Bernard, Bob Burns. Assigned to look into the murders of a prospector and a comrade, a Mountie works undercover in a town plagued by claim jumpers. Okay compact drama with nice scenic values and a good cast.\n\n**2766** _ **'Neath the Arizona Skies**_ **** Monogram, 1934. 52 min. D: Harry Fraser. SC: B.R. (Burl) Tuttle. With John Wayne, Sheila Terry, Shirley Jane Ricketts (Shirley Jean Rickert), George Hayes, Jack Rockwell, Yakima Canutt, Weston Edwards (Harry Fraser), Buffalo Bill, Jr., Phil (Kieffer) Keefer, Frank Hall Crane, Earl Dwire, Artie Ortego, Tex Phelps, Eddie Parker, Billy Franey, George Morrell, Herman Hack, Allen Pomeroy. A cowboy is the guardian of a small Indian girl who is the heir to oil land and when outlaws kidnap the child he tries to free her. Shoddy production values hurt this otherwise fun John Wayne-Lone Star entry, filmed the same year by Victor Adamson as _**Circle Canyon**_ (q.v.); also known as _**'Neath Arizona Skies**_.\n\n**2767** _ **'Neath Western Skies**_ **** Syndicate, 1930. 60 min. D: J.P. McGowan. SC: Sally Winters. With Tom Tyler, Lotus Thompson, J.P. McGowan, Harry Woods, Hank Bell, Bobby Dunn, Alfred Hewston, Barney Furey. An outlaw gang tries to sabotage an oil driller's operations and kidnaps his girlfriend. Early Tom Tyler outdoor talkie that should please his fans.\n\n**2768** _ **The Nebraskan**_ **** Columbia, 1953. 68 min. Color. D: Fred F. Sears. SC: David Lang and Martin Berkeley. With Phil(ip) Carey, Roberta Haynes, Wallace Ford, Richard Webb, Lee Van Cleef, Maurice Jara, Regis Toomey, Jay Silverheels, Pat Hogan, Dennis Weaver, Boy \"Red\" Morgan. An Indian scout is blamed for a murder and the accusation almost sets off a war. Cheaply made oater originally issued in 3-D.\n\n**2769** _ **Ned Blessing:**_ _**The True Story of My Life**_ **** CBS-TV, 1992. 97 min. Color. D: Peter Werner. SC: William D. Wittliff. With Daniel Baldwin, Luis Avalos, Chris Cooper, Sean Baca, Taylor Fry, Julia Campbell, Rene Auberjonois, Tim Scott, Bob Gunton, Miguel Sandoval, Jeff Kober, Tony Frank, Jimmie F. Skaggs, Vince Davis, M.C. Gainey, Michael Harris, Sonny Carl Davis, Julius Tennon, Annalee Jeffries, Jill Parker-Jones, Blue Deckert, Mark Voges, Dennis Hill, John Martin, Harvey Christiansen, Brady Coleman, James Prince, Richard Jones. While awaiting execution a cowboy and ex-lawman reflects on his adventurous life. Average telefilm that served as the pilot for the 1993 CBS-TV series \"Ned Blessing: The Story of My Life and Times\" starring Brad Johnson, from which three paste-up features were derived: _**The Adventures of Ned Blessing:**_ _**Dead Man's Revenge**_ , _**The Adventures of Ned Blessing:**_ _**Return of the Hooded Man**_ and _**The Adventures of Ned Blessing:**_ _**Return to Plum Creek**_ (qq.v.).\n\n**2770** _ **Ned Kelly**_ **** United Artists, 1970. 103 min. Color. D: Tony Richardson. SC: Tony Richardson and Neil Hartley. With Mick Jagger, Allen Bickford, Geoff Gilmour, Mark McManus, Serge Lazareff, Peter Sumner, Ken Shorter, James Elliott, Clarissa Kaye, Diane Craig, Susan Lloyd, Bruce Barry, Janne Wesley, Ken Goodlet, Nigel Lovell, John Gray, Anne Harvey, Frank Thring, Gordon McDougall. A cowboy ends up being Australia's most famous and wanted outlaw in the 1870s. Worth watching if you are a fan of Mick Jagger, who also did the music along with Waylon Jennings; re-titled _**Ned Kelly, Outlaw**_.\n\n**2771** _ **Ned Kelly**_ **** Focus Features, 2003. 110 min. Color. D: Gregor Jordan. SC: John Michael McDonagh. With Heath Ledger, Orlando Bloom, Geoffrey Rush, Naomi Watts, Joel Edgerton, Laurence Kinlan, Philip Barantini, Kerry Condon, Kris McQuade, Emily Browning, Kiri Paramore, Rachel Griffiths, Geoff Morrell, Charles \"Bud\" Tingwell, Saskia Burmeister, Peter Phelps, Russell Dykstra, Nick Farrell, Russell Gilbert, Brooke Harman, Molly McCaffrey, Tim Wright, Nicholas Bell, Anthony Hayes, Jonathan Hardy, Karen Davitt, Declan Simpson, Andrew S. Gilbert, John Muirhead, Eddy McShortall, Peter O'Shea, Nick Bourke, Christopher Baker, Brian Wray, Thea Gumbert, Gregan O'Leary, Chris Wilson, Graham Jahne, Clayton Jacobson, Cody O'Prey, Tasman Vaughan, Greg Saunders, Victoria Eagger, Peter Young, Alexander Ramsey, Samuel Shepherd, Talia Zucker, Laurie Jensen. After falsely being accused of a crime, Australian bandit Ned Kelly and his gang seek refuge in the outback. Okay retelling of the Ned Kelly saga.\n\n_**Ned Kelly, Outlaw**_ see _**Ned Kelly**_ (1970)\n\n**2772** _ **Neeka**_ **** Wrather Corporation, 1968. 100 min. Color. D: Jack B. Hively and Dick Moder. SC: Eric Freiwald and Robert Schaefer. With Lassie, Jed Allan, Robert Rockwell, Mark Miranda, Jeff Pomerantz, Philip Pine, Douglas Henderson, John Harmon, Carlo Rizzo, William Bramley. Lassie and her Forest Ranger master head to Alaska to search for a deranged hunter and soon meet an adopted Indian boy and his foster father. Okay family fare filmed in Alaska and made up of four segments of \"Lassie\" (CBS-TV, 1954\u201371): \"Day of the Wolf,\" \"Eagle's Dynasty,\" \"Glacier Canyon\" and \"Patsy.\" Also called _**The Adventures of Neeka**_.\n\n_**Nelson Nye's Seven Sixgunners**_ see _**The Seven Sixgunners**_\n\n**2773** _ **Nevada**_ **** Paramount, 1927. 65 min. D: John Waters. SC: John Stone and L.G. Rigby. With Gary Cooper, Thelma Todd, William Powell, Philip Strange, Ernie Adams, Christian Frank, Ivan Christy, Guy Oliver. Two outlaws try to go straight but get mixed up with cattle rustlers. Interesting silent version of the Zane Grey book teaming Gary Cooper and Ernie Adams as the two good-bad guys with William Powell as the villain; remade in 1935 (q.v.).\n\n**2774** _ **Nevada**_ **** Paramount, 1935. 70 min. D: Charles Barton. SC: Garnett Weston and Stuart Anthony. With Larry \"Buster\" Crabbe, Kathleen Burke, Monte Blue, Syd Saylor, William Duncan, Richard Carle, Stanley Andrews, Frank Sheridan, Raymond Hatton, Glenn (Leif) Erickson, Jack Kennedy, Henry Roquemore, William Desmond, Frank Rice, Barney Furey, William L. Thorne. A gunman and his pal become cowboys on an Englishman's ranch but due to their pasts they get involved with outlaws. Program feature remake of the Zane Grey book with good work by Buster Crabbe as the reformed gunfighter.\n\n**2775** _ **Nevada**_ **** RKO Radio, 1944. 62 min. D: Edward Killy. SC: Norman Houtston. With Robert Mitchum, Anne Jeffreys, Guinn Williams, Nancy Gates, Richard Martin, Craig Reynolds, Harry Woods, Russell Hopton, Edmund Glover, Alan Ward, Harry McKim, Larry Wheat, Jack Overman, Emmett Lynn, Wheaton Chambers, Philip Morris, Mary Halsey, Patti Brill, Bryant Washburn, Bert Moorhouse, George DeNormand, Sammy Blum, Margie Stewart. A man is nearly lynched for a murder he did not commit and attempts to prove a gang of claim jumpers are the real culprits. Robert Mitchum is fine in this \"B\" outing, a good third filming of Zane Grey's novel.\n\n_**Nevada**_ (1971) see _**The Boldest Job in the West**_\n\n**2776** _ **Nevada Badmen**_ **** Monogram, 1951. 58 min. D: Lewis D. Collins. SC: Joseph O'Donnell. With Whip Wilson, Fuzzy Knight, Phyllis Coates, Jim Bannon, I. Stanford Jolley, Kenne Duncan, Bill Kennedy, Marshall Reed, Earle Hodgins, Riley Hill, Lee Roberts, Pierce Lyden, Bud Osborne. Three cattlemen attempt to find out who murdered the brother of one of them for his hidden gold claim. A good plot helps this otherwise mediocre Whip Wilson vehicle.\n\n**2777** _ **Nevada Buckaroo**_ **** Tiffany, 1931. 59 min. D: John P. McCarthy. SC: Wellyn Totman. With Bob Steele, Dorothy Dix, George Hayes, Ed Brady, Glen Cavender, Billy Engle, Artie Ortego, Blackie Whiteford, Gordon DeMain, Arthur Millett, Merrill McCormick, John Elliott, Phil Dunham, Charles Le Moyne, Tina Menard, Perry Murdock, Frank Lanning, William McCall, F.R. Smith, Rose Plummer. Once on the wrong side of the law, a man changes his ways for a girl but runs into trouble when his old gang murders his sidekick. Well written and quick paced Bob Steele film.\n\n**2778** _ **Nevada City**_ **** Republic, 1941. 58 min. D: Joseph Kane. SC: James R. Webb. With Roy Rogers, George \"Gabby\" Hayes, Sally Payne, Fred Kohler, Jr., George Cleveland, Billy Lee, Joseph Crehan, Pierre Watkin, Yakima Canutt, Rex Lease, Art Mix, Jack Ingram, Syd Saylor, Hank Bell, Henry Wills, Bob Woodward, Jack Kirk, Fred Burns, Chuck Baldra, Jack C. Smith. When a crooked businessman attempts to control freight traffic in California, a cowboy tries to stop him. Fast moving, entertaining Roy Rogers feature.\n\n**2779** _ **Nevada Smith**_ **** Paramount, 1966. 120 min. Color. D: Henry Hathaway. SC: John Michael Hayes. With Steve McQueen, Karl Malden, Suzanne Pleshette, Brian Keith, Arthur Kennedy, Raf Vallone, Janet Margolin, Howard Da Silva, Pat Hingle, Martin Landau, Paul Fix, Gene Evans, Josephine Hutchinson, John Doucette, Val Avery, Lyle Bettger, Bert Freed, David McLean, Ric Roman, John Litel, Ted De Corsia, Stanley Adams, George Mitchell, Sheldon Allman, Strother Martin, Holly Bane, Iron Eyes Cody, Henry Wills, Chuck Roberson, Sandy Kenyon, John Lawrence, Merritt Bohn. A man seeks revenge on the trio of outlaws who brutally murdered his parents. Well acted and produced follow-up to _**The Carpetbaggers**_ (Paramount, 1964).\n\n**2780** _ **Nevada Smith**_ **** NBC-TV\/Metro-Goldwyn-Mayer, 1975. 74 min. Color. D: Gordon Douglas. SC: Martin Rackin and John Michael Hayes. With Cliff Potts, Lorne Greene, Adam West, Warren Vanders, Jorge Luke, Jerry Gatlin, Eric Cord, Lorraine Chanel, John McKee, Alan George, Roger Cudney. Two old friends, a half-breed cowboy and his former teacher, band together to carry a shipment of explosives. So-so TV movie based on the 1966 feature (q.v.) and the pilot for an unsold series.\n\n**2781** _ **The Nevadan**_ **** Columbia, 1950. 81 min. Color. D: Gordon Douglas. SC: George W. George and George P. Slavin. With Randolph Scott, Dorothy Malone, Forrest Taylor, Frank Faylen, George Macready, Charles Kemper, Jeff Corey, Tom Powers, Jack O'Mahoney (Jock Mahoney), Stanley Andrews, James Kirkwood, Kate Drain Lawson, Olin Howlin, Louis Mason. A government agents works undercover with an outlaw to retrieve gold the latter stole but which is now in the possession of a gang. Good Randolph Scott fare with plenty of action coupled with an entertaining story.\n\n**2782** _ **Never a Dull Moment**_ **** RKO Radio, 1950. 89 min. D: George Marshall. SC: Lou Breslow and Doris Anderson. With Irene Dunne, Fred MacMurray, William Demarest, Andy Devine, Gigi Perreau, Natalie Wood, Philip Ober, Jack Kirkwood, Ann Doran, Lela Bliss, Irving Bacon, Chester Conklin, Jacqueline de Wit, Gene Evans, Jimmy Hawkins, Olin Howlin, Harry Tyler, Ralph Peters, Paul Newlan, Anne O'Neal, Janine Perreau, Margaret Gibson, Jack Jackson, George Leigh, Alan Dinehart III, Jo Ann Marlowe, Art Dupuis, Kermit Maynard, Helen Dickson, Victoria Horne, Edna Holland, Virginia Mullen, Frank Yaconelli, Dan White, Carl Sklover, Bob Thom, Connie Van. An attractive composer marries a rancher and moves to his home only to miss city life besides being at odds with his children. Slow going Western romantic comedy misfire from producer Harriet Parsons.\n\n**2783** _ **Never Cry Wolf**_ **** Buena Vista, 1983. 105 min. Color. D: Carroll Ballard. SC: Curtis Hanson, Sam Hamm and Richard Kletter. With Charles Martin Smith, Brian Dennehy, Zachary Ittimangnaq, Samson Jorah, Hugh Webster, Martha Ittimangnaq, Tom Dalgren, Walker Start; C.M. Smith, Eugene Corr, Christina Luescher (narrators). A biologist learns to survive in the Arctic while studying the habits of the white wolf to see if they are responsible for the disappearance of caribou herds. Location shooting, a good story and fine work by Charles Martin Smith as the biologist make this pleasant viewing.\n\n_**Never Give an Inch**_ see _**Sometimes a Great Notion**_\n\n**2784** _ **The New Daughters of Joshua Cabe**_ **** ABC-TV, 1976. 74 min. D: Bruce Bilson. SC: Paul Savage. With John McIntire, Jack Elam, Jeanette Nolan, John Dehner, Liberty Williams, Renne Jarrett, Lezlie Dalton, Geoffrey Lewis, Sean McClory, Joel Gabiani, Ford Rainey, Larry Hovis, James Lydon, Randall Carver. When a sheriff is falsely jailed on a murder charge, a trio of young women, who he previously palmed off as his daughters, come to the rescue. Anemic telefeature preceded by _**The Daughters of Joshua Cabe**_ and _**The Daughters of Joshua Cabe Return**_ (qq.v.).\n\n**2785** _ **The New Frontier**_ **** Republic, 1935. 55 min. D: Carl L. Pierson. SC: Robert Emmett (Tansey). With John Wayne, Muriel Evans, Warner Richmond, Alan Bridge, Murdock MacQuarrie, Allan Cavan, Sam Flint, Mary MacLaren, Theodore Lorch, Glenn Strange, Phil Kiefer, Jack Montgomery, Earl Dwire, Hooper Atchley, Jack Kirk, Frank Ball, Sherry Tansey, Herman Hack, Art Dillard, Pat Harmon, Chuck Baldra, Tex Phelps, Perry Murdock, John Ince, Cactus Mack, Eddie Parker, Tex Palmer, Fred Parker, Jack Evans, Buck Moulton. A cowboy finds out his sheriff father has been murdered by a corrupt saloon owner and he enlists the help of a bandit gang in opposing the killer and his followers. Very good John Wayne feature with a terrifically staged shootout finale.\n\n**2786** _ **The New Frontier**_ **** Republic, 1939. 57 min. D: George Sherman. SC: Betty Burbridge and Luci Ward. With John Wayne, Ray Corrigan, Raymond Hatton, Phyllis Isley (Jennifer Jones), Eddy Waller, Sammy McKim, LeRoy Mason, Harrison Greene, Reginald Barlow, Burr Caruth, Dave O'Brien, Hal Price, Jack Ingram, Bud Osborne, Slim Whitaker, Wilbur Mack, George Chesebro, Frankie Marvin, Oscar Gahan, Jody Gilbert, Herman Hack, Charles Murphy, Curley Dresden, Fred Burns, Cactus Mack, Bill Nestell, Victor Cox, Bob Reeves, John Elliott, Frank Ellis, Walt LaRue, Bud McClure, Bill Wolfe, Jim Corey, Bob Burns, Chuck Baldra, George Plues. Three cowboys convince settlers to move to a new range only to find out they have been cheated by crooked land speculators. John Wayne's last \"Three Mesquiteers\" adventure is okay but not up to the standard of earlier entries; TV title: _**Frontier Horizon**_.\n\n**2787** _ **The New Land**_ **** Warner Bros., 1973. 161 min. Color. D: Jan Troell. SC: Bengi Forslund and Jan Troell. With Max von Sydow, Liv Ullman, Eddie Axberg, Hans Alfredson, Halvar Bjork, Allan Edwall, Peter Lindgren, Oscar Ljung. A Swedish immigrant, along with his wife and brother, settle in Minnesota in the 1850s and face a series of hardships. Somewhat overlong but engaging follow-up to director Jan Troell's _**The Emigrants**_ (Warner Bros., 1971); well worth viewing.\n\n**2788** _ **The New Maverick**_ **** ABC-TV\/Warner Bros., 1978. 100 min. Color. D: Hy Averback. SC: Juanita Bartlett. With James Garner, Jack Kelly, Charles Frank, Susan Blanchard, Eugene Roche, Susan Sullivan, George Loros, Woodrow Parfrey, Gary Allen, Henel Paige Camp, Jack Garner, Graham Jarvis. The British nephew of the notorious Maverick brothers enlists the help of his famous uncles, resulting in a series of misadventures. Nostalgic TV feature recreation of \"Maverick\" (ABC-TV, 1957\u201362) that served as the pilot for \"Young Maverick\" (CBS-TV, 1979\u201380).\n\n**2789** _ **New Mexico**_ **** United Artists, 1951. 78 min. Color. D: Irving Reis. SC: Max Trell. With Lew Ayres, Marilyn Maxwell, Andy Devine, Robert Hutton, Raymond Burr, Jeff Corey, Lloyd Corrigan, Verna Felton, Ted De Corsia, John Hoyt, Donald Buka, Robert Osterloh, Ian MacDonald, William Tannen, Hans Conreid, Walter Greaza, Jack Kelly, Bob Duncan. A captain in the U.S. cavalry tries to prevent warfare in New Mexico with Indians led by Acoma. Mundane plot is given a good shot in the arm by fine production values, good direction and cast.\n\n**2790** _ **New Moon**_ **** Metro-Goldwyn-Mayer, 1940. 105 min. D: Robert Z. Leonard. SC: Jacquel Deval and Robert Arthur. With Jeanette MacDonald, Nelson Eddy, Mary Boland, George Zucco, H.B. Warner, Grant Mitchell, Stanley Fields, Dick Purcell, John Miljan, Ivan Simpson, William Tannen, Bunty Cutler, Claude King, Cecil Cunningham, Joe Yule, George Irving, Robert Warwick, Hillary Brooke, Rafael Storm, Winifred Harris, Edwin Maxwell, Paul E. Burns, Trevor Bardette, LeRoy Mason, Ray Walker, Gayne Whitman, Jack Perrin, Claire Rochelle, Alden Chase, Nat Pendleton, Buster Keaton, Edward Hearn, Ralph Dunn, Gino Corrado, Christian J. Frank, Ray Teal, Fred Graham, Harry Strang, Ted Oliver, Dorothy Granger, Forbes Murray, June Gittelson, Warren Rock, George Magrill, Ed O'Neill, Sarah Edwards, Max Marx, Arthur Belasco, Nick Copeland, David Alison. During the reign of King Louis XVI a woman arrives in New Orleans to look over property she has inherited and falls in love with a French political fugitive who is a bondsman. Pleasant teaming of Jeanette MacDonald and Nelson Eddy with a good Sigmund Romberg score as added dressing for this period piece. The 1930 MGM version, called _**Parisian Belle**_ on TV, with Grace Moore and Lawrence Tibbett, is set in Russia and not frontier Louisiana.\n\n**2791** _ **Nido de Aguilas**_ (Nest of Eagles). Almada Films, 1965. 109 min. Color. D-SC: Vicente Orona. With Fernando Almada, Jose Elias Moreno, Jaime Fernandez, Dacia Gonzalez, Jorge Martinez de Hoyos, Xavier Loya, J. Antonio Brillas, Martha Elena Cervantes, Noe Murayama, Sadi Dupeyron, Manuel Arvide, Jose Raul Mena, Edmundo Espino. The female member of an outlaw gang falls in love with a farmer but a rival bandit kidnaps her. Well done Mexican Western.\n\n**2792** _ **The Night Cry**_ **** Warner Bros., 1926. 55 min. D: Herman C. Raymaker. SC: Ewart Adamson, Paul Klein and Edward Meagher. With Rin Tin Tin, John Harron, June Marlowe, Gayne Whitman, Heine Conklin, Don Alvarado, Mary Louise Miller. A dog is unjustly accused of killing sheep when the culprit is a giant condor that steals his master's baby and the canine sets out to rescue her. Top notch Rin Tin Tin silent feature.\n\n**2793** _ **Night Games**_ **** NBC-TV\/Paramount, 1974. 74 min. Color. D: Don Taylor. SC: E. Jack Neumann. With Barry Newman, Susan Howard, Stefanie Powers, Anjanette Comer, Joanna Cameron, Albert Salmi, Luke Askew, Jon Cypher, Henry Darrow, Ralph Meeker, William Prince, Dennis Patrick, Robert Emhardt, William Hanson, Larry Thor. A lawyer in a modern-day Arizona cattle town defends a young, socially prominent woman accused of killing her husband. Fair telefilm that served as the pilot for \"Petrocelli\" (NBC-TV, 1974\u201376).\n\n**2794** _ **The Night Hawk**_ **** W.W. Hodkinson, 1924. 60 min. D: Stuart Paton. SC: Joseph Poland. With Harry Carey, Claire Adams, Joseph Girard, Fred Malatesta, Nicholas De Ruiz, Lee Shumway, Myles McCarthy, Fred Kelsey. Hired to murder a sheriff in the West, a New York City crook arrives on the scene only to fall in love with the lawman's daughter. Sturdy Harry Carey silent feature.\n\n_**The Night of the Desperado**_ see _**Ringo's Big Night**_\n\n**2795** _ **The Night of the Grizzly**_ **** Paramount, 1966. 102 min. Color. D: Joseph Pevney. SC: Warren Douglas. With Clint Walker, Martha Hyer, Keenan Wynn, Nancy Kulp, Kevin Brodie, Ellen Corby, Jack Elam, Ron Ely, Med Florey, Leo Gordon, Don Haggerty, Sammy Jackson, Victoria Paige Meyerink, Candy Moore, Regis Toomey. Trying to settle down to ranching in Wyoming in the 1880s, a former lawman finds himself up against unexpected bills, former foes and a killer bear. Well made action melodrama enhanced by a fine cast.\n\n**2796** _ **Night of the Wolf**_ **** Animal Planet, 2002. 89 min. Color. D: David S. Cass, Sr. SC: Paul Cooper. With Anne Archer, Robert Urich, Michael Shamus Wiles, Peter Dobson, Sally Kirkland, C. Thomas Howell, Zach Bostrom, Stephen Bridgewater, Norman Alden, James Lashly, Michele Nordin, Monty Stuart, David Atkinson. A widowed rancher is stranded in the wilderness with a trapped wolf and as her son searches for her he comes across murderous poachers. Mediocre modern-day TV fare.\n\n**2797** _ **Night Passage**_ **** Universal-International, 1957. 90 min. Color. D: James Neilson. SC: Borden Chase. With James Stewart, Audie Murphy, Dan Duryea, Dianne Foster, Elaine Stewart, Brandon de Wilde, Jay C. Flippen, Herbert Anderson, Robert Wilkie, Hugh Beaumont, Jack Elam, Tommy Cook, Paul Fix, Olive Carey, James Flavin, Donald Curtis, Ellen Corby, Ted Mapes, Patsy Novak, Chuck Roberson, Kenne Duncan, John Davis, Paul Spahn, Jack Lowell, Herman Pulver. A railroad troubleshooter learns an outlaw gang, that includes his younger brother, plans to rob a train of its payroll shipment. Expansive and well done entertainment, with Dan Duryea especially good as the outlaw leader, Whitey Harbin.\n\n**2798** _ **Night Raiders**_ **** Monogram, 1952. 52 min. D: Howard Bretherton. SC: Maurice Tombragel. With Whip Wilson, Fuzzy Knight, Lois Hall, Tommy Farrell, Terry Frost, Marshall Reed, Lane Bradford, Steve Clark, Boyd Stockman, Forrest Taylor, Iron Eyes Cody, Carol Henry, Ed Cassidy, Roy Butler, Stanley Price. Two lawmen investigate mysterious raids on ranches during which nothing is stolen. The mystery angle greatly helps this otherwise pedestrian Whip Wilson adventure.\n\n**2799** _ **The Night Rider**_ **** Artclass, 1932. 54 min. D: Fred Newmeyer. SC: Harry (Fraser) P. Crist. With Harry Carey, Elinor Fair, George Hayes, Robert Kortman, Walter Shumway, Julian Rivero, Jack Weatherby, Tom London, Slim Whitaker, Jack Kirk, Hank Bell, Ben Corbett, Bart Carre, Cliff Lyons. A lawman pretends to be a gunfighter to stop a murderous outlaw gang. Defective production values hurt this otherwise entertaining Harry Carey vehicle.\n\n**2800** _ **The Night Rider**_ **** ABC-TV\/Universal, 1979. 78 min. Color. D: Hy Averback. SC: Stephen J. Cannell. With David Selby, Kim Cattrall, Percy Rodrigues, George Grizzard, Harris Yulin, Pernell Roberts, Anthony Herrera, Anna Lee, Michael Sharrett, Hildy Brooks, Curt Lowens, Van Williams, Stuart Nisbet, Gary Allen, Whit Bissell, Edward Knight, Susan Davis, Maria Diane, Sydney Penny. When crooks kill his family for its silver mine a New Orleans gentleman turns masked avenger. Passable TV movie.\n\n**2801** _ **The Night Riders**_ **** Second National Pictures, 1920. 63 min. D-SC: Alexander Butler. With Maudie Dunham, Albert Ray, Andrea Beaulieu, Russell Gordon, C. McCarthy, Jose De La Cruz, Goober Glen, William Ryno. In the Canadian Northwest, a man falls in love with a blind rancher's daughter and organizes a posse to purse a raider gang led by the Red Mask. A neat plot is not executed well in this Canadian made silent yarn.\n\n**2802** _ **The Night Riders**_ **** Republic, 1939. 58 min. D: George Sherman. SC: Betty Burbridge and Stanley Roberts. With John Wayne, Ray Corrigan, Max Terhune, Doreen McKay, Ruth Rogers, George Douglas, Tom Tyler, Kermit Maynard, Sammy McKim, Walter Wills, Ethan Laidlaw, Ed Peil, Sr., Tom London, Jack Ingram, Bill Nestell, Yakima Canutt, Glenn Strange, David Sharpe, Bud Osborne, Lee Shumway, Cactus Mack, Hal Price, Hank Worden, Roger Williams, Olin Francis, Francis Walker, Hugh Prosser, Jack Kirk, Georgia Summers, Horace Murphy, Francis Sayles, John Ince, Curley Dresden, Art Dillard, George (Montgomery) Letz, Bob Card, Eva McKenzie, Jane Keckley, David McKim, Allan Cavan, Frank O'Connor, Al Taylor. A crook uses a forged land grant to make himself the ruler of thousands of acres of land but when he forces settlers, who cannot pay his high taxes, off their properties three cowboys come to the rescue. Very good \"Three Mesquiteers\" series segment. Remade as _**Arizona Terrors**_ (q.v.).\n\n**2803** _ **Night Riders**_ **** Alameda Films, 1959. 77 min. Color. D: Fernando Mendez. SC: Ramon Obon. With Gaston Santos, Alma Rosa Aguirre, Carlos Ancira, Pedro de Aguillon, Quintin Bulnes, Jose Chavez, Antonio Raxel, Guillermo Alvarez Dianch. A gang of masked riders in the shape of demons terrorize a remote town and a government agent and his sidekick try to stop them. Fast paced Mexican Western issued in that country as _**Los Diablos del Terror**_ (The Devils of Terror).\n\n**2804** _ **Night Riders of Montana**_ **** Republic, 1951. 60 min. D: Fred C. Brannon. SC: M. Coates Webster. With Allan \"Rocky\" Lane, Claudia Barrett, Roy Barcroft, Chubby Johnson, Arthur Space, Myron Healey, Mort Thompson, Lester Dorr, Ted Adams, George Chesebro, Don C. Harvey, Zon Murray, John Hamilton, Bud Osborne, Marshall Bradford. An outlaw gang plagues ranchers and a ranger working for the state is sent to halt their activities. Another action filled Allan Lane effort.\n\n**2805** _ **Night Stage to Galveston**_ **** Columbia, 1952. 60 min. D: George Archainbaud. SC: Norman S. Hall. With Gene Autry, Pat Buttram, Virginia Houston, Thurston Hall, Judy Nugent, Robert Livingston, Harry Cording, Robert Bice, Frank Sully, Clayton Moore, Frank Rawls, Steve Clark, Harry Lauter, Robert Peyton, Lois Austin, Kathleen O'Malley, Riley Hill, Richard Alexander, Boyd Stockman, Bob Woodward, Sandy Sanders, Ben Weldon, Gary Goodwin. Two ex-rangers, now newspapermen, work on a story about Texas state police corruption and nearly get killed when they try to save the kidnapped daughter of their publisher from crooked officials. A good story highlights this well made Gene Autry vehicle.\n\n**2806** _ **Night Time in Nevada**_ **** Republic, 1948. 67 min. Color. D: William Witney. SC: Sloan Nibley. With Roy Rogers, Andy Devine, Adele Mara, Grant Withers, Marion Harmon, Joseph Crehan, Holly Bane, Bob Nolan and The Sons of the Pioneers (Tim Spencer, Lloyd Perryman, Ken Carson, Pat Brady, Hugh Farr, Karl Farr), George Carleton, Steve Darrell, Hank Patterson, Rex Lease, Forrest Taylor, Bob Reeves, Jim Nolan. To cover up a murder he committed sixteen years before, a man plans to steal cattle belonging to Roy Rogers and the Sons of the Pioneers to pay off the victim's daughter. A good, exciting and well directed Roy Rogers feature dominated by Grant Withers as the bad guy.\n\n**2807** _ **Nightwing**_ **** Columbia, 1979. 103 min. Color. D: Arthur Hiller. SC: Steve Shagan, Bud Shrake and Martin Cruz Smith. With Nick Mancuso, David Warner, Stephen Macht, Kathryn Harrold, Strother Martin, Ben Piazza, George Clutesi, Donald Hotton, Judith Novgrod, Charles Hallahan, Pat Corley, Alice Hirson, Danny Dapien, Jose Toledo, Charlie Bird, Peter Prouse, Richard Romacito, Flavio Martinez III. A mysterious man arrives in the Arizona desert intent on killing an army of disease carrying vampire bats. Overlong and basically boring horror Western.\n\n**2808** _ **Nikki, Wild Dog of the North**_ **** Buena Vista, 1961. 74 min. Color. D: Jack Couffer. SC: Ralph Wright and Winston Hibler. With Jean Coutu, Emile Genest, Uriel Luft, Robert Rivard, Nikki (dog), Neewa (bear); Jacques Fauteux, Dwight Hauser (narrators). A young wolf dog and a bear cub, separated from their master, are forced to survive in the wilds. Satisfying adaptation of James Oliver Curwood's novel _Nomads of the North_.\n\n**2809** _ **The Nine Lives of Elfego Baca**_ **** Buena Vista, 1960. 80 min. Color. D-SC: Norman Foster. With Robert Loggia, James Dunn, Lisa Montell, Robert F. Simon, Leonard Strong, Rico Alaniz. A New Mexico lawman defies an 80 man lynch posse in order to protect a prisoner. Issued abroad theatrically, this feature was first shown as two episodes of Walt Disney's ABC-TV series in 1958; good entertainment.\n\n_**No Man's Land**_ see _**No Man's Range**_\n\n**2810** _ **No Man's Land**_ **** NBC-TV, 1984. 104 min. D: Rod Holcomb. SC: Juanita Bartlett. With Stella Stevens, Terri Garber, Melissa Michaelsen, Donna Dixon, Estelle Getty, Sam J. Jones, Frank Bonner, John Rhys-Davies, Janis Paige, Dack Rambo, John Quade, Buck Taylor, Jack Garner, Tony Swartz, Will Albert, Marc Alaimo, Jeremy Ross, Roz Witt, Eldon Quick. A beautiful frontier marshal and her three comely daughters fight lawlessness in the Old West. Forced TV Western comedy.\n\n**2811** _ **No Man's Law**_ **** Path\u00e9, 1927. 66 min. D: Fred Jackman. SC: Frank Butler. With Rex (horse), Barbara Kent, Theodore Von Eltz, Oliver Hardy, Jimmy Finlayson. Two crooks stumble onto a remote cabin where a young woman and her uncle are protected by a wild stallion. Another silent adventure with Rex the Wonder Horse and a pretty good one.\n\n**2812** _ **No Man's Range**_ **** Supreme, 1935. 56 min. D: Robert North Bradbury. SC: Forbes Parkhill. With Bob Steele, Roberta Gale, Buck Connors, Steve Clark, Charles K. French, Jack Rockwell, Roger Williams, Earl Dwire, Ed Cassidy, Jim Corey, Forrest Taylor, Herman Hack, Art Dillard, Clyde McClary. When outlaws rule the range a cowboy pretends to be one of them in order to bring a close to their activities. Average entry in Bob Steele's lengthy series for producer A.W. Hackel; also called _**No Man's Land**_.\n\n**2813** _ **No Name on the Bullet**_ **** Universal, 1959. 77 min. Color. D: Jack Arnold. SC: Gene L. Coon. With Audie Murphy, Joan Evans, Charles Drake, R.G. Armstrong, Virginia Grey, Warren Stevens, Whit Bissell, Karl Swenson, Willis Bouchey, Edgar Stehli, Jerry Paris, Charles Watts, Simon Scott, John Alderson, Russ Bender, Jim Hyland, Bob Steele, Hank Patterson, Edgar Dearing, Harold Goodwin, William Mims, Jess Kirkpatrick, Marjorie Bennett, Charles Cane, Guy Wilkerson, Dennis Rush, Vincent Perry, Hugh Corcoran, Helen Jay, Fern Barry. Each of the citizens of a small town feel they may be the intended victim when a hired killer arrives. Okay action Audie Murphy outing.\n\n**2814** _ **No Room to Die**_ **** Junior Film, 1969. 88 min. Color. D: Willy S. Regan (Sergio Garrone). SC: Sergio Garrone. With Antonio De Teffe (Anthony Steffen), William Berger, Nicoletta Machiavelli, Mario Brega, Riccardo Garrone, Maria Angela Giordano, Giancarlo Sisti, Franco Ukmar, Guilio Mauroni, Gabriele Torrei, Giorgio Dolfin, Darar Gilberto Galimberti, Claudio Ruffini, Roberto Messina, Emilio Messina, Renzo Paarello, Angelo Susani. A preacher (Ringo) and a bounty hunter (Django) join forces to get the reward money posted on gang smuggling aliens across the border. Well made, confusing and very violent Italian oater made as _**Una Lunga Fila di Croci**_ , running 97 minutes; also called _**Hanging for Django**_ and _**Noose for Django**_.\n\n**2815** _ **Nob Hill**_ **** 20th Century\u2013Fox, 1945. D: Henry Hathaway. SC: Norman Reilly Raine and Wanda Tuchock. With George Raft, Joan Bennett, Vivian Blaine, Peggy Ann Garner, Alan Reed, B.S. Pully, Emil Coleman, Edgar Barrier, Smith and Dale (Joe Smith, Charles Dale), J. Farrell MacDonald, Chick Chandler, Paul Hurst, Don Costello, Edward Keane, Arthur Loft, William Haade, Ralph Peters, Chief Thundercloud, Harry Strang, Otto Reichow, Harry Shannon, Syd Saylor, Claire Rochelle, Grandon Rhodes, Will Stanton, Alphonse Martell, Bud Jamison, Forbes Murray, Nestor Paiva, Benson Fong, Sven Hugo Borg, Sam Flint, Harrison Greene, Neal Hart, Sam Ash, Rory Calhoun, Freddie Chapman, Robert Greig, Joseph J. Greene, Dorothy Ford, Robert Filmer, Ralph Sanford, Julius Tannen, The Three Swifts, Arthur Thalasso, George Anderson, George Lloyd, Olive Blakeney, Charles Cane, George Reed, Barbara Sears, Susan Scott, Priscilla White, Virginia Walker, The Troupers, Tom Dillon, Paul Graeff, Vincent Graeff, Irving Gump, Eddie Hart, Freeman High, Brooks Hunt, Helen O'Hara, Virginia Lyndon, George Leigh, Eddie Lee, George T. Lee, John Kelly, Jane Jones, Edna Mae Jones, Ben Jade, William Hunter. In frontier San Francisco, a Barbary Coast saloon owner longs for respectability and dates a socialite but is loved by a singer. Slick production with a mundane plot, highlighted by Vivian Blaine singing \" I Don't Care.\"\n\n**2816** _ **Nomads of the North**_ **** Associated First National, 1920. 50 min. D: David M. Hartford. SC: David M. Hartford and James Oliver Curwood. With Lewis Stone, Betty Blythe, Lon Chaney, Francis McDonald, Milbourne MacDonald, Spottiswoode Aitken, Gordon Muller, Charles H. Simly. A man in the north country tries to win a girl by making her think her trapper lover is dead but trouble ensues when the latter returns. Somewhat overacted but still pleasing silent adaptation of a James Oliver Curwood (he co-scripted) novel with well staged storm and forest fire scenes.\n\n**2817** _ **A Noose Is Waiting for You Trinity**_ **** Dora Film\/Balcazar, 1971. D: George Martin. SC: S. Giovanni (Simonelli). With George Martin, Marina Malfatti, Klaus Kinski, Daniel Martin, Augusto Pesarini, Francisco Jose Huetos, Susanne Atkinson, Willi Colombini, Luis Ponciado, Indio Gonzales, Pajarito (Manuel Muniz), Manuel Sas, Ricardo Moyan, Manuel Brochud, Gustavo Re, Luis Induni, Alfonso Alises, Miguel Muniesa. After being hunted for six years for shooting the man who murdered his brother, a rancher returns home and promises his wife he will give up gun fighting but is soon forced to face a grasping land baron and a vicious bounty hunter. Pleasing follow-up to _**Clint the Nevada Loner**_ (q.v.), utilizing footage from that feature; made as _**Il Ritorno de Clint il Solitario**_ (The Return of Clint the Stranger).\n\n**2818** _ **Noose for a Gunman**_ **** United Artists, 1960. 90 min. D: Edward L. Cahn. SC: James B. Gordon. With Jim Davis, Lyn Thomas, Ted De Corsia, Walter Sande, Barton MacLane, Harry Carey, Jr., Lane Chandler, John Hart, Leo Gordon, William Tannen, Jan Arvan, William Remick, Bob Tetrick, Kermit Maynard, William Challee, Cecil Weston. An honest gunman, banished after killing a corrupt land baron's two sons, returns to tell the citizens that an outlaw, in cahoots with the rancher, is planning a robbery. Well made program feature with a good performance by Jim Davis as the shootist.\n\n_**Noose for Django**_ see _**No Room to Die**_\n\n**2819** _ **North Beach and Rawhide**_ **** CBS-TV, 1985. 104 min. Color. D: Harry Falk. SC: Jimmy Sangster, John Beaird and George Yanok. With William Shatner, Tate Donovan, Christopher Penn, James Olson, Lori Loughlin, Gretchen Corbett, Beau Dremann O'Neil, Conchata Farrell, G.W. Bailey, Leo Penn, Grace Zabriskie, Geoffrey Blake, Dean Devlin, David Raynr, Nicholas Guest, J.C. Quinn, Lenny Hicks, Hugh Gillin. After getting out of prison, a man establishes a ranch to help troubled youth. Passable modern-day TV Western.\n\n**2820** _ **North Country**_ **** American National Enterprises, 1969. 105 min. Color. D: Ron Hayes. With Jeff Graham. A woodsman makes a life for himself in the remote Alaskan wilderness. Pleasant documentary filmed on location will appeal to nature lovers; 1973 reissue runs 94 minutes.\n\n**2821** _ **North from the Lone Star**_ **** Columbia, 1941. 58 min. D: Lambert Hillyer. SC: Charles Francis Royal. With Bill Elliott, Dorothy Fay, Dub Taylor, Richard Fiske, Arthur Loft, Jack Roper, Chuck Morrison, Claire Rochelle, Al Rhein, Edmund Cobb, Steve Clark, Art Mix, Hank Bell, Richard Botiller, Francis Walker, Lane Bradford, Oscar Gahan, Ray Jones, Jack Evans, Barney Beasley, Joe Garcia, Clem Horton, George Morrell, Tex Cooper. A crook takes over Deadwood and appoints Wild Bill Hickok its marshal but he sets out to clean up the town. Action filled \"Wild Bill Hickok\" series affair with a well staged saloon fight.\n\n**2822** _ **North of Arizona**_ **** Reliable, 1935. 60 min. D: Harry S. Webb. SC: Carl Krusada. With Jack Perrin, Blanche Mehaffey, Lane Chandler, Alan Bridge, Murdock MacQuarrie, George Chesebro, Artie Ortego, Budd Buster, Steve Clark, Frank Ellis, Blackie Whiteford, Oscar Gahan, George Morrell, Hank Bell, Ray Henderson, Barney Beasley. A cowboy joins a gang and pretends to be a part of their plans although he really wants to stop them from cheating Indians of gold ore and shipments. Not bad, with Jack Perrin a likable hero.\n\n**2823** _ **North of Nome**_ **** Columbia, 1936. 62 min. D: William Nigh. SC: Albert DeMond. With Jack Holt, Evelyn Venable, Guinn Williams, John Miljan, Roger Imhoff, Dorothy Appleby, Paul Hurst, Frank McGlynn, George Cleveland, Ben Hendricks, Robert Glecker, Mike Morita. A seal poacher, on the run from both the law and hijackers, comes across a shipwreck and tries to rescue the survivors. Average action drama that will appeal to Jack Holt followers.\n\n**2824** _ **North of the Border**_ **** Screen Guild, 1946. 40 min. D: B. Reeves Eason. SC: Arthur V. Jones. With Russell Hayden, Inez Cooper, Lyle Talbot, Douglas Fowley, Anthony Warde, Jack Mulhall, Guy Beach, I. Stanford Jolley, Richard Alexander. A cowboy crosses into Canada and finds he is suspected of murdering his partner, a deed done by a gang of fur thieves and smugglers. Cheaply made but entertaining featurette.\n\n**2825** _ **North of the Great Divide**_ **** Republic, 1950. 67 min. Color. D: William Witney. SC: Eric Taylor. With Roy Rogers, Penny Edwards, Gordon Jones, Roy Barcroft, Foy Willing and The Riders of the Purple Sage, Jack Lambert, Keith Richards, Douglas Evans, Noble Johnson, Iron Eyes Cody, Holly Bane, Alan Bridge, Stephen Chase, Frank Lackteen, George Sowards, Al Sloey. Government agent Roy Rogers tries to protect Indian salmon rights from a murderous crook who dams up a river, catches the fish and illegally ships them to Canadian canneries. Well done Roy Rogers film badly butchered for TV with some prints in black and white.\n\n**2826** _ **North of the Rio Grande**_ **** Paramount, 1937. 70 min. D: Nate Watt. SC: Joseph O'Donnell. With William Boyd, George Hayes, Russell Hayden, Stephen Morris (Morris Ankrum), Bernadene Hayes, John Rutherford, Lorraine Randall, Walter Long, Lee Colt (Lee J. Cobb), John Beach, Al Ferguson, Lafe McKee, Richard Cramer, George Plues, Harry Bernard, Lee Brooks, Bill Nestell, Horace B. Carpenter, Fred Burns, Silver Tip Baker, Hank Bell, William H. O'Brien, Ted Billings, Al Haskell, Herman Hack, Carl Mathews, Cliff Lyons, Buck Morgan, Charles Murphy, Cliff Parkinson, George Morrell. Hopalong Cassidy poses as a bad man to uncover the identity of \"The Lone Wolf,\" the leader of a robbery gang that killed his brother. Slow moving series entry which livens up at the finale.\n\n**2827** _ **North of the Rockies**_ **** Columbia, 1942. 60 min. D: Lambert Hillyer. SC: Herbert Dalmas. With Bill Elliott, Tex Ritter, Shirley Patterson, Frank Mitchell, Larry Parks, John Miljan, Ian MacDonald, Lloyd Bridges, Gertrude F. Hoffman, Earl Gunn, Boyd Irwin, Art Dillard, David Harper, Francis Sayles. A Canadian Mountie and a U.S. marshal find themselves at odds as they try to capture a gang of fur smugglers. Well made outing hurt by having its two stars spending most of their screen time as opponents.\n\n**2828** _ **North of the Yukon**_ **** Columbia, 1939. 59 min. D: Sam Nelson. SC: Bennett Cohen. With Charles Starrett, Linda Winters (Dorothy Comingore), The Sons of the Pioneers (Bob Nolan, Tim Spencer, Lloyd Perryman, Pat Brady, Hugh Farr, Karl Farr), Lane Chandler, Paul Sutton, Robert Fiske, Vernon Steele, Edmund Cobb, Tom London, Kenne Duncan, Hal Taliaferro, Richard Botiller, Harry Cording, Ed Brady. Royal Canadian Mounted policemen brothers search for the fur thieves who murdered a trader. Action filled, well made north woods drama.\n\n**2829** _ **North to Alaska**_ **** 20th Century\u2013Fox, 1960. 122 min. Color. D: Henry Hathaway. SC: John Lee Mahin, Martin Rackin and Claude Binyon. With John Wayne, Stewart Granger, Capucine, Fabian, Ernie Kovacs, Mickey Shaughnessy, Karl Swenson, Kathleen Freeman, John Qualen, Stanley Adams, Stephen Courtleigh, Douglas Dick, Jerry O'Sullivan, Ollie O'Toole, Tudor Owen, Lilyan Chauvin, Marcel Hillaire, Richard Deacon, James Griffith, Max Hellinger, Richard Collier, Esther Dale, Fortune Gordien, Roy Jenson, Charles Seel, Rayford Barnes, Fred Graham, Alan Carney, Peter Bourne, Tom Dillon, Arlene Harris, Paul Maxey, Oscar Beregi, Kermit Maynard, Maurice Delamore. Two prospectors strike it rich in Alaska and one sends the other south to claim his fiancee who turns out to be married, so he tries to find a substitute. Big, brawling, entertaining tongue-in-cheek adventure feature.\n\n**2830** _ **North to the Klondike**_ **** Universal, 1942. 60 min. D: Erle C. Kenton. SC: Clarence Upson Young, Lou Sarecky and George Bricker. With Broderick Crawford, Lon Chaney, Evelyn Ankers, Andy Devine, Stanley Andrews, Willie Fung, Keye Luke, Dorothy Granger, Lloyd Corrigan, Riley Hill, Paul Dubov, Armand Cortes, Fred Cordova, Monte Blue, Spade Cooley, Tony Paton, Jeff Corey, Robert Homans, Lee Phelps, William Ruhl. A mining engineer joins forces with farmers in Alaska who are being run off their lands by a trader who thinks there is gold in the area. Good adaptation of Jack London's _Gold Hunters of the North_ with a dilly of a brawl between good guy Broderick Crawford and baddie Lon Chaney.\n\n**2831** _ **North West Mounted Police**_ **** Paramount, 1940. 125 min. Color. D: Cecil B. DeMille. SC: Alan LeMay, Jesse Lasky, Jr. and C. Gardner Sullivan. With Gary Cooper, Madeleine Carroll, Paulette Goddard, Preston Foster, Robert Preston, George Bancroft, Lynne Overman, Akim Tamiroff, Walter Hampden, Lon Chaney, Jr., Montagu Love, Francis McDonald, George E. Stone, Willard Robertson, Regis Toomey, Richard Denning, Douglas Kennedy, Clara Blandick, Ralph Byrd, Lane Chandler, Julia Faye, Jack Pennick, Rod Cameron, James Seay, Jack Chapin, Eric Alden, Wallace Reid, Jr., Bud Geary, Evan Thomas, Davison Clark, Chief Thundercloud, Harry Burns, Lou Merrill, Ynez Seabury, Philip Terry, Soledad Jiminez, Kermit Maynard, Anthony Caruso, Paul Sutton, James Flavin, Archie Twitchell, Nestor Paiva, Ray Mala, Monte Blue, Chief Yowlachie, David Newell, Robert Ryan, Eva Puig, Weldon Heyburn, Emory Parnell, George Regas, Norma Nelson, John Laird, James Dundee. A Texas Ranger arrives in Canada on the trail of a wanted man and becomes involved with the Mounties in stopping an Indian uprising. Much maligned but very entertaining and well made Cecil B. DeMille production.\n\n**2832** _ **Northern Frontier**_ **** Ambassador, 1935. 57 min. D: Sam Newfield. SC: Barry Barringer. With Kermit Maynard, Eleanor Hunt, J. Farrell MacDonald, LeRoy Mason, Charles King, Ben Hendricks, Jr., Russell Hopton, Nelson McDowell, Walter Brennan, Gertrude Astor, Dick Curtis, Henry Hall, Kernan Cripps, Jack Chisholm, Lloyd Ingraham, Lafe McKee, Tyrone Power, Jr., Artie Ortego. A Mountie is after a murderous outlaw gang engaged in stealing furs and counterfeiting currency. Not the best in Kermit Maynard's series for producer Maurice Conn but still worth seeing, mainly for its fine cast and nice scenery; look for Tyrone Power in a bit as a mounted policeman.\n\n**2833** _ **Northern Lights**_ **** Cine Manifest, 1979. 90 min. Color. D-SC: John Hanson and Rob Nilsson. With Robert Behling, Susan Lynch, Henry Martinson, Joe Spano, Ray Ness, Helen Ness, Marianne Astrom-DeFina, Gary Hanish, Joe Ness, Thorbjorn Rue, Nick Eldridge. Farmers in North Dakota in the 1910s fight railroads and market monopolies while trying to form a grange. Interesting independent historical drama.\n\n**2834** _ **Northern Patrol**_ **** Allied Artists, 1953. 62 min. D: Rex Bailey. SC: Warren Douglas. With Kirby Grant, Marian Carr, Emmett Lynn, Bill Phipps, Claudia Drake, Frank Sully, Dale Van Sickel, Gloria Talbott, Richard Walsh, Frank Lackteen, Chinook (dog). When crooks plan to plunder a sacred Indian burial ground they are opposed by a lone mounted policeman and his loyal dog. Cheaply made but enjoyable north woods tale supposedly based on a James Oliver Curwood's _Nomads of the North_.\n\n**2835** _ **Northern Pursuit**_ **** Warner Bros., 1943. 94 min. D: Raoul Walsh. SC: Frank Gruber and Alvah Bessie. With Errol Flynn, Julie Bishop, Helmut Dantine, John Ridgely, Gene Lockhart, Tom Tully, Bernard Nedell, Warren Douglas, Monte Blue, Alec Craig, Tom Fadden, Carl Harbaugh, Fred Kelsey, Herbert Heywood, Arno Frey, Robert Hutton, Robert Kent, John Forsythe, Jay Silverheels, Russell Hicks, Milton Kibbee, Lester Mathews, George Urchel, Joe Herrera. A Mountie pretends to be a turncoat in order to infiltrate Nazis working around Hudson Bay during World War II. Surprisingly none-too-entertaining Errol Flynn feature.\n\n**2836** _ **Northwest Outpost**_ **** Republic, 1947. 91 min. D: Allan Dwan. SC: Elizabeth Meehan and Richard Sale. With Nelson Eddy, Ilona Massey, Joseph Schildkraut, Elsa Lanchester, Hugo Haas, Erno Verebes, Lenore Ulric, Peter Whitney, Tamara Shayne, George Sorel, Rick Vallin, Henry Brandon, Michael Visaroff, Muni Seroff, Nina Hansen, Eugene Sigaloff, Michael Mark, Richard Alexander, George Paris, Ray Teal, Inna Guest, John Bleifer, John Peters, Jay Silverheels, Constantine Romanoff, Peter Seal, The American G.I. Chorus. In pioneer California a Russian woman tries to defeat the plans of her evil husband and ends up falling in love with a dashing ranger. Republic's attempt to revive the romantic operetta with a new score by Rudolf Friml was defeated by this lumbering production.\n\n**2837** _ **Northwest Passage**_ **** Metro-Goldwyn-Mayer, 1940. 126 min. Color. D: King Vidor. SC: Lawrence Stallings and Talbot Jennings. With Spencer Tracy, Robert Young, Walter Brennan, Ruth Hussey, Nat Pendleton, Louis Hector, Robert Barrat, Lumsden Hare, Donald MacBride, Isabel Jewell, Douglas Walton, Addison Richards, Hugh Sothern, Regis Toomey, Montagu Love, Lester Mathews, Truman Bradley, Andrew Pena, Tom London, Eddie Parker, Hank Worden, Don Castle, Rand Brooks, Kent Rogers, Verna Felton, Richard Cramer, Ray Teal, Edward Gargan, John Merton, Gibson Gowland, Frank Hagney, Gwendolen Logan, Addie McPhail, Helen MacKellar, Arthur Aylesworth, Ted Oliver, Lawrence Porter, Tony Guerrero, Ferdinand Munier, George Eldredge, Fredric Worlock. Major Robert Rogers leads his Rangers in an arduous trek to stop the Indians at St. Francis in Canada to break the French hold on the area in 1759. Colorful and entertaining feature based on Kenneth Roberts' best seller; film contains some fine character performances, especially Addison Richards as a mad ranger.\n\n**Truman Bradley, Spencer Tracy, in front, and Robert Young in** _**Northwest Passage**_ **(Metro-Goldwyn-Mayer, 1940).**\n\n**  \n**\n\n**2838** _ **Northwest Rangers**_ **** Metro-Goldwyn-Mayer, 1942. 64 min. D: Joe Newman. SC: Gordon Kahn and David Lang. With James Craig, William Lundigan, Patricia Dane, John Carradine, Jack Holt, Keenan Wynn, Grant Withers, Darryl Hickman, Drew Roddy, John Butler, Philip Van Zandt, Michael Brown, Luis Alberni, Jim Farley, Alec Craig, Kay Medford, Hugh Beaumont, Alexander Granach, Mitchell Lewis, Ray Teal, Al Hill, George Carleton, Howard Hickman, Herbert Heyes, Emmett Vogan, Patrick McVey, William Tannen, Roy Barcroft, Ivan \"Dusty\" Miller, Hubert Brill, LeRoy Mason, Mark Daniels, Hooper Atchley, Howard Mitchell, Dick Rush, Murdock MacQuarrie, Robert Winkler. Two boys grow up together, one becomes a gambler and the other a ranger, and eventually they are forced into a showdown. Great cast but low budget affair that is fairly entertaining.\n\n**2839** _ **Northwest Stampede**_ **** Eagle-Lion, 1948. 76 min. Color. D: Albert S. Rogell. SC: Art Arthur and Lillie Hayward. With James Craig, Joan Leslie, Jack Oakie, Chill Wills, Victor Kilian, Stanley Andrews, Lane Chandler, Ray Bennett, Harry Shannon, Lane Bradford, Kermit Maynard, Harry V. Cheshire, Eddie Acuff, Lee Roberts, Flame (dog). A lady rancher is at odd with her rodeo champion foreman who wants to corral a wild stallion. The cast and the scenery help breathe life into this average feature.\n\n**2840** _ **Northwest Territory**_ **** Monogram, 1951. 61 min. D: Frank McDonald. SC: Bill Raynor. With Kirby Grant, Gloria Saunders, Warren Douglas, Pat Mitchell, Tristram Coffin, John Crawford, Duke York, Don C. Harvey, Sam Flint, Chinook (dog). Outlaws murder an old man for his oil claim and a Mountie, who brought the victim's grandson-heir to the area, sets out to track down the killers. Action filled, compact program film with good work by Kirby Grant as Corporal Rod Webb.\n\n**2841** _ **Northwest Trail**_ **** Screen Guild, 1945. 66 min. Color. D: Derwin Abrahams. SC: Harvey Gates and L.J. Swabacher. With Bob Steele, Joan Woodbury, John Litel, Ian Keith, Raymond Hatton, Madge Bellamy, Poodles Hanneford, Grace Hanneford, George Meeker, Charles Middleton, John Hamilton, Al Ferguson, Bud Osborne, Bob Duncan, Bill Hammond, Josh (John) Carpenter. When a woman brings money to her uncle for the purchase of timberland it is stolen and a Mountie tries to find the thieves. Colorful action outing with a fine performance by Bob Steele as the law officer.\n\n**2842** _ **Not Above Suspicion**_ **** Wrather Corporation, 1956. 75 min. Color. D: Earl Bellamy and Oscar Rudolph. SC: Herbert Purdom, Robert Leslie Bellem, Thomas Seller and Charles Carson. With Clayton Moore, Jay Silverheels, Allen Pinson, Wayne Burson, Dennis Moore, Tristram Coffin, Roy Barcroft, Richard Benedict, Francis McDonald, Florence Lake, Tyler MacDuff, Harry Strang, Rick Vallin, Jason Johnson, Alan Wells, Robert Burton, Gregg Barton, Melinda Byron, Joseph Sargent. The Lone Ranger and Tonto try to stop a crook from taking over a town, fight renegade Indians and disrupt a plot to murder a rancher. Okay telefeature from \"The Lone Ranger\" (ABC-TV, 1949\u201357) episodes \"The Avenger,\" \"Journey to San Carlos\" and \"Mission for Tonto.\"\n\n**2843** _ **Not Exactly Gentlemen**_ **** Fox, 1931. 70 min. D: Benjamin Stoloff. SC: William Counselman, Dudley Nichols and Emmett Flynn. With Victor McLaglen, Fay Wray, Lew Cody, Robert Warwick, Eddie Gribbon, David Worth, Joyce Compton, Louise Huntington, Franklyn Farnum, Carol Wines, James Farley. Three rascals raid a wagon train and ride off with a pretty girl whose father has a map to the gold mine they seek. Average early talkie with some well staged land rush scenes; originally called _**Three Rogues**_.\n\n**2844** _ **Notch Number One**_ **** Arrow, 1924. 55 min. D: Ben Wilson. SC: Daniel F. Whitcomb. With Ben Wilson, Marjorie Daw, Reed Howes, Merrill McCormick, Arthur Mackley, Yakima Canutt. A cowboy, who thinks he killed his boss after smoking loco weed, is helped by the ranch foreman who takes the blame for the crime, actually committed by a fired cowhand. Only one and one-half reels exist from this interesting poverty row production.\n\n**2845** _ **Nothing Too Good for a Cowboy**_ **** Canadian Broadcasting Corporation (CBC), 1998. 90 min. Color. D: Kari Skogland. SC: David Barlow and Charles Lazer. With Chad Willett, Ted Atherton, Sarah Chalke, Falconer Abraham, Zachary Bennett, Marion Bennett, Ryan Gosling, Dan MacDonald, Jonathan Whittaker, Robin Brule, John Keller, Richard McMillan, Graham McPhearson, Neil Crone, W.J. Matheson, Mark Lutz, Andrew Smith, Leslie Urquhart, Paul O'Sullivan, Vincent Corazza, Curtis Parker, Nigel Hamer, Bob Martin, Jen Cohen, Sheree Jeacocke. A New York City stockbroker realizes his goal of owning ranch in British Columbia and marries his dream girl but faces trouble from a banker who wants the land. Charming Canadian Western set in the pre\u2013World War II era; it was followed by a TV series of the same title telecast on the Canadian Broadcasting Company (CBC) during the 1999\u20132000, with Ted Atherton and Sarah Chalke repeating their film roles.\n\n**2846** _ **Now They Call Him Sacramento**_ **** Filmax, 1972. 90 min. Color. D: Al Bagrain (Alfonso Balcazar). SC: Alfonso Balcazar and Giovanni Simonelli. With Michael Forest, Fred Harrison (Fernando Bilboa), Malisa Longo, Paolo Gozlino, Luis Bonos, Gaspar Gonzalez, Pajarito (Manuel Muniz), Antonio Almoros, Antonio Molino Rojo, Fernando Rubio, Johnny (Juan) Fairen, Luigi Antonio Guerra, Juan Torres, Manuel Bronchud, Mario Del Vago, Irene D'Astrea, Cesar Ojinaga. Three bandits steal settlers' money for their banker boss so he can get their lands but the trio soon retrieve it after falling for some pretty homesteaders. Fairly pleasing, light hearted Spaghetti Western, filmed as _**I Bandoleros della Dodicesima Ora**_ (The Bandits of the Twelfth Hour) and also called _**Desperado**_.\n\n**2847** _ **Oath of Vengeance**_ **** Producers Releasing Corporation, 1944. 57 min. D: Sam Newfield. SC: Fred Myton. With Buster Crabbe, Al St. John, Mady Lawrence, Karl Hackett, Marin Sais, Jack Ingram, Charles King, Kermit Maynard, Frank Ellis, Hal Price, Budd Buster, Jimmy Aubrey, John Cason, Frank McCarroll, Augie Gomez, Herman Hack, Jack Kenney, Rose Plummer, Hank Bell, Jack Evans, Morgan Flowers, Wally West, Ray Henderson, Ralph Bucko, Tex Palmer. When Fuzzy Q. Jones purchases a ranch and finds he cannot settle down, he and Billy Carson try to prove the innocence of a young man accused of murder. Another tacky entry in the long running \"Billy Carson\" series.\n\n_**Oath of Zorro**_ see _**Behind the Mask of Zorro**_\n\n**2848** _ **Oblivion**_ **** Sequential One Filmworks, 1994. 94 min. Color. D: Sam Irvin. SC: Peter David. With Richard Joseph Paul, Jackie Swanson, Andrew Divoff, Meg Foster, Isaac Hayes, Julie Newmar, Carel Struycken, George Takei, Musetta Vander, Jimmie F. Skaggs, Irwin Keyes, Mike Genoevese, Frank Roman, Jeff Moldovan, Joe Muzio, Craig Anthony Muzio, Tim Miller, Peter David, Nadine Emilie Voindrouh, Sam Irvin. In a frontier town of the future, the citizens are cowed by a lizard man and his gang who have murdered the sheriff but are opposed by the victim's son and a female cyborg deputy. Confusing sci-fi Western, followed by _**Oblivion 2:**_ _**Backlash**_ (q.v.).\n\n**2849** _ **Oblivion 2:**_ _**Backlash**_ **** Full Moon Pictures, 1996. 83 min. Color. D: Sam Irvin. SC: Peter David. With Richard Joseph Paul, Jackie Swanson, Andrew Divoff, Meg Foster, Isaac Hayes, Julie Newmar, Carel Struycken, George Takaei, Musetta Vander, Jimmie F. Skaggs, Irwin Keyes, Maxwell Caufield, Mike Genovese, Jeff Weston, Frank Roman, Brent Huff, Jeff Moldovan, Michael C. Mahon, Craig Muzio, Joe Muzio, Tim Miller, Peter David, Nadine Emilie Voindrouh, Sam Irvin. In the future world town of Oblivion, a bounty hunter arrives to capture a beautiful outlaw who has won a valuable mine that is coveted by a half-lizard outlaw. Futuristic Western that is better than its predecessor, _**Oblivion**_ (q.v.).\n\n**2850** _ **Oh, Susanna!**_ **** Republic, 1936. 56 min. D: Joseph Kane. SC: Oliver Drake. With Gene Autry, Smiley Burnette, Frances Grant, Donald Kirke, Earle Hodgins, The Light Crust Doughboys, Clara Kimball Young, Boothe Howard, Ed Peil, Sr., Frankie Marvin, Carl Stockdale, Gerald Roscoe, Roger Gray, Fred Burns, Walter James, Fred \"Snowflake\" Toones, Earl Dwire, Bruce Mitchell, Jack Kirk, George Morrell, Horace B. Carpenter, Tommy Coats, Pascale Perry, William McCall, Silver Tip Baker. After outlaws knock him out and steal his clothes, Gene Autry is mistaken for a bad man and when his pals help him to escape he becomes the object of a manhunt. More music and pseudo-Western nonsense that made Gene Autry the era's top genre star.\n\n**2851** _ **Oh, Susanna**_ **** Republic, 1951. 90 min. Color. D: Joseph Kane. SC: Charles Marquis Warren. With Rod Cameron, Adrian Booth, Forrest Tucker, Chill Wills, Jim Davis, William Ching, Wally Cassell, Douglas Kennedy, James Lydon, William Haade, John Compton, James Flavin, Charles Stevens, Alan Bridge, Marshall Reed, John Pickard, Ruth Brennan, Louise Kane, Marion Randolph. At a frontier outpost, two Army rivals battle each other and the possibility of a Sioux uprising. Melodrama leans more toward dialogue than action but the fine cast carries it off okay.\n\n**2852** _ **The Oil Raider**_ **** Mayfair, 1934. 59 min. D: Spencer Gordon Bennet. SC: George Morgan and Homer King Gordon. With Buster Crabbe, Gloria Shea, George Irving, Max Wagner, Emmett Vogan, Harold Minjir, Tom London, Wally Wales, Chuck Morrison, Tetsu Komai. A wildcatter borrows money from an investment banker to complete an oil drilling project but when the banker suffers marked reverses and needs funds he hires a dishonest rival driller to sabotage the operation. Fairly good low budget program feature from producer Lester Scott, Jr.\n\n**2853** _ **Oklahoma!**_ **** Magna Corporation, 1955. 143 min. Color. D: Fred Zinneman. SC: Sonya Levien and William Ludwig. With Gordon MacRae, Shirley Jones, Gloria Grahame, Gene Nelson, Charlotte Greenwood, Eddie Albert, James Whitmore, Rod Steiger, Barbara Lawrence, Jay C. Flippen, Roy Barcroft, James Mitchell, Bambi Lynn, Marc Platt, Rex Lease, Al Ferguson, Russell Simpson, Buddy Roosevelt, Ben Johnson, Rory Mallinson, Donald Kerr, Jane Fischer, Jennie Workman, Lizanne Truex, Evelyn Taylor, Virginia Bosler, Kelly Brown, Dolores Starr, Nancy Kilgas, Jerry Dealey. A cowboy falls in love with a pretty girl and asks her to a dance but has trouble with a rival. Enjoyable screen adaptation of the popular Broadway musical.\n\n**2854** _ **Oklahoma!**_ **** BBC-TV, 1999. 180 min. Color. D: Trevor Nunn. SC: Oscar Hammerstein II and Lynn Riggs. With Hugh Jackman, Josefina Gabrielle, David Shelmerdine, Jimmy Johnston, Shuler Hensley, Maureen Lipman, Vicki Simon, Peter Polycarpou, Rebecca Thornhill, Sidney Livingston, Stuart Milligan. Well done, stylish filming of the London stage revival of the 1943 Oscar Hammerstein II-Lynn Riggs musical.\n\n**2855** _ **Oklahoma!**_ **** UNC-TV, 2011. 180 min. Color. D: David Stern. SC: Oscar Hammerstein II and Lynn Riggs. With Kyle Guglielmo, Rebecca Moyes, Devon Diffenderfer, Caroline Kava, Braxton Molinaro, Jennifer Webb, Charles Osborne, Jillian Ratledge, Tommy Burnett, Gabriel Arant, Andrew Robert Bodd, Diandra Langenbach, Benjamin Rush. More than passable filmed stage production of the famed musical made by the University of North Carolina Center for Public Television.\n\n**Shirley Jones and Gordon MacRae in** _**Oklahoma!**_ **(Magna Corporation, 1955).**\n\n**  \n**\n\n**2856** _ **Oklahoma Annie**_ **** Republic, 1952. 90 min. D: R.G. Springsteen. SC: Jack Townley. With Judy Canova, John Russell, Grant Withers, Roy Barcroft, Emmett Lynn, Frank Ferguson, Minerva Urecal, Houseley Stevenson, Almira Sessions, Allen Jenkins, Maxine Gates, Emory Parnell, Denver Pyle, House Peters, Jr., Andrew Tombes, Fuzzy Knight, Si Jenks, Marion Martin, Herbert Vigran, Hal Price, Fred Hoose, Lee Phelps, Bobby Taylor, William Fawcett, Bob Reeves. A backwoods girl running a gun shop falls for the town's new sheriff and when she wants him to arrest a saloon owner he makes her his deputy. Fun Judy Canova vehicle.\n\n**2857** _ **Oklahoma Badlands**_ **** Republic, 1948. 59 min. D: Yakima Canutt. SC: Bob Williams. With Allan \"Rocky\" Lane, Eddy Waller, Mildred Coles, Roy Barcroft, Gene (Roth) Stutenroth, Earle Hodgins, Jay Kirby, Terry Frost, Hank Patterson, House Peters, Jr., Jack Kirk, Bob Woodward, Claire Whitney, Dale Van Sickel. When a man is about to lose his ranch due to rustlers, a lawman pretends to be a friend from the East in order to help him. Director Yakima Canutt keeps this Allan Lane vehicle moving at a good clip.\n\n**2858** _ **Oklahoma Blues**_ **** Monogram, 1948. 56 min. D: Lambert Hillyer. SC: Bennett Cohen. With Jimmy Wakely, Dub Taylor, Virginia Belmont, I. Stanford Jolley, Zon Murray, George J. Lewis, Steve Clark, Frank LaRue, Milburn Morante, Charles King, Bob Woodward, J.C. Lytton, Dick Reinhart, Don Weston, Arthur \"Fiddlin'\" Smith, George Morrell, Artie Ortego, Victor Cox, Jack Hendricks. A singing cowboy finds himself in between two towns fighting for the location of the county seat. A bit more action than usual for a Jimmy Wakely musical, but still none too good.\n\n**Arthur Smith, Frank LaRue, George Morrell, Dick Reinhart, Virginia Belmont, Jimmy Wakely, Don Weston and Dub Taylor in** _**Oklahoma Blues**_ **(Monogram, 1948).**\n\n**  \n**\n\n**2859** _ **Oklahoma Crude**_ **** Columbia, 1973. 105 min. Color. D: Stanley Kramer. SC: Marc Norman. With George C. Scott, Faye Dunaway, Jack Palance, John Mills, William Lucking, Harvey James, Ted Gehring, Cliff Osmond, Rafael Campos, Woodrow Parfrey, John Hudkins, Harvey Parry, Bob Herron, Jerry Brown, Jim Burk, Henry Wills, Hal Smith, Cody Bearpaw, James Jester, Larry D. Mann, John Dierkes, Karl Lucas, Wayne Storm, Billy Varga. A drifter is hired by the woman owner of an oil well to help her fight the encroachment of a large petroleum company. Modern-day drama cannot decide if it is serious or comedic.\n\n**2860** _ **Oklahoma Cyclone**_ **** Tiffany, 1930. 60 min. D: John P. McCarthy. SC: Ford Beebe. With Bob Steele, Nita Ray, Al St. John, Charles L. King, Slim Whitaker, Shorty Hendricks, Emilio Fernandez, Hector Sarno, Fred Burns, Cliff Lyons, John Ince. A cowboy infiltrates an outlaw gang as he searches for his missing sheriff father and promptly falls for a pretty senorita. Early talkie is a bit shaky in its production values but Bob Steele's (who croons a couple of tunes) fans will not mind.\n\n**2861** _ **Oklahoma Frontier**_ **** Universal, 1939. 59 min. D-SC: Ford Beebe. With Johnny Mack Brown, Anne Gwynne, Fuzzy Knight, Bob Baker, James Blaine, Lane Chandler, Anthony Warde, Robert Kortman, Harry Tenbrook, Charles King, Horace Murphy, George Chesebro, Jose De La Cruz, Lloyd Ingraham, The Texas Rangers, Alan Bridge, Hank Worden, Hank Bell, Blackie Whiteford, Roy Harris, George Magrill, Tom Smith, Robert Cummings, Frank Mayo, Dick Rush. Crooks after water rights frame a cowboy for the murder of his best friend and the accused man's girl tries to save him. Somewhat complicated oater with plenty of action.\n\n**2862** _ **Oklahoma Jim**_ **** Monogram, 1931. 61 min. D: Harry Fraser. SC: Harry Fraser and G.A. Durlam. With Bill Cody, Marion Burns, Andy Shuford, William Desmond, Si Jenks, Franklyn Farnum, John Elliott, Ed Brady, G.D. Wood (Gordon DeMain), Earl Dwire, Iron Eyes Cody, Ann Ross, Artie Ortego. A saloon owner causes the death of an Indian maiden and places the blame on a gambler who has fallen in love with the man's late partner's daughter. Laggard Bill Cody outing, poorly recorded with lots of stock footage.\n\n**2863** _ **Oklahoma Justice**_ **** Monogram, 1951. 56 min. D: Lewis D. Collins. SC: Joseph O'Donnell. With Johnny Mack Brown, James Ellison, Phyllis Coates, Barbara Allen, Kenne Duncan, Lane Bradford, Marshall Reed, Zon Murray, I. Stanford Jolley, Stanley Price, Bruce Edwards, Richard Avonde, Lyle Talbot, Carl Mathews, Ed Cassidy, George DeNormand. With the assistance of his stagecoach driver pal, a lawman pretends to be an outlaw to locate a robbery gang. First feature teaming Johnny Mack Brown and James Ellison is only average.\n\n**2864** _ **The Oklahoma Kid**_ **** Warner Bros., 1939. 85 min. D: Lloyd Bacon. SC: Warren Duff, Robert Buckner and Edward E. Paramore. With James Cagney, Rosemary Lane, Humphrey Bogart, Donald Crisp, Harvey Stephens, Hugh Sothern, Charles Middleton, Edward Pawley, Ward Bond, Lew Harvey, Trevor Bardette, John Miljan, Arthur Aylesworth, Irving Bacon, Joe Devlin, Wade Boteler, Dan Wolheim, Ray Mayer, Robert Kortman, Tex Cooper, John Harron, Stuart Holmes, Jeffrey Sayre, Frank Mayo, Jack Mower, Alan Bridge, Don Barclay, Horace Murphy, Robert Homans, George Lloyd, Soledad Jiminez, Clem Bevans, Ed Brady, Tom Chatterton, Elliott Sullivan, George Chesebro, Olin Francis, Al Jennings, Blackjack Ward, Jack Kenney, Gene Alsace, Jess Cavin, Kit Guard, Morgan Flowers. In the 1890s a daredevil bandit in Oklahoma Territory robs from the rich and gives to the poor as he tries to avenge his father's murder. Rather strange Western with James Cagney as a singing hero and Humphrey Bogart as a dastardly villain; best viewed as tongue-in-cheek.\n\n**2865** _ **Oklahoma Passage**_ **** Oklahoma Educational Television Authority, 1989. 300 min. Color. D: Kenneth A. Meyer. SC: Kevin Meyer. With Jeanette Nolan, Robin Brooks, Chris Todd, Charles Benton, Eldon G. Hallum, Byron Bourg, James Fields, Lou Michaels, Thesa Rogers Loving, Charles Ballinger, Carter Mullally, Jr., Daniel Kamit, Melvin Holt, Rex Linn, Frank Otterman, Whitman Mayo, Megan Mullally, Paul Newsom, Jeff MacKay, Tom Ward; Hoyt Axton, Ben Johnson, Dale Robertson, G.D. Spradlin and General Thomas Stafford (hosts). TV miniseries detailing the removal of the Five Civilized Tribes to the Oklahoma Territory and its resulting history. A fine historical production incorporating footage from the 1931 version of _**Cimarron**_ (q.v.).\n\n**2866** _ **Oklahoma Raiders**_ **** Universal, 1944. 56 min. D: Lewis D. Collins. SC: Betty Burbridge. With Tex Ritter, Jennifer Holt, Fuzzy Knight, Dennis Moore, Jack Ingram, George Eldredge, John Elliott, Slim Whitaker, I. Stanford Jolley, Richard Alexander, Herbert Rawlinson, Ethan Laidlaw, Johnny Bond and His Red River Valley Boys (Wesley Tuttle, Jimmie Dean, Paul Sells), Steve Keyes, William Desmond, Bob Baker, Lane Chandler, Frank Ellis, Michael Vallon, Gil Patric, Bill Sloan. A Union Army lieutenant in the Civil War is sent to Oklahoma in the guise of a drifter to stop a masked bandit called El Vengador, who has been leading raids on the cavalry. Nice Tex Ritter vehicle with plenty of action, a good story and supporting cast plus five pleasant songs, including \"Cowboy's Dream\" and \"Starlight on the Prairie.\" British title: _**Riders of Oklahoma**_.\n\n**2867** _ **Oklahoma Renegades**_ **** Republic, 1940. 57 min. D: Nate Watt. SC: Earle Snell and Doris Schroeder. With Robert Livingston, Raymond Hatton, Duncan Renaldo, Florine McKinney, Lee \"Lasses\" White, Al Herman, William Ruhl, Eddie Dean, James Seay, Harold Daniels, Jack Lescoulie, Frosty Royce, Yakima Canutt, Art Dillard, Harry Strang, Ken Terrell, Frankie Marvin, Hank Bell, Jack Lawrence, Tom Smith, Al Taylor, Ted Mapes, Pascale Perry. Coming home after serving in the Spanish-American War, three cowboys help veterans trying to homestead but are being harassed by local ranchers. Fairly pleasing effort in \"The Three Mesquiteers\" series; Raymond Hatton and Duncan Renaldo's last film as part of the Mesquiteers.\n\n**2868** _ **Oklahoma Territory**_ **** United Artists, 1960. 67 min. D: Edward L. Cahn. SC: Orville Hampton. With Bill Williams, Gloria Talbott, Ted De Corsia, Grant Richards, Walter Sande, X Brands, Walter Baldwin, Grandon Rhodes, Kenne Duncan, Charles Stevens, Boyd \"Red\" Morgan, Bob Woodward, John Cliff. When the local Indian commissioner is murdered a chief is charged with the crime but the district attorney believes him innocent and tries to find the real killer. Low budget but adequate entertainment.\n\n**2869** _ **Oklahoma Terror**_ **** Monogram, 1938. 50 min. D: Spencer Gordon Bennet. SC: Joseph West (George Waggner). With Jack Randall, Al St. John, Virginia Carroll, Davison Clark, Norman Willis, Glenn Strange, Warren McCollum, Tristram Coffin, Ralph Peters, Slim Whitaker, Nelson McDowell, Don Rowan, Brandon Beach. Following the Civil War, a cowboy tries to learn who killed his stage line manager father and organizes vigilantes to clean up lawlessness. Pretty good Jack Randall film.\n**2870** _ **The Oklahoma Woman**_ **** American Releasing Corporation, 1956. 72 min. D: Roger Corman. SC: Lou Russoff. With Richard Denning, Peggie Castle, Cathy Downs, Tudor Owen, Martin Kingsley, Touch (Michael) Connors, Jonathan Haze, Richard (Dick) Miller, Tom Dillon, Edmund Cobb, Bruno Ve Sota, Aaron Saxon, Joe Brown. Released from prison, a former gunfighter returns to his Oklahoma ranch where a former girlfriend tries to frame him on a murder charge. Roger Corman's fans will find this early effort of interest but others beware.\n\n**2871** _ **The Oklahoman**_ **** Allied Artists, 1957. 81 min. Color. D: Francis D. Lyon. SC: Daniel B. Ullman. With Joel McCrea, Barbara Hale, Gloria Talbott, Brad Dexter, Michael Pate, Verna Felton, Douglas Dick, Anthony Caruso, Esther Dale, Adam Williams, Ray Teal, Peter Votrian, John Pickard, Mimi Gibson, I. Stanford Jolley, Jody Williams, Earle Hodgins, Sheb Wooley, Harry Lauter, Diane Brewster, Mimi Gibson, Robert Hinkle, Doris Kemper, Dorothy Neumann, Gertrude Astor, Wheaton Chambers, Laurie Mitchell, Scotty Beckett, Kermit Maynard, Don Marlowe, Rankin Mansfield, Jennifer Lea, Bill Foster, Al Kramer. When cattlemen try to cheat an Indian out of his oil lands, a doctor comes to his defense. Somewhat predictable but okay drama.\n\n**2872** _ **The Old Barn Dance**_ **** Republic, 1938. 60 min. D: Joseph Kane. SC: Bernard McConville and Charles Francis Royal. With Gene Autry, Smiley Burnette, Helen Valkis, Sammy McKim, Ivan Miller, Earl Dwire, Hooper Atchley, Ray Bennett, Carleton Young, Frankie Marvin, Earle Hodgins, Gloria Rich, Dick Weston (Roy Rogers), Walt Shrum and His Colorado Hillbillies, The Maple City Four, The Stafford Sisters, Bill Nestell, Chuck Baldra, Jack Kenney, Denver Dixon. Gene Autry and his singing group try to help farmers cheated by a tractor company but he is accused of a double cross when he goes to work for a radio station owned by the outfit. Entertaining Gene Autry opus with plenty of good music.\n\n**2873** _ **The Old Chisholm Trail**_ **** Universal, 1942. 61 min. D-SC: Elmer Clifton. With Johnny Mack Brown, Tex Ritter, Fuzzy Knight, Jennifer Holt, Mady Correll, Earle Hodgins, Roy Barcroft, Edmund Cobb, Budd Buster, Michael Vallon, Scoop Martin, The Jimmy Wakely Trio (Jimmy Wakely, Johnny Bond, Scotty Harrell), George Sherwood, Roy Butler, Frosty Royce. The female owner of a trading post opposes a woman gambler who tries to run a cowboy out of town over water rights. The two male stars are at odds over Jennifer Holt, who is fighting Mady Correll, and Jimmy Wakley and his boys do three songs\u2014all in one hour!\n\n**2874** _ **The Old Corral**_ **** Republic, 1936. 56 min. D: Joseph Kane. SC: Joseph Poland and Sherman Lowe. With Gene Autry, Smiley Burnette, Hope (Irene) Manning, The Sons of the Pioneers (Bob Nolan, Len Slye [Roy Rogers], Tim Spencer, Hugh Farr, Karl Farr), Cornelius Keefe, Lon Chaney, Jr., John Bradford, Milburn Morante, Abe Lefton, Merrill McCormick, Charles Sullivan, Buddy Roosevelt, Lynton Brent, Frankie Marvin, Oscar and Elmer (Ed Platt, Lou Fulton), Jack Ingram. A singer witnesses a Chicago gangland murder and flees West to a town where a gambler recognizes her and informs the gangsters who want her silenced. Gene Autry versus gangsters and the result is pretty entertaining.\n\n**2875** _ **The Old Frontier**_ **** Republic, 1950. 60 min. D: Philip Ford. SC: Bob Williams. With Monte Hale, Paul Hurst, Claudia Barrett, William Henry, Tristram Coffin, William Haade, Victor Kilian, Lane Bradford, Denver Pyle, Tom London, Almira Sessions, Ted Mapes, Chick Hannon. The town's new lawman is after an outlaw gang clandestinely led by a local attorney. Average Monte Hale outing.\n\n**2876** _ **Old Gringo**_ **** Columbia, 1989. 119 min. Color. D: Luis Puenzo. SC: Luis Puenzo and Aida Bortnik. With Jane Fonda, Gregory Peck, Jimmy Smits, Patricio Conteras, Jenny Gago, Jim Meltzler, Grabriela Roel, Anne Pitoniak, Pedro Armendariz, Jr., Sergio Calderon, Guillermo Rios, Samuel Valadez, Stanley Grover, Josefina Echanove, Pedro Damian, Maya Zapata, Jose Olivares. Writer Ambrose Bierce is in Mexico covering the 1913 Pancho Villa uprising and becomes involved with an American woman and a Mexican general. Big budget affair is a half-hearted attempt to portray an involved historical event.\n\n**2877** _ **The Old Homestead**_ **** Liberty, 1935. 73 min. D: William Nigh. SC: W. Scott Darling. With Mary Carlisle, Lawrence Gray, Willard Robertson, Dorothy Lee, Eddie Nugent, Lillian Miles, Fuzzy Knight, Eddie Kane, Harry Conley, The Sons of the Pioneers (Bob Nolan, Len Slye [Roy Rogers], Hugh Farr, Karl Farr, Vern [Tim] Spencer), Sally Sweet, George Lloyd, Gayne Whitman, Horace B. Carpenter, Alec Craig, William H. O'Brien. A young woman encourages her bashful Missouri boyfriend to pursue a singing career but both are overwhelmed when they arrive in New York City. Pleasant bucolic affair highlighted by Lawrence Gray's singing and the feature film debut of The Sons of the Pioneers, including Roy Rogers; previously filmed in 1915 and 1922.\n\n**2878** _ **Old Los Angeles**_ **** Republic, 1948. 87 min. D: Joseph Kane. SC: Clement Riley and Gerald Adams. With William Elliott, John Carroll, Catherine McLeod, Joseph Schildkraut, Andy Devine, Estelita Rodriguez, Grant Withers, Virginia Brissac, Tito Renaldo, Roy Barcroft, Henry Brandon, Julian Rivero, Earle Hodgins, House Peters, Jr., Augie Gomez, Franklyn Farnum, Tex Terry, Sam Flint, Hank Bell, Chris-Pin Martin, Lucio Villegas, Alex Montoya, Rosa Turich, Lynn Farr. When a Missouri lawman arrives in frontier Los Angeles to prospect for gold he learns his brother and several other miners have been murdered and he tries to find the killers. Fast paced and very entertaining production also known as _**In Old Los Angeles**_ and reissued as _**California Outpost**_.\n\n**2879** _ **Old Louisiana**_ **** Crescent, 1937. 60 min. D: Irvin V. Willat. SC: Mary Ireland. With Tom Keene, Rita (Hayworth) Cansino, Will Morgan, Robert Fiske, Ray Bennett, Budd Buster, Allan Cavan, Carlos De Valdez, Wally Albright, Ramsey Hill, J. Louis Johnson, Iron Eyes Cody. A frontiersman helps American settlers in the Upper Mississippi Valley when a dishonest trader tries to stir up trouble between them and the Spanish so he can have the territory for himself. Cheaply made but rather interesting pseudo-historical drama revolving around the Louisiana Purchase; reissued as _**Louisiana Gal**_.\n\n**Tom Keene and Rita (Hayworth) Cansino in** _**Old Louisiana**_ **(Crescent, 1937).**\n\n**  \n**\n\n**2880** _ **Old Oklahoma Plains**_ **** Republic, 1952. 60 min. D: William Witney. SC: Milton Raison. With Rex Allen, Slim Pickens, Elaine Edwards, Roy Barcroft, John Crawford, Joel Marston, Russell Hicks, Fred Graham, Stephen Chase, The Republic Rhythm Riders, Chick Hannon, Cactus Mack. During the 1920s a former cavalry officer returns to duty to stop outlaws terrorizing the plains and comes up with the idea of using tanks to stop them. More than passable Rex Allen affair hampered by the overuse of footage from _**Army Girl**_ (Republic, 1938).\n\n**2881** _ **The Old Oregon Trail**_ **** Art Mix Productions, 1928. 40 min. D: Victor Adamson (Denver Dixon). SC: Denver Dixon. With Art Mix (Victor Adamson\/Denver Dixon), Delores Booth, F.C. Rose, Grace Underwood, Art Seales, Sid Seales. Ten years after he and his two pals help settlers oppose outlaws, they return to a now thriving Oregon town where their benefactor rancher is beset by rebellious workers. Sturdy and beautifully photographed featurette, probably Victor Adamson's best film; made on location in Oregon's John Day River Valley.\n\n**2882** _ **Old Overland Trail**_ **** Republic, 1953. 60 min. D: William Witney. SC: Milton Raison. With Rex Allen, Slim Pickens, Virginia Hall, Roy Barcroft, The Republic Rhythm Riders, Zon Murray, Harry Harvey, Gil Herman, Wade Crosby, Leonard Nimoy. A cowboy attempts to avert warfare between immigrant settlers and Apaches. Standard Rex Allen vehicle hurt by the decline of the \"B\" Western.\n\n**2883** _ **Old Shatterhand**_ **** Constantin, 1964. 122 min. Color. D: Hugo Fregonese. SC: Ladislas Fodor and Robert A. Stemmle. With Lex Barker, Pierre Brice, Guy Madison, Daliah Lavi, Rik Battaglia, Gustavo Rojo, Ralf Wolter, Kitty Mattern, Bill Ramsey, Alan Tissier, Charles Fawcett, Nikola Popovic, Mirko Ellis, Burschi Putzgruber, Jim Burke, Dusko Radojcic, Zivojin Denic, Nikola Illic, Stevo Petrovic, Uwe Rehse, Vladimir Sovanovic, George Attifeliner, Mirko Boman, Gojko Mitic, Vojkan Pavlovic, Andrea Scotti (Andrew Scott), Milivoje Popovic-Mavid, Ulla Moritz, Dustan Tadic. White renegades try to discredit Apaches by attacking ranchers in hopes of getting Indian lands, but Old Shatterhand and his blood brother Winnetou uncover the plot. Filmed in Yugoslavia in 70mm Superpanorama, this West German Western is well done, highly entertaining and will appeal to genre fans due to stars Lex Barker and Guy Madison. Called _**La Battaglia di Fort Apache**_ (The Battle of Fort Apache) in Italy, it was released in the U.S. in 1967 by Goldstone Film Enterprises as _**Shatterhand**_ but cut by 33 minutes; British and TV title: _**Apache's Last Battle**_.\n\n**2884** _ **The Old Texas Trail**_ **** Universal, 1944. 60 min. D: Lewis D. Collins. SC: William Lively. With Rod Cameron, Fuzzy Knight, Eddie Dew, Marjorie Clements, Ray Whitley, Virginia Christine, Edmund Cobb, Joseph J. Greene, George Eldredge, Jack Clifford, Dick Purcell, Harry Strang, Ray Jones, Merle Travis, William Desmond, George Turner, Art Fowler, Henry Wills, Terry Frost, Ray Whitley's Bar-6 Cowboys, Michael Vallon, Herman Hack, Frank McCarroll, George Plues. Three cowboys help a young woman about to lose the option to her stage line thanks to a crook and his gang who are after the contract. Strong Rod Cameron starring effort.\n\n**2885** _ **The Old West**_ **** Columbia, 1952. 61 min. D: George Archainbaud. SC: Gerald Geraghty. With Gene Autry, Pat Buttram, Gail Davis, Lyle Talbot, Louis Jean Heydt, House Peters, House Peters, Jr., Dick Jones, Kathy Johnson, Don C. Harvey, Dee Pollock, James Craven, Tom London, Frankie Marvin, Syd Saylor, Bob Woodward, Buddy Roosevelt, Tex Terry, John Merton, Pat O'Malley, Bobby Clark, Frank Ellis. After a traveling minister comes to the aid of a horse wrangler when he is attacked, the cowboy helps the sky pilot bring religion to a small town. Good, somewhat offbeat, Gene Autry feature.\n\n**2886** _ **The Old Wyoming Trail**_ **** Columbia, 1937. 56 min. D: Folmer Blangsted. SC: Ed Earl Repp. With Charles Starrett, Donald Grayson, Barbara Weeks, The Sons of the Pioneers (Bob Nolan, Len Slye [Roy Rogers], Lloyd Perryman, Hugh Farr, Karl Farr), Dick Curtis, Ed LeSaint, Guy Usher, George Chesebro, Art Mix, Slim Whitaker, Alma Chester, Ernie Adams, Richard Botiller, Frank Ellis, Joe Yrigoyen, Charles Brinley, Fred Burns, Si Jenks, Curley Dresden, Ray Whitley, Blackie Whiteford, Tom London, Art Dillard, Ray Jones, Jerome (Blackjack) Ward, Tex Cooper. Two cowboys try to stop crooks from forcing a rancher to sell his spread, needed for a railroad, for too little money. Well done Charles Starrett film with emphasis on music from co-star Donald Grayson and The Sons of the Pioneers, including Roy Rogers.\n\n**2887** _ **Old Yeller**_ **** Buena Vista, 1957. 83 min. Color. D: Robert Stevenson. SC: Fred Gipson and William Tunberg. With Dorothy McGuire, Fess Parker, Tommy Kirk, Kevin Corcoran, Jeff York, Beverly Washburn, Chuck Connors, Spike (dog). In 1869 Texas a stray dog ingratiates himself into the lives of a frontier family. Very entertaining Walt Disney feature.\n\n**2888** _ **Ole Rex**_ **** Universal-International, 1961. 40 min. Color. D-SC: Robert Hinkle. With Billy Hughes, Rex (dog), William Foster, Robert Hinkle, Whitey Hughes, William Hughes, Richard McCarthy, Red Bray, Dale Marlow, Jr., Dale Terry. The adventures of a young boy and his German shepherd dog. Fair featurette.\n\n**2889** _ **The Omaha Trail**_ **** Metro-Goldwyn-Mayer, 1942. 62 min. D: Edward Buzzell. SC: Jesse Lasky, Jr. and Hugh Butler. With James Craig, Pamela Blake, Dean Jagger, Edward Ellis, Chill Wills, Donald Meek, Howard Da Silva, Henry (Harry) Morgan, Morris Ankrum, Kermit Maynard, Iron Eyes Cody, Al Ferguson, Joe Yule, Hooper Atchley, Henry Roquemore, Ethan Laidlaw, Robert Emmett O'Connor, Murdock MacQuarrie, Edward Hearn, Tom London, Bud Geary, J.W. Cody, Harry Fleischmann, Henry Sylvester, Fred Alrich, Jack Lorenz. When an ox train race causes the deaths of two Indians, the result in a tribal uprising. Cheaply made, mediocre oater with excessive stock footage.\n\n**2890** _ **O'Malley of the Mounted**_ **** 20th Century\u2013Fox, 1936. 59 min. D: David Howard. SC: Dan Jarrett and Frank Howard Clark. With George O'Brien, Irene Ware, Crauford Kent, James Bush, Victor Potel, Charles King, Stanley Fields, Tom London, Reginald Barlow, Richard Cramer, Olin Francis, Blackjack Ward, Frank Ellis, Al Taylor. A mounted police officer infiltrates an outlaw gang terrorizing U.S.-Canadian border towns and initiates a plan that leads the bad men into a robbery, resulting in their capture. Nicely done George O'Brien feature that should please north of the border film fans.\n\n_**On Boot Hill**_ see _**Last Days of Boot Hill**_\n\n**2891** _ **On the Great White Trail**_ **** Grand National, 1938. 59 min. D: Al Herman. SC: Charles Logue and Joseph F. Poland. With James Newill, Terry Walker, Robert Frazer, Richard Alexander, Richard Tucker, Robert Terry, Eddie Gribbon, Walter McGrail, Philo McCullough, Charles King, Juan Duval, Victor Potel, Carl Mathews, Bruce Warren, Roger Williams, Herman Hack, Wally West, Gene Alsace, Jimmy Aubrey, Silver King (dog). When his girl's father is falsely accused of robbery and murder, a Royal Canadian Mounted Policeman heads into the wilderness to capture the real criminal. Picturesque second entry in the \"Renfrew of the Royal Mounted\" series; also called _**On the Trail**_ , _**Renfrew of the Royal Mounted on the Great White Trail**_ and _**Renfrew on the Great White Trail**_.\n\n**2892** _ **On the Night Stage**_ **** Mutual, 1915. 60 min. D: Reginald Barker. SC: C. Gardner Sullivan and Thomas H. Ince. With Robert Edeson, Rhea Mitchell, William S. Hart, Herschel Mayall, Gladys Brockwell, Shorty Hamilton. A good-bad man steps aside when the girl he loves weds the parson who helped him in a fight but when she is blackmailed by a former cohort, he comes to her rescue. Probably the best known of William S. Hart's early films, this silent feature provides good entertainment.\n\n**2893** _ **On the Old Spanish Trail**_ **** Republic, 1947. 75 min. Color. D: William Witney. SC: Sloan Nibley. With Roy Rogers, Tito Guizar, Bob Nolan and The Sons of the Pioneers (Tim Spencer, Lloyd Perryman, Pat Brady, Hugh Farr, Karl Farr), Jane Frazee, Andy Devine, Estelita Rodriguez, Charles McGraw, Fred Graham, Steve Darrell, Marshall Reed, Wheaton Chambers, Ed Cassidy, Jack O'Shea, Edward Keane, Shug Fisher, Billy Mitchell. When he helps the Sons of the Pioneers' flagging road show, Roy Rogers runs into \"The Gypsy,\" a mysterious figure wanted in connection with oil company robberies. Tito Guizar is about as much of a star in this film as Roy Rogers but while it has some nice songs it tends to drag; cut by over 20 minutes for television.\n\n_**On the Trail**_ see _**On the Great White Trail**_\n\n**2894** _ **On Top of Old Smoky**_ **** Columbia, 1953. 59 min. D: George Archainbaud. SC: Gerald Geraghty. With Gene Autry, Smiley Burnette, Gail Davis, Sheila Ryan, Cass County Boys (Jerry Scoggins, Bert Dodson, Fred Martin), Grandon Rhodes, Kenne Duncan, Robert Bice, Zon Murray, Pat O'Malley, Art Dillard, Frankie Marvin, Mathew McCue, Jack Gargan, Jack Tornek. Mistaken for a ranger, a singing star comes assists a woman whose ranch is coveted by a crook for its rich mica deposits. Good songs and an interesting plot bring life to this Gene Autry outing.\n\n**2895** _ **Once Upon a Horse**_ **** Universal-International, 1958. 85 min. D-SC: Hal Kantor. With Dan Rowan, Dick Martin, Martha Hyer, Leif Erickson, Nita Talbot, James Gleason, John McGiver, David Burns, Dick Ryan, Max Baer, Buddy Baer, Bob Steele, Bob Livingston, Tom Keene, Kermit Maynard, Steve Pendleton, Paul Anderson, Tom London, Ingrid Goude, Joe Oakie, Ray Jones, Sam Hearn. Two cowpokes rustle herds belonging to a ruthless cattle queen and then find out they do have not have the money to feed the stock. Comedy does not offer much except for seeing veteran genre stars (Bob Steele, Bob Livingston, Tom Keene, Kermit Maynard) as themselves in a brief scene; reissued as _**The Hot Horse**_.\n\n**2896** _ **Once Upon a Starry Night**_ **** NBC-TV, 1978. 78 min. Color. D: Jack B. Hively. SC: Brian Russell and James Simmons. With Dan Haggerty, Denver Pyle, Ken Curtis, Jack Kruschen, Diane McBain, Don Galloway, Linda Arbizu, Stephen Robertson, Bozo (bear). At Christmas time, when the parents of young children are separated from them by an avalanche, a trapper goes into a blizzard to attempt a rescue. Family oriented affair originally telecast as an episode of \"The Life and Times of Grizzly Adams\" (NBC-TV, 1977\u201378).\n\n**2897** _ **Once Upon a Texas Train**_ **** CBS-TV, 1988. 96 min. Color. D-SC: Burt Kennedy. With Willie Nelson, Richard Widmark, Shaun Cassidy, Chuck Connors, Ken Curtis, Royal Dano, Jack Elam, Gene Evans, Kevin McCarthy, Dub Taylor, Stuart Whitman, Angie Dickinson, Jeb Stuart Adams, Clare Carey, Harry Carey (Jr.), David Michael O'Neill, Red West, Hank Worden, John Calkins, Lisa Cloud, Don Collier, Dennis Fimple, John Purlong. An ex\u2013Texas Ranger gets on the trail of a famous bank robber he sent to prison two decades earlier. Many familiar faces makes this TV movie enjoyable; video title: _**Texas Guns**_.\n\n**2898** _ **Once Upon a Time in the West**_ **** Paramount, 1969. 168 min. Color. D: Sergio Leone. SC: Sergio Leone and Sergio Donati. With Henry Fonda, Claudia Cardinale, Jason Robards, Charles Bronson, Frank Wolff, Gabriele Ferzetti, Keenan Wynn, Paolo Stoppa, Marco Zuanelli, Robert Hossein, Aldo Sambrell, Fabio Testi, Lionel Stander, Jack Elam, John Frederick, Woody Strode, Al Mulock, Spartaco Conversi, Enzo Santinello, Dino Mele, Benito Stefanelli, Salvo Basile. In late 1800s Kansas a mysterious harmonica playing stranger shows up to avenge the murder of his father as the ruthless killer and his hired guns try to take land containing water needed by the railroad. Long, leisurely and violent European production which has a cult following and is well worth seeing, especially for Charles Bronson as the avenger; cut by 24 minutes for U.S. release. Italian title: _**C'era una Volta il West**_.\n\n**2899** _ **One Desire**_ **** Universal-International, 1955. 94 min. Color. D: Jerry Hopper. SC: Lawrence Roman and Robert Blees. With Anne Baxter, Rock Hudson, Julie Adams, Carl Benton Reid, Natalie Wood, William Hopper, Betty Garde, Barry Curtis, Adrienne Marden, Fay Morley, Vici Raaf, Lynn Millan, Smoki Whitfield, Howard Wright, Edward Earle, Terry Frost, Dennis Moore, William Forrest, Edmund Cobb, Guy Wilkerson, Holly Bane, Marshall Bradford, Alan DeWitt, Joel Allen, John Close, Forbes Murray, Jane Howard, Steve Pendleton, Paul Keast, Rory Mallinson, Paul McGuire, Joseph Mell, Lana Wood, Damian O'Flynn, John Daheim, Robert F. Hoy, Harvey B. Dunne, Jack Chefe, Mack Williams, Charles H. Gray, Don House, Betty Jane Howarth, Donald Kerr, Paul Levitt, Mike Mahoney, Rankin Mansfield, Donald Moore, Clarence Straight, Paul Weber, Major Sam Harris, Kenner G. Kemp. At the turn of the 20th Century, an ex-gambler, now a bank clerk, is pursed by his boss' daughter although he loves a former madam who has adopted an orphan girl. Well acted boom town drama based on Conrad Richter's book _Tacey Cromwell_.\n\n**2900** _ **One Dollar Too Many**_ **** Titanus, 1968. 95 min. Color. D: Enzo G. Castellari. SC: Augusto Finocchi and Vittorio Metz. With Antonio Sabato, John Saxon, Frank Wolff, Agata Flori, Leo Anchoriz, Antonio Vico, Rosella Bergamonti, Hercules Cortez, Tito Garcia, Edy Biagetti, Josefina Serratosa, Leonardo Scavino, Kathleen Trentini, Paolo Magalotti, Margareth Horowitz, Roberto Fuentes, Pilar Vela, Claudio Castelli, Jose Maria Tasso, Ivan Scartuglia, Luis Barboo, Jesus Guzman, Victor Israel. A gambler, sharpshooter and an actor vie for a cache of stolen loot but unite to oppose Mexican outlaws also after the money. The three leads, a seriocomic approach and not much violence make this a more than passable Spaghetti Western; an Italian-Spanish co-production released in Europe as _**I Tre Che Sconvolsero il West**_ (The Three That Upset the West) and _**Vado, Vedo e Sparo**_ (I Came, I Saw and I Shot).\n\n**2901** _ **One Eyed Jacks**_ **** Paramount, 1961. 141 min. Color. D: Marlon Brando. SC: Guy Trosper and Calder Willingham. With Marlon Brando, Karl Malden, Pina Pellicer, Katy Jurado, Ben Johnson, Slim Pickens, Elisha Cook (Jr.), Rodolfo Acosta, Larry Duncan, Sam Gilman, Timothy Carey, Miriam Colon, Ray Teal, John Dierkes, Hank Worden, Nina Martinez, Margarita Cordova, William Forrest, Nacho Galindo, Philip Ahn, Henry Wills, Felipe Turich, Mickey Finn, Joan Petrone, Francy Scott, Margarita Martin, Clem Harvey. Released from prison, an outlaw seeks revenge on the partner who betrayed him and finds he is now a sheriff. Lots of psychological overtones in this overblown feature that was cut prior to release, which may explain why it is not as satisfying as it should be.\n\n**2902** _ **One Foot in Hell**_ **** 20th Century\u2013Fox, 1960. 90 min. Color. D: James B. Clark. SC: Aaron Spelling. With Alan Ladd, Don Murray, Dan O'Herlihy, Dolores Michaels, Larry Gates, Karl Swenson, Barry Coe, John Alexander, Rachel Stephens, Edmund Cobb, Harry Carter, Stanley Adams, Ann Morriss, Charles Watts, Henry Norell, Robert Adler, I. Stanford Jolley, Charles Wagenheim, William Challee, Lyle Latell, Ned Wever, Kermit Maynard, Harry Seymour, Max Wagner, Harry Strang, Fred Aldrich, Charles Sullivan, George Cisar. When his wife is killed, a deputy sheriff is determined to get even with the three businessmen he feels are responsible. A different kind of Western that will appeal to Alan Ladd fans.\n\n**2903** _ **100 Rifles**_ **** 20th Century\u2013Fox, 1969. 110 min. Color. D: Tom Gries. SC: Clair Huffaker and Tom Gries. With Jim Brown, Raquel Welch, Burt Reynolds, Fernando Lamas, Dan O'Herlihy, Michael Forest, Soledad Miranda, Alberto Dalbes, Jose Manuel Martin, Hans Guedgast, Aldo Sambrell, Carlos Bravo. A lawman after a bank robber ends up helping a female revolutionary in Mexico defend an Indian village against a tyrant. Fair action oater, but nothing special.\n\n**2904** _ **100,000 Dollars for Lassiter**_ **** P.E.A.\/Centauro Films, 1966. 97 min. Color. D: Joaquin Romero Marchant. SC: Sergio Donati and Joaquin Romero Hernandez. With Robert Hundar (Claudio Undari), Pamela Tudor, Peter Martell, Andrew Ray (Andea Aureli), Luigi Pistilli, Jose Bodalo, Jesus Puente, Roberto Camardiel, Aldo Sambrell, Benito Stefanelli, Robert Johnson, Jr., Luis Gasper. A wheelchair bound rancher, who controls the area's water supply with the help of an outlaw gang in his employ, is opposed by a widow and her hired gunman. Typically vicious Spaghetti Western, an Italian-Spanish co-production filmed as _**100.000 Dollari per Lassiter**_ (100,000 Dollars for Lassiter) and shown on U.S. TV as _**Dollars for a Fast Gun**_.\n\n**2905** _ **$100,000 for Ringo**_ **** Balcazar, 1965. 106 min. Color. D: Alberto De Martino. SC: Alberto De Martino, Giovanni Simonelli, Alfonso Balcacar and Vincenzo Flamini. With Richard Harrison, Fernando Sancho, Eleanora Bianchi, Luis Induni, Monica Randall, Gerard Tichy, John Barracuda, Loris Lotty, Lee Burton (Guido Lollobrigida). A stranger rides into a town and tries to bring order between factions after buried treasure. Solid Italian production released there as _**100.000 Dollari per Ringo**_ (100,000 Dollars for Ringo).\n\n**2906** _ **One Little Indian**_ **** Buena Vista, 1973. 90 min. Color. D: Bernard McEveety. SC: Harry Spalding. With James Garner, Vera Miles, Pat Hingle, Morgan Woodward, Jim Davis, John Doucette, Clay O'Brien, Robert Pine, Bruce Glover, Ken Swofford, Jay Silverheels, Andrew Prine, Jodie Foster, Walter Brooke, Rudy Diaz, John Flynn, Tom Simcox, Lois Red Elk, Hal Baylor, Terry Wilson, Paul Sorenson, Boyd \"Red\" Morgan. Falsely accused of crimes by his superiors, a cavalryman escapes from prison, heads into the desert on a camel and befriends an runaway Indian boy. Light oater comedy-drama from Walt Disney studios should please genre fans.\n\n**2907** _ **One Man Justice**_ **** Columbia, 1937. 59 min. D: Leon Barsha. SC: Paul Perez. With Charles Starrett, Barbara Weeks, Hal Taliaferro, Jack Clifford, Alan Bridge, Walter Downey, Mary Gordon, Jack Lipson, Edmund Cobb, Dick Curtis, Maston Williams, Art Mix, Hank Bell, Steve Clark, Frank Ellis, Ethan Laidlaw, Eddie Laughton, Ted Mapes, Lew Meehan, Merrill McCormick, Harry Fleischman. Arriving in town, a cowboy learns he is the look-a-like of a supposedly dead rancher and he agrees to impersonate the man to help the sheriff catch a gang rustling cattle belonging to the deceased's pretty widow. Strong Charles Starrett vehicle making good use of the amnesia gimmick; a remake of _**Texas Cyclone**_ (q.v.).\n\n**2908** _ **One Man Law**_ **** Columbia, 1932. 60 min. D-SC: Lambert Hillyer. With Buck Jones, Shirley Grey, Robert Ellis, Murdock MacQuarrie, Harry Todd, Henry Sedley, Ernie Adams, Richard Alexander, Wesley Girard, Ed LeSaint, Fred Burns, Jim Corey, Tex Phelps, Roy Bucko. A crooked land speculator convinces a cowboy to become a sheriff so his dishonest activities can be hidden but the new lawman soon gets wise. Well done Buck Jones feature.\n\n**2909** _ **One Man's Law**_ **** Republic, 1940. 57 min. D: George Sherman. SC: Bennett Cohen and Jack Natteford. With Don \"Red\" Barry, Janet Waldo, George Cleveland, Dub Taylor, Rex Lease, Carleton Young, Edmund Cobb, Robert Frazer, Charles King, Dick Elliott, Jack Ingram, Roy Barcroft, Stanley Price, Ed Peil, Sr., Fred \"Snowflake\" Toones, Bud Osborne, Horace B. Carpenter, Jack Kirk, Cactus Mack, Jim Corey, Curley Dresden, Roy Brent, Guy Usher, William Kellogg, James H. MacNamara. When crooks try to stop a town from getting a railroad franchise a cowboy comes to the rescue. Fast paced and nicely produced Don Barry entry.\n\n**2910** _ **One Mask Too Many**_ **** Wrather Corporation, 1956. 75 min. Color. D: Earl Bellamy and Oscar Rudolph. SC: Doane Hoag, Thomas Seller, Edmund Kelso and Orville Hampton. With Clayton Moore, Jay Silverheels, Allen Pinson, Wayne Burson, Tristram Coffin, Jim Bannon, Roy Barcroft, Virginia Christine, William Challee, Paul Engle, Sydney Mason, John Cliff, Louise Lewis, Sandy Sanders, Walt LaRue, Charles Wagenheim, Robert Closson, Saul M. Gorss, Michael Winkelman, John Ira Hudkins, Paul Stader, Gabor Curtiz, Richard Benedict, Peter Miles, Jason Johnson. The Lone Ranger tries to clear his name when crimes are committed by a masked bandit, then he and Tonto attempt to stop the assassination of a European nobleman as well as oppose a brutal outlaw gang. Fast paced paste-up from the \"Canuck,\" \"Counterfeit Mask\" and \"Prince of Buffalo Gap\" episodes of \"The Lone Ranger\" (ABC-TV, 1949\u201357).\n\n**2911** _ **One More Train to Rob**_ **** Universal, 1971. 106 min. Color. D: Andrew V. McLaglen. SC: Don Tait and Dick Nelson. With George Peppard, Diana Muldaur, John Vernon, France Nuyen, Steve Sandor, Soon-Talk On, Richard Loo, C.K. Yang, John Doucette, Robert Donner, George Chandler, Pamela McMyler, Merlin Olsen, Phil Olsen, Marie Windsor, Joan Shawlee, Harry Carey, Jr., Ben Cooper, Walter Reed, Andy Albin, Charles Seel, Mike Henry, Don Barry, Larry J. Blake, Lane Chandler, Timothy Scott, Hal Needham, Jim Burk, Ray Limas, Guy Lee. After being released from prison a man tries to locate the partner who double crossed him and learns he is the leading citizen of a town trying to cheat Chinese out of a fortune in gold. Average action outing somewhat helped by its supporting cast of veteran players.\n\n**2912** _ **One of the Missing**_ **** Feigelson Productions, 1971. 56 min. Color. D: Julius D. Feigelson. With Todd Armstrong, Gordon Baxter. During the Civil War a sharpshooter is pinned by heavy beams after an explosion with his cocked rifle pointed directly between his eyes. Interesting short feature based on the story by Ambrose Bierce.\n\n**2913** _ **One Punch O'Day**_ **** Rayart, 1926. 55 min. D: Harry J. Brown. SC: Henry R. Symonds. With Billy Sullivan, Charlotte Merriam, Jack Herrick, William Malan, J.C. Fowler, Eddie Diggins. A boxing champion defeats the machinations of a dishonest businessman when he stages a fight to get money to pay for an oil land lease and brings in a well, thus saving the fortunes of the father of the girl he loves. Typical silent action program feature starring boxing champion John L. Sullivan's nephew, Billy Sullivan.\n\n**2914** _ **$1,000 Reward**_ **** Aywon, 1923. 60 min. D: Charles R. Seeling. With Guinn \"Big Boy\" Williams. Falsely accused of murder, a cowboy escapes to another locale where he becomes a deputy sheriff and arrests his accuser. Here is a chance to see Guinn Williams in one of his silent outings, an a fairly good one, for Charles R. Seeling Productions.\n\n**2915** _ **One Way Trail**_ **** Columbia, 1931. 60 min. D: Ray Taylor. SC: George Plympton. With Tim McCoy, Doris Hill, Polly Ann Young, Al Ferguson, Carroll Nye, Robert Homans, Bud Osborne, Slim Whitaker, Blackjack Ward, Herman Hack, Jack Long, Artie Ortego, Bob Burns, Art Mix, Silver Harr, Charles Le Moyne, George Sowards, Al Taylor, Bud McClure, Barney Beasley, Jack King, Jack Evans, Charles Murphy, Archie Ricks, Ralph Bucko, Roy Bucko. A cowboy suspects a rancher murdered his brother and sets out to ruin the man. Tim McCoy's first Columbia series entry is okay but not up to the standard of some of his later movies.\n\n**2916** _ **One's the Same as Another**_ **** Producciones Rosas Priego, 1959. 87 min. D-SC: Jaime Salvador. With Luis Aguilar, Demetrio Gonzalez, Flor Silvestre, Rosa de Castilla, Carlos Riquelme, Jose Jasso, Armando Soto La Marina \"El Chicote,\" Joaquin Garcia Vargas \"Borales,\" Jose Eduardo Perez, Hilda Moreno, Aurora Walker, Humberto Rodriguez, Emilio Garibay, Jesus Gomez. Two girl friends sneak out of school, go to a fiesta and plan to seduce each other's cowboy brother. Silly Mexican Western musical comedy, released there as _**Tan Bueno el Giro Como el Colorado**_ (As Good as the Colorado Flows).\n\n**2917** _ **Only Birds and Fools**_ **** NBC-TV\/Universal, 1974. 76 min. Color. D: Harry Morgan. SC: Richard Fiedler. With Richard Boone, Robert Foxworth, Cliff Potts, Harry Morgan, Rick Lenz, Charles Aidman, Harold J. Stone, Dennis Rucker, Fionnula Flanagan, John Daheim, John Hart, Katherine Helmond. When a stranger is murdered a lawman investigates and is led to two aviators seeking funds from the town council. Turn-of-the-century Western mystery is pleasant; originally telecast as the final episode of \"Hec Ramsey\" (NBC-TV, 1972\u201374) on April 7, 1974.\n\n**2918** _ **Only the Brave**_ **** Paramount, 1930. 71 min. D: Frank Tuttle. SC: Edward E. Paramore, Jr. With Gary Cooper, Mary Brian, Phillips Holmes, James Neill, Morgan Farley, Guy Oliver, John Elliott, E.H. Calvert, Virginia Bruce, Elda Voelkel, William LeMaire, Freeman S. Wood, Lalo Encinas, Clinton Rosemond. During the Civil War a Union officer is rejected by his girlfriend and volunteers to work as a spy. Dated melodrama with its main appeal for Gary Cooper fans.\n\n**2919** _ **Only the Valiant**_ **** Warner Bros., 1951. 107 min. D: Gordon Douglas. SC: Charles Marquis Warren. With Gregory Peck, Barbara Payton, Gig Young, Ward Bond, Lon Chaney, Warner Anderson, Jeff Corey, Steve Brodie, Neville Brand, Terry Kilburn, Herbert Heyes, Art Baker, Hugh Sanders, Michael Ansara, Nana Bryant, Harvey Udell, Claire James, Clark Howat, Harlan Howe, John Halloran, David Clarke, William Newell, John Doucette, William Phillips. A disciplinarian cavalry officer leads a small group of men who hate him through Indian country, fighting Apaches and feuding among themselves over a beautiful woman who has joined them. Somewhat offbeat psychological Western is pretty good, especially Lon Chaney as a murderous, mercenary Arab.\n\n**2920** _ **Open Range**_ **** Buena Vista, 2003. 139 min. Color. D: Kevin Costner. SC: Craig Storper. With Robert Duvall, Kevin Costner, Annette Bening, Michael Gambon, Michael Jeter, Diego Luna, James Russo, Abraham Benrubi, Dean McDermott, Kim Coates, Herb Kohler, Peter MacNeill, Cliff Saunders, Pat Stutz, Julian Richards, Ian Tracey, Rod Wilson, Diego Del Mar, Patricia Benedict, Tim Koetting, Tom Carey, Kurtis Sanheim, Billy Morton, Alex Zahara, Chad Camilleri, Greg Schlosser, Guy Bews, Loretta Clow, Alexis Cerkiewicz. A dishonest sheriff and a land baron attempt to take over an ex-gunman's friend's cattle herd. Nicely made and very entertaining medium budget Western, a moneymaker at the box office.\n\n**2921** _ **The Open Switch**_ **** Rayart, 1926. 50 min. D: J.P. McGowan. With Helen Holmes, Jack Perrin, Charles \"Slim\" Whitaker, Mack V. Wright, Arthur Millet, Henry Roquemore, Max Asher, J.P. McGowan, J. Carrol Naish. After the theft of an express package, a crook takes the identity of a railroad agent in order to get the reward but is opposed by a woman and her partner. The teaming of silent serial queen Helen Holmes and oater star Jack Perrin brings some life to this silent program feature.\n\n**2922** _ **Operation Haylift**_ **** Lippert, 1950. 75 min. D: William Berke. SC: Joseph Sawyer and Dean Riesner. With Bill Williams, Ann Rutherford, Tom Brown, Jane Nigh, Joseph Sawyer, Richard Travis, Raymond Hatton, James Conlin, Tommy Ivo, Dick Dean, Joanna Armstrong, M'liss McClure, Frank Jaron, H.G. Fisher, Roger Norton. When cattle are stranded and starving during a blizzard in Montana, the brother of a rancher helps the Air Force in dropping hay to feed the stock. Very entertaining modern Western based on an actual event; actor Joseph Sawyer not only appeared in the film but also co-wrote and co-produced it.\n\n**2923** _ **Operator 13**_ Metro-Goldwyn-Mayer, 1934 86 min. D: Richard Boleslavsky. SC: Harvey Thew, Zelda Sears and Eve Greene. With Marion Davies, Gary Cooper, Jean Parker, Katharine Alexander, Ted Healy, Russell Hardie, Henry Wadsworth, Douglass Dumbrille, Willard Robertson, Fuzzy Knight, Sidney Toler, Robert McWade, Marjorie Gateson, Wade Boteler, Walter Long, Hattie McDaniel, Francis McDonald, William H. Griffith, James Marcus, The Mills Brothers, Sam McDaniel, Buddy Roosevelt, Frank McGlynn, Jr., Wheeler Oakman, Don Douglas, Si Jenks, Reginald Barlow, Ernie Alexander, Richard Powell, Wilfred Lucas, William Henry, Richard Tucker, Arthur Grant, Sherry Tansey, George Lloyd, Sam Ash, Claudia Coleman, Sterling Holloway, Douglas Fowley, Samuel S. Hinds, Frank Marlowe, DeWitt Jennings, Ernie Adams, Clarence Wilson, Franklin Parker, Claudia Coleman, Sherry Hall, James C. Morton, John Larkin, Wally Howe, Bob Stevenson, Lia Lance, Charles Lloyd. During the Civil War an actress heads South in order to spy for the North. Hokey historical melodrama wastes a good cast.\n\n**2924** _ **Ordeal**_ **** ABC-TV\/20th Century\u2013Fox, 1973. 73 min. Color. D: Lee H. Katzin. SC: Francis M. Cockrell and Leon Tokatian. With Arthur Hill, Diana Muldaur, James Stacy, Macdonald Carey, Michael Ansara, Arch Whiting, Bill Catching, Len Felber. After his leg is broken a man is left to die in the Mojave Desert by his wife and her boyfriend as he struggles to survive. Well done TV movie; remake of _**Inferno**_ (q.v.).\n\n**2925** _ **The Ordeal of Dr. Mudd**_ **** CBS-TV, 1980. 104 min. Color. D: Paul Wendkos. SC: Douglas Schwartz and Michael Berk. With Dennis Weaver, Susan Sullivan, Richard Dysart, Michael McGuire, Nigel Davenport, Arthur Hill, Mary Nell Santacroce, Larry Larson, Bill Gribble, Luke Halpin, Don Kovacs, Lawrence Montaigne, Harold Bergman, Fred Covington, Joe Dorsey, Jim Peck, Gregg Oliver, Bill Eudaly, Don Devendorf, Tony Kish, Wallace Wilkinson, Anthony Edenfield, Jere Beery, Kent Stephens, Stuart Culpepper, Richard Andrew, Dan Chandler, Skip Foster, Charles Kaufman, Tommy Lane, Bill Hindman, George De Vries, Richard Andrew. Dr. Samuel Mudd, for setting John Wilkes Booth's broken leg after the Lincoln assassination, is sentenced to life at the prison in Dry Tortugas and later becomes a hero during a Yellow Fever outbreak. More than acceptable TV fare already covered theatrically in _**Hellgate**_ and _**The Prisoner of Shark Island**_ (qq.v.).\n\n**2926** _ **Oregon Passage**_ **** Allied Artists, 1958. 82 min. Color. D: Paul Landres. SC: Jack DeWitt. With John Ericson, Lola Albright, Toni Gerry, Edward Platt, Harvey Stephens, Judith Ames, H.M. Wynant, Jon Shepodd, Walter Barnes, Paul Fierro. A cavalry officer innocently incurs the wrath of a Shoshone Indian chief when he rescues a maiden from a tribal ceremony. Despite an interesting plot, this film is on the bland side.\n\n**2927** _ **The Oregon Trail**_ **** Universal, 1939. 15 Chapters. D: Ford Beebe and Saul A. Goodkind. SC: George H. Plympton, Basil Dickey, Edmund Kelso, W.W. Watson and Dorothy Cormack. With Johnny Mack Brown, Louise Stanley, Bill Cody, Jr., Fuzzy Knight, Forrest Taylor, Ed LeSaint, James Blaine, Jack C. Smith, Roy Barcroft, Colin Kenny, Charles King, Charles Stevens, Jim Torney, Karl Hackett, Warner Richmond, Kenneth Harlan, Horace Murphy, Helen Gibson, Frank LaRue, Frank Ellis, Richard Alexander, Lafe McKee, Jim Thorpe, Chick Hannon, Tom Smith, Cactus Mack, George Plues, Iron Eyes Cody, Tex Young. The government hires a scout to stop Indian attacks on a wagon train headed for Oregon Territory. Action packed, but juvenile, cliffhanger.\n\n**2928** _ **The Oregon Trail**_ **** Republic, 1945. 55 min. D: Thomas Carr. SC: Betty Burbridge. With Sunset Carson, Peggy Stewart, Frank Jaquet, Si Jenks, Mary Carr, Lee Shumway, Bud Geary, Kenne Duncan, Steve Winston, Tex Terry, Tom London, Earle Hodgins, Monte Hale, Rex Lease, Tommy Coats, Horace B. Carpenter, George Magrill, Bud Osborne, Sheila Stuart, Henry Wills. A railroad detective goes undercover to infiltrate an outlaw gang and also helps a rancher and his pretty daughter when a crook tries to take over the area because it is wanted by the railroad. Fairly interesting Sunset Carson vehicle.\n\n**2929** _ **The Oregon Trail**_ **** 20th Century\u2013Fox, 1959. 82 min. Color. D: Gene Fowler, Jr. SC: Gene Fowler, Jr. and Louis Vittes. With Fred MacMurray, Gloria Talbott, William Bishop, Nina Shipman, Henry Hull, John Carradine, John Dierkes, Elizabeth Patterson, James Bell, Ralph Sanford, Tex Terry, Oscar Bergei, Addison Richards, Lumsden Hare, Gene H. Fowler, Sherry Spalding, Roxene Wells. A newspaper reporter joins a wagon train heading for Oregon to investigate reports the government has sent troops there to fight the British in a dispute over the territory. Film has a lot of promise but not enough budget to fulfill it.\n\n**2930** _ **The Oregon Trail**_ **** NBC-TV\/Universal, 1976. 100 min. Color. D: Boris Sagal. SC: Michael Gleason. With Rod Taylor, Blair Brown, David Huddleston, Douglas V. Fowley, Andrew Stevens, Linda Purl, G.D. Spradlin, Tony Becker, Gina Maria Smika, George Keymas, Eddie Little Sky, Robert Karnes, Jerry Hardin, Wilford Brimley, Hoke Howell, Walter Edmiston, John Wyler. An Eastern family gives up their farm to move West for free land and a new life. Well done wagon train drama that led to the series of the same title that ran on NBC-TV in the fall of 1977.\n\n**2931** _ **Oregon Trail Scouts**_ **** Republic, 1947. 58 min. D: R.G. Springsteen. SC: Earle Snell. With Allan \"Rocky\" Lane, Bobby Blake, Martha Wentworth, Roy Barcroft, Emmett Lynn, Edmund Cobb, Earle Hodgins, Ed Cassidy, Frank Lackteen, Jack Kirk, Jack O'Shea, Chief Yowlachie, Billy Cummings, John War Eagle, Forrest Burns, Jack Sparks, Ted Elliott, Ernest \"Tex\" Young. Red Ryder helps Indians whose trapping rights are sought by a gang of hoodlums who kidnap one of the tribe's young boys. Action filled entry in the \"Red Ryder\" series.\n\n**2932** _ **Orphan of the North**_ **** Monogram, 1940. 56 min. D-SC: Norman Dawn. With Bob Webster, Mary Joyce, Ann Hemming, Eleanor Phillips, John Pool. When a small girl's father fails to return home from a gold hunt she sets out to find him and a rescue party follows. Low grade, but picturesque, semi-documentary drama; the title refers to bear cubs.\n\n**2933** _ **Orphan of the Pecos**_ **** Victory, 1937. 55 min. D: Sam Katzman. SC: Basil Dickey. With Tom Tyler, Jeanne Martel, Forrest Taylor, Lafe McKee, Theodore (Ted) Lorch, Slim Whitaker, John Elliott, Marjorie Beebe, Roger Williams, Milburn Morante, George Morrell, Frank Wayne, Bud Pope, Howard Bryant. A cowboy tries to find out who killed his pal while outlaws are after a beautiful woman's ranch. Producer Sam Katzman directed this cheap Tom Tyler vehicle with Jeanne Martel making a comely orphan as a neat plot twist has ventriloquism used to reveal the murderer.\n\n**2934** _ **Orphan Train**_ **** CBS-TV, 1979. 150 min. Color. D: William A. Graham. SC: Millard Lampell. With Jill Eikenberry, Kevin Dobson, Linda Manz, Melissa Michaelson, Graham Fletcher-Cook, Glenn Close, Morgan Farley, Severn Darden, Charlie Fields, Peter Neumann, Joe Femia, Sara Ingliss, Andreas Manske, Scott Rogers, Justine Johnson, Sue Ann Gilfillon, Barbara Hallie-Foote, Mike Hammett. A novice social worker teams with a newspaper photographer to lead a group of slum children out of New York City to start new lives in the West. Well done TV drama based on Dorothea G. Petrie's book.\n\n**2935** _ **Out California Way**_ **** Republic, 1946. 67 min. Color. D: Lesley Selander. SC: Betty Burbridge. With Monte Hale, Adrian Booth, Bobby Blake, John Dehner, Nolan Leary, Fred Graham, Tom London, Jimmy Starr, Edward Keane, Robert Wilke, Brooks Benedict, Roy Rogers, Dale Evans, Don \"Red\" Barry, Allan \"Rocky\" Lane, Foy Willing and The Riders of the Purple Sage, St. Luke's Choristers. A cowboy arrives in Hollywood hoping to become a film star but is hindered by a jealous fading genre hero. Fans will enjoy this Monte Hale movie, both for its plot and guest stars.\n\n**2936** _ **Out West with the Hardys**_ **** Metro-Goldwyn-Mayer, 1938. 90 min. D: George B. Seitz. SC: Kay Van Riper, Agnes Christine Johnson and William Ludwig. With Lewis Stone, Mickey Rooney, Cecilia Parker, Ann Rutherford, Fay Holden, Sara Haden, Don Castle, Virginia Weidler, Gordon Jones, Ralph Morgan, Nana Bryant, Thurston Hall, Tom Neal, Anthony Allen (John Hubbard), Eddy Waller, Erville Alderson, George Douglas, Joe Dominguez, Charles Grove, Mary Bovard, Marilyn Stuart, Jesse Graves. The Hardy family takes a vacation on a ranch where the owner is having problems with water rights. Delightful \"Hardy Family\" fare for fans of the series.\n\n**2937** _ **Out West with the Peppers**_ **** Columbia, 1940. 63 min. D: Charles Barton. SC: Harry (Sauber) Rebuas. With Edith Fellows, Dorothy Peterson, Dorothy Ann Seese, Tommy Bond, Charles Peck, Bobby Larson, Victor Kilian, Helen Brown, Emory Parnell, Pierre Watkin, Ronald Sinclair, Walter Soderling, Roger Gray, Hal Price. A widow takes her five children to redwood country where the youngsters built a raft and while taking it down river they almost lose their lives. Pleasing family program picture in the series based on Margaret Sidney's books.\n\n_**Outback**_ see _**Wrangler**_\n\n**2938** _ **The Outcast**_ **** Republic, 1954. 90 min. Color. D: William Witney. SC: John K. Butler and Richard Wormser. With John Derek, Joan Evans, Jim Davis, Catherine McLeod, Ben Cooper, Taylor Holmes, Nana Bryant, Slim Pickens, Frank Ferguson, James Millican, Bob Steele, Nacho Galindo, Harry Carey, Jr., Buzz Henry, Hank Worden, Bill Walker, Nicholas Coster, Marc Hamilton. A man returns home to claim an inheritance but finds his vicious, crooked uncle is trying to cheat him out of it. Fast moving and well produced feature with especially good work by Jim Davis as the uncle, Taylor Holmes as his corrupt partner and Bob Steele as their gunman.\n\n**Joan Evans and Jim Davis in** _**The Outcast**_ **(Republic, 1954).**\n\n**  \n**\n\n**2939** _ **Outcasts of Black Mesa**_ **** Columbia, 1950. 54 min. D: Ray Nazarro. SC: Barry Shipman. With Charles Starrett, Smiley Burnette, Martha Hyer, Richard Bailey, Stanley Andrews, William Haade, Lane Chandler, William Gould, Robert Wilke, Chuck Roberson, Ozie Waters. The Durango Kid helps a young woman whose father and two partners were murdered over their mine. Good action entry in the long running series. British title: _**The Clue**_.\n\n**2940** _ **The Outcasts of Poker Flat**_ **** RKO Radio, 1937. 68 min. D: Christy Cabanne. SC: John Twist and Harry Segall. With Preston Foster, Jean Muir, Van Heflin, Virginia Weidler, Margaret Irving, Frank M. Thomas, Si Jenks, Dick Elliott, Al St. John, Bradley Page, Richard Lane, Monte Blue, Billy Gilbert, Al Ferguson, George Irving, Dudley Clements, Bryant Washburn, Barbara Pepper, Georgia Caine, Otto Hoffman, Tex Driscoll. A diverse group of people are snowbound in a cabin with some of them finding new meaning for life. A combination of Bret Harte's stories \"Outcasts of Poker Flat\" and \"The Luck of Roaring Camp,\" the film does a good job in retaining the flavor of the author's work. First filmed in 1919 by Universal with Harry Carey, Cullen Landis, Gloria Hope and J. Farrell MacDonald, directed by John Ford.\n\n**2941** _ **The Outcasts of Poker Flat**_ **** 20th Century\u2013Fox, 1952. 81 min. D: Joseph M. Newman. SC: Edmund H. North. With Anne Baxter, Dale Robertson, Miriam Hopkins, Cameron Mitchell, Craig Hill, Barbara Bates, Billy Lynn, Dick Rich, Russ Conway, Robert Adler, John Ridgely, Harry Shannon, Lee Phelps, Harry Carter, Tom Greenway, Harry Harvey, Jr., Frosty Royce, Jack Byron, Albert Schmidt, Joe P. Smith, Joe Haworth, Kit Carson. Several people are tossed out of a mining town and take refuse in a mountain cabin during a blizzard. Okay retelling of the Bret Harte story with good work by Miriam Hopkins as Duchess, a has-been saloon singer.\n\n**2942** _ **Outcasts of the Trail**_ **** Republic, 1949. 60 min. D: Philip Ford. SC: Olive Cooper. With Monte Hale, Jeff Donnell, Paul Hurst, Roy Barcroft, John Gallaudet, Milton Parsons, Tommy Ivo, Minerva Urecal, Ted Mapes, George Lloyd, Steve Darrell, Tom Steele, Hank Patterson, Hank Bell. Wandering into town, two cowpokes find they are unwelcome and try to find out why. Good entry in Monte Hale's Republic series with an especially interesting plot.\n\n_**The Outing**_ see _**Scream**_\n\n**2943** _ **The Outlaw**_ **** United Artists, 1943. 126 min. D: Howard Hughes, Howard Hawks and Otto Lovering. SC: Jules Furthman. With Jane Russell, Jack Beutel, Walter Huston, Thomas Mitchell, Mimi Aguglia, Joseph Sawyer, Gene Rizzi, Frank Darien, Pat West, Carl Stockdale, Nena Quartero, Martin Garralaga, Julian Rivero, Dickie Jones, Ethan Laidlaw, Ed Brady, William Steele, Wallace Reid, Jr., Ed Peil, Sr., Lee \"Lasses\" White, Ted Mapes, Willam Newell, Lee Shumway, Emory Parnell, Arthur Loft, Dick Elliott, John Sheehan, Frank Ward, Bobby Callahan, Cecil Kellogg. Doc Holliday befriends the notorious Billy the Kid, saving him from Sheriff Pat Garrett and later when Billy is shot he leaves him in the care of the beautiful Rio and the two fall in love. Made in 1941 and given brief release in 1943, this Howard Hughes production was reissued in 1946 by RKO Radio and cut by nine minutes due to censorship problems. The film is historically inaccurate and not overly entertaining but Jane Russell is a knockout as Rio.\n\n**2944** _ **Outlaw Brand**_ **** Monogram, 1948. 60 min. D: Lambert Hillyer SC: J. Benton Cheney. With Jimmy Wakely, Dub Taylor, Kay Morley, Christine Larson, Ray Whitley, John James, Bud Osborne, Nolan Leary, Eddie Majors, Tom Chatterton, Boyd Stockman, Leonard Penn, Frank McCarroll, Jay Kirby, Dick Reinhart. An wild stallion torments area ranchers and when a singing cowpoke tries to tame him he uncovers the activities of a crook. Passable Jimmy Wakely feature with the star better with a guitar than a gun.\n\n**2945** _ **The Outlaw Breaker**_ **** Goodwill, 1926. 55 min. D: Jacques Jaccard. SC: Jacques Jaccard and Yakima Canutt. With Yakima Canutt, Alma Rayford, Nelson McDowell, Harry Northrub, Dick La Reno, Florence Lee, William Bertram, Frank Ellis, Boy (horse). Carrying on a feud with sheepherders began by his late rancher father, a cowboy finds himself framed on a murder charge. Yakima Canutt fans will enjoy this fast paced silent feature.\n\n**2946** _ **Outlaw Country**_ **** Screen Guild, 1949. 76 min. D: Ray Taylor. SC: Ron Ormond and Ira Webb. With Lash LaRue, Al St. John, Nancy Saunders, Dan White, House Peters, Jr., Steve Dunhill, Lee Roberts, Ted Adams, John Merton, Dee Cooper, Jack O'Shea, Sandy Sanders, Bob Duncan, Herman Hack, Artie Ortego. While working to break up a counterfeiting gang, Lash and Fuzzy meets an outlaw called the Frontier Phantom, who turns out to be Lash's long lost brother. Overlong but still entertaining Lash LaRue vehicle with footage later used in _**The Frontier Phantom**_ (q.v.).\n\n**2947** _ **The Outlaw Deputy**_ **** Puritan, 1935. 56 min. D: Otto Brower. SC: Dell Andrews. With Tim McCoy, Nora Lane, Bud Osborne, George Offerman, Jr., Si Jenks, Joe Girard, Hooper Atchley, Richard Botiller, Charles Brinley, Jack Montgomery, Jim Corey, Hank Bell, Eddie Gribbon, Tex Cooper, Bud Pope, Tom Smith, Ray Jones, Buck Morgan, Bob Card, George Holtz. A cowboy heads to a lawless town to get revenge for the murder of his friend. Tim McCoy's first Puritan film is an action laced affair, sure to please his fans.\n\n**2948** _ **Outlaw Express**_ **** Universal, 1938. 57 min. D: George Waggner. SC: Norton S. Parker. With Bob Baker, Cecilia Callejo, Don Barclay, LeRoy Mason, Forrest Taylor, Nina Campana, Martin Garralaga, Carleton Young, Carlyle Moore, Jr., Jack Kirk, Ed Cassidy, Jack Ingram, Julian Rivero, Tex Palmer, Chief Many Treaties, Ray Jones, Joe Dominguez, William McCauley. The government assigns a cavalry captain to investigate the murders of Pony Express riders and the theft of mail. Pleasant Bob Baker affair, well directed by George Waggner.\n\n_**Outlaw Fury**_ see _**Hostile Country**_\n\n_**The Outlaw Gang**_ see _**The Dalton Gang**_\n\n**2949** _ **Outlaw Gold**_ **** Monogram, 1950. 51 min. D: Wallace Fox. SC: Jack Lewis. With Johnny Mack Brown, Jane Adams, Myron Healey, Milburn Morante, Marshall Reed, Hugh Prosser, Carol Henry, Bud Osborne, George DeNormand, Frank Jaquet, Carl Mathews, Ray Jones, Steve Clark, Bob Woodward, Merrill McCormick. A ranger and his pal look into the thefts of Mexican gold shipments and find a newspaper publisher is the culprit. Short, but fairly interesting oater enhanced by Myron Healey's work as a good, bad man.\n\n**2950** _ **The Outlaw Josey Wales**_ **** Warner Bros., 1976. 137 min. Color. D: Clint Eastwood. SC: Paul Kaufman and Sonia Clemens. With Clint Eastwood, Chief Dan George, Sondra Locke, Bill McKinney, John Vernon, Paula Trueman, Sam Bottoms, Geraldine Keams, Woodrow Parfrey, Joyce Jameson, Sheb Wooley, Royal Dano, Matt Clarke, John Verros, Will Sampson, William O'Connell, John Quade, Frank Schofield, Buck Kartalian, Len Lesser, Doug McGrath, John Russell, Charles Tyner, Bruce M. Fisher, John Mitchum, John Davis Chandler, Tom Roy Lowe, Clay Tanner, Robert Hoy, Madeleine Taylor Holmes, Erik Holland, Cissy Wellman, Faye Hamblin, Danny Green, Richard Farnsworth, Frank Cockrell, Walter Scott, Kyle Eastwood. A Civil War veteran with a price on his head takes revenge on the soldiers who murdered his family. Overlong and too violent, this Western is still more interesting than some of Clint Eastwood's later efforts.\n\n**Clint Eastwood in** _**The Outlaw Josey Wales**_ **(Warner Bros., 1976).**\n\n**  \n**\n\n**2951** _ **Outlaw Justice**_ **** Majestic, 1932. 61 min. D: Armand Schaefer. SC: Oliver Drake. With Jack Hoxie, Dorothy Gulliver, Donald Keith, Chris-Pin Martin, Charles King, Kermit Maynard, Jack Rockwell, Walter Shumway, Tom London, Jack Trent, Slim Whitaker, Frank Ellis, Jack Kirk, Tex Palmer, Pete Morrison, Jim Corey, Hank Bell, Horace B. Carpenter, Blackjack Ward. A cowboy takes on the guise of a notorious outlaw in order to bring in a crook. Jack Hoxie's first sound feature is slow but well made.\n\n**2952** _ **Outlaw of Red River**_ **** Fenix\/Harold Goldman, 1966. 76 min. Color. D: Maury Dexter. SC: Eduardo Brochero. With George Montgomery, Elisa Montes, Joseph (Jose) Nieto, Miguel Castillo, Jesus Todesillas, Raf Baldassare, Anna Custodio, Gloria Camara, Ricardo Valle, Carmen Procel, Jose Villasante, Franco Brano, Rafael Yaquero. A wanted gunman who works for a Mexican colonel must oppose the lawlessness of his boss' fiancee's brother as well as that of a bandit and his gang. Lumbering European oater with star George Montgomery, as the gunfighter O'Brien, its main asset; filmed in Spain as _**El Proscrito del Rio Colorado**_ (The Exile of the Colorado River) and re-released in 1968 as _**Django Killer per l'Onore**_ (Django the Honorable Killer).\n\n**2953** _ **Outlaw Roundup**_ **** Producers Releasing Corporation, 1944. 56 min. D: Harry Fraser. SC: Elmer Clifton. With Dave O'Brien, Jim Newill, Guy Wilkerson, Helen Chapman, Jack Ingram, I. Stanford Jolley, Charles King, Reed Howes, Bud Osborne, Frank Ellis, Budd Buster, Frank McCarroll, Jimmy Aubrey, Cal Shrum, Dan White, Jack Ternak, Aleth Hanson, Jess Cavin. Three Texas Rangers devise a plan to arrest a gang by starting the rumor than an outlaw who buried stolen loot in the area has escaped from jail and one of them accuses the other of being the convict. Low grade \"Texas Rangers\" series feature.\n\n**2954** _ **Outlaw Rule**_ **** Willis Kent, 1935. 61 min. D: S. Roy Luby. SC: E.B. Mann. With Reb Russell, Betty Mack, Alan Bridge, Yakima Canutt, John McGuire, Henry Hall, Ralph Lewis, Joseph Girard, Jack Rockwell, Jack Kirk, Bart Carre, Murdock MacQuarrie, Marin Sais, Ed Porter, Tommy Bupp, Rose Plummer, Bud Pope, Lew Meehan, Richard Botiller, Budd Buster, Bill Patton, Silver Tip Baker, Oscar Gahan, Olin Francis, Bud McClure, Chuck Baldra, Jack Jones, Barney Beasley, Jack Hendricks. When a rancher is falsely accused of murdering a lawman he is helped by a gunman called The Whistler, who is really a cattlemen's association operative after a notorious outlaw. Standard, rough hewn Reb Russell vehicle.\n\n**2955** _ **The Outlaw Stallion**_ **** Columbia, 1954. 64 min. Color. D: Fred F. Sears. SC: David Lang. With Phil(ip) Carey, Dorothy Patrick, Billy Gray, Gordon Jones, Roy Roberts, Trevor Bardette, Morris Ankrum, Chris Alcaide, Robert Anderson, Harry Harvey, Guy Teague. Thieves pretend to befriend a woman and her young son in order to steal their horse herd. Non-ambitious juvenile fare from producer Wallace MacDonald.\n\n**2956** _ **The Outlaw Tamer**_ **** Empire, 1935. 60 min. D: J.P. McGowan. SC: J. Wesley Patterson and Kaye Northrup With Lane Chandler, Janet Morgan (Blanche Mehaffey), Charles (Slim) Whitaker, Ben Corbett, George Hayes, J.P. McGowan, Tex Palmer, Herman Hack, Maston Williams, Jack Evans, Ed Gyton, Blackjack Ward, Richard Cramer, Wally West, Art Dillard, Jack Hendricks, Tex Phelps, Ed Carey, Gertrude Chorre, George Hazel, The Range Riders (Hugh Farr, Johnny Luther, Chuck Baldra, Al Haskell, Jack Jones). A cowboy known as the Phantom Rider discovers a murdered man and attempts to find his killer. Well made low budget adventure; Lane Chandler's final series starring film.\n\n_**Outlaw Territory**_ see _**Hannah Lee**_\n\n**2957** _ **Outlaw Trail**_ **** Monogram, 1944. 54 min. D: Robert Tansey. SC: Alvin J. Neitz (Alan James). With Hoot Gibson, Bob Steele, Chief Thundercloud, Jennifer Holt, Cy Kendall, Rocky Camron, George Eldredge, Charles King, Hal Price, John Bridges, Bud Osborne, Jim Thorpe, Warner Richmond, Frank Ellis, Al Ferguson, Tex Palmer, Charles Murray, Jr., Lee Roberts, Fred Hoose, Evelyn Selbie, Rose Plummer, Bert Dillard, Artie Ortego, Herman Hack, Denver Dixon, Lynton Brent, Roy Bucko. When a cattle buyer disappears on a visit to a town run by a banker, The Trail Blazers are sent to investigate. Good, fast paced penultimate entry in the popular \"Trail Blazers\" series.\n\n**2958** _ **Outlaw Trail:**_ _**The Treasure of Butch Cassidy**_ **** Allumination Filmworks, 2006. 53 min. Color. D: Ryan Little. SC: David Piller. With Brian Wimmer, Michael Van Wagenen, Brock Richards, James Karen, Ryan Kelley, James Hardy, Arielle Kebbel, Brent Weber, Rick Mac, Shaunna Thompson, David Piller, Dan Byrd, Brian Peck, Ron Melendez, Ian Lonsdale, Steve Anderson, Bruce McGill, James Gammon, Scott Wilkinson, Steven A. Lee, Chris Kendrick, Nancy Everhard, Andrew Roach, Noah Sunday, Matthew Brown. Four young people, one of them the great nephew of Butch Cassidy, hunt for the outlaw's treasure only to be pursued by bad men who are also after it. Award winning, action filled juvenile oriented short feature.\n\n**2959** _ **Outlaw Treasure**_ **** American Releasing Corporation, 1955. 67 min. D: Oliver Drake. SC: John Carpenter. With John (Carpenter) Forbes, Adele Jergens, Glenn Langan, Michael Whalen, Harry Lauter, Frank Jenks, Hal Baylor, Frank \"Red\" Carpenter, Bob Hinkle, Whitey Hughes. When gold shipments disappear an Army troubleshooter tries to get to the bottom of the problem. Sub-par production from producer-writer-star John Carpenter.\n\n**2960** _ **Outlaw Women**_ **** Howco International, 1952. 76 min. Color. D: Sam Newfield and Ron Ormond. SC: Orville Hampton. With Marie Windsor, Richard Rober, Allan Nixon, Carla Balenda, Jackie Coogan, Maria Hart, Jacqueline Fontaine, Billy House, Richard Avonde, Leonard Penn, Lyle Talbot, Brad Johnson, Tom Tyler, Angela Stevens, Ted Cooper, Riley Hill, Kermit Maynard, Bud Osborne, Lou Lubin, Cliff Taylor, Connie Cezona. A town is controlled by Iron Mae McLeod and her gang of female hellions but they are opposed by a gambler who is appointed U.S. marshal. Just as bad as it sounds and Cinecolor does not help; sad to see Tom Tyler gunned down early in the proceedings.\n\n**2961** _ **Outlawed Guns**_ **** Universal, 1935. 62 min. D: Ray Taylor. SC: Jack Neville. With Buck Jones, Ruth Channing, Frank McGlynn, Roy D'Arcy, Joseph Girard, Pat J. O'Brien, Joan Gale, Lee Shumway, Charles King, Jack Rockwell, Monte Montague, Bob Walker, Carl Stockdale, Cliff Lyons, Jack Montgomery. When his younger brother becomes involved with outlaws, a cowboy plans to stop the gang and save his sibling. Very picturesque Buck Jones oater enhanced by a pleasing story.\n\n**2962** _ **Outlaws**_ **** CBS-TV, 1986. 104 min. Color. D: Peter Werner. SC: Nicholas Corea. With Rod Taylor, William Lucking, Richard Roundtree, Charles Napier, Patrick Houser, Christina Belford, Lewis Van Bergen, Wendy Girand, Grand L. Bush, Avery Schreiber, Ron Josephs, George American Horse. A quartet of old West cowboys are propelled into modern times where they must contend with new technology and a variety of criminals. Fairly intriguing pilot for the series of the same title that ran on CBS-TV in 1987.\n\n**2963** _ **The Outlaw's Daughter**_ **** 20th Century\u2013Fox, 1954. 75 min. D: Wesley Barry. SC: Sam Roeca. With Jim Davis, Bill Williams, Kelly Ryan, George Cleveland, Elisha Cook, Jr., Guinn Williams, Sara Haden, Nelson Leigh, George Barrows, Zon Murray, Dick (Richard) Powers, Regina Gleason, Sam Flint, Paul Stader, Danny Fisher, Eugene Anderson, Jr. A young woman is implicated in a stagecoach holdup when the robbers leave a trail leading to her grandfather's ranch, because the old man was once a famous outlaw. Fair program feature done on a modest budget.\n\n_**Outlaw's Highway**_ see _**Fighting Fury**_\n\n**2964** _ **The Outlaws Is Coming!**_ **** Columbia, 1965. 89 min. D: Norman Maurer. SC: Elwood Ullman. With The Three Stooges (Moe Howard, Larry Fine, Joe Da Rita), Adam West, Nancy Kovack, Mort Mills, Don Lamond, Rex Holman, Emil Sitka, Henry Gibson, Murray Alper, Tiny Brauer, Joe Bolton, Hal Fryar (Harlow Hickenlooper), Johnny Ginger, Wayne Mack, Bruce Sedley, Paul Shannon, Sally Starr. Three newsroom zanies accompany a reporter West to do a story on the slaughter of buffalo and they become involved with a legion of gunslingers. Very funny Three Stooges feature that will appeal to both adults and kiddies.\n\n**2965** _ **Outlaws of Boulder Pass**_ **** Producers Releasing Corporation, 1942. 61 min. D: Sam Newfield. SC: Steve Braxton (Sam Robins). With George Houston, Al St. John, Smoky (Dennis) Moore, Marjorie Manners, Charles King, I. Stanford Jolley, Karl Hackett, Ted Adams, Kenne Duncan, Frank Ellis, Steve Clark, Jimmy Aubrey, Budd Buster, Milburn Morante, George Morrell, Tex Palmer, Ray Henderson, Charley Murray, Jr., Bert Dillard, Art Dillard, Art Fowler. An outlaw gang is charging illegal tolls for cattle drives and the Lone Rider and his pals try to stop them. Passable entry in the \"Lone Rider\" series. TV title: _**The Lone Rider and the Outlaws of Boulder Pass**_.\n\n**2966** _ **Outlaws of Pine Ridge**_ **** Republic, 1942. 56 min. D: William Witney. SC: Norman S. Hall. With Don \"Red\" Barry, Lynn Merrick, Noah Beery, Emmett Lynn, Clayton Moore, Donald Kirke, Forrest Taylor, Stanley Price, Francis Ford, Wheaton Chambers, George J. Lewis, Roy Brent, Kenneth Terrell, Al Taylor, Tex Terry, Jack O'Shea, Cactus Mack, Tom Steele, Horace B. Carpenter, Duke Green. When he thwarts a holdup, a gambler finds himself a hero and he ends up bringing in an outlaw gang. William Witney's first directorial effort in Don Barry's Republic series is a fast moving and exciting one.\n\n**2967** _ **Outlaws of Santa Fe**_ **** Republic, 1944. 56 min. D: Howard Bretherton. SC: Norman S. Hall. With Don \"Red\" Barry, Helen Talbot, Wally Vernon, Twinkle Watts, Charles Morton, Herbert Heyes, Bud Geary, LeRoy Mason, Kenne Duncan, Nolan Leary, Walter Soderling, Edmund Cobb, Frank McCarroll, Robert Kortman, Emmett Lynn, Ernie Adams, Jack Kirk, Pierce Lyden, Forrest Taylor, Bob Burns, Jack O'Shea, Fred Graham. Finding out he is the son of a murdered lawman, and not the offspring of an outlaw, a bandit agrees to take the job of sheriff to stop a crook and his gang that control Santa Fe. Pretty good action entry for Don Barry, the final outing in his Republic series.\n\n**2968** _ **Outlaws of Sonora**_ **** Republic, 1938. 56 min. D: George Sherman. SC: Betty Bur bridge and Edmund Kelso. With Robert Livingston, Ray Corrigan, Max Terhune, Jack Mulhall, Jean Joyce, Sterlita Peluffo, Otis Harlan, Tom London, Gloria Rich, Ralph Peters, George Chesebro, Frank LaRue, Jack Ingram, Merrill McCormick, Curley Dresden, Jim Corey, George Cleveland, Earl Dwire, Jack Kirk, Edwin Mordant, Bob Reeves, Tommy Coats, Art Dillard, Horace B. Carpenter, George (Montgomery) Letz, Bob Burns, Fred Burns, Bob Card, Herman Willingham, Jack O'Shea, Blackjack Ward. When an outlaw impersonates Stony Brooke, who is blamed for a series of robberies, he and his pals Tucson and Lullaby try to capture the culprit. Another fine outing in \"The Three Mesquiteers\" series.\n\n**2969** _ **Outlaws of Stampede Pass**_ **** Monogram, 1943. 58 min. D: Wallace Fox. SC: Jess Bowers (Adele Buffington). With Johnny Mack Brown, Raymond Hatton, Harry Woods, Ellen Hall, Edmund Cobb, Charles King, Milburn Morante, Sam Flint, Mauritz Hugo, Art Mix, Cactus Mack, Artie Ortego, Hal Price, Dan White, Tex Cooper, Bud Wolfe, Jon Dawson, Herman Hack, Eddie Burns, Kansas Moehring, Denver Dixon, Curley Dresden, Rube Dalroy. Two lawmen are helped by a blacksmith's daughter as they try to find out who is behind the rustling of a rancher's cattle. Standardized but entertaining \"Nevada Jack McKenzie\" film from a story by Johnston McCulley, the creator of \"Zorro.\"\n\n**2970** _ **Outlaws of Texas**_ **** Monogram, 1950. 56 min. D: Thomas Carr. SC: Daniel Ullman. With Whip Wilson, Andy Clyde, Phyllis Coates, Terry Frost, Tommy Farrell, Zon Murray, George DeNormand, Stanley Price, Steve Carr, Tom Tyler, Rex Lease, Sam Flint, Clarke Stevens, Ray Jones, George Sowards. Two U.S. marshals work undercover to capture a gang of bank robbers. Good Whip Wilson vehicle.\n\n**2971** _ **Outlaws of the Cherokee Trail**_ **** Republic, 1941. 56 min. D: Lester Orlebeck. SC: Albert DeMond. With Bob Steele, Tom Tyler, Rufe Davis, Lois Collier, Rex Lease, Tom Chatterton, Roy Barcroft, Joel Friedkin, Philip Trent, Peggy Lynn, Bud Osborne, Chief Yowlachie, John James, Lee Shumway, Karl Hackett, Billy Curtis, Griff Barnett, Bud Geary, Al Taylor, Henry Wills, Sarah Padden, Iron Eyes Cody, Cactus Mack, Chuck Morrison, Eddie Dean, Lloyd Ingraham, Wally West, Ernest Sarracino, Ethyl May Halls. When the daughter of a ranger captain is kidnapped, The Three Mesquiteers attempt to rescue her. Precise action entry in the popular series.\n\n**2972** _ **Outlaws of the Desert**_ **** Paramount, 1941. 66 min. D: Howard Bretherton. SC: J. Benton Cheney and Bernard McConville. With William Boyd, Andy Clyde, Brad King, Duncan Renaldo, Forrest Stanley, Jean Phillips, Nina Guilbert, Luli Deste, Alberto Morin, George J. Lewis, Jean Del Val, George Woolsey, Jamiel Hasson, Mickey Eissa, Ted Wells, Charles Murphy, Bill Nestell. The Bar 20 pals go to Arabia to buy horses for a rancher and after he is kidnapped they become involved in desert warfare. Nice photography helps this meandering and only fair \"Hopalong Cassidy\" adventure.\n\n**2973** _ **Outlaws of the Panhandle**_ **** Columbia, 1941. 60 min. D: Sam Nelson. SC: Paul Franklin. With Charles Starrett, Frances Robinson, The Sons of the Pioneers (Bob Nolan, Tim Spencer, Lloyd Perryman, Pat Brady, Hugh Farr, Karl Farr), Richard Fiske, Ray Teal, Lee Prather, Bud Osborne, Steve Clark, Eddie Laughton, Norman Willis, Blackie Whiteford, Stanley Brown, Jack Low. A cowboy helps cattlemen trying to construct a railroad spur but the project is hampered by a gambler who robs their gold shipments. Good Charles Starrett vehicle.\n\n**2974** _ **Outlaws of the Plains**_ **** Producers Releasing Corporation, 1946. 58 min. D: Sam Newfield. SC: Elmer Clifton. With Buster Crabbe, Al St. John, Patti McCarthy, Charles King (Jr.), Karl Hackett, Jack O'Shea, Bud Osborne, Roy Brent, Slim Whitaker, John Cason, Budd Buster, Jimmy Aubrey, Lane Bradford, Al Ferguson, Frank Ellis, George Morrell, Ray Henderson. Crooks convince Fuzzy that worthless land contains gold so he persuades others to join him in purchasing it with Billy Carson trying to stop them. Standard, and last, entry in the \"Billy Carson\" series.\n\n**2975** _ **Outlaws of the Prairie**_ **** Columbia, 1937. 59 min. D: Sam Nelson. SC: Ed Earl Repp. With Charles Starrett, Donald Grayson, Iris Meredith, The Sons of the Pioneers (Bob Nolan, Lloyd Perryman, Pat Brady, Hugh Farr, Karl Farr), Ed LeSaint, Hank Bell, Dick Curtis, Norman Willis, Edmund Cobb, Art Mix, Steve Clark, Earle Hodgins, Richard Alexander, Frank Shannon, Fred Burns, Jack Rockwell, Jack Kirk, George Chesebro, Frank Ellis, Charles LeMoyne, Frank McCarroll, Curley Dresden, Vernon Dent, George Morrell, Ray Jones, Jim Corey, Blackie Whiteford, Lee Shumway, Bob Burns, Charles Brinley, Delmar Watson, Lambert Rogers, Joe Yrigoyen, Chuck Baldra, Art Dillard, Bert Dillard, Jack Shannon, Frank Austin, Buck Moulton, E.L. Dale, Buel Bryant. Two Texas Rangers are sent to a town to investigate a series of stagecoach holdups while one of them is also after the man who murdered his father and branded him when he was a child. Pretty good Charles Starrett feature with a grand supporting cast.\n\n**2976** _ **Outlaws of the Range**_ **** Spectrum, 1936. 60 min. D: Al Herman. SC: Zara Tazil. With Bill Cody, Catherine Cotter, Bill Cody, Jr., William McCall, Gordon Griffith, Dick Strong, Wally West, Hank Bell, Buck Morgan, Curley Baldwin. A drifting cowboy, who saves a rancher's daughter from being dragged by a horse, is blamed when the man is murdered although his foreman is in cahoots with another cattleman who wants the dead man's land for its oil deposits. Complicated but formula Bill Cody affair; poorly edited.\n\n**2977** _ **Outlaws of the Rio Grande**_ **** Producers Releasing Corporation, 1941. 63 min. D: Peter Stewart (Sam Newfield). SC: George H. Plympton. With Tim McCoy, Virginia Carpenter, Charles King, Ralph Peters, Karl Hackett, Rex Lease, Philip (Felipe) Turich, Frank Ellis, Kenne Duncan, Thornton Edwards, Joe Dominguez, George Chesebro, Sherry Tansey. A gang of counterfeiters forces an engraver to work for them as a U.S. marshal tries to break up the operation. Well made, nicely paced Tim McCoy film.\n\n**2978** _ **Outlaws of the Rockies**_ **** Columbia, 1945. 55 min. D: Ray Nazarro. SC: J. Benton Cheney. With Charles Starrett, Tex Harding, Dub Taylor, Carole Mathews, Spade Cooley, Carolina Cotton, Philip Van Zandt, I. Stanford Jolley, George Chesebro, Steve Clark, Jack Rockwell, Frank LaRue, James T. \"Bud\" Nelson, Kermit Maynard, Ted Mapes, Frank O'Connor, Tex Williams, Deuce Spriggins, Frank Lanning, Nolan Leary, John Tyrrell, Victor Travers, Horace B. Carpenter, Herman Hack, Roy Bucko. The Durango Kid comes to the side of two lawmen accused of helping an outlaw gang who are forced out of town only to be opposed by the crooks. Fast moving but somewhat hard to follow \"Durango Kid\" episode. British title: _**A Roving Rogue**_.\n\n**2979** _ **Outlaws' Paradise**_ **** Victory, 1938. 62 min. D: Sam Newfield. SC: Basil Dickey. With Tim McCoy, Joan Barclay, Ben Corbett, Ted Adams, Forrest Taylor, Bob Terry, Don Gallaher, Dave O'Brien, Jack Mulhall, Lloyd Whitlock, Ed Cassidy, Wally West, Carl Mathews, Jack C. Smith, George Morrell, Jack \"Tiny\" Lipson, Frank Wayne. Federal investigator Lightning Bill Carson, who closely resembles an imprisoned outlaw, decides to take on the identity of the bad man to infiltrate his gang. Tim McCoy's handling of dual roles is the most interesting aspect of this low budget Sam Katzman production.\n\n**2980** _ **Outlaw's Son**_ **** Allied Artists, 1954. 54 min. D: Frank McDonald. SC: Maurice Tombragel. With Guy Madison, Andy Devine, Ralph Reed, Anne Kimball, Steve Darrell, Dan White, Sally Fraser, Bobby Hyatt, Pierce Lyden, Frank Fenton, Fred Kelsey, Guy Wilkerson, Wes Hudman. Wild Bill Hickok and Jingles deal with a robber who would rather see his son dead than leading a life of crime and they also try to find a prospector who has a stolen map. Okay theatrical compilation feature made up of the \"Outlaw's Son\" and \"Savvy, the Smart Little Dog\" episodes of \"The Adventures of Wild Bill Hickok\" (1951\u201358).\n\n**2981** _ **Outlaw's Son**_ **** United Artists, 1957. 88 min. D: Lesley Selander. SC: Richard Alan Simmons. With Dane Clark, Ben Cooper, Lori Nelson, Ellen Drew, Charles Watts, Cecile Rogers, Joseph \"Bucko\" Stafford, Eddie Foy III, John Pickard, Robert Knapp, Guy Preston, George Pembroke, Jeff Daley, James Parnell. A young man helps his outlaw father, who deserted him years before, when he is falsely accused of committing a robbery. Okay melodrama, well acted by its lead players.\n\n**2982** _ **Outlaws:**_ _**The Legend of O.B. Taggart**_ **** Hannover House, 1994. 98 min. Color. D: Rupert Hitzig. SC: Mickey Rooney. With Mickey Rooney, Ernest Borgnine, Ben Johnson, Ned Beatty, Randy Travis, Larry Gatlin, Gloria De Haven, Pamela Guest, Billy Barty, Christopher Aber, Cliff Gravel, Brandon Maggart, Rob Word, James Pollard, Willie Rack. A robber returns home from prison to find his family torn apart from trying to find the loot he hid. Star Mickey Rooney scripted this lumbering affair, a sad waste of a good cast.\n\n**2983** _ **Outpost of the Mounties**_ **** Columbia, 1939. 63 min. D: C.C. Coleman. SC: Charles Francis Royal. With Charles Starrett, Iris Meredith, The Sons of the Pioneers (Bob Nolan, Tim Spencer, Lloyd Perryman, Pat Brady, Hugh Farr, Karl Farr), Stanley Brown, Kenneth MacDonald, Edmund Cobb, Lane Chandler, Dick Curtis, Alberto Morin, Hal Taliaferro, Pat O'Hara, Harry Cording, Roger Gray. A Royal Canadian Mounted policeman is forced to arrest his girl's brother for the murder of a trading company co-owner but doubts his guilt and tries to find the real killer. Another adventure in the north woods with Charles Starrett and a fairly exciting one.\n\n**2984** _ **The Outrage**_ **** Metro-Goldwyn-Mayer, 1964. 97 min. D: Martin Ritt. SC: Michael Kanin. With Paul Newman, Laurence Harvey, Claire Bloom, Edward G. Robinson, William Shatner, Howard Da Silva, Albert Salmi, Thomas Chalmers, Paul Fix. Three stories are told about the incident of an outlaw capturing a man and his wife, with the woman raped and her spouse dying. Surprisingly sturdy Western adaptation of the Japanese film _**Rashomon**_ (1951).\n\n**2985** _ **Outride the Devil:**_ _**A Morning with Doc Holliday**_ **** Greeve HD Productions, 2007. 87 min. Color. D: Gayle Hussey. SC: Kit Hussey. With Kit Hussey. A live performance with Kit Hussey as Doc Holliday, relating the story of the famed Western icon. Good entertainment.\n\n**2986** _ **The Outriders**_ **** Metro-Goldwyn-Mayer, 1950. 93 min. D: Roy Rowland. SC: Irving Ravetch. With Joel McCrea, Arlene Dahl, Barry Sullivan, Claude Jarman, Jr., James Whitmore, Ramon Novarro, Jeff Corey, Ted De Corsia, Martin Garralaga, Dick Curtis, Gregg Barton, Frank Richards, Russell Simpson, William \"Bill\" Phillips, Dorothy Adams, Robert B. Williams, Alex Montoya, Joe Dominguez, Billy Dix, Gene Coogan, Gil Herman, Warren MacGregor, Charles Rivero, George Tyne, Buck Bucko. During the Civil War three Confederate spies join a wagon train in Santa Fe with plans to hijack its one million dollars in gold for the Southern cause. Glossy action feature.\n\n**2987** _ **The Outsider**_ **** Showtime, 2002. 119 min. Color. D: Randa Haines. SC: Jenny Wingfield. With Tim Daly, Naomi Watts, Keith Carradine, David Carradine, Thomas Curtis, Brett Tucker, John Noble, Grant Piro, Peter McCauley, Jason Clarke, Todd Leigh, Aaron Cash, Simon Watts, Eamon Farren, Kim Knuckey, Mick Roughan, Kathryn Smith, Barry Thurlow, David White, Geoff Lyall, Capkin Van Alphen, Imogen Annesley, Leonie Plummer, David Pitstock, Stuart Maybury, Susan Dowideit, Alison Goode, Demis Lyall-Wilson. While caring for a wounded gunman, a religious young widow falls in love with him. Too long, but worth watching, romantic TV Western.\n\n**2988** _ **Over the Border**_ **** Monogram, 1950. 58 min. D: Wallace Fox. SC: J. Benton Cheney. With Johnny Mack Brown, Wendy Waldron, Myron Healey, Marshall Reed, Mike Ragan, House Peters, Jr., Pierre Watkin, Hank Bell, George DeNormand, Milburn Morante, Frank Jaquet, Buck Bailey, George Sowards, Carol Henry, Frank McCarroll, Bud Osborne, Artie Ortego, Herman Hack, Ray Jones, Bob Woodward. A Well Fargo guard uncovers a plot by a businessman to smuggle silver into the country from Mexico and sell it for a profit. Okay Johnny Mack Brown entry enhanced by Myron Healey's villainy.\n\n**2989** _ **The Over-the-Hill Gang**_ **** ABC-TV\/Paramount, 1969. 74 min. Color. D: Jean Yarbrough. SC: Jameson Brewer. With Walter Brennan, Edgar Buchanan, Andy Devine, Jack Elam, Gypsy Rose Lee, Rick Nelson, Kris Nelson, Pat O'Brien, Chill Wills, Edward Andrews, William Smith, Dennis Cross, Rex Holman, Burt Mustin, Almira Sessions. Three former Texas Rangers try to help a buddy and end up defending a town against a dishonest lawyer and his outlaw gang. Very amusing and well done TV Western comedy, expertly directed by Jean Yarbrough and nicely acted by its veteran cast.\n\n**2990** _ **The Over-the-Hill Gang Rides Again**_ **** ABC-TV\/Paramount, 1970. 74 min. Color. D: George McGowan. SC: Richard Carr. With Walter Brennan, Fred Astaire, Edgar Buchanan, Andy Devine, Chill Wills, Lana Wood, Paul Richards, Parley Baer, Walter Burke, Jonathan Hole, Lillian Bronson, Burt Mustin, Pepper Martin, Don Wilbanks. After three ex\u2013Texas Rangers help an alcoholic friend reform, the four team to clean up the town of Waco. Tired follow-up to the delightful _**The Over-the-Hill Gang**_ (q.v.).\n\n**Jack Elam and Edgar Buchanan in** _**The Over-the-Hill Gang**_ **(ABC-TV\/Paramount, 1969).**\n\n**  \n**\n\n**2991** _ **Over the Santa Fe Trail**_ **** Columbia, 1947. 63 min. D: Ray Nazarro. SC: Louise Rousseau. With The Hoosier Hot Shots (Paul \"Hezzie\" Trietsch, Charles \"Gabe\" Ward, Ken Trietsch, Gil Taylor), Ken Curtis, Jennifer Holt, Guy Kibbee, Guinn Williams, Noel Neill, Holmes Herbert, George Chesebro, Jim Diehl, Frank LaRue, Steve Clark, Julian Rivero, Nolan Leary, Bud Osborne, The DeCastro Sisters (Babette, Cherie, Peggy), Art West and His Sunset Riders, George Chesebro, Julian Rivero, Jock Mahoney, Syd Saylor, John Cason, Jim Diehl, Herman Hack. An refined outlaw uses a traveling medicine show as a front for his gang to rob the towns where they appear. Tame oater full of musical numbers and comedy but not much action; for fans of The Hoosier Hot Shots and The DeCastro Sisters. Smiley Burnette can be spotted as a posse member in stock footage.\n\n**2992** _ **The Overland Express**_ **** Columbia, 1938. 55 min. D: Drew Eberson. SC: Monroe Shaff. With Buck Jones, Marjorie Reynolds, Carlyle Moore, Maston Williams, William Arnold, Lew Kelly, Bud Osborne, Ben Taggart, Ben Corbett, Gene Alsace, Blackie Whiteford, Bob Woodward. At the outbreak of the Civil War, a man sets up the cross country pony express but the operation is opposed by renegades planning to start an Indian uprising. Exciting Buck Jones vehicle.\n\n**2993** _ **Overland Mail**_ **** Monogram, 1939. 51 min. D: Robert Hill. SC: Robert Emmett (Tansey). With Jack Randall, Vince Barnett, Jean Joyce, Tristram Coffin, Glenn Strange, Dennis Moore, Merrill McCormick, Joe Garcia, Sherry Tansey, Hal Price, Maxine Leslie, Harry Semels, George Cleveland, Iron Eyes Cody, George Morrell, Hank Bell. A mail rider and a federal agent team to capture a counterfeiting gang responsible for the death of an Indian in hopes of preventing tribal warfare. Fast moving Jack Randall short feature.\n\n**2994** _ **Overland Mail**_ **** Universal, 1942. 15 Chapters. D: Ford Beebe and John Rawlins. SC: Paul Huston. With Lon Chaney, Helen Parrish, Noah Beery, Noah Beery, Jr., Don Terry, Roy Harris, Robert Barron, Jack Clifford, Tom Chatterton, Harry Cording, Charles Stevens, Carleton Young, Bob Baker, Ethan Laidlaw, William Gould, Ben Taggart, Frank Pershing, Tom Steele, Forrest Taylor, Chief Thundercloud, Jack Rockwell, Bill Moss, Marguerite De La Motte, Ruth Ricksby, Charles Phipps, Eddie Polo, Henry Hall, Edmund Cobb, Curley Dresden, Frosty Royce, George Sherwood, Gene O'Donnell, Jack Shannon. A frontiersman and his pal are assigned to find out who is sabotaging mail shipments in a remote territory and learn a renegade is dressing his gang as Indians when making the raids. Lon Chaney is an appealingly athletic hero in this fast paced chapter play.\n\n**2995** _ **Overland Mail Robbery**_ **** Republic, 1943. 56 min. D: John English. SC: Bob Williams and Robert Yost. With Wild Bill Elliott, George \"Gabby\" Hayes, Anne Jeffreys, Weldon Heyburn, Nancy Gay, Kirk Alyn, Roy Barcroft, Bud Geary, Tom London, Alice Fleming, Jack Kirk, Kenne Duncan, Jack Rockwell, Frank McCarroll, Jack O'Shea, LeRoy Mason, Hank Bell, Cactus Mack, Ray Jones, Tom Steele, Frank Ellis, Maxine Doyle, Peter Michael, Diane Henry, Al Taylor. A man from Boston travels West to claim an inheritance and is helped by Wild Bill Elliott when a crook tries to stop him. Standard but fast paced and action laden Saturday matinee fare.\n\n**2996** _ **Overland Pacific**_ **** United Artists, 1954. 73 min. Color. D: Fred F. Sears. SC: J. Robert Bren, Gladys Atwater and Martin Goldsmith. With Jack (Jock) Mahoney, Peggie Castle, Adele Jergens, William Bishop, Walter Sande, Chubby Johnson, Pat Hogan, Chris Alcaide, Phil Chambers, George Eldredge, Dick Rich, House Peters, Jr, Fred Graham. A railroad investigator works incognito as he looks into reports of Indian raids on trains. Low budget but effective action drama.\n\n**2997** _ **Overland Riders**_ **** Producers Releasing Corporation, 1946. 54 min. D: Sam Newfield. SC: Ellen Coyle. With Buster Crabbe, Al St. John, Patti McCarty, Slim Whitaker, Bud Osborne, Jack O'Shea, Frank Ellis, Al Ferguson, John Cason, George Chesebro, Lane Bradford, Wally West, Jimmy Aubrey. Billy Carson and Fuzzy Q. Jones investigate a stagecoach robbery and find the money taken was to be used to pay the mortgage on property where a railroad line will intersect. Typically cheap but fast moving penultimate \"Billy Carson\" entry.\n\n**2998** _ **Overland Stage Raiders**_ **** Republic, 1938. 55 min. D: George Sherman. SC: Luci Ward. With John Wayne, Ray Corrigan, Max Terhune, Louise Brooks, Anthony Marsh, Ralph Bowman (John Archer), Gordon Hart, Roy James, Olin Francis, Fern Emmett, Henry Otho, George Sherwood, Archie Hall, Frank LaRue, Yakima Canutt, Milton Kibbee, Jack Kirk, Slim Whitaker, Bud Osborne, Dirk Thane, Bud McClure, John Beach, Curley Dresden, George Plues, Edwin Gaffney, Tommy Coats, Burr Caruth, Chuck Baldra, Duke R. Lee, Fred Burns, Charles Brinley, George Morrell, Bill Wolfe. Three cowpokes invest an operator flying ore out of a remote gold mine but a partner in the enterprise is blackmailed by the corrupt owner of a stage line. Modern-day \"Three Mesquiteers\" outing moves along at a fast clip but is best remembered as silent film siren Louise Brooks' final movie.\n\n**2999** _ **Overland Stagecoach**_ **** Producers Releasing Corporation, 1942. 61 min. D: Sam Newfield. SC: Fred Myton and Steve Braxton (Sam Robins). With Robert Livingston, Al St. John, Smoky (Dennis) Moore, Julie Duncan, Glenn Strange, Charles King, Art Mix, Budd Buster, Ted Adams, Julian Rivero, John Elliott, Milburn Morante, Tex Cooper, George Morrell, Kenne Duncan, Jimmy Aubrey, Chick Hannon, Art Dillard, Bert Dillard, Jack Casey, Lew Morphy, Rose Plummer, Herman Hack, Roy Brent, Dan White, Jack Evans, Augie Gomez, Barney Beasley, Frank McCarroll, Jack Tornek. The Lone Rider helps his friend Fuzzy, a driver who works for a stage line jeopardized by a railroad and the machinations of his late boss' partner, who wants the business for himself. Robert Livingston's first, and Dennis Moore's last, entry in \"The Lone Rider\" series is action filled enough to satisfy fans.\n\n**3000** _ **Overland Telegraph**_ **** RKO Radio, 1951. 60 min. D: Lesley Selander. SC: Adele Buffington. With Tim Holt, Richard Martin, Gail Davis, George Nader, Mari Blanchard, Hugh Beaumont, Robert Wilke, Robert Bray, Fred Graham, Cliff Clark, Russell Hicks, Jack O'Shea. A man is falsely blamed of the murder of a worker installing a telegraph line and a cowboy tries to prove his innocence. Typical entry in Tim Holt's fine RKO series.\n\n**3001** _ **Overland to Deadwood**_ **** Columbia, 1942. 59 min. D: William Berke. SC: Paul Franklin. With Charles Starrett, Russell Hayden, Cliff Edwards, Leslie Brooks, Norman Willis, Francis Walker, Lynton Brent, Matt Willis, June Pickrell, Gordon DeMain, Art Mix, Herman Hack, Bud Osborne, Bud Geary. Two cowboys help a young woman whose hauling operation is being sabotaged by a rival wanting to obtain an important railroad franchise. The final teaming of Charles Starrett and Russell Hayden is on the mediocre side although Norman Willis is very good as the villain.\n\n_**Overland Trail**_ see _**Trail Riders**_ (1942)\n\n**3002** _ **Overland Trails**_ **** Monogram, 1948. 60 min. D: Lambert Hillyer. SC: Jess Bowers (Adele Buffington). With Johnny Mack Brown, Raymond Hatton, Virginia Belmont, Steve Darrell, Bill Kennedy, Holly Bane, Ted Adams, Boyd Stockman, Virginia Carroll, Carl Mathews, Milburn Morante, Bob Woodward, Tom London, Pierce Lyden, Roy Butler, Post Park, Marshall Reed, Artie Ortego, George Peters. A cowboy falls in love with the daughter of a man, who with his partner, grubstakes prospectors and then kills them to get their claims. Some interesting plot twists add life to this Johnny Mack Brown-Raymond Hatton vehicle.\n\n**3003** _ **Overland with Kit Carson**_ **** Columbia, 1939. 15 Chapters. D: Sam Nelson and Norman Deming. SC: Morgan Cox, Joseph Poland and Ned Dandy. With Bill Elliott, Iris Meredith, Richard Fiske, Bobby Clark, James Craig, Hal Taliaferro, Trevor Bardette, LeRoy Mason, Olin Francis, Francis Sayles, Kenneth MacDonald, Dick Curtis, Richard Botiller, Ernie Adams, Ben Campbell, Joe Garcia, Stanley Brown, Hank Bell, Art Mix, John Tyrrell, Lee Prather, Jack Rockwell, Ed LeSaint, Martin Garralaga, Iron Eyes Cody, Carl Stockdale, Robert Fiske, Eddie Foster, Irene Herndon, J.W. Cody, Del Lawrence. Frontier scout Kit Carson tries to find a mysterious outlaw called Pegleg and his Black Raiders who raid settlements west of the Mississippi River in an attempt to run off homesteaders to set up an empire. There is nothing special about this cliffhanger other than Bill Elliott holding it together in good fashion as he essays the title role.\n\n**3004** _ **The Overlanders**_ **** Universal-International\/J. Arthur Rank\/Associated British-Path\u00e9, 1946. 91 min. D-SC: Harry Watt. With Chips Rafferty, John Nugent Howard, Daphne Campbell, Jean Blue, Helen Grieve, John Fernside, Peter Pagan, Frank Ransome, Stan Tolhurst, Marshall Crosby, Clyde Combo, Henry Murdock. In 1942 Australia, cattlemen decide to drive their herds south across the continent to keep them from falling into the hands of possible Japanese invaders. Well done Australian featured based on a true story.\n\n**3005** _ **The Ox-Bow Incident**_ **** 20th Century\u2013Fox, 1943. 76 min. D: William A. Wellman. SC: Lamar Trotti. With Henry Fonda, Dana Andrews, Mary Beth Hughes, Anthony Quinn, William Eythe, Henry (Harry) Morgan, Jane Darwell, Francis Ford, Harry Davenport, Matt Briggs, Frank Conroy, Marc Lawrence, Victor Kilian, Chris-Pin Martin, Paul Hurst, Ted North, George Meeker, Almira Sessions, Margaret Hamilton, Dick Rich, Stanley Andrews, Billy Benedict, Rondo Hatton, Paul Burns, Leigh Whipper, George Lloyd, George Chandler, Hank Bell, Forrest Dillon, Willard Robertson, Tom London, George Plues. Two drifters get involved with a lynch mob wanting to hang three men accused of cattle rustling and murder. Classic adaptation of Walter Van Tilburg Clark's novel, a must see! Also done for TV as _**Lynch Mob**_ (q.v.).\n\n**3006** _ **Pack Train**_ **** Columbia, 1953. 57 min. D: George Archainbaud. SC: Norman S. Hall. With Gene Autry, Smiley Burnette, Gail Davis, Kenne Duncan, Sheila Ryan, Tom London, Harry Lauter, Melinda Plowman, Louise Lorimer, Frankie Marvin, Tex Terry, Kermit Maynard, Frank Ellis, Richard Alexander, Herman Hack. A cowboy tries to get supplies needed by settlers but crooks want to sell the commodities at inflated prices to miners at a gold strike. Fair Gene Autry vehicle.\n\n**3007** _ **Packin' It In**_ **** CBS-TV, 1983. 100 min. Color. D: Jud Taylor. SC: Patricia Jones and Donald Reiker. With Richard Benjamin, Paula Prentiss, Andrea Marcovicci, Tony Roberts, Molly Ringwald, David Hollander, Mari Gorman, Kenneth McMillan, Sam Whipple, Clinton Dean, Susan Ruttan, Laura Bruneau. A city family, fed up with urban life, join their former neighbors in the Oregon high country and find a truckers' strike is causing food hoarding. Pleasant TV movie along the lines of the theatrical \"Wilderness Family\" trilogy.\n\n**3008** _ **El Padre Pistolas**_ (Father Pistols) **** Producciones Sotomayer, 1961. D: Julian Soler. SC: Alfredo Varela, Jr. With Eulalio Gonzalez \"Piporro,\" Christiane Martel, Carlos Lopez Moctezuma, Oscar Pulido, Jaime Fernandez, Domingo Soler, Alejandro Reyna Garcia, Norma Angelica, Jose Chavez, Edmund Espino, Emilio Garibay, Carlos Leon, Arturo Castro, Manuel Alvarado. An outlaw takes on the guise of a priest to help rid a town of a tyrant and his gang. Pretty fair Mexican Western.\n\n**3009** _ **Paint Your Wagon**_ **** Paramount, 1969. 151 min. Color. D: Joshua Logan. SC: Alan Jay Lerner. With Clint Eastwood, Jean Seberg, Lee Marvin, Harve Presnell, Ray Walston, Tom Ligon, Ben(ny) Baker, Alan Baxter, Alan Dexter, William O'Connell, Paula Trueman, Robert Easton, Geoffrey Morgan, H.B. Haggerty, Terry Jenkins, Karl Bruck, John Mitchum, Sue Casey, Eddie Little Sky, Harvey Parry, H.W. Gim, William Mims, Roy Jenson, Pat Hawley, The Nitty Gritty Dirt Band. After a Mormon woman is bought by a gold miner she falls in love with his partner and decides to marry both men to the consternation of the locals. Alan Jay Lerner wrote and produced this overlong musical from his Broadway play and the end result is fair entertainment, although it should probably have been done years before with Nelson Eddy or Howard Keel.\n\n**3010** _ **Painted Angels**_ **** Lions Gate Films, 1998. 110 min. Color. D: Jon Sanders. SC: Jon Sanders and Anna Mottram. With Brenda Fricker, Kelly McGillis, Meret Becker, Bronagh Gallagher, Lisa Jakub, Anna Mottram, Kent Allen, Bruce McFee, Greg Lawson, Alan Bratt, Andrea Rodrigue, Robert Wu, Dwayne Brenna, Michael Burns, Joseph Griffin, Michelle Sereda, Keiran Semple, Jodi Sadowsky, Elyssa Dombowsky, Bob Clout. In a remote Western town a madam and her girls fight back when lawless elements try to take over. Somewhat ingratiating, stark drama also called _**The Wicked, Wicked West**_.\n\n**3011** _ **The Painted Desert**_ **** Path\u00e9, 1931. 75 min. D: Howard Higgin. SC: Howard Higgins and Tom Buckingham. With William Boyd, Helen Twelvetrees, William Farnum, J. Farrell MacDonald, Clark Gable, William Walling, Wade Boteler, William LeMaire, Richard Cramer, Jim Mason, Charles Sellon, Edward Hearn, James Donlan, Al St. John, George Burton. An infant boy is found in the desert by two drifters and abducted by one of them; he grows up as a thirty year feud between his adopted father and his rival continues. Okay early sound oater mainly known because of Clark Gable in a supporting role.\n\n**3012** _ **The Painted Desert**_ **** RKO Radio, 1938. 59 min. D: David Howard. SC: John Rathmell and Oliver Drake. With George O'Brien, Laraine (Day) Johnson, Ray Whitley, Fred Kohler, Max Wagner, Stanley Fields, Harry Cording, Lee Shumway, Lloyd Ingraham, Maude Allen, William V. Mong, Lew Kelly, Jim Mason, Jack O'Shea, Ray Jones. A man returns to his feuding home range only to find out a crook is trying to steal a tungsten mine. Good remake of the 1931 (q.v.) film as a George O'Brien vehicle deftly using stock footage from the earlier feature; Ray Whitley sings the title song.\n\n**3013** _ **Painted Hero**_ **** Astra Cinema, 1997. 105 min. Color. D: Terry Benedict. SC: Stan Berthoud and Terry Benedict. With Dwight Yoakam, Michelle Joyner, Kiersten Warren, Cindy Pickett, John Getz, Bo Hopkins, Walton Goggins, Terry McIlvain, Peter Fonda, Brent Anderson, Toby Metcalf, Bill Thurman, Brad Leland, Rick Herod, Richard Phillips, Brandon Lilly. Returning to his hometown of Waco, a circus clown has a one night stand with a woman who believes she is a vampire and when she is found dead the next day he is blamed and tries to prove his innocence. Offbeat but likable modern-day Western mystery.\n\n**3014** _ **The Painted Hills**_ **** Metro-Goldwyn-Mayer, 1951. 65 min. Color. D: Harold F. Kress. SC: True Boardman. With Lassie, Paul Kelly, Bruce Cowling, Gary Gray, Art Smith, Ann Doran, Chief Yowlachie, Andrea Virginia Lester, Brown Jug (Don Kay) Reynolds. In the California gold fields of the 1880s a boy and his Collie dog try to outwit a crook. Well made, pleasing family fare; also called _**Lassie's Adventures in the Gold Rush**_.\n\n**3015** _ **The Painted Stallion**_ **** Republic, 1937. 12 Chapters. D: William Witney, Alan James and Ray Taylor. SC: Barry Shipman and Winston Miller. With Ray Corrigan, Hoot Gibson, Julia Thayer (Jean Carmen), LeRoy Mason, Duncan Renaldo, Jack Perrin, Sammy McKim, Hal Taliaferro, Oscar and Elmer (Ed Platt and Lou Fulton), Yakima Canutt, Maston Williams, Duke Taylor, Loren Riebe, George DeNormand, Gordon DeMain, Charles King, Vinegar Roan, Lafe McKee, Frankie Marvin, Chief Big Tree, Pascale Perry, Henry Hall, Ed Peil, Sr., Horace B. Carpenter, Joe Yrigoyen, Monte Montague, Roy Bucko, Joe Dominguez, Jack Padjan, Al Haskell, Augie Gomez, Frank Leyva, Gregg Star Whitespear, Lee White, Paul Lopez, Don Orlando, Curley Dresden, Ralph Bucko, Leo Dupree, Al Haskell, Babe DeFreest. A crooked politician tries to stop a wagon train on its way to New Mexico Territory, hoping to sabotage a trade agreement between the U.S. and Mexico. Top notch Republic serial with beautiful Julia Thayer (Jean Carmen) as the mysterious rider; great fun.\n\n**3016** _ **The Painted Trail**_ **** Monogram, 1938. 50 min. D: Robert Hill. SC: Robert Emmett (Tansey). With Tom Keene, Eleanor Stewart, LeRoy Mason, Walter Long, James Eagles, Forrest Taylor, Harry Harvey, Ernie Adams, Bud Osborne, Glenn Strange, Frank Campeau, Robert Kortman, Richard Cramer, Tom London, Ed Cassidy, Jimmy Aubrey. Masquerading as an outlaw called The Pecos Kid, a federal agent infiltrates an outlaw gang to arrest them and stop their smuggling and rustling activities. Compact but sturdy action outing providing good entertainment.\n\n**3017** _ **Pair of Aces**_ **** CBS-TV, 1990. 100 min. Color. D: Aaron Lipstadt. SC: Bud Shrake and Gary Cartwright. With Willie Nelson, Kris Kristofferson, Rip Torn, Helen Shaver, Emily Warfield, Lash LaRue, Doris Hargrave, William Fair, Jeff Miller, Weasel Forshaw, John Swasey, Sonny Carl Davis, Jane Cameron, Erich Anderson, Michael Mariach, Steven Chester Prince, J. David Moeller, Bethlyn Gerard, W.T. Bryant, Turk Pipkin, Shannon Sedwick. A Texas Ranger teams with a veteran robber to protect the lawman's daughters from a serial killer with a penchant for cheerleaders. Mediocre modern-day TV oater mystery that spawned a sequel, _**Another Pair of Aces**_ (q.v.); Lash LaRue's last Western.\n\n**3018** _ **The Pal from Texas**_ **** Metropolitan, 1939. 55 min. D: Harry S. Webb. SC: Carl Krusada. With Bob Steele, Claire Rochelle, Jack Perrin, Josef Swickard, Ted Adams, Betty Mack, Carleton Young, Jack Ingram, Robert Walker, Reed Howes, Art Davis, Milburn Morante, Lew Porter, Bud McClure, Tex Palmer. When a crook tries to cheat the rightful owner out of his gold mine a cowboy comes to the rescue. Tawdry, low grade affair; Bob Steele deserved far better.\n\n**3019** _ **Pale Rider**_ **** Warner Bros., 1985. 115 min. Color. D: Clint Eastwood. SC: Michael Butler and Fritz Manes. With Clint Eastwood, Michael Moriarty, Carrie Snodgrass, Christopher Penn, John Russell, Richard Dysart, Sydney Penny, Richard Kiel, Doug McGrath, Charles Hallahan, Marvin J. McIntyre, Fran Ryan, Richard Hamilton, Graham Paul, Chuck LaFont, Jeffrey Weissman, Allen Keller, Tom Oglesby, Herman Poppe, Kathleen Wygle, Terrence Evans, Jim Hitson, Loren Adkins, Tom Friedkin, Billy Drago, Mike Adams, Clay Lilley, Larry Randles, Jerry Gatlin, Lloyd Nelson, Mike Munsey, Wayne Van Horn, Glenn Wright, Walt LaRue, Bob Herron, Kerrie Cullen. A drifter tries to help gold prospectors whose land is coveted by a grasping tycoon bringing in hired guns to help his cause. It has all been done before but this mid\u20131980s genre revival attempt is worth watching.\n\n**3020** _ **The Paleface**_ **** Paramount, 1948. 91 min. Color. D: Norman Z. McLeod. SC: Edmund Hartman and Frank Tashlin. With Bob Hope, Jane Russell, Robert Armstrong, Iris Adrian, Robby (Bobby) Watson, Jackie Searl, Joseph Vitale, Henry Brandon, Charles Trowbridge, Clem Bevans, Jeff York, Stanley Andrews, Wade Crosby, Chief Yowlachie, Iron Eyes Cody, John Maxwell, Tom Kennedy, Francis McDonald, Frank Hagney, Skelton Knaggs, Olin Howlin, George Chandler, Nestor Paiva, Earle Hodgins, Arthur Space, Edgar Dearing, Dorothy Granger, Charles Cooley, Eric Alden, Jody Gilbert, Al Hill, Harry Harvey, Hall Bartlett, Stanley Blystone, Robert Kortman, Oliver Blake, Lane Chandler, Syd Saylor, Paul E. Burns, Dick Elliott, Sharon McManus. Calamity Jane and a correspondence school dentist team to take on a notorious outlaw gang. Fun spoof of Westerns, followed by _**Son of Paleface**_ (q.v.) and remade as _**The Shakiest Gun in the West**_ (q.v.).\n\n**3021** _ **Palenque Sangriento**_ (Bloody Arena) **** Madera, 1980. 90 min. Color. D-SC: Fernando Gou. With Pedro Infante, Beatriz Adriana, Felipe Arriaga, Bruno Rey, Luciana, Leonor Llausas, Humberto Cabanas, Robert Guzman. Two violent cowboys vie for the love of the same woman. Brutal Mexican Western that deals with cockfighting.\n\n**3022** _ **Palm Springs**_ **** Paramount, 1936. 72 min. D: Aubrey Scotto. SC: Joseph Fields. With Frances Langford, Sir Guy Standing, David Niven, Smith Ballew, Ernest Cossart, E.E. Clive, Spring Byington, Sterling Holloway, Grady Sutton, Sarah Edwards, Ed Moose, Mary Jane Temple, June Horn, Ann Doran, Ella McKenzie, Fred \"Snowflake\" Toones, Frances Morris, David Worth, Annabelle and Marianne Brudie, Lee Phelps, Maidel Turner, Bert Gale, Cyril Ring. Although the daughter of a wealthy man is supposed to wed an English nobleman she falls in love with a cowboy. Musical comedy of interest because it gave Smith Ballew his first leading role in a Western, although a peripheral one at best.\n\n**3023** _ **The Palomino**_ **** Columbia, 1950. 73 min. Color. D: Ray Nazarro. SC: Tom Kilpatrick. With Jerome Courtland, Beverly Tyler, Joseph Calleia, Roy Roberts, Gordon Jones, Robert Osterloh, Trevor Bardette, Tom Trout, Harry Garcia, Juan Duval. After crooks steal a girl's prize horse so she will lose her ranch, a cattle buyer tries to help retrieve the steed. Okay juvenile drama helped somewhat by Technicolor.\n\n**3024** _ **Pals of the Golden West**_ **** Republic, 1951. 68 min. D: William Witney. SC: Robert DeMond and Eric Taylor. With Roy Rogers, Dale Evans, Estelita Rodriguez, Pinky Lee, Roy Barcroft, Anthony Caruso, Eduardo Jiminez, Kenneth Terrell, Emmett Vogan, Roy Rogers Riders, Maurice Jara. When diseased cattle are smuggled into the country, the Border Patrol assigns agent Roy Rogers the task of stopping the illegal operations. This competent affair was a good finale to Roy Rogers' Republic tenure.\n\n**3025** _ **Pals of the Pecos**_ **** Republic, 1941. 56 min. D: Lester Orlebeck. SC: Oliver Drake and Herbert Dalmas. With Robert Livingston, Bob Steele, Rufe Davis, June Johnson, Dennis Moore, Roy Barcroft, Pat O'Malley, Robert Frazer, John Holland, Tom London, Robert Winkler, George Chesebro, Chuck Morrison, Bud Osborne, Jack Kirk, Forrest Taylor, Frank Ellis, Eddie Dean, Neal Hart, William Nestell, Tom Smith. Rivalry develops between two stagecoach lines with the Three Mesquiteers helping the one who is being cheated by the other. Only average outing in the long running series.\n\n**3026** _ **Pals of the Range**_ **** Superior, 1935. 55 min. D: Elmer Clifton. SC: Elmer Clifton and George Merrick. With Rex Lease, Frances (Morris) Wright, Art Mix, George Chesebro, Yakima Canutt, Blackie Whiteford, Bill Patton, Artie Ortego, Milburn Morante, Tom Forman, Bud Osborne, Ben Corbett, George Morrell, Joey Ray. A rancher falsely accused of stealing cattle escapes from jail to find the real thieves. Low grade entry in Rex Lease's series for producer Louis Weiss.\n\n**3027** _ **Pals of the Saddle**_ **** Republic, 1938. 55 min. D: George Sherman. SC: Stanley Roberts and Betty Burbridge. With John Wayne, Ray Corrigan, Max Terhune, Doreen McKay, George Douglas, Josef (Joe) Forte, Frank Milan, Ted Adams, Harry Depp, Dave Weber, Don Orlando, Charles Knight, Jack Kirk, Monte Montague, Olin Francis, Curley Dresden, Art Dillard, Tex Palmer, Phil Kieffer, Bob Burns, Yakima Canutt, George Plues, Herman Nowlin, George (Montgomery) Letz, Otto Hoffman, Kenner G. Kemp. The Three Mesquiteers become involved with a female secret agent on the trail of foreign spies planning to smuggle a secret chemical out of the country. John Wayne's first outing in \"The Three Mesquiteers\" series is a fast paced, action packed affair. Remade as _**Song of the Range**_ (q.v.).\n\n**3028** _ **Pals of the Silver Sage**_ **** Monogram, 1940. 52 min. D: Al Herman. SC: George Martin. With Tex Ritter, Sugar Dawn, Arkansas Slim Andrews, Clarissa Curtis, Carleton Young, Glenn Strange, Joe McGuinn, Chester Gan, Warner Richmond, Gene Alsace, Chick Hannon, Harry Harvey, Sherry Tansey. Two cowhands help a little girl who will lose the ranch she inherited if her cattle cannot get to market on time. Pretty fair Tex Ritter vehicle; called _**Roundup Time**_ in England.\n\n**3029** _ **Panamint's Bad Man**_ **** Principal\/20th Century\u2013Fox, 1938. 60 min. D: Ray Taylor. SC: Luci Ward and Charles A. Powell. With Smith Ballew, Evelyn Daw, Noah Beery, Stanley Fields, Harry Woods, Pat J. O'Brien, Armand Wright, Charles King, Lew Meehan, Ed Cassidy, Robert Kortman, Horace B. Carpenter, Curley Dresden, Budd Buster, Frank Ellis, Bud McClure, Blackjack Ward, Ray Henderson, Charles Murphy. A marshal goes undercover to expose a gang of robbers and their boss. Smith Ballew's final starring Western is a good one; remade as _**Frontier Revenge**_ (q.v.), also directed by Ray Taylor.\n\n**3030** _ **Pancho Villa**_ **** Scotia International, 1972. 92 min. Color. D: Gene (Eugenio) Martin. SC: Julian Halvey. With Telly Savalas, Clint Walker, Anne Francis, Chuck Connors, Jose Maria Prada, Angel Del Pozo, Luis Davilla, Monica Randall, Antonio Casas, Alberto Dalbes, Berta Barri, Eduardo Calvo, Dan Van Husen, Norman Bailey, Tony Ross, Art Larkin, Gene Collins, Ralph Neville, Walter Coy. Mexican bandit leader Pancho Villa leads a revolution of peons against the government and invades the United States, attacking a border town. Fairly standard European made biopic.\n\n**3031** _ **Pancho Villa Returns**_ **** Hispano Continental Films, 1950. 96 min. D-SC: Miguel Contreras. With Leo Carrillo, Jeanette Combs, Esther Fernandez, Rodolfo Acosta, Rafael Alcayde, Jorge Trevino, Eduardo Gonzales Pliego. Pancho Villa leads the Mexican people in revolt against the government but must contend with personal problems including ordering a firing squad for a respected officer who broke an order. Mexican made drama has its main interest in the performance of Leo Carrillo in the title role; a Spanish language version, _**Pancho Villa Vuelve**_ (Pancho Villa Returns), was filmed with Pedro Armendariz as Villa.\n\n**3032** _ **Panhandle**_ **** Allied Artists, 1948. 85 min. D: Lesley Selander. SC: John C. Champion and Blake Edwards. With Rod Cameron, Cathy Downs, Reed Hadley, Anne Gwynne, Blake Edwards, Dick Crockett, Charles Judels, Alex Gerry, Francis McDonald, J. Farrell MacDonald, Henry Hall, Stanley Andrews, Jeff York, James Harrison, Charles La Torre, Frank Dae, Bud Osborne. An ex-gunman takes up his six-shooters to get revenge for the murder of his brother. Top notch action feature with fine work by hero Rod Cameron and villain Blake Edwards. Remade by director Lesley Selander as _**The Texican**_ (q.v.).\n\n**3033** _ **Panhandle .38**_ **** Scotia American, 1972. 93 min. Color. D: Antonio Secchi. SC: Mario Amendola, Massimo Franciosa and Antonio Secchi. With Keenan Wynn, Scott Holden, Delia Boccardo, Giorgio Trestini, Ray O'Connor (Remo Capitani), Philippe Leroy, Mimmo Palmara (Dick Palmer), Giorgio White, Nello Pazzafini, Gino Marturano, Osiride Pevarello, Franco Fabrizi, Gino Cagna, Carla Mancini, Alberto Dell'Acqua (Robert Widmark), Roberto Dell'Acqua, Riccardo Donzelli. An aging gunman's son returns home from school with a cache of Confederate gold and the locals do not trust him. Limp attempt at comedy in this Italian production released there as _**Padella Calibro .38**_ (Frying Pan Calibre .38).\n\n_**Panhandle Trail**_ see _**The Mysterious Rider**_ (1942)\n\n**3034** _ **Paper Hearts**_ **** Trimark, 1993. 88 min. Color. D-SC: Rod McCall. With James Brolin, Sally Kirkland, Pamela Gidley, Laura Johnson, Michael Moore, Rene Estevez, Kris Kristofferson, Mickey Cottrell, Helen Evans, Mark Voltura, Jim Magee, Gary Naylor, Jackson D. Kane, Paula Baz, Ed Ostertag, Lewis Goggin, Martha Ostertag, Francisco Topele. A year after deserting his family, a man returns home with his new girlfriend to attend his daughter's wedding and finds out his estranged wife may lose their home due to his outstanding debts. Standard modern-day oater issued on video as _**Cheatin' Hearts**_.\n\n**3035** _ **Parade of the West.**_ Universal, 1930, 75 min. D: Harry Joe Brown. SC: Bennett Cohen and Lesley Mason. With Ken Maynard, Gladys McConnell, Otis Harlan, Frank Rice, Bobby Dunn, Jackie Hanlon, Fred Burns, Frank Yaconelli, Stanley Blystone, Edgar \"Blue\" Washington. A cowboy, the guardian of a small boy, appears in a wild west show and romances one of the performers but the owner's right hand man resents his attentions to the girl and plans to sabotage his ride on a wild horse. Ken Maynard part talkie that will be of interest to his fans.\n\n**3036** _ **Paradise Canyon**_ **** Monogram, 1935. 55 min. D: Carl L. Pierson. SC: Lindsley Parsons and Robert Emmett (Tansey). With John Wayne, Marion Burns, Earle Hodgins, Yakima Canutt, Reed Howes, Perry Murdock, Gordon Clifford, Henry Hall, Gino Corrado, Tex Palmer, Earl Dwire, John Goodrich, Herman Hack, Bob Burns, George Morrell, Horace B. Carpenter, Fred Parker, Tex Phelps, Sherry Tansey, Chuck Baldra, Joe De La Cruz, Joe Dominquez, George Hazel. An undercover agent works along the Mexican border trying to capture a gang of counterfeiters. The final entry in John Wayne's Monogram\u2013Lone Star series is a pretty good one, highlighted by Earle Hodgins' barker scenes; colorized as _**Guns Along the Trail**_.\n\n**3037** _ **The Paradise Trail**_ **** Mark IV Pictures, 1979. 90 min. Color. D: Donald W. Thompson. SC: Russell S. Doughten, Jr. and Donald W. Thompson. With Burt Douglas, Robert Somers, Gene Otis, Teri Hernandez, Deborah Trissel, Richard Jury, Russell Porter, John Isaacs, Ron Kaye, Jim Koch, Terry Wagner, The Chuckwagoneers, Dusty (mule). A blind preacher and a crippled gunfighter meet with both finding religious salvation. Obscure faith based Western filmed in Iowa.\n\n**3038** _ **Paradise Valley**_ **** Imperial, 1936. 51 min. D: James P. Hogan. SC: G.A. Durlam and Francis Wheeler. With Sam Pierce, Jean Chadbourne, Wheeler Oakman, Arthur Loft, Jimmy Aubrey, Si Jenks, Donny Baker, Aleth \"Speed\" Hansen, Walter Brennan, Don Paul, The Beverly Hill Billies, Zandra (dog). After losing his radio crooning job to drink, a singer heads West where he befriends a dog after freeing him from a trap and the two settle in an area plagued by cattlemen-sheepherder warfare. Rawboned feature from producer-director James P. Hogan, made in 1934 and highlighted by Brydon Baker's photography.\n\n**3039** _ **Parasite**_ **** Embassy, 1982. 85 min. Color. D: Charles Band. SC: Alan Adler, Michael Shoob and Frank Leverny. With Robert Claudini, Demi Moore, Luca Bercovini, Vivian Blaine, James Davidson, Al Fann, Cherie Currie, Tom Villard, Scott Thomson, James Cavan, Joanelle Romero. In the savage West of the future a terrible giant parasite destroys people while a doctor tries to find a way to combat the menace. Absolutely awful 3-D effort.\n\n**3040** _ **Pardners**_ **** Paramount, 1956. 90 min. Color. D: Norman Taurog. SC: Sidney Sheldon. With Dean Martin, Jerry Lewis, Lori Nelson, Jackie Loughery, John Baragrey, Jeff Morrow, Agnes Moorehead, Lon Chaney, Mickey Finn, Douglas Spencer, Philip Tonge, Bob Steele, Jack Elam, Lee Van Cleef, Stuart Randall, Richard Aherne, Milton Frome, Scott Douglas, Emory Parnell, Mary Newton, Gavin Gordon, Dorothy Ford, William Forrest, Frances Mercer, Elaine Riley, Dorothy Abbott, Stanley Blystone, Charles Stevens, Matt Moore, Hank Mann, Frank Cordell, Bobby Barber, Len Hendry, Tony Michael, Don House, Bill Baldwin. Two Easterners travel to the Western community where their fathers were gunned down and attempt to clean up the lawless element. Average genre comedy based on _**Rhythm on the Range**_ (q.v.), also directed by Norman Taurog.\n\n**3041** _ **Pardon My Gun**_ **** Path\u00e9, 1930. 70 min. D: Robert De Lacy. SC: Hugh Cummings. With Sally Starr, George Duryea (Tom Keene\/Richard Powers), Lee Moran, Robert Edeson, Frank MacFarlane, Tom MacFarlane, Harry Woods, Lew Meehan, Ethan Laidlaw, Harry Watson, Al Norman, Ida May Chadwick, Abe Lyman and His Band. A cowboy loves the boss' daughter who is also courted by a rival rancher, with the two men at odds during an annual relay race. Stupefying, silly early talkie that lends its second half to a series of musical numbers in a barn dance setting; mainly for Tom Keene fans.\n\n**3042** _ **Pardon My Gun**_ **** Columbia, 1942. 57 min. D: William Berke. SC: Wyndham Gittens. With Charles Starrett, Alma Carroll, Noah Beery, Arthur Hunnicutt, Texas Jim Lewis and His Lone Star Cowboys, Dick Curtis, Ted Mapes, Lloyd Bridges, Dave Harper, Robert Graves, Guy Usher, Jack Kirk, Steve Clark, Art Mix, Robert (Kellard) Stevens, George Morrell, Joel Friedkin, Denver Dixon, Rube Dalroy, Jim Corey, Jessie Arnold, Lyle Clement. A surveyor and a sheep rancher's daughter find themselves accused of murder when a man is bushwhacked and robbed of a large amount of money. Fast moving and well written Charles Starrett vehicle with some good western swing music by Texas Jim Lewis and his group.\n\n**3043** _ **Park Avenue Logger**_ **** RKO Radio, 1937. 67 min. D: David Howard. SC: Dan Jarrett. With George O'Brien, Beatrice Roberts, Willard Robertson, Ward Bond, Bert Hanlon, Gertrude Short, Lloyd Ingraham, George Rosener, Robert Emmett O'Connor, Al Baffert, Dave Wengren. A playboy sent to work in a lumber camp finds out the foreman is a crook. Another good example of the high quality George O'Brien RKO series; TV title: _**Tall Timber**_.\n\n**3044** _ **Paroled to Die**_ **** Republic, 1938. 55 min. D: Sam Newfield. SC: George Plympton. With Bob Steele, Kathleen Elliot, Karl Hackett, Horace Murphy, Steve Clark, Budd Buster, Sherry Tansey, Frank Ball, Jack C. Smith, Horace B. Carpenter, Buzz Barton. A rancher is blamed for a bank robbery and a series of killings but the real culprit is a local businessman. Quality entry in Bob Steele's series for producer A.W. Hackel.\n\n**3045** _ **The Parson and the Outlaw**_ **** Columbia, 1957. 71 min. Color. D: Oliver Drake. SC: Oliver Drake and John Mantley. With Anthony Dexter, Charles \"Buddy\" Rogers, Marie Windsor, Sonny Tufts, Robert Lowery, Jean Parker, Madalyn Trahey, Bob Steele, Bob Duncan, Bob Gilbert, Jack Owell, John Davis, Joe Sodja, Paul Spahn, Herman Pulver, Richard Reeves. Escaping death at the hands of Sheriff Pat Garrett, Billy the Kid tries to lead a peaceful life but becomes involved with a minister fighting a corrupt land baron and his henchmen. Tacky outing co-produced by Charles \"Buddy\" Rogers.\n\n**3046** _ **The Parson of Panamint**_ **** Paramount, 1941. 84 min. D: William McGann. SC: Harold Shumate and Adrian Scott. With Charles Ruggles, Ellen Drew, Philip Terry, Joseph Schildkraut, Porter Hall, Henry Kolker, Janet Beecher, Clem Bevans, Douglas Fowley, Paul Hurst, Frank Puglia, Minor Watson, Harry Hayden, Russell Hicks, Hal Price, The Guardsmen Quartet, Rod Cameron, Frances Morris, Suzanne Ridgeway, Tom London, Paul Maxey, Dan White. A young reverend comes to a brawling mining town and tries to reform its citizens. Producer Harry Sherman's very pleasant screen adaptation of the Peter B. Kyne story, previously filmed in 1916 by Paramount with Dustin Farnum and remade by the same studio in 1922 starring Jack Holt.\n\n**3047** _ **The Parting of the Trails**_ **** Syndicate, 1930. 60 min. D: J.P. McGowan. SC: Sally Winters. With Bob Custer, Vivian Ray, Bobby Dunn, Henry Roquemore, George A. Miller, Tommy Bay. Two drifters assist a young woman whose millionaire father has been kidnapped by outlaws. Vapid Bob Custer silent effort also issued with a music score.\n\n**3048** _ **Partners**_ **** RKO Radio, 1932. 57 min. D: Fred Allen. SC: Donald W. Lee. With Tom Keene, Nancy Drexel, Otis Harlan, Victor Potel, Bobby Nelson, Lee Shumway, Billy Franey, Carleton King, Ben Corbett, Ed Cassidy, Slim Whitaker, Fred Burns, Jack Kirk, Jim Corey, Frank Ellis, S.S. Simon, Tracy Layne, Bill Nestell. A horse raiser is blamed for the murder of the man who loaned him the money to buy a ranch and he tries to find the killer. Somewhat lumbering Tom Keene affair with pleasant scenery; Lafe McKee dubbed Ed Cassidy's voice.\n\n**Poster for** _**Partners**_ **(RKO Radio, 1932).**\n\n**  \n**\n\n**3049** _ **Partners of the Plains**_ **** Paramount, 1938. 70 min. D: Lesley Selander. SC: Harrison Jacobs. With William Boyd, Russell Hayden, Gwen Gaze, Harvey Clark, Hilda Plowright, John Warburton, Alan Bridge, Al Hill, Earle Hodgins, John Beach, Jim Corey, Bud McClure, Herman Hack, Hank Bell. Hoppy and the Bar 20 boys attempt to help a snobbish young lady by saving her cattle and land from the evil Scar Lewis. Well photographed and entertaining entry in the \"Hopalong Cassidy\" series.\n\n**3050** _ **Partners of the Sunset**_ **** Monogram, 1948. 53 min. D: Lambert Hillyer. SC: J. Benton Cheney. With Jimmy Wakely, Dub Taylor, Christine Larson, Steve Darrell, Marshall Reed, Jay Kirby, Leonard Penn, Bob Woodward, Carl Mathews, Carl Sepulveda, J.C. Lytton, Boyd Stockman, Arthur \"Fiddlin'\" Smith, Don Weston. A woman plans to murder her husband but the plot is uncovered by a singing cowboy. Mystery element and songs help this otherwise laggard Jimmy Wakely feature.\n\n**3051** _ **Partners of the Trail**_ **** Monogram, 1931. 63 min. D: Wallace Fox. SC: G.A. Durlam. With Tom Tyler, Betty Mack, Lafe McKee, Reginald Sheffield, Pat Rooney, Alan Bridge, Stanley Blystone, Marguerite McWade, Horace B. Carpenter, Jack Richardson, C.V. Bussey. A man kills his wife's lover only to have his buddy convicted of the crime. Very low grade Tom Tyler outing; remade as _**Sagebrush Trail**_ (q.v.).\n\n**3052** _ **Partners of the Trail**_ **** Monogram, 1944. 59 min. D: Lambert Hillyer. SC: Frank Young. With Johnny Mack Brown, Raymond Hatton, Christine McIntyre, Craig Woods, Robert Frazer, Lloyd Ingraham, Marshall Reed, Jack Ingram, Lynton Brent, Steve Clark, Ben Corbett, Ted Mapes, Joe Eggerton, Hal Price, Slim Whitaker, Wally West, Chick Hannon, Al Taylor, Kanasa Moehring, Bill Wolfe. Two lawmen ride to a small town to discover why ranchers are being murdered and uncover a plot to obtain a rich gold claim. Well written \"Nevada Jack McKenzie\" series production.\n\n**3053** _ **Passage West**_ **** Paramount, 1951. 81 min. Color. D-SC: Lewis R. Foster. With John Payne, Dennis O'Keefe, Arleen Whelan, Frank Faylen, Mary Anderson, Peter Hansen, Richard Rober, Griff Barnett, Dooley Wilson, Mary Field, Richard Travis, Mary Beth Hughes, Arthur Hunnicutt, Lillian Bronson, Ilka Gruning, Estelle Carr, Susan Whitney, Walter Reed, Paul Fierro, Clint Stuart, Earle Hodgins, Howard Negley, Victor Kilian, Kit Guard, Guy Wilkerson, Tim Graham, Hank Mann. A half dozen escaped convicts take refuge in a wagon train belonging to a religious sect heading West. Average oater, but well produced.\n\n**3054** _ **Passion**_ **** RKO Radio, 1954. 84 min. Color. D: Allan Dwan. SC: Josepth Leytes, Beatrice A. Dresher and Howard Estabrook. With Cornel Wilde, Yvonne De Carlo, Raymond Burr, Lon Chaney, John Qualen, Rodolfo Acosta, Anthony Caruso, Frank De Kova, Peter Coe, Clayton Moore, John Dierkes, Richard Hale, Rosa Turich, Stuart Whitman, James Kirkwood, Robert Warwick, Belle Mitchell, Alex Montoya, Zon Murray, Gil Frye, Rozene Kemper. In Spanish California a man learns his wife and daughter have been murdered by a land hungry Army officer and his thugs and he teams with his pretty sister-in-law to get revenge. Interesting, violent Western with good work by Yvonne De Carlo in dual roles and Lon Chaney as the vicious henchman Castro; concluding gunfight sequence in the snowy Sierras is a knockout.\n\n**3055** _ **Pat Garrett and Billy the Kid**_ **** Metro-Goldwyn-Mayer, 1973. 106 min. Color. D: Sam Peckinpah. SC: Rudolph Wurlitzer. With James Coburn, Kris Kristofferson, Richard Jaeckel, Katy Jurado, Chill Wills, Jason Robards, Bob Dylan, R.G. Armstrong, Luke Askew, John Beck, Richard Bright, Matt Clark, Rita Coolidge, Jack Dodson, Jack Elam, Emilio Fernandez, Paul Fix, L.Q. Jones, Slim Pickens, Jorge Russek, Charles Martin Smith, Harry Dean Stanton, John Chandler, Rudy Wurlitzer, Elisha Cook, Jr., Gene Evans, Dub Taylor, Don Levy, Sam Peckinpah, Rutanya Alda, Walter Kelly, Claudia Bryar, Mike Mikler, Aurora Clavel, Donnie Fritts. Sheriff Pat Garrett gets on the trail of his ex-pal Billy the Kid after the outlaw refuses his orders to leave New Mexico Territory. Stagnant version of the final days of the famous outlaw with self-indulgent direction and a lifeless music score by Bob Dylan. Heavily re-cut for television in a 122-minute version that reinstates footage with Barry Sullivan.\n\n**3056** _ **The Pathfinder**_ **** Columbia, 1953. 78 min. Color. D: Sidney Salkow. SC: Robert E. Kent. With George Montgomery, Helena Carter, Jay Silverheels, Walter Kingsford, Rodd Redwing, Elena Verdugo, Chief Yowlachie, Ross Conklin, Bruce Lester, Ed Coch, Jr., Stephen Bekassy, Vi Ingraham, Adele St. Maur. When the French attack his tribe, an Englishman raised by the Indians tries to help the British. Tepid retelling of James Fenimore Cooper's \"Leatherstocking\" tales, embellished by color.\n\n**3057** _ **The Pathfinder and the Mohican**_ **** International Television Corporation (ITC), 1964. 90 min. D: Sam Newfield. SC: Nat Tanchuck. With John Hart, Lon Chaney, Jonathan White, Angela Fusco, Larry Solway. Delaware Indians are falsely accused of crimes against the settlers with Hawkeye and Chingachgook trying to find out the truth. Okay paste-up telefeature from three segments of \"Hawkeye and the Last of the Mohicans\" (Syndicated, 1956\u201357), filmed in Canada.\n\n_**Paths of Hate**_ see _**Bullet and the Flesh**_\n\n**3058** _ **Pawnee**_ **** Republic, 1957. 80 min. Color. D-SC: George Waggner. With George Montgomery, Lola Albright, Bill Williams, Francis McDonald, Robert E. Griffin, Dabbs Greer, Kathleen Freeman, Charlotte Austin, Ralph Moody, Anne Barton, Raymond Hatton, Charles Horvath, Robert Nash. A white man raised by the Indians must choose between his own race and his adopted one when corrupt whites try to steal Indian lands. Pretty fair action outing.\n\n**3059** _ **Payment in Blood**_ **** Columbia, 1968. 89 min. Color. D: E.G. Rowland (Enzo Girolami). SC: Tito Carpi and E.G. Rowland (Enzo Girolami). With Guy Madison, Edd Byrnes, Louise Barrett, Enio Girolami, Marion Donen, Rick Boyd (Federico Boido), Rosella Bergamonti, Alfred Facchetti, Attillio Severini, Pedro Sanchez, Guilio Maculani, Mirella Pamphilio, Piero Vida. After the Civil War a bounty hunter infiltrates a renegade band of Confederates in Texas after a hidden treasure. Fierce and bloody Italian Western, likely to appeal to Guy Madison fans; made in 1967 by Circus Film\/Rono Roma\/St. Regis Films as _**7 Winchester per un Massacre**_ (7 Winchesters for a Massacre) and re-titled _**Winchester for Hire**_ for television and _**Blake's Marauders**_ on video.\n\n**3060** _ **Peace for a Gunfighter**_ **** Crown International, 1965. 82 min. Color. D: Raymond Boley. SC: Michael W. Fuller. With Burt Berger, JoAnne Meredith, Everett King, Sterling Walker, Danny Zapien, John Scovern, Mark Farrington, Ray Odom, Mark Sanchez, Allen Wood. A gunman called \"The Preacher\" tries to give up his trade but meets with resistance in a frontier town. Obscure, low grade oater.\n\n**3061** _ **The Peacemaker**_ **** United Artists, 1958. 83 min. D: Ted Post. SC: Hal Richards and Jay Ingram. With James Mitchell, Rosemarie Bowe, Jan Merlin, Jess Barker, Hugh Sanders, Herbert Patterson, Dorothy Patrick, Taylor Holmes, Robert Armstrong, Philip Tonge, Wheaton Chambers, Harry Shannon, Jack Holland, Nancy Evans. An ex-gunfighter turned preacher arrives in an area to find a feud between settlers and ranchers. Vapid affair; typical example of the genre's decline in the late 1950s.\n\n**3062** _ **The Pecos Kid**_ **** Commodore, 1935. 56 min. D: William Berke. SC: Henry Hess. With Fred Kohler, Jr., Ruth Findlay, Wally Wales, Roger Williams, Francis Walker, Ed Cassidy, Budd Buster, Robert Walker, Clyde McClary, Rose Plummer, Earl Dwire, Jack Evans, Milburn Morante, Phil Dunham, Tex Palmer, Ray Henderson. After his family is murdered by outlaws, a boy grows up determined to get revenge on the culprits. Cheaply made but rather interesting Fred Kohler, Jr. vehicle; worth a look.\n\n**3063** _ **Pecos River**_ **** Columbia, 1951. 55 min. D: Fred F. Sears. SC: Barry Shipman. With Charles Starrett, Smiley Burnette, Jack (Jock) Mahoney, Delores Sidener, Steve Darrell, Harmonica Bill (William Russell), Edgar Dearing, Frank Jenks, Paul Campbell, Zon Murray, Maudie Prickett, Eddie Fetherston, Frank McCarroll, Al Haskell, Blackie Whiteford. A post office investigator masquerades as a stage driver while trying to find out who has been carrying out a series of mail robberies. Okay entry in the \"Durango Kid\" series. British title: _**Without Risk**_.\n\n**3064** _ **Per un Pugno nell'Occhio**_ (For a Fist in the Eye) **** Fenix Film\/Ramo Film, 1965. 101 min. Color. D: Michele Lupo. SC: Eduardo M. Brochero. With Franco Franchi, Ciccio Ingrassia, Aurora Julia (Monica Randall), Paco (Francisco Moran), Carmen Esbri, Jesus Puente, Lina Morales, Jesus Tordesillas, Romano Giomini, Maria Badmayev, Emilio Rodriguez, Rafael Albaicin, Guillermo Mendez, Jose Riesgo, Alvaro de Luna, Francisco Camorias, Tito Garcia, Simon Arriaga, Rafael Hernandez, Jose Canalejas, Luis Barboo. Two zanies arrives in a border town where rival gangs are out to control the restaurant business. Unintelligible Italian-Spanish Western \"comedy\" from the inane team of Franco and Ciccio, also known as _**Fistful of Knuckles**_.\n\n**3065** _ **The Perfect Alibi**_ **** Photo Drama, 1924. 55 min. D: Ford Beebe. SC: Frances Beebe and Ford Beebe. With Leo Maloney, Josephine Hill, Leonard Clapham (Tom London), Jim Corey, Earl Close, Whitehorse, Bullet (dog). A ranger, who refuses to pursue charges against his girl's brother, is dismissed from the service but tries to find out who really pulled off a robbery for which the young man blamed. This quickly made and fast moving William Steiner production provides a chance to see Leo Maloney in one of his silent vehicles.\n\n**3066** _ **A Perilous Journey**_ **** Republic, 1953. 90 min. D: R.G. Springsteen. SC: Richard Wormser. With Vera Ralston, David Brian, Scott Brady, Charles Winninger, Hope Emerson, Eileen Christy, Leif Erickson, Veda Ann Borg, Virginia Grey, Dorothy Ford, Ben Cooper, Kathleen Freeman, Barbara Hayden, Paul Fierro, Angela Greene, John Dierkes, Alden Aldrich, Fred Graham, Trevor Bardette, Richard Reeves, Bob Carney, Charles Evans, Philip Van Zandt, Byron Foulger, Denver Pyle, Harry Tyler, Emil Sitka, Jack O'Shea, Brandon Beach, Frank Hagney, Stanley Blystone, Richard Alexander, Charles Cane, Gloria Clark. Joining four dozen women, all mail order brides on the way to California on a ship via Panama, a woman seeks to locate her gambler husband. Standard drama with Hope Emerson stealing the show as the stern willed chaperone.\n\n**3067** _ **Perils of the Royal Mounted**_ **** Columbia, 1942. 15 Chapters. D: James W. Horne. SC: Basil Dickey, Scott Littleton, Jesse A. Duffy and Louis Heifetz. With Robert (Kellard) Stevens, Nell O'Day, Herbert Rawlinson, Kenneth MacDonald, John Elliott, Nick Thompson, Art Miles, Richard Fiske, Rick Vallin, Forrest Taylor, Kermit Maynard, George Chesebro, Jack Ingram, Charles King, I. Stanford Jolley, Al Ferguson, Iron Eyes Cody. A Mounted Policeman learns that supposed Indian attacks are being carried out by white men led by a renegade in cahoots with a corrupt medicine man. Standard Columbia cliffhanger.\n\n**3068** _ **Perils of the Wilderness**_ **** Columbia, 1956. 15 Chapters. D: Spencer Gordon Bennet. SC: George H. Plympton. With Dennis Moore, Richard Emory, Eve Anderson (Evelyn Finley), Kenneth MacDonald, Rick Vallin, John Elliott, Don C. Harvey, Terry Frost, Al Ferguson, Bud Osborne, Rex Lease, Pierce Lyden, John Mitchum, Lee Roberts, Stanley Price, Kermit Maynard, John Hart, Frank Lackteen, I. Stanford Jolley, Robert Bice, Jack Ingram, Wally West, Dan White, Ed Coch. A man poses as an outlaw to capture a ruthless crime baron in the Canadian north country and is helped by a Royal Canadian Mounted policeman and a pretty girl. The penultimate serial is a sadly cheap affair with an unbelievable plot; interpolates footage from _**The Mysterious Pilot**_ (1937) and _**Perils of the Royal Mounted**_ (q.v.).\n\n**3069** _ **The Persuader**_ **** Allied Artists, 1957. 72 min. D: Dick Ross. SC: Curtis Kenyon. With James Craig, Kristine Miller, William Talman, Darryl Hickman, Georgia Lee, Alvy Moore, Rhoda Williams, Gregory Walcott, Paul Engel, Nolan Leary, Frank Richards, Jason Johnson, Joyce Compton, Wendy Stuart, Leilani Sorenson. A minister comes to the Oklahoma town where outlaws killed his brother and helps the citizens stand up to lawlessness. Pretty fair program feature.\n\n**3070** _ **Peter Lundy and the Medicine Hat Stallion**_ **** NBC-TV, 1977. 85 min. Color. D: Michael O'Herlihy. SC: Jack Turley. With Leif Garrett, Milo O'Shea, John Anderson, Bibi Besch, John Quade, Ann Doran, Brad Rearden, Mitch Ryan, Charles Tyner, Ned Romero, James Lydon, Phil Mead, Bill Hicks, Robert Tzudiker. Prior to the Civil War, a teenager becomes a pony express rider in Nebraska Territory and confronts a road agent. Well made and entertaining juvenile fare for television.\n\n**3071** _ **The Petrified Forest**_ **** Warner Bros., 1936. 83 min. D: Archie Mayo. SC: Charles Kenyon and Delmer Daves. With Leslie Howard, Bette Davis, Genevieve Tobin, Dick Foran, Humphrey Bogart, Joseph Sawyer, Porter Hall, Charley Grapewin, Paul Harvey, Eddie Acuff, Adrian Morris, Nina Campana, Slim Thompson, John Alexander, Addison Richards (voice). A diverse group of people are held prisoners at a desert way station by a fleeing hoodlum and his gang. Top notch screen adaptation of Robert Emmet Sherwood's play; a gangster drama in a modern Western setting.\n\n**Humphrey Bogart, Leslie Howard and Bette Davis in** _**The Petrified Forest**_ **(Warner Bros., 1936).**\n\n**  \n**\n\n**3072** _ **The Phantom Bullet**_ **** Universal, 1926. 57 min. D: Clifford Smith. SC: Curtis Benton. With Hoot Gibson, Eileen Percy, Allan Forrest, Pat Harmon, William H. Turner, Nelson McDowell, John T. Price, Pee Wee Holmes, Rosemary Cooper. After his father is mysteriously killed a man returns home and takes on the guise of a bungler to find out who did the shooting. Well photographed Hoot Gibson silent Western with lots of comedy and an exciting car chase sequence.\n\n**3073** _ **The Phantom Cowboy**_ **** Aywon, 1935. 56 min. D: Robert J. Horner. SC: Carl Krusada. With Ted Wells, Doris Brook, George Chesebro, Jimmy Audrey, Lew Meehan, Allen Greer, Oscar Gahan, Milburn Morante, Herman Hack, Sherry Tansey, Richard Cramer, Frank Clark, Rosamond Wagman, William Desmond, Wally West, Edna Aslin, Carl Mathews, Tex Palmer, Fred Parker, Al Haskell, Charles Le Moyne, Jack Evans, Barney Beasley. A cowboy and his pal join forces with a prospector, really a masked highwayman, in trying to discover who is out to steal the man's mining claim. Appallingly bad poverty row affair that has star Ted Wells equally inept in dual roles.\n\n**3074** _ **The Phantom Cowboy**_ **** Republic, 1941. 56 min. D: George Sherman. SC: Doris Schroeder. With Don \"Red\" Barry, Virginia Carroll, Milburn Stone, Neyle Marx, Rex Lease, Nick Thompson, Bud Osborne, Ernest Wilson, Burr Caruth, Frank Ellis, Art Dillard, Jack O'Shea, Chuck Baldra, Leander De Cordova, Matty Roubert, Jim Corey, Hank Patterson, Hank Bell. When a rancher is murdered and his niece is cheated out of her inheritance, a cowboy becomes a masked avenger to restore her property. Star Don Barry and director George Sherman do their best in the confines of a script that lacks action.\n**3075** _ **The Phantom Empire**_ **** Mascot, 1935. 12 Chapters. D: Otto Brower and B. Reeves Eason. SC: John Rathmell, Armand L. Schaefer, Wallace MacDonald, Gerald Geraghty and Hy Freedman. With Gene Autry, Frankie Darro, Betsy King Ross, Dorothy Christy, Wheeler Oakman, Charles K. French, Warner Richmond, J. Frank Glendon, Smiley Burnette, William Moore, Ed Peil, Sr., Jack Carlyle, Frank Ellis, Wally Wales, Buffalo Bill, Jr., Fred Burns, Stanley Blystone, Richard Talmadge, Bob Card, Bruce Mitchell, Frankie Marvin, Slim Whitaker, Wally West, Bob Burns, George Magrill, Henry Hall, Peter Potter, Ray (Corrigan) Bernard. Crooks are after a valuable mineral deposit on a radio singer's ranch and while fighting them he and his two juvenile pals find a secret underground civilization. Gene Autry's first starring film is a flavorful combination of the Western and sci-fi genres; later released in feature versions as _**Men with Steel Faces**_ and _**Radio Ranch**_.\n\n**3076** _ **The Phantom Flyer**_ **** Universal, 1928. 45 min. D: Bruce Mitchell. SC: Bruce Mitchell and Gardner Bradford. With Al Wilson, Lillian Gilmore, Buck Connors, Billy \"Red\" Jones, Don Fuller, Myrtis Crinley, Mary Cornwallis, Larry Steers. A homesteader and his family find themselves at odds with a female cattle rancher after their water rights. Colorful silent action feature with lots of exciting aerial footage of star Al Wilson, here playing a border patrol aviator. Also called _**The Phantom Ranger**_.\n\n**3077** _ **Phantom Gold**_ **** Columbia, 1938. 56 min. D: Joseph Lovering. SC: Nate Gatzert. With Jack Luden, Beth Marion, Barry Downing, Slim Whitaker, Hal Taliaferro, Art Davis, Jack Ingram, Marin Sais, Buzz Barton, Jimmy Robinson, Forrest Taylor, Jack Rockwell, Harry Harvey, Charles King, George Morrell, Jack O'Shea, Tex Palmer, Tuffy (dog). Outlaws plan a gold rush by salting an old mine but are opposed by a cowboy, his two buddies and the young boy and dog they rescued. Jack Luden's final series film is mediocre at best.\n\n**3078** _ **The Phantom of Santa Fe**_ **** Burroughs-Tarzan Enterprises, 1936. 87 min. Color. D: Jacques Jaccard. SC: Charles F. Royal. With Norman Kerry, Nina Quartero, Frank Mayo, Monte Montague, Tom O'Brien, Carmelita Geraghty, Jack Mower, Frank Ellis, Merrill McCormick, Fernando Valdez. A man pretends to be a coward to disguise himself as the \"Hawk,\" a masked avenger opposed to a crook and his gang who have stolen a treasure. Made in 1931 as _**The Hawk**_ , this feature was re-recorded and re-edited before receiving theatrical release five years later; pretty poor stuff.\n\n**3079** _ **The Phantom of the Desert**_ **** Syndicate, 1930. 55 min. D: Harry S. Webb. SC: Carl Krusada. With Jack Perrin, Eva Novak, Josef Swickard, Lila Eccles, Ben Corbett, Edward Earle, Robert Walker, Pete Morrison. Two cowpokes go to work for a rancher whose horses are supposedly being rustled by a wild stallion and one of them tries to get to the bottom of the trouble. Likable Jack Perrin early talkie spotlighting his beautiful steed Starlight.\n\n**3080** _ **Phantom of the Plains**_ **** Republic, 1945. 58 min. D: Lesley Selander. SC: Earle Snell and Charles Kenyon. With Wild Bill Elliott, Bobby Blake, Alice Fleming, Ian Keith, William Haade, Virginia Christine, Jack Rockwell, Tom London, Earle Hodgins, Bud Geary, Henry Hall, Fred Graham, Neal Hart, Jack Kirk, Bob Burns, Jack Tornek, Rose Plummer. When the Duchess falls in love with an Englishman, Red Ryder does not trust the man and finds out he is a wife murderer. So-so \"Red Ryder\" episode.\n\n**3081** _ **Phantom of the Range**_ **** Victory, 1936. 58 min. D: Robert Hill. SC: Basil Dickey. With Tom Tyler, Beth Marion, Sammy Cohen, Forrest Taylor, Soledad Jiminez, Charles King, John Elliott, Richard Cramer, Steve Clark, Robert Hill, Johnny Luther, Francis Walker, Denver Dixon, Clyde McClary, Jack Evans, Bud Pope, Tiny Sanford, Tex Phelps. A cattlemen's association investigator is after a gang seeking secret treasure. Shoddy Sam Katzman production and one that hardly enhances Tom Tyler's genre reputation.\n\n**3082** _ **The Phantom of the West**_ **** Mascot, 1931. 10 Chapters. D: D. Ross Lederman. SC: Ford Beebe. With Tom Tyler, Dorothy Gulliver, William Desmond, Tom Santschi, Tom Dugan, Philo McCullough, Joe Bonomo, Kermit Maynard, Frank Lanning, Frank Hagney, Dick Dickinson, Hallee Sullivan, Al Taylor, Ernie Adams. A phantom murders several citizens while a rancher tries to solve the mystery of who killed his father. Mascot's second sound serial, and Tom Tyler's talkie debut, is an interesting affair with a good mystery angle and the usual genre thrills.\n\n**3083** _ **Phantom Patrol**_ **** Ambassador, 1936. 60 min. D: Charles Hutchison. SC: Joseph O'Donnell. With Kermit Maynard, Joan Barclay, Dick Curtis, Harry Worth, George Cleveland, Paul Fix, Julian Rivero, Eddie Phillips, Roger Williams, Lester Dorr. A Mountie impersonates an American detective story writer in order to round up a gang of crooks. Average Kermit Maynard vehicle for producer Maurice Conn, supposedly based on James Oliver Curwood's _Fatal Note_.\n\n_**Phantom Pinto**_ see _**Buzzy and the Phantom Pinto**_\n\n**3084** _ **The Phantom Plainsmen**_ **** Republic, 1942. 57 min. D: John English. SC: Robert Yost and Barry Shipman. With Bob Steele, Tom Tyler, Rufe Davis, Lois Collier, Robert O. Davis (Rudolph Anders), Charles F. Miller, Alex Callam, Monte Montague, Henry Rowland, Richard Crane, Jack Kirk, Ed Cassidy, Vince Barnett, Lloyd Ingraham, Al Taylor, Bud Geary, Herman Hack. Three cowboys work for a horse rancher whose son is abducted by Nazis so he will sell them his herd. Interesting, patriotic \"Three Mesquiteers\" series entry.\n\n**3085** _ **Phantom Rancher**_ **** Colony, 1940. 61 min. D: Harry Fraser. SC: William Lively. With Ken Maynard, Dorothy Short, Harry Harvey, Ted Adams, Dave O'Brien, Tom London, John Elliott, Reed Howes, Steve Clark, Carl Mathews, Sherry Tansey, Wally West, George Morrell, Herman Hack. A cowboy arrives to take over his late uncle's ranch and becomes a masked phantom to oppose a land grabber. Ken Maynard was getting hefty when he made this entertaining oater but he was still quite agile.\n\n_**The Phantom Ranger**_ (1928) see _**The Phantom Flyer**_\n\n**3086** _ **Phantom Ranger**_ **** Monogram, 1938. 54 min. D: Sam Newfield. SC: Joseph O'Donnell. With Tim McCoy, Suzanne Kaaren, John Merton, Charles King, Karl Hackett, Tom London, Richard Cramer, John St. Polis, Edward Earle, Harry Strang, Bruce Warren, Bob McKenzie, Jimmy Aubrey, Donald Dean, Herb Holcombe, Wally West, Horace B. Carpenter, Sherry Tansey, George Morrell, Herman Hack, Frank Ellis, Victor Cox, Ray Henderson, Clyde McClary. A Secret Service is sent to round up a gang responsible for flooding an area with counterfeit currency. Low grade but okay modern sagebrush yarn.\n\n**3087** _ **The Phantom Rider**_ **** Universal, 1936. 15 Chapters. D: Ray Taylor. SC: George Plympton, Basil Dickey and Ella O'Neill. With Buck Jones, Marla Shelton, Diana Gibson, Joey Ray, Harry Woods, Frank LaRue, George Cooper, Eddie Gribbon, Helen Shipman, Jim Mason, Charles Lemoyne, Charles King, Jim Corey, Lee Shumway, Clem Bevans, Cecil Weston, Matt McHugh, Jim Thorpe, Cactus Mack, Charles K. French, Tom London, Slim Whitaker, Frank Ellis, Drew Stanfield, Art Mix, Bob Reeves, Buffalo Bill, Jr., Glenn Strange, Blackjack Ward, Horace Murphy, Allen Holbrook, Hank Bell, Lafe McKee, Priscilla Lawson, George Plues, Olin Francis, Iron Eyes Cody, Paul Regas, Eva McKenzie, Bob Card. With crooks trying to steal a young woman's ranch, a government agent takes on the guise of a masked rider to thwart them. Well paced serial sure to delight Buck Jones fans.\n\n**3088** _ **The Phantom Rider**_ **** Republic, 1946. 12 Chapters. D: Spencer Gordon Bennet and Fred C. Brannon. SC: Albert DeMond, Basil Dickey, Jesse Duffy, Lynn Perkins and Barney A. Sarecky. With Robert Kent, Peggy Stewart, LeRoy Mason, George J. Lewis, Kenne Duncan, Hal Taliaferro, Chief Thundercloud, Monte Hale, Tom London, Roy Barcroft, John Hamilton, Hugh Prosser, Jack Kirk, Rex Lease, Tommy Coats, Joe Yrigoyen, Bill Yrigoyen, Jack O'Shea, Cliff Lyons, Walt LaRue, Cliff Parkinson, Carl Sepulveda, George Carleton, Dave Van Sickel, Tom Steele, George Chesebro, Wayne Burson, Post Park, Fred Graham, Bob Duncan, Augie Gomez, Robert Wilke, John Roy, Cactus Mack, Eddie Parker, Ted Mapes, Duke Taylor, Hal Price, Henry Wills, Tex Cooper, Bud Bailey, James Linn. A frontier doctor becomes a masked phantom to bring justice to an area threatened by outlaws inciting Indians to go on the warpath. Pretty fair cliffhanger; issued in a feature version as _**Ghost Riders of the West**_.\n\n**3089** _ **The Phantom Stage**_ **** Universal, 1939. 58 min. D: George Waggner. SC: Joseph West (George Waggner). With Bob Baker, Marjorie Reynolds, George Cleveland, Forrest Taylor, Reed Howes, Tex Palmer, Murdock MacQuarrie, Glenn Strange, Jack Kirk, Ernie Adams, Dick Rush. Two cowpokes try to help a young woman about to lose her stage line because of gold shipment robberies. Only fair Bob Baker vehicle.\n\n**3090** _ **The Phantom Stagecoach**_ **** Columbia, 1957. 69 min. D: Ray Nazarro. SC: David Lang. With William Bishop, Richard Webb, Kathleen Crowley, Hugh Sanders, John Doucette, Frank Ferguson, Ray Teal, Percy Helton, Maudie Prickett, Lane Bradford, Eddy Waller, Robert Anderson, John Lehmann, Dennis Moore, Kermit Maynard. Stage line owners have a dispute over right-of-way, leading to gunplay. Only average \"B\" second bill feature.\n\n**3091** _ **Phantom Stallion**_ **** Republic, 1954. 54 min. D: Harry Keller. SC: Gerald Geraghty. With Rex Allen, Slim Pickens, Carla Balenda, Harry Shannon, Don Haggerty, Peter Price, Rosa Turich, Zon Murray, Rocky Shahan, Charles La Torre. A cowboy helps a ranch owner whose best horses are disappearing, the culprits supposedly a stallion and his herd. Standard Rex Allen outing with the plot twist of having the rancher's niece as the main villain.\n\n**3092** _ **The Phantom Thunderbolt**_ **** World Wide\/Fox, 1933. 63 min. D-SC: Alan James. With Ken Maynard, Frances Dade, Frank Rice, Robert Kortman, William Gould, Harry Holman, Frank Beal, Wilfred Lucas, William Robyns, Nelson McDowell, Lew Meehan, Jack Rockwell, Frank Ellis, Robert Walker, Horace B. Carpenter, Johnny Luther, Bud McClure, Archie Ricks, Blackjack Ward, Silver Tip Baker. The Thunderbolt Kid is hired by the leaders of Coyote Gulch to stop a lawless gang that is keeping a railroad from going through the area. Rawboned but action filled Ken Maynard vehicle.\n\n**3093** _ **Phantom Trails**_ **** Allied Artists, 1955. 54 min. D: Frank McDonald and Wesley Barry. SC: Maurice Tombragel and William Raynor. With Guy Madison, Andy Devine, Steve Brodie, Harry Harvey, Byron Foulger, Robert Filmer, Burt Wendland, Steve Pendleton, Hank Patterson, Ethan Laidlaw, Paul Bryar, William Vedder. Wild Bill Hickok has his partner Jingles pretend to be a ghost to capture a gang attacking local farmers and Hickok joins looters in order to arrest them. Pretty good big screen feature made up of \"A Close Shave for the Marshal\" and \"Ghost Rider\" episodes of \"The Adventures of Wild Bill Hickok\" that was on TV from 1951 to 1958.\n\n**3094** _ **Phantom Valley**_ **** Columbia, 1948. 56 min. D: Ray Nazarro. SC: J. Benton Cheney. With Charles Starrett, Smiley Burnette, Virginia Hunter, Sam Flint, Ozie Waters and His Colorado Rangers, Joel Friedkin, Robert Filmer, Mikel Conrad, Zon Murray, Fred F. Sears, Teddy Infuhr, Jerry Jerome, Denver Dixon. A new sheriff arrives in a town where ranchers and homesteaders are at war due to attacks by an outlaw gang. Better than average \"Durango Kid\" whodunit feature.\n\n**3095** _ **Pierre of the Plains**_ **** Metro-Goldwyn-Mayer, 1942. 66 min. D: George B. Seitz. SC: Lawrence Kimble. With John Carroll, Ruth Hussey, Bruce Cabot, Phil Brown, Reginald Owen, Evelyn Ankers, Henry Travers, Raymond Hatton, Patrick McVey, Sheldon Leonard, Lois Ransom, Charles Stevens, Frederic Worlock, Iron Eyes Cody. A Canadian Mounted Police officer in love with a pretty innkeeper tries to foil lawlessness in the Northwest. John Carroll is okay as the daredevil title hero but he has to carry this otherwise tame effort.\n\n**3096** _ **Pillars of the Sky**_ **** Universal-International, 1956. 95 min. Color. D: George Marshall. SC: Sam Rolfe. With Jeff Chandler, Dorothy Malone, Ward Bond, Keith Andes, Lee Marvin, Sydney Chaplin, Willis Bouchey, Michael Ansara, Olive Carey, Charles Horvath, Orlando Rodriguez, Glen Kramer, Floyd Simmons, Pat Hogan, Felix Noriega, Paul Smith, Martin Milner, Robert Ellis, Robert Botrian, Walter Coy, Alberto Morin, Richard Hale, Frank De Kova, Terry Wilson, Phil Kieffer, Gilbert Connor. A hard drinking Army sergeant is forced to fight off Indians with men he does not like but soon learns to respect their skills and bravery. Another of the seemingly aimless 1950s adult oaters about a man reformed by responsibility and a good woman; only fair.\n\n**3097** _ **The Pinto Bandit**_ **** Producers Releasing Corporation, 1944. 57 min. D-SC: Elmer Clifton. With Dave O'Brien, Jim Newill, Guy Wilkerson, Mady Lawrence, James Martin, Jack Ingram, Ed Cassidy, Budd Buster, Karl Hackett, Robert Kortman, Charles King, Jimmy Aubrey, Kermit Maynard, Don Weston, Herman Hack, Carl Mathews, Ray Henderson. A ranger trio enters a three man relay race for a mail contract in hopes of capturing the masked bandit stealing postal shipments between two towns. Okay entry in \"The Texas Rangers\" series.\n\n**3098** _ **Pinto Canyon**_ **** Metropolitan, 1940. 55 min. D: Raymond Johnson. SC: Carl Krusada. With Bob Steele, Louise Stanley, Kenne Duncan, Ted Adams, Steve Clark, Budd Buster, Murdock MacQuarrie, George Chesebro, Jimmy Aubrey, Carl Mathews, Buzz Barton, Silver Tip baker, Ray Jones, Denver Dixon. A lawman is on the trail of a gang of cattle thieves. Badly made and boring Bob Steele vehicle for producer Harry S. Webb.\n\n**3099** _ **The Pinto Kid**_ **** Film Bookings Office (FBO), 1928. 55 min. D: Louis King. SC: Oliver Drake, Jean Dupont, John Twist and Frank T. Daugherty. With Buzz Barton, Frank Rice, James Welsh, Gloria Lee, Milburn Morante, Hugh Trevor, Bill Patton, Walter Shumway. A young boy and his adult pal save a pretty ranch owner from quicksand and stop a crook from taking her property. A welcome chance to see juvenile star Buzz Barton in one of his starring silent Westerns.\n\n**3100** _ **The Pinto Kid**_ **** Columbia, 1941. 61 min. D: Lambert Hillyer. SC: Fred Myton. With Charles Starrett, Louise Currie, The Sons of the Pioneers (Bob Nolan, Tim Spencer, Hugh and Karl Farr, Tim Spencer, Lloyd Perryman, Pat Brady), Paul Sutton, Hank Bell, Francis Walker, Ernie Adams, Jack Rockwell, Roger Gray, Richard Botiller, Steve Clark, Frank Ellis, Art Dillard. After a cattleman is framed by a gang leader for a bank robbery he takes refuge at a young woman's ranch but she is kidnapped by the outlaws. Standard Charles Starrett adventure.\n\n**3101** _ **Pinto Rustlers**_ **** Reliable, 1936. 56 min. D: Henri Samuels (Harry S. Webb). SC: Robert Tansey. With Tom Tyler, George Walsh, Al St. John, Catherine Cotter, Earl Dwire, William Gould, George Chesebro, Roger Williams, Bud Osborne, Murdock MacQuarrie, Charles King, Slim Whitaker, Milburn Morante, Sherry Tansey, Richard Cramer, Wally West, Bob Burns. Rustlers kill his father and a man pretends to be an outlaw so he can join the gang to get the goods on them. Pretty fair Tom Tyler oater for which R.G. Springsteen was the assistant director.\n\n_**Pioneer Builders**_ see _**The Conquerors**_\n\n**3102** _ **Pioneer Days**_ **** Monogram, 1940. 51 min. D: Harry S. Webb. SC: Bennett Cohen. With Jack Randall, June Wilkins, Frank Yaconelli, Ted Adams, Nelson McDowell, Bud Osborne, Robert Walker, Glenn Strange, George Chesebro, Lafe McKee, Richard Cramer, Jimmy Aubrey, Denver Dixon. A cowboy helps a woman who has inherited half interest in a saloon while her partner wants the place for himself and tries to cheat her. Average Jack Randall vehicle.\n\n**3103** _ **Pioneer Justice**_ **** Producers Releasing Corporation, 1947. 56 min. D: Ray Taylor. SC: Adrian Page. With Lash LaRue, Al St. John, Jennifer Holt, William Fawcett, Jack Ingram, Dee Cooper, Lane Bradford, Henry Hall, Terry Frost, Slim Whitaker, Wally West, Bob Woodward, Steve Drake. U.S. marshals Cheyenne Davis and Fuzzy Q. Jones help homesteaders who have been victims of killings and property seizures. Very good Lash LaRue feature; non-stop action from the start.\n\n**3104** _ **Pioneer Marshal**_ **** Republic, 1949. 60 min. D: Philip Ford. SC: Bob Williams. With Monte Hale, Paul Hurst, Nan Leslie, Damian O'Flynn, Roy Barcroft, Myron Healey, Ray Walker, John Hamilton, Marshall Reed, Clarence Straight, Robert Williams. A lawman infiltrates a town used by outlaws as he tracks down a notorious gunman. Well made Monte Hale film with an exciting climactic gunfight.\n\n**3105** _ **Pioneer Trail**_ **** Columbia, 1938. 55 min. D: Joseph Lovering. SC: Nate Gatzert. With Jack Luden, Joan Barclay, Hal Taliaferro, Marin Sais, Slim Whitaker, Leon Beaumont, Eva McKenzie, Hal Price, Richard Botiller, Tom London, Bud Osborne, Bob McKenzie, Art Mix, Fred Burns, Peter Palmer, Tuffy (dog). A foreman convinces area ranchers to take their herds to market in one big drive in order to get better prices but he is captured by a female outlaw and her gang. Low grade Jack Luden vehicle with more heroics from Tuffy the dog than the star.\n\n**3106** _ **Pioneer Woman**_ **** ABC-TV\/Filmways, 1973. 74 min. D: Buzz Kulik. SC: Suzanne Clauser. With Joanna Pettet, William Shatner, David Janssen, Lance LeGault, Helen Hunt, Russell Baer, Linda Kupecek, Lloyd Berry, Robert Koons, Agatha Mercer. In 1867 a woman struggles to keep her family together in Wyoming Territory and continue homesteading after her husband is killed. Fair made-for-TV feature.\n\n**3107** _ **The Pioneers**_ **** Monogram, 1941. 59 min. D: Al Herman. SC: Charles Alderson. With Tex Ritter, Wanda McKay, Red Foley and His Saddle Pals, Arkansas Slim Andrews, Doye O'Dell, George Chesebro, Del Lawrence, Post Park, Karl Hackett, Lynton Brent, Chick Hannon, Gene Alsace, Jack C. Smith, Chief Many Treaties, Art Dillard, Charles Soldani, Tex Palmer, Sherry Tansey. A singing cowboy leads a group of settlers to their destination, fighting Indians and outlaws along the way. Poor Tex Ritter feature, supposedly based on James Fenimore Cooper, that is full of stock footage and too much music; the Indiana attack sequence is from _**Fighting with Kit Carson**_ (q.v.).\n\n**3108** _ **Pioneer's Gold**_ **** Sanford Productions, 1924. 61 min. D-SC: Denver Dixon (Victor Adamson). With Pete Morrison, Kathryn McGuire, Virginia Warwick, Spottiswoode Aitken, Louise Emmons, Madge Lorese Bates, Merrill McCormick, Les Bates, George King, George Sowards. A rancher plans to leave his wealth to relatives, hoping they will marry but the two are kidnapped and imposters take their place. Meandering, low grade poverty row silent affair from producer F.M. Sanford.\n\n**3109** _ **Pioneers of the Frontier**_ **** Columbia, 1940. 58 min. D: Sam Nelson. SC: Fred Myton. With Bill Elliott, Linda Winters (Dorothy Comingore), Dub Taylor, Dick Curtis, Lafe McKee, Stanley Brown, Richard Fiske, Carl Stockdale, Ralph McCullough, Alan Bridge, Edmund Cobb, George Chesebro, Lynton Brent, Jack Kirk, Ralph Peters, Blackjack Ward, Ed Coxen, Hank Bell, Jay Wilsey (Buffalo Bill, Jr.), Francis Walker, Buddy Cox, George Morrell, Art Dillard, Jim Corey, Ray Jones, Bob Card. A ruthless gunman murders his kindly land baron boss and takes over the range but the dead man's nephew arrives to help local settlers oppose his tyranny. Very good \"Wild Bill Saunders\" series thriller with an excellent performance by Dick Curtis as the brutal bully.\n\n**3110** _ **Pioneers of the West**_ **** Bill Mix Productions, 1927. 50 min. D-SC: Marcel Perez. With Dick Carter, Dorothy Earle, Bud Osborne, Gene Crosby, Olin Francis. A Pony Express rider tries to stop an Indian uprising led by a renegade, his rival for a pretty white squaw also wanted by the tribe's chief. Cheap silent that is fun to watch; produced by Bill Mix, star Dick Carter's real name.\n\n**3111** _ **Pioneers of the West**_ **** Republic, 1940. 56 min. D: Lester Orlebeck. SC: Jack Natteford, Karen DeWolf and Gerald Geraghty. With Robert Livingston, Raymond Hatton, Duncan Renaldo, Noah Beery, Beatrice Roberts, Lane Chandler, George Cleveland, Hal Taliaferro, Yakima Canutt, John Dilson, Joe McGuinn, Earl Askam, George Chesebro, Jack Kirk, Herman Hack, Bob Burns, Tex Terry, Art Dillard, Ray Jones, Artie Ortego, Chuck Baldra, Chief Big Tree, Frankie Marvin, Duke Taylor, Jane Keckley, Hansel Warner, Cecil Weston, Iron Eyes Cody. Settlers swindled out of their lands are helped by the Three Mesquiteers as they cross Indian territory by wagon train. Good \"Three Mesquiteers\" entry enhanced by a fine cast of veteran players.\n\n**3112** _ **Pirates of Monterrey**_ **** Universal-International, 1947. 75 min. Color. D: Alfred Werker. SC: Sam Hellman and Margaret Buell Wilder. With Maria Montez, Rod Cameron, Mikhail Rasumny, Philip Reed, Gilbert Roland, Gale Sondergaard, Tamara Shayne, Robert Warwick, Michael Raffetto, Neyle Morrow, Victor Varconi, Charles Wagenheim, George J. Lewis, Joe Barnard, George Navarro, Victor Romito, Don Driggers, George Magrill, Lucio Villegas, Chris-Pin Martin, Julia Andre, Fred Cordova, Dick Dickinson. In 1840s California a Spanish noblewoman arrives to wed a soldier but falls for an American aiding the Mexican government is trying to put down a rebellion. Dull Maria Montez outing with good support from Rod Cameron and Gilbert Roland.\n\n**3113** _ **The Pirates of the Mississippi**_ **** Rapid Film, 1963. 95 min. Color. D: Jurgen Roland. SC: Werner P. Zibaso and Johannes Kas. With Horst Frank, Brad Harris, Hansjorg Felmy, Sabine Sinjen, Dorothea Parker, Tony Kendall, Dan Vadis, Jeanette Batti, Paolo Solvay. River pirates steal an Indian land grant during a mail robbery and take over a town but the Cherokees save the settlers and take revenge on the marauders. Far out, but sumptuous, West German oater for fans of this type of fare; made as _**Die Flusspiraten vom Mississippi**_ (The River Pirates of the Mississippi).\n\n**3114** _ **Pirates of the Plains.**_ Colorado Motion Picture Company, 1914. 36 min. D: Otis Thayer. With Colin Chase, Josephine Wells, Joe Ryan. A horse thief frames his rodeo rider champion brother on a murder charge but after being wounded in a gunfight confesses and the victim's fiancee rushes to save him from the gallows. Well made and photographed silent, filmed in Colorado.\n\n**3115** _ **Pirates of the Prairie**_ **** RKO Radio, 1942. 57 min. D: Howard Bretherton. SC: Doris Schroeder and J. Benton Cheney. With Tim Holt, Nell O'Day, Cliff Edwards, Roy Barcroft, John Elliott, Karl Hackett, Richard Cramer, Ed Cassidy, Eddie Dew, Merrill McCormick, Reed Howes, Charles King, Bud Geary, Lee Shumway, Russell Wade, Ben Corbett, Frank McCarroll, Artie Ortego, George Morrell. A U.S. marshal is after a gang of masked riders stealing land in order to sell it for a big profit when the railroad comes through. Good Tim Holt affair with plenty of action.\n\n**3116** _ **Pirates on Horseback**_ **** Paramount, 1941. 66 min. D: Lesley Selander. SC: Ethel La Blanche and J. Benton Cheney. With William Boyd, Russell Hayden, Andy Clyde, Eleanor Stewart, Morris Ankrum, William Haade, Dennis Moore, Henry Hall, Britt Wood, Chief Thundercloud, Bruce Mitchell, Wen Wright, Henry Wills, George Sowards, Chuck Morrison, Tom Smith, Ray Henderson, Tex Harper, Silver Tip Baker. Hopalong Cassidy and his pals trail an outlaw gang trying to locate a hidden gold mine. Another well made entry in the popular \"Hopalong Cassidy\" series, this one enhanced by a mystery plot.\n\n**3117** _ **Pistol for a Hundred Coffins**_ **** Sanchez Ramada, 1968. 83 min. Color. D: Umberto Lenzi. SC: Marco Leto and Vittorio Salerno. With Peter Lee Lawrence, John Ireland, Gloria Osuna, Eduardo Fajardo, Julio Pena, Raf Baldassare, Piero Lulli, Franco Pesce, Andrea Scotti (Andrew Scott), Calisto Calisti, Franco Narducci, Ivan Scratuglia, Paola Natale, Giovanni Petti, Francisco (Frank) Brana, Luis de Jejada, Armando Calvo, Victor Israel, Alfonso Rojas, Jose Jaspe, Giovanni Petrucci, Miguel Del Castillo, Miguel Guzman, Joaquin Parra, Gino Turini, Leonidas Guerra. Coming home from the Civil War, a soldier finds his parents have been murdered and he takes the job as local sheriff to get the gang that committed the killings. Better than usual Italian-Spanish co-production with an especially good performance by John Ireland as a gunslinger minister; released in Europe as _**Una Pistola per Cento Bare**_ (A Pistol for One Hundred Coffins).\n\n**3118** _ **A Pistol for Ringo**_ **** Embassy, 1966. 97 min. Color. D-SC: Duccio Tessari. With Montgomery Wood (Giuliano Gemma), Fernando Sancho, Hally Hammond, George Martin, Nieves Navarro, Antonio Casas, Jose Manuel Martin, Paco Sanz, Parajito. A sheriff enlists the help of a gunman to save the ranch where his fiancee and her father are held prisoners by a wounded Mexican bandit and his gang. Another brutal Spaghetti Western with a plot and cast similar to its successor, _**The Return of Ringo**_ (q.v.); released in Italy as _**Una Pistola per Ringo**_ (A Pistol for Ringo) and on video as _**Ballad of Death Valley**_ and _**A Gun for Ringo**_.\n\n**3119** _ **Pistol Harvest**_ **** RKO Radio, 1951. 60 min. D: Lesley Selander. SC: Norman Houston. With Tim Holt, Richard Martin, Joan Dixon, Guy Edward Hearn, Mauritz Hugo, Robert Clarke, William Griffith, Lee Phelps, Robert Wilke, Joan Freeman. When their boss is murdered by rustlers, two cowboys try to track down the killers. Average Tim Holt feature.\n\n**3120** _ **Pistol Packin' Mama**_ **** Republic, 1943. 64 min. D: Frank Woodruff. SC: Edward Dein and Fred Schiller. With Ruth Terry, Robert Livingston, Wally Vernon, Jack LaRue, Kirk Alyn, Helen Talbot, Eddie Parker, Lydia Bilbrook, George Lessey, Joseph Kirk, The (Nat) King Cole Trio. After being cheated by a slick East coast night club proprietor, a Nevada gambling house owner goes to New York City, changes her name and goes up against him and gangsters. Peripheral modern-day Western inspired by Al Dexter's hit song.\n\n_**Pistol Packin' Preacher**_ see _**Leave Your Guns at the Door!**_\n\n_**Pistolero of Red River**_ see _**The Last Challenge**_\n\n**3121** _ **Pistoleros**_ **** Constantin Film, 1967. 98 min. Color. D-SC: Alfio Caltabinao. With Anthony Ghidra, Angelo Infanti, Anthony Freeman, Al Northon, Dan May, Monica Teuber, Ivan Scraguglia, Ellen Schwiers, Peter Jacob, Lanfranco Ceccarelli, Giovanni Gianfriglia, Nicola Balini. Two outlaw siblings attempt to rob a train but are beaten to it by a gang who abducts a young woman as a hostage with the brothers taking revenge. Retread Italian-West German oater that moves at a fair clip; released in Europe as _**Ballata per un Pistolero**_ (Ballad of a Gunman) with actor Hermann Nehlsen credited in the West German version.\n\n**3122** _ **Pistoleros Asesinos**_ (Killer Gunmen) **** Peraco, 1982. 85 min. Color. D: Angel Rodriguez Vaquez. With Rosa Gloria Chagoyan, Federico Villa, Victor Alcocer, Polo Ortin, Armando Soto La Marina, Gilberto Trujillo, Dolores Camarillo, Maria de Lourdes. A cowboy falls in love with the daughter of a rancher who is being forced to marry her father's blackmailer. Dour Mexican Western melodrama.\n\n**3123** _ **Pistoleros Bajo el Sol**_ (Gunmen Under the Sun) **** Filmadora Chapultepec, 1972. 93 min. Color. D: Ruben Galindo. With Fernando Almada, Sasha Montenegro, Juan Gallardo, Maria Fernanda, Raul Hernandez, Bruno Rey, Marco Antonio Campos, Alfredo Gutierrez, Federico Falcon, Marcela Mozzarella, Marcelo Vittamil, Mario Cit, Regino Herrera, Carlos Jordan, Hernando Name, Jorge Patino. After outlaws murder his brother, a man hunts them down in revenge, is left by the gang to die in the desert and is rescued by a widow later raped by one of the outlaws. Violent Mexican oater.\n\n**3124** _ **Pistoleros del Oeste**_ (Gunmen of the West) **** Alameda Films, 1965. 95 min. D: Rene Cordona. SC: Abel Salazar. With Abel Salazar, Rosa de Castilla, Luz Maria Aguilar, Conrado Cortes, Eleazar Garcia, Hector Suarez, Emilio Brillas, Enrique Lucero, Mario Alberto Rodriguez, Enrique Rocha, Nathaniel (Frankenstein) Leon, Luis Aragon, Emilio Garibay, Armando Gutierrez, Antonio Raxel, Sergio Ramos, Pascual Garcia Pena, Evangelina Elizondo. Kidnapping the wrong woman, two gunslingers find themselves in deep trouble when they try to exchange her for the ransom money. Fun Mexican Western comedy produced and written by star Abel Salazar.\n\n**3125** _ **A Place Called Glory**_ **** Embassy, 1966. 92 min. Color. D: Sheldon Reynolds. SC: Edward Di Lorenzo, Jerold Hayden Boyd and Fernando Lamas. With Lex Barker, Pierre Brice, Marianne Koch, Jorge Rigaud, Gerard Tichy, Angel Del Pozo, Aldo Sambrell, Santiago Intanon, Hans Nielsen, Wolfgang Lukschy, Victor Israel. In the town of Glory, two gunmen plan to oppose each other in a duel but end up joining forces to stop an bandit gang. Pretty good West German Western (co-written by actor Fernando Lamas) with fine work by Lex Barker and Pierre Brice in the leads; released in its homeland as _**Die Holle von Manitoba**_ (The Hell of Manitoba) by Omnia Deutsche Film Export.\n\n_**A Place Called Trinity**_ see _**Jesse and Lester**_\n\n**3126** _ **The Plainsman**_ **** Paramount, 1936. 115 min. D: Cecil B. DeMille. SC: Waldemar Young, Harold Lamb and Lynn Riggs. With Gary Cooper, Jean Arthur, James Ellison, Charles Bickford, Porter Hall, Helen Burgess, John Miljan, Victor Varconi, Paul Harvey, Frank McGlynn, Granville Bates, Purnell Pratt, Pat Moriarty, Charles Judels, Harry Woods, Anthony Quinn, Francis McDonald, George MacQuarrie, George Hayes, Fuzzy Knight, George Ernest, Fred Kohler, Frank Albertson, Francis Ford, Irving Bacon, Edgar Dearing, Edwin Maxwell, John Hyams, Bruce Warren, Mark Strong, Charles Stevens, Arthur Aylesworth, George Cleveland, Lona Andre, Leila McIntyre, Harry Stubbs, Davison Clark, C.W. Herzinger, William Humphrey, Sidney Jarvis, Wadsworth Harris, Bud Flanagan (Dennis O'Keefe), Gail Sheridan, Lane Chandler, Franklyn Farnum, Douglas Wood, Bud Osborne, Noble Johnson, Ted Oliver, Jim Mason, William Royle, Tex Driscoll, Francis Sayles, Hank Bell, Jonathan Hale, Hank Worden, Earl Askam, Paul Newlan, Chief Thundercloud, Edgar \"Blue\" Washington, Richard Alexander, Hooper Atchley, Frank Ellis, Max Davidson, Buck Connors, Ken Cooper, Frank Cordell, Bob Burns, Jess Cavin, Jack Clifford, Sonny Chorre, Richard Robles, David Clyde, Walter McGrail, Wesley Giraud, Nelson McDowell, Chuck Hamilton, Philo McCullough, Ben Hendricks, Jr., Duke R. Lee, Jane Keckley, Cecil Kellogg, Slim Hightower, Wilbur Mack, Frank Layton, Carl Miller, Jimmy Phillips, Oscar Rudolph, Louise Stuart, Jack Walters, Frank Watson, Robert Wilber, Don Rowan. Wild Bill Hickok, Calamity Jane and Buffalo Bill Cody team to oppose a gun runner selling to the Indians out to get General Custer. Highly inaccurate but quite entertaining Cecil B. DeMille epic.\n\n**3127** _ **The Plainsman**_ **** Universal, 1966. 92 min. Color. D: David Lowell Rich. SC: Michael Blankfort. With Don Murray, Abby Dalton, Guy Stockwell, Bradford Dillman, Henry Silva, Simon Oakland, Leslie Nielsen, Edward Binns, Michael Evans, Percy Rodriguez, Terry Wilson, Walter Burke, Emily Banks. A renegade white man is selling guns to the Indians planning to attack General Custer with Wild Bill Hickok, Calamity Jane and Buffalo Bill Cody trying to stop him. Bland remake of the 1938 (q.v.) film with little to recommend it.\n\n**3128** _ **The Plainsman and the Lady**_ **** Republic 1946. 82 min. D: Joseph Kane. SC: Richard Wormser. With William Elliott, Vera Ralson, Gail Patrick, Joseph Schildkraut, Donald Barry, Andy Clyde, Raymond Walburn, Reinhold Schunzel, Paul Hurst, William B. Davidson, Charles Judels, Eva Puig, Jack Lambert, Stuart Hamblen, Noble Johnson, Hal Taliaferro, Byron Foulger, Pierre Watkin, Eddy Waller, Charles Morton, Martin Garralaga, Guy Beach, Joseph Crehan, Grady Sutton, Eddie Parks, Norman Willis, Tex Terry, Chuck Roberson, Rex Lease, Henry Wills, Hank Bell, Roy Barcroft, Jack O'Shea, Carl Sepulveda, Daniel Day Tolman, David Williams, Lola and Fernando. Before the Civil War a rich cattleman helps a banker and his pretty daughter establish the Pony Express between St. Joseph, Missouri, and Sacramento, California, despite the machinations of a crooked stagecoach operator and his murderous henchman. Slick Bill Elliott \"A\" film that benefits from good production values with nice support from Don Barry as the gunman and Gail Patrick as the heroine's social climbing sister.\n\n**3129** _ **Plainsong**_ **** Ed Stabile, 1983. 88 min. Color. D-SC: Ed Stabile. With Teresanne Joseph, Jessica Nelson, Lyn Traverse, Steve Geiger, Sandon McCall, Carl Kielblock. A man and a group of women arrive in Kansas in the 1880s and find themselves in the middle of a range war. Mini-budget New Jersey filmed drama has little to offer genre fans.\n\n**3130** _ **The Pledge**_ **** Hallmark Channel, 2008. 79 min. Color. D: Armand Mastroianni. SC: Jim Byrnes. With Luke Perry, C. Thomas Howell, Kim Coates, Jaclyn DeSantis, Francesco Quinn, Jorge-Luis Pallo, Wyatt Smith, Nicholas Guest, Alex Paez, Johann Urb, Daniel Wisler, James Keane, Chip Sickler, Jeffrey Markle, Lisa Brenner, Franc Ross, Laci Greenfield, Jordan Timsit. A lawman seeks revenge against a land baron who ordered the murder of his wife and son so he could get his family's land. Nice photography highlights this more than passable TV oater; also called _**A Gunfighter's Pledge**_.\n\n**3131** _ **Plunder of the Sun**_ **** Warner Bros., 1953. 81 min. D: John Farrow. SC: Jonathan Latimer. With Glenn Ford, Diana Lynn, Patricia Medina, Francis L. Sullivan, Sean McClory, Eduardo Noriega, Julio Villareal, Charles Rooner, Douglass Dumbrille, Mona Barrie, Carlos Muzquiz, Juan Garcia, Margarito Luna, Victorio Blanco. An expedition in Mexico sets out to find Aztec treasure. Mystery writer Jonathan Latimer's script would have benefited from more action and less talk in an otherwise okay melodrama.\n\n**3132** _ **The Plunderers**_ **** Republic, 1948. 87 min. Color. D: Joseph Kane. SC: Gerald Geraghty and Gerald Adams. With Rod Cameron, Ilona Massey, Adrian Booth, Forrest Tucker, George Cleveland, Grant Withers, Taylor Holmes, Paul Fix, Francis Ford, James Flavin, Maude Eburne, Russell Hicks, Mary Ruth Wade, Clayton Moore, Roy Barcroft, Hank Bell, Rex Lease, Louis R. Faust, John Hart, Bud Osborne, Kenneth MacDonald, Steve Clark, Tex Terry, Forrest Taylor, Guy Wilkerson, House Peters, Jr., Monte Montague, Hugh Prosser, Jack O'Shea, Augie Gomez, Hank Patterson, Al Murphy, Wheaton Chambers, Kenneth Terrell, Tex Cooper, Craig Lawrence, Dewey Troub, Garrett Craig, John Hilton. An Army officer is sent to bring in a young outlaw but they join forces when attacked by rampaging Indians. Fairly good action laden production.\n\n**3133** _ **The Plunderers**_ **** Allied Artists, 1960. 94 min. D: Joseph Pevney. SC: Bob Barbash. With Jeff Chandler, Dolores Hart, Marsha Hunt, John Saxon, Jay C. Flippen, Ray Stricklyn, James Westerfield, Harvey Stephens, Vaughn Taylor, William Challee, Ken Patterson, Dee Pollack, Roger Torrey, Joseph Hamilton, Ray Ferrell, Ella Ethridge. A quartet of hellions attempt to take over a Western town but are opposed by a Civil War veteran. Nicely done oater from producer Lindsley Parsons.\n\n**3134** _ **The Plunderers of Painted Flats**_ **** Republic, 1959. 70 min. D: Albert C. Gannaway. SC: Phil Shaken and John Greene. With John Carroll, Corinne Calvet, Skip Homeier, Edmund Lowe, George Macready, Bea Benadaret, Madge Kennedy, Joe Besser, Allan Lurie, Candy Candido, Herbert Vigran, Burt Topper, Roy Gordon, Bob Kline, Bill Foster, Lee Redman, Wade Lane, David Waldor, John Kidd. A town boss wants to run settlers out of the area and hires a notorious gunman to do his bidding, but one of the intended victims is a young man out to kill the gunslinger for murdering his father. Mediocre production greatly helped by fine performances, especially Edmund Lowe as an aged shootist and Corinne Calvet as a woman with a past; Republic Pictures final production.\n\n**3135** _ **The Pocatello Kid**_ **** Tiffany, 1931. 61min. D: Phil Rosen. SC: W. Scott Darling. With Ken Maynard, Marceline Day, Richard Cramer, Charles King, Lafe McKee, Lew Meehan, Jack Rockwell, Bert Lindley, Bob Reeves, Bud Osborne, Jack (Blackjack) Ward, Arthur Millett, Jack King, Archie Ricks, Jim Corey, Al Taylor, Bob Card. Falsely thinking he is responsible for the death of his dishonest lawman brother, a cowboy takes his place and plans to stop a rustling gang. Slow moving Ken Maynard feature, hardly one of the star's better efforts.\n\n**3136** _ **Pocket Money**_ **** National General, 1972. 102 min. Color. D: Stuart Rosenberg. SC: Terry Malick. With Paul Newman, Lee Marvin, Strother Martin, Christine Belford, Kelly Jean Peters, Fred Graham, Wayne Rogers, Hector Elizondo, Mickey Gilbert, Gregg Sierra, Matt Clark, Claudio Miranda, Richard Farnsworth, Ken Freehill, Bruce Davis Bayne, Terry Malick. A down-on-his-luck cowpoke heads to Mexico to buy cattle from a crooked dealer and enlists the help of a shiftless pal in getting the goods on the man. Poor genre \"comedy\" somewhat saved by Lee Marvin's mugging.\n\n**Paul Newman in** _**Pocket Money**_ **(National General, 1972).**\n\n**  \n**\n\n**3137** _ **Poco**_ **** Cinema Shares, 1977. 88 min. Color. D: Dwight Brooks. SC: William E. Carville. With Chill Wills, Michaelle Ashburn, Clint Ritchie, Sherry Bain, John Steadman, Tom Roy Lowe. A small dog, lost after a car wreck, treks through the desert in search of his owner, a disabled little girl. Pleasing family feature.\n\n**3138** _ **Poker Alice**_ **** CBS-TV, 1987. 104 min. Color. D: Arthur Allan Seidelman. SC: James Lee Barrett. With Elizabeth Taylor, George Hamilton, Tom Skerritt, Richard Mulligan, Paul Drake, Susan Tyrrell, Pat Corley, David Wayne, Merrya Small, Liz Torres, Gary Grubbs, Annabella Price, Gary Bisig, John Bennett Perry, Ed Adams, Sid Dawson, Jack Dunlop, Marten Goslins, William M. Hannah, Henry Max Kendrick, Stephen Jace Kent, Gloria Manon, John Pearce, Caroline Reed, Bob Shelton. A beautiful woman gambler outwits a variety of male gamers before finding true love with a bounty hunter. Glossy TV movie centered around its glamorous star.\n\n**3139** _ **The Pony Express**_ **** Paramount, 1925. 90 min. D: James Cruze. SC: Walter Woods. With Betty Compson, Ricardo Cortez, Ernest Torrance, Wallace Beery, George Bancroft, Frank Lackteen, Ed Peil, Sr., William Turner, Al Hart, Charles Gerson, Rose Tapley, Vondell Darr, Hank Bell, Ernie Adams, Toby Wing. During the Civil War the Knights of the Golden Circle try to get California to secede from the Union with the plan opposed by a Pony Express rider. Fine silent feature with a good story and plenty of action.\n\n**3140** _ **The Pony Express**_ **** Paramount, 1953. 101 min. Color. D: Jerry Hopper. SC: Charles Marquis Warren. With Charlton Heston, Rhonda Fleming, Forrest Tucker, Jan Sterling, Michael Moore, Porter Hall, Richard Shannon, Henry Brandon, Stuart Randall, Lewis Martin, Pat Hogan, James Davies, Eric Alden, Willard Willingham, Frank Wilcox, Len Hendry, Charles Hamilton, Bob Templeton. Buffalo Bill Cody and Wild Bill Hickok team to establish the Pony Express across the West and stop the secession of California from the Union during the Civil War. Okay yarn has colorful characters and good movement.\n\n**3141** _ **Pony Express Rider**_ **** Doty-Dayton, 1976. 100 min. Color. D: Robert Totten. SC: Dan Greer, Hal Harrison, Jr. and Robert Totten. With Stewart Petersen, Jack Elam, Henry Wilcoxon, Joan Caulfield, Slim Pickens, Dub Taylor, Buck Taylor, Maureen McCormack, Ace Reis. While looking for his father's killer, a young man finds a murdered Pony Express rider and decides to finish his mail run. Okay action film helped by a good cast.\n\n**3142** _ **Pony Post**_ **** Universal, 1940. 61 min. D: Ray Taylor. SC: Sherman Lowe. With Johnny Mack Brown, Nell O'Day, Dorothy Short, Ray Teal, Tom Chatterton, Kermit Maynard, Stanley Blystone, Jack Rockwell, Edmund Cobb, Lloyd Ingraham, Iron Eyes Cody, Charles King, Worth Crouch, Jimmy Wakely and His Rough Riders (Johnny Bond, Dick Reinhart), Lane Chandler, Frank McCarroll, Chuck Morrison, George Hazel, Frank McCarroll, Bill Wolfe. A Pony Express operator runs into opposition from outlaws and Indians when he tries to open a relay station in an isolated valley. More than passable Johnny Mack Brown vehicle.\n\n**3143** _ **Pony Soldier**_ **** 20th Century\u2013Fox, 1952. 82 min. Color. D: Joseph M. Newman. SC: John D. Higgins. With Tyrone Power, Cameron Mitchell, Penny Edwards, Thomas Gomez, Robert Horton, Anthony Numken, Adeline De Walt Reynolds, Howard Petrie, Stuart Randall, Richard Shackelford, James Hayward, Muriel Landers, Frank De Kova, Louis Heminger, John War Eagle. A Mountie tries to keep Cree Indians from going on the warpath as he escorts them back to the reservation. Colorful but not exceptional Northwest Mounted Police drama.\n\n**3144** _ **Por Mis Pistolas**_ (By My Pistols) **** Columbia\/Posa Films, 1968. 123 min. Color. D: Miguel M. Delgado. SC: Mario Moreno (Cantinflas) and Marco Antonio Almazon. With Cantinflas (Mario Moreno), Isela Vega, Jorge Rado, Alfonso Mejia, Gloria Coral, Quintin Bulnes, Manuel Alvarado, Rhea (Frichina), Carlos Cardon, John Kelly, Eduardo Alcarez, Pedro Galvan, Agustin Isunza, Carlos Pouliot, Jose Torvay. An easygoing druggist heads West where he gets mixed up with Indians, outlaws and a lost mine. Action packed comedy from Mexican star Cantinflas.\n\n**3145** _ **El Porto Salvaje**_ (The Wild Appearance) **** Alameda Films, 1958. 90 min. D: Rafael Baledon. With Gaston Santos, Rodolfo Landa, Lilia Martinez, Carmen Montejo. When a wild stallion is accused of several killings, a cowboy and his pal try to prove his innocence by finding the real culprits. Routine south of the border oater.\n\n_**Portrait of a Dead Girl**_ see _**McCloud: Who Killed Miss U.S.A.?**_\n\n**3146** _ **Posse**_ **** Paramount, 1975. 94 min. Color. D: Kirk Douglas. SC: William Roberts and Christopher Knof. With Kirk Douglas, Bruce Dern, Bo Hopkins, James Stacy, Alfonso Arau, David Canary, Luke Askew, Beth Brickell, Katherine Woodville, Mark Roberts, Dick O'Neill, Bill Burton, Allan Warnick, Roger Behrstock. An ambitious, dishonest politician tries to get himself elected to the senate by hunting a wanted man, but the outlaw turns the tables on him. Offbeat and interesting oater from producer-director-star Kirk Douglas, with the good guy and the bad man changing roles at the finale.\n\n**3147** _ **Posse**_ **** Gramercy Pictures, 1993. 111 min. Color. D: Mario Van Pebbles. SC: Sy Richardson and Dario Scardapane. With Mario Van Pebbles, Stephen Baldwin, Charles Lane, Tiny Lister, Big Daddy Kane, Billy Zane, Blair Underwood, Melvin Van Pebbles, Salli Richardson, Tone Loc, Pam Grier, Vesta (Williams), Isaac Hayes, Richard Jordan, Paul Bartel, Stephen J. Cannell, Richard Edson, Nipsey Russell, Reginald VelJohnson, Woody Strode, Reginald Hudlin, Warrington Hudlin, Aaron Neville, James Bigwood, Mark Buntzman, Ismael Calderon, Tracy Lee Chavis, James E. Christopher, Lawrence Cook, Richard Grant, Thomas Steven Hall, Robert Hooks, Sandra Ellis Lafferty, Jeffrey Lloyd Layne, Robert May, T.J. McClain, Christopher Michael, Bob Minor, Steve Reevis, Sy Richardson, Dario Scardapane, Frank A. Soto, David Jean Thomas, Mark Twogood, Karen Williams. The black leader of a group of infantry men prowls the West looking for the gang that hung their father. Confusing revenge motif dampens this otherwise moderately okay outing.\n\n**3148** _ **Posse from Heaven**_ **** P.M. Films, 1975. 87 min. Color. D: Phillip Pine. SC: Ward Wood and Phillip Pine. With Fanne Foxe, Todd Compton, Sherry Bain, Ward Wood, Dick Burch. A non-too-bright young man is sent to the Old West by God to save it from sin and the Archangel Gabriel comes along, reincarnated as a horse, to protect him. Hard-to-believe Western fantasy made on the cheap.\n\n**3149** _ **Posse from Hell**_ **** Universal-International, 1961. 89 min. Color. D: Herbert Coleman. SC: Clair Huffaker. With Audie Murphy, John Saxon, Zohra Lampert, Vic Morrow, Robert Keith, Ward Ramsey, Rodolfo Acosta, Royal Dano, Frank Overton, James Bell, Paul Carr, Lee Van Cleef, Ray Teal, Forrest Lewis, Charles Horvath, Harry Lauter, Henry Wills, Stuart Randell, Allan Lane, Rand Brooks, I. Stanford Jolley, Kenneth MacDonald, Steve Darrell. When four escaped convicts murder the town marshal, an ex-gunman friend of the lawman forms a posse to catch the killers. Clair Huffaker adapted his novel for this film but the end result is nothing exceptional.\n\n**3150** _ **Powder River**_ **** 20th Century\u2013Fox, 1953. 77 min. Color. D: Louis King. SC: Geoffrey Homes. With Rory Calhoun, Corinne Calvet, Cameron Mitchell, Penny Edwards, Carl Betz, John Dehner, Raymond Greenleaf, Victor Sutherland, Ethan Laidlaw, Robert Wilke, Harry Carter, Robert Adler, Post Park, Richard Garrick, James Griffith, Stanley Andrews, Frank Ferguson, Henry Kulky, Walter Sande, Zon Murray, Ray Bennett, Arthur MacDonald, Val Setz, Harry Hines, Hank Worden, Robert Foulk, Doodles Weaver, Mae Marsh, Emmett Vogan, Eddy Waller, George Lynn, Edward Hearn, John Berardino, Paul E. Burns, Phil Chambers, Ruth Warren, Paul Newlan, Harry Seymour. After his buddy is murdered, a man takes the job of town sheriff in order to capture the gambler he suspects of the crime, but he too is killed. Minor, but rather interesting oater with a good cast.\n\n**3151** _ **Powder River Rustlers**_ **** Republic, 1949. 60 min. D: Philip ford. SC: Richard Wormser. With Allan \"Rocky\" Lane, Eddy Waller, Gerry Ganzer, Roy Barcroft, Bud Geary, Cliff Clark, Francis McDonald, Douglas Evans, Bruce Edwards, Stanley Blystone, Eddie Parker, Herman Hack. A government agent investigates a scheme to bilk locals through a bridge building contract. Although it contains some action, this is one of the more drab \"Famous Westerns\" entries.\n\n**3152** _ **Powderkeg**_ **** CBS-TV\/Filmways, 1971. 93 min. Color. D-SC: Douglas Heyes. With Rod Taylor, Dennis Cole, Fernando Lamas, Luciana Paluzzi, John McIntire, Michael Ansara, Tisha Sterling, Reni Santoni, Melodie Johnson, William Bryant, Joe DeSantis, Jay Novello, Jim L. (James) Brown, Roy Jenson. In 1914 outlaws hijack a train and hold its passenger hostages as two troubleshooters are hired to capture the gang. Pretty fair action drama that resulted in the brief series \"The Bearcats\" (CBS-TV, 1971).\n\n**3153** _ **Powdersmoke Range**_ **** RKO Radio, 1935. 72 min. D: Wallace Fox. SC: Adele Buffington. With Harry Carey, Hoot Gibson, Guinn Williams, Bob Steele, Tom Tyler, Boots Mallory, Ray Mayer, Sam Hardy, Adrian Morris, Wally Wales, Art Mix, Buddy Roosevelt, Buffalo Bill, Jr., Franklyn Farnum, William Desmond, William Farnum, Buzz Barton, Ethan Laidlaw, Irving Bacon, Henry Roquemore, Bob McKenzie, Frank Rice, Eddie Dunn, Barney Furey, Jim Mason, Nelson McDowell, Frank Ellis, Phil Dunham, Tex Palmer, George Sowards, Lem Sowards, Silver Tip Baker, Silver Harr, Charles Murphy. When their pal is framed on a murder charge by a crook and his lawman cohort, three cowboys come to his defense but are forced to go up against a notorious hired gunman. Slick, well acted and produced version of William Colt MacDonald's novel, this initial film to include all \"Three Mesquiteers\" is a who's who of genre stars; well worth viewing.\n\n**Guinn Williams, Buzz Barton, Hoot Gibson, Art Mix and Harry Carey in** _**Powdersmoke Range**_ **(RKO Radio, 1935).**\n\n**  \n**\n\n_**Power of Justice**_ see _**Beyond the Sacramento**_\n\n_**Power of Possession**_ see _**Lawless Empire**_\n\n**3154** _ **The Prairie**_ **** Screen Guild, 1947. 80 min. D: Frank Wisbar. SC: Arthur St. Claire. With Lenore Aubert, Alan Baxter, Russ Vincent, Jack (John) Mitchum, Charles Evans, Edna Holland, Chief Thundercloud, Fred Coby, Bill Murphy, David Gerber, George Morrell, Don Lynch, Chief Yowlachie, Jay Silverheels, Beth Taylor, Frank Hemingway (narrator). The story of the trials and tribulations of pioneers as they settle upstate New York in the pre\u2013Revolution period, battling Indians and the elements. Paltry adaptation of the James Fenimore Cooper work.\n\n**3155** _ **Prairie Badmen**_ **** Producers Releasing Corporation, 1946. 55 min. D: Sam Newfield. SC: Fred Myton. With Buster Crabbe, Al St. John, Patricia Knox, Charles King, Ed Cassidy, Kermit Maynard, John Cason, Steve Clark, Frank Ellis, John L. (Budd) Buster. Billy Carson and Fuzzy Q. Jones try to help a medicine showman return gold he found to its owner but an outlaw gang also wants it. Typically shoddy entry in the \"Billy Carson\" series.\n\n**3156** _ **Prairie Chickens**_ **** United Artists, 1943. 46 min. D: Hal Roach, Jr. SC: Arnold Belgrade and Earle Snell. With Noah Beery, Jr., Jimmy Rogers, Marjorie Reynolds, Joseph Sawyer, Raymond Hatton, Rosemary LaPlanche, Jack Norton, Edward Gargan, Frank Faylen, Dudley Dickerson, Mary Ann Deighton, Mike Mazurki, Noel Neill, Ray Teal, Ethan Laidlaw, Glenn Strange, Jayne Hazzard, Nancy Brinkman. Two cowpokes get involved with a man who has just inherited a ranch. Last of the trio of mediocre featurettes Hal Roach produced with Noah Beery, Jr., and Jimmy Rogers (Will's son); the highlight is Jack Norton's usual tipsy portrayal of the rancher.\n\n**3157** _ **Prairie Express**_ **** Monogram, 1947. 60 min. D: Lambert Hillyer. SC: Anthony Coldeway and J. Benton Cheney. With Johnny Mack Brown, Raymond Hatton, Virginia Belmont, Robert Winkler, William Ruhl, Marshall Reed, Gary Garrett, Ted Adams, Curly Gibson, Frank LaRue, Steve Darrell, Hank Worden, Carl Mathews, Boyd Stockman, Steve Clark, Artie Ortego, I. Stanford Jolley, Jack Hendricks. A respected citizen is behind a gang out to force a family off their ranch so the land can be bought cheap and sold to the railroad. Average Johnny Mack Brown\u2013Raymond Hatton series affair.\n\n**3158** _ **Prairie Fever**_ **** ION Television, 2008. 81 min. Color. D: Stephen Bridgewater and Davis S. Cass, Sr. SC: Steven H. Berman. With Kevin Sorbo, Lance Henriksen, Dominique Swain, Jamie Anne Allman, Jillian Armenante, Felicia Day, Lucy Lee Flippin, Robert Norsworthy, Blake Gibbons, Don Swayze, Richard Clarke Larsen, Silas Weir Mitchell, Ken Magee, Chris McKenna, E.E.Bell, Michael Bonnabel, Michael Ensign, Jacob Bruce, Caryn Mower, Mark Brooks, Monty Stuart, Jerry Wills, Richard Bucher, Gianfranco Tordi, Caroline Neville, Mark Kulka. A one time lawman, now a drunk, agrees to take three unstable mail order brides to Carson City and he joins forces with a female crook when one of the women's mean minded husbands pursues them. The title refers to mental illness in this okay video Western.\n\n**3159** _ **Prairie Gunsmoke**_ **** Columbia, 1942. 56 min. D: Lambert Hillyer. SC: Fred Myton. With Bill Elliott, Tex Ritter, Virginia Carroll, Frank Mitchell, Hal Price, Tristram Coffin, Joe McGuinn, Frosty Royce, Rick Anderson, Art Mix, Francis Walker, Ray Jones, Ted Mapes, Glenn Strange, Steve Clark, Paul Conrad, Herman Hack, Horace B. Carpenter, Milburn Morante, George Morrell, Fred Parker, Jack Evans, Tex Cooper. Although Wild Bill Hickok comes to a frontier town to help citizens and ranchers harassed by rustlers, he finds he is distrusted by the locals. Good action filled \"Wild Bill Hickok\" entry, the last in the series.\n\n**3160** _ **Prairie Justice**_ **** Universal, 1938. 58 min. D: George Waggner. SC: Joseph West (George Waggner). With Bob Baker, Dorothy Fay, Hal Taliaferro, Jack Rockwell, Carleton Young, Jack Kirk, Forrest Taylor, Glenn Strange, Tex Palmer, Slim Whitaker, Murdock MacQuarrie, Chuck Baldra, Dick Dickinson, Jimmy Phillips, Archie Ricks, Victor Cox, George Plues, George Hazel, Tex Phelps, Wimpy (dog). When cattle rustlers murder his father a cowboy gets on their trail for revenge. Routine Bob Baker series vehicle.\n\n**3161** _ **The Prairie King**_ **** Universal, 1927. 57 min. D: B. Reeves Eason. SC: Frank Howard Clark and William Wallace Cook. With Hoot Gibson, Barbara Worth, Albert Prisco, Charles Sellon, Rosa Gore, Sidney Jarvis, George Periolat, Robert Homans, Jim Corey, Jack Randall, Andrew Waldron. A mine owner leaves his property to a cowboy, a young woman and a man who will go to any lengths to get the total bequest for himself. Typically breezy Hoot Gibson silent affair.\n\n**3162** _ **Prairie Law**_ **** RKO Radio, 1940. 58 min. D: David Howard. SC: Doris Schroeder and Arthur V. Jones. With George O'Brien, Virginia Vale, J. Farrell MacDonald, Slim Whitaker, Dick Hogan, Cy Kendall, Paul Everton, Henry Hall, Monte Montague, Quen Ramsey, Lloyd Ingraham, Bud Osborne, Ferris Taylor, Ben Corbett, Hank Bell, Cactus Mack, Frank O'Connor, Jack O'Shea, Ed Brady, Hank Worden, Frank Ellis, Billy Benedict, Jack Henderson. Crooks bring settlers to range land with false promises of plenty of water but the scheme is soon opposed by a cattleman who wants to help the homesteaders. Fine George O'Brien film with a good plot and cast.\n\n**3163** _ **Prairie Moon**_ **** Republic, 1938. 58 min. D: Ralph Staub. SC: Betty Burbridge and Stanley Roberts. With Gene Autry, Smiley Burnette, Shirley Deane, Tommy Ryan, Warner Richmond, Tom London, William Pawley, Walter Tetley, David Gorcey, Stanley Andrews, Peter Potter, Bud Osborne, Ray Bennett, Jack Rockwell, Merrill McCormick, Hal Price, Lew Meehan, Jack Kirk, Frankie Marvin, Mira McKinney, Dan White, Al Taylor, Fred Burns, Chuck Baldra, Art Baker, Buster Slaven. Gene and Frog become the guardians of three youths whose father was a gangster and the boys help rustlers hide cattle on their ranch. Pretty fair Gene Autry musical with the old ploy of having cattle sequestered behind a waterfall.\n\n_**Prairie Outlaws**_ see _**Wild West**_\n\n**3164** _ **Prairie Pals**_ **** Producers Releasing Corporation, 1942. 60 min. D: Peter Stewart (Sam Newfield). SC: Patricia Harper. With Bill \"Cowboy Rambler\" Boyd, Art Davis, Lee Powell, Esther Estrella, Charles King, John Merton, Kermit Maynard, I. Stanford Jolley, Karl Hackett, Bob Burns, Al St. John, Art Dillard, Curley Dresden, Frank McCarroll, Bill Patton, Carl Mathews, Frank Ellis, Jack (J. Merrill) Holmes, Bert Dillard, Al Taylor, George Morrell, Jack Kenney, Morgan Flowers. Outlaws kidnap a scientist working on a formula to extract gold from rock and a trio of lawmen try to rescue him. Tacky next to the last outing in the low grade \"Frontier Marshals\" series.\n\n**3165** _ **Prairie Pioneers**_ **** Republic, 1941. 58 min. D: Lester Orlebeck. SC: Barry Shipman. With Robert Livingston, Bob Steele, Rufe Davis, Esther Estrella, Robert Kellard, Guy D'Ennery, Davison Clark, Jack Ingram, Kenneth MacDonald, Lee Shumway, Mary MacLaren, Yakima Canutt, Wheaton Chambers, Jack Kirk, Carleton Young, Frank Ellis, Cactus Mack, Curley Dresden, Frank McCarroll, Leander De Cordova, Rosa Turich, Pascale Perry, Bob Burns, Dan White, Al Taylor, Tom Smith, Silver Tip Baker, Roy Bucko, Chuck Baldra, Jim Corey, George Plues, Ray Henderson. A half-breed outlaw is trying to steal a gold mine and three cowboys are out to stop him. Not one of the better \"Three Mesquiteers\" features despite a fine supporting cast of genre favorites.\n\n**3166** _ **The Prairie Pirate**_ **** Producers Distributing Corporation, 1925. 60 min. D: Edmund Mortimer. SC: Anthony Dillon. With Harry Carey, Jean Dumas, Lloyd Whitlock, Trilby Clark, Robert Edeson, Tote Du Crow, Evelyn Selbie, Fred Kohler. After his sister is murdered, a cowboy becomes an outlaw in order to track down the killer. Entertaining Harry Carey silent oater.\n\n**3167** _ **Prairie Raiders**_ **** Columbia, 1947. 55 min. D: Derwin Abrahams. SC: Ed Earl Repp. With Charles Starrett, Smiley Burnette, Nancy Saunders, Robert Scott, Ozie Waters and His Colorado Rangers, Hugh Prosser, Lane Bradford, Ray Bennett, Doug Coppin, Steve Clark, Tommy Coats, Frank LaRue, John Cason, Sam Flint, Scotty Harrell, Eddie Kirk. A rancher leases land from the Interior Department so he can round up and sell horses but is faced with competition from outlaws. Another look-a-like \"Durango Kid\" film.\n\n**3168** _ **Prairie Roundup**_ **** Columbia, 1951. 53 min. D: Fred F. Sears. SC: Joseph O'Donnell. With Charles Starrett, Smiley Burnette, Mary Castle, Frank Fenton, The Sunshine Boys (Eddie Wallace, Freddie Daniel, M.H. Richman, J.D. Sumner), Lane Chandler, Frank Sully, Paul Campbell, Forrest Taylor, Don C. Harvey, George Baxter, John Cason, Al Wyatt, Allan Sears, Glenn Thompson, Ace Richmond, Blackie Whiteford. Falsely accused of murder by an outlaw gang, a cowboy escapes from jail with the help of his pal and the two take jobs on a ranch owned by a young woman whose cattle the bad men plan to rustle. Fast moving \"Durango Kid\" affair.\n\n**3169** _ **Prairie Rustlers**_ **** Producers Releasing Corporation, 1945. 56 min. D: Sam Newfield. SC: Fred Myton. With Buster Crabbe, Al St. John, Evelyn Finley, Karl Hackett, Bud Osborne, Marin Sais, I. Stanford Jolley, Kermit Maynard, Herman Hack, George Morrell, Tex Cooper, Dorothy Vernon, Tex Williams, Wally West, Jimmy Aubrey, John Cason, Al Ferguson, Dean Eaker, George Bamby, Carl Mathews, Ray Jones, Al Haskell, Jack Evans. Because of a close resemblance to his outlaw cousin, Billy Carson is accused of the desperado's crimes. No better or worse than most of the \"Billy Carson\" features.\n\n**3170** _ **Prairie Schooners**_ **** Columbia, 1940. 58 min. D: Sam Nelson. SC: Robert Lee Johnson and Fred Myton. With Bill Elliott, Evelyn Young, Dub Taylor, Kenneth Harlan, Ray Teal, Bob Burns, Netta Parker, Richard Fiske, Edmund Cobb, Jim Thorpe, George Morrell, Merrill McCormick, Sammy Stein. Wild Bill Hickok leads a wagon train of settlers, who have lost their lands due to foreclosures, west to search for gold and along the way they are attacked by Indians. Action filled initial entry in the \"Wild Bill Hickok\" series starring Bill Elliott.\n\n**3171** _ **Prairie Stranger**_ **** Columbia, 1941. 58 min. D: Lambert Hillyer. SC: Winston Miller. With Charles Starrett, Cliff Edwards, Patti McCarty, Lew Preston and His Ranch Hands, Forbes Murray, Frank LaRue, Archie Twitchell, Francis Walker, Edmund Cobb, Jim Corey, Russ Powell, George Morrell, Chester Conklin, Monty Collins, Hank Bell, George Sherwood, Edward Hearn, Lynn Lewis, John Tyrrell, Ray Jones, Jack Tornek, Rube Dalroy. A frontier doctor opens a practice in a Nevada town and finds he is opposed by a rival and accused of poisoning cattle. Nicely done third and final entry in the all-too-brief \"Doc Monroe\" series, based on the books by James L. Rubel.\n\n**3172** _ **Prairie Thunder**_ **** Warner Bros., 1937. 54 min. D: B. Reeves Eason. SC: Ed Earl Repp. With Dick Foran, Ellen Clancy (Janet Shaw), Wilfred Lucas, Frank Orth, Frank Ellis, Yakima Canutt, Arthur J. Smith, George Chesebro, J.P. McGowan, John Harron, Frank McCarroll, Slim Whitaker, Henry Otho, Art Mix, Jim Corey, Iron Eyes Cody. A cavalry scout learns a freight operator has been inciting Indians to disrupt the construction of telegraph lines. Dick Foran's final Warner Bros.' series Western is a pretty good one.\n\n**3173** _ **Prescott Kid**_ **** Columbia, 1934. 60 min. D: David Selman. SC: Ford Beebe. With Tim McCoy, Sheila Mannors, Alden Chase, Hooper Atchley, Joseph (Sawyer) Sauers, Albert J. Smith, Carlos De Valdez, Ernie Adams, Steve Clark, Slim Whitaker, Charles King, Bud Osborne, Art Mix, Tom London, Edmund Cobb, Walter Brennan, Lew Meehan, Jack Rockwell, Blackie Whiteford, Bill Patton, Bob Burns, Al Haskell, Fred Burns, Rose Plummer, Lionel Backus, Roy Bucko, Bob Card. Riding into a small town, a man is mistaken for the expected marshal and runs up against a gang of crooks. Exceedingly well done Tim McCoy film; worth watching.\n\n_**The Price of Crime**_ see _**Law of the Canyon**_\n\n**3174** _ **The Price of Death**_ **** Overseas Film Company, 1971. 91 min. Color. D: Vincent Thomas (Lorenzo Gicca Palli). SC: Enzo Gicca (Lorenzo Gicca Palli). With Gianni Garko, Klaus Kinski, Gely Genka, Franco Abbina, Luciano Lorcas, Laura Gianoli, Giancarlo Prete, Luigi Castellato, Alan Collins (Luciano Pigozzi), Franca De Stratis, Andrea Scotti (Andrew Scott), Alfredo Rizzo, Giuseppe Castellano, Silvana Bacci, Fortunato Arena, Osiride Pevarello, Tchang Yu, Augusto Funari, Fulvio Pellegrino. An unpopular citizen is accused of killing several people in a robbery and a gunman is hired by a lawyer and a hooker to prove his innocence. Better than average Spaghetti Western with an intriguing mystery angle; produced in Italy by Mida Cinematographic as _**Il Venditore di Morte**_ (The Seller of Death).\n\n**3175** _ **The Price of Power**_ **** Golden Era, 1969. 96 min. Color. D: Tonino Valerii. SC: Massimo Patrizi. With Giuliano Gemma, Van Johnson, Warren Vanders, Fernando Rey, Jose Suarez, Benito Stefanelli, Maria Jesus Cuadra, Ray Saunders, Maria Luisa Sala, Antonio Casas, Manolo Zarzo, Frank Brana, Jose Calvo, Angel Alvarez, Julio Pena, Francisco Sanz. In post\u2013Civil War Texas a man wants revenge on those who falsely accused his buddy of killing the governor, resulting in his friend's death by a mob. Interesting pseudo-historical Italian action feature issued in that country by Patry Film\/Film Montana as _**Il Prezzo de Potere**_ (The Price of Power), running 122 minutes.\n\n**3176** _ **Pride of the Plains**_ **** Republic, 1944. 56 min. D: Wallace Fox. SC: John K. Butler and Bob Williams. With Robert Livingston, Smiley Burnette, Nancy Gay, Stephen Barclay, Kenneth MacDonald, Charles Miller, Kenne Duncan, Jack Kirk, Bud Geary, Yakima Canutt, Budd Buster, Bud Osborne, Horace B. Carpenter, Kansas Moehring. A lawman is on the trail of an outlaw gang rustling cattle sold to be canned for animal food. Bob Livingston's first of two \"John Paul Revere\" films is a good one.\n\n**3177** _ **Pride of the West**_ **** Paramount, 1938. 56 min. D: Lesley Selander. SC: NateWatt. With William Boyd, Russell Hayden, George Hayes, Charlotte Field, Earle Hodgins, Billy King, Kenneth Harlan, Art Mix, Glenn Strange, James Craig, Bruce Mitchell, Willie Fung, George Morrell, Earl Askam, Jim Toney, Horace B. Carpenter, Henry Otho, Leo J. McMahon, Wen Wright, Jess Cavin, Johnny Luther, Charles Murphy. A realty agent uses a stagecoach robber to help him defraud citizens and Hopalong Cassidy gets on their trail. Action packed entry in the popular series based on the Clarence H. Mulford characters.\n\n**3178** _ **Prince of the Plains**_ **** Republic, 1949. 60 min. D: Philip Ford. SC: Louise Rousseau and Albert DeMond. With Monte Hale, Paul Hurst, Shirley Davis, Roy Barcroft, Rory Mallinson, Harry Lauter, Lane Bradford, George Carleton, Edmund Cobb, Holly Bane, Frank Jaquet, Tom Chatterton. An outlaw gang raids and terrorizes ranchers while a cowboy tries to stop them. More than passable Monte Hale feature that should please his fans.\n\n**3179** _ **The Prisoner of Shark Island**_ **** 20th Century\u2013Fox, 1936. 95 min. D: John Ford. SC: Nunnally Johnson. With Warner Baxter, Gloria Stuart, Claude Gillingwater, Arthur Byron, O.P. Heggie, Harry Carey, Francis Ford, John McGuire, Francis McDonald, Douglas Wood, John Carradine, Joyce Kay, Fred Kohler, Ernest Whitman, Paul Fix, Frank Shannon, Frank McGlynn, Sr., Leila McIntyre, Etta McDaniel, J.M. Kerrigan, Arthur Loft, Paul McVey, Maurice Murphy, Paul Stanton, Wilfred Lucas, Stanley Blystone, Paul Kruger, Vester Pegg, Merrill McCormick, J.P. McGowan, Harry Strang, Arthur Millett, Jack Pennick, Jan Dugan, James Marcus, Lloyd Whitlock, Murdock MacQuarrie, Dick Elliott, Bud Geary, Robert Homans, Cecil Weston, Beulah Hall Jones, Duke R. Lee, Paul McAllister, John Lester Johnson, Raymond Turner, Gus Reed, Earl Eby, Robert Dudley, Cyril Thornton, Charles Haefeli, Henry Washington. Dr. Samuel Mudd is accused of helping John Wilkes Booth after the Lincoln assassination and is sentenced to life in the hellhole at Dry Tortugas Island but after a failed escape attempt he redeems himself during a yellow fever epidemic. Grand historical drama highlighted by Warner Baxter's wonderful portrayal of Dr. Mudd.\n\n**John McGuire, Fred Kohler, Jr., Gloria Stuart, Joyce Kay, Warner Baxter and Claude Gillingwater, Sr., in** _**The Prisoner of Shark Island**_ **(20th Century** **\u2013** **Fox, 1936).**\n\n**  \n**\n\n**3180** _ **The Professionals**_ **** Columbia, 1966. 117 min. Color. D-SC: Richard Brooks. With Burt Lancaster, Lee Marvin, Robert Ryan, Jack Palance, Claudia Cardinale, Ralph Bellamy, Woody Strode, Joe De Santis, Rafael Bertrand, Jorge Martinez De Hoyos, Maria Gomez, Jose Chavez, Carlos Romero, Vaughn Taylor, Robert Conteras, Don Carlos, John Lopez, John McKee, Eddie Little Sky, Leigh Chapman, Elizabeth Campbell, Phil Parslow. A wealthy man hires a quartet of mercenaries to return his young wife kidnapped by a bandit outlaw during the 1917 Mexican Revolution. More than adequate action melodrama.\n\n_**Promise Fulfilled**_ see _**The Wildcat of Tucson**_\n\n**3181** _ **Promise the Moon**_ **** Sullivan Entertainment, 1997. 94 min. Color. D: Ken Jubenvill. SC: Kevin Sullivan and Peter Behren. With Henry Czemy, Colette Stevenson, Shawn Ashmore, Aidam Devine, Richard Donat, Ken James, Gloria May Eshkibook, David Fox, Gordon Michael Woolveth, Frank Crudele, Richard McMillan, J.W. Caroll, James Nicholson, Maja Ardal, Robert Haley. After his rancher boss dies, a worker looks after the man's deaf son and his Indian woman guardian as he tries to stop a banker from foreclosing on the property. Heartwarming modern TV drama filmed in Canada.\n\n**3182** _ **The Proposition**_ **** The Works, 2005. 104 min. Color. D: John Hillcoat. SC: Nick Cave. With Richard Wilson, Noah Taylor, Jeremy Madrona, Jae Mamuyac, Guy Pearce, Mick Roughan, Shane Watt, Ray Winstone, Robert Morgan, David Gulpilli, Bryan Probets, Oliver Ackland, Danny Huston, David Vallon, Daniel Parker, Carl Rush, Gary Waddell, Iain Gardiner, Emily Watson, Bogdan Koca, Sue Dwyer, Lance Medlin, John Hurt, David Wenham, Rodney Boschman, Boris Brkic, Ned Rose, Leah Purcell, Tom Budge, Tom E. Lewis, Ralph Cotterill, Max Age, Jerry Solomon. A lawman promises to pardon two outlaw brothers if they will kill their older sibling within nine days after their gang massacred a farm family. Violent, well made Australian Western about the early settlement of that nation.\n\n**3183** _ **The Proud and the Damned**_ **** Prestige, 1972. 97 min. Color. D-SC: Ferde Grofe, Jr. With Chuck Connors, Aron Kincaid, Cesar Romero, Jose Greco, Henry Capps, Peter Ford, Smoky Roberts, Maria Grimm, Dana Lorca, Anita Quinn, Conrad Parkman, Alvaro Ruiz. Four Civil War veterans drift into Latin America and are forced to help a military dictator. Fair drama helped by Chuck Connors and Cesar Romero.\n\n**3184** _ **Proud Men**_ **** ABC-TV, 1987. 96 min. Color. D: William A. Graham. SC: Jeff Andrus. With Charlton Heston, Peter Strauss, Nan Martin, Alan Autry, Belinda Balaski, Maria Mayenzet, Red West, Gregory Kupiec, Buck Taylor, Mark McIntire, Darren Prentice, Billy Ray Sharkey, Dale Swan, Bud Walls, Steve Whittaker, John Woodbridge. A dying cattle rancher and his Vietnam War deserter son try to resolve their personal bitterness before it is too late. Thoughtful modern-day Western.\n\n**3185** _ **The Proud Ones**_ **** 20th Century\u2013Fox, 1956. 94 min. Color. D: Robert D. Webb. SC: Edmund North and Joseph Petracca. With Robert Ryan, Virginia Mayo, Jeffrey Hunter, Robert Middleton, Walter Brennan, Arthur O'Connell, Ken Clark, Rodolfo Acosta, George Mathews, Fay Roope, Edward Platt, Whit Bissell, Paul Burns, Richard Deacon, Lois Ray, Jack Low, Kenneth Terrell, Don Brodie, Jackie Coogan, Juanita Close, I. Stanford Jolley, Jack Mather, Steve Darrell. A young man, bent on getting revenge on the lawman who killed his father, arrives in a small town with two hired guns to carry out his plan. Fairly interesting psychological yarn, enhanced by good performances.\n\n**3186** _ **The Proud Rebel**_ **** Buena Vista, 1958. 103 min. Color. D: Michael Curtiz. SC: Joseph Petracca and Lillie Hayward. With Alan Ladd, Olivia de Havilland, Dean Jagger, David Ladd, Cecil Kellaway, Dean Stanton, Tom Pittman, Henry Hull, Eli Mintz, James Westerfield, John Carradine, Mary Wickes, Percy Helton, Dan White, King (dog). A woman who refuses to sell out to a rich sheep farmer hires a convict, falsely imprisoned by the man, to help her work the place and fight their mutual enemy. Rather interesting Disney feature with a good cast and plot.\n\n**3187** _ **Public Cowboy No. 1**_ **** Republic, 1937. 59 min. D: Joseph Kane. SC: Oliver Drake. With Gene Autry, Smiley Burnette, Ann Rutherford, William Farnum, James C. Morton, Maston Williams, Arthur Loft, Frankie Marvin, House Peters, Jr., Frank LaRue, Milburn Morante, Hal Price, Jack Ingram, Ray Bennett, Frank Ellis, George Plues, Jim Mason, Bob Burns. Ranchers are stymied by the loss of cattle until a singing cowboy discovers that rustlers are using modern methods like radios, airplanes and refrigerated trucks. Popular and well done Gene Autry vehicle.\n\n**3188** _ **Pueblo Terror**_ **** Artclass, 1931. 59 min. D: Alvin J. Neitz (Alan James). SC: L.V. Jefferson. With Buffalo Bill, Jr., Wanda Hawley, Jack Harvey, Jim Spencer, Aline Goodwin, Art Mix, Yakima Canutt, Horace B. Carpenter, Al Ferguson, Hank Bell, Robert Walker, Herman Hack, Frank Ball, Chuck Baldra, Ralph Bucko, Roy Bucko. A man, returning home to Paradise Valley after being gone three years, must prove his innocence when framed for a murder committed by a ranch foreman in cahoots with a crook out to take over area ranches. Feeble poverty row effort headlining laconic Buffalo Bill, Jr. (Jay Wilsey).\n\n**3189** _ **El Puma**_ **** Filmadora Independiente, 1958. 77 min. D: Rene Cardona. SC: Jesus Cardenas. With Rene Cardona, Jr., Sofia Alvarez, Lorena Velazquez, Dagoberto Rodriguez, Juan Manuel Guerrero, Rene Cardona, Andres Soler, Miguel Manzano, David Reynoso, Jorge Alzaga, Victor Velazquez, Ada Carrasco, Armando Gutierrez, Rafael Estrada, Emilio Garibay, Dacia Gonzalez. Alienated from his rancher father, a frontier lawyer is forced to become a gunman when opposed by a bandit. Pretty fair Mexican Western that spawned two sequels, _**La Ley del Mas Rapido**_ and _**A Tiro Limpio**_ (qq.v.).\n\n**3190** _ **Punos de Roca**_ (Fists of Rock) **** Alameda Films, 1960. 90 min. D: Rafael Baledon. With Rafael Bertrand, Olivia Michel, Alfonso Mejia, Pedro de Aguillon, Quintin Buines, Guillermo Cramer, Jose Chavez, Maria Idalia, Jose Baviera. A cowboy comes across massacred settlers and assists the Mexican army in its struggle with marauding Indians in hopes of finding the real killers. Gritty Mexican oater.\n\n**3191** _ **Pure Country**_ **** Warner Bros., 1992. 112 min. Color. D: Christopher Cain. SC: Rex McGee. With George Strait, Lesley Ann Warren, Isabel Glasser, Kyle Chandler, John Doe, Rory Calhoun, Molly McClure, James Terry McIlvain, Toby Metcalf, Mark Walters, Sharon Thomas, Gil Glasgow, Julie Johnson, Fred Ellis, Fred Fontana, Kristen Michaels, Jeff Prettyman, David Anthony, Mike D. Daily, Gene Elders, Terry Hale, Rondel Huckaby, Mike A. Kennedy, Benny McArthur, Rick McRae, Tom Christopher, Jeffrey R. Fontana, Evelyn Furtak, Eric Randall, Loretta Holloway, Roy Kieffer, Bob Tallman. A burned out country music star finds romance as he seeks to return to his Western heritage. Pleasant George Strait vehicle in which he demonstrates his riding and roping skills in addition to singing.\n\n**3192** _ **Purgatory**_ **** Turner Network Television (TNT), 1999. 95 min. Color. D: Uli Edel. SC: Gordon Dawson. With Sam Shepard, Eric Roberts, Randy Quaid, Peter Stormare, Brad Rowe, Donnie Whalberg, John David Souther, Amelia Heinie, Shannon Kenny, John Dennis Johnston, Saginaw Grant, Richard Edson, Gregory Scott Cummins, John Diehl, R.G. Armstrong, Michael Shaner, Les Lannon, Phil Hawn. Trailed by a posse, a gang arrives in a remote town where famous dead outlaws await judgment as to whether they will go to heaven or hell. Curio TV horror Western, well done but somewhat confusing.\n\n**3193** _ **The Purple Hills**_ **** 20th Century\u2013Fox, 1961. 60 min. Color. D: Maury Dexter. SC: Edith Cash Pearl and Russell (Russ) Bender. With Gene Nelson, Kent Taylor, Joanna Barnes, Russ Bender, Jerry Summers, Jack Carr, Danny Zapien, Jack Riggs, Medford Salway. When a cowboy kills an outlaw in Indian Territory he finds he is hunted by tribesmen as he attempts to take the body in for the reward. Compact little \"B\" effort.\n\n**Jerry Summers, Joanna Barnes, Russ Bender, Kent Taylor and Gene Nelson in** _**The Purple Hills**_ **(20th Century** **\u2013** **Fox, 1961).**\n\n**  \n**\n\n**3194** _ **The Purple Vigilantes**_ **** Republic, 1938. 58 min. D: George Sherman. SC: Betty Burbridge and Oliver Drake. With Robert Livingston, Ray Corrigan, Max Terhune, Joan Barclay, Earle Hodgins, Earl Dwire, Jack Perrin, Francis Sayles, George Chesebro, Robert Fiske, Ernie Adams, William Gould, Harry Strang, Ed Cassidy, Frank O'Connor, Jason Robards, Niles Welch, Allan Cavan, Jack Kirk, George (Montgomery) Letz, Lee Shumway, Ed Peil, Sr., Frank Ellis, Curley Dresden, Murdock MacQuarrie, Frankie Marvin, Bob Burns, Merrill McCormick, Dot Farley, Bill Patton, Wally West, Jim Corey, Fred Burns, Herman Hack, Tom Smith, Brandon Beach, Billy Bletcher (voice). Outlaws use the guise of a vigilante group to terrorize the locals until three cowboy pals get on their trail. Well made entry in the popular \"Three Mesquiteers\" series.\n\n**Pursued** (1928) see _**The Arizona Kid**_ (1928)\n\n**3195** _ **Pursued**_ **** Warner Bros., 1947. 100 min. D: Raoul Walsh SC: Niven Busch. With Teresa Wright, Robert Mitchum, Judith Anderson, Dean Jagger, Alan Hale, John Rodney, Harry Carey, Jr., Clifton Young, Ernest Severn, Charles Bates, Peggy Miller, Norman Jolley, Lane Chandler, Elmer Ellingwood, Jack Montgomery, Ian MacDonald, Kathy Jeanne Johnson, Mickey Little, Scotty Hugenberg, Ray Teal, Eddy Waller, Russ Clark, Jack Davis, Crane Whitley, Carl Harbough, Lester Dorr, Bill Sundholm, Paul Scardon, Harry Lamont, Erville Alderson, Sherman Saunders, Al Kundee, Ben Corbett, Charles Miller, Tom Fadden, Virginia Brissac, Ervin Richardson, Louise Volding, Ian Wolfe, Ian Coffey. A young man haunted by his past sets out to find his father's murderer. Psychological Western that is a bit hard to follow but is fairly entertaining with fine work by Robert Mitchum as a Spanish-American War veteran.\n\n**3196** _ **Pursuit**_ **** Key International, 1975. 86 min. Color. D: Thomas Quillen. SC: DeWitt Lee and Jack Lee. With Ray Danton, DeWitt Lee, Troy Nabors, Diane Taylor, Eva Kovacs, Jason Clark. An Army scout wounded by a bear is tracked through the desert by an Indian brave who wants to kill him. Suspenseful R-rated thriller.\n\n**3197** _ **Pursuit Across the Desert**_ **** Cinematografica Intercontinental, 1960. 75 min. Color. D: Gilberto Gazcon. SC: Gilberto Gazcon, Fernando Mendez and Raul de Anda. With Pedro Armendariz, Teresa Velasquez (Tere Velazquez), Sonia Furio, Agustin de Anda, Andres Soler, Carlos Lopez Moctezunna, Felix Gonzales, Jose Chavez. Although he knows he is innocent, a lawman attempts to return an accused murderer who escaped from jail. Well done Mexican drama, originally called _**La Carcel de Cananea**_ (The Jail of Cananea).\n\n_**Put on the Spot**_ see _**Rio Grande Romance**_\n\n**3198** _ **Pyramid of the Sun God**_ **** Gloria Film, 1965. 98 min. D: Robert Siodmak. SC: Ladislas Fodor, R.A. Stemmle and Georg Marischka. With Lex Barker, Gerard Barray, Michele Girardon, Hans Nielsen, Rik Battaglia, Gustavo Rojo, Teresa Lorca, Ralf Wolter, Kelo Henderson, Alessandra Panaro, Jean-Roger Caussimon, Antun Nalis, Vladimir Popovic, Branimir Tori Jankovic, Nada Radovic, Peter Buntic, Petar Obradovic, Jovan Rancic, Willy Egger, Rolf Rolphs, John Kirby, Jeff Corey, Fausto Tozzi, Jovan Nikolic. In Mexico in the 1860s two German emissaries originally allied with Emperor Maxmilian change sides and support the revolution led by Juarez, as both groups seek ancient Aztec treasure. Action filled West German follow up to _**The Treasure of the Aztecs**_ (q.v.), filmed back-to-back with that feature as _**Die Pyramide des Sonnengottes**_ (The Pyramid of the Sun Gods).\n\n**3199** _ **Quantez**_ **** Universal-International, 1957. 80 min. Color. D: Harry Keller. SC: R. Wright Campbell. With Fred MacMurray, Dorothy Malone, John Gavin, James Barton, Sydney Chaplin, John Larch, Michael Ansara. Several people are held prisoner in a saloon by a group of bank robbers who are heading for Mexico. Compact little melodrama that is well acted.\n\n**3200** _ **Quantrill's Raiders**_ **** Allied Artists, 1958. 71 min. Color. D: Edward Bernds. SC: Polly James. With Steve Cochran, Diane Brewster, Leo Gordon, Gale Robbins, Will Wright, Kim Charney, Myron Healey, Robert Foulk, Glenn Strange, Lane Chandler, Guy Prescott, Thomas Browne Henry, Dan White, Robert Colbert. General Robert E. Lee sends a Confederate captain to contact Quantrill about raiding a Kansas arsenal but the emissary soon turns against the guerrilla leader. Not much historical fact here but there is some action with Leo Gordon believable as Quantrill.\n\n**3201** _ **Quebec**_ **** Paramount, 1951. 85 min. Color. D: George Templeton. SC: Alan LeMay. With John Barrymore, Jr., Corinne Calvet, Barbara Rush, Patric Knowles, John Hoyt, Arnold Moss, Don Haggerty, Patsy Ruth Miller, Howard Joslin, Paul Guevremont, Adrian Belanger. During the 1837 Canadian rebellion against England a rebel leader falls in love with the wife of the British commander. Average historical effort.\n\n**3202** _ **Queen of the Yukon**_ **** Monogram, 1940. 73 min. D: Phil Rosen. SC: Joseph West (George Waggner). With Charles Bickford, Irene Rich, Melvin Lang, George Cleveland, Guy Usher, June Carlson, Dave O'Brien, Tristram Coffin, John Merton, I. Stanford Jolley, J. Merrill Holmes, Gene O'Donnell, Jack Daley, Johnny Morris, C.E. Anderson. An aging dance hall hostess tries to protect her daughter from her way of life during the gold rush days. Taken from a Jack London story, this program feature offers a fine performance by veteran star Irene Rich in the title role.\n\n**3203** _ **The Quest**_ **** NBC-TV\/Columbia, 1976. 100 min. Color. D: Lee H. Katzin. SC: Tracy Keenan Wynn. With Tim Matheson, Kurt Russell, Brian Keith, Keenan Wynn, Will Hutchins, Neville Brand, Cameron Mitchell, Morgan Woodward, Art Lund, Mark Lambert, Gregory Walcott, Iron Eyes Cody, Luke Askew, Irene Yah-Ling Sun, Nick Ramus, Nathan Jung, Michael Swan. Two brothers search for their sister who was taken from them as a child and now lives with the Indians. Pretty fair TV movie that resulted in a series of the same title which had a brief run on NBC-TV in 1976.\n\n**Kurt Russell and Tim Matheson in** _**The Quest**_ **(NBC-TV, 1976).**\n\n**  \n**\n\n**3204** _ **The Quest:**_ _**The Longest Drive**_ **** Columbia Pictures Television, 1976. 89 min. Color. D: Bernard McEveety. SC: Michael Michaelian and Katharyn Michaelian Powers. With Kurt Russell, Tim Matheson, Dan O'Herlihy, Keenan Wynn, Woody Strode, Erik Estrada, Sander Johnson, Cooper Huckabee, John Rubinstein, Gary Lockwood, Dick Davalos, Angela May, Meegan King, John Alvin, Duncan McLeod, Judith Hanson, Glenn Buttkus, Bill Smillie, Frank Salsedo, Peter Haas, Reid Rondell, Mary Angela, Jane Kellem. A pair of siblings go on a cattle drive to Colorado and have a series of adventures with Indians, rustlers, homesteaders, thirst and a stampede. Okay TV movie derived from the \"The Quest\" (NBC-TV, 1976).\n\n**3205** _ **The Quick and the Dead**_ **** Home Box Office (HBO), 1987. 91 min. Color. D: Robert Day. SC: James Lee Barrett. With Sam Elliot, Kate Capshaw, Tom Conti, Matt Clark, Kenny Morrison, Patrick Kilpatrick, Jerry Potter, Billy Streater, Del Shores, R.K. Tolbert, Jeffrey M. Meyer, Kurt D. Lott, Hardy Rawls, Larry Sellors, Bill Stedman. A frontiersman assists newly arrived settlers besieged by outlaws with the wife finding herself attracted to the defender. Entertaining TV movie adaptation of Louis L'Amour's novel.\n\n**3206** _ **The Quick and the Dead**_ **** Tri-Star Pictures, 1995. 107 min. Color. D: Sam Raimi. SC: Simon Moore. With Sharon Stone, Gene Hackman, Russell Crowe, Leonardo DiCaprio, Tobin Bell, Roberts Blossom, Kevin Conway, Keith David, Lance Henriksen, Pat Hingle, Gary Sinise, Mark Boone Junior, Olivia Burnette, Fay Masterson, Raynor Scheine, Woody Strode, Jerry Swindall, Scott Spiegel, Jonothon Gill, Sven-Ole Thorsen, Lennie Loftin, Matthew Gold, Arturo Gastelu, David Cornell, Josef Rainer, Stacey Ramsower, Tony Boggs, Scott Ryder, Timothy Patrick Quill, Solomon Abrams, John Cameron, Michael Stone, Butch Molina, Gregory Goossen, Mike Garris, Oliver Dear. A female shootist arrives in a town where the boss has set up a contest trying to force a reformed gunslinger, now a minister, to again take up shooting irons. Money making but confusing Western.\n\n**3207** _ **The Quick and the Undead**_ **** North Entertainment, 2006. 78 min. Color. D-SC: Gerald Nott. With Clint Glenn, Parrish Randall, Nicola Giacobbe, Dion Day, Jeff Swarthout, Derik Van Derbeken, Erin McCarthy, Elysia Skye, Paul Molnar, Kim Solow, Toar Campbell, Brian Koehler, Vito La Morte, John Reynolds, Jason Rogel, Jarod Scott. After a virus effected the world's population eighty-five years before, a bounty hunter makes a living destroying zombies but he also tracks the human gang who tried to kill him. Cheap, poorly acted video horror Western.\n\n**3208** _ **The Quick Gun**_ **** Columbia, 1964. 88 min. Color. D: Sidney Salkow. SC: Robert E. Kent. With Audie Murphy, Merry Anders, James Best, Ted De Corsia, Walter Sande, Rex Holman, Charles Meredith, Frank Ferguson, Mort Mills, Gregg Palmer, Frank Gerstle, Stephen Roberts, Paul Bryar, Raymond Hatton, William Fawcett, Rick Vallin, William Tannen. A cowboy returns home to find rejection because two years before he was forced to kill the area land baron's son in self defense. Nothing special about this redemption plotted Audie Murphy feature.\n\n**3209** _ **Quick on the Trigger**_ **** Columbia, 1948. 55 min. D: Ray Nazarro. SC: Elmer Clifton. With Charles Starrett, Smiley Burnette, Lyle Talbot, Helen Parrish, George Eldredge, The Sunshine Boys (Eddie Wallace, J.D. Sumner, Freddie Daniel, M.H. Richman), Ted Adams, Alan Bridge, Russell Arms, Budd Buster, Blackie Whiteford, Tex Cooper, Bud Osborne, Russell Meeker, George Morrell, Sandy Sanders. When outlaws attack a young woman's stage line the sheriff captures one of them and he turns out to be her brother but when the man is murdered in his cell the lawman is blamed. Well written \"Durango Kid\" entry. British title: _**Condemned in Error**_.\n\n**3210** _ **Quick Trigger Lee**_ **** Big 4, 1931. 60 min. D: J.P. McGowan. SC: George Morgan. With Bob Custer, Caryl Lincoln, Monte Montague, Lee De Cordova, Richard Carlyle, Frank Ellis, Al Taylor, J.P. McGowan, Chuck Baldra, Ray Henderson. A gunman helps a nearly blind prospector about to be swindled out of his mine by a crook and his son. Pedestrian low grade oater not helped by stoic star Bob Custer.\n\n**3211** _ **The Quiet Gun**_ **** 20th Century\u2013Fox, 1957. 79 min. D: William F. Claxton. SC: Eric Norden. With Forrest Tucker, Mara Corday, Jim Davis, Cleo Moore, Kathleen Crowley, Lee Van Cleef, Tom Brown, Lewis Martin, Hank Worden, Everett Glass, Edith Evanson, Vince Barnett, Gene Roth, Gerald Milton. A saloon owner and his girlfriend hatch a plot that forces a rancher into committing murder. Strangely violent Western with a good cast and direction.\n\n**3212** _ **Quigley Down Under**_ **** Metro-Goldwyn-Mayer, 1990. 119 min. Color. D: Simon Wincer. SC: John Hill. With Tom Selleck, Laura San Giacomo, Alan Rickman, Chris Haywood, Ron Haddrick, Tony Bonner, Jerome Ehlers, Conor McDermottoe, Roger Ward, Ben Mendelsohn, Steve Dodd, Karen Davitt, Kylie Foster, William Zappa, Jonathan Sweet, Jon Ewing, Tim Hughes, David Slingsby, Danny Adcock, Maeliosa Stafford, Ollie Hall, Danny Baldwin, Jim Willoughby, Spike Cherrie, Gerald Egan, Guy Norris, Mark Minchinton, Brian Ellison, Mark Pennell, Everlyn Krape, Eamon Kelly. Hired to work for an Australian rancher, an American sharpshooter becomes a fugitive after he refuses to kill aborigines. Above average Aussie Western that looks at the dark side of continental settlement.\n\n**3213** _ **Quincannon, Frontier Scout**_ **** United Artists, 1956. 83 min. Color. D: Lesley Selander. SC: John C. Higgins and Don Martin. With Tony Martin, Peggie Castle, John Bromfield, John Smith, Ron Randell, John Doucette, Morris Ankrum, Peter Mamakos, Ed Hashim, Tom London. A former Army officer, now a scout, agrees to lead an expedition into hostile territory in order to find stolen rifles. Tony Martin is very good in the title role of this pleasant program feature, although, surprisingly, he does not sing the title song.\n\n**Spanish lobby card for** _**Quincannon, Frontier Scout**_ **(United Artists, 1956).**\n\n**  \n**\n\n**3214** _ **Rachel and the Stranger**_ **** RKO Radio, 1948. 92 min. D: Norman Foster. SC: Waldo Salt. With Loretta Young, William Holden, Robert Mitchum, Gary Gray, Tom Tully, Sara Haden, Frank Ferguson, Walter Baldwin, Regina Wallace, Fred Conlan. A farmer buys a bond servant for a wife but finds she is attracted to his vagabond hunter pal. Well modulated frontier fare with just the right amount of drama and humor plus excellent work from its trio of stars.\n\n**3215** _ **Racing Blood**_ **** 20th Century\u2013Fox, 1954. 76 min. Color. D: Wesley Barry. SC: Sam Roeca and Wesley Barry. With Bill Williams, Jean Porter, Jimmy Boyd, George Cleveland, Frankie Darro, John Eldredge, Sam Flint, Fred Kohler, Jr., Fred Kelsey, George Steele, Bobby Johnson. A colt, which was supposed to have been destroyed at birth due to a split hoof, is raised by a stable boy and his uncle. Fair family film trading on the popularity of child singing star Jimmy Boyd.\n\n_**Racketeer Round-Up**_ see _**Gunners and Guns**_\n\n**3216** _ **Racketeers of the Range**_ **** RKO Radio, 1939. 62 min. D: D. Ross Lederman. SC: Oliver Drake. With George O'Brien, Marjorie Reynolds, Chill Wills, Ray Whitley, Gay Seabrook, Robert Fiske, Ben Corbett, Bud Osborne, John Dilson, Monte Montague, Cactus Mack, Frankie Marvin, Ed Peil, Sr., Frank O'Connor, Mary Gordon, Stanley Andrews, Wilfred Lucas, Harry Cording, Dick Hunter. A dishonest lawyer tries to cheat a woman out of her packing plant and a rival takes over the operation to save ranchers from being fleeced by the crook. Well done and every entertaining George O'Brien vehicle.\n\n_**Radio Ranch**_ see _**The Phantom Empire**_\n\n**3217** _ **Rage**_ **** Columbia, 1967. 103 min. Color. D: Gilberto Gazcon. SC: Teddi Sherman and Gilberto Gazcon. With Glenn Ford, Stella Stevens, David Reynoso, Armando Silvestre, Ariadna Welter, Jose Elias Moreno, David Silva, Valentin Trujillo, Jorge Russek, Raul Martinez. A physician makes a desperate flight across the Mexican desert as he tries to reach a medical clinic after being bitten by a rabid dog. Good screen fare, enhanced by Glenn Ford's fine work as the doctor.\n\n**3218** _ **Rage**_ **** Warner Bros., 1972. 100 min. Color. D: George C. Scott. SC: Philip Friedman and Dan Kleinman. With George C. Scott, Richard Basehart, Martin Sheen, Barnard Hughes, Nicholas Beauvy, Paul Stevens, Stephen Young, Kenneth Tobey, Robert Walden, William Jordan, Dabbs Greer, John Dierkes, Lou Frizzell, Ed Lauter, Terry Wilson, Fielding Greaves. A rancher wants revenge for the death of his son, who was killed as a result of an Army chemical experiment. Fair melodrama promising more than it delivers.\n\n**3219** _ **Rage at Dawn**_ **** RKO Radio, 1955. 87 min. Color. D: TimWhelan. SC: Horace McCoy. With Randolph Scott, Forrest Tucker, Mala Powers, J. Carrol Naish, Edgar Buchanan, Kenneth Tobey, Howard Petrie, Myron Healey, Ray Teal, Ralph Moody, Guy Prescott, Mike Ragan, Phil Chambers, George Wallace, Dennis Moore, James Lydon, Arthur Space, William Forrest, Denver Pyle, Trevor Bardette, Henry Wills, William Phipps, Holly Bane, Richard Garland, Dan White, Chubby Johnson. Two undercover agents pose as outlaws to capture the Reno Brothers as they arrive at the ranch of the gang's sister, who is shielding them against her better judgment. Fine Randolph Scott feature with a good script, direction and cast; originally called _**Seven Bad Men**_.\n\n**3220** _ **Ragtime Cowboy Joe**_ **** Universal, 1940. 60 min. D: Ray Taylor. SC: Sherman Lowe. With Johnny Mack Brown, Fuzzy Knight, Nell O'Day, Marilyn (Lynn) Merrick, Dick Curtis, Walter Soderling, Roy Barcroft, Harry Tenbrook, Wilfred Lucas, Harold Goodwin, Ed Cassidy, Buck Moulton, George Plues, Viola Vonn, Kermit Maynard, Jack Clifford, William Gould, Bud Osborne, Bob O'Connor, Eddie Parker, Slim Whitaker, Frank McCarroll, The Texas Rangers. A cattlemen's association detective is after a dishonest lawyer and his cohort who are rustling from a ranch so they can obtain it to sell to the railroad. The hackneyed plot gets good service in this Johnny Mack Brown opus.\n\n**3221** _ **The Raid**_ **** 20th Century\u2013Fox, 1954. 83 min. Color. D: Hugo Fregonese. SC: Sidney Boehm. With Van Heflin, Anne Bancroft, Richard Boone, Lee Marvin, Tommy Rettig, Peter Graves, Douglas Spencer, Paul Cavanagh, Will Wright, James Best, John Dierkes, Helen Ford, Harry Hines, Simon Scott, Claude Akins, Edmund Cobb, Roy Glenn, Lee Aaker, Richard Eyer, Robert Easton, Ethan Laidlaw, Kermit Maynard, Howard Wright, William Schallert, Kenneth Terrell, Frank McClure, James Stone, Stanley Blystone, George Keymas, Jack Low, John Berardino, Arthur Tovey, Dolores Fuller. During the Civil War a young widow and her son try to thwart the plans of several Confederate soldiers who have escaped from a military prison and plan to loot their town. Well done drama.\n\n**3222** _ **The Raiders**_ **** Universal-International, 1952. 80 min. Color. D: Lesley Selander. SC: Polly James and Lillie Hayward. With Richard Conte, Viveca Lindfors, Barbara Britton, Hugh O'Brian, Richard Martin, Palmer Lee (Gregg Palmer), William Reynolds, William Bishop, Morris Ankrum, Dennis Weaver, Margaret Field, John Kellogg, Frank Wilcox, Lane Bradford, Riley Hill, Neyle Morrow, Carlos Rivero, George J. Lewis, Francis McDonald, I. Stanford Jolley, Clayton Moore, Dennis Weaver, Dennis Ross, Edmund Cobb, Edward Earle, Riley Hill, Sydney Mason, Virginia Mullen, Max Wagner, Buddy Roosevelt, Clem Fuller, Ethan Laidlaw, Paul Kruger, Larry Hudson, Philo McCullough, Frank Ellis, Monte Montague, William Fawcett, Paul Newlan, Lee Morgan, Rush Williams, Eddie Parker, Leo Curley, Marvin Press. In 1849 California two men who have been wronged by local authorities team to destroy a crooked judge, the leader of an outlaw gang. Feature delivers in the entertainment department; reissue and TV title: _**Riders of Vengeance**_.\n\n**3223** _ **The Raiders**_ **** Universal, 1964. 75 min. Color. D: Herschel Daugherty. SC: Gene L. Coon. With Brian Keith, Robert Culp, Judi Meredith, James McMullan, Alfred Ryder, Simon Oakland, Ben Cooper, Trevor Bardette, Harry Carey, Jr., Richard Cutting, Addison Richards, Cliff Osmond, Paul Birch, Richard Deacon, Michael Burns. Texans try to drive cattle herds to the Kansas railheads but are ambushed as Wild Bill Hickok, Buffalo Bill Cody and Calamity Jane come to their rescue. Action filled little oater that has the look of a TV movie.\n\n**3224** _ **Raiders of Ghost City**_ **** Universal, 1944. 13 Chapters. D: Ray Taylor and Lewis D. Collins. SC: Luci Ward and Morgan Cox. With Dennis Moore, Wanda McKay, Lionel Atwill, Joseph Sawyer, Regis Toomey, Virginia Christine, Eddy Waller, Emmett Vogan, Addison Richards, Charles Wagenheim, Edmund Cobb, Jack Ingram, Jack Rockwell, Ernie Adams, George Eldredge, Rex Lease, Gene Garrick, Chief Thundercloud, Herman Hack, Chick Hannon, Denny Morton, Richard Hunter. Near the end of the Civil War a Union Secret Service operative gets on the trail of a gang of supposed Confederates who have been hijacking California gold shipments. Pretty good Universal cliffhanger enhanced by a fine cast.\n\n**3225** _ **Raiders of Old California**_ **** Republic, 1957. 72 min. D: Albert C. Gannaway. SC: Sam Roeca and Thomas C. Hubbard. With Jim Davis, Arleen Whelan, Faron Young, Marty Robbins, Louis Jean Heydt, Harry Lauter, Douglas Fowley, Lee Van Cleef, Larry Dobkin, Bill Coontz, Don Diamond, Rick Vallin, Tom Hubbard. As the Mexican War comes to a close a group of cavalry officers in California attempt to set up their own empire. Low budget not uninteresting drama; okay for action fans.\n\n**3226** _ **Raiders of Red Gap**_ **** Producers Releasing Corporation, 1943. 57 min. D: Sam Newfield. SC: Joseph O'Donnell. With Robert Livingston, Al St. John, Myrna Dell, Ed Cassidy, Charles King, Kermit Maynard, Roy Brent, Frank Ellis, George Chesebro, Reed Howes, Bud Osborne, Jimmy Aubrey, Merrill McCormick, George Morrell, Wally West, Slim Whitaker, Curley Dresden, Pascale Perry. A crooked, greedy rancher wants all the area's cattle so he hires the Lone Rider, thinking he is an outlaw, to kill off his neighbors but has the tables turned on him. Pretty fair \"Lone Rider\" entry, the last in the series.\n\n_**Raiders of Red Rock**_ see _**Fugitive of the Plains**_\n\n**3227** _ **Raiders of San Joaquin**_ **** Universal, 1943. 60 min. D: Lewis D. Collins. SC: Elmer Clifton and Morgan Cox. With Johnny Mack Brown, Tex Ritter, Fuzzy Knight, Jennifer Holt, Henry Hall, Joseph Bernard, George Eldredge, Henry Roquemore, John Elliott, Michael Vallon, Jack O'Shea, Jack Ingram, Carl Sepulveda, Budd Buster, The Jimmy Wakely Trio (Jimmy Wakely, Johnny Bond, Scotty Harrell), Slim Whitaker, Roy Brent, Earle Hodgins. A man becomes a fugitive when his father is murdered by railroaders trying to burn out area ranchers but he gets help from the son of the company's vice president. Action filled teaming of Johnny Mack Brown and Tex Ritter plus some nice songs composed by Oliver Drake.\n\n**3228** _ **Raiders of Sunset Pass**_ **** Republic, 1943. 57 min. D: John English. SC: John K. Butler. With Eddie Dew, Smiley Burnette, Jennifer Holt, Roy Barcroft, Charles Miller, LeRoy Mason, Maxine Doyle, Kenne Duncan, Jack Kirk, Jack Rockwell, Hank Bell, Budd Buster, Jack Ingram, Frank McCarroll, Fred Burns, Al Taylor, Mozelle Cravens, Nancy Worth, Isabel La Mal, Dorothy Andre, Kansas Moehring, George Byron, Larry Sewart. During World War II there is a manpower shortage on the range so a lawman gets cowgirls to round up needed cattle but they are opposed by an outlaw gang. Novel idea is used to good advantage in this \"John Paul Revere\" series oater but a weak hero does not help.\n\n**3229** _ **Raiders of the Border**_ **** Monogram, 1944. 58 min. D: John P. McCarthy. SC: Jess Bowers (Adele Buffington). With Johnny Mack Brown, Raymond Hatton, Craig Woods, Ellen Hall, Raphael (Ray) Bennett, Edmund Cobb, Ernie Adams, Richard Alexander, Lynton Brent, Stanley Price, Kermit Maynard, Ben Corbett, Herman Hack, Kansas Moehring. Two lawmen track outlaws along the Mexican border who rustle cattle and trade them for stolen jewels. Well scripted and action filled \"Nevada Jack McKenzie\" feature.\n\n_**Raiders of the Frontier**_ see _**Gangsters of the Frontier**_\n\n**3230** _ **Raiders of the Range**_ **** Republic, 1942. 55 min. D: John English. SC: Barry Shipman. With Bob Steele, Tom Tyler, Rufe Davis, Lois Collier, Frank Jaquet, Fred Kohler, Jr., Dennis Moore, Tom Chatterton, Charles Miller, Max Waizmann, Hal Price, Bud Geary, Jack Ingram, Al Taylor, Chuck Morrison, Bob Woodward, Monte Montague, Tom Steele, Kenneth Terrell, Richard Alexander, Cactus Mack, John Cason, Charles Phillips, Joel Friedkin, David Sharpe, Frank McCarroll, John Tyrrell, Pascale Perry, Bill Nestell. Outlaws are after a man's property because it contains rich oil deposits and a trio of cowboys come to his rescue when the crooks sabotage his drilling efforts. Fair entry from the latter days of \"The Three Mesquiteers\" series.\n\n**3231** _ **Raiders of the South**_ **** Monogram, 1947. 55 min. D: Lambert Hillyer. SC: J. Benton Cheney. With Johnny Mack Brown, Raymond Hatton, Evelyn Brent, Marshall Reed, Reno Blair, John Merton, John Hamilton, Pierce Lyden, Cactus Mack, Eddie Parker, Ted Adams, Frank LaRue, George Morrell, Curt Barrett and The Trailsmen, Ray Jones, Artie Ortego, Dee Cooper. In Texas during Reconstruction a Secret Service agent poses as an ex\u2013Confederate to stop a lawyer's scheme to start an empire by a land grab. Interesting low budget drama greatly helped by series stars Johnny Mack Brown and Raymond Hatton and leading lady Evelyn Brent.\n\n**3232** _ **Raiders of the West**_ **** Producers Releasing Corporation, 1942. 64 min. D: Peter Stewart (Sam Newfield). SC: Oliver Drake. With Bill \"Cowboy Rambler\" Boyd, Art Davis, Lee Powell, Virginia Carroll, Rex Lease, Glenn Strange, Charles King, Slim Whitaker, Milton Kibbee, Lynton Brent, John Elliott, Eddie Dean, Curley Dresden, William Desmond, Dale (Gale) Sherwood, Kenne Duncan, Bill Cody, Jr., Reed Howes, Hal Price, Fred \"Snowflake\" Toones, Carl Sepulveda, Frank Ellis, John Cason, Carl Mathews, Wally West, George Morrell, Arch Hall, Lane Bradford, Tex Palmer, Hank Bell, Fred MacKaye. Trying to capture outlaws, two range detectives pretend to be entertainers and get a job with the man they suspect is the gang leader. Vapid \"Frontier Marshals\" outing, wasting a fine supporting cast.\n\n**3233** _ **Raiders of Tomahawk Creek**_ **** Columbia, 1950. 55 min. D: Fred F. Sears. SC: Barry Shipman. With Charles Starett, Smiley Burnette, Kay Buckley, Edgar Dearing, Billy Kimbley, Paul Marion, Paul McGuire, Bill Hale, Ted Mapes, Lee Morgan. A new Indian agent tries to find out the reason for the killings of several area ranchers, deeds committed by his predecessor who wants native lands because he has discovered a valuable silver deposit. Okay \"Durango Kid\" adventure.\n\n**3234** _ **Rails into Laramie**_ **** Universal-International, 1954. 81 min. Color. D: Jesse Hibbs. SC: D.D. Beauchamp and Joseph Hoffman. With John Payne, Mari Blanchard, Dan Duryea, Joyce MacKenzie, Barton MacLane, Harry Shannon, Ralph Dumke, Lee Van Cleef, Myron Healey, Douglas Kennedy, James Griffith, Alexander Campbell, George Chandler, Charles Horvath, Steve (Stephen) Chase, Rex Lease, Ric Roman, Forrest Taylor, Gilbert Fallman, Tim Graham, Frank J. Scannell, Gayne Whitman, Bruno Ve Sota, Harry Wilson, Max Wagner, Ferris Taylor, Jack Stoney, Franklyn Farnum, Charles Sherlock, Paul Brinegar, Kernan Cripps, Roy Butler, Kenneth MacDonald, Hal K. Dawson, Kermit Maynard, Dean Fredericks, Christiane Martel, Race Gentry, Sol Gorss, Paul McGuire, John Harmon, Don Nagel, Larry Hudson, Eddie Parker, Anthony Jochim, Ethan Laidlaw, Kenner G. Kemp, Donald Kerr, Jack Lomas, Robert Keys, Brick Sullivan, Dale Van Sickel, Rusty Wescoatt, John Cliff. In the 1870s an Army sergeant tries to get a rail line built to Laramie, Wyoming, in spite of local crooks and sabotage. Entertaining and colorful oater; Rex Allen sings the title song.\n\n_**Rainbow**_ see _**Gringo**_\n\n**3235** _ **Rainbow Over Texas**_ **** Monogram, 1940. 62 min. D: Al Herman. SC: Roland Lynch, Roger Merton and Robert Emmett (Tansey). With Tex Ritter, Dorothy Fay, Warner Richmond, Dennis Moore, Arkansas Slim Andrews, Jim Pierce, Chuck Morrison, John Merton, Romaine Loudermilk and His Ranch House Cowboys, Tommy Southworth, Steve Lorber. Outlaws take over a town and close its school with a singing cowboy coming to the rescue. Fairly good Tex Ritter vehicle in which he performs a trio of songs, including the title tune.\n\n**3236** _ **Rainbow Over Texas**_ **** Republic, 1946. 65 min. D: Frank McDonald. SC: Gerald Geraghty. With Roy Rogers, George \"Gabby\" Hayes, Dale Evans, Bob Nolan and The Sons of the Pioneers (Tim Spencer, Ken Carson, Shug Fisher, Hugh Farr, Karl Farr), Sheldon Leonard, Robert Emmett Keane, Gerald Oliver Smith, Minerva Urecal, George J. Lewis, Kenne Duncan, Pierce Lyden, Dick Elliott, Bud Osborne, George Chesebro, Jo Ann Dean. Movie star Roy Rogers, along with the Sons of the Pioneers, returns to a Texas town and tries to rid it of crooks. Mediocre Western musical.\n\n**3237** _ **Rainbow Over the Range**_ **** Monogram, 1940. 60 min. D: Al Herman. SC: Roger Merton and Robert Emmett (Tansey). With Tex Ritter, Dorothy Fay, Arkansas Slim Andrews, Gene Alsace, Warner Richmond, Jim Pierce, Chuck Morrison, Dennis Moore, Art Wilcox and His Arizona Rangers, Tommy Southworth, Sherry Tansey, Tex Palmer. Outlaws rustle cavalry horses and an U.S. marshal and his pal are assigned to investigate. A pleasant Tex Ritter musical Western drama filmed in Arizona.\n\n**3238** _ **Rainbow Over the Rockies**_ **** Monogram, 1947. 54 min. D: Oliver Drake. SC: Elmer Clifton. With Jimmy Wakely, Lee \"Lasses\" White, Dennis Moore, Pat Starling, Wesley Tuttle and His Texas Stars, Budd Buster, Zon Murray, Carl Sepulveda, Bob Gilbert, Billy Dix, Jack Baxley. Two ranchers are pushed into a feud instigated by rustlers wanting their herds. Low grade Jimmy Wakely musical vehicle.\n\n**3239** _ **Rainbow Ranch**_ **** Monogram, 1933. 55 min. D: Harry Fraser. SC: Phil Dunham. With Rex Bell, Cecilia Parker, Robert Kortman, Henry Hall, George Nash, Gordon DeMain, Phil Dunham, Tiny Sanford, Jerry Storm, Tex Palmer, Archie Ricks, Vane Calvert, Harry Bowen. A cowboy returns home to find his father murdered and his girl and water rights stolen by a crook. So-so Rex Bell outing.\n\n**3240** _ **The Rainbow Trail**_ **** Fox, 1925. 58 min. D-SC: Lynn Reynolds. With Tom Mix, Anne Cornwall, George Bancroft, Lucien Littlefield, Mark Hamilton, Vivian Oakland, Thomas Delmar, Fred De Silva, Steve Clemento, Carol Holloway, Diana Muller. A man tries to free his uncle who has been trapped in a canyon by an outlaw and his gang. Tom Mix plays dual roles in this follow-up to Zane Grey's _**Riders of the Purple Sage**_ (q.v.), an entertaining silent initially filmed in 1918 by Fox starring William Farnum.\n\n**3241** _ **The Rainbow Trail**_ **** Fox, 1932. 65 min. D: David Howard. SC: Barry Connors and Philip Klein. With George O'Brien, Cecilia Parker, Minna Gombell, Roscoe Ates, J.M. Kerrigan, James Kirkwood, W.L. Thorne, Robert Frazer, Ruth Donnelly, Niles Welch, Landers Stevens, Laska Winters, Edward Hearn, Alice Ward, George Burton, Iron Eyes Cody, Tom Smith, Vinegar Roan, Edward Burns, Ralph Bucko, Johnny Luther, Cliff Lyons, Frank McGrath, Herman Nowlin, Cy Clegg, Dick Hunter, Little Pine, Clint Sharp. A cowboy joins a convoy searching for travelers trapped years before in a valley filled with gold as they tried to escape from a notorious outlaw. Polished third version of 1915 Zane Grey novel.\n\n**3242** _ **Rainbow Valley**_ **** Monogram, 1935. 52 min. D: Robert North Bradbury. SC: Lindsley Parsons. With John Wayne, Lucille Brown, LeRoy Mason, George Hayes, Buffalo Bill, Jr., Bert Dillard, Lloyd Ingraham, Lafe McKee, Frank Ellis, Art Dillard, Frank Ball, Fern Emmett, Henry Roquemore, Eddie Parker, Herman Hack, Artie Ortego, Jack Evans, Tex Palmer, Tommy Coats, Buck Morgan, Tex Phelps. An undercover agent pretends to be an escaped convict to get the goods on a gang after a tract of valuable land. A rather complicated plot does not hurt the overall entertainment value of this pleasant Lone Star Western from producer Paul Malvern that contains a terrific finale shootout.\n\n**3243** _ **Rainbow's End**_ **** First Division, 1935. 59 min. D: Norman Spencer. SC: Rollo Ward. With Hoot Gibson, June Gale, Oscar Apfel, Warner Richmond, Buddy Roosevelt, Ada Ince, Stanley Blystone, John Elliott, Henry Roquemore, Fred Gilman, Jerry Mandy. After a falling out with his businessman father, a cowpoke becomes foreman of a ranch where the old man holds the mortgage and a crook tries to get him to foreclose so he can get possession of the property. Modern Western with a good story and lots of comedy, the latter typical for Hoot Gibson.\n\n**3244** _ **The Rainmaker**_ **** Paramount, 1956. 121 min. Color. D: Joseph Anthony. SC: N. Richard Nash. With Katharine Hepburn, Burt Lancaster, Wendell Corey, Lloyd Bridges, Earl Holliman, Cameron Prud'Homme, Wallace Ford, Yvonne Lime, Dottie Bee Baker, Dan White, Stan Jones, John Benson, James Stone, Tony Merrill, Joe Brown, Ken Becker. A fake rainmaker comes to a drought stricken ranch in the Southwest and remains to romance a lonely spinster. Talkative but pleasant offbeat drama.\n\n**3245** _ **Ramona**_ **** 20th Century\u2013Fox, 1936. 94 min. Color. D: Henry King. SC: Lamar Trotti. With Loretta Young, Don Ameche, Kent Taylor, Pauline Frederick, Jane Darwell, Katherine De Mille, J. Carrol Naish, Victor Kilian, John Carradine, Pedro de Cordoba, Charles Waldron, Claire DuBrey, Russell Simpson, William Benedict, Chief Thundercloud, Erville Alderson, Donald Reed, Cecil Weston, D'Arcy Corrigan, Ethan Laidlaw, Kathryn Sheldon, Charles Middleton, Tom London, Richard Botiller, Sam Appel, Anita Ray, Carmen Bailey, Solidad Gonzales, Allan Jones, Robert Spindola, Martin Faust, Del Campo, Lee Kohlmar, D'Arcy Corrigan, Carmen La Roux, Tito Renaldo, Joe De La Cruz, Gertrude Chorre, Fred Godoy, Manuel Lopez. A half-breed Indian girl and a chief's son marry in Old California but find themselves the victims of prejudice. Colorful but somewhat miscast fourth screen version of the Helen Hunt Jackson novel; first filmed in 1910 by director D.W. Griffith with Mary Pickford and Henry B. Walthall, followed by a 1916 Clune Producing Company release starring Adda Gleason and Monroe Salisbury and a 1928 United Artists outing with Dolores Del Rio and Warner Baxter containing synchronized sound effects.\n\n**3246** _ **Rampage at Apache Wells**_ **** Columbia, 1966. 91 min. Color. D: Harald Philipp. SC: Fred Denger and Harald Philipp. With Stewart Granger, Pierre Brice, Macha Meril, Harald Leipnitz, Antje Weissgerber, Mario Girotti (Terence Hill), Walter Barnes, Heinz Erhart, Gerd Frickhoffer, Petar Poetrovic, Paddy Fox (Milan Srdoc), Milivoje Popovic-Mavid, Slobodan Dimitrijevic, Dusan Janicijevic, Davor Antolic. Frontiersman Old Surehand and his Indian friend Winnetou oppose an outlaw and his gang who have been cheating whites and Comanches out of their lands. One of the better European Westerns of the 1960s with fine work by Stewart Granger as Old Surehand; produced in West Germany in 1965 by Rialto-Film\/Jadran-Film as _**Der Olprinz**_ (The Oil Prince).\n\n**3247** _ **Ramrod**_ **** United Artists, 1947. 94 min. D: Andre De Toth. SC: John Moffitt, Graham Baker and Cecile Kramer. With Joel McCrea, Veronica Lake, Preston Foster, Charles Ruggles, Arleen Whelan, Donald Crisp, Lloyd Bridges, Don DeFore, Ian MacDonald, Sarah Padden, Nestor Paiva, Trevor Bardette, Hal Taliaferro, Wally Cassell, Ray Teal, Jeff Corey, Rose Higgens, Chic York, Cliff Parkinson, Ward Wood, John Powers, Victor Potel, Holly Bane, Houseley Stevenson, Robert Wood. The rebellious young female owner of a sheep ranch feuds with her father and hires a cowboy to do her bidding. Well done oater with good acting by the three leads.\n\n**3248** _ **The Ramrodder**_ **** Entertainment Ventures, 1969. 92 min. Color. D-SC: Van Guylder. With Jim (Roger) Gentry, Julia Blackburn, Brave Eagle (Robert Aiken), Kathy Williams, David Rosenkranz, Bob Beausoleil, Kathy Share, Kedric Wolfe, Marcia (Marsha) Jordan. An Indian chief's daughter comes to the rescue of a cowboy falsely accused of raping and killing a young maiden. Poorly done adult Western filmed at the Spahn Ranch in Chatsworth, California.\n\n**3249** _ **Ramsbottom Rides Again**_ **** British Lion, 1956. 93 min. D: John Baxter. SC: John Baxter, Basil Thomas, Geoffrey Orme, Arthur Askey and Glenn Melvyn. With Arthur Askey, Glenn Melvyn, Betty Marsden, Shani Wallis, Frankie Vaughan, Jerry Desmonde, Danny Ross, Anthea Askey, June Grant, Sabrina, Donald Stewart, Billy Percy, Dennis Wyndham, Gary Wayne, Campbell Singer, Marne Maitland, Beckett Bould, Sam Kydd, Deryck Guyler, Edie Martin, Leonard Williams, John Carson. After inheriting land in Canada, a pub owner moves there with his family and finds himself up against a bad man who controls the area. Typical over the top British comedy set in the wilds of Canada.\n\n**3250** _ **Rana:**_ _**The Legend of Shadow Lake**_ **** Titan International, 1980. 96 min. Color. D: Bill Rebane and Jerry Gregoris. SC: Lyona Oenez, Jerry Gregoris and Mike Landers. With Karen Diarmid, Alan Ross, Brad Ellingson, Julie Wheaton, Glenn Sherer, Doreen Moze, Jerry Gregoris, Jim Iquiente, Bruno Aclin, Lorry Getz, Michael J. Skewes, Paul Callaeay, Richard Lange, Angel Rebane. On a remote island an ancient Indian god takes revenge on those who try to steal his gold. Picturesque Georgia filmed cheapie with a scary monster resembling the Creature from the Black Lagoon.\n\n**3251** _ **Ranchers and Rascals**_ **** William Steiner, 1925. 57 min. D: Leo D. Maloney. With Leo D. Maloney, Josephine Hill, Whitehorse, Evelyn Thatcher, Barney Furey, Patricia Darling, Tom London, Bud Osborne, Bullet (dog). A cowboy, who only wants a peaceful life as he plans to marry, gets involved with a runaway wife, two malicious neighbors and a small baby. Amusing silent Leo Maloney feature.\n\n**3252** _ **Rancho Deluxe**_ **** United Artists, 1975. 93 min. Color. D: Frank Perry. SC: Thomas McGuane. With Jeff Bridges, Sam Waterston, Elizabeth Ashley, Slim Pickens, Clifton James, Charlene Dallas, Harry Dean Stanton, Richard Bright, Patti D'Arbanville, Maggie Wellman, Bert Conway, Anthony Palmer, Sandy Kenyon, Helen Craig, Joseph (Joe) Spinell, Richard McMurray, Danna Hansen, Doria Cooke, Richard Cavanaugh, Patti Jerome, Arnold Huppert, Esther Black, Ronda Copland, Jimmy Buffett, Dwight Riley, Jim Melin, Tim Schaeffer, Warren Oates, John Quade, Wilma Riley, Bob Wetzel, John Rogers, Joseph Sullivan, Ben Mar, Jr. Two pals pick off cattle from a rich rancher and then decide to rustle his entire herd. Passable low key comedy oater with several defects although Slim Pickens is a sheer delight as the supposedly decrepit, bumbling range investigator.\n\n**3253** _ **Rancho Grande**_ **** United Artists, 1936. 95 min. D: Fernando de Fuentes. SC: Guz Aguila and Fernando de Fuentes. With Tito Guizar, Rene Cardona, Esther Fernandez, Lorenzo Barcelata, Emma Roldan, Carlos Lopez \"Chaflan,\" Margarita Cortes, Dolores Camarillo, Manolo Noriega, Hernan Vera, Alfonso Sanchez Tello, Armando Aleman, Gaspar Nunez, Lucha Avila, Emilio Fernandez, Olga Falcon. The owner of a large rancho and his general manager both love the same woman. Pleasant Mexican Western romantic musical released in the U.S. as _**Out on the Big Ranch**_ in a dubbed version by Cinexport Distributing. Director Fernando de Fuentes remade it in 1949 starring Jorge Negrete, Lilia de Valle and Eduardo Noriega.\n\n**3254** _ **Rancho Grande**_ **** Republic, 1940. 68 min. D: Frank McDonald. SC: Bradford Ropes. With Gene Autry, Smiley Burnette, June Storey, Mary Lee, Dick Hogan, Ellen Lowe, Ferris Taylor, Joseph De Stefani, Roscoe Ates, Rex Lease, Ann Baldwin, Roy Barcroft, The Pals of the Golden West, Edna Lawrence, Jack Ingram, Bud Osborne, Slim Whitaker, Richard Webb, Hank Bell, Eddie Parker, Horace B. Carpenter, Jim Corey, Frankie Marvin, Chuck Baldra, The Brewer Kids, St. Joseph's School Boys' Choir. A foreman and his pal try to help the heirs of the ranch where they work with crooks who want the land for an irrigation project. Too much music and not enough action hamper this Gene Autry vehicle. TV title: _**El Rancho Grande**_.\n\n**3255** _ **Rancho Notorious**_ **** RKO Radio, 1952. 89 min. D: Fritz Lang. SC: Daniel Taradash. With Marlene Dietrich, Arthur Kennedy, Mel Ferrer, Gloria Henry, William Frawley, Lisa Ferraday, John Raven, Jack Elam, George Reeves, Frank Ferguson, Francis McDonald, Dan Seymour, John Kellogg, Redd Redwing, Stuart Randall, Roger Anderson, I. Stanford Jolley, Felipe Turich, John Doucette, Jose Dominguez, Lane Chandler, Fuzzy Knight, Lloyd Gough, Harry Woods, William Haade, Kermit Maynard, Fred Graham, Russell Johnson, Dick Elliott, Ray Jones. A man hunts for the killer of his girlfriend and ends up at a place run by a woman who protects outlaws. There is not much to recommend this attempt to re-establish Marlene Dietrich's image from _**Destry Rides Again**_ (1939) [q.v.].\n\n**3256** _ **Randy Rides Alone**_ **** Monogram, 1934. 53 min. D: Harry Fraser. SC: Lindsley Parsons. With John Wayne, Alberta Vaughn, George Hayes, Yakima Canutt, Earl Dwire, Tex Phelps, Artie Ortego, Herman Hack, Mack V. Wright, Horace B. Carpenter, Perry Murdock, Tommy Coats, Tex Palmer. A drifter is falsely accused of robbery and murder but with the help of a young woman he tries to find the real culprits. Rawboned Monogram Lone Star Western greatly helped by George \"Gabby\" Hayes as the villain.\n\n**3257** _ **Range Beyond the Blue**_ **** Producers Releasing Corporation, 1947. 53 min. D: Ray Taylor. SC: Patricia Harper. With Eddie Dean, Roscoe Ates, Helen Mowery, Ted Adams, Bob Duncan, Bill Hammond, George Turner, Ted French, Brad Slavin, Steve Clark, The Sunshine Boys (Eddie Wallace, J.D. Sumner, M.H. Richman, Freddie Daniel). A stage line is being robbed only when gold shipments are aboard so an investigator is called in to corral the thieves. Dreary oater, except for Eddie dean singing the title song and the novelty ditty \"The Pony with the Uncombed Hair.\"\n\n**3258** _ **The Range Busters**_ **** Monogram, 1940. 55 min. D: S. Roy Luby. SC: John Rathmell. With Ray Corrigan, John King, Max Terhune, Luana Walters, LeRoy Mason, Earle Hodgins, Frank LaRue, Kermit Maynard, Bruce King, Duke (Carl) Matthews, Horace Murphy, Karl Hackett, Herman Hack, Ed Brady, Hank Worden, Jimmie Widener. A crook kills a rancher for his land and gold mine while a trio of cowboys arrive looking for a mysterious figure called \"The Phantom,\" who hides out on the spread. The first in \"The Range Busters\" series provides fast action and mystery.\n\n**3259** _ **Range Defenders**_ **** Republic, 1937. 56 min. D: Mack V. Wright. SC: Joseph Poland. With Robert Livingston, Ray Corrigan, Max Terhune, Eleanor Stewart, Harry Woods, Yakima Canutt, Earle Hodgins, Thomas Carr, John Merton, Harrison Greene, Horace B. Carpenter, Frank Ellis, Fred \"Snowflake\" Toones, Jack O'Shea, Ernie Adams, Jack Rockwell, Merrill McCormick, Curley Dresden, Jack Kirk, George Morrell, Donald Kirke, Milburn Morante, Al Taylor, Hank Bell, Lafe McKee, Lew Meehan, Charles Brinley, Jack Lowe. Crooks cause a feud between cattle ranchers and sheep men as three cowpoke pals try to calm the situation. Action packed and entertaining \"Three Mesquiteers\" outing, well directed by Mack V. Wright.\n\n**3260** _ **Range Feud**_ **** Columbia, 1931. 64 min. D: D. Ross Lederman. SC: George Plympton. With Buck Jones, Susan Fleming, John Wayne, Ed Le Saint, William Walling, Wallace MacDonald, Harry Woods, Frank Austin, Glenn Strange, Lew Meehan, Jim Corey, Frank Ellis, Bob Reeves, Merrill McCormick, Archie Ricks, Hank Bell, Blackjack Ward, Rube Dalroy, William McCall, Al Taylor, Bob Burns, Jack Curtis, Jack Low. The town's new sheriff arrests his foster brother who is accused of killing his girl's rancher father. Nicely done Buck Jones vehicle with John Wayne as the accused.\n\n**3261** _ **Range Justice**_ **** Monogram, 1949. 57 min. D: Ray Taylor. SC: Ronald Davidson. With Johnny Mack Brown, Max Terhune, Sarah Padden, Felice Ingersoll, Riley Hill, Tristram Coffin, Fred Kohler, Jr., Eddie Parker, Kenne Duncan, Bill Hale, Myron Healey, Bill Potter, Bob Woodward, Carl Mathews. A ranch foreman joins an outlaw gang to get the goods on the hoodlums rustling his female boss' cattle. Fair Johnny Mack Brown entry from the latter days of his long running Monogram series.\n\n**3262** _ **Range Land**_ **** Monogram, 1949. 60 min. D: Lambert Hillyer. SC: Adele Buffington. With Whip Wilson, Andy Clyde, Reno Browne, Reed Howes, Kenne Duncan, Kermit Maynard, Stanley Blystone, Steve Clark, Leonard Penn, John Cason, Carol Henry, Carl Mathews, Dee Cooper. A cowboy tries to stop an outlaw gang from stealing a vast amount of range land. Below average Whip Wilson oater; remake of _**Gun Packer**_ (q.v.).\n\n**3263** _ **Range Law**_ **** Tiffany, 1931. 60 min. D: Phil Rosen. SC: Earle Snell. With Ken Maynard, Frances Dade, Lafe McKee, Frank Mayo, William Duncan, Charles King, Jack Rockwell, Tom London, Blackjack Ward, Aileen Manning, Robert Dudley, Bob Burns, Bud McClure, Ralph Bucko, Roy Bucko. A cowboy is falsely put in jail for a crime he did not commit but his friends arrange his rescue so he can roundup the real culprits. Fairly action filled Ken Maynard early talkie with Lafe McKee in a comedy role for a change.\n\n**3264** _ **Range Law**_ **** Monogram, 1944. 57 min. D: Lambert Hillyer. SC: Frank H. Young. With Johnny Mack Brown, Raymond Hatton, Ellen Hall, Sarah Padden, Lloyd Ingraham, Marshall Reed, Steve Clark, Jack Ingram, Hugh Prosser, Stanley Price, Art Fowler, Hal Price, Ben Corbett, Bud Osborne, Tex Palmer, George Morrell, Lynton Brent, Forrest Taylor, Horace B. Carpenter, Kansas Moehring, Milburn Morante, Foxy Callahan, Denver Dixon, Chick Hannon, Artie Ortego. Two marshals come to the aid of a woman whose friend has been falsely accused of cattle rustling. Average \"Nevada Jack McKenzie\" series effort.\n\n**3265** _ **Range Renegades**_ **** Monogram, 1948. 54 min. D: Lambert Hillyer. SC: Ronald Davidson and William Lively. With Jimmy Wakely, Dub Taylor, Jennifer Holt, Dennis Moore, Riley Hill, John James, Frank LaRue, Steve Clark, Milburn Morante, Bob Woodward, Carl Mathews, Roy Garrett. A marshal is on the trail of an outlaw gang led by a woman. Typically low grade Jimmy Wakely singing oater.\n\n**3266** _ **Range Riders**_ **** Superior, 1934. 46 min. D: Victor Adamson (Denver Dixon). SC: L.V. Jefferson. With Buddy Roosevelt, Barbara Starr, Lew Meehan, William (Merrill) McCormick, Horace B. Carpenter, Herman Hack, Clyde McClary, Fred Parker, Lionel Belmore, Allen Holbrook, Steve Clemente, Bob McKenzie, Ed Gyton, Sam Pierce, Denver Dixon. An agricultural college student returns home to thwart the machinations of bully Buck Crawford and his outlaw gang. About as low grade as a Western can go, with rag tag production values.\n\n**3267** _ **Range War**_ **** Paramount, 1939. 65 min. D: Lesley Selander. SC: Sam Robins. With William Boyd, Russell Hayden, Willard Robertson, Matt Moore, Pedro de Cordoba, Betty Moran, Britt Wood, Kenneth Harlan, Eddie Dean, Earle Hodgins, Glenn Strange, Jason Robards, Stanley Price, George Chesebro, Raphael (Ray) Bennett, Don Latorre, Wen Wright, Rad Robinson, Tom Smith, Herman Hack, George Morrell, Pascale Perry. To help a young woman stop the destruction of a railroad, Hopalong Cassidy takes money from a stagecoach so outlaws will not steal it, gets arrested and joins the gang to bring them to justice. Scenic locations and good photography highlight this \"Hopalong Cassidy\" feature.\n\n**3268** _ **Range Warfare**_ **** Willis Kent, 1935. 55 min. D: S. Roy Lucy. SC: E.B. Mann. With Reb Russell, Lucille Lund, Wally Wales, Lafe McKee, Roger Williams, Slim Whitaker, Ed Boland, Richard Botiller, Chief Black Hawk, Ed Porter, Gene Alsace, Bart Carre, George Morrell, Jack Kirk, Artie Ortego, Chuck Baldra, Bud Pope, Jack Hendricks, Clyde McClary, Jack King, Jack Jones. A cowboy is after an outlaw gang wanted for cattle theft and murder. Pretty fair Reb Russell film that will please his fans; reissued as _**Vengeance**_.\n\n_**Rangeland Empire**_ see _**West of the Brazos**_\n\n**3269** _ **The Ranger and the Lady**_ **** Republic, 1940. 59 min. D: Joseph Kane. SC: Stuart Anthony and Gerald Geraghty. With Roy Rogers, George \"Gabby\" Hayes, Jacqueline Wells (Julie Bishop), Harry Woods, Henry Brandon, LeRoy Mason, Tom London, Noble Johnson, Si Jenks, Ted Mapes, Yakima Canutt, Herman Hack, Art Dillard, Lloyd Ingraham, Henry Wills, Davison Clark, Fred Burns, Al Taylor, Bud McClure, Bill Nestell, Victor Cox. In Texas ranger Roy Rogers fights outlaws trying to hijack settler's wagons so they can take over the country, until General Sam Houston comes to the rescue. Lots of action, a good plot and pleasant songs in this Roy Rogers vehicle.\n\n**3270** _ **Ranger Courage**_ **** Columbia, 1937. 59 min. D: Spencer Gordon Bennet. SC: Nate Gatzert. With Bob Allen, Martha Tibbetts, Walter Miller, Buzzy Henry, Bud Osborne, Robert Kortman, Harry Strang, William Gould, Horace Murphy, Franklyn Farnum, Buffalo Bill, Jr., Gene Alsace, George Morrell, Oscar Gahan, Rudy Sooter, Lloyd Perryman, Robert Hoag, Cactus Mack, J.W. Cody, Lafe McKee, Buck Moulton, Bob Reeves, Frank Ball, Nate Gatzert, Jack King, Horace B. Carpenter, Bob Burns, Jack Evans, Silver Tip Baker, Jack Tornek, Tex Palmer, Al Taylor, George Hazel, Jim Corey, Eva McKenzie. After helping wagon train passengers attacked by outlaws disguised as Indians, a ranger sets out to round up the gang. Mediocre entry in Bob Allen's brief Columbia series.\n\n**3271** _ **Ranger of Cherokee Strip**_ **** Republic, 1949. 60 min. D: Philip Ford. SC: Bob Williams. With Monte Hale, Alice (Alix) Talton, Paul Hurst, Roy Barcroft, Douglas Kennedy, George Meeker, Frank Fenton, Monte Blue, Lane Bradford, Arthur Walsh, George Chesebro, Herman Hack, Tom Steele, Tommy Coats. In the Cherokee Indian Nation in the 1890s, a ranger tries to stop trouble caused by a renegade blamed for the death of his chief, the culprits being cattlemen who want to lease the tribe's land. Average program Western with Douglas Kennedy more colorful than hero Monte Hale.\n\n**3272** _ **Ranger of the Law**_ **** American, 1935. 50 min. D: R.J. Renroh (Robert J. Horner). SC: Royal Hampton. With Buffalo Bill, Jr., Jeanne (Genee) Boutell, George Chesebro, Jack Long, Boris Bullock, Ben Corbett, Frank Clark, Duke R. Lee, Lake Reynolds, Herman Hack, Tex Palmer, Oscar Gahan, Al Haskell, Clyde McClary. A rodeo rider opposes a crook out to steal a ranch from a pretty girl. Very cheaply made with lots of stock rodeo footage and a poor soundtrack with muffled dialogue. Also called _**Whirlwind Rider**_.\n\n**3273** _ **The Ranger, the Cook and a Hole in the Sky**_ **** ABC-TV, 1995. 94 min. Color. D: John Kent Harrison. SC: Robert Wayne. With Sam Elliott, Jerry O'Connell, Ricky Jay, Molly Parker, Don S. Davis, Robert Wisden, Michael Tayles, Tom Butler, Jay Brazeau, Callum Keith Rennie, Alan C. Peterson, Campbell Lane, Frank Cassini. A legendary forest ranger serves as a mentor to a teenager in 1919 Montana. Pretty fair TV movie called _**A Hole in the Sky**_ on video.\n\n**3274** _ **The Rangers**_ **** NBC-TV\/Universal, 1974. 74 min. Color. D: Christian Nyby II. SC: Robert A. Cinader, Michael Donavan and Preston Wood. With James G. Richardson, Colby Chester, Jim B. Smith, Laraine Stephens, Laurence Delaney, Michael Conrad, Roger Bowen, Carl Roger Breedlove, David Birkoff. U.S. Forest Service park rangers work to preserve the environment and wildlife as well as rescue those in danger. Passable telefilm that evolved into the brief \"Sierra\" (NBC-TV, 1974) series.\n\n**3275** _ **The Ranger's Code**_ **** Monogram, 1933. 60 min. D: Robert North Bradbury. SC: Harry O. (Fraser) Jones. With Bob Steele, Doris Hill, George Hayes, George Nash, Frank Ball, Ed Brady, Hal Price, Ernie Adams, Dick Dickinson, Tex Phelps, Joe Dominguez. A lawman learns his girl's brother is hooked up with an outlaw gang. So-so Bob Steele vehicle.\n\n_**Rangers Go West**_ see _**Three Men from Texas**_\n\n**3276** _ **Rangers of Fortune**_ **** Paramount, 1940. 80 min. D: Sam Wood. SC: Frank Butler. With Fred MacMurray, Albert Dekker, Gilbert Roland, Patricia Morison, Joseph Schildkraut, Dick Foran, Betty Brewer, Arthur Allen, Bernard Nedell, Brandon Tynan, Minor Watson, Rosa Turich, Frank Puglia, Frank Milan, Matt McHugh, Erville Alderson, Fern Emmett, Joseph Eggenton, Ed Le Saint, Rod Cameron, Fred Malatesta, Harry Fleischmann, Martin Garralaga, Paul \"Tiny\" Newlan, Charles Middleton, Richard Alexander, Charles Irwin, Frank Hagney, Dewey Robinson, Jack Robinson. Three desperadoes on the run from the law arrive in a town where they befriend a newspaper editor and a small girl and help get rid of area outlaws. Breezy action film with the three leads making a good team.\n\n**3277** _ **The Rangers Ride**_ **** Monogram, 1948. 56 min. D: Derwin Abrahams. SC: Basil Dickey. With Jimmy Wakely, Dub Taylor, Virginia Belmont, Riley Hill, Marshall Reed, Steve Clark, Pierce Lyden, Milburn Morante, Jim Diehl, Cactus Mack, Carol Henry, Bud Osborne, Bob Woodward, Boyd Stockman. When a former Texas Ranger is accused of murder a friend comes to his defense. There is not much to brag about in this musical oater.\n\n**3278** _ **The Ranger's Round-Up**_ **** Spectrum, 1938. 57 min. D: Sam Newfield. SC: George Plympton. With Fred Scott, Al St. John, Christine McIntyre, Earle Hodgins, Steve Ryan, Karl Hackett, Robert Owen, Syd Chatan, Carl Mathews, Richard Cramer, Jimmy Aubrey, Lew Porter, Cactus Mack, Steve Clark, Chick Hannon, Milburn Morante, Oscar Gahan, Sherry Tansey, Olin Francis, Tex Palmer. An undercover agent joins a medicine show that outlaws have been using as a front for their illegal activities, a fact not known by its proprietor. Pretty fair Fred Scott vehicle; this one includes the classic song \"The Terror of Termite Valley.\"\n\n**3279** _ **The Rangers Step In**_ **** Columbia, 1937. 58 min. D: Spencer Gordon Bennet. SC: Nate Gatzert. With Bob Allen, Eleanor Stewart, Hal Taliaferro, John Merton, Jack Ingram, Jack Rockwell, Jay Wilsey (Buffalo Bill, Jr.), Lafe McKee, Robert Kortman, Harry Harvey, Joseph Girard, Herman Hack, Harry Tenbrook, Richard Cramer, Arthur Millett, Lew Meehan, Ray Jones, Jack King, George Plues, Francis Walker, Eddie Jarequi, Billy Townsend, Tex Palmer, Artie Ortego, Bert Dillard, Eva McKenzie, Jack Evans, Art Dillard, Ray Henderson, Al Taylor. A sheriff calls in a Texas Ranger to investigate trouble caused by a crook reviving a feud between two families so he can obtain a ranch. Bob Allen's final series film is a pleasant affair.\n**3280** _ **The Rangers Take Over**_ **** Producers Releasing Corporation, 1942. 62 min. D: Al Herman. SC: Elmer Clifton. With Dave O'Brien, Jim Newill, Guy Wilkerson, Iris Meredith, Forrest Taylor, I. Stanford Jolley, Charles King, Carl Mathews, Harry Harvey, Lynton Brent, Bud Osborne, Cal Shrum and The Rhythm Rangers (Rusty Cline, Don Weston, Art Wenzell), Slim Whitaker, Hank Bell, Jess Cavin, Rube Dalroy, Art Dillard, George Morrell, Jack Tornek. Following his discharge from the Texas Rangers, a man joins an outlaw gang working as an informant. The first entry in \"The Texas Rangers\" series is a mediocre affair, a portent of things to come.\n\n**3281** _ **Rangle River**_ **** J.H. Hoffberg, 1936. 75 min. D: Clarence Badger. SC: Charles Chauvel and Elsa Chauvel. With Victor Jory, Margaret Dare, Robert Coote, George Bryant, Rita Paucefort, Leo Crackwell, Cecil Perry, Georgia Sterling, Stewart McColl, Phil Smith. Crooks try to put a rancher out of business by cheating him of his water rights. Interesting Australian production based on Zane Grey's novel; TV title: _**Men with Whips**_.\n\n**3282** _ **Ransom for Alice**_ **** NBC-TV\/Universal, 1977. 78 min. Color. D: David Lowell Rich. SC: Jim Byrnes. With Yvette Mimieux, Gil Gerard, Charles Napier, Gene Barry, John Dennan, Lauire Prange, Barnard Hughes, Robert Logan, Harris Yulin, Marc Vahanian, Mills Watson, Gavin MacLeod, Anthony James. In 1890s Seattle a deputy marshal and his pretty partner try to find a young girl caught in a white slavery ring. Average TV movie with a good cast.\n\n**3283** _ **The Rare Breed**_ **** Universal, 1966. 97 min. Color. D: Andrew V. McLaglen. SC: Ric Hardman. With James Stewart, Maureen O'Hara, Brian Keith, Juliet Mills, Don Galloway, David Brian, Jack Elam, Ben Johnson, Harry Carey, Jr., Perry Lopez, Larry Domasin, Alan Caillou, Bob Gravage, Wayne Van Horn, Leroy Johnson, John Harris, Ted Mapes, Larry Blake, Charles Lampkin, Tex Armstrong. A woman cattle breeder and her daughter bring a prize bull to the U.S. to start a new line but she becomes involved with an ex-rancher and his pal. Well made production that is on the dull side.\n\n**3284** _ **El Ratero de las Pobres**_ (The Pickpocket of the Poor) Conacite Dios, 1982. 103 min. Color. D: Francisco Guerrero. SC: Jorge Patino and Alfredo Gurrola. With Hector Suarez, Bruno Rey, Dacia Gonzalez, Blanca Guerra, Carlos Cardan, Teresa Alvarez, Sergio Calderon, Susana Cabrera, Armando Soto La Marina, Marcela Rubiales, Arsenio Campos, Jorge Fegan, Alfred Espinoza, Arlette Pacheco, Jorge Reynoso, Jorge Patino. A handsome outlaw with an eye for the women, and three cohorts, loot a village and give some of the money to the poor but spend most of it gambling. Fair Mexican Western comedy originally titled _**Valentine Lazana, el Ratero de las Pobres**_ (Valentine Lazana, the Pickpocket of the Poor).\n\n**3285** _ **Raton Pass**_ **** Warner Bros., 1951. 84 min. D: Edwin L. Marin. SC: Tom Blackburn and James Webb. With Dennis Morgan, Patricia Neal, Steve Cochran, Dorothy Hart, Scott Forbes, Basil Ruysdael, Louis Jean Heydt, Roland Winters, James Burke, Elvira Curci, Carlos Conde, John Crawford, Rodolfo Hoyos, Jr. A married couple are at odds over their cattle empire and when the wife gets the upper hand her husband organizes area homesteaders against her. More than passable melodrama with good work by its leads.\n\n**3286** _ **The Rattler Kid**_ **** Copercines\/Nike Cinematografica, 1967. 83 min. Color. D: Leon Klimovsky. SC: Odoardo Fiory and Luigi Mondello. With Richard Wyler, Brad Harris, William Spolt (Guglielmo Spoletini), Jesus Puente, Femi Benussi, Aurora de Alba, Simon Arriaga, Luis Barboo, Luis Induni, Miguel Del Castillo, Frank Brana, Conny Caracciolo, Jose Maria Caffarel, Lucio De Santis, Rafael Albaicin, Santiago Rivero. After being framed for an Army money robbery and murder, a sergeant escapes and becomes a gunman to find the men who betrayed him. Passable Spaghetti Western called _**Un Hombre Vino a Matar**_ in Spain and _**L'Uomo Ventuo per Uccidere**_ in Italy.\n\n**3287** _ **Raw Edge**_ **** Universal-International, 1956. 76 min. Color. D: John Sherwood. SC: Harry Essex and Robert Hill. With Rory Calhoun, Yvonne De Carlo, Mara Corday, Rex Reason, Neville Brand, Emile Meyer, Herbert Rudley, Robert Wilke, John Gilmore, Gregg Barton, Ed Furey, Francis McDonald, Julia Montoya, Paul Fierro, William Schallert, Richard James, Robert Hoy. A beautiful woman, taken by the first man who claims her in Oregon in the 1840s, finds herself attracted to a another out to kill her husband in revenge for his brother's murder. Complicated Albert Zugsmith production saved by good photography and fetching Yvonne De Carlo.\n\n**3288** _ **Raw Timber**_ **** Crescent, 1937. 63 min. D: Ray Taylor. SC: Bennett Cohen and John T. Neville. With Tom Keene, Peggy Keys, Budd Buster, Robert Fiske, Lee Phelps, John Rutherford, Rafael (Ray) Bennett, Slim Whitaker, Bart Carre, Dorothy Vernon, Fred Parker. When a timber baron murders a ranger who found out he was destroying the forest for his own gain another lawman shows up to investigate. Pretty fair entry in Tom Keene's historical series for producer E.B. Derr with nice locales and good photography by Arthur Martinelli.\n\n**3289** _ **Rawhide**_ **** Principal\/20th Century\u2013Fox, 1938. 60 min. D: Ray Taylor. SC: Dan Jarrett and Jack Natteford. With Smith Ballew, Lou Gehrig, Evelyn Knapp, Carl Stockdale, Cy Kendall, Slim Whitaker, Arthur Loft, Si Jenks, Lafe McKee, Lee Shumway, Dick Curtis, Tom Forman, Cliff Parkinson, Harry Tenbrook, Ed Cassidy, Ray Whitley, Carleton Young, Ed Peil, Sr., Bill Patton, Fred Burns, George Plues, Donald Kirke, Merrill McCormick, George Morrell, Al Haskell, Ray Henderson, Charles Brinley, Charles Murphy, Sid Kibrick. Baseball star Lou Gehrig finds crooks are trying to steal his sister's ranch so he teams with her lawyer to stop them. Interesting curio with Lou Gehrig's sturdy performance somewhat overshadowing star Smith Ballew.\n\n**3290** _ **Rawhide**_ **** 20th Century\u2013Fox, 1951. 86 min. D: Henry Hathaway. SC: Dudley Nichols. With Tyrone Power, Susan Hayward, Hugh Marlowe, Dean Jagger, Edgar Buchanan, Jack Elam, George Tobias, Jeff Corey, James Millican, Louis Jean Heydt, William Haade, Milton Corey, Sr., Kenneth Tobey, Dan White, Max Terhune, Robert Adler, Judy Ann Dunn, Vincent Neptune, Walter Sande, Si Jenks, Dick Curtis, Edith Evanson. Outlaws hole up at a lonely way station, kill the owner and hold his assistant and a woman with a small child hostage. Well modulated and entertaining Western with fine work by Hugh Marlowe as the gang leader. Alternate TV title: _**Desperate Siege**_.\n\n**3291** _ **Rawhide Rangers**_ **** Universal, 1941. 56 min. D: Ray Taylor. SC: Ed Earl Repp. With Johnny Mack Brown, Fuzzy Knight, Nell O'Day, Kathryn Adams, Roy Harris, Harry Cording, Alan Bridge, Frank Shannon, Ed Cassidy, Robert Kortman, Chester Gan, James Farley, Jack Rockwell, Frank Ellis, Fred Burns, Tex Palmer, Tex Terry, The Pickard Family, The Texas Rangers. After the murder of his brother, a Texas Ranger supposedly resigns and becomes an outlaw so he can infiltrate the gang responsible for the killing. Standard, but slick, Johnny Mack Brown vehicle.\n\n**3292** _ **Rawhide Romance**_ **** Superior, 1934. 47 min. D: Denver Dixon (Victor Adamson). SC: L.V. Jefferson. With Buffalo Bill, Jr., Si Jenks, Lafe McKee, Boris Bullock, Genee Boutell, Jack Evans, Marin Sais, Clyde McClary, Ken Brocker, Bart (Carre) Carey, Herman Hack, Hamilton Steele, Denver Dixon. A cowboy gets involved with a pretty girl and her parents at a rustic guest ranch plagued by a robbery gang. A bit better than the usual Victor Adamson production, highlighted by Brydon Baker's photography; Buffalo Bill, Jr., briefly sings \"I've Got No Use for the Women.\"\n\n**3293** _ **The Rawhide Terror**_ **** Security, 1934. 52 min. D: Jack Nelson and Bruce Mitchell. SC: Jack Nelson. With Art Mix, William Desmond, Edmund Cobb, William Barrymore (Boris Bullock), Frances Morris, Bill Patton, Tommy Bupp, Herman Hack, George Holt, George Gyton, Ed Carey, Ernest Scott, Fred Parker, Clyde McClary, Joe Weaver, Denver Dixon. A lawman is on the trail of a sadistic, demented outlaw who is really his orphaned brother. This Victor Adamson (Denver Dixon) production has to be seen to be believed; it is a shoddy vintage horror Western and a treat for grade-Z movie followers. Their fans will appreciate seeing three genre greats, Art Mix, William Desmond and Edmund Cobb, in the starring roles, and Boris Bullock's madman predates the 1970s superhuman horror film fiends.\n\n**3294** _ **The Rawhide Trail**_ **** Allied Artists, 1958. 67 min. D: Robert Gordon. SC: Alexander Wells. With Rex Reason, Nancy Gates, Richard Erdman, Rusty Lane, Frank Chase, Ann Doran, Robert Knapp, Richard Warren, Al Wyatt, John Dierkes, Sam Buffington, Jana Davi, William Murphy, Richard Greary, Chet Sampson. Two men falsely accused of leading settlers into an Indian ambush try to prove their innocence as they await hanging and the tribe attacks the fort where they are imprisoned. Low budget affair that is nothing to shout about; title song sung by the Guardsmen.\n\n**3295** _ **The Rawhide Years**_ **** Universal-International, 1956. 85 min. Color. D: Rudolph Mate. SC: Earl Fenton, Robert Presnell, Jr. and D.D. Beauchamp. With Tony Curtis, Colleen Miller, Arthur Kennedy, William Demarest, William Gargan, Peter Van Eyck, Minor Watson, Donald Randolph, Chubby Johnson, James Anderson, Robert Wilke, Trevor Bardette, Robert Foulk, Leigh Snowden, Don Beddoe, Malcolm Atterbury, Charles Evans, I. Stanford Jolley, Rex Lease, Chuck Roberson, Marlene Felton, Clarence Lung, Lane Bradford. A reformed gambler in the 1870s is falsely accused of a riverboat murder. More than passable melodrama.\n\n_**R.C.M.P. and the Treasure of Genghis Khan**_ see _**Dangers of the Canadian Mounted**_\n\n**3296** _ **The Reason Nobody Hardly Ever Saw a Fat Outlaw in the Old West Is as Follows:**_ **** NBC-TV\/Universal, 1967. 49 min. Color. D: Hal Kantor. With Don Knotts, Arthur Godfrey, Percy Helton, Mary-Robin Redd, Jack Lambert, Herbert (Herb) Edelman, Bob Hope (host). The bumbling Curly Kid finds he cannot get himself arrested or even break the law despite a desire to be the most famous outlaw in the West. Fair comedy Western originally telecast on \"Bob Hope Chrysler Theatre\" (NBC-TV, 1963\u201367).\n\n**3297** _ **A Reason to Live, A Reason to Die!**_ **** K-Tel, 1974. 92 min. Color. D-SC: Tonino Valerii. With James Coburn, Telly Savalas, Bud Spencer, Georges Geret, Robert Burton, Jose Suarez, Ralph Goodwin, Paco Sanz, Joseph Mitchell. During the Civil War a Union officer and seven prisoners try to capture a fort held by Confederates. Violent European co-production that must depend on the name value of its stars rather than any innate quality; released in Europe in 1972. TV title: _**Massacre at Fort Holman**_.\n\n**3298** _ **Rebel City**_ **** Allied Artists, 1953. 60 min. D: Thomas Carr. SC: Sid Theil. With Bill Elliott, Marjorie Lord, Robert Kent, Ray Walker, I. Stanford Jolley, Keith Richards, Henry Rowland, Denver Pyle, John Crawford, Otto Waldis, Stanley Price, Michael Vallon, Pierce Lyden, Gregg Barton. Arriving in a Kansas Town intent on finding his father's killer, a man uncovers a Copperhead conspiracy to aid the Confederacy. Compact and entertaining Bill Elliott outing.\n\n**3299** _ **Rebel in Town**_ **** United Artists, 1956. 78 min. Color. D: Alfred Werker. SC: Danny Arnold. With John Payne, Ruth Roman, J. Carrol Naish, Ben Cooper, John Smith, James Griffith, Mary Adams, Bobby Clark, Mimi Gibson, Ben Johnson, Joel Ashley, Jack Perrin, Kermit Maynard, Sterling Franck. Returning home from the Civil War with his father and brothers, an ex-soldier accidentally kills a small boy but ends up having his life saved by the victim's father. A different kind of plot for this type of fare; above average.\n\n**3300** _ **Rebellion**_ **** Crescent, 1936. 62 min. D: Lynn Shores. SC: John T. Neville. With Tom Keene, Rita (Hayworth) Cansino, Duncan Renaldo, William Royle, Gino Corrado, Roger Gray, Bob McKenzie, Allen Cavan, Jack Ingram, Lita Cortez, Theodore Lorch, M.W. (Merrill) McCormick, George Regas, Allen Greer, Al Haskell, Ralph Bucko. President Zachary Taylor sends an Army officer to California after its acquisition from Mexico to stop lawlessness against Spanish landowners. Pretty fair entry in the historical film series Tom Keened made for producer E.B. Derr; reissued in 1946 as _**Frisco Lady**_.\n\n**3301** _ **Rebels on the Loose**_ **** Fenix Film, 1966. 92 min. Color. D: Bruno Corbucci. SC: Vittorio Vighi, Ugo Guerra, Scarnicci and Tarabusi. With Raimondo Vianello, Lando Buzzanca, Maria Martinez, Monica Randall, Gino Buzzanca, Alfonso Rojos, Emilio Rodrigues, Giovanni Lenzi, Miguel De Castillo, Santiago Rivero, Mario Castellani, Mario De Simone, Antonio Albaisin. Eight years after the Civil War ends two Southern soldiers at an isolated fort still think the conflict is going on and they meet two lawless women who urge them to continue their sabotage activities. Limp Spaghetti Western takeoff released in Italy as _**Ringo e Gringo Contro Tutti**_ (Ringo and Gringo Against All).\n\n**3302** _ **The Reckless Buckaroo**_ **** Spectrum, 1937. 57 min. D: Harry Fraser. SC: Zarah Tazil. With Bill Cody, Bill Cody, Jr., Betty Mack, Buzz Barton, Roger Williams, Ed Cassidy, Lew Meehan, Milburn Morante, Budd Buster, Francis Walker, Allen Greer, Jack Nelson. A prospector is enlisted by a wounded lawman to bring in smugglers who are really being led by his deputy. Passable teaming of Bill Cody and his son for producer Ray Kirkwood. Also called _**Reckless Buckaroos**_.\n\n_**Reckless Buckaroos**_ see _**The Reckless Buckaroo**_\n\n**3303** _ **Reckless Ranger**_ **** Columbia, 1937. 56 min. D: Spencer Gordon Bennet. SC: Nate Gatzert. With Bob Allen, Louise Small, Jack Perrin, Mary MacLaren, Harry Woods, Buddy Cox, Jack Rockwell, Slim Whitaker, Roger Williams, Lane Chandler, Dirk Thane, Bud Osborne, Jim Corey, Tom London, Hal Price, Al Taylor, Tex Cooper, Bob McKenzie, Frank Ball, George Plues, Lafe McKee, Tex Palmer, Chick Hannon, Oscar Gahan, Rudy Sooter, Lloyd Perryman, Jack Tornek, Arthur Millett, Tommy Coats, Bud Pope, Archie Ricks, Jack King, Jack Evans, Eva McKenzie, Buck Morgan, Victor Cox. A ranger investigates his brother's killing that was carried out by a gang under the control of a crook who wants to run off all sheep men so he can use government grazing lands for his cattle. Fast moving Bob Allen affair, although Jack Perrin just about steals the show as the secondary hero.\n\n**3304** _ **Reckoning**_ **** Lincoln Media Group, 2002. 110 min. Color. D-SC: Jason Rodriguez. With Jason Rodriguez, Stacy Cunningham, Cheryl Lawson, Alan Waserman, Kim Jackson, Kent Smith, Craig Davis, Ritchie Copenhaver, Shiu Tong Lee, Mark Rodriguez, John Lamble, Ramon Becerra, Cosmo Segurson. A cowboy joins forces with a hooker to find his sister who has been forced into prostitution by a gunman. Slow moving adventure yarn.\n\n**3305** _ **The Red Badge of Courage**_ **** Metro-Goldwyn-Mayer, 1951. 69 min. D-SC: John Huston. With Audie Murphy, Bill Mauldin, Douglas Dick, Royal Dano, John Dierkes, Arthur Hunnicutt, Andy Devine, Robert Easton Burke, Smith Ballew, Glenn Strange, Dan White, Frank McGraw, Tim Durant, Emmett Lynn, I. Stanford Jolley, William Phillips, House Peters, Jr., Frank Sully, George Offerman, Jr., Joe Marston, Robert Nichols, Lou Nova, Fred Kohler, Jr., Dick Curtis, Guy Wilkerson, Buddy Roosevelt, Jim Hayward, Gloria Eaton, Robert Cherry, Whit Bissell, William Phipps, Ed Hinton, Lynn Farr. During the Civil War a raw recruit panics during his first battlefield encounter but later garners the courage to fight and become a hero. John Huston's truncated version of the Stephen Crane novel is fairly interesting, especially for showing the effects of battle on individuals; look for Andy Devine's dynamic cameo as the optimistic soldier.\n\n**3306** _ **The Red Badge of Courage**_ **** NBC-TV\/20th Century\u2013Fox, 1974. 78 min. Color. D: Lee Phillips SC: John Gay. With Richard Thomas, Michael Brandon, Wendell Burton, Charles Aidman, Warren Berlinger, Lee DeBroux, Francesca Jarvis, George Sawaya, Hank Hendrick, John Cox, Tiny Wells, Norman Stone, Jack DeLeon (narrator). A frightened Union soldier learns the meaning of bravery after running from the enemy during his first battle. Okay TV adaptation of Stephen Crane's book.\n\n**3307** _ **Red Blood**_ **** Anchor, 1926. 50 min. D: J.P. McGowan. SC: G.A. Durlam. With Al Hoxie, Nayone Warfield, Lew Meehan, Eddie Barry, J.P. McGowan, Frances Kellogg, Walter Patterson, Lem Sowards. A cowboy, who is always in trouble, loves the boss' daughter who is also sought by a crooked gambler blackmailing her brother. Although it contains lots of fights, this Al Hoxie silent horse opera is pretty dull going.\n\n**3308** _ **Red Blood of Courage**_ **** Ambassador, 1935. 55 min. D: Jack (John) English. SC: Barry Barringer. With Kermit Maynard, Ann Sheridan, Reginald Barlow, Charles King, Ben Hendricks, Jr., George Regas, Nat Carr, Milburn Morante, Art Dillard, Carl Mathews. A Mountie uncovers a plot with crooks kidnapping a man for his land while one of them impersonates him to fool his visiting niece. Entertaining Kermit Maynard north woods drama; first filmed by Selig in 1915 with Thomas Santschi, Bessie Eyton and Lafe McKee, from James Oliver Curwood's scenario.\n\n**3309** _ **Red Blood, Yellow Gold**_ **** Hispamex, 1967. 89 min. Color. D: Nando Cicero. SC: Jaime Jesus Balcazar, Jose Antonio de la Loma, Enzo Dell'Aquilla and Roberto Gianviti. With George Hilton, Edd Byrnes, George Martin, Milo Quesada, Monica Randall, Gerard Herter, Jose Badalo, Gisella Monaldi. A former priest, a bandit and an Mexican outlaw team to oppose marauders, Union and Rebel soldiers, a woman who says they murdered her folks and Confederate deserters. Above average Italian-Spanish co-production filmed as _**Professionisti per un Massacro**_ (Professionals for a Massacre).\n\n**3310** _ **Red Canyon**_ **** Universal-International, 1949. 82 min. Color. D: George Sherman. SC: Maurice Geraghty. With Ann Blyth, Howard, Duff, George Brent, Edgar Buchanan, John McIntire, Chill Wills, Jane Darwell, Lloyd Bridges, James Seay, Edmund MacDonald, David Clarke, Denver Pyle, Hank Patterson, Ray Bennett, Hank Worden, Sonny Chorre, Edmund Cobb, Willard W. Willingham, John Carpenter. Wanting to enter a horse in a race, a girl and an ex-outlaw try to tame a wild stallion called Black Velvet. Fair adaptation of Zane Grey's novel _Wildfire_.\n\n**3311** _ **Red Canyon**_ **** Fireside Releasing, 2008. 92 min. Color. D: Giovanni Rodriguez. SC: Laura Pratt and Giovanni Rodriguez. With Charistine Lakin, Tim Draxl, Katie Maguire, Norman Reedus, Justin Hartley, Noah Fleiss, Ankur Bhatt, Richard T. Pratt, Andy Mackenzie, Walter Rodriguez. Two siblings, along with three friends, return to a Western town where they survived a brutal attack and learn a terrible truth. Better than average modern-day horror Western.\n\n**3312** _ **Red Desert**_ **** Lippert, 1949. 60 min. D: Ford Beebe. SC: Daniel B. Ullman and Ron Ormond. With Don Barry, Tom Neal, Jack Holt, Margia Dean, Byron Foulger, Joseph Crehan, John Cason, Tom London, Holly Bane, Hank Bell, George Slocum, Reed Hadley (narrator). Two gambling house operators use their business as a front for the sale of stolen government money and President Grant assigns the Pecos Kid to uncover the culprits. Action laden, well acted Don Barry vehicle; ten times better than the 1964 Michelangelo Antonioni Italian film with the same title.\n\n**Tom Neal, Hank Bell, Byron Foulger and Don Barry in** _**Red Desert**_ **(Lippert, 1949).**\n\n**  \n**\n\n**3313** _ **Red Earth, White Earth**_ **** CBS-TV, 1989. 96 min. Color. D: David Greene. SC: Michael De Guzman. With Timothy Daly, Genevieve Bujold, Ralph Waite, Richard Farnsworth, Billy Merasty, Alberto Watson, Danette Mackay, Joseph Cazalet, Norris Domingue, Patricia Ann Eshibok, Ian Finley, Dean Hagopian, Francois Klanfer, Ron Lea, Jordan Marchand, Walter Massey, Ritchie Nadeau, Michael Sandy, Philip Spensley, Harry Standjofski, Doreen Stevens, Vlasi Vrana, Billy Two Rivers. A man returns home to find his family in disorder because their homestead is being claimed by an Indian tribe. Not particularly ingratiating modern-day TV movie.\n\n**3314** _ **Red Fork Range**_ **** Big 4, 1931. 60 min. D-SC: Alvin J. Neitz (Alan James). With Wally Wales, Ruth Mix, Al Ferguson, Cliff Lyons, Bud Osborne, Lafe McKee, Fred Gilman, Jim Corey, George Gerwing. Will Armstrong, Chief Big Tree, Slim Whitaker, Herman Hack, Bob Burns, Tex Phelps, Charles Le Moyne, Barney Beasley, Ralph Bucko, Starlight (horse). A cowboy tries to win a stagecoach race but gets opposition from an outlaw gang. Bottom rung, torpid Wally Wales outing.\n\n**3315** _ **The Red Fury**_ **** Dayton Films, 1984. 105 min. Color. D: Lyman Dayton. SC: Joe Elliott. With Alan Hale, Katherine Cannon, Diane McBain, Cindy Roberts, Juan Gonzales, Jason Wingreen, Calvin Barlett, Paul Stahell, Mary Ethel Gregory, Addie Marsden, Al Hanson, Ronald Hatch. A young Indian boy sacrifices his beloved horse to bring tolerance to a Western town. Nicely done family film.\n\n**3316** _ **Red Garters**_ **** Paramount, 1954. 91 min. Color. D: George Marshall. SC: Michael Fessier. With Rosemary Clooney, Jack Carson, Guy Mitchell, Pat Crowley, Gene Barry, Buddy Ebsen, Cass Daley, Reginald Owen, Frank Faylen, Joanne Gilbert, Richard Hale, Herbert N. Golden, Anthony Earl Numkena, Ethan Laidlaw, Sylvia Lewis, Maxine Gates, Rand Harper, Elizabeth Slifer, Marla English, Walter Tetley (voice). A man rides into a small town looking for his brother's killer only to find the citizens celebrating the event. Strange conglomerate of music and drama; definitely a curio.\n\n**3317** _ **Red Headed Stranger**_ **** Alive Films, 1986. 105 min. D-SC: Bill Wittliff. With Willie Nelson, Morgan Fairchild, R.G. Armstrong, Royal Dano, Katharine Ross, Sonny Carl Davis, Ted J. Crum, Marinell Madden, Bryan Fowler, Paul English, Bee Spears, Dennis Hill, Mark Jenkins, Berkley Garrett, Elberta Hunter, Mark Voges, John Dodson, John Browning, Julius Tennon, Joanne Russell, Bob Boothe, Bill Richardson, Robert Kuhn, Ralph Ware, Joe K. Longley, Steve Uzzell, Jubal Clark, James Wong, Martha Fowler, Keith Larsen, Army McMichael, Allison Wittliff, Ada Harden. A Montana parson finds out his bride is two timing him and he becomes attracted to another woman while taking up arms against a gang of ruffians. Lethargic screen version of Willie Nelson's 1975 Columbia Records album.\n\n**3318** _ **Red Mountain**_ **** Paramount, 1951. 84 min. Color. D: William Dieterle. SC: John Meredyth Lucas, George F. Slavin and George W. George. With Alan Ladd, Lizabeth Scott, Arthur Kennedy, John Ireland, Jeff Corey, James Bell, Bert Freed, Walter Sande, Neville Brand, Carleton Young, Whit Bissell, Jay Silverheels, Francis McDonald, Iron Eyes Cody, Dan White, Ralph Moody, Crane Whitley, Herbert Belles. Quantrill and his followers, pretending to fight for the Confederacy, go on looting and murder raids in Kansas and Missouri during the Civil war. Action filled account of the notorious marauder, well played by John Ireland.\n\n**3319** _ **The Red Pony**_ **** Republic, 1949. 89 min. Color. D: Lewis Milestone. SC: John Steinbeck. With Robert Mitchum, Myrna Loy, Louis Calhern, Shepperd Strudwick, Peter Miles, Margaret Hamilton, Patty King, Jackie Jackson, Beau Bridges, Don Kay Reynolds, Wee Willie Davis, Tommy Sheridan, George Tyne, Nino Tempo, Poodles Hanneford, Grace Hanneford, Eddie Borden, Max Wagner, Alvin Hammer, Dolores Castle, William Quinan. A young boy longs for his own pony and after his stern father gets him one he neglects it and the animal wanders away and dies. John Steinbeck adapted his uneven version of his novella, although the film makes for pleasant entertainment.\n\n**3320** _ **The Red Pony**_ **** NBC-TV\/Universal, 1973. 100 min. Color. D: Robert Totten. SC: Robert Totten and Ron Bishop. With Henry Fonda, Maureen O'Hara, Ben Johnson, Jack Elam, Richard Jaeckel, Clint Howard, Julian Rivero, Roy Jenson, Woodrow Chambliss, Warren Douglas, Yvonne Wood, Victor Sun Yung, Lieux Dressler, Link Wyler, Rance Howard, Sally Carter-Ihnat, Heather Totten, Kurt Sled. A young boys feels more kinship with his pony than with his hard to understand father. Well made TV version of the John Steinbeck work, although the character of Billy Buck (played by Robert Mitchum in the 1949 screen version [q.v.]) is deleted.\n\n**3321** _ **The Red Raiders**_ **** First National, 1927. 63 min. D: Albert Rogell. SC: Marion Jackson. With Ken Maynard, Ann Drew, J.P. McGowan, Paul Hurst, Harry Shutan, Ben Corbett, Chief Yowlachie, Tom Bay, Lafe McKee, Hal Salter. An Army lieutenant is assigned to a fort in Sioux Territory and there he manages to subdue a wild horse, fall in love with a beautiful girl and thwart the war intentions of an Indian chief who does not want his people on a reservation. One of Ken Maynard's very best films in which he does impressive stunt work; highly entertaining.\n\n**3322** _ **The Red Rider**_ **** Universal, 1934. 15 Chapters. D: Louis Friedlander (Lew Landers). SC: George Plympton, Vin Moore, Ella O'Neill and George Morgan. With Buck Jones, Marion Shilling, Grant Withers, Walter Miller, J.P. McGowan, Richard Cramer, Margaret La Marr, Charles K. French, Edmund Cobb, William Desmond, Mert Lavarre (John Merton), Frank Rice, Jim Thorpe, Monte Montague, Denny Meadows (Dennis Moore), Jim Corey, Bud Osborne, Al Ferguson, Artie Ortego, Tom Ricketts, J. Frank Glendon, King Baggott, Charles Brinley, Bill Steele, Fred Burns, Hank Bell, Chester Gan, Jim Tony, Art Mix, Jack Rockwell, Jack O'Shea, Frank Ellis, Ben Hendricks, Harry Royer, Charles McMurphy, Frank Hagney, Jack Shannon, Chet Ryan, Cliff Lyons, Tom Steele, Eddie Woehler, Jr, Rose Plummer. A sheriff loses his job because he refuses to believe his pal committed a murder and while investigating the case along the Mexican border he comes across clues to support this opinion. Top notch Buck Jones cliffhanger.\n\n**3323** _ **Red River**_ **** United Artists, 1948. 125 min. D: Howard Hawks. SC: Borden Chase and Charles Schnee. With John Wayne, Montgomery Clift, Joanne Dru, Walter Brennan, Coleen Gray, John Ireland, Noah Beery, Jr., Harry Carey, Harry Carey, Jr., Chief Yowlachie, Mickey Kuhn, Paul Fix, Hank Worden, Ivan Parry, Hal Taliaferro, Paul Fierro, Ray Hyke, Glenn Strange, Tom Tyler, Lane Chandler, Dan White, Lee Phelps, George Lloyd, John Merton, Pierce Lyden, Shelley Winters, Davison Clark, William Self, Harry Cording, Jack Montgomery, Chief Sky Eagle, Richard Farnsworth. A cattle baron leads a rough drive forming the Chisholm Trail but along the way his methods are questioned by his foster son who takes command away from him. Classic Western, one of the very best.\n\n**3324** _ **Red River**_ **** CBS-TV, 1988. 96 min. Color. D: Richard Michaels. SC: Richard Fielder. With James Arness, Bruce Boxleitner, Gregory Harrison, Ray Walston, Stan Shaw, Laura Johnson, Zachary Ansley, Ty Hardin, Robert Horton, John Lupton, Guy Madison, L.Q. Jones, Burton Gilliam, Jerry Potter, Johnmark Bradley, Donnie Jeffcoast, James Oscar Lee, Bud Stout, Travis Swords, Bob Terhune, Temple Williams. Tension brews between an cattleman and his young prot\u00e9g\u00e9 during a long, trouble filled trail drive. Pale TV remake of the 1948 classic (q.v.) although James Arness is good in the John Wayne role.\n\n**3325** _ **Red River Range**_ **** Republic, 1938. 56 min. D: George Sherman. SC: Stanley Roberts, Betty Burbridge and Luci Ward. With John Wayne, Ray Corrigan, Max Terhune, Polly Moran, Lorna Gray (Adrian Booth), Kirby Grant, Sammy McKim, William Royle, Perry Ivins, Stanley Blystone, Lenore Bushman, Burr Caruth, Roger Williams, Earl Askam, Olin Francis, Ed Cassidy, Fred \"Snowflake\" Toones, Bob McKenzie, Theodore Lorch, Al Taylor, Curley Dresden, John Beach, Joe Whitehead, Jack Montgomery, Chuck Baldra. Rustlers use a dude ranch as their base for cattle thefts via trucks and three cowboys are called in to solve the mystery with one of them pretending to be an outlaw to get in with the gang. Fast moving, slick \"Three Mesquiteers\" series addition.\n\n**3326** _ **Red River Renegades**_ **** Republic, 1946. 55 min. D: Thomas Carr. SC: Norman S. Hall. With Sunset Carson, Peggy Stewart, Tom London, Bruce Langley, Kenne Duncan, LeRoy Mason, Ted Adams, Edmund Cobb, Stanley Price, Fred Graham, Jack Rockwell, Tex Terry. Two postal inspectors investigate a rash of stagecoach robberies and disappearances, with one of them being murdered. Typically action filled Sunset Carson affair with speed making up for lack of finesse.\n\n**3327** _ **Red River Robin Hood**_ **** RKO Radio, 1943. 57 min. D: Lesley Selander. SC: Bennett Cohen. With Tim Holt, Cliff Edwards, Barbara Moffett, Eddie Dew, Otto Hoffman, Russell Wade, Tom London, Earle Hodgins, Bud McTaggart, Reed Howes, Kenne Duncan, David Sharpe, Bob McKenzie, Jack Rockwell, Jack Montgomery. A masked figure called \"Mr. Justice\" fights for the rights of ranchers being bilked in taxes by crooks with a fake land grant claim. Satisfying Tim Holt vehicle.\n\n**3328** _ **Red River Shore**_ **** Republic, 1953. 54 min. D: Harry Keller. SC: Arthur Orloff and Gerald Geraghty. With Rex Allen, Slim Pickens, Lyn Thomas, Bill Phipps, Douglas Fowley, Trevor Bardette, William Haade, Emmett Vogan, John Cason, Rayford Barnes, Jack Perrin. Forced to kill a crooked businessman in a gunfight, a marshal vows to keep the man's guilt a secret but trouble soon develops between him and the dead man's son over a bogus oil drilling operation. A complicated plot keeps this Rex Allen outing moving along despite dropping production values.\n\n**3329** _ **Red River Valley**_ **** Republic, 1936. 56 min. D: B. Reeves Eason. SC: Dorrell McGowan and Stuart McGowan. With Gene Autry, Smiley Burnette, Frances Grant, Boothe Howard, George Chesebro, Charles King, Frankie Marvin, Lloyd Ingraham, Hank Bell, Earl Dwire, Jack Kenney, Sam Flint, Eugene Jackson, Edward Hearn, Frank LaRue, Ken Cooper, C.E. \"Cap\" Anderson, George Morrell. A singing cowpoke tries to find out who has been dynamiting ditches, the action endangering a big irrigation project. Colorful, fast moving Gene Autry musical opus; alternate title: _**Man of the Frontier**_.\n\n**3330** _ **Red River Valley**_ **** Republic, 1941. 62 min. D: Joseph Kane. SC: Malcolm Stuart Boylan. With Roy Rogers, George \"Gabby\" Hayes, Sally Payne, Gale Storm, Hal Taliaferro, Bob Nolan and The Sons of the Pioneers (Tim Spencer, Lloyd Perryman, Pat Brady, Hugh Farr, Karl Farr), Trevor Bardette, Robert Homans, Lynton Brent, Ed Peil, Sr., Dick Wessel, Jack Rockwell, Ted Mapes, Jack Kirk, Bob Burns, Chuck Baldra, Hank Bell. Racketeers attempt to obtain water rights and cattle in an area where a dam is being built. Fair Roy Rogers feature.\n\n**3331** _ **Red Rock Outlaw**_ **** Friedgen, 1950. 56 min. D-SC: Elmer (Clifton) Pond. With Bob Gilbert, Lee \"Lasses\" White, Ione Nixon, Forrest Matthews, Virginia Jackson, Wanda Cantlon, Billy Dix, Reno Browne, Tennessee Jim, Joyce Gardner, Pinky Patek, Ewing \"Lucky\" Brown, Eddie Majors, Billy McCoy, Clint Johnson, Johnny Bias. A murderous outlaw tries to kill his honest rancher twin brother and take his place. Rock bottom.\n\n**3332** _ **The Red Rope**_ **** Republic, 1937. 60 min. D: S. Roy Luby. SC: George Plympton. With Bob Steele, Lois January, Horace Murphy, Charles King, Bobby Nelson, Ed Cassidy, Lew Meehan, Frank Ball, Karl Hackett, Jack Rockwell, Forrest Taylor, Lionel Belmore, Richard Cramer, Horace B. Carpenter, Willie Fung, Wally West, Oscar Gahan, Tex Palmer, Emma Tansey, Sherry Tansey, Ray Henderson, Fred Parker. A cowboy helps a young couple who want to marry but whose plans are altered by a man who holds the mortgage on the girl's father's ranch and wants her for himself. Well written Bob Steele vehicle.\n\n**3333** _ **Red Skies of Montana**_ **** 20th Century\u2013Fox, 1952. 96 min. Color. D: Joseph M. Newman. SC: Harry Kleiner. With Richard Widmark, Constance Smith, Jeffrey Hunter, Richard Boone, Warren Stevens, James Griffith, Joseph Sawyer, Gregory Walcott, Richard Crenna, Bob Nichols, Ralph Reed, Walter Murphy, Robert Adler, Charles (Bronson) Buchinsky, Mike Mahoney, Larry Dobkin, John Close, Grady Galloway, Henry Kulky, Harry Carter, Charles Tannen, Ron Hargrave, Robert Osterloh, Ted Ryan, John Kennedy, Parley Baer, Barbara Woodell, Ray Hyke, Wilson Hood, Ann Morrison. A firefighter for the U.S. Forestry Service plans revenge on a superior who he feels caused his father's death during a mission. A dull plot hampers some well staged fire sequences in this action melodrama, also called _**Smoke Jumpers**_.\n\n**3334** _ **The Red Stallion**_ **** Eagle-Lion, 1947. 82 min. Color. D: Lesley Selander. SC: Robert E. Kent and Crane Wilbur. With Robert Paige, Noreen Nash, Ted Donaldson, Jane Darwell, Ray Collins, Guy Kibbee, Willie Best, Robert Bice, Pierre Watkin, Bill Carledge, Emmett Vogan, Big Red (horse), Daisy (dog). When his grandmother is about to lose her ranch, a young boy desperately tries to use his beloved horse to make the money to save it. Well made and entertaining family fare.\n\n**3335** _ **Red Stallion in the Rockies**_ **** Eagle-Lion, 1949. 85 min. Color. D: Ralph Murphy. SC: Francis Rosenwald. With Arthur Franz, Jean Heather, Wallace Ford, Jim Davis, Ray Collins, Leatrice Joy, James Kirkwood, Ray Bennett, Guy Wilkerson, John Doucette, Howard Negley, Lyle Latell, Joseph J. Greene, Gustino Loyal, Dynamite (horse). Two circus performers go to work on a ranch and one of them falls in love with the owner's niece. Good drama enhanced by Jim Davis' performance as the rancher's grasping son and the presence of silent screen star Leatrice Joy.\n\n**3336** _ **Red Sun**_ **** National General, 1972. 112 min. Color. D: Terence Young. SC: Laird Koenig, Denne Bart Petitclerc, William Roberts and Lawrence Roman. With Charles Bronson, Toshiro Mifune, Alain Delon, Ursula Andress, Capucine, Satoshi Nakamoura, Bart Barry, Lee Burton, Anthony Dawson, Hiroshi Tanaka, John Hamilton, George W. Lycan, Jose Nieto, Julia Pena, Monica Randall, Luc Merenda, John Vermont. A Japanese Sumurai travels to the U.S. and eventually joins forces with an outlaw in retrieving a valuable sword stolen by outlaws. The teaming of Charles Bronson and Toshiro Mifune adds zest to this picture that was issued in Europe in 1971 as _**Soleil Rouge**_ (Red Sun) by Corona Films\/Oceania Films\/Balcazar Films.\n\n**3337** _ **Red Sundown**_ **** Universal-International, 1956. 81 min. Color. D: Jack Arnold. SC: Martin Berkeley. With Rory Calhoun, Martha Hyer, Dean Jagger, Robert Middleton, James Millican, Lita Baron, Grant Williams, Trevor Bardette, David Kasday, Leo Gordon, Steve Darrell, Stevie Wootton, John Carpenter, Henry Wills, Alex Sharp, Lee Van Cleef. An ex-gunman becomes a deputy sheriff in a frontier town and opposes a land baron and his hired killer. There is nothing exceptional in this Albert Zugsmith production.\n\n**3338** _ **The Red Tomahawk**_ **** Paramount, 1967. 82 min. Color. D: R.G. Springsteen. SC: Steve Fisher. With Howard Keel, Joan Caulfield, Broderick Crawford, Scott Brady, Wendell Corey, Richard Arlen, Tom Drake, Ben Cooper, Tracy Olsen, Donald Barry, Reg Parton, Roy Jenson, Dan White, Henry Wills, Gerald Jann, Sol Gorss, Sailor Vincent, Kenner G. Kemp, Joe Ploski. Following the Little Big Horn massacre, an Army captain tries to warn the citizens of Deadwood of a possible Indian attack and he uncovers four Gatling guns to hold off the marauders. The veteran cast tries hard but is stymied by an indifferent script and laggard production values.\n\n_**The Red, White and Black**_ see _**Soul Soldier**_\n\n**3339** _ **The Redhead and the Cowboy**_ **** Paramount, 1951. 82 min. D: Leslie Fenton. SC: Jonathan Latimer and Liam O'Brien. With Glenn Ford, Rhonda Fleming, Edmond O'Brien, Alan Reed, Morris Ankrum, Edith Evanson, Perry Ivins, Janine Perreau, Douglas Spencer, Ray Teal, Ralph Byrd, King Donovan, Jim Bannon, Tom Moore, Robert Kortman, Jeff York, Paul Lees, Don Dunning, Lester Dorr, Len Hendry, Rodd Redwing, Richard Karlan, Rory Mallinson, Henry Wills, Eric Alden, Iron Eyes Cody, Gertrude Astor, Lupe Gonzalez, Charles Quirk. A woman becomes a courier for the Confederacy in its waning days and is pursued by a cowboy, who needs her testimony to clear him of a murder charge, and a Union spy. Pretty fair screen entertainment.\n\n**3340** _ **The Redhead from Wyoming**_ **** Universal-International, 1953. 80 min. Color. D: Lee Sholem. SC: Polly James and Herb Meadow. With Maureen O'Hara, Alex Nicol, Robert Strauss, Jeanne Cooper, William Bishop, Alexander Scourby, Jack Kelly, Palmer Lee (Gregg Palmer), Claudette Thornton, Ray Bennett, Joe Bailey, Rush Williams, Dennis Weaver, Stacy Harris, Larry Hudson, Jack Perrin, Edmund Cobb, Buddy Roosevelt, Syd Saylor, Philo McCullough, Harold Goodwin, Henry Wills, Boyd Morgan, David Sharpe, George Taylor, Bob Merrick, Jack Hyde. An attractive woman operates a front shielding cattle rustlers but finds herself falling in love with the lawman after an outlaw she is protecting. Okay action feature with Maureen O'Hara doing a good job in the title role.\n\n**3341** _ **The Redmen and the Renegades**_ **** International Television Corporation, 1964. 89 min. D: Sam Newfield. With John Hart, Lon Chaney, George Barnes, John Vernon, Brian Smyth. Ethan Allen is accused of treason and Hawkeye and Chingachgook try to prove his innocence. Average television feature made up of three episodes of \"Hawkeye and the Last of the Mohicans\" (Syndicated, 1956).\n\n**3342** _ **Redwood Forest Trail**_ **** Republic, 1950. 68 min. D: Philip Ford. SC: Bradford Ropes. With Rex Allen, Jeff Donnell, Jane Darwell, Marten Lamont, Carl \"Alfalfa\" Switzer, Pierre Watkin, Jimmy Ogg, Dick Jones, John Cason, Jack Larson, Robert Burns, Joseph Granby. When boys staying at a ranch for the rehabilitation of delinquents are accused of being involved in a murder a cowboy tries to prove their innocence. Pretty good Rex Allen vehicle.\n\n_**The Refugee**_ see _**Three Faces West**_\n\n**3343** _ **El Regreso del Monstruo**_ (The Return of the Monster). Filmadora Mexicana, 1958. 63 min. D: Joselito Rodriguez. SC: Luis Manrique, Antonio Oreland and Fernando Oses. With Luis Aguilar, Pascual Garcia Pena, Teresa Velazquez, Jaime Fernandez, Yolanda del Valle, Arturo Martinez, Fanny Shiller, Roger Lopez, Sergio Murrietta. A masked avenger, the Scarlet Fox, and his pal battle a mad doctor planning to mate a young woman with a resurrected monster. Chilling Mexican horror Western released on video as _**Zorro vs. the Teenage Monster**_.\n\n**3344** _ **Relentless**_ **** Columbia, 1948. 93 min. Color. D: George Sherman. SC: Winston Miller. With Robert Young, Marguerite Chapman, Willard Parker, Akim Tamiroff, Barton MacLane, Mike Mazurki, Robert Barrat, Clem Bevans, Frank Fenton, Hank Patterson, Paul E. Burns, Emmett Lynn, Will Wright, Earle Hodgins, John Cason, John Carpenter, Joseph Crehan, Harry Tyler, William Desmond, Ethan Laidlaw, Bob Reeves, Olin Howlin, Nacho Galindo, Byron Foulger, Wade Crosby, Victor Potel, Ernie Adams, Robert Barron, Roy Brent, Victor Cox. Framed for murder he did not commit and chased by a posse, a cowboy gets help from a young woman. Well done pursuit melodrama.\n\n**3345** _ **Relentless**_ **** CBS-TV, 1977. 74 min. Color. D: Lee H. Katzin. SC: Sam Rolfe. With Will Sampson, Monte Markham, John Hillerman, Marianna Hill, Larry Wilcox, Antony Ponzini, John Lawlor, Ted Markland, David Pendleton, Ron Foster, Don Starr, Danny Zapien, Mel Todd, Dick Armstrong, Teddy (dog). An Arizona state policeman tracks a gang of robbers who have pulled a heist, murdered his uncle and taken a woman hostage into the mountains. Action filled and well produced telefeature.\n\n**Antony Ponzini and John Hillerman in** _**Relentless**_ **(CBS-TV, 1979).**\n\n**  \n**\n\n**3346** _ **The Relentless Four**_ **** Astor, 1965. 90 min. Color. D: Primo Zeglio. SC: Sebares De Caso and Primo Zeglio. With Adam West, Robert Hundar (Claudio Undari), Pauline Bards (Paola Barbara), Red Ross, Roberto Camardiel, Ralph Baldwyn (Raf Baldassare), Cris Huerta, John Bartha, Robert Johnson, Jr., Dina Loy, Luis Induni, Jose Jaspe, Francisco Sanz. A quartet of bandits, secretly supported by a corrupt deputy sheriff, commit a series of lawless acts and place the blame on a lawman. Better than average Spaghetti Western with star Adam West for domestic appeal. Released in Italy by P.E.A.\/Astorfilms as _**I Quattro Inesorabili**_ (The Inexorable Four).\n\n_**Remember Me**_ see _**Horsemen of the Sierras**_\n\n_**Remember the Alamo**_ see _**Heroes of the Alamo**_\n\n**3347** _ **Remolino**_ (Whirlpool) Cinematograpfica Intercontinental, 1961. 90 min. D: Gilbert Garzcon. SC: Raul de Anda. With Luis Aguilar, Maria Duval, Miguel Arenas, Agustin de Anda, Sonia Furio, Armando Arriola, Dolores Finoco, Jose Elias Moreno. A ranch owner, his son and one of their cowboys become jealous of each other over the affections of a newly arrived young woman. Entertaining Mexican Western drama.\n\n**3348** _ **The Renegade**_ **** Producers Releasing Corporation, 1943. 58 min. D: Sam Newfield. SC: George Milton (George W. Sayre and Milton Raison). With Buster Crabbe, Al St. John, Lois Ranson, Karl Hackett, Ray Bennett, Frank Hagney, Jack Rockwell, Tom London, George Chesebro, Jimmy Aubrey, Dan White, Carl Sepulveda, Wally West, Jack Montgomery, Milburn Morante, Art Dillard, Silver Harr, Jack Evans, Jack Tornek. Billy the Kid and his sidekick Fuzzy Jones are on the trail of a notorious lawbreaker. Standard \"Billy the Kid\" series fare; reissued by Eagle-Lion in 1947 in a 38 minute re-edited version called _**Code of the Plains**_.\n\n**3349** _ **Renegade**_ **** Cinecitta\/Paloma Films, 1987. 90 min. Color. D: E.B. Clucher (Enzo Barboni). SC: Mark (Marco) Barboni. With Terence Hill, Robert Vaughn, Ross Hill, Norman Bowler, Donald Hodson, Beatrice Palmer, Lisa Ann Rubin, Luisa Maneri, Valeria Sabel, Cole S. McKay, Curt Bortel, Joe Krieg, Jannel Robinson, Matthew Uriarte, Royce Clark, Cyrus Elias, Moe Mosley, Sandy Gibbons, Gigi Bonos, Carolyn Jacobs, Ron Nix. A defrauding drifter makes a deal with a convict to become the guardian of the man's teenage punk son. Fun modern-day Western action comedy with star Terence Hill co-writing the story on which it is based; also titled _**They Call Me Renegade**_.\n\n**3350** _ **Renegade**_ **** Columbia Tri Star, 2004. 124 min. Color. D: Jan Kounen. SC: Cassidy Pope. With Vincent Cassel, Juliette Lewis, Michael Madsen, Temuera Morrison, Ernest Borgnine, Diimon Hounsou, Hugh O'Connor, Geoffrey Lewis, Nicole Hiltz, Kateri Walker, Vahina Giocante, Kestenbetsa, Tcheky Karyo, Eddie Izzard, Colm Meaney, Dominique Bettenfeld, Antonio Monroy, William Lightning, Jan Kounen, Francois Levantal, Joel Gonzales, Panshin Biri, Juan Manual Bernal, Francois Bercovici, Richard Jones, Val Avery, Pascal Demolon, Leticia Gutierrez, Tetsu Nagata, Cyril Dupuy, Paul Rodden, Javier Clave, Jose Gomez Parcero, Karl H. Braun. After being saved by a Chiricahua Apache family, a man changes his ways and becomes a peace officer who tries to protect the tribe when gold is found on their land. Hard to follow French production filmed in Mexico.\n\n**3351** _ **Renegade Girl**_ **** Screen Guild, 1948. 65 min. D: William Berke. SC: Edwin K. Westrate. With Alan Curtis, Ann Savage, Jack Holt, Edward Brophy, Russell Wade, Ray Corrigan, John King, Chief Thundercloud, Edmund Cobb, Claudia Drake, Dick Curtis, Nick Thompson, James Martin, Harry Cording, Ernie Adams, Forrest Taylor, Kermit Maynard. During the Civil War the woman leader of a band of raiders is stalked by a special Union investigator. Fair action program feature with a supporting cast of ex-genre stars.\n\n_**Renegade Gun**_ see _**Shoot the Living...Pray for the Dead**_\n\n**3352** _ **Renegade Gunfighter**_ **** Tirso\/Petruka Film, 1966. 76 min. Color. D: Silvio Amadio. SC: Silvio Amadio, Tito Carpi and Luciano Gregoretti. With Zachary Hatcher, Dick Palmer (Mimmo Palmara), Pier Angeli (Annamaria Pierangeli), Ruben Rojo, Mirko Ellis, Manuel Gil, Jose Calvo, Bruno Scipioni. A peace lover becomes a vengeful killer after his parents are murdered by two evil land grabbing brothers. Typically violent, mediocre dubbed oater from Italy initially called _**Per Mille Dollari al Giorno**_ (For One Thousand Dollars Per Day).\n\n**3353** _ **Renegade Ranger**_ **** RKO Radio, 1939. 60 min. D: David Howard. SC: Oliver Drake. With George O'Brien, Rita Hayworth, Tim Holt, Ray Whitley, William Royle, Neal Hart, Monte Montague, Robert Kortman, Charles Stevens, Jim Mason, Tom London, Guy Usher, Lucio Villegas, Cecilia Callejo, The Phelps Brothers, Hank Bell, Al Haskell, Pete Morrison, Frank Ellis, Carl Mathews, Victor Cox, Jack O'Shea, Ray Jones, Buck Bucko. A Texas Ranger, sent to capture a female bandit leader, ends up saving her life and discovers she and other ranchers have been forced into crime by a crooked tax collector. Good production values and an entertaining story make this George O'Brien vehicle worth viewing, with lovely Rita Hayworth co-starring as the bandit queen. A remake of _**Come On, Danger**_ (1932) [q.v.] it was done a third time under that title in 1942 (q.v.) with Tim Holt and Ray Whitley again in the cast, this time Holt having the lead role.\n\n**3354** _ **The Renegade Trail**_ **** Paramount, 1939. 61 min. D: Lesley Selander. SC: John Rathmell and Harrison Jacobs. With William Boyd, George Hayes, Russell Hayden, Charlotte Wynters, Russell Hopton, Sonny Bupp, Jack Rockwell, Roy Barcroft, John Merton, Robert Kortman, Eddie Dean, The King's Men (Ken Darby, Rad Robinson, Bud Linn, Jon Dodson), John Wallace, Leo J. McMahon, Blackjack Ward, Cliff Lyons. The Bar 20 trio come to the aid of a woman and her son whose cattle are the target of an outlaw gang that includes her ex-convict husband, who the boy thinks died a good man. A bit slower than the average \"Hopalong Cassidy\" outing and somewhat less scenic but still pretty good with a pleasant musical interlude by The King's Men.\n\n**3355** _ **Renegades**_ **** Columbia, 1946. 88 min. Color. D: George Marshall. SC: Melvin Levy and Francis Edwards Faragoh. With Evelyn Keyes, Willard Parker, Larry Parks, Edgar Buchanan, Forrest Tucker, Jim Bannon, Ludwig Donath, Willard Robertson, Paul E. Burns, Frank Sully, Eddy Waller, Virginia Brissac, Francis Ford, Vernon Dent, Addison Richards, Syd Saylor, John Hamilton, William Haade, Eileen Janssen, Hermine Sterler. The youngest son of a family of outlaws tries to lead a peaceful life but finds his clan's reputation too great to conquer. Okay oater, well acted by a good cast.\n\n**3356** _ **Renegades of Sonora**_ **** Republic, 1948. 60 min. D: R.G. Springsteen. SC: M. Coates Webster. With Allan \"Rocky\" Lane, Eddy Waller, William Henry, Douglas Fowley, Roy Barcroft, George J. Lewis, Frank Fenton, Mauritz Hugo, Marshall Reed, Holly Bane, Dale Van Sickel, Art Dillard, House Peters, Jr. On his way to Wyoming to buy a ranch, a cowboy stops in a frontier town and crooks frame him for murder. Nothing special here but still entertaining.\n\n**3357** _ **Renegades of the Rio Grande**_ **** Universal, 1945. 56 min. D: Howard Bretherton. SC: Ande Lamb. With Rod Cameron, Fuzzy Knight, Jennifer Holt, Eddie Dew, Ray Whitley and His Bar-6 Cowboys, Glenn Strange, Ethan Laidlaw, Edmund Cobb, Richard Alexander, John James, Richard Botiller, Iris Clive, Larry McGrath, Roy Butler, Jack Casey, Virgil Drake, Hal Hart. A ranger is assigned to stop an outlaw gang rustling cattle along the Mexican border. Solid Rod Cameron vehicle, showing why he went on to bigger things.\n\n**3358** _ **Renegades of the Sage**_ **** Columbia, 1949. 56 min. D: Ray Nazarro. SC: Earle Snell. With Charles Starrett, Smiley Burnette, Leslie Banning, Trevor Bardette, Douglas Fowley, Jock (Mahoney) O'Mahoney, Fred F. Sears, George Chesebro, Jerry Hunter, Frank McCarroll, Selmer Jackson. A Secret Service agent is after a gang destroying territorial telegraph lines. Tepid entry in the \"Durango Kid\" series. British title: _**The Fort**_.\n\n**3359** _ **Renegades of the West**_ **** RKO Radio, 1932. 55 min. D: Casey Robinson. SC: Albert LeVine. With Tom Keene, Betty Furness, Rosco(e) Ates, Rockcliffe Fellows, Jim Mason, Jack Pennick, Max Wagner, Joseph Girard, Billy Franey, Roland Southern, Jules Cowles, Frank O'Connor, Josefina Ramos, Chuck Baldra, Tex Palmer, Mike Morita, Fred Parker. When cattle rustlers murder his rancher father a cowboy plans to get even with them. Well directed Tom Keene vehicle; above average.\n\n**3360** _ **El Renegado Blanco**_ (The White Renegade). Alameda Filma, 1960. 90 min. D: Fernando Mendez. SC: Alfredo Salazar. With Mauricio Garces, Abel Salazar, Rafael Baledon, Martha Roth, Luis Aragon, Renee Dumas, Begona Palacios, Carlos Nieto, Guillermo Rivas, Eduardo Arcaraz, David Reynoso, Virma Gonzalez, Angel D'Stefani, Jose Dupeyron, Tito Novaro, Salvador Terroba, Antonio Badu, Fernando Galiana. Three brothers think Sitting Bull and his braves are behind a raid really the work of an outlaw called The White Renegade. Well done Mexican oater.\n\n**3361** _ **Renfrew of the Royal Mounted**_ **** Grand National, 1937. 57 min. D: Al Herman. SC: Charles Logue. With James Newill, Carol Hughes, William Royle, Herbert Corthell, Kenneth Harlan, Dickie Jones, Chief Thundercloud, William Austin, Donald Reed, Bob Terry, William Gould, David Barclay (Dave O'Brien), Dwight Frye, Forrest Taylor, Earl Douglas, Marin Sais, Arthur Millett, Otto Hoffman, Buck Morrison, Lighting (dog). A Mounted Policeman is assigned to look into the smuggling of bogus currency across the U.S.-Canadian border and learns a former counterfeiter, who has gone straight, is being held prisoner and forced to engrave plates. Good opener for the popular \"Renfrew of the Royal Mounted\" series, a pleasing film highlighted by excellent baritone James Newill in the title role\n\n_**Renfrew of the Royal Mounted in Fighting Mad**_ see _**Fighting Mad**_\n\n_**Renfrew of the Royal Mounted in Murder on the Yukon**_ see _**Murder on the Yukon**_\n\n_**Renfrew of the Royal Mounted in Yukon Flight**_ see _**Yukon Flight**_\n\n_**Renfrew of the Royal Mounted on the Great White Trail**_ see _**On the Great White Trail**_\n\n_**Renfrew on the Great White Trail**_ see _**On the Great White Trail**_\n\n**3362** _ **Reno**_ **** RKO Radio, 1939. 73 min. D: John Farrow. SC: John Twist. With Richard Dix, Gail Patrick, Anita Louise, Paul Cavanagh, Laura Hope Crews, Louis Jean Heydt, Hobart Cavanaugh, Charles Halton, Astrid Allwyn, Joyce Compton, Frank Faylen, William Haade, Anthony Averill, Carole Landis, Billie Seward, Max Wagner, Steve Pendleton, John Dilson, Paul E. Burns, Lloyd Ingraham, Bob McKenzie, Hank Worden, Larry Steers, George Watts, Blackie Whiteford, Fern Emmett, Wedgwood Newell, Selmer Jackson, Jim Farley, Donald Kerr, Bob Perry. A smart lawyer turns Reno, Nevada, from a rough mining town into the country's divorce capital but lives to regret it. Entertaining medium budget feature.\n\n**3363** _ **Reprisal!**_ **** Columbia, 1956. 74 min. Color. D: George Sherman. SC: David P. Harmon, Raphael Hayes and David Dortort. With Guy Madison, Felicia Farr, Kathryn Grant, Michael Pate, Edward Platt, Otto Mulett, Wayne Mallory, Robert Burton, Ralph Moody, Frank De Kova, Paul McGuire, Don Rhodes, Philip Breedlove, Malcolm Atterbury, Eve McVeagh, Addison Richards, Jack Lomas, John Zaremba, Kermit Maynard, Eddie Parker. Falsely accused of murdering a cattle baron, a half-breed is saved from a lynch mob by the two women who love him. Compact action film provides good viewing.\n\n**Guy Madison in** _**Reprisal!**_ **(Columbia, 1956).**\n\n**  \n**\n\n**3364** _ **Requiem for a Bounty Killer**_ **** Cineproduzionne Daunia\/Universalia Vision, 1972. 84 min. Color. D: Mark Welles (Angelo Pannaccio). SC: Craig Marina and Angelo Pannaccio. With Ray O'Conner (Remo Capitani), Michael Forest, Michele Branca, Lawrence Bien, Thomas Rudy, Steven Tedd, Anna Bacchi, Giovanni Petti, Italo Guitto, Iro Fantini, Benito Pacifici, Ivi D'Annunzio, Alceste Bogart, Chet Davis (Michelle Borelli), Antonio Molino Rojo, Susanna Levi, Luciano Conti. After his family is brutalized and murdered by outlaws, a rancher teams with a mysterious gunman to track them down. Above average, but violent, Italian production issued there as _**Requiem per un Bounty Hunter**_ (Requiem for a Bounty Hunter); some sources claim Mark Welles is actor-director Mel Welles.\n\n**3365** _ **Requiem for a Gunfighter**_ **** Embassy, 1965. 91 min. Color. D: Spencer Gordon Bennet. SC: R. Alexander (Ruth Gordon). With Rod Cameron, Stephen McNally, Mike Mazurki, Tim McCoy, Olive Sturgess, Bob Steele, Johnny Mack Brown, Lane Chandler, Raymond Hatton, Dick Jones, Rand Brooks, Dale Van Sickel, Frank Lackteen, Zon Murray, Edmund Cobb, Richard Alexander, Boyd \"Red\" Morgan, Fred Carson, Chet Douglas, Chris Hughes. When a judge is murdered to prevent a trial, a gunman mistaken for him makes plans to see that justice is carried out. A cast of genre veterans add zest to this pleasant Alex Gordon production.\n\n**3366** _ **Requiescant**_ (Let Them Rest) **** Accord-Film\/CPF\/Iris-Film, 1967. 110 min. Color. D: Carlo Lizzani. SC: Franco Bucceri, Adriano Bolzoni, Armando Crispino and Lucio Battistrada. With Lou Castel, Mark Damon, Pier Paolo Pasolini, Barbara Frey, Rosanna Krisman, Mirella Maravidi, Franco Citti, Luis Baratto, Nino Davoli, Nino Musco, Charles Palmset (Carlo Palmucci), Anne Carter, Lorenza Guerrieri, Victor Duse, Dean Light (Feruccio Viotti), Massimo Sarchielli, Pier Annibale Danovi, Ivan Scratuglia, Renato Terra, Aldo Marianecci, Peter Jacob, Sparataco Conversi, Henry Danby (Hermann Nehlsen), Fulvio Mingozzi, Max Guthner. After being raised by Quakers, a Mexican vows revenge against the aristocrat who stole his land as well as those responsible for turning his benefactor's daughter into a prostitute. Bizarre, sadistic Italian-West German co-production, also called _**Kill and Pray**_ and _**Let Them Rest**_. Some prints run 92 minutes.\n\n**3367** _ **The Restless Breed**_ **** 20th Century\u2013Fox, 1957. 81 min. Color. D: Allan Dwan. SC: Steve Fisher. With Scott Brady, Anne Bancroft, Jim Davis, Jay C. Flippen, Leo Gordon, Rhys Williams, Myron Healey, Scott Marlowe, Eddy Waller, Harry V. Cheshire, Gerald Milton, Dennis King, Jr., James Flavin, Billy Miller, Evelyn Rudie, Clegg Hoyt, Joe Devlin, Fred Graham, Tom Steele. Following the killing of his Secret Service agent father, a man sets out for revenge. Average melodrama, well made but nothing exciting.\n\n**3368** _ **Retribution Road**_ **** Grindstone Entertainment Group, 2007. 75 min. Color. D-SC: Chuck Walker. With Michael Gregory, John Castellanos, Leslie Easterbrook, Burton Gilliam, Peter Sherayko, Corbin Timbrook, Hallie Pierce, Alan Arnold, Al Hayter, Molly Elswick, Terry Mann, Eduardo Enriquez, Jr., Mark Enriquez, Carlos Rodriguez, Tony Rowe, John Rouse, Heidi Smith, Duke Meek, Jimmy Phillis, Doc Sheedy, Wilton Stewart, Yankie Grant, James B. Lewis, Toledo Boulware Hues, Leeza Zimmerman. In South Texas a sheriff waits to confront the family of a notorious outlaw he has locked in jail. Texas filmed, low budget _**High Noon**_ (q.v.) clone makes for pretty fair viewing; also called _**Blue Eyes**_.\n\n**3369** _ **The Return of a Man Called Horse**_ **** United Artists, 1976. 129 min. Color. D: Irvin Kerschner. SC: Jack DeWitt. With Richard Harris, Ana DeSade, Gale Sondergaard, Geoffrey Lewis, Bill Lucking, Jorge Luke, Claudio Brook, Enrique Lucerno, Jorge Russek, Pedro Damien. A title Englishman, who became a Sioux after many trials, returns to the tribe to help them in their struggle against whites. A bit long but still a good sequel to _**A Man Called Horse**_ (q.v.), followed by the vapid _**Triumphs of a Man Called Horse**_ (q.v.).\n\n**3370** _ **Return of Daniel Boone**_ **** Columbia, 1941. 56 min. D: Lambert Hillyer. SC: Paul Franklin and Joseph Hoffman. With Bill Elliott, Dub Taylor, Betty Miles, Ray Bennett, Walter Soderling, Carl Stockdale, Bud Osborne, Francis Walker, Lee Powell, Tom Carter, Edmund Cobb, Roy Butler, Art Miles, Edwin Bryant, Steve Clark, Murdock MacQuarrie, Hank Bell, Rodik Twins. Daniel Boone's grandson goes to work as a tax collector but soon deduces his boss is a crook cheating settlers of their properties. Okay Bill Elliott vehicle.\n\n**3371** _ **The Return of Desperado**_ **** NBC-TV, 1988. 96 min. Color. D: E.W. Swackhamer. SC: John Mankiewicz, Daniel Pyne and Charles Grant Craig. With Alex McArthur, Robert Foxworth, Billy Dee Williams, Marcy Walker, Vanessa Bell, Charles Boswell, Hal Havins, J. Jay Saunders, Victor Jove, John Barks, Vivian Bonnell, Gregg Brinkley, Rahda Delamarter, Rusty Dillen, Jerry Gardner, Darlah Rusch Gathings, Sam Gauny, Dan Kamin, Shelby Livingston, Ivy Price, Adam Taylor, Tommy Townsend, Marvin Walters. When black homesteaders are threatened by a corrupt rancher an outlaw joins forces with a woman reporter to help them. Mundane TV Western feature.\n\n**3372** _ **Return of Django**_ **** Denwer Films, 1967. 95 min. Color. D: Osvaldo Civirani. SC: Tito Carpi, Alessandro Ferrau and Osvaldo Civirani. With Guy Madison, Gabriele Tinti, Ingrid Schoeller, Daniele Vargas, Pedro Sanchez, Andrew Scott (Andrea Scotti), Bob Messenger (Roberto Messina), Christl Penz, Ivan Scratt, Luis Chavarro, Franco Gula, Luciano Rossi, Piero Morgia, Giorgio Dionisio, Renato Mambor, John Bartha, Bob Johnson, Giuseppe Castellano, The Wilder Brothers. Years after his mother and father are slaughtered by outlaws, a man teams with an evangelist, a former gunslinger, to take revenge on the town boss behind the killings. Pretty fair Italian oater also called _**Django Strikes Back**_ , _**Son of Django**_ and _**Vengeance Is a Colt 45**_.\n\n**3373** _ **The Return of Draw Egan**_ **** Triangle, 1916. 55 min. D: William S. Hart. SC: C. Gardner Sullivan. With William S. Hart, Louise Glaum, Margery Wilson, Robert McKim, J.P. Lockney. A bad man becomes the reform sheriff of a frontier town and brings about law and order only to be blackmailed by his old gang. Very entertaining William S. Hart silent effort often reworked plot wise in the sound era; it benefits from the presence of beautiful Margery Wilson.\n\n**3374** _ **The Return of Frank James**_ **** 20th Century\u2013Fox, 1940. 92 min. Color. D: Fritz Lang. SC: Sam Hellman. With Henry Fonda, Gene Tierney, Jackie Cooper, Henry Hull, J. Edward Bromberg, Donald Meek, John Carradine, Eddie Collins, George Barbier, Ernest Whitman, Charles Tannen, Lloyd Corrigan, Russell Hicks, Victor Kilian, Edward McWade, George Chandler, Irving Bacon, Frank Shannon, Barbara Pepper, Louis Mason, Matthew \"Stymie\" Beard, William Pawley, Frank Sully, Davison Clark, Nelson McDowell, Lee Phelps, Lillian Yarbo, Adrian Morris, Lester Dorr, Milton Kibbee, Frank Melton, Almeda Fowler, Lew Meehan, Bob McKenzie, Budd Fine, Kernan Cripps, Russ Powell, Dale Van Sickel, James C. Morton, Sherry Hall, Edmund Elton, Hattie Noel, Tex Phelps, Kermit Maynard, A.S. Byron, Bob Battier. When a judge sets Bob Ford free on the charge of murdering Jesse James, the victim's brother Frank vows revenge. Not much history but lots of entertainment in this follow-up to _**Jesse James**_ (q.v.), which contains footage of Tyrone Power from it; John Carradine makes the film as the cowardly villain Bob Ford.\n\n**3375** _ **The Return of Grey Wolf**_ **** Ambassador, 1925. 60 min. D: Jacques Rollens. SC: Jay Arr. With Leader (dog), James Pierce, Helen Lynch, Walter Shumway, Edward (Ed) Coxen, Harry Belmore, Whitehorse. A dog helps a man and woman fight crooks in the Canadian Rockies. The scenery is the best part of this silent effort, although Jim Pierce fans will enjoy seeing him in a starring role.\n\n**3376** _ **The Return of Jack Slade**_ **** Allied Artists, 1955. 79 min. D: Harold Schuster. SC: Warren Douglas. With John Ericson, Mari Blanchard, Neville Brand, Casey Adams (Max Showalter), Jon Shepodd, Howard Petrie, John Dennis, Angie Dickinson, Donna Drew, Mike Ross, Alan Wells, Raymond Bailey, Lyla Graham. The son of the notorious Jack Slade becomes a lawman to redeem the family name and plans to capture a gang of outlaws. Fair low budget affair with good work by John Ericson as Slade's son.\n\n**3377** _ **The Return of Jesse James**_ **** Lippert, 1950. 77 min. D: Arthur Hilton. SC: Carl K. Hittleman. With John Ireland, Ann Dvorak, Henry Hull, Reed Hadley, Hugh O'Brian, Carleton Young, Barbara Woodell, Margia Dean, Sid Melton, Victor Kilian, Byron Foulger, Sam Flint, Robin York, Paul Maxey, I. Stanford Jolley, Earle Hodgins, Hank Patterson. A Jesse James look-a-like takes on the guise of the dead outlaw and helps a gang rob banks but is overcome by his own greed and the ambition of the saloon singer he loves. An odd kind of oater, more psychological than action filled, but pretty fair anyway.\n\n**3378** _ **The Return of Josey Wales**_ **** Magnum Entertainment, 1986. 90 min. Color. D: Michael Parks. SC: Forrest Carter. With Michael Parks, Rafael Campos, Charlie McCoy, Suzie Humphreys, Bob Magruder, Paco Vela, Everett Sifuentes, John Galt, Joe Kurtzo, Benita Faulkner, Charles Escamilla, Arturo R. Tamez, Manuel Valdes, Paul Flores, Mary Ellen Averett, Ron Bledsoe, Mike Bledsoe, Doug Bledsoe, Happy Shahan, Larry Melton, John Burkhead, Buddy Harper, Russ Taylor, Donny Fountain, Ron Taylor. An outlaw heads to Mexico to get a friend out of jail and also rescues an Indian woman and two other prisoners but they are pursued by a corrupt police chief. Puny follow-up to _**The Outlaw Josey Wales**_ (q.v.), directed by star Michael Parks and adapted by Forrest Carter from his novel _Vengeance Trail of Josey Wales_.\n\n**3379** _ **The Return of Rin Tin Tin**_ **** Eagle-Lion, 1947. 68 min. Color. D: Max Nosseck. SC: Jack DeWitt. With Rin Tin Tin III, Donald Woods, Bobby Blake, Claudia Drake, Gaylord (Steve) Pendleton, Earle Hodgins. A priest brings a boy from Europe to his Western mission hoping to restore his faith and the little fellow becomes attached to a dog whose return is demanded by its master. Pleasant family fare made by Producers Releasing Corporation. ****\n\n**3380** _ **The Return of Ringo**_ **** Rizzoli Film, 1965. 104 min. Color. D: Duccio Tessari. SC: Duccio Tessari and Fernando Di Leo. With Montgomery Wood (Giuliano Gemma), Fernando Sancho, Hally Hammond, George Martin, Nieves Navarro, Antonio Casas, Pajarito, Jose Manuel Martin. Returning home from the Civil War, a Northern army captain takes revenge on the Mexican who took his family hostage and stole his land. Very violent sequel to _**A Pistol for Ringo**_ (q.v.); issued in Europe as _**Il Ritorno di Ringo**_ (The Return of Ringo) and on video as _**Ballad of Death Valley**_ , running 91 minutes.\n\n**3381** _ **Return of Sabata**_ **** United Artists, 1972. 106 min. Color. D: Frank Kramer (Gianfranco Parolini). SC: Renato Izzo and Gianfranco Parolini. With Lee Van Cleef, Reiner Schone, Annabella Incontrera, Gianni Rizzo, Giampiero Albertini, Jacqueline Alexandre, Pedro Sanchez, Mario Brega, Nick Jordan, Gunther Stoll, Vasili Karis. An ex\u2013Confederate major, now a sharpshooter with a circus, arrives in a town to get even with those who bilked him out of money and settles the score by exposing a corrupt banker. Lee Van Cleef's fans will like this okay follow-up to _**Sabata**_ (q.v.).\n\n**Charles Starrett and Raider in** _**The Return of the Durango Kid**_ **(Columbia, 1945).**\n\n**  \n**\n\n**3382** _ **Return of Shanghai Joe**_ **** Divina-Film, 1975. Min. Color. D: Bitto Albertini. SC: Bitto Albertini and Carlo Alberto Alfieri. With Klaus Kinski, Cheen Lie, Tommy Polgar, Karin Field, Fausto Ulisse, Primiano Muratori, Paul Sholer, Fortunato Arena, Claudio Giorgi, Tom Felleghy, Attilio Dottesio, Claudio Ruffini, Roberto Dell'Acqua, Luigi Ruso, Dante Cleri. A crook teams with a kung fu artist to oppose a dishonest politician in a frontier community. Toned down Spaghetti Western, a pale follow-up to _**Fighting Fists of Shanghai Joe**_ (q.v.).\n\n**3383** _ **Return of the Bad Men**_ **** RKO Radio, 1948. 90 min. D: Ray Enright. SC: Charles O'Neal, Jack Natteford and Luci Ward. With Randolph Scott, Robert Ryan, Anne Jeffreys, George \"Gabby\" Hayes, Jacqueline White, Steve Brodie, Richard Powers (Tom Keene), Robert Armstrong, Robert Bray, Lex Barker, Walter Reed, Michael Harvey, Dean White, Tom Tyler, Lew Harvey, Gary Gray, Walter Baldwin, Minna Gombell, Warren Jackson, Robert Clarke, Jason Robards, Harry Shannon, Charles McAvoy, Larry McGrath, Billy Vincent, Ernie Adams, Sam Flint, Lane Chandler, Earle Hodgins, Charles Stevens, Kenneth MacDonald, John Hamilton, Frank O'Connor, Cy King, Dan Foster, Ida Moore. In the 1880s an Oklahoma town marshal falls in love with a female desperado as he fights outlaws like Billy the Kid, the Dalton and Younger brothers, the Sundance Kid and Wild Bill Doolin. Nicely produced \"A\" feature; quasi-sequel to _**Badman's Territory**_ (q.v.) and followed by _**Best of the Bad Men**_ (q.v.).\n\n**3384** _ **Return of the Cisco Kid**_ **** 20th Century\u2013Fox, 1939. 70 min. D: Herbert I. Leeds. SC: Milton Sperling. With Warner Baxter, Lynn Bari, Cesar Romero, Henry Hull, Kane Richmond, C. Henry Gordon, Robert Barrat, Chris-Pin Martin, Adrian Morris, Soledad Jiminez, Harry Strang, Arthur Aylesworth, Paul E. Burns, Victor Kilian, Eddy Waller, Ruth Gillette, Ward Bond, Gino Corrado, Ralph Dunn, Herbert Heywood, Ethan Laidlaw, Charles Tannen, Lee Shumway. The Cisco Kid and two pals help a man and his daughter who are about to cheated out of their newly purchased ranch by a dishonest lawman. Warner Baxter's third and final feature film portrayal of the Cisco Kid is an enjoyable affair.\n\n**3385** _ **The Return of the Durango Kid**_ **** Columbia, 1945. 58 min. D: Derwin Abrahams. SC: J. Benton Cheney. With Charles Starrett, Tex Harding, Jean Stevens, John Calvert, Betty Roadman, Britt Wood, The Jesters, Hal Price, Richard Botiller, Ray Bennett, Elmo Lincoln, Paul (Conrad) Zaremba, Steve Clark, Carl Sepulveda, Ted Mapes, Herman Hack, James T. \"Bud\" Nelson, Dan White, Francis Walker, Tex Palmer, William Desmond, Carl Mathews, Wally West, Lew Morphy, Jack Evans, Philip Kieffer, Robert Walker. A passenger on a stagecoach robbed by outlaws takes on the guise of a masked avenger and steals the money back from the bad men. Mediocre production that launched Charles Starrett as \"The Durango Kid\" in a popular series of more than three score features; despite its title it is not a sequel to _**The Durango Kid**_ (q.v.).\n\n**3386** _ **Return of the Frontiersman**_ **** Warner Bros., 1950. 74 min. Color. D: Richard L. Bare. SC: Edna Anhalt. With Gordon MacRae, Julie London, Rory Calhoun, Jack Holt, Fred Clark, Edwin Rand, Raymond Bond, Matt McHugh, Britt Wood, Richard Egan, John Doucette, Guy Wilkerson, Dan White, Steve Pendleton, Jack Mower, Harlan Briggs, Tex Cooper, Doris Kemper, Bobby Henshaw, Ruth Warren, William Sundholm. A sheriff is forced to jail his son when he is falsely accused of murder, but the offspring escapes, aided by the real killer. Okay production with fine work by Jack Holt as the lawman.\n\n**3387** _ **Return of the Gunfighter**_ **** ABC-TV\/Metro-Goldwyn-Mayer, 1967. 98 min. Color. D: James Neilson. SC: Robert Buckner. With Robert Taylor, Chad Everett, Ana Martin, Lyle Bettger, Mort Mills, John Davis Chandler, Michael Pate, Barry Atwater, John Crawford, Willis Bouchey, Rodolfo Hoyos, Read Morgan, Henry Wills, Robert Shelton. When a young Mexican woman's parents are murdered for their land she enlists the help of an aging gunman and a wounded cowboy to get revenge. One of the first, and best, movies made for network television; Robert Taylor is outstanding as the gunslinger. Alternate title: _**As I Rode to Laredo**_.\n\n**Robert Taylor in** _**The Return of the Gunfighter**_ **(ABC-TV\/Metro-Goldwyn-Mayer, 1967).**\n\n**  \n**\n\n**3388** _ **Return of the Lash**_ **** Producers Releasing Corporation, 1947. 53 min. D: Ray Taylor. SC: Joseph O'Donnell. With Lash LaRue, Al St. John, Mary Maynard, Brad Slaven, George Chesebro, Lane Bradford, Bud Osborne, George DeNormand, Lee Morgan, Carl Mathews, Slim Whitaker, Kermit Maynard, Frank Ellis, Bob Woodward, Budd Buster, Tex Palmer, Roy Butler. Two marshals arrive in a town to stop a range war caused by a crook after the land because a railroad is coming through. Mediocre \"Cheyenne Davis\" entry, but not as dull as some in the series.\n\n_**Return of the Mohicans**_ see _**The Last of the Mohicans**_ (1932)\n\n**3389** _ **Return of the Outlaws**_ **** Grindstone Entertainment Group, 2009. 96 min. Color. D-SC: Chuck Walker. With Lorenzo Lamas, John Castellanos, J. Eddie Peck, Michael Gregory, Corbin Timbrook, Samantha Lockwood, Kim Waltrip, Molly Elswick, Peter Sherayko, Eduardo Enriquez, Jr., Shane Ryan Savage, \"Cowboy\" Bill Sallas, Yankie Grant, Leslie Harlton, Cliff Miller, Al Hayter, Terry Mann. After nearly being hanged, a violent outlaw forms a gang, including a saloon girl, to stage a robbery but is opposed by a lawman. Violent oater filmed in Texas and also called _**Mexican Gold**_.\n\n**3390** _ **The Return of the Rangers**_ **** Producers Releasing Corporation, 1943. 61 min. D-SC: Elmer Clifton. With Dave O'Brien, Jim Newill, Guy Wilkerson, Nell O'Day, Glenn Strange, Emmett Lynn, Robert Barron, Henry Hall, Harry Harvey, I. Stanford Jolley, Richard Alexander, Charles King, Art Fowler, Wally West, Hank Bell, Tex Cooper, Curley Dresden, Herman Hack, Horace B. Carpenter, Rose Plummer, Art Wenzel. In order to capture a rustling gang, three lawmen arrive in town incognito with one pretending to be a criminal and another the judge planning to try his case. A good script puts some life in this \"Texas Rangers\" series outing.\n\n**3391** _ **Return of the Seven**_ **** United Artists, 1966. 96 min. Color. D: Burt Kennedy. SC: Larry Cohen. With Yul Brynner, Robert Fuller, Warren Oates, Julian Mateos, Jordan Christopher, Claude Akins, Fernando Rey, Emilio Fernandez, Elisa Montes, Virgilio Texiera, Rudy (Rodolfo) Acosta, Francisco Anton. When outlaws kidnap a former gang member, the man's wife asks two of his former comrades to help him and they enlist four other men to assisr them. Adequate sequel to _**The Magnificent Seven**_ (q.v.).\n\n**3392** _ **Return of the Texan**_ **** 20th Century\u2013Fox, 1952. 87 min. D: Delmer Daves. SC: Dudley Nichols. With Dale Robertson, Joanne Dru, Walter Brennan, Richard Boone, Tom Tully, Robert Horton, Helen Westcott, Lonnie Thomas, Robert Adler, Kathryn Sheldon, Dennis Ross, Willis Bouchey, Jim Hayward, Brad Morrow, Linda Green, Aileen Carlyle, Sherman Sanders. A widower returns to his family home hoping to resume ranching but finds he is opposed by local crooks. Fair oater for Dale Robertson fans.\n\n**3393** _ **The Return of Wild Bill**_ **** Columbia, 1940. 59 min. D: Joseph H. Lewis. SC: Robert Lee Johnson and Fred Myton. With Bill Elliott, Iris Meredith, Dub Taylor, Luana Walters, George Lloyd, Ed LeSaint, Frank LaRue, Francis Walker, Chuck Morrison, Buel Bryant, William Kellogg, John Merton, Jack Rockwell, Jim Corey, John Ince, Donald Haines, Bill Nestell, Tex Cooper. Going home, a cowboy finds his father has been murdered by two outlaw brothers harassing local ranchers and he vows revenge. Sturdy \"Wild Bill Saunders\" affair, the fourth and last in the series.\n\n**3394** _ **The Return of Wildfire**_ **** Screen Guild, 1948. 83 min. Color. D: Ray Taylor. SC: Betty Burbridge and Carl K. Hittleman. With Richard Arlen, Patricia Morison, Mary Beth Hughes, Reed Hadley, James Millican, Stanley Andrews, Edmund Cobb, Chris-Pin Martin, Holly Bane, Highland Dale (horse). A drifter gets a job at a ranch trying to capture a wild stallion but becomes involved with a gambler and falls for one of the owner's daughters. Well done low budget effort with plenty of excitement and a nice performance by Richard Arlen as the cowboy; Patricia Morison sings \"Just an Old Sombrero.\"\n\n**3395** _ **Return to Lonesome Dove**_ **** CBS-TV, 1993. 322 min. Color. D: Mike Robe. SC: John Wilder. SC: Jon Voight, Barbara Hershey, Rick Schroder, Louis Gossett, Jr., William L. Petersen, Oliver Reed, Dennis Haysbert, Reese Witherspoon, Tim Scott, Chris Cooper, CCH Pounder, Nia Peeples, Barry Tubb, William Sanderson, David Carpenter, Leon Singer, Jack Caffrey, Reginald T. Dorsey, John Quade, Chet Carlin, Dylan Baker, Jamar Curtis, John Speredakos, Jessica Drake, Adrian Sparks, Colin Fox, Richard Slaughter, Douglas Sebern, Peter Gerety, Eddie Gomez, Sofia Gomez, Nick Searcy, Siobodan Guerra, Veronica Lauren, Bendigo Quade, Stephen C. Prince, Jane Lind, Anner Marble, Stephen C. Prince, R.J. Preston, Steven Mayville, Gina Minervini, Thomas J. Perlman, Bill Neff, Maria Owens, Kip Niven, Molly Orr, Jeff O'Haco, Douglas Roberts. A tough former Texas Ranger drives a mustang herd north before meeting a former love and an illegitimate son. Well made but conventional follow-up to _**Lonesome Dove**_ (q.v.).\n\n**3396** _ **Return to Snowy River**_ **** Hoyle Entertainment, 1988. 105 min. Color. D: Geoff Burrows. SC: Geoff Burrows and John Dixon. With Tom Burlinson, Sigrid Thornton, Brian Dennehy, Nicholas Eadie, Byran Marshall, Rhys McConnochie, Mark Hembrow, Peter Cummins, Cornelia Francis, Tony Barry. After becoming engaged to the daughter of a wealthy enemy, an Australian cowboy must prove he is worthy of her love. Slim sequel to _**The Man from Snowy River**_ (q.v.), enhanced by scenic locales. Also called _**Return to Snowy River Part II:**_ ****_**The Legend Continues**_.\n\n_**Return to Snowy River Part II: The Legend Continues**_ see _**Return to Snowy River**_\n\n**3397** _ **Return to Warbow**_ **** Columbia, 1958. 67 min. D: Ray Nazarro. SC: Les Savage, Jr. With Phil(ip) Carey, Catherine McLeod, Andrew Duggan, William Leslie, Robert Wilke, James Griffith, Jay Silverheels, Chris Olsen, Francis De Sales, Harry Lauter, Paul Picerni, Joe Forte. Three outlaws return to the spot where they buried stolen loot only to learn the brother of one of them has taken the money. Fair low budget oater from producer Wallace MacDonald, with Les Savage, Jr., adapting his novel to the screen.\n\n**3398** _ **The Returning**_ **** Willow Films, 1983. 80 min. Color. D: Joel Bender. SC: Patrick Nash. With Gabriel Walsh, Susan Strasberg, Brian Foleman, Victor Arnold, Ruth Warrick, H.E.D. Redford, Mostea Oshley, Rick Barker. While on a trip to the Mojave Desert, a Utah family acquires a rock with strange powers controlled by the spirits of dead Indian warriors. Fair combination of modern-day Western and horror genres; on the cerebral side.\n\n_**Revenge in El Paso**_ see _**Ace High**_\n\n_**Revenge of a Gunfighter**_ see _**The Mercenary**_\n\n**3399** _ **Revenge of the Virgins**_ **** RVA, 1959. 53 min. D: Peter Perry, Jr. SC: Pete La Roche. With Charles Veltman, Jodean Russo, Stan Pritchard, Hank Delgado, Lou Massad, Jewell Morgan, Ralph Cookson, Betty Shay, Del Monroe, Jan Lee, Hugo Stanger, Nona Carver, Joanne Bowers, Ramona Rogers, Pat O'Connell, Kenne Duncan (narrator). A man, his wife, an old prospector and two gunmen search for gold in a mountain area where a treasure is guarded by topless Indian maidens. Bottom of the barrel sexploitation feature.\n\n_**Revenge of the Wild Bunch**_ see _**40 Graves for 40 Guns**_\n\n**3400** _ **Revenge of Trinity**_ **** Excisa S.A.\/Suevia Films, 1970. 105 min. Color. D: Mario Camus. SC: Manuel Marinero, Miguel Rubio, Jose Vincent Puente, Mario Camus, Mario Cecchi Gori, Alberto Silvestri and Franco Verucci. With Terence Hill, Maria Grazia Buccella, Fernando Rey, Mario Pardo, Carlo Alberto Contina, Maximo Valverde, Angel Lombarte, William Layton, Jose Manuel Martin, Manuel de Blas, Manuel Alexandre, Carlo Otero, Andres Resino, Fernando Sanchez Polack. A morose, self centered gunman takes up the cause of peons badly exploited by a cruel land baron. Moody, atmospheric Spanish-Italian co-production, cut for U.S. release to 93 minutes; filmed as _**La Collera del Vento**_ (The Anger of the Wind) and also called _**Trinity Sees Red**_ and _**The Wind's Fierce**_.\n\n**3401** _ **The Revenge Rider**_ **** Columbia, 1935. 60 min. D: David Selman. SC: Ford Beebe. With Tim McCoy, Robert Allen, Billie Seward, Edward Earle, Frank Sheridan, Jack Clifford, Jack Mower, George Pierce, Allan Sears, Harry Semels, Joseph (Sawyer) Sauers, Lafe McKee, Tom London, Charles King, Ed Peil, Sr., Stanley Blystone, Edward Hearn, Steve Clark, Bob Reeves, Bud McClure, Jack Evans. Returning home, a man finds his sheriff brother has been murdered and the culprits appear to belong to the local cattleman's association. Good Tim McCoy vehicle with a dandy finale having the star piece clues together \u00e0 la Charlie Chan, plus a corker of a shootout; remade as _**Riders of Black River**_ (q.v.).\n\n**3402** _ **The Revengers**_ **** National General, 1971. 107 min. Color. D: Daniel Mann. SC: Wendell Mayes. With William Holden, Susan Hayward, Ernest Borgnine, Woody Strode, Roger Hanin, Rene Koldehoff, Jorge Luke, Jorge Martinez De Hoyos, Arthur Hunnicutt, Warren Vanders, Larry Pennell, John Kelly, Scott Holden, James Daughton, Lorraine Chanel, Paul Prieto. Following the massacre of his wife and children by renegade whites and Indians, a rancher sets out for revenge. Pretty fair action feature unjustly overlooked when first released.\n\n**3403** _ **Reverend Colt**_ **** R.M. Films, 1971. 90 min. Color. D: Leon Klimovsky. SC: Tito Carpi and Manuel Martinez Remis. With Guy Madison, Richard Harrison, Thomas Moore, Maria Martin, German Cobos, Pedro Sanchez, Perla Cristina, Alfonso Rojas, Marta Moterei, Steven Tedd, Vidal Molina, Cris Huerta. A minister, an ex\u2013bounty hunter, comes to a frontier town to build a church but is blamed when outlaws rob the bank and he has to clear his name by tracking down the desperadoes. Guy Madison is good in the title role of this better than average Italian oater issued there as _**Reverendo Colt**_ (Reverend Colt).\n\n**3404** _ **Revolt at Fort Laramie**_ **** United Artists, 1957. 72 min. D: Lesley Selander. SC: Robert C. Dennis. With John Dehner, Frances Helm, Gregg Palmer, Don Gordon, Robert Keys, William Phillips, Robert Knapp, Cain Mason, Eddie Little, Dean Stanton, Bill Barker, Kenne Duncan, Clay Randolph. When the Civil War begins, Southern soldiers are at remote fort want to join the Confederacy despite the threat of an Indian attack. An interesting story and good direction highlight this program oater.\n\n**3405** _ **Revolt in Canada**_ **** Embassy, 1964. 107 min. Color. D-SC: Armando De Osssorio. With George Martin, Pamela Tudor, Luis Marin, Diana Lorys, Raf Baldassare, Mirko Ellis, Franco Fantasia, Santiago Rivero, Francisco Nieto, Giovanni Petti. In frontier Canada a trapper protected by the British commits a series of crimes and places the blame on rebels. Typical European made dubbed frontier action drama made as _**Canada Salvaje**_ (Savage Canada) and _**Rebeldes en Canada**_ (Rebellion in Canada).\n\n**3406** _ **The Reward**_ **** 20th Century\u2013Fox, 1965. 92 min. Color. D: Serge Bourguignon. SC: Serge Bourguignon and Oscar Millard. With Max von Sydow, Yvette Mimieux, Efrem Zimbalist, Jr., Gilbert Roland, Nino Castelnuovo, Emilio Fernandez, Henry Silva, Rodolfo Acosta, Julian Rivero, Rafael Lopez. Bounty hunters capture a wanted criminal but during a trek through the desert they have a falling out over the division of the reward money. Fairly interesting drama enhanced by a good cast.\n\n_**The Reward's Yours, The Man's Mine**_ see _**El Puro**_\n\n_**Rex, King of the Wild Horses**_ (1924) see _**King of the Wild Horses**_ (1924)\n\n_**Rex, King of the Wild Horses**_ (1933) see _**King of the Wild Horses**_ (1933)\n\n**3407** _ **Rhythm of the Rio Grande**_ **** Monogram, 1940. 53 min. D: Al Herman. SC: Robert Emmett (Tansey). With Tex Ritter, Suzan Dale, Warner Richmond, Martin Garralaga, Arkansas Slim Andrews, Frank Mitchell, Tristram Coffin, Earl Douglas, Forrest Taylor, Mike Rodriguez, Glenn Strange, James McNally, Juan Duval, Chick Hannon, Sherry Tansey. A cowboy teams with a Mexican outlaw to prove that a respected rancher is behind the lawlessness in the territory. Okay Tex Ritter musical that includes the song \"Mexicali Moon.\" British title: _**Lonesome Trail to the Rio Grande**_.\n\n**3408** _ **Rhythm of the Saddle**_ **** Republic, 1938. 58 min. D: George Sherman. SC: Paul Franklin. With Gene Autry, Smiley Burnette, Peggy Moran, Pert Kelton, LeRoy Mason, Arthur Loft, Ethan Laidlaw, Walter De Palma, Archie Hall, Eddie Hart, Eddie Acuff, Douglas Wright, Kelsey Sheldon, Lola Monte, Alan Gregg, Rudy Sooter, William Norton Bailey, Tom London, Roger Williams, Curley Dresden, Jim Mason, Jack Kirk, Emmett Vogan, Frankie Marvin, Karl Hackett, Horace B. Carpenter. A dishonest nightclub owner bribes the chairman of the annual rodeo into helping him get the next year's contract and he fixes a halter rope that causes an injury to a rider with Gene Autry investigating. A strong plot greatly helps this Autry vehicle.\n\n**3409** _ **Rhythm on the Range**_ **** Paramount, 1936. 87 min. D: Norman Taurog. SC: John C. Moffitt, Sidney Salkow, Walter De Leon and Francis Martin. With Bing Crosby, Frances Farmer, Bob \"Bazooka\" Burns, Martha Raye, Samuel S. Hinds, Warren Hymer, Lucille Webster Gleason, George E. Stone, James Burke, Martha Sleeper, Clem Bevans, The Sons of the Pioneers (Bob Nolan, Len Slye [Roy Rogers], Tim Spencer, Hugh Farr, Karl Farr), Leonid Kinsky, Charles Williams, Beau Baldwin, Emmett Vogan, Billy Bletcher, Eddy Waller, Bud Flanagan (Dennis O'Keefe), Duke York, James Blaine, Herbert Ashley, James \"Slim\" Thompson, Robert E. Homans, Jim Toney, Ed LeSaint, Sam McDaniel, Syd Saylor, Oscar Smith, Charles Arnt, Harry C. Bradley, Otto Yamaoka, Bob McKenzie, Irving Bacon, Heinie Conklin, Frank Dawson. The owners of a dude ranch in California, on their way West with a prize bull they have won, meet an heiress on the run from her fiancee and they ask her to join them. Very pleasant musical comedy, remade as _**Pardners**_ (q.v.) with Norman Taurog again directing.\n\n**3410** _ **Rhythm Round-up**_ **** Columbia, 1945. 66 min. D: Vernon Keays. SC: Charles Marion. With Ken Curtis, Cheryl Walker, Guinn Williams, Raymond Hatton, The Hoosier Hot Shots (Charles \"Gabe\" Ward, Paul \"Hezzy\" Trietsch, Ken Trietsch, Gil Taylor), The Pied Pipers, Bob Wills and The Texas Playboys, Victor Potel, Arthur Loft, Walter Baldwin, Vera Lewis, Eddie Bruce, Marilyn Johnson. When they are unable to secure a radio contract, the Hoosier Hot Shots get title to an Arizona hotel but there are rumors that it is haunted. Fast paced musical Western program feature. ****\n\n_**Riata**_ see _**The Deadly Trackers**_\n\n**3411** _ **Ricochet Romance**_ **** Universal-International, 1954. 80 min. D: Charles Lamont. SC: Kay Lenard. With Marjorie Main, Chill Wills, Rudy Vallee, Benay Venuta, Pedro Gonzalez Gonzalez, Alfonso Bedoya, Marjorie Bennett, Judith (Rachel) Ames, Darryl Hickman, Lee Aaker, Irene Ryan, Philip Tonge, Phillip Chambers, Charles Watts, Ruth Hampton, Claire Du Brey, Madge Blake, Hal Smith, Jack Daley, Pitt Herbert, Fred Graham, Arthur Lovejoy, Yvonne Peattie, The Guadalajara Trio. Hired as a cook at a dude ranch, a woman soon involves herself in everyone's business. Attempt to team Marjorie Main and Chill Wills is a bit forced but Main's fans will like it and Rudy Vallee is great, as usual, portraying a stuffed-shirt guest.\n\n**3412** _ **Riddle Gawne**_ **** Paramount-Artcraft, 1918. 48 min. D: William S. Hart. SC: Charles Alden Seltzer. With William S. Hart, Katherine MacDonald, Lon Chaney, Gertrude Short, Gretchen Lederer, E.B. Tilton, Milton Ross, George Field, Leon Kent. Seeking revenge on the outlaw who killed his brother and took his wife, a man rescues a woman from gang members and they fall in love. Two reels remain of this classic teaming of William S. Hart and Lon Chaney, who plays the villain.\n\n**3413** _ **Riddle Ranch**_ **** Beaumont, 1935. 63 min. D: Charles Hutchison. SC: E.J. Thompson. With Black King (horse), David Worth, June Marlowe, Baby Charlene Barry, Julian Rivero, Richard Cramer, Fred \"Snowflake\" Toones, Budd Buster, Art Felix, Henry Sylvester, Ray Gallagher, Ace Cain, Sue Milford, Larry Francis, May Brinker. The owner of a beautiful stallion is framed for murder by a Mexican outlaw posing as a horse buyer wanting the animal for himself. Bottom rung production.\n\n**3414** _ **Ride a Crooked Mile**_ **** Paramount, 1938. 78 min. D: Alfred E. Green. SC: Ferdinand Reyher and John C. Moffitt. With Akim Tamiroff, Frances Farmer, Leif Erickson, Lynne Overman, John Miljan, J.M. Kerrigan, Vladimir Sokoloff, Genia Nikolaieva, Wade Crosby, Robert Glecker, Nestor Paiva, Archie Twitchell, Steve Pendleton, Fred Kohler, Jr., Eva Novak, Ethel Clayton, John Bleifer, James Flavin, Joseph Crehan, William Newell, Dewey Robinson, Barry Macollum, Leonid Snegoff, Alex Woloshin, Eddie Acuff, Ernie Adams, Robert Homans, Fred \"Snowflake\" Toones, Charles Anthony Hughes, Sam Ash, T.C. Jack, Harry Fleischmann, Eddie Borden, Hal Craig, George Magrill, Lee Shumway, Michael Mark, Ralph Sanford, Dick Rush, Gloria Williams. A Russian Cossack flees his homeland and becomes a cattle rustler in the West until he discovers he has a grown son jailed for the same crime. The three stars do their best but get bogged down by the script.\n\n**3415** _ **Ride a Crooked Trail**_ **** Universal-International, 1958. 87 min. Color. D: Jesse Hibbs. SC: Borden Chase. With Audie Murphy, Gia Scala, Walter Matthau, Henry Silva, Joanna Moore, Eddie Little, Mary Field, Leo Gordon, Mort Mills, Frank Chase, Bill Walker, Ned Weaver, Richard Cutting, Morgan Woodward, Rayford Barnes, Henry Wills, Eddie Parker. An outlaw takes on the identity of a dead peace officer in order to pull off a bank robbery. Fairly good Audie Murphy vehicle.\n\n**3416** _ **Ride a Northbound Horse**_ **** Buena Vista, 1969. 79 min. Color. D: Robert Totten. SC: Herman Groves. With Michael Shea, Carroll O'Connor, Ben Johnson, Andy Devine, Edith Atwater, Jack Elam, Harry Carey, Jr., Dub Taylor. A young man wins and loses a horse in the Old West and sets out to regain the steed. Minor, but entertaining, version of Richard Wormser's novel, originally shown as a two part episode of NBC-TV's \"Walt Disney's Wonderful World of Color.\"\n\n**3417** _ **Ride a Violent Mile**_ **** 20th Century\u2013Fox, 1957. 79 min. D-SC: Charles Marquis Warren. With John Agar, Penny Edwards, John Pickard, Richard Shannon, Charles Gray, Bing Russell, Helen Wallace, Richard Gilden, Sheb Wooley, Patrick O'Moore, Rush Williams, Roberto Contreras, Eva Novak, Mary Townsend, Rocky Shahan. During the Civil War a cowboy and a beautiful woman become involved in an attempt to break the Union blockade of Confederate seaports. Fair action drama.\n\n**3418** _ **Ride a Wild Pony**_ **** Buena Vista, 1976. 91 min. Color. D: Don Chaffey. SC: Rosemary Anne Sisson. With Michael Craig, Robert Bettles, Eva Griffith, John Meillon, Jr., Graham Rouse, Alfred Bell, Roy Haddrick, Peter Gywnne, Melissa Jaffer, Loraine Bayly, Kate Clarkson. In Australia the poor son of homesteaders battles a rich polio-crippled girl for the possession of a beautiful Welsh pony. Delightful Walt Disney family feature filmed in Australia.\n\n**3419** _ **Ride a Wild Stud**_ **** Vega International, 1969. 76 min. Color. D: Revilo Ekard (Oliver Drake). SC: William Edwards and Rachel Edwards. With Hale Williams, Josie Kirk, Frenchy Le Boyd, Cliff Alexander, William Fosterwick, C.C. Chase, Barbara Parks, Bill Ferrill, Burke Reynolds, Helga Honshue, Richard Smedley, Bill Johnson, Chuck Alford, Tex Gates, S.T. Alexander, Sr., Bob Goldfarb. Qunatrill and his men raid towns and force beautiful women to serve as prostitutes while a lawman, incognito as an outlaw, tries to bring down the gang. Ridiculous soft core Western helmed by genre veteran Oliver Drake under a pseudonym and made as _**Quantrill's Raiders**_ ; released on video as _**A Wild Ride**_.\n\n**3420** _ **Ride and Kill**_ **** P.E.A.\/Fenix Film, 1964. 94 min. Color. D: J.L. Boraw (Jose Luis Borau). SC: Jose Mallorqui (Mario Caiano). With Alex Nicol, Robert Hundar (Claudio Undari), Margaret Grayson, Lawrence Palmer, Pauline Baards, John MacDouglas (Giuseppe Addobbati), Anthony Gradwell (Antonio Gradoli), Luis Induni, Antonio Casas, Jorge Ringaud (George Rigaud), Natalia Silva. When outlaws take over an Arizona town and murder the sheriff, a drunk puts down the bottle and takes up a badge to defend the woman he loves. Fair Italian-Spanish co-production made as _**Cavalco e Uccidi**_ (Ride and Kill); it contains a fine music score by Riz Ortolani.\n\n**3421** _ **The Ride Back**_ **** United Artists, 1957. 79 min. D: Allen H. Miner. SC: Anthony Ellis. With Anthony Quinn, William Conrad, Lita Milan, George Trevino, Victor Millan, Ellen Hope Monroe, Joe Dominguez, Louis Towers. A lawman and his prisoner travel through hostile Indian country and learn they need each other to survive. William Conrad produced this interesting low budget suspense Western; worth viewing.\n\n**3422** _ **Ride Beyond Vengeance**_ **** Columbia, 1966. 100 min. Color. D: Bernard McEveety. SC: Andrew J. Fenady. With Chuck Connors, Michael Rennie, Kathryn Hays, Joan Blondell, Gloria Grahame, Gary Merrill, Bill Bixby, Claude Akins, Paul Fix, Marissa Mathes, Harry Harvey, Sr., William Bryant, Jamie Farr, Larrie Domasin, William Catching, James MacArthur, Ruth Warrick, Parley Baer, Frank Gorshin, Robert Q. Lewis, Chuck Hamilton, Bill Coontz, Arthur O'Connell (narrator). After being separated from his wife for eleven years, a buffalo hunter returns home only to be attacked and branded with her then rejecting him, so he plans revenge on his assailants. Fair action film with a good cast.\n\n**3423** _ **Ride Clear of Diablo**_ **** Universal-International, 1953. 80 min. Color. D: Jesse Hibbs. SC: George Zuckerman. With Audie Murphy, Dan Duryea, Susan Cabot, Abbe Lane, Russell Johnson, Paul Birch, William Pullen, Jack Elam, Lane Bradford, Mike Ragan, Denver Pyle, James Griffith, Ray Bennett, Hank Patterson, Carol Henry. To avenge the murders of his father and brother, a man becomes a deputy to a sheriff in cahoots with the killers. Better than average Audie Murphy film, mainly thanks to Dan Duryea's villainy.\n\n**3424** _ **Ride 'Em Cowboy**_ **** Universal, 1936. 59 min. D: Lesley Selander. SC: Frances Guihan. With Buck Jones, Luana Walters, George Cooper, William Lawrence, J.P. McGowan, Joseph Girard, Donald Kirke, Charles Le Moyne, Edmund Cobb, Lester Dorr, William Lawrence, Bob McKenzie, Ed Peil, Sr., Dick Rush, Burr Caruth, Fay McKenzie, Hal Price, George Plues, Blackjack Ward, Clyde McClary, Edward Hearn, Eva McKenzie, Bud McClure, Hank Bell, Billy Franey, Al Taylor, Francis Walker, George Sowards, Bud Pope . A happy-go-lucky cowboy becomes a race driver to help a pal and also save a woman from being forced to marry a rich man to save her father's ranch. Buck Jones wrote the original story and produced this outing which shows his penchant for comedy and auto racing but it is somewhat disappointing. TV title: _**Cowboy Roundup**_.\n\n**3425** _ **Ride 'Em Cowboy**_ **** Universal, 1942. 82 min. D: Arthur Lubin. SC: True Boardman and John Grant. With Bud Abbott, Lou Costello, Johnny Mack Brown, Dick Foran, Anne Gwynne, Samuel S. Hinds, Richard Lane, Douglass Dumbrille, Charles Lane, Ella Fitzgerald, Judd McMichael, Ted McMichael, Joe McMichael, Mary Lou Cook, Jody Gilbert, Morris Ankrum, Russell Hicks, Wade Boteler, James Flavin, Boyd Davis, Eddie Dunn, Isabel Randolph, Tom Hanlon, James Seay, Harold Daniels, Ralph Peters, Linda Brent, Lee Sunrise, Chief Yowlachie, Harry Monty, Sherman E. Sanders, Carmelo Cansino, Iron Eyes Cody, Katherine Marlowe, James Flavin, Harry Cording. Two hot dog vendors head West to a dude ranch and get involved in the romance between a pseudo-cowboy and the rodeo rider daughter of the owner. Good Abbott and Costello comedy with Dick Foran singing \"I'll Remember April.\"\n\n**3426** _ **Ride 'Em Cowgirl**_ **** Grand National, 1939. 52 min. D: Samuel Diege. SC: Arthur Hoerl. With Dorothy Page, Milton Frome, Vince Barnett, Warner Richmond, Lynn Mayberry, Joe Girard, Frank Ellis, Merrill McCormick, Fred Berhle, Harrington Reynolds, Pat Henning, Fred Cordova, Lester Dorr, Stanley Price, Lloyd Ingraham, Eddie Gordon. A cowgirl is blamed for having contraband silver but escapes from jail to prove a smuggler has framed her father to acquire his ranch as a front for his operations. Okay entry in Dorothy Page's brief starring series.\n\n**3427** _ **Ride Him Cowboy**_ **** Warner Bros., 1932. 55 min. D: Fred Allen. SC: Scott Mason. With John Wayne, Ruth Hall, Henry B. Walthall, Harry Gribbon, Otis Harlan, Frank Hagney, Charles Sellon, Lafe McKee, Frank Fanning, Ben Corbett, Glenn Strange, Fred Burns, Bob Burns, Bud McClure, Jack Kirk, Slim Whitaker, Bud Osborne, Edmund Cobb, Wally Wales, Murdock MacQuarrie, Hal Price, Blackjack Ward, Jim Corey, Chuck Baldra, Rose Plummer, Ada Belle Driver, Helen Dickson, Tiny Jones, William Gillis, F.R. Smith, Edward Burns. A cowboy, after saving a horse from being shot, sets out to capture a notorious bandit leader known as \"The Hawk,\" but ends up accused of being the outlaw. John Wayne's first starring \"B\" series Western is a good one that moves along at a steady clip; a remake of _**The Unknown Cavalier**_ (First National, 1926) starring Ken Maynard, who can be seen in stock footage with his horse Tarzan. Frank Hagney is especially good as the villain. British title: _**The Hawk**_.\n\n**3428** _ **Ride in the Whirlwind**_ **** Jack H. Harris Enterprises, 1971. 83 min. Color. D: Monte Hellman. SC: Jack Nicholson. With Cameron Mitchell, Millie Perkins, Jack Nicholson, Harry Dean Stanton, Rupert Crosse, Katherine Squire, George Mitchell. A trio of cowpokes are mistaken for outlaws by a posse after they meet the real bad guys on their way home from a long trail drive. Gritty feature originally made in 1966 and issued to television before the advent of Jack Nicholson's popularity brought it to theatres.\n\n**3429** _ **Ride, Kelly, Ride**_ **** 20th Century\u2013Fox, 1941. 59 min. D: Norman Foster. SC: William Counselman, Jr. and Irving Cummings, Jr. With Eugene Pallette, Marvin Stephens, Rita Quigley, Mary Healy, Richard Lane, Charles D. Brown, Chick Chandler, Dorothy Peterson, Lee Murray, Frankie Burke, Cy Kendall, Hamilton MacFadden, Walter O'Donnell, Ernie Adams. With the help of a young girl, a cowboy is taught to become a top jockey. Action filled program feature.\n\n**3430** _ **Ride Lonesome**_ **** Columbia, 1959. 73 min. Color. D: Budd Boetticher. SC: Burt Kennedy. With Randolph Scott, Karen Steele, Pernell Roberts, James Best, Lee Van Cleef, James Coburn, Dyke Johnson, Boyd Stockman, Roy Jenson, Boyd \"Red\" Morgan, Bonnie Dubbins. A lawman taking a prisoner across the desert enlists the aid of two bounty hunters for a share of the reward since they all are being hunted by the felon's brother and his gang. Taut Randolph Scott film; well written, produced and directed.\n\n**3431** _ **Ride on Vaquero**_ **** 20th Century\u2013Fox, 1941. 64 min. D: Herbert I. Leeds. SC: Samuel G. Engel. With Cesar Romero, Mary Beth Hughes, Lynne Roberts, Chris-Pin Martin, Robert Lowery, William Demarest, Ben Carter, Robert Shaw, Edwin Maxwell, Paul Sutton, Don Costello, Arthur Hohl, Irving Bacon, Joan Woodbury, Paul Harvey, Dick Rich, Hector V. Sarno, Frank Orth, Paco Moreno, Joe Whitehead, Paul Kruger, Alec Craig, Victor Potel, Max Wagner, Edgar Edwards, James Flavin, Eva Puig, Philip Van Zandt. After his lady love turns the Cisco Kid and his pal Gordito into the law for a reward the duo is offered a chance to find kidnappers who may have killed a family friend and taken his son. Cesar Romero's final \"Cisco Kid\" adventure is only average.\n\n**3432** _ **Ride Out for Revenge**_ **** United Artists, 1957. 79 min. D-SC: Norman Retchin. With Rory Calhoun, Gloria Grahame, Lloyd Bridges, Vincent Edwards, Joanne Gilbert, Richard Shannon, Cyril Delevanti, John Merrick. A sheriff tries to help Indians being forced off their lands by a corrupt Army officer who has discovered gold there. Run-of-the mill oater for Rory Calhoun fans.\n\n**3433** _ **Ride, Ranger, Ride**_ **** Republic, 1936. 56 min. D: Joseph Kane. SC: Dorrell McGowan and Stuart McGowan. With Gene Autry, Smiley Burnette, Kay Hughes, Monte Blue, Max Terhune, George J. Lewis, Robert Homans, Chief Thundercloud, The Tennessee Ramblers, Frankie Marvin, Iron Eyes Cody, Bud Pope, Nelson McDowell, Robert Thomas. Former Texas Rangers become scouts for the cavalry and attempt to stop a half-breed's attempts to start an Indian uprising. Lots of action, music and comedy highlight this pretty good Gene Autry vehicle.\n\n**3434** _ **Ride, Tenderfoot, Ride**_ **** Republic, 1940. 65 min. D: Frank McDonald. SC: Winston Miller. With Gene Autry, Smiley Burnette, June Storey, Warren Hull, Mary Lee, Si Jenks, Forbes Murray, Joe Frisco, Joe McGuinn, Isabel Randolph, Herbert Clifton, Mildred Shay, Cindy Walker, Jack Kirk, The Pacemakers, Bob Burns, Fred \"Snowflake\" Toones, Chuck Morrison, Frank O'Connor, Curley Dresden, Frankie Marvin, Cactus Mack. When Gene Autry becomes the owner of a packing company the boyfriend of a rival tries to merge the two businesses and dissolve Gene's operations. Musical Western does not have enough action to sustain its storyline.\n\n**3435** _ **Ride the High Country**_ **** Metro-Goldwyn-Mayer, 1962. 94 min. Color. D: Sam Peckinpah. SC: N.B. Stone, Jr. With Randolph Scott, Joel McCrea, Mariette Hartley, Ronald Starr, Edgar Buchanan, R.G. Armstrong, John Anderson, L.Q. Jones, Warren Oates, James Drury, John Davis Chandler, Jenie Jackson, Byron Foulger, Carmen Phillips. Two aging former lawmen are hired to protect a gold shipment and are joined by a young hellion and a girl trying to escape her new husband and his lustful brothers. This teaming of genre stars Randolph Scott (his final film) and Joel McCrea is a near-classic Western; very, very good and a must see.\n\n**3436** _ **Ride the Man Down**_ **** Republic, 1953. 90 min. Color. D: Joseph Kane. SC: Mary McCall, Jr. With Rod Cameron, Brian Donlevy, Ella Raines, Forrest Tucker, Barbara Britton, Chill Wills, J. Carrol Naish, Jim Davis, Taylor Holmes, James Bell, Paul Fix, Roy Barcroft, Douglas Kennedy, Chris-Pin Martin, Jack LaRue, Al Caudebec, Claire Carleton, Roydon Clark. When a rich rancher dies a feud develops between his daughter and land grabbers while the foreman tries to protect the property. Fine production and cast makes this \"A\" feature a good one to watch.\n\n**3437** _ **Ride the Wind**_ **** NBC-TV, 1966. 120 min. Color. D: William Witney. SC: Paul Schneider. With Lorne Greene, Rod Cameron, Victor Jory, Michael Landon, Dan Blocker, DeForrest Kelley, Ray Teal, Victor Sen Yung, Wolfe Brazell, Stewart Moss, Warren Vanders, Bill Edwards, Ben Wright, Richard Hale, Clay Tanner, Jack Bighead, James Novak, Tom Lowell, Robert Brubaker, Bill Clark, Roger Etienne, Tom Lutz, Bob La Wandt, Gilbert Green, Raymond Guth, Peter Ritter, Whitey Hughes, James Noah, David Pritchard, S. Newton Anderson. A rancher tries to help a man set up the Pony Express despite opposition, including Indian attacks, on the final leg of the route. Well made and entertaining TV feature made up of a two part episode of \"Bonanza\" (NBC-TV, 1959\u201373); issued theatrically in Europe.\n\n**3438** _ **Ride to Glory**_ **** NBC-TV\/Columbia, 1965. 90 min. Color. D: Allen Riesner. SC: John Wilder, Jerry Ziegman and Larry Cohen. With Chuck Connors, Robert Lansing, David Brian, Kathie Browne, Noah Berry, Jr., H.M. Wynant, Michael Pate, Lee Van Cleef, William Bryant, John Pickard, Vaughn Taylor, Richard Tarto, Jacquelyn Hyde, Garry New, John Howard, James Hurst. A former Army officer, falsely accused of cowardice, tries to prove his innocence as well as prevent an Indian uprising. Sturdy TV film from three episodes of \"Branded\" (NBC-TV, 1965\u201366) and released abroad to theatres under its original title, _**Call to Glory**_.\n\n**3439** _ **Ride to Hangman's Tree**_ **** Universal, 1967. 90 min. Color. D: Alf Rafkin. SC: Luci Ward, Jack Natteford and William Bowers. With Jack Lord, James Farentino, Don Galloway, Melodie Johnson, Richard Anderson, Robert Yuro, Ed Peck, Paul Reed, Richard Cutting, Bing Russell, Virginia Capers, Robert Sorrells, Robert Cornthwaite. A notorious road agent terrorizes citizens after escaping from a lynch mob. Vapid remake of _**Black Bart**_ (q.v.).\n\n**3440** _ **Ride, Vaquero!**_ **** Metro-Goldwyn-Mayer, 1953. 90 min. Color. D: John Farrow. SC: Frank Fenton. With Robert Taylor, Ava Gardner, Howard Keel, Anthony Quinn, Kurt Kasznar, Ted De Corsia, Charlita, Jack Elam, Frank McGrath, Joe Dominguez, Walter Baldwin, Charles Stevens, Rex Lease, Tom Greenway. A couple try to settle a ranch but are opposed by half-brothers, who both fall in love with the wife. A good cast does its best but this oater is still nothing to brag about.\n\n**3441** _ **Rider from Tucson**_ **** RKO Radio, 1950. 60 min. D: Lesley Selander. SC: Ed Earl Repp. With Tim Holt, Richard Martin, Elaine Riley, Douglas Fowley, Veda Ann Borg, Robert Shayne, William Phipps, Harry Tyler, Marshall Reed, Stuart Randall, Dorothy Vaughn. In order to obtain a valuable gold claim, crooks turn to murder and kidnapping but are opposed by a cowpoke. Average Tim Holt vehicle.\n\n**3442** _ **The Rider of Death Valley**_ **** Universal, 1932. 78 min. D: Albert Rogell. SC: Jack Cunningham. With Tom Mix, Lois Wilson, Fred Kohler, Forrest Stanley, Willard Robertson, Edith Fellows, Mae Busch, Max Asher, Pete Morrison, Edmund Cobb, Otis Harlan, Francis Ford, Richard Cramer, Bob McKenzie, Lloyd Whitlock, Iron Eyes Cody. A man is murdered for his desert gold claim and a rancher friend tries to protect the victim's small daughter's interest in the mine and bring the killers to justice. One of Tom Mix's best sound features, this \"A\" outing is highly entertaining and contains beautiful Death Valley photography by Daniel Clark; well worth seeing. TV title: _**Riders of the Desert**_.\n\n**3443** _ **Rider of the Law**_ **** Supreme, 1935. 56 min. D: Robert North Bradbury. SC: Jack Natteford. With Bob Steele, Gertrude Messinger, Si Jenks, Earl Dwire, Forrest Taylor, Lloyd Ingraham, John Elliott, Sherry Tansey, Tex Palmer, Chuck Baldra, Steve Clark, Bud Osborne, Art Dillard, Jack Kirk, Ray Henderson. Posing as a dude, a government agent attempts to round up an outlaw gang. Typically entertaining Bob Steele entry for producer A.W. Hackel.\n\n**3444** _ **Rider of the Plains**_ **** Syndicate, 1931. 57 min. D: J.P. McCarthy. SC: Wellyn Totman. With Tom Tyler, Andy Shuford, Lillian Bond, Alan Bridge, Gordon DeMain, Jack Perrin, Orbie (Slim) Whitaker, Ted Adams, Fern Emmett, Frank Lanning, William Bertram, George Offerman, Jr., Nina Campana, Artie Ortego, Jim Corey, Bob Woodward, Jess Cavin, Eddie Fetherston. An outlaw befriends a young boy and is reformed by him and a pretty girl. Slow paced Tom Tyler feature; not much.\n\n_**Rider of the Plains**_ (1941) see _**The Lone Rider Rides On**_\n\n**3445** _ **Rider on a Dead Horse**_ **** Allied Artists, 1962. 67 min. D: Herbert L. Strock. SC: Stephen Longstreet. With John Vivyan, Lisa Lu, Bruce Gordon, Kevin Hagen, Charles Lampkin. After three miners divide their diggings, one is murdered and the culprit tries to blame the third man but he finds out that love is more important than riches. Tacky, talky feature.\n\n_**Riders for Justice**_ see _**Westward Ho**_ (1942)\n\n**3446** _ **Riders from Nowhere**_ **** Monogram, 1940. 47 min. D: Raymond K. Johnson. SC: Carl Krusada. With Jack Randall, Margaret Roach, Charles King, Ernie Adams, Tom London, Nelson McDowell, George Chesebro, Ted Adams, Dorothy Adams, Carl Mathews, Jack Evans, Herman Hack, Archie Ricks, Ray Henderson, Kit Guard, Jimmy Aubrey, Tex Palmer, Denver Dixon, Clyde McClary, Rube Dalroy. A stranger tries to learn who murdered a lawman and robbed a gold shipment. Average Jack Randall affair.\n\n**3447** _ **Riders in the Sky**_ **** Columbia, 1949. 70 min. D: John English. SC: Gerald Geraghty. With Gene Autry, Pat Buttram, Gloria Henry, Mary Beth Hughes, Robert Livingston, Steve Darrell, Alan Hale, Jr., Tom London, Hank Patterson, Ben Welden, Kenne Duncan, Dennis Moore, Joe Forte, Frank Jaquet, Roy Gordon, Boyd Stockman, Pat O'Malley, John Parrish, Kermit Maynard, Bud Osborne, Lynton Brent, Isabel Withers, Sandy Sanders, Denver Dixon, Robert Walker, Loie Bridge, Vernon Johns, Stan Jones, Cactus Mack, Tom Smith, Herman Hack, Lee Phelps, Jack Evans. A rancher is framed on false charges by a gambler and Gene Autry plans to clear him and bring the crook to justice. Fine Gene Autry feature with good use of songs, including the title tune made famous by Vaughn Monroe.\n\n**3448** _ **Riders of Black Mountain**_ **** Producers Releasing Corporation, 1940. 59 min. D: Peter Stewart (Sam Newfield). SC: Joseph O'Donnell. With Tim McCoy, Pauline Haddon, Rex Lease, Ed Peil, Sr., Frank LaRue, Ralph Peters, Ted Adams, Julian Rivero, Jack Rutherford, George Chesebro, Dirk Thane, Carl Mathews, Alden Chase, Steve Clark, Lane Bradford, Herman Hack, Pascale Perry, Al Haskell, Tex Palmer, Ray Henderson, Augie Gomez. A banker, involved in an insurance fraud, is behind a series of stagecoach robberies and his gang is pursued by a federal marshal disguised as a gambler. Okay Tim McCoy PRC outing; also called _**Black Mountain Stage**_.\n\n**3449** _ **Riders of Black River**_ **** Columbia, 1939. 59 min. D: Norman Deming. SC: Bennett Cohen. With Charles Starrett, Iris Meredith, The Sons of the Pioneers (Bob Nolan, Tim Spencer, Lloyd Perryman, Pat Brady, Hugh Farr, Karl Farr), Dick Curtis, Edmund Cobb, Stanley Brown, Francis De Sayles, Forrest Taylor, George Chesebro, Olin Francis, Lew Meehan, Maston Williams, Carl Sepulveda. A former Texas Ranger returns home with plans to marry a girl, not knowing her brother is involved with an outlaw gang rustling cattle. Competent remake of _**The Revenge Rider**_ (q.v.).\n\n**3450** _ **Riders of Death Valley**_ **** Universal, 1941. 15 Chapters. D: Ray Taylor and Ford Beebe. SC: Sherman Lowe, Basil Dickey, George Plympton and Jack Connell. With Dick Foran, Leo Carrillo, Buck Jones, Charles Bickford, Lon Chaney, Jr., Noah Beery, Jr., Guinn Williams, Jeanne Kelly (Jean Brooks), Monte Blue, James Blaine, Glenn Strange, Roy Barcroft, Ethan Laidlaw, Richard Alexander, Jack Rockwell, Frank Austin, Charles Thomas, William Hall, James Guilfoyle, Ernie Adams, Edmund Cobb, William Pagan, Jack Clifford, Richard Travis, Ivar McFadden, Jack Perrin, Slim Whitaker, Bud Osborne, Frank Brownlee, Art Miles, Ed Payson, James Farley, Ted Adams, Dick Rush, Gil Perkins, Duke York, Jerome Harte, Ruth Rickaby, Don Rowan, Alonzo Price, Ken Nolan, Jay Michael. A vigilante group helps miners fighting a protection racket and end up opposing a notorious outlaw after all the gold mines in the district. A top notch cast is the chief asset of this otherwise mediocre serial.\n\n**Richard Alexander and Lon Chaney in** _**Riders of Death Valley**_ **(Universal, 1941).**\n\n**  \n**\n\n**3451** _ **Riders of Destiny**_ **** Monogram, 1933. 55 min. D-SC: Robert North Bradbury. With John Wayne, Cecilia Parker, George Hayes, Forrest Taylor, Al St. John, Heinie Conklin, Earl Dwire, Yakima Canutt, Lafe McKee, Addie Foster, Fern Emmett, Hal Price, Si Jenks, Horace B. Carpenter, Tex Palmer, Silver Tip Baker, William Dyer, Bert Lindley, Herman Nowlin. Posing as a notorious bandit, a government agent tries to get the goods on a businessman robbing settlers of their land by denying them water rights. The _only_ \"Singin' Sandy\" film, this initial entry in John Wayne's Monogram-Lone Star series for producer Paul Malvern in a good one, when the star is not singing, dubbed or not.\n\n**3452** _ **Riders of Pasco Basin**_ **** Universal, 1940. 56 min. D: Ray Taylor. SC: Ford Beebe. With Johnny Mack Brown, Bob Baker, Fuzzy Knight, Frances Robinson, Lafe McKee, Arthur Loft, Frank LaRue, James Guifoyle, Chuck Morrison, Ed Cassidy, Robert Winkler, William Gould, Ted Adams, Kermit Maynard, David Sharpe, Rudy Sooter's Californians, George Chesebro, Ed Peil, Sr., Hank Bell, Gordon Hart, Slim Whitaker, Lynton Brent, Frank Ellis, Jim Corey, Hank Worden, Bob Burns, Al Taylor, Blackie Whiteford. A rodeo rider returns home to find promoters of an irrigation project trying to force themselves on the area and he leads vigilantes in stopping them. Well done Johnny Mack Brown feature.\n\n**3453** _ **Riders of the Badlands**_ **** Columbia, 1941. 57 min. D: Howard Bretherton. SC: Betty Burbridge. With Charles Starrett, Russell Hayden, Cliff Edwards, Ilene Brewer, Kay Hughes, Roy Barcroft, Rick Anderson, Edith Leach, Ethan Laidlaw, Harry Cording, Hal Price, Ted Mapes, George J. Lewis, John Cason, Edmund Cobb, Francis Walker. A ranger and his dentist pal try to bring in an outlaw and his gang but the lawman is a look-a-like for the crook and ends up being arrested by another ranger whose wife was murdered by the bandit. The plot may be complicated but this Charles Starrett-Russell Hayden opus is action filled from start to finish.\n\n**3454** _ **Riders of the Black Hills**_ **** Republic, 1938. 55 min. D: George Sherman. SC: Betty Burbridge and Bernard McConville. With Robert Livingston, Ray Corrigan, Max Terhune, Ann Evers, Roscoe Ates, Maude Eburne, Frank Melton, Johnny Lang Fitzgerald, Jack Ingram, John P. Wade, Fred \"Snowflake\" Toones, Edward Earle, Monte Montague, Ben Hall, Frank O'Connor, Tom London, Bud Osborne, Milburn Morante, Jack O'Shea, Art Dillard, John Merton, Slim Whitaker, Dick Elliott, Gloria Rich, Lester Sharpe, Jette White, George Magrill, David Sharpe. When a young woman's valuable race horse is stolen, three cowboys help her find it and capture the thieves. High grade entry in the popular \"Three Mesquiteers\" series.\n\n**3455** _ **Riders of the Cactus**_ **** Big 4, 1931. 60 min. D-SC: David Kirkland. With Wally Wales, Buzz Barton, Lorraine La Val, Fred Church, Ed Cartwright, Don Wilson, Joe Lawliss, Tete Brady, Etta Delmar, Gus Anderson, Sam J. Garrett. A cowboy is on the trail of an outlaw gang stalking a man trying to find buried treasure. Tattered poverty row Wally Wales vehicle.\n\n**3456** _ **Riders of the Dawn**_ **** Monogram, 1937. 53 min. D: Robert North Bradbury. SC: Robert Emmett (Tansey). With Jack Randall, Peggy Keys, Warner Richmond, James Sheridan (Sherry Tansey), George Cooper, Earl Dwire, Lloyd Ingraham, Ed Brady, Tim Davis, Yakima Canutt, Frank Hagney, Tex Cooper, Oscar Gahan, Forrest Taylor, Chick Hannon, Ella McKenzie, Ed Coxen, Jim Corey, Augie Gomez. Two lawmen are assigned to clean up a lawless town plagued by an outlaw gang led by the notorious gunman Danti. Jack Randall's first series film is a good one, greatly helped by Warner Richmond's usual excellence as the villain.\n\n**3457** _ **Riders of the Dawn**_ **** Monogram, 1945. 58 min. D: Oliver Drake. SC: Louise Rousseau. With Jimmy Wakely, Lee \"Lasses\" White, John James, Phyllis Adair, Sarah Padden, Horace Murphy, Richard Alexander, Jack Baxley, Bob Shelton, Wesley Tuttle and His All Stars, Arthur \"Fiddlin'\" Smith, Bill Hammond, Dad Pickard, Eddie Majors, Brooke Temple. Three medicine show entertainers find a young couple murdered and rescue their baby, the culprit being a doctor after the victim's oil rich property. The plot is okay but overall this is just another dull Jimmy Wakely outing except for the many song interludes; a reworking of _**Gunman from Bodie**_ (q.v.).\n\n**3458** _ **Riders of the Deadline**_ **** United Artists, 1943. 70 min. D: Lesley Selander. SC: Bennett Cohen. With William Boyd, Andy Clyde, Jimmy Rogers, Richard Crane, Frances Woodward, William Harrigan, Tony (Anthony) Warde, Robert Mitchum, Jim Bannon, Hugh Prosser, Herbert Rawlinson, Montie Montana, Jack Rockwell, Earle Hodgins, Bill Beckford, Pierce Lyden, Herman Hack, Art Felix, Robert Walker, Cliff Parkinson, Roy Bucko. Gun runners murder a Texas Ranger and Hopalong Cassidy, disguised as an outlaw, tries to find them. Average outing in the long running series; remake of _**Desert Bandit**_ (q.v.).\n\n**3459** _ **Riders of the Desert**_ **** World Wide, 1932. 57 min. D: Robert North Bradbury. SC: Wellyn Totman. With Bob Steele, Gertrude Messinger, George Hayes, Al St. John, Horace B. Carpenter, Louise Carter, Earl Dwire, Joe Dominguez, Greg Whitespear, John Elliott. An Arizona ranger is on the trail of an outlaw gang terrorizing the vicinity. Fine Bob Steele affair with plenty of action for his fans.\n\n_**Riders of the Desert**_ (1932) see _**The Rider of Death Valley**_\n\n**3460** _ **Riders of the Dusk**_ **** Monogram, 1949. 60 min. D: Lambert Hillyer. SC: Jess Bowers (Adele Buffington). With Whip Wilson, Andy Clyde, Reno Browne, Tristram Coffin, Marshall Reed, Myron Healey, John Merton, Holly Bane, Lee Roberts, Dee Cooper, Thornton Edwards, Ray Jones, John Cason. A deputy marshal rides to a town to help the sheriff capture a mysterious cattle rustler but along the way he is mistaken for the bad man. Pretty fair Whip Wilson vehicle.\n\n**3461** _ **Riders of the Frontier**_ **** Monogram, 1939. 58 min. D: Spencer Gordon Bennet. SC: Jesse Duffy and Joseph Lovering. With Tex Ritter, Jean Joyce, Hal Taliaferro, Jack Rutherford, Mantan Moreland, Marin Sais, Olin Francis, Nolan Willis, Roy Barcroft, Merrill McCormick, Edward Cecil, Bruce Mitchell, Maxine Leslie, Charles King, Forrest Taylor, Nelson McDowell. Outlaws hold a woman prisoner at her ranch as a lawman pretends to be a wanted criminal to gain access to the place and save her. One of the better Tex Ritter Monogram efforts and it includes the traditional folk song \"Boll Weevil,\" a tune closely associated with the star. British title: _**Ridin' the Frontier**_.\n\n**3462** _ **Riders of the Golden Gulch**_ **** West Coast, 1932. 52 min. D: Clifford Smith. SC: Robert J. Horner. With Buffalo Bill, Jr., Mary Dunn, Yakima Canutt, Edmund Cobb, Pete Morrison, Jack Harvey. A cowboy gets involved in a plot to rob his girl's father. Just about as bad as they get, another dud from producer Robert J. Horner based on a story by Yakima Canutt.\n\n**3463** _ **Riders of the Law**_ **** Sunset, 1922. 55 min. D-SC: Robert North Bradbury. With Jack Hoxie, Marin Sais, Pat Harmon, Tom Lingham, Jack Pierce, Frank Jonasson, Sonny Hicks. A government ranger, working undercover, saves the life of a sheriff who has been wounded by a gang of liquor smugglers. Fast moving Jack Hoxie silent feature.\n\n**3464** _ **Riders of the Lone Star**_ **** Columbia, 1947. 55 min. D: Derwin Abrahams. SC: Barry Shipman. With Charles Starrett, Smiley Burnette, Virginia Hunter, Steve Darrell, Curly Williams and His Georgia Peach Pickers, Edmund Cobb, Mark Dennis, Lane Bradford, Ted Mapes, George Chesebro, Peter Perkins, Eddie Parker, Bud Osborne, Nolan Leary. Two Texas Rangers are after a notorious bad man who has returned to an area he once terrorized in order to stop the re-opening of a mine. Pretty good \"Durango Kid\" adventure.\n\n**3465** _ **Riders of the North**_ **** Syndicate, 1931. 59 min. D: J.P. McGowan. SC: G.A. Durlam. With Bob Custer, Blanche Mehaffey, Al Ferguson, Frank Rice, Eddie Dunn, George Regas, Buddy Shaw, William Walling, George Hackathorne, Horace B. Carpenter, Blackie Whiteford, Tom Smith, Carl de Loro. A Canadian Mounted Policeman is on the trail of an outlaw gang in the north woods. Shoddy Bob Custer vehicle with the star his usual stoical self.\n\n**3466** _ **Riders of the Northland**_ **** Columbia, 1942. 58 min. D: William Berke. SC: Paul Franklin. With Charles Starrett, Russell Hayden, Cliff Edwards, Shirley Patterson, Lloyd Bridges, Bobby Larson, Kenneth MacDonald, Paul Sutton, Robert O. Davis, Joe McGuinn, Francis Walker, George Piltz, Blackjack Ward, Dick Jensen. A trio of Texas Rangers are sent to Alaska to investigate enemy activities and find a group of saboteurs trying to construct a runway for Axis planes. Sturdy Charles Starrett-Russell Hayden vehicle.\n\n**3467** _ **Riders of the Northwest Mounted**_ **** Columbia, 1943. 57 min. D: William Berke. SC: Fred Myton. With Russell Hayden, Dub Taylor, Bob Wills and His Texas Playboys, Adele Mara, Dick Curtis, Richard Bailey, Jack Ingram, Wen Wright, Vernon Steele. A Mounted Policeman is assigned to stop fur thieves working in the Red River district, the gang being led by a corrupt and vicious trading post operator. Probably the best of Russell Hayden's Columbia series with nice photography by Benjamin Kline, pretty Adele Mara as the leading lady and the grand nastiness from Dick Curtis as the villain.\n\n**3468** _ **Riders of the Pony Express**_ **** Screencraft, 1949. 61 min. D-SC: Michael Salle. With Ken Curtis, Shug Fisher, Cathy Douglas, Billy Benedict, Billy Hammond, Eddie McLean, Truman Van Dyke, John Dehner, Lou Marcelle, Rodd Redwing. On the run from the law, a rancher takes a new name and becomes a Pony Express rider not knowing the district supervisor is a half-breed trying to sabotage the operation. Low budget oater with songs.\n\n**3469** _ **Riders of the Purple Sage**_ **** Fox, 1925. 56 min. D: Lynn Reynolds. SC: Edfrid Bingham. With Tom Mix, Beatrice Burnham, Arthur Morrison, Seesel Ann Johnson, Warner Oland, Fred Kohler, Wilfred Lucas, Charles Newton, Joe Rickson, Mabel Ballin, Charles Le Moyne, Harold Goodwin, Marion Nixon, Dawn O'Day (Anne Shirley). After a dishonest lawyer kidnaps his sister and niece a Texas Ranger sets out to find them. Great photography (by Dan Clark) and a fine story make this Tom Mix silent feature a good viewing bet. Followed by _**The Rainbow Trail**_ (1925) [q.v.], this Zane Grey work was first filmed in 1918 by Fox with William Farnum.\n\n**3470** _ **Riders of the Purple Sage**_ **** Fox, 1931. 58 min. D: Hamilton MacFadden. SC: John F. Goodrich, Philip Klein and Barry Connors. With George O'Brien, Marguerite Churchill, Noah Beery, Yvonne Pelletier, James Todd, Stanley Fields, Lester Dorr, Frank McGlynn, Jr., Shirley Nails. A cowboy becomes an outcast for killing the man who kidnapped his sister and her daughter but he later saves a young woman and her ranch from outlaws. Fox's third screen version of the Zane Grey novel is a good one, again followed by its sequel _**The Rainbow Trail**_ (1932) [q.v.].\n\n**3471** _ **Riders of the Purple Sage**_ **** 20th Century\u2013Fox, 1941. 56 min. D: James Tinling. SC: William Buckner and Robert Metzler. With George Montgomery, Mary Howard, Robert Barrat, Lynne Roberts, Kane Richmond, Patsy Peterson, Richard Lane, Oscar O'Shea, James Gillette, Frank McGrath, Leroy Mason, George Cleveland, Francis Ford, Ethan Laidlaw, Frank McCarroll. A cowboy helps a pretty rancher fight a gang of vicious vigilantes. The fourth screen adaptation of Zane Grey's book and it makes for fine entertainment.\n\n**3472** _ **Riders of the Range**_ **** Truart, 1923. 50 min. D-SC: Otis B. Thayer. With Edmund Cobb, Clare Hatton, Frank Gallagher, Roy Langdon, Harry Ascher, E. Glendower, B. Bonaventure, Levi Simpson, Dolly Dale, Helen M. Hayes, Mae Dean, Ann Drew. The president of the cattlemen's association believes sheep men are behind a series of rustlings but he changes his mind after falling in love with daughter of a sheepherder. Okay silent action film that lets us see the great Edmund Cobb in a starring role.\n\n**3473** _ **Riders of the Range**_ **** RKO Radio, 1950. 60 min. D: Lesley Selander. SC: Norman Houston. With Tim Holt, Richard Martin, Jacqueline White, Reed Hadley, Robert Barrat, Tom Tyler, Robert Clarke, William Tannen, Holly Bane. Two cowpokes help a young woman whose cattle are stolen by a crook blackmailing her brother. Nice Tim Holt outing with a fine supporting cast.\n\n**3474** _ **Riders of the Rio Grande**_ **** Syndicate, 1929. 55 min. D: J.P. McGowan. SC: Sally Winters. With Bob Custer, Edna Aslin, Horace B. Carpenter, Kip Cooper, Bob Erickson, Martin Cichy, Merrill McCormick. A cowboy tries to help a woman and an engraver who have been kidnapped by the Quantrill gang. Slow moving Bob Custer silent also issued with a music score.\n\n**3475** _ **Riders of the Rio Grande**_ **** Republic, 1943. 55 min. D: Howard Bretherton. SC: Albert DeMond. With Bob Steele, Tom Tyler, Jimmie Dodd, Lorraine Miller, Edward Van Sloan, Rick Vallin, Harry Worth, Roy Barcroft, Charles King, Jack Ingram, John James, Jack O'Shea, Henry Hall, Bud Osborne, Yakima Canutt, Chester Conklin, Curley Dresden, Budd Buster, Robert Kortman, Charles Sullivan. Outlaws threaten a town and its leading citizens as a trio of cowboy peacemakers come to the rescue. Final entry in the popular and long running \"The Three Mesquiteers\" series; nothing exception but it is entertaining.\n\n**3476** _ **Riders of the Rockies**_ **** Grand National, 1937. 59 min. D: Robert North Bradbury. SC: Robert Emmett (Tansey). With Tex Ritter, Louise Stanley, Horace Murphy, Snub Pollard, Heber Snow (Hank Worden), Charles King, Yakima Canutt, Earl Dwire, Martin Garralaga, Jack Rockwell, Paul Lopez, Tex Palmer, Clyde McClary, The Texas Tornados. Two Texas Rangers are falsely accused of a crime and a fellow officer resigns his commission and tries to prove their innocence. Average Tex Ritter film with a quartet of songs, including the title tune, \"Home on the Range\" and \"Song of the Open Range.\"\n\n**3477** _ **Riders of the Sage**_ **** Metropolitan, 1939. 55 min. D: Harry S. Webb. SC: Carl Krusada. With Bob Steele, Claire Rochelle, Ralph Hoopes, James Whitehead, Earl Douglas, Ted Adams, Dave O'Brien, Frank LaRue, Bruce Dane, Jerry Sheldon, Reed Howes, Bud Osborne, Gordon Roberts (Carleton Young). A cowboy stops two outlaws from killing a man and finds himself in the middle of a sheep men versus cattle ranchers feud. Pretty tattered Bob Steele vehicle.\n\n**3478** _ **Riders of the Santa Fe**_ **** Universal, 1944. 60 min. D: Wallace Fox. SC: Ande Lamb. With Rod Cameron, Eddie Dew, Fuzzy Knight, Jennifer Holt, Lane Chandler, Earle Hodgins, Ray Whitley, The Bar-6 Cowboys (Aleth Hansen, Ezra Paulette, Charley Quirt), Richard Alexander, Budd Buster, Ida Moore, Jack Hendricks, Scotty Harrell, George Douglas, Henry Wills, Ethan Laidlaw, Curley Dresden, Roy Bucko, George Sowards, Jack Tornek, George Plues, Ray Jones, George Hazel, Bob Burns, Ralph Bucko. A new sheriff tries to prove a town boss killed the previous mayor and changed survey maps in order to control the area water supply. Sturdy Rod Cameron series entry.\n\n**3479** _ **Riders of the Timberline**_ **** Paramount, 1941. 59 min. D: Lesley Selander. SC: J. Benton Cheney. With William Boyd, Andy Clyde, Brad King, Victor Jory, Eleanor Stewart, J. Farrell MacDonald, Anna Q. Nilsson, Edward Keane, Hal Taliaferro, Tom Tyler, Mickey Elissa, Hank Bell, The Guardsmen Quartet, Frank Miller, Herman Hack, Tex Phelps, Tex Cooper. The Bar 20 trio help the owner of a logging operation when a crook plans to blow up a dam to ruin the man's business. A fast moving story, nice north woods locations and photography (by Russell Harlan), plus an exciting finale, highlight this \"Hopalong Cassidy\" feature.\n\n**3480** _ **Riders of the West**_ **** Monogram, 1942. 58 min. D: Howard Bretherton. SC: Jess Bowers (Adele Buffington). With Buck Jones, Tim McCoy, Raymond Hatton, Christine McIntyre, Dennis Moore, Harry Woods, Sarah Padden, Walter McGrail, Robert Frazer, Bud Osborne, Charles King, Lee Phelps, Kermit Maynard, Milburn Morante, Ed Peil, Sr., Lynton Brent, George Morrell, Tom London, J. Merrill Holmes, Lee Phelps, Herman Hack, Jack Kirk, Jimmy Aubrey, Denver Dixon, Roy Bucko. As a trio of crooks try to steal a woman's ranch, three lawmen help her with one of them infiltrating the gang. Pleasant \"Rough Riders\" series entry.\n\n**3481** _ **Riders of the Whistling Pines**_ **** Columbia, 1949. 70 min. D: John English. SC: Jack Townley. With Gene Autry, Patricia (Barry) White, Jimmy Lloyd, Douglass Dumbrille, Damian O'Flynn, Clayton Moore, Harry V. Cheshire, Leon Weaver, Loie Bridge, The Cass County Boys (Jerry Scoggins, Fred S. Martin, Bert Dodson), Roy Gordon, Jason Robards, Britt Wood, Len Torrey, The Pinafores, Lane Chandler, Lynn Farr, Al Thompson, Emmett Vogan, Virginia Carroll, Nolan Leary, Steve Benton. A man mistakenly thinks he accidentally killed a forest ranger when the official was really murdered because he discovered a moth infestation that would have profited two crooked businessmen. Colorful and entertaining Gene Autry opus.\n\n**3482** _ **The Riders of the Whistling Skull**_ **** Republic, 1937. 56 min. D: Mack V. Wright. SC: Oliver Drake and John Rathmell. With Robert Livingston, Ray Corrigan, Max Terhune, Mary Russell, Fern Emmett, Roger Williams, C. Montague Shaw, Yakima Canutt, John Ward, George Godfrey, Frank Ellis, Earle Ross, Chief Thundercloud, John Van Pelt, Ed Peil, Sr., Jack Kirk, Iron Eyes Cody, Tom Steele, Wally West, Tracy Layne, Ken Cooper. An archeologist searching for a lost Indian city is missing and three cowboys join an expedition, led by his daughter, trying to find him. One of the most satisfying entries in \"The Three Mesquiteers\" series and based on the novel by William Colt MacDonald; reworked as a Charlie Chan feature, _**The Feathered Serpent**_. (q.v.).\n\n_**Riders of Vengeance**_ see _**The Raiders**_ (1952)\n\n**3483** _ **Ridin' Down the Canyon**_ **** Republic, 1942. 56 min. D: Joseph Kane. SC: Albert DeMond and Robert Williams. With Roy Rogers, George \"Gabby\" Hayes, Bob Nolan and The Sons of the Pioneers (Tim Spencer, Pat Brady, Hugh Farr, Karl Farr, Lloyd Perryman), Buzzy Henry, Linda Hayes, Addison Richards, Lorna Gray (Adrian Booth), Olin Howlin, James Seay, Hal Taliaferro, Forrest Taylor, Roy Barcroft, Art Mix, Tom London, Jack Kirk, Ed Cassidy, Art Dillard, Tommy Coats, Major Sam Harris. A young boy enlists the help of Roy Rogers and the Sons of the Pioneers in rounding up a gang of horse rustlers interfering with the war effort. Seven songs and Buzzy Henry's riding add zest to this compact Roy Rogers film set in the contemporary West.\n\n**3484** _ **Ridin' Down the Trail**_ **** Monogram, 1947. 53 min. D: Howard Bretherton. SC: Bennett Cohen. With Jimmy Wakely, Dub Taylor, Beverly Jons, Douglas Fowley, John James, Doug Aylesworth, Charles King, Matthew (Brad) Slaven, Kermit Maynard, Harry Carr, Milburn Morante, Ted French, Post Park, Dick Reinhart, Don Weston. Arriving at a ranch, members of a medicine show find the inhabitants murdered and end up being blamed for the crime. One of the better Jimmy Wakely features.\n**3485** _ **The Ridin' Fool**_ **** Tiffany, 1931. 58 min. D: J.P. McCarthy. SC: Wellyn Totman. With Bob Steele, Frances Morris, Florence Turner, Ted Adams, Alan Bridge, Eddie Fetherston, Jack Henderson, Gordon DeMain, Josephine Velez, Fern Emmett, Artie Ortego. A cowboy saves a gambler from being hung for a crime he did not commit but later, in another town, they both fall for the same girl and are accused of robbing a stagecoach and killing the driver. While a bit on the slow side with a complicated plot, this Bob Steele film should satisfy his fans; Ted Adams is fine as the good-bad man while Steele croons \"I Fell in Love with You, Can't You Fall in Love with Me?\"\n\n**3486** _ **Ridin' for Justice**_ **** Columbia, 1932. 64 min. D: D. Ross Lederman. SC: Harold Shumate. With Buck Jones, Mary Doran, Russell Simpson, Walter Miller, Bob McKenzie, William Willing, Billy Engle, Hank Mann, Robert Kortman, Lee Phelps, Archie Ricks, Nancy Drexel, Art Mix, Buffalo Bill, Jr., Ben Corbett, Al Taylor, Bob Burns, Jack Low, Jack King, Ken Cooper, Al Haskell, Lafe McKee, Bud Pope, William McCall, Tex Phelps, Bob Roper. A cowboy gets in trouble with the law after a saloon brawl and is given sanctuary by the marshal's wife but she is attacked by his deputy and when he is shot the cowpoke is blamed. Top notch Buck Jones feature, surprisingly adult in its theme for the time and given audience.\n\n_**Ridin' Home to Texas**_ see _**Rollin' Home to Texas**_\n\n**3487** _ **Ridin' Law**_ **** Big 4\/Biltmore, 1930. 55 min. D: Harry S. Webb. SC: Carl Krusada. With Jack Perrin, Renee Borden, Yakima Canutt, Jack Mower, Ben Corbett, Robert Walker, Pete Morrison, Fern Emmett, Olive Young. While searching for the killer of his father in Mexico a cowboy is captured by a gang of smugglers. Tacky early talkie.\n\n**3488** _ **Ridin' Mad**_ **** Arrow, 1924. 60 min. D-SC: Jacques Jaccard. With Yakima Canutt, Lorraine Eason, Helen Rosson, Annabelle Lee, Dick LaReno. Forced to kill a man in self defense, a cowboy learns his sister is in love with a crooked oil promoter who plans to leave her. Low grade but fast paced Yakima Canutt silent feature from producer Ben Wilson; Canutt's first starring film.\n\n**3489** _ **Ridin' On**_ **** Reliable, 1936. 60 min. D: Bernard B. Ray. SC: John T. Neville. With Tom Tyler, Germaine Greear (Joan Barclay), Rex Lease, John Elliott, Earl Dwire, Bob McKenzie, Roger Williams, Slim Whitaker, Jimmy Aubrey, Francis Walker, Wally West, Richard Cramer, Milburn Morante, Jack Evans, Chuck Morrison. Two range families engage in a feud but romance complicates things when the son of one family falls for the daughter of the other. Mediocre Tom Tyler outing.\n\n**3490** _ **Ridin' on a Rainbow**_ **** Republic, 1941. 74 min. D: Lew Landers. SC: Bradford Ropes and Doris Malloy. With Gene Autry, Smiley Burnette, Mary Lee, Carol Adams, Ferris Taylor, Georgia Caine, Byron Foulger, Rolf Harolde, James Conlin, Guy Usher, Anthony Warde, Forrest Taylor, Burr Caruth, Ed Cassidy, Ben Hall, Tom London, William V. Mong. After completing a cattle drive, a rancher puts the profits in a bank only to have it robbed and while investigating the holdup he joins a entertainment group aboard a steamboat where an old time performer, suspected in the heist, works. A good script highlights this Gene Autry action musical Western.\n\n**3491** _ **Ridin' the Cherokee Trail**_ **** Monogram, 1941. 60 min. D: Spencer Gordon Bennet. SC: Edmund Kelso. With Tex Ritter, Arkansas Slim Andrews, The Tennessee Ramblers, Betty Miles, Forrest Taylor, Jack Roper, Fred Burns, Bruce Nolan, Gene Alsace, Ed Cassidy, Bob Card, Nolan Willis, Chuck Baldra, Sherry Tansey. A Texas Ranger heads into the Cherokee Strip to stop a crooked empire builder planning to control the land before it is open to settlement. Nicely done Tex Ritter affair with several good songs. British title: _**Cherokee Trail**_.\n\n**3492** _ **Ridin' the Lone Trail**_ **** Republic, 1937. 56 min. D: Sam Newfield. SC: E.B. Mann. With Bob Steele, Claire Rochelle, Charles King, Ernie Adams, Lew Meehan, Julian Rivero, Steve Clark, Hal Price, Frank Ball, Jack Kirk, Horace Murphy, Jack Evans, Bob Roper. A Texan helps a lawman in trying to apprehend a band of road agents who use a ranch owner's daughter's white horse in their killing and robbery sprees. Bob Steele is great in this very exciting and well done film which contains underplayed comedy and well stage fights.\n\n**3493** _ **Ridin' the Outlaw Trail**_ **** Columbia, 1951. 56 min. D: Fred F. Sears. SC: Victor Arthur. With Charles Starrett, Smiley Burnette, Sunny Vickers, Jim Bannon, Pee Wee King and the Golden West Cowboys, Edgar Dearing, Peter Thompson, Lee Morgan, Chuck Roberson, Ethan Laidlaw, Frank McCarroll, Guy Teague. The Durango Kid is after a man who stole gold pieces worth $20,000 but the thief is murdered by a crook who plans to have the gold melted so he can claim it was recently discovered. Fairly complicated, but okay, \"Durango Kid\" series entry.\n\n**3494** _ **Ridin' the Trail**_ **** Arthur Ziehm, 1940. 57 min. D: Raymond K. Johnson. SC: Phil Dunham. With Fred Scott, Iris Lancaster, Harry Harvey, Jack Ingram, John Ward, Bud Osborne, Carl Mathews, Gene Howard, Ray Lenhart, Buddy Kelly, Elias Gamboa, Cactus Mack, Denver Dixon. A masked avenger assists the side of the law when he takes after an outlaw gang. Follow-up to _**Two Gun Troubador**_ (q.v.), this was originally made by Spectrum but the studio folded before its release; fairly good Fred Scott singing oater.\n\n**3495** _ **Ridin' Thru**_ **** Reliable, 1934. 55 min. D: Harry S. Webb. SC: Rose Gordon and Carl Krusada. With Tom Tyler, Ruth Hiatt, Lafe McKee, Philo McCullough, Ben Corbett, Lew Meehan, Bud Osborne, Colin Chase, Jayne Regan, Buck Morgan. Two cowpokes investigate a series of thefts that have forced a cattleman to turn his place into a dude ranch. There is not much to brag about in this below average Tom Tyler vehicle.\n\n**3496** _ **Ridin' Wild**_ **** Aywon, 1925. 48 min. D: Leon De La Mothe. SC: Robert J. Horner and Matilda Smith. With Kit Carson (Boris Bullock), Pauline Curley, Jack Richardson, Walter Maly, C.L. James. Coming West for his health, a man is mistreated by outlaws but befriended by a young woman who he later saves when the gang attacks the stagecoach on which she is a passenger. Tacky silent effort from Robert J. Horner Productions; also called _**Riding Wild**_.\n\n**3497** _ **The Riding Avenger**_ **** Diversion, 1936. 58 min. D: Harry Fraser. SC: Norman Houston. With Hoot Gibson, Ruth Mix, June Gale, Buzz Barton, Stanley Blystone, Roger Williams, Francis Walker, Slim Whitaker, Budd Buster, Blackie Whiteford, Jack Evans, Ed Cassidy, Herman Hack, Tom London, Art Dillard, Allen Greer. Appointed by the governor to round up a notorious rustler and his gang, a marshal takes on the guise of a gunman. Hoot Gibson does his best but poor production values hurt this modest affair.\n\n**3498** _ **Riding for Life**_ **** Rayart, 1926. 50 min. D: Mack V. Wright. SC: Joseph Kane. With Bob Reeves, Aline Goodwin, Hal Walters, Bob Fleming, Bud Pope, Frank Austin, Barney Furey, Andrew Waldron. Falsely accused of robbing an express office safe, a cowboy trails the gang that committed the crime and kidnapped his brother, the depot clerk. Limited action silent Western written by future genre director Joseph Kane.\n\n**3499** _ **Riding High**_ **** Paramount, 1944. 88 min. D: George Marshall. SC: Walter De Leon, Arthur Phillips and Art Arthur. With Dorothy Lamour, Dick Powell, Victor Moore, Gil Lamb, Bill Goodwin, Cass Daley, Rod Cameron, Glenn Langan, Milt Britton and His Band, George Carleton, Andrew Tombes, Douglas Fowley, Pierre Watkin, James Burke, Roscoe Karns, Patricia Mace, Gwen Kenyon, Lorraine Miller, Stanley Andrews, Lane Chandler, Ray Spiker, Charles Soldani, Hal K. Dawson. A mining engineer, trying to capture a gang of counterfeiters, romances a burlesque dancer, the daughter of a local miner. Not one of the better Western musical comedies.\n\n_**Riding Rivals**_ see _**Thundering Through**_\n\n**3500** _ **Riding Shotgun**_ **** Warner Bros., 1954. 75 min. Color. D: Andre De Toth. SC: Tom Blackburn. With Randolph Scott, Wayne Morris, Joan Weldon, Joseph Sawyer, James Millican, Charles (Bronson) Buchinsky, James Bell, Fritz Feld, Richard Garrick, Victor Perrin, John Baer, William Johnstone, Ken Dibbs, Alvin Freeman, Edward Coch, Jr., Lonnie Pierce, Mary Lou Holloway, Boyd \"Red\" Morgan, Richard Benjamin, Jack Kenney, Dub Taylor, Jack Woody, Frosty Royce, Ruth Whitney, Phil Chambers, Clem Fuller, Bud Osborne, Frank Ferguson, Budd Buster, Dick Dickinson, Buddy Roosevelt, Mira McKinney, Carol Henry, Ned Young, Paul Picerni, Evan Lowe, Holly Brooke, Allegra Varron, Jimmy Mobley, George Ross, Maura Murphy, Harry Hines, Ray Bennett, Jock Brockman, Opan Evard, George Selk, Merry Townsend, Morgan Brown, Bob Stephenson. In an attempt to find the outlaw responsible for his wife's death, a man takes the job of stagecoach guard hoping to locate the killer when he pulls a robbery. Well paced and entertaining Randolph Scott vehicle.\n\n**3501** _ **Riding Speed**_ **** Superior, 1934. 50 min. D: Jay Wilsey (Buffalo Bill, Jr.). SC: Delores Booth. With Buffalo Bill, Jr., Joile Benet, Bud Osborne, Lafe McKee, Clyde McClary, Allen Holbrook, Ernest Scott, Denver Dixon. A cowboy opposes a gang of smugglers working along the Mexican border. Rock bottom, but fun, Victor Adamson (Denver Dixon) production, written by Mrs. Adamson. Although star Jay Wilsey is listed as the director, this one has all the looks of an Adamson concoction.\n\n**3502** _ **Riding the California Trail**_ **** Monogram, 1947. 59 min. D: William Nigh. SC: Clarence Upson Young. With Gilbert Roland, Teala Loring, Inez Cooper, Frank Yaconelli, Martin Garralaga, Ted Hecht, Marcelle Grandville, Eva Whitney, Frank Marlo, Alex Montoya, Rosa Turich, Julia Kent, Gerald Echaverria, Tony Roux. The Cisco Kid and his pal Baby try to stop a kindly woman from marrying a man who is secretly being paid by her uncle who is after the family rancho. Typically pleasant \"Cisco Kid\" series adventure.\n\n**3503** _ **Riding the Sunset Trail**_ **** Monogram, 1941. 56 min. D: Robert Emmett Tansey. SC: Robert Emmett (Tansey) and Frances Kavanaugh. With Tom Keene, Betty Miles, Frank Yaconelli, Sugar Dawn, Arkansas Slim Andrews, Kenne Duncan, Tom London, Tom Seidel, James Sheridan (Sherry Tansey), Earl Douglas, Gene Alsace, Fred Hoose. A crook tries to murder his half-brother for the family ranch but a cowboy and his pal find the wounded man and set out to stop the culprit. Film has stronger plot than production values.\n\n**3504** _ **Riding the Wind**_ **** RKO Radio, 1942. 60 min. D: Edward Killy. SC: Morton Grant and Earle Snell. With Tim Holt, Ray Whitley, Lee \"Lasses\" White, Eddie Dew, Mary Douglas, Ernie Adams, Earle Hodgins, Kate Harrington, Charles Phipps, Bud Osborne, Karl Hackett, Hank Worden, Frank McCarroll, Bob Burns, Larry Steers, Spade Cooley. A rancher and his friends help a fellow cattleman trying to build a windmill to provide water for his herd after crooks block his supply with a dam. Pretty good Tim Holt series entry.\n\n**3505** _ **Riding Through Nevada**_ **** Columbia, 1942. 55 min. D: William Berke. SC: Gerald Geraghty. With Charles Starrett, Arthur Hunnicutt, Shirley Patterson, Jimmie Davis and His Rainbow Ramblers, Davison Clark, Clancy Cooper, Minerva Urecal, Edmund Cobb, Ethan Laidlaw, Kermit Maynard, Art Mix, Stanley Brown. A postal inspector investigates a series of stagecoach stickups and takes the job of shotgun guard to try and catch the robbers. Pretty thin Charles Starrett vehicle.\n\n**3506** _ **Riding Tornado**_ **** Columbia, 1932. 59 min. D: D. Ross Lederman. SC: Burt Kempler. With Tim McCoy, Shirley Grey, Wallace MacDonald, Wheeler Oakman, Russell Simpson, Montagu Love, Lafe McKee, Art Mix, Vernon Dent, Bud Osborne, Hank Bell, Silver Tip Baker, Tex Palmer, Artie Ortego. A championship rodeo rider is at odds with a local boss who he believes is the mastermind behind a rustling gang. Very good and action filled Tim McCoy feature.\n\n**3507** _ **Riding West**_ **** Columbia, 1944. 58 min. D: William Berke. SC: Luci Ward. With Charles Starrett, Arthur Hunnicutt, Shirley Patterson, Wheeler Oakman, Clancy Cooper, Steve Clark, Ernest Tubb and His Singing Cowboys (Cal Shrum, Wesley Tuttle, Art Wenzel), Johnny Bond, Stanley Brown, Lloyd Bridges, Tom London, Ted Mapes, Frosty Royce, Blackie Whiteford, Billy Wilkerson, George Fiske. A gambler tires to prevent a man from setting up a Pony Express operation. Good Charles Starrett action film with some pleasing music from Ernest Tubb and Johnny Bond.\n\n_**Riding Wild**_ (1925) see _**Ridin' Wild**_\n\n**3508** _ **Riding Wild**_ **** Columbia, 1935. 57 min. D: David Selman. SC: Ford Beebe. With Tim McCoy, Billie Seward, Niles Welch, Ed Le Saint, Richard Alexander, Richard Botiller, Eddie (Edmund) Cobb, Jack Rockwell, Bud Osborne, Wally West, Al Haskell, Si Jenks, Lafe McKee. A crooked rancher, needing money, sells land to nesters and then tries to run them off but is opposed by a foreman. Good Tim McCoy drama with some brief, but well done, night riding sequences.\n\n**Poster for** _**Riding Wild**_ **(Columbia, 1935).**\n\n**  \n**\n\n**3509** _ **Riding with Buffalo Bill**_ **** Columbia, 1954. 15 Chapters. D: Spencer Gordon Bennet. SC: George H. Plympton. With Marshall Reed, Rick Vallin, Joanne Rio, Shirley Whitney, Jack Ingram, William Fawcett, Gregg Barton, Ed Coch, Steven Ritch, Pierce Lyden, Michael Fox, Lee Roberts, Zon Murray, Al Ferguson, John Truex, Al Cantor, Terry Frost, Ray Jones. Buffalo Bill Cody helps a miner-rancher whose property has been attacked by a gang led by a man trying to stop railroad expansion. Latter day Sam Katzman serial has little to recommend it; for genre fans only.\n\n**3510** _ **Rim of the Canyon**_ **** Columbia, 1949. 70 min. D: John English. SC: John K. Butler. With Gene Autry, Nan Leslie, Thurston Hall, Clem Bevans, Walter Sande, Jock Mahoney, Francis McDonald, Alan Hale, Jr., Amelita Ward, Denver Pyle, Bobby Clark, Boyd Stockman, Sandy Sanders, Rory Mallinson, Frankie Marvin, John R. McKee, Lynn Farr. Gene Autry and his pals try to defend a small town against three ex-convicts who return there for revenge. More than adequate Gene Autry opus.\n\n**3511** _ **Rimfire**_ **** Screen Guild, 1949. 66 min. D: B. Reeves Eason. SC: Arthur St. Clair and Fred Wisbar. With James Millican, Mary Beth Hughes, Henry Hull, Reed Hadley, Fuzzy Knight, Glenn Strange, Chris-Pin Martin, Richard Alexander, George Cleveland, John Cason, Ray Bennett, Margia Dean, I. Stanford Jolley, Victor Kilian, Jason Robards, Don C. Harvey, Lee Roberts, Stanley Price. In Texas after the Civil War a cavalry officer opposes crooked gamblers in a boom town. Fairly interesting action feature with a fine cast.\n\n**3512** _ **Rin Tin Tin:**_ ****_**Hero of the West**_. Monterey Home Video, 1991. 80 min. Color. D: Robert G. Walker, Don McDougall and Douglas Heyes. SC: Douglas Heyes, John O'Dea and Jerry Thomas. With Rin Tin Tin III, James Brown, Lee Aaker, Joseph Sawyer, Rand Brooks, Pierre Watkin, Leo Gordon, Louis Lettieri, Steven Ritch, Norman Frederic (Dean Fredericks), George J. Lewis, Chief Thundercloud, Hal Hopper, Richard Reeves, Francis McDonald, Norman Leavitt, Denver Dixon. Rin Tin Tin comes to the rescue of a wild mustang attacked by a renegade stallion, he and his master Rusty aid an Indian boy endangered by a hostile tribe and then try to stop a buffalo hunter from breaking a treaty. Enjoyable telefilm made up of three colorized episodes of \"The Adventures of Rin Tin Tin\" (ABC-TV, 1954\u201359); James Brown sings \"The White Buffalo\" with vocal effects by Hal Hopper.\n\n_**Ring of Fire**_ see _**Cowboy Up**_\n\n_**Ringo Against Johnny Colt**_ see _**God Holds the Bullet**_\n\n**3513** _ **Ringo and His Golden Pistol**_ **** Metro-Goldwyn-Mayer, 1967. 88 min. Color D: Sergio Corbucci. SC: Adriano Belzoni and Franco Rossetti. With Mark Damon, Valeria Fabrizi, Franco De Rosa, Ettore Manni, Andrea Aureli, John Bartha, Ken Wood (Giovanni Cianfriglia), Giulia Rubini. In Mexico a deadly bounty hunter is on the trail of several outlaws with prices on their heads. No better and no worse than most of its ilk, issued in Italy in 1966 as _**Johnny Oro**_.\n\n**3514** _ **Ringo, Face of Revenge**_ **** Estela Films, 1967. 102 min. Color. D: Mario Caiano. SC: Eduardo Manzanos. With Anthony Steffen, Frank Wolff, Eduardo Fajardo, Armando Calvo, Alejandra Nilo, Alfonso Goda, Manuel Bermudez \"Boliche,\" Amedeo Trilli, Ricardo Canales, Rafael Vaquero, Antonio Orengo. When two cowpokes save a man's life and find half of a treasure map tattooed on his back, the trio try to locate the lawman with the rest of the drawing. Despite an involved plot (but no Ringo), this is only an average Spaghetti Western, a Italian-Spanish co-production released in Europe as _**Los Cuatro Salvajes**_ (The Four Savages) and _**Ringo Volto della Vendetta**_ (Ringo Face of Revenge).\n\n**3515** _ **Ringo the Lone Rider**_ **** Hispamex, 1968. 87 min. Color. D: Rafael Romero Marchent. SC: Mario Caiano. With Peter Martell, Piero Lulli, Dianik Zurakovska, Armando Calvo, Paolo Hertz, Jose Jaspe, Jesus Puente, Antonio Pica, Angel Mendndez, Miguel del Castillo, Jesus Tordesillas, Alfonso Rojas, Francisco (Frank) Brana, Guillermo Mendez, Luis Barboo, Pedro Fenollar, Mario Morales, Antonio Peral, Alfonso de la Vega, Joaquin Burgos, Jose Sepulveda. Two bounty hunters team to capture a gang of marauding former soldiers. Well made and none too violent European Western released there as _**Dos Hombres van a Morir**_ (Two Brothers, One Death) and _**Ringo il Cavaliere Solitario**_ (Ringo the Solitary Cavalier).\n\n**3516** _ **Ringo's Big Night**_ **** Fenix Film, 1965. 95 min. Color. D: Mario Maffei. SC: Emo Bistolfi. With William Berger, Adriana Ambesi, Eduardo Fajardo, Walter Maestosi, Guido Da Salvi, Tom Felleghy, Jorge (George) Rigaud, Jose Bodalo, Armando Calvo, Francisco Moran. A federal agent is arrested along with an outlaw in order to find the hiding lace of $200,000 stolen on its way to Tombstone. Typically violent and action filled Italian oater made as _**Grande Notte di Ringo**_ (Grand Night of Ringo) and _**La Notte del Desperado**_ (The Night of the Desperado); interesting music score by Carlo Rustichelli.\n\n**3517** _ **Rio Bravo**_ **** Warner Bros., 1959. 141 min. Color. D: Howard Hawks. SC: Jules Furthman and Leigh Brackett. With John Wayne, Dean Martin, Ricky Nelson, Walter Brennan, Angie Dickinson, Ward Bond, John Russell, Pedro Gonzalez Gonzalez, Estelita Rodriguez, Claude Akins, Harry Carey, Jr., Malcolm Atterbury, Bob Steele, Bing Russell, Myron Healey, Eugene Iglesias, Fred Graham, Tom Monroe, Riley Hill, Walter Barnes, Sheb Wooley, Joseph Shimada, Chuck Roberson, Dean Smith, Yakima Canutt, George B Bruggerman, Jose Cuchillo, Joe Gray, Gordon Mitchell. A sheriff, aided only by four others, tries to keep a murderer in jail while the prisoner's powerful rancher brother and his hired guns plan to break him out. Classic Western is well worth watching; Walter Brennan steals the show as the cantankerous Stumpy.\n\n**John Wayne and Angie Dickinson in** _**Rio Bravo**_ **(Warner Bros., 1959).**\n\n**  \n**\n\n**3518** _ **Rio Conchos**_ **** 20th Century\u2013Fox, 1964. 107 min. Color. D: Gordon Douglas. SC: Joseph Landon and Clair Huffaker. With Richard Boone, Stuart Whitman, Tony Franciosa, Edmond O'Brien, Wende Wagner, Warner Anderson, Jim Brown, Rodolfo Acosta, Barry Kelley, Vito Scotti, House Peters, Jr., Kevin Hagen, Timothy Carey, Mickey Simpson, Robert Adler, Abel Fernandez. Following the Civil War, four men trek across the Texas desert in search of stolen rifles and are attacked by Indians and outlaws. Well acted and entertaining feature.\n\n**3519** _ **Rio Diablo**_ **** CBS-TV, 1993. 92 min. Color. D: Rod Hardy. SC: Frank Q. Dobbs, David S. Cass, Sr. and Stephen Lodge. With Kenny Rogers, Travis Tritt, Naomi Judd, Brion James, Bruce Greenwood, Laura Harring, Michael G. Hagerty, Luis Contreras, Casey Sander, Stacy Keach, Kelly Junkerman, Marc Alaimo, Tommy Townsend, Arnold Johnson, David S. Cass, Sr., Lupe Ontiveros, Maria Diaz, Richard Yniguez, Richard Curilla, David Samoya, James Crittenden, Kenny Rogers, Jr., Monty L. Simons, Jorge Cervera, Jr. Outlaws kidnap a bride on her wedding day with her husband teaming with a bounty hunter, who is after the marauders, to rescue her. Average TV Western produced by star Kenny Rogers.\n\n**3520** _ **Rio Grande**_ **** Columbia, 1939. 58 min. D: Sam Nelson. SC: Charles Francis Royal. With Charles Starrett, Ann Doran, The Sons of the Pioneers (Bob Nolan, Lloyd Perryman, Pat Brady, Hugh Farr, Karl Farr), Dick Curtis, Hal Taliaferro, Stanley Brown, Hank Bell, Forrest Taylor, Harry Strang, Ed Le Saint, Ed Peil, Sr., Ted Mapes, Art Mix, George Chesebro, Lee Prather, Fred Burns, George Morrell, John Tyrrell. A cowboy and his pals help a young woman being forced off her ranch by land grabbers. Average Charles Starrett action feature.\n\n**3521** _ **Rio Grande**_ **** Astor, 1949. 70 min. D: Norman Sheldon. SC: Hugh Jamison and Norman Sheldon. With Sunset Carson, Evohn Keyes, Lee Morgan, Bobby Clark, Bob Deats, Henry Garcia, Walter Calmbach, Jr., Maria Louisa Marulanda, Don Gray, Curley Rucker. A cowboy helps a rancher and his pretty sister when two crooked brothers attempt to cheat them of their water rights. Sunset Carson's final starring \"B\" Western is rock bottom all the way; filmed in Texas by Lautem Productions.\n\n**3522** _ **Rio Grande**_ **** Republic, 1950. 105 min. D: John Ford. SC: James Kevin McGuinness. With John Wayne, Maureen O'Hara, Ben Johnson, Claude Jarman, Jr., Harry Carey, Jr., Chill Wills, J. Carrol Naish, Victor McLaglen, Grant Withers, Peter Ortiz, Steve Pendleton, Karolyn Grimes, Alberto Morin, Stan Jones, Jack Pennick, Fred Kennedy, The Sons of the Pioneers (Ken Curtis, Hugh Farr, Karl Farr, Lloyd Perryman, Shug Fisher, Tommy Doss), Chuck Roberson, Patrick Wayne, Cliff Lyons, Eve March, Barlow Simpson. A cavalry lieutenant stationed near the Mexican border must deal with raiding Apaches as well as his estranged wife and new recruit son. Entertaining John Ford feature, but not as good as his other cavalry films of the period, _**Fort Apache**_ and _**She Wore a Yellow Ribbon**_ (qq.v.).\n\n**3523** _ **Rio Grande Patrol**_ **** RKO Radio, 1950. 60 min. D: Lesley Selander. SC: Norman Houston. With Tim Holt, Richard Martin, Jane Nigh, Douglas Fowley, Cleo Moore, Tom Tyler, Rick Vallin, John Holland, Larry Johns, Harry Harvey, Forrest Burns. A Border Patrol official learns that two members of the service are in league with Mexican bandits in a gun smuggling scheme. Pretty good Tim Holt vehicle.\n\n**3524** _ **Rio Grande Raiders**_ **** Republic, 1946. 56 min. D: Thomas Carr. SC: Norton S. Parker. With Sunset Carson, Linda Stirling, Bob Steele, Tom London, Tristram Coffin, Edmund Cobb, Jack O'Shea, Tex Terry, Kenne Duncan, Al Taylor, Blackie Whiteford, Bob Burns, Roy Bucko, Frank O'Connor, Bobby Barber. A cowboy tries to help is ex-convict brother who is being used as a pawn in a battle between two stage lines for a mail contract. Sunset Carson's final Republic outing is pretty good, somewhat hampered by stock footage and the casting of the star as Bob Steele's older brother!\n\n**3525** _ **Rio Grande Ranger**_ **** Columbia, 1936. 54 min. D: Spencer Gordon Bennet. SC: Nate Gatzert. With Bob Allen, Iris Meredith, Hal Taliaferro, Paul Sutton, Buzz Henry, John Elliott, Tom London, Slim Whitaker, Jack Rockwell, Richard Botiller, Art Mix, Frank Ellis, Jack Ingram, Al Taylor, Jim Corey, Henry Hall, Jack C. Smith, Ed Cassidy, Ray Jones, Jim Corey, Bud McClure, Jack King, George Plues, Art Dillard. Two Texas rangers are assigned to a border town to round up an outlaw gang terrorizing the area. Bob Allen and Hal Taliaferro (Wally Wales) make a good team in this more than passable action drama; a remake of _**Border Law**_ (q.v.).\n\n**3526** _ **Rio Grande Romance**_ **** Victory, 1936. 70 min. D: Robert Hill. SC: Al Martin. With Eddie Nugent, Maxine Doyle, Fuzzy Knight, Don Alvarado, Nick Stuart, George Walsh, Forrest Taylor, Lucille Lund, Ernie Adams, George Cleveland, Joyce Kay, Ed Cassidy, Ivo Henderson, John Cowell, Richard Cramer. An FBI agent, trying to prove his brother-in-law innocent of murder and bond theft, is on the trail of a gang of crooks. Pleasant, but average, program dual bill item, reissued by Principal Pictures as _**Put on the Spot**_.\n\n**3527** _ **Rio Lobo**_ **** Cinema Center, 1970. 114 min. Color. D: Howard Hawks. SC: Burton Wohl and Leigh Brackett. With John Wayne, Jennifer O'Neill, Jorge Rivero, Jack Elam, Victor French, Christopher Mitchum, Susana Dosamantes, Mike Henry, David Huddleston, Bill Williams, Edward Faulkner, Sherry Lansing, Dean Smith, Robert Donner, Jim Davis, Peter Jason, Robert Rothwell, Chuck Courtney, George Plympton, Bob Steele, Boyd \"Red\" Morgan, Hank Worden, Chuck Roberson, Ethan Wayne, Don \"Red\" Barry, Gregg Palmer, Sondra Currie, Conrad Hool, Lance Hool, John McKee, John Hudkins, William H. O'Brien, Tommy Tedesco. Betrayed during the Civil War, an ex\u2013Union colonel sets out to find the culprits and discovers one of them trying to cheat an old man out of his ranch. Colorful reworking of _**Rio Bravo**_ (q.v.) with good second unit direction by Yakima Canutt.\n\n**3528** _ **Rio Rattler**_ **** Reliable, 1935. 55 min. D: Franklin Shamray (Bernard B. Ray). SC: Carl Krusada. With Tom Tyler, Eddie Gribbon, Marion Shilling, William Gould, Tom London, Charles (Slim) Whitaker, Lafe McKee, Ace Cain, Frank Ellis, Jimmy Aubrey, Philo McCullough, Nelson McDowell, Tom Browner, Sherry Tansey, Buck Morgan, Herman Hack, Barney Beasley, Bob Card, Al Haskell, Blackie Whiteford, Jack Evans, Rube Dalroy, Clyde McClary, John Ince, Fred Parker, S.S. Simon. A cowboy pretends to be a dead lawman only to be accused of his murder by the real killer and his boss. Cheap Tom Tyler affair highlighted by Slim Whitaker's portrayal of the vicious title character.\n\n**3529** _ **Rio Rita**_ **** Radio Pictures, 1929. 135 min. Part-Color. D: Luther Reed. SC: Luther Reed and Russell Mack. With Bebe Daniels, John Boles, Don Alvarado, Bert Wheeler, Robert Woolsey, Dorothy Lee, Georges Renevant, Helen Kaiser, Tiny Sanford, Nick De Ruiz, Sam Nelson, Fred Burns, Eva Rosita, Sam Blum. A gringo wins the love of a beautiful woman but a rival for her affections tells her the American is really a Texas Ranger out to arrest her bandit brother. Dated, but still fun, musical partially filmed in Technicolor.\n\n**3530** _ **Rio Rita**_ **** Metro-Goldwyn-Mayer, 1942. 91 min. D: S. Sylvan Simon. SC: Richard Connell and Gladys Lehman. With Bud Abbott, Lou Costello, Kathryn Grayson, John Carroll, Patricia Dane, Tom Conway, Peter Whitney, Arthur Space, Joan Valerie, Dick Rich, Barry Nelson, Eva Puig, Mitchell Lewis, Eros Volusia, Julian Rivero, Douglass Newland, Lee Murray, Inez Cooper, Frank Penny, William Tannen, David Oliver, Robert Bradford, J.D. Jewkes, Nacho Galindo, Alfredo Garmo, The Guadalajara Trio, Flores Brothers, Tudor Williams, Morton Scott, Mercedes Ruffino, Jenny Mac, Ruth Cherrington, Vangie Beilby. Two pet shop workers get stranded on a ranch being used as the headquarters for Nazi spies. This pleasant Abbott and Costello musical comedy is a remake of the 1929 film (q.v.); better than average.\n\n**3531** _ **Rip Roarin' Buckaroo**_ **** Victory, 1936. 51 min. D: Robert Hill. SC: William Buchanan. With Tom Tyler, Beth Marion, Sammy Cohen, Charles King, Forrest Taylor, Richard Cramer, John Elliott, Theodore Lorch, Wally West, Bud Pope, Wimpy (dog). A boxer, framed in a crooked match, voluntarily leaves the ring and heads West to get the culprits. Very poor production values make this Tom Tyler vehicle a low grade affair.\n\n**3532** _ **Rita of the West**_ **** Euro International Films, 1967. 90 min. Color. D: Ferdinando Baldi. SC: Ferdinando Baldi and Franco Rossetti. With Rita Pavone, Terence Hill, Lucio Dalla, Nina Larker, Teddy Reno, Kirk Morris, Pinuccio Ardia, Gordon Mitchell, Fernando Sancho, Nini Rosso, Luigi Pernice, Romano Puppo, Mirella Pamfili, Franco Gula, Enzo Di Natale, Livio Lorenzon. A young girl wants to destroy all the world's gold because she thinks it is evil and is helped by a friend and an Indian chief. Silly Italian production that mocks Spaghetti Western characters like Django and Ringo; made as _**Little Rita nel Far West**_ (Little Rita of the Far West) and also called _**Crazy Westerners**_.\n\n**3533** _ **Rivales a Muerte**_ (Rivals to Death) **** Laguna Productions, 2003. 90 min. Color. D: Enrique Murillo. SC: Carlos Valdemar. With Alfredo Estrella, Diana Golden, Luis Reynoso, Jorge Aldama, Robert Munguia, Bibelot Mansur, Jose Luis Chavez, Angelica Lara, Fernando Sieber, Eduardo Mendizabal, Jorge Aldama, Jr. Victor Bejarano, Carlos Camacho, Dario Figueroa. A couple begin their married life terrorized by two men and find they must defend themselves to live. Violent Mexican Western.\n\n**3534** _ **River Lady**_ **** Universal-International, 1948. 78 min. Color. D: George Sherman. SC: D.D. Beauchamp and William Bowers. With Yvonne De Carlo, Rod Cameron, Dan Duryea, Helena Carter, Lloyd Gough, Florence Bates, John McIntire, Jack Lambert, Esther Somers, Anita Turner, Edmund Cobb, Dewey Robinson, Eddy Waller, Milton Kibbee, Billy Wayne, Jimmy Ames, Edward Earle, Paul Maxey, Dick Wessel, Charles Sullivan, Mickey Simpson, Reed Howes, George Magrill, Carl Sepulveda, John McGuire, Howard Negley, Charles Wagenheim, Robert Wilke, Perc Launders, Al Hill, Harold Goodwin, Paul Fierro, Beverly Warren, Jack Shutta, Jerry Jerome, Frank Hagney, Philip Van Zandt, Charles Morton, Don MacCracken. The beautiful owner of a Mississippi River gambling ship wants a lumberman who is in love with a timber king's daughter, so to get her man she forms a syndicate to buy all the forest. Colorful, brawling action drama.\n\n_**River of Destiny**_ see _**Forlorn River**_\n\n**3535** _ **River of No Return**_ **** 20th Century\u2013Fox, 1954. 91 min. Color. D: Otto Preminger. SC: Frank Fenton. With Robert Mitchum, Marilyn Monroe, Rory Calhoun, Tommy Rettig, Murvyn Vye, Will Wright, Douglas Spencer, Ed Hinton, Don Beddoe, Claire Andre, Jack Mather, Edmund Cobb, Jarma Lewis, Hal Baylor, Barbara Nichols, Fay Morley, John Doucette, Arthur Shields, Geneva Gray, Larry Chance, Paul Newlan. A beautiful woman hires a man and his young son to take her on a dangerous river voyage in pursuit of her husband and they are tracked by a gambler and attacked by Indians. Director Otto Preminger smartly keeps this picture moving, otherwise it is only average.\n\n_**River of Poison**_ see _**South of Death Valley**_\n\n**3536** _ **The River's Edge**_ **** 20th Century\u2013Fox, 1957. 87 min. Color. D: Allan Dwan. SC: Harold Jacob Smith and James Leicester. With Ray Milland, Debra Paget, Anthony Quinn, Harry Carey, Jr., Chubby Johnson, Byron Foulger, Tom McKee, Frank Gerstle. A crook enlists the assistance of his ex-girlfriend's Mexican farmer husband in helping him cross the border with a cache of stolen money. Sadly underrated melodrama; well worth viewing.\n\n**3537** _ **River's End**_ **** Warner Bros., 1930. 74 min. D: Michael Curtiz. SC: Charles Kenyon. With Charles Bickford, Evelyn Knapp, J. Farrell MacDonald, ZaSu Pitts, Walter McGrail, David Torrence, Junior Coghlan, Tom Santschi, Lionel Belmore, Frank Hagney, Willie Fung, Cliff Saum, Tom London. A man falsely accused of a crime takes the identity of the sheriff sent to capture him and falls in love with the dead man's girlfriend, but trouble ensues. Early talkie still holds good entertainment value, especially for Charles Bickford in dual roles.\n\n**3538** _ **River's End**_ **** Warner Bros., 1940. 69 min. D: Ray Enright. SC: Barry Trivers and Bertram Millhauser. With Dennis Morgan, Elizabeth Earl, Victor Jory, George Tobias, James Stephenson, Steffi Duna, Edward Pawley, John Ridgely, Frank Wilcox, David Bruce, Gilbert Emery, Stuart Robertson, Frank Mayo, Stuart Holmes, Pat O'Malley, Jim Mason, Milton Kibbee, Jack Mower, Glen Cavender, Sailor Vincent, Paul Panzer, Cliff Saum, Jack Wise, Tom Wilson. Falsely convicted of a crime, a man escapes from prison, takes the identity of his dead Mountie brother and tries to find the real killer. Okay remake of the James Oliver Curwood story. TV title: _**Double Identity**_.\n\n**3539** _ **The Road Agent**_ **** Rayart, 1926. 50 min. D: J.P. McGowan. SC: Charles Saxton. With Al Hoxie, Ione Reed, Lew Meehan, Leon de la Mothe, Florence Lee, Cliff Lyons, Frank Ellis. Running from the law, a cowpoke is hired by a crook to impersonate a man about to inherit a ranch. Bottom of the barrel silent feature with Al Hoxie in dual roles.\n\n**3540** _ **Road Agent**_ **** Universal, 1941. 69 min. D: Charles Lamont. SC: Morgan Cox, Arthur Strawn and Maurice Tombragel. With Dick Foran, Leo Carrillo, Andy Devine, Anne Gwynne, Samuel S. Hinds, Richard Davies, Anne Nagel, Morris Ankrum, John Gallaudet, Reed Hadley, Ernie Adams, Lew Kelly, Luana Walters, Chuck Morrison, Jack Rockwell, George J. Lewis, Eddy Waller, Emmett Lynn, William Ruhl, Alan Bridge, Harry Strang, Leyland Hodgson. Three pals arrive in a small town and are promptly jailed on a fake murder charge but they are later released to fight outlaws. Typically fast moving, action laden and slick Universal program Western. Reissued by Realart in 1951 as _**Texas Road Agent**_ ; remade as _**Gunman's Code**_ (q.v.).\n\n**3541** _ **Road Agent**_ **** RKO Radio, 1952. 60 min. D: Lesley Selander. SC: Norman Houston. With Tim Holt, Richard Martin, Noreen Nash, Mauritz Hugo, Dorothy Patrick, Robert Wilke, Tom Tyler, Guy Edward Hearn, William Tannen, Sam Flint, Forbes Murray, Stanley Blystone, Tom Kennedy. When crooks steal money from local citizens, a cowboy takes on the guise of a Robin Hood-type character to right their wrongs. Pretty good Tim Holt vehicle, one of the last of his long running RKO series.\n\n**Leo Carrillo and Andy Devine in** _**Road Agent**_ **(Universal, 1941).**\n\n**  \n**\n\n**3542** _ **The Road to Denver**_ **** Republic, 1955. 90 min. Color. D: Joseph Kane. SC: Horace McCoy and Allen Rivkin. With John Payne, Mona Freeman, Lee J. Cobb, Ray Middleton, Skip Homeier, Andy Clyde, Lee Van Cleef, Karl Davis, Glenn Strange, Buzz Henry, Dan White, Robert Burton, Anne Carroll, Tex Terry, William Haade, Hank Worden, Fred Graham. A stage line operator tries to convince his younger brother the man he works for is a crook with the siblings eventually meeting in a showdown. Well produced drama that provides good entertainment.\n\n**3543** _ **The Road to Fort Alamo**_ **** World Entertainment Corporation, 1966. 82 min. Color. D: John M. Old (Mario Bava). SC: Vincent Thomas, Charles Price and Jane Brisbane. With Ken Clark, Jany Clair, Michel Lemoine, Andreina Paul, Kirk Bert, Antonio Gratoldi, Dean Ardow. Following a disagreement with fellow gang members, an outlaw is left to die in the desert but after his rescue he pretends to be a federal officer only to become a hero when he tires to save a wagon train from Indians. Action filled Spaghetti Western greatly helped by Mario Bava's stylish direction. Issued in Italy in 1965 by Protor\/Piazzi\/Comptori as _**La Strada per Fort Alamo**_ (The Road to Fort Alamo) and released in France as _**Arizona Bill**_.\n\n**3544** _ **Road to Utopia**_ **** Paramount, 1945. 89 min. D: Hal Walker. SC: Norman Panama and Melvin Frank. With Bing Crosby, Bob Hope, Dorothy Lamour, Hillary Brooke, Douglass Dumbrille, Jack LaRue, Robert Barrat, Nestor Paiva, Will Wright, Jimmy Dundee, Billy Benedict, Arthur Loft, Stanley Andrews, Alan Bridge, Romaine Callender, Paul Newlan, Jack Rutherford, Al Hill, Edward Emeron, Ronnie Rondell, Allen Pomeroy, Jack Stoney, George McKay, Larry Daniels, Charles Gemora, Claire James, Maxine Fife, Ferdinand Munier, Edgar Dearing, Charles C. Wilson, Jim Thorpe, Robert Benchley (narrator). Two vaudevillians head to the Klondike where they get mixed up with a gold claim map, crooks after it and a pretty dance hall girl. One of the better \"Road\" series outings.\n\n**3545** _ **Roamin' Wild**_ **** Reliable, 1936. 58 min. D: Bernard B. Ray. SC: Robert Emmett Tansey. With Tom Tyler, Carol Wyndham, Max Davidson, Al Ferguson, George Chesebro, Fred Parker, Slim Whitaker, Bud Osborne, Wally West, Earl Dwire, Lafe McKee, Sherry Tansey, Frank Ellis, John Elliott, Jimmy Aubrey, Buck Morgan. Thieves posing as government men try to bilk miners out of their earnings but a U.S. marshal investigates. Tacky Tom Tyler film with a maximum of outdoor activity supplemented by fights, gunplay, etc., to cover up the lack of script and budget.\n\n**3546** _ **The Roaming Cowboy**_ **** Spectrum, 1937. 60 min. D: Robert Hill. SC: Fred Myton. With Fred Scott, Al St. John, Lois January, Forrest Taylor, Roger Williams, Buddy Cox, Art Miles, George Morrell, George Chesebro, Carl Mathews, Richard Cramer, Lew Meehan, Oscar Gahan, Ed Cassidy, Slim Whitaker, Jack Evans. After finding a rancher murdered and his son orphaned, two cowpokes join an outfit and get involved in a range war caused by a crook out to buy all the area land. Good low budget Fred Scott series film highlighted by his fine singing of several Stephen Foster songs.\n\n**3547** _ **Roar of the Iron Horse**_ **** Columbia, 1950. 15 Chapters. D: Spencer Gordon Bennet. SC: George H. Plympton, Sherman L. Lowe and Royal K. Cole. With Jock (Mahoney) O'Mahoney, Virginia Herrick, William Fawcett, Hal Landon, Jack Ingram, Mickey Simpson, George Eldredge, Myron Healey, Rusty Wescoatt, Frank Ellis, Pierce Lyden, Dick Curtis, Hugh Prosser, Rick Vallin, Bud Osborne, Tommy Farrell, Milton Kibbee, Charles Horvath, Wally West, Knox Manning (narrator). A special investigator from Washington is assigned to find out who is behind a series of construction mishaps on a government financed railroad. Jock Mahoney fans will enjoy this action packed cliffhanger, reworked from _**The Vigilante**_ (q.v.).\n\n**3548** _ **Roarin' Guns**_ **** Puritan, 1936. 60 min. D: Sam Newfield. SC: Joseph O'Donnell. With Tim McCoy, Rosalinda Rice, Wheeler Oakman, Karl Hackett, John Elliott, Tommy Bupp, Jack Rockwell, Lee Meehan, Rex Lease, Frank Ellis, Ed Cassidy, Richard Alexander, Roger Williams, Milburn Morante, Slim Whitaker, Artie Ortego, Wally West, Tex Phelps, Al Taylor, Art Dillard, Hank Bell. A cowboy assists several ranchers being cheated by a group fronting a cattle combine. Low budget but appealing Tim McCoy vehicle.\n\n**Poster for** _**Roarin' Guns**_ **(Puritan, 1936).**\n\n**  \n**\n\n**3549** _ **Roarin' Lead**_ **** Republic, 1936. 57 min. D: Mack V. Wright and Sam Newfield. SC: Oliver Drake and Jack Natteford. With Robert Livingston, Ray Corrigan, Max Terhune, Christine Maple, Hooper Atchley, Yakima Canutt, George Chesebro, Tommy Bupp, Mary Russell, Grace Kern, George Plues, Harry Tenbrook, Newt Kirby, Pascale Perry, Jane Keckley, Tamara Lynn Kauffman, Beverly Luff, Theodore Frye, Katherine Frye, Frank Austin, The Meglin Kiddies, Burr Caruth, Maston Williams, Bob Burns, Murdock MacQuarrie, Forbes Murray, Jack Kirk. A crook uses his outlaw rustling gang like a military unit but their activities are opposed by three cowboys, the trustees of an estate involved with a cattlemen's protection organization and an orphanage. Typically fast paced, entertaining action entry in \"The Three Mesquiteers\" series.\n\n**3550** _ **Roaring Frontiers**_ **** Columbia, 1941. 61 min. D: Lambert Hillyer. SC: Robert Lee Johnson. With Bill Elliott, Tex Ritter, Ruth Ford, Frank Mitchell, Hal Taliaferro, Bradley Page, Tristram Coffin, Francis Walker, Joe McGuinn, George Chesebro, Charles Stevens, Charles King, Lew Meehan, Hank Bell, George Eldredge, Fred Burns, Ernie Adams, Rick Anderson, Steve Clark, Jim Corey, Richard Botiller, George Hazel, Tom Moray, Clem Horton, Earl Gunn, Sammy Stein, Jess Cavin, Tex Cooper. A marshal sent to a town to arrest a cowboy for killing the sheriff ends up saving him from a lynch mob instigated by the real murderer. There is solid entertainment in this \"Wild Bill Hickok\" series entry with songs by Johnny Marvin.\n\n_**Roaring Mountain**_ see _**Thunder Mountain**_ (1935)\n\n**3551** _ **Roaring Ranch**_ **** Universal, 1930. 70 min. D-SC: B. Reeves Eason. With Hoot Gibson, Sally Eilers, Wheeler Oakman, Bobby Nelson, Frank Clark, Leo White, Baby (Marilyn) Walker, Mary Gordon, Bob Burns, Fred Gilman, John Oscar, Jim Corey. A geologist and a rancher both love the same girl but when the former discovers oil on his rival's spread he schemes to get the property cheap. Well done Hoot Gibson (he was also the producer) early talkie and one that should please his fans.\n\n**3552** _ **Roaring Rangers**_ **** Columbia, 1946. 58 min. D: Ray Nazarro. SC: Barry Shipman. With Charles Starrett, Smiley Burnette, Adelle Roberts, Merle Travis and His Bronco Busters (Slim Duncan, Alan Reinhart, Red Murrell), Jack Rockwell, Ed Cassidy, Mickey Kuhn, Edmund Cobb, Ted Mapes, Robert Wilke, Herman Hack, Gerald Mackey, Teddy Infuhr, Roger Williams, John Tyrrell, Nolan Leary, Ethan Laidlaw, Frank Fanning, Frank O'Connor, Jack Kirk, Kermit Maynard, Tommy Coats, Chick Hannon, Carol Henry, Chuck Baldra, George Morrell, Blackie Whiteford, Robert Williams, Tex Harper, Jack Tornek, Lew Morphy, Roy Bucko. Upon the request of a sheriff's young son, the Durango Kid investigates a series of lawless acts and learns the lawman's brother is behind the activities. Pretty good \"Durango Kid\" segment; also called _**False Hero**_.\n\n_**Roaring Rider**_ see _**Wyoming Whirlwind**_\n\n**3553** _ **Roaring Six Guns**_ **** Ambassador, 1937. 55 min. D: J.P. McGowan. SC: Arthur Everett. With Kermit Maynard, Mary Hayes, Sam Flint, John Merton, Budd Buster, Robert Fiske, Ed Cassidy, Curley Dresden, Dick Moorehead, Slim Whitaker, Earle Hodgins, Rene Stone, J.P. McGowan, Oscar Gahan, Bob Woodward. A cattleman falls in love with a neighbor's daughter but the man opposes the match and joins forces with a crook in trying to cause the rancher to lose his government grazing land lease. Standard Kermit Maynard vehicle, but not up to par with his previous starring features.\n\n**3554** _ **Roaring Timber**_ **** Columbia, 1937. 65 min. D: Phil Rosen. SC: Franklin Cosgriff and Robert James Cosgriff. With Jack Holt, Grace Bradley, Ruth Donnelly, Raymond Hatton, Willard Robertson, J. Farrell MacDonald, Charles Wilson, Fred Kohler, Jr., Tom London, Philip Ahn, Ben Hendricks, Ernest Wood. A timber boss struggles to complete a job, in spite of his opposition's sabotage, to get a bonus and win the heart of his pretty boss. Rugged, action filled Jack Holt feature.\n\n**3555** _ **The Roaring West**_ **** Universal, 1935. 15 Chapters. D: Ray Taylor. SC: George Plympton, Nate Gatzert, Basil Dickey, Robert C. Rothafel and Ella O'Neill. With Buck Jones, Muriel Evans, Walter Miller, Frank McGlynn, Sr., Harlan E. Knight, William Desmond, William Thorne, Eole Galli, Pat J. O'Brien, Charles King, Slim Whitaker, Tom London, Edmund Cobb, Dick Rush, Cecil Kellogg, Paul Palmer, Harry Tenbrook, Buffalo Bill, Jr., Tiny Skelton, George Ovey, Fred Humes, Cliff Lyons, John Bose, Lafe McKee, Hank Bell, Horace B. Carpenter, Jack Rockwell, Fred Santley, Bud McClure, Bobby Dunn, Buck Bucko, Roy Bucko. Two men plan to file a claim on a mineral rich area during a land rush but crooks steal their map only to learn the chart is bogus and they begin a reign of terror to find the real one. Fast paced Buck Jones serial, sure to please his fans.\n\n**3556** _ **Roaring Westward**_ **** Monogram, 1949. 55 min. D: Oliver Drake. SC: Ronald Davidson. With Jimmy Wakely, Dub Taylor, Lois Hall, Dennis Moore, Jack Ingram, Claire Whitney, Kenne Duncan, Buddy Swan, Holly Bane, Marshall Reed, Nolan Leary, Bud Osborne, Bob Woodward, Al Haskell, Denver Dixon, Tom Smith. A singing cowboy is after thieves who have stolen money intended for a school sponsored by a sheriff's association. Jimmy Wakely's penultimate series oater is nothing to roar about, despite its title.\n\n**3557** _ **Robbers of the Range**_ **** RKO Radio, 1941. 61 min. D: Edward Killy. SC: Morton Grant and Arthur V. Jones. With Tim Holt, Ray Whitley, Virginia Vale, Emmett Lynn, LeRoy Mason, Howard Hickman, Ernie Adams, Frank LaRue, Ray Bennett, Tom London, Ed Cassidy, Bud Osborne, George Melford, Bud McTaggart, Harry Harvey, Lloyd Ingraham. When a rancher refuses to sell out to a corrupt railroad land agent he is framed on a murder charge but escapes to help a neighbor get the money to pay off his mortgage. Very good Tim Holt film.\n\n**3558** _ **Robbers' Roost**_ **** Fox, 1933. 64 min. D: Louis King. SC: Dudley Nichols. With George O'Brien, Maureen O'Sullivan, Walter McGrail, Reginald Owen, Doris Lloyd, Maude Eburne, Walter Pawley, Ted Oliver, Frank Rice, Bill Nestell, Clifford Santley, Gilbert \"Pee Wee\" Holmes, Vinegar Roan. Following the rustling of cattle, a ranch hand, who has fallen for his boss' sister, comes to suspect the foreman is the culprit. Entertaining George O'Brien action feature, remade in 1955 (q.v.).\n\n**3559** _ **Robber's Roost**_ **** United Artists, 1955. 82 min. Color. D: Sidney Salkow. SC: John O'Dea, Sidney Salkow and Maurice Geraghty. With George Montgomery, Richard Boone, Bruce Bennett, Peter Graves, Sylvia Findley, Warren Stevens, William Hopper, Leo Gordon, Tony Romano, Stanley Clements, Joe Bassett, Leonard Geer, Al Wyatt, Boyd \"Red\" Morgan. Two outlaw gangs are hired by a handicapped cattle baron to protect his valuable range land. Standard redo of the 1933 (q.v.) feature, supposedly based on Zane Grey.\n\n**3560** _ **Robbery Under Arms**_ **** Rank, 1958. 99 min. Color. D: Jack Lee. SC: Alexander Baron and W.P. Lipscomb. With Peter Finch, Ronald Lewis, Maureen Swanson, David McCallum, Jill Ireland, Laurence Naismith, Vincent Ball, Dudy Nimmo, Jean Anderson, Ursula Finlay, Johnny Dascell, Larry Taylor, Russell Napier, Yvonne Buckingham, George Cormack, Doris Goddard, Johnny Cadell, Max Wagner, Bill Pepper, Edna Morris, Bartlett Mullins, Colin Ballantyne, Ewen Solon, S. Scrutton, Robert Reardon, Pat Hagen, Sergeant Holmes, John Hargreaves, Ivor Broley, Rita Ponsford, Laune Pumpa, Phillipa Morgan. In 1870s frontier Australia two brothers become involved in cattle rustling while romancing pretty sisters. Pleasantly paced British production filmed in Australia, based on Rolf Boldrewood's novel.\n\n**3561** _ **Robin Hood of El Dorado**_ **** Metro-Goldwyn-Mayer, 1936. 86 min. D: William A. Wellman. SC: William A. Wellman, Joseph Calleia and Melvin Levy. With Warner Baxter, Ann Loring, Bruce Cabot, Margo, J. Carrol Naish, Soledad Jiminez, Carlos De Valdez, Eric Linden, Edgar Kennedy, Charles Trowbridge, Harvey Stephens, Ralph Remley, George Regas, Harry Woods, Francis McDonald, Kay Hughes, Paul Hurst, Boothe Howard, Lou Yaconelli (Earl Douglas), J.P. McGowan, Jason Robards, Marc Lawrence, Pedro de Cordoba, Harold Goodwin, Frank Campeau, Frank Hagney, Al Ferguson, George MacQuarrie, Cully Richards, Tom Moore, Nigel De Brulier, Lee Shumway, Frank Yaconelli, Richard Botiller, Sam Ash, Hank Bell, Si Jenks, Carlotta Monti, Lee Phelps, Morgan Wallace, Ben Taggart, Bob Burns, G. Pat Collins, Pedro Regas, Lew Harvey, Ivan Miller, Herbert Heywood, Richard Cramer, Joe Dominguez, Nick De Ruiz, Duke Green. A Mexican farmer becomes a notorious bandit and rebel leader after his wife dies from a beating when they are thrown off their land. Well made and exciting, if a bit romantic, account of the life of Joaquin Murieta.\n\n**3562** _ **Robin Hood of Monterey**_ **** Monogram, 1947. 57 min. D: Christy Cabanne. SC: Bennett Cohen. With Gilbert Roland, Chris-Pin Martin, Evelyn Brent, Jack LaRue, Pedro de Cordoba, Donna (Martell) De Mario, Travis Kent, Thornton Edwards, Nestor Paiva, Ernie Adams, Julian Rivero, Alex Montoya, Fred Cordova, Felipe Turich, George Navarro. The Cisco Kid and Pancho help a young man accused of killing his father, a Spanish rancher. Star Gilbert Roland is credited with additional dialogue in this credible \"Cisco Kid\" outing that includes fine villainous work by Evelyn Brent and Jack LaRue.\n\n**3563** _ **Robin Hood of Texas**_ **** Republic, 1947. 71 min. D: Lesley Selander. SC: John Butler and Earle Snell. With Gene Autry, Lynne Roberts, Adele Mara, Sterling Holloway, The Cass County Boys (Jerry Scoggins, Bert Dodson, Fred Martin), James Cardwell, John Kellogg, Ray Walker, Michael Brandon, Paul Bryar, James Flavin, Dorothy Vaughn, Stanley Andrews, Alan Bridge, Hank Patterson, Edmund Cobb, Lester Dorr, William Norton Bailey, Irene Mack, Eva Novak, Frankie Marvin, Billy Wilkerson, Kenneth Terrell, Joe Yrigoyen. Gene Autry and his pals fix up an old spread and turn it into a dude ranch while helping the sheriff capture a bank robbery gang. Very pleasant and entertaining Gene Autry opus.\n\n**3564** _ **Robin Hood of the Pecos**_ **** Republic, 1941. 59 min. D: Joseph Kane. SC: Olive Cooper. With Roy Rogers, George \"Gabby\" Hayes, Marjorie Reynolds, Cy Kendall, Leigh Whipper, Sally Payne, Eddie Acuff, Robert Strange, Jay Novello, William Haade, Roscoe Ates, Jim Corey, Chick Hannon, Art Mix, Bob Burns, Ted Mapes, Frank McCarroll, Al Taylor, Chuck Baldra. With a tyrant running the territory in post\u2013Civil War Texas, the citizens band together and form a group of masked night riders to combat the carpetbaggers harassing them. Nicely done pseudo-historical Roy Rogers film.\n\n**3565** _ **Robin Hood of the Range**_ **** Columbia, 1943. 57 min. D: William Berke. SC: Betty Burbridge. With Charles Starrett, Kay Harris, Arthur Hunnicutt, The Jimmy Wakely Trio (Jimmy Wakely, Johnny Bond, Dick Reinhart), Stanley Brown, Kenneth MacDonald, Douglas Drake, Bud Osborne, Ed Peil, Sr., Frank LaRue, Frank McCarroll, Ray Jones, Merrill McCormick, Steve Clark, Hal Price, Herman Hack, Frank O'Connor, Ernie Adams, John Tyrrell, Jessie Arnold, Bobby Larson, Marcia Raport, Cara Raport, Bessie Ward. The mysterious Vulcan rides to the aid of homesteaders about to lose their homes to the railroad. Action filled Charles Starrett vehicle.\n\n**3566** _ **El Robo al Tren Correo**_ (The Mail Train Robbery) **** Alameda Films, 1964. 90 min. D: Chano Ureta. SC: Ramon Obon. With Carlos Cortes, Noe Murayama, Julissa, Bertha Moss, Ines Murillo. To prove he did not steal a gold shipment a cowboy must do battle with an outlaw who is also after his girl. Pretty good Mexican Western.\n\n**3567** _ **Rock Island Trail**_ **** Republic, 1950. 90 min. D: Joseph Kane. SC: James Edward Grant. With Forrest Tucker, Adele Mara, Adrian Booth, Bruce Cabot, Chill Wills, Jeff Corey, Grant Withers, Barbara Fuller, Roy Barcroft, Pierre Watkin, Valentine Perkins, Jimmy Hunt, Olin Howlin, Dick Curtis, Sam Flint, John Holland, Emory Parnell, Richard Alexander, William Haade, Dick Elliott, Jack Pennick, Billy Wilkerson, Kate Lawson. A railroad engineer, attempting to move his line out of Illinois, battles competition from a rival stagecoach operation. Fairly good Republic \"A\" film elevated by a typically fine studio cast; issued in England as _**Transcontinent Express**_.\n\n**3568** _ **Rock River Renegades**_ **** Monogram, 1942. 59 min. D: S. Roy Luby. SC: John Vlahos and Earle Snell. With Ray Corrigan, John King, Max Terhune, Christine McIntyre, John Elliott, Weldon Heyburn, Kermit Maynard, Frank Ellis, Carl Mathews, Richard Cramer, Tex Palmer, Hank Bell, Budd Buster, Steve Clark. Three pals help a sheriff fight a band of mysterious road agents as well as the saloon owner after his girl. One of the lesser \"Range Busters\" outings due to a lack of plot development.\n\n**3569** _ **Rockin' in the Rockies**_ **** Columbia, 1945. 69 min. D: Vernon Keays. SC: J. Benton Cheney and John Gray. With Mary Beth Hughes, The Three Stooges (Moe Howard, Larry Fine, Jerry \"Curly\" Howard), Jay Kirby, Gladys Blake, Tim Ryan, Vernon Dent, The Hoosier Hot Shots (Gabe Ward, Paul \"Hezzie\" Triesch, Ken Triesch, Frank Kettering), The Cappy Barra Boys, Spade Cooley, Forrest Taylor, Jack Clifford, Steve Clark, Snub Pollard, Hal Price, John Tyrrell, James T. \"Bud\" Nelson, Louis Manley, Edward Howard, Eddie Bruce, Tex Williams, Deuce Spriggins, Smokey Rogers, Johnny Weiss, Joaquin Murphy, Spike Featherstone. A rancher trying to sell his spread gets mixed up with three zanies and show business folk trying to make it to Broadway. Broad musical Western farce, but lots of fun.\n\n**3570** _ **Rockwell**_ **** Inspired Corporation, 1994. 105 min. Color. D-SC: Robert Lloyd Dewey. With Randy Gleave, Scott Christopher, Michael Rudd, Linda Gilbert, Shantal Hiatt, Scott M. Milias, George Sullivan, Laves C. Williams, Scott Claflin, Michael Flynn, Karl Malone, Meilani Paul, Paul Mugerias, John Bozung. After Mormon leader Joseph Smith is murdered, his friend goes West, become a lawman and takes revenge on claim jumpers who killed others in the sect. Tacky, R-rate fare re-titled _**Rockwell\u2014A Legend of the Wild West**_ for video.\n\n_**Rockwell\u2014A Legend of the Wild West**_ see _**Rockwell**_\n\n**3571** _ **Rocky**_ **** Monogram, 1946. 76 min. D: Phil Karlson. SC: Jack DeWitt. With Roddy McDowall, Edgar Barrier, Nita Hunter, Gale Sherwood, Jonathan Hale, William Ruhl, Claire Whitney, Irving Bacon, John Alvin, Ben Corbett. The adventures of a young boy and his dog in the early days of the West. Pleasing program feature with star Roddy McDowall and editor Ace Herman the associate producers.\n\n**3572** _ **Rocky Mountain**_ **** Warner Bros., 1950. D: William Keighley. SC: Winston Miller and Alan Le May. With Errol Flynn, Patrice Wymore, Scott Forbes, Guinn Williams, Dick Jones, Howard Petrie, Slim Pickens, Chubby Johnson, Buzz Henry, Sheb Wooley, Peter Coe, Rush Williams, Steve Dunhill, Alex Sharp, Yakima Canutt, Nakai Snez. A Confederate officer and his men are ordered to get outlaws on their side for the South to get control of California. Errol Flynn's final Western is a fairly good one with nice locales and some well staged action sequences.\n\n**3573** _ **Rocky Mountain Mystery**_ **** Paramount, 1935. 63 min. D: Charles Barton. SC: Edward E. Paramore, Jr. and Ethel Doherty. With Randolph Scott, Charles \"Chic\" Sale, Mrs. Leslie Carter, Kathleen Burke, George Marion, Ann Sheridan, James C. Eagles, Howard Wilson, Willie Fung, Florence Roberts. A novice lawman joins forces with an aged sheriff to solve several murders at an isolated radium mine. A spooky setting adds zest to this nicely entertaining \"B\" effort allegedly based on Zane Grey's _Golden Dreams_ ; it affords a chance to see the famous stage actress Mrs. Leslie Carter, in a surprisingly villainous role. Alternate titles: _**The Fighting Westerner**_ and _**Vanishing Frontier**_.\n\n**3574** _ **Rocky Mountain Rangers**_ **** Republic, 1940. 58 min. D: George Sherman. SC: Barry Shipman and Earle Snell. With Robert Livingston, Raymond Hatton, Duncan Renaldo, Rosella Towne, Sammy McKim, LeRoy Mason, Pat O'Malley, Dennis Moore, John St. Polis, Robert Blair, Burr Caruth, Jack Kirk, Hank Bell, Budd Buster, Bud Osborne, Mary MacLaren, Kernan Cripps, Brandon Beach, Ted Mapes, Carey Loftin, Fred Burns, Frank Ellis, Silver Tip Baker, Frankie Marvin, Chuck Baldra, Tommy Coats, Curley Dresden, Buck Morgan, Pascale Perry, Lew Morphy, Vinegar Roan, Tex Harper, Augie Comez, Pat McKee, Bill Nestell, Betty Roadman. Three Texas Rangers are on the trail of a notorious outlaw gang working the Panhandle and they are joined by a teenage boy, the survivor of a wagon raid. Okay action entry in \"The Three Mesquiteers\" series, but nothing special.\n\n**3575** _ **Rocky Rhodes**_ **** Universal, 1934. 64 min. D: Al Raboch. SC: Edward Churchill. With Buck Jones, Sheila Terry, Stanley Fields, Walter Miller, Alf P. James, Paul Fix, Lydia Knott, Lee Shumway, Jack Rockwell, Al Ferguson, Carl Stockdale, Monte Montague, Bud Osborne, Harry Samuels, Bob Reeves. A cowboy and a Chicago hoodlum team to stop land grabbers in a small Arizona town. Nice blend of action and comedy make this a very good Buck Jones outing; reissued by Realart.\n\n**3576** _ **Rodeo**_ **** Monogram, 1952. 70 min. D: William Beaudine. SC: Charles R. Marion. With John Archer, Jane Nigh, Wallace Ford, Gary Gray, Frances Rafferty, Sara Haden, Frank Ferguson, Myron Healey, Fuzzy Knight, Robert Karnes, Jim Bannon, I. Stanford Jolley, Ann Doran, Russell Hicks, Milton Kibbee, Dave Willock. When dishonest promoters run out on a rodeo owing her father money, a young woman takes over the show and makes it a success. Competently made and entertaining \"B\" drama.\n\n**3577** _ **Rodeo Girl**_ **** CBS-TV, 1980. 100 min. Color. D: Jackie Cooper. SC: Kathryn Powers. With Katharine Ross, Bo Hopkins, Candy Clark, Jacqueline Brookes, Wilford Brimley, Parley Baer, Elise Caitlin, Savannah Bentley, Nancy Priddy, Buchlind Beery, Dee Croton, Arlene Banar, June Evett, Pamela Earnhardt, Bob Tallman, Lex Connelly. Bored with her rodeo husband's way of life, a woman goes into business for herself by forming an all-female troupe and becomes a world champion. Enjoyable TV flick based on the actual experiences of rodeo queen Sue Pirtle.\n\n**3578** _ **Rodeo King and the Senorita**_ **** Republic, 1951. 67 min. D: Philip Ford. SC: John K. Butler. With Rex Allen, Mary Ellen Kay, Buddy Ebsen, Roy Barcroft, Tristram Coffin, Bonne DeSimone, Don Beddoe, Jonathan Hale, Harry Harvey, Rory Mallinson, Joe Forte, Buff Brady. A rodeo star tries to help an attractive woman save her show from a crooked partner trying to bankrupt it. Enjoyable remake of _**My Pal Trigger**_ (q.v.).\n\n_**Rodeo Racketeers**_ see _**The Man from Utah**_\n\n**3579** _ **Rodeo Rhythm**_ **** Producers Releasing Corporation, 1942. 68 min. D: Fred Newmeyer. SC: Gene Tuttle and Eugene Allen. With Fred Scott, The Red Knapp Rough Riders, Patricia Redpath, Lori Bridge, Pat Dunn, Jack Cooper, John Frank, Doc Hartley, Landon Laird, Raylene Smith, Vernon Brown, Donna Lee Meinke, Gloria Morse. In danger of losing their home, a group of youngsters appear in a rodeo to get the money to save the orphanage. Although top billed Fred Scott sings a few songs, this Kansas City filmed outing is basically a vehicle for a troupe of rodeo kids.\n\n**3580** _ **The Rogue and the Grizzly**_ **** American National Enterprises, 1982. 96 min. Color. D: Kent Bateman and Dick Robinson. SC: Ken Bateman and James Bryan. With Dick Robinson, Don Shanks, Garlan Wilde, Carol Elasz, Tom Drury. In 1855 a New England man travels to the High Sierras where he lives a rugged life, eventually adopting an Indian boy and a bear cub. Okay family adventure yarn with eye catching scenery.\n\n**3581** _ **Rogue of the Range**_ **** Supreme, 1936. 60 min. D: S. Roy Luby. SC: Earle Snell. With Johnny Mack Brown, Lois January, Phyllis Hume, Alden (Stephen) Chase, George Ball, Jack Rockwell, Horace Murphy, Frank Ball, Lloyd Ingraham, Fred Hoose, Forrest Taylor, George Morrell, Blackie Whiteford, Slim Whitaker, Tex Palmer, Horace B. Carpenter, Max Davidson, Art Dillard, Wally West, Slim Whitaker, Oscar Gahan, Herman Hack. A Secret Service agent, after rescuing a woman preacher from a runaway wagon, hires out as a gunman to get the goods on outlaws and ends up romancing a saloon girl. A meandering plot detracts from this Johnny Mack Brown entry from producer A.W. Hackel.\n\n**3582** _ **Rogue of the Rio Grande**_ **** Sono Art-World Wide, 1930. 70 min. D: Spencer Gordon Bennet. SC: Oliver Drake. With Jose Bohr, Raymond Hatton, Myrna Loy, Carmelita Geraghty, Walter Miller, Gene Morgan, William P. Burt, Florence Dudley, Fred Parker, Merrill McCormick, Jack Clifford, Blackjack Ward, Fred Burns, Tex Phelps. A dashing Mexican bandit, El Malo, falls in love with a pretty cantina dancer and she urges him to reform. Dated, labored early talkie that includes the song \"Argentine Moon.\"\n\n**3583** _ **Rogue River**_ **** Eagle-Lion, 1950. 84 min. Color. D: John Rawlins. SC: Louis Lantz. With Rory Calhoun, Peter Graves, Ellye Marshall, Frank Fenton, Ralph Sanford, George Stern, Roy Engel, Jane Liddell, Robert Rose, Stephen Roberts, Duke York. A state policeman and his no account cousin get involved in a robbery. Somewhat talkative but none-the-less well sustained melodrama.\n\n**3584** _ **Roll Along, Cowboy**_ **** 20th Century\u2013Fox\/Principal, 1937. 57 min. D: Gus Meins. SC: Dan Jarrett. With Smith Ballew, Cecilia Parker, Stanley Fields, Gordon (Bill) Elliott, Wally Albright, Ruth Robinson, Frank Milan, Monte Montague, Bud Osborne, Harry Bernard, Budd Buster, Syd Saylor, Herman Hack, Frank Ellis, Buster Fite and His Six Saddle Tramps. A cowboy inherits a ranch, falls in love with a pretty girl and is plagued by rustlers. Smith Ballew's second series film is none too good, a poor remake of _**The Dude Ranger**_ (q.v.).\n\n_**Roll, Covered Wagon**_ see _**Roll Wagon Roll**_\n\n**3585** _ **Roll On, Texas Moon**_ **** Republic, 1946. 68 min. D: William Witney. SC: Paul Gangelin and Mauri Grashin. With Roy Rogers, George \"Gabby\" Hayes, Dale Evans, Dennis Hoey, Elisabeth Risdon, Bob Nolan and The Sons of the Pioneers (Lloyd Perryman, Pat Brady, Shug Fisher, Hugh Farr, Karl Farr), Francis McDonald, Edward Keane, Kenne Duncan, Harry Strang, Lee Shumway, Tom London, Ed Cassidy, Steve Darrell, Pierce Lyden. Roy Rogers is sent by a cattle syndicate to stop a long time feud between ranchers and sheepherders. Complicated plot, but fairly interesting oater with lots of action.\n\n**3586** _ **Roll, Thunder, Roll**_ **** Eagle-Lion, 1949. 58 min. Color. D: Lewis D. Collins. SC: Paul Franklin. With Jim Bannon, Little Brown Jug (Don Kay Reynolds), Nancy Gates, Marin Sais, Emmett Lynn, Glenn Strange, I. Stanford Jolley, Lee Morgan, Lane Bradford, Steve Pendleton, George Chesebro, Charles Stevens, William Fawcett, Rocky Shanhan, Carol Henry, Jack O'Shea, Dorothy Latta, Joe Green, Frank Ellis, Frank O'Connor, Fess Reynolds. Bandits try to blame El Coujo, a Mexican Robin Hood, for a series of robberies but Red Ryder believes he is innocent. Okay \"Red Ryder\" segment.\n\n**3587** _ **Roll Wagons Roll**_ **** Monogram, 1939. 52 min. D: Al Herman. SC: Victor Adamson (Denver Dixon). With Tex Ritter, Muriel Evans, Nelson McDowell, Tom London, Nolan Willis, Steve Clark, Reed Howes, Frank Ellis, Kenne Duncan, Frank LaRue, Chick Hannon, Charles King, Gene Alsace, Denver Dixon. An Army officer, trying to find out who is supplying Indians with weapons, joins a wagon train headed for Oregon. Action filled Tex Ritter film; British title: _**Roll, Covered Wagon**_.\n\n**3588** _ **Rollin' Home to Texas**_ **** Monogram, 1940. 53 min. D: Al Herman. SC: Robert Emmett (Tansey). With Tex Ritter, Virginia Carpenter, Arkansas Slim Andrews, Eddie Dean, Cal Shrum and His Rhythm Rangers, I. Stanford Jolley, Harry Harvey, Gene Alsace, John Rutherford, Olin Francis, Sherry Tansey, Charles Phipps, Minta Durfee. Two cowboys are asked by a prison warden to find out how convicts are escaping to pull off area robberies. Okay Tex Ritter action feature called _**Ridin' Home to Texas**_ in England.\n\n**3589** _ **Rollin' Plains**_ **** Grand National, 1938. 57 min. D: Al Herman. SC: Lindsley Parsons and Edmond Kelso. With Tex Ritter, Horace Murphy, Snub Pollard, Harriet Bennet, Hobart Bosworth, Edward (Ed) Cassidy, Karl Hackett, Charles King, Ernest (Ernie) Adams, Lynton Brent, Horace B. Carpenter, The Beverly Hill Billies, Hank Worden, Augie Gomez, Robert Walker, Oscar Gahan, Clyde McClary, Bud Pope, Carl Mathews, Jack Hendricks, Denver Dixon, Johnny Luther, George Morrell, Ralph Bucko, Roy Bucko. A ranger tries to halt a range war between sheep and cattle men that is secretly promoted by a town boss and his hired gun. Standard Tex Ritter affair highlighted by a great title song and the comedy sidekick antics of Snub Pollard and Horace Murphy.\n\n_**Rollin' West**_ see _**Rollin' Westward**_\n\n**3590** _ **Rollin' Westward**_ **** Monogram, 1939. 55 min. D: Al Herman. SC: Fred Myton. With Tex Ritter, Dorothy Fay, Horace Murphy, Slim Whitaker, Herbert Corthell, Harry Harvey, Charles King, Hank Worden, Dave O'Brien, Tom London, Bob Terry, Rudy Sooter, Estrellita Novarro. A singing cowboy opposes the unfair tactics of a land baron trying to force small ranchers off the range. Fair Tex Ritter outing, issued as _**Rollin' West**_ in Great Britain.\n\n**3591** _ **Rolling Caravans**_ **** Columbia, 1938. 55 min. D: Joseph Lovering. SC: Nate Gatzert. With Jack Luden, Eleanor Stewart, Harry Woods, Slim Whitaker, Lafe McKee, Buzz Barton, Bud Osborne, Richard Cramer, Jack Rockwell, Franklyn Farnum, Cactus Mack, Tex Palmer, Sherry Tansey, Oscar Gahan, Curley Dresden, Horace Murphy, Francis Walker, Harry Harvey, Tuffy (dog). Pioneers plan to settle a new area but outlaws try to stop them and a cowboy comes to the rescue. Flat oater with a bland hero (who even uses a drab ventriloquist doll) although Eleanor Stewart is a fetching heroine and Harry Woods a dastardly villain.\n\n**3592** _ **Rolling Down the Great Divide**_ **** Producers Releasing Corporation, 1942. 59 min. D: Peter Stewart (Sam Newfield). SC: George Milton (Milton Raison and George W. Sayre). With Bill \"Cowboy Rambler\" Boyd, Art Davis, Lee Powell, Wanda McKay, Glenn Strange, Karl Hackett, J. Merrill (Jack) Holmes, Ted Adams, Jack Ingram, John Elliott, George Chesebro, Horace B. Carpenter, Jack Roper, Curley Dresden, Dennis Moore, Tex Palmer, Hank Bell, Blackie Whiteford. A marshal is helped by two musical cowboys in trying to find a short wave station employed by rustlers in their illegal operations. Tattered entry in the \"Frontier Marshals\" series.\n\n**3593** _ **Rolling Home**_ **** Screen Guild, 1946. 69 min. D-SC: William Berke. With Jean Parker, Russell Hayden, Pamela Blake, Raymond Hatton, Jo Ann Marlowe, Buzz Henry, James Conlin, William Farnum, Jonathan Hale, Milton Parsons, Elmo Lincoln, Jimmie Dodd, Harry Carey, Jr., Andre Charlot. A minister about to lose his church befriends an old cowboy and his grandson, along with their lame horse. Basically an equestrian drama this film includes scenes of Raymond Hatton riding the range and Jimmie Dodd singing a campfire cowboy song; very pleasant.\n\n_**The Romance of Lone Valley**_ see _**The Grub Stake**_\n\n**3594** _ **Romance of the Redwoods**_ **** Columbia, 1939. 67 min. D: Charles Vidor. SC: Michael L. Simmons. With Charles Bickford, Jean Parker, Alan Bridge, Gordon Oliver, Alan Mowbray, Lloyd Hughes, Pat O'Malley, Anne Shoemaker, Marc Lawrence, Don Beddoe, Earl Gunn. Two loggers both love the camp's pretty dishwasher but when one of them is killed under mysterious circumstances the other is blamed. Fair program feature from a Jack London story, with fine photography.\n\n**3595** _ **Romance of the Rio Grande**_ **** Fox, 1929. 90 min. D: Alfred Santell. SC: Marion Orth. With Warner Baxter, Mona Maris, Antonio Moreno, Mary Duncan, Robert Edeson, Agostino Borgato, Albert Roccardi, Soledad Jiminez, Majel Coleman, Charles Byer, Merrill McCormick. Following an attack by bandits, a railroad construction supervisor is injured and taken to the home of his grandfather, who he has always disliked, and there he meets a girl with whom he falls in love. Dated early talkie.\n\n**3596** _ **Romance of the Rio Grande**_ **** 20th Century\u2013Fox, 1941. 73 min. D: Herbert I. Leeds. SC: Harold Shumate and Samuel G. Engel. With Cesar Romero, Patricia Morison, Ricardo Cortez, Lynne Roberts, Chris-Pin Martin, Pedro de Cordoba, Richard Lane, Ray Bennett, Trevor Bardette, Joseph McDonald, Aldrich Bowker, Inez Palange, Tom London, Eva Puig, Francis Ford. The Cisco Kid impersonates the nephew of an aged ranch owner whose property is coveted by the young man's conniving fiancee and her lover. Average \"Cisco Kid\" series outing with good work by Patricia Morison, as the deceptive bride-to-be, and Lynne Roberts playing the rancho owner's ward.\n\n**3597** _ **Romance of the Rockies**_ **** Monogram, 1937. 53 min. D: Robert North Bradbury. SC: Robert Emmett (Tansey). With Tom Keene, Beryl Wallace, Don Orlando, Bill Cody, Jr., Franklyn Farnum, Earl Dwire, Russell Paul, Steve Clark, Jim Corey, Tex Palmer, Jack C. Smith, Blackie Whiteford, Frank Ellis, Blackjack Ward, Oscar Gahan, Denver Dixon. In cattle country a newcomer doctor gets mixed up in a battle over water rights. Good Tom Keene vehicle; well made.\n\n**3598** _ **Romance of the Wasteland**_ **** Aywon, 1924. 50 min. D: Victor Adamson (Denver Dixon). SC: Milburn Morante. With Art Mix, Alma Rayford, Wilbur McGaugh, Virginia Marshall, Clifford Davidson, Princess Neola. A cowboy finds a lost little girl and takes care of her until he locates her mother. Standard and somewhat sentimental low grade silent oater starring the great Art Mix (George Kesterson).\n\n**3599** _ **Romance of the West**_ **** Producers Releasing Corporation, 1946. 58 min. D: Robert Emmett (Tansey). SC: Frances Kavanaugh. With Eddie Dean, Joan Barton, Emmett Lynn, Bob McKenzie, Forrest Taylor, Jerry Jerome, Stanley Price, Chief Thundercloud, Don Reynolds, Laurie Harrison, Al Ferguson, John Carpenter, Rocky Camron, Lee Roberts, Don Williams, Jack Richardson, Matty Roubert, Forbes Murray, Jack O'Shea, Lee Bennett, Tex Cooper. An Indian agent investigates renegade attacks and learns that outlaws are encouraging them in order to steal tribal lands. Average Eddie Dean vehicle with some good songs.\n\n**3600** _ **Romance on the Range**_ **** Republic, 1942. 63 min. D: Joseph Kane. SC: J.Benton Cheney. With Roy Rogers, George \"Gabby\" Hayes, Linda Hayes, Bob Nolan and The Sons of the Pioneers (Tim Spencer, Lloyd Perryman, Pat Brady, Hugh Farr, Karl Farr), Sally Payne, Ed Pawley, Hal Taliaferro, Harry Woods, Glenn Strange, Roy Barcroft, Jack Kirk, Jack O'Shea, Dick Wessel, Richard Alexander, Chester Conklin, Art Mix, Monte Montague, Frank Brownlee, Selmer Jackson, Henry Wills, Jack Montgomery. A singing cowboy is on the trail of an outlaw gang leader and finds out he is after a supposed highly respectable citizen. Typical Roy Rogers opus with the advantage of more action than music or romance.\n\n**3601** _ **Romance Rides the Range**_ **** Spectrum, 1936. 59 min. D: Harry Fraser. SC: Tom Gibson. With Fred Scott, Marion Shilling, Cliff Nazarro, Buzz Barton, Robert Kortman, Theodore Lorch, Frank Yaconelli, William (Steele) Steurer, Wally West, Horace B. Carpenter, Oscar Gahan, George Morrell, Allen Greer, Jack Evans, Phil Dunham, Carl Mathews, Ed Carey. An opera star goes West and stops crooks from swindling a young woman and her brother out of their money. Pleasing Fred Scott outing with the star singing \"Only You\" and the villains more inept than evil.\n\n**3602** _ **Rooster Cogburn**_ **** Universal, 1975. 107 min. Color. D: Stuart Millar. SC: Martin Julien (Martha Hyer). With John Wayne, Katharine Hepburn, Anthony Zerbe, Richard Jordan, John McIntire, Paul Koslo, Strother Martin, Jack Colvin, Jon Lormer, Richard Romancito, Lane Smith, Warren Vanders, Jerry Gatlin, Mickey Gilbert, Chuck Hayward, Gary McLarty, Tommy Lee. An aging missionary teams with a hard-drinking, one-eyed lawman to find the outlaws who killed her father. This sequel to _**True Grit**_ (1969) [q.v.] is not nearly as good but the teaming of John Wayne and Katharine Hepburn, plus some fine action sequences, makes it entertaining.\n\n**John Wayne and Katharine Hepburn in** _**Rooster Cogburn**_ **(Universal, 1975).**\n\n**  \n**\n\n**3603** _ **Rootin' Tootin' Rhythm**_ **** Republic, 1937. 60 min. D: Mack V. Wright. SC: Jack Natteford. With Gene Autry, Smiley Burnette, Armida, Monte Blue, Ann Pendleton, Hal Taliaferro, Charles King, Max Hoffman, Jr., Frankie Marvin, Nina Campana, Charles Mayer, Karl Hackett, Jack Rutherford, Henry Hall, Curley Dresden, Art Davis, Al Clauser and His Oklahoma Outlaws, Milburn Morante, George Morrell, Pascale Perry. The head of a cattlemen's association clandestinely heads an outlaw gang but his secret is uncovered by a singing cowboy and his partner. Standard Gene Autry vehicle with more emphasis on music than action.\n\n**3604** _ **The Rope and the Colt**_ **** Fono Roma\/Copernic, 1969. 87 min. Color. D: Robert Hossein. SC: Dario Argento, Claude Desailly and Robert Hossein. With Michele Mercier, Robert Hossein, Lee Burton (Guido Lollobrigida), Daniele Vargas, Serge Marquand, Pierre Hatet, Philippe Baronnet, Pierre Collet, Ivano Staccioli, Beatrice Altariba, Michel Lemoine, Anne-Marie Balin, Angel Alvarez, Simon Arriaga, Charley Bravo, Cris Huerta, Benito Stefanelli, Alvaco de Luna, Lorenzo Robeldo, Jose Canalejas. A gunman is urged by his former lover to take part in a kidnapping so she can get revenge against her husband's killer. Offbeat French-Italian co-production filmed in Spain as _**Un Corde, Un Colt**_ (A Rope, a Colt); highly visual with lots of desert scenery and little dialogue. Title song sung by Scott Walker. ****\n\n**3605** _ **Rose Hill**_ **** CBS-TV, 1997. 90 min. Color. D: Christopher Cain. SC: Earl W. Wallace. With Jennifer Garner, Jeffrey D. Sams, Zak Orth, Justin Chambers, Tristan Tait, David Newsom, Casey Siemaszko, Stuart Wilson, Kristin Griffin, Courtney Chase, Michael Alexander Jackson, David Klein, Kevin Zegers, Blair Slater, Vanya Rose, David Aaron Baker, Vera Farmiga, Peggy Ann Adams, Carmen Moore, Addison Bell, James MacDonald, Donovan Workun. A young woman learns that as an infant she was taken West by four orphan teenage boys she thought were her brothers and she goes back East to find her heritage not knowing her family is coming apart. Enjoyable TV Western.\n\n**3606** _ **Rose Marie**_ **** Metro-Goldwyn-Mayer, 1936. 113 min. D: W.S. Van Dyke. SC: Frances Goodrich, Albert Hackett and Alice Duer Miller. With Jeannette MacDonald, Nelson Eddy, Allan Jones, Gilda Gray, Reginald Owen, James Stewart, George Regas, Robert Freig, Una O'Connor, Lucien Littlefield, Alan Mowbray, David Niven, Herman Bing, James Conlin, Dorothy Gray, Mary Anita Loos, Aileen Carlyle, Halliwell Hobbes, Paul Porcasi, Edgar Dearing, Pat West, David Clyde, Russell Hicks, Milton Owen, Rolfe Sedan, Jack Pennick, Leonard Carey, Major Sam Harris, Jim Mason, Agostino Borgato, Fred Graham, Lee Phelps, David Robel, Rinaldo Alacorn. An opera singer arrives in the northwest to see her brother who has committed a murder and is pursued by a Royal Mounted Policeman with whom she falls in love. Classic Rudolf Friml-Otto A. Harbach-Oscar Hammerstein II operetta is successfully brought to the screen with Jeannette MacDonald and Nelson Eddy singing their famous duet, \"Indian Love Call.\" TV title: _**Indian Love Call**_.\n\n**3607** _ **Rose Marie**_ **** Metro-Goldwyn-Mayer, 1954. 104 min. Color. D: Mervyn LeRoy. SC: Ronald Miller and George Froeschel. With Ann Blyth, Howard Keel, Fernando Lamas, Bert Lahr, Marjorie Main, Joan Taylor, Ray Collins, Chief Yowlachie, James Logan, Turl Ravenscroft, Abel Fernandez, Billy Dix, Al Ferguson, Frank Hagney, Marshall Reed, Sheb Wooley, Dabbs Greer, John Pickard, John Damler, Sally Yarnell, Gordon Richards, Lumsden Hare, Mickey Simpson, Paul Lanzi. A Mountie tries to civilize a wild backwoods Canadian girl and ends up vying for her affections with an adventurer. Glossy screen remake of the aging operetta with the songs better than the plot.\n\n**Nelson Eddy and Jeanette MacDonald in** _**Rose Marie**_ **(Metro-Goldwyn-Mayer, 1936).**\n\n**  \n**\n\n**3608** _ **Rose of Cimarron**_ **** 20th Century\u2013Fox, 1952. 77 min. Color. D: Harry Keller. SC: Maurice Geraghty. With Mala Powers, Jack Buetel, Bill Williams, Jim Davis, Dick Curtis, Lane Bradford, William Phipps, Bob Steele, Alex Gerry, Lillian Bronson, Art Smith, Monte Blue, Argentina Brunetti, John Doucette, Byron Foulger, Kenneth MacDonald, Irving Bacon, George Chandler, William Schallert, Frank Ferguson, Wade Crosby, Tommy Cook, William Fawcett, Charles Stevens, Tom Monroe, Tom Steele. A young woman raised by Indians plans to take revenge on the white outlaws who killed her foster parents. Fair action film with a good cast.\n\n**3609** _ **Rose of the Rancho**_ **** Paramount, 1936. 85 min. D: Marion Gering. SC: Frank Partos, Charles Brackett, Arthur Sheekmand and Nat Perrin. With Gladys Swarthout, Jon Boles, Charles Bickford, Willie Howard, Benny Baker, Grace Bradley, H.B. Warner, Charlotte Granville, Don Alvarado, Herb Williams, Minor Watson, Louise Carver, Pedro de Cordoba, Paul Harvey, Arthur Aylesworth, Harry Woods, Russell Hopton, Charles Middleton, Robert Kortman, Merrill McCormick, James Marcus, Harry Semels, Ernie Adams, Robert E. Homans, Edgar Dearing, Russ Powell, Jack Norton, Eddie Dunn, Nelson McDowell, Eddie Borden, Lester Sharpe, Sam Lufkin, Edwin J. Brady, Charles Stevens, Frank Lackteen, Olin Francis, Jules Cowles, Sam Appel, Nick Thompson, Evelyn Selbie, Eleanor Virzie, Lew Kelly, Sam Blum, Redmond Flood, S.S. Simon, Paul Sotoff, Ivan Christy, Lillian Pearl, Jack Perry, Harry Lamont, John Nasborough, George Bookasta, Lalo Encinas, Charles Morris. In Old California a government agent is assigned to capture a masked guerrilla leader who in reality is the daughter of a nobleman working against a land grabber. Pleasant operetta more for music fans than Western addicts; filmed in 1914 by Paramount starring Bessie Barriscale, J.W. Johnston, Jane Darwell, Dick La Reno and Monroe Salisbury, produced, directed and scripted by Cecil B. DeMille.\n\n_**Rose of the Rio Grande**_ (1932) see _**God's Country and the Man**_ (1932)\n\n**3610** _ **Rose of the Rio Grande**_ **** Monogram, 1938. 60 min. D: William Nigh. SC: Dorothy (Davenport) Reid and Ralph Bettinson. With John Carroll, Movita, Antonio Moreno, Don Alvarado, Lina Basquette, George Cleveland, Duncan Renaldo, Gino Corrado, Martin Garralaga, Rosa Turich, Carlos Villarias. A man takes on the guise of El Gato, a bandit, in order to find those who murdered his family. Mediocre costume feature with John Carroll as a singing (dubbed) cowboy.\n\n**3611** _ **Rose of the Yukon**_ **** Republic, 1948. 61 min. D: George Blair. SC: Norman S. Hall. With Steve Brodie, Myrna Dell, William Wright, Benny Baker, Emory Parnell, Jonathan Hale, Gene Gary, Dick Elliott, Francis McDonald, Lotus Long, Wade Crosby, Eugene Sigaloff, Rex Lease, House Peters, Jr., Stanley Blystone, Charles Soldani, Brandon Beach, Charles Griffin. While prospecting for gold two men are framed on a murder charge after they find a rich pitchblende deposit. Fair action drama.\n\n**3612** _ **Rough Night in Jericho**_ **** Universal, 1967. 104 min. Color. D: Arnold Laven. SC: Sidney Boehm and Marvin H. Albert. With Dean Martin, Jean Simmons, George Peppard, John McIntire, Slim Pickens, Don Galloway, Brad Weston, Richard O'Brien, Carol Anderson, John Napier, Steve Sandor, Warren Vanders, Med Flory, Dean Paul Martin, Kenner G. Kemp, Ray Kellogg, Sydney Smith, Wallace Earl. After a former lawman takes over a town and proves to be ruthless, he is forced into a showdown by a female stage line operator. Very violent affair with only average entertainment value.\n\n**3613** _ **Rough Riders of Cheyenne**_ **** Republic, 1945. 56 min. D: Thomas Carr. SC: Elizabeth Beecher. With Sunset Carson, Peggy Stewart, Mira McKinney, Monte Hale, Wade Crosby, Michael Sloane, Kenne Duncan, Tom London, Eddy Waller, Jack O'Shea, Robert Wilke, Tex Terry, Jack Rockwell, Rex Lease, Hank Bell, Henry Wills, Cactus Mack, Artie Ortego, Jack Luden, Carl Mathews. A mysterious figure starts a feud between two families so he can grab their ranches for a cattle rustling scheme once they have wiped each other out. Fair Sunset Carson vehicle with a good script.\n\n**3614** _ **Rough Riders of Durango**_ **** Republic, 1951. 60 min. D: Fred C. Brannon. SC: M. Coates Webster. With Allan \"Rocky\" Lane, Aline Towne, Walter Baldwin, Steve Darrell, Ross Ford, Denver Pyle, Stuart Randall, Tom London, Hal Price, Russ Whiteman, Dale Van Sickel, Bob Burns. A special courier comes to the rescue when outlaws hijack ranchers' grain shipments and money. Some well staged fights, but productions values were declining in this \"Famous Westerns\" entry which utilizes lots of stock footage.\n\n**3615** _ **Rough Riders Roundup**_ **** Republic, 1939. 58 min. D: Joseph Kane. SC: Jack Natteford. With Roy Rogers, Mary Hart, Raymond Hatton, Eddie Acuff, Edward Pawley, Dorothy Sebastian, George Meeker, Guy Usher, Duncan Renaldo, George Chesebro, Glenn Strange, Jack Rockwell, Jack Kirk, Hank Bell, Dorothy Christy, Fred Kelsey, Eddy Waller, John Merton, George (Montgomery) Letz, Frank Ellis, Frank McCarroll, Dan White. A band of ex\u2013Rough Riders regroup to combat a gang involved in a gold shipment robbery. Plenty of action and a good story make this a fine Roy Rogers entry.\n\n**3616** _ **Rough Ridin' Justice**_ **** Columbia, 1945. 58 min. D: Derwin Abrahams. SC: Elizabeth Beecher. With Charles Starrett, Dub Taylor, Betty Jane Graham, Jimmy Wakely and His Oklahoma Cowboys, Wheeler Oakman, Jack Ingram, Forrest Taylor, Jack Rockwell, Edmund Cobb, Dan White, Robert Kortman, George Chesebro, Steve Clark, Kermit Maynard, Ethan Laidlaw, Bud Osborne, Robert Ross, Don Weston. The leader of a gang harassing cattlemen is hired by the ranchers when they are up against outlaws. Sturdy Charles Starrett film; well done.\n\n**3617** _ **Rough Riding Ranger**_ **** Superior, 1935. 57 min. D: Elmer Clifton. SC: Elmer Clifton and George M. Merrick. With Rex Lease, Janet Chandler, Bobby Nelson, Yakima Canutt, Mabel Strickland, David Horsley, George Chesebro, William Desmond, Robert Walker, Carl Mathews, Artie Ortego, Allen Greer, George Morrell, Milburn Morante, Johnny Luther's Cowboy Band, Jack Kirk, Jack Evans, Clyde McClary, Cactus Mack. A cowpoke tries to help a family being harassed by a mysterious letter writer and outlaw attacks. Low grade Rex Lease vehicle.\n\n**3618** _ **Rough Romance**_ **** Fox, 1930. 55 min. D: A.F. Erickson. SC: Elliott Lester and Donald Davis. With George O'Brien, Helen Chandler, Antonio Moreno, Roy Stewart, Harry Cording, David Hartford, Eddie Borden, Noel Francis, Frank Lanning, John Wayne, Elliott Lester. In the Oregon timber country a lumberjack fights with a notorious outlaw over the girl he loves. Poorly done George O'Brien early talkie interspersed with musical interludes.\n\n**3619** _ **The Rough, Tough West**_ **** Columbia, 1952. 54 min. D: Ray Nazarro. SC: Barry Shipman. With Charles Starrett, Smiley Burnette, Jack (Jock) Mahoney, Carolina Cotton, Pee Wee King and the Golden West Cowboys, Marshal Reed, Fred F. Sears, Bert Arnold, Tommy Ivo, Boyd \"Red\" Morgan, Valerie Fisher, Tommy Kingston, Redd Stewart, Hank Garland, Ethan Laidlaw, Bob Woodward, Ben Corbett, Frank Ellis. When his pal, the local saloon owner, makes a man the town's sheriff he beings to suspect his friend may he part of a scheme to cheat miners out of their property. Fast moving and action filled \"Durango Kid\" segment.\n\n**3620** _ **Roughnecks**_ **** Metromedia, 1980. 240 min. Color. D: Bernard McEveety. SC: Michael Michaelian. With Vera Miles, Steve Forrest, Stephen McHattie, Ana Alicia, Wilford Brimley, Cathy Lee Crosby, Kevin Geer, Harry Morgan, A. Martinez, Andrew Rubin, Sam Melville, Sara Rush, Timothy Scott, Louise Heath, William Marquez, Rockne Tarkington, Janice Carroll. Conflict ensues when an Oklahoma woman rancher in need of money permits oil drilling on her land. Overlong, padded TV drama.\n\n**3621** _ **Roughshod**_ **** RKO Radio, 1949. 88 min. D: Mark Robson. SC: Geoffrey Homes and Hugo Butler. With Robert Sterling, Gloria Grahame, Claude Jarman, Jr., John Ireland, Jeff Donnell, Myrna Dell, Martha Hyer, George Cooper, Jeff Corey, Sara Haden, James Bell, Sean McClory, Robert B. Williams, Steve Savage, Ed Cassidy. A farmer, in love with a saloon girl, fears three outlaws on the run from the law are after him. Pretty good action drama.\n\n**3622** _ **The Rounders**_ **** Metro-Goldwyn-Mayer, 1965. 85 min. Color. D-SC: Burt Kennedy. With Glenn Ford, Henry Fonda, Sue Anne Langdon, Hope Holiday, Chill Wills, Edgar Buchanan, Kathleen Freeman, Joan Freeman, Denver Pyle, Barton MacLane, Doodles Weaver, Allegra Varron, Warren Oates, Chuck Roberson. Two aging cowboys attempt to tame a horse and two equally wild women. Very pleasant genre action comedy.\n\n**3623** _ **Rounding Up the Law**_ **** Aywon, 1922. 50 min. D: Charles R. Seeling. SC: W.H. Allen. With Guinn Williams, Russell Gordon, Patricia Palmer, Chet Ryan, William McCall. A cowboy wins a sheriff's ranch in a poker game with the lawman and his crooked pal planning to run the cowpoke out of the country. Low budget Guinn \"Big Boy\" Williams offering; it should please his fans.\n\n**3624** _ **The Roundup**_ **** Paramount, 1941. 90 min. D: Lesley Selander. SC: Harold Shumate. With Richard Dix, Patricia Morison, Preston Foster, Don Wilson, Ruth Donnelly, Betty Brewer, Douglass Dumbrille, Jerome Cowan, William Haade, Morris Ankrum, Clara Kimball Young, Dick Curtis, Weldon Heyburn, Lane Chandler, Lee \"Lasses\" White, The King's Men. A rancher plans to marry the woman he loves only to find her ex-lover, who she thought was dead, has returned on their wedding day. None too action filled romantic oater, filmed first in 1920 with Roscoe \"Fatty\" Arbuckle.\n\n**3625** _ **Roundup Time in Texas**_ **** Republic, 1937. 58 min. D: Joseph Kane. SC: Oliver Drake. With Gene Autry, Smiley Burnette, Maxine Doyle, LeRoy Mason, Buddy Williams, Earle Hodgins, Dick Wessel, Cornie Anderson, Frankie Marvin, Ken Cooper, Elmer Fain, Al Ferguson, Slim Whitaker, Al Knight, Carleton Young, Jack C. Smith, Jim Corey, Jack Kirk, George Morrell, The Cabin Kids. Two cowboys take a horse herd to South Africa where they discover a mine and a smuggling operation. Offbeat plot adds some color to this Gene Autry musical.\n\n**3626** _ **Rovin' Tumbleweeds**_ **** Republic, 1939. 64 min. D: George Sherman. SC: Betty Burbridge, Dorrell McGowan and Stuart McGowan. With Gene Autry, Smiley Burnette, Mary Carlisle, Douglass Dumbrille, Pals of the Golden West, William Farnum, Lee \"Lasses\" White, Ralph Peters, Victor Potel, Jack Ingram, Sammy McKim, Gordon Hart, Horace Murphy, Fred \"Snowflake\" Toones, Forrest Taylor, Reginald Barlow, Eddie Kane, Guy Usher, David Sharpe, Jack Kirk, Bob Burns, Art Mix, Horace B. Carpenter, Frank Ellis, Fred Burns, Ed Cassidy, Tom Chatterton, Crauford Kent, Maurice Costello, Charles K. French, Lee Shumway, Bud Osborne, Harry Semels, Chuck Morrison, Rose Plummer, Nora Lou Martin, Hal Taliaferro. Corrupt politicians fail to pass a flood control bill and a deluge causes great damage to farm land with singer Gene Autry being elected to Congress to get the legislation pushed into law. A very good Gene Autry film that has the star getting married at the finale; it introduced Autry's theme song, \"Back in the Saddle Again,\" which he co-wrote with Ray Whitley, and contains a takeoff on the popular \"Lum 'n Abner\" radio program. Also called _**Washington Cowboy**_.\n\n_**A Roving Rogue**_ see _**Outlaws of the Rockies**_\n\n**3627** _ **The Rowdy Girls**_ **** Troma Entertainment, 2000. 86 min. Color. D: Steve Nevius. SC: India Allen and Khara Bromiley. With Shannon Tweed, Julie Strain, Deanna Brooks, Richie Varga, Daniel Henry Murray, Todd Eckert, Rick Williams, Mink Stole, David Wilson, Sita Thompson, Gonzalo Menendez, Eduardo Rodriguez, Myla Martin, Tom Poster, Mark C. Adams, Julie Nevius, Bobby Weinberg, Brad Buckman, Chanda Fuller, Tracy Cranshaw. Three women, a robber, a sharpshooter and a runaway wife, are kidnapped by an outlaw gang and must fight for their freedom. Somewhat amusing R-rated effort.\n\n**3628** _ **Roy Colt and Winchester Jack**_ **** Libert, 1970. 90 min. Color. D: Mario Bava. SC: Mario Di Nardo and Roberto Agrin. With Brett Halsey, Marilu Tolo, Charles Southwood, Teodoro Corra, Lee Burton (Guido Lollobrigida), Rick Boyd (Federico Boido), Bruno Corazzari, France Pesce. Two friends vie for the leadership of their outlaw gang and when one wins the other becomes a lawman who must hunt down his former comrades after they team with a renegade searching for a treasure map. A violent Spaghetti Western with good direction by Mario Bava; issued in Europe as _**Roy Colt e Winchester Jack**_ (Roy Colt and Winchester Jack) by P.A.C.\/Tigielle 33.\n\n**3629** _ **The Royal Mounted Patrol**_ **** Columbia, 1941. 59 min. D: Lambert Hillyer. SC: Winston Miller. With Charles Starrett, Russell Hayden, Wanda McKay, Donald Curtis, Lloyd Bridges, Kermit Maynard, Evan Thomas, Ted Adams, Harrison Greene, Ted Mapes, George Morrell. Two Mounties both fall for the same gal, a teacher at a remote post who is the sister of a corrupt lumber camp boss. The initial teaming of Charles Starrett and Russell Hayden is a likable effort.\n\n**3630** _ **The Royal Mounted Rides Again**_ **** Universal, 1945. 13 Chapters. D: Ray Taylor and Lewis D. Collins. SC: Joseph O'Donnell, Tom Gibson and Harold C. Wire. With Bill Kennedy, Daun Kennedy, George Dolenz, Paul E. Burns, Milburn Stone, Robert Armstrong, Danny Morton, Addison Richards, Tom Fadden, Joseph Haworth, Helen Bennett, Joseph Crehan, Selmer Jackson, Daral Hudson, George Lloyd, George Eldredge, Rondo Hatton, Richard Alexander, Lane Chandler, Guy Wilkerson, William Haade. A Royal Canadian Mounted Police officer is assigned to find out who murdered a mill owner and discovers his mining operator father is the chief suspect. Fair cliffhanger.\n\n**3631** _ **The Royal Rider**_ **** First National, 1929. 67 min. D: Harry Joe Brown. SC: Nate Gatzert. With Ken Maynard, Olive Hasbrouck, Philippe De Lacey, Theodore Lorch, Joseph Burke, Harry Semels, William (Billy) Franey, Frank Rice, Bobby Dunn, Johnny Sinclair, Ben Corbett. Members of a Wild West show become palace guards for a Balkan boy king and help him put down a revolt. Ken Maynard silent (a version was issued with music and sound effects) with a Graustarkian background finds the star a bit out of place although the action is plentiful.\n\n**3632** _ **Ruby Jean and Joe**_ **** Showtime, 1996. 99 min. Color. D: Geoffrey Sax. SC: James Lee Barrett. With Tom Selleck, Rebekah Johnson, JoBeth Williams, Ben Johnson, Eileen Seeley, John Diehl, Margo Martindale, Larry Soller, Robert Guajardo, Robert Starr, Forrie J. Smith, Ed Adams, Darwin Hall, Warner McKay, Boots Southerland, Glen Gold, Shawn Howell, Shane McCabe, Stan Sessums, Bob Tallman. An aging rodeo star meets a pretty hitchhiker and despite age and personality differences they develop a close relationship. Okay contemporary Western drama co-produced by star Tom Selleck.\n\n**3633** _ **Rugged Gold**_ **** The Family Channel, 1994. 120 min. Color. D: Michael Anderson. SC: Sarah James. With Jill Eikenberry, Art Hindle, Ari Magder, Graham Greene, Davina Whitehouse, Tony Groser, June Bishop, Helen Moulder, Sam Tyson-Hogg, Christopher Douglas. A widow with a young son remarries and travels with her new husband to Alaska to pan for gold and when the boy disappears his stepfather goes looking for him while she must face delivering their new baby alone. Enjoyable family fare filmed in Canada.\n\n**3634** _ **Ruggles of Red Gap**_ **** Paramount, 1935. 90 min. D: Leo McCarey. SC: Walter De Leon, Harlan Thomson and Humphrey Pearson. With Charles Laughton, Charles Ruggles, Mary Boland, ZaSu Pitts, Roland Young, Leila Hyams, Maude Eburne, Lucien Littlefield, Leota Lorraine, James Burke, Dell Henderson, Richard Cezon, Brenda Fowler, Augusta Anderson, Sarah Edwards, Clarence H. Wilson, Rafael Storm, George Burton, Victor Potel, Frank Rice, William J. Welsh, Lee Kohlmar, Harry Bernard, Alice Ardell, Rolfe Sedan, Jack Norton, Willie Fung, Libby Taylor, Armand Kaliz, Henry Roquemore, Heinie Conklin, Ed Le Saint, Charles Fallon, Isabella La Mal, Ernie Adams, Frank O'Connor, Ian Welch, Genaro Spangoli, Albert Petit, Carrie Daumery. An uncouth Western family wins a debonair British butler in a poker game and he makes a big change in their lives. Very pleasant comedy; Charles Laughton is grand in the title role. Filmed in 1918 by Essanay and in 1923 by Paramount and remade in 1950 as _**Fancy Pants**_ (q.v.).\n\n_**The Rumpo Kid**_ see _**Carry on Cowboy**_\n\n**3635** _ **Run, Cougar, Run**_ **** Buena Vista, 1973. 100 min. Color. D: Jerome Courtland. SC: Louis Pelletier. With Stuart Whitman, Alfonso Arau, Harry Carey, Jr., Douglas V. Fowley, Frank Aletter, Lonny Chapman. An easy going sheepherder opposes a hunter out to kill a cougar. Fine Walt Disney production originally telecast on NBC-TV.\n\n**3636** _ **Run for Cover**_ **** Paramount, 1955. 92 min. Color. D: Nicholas Ray. SC: Winston Miller. With James Cagney, John Derek, Viveca Lindfors, Jean Hersholt, Grant Withers, Jack Lambert, Ernest Borgnine, Irving Bacon, Trevor Bardette, Ray Teal, John Miljan, Denver Pyle, Emerson Tracey, Gus Schilling, Phil Chambers, Harold Kennedy, Joe Hayworth, Henry Wills. A former outlaw becomes the sheriff of a frontier town, much to the chagrin of his young friend. Rather interesting psychological Western from action producers William H. Pine and William C. Thomas.\n\n**3637** _ **Run for the Hills**_ **** Realart, 1953. 72 min. D: Lew Landers. SC: Leonard Neubrauner. With Sonny Tufts, Barbara Payton, John Harmon, Mauritz Hugo, Vick Raaf, Jack Wrightson, Paul Maxey, John Hamilton, Byron Foulger, Charles Victor, William Fawcett, Ray Parsons, Jean Willes, Richard Benedict, Michael Fox. An insurance actuary becomes worried about an H-bomb attack with he and his wife taking refuge in a cave. A silly script and unfunny situations make this a dud.\n\n**3638** _ **Run Home Slow**_ **** Emerson, 1965. 66 min. D: Tim Sullivan (Ted Brenner). SC: Donald Ceveris. With Mercedes McCambridge, Linda Gaye Scott, Allan Richards, Gary Kent, Jim Hogan, Ted Brenner, Brian Casey, Leah Cooper, John \"Bud\" Cardos, Jeff Masters, Pat Raines, Jesse Bates. To avenge their father's hanging, a woman organizes her family into a brutal outlaw gang. Obscure feature, with music score by Frank Zappa, which will not appeal to the average genre viewer.\n\n**3639** _ **Run, Man, Run**_ **** Adria\/Compagnie Francaise, 1968. 120 min. Color. D: Sergio Sollima. SC: Fabrizio De Angelis and Sergio Sollima. With Tomas Milian, Donald O'Brien, John Ireland, Linda Veras, Marco Guglilmi, Jose Torres, Edward Ross (Luciano Rossi), Nello Pazzafini, Gianni Rizzo, Dan May (Dante Maggio), Umberto Di Grazia, Noe Murayama, Attilio Dottesio, Orso Maria Guerrini, Federico Boldo, Calisto Calisti, Chelo Alonso, Goeffredo Unger, Joe Marco, Pietro Tordi, Osiride Pevarello, Ricardo Palacios. Three million dollars in Mexican revolutionary gold is sought by several gunman plus a petty criminal. Average follow-up to _**The Big Gundown**_ (q.v.); a French-Italian co-production filmed as _**Corri, Uomo, Corri**_ (Run, Man, Run) and also called _**Big Gundown 2**_.\n\n**3640** _ **Run of the Arrow**_ **** RKO Radio\/Universal-International, 1957. 85 min. D-SC: Samuel Fuller. With Rod Steiger, Sarita Montiel, Brian Keith, Ralph Meeker, Jay C. Flippen, Charles Bronson, Tim McCoy, Olive Carey, H.M. Wynant, Neyle Morrow, Frank DeKova, Stuart Randall, Frank Warner, Billy Miller, Chuck Hayward, Chuck Roberson, Carleton Young, Don Orlando, Bill White, Jr., Frank Baker, Emile Avery, Tex Holden, Roscoe Ates, Frank O'Connor, Ray Stevens. An embittered Confederate soldier joins the Sioux Indians in their fight against the U.S. government. Rather strange psychological Western with solid performances to give it strength. Angie Dickinson dubbed star Sarita Montiel.\n\n_**Run or Burn**_ see _**White-Water Sam**_\n\n**3641** _ **Run, Simon, Run**_ **** ABC-TV, 1970. 74 min. Color. D: George McGowan. SC: Lionel E. Siegel. With Burt Reynolds, Inger Stevens, Royal Dano, James Best, Rodolfo Acosta, Don Dubbins, Joyce Jameson, Barney Phillips, Herman Rudin, Eddie Little Sky, Marsha Moore, Ken Lynch, Martin G. Soto, Rosemary Eliot. Falsely sent to prison, a Papago Indian returns home after a decade to find his brother's murderer. Fairly good drama made for television. Video title: _**Savage Run**_.\n\n**3642** _ **Run to the High Country**_ **** Sun International, 1972. 97 min. Color. D-SC: Keith Larsen. With Erik Larsen, Keith Larsen, Karen Steele, Alvin Redmond, Rodney Burt. A young boy tries to protect wildlife from hunters. Filmed on location in Utah, this feature is short on plot but heavy on scenery.\n\n**3643** _ **The Runaway Barge**_ **** NBC-TV, 1975. 75 min. Color. D: Boris Sagal. SC: Stanford Whitmore. With Bo Hopkins, Tim Matheson, Jim Davis, Nick Nolte, Devon Ericson, Christina Hart, James Best, Lucille Henson, Clifton James, Dom Plumley, Beau Gibson, Bill Rowley. Three boatmen try to make a living on a modern-day Mississippi riverboat and find themselves involved in a hijacking and kidnapping. Okay action drama made for the small screen.\n\n**3644** _ **Running Target**_ **** United Artists, 1956. 83 min. Color. D: Marvin R. Weinstein. SC: Marvin R. Weinstein, Jack Couffer and Conrad Hall. With Arthur Franz, Doris Dowling, Richard Reeves, Myron Healey, James Parnell, Charles Delaney, Gene Roth, James Anderson, Frank Richards. Four escaped convicts head into the Colorado Rockies pursued by a sheriff and his posse. Standard action yarn with the plot twist of having the lawman opposed to killing.\n\n**3645** _ **Running Wild**_ **** Golden Circle, 1973. 104 min. Color. D: Robert McChaon. With Robert McChaon, Maurice Tombragel and Finley Hunt. With Lloyd Bridges, Dina Merrill, Gilbert Roland, Pat Hingle, Morgan Woodward, R.G. Armstrong, Lonny Chapman, Fred Betts, Slavio Martinez. While in Colorado doing a photo story, a woman journalist becomes alarmed at the treatment given wild horses. Sturdy drama, reissued in 1976 by Dimension Pictures as _**Deliver Us from Evil**_.\n\n**3646** _ **The Rustlers**_ **** RKO Radio, 1949. 61 min. D: Lesley Selander. SC: Jack Natteford. With Tim Holt, Richard Martin, Martha Hyer, Lois Andrews, Steve Brodie, Francis McDonald, Harry Shannon, Addison Richards, Frank Fenton, Robert Bray, Don Haggerty, Monte Montague, Stanley Blystone, Pat Patterson, George Ross. Two cowpokes are framed on charges of robbery and murder and set out to find the real culprits. Another entertaining Tim Holt series feature.\n\n**3647** _ **The Rustler's End**_ **** Krelbar Pictures, 1928. 50 minutes. D-SC: Robert J. Horner. With Al Hoxie, Betty Gates, Bill Nestell, Carl Berlin, Herbert Walter, Jack Dailey. Rustlers abduct the girl friend of a Texas Ranger who is trying to bring them to justice. Cheap but fast paced Al Hoxie vehicle for William M. Pizor Productions, Hoxie's last starring effort.\n\n**3648** _ **Rustlers' Hideout**_ **** Producers Releasing Corporation, 1944. 60 min. D: Sam Newfield. SC: Joseph O'Donnell. With Buster Crabbe, Al St. John, Patti McCarty, Charles King, John Merton, Lane Chandler, Terry Frost, Hal Price, Al Ferguson, Frank McCarroll, Ed Cassidy, Bud Osborne, Steve Clark, John Cason, Lane Chandler, Wally West, Ed Peil, Sr. Billy Carson and Fuzzy Q. Jones lead a large Wyoming cattle herd to market and come up against outlaws wanting to take the beef to start their own business. A good storyline and nice camera work make this a better than average \"Billy Carson\" entry.\n\n_**Rustler's Hideout**_ (1946) see _**Rustler's Roundup**_\n\n**3649** _ **Rustlers of Devil's Canyon**_ **** Republic, 1947. 58 min. D: R.G. Springsteen. SC: Earle Snell. With Allan \"Rocky\" Lane, Peggy Stewart, Bobby Blake, Martha Wentworth, Arthur Space, Emmett Lynn, Roy Barcroft, Tom London, Harry Carr, Pierce Lyden, Forrest Taylor, Bob Burns, Frank O'Connor, Bob Reeves, Art Dillard, Pascale Perry, Cactus Mack, Jack Montgomery, Tom Smith. Red Ryder tries to stop hostilities between ranchers and homesteaders in a range war instigated by rustlers. Average \"Red Ryder\" affair.\n\n**3650** _ **Rustlers of Red Dog**_ **** Universal, 1935. 12 Chapters. D: Louis Friedlander (Lew Landers). SC: George H. Plympton, Basil Dickey, Ella O'Neill, Nate Gatzert and Vin Moore. With Johnny Mack Brown, Joyce Compton, Walter Miller, Raymond Hatton, Harry Woods, Frederick MacKaye, Charles K. French, Lafe McKee, William Desmond, J.P. McGowan, Edmund Cobb, Al Ferguson, Bud Osborne, Monte Montague, Jim Thorpe, Chief Thundercloud, Wally Wales, Slim Whitaker, Art Mix, Bill Patton, Cliff Lyons, Tex Cooper, Ben Corbett, Hank Bell, Artie Ortego, Ann D'Arcy, Fritzi Burnette, Grace Cunard, Virginia Ainsworth, Iron Eyes Cody, Chief Many Treaties, Jim Corey, Bob Card, William McCall, Jerry Frank, Horace B. Carpenter, Jack Rockwell, George Magrill, Frank McCarroll, John Ince, Harry Harvey, Nelson McDowell, Harry Tenbrook, Ted Billings, Eddie Bugard, Buddy Wacker, Bernadet Sebastian, Charles Murphy, B.J. Wilson. Three cowboys try to protect a wagon train from ruthless outlaws and marauding Indians. Action packed and entertaining Johnny Mack Brown cliffhanger, his second serial.\n\n**3651** _ **Rustlers of the Badlands**_ **** Columbia, 1945. 55 min. D: Derwin Abrahams. SC: J. Benton Cheney. With Charles Starrett, Tex Harding, Dub Taylor, Sally Bliss, George Eldredge, Edward Howard, Ray Bennett, Al Trace and His Silly Symphonists, Ted Mapes, Karl Hackett, James T. \"Bud\" Nelson, Frank McCarroll, Carl Sepulveda, Steve Clark, Ted French, Frank LaRue, Bud Osborne, Edmund Cobb, Nolan Leary, Frank Ellis, Jack Ingram. Three Army scouts are assigned to find out who murdered a lieutenant and they comes across a rash of cattle thefts that seem tied to the killing. Standard \"Durango Kid\" offering. British title: _**By Whose Hand?**_\n\n**3652** _ **Rustlers on Horseback**_ **** Republic, 1950. 60 min. D: Fred C. Brannon. SC: Richard Wormser With Allan \"Rocky\" Lane, Eddy Waller, Roy Barcroft, Claudia Barrett, John Eldredge, George Nader, Forrest Taylor, John Cason, Stuart Randall, Douglas Evans, Tom Monroe, Marshall Reed, George Lloyd. A lawman and his peddler pal infiltrate a gang secretly run by a book salesman out to fleece a man of his life savings to finance plans to loot the area. Typically well mounted Allan Lane series feature.\n\n**3653** _ **Rustlers' Paradise**_ **** Ajax, 1935. 61 min. D: Harry Fraser. SC: Weston Edwards. With Harry Carey, Gertrude Messinger, Edmund Cobb, Carmen Bailey, Theodore Lorch, Charles (Slim) Whitaker, Roger Williams, Chuck Morrison, Allen Greer, Chief Thundercloud, Jimmy Aubrey, Tex Palmer, Barney Beasley. A man searches for his wife and daughter who were abducted years before by an outlaw and he poses as a crook to infiltrate a gang whose leader he suspects of the kidnapping. Austere Harry Carey vehicle highlighted by Theodore Lorch's work as the oily villain.\n\n**3654** _ **Rustlers' Rhapsody**_ **** Paramount, 1985. 88 min. Color. D-SC: Hugh Wilson. With Tom Berenger, G.W. Bailey, Marilu Henner, Andy Griffith, Fernando Rey, Sela Ward, Patrick Wayne, Brant Van Hoffman, Christopher Malcolm, Jim Carter, Billy J. Mitchell. A singing cowboy and his pal wander through the West and stopping at a town they help sheepherders fight a corrupt cattle baron. Pleasant satire of the musical Westerns of yore.\n\n**3655** _ **Rustlers' Roundup**_ **** Universal, 1933. 56 min. D: Henry MacRae. SC: Frank Clark and Jack Cunningham. With Tom Mix, Dianne Sinclair, Noah Beery, Jr., Douglass Dumbrille, Roy Stewart, William Desmond, Gilbert \"Pee Wee\" Holmes, Bud Osborne, Edmund Cobb, Frank Lackteen, William Wagner, Nelson McDowell, Walter Brennan, Fred Humes, Cliff Lyons, Al Haskell. A rancher tries to help a man and his sister when outlaws want to take over their land which has an underground spring they want to use for their rustling operation. Tom Mix's final Universal series film is not up to some of the others in the series but is still above average.\n\n**3656** _ **Rustler's Roundup**_ **** Universal, 1946. 57 min. D: Wallace Fox. SC: Jack Natteford. With Kirby Grant, Jane Adams, Fuzzy Knight, Edmund Cobb, Ethan Laidlaw, Earle Hodgins, Charles Miller, Mauritz Hugo, Eddy Waller, Roy Brent, Frank Marlo, Hank Bell, Rex Lease, Budd Buster, Steve Clark, Bud Osborne, Jack Curtis, George Morrell, Carl Mathews, Ray Spiker, Alfred Wagstaff. A cowboy and his pals set out to round up an outlaw gang harassing area ranchers. Nicely paced and action laden Kirby Grant vehicle; TV title: _**Rustler's Hideout**_.\n\n**3657** _ **Rustler's Valley**_ **** Paramount, 1937. 61 min. D: Nate Watt. SC: Harry O. Hoyt. With William Boyd, George Hayes, Russell Hayden, John St. Polis, Lee Colt (Lee J. Cobb), Stephen Morris (Morris Ankrum), Muriel Evans, Ted Adams, Al Ferguson, John Beach, Oscar Apfel, Bernadene Hayes, Leo J. McMahon, Dot Farley, John Powers, Horace B. Carpenter, Ben Corbett. Hoppy helps his pal Lucky when he is accused of a bank robbery really masterminded by a crooked lawyer out to get a ranch from his fiancee and her father. A bit slow for a \"Hopalong Cassidy\" feature but the scenery is nice and the finale (reused in _**Lost Canyon**_ [q.v.]) exciting.\n\n**3658** _ **Rusty Rides Alone**_ **** Columbia, 1933. 58 min. D: D. Ross Lederman. SC: Robert Quigley. With Tim McCoy, Barbara Weeks, Dorothy Burgess, Wheeler Oakman, Edmund Cobb, Ed Burns, Rockliffe Fellows, Clarence Geldert, Wally Wales, Buffalo Bill, Jr., Frank Ellis, Richard Botiller, Slim Whitaker, Hank Bell, Artie Ortego, Blackjack Ward, Bud McClure, Ray Jones, Jack Evans, Charles Brinley, Jack King, Barney Beasley, Silver King (dog). A cowboy opposes a dishonest sheep man who wants to set up an empire for himself by driving all the cattle ranchers from their ranges. A weak script hurts this Tim McCoy series effort; based on a story by Walt Coburn.\n\n**3659** _ **The Ruthless Four**_ **** Metro-Goldwyn-Mayer, 1968. 97 min. Color. D: Giorgio Capitani. SC: Fernando Di Leo. With Van Heflin, Gilbert Roland, George Hilton, Klaus Kinski, Sarah Ross, Rick Boyd (Federico Boido), Sergio Doria, Ivan Scratuglia, Giorgio Groden. A veteran prospector teams with his adopted son and a friend along with an old comrade in trekking to his rich claim but they fall out among themselves over the gold. Very good Italian-Spanish co-production that will appeal to the two stars' fans; also called _**Each Man for Himself**_ and _**Sam Cooper's Gold**_ and released in Italy as _**Ognuno per Se**_ (Each One for Himself).\n\n**3660** _ **Sabata**_ **** United Artists, 1970. 106 min. Color. D: Frank Kramer (Gianfranco Parolini). SC: Gianfranco Parolini and Renato Izzo. With Lee Van Cleef, William Berger, Pedro Sanchez, Nick Jordan, Franco Ressel, Linda Veras, Antonio Gradoli, Robert Hundar (Claudio Undari), Gianno Rizzo, Spean Covery, Marco Zuanelli, John Bartha, Romano Puppo, Ken Wood (Giovanni Cianfriglia), Alan Collins (Luciano Pignozzi). Three corrupt businessmen hire a gunman to steal a safe for them and then try to double cross him when he demands a higher payment. Lee Van Cleef is very good in the title role of this ultra violent European oater released in Italy in 1969 as _**...Ehi, Amico, C'e Sabata...Hai Chiuso!**_ (Hey Friend, Here's Sabata...You're Finished) and followed by two sequels, _**Adios Sabata**_ and _**The Return of Sabata**_ (qq.v.).\n\n**3661** _ **The Sacketts**_ **** NBC-TV, 1979. 200 min. Color. D: Robert Totten. SC: Jim Byrnes. With Sam Elliott, Tom Selleck, Jeff Osterhage, Glenn Ford, Ben Johnson, Gilbert Roland, John Vernon, Ruth Roman, Jack Elam, Gene Evans, L.Q. Jones, Paul Koslo, Mercedes McCambridge, Slim Pickens, Pat Buttram, James Gammon, Buck Taylor, Lee DeBroux, Marcy Hanson, Ana Alicia, Wendy Rastatter, Shug Fisher, Frank Ramirez, Ramon Chavez, Don Collier, Billy Cardi, Rusty Lane. Following the Civil War three brothers attempt to bring law and order to New Mexico Territory after avenging a family murder. Glossy TV adaptation of Louis L'Amour's work; entertaining for the author's followers.\n\n**3662** _ **Sacred Ground**_ **** Pacific International, 1983. 100 min. Color. D-SC: Charles B. Pierce. With Tim McIntire, Jack Elam, L.Q. Jones, Mindi Miller, Serene Hedin, Eloy Phil Casados. In the 1860s trouble results when a frontiersman and his Indian wife settle in sacred Paiute territory with their new baby. Fairly engrossing drama.\n\n**3663** _ **The Sad Horse**_ **** 20th Century\u2013Fox, 1959. 81 min. Color. D: James B. Clark. SC: Charles Hoffman. With David Ladd, Chill Wills, Rex Reason, Patrice Wymore, Gregg Palmer, Eve Brent, Leslie Bradley, William Yip, Dave De Paul. A lonely little boy develops a close relationship with a racehorse. Pleasant family fare.\n\n**3664** _ **Saddle Aces**_ **** Resolute, 1935. 56 min. D: Harry Fraser. SC: Harry C. (Fraser) Crist. With Rex Bell, Ruth Mix, Buzz Barton, Stanley Blystone, Earl Dwire, Chuck Morrison, Mary MacLaren, John Elliott, Roger Williams, Chief Thundercloud, Allen Greer, Bud Osborne, Francis Walker, Bob Burns, Blackjack Ward, Chief Standing Bear, Guate Mozin. Two falsely convicted men escape from a prison train and help a woman whose ranch is sought by the crook responsible for the crimes for which they are accused. Last of the Rex Bell-Ruth Mix-Buzz Barton series; cheap and it shows.\n\n**3665** _ **The Saddle Buster**_ **** RKO Radio, 1932. 60 min. D: Fred Allen. SC: Oliver Drake. With Tom Keene, Helen Forest, Charles Quigley, Marie Quillan, Ben Corbett, Fred Burns, Richard Carlyle, Robert Frazer, Harry Bowen, Al Taylor, Slim Whitaker, Montie Montana, Yakima Canutt, Murdock MacQuarrie, Jack Kirk, Edgar \"Blue\" Washington. A Montana cowboy leaves home to become a rodeo star but his success is almost thwarted by the jealousy of a rival. Very fine, well made Tom Keene feature; quite exciting.\n\n**3666** _ **Saddle Leather Law**_ **** Columbia, 1944. 55 min. D: Benjamin Kline. SC: Elizabeth Beecher. With Charles Starrett, Dub Taylor, Vi Athens, Lloyd Bridges, Jimmy Wakely and His Saddle Pals, Salty Holmes, Reed Howes, Robert Kortman, Frank LaRue, Ted French, Ed Cassidy, Steve Clark, Frank O'Connor, Budd Buster, Franklyn Farnum, Nolan Leary, Joseph Eggenton, Netta Parker. When a rancher is killed, two cowboys are blamed but they learn a young woman, working for a syndicate after the dead man's spread for a dude ranch, is the culprit. Pretty fair Charles Starrett pic with the leading lady as the chief villain.\n\n**3667** _ **Saddle Legion**_ **** RKO Radio, 1951. 61 min. D: Lesley Selander. SC: Ed Earl Repp. With Tim Holt, Richard Martin, Dorothy Malone, Robert Livingston, James Bush, Mauritz Hugo, Cliff Clark, Joseph J. Lewis, Robert Wilke, Stanley Andrews, Reed Howes, Monte Montague, Clem Fuller. Outlaws try to obtain cattle by falsely making the herd appear to be diseased but two cowpokes see through the ruse. Well done Tim Holt series entry.\n\n**3668** _ **Saddle Mountain Roundup**_ **** Monogram, 1941. 61 min. D: S. Roy Luby. SC: Earle Snell and John Vlahos. With Ray Corrigan, John King, Max Terhune, Jack Mulhall, Lita Conway, Willie Fung, John Elliott, George Chesebro, Jack Holmes, Cousin Harold Goodwin, Carl Mathews, Al Ferguson, Steve Clark, Slim Whitaker, Tex Palmer. A crusty rancher hires three cowboys to protect him from death threats and after he is murdered they try to find the killer. Dandy outing in \"The Range Busters\" series, enhanced by its spooky mystery element; remake of _**Big Boy Rides Again**_ (q.v.).\n\n**3669** _ **Saddle Pals**_ **** Republic, 1947. 72 min. D: Lesley Selander. SC: Bob Williams and Jerry Sackheim. With Gene Autry, Lynne Roberts, Sterling Holloway, Irving Bacon, Damian O'Flynn, The Cass County Boys (Jerry Scoggins, Bert Dodson, Fred Martin), Charles Arnt, Jean Van, Tom London, Charles Williams, Francis McDonald, Edward Gargan, Carl Sepulveda, George Chandler, LeRoy Mason, Paul E. Burns, Joel Friedkin, Larry Steers, Nolan Leary, Edward Keane, Maurice Cass, Minerva Urecal, Sam Ash, Frank O'Connor, Neal Hart, Ed Peil, Sr., Bob Burns, Bob Yrigoyen. Gene Autry comes to the aid of area ranchers whose rents have been suddenly raised by the local land company. Tired Gene Autry film with little action and limp comedy.\n\n**3670** _ **Saddle Serenade**_ **** Monogram, 1945. 60 min. D: Oliver Drake. SC: Frances Kavanaugh. With Jimmy Wakely, Lee \"Lasses\" White, John James, Nancy Brinckman, Foy Willing and The Riders of the Purple Sage, Jack Ingram, Claire James, Pat Gleason, Gay Deslys, Roy Butler, Alan Foster, Elmer Napier, Frank McCarroll, Dee Cooper, Jack Hendricks, Jack Spear, Carl Mathews. Two cowpokes go to work at a dude ranch not knowing it is a front for Eastern jewel thieves. Tepid Jimmy Wakely opus with some good songs.\n\n**3671** _ **Saddle the Wind**_ **** Metro-Goldwyn-Mayer, 1958. 84 min. Color. D: Robert Parrish. SC: Thomas Thompson. With Robert Taylor, Julie London, John Cassavetes, Donald Crisp, Charles McGraw, Royal Dano, Richard Erdman, Douglas Spencer, Ray Teal, Stanley Adams, Nacho Galindo, Lars Henderson, Irene Tedrow, Jay Adler, Henry Wills, Kelo Henderson, William Challee, Wes Fuller. A one-time gunman settles down to a peaceful life as a rancher only to be forced into a showdown by his gunslinger younger brother. Well done and entertaining; Robert Taylor is especially good as the rancher.\n\n**3672** _ **Saddle Tramp**_ **** Universal-International, 1950. 76 min. Color. D: Hugo Fregonese. SC: Harold Shumate. With Joel McCrea, Wanda Hendrix, John Russell, John McIntire, Jeanette Nolan, Russell Simpson, Antonio Moreno, Ed Begley, Jimmy Hunt, Orley Lindgren, Gordon Gebert, Gregory Moffett, John Ridgely, Walter Coy, Joaquin Garay, Peter Leeds, Michael Steele, Paul Picerni. A saddle tramp becomes the guardian of four orphans and after getting a job as a ranch hand he becomes involved in lots of trouble. Good viewing.\n\n**3673** _ **Saddlemates**_ **** Republic, 1941. 56 min. D: Lester Orlebeck. SC: Albert DeMond and Herbert Dalmas. With Robert Livingston, Bob Steele, Rufe Davis, Gale Storm, Forbes Murray, Cornelius Keefe, Peter George Lynn, Marin Sais, Glenn Strange, Iron Eyes Cody, Chief Yowlachie, Henry Wills, Matty Faust, Ellen Lowe, Rex Lease, Ed Cassidy, Slim Whitaker, Yakima Canutt, Bill Keefer, Jack Kirk, Space Cooley, Art Dillard, Kansas Moehring, Bob Woodward, Bill Hazlett, Chick Hannon, Tex Cooper, Roy Bucko, Bill Nestell, Herman Hack, Jess Cavin, Victor Cox, Lew Meehan, Bert Dillard. The Three Mesquiteers help the Army in trying to control a band of hostile Indians led by a half-breed. Fair entry in the popular series.\n\n**3674** _ **Saddles and Sagebrush**_ **** Columbia, 1943. 57 min. D: William Berke. SC: Ed Earl Repp. With Russell Hayden, Dub Taylor, Bob Wills and The Texas Playboys, Ann Savage, William Wright, Frank LaRue, Wheeler Oakman, Edmund Cobb, Jack Ingram, Joe McGuinn, Ray Jones, Art Mix, Blackie Whiteford, Ben Corbett, Bob Burns. A cowboy and his pals help a rancher and his daughter being victimized by crooks. Pretty sturdy Russell Hayden vehicle.\n\n**3675** _ **Saga of Death Valley**_ **** Republic, 1939. 58 min. D: Joseph Kane. SC: Karen DeWolf and Stuart Anthony. With Roy Rogers, George \"Gabby\" Hayes, Donald Barry, Doris Day, Frank M. Thomas, Jack Ingram, Hal Taliaferro, Lew Kelly, Fern Emmett, Tommy Baker, Buzz Buckley, Horace Murphy, Lane Chandler, Fred Burns, The Jimmy Wakely Trio (Jimmy Wakely, Johnny Bond, Dick Reinhart), Ed Brady, Pasquale Perry, Cactus Mack, Art Dillard, Horace B. Carpenter, Hooper Atchley, Frankie Marvin. A man returns to the ranch where his father was murdered to take revenge on the killer and finds the culprit's chief henchman is his younger brother. Very good Roy Rogers film; well worth viewing.\n\n**3676** _ **The Saga of Hemp Brown**_ **** Universal-International, 1959. 80 min. Color. D: Richard Carlson. SC: Bob Williams. With Rory Calhoun, Beverly Garland, John Larch, Russell Johnson, Fortunio Bonanova, Marjorie Stapp, Morris Ankrum, Yvette Vickers, Charles Boaz, Allan Lane, Victor Sen Yung, Trevor Bardette, Addison Richards, Francis McDonald, Theodore Newton, I. Stanford Jolley, Tom London. Framed on a payroll robbery charge, a soldier is dismissed from the Army and tries to find out who committed the crime. Average outing helmed by actor Richard Carlson.\n\n_**Saga of the West**_ see _**When a Man's a Man**_\n\n**3677** _ **The Sagebrush Family Trails West**_ **** Producers Releasing Corporation, 1940. 62 min. D: Peter Stewart (Sam Newfield). SC: William Lively. With Bobby Clark, Earle Hodgins, Nina Guilbert, Joyce Bryant, Minerva Urecal, Arch Hall, Kenneth (Kenne) Duncan, Forrest Taylor, Carl Mathews, Wally West, Byron Vance, Augie Gomez. The young son of an inventor comes to his rescue when a gang is after the scientist's secret formula. Shoddy effort which mercifully did not make it as a series; Bobby Clark is not the famous comedian but a teenage world's junior champion cowboy.\n\n**3678** _ **Sagebrush Heroes**_ **** Columbia, 1945. 55 min. D: Benjamin Kline. SC: Luci Ward. With Charles Starrett, Dub Taylor, Constance Worth, Jimmy Wakely, Ozie Waters and His Saddle Pals, Elvin Field, Bobby Larson, Forrest Taylor, Joel Friedkin, Lane Chandler, Paul (Conrad) Zaremba, Eddie Laughton, John Tyrrell, Vernon Dent, Davison Clark, Edmund Cobb, Budd Buster, Jessie Arnold, Ted French. A radio actor finds out a ranch for boys is a front used by cattle thieves and child labor violators. Fast paced affair that has Charles Starrett portraying a radio hero called the Durango Kid prior to his long running series about the character.\n\n**3679** _ **Sagebrush Law**_ **** RKO Radio, 1943. 56 min. D: Sam Nelson. SC: Bennett Cohen. With Tim Holt, Joan Barclay, Cliff Edwards, John Elliott, Ed Cassidy, Karl Hackett, Roy Barcroft, Ernie Adams, John Merton, Bud McTaggart, Edmund Cobb, Otto Hoffman, Cactus Mack, Ben Corbett, Frank McCarroll, Bob McKenzie, Dick Rush, Chester Conklin, Russell Wade, Richard Cramer, David Sharpe, Merlyn Nelson. When his town banker father is falsely accused of embezzlement and then murdered, a man tries to prove his innocence and find the killers. A fine script and cast make this Tim Holt vehicle a good one.\n\n**3680** _ **Sagebrush Trail**_ **** Monogram, 1933. 54 min. D: Armand L. Schaefer. SC: Lindsley Parsons. With John Wayne, Nancy Shubert, Lane Chandler, Yakima Canutt, Henry Hall, Wally Wales, Art Mix, Bob Burns, Bill (William) Dwyer, Ted Adams, Earl Dwire, Hank Bell, Slim Whitaker, Hal Price, Blackjack Ward, Archie Ricks, Tex Phelps, Robert Walker, Tex Phelps, Silver Tip Baker, Julie Kingdon. Falsely accused of murder, a cowboy escapes from jail and joins outlaws hoping to find the real killer not realizing the gang member who befriends him is the man he is seeking. John Wayne's second \"Lone Star\" oater for producer Paul Malvern is a speedy affair that will appeal to Duke's fans. Colorized as _**An Innocent Man**_ ; a remake of _**Partners of the Trail**_ (q.v.).\n\n**Poster for** _**Sagebrush Trail**_ **(Monogram, 1933).**\n\n**  \n**\n\n**3681** _ **The Sagebrush Troubadour**_ **** Republic, 1935. 54 min. D: Joseph Kane. SC: Oliver Drake and Joseph Poland. With Gene Autry, Smiley Burnette, Barbara Pepper, J. Frank Glendon, Hooper Atchley, Dennis Moore, Fred Kelsey, Julian Rivero, Tom London, Frankie Marvin, Art Davis, Wes Warner. When an elderly, nearly blind man is murdered, a singing cowboy tries to discover who committed the crime. Pretty fair Gene Autry affair with just the right blend of music, action and mystery.\n\n**3682** _ **Saginaw Trail**_ **** Columbia, 1953. 56 min. D: George Archainbaud. SC: Dorothy Yost and Dwight Cummings. With Gene Autry, Smiley Burnette, Connie Marshall, Eugene Borden, Myron Healey, John Merton, Ralph Reed, Henry Blair, Mickey Simpson, John War Eagle, Rodd Redwing, Billy Wilkerson, Gregg Barton, John Parrish. In 1827 Michigan, the captain of Hamilton's Rangers is out to stop a fur magnate from murdering settlers. Pretty good, and different, Gene Autry film; his penultimate starring effort.\n\n**3683** _ **The Sagittarius Mine**_ **** Gold Key, 1972. 91 min. Color. With Steve Forrest, Diane Baker, Ray Danton, Richard Basehart. A sheriff suspects an invisible force is stopping prospectors from finding the Lost Dutchman gold mine. Obscure sci-fi Western.\n\n**3684** _ **Salome, Where She Danced**_ **** Universal, 1945. 90 min. Color. D: Charles Lamont. SC: Laurence Stallings. With Yvonne De Carlo, Rod Cameron, David Bruce, Walter Slezak, Albert Dekker, Marjorie Rambeau, J. Edward Bromberg, Abner Biberman, John Litel, Kurt Katch, Arthur Hohl, Nestor Paiva, Gavin Muir, Will Wright, Joseph Haworth, Matt McHugh, Jane Adams, Barbara Bates, Daun Kennedy, Kathleen O'Malley, Karen Randle, Jean Trent, Kerry Vaughn, Jan Williams, Doreen Tryden, Bert Dole, Emmett Casey, Eddie Dunn, Charles Wagenheim, Gene Garrick, Eric Feldary, George Sherwood, Colin Campbell, Charles McAvoy, Al Ferguson, Edmund Cobb, Jack Clifford, Bud Osborne, George Morrell, Hank Bell, George Chesebro, Budd Buster, Richard Alexander, Cecilia Callejo, Sylvia Field, Richard Ryden, Alan Edwards, George Leigh, Ina Owenbey, Jimmy Lung, Peter Seal, Jasper Palmer. A beautiful dancer escapes from Europe, begins a tour of the West and in an Arizona town convinces an outlaw gang to go straight. Amusing tongue-in-cheek drama with Yvonne De Carlo a knockout in the title role.\n\n**3685** _ **Salt Lake Raiders**_ **** Republic, 1950. 60 min. D: Fred C. Brannon. SC: M. Coates Webster. With Allan \"Rocky\" Lane, Eddy Waller, Martha Hyer, Roy Barcroft, Byron Foulger, Myron Healey, Clifton Young, Stanley Andrews, George Chesebro, Kenneth MacDonald, Rory Mallinson. When a convict escapes from prison, a U.S. marshal is assigned to track him down while outlaws are after him thinking he has hidden gold. Pretty entertaining Allan Lane vehicle.\n\n**3686** _ **Sam Cade**_ **** CBS-TV\/20th Century\u2013Fox, 1972. 100 min. Color. D: Marvin J. Chomsky and Leo Penn. SC: Eric Berovici, Cliff Gould and Jerrold L. Ludwig. With Glenn Ford, Edgar Buchanan, Darren McGavin, Loretta Swit, Edward Asner, Shelley Fabares, H.M. Wynant, Richard Anders, Taylor Lacher, Victor Campos, Peter Ford, Betty Ann Carter, Myron Healey, Jean Fowler, Ralph James, Ed Flanders, Larry Casey, William H. Bassett, Felice Orlandi, Philip Kenneally, Anne Randall, Gene Lebell, Sandra Ego. A New Mexico sheriff must face a wartime friend who comes home to kill him and then stop the proposed assassination of an ex-syndicate boss. Two episodes (\"The Fake\" and \"Homecoming\") of \"Cade's County\" (CBS-TV, 1971\u201372) tied together into a feature make for good viewing.\n\n**3687** _ **Sam Hill:**_ _**Who Killed the Mysterious Mr. Foster?**_ **** NBC-TV\/Universal, 1971. 100 min. Color. D: Fiedler Cook. SC: Richard Levinson and William Link. With Ernest Borgnine, Judy Geeson, Stephen Hudis, Will Geer, J.D. Cannon, Bruce Dern, Sam Jaffe, Carmen Matthews, John McGiver, Slim Pickens, G.D. Spradlin, Jay C. Flippen, Woodrow Parfrey, George Furth, Dub Taylor, Milton Selzer, Ted Gehring, Dennis Fimple, Robert Gooden. In order to win re-election a small town sheriff must find the man who murdered a minister. Mediocre TV Western re-titled _**Who Killed the Mysterious Mr. Foster?**_\n\n**3688** _ **Sam Whiskey**_ **** United Artists, 1969. 95 min. Color. D: Arnold Laven. SC: William N. Norton. With Burt Reynolds, Clint Walker, Angie Dickinson, Ossie Davis, Del Reeves, Rick Davis, William Schallert, Woodrow Parfrey, Anthony James, Bud Adler, Ayllene Gibbons, Amanda Harley, Tracey Roberts, Virgil Warner, William Boyett, Sidney Clute, Chubby Johnson, John Damler. A rogue comes under the spell of a beautiful widow who convinces him to take a million dollars in gold bars from a sunken riverboat and return it to the U.S. mint before the theft, which was perpetrated by her late husband, is discovered. Burt Reynolds fans may like this stale film but others beware.\n\n**3689** _ **Samson and the Slave Queen**_ **** American International, 1964. 86 min. Color. D: Umberto Lenzi. SC: Guido Malatesta and Umberto Lenzi. With Pierre Brice, Alan Steel (Sergio Ciani), Massimo Serato, Moira Orfel, Maria Grazia Spina, Andrea Aureli, Antonio Corevi, Loris Gizzi, Rosy di Leo, Attilio Dottsio, Nello Pazzafini, Andrea Scotti (Andrew Scott), Amedeo Trilli, Nazzareno Zamperla, Gianni Gaghino, Ignazio Ballsamo, Gianni Baghino, Aldo Bufi Landi. Two women want to become the queen of Navarre after the death of their uncle and one of them enlists the aid of Zorro while the other seeks help from mighty man Samson. Sub-par swashbuckler issued in Europe in 1963 by Romana Film as _**Zorro Contra Maciste**_ (Zorro Against Maciste) and included here because of the Zorro character.\n**3690** _ **San Antone**_ **** Republic, 1953. 90 min. D: George Sherman. SC: Steve Fisher. With Rod Cameron, Arleen Whelan, Forrest Tucker, Katy Jurado, Rodolfo Acosta, Roy Roberts, Bob Steele, Harry Carey, Jr., James Lilburn, Andrew Brennan, Richard Hale, Martin Garralaga, Argentina Brunetti, Douglas Kennedy, Paul Fierro, George Cleveland, Francis McDonald, Marshall Reed, James Craven, William Haade, Joseph Crehan, Chris-Pin Martin, Lee Shumway, Steve Darrell, Chuck Hayward, James Harrison, Charles Cane, Jack O'Shea, Carleton Young, William Haade, John Halloran, Pepe Hern, Charles Stevens, Peter Ortiz, Alex Montoya, Robert Keys, Ralph Clanton (narrator). During the Civil War a rancher agrees to lead a cattle drive through enemy country not knowing he his being used by his so-called friends. Sedate oater provides good entertainment.\n\n**3691** _ **San Antone Ambush**_ **** Republic, 1949. 60 min. D: Philip Ford. SC: Norman S. Hall. With Monte Hale, Paul Hurst, Roy Barcroft, Bette Daniels, James Cardwell, Trevor Bardette, Lane Bradford, Tommy Coats, Francis Ford, Tom London, Edmund Cobb, Carl Sepulveda. After the robbery of an Army pay wagon a cavalry officer is falsely accused of being the tip-off man for the job. Monte Hale fans will enjoy this fairly exciting outing.\n\n**3692** _ **San Antonio**_ **** Warner Bros., 1945. 111 min. Color. D: David Butler. SC: Alan LeMay and W.R. Burnett. With Errol Flynn, Alexis Smith, S.Z. Sakall, Victor Francen, Florence Bates, John Litel, Paul Kelly, Robert Shayne, John Alvin, Monte Blue, Robert Barrat, Pedro de Cordoba, Tom Tyler, Chris-Pin Martin, Charles Stevens, Poodles Hannaford, Doodles Weaver, Dan White, Ray Spiker, William Gould, Harry Seymour, Norman Willis, Eddy Waller, James Flavin, Henry Hall, Al Hill, Harry Cording, Chalky Williams, Wallis Clark, Bill Steele, Allen E. Smith, Howard Hill, Arnold Dent, Francis Ford, Lane Chandler, Hal Taliaferro, Jack Mower, Joe Dominguez, Dan Seymour, Eva Puig, Eddie Acuff, Si Jenks, Brandon Hurst, Fred Kelsey, Francis Ford, Brad King, Don McGuire, John Compton, Jasper Palmer. Returning to Texas from Mexico in 1877, a cattleman brings proof a saloon owner is behind a gang of rustlers. Colorful Errol Flynn vehicle.\n\n**3693** _ **The San Antonio Kid**_ **** Republic, 1944. 59 min. D: Howard Bretherton. SC: Norman S. Hall. With Wild Bill Elliott, Bobby Blake, Alice Fleming, Linda Stirling, Tom London, Earle Hodgins, Glenn Strange, Duncan Renaldo, LeRoy Mason, Jack Kirk, Robert Wilke, Jack O'Shea, Tex Terry, Bob Woodward, Herman Hack, Henry Wills, Tom Steele, Billy \"Sailor\" Vincent, Bud Geary, Cliff Parkinson, Joe Garcia, Roy Bucko, Lew Morphy, Herman Howlin. Crooks try to run ranchers off their spreads before news of an oil strike is announced but Red Ryder arrives to get to the bottom of the trouble. Average outing in the popular series.\n\n**3694** _ **San Fernando Valley**_ **** Republic, 1944. 74 min. D: John English. SC: Dorrell McGowan and Stuart McGowan. With Roy Rogers, Dale Evans, Jean Porter, Bob Nolan and The Sons of the Pioneers (Tim Spencer, Ken Carson, Shug Fisher, Hugh Farr, Karl Farr), Andrew Tombes, Edward Gargan, Dot Farley, LeRoy Mason, Charles Smith, Pierce Lyden, Maxine Doyle, Helen Talbot, Pat Starling, Kay Forrester, Kenne Duncan, Ed Cassidy, Hank Bell, Marguerite Blount, The Morell Trio, Vernon and Draper. When outlaws plague the San Fernando Valley, Roy Rogers tries to stop them and helps a lady rancher who has fired all her male hands and replaced them with females. Top notch Roy Rogers film, the one where he gets his first screen kiss from Jean Porter. ****\n\n**3695** _ **San Francisco**_ **** Metro-Goldwyn-Mayer, 1936. 115 min. D: W.S. Van Dyke. SC: Anita Loos. With Clark Gable, Jeanette MacDonald, Spencer Tracy, Jack Holt, Ted Healy, Margaret Irving, Jessie Ralph, Shirley Ross, Harold Huber, Al Shean, William Ricciardi, Kenneth Harlan, Roger Imhof, Frank Mayo, Tom Dugan, Charles Judels, Russell Simpson, Bert Roach, Warren Hymer, Edgar Kennedy, Adrienne d'Ambricourt, Nigel de Brulier, Mae Digges, Tudor Williams, Tandy McKenzie, Myas Beery, Tom Mahoney, Gertrude Astor, Jason Robards, Vernon Dent, Jack Baxley, Anthony Jowitt, Carl Stockdale, Richard Carle, Oscar Apfel, Frank Sheridan, Ralph Lewis, Chester Gan, Jack Kennedy, Cy Kendall, Don Rowan, Jim Farley, Belle Mitchell, Billy Newell, Irving Bacon, John \"Skins\" Miller, George Guhl, Edward Earle, Wilbur Mack. In 1905 San Francisco a Barbary Coast saloon owner and a priest both keep an eye on a beautiful singer. This big production feature still packs a punch but the plot is secondary to the grand special effects of the San Francisco earthquake.\n\n**3696** _ **The San Francisco Story**_ **** Warner Bros., 1952. 80 min. D: Robert Parrish. SC: D.D. Beauchamp. With Joel McCrea, Yvonne De Carlo, Sidney Blackmer, Richard Erdman, Florence Bates, Onslow Stevens, John Raven, O.Z. Whitehead, Ralph Dumke, Robert Foulk, Lane Chandler, Trevor Bardette, John Doucette, Peter Virgo, Tor Johnson, Ted Adams, Mickey Simpson, Frank Hagney, Fred Graham, Fred Graham, Ray Jones. The pretty mistress of a corrupt politician in 1856 San Francisco falls in love with a mine owner out to lock up her keeper. Fairly good action drama; Tor Johnson is the scariest bartender in the history of the genre.\n\n**3697** _ **Sand**_ **** Paramount-Artcraft, 1920. 49 min. D-SC: Lambert Hillyer. With William S. Hart, Mary Thurman, G. Raymond Nye, Patricia Palmer, Bill Patton, Lon Poff, Hugh Jackson. A railroad station agent is at odds with a crooked stockholder over the woman they both love and after getting fired he tries to locate train robbers. Well done western railroad melodrama featuring William S. Hart's beloved pinto, Fritz.\n\n**3698** _ **Sand**_ **** 20th Century\u2013Fox, 1949. 77 min. Color. D: Louis King. SC: Martin Berkeley and Jerome Cady. With Mark Stevens, Coleen Gray, Rory Calhoun, Charley Grapewin, Bob Patten, Mikel Conrad, Tom London, Paul Hogan, Jack Gallagher, William Walker, Davison Clark, Ben Erway, Harry V. Cheshire, Iron Eyes Cody, Jay Silverheels, Joseph Cody. A show horse escapes during a fire and his trainer tries to find him before he becomes wild. Mediocre outing that may appeal to juvenile fans. Also called _**Will James' Sand**_.\n\n**3699** _ **Sandflow**_ **** Universal, 1937. 58 min. D: Lesley Selander. SC: Frances Guihan. With Buck Jones, Lita Chevret, Robert Kortman, Arthur Aylesworth, Robert Terry, Enrique De Rosas, Josef Swickard, Lee Phelps, Harold Hodge, Tom Chatterton, Arthur Van Slyke, Malcolm Graham, Ben Corbett. The two sons of a cattle rustler try to make good on losses to ranchers whose herds their father stole but one of them is falsely accused of killing a lawman. Somewhat meandering Buck Jones feature.\n\n**3700** _ **Sandy Burke of the U-Bar-U**_ **** Betzwood Films, 1919. 55 min. D: Ira M. Morgan. SC: J. Allen Dunn. With Leslie Bennison, Virginia Lee, Alphonse Ethier, Herbert Horton Patlee, Echlin C. Gayer, Lucy Beaumont, Wilma Bayley, Nadia Gery. A cowboy fights a gang of rustlers who have abducted his boss' pretty daughter. Silent feature, filmed in Pennsylvania, is worth a look.\n\n**3701** _ **Sangaree**_ **** Paramount, 1953. 95 min. Color. D: Edward Ludwig. SC: David Doven. With Fernando Lamas, Arlene Dahl, Patricia Medina, Francis L. Sullivan, Charles Korvin, Tom Drake, John Sutton, Willard Parker, Charles Evans, Lester Mathews, Russell Gaige, William Walker, Felix Nelson, Voltaire Perkins. In 1781 frontier Georgia, a doctor, managing his late friend's estate, uncovers piracy and battles the plague. Handsomely mounted, but often dull, costumer.\n\n**3702** _ **Santa Fe**_ **** Columbia, 1951. 89 min. Color. D: Irving Pichel. SC: Kenneth Gamet. With Randolph Scott, Janis Carter, Jerome Courtland, Peter Thompson, John Archer, Warner Anderson, Roy Roberts, Billy House, Olin Howlin, Alice Roberts, Jack O'Mahoney (Jock Mahoney), Harry Cording, Sven Hugo Borg, Frank Ferguson, Irving Pichel, Harry Tyler, Chief Thundercloud, Paul E. Burns, Reed Howes, Charles Meredith, Paul Stanton, Richard Cramer, William Haade, Francis McDonald, Frank O'Connor, Harry Tenbrook, Jim Mason, Guy Wilkerson, Frank Hagney, William Tannen, James Kirkwood, Stanley Blystone, Edgar Dearing, Al Junde, Art Loeb, Blackie Whiteford, Bud Fine, Lane Chandler, Charles Evans, George Sherwood, Louis Mason, Roy Butler, Ralph Sanford, William McCormack, Chuck Hamilton. Following the Civil War several brothers head West with one going to work helping build the Santa Fe Railroad while the others become outlaws. Average action fare, with a great cast, for Randolph Scott fans.\n\n**3703** _ **Santa Fe Bound**_ **** Reliable, 1936. 56 min. D: Henri Samuels (Harry S. Webb). SC: Carl Krusada. With Tom Tyler, Jeanne Martel, Richard Cramer, Charles King, Slim Whitaker, Ed Cassidy, Lafe McKee, Wally West, Earl Dwire, Dorothy Woods, Ray Henderson. Falsely accused of murdering an old man bushwhacked by bandits, a cowboy pretends to be a crook so he can infiltrate and capture the gang responsible for the crime. A trifle better than most of his Reliable outings, this was Tom Tyler's final series film for that outfit.\n\n**3704** _ **Santa Fe Marshal**_ **** Paramount, 1940. 68 min. D: Lesley Selander. SC: Harrison Jacobs. With William Boyd, Russell Hayden, Marjorie Rambeau, Bernadene Hayes, Earle Hodgins, Britt Wood, Kenneth Harlan, William Pagan, George Anderson, Jack Rockwell, Eddie Dean, Fred Graham, Matt Moore, Tex Phelps, Cliff Parkinson, Horace B. Carpenter, Frank Ellis, Bob McKenzie, George Morrell, Duke Green, Billy Jones. Hopalong Cassidy goes undercover as a doctor to expose an outlaw gang. Okay series entry but nothing special.\n\n**3705** _ **Santa Fe Passage**_ **** Republic, 1955. 90 min. Color. D: William Witney. SC: Lillie Hayward. With John Payne, Rod Cameron, Faith Domergue, Slim Pickens, Anthony Caruso, Leo Gordon, Irene Tedrow, George Keymas, Tyler McVey, John Patrick, Hal Smith, Edward Colmans, Tom Monroe, Howard Negley, John Patrick, Earl Robie. A wagon train heading to Santa Fe is menaced by Kiowas and gun runners with the trail boss, despite his hatred of Indians, falling in love with a half-breed woman passenger. Standard, but action filled, Republic \"A\" effort.\n\n**3706** _ **Santa Fe Rides**_ **** Reliable, 1937. 58 min. D: Raymond Samuels (Bernard B. Ray). SC: Pliny Goodfriend. With Bob Custer, Eleanor Stewart, Ed Cassidy, David Sharpe, Roger Williams, Slim Whitaker, Lafe McKee, Snub Pollard, The Singing Cowboys (Lloyd Perryman, Rudy Sooter, Curley Hoag), Nelson McDowell, John Elliott. A rival tries to stop a cowboy and his musical group from getting a radio contract, also framing a woman's father and brother on a cattle theft charge. Bob Custer's final film tries to interpolate the then popular fad of having music in Westerns but the overall result is dismal.\n\n**3707** _ **Santa Fe Saddlemates**_ **** Republic, 1945. 56 min. D: Thomas Carr. SC: Bennett Cohen. With Sunset Carson, Linda Stirling, Olin Howlin, Roy Barcroft, Bud Geary, Kenne Duncan, George Chesebro, Robert Wilke, Henry Wills, Forbes Murray, Frank Jaquet, Josh (John) Carpenter, Rex Lease, Edmund Cobb, Nolan Leary, Fred Graham, George Magrill, Jack O'Shea, Carol Henry, Billy Vincent, Horace B. Carpenter, Bill Nestell, William McCall, Bob Reeves, Kansas Moehring, Rose Plummer, Bill Wolfe. The government sends an investigator to the U.S.-Mexican border to locate a diamond smuggling ring believed to be headquartered at an area ranch. Action from start to finish make this one of Sunset Carson's best movies.\n\n**3708** _ **Santa Fe Scouts**_ **** Republic, 1943. 55 min. D: Howard Bretherton. SC: Morton Grant and Betty Burbridge. With Bob Steele, Tom Tyler, Jimmie Dodd, Lois Collier, John James, Tom Chatterton, Elizabeth Valentine, Tom London, Budd Buster, Jack Ingram, Kermit Maynard, Rex Lease, Ed Cassidy, Yakima Canutt, Jack Kirk, Curley Dresden, Reed Howes, Edmund Cobb, Bud Geary, Carl Sepulveda, Kenne Duncan, Al Taylor. Three pals work for a rancher whose son has been framed on a murder charge and they try to obtain his freedom. The penultimate \"Three Mesquiteers\" film is more than passable entertainment.\n\n**3709** _ **Santa Fe Stampede**_ **** Republic, 1938. 56 min. D: George Sherman. SC: Luci Ward and Betty Burbridge. With John Wayne, Ray Corrigan, Max Terhune, William Farnum, June Martel, LeRoy Mason, Martin Spellman, Genee Hall, Walter Wills, Ferris Taylor, Tom London, Dick Rush, James T. Cassidy, George Chesebro, Charles King, Yakima Canutt, Bud Osborne, Richard Alexander, Griff Barnett, Nelson McDowell, Curley Dresden, George Morrell, Ralph Peters, Marin Sais, Charles Murphy, Robert Milasch, Chick Hannon, Blackjack Ward, Cliff Parkinson, Horace B. Carpenter, John Elliott, Jim Corey, Russ Powell, Frank O'Connor, George Sowards, Tex Driscoll, Murdock MacQuarrie, Duke R. Lee, Tex Phelps, Fred Parker, Bud McClure, Bill Wolfe. When crooks kill a miner whose successful claim was grubstaked by the Three Mesquiteers, Stony Brooke is accused of the crime and his pals Tucson and Lullaby try to clear him. Action packed entry in \"The Three Mesquiteers\" series with the stark murder of the miner and his little daughter in a buckboard well staged and heart rending.\n\n**3710** _ **The Santa Fe Trail**_ **** Paramount, 1930. 80 min. D: Edwin Knopf and Otto Brower. SC: Sam Mintz and Edward E. Paramore, Jr. With Richard Arlen, Rosita Moreno, Eugene Pallette, Mitzi Green, Junior Durkin, Hooper Atchley, Luis Alberni, Lee Shumway, Chief Yowlachie, Jack Byron, Blue Cloud, Chief Standing Bear. Three men lead a large sheep herd tended by Indians and arrange for them to graze on a Spaniard's ranch but when his barn burns he blames the tribesmen. Early talkie of interest to Richard Arlen fans.\n\n**3711** _ **Santa Fe Trail**_ **** Warner Bros., 1940. 110 min. D: Michael Curtiz. SC: Robert Buckner. With Errol Flynn, Olivia de Havilland, Raymond Massey, Ronald Reagan, Alan Hale, William Lundigan, Van Heflin, Gene Reynolds, Henry O'Neill, Guinn Williams, Alan Baxter, John Litel, Moroni Olsen, David Bruce, Hobart Cavanaugh, Charles D. Brown, Joseph Sawyer, Frank Wilcox, Ward Bond, Russell Simpson, Charles Middleton, Douglas Fowley, Erville Alderson, Spencer Charters, Suzanne Carnahan (Susan Peters), William Marshall, George Haywood, Wilfred Lucas, Russell Hicks, Napoleon Simpson, Roy Barcroft, Lane Chandler, Richard Kipling, Nestor Paiva, Trevor Bardette, Eddy Waller, Libby Taylor, Edmund Cobb, Creighton Hale, William Hopper, Addison Richards, the Rev. Neal Dodd, Harry Strang, Emmett Vogan, Selmer Jackson, Joseph Crehan, Clinton Rosemond, Theresa Harris, Lafe McKee, Grace Stafford, Bernice Pilot, Libby Taylor, Mildred Gover, Frank Mayo, Louis Jean Heydt, Jack Mower, Mira McKinney, Harry Cording, James Farley, Alan Bridge, John Meyer, Maris Wrixon, Lucia Carroll, Mildred Coles, Georgia Caine, Arthur Aylesworth, Walter Soderling, Henry Hall, Victor Kilian, Eddy Chandler, Ed Peil, Sr., Jess Lee Brooks. West Point graduates Jeb Stuart and George A. Custer are stationed in Kansas during the fight over the free soil question with both falling in love with the same woman and eventually becoming involved in the capture of abolitionist John Brown. Pseudo-historical drama makes for big scale entertainment; some video releases run 90 minutes.\n\n**3712** _ **Santee**_ **** Crown International, 1973. 93 min. Color. D: Gary Nelson. SC: Tom Blackburn. With Glenn Ford, Dana Wynter, Michael Burns, Jay Silverheels, Harry Townes, John Larch, Robert Wilke, Robert Donner, Taylor Lacher, Lindsay Crosby, Chuck Courtney, X Brands, John Hart, Boyd \"Red\" Morgan, Robert Mellard, Ben Zeller, William Ford, John Bailey, Caruth C. Byrd. After his son is killed, a bounty hunter becomes the mentor of a young man whose outlaw father was shot by the gunman. Glenn Ford's work is the highlight of this otherwise average feature.\n\n_**Santo and the Border of Terror**_ see _**Santo in the Frontier of Terror**_\n\n**3713** _ **Santo in the Frontier of Terror**_ **** Producciones Geminis\/Cinematografica R.A., 1979. 85 min. Color. D-SC: Rafael Perez Grovas. With Santo, Carmen del Valle, Jean Safont, Federico Falcon, Sarita Gomez, Gerardo Reyes, Carlos Suarez, Cesar Gomez, Enrique Estrada, Lilia Landua, Victor Manuel Mar, Miguel Angel Fuentes, Armando Garcia Vaca, Angelica Sierra, Guillermo Inclan, Oscar Ricci, Carnicero Aguilar, sangre Chicana, Jungla, Mocho Kotta, Karloff Lagarde, Bobby Lee, Ringo Mendoza. In an effort to find two missing men who crossed the border into Texas to make money to help a girl regain her sight, a masked wrestler ends up opposing a mad scientist and his zombies. Not one of wrestling great Santo's better outings, released in Mexico as _**Santo en la Frontera del Terror**_ (Santo on the Frontier of Terror) and on video as _**Santo and the Border of Terror**_.\n\n**3714** _ **Santo vs. the Riders of Terror**_ **** Cinematographica Calderon, 1970. 85 min. Color D: Rene Cordona. SC: Rene Cordona and Jesus Valezquez Quintero. With Santo, Mary Montiel, Armando Silvestre, Julia Aldama, Gregorio Cassals, Ivonne Govea, Carlos Agosti, Carlos Suarez, Nathaniel \"Frankenstein\" Leon, Gloria Chavez, Ruben Marquez, Rene Barrera, Margarito Luna, Victor Blanco, Felix Gonzalez, Armando Acosta, Adolfo Aguilar, Regino Herrera, Jesus Gomez, Alfred Gutierrez. Crime fighting masked wrestler Santo helps a lawman combat a gang of lepers looting and terrorizing the countryside. Santo's fans will like this Mexican horror Western but others beware; released in its homeland as _**Santo Contra los Jinetes del Terror**_ (Santo Against the Terror Riders).\n\n_**Sartana**_ see _**If You Meet Sartana, Pray for Your Death**_\n\n**3715** _ **Sartana Does Not Forgive**_ **** Balcazar\/FIDA, 1968. 92 min. Color. D: Alfonso Balcazar. SC: Giovanni Simonelli. With Gilbert Roland, George Martin, Jack Elam, Tony Norton (Alfio Caltabiano), Hugo Blanco, Gerard Tichy, Diana Lorys, Donatella Turri, Rosalba Neri, Tomas Torres, Gustavo Re, Miguel del la Riva. Sartana, searching for the man who raped and murdered his fiancee, forms an uneasy alliance with an aging gunman. Exciting adventure in the \"Sartana\" series issued in Europe as _**Sonora**_.\n\n_**Sartana in the Valley of Death**_ see _**Sartana in the Valley of Vultures**_\n\n**3716** _ **Sartana in the Valley of Vultures**_ **** Cire Films, 1970. 95 min. Color. D-SC: Roberto Mauri. With William Berger, Wayde Preston, Pamela Tudor, Jolanda Modio, Alan Collins (Luciano Pignozzi), Aldo Berti, Carlo Giordana, Franco De Rosa, Josiane Tanzilli, Franco Ressel, Betsy Bell, Federico Boldo, Bruno Ukmar, Claudio Aponte, Brune Are, Gaetano Imbro. A gunman helps three brothers get out of jail in return for the gold they stole only to be double crossed and left in the desert where he is rescued by a woman after the bounty offered on him. Well filmed \"Sartana\" segment made in Italy as _**Sartana nella Valle degli Avvoltoi**_ (Sartana in the Valley of Vultures); also called _**Ballad of Death of Valley**_ and _**Sartana in the Valley of Death**_.\n\n**3717** _ **Sartana Kills Them All**_ **** Hispamex, 1971. 95 min. Color. D: Rafael Romero Marchent. SC: Joaquin Romero Marchent and Santiago Moncada. With John (Gianni) Garko, William Bogart (Guglielmo Spoletini), Maria Silva, Carlos Romero Marchent, Luis Induni, Raf Baldassare, Paco Sanz, Cris Huerta, Carlos (Charly) Bravo, Maria Martin, Andres Mejuto, Alvardo de Luna, Alejandro de Enciso, Lorenzo Robledo, Jesus Guzman, Cristina Iosani. Two outlaws slay everyone in their path as they seek $100,000 but meet their match in a widow who wants the money to buy a saloon. Mediocre outing in the \"Sartana\" series, originally released in Spain as _**Un Par des Asesinos**_ (A Pair of Killers).\n\n_**Sartana the Gravedigger**_ see _**I Am Sartana, Your Angel of Death**_\n\n**3718** _ **Saskatchewan**_ **** Universal-International, 1954. 87 min. Color. D: Raoul Walsh. SC: Gil Doud. With Alan Ladd, Shelley Winters, Robert Douglas, J. Carrol Naish, Hugh O'Brian, Richard Long, Jay Silverheels, Antonio Moreno, Lowell Gilmore, George J. Lewis, Frank Chase, John Cason, Henry Wills, Robert D. Herron, Russell Saunders, Jonas Applegate, Rex Reason (narrator). A Canadian Mounted Policeman tries to prevent Sioux Indians from forcing a peaceful Cree tribe in joining them in a rebellion. The story is not much but the scenic locales and fine photography (by John Seitz) make up for it. British title: _**O'Rourke of the Royal Mounted**_.\n\n**3719** _ **Sasquatch**_ **** North American Film Enterprises, 1976. 94 min. Color. D: Ed Ragozzini. SC: Edward H. Hawkins. With George Lauris, Jim Bradford, William Emmons, Steve Boergadine, Joe Morello, Ken Kenzle. Several men go into the wilds of British Columbia in search of the legendary Bigfoot creature. Average semi-documentary speculation feature. Alternate title: _**Sasquatch, the Legend of Bigfoot**_.\n\n_**Sasquatch, the Legend of Bigfoot**_ see _**Sasquatch**_\n\n**3720** _ **Satan's Cradle**_ **** United Artists, 1949. 60 min. D: Ford Beebe. SC: J. Benton Cheney. With Duncan Renaldo, Leo Carrillo, Ann Savage, Douglas Fowley, Byron Foulger, Buck Bailey, George DeNormand, Claire Carleton, Wes Hudman. In a frontier town the Cisco Kid and Pancho find themselves up against a crooked lawyer and his pretty saloon owner accomplice. More than adequate \"Cisco Kid\" program outing.\n\n**3721** _ **Satan's Harvest**_ **** Killarney Studios, 1970. 104 min. Color. D: George Montgomery. With George Montgomery, Tippi Hedren, Matt Munro, Davy Kaye, Brian O'Shaughnessy, Roland Robinson, Tromp Terreblanche, Melody O'Brian, Don Barrigo, George Peters, Simon Sabeia. An American detective goes to South Africa to take over a ranch he has inherited and finds it is being used as headquarters for drug smugglers. Colorful modern-day action drama filmed in South Africa and Rhodesia.\n\n**3722** _ **The Savage**_ **** Paramount, 1952. 95 min. Color. D: George Marshall. SC: Sidney Boehm. With Charlton Heston, Susan Morrow, Joan Taylor, Peter Hansen, Don Porter, Ted De Corsia, Milburn Stone, Richard Rober, Howard Negley, Ian MacDonald, Angela Clarke, Orley Lindgren, Michael Tolan, Frank Richards, John Miljan, Henry Wills, Roger Creed, Kirk Alyn, Marion Gray, David Miller, John S. Peters, Jimmie Dundee, Jim Hayward, Willard W. Willingham, James Van Horn, Iron Eyes Cody, Frank Cordell, Chief American Horse, Ben Black Elk, Sr. A white man, raised by Indians, is torn between loyalties when war breaks out between the two peoples. Pretty engrossing tale with Charlton Heston handling the lead role in good fashion.\n\n_**The Savage American**_ see _**The Talisman**_\n\n**3723** _ **The Savage Eye**_ **** NBC-TV\/Universal, 1971. 74 min. Color. D: Leo Penn. SC: Leon Tokatyan. With Robert Stack, Jim Hutton, Peter Duel, Mariana Hill, Susan Saint James, Geoffrey Duel, John Randolph, Robert Foulk, Kelly Thordsen. An investigator for a large publishing company looks into reports that an ecology documentary film made by his employers has caused trouble among lumberjacks. Average drama originally telecast February 19, 1971, as a segment of \"The Name of the Game\" (NBC-TV, 1968\u201371).\n\n**3724** _ **Savage Frontier**_ **** Republic, 1953. 54 min. D: Harry Keller. SC: Dwight Babcock and Gerald Geraghty. With Allan \"Rocky\" Lane, Eddy Waller, Bob Steele, Dorothy Patrick, Roy Barcroft, Richard Avonde, William Phipps, Jimmy Hawkins, Lane Bradford, John Cason, Kenneth MacDonald, Bill Henry, John Hamilton, Art Dillard, Gerry Flash. A former convict, now a farmer, risks losing his parole when he tries to prove to a U.S. marshal that a prominent citizen is behind an outlaw gang. Solid entry near the end of Allan Lane's \"Famous Westerns\" series highlighted by Bob Steele as the reformed gunman.\n\n**3725** _ **Savage Gringo**_ **** Italian International Film\/Castilla Cinematografica, 1965. 82 min. Color. D-SC: Antonio Roman. With Ken Clark, Yvonne Bastien, Piero Lulli, Renato Rossini, Alfonso Rojas, Antonio Gradoli, Angel Ortiz, Livio Lorenzon, Aldo Sambrell, Renato Terra, Paco Senz. A cowboy, who goes to work for a rancher hated by both his wife and a rival, is accused of killing the local sheriff. Star Ken Clark adds some life to this Italian oater issued there as _**Nebraska il Pistolero**_ (Nebraska the Gunman); some sources claim Mario Bava co-directed.\n\n**3726** _ **The Savage Guns**_ **** Metro-Goldwyn-Mayer, 1961. 83 min. Color. D: Michael Carreras. SC: Jimmy Sangster and Edmund Morris. With Richard Basehart, Don Taylor, Alex Nicol, Pacquita Rico, Maria Granada, Jose Nieto, Fernando Rey, Felix Fernandez, Francisco Camoiras, Antonio Fuentas, Sergio Mendizabal, Jose Manuel Martin, Pilar Caballero, Rafael Albaicin, Victor Israel. A Civil War veteran, tired of violence, rides into a Mexican village and teams with a former Confederate officer to combat an evil land grabber and his gang. Adequate action feature made by the British in Spain as _**Tierra Brutal**_ (Brutal Land).\n\n**3727** _ **Savage Guns**_ **** Demofilo Fidani, 1971. 85 min. Color. D: Miles Deem (Demofilio Fidani). SC: Miles Deem (Demofilo Fidani) and Mila Vitelli. With Robert Wood, Dean Stratford, Dennis Colt, Custer Gail, Simone Blondell, Gordon Mitchell, Peter Martell, Lincoln Tate, Marina Malfatti, Pietro Furnelli, Piera Bruni, Attilio Severini. After seeing a gunman kill his brother after a holdup, a wounded man plots vengeance. Somewhat appealing Spaghetti Western with more humor than most of its ilk; made in Italy by Glassia Cinematografica as _**Era Sam Walbash**_ **...** _ **lo Chiamavano \"Cosi Sia\"**_ (It was Sam Walbash, They Called Him Thus) and also titled _**His Name Was Sam Walbash, but They Call Him Amen**_.\n\n**3728** _ **The Savage Horde**_ **** Republic, 1950. 90 min. D: Joseph Kane. SC: Kenneth Gamet. With William Elliott, Adrian Booth, Grant Withers, Jim Davis, Barbara Fuller, Noah Beery, Jr., Douglas Dumbrille, Bob Steele, Will Wright, Roy Roberts, Earle Hodgins, Stuart Hamblen, Hal Taliaferro, Lloyd Ingraham, Marshall Reed, Crane Whitley, Charles Stevens, James Flavin, Ed Cassidy, Kermit Maynard, George Chesebro, Jack O'Shea, Monte Montague, Bud Osborne, Reed Howes, Chick Hannon, Bob Burns, Frank O'Connor, Foxy Callahan, Chuck Baldra. A reformed gunfighter sides with ranchers menaced by a land grabber while members of his former gang ride with the crook. William Elliott's last big budget Western is a fine action drama with a great cast.\n\n_**Savage Hunter**_ see _**Mission to Glory: A True Story**_\n\n**3729** _ **The Savage Innocents**_ **** Paramount, 1960. 110 min. Color. D-SC: Nicholas Ray. With Anthony Quinn, Yoko Tani, Peter O'Toole, Anna May Wong, Carlo Guistini, Marie Yang, Marco Guglielmi, Lee Montague, Andy Ho, Anthony Chin. Two Canadian Mounties are assigned to bring in an Eskimo who accidentally killed a missionary. Good photography highlights this mundane story.\n\n_**Savage Journey**_ see _**Brigham**_\n\n_**Savage Justice**_ see _**Bitter Springs**_\n\n_**The Savage Land**_ see _**This Savage Land**_\n\n**3730** _ **Savage Pampas**_ **** Comet, 1967. 99 min. Color. D: Hugo Fregonese. SC: Hugo Fregonese and John Melson. With Robert Taylor, Ron Randell, Ty Hardin, Rosenda Monteros, Marc Lawrence, Felicia Roc (Fela Roque), Angel Del Pozo. Mario Lozano, Enrique Avila, Laura Granados, Milo Quesada, Charles Fawcett, Julia Pena, Jose Nieto, Lucia Prado, George Rigaud. In Argentina, an Arm captain tracks an outlaw gang made up of deserters and Indians. Pretty fair action drama filmed in South America; a remake of the 1946 Argentine feature _**Pampa Barbara**_ (Barbarous Pampas), also helmed by Hugo Fregonese.\n\n_**Savage Red\u2014Outlaw White**_ see _**40 Graves for 40 Guns**_\n\n_**Savage Run**_ see _**Run, Simon, Run**_\n\n**3731** _ **Savage Sam**_ **** Buena Vista, 1963. 103 min. Color. D: Norman Tokar. SC: Fred Gipson and William Tunberg. With Brian Keith, Tommy Kirk, Kevin Corcoran, Dewey Martin, Jeff York, Royal Dano, Marta Kristen, Rafael Campos, Slim Pickens, Rodolfo Acosta, Pat Hogan, Dean Fredericks, Brad Weston. Two boys, along with a neighbor girl, are kidnapped by Indians and it is up to the brothers' dog to lead a rescue party to free them. Standard Walt Disney follow-up to _**Old Yeller**_ (q.v.).\n\n**3732** _ **The Savage Seven**_ **** American International, 1968. 96 min. Color. D: Richard Rush. SC: Michael Fisher. With Robert Walker, Larry Bishop, Adam Roarke, Joanna Frank, John Garwood, Max Julien, Richard Anders, Duane Eddy, Chuck Bail, Mel Berger, Billy Rush, John \"Bud\" Cardos, Susanna Darrow, Beach Dickerson, Gary Kent, Penny Marshall, Walt Robles. Indians ally themselves with a motorcycle gang to oppose the town boss who controls their lives. Violent combination of the modern Western and cycle genres by producer Dick Clark; the result is nothing special for fans of either type of film.\n\n**3733** _ **The Savage Wild**_ **** American International, 1970. 103 min. Color. D-SC: Gordon Eastman. With Gordon Eastman, Carl Spore, Maria Eastman, Arlo Curtis, Jim Timiaough, Robert Wellington Kirk, John Payne, Charles Abou, Alex Dennis, Charley Davis, Wilber O'Brien. A documentary maker and his crew film the wildlife in Northern Canada, just below the Arctic Circle, and raise baby wolves. Well made and entertaining British docudrama.\n\n**3734** _ **Savages**_ **** ABC-TV, 1974. 74 min. Color. D: Lee H. Katzin. SC: William Wood. With Andy Griffith, Sam Bottoms, Noah Beery, James Best, Randy Boone, Jim Antonio, Jim Chandler. A New York City attorney accidentally kills an old prospector while on a hunting trip and to cover up the crime he tries to murder a guide. The story has been told many times before with all kinds of variations but this TV movie is nonetheless enjoyable.\n\n**3735** _ **Scalawag**_ **** Paramount, 1973. 93 min. Color. D: Kirk Douglas. SC: Albert Maltz and Sid Fleishman. With Kirk Douglas, Mark Lester, Neville Brand, George Eastman, Don Stroud, Lesley Ann Down, Danny DeVito, Mel Blanc, Phil Brown, Davor Antholic, Stole Aradjelovic, Fabijan Sovagovic, Shaft Douglas. In the 1840s a lovable peg-legged pirate leads his band of cutthroats on a treasure hunt in California. Less than average family film made in Yugoslavia, burdened by too many songs.\n\n**3736** _ **The Scalphunters**_ **** United Artists, 1968. 102 min. Color. D: Sydney Pollack. SC: William Norton. With Burt Lancaster, Shelley Winters, Telly Savalas, Ossie Davis, Armando Silvestre, Dan Vadis, Dabney Coleman, Paul Picerni, Nick Cravat, John Epper, Jack Williams, Chuck Roberson, Tony Epper, Agapito Roldan, Gregorio Acosta, Marco Antonio, Raul Hernandez, Alejandro Lopez, Pedro Aguilar, Antonio Arzate, Cuco Velazquez. A trapper, whose furs are stolen by Indians, joins forces with a runaway slave to get them back. Fair action oater comedy with plenty of plot twists.\n\n**3737** _ **Scaplock**_ **** ABC-TV\/Columbia, 1966. 100 min. Color. D: James Goldstone. SC: Steven Kandel. With Dale Robertson, Diana Hyland, Lloyd Bochner, Gary Collins, David Sheiner, Steve Ihnat, Robert Random, Roger Torrey, Sandra Smith, James Westerfield, John Anderson, Todd Armstrong, Robert Cinder, Cliff Hall, Woodrow Parfrey, James Doohan, Herbert Voland, Eddie Firestone, Stephanie Hill, Harry Bausch, Paul Sorensen, Jerry Summers. A notorious gambler wins a railroad in a poker game and learns running a business is more than giving orders. One of the first movies made for network television and a mediocre pilot for \"The Iron Horse\" (ABC-TV, 1966\u201368) series.\n\n**3738** _ **Scalps**_ **** 21st Century, 1983. 82 min. Color. D-SC: Fred Olen Ray. With Kirk Alyn, Carroll Borland, Ann Robinson, Richard Hench, Barbara Magnuson, Frank MacDonald, Roger Maycock, Forrest J. Ackerman, Carol Flockart. A group of students on an archaeological dig in the desert are possessed by the spirit of an Indian sorcerer. Low budget but fairly interesting modern Western starring Kirk Alyn, the screen's original Superman.\n\n**3739** _ **Scandalous John**_ **** Buena Vista, 1971. 117 min. Color. D: Robert Butler. SC: Bill Walsh and Don DaGradi. With Brian Keith, Michele Carey, Alfonso Arau, Rick Lenz, Harry Morgan, Simon Oakland, Bill Williams, Christopher Dark, Fran Ryan, Bruce Glover, Richard Hale, James Lydon, John Ritter, Iris Adrian, Larry D. Mann, Jack Raine, Booth Colman, Edward Faulkner, Bill Zuckert, John Zaremba, Robert Padilla, Ben(ny) Baker, Alex Tinne, Paul Koslo, William O'Connor, Sam Edwards, Lenore Stevens, Jose Nieto, Margarita Mendoza, Joseph Gutierrez, Freddie Hernandez. An aging man living in a fantasy world goes up against a land baron trying to take over and flood his property. Overlong, minor league Disney Western.\n\n**3740** _ **Scar Tissue**_ **** NBC-TV\/Universal, 1974. 76 min. Color. D: Andrew V. McLaglen. SC: Mann Rubin. With Richard Boone, Kurt Russell, Dick Haymes, Chill Wills, Tom Drake, Rick Lenz, Harry Morgan, Dennis Rucker, William Campbell, Jason Evers, Hilarie Thompson, Albert Salmi, Terry Wilson, Charles Aidman, Jim Burk, Terry Leonard. A sheriff and his deputies hunt for a young man planning to kill the father who deserted him as an infant. Entertaining drama initially telecast as an episode of \"Hec Ramsey\" (NBC-TV, 1972\u201374).\n\n**3741** _ **Scarlet Angel**_ **** Universal-International, 1952. 81 min. Color. D: Sidney Salkow. SC: Oscar Brodney. With Yvonne De Carlo, Rock Hudson, Richard Denning, Bodil Miller, Amanda Blake, Henry O'Neill, Henry Brandon, Maude Wallace, Dan Riss, Whitfield Connor, Tol Avery, Arthur Page, George Hamilton, Dale Van Sickel, Mickey Pfleger, Harry Harvey, George Spaulding, Thomas Browne Henry, Fred Graham, Fred Coby, Eddie Dew, Nolan Leary, Wilma Francis, Leo Curley, Dabbs Greer, Joe Forte, Coleman Francis, Charles Horvath, Bud Wolfe, Creighton Hale, Carl Saxe. After stealing money from a sea captain, a pretty girl befriends a woman and her baby and when the woman dies she cares for the child, assumes the mother's name and becomes a member of San Francisco's Hob Hill society. Fair remake of _**The Flame of New Orleans**_ (q.v.).\n\n**3742** _ **The Scarlet Brand**_ **** Big 4, 1932. 58 min. D: J.P. McGowan. SC: Oliver Drake and Ethel Hill. With Bob Custer, Betty Mack, Robert Walker, Frank Ball, Duke R. Lee, Nelson McDowell, Blackie Whiteford, Frederick Ryter, William Nolte, Jack Long, Jim Corey, Bob Burns, Bud McClure, Rube Dalroy. Branded after being framed for cattle rustling, a cowboy vows revenge and gets a job on a ranch owned by the man he thinks is the culprit. Impassive Bob Custer strikes again in this tame affair.\n\n**3743** _ **Scarlet Days**_ **** Paramount-Artcraft, 1919. 77 min. D: D.W. Griffith. SC: Stanner E.V. Taylor. With Richard Barthelmess, Carol Dempster, Clarine Seymour, Ralph Graves, Eugenie Besserer, George Fawcett, Walter Long, Kate Bruce, Rhea Haines, Adolph Lestina, Herbert Sutch, J. Wesley Warner. In 1849 California a Mexican bandit tries to help a young woman whose saloon gal mother is about to be hanged for killing a thief. Too many plot ploys hamper this D.W. Griffith Production; Richard Barthelmess is grand as Alvarez the bandit.\n\n**3744** _ **The Scarlet Horseman**_ **** Universal, 1946. 13 Chapters. D: Ray Taylor and Lewis D. Collins. SC: Joseph O'Donnell, Tom Gibson and Patricia Harper. With Peter Cookson, Victoria Horne, Paul Guifoyle, Virginia Christine, Danny Morton, Fred Coby, Janet Shaw, Jack Ingram, Edward M. Howard, Harold Goodwin, Ralph Lewis, Edmund Cobb, Cy Kendall, Helen Bennett, Guy Wilkerson, Al Woods, Frank Lackteen, Jack Rockwell, Rex Lease, William Desmond, Marshal Reed, Hal Taliaferro, Mauritz Hugo, Ellen Corby, Ernie Adams, Pierce Lyden, Budd Buster, Dick Curtis, Lee Roberts, Frank McCarroll, Bob Duncan, Jack Kirk, Ralph Moody, Hank Patterson, Paul Birch. A government agent becomes \"The Scarlet Horseman,\" an ancient Indian idol, to prevent a tribal uprising and save female relatives of officials kidnapped and held prisoner by a chief. Slow moving chapter play.\n\n**3745** _ **Scarlet River**_ **** RKO Radio, 1933. 57 min. D: Otto Brower. SC: Harold Shumate. With Tom Keene, Dorothy Wilson, Roscoe Ates, Edgar Kennedy, Creighton (Lon, Jr.) Chaney, Hooper Atchley, Betty Furness, Billy Butts, Yakima Canutt, Jack Mower, Jim Mason, Perry Ivins, Paddy O'Flynn, Jack Raymond, Joel McCrea, Myrna Loy, Bruce Cabot, Julie Haydon, Rochelle Hudson. On location shooting a Western, a cowboy star tries to help a pretty ranch owner being swindled by her corrupt foreman. Fast paced and entertaining Tom Keene series entry, with a look at movie making (including guest stars) as well as the usual genre fare.\n\n**3746** _ **Scorching Fury**_ **** Fraser Productions, 1952. 64 min. D: Rick Freers. SC: James Craig and Richard Devon. With Richard Devon, William Leslie, Peggy Nelson, Sherwood Price, Phyllis Coates, Rory Mallinson, Charles Morton, Arthur Dineen, Eddie McLean, Allen Windsor, Twyla Paxton. Outlaws ambush a stagecoach and leave the passengers stranded in the desert, one of them an lawman out to bring in the gang leader. Bedraggled, low grade obscure production; really bad.\n\n**3747** _ **Scott Free**_ **** NBC-TV\/Universal, 1976. 74 min. Color. D: William Wiard. SC: Stephen J. Cannell. With Michael Brandon, Stephan Nathan, Susan Saint James, Robert Loggia, Ken Swofford, Allan Rich, Paul Koslo, Cal Bellini, Michael Lerner. A gambler wins a few acres of land in a poker game but soon finds to his dismay it is sought by gangsters and Indians. Mundane pilot for a television series that did not sell.\n\n**3748** _ **Scream**_ **** Cal-Com Releasing, 1985. 81 min. Color. D-SC: Byron Quisenberry. With Pepper Martin, Ethan Wayne, Alvy Moore, Woody Strode, Hank Worden, Gregg Palmer, Julie Marine, Bobby Diamond, Joseph Alvarado, Anna Bronson, Nancy St. Marie. A group of hikers arrive in a Western ghost town where they are stalked by an insane killer. Like watching paint, or in this case blood, dry; made in 1981 as _**The Outing**_.\n\n**3749** _ **Scream of the Wolf**_ **** ABC-TV\/Metromedia, 1974. 74 min. Color. D: Dan Curtis. SC: Richard Matheson. With Peter Graves, Clint Walker, Jo Ann Pflug, Philip Carey, Don Megowan, Brian Richards, Lee Paul, James Storm, Bonnie Van Dyke, Dean Smith, Orville Sherman, Grant Owens, William Baldwin. A noted hunter emerges from retirement to destroy a man killing beast and evidence mounts his quarry might be a werewolf. Fair horror TV Western with Clint Walker dominating as the hero's friend turned foe.\n\n**3750** _ **Sea of Grass**_ **** Metro-Goldwyn-Mayer, 1947. 131 min. D: Elia Kazan. SC: Marguerite Roberts. With Spencer Tracy, Katharine Hepburn, Melvyn Douglas, Robert Walker, Phyllis Thaxter, Edgar Buchanan, Harry Carey, Ruth Nelson, William \"Bill\" Phipps, Robert Armstrong, James Bell, Robert Barrat, Charles Trowbridge, Russell Hicks, Trevor Bardette, Morris Ankrum, Dan White, Glenn Strange, Douglas Fowley, Guy Wilkerson, Buddy Roosevelt, Earle Hodgins, Robert Bice, John Rice, Hank Worden, George Reed, Dorothy Vaughn, Vernon Dent, Erville Alderson, Leota Lorraine, Wyndham Standing, William Holmes, Henry Adams, Joseph Crehan, John Hamilton, John Vosper, Budd Fine, Chief Many Treaties, Nora Cecil, Fred Graham, Frank Hagney, Frank Austin, Ray Teal, Eddie Acuff, Davison Clark, Fred Gilman, Dick Rush, Charles Middleton, Carol Nugent, Jimmy Hawkins, Wheaton Chambers, George Magrill, Nolan Leary, Charles McAvoy, Eddy Waller, Forrest Taylor, Gene (Roth) Stutenroth, Joe Bernard, Frank Darien, William Challee, Stanley Andrews, Ralph Littlefield, Whit Bissell, Gertrude Chorre, Patty Smith, Jack Baxley, Dick Barron. A feud develops over grasslands between ranchers while a cattle baron learns his son was fathered by a long time rival. Ponderous Western, well made and acted, but dull.\n\n**3751** _ **The Search**_ **** Wrather Corporation, 1956. 75 min. Color. D: Earl Bellamy. SC: Thomas Seller, Robert E. Schaefer, Eric Friewald, Hilary Creston Rhodes and Robert Leslie Bellem. With Clayton Moore, Jay Silverheels, Allen Pinson, Wayne Burson, Aline Towne, Richard Crane, Denver Pyle, Lane Bradford, John Crawford, Jeanne Bates, Terry Frost, Bill Henry, Keith Richards, Charles Wagenheim, House Peters, Jr., Tom Steele, Brad Morrow, Don Turner, David T. Armstrong, Baynes Barron, Steve Raines, James Baird, Mary Newton, Gregg Barton, Robert Burton, Ric Roman, Larry Jans. At Christmas time, the Lone Ranger and Tonto help a boy in protecting his dog, look for a lost father and try to find a stolen jewel encrusted cross. Nicely done holiday telefeature from the episodes \"The Breaking Point,\" \"The Christmas Story\" and \"The Cross of Santo Domingo\" of \"The Lone Ranger\" (ABC-TV, 1949\u201357) series\n\n**3752** _ **The Searchers**_ **** Warner Bros., 1956. 119 min. Color. D: John Ford. SC: Frank S. Nugent. With John Wayne, Jeffrey Hunter, Vera Miles, Ward Bond, Natalie Wood, John Qualen, Olive Carey, Henry Brandon, Ken Curtis, Harry Carey, Jr., Antonio Moreno, Hank Worden, Lana Wood, Walter Coy, Dorothy Jordan, Pippa Scott, Patrick Wayne, Beulah Archuletta, Jack Pennick, Peter Mamakos, Chuck Roberson, Nacho Galindo, Robert Lyden, Mae Marsh, Dan Borgaze, Cliff Lyons, Terry Wilson, Frank McGrath, Chuck Hayward, Fred Kennedy, Slim Hightower, Billy Cartledge, Dale Van Sickel, Henry Wills, Billy Yellow. A Civil War veteran and his niece's fiance spend years searching for the woman's little sister who was kidnapped by Indians. One of the all-time great classic Westerns; a must see feature film.\n\n**3753** _ **Second Chance**_ **** ABC-TV\/Metromedia, 1972. 74 min. Color. D: Peter Tewksbury. SC: Michael Morris. With Brian Keith, Elizabeth Ashley, Kenneth Mars, William Windom, Pat Carroll, Avery Schreiber, Rosey (Roosevelt) Grier, Juliet Prowse, Ann Morgan Builbert, Mark Savage, Ned Wertimer, Bret Parker, Emily Yancy. A stockbroker guys a Nevada ghost town and turns it into a resort for people who never had a chance in life. Passable modern-day TV feature.\n\n**3754** _ **The Second Greatest Sex**_ **** Universal-International, 1955. 87 min. Color. D: George Marshall. SC: Charles Hoffman. With Jeanne Crain, George Nader, Kitty Kallen, Bert Lahr, Mamie Van Doren, Keith Andes, Kathleen Case, Paul Gilbert, Tommy Rall, Edna Skinner, Jimmy Boyd, Cynthia May Carver (Cousin Emmy), The Midwesterners, Ward Ellis, Mary Marlo, Sheb Wooley, George Wallace, Harry Harvey, Sharon Bell, Barrie Chase, Diana Darrin. When the men of two Kansas towns spend all their time fighting over which will become the county seat their wives revolt. Bizarre, jaw dropping Western musical comedy based on Aristophanes' play _Lysistrata_.\n\n**3755** _ **The Second Time Around**_ **** 20th Century\u2013Fox, 1961. 99 min. Color. D: Vincent Sherman. SC: Oscar Paul and Dan Hansen. With Debbie Reynolds, Steve Forrest, Andy Griffith, Thelma Ritter, Juliet Prowse, Ken Scott, Isobel Elsom, Rodolfo Acosta, Timothy Carey, Tom Greenway, Eleanor Audley, Blossom Rock, Tracy Stratford, Jimmy Garrett, Lisa Pons, Nicky Blair. A young widow and her children come to Arizona in 1912 and she is quickly romanced by two men, including a sheriff. Very pleasant Western comedy.\n\n_**Secret Barriers**_ see _**The Great Barrier**_\n\n**3756** _ **Secret of Captain O'Hara**_ **** Lacy International\/Cire Films, 1968. 99 min. Color. D-SC: Arturo Ruiz Castillo. With German Cobos, Marta Padovan, Vidal Molina, Frank Brana, Charo Tejeiro, Jose Canalejas, Tomas Blanco, Angel Ter, Rafael Hernandez, Jorge Vico, Emilio Sanchez, Rafael Albaicin. After being court-martialed and demoted, an Army captain leads a wagon train to the fort where the officer who gave false evidence against him is in charge and he falls in love with the man's wife while the stockade is about to be attacked by marauding Indians. Complicated Italian-Spanish Spaghetti Western with well staged Indian attacks. Original title: _**El Secreto del Captain O'Hara**_ (The Secret of Captain O'Hara).\n\n**3757** _ **The Secret of Convict Lake**_ **** 20th Century\u2013Fox, 1951. 83 min. D: Michael Gordon. SC: Oscar Paul and Victor Trivas. With Glenn Ford, Gene Tierney, Ethel Barrymore, Zachary Scott, Ann Dvorak, Barbara Bates, Cyril Cusack, Richard Hylton, Helen Westcott, Jeanette Nolan, Ruth Donnelly, Harry Carter, Jack Lambert, Mary Carroll, Houseley Stevenson, Charles Flynn, David Post, Max Wagner, Raymond Greenleaf, Ray Teal, Tom London. A group of escaped convicts arrive in a town populated by women with one of them after the man who sent him to jail while another is after hidden money and the accuser's sister. Murky melodrama, well acted and fairly entertaining.\n\n**3758** _ **Secret of Navajo Cave**_ **** Key International, 1976. 87 min. Color. D-SC: James T. Flocker. With Rex Allen (narrator), Holger Kasper, Steve Benally, Jr., Johnny Guerro. Two boys fight with a cougar while pursuing their stray goat. Pleasant outdoor film with a long prologue that somewhat detracts from its adventure aspects; also called _**Legend of Cougar Canyon**_.\n\n**3759** _ **Secret of Outlaw Flats**_ **** Allied Artists, 1953. 54 min. D: Frank McDonald. SC: Bill Raynor. With Guy Madison, Andy Devine, Kristine Miller, Richard Avonde, John Crawford, Bobby Jordan, Tristram Coffin, Ed Clarke, William Haade, Jane Adams, Wade Crosby, Reed Howes, Riley Hill, Lennie Geer. U.S. marshals Wild Bill Hickok and Jingles P. Jones get on the trail of hooded outlaws stealing from ranchers at the behest of a corrupt cattle buyer and oppose a silver smelter who uses a gunman to rob his shipments. Okay theatrical feature made up of the \"Outlaw Flats\" and \"Silver Stage Holdup\" episodes of \"The Adventures of Wild Bill Hickok\" (1951\u201358) TV series.\n\n**3760** _ **The Secret of the Pueblo**_ **** William Steiner, 1923. 55 min. D: Neal Hart. SC: Alvin J. Neitz (Alan James). With Neal Hart, Hazel Deane, Tom Grimes, Monte Montague, John Blake. After a crooked lawyer and a mining engineer connive to obtain a ranch because it houses a lost mine and a water source, the owner's daughter is kidnapped by the Indian guardians of the treasures and taken to their secret altar room, but a cowboy comes to her rescue. Typical low budget silent oater of the 1920s; one of the few available Neal Hart vehicles which he made for his production company.\n\n**3761** _ **Secret of the Wastelands**_ **** Paramount, 1941. 66 min. D: Derwin Abrahams. SC: Gerald Geraghty. With William Boyd, Andy Clyde, Brad King, Barbara Britton, Douglas Fowley, Keith Richards, Soo Young, Gordon Hart, Hal Price, Earl Gunn, Ian MacDonald, Richard Loo, Jack Rockwell, John Rawlins, Lee Tung Foo, Roland Got, Bill Nestell, Charles Murphy. A group of Chinese try to stop an expedition to a lost ruins where they have a hidden city as Hopalong Cassidy, the caravan's leader, attempts to rescue a young woman who has been kidnapped. _**Lost Horizon**_ (Columbia, 1937) out West with an interesting story and nice scenery that make for a better than average series entry; based on the novel by Bliss Lomax (Harry Sinclair Drago).\n\n**3762** _ **Secret of Treasure Mountain**_ **** Columbia, 1956. 68 min. D: Seymour Friedman. SC: David Lang. With Valerie French, Raymond Burr, William Prince, Lance Fuller, Susan Cummings, Pat Hogan, Reginald Sheffield, Rodolfo Hoyos, Paul McGuire, Tom Hubbard, Boyd Stockman. Several men search for Indian treasure in the desert and find an old prospector and his daughter living next to the guardian of the riches. Producer Wallace MacDonald made a fairly interesting film although its low budget is evident; a reworking of _**Lust for Gold**_ (q.v.), also with William Prince.\n\n**3763** _ **Secret Patrol**_ **** Columbia, 1936. 60 min. D: David Selman. SC: J.P. McGowan and Robert Watson. With Charles Starrett, Finis Barton, J.P. McGowan, Henry Mollinson, LeStrange Millman, James McGrath, Arthur Kerr, Reginald Hincks, Ted Mapes. A Mounted Policeman works undercover as a woodsman to find the killer of a comrade as well as the wrecker trying to sabotage a lumber mill. Colorful, entertaining Charles Starrett film.\n\n_**Secret Rancher**_ see _**The Dude Cowboy**_ (1926)\n\n**3764** _ **Secret Valley**_ **** 20th Century\u2013Fox, 1936. 60 min. D: Howard Bretherton. SC: Earle Snell, Dan Jarrett and Paul Franklin. With Richard Arlen, Virginia Grey, Jack Mulhall, Syd Saylor, Russell Hicks, Tom London, Norman Willis, Maude Allen, Willie Fung, Charles Delaney, Al Hill. A western farmer decides to raise horses and soon finds them coveted by an outlaw gang. Standard program feature from the Harold Bell Wright story.\n\n**3765** _ **El Secreto de Pancho Villa**_ (The Secret of Pancho Villa) **** Filmadora Mexicana, 1957. 93 min. D: Rafael Baledon. SC: Ramon Obon. With La Sombra Vengadora (Fernando Oses), Alicia Caro, Rodolfo Landa, Pascual Garcia Pena, Victor Alcocer, Carlos Munquiz, Guillermo Hernandez \"Lobo Negro,\" Rafael Banquells, Robert G. Rivera, Felipe Montoya, Gabriel Sanchez Tapia, Guillermo Bravo Sosa, Carlos Suarez, Georgina Barragan, Vicente Lara \"Cacama,\" Guillermo Cramer, Roger Lopez, Indio Cacama. A masked wrestler leads the search for the lost treasure of Pancho Villa. Passable Mexican horror Western.\n\n**3766** _ **Secrets**_ **** United Artists, 1933. 90 min. D: Frank Borgaze. SC: Frances Marion. With Mary Pickford, Leslie Howard, C. Aubrey Smith, Blanche Frederici, Doris Lloyd, Herbert Evans, Ned Sparks, Allan Sears, Mona Maris, Huntley Gordon, Ethel Clayton, Bessie Barriscale, Theodore Von Eltz, Virginia Grey, Lyman Williams, Ellen Johnson, Randolph Connelly, King Baggott, Florence Lawrence, Francis Ford, Paul Panzer, Jerry Stewart. A couple elope, take a wagon train West and settle down to cattle ranching with the husband rising in politics until news of a love affair ruins his career. Heavy, but well made, melodrama; best remembered as Mary Pickford's screen swan song.\n\n**3767** _ **The Seekers**_ **** Rank\/Universal, 1966. 75 min. Color. D: Ken Annakin. SC: William Fairchild. With Jack Hawkins, Glynis Johns, Noel Purcell, Laya Raki, Inia Te Wiata, Patrick Warbrick, Kenneth Williams, Tony Estrich, Edward Baker. An Englishman, falsely convicted of smuggling, and his school teacher wife are sent to New Zealand in the 1820s and endure the hardships of pioneer life. Good British made drama released in the U.S. as _**Island of Fury**_.\n\n**3768** _ **Seminole**_ **** Universal-International, 1953. 86 min. Color. D: Budd Boetticher. SC: Charles K. Peck, Jr. With Rock Hudson, Barbara Hale, Anthony Quinn, Richard Carlson, Hugh O'Brian, Russell Johnson, Lee Marvin, Ralph Moody, James Best, Dan Poore, Frank Chase, Earl Spainard, Scott Lee, Fay Roope, Don Gibson, John Day, Howard Erskine, Duane Thorsen, Walter Reed, Robert Karns, Robert Dane, John Phillips, Soledad Jiminez, Don Garrett, Robert Bray, Alex Sharp, William Janssen, Jack Finlay, Jody Hutchinson. In Florida, Seminole Indians refuse to sign a treaty with whites, preferring to live their own lives, and a West Point graduate returns to find his girlfriend engaged to a member of the tribe. Fairly interesting affair bolstered by a fine cast.\n\n**3769** _ **Seminole Uprising**_ **** Columbia, 1955. 74 min. Color. D: Earl Bellamy. SC: Robert E. Kent. With George Montgomery, Karin Booth, William Fawcett, Steven Ritch, Ed Hinton, John Pickard, Jim Maloney, Rory Mallinson, Howard Wright, Russ Conklin, Richard Cutting, Paul McGuire, Kenneth MacDonald, Jonni Paris, Joanne Rio, Fritz Ford, Paul McGuire, Ed Coch, Charles Schaeffer. An Army man raised by Indians is torn between orders to bring in the tribe's chief and the safety of his girl who the braves have kidnapped. Sam Katzman produced this one and the threadbare budget shows.\n\n**3770** _ **Senor Americano**_ **** Universal, 1929. 71 min. D: Harry Joe Brown. SC: Bennett Cohen and Lesley Mason. With Ken Maynard, Kathryn Crawford, Gino Corrado, J.P. McGowan, Frank Yaconelli, Frank Beal. The government sends an Army lieutenant to California to investigate land grabbers and there he wins a golden bridle in a riding contest and learns of plans to steal a man's land. Action filled Ken Maynard part-talkie.\n\n**3771** _ **Senor Jim**_ **** Beaumont, 1936. 61 min. D: Jacques Jaccard. SC: Celia Jaccard. With Conway Tearle, Barbara Bedford, Alberta Dugan, Fred Malatesta, Betty Mack, Richard (Dirk) Thane, Evelyn Hagara, Bob McKenzie, Harrison Greene, Ashton and Co'ena, Tove Linden, Lloyd Brooks, Budd Buster, Jack Evans. A Louisiana rancher acts as the lawyer for a woman who is about to lose her child due persecution by his wife, the little girl's real mother, who abandoned her before she married him. A contrived plot and poor production values hurt this Conway Tearle vehicle, but, as usual, he is good in the title role.\n\n**3772** _ **Senorita from the West**_ **** Universal, 1945. 63 min. D: Frank L. Strayer. SC: Howard Dinsdale. With Allan Jones, Bonita Granville, Jess Barker, George Cleveland, Fuzzy Knight, Spade Cooley and Orchestra, Oscar O'Shea, Benny McEvoy, Olin Howlin, Danny Mummert, Bob Merrill, Emmett Vogan, Billy Nelson, Jack Clifford, Gwen Donovan, Ralph Dunn, Ann Lawrence, Richard Alexander, Al Ferguson, Frank Hagney, Lane Chandler, Cyril Ring. A pretty girl from the West, wanting to become a singer, runs away from home and meets the \"ghost voice\" for a famous radio star. Typically amusing and glossy Universal mid\u20131940s product.\n\n**3773** _ **September Gun**_ **** CBS-TV, 1983. 100 min. Color. D: Don Taylor. SC: William Norton. With Robert Preston, Patty Duke Astin, Christopher Lloyd, Geoffrey Lewis, Sally Kellerman, David Knell, Jacques Aubuchon, Jonathan Gries, Clayton Landey, Pat Anderson. An aging gunfighter reluctantly becomes the protector of a nun and her orphaned Apache charges in a wild Colorado town. Engaging TV movie vehicle for Robert Preston as the good hearted gunman.\n\n**3774** _ **Sequoia**_ **** Metro-Goldwyn-Mayer, 1935. 73 min. D: Chester M. Franklin. SC: Anna Cunningham, Sam Armstrong and Carey Wilson. With Jean Parker, Russell Hardie, Samuel S. Hinds, Paul Hurst, Ben Hall, Willie Fung, Harry Lowe, Jr., Malibu (deer), Gato (puma). In the High Sierras a young girl who loves animals protects a puma cub and a fawn from hunters. Pretty fair outdoor drama.\n\n**3775** _ **Seraphim Falls**_ **** Samuel Goldwyn Films, 2006. 115 min. Color. D: David Von Ancken. SC: David Von Ancken and Abby Everett Jaques. With Liam Neeson, Pierce Brosnan, Michael Wincott, Xander Berkeley, Ed Lauter, Tom Noonan, Kevin J. O'Connor, John Robinson, Anjelica Huston, Angie Harmon, Robert Baker, Wes Studie, Jimmy Simpson, James Jordan, Nate Mooney, Justin Tate, Shannon Zeller, Lewie Wickham, Boots Southerland, Adam Houlton, Darren Gibson, Hugh Elliot. A colonel and his four hirelings track a man through the Nevada desert as they plan to carry out a vendetta. Passable, methodic box office bust.\n\n**3776** _ **Sergeant Rutledge**_ **** Warner Bros., 1960. 111 min. Color. D: John Ford. SC: Willis Goldbeck and James Warner Bellah. With Jeffrey Hunter, Constance Towers, Woody Strode, Billie Burke, Juano Hernandez, Willis Bouchey, Carleton Young, Judson Pratt, Bill Henry, Walter Reed, Chuck Hayward, Mae Marsh, Fred Libby, Toby Richards, Jan Styne, Cliff Lyons, Charles Seel, Jack Pennick, Hank Worden, Chuck Roberson, Eva Novak, Estelle Winwood, Shug Fisher, Rafer Johnson, Jack Lewis, Mike Lally, Sam Harris, Jack Mower, Ed Shaw, Toby Michaels. When a black cavalry officer is falsely accused of rape and murder, a lieutenant defends him at his court martial. Tense and well acted melodrama from director John Ford.\n\n**3777** _ **Sergeants 3**_ **** United Artists, 1962. 112 min. Color. D: John Sturges. SC: W.R. Burnett. With Frank Sinatra, Dean Martin, Sammy Davis, Jr., Peter Lawford, Joey Bishop, Henry Silva, Ruta Lee, Buddy Lester, Philip Crosby, Dennis Crosby, Lindsay Crosby, Hank Henry, Richard Simmons, Michael Pate, Richard Hale, Mickey Finn, Sonny King, Eddie Little Sky, Rodd Redwing, Madge Blake, Dorothy Abbott, Walter Merrill. A former slave is rescued from Indians by a trio of Army sergeants and they become heroes when the tribe plans to murder incoming settlers. Comedy re-working of _**Gunga Din**_ (RKO Radio, 1939); not much.\n\n**3778** _ **Seven Alone**_ **** Doty-Dayton, 1974. 97 min. Color. D: Earl Bellamy. SC: Eleanor Lamb and Douglas C. Stewart. With Dewey Martin, Aldo Ray, Stewart Petersen, Anne Collings, James Griffith, Dehl Berti, Bea Morris, Dean Smith. On the way to Oregon in 1843, seven children are orphaned when their parents are killed and the oldest, a 13 year old boy, leads them on 2,000 mile trek from Missouri to their destination. Well staged and acted family Western; quite entertaining.\n\n**3779** _ **Seven Angry Men**_ **** Allied Artists, 1958. 90 min. D: Charles Marquis Warren. SC: Daniel B. Ullman. With Raymond Massey, Debra Paget, Jeffrey Hunter, Larry Pennell, Leo Gordon, John Smith, James Best, Dennis Weaver, Guy Williams, Tom Irish, James Anderson, James Edwards, John Pickard, Smoki Whitfield, Jack Lomas, Robert Simon, Dabbs Greer, Ann Tyrell, Robert Osterloh, Kenneth MacDonald, Jack Perrin, Lane Bradford, I. Stanford Jolley, Don C. Harvey. Abolitionist John Brown and his sons fight to free the slaves and become hunted fugitives after massacring slave holders. Fine drama with Raymond Massey giving an excellent performance as John Brown, the role he also portrayed in _**Santa Fe Trail**_ (q.v.).\n\n_**Seven Bad Men**_ see _**Rage at Dawn**_\n\n**3780** _ **Seven Brides for Seven Brothers**_ **** Metro-Goldwyn-Mayer, 1954. 103 min. Color. D: Stanley Donan. SC: Albert Hackett, Frances Goodrich and Dorothy Kingsley. With Howard Keel, Jane Powell, Jeff Richards, Russ Tamblyn, Tommy Rall, Marc Platt, Julie (Newmar) Newmeyer, Nancy Kilgas, Betty Carr, Virginia Gibson, Matt Mattox, Jacques d'Amboise, Ruta Kilmonis, Norma Doggett, Ian Wolfe, Howard Petrie, Earl Burton, Dante Diapolo, Kelly Brown, Matt Moore, Dick Rich, Marjorie Wood, Russell Simpson, Anna Q. Nilsson, Larry Blake, Phil Rich, Lois Hall, Russ Anders, Terry Wilson, George Robotham, Walter Beaver, Jarma Lewis, Sheila James, I. Stanford Jolley, Tim Graham. After an Oregon farmer brings home a pretty bride, his six brothers go out and kidnap girls for themselves. Delightful M-G-M Western musical with good songs by Johnny Mercer and Gene de Paul.\n\n**Howard Keel and Jane Powell in** _**Seven Brides for Seven Brothers**_ **(Metro-Goldwyn-Mayer, 1954).**\n\n**  \n**\n\n**3781** _ **Seven Cities of Gold**_ **** 20th Century\u2013Fox, 1955. 103 min. Color. D: Robert D. Webb. SC: Richard L. Breen and John C. Higgins. With Richard Egan, Anthony Quinn, Michael Rennie, Jeffrey Hunter, Rita Moreno, Eduardo Noriega, Leslie Bradley, John Doucette, Kathleen Crowley, Victor Junco, Julio Villareal, Yerye Beirute, Jack Mower. In the 18th century, a Spanish expedition tries to find the legendary seven cities of gold in the Southwest. Average adventure yarn highlighted by Michael Rennie's performance as Father Junipero Serra.\n\n**3782** _ **Seven Dollars on the Red**_ **** Gloria Film\/Brepi Film, 1966. 95 min. Color. D: Albert Cardiff (Alberto Cardone). SC: Juan Cobos and Mel Collins (Melchiade Coletti). With Anthony Steffen, Elisa Montes, Fernando Sancho, Jerry Wilson (Roberto Miali), Loredana Nusciak, Carroll Brown (Bruno Carotento), Jose Manuel Martin, Spean Convery (Spartaco Conversi), Fred Warrel (Alberto Varelli), Gianni Manera, Frank Farrell (Franco Fantasia), Annie Giss, Franco Gula, Renato Terra, Ninco Musco, Miriam Salonicchio, David Mancori, Fortunato Arena. Years pass as a gunman searches for his son abducted by the outlaw gang who massacred other family members. Well made and entertaining Spaghetti Western released in Italy as _**Sette Dollari sui Rosso**_ (Seven Dollars on the Red).\n\n**3783** _ **The 7 Faces of Dr. Lao**_ **** Metro-Goldwyn-Mayer, 1964. 100 min. Color. D: George Pal. SC: Charles Beaumont. With Tony Randall, Barbara Eden, Arthur O'Connell, John Ericson, Kevin Tate, Argentina Brunetti, Noah Beery, Royal Dano, John Doucette, Frank Cady, Lee Patrick, John Qualen, Douglas Fowley, Minerva Urecal, Eddie Little Sky, Peggy Rae, Dal McKennon, Chubby Johnson. The crooked citizens in a Western town are brought to their senses by the acts in a small touring circus. Tony Randall has a field day playing the various characters in this excellent fantasy based on Charles G. Finney's sadly neglected novel _The Circus of Dr. Lao_.\n\n**3784** _ **Seven from Texas**_ **** Centauro\/PEA, 1964. 93 min. Color. D-SC: Joaquin L. Romero Marchent. With Paul Piaget, Robert Hundar (Claudio Undari), Gloria Milland, Fernando Sancho, Jesus Puente, Raf Baldasssare, Panco Sanz, Joe Kamel, Gregory Wu. A bride sways between her husband and feelings for a former lover, a gunman who is part of an escort taking them to a settlement through Indian country. More dramatic than violent European oater, originally titled _**Camino del Sur**_ (Road to the South) and also called _**Seven Guns from Texas**_.\n\n**3785** _ **Seven Guns for the MacGregors**_ **** Columbia, 1968. 97 min. Color. D: Frank Garfield (Franco Giraldi). SC: Enzo Dell'Aquila, Fernando Di Leo, David Moreno and Duccio Tessari. With Robert Wood, Fernando Sancho, Manolo (Manuel) Zarzo, Nick Anderson, Paul Carter, Nazzareno Zamperla, Paolo Magalotti, Perla Cristal, Jorge Rigaud, Alberto Dell'Acqua (Robert Widmark), Julio Perez Tabernero, Saturno Cerra, Albert Waterman, Agata Flori, Leo Anchoriz, Harold Cotton, Anne-Marie Noe, Margaret Horowitz, Raphael Bardem, Antonio Molino Rojo, Cris Huerta, Max Dean (Massimo Righi). The seven sons of two Scot pioneers are arrested after a cattle drive by a crooked sheriff in cahoots with a bandit and after escaping they plot revenge. There is nothing special about this Italian-Spanish co-production other than it has more humor and less violence than most of its ilk; followed by the unrelated _**Up the MacGregors**_ (q.v.) and issued in Italy in 1965 as _**Sette Pistole per I Macgregor**_ (Seven Pistols for the MacGregors).\n\n**3786** _ **Seven Guns for Timothy**_ **** Filmax, 1966. 100 min. Color. D: Rod Gilbert (Romolo Guerrieri). Jose Antonio de la Loma and Giovanni Simonelli. With Sean Flynn, Fernando Sancho, Evelyn Stewart (Ida Galli), Daniel Martin, Frank Oliveras, Poldo Bendandi, Spartaco Conversi, Tito Garcia, Anita Todesco, Ivan Basta, Antonio Almoros, Silvana Bacci, William Conroy, Osvaldo Genazzani, Maruska Rosetti. After being trained to take care of himself, a man joins his six mentors in forming a gang opposed to the bandit who is after a gold mine. Average Italian-Spanish film originally called _**Sette Magnifiche Pistole**_ (Seven Magnificent Guns).\n\n_**Seven Guns from Texas**_ see _**Seven from Texas**_\n\n**3787** _ **Seven Guns to Mesa**_ **** Allied Artists, 1958. 69 min. D: Edward Dein. SC: Myles Wilder, Edward Dein and Mildred Dein. With Charles Quinlivan, Lola Albright, James Griffith, Jay Adler, John Cliff, Burt Nelson, John Merrick, Charles Keane, Jack Carr, Don Sullivan, Rush Williams, Neil Grant, Reed (Howes) Hawes, Mauritz Hugo, Harvey Russell, Gerald Frank. Stagecoach passengers are taken prisoners by outlaws planning to rob a gold shipment. Dull oater.\n\n_**Seven Magnificent Guns**_ see _**Seven Guns for Timothy**_\n\n**3788** _ **7 Men from Now**_ **** Warner Bros., 1956. 78 min. Color. D: Budd Boetticher. SC: Burt Kennedy. With Randolph Scott, Gail Russell, Lee Marvin, Walter Reed, John Larch, Donald Barry, Fred Graham, John Berardino, John Phillips, Chuck Roberson, Steve Mitchell, Pamela Duncan, Stuart Whitman, Fred Sherman, Cliff Lyons. An ex-lawman hunts for the gunmen who murdered his wife during a robbery. Entertaining, and well written, action drama, sure to please Randolph Scott fans.\n\n_**7 Mummies**_ see _**Seven Mummies**_\n\n**3789** _ **Seven Mummies**_ **** American World Pictures, 2006. 90 min. Color. D: Nick Quested. SC: Thadd Turner. With Matt Schulze, Cerina Vincent, Billy Wirth, Billy Drago, Martin Kove, Andrew Bryniarski, Danny Trejo, James Intveld, Noel Gugliemi, Max Perlich, Victor \"Nore\" Santiago, Adrianne Palicki, Thadd Turner, Michela Fruest, David Katner, Vic Roych, Lance Elchlepp. After finding a gold medallion in the desert, sex escaped convicts and their female guard prisoner are directed by an old Indian to a ghost town supposedly possessing treasure but really inhabited by the undead. Slack horror Western with a convoluted plot; also titled _**7 Mummies**_.\n\n**3790** _ **The Seven Sixgunners**_ **** Western Movies, 1987. 91 min. Color. D: George Potter. SC: Mike Rom. With Christopher Lucas Rhodes, Jay Gammon, Catherine Lynn Lane, Corrine Morrison, Larry Brooks, Paul Fogle, Oscar Stevens, Jr., Ken Gardiner, David McCauley, Maria Elena Acuna, Mike Rom, Mike Robertson, Beto Stevens, Lora Joann Soto, Carmen Gastelum, Christianne Stevens, Rick Harker, Dan Herrigan, David Herren, Roy Leyvar, John Morgia. A gunman is hired by a woman to protect her lover who fears his former partners are after a cache of gold he is seeking. Tacky, amateurish direct to video pseudo-historical Western made in Arizona; also called _**Nelson Nye's Seven Sixgunners**_.\n\n**3791** _ **Seven Ways from Sundown**_ **** Universal-International, 1960. 86 min. Color. D: Harry Keller. SC: Clair Huffaker. With Audie Murphy, Barry Sullivan, Venetia Stevenson, John McIntire, Kenneth Tobey, Mary Field, Teddy Rooney, Suzanne Lloyd, Ken Lynch, Wade Ramsey, Don Collier, Jack Kruschen, Claudia Barrett, Don Haggerty, Robert Burton, Fred Graham, Dale Van Sickel. A Texas Ranger becomes fast friends with a murderous outlaw but eventually realizes he will have to hunt him down. Pretty fair oater with Audie Murphy and Barry Sullivan good in the leads.\n\n**3792** _ **7th Cavalry**_ **** Columbia, 1956. 75 min. Color. D: Joseph H. Lewis. SC: Peter Packer. With Randolph Scott, Barbara Hale, Jay C. Flippen, Jeanette Nolan, Frank Faylen, Leo Gordon, Denver Pyle, Harry Carey, Jr., Michael Pate, Donald Curtis, Frank Wilcox, Pat Hogan, Russell Hicks, Peter Ortiz, William Leslie, Jack Parker, Al Wyatt. An officer returns from a furlough to find his regiment, Custer's 7th Cavalry, has been wiped out at the Little Big Horn and he tries to determine the cause of the massacre. Very fine feature with Randolph Scott excellent in the lead.\n\n**Advertisement for** _**7th Cavalry**_ **(Columbia, 1956).**\n\n**  \n**\n\n_**The Shadow Gang**_ see _**The Star Packer**_\n\n**3793** _ **The Shadow of Chikara**_ **** Howco International, 1977. 114 min. Color. D-SC: Earl E. Smith. With Joe Don Baker, Sondra Locke, Slim Pickens, Ted Neeley, Dennis Fimple, John Chandler, Joy Houck, Jr., Linda Dano, Grady Wyatt, Robert Ginnaven, Bud Davis, Roger Manning, Cory Kelly, Don Kellams, William Kerwin, Linda Davis, Steve Wobecky, Tom Grozis, Jody Ratliff, Fred Judkins. After being joined by a mysterious young woman, the survivors of the final battle of the Civil War attempt to find a hidden treasure not knowing it is protected by demon hawks. Fair horror Western, originally titled _**Wishbone Cutter**_ and also called _**Curse of the Demon Mountain**_.\n\n**3794** _ **Shadow of Terror**_ **** Producers Releasing Corporation, 1945. 64 min. D: Lew Landers. SC: Arthur St. Clair. With Richard Fraser, Grace Gillern, Cy Kendall, Emmett Lynn, Kenneth MacDonald, Eddie Acuff, Sam Flint, Emmett Vogan, John Harmon. A female rancher and her foreman save a man who has amnesia after being thrown off a train not knowing he is a scientist who carries with him the secret of the A-bomb. Cheap but exciting topical program feature with atomic explosion footage tacked on to the finale.\n\n**3795** _ **Shadow of the Hawk**_ **** Columbia, 1976. 92 min. Color. D: George McGowan. SC: Norman Thaddeus Vane and Herbert J. Wright. With Jan-Michael Vincent, Marilyn Hassett, Chief Dan George, Pia Shandel, Marianne Jones, Jacques Hubert, Cindi Griffith, Anna Hagan, Murray Lowry, Terry York. The grandson of an Indian chief and a newspaperwoman go to the man's reservation where his grandfather wants him to use tribal rituals to combat evil forces in the form of a 200-year-old sorceress. Rather unpleasant horror Western filmed in the backwoods of Vancouver, British Columbia.\n\n**3796** _ **Shadow of Zorro**_ **** Centauro Film, 1962. 87 min. Color. D: Joaquin L. Romero Marchant. SC: Joaquin L. Romero Marchant, Jose Mallorqui Figueroa, Rafael Romero Marchent and Jesus Franco Manera (Jess Franco). With Frank Latimore, Marie Gale (Maria Luz Galicia), Paul Piaget, Robert Hundar (Claudio Undari), Ralph (Raf) Baladassare, Howard Vernon, Gianni Santuccio, Marco Feliciani, Maria Silva, Marco Tulli, Xan das Bolas, Piero Lulli, Diana Lorys, Miguel Merino, Jose Marco Davo, Jesus Tordesillas, Jose Marco. A revolutionary tries to capture Zorro by committing acts of violence and blaming them on the masked avenger. Pretty good adventure issued in Spain as _**La Sombra de Zorro**_ (The Shadow of Zorro) and _**La Venganza del Zorro**_ (The Vengeance of Zorro), in France as _**L'Ombra di Zorro**_ (The Shadow of Zorro) and in the U.S. as _**Zorro the Avenger**_.\n\n**3797** _ **Shadow Ranch**_ **** Columbia, 1930. 55 min. D: Louis King. SC: Frank Clark. With Buck Jones, Marguerite De La Motte, Kate Price, Al Smith, Frank Rice, Slim Whitaker, Ben Wilson, Bob McKenzie, Lafe McKee, Fred Burns, Ben Corbett, Frank Ellis, Hank Bell. When a saloon owner murders a foreman in an attempt to get a woman's ranch to control a valley's water supply, the victim's pal seeks revenge. Slow moving and poorly recorded early Buck Jones talkie with some nice locations and the songs \"When It's Roundup Time in Texas\" and \"Ragtime Cowboy Joe.\"\n\n**3798** _ **The Shadow Riders**_ **** CBS-TV, 1982. 100 min. Color. D: Andrew V. McLaglen. SC: Jim Byrnes. With Tom Selleck, Sam Elliott, Katharine Ross, Ben Johnson, Geoffrey Lewis, Gene Evans, Jeff Osterhage, Jane Greer, R.G. Armstrong, Harry Carey, Jr., Scanlon Gail, Marshall Teague, Ben Fuhrman, Natalie May, Jeanetta Arnett, Owen Orr, Kristina David, Joe Capone, Robert B. Craig. Two brothers who fought on opposite sides in the Civil War unite to oppose renegade soldiers in Texas. Very well done television feature from the work by Louis L'Amour.\n\n**3799** _ **Shadow Valley**_ **** Eagle-Lion, 1947. 58 min. D: Ray Taylor. SC: Arthur Sherman. With Eddie Dean, Roscoe Ates, Jennifer Holt, Andy Parker and The Plainsmen, George Chesebro, Eddie Parker, Lee Morgan, Lane Bradford, Carl Mathews, Budd Buster, Forrest Taylor. A cowboy tries to help a woman whose ranch is coveted by a train robber masquerading as a lawyer. Fair Eddie Dean musical opus.\n\n**3800** _ **Shadowheart**_ **** Koan, 2009. 110 min. Color. D: Dean Alioto. SC: Dean Alioto, Peter Vanderwall and Brad Goodman. With Angus Macfadyen, Justin Ament, Marnie Alton, Tonantzin Carmelo, Michael Spears, William Sadler, Dean Alioto, Ines Dali, Anthony Michael Jones, Courtney Gains, Shawn Reaves, Charles Napier, Daniel Baldwin, Steve Pink, Rance Howard, Devin Brochu, Timothy Patrick Cavanaugh, Larry Zeug, Zach Selwyn, Ruby Alioto, Ross Hagen, Justin Rodgers Hall, Trish Moreno, Nigel Daly, Christian Fortune, Fred Griffith, Stephanie Patton, Peter Sherayko, Matt Silver, Kevin McNiven, Tommy Giavocchini, Jonathan Eisley. A man must chose between being reunited with the woman he loves and taking revenge on the man who murdered his father. So-so medium budget video production.\n\n**3801** _ **Shadows of Death**_ **** Producers Releasing Corporation, 1945. 59 min. D: Sam Newfield. SC: Fred Myton. With Buster Crabbe, Al St. John, Donna Dax, Edward Hall, Charles King, Frank Ellis, Emmett Lynn, Karl Hackett, Ed Peil, Sr., Bob (John) Cason, Frank McCarroll, Bud Osborne, Budd Buster, Jack Baxley, Wally West, Jimmy Aubrey, George Morrell, Ray Henderson, Rube Dalroy, Jack Evans, Art Dillard, Lew Morphy, Jack Tornek. Marshals Billy Carson and Fuzzy Q. Jones learn outlaws have murdered a man in order to get land they plan to sell to a railroad. Typically mediocre, shoddy \"Billy Carson\" entry.\n\n**3802** _ **Shadows of the West**_ **** Monogram, 1949. 60 min. D: Ray Taylor. SC: Adele Buffington. With Whip Wilson, Andy Clyde, Reno Browne, Riley Hill, Bill Kennedy, Pierce Lyden, Keith Richards, William Ruhl, Ted Adams, Kenne Duncan, Frank Ellis, Curt Barrett, Red Egner, Lee Phelps, Bert Hamilton, Bud Osborne, Donald Kerr, Billy Hammond, Clem Fuller, Carol Henry, Bob Woodward, Edmund Glover, Dee Cooper. A lawman takes a vacation in a town where his pal is the ex-sheriff and the new peacemaker appears to be involved with an outlaw gang. Okay Whip Wilson series vehicle.\n\n**3803** _ **Shadows of Tombstone**_ **** Republic, 1953. 54 min. D: William Witney. SC: Gerald Geraghty. With Rex Allen, Slim Pickens, Jeanne Cooper, Roy Barcroft, Emory Parnell, Ric Roman, Richard Avonde, Julian Rivero, Rex Lease, Clarence Straight, Chick Hannon, Art Dillard. A cowboy running for sheriff gets help from a woman newspaper editor. Fair Rex Allen outing, well written and directed.\n\n**3804** _ **Shadows on the Range**_ **** Monogram, 1946. 56 min. D: Lambert Hillyer. SC: Jess Bowers (Adele Buffington). With Johnny Mack Brown, Raymond Hatton, Jan Bryant, John Merton, Marshall Reed, Steve Clark, Ted Adams, Terry Frost, Pierce Lyden, Cactus Mack, Roy Butler, Jack Perrin, Lane Bradford. A cattlemen's association investigator works undercover to expose rustlers by taking a job as a ranch foreman and pretending to join the outlaws. Compact and fast Johnny Mack Brown-Raymond Hatton entry with fine direction by Lambert Hillyer.\n\n**3805** _ **Shadows on the Sage**_ **** Republic, 1942. 55 min. D: Lester Orlebeck. SC: J. Benton Cheney. With Bob Steele, Tom Tyler, Jimmie Dodd, Bryant Washburn, Cheryl Walker, Harry Holman, Tom London, Griff Barnett, Yakima Canutt, Freddie Mercer, Rex Lease, Curley Dresden, Eddie Dew, Horace B. Carpenter, Frank Brownlee, John Cason, Pascale Perry, Johnnie Morris, Bill Nestell, Fred Burns, Burr Caruth, Jack Rockwell, Tommy Coats, Betty Farrington, Cactus Mack. Outlaws have been stealing from miners and the Three Mesquiteers attempt to find out who is the operation's mastermind. Well written and action packed, this is a good entry from the later stages of the popular series.\n\n**3806** _ **The Shakiest Gun in the West**_ **** Universal, 1966. 101 min. Color. D: Alan Rafkin. SC: Jim Fritzell and Everett Greenbaum. With Don Knotts, Barbara Rhodes, Jackie Coogan, Donald Barry, Ruth McDevitt, Frank McGrath, Terry Wilson, Carl Ballantine, Pat Morita, Robert Yuro, Herbert Voland, Fay DeWitt, Dub Taylor, Hope Summers, Dick Wilson, Vaughn Taylor, Ed Peck, Edward Faulkner, Arthur Space, Greg Mullavey, Benny Rubin, E.J. Andre, Myron Healey, I. Stanford Jolley. A meek dentist goes West to set up practice and gets involved with a woman bandit working for the government in trying to capture gun smugglers. Okay Don Knotts vehicle, a remake of _**The Paleface**_ (q.v.).\n\n**3807** _ **Shalako**_ **** Cinerama, 1968. 116 min. Color. D: Edward Dmytryk. SC: J. J. Griffith, Hal Hopper and Scott Finch. With Sean Connery, Brigitte Bardot, Stephen Boyd, Jack Hawkins, Peter Van Eyck, Honor Blackman, Woody Strode, Eric Sykes, Alexander Knox, Valerie French, Julian Mateos, Donald Barry, Rodd Redwing, Chief Tug Smith, Hans De Vries, Walter Brown, Charles Stalnaker, Bob Cunningham, John Clark, Bob Hall. A group of European nobles on a hunting trip in the West are attacked by Indians with a loner trying to rescue them. Well made but slow moving and not very interesting.\n\n**3808** _ **Shane**_ **** Paramount, 1953. 118 min. Color. D: George Stevens. SC: A.B. Guthrie, Jr. With Alan Ladd, Jean Arthur, Van Heflin, Brandon De Wilde, Jack Palance, Ben Johnson, Edgar Buchanan, Emile Meyer, Elisha Cook, Jr., Douglas Spencer, John Dierkes, Ellen Corby, Paul McVey, John Miller, Edith Evanson, Leonard Strong, Ray Spiker, Janice Carroll, Martin Mason, Helen Brown, Nancy Kulp, Howard Negley, Beverly Washburn, George J. Lewis, Robert Wilke, George Quirk, Jack Sterling, Henry Wills, Rex Moore, Ewing Brown, Alana Ladd, David Ladd, Laddie Ladd. A one-time gunman, wanting to lead a peaceful life as a ranch hand in Wyoming, is forced to take up his trade again when homesteaders are threatened by range warfare. One of the all time classic Westerns; grand performances by Alan Ladd, Van Heflin and Jack Palance.\n\n_**Shanghai Gold**_ see _**Have a Nice Funeral, My Friend**_\n\n_**Shanghai Joe**_ see _**Fighting Fists of Shanghai Joe**_\n\n**3809** _ **Shanghai Noon**_ **** Buena Vista, 2000. 110 min. Color. D: Tom Dey. SC: Miles Millar and Alfred Gough. With Jackie Chan, Owen Wilson, Lucy Lu, Brandon Merrill, Roger Yuan, Xander Berkeley, Rong Guang Yu, Cui Ya Hi, Eric Chi Cheng Chen, Jason Connery, Walton Goggins, P. Adrien Dorval, Rafael Baez, Stacy Grant, Kate Luyben, Henry O, Russell Badger, Simon Baker, Sam Simon, Alan C. Peterson, Rad Daly, Lee Jay Bamberry, Stephen Strachan, Tim Koetting, Rick Ash, Valerie Planche, Tom Heaton, Jody Thompson, Chang Tseng, Sherman Chao, Regent Or, Melvin Skales, Cliff Solomon. A Chinese martial arts expert teams wit an outlaw to find a kidnapped princess and her abductors. Fun Western Kung Fu adventure with comic undertones.\n\n**3810** _ **Shango**_ **** PAC, 1970. 81 min. Color. D: Edward G. Muller (Edoardo Mulargia). SC: Edoardo Mulargia and Antonio De Teffe (Anthony Steffen). With Anthony Steffen, Eduardo Fajardo, Maurice Poli, Barbara Nelli, Giusva Fioravati, Attilio D'Ottesio, Massimo Carocci, Spartaco Conversi, Liana Del Blazo, Angelo Dessi, Adriana Giuffre, Franco Pesce, Mirella Pomili, Andrea Scotti (Andrew Scott), Gabriella Giorgelli, Claudio Ruffini, Franco Ukmar, Angelo Susani, Renzo Pevarello, Gilberto Galimberti, Pietro Torrisi. A Texas Ranger opposes a Rebel major who tries to rally the locals in reviving the Confederacy. Typical Spaghetti Western made in Italy as _**Shango, la Pistola Infallible**_ (Shango, The Infallible Pistol); co-written by star Anthony Steffen.\n\n**3811** _ **Shark River**_ **** United Artists, 1953. 80 min. Color. D: John Rawlins. SC: Joseph Carpenter and Lewis Meltzer. With Steve Cochran, Carole Mathews, Warren Stevens, Robert Cunningham, Spencer Fox, Ruth Foreman, Bill Piper. After the Civil War, a wanted man tries to elude the law in the Florida Swamps. Okay action romance drama.\n\n_**Shatterhand**_ see _**Old Shatterhand**_\n\n**3812** _ **Shaughnessy**_ **** Allumination Fireworks, 1996. 95 min. Color. D: Michael Rhodes. SC: William Blinn. With Matthew Settle, Linda Kozlowski, Tom Bower, Sarah Paulson, Stuart Whitman, Michael Jai White, Tim Grimm, John Carroll Lynch, John Hawkes, Bo Hopkins, O'Neal Compton, Cari Shayne, Daragh O'Malley, Norman D. Golden II, Michael Ray Wisely, Robert Elross, Velina Brown, Nina Marie Boston, Christopher Shaw, Shawn Cady, Adam Hervey, Marcy Goodnow Swift, Dwight Hicks, Sherri Young, Robert Ernst, Bradley J. Bovee, Phil Culotta, Daniel W. Barringer, Don William Owen, Gene LeBell, Troy Bishop, Paul Ennis, Larry Holt, Bill McIntosh. After migrating to Kansas, an Irish immigrant becomes the sheriff of a frontier town. Average TV feature adaptation of _The Iron Marshal_ by Louis L'Amour.\n\n**3813** _ **She Came to the Valley**_ **** R.G.V. Pictures, 1979. 92 min. Color. D: Albert Band. SC: Albert Band and Frank Ray Perilli. With Ronee Blakley, Dean Stockwell, Scott Glenn, Freddy Fender, Anna Jones, Jennifer Jones, Rafael Flores, Jr., Les Brecht, Frrank Benedetto, Sol Marroquin, Everlyn Guerrero, Ruth Reeves, Klaus Eggers, Detley Nitche, Michael Hart, Dan Willis, John Hayes, Jesus Saenz, Juanita Rutledge, Cindy Klein, Miriam Moroles, Martin Sanchez, Cedric Wood, Robby Romero, Mary Alice Artes, Frank Ray Perilli, Cleo Dawson, W.T. Ellis, Maurine Duncan, T.L. Duncan. A married couple and their two little daughters homestead near the Mexican border and become involved in the dispute between Pancho Villa's rebels and government forces. Mediocre pseudo-historical drama, casting country music singer Freddy Fender as Villa; released on video as _**Texas in Flames**_.\n\n**3814** _ **She Wore a Yellow Ribbon**_ **** RKO Radio, 1949. 103 min. Color. D: John Ford. SC: Frank S. Nugent and Laurence Stallings. With John Wayne, Joanne Dru, John Agar, Ben Johnson, Harry Carey, Jr., Victor McLaglen, Mildred Natwick, George O'Brien, Arthur Shields, Francis Ford, Harry Woods, Chief Big Tree, Cliff Lyons, Noble Johnson, Tom Tyler, Michael Dugan, Mickey Simpson, Fred Graham, Frank McGrath, Don Summers, Fred Libby, Jack Pennick, Billy Jones, Bill Gettinger, Post Park, Fred Kennedy, Rudy Bowman, Ray Hyke, Lee Bradley, Chief Sky Eagle, Dan White, Irving Pichel (narrator). An Army captain, on his last mission, tries to prevent Indian warfare while escorting his commanding officer's wife and daughter out of dangerous territory. One of the more enduring of Western classics; well worth viewing.\n\n**3815** _ **The Sheepman**_ **** Metro-Goldwyn-Mayer, 1958. 85 min. Color. D: George Marshall. SC: William Bowers and James Edward Grant. With Glenn Ford, Shirley MacLaine, Leslie Nielsen, Mickey Shaughnessy, Edgar Buchanan, Willis Bouchey, Pernell Roberts, Slim Pickens, Buzz Henry, Pedro Gonzalez Gonzalez, Roscoe Ates, Richard Alexander, Harry Woods, Percy Helton, Tom London, Kermit Maynard. A cattle baron tries to destroy a sheep farmer who has moved into the area and attracted the attentions of his fiancee. Likable oater with more than a touch of humor.\n\n**3816** _ **Shenandoah**_ **** Universal, 1965. 105 min. Color. D: Andrew V. McLaglen. SC: James Lee Barrett. With James Stewart, Doug McClure, Glenn Corbett, Patrick Wayne, Rosemary Forsyth, Phillip Alford, Katharine Ross, Charles Robinson, Paul Fix, Denver Pyle, George Kennedy, Tim McIntire, James McMullan, James Best, Warren Oates, Strother Martin, Dabbs Greer, Harry Carey, Jr., Kevin Hagen, Tom Simcox, Berkeley Harris, Edward Faulkner, Peter Wayne, Gregg Palmer, Bob Steele, James Heneghan, Jr., Rae Miller, Rayford Barnes, Dave Cass, Hoke Howell, Kelly Thordsen, Lane Bradford, Shug Fisher, John Daheim, Joe Yrigoyen, Henry Wills, Buzz Henry, James Carter, Leroy Johnson. During the Civil War a Virginia farmer tries not to get involved in the conflict but this ends in tragic results for his family. Well made, interesting and poignant feature with James Stewart giving a powerful performance as the patriarch.\n\n**3817** _ **The Shepherd of the Hills**_ **** Paramount, 1941. 98 min. Color. D: Henry Hathaway. SC: Grover Jones and Stuart Anthony. With John Wayne, Betty Field, Harry Carey, Beulah Bondi, James Barton, Marjorie Main, Samuel S. Hinds, John Qualen, Marc Lawrence, Tom Fadden, Ward Bond, Dorothy Adams, Olin Howland, Fuzzy Knight, John Harmon, Carl Knowles, Fern Emmett, Vivita Campbell, William Haade, Robert Kortman, Henry Brandon, Jim Corey, Selmer Jackson. A stranger arrives in the Ozark Mountains resulting in changes in people's lives, including that of his long unseen son. Intelligent screen adaptation of Harold Bell Wright's novel, highlighted by good photography (by Charles Lang and W. Howard Greene) with especially fine performances by John Wayne, Harry Carey, Beulah Bondi and Marjorie Main. First filmed in 1919 by Wright Films featuring George Hackathorne and remade in 1928 by First National with Alec B. Francis, Molly O'Day and John Boles, and done again in 1964 (q.v.) starring Richard Arlen.\n\n**3818** _ **The Shepherd of the Hills**_ **** Howco International, 1964. 105 min Color. D-SC: Ben Parker. With Richard Arlen, James W. Middleton, Sherry Lynn, James Collie, Lloyd Durre, Hal Meadows, James Bradford, Joy N. Houck, Jr., Gilbert Elmore, George Jackson, Delores James, Danny Spurlock, Reubin Egan, Tom Pope, Roy Idom, Jim Teague, Roger Nash, Jim Greene. A man tries to end a feud between two mountain families while helping a drought stricken community. Cheaply made production of the 1907 Harold Bell Wright work with Richard Arlen very good as Old Matt. Reissue and TV title: _**Thunder Mountain**_.\n\n**3819** _ **The Sheriff and the Satellite Kid**_ **** Tobis Filmkunst, 1979. 88 min. Color. D: Michele Lupo. SC: Marcello Fondato and Francesco Scardamaglia. With Bud Spencer, Raimund Harmstorf, Cary Guffey, Joe Bugner, Carlo Reali, Gigi (Luigi) Bonos, Harold E. Finch, Bernardino Emanueli, Claudio Ruffini, Roberto Dell'Acqua, Osiride Pevarello, Amedeo Luerini, Guilio Maculani, Raffaele Mottola, Giovanni Cianfrigilia, Sergio Smacchi, Ottaviano Dell'Acqua, Marco Stefanelli. After landing on Earth an alien boy tries to help a sheriff solve a crime in a Western town. Pleasant West German sci-fi action comedy followed by _**Everything Happens to Me**_ (q.v.).\n\n_**Sheriff Brandy**_ see _**Ride and Kill**_\n\n**3820** _ **Sheriff of Cimarron**_ **** Republic, 1945. 54 min. D: Yakima Canutt. SC: Bennet Cohen. With Sunset Carson, Linda Stirling, Olin Howlin, Riley Hill, Jack Ingram, Tom London, Jack Kirk, Robert Wilke, Jack O'Shea, Ed Cassidy, George Chesebro, Hal Price, Carol Henry, Herman Hack, Horace B. Carpenter, Tommy Coats, Post Park, Dee Cooper. A town's new sheriff is framed for a crime by his brother, the one responsible for the area's lawlessness. Nicely directed, action filled outing greatly helped by pretty Linda Stirling, in deference to Sunset Carson's acting.\n\n**3821** _ **Sheriff of Fractured Jaw**_ **** 20th Century\u2013Fox, 1959. 103 min. Color. D: Raoul Walsh. SC: Arthur Dales. With Kenneth More, Jayne Mansfield, Henry Hull, William Campbell, Bruce Cabot, Robert Morley, Ronald Squire, David Horne, Eynon Evans, Sidney James, Donald Stewart, Reed De Rouen, Clancy Cooper, Charles Irwin, Gordon Tamer, Tucker McGuire, Nick Brady, Larry Taylor, Jack Lester, Nicholas Stuart, Sheldon Lawrence, Susan Denny, Charles Farrell, Chief Joneas Applegarth, Chief Joe Buffalo. An unsuccessful British inventor heads to the American West to sell weapons, is mistaken for a gunman and ends up the star packer in a rowdy town. Fairly pleasant British made genre satire.\n\n**3822** _ **Sheriff of Las Vegas**_ **** Republic, 1944. 55 min. D: Lesley Selander. SC: Norman S. Hall. With Wild Bill Elliott, Bobby Blake, Alice Fleming, Peggy Stewart, Selmer Jackson, William Haade, Jay Kirby, John Hamilton, Kenne Duncan, Bud Geary, Jack Kirk, Frank McCarroll, Dickie Dillon, Freddie Chapman, Robert Wilke. A man estranged from his judge father is blamed for his murder and Red Ryder tries to prove his innocence. Well written series entry with a bang-up finale; remade as _**Beyond the Purple Hills**_ (q.v.).\n\n**3823** _ **The Sheriff of Medicine Bow**_ **** Monogram, 1948. 55 min. D: Lambert Hillyer. SC: J. Benton Cheney. With Johnny Mack Brown, Raymond Hatton, Max Terhune, Evelyn Finley, George J. Lewis, Bill Kennedy, Frank LaRue, Peter Perkins, Carol Henry, Bob Woodward, Ted Adams, Herman Hack, John Carpenter, Ray Jones, Bob McFlory. A sheriff helps a parolee when crooks try to steal gold hidden on his ranch. Okay Johnny Mack Brown vehicle, helped by co-stars Raymond Hatton and Max Terhune.\n\n**3824** _ **The Sheriff of Redwood Valley**_ **** Republic 1946. 57 min. D: R.G. Springsteen. SC: Earle Snell. With Wild Bill Elliott, Bobby Blake, Alice Fleming, Bob Steele, Peggy Stewart, Arthur Loft, James Craven, Tom London, Kenne Duncan, Bud Geary, Tom Chatterton, Budd Buster, Frank McCarroll, John Wayne Wright, Frank Linn, Jack Kirk, James Linn, Tex Cooper. Red Ryder uncovers an outlaw gang when he comes to the aid of a rancher falsely accused of a crime. More than competent \"Red Ryder\" feature, buoyed by Bob Steele as the Reno Kid.\n\n**3825** _ **Sheriff**_ _**of Sage Valley**_ **** Producers Releasing Corporation, 1942. 57 min. D: Sherman Scott (Sam Newfield). SC: Milton Raison and George W. Sayre. With Buster Crabbe, Al St John, Maxine Leslie, Tex (Dave) O'Brien, Charles King, John Merton, Kermit Maynard, Hal Price, Curley Dresden, Lynton Brent, Jack Kirk, Budd Buster, Jimmy Aubrey, Al Taylor, Merrill McCormick, Art Dillard, Carl Mathews, Dan White, Ray Henderson, Jack Evans, Bert Dillard. Billy the Kid is asked by a town's mayor to round up an outlaw gang who shot the sheriff during a holdup. Average \"Billy the Kid\" series affair called _**Billy the Kid, Sheriff of Sage Valley**_ on TV.\n\n**3826** _ **Sheriff of Sundown**_ **** Republic, 1944. 57 min. D: Lesley Selander. SC: Norman S. Hall. With Allan Lane, Linda Stirling, Max Terhune, Duncan Renaldo, Roy Barcroft, Herbert Rawlinson, Bud Geary, Jack Kirk, Twinkle Watts, Tom London, Robert Wilke, Kenne Duncan, Rex Lease, Herman Hack, Jack O'Shea, Carl Sepulveda, Nolan Leary, Horace B. Carpenter, Cactus Mack, Neal Hart, Foxy Callahan, Chick Hannon, Duke Green. Cowboys leading a cattle herd to market find they are up against a crooked town boss. Well done Allan Lane vehicle.\n\n**3827** _ **Sheriff of Tombstone**_ **** Republic, 1941. 56 min. D: Joseph Kane. SC: Olive Cooper. With Roy Rogers, George \"Gabby\" Hayes, Elyse Knox, Addison Richards, Sally Payne, Harry Woods, Hal Taliaferro, Jay Novello, Roy Barcroft, Jack Rockwell, Zeffie Tilbury, Jack Ingram, George Rosenor, Jack Kirk, Frank Ellis, Art Dillard, Herman Hack, Vester Pegg, Al Haskell, Ray Jones, Jess Cavin, Chuck Baldra, Oscar Gahan, Bob Reeves, Jim Corey, Fred Burns, Al Taylor. Crooked businessmen hire a tough cowpoke to be Tombstone's lawman but he turns on them when they try to cheat an old lady and her family out of their mine. Action filled and entertaining Roy Rogers opus chock full of favorite villains.\n\n**3828** _ **Sheriff of Wichita**_ **** Republic, 1949. 60 min. D: R.G. Springsteen. SC: Bob Williams. With Allan \"Rocky\" Lane, Eddy Waller, Lyn Wilde, Clayton Moore, Roy Barcroft, Gene Roth, Trevor Bardette, Edmund Cobb, House Peters, Jr., Earle Hodgins, John Hamilton, Jack O'Shea, Dick Curtis, Lane Bradford, Steve Raines, Stanley Price. A sheriff agrees to help a young woman find out who murdered her father. Too much obvious indoor for outdoor scenery but still a fair Allan Lane film with a good mystery flavor.\n\n**3829** _ **The Sheriff Was a Lady**_ **** Arthur Brauner, 1965. 88 min. Color. D: Sobey Martin (Carlo Croccolo). SC: Gustav Kampendonk. With Freddy Quinn, Mamie Van Doren, Rik Battalgia, Carlo Croccolo, Otto Waldis, Klaus Dahlen, Beba Lancar, Trude Herr. Pretending to be a greenhorn, a cowboy hunts the gang who murdered his parents and enlists the assist of a pretty saloon hostess. West German vehicle for popular singer Freddy Quinn with stateside appeal via co-star Mamie Van Doren; okay dubbed oater. West German title: _**Freddy und das Lied per Prarie**_ (Freddy and the Song of the Prairie), released there by CCC Filmkunst\/Avala-Film.\n\n**3830** _ **Sheriff with the Gold**_ **** Fono Roma, 1966. 89 min. Color. D: Richard Kean (Osvaldo Civirani). SC: Enzo Dell'Aquila and Roberto Gianviti. With Kathleen Parker (Caterina Trentini), Jacques Berthier, Louis McJulian (Luigi Giuliani), Bob Messenger (Roberto Messina), Luciano Rossi, Piero Morgia, Nando Angelini, Ivan Scratuglia, Franco Pesce, Renzo Pevarello, Amerigo Santarelli, Ettore Arena, Ares Lucky (Fortunato Arena), Claudio Biava, Franco Etella, Aldo Redine, Cristina Gallo. A crooked lawman robs a gold shipment but it is stolen by a beautiful woman who then loses it to outlaws, with the marshal changing his ways and helping her. There is not much to recommend this Italian-Spanish co-production originally called _**Uno Sceriffo Tutto d'Oro**_ (The Sheriff with All the Gold). Don Powell sings the title song.\n\n**3831** _ **Sheriff Won't Shoot**_ **** Hispamer, 1965. 85 min. Color. D: J. Luis Monter. SC: Robert Montero and Franco Verucci. With Mickey Hargitay, Pilar Clemens, Vincent Cashino, Alche Nana, Dan Clark, Manuel Zarzo Nana, Angel Ter, Sancho Garcia. A lawman is forced into a showdown when he discovers his younger brother is the brains behind an outlaw band. Passable English-French-Italian co-production issued in the latter country as _**Lo Sceriffo Che Non Spara**_ (The Sheriff Who Will Not Shoot).\n\n**3832** _ **Los Sheriffs de la Frontera**_ (The Frontier Sheriffs) **** Alameda Films, 1965. 82 min. D: Rene Cardona. With Fernando Casanova, Juan Gallardo, Dagoberto Rodriguez, Jorge Russek, Eva Calvo, Irma Serrano, Aurora Alvarado, Enrique Lucero, Pedro de Aguillon, Enrique Rocha. Two marshals attempt to clean out an outlaw gang looting a frontier community. Action filled Mexican Western.\n\n**3833** _ **Shiloh Falls**_ **** Radio London Films, 2007. 90 min. Color. D-SC: Adrian Fulle. With Esteban Powell, Forrest Witt, Patrick Graves, Marvin Campbell, Jack Littman, Greg Littman, Brad Greenquist, John Myers, Eric John Scialo, Steve Bannos, Amber Mellot, Roddy Mancuso, Rudy Verwey, Desiree Carey, Nikita Lea, Elizabeth Rypel, Art LaFleur, John Bader, Ellie Araiza, Renato Avenia, Ryan Bernstein, Alexander Emmert, Dave Keefer (narrator). As a lawman corners an outlaw gang in remote town, a man with enormous powers arrives and the sheriff must joins forces with his enemies to combat the stranger. Poor low budget production.\n\n**3834** _ **Shine on Harvest Moon**_ **** Republic, 1938. 57 min. D: Joseph Kane. SC: Jack Natteford. With Roy Rogers, Mary Hart, William Farnum, Lulu Belle and Scotty (Wiseman), Stanley Andrews, Frank Jaquet, Chester Gunnels, Matty Roubert, Pat Henning, Jack Rockwell, Joe Whitehead, David Sharpe, George (Montgomery) Letz, Lloyd Ingraham, Art Mix, Slim Whitaker, Jack Ingram, Jack Kirk, Horace B. Carpenter, Blackjack Ward, Bill Nestell, Dan White, George Plues, Al Taylor, Tom Smith, Rose Plummer, Herman Hack, Frank McCarroll, George DeNormand, Jim Corey, Eva McKenzie, Bob Card, Chick Hannon, Roy Bucko, Charls Hogg. An outlaw tries to convince his former partner to join him in a cattle rustling scheme and when he refuses the crook frames him for the crimes he committed. Too many novelty tunes bog down this outing which fortunately is dominated by Willliam Farnum, as the ex-outlaw turned rancher, and Stanley Andrews as the villain.\n\n**3835** _ **Shipwreck!**_ **** Pacific International, 1979. 102 min. Color. D-SC: Stewart Raffill. With Robert Logan, Heather Rattray, Shannon Saylor, Mikki-Jamison Olsen, Cjon Damitri Patterson. A man, his two young daughters, one of their friends and a black stowaway set out on a boat trip and become marooned on a remote Alaskan island. Well done family feature from the makers of \"The Wilderness Family\" trilogy.\n\n**3836** _ **Shoot First and Pray You Live (Because Luck Has Nothing to Do with It)**_ **** Grindstone Entertainment Group, 2008. 110 min. Color. D-SC: Lance Doty. With Alexandra Krizman, JoAnne Blackstone, Erik J. Bockemeier, Jason Bowen, Chris Browning, Stephen Chomko, Andrea Cypress, John Doman, Jim Gaffigan, Robert Nathan Gleason, Tom B. Gleason, Jeff Hephner, Tamara Hope, Frederick Lopez, Jennifer Miller, Carlos A. Montoya, Chuck Paul, Stephen Payne, Luce Rains, Chris Ranney, James Russo, Clark Sanchez, Neil Summers, Shannon Zeller, Art Usher, Clay Wilcox, Jay Winton, Gabriel Sanchez, William Sterchi, Rick Thompson, Richard Tyson. A cowpoke trails the gunfighter who killed his parents in order to avenge them. Fair attempt at the old revenge ploy.\n\n_**Shoot, Gringo, Shoot**_ see _**Gringo**_\n\n**3837** _ **Shoot Out**_ **** Universal, 1971. 94 min. Color. D: Henry Hathaway. SC: Marguerite Roberts. With Gregory Peck, Pat Quinn, Robert F. Lyons, Susan Tyrell, Jeff Corey, James Gregory, Rita Gam, Dawn Lyn, Pepe Serna, John Chandler, Paul Fix, Arthur Hunnicutt, Nicholas Beauvy, Arthur Space, Lane Bradford, Willis Bouchey, Elizabeth Harrower, Karen Kiett. After six years in prison a man seeks revenge on the now prosperous partner who betrayed him and along the way he is adopted by a small girl. Interesting premise goes awry due to lack of action; remake of _**Lone Cowboy**_.\n\n**3838** _ **Shoot Out at Big Sag**_ **** Parallel, 1962. 64 min. D-SC: Roger Kay. With Walter Brennan, Leif Erickson, Luana Patten, Chris Robinson, Constance Ford, Virginia Gregg, Les Tremayne, Don O'Kelly, Andy Brennan, William Foster, Robert Beecher, Lennie Geer. In a remote Montana area a cowardly preacher tries to run off a recently settled Texan and his son. Okay film that was the pilot for the unsold TV series \"Barbed Wire\" and based on the 1931 Walt Coburn novel of the same title.\n\n**3839** _ **Shoot-Out at Medicine Bend**_ **** Warner Bros., 1957. 87 min. D: Richard L. Bare. SC: John Tucker Battle and D.D. Beauchamp. With Randolph Scott, James Craig, Angie Dickinson, James Garner, Dani Crayne, Gordon Jones, Trevor Bardette, Don Beddoe, Myron Healey, John Alderson, Harry Harvey, Sr., Robert Warwick, Howard Negley, Marshall Bradford, Ann Doran, Daryn Hinton, Dickie Bellis, Edward Hinton, Lane Bradford, Frances Morris, Robert Lynn, Sam Flint, Philip Van Zandt, Guy Wilkerson, Syd Saylor, Harry Rowland, Marjorie Bennett, Jesslyn Fax, Marjorie Stapp, Nancy Kulp, George Meader, Rory Mallinson, Dee Carroll, Dale Van Sickel, Gil Perkins, Harry Lauter, Carol Henry, George Pembroke, Tom Monroe, Buddy Roosevelt, George Bell. Three men, whose families were massacred by Indians due to faulty ammunition, set out to get revenge on the trader who sold them the defective merchandise. Fairly good horse opera but not up to Randolph Scott's usual 1950s fare.\n\n**3840** _ **Shoot Out in a One Dog Town**_ **** ABC-TV, 1974. 74 min. Color. D: Burt Kennedy. SC: Larry Cohen and Dick Nelson. With Richard Crenna, Stefanie Powers, Richard Egan, Arthur O'Connell, Michael Ansara, Dub Taylor, Gene Evans, Michael Anderson, Jr., John Pickard, Jay Ripley, Jerry Gatlin, Henry Wills. When outlaws threaten to rob his establishment, a small town banker takes drastic measures to protect the money. Okay action TV Western with a cast of familiar faces.\n\n**3841** _ **Shoot the Living...Pray for the Dead**_ **** Castor Film, 1971. 90 min. Color. D: Joseph Warren (Giuseppe Vari). SC: Adriano Bolzoni. With Klaus Kinski, Victoria Zinny, Paul Sullivan (Paolo Casella), Dean Stratford (Dino Strano), Patrizia Adiutori, John Ely, Anthony Rock, Dan May (Dante Maggio), Ares Lucky (Fortunato Arena), Anna Zimmerman, Adriana Giuffre, Gianni Pulone, Aldo Barberito, Freddy (Goffredo) Unger. A stranger agrees to lead an outlaw gang into Mexico to escape Texas Rangers but he demands the gold they stole. This character oriented Italian oater is pretty good, released there as _**Prega il Morto e Ammazza il Vivo**_ (Pray for the Dead and Kill the Living); also called _**Pray to Kill and Return Alive**_ and _**Renegade Gun**_.\n\n**3842** _ **Shoot the Sun Down**_ **** JAD Films International, 1981. 93 min. Color. D: David Leeds. SC: Richard Rothstein and David Leeds. With Margot Kidder, Christopher Walken, Geoffrey Lewis, Bo Brundin, A. Martinez, Sacheen Littlefeather. In 1836 a gunman joins a bounty hunter and a retired sea captain in searching for lost gold and he falls in love with the old salt's pretty indentured girl. Slow moving, uninteresting and rather pointless effort.\n\n**3843** _ **The Shooter**_ **** Royal Oaks Entertainment, 1997. 91 min. Color. D: Fred Olen Ray. SC: Tony Giglio. With Michael Dudikoff, Randy Travis, Valerie Wildman, Andrew Stevens, William Smith, Eric Lawson, Robert Donovan, Carl Bartlett, Hoke Howell, Libby George, Matt Anderson, Peter Sherayko, William L. Monroe, Ryan Latshaw, Marv Vahanian, Pete Walsh, Robert Quarry, Neal DeLama, James Assi, Kane Hodder. After being beaten and left for dead, a gunfighter is saved by a hooker and they find themselves up against a corrupt family's hired gun. Pretty fair low budget action drama, issued on video as _**Deadly Shooter**_.\n\n_**Shootin' Irons**_ see _**West of Texas**_\n\n**3844** _ **Shootin' Square**_ **** Anchor, 1924. 50 min. D-SC: Robert J. Horner. With Jack Perrin, Peggy O'Day, Bud Osborne, Alfred Hewston, S.J. Bingham, Horace B. Carpenter, Milburn Morante, David Dunbar, Harry Pringle, Martin Turner, Starlight (horse). A crooked foreman, wanted for murder, is at odds with a cowboy over the affections of the ranch owner's lovely daughter. The lightheartedness of this otherwise competent Jack Perrin silent film is somewhat hurt by a complicated plot.\n\n**3845** _ **The Shooting**_ **** Jack H. Harris, 1971. 82 min. Color. D: Monte Hellman. SC: Adrien Joyce. With Jack Nicholson, Millie Perkins, Warren Oates, Will Hutchins, B.J. Merholz, Charles Eastman, Guy El Tsosie. A woman persuades two miners to be her guides on a journey that leads to revenge. Passable feature that, like _**Ride in the Whirlwind**_ (q.v.), was given TV release before making it to theatres.\n\n**3846** _ **Shooting High**_ **** 20th Century\u2013Fox, 1940. 65 min. D: Alfred E. Green. SC: Lou Breslow and Owen Francis. With Jane Withers, Gene Autry, Marjorie Weaver, Frank M. Thomas, Robert Lowery, Katharine (Kay) Aldridge, Hobart Cavanaugh, Jack Carson, Hamilton MacFadden, Charles Middleton, Ed Brady, Tom London, Eddie Acuff, Pat O'Malley, George Chandler, Carl Stockdale, LeRoy Mason, Emmett Vogan, Kathryn Sheldon, Harold Goodwin, Lee Moore, Lew Kelly, Ivan Miller, Paul E. Burns, Georgia Simmons. The grandson of a famous outlaw plays the part of his grandfather in a movie and ends up winning the girl he loves and capturing bank robbers. Gene Autry's loan out to 20th Century\u2013Fox for teaming with moppet Jane Withers results in an uneven film.\n\n**3847** _ **The Shootist**_ **** Paramount, 1976. 100 min. Color. D: Don Siegel. SC: Miles Hood Swarthout and Scott Hale. With John Wayne, Lauren Bacall, James Stewart, Ron Howard, Harry Morgan, Richard Boone, John Carradine, Hugh O'Brian, Sheree North, Richard Lenz, Scatman Crothers, Bill McKinney, Gregg Palmer, Alfred Dennis, Dick Winslow, Melody Thomas, Kathleen O'Malley. At the turn of the century, a famous gunfighter with terminal cancer finds his reputation getting in the way of his wish to die peacefully. The best Western of the 1970s and an all time genre classic; had the political climate of Hollywood not been so hostile, John Wayne would have won a second Oscar for this film, as it was he was not even nominated.\n\n**John Wayne in** _**The Shootist**_ **(Paramount, 1976).**\n\n**  \n**\n\n_**The Short and Happy Life of the Brothers Blue**_ see _**Brothers Blue**_\n\n**3848** _ **Short Grass**_ **** Allied Artists, 1950. 82 min. D: Lesley Selander. SC: Tom W. Blackburn. With Rod Cameron, Cathy Downs, Johnny Mack Brown, Alan Hale, Jr., Morris Ankrum, Jeff York, Raymond Walburn, Jonathan Hale, Riley Hill, Harry Woods, Stanley Andrews, Tristram Coffin, Myron Healey, Jack Ingram, Rory Mallinson, Marlo Dwyer, Felipe Turich, George J. Lewis, Lee Tung Foo, Lee Roberts, Frank Ellis, Tom Monroe, Kermit Maynard. A sheriff joins forces with a rancher to stop a crooked land scheme. A good script, direction and cast add up to a very fine little oater.\n\n**3849** _ **Shotgun**_ **** Allied Artists, 1955. 80 min. Color. D: Lesley Selander. SC: John Champion, Clark E. Reynolds and Rory Calhoun. With Sterling Hayden, Yvonne De Carlo, Zachary Scott, Robert Wilke, Guy Prescott, Ralph Sanford, John Pickard, Ward Wood, Rory Mallinson, Paul Marion, Harry Harvey, Jr., Lane Chandler, Angela Greene, Robert E. Griffin, Al Wyatt, Bob Morgan, Peter Coe, Charles Morton, James Parnell, Richard Cutting, Fiona Hale, Francis McDonald. A showgirl joins a sheriff and a bounty hunter to track a killer only to find themselves stalked by Apaches. Co-writer Rory Calhoun was originally scheduled to star in this minor oater, best recommended for Yvonne De Carlo's bathing sequence.\n\n**3850** _ **Shotgun Pass**_ **** Columbia, 1931. 60 min. D: J.P. McGowan. SC: Robert Quigley. With Tim McCoy, Virginia Lee Corbin, Frank Rice, Dick Stewart, Joe Marba, Monty Vandergrift, Ben Corbett, Albert J. Smith, Archie Ricks. Two dishonest brothers own a pass and refuse to let a cowboy, with an Army contract, lead a herd of horses through it and trouble follows, including murder. There is enough action in this Tim McCoy film to delight his fans.\n\n_**Shots Ring Out!**_ see _**Four Bullets for Joe**_\n\n**3851** _ **The Showdown**_ **** Paramount, 1940. 65 min. D: Howard Bretherton. SC: Howard Kusel and Donald Kusel. With William Boyd, Russell Hayden, Britt Wood, Morris Ankrum, Jane (Jan) Clayton, Wright Kramer, Donald Kirke, Roy Barcroft, Eddie Dean, Kermit Maynard, Walter Shumway, The King's Men (Ken Darby, Rad Robinson, Bud Linn, Jon Dodson), Snub Pollard, Eddy Chandler, Murdock MacQuarrie, George Morrell, Jim Corey, Ray Jones. Hoppy and Lucy are at odds when the latter sides with a woman ranch owner while Cassidy believes a fake European baron plans to rustle her cattle. Average \"Hopalong Cassidy\" feature without much action until the finale.\n\n**3852** _ **The Showdown**_ **** Republic, 1950. 86 min. D-SC: Dorrell McGowan and Stuart McGowan. With William Elliott, Walter Brennan, Marie Windsor, Harry Morgan, Rhys Williams, Jim Davis, William Ching, Nacho Galindo, Leif Erickson, Henry Rowland, Charles Stevens, Victor Kilian, Yakima Canutt, Guy Teague, William Steele, Jack Sparks. A cattle herd driver, a former outlaw, tracks down the bad man who murdered his brother. Distinguished adult Western starring William Elliott, who also co-produced; his final Republic feature.\n\n**3853** _ **Showdown**_ **** Universal, 1963. 79 min. D: R.G. Springsteen. SC: Bronson Howitzer. With Audie Murphy, Kathleen Crowley, Charles Drake, Skip Homeier, Harold J. Stone, L.Q. Jones, Strother Martin, Charles Horvath, John McKee, Henry Wills, Joe Haworth, Kevin Brodie, Carol Thurston, Dabbs Greer, Harry Lauter, Dale Van Sickel. Two convicts escape from prison and head to the Mexican border where they get involved in a robbery. Audie Murphy fans will like this program feature.\n\n**3854** _ **The Showdown**_ **** NBC-TV\/Universal, 1971. 74 min. Color. D: Daniel Petrie. SC: Dick Nelson. With Gene Barry, Jessica Walter, Warren Oates, Jack Albertson, Albert Salmi, Ron Turbeville, Eve Bruce, Jack Garner, Daniel Kemp, William Bramley, Jack Collins, Martin Garralaga. A magazine editor is tipped off to the fake account of a famous gunfight and during the investigation, in flashbacks, returns to the Old West. Originally a segment of \"The Name of the Game\" (NBC-TV, 1968\u201371), this telefilm is for avid fans of the series.\n\n**3855** _ **Showdown**_ **** Universal, 1973. 90 min. Color. D: George Seaton. SC: Theodore Taylor. With Dean Martin, Rock Hudson, Susan Clark, Donald Moffatt, John McLiam, Charles Baca, Jackson Kane, Ben Zeller, John Richard Gill, Phillip L. Mead, Rita Rogers, Vic Mohica, Raleigh Gardenhire, Ed Begley, Jr., Dan Boydston. Two men who once loved the same woman find themselves at odds again when one of them, a lawman, hunts the other, an outlaw. Fair teaming of Dean Martin and Rock Hudson makes for an okay time passer.\n\n**3856** _ **Showdown at Abilene**_ **** Universal-International, 1956. 80 min. Color. D: Charles Haas. SC: Bernie Giler. With Jock Mahoney, Martha Hyer, David Janssen, Lyle Bettger, Grant Williams, Ted De Corsia, Harry Harvey, Sr., Dayton Lummis, Richard Cutting, Robert G. Anderson, John Maxwell, Lane Bradford, Kenneth MacDonald, Tom Steele, Rusty Wescoatt. Returning home to Texas after the Civil War, a gun shy ex-lawman finds his girl, who thought him dead, engaged to a dishonest cattleman. Nicely done Jock Mahoney feature, remade as _**Gunfight in Abilene**_ (q.v.).\n\n**3857** _ **Showdown at Boot Hill**_ **** 20th Century\u2013Fox, 1957. 72 min. D: Gene Fowler, Jr. SC: Louis Vittes. With Charles Bronson, Robert Hutton, John Carradine, Carole Mathews, Fintan Meyler, Paul Maxey, Thomas Browne Henry, William Stevens, Martin Smith, Joseph McGuinn, George Douglas, Michael Mason, George Pembroke, Argentina Brunetti, Ed Wright, Dan Simmons, Barbara Woodell, Norman Leavitt. A bounty hunter arrives in town, kills a wanted man and faces the wrath of its citizens, while falling in love with a shy young woman. Sadly neglected \"B\" feature, enhanced by good direction and top flight performances by Charles Bronson (his first starring Western), John Carradine, Carole Mathews and Fintan Meyler.\n\n**3858** _ **Showdown at Williams Creek**_ **** Crescent Entertainment, 1991. 97 min. Color. D: Allan Kroeker. SC: John Gray. With Tom Burlinson, Stephen E. Miller, Michelle Thrush, Pascal Bernier, Raymond Burr, Betty Phillips, John Pyper-Ferguson, William Samples, Donnelly Rhodes, Jay Brazeau, John Gray, Patti Allen, John \"Bear\" Curtis, Frank C. Turner, Brent Strait, Rick Poltaruk, Dale Wilson, Barrett Reid, Tamsin Kelsey, Alex Bruhanski, Pierce Bros. Band, Don MacKay, Tim Henry, Eric Armstrong, Bill Croft, David Longworth, Buffalo Child, Andrew Seebaran. On trial for murder in 1870 Montana, a former British soldier relates how he came to Canada, was left for dead by his partner and saved by Indians, falling in love with a maiden with the two having a son. Competently done Canadian feature.\n\n_**Showdown in Silver City**_ see _**Longarm**_\n\n**3859** _ **Shut My Big Mouth**_ **** Columbia, 1942. 71 min. D: Charles Barton. SC: Oliver Drake, Karen DeWolf and Francis Martin. With Joe E. Brown, Adele Mara, Victor Jory, Don Beddoe, Lloyd Bridges, Forrest Tucker, Earle Hodgins, Fritz Feld, Russell Simpson, Pedro de Cordoba, Joan Woodbury, Ralph Peters, Joe McGuinn, Noble Johnson, Chief Thundercloud, Art Mix, Edmund Cobb, Dick Curtis, Eddy Waller, Will Wright, Fern Emmett, Ed Peil, Sr., Frank McCarroll, John Tyrrell, Hank Bell, Blackjack Ward. A timid man from the East accidentally knocks out a desperado, is made sheriff of a frontier town and has to stand up to an outlaw gang leader. Amusing Joe E. Brown feature with a good story and direction.\n\n**3860** _ **Sidekicks**_ **** CBS-TV\/Warner Bros., 1974. 75 min. Color. D: Burt Kennedy. SC: William Bowers. With Larry Hagman, Lou Gossett, Blythe Danner, Jack Elam, Harry Morgan, Gene Evans, Noah Beery, Hal Williams, Dick Peabody, Denver Pyle, John Beck, Dick Haynes, Tyler McVey, Bill Shannon. After the Civil War two con men out West try to collect a bounty on an outlaw. Tepid television version of _**Skin Game**_ (q.v.) that never made it as a series.\n\n**Larry Hagman and Lou Gossett in** _**Sidekicks**_ **(CBS-TV\/Warner Bros., 1974).**\n\n**  \n**\n\n**3861** _ **The Siege at Red River**_ **** 20th Century\u2013Fox, 1954. 86 min. Color. D: Rudolph Mate. SC: Sydney Boehm. With Van Johnson, Joanne Dru, Richard Boone, Milburn Stone, Jeff Morrow, Craig Hill, Rico Alaniz, Robert Burton, Pilar Del Rey, Ferris Taylor. During the Civil War a Confederate spy masquerades as a showman in order to steal a Gatling gun and becomes involved with a pretty Yankee nurse, a Pinkerton agent and marauding Indians. Fairly entertaining oater also called _**The Siege of Red River**_.\n\n_**The Siege of Red River**_ see _**The Siege at Red River**_\n\n**3862** _ **Sierra**_ **** Universal-International, 1950. 83 min. Color. D: Alfred E. Green. SC: Edna Anhalt. With Audie Murphy, Wanda Hendrix, Dean Jagger, Burl Ives, Richard Rober, Anthony Caruso, Houseley Stevenson, Elliott Reid, Griff Barnett, Elisabeth Risdon, Roy Roberts, Gregg Martell, Sara Allgood, James Arness, Ted Jordan, I. Stanford Jolley, Jack Ingram. A female lawyer accidentally stumbles across the hideout of a man and his son, the former on the run from the law for a crime he did not commit. Pretty good remake of _**Forbidden Valley**_ (q.v.).\n\n**3863** _ **Sierra Baron**_ **** 20th Century\u2013Fox, 1958. 80 min. Color. D: James B. Clark. SC: Houston Branch. With Brian Keith, Rick Jason, Rita Gam, Mala Powers, Allan Lewis, Pedro Calvan, Fernando Wagner, Steve Brodie, Carlos Muzquiz, Lee Morgan, Reed Howes, Albert Mariscal. In nineteenth century California a ruthless man hires a gunman to kill a Mexican in order to get his vast land holdings. More than passable film, enhanced by good photography.\n\n**3864** _ **Sierra Passage**_ **** Monogram, 1951. 81 min. D: Frank McDonald. SC: Tom W. Blackburn, Warren D. Wandberg and Sam Roeca. With Wayne Morris, Lola Albright, Lloyd Corrigan, Alan Hale, Jr., Roland Winters, Jim Bannon, Billy Gray, Paul McGuire, Richard Karlan, George Eldredge, John Doucette, Zon Murray, Paul Bryar. A cowboy postpones his marriage to hunt down the man who murdered his father. Grade B action feature that provides its allotted modicum of entertainment.\n\n**3865** _ **Sierra Stranger**_ **** Columbia, 1957. 78 min. D: Lee Sholem. SC: Richard J. Dorso. With Howard Duff, Gloria McGhee, Dick Foran, John Hoyt, Barton MacLane, George E. Stone, Ed Kemmer, Robert Foulk, Eve McVeagh, Henry Kulky, Byron Foulger. A wild young man is saved from lynching by a prospector and the two end up in still more trouble. Barely passable second bill item although it is nice to see Dick Foran in a co-starring role.\n\n**3866** _ **Sierra Sue**_ **** Republic, 1941. 64 min. D: William Morgan. SC: Earl Fenton and Julian Zimet. With Gene Autry, Smiley Burnette, Fay McKenzie, Frank M. Thomas, Robert Homans, Earle Hodgins, Dorothy Christy, Jack Kirk, Eddie Dean, Kermit Maynard, Budd Buster, Rex Lease, Hugh Prosser, Vince Barnett, Hal Price, Syd Saylor, Roy Butler, Sammy Stein, Bob McKenzie, Marin Sais, Ray Davis, Frankie Marvin, Art Dillard. The state agricultural commission sends inspector Gene Autry to a devil weed infested area to study the problem but he meets opposition from the head of the local cattlemen's association. Pretty dreary Gene Autry vehicle except for a few songs.\n\n**3867** _ **El Siete de Copas**_ (The Seven Cups) **** Filmadora Independiente, 1960. 95 min. D-SC: Roberto Gavaldon. With Antonio Aguilar, Elvira Quintana, Julio Adama, Pedro de Aguillon, Aurora Walker, Jose Carlos Mendez, David Reynoso, Tito Novaro, Javier Gomez, Jose Luis Fernandez, Edmundo Espino, Roberto Meyer, Salvador Lozano, Jose Pardave, Pepe Hernandez, Guillermo Bravo Sosa, Margarito Luna, Chel Lopez, Jesus Gomez, Jose Chavez. A gambler, who has had good luck since he was a boy, comes to feel he has lost his touch and makes a major effort to regain his gaming prowess. Only average Mexican Western from producer Rafael Baledon.\n\n**3868** _ **Siete Leguas**_ (Seven Leagues) **** Cinematografica Intercontinental, 1955. 90 min. D-SC: Raul de Anda. With Luis Aguilar, Yolanda Varela, Linda Cristal, Arturo Martinez, Jose Elias Moreno, Dagoberto Rodriguez, Luis Aldas, Fernando Casanova, Victor Alcocer, Conchita Gentil Arcos, America Martin, Lola Casanova, Armando Soto La Marina. A emissary for Pancho Villa arrives in a town to prepare for an attack and falls for the mayor's daughter who warns him of her father's love for a rival general's sister. Well made pseudo-historical drama from Mexico.\n\n_**Sign of the Beaver**_ see _**Keeping the Promise**_\n\n_**Sign of the Otter**_ see _**The Little Patriot**_\n\n**3869** _ **The Sign of the Wolf**_ **** Metropolitan, 1931. 10 Chapters. D: Harry S. Webb and Forrest Sheldon. SC: Carl Krusada. With King (dog), Rex Lease, Virginia Brown Faire, Joe Bonomo, Jack Mower, Josephine Hill, Al Ferguson, Robert Walker, Edmund Cobb, Harry Todd, Jack Perrin, Billy O'Brien. In Tibet, an explorer steals chains that can turn sand into jewels and years later crooks in the West try to take them from him and his daughter. Very low grade, but fun, cliffhanger; issued in a feature version in 1932 by Syndicate called _**The Lone Trail**_.\n\n**3870** _ **Sign of the Wolf**_ **** Monogram, 1941. 69 min. D: Howard Bretherton. SC: Elizabeth Hopkins and Edmund Kelso. With Michel Whelan, Grace Bradley, Mantan Moreland, Darryl Hickman, Louise Beavers, Wade Crosby, Tony Paton, Smoky and Shadow (dogs). Two canines are raised together in the north country with one of them becoming a thief while the other remains loyal to his master who is beset by fur grabbers. Fair adaptation of a Jack London story from producer Paul Malvern.\n\n**3871** _ **The Sign of Zorro**_ **** Buena Vista, 1959. 91 min. Color. D: Norman Foster and Lewis R. Foster. SC: Norman Foster, Lowell S. Hawley, Bob Wehling and John Meredyth Lucas. With Guy Williams, George J. Lewis, Henry Calvin, Gene Sheldon, Britt Lomond, Tony Russo, John Dehner, Lisa Gaye, Romney Brent, Than Wyenn, Elvira Corona, Eugenia Paul, Jan Arvan, Nestor Paiva, Madeleine Holmes. In 1820 the son of a California rancher arrives home from Spain and pretends to be a fop in order to disguise himself as the masked avenger Zorro, out to right wrongs of a local despot. Fun compilation of episodes of \"The Adventures of Zorro\" (ABC-TV, 1957\u201359) and issued theatrically both in the U.S. and abroad.\n\n_**The Sign of Zorro**_ (1963) see _**Duel at the Rio Grande**_\n\n**3872** _ **Silence of the North**_ **** Universal, 1981. 94 min. Color. D: Allan Winton King. SC: Patricia Louisiana Knop. With Ellen Burstyn, Tom Skerritt, Gordon Pinsent, Jennifer McKinney, Donna Dobrijevic, Colin Fox, Chapelle Jaffe, Ken Pogue, Tom Hauff, Murray Westgate, Ken James, Booth Savage, Louis Banks, Sean McCann, Frank Adamson. A woman marries a vagabond trapper in 1919 and goes to live with him in the wilds of Canada. Based on a true story, this drama holds some interest but the scenery is better than the plot.\n\n**3873** _ **El Silencioso**_ (The Silent) **** Antenea Film, 1967. 85 min. Color. D: Alberto Mariscal. SC: Jose Maria Fernandez Unsain and Fernando Galiano. With Gaston Santos, Luis Aguilar, Adriana Roel, Mari Carmen Gonzalez, Emilio Fernandez, Roberto Canedo, Manuel Donde, Guillermo Alvarez Bianchi, Jesus Gomez, Carlos Leon, Jose Eduardo Perez, Fernando Seshum. A child is the only witness to the murder of his parents and grows up planning to take revenge on the killers. Entertaining Mexican Western.\n\n**3874** _ **Silent Barriers**_ **** Gaumont-British, 1937. 84 min. D: Milton Rosmer. SC: Michael Barringer and Milton Rosmer. With Richard Arlen, Lilli Palmer, Antoinette Cellier, J. Farrell MacDonald, Barry MacKay, Roy Emerton, Ben Welden, Jock MacKay, Ernest Sefton, Henry Victor, Reginald Barlow, Arthur Loft, Frank McGlynn, Gilbert Emery, Howard Hickman, William Kuhl, LeStrange Millman. A former gambler becomes a railroad builder and helps in the construction of a line across the Canadian wilderness. British financed and Canadian filmed, this movie combines trite drama with good action sequences. British title: _**The Great Barrier**_.\n\n**3875** _ **The Silent Call**_ **** 20th Century\u2013Fox, 1961. 63 min. D: John Bushelman. SC: Tom Maruzzi. With Gail Russell, Roger Mobley, David McLean, Joe Besser, Jack Younger, Rusty Wescoatt, Roscoe Ates, Sherwood Keith, Milton Parsons, Dal McKennon. Separated from his young master, a faithful dog makes the dangerous 600 mile trek from Reno to Los Angeles. Pleasant family program feature.\n\n**3876** _ **The Silent Code**_ **** Stage and Screen, 1935. 55 min. D: Stuart Paton. SC: George Morgan. With Kane Richmond, Blanche Mehaffey, J.P. McGowan, Joe Girard, Barney Furey, Pat Harmon, Ben Corbett, Carl Mathews, Ed Coxen, Bud Osborne, Ted Mapes, Clarence Davis, Douglas Ross, Rose Higgins, Wolfgang (dog). A miner is murdered and the blame for the crime is placed on a member of the Royal Canadian Mounted Police. Low grade action dual bill item with nice locales.\n\n**3877** _ **Silent Conflict**_ **** United Artists, 1948. 61 min. D: George Archainbaud. SC: Charles Earl Belden. With William Boyd, Andy Clyde, Rand Brooks, Virginia Belmont, Earle Hodgins, James Harrison, Forbes Murray, John Butler, Herbert Rawlinson, Richard Alexander, Don Haggerty, Leo J. McMahon, George Magrill. A fake doctor uses hypnotism to make Lucky steal cattle association funds from Hopalong Cassidy and when they find out, Hoppy and California get on his trail. Dull entry from the tail end of the long running series.\n\n**3878** _ **The Silent Gun**_ **** ABC-TV\/Paramount, 1969. 74 min. Color. D: Michael Caffey. SC: Clyde Ware. With Lloyd Bridges, John Beck, Ed Begley, Edd Byrnes, Pernell Roberts, Susan Howard, Michael Forest, Trace Evans, Bob Diamond, Barbara Rhodes. A once famous gunman, who vowed never again to take up arms, is made the town sheriff and has to deal with the hatred between a politician and a settler. Mediocre made-for-television oater with more talk than action.\n\n**John Beck and Lloyd Bridges in** _**The Silent Gun**_ **(ABC-TV\/Paramount, 1969).**\n\n**  \n**\n\n**3879** _ **The Silent Man**_ **** Paramount-Artcraft, 1917. 60 min. D: William S. Hart. SC: Charles Kenyon. With William S. Hart, Vola Vale, Robert McKim, Harold Goodwin, J.P. Lockney, George P. Nichols, Gertrude Claire, Milton Ross, Dorcas Matthews. A miner, cheated out of his rich claim by a corrupt saloon owner, plans revenge. Entertaining William S. Hart silent feature.\n\n**3880** _ **Silent Men**_ **** Columbia, 1933. 60 min. D: D. Ross Lederman. SC: Jack Cunningham, Stuart Anthony and Gerald Geraghty. With Tim McCoy, Florence Britton, Wheeler Oakman, J. Carrol Naish, Matthew Betz, Lloyd Ingraham, Steve Clark, William V. Mong, Walter Brennan, Syd Saylor, Joseph Girard, Lafe McKee, Art Mix, Frank Ellis, Lew Meehan, Glenn Strange, Slim Whitaker, Artie Ortego, Richard Botiller, Archie Ricks, Charles Brinley. A special agent for cattlemen loses his job when it is found out he is an escaped convict and suspected of being the leader of a gang of rustlers. Eerie and atmospheric Tim McCoy vehicle with a complicated, and sometimes hard to follow, plot.\n\n**3881** _ **Silent Rage**_ **** Columbia, 1982. 103 min. Color. D: Michael Miller. SC: Joseph Fraley. With Chuck Norris, Ron Silver, Steven Keats, Toni Kalem, William Finley, Brian Libby, Stephen Furst, Stephanie Dunnam, Joyce Ingle, Jay De Plano, Lillette Zoe Raley, Mike Johnson, Linda Tatum, Kathleen Lee, James Bodeen, John Barrett, Desmond Dhooge, Joe Farago, Russell Higginbotham, Eddie Galt, David Unger, Sonny Jones, Sandy Lang, Paul Selzer. The sheriff of a Texas community tries to protect its citizens, including his ex-girlfriend, from an all powerful psychotic killer. Fair horror movie, in a Western setting, starring six time World Karate Champion Chuck Norris.\n\n**3882** _ **Silent Sentence**_ **** Bryanston\/National General, 1973. 85 min. Color. D: Larry G. Spangler. SC: George Arthur Bloom and Seton I. Miller. With Jack Elam, Ruth Roman, Jeff Cooper, Gene Evans, John Kellogg, Richard Shaal, Diana Ewing, Joe Spangler, Derek Sanderson, Joe Santos. Three prostitutes are horribly murdered in a remote community where the banker hires a detective to investigate, much to the chagrin of an old time sheriff. This Jack the Ripper out West affair, filmed at Arizona's Old Tucson, is dull, without much action; originally called _**A Knife for the Ladies**_.\n\n_**The Silent Stranger**_ see _**A Stranger in Japan**_\n\n**3883** _ **Silent Tongue**_ **** Trimark Pictures, 1994. 102 min. Color. D-SC: Sam Shepard. With Richard Harris, Sheila Tousey, Alan Bates, River Phoenix, Dermot Mulroney, Jeri Arredondo, Tantoo Cardinal, Bill Irwin, David Shiner, Tim Carroll, Nicholas Ortiz y Pino, Robert Harnsberger, Fred Maio, Leslie Flemming, Jill Momaday, Lynn Davis, Tim Scott, Tommy Thompson, Jack Herrick, Bland Simpson, Clay Buckner, Chris Frank, Arturo Gil, Joseph Griffo, Billy Beck, Phillip Attmore, Al Lujan, Devino Tricoche, April Tatro. A cowboy abducts his son's sister-in-law, a half-breed, so she can try to help the man out of the grief he has for the loss of his young Indian wife. Eerie, surreal Western, not for all tastes.\n\n**3884** _ **Silent Valley**_ **** Reliable, 1935. 56 min. D: Bernard B. Ray. SC: Rose Gordon. With Tom Tyler, Nancy DeShon, Alan Bridge, Wally Wales, Charles King, Charles \"Slim\" Whitaker, Art Miles, Murdock MacQuarrie, Jimmy Aubrey, Frank Ellis, Lew Meehan, Herman Hack, Budd Buster, George Morrell, Art Dillard, Tex Palmer, Oscar Gahan, Bob Reeves, George Hazel, Barney Beasley, Ray Jones, Jack Hendricks, Art Felix. A lawman hunting cattle rustlers suspects his girl's brother of being in the gang that is led by a supposedly respectable citizen. Alan Bridge is very good as the slick villain as is Slim Whitaker as his hired gunman, but overall this Tom Tyler affair is just fair.\n\n**3885** _ **Silent Wilderness**_ **** Ted Leverfech, 1976. 92 min. Color. With Dr. Roger Latham. A naturalist explores Alaska, from abandoned gold mines to oil pipelines, and encounters a grizzly bear near Mt. McKinley and a whale close to the Arctic. Well made and enjoyable documentary.\n\n**3886** _ **Silly Billies**_ **** RKO Radio, 1936. 64 min. D: Fred Guiol. SC: Al Boasberg and Jack Townley. With Bert Wheeler, Robert Woolsey, Dorothy Lee, Harry Woods, Ethan Laidlaw, Chief Thundercloud, Delmar Watson, Richard Alexander, Lafe McKee, Tommy Bond, Willie Best, Maurice Black, John Ince, Nelson McDowell, Jim Thorpe. Two zany dentists go West on a wagon train and end up saving it from an Indian attack. Very weak Bert Wheeler-Robert Woolsey comedy.\n\n**3887** _ **Silver Bandit**_ **** Friedgen, 1950. 54 min. D: Elmer Clifton. SC: Elmer S. :Pond (Clifton). With Spade Cooley, Bob Gilbert, Virginia Jackson, Richard Elliott, Billy Dix, Jene Gray. After silver is stolen from a mine, its owner sends a bookkeeper to investigate and he uncovers an outlaw gang is the culprit. Bottom of the barrel Spade Cooley musical fare.\n\n**3888** _ **The Silver Bullet**_ **** Reliable, 1935. 53 min. D: Bernard B. Ray. SC: Rose Gordon and Carl Krusada. With Tom Tyler, Jayne Regan, Lafe McKee, Charles King, George Chesebro, Slim Whitaker, Lew Meehan, Franklyn Farnum, Walt Williams (Wally Wales), Blackie Whiteford, Hank Bell, Nelson McDowell, Robert Brower, Allen Smith, Tom Smith, Tex Palmer, Fern Emmett, Jack Evans, Robert Walker, Herman Hack, Barney Beasley, Bill Patton, Jimmy Aubrey, George Morrell, Bruce Mitchell, Ray Henderson, Murray Horn. A prospector agrees to become the sheriff of a town plagued by outlaws and tries to find out who is the leader of the gang. Cheaply made but more than passable Tom Tyler oater.\n\n**3889** _ **The Silver Bullet**_ **** Universal, 1942. 56 min. D: Joseph H. Lewis. SC: Elizabeth Beecher. With Johnny Mack Brown, Fuzzy Knight, William Farnum, Jennifer Holt, LeRoy Mason, Rex Lease, Grace Lenard, Claire Whitney, Slim Whitaker, William Desmond, Merrill McCormick, Michael Vallon, James Farley, Lloyd Ingraham, The Pals of the Golden West and Nora Lou Martin, Harry Holman, Hank Bell, Tex Phelps. A cowboy searches for the outlaw who shot him in the back with a silver bullet and murdered his father. A bit complicated but well done Johnny Mack Brown outing.\n\n**3890** _ **Silver Canyon**_ **** Columbia, 1951. 70 min. D: John English. SC: Gerald Geraghty. With Gene Autry, Pat Buttram, Gail Davis, Jim Davis, Bob Steele, Edgar Dearing, Richard Alexander, Terry Frost, Peter Mamakos, Steve Clark, Stanley Andrews, Duke York, Eugene Borden, Bobby Clark, Frankie Marvin, Boyd Stockman, Sandy Sanders, Kenne Duncan, Bill Hale, Jack O'Shea, Stanley Blystone, John Merton, Jack Pepper, Pat O'Malley, Jim Magill, John Daheim, Eddie Parker. During the Civil War, an Army scout is on the trail of a Union renegade leader and his band, whose activities have also been denounced by the Confederacy. Top notch Gene Autry feature with splendid work by Jim Davis as the villain.\n\n_**Silver Chains**_ see _**The Kid from Amarillo**_\n\n**3891** _ **Silver City**_ **** Paramount, 1951. 90 min. Color. D: Byron Haskin. SC: Frank Gruber. With Yvonne De Carlo, Edmond O'Brien, Richard Arlen, Barry Fitzgerald, Gladys George, Laura Elliot, Edgar Buchanan, Michael Moore, John Dierkes, Don Dunning, Warren Earl Fisk, James Van Horn, John Mansfield, Harvey Parry, Boyd \"Red\" Morgan, Frank Cordell, Leo J. McMahon, Howard Joslin, Robert G. Anderson, Frank Fenton, Myron Healey, James R. Scott, Paul E. Burns, Cliff Clark, Billy House, Howard Negley, Ray Hyke, Slim Gault. A miner attempts to help a pretty girl and her father develop their claim but they are opposed by a rich rancher who wants the gold and the woman for himself. A good cast does what it can with a mediocre story. British title: _**High Vermillion**_.\n\n**Richard Arlen, Barry Fitzgerald, Michael Moore, Gladys George, Edmond O'Brien, Myron Healey and Laura Elliot in** _**Silver City**_ **(Paramount, 1951).**\n\n**  \n**\n\n**3892** _ **Silver City Bonanza**_ **** Republic, 1951. 67 min. D: George Blair. SC: Bob Williams. With Rex Allen, Buddy Ebsen, Mary Ellen Kay, Billy Kimbley, Bill Kennedy, Alix Ebsen, Gregg Barton, Clem Bevans, Frank Jenks, Hank Patterson, Harry Lauter, Harry Harvey, Edmund Cobb, Marshall Reed, Ted Mapes, Tom Steele. When a blind man is murdered at a supposedly haunted ranch, a cowboy tries to find the killer. Fast paced and well written Rex Allen feature.\n\n**3893** _ **Silver City Kid**_ **** Republic, 1944. 56 min. D: John English. SC: Taylor Cavan. With Allan Lane, Peggy Stewart, Wally Vernon, Twinkle Watts, Frank Jaquet, Harry Woods, Glenn Strange, Lane Chandler, Bud Geary, Tom London, Tom Steele, Jack Kirk, Sam Flint, Frank McCarroll, Hal Price, Ed Peil, Sr., Fred Graham, Frank O'Connor, Horace B. Carpenter. Valuable ore is stolen by an outlaw gang and a ranch foreman suspects the town judge is behind the operation. Allan Lane's first series Western is a quick moving affair.\n\n**3894** _ **Silver City Raiders**_ **** Columbia, 1943. 55 min. D: William Berke. SC: Ed Earl Repp. With Russell Hayden, Dub Taylor, Alma Carroll, Bob Wills and The Texas Playboys, Paul Sutton, Edmund Cobb, Jack Ingram, Art Mix, Luther Wills, Jack Rockwell, John Tyrrell, Merrill McCormick, George Morrell, Tex Palmer, Horace B. Carpenter, Jose De La Cruz, Curley Gibson. Ranchers learn they may lose their spreads to a land office operator with a Spanish grant giving him claim to the area. Better than average outing in Russell Hayden's Columbia series.\n\n_**Silver Devil**_ see _**Wild Horse**_\n**3895** _ **Silver Dollar**_ **** First National, 1932. 84 min. D: Alfred E. Green. SC: Carl Erickson and Harvey Thew. With Edward G. Robinson, Bebe Daniels, Aline MacMahon, Jobyna Howland, De Witt Jennings, Robert Warwick, Russell Simpson, Harry Holman, Charles Middleton, John Marston, Marjorie Gateson, Emmett Corrigan, Wade Boteler, William Le Maire, David Durand, Lee Kohlmar, Theresa Conover, Leon Ames, Virginia Edwards, Christian Rub, Walter Rogers, Niles Welch, Wilfred Lucas, Herman Bing, Bonita Granville, Walter Long, Charles Coleman, Frederick Burton, Willard Robertson, Alice Wetherfield. A Kansas farmer moves to Colorado for a gold rush but goes broke only to get rich with a silver strike and rise politically before becoming involved in a scandal. Entertaining soap opera.\n\n**3896** _ **The Silver Horde**_ **** RKO Radio, 1930. 75 min. D: George Archainbaud. SC: Wallace Smith. With Evelyn Brent, Joel McCrea, Louis Wolheim, Raymond Hatton, Jean Arthur, Gavin Gordon, Blanche Sweet, Purnell Pratt, William B. Davidson, Ivan Linow, Robert Homans, William H. O'Brien, Dick Curtis, Bud Flanagan (Dennis O'Keefe). A saloon entertainer helps a young salmon fisherman whose business is threatened by a crook. Good production values highlight this Rex Beach adaptation.\n\n**3897** _ **Silver Lode**_ **** RKO Radio, 1954. 80 min. Color. D: Allan Dwan. SC: Karen De Wolfe. With John Payne, Lizabeth Scott, Dan Duryea, Dolores Moran, Emile Meyer, Harry Carey, Jr., Morris Ankrum, John Hudson, Robert Warwick, Stuart Whitman, Alan Hale, Jr., Frank Sully, Paul Birch, Florence Auer, Roy Gordon, Edgar Barrier, John Dierkes, Myron Healey, Hugh Sanders, Lane Chandler, Byron Foulger, Gene Roth, Roy Jordan, Ray Jones. On his wedding day a man is accused of murder and he runs away to prove his innocence. Sturdy action melodrama.\n\n**3898** _ **Silver on the Sage**_ **** Paramount, 1939. 68 min. D: Lesley Selander. SC: Maurice Geraghty. With William Boyd, George Hayes, Russell Hayden, Stanley Ridges, Ruth Rogers, Frederick Burton, Jack Rockwell, Roy Barcroft, Ed Cassidy, Jim Corey, Sherry Tansey, Bruce Mitchell, Wen Wright, George Morrell, Frank O'Connor, Buzz Barton, Herman Hack, Dick Dickinson, Hank Bell, Bud McClure. Lucky is falsely accused of murder and Hoppy and Windy help him escape from the law as Cassidy seeks to uncover the brains behind a cattle rustling gang. Action filled \"Hopalong Cassidy\" feature with an amusing finale.\n\n**3899** _ **Silver Queen**_ **** United Artists, 1942. 81 min. D: Lloyd Bacon. SC: Bernard Schubert and Cecile Kramer. With George Brent, Priscilla Lane, Bruce Cabot, Lynne Overman, Eugene Pallette, Janet Beecher, Guinn Williams, Roy Barcroft, Eleanor Stewart, Arthur Hunnicutt, Sam McDaniel, Spencer Charters, Cy Kendall, Georges Renavent, Francis X. Bushman, Franklyn Farnum, Marietta Canty, Herbert Rawlinson, George Eldredge, Fred \"Snowflake\" Toones, Frederick Burton, Ed Cassidy, Jason Robards, Wilbur Mack, Henry Otho. A young woman gambles to raise money to pay her father's debts while her fiance invests her winnings in a silver mine. Passable viewing\u2014nothing more.\n\n**3900** _ **Silver Raiders**_ **** Monogram, 1950. 55 min. D: Wallace Fox. SC: Dan Ullman. With Whip Wilson, Andy Clyde, Virginia Herrick, Leonard Penn, Dennis Moore, Patricia Rice, Reed Howes, Riley Hill, Marshall Reed, George DeNormand, Kermit Maynard, Ed Cassidy, Frank Hagney, Frank Ellis. A Texas Ranger infiltrates a gang smuggling silver ore across the Mexican border into the United States. Pretty good Whip Wilson vehicle.\n\n**3901** _ **Silver Range**_ **** Monogram, 1946. 53 min. D: Lambert Hillyer. SC: J. Benton Cheney. With Johnny Mack Brown, Raymond Hatton, Jan Bryant, I. Stanford Jolley, Terry Frost, Eddie Parker, Ted Adams, Frank LaRue, Cactus Mack, Lane Bradford, Bill Willmering, George Morrell, Dee Cooper. A cattle rancher helps a former peace office in finding a kidnapped man and those behind a silver smuggling operation. Nicely done, compact dual bill item.\n\n**3902** _ **Silver River**_ **** Warner Bros., 1948. 110 min. D: Raoul Walsh. SC: Stephen Long- street and Harriet Frank, Jr. With Errol Flynn, Ann Sheridan, Thomas Mitchell, Bruce Bennett, Tom D'Andrea, Barton MacLane, Monte Blue, Jonathan Hale, Alan Bridge, Arthur Space, Art Baker, Joseph Crehan, Harry Woods, Franklyn Farnum, Otto Reichow, Rose Ford, Bud Osborne, Henry (Harry) Morgan, Russell Hicks, Jerry Jerome, Frank McCarroll, Ian Wolfe, Fred Kelsey, Ben Corbett, Eddie Parker, Dan White, Norman Jolley, Harry Strang, James Harrison, Norman Willis, Bob Stephenson, Marjorie Bennett. An ex\u2013Civil War officer becomes a gambler and then a rancher who almost loses everyone he cares for due to greed. Okay Errol Flynn feature, but not up to his earlier Westerns.\n\n**3903** _ **Silver Spurs**_ **** Universal, 1936. 60 min. D: Ray Taylor. SC: Joseph Poland. With Buck Jones, Muriel Evans, J.P. McGowan, George Hayes, Dennis Moore, Beth Marion, Robert Frazer, Bruce Lane, Charles K. French, William Lawrence, Earl Askam, Kernan Cripps, Eddy Waller, Helen MacKeller. A cowboy comes to the defense of a rancher harassed by rustlers. Very good Buck Jones film, produced by the star.\n\n**3904** _**Silver Spurs**_ **** Republic, 1943. 65 min. D: Joseph Kane. SC: John K. Butler and J. Benton Cheney. With Roy Rogers, Smiley Burnette, John Carradine, Phyllis Brooks, Jerome Cowan, Joyce Compton, Dick Wessel, Hal Taliaferro, Forrest Taylor, Charles Wilson, Byron Foulger, Bob Nolan and The Sons of the Pioneers (Tim Spencer, Lloyd Perryman, Pat Brady, Hugh Farr, Karl Farr), Kermit Maynard, Tom London, Jack Kirk, Jack O'Shea, Slim Whitaker, Arthur Loft, Eddy Waller, Bud Osborne, Fred Burns, Henry Wills. Roy Rogers is accused of killing is ex-boss by the swindler responsible for the crime. Villain John Carradine adds class to this entertaining Roy Rogers entry that boasts an exciting shootout finale and good stunt work.\n\n**3905** _ **Silver Stallion**_ **** Monogram, 1941. 59 min. D: Edward Finney. SC: Robert Emmett (Tansey). With David Sharpe, LeRoy Mason, Chief Thundercloud, Walter Long, Janet Waldo, Thornton Edwards, Fred Hoose, Thunder (horse), Captain Boots (dog). Three partners are forced to become thieves as one of them tries to find the man who framed his brother. Modest action film marred by too much stock footage, although David Sharpe, LeRoy Mason and Chief Thundercloud are likable triad heroes.\n\n**3906** _ **The Silver Star**_ **** Lippert, 1955. 73 min. D: Richard Bartlett. SC: Richard Bartlett and Ian MacDonald. With Edgar Buchanan, Marie Windsor, Lon Chaney, Earle Lyons, Richard Bartlett, Barton MacLane, Morris Ankrum, Edith Evanson, Michael Whalen, Steve Rowland, Robert Kramer, Earl Hansen, Jill Richards, Charles Knapp, Tim Graham. A pacifistic sheriff does not want to face three gunmen hired to kill him with an old time lawman coming to his defense. Low grade production is highlighted by the work of veterans Edgar Buchanan, Marie Windsor and Lon Chaney, plus Jimmy Wakely singing the title song.\n\n**3907** _ **The Silver Trail**_ **** Reliable, 1937. 58 min. D: Raymond Samuels (Bernard B. Ray). SC: Bennett Cohen and Forrest Sheldon. With Rin Tin Tin, Jr., Rex Lease, Mary Russell, Ed Cassidy, Roger Williams, Steve Clark, Slim Whitaker, Oscar Gahan, Sherry Tansey, Tom London, Snub Pollard, Margaret Mann. A man and a dog join forces to stop crooks who are murdering miners in order to put together a silver combine. Cheap dual bill item allegedly based on the James Oliver Curwood story \"Mystery of the Seven Chests.\"\n\n**3908** _ **Silver Trails**_ **** Monogram, 1948. 53 min. D: Christy Cabanne. SC: J. Benton Cheney. With Jimmy Wakely, Dub Taylor, Christine Larson, George J. Lewis, Pierce Lyden, Whip Wilson, William Norton Bailey, Fred Edwards, Robert Strange, Bob Woodward, Bud Osborne. In California crooks try to steal land by causing a feud between settlers and ranchers. One of the better Jimmy Wakely films thanks to good direction and a flashy performance by Whip Wilson in a supporting role.\n\n**3909** _ **The Silver Whip**_ **** 20th Century\u2013Fox, 1953. 73 min. D: Harmon Jones. SC: Jesse Lasky, Jr. With Dale Robertson, Rory Calhoun, Robert Wagner, Kathleen Crowley, James Millican, Lola Albright, J.M. Kerrigan, John Kellogg, Harry Carter, Ian MacDonald, Robert Adler, Clancy Cooper, Burt Mustin, Dan White, Paul Wexler, Bobby Diamond, Jack Rice, Charles Watts. Wanting to be like a sheriff and stage guard he admires, a young man takes the job of stagecoach driver and runs into outlaws. A good cast adds some life to this otherwise average outing.\n\n**3910** _ **Silver Wolf**_ **** Blue Rider Pictures, 1999. 90 min. Color. D: Peter Svatek. SC: Michael Amo. With Michael Biehn, Roy Scheider, Shane Meier, Kimberely Warnat, Lynda Boyd, Trevor Roberts, Ron Sauve, T.J. Shanks, Reg Tupper, Shaun Johnston, Don MacKay, Jade Pawluk, John Hawkes. Going to live with his uncle in the wilderness following the death of his father during a ski strip, a teenager helps an injured wolf and forms a bond with the animal. So-so outdoor adventure with scenic Canadian Rockies locales.\n\n**3911** _ **Silverado**_ **** Columbia, 1985. 132 min. Color. D: Lawrence Kasdan. SC: Lawrence Kasdan and Mark Kasdan. With Kevin Kline, Scott Glenn, Kevin Costner, Danny Glover, John Cleese, Rosanna Arquette, Brian Dennehy, Linda Hunt, Jeff Goldblum, Marvin J. McIntyre, Brad Williams, Sheb Wooley, Jon Kasdan, Todd Allen, Kenny Call, Bill Thurman, Meg Kasdan, Dick Durock, Gene Hartline, Autry Ward, Jacob Kasdan, Rusty Meyers, Zeke Davidson, Lois Geary, James Gammon, Troy Ward, Roy McAdams, Ray Baker, Joe Seneca, Lynn Whitfield, Jeff Fahey, Patricia Gaul, Amanda Wyss, Earl Hindman, Tom Brown, Jim Haynie, Richard Jenkins, Jerry Biggs, Sam Gauny, Ken Farmer, Bill McIntosh, Charles Seybert, Jane Beauchamp, Jerry Block, Ben Zeller, Pepe Serna, Ted White, Ross Loney, Walter Scott, Bob Terhune, Brion James, Bob Morgan, Mark Kasdan, Richard Lester, Matthew Hotsinpiller. Two drifters join forces and arrive in the town of Silverado where they oppose a corrupt boss who controls the law. Overlong but pretty good look at the mythical West.\n\n**3912** _ **Sin Town**_ **** Universal, 1942. 74 min. D: Ray Enright. SC: W. Scott Darling and Gerald Geraghty. With Constance Bennett, Broderick Crawford, Anne Gwynne, Patric Knowles, Andy Devine, Leo Carrillo, Ward Bond, Arthur Aylesworth, Ralf Harolde, Charles Wagenheim, Billy Wayne, Hobart Bosworth, Jack Mulhall, Paul Bryar, Rebel Randall, Jean Trent, Oscar O'Shea, Eddy Waller, Clarence Muse, Ben Erdway, Ed Peil, Sr., Harry Strang, Guy Usher, Victor Zimmerman, George J. Lewis, Larry McGrath, Murray Parker, Frank Hagney, Neeley Edwards, Jack C. Smith, Kernan Cripps, Art Miles, Charles Marsh, Frank Coleman. A pair of confidence operators arrive in a town where a lynch mob is after the murderer of a newspaper editor. A good cast and a fast pace make this potboiler an adequate diversion.\n\n**3913** _ **Sing, Cowboy, Sing**_ **** Grand National, 1937. 60 min. D: Robert North Bradbury. SC: Robert Emmett (Tansey). With Tex Ritter, Louise Stanley, Al St. John, Karl Hackett, Charles King, Bob McKenzie, Budd Buster, Heber Snow (Hank Worden), Chick Hannon, Horace Murphy, Snub Pollard, Tex Palmer, Jack C. Smith, Oscar Gahan, Herman Hack, Milburn Morante, Rudy Sooter, Clyde McClary, Jack Evans, Sherry Tansey, Buck Morgan, Rube Dalroy, Tex Ritter's Tornadoes. Two cowpokes masquerade as entertainers to find out who killed a young woman's father, the proprietor of a shipping franchise. Good scenic locations and an action filled climax help this Tex Ritter vehicle which also benefits from a quartet of songs, including the very good title tune.\n\n**3914** _ **Sing Me a Song of Texas**_ **** Columbia, 1945. 66 min. D: Vernon Keays. SC: Elizabeth Beecher and J. Benton Cheney. With Rosemary Lane, Hal McIntyre and His Orchestra, The Hoosier Hot Shots (Charles \"Gabe\" Ward, Ken Trietsch, Paul \"Hezzie\" Trietsch, Gil Taylor), Tom Tyler, Guinn Williams, Slim Summerville, Carole Mathews, Noah Beery, Pinky Tomlin, Marie Austin, Foy Willing and The Riders of the Purple Sage, James T. \"Bud\" Nelson, Reed Howes, Kermit Maynard, Forrest Taylor, Davison Clark, Vernon Dent, Victor Travers, John Tyrrell, Joel Friedkin. A rich rancher invites his two nieces to visit and then poses as a cook in order to decide which one will inherit his fortune. Light hearted Western with a number of musical interludes.\n\n**3915** _ **The Singer Not the Song**_ **** Warner Bros.\/Rank, 1961. 129 min. Color. D: Roy Baker. SC: Nigel Balchin. With Dirk Bogarde, John Mills, Mylene Demongeot, Laurence Naismith, John Bentley, Leslie French, Eric Pohlmann, Nyall Florenz, Roger Delgado, Philip Gilbert, Shelia Gallagher, Selma Vaz Dias, Laurence Payne, Jacqueline Evans, Lee Montague, Serafina Di Leo. In a Mexican village the new priest vies for control of the people with a bandit while trying to fend off the lust of a local beauty. Overlong, dull melodrama.\n\n**3916** _ **Singin' in the Corn**_ **** Columbia, 1946. 65 min. D: Del Lord. SC: Elwood Ullman, Monte Brice, Isabel Dawn and Richard Weil. With Judy Canova, Allen Jenkins, Guinn Williams, Alan Bridge, Charles Halton, Robert Dudley, Nick Thompson, George Chesebro, Ethan Laidlaw, Frances Rey, Frank Lackteen, Guy Beach, Jay Silverheels, Rodd Redwing, Dick Stanley, Charles Reynolds, Si Jenks, Pat O'Malley, Chester Conklin, Mary Gordon, The Singing Indian Braves, Chief Yowlachie. A carnival mind reader inherits a ranch that is supposedly haunted as crooks try to steal it from local Indians, the rightful owners. More rustic comedy hokum from Judy Canova, sure to please her fans.\n\n**3917** _ **Singin' Spurs**_ **** Columbia, 1948. 62 min. D: Ray Nazarro. SC: Barry Shipman. With The Hoosier Hot Shots (Charles \"Gabe\" Ward, Ken Trietsch, Paul \"Hezzie\" Trietsch, Frank Kettering), Kirby Grant, Patricia (Barry) White, Lee Patrick, Jay Silverheels, Dick Elliott, Billy Wilkerson, Chester Clute, Marion Colby, Red Enger, Riley Hill, Patricia Knox, Billy Cypert, The Shamrock Cowboys. One of the Hoosier Hot Shots marries a rich widow to get money needed to stage a campaign to help Indians irrigate their lands. Silly cornpone Western musical comedy. Also called _**Singing Spurs**_.\n\n**3918** _ **The Singing Buckaroo**_ **** Spectrum, 1937. 60 min. D-SC: Tom Gibson. With Fred Scott, Cliff Nazarro, Victoria Vinton, William Faversham, Howard Hill, Roger Williams, Rosa Caprino, Carl Mathews, Dick Curtis, Augie Gomez, Shorty Miller, Wade Walker, Oscar Gahan, The Singing Buckaroos. A cowboy helps a young woman who has taken money for safekeeping since her father is the hostage of crooks after the currency. Entertaining Fred Scott musical opus.\n\n**3919** _ **The Singing Cowboy**_ **** Republic, 1936. 56 min. D: Mack V. Wright. SC: Dorrell McGowan and Stuart McGowan. With Gene Autry, Smiley Burnette, Lois Wilde, Lon Chaney, Jr., Ann Gillis, John Van Pelt, Earle Hodgins, Earl Eby, Ken Cooper, Harrison Greene, Wes Warner, Jack Rockwell, Tracy Layne, Fred \"Snowflake\" Toones, Oscar Gahan, Frankie Marvin, Jack Kirk, Audrey Davis, George Pearce, Charlie McAvoy, Alfred P. James, Pat Carson, Harvey Clark. A cowboy crooner becomes the guardian of a girl, who needs an operation, after her mine owner father is murdered and he tries to raise the money. Fine interpolation of music, story and action make this Gene Autry film a good one.\n\n**3920** _ **The Singing Cowgirl**_ **** Grand National, 1939. 59 min. D: Samuel Diege. SC: Arthur Hoerl. With Dorothy Page, Dave O'Brien, Vince Barnett, Ed Peil, Sr., Dix Davis, Stanley Price, Warner Richmond, Dorothy Short, Paul Barrett, Lloyd Ingraham, Ethan Allen, Eddie Gordon, Merrill McCormick, Leonard Trainor. A woman rancher takes in a boy who parents were killed by rustlers and she sets out to help round up the gang. This third, and last, musical Western starring Dorothy Page is average.\n\n**3921** _ **Singing Guns**_ **** Republic, 1950. 91 min. Color. D: R.G. Springsteen. SC: Dorrell McGowan and Stuart McGowan. With Vaughn Monroe, Ella Raines, Walter Brennan, Ward Bond, Jeff Corey, Barry Kelley, Harry Shannon, Tom Fadden, Ralph Dunn, Rex Lease, Mary Baer, Jimmie Dodd, Denver Pyle, John Doucette, Richard Emory, George Chandler, Billy Gray, Mary Eleanor (Elinor) Donahue, Douglas Hughes, Stanley Blystone, Wallace Scott, Roy Barcroft (voice). A notorious outlaw saves the life of the lawman tracking him and with a new identity becomes the sheriff of a small town. Bandleader-singer Vaughn Monroe makes a very convincing Western star in this entertaining \"A\" production in which he sings \"Mule Train.\"\n\n**3922** _ **The Singing Hill**_ **** Republic, 1941. 61 min. D: Lew Landers. SC: Olive Cooper. With Gene Autry, Smiley Burnette, Virginia Dale, Mary Lee, Spencer Charters, Gerald Oliver Smith, George Meeker, Wade Boteler, Harry Stubbs, Cactus Mack, Jack Kirk, Chuck Morrison, Monte Montague, Hal Price, Fred Burns, Herman Hack, Jack O'Shea, Frankie Marvin, Forrest Taylor, Dan White. When a young woman wants to sell the ranch she inherited her neighbors fear it will bring an end to the open range. Average Gene Autry opus. Also known as _**Singing Hills**_.\n\n_**Singing Hills**_ see _**The Singing Hill**_\n\n**3923** _ **Singing on the Trail**_ **** Columbia, 1946. 65 min. D: Ray Nazarro. SC: J. Benton Cheney. With Ken Curtis, Jeff Donnell, Guy Kibbee, Dusty Anderson, Guinn Williams, Four Chicks and a Chuck (Chuck Goldstein), Deuce Spriggins and His Band, The Plainsmen, The Hoosier Hot Shots (Charles \"Gabe\" Ward, Ken Trietsch, Paul \"Hezzie\" Trietsch, Gil Taylor), Ian Keith, Matt Willis, Sam Flint, Joe Haworth, Eddy Waller, Carolina Cotton, Jody Gilbert, Coulter Irwin. The Hoosier Hot Shots are cheated out of a ranch by a crook and two cowboys come to their rescue. Another featherweight Western musical comedy from Columbia's \"B\" unit.\n\n**3924** _ **The Singing Outlaw**_ **** Universal, 1938. 56 min. D: Joseph H. Lewis. SC: Harry O. Hoyt. With Bob Baker, Joan Barclay, Fuzzy Knight, Carl Stockdale, Harry Woods, LeRoy Mason, Ralph Lewis, Glenn Strange, Georgia O'Dell, Jack Rockwell, Ed Peil, Sr., Jack Kirk, Bob McKenzie, Budd Buster, Lafe McKee, Hank Worden, Art Mix, Chick Hannon, Herman Hack, Jack Montgomery, Curley Gibson, Francis Walker. In order to find out who killed a U.S. marshal, a cowboy takes on his identity and goes after a rustling gang. Despite a fine supporting cast, this Bob Baker musical outing is sub-standard.\n\n**3925** _ **The Singing Sheriff**_ **** Universal, 1944. 60 min. D: Leslie Goodwins. SC: Henry Blankfort and Eugene Conrad. With Bob Crosby, Fay McKenzie, Fuzzy Knight, Iris Adrian, Samuel S. Hinds, Edward Norris, Andrew Tombes, Joseph Sawyer, Walter Sande, Doodles Weaver, Jean Trent, Donald Kerr, Pat Starling, Louis Da Pron, Spade Cooley and Orchestra. The son of a prominent citizen arrives in town incognito and immediately gets involved with the sheriff and outlaws. Slim musical vehicle for bandleader Bob Crosby.\n\n_**Singing Spurs**_ see _**Singin' Spurs**_\n\n**3926** _ **The Singing Vagabond**_ **** Republic, 1935. 54 min. D: Carl L. Pierson. SC: Oliver Drake and Betty Burbridge. With Gene Autry, Smiley Burnette, Ann Rutherford, Barbara Pepper, Warner Richmond, Frank LaRue, Grace Goodall, Niles Welch, Tom Bower, Robinson Neeman, Henry Roquemore, Ray (Corrigan) Bernard, Allan Sears, Bob Burns, Charles King, Chief Big Tree, Chief Thundercloud, Marie Quillan, Elaine Shepard, Edmund Cobb, George (Montgomery) Letz, Celia McCanon. June Thompson, Janice Thompson, Marion O'Connell. A young woman, who has run away from home to join a traveling show, is rescued by a cavalry captain when her wagon train is attacked. Okay Gene Autry outing, with more music than action.\n\n**3927** _ **Single Handed Sanders**_ **** Monogram, 1932. 61 min. D: Lloyd Nosler. SC: Charles A. Post. With Tom Tyler, Margaret Morris, Loie Bridge, Robert Manning, G.D. Wood (Gordon DeMain), John Elliott, Hank Bell, Fred \"Snowflake\" Toones, Theodore Lorch, Lafe McKee, Glenn Strange, Frank Ellis, Helen Gibson, Al Haskell, Barney Beasley, F.R. Smith, Rose Plummer, S.S. Simon. A blacksmith, whose brother is a crook, tries to save his girl and their town from a gang led by a senator. Poorly made, creaky Tom Tyler feature.\n\n_**Single Shot Poker**_ see _**The Heart of Texas Ryan**_\n\n**3928** _ **Sinister Journey**_ **** United Artists, 1948. 59 min. D: George Archainbaud. SC: Doris Schroeder. With William Boyd, Andy Clyde, Rand Brooks, Elaine Riley, John Kellogg, Don Haggerty, Stanley Andrews, Harry Strang, Herbert Rawlinson, John Butler, Wayne Treadway, Snub Pollard, Will Orlean. The Bar 20 trio tries to help a man clear himself of murder charge and get him reunited with his girl friend. Mild \"Hopalong Cassidy\" film that plays better as a truncated segment of the Hoppy TV series.\n\n**3929** _ **Sioux City Sue**_ **** Republic, 1946. 68 min. D: Frank McDonald. SC: Olive Cooper. With Gene Autry, Lynne Roberts, Sterling Holloway, The Cass County Boys Jerry Scoggins, Bert Dodson, Fred Martin), Richard Lane, Ralph Sanford, Ken Lundy, Helen Wallace, Pierre Watkin, Edwin Wills, Minerva Urecal, Frank Marlowe, LeRoy Mason, Kenne Duncan, Harry V. Cheshire, George Carleton, Sam Flint, Tex Terry, Tristram Coffin, Frankie Marvin. In order to pay off debts and save his ranch, a singing cowboy is persuaded by a pretty talent scout to make a movie only to find out his voice is used in an animated feature for a donkey. Gene Autry's first feature after World War II service is a pleasant affair, although more for comedy and music than traditional genre values.\n\n**3930** _ **Siringo**_ **** Rysher Entertainment, 1994. 90 min. Color. D: Kevin G. Cremin. SC: Peter A. Kinloch. With Brad Johnson, Chad Lowe, Stephen Macht, Keith Szarabajka, Floyd \"Red Crow\" Westerman, William Sanderson, George Aguilar, Maggie Baird, Michael Horton, Apesanahkwat, Barry Corbin, Crystal Bernard, Jerry Wayne Bernard, Roger Rook, Logan Senn, Sonny Skyhawk, Thomas Sminkey, Brigitta Stenberg, Patricia Van Ingen. A lawman, his rookie deputy and a former hooker team to bring in an escaped convict who has committed armed robbery. Much gunplay but little interest here.\n\n**3931** _ **Sitting Bull**_ **** United Artists, 1954. 105 min. Color. D: Sidney Salkow. SC: Jack De Witt and Sidney Salkow. With Dale Robertson, Mary Murphy, J. Carrol Naish, Iron Eyes Cody, John Litel, William Hopper, Douglas Kennedy, Bill Tannen, Joel Fluellen, John Hamilton, Thomas Browne Henry, Felix Gonzalez, Al Wyatt. A pro\u2013Indian soldier is falsely accused of helping Chief Sitting Bull at the time of the Custer massacre. Silly, boring and over long, this pseudo-historical piece is notable only for J. Carrol Naish's fine performance in the title role.\n\n**3932** _ **Sitting Bull at the Spirit Lake Massacre**_ **** Sunset, 1927. 72 min. D: Robert North Bradbury. SC: Ben Allah. With Bryant Washburn, Ann Schaeffer, Jay Morley, Shirley Palmer, Thomas Lingham, Chief Yowlachie, James O'Neil, Bob (Steele) Bradbury, Jr., Fred Warren, Leon Kent, Lucille Ballart. A scout falls for a pretty girl but their romance is interrupted by an Indian uprising led by Chief Sitting Bull. Pretty good low budget affair, from producer Anthony J. Xydias, with a chance to see Bob Steele in his pre-series days. Also called _**With Sitting Bull at the Spirit Lake Massacre**_.\n\n**3933** _ **Six Black Horses**_ **** UniversalInternational, 1962. 80 min. Color. D: Harry Keller. SC: Burt Kennedy. With Audie Murphy, Dan Duryea, Joan O'Brien, George Wallace, Roy Barcroft, Bob Steele, Henry Wills, Phil Chambers, Richard Pasco, Charles Regis, Dale Van Sickel. A woman hires two men to lead her through Indian territory as she plans to kill one of them, a gunman who murdered her husband. Good drama highlighted by Dan Duryea's slick villain portrayal.\n\n**3934** _ **Six Feet Four**_ **** Path\u00e9, 1919. 70 min. D: Henry King. SC: Stephen Fox (Jules Furthman). With William Russell, Vola Vale, Harvey Clark, Al Ernest Garcia, Charles K. French, Jack Brammall, Jack Collins, John Gough, Clarence Burton, Calvert Carter, Perry Banks, Ann Schaffer, John Carter. A crooked lawman teams with a rancher in trying to blame two robberies on another cattleman whose spread they want. Action filled silent galloper starring William Russell at his peak.\n\n**3935** _ **Six Gun**_ **** Wood Entertainment, 2008. 91 min. Color. D: Scott Perry. SC: Luke Hill. With Tommy Hill, Bill Wise, Sue Rock, Robert Graham, Michael Hankin, Marlene Peralez, Maurice Ripke, Matthew Rimmer, Kerry Awn, Gordon Capps, Yvonne Lynn, DanI Marco, Dahell Hall, Kevin Flood, Joe King Carrasco, Sarah Agor, Tom Adkins, Mark Jeffrey Miller, Eric Perry, Ben Morrison, Douglas Taylor. An aging bounty hunter, needing money to save his ranch, goes after the killer of three cowboys. Tacky, low grade affair filmed in Texas.\n\n**3936** _ **Six Gun Decision**_ **** Allied Artists, 1953. 54 min. D: Frank McDonald. SC: Maurice Tombragel and William Raynor. With Guy Madison, Andy Devine, Lyle Talbot, Gloria Saunders, David Sharpe, Robert Bice, Fred Hoose, Zon Murray, Jim Connell, Peggy Stewart, Fred Kohler, Jr., Tom Steele, Michael Vallon, Don Hayden, Hank Patterson, Parke MacGregor. Wild Bill Hickok and his pal Jingles look into the pre-election killing of a newspaper editor and stop angry citizens from hanging a Pony Express rider. Passable theatrical compilation from \"The Adventures of Wild Bill Hickok\" (1951\u201358) TV episodes \"Border City Election\" and \"Pony Express vs. Telegraph.\"\n\n**3937** _ **Six Gun Gold**_ **** RKO Radio, 1941. 57 min. D: David Howard. SC: Norton S. Parker. With Tim Holt, Ray Whitley, Jan Clayton, Lee \"Lasses\" White, Lane Chandler, LeRoy Mason, Eddy Waller, Davison Clark, Harry Harvey, Jr., Slim Whitaker, Jim Corey, Fern Emmett, Lew Meehan, Ethan Laidlaw, David Sharpe, Ken Card. A young man, whose brother has been kidnapped by gold thieves, tries to find him and get to the bottom of the robberies. Interesting Tim Holt vehicle.\n\n**3938** _ **Six Gun Gospel**_ **** Monogram, 1943. 55 min. D: Lambert Hillyer. SC: Jess Bowers (Adele Buffington) and Ed Earl Repp. With Johnny Mack Brown, Raymond Hatton, Inna Gest, Kenneth MacDonald, Roy Barcroft, Edmund Cobb, Mary MacLaren, Isabel Withers, Eddie Dew, Bud Osborne, Milburn Morante, Artie Ortego, Lynton Brent, Kernan Cripps, Tom London, Lew Porter, Jack Evans, Chick Hannon, Lew Morphy, Jack Daley, Rube Dalroy, Jack Tornek. Two lawmen, one masquerading as a preacher, try to find out who is behind the hijacking of gold shipments. The same old plot is given a fairly good treatment in this \"Nevada Jack McKenzie\" segment.\n\n**3939** _ **Six Gun Justice**_ **** Spectrum, 1935. 57 min. D: Robert Hill. SC: Oliver Drake. With Bill Cody, Ethel Jackson, Wally Wales, Budd Buster, Donald Reed, Roger Williams, Frank Moran, Ace Cain, Bert Young, Buck Morgan, Jimmy Aubrey, Blackie Whiteford, Bud Pope. A U.S. marshal is injured trying to help a former outlaw who has double crossed his gang and plans to return the money they stole. Slim production values from producer Ray Kirkwood mar this otherwise interesting tale.\n\n**3940** _ **Six Gun Law**_ **** Columbia, 1948. 54 min. D: Ray Nazarro. SC: Barry Shipman. With Charles Starrett, Smiley Burnette, Nancy Saunders, Paul Campbell, Hugh Prosser, George Chesebro, Curley Clements and His Rodeo Rangers, Billy Dix, Robert Wilke, Bob (John) Cason, Ethan Laidlaw, Pierce Lyden, Bud Osborne, Budd Buster, Slim Gault. A rancher is falsely accused of murdering the local sheriff and forced to sign a confession by the gang leader who makes him the new sheriff, thinking he can control him, not realizing the accused is the Durango Kid. Fair series outing.\n\n**3941** _ **Six-Gun Law**_ **** Buena Vista, 1963. 78 min. Color. D: Christian Nyby. SC: Maurice Tombragel. With Robert Loggia, James Dunn, Lynn Bari, Annette (Funicello), Jay C. Flippen, Patric Knowles, Audrey Dalton, James Drury, Kenneth Tobey, R.G. Armstrong, Grant Withers, Edward Colmans. When a notorious rustler is murdered an English rancher is charged with the shooting. Entertaining affair issued in Europe theatrically; originally a segment of Walt Disney's show on ABC-TV, telecast February 6, 1959, as \"Attorney at Law\" in \"The Nine Lives of Elfego Baca\" mini-series.\n\n**3942** _ **Six Gun Man**_ **** Producers Releasing Corporation, 1946. 59 min. D-SC: Harry Fraser. With Bob Steele, Syd Saylor, Jean Carlin, Bud Osborne, Brooke Temple, I. Stanford Jolley, Budd Buster, Roy Brent, Jimmie Martin, Stanley Blystone, Steve Clark, Dorothy Whitmore, Ray Jones, Jimmy Aubrey, Buck Morgan. Rustlers are terrorizing the citizens of a small town and two U.S. marshals try to stop them. Ragged PRC effort, mainly for Bob Steele fans.\n\n**3943** _ **Six Gun Mesa**_ **** Monogram, 1950. 57 min. D: Wallace Fox. SC: Adele Buffington. With Johnny Mack Brown, Gail Davis, Riley Hill, Leonard Penn, Marshall Reed, Steve Clark, Milburn Morante, Carl Mathews, Bud Osborne, George DeNormand, Stanley Blystone, Holly Bane, Frank Jaquet, Artie Ortego, Merrill McCormick. A town boss tries to blame the foreman of a cattle herd for the murder of his wranglers but a lawman suspects the plot. Fair Johnny Mack Brown vehicle; shows the wear on the genre with the coming of the 1950s.\n\n**3944** _ **Six-Gun Rhythm**_ **** Grand National\/Arcadia, 1939. 55 min. D: Sam Newfield. SC: Fred Myton. With Tex Fletcher, Joan Barclay, Ralph Peters, Reed Howes, Bud McTaggart, Ted Adams, Walter Shumway, Slim Hacker, Carl Mathews, Art Davis, Robert Frazer, Sherry Tansey, Kit Guard, Art Mix, Jack O'Shea, Frank Ellis, George Morrell, Tex Phelps, David Sharpe, Jack C. Smith. A singing football player returns home to Texas to find his sheriff father missing and the area infested with outlaws. Tex Fletcher's only oater is a fairly pleasant affair except for an obtrusive canned music track although the star does warble several tunes, including \"Git Along Little Doggies\" and \"Lonesome Cowboy.\"\n\n**3945** _ **Six Gun Serenade**_ **** Monogram, 1947. 55 min. D: Ford Beebe. SC: Bennett Cohen. With Jimmy Wakely, Lee \"Lasses\" White, Jimmie Martin, Kay Morley, Steve Clark, Pierce Lyden, Bud Osborne, Chick Hannon, Cactus Mack, Ray Jones. Ranchers get a group of cowboys out of jail in order to stop a gang of cattle thieves. Good direction and script lift this Jimmy Wakely vehicle a bit above the usual for the singing star.\n\n**3946** _ **Six Gun Trail**_ **** Victory, 1938. 59 min. D: Sam Newfield. SC: Joseph O'Donnell. With Tim McCoy, Nora Lane, Alden Chase, Ben Corbett, Karl Hackett, Donald Gallaher, Ted Adams, Kenne Duncan, Sherry Tansey, Bob Terry, Jimmy Aubrey, George Morrell, Frank Wayne, Hal Carey, Jack \"Tiny\" Lipson, Lew Porter, Ray Henderson, Herman Hack, James Sheridan (Sherry Tansey), Buck Morgan, Wally West, Oscar Gahan, Artie Ortego, Clyde McClary, Barney Beasley, Rube Dalroy. A Justice Department investigator, masquerading as a Chinese, heads to a small town to capture a gang trying to sell stolen gems. Surprisingly good Sam Katzman production, although more for plot, acting and direction than budget; the second entry in Victory's \"Lightning Bill Carson\" series.\n\n**3947** _ **6 Guns**_ **** Asylum Home Entertainment, 2010. 95 min. Color. D: Shane Van Dyke. SC: Geoff Meed. With Barry Van Dyke, Sage Mears, Greg Evigan, Brian Wimmer, Geoff Meed, Shane Van Dyke, Carey Van Dyek, Jason Ellefson, Jonathan Nation, Erin Marie Hogan, Peter Sherayko, Anya Benton, Valerie Garcia, Kenny A. Remmel, Riley Polanski, Becky Byrum, Don Harrington, Rick Roat, Tom Troutman, Cathi Harrington, Cody Williams. A gunman trains a young girl who wants to take revenge on the gang who murdered her family. So-so direct to video Western.\n\n**3948** _ **Six Reasons Why**_ **** ThinkFilm, 2008. 88 min. Color. D-SC: Jeff Campagna and Matthew Campagna. With Dan Wooster, Mads Koudal, Christopher Harrison, Colm Feore, Jeff Campagna, Matthew Campagna, Aaron Harrison, Romas Stanullis, Geoff Kolomahz, Anastasia Tubanos, Mike Celia, Rudy Jahchan, Casey McKinnon, Rosie (horse). A quartet of men, all with pasts, unite to bring law and order to the town of Badland. Ethereal, but confusing, futuristic oater.\n\n**3949** _ **Six Shootin' Sheriff**_ **** Grand National, 1938. 59 min. D: Harry Fraser. SC: Weston Edwards. With Ken Maynard, Marjorie Reynolds, Warner Richmond, Jane Keckley, Bob Terry, Harry Harvey, Sr., Walter Long, Earl Dwire, Ben Corbett, Lafe McKee, Tom London, Richard Alexander, Glenn Strange, Roger Williams, Bud Osborne, Ed Peil, Sr., Milburn Morante, Carl Mathews, Richard Cramer, George Morrell, Jim Corey, Herb Holcombe, Buck Morgan, Jack Evans, Fred Parker, Bud Pope. An outlaw is accidentally made the town's sheriff and when his old gang arrives on the scene he tries to warn them away but when they pull a robbery he gets on their trail. Poor production values hurt this Ken Maynard film; hardly one of his better efforts.\n\n**3950** _ **Skin Game**_ **** Warner Bros., 1971. 102 min. Color. D: Paul Bogart. SC: Pierre Marton. With James Garner, Lou Gossett, Susan Clark, Brenda Sykes, Edward Asner, Andrew Duggan, Henry Jones, Neva Patterson, Parley Baer, George Tyne, Royal Dano, Pat O'Malley, Joel Fluellen, Napoleon Whiting, Juanita Moore, Cort Clark, Jim Boles, George Wallace, Robert Foulk, Bill Henry, Tom Monroe, Don Haggerty, Claude Stroud, Forrest Lewis, James McCallion, Dan Borgaze, Reg Parton, Bob Steele. Before the Civil War, two con men, one black and the other white, travel through the South with the latter \"selling\" the former and the two splitting the profits. Entertaining genre comedy remade for TV as _**Sidekicks**_ (q.v.).\n\n**James Garner in** _**Skin Game**_ **(Warner Bros., 1971).**\n\n**  \n**\n\n**3951** _ **Skipalong Rosenbloom**_ **** United Artists, 1951. 72 min. D: Sam Newfield. SC: Dean Riesner and Eddie Forman. With Maxie Rosenbloom, Max Baer, Jackie Coogan, Hillary Brooke, Fuzzy Knight, Jacqueline Fontaine, Raymond Hatton, Ray Walker, Sam Lee, Al Shaw, Joseph Greene, Dewey Robinson, Whitey Haupt, Carl Mathews, Artie Ortego. An Eastern gunslinger plans to put an end to lawlessness caused by a bad man. Broad genre take-off, starring the two boxing greats, with lots of slapstick; reissued as _**The Square Shooter**_.\n\n**3952** _ **Skull and Crown**_ **** Reliable, 1935. 60 min. D: Elmer Clifton. SC: Bennett Cohen and Carl Krusada. With Rin Tin Tin, Jr., Regis Toomey, Jack Mulhall, Molly O'Day, James Murray, Lois January, Jack Mower, Tom London, George Chesebro, Robert Walker, John Elliott, Milburn Morante, Jack Evans, Jimmy Aubrey, George Hazel. A Customs Patrol officer and his dog are after smugglers with the lawman finding out one of them murdered his sister. A somewhat stark but nevertheless interesting poverty row affair with a fine cast.\n\n**3953** _ **Sky Bandits**_ **** Monogram, 1940. 62 min. D: Ralph Staub. SC: Edward Halperin. With James Newill, Louise Stanley, Dave O'Brien, William Pawley, Ted Adams, Bob Terry, Dwight Frye, Joseph Stefani, Dewey Robinson, Jack Clifford, Kenne Duncan, James Farley, Karl Hackett, Eddie Fetherston, Don Brodie, Harry Harvey, Snub Pollard, Marin Sais, Earl Douglas. Looking into the disappearance of a plane carrying gold from a Yukon mine, two Canadian Mounties uncover crooks using a mysterious ray. The use of sci-fi makes this \"Renfrew of the Royal Mounted\" entry (the last in the series) interesting although its plot is similar to _**Yukon Flight**_ (q.v.).\n\n**3954** _ **Sky Full of Moon**_ **** Metro-Goldwyn-Mayer, 1952. 73 min. D-SC: Norman Foster. With Carleton Carpenter, Jan Sterling, Keenan Wynn, Robert Burton, Elaine Stewart, Emmett Lynn, Douglass Dumbrille, Jonathan Cott, Rex Bell, Fred Kohler, Jr., Syd Saylor, Sara Taft, G. Pat Collins, Cliff Clark, Chubby Johnson, Jean Ransome, Mitchell Lewis, Leon Alton, Joe Dominguez, Paul Kruger, Sheb Wooley (voice). A young rodeo cowboy falls for a jaded Las Vegas gambling house hostess. Okay modern-day comedy with only the second half taking place in the Nevada countryside; Rex Bell appears as himself and Sheb Wooley provides the signing voice of the Balladeer.\n\n**3955** _ **A Sky Full of Stars for a Roof**_ **** Documento Film, 1968. 100 min. Color. D: Giulio Petroni. SC: Alberto Areal and Franceso Martino. With Guiliano Gemma, Mario Adorf, Madga Konopka, Rick Boyd (Federico Boido), Cris Huerta, Julia Menard, Anthony M. Dawson, Sandro Dori, Franco Baludcci, Peter Branco, Franco Lantieri, Ivan Scarauglia, Angiolino Rizzieri, John Bartha, Mimmo Poli, Victor Israel, Benito Stefanelli, Alberto Dell'Acqua (Robert Widmark), Luciano Bonanni, Piero Magalotti, Maria Gustafson, Alfonso de la Vega. After a stagecoach massacre, a gunman joins forces with a gullible bungler as they try to avoid a vicious man out to kill the gunfighter for his father. A top notch Ennio Morricone score and quite a bit of comedy highlight his Italian production originally called _**...E per Tetto un Cielo di Stelle**_ (And for a Roof a Heaven of Stars).\n\n**3956** _ **Sky High**_ **** Fox, 1922. 72 min. D-SC: Lynn Reynolds. With Tom Mix, Eva Novak, J. Farrell MacDonald, Sid Jordan, William Buckley, Adele Warner, Wynn Mace, Pat Chrisman. An immigration officer is assigned to find out who is smuggling Chinese across the Mexican border into the United States. Thrill packed, entertaining Tom Mix silent feature.\n\n**3957** _ **The Sky Pilot**_ **** Associated First National, 1921. 68 min. D: King Vidor. SC: John McDermott. With John Bowers, Colleen Moore, David Butler, Harry Todd, James Corrigan, Donald MacDonald, Kathleen Kirkham. In the Canadian northwest a minister gets a rough reception from the local cowboys but he later saves the life of a ranch owner's daughter. This silent melodrama holds up rather well; some prints run 45 minutes.\n\n**3958** _ **Slaughter Trail**_ **** RKO Radio, 1951. 78 min. Color. D: Irving Allen. SC: Sid Kuller. With Brian Donlevy, Gig Young, Virginia Grey, Andy Devine, Robert Hutton, Terry Gilkyson, Lew Bedell, Myron Healey, Emmett Lynn, Ken Koutnik, Eddie Parks, Ralph Peters, Ric Roman, Lois Hall, Rosemary Clooney, Boyd \"Red\" Morgan, Robin Fletcher, Jody Gilbert, Ralph Volkie, Fenton Jones, Kenneth Otto, Toni Whaethel. Escaping after a robbery, and assisted by a woman, an outlaw gang murders three Indians and a fort commander. Passable action melodrama with \"A\" trappings.\n\n**3959** _ **Slay Ride**_ **** CBS-TV\/20th Century\u2013Fox, 1972. 100 min. Color. D: Robert Day. SC: Anthony Wilson and Rick Husky. With Glenn Ford, Edgar Buchanan, Victor Campos, Peter Ford, Leslie Parrish, Gerald S. O'Loughlin, Tony Bill, John Schuck, Anne Seymour, Sam Chew, Harry Lauter, Bernie Casey, Hunter Von Leer, Mark Jenkins, Jill Banner, Dehl Berti. The sheriff of a southwestern community tries to solve a murder case but find the situation is complicated by an Apache, a chronic confessor. Well done feature originally telecast as two episodes of \"Cade's County\" (CBS-TV, 1971\u201372).\n\n**3960** _ **Slim Carter**_ **** Universal-International, 1957. 82 min. Color. D: Richard Bartlett. SC: Montgomery Pittman. With Jock Mahoney, Julie Adams, Tim Hovey, William Hopper, Ben Johnson, Joanna Moore, Walter Reed, Bill Williams, Barbara Hale, Maggie Mahoney, Roxanne Arlen, Jean Moorehead, Donald Kerr, Jim Healey. A young orphan, who has won a contest, spends a month with his favorite cowboy star, with the actor changing from an egotist who eventually wants to adopt the boy. Surprisingly good Hollywood satire with a fine performance by Jock Mahoney as the flawed Western hero.\n\n**3961** _ **A Small Town in Texas**_ **** American International, 1976. 96 min. Color. D: Jack Starrett. SC: William Norton. With Timothy Bottoms, Susan George, Bo Hopkins, Morgan Woodward, John Karlen, Art Hindle, Hank Rolike, George \"Buck\" Flower, Clay Tanner, Mark Silva, Santos Reyes, Debi Bieberly, Claude Ennis (Jack) Starrett, Jr., James N. Harrell, Ron McPherson, Leotis Duffie, Kathryn Lacy, Starrett Berry, L.B. Stele, Joe Michel, Fay Armstrong, James Brewer, Amy Andrewartha, Randee Lynne Jensen. Following a five year hitch in prison, a man returns home to find his wife involved with the lawman who framed him. Standard modern-day Western with lots of auto and bike action.\n\n**3962** _ **Smith!**_ **** Buena Vista, 1969. 102 min. D: Michael O'Herlihy. SC: Louis Pelletier. With Glenn Ford, Nancy Olson, Dean Jagger, Keenan Wynn, Warren Oates, Chief Dan George, Frank Ramirez, Jay Silverheels, James Westerfield, Christopher Shea, Roger Ewing, Ricky Cordell, Gregg Palmer, William Bryant, Fred Aldrich, Eric Clavering. A rancher tries to help an Indian boy falsely accused of murder. Easygoing Disney feature filmed in Oregon and Washington.\n\n**3963** _ **Smoke in the Wind**_ **** Adelphi Film Distributors, 1976. 94 min. Color. D: Joseph Kane. SC: Eric Allen. With Walter Brennan, John Ashley, John Russell, Myron Healey, Susan Houston, Linda Weld, Henry Kingi, Adair Jameson, Dan White, Lorna Thayer, Billy Hughes, Jr., Bill Foster, Jack Horton, Bill McKenzie. Following the Civil War, Confederate veterans return to their Arkansas mountain homeland only to be denounced as traitors to the cause. Made in 1971, this pedestrian feature's main interest is Joseph Kane's direction (it was his final film) and a fine cast.\n\n**3964** _ **Smoke Jumpers**_ **** CBS-TV\/20th Century\u2013Fox, 1956. 45 min. D: Albert S. Rogell. SC: Art Cohn. With Dan Duryea, Joan Leslie, Dean Jagger, Richard Jaeckel, Robert Armstrong, Brett Halsey, Lawrence Dobkin, Robert Bray, Bobs Watson, Don Kennedy, John Conte (host). When the leader of a fire fighting unit is the only survivor of a forest blaze, he is accused of sacrificing his men in order to save himself. Well acted drama, a small screen remake of _**Red Skies of Montana**_ (q.v.), shown as an episode of \"The 20th Century\u2013Fox Hour\" (CBS-TV, 1955\u201357) on November 14, 1956.\n\n**3965** _ **Smoke Lightning**_ **** Fox, 1933. 63 min D: David Howard. SC: Gordon Rigby and Sidney Mitchell. With George O'Brien, Nell O'Day, Betsy King Ross, Frank Atkinson, Virginia Sale, Douglass Dumbrille, Morgan Wallace, Clarence Wilson, George Burton, Fred Wilson. A small orphan girl is about to be cheated by her crooked uncle and his lawman cohort but a cowboy protects her. Entertaining adaptation of Zane Grey's novel _Canyon Walls_.\n\n**3966** _ **Smoke Signal**_ **** Universal-International, 1955. 88 min. Color. D: Jerry Hopper. SC: George F. Slavin and George W. George. With Dana Andrews, Piper Laurie, Rex Reason, William Talman, Douglas Spencer, Gordon Jones, William Schallert, Bill Phipps, Robert Wilke, Pat Hogan, John Day. After Indians attack and destroy a frontier post, the survivors head for safety aboard flatboats on the Colorado River. Action filled and enjoyable drama.\n\n**3967** _ **Smoke Tree Range**_ **** Universal, 1937. 60 min. D: Lesley Selander. SC: Arthur Henry Gordon. With Buck Jones, Muriel Evans, Edmund Cobb, John Elliott, Robert Kortman, Donald Kirke, Ted Adams, Ben Hall, Dickie Jones, Lee Phelps, Charles King, Earle Hodgins, Mabel Concord, Eddie Phillips, Bob McKenzie, Slim Whitaker. A cowboy helps an orphaned girl whose cattle are being rustled by outlaws. Fine Buck Jones vehicle, produced by the star.\n\n**3968** _ **Smokey Smith**_ **** Supreme, 1935. 58 min. D-SC: Robert North Bradbury. With Bob Steele, Mary Kornman, George Hayes, Warner Richmond, Earl Dwire, Horace B. Carpenter, Tex Phelps, Archie Ricks, Herman Hack Vane Calvert, Bert Dillard, Tex Palmer. A cowboy is determined to find the outlaws who murdered his parents. Another revenge angle oater from Bob Steele and his director-writer father, Robert North Bradbury, and a good one.\n\n**3969** _ **Smoking Guns**_ **** Universal, 1934. 65 min. D: Alan James. SC: Ken Maynard. With Ken Maynard, Gloria Shea, Walter Miller, Jack Rockwell, William Gould, Harold Goodwin, Robert Kortman, Ed Coxen, Edgar \"Blue\" Washington, Etta McDaniel, Slim Whitaker, Bob Reeves, Jim Corey, Wally Wales, Edmund Cobb, Fred McKaye, Martin Turner, Hank Bell, Horace B. Carpenter, Roy Bucko, Buck Bucko, Ben Corbett, Blackjack Ward, Bud McClure, Cliff Lyons. Falsely accused of a crime, a cowboy heads to the jungles of South America and is followed by a Texas Ranger with the two becoming friends and when the lawman is attacked by crocodiles and later dies, the cowpoke takes his identity and returns home to clear himself. Ken Maynard's final Universal film, which he also wrote, is fun if taken tongue-in-cheek.\n\n_**Smoking Guns**_ (1942) see _**Billy the Kid's Smoking Guns**_\n\n**3970** _ **Smoking Trails**_ **** Madoc Sales, 1924. 63 min. D: William Bertram. With Bill Patton, Alma Rayford, William Bertram, Tom Ross, Jack House, Adrian Rayford, Maine (Bud) Geary, Horace B. Carpenter. A Texas Ranger works at a ranch where the owner's cattle are rustled by a gang working for a banker out to get the man's spread. Drawn out, cheap production spotlighting vapid hero Bill Patton.\n\n**3971** _ **Smoky**_ **** Fox, 1933. 69 min. D: Eugene Forde. SC: Stuart Anthony and Paul Perez. With Victor Jory, Irene Manning, LeRoy Mason, Hank Mann, Frank Campeau, Leonard Snegoff, Will James. A cowboy befriends and tames a beautiful stallion who has been made hostile by bad men. Okay program feature, the first of a trio of films based on the Will James book with the author appearing in this version.\n\n**3972** _ **Smoky**_ **** 20th Century\u2013Fox, 1946. 87 min. Color. D: Louis King. SC: Lillie Hayward, Dwight Cummings and Dorothy Yost. With Fred MacMurray, Anne Baxter, Burl Ives, Bruce Cabot, Esther Dale, Roy Roberts, J. Farrell MacDonald, Max Wagner, Guy Beach, Howard Negley, Bud Geary, Harry Carter, Bob Adler, Victor Kilian, Herbert Heywood, Douglas Spencer, Stanley Andrews. A man befriends, tames and trains a wild stallion who has a hatred of humans. Satisfying second screen version of Will James' book.\n\n**3973** _ **Smoky**_ **** 20th Century\u2013Fox, 1966. 103 min. Color. D: George Sherman. SC: Howard Medford. With Fess Parker, Diana Hyland, Katy Jurado, Hoyt Axton, Robert Wilke, Armando Silvestre, Jose Hector, Ted White, Chuck Roberson, Bob Terhune. A wrangler captures and trains a black stallion but his brother beats the animal who tramples him and escapes. Fair third production of the Will James work.\n\n**3974** _ **Smoky Canyon**_ **** Columbia, 1952. 55 min. D: Fred F. Sears. SC: Barry Shipman. With Charles Starrett, Smiley Burnette, Jack (Jock) Mahoney, Dani Sue Nolan, Tristram Coffin, Larry Hudson, Chris Alcaide, Sandy Sanders, Forrest Taylor, Charles Stevens, Leroy Johnson, Boyd \"Red\" Morgan, Frank O'Connor, Blackie Whiteford, Chick Hannon, Dick Botiller, Gerald Mohr (narrator). Sheep men are blamed for the slaughter of cattle with a government agent investigating and learning that crooked ranchers are behind the trouble by trying to deplete their herds in order to raise prices. Fast moving, compact \"Durango Kid\" episode.\n\n**3975** _ **Smoky Mountain Melody**_ **** Columbia, 1948. 61 min. D: Ray Nazarro. SC: Barry Shipman. With Roy Acuff, Guinn Williams, Russell Arms, Sybill Merritt, Jason Robards, Harry V. Cheshire, Fred F. Sears, Trevor Bardette, Carolina Cotton, Tommy Ivo, Jack (Jock) Mahoney, John Elliott, Sam Flint, Ralph Littlefield, Eddie Acuff, Heinie Conklin, Olin Howlin, The Smoky Mountain Boys. A singer gets a three month trial at running a ranch but the late owner's son tries to sabotage his chances. Pleasant country music Western starring the great Roy Acuff and His Smoky Mountain Boys, including Bashful Brother Oswald (Pete Kirby).\n\n**3976** _ **Smoky River Serenade**_ **** Columbia, 1947. 67 min. D: Derwin Abrahams. SC: Barry Shipman. With Ruth Terry, Paul Campbell, Guinn Williams, Virginia Hunter, The Hoosier Hot Shots (Charles \"Gabe\" Ward, Ken Trietsch, Paul \"Hezzie\" Trietsch, Gil Taylor), Carolina Cotton, Cottonseed Clark, Paul E. Burns, Russell Hicks, Emmett Vogan, Michael Towne, Sandy Sanders, Lulu Mae Bohrman, The Boyd Triplets, The Sunshine Boys (Freddie Daniel, M.H. \"Ace\" Richman, J.D. Summer, Eddie Wallace), Texas Rose Bascom, Billy Williams. Several entertainers come to the aid of a good hearted ranch owner whose land is sought by a conniving businessman. Ten songs are the highlight of this hayseed musical comedy Western.\n\n**3977** _ **Smoky Trails**_ **** Metropolitan, 1939. 55 min. D: Bernard B. Ray. SC: George Plympton. With Bob Steele, Jean Carmen, Murdock MacQuarrie, Jimmy Aubrey, Frank LaRue, Ted Adams, George Chesebro, Carleton Young, Steve Clark, Montie Montana, Frank Wayne, Bob Terry, Bruce Dane, Rube Dalroy. A cowpoke tries to capture an outlaw gang before they commit murder. Barely passable Bob Steele vehicle from producer Harry S. Webb.\n\n**3978** _ **Snake River Desperadoes**_ **** Columbia, 1951. 55 min. D: Fred F. Sears. SC: Barry Shipman. With Charles Starrett, Smiley Burnette, Monte Blue, Don Kay \"Brown Jug\" Reynolds, Tommy Ivo, George Chesebro, Boyd \"Red\" Morgan, John Pickard, Charles Horvath, Sam Flint, Duke York, Herman Hack, Al Wyatt. The Durango Kid tries to keep peace when a series of Indian raids, actually perpetrated by white men dressed as braves, take place. Well done action film in the \"Durango Kid\" series.\n\n**3979** _ **Snarl of Hate**_ **** Bischoff, 1927. 60 min. D: Noel Mason Smith. SC: Ben Bellah. With Johnnie Walker, Mildred June, Jack Richardson, Wheeler Oakman, Silverstreak (dog). When his prospector brother is murdered a man sets out to track down the killer. Johnnie Walker plays both the hero and victim in this average silent drama.\n\n**3980** _ **Snow Dog**_ **** Monogram, 1950. 63 min. D: Frank McDonald. SC: William Raynor. With Kirby Grant, Elena Verdugo, Rick Vallin, Milburn Stone, Richard Karlan, Jane Adrian, Hal Gerard, Richard Avonde, Duke York, Guy Zanette, Chinook (dog). While trailing the killer of a fellow officer, a Northwest Mounted Policeman uncovers an outlaw gang using wolves to murder their victims. Average outing in Kirby Grant's north woods series, supposedly based on the works of James Oliver Curwood.\n\n**3981** _ **Snowbeast**_ **** NBC-TV, 1977. 96 min. Color. D: HerbWallerstein. SC: Joseph Stefano. With Bo Svenson, Yvette Mimieux, Clint Walker, Robert Logan, Sylvia Sidney, Michael J. London, Thomas Babson, Kathy Christopher, Ann McEncroe, Richard Jamison, Prentiss Rowe. At a Western ski resort during a winter carnival, the owners try to suppress evidence that a murderous monster is on the rampage. Pretty fair TV horror thriller.\n\n**3982** _ **Snowfire**_ **** Allied Artists, 1958. 73 min. Color. D-SC: Dorrell McGowan and Stuart McGowan. With Don Megowan, Molly McGowan, Claire Kelly, John Cason, Michael Vallon, Melody McGowan, Rusty Wescoatt, Bill Hale, Paul Keast. After her father captures a wild white stallion, a young girl sets him free and earns his friendship. Nice low budget family film.\n\n_**Snowman**_ see _**Land of No Return**_\n\n**3983** _ **So This Is Arizona**_ **** Big 4, 1931. 55 min. D: J.P. McGowan. SC: Joe Lawliss and David Kirkland. With Wally Wales, Buzz Barton, Fred Church, Lorraine La Val, Tete Brady, Don Wilson, Gus Anderson, Joe Lawliss, Jack Russell. An Arizona Ranger reluctantly has to bring in his girl's outlaw brother, causing her to reject him. Rawboned, ragtag galloper from producer John R. Freuler.\n\n**3984** _ **Sodbusters**_ **** Atlantis Films\/Showtime, 1994. 98 min. Color. D: Eugene Levy. SC: Eugene Levy and John Hemphill. With Kris Kristofferson, John Vernon, Fred Willard, Wendel Meldrum, Max Gail, Steve Landesberg, John Hemphill, Don Lake, Lou Wagner, George Buza, Cody Jones, Lela Ively, Maria Vacratsis, Earl Pastko, James Pickens, Jr., Henry Ramer, Jack Duffy, John Friesen, Ronnie Hawkins, Chandler Nicol, Natalie Radford, Dean McDermott, Gerry Quigley, Wayne Robson, Jonathan Scarfe, Robert Mayor. In 1875 Colorado homesteaders find themselves at odds with a corrupt cattle baron and his railroad allies. Okay TV Western comedy.\n\n**3985** _ **Soft Boiled**_ **** Fox, 1923. 78 min. D-SC: John G. Blystone. With Tom Mix, Billie Dove, Joseph Girard, L.C. Shumway, Tom Wilson, Frank Beal, Jack Curtis, Charles Hill Mailes, Harry Dunkinson, Wilson Hummell. A cowboy tries to control his temper with his uncle betting him he cannot do so for a month and during that time he has to endure insults to his sweetheart without becoming angry. Humorous Tom Mix outing that will please his fans.\n\n**3986** _ **Sol en Llamas**_ (Flaming Sun). Rosas Films, S.A., 1962. 95 min. D: Alfredo B. Crevenna. SC: Edmundo Baez and Alfredo B. Crevenna. With Antonio Aguilar, Maricruz Olivier, Irma Dorantes, Domingo Soler, Beatriz Aguirre, Fernando Soler, Hector Godoy, Jose Chavez, Antonio Raxel, Lidia Franco, Manuel Arvide, Eduardo Moreno, Gloria Leticia Ortiz, Raul Mena, Jose Dupeyron. The son of a wealthy man takes the side of poor workers being abused by local tyrants. This Mexican Robin Hood imitation is fairly entertaining.\n\n**3987** _ **Soldier Blue**_ **** Avco-Embassy, 1970. 114 min. Color. D: Ralph Nelson. SC: John Gay. With Candice Bergen, Peter Strauss, Donald Pleasence, Bob Carraway, Mort Mills, Jorge Rivero, Dana Elcar, John Anderson, Martin West, Jorge Russek, Marco Antonio Arzate, Ron Fletcher, Barbara Turner, Aurora Clavell. Indians attack a paymaster's detachment and leave only two survivors, a private and a woman planning to marry a lieutenant for his money. Pro-Indian feature is excessively violent with little to offer.\n\n**Candice Bergen and Peter Strauss in** _**Soldier Blue**_ **(Avco Embassy, 1970).**\n\n**  \n**\n\n**3988** _ **The Soldiers of Pancho Villa**_ **** Unifilms-Cimex, 1958. 90 min. Color. D: Ismael Rodriguez. SC: Ricardo Garibay and Jose Luis de Celis. With Dolores Del Rio, Maria Felix, Emilio Fernandez, Pedro Armendariz, Antonio Aguilar, Flor Silvestre, Tito Novaro, David Reynoso, Ignacio Lopez Taro, Cuco Sanchez, Irma Torres, Miguel Manzano, Lupe Carriles, Manuel Trejo Morales, Jose Carlos Mendez, Armando Gutierrez, Jose Munoz, Antonio Haro Oliva, Humberto Almazan. During the Mexican Revolution two women of different social classes love a peasant general follower of Pancho Villa. Fine Mexican feature with several well staged battle sequences, but hurt by mediocre dubbing; originally released as _**La Cucaracha**_.\n\n**3989** _ **The Sombrero Kid**_ **** Republic, 1942. 56 min. D: George Sherman. SC: Norman S. Hall. With Don \"Red\" Barry, Lynn Merrick, Robert Homans, John James, Joel Friedkin, Rand Brooks, Stuart Hamblen, Bob McKenzie, Lloyd \"Slim\" Andrews, Anne O'Neal, Kenne Duncan, I. Stanford Jolley, Bud Geary, Frank Brownlee, Bill Nestell, Hank Bell, Curley Dresden, Jack O'Shea, Pascale Perry, Griff Barnett, Chick Hannon, Merrill McCormick, Ed Cassidy, Jack Evans, Tommy Coats, Roy Bucko, Buck Bucko, Rose Plummer, Bill Wolfe. A cowboy is forced to become an outlaw by the man who murdered the peace officer he thought was his father. Rather complicated and grim Don Barry vehicle, but full of action nonetheless. Bob McKenzie just about steals the show as the jovial Judge Tater.\n\n**3990** _ **Something Big**_ **** National General, 1971. 108 min. Color. D: Andrew V. McLaglen. SC: James Lee Barrett. With Dean Martin, Brian Keith, Honor Blackman, Carol White, Ben Johnson, Albert Salmi, Don Knight, Joyce Van Patten, Denver Pyle, Merlin Olsen, Robert Donner, Harry Carey, Jr., Judi Meredith, Edward Faulkner, Paul Fix, David Huddleston, Bob Steele, Chuck Hicks, John Kelly. During the Mexican War outlaws battle each other for the possession of a Gatling Gun. There is nothing to brag about in this big budget affair.\n\n**Dean Martin in** _**Something Big**_ **(National General, 1971).**\n\n**  \n**\n\n**3991** _ **Something for a Lonely Man**_ **** NBC-TV\/Universal, 1968. 98 min. Color. D: Don Taylor. SC: John Fante and Frank Fenton. With Dan Blocker, Susan Clark, John Dehner, Warren Oates, Paul Petersen, Don Stroud, Henry Jones, Stanley Kenyon, Edgar Buchanan, Tom Nolan, Dub Taylor, Grady Sutton, Joan Shawlee, Iron Eyes Cody, Ralph Neff, Conlan Carter. A blacksmith tries to redeem himself in the eyes of the people he brought West only to have their town bypassed by the railroad. Dan Blocker fans will like this light weight feature made for TV.\n\n_**Something Is Out There**_ see _**Day of the Animals**_\n\n**3992** _ **Something New**_ **** Nell Shipman Productions, 1920. 57 min. D-SC: Nell Shipman and Bert Van Tuyle. With Nell Shipman, Bert Van Tuyle, L.M. Wells, William (Merrill) McCormick, Laddie (dog). Unable to locate a horse, a man uses a car to travel across the desert to rescue his writer girlfriend from Mexican bandits. Fun action fest with harrowing driving sequences; filmed in the Mojave Desert and financed by the Maxwell Motor Company to show off its 1920 model.\n\n**3993** _ **Sometimes a Great Notion**_ **** Universal, 1971. 113 min. Color. D: Paul Newman. SC: John Gay. With Paul Newman, Henry Fonda, Lee Remick, Michael Sarrazin, Richard Jaeckel, Linda Lawson, Cliff Potts, Sam Gilman, Lee De Broux, Jim Burk, Roy Jenson, Joe Maross, Roy Poole, Charles Tyner, Hal Needham, Dean Smith. An old time logging baron refuses to take part in a strike against a big lumber company and ends up in a great deal of trouble as a result. Poor modern-day action feature, although its cast tries hard. TV title: _**Never Give an Inch**_.\n\n**3994** _ **Somewhere in Sonora**_ **** Warner Bros., 1933. 57 min. D: Mack V. Wright. SC: Joe Roach. With John Wayne, Henry B. Walthall, Shirley Palmer, J.P. McGowan, Ann Fay, Frank Rice, Billy Franey, Paul Fix, Ralph Lewis, Slim Whitaker, Jim Corey, Blackie Whiteford, Bob Fleming, Frank Ellis, Pat Harmon, Joe Dominguez, Glenn Strange, Bud Osborne, G. Raymond Nye, William McCall, Jack Hendricks, Jack Evans, Jim Corey, Art Dillard, Richard Botiller, Barney Beasley, Charles Le Moyne. Falsely accused of wrongdoing during a stagecoach race, a cowboy goes to Mexico where he uncovers a plot to rob a mine belonging to his girl's father. John Wayne fans will enjoy this remake of the 1927 Ken Maynard (who is seen in stock shots), First National film of the same title.\n\n**3995** _ **Son of a Bad Man**_ **** Screen Guild, 1949. 64 min. D: Ray Taylor. SC: Ron Ormond and Ira Webb. With Lash LaRue, Al St. John, Noel Neill, Michael Whalen, Zon Murray, Frank Lackteen, Francis McDonald, Jack Ingram, Steve Raines, Chuck (Bob\/John) Cason, Don C. Harvey, Edna Holland, William Norton Bailey, Sandy Sanders, Doye O'Dell. U.S. marshals Lash LaRue and Fuzzy Q. Jones head for a town whose citizens have been beset by a gang led by the mysterious El Sombre. Lash LaRue's last Screen Guild series film is fast on action and sure to please his followers.\n\n**3996** _ **The Son of a Gun**_ **** William L. Sherry Service, 1919. 68 min. D: G.M. Anderson. SC: G.M. Anderson and Jesse J. Robbins. With G.M. (Broncho Billy) Anderson, Joy Lewis, Fred Church, Frank Whitson, A.E. Whiting, Mrs. A.E. Whiting, Paul Willis, Harry Todd. The town's lovable no good becomes a hero when he stops a gang of swindlers. Broncho Billy Anderson's final starring film is worth a look just to see the screen first cowboy's star; otherwise an average silent Western with an ingratiating star.\n\n**3997** _ **Son of a Gunfighter**_ **** Metro-Goldwyn-Mayer, 1966. 92 min. Color. D: Paul Landres. SC: Clarke Reynolds. With Russ Tamblyn, Kieron Moore, James Philbrook, Fernando Rey, Maria Granada, Aldo Sambrell, Antonio Casas, Ralph Browne, Julio Perez Tabernero, Barta Barri. A cowboy teams with a bounty hunter to get revenge on his father but eventually the boy and his old man join forces to fight an outlaw gang after a woman's ranch. Fair Spanish oater, released there as _**El Hijo del Pistolero**_ (The Son of a Gunfighter).\n\n**3998** _ **Son of Belle Starr**_ **** Allied Artists, 1953. 70 min. Color. D: Frank McDonald. SC: D.D. Beauchamp and William Raynor. With Keith Larsen, Dona Drake, Peggie Castle, Regis Toomey, James Seay, Myron Healey, Frank Puglia, Robert Keys, I. Stanford Jolley, Paul McGuire, Lane Bradford, Mike Ragan, Joe Dominguez, Alex Montoya. Growing to adulthood, Belle Starr's son attempts to prove he is not an outlaw like his famous mother. There is very little to recommend this tired outing.\n\n**3999** _ **Son of Billy the Kid**_ **** Screen Guild, 1949. 64 min. D: Ray Taylor. SC: Ron Ormond and Ira Webb. With Lash LaRue, Al St. John, Marion Colby, June Carr, George Baxter, Terry Frost, John James, House Peters, Jr., Clarke Stevens, Bob Duncan, Cliff Taylor, William Perrott, Felipe Turich, Rosa Turich, I. Stanford Jolley, Bud Osborne, Eileen Dixon, Jerry Riggio, Frazer McMinn. A special U.S. marshal is assigned to a community terrorized by a gang after mortgages on land wanted by an incoming railroad. Lash LaRue vehicle moves along fast enough to cover its budget and script deficits.\n\n**4000** _ **The Son of Davy Crockett**_ **** Columbia, 1941. 59 min. D-SC: Lambert Hillyer. With Bill Elliott, Iris Meredith, Dub Taylor, Richard Fiske, Kenneth MacDonald, Eddy Waller, Don Curtis, Edmund Cobb, Steve Clark, Harrison Greene, Lloyd Bridges, Curley Dresden, Paul Scanlon, Frank Ellis, Richard Botiller, Tom London, Merrill McCormick, Martin Garralaga, Lew Meehan, Jack Ingram, Frank LaRue, Sven Hugo Borg, Emmett Lynn, Herman Hack, Horace B. Carpenter, Eddie Laughton, Oscar Gahan, Russ Powell, Hank Bell, Fred Parker, Jack Evans, Ray Jones, Roy Bucko, Jack Montgomery. President Ulysses S. Grant sends Davy Crockett's son to the unclaimed territory of Yucca Valley as the government's unofficial representative to overthrow a tyrant and his hired killers. Well written and directed segment in the pseudo-historical series Bill Elliott did for Columbia.\n\n**Son of Django** see _**Return of Django**_\n\n**4001** _ **Son of Geronimo**_ **** Columbia, 1952. 15 Chapters. D: Spencer Gordon Bennet. SC: Arthur Hoerl, Royal K. Cole and George H. Plympton. With Clay(ton) Moore, Bud Osborne, Tommy Farrell, Rodd Redwing, Marshall Reed, Eileen Rowe, John Crawford, Zon Murray, Rick Vallin, Lyle Talbot, Chief Yowlachie, Sandy Sanders, Bob (John) Cason, Wally West, Frank Matts, Frank Ellis, Anthony Dante, Al Cantor. A frontier scout tries to bring peace between settlers and Indians but the latter are led by a brave who claims to be Geronimo's son and who is in cahoots with renegade whites. Clayton Moore is the main asset of this anemic cliffhanger although it is nice to see Bud Osborne (as wagon trail boss Tulsa) in a major role. ****\n\n**4002** _ **Son of God's Country**_ **** Republic, 1948. 60 min. D: R.G. Springsteen. SC: Paul Gangelin. With Monte Hale, Pamela Blake, Paul Hurst, Jason Robards, Jay Kirby, Jim Nolan, Steve Darrell, Francis McDonald, Fred Graham, Herman Hack. In order to capture an outlaw gang, a U.S. marshal pretends to be a crook so he can locate the bad men. Pretty fair Monte Hale vehicle.\n\n**4003** _ **Son of Jesse James**_ **** P.E.A.\/Apolofilm, 1965. 90 min. Color. D: Antonio Del Amo (Adrian Hoven). SC: Pino Passalacqua and Marcello Fondato. With Robert Hundar (Claudio Undari), Mercedes Alonso, Adrian Hoven, Luis Induni, Ralph Baldwin (Raf Baldassare), Roberto Camardiel, Janos (John) Bartha, Joe Kamel, Pier Caminnecci, Jose Jaspe, Robert Johnson, Jr. Jesse James's grown son is falsely accused of murder by Bob Ford, the man who killed his famous father. As pseudo-historical fare this Italian production, made as _**Solo Contro Tutti**_ (One Against All), may hold some interest.\n\n**4004** _ **Son of Oklahoma**_ **** World Wide, 1932. 57 min. D: Robert North Bradbury. SC: Burl Tuttle and George Hull. With Bob Steele, Josie Sedgwick, Julian Rivero, Carmen LaRoux, Earl Dwire, Robert Homans, Henry Roquemore, Jack Perrin, Si Jenks, Dick Dickinson, Jack Kirk, Herman Hack, Jack Evans, Silver Tip Baker. A cowboy, separated from his family for seventeen years by an outlaw, sets out to find them. Pretty good Bob Steele feature with plenty of action.\n\n**4005** _ **Son of Paleface**_ **** Paramount, 1952. 95 min. Color. D: Frank Tashlin. SC: Frank Tashlin, Robert L. Welch and Joseph Quillan. With Bob Hope, Jane Russell, Roy Rogers, Bill Williams, Lloyd Corrigan, Paul E. Burns, Douglass Dumbrille, Harry Von Zell, Iron Eyes Cody, Wee Willie Davis, Charley Cooley, Charles Morton, Don Dunning, Oliver Blake, Al Ferguson, Leo J. McMahon, Felice Richmond, Charmeinne Harker, Isabel Cushin, Jane Easton, Homer Dickinson, Lyle Moraine, Hank Mann, Michael A. Cirillo, Chester Conklin, Flo Stanton, John George, Charles Quirk, Frank Cordell, Willard Willingham, Warren Fiske, Jean Willes, Jonathan Hale, Cecil B. DeMille, Bing Crosby, Robert L. Welch, Rose Plummer, Geraldine Farnum, Flo Stanton, Marie Shaw. A tenderfoot goes West to collect an inheritance and ending up with only debts he decides to marry a buxom, and rich, young woman. Amusing follow-up to _**The Paleface**_ (q.v.).\n\n**4006** _ **Son of Roaring Dan**_ **** Universal, 1940. 63 min. D: Ford Beebe. SC: Clarence Upson Young. With Johnny Mack Brown, Fuzzy Knight, Nell O'Day, Jeanne Kelly (Jean Brooks), Robert Homans, Tom Chatterton, John Eldredge, Ethan Laidlaw, Lafe McKee, Richard Alexander, Eddie Polo, Bob Reeves, Frank McCarroll, The Texas Rangers, Chuck Morrison, Lloyd Ingraham, Jack Shannon, Ben Taggart, Ralph Peters, Ralph Dunn, Jack Montgomery. When his father is murdered a cowboy pretends to be the tenderfoot son of a fellow rancher to catch the killers. Well done Johnny Mack Brown film with a trio of well interpolated songs.\n\n**4007** _ **Son of the Border**_ **** RKO Radio, 1933. 55 min. D: Lloyd Nosler. SC: Wellyn Totman and Harold Shumate. With Tom Keene, Julie Haydon, Edgar Kennedy, David Durand, Creighton (Lon, Jr.) Chaney, Charles King, Alan Bridge, Claudia Coleman, Yakima Canutt, Lew Meehan, Murdock MacQuarrie, George Sowards, Bud Pope, Ken Cooper, Nick Cogley. After her fiance is shot by a cowboy during a gunfight with bank robbers, a young woman vows revenge and later uses the man's younger brother to carry out her plan. Well made galloper with strong performances from the cast.\n\n**Poster for** _**Son of the Border**_ **(RKO Radio, 1933).**\n\n**  \n**\n\n**4008** _ **Son of the Morning Star**_ **** ABC-TV, 1991. 187 min. Color. D: Mike Robe. SC: Melissa Mathison. With Gary Cole, Rosanna Arquette, Stanley Anderson, Edward Blatchford, George Dickerson, Rodney A. Grant, Tom O'Brien, Terry O'Quinn, Nick Ramus, Floyd \"Red Crow\" Westerman, Tim Ranson, Robert Schenkkan, David Strathairn, Dean Stockwell, Bryce Chamberlain, Peter Leitner, George K. Sullivan, Demina Becker, George American Horse, Ron Hunter, Sheldon Peters Wolfchild, Michael Medeiros, Mike Casey, Sav Farrow, Wendy Feder, Patrick Johnston, Eric Lawson, Jay Bernard, Kimberly Norris, Russ Walks, Buffy Sainte-Marie (voice). Two women tell their sides of the story regarding General Custer and the events leading up to the Battle of the Little Big Horn. Overlong TV movie.\n\n**4009** _ **A Son of the Plains**_ **** Syndicate, 1931. 59 min. D-SC: Robert North Bradbury. With Bob Custer, Doris Phillips, J.P. McGowan, Edward Hearn, Gordon DeMain, Al St. John, Art Mix, Bob Burns, Jack Evans, Blackie Whiteford, Artie Ortego, Jane Crowley, Eve Humes. A deputy sheriff is torn between his duty and the love of a woman whose father helps another man in a holdup. Very poor movie made worse by Bob Custer's BAD acting. Film does contain an amusing sequence at the Yucca Saloon where a jaded gal sings its theme song, \"On the Banks of the Wabash\"(!), and Al St. John has a few good comic moments as a drunk. Reworked as _**Blue Steel**_ (q.v.).\n\n**4010** _ **Son of the Renegade**_ **** United Artists, 1953. 57 min. D: Reg Browne. SC: John Carpenter. With John Carpenter, Lori Irving, Joan McKellan, Valley Keene, Jack Ingram, Henry Wills, Verne Teters, Bill Coontz, Bill Ward, Roy Canada, Whitney Hughes, Ewing Brown, Freddie Carson, Pat McGeehan (narrator). Returning home, the son of a notorious outlaw meets with resentment until he uncovers a robbery plan. Low grade John Carpenter outing, but worth a look for his followers.\n\n**4011** _ **Son of Zorro**_ **** Republic, 1947. 13 Chapters. D: Spencer Gordon Bennet and Fred C. Brannon. SC: Franklin Adreon, Basil Dickey, Jesse Duffy and Sol Shor. With George Turner, Peggy Stewart, Roy Barcroft, Ed Cassidy, Ernie Adams, Stanley Price, Edmund Cobb, Kenneth Terrell, Wheaton Chambers, Fred Graham, Eddie Parker, Si Jenks, Jack O'Shea, Jack Kirk, Tom Steele, Dale Van Sickel, Tom London, Mike J. Frankovich, Pierce Lyden, Rocky Shahan, Charles King, Ted Adams, John Daheim, Pascale Perry, Gil Perkins, Ted Mapes, Tex Terry, Art Dillard, Joe Phillips, George Bell, Duke Taylor, Post Park, Al Ferguson, Cactus Mack, Bud Wolfe, Newton House, Frank O'Connor, Tommy Ryan, Carl Sepulveda, George Chesebro, Howard Mitchell, Frank Ellis, Tommy Coats, Silver Harr, Ralph Bucko, Roy Bucko, Doc Adams, Joe Balch. After the Civil War, a cavalry officer returns home to find the area controlled by dishonest politicians and he revives the character of Zorro to stop them. Pretty good pseudo-Zorro cliffhanger that was reissued in 1956; the previous year Republic used the Zorro plot motif in an entirely modern-day detective serial, _**Daughter of Don Q**_ , starring Adrian Booth and Kirk Alyn.\n\n**4012** _ **Son of Zorro**_ **** Films Triunfosa, 1974. 86 min. Color. D: Franck G. Carroll (Gianfranco Baldanello). SC: Mario DeRiso, Joaquin Luis Romero Marchent, Guido Zurli and Gianfranco Baldanello. With Robert Widmark (Alberto Dell'Acqua), Fernando Sancho, Elisa Ramirez, William Berger, George Wang, Marina Malfatti, Marco Zuanelli, Franco Fantasia, Giorgio Dolfin, Marcello Monti, Mario Dardanelli, Marcello Simoni, Lorenzo Piani, Pietro Riccione, Andrea Fantasia, Carlos Bravo. A nobleman takes on the guise of Zorro and teams with an Army colonel to subdue a tyrant in Old California. Respectable Italian-Spanish co-production filmed as _**Il Figlio di Zorro**_ (The Son of Zorro) and also called _**Man with the Golden Winchester**_.\n\n**4013** _ **Song of Arizona**_ **** Republic, 1946. 68 min. D: Frank McDonald. SC: M. Coates Webster. With Roy Rogers, George \"Gabby\" Hayes, Dale Evans, Lyle Talbot, Tommy Cook, Bob Nolan and The Sons of the Pioneers (Tim Spencer, Ken Carson, Shug Fisher, Hugh Farr, Karl Farr), Edmund Cobb, Johnny Calkins, Sarah Edwards, Tommy Ivo, Michael Chapin, Dick Curtis, Tom Quinn, Noble \"Kid\" Chissel, Don Kay Reynolds, The Robert Mitchell Boychoir. Before dying, a bank robber leaves stolen loot with his son at a homeless boys' ranch and when the gang members come to retrieve it they are opposed by Roy Rogers and the Sons of the Pioneers. Lively Roy Rogers affair with Gabby Hayes dominating the proceedings.\n\n**4014** _ **The Song of Hiawatha**_ **** Hallmark Home Entertainment, 1997. 114 min. Color. D: Jeffrey Shore. SC: Earl W. Wallace. With Graham Greene, Litefoot, Irene Bedard, Russell Means, Sheila Tousey, Adam Beach, Michael Rooker, David Strathaim, Gordon Tootosis, Tina Louise Bomberry, Peter Kelly Gaudreault, Mike Kanentakeron, Adrian Jamieson, Flint Eagle, Vern Harper, Sid Bobb, Shirley Cheechoo, Madeleine Bergeron. The story of Indian brave Hiawatha and his true love, the beautiful maiden Minnehaha. Well photographed and acted tale of the early American frontier based on the poem by Henry Wadsworth Longfellow.\n\n**4015** _ **Song of Idaho**_ **** Columbia, 1948. 70 min. D: Ray Nazarro. SC: Barry Shipman. With Kirby Grant, The Hoosier Hot Shots (Charles \"Gabe\" Ward, Ken Trietsch, Paul \"Hezzie\" Trietsch, Gil Taylor), June Vincent, The Sunshine Boys (Freddie Daniel, M.H. Richman, J.D. Sumner, Eddie Wallace), Tommy Ivo, Emory Parnell, The Starlighters, Mary Newton, Eddie Acuff, Dorothy Vaughn, Maudie Prickett, Fred F. Sears, George Lloyd. A radio singer has to please his sponsor's brat son in order to get his contract renewed. Average Columbia country music Western program feature.\n\n**4016** _ **Song of Nevada**_ **** Republic, 1944. 75 min. D: Joseph Kane. SC: Gordon Kahn and Olive Cooper. With Roy Rogers, Dale Evans, Mary Lee, Bob Nolan and The Sons of the Pioneers (Tim Spencer, Ken Carson, Shug Fisher, Hugh Farr, Karl Farr), Thurston Hall, Lloyd Corrigan, John Eldredge, Forrest Taylor, LeRoy Mason, George Meeker, Emmett Vogan, William B. Davidson, Kenne Duncan, Si Jenks, Frank McCarroll, Henry Wills, Jack O'Shea, Helen Talbot, Jack Perrin, Tom Steele. A millionaire tries to stop his daughter from marrying a man he dislikes by pretending to be dead and having a cowboy romance her. Music outweighs action in this mediocre Roy Rogers affair that is somewhat saved by Thurston Hall as the rich man; TV version is badly butchered at 54 minutes.\n\n**4017** _ **Song of Old Wyoming**_ **** Producers Releasing Corporation, 1945. 65 min. Color. D: Robert Emmett (Tansey). SC: Frances Kavanaugh. With Eddie Dean, Jennifer Holt, Sarah Padden, Al (Lash) LaRue, Emmett Lynn, Ray Elder, John Carpenter, Ian Keith, Robert Barron, Horace Murphy, Rocky Camron, Richard Cramer, Steve Clark, Lee Bennett. A crooning cowboy helps his woman rancher-newspaper owner boss when rustlers take her cattle and try to bankrupt her. Lash LaRue steals the show as the woman's outlaw nephew but attractive Cinecolor and Eddie Dean's singing also help make this good viewing.\n\n**4018** _ **Song of Texas**_ **** Republic, 1943. 69 min. D: Joseph Kane. SC: Winston Miller. With Roy Rogers, Sheila Ryan, Bob Nolan and The Sons of the Pioneers (Tim Spencer, Lloyd Perryman, Pat Brady, Hugh Farr, Karl Farr), Barton MacLane, Harry Shannon, Arline Judge, William Haade, Hal Taliaferro, Yakima Canutt, Tom London, Forrest Taylor, Maxine Doyle, Jack O'Shea, Eve March. Roy Rogers tries to help a drunken once famous rodeo star, who is being used by crooks, in fooling his visiting daughter into thinking he is still a success. Reworking of _**Lady for a Day**_ (Warner Bros., 1932), this Roy Rogers vehicle is very good with Harry Shannon quite effective as the down on this luck rodeo performer.\n\n**4019** _ **Song of the Buckaroo**_ **** Monogram, 1938. 58 min. D: Al Herman. SC: John Rathmell. With Tex Ritter, Jinx Falkenberg, Mary Ruth, Tom London, Frank LaRue, Charles King, Bob Terry, Horace Murphy, Snub Pollard, Dave O'Brien, Dorothy Fay, George Chesebro, Ernie Adams, Rudy Sooter, Bud Osborne, George Morrell. A desperado finds a dead man and assumes his identity, raises his little girl and becomes a respected citizen but his old gang recognizes him and plots blackmail. Pretty fair Tex Ritter opus that includes the Carson Robison tune \"Texas Dan.\"\n\n**4020** _ **Song of the Caballero**_ **** Universal, 1930. 72 min. D: Harry Joe Brown. SC: Bennett Cohen and Lesley Mason. With Ken Maynard, Doris Hill, Francis Ford, Gino Corrado, Evelyn Sherman, Josef Swickard, Frank Rice, William Irving, Jozelle Joyner. Because of abuse to his mother, a cowboy becomes a bandit who only preys on a rich family until he saves the life of their pretty daughter. Okay Ken Maynard early talkie set in Mexico.\n\n**4021** _ **Song of the Drifter**_ **** Monogram, 1948 55 min. D: Lambert Hillyer. SC: Frank Young. With Jimmy Wakely, Dub Taylor, Mildred Coles, Patsy Moran, William Ruhl, Marshall Reed, Frank LaRue, Carl Mathews, Jimmie Martin, Steve Clark, Wheaton Chambers, Bud Osborne, Bob Woodward, Dick Reinhart, Cliffie Stone. A singing cowboy combats crooks trying to pollute water so they can get range land for themselves. Lambert Hillyer's direction tries hard but cannot help this Jimmy Wakely opus.\n\n**4022** _ **Song of the Gringo**_ **** Grand National, 1936. 57 min. D: John McCarthy. SC: Robert Emmett (Tansey) and Al Jennings. With Tex Ritter, Joan Woodbury, Monte Blue, Fuzzy Knight, Richard (Ted) Adams, Warner Richmond, Al Jennings, Martin Garralaga, William Desmond, Glenn Strange, Budd Buster, Murdock MacQuarrie, Ethan Laidlaw, Slim Whitaker, Ed Cassidy, Earl Dwire, Jack Kirk, Bob Burns, Forrest Taylor, Robert Fiske, Rosa Rey, Jose Pacheco and His Continental Orchestra. Cowpoke Tex infiltrates an outlaw gang posing as cowboys and ends up being accused of murdering a ranch owner. More drama and music than action in Tex Ritter's film debut but it contains an exciting finale with a courtroom shootout.\n\n**Advertisement for** _**Song of the Gringo**_ **(Grand National, 1936).**\n\n**  \n**\n\n**4023** _ **Song of the Prairie**_ **** Columbia, 1945. 62 min. D: Ray Nazarro. SC: J. Benton Cheney. With Ken Curtis, June Storey, Guinn Williams, Jeff Donnell, Andy Clyde, Grady Sutton, Thurston Hall, Robert Williams, John Tyrrell, Deuce Spriggins, Carolina Cotton, The Hoosier Hot Shots (Charles \"Gabe\" Ward, Ken Trietsch, Paul \"Hezzie\" Trietsch, Gil Taylor), The Town Criers, Rudy Sooter, Dick Curtis, Reed Howes, Heinie Conklin, Vernon Dent, William Gould, Curt Barrett, Warren Jackson, Robert Walker, Charles Coleman, Sam Flint, Donald Kerr, Paul Bradley, Matt Roubert, Tex Cooper. When a rancher wants to become a bandleader and start a show club, he is helped by the Hoosier Hot Shots and other entertainers. Average Western musical with the usual modicum of songs.\n\n**4024** _ **Song of the Range**_ **** Monogram, 1944. 55 min. D: Wallace Fox. SC: Betty Burbridge. With Jimmy Wakely, Dennis Moore, Lee \"Lasses\" White, Kay Forrester, Sam Flint, Hugh Prosser, George Eldredge, Steve Clark, Johnny Bond and The Red River Valley Boys, Edmund Cobb, Pierre Watkin, Bud Osborne, Kenneth Terrell, Carl Mathews, Carl Sepulveda, The Sunshine Girls, Frankie Marvin, Carl Mathews, Sam Flint, Roy Brent, Wesley Tuttle, Jimmie Dean, Paul Sells, Colleen Summers (Mary Ford) Cedric Stevens. Falsely accused of murder, a cowboy escapes from jail and assumes the guise of a federal agent to capture a gang of gold smugglers. Jimmy Wakely's first starring Western gives him little to do since Dennis Moore dominates throughout as the wrongly accused cowpoke; okay production that moves fairly quickly but contains too much music. Remake of _**Pals of the Saddle**_ (q.v.).\n\n**4025** _ **Song of the Saddle**_ **** Warner Bros., 1936. 58 min. D: Louis King. SC: William Jacobs. With Dick Foran, Alma Lloyd, Charles Middleton, Addison Richards, Eddie Schubert, Monte Montague, Victor Potel, Kenneth Harlan, Myrtle Stedman, George Ernest, Pat West, James Farley, Julian Rivero, William Desmond, Bud Osborne, Robert Kortman, Bonita Granville, The Sons of the Pioneers (Bob Nolan, Len Slye [Roy Rogers], Tim Spencer, Hugh Farr, Karl Farr). Fifteen years after the murder of his father, a cowboy crooner returns to find the killers. Typically good Dick Foran vehicle.\n\n**4026** _ **Song of the Sierras**_ **** Monogram, 1946. 55 min. D: Oliver Drake. SC: Elmer Clifton. With Jimmy Wakely, Lee \"Lasses\" White, Jean Carlin, Jack Baxley, Iris Clive, Zon Murray, Budd Buster, Bob Duncan, Brad Slaven, Ben Corbett, Ray Jones, Carl Sepulveda, Wesley Tuttle and His Texas Stars, Artie Ortego, George Morrell, Arthur Smith, Forrest Matthews, Bob Gilbert, Jesse Ashlock. In order to win a big race, a cowboy trains wild horses but his activities are opposed by crooks. Another anemic Jimmy Wakely film.\n\n**4027** _ **Song of the Trail**_ **** Ambassador, 1936. 60 min. D: Russell Hopton. SC: George Sayre and Barry Barrington. With Kermit Maynard, Evelyn Brent, Fuzzy Knight, George Hayes, Antoinette Lees (Andrea Leeds), Wheeler Oakman, Lee Shumway, Roger Williams, Ray Gallagher, Charles McMurphy, Horace Murphy, Lynette London, Bob McKenzie, Frank McCarroll, Artie Ortego. A rodeo star falls in love with woman whose father is being harassed by crooks after his valuable mine. One of the best Kermit Maynard features for producer Maurice Conn; full of action and well made.\n\n**4028** _ **Song of the Wasteland**_ **** Monogram, 1947. 58 min. D: Thomas Carr. SC: J. Benton Cheney. With Jimmy Wakely, Lee \"Lasses\" White, Dottye Brown, Holly Bane, John James, Henry Hall, Marshall Reed, Gary Garrett, Ted Adams, Pierce Lyden, George Chesebro, Chester Conklin, John Carpenter, Ray Jones, The Saddle Pals (Johnny Bond, Dick Reinhart, River Lewis). Vigilantes are formed to combat outlaws but a singing cowpoke learns they are the real cause of the area's lawlessness. A fine supporting cast can do little to retrieve this Jimmy Wakley songfest from boredom.\n\n**4029** _ **Song of the West**_ **** Warner Bros., 1930. 82 min. Color. D: Ray Enright. SC: Harry Thew. With John Boles, Vivienne Segal, Joe E. Brown, Marie Wells, Sam Hardy, Marion Byron, Eddie Gribbon, Edward Martindel, Rudolph Cameron, Jack Kirk, Harriett Lake (Ann Sothern). A lieutenant, falsely accused of murder, heads West with a wagon train housing a colonel's daughter, the girl he loves. Stilted early sound musical based on Oscar Hammerstein II and Laurence Stallings' operetta _Rainbow_ , buoyed by John Boles' singing.\n\n**4030** _ **Songs and Bullets**_ **** Spectrum, 1938. 58 min. D: Sam Newfield. SC: Joseph O'Donnell and George H. Plympton. With Fred Scott, Al St. John, Alice Ardell, Karl Hackett, Charles King, Frank LaRue, Richard Cramer, Carl Mathews, Jimmy Aubrey, Budd Buster, Lew Porter, Sherry Tansey, Wally West, Tom Smith. A corrupt, murdering businessman leads a gang, including the local sheriff, which pulls off a robbery but he is trailed by a singing lawman and his partner. Better than average Fred Scott outing with the lovely theme song \"Prairie Moon\" deftly sung by the star who also belts out \"My Old Ten Gallon Hat\" and \"Back in Arkansas.\" When villain Karl Hackett meets pretty French schoolmarm Alice Ardell one of his cohorts exclaims, \"That's the first time Shelton's smiled since he dispossessed the Higgins family.\"\n\n**4031** _ **Songs and Saddles**_ **** Colony, 1938. 65 min. D: Harry Fraser. SC: Wayne Carter. With Gene Austin, Joan Brooks, Lynne Barkeley, Henry Roquemore, Walter Wills, Charles King, Karl Hackett, Ted Claire, John Merton, Ben Corbett, Bob Terry, John Elliott, Lloyd Ingraham, Russell \"Candy\" Hall, Otto \"Coco\" Heimel, Darryl Harper. An entertainer and his troupe find themselves captured by a gang of outlaws. Gene Austin's solo starring Western is a low budget affair but his fans will like him as a singing cowboy and it is loaded with good songs.\n\n_**Sonny and Jed**_ see _**Bandera Bandits**_\n\n**4032** _ **Sonora Stagecoach**_ **** Monogram, 1944. 61 min. D: Robert Emmett Tansey. SC: Frances Kavanaugh. With Hoot Gibson, Bob Steele, Chief Thundercloud, Rocky Camron, Betty Miles, Glenn Strange, George Eldredge, Karl Hackett, Henry Hall, Charles King, Bud Osborne, Charles Murray, Jr., John Bridges, Al Ferguson, Forrest Taylor, Frank Ellis, Hal Price, Rodd Redwing, John Cason, Horace B. Carpenter, Fred Hoose, Augie Gomez. The Trail Blazers are assigned to take an accused killer to stand trial with the men who really committed the crime trying to ambush them. Fast paced and well done, the last in \"The Trail Blazers\" series.\n\n**4033** _ **Sons of Adventure**_ **** Republic, 1948. 68 min. D: Yakima Canutt. SC: Franklin Adreon and Sol Shor. With Lynne Roberts, Russell Hayden, Gordon Jones, Grant Withers, George Chandler, Roy Barcroft, John Newland, Stephanie Bachelor, John Holland, Gilbert Frye, Richard Irving, Joan Blair, John Crawford, Keith Richards, James Dale. After a Western star is murdered on the set of his new film a stuntman is blamed and his pals try to find the real killer. Out-of-the-ordinary feature that is well directed by Yakima Canutt with an added bonus of a look at the Republic film factory.\n\n**4034** _ **The Sons of Great Bear**_ **** VEB Progress Film-Vertrief, 1966. 92 min. Color. D: Josef Mach. SC: Lisolette Welskopf-Henrich. With Gojko Mitic, Jiri Vrstala, Rolf Romer, Hans Hardt-Hardtloff, Gerhard Rachold, Horst Jonischkan, Jozef Majercik, Jozef Adamovic, Milan Jablonsky, Hannjo Haasse, Helmut Schreiber, Jose Lepetic, Rolf Ripperger, Brigitte Krause, Karin Beewen, Ruth Kommerell, Kati Szekel, Zofia Slaboszowska, Slabodanka Markovic, Hans Finohr, A.P. Hoffmann, Martin Tapak, Horst Kube, Walter E. Fuss, Sepp Klose. When gold is discovered in the Black Hills, Indian tribes not only face invasion and resettlement by whites but also friction among themselves. Ground breaking Eastern Bloc Western that is a bit stilted but still worth seeing; filmed in East Germany as _**Die Sohne der Grossen Barin**_ (The Son of the Great Bear).\n\n**4035** _ **The Sons of Katie Elder**_ **** Paramount, 1965. 122 min. Color. D: Henry Hathaway. SC: William H.Wright, Allen Weiss and Harry Essex. With John Wayne, Dean Martin, Martha Hyer, Michael Anderson, Jr., Earl Holliman, Jeremy Slate, James Gregory, Paul Fix, George Kennedy, Dennis Hopper, Sheldon Allman, John Litel, John Doucette, James Westerfield, Rhys Williams, John Qualen, Rodolfo Acosta, Strother Martin, Percy Helton, Karl Swenson, Chuck Roberson, Henry Wills, Chuck Hayward. Four brothers return home after their mother's death only to be falsely blamed for a killing by the crooks who cheated their father. Very entertaining John Wayne outing.\n\n**4036** _ **Sons of New Mexico**_ **** Columbia, 1950. 71 min. D: John English. SC: Paul Gangelin. With Gene Autry, Gail Davis, Robert Armstrong, Dick Jones, Clayton Moore, Frankie Darro, Irving Bacon, Russell Arms, Marie Blake, Sandy Sanders, Roy Gordon, Frankie Marvin, Pierce Lyden, Paul Raymond, Kenne Duncan, Harry Mackin, Bobby Clark, Gaylord (Steve) Pendleton, Billy Lechner. After being appointed the executor of an estate, Gene Autry tries to get the young heir from under the influence of a dishonest rancher by sending him to military school. Pleasant Gene Autry opus, mainly geared to juveniles.\n\n**4037** _ **Sons of the Pioneers**_ **** Republic, 1942. 61 min. D: Joseph Kane. SC: M. Coates Webster, Mauri Grashin and Robert T. Shannon. With Roy Rogers, George \"Gabby\" Hayes, Maris Wrixon, Bob Nolan and The Sons of the Pioneers (Tim Spencer, Lloyd Perryman, Pat Brady, Hugh Farr, Karl Farr), Bradley Page, Hal Taliaferro, Tom London, Minerva Ureval, Jack O'Shea, Frank Ellis, Bob Woodward, Fern Emmett, Chester Conklin, Karl Hackett, Fred Burns, Art Mix, Sarah Edwards, Horace B. Carpenter, Neal Hart, Frank Brownlee, Bud Osborne, Pascale Perry. Roy Rogers returns home to help Gabby and the townspeople in stopping ruthless land grabbers who know that valuable minerals are beneath the soil. Fairly good Roy Rogers vehicle although the title singing group has little to do in the proceedings.\n\n**4038** _ **Sons of the Saddle**_ **** Universal, 1930. 76 min. D: Harry Joe Brown. SC: Bennett Cohen and Lesley Mason. With Ken Maynard, Doris Hill, Joseph Girard, Carroll Nye, Francis Ford, Harry Todd, Frank Rice, William Gillis. A ranch foreman loves the boss' daughter, but so does his pal, and when she rejects the latter he joins an outlaw gang planning to rustle the spread's cattle herd. Entertaining Ken Maynard early talkie.\n\n_**Sons of Vengeance**_ see _**Gunfight at High Noon**_\n\n**4039** _ **Sota, Caballo y Rey**_ (Jack, Horse and King) **** Produccione Raul de Anda, 1944. 90 min. D: Roberto O'Quigley. SC: Roberto O'Quigley and Raul de Anda. With Luis Aguilar, Susana Cora, Domingo Soler, Jose Torvay, Conchita Gentil Arcos, Gilberto Gonzalez, Lupe Inclan, Agustin Isunza, Manuel Donde, Meche Barba, Julio Ahuet, Jorge Arriaga, Carlos Lopez Moctezuma, Jose Eduardo Perez, Armando Soto la Marina, Alfonso Jiminez. After receiving payment of a debt a wealthy rancher is murdered and the man he was supposed to meet finds the body and is suspected of the crime. Pretty fair Mexican Western mystery from producer-writer Raul de Anda.\n\n**4040** _ **The Soul of Nigger Charley**_ **** Paramount, 1973. 104 min. Color. D: Larry G. Spangler. SC: Harold Stone. With Fred Williamson, D'Urville Martin, Denise Nicholas, Pedro Armendariz, Jr., Kirk Calloway, George Allen, Kevin Hagen, Michael Cameron, Johnny Greenwood, James Garbo, Nai Bonet, Robert Minor, Fred Lerner, Joe Hendeson, Richard Farnsworth, Tony Brubaker, Boyd \"Red\" Morgan, Al Hassan, Ed Hice, Henry Wills, Phil Aventetti. A former slave goes to Mexico to free a group of his people held there by an ex\u2013Confederate officer. Overlong and none-too-entertaining sequel to _**The Legend of Nigger Charley**_ (q.v.).\n\n**4041** _ **Soul Soldier**_ **** Fanfare\/Metromedia, 1971. 78 min. Color. D: John \"Bud\" Cardos. SC: Marlene Weed. With Rafer Johnson, Barbara Hale, Cesar Romero, Robert Doqui, Isaac Fields, Lincoln Kilpatrick, Isabel Sanford, Otis Taylor, Steve Drexel, Robert Dix, James Michelle, Bobby Clark, Byrd Holland, Bill Collins, John Fox, Russ Nannarello, Jr., Bernard Brown, Clarence Comas, Donald Diggs, Jeff Everett, Cal Fields, Perry Fluker, Noah Hobson, Earl Humphrey, DeVaughn LaBon, Rod Law, John Nettles, Jim Pace, Eric Richmond, John Ramsey, Charles Wells, Paul Wheaton, Dave White. Following the Civil War, a regiment of black soldiers is assigned to border duty in Texas but find themselves hated by both whites and Indians. Bombed out effort originally issued in 1970 by Hirschman Northern as _**The Red, Black and White**_ , running 97 minutes. Video title: _**Buffalo Soldier**_.\n\n**4042** _ **South of Arizona**_ **** Columbia, 1938. 55 min. D: Sam Nelson. SC: Bennett Cohen. With Charles Starrett, Iris Meredith, The Sons of the Pioneers (Bob Nolan, Hugh Farr, Karl Farr, Lloyd Perryman, Pat Brady), Dick Curtis, Robert Fiske, Edmund Cobb, Art Mix, Richard Botiller, Lafe McKee, Ed Coxen, Hank Bell, Hal Taliaferro, John Tyrrell, Merrill McCormick, Steve Clark, George Morrell. Crooks wanting land for themselves rustle ranchers' cattle and murder a government ranger sent to stop them. Another streamlined Columbia Charles Starrett effort.\n\n**4043** _ **South of Caliente**_ **** Republic, 1951. 67 min. D: William Witney. SC: Eric Taylor. With Roy Rogers, Dale Evans, Pinky Lee, Douglas Fowley, Pat Brady, Charlita, Ric Roman, Leonard Penn, Willie Best, Frank Richards, George J. Lewis, Roy Rogers Riders, Lillian Molieri, Marguerite McGill. A racehorse needed by a woman rancher to sell in order to save her place is stolen and Roy Rogers tries to recover it. Pretty good Roy Rogers vehicle if you can overlook the \"comedy\" of Pinky Lee and Pat Brady.\n\n**4044** _ **South of Death Valley**_ **** Columbia, 1949. 55 min. D: Ray Nazarro. SC: Earle Snell. With Charles Starrett, Smiley Burnette, Gail Davis, Clayton Moore, Fred F. Sears, Lee Roberts, Tommy Duncan and His All Stars, Richard Emory, Jason Robards, Chuck Hamilton, Kermit Maynard, Jack Evans, Blackie Whiteford, George Morrell, George Sowards. Trying to learn who murdered his mine owner brother-in-law, a cowboy rides into an area and finds himself in the middle of a range war. Another assembly line \"Durango Kid\" job, this one on the slow side. British title: _**River of Poison**_.\n\n**4045** _ **South of Heaven, West of Hell**_ **** Blue Steel Releasing, 2000. 127 min. Color. D-SC: Dwight Yoakam. With Dwight Yoakam, Vince Vaughn, Billy Bob Thornton, Bridget Fonda, Peter Fonda, Paul Reubens, Bud Cort, Michael Jeter, Bo Hopkins, Luke Askew, Joe Unger, Matt Clark, Nobel Willingham, Scott Wilson, Ritchie Montgomery, Matt Malloy, Natalie Canerday, Otto Felix, Joe Ely, Terry McIlvain, Amber Taylor, Charles Burba, Audrey Lowe, Maria Daleo, Marta Santamaria, Warren Zevon, Corky Wimberly, Flecia Beard, Rudy Ugland, Jim Clark, Warner McKay, Glen Gold, Forrie Smith, Claude Aichele, Rose Duarte, George Salazar, Maurice Orozco, Gina Carizoza, Richard Smith, George Dobbs, Thadd Turner, Joseph Romanov. Not only does a lawman have to face his past when his family shows up on Christmas Eve, but he and a pal must take on the outlaw gang who abducted the sidekick's girl friend. Mixed up attempt at a Western, but it made money thanks to the popularity of country music star Dwight Yoakam, who wrote, directed and starred in it, along with composing the score; released on video at 104 minutes.\n\n**4046** _ **South of Hell Mountain**_ **** Cannon Films, 1971. 92 min. Color. D-SC: William Sachs and Louis Lehman. With Anna Stewart, Sam Hall, Nicol Britton, Elsa Raven, David Willis, Paul Haller, John Martin Kelly, Mark Hellett. A young woman and her stepmother are held hostage in their cabin by three outlaws but the leader of the band and the girl fall in love. Somewhat obscure, violent drama.\n\n**4047** _ **South of Monterey**_ **** Monogram, 1946. 63 min. D: William Nigh. SC: Charles Belden. With Gilbert Roland, Martin Garralaga, Frank Yaconelli, Marjorie Riordan, George J. Lewis, Terry Frost, Harry Woods, Iris Flores, Wheaton Chambers, Rosa Turich, Nick Thompson, Felipe Turich, Drew Allen (Gil Frye), Joe Dominguez, Lane Bradford, Wally West, Blackie Whiteford, Ray Jones, Roy Bucko, George DeNormand. The Cisco Kid tries to stop two crooked officials from carrying out a land swindle. Pleasant \"Cisco Kid\" series offering.\n\n**4048** _ **South of Rio**_ **** Republic, 1949. 60 min. D: Philip Ford. SC: Norman S. Hall. With Monte Hale, Kay Christopher, Paul Hurst, Roy Barcroft, Douglas Kennedy, Don Haggerty, Rory Mallinson, Lane Bradford, Emmett Vogan, Myron Healey, Tom London, Edmund Cobb, George Lloyd, Tommy Coats. Outlaws terrorize a frontier area with a newly appointed ranger trying to stop them. Okay Monte Hale feature.\n\n**4049** _ **South of St. Louis**_ **** Warner Bros., 1949. 88 min. Color. D: Ray Enright. SC: Zachary Gold and James R. Webb. With Joel McCrea, Alexis Smith, Zachary Scott, Dorothy Malone, Douglas Kennedy, Alan Hale, Victor Jory, Bob Steele, Art Baker, Monte Blue, Nacho Galindo, Warren Jackson, Russell Hicks, Harry Woods, Art Smith, Alan Bridge, Holmes Herbert, Forrest Taylor, Paul Maxey, John Goldsworthy, Jack Mower, William Ruhl, Ray Spiker, Ray Montgomery, Mikel Conrad, Julian Rivero, Lew Harvey, Sailor Vincent, Dan White, Frank Wilcox, Tex Parker, Tony Romano. During the Civil War three ranchers run blockades for the South but they break up when one gets greedy and kills several soldiers for a gun shipment. Well made action drama.\n\n**4050** _ **South of Santa Fe**_ **** World Wide, 1932. 60 min. D: Bert Glennon. SC: G.A. Durlam. With Bob Steele, Janis Elliott, Jack Clifford, Eddie Dunn, Bob Burns, Hank Bell, Allan Garcia, Slim Whitaker, John Elliott, Ed Brady, Buddy Wood (Gordon DeMain), Chris-Pin Martin, Perry Murdock, Archie Ricks, Al Haskell, Jack Evans, F.R. Smith. A cowboy tries to combat an outlaw gang operating along the U.S.-Mexican border. Bob Steele's first World Wide release is a good one, thanks to fine direction and a good script.\n\n**4051** _ **South of Santa Fe**_ **** Republic, 1942. 56 min. D: Joseph Kane. SC: James R. Webb. With Roy Rogers, George \"Gabby\" Hayes, Linda Hayes, Bob Nolan and The Sons of the Pioneers (Tim Spencer, Lloyd Perryman, Pat Brady, Hugh Farr, Karl Farr), Paul Fix, Judy Clark, Bobby Beers, Arthur Loft, Charles Miller, Sam Flint, Jack Kirk, Jack Ingram, Hank Bell, Carleton Young, Lynton Brent, Robert Strange, Henry Wills, Jack O'Shea, Merrill McCormick, Spade Cooley. Roy Rogers invites three industrialists to appear in a town celebration, hoping they will back the opening of a gold mine that will keep the town from ruin, but the trio is kidnapped by gangsters and the blame is placed on Rogers. There is plenty of action in this fast paced Roy Rogers vehicle.\n\n**4052** _ **South of the Border**_ **** Republic, 1939. 71 min. D: George Sherman. SC: Dorrell McGowan and Stuart McGowan. With Gene Autry, Smiley Burnette, June Storey, Lupita Tovar, Mary Lee, Duncan Renaldo, Frank Reicher, Alan Edwards, Claire DuBrey, Richard Botiller, William Farnum, Selmer Jackson, Sheila Darcy, Rex Lease, Charles King, Reed Howes, Jack O'Shea, Slim Whitaker, Hal Price, Julian Rivero, Curley Dresden, The Checkerboard Band, Art Wenzel. Gene Autry is sent to Mexico to squelch a revolution and when he arrives he learns foreign agents are at work. Although one of Gene Autry's best known features, this outing is nothing to brag about except for the title song.\n\n**4053** _ **South of the Chisholm Trail**_ **** Columbia, 1947. 58 min. D: Derwin Abrahams. SC: Michael Simmons. With Charles Starrett, Smiley Burnette, Nancy Saunders, Frank Sully, Hank Newman and The Georgia Crackers, Jim Diehl, Jack Ingram, George Chesebro, Frank LaRue, Jacques O'Mahoney (Jock Mahoney), Eddie Parker, Kit Guard, Ray Elder, Victor Holbrook, Fred F. Sears, Thomas Kingston, Peter Perkins, Pierce Lyden, Lane Bradford, Chuck Hamilton, Joseph Palma, Victor Travers, Cy Malis, Ethan Laidlaw, Merrill McCormick, Kermit Maynard, Sam Lufkin, Kernan Cripps, John Tyrrell, John Cason, Robert Barron, Milton Kibbee, Steve Clark, Herman Hack, Rube Dalroy, Jack Evans, Blackie Whiteford. When members of a musical troupe find money stolen by an outlaw gang they are mistaken for the thieves and nearly hung until rescued by the Durango Kid. Complicated and hard to follow \"Durango Kid\" entry, a reworking of _**Texas**_ (q.v.).\n\n**4054** _ **South of the Rio Grande**_ **** Columbia, 1932. 60 min. D: Lambert Hillyer. SC: Harold Shumate. With Buck Jones, Mona Maris, George J. Lewis, Doris Hill, Philo McCullough, Paul Fix, Charles Reque, James Durkin, Harry Semels, Charles Stevens, Merrill McCormick. A lawman helps a pal who is in love with the woman who caused the death of the peacekeeper's brother. Well produced and directed Buck Jones feature but the star is not as good in a Mexican getup as Tim McCoy.\n\n**4055** _ **South of the Rio Grande**_ **** Monogram, 1945. 62 min. D: Lambert Hillyer. SC: Victor Hammond and Ralph Bettinson. With Duncan Renaldo, Martin Garralaga, George J. Lewis, Armida, Francis McDonald, Lillian Molieri, Charles Stevens, Pedro Regas, Soledad Jiminez, The Guadalajara Trio, Tito Renaldo, Joe Dominguez. The Cisco Kid is opposed to a corrupt military leader trying to control the countryside. Passable \"Cisco Kid\" dual bill item from a story by Johnston McCulley; TV prints dub the Cisco Kid and his sidekick Pancho as Chico and Pablo. Also called _**The Cisco Kid in South of the Rio Grande**_.\n\n**4056** _ **South Pacific Trail**_ **** Republic, 1952. 60 min. D: William Witney. SC: Arthur Orloff. With Rex Allen, Estelita Rodriguez, Slim Pickens, Roy Barcroft, Nestor Paiva, Douglas Evans, Forrest Taylor, Joe McGuinn, The Republic Rhythm Riders, Chick Hannon. A cowboy deduces his ranch foreman is planning a gold hijack scheme. Fair Rex Allen film.\n\n**4057** _ **A Southern Yankee**_ **** Metro-Goldwyn-Mayer, 1948. 90 min. D: Edward Sedgwick. SC: Melvin Frank and Harry Tugend. With Red Skelton, Brian Donlevy, Arlene Dahl, George Coulouris, Lloyd Gough, John Ireland, Minor Watson, Charles Dingle, Art Baker, Reed Hadley, Arthur Space, Joyce Compton, Susan Simon, Henry Hall, Lane Chandler, Stanley Andrews, Cliff Clark, Richard Alexander, Louise Beavers, Harry Cording, Gene Roth, Jeff Corey, Buddy Roosevelt, Wade Crosby, George DeNormand, William \"Bill\" Phillips, Garry Owen, Sam Flint, Frank McGrath, Frank Hagney, Paul Newlan, David Newell, Christian J. Frank, Paul Harvey, Forbes Murray, Weldon Heyburn, John Merton, Kermit Maynard, Paul Kruger, Ian MacDonald, George Magrill, Harry Woods, Ann Staunton, Richard Simmons, Duke York, Ralph Sanford, William Tannen, Carl Saxe, Dick Wessel, Glenn Strange, Ralph Volkie, Robert Wilkie, Jack Worth, Harry Woods, David Sharpe, Jack Stoney, Harry Wilson, Jack Lee. A nitwit becomes a Union spy during the Civil War so he can see a Southern belle with whom he has fallen in love. Very funny Red Skelton costume comedy.\n\n**4058** _ **The Southerner**_ **** United Artists, 1945. 92 min. D: Jean Renoir. SC: Jean Renoir and Hugo Butler. With Zachary Scott, Betty Field, J. Carrol Naish, Beulah Bondi, Percy Kilbride, Charles Kemper, Blanche Yurka, Norman Lloyd, Estelle Taylor, Paul Harvey, Noreen Nash, Jack Norworth, Nestor Paiva, Paul E. Burns, Jay Gilpin, Jean Vanderwill, Earle Hodgins, Wheaton Chambers, Almira Sessions, Glen Walters, Dorothy Granger, Anne Cornwall, Ann Kunde, Grace Christy, Bunny Sunshine. A south Texas cotton farmer struggles to bring about a better life for himself and his family. Fine contemporary social drama.\n\n**4059** _ **Southward Ho!**_ **** Republic, 1939. 57 min. D: Joseph Kane. SC: Jack Natteford and John Rathmell. With Roy Rogers, Mary Hart, George \"Gabby\" Hayes, Wade Boteler, Arthur Loft, Lane Chandler, Tom London, Charles Moore, Edwin Brady, Hal Taliaferro, George Chesebro, Fred Burns, Frank Ellis, Jack Ingram, Frank McCarroll, Curley Dresden, Jim Corey, Earl Dwire, Harry Strang, Nicodemus Stewart, Bob Woodward, Art Dillard. In post Civil War Texas Roy and Gabby are at odds with renegade Yankee soldiers ransacking the area while supposedly working for the military governor. Nicely done early Roy Rogers starring effort.\n\n**4060** _ **Southwest Passage**_ **** United Artists, 1954. 75 min. Color. D: Ray Nazarro. SC: Harry Essex. With Rod Cameron, Joanne Dru, John Ireland, Guinn Williams, John Dehner, Darryl Hickman, Stuart Randall, Morris Ankrum, Douglas Fowley, Kenneth MacDonald, Stanley Andrews, Mark Hanna, Hank Patterson. A caravan testing the use of camels in the West is joined by a banker, along with an outlaw and his girl, with the group facing an Indian attack. A different kind of oater providing good entertainment; made in 3-D.\n\n_**Spaghetti Western**_ see _**Cipolla Colt**_\n\n**4061** _ **Spawn of the North**_ **** Paramount, 1938. 110 min. D: Henry Hathaway. SC: Jules Furthman and Talbot Jennings. With George Raft, Dorothy Lamour, Henry Fonda, Akim Tamiroff, Lynne Overman, John Barrymore, Louise Platt, Fuzzy Knight, Vladimir Sokoloff, Duncan Renaldo, John Wray, Michio Ito, Stanley Andrews, Richard Ung, Alex Woloshin, Archie Twitchell, Lee Shumway, Wade Boteler, Galan Galt, Arthur Aylesworth, Rollo Lloyd, Guy Usher, Henry Brandon, Egon Brecher, Harvey Clark, Monte Blue, Irving Bacon, Robert Middlemass, Eddie Marr, Frank Puglia, Leonid Snegoff, Edmund Elton, Aids Kutzenoff, Slicker (seal). Two Alaskan fisher pals have a falling out when one of them joins forces with Russian pirates. Sturdy melodrama in which George Raft out-acts Henry Fonda, but it is sad to see John Barrymore wasted in a supporting role.\n\n**4062** _ **Special Agent**_ **** Paramount, 1949. 70 min. D: William C. Thomas. SC: Milton Raison. With William Eythe, George Reeves, Laura Elliot, Paul Valentine, Carole Mathews, Tom Powers, Raymond Bond, Frank Puglia, Walter Baldwin, Jeff York, Virginia Christine, Robert Williams, Morgan Farley, Joseph Granby, John Hilton, Peter Miles, Jimmy Hunt, Arthur Stone, Truman Bradley (narrator). An unhappy investigator in a cattle town suddenly finds himself in the middle of a crime wave. Fair program feature from the Pine-Thomas unit.\n\n**4063** _ **Speeding Hoofs**_ **** Rayart, 1927. 50 min. D: Louis Chaudet. With Dick Hutton, Elsa Benham, Roy Watson, William Ryno, Bud Osborne, Raymond Turner. Crooks hide treasure on a ranch and then start rumors that the spread is haunted while the rightful owner and her boyfriend search for the riches. Standard low grade silent affair from producer Ben Wilson; also called _**Lure of the Range**_.\n\n**4064** _ **Spencer's Mountain**_ **** Warner Bros., 1963. 118 min. Color. D-SC: Delmer Daves. With Henry Fonda, Maureen O'Hara, James MacArthur, Donald Crisp, Wally Cox, Mimsy Farmer, Virginia Gregg, Lillian Bronson, Whit Bissell, Hayden Rorke, Kathy Bennett, Dub Taylor, Hope Summers, Ken Mayer, Bronwyn Fitzsimmons, Barbara McNair, Larry Mann, Buzz Henry, Jim O'Hara, Victor French, Michael Greene, Med Flory, Ray Savage, Mike Henry, Gary Young, Michael Young, Veronica Cartwright, Ricky Young, Susan Young, Rocky Young, Kym Karath, Michelle Daves, William Breen. A Wyoming mountain couple, with a large family, decide to use the money intended to build their dream home for their eldest son's college education. Folksy, pleasant drama from Earl Hamner, Jr.'s novel, later the basis for the fine TV series \"The Waltons\" (CBS-TV, 1972\u201381).\n\n**4065** _ **The Spikes Gang**_ **** United Artists, 1974. 96 min. Color. D: Richard Fleischer. SC: Irving Ravetch and Harriet Frank, Jr., With Lee Marvin, Gary Grimes, Ron Howard, Charlie Martin Smith, Arthur Hunnicutt, Noah Beery, Marc Smith, Don Fellows, Elliott Sullivan, Robert Beatty, Ralph Brown, Bill Curran, Bert Conway, Frances O'Flynn. Three farm boys nurse a wounded bank robber back to health and he takes them along with him on his robbery exploits. Okay pass time viewing but nothing special.\n\n**4066** _ **Spirit of the Eagle**_ **** Amsell Entertainment, 1991. 93 min. Color. D-SC: Boon Collins. With Dan Haggerty, William Smith, Trever Yarrish, Jeri Arredondo, Taylor Lacher, Ken Carpenter, Don Shanks, Jack Gamby, Paul Booth, Priscilla Bettles, Reed David, Bill Reid, Bruce Sanders, Ivan Sherk, Suzette L. Canonizado. After is small son is kidnapped by a river pirate and sold to Indians, a mountain man, aided by a bear and a golden eagle, sets out to rescue him. Less than average scenic family fare filmed in Oregon.\n\n**4067** _ **The Spirit of the West**_ **** Allied, 1932. 59 min. D: Otto Brower. SC: Jack Natteford. With Hoot Gibson, Doris Hill, Hooper Atchley, Alan Bridge, George Mendoza, Lafe McKee, Walter Perry, Charles Brinley, Tiny Sanford, Gordon DeMain, Lucio Villegas, Hank Bell, Tex Palmer. A rodeo champion helps his brother, the foreman of a ranch whose owner is murdered by a corrupt banker and his sheriff pal for his range. Typical sound era Hoot Gibson vehicle with the star masquerading as a simpleton in order to capture the bad guys.\n\n**4068** _ **Spirit of the Wind**_ **** Raven Pictures, 1980. 103 min. Color. D: Ralph Liddle. SC: Ralph Liddle and John Logue. With Pius Savage, Chief Dan George, Slim Pickens, George Clutesi, Rosa Attla Ambrose, William Ambrose, Rudy Wiehl, Eileen Newman, Louise Ambrose, Phyllis Atta, Curtis Erhart, Ace Robbins, Charlie Peters, Lee Salisburg, Melinda Matsen, John Adams, Leonard Kriska, John Allen. A handicapped young boy tries to fulfill his goals without letting physical problems stop him. Fair outdoor drama.\n\n**4069** _ **Split Second**_ **** RKO Radio, 1953. 81 min. D: Dick Powell. SC: William Bowers and Irving Wallace. With Stephen McNally, Alexis Smith, Jan Sterling, Keith Andes, Arthur Hunnicutt, Paul Kelly, Robert Paige, Richard Egan, Frank DeKova, William Forrest, Nestor Paiva, Dick Crockett, Fred Graham, Nelson Leigh, Clark Howat, Karen Hale, Frank Marlowe, John Diggs, John Cliff, Benny Burt, David McMahon, Fred Alrich, Alex Sharp, Bill Wallace. An escaped killer holds hostages in a Nevada ghost town knowing the area is about to be hit by an A-bomb test. Well acted suspense drama.\n\n**4070** _ **The Spoilers**_ **** Selig-Poliscope, 1914. 90 min. D-SC: Colin Campbell. With William Farnum, Kathlyn Williams, Bessie Eyton, Tom Santschi, Frank Clark, Jack McDonald, Wheeler Oakman, Norvel MacGregor, William Ryno. Two gold prospectors in the Klondike are cheated out of their valuable claim by a crooked politician. The first, and still best, of the five versions of the Rex Beach novel, famous for its reel long fight between William Farnum and Tom Santschi; a true cinema classic. A second version was done in 1923 by Goldwyn with Lambert Hillyer directing Milton Sills, Anna Q. Nilsson, Noah Beery and Barbara Bedford in the lead roles.\n\n**4071** _ **The Spoilers**_ **** Paramount, 1930. 81 min. D: Edwin Carewe. SC: Agnes Brand Leahy and Bartlett Cormack. With Gary Cooper, Kay Johnson, Betty Compson, William \"Stage\" Boyd, Harry Green, James Kirkwood, Slim Summerville, Lloyd Ingraham, Oscar Apfel, Edward Coxen, Jack Trent, Edward Hearn, Hal David, Knute Erickson, John Beck, Jack N. Holmes. In the Klondike gold rush days a man romances two women while he and his partners fight claim jumpers. Initial sound version of Rex Beach's novel is a fairly good screen adaptation. Gary Cooper replaced George Bancroft in the lead and the stars of the 1914 version, William Farnum and Tom Santschi, were technical advisors for the restaging of the big fight scene. Ironically Farnum and Santschi did a better job redoing the brawl the next year in the non-Western _**Ten Nights in a Bar Room**_ (Road Show Productions, 1931).\n\n**4072** _ **The Spoilers**_ **** Universal, 1942. 87 min. D: Ray Enright. SC: Lawrence Hazard and Tom Reed. With Marlene Dietrich, Randolph Scott, John Wayne, Margaret Lindsay, Harry Carey, Richard Barthelmess, George Cleveland, Samuel S. Hinds, Russell Simpson, William Farnum, Marietta Canty, Jack Norton, Ray Bennett, Forrest Taylor, Charles Halton, Bud Osborne, Drew Demarest, Robert W. Service, Glenn Strange, Richard Cramer, Earle Hodgins, Lloyd Ingraham, Harry Woods, William Gould, Harry Cording, Charles McMurphy, Art Miles, William Haade, Robert Homans, Irving Bacon, Bob McKenzie, Emmett Lynn, Frank Austin, Chester Clute, Willie Fung, Mickey Simpson, Paul Newlan, Duke York, Robert Barron, Ben Taggart, Dick Rush, Kitty O'Neill. Two prospectors are at odds with a corrupt gold commissioner in the Klondike in the 1890s and they are helped by an exotic saloon entertainer. The slickest of the five screen versions of the Rex Beach book with Harry Carey and Richard Barthelmess stealing the acting honors; famous poet Robert W. Service is seen reciting some of his work.\n\n**4073** _ **The Spoilers**_ **** Universal-International, 1955. 84 min. Color. D: Jesse Hibbs. SC: Oscar Brodney and Charles Hoffman. With Anne Baxter, Jeff Chandler, Rory Calhoun, Ray Danton, Barbara Britton, John McIntire, Wallace Ford, Carl Benton Reid, Raymond Walburn, Ruth Donnelly, Willis Bouchey, Forrest Lewis, Roy Barcroft, Robert Foulk, Dayton Lummis, John Harmon, Paul McGuire, Frank Sully, Bob Steele, Byron Foulger, Arthur Space, Lane Bradford, Terry Frost, Harry Seymour, Eddie Parker, Lee Roberts, John Close, Joe Haworth, John McKee, Billy Wayne, Henry Rowland, Charles Morton, John Phillips, Henry Wills, Harry Tenbrook, Richard Alexander, Heinie Conklin, Holly Bane, Emil Sitka, William Fawcett, Donald Kerr, Frank Chase, John Epper, Chuck Hamilton, Jack Kenny, Mike Lally, Joseph Mell, Sailor Vincent, Harry Wilson, William Yip, Tim Graham, Frank Mills, Robert Strong, Patsi Donahue, Peggy Gordon, Lucille Lamarr, Ila McAvoy, Patti McKay. A woman saloon owner vies for the attentions of a Klondike prospector with a respectable woman who works for the crook trying to cheat the man and his partner out of their gold claim. Color is the only asset to this final screen version of the Rex Beach work; mediocre.\n\n**Randolph Scott, John Wayne and Marlene Dietrich in** _**The Spoilers**_ **(Universal, 1942).**\n\n**  \n**\n\n**4074** _ **Spoilers of the Forest**_ **** Republic, 1957. 70 min. Color. D: Joe (Joseph) Kane. SC: Bruce Manning. With Rod Cameron, Vera Ralston, Ray Collins, Hillary Brooke, Edgar Buchanan, Carl Benton Reid, Sheila Bromley, Hank Worden, John Compton, Angela Greene, Paul Stader, Mary Alan Hokanson, Raymond Greenleaf, Eleanor Audley, Don Haggerty, William Haade, Jo Ann Lilliquist, Bucko Stafford, Robert Karns, Ken Dibbs, Rory Mallinson, Virginia Carroll, John Patrick, Bob Swan, Mack Williams, Theresa Harris, Helen Wallace, Pauline Moore, Judd Holdren. A lumber company owner uses his handsome foreman to woo a woman so he can obtain timber from her Montana ranch. Cheaply made but entertaining.\n\n**4075** _ **Spoilers of the North**_ **** Republic, 1947. 66 min. D: Richard Sale. SC: Milton Raison. With Paul Kelly, Adrian Booth, Evelyn Ankers, James Millican, Roy Barcroft, Louis Jean Heydt, Ted Hecht, Francis McDonald, Neyle Morrow, Maurice Cass, Harlan Briggs. A crooked salmon tycoon and his Indian girl friend enlist the help of an unsuspecting city woman in their schemes to defraud fishermen. Standard, entertaining Republic program picture.\n\n**4076** _ **Spoilers of the Plains**_ **** Republic, 1951. 68 min. D: William Witney. SC: Sloan Nibley. With Roy Rogers, Penny Edwards, Gordon Jones, Foy Willing and The Riders of the Purple Sage, Grant Withers, Fred Kohler, Jr., William Forrest, Don Haggerty, House Peters, Jr., George Meeker, Keith Richards, Rex Lease, James Craven, Lee Shumway, John Daheim, Phyllis Kennedy. The foreman of an oil supply company suspects a rival firm has planted spies in his operation. There is lots of action in this fast moving Roy Rogers film.\n\n**4077** _ **Spoilers of the Range**_ **** Columbia, 1939. 58 min. D: C.C. Coleman, Jr. SC: Paul Franklin. With Charles Starrett, Iris Meredith, The Sons of the Pioneers (Bob Nolan, Tim Spencer, Lloyd Perryman, Pat Brady, Hugh Farr, Karl Farr), Dick Curtis, Kenneth MacDonald, Hank Bell, Ed Le Saint, Forbes Murray, Art Mix, Edmund Cobb, Ed Peil, Sr., Horace B. Carpenter, Charles Brinley, Carl Sepulveda, Ethan Laidlaw, Joe Weaver. Crooks try to keep ranchers' cattle from going to market so they cannot repay a loan that will save their spreads. Average Charles Starrett vehicle.\n\n**4078** _ **Spook Town**_ **** Producers Releasing Corporation, 1944. 59 min. D-SC: Elmer Clifton. With Dave O'Brien, Jim Newill, Guy Wilkerson, Mady Lawrence, Dick Curtis, Harry Harvey, Ed Cassidy, Charles King, Robert Barron, Richard Alexander, John Cason, Bert Dillard, Kermit Maynard, Chick Hannon, John Elliott, Jack Tornek. A ranger captain is forced to resign when money entrusted to him by ranchers is stolen but a trio of his comrades search for the real thief. The mystery element adds some flavor to this otherwise mundane \"Texas Rangers\" series film.\n\n**4079** _ **Springfield Incident**_ **** CBS-TV\/20th Century\u2013Fox, 1957. 45 min. With Ann Harding, Tom Tryon, Marshall Thompson, Alan Hale, Craig Hill, Walter Coy, Helen Wescott, Carl Benton Reid, Lloyd Corrigan, Kathleen Case, Frank Sully, Ray Teal, Alex Gerry, John Conte (host). In frontier Illinois, lawyer Abraham Lincoln defends a widow's two sons accused of murder. Adequate television adaptation of _**Young Mr. Lincoln**_ (q.v.), originally telecast February 6, 1957, as \"Young Man from Kentucky\" on \"The 20th Century\u2013Fox Hour\" (CBS-TV, 1955\u201357).\n\n**4080** _ **Springfield Rifle**_ **** Warner Bros., 1952. 93 min. Color. D: Andre De Toth. SC: Charles Marquis Warren and Frank Davis. With Gary Cooper, Phyllis Thaxter, David Brian, Paul Kelly, Philip Carey, Lon Chaney, James Millican, Martin Milner, Guinn Williams, James Brown, Jack Woody, Alan Hale, Vince Barnett, Fess Parker, Richard Lightner, Ewing Mitchell, Poodles Hanneford, George Ross, Eric Hoeg, Wilton Graff, Ned Young, William Fawcett, Richard Hale, Ben Corbett, Guy E. (Edward) Hearn, George Eldredge, Ralph Sanford, Rory Mallinson, Ric Roman, Jack Mower, Mike Ragan (Holly Bane), Michael Chapin, Ray Bennett, Paula Souel, Richard Benjamin. Renegades rustle horses intended for the Union cause and an Army officer is sent to solve the problem. Slightly better than average Civil War Western yarn.\n\n**4081** _ **Springtime in Texas**_ **** Monogram, 1945. 55 min. D: Oliver Drake. SC: Frances Kavanaugh. With Jimmy Wakely, Dennis Moore, Lee \"Lasses\" White, Marie Harmon, Rex Lease, The Callahan Brothers and Their Blue Ridge Mountain Folks, Pearl Early, Horace Murphy, I. Stanford Jolley, Hal Taliaferro, Budd Buster, Roy Butler, Ted French, Johnny Bond, Frankie Marvin, Lloyd Ingraham, Pat Patterson, Rusty McDonald, Spud Goodall, Robert Barron, Bob Duncan, Chick Hannon. Three pals are suspected of murdering one of the candidates for mayor of a small town. Jimmy Wakely's second starring film is not much but it does provide a chance to see country music veterans The Callahan Brothers.\n\n**4082** _ **Springtime in the Rockies**_ **** Republic, 1937. 60 min. D: Joseph Kane. SC: Betty Burbridge and Gilbert Wright. With Gene Autry, Smiley Burnette, Polly Rowles, Ula Love, Ruth Bacon, Jane Hunt, George Chesebro, Lew Meehan, Edmund Cobb, Jack Rockwell, Alan Bridge, Tom London, Edward Hearn, Frankie Marvin, William Hole, Fred Burns, Art Davis, Jack Kirk, Frank Ellis, George (Montgomery) Letz, Oscar Gahan, Jim Corey, Robert Dudley, Victor Cox, Jimmy LeFeur and His Saddle Pals. Singer Gene Autry gets involved in a range feud between cattle ranchers and sheepherders by telling a female rancher that her place is rundown. Entertaining Gene Autry outing enhanced by the title song and \"You're the Only Star in My Blue Heaven.\"\n\n**4083** _ **Springtime in the Sierras**_ **** Republic, 1947. 75 min. D: William Witney. SC: Sloan Nibley. With Roy Rogers, Andy Devine, Jane Frazee, Bob Nolan and The Sons of the Pioneers (Pat Brady, Shug Fisher, Hugh Farr, Karl Farr), Roy Barcroft, Stephanie Bachelor, Hal Landon, Harry V. Cheshire, Chester Conklin, Hank Patterson, Bob Woodward, Whitey Christy, Pascale Perry, Buck Moulton, Milton Kibbee, Frank Dae. Roy Rogers gets on the trail of professional hunters who are killing outlawed game after they murder his long time friend, a game warden. Only so-so Roy Rogers entry highlighted by Stephanie Bachelor as the villain and her fight with Jane Frazee at the finale.\n\n**4084** _ **Spurs**_ **** Universal, 1930. 59 min. D-SC: B. Reeves Eason. With Hoot Gibson, Helen Wright, Buddy Hunter, Pee Wee Holmes, Robert Homans, Frank Clark, William Bertram, Philo McCullough, Pete Morrison, Art Ardigan, Cap Anderson. A cowboy teams with a young boy to track an outlaw gang while the cowpoke vies for a large purse and silver spurs in a rodeo and romances a pretty girl. Exciting and well done early Hoot Gibson sound feature, one sure to appeal to his fans.\n\n**4085** _ **Square Dance Jubilee**_ **** Lippert, 1949. 80 min. D: Paul Landres. SC: Ron Ormond and Dan Ullman. With Don Barry, Wally Vernon, Mary Beth Hughes, Max Terhune, Thurston Hall, Britt Wood, Spade Cooley and His Band, John Eldredge, Marshall Reed, Tom Tyler, Tom Kennedy, Chester Clute, Clarke Stevens, Lee Roberts, Slim Gault, Cliff Taylor, Ralph Moody, Hazel Nilsen, Alex Montoya, Hal King, Lloyd \"Cowboy\" Copas, Johnny Downs, The Broome Brothers, Smiley and Kitty, Herman the Hermit, Ray Vaughn, The Tumbleweed Tumblers, The Elder Lovelies, Claude Casey, Buddy McDowell, Dana Gibson, Dot Remly. A TV promoter sends two scouts West to find authentic talent for his program and they run across a gang trying to cheat a woman out of her ranch. Interesting curio full of old time country music acts with Don Barry even singing a song; worth a look.\n\n**4086** _ **Square Dance Katy**_ **** Monogram, 1950. 76 min. D: Jean Yarborough. SC: Warren Wilson. With Vera Vague (Barbara Jo Allen), Jimmie Davis and His Sunshine Band, Phil Brito, Virginia Welles, Warren Douglas, Sheila Ryan, Dorothy Vaughan, Harry V. Cheshire, Fenton Jones, Russell Hicks, Ray Walker, William Forrest, Tristram Coffin, Jon Riffel, Warren Jackson, Donald Kerr, Paul Bryar, Earle Hodgins, Frank Sully, Stanley Blystone, Lee Phelps, Edward Gargan, Joseph Crehan. A woman promoting her singer boyfriend ends up a TV star but is eventually able to bring her man to stardom. Standard bucolic musical comedy with Western tinges.\n\n**4087** _ **The Square Deal Man**_ **** Triangle, 1917. 45 min. D: William S. Hart. SC: J.G. Hawks. With William S. Hart, Mary McIvor, Joseph J. Dowling, Mary Jane Irving, J. Frank Burke, Darrell Foss, Thomas Kirihara, Milton Ross, Charles O. Rush. An honest gambler wins a ranch from a man who is murdered and the victim's daughter is made to think the gambling man killed her father when the culprit is his rival for her affections. Entertaining William S. Hart silent feature.\n\n**4088** _ **Square Deal Sanderson**_ **** Paramount-Artcraft, 1919. 60 min. D: William S. Hart and Lambert Hillyer. SC: Lambert Hillyer. With William S. Hart, Ann Little, Lloyd Bacon, Frank Whitson, Andrew Robson, Edwin Wallach. A man finds the bodies of two murder victims and learns from a letter that one of them is the long lost brother of a girl who is being persecuted by a rejected suitor, so he sets out to defend her and she mistakes him for her sibling. Melodramatic William S. Hart silent effort; his fans will enjoy it.\n\n**4089** _ **Square Shooter**_ **** Columbia, 1935. 57 min. D: David Selman. SC: Harold Shumate. With Tim McCoy, Jacqueline Wells (Julie Bishop), Wheeler Oakman, J. Farrell MacDonald, Charles Middleton, John Darrow, Erville Alderson, Steve Clark, William V. Mong, Eddy Chandler, Ernie Adams, Bud Osborne, Art Mix, Jack Evans, Roy Bucko, Buck Bucko. Returning home after five years of being falsely imprisoned for his uncle's murder, a cowboy intends to find the real killers. Another finely written, directed and acted Tim McCoy vehicle.\n\n_**The Square Shooter**_ (1951) see _**Skipalong Rosenbloom**_\n\n**4090** _ **Squares**_ **** CBS-TV, 1972. 92 min. Color. D: Patrick J. Murphy. SC: Mary Ann Saxon. With Andrew Prine, Gilmer McCormick, Robert Easton, Harriet Medin, Jack Mather, Dean Smith, Tom Hennessy, Tom Basham, William Wintersole, Patty Sauers, San Christopher. A down on his luck rodeo rider becomes involved with a college coed dropout. Tepid modern-day TV drama.\n\n**4091** _ **The Squaw Man**_ **** Jesse L. Lasky Feature Play Co., 1914. 74 min. D-SC: Cecil B. DeMille and Oscar Apfel. With Dustin Farnum, Winifred Kingston, Monroe Salisbury, Red Wing (Lillian St. Cyr), Billy Elmer, Dick La Strange, Foster Knox, Joe E. Singleton, Dick La Reno, Fred Montague, Baby de Rue, Mrs. A.W. Filson, Haidee Fuller, Art Acord. Falsely accused of embezzlement, an English officer becomes a rancher and marries the Indian maiden who saves his life when a bad man tries to kill him. Cecil B. DeMille filmed the Edwin Milton Royle play again in 1918 and 1931 (qq.v.) but this well staged initial version is worth a look.\n\n**4092** _ **The Squaw Man**_ **** Famous Players-Lasky\/Paramount, 1918. 60 min. D: Cecil B. DeMille. SC: Edwin Milton Royle. With Elliott Dexter, Ann Little, Katherine MacDonald, Theodore Roberts, Jack Holt, Thurston Hall, Tully Marshall, Herbert Standing, Edwin Stevens, Helen Dunbar, Winter Hall, Julia Faye, Noah Beery, Pat Moore, Jim Mason, Monte Blue, William Brunton, Charles Ogle, Guy Oliver, Jack Herbert, M. Hallward, Clarence Geldert. After becoming a rancher, an Englishman marries an Indian girl who saves his life when he is threatened by a rival. Director Cecil B. DeMille's successful remake of his 1914 (q.v.) feature; only the last reel has survived.\n\n**4093** _ **The Squaw Man**_ **** Metro-Goldwyn-Mayer, 1931. 106 min. D: Cecil B. DeMille. SC: Lucien Hubbard, Lenore Coffee and Elsie Janis. With Warner Baxter, Eleanor Boardman, Paul Cavanagh, Lawrence Grant, Roland Young, Charles Bickford, Desmond Roberts, Mitchell Lewis, Luke Cosgrove, J. Farrell MacDonald, DeWitt Jennings, Frank Rice, Raymond Hatton, Frank Hagney, Victor Potel, Dickie Moore, Harry Northrup, Julia Faye, Eva Dennison, Ed Brady, Lillian Bond. After being disinherited, an Englishman moves to Wyoming where he establishes a cattle empire and marries an Indian maiden who has his child. Early sound version of Edwin Milton Royle's famous play is a bit creaky but genre fans will still want to watch it, especially for Warner Baxter's fine work in the title role. This is the third screen version of the story by director Cecil B. DeMille, who also made it in 1914 and 1918 (qq.v.).\n\n**4094** _ **Stacked Cards**_ **** Circle Productions\/Fred J. Balshofer, 1926. 55 min. D: Robert Eddy. SC: Guy C. Cleveland and William De Geiger. With Fred Church, Kathryn McGuire, Robert Thurston, John Watson, Artie Ortego. A dishonest ranch foreman tries to cheat a young woman out of her rightful inheritance but a cowboy comes to the rescue. Low grade, but action filled, silent quickie.\n\n**4095** _ **Stage to Blue River**_ **** Monogram, 1951. 55 min. D: Lewis D. Collins. SC: Joseph Poland. With Whip Wilson, Fuzzy Knight, Phyllis Coates, Lee Roberts, Lane Bradford, Pierce Lyden, John Hart, Terry Frost, I. Stanford Jolley, William Fawcett, Steve Clark, Stanley Price, Bud Osborne, Boyd Stockman. U.S. marshals try to help a woman whose stage line is coveted by a crook and his lawman henchman. Average Whip Wilson affair.\n\n**4096** _ **Stage to Chino**_ **** RKO Radio, 1940. 59 min. D: Edward Killy. SC: Morton Grant and Arthur V. Jones. With George O'Brien, Virginia Vale, Hobart Cavanaugh, Roy Barcroft, William Haade, Carl Stockdale, Glenn Strange, Harry Cording, Martin Garralaga, Ethan Laidlaw, Tom London, Billy Benedict, John Dilson, Bob Burns, Frank Ellis, Hank Bell, Jack O'Shea, The Pals of the Golden West. A postal inspector stops a stagecoach holdup and takes the job as driver for the woman owner of the line in order to investigate a series of robberies. Very good George O'Brien feature.\n\n**4097** _ **Stage to Mesa City**_ **** Eagle Lion, 1947. 52 min. D: Ray Taylor. SC: Joseph Poland. With Lash LaRue, Al St. John, Jennifer Holt, George Chesebro, Brad Slaven, Marshall Reed, Terry Frost, Carl Mathews, Bob Woodward, Steve Clark, Frank Ellis, Lee Morgan, Wally West, Russell Arms, Dee Cooper. U.S. marshals Cheyenne Davis and Fuzzy Q. Jones are sent to Mesa City to investigate a stage line being harassed by bandits. Very good Lash LaRue film, fast moving with realistic fight sequences.\n\n**4098** _ **Stage to Thunder Rock**_ **** Paramount, 1964. 82 min. Color. D: William F. Claxton. SC: Charles Wallace. With Barry Sullivan, Marilyn Maxwell, Scott Brady, Lon Chaney, Anne Seymour, John Agar, Keenan Wynn, Wanda Hendrix, Ralph Taeger, Allan Jones, Laurel Goodwin, Robert Strauss, Robert Lowery, Rex Bell, Jr., Argentina Brunetti, Suzanne Cupito, Paul E. Burns, Wayne Peters, Roy Jenson. A lawman arrives at a stagecoach station with a prisoner and learns the man's father is planning to rescue his son and kill the sheriff. The best of producer A.C. Lyles' Westerns for Paramount in the 1960s, well made with a good cast and an especially fine performance by Lon Chaney as the drunken way station owner.\n\n**4099** _ **Stage to Tucson**_ **** Columbia, 1951. 82 min. Color. D: Ralph Moody. SC: Bob Williams, Frank Burt and Robert Libott. With Rod Cameron, Wayne Morris, Kay Buckley, Sally Eilers, Carl Benton Reid, Roy Roberts, Harry Bellaver, Douglas Fowley, John Pickard, Olin Howlin, Boyd Stockman, John Sheehan, Reed Howes, James Kirkwood, Stanley Andrews, John Cason, Francis McDonald, Guy Wilkerson, Frank Moran, Charles Evans, Joe Dominguez, Fred Essler, Paul E. Burns, Hank Mann, Rusty Wescoatt, Bob Woodward, Frank O'Connor, Frank Hagney, George Magrill, Edward Clark, Cactus Mack, Roy Bucko. The government sends two agents to the Southwest to find out about numerous stagecoach hijackings and they learn secessionists are causing the problem. Action filled drama highlighted by the work of its two likable stars.\n**4100** _ **Stagecoach**_ **** United Artists, 1939. 96 min. D: John Ford. SC: Dudley Nichols. With Claire Trevor, John Wayne, Thomas Mitchell, Andy Devine, George Bancroft, John Carradine, Donald Meek, Louise Platt, Tim Holt, Berton Churchill, Tom Tyler, Chris-Pin Martin, Francis Ford, Elvira Rios, Yakima Canutt, Chief Big Tree, Harry Tenbrook, Jack Pennick, Paul McVey, Walter McGrail, Brenda Fowler, Florence Lake, Cornelius Keefe, Vester Pegg, Bryant Washburn, Nora Cecil, Bill Cody, Buddy Roosevelt, Chief White Horse, Duke R. Lee, Mary Kathleen Walker, Helen Gibson, Dorothy Appleby, Joe Rickson. An assorted group of passengers on a stagecoach bound for Lordsburg learn they are in the path of Geronimo's warring apaches. One of the all time great classic Westerns and a must-see for genre followers. The entire cast is superb, especially Andy Devine as the stage driver, John Carradine's gambler, Louise Platt as the pregnant passenger and Tom Tyler as Luke Plummer. Excellent.\n\n**4101** _ **Stagecoach**_ **** 20th Century\u2013Fox, 1966. 114 min. Color. D: Gordon Douglas. SC: Joseph Landon. With Alex Cord, Ann-Margret, Red Buttons, Michael (Mike) Connors, Bing Crosby, Bob (Robert) Cummings, Van Heflin, Slim Pickens, Stefanie Powers, Keenan Wynn, Brad Weston, Joseph Hoover, Oliver McGowan, David Humphreys Miller, Bruce Mars, Edwin Mills, Hal Lynch, Norman Rockwell, Muriel Davidson, Brett Pearson, John Gabriel. A saloon gal, a gambler, a drunken doctor, a pregnant woman and a wanted outlaw are among the passengers on a stagecoach heading into Indian country. Bland remake of the 1939 (q.v.) feature; its only compensation is fine performances by Van Heflin as the stage driver and Slim Pickens as his shotgun rider.\n\n**4102** _ **Stagecoach**_ **** CBS-TV, 1986. 104 min. Color. D: Ted Post. SC: James Lee Barrett. With Willie Nelson, Kris Kristofferson, Johnny Cash, Waylon Jennings, Elizabeth Ashley, Mary Crosby, John Schneider, Anthony Franciosa, Anthony Newley, Alex Kubic, June Carter Cash, Merritt Butrick, John Carter Cash, Jessi Colter, David Allan Coe, Lash LaRue, Bob McLean, Anthony Russell, Joe Unger, Kal Roberts, Ed Adams, Michael Hayes, Billy Swan, Sonny Carl Davis, Glen Clark, Tim Gilbert, Dave Adams, Norman Stone, Jack Dunlap, Bob Cota. An assorted group of passengers take a stagecoach to the town of Lordsburg although threatened by Geronimo and his rampaging braves. A bit better than the 1966 (q.v.) version of the Ernest Haycox story, this outing is helped by Gary Graver's fine photography and Waylon Jennings as gambler Hatfield; Lash LaRue has a nice cameo as an outlaw turncoat.\n\n**4103** _ **Stagecoach Buckaroo**_ **** Universal, 1942. 58 min. D: Ray Taylor. SC: Al Martin. With Johnny Mack Brown, Fuzzy Knight, Nell O'Day, Anne Nagel, Herbert Rawlinson, Glenn Strange, Ernie Adams, Henry Hall, Lloyd Ingraham, Kermit Maynard, Frank Brownlee, Jack C. Smith, Harry Tenbrook, Frank Ellis, Blackie Whiteford, Hank Bell, Ray Jones, Jim Corey, Bill Nestell, Carl Sepulveda, The Guardsmen. A woman tries to carry on her father's stagecoach operation after he is killed by outlaws and hires two men to help her. Action laden Johnny Mack Brown affair with the unique plot device of using a bulletproof stagecoach to thwart holdups.\n\n**4104** _ **Stagecoach Days**_ **** Columbia, 1938. 58 min. D: Joseph Levering. SC: Nate Gatzert. With Jack Luden, Eleanor Stewart, Hal Taliaferro, Harry Woods, Slim Whitaker, Jack Ingram, Lafe McKee, Robert Kortman, Richard Botiller, Blackjack Ward, Tom London, Buzz Barton, Ernie Adams, Herman Hack, Jim Mason, Hal Price, Bob Burns, Oscar Gahan, Chick Hannon, Tex Palmer, Jack C. Smith, George Plues, Tuffy (dog). A cowboy helps a woman and her father acquire a government mail contract for their stage line. Crude, low grade, but passable, Jack Luden vehicle with good work by Hal Taliaferro (Wally Wales) as the father.\n\n**4105** _ **Stagecoach Driver**_ **** Monogram, 1951. 52 min. D: Lewis D. Collins. SC: Joseph O'Donnell. With Whip Wilson, Fuzzy Knight, Jim Bannon, Gloria Winters, Lane Bradford, Marshall Reed, Barbara Allen, Leonard Penn, John Hart, Stanley Price, George DeNormand. A star packer and his pals try to stop the lawlessness caused when the telegraph begins putting the Pony Express and freight lines out of business. Pleasant effort in the Whip Wilson series.\n\n**4106** _ **Stagecoach Express**_ **** Republic, 1942. 56 min. D: George Sherman. SC: Doris Schroeder. With Don \"Red\" Barry, Lynn Merrick, Charles King, Al St. John, Robert Kent, Emmett Lynn, Guy Kingsford, Ethan Laidlaw, Eddie Dean, Wheaton Chambers, Eddie Phillips, Mary MacLaren, Frank O'Connor, Freddie Steele, Tommy Coats, Francis Sayles, Bill Nestell, Al Taylor, Marty Faust, Cyclone (horse), Duke (dog). A man and his pal agree to help a young woman by driving her stagecoach which has been attacked by bandits. Okay but slow Don Barry film that includes several well staged riding and chase sequences involving holdups.\n\n**4107** _ **Stagecoach Kid**_ **** RKO Radio, 1949. 60 min. D: Lew Landers. SC: Norman Houston. With Tim Holt, Richard Martin, Jeff Donnell, Joseph Sawyer, Thurston Hall, Carol Hughes, Robert Bray, Robert B. Williams, Kenneth MacDonald, Harry Harvey. Outlaws plan to murder a wealthy man and kidnap his daughter but their plot is foiled by a stage line owner. Another good Tim Holt Western.\n\n**4108** _ **Stagecoach Outlaws**_ **** Producers Releasing Corporation, 1945. 58 min. D: Sam Newfield. SC: Fred Myton. With Buster Crabbe, Al St. John, Frances Gladwin, Ed Cassidy, Kermit Maynard, I. Stanford Jolley, Steve Clark, Robert Kortman, Bob (John) Cason, George Chesebro, Hank Bell, Wally West, Victor Cox, Herman Hack, Roy Bucko, Jimmy Aubrey, Frank McCarroll, Tex Cooper, George Morrell, Jack Evans, Rube Dalroy, Rose Plummer. Marshal Billy Carson pretends to be a wanted man to stop a gang from destroying a stage operation. Typically cheap, but fun, \"Billy Carson\" outing.\n\n**4109** _ **Stagecoach to Dancer's Rock**_ **** Universal-International, 1962. 72 min. D: Earl Bellamy. SC: Kenneth Darling. With Warren Stevens, Jody Lawrence, Martin Landau, Judy Dan, Del Moore, Don Wilbanks, Bob Anderson, Rand Brooks, Gene Roth, Charles Tannen, Mike Ragan (Holly Bane), Mauritz Hugo, Tim Bolton. When a stage driver finds one of his passengers has smallpox he leaves all of them stranded in the desert. Nothing special but okay viewing.\n\n**4110** _ **Stagecoach to Denver**_ **** Republic, 1946. 56 min. D: R.G. Springsteen. SC: Earle Snell. With Allan Lane, Bobby Blake, Martha Wentworth, Peggy Stewart, Roy Barcroft, Emmett Lynn, Ted Adams, Edmund Cobb, Tom Chatterton, Bobby Hyatt, George Chesebro, Ed Cassidy, Wheaton Chambers, Forrest Taylor, Britt Wood, Tom London, Stanley Price, Frank O'Connor, Marin Sais, Budd Buster, Lew Morphy, Herman Hack, Chuck Baldra, Chick Hannon, Cactus Mack. Wanting a woman's property, a supposedly good citizen has her kidnapped and then murders a local official, putting his henchman in his place. Fairly action filled \"Red Ryder\" adventure.\n\n**4111** _ **Stagecoach to Fury**_ **** 20th Century\u2013Fox, 1956. 76 min. D: William F. Claxton. SC: Eric Norden. With Forrest Tucker, Mari Blanchard, Wallace Ford, Rodolfo Hoyos, Paul Fix, Rico Alaniz, Wright King, Margia Dean, Ian MacDonald, Steven Geray, Ellen Corby, Paul Fierro, Leslie Banning, Rayford Barnes, Robert Karnes, Norman Leavitt, Alex Montoya, William \"Bill\" Phillips. Passengers aboard a stage are held hostage by Mexican bandits who await the arrival of the next coach in order to steal its gold shipment. Dreary feature with too much talk.\n\n_**Stagecoach Run**_ see _**Winds of the Wasteland**_\n\n**4112** _ **Stagecoach War**_ **** Paramount, 1940. 63 min. D: Lesley Selander. SC: Norman Houston. With William Boyd, Russell Hayden, Julie Carter, Harvey Stephens, J. Farrell MacDonald, Rad Robinson, Eddy Waller, Frank Lackteen, Jack Rockwell, Eddie Dean, Robert Kortman, The King's Men (Ken Darby, Bud Linn, Jon Dodson), Rod Cameron, Johnny Luther, Frank Ellis, Merrill McCormick, Hank Bell, Denver Dixon, George Morrell, Victor Cox, Tex Palmer, George Sowards. Hopalong Cassidy and the Bar 20 men find themselves in the middle of a contract war between two stagecoach lines. Good outing, made when the \"Hopalong Cassidy\" series was near its peak.\n\n**4113** _ **The Stalking Moon**_ **** National General, 1968. 109 min. Color. D: Robert Mulligan. SC: Wendell Mayes and Alvin Sargent. With Gregory Peck, Eva Maria Saint, Robert Forster, Nolan Clay, Russell Thorson, Frank Silvera, Lonny Chapman, Lou Frizzell, Henry Beckman, Charles Tyner, Richard Bull, Sandy Wyeth, Joaquin Martinez, Boyd \"Red\" Morgan. An Apache warrior comes in search of the white woman and his son taken from him by a man who has settled with them on his New Mexico ranch. Well produced oater is basically dull due to lack of suspense.\n\n**4114** _ **Stallion Canyon**_ **** Astor, 1949. 72 min. Color. D: Harry Fraser. SC: Hy Heath. With Ken Curtis, Carolina Cotton, Shug Fisher, Forrest Taylor, Ted Adams, Billy Hammond, Roy Butler. A cowboy tries to help an Indian framed on a murder charge in addition to attempting to win the big purse at an annual race. Low grade program feature.\n\n**4115** _ **Stallion Road**_ **** Warner Bros., 1947. 97 min. D: James V. Kern. SC: Stephen Longstreet. With Ronald Reagan, Alexis Smith, Zachary Scott, Peggy Knudsen, Patti Brady, Harry Davenport, Angela Greene, Frank Puglia, Ralph Byrd, Lloyd Corrigan, Fernando Alvarado, Matthew Boulton, Mary Gordon, Nina Campana, Dewey Robinson, Paul Panzer, Bobby Valentine, Ralph Littlefield, Tom Wilson, Oscar O'Shea, Leon Lenoir, Monte Blue, Fred Kelsey, Major Sam Harris, Joan Winfield, Danny Dowling, Douglas Kennedy, Creighton Hale, Elaine Lange, Roxanne Stark, Vera Lewis. A veterinarian and his novelist pal both fall in love with a woman who breeds horses but the doctor almost loses her when the herd contracts anthrax. Only fair modern-day melodrama; not even un-credited direction by Raoul Walsh helps.\n\n**4116** _ **The Stampede**_ **** Victor Kremer Films, 1921. 50 min. D: Francis Ford. SC: Kingsley Benedict and Eugenie Kremer. With Texas Guinan, Francis Ford, Frederick Moore, Jean Carpenter, Vale Rio, Fred Kohler, Cecil McLean, Kingsley Benedict, Snowflake (horse). A woman, in love with a cowboy who does not return her affections, tries to claim a section of government land only to be opposed by crooks who want it for themselves. This action filled silent gives viewers a chance to see famous night club hostess Texas Guinan in one of her several starring Westerns.\n\n**4117** _ **Stampede**_ **** Columbia, 1936. 58 min. D: Ford Beebe. SC: Robert Watson. With Charles Starrett, Finis Barton, J.P. McGowan, LeStrange Millman, Reginald Hincks, James McGrath, Arthur Kerr, Jack Atkinson, Michael Heppell, Ted Mapes. A rancher wants the spread of a rival and kills a buyer for the man's cattle but the victim's brother arrives on the scene looking for the murderer. Mediocre Charles Starrett vehicle, not up to the usual standard of his Columbia series.\n\n**4118** _ **Stampede**_ **** Allied Artists, 1949. 78 min. D: Lesley Selander. SC: John C. Champion and Blake Edwards. With Rod Cameron, Gale Storm, Don Castle, Johnny Mack Brown, Don Curtis, John Eldredge, John Miljan, Jonathan Hale, James Harrison, Ted Elliott, Jack Parker, Chuck Roberson, Tim Ryan, Kenne Duncan, Carol Henry, Adrian Wood, I. Stanford Jolley, Marshall Reed, Philo McCullough, Charles King, Duke York, Wes Christensen, Bud Osborne, Henry Hall, Boyd Stockman. Two feuding cattlemen brothers become involved with settlers who are being cheated out of their water rights. This nicely made compact oater moves along at a fast clip.\n\n**4119** _ **Stampede at Bitter Creek**_ **** Buena Vista, 1966. 81 min. Color. D: Harry Keller. SC: D.P. Harmon. With Tom Tryon, Stephen McNally, Sidney Blackmer, Bill Williams, John Larch, Harold J. Stone, Norma Moore, Grant Williams, H.M. Wynant, Don Kelly. After marrying the girl he loves, a Texas Ranger finds himself caught between Indians on the warpath and outlaws, ending up a wanted man. Action filled Walt Disney Western first shown on his TV program on March 6, 1959, as \"The Man from Bitter Creek\" segment of the \"Texas John Slaughter\" miniseries.\n\n**4120** _ **The Stand at Apache River**_ **** Universal-International, 1953. 77 min. Color. D: Lee Sholem. SC: Arthur Ross. With Stephen McNally, Julia (Julie) Adams, Hugh Marlowe, Jaclynne Greene, Hugh O'Brian, Russell Johnson, Jack Kelly, Edgar Barrier, Forrest Lewis, Frankie Darro, Henry Wills. Eight people are stranded at a way station with Apaches about to attack. Mediocre feature that may appeal to diehard genre fans.\n\n**4121** _ **Stand Up and Fight**_ **** Metro-Goldwyn-Mayer, 1939. 97 min. D: W.S. Van Dyke. SC: James M. Cain, Jane Murtin and Harvey Fergusson. With Wallace Beery, Robert Taylor, Florence Rice, Helen Broderick, Charles Bickford, Barton MacLane, Charles Grapewin, John Qualen, Robert Glecker, Clinton Rosemond, Cy Kendall, Paul Everton, Claudia Morgan, Selmer Jackson, Robert Middlemass, Al Ferguson, Frank Jaquet, Theodore Lorch, Claire McDowell, Jonathan Hale, Edward Hearn, Clem Bevans, Syd Saylor, Minor Watson, Frank Darien, Edward Keane, John Dilson, William Tannen, Sam Ash, Hal Price, Forrest Taylor, Ben Welden, Eddy Waller, Victor Potel, Walter Soderling, Mitchell Lewis, Harry Cording, Trevor Bardette, John Ince, Murdock MacQuarrie, Everett Brown, Henry Hastings, Ted Oliver, Sidney D'Albrook, Louise Springer, Jack Grey, Lee Tung Foo, James Kilgannon, George Cooper, George Ovey. A man is hired by a railroad to investigate slave running and he becomes at odds with a stagecoach operator involved in the trade. Slick pre\u2013Civil War drama sure to delight Wallace Beery fans.\n\n**4122** _ **Standing Tall**_ **** NBC-TV, 1978. 100 min. Color. D: Harvey Hart. SC: Franklin Thompson. With Robert Forster, Will Sampson, L.Q. Jones, Robert Donner, Ron Hayes, Buck Taylor, Linda Evans, Chuck Connors, Faith Quabius, Dani Jannssen, Robert Gentry, Eddie Firestone, David Lewis. During the Depression a half-breed rancher runs into trouble when he refuses to sell his spread to a ruthless cattle baron. Typically mediocre made-for-TV Western.\n\n**4123** _ **Star in the Dust**_ **** Universal-International, 1956. 80 min. Color. D: Charles Haas. SC: Oscar Brodney. With John Agar, Mamie Van Doren, Richard Boone, Coleen Gray, Leif Erickson, James Gleason, Randy Stuart, Terry Gilkyson, Paul Fix, Harry Morgan, Stuart Randall, Robert Osterloh, Stanley Andrews, John Day, Stafford Repp, Lewis Martin, Renny McEvoy, Jesse Kirkpatrick, James Parnell, Anthony Jochim, Kenneth MacDonald, George Wallace, Clint Eastwood, Jack Ingram, Kermit Maynard, Chuck Hamilton, Frank Mills. When a gunman kills three farmers, a sheriff plans to hang him for the crimes but the town's citizens refuse to go along with the execution. Passable Albert Zugsmith production but nothing special.\n\n**4124** _ **Star of Texas**_ **** Allied Artists, 1953. 68 min. D: Thomas Carr. SC: Dan Ullman. With Wayne Morris, Paul Fix, Rick Vallin, Robert Bice, Frank Ferguson, Jack Larson, James Flavin, Lyle Talbot, William Fawcett, Mickey Simpson, George Wallace, John Crawford, Stanley Price, Pierce Lyden, Frank Ellis, Jack O'Shea, Ray Jones. An outlaw gang recruits new members from prison and a Texas Ranger pretends to be an escaped convict to track them down. Documentary-like atmosphere makes this Wayne Morris film (the first in his Allied Artists series, the final official \"B\" Western grouping) good entertainment; remade as _**Gunfight at Comanche Creek**_ and _**Last of the Badmen**_ (qq.v.).\n\n**4125** _ **The Star Packer**_ **** Monogram, 1934. 54 min. D-SC: Robert North Bradbury. With John Wayne, Verna Hillie, George Hayes, Yakima Canutt, Earl Dwire, Ed (Eddie) Parker, George Cleveland, Tom Lingham, Artie Ortego, Davie Aldrich, Tex Palmer, Billy Franey, Glenn Strange, Arthur Millett, Frank Ball. A government agent, helped by his Indian pal, takes the job of sheriff in a town plagued by an outlaw gang led by a mysterious figure called The Shadow. Hidden tunnels, a hooded gang kingpin and lots of action makes this Lone Star production a good one; colorized as _**The Shadow Gang**_.\n\n**4126** _ **Starbird and Sweet William**_ **** Howco International, 1975. 95 min. Color. D: Jack B. Hively. SC: Axel Gruenberg. With Dan Haggerty, Skip Homeier, A. Martinez, Louise Fitch, Skeeter Vaughn, Roger Bear, Ancil Cook. An Indian youth takes an unauthorized solo flight in a plane and ends up crashing it in the wilderness where he has to fight to survive and is befriended by a bear cub and other animals. Pleasant outdoor drama; well directed.\n\n_**Starblack**_ see _**Black Star**_\n\n**4127** _ **Stardust on the Sage**_ **** Republic, 1942. 65 min. D: William Morgan. SC: Stuart McGowan and Dorrell McGowan. With Gene Autry, Smiley Burnette, Edith Fellows, Bill Henry, Louise Currie, George Ernest, Emmett Vogan, Vince Barnett, Betty Farrington, Roy Barcroft, Frankie Marvin, Tom London, Rex Lease, Frank Ellis, Ed Cassidy, Fred Burns, Frank LaRue, Franklyn Farnum, Edmund Cobb, Merrill McCormick, Monte Montague, George DeNormand, George Sherwood, Bill Nestell, Frank O'Connor, Griff Barnett, Lee Shumway. When crooks frame a young man on an embezzlement charge, Gene Autry comes to the rescue. Note even some good songs, including \"Deep in the Heart of Texas,\" and an excellent supporting cast can overcome the tedium of this Gene Autry effort.\n\n**4128** _ **Starlight Over Texas**_ **** Monogram, 1938. 56 min. D: Al Herman. SC: John Rathmell. With Tex Ritter, Carmen LaRoux, Snub Pollard, Horace Murphy, Karl Hackett, Charles King, Martin Garralaga, George Chesebro, Carlos Villarias, Ed Cassidy, Sherry Tansey, Bob Terry, Horace B. Carpenter, Dave O'Brien, Denver Dixon, Chick Hannon, Tex Palmer, Rosa Turich, Carmen Alvarez, Jerry Comez, The Northwesterners, Martin Cazares. With outlaws attacking Spanish ranchers, a cowboy and his pal try to bring them to justice. Standard Tex Ritter vehicle hurt by a long mid-way musical segment, although it is well staged, with the star singing the rousing \"A Viva Tequila.\" British title: _**Moonlight Over Texas**_.\n\n**4129** _ **Stars in My Crown**_ **** Metro-Goldwyn-Mayer, 1950. 89 min. D: Jacques Tourneur. SC: Margaret Fitts. With Joel McCrea, Ellen Drew, Dean Stockwell, Alan Hale, Lewis Stone, James Mitchell, Amanda Blake, Juano Hernandez, Charles Kemper, Connie Gilchrist, Ed Begley, Jack Lambert, Arthur Hunnicutt, James Arness, Snub Pollard, Victor Kilian, Chuck Courtney, Wilson Wood, Ralph Hodges, Polly Bailey, Adeline de Walt Reynolds, Norman Ollestad, Jr., Ben Watson, Jimmy Moss, Jessie Grayson, Philo McCullough, James Pierce, Buddy Roosevelt, Howard Mitchell, Tex Terry, Frank Pharr, Carl Petti, Helen Eby-Rock, Bill Clauson, Rhea Mitchell, Patricia Miller, Al Kundel, Jessie Arnold (voice), Marshall Thompson (narrator). A new minister comes to the pulpit in a rural community and finds he needs a gun to carry out his mission. Homey, episodic film about life in rural nineteenth century America; good viewing.\n\n**4130** _ **Stars Over Arizona**_ **** Monogram, 1937. 62 min. D: Robert North Bradbury. SC: Robert Emmett (Tansey). With Jack Randall, Kathleen Eliot, Horace Murphy, Warner Richmond, Tom Herbert, Hal Price, Earl Dwire, Glenn Strange, Ernie Adams, Chick Hannon, Jack Rockwell, Forrest Taylor, Bob McKenzie, Sherry Tansey, Tex Palmer. A crooked town boss tries to stop a female rancher from selling her cattle and she is helped by a federal marshal sent to the area to halt lawlessness. Okay Jack Randall vehicle.\n\n**4131** _ **Stars Over Texas**_ **** Producers Releasing Corporation, 1946. 59 min. D: Robert Emmett Tansey. SC: Frances Kavanaugh. With Eddie Dean, Shirley Patterson, Roscoe Ates, Lee Bennett, Lee Roberts, Kermit Maynard, Jack O'Shea, Hal Smith, Matty Roubert, Carl Mathews, William Fawcett, Frank Ellis, Hal Smith, The Sunshine Boys (Eddie Wallace, J.D. Sumner, M.H. Richman, Freddie Daniel). Outlaws are murdering citizens and rustling cattle with area ranchers hiring a detective to stop them. Plot wise this Eddie Dean film is not much but it does include two of the star's lovely compositions, the title song and \"Sands of the Old Rio Grande,\" the latter written with Glenn Strange. A remake of _**Driftin' Kid**_ (q.v.).\n\n_**State Police**_ see _**Whirlwind Raiders**_\n\n**4132** _ **Station West**_ **** RKO Radio, 1948. 92 min. D: Sidney Lanfield. SC: Frank Fenton and Winston Miller. With Dick Powell, Jane Greer, Agnes Moorehead, Burl Ives, Tom Powers, Steve Brodie, Joseph Sawyer, Gordon Oliver, Guinn Williams, Raymond Burr, Regis Toomey, Michael Steele, John Kellogg, Charles Middleton, John Doucette, Suzi Crandall, Robert Gates, Marie Thomas, Lomax Study, Robert Jefferson, Bill Phipps, Stanley Blystone, Monte Montague, Ethan Laidlaw, Joey Ray, Al Hill, Bud Osborne, Jimmy Aubrey, Jack Stoney, Leo McMahon. An Army officer works undercover to find out who is behind a series of hijackings that have led to murder. Fast paced action melodrama. Some prints run 80 minutes and there is also a colorized version.\n\n**Dick Powell, Joseph Sawyer and Agnes Moorehead in** _**Station West**_ **(RKO Radio, 1948).**\n\n**  \n**\n\n**4133** _ **Stay Away, Joe**_ **** Metro-Goldwyn-Mayer, 1968. 101 min. Color. D: Peter Tewksbury. SC: Burt Kennedy and Michael A. Hoey. With Elvis Presley, Burgess Meredith, Joan Blondell, Katy Jurado, Thomas Gomez, Henry Jones, L.Q. Jones, Quentin Dean, Anne Seymour, Angus Duncan, Douglas Henderson, Michael Lane, Susan Trustman, Warren Vanders, Buck Kartalian, Maurishka, Caitlin Wyles, Marya Christian, Del \"Sonny\" West, Jennifer Peak, Brett Parker, Michael Keller, Dick Wilson, David Cadiente, Harry Harvey, Joe Esposito, Robert Lieb, The Jordanaires. A half-blood Cree Indian rodeo rider returns to the reservation to help his people with a government rehabilitation program to set them up as cattle ranchers. Pretty poor Elvis Presley film.\n\n**4134** _ **Steel Cowboy**_ **** NBC-TV\/EMI Television, 1978. 100 min. Color. D: Harvey Laidman. SC: Douglas Wheeler and Bill Kerby. With James Brolin, Jennifer Warren, Rip Torn, Melanie Griffith, Julie Cobb, Lou Frizzell, Strother Martin, Albert Popwell, John Dennis Johnston, Bob Schott, Don Calfa, Rudy Diaz, Bob Hoy, Scott Thompson, Larry Spaulding. Despite opposition from his wife, a trucker tries to save his rig by agreeing to haul stolen cattle for a pal. Mediocre TV movie.\n\n**4135** _ **Stick to Your Guns**_ **** Paramount, 1941. 63 min. D: Lesley Selander. SC: J. Benton Cheney. With William Boyd, Andy Clyde, Brad King, Jacqueline (Jennifer) Holt, Dick Curtis, Weldon Heyburn, Henry Hall, Joe Whitehead, Bob Card, Jack C. Smith, Herb Holcombe, Tom London, Kermit Maynard, Frank Ellis, Jack Rockwell, The Jimmy Wakely Trio (Jimmy Wakely, Johnny Bond, Dick Reinhart), Ian MacDonald, Charles Middleton, Joe Whitehead, Jack C. Smith, Jack Trent, Homer Holcomb, Tom Ung, Mickey Eissa, Robert Kortman, Frank Mills, Robert Barron, Herman Hack, Charles Murphy, Lew Morphy, Roy Bucko, Silver Tip Baker. When a gang of rustlers proves elusive, Hopalong Cassidy takes on the guise of a wanted man in order to infiltrate them. Rather slow Hoppy series entry with William Boyd also playing gambler Tex Riley, a disguise he used earlier in _**Bar 20**_ _**Rides Again**_ (q.v.).\n\n**4136** _ **The Still Trumpet**_ **** CBS-TV\/20th Century\u2013Fox, 1957. 45 min. With Dale Robertson, Victor Jory, Regis Toomey, Carol Ohmart, Patrick McVey, James Griffith, Douglas Dick, Ed Kemmer, Grandon Rhodes, Dan Riss, Forrest Taylor, Paul McGuire, Tommy Farrell, William Challee, Jim Hayward, Morgan Jones, Jack Tornek, Abel Fernandez, Eddie Little Sky, John Conte (host). Indians threaten the denizens of a remote Western fort with imprisoned Confederate soldiers enlisted to protect them. Okay TV remake of _**Two Flags West**_ (q.v.), originally telecast as a segment of \"The 20th Century\u2013Fox Hour\" (CBS-TV, 1955\u201357) on April 3, 1957.\n\n_**Sting of the West**_ see _**Tedeum**_\n\n_**Stolen Goods**_ see _**Blue Steel**_\n\n**4137** _ **Stone Fox**_ **** NBC-TV, 1987. 104 min. Color. D: Harvey Hart. SC: Walter Halsey Davis. With Buddy Ebsen, Joey Cramer, Belinda Montgomery, Gordon Tootoosis, Jason Michas, Charles Siegel, Nikky Jensen, Franklin Johnson, J.C. \"Jim\" Roberts, Joel Dacks, Gordon McIntosh, Larry Musser, Frank C. Turner, Jerry Wasserman, Sherry Wells, Dale Wilson, O.J. (dog). In order to help save his grandfather's farm, an orphan boy competes in a dog sled race with a previously unbeaten Indian. TV family film is on the pale side.\n\n**4138** _ **Stone of Silver Creek**_ **** Universal, 1935. 61 min. D: Nick Grinde. SC: Earle Snell. With Buck Jones, Noel Francis, Niles Welch, Murdock MacQuarrie, Marion Shilling, Peggy Campbell, Rodney Hildebrand, Harry Semels, Grady Sutton, Bob McKenzie, Lew Meehan, Frank Rice, Kernan Cripps, Eddie Gribbon, Horace B. Carpenter, Bill Patton, Hank Bell, Charles Brinley. A saloon owner gets religion but finds he is at odds with the town's preacher over a pretty girl. A different kind of Buck Jones action film with touches of the type of fare William S. Hart did in the silent days, although not so austere.\n\n**4139** _ **The Storm**_ **** Universal, 1930. 90 min. D: William Wyler. SC: Wells Root and Tom Reed. With Lupe Velez, Paul Cavanagh, William Boyd, Alphonse Ethier, Ernie Adams, Tom London, Nick Thompson, Erin La Bissoniere. Two war buddies fall in love with a girl who they get snowbound with in the wilds of Canada and the men become bitter enemies for her hand in marriage. Standard screen version of the 1919 Landgon McCormick novel with a well staged avalanche sequence; first filmed in 1922 by Universal with Matt Moore, House Peters and Virginia Valli.\n\n**4140** _ **Storm Over Wyoming**_ **** RKO Radio, 1950. 60 min. D: Lesley Selander. SC: Ed Earl Repp. With Tim Holt, Richard Martin, Noreen Nash, Richard Powers (Tom Keene), Betty Underwood, Kenneth MacDonald, Leo McMahon, Bill Kennedy, Holly Bane, Don Haggerty, Richard Kean. A dishonest ovine ranch foreman causes trouble between cattlemen and sheep herders. Despite more than adequate production values, this Tim Holt vehicle is only average.\n\n**4141** _ **The Storm Rider**_ **** 20th Century\u2013Fox, 1957. 70 min. D: Edward Bernds. SC: Edward Bernds and Don Martin. With Scott Brady, Mala Powers, Bill Williams, John Goddard, William Fawcett, Roy Engel, George Keymas, Olin Howlin, Hank Patterson, James Dobson, John Close, Jim Hayward, Rocky Shahan, Frank Richards, Rick Vallin, Lane Chandler, Tom London, I. Stanford Jolley, Britt Wood, John Cason, Bud Osborne, Al Baffert, Ron Foster, Jean Ann Lewis, Wayne Mallory, Cortland Shepard, Rocky Lundy. A big rancher, in order to stop competition, hires a vicious gunman while the Cattle Association sends an agent to help those being attacked. Moody and fairly action filled oater highlighted by Byrdon Baker's photography; it contains an especially good early sequence of the agent riding into town during a dust storm.\n\n_**Storm Rider**_ (1972) see _**The Grand Duel**_\n\n**4142** _ **Stormy**_ **** Universal, 1935. 68 min. D: Louis Friedlander (Lew Landers). SC: George Plympton and Ben Grauman Kohn. With Noah Beery, Jr., Jean Rogers, J. Farrell MacDonald, Raymond Hatton, Walter Miller, Fred Kohler, Harry Woods, James P. Burtis, The Arizona Wranglers (Charles Hunter, L.F. Costello, Cal Short, John Jackson, Glenn Strange, Johnny Luther), Bud Osborne, Kenny Cooper, James Phillips, Jack Sanders, Cecil Kellogg, Jack Shannon, Robert E. Homans, Wilfred Lucas, Samuel R. McDaniel, Eddie (Edmund) Cobb, Charles Murphy, James Welch, Shirley Marks, Chester Gan, William Welsh, Jack Leonard, Monte Montague, W.H. Davis, Rex (horse). A young man searches for a beautiful stallion lost during a train wreck and ends up saving a herd of wild horses. Well made and entertaining action drama.\n\n**4143** _ **Stormy Trails**_ **** Colony\/Grand National, 1937. 59 min. D: Sam Newfield. SC: Phil Dunham. With Rex Bell, Lois Wilde, Bob Hodges, Lane Chandler, Earl Dwire, Lloyd Ingraham, Karl Hackett, Earle Ross, Murdock MacQuarrie, Jimmy Aubrey, Roger Williams, George Morrell. Two brothers own a ranch with a mortgage and bad men are after the land for the gold it contains. Low grade production with a complicated plot and some fast action.\n\n**4144** _ **Straight Shooter**_ **** Victory, 1939. 60 min. D: Sam Newfield. SC: Basil Dickey and Joseph O'Donnell. With Tim McCoy, Julie Sheldon, Ben Corbett, Forrest Taylor, Carl Mathews, Ted Adams, Budd Buster, Reed Howes, Wally West, Jack Ingram, Dan White, George Morrell. A lawman pretends to be a rancher trying to buy property where he suspects an outlaw gang has hidden stolen loot. This \"Lightning Bill Carson\" outing is not one of Tim McCoy's best, mainly due to budget limitations.\n\n**4145** _ **Straight Shooting**_ **** Universal\/Butterfly, 1917. 53 min. D: Jack (John) Ford. SC: George Hively. With Harry Carey, Molly Malone, Duke R. Lee, Vester Pegg, Hoot Gibson, George Berrell, Ted Brooks, Milt Brown. Cattlemen hire an outlaw to help them in their fight with settlers but he soon changes sides due to the brutality of his employers and the love of a nester girl. John Ford's first feature film is crude and simplistic by today's standards but it is still well worth viewing. Reissued in two reels in 1925 and called _**Straight Shooting**_. Also known as _**Joan of Cattle Country**_.\n\n**4146** _ **Straight to Hell**_ **** Initial Pictures, 1987. 86 min. Color. D: Alex Cox. SC: Alex Cox and Dick Rude. With Dennis Hopper, Courtney Love, Elvis Costello, Grace Jones, Sy Richardson, Joe Strummer, Dick Rude, Bill Yeager, Sara Sugarman, Jim Jarmusch, Zander Schloss, Juan Torres, The Pogues. Bank robbers bury money they stole in a holdup and end up in a town where they eventually have a showdown with a mysterious stranger. Mixed up take off of Spaghetti Westerns not likely to appeal to genre fans.\n\n**4147** _ **Strange Gamble**_ **** United Artists, 1948. 61 min. D: George Archainbaud. SC: J. Benton Cheney, Bennett Cohen and Ande Lamb. With William Boyd, Andy Clyde, Rand Brooks, Elaine Riley, Francis McDonald, Paul Fix, William F. Leicester, Joan Barton, James Craven, Joel Friedkin, Herbert Rawlinson, Robert B. Williams, Alberto Morin, Lee Tung Foo, Dewey Robinson, George Sowards. The government hires Hopalong Cassidy and his pals to investigate the appearance of counterfeit money in a border town. Turgid final \"Hopalong Cassidy\" series entry.\n\n**4148** _ **Strange Lady in Town**_ **** Warner Bros., 1955. 112 min. Color. D: Mervyn LeRoy. SC: Frank Butler. With Greer Garson, Dana Andrews, Cameron Mitchell, Lois Smith, Walter Hampden, Pedro Gonzalez-Gonzalez, Joan Camden, Jose Torvay, Adele Jergens, Robert Wilke, Frank De Kova, Russell Johnson, Gregory Walcott, Douglas Kennedy, Ralph Moody, Nick Adams, Jack Williams, The Trianas. A woman doctor arrives in Santa Fe in 1879 to find her brother is an outlaw. Overlong and none-too-successful vehicle for Greer Garson; Frankie Laine sings the title song.\n\n**4149** _ **The Strange Vengeance of Rosalie**_ **** 20th Century\u2013Fox, 1972. 107 min. Color. D: Jack Starrett. SC: Anthony Greville-Bell and John Kohn. With Bonnie Bedelia, Ken Howard, Anthony Zerbe. A young Indian girl forces a traveling salesman to keep her company in an isolated New Mexico desert home until they are terrorized by a crazed biker. Offbeat, but unappealing, melodrama.\n\n**4150** _ **The Stranger and the Gunfighter**_ **** Columbia, 1976. 107 min. Color. D: Anthony M. Dawson (Antonio Margheriti). SC: Barth Jules Sussman. With Lee Van Cleef, Lo Lieh, Julian Ugarte, Patty Shepard, Karen Yeh, Femi Benussi, Erika Blanc, George Rigaud, Richard Palacios, Goyo (Gregorio) Peralta, Al Tung, Alfred Boreman, Bart Barry, Paul Costello. In order to obtain a fortune once belonging to a Chinese war lord, a gunman joins forces with a kung fu expert as they search for clues tattooed on the backsides of four comely young ladies. Elaborate genre put-on is lots of fun for Spaghetti Western fans. Video title: _**Blood Money\u2014Stranger and the Gunfighter**_.\n\n**4151** _ **Stranger at My Door**_ **** Republic, 1956. 85 min. D: Willian Witney. SC: Barry Shipman. With Macdonald Carey, Patricia Media, Skip Homeier, Stephen Wootton, Louis Jean Heydt, Howard Wright, Slim Pickens, Fred Sherman, Malcolm Atterbury, Virginia Carroll, Helen Wallace, Nancy Howard, Bernadette Withers, Paul E. Burns, Tom Black, Peter Brocco, Penny Carpenter. A minister trying to help an outlaw ends up putting his family in danger. Good, upbeat drama with an especially find performance by Macdonald Carey as the preacher.\n\n**4152** _ **The Stranger from Arizona**_ **** Columbia, 1938. 60 min. D: Elmer Clifton. SC: Monroe Shaff. With Buck Jones, Dorothy Fay, Hank Mann, Roy Barcroft, Hank Worden, Bob Terry, Horace Murphy, Budd Buster, Dot Farley, Stanley Blystone, Ralph Peters, Horace B. Carpenter, Walter Anthony. A fast talking cowpoke is really a railroad detective looking into a series of robberies and killings. Although well done, this Buck Jones film is not up to his usual standards.\n\n**4153** _ **The Stranger from Pecos**_ **** Monogram, 1943. 58 min. D: Lambert Hillyer. SC: Jess Bowers (Adele Buffington). With Johnny Mack Brown, Raymond Hatton, Kirby Grant, Christine McIntyre, Steve Clark, Sam Flint, Roy Barcroft, Robert Frazer, Edmund Cobb, Charles King, Bud Osborne, Artie Ortego, Tom London, Kermit Maynard, Milburn Morante, Lynton Brent, Carol Henry, George Morrell, Frosty Royce, Herman Hack, Chick Hannon, Lew Morphy, Roy Bucko, Ralph Bucko. Helping to investigate a series of robberies, two lawmen discover the local sheriff and banker are the culprits. This second \"Nevada Jack McKenzie\" outing is action filled.\n\n**4154** _ **The Stranger from Ponca City**_ **** Columbia, 1947. 56 min. D: Derwin Abrahams. SC: Ed Earl Repp. With Charles Starrett, Smiley Burnette, Virginia Hunter, Paul Campbell, Texas Jim Lewis and His Lone Star Cowboys, Harmonica Bill (William Russell), Jim Diehl, Forrest Taylor, Ted Mapes, Jacques O'Mahoney (Jock Mahoney), Tom McDonough, John Carpenter, Charles Hamilton, Ted Wells, Herman Hack, Bud Osborne, Kermit Maynard, Roy Butler. A cowboy arrives in a community torn between peaceful and lawless elements. Fair \"Durango Kid\" series segment.\n\n**4155** _ **Stranger from Santa Fe**_ **** Monogram, 1945. 56 min. D: Lambert Hillyer. SC: Frank Young. With Johnny Mack Brown, Raymond Hatton, Beatrice Gray, Lewis Hart, Jack Ingram, Jimmie Martin, Bud Osborne, Tom Quinn, Hal Price, Steve Clark, Jack Rockwell, Eddie Parker, Joann Curtis, John Merton, Dick Dickinson, Ray Elder, Louis Hart, Henry Wills, Horace B. Carpenter. Impersonating a cowpoke, a lawman is forced by an outlaw gang to take part in a stage holdup and is framed for a guard's murder. A good script enhances this \"Nevada Jack McKenzie\" film.\n\n**4156** _ **The Stranger from Texas**_ **** Columbia, 1939. 54 min. D: Sam Nelson. SC: Paul Franklin. With Charles Starrett, Lorna Gray, Richard Fiske, The Sons of the Pioneers (Bob Nolan, Tim Spencer, Lloyd Perryman, Pat Brady, Hugh Farr, Karl Farr), Dick Curtis, Edmund Cobb, Alan Bridge, Jack Rockwell, Hal Taliaferro, Art Mix, Ed LeSaint, George Chesebro, Buel Bryant, Frank Ellis, Richard Botiller, Edward Hearn, Ethan Allen. A rancher's son, a lawman working incognito, tries to find who is behind a series of fence cuttings and cattle rustling with a neighbor blaming his father. Passable remake of an earlier Charles Starrett feature, _**The Mysterious Avenger**_ (q.v.).\n\n**4157** _ **Stranger in Japan**_ **** Metro-Goldwyn-Mayer, 1968. 90 min. Color. D: Vance Lewis (Luigi Vanzi). SC: Tony Anthony, Giancarlo Fernando and Vincenzo Cerami. With Tony Anthony, Lloyd Battista, Kim Omae, Rita Maura, Kanki Ohara, Raf Baldassare, Oshio Nukano, William Conroy. A mysterious gunman travels to Japan in an attempt to obtain a huge gold consignment. Star and co-author Tony Anthony's third \"Stranger\" feature, preceded by _**Stranger in Town**_ and _**The Stranger Returns**_ (qq.v.), is somewhat confusing after having been edited for 1975 stateside showings, seven years after its European release as _**Lo Straniero de Silenzio**_ (The Silent Stranger).\n\n**4158** _ **Stranger in Town**_ **** Metro-Goldwyn-Mayer, 1968. 86 min. Color. D: Vance Lewis (Luigi Vanzi). SC: Warren Garfield. With Tony Anthony, Frank Wolff, Iolenda Modio, Gia Sandri, Raf Baldassare, Aldo Berti, Antonio Marsina, Enrico Capoleoni, Arturo Corso. A mystical loner is double crossed by a ruthless bandit over a stolen gold shipment and seeks revenge. Empty and violent Italian Western, but one of the few of its ilk that made money in the U.S.; issued in Italy in 1966 by Primex\u2013Italiana as _**Un Dollaro Tia I Denti**_ (A Dollar Between the Teeth). Followed by _**The Stranger Returns**_ and _**Stranger in Japan**_ (q.v.).\n\n**4159** _ **A Stranger in Town**_ **** National Educational Television, 1969. 90 min. Color. D: Earl J. Miller. SC: Robert Angus. With Lon Chaney, Jimmy Miller, Chuy Sanchez, Lance Roselle, Scott Smith, Anne McAdams, Bob Harris, Bud Breen, Happy Shahan, Sieker Fisher, Anges Vonde, Anna de la Garza, Edmundo Trevino, Jack Carney, Ron Walker, Roy Langston, Buck McCulley, Stan Schooler, Nakai. Two young boys befriend a doctor acquitted in the killing of his wife but believed guilty by the townspeople. Above average TV film produced by Southwest Texas Educational Television Council and filmed at Alamo Village in Brackettville, Texas; also called _**The Children's West**_.\n\n**4160** _ **Stranger on Horseback**_ **** United Artists, 1955. 66 min. Color. D: Jacques Tourneur. SC: Herb Meadow and Don Martin. With Joel McCrea, Miroslava, Kevin McCarthy, John McIntire, Nancy Gates, John Carradine, Emile Meyer, Robert Cornthwaite, James Bell, Jaclynne Greene. A circuit rider judge goes to a town to restore law and order and arrests the son of a local cattle baron on a murder charge. Entertaining, compact feature with good work by Joel McCrea as the peacemaker.\n\n**4161** _ **Stranger on the Run**_ **** NBC-TV\/Universal, 1967. 97 min. Color. D: Donald Siegel. SC: Dean Riesner. With Henry Fonda, Anne Baxter, Dan Duryea, Michael Parks, Sal Mineo, Lloyd Bochner, Michael Burns, Tom Reese, Bernie Hamilton, Madlyn Rhue, Zalman King, Walter Burke, Rodolfo Acosta, George Dunn, Pepe Hern. A drifter, taking a message from a prisoner to his sister, is falsely accused of murder and chased into the desert by a sheriff and his posse. Don Siegel followers will like this TV movie but others will find it nothing special.\n\n**4162** _ **The Stranger Returns**_ **** Metro-Goldwyn-Mayer, 1968. 90 min. Color. D: Vance Lewis (Luigi Vanzi). SC: Bob Ensescalle, Jr. and Jone Maug. With Tony Anthony, Dan Vadis, Daniele Vargas, Marc Guglielmi, Jill Banner, Ettore Manni, Marina Berti, Ralf Baldassare, Anthony Freeman, Luciano Catenacci. A mysterious stranger finds a murdered postal inspector and impersonates him in order to track an outlaw band who stole a gold plated stagecoach. Star Tony Anthony wrote the story for the violence-for-violence's sake oater, a sequel to _**Stranger in Town**_ (q.v.) and followed by _**Stranger in Japan**_ (q.v .). Issued in Italy by Primex\/Juventus\/Reverse as _**Un Uomo, un Cavallo, una Pistola**_ (A Man, a Horse and a Pistol).\n\n**4163** _ **The Stranger Wore a Gun**_ **** Columbia, 1953. 83 min. Color. D: Andre De Toth. SC: Kenneth Gamet. SC: Randolph Scott, Claire Trevor, Joan Weldon, George Macready, Alfonso Bedoya, Lee Marvin, Clem Bevans, Roscoe Ates, Ernest Borgnine, Pierre Watkin, Joseph Vitale, Paul Maxey, Frank Scannell, Reed Howes, Edward Earle, Guy Wilkerson, Mary Newton, Franklyn Farnum, Barry Brooks, Tap Canutt, Al Haskell, Frank Hagney, Frank Ellis, Francis McDonald, Al Hill, Terry Frost, Herbert Rawlinson, Britt Wood, James Millican, Jack Woody, Rayford Barnes, Edith Evanson, Guy Teague. After his life is saved by an outlaw, a stage line employee must choose between loyalty to his benefactor and his job when the bandit plans to rob a gold shipment. An interesting script and good production values, plus a fine cast, help this Randolph Scott film add up to good entertainment.\n\n**4164** _ **Strangers at Sunrise**_ **** Commonwealth United, 1971. 91 min. Color. D: Percival Roberts. SC: Lee Marcus and Percival Roberts. With George Montgomery, Deanna Martin, Brian O'Shaughnessy, Tromp Terreblanche, Beryl Gresak, Simon Sabela, Avron Pearson, Roland Robinson, Helen Braithwaite, Bess Finney. A fugitive American mining engineer in South Africa during the Boer War tries to help a family threatened by three British Army deserters. George Montgomery is quite good in the lead in this well made and enjoyable South African production.\n\n_**Stranger's Gold**_ see _**Have a Good Funeral My Friend**_\n\n_**The Stranger's Gundown**_ see _**Django the Avenger**_\n\n**4165** _ **The Strawberry Roan**_ **** Universal, 1933. 60 min. D: Alan James. SC: Nate Gatzert. With Ken Maynard, Ruth Hall, Harold Goodwin, Frank Yaconelli, Charles King, William Desmond, James Marcus, Jack Rockwell, Robert Walker, Ben Corbett, Art Mix, Bill Patton, Bud McClure. A cowboy defends a beautiful stallion accused of horse rustling, an activity being carried out by a local citizen. This film is said to have been Ken Maynard's personal favorite and it is easy to understand why as it is full of action, music and good humor.\n\n**4166** _ **The Strawberry Roan**_ **** Columbia, 1948. 79 min. Color. D: John English. SC: Dwight Cummings and Dorothy Yost. With Gene Autry, Pat Buttram, Gloria Henry, Jack Holt, Dick Jones, Rufe Davis, Eddy Waller, John McGuire, Redd Harper, Jack Ingram, Ted Mapes, Eddie Parker, Sam Flint. A horse breaker tries to protect one of his stock from being killed by a ranch owner whose son was injured by the animal. One of Gene Autry's best Columbia outings.\n\n**4167** _ **Streets of Ghost Town**_ **** Columbia, 1950. 54 min. D: Ray Nazarro. SC: Barry Shipman. With Charles Starrett, Smiley Burnette, Mary Ellen Kay, Ozie Waters and His Colorado Rangers, George Chesebro, Stanley Andrews, Frank Fenton, John Cason, Little Brown Jug (Don Kay Reynolds), Jack Ingram, Nolan Leary, Robert Kortman, Emmett Lynn, Doris Houck, Dick Rush, John Tyrrell. Three lawmen come to a deserted community to investigate a series of mysterious happenings caused by a blind outlaw using his young nephew to help him find the stolen loot he once hit there. The mystery element adds some life to this \"Durango Kid\" effort, but it is a tattered affair, made up of footage from several other series entries, including _**Gunning for Vengeance**_ and _**Landrush**_ (qq.v.).\n\n**4168** _ **Streets of Laredo**_ **** Paramount, 1948. 92 min. Color. D: Leslie Fenton. SC: Charles Stevens and Elizabeth Hill. With William Holden, Macdonald Carey, Mona Freeman, William Bendix, Stanley Ridges, Alfonso Bedoya, Ray Teal, Clem Bevans, James Bell, Dick Foote, Joe Dominguez, Grandon Rhodes, Perry Ivins, James Davies, Robert Kortman, Byron Foulger, Wade Crosby, Carl Andre, Hank Worden, Julian Rivero, Alex Montoya, Marguerite Martin, Frank Cordell, Frank Hagney, Pat Lane, Joaquin Elizondo, Mike Lally, William Hamel. Two outlaws join the Texas Rangers after being converted to the side of the law and are forced to hunt down their ex-partner. Mediocre remake of _**The Texas Rangers**_ (q.v.).\n\n**4169** _ **Streets of Laredo**_ **** CBS-TV, 1995. 300 min. Color. D: Joseph Sargent. SC: Larry McMurtry and Diana Ossana. With James Garner, Sissy Spacek, Sam Shepard, Ned Beatty, Randy Quaid, Wes Studi, Charles Martin Smith, George Carlin, Alexis Cruz, Kevin Conway, James Gammon, Tristan Tait, Sonja Braga, Miriam Colon, Anjanette Comer, David S. Cass, Sr., James Victor, Doran Atherton, Rutherford Cravens, Stephen Bridgewater, Emily Courtney, Helen Cates, Wally Welch, Cameron Finley, Christopher Wagner, Weasel Forshaw, Tony Frank, Angelina Calderon Torres, Billy Tolson, Bill Gribble, Joe Stevens, Joanna Sanchez, Roland Rodriguez, Kirk Griffith, Karen Jones, Nik Hagler, William Hardy, Peyton Park, Lidia Porto, Lanell Pena, Jill Parker-Jones, Frederick Lopez, Richard Nace, Renee Olstead, Richard Norsworthy, Frank Q. Dobbs, Bunk Duncan, John L. Martin, Vanessa Martinez. Two retired Texas Rangers are hired to bring in a sadistic teenage killer whose mother was once involved with one of them. Fast moving, well done TV mini-series which Larry McMurtry co-adapted from his novel.\n\n**4170** _ **Strictly in the Groove**_ **** Universal, 1943. 60 min. D: Vernon Keays. SC: Kenneth Higgins and Warren Wilson. With Richard Davies, Mary Healy, Leon Errol, Franklin Pangborn, Ozzie Nelson, Jimmie Davis, The Dinning Sisters, The Jimmy Wakely Trio (Jimmy Wakely, Johnny Bond, Eddie Snyder), Diamond's Solid-Aires, Russell Hicks, Martha Tilton, Shemp Howard, Grace MacDonald, Eddie Johnson, Charles Lang, Holmes Herbert, Tim Ryan, Ralph Dunn, Ken Stevens, Lloyd Ingraham, Neeley Edwards, Frances Morris, Drew Demarest, Grace Lenard, Jim Lucas, Joey Ray, Francis Sayles, Jack Gardner. When a man refuses to join his father in the restaurant business and instead forms a band, the old man banishes him to a Western dude ranch. Fair Universal program musical Western, mainly of interest because of its vocalists and songs like \"You Are My Sunshine,\" \"Chisholm Trail,\" \"Happy Cowboy,\" etc.\n\n**4171** _ **Strike It Rich**_ **** Allied Artists, 1949. 81 min. D: Lesley Selander. SC: Francis Rosenwald. With Rod Cameron, Bonita Granville, Don Castle, Stuart Erwin, Lloyd Corrigan, Ellen Corby, Emory Parnell, Harry Tyler, Virginia Dale, William Haade, Edward Gargan, Robert Dudley. Two Texas drillers make a big strike and then fight the law limiting the amount of oil they can produce. Action filled, well directed comedy drama produced by Jack Wrather.\n\n**4172** _ **Strike Me Deadly**_ **** Medallion, 1963. 81 min. D: Herbert L. Strock. SC: Steve Ihnat and Ted V. Mikels. With Gary Clarke, Jeannine Riley, Steve Ihnat, Gordon Mauser. A forest ranger and his young bride are stalked by a man who has murdered a hunter, with the three getting caught in an out-of-control fire. Low budget but fast paced drama, produced, co-written and edited by Ted V. Mikels.\n\n**4173** _ **Strong Medicine**_ **** NBC-TV, 1956. 54 min. D: Walter Grauman. SC: William Mourne. With Patrick O'Neal, Mary Webster, Myron Healey, Joe Maross, John Conte (host). An Easterner has trouble taking possession of a ranch he inherited. Fair telefeature first shown as a segment of \"Matinee Theatre\" (NBC-TV, 1955\u201358) on December 28, 1956.\n\n**4174** _ **Strongheart**_ **** Biograph, 1914. 45 min. D: James Kirkwood. SC: Frank E. Woods. With Blanche Sweet, Henry B. Walthall, Antonio Moreno, Lionel Barrymore, Alan Hale, Gertrude Robinson, Tom McEvoy, William J. Butler, W.C. Robinson, Jack Mulhall, James Kirkwood. An Indian brave saves a man's life and the latter's sister falls in love with him. Supervised by D.W. Griffith, this silent melodrama is quite entertaining, especially for its stars; adapted from William C. DeMille's play.\n\n**4175** _ **Stronghold**_ **** Lippert, 1952. 72 min. D: Steve Sekeley. SC: Wells Root. With Veronica Lake, Zachary Scott, Arturo de Cordova, Rita Lacedo, Alfonso Bedoya, Yadiro Jiminez, Fanny Schiller, Gilberto Gonzales, Carlos Muzquiz. The pretty owner of several Mexican silver properties is kidnapped by a bandit leader and she soon warms to his cause but her dishonest foreman plans to dynamite the mines to insure the rebel's capture. The trio of stars do a lot to keep this low budget action item moving.\n\n**4176** _ **Su Precio...Unos Dolares**_ (His Cost...One Dollar) **** Radaent Films, 1970. 85 min. Color. D-SC: Raul de Anda. With Rodolfo de Anda, Pedro Armendariz, Jr., Dagoberto Rodriguez, Sonia Furio, Jorge Russek, Mario Almada, Rafael Baledon, Juan Gallardo, Jose L. Murillo, Manuel Donde, Mario Cid, Victorio Blanco, Julian Bravo, Hernando Name, Juan Garza. A dance hall girl seduces a banker to get information about a gold shipment and passes it along to her boyfriend and his outlaw gang not knowing the money is protected by Texas Rangers. Action filled Mexican Western.\n\n**4177** _ **Sucedio en Jalisco**_ (It Happened in Jalisco) **** Radaent Films, 1972. 90 min. Color. D-SC: Raul de Anda. With Rodolfo de Anda, Pedro Armendariz, Jr., Patricia Aspillaga, Hector Suarez, Alicia Bonet, Carlos Lopez Moctezuma, Juan Gallardo, Julio Almada, Jorge Lavat, Pancho Cordova, Pascual Garcia Pena, Consuelo Frank, Federico Falcon, Tito Novaro, Jose L. Murillo, Cecilia Leger, Bernardina Green, Luciano Hernandez de la Vega. A doctor returns to his village during the Mexican Revolution and falls in love with a rancher's beautiful daughter, causing a cowboy to become jealous of him. Well done romantic drama from prolific Mexican filmmaker Raul de Anda.\n\n**4178** _ **Sudden Bill Dorn**_ **** Universal, 1938. 60 min. D: Ray Taylor. SC: Frances Guihan. With Buck Jones, Evelyn Brent, Noel Francis, Frank McGlynn, Harold Hodge, Ted Adams, William Lawrence, Lee Phelps, Tom Chatterton, Carlos Valdez, Ezra Pallette, Red Hightower, Charles LeMoyne, Adolph Milar. A cowboy gets on the trail of crooks who arrive in a small town after the discovery of gold. Buck Jones produced his fair action film made at the close of his Universal tenure.\n\n_**Sudden Death**_ see _**Fast on the Draw**_\n\n**4179** _ **Sugarfoot**_ **** Warner Bros., 1951. 80 min. Color. D: Edwin L. Marin. SC: Russell Hughes. With Randolph Scott, Raymond Massey, Adele Jergens, S.Z. Sakall, Robert Warwick, Gene Evans, Hugh Sanders, Hope Landin, Hank Worden, Arthur Hunnicutt, Edward Hearn, John Hamilton, Cliff Clark, Kenneth MacDonald, Dan White, Paul Newlan, Philo McCullough, Ben Corbett. A former Confederate officer tries to settle down peacefully as a rancher in Arizona but soon find he is the sworn enemy of a local crook, a one time rival. Fast paced Randolph Scott vehicle. TV title: _**A Swirl of Glory**_.\n\n**4180** _ **The Sugarland Express**_ **** Universal, 1974. 109 min. Color. D: Steven Spielberg. SC: Hal Barwood and Matthew Boulton. With Goldie Hawn, Ben Johnson, Michael Sacks, William Atherton, Gregory Walcott, Harrison Zanuck, Steve Kanaly, Louise Latham, A.L. Camp, Jessie Lee Fuller, Dean Smith, Ted Grossman, Bill Thurman, Kenneth Hudgins, Buster Daniels, Jim Harrell, Frank Steggal, Roger Ernest, Gene Rader, Gordon Hurst, George Hagy, John Hamilton. A police official leads a chase across Texas in 1968 after a fugitive couple who have escaped from prison to locate their small daughter who has been adopted. Overrated thriller, although Ben Johnson is good as the lawman.\n\n**4181** _ **Sun Valley Cyclone**_ **** Republic, 1946. 56 min. D: R.G. Springsteen. SC: Earle Snell. With Wild Bill Elliott, Bobby Blake, Alice Fleming, Roy Barcroft, Monte Hale, Kenne Duncan, Eddy Waller, Tom London, Edmund Cobb, Ed Cassidy, George Chesebro, Rex Lease, Hal Price, Jack Kirk, Frank O'Connor, Jack Sparks, Jack Rockwell, Horace B. Carpenter, Bob Burns, Silver Tip Baker, Tommy Coats, Tom Steele, LeRoy Mason (voice). An outlaw gang is stealing horses intended for the Army and Red Ryder tries to stop them. Well written and paced series entry.\n\n**4182** _ **Sundance and the Kid**_ **** Hisperia\/Ultra Film, 1969. 78 min. D: Duccio Tessari. SC: Ennio Flaiano. With Giuliano Gemma, Nino Benvenuti, Sydne Rome, Julia Pena, Antonio Casas, Cris Huerta, George Rigaud, Dan Van Husen, Luis Barboo, Victor Israel, Vicente Roca, Juan Olaquivel, Arthuro Pallandino, Brizio Montinaro. In order to inherit $300,000, two brothers with opposite personalities must live together for six months, but both are soon attracted to the same girl. Fairly amusing Spaghetti Western comedy co-starring two time world's middleweight boxing champion Nino Benvenuti; released in Italy as _**Vivi o, Preferibilmente, Morti**_ (Alive or Preferably Dead) and also called _**Sundance Cassidy and Butch the Kid**_. Originally running 103 minutes, it was badly cut for U.S. video release.\n\n_**Sundance Cassidy and Butch the Kid**_ see _**Sundance and the Kid**_\n\n_**Sundown Fury**_ see _**Jesse James Jr.**_\n\n**4183** _ **Sundown in Santa Fe**_ **** Republic, 1948. 60 min. D: R.G. Springsteen. SC: Norman S. Hall. With Allan \"Rocky\" Lane, Eddy Waller, Roy Barcroft, Jean Dean, Russell Simpson, Minerva Urecal, Rand Brooks, Trevor Bardette, Lane Bradford, Joseph Crehan, Kenne Duncan, Robert Wilke. An Army intelligence agent tries to find out who is the leader of an outlaw band. Fairly interesting \"Famous Westerns\" outing with a good plot making it above average.\n\n**4184** _ **Sundown Jim**_ **** 20th Century\u2013Fox, 1942. 58 min. D: James Tinling. SC: Robert F. Metzler and William Bruckner. With John Kimbrough, Virginia Gilmore, Arleen Whelan, Moroni Olsen, Paul Hurst, Cliff Edwards, Joseph Sawyer, Don Costello, Tom Fadden, Frank McGrath, LeRoy Mason, James Bush, Lane Chandler, Charles Tannen, Paul Sutton, Eddy Waller, Glenn Strange, Syd Saylor, Frank McCarroll, Kermit Maynard. A new sheriff learns the town's citizens do not support him when he attempts to bring a land baron to justice after his gang commits murder. There is nothing special about John Kimbrough's second, and last, Western.\n\n**4185** _ **The Sundown Kid**_ **** Republic, 1942. 55 min. D: Elmer Clifton. SC: Norman S. Hall. With Don \"Red\" Barry, Linda Johnson, Ian Keith, Helen MacKellar, Emmett Lynn, Wade Crosby, Robert Kortman, Ted Adams, Kenne Duncan, Bud Geary, Fern Emmett, Kenneth Harlan, Jack Ingram, Jack Rockwell, Joe McGuinn, Cactus Mack. A Pinkerton agent teams with a pretty newspaper reporter to oppose a gang of counterfeiters. Engaging Don Barry series feature.\n\n**4186** _ **Sundown on the Prairie**_ **** Monogram, 1939. 53 min. D: Al Herman. SC: William Nolte and Edmund Kelso. With Tex Ritter, Dorothy Fay, Horace Murphy, Hank Worden, Charles King, Dave O'Brien, Karl Hackett, Bob Terry, Frank LaRue, Ed Peil, Sr., Bud Osborne. When rustlers run rampant in the area around Santa Fe, two government men are sent to stop them. Fair Tex Ritter action songfest. British title: _**Prairie Sundown**_.\n\n**4187** _ **The Sundown Rider**_ **** Columbia, 1933. 56 min. D-SC: Lambert Hillyer. With Buck Jones, Barbara Weeks, Wheeler Oakman, Pat O'Malley, Niles Welch, Bradley Page, Frank LaRue, Ward Bond, Ed Brady, Harry Todd, George Chesebro, Glenn Strange, Richard Alexander, Jack Kirk, Arthur Wanzer. A cowboy falsely accused of rustling uncovers a plot by crooks to steal a woman's ranch because it contains oil deposits. Really good Buck Jones feature, well written and helmed by Lambert Hillyer, one of the most underrated of film directors.\n\n**4188** _ **Sundown Riders**_ **** Film Enterprises, 1948. 60 min. D: Lambert Hillyer. SC: Rodney J. Graham. With Russell Wade, Andy Clyde, Jay Kirby, Evelyn Finley, Marshall Reed, Jack Ingram, Steve Clark, Hal Price, Ted Mapes, Bud Osborne, Ted Wells, Henry Wills, Cliff Parkinson, Cactus Mack, Chief Many Treaties. A trio of cowpokes get involved with an outlaw gang and end up nearly being hanged. Okay action feature filmed in 16mm in 1944 for non-theatrical release and issued four years later.\n\n**4189** _ **Sundown Saunders**_ **** Supreme, 1936. 64 min. D-SC: Robert North Bradbury. With Bob Steele, Catherine Cotter, Earl Dwire, Milburn Morante, Ed Cassidy, Jack Rockwell, Frank Ball, Hal Price, Charles King, Horace Murphy, Edmund Cobb, Robert McKenzie, Jack Kirk, Herman Hack. After winning a big horse race a cowboy ends up with a ranch but crooks try to cheat him out of it. Action filled Bob Steele vehicle.\n\n**4190** _ **Sundown Trail**_ **** RKO Path\u00e9, 1931. 55 min. D-SC: Robert Hill. With Tom Keene, Marion Shilling, Nick Stuart, Hooper Atchley, Louise Beavers, Stanley Blystone, William Welsh, Murdock MacQuarrie, Alma Chester, William Gillis, Ben Corbett, Tommy Coats, Slim Whitaker, Jim Corey, Hank Bell, Blackjack Ward, Bill Nestell, Bud McClure, Bob Burns, Buck Moulton, Fred Burns, Bob Card, Ralph bucko, Roy Bucko, Rose Plummer, Tiny Jones. A cowpoke and a crook are at odds over the same pretty girl. Tom Keene's first series film is a good one.\n\n**4191** _ **Sundown Valley**_ **** Columbia, 1944. 55 min. D: Benjamin Kline. SC: Luci Ward. With Charles Starrett, Dub Taylor, Jeanne Bates, Jimmy Wakely and His Saddle Pals, Clancy Cooper, Jessie Arnold, Wheeler Oakman, Jack Ingram, Forrest Taylor, Joel Friedkin, Grace Lenard, Eddie Laughton, The Tennessee Ramblers, Ted Mapes, Blackie Whiteford. A war hero wants to close a gambling den in his town because its owners are causing absenteeism at the local gun manufacturing plant. Fairly good Charles Starrett yarn with a different kind of plot, one geared to World War II audiences.\n\n**4192** _ **The Sundowners**_ **** Warner Bros., 1960. 141 min. Color. D: Fred Zinneman. SC: Isobel Lennart. With Deborah Kerr, Robert Mitchum, Peter Ustinov, Glynis Johns, Dina Merrill, Chips Rafferty, Michael Anderson, Jr., Lola Brooks, Wylie Watson, John Meillon, Ronald Fraser, Mervyn Johns, Molly Urquhart, Ewen Solon. An Australian sheepherder must choose between his penchant for wanderlust and the love of his wife and son. Excellent on-location filming of John Cleary's novel with a magnificent performance by Robert Mitchum as the sheep man, Paddy Carmody.\n\n**4193** _ **Sunrise Trail**_ **** Tiffany, 1931. 65 min. D: J.P. McCarthy. SC: Wellyn Totman. With Bob Steele, Blanche Mehaffey, Jack Clifford, Richard Alexander, Eddie Dunn, Fred Burns, Germaine De Neel, Jimmy Aubrey, Emilio Fernandez, William Gould, Curley Baldwin, Carlton Griffin, Wally West, George Hazel, Jack Richardson, Harry Allen. A cowboy, secretly working for the local sheriff, pretends to be a gunman to expose a rustling gang. Pretty good Bob Steele early talkie.\n\n**4194** _ **Sunscorched**_ **** Creole\/Production Cinema, 1964. 77 min. Color. D: Mark Stevens. SC: Mark Stevens and Irving Dennis. With Mark Stevens, Marianne Koch, Mario Adorf, Vivien Dobbs, Albert Bessler, Antonio Iranzo, Frank Oliveras, Oscar Pellicer. Four outlaws terrorize a town where the sheriff, once a member of their gang, is helpless in stopping them. West German made oater, directed and co-written by star Mark Stevens, is a well made action affair, originally called _**Vergeltung in Catano**_ (Retaliation in Catano).\n\n**4195** _ **Sunset**_ **** TriStar, 1988. 102 min. Color. D-SC: Blake Edwards. With Bruce Willis, James Garner, Malcolm McDowell, Mariel Hemingway, Kathleen Quinlan, Jennifer Edwards, Patricia Hodge, Richard Bradford, M. Emmett Walsh, Joe Dallesandro, Andreas Katsulas, Dann Florek, Bill Marcus, Michael C. Gwynne, Dermot Mulroney, Miranda Garrison, Liz Torres, Castulo Guerra, Dakin Mathews, Vernon Wells, Dennis Rucker, John Dennis Johnston, Kenny Call, Jack Garner, Jerry Tullos, Steem Tanney, Peter Jason, Richard Fancy, Glenn Shadix, Jon Van Ness, Randy Bowers, Maureen Teely, Arnold Johnson, Rod McCary, John Fountain, F. William Parker. While working as an advisor on a movie about his life, famed lawman Wyatt Earp teams with star Tom Mix to investigate a murder. Pleasant, nostalgic Western mystery from Rod Amateau's story.\n\n**4196** _ **Sunset Carson Rides Again**_ **** Astor, 1948. 63 min. Color. D: Oliver Drake. SC: Elmer Clifton. With Sunset Carson, Pat Starling, Al Terry, Bob (John) Cason, Dan White, Pat Gleason, Steven Keyes, Ron Ormond, Bob Curtis, Joe Hiser, Forrest Matthews, The Rodeo Revelers. A cowboy's ranch partner is the secret head of an outlaw gang trying to cheat him out of his spread and the steal money he raised for a new school. Sunset Carson's first film for Astor is bottom rung, mainly due to a limited budget.\n\n**4197** _ **Sunset in El Dorado**_ **** Republic, 1945. 65 min. D: Frank McDonald. SC: John K. Butler. With Roy Rogers, George \"Gabby\" Hayes, Dale Evans, Hardie Albright, Margaret Dumont, Roy Barcroft, Tom London, Hal Price, Robert Wilke, Ed Cassidy, Dorothy Granger, Bob Nolan and The Sons of the Pioneers (Tim Spencer, Ken Carson, Shug Fisher, Hugh Farr, Karl Farr), Edmund Cobb, Hank Bell, Jack Kirk, Gino Corrado, Frank Ellis, Tex Cooper, Tex Terry, Bud Osborne, Bert Moorhouse, Joe McGuinn, Bob Reeves. The granddaughter of a famous saloon singer dreams of how her ancestor threw over a crooked partner for a cowboy framed for murder. Only fair, with too many musical numbers.\n\n**4198** _ **Sunset in the West**_ **** Republic, 1950. 67 min. Color. D: William Witney. SC: Gerald Geraghty. With Roy Rogers, Penny Edwards, Estelita Rodriguez, Gordon Jones, Foy Willing and The Riders of the Purple Sage, Will Wright, Pierre Watkin, Charles La Torre, William Tannen, Gaylord (Steve) Pendleton, Paul E. Burns. An outlaw gang is wrecking trains as they smuggle weapons out of the country and Roy Rogers tries to help an elderly sheriff stop them. There is lots of action and good fun in this Roy Rogers songfest.\n\n**4199** _ **Sunset in Wyoming**_ **** Republic, 1941. 65 min. D: William Morgan. SC: Ivan Goff and Anne Morrison Chapin. With Gene Autry, Smiley Burnette, Maris Wrixon, George Cleveland, Robert Kent, Sarah Edwards, Monte Blue, Dick Elliott, John Dilson, Stanley Blystone, Earle Hodgins, Eddie Dean, Reed Howes, Fred Burns, Ralph Peters, Syd Saylor, Tex Terry, Lloyd Whitlock, Herman Hack, Bob Woodward. Gene Autry attempts to get a mountain converted into a state park after a lumber company cuts too much timber, causing floods. Good Gene Autry film with nice scenery and photography.\n\n**4200** _ **The Sunset Legion**_ **** Paramount, 1928. 70 min. D: Lloyd Ingraham and Alfred L. Werker. SC: Frank M. Clifton and Garrett Graham. With Fred Thompson, Edna Murphy, William Courtright, Harry Woods, Slim Whitaker. In order to capture an outlaw gang a Texas Ranger takes on the guise of a masked man as well as a gun peddler. Entertaining Fred Thompson silent opus with his horse Silver King in dual roles!\n\n**4201** _ **Sunset of Power**_ **** Universal, 1936. 66 min. D: Ray Taylor. SC: Earle Snell. With Buck Jones, Dorothy Dix, Charles Middleton, Donald Kirke, Charles King, Ben Corbett, William Lawrence, Joe de la Cruz, Nina Campana, Murdock MacQuarrie, Allan Sears, Glenn Strange, Monty Vandergrift, Eumenco Blanco. A ranch foreman tries to find out who is stealing a rancher's cattle while romancing the granddaughter of his boss, who resents his only grandchild being female. Interesting and well made Buck Jones vehicle.\n\n**4202** _ **Sunset on the Desert**_ **** Republic, 1942. 63 min. D: Joseph Kane. SC: Gerald Geraghty. With Roy Rogers, George \"Gabby\" Hayes, Lynne Carver, Bob Nolan and The Sons of the Pioneers (Tim Spencer, Lloyd Perryman, Pat Brady, Hugh Farr, Karl Farr), Frank M. Thomas, Beryl Wallace, Glenn Strange, Douglas Fowley, Fred Burns, Roy Barcroft, Henry Wills, Forrest Taylor, Bob Woodward, Ed Cassidy, Cactus Mack. When a group of wicked land grabbers try to take over Roy Rogers' hometown, he sets out to stop them. Average Roy Rogers action musical in which the star has a dual role.\n\n**4203** _ **Sunset Pass**_ **** Paramount, 1933. 64 min. D: Henry Hathaway. SC: Jack Cunningham and Gerald Geraghty. With Randolph Scott, Tom Keene, Kathleen Burke, Harry Carey, Noah Beery, Leila Bennett, Fuzzy Knight, Kent Taylor, George Barbier, Vince Barnett, Patricia Farley, Charles Middleton, Christian J. Frank, Tom London, Frank Beal, Alan Bridge, Robert Kortman, Jim Mason, Nelson McDowell. A government agent, working undercover as a cowpoke, falls in love with a woman whose brother is suspected of being behind a rustling operation. A top flight cast highlights this sound remake of Zane Grey's novel, first filmed by Paramount in 1929 with Jack Holt and remade (q.v.) by RKO Radio in 1946.\n\n**4204** _ **Sunset Pass**_ **** RKO Radio, 1946. 59 min. D: William Berke. SC: Norman Houston. With James Warren, Nan Leslie, Jane Greer, Steve Brodie, John Laurenz, Robert Clarke, Harry Woods, Harry Harvey, Slim Balch, Roy Bucko, Steve Stevens, George Plues, Clem Fuller, Artie Ortego, Buck Bucko, Bob Dyer, Slim Hightower, Boyd Stockman, Frank O'Connor, Robert Bray, Florence Pepper, Vonne Lester, Dennis Waters, Marcia Dodd, Dorothy Curtis. Agents for an express company are trying to locate a stolen gold shipment. Fair third screen version of the Zane Grey work.\n\n**4205** _ **Sunset Range**_ **** First Division, 1935. 59 min. D: Ray McCarey. SC: Paul Schofield. With Hoot Gibson, Mary Doran, James Eagles, Walter McGrail, John Elliott, Ralph Lewis, Eddie Lee, Kitty McHugh, Lee Fong, Martha Sleeper, Fred Gilman, Slim Whitaker, Fred Humes, Horace B. Carpenter, George Sowards, Lem Sowards, Jim Corey, Bill Hickey, Joe Jackson. A man involved with a robbery gang gives his sister title to a ranch and a cowboy hunts the outlaws after the brother is shot trying to protect his sibling. Pleasant film with heavy emphasis on comedy, including a sequence where the female ranch owner tries to dress her ranch hands like Hollywood cowboys.\n\n**4206** _ **Sunset Serenade**_ **** Republic, 1942. 58 min. D: Joseph Kane. SC: Earl Fenton. With Roy Rogers, George \"Gabby\" Hayes, Joan Woodbury, Helen Parrish, Onslow Stevens, Frank M. Thomas, Bob Nolan and The Sons of the Pioneers (Tim Spencer, Lloyd Perryman, Pat Brady, Hugh Farr, Karl Farr), Roy Barcroft, Jack Kirk, Dick Wessell, Rex Lease, Jack Ingram, Fred Burns, Budd Buster, Jack Rockwell, Art Mix, Rex Lease, Karl Hackett, Ed Peil, Sr., Steve Clark, Frank Ellis, Curley Dresden, Mary MacLaren, Lynton Brent, Monte Montague, Jack O'Shea, Pascale Perry, Eddie Juarequi, Charles R. Moore. Roy Rogers suspects a housekeeper and her boyfriend are trying to murder a young heir and his pretty guardian. Fair Roy Rogers opus.\n\n**4207** _ **The Sunset Trail**_ **** Tiffany, 1932. 62 min. D: B. Reeves Eason. SC: Bennett Cohen. With Ken Maynard, Ruth Hiatt, Frank Rice, Philo McCullough, Buddy Hunter, Richard Alexander, Frank Ellis, Slim Whitaker, Jack Rockwell, Lew Meehan, Bud Osborne, Bud McClure. Two cowboys are in love with the same girl whose ranch is being sought by crooks. Pretty good Ken Maynard action feature.\n\n**4208** _ **Sunset Trail**_ **** Paramount, 1939. 69min. D: Lesley Selander. SC: Norman Houston. With William Boyd, George Hayes, Russell Hayden, Charlotte Wynters, Jan Clayton, Robert Fiske, Kathryn Sheldon, Maurice Cass, Anthony Nance, Kenneth Harlan, Alphonse Ethier, Glenn Strange, Jack Rockwell, Tom London, Claudia Smith, Jerry Jerome, Al Ferguson, Wen Wright, Frank Ellis, Jim Corey, Fred Burns, Horace B. Carpenter, Charles Murphy, Bob Woodward, Ralph Bucko, Roy Bucko. Hopalong Cassidy poses as a dude to thwart a villain who is trying to steal a guest ranch from a woman and her daughter. Very good Hoppy series entry.\n\n**4209** _ **Super Colt 38**_ **** Columbia, 1969. 91 min. Color. D: Federico Curiel. SC: Federico Curiel and Armando Le Molle. With Jeffrey Hunter, Rosa Maria Vazquez, Pedro Armendariz, Jr., Andres Garcia, Quintin Bulnes, Dagoberto Rodriguez, Chano Urueta, Francisco Reiguera, Carlos Riquelme, Rene Barrera, Pascal Garcia Pena, Jose Eduardo Perez, Victor Alcocer, Raul Perez Prieto. A lawman, who gave up his profession after being forced to kill an outlaw who was a childhood friend, must decide whether to take up his guns again. Competent Mexican Western produced by Luis Enrique Vergara.\n\n**4210** _ **Support Your Local Gunfighter**_ **** United Artists, 1971. 92 min. Color. D: Burt Kennedy. SC: James Edward Grant. With James Garner, Suzanne Pleshette, Jack Elam, Joan Blondell, Harry Morgan, Marie Windsor, Henry Jones, John Dehner, Chuck Connors, Dub Taylor, Kathleen Freeman, Willis Bouchey, Walter Burke, Gene Evans, Dick Haynes, John (Day) Daheim, Ellen Corby, Ben Cooper, Grady Sutton, Herbert Vigran, Terry Wilson, Jim Nolan, Guy Way. A con man escapes from his planned wedding, is mistaken for a gunman and gets involved with a town whose citizens are divided over rival mine operations. Lukewarm sequel to _**Support Your Local Sheriff**_ (q.v.); mediocre except for a fine supporting cast.\n\n**James Garner and Suzanne Pleshette in** _**Support Your Local Gunfighter**_ **(United Artists, 1971).**\n\n**  \n**\n\n**4211** _ **Support Your Local Sheriff**_ **** United Artists, 1969. 93 min. Color. D: Burt Kennedy. SC: William Bowers. With James Garner, Joan Hackett, Walter Brennan, Harry Morgan, Jack Elam, Bruce Dern, Henry Jones, Walter Burke, Dick Peabody, Gene Evans, Willis Bouchey, Kathleen Freeman, Gayle Rogers, Dick Haynes, Richard Hoyt, Marilyn Jones, Chubby Johnson. A soldier of fortune is hired to be the sheriff of a small town and oppose a local family charging heavy tolls to use its road because gold has been discovered there. Clever genre comedy that proves amusing. Sequel: _**Support Your Local Gunfighter**_ (q.v.).\n\n**4212** _ **Surrender**_ **** Republic, 1951. 90 min. D: Allan Dwan. SC: James Edward Grant and Sloan Nibley. With Vera Ralston, John Carroll, Walter Brennan, Francis Lederer, Maria Palmer, William Ching, Jane Darwell, Roy Barcroft, Paul Fix, Esther Dale, Edward Norris, Howard Chamberlin, Norman Budd, Nacho Galindo, Jeff York, Mickey Simpson, Kenne Duncan, Dick Elliott, Ralph Dunn, Virginia Farmer, J. Louis Johnson, Glenn Strange, Tex Terry, Elizabeth Dunne, Cecil Elliott, Paul Stander, Wesley Hopper, Fred Hoose, Shelby Bacon, Tina Menard, Charles Morton, Doris Cole, Tony Roux, Petra Silva, Frank Dae, Al Murphy, Al Rhein. Wanted by the law, a woman weds a newspaper man, kills her first husband when he threatens to expose her bigamy and runs off with a gambler with both pursued by a sheriff. Somber, but well made and acted, Republic \"A\" production.\n\n**4213** _ **Susanna Pass**_ **** Republic, 1949. 67 min. Color. D: William Witney. SC: Sloan Nibley and John K. Butler. With Roy Rogers, Dale Evans, Estelita Rodriguez, Foy Willing and The Riders of the Purple Sage, Martin Garralaga, Robert Emmett Keane, Lucien Littlefield, Douglas Fowley, David Sharpe, Robert Bice. A game warden tries to stop the destruction of local animal life by crooks out to sabotage a fish hatchery. Entertaining Roy Rogers vehicle.\n\n**4214** _ **Susannah of the Mounties**_ **** 20th Century\u2013Fox, 1939. 78 min. D: William A. Seiter. SC: Fidel LaBarba, Walter Ferris, Robert Ellis and Helen Logan. With Shirley Temple, Randolph Scott, Margaret Lockwood, J. Farrell MacDonald, Maurice Moscovitch, Martin Good Rider, Moroni Olsen, Victor Jory, Lester Mathews, Leyland Hodgson, Herbert Evans, Jack Luden, Charles Irwin, John Sutton, Chief Big Tree, Larry Dobbs, Harold Goodwin, Chief Thunderbird, Bill Wilkerson, Russ Clark, Herbert Heywood. A little orphan girl is raised by a Mountie at a remote post where she aids in his romance with a pretty girl as well as warning of an Indian attack. This Shirley Temple feature is a pleasant affair that will also appeal to Randolph Scott fans.\n\n_**Suspected**_ see _**Texas Dynamo**_\n\n**4215** _ **Sutter's Gold**_ **** Universal, 1936. 94 min. D: James Cruze. SC: Jack Kirkland, Walter Woods and George O'Neil. With Edward Arnold, Lee Tracy, Binnie Barnes, Katherine Alexander, Addison Richards, Montagu Love, John Miljan, Robert Warwick, Harry Carey, Mitchell Lewis, William Janney, Ronald Cosbey, Nan Grey, Joanne Smith, Billy Gilbert, Aura De Silva, Allen Vincent, Harry Cording, Sidney Bracey, Bryant Washburn, Gaston Glass, Frank Reicher, Harry Stubbs, William Gould, George Irving, William Gilbert, William Ruhl, Russell Hopton, John King, George Lloyd, Walter Long, Ed Brady, John Bleifer, Clarence H. Wilson, Russ Powell, Priscilla Lawson, Don Briggs, Oscar Apfel, Albert J. Smith, Neely Edwards, Jose Rubio, Jim Thorpe, Paul Weigel, Pedro Regas. Swiss immigrant Johann August Sutter builds an empire in the West only to have it destroyed when gold is discovered at his California mill in 1849. Gigantic screen effort that was a financial failure in its time but a film that deserves viewing.\n\n**4216** _ **Swamp of Lost Monsters**_ **** Trans-International, 1965. 80 min. Color. D: Rafael Baledon. SC: Ramon Obon. With Gaston (Santos) Sands, Manola Saavedra, Pedro de Aguillon, Manuel Donde, Sara Cabrera, Salvador Godinez, Lupe Carrilles, Jose Dupeyron, Herman Vera, Gabriel Alvarez, Arturo Corona, Rayo de Plata (horse). In a remote area of Mexico the inhabitants are terrorized by a monster from a deep lake, with a cowboy and is sidekick investigating. Mexican horror Western with cheesy monsters made worse by its dubbing; released in its homeland as _**El Pantano de las Animas**_ (The Swamp of the Lost Souls) by Alameda Films. TV title: _**Swamp of the Lost Souls**_.\n\n_**Swamp of the Lost Souls**_ see _**Swamp of Lost Monsters**_\n\n**4217** _ **The Sweet Creek County War**_ **** Key International, 1979. 99 min. Color. D-SC: J. Frank James. With Richard Egan, Albert Salmi, Nita Talbot, Slim Pickens, Robert Wilke, Joe Orton, Ray Cardi, Tom Jackman. Two adversaries team to help settlers harassed by outlaws. Well done independent production; Richard Egan's last film.\n\n**4218** _ **Sweet Georgia**_ **** Boxoffice International, 1972. 60 min. Color. D-SC: Edward Boles. With Marsha Jordan, Barbara Caron, Gene Drew, Chuck Lawson, Bill King, Jr., Al Wilkins. A sadistic fat drunk tries to strike gold on his ranch while his abused, promiscuous young wife seduces anyone who gets near her. Tawdry, cheap soft core romp.\n\n**4219** _ **Swifty**_ **** Diversion\/Grand National, 1935. 62 min. D: Alan James. SC: Bennett Cohen. With Hoot Gibson, June Gale, George Hayes, Ralph Lewis, Wally Wales, Robert Kortman, William Gould, Lafe McKee, Art Mix, Duke R. Lee, Starlight (horse). A drifter is falsely accused of murdering a rancher and escapes to find the killer. The lack of good production values greatly hinders this otherwise fair Hoot Gibson vehicle.\n\n**4220** _ **Swing, Cowboy, Swing**_ **** Three Crown, 1946. 60 min. D-SC: Elmer Clifton. With Max Terhune, Cal Shrum and His Rhythm Rangers, Alta Lee, Walt Shrum and His Colorado Hillbillies, I. Stanford Jolley, Frank Ellis, Ed Cassidy, Ted Adams, Tom Hubbard, Shorty Woodward, Don Weston, Ann Roberts, Phil Dunham, Ace Dehne. While appearing in a small town, a musical troupe is the target of a mysterious killer. Fans of old time country music will go for this low budget affair (others beware) that Astor reissued in 1949 as _**Bad Man from Big Bend**_.\n\n**4221** _ **Swing in the Saddle**_ **** Columbia, 1944. 68 min. D: Lew Landers. SC: Elizabeth Beecher, Morton Grant and Radford Ropes. With Jane Frazee, Guinn Williams, Red River Dave, Slim Summerville, Mary Treen, Sally Bliss (Carla Balenda), Carole Mathews, Byron Foulger, The Hoosier Hot Shots (Charles \"Gabe\" Ward, Ken Trietsch, Paul \"Hezzie\" Trietsch, Gil Taylor), The (Nat) King Cole Trio, Jimmy Wakely and His Oklahoma Cowboys (Foy Willing, Fiddlin' Arthur Smith, Art Wenzel, Don Weston, Jack Statham), Cousin Emmy, Emmett Lynn, Virginia Sale, Earle Hodgins. A pretty girl, who comes to a ranch through a misunderstanding, ends up engaged to the foreman and winning a local singing contest. Fun film with far more emphasis on music than action. This is the only feature film starring the sadly neglected singer-songwriter Red River Dave McEnery, who also starred in two 1948 Universal Western featurettes, _**Echo Ranch**_ and _**Hidden Valley Days**_ , as well as the short _**Pretty Women**_ (Sack Amusements, 1949), plus 14 three-minute Western songfests for the Soundies Corporation of America between 1942 and 1946.\n\n**4222** _ **Swing the Western Way**_ **** Columbia, 1947. 66 min. D: Derwin Abrahams. SC: Barry Shipman. With Jack Leonard, Mary Dugan, Thurston Hall, Regina Wallace, Jerry Wald and His Orchestra, Johnny Bond, The Hoosier Hot Shots (Charles \"Gabe\" Ward, Ken Trietsch, Paul \"Hezzie\" Trietsch, Gil Taylor), The Crew Chiefs, Tristram Coffin, Kenneth MacDonald, Ralph Littlefield, Sam Flint, George Lloyd, Jacques O'Mahoney (Jock Mahoney), Eddie Acuff, Lane Bradford, Ralph Dunn, Gary Gray, Ted Stanhope, Douglas D. Coppin, Lynn Craft, Rube Schaffer, Earl Brown, George Dockstader. The Hoosier Hot Shots get involved with a rodeo star and a windbag who plans to marry a rich spinster so he can buy a ranch the mob wants for a casino site. Convoluted program Western songfest headlining band singer Jack Leonard.\n\n_**A Swirl of Glory**_ see _**Sugarfoot**_\n\n_**Sword of Zorro**_ see _**The Three Swords of Zorro**_\n\n**4223** _ **Taggart**_ **** Universal, 1964. 85 min. Color. D: R.G. Springsteen. SC: Robert Creighton Williams. With Tony Young, Dan Duryea, Dick Foran, Elsa Cardenas, Emile Meyer, Jean Hale, Peter Duryea, David Carradine, Harry Carey, Jr., Bob Steele, Ray Teal, Arthur Space, Sarah Selby, Stuart Randall, Bill Henry, Tom Reese, George Murdock. A man hunting for the outlaws who murdered his folks is tracked through Indian country by gunmen. Well made and entertaining adaptation of the Louis L'Amour novel.\n\n**4224** _ **Take a Hard Ride**_ **** 20th Century\u2013Fox, 1975. 108 min. Color. D: Anthony M. Dawson (Antonio Marghertitti). SC: Eric Bercovici and Jerry Ludwig. With Jim Brown, Lee Van Cleef, Fred Williamson, Jim Kelly, Catherine Spaak, Dana Andrews, Barry Sullivan, Harry Carey, Jr., Robert Donner, Charles McGregor, Leonard Smith, Ronald Howard, Ricardo Palacios, Robin Levitt, Buddy Joe Hooker, Hal Needham, Paul Costello. After his boss is murdered by outlaws, a black cowboy teams with a gambler and an Indian scout in taking money across the desert to be delivered to its rightful owners in Mexico. Fairly exciting action feature made in the Canary Islands.\n\n**4225** _ **Take It Big**_ **** Paramount, 1944. 75 min. D: Frank McDonald. SC: Howard J. Green. With Jack Haley, Harriet Hilliard, Mary Beth Hughes, Richard Lane, Arline Judge, Fritz Feld, Lucille Gleason, Fuzzy Knight, Frank Forest, George Meeker, Nils T. Granlund, Ozzie Nelson and His Orchestra, Ralph Peters, Rochelle and Beebe, Pansy the Horse (Andy Mayo), Montie Montana, Evelyn Finley, Byron Foulger, Johnny Arthur, Billy Nelson, Will Wright, Sam Flint, Edward Keane, Art Mix, Lee Phelps, Eddie Kane, Tom Kennedy, Dewey Robinson, Donald Kerr, George McKay, Bob Burns, Victor Cox, Hal K. Dawson, John Dilson, Margia Dean, Helene Bank, Clancy Cooper, Clyde Dilson. A third rate entertainer inherits a posh dude ranch and ignores the singer who loves him for a gold digger trying to make her boyfriend jealous. So-so contemporary Western musical comedy.\n\n**4226** _ **Take Me Back to Oklahoma**_ **** Monogram, 1940. 57 min. D: Al Herman. SC: Robert Emmett (Tansey). With Tex Ritter, Terry Walker, Arkansas Slim Andrews, Bob Wills and The Texas Playboys (Leon McAuliffe, Johnnie Lee Wills, Wayne Johnson, Son Caz Lansford, Eldon Shamblin), Bob McKenzie, Karl Hackett, Donald Curtis, Gene Alsace, Olin Francis, Carleton Young, George Eldredge, Sherry Tansey, Jack C. Smith, Victor Cox, Bob Card, Chick Hannon, Herman Nowlin, Tex Cooper, Tex Phelps, Rose Plummer. A singing cowboy joins with a hired gunman to double cross a crook after a friend's stage line. Outside of the Tex Ritter and Bob Wills' songs there is not much to recommend this pedestrian effort.\n\n**4227** _ **Take Me to Town**_ **** Universal-International, 1953. 81 min. Color. D: Douglas Sirk. SC: Richard Morris. With Ann Sheridan, Sterling Hayden, Philip Reed, Lee Patrick, Lee Aaker, Phyllis Stanley, Harvey Grant, Dusty Henley, Guy Williams, Alice Kelley, Lane Chandler, Larry Gates, Frank Sully, Forrest Lewis, Ann Tyrell, Dorothy Neumann, Robert Anderson. A saloon singer escapes from a lawman and ends up at a logging camp where she is taken in by a lumberjack minister and his three orphan boys. Likable musical comedy vehicle for Ann Sheridan.\n\n**4228** _ **Tale of Gold**_ **** Wrather Corporation, 1956. 75 min. Color. D: Earl Bellamy. SC: Jack Natteford, Thomas Seller and Herbert Purdom. With Clayton Moore, Jay Silverheels, Allen Pinson, Wayne Burson, Harry Lauter, Louise Lewis, Trevor Bardette, Charles Stevens, Mike Ragan (Holly Bane), John Cason, Pat O'Malley, Ralph Sanford, Pierce Lyden, Robert Roark, Bill Ward, Walt LaRue, Jerry Brown, Sandy Sanders, Robert Swan, George Mather, William J. Tanner, Mae Morgan. The Lone Ranger and Tonto try to prevent violence resulting from a horse race wager between the citizens of a small town and a Cheyenne Indian tribe in addition to going after gold thieves and helping a farmer get back his savings. Pleasant telefilm from \"The Lone Ranger\" (ABC-TV, 1949\u201357) episodes \"Decision for Chris McKeever,\" \"A Harp for Hannah\" and \"Quarterhorse War.\"\n\n**4229** _ **Tales of Adventure**_ **** Path\u00e9, 1954. 80 min. D: Herbert Kline. SC: Jerome Gruskin, Nord Riley, Halsted Welles and Richard Wormser. With Lon Chaney, Don De Fore, Rita Moreno, Robert Hutton, Robert Lowery, Coleen Gray, Eve McVeagh, Frank Silvera, Armando Silvestre, Jonathan Hale. Three Jack London stories, including the trial of an evil man in the north woods and the marriage of an Indian maiden, are recounted in this film issued only to television and made up of a trio (\"The House of Pride,\" \"The Marriage of Lit-Lit\" and \"The Trial\") of 1952 segments from \"The Schlitz Playhouse of Stars\" (CBS-TV, 1951\u201355); the episode with Lon Chaney was filmed near Mexico City. Also called _**Flight from Adventure**_ and _**Jack London's Tales of Adventure**_.\n\n_**Talion**_ see _**An Eye for an Eye**_\n\n**4230** _ **The Talisman**_ **** Universal Entertainment\/Gillman Film Corporation, 1970. 93 min. Color. D-SC: John Carr. With Ned Romero, Linda Hawkins, Richard Thies, Jerald Cormier, Raymond Brown, Raymonda de Anda, Louis Bacigalupi. An Indian warrior and a white woman, the survivors of a wagon train massacre, form an uneasy alliance until three renegade Confederates rape and murder the woman and the brave swears revenge. Violent, and somewhat obscure, oater also called _**The Savage American**_. Average.\n\n**4231** _ **Tall in the Saddle**_ **** RKO Radio, 1944. 87 min. D: Edwin L. Marin. SC: Michael Hogan and Paul Fix. With John Wayne, Ella Raines, Ward Bond, George \"Gabby\" Hayes, Audrey Long, Elisabeth Risdon, Russell Wade, Don Douglas, Frank Puglia, Emory Parnell, Raymond Hatton, Paul Fix, Harry Woods, Cy Kendall, Bob McKenzie, Wheaton Chambers, Walter Baldwin, Russell Simpson, Frank Orth, Russell Hopton, George Chandler, Eddy Waller, Frank Darien, Clem Bevans, Erville Alderson, William Desmond, Hank Bell, Tom Smith, Victor Cox, Ben Johnson, Denver Dixon. A cowboy is hired work at a ranch where the owner has been shot and a young woman with a guardian aunt inherits the place but is manipulated by a corrupt lawyer Rugged, well done John Wayne adventure with a good blend of action, comedy and mystery.\n\n**4232** _ **Tall Man Riding**_ **** Warner Bros., 1955. 83 min. Color. D: Lesley Selander. SC: Joseph Hoffman. With Randolph Scott, Dorothy Malone, Peggie Castle, Robert Barrat, Willing Ching, John Baragrey, John Dehner, Paul Richards, Mickey Simpson, Lane Chandler, Joe Bassett, Charles Watts, Russ Conway, Mike Ragan (Holly Bane), Carl Andre, John Logan, Guy (Edward) Hearn, William Fawcett, Nolan Leary, Phil Rich, Eva Novak, Buddy Roosevelt, Jack Henderson, Bob Peeples, Dub Taylor, William Norton Bailey, Bob Stephenson, Roger Creed, Vernon Rich. After fourteen years a man returns home to take revenge on the land baron who stole his land and ruined his intended marriage. Entertaining Randolph Scott feature with a good script.\n\n**4233** _ **The Tall Men**_ **** 20th Century\u2013Fox, 1955. 122 min. Color. D: Raoul Walsh. SC: Sydney Boehm and Frank Nugent. With Clark Gable, Jane Russell, Robert Ryan, Cameron Mitchell, Juan Garcia, Emile Meyer, Harry Shannon, Steve Darrell, Will Wright, Robert Adler, Russell Simpson, Tom Wilson, Tom Fadden, Tom White, Argentine Brunetti, Doris Kemper, Carl Harbaugh, Post Park, Jack Mather. Following the Civil War two brothers head West, become involved with a dishonest cattleman and rescue a pretty woman from Indians. Rugged, lusty melodrama with the fine teaming of Clark Gable and Jane Russell.\n\n**4234** _ **The Tall Stranger**_ **** Allied Artists, 1957. 83 min. Color. D: Thomas Carr. SC: Christopher Knopf. With Joel McCrea, Virginia Mayo, Barry Kelley, Michael Ansara, Whit Bissell, James Dobson, George Neise, Adam Kennedy, Michael Pate, Leo Gordon, Ray Teal, Robert Foulk, George J. Lewis, Guy Prescott, Mauritz Hugo, William Haade, Pierce Lyden, Tom London. After having his life saved by the passengers of a wagon train, a man agrees to help them settle their land in opposition to crooks. Colorful, if not overly exciting, Joel McCrea film that will please his fans.\n\n**4235** _ **The Tall T**_ **** Columbia, 1957. 78 min. Color. D: Budd Boetticher. SC: Burt Kennedy. With Randolph Scott, Maureen O'Sullivan, Richard Boone, Arthur Hunnicutt, Skip Homeier, Henry Silva, John Hubbard, Robert Burton, Robert Anderson, Fred E. Sherman, Chris Olsen. Three outlaws kidnap the daughter of a rich cattleman and hold her hostage at a way station where a rancher tries to help her. Exceedingly fine Randolph Scott outing, well written and taut.\n\n**4236** _ **The Tall Texan**_ **** Lippert, 1953. 81 min. D: Elmo Williams. SC: Sam Roeca. With Lloyd Bridges, Lee J. Cobb, Marie Windsor, Luther Adler, Syd Saylor, Samuel Herrick, George Steele, Dean Train. A group of wagon passengers, including a lady of easy virtue and a lawman with a prisoner, head into sacred Indian lands in search of gold. Low budget but enjoyable action drama.\n\n_**Tall Timber**_ see _**Park Avenue Logger**_\n\n**4237** _ **The Tall Women**_ **** Allied Artists, 1967. 95 min. Color. D: Sidney Pink. SC: Mino Rolli. With Anne Baxter, Maria Perschy, Gustavo Rojo, Rossella Como, Adriana Ambesi, Mara Cruz, Christa Linder, Luis Prendes, Maria Mahor, Fernando Hilbeck, John Clarke. Several women who survive an Indian massacre of their wagon train trek across the desert and learn to fight to stay alive. Better-than-average European made Western first issued in 1966 as _**Donne alla Frontiera**_ (Women at the Frontier) by Danny\/L.M.\/Danubia Film.\n\n**4238** _ **The Taming of the West**_ **** Columbia, 1939. 55 min. D: Norman Deming. SC: Robert Lee Johnson and Charles Francis Royal. With Bill Elliott, Iris Meredith, Dub Taylor, Dick Curtis, James Craig, Stanley Brown, Kenneth MacDonald, Ethan Allen, Victor Wong, Charles King, Lane Chandler, Jack Kirk, George Morrell, Art Mix, Don Beddoe, Richard Fiske, John Tyrrell, Bob Woodward, Hank Bell, Stella Le Saint, Irene Herndon, Francis Walker, Fred Burns, Horace B. Carpenter, Fred Parker, Ray Jones, Al Haskell, Jack Evans. The new sheriff of a lawless town tries to bring peace but is opposed by supposedly honest businessmen who are behind the outlaws terrorizing the area. Bill Elliott fans will enjoy this action filled outing, the first of four \"Wild Bill Saunders\" adventures.\n\n**4239** _ **Tangled Trails**_ **** William Steiner, 1921. 56 min. D-SC: Charles Bartlett. With Neal Hart, Violet Palmer, Gladys Hampton, Jean Barry, Jules Cowles, Ed Roseman. A Canadian Mounted Policeman is after a dishonest stock promoter who murdered his partner, and the chase leads to Gotham and back, along with his becoming involved with two sisters. Complicated silent drama produced by star Neal Hart.\n\n**4240** _ **Tarantula**_ **** Universal-International, 1955. 80 min. D: Jack Arnold. SC: Martin Berkeley. With John Agar, Mara Corday, Leo G. Carroll, Nestor Paiva, Ross Elliott, Ed Rand, Raymond Bailey, Clint Eastwood, Jane Howard, Billy Wayne, Hank Patterson, Dee Carroll, Bert Holland, Steve Darrell, Tom London, Edgar Dearing, James J. Hyland, Stuart Wade, Vernon Rich, Bob Nelson, Eddie Parker, Bing Russell, Ray Quinn, Robert R. Stephenson, Don Dillaway, Bud Wolfe, Jack Stoney, Rusty Wescoatt. In the desert a scientist works on a serum to grow gigantic crops and it causes a tarantula to mutate with the huge creature going on a rampage. Another big bug monster caper in the West, but nicely made and satisfying for sci-fi fans.\n\n**4241** _ **Target**_ **** RKO Radio, 1952. 60 min. D: Stuart Gilmore. SC: Norman Houston. With Tim Holt, Richard Martin, Linda Douglas, Walter Reed, Harry Harvey, John Hamilton, Lane Bradford, Riley Hill, Mike Ragan (Holly Bane). Two cowboys find themselves opposing a dishonest land agent and his outlaw gang. One of the last in Tim Holt's RKO series; it is fast on action from beginning to end.\n\n**4242** _ **Taste of Death**_ **** Esterofilm, 1968. 92 min. Color. D: Sergio Merolle. SC: Biagio Proietti. With Andrea Giordano (Chip Gorman), John Ireland, Raymond Pellegrin, Betsy Bell, Bruno Corazzari, Giovanni Petrucci, Fulvio Pellegrino, Ruggero Chessa, Claudio Scarchilli, Giuseppe Altamurra, Mireille Granelli, Sergio Scarchilli. An ex-lawman teams with a cowboy to combat a marauding outlaw gang, one of them being the young man's father. Supposed Colorado locales help this standard Spaghetti Western originally called _**Quanto Costa Morire**_ (Cost of Dying).\n\n**4243** _ **Taste of Killing**_ **** Altamira Films, 1966. 86 min. Color. D: Tonino Valerii. SC: Victor Auz. With Craig Hill, George Martin, Peter Carter (Piero Lulli), Fernando Sancho, Franco Ressel, George Wang, Diana Martin, Graham Sooty (Franco Pesce), Rada Rassimov, Jose Marco, Lorenzo Robledo, Sancho Garcia, Jose Canalejas, Jose Manuel Martin, Dario De Grassi, Frank Brana, Olga Karlatos. A bounty hunter agrees to a double reward if he can stop a planned stagecoach robbery. Pretty fare Italian-Spanish co-production made as _**Per il Gusto di Uccidere**_ (For the Taste of Killing).\n\n**4244** _ **Taza, Son of Cochise**_ **** Universal-International, 1954. 79 min. Color. D: Douglas Sirk. SC: Gerald Drayson Adams. With Rock Hudson, Barbara Rush, Gregg Palmer, Bart Roberts, Morris Ankrum, Ian MacDonald, Richard Cutting, Joseph Sawyer, Robert Burton, Eugene Iglesias, Lance Fuller, Brad Jackson, James Van Horn, Charles Horvath, Robert Hoy, William Leslie, Dan White, Edna Parrish, Seth Bigman, John Kay Hawks, Barbara Burck, Jeff Chandler. Cochise's son is made head of the Apaches after his father's death but finds himself at odds with his brother over how to deal with whites as well as a pretty maiden. Passable program feature with Jeff Chandler briefly reprising his Cochise role from _**Broken Arrow**_ and _**The Battle at Apache Pass**_ (qq.v.).\n\n**4245** _ **Tearin' Loose**_ **** Artclass, 1925. 49 min. D: Richard Thorpe. SC: Frank L. Inghram. With Wally Wales, Jean Arthur, Charles (Slim) Whitaker, Alfred Hewston, Polly Vann, Harry Belmour, William Ryno, Frank Ellis, Vester Pegg. An imposter tries to take over an old man's ranch and blames his nephew, the rightful heir, for a series of crimes. Only a 12 minute edited version of this silent Wally Wales vehicle exists.\n\n**4246** _ **Tedeum**_ **** Film Ventures, 1972. 99 min. Color. D: Enzo G. Castellari. SC: Gianni Simonelli. With Jack Palance, Timothy Brent, Lionel Stander, Francesca Romana Coluzzi, Mabel Karin, Eduardo Fajardo, Riccardo Garrone, Miguel Pedregosa, Maria Vico, Rocco Lero, Renzo Palmer, Carlo Ruffini, Angel Alvarez, Franco Borelli, Fidel Gonzales, Ana Suriani, Miguel Pedrosa, Guillermo Mendez, Brunco Boschetti, Karl Braun, Dante Cleri, Ivan Palance. A crook thinks the mine he inherited is worthless and plans to get rid of it but his equally dishonest family finds it is rich in ore and tries to get it away from him. Fairly funny Spaghetti Western spoof also called _**Father Jackleg**_ and _**Sting of the West**_.\n\n**4247** _ **Teenage Monster**_ **** Howco International, 1957. 73 min. D: James (Jacques) Marquette. SC: Ray Buffum. With Anne Gwynne, Stuart Wade, Gloria Castillo, Charles Courtney, Gilbert Perkins, Frank Davis, Stephen Parker, Norman Leavitt, Jim McCullough, Gabye Mooradian, Arthur Berkeley. Near a remote Western town a boy is struck by a crashing meteor and grows up to be a rampaging monster hunted by a posse. Low grade mixture of horror and Western themes; mediocre. TV title: _**Meteor Monster**_.\n\n**4248** _ **The Telegraph Trail**_ **** Warner Bros., 1933. 55 min. D: Tenny Wright. SC: Kurt Kempler. With John Wayne, Marceline Day, Frank McHugh, Otis Harlan, Albert J. Smith, Yakima Canutt, Lafe McKee, Clarence Geldert, Slim Whitaker, Frank Ellis, Jack Kirk, Bud Osborne, Ben Corbett, Al Taylor, Jack Jones, Chuck Baldra, Bob Fleming, Artie Ortego, Chief Big Tree, Bud McClure, Bob Burns. An Army scout organizes the citizens of a town in stringing a telegraph wire after his best friend, in trying to complete the job, is killed by Indians. Patchwork John Wayne film with most of his best action footage culled from the Ken Maynard silent _**The Red Raiders**_ (q.v.); below average.\n\n**4249** _ **Tell Them Willie Boy Is Here**_ **** Universal, 1970. 98 min. Color. D-SC: Abraham Polonsky. With Robert Redford, Katharine Ross, Robert Blake, Susan Clark, Barry Sullivan, Charles McGraw, Charles Aidman, John Vernon, Shelly Novack, Ned Romero, John Day, Lee De Broux, George Tyne, Robert Lipton, Steve Shemayne, Lloyd Gough, John Hudkins, Jerry Velasco, Gary Walberg, Jerome Raphel, Johnny Coons, Stanley Torres, Kenneth Holzman, Joseph Mandel, Spencer Lyons, Everett Creach. In 1909 a Paiute Indian accidentally kills the father of the girl he loves and the two try to elude a posse led by an assistant sheriff. Oater with all kinds of political undertones, but only average.\n\n**4250** _ **Ten Days to Tulara**_ **** United Artists, 1958. 77 min. D: George Sherman. SC: Lawrence Mascott. With Sterling Hayden, Grace Raynor, Rodolfo Hoyos, Carlos Muzquiz, Tony Caravaijal, Juan Garcia. Police pursue an American pilot and his outlaw friend through the Mexican desert to retrieve the gold the two are carrying. Passable melodrama filmed in Mexico.\n\n**4251** _ **$10,000 Blood Money**_ **** Golden Era, 1967. 97 min. Color. D: Romolo Guerrieri. SC: Franco Fogagnolo, Ernesto Gastaldi, Luciano Martino and Sauro Scavolin. With Gary Hudson (Gianni Garko), Claudio Camaso, Fernando Sancho, Lorenda Nusciak, Adriana Ambesi, Pinuccio Ardia, Fidel Gonzales, Franco Lantieri. A bounty hunter and a kidnapper are forced into an uneasy alliance with each trying to double cross the other. Very violent Italian \"Django\" series Western first issued there in 1966 as _**10,000 Dollari per un Massacro**_ ($10,000 for a Massacre) and on video as _**Ten Thousand Dollars for a Massacre**_.\n\n_**Ten Thousand Dollars for a Massacre**_ see _**$10,000 Blood Money**_\n\n**4252** _ **Ten Wanted Men**_ **** Columbia, 1955. 80 min. Color. D: H. Bruce Humberstone. SC: Kenneth Gamet. With Randolph Scott, Jocelyn Brando, Richard Boone, Alfonso Bedoya, Donna Martell, Skip Homeier, Clem Bevans, Leo Gordon, Minor Watson, Lester Mathews, Tom Powers, Dennis Weaver, Lee Van Cleef, Louis Jean Heydt, Kathleen Crowley, Boyd \"Red\" Morgan, Denver Pyle, Francis McDonald, Pat Collins, Paul Maxey, Julian Rivero, Edna Holland, Reed Howes, Jack Perrin, Terry Frost, Franklyn Farnum, George Boyce. A crook frames a cattle baron's nephew on a murder charge because he wants the young man's girl. Better-than-average, with an impressive cast.\n\n**4253** _ **Ten Who Dared**_ **** Buena Vista, 1960. 92 min. Color. D: William Beaudine. SC: Lawrence E. Watkin. With Brian Keith, John Beal, James Drury, R.G. Armstrong, Ben Johnson, L.Q. Jones, Dan Sheridan, David Stollery, Stan Jones, David Frankham, Pat Hogan, Ray Walker, Jack Bighead, Roy Barcroft, Dawn Little Sky. A motley crew of men join Major John Wesley Powell on his expedition exploring the Colorado River in 1869. Surprisingly poor historical film from Walt Disney.\n\n**4254** _ **The Tenderfoot**_ **** First National, 1932. 70 min. D: Ray Enright. SC: Earl Baldwin, Monty Banks and Arthur Caesar. With Joe E. Brown, Ginger Rogers, Lew Cody, Vivien Oakland, Robert Greig, Ralph Ince, Marion Byron, Spencer Charters, Douglas Gerrard, Wilfred Lucas, Nat Pendleton, Richard Cramer, George Chandler, Al Hill, Charles Sullivan, Mae Madison, Dorothy Vernon, Harry Seymour, Herman Bing, John Larkin, Theodore Lorch, Allan Lane, Eddie Kane, George Davis, Joe Barton, Ben Hall, Edith Allen, Jill Dennett, Lee Kohlmar, Gus Leonard, Charlotte Mernam, Zita Moulton, Walter Percival, Bob Perry. A pretty secretary decides to help a cowboy who her boss wants to swindle by having him invest in a Broadway musical. Amusing teaming of Joe E. Brown and Ginger Rogers, based on the play by Richard Carle and George S. Kaufman.\n\n**4255** _ **Tenderfoot**_ **** Buena Vista, 1966. 80 min. Color. D: Byron Paul. SC: Maurice Tombragel. With Brandon de Wilde, James Whitmore, Richard Long, Donald May, Christopher Dark, Judson Pratt, Carlos Romero, Angela Dorian (Victoria Vetri), Rafael Campos, Harry Harvey, Jr. A young man learns the values of growing up in Arizona in the 1850s. Pretty good family drama originally telecast in three parts on Walt Disney's ABC-TV program in 1964.\n\n**4256** _ **A Tenderfoot Goes West**_ **** J.H. Hoffberg, 1936. 65 min. D: Maurice O'Neill. With Jack LaRue, Virginia Carroll, Russell Gleason, Ralph Byrd, Chris-Pin Martin, Si Jenks, John Merton, Joseph Girard, John Ince, Ray Turner, Glenn Strange, Charles Sargent, Oscar Gahan, Merrill McCormick, Bill Patton, Bud Pope, Gertrude Chorre. An outlaw saves an Easterner mistaken for him from a lynch mob. Fair oater comedy from poverty row.\n\n**4257** _ **Tennessee Johnson**_ **** Metro-Goldwyn-Mayer, 1942. 100 min. D: William Dieterle. SC: John Balderson and Wells Root. With Van Heflin, Ruth Hussey, Lionel Barrymore, Marjorie Main, Regis Toomey, J. Edward Bromberg, Grant Withers, Alec Craig, Charles Dingle, Carl Benton Reid, Russell Hicks, Noah Beery, Robert Warwick, Montagu Love, Lloyd Corrigan, William Farnum, Charles Trowbridge, Morris Ankrum, Sheldon Leonard, Harry Worth, Dane Clark, Robert Emmett O'Connor, Lee Phelps, Brandon Hurst, Charles Ray, Harlan Briggs, Hugh Sothern, Frederick Burton, Allen Pomeroy, Duke York, Roy Barcroft, Ed O'Neill, Jack Norton, Russell Simpson, Louise Beavers, James (Jim) Davis, William Roberts, Frank Jaquet, Emmett Vogan, James Kirkwood, Pat O'Malley, Will Wright, William B. Davidson, John Hamilton, Harry Cording, Tom Herbert, Mark Daniels, Harrison Greene, Richard Hall, James Warren, Harry Holman, Alberto Morin, Lloyd Ingraham, Roger Imhof, Patsy Nash, Walter Baldwin, Davison Clark, Herbert Heyes, Frank Hagney, Gibson Gowland, Jeff Corey, Cliff Clark, Frank Austin, Arthur Space, Ray Teal, Ludwig Stossel, Gayne Whitman, Harry Strang, Carl Stockdale, Si Jenks, Al Hill, Henry Roquemore, John Ince, Syd Saylor, Dewey Robinson, Milton Kibbee, Louis Mason, Lee Prather, Bert Roach, Ruth Robinson, Jack Norton, Gladden James, Jack Kenny, Jerry Jerome, Ivan Miller, Ferdinand Munier, Christian J. Frank, Sonny Bupp, Jesse Graves, Frank Dawson, Anna Chandler, Robert Dudley, Paul Everton, Arthur Belasco, Jules Cowles, Albert Godderis, George M. Carleton, Leila McIntyre, Eric Mayne, Bob Stebbins, Jack Zeller, Ernie Alexander, Francis Stevens, Henry Sylvester, Michael Audley. Runaway frontier bond servant Andrew Johnson works his way up the political ladder to eventually become one of the most controversial of U.S. presidents. Excellent biographical film with great work by Van Helfin in the title role, plus an excellent supporting cast, including Morris Anrkum's memorable cameo as Jefferson Davis.\n\n**4258** _ **Tennessee's Partner**_ **** RKO Radio, 1955. 87 min. Color. D: Allan Dwan. SC: Allan Dwan, Milton Krims and D.D. Beauchamp. With John Payne, Rhonda Fleming, Ronald Reagan, Coleen Gray, Anthony Caruso, Leo Gordon, Myron Healey, Morris Ankrum, Chubby Johnson, Joe Devlin, John Mansfield, Angie Dickinson. A cowboy saves a crook from being bushwhacked and the two become pals, with the latter after the former's girl. Mediocre Western that is a bit on the dull side.\n\n**4259** _ **Tension at Table Rock**_ **** RKO Radio, 1956. 93 min. Color. D: Charles Marquis Warren. SC: Winston Miller. With Richard Egan, Dorothy Malone, Cameron Mitchell, Billy Chapin, Royal Dano, Edward Andrews, John Dehner, DeForrest Kelley, Angie Dickinson, Joe De Santis, Harry Lauter, Tom Steele. After killing his partner in self defense, an outlaw is forced to change his identity. Fair screen adaptation of Frank Gruber's _Bitter Sage_.\n\n**4260** _ **Tentacles of the North**_ **** Rayart, 1926. 50 min. D: Louis Chaudet. SC: Leslie Curtis. With Gaston Glass, Alice Calhoun, Al Ferguson, Albert Roscoe, Joseph Girard, T. Hohai. A man finds a young woman stranded on a ship where the crew has died and the men on his vessel chase them into the Arctic. Acceptable low budget silent adaptation of James Oliver Curwood's \"In the Tentacles of the North.\"\n\n**4261** _ **Tenting Tonight on the Old Camp Ground**_ **** Universal, 1943. 59 min. D: Lewis D. Collins. SC: Elizabeth Beecher. With Johnny Mack Brown, Tex Ritter, Fuzzy Knight, Jennifer Holt, John Elliott, Earle Hodgins, The Jimmy Wakely Trio (Jimmy Wakely, Johnny Bond, Scotty Harrell), Rex Lease, Lane Chandler, Alan Bridge, Dennis Moore, Tom London, Reed Howes, Bud Osborne, Lynton Brent, Hank Worden, George Plues, Ray Jones, George Eldredge. A dishonest saloon keeper tries to tempt workers, who are really prisoners, from their jobs building a bridge for a stagecoach route carrying the mail. Sufficient Universal program Western.\n\n_**Tepea**_ see _**Blood and Guns**_\n\n**4262** _ **Tequilla Joe**_ **** CR Cine\/PEA, 1968. 95 min. Color. D: Vincent Eagle (Enzo Dell'Aquila). SC: Fernando Di Leo and Enzo Dell'Aquila. With Antonio Ghidra, Jean Sobieski, Dick Palmer (Mimmo Palmara), Furio Meniconi, Felicita Fanni, Mimmo Bill, Fidel Gonzales, Claudio Ruffini, Gilberto Galimberti, Frank Fargas, Fortunato Arena, Max Fraser, Luciano Doria, Raoul Amari, Remo Capitani, Ivan G. Scratuglia, Leonora Ruffo, Paolo Figlia. A greenhorn deputy allies himself with an old sheriff in fighting corrupt town bosses. Well made Spaghetti Western, produced in Italy as _**...e Venne il Tempo di Uccidere**_ (And Then a Time for Killing).\n\n**4263** _ **Terrible Sheriff**_ **** J.J. Films S.A., 1962. 80 min. Color. D: Alberto De Martino and Antonio Momplet. SC: Mario Guerra, Ettore Scola, Guilio Scarnicci, Vittorio Vighi, Jose Mallorqui and Ruggero Maccari. With Walter Chiari, Licia Calderon, Maria Silva, Raimondo Vianello, Aroldo Tieri, Antonio Vico, Felix Fernandez, Antonio Molino Rojo, Angela Pia, Jose Villasante, Maruja Tamayo, Xan das Bolas, Emilio Rodriguez, Alfonso Rojas, Fernando Hilbeck. Two cowardly brothers consume drugged chickens and become super fighters, getting appointed dual sheriffs of a town harassed by a masked bandit the siblings know to be dead. Strange Spaghetti Western concoction made as _**Due Contro Tutti**_ (Two Against All).\n\n**4264** _ **Territorial Men**_ **** BCI, 1976. 98 min. D: Lawrence Dobkin and William F. Claxton. SC: James Dixon. With Brenda Vaccaro, Sam Groom, Jerry Hardin, Michael St. Clair, Raymond Singer, Robert Donner, Christian Grey, Bert Kramer, Albert Stratton, William Phipps, Mariclare Costello, Louise Latham, Kraig Metzinger, Debbie Lytton, Hallie Morgan. A novice frontier schoolmarm tries to cope with unruly students and local prejudice at the same time falling in love with a recently arrived businessman. Okay telefilm comprised of episodes of \"Sara\" (CBS-TV, 1976).\n\n**4265** _ **The Territory of Others**_ **** Gold Key, 1974. 93 min. Color. D: Francois Bel, Jacqueline Lecompte, Michael Fano and Gerard Vienne. Wildlife in the American desert have developed interdependent relationships over thousands of years and among those presented are a jaguar, poisonous lizard and the rattlesnake. Well done French documentary.\n\n**4266** _ **Terror at Black Falls**_ **** Beckman, 1962. 76 min. D-SC: Richard Sarafian. With House Peters, Jr., Sandra Knight, John Alonzo, Peter Mamakos, Gary Gray, I. Stanford Jolley, Marshall Bradford, Jim Hayward, Maureen Cookson, Randy Clark, Madeleine Holmes, Jim Bysel, Joe Adelman, Armand Alzamora, Mickey Finn, George Cisar, King Moody, Charlotte Johnson, C.B. Lash, William L. (Bill) Erwin, John Quihas. Seeking revenge for the death of his son and the loss of a hand, a madman takes hostages in a remote area before doing battle with the local sheriff. Slow moving and not very interesting poverty row melodrama.\n\n**4267** _ **Terror in a Texas Town**_ **** United Artists, 1958. 81 min. D: Joseph H. Lewis. SC: Ben L. Perry. With Sterling Hayden, Carol Kelly, Sebastian Cabot, Victor Millan, Eugene Martin, Ned Young, Ann Verela, Sheb Wooley, Fred Kohler, Jr., Steve Mitchell, Tyler McVey, Ted Stanhope, Gil Lamb, Frank Ferguson, Hank Patterson. Returning home to his father's Texas ranch a man finds the area terrorized by a land baron after rich oil deposits. Standard oater given some atmosphere by Joseph H. Lewis' direction.\n\n**4268** _ **Terror of the Black Mask**_ **** Embassy, 1967. 97 min. Color. D: Umberto Lenzi. SC: Gino De Santis, Guido Malatesta and Umberto Lenzi. With Pierre Brice, Helene Chanel, Daniele Vargas, Adolf Bufi Landi, Carlo Latimer, Gisela Arden, Massimo Serato, Nero Bernardi, Romano Ghini, Tulio Alamura, Attilo Torelli, Amedeo Trilli, Salvatore Campochiaro, Guido Celano, Gino Marturano, Eleanora Morana, Giovanni Pazzafini, Gino Soldi. The timid stepson of a 17th century Spanish despot is actually the masked Don Diego, alias Zorro, who attacks his stepfather's army and weakens his control over the people. Typical dubbed European oater included here because of the Zorro character. Issued in Europe in 1963 by Romana Film _**L'Invincible Cavaliere Mascherato**_ (The Invincible Masked Cavalier).\n\n**4269** _ **Terror of the Plains**_ **** Reliable, 1935. 57 min. D: Harry S. Webb. SC: Jayne Regan and Carl Krusada. With Tom Tyler, Roberta Gale, William Gould, Charles (Slim) Whitaker, Ralph Lewis, Fern Emmett, Murdock MacQuarrie, Frank Rice, Robert Walker, Nelson McDowell, Jack Kirk, Budd Buster, Jimmy Aubrey, Herman Hack, Jack Cross. A cowboy and his pal go to a ghost town used as the headquarters of an outlaw gang led by a man who committed a murder for which the cowpoke's father is blamed. Another tacky entry in Tom Tyler's Reliable series for producers Bernard B. Ray and Harry S. Webb; the star deserved better.\n\n**4270** _ **The Terror of Tiny Town**_ **** Columbia, 1938. 62 min. D: Sam Newfield. SC: Fred Myton. With Billy Curtis, Yvonne Moray, Little Billy, Bill Platt, John Bambury, Charles Becker, Joseph Herbst, Nita Krebs, George Ministeri, Karl Casitzky, Fern McDill, W.H. O'Docharty, Johnnie Ferr. A bad man pits two families against one another so they will kill each other off and he can get their lands. Outside the novelty of having an all-midget cast, there is not much here for the viewer in this musical opus from producer Jed Buell.\n\n**4271** _ **Terror Trail**_ **** Universal, 1933. 58 min. D: Armand L. Schaefer. SC: Jack Cunningham. With Tom Mix, Naomi Judge, Arthur Rankin, Raymond Hatton, Francis McDonald, Robert Kortman, John St. Polis, Francis Brownlee, Harry Tenbrook, Lafe McKee, W.J. Holmes, Hank Bell, Leonard Trainer, Jim Corey, Jay Wilsey (Buffalo Bill, Jr.). A cowboy sets out to retrieve his horse that has been stolen by an outlaw gang led by the local sheriff. Entertaining Tom Mix feature but not quite up to some of his other Universal work.\n\n**4272** _ **Terror Trail**_ **** Columbia, 1946. 56 min. D: Ray Nazarro. SC: Ed Earl Repp. With Charles Starrett, Smiley Burnette, Barbara Pepper, Lane Chandler, Zon Murray, Elvin Feld, Ozie Waters and His Colorado Rangers, Tommy Coats, George Chesebro, Robert Barron, Budd Buster, Bill Clark, Ted Mapes, Wesley Tuttle, Jack Evans, Matty Roubert, Billy Dix, Edward Howard, George Morrell. When a dishonest rancher tries to start a range war by secretly placing sheep on cattlemen's land, the Durango Kid tries to stop the hostilities. Pretty fair series episode with good supporting work by Lane Chandler and Barbara Pepper.\n\n**4273** _ **Terrors on Horseback**_ **** Producers Releasing Corporation, 1946. 55 min. D: Sam Newfield. SC: George Milton (Milton Raison and George W. Sayre). With Buster Crabbe, Al St. John, Patti McCarthy, I. Stanford Jolley, Henry Hall, Kermit Maynard, Karl Hackett, Marin Sais, Budd Buster, Steve Darrell, Steve Clark, Bud Osborne, Al Ferguson, George Chesebro, Frank Ellis, Jack Kirk, Lane Bradford, George Morrell, Herman Hack, Jack Evans. To avenge the murder of his niece, a cowboy plans to kill off the outlaw gang responsible for her death during a stagecoach robbery. One of the last \"Billy Carson\" films and one of the best; action filled from start to finish.\n\n**4274** _ **The Test**_ **** Reliable, 1935. 55 min. D: Bernard B. Ray. SC: L.V. Jefferson. With Rin Tin Tin, Jr., Grant Withers, Grace Ford, Monte Blue, Lafayette (Lafe) McKee, James (Jimmy) Aubrey, Artie Ortego, Dorothy Vernon, Jack Evans, Tom London, George Rosener, Nanette (dog). Two north woods fur trappers vie for the daughter of a trading post owner and one of them has his men steal the other's valuable furs. Low budget action program feature from producer-director Bernard B. Ray, based on a James Oliver Curwood story; picturesque due to its mostly outdoor setting.\n\n**4275** _ **The Test of Donald Norton**_ **** Chadwick, 1926. 68 min. D: B. Reeves Eason. SC: Adele Buffington. With George Walsh, Tyrone Power (Sr.), Eugenia Gilbert, Robert Graves, Evelyn Selbie, Michael D. Moore, Virginia True Boardman, John Francis Dillon, Virginia Marshall, Ed Coxen. In the north country, a half breed young man fights prejudice while struggling to become an accepted fur trader all the while realizing he cannot marry the girl he loves. Good silent drama.\n\n**4276** _ **The Testing Block**_ **** Paramount-Artcraft, 1920. 60 min. D-SC: Lambert Hillyer. With William S. Hart, Eva Novak, Gordon Russell, Florence Carpenter, Richard Headrick, Ira McFadden. An outlaw gang leader falls in love with a pretty entertainer and after they marry and settle down one of his henchmen arrives planning to steal the wife. Melodramatic and somber, this silent drama was based on star William S. Hart's original story.\n\n**4277** _ **Tex and the Lord of the Deep**_ **** Cinecitta\/Titanus, 1985. 103 min. D: Duccio Tessari. SC: Duccio Tessari, Gianfranco Clerici, Marcello Coscia and Giorgio Bonelli. With Giulian Gemma, William Berger, Isabella Russivora, Aldo Sambrell, Carlo Mucari, Flavio Bucci, Giovanni Luigi Bonelli, Peter Berling. A frontier investigator uncovers a plot by a Yaqui Indian shaman to use a mysterious rock to mummify a rival tribe. The combination of Spaghetti Western and horror does not blend well; released in Italy, where it was made, as _**Tex il Signore degli Abissi**_ (Tex and the Lord of the Abyss).\n\n**4278** _ **Tex Granger**_ **** Columbia, 1948. 15 Chapters. D: Derwin Abrahams. SC: Arthur Hoerl, Lewis Clay, Harry Fraser and Royal K. Cole. With Robert (Stevens) Kellard, Peggy Stewart, Buzz Henry, Smith Ballew, Jack Ingram, I. Stanford Jolley, Terry Frost, Jim Diehl, Britt Wood, William Fawcett, Charles King, Slim Whitaker, Bill Brauer, Stanley Blystone, Al Ferguson, John Hart, Edmund Cobb, Eddie Parker, Duke the Wonder Dog. A cowpoke joins a boy and a young woman in opposing the activities of a crook who has become town sheriff. Standard cliffhanger highlighted by Smith Ballew's portrayal of Blaze Talbot.\n\n**4279** _ **Tex Rides with the Boy Scouts**_ **** Grand National, 1937. 66 min. D: Ray Taylor. ED: Edmund Kelso. With Tex Ritter, Marjorie Reynolds, Horace Murphy, Snub Pollard, Tommy Bupp, Charles King, Forrest Taylor, Karl Hackett, Lynton Brent, Philip Ahn, Ed Cassidy, Timmy Davis, Heber Snow (Hank Worden), The Beverly Hill Billies. A mine ore geologist and his two pals are helped by a Boy Scout troop in capturing a gang of train robbers masquerading as gold mine operators. Pretty good Tex Ritter film with lots of action and good music, with the introduction relating the history of the Boy Scouts, to whom the film is dedicated.\n\n**4280** _ **Tex Takes a Holiday**_ **** First Division, 1932. 60 min. Color. D: Alvin J. Neitz (Alan James). SC: Alan James. With Wallace MacDonald, Virginia Brown Faire, George Chesebro, Ben Corbett, Jack Perrin, James Dillon, Claude Peyton, George Gerwing, Sheldon Lewis, Olin Francis, Steve Clemente, Tom London, Charles Stevens, Mme. DeLatta. A mysterious stranger is blamed for a series of local crimes but uncovers the real culprit. Poor Natural Color mystery Western from producer Robert J. Horner.\n\n**4281** _ **The Texan**_ **** Paramount, 1930. 79 min. D: John Cromwell. SC: Daniel N. Rubin. With Gary Cooper, Fay Wray, Emma Dunn, Oscar Apfel, James Marcus, Donald Reed, Soledad Jiminez, Veda Buckman, Cesar Vanoni, Edwin J. Brady, Enrique Acosta, Romualdo Tirado, Russ Columbo. The Llano Kid, a wanted outlaw, claims to be the long lost son of an old woman. Gary Cooper early talkie is dated and fairly bland; remade as _**The Llano Kid**_ (q.v.).\n\n**4282** _ **The Texan**_ **** Principal Attractions, 1932. 64 min. D: Cliff Smith. With Buffalo Bill, Jr., Lucille Brown, Bobby Nelson, Lafe McKee, Jack Mower, Art Mix, Yakima Canutt, Duke R. Lee, Lew Meehan, Bud Pope, Allen Holbrook, Harry Keaton. A fugitive joins two crooks in cheating citizens out of their money in a fixed horse race but is redeemed by the love of a local girl. Better than might be expected.\n\n**Poster for** _**The Texan**_ **(Principal Attractions, 1932).**\n\n**  \n**\n\n**4283** _ **The Texan Meets Calamity Jane**_ **** Columbia, 1950. 71 min. Color. D-SC: Ande Lamb. With James Ellison, Evelyn Ankers, Lee \"Lasses\" White, Jack Ingram, Ruth Whitney, Frank Pharr, Sally Weidman, Rudy de Saxe, Hugh Hooker, Ray Jones, Paul Barney, Ferrell Lester, Ronald Marriott, Bill Orisman, Lou W. Pierce, Elmer Herzberg. A cowpoke helps Calamity Jane as she tries to prove the validity of her ownership of a saloon. Unbelievably poor oater that is neither straight drama nor satire.\n\n**4284** _ **The Texans**_ **** Paramount, 1938. 90 min. D: James Hogan. SC: Paul Sloane and William Wister Haines. With Joan Bennett, Randolph Scott, May Robson, Walter Brennan, Robert Cummings, Robert Barrat, Harvey Stephens, Francis Ford, Bill Roberts, Clarence Wilson, Raymond Hatton, Jack Moore, Francis McDonald, Alan Ladd, Chris-Pin Martin, Anna Demetrio, Richard Tucker, Edward Gargan, Otis Harlan, Spencer Charters, Archie Twitchell, William Haade, Irving Bacon, Jack Perrin, Richard Denning, Wheeler Oakman, Harry Woods, Esther Howard, Ed LeSaint, John Qualen, Ed Brady, Margaret McWade, Vera Steadman, James Burtis, Lon Poff, Kay Whitehead, Frank Cordell, Slim Hightower, Slim Talbot, Oscar Smith, Everett Brown, James Kelso, Philip Morris, Ralph Remley, Pat West. A cowboy leads a cattle drive, which includes the pretty daughter of the herd's owner, from Texas to Kansas after the Civil War. Too much stock footage mars this remake of Emerson Hough's novel _North of '36_ which Paramount first filmed in 1924 with Jack Holt under its original title.\n\n**4285** _ **Texans Never Cry**_ **** Columbia, 1951. 70 min. D: Frank McDonald. SC: Norman S. Hall. With Gene Autry, Pat Buttram, Gail Davis, Mary Castle, Russell Hayden, Richard Powers (Tom Keene), Don C. Harvey, Mike Ragan, Roy Gordon, I. Stanford Jolley, Frank Fenton, Sandy Sanders, John McKee, Harry McKim, Minerva Urecal, Duke York. A Texas Ranger is after a gang counterfeiting Mexican lottery tickets. Fair Gene Autry opus with good work by Richard Powers (Tom Keene) as the villain.\n\n**4286** _ **Texas**_ **** Columbia, 1941. 94 min. D: George Marshall. S: Horace McCoy, Lewis Meltzer and Michael Blankfort. With William Holden, Glenn Ford, Claire Trevor, George Bancroft, Edgar Buchanan, Don Beddoe, Andrew Tombes, Addison Richards, Edmund MacDonald, Joseph Crehan, Willard Robertson, Pat Moriarity, Edmund Cobb, Lyle Latell, Raymond Hatton, Ralph Peters, Duke York, James Flavin, Carleton Young, Jack Ingram, Ethan Laidlaw, William Gould, Art Mix, Clem Bevans, George Morrell, Hank Bell, Dan White. After the Civil War, two ex\u2013Confederates head to Texas, one going to work for a woman cattle rancher while the other joins an outlaw gang. Entertaining and action filled oater, originally filmed in Sepia; reworked as _**South of the Chisholm Trail**_ (q.v.).\n\n**4287** _ **Texas Across the River**_ **** Universal, 1966. 101 min. Color. D: Michael Gordon. SC: Wells Root, Harold Green and Ben Starr. With Dean Martin, Alain Delon, Joey Bishop, Rosemary Forsyth, Tina (Aumont) Marquand, Peter Graves, Michael Ansara, Andrew Prine, Linden Chiles, Roy Barcroft, Stuart Anderson, George Wallace, Richard Farnsworth, John Harmon. When he is accused of murdering the fiancee of the woman he loves, a Spanish nobleman rides to Texas and falls for an Indian maiden while his ex-love follows him and is attracted to a cattleman. Fairly amusing genre satire.\n\n**4288** _ **Texas, Adios**_ **** Constantin Film\/Jeme Film, 1966. 93 min. Color. D: Ferdinando Baldi. SC: Franco Rossetti and Ferdinando Baldi. With Franco Nero, Cole Kitsch (Alberto Dell'Acqua\/Robert Widmark), Elisa Montes, Jose Guardiola, Livio Lorenzon, Hugo Blanco, Luigi Pistilli, Antonella Murgia, Gino Pernice, Ivan Scratuglia, Silvana Bacci, Remo De Angelis, Mario Novelli, Jose Suarez. A lawman and his brother go to Mexico to bring back the land baron who murdered their father years before. Violent but exciting Italian-Spanish Western originally called _**Texas, Addio**_ (Texas, Goodbye).\n\n**4289** _ **Texas Bad Man**_ **** Universal, 1932. 60 min. D: Edward Laemmle. SC: Jack Cunningham. With Tom Mix, Lucille Powers, Fred Kohler, Ed LeSaint, Willard Robertson, Richard Alexander, C.E. Anderson, Lynton Brent, Frankly Farnum, Joe Girard, Buck Moulton, James Burtis, Slim Cole, Boothe Howard, Francis Sayles, Theodore Lorch, George Magrill, Bud Osborne, Buck Bucko. A lawman pretends to be dishonest to infiltrate an outlaw gang. Very good Tom Mix feature.\n\n**4290** _ **Texas Buddies**_ **** World Wide, 1932. 57 min. D-SC: Robert North Bradbury. With Bob Steele, Nancy Drexel, Francis McDonald, Harry Semels, George Hayes, Bill Dyer, Dick Dickinson, Earl Dwire, Si Jenks, Henry Roquemore, Herman Hack, Artie Ortego. In 1919 a man returns home from the war and teams with his late father's friend to work a mine as they attempt to trap crooks who tried to rob a payroll and murder a pilot. Entertaining, but somewhat rambling Bob Steele film with such disparate plot elements as aviation, a horse race, robbery, murder and a runaway tin lizzie.\n\n**4291** _ **Texas Carnival**_ **** Metro-Goldwyn-Mayer, 1951. 77 min. Color. D: Charles Walters. SC: Dorothy Kingsley. With Esther Williams, Red Skelton, Howard Keel, Ann Miller, Paula Raymond, Keenan Wynn, Tom Tully, Red Norvo Trio, Foy Willing and The Riders of the Purple Sage, Glenn Strange, Dick Wessel, Donald MacBride, Marjorie Wood, Hans Conreid, Thurston Hall, Duke Johnson, Wilson Wood, Michael Dugan. The managers of a large Texas resort hotel mistakenly believe a circus bum is a big cattle and oil tycoon. There is not much to recommend this glossy musical comedy outside the presence of Red Skelton.\n\n**4292** _ **Texas City**_ **** Monogram, 1952. 54 min. D: Lewis D. Collins. SC: Joseph Poland. With Johnny Mack Brown, James Ellison, Lois Hall, Lorna Thayer, Lane Bradford, Marshall Reed, Terry Frost, Lyle Talbot, Pierce Lyden, John Hart, Lennie Osborne, Stanley Price. An ex\u2013cavalry officer is falsely accused of giving secret information about Army gold shipments to outlaws and a U.S. marshal tries to find the real culprit. Okay Johnny Mack Brown effort from near the end of his long term Monogram series.\n\n**4293** _ **Texas Cowboy**_ **** Syndicate, 1929. 50 min. D: J.P. McGowan. SC: Sally Winters. With Bob Steele, Edna Aslin, J.P. McGowan, Grace Stevens, Bud Osborne, Perry Murdock, Alfred Hewston, Merrill McCormick, Cliff Lyons. A young man returns to his California ranch home to find his mother has married a brute who is trying to control her property and cheat him out of his rightful inheritance. Nicely done Bob Steele silent enhanced by pretty Edna Aslin as the neighbor girl who loves the hero.\n\n**4294** _ **Texas Cyclone**_ **** Columbia, 1932. 57 min. D: D. Ross Lederman. SC: Randall Faye. With Tim McCoy, Shirley Grey, Wheeler Oakman, John Wayne, Wallace MacDonald, James Farley, Harry Cording, Vernon Dent, Walter Brennan, Mary Gordon, Monte Montague, Glenn Strange, Alfred P. James, Al Haskell, F.R. Smith, Bud Osborne, Bob Reeves, Dick Dickinson, Frank Ellis, Tex Palmer, Herman Hack, Jack Kirk, Bud McClure, Jack Evans, Clyde McClary, Jack Hendricks, Al Taylor, Jack King, Ken Cooper. A cowboy rides into an Arizona town where he is mistaken for another man, almost gets killed and then tries to help a beautiful woman save her ranch from a crooked saloon keeper. Interesting Tim McCoy feature with John Wayne in a supporting role; remade as _**One Man Justice**_ (q.v.).\n\n_**Texas Desperadoes**_ see _**Drift Fence**_\n\n**4295** _ **Texas Detour**_ **** Arista Films, 1978. 92 min. Color. D-SC: Hikmet Avedis. With Patrick Wayne, Mitch Vogel, Priscilla Barnes, Cameron Mitchell, Lindsay Bloom, R.G. Armstrong, Anthony James, Michael Mullins, Kate O'Dare, Jeri Blenders, Gary Davis, Alan Sands, Mike Honaker, Blau Gibson, Dee Cooper, Paul Nuckles, Larry Dunn, Denver Mattson, Pam Harvey, George Harvey, Marlene Schmidt. A stuntman in the southwest joins forces with a pretty woman as they are chased by a gang of crooks. Cheaply made action melodrama.\n\n**4296** _ **Texas Dynamo**_ **** Columbia, 1950. 54 min. D: Ray Nazarro. SC: Barry Shipman. With Charles Starrett, Smiley Burnette, Lois Hall, John Dehner, Jock (Mahoney) O'Mahoney, Marshall Reed, George Chesebro, Lane Bradford, Slim Duncan, Emil Sitka, Fred F. Sears, Gregg Barton. The Durango Kid takes on the guise of notorious gunman Texas Dynamo as he infiltrates a gang led by a ruthless town boss. Pretty fair series entry with strong work by John Dehner as the villain. British title: _**Suspected**_.\n\n**4297** _ **Texas Gunfighter**_ **** Tiffany, 1932. 63 min. D: Phil Rosen. SC: Bennett Cohen. With Ken Maynard, Sheila Mannors, Harry Woods, Bob Fleming, Jim Mason, Edgar Lewis, Lloyd Ingraham, Jack Rockwell, Frank Ellis, Steve Clark, Blackjack Ward, Bob Burns, Bud McClure. An outlaw gang member tries to go straight and becomes a lawman but his former cohorts want him to rob a safe. Mediocre Ken Maynard vehicle.\n\n_**Texas Guns**_ see _**Once Upon a Texas Train**_\n\n_**Texas in Flames**_ see _**She Came to the Valley**_\n\n**4298** _ **Texas Jack**_ **** Reliable, 1932. 52 min. D: Bernard B. Ray. SC: Carl Krusada. With Jack Perrin, Jayne Regan, Nelson McDowell, Robert Walker, Lew Meehan, Cope Borden, Blackie Whiteford, Budd Buster, Oscar Gahan, Jim Oates, Steve Clark, Clyde McClary, Jack Evans, Buck Morgan. Using a medicine show as a front, a cowboy searches for the crook who lured his sister south of the border in a shady operation that resulted in her suicide. Low grade Jack Perrin feature; a poor proposition.\n\n_**Texas Justice**_ see _**The Lone Rider in Texas Justice**_\n\n**4299** _ **The Texas Kid**_ **** Monogram, 1943. 57 min. D: Lambert Hillyer. SC: Jess Bowers (Adele Buffington). With Johnny Mack Brown, Raymond Hatton, Marshall Reed, Shirley Patterson, Robert Fiske, Edmund Cobb, Stanley Price, Lynton Brent, Bud Osborne, Kermit Maynard, John Judd, Cyril Ring, George J. Lewis, Charles King, Horace B. Carpenter, Fred Hoose, Harry Tenbrook, Joe Phillips. Two lawmen help an ex-outlaw whose old gang is after a gold shipment carried by his stage line. Pretty good \"Nevada Jack McKenzie\" outing, based on a story by Lynton Brent, who is in the cast. Also known as _**The Adventures of the Texas Kid**_.\n\n**4300** _ **Texas Lady**_ **** RKO Radio, 1955. 86 min. Color. D: Tim Whelan. SC: Horace McCoy. With Claudette Colbert, Barry Sullivan, Ray Collins, Gregory Walcott, Walter Sande, James Bell, Horace McMahon, John Litel, Douglas Fowley, Don Haggerty, Celia Lovsky, Alexander Campbell, Kathleen Mulqueen, Robert Lynn, Raymond Greenleaf, Francis McDonald, Buzz Henry, Stuart Randall, Grandon Rhodes, Harry Tyler, Bruce Payne, Paul Wexler, George Brand, Jim Hayward, Jeffrey Sayre, Leroy Johnson. After winning big gambling and paying off her father's debts, a woman takes over a newspaper and opposes a local crook. Claudette Colbert out West provides some charm in this otherwise mundane drama.\n\n**4301** _ **Texas Lawmen**_ **** Monogram, 1951. 57 min. D: Lewis D. Collins. SC: Joseph Poland. With Johnny Mack Brown, James Ellison, I. Stanford Jolley, Lee Roberts, Lane Bradford, Marshall Reed, Terry Frost, Lyle Talbot, Pierce Lyden, Stanley Price, John Hart, Roy Butler, Jack Hendricks. A federal marshal enlists the help of a sheriff in rounding up the outlaws who robbed a mine payroll. Stale Johnny Mack Brown-James Ellison teaming; also called _**Lone Star Lawmen**_.\n\n_**Texas Layover**_ see _**Blazing Stewardesses**_\n\n_**Texas Legionnaires**_ see _**Man from Music Mountain**_ (1943)\n\n**4302** _ **Texas Manhunt**_ **** Producers Releasing Corporation, 1942. 61 min. D: Peter Stewart (Sam Newfield). SC: William Lively. With Bill \"Cowboy Rambler\" Boyd, Art Davis, Lee Powell, Julie Duncan, Dennis Moore, Frank Hagney, Karl Hackett, Frank Ellis, Arno Frey, Eddie Phillips, Kenne Duncan, Forrest Taylor, Frank LaRue. When ranchers are sabotaged, federal marshals suspect two cattlemen are enemy agents. The plot of spies on the range adds some life to this \"Frontier Marshals\" series opener.\n\n**4303** _ **The Texas Marshal**_ **** Producers Releasing Corporation, 1941. 58 min. D: Peter Stewart (Sam Newfield). SC: William Lively. With Tim McCoy, Art Davis, Kay Leslie, Karl Hackett, Ed Peil, Sr., Charles King, Dave O'Brien, Budd Buster, John Elliott, Frank Ellis, Byron Vance, Wilson Edwards, Kenne Duncan, Horace B. Carpenter, Herman Hack, Carl Mathews, George Morrell, Tex Palmer, Oscar Gahan, Art Dillard, Ray Henderson, Art Davis' Rhythm Riders. A lawman is called in to investigate terrorism against local ranchers caused by three crooks who use a legitimate business as a front, and his problems are compounded when his singing partner is taken in by the organization. Mediocre PRC oater with too much music and too little action; Tim McCoy's final solo starring series film.\n\n**4304** _ **Texas Masquerade**_ **** United Artists, 1944. 59 min. D: George Archainbaud. SC: Norman Houston and Jack Lait, Jr. With William Boyd, Andy Clyde, Jimmy Rogers, Mady Correll, Don Costello, Russell Simpson, Nelson Leigh, Francis McDonald, J. Farrell MacDonald, June Pickrell, John Merton, Pierce Lyden, Robert McKenzie, Bill Hunter, Snub Pollard, George Morrell, Keith Richards, Bob Burns, Ralph Bucko, Roy Bucko. Hoppy pretends to be an Eastern dude lawyer to get the goods on a murderous band of outlaws. Novel idea wears thin as the film progresses although this \"Hopalong Cassidy\" entry does have a different ending with the villain dying in quicksand.\n**4305** _ **Texas Panhandle**_ **** Columbia, 1945. 58 min. D: Ray Nazarro. SC: Ed Earl Repp. With Charles Starrett, Tex Harding, Dub Taylor, Nanette Parks, Carolina Cotton, The Spade Cooley Band, Forrest Taylor, Edward Howard, Ted Mapes, George Chesebro, Jody Gilbert, William Gould, Jack Kirk, Budd Buster, Tex Palmer, Hugh Hooker, Tex Williams, Robert Walker, Ray Jones. An ex\u2013Secret Service agent joins a wagon train to investigate a robbery, using his guise as the Durango Kid to round up the outlaws. Okay entry in the long running series.\n\n**4306** _ **Texas Pioneers**_ **** Monogram, 1932. 58 min. D: Harry Fraser. SC: Wellyn Totman and Harry Fraser. With Bill Cody, Andy Shuford, Sheila Mannors, Harry Allen, LeRoy Mason, Frank Lackteen, John Elliott, Ann Ross, Hank Bell, Iron Eyes Cody, Chief Standing Bear. When an outlaw gang attacks a remote frontier post, a scout tries to defend it. Just passable Bill Cody vehicle.\n\n**4307** _ **The Texas Rambler**_ **** Spectrum, 1935. 59 min. D: Robert Hill. SC: Oliver Drake. With Bill Cody, Catherine Cotter, Earle Hodgins, Stuart James, Mildred Rogers, Budd Buster, Ace Cain, Roger Williams, Buck Morgan, Colin Chase, Allen Greer, Bud Pope. A mysterious figure enlists the aid of a cowboy called \"The Rambler\" to help a young woman who crooks want to kidnap for her inheritance, one-half interest in a ranch. Better than average Bill Cody film due to Bob Hill's direction, Oliver Drake's script and Earle Hodgins' villainy.\n\n**4308** _ **The Texas Ranger**_ **** Columbia 1931. 60 min. D: D. Ross Lederman. SC: Forrest Sheldon. With Buck Jones, Carmelita Geraghty, Harry Woods, Ed Brady, Nelson McDowell, Billy Bletcher, Harry Todd, Budd D. Fine, Ed Peil, Sr., Blackie Whiteford, Lew Meehan, Bert Woodruff, Edward Hearn, Dorothy Vernon, Lafe McKee, Ben Corbett, Bud Osborne, Fred Burns, Jim Corey, Blackjack Ward. A ranger tries to stop a feud between two families in Texas cattle country, the trouble being caused by a crook and his gang. Exciting and fast moving Buck Jones early talkie.\n\n**4309** _ **The Texas Rangers**_ **** Paramount, 1936. 98 min. D: King Vidor. SC: Louis Stevens. With Fred MacMurray, Jack Oakie, Jean Parker, Lloyd Nolan, Edward Ellis, Bennie Bartlett, Frank Shannon, Frank Cordell, Richard Carle, Jed Prouty, Fred Kohler, George Hayes, Elena Martinez, Kathryn Bates, Rhea Mitchell, Charles Middleton, Stanley Andrews, Irving Bacon, Del Henderson, Hank Bell, Neal Hart, Jack Montgomery, Howard Joslin, Joe Dominguez, Joseph Rickman, Frank Ellis, Bill Gillis, Cecil Kellogg, Lloyd A. Saunders, Homer Farra, Ray Burgess, Gayne Whitman, Bobby Caldwell. Two outlaws reform and join the Texas Rangers, eventually taking part in a manhunt for their ex-partner. Sturdy saga, remade as _**Streets of Lardeo**_ (q.v.).\n\n**4310** _ **The Texas Rangers**_ **** Columbia, 1951. 74 min. D: Phil Karlson. SC: Richard Schayer. With George Montgomery, Gale Storm, Jerome Courtland, Noah Beery, Jr., John Litel, William Bishop, Douglas Kennedy, John Dehner, Ian MacDonald, John Doucette, Jock (Mahoney) O'Mahoney, Joseph Fallon, Myron Healey, Julian Rivero, Trevor Bardette, Stanley Andrews, Edward Earle, Jim Bannon, Kenne Duncan, George Chesebro, Dick Curtis, William Haade, John Cason. Two former outlaws join the Texas Rangers in an attempt to bring in a notorious gang. Dull George Montgomery film.\n\n**4311** _ **Texas Rangers**_ **** Miramax, 2001. 92 min. Color. D: Steve Miner. SC: Scott Busby and Martin Copeland. With James Van Der Beek, Dylan McDermott, Usher Raymond, Ashton Kutcher, Rachael Leigh Cook, Tom Skerritt, Randy Travis, Leonor Varela, Brian Martel, Alfred Molina, Billy Morton, Kate Newby, Gordon Michaels, Joe Renteria, Robert Patrick, Joe Spano, Jordan Brower, Maro Leonardi, William Prael, Vincent Spano, Jon Abrahams, Oded Fehr, Derek S. Flores, Breon Gorman, Eric Johnson, Jesse G. Thompson, Jim Shield, David Millbern, Matt Keeslar, Steve Bridgewater, James Coburn (narrator). During the Civil War a group of Texas Rangers try to maintain the peace in the face of not only civil conflict but marauding Indians, Mexicans and outlaws. Historically inaccurate but pleasant viewing.\n\n**4312** _ **The Texas Rangers Ride Again**_ **** Paramount, 1940. 68 min. D: James Hogan. SC: William Lipman and Horace McCoy. With John Howard, Ellen Drew, Akim Tamiroff, Broderick Crawford, May Robson, Charles Grapewin, John Miljan, Anthony Quinn, Tom Tyler, Donald Curtis, Eddie Acuff, Ruth Rogers, Robert Ryan, Eva Puig, Monte Blue, James Pierce, William Duncan, Harvey Stephens, Harold Goodwin, Edward Pawley, Eddie Foy, Jr., Joseph Crehan, Stanley Price, Charles Lane, Jack Perrin, Gordon Jones, John Miller, Henry Roquemore, Franklin Parker, Chuck Hamilton, Paul Kruger. Members of the Texas Rangers pretend to be rustlers so they can infiltrate an outlaw gang. Slick \"B\" production enhanced by a fine cast; a sequel to _**The Texas Rangers**_ (1936) [q.v.].\n\n**4313** _ **Texas Renegades**_ **** Producers Distributing Corporation, 1940. 56 min. D: Peter Stewart (Sam Newfield). SC: Joseph O'Donnell. With Tim McCoy, Nora Lane, Harry Harvey, Kenne Duncan, Lee Prather, Earl Dunn, Hal Price, Joe McGuinn, Ray Bennett, Bud McClure, Buel Bryant, Arnold Clark. Posing as an outlaw, a lawman attempts to clean up a wild western community. Tim McCoy's initial film for producer Sigmund Neufeld at PDC (soon to become Producers Releasing Corporation) is pretty good.\n\n_**Texas Road Agent**_ see _**Road Agent**_\n\n_**Texas Sandman**_ see _**Arkansas Swing**_\n\n**4314** _ **Texas Stagecoach**_ **** Columbia, 1940. 59 min. D: Joseph H. Lewis. SC: Fred Myton. With Charles Starrett, Iris Meredith, The Sons of the Pioneers (Bob Nolan, Tim Spencer, Lloyd Perryman, Pat Brady, Hugh Farr, Karl Farr), Dick Curtis, Kenneth MacDonald, George Chesebro, Don Beddoe, Ed LeSaint, Harry Cording, George Morrell, Francis Walker, Blackie Whiteford, Carl Stockdale, Eddie Laughton, Lillian Lawrence, Fred Burns, George Becinita. A crooked banker promotes trouble between two rival stage lines in hopes of taking over both operations. Fairly good Charles Starrett feature.\n\n**4315** _ **Texas Stampede**_ **** Columbia, 1939. 57 min. D: Sam Nelson. SC: Charles Francis Royal. With Charles Starrett, Iris Meredith, The Sons of the Pioneers (Bob Nolan, Tim Spencer, Lloyd Perryman, Pat Brady, Hugh Farr, Karl Farr), Fred Kohler, Jr., Edmund Cobb, Lee Prather, Ray Bennett, Edward Hearn, Ernie Adams, Ed Coxen, Blackjack Ward, Hank Bell, Blackie Whiteford, Charles Brinley, Horace B. Carpenter, Richard Botiller, Fred Parker. A lawman attempts to keep the peace between cattle ranchers and sheep men when the latter shut off water during a drought. Nicely done remake of _**The Dawn Trail**_ (q.v.).\n\n**4316** _ **The Texas Streak**_ **** Universal, 1926. 52 min. D-SC: Lynn Reynolds. With Hoot Gibson, Blanche Mehaffey, Slim Summerville, Alan Roscoe, James Marcus, Jack Curtis, Les Bates, Jack Murphy, William H. Turner. A broke movie cowboy extra gets a job as a surveyor's guard for a power and water company and uncovers a plot by a crook to turn ranchers against his employer as he romances a cattleman's pretty daughter. Breezy, fun Hoot Gibson silent feature.\n\n**4317** _ **Texas Terror**_ **** Monogram, 1935. 58 min. D-SC: Robert North Bradbury. With John Wayne, Lucille Brown, LeRoy Mason, Fern Emmett, John Ince, George Hayes, Henry Roquemore, Buffalo Bill, Jr., Bert Dillard, Jack Duffy, Lloyd Ingraham, Bobby Nelson, Yakima Canutt, Bert O'Hara, Julia Griffith, Bud Pope, Artie Ortego, Frank Ball, Herman Hack, Tex Palmer, Jack Kenny, Tex Phelps, Tom Lingham, George Ovey, Jack Jones. Falsely believing he killed his best pal, a lawman resigns his job and becomes a prospector but after he saves the dead man's sister in a stage holdup attempt he begins to realize the truth. Average John Wayne-Lone Star feature, but entertaining none-the-less. Remade as _**Guilty Trails**_ (q.v.).\n\n**4318** _ **Texas Terrors**_ **** Republic, 1940. 57 min. D: George Sherman. SC: Doris Schroeder and Anthony Coldeway. With Don \"Red\" Barry, Julie Duncan, Al St. John, Arthur Loft, Ann Pennington, Eddy Waller, William Ruhl, Sammy McKim, Reed Howes, Robert Fiske, Fred \"Snowflake\" Toones, Hal Taliaferro, Edmund Cobb, Al Haskell, Jack Kirk, Ruth Robinson, Blackjack Ward, Curley Dresden, Jimmy Wakely and His Rough Riders (Johnny Bond, Dick Reinhart). A western lawyer is on the trail of the outlaws who murdered his folks. Another fast paced and well made Don Barry film; the plot is nothing new but the star and director keep it moving. Look for one-time Broadway and early talkies star Ann Pennington in a brief production number.\n\n**4319** _ **Texas to Bataan**_ **** Monogram, 1942. 58 min. D: Robert (Emmett) Tansey. SC: Arthur Hoerl. With John King, David Sharpe, Max Terhune, Marjorie Manners, Budd Buster, Escolastico Baucin, Kenne Duncan, Frank Ellis, Carl Mathews, Guy Kingsford, Steve Clark, Al Ferguson, Tom Steele, Tex Palmer. Assigned to take a shipment of horses to the Philippine Islands for the Army, the Range Busters find they are opposed by enemy agents. Fans of the series will enjoy the patriotic romp; also called _**The Long, Long Trail**_.\n\n**4320** _ **The Texas Tornado**_ **** Film Bookings Offices (FBO), 1928. 55 min. D: Frank Howard Clark. SC: Frank Howard Clark and Randolph Bartlett. With Tom Tyler, Frankie Darro, Nora Lane, Jack Anthony, Frank Whitson, Bob Burns, Bob Reeves, Beans (dog). A villainous rancher tries to get an oil lease by stopping his neighbors from renewing it but is thwarted by a mysterious stranger, the uncle of an orphan boy who is abducted by the crook. One of the few surviving (at 40 minutes) Tom Tyler-Frankie Darro series films from FBO moves at a fast clip and is filled with lots of action.\n\n**4321** _ **Texas Tornado**_ **** Willis Kent, 1932. 55 min. D-SC: Oliver Drake. With Lane Chandler, Doris Hill, Buddy Roosevelt, Yakima Canutt, Robert Hale, Ben Corbett, Edward Hearn, Bart Carre, Mike Brand, Fred Burns, J. Frank Glendon, Wes Warner, Slim Whitaker, Pat Herly. A Texas Ranger gets involved with gangsters who have kidnapped a young woman and killed her father. Okay poverty row outing in Lane Chandler's series for producer Willis Kent.\n\n**4322** _ **Texas Trail**_ **** Paramount, 1937. 59 min. D: David Selman. SC: Joseph O'Donnell. With William Boyd, George Hayes, Russell Hayden, Judith Allen, Billy King, Alexander Cross, Karl Hackett, Robert Kortman, Jack Rockwell, John Beach, Ray Bennett, Phil McCullough, Earle Hodgins, Ben Corbett, John Judd, Clyde Kinney, Leo McMahon, John Powers, Cliff Parkinson. A fort commander asks Hopalong Cassidy to round up 500 horses for government use in the Spanish-American War while an outlaw leader and his gang plan to rustle the herd. Compact and entertaining series entry.\n\n_**Texas Trouble**_ see _**Billy the Kid's Range War**_\n\n**4323** _ **Texas Trouble Shooters**_ **** Monogram, 1942. 55 min. D: S. Roy Luby. SC: Arthur Hoerl. With Ray Corrigan, John King, Max Terhune, Julie Duncan, Roy Harris (Riley Hill), Eddie Phillips, Frank Ellis, Ted Mapes, Kermit Maynard, Gertrude W. Hoffman, Steve Clark, Jack Holmes, Glenn Strange, Richard Cramer, Carl Mathews. Three cowboys try to help a man who has been attacked while trying to claim a ranch he inherited. Action filled \"Range Busters\" entry, including such song favorites as \"Deep in the Heat of Texas\" and \"Light of the Western Skies.\"\n\n**4324** _ **Texas Wildcats**_ **** Victory, 1939. 57 min. D: Sam Newfield. SC: George Plympton. With Tim McCoy, Joan Barclay, Forrest Taylor, Ted Adams, Dave O'Brien, Frank Ellis, Carl Mathews, Bob Terry, Slim Whitaker, Reed Howes, George Morrell, Avando Reynaldo, Wally West, Frank Wayne, Sherry Tansey, Herman Hack, Milburn Morante, Clyde McClary, Denver Dixon. Taking the guise of the mysterious Phantom, a lawman seeks to trap the man who murdered his pal and he also helps siblings whose mortgage is held by a crook. Tim McCoy fans should enjoy this \"Lighting Bill Carson\" episode although it is slight on production values.\n\n**4325** _ **The Texican**_ **** Columbia, 1966. 90 min. Color. D: Lesley Selander. SC: John C. Champion. With Audie Murphy, Broderick Crawford, Diana Lorys, Luz Marauez, Antonio Casas, Antonio Molino Rojo, Aldo Sambrell, Victor Israel, Antonio Peral, Jorge (George) Rigaud, Martha May, Juan Carlos Torres, Gerald Tichy, Luis Induni, Helga Genth. A cowboy seeks revenge against the ruthless town boss who falsely accused him of murder. Director Lesley Selander's mediocre remake his earlier success _**Panhandle**_ (q.v.), filmed in Spain.\n\n**4326** _ **That Girl Montana**_ **** Path\u00e9, 1921. 67 min. D: Robert Thornby. SC: George H. Plympton. With Blanche Sweet, Mahlon Hamilton, Frank Lanning, Ed Peil, Sr., Charles Edler, Claire Du Brey, Kate Price, Jack Roseleigh. A young woman, raised as a tomboy, is romanced by a married prospector and denounced by a mine owner who later makes her his partner when he finds out she is his long lost daughter. Complicated silent dramatization of Marah Ellis Ryan's novel.\n\n**4327** _ **That Texas Jamboree**_ **** Columbia, 1946. 67 min. D: Ray Nazarro. SC: J. Benton Cheney. With Ken Curtis, Jeff Donnell, Andy Clyde, Guinn Williams, Robert (Kellard) Stevens, The Dinning Sisters (Lou, Jean and Ginger Dinning), Andy Parker and The Plainsmen, Carolina Cotton, The Hoosier Hot Shots (Charles \"Gabe\" Ward, Paul \"Hezzie\" Trietsch, Ken Trietsch, Gil Taylor), Curt Barrett and The Trailsmen, Deuce Spriggins and His Orchestra, George Chesebro, Kenneth MacDonald, Claire Carleton, Vernon Dent, Forrest Taylor, Frank Ellis, Dick Elliott, Nolan Leary, John Ince, Frank O'Connor, Hank Bell, Ray Jones, Paul Birch. The daughter of a medicine show operator renews her romance with a town sheriff out to stop gamblers in his Western town. Not much plot but a dozen songs and some noted country-western stars make this a pleasant offering.\n\n**4328** _ **Their Only Chance**_ **** Ellman Film Enterprises, 1975. 84 min. Color. D-SC: J. David Siddon. With Jock Mahoney, Steve Hoddy, Chris Jeffers, Mildred Watt, Ross Goddard, Jack Goddard, Greg Ruppel, Ron Lowe, Ken Ebert, Jim Swaggerty. A man braves the harsh wilderness to free a trio of wild animals. Pleasant low budget outdoor drama from producer-director-writer J. David Siddon.\n\n**4329** _ **Them!**_ **** Warner Bros., 1954. 93 min. D: Gordon Douglas. SC: Ted Sherdeman. With James Whitmore, Edmund Gwenn, James Arness, Joan Weldon, Onslow Stevens, Chris Drake, Sean McClory, Sandy Descher, Mary Alan Hokanson, Frederick J. Foote, Olin Howlin, Scott Correll, Richard Bellis, Joel Smith, John Close, William Schallert, Cliff Ferre, Matthew McCue, Marshall Bradford, Joe Forte, Ann Doran, Willis Bouchey, John Maxwell, Leonard Nimoy, Fess Parker, Dick Wessel, Dub Taylor, Russell Gage, Robert Burger, Harry Tyler, Harry Wilson, Eddie Dew, Dorothy Green, Dean Cromer, Lawrence Dobkin, James Cardwell, Booth Colman, Walter Coy, Victor Sutherland, Jack Perrin, Royden Clark, Hubert Kerns. Due to atomic testing, ants become huge giants terrorizing the Arizona desert. One of the all-time best science fiction films, in a Western setting.\n\n**4330** _ **There Was a Crooked Man**_ **** Warner Bros.-Seven Arts, 1970. 126 min. Color. D: Joseph L. Mankiewicz. SC: David Newman and Robert Benton. With Kirk Douglas, Henry Fonda, Hume Cronyn, Warren Oates, Burgess Meredith, John Randolph, Arthur O'Connell, Martin Gabel, Michael Blodgett, Claude McNeil, Alan Hale, Victor French, Lee Grant, C.K. Yank, Barbara Hensley, Bert Freed, Barbara Rhodes, J. Edward McKinley, Gene Evans, Jeanne Cooper, Byron Foulger, Guy Wilkerson, James Seay, Boyd \"Red\" Morgan. A lawman becomes the warden of an Arizona territorial prison and matches wits with an inmate who has hidden away $500,000 from a robbery. Sturdy comedy-drama with a twist ending.\n\n**4331** _ **There Will Be Blood**_ **** Paramount Vantage, 2007. 158 min. Color. D-SC: Paul Thomas Anderson. With Daniel Day-Lewis, Paul Dano, Ciaran Hinds, Kevin J. O'Connor, Dillon Freasier, Kellie Hill, Robert Arber, David Williams, Barry Del Sherman, Paul F. Tompkins, Randal Carver, Coco Leigh, Sydney McCallister, David Willis, James Downey, Dan Swallow, Hope Elizabeth Reeves, Irene G. Hunter, David Warshofsky, Tom Doyle, Collon Woodward, John Burton, John Chitwood, Hans Howes, Robert Barge, the Rev. Bob Bock, Colleen Fox, Irene G. Hunter, Beau Smith. In the early 1900s a ruthless Western oil tycoon pretends to be a family man so he can fool others as he claws his way to the top. Lengthy, well etched character study based on Upton Sinlair's novel _Oil_.\n\n**4332** _ **There's a Noose Waiting for You**_ **** Doria Film\/Balcazar, 1972. 89 min. Color. D: George Martin (Alfonso Balcazar). SC: S. Giovanni (Giovanni Simonelli). With George Martin, Marina Malfatti, Klaus Kinski, Daniel Martin, Augusto Pesarini, Francisco Jose Huetos, Billy (Susanna Atkinson), Willi Columbini, Luis Ponciado, Indio Gonzales, Manuel Muniz, Manuel Sas, Ricardo Moyan, Manuel Bronchud, Gustavo Re, Adolfo Alises, Miguel Muniesa, Vittorio Fanfoni, Luigi Antonio Guerra, Luis Induni, Mara Krupp. After serving a prison sentence for shooting his brother's killer, a man returns home vowing to give up his guns but is soon beset by land grabbers and a bounty hunter. Average sequel to _**Clint the Stranger**_ (q.v)., an Italian-Spanish co-production originally called _**Il Ritorno di Clint il Solitario**_ (The Return of Clint the Stranger).\n\n**4333** _ **These Thousand Hills**_ **** 20th Century\u2013Fox, 1959. 96 min. Color. D: Richard Fleischer. SC: Alfred Hayes. With Don Murray, Richard Egan, Lee Remick, Patricia Owens, Stuart Whitman, Albert Dekker, Harold J. Stone, Royal Dano, Jean Willes, Douglas Fowley, Fuzzy Knight, Robert Adler, Barbara Morrison, Ned Weaver, Steve Darrell, Tom Greenway, Ben Wright, Jess Kirkpatrick, John Epper, Fred Graham, Nelson Leigh, Ken Renard, Frank Lavier, Cap Somers. An ambitious rancher deserts the girl he loves to marry a banker's daughter but eventually learns the meaning of loyalty and responsibility. Well written, directed and acted drama.\n\n_**They Call Him Cemetery**_ see _**A Bullet for a Stranger**_\n\n_**They Call Me Hallelujah**_ see _**Deep End**_\n\n_**They Call Me Renegade**_ see _**Renegade**_ (1987)\n\n**4334** _ **They Call Me Trinity**_ **** Avco-Embassy, 1971. 117 min. D-SC: E.B. Clucher (Enzo Barboni). With Terence Hill, Bud Spencer, Farley Granger, Steffan Zacharias, Dan Sturkie, Gisela Hahn, Elena Pedemonte, Ezio Marano, Luciano Rossi, Michelle Spaeara, Remo Capitani, Michele Cimarosa. Two bungling brothers agree to protect a Mormon town against a band of marauding bandits. Funny take-off of Spaghetti Westerns, first released in Italy as _**Lo Chiamavano Trinita**_ (They Called Him Trinity) followed by _**Trinity Is Still My Name**_ (q.v.).\n\n**4335** _ **They Came to Cordura**_ **** Columbia, 1959. 123 min. Color. D: Robert Rossen. SC: Ivan Moffat and Robert Rossen. With Gary Cooper, Rita Hayworth, Van Heflin, Richard Conte, Tab Hunter, Michael Callan, Dick York, Robert Keith, Carlos Romero, Jim Bannon, Edward Platt, Maurice Jara, Sam Buffington, Arthur Hanson. During the conflict with Pancho Villa, a demoted Army officer is assigned to find recipients for the Congressional Medal of Honor and during a trek across the Mexican desert his party is accompanied by a woman accused of giving aid to the rebels. Overlong and complicated, this unsatisfying film was Gary Cooper's final Western.\n\n**4336** _ **They Died with Their Boots On**_ **** Warner Bros., 1941. 140 min. D: Raoul Walsh. SC: Wally Kline and Aeneas MacKenzie. With Errol Flynn, Olivia de Havilland, Arthur Kennedy, Charles Grapewin, Gene Lockhart, Anthony Quinn, Sydney Greenstreet, Stanley Ridges, John Litel, Walter Hampden, Regis Toomey, Hattie McDaniel, G.P. Huntley, Jr., Frank Wilcox, Joseph Sawyer, Minor Watson, Gig Young, John Ridgely, Joseph Crehan, Aileen Pringle, Anna Q. Nilsson, Harry Lewis, Tod Andrews, William Hopper, Selmer Jackson, Patrick McVey, Renie Riano, Minerva Urecal, Virginia Sale, Vera Lewis, Frank Orth, Hobart Bosworth, Irving Bacon, Roy Barcroft, Lane Chandler, Ed Keane, Francis Ford, Frank Ferguson, Herbert Heywood, Walter Brooke, Sam McDaniel, Addison Richards, Eddie Parker, George Reed, William Forrest, James Seay, George Eldredge, John Hamilton, Edna Holland, Spencer Charters, Ray Teal, Harry Strang, Max Hoffman, Jr., Dick Wessel, Weldon Heyburn, Frank Mayo, Irving Bacon, Steve Darrell, Russell Hicks, Ian MacDonald, Jack Mower, Hugh Sothern, Arthur Loft, G. Pat Collins, Virginia Brissac, Walter Baldwin, Fred Kelsey, Wade Crosby, Joe Devlin, Joseph King, Matty Faust, Paul Kruger, Victor Zimmerman, Dick French, Garland Smith, Bob Perry, Sol Gorss, Alberta Gary, Annabelle Jones, Carl Harbaugh. The story of George Armstrong Custer, from his graduation from West Point, through the Civil War and his final stand at the Little Big Horn River. Overlong and historically inaccurate, but still fun.\n\n**4337** _ **They Ran for Their Lives**_ **** Columbia\/Masterpiece, 1969. 92 min. Color. D: John Payne (and uncredited Oliver Drake). SC: Monroe Mowsley. With John Payne, Luana Patten, Scott Brady, John Carradine, Jim Davis, Anthony Eisley, Darwin Lamb, Boyd Stockman, Bill Koontz, Bravo (dog). A man camping with his dog assists a young woman being tracked by three crooks after important papers she is carrying. A good cast helps cover the production deficiencies in this little seen feature made near Las Vegas in 1967.\n\n**4338** _ **They Rode West**_ **** Columbia, 1954. 84 min. Color. D: Phil Karlson. SC: DeVallon Scott and Frank Nugent. With Robert Francis, Donna Reed, May Wynn, Phil(ip) Carey, Onslow Stevens, Peggy Converse, Roy Roberts, Jack Kelly, Stuart Randall, Eugene Iglesias, Frank De Kova, Ralph Dumke, James Best, George Keymas, Maurice Jara, John War Eagle. An Army camp commander and his post doctor are at odds when the physician wants to treat a local Indian tribe during a malaria outbreak. Petty fair action melodrama.\n\n_**They Were Called Graveyard**_ see _**Twice a Judas**_\n\n**4339** _ **Thirteen Fighting Men**_ **** 20th Century\u2013Fox, 1960. 71 min. D: Harry Gerstad. SC: Robert Hammer and Jack Thomas. With Guy Williams, Carole Mathews, Brad Dexter, Robert Dix, Richard Garland, Rayford Barnes, John Erwin, Richard Crane, Rex Holman, Bob Palmer, Mauritz Hugo, Dick Monohan, Ted Knight, Fred Kohler, Jr., I. Stanford Jolley, Walter Reed, John Merrick, Brad Harris. Near the end of the Civil War, a Union patrol tries to prevent Confederates from taking a fortune in gold they are transporting. Standard action feature with a good supporting cast.\n\n**4340** _ **30 Winchesters for El Diablo**_ **** Foreign Studios, 1965. 91 min. Color. D: Frank G. Carrol (Gianfranco Baldanello). SC: Al Bradly (Alfonso Brescia). With Carl Mohner, Topsy Collins (Alessandra Panaro), John Heston (Ivano Staccioli), Anthony Garof (Antonio Garisa), Jose Torres, Mila Stanic, Gay Gallwey (Renato Chiantoni), William Spoletin (Guglielmo Spoletini), William Burke (Attilio Dottesio), Max Darnell (Mario Dardanelli), Jean Mean (Dante Maggio). The mysterious El Diablo heads a cattle rustling gang that operates around Canyon City and a federal agent is sent to stop him. Average Italian produced oater made there as _**Trenta Winchester per El Diablo**_ (Thirty Winchesters for El Diablo) and also called _**Gold Train**_.\n\n**4341** _ **This Is My Alaska**_ **** Alaskan Adventure, 1969. 120 min. Color. D-SC: Leroy Shebal. With Leroy Shebal, Vivian Shebal, Gary Okahal. Leory Shebal photographed and narrates this sportsman's guide to Alaska, including exploration by bush plane, wolf hunting, snowmobile racing and Eskimos after polar bear. For fans of this type of fare.\n\n**4342** _ **This Man Can't Die**_ **** Fine Products, 1970. 90 min. Color. D: Gianfranco Baldanello. SC: Luigi Emmanuele and Gino Mangini. With Guy Madison, Peter Martell, Rik Battaglia, Lucienne Birdou, Steve Merrick, Rosalba Neri, Robert Widmark (Alberto Dell'Acqua), John Bartha. Working for the government, two adventurers pose as outlaws to infiltrate a gang of gun runners. Pretty good Spaghetti Western that should please Guy Madison fans; made in Italy as _**I Lunghi Giorni dell'Odio**_ (The Long Days of Hate).\n\n**4343** _ **This Rugged Land**_ **** Columbia, 1965. 72 min. D: Arthur Hiller. SC: Frank S. Nugent. With Richard Egan, Charles Bronson, Terry Moore, Ryan O'Neal, Anne Seymour, Denver Pyle, Oliver McGowan, Vic Perrin, Paul Tripp. A New Mexico ranch hand is accused of the murder of a co-worker's daughter. Shown in Europe in 1965 as a feature film, this was the debut episode of \"Empire\" (NBC-TV, 1962\u201363); issued on video as _**Mean Justice**_.\n\n**4344** _ **This Savage Land**_. Universal, 1969. 98 min. Color. D: Vincent McEveety. SC: Richard Fiedler. With Barry Sullivan, George C. Scott, Kathryn Hays, Brenda Scott, Andrew Prine, Kelly Corcoran, Katherine Squire, Glenn Corbett, Charles Seel, John Drew Barrymore, Roy Roberts, Rex Holman. A widower moves his family West from Ohio to a frontier town where they are terrorized by vigilantes. Strong melodrama, issued theatrically, although it was first a two part episode of \"The Road West\" (NBC-TV, 1966\u201367), telecast September 12 and 19, 1966. Also called _**The Savage Land**_.\n\n**4345** _ **This Was the West That Was**_ **** NBC-TV\/Universal, 1974. 74 min. Color. D: Fiedler Cook. SC: Sam H. Rolfe. With Ben Murphy, Kim Darby, Matt Clark, Jane Alexander, Anthony Franciosa, Stuart Margolin, Stefan Gierasch, Bill McKinney, M.L. LeGaut, Roger Robinson, Luke Askew, Woodrow Parfrey, Milton Selzer, Bruce Glover, Wayne Sutherlin, Ronnie Clair Edwards, Dimitra Arliss, Roger Davis (narrator). Gunmen are out to get even with Wild Bill Hickok, who must also contend with a romantic Calamity Jane. There is not much to recommend the satire of Western legends.\n\n**4346** _ **Thomasine and Bushrod**_ **** Columbia, 1974. 95 min. Color. D: Gordon Parks, Jr. SC: Max Julien and Harvey Bernard With Max Julien, Vonetta McGee, George Murdock, Glynn Turman, Juanita Moore, Joel Fluellen, Jackson D. Kane, Ben Zeller, Jason Bernard, Bud Conlan, Kip Allen, Herb Robins, Harry Luck, Jason Bernard. In 1911 Texas a black man and woman form a Robin Hood-type robbery team and are hunted by the law. Passable exploitation drama.\n\n**4347** _ **Thorobred**_ **** Clark-Cornelius Corporation, 1922. 50 min. D-SC: George Halligan. With Helen Gibson, Bob Burns, Otto Nelson, Jack Ganzhorn. A woman takes over her father's duties as sheriff and tracks an outlaw, eventually using the guise of a saloon girl to catch him. This silent Helen Gibson vehicle is lots of fun.\n\n**4348** _ **Those Dirty Dogs**_ **** Cinema Financial of America, 1974. 89 min. Color. D: Giuseppe Rosati. SC: Carl (Carlos) Veo, Giuseppe Rosati and Henry Lovett (Enrique Llovet). With Stephen Boyd, Johnny (Gianni) Garko, Helga Line, Simon Andreu, Howard Ross, Harry Baird, Teresa Gimpera, Alfredo Mayo, Mirella Dogan, Enzo Fiermann, Gabriella Giorgelli, Andrea Scotti (Andrew Scott), Daniele Vargas, Lee Burton (Guido Lollobrigida), Furio Meniconi. Three officers, whose convoy has been mostly wiped out by Mexican bandits, try to exchange the kidnapped daughter of a fort doctor for the gang's leader, who they have captured. Well made but very brutal Italian-Spanish co-production, issued in Spain by Plata Films\/San Bernardo\/Horse Films as _**Los Cuatro de Fort Apache**_ (The Four of Fort Apache) and in Italy as _**Campa Carogna...La Taglia Cresce**_ ; also called _**Charge!**_\n\n**4349** _ **Those Redheads from Seattle**_ **** Paramount, 1953. 90 min. Color. D: Lewis R. Foster. SC: Lewis R. Foster, Geoffrey Homes and George Worthing Yates. With Rhonda Fleming, Gene Barry, Agnes Moorehead, Guy Mitchell, Teresa Brewer, Jean Parker, Cynthia Bell, Kay Bell, Bill Pullen, John Kellogg, Frank Wilcox, Roscoe Ates, Michael Ross, Walter Reed, Ed Rand. A woman and her four lovely daughters head to Alaska during the Gold Rush to join her new husband, a newspaper editor, and find he has been murdered. Originally issued in 3-D, this light affair is fairly enjoyable and provides a chance to see two of the all-time top recording artists, Guy Mitchell and cute Teresa Brewer, who is quite good as the youngest sister.\n\n**4350** _ **Thousand Pieces of Gold**_ **** Greycat Films, 1991. 105 min. Color. D: Nancy Kelly. SC: Anne Makepeace. With Rosalind Chao, Chris Cooper, Michael Paul Chan, Dennis Dun, Jimmie F. Skaggs, Will Oldham, David Hayward, Beth Broderick, Kim Chan, Chris Evans, Weili Fan, Evan C. Kim, Freda Foh Shen, John M. Hosking, Mary Vatvy, Alvert J. Kalanick, Mary Lee, Jianli Zhang, Ron Dorn, George Kee Cheung, James Lortz, Saachiko, Brien D. Sankey. A Chinese girl comes to an Idaho mining community where a saloon keeper tries to sell her as a prostitute but she finds love with a cowboy. Fair program Western based on Ruthanne Lum McCunn's book.\n\n**4351** _ **Three Amigos**_ **** Orion, 1986. 105 min. Color. D: John Landis. SC: Steve Martin, Lorne Michaels and Randy Newman. With Chevy Chase, Steve Martin, Martin Short, Patrice Martinez, Alfonso Arau, Tony Plana, Joe Mantegna, Jon Lovitz, Phil Hartman, Jorge Cervera, Kai Wulff, Abel Franco, Fred Asparagus, Philip Gordon, Michael Wren, Gene Hartline, William Kaplan, Sophia Lamour, Santos Morales, Tino Insana, Craig Berenson, Josh Gallegos, Norbert Weisser, Brian Thompson, Hector Elias, Hector Morales, Betty Carvalho, Brinke Stevens, Randy Newman (voice). Going to Mexico to make personal appearances, three faded cowboy silent film heroes find they are expected to live up to their screen images by ridding a Mexican village of bandits. Somewhat amusing takeoff on the Western genre.\n\n**4352** _ **Three Bad Men**_ **** Fox, 1926. 87 min. D: John Ford. SC: John Stone. With George O'Brien, Olive Borden, Lou Tellegen, J. Farrell MacDonald, Tom Santschi, Frank Campeau, George Harris, Jay Hunt, Priscilla Bonner, Otis Harlan, Walter Perry, Grace Gordon, Alec B. Francis, George Irving, Phyllis Haver, Vester Pegg. During the 1876 Dakota land rush a former West Point cadet joins a family planning to settle there and helps them oppose outlaws and a crooked sheriff. Very good John Ford silent effort with just the right mixture of action, romance and sentiment.\n\n**4353** _ **Three Bad Men**_ **** Iron Horse Entertainment, 2005. 118 min. Color. D-SC: Jeff Hathcock. With George Kennedy, Peter Brown, Mike Moroff, Chris Gann, John Dixon, Don Mack, David Orton, June Wilkinson, Megan McNally, Curtis Pierson, Tim Cable, Jerry Banks, Spero Stamboulis, Craig Kolkebeck, Troy Hardin, Tyrone Loukas, Wendy Miklovic, Jon K. Farless, Ben Drury, Monica Zamora. As they flee after a bank heist, three robbers agree to a dying man's request to save his wife who has been kidnapped by an outlaw gang. Overlong, less than mediocre oater.\n\n**4354** _ **Three Bullets for a Long Gun**_ **** Avco-Embassy, 1973. 89 min. Color. D-SC: Peter Henkel. With Beau Brummell, Keith Van Der Wat, Patrick Mynhardt, Tullio Moneta, Don McCorkindale, Gaby Getz, Jose De Sousa. Two men, a gunman and a Mexican bandit, join forces to find a hidden Confederate treasure only to become enemies during the quest. Okay action Western, a South African-West German co-production, filmed in Africa as _**Friss den Staub von Meinen Stiefeln**_ in 1970.\n\n**4355** _ **Three Bullets for Ringo**_ **** Profilm, 1966. 100 min. Color. D: Emimmo Salvi. SC: Ambrogio Molteni and Emimmo Salvi. With Gordon Mitchell, Mickey Hargitay, Milla Sannoner, John Heston (Ivano Staccioli), Mike Moore (Amedeo Trilli), Spean Convery (Sparataco Conversi), Dante Maggio, Margherita Horowitz, Isarco Ravaioli, Nino Fuscagni, Bruno Arie, Willy Miniver. After losing his sight, a gunman plots to get revenge for the murder of his mother and save his wife and son who have been kidnapped by a land grabber and his gang. Average Italian oater originally called _**Tre Colpi de Winchester per Ringo**_ (Three Shots of a Winchester for Ringo) and titled _**Three Graves for a Winchester**_ in the U.S.\n\n**4356** _ **The Three Burials of Melquiades Estrada**_ **** Sony Pictures Classics, 2005. 121 min. Color. D: Tommy Lee Jones. SC: Guillermo Arriaga. With Tommy Lee Jones, Barry Pepper, Julio Cesar Cedillo, Dwight Yoakam, January Jones, Melissa Leo, Levon Helm, Mel Rodriguez, Cecilia Suarez, Ignacio Guadalupe, Vanessa Bauche, Irineo Alvarez, Guillermo Arriaga, Josh Berry, Rodger Boyce, Edwin \"Bubba\" Broussard, Rene Campero, Ariel Castro, Sonny Carl Davis, Jesse De Luna, Richard Dillard, Guillermo von Son, Sean Hennigan, Richard Jones, Barry Tubb, Adrian Navarette, Angelina Torres, Terry Parks, Gustavo Sanchez Parra, Brent Smiga, Charles Sanders. After a friend has been buried twice, a ranch foreman tries to keep a promise he made to the deceased by forcing a lawman to dig up the body a third time and take it by mule into Mexico for final interment. Disoriented modern Western, not for all tastes.\n\n**4357** _ **Three Desperate Men**_ **** Lippert, 1951. 71 min. D: Sam Newfield. SC: Orville Hampton. With Preston Foster, Jim Davis, Virginia Grey, Monte Blue, Ross Latimer, Sid Melton, Rory Mallinson, John Brown, Margaret Seddon, House Peters, Jr., Joel Newfield, Lee Bennett, Steve Belmont, Carol Henry, Kermit Maynard, Bert Dillard, Milton Kibbee, William Norton Bailey, Gene Randall, William Haade, Denver Dixon. Three brothers who were once on the side of the law are forced to become criminals with prices on their heads. Pretty fair action drama with an especially good performance by Monte Blue as the lawman forced to hunt down his pals.\n\n**4358** _ **Three Faces West**_ **** Republic, 1940. 83 min. D: Bernard Vorhaus. SC: F. Hugh Herbert, Joseph Moncure March and Samuel Ornitz. With John Wayne, Sigrid Gurie, Charles Coburn, Spencer Charters, Roland Varno, Trevor Bardette, Helen MacKellar, Sonny Bupp, Wade Boteler, Russell Simpson, Charles Waldron, Wendell Niles, Dewey Robinson, Francis Ford, Manuel Paris, Wolfgang Zilzer, Frederick Vogeding, Frank Brownlee, Byron Foulger, Si Jenks, Stuart Holmes, Mary Field, Jack Montgomery, Bob Burns, Arthur Millett, Douglas Evans, Jim Corey, Horace B. Carpenter, Victor Potel, Bill Nestell, John Sheehan, Hank Patterson, Ted Stanhope, Bill Wolfe. A Viennese refugee doctor and his daughter arrive in a North Dakota town and eventually help a farmer in moving the community to Oregon after drought ruins their crops. Underrated John Wayne feature provides good entertainment; originally called _**The Refugee**_.\n\n**4359** _ **Three Godfathers**_ **** Metro-Goldwyn-Mayer, 1936. 82 min. D: Richard Boleslawski. SC: Edward E. Paramore and Manuel Self. With Chester Morris, Lewis Stone, Walter Brennan, Irene Hervey, Dorothy Tree, Robert Livingston, Joseph Marievsky, Jean Kirchner, Sidney Toler, Roger Imhoff, Willard Robertson, John Sheehan, Victor Potel, Harvey Clark, Helen Brown, Virginia Brissac. Escaping into the desert after robbing a frontier town bank of its Christmas savings, three outlaws find a dying woman and her baby and after their horses drink poisoned water they attempt to return the infant to civilization. Well done, glossy version of the Peter B. Kyne story that was first filmed in 1908 as _**Broncho Billy and the Baby**_ starring G.M. \"Broncho Billy\" Anderson. In 1916 Universal made a six reel version of _**The Three Godfathers**_ starring Harry Carey, with Hart (Jack) Hoxie in a supporting role; Edward J. LeSaint, later a character actor in Westerns, directed. The first sound version was _**Hell's Heroes**_ (q.v.). TV title: _**Miracle in the Sand**_.\n\n**4360** _ **3 Godfathers**_ **** Metro-Goldwyn-Mayer, 1948. 108 min. Color. D: John Ford. SC: Laurence Stallings and Frank S. Nugent. With John Wayne, Pedro Armendariz, Harry Carey, Jr., Ward Bond, Mildred Natwick, Guy Kibbee, Jane Darwell, Mae Marsh, Charles Halton, Dorothy Ford, Ben Johnson, Michael Dugan, Don Summers, Fred Libby, Hank Worden, Jack Pennick, Francis Ford. After robbing a small town bank, three outlaws flee into the desert where they find a woman and her newborn infant and agree to her dying wish that they take it to safety. Filmed in Monument Valley and dedicated to the memory of Harry Carey, this John Ford classic is one very fine Western.\n\n**John Wayne and Pedro Armendariz in** _**3 Godfathers**_ **(Metro-Goldwyn-Mayer, 1948).**\n\n**  \n**\n\n_**Three Graves for a Winchester**_ see _**Three Bullets for Ringo**_\n\n**4361** _ **Three Guns for Texas**_ **** Universal, 1968. 99 min. Color. D: David Lowell Rich, Paul Stanley and Earl Bellamy. SC: John D.F. Black. With Neville Brand, Peter Brown, William Smith, Martin Milner, Philip Carey, Albert Salmi, Cliff Osmond, Michael Conrad, Shelley Morrison, John Abbott, Richard Devon, Ralph Manza, Dub Taylor, Roy Barcroft, Chuck Courtney, John Mitchum, Mike Ragan (Holly Bane), X Brands, Bill Walker, John Cliff, Sam Edwards, Richard Collier, Russ McCubbin, William Vaughan, Marianne Gordon. Texas Rangers are on the trail of an outlaw gang led by an Indian woman with one of the lawmen becoming under the romantic spell of a pretty tribal maiden. Passable theatrical dual bill item sewn together from three episodes of \"Laredo\" (NBC-TV, 1965\u201367).\n\n**4362** _ **Three Hours to Kill**_ **** Columbia, 1954. 77 min. Color. D: Alfred Werker. SC: Richard Alan Simmons, Roy Huggins and Maxwell Shane. With Dana Andrews, Donna Reed, Dianne Foster, Stephen Elliott, Richard Coogan, Laurence Hugo, James Westerfield, Richard Webb, Carolyn Jones, Charlotte Fletcher, Whit Bissell, Felipe Turich, Arthur Fox, Francis McDonald, Frank Hagney, Syd Saylor, Paul E. Burns. Years after he is falsely accused of killing his fiancee's brother, a man returns home to prove his innocence. Taut melodrama that is well acted, especially by Dana Andrews as the protagonist trying to find a murderer.\n\n**4363** _ **Three in the Saddle**_ **** Producers Releasing Corporation, 1945. 61 min. D: Harry Fraser. SC: Elmer Clifton. With Tex Ritter, Dave O'Brien, Guy Wilkerson, Lorraine Miller, Charles King, Edward Howard, Ed Cassidy, Bud Osborne, Frank Ellis, Bob Duncan, Art Fowler, Jimmy Aubrey, Herman Hack, Ray Henderson, Jack Tornek, Post Park. A trio of lawmen attempt to help a young woman whose ranch is being sought by a land grabber. Ragged \"Texas Rangers\" series entry with Tex Ritter singing a couple of fair tunes.\n\n**4364** _ **Three Men from Texas**_ **** Paramount, 1940. 73 min. D: Lesley Selander. SC: Norton S. Parker. With William Boyd, Russell Hayden, Andy Clyde, Morris Ankrum, Morgan Wallace, Thornton Edwards, Esther Estrella, Davison Clark, Dick Curtis, Glenn Strange, Bob Burns, Jim Corey, George Morrell, Frank McCarroll, George Lollier, Nely Marx (Neyle Morrow), Wen Wright, Carlos De Valdez, Lucio Villegas, Roy Butler, Frank Ellis, Cliff Markson, Michael Vallon, Charles Murphy, John \"Skins\" Miller, Tex Phelps, Chuck Morrison, Milburn Morante, Herman Hack, Bill Nestell, Jack King, Ralph Bucko, Roy Bucko, Fred Burns. Hopalong Cassidy attempts to reform petty crook California Carlson and with Lucky Jenkins they travel to a town where a criminal is trying to steal land from its owners. A good story and an exciting climax makes this a top notch entry in the \"Hopalong Cassidy\" series, although it is more brutal than most Hoppy features. Well worth watching, this one introduced Andy Clyde as California Carlson. British title: _**Rangers Go West**_.\n\n**4365** _ **The Three Mesquiteers**_ **** Republic, 1936. 61 min. D: Ray Taylor. SC: Jack Natteford. With Robert Livingston, Ray Corrigan, Syd Saylor, Kay Hughes, J.P. McGowan, Frank Yaconelli, Alan Bridge, Stanley Blystone, John Merton, Gene Marvey, Milburn Stone, Duke York, Allen Connor, Nena Quartero, George Plues, Wally West, Tracy Layne, Ray Henderson, Ralph Bucko, Roy Bucko, Oscar Gahan, Cactus Mack, Rudy Sooter, John Ince, Jack Evans, Rose Plummer. Three cowboy pals find themselves in the middle of a feud between rival cattlemen. Fast paced initial entry in the popular and long running \"The Three Mesquiteers\" series, based on the works of William Colt MacDonald.\n\n**4366** _ **Three Musketeers of the West**_ **** Star Films S.A., 1973. 87 min. Color. D: Bruno Corbucci. SC: Tito Carpi, Bruno Corbucci, Leonardo Martin and Peter Berling. With George Eastman (Luigi Montefiori), Timothy Brent (Giancarlo Prete), Eduardo Fajardo, Karin Schubert, Chris Huerta, Leo Anchoriz, Carlo Rustichelli, Pietro Tordi, Chen Lee, Peter Berling, Osiride Pevarello, Max Turilli, Vittorio Congia, Eleanora Giorgi. A trio of Texas Rangers and a hillbilly agree to escort a female doctor, who is really a spy, into hostile Mexican territory. Fair Spaghetti Western, an Italian-Spanish-West German co-production made as _**Tutti per Uno, Botte per Tutti**_ (All for One, Cash for All).\n\n**4367** _ **Three on the Trail**_ **** Paramount, 1936. 67 min. D: Howard Bretherton. SC: Doris Schroeder and Vernon Smith. With William Boyd, James Ellison, Onslow Stevens, George Hayes, Muriel Evans, Claude King, William Duncan, Clara Kimball Young, Ernie Adams, Ted Adams, Lew Meehan, John St. Polis, Al Hill, Jack Rutherford, Lita Cortez, Artie Ortego, Franklyn Farnum, Joe Rickson, Hank Bell, Jack Montgomery. The Bar 20 trio oppose a ruthless saloon owner who is in cahoots with the local sheriff in cattle rustling and stage holdups. Sturdy \"Hopalong Cassidy\" feature.\n\n**4368** _ **The Three Outlaws**_ **** Associated Film Releasing, 1956. 75 min. D: Sam Newfield. SC: Orville Hampton. With Neville Brand, Bruce Bennett, Alan Hale, Jeanne Carmen, Jose Gonzales Gonzales, Rodolfo Hoyos, Robert Tafur, Bill Henry, Harry Lauter, Jonathan Hale, Lillian Molieri, Robert Christopher, Vincent Padula, Henry Escalante. A federal lawman is after Butch Cassidy and his gang, who have gone to Mexico pretending to be honest citizens. Pretty dreary.\n\n_**Three Rogues**_ see _**Not Exactly Gentlemen**_\n\n**4369** _ **The Three Swords of Zorro**_ **** Hispamer\/Roder\/Cepicsa Italia Films, 1963. 89 min. Color. D: Richard (Ricardo) Blasco. SC: Jacques Dumas, Mario Amendola, Luis Lucas Ojeda, Daniel Ribera, Ricardo Blasco and Jose Gallardo. With Guy Stockwell, Gloria Milland, Mikaela Wood, Antonio Prieto, John MacDouglas (Giuseppe Addobbati), Franco Fantasia, Rafael Vaquero, Felix Fernandez, Robert Dean, Antonio Gradoli. In 1830s Spanish California, Zorro's son and daughter help him in fighting a corrupt governor. Action filled Italian-Spanish co-production released in Spain as _**Las Tres Espadas del Zorro**_ (The Three Swords of Zorro) and in Italy as _**Le Tre Spade di Zorro**_ (The Three Swords of Zorro); U.S. title: _**Sword of Zorro**_.\n\n**4370** _ **3:10 to Yuma**_ **** Columbia, 1957. 92 min. D: Delmer Daves. SC: Halstead Welles. With Glenn Ford, Van Heflin, Felicia Farr, Leora Dana, Henry Jones, Richard Jaeckel, Robert Emhardt, Sheridan Comerate, George Mitchell, Robert Ellenstein, Ford Rainey, Barry Curtis, Jerry Hartleben. When a peaceable cowboy witnesses a holdup he tries to keep the gang leader prisoner until a train arrives with a lawman who will take him to trial. Sturdy action film that is good entertainment; Frankie Laine sings the haunting title song.\n\n**4371** _ **3:10 to Yuma**_ **** Lionsgate, 2007. 122 min. Color. D: James Mangold. SC: Halstead Welles, Michael Brandt and Derek Haas. With Russell Crowe, Christian Bale, Logan Lerman, Dallas Roberts, Ben Foster, Peter Fonda, Vinessa Shaw, Alan Tudyk, Luce Rains, Gretchen Mol, Lennie Loftin, Rio Alexander, Johnny Whitworth, Shawn D. Howell, Pat Ricotti, Ramon Frank, Deryle Lujan, James Augure, Brian Duffy, Jason Rodriguez, Kevin Durand, Chris Browning, Chad Brummett, Forrest Frye, Luke Wilson, Ben Petry, Arron Shiver, Sean Hennigan, Girard Swan, Christopher Berry, David Oliver, Jason Henning, Trevor Coppola, Brent Lambert, Brian T. Short, James Blackburn, Billy Lockwood, Hugh Elliot, Darren Gibson. An outlaw and a rancher match wits on the way to catch the train that is to take the bad man to trial. Acceptable remake of the 1957 film (q.v.); a big moneymaker.\n\n**4372** _ **Three Texas Steers**_ **** Republic, 1939. 57 min. D: George Sherman. SC: Betty Burbridge and Stanley Roberts. With John Wayne, Ray Corrigan, Max Terhune, Carole Landis, Ralph Graves, Roscoe Ates, Collette Lyons, Billy Curtis, Ted Adams, Stanley Blystone, David Sharpe, Ethan Laidlaw, Lew Kelly, Dave Willock, John Merton, Ted Mapes, Jack Kirk, Bob Burns, John Beach, Naba the Gorilla (Ray Corrigan). Crooks sabotage a woman's circus to force her to sell a ranch so they can use it to build a dam to control the local water supply. Fair entry in \"The Three Mesquiteers\" series, but not one of the best. Ray Corrigan played both Tucson Smith and Willie the Gorilla, the latter in his Naba costume. British title: _**Danger Rides the Range**_.\n\n**4373** _ **Three Violent People**_ **** Paramount, 1957. 100 min. Color. D: Rudolph Mate. SC: James Edward Grant. With Charlton Heston, Anne Baxter, Gilbert Roland, Forrest Tucker, Bruce Bennett, Tom Tryon, Elaine Stritch, Barton MacLane, Peter Hansen, John Harmon, Ross Bagdasarian, Bobby (Robert) Blake, Raymond Greenleaf, Don Devlin, Roy Engel, Argentina Brunetti, Leo Castillo. In Texas during Reconstruction an ex\u2013Confederate and his brother oppose carpetbaggers while facing a romantic triangle involving one of their wives. Okay action melodrama.\n\n**4374** _ **Three Warriors**_ **** United Artists\/Fantasy Films, 1978. 100 min. Color. D: Keith Merrill. SC: Sy Gomberg. With Randy Quaid, Byron Patt, Charlie White Eagle, Lois Red Elk, McKee Redwing. A 14-year-old Indian boy leaves his mother and two sisters to learn the ways of his people and meets a rawboned Indian agent recruit trying to adapt to his new job. Overlong but fairly pleasing drama.\n\n**4375** _ **Three Word Brand**_ **** Paramount-Artcraft, 1921. 70 min. D-SC: Lambert Hillyer. With William S. Hart, Jane Novak, Gordon Russell, S.J. Bingham, George C. Pearce, Colette Forbes, Ivor McFadden, Herschel Mayall, Leo Willis, Bill Patton. After their father is killed fighting Indians two brothers grow up as strangers, one being the governor of the state where the other, a rancher, opposes a dishonest water rights bill. Well produced William S. Hart silent entertainment with the star performing three roles, that of the two grown siblings and their father.\n\n**4376** _ **Three Young Texans**_ **** 20th Century\u2013Fox, 1954. 76 min. Color. D: Henry Levin. SC: Gerald Drayson Adams. With Mitzi Gaynor, Jeffrey Hunter, Keefe Brasselle, Harvey Stephens, Dan Riss, Michael Ansara, Aaron Spelling, Morris Ankrum, Frank Wilcox, Helen Wallace, John Harmon, Alex Montoya, Vivian Marshall. When outlaws try to force his father to take part in a robbery a young man commits the crime himself but cannot return the money because his friend hides it. Average oater with nothing special to offer.\n\n**4377** _ **The Thrill Hunter**_ **** Columbia, 1933. 58 min. D: George B. Seitz. SC: Harry O. Hoyt. With Buck Jones, Dorothy Revier, Ed LeSaint, Eddie Kane, Alice Dahl, Arthur Rankin, Frank LaRue, Robert Ellis, Harry Semels, Al Smith, John Ince, Alf James, Harry Todd, Willie Fung, Jim Corey, Frank Ellis, Art Mix, Glenn Strange, Buddy Roosevelt, Buffalo Bill, Jr., Hank Bell, Joe Ryan, Billy Sullivan, Charles Brinley. A long-winded cowboy gets to star in a movie and ends up capturing the remaining members of a notorious outlaw gang. Speedy adventure that pokes fun at Westerns.\n\n**4378** _ **Throw a Saddle on a Star**_ **** Columbia, 1946. 60 min. D: Ray Nazarro. SC: J. Benton Cheney. With Ken Curtis, Jeff Donnell, Adele Roberts, Guinn Williams, Andy Clyde, Frank Sully, The Dinning Sisters (Ginger, Jean and Lou Dinning), Foy Willing and the Riders of the Purple Sage (Darol Rice, Al Sloey, George Bamby), The Hoosier Hotshots (Charles \"Gabe\" Ward, Ken Trietsch, Paul \"Hezzie\" Trietsch, Gil Taylor), Maxine Doyle, Robert (Scott) Kellard, Emmett Lynn, Ed Peil, Sr., George Morrell, Roy Butler, Eddie Bruce, Rube Dalroy, Billy Gray, Ace Williams, Earl Duane, Jack Parker, Larry Prescott, Buck Shaw, Danny Weir. A ranch owner tries to get his rodeo rider son to reconcile with his girlfriend, fearing the breakup will make the young man lose in the championship competition thus causing the father to lose his ranch which he bet on his boy to win. Another Columbia Pictures' western music fest under the guise of being an oater.\n\n**4379** _ **The Throwback**_ **** Universal, 1935. 61 min. D: Ray Taylor. SC: Frances Guihan. With Buck Jones, Muriel Evans, George Hayes, Eddie Phillips, Paul Fix, Frank LaRue, Earl Pinegree, Robert Walker, Charles K. French, Bryant Washburn, Lafe McKee, Allan Ramsey, Margaret Davis, Bobby Nelson, Mickey Martin. Fifteen years after his father was falsely accused of thievery, a man returns home to find himself framed on the same charge. Buck Jones fans will go for this well scripted action melodrama.\n\n**4380** _ **Thunder at the Border**_ **** Columbia, 1967. 98 min. Color. D: Alfred Vohrer. SC: David De Reszke, C.B. Taylor and Harald G. Petersson. With Rod Cameron, Pierre Brice, Marie Versini, Harald Leipnitz, Todd Armstrong, Viktor de Kowa, Nadia Gray, Rik Battaglia, Jorg Marquardt, Vladimir Medar, Miha Baloh, Aleksandar Gavric, Emil Kutijaro, Illja Ivezic, Dusan Antonijevic, Walter Wilz, Milan Bosiljcic, Aleksanadar Stojkovic, Marija Cronbori, Tana Mascarelli, Aleksandar Belaric, Adela Podjed, Boris Dvornik, Dado Habazin, Nikola Gec, Emil Mikuljan, Stejepan Spoljaric, Ivo Kristof, Vladimir Simac, Franc Ursic, Jovan Yukovic, Zvonko Dobrin, Sime Jagarinac. Frontiersman Old Firehand and his blood brother, Apache chief Winnetou, help the settlers of a remote village besieged by a gang of cutthroats. Nifty, fierce West German production with a fine performance by Rod Cameron as Old Firehand. Issued in Europe in 1966 by Rialto\/Jadran-Film as _**Winnetou und Sein Freund Old Firehand**_ (Winnetou and His Friend Old Firehand).\n\n_**Thunder Cloud**_ see _**Colt .45**_\n\n**4381** _ **Thunder in God's Country**_ **** Republic, 1951. 67 min. D: George Blair. SC: Arthur Orloff. With Rex Allen, Buddy Ebsen, Mary Ellen Kay, Ian MacDonald, Paul Harvey, Harry Lauter, John Doucette, Harry V. Cheshire, John Ridgely, Frank Ferguson, Wilson Wood. An escaped convict arrives in a small town and begins taking advantage of the citizens until he is opposed by a cowboy and his pal. Rex Allen fans will go for this one, with a good plot and lots of action.\n\n**4382** _ **Thunder in the Desert**_ **** Republic, 1937. 60 min. D: Sam Newfield. SC: George Plympton. With Bob Steele, Louise Stanley, Don Barclay, Ed Brady, Charles King, Horace Murphy, Steve Clark, Lew Meehan, Ernie Adams, Richard Cramer, Budd Buster, Sherry Tansey. Pals Bob and Rusty join an outlaw gang while searching for the man who killed Bob's uncle, the rightful owner of a ranch the crooks want. Bob Steele is great in this otherwise fair Western with some well done comedy that even has villain Charles King as an early romantic interest for the leading lady! \"Thunder\" in the title refers to dynamite used to blow up waterholes.\n\n**4383** _ **Thunder in the Pines**_ **** Screen Guild, 1948. 62 min. D: Robert (Gordon) Edwards. SC: Maurice Tombragel. With George Reeves, Ralph Byrd, Denise Darcel, Greg McClure, Michael Whalen, Marion Martin, Lyle Talbot, Vince Barnett, Roscoe Ates, Tom Kennedy, Millicent Patrick, Frank Hagney, Jack Tornek, Joey Ray, Arno Tanney. Two rival lumberjacks put their hostilities aside when outsiders endanger their interests. Okay north woods action feature with a chance to see Denise Darcel early in her career.\n\n**4384** _ **Thunder in the Sun**_ **** Paramount, 1959. 82 min. Color. D-SC: Russell Rouse. With Susan Hayward, Jeff Chandler, Jacques Bergerac, Blanche Yurka, Carl Esmond, Fortunio Bonanova, Felix Locher, Bertrand Castelli, Albert Carrier, Michele Marty, Alberto Villa, Veda Ann Borg. In 1850 a wagon train of French Basques heads to California to plant vineyards and along the way a pretty woman is romanced by two men, including the wagon master. The plot is not much but the overall feature is colorful and entertaining.\n\n**4385** _ **Thunder Mountain**_ **** Fox, 1932. 68 min. D: David Howard. SC: Dan Jarrett and Don Swift. With George O'Brien, Barbara Fritchie, Frances Grant, Morgan Wallace, George Hayes, Ed Le Saint, Dean Benton, William Norton Bailey, Carl Stockdale, Lloyd Ingraham, Lafe McKee, Neal Hart, Hal Price, Clyde McClary, Harry Tenbrook, Harry Baven, Hank Bell, Arthur Thalasso, Sid Jordan, Denver Dixon. In the north country, a prospector is cheated out of his portion of a mining claim and plans to right the wrong. Okay George O'Brien vehicle that tends to drag a bit. TV title: _**Roaring Mountain**_.\n\n**4386** _ **Thunder Mountain**_ **** RKO Radio, 1947. 60 min. D: Lew Landers. SC: Norman Houston. With Tim Holt, Martha Hyer, Richard Martin, Steve Brodie, Richard Powers (Tom Keene), Virginia Owen, Harry Woods, Jason Robards, Robert Clarke, Harry Harvey, Allen Lee, Dick Elliott, Jim Corey, Dick Foote, Cactus Mack, Jack Tornek, Tom Smith, Bob Reeves, Denver Dixon. Returning home from college a man finds himself in the middle of a feud instigated by a dishonest saloon owner and his cohort, the sheriff. Supposed based on the Zane Grey novel, this film has little to do with that source but is still a rip-roaring good time.\n\n_**Thunder Mountain**_ (1964) see _**The Shepherd of the Hills**_ (1964)\n\n**4387** _ **A Thunder of Drums**_ **** Metro-Goldwyn-Mayer, 1961. 97 min. Color. D: Joseph M. Newman. SC: James Warner Bellah. With Richard Boone, George Hamilton, Luana Patten, Arthur O'Connell, Charles Bronson, Richard Chamberlain, Duane Eddy, James Douglas, Tammy Marihugh, Carole Wells, Slim Pickens, Clem Harvey, Casey Tibbs, Irene Tedrow, Marjorie Bennett, J. Edward McKinley. A young lieutenant is assigned to a remote cavalry post where he has troubles with a tough captain. Fans of the leads will enjoy this feature although it is only average.\n\n**4388** _ **Thunder Over Arizona**_ **** Republic, 1956. 75 min. Color. D: Joseph Kane. SC: Sloan Nibley. With Skip Homeier, Kristine Miller, George Macready, Wallace Ford, Jack Elam, Gregory Walcott, Nacho Galindo, George Keymas, John Doucette, John Compton, Bob Swain, Julian Rivero, Francis McDonald, Fred Graham. When a rich silver strike is discovered a dishonest politician tries to get control of the mine. Well made and entertaining.\n\n**4389** _ **Thunder Over Texas**_ **** Beacon, 1934. 61 min. D: John Warner (Edgar G. Ulmer). SC: Eddie Granemann. With Guinn Williams, Marion Shilling, Helen Wescott, Richard Botiller, Philo McCullough, Ben Corbett, Bob McKenzie, Victor Potel, Jack Kirk, Hank Bell, Tiny Skelton, Claude Peyton, Eva McKenzie, Lionel Backus, Rose Plummer, Al Haskell, George Morrell, Bob Card, Silver Tip Baker, Jack Jones. A little girl, whose father is murdered over valuable railroad rights-of-way maps, is kidnapped by outlaws but she is rescued by a cowboy. Okay, but nothing exceptional in this Guinn \"Big Boy\" Williams series vehicle directed by Edgar G. Ulmer using a pseudonym with the original story by his wife Sherle Castle.\n\n**4390** _ **Thunder Over the Plains**_ **** Warner Bros., 1953. 82 min. Color. D: Andre De Toth. SC: Russell Hughes. With Randolph Scott, Phyllis Kirk, Lex Barker, Henry Hull, Elisha Cook, Jr., Charles McGraw, Hugh Sanders, James Brown, Lane Chandler, Fess Parker, Richard Benjamin, Trevor Bardette, Earle Hodgins, Jack Woody, John Cason, Monte Montague, Carl Andre, Charles Horvath, John McKee, Gail Robinson, Boyd \"Red\" Morgan, Gayle Kellogg. In post\u2013Civil War Texas an Army officer is assigned to bring in a bandit terrorizing carpetbaggers although the soldier sympathizes with him. Pretty good Randolph Scott action feature.\n\n**4391** _ **Thunder Over the Prairie**_ **** Columbia, 1941. 60 min. D: Lambert Hillyer. SC: Betty Burbridge. With Charles Starrett, Eileen O'Hearn, Cliff Edwards, Cal Shrum and His Rhythm Rangers, Stanley Brown, Danny Mummert, David Sharpe, Joe McGuinn, Donald Curtis, Ted Adams, Jack Rockwell, Steve Clark, Murdock MacQuarrie, Eddie Laughton, John Tyrrell, Francis Sayles, Budd Buster, Horace B. Carpenter, Denver Dixon. A frontier doctor tries to help an Indian medical student falsely accused of murder and dynamiting a dam. Nicely done entry in the all-too-brief \"Dr. Monroe\" series starring Charles Starrett.\n\n**4392** _ **Thunder Pass**_ **** Lippert, 1954. 80 min. D: Frank McDonald. SC: Tom Hubbard and Fred Eggers. With Dane Clark, Dorothy Patrick, Andy Devine, Raymond Burr, John Carradine, Mary Ellen Kay, Raymond Hatton, Monte Blue, Rick Vallin, Nestor Paiva, Charles Fredericks, Tom Hubbard. A cavalry captain has two days to lead settlers out of Indian territory but his convoy is surrounded by tribesmen supported by a gun runner. Standard oater sporting a fine cast.\n\n**4393** _ **Thunder River Feud**_ **** Monogram, 1942. 58 min. D: S. Roy Luby. SC: John Vlahos and Earle Snell. With Ray Corrigan, John King, Max Terhune, Jan Wiley, Jack Holmes, Rick Anderson, Carleton Young, George Chesebro, Carl Mathews, Budd Buster, Ted Mapes, Steve Clark, Richard Cramer, Rudy Sooter, Hal Price, Jimmy Aubrey, Tex Palmer. Three cowpokes follow a pretty girl to her Wyoming ranch where they get mixed up in a feud instigated by a crook and his pals. Fans of \"The Range Busters\" may like this one but otherwise it is on the poor side, both in plot and execution.\n\n**4394** _ **Thunder Town**_ **** Producers Releasing Corporation, 1946. 57 min. D: Harry Fraser. SC: James Oliver. With Bob Steele, Ellen Hall, Syd Saylor, Bud Geary, Charles King, Edward Howard, Steve Clark, Bud Osborne, Jimmy Aubrey, Pascale Perry, Don Weston and His Band. Released from prison after serving time when framed for a crime, a man returns home to save the woman he loves from one of the men responsible for sending him behind bars. While nothing to brag about, this Bob Steele vehicle (from his final starring series) should please his legion of fans; remake of _**Lawless Valley**_ (q.v.).\n\n**4395** _ **Thunder Trail**_ **** Paramount, 1937. 58 min. D: Charles Barton. SC: Robert Yost and Stuart Anthony. With Gilbert Roland, Charles Bickford, Marsha Hunt, J. Carrol Naish, James Craig, Monte Blue, William Duncan, Billy Lee, Gene Reynolds, Reginald Barlow, Lucien Littlefield, Lee Shumway, Vester Pegg, Ed Coxen, Frank Cordell, Tommy Coats, Jack Daley, Mary Foy, Carol Holloway, Bob Clark, Slim Hightower, Ed Warren, Danny Morgan, Ray Hanford, Gertrude Simpson. Two brothers are separated by outlaws and fifteen years later one of them rides the desperado trail until he learns who was really behind the event. Outstanding \"B\" adaptation of Zane Grey's _Arizona Ames_ ; action packed and very entertaining.\n\n**4396** _ **Thunderbolt**_ **** Regal, 1935. 55 min. D: Stuart Paton. SC: Jack Jevne. With Kane Richmond, Lobo the Marvel Dog, Fay McKenzie, Bobby Nelson, Hank Bell, Frank Hagney, Barney Furey, Lafe McKee, Frank Ellis, George Morrell, Wally West, Jack Kirk, Blackie Whiteford, Bob Burns. A boy and his dog attempt to stop a murderous outlaw gang whose leader is trying to force a pretty girl to marry him after a prospector is falsely arrested for his crimes. Unbelievably bad independent effort from producer Sherman S. Krellberg.\n\n**4397** _ **Thunderbolt's Tracks**_ **** Rayart, 1927. 55 min. D: J.P. McGowan. SC: Bennett Cohen. With Jack Perrin, Pauline Curley, Jack Henderson, Billy Lamar, Harry Tenbrook, Ethan Laidlaw, Ruth Royce. Two Marines find the family of a dead friend in Mexico and while there they are duped into buying a worthless ranch. Standard independent silent oater.\n\n_**Thundergap Outlaws**_ see _**Bad Men of Thunder Gap**_\n\n**4398** _ **Thunderhead, Son of Flicka**_ **** 20th Century\u2013Fox, 1945. 78 min. Color. D: Louis King. SC: Dwight Cummings and Dorothy Yost. With Preston Foster, Rita Johnson, Roddy McDowall, James Bell, Diana Hale, Carleton Young, Ralph Sanford, Robert Filmer, Alan Bridge. A young boy trains a colt, wanting to make his show horse a champion. Colorful sequel to _**My Friend Flicka**_ (q.v.).\n\n**4399** _ **Thunderhoof**_ **** Columbia, 1948. 77 min. D: Phil Karlson. SC: Hal Smith. With Preston Foster, Mary Stuart, William Bishop. Three people, two men and a woman, trek into Mexico in search of a wild stallion. Compact cast and good scenic values make this pretty fair entertainment.\n\n**4400** _ **Thundering Caravans**_ **** Republic, 1952. 54 min. D: Harry Keller. SC: M. Coates Webster. With Allan \"Rocky\" Lane, Eddy Waller, Mona Knox, Roy Barcroft, Isabel Randolph, Richard Crane, Bill Henry, Edward Clark, Pierre Watkin, Stanley Andrews, Boyd \"Red\" Morgan, Marshall Reed, Tex Terry, Art Dillard, Dale Van Sickel. When valuable ore shipments are hijacked a U.S. marshal is called in to investigate. Action laden entry in Allan \"Rocky\" Lane's \"Famous Westerns\" series.\n\n**4401** _ **Thundering Frontier**_ **** Columbia, 1940. 57 min. D: D. Ross Lederman. SC: Paul Franklin. With Charles Starrett, Iris Meredith, Raphael (Ray) Bennett, Alex Callam, Carl Stockdale, Fred Burns, The Sons of the Pioneers (Bob Nolan, Tim Spencer, Lloyd Perryman, Pat Brady, Hugh Farr, Karl Farr), John Tyrrell, Francis Walker, John Dilson, George Chesebro, Eddie Laughton, Blackie Whiteford. Two brothers have a falling out after their rancher father dies with one of them wanting to help a man and his daughter build a telegraph line while the other is in cahoots with a saloon owner out to stop them. Vapid Charles Starrett outing.\n\n**4402** _ **Thundering Gun Slingers**_ **** Producers Releasing Corporation, 1944. 61 min. D: Sam Newfield. SC: Fred Myton. With Buster Crabbe, Al St. John, Frances Gladwin, George Chesebro, Karl Hackett, Charles King, Jack Ingram, Kermit Maynard, Budd Buster, Hank Bell, Cactus Mack, Ray Henderson, Augie Gomez, Herman Hack, George Morrell, Roy Bucko. Billy Carson's uncle is murdered and he and his pal Fuzzy try to find the killer but when a suspect is also bumped off Billy gets the blame. Average entry in the popular, but shoddy, PRC series.\n\n**4403** _ **The Thundering Herd**_ **** Paramount, 1933. 62 min. D: Henry Hathaway. SC: Jack Cunningham and Mary Flannery. With Randolph Scott, Judith Allen, Larry \"Buster\" Crabbe, Noah Beery, Raymond Hatton, Blanche Frederici, Harry Carey, Monte Blue, Barton MacLane, Alan Bridge, Dick Rush, Frank Rice, Buck Connors, Charles McMurphy. A buffalo hunter joins a wagon train heading West and tries to help them when attacked by Indians who have been incited to war by the needless slaughter of bison. Program feature remake of the 1925 film starring Jack Holt and Lois Wilson, this version is hurt by too much footage from the original. Noah Berry, repeating his role from the 1925 feature, is great as the lecherous villain Randall Jett and he is equaled by Blanche Frederici as his vengeful wife. Reissued as _**Buffalo Stampede**_.\n\n**4404** _ **Thundering Hoofs**_ **** Film Booking Offices (FBO), 1924. 47 min. D: Al (Albert S.) Rogell. SC: Marion Jackson. With Fred Thompson, Ann May, Charles Mailes, Fred Huntely, Charles De Revenna, Carrie Clark Ward, William Lowery, Willie Fung, Silver King (horse). A rancher's son wins a prize horse from a man who mistreats him and then thwarts the villain's plans to steal money from a Spanish land baron. Good photography, lots of action and nice comedy relief highlight his silent Fred Thompson feature in which his horse, Silver King, plays a big part in the story. A very good film.\n\n**4405** _ **Thundering Hoofs**_ **** RKO Radio, 1942. 61 min D: Lesley Selander. SC: Paul Franklin. With Tim Holt, Luana Walters, Lee \"Lasses\" White, Fred Scott, Archie Twitchell, Gordon De Main, Charles Phipps, Monte Montague, Joe Bernard, Frank Fanning, Frank Ellis, Robert Kortman, Lloyd Ingraham, Spade Cooley. A man would rather be a rancher than run his father stage line but he comes to the aid of a rival operator when the family business is threatened by outlaws. Average Tim Holt series entry.\n\n**4406** _ **Thundering Thompson**_ **** Anchor, 1929. 46 min. D: Benjamin (Ben) Franklin Wilson. SC: Robert Dillon. With Cheyenne Bill, Neva Gerber, Al Ferguson, Cliff Lyons, Ed La Niece, Buffalo Bill, Jr., Silver Tip Baker, Chuck Baldra, Bud Pope, Jack Jones. A deputy sheriff turns on his cattleman boss when he falls for the daughter of a sheepherder he is hired to evict from his ranch. Fast paced silent effort from producer Morris R. Schlank starring long forgotten Cheyenne Bill (Harry William McKechrie).\n\n**4407** _ **Thundering Through**_ **** Artclass, 1925. 50 min. D: Fred Bain. SC: Barr Cross. With Buddy Roosevelt, Jean Arthur, Charles Colby, Lew Meehan, Frederick Lee, L.J. O'Connor, Lawrence Underwood. After falling in love with the daughter of a neighbor, a rancher tries to foil the machinations of a banker and his outlaw cohorts trying to get their spreads for a railroad right-of-way. This silent poverty row affair is available only in a 13-minute version called _**Riding Rivals**_.\n\n**4408** _ **Thundering Trails**_ **** Western Adventure, 1951. 55 min. D: Ron Ormond. SC: Alexander White. With Lash LaRue, Al St. John, Sally Anglim, Archie Twitchell, Ray Bennett, Reed Howes, Bud Osborne, George Chesebro, I. Stanford Jolley, John Cason, Lee Roberts, Clarke Stevens, Jimmie Martin, Mary Lou Webb, Sue Hussey, Ray Broome, Cliff Taylor. Marshals Lash and Fuzzy are assigned to protect the new territorial governor whose life is threatened by an outlaw gang. Tacky production with long, boring sequences of little action; lots of footage from previous Lash LaRue Screen Guild efforts.\n\n**4409** _ **Thundering Trails**_ **** Republic, 1943. 56 min. D: John English. SC: Norman S. Hall and Robert Yost. With Bob Steele, Tom Tyler, Jimmie Dodd, Nell O'Day, Vince Barnett, Karl Hackett, Sam Flint, Charles F. Miller, John James, Forrest Taylor, Ed Cassidy, Forbes Murray, Reed Howes, Bud Geary, Budd Buster, Lane Bradford, Cactus Mack, Eddie Parker, Art Mix, Al Taylor, Jack O'Shea, John Carpenter, George DeNormand, Tex Cooper. The Three Mesquiteers assist a Texas Ranger whose brother is hooked up with a bunch of outlaws. There is lots of action is this fast paced entry from the near the end of the long running series.\n\n**4410** _ **The Thundering West**_ **** Columbia, 1939. 58 min. D: Sam Nelson. SC: Bennett Cohen. With Charles Starrett, Iris Meredith, The Sons of the Pioneers (Bob Nolan, Tim Spencer, Lloyd Perryman, Pat Brady, Hugh Farr, Karl Farr), Hal Taliaferro, Dick Curtis, Hank Bell, Art Mix, Ed LeSaint, Robert Fiske, Edmund Cobb, Slim Whitaker, Blackie Whiteford, Steve Clark, Fred Burns, Ed Peil, Sr., Art Dillard. A reformed outlaw becomes the sheriff of a community but is soon blackmailed by his former gang. Pretty fair third version of two previous Buck Jones features, _**The Lone Rider**_ (1930) and _**The Man Trailer**_ (qq.v.).\n\n**4411** _ **A Ticket to Tomahawk**_ **** 20th Century\u2013Fox, 1950. 90 min. Color. D: Richard Sale. SC: Mary Loos and Richard Sale. With Dan Dailey, Anne Baxter, Rory Calhoun, Walter Brennan, Charles Kemper, Connie Gilchrist, Arthur Hunnicutt, Will Wright, Chief Yowlachie, Victor Sen Yung, Mauritz Hugo, Raymond Greenleaf, Harry Carter, Harry Seymour, Robert Adler, Chief Thundercloud, Marion Marshall, Joyce McKenzie, Marilyn Monroe, John Merton, Jack Elam, Olin Howlin, John War Eagle, Lee MacGregor. Upon his arrival in a Western town, a drummer gets caught up in the middle of a fight for a railroad franchise. Zesty musical comedy.\n\n**4412** _ **Tickle Me**_ **** Allied Artists, 1965. 90 min. Color. D: Norman Taurog. SC: Elwood Ullman and Edward Bernds. With Elvis Presley, Julie Adams, Jocelyn Lane, Jack Mullaney, Merry Anders, Connie Gilchrist, Edward Faulkner, Bill Williams, Louis Elias, John Dennis, Laurie Benton, Linda Rogers, Ann Morrell, Lilya Chauvin, Jean Ingram, Francine York, Eve Bruce, Jackie Russell, Angela Greene, Peggy Ward, Dorian Brown, Inez Pedroza, Grady Sutton, Dorothy Conrad, Barbara Werle, Allison Hayes. A rodeo stars seeks refuge as a wrangler at a dude ranch for women. The songs are okay but the plot is not much in this Elvis Presley vehicle.\n\n**4413** _ **Tide of Empire**_ **** Metro-Goldwyn-Mayer, 1929. 73 min. D: Allan Dwan. SC: Waldemar Young and Joe Farnham. With Renee Adoree, George Duryea (Tom Keene\/Richard Powers), Fred Kohler, George Fawcett, William Collier, Jr., James Bradbury, Jr., Harry Gribbon, Paul Hurst, Buster Keaton, Gino Corrado, Eric Mayne. During the California Gold Rush a cowboy wins a ranch in a horse race and then gives it to the former owner's daughter, with whom he has fallen in love. Not uninteresting silent feature with sound effects; Fred Kohler is quite good as the villain. Some footage was shot with Joan Crawford in the lead before she was replaced by Renee Adoree.\n\n**4414** _ **Tierra Baja**_ (Low Land) **** Laguna Films, 1951. 90 min. D: Miguel Zacarias. With Pedro Armendariz, Zully Moreno, Luis Aldas, Barbara Gil, Gustavo Reviere, Angel Infante, Julio Villareal, Luis Mossot, Queta Lavat, Kika Meyer, Chel Lopez. A wealthy mill owner and his fiancee cheat a poor shepherd out of his property by having the peon marry the beautiful woman. Torrid Mexican Western melodrama.\n\n**4415** _ **Tierre de Violencia**_ (Land of Violence) **** Radaent Films, 1966. 87 min. D: Raul de Anda, Jr. SC: Raul de Anda. With Rodolfo de Anda, Lorena Velaquez, David Reynoso, Sonia Furo, Manuel Donde, Quintin Bulnes, Victor Alcocer, Jose Luis Caro, Cecilia Leger, Tito Novaro, Rogelio Guerra, Guillermo Orea, Agustin Isunza. The coming of a railroad spurs local businessmen to hire gunslingers to sabotage the operation but the bad men take over the area. Fast paced Mexican Western, produced and written by Raul de Anda, directed by his son, Raul de Anda, Jr., and starring another son, Rodolfo de Anda.\n\n**4416** _ **Tierra Sangrienta**_ (Bloody Land) **** Peliculas Internacionales, S.A., 1965. 92 min. Color. D: Rafael Portillo. SC: Fernando Oses. With Armando Silvestre, Jaime Fernandez, Anthony Caruso, Laura Leon, Elizabeth Dupeyron, Angel Aragon, Arturo Salvador, Jose Luis Llamas, Ricardo Gomez, Lorenzo de Monteclaro, Antono de Hud, Valentine Leyva. Two brothers vow revenge against the cruel outlaw band that murdered their younger sister. Violent south of the border oater.\n\n**4417** _ **A Tiger Walks**_ **** Buena Vista, 1964 91 min. Color. D: Norman Tokar. SC: Lowell S. Hawley. With Brian Keith, Vera Miles, Pamela Franklin, Sabu, Kevin Corcoran, Peter Brown, Edward Andrews, Una Merkel, Arthur Hunnicutt, Connie Gilchrist, Theodore Marcuse, Merry Anders, Frank McHugh, Doodles Weaver, Frank Aletter, Jack Albertson, Donald May, Robert Shayne, Hal (Harold) Peary, Ivor Francis, Michael Fox, Richard O'Brien. A tiger escapes from a circus truck and panics a small town where the sheriff's daughter tries to save the animal. Often overlooked, but well done, Disney feature.\n\n**4418** _ **El Tigre Enmascarado**_ (The Masked Tiger) **** Clasa-Mohme, 1951. 90 min. D: Zacarias Gomez Urquizo. With Luis Aguilar, Flor Silvestre, Aurora Segura, Francisco Avitia, Emma Roldan, Pascual Garcia Pena, Carlos Valdez, Robert G. Rivera, Jose L. Murillo, Agustin Fernandez. Following the murder of his priest brother, a singer gives up his studies in Mexico City and returns home, becoming a masked avenger out to find the killer. Luis Aguilar as another Mexican masked hero highlights this fair comedy-drama-musical feature.\n\n**4419** _ **Timber**_ **** Universal, 1942. 60 min. D: Christy Cabanne. SC: Griffin Jay. With Leo Carrillo, Andy Devine, Dan Dailey, Jr., Marjorie Lord, Edmund MacDonald, Wade Boteler, Nestor Paiva, Paul E. Burns, James Seay, Jean Phillips, William Hall, Walter Sande, Guy Usher, Lloyd Ingraham, Murdock MacQuarrie, Ernie Adams, Anthony Warde, Ethan Laidlaw, Eddie Dew, Stanley Blystone, Mickey Simpson, Frank McCarroll, Frank Hagney, Bob Reeves, Jack C. Smith, James Westerfield, Eddie Parker. When sabotage takes place at a lumber camp two FBI agents are called into the case. Standard, but well made, Universal program feature.\n\n**4420** _ **Timber Country Trouble**_ **** Allied Artists, 1955. 54 min. D: Wesley Barry. SC: William Raynor and Sam Roeca. With Guy Madison, Andy Devine, Edmund Cobb, Harry Lauter, Kenne Duncan, Buddy Roosevelt, Bruce Edwards, John Merton, Frances Charles, Michael Vallon, George Barrows, Fred Krone, Henry Blair. Two U.S. marshals look into claims a wild stallion is behind crimes caused by rustlers and they pretend to be timber tramps to investigate lumber camp thefts. More than passable theatrical paste-up composed of the \"Lumber Camp Story\" and \"Rustling Stallion\" episodes of \"The Adventures of Wild Bill Hickok\" (1951\u201358).\n\n**4421** _ **Timber Fury**_ **** Eagle-Lion, 1950. 63 min. D: Bernard B. Ray. SC: Michael Hanson. With David Bruce, Laura Lee, Nichle Di Bruno, Sam Flint, George Slocum, Lee Phelps, Gilbert Gryl, Paul Hoffman, Spencer Chan. In the north woods, a young girl and her father fight crooks trying to steal their valuable timber. Fair low budget affair.\n\n**4422** _ **Timber Queen**_ **** Paramount, 1944. 66 min. D: Frank McDonald. SC: Maxwell Shane and Edward T. Lowe. With Richard Arlen, Mary Beth Hughes, June Havoc, Sheldon Leonard, George E. Stone, Dick Purcell, Tony Hughes, Edmund McDonald, Horace McMahon, William Haade, Clancy Cooper, Dewey Robinson, Jimmy Ames, George Chandler. When thieves threaten to steal the timberland of his dead buddy's widow, an ex-pilot tries to help her save the business. Rowdy action drama from the Pine-Thomas unit.\n\n**4423** _ **Timber Stampede**_ **** RKO Radio, 1939. 59 min. D: David Howard. SC: Morton Grant. With George O'Brien, Marjorie Reynolds, Chill Wills, Morgan Wallace, Guy Usher, Earl Dwire, Frank Hagney, Monte Montague, Robert Fiske, Bob Burns, Tom London, Billy Benedict, Bud Osborne, Robert Kortman, Ben Corbett, Cactus Mack, Hank Worden, Elmo Lincoln, Frank O'Connor, Sid Jordan, Herman Nolan. A cattleman opposes the plans of a lumber baron and a railroad magnate to build a line through his spread. Okay George O'Brien film but not as good as some of his other efforts.\n\n**4424** _ **Timber Terrors**_ **** Stage and Screen, 1935. 50 min. D-SC: Robert Emmett (Tansey). With John Preston, Marla Bratton, William Desmond, James Sheridan (Sherry Tansey), Tiny Skelton, Fred Parker, Tom London, Clyde McClary, Harry Beery, Tex Jones, Captain, King of the Dogs, Dynamite the Wonder Horse. Following the brutal murder of his partner, a Canadian Mounted Policeman tries to bring in the killers. Bottom rung \"Morton of the Mounted\" drama with fine camerawork by Brydon Baker.\n\n**4425** _ **The Timber Trail**_ **** Republic, 1948. 67 min. Color. D: Philip Ford. SC: Bob Williams. With Monte Hale, Lynne Roberts, James Burke, Roy Barcroft, Foy Willing and The Riders of the Purple Sage, Francis Ford, Robert Emmett Keane, Steve Darrell, Fred Graham, Wade Crosby, Eddie Acuff. A Mountie helps a pretty girl whose property is sought by a power mad gunslinger. Colorful Monte Hale vehicle.\n\n**4426** _ **Timber Tramps**_ **** Howco International, 1975. 90 min. Color. D: Tay Garnett. SC: Chuck Keen. With Claude Akins, Joseph Cotten, Patricia Medina, Cesar Romero, Tab Hunter, Leon Ames, Eve Brent, Rosie (Roosevelt) Grier, Bob (Robert) Easton, Stash Clemmens, Hal Baylor, Shug Fisher. Timber men fight the elements and crooks in the Alaskan wilderness to complete a lumber contract. Colorful action feature filmed on location in 1972 by Alaska Pictures.\n\n**4427** _ **Timber War**_ **** Ambassador, 1935. 58 min. D: Sam Newfield. SC: Joseph O'Donnell. With Kermit Maynard, Lucille Lund, Lawrence Grant, Robert Warwick, Lloyd Ingraham, Wheeler Oakman, Roger Williams, George Morrell, James Pierce, Patricia Royale, Horace Murphy, Sydney Jarvis, Horace B. Carpenter, Ed Cassidy, Ted Mapes, Tex Phelps, Fred Parker. At a remote lumber camp a logger tries to help a young woman whose sawmill is the object of saboteurs. Weak effort in Kermit Maynard's James Oliver Curwood story series for producer Maurice Conn.\n\n**4428** _ **Timberjack**_ **** Republic, 1955. 94 min. Color. D: Joe (Joseph) Kane. SC: Allen Rivkin. With Sterling Hayden, Vera Ralston, David Brian, Adolphe Menjou, Hoagy Carmichael, Chill Wills, Jim Davis, Howard Petrie, Ian MacDonald, Wally Cassell, Elisha Cook, Jr., Karl \"Killer\" Davis, Tex Terry, George Marshall, Chuck Roberson, Ric Roman, John Dierkes, Emil Sitka, Boyd \"Red\" Morgan, John Halloran, Frank Jaquet, Richard Alexander, Joe Evans, William Fawcett, Max Wagner, Paul Savage, Frank Hagney, Margaret Cahill, Esther Ying Lee. A ruthless land baron tries to cheat a man out of the timberland he inherited but the victim's cause is helped by a saloon singer. Fair Republic \"A\" effort, but not up to the studio's usual standards. ****\n\n**4429** _ **A Time for Dying**_ **** Corinth Films, 1982. 73 min. Color. D-SC: Budd Boetticher. With Richard Lapp, Anne Randall, Bob Random, Victor Jory, Audie Murphy, Beatrice Kay, Ron Masak, Bert Mustin, Peter Brocco, Walter Reed, Louis Ojena, Jorge Rada, Walt LaRue, Charles Wagenheim, Ira Augustan, Terry M. Murphy, Skip Murphy, Randy Shields, Bob Herron, William Bassett, Casey Tibbs, Willard Willingham, J.N. Roberts. A man trained by his father to be a fast gun travels though the West, saving a young woman from a prostitution ring only to be forced to marry her by Judge Roy Bean. Filmed in 1969 and produced by Audie Murphy (who does an excellent cameo as Jesse James), this obscure oater got some release in 1982 and is interesting viewing for Budd Boetticher followers; Victory Jory is enjoyably hammy as Bean.\n\n**4430** _ **A Time for Every Season**_ **** Gold Key, 1972. 95 min. Color. A man and a young boy become the first to explore the awesome Alaskan Tundra, marveling at its dangers and beauty. Picturesque documentary.\n\n**4431** _ **A Time for Killing**_ **** Columbia, 1967. 88 min. Color. D: Phil Karlson. SC: Halsted Welles. With Glenn Ford, Inger Stevens, George Hamilton, Kenneth Tobey, Paul Petersen, Timothy Carey, Richard X. Slattery, Harrison J. Ford, Kay E. Kuter, Dick Miller, Emile Meyer, Marshal Reed, Max Baer (Jr.), Todd Armstrong (Harry) Dean Stanton, Charlie Briggs, James Davidson. Near the end of the Civil War several Confederate prisoners escape from their Union captors and take the captain's fiancee as a hostage. Average action feature also called _**The Long Ride Home**_.\n\n**4432** _ **A Time to Revenge**_ **** Ardustry Home Entertainment, 1997. 96 min. Color. D: John Harwood. SC: John Harwood and Dale Gibson. With Ken Olandt, Julie Michaels, Paul Gleason, Leslie Ryan, William O'Leary, Dewey Weber, Mike Moroff, Larry Mahan, Heather Burton, Dale Gibson, Robert Phillips, Christopher Michael, Mark Nearing, Taylor Kowalski. When his father is murdered, a rancher goes after the gang who killed him. Average modern-day revenge Western; scenes with Elizabeth Berkley were deleted before its release.\n\n_**A Time to Run**_ see _**The Female Bunch**_\n\n**4433** _ **Timerider**_ **** Manson International, 1983. 93 min. Color. D: William Dear. William Dear and Michael Nesbitt. With Fred Ward, Belinda Bauer, Peter Coyote, L.Q. Jones, Ed Lauter, Tracey Walker, Bruce Gordon, Richard Masur, Chris Mulkey. While competing in a cross country bike contest, a rider gets caught in a time transference experiment and is sent back to the West of the 1870s where he encounters outlaws. Fairly interesting combination of the sci-fi and Western genres, but it should have been better.\n\n**4434** _ **Timestalkers**_ **** Fries Entertainment, 1987. 104 min. Color. D: Michael Schutz. SC: Brian Clemens. With William Devane, Lauren Hutton, Klaus Kinski, Forrest Tucker, John Ratzenberger, John Considine, Gail Youngs, James Avery, Patrick Baldauff, Buck Taylor, Ritch Brinkley, Burke Dennis, Joshua Devane, J. Michael Flynn, A.J. Freeman, Terry Funk, Tommy Lamey, Deborah Levin, Danny Pintauro, Begona Plaza, Tim Russ, Michael Strasser, John Wesley, Tracy Walter. A woman from the future joins a scientist, whose wife and son have been killed in a car accident, in tracking down an evil gunman who travels back and forth in time from the Old West. Involved, but not overly interesting, TV sci-fi effort.\n\n**4435** _ **The Tin Star**_ **** Paramount, 1957. 93 min. D: Anthony Mann. SC: Dudley Nichols. With Henry Fonda, Anthony Perkins, Betsy Palmer, Michael Ray, Neville Brand, John McIntire, Mary Webster, Peter Baldwin, Richard Shannon, Lee Van Cleef, James Bell, Howard Petrie, Russell Simpson, Hal K. Dawson, Jack Kenney, Mickey Finn, Frank Cady, Frank Kensaton, Frank Cordell, Frank McGrath, Tim Sullivan, Allan Gettel. A bounty hunter arrives in town with a dead outlaw and finds the inexperienced sheriff unable to cope with an gang terrorizing the community. Solid, entertaining melodrama.\n\n**4436** _ **Tin Star Void**_ **** Double Helix Films, 1988. 92 min. Color. D: Tom (Garrett) Gniazdowski. SC: John McLaughlin. With Daniel Chapman, Ruth Collins, Loren Blackwell, John Pierce, Karen Rizzo, Phillip Nutman, Frank Stewart, Tom Gniazdowski, Guy Perrotta, John Skipp, Craig Spector, Sherman Backus, Brian Edwards, Debi Thiebeault, John Geisler, Susan Griffiths, Tim Metzger, Michael D. Lang, Fern Feller, Will Dejesus, Carl Barratte, Stuart Chapin. In the Old West of the future, a cowboy hunts for the gang that killed his sheriff brother. Only for fans of bizarre Westerns.\n\n**4437** _ **The Tioga Kid**_ **** Eagle-Lion, 1948. 54 min. D: Ray Taylor. SC: Ed Earl Repp. With Eddie Dean, Roscoe Ates, Jennifer Holt, Andy Parker and The Plainsmen, Dennis Moore, Lee Bennett, Terry Frost, William Fawcett, Eddie Parker, Bob Woodward, Tex Palmer, Lee Roberts, Carl Mathews, Ray Henderson, Ray Jones. A Texas Ranger pretends to be a notorious outlaw as he tries to horn in on a gang's operations to bring them to justice. Surprisingly good Eddie Dean film refashioned from _**Driftin' River**_ (q.v.).\n\n**4438** _ **A Tiro Limpio**_ (Fire at Will) **** Filmadora Independiente, 1958. 73 min. D: Rene Cardona, Sr. SC: Jesus Cardenas. With Rene Cardona, Jr., Sofia Alvarez, Lorena Velazquez, Rodolfo Landa, Juan Manuel Guerrero, Yolanda del Valle, Guillermo Orea, Jorge Alzaga, Victor Velazquez, Ada Carrasco, Armando Gutierrez, Rafael Estrada, Dacia Gonzalez, Dagoberto Rodriguez, Rene Cardona, Wally Barron, David Reynoso, Emilio Garibay, Savador Lozano. Trying to collect a debt for a client, a lawyer learns an old enemy may have killed his father. Standard follow-up to _**El Puma**_ and _**La Ley del Mas Rapido**_ (qq.v.), containing much footage from both features.\n\n**4439** _ **The Titled Tenderfoot**_ **** Allied Artists, 1955. 52 min. D: Frank McDonald. SC: Bill Raynor and Maurice Tombragel. With Guy Madison, Andy Devine, Jeanne Cagney, Clayton Moore, Marshall Reed, I. Stanford Jolley, David Cavendish, Hal Gerard, James Bell, Dick Elliott, Jack Reynolds, Gerald O. Smith, Parke MacGregor, Russ Whiteman, Guy Teague. Wild Bill Hickok and his deputy Jingles oppose a gang of fur thieves in the north woods and assist a diplomat who brags about his Asian acquired fighting skills. Adequate program feature made up of two 1952 episodes (\"A Joke on Sir Anthony\" and \"Trapper Story\") of the popular \"The Adventures of Wild Bill Hickok\" (1951\u201358) series.\n\n**4440** _ **To Find a Rainbow**_ **** American National Enterprises\/Gold Key, 1972. 90 min. Color. D-SC: The Staff of American National Enterprises. With Lawrence Dobkin (narrator), Jerry, Lucille, Jeff and Jenny Romney, Jerry, Angela, Donna, David and Danny Pimm. Two Utah families explore the Teton Mountains of Wyoming and Bryce Canyon, the stomping grounds of Butch Cassidy and his gang. Pleasant enough family oriented documentary with nice footage of Zion National Park.\n\n**4441** _ **To Hell You Preach**_ **** Modern Art Productions, 1972. 76 min. Color. D: Richard Robinson. SC: David Allen Russell (Hagen Smith). With Hagen Smith, Michael Christian, Tim Scott, Kitty Vallacher, Richard Hurst, Orville Sherman, Ellen Brown, Ivy Jones, Rance Howard, Emile Meyer, Hank Worden, Tom Monroe, Howard Wright, James Bacon. A gunman escapes from a posse and pretends to be a preacher and after a town accepts him he steals the church's money. Tawdry production, shown in England as _**Vengeance of a Gunfighter**_ , later re-edited with new footage and released as _**The Legend of Frank Woods**_ (q.v.).\n\n_**To Kill a Jackal**_ see _**Shoot the Living...Pray for the Dead**_\n\n**4442** _ **To the Last Man**_ **** Paramount, 1933. 70 min. D: Henry Hathaway. SC: Jack Cunningham. With Randolph Scott, Esther Ralston, Noah Beery, Jack LaRue, Larry \"Buster\" Crabbe, Fuzzy Knight, Barton MacLane, Gail Patrick, Muriel Kirkland, James Eagles, Eugenie Besserer, Harlan Knight, Shirley Temple, John Carradine, Delmar Watson, Egon Brecher, Jim Mason, Russ Powell, Erville Alderson, James Burke, Ethan Laidlaw, Harry Cording, Maston Williams, Dick Rush, Blackjack Ward, Tom Bay, Rosita Butler, Jay Wright, William Gillis, Cullen Johnson. Two families carry out a long standing feud from Kentucky to Nevada. Excellently told, well made drama based on Zane Grey's novel. Among the highlights are Esther Ralston's raw sex appeal and the murder of a family member played by Buster Crabbe; recommended. Reissued as _**Law of Vengeance**_.\n\n**4443** _ **Toby McTeague**_ **** International Spectrafilm, 1986. 96 min. Color. D: Jean-Claude Lord. SC: Jeff Maguire and Diordie Millecevic. With Yannick Bisson, Winston Rekert, Andrew Bednarski, Stephanie Morgenstern, Timothy Webster, Lilian Clune, Evan Adams, George Clutesi, Hamish McEwan, Tom Rack, Anthony Levinson, Mark Kulik, Joanne Vannicola, Doug Price, Ian Finlay. A Canadian boy who raises and trains sled dogs becomes alienated from his father, runs away from home and is counseled by a mystical old Indian who was once his dad's spiritual advisor. Mediocre Canadian family feature.\n\n**4444** _ **Today We Kill, Tomorrow We Die!**_ **** Cinerama, 1971. 95 min. Color. D: Tonino Cervi. SC: Dario Argento and Tonino Cervi. With Montgomery Ford (Brett Halsey), Bud Spencer, Tatsuya Nakadai, William Berger, Wayde Preston, Jeff Cameron (Geoffredo Scarciofolo), Stanley Gordon. After outlaws murder his wife and frame a farmer, sending him to prison, the man gets out and hires mercenaries to join him in revenge. Another in the long line of brutal Westerns from Italy released there as _**Oggi a Me...Domani a Te**_ (Today It's Me...Tomorrow It's You), highlighted by a fine script.\n\n**4445** _ **Toklat**_ **** Sun International, 1971. 100 min. Color. D: Robert W. Davidson. SC: Hugh Hogle and Bette Bennett Penney. With Leon Ames, Dick Robinson, Bette Bennett Penney (narrator). A sheepherder, who has watched a bear grow from a cub, must hunt the animal when he thinks it is responsible for his brother's death. Typical family oriented outdoor adventure of the 1970s, shy on plot but heavy on beautiful scenery.\n\n**4446** _ **Told in the Hills**_ **** Paramount-Artcraft, 1919. 60 min. D: George Melford. SC: Will M. Ritchey. With Robert Warwick, Ann Little, Tom Forman, Wanda Hawley, Charles Ogle, Monte Blue, Margaret Loomis, Eileen Percy, Hart (Jack) Hoxie, Jack Herbert, Guy Oliver, Joe Kentuck, James Whitebird, Peo-Peo-at-likt, John Moses. Leaving behind the woman he married to give her child his name, although it belongs to his brother, a man becomes a guide and prospector in Montana and falls in love with a settler's daughter. Interesting silent version of Mariah Ellis Ryan's novel, filmed on location in Idaho.\n\n**Advertisement for** _**Told in the Hills**_ **(Paramount-Artcraft, 1919).**\n\n**  \n**\n\n**4447** _ **The Toll Gate**_ **** Paramount-Artcraft, 1920. 55 min. D: Lambert Hillyer. SC: Lambert Hillyer and William S. Hart. With William S. Hart, Anna Q. Nilsson, Jack Richardson, Joseph Singleton, Richard Headrick. Finding out one of his men betrayed him for a reward, a gang leader traces him to a frontier town, burns the man's cantina and is chased by both his adversary and the law but is saved by a woman whose small child he rescued. William S. Hart wrote the story on which this somber, but entertaining, silent feature is based.\n\n**4448** _ **Toll of the Desert**_ **** Commodore, 1935. 55 min. D: Lester Williams (William Berke). SC: Miller Easton. With Fred Kohler, Jr., Betty Mack, Roger Williams, Tom London, George Chesebro, Earl Dwire, Ted Adams, Ed Cassidy, Billy Stevens, John Elliott, Steve Clark, Ace Cain, Blackie Whiteford, Iron Eyes Cody, Herman Hack, Budd Buster. A novice lawman is forced to hunt an outlaw whose personal code he has always admired, unaware the man is his father. Dandy little production, a gem from poverty row; highly recommended.\n\n**4449** _ **Tom Horn**_ **** Warner Bros., 1980. 97 min. Color. D: William Wiard. SC: Thomas McGuane and Bud Shrake. With Steve McQueen, Linda Evans, Richard Farnsworth, Billy Green Bush, Slim Pickens, Peter Canon, Elisha Cook, Roy Jenson, James Kline, Geoffrey Lewis, Harry Northrup, Steve Oliver. Now a drifter, the scout who captured Geronimo is hired by ranchers to stop cattle rustlers and ends up being framed on a murder charge. Slow moving biopic, not as good as the TV movie _**Mr. Horn**_ (q.v.).\n\n**4450** _ **Tomahawk**_ **** Universal-International, 1951. 82 min. Color. D: George Sherman. SC: Silvia Richards and Maurice Geraghty. With Van Heflin, Yvonne De Carlo, Preston Foster, Jack Oakie, Alex Nicol, Tom Tully, Ann Doran, Rock Hudson, Susan Cabot, Arthur Space, Stuart Randall, John Peters, Russell Conway, Ray Montgomery, David Sharpe, David H. Miller, John War Eagle, Regis Toomey, Sheila Darcy, Chief American Horse, Chief Bad Bear, Harry Townes, Floyd Sparks, David Miller, Harry Peterson. Scout Jim Bridger tries to avoid violence when the government fails to heed his warnings about letting settlers into Sioux Indian territory. Weak plot detracts from the cast and fine use of color in this \"A\" drama. British title: _**Battle of Powder River**_.\n\n**4451** _ **Tomahawk Trail**_ **** United Artists, 1957. 60 min. D: Lesley Selander. SC: David Chandler. With Chuck Connors, John Smith, Susan Cummings, Lisa Montell, George N. Neise, Robert Knapp, Eddie Little Sky, Frederick Ford, (Harry) Dean Stanton, Boyd \"Red\" Morgan. Due to the arrogance and incompetence of their new West Point commander, an Army troop is left stranded in the desert after their horses, ammunition and supply wagons are stolen by Indians. Standard, talky dual bill feature also called _**Mark of the Apache**_.\n\n_**Tomb for the Sheriff**_ see _**Lone and Angry Man**_\n\n**4452** _ **Tomboy and the Champ**_ **** Universal-International, 1962. 77 min. Color. D: Francis D. Lyon. SC: Virginia M. Cooke. With Candy Moore, Ben Johnson, Christine Smith, Jess Kirkpatrick, Jesse White, Casey Tibbs, Jerry Naill. A young girl raises a calf until it is grown and wins first prize at a cattle show before realizing her pet will be killed. Standard program film for the family trade.\n\n**4453** _ **Tombstone**_ **** Cinergi\/Hollywood Pictures, 1993. 134 min. Color. D: George P. Cosmatos. SC: Kevin Jarre. With Kurt Russell, Val Kilmer, Sam Elliott, Bill Paxton, Powers Boothe, Michael Biehn, Charlton Heston, Jason Priestley, Jon Tenney, Stephen Lang, Thomas Haden Church, Dana Delany, Paula Malcomson, Lisa Collins, Dana Wheeler-Nicholson, Joanna Pacula, Michael Rooker, Harry Carey, Jr., Billy Bob Thornton, Tomas Arana, Pat Brady, Paul Ben-Victor, John Philbin, Robert Burke, Billy Zane, John Corbett, Bo Greigh, Forrie J. Smith, Peter Sherayko, Buck Taylor, Terry O'Quinn, Charles Schneider, Gary Clarke, Billy Joe Patton, Frank Stallone, Bobby Joe McFadden, Pedro Armendariz, Jr., Michael N. Garcia, Grant Wheeler, Jim Dunham, Stephen Foster, Grant James, Don Collier, Cecil Hoffmann, Charlie Ward, Clark Ray, Chris Mitchum, Sandy Gibbons, Evan Osborne, Shana McCabe, Jerry Whittington, Jim Flowers, Frank P. Costanza, Michelle Beauchamp, Robert Mitchum (narrator). Returning to Tombstone, famed lawman Wyatt Earp teams with his brother Virgil and Doc Holliday for a showdown with the Clanton bunch. Sumptuous rehash of the OK Corral shootout that made a lot of money but is hard to follow.\n\n**4454** _ **Tombstone Canyon**_ **** World Wide, 1932. 62 min. D: Alan James. SC: Claude Rister and Earle Snell. With Ken Maynard, Cecilia Parker, Sheldon Lewis, Frank Brownlee, Bob Burns, George Gerwing, Lafe McKee, Jack Clifford, Ed Peil, Sr., George Chesebro, Jack Kirk, Merrill McCormick, Bud McClure. Looking into the secrecy behind his parentage, a cowboy arrives in an area terrorized by a mysterious hooded phantom. Action filled Ken Maynard feature benefiting from the use of the mystery-horror angle in its interesting plot.\n\n**4455** _ **Tombstone Terror**_ **** Supreme, 1935. 55 min. D-SC: Robert North Bradbury. With Bob Steele, Kay McCoy, John Elliott, George Hayes, Earl Dwire, Hortense Petro, Ann Howard, Frank McCarroll, Artie Ortego, George Morrell, Herman Hack, Nancy DeShon, Tex Phelps, Ray Henderson. A cowpoke is mistaken for an outlaw and tries to prove his rightful identity. Average Bob Steele outing for producer A.W. Hackel, which means lots of action and good entertainment for his fans.\n\n**4456** _ **Tombstone, the Town Too Tough to Die**_ **** Paramount, 1942. 80 min. D: William McGann. SC: Albert Shelby Le Vino and Edward E. Paramore. With Richard Dix, Kent Taylor, Frances Gifford, Don Castle, Edgar Buchanan, Victor Jory, Rex Bell, Harvey Stephens, Clem Bevans, Charles Halton, Beryl Wallace, Chris-Pin Martin, Jack Rockwell, Charles Stevens, Hal Taliaferro, Wallis Clark, Paul Sutton, Dick Curtis, Charles Middleton, Donald Curtis, James Farrera. When a gunfight accidentally results in the killing of a child, Wyatt Earp agrees to become the sheriff of Tombstone and clean up the town. Another re-telling of the Earp saga, no more historically accurate than the others but still worth watching.\n\n**4457** _ **Tonka**_ **** Buena Vista, 1958. 97 min. Color. D: Lewis R. Foster. SC: Lewis R. Foster and Lillie Hayward. With Sal Mineo, Philip Carey, Jerome Courtland, Rafael Campos, H.M. Wynant, Joy Page, Britt Lomond, Herbert Rudley, Sydney Smith, John War Eagle, Gregg Martell, Slim Pickens, Robert \"Buzz\" Henry. A young Indian brave captures and tames a wild horse but the animal is claimed by his cruel cousin and sold to the cavalry. Fair Disney film that evolves into a tepid retelling of the Battle of the Little Big Horn. TV title: _**A Horse Called Comanche**_.\n\n**4458** _ **Tonto Basin Outlaws**_ **** Monogram, 1941. 60 min. D: S. Roy Luby. SC: John Vlahos. With Ray Corrigan, John King, Max Terhune, Tristram Coffin, Jan Wiley, Ted Mapes, Rex Lease, Reed Howes, Art Fowler, Carl Mathews, Budd Buster, Ed Peil, Sr., Frank Ellis, Tex Palmer, Jim Corey, Art Dillard, Bud McClure, Bert Dillard, Hank Bell, Rube Dalroy, George Morrell, Jack Evans, Jack Tornek, Chick Hannon, Denver Dixon, Foxy Callahan. During the Spanish-American War, the Range Busters are assigned to find rustlers in Montana stealing cattle contracted to feed government troops. Standard entry in the popular series.\n\n**4459** _ **The Tonto Kid**_ **** Resolute, 1934. 61 min. D: Harry Fraser. SC: Harry C. (Fraser) Crist. With Rex Bell, Ruth Mix, Buzz Barton, Theodore Lorch, Joseph Girard, Barbara Roberts, Jack Rockwell, Murdock MacQuarrie, Bert Lindsley, Jane Keckley, Stella Adams, Bud Pope. Wanting a man's ranch, a crooked lawyer has him shot and places the blame on a cowboy while hiring a circus performer to pose as the dying man's missing daughter. Poor first effort in the quartet of films headlining Rex Bell, Ruth Mix and Buzz Barton from producer Arthur T. Mannon.\n\n**4460** _ **Too Much Beef**_ **** First Division\/Grand National, 1936. 66 min. D: Robert Hill. SC: Rock Hawkey (Robert Hill). With Rex Bell, Connie Bergen, Forrest Taylor, Lloyd Ingraham, Jimmy Aubrey, Jack Cowell, Peggy O'Connell, Horace Murphy, George Ball, Fred Burns, Steve Clark, Jack Kirk, Denny Meadows (Dennis Moore), Frank Ellis. Crooks are after a rancher's spread because it is in the path of a railroad and a cowboy helps him save it. Rex Bell's fans should like this okay outing he did for producers Max and Arthur Alexander.\n\n**4461** _ **Top Gun**_ **** United Artists, 1955. 73 min. D: Ray Nazarro. SC: Richard Shayer and Steve Fisher. With Sterling Hayden, William Bishop, Karin Booth, James Millican, Regis Toomey, Hugh Sanders, John Dehner, Rod Taylor, Denver Pyle, William Phillips, Richard Reeves, John Cason, Tom London, Frank O'Connor, George Eldredge, William Tannen, Florence Auer, Carl Mathews, Herman Hack, Ray Jones, Wheaton Chambers, Jack Kenny, Suzanne Ridgeway. After a man is cleared of a murder charge he becomes the town's sheriff but the job brings about a change in his character. More than passable melodrama.\n\n**4462** _ **Topeka**_ **** Allied Artists, 1953. 60 min. D: Thomas Carr. SC: Milton Raison. With Bill Elliott, Phyllis Coates, Rick Vallin, Fuzzy Knight, John James, Denver Pyle, Dick Crockett, Harry Lauter, Dale Van Sickel, Ted Mapes, Henry Rowland, Edward Clark, I. Stanford Jolley, Stanley Price, Michael Vallon. When an outlaw becomes the star packer of a lawless town his former gang members help him restore peace. Well done little action film with a strong script and performances.\n\n**4463** _ **Topeka Terror**_ **** Republic, 1945. 55 min. D: Howard Bretherton. SC: Patricia Harper and Norman S. Hall. With Allan Lane, Linda Stirling, Twinkle Watts, Roy Barcroft, Earle Hodgins, Bud Geary, Frank Jaquet, Jack Kirk, Tom London, Eva Novak, Hank Bell, Robert Wilke, Monte Hale, Jess Cavin, Fred Graham, Jack O'Shea, Horace B. Carpenter, Herman Hack, Bill Wolfe, Tom Smith, Herman Nolan. On the trail of outlaws, a special investigator pretends to be a vagabond cowboy to conceal his identity. Typically fine action filled Allan Lane series entry.\n\n**4464** _ **The Torch**_ **** Eagle-Lion, 1950. 90 min. D: Emilio Fernandez. SC: Ingio de Martino Noriega and Emilio Fernandez. With Paulette Goddard, Pedro Armendariz, Gilbert Roland, Walter Reed, Julio Villareal, Carlos Muzquiz, Margarito Luna, Jose Torvay, Pascual Garcia Pena, Antonia Daneem, Jorge Trevino, Rosaura Revueltas, Eduardo Arozamena, Guillermo Calles. A Mexican village is captured by a rebel leader and his army with the man falling in love with a nobleman's pretty daughter. Fairly interesting remake of director Emilio Fernandez's 1946 Mexican feature _**Enamorada**_ (In Love).\n\n**4465** _ **A Tornado in the Saddle**_ **** Columbia, 1942. 59 min. D: William Berke. SC: Charles Francis Royal. With Russell Hayden, Bob Wills and The Texas Playboys, Dub Taylor, Alma Carroll, Tristram Coffin, Don Curtis, Jack Baxley, Tex Cooper, John Merton, Ted Mapes, Blackie Whiteford, Art Mix, Carl Sepulveda, Jack Kirk, Jack Evans, George Morrell. A new lawman is at odds with a crooked saloon keeper who tries to steal a gold claim. Threadbare oater with it only asset being Bob Wills and his gang performing \"Dusty Skies.\"\n\n**4466** _ **Tornado Range**_ **** Eagle-Lion, 1948. 56 min. D: Ray Taylor. SC: William Lively. With Eddie Dean, Roscoe Ates, Jennifer Holt, Andy Parker and The Plainsmen, George Chesebro, Brad Slavin, Marshall Reed, Terry Frost, Lane Bradford, Russell Arms, Steve Clark, Hank Bell, Jack Hendricks, Ray Jones. The U.S. Land Office assigns an agent to stop warfare between ranches and homesteaders. Minor league Eddie Dean musical.\n\n**4467** _ **Tough Assignment**_ **** Lippert, 1949. 66 min. D: William Beaudine. SC: Carl K. Hittleman. With Don Barry, Marjorie Steele, Steve Brodie, Marc Lawrence, Sid Melton, Ben Welden, Iris Adrian, Michael Whalen, Fred Kohler, Jr., Dewey Robinson, J. Farrell MacDonald, John Cason, Frank Richards, Stanley Andrews, Leander de Cordova, Stanley Price, Gayle Kellogg, Hugh Simpson. An outlaw gang forces meat suppliers to buy an inferior product and a newspaper reporter and his bride go after them. Fairly efficient \"B\" drama with good direction; produced by Don Barry's company.\n\n**4468** _ **The Tougher They Come**_ **** Columbia, 1950. 69 min. D: Ray Nazarro. SC: George Bricker. With Wayne Morris, Preston Foster, Kay Buckley, William Bishop, Frank McHugh, Joseph Crehan, Mary Castle, Frank O'Connor, Al Thompson, Alan Bridge. Attempts by a lumberjack to work a track of forest land he has inherited are foiled by a gang of timber thieves. Standard north woods program feature from producer Wallace MacDonald.\n\n**4469** _ **Toughest Gun in Tombstone**_ **** United Artists, 1958. 72 min. D: Earl Bellamy. SC: Orville Hampton. With George Montgomery, Beverly Tyler, Jim Davis, Don Beddoe, Scott Morrow, Harry Lauter, Charles Wagenheim, Jack Kenney, John Merrick, Al Wyatt, Lane Bradford, Gregg Barton, Tex Terry, Hank Worden, Rodolfo Hoyos, Alex Montoya, Rico Alaniz, Jack Carr, William Forrest, Harry Strang, Mary Newton, Joey Ray, Gerald Milton. A Texas Ranger pretends to be an outlaw so he can formulate a plan to capture the Johnny Ringo gang. Plot possibilities are not carried out well in this mediocre outing.\n\n**4470** _ **Toughest Man in Arizona**_ **** Republic, 1952. 90 min. Color. D: R.G. Springsteen. SC: John K. Butler. With Vaughn Monroe, Joan Leslie, Edgar Buchanan, Victor Jory, Jean Parker, Henry (Harry) Morgan, Ian MacDonald, Diana Christian, Lee MacGregor, Bobby Hyatt, Charlita, Nadine Ashdown, Francis Ford, Paul Hurst, John Doucette, Edmund Cobb, Rex Lease, Sheb Wooley, Cliff Clark, Carleton Young, James Burke, Tom Fadden, George Plues, Paul E. Burns, Andy Brennan. A widower sheriff on the trail of a notorious outlaw falls in love with a beautiful woman. Vaughn Monroe's second \"A\" film for Republic is a pleasant affair.\n\n_**A Town Called Bastard**_ see _**A Town Called Hell**_\n\n**4471** _ **A Town Called Hell**_ **** Scotia International, 1971. 95 min. Color. D: Robert Parrish. SC: Richard Aubrey. With Robert Shaw, Stella Stevens, Telly Savalas, Martin Landau, Fernando Rey, Michael Craig, Al Lettieri, Dudley Sutton, Aldo Sambrell, Charley Bravo, Cris Huerta. A Mexican village ruled by a cruel bandit is the scene of a manhunt for a revolutionary leader as well as the killer of a woman's husband. Exceedingly violent Spanish-made horse opera without much appeal to genre fans. Original title: _**A Town Called Bastard**_.\n\n**4472** _ **Town Tamer**_ **** Paramount, 1965. 89 min. Color. D: Lesley Selander. SC: Frank Gruber. With Dana Andrews, Terry Moore, Lon Chaney, Bruce Cabot, Lyle Bettger, Richard Arlen, Barton MacLane, Richard Jaeckel, Philip Carey, Sonny Tufts, Coleen Gray, Jeanne Cagney, Roger Torrey, Don Barry, Robert Ivers, James Brown, Richard Webb, Bob Steele, DeForrest Kelley, Dale Van Sickel, Dinny Powell, Frank Gruber. After a gunslinger murders his wife, a lawman travels from town to town bringing peace and searching for the killer, who he finds as the marshal of a small community. A top notch cast of veteran players highlight this screen version of Frank Gruber's novel.\n\n**4473** _ **Track of the Cat**_ **** Warner Bros., 1954. 102 min. Color. D: William A. Wellman. SC: A.I. Bezzerides. With Robert Mitchum, Teresa Wright, Diana Lynn, Tab Hunter, Beulah Bondi, Philip Tonge, William Hopper, Carl (\"Alfalfa\") Switzer. A mountain family, torn by bitter personal conflicts, is harassed by a cougar and the two sons attempt to kill the animal. Strangely compelling psychological melodrama.\n\n**4474** _ **Track of the Moon Beast**_ **** Derio, 1977. 90 min. Color. D: Dick Ashe. SC: William Finger and Charles Sinclair. With Chase Cordell, Donna Leigh Drake, Gregorio Sala, Patrick Wright, Francine Kessler, Timothy Wayne Brown, Crawford MacCallum, Jeanne Swain, Alan Swain, Fred McCaffrey, Tim Butler, Gary Kanin, Frank Larrabee, Joe Blasco. A mineralogist turns into a giant lizard after being hit by particles of a meteor. Low budget, but effective, thriller filmed in the southwest.\n\n**4475** _ **Tracked**_ **** Film Booking Offices (FBO), 1928. 50 min. D: Jerome Storm. SC: Frank Howard Clark, John Stuart Twist and Helen Gregg. With Ranger (dog), Caryl Lincoln, Sam Nelson, Al (Albert J.) Smith, Jack Henderson, Art Robbins, Clark Comstock. When his dog is falsely accused of killing sheep, a man hides him in a cave and later the two save a woman from a runaway rig. Ranger the dog, a rival to Rin Tin Tin, is the highlight of this better than average silent canine epic.\n\n**4476** _ **Tracked by the Police**_ **** Warner Bros., 1927. 60 min. D: Ray Enright. SC: John Grey. With Rin Tin Tin, Jason Robards, Virginia Brown Faire, Tom Santschi, David Morris, Theodore Lorch, Ben Walker, Wilfred North, Nanette (dog). An escaped police dog helps the foreman of an Arizona irrigation project being sabotaged by a rival camp. Very good, action filled Rin Tin Tin vehicle.\n\n**4477** _ **The Tracker**_ **** Home Box Office (HBO), 1988. 98 min. Color. D: John Guillermin. SC: Kevin Jarre. With Kris Kristofferson, Scott Wilson, Mark Moses, David Huddleston, John Quade, Don Swayze, Geoffrey Blake, Leon Rippy, Ernie Lively, Karen Kopins, Celia Xavier, Jeff Weston, Jennifer Snyder, Brynn Thayer, Jose Rey Toledo, Kip Allen, John Barks, Michael D. Blum, Forrest Broadley, Jake Dengel, Brook Gamble, Jerry Gardner, Lois Geary, Neil Summers, Stephen Parks, Ron Kathman, Phil Mead, Adan Sanchez. A retired tracker and his son are hired to bring in a vicious outlaw, the head of a marauding gang. Fine TV movie filmed in Utah.\n\n**4478** _ **Trackers**_ **** Wrather Corporation, 1956. 75 min. Color. D: Earl Bellamy and Oscar Rudolph. SC: Robert E. Schaefer, Eric Friewald, Charles Larsay and Melvin Levy. With Clayton Moore, Jay Silverheels, Mary Ellen Kay, Harry Lauter, Mike Ragan, Charles Stevens, Allen Pinson, Wayne Burson, Terry Frost, Bill Henry, Frank Hagney, Tyler McDuff, Francis McDonald, John Pickard, Robert Burton, Gregg Barton, Molly Wrather, Don Turner, Steve E. Raines, Charles Aldridge, Tom Brocon, Ben Wilder, Edmond Hashim, Robert Swan. The Lone Ranger and Tonto stop a lynch mob, get on the trail of renegade Confederates charged with murder and try to find out who is behind the mysterious goings-on in a ghostly canyon. Fans of \"The Lone Ranger\" (ABC-TV, 1949\u201357) will like this telefilm made up of three episodes of the series: \"Ghost Canyon,\" \"The Trouble at Tylerville\" and \"Twisted Track.\"\n\n**4479** _ **The Trackers**_ **** ABC-TV, 1971. 73 min. Color. D: Earl Bellamy. SC: Gerald Gaiser. With Sammy Davis, Jr., Ernest Borgnine, Julie Adams, Jim Davis, Connie Kreski, Arthur Hunnicutt, Caleb Brooks, Norman Alden, Leo Gordon, Ross Elliott, David Reynard. A rancher teams with a black tracker to find out who murdered his son and kidnapped his daughter. Ordinary TV movie.\n\n**4480** _ **Tracy Rides**_ **** Reliable, 1935. 59 min. D: Harry S. Webb. SC: Rose Gordon and Betty Burbridge. With Tom Tyler, Virginia Brown Faire, Edmund Cobb, Charles K. French, Carol Shandrew, Lafe McKee, Jimmy Aubrey, Art Dillard, Jack Evans, Richard Botiller, Harry S. Webb, Robert Walker, Frank Ellis, Chuck Baldra, Rube Dalroy, S.S. Simon. A lawman gets caught in the middle of a feud between cattlemen and sheep raisers and when one of the latter is shot he is forced to arrest his girl's rancher father. Poor Tom Tyler vehicle benefiting from the presence of the lovely and talented Virginia Brown Faire.\n\n**4481** _ **Tracy the Outlaw**_ **** New-Cal Film Corporation, 1928. 60 min. D: Otis B. Thayer. SC: Merritt Crawford. With Jack Hoey, Rose Chadwick, Jane LaRue, Howard Chandler. After branded an outlaw, Harry Tracy attempts to find security but is always hounded by peace officers. Whitewashed silent biopic of the notorious bandit who once rode with the Butch Cassidy's Wild Bunch.\n\n**4482** _ **The Trail Beyond**_ **** Monogram, 1934. 55 min. D: Robert North Bradbury. SC: Lindsley Parsons. With John Wayne, Verna Hillie, Noah Berry, Noah Beery, Jr., Iris Lancaster, Robert Frazer, Earl Dwire, Eddie Parker, James Marcus, Reed Howes, Artie Ortego, Tex Palmer. A man saves his buddy from cardsharps and uncovers a map to a hidden gold mine but they are pursued by crooks also after its location. Perhaps the best of Paul Malvern's Lone Star productions, this adaptation of James Oliver Curwood's _The Wolf Hunters_ is very entertaining and sports beautiful photography by Archie Stout. First filmed by Ben Wilson Productions for Rayart in 1926 with Robert McKim and Virginia Brown Faire as _**The Wolf Hunters**_ and that title was used for a third version (q.v.) in 1949.\n\n**4483** _ **The Trail Blazers**_ **** Republic, 1940. 58 min. D: George Sherman. SC: Barry Shipman. With Robert Livingston, Bob Steele, Rufe Davis, Pauline Moore, Rex Lease, Weldon Heyburn, Carroll Nye, Tom Chatterton, Si Jenks, Mary Field, John Merton, Bob Blair, Pascale Perry, Harry Strang, Barry Hays, Forrest Taylor, Harrison Greene, Horace B. Carpenter, Brandon Beach, Jack Kirk, Bud Osborne, Post Park, Cactus Mack, Herman Hack, Merrill McCormick, Ray Teal, Bill Nestell, Tom Smith, Matty Roubert, Al Taylor, Curley Dresden, Chuck Baldra, Roy Bucko. The Three Mesquiteers attempt to stop an outlaw gang trying to sabotage the stringing of a telegraph line. Typically swift segment in the popular series.\n\n**4484** _ **The Trail Drive**_ **** Universal, 1933. 65 min. D: Alan James. SC: Nate Gatzert and Alan James. With Ken Maynard, Cecilia Parker, William Gould, Lafe McKee, Robert Kortman, Alan Bridge, Frank Rice, Fern Emmett, Jack Rockwell, Slim Whitaker, Frank Ellis, Hank Bell, Wally Wales, Ben Corbett, Cliff Lyons. The foreman of a large ranch agrees to lead a cattle drive with the combined herds of all the local ranchers but after it is underway he discovers his boss is a crook out to cheat his neighbors. A good story and typically Ken Maynard high standards for action makes this top notch feature a must for his fans.\n\n**4485** _ **Trail Dust**_ **** Paramount, 1936. 77 min. D: Nate Watt. SC: Al Martin. With William Boyd, James Ellison, George Hayes, Gwynne Shipman, Stephen Morris (Morris Ankrum), Britt Wood, Dick Dickinson, Earl Askam, Alan Bridge, John Beach, Ted Adams, Al St. John, Harold Daniels, Kenneth Harlan, John Elliott, George Chesebro, Robert Drew, Tom Halligan, Dan Wolheim, Emmett Daly, Leo McMahon. During a cattle drive the Bar 20 boys find themselves up against an outlaw gang planning to steal their herd by dynamiting a pass. A bit overlong (it was cut by 23 minutes for TV), but still satisfying \"Hopalong Cassidy\" feature.\n\n**4486** _ **Trail Guide**_ **** RKO Radio, 1952. 60 min. D: Lesley Selander. SC: William Lively. With Tim Holt, Richard Martin, Linda Douglas, Frank Wilcox, Robert Sherwood, John Pickard, Kenneth MacDonald, Tom London, John Merton, Hank Bell. A scout leading a wagon train learns crooks are planning to steal land intended for the homesteaders. Average Tim Holt vehicle from near the end of his RKO tenure.\n\n**4487** _ **Trail of Kit Carson**_ **** Republic, 1945. 56 min. D: Lesley Selander. SC: Albert DeMond and Jack Natteford. With Allan Lane, Helen Talbot, Twinkle Watts, Roy Barcroft, Tom London, Kenne Duncan, Jack Kirk, Bud Geary, Tom Dugan, George Chesebro, Robert Wilke, Herman Hack, John Carpenter, Henry Wills, Tom Steele. Although the death of his partner appears to be an accident, Kit Carson tries to find the truth. Fair Allan Lane film, the final entry in his \"Action Westerns\" series.\n\n**4488** _ **Trail of Robin Hood**_ **** Republic, 1950. 67 min. D: William Witney. SC: Gerald Geraghty. With Roy Rogers, Penny Edwards, Gordon Jones, Jack Holt, Foy Willing and The Riders of the Purple Sage, Rex Allen, George Chesebro, Ray Corrigan, Monte Hale, William Farnum, Allan \"Rocky\" Lane, Tom Keene, Kermit Maynard, Tom Tyler, Emory Parnell, Clifton Young, James Magrill, Carol Nugent, Ed Cassidy, Lane Bradford, Stanley Blystone, Kenneth Terrell. When thieves make trouble for a once famous Western film star, now a Christmas tree rancher, Roy Rogers and other screen cowboys come to his rescue. Sturdy outing with pleasant nostalgia value.\n\n**Poster for** _**Trail of Robin Hood**_ **(Republic, 1950).**\n\n**  \n**\n\n**4489** _ **Trail of Terror**_ **** Supreme, 1935. 59 min. D-SC: Robert North Bradbury. With Bob Steele, Beth Marion, Forrest Taylor, Charles King, Lloyd Ingraham, Charles K. French, Richard Cramer, Budd Buster, Ed Cassidy, Bob McKenzie, Herman Hack, Wally West, Clyde McClary. Trying to get evidence to convict an outlaw gang, a federal man pretends to be an escaped convict. Entertaining Bob Steele adventure.\n\n**4490** _ **Trail of Terror**_ **** Producers Releasing Corporation, 1943. 64 min. D-SC: Oliver Drake. With Dave O'Brien, Jim Newill, Guy Wilkerson, Patricia Knox, Jack Ingram, I. Stanford Jolley, Budd Buster, Kenne Duncan, Frank Ellis, Robert Hill, Dan White, Jimmy Aubrey, Tom London, Slim Whitaker, Wally West, Artie Ortego, Rose Plummer, Tom Smith. After his twin brother is killed helping robbers hold up a stage, a lawman takes his place to capture the gang. Low grade entry in \"The Texas Rangers\" series.\n\n**4491** _ **Trail of the Arrow**_ **** Monogram, 1952. 54 min. D: Thomas Carr. SC: Maurice Tombragel. With Guy Madison, Andy Devine, Monte Blue, Raymond Hatton, Francis Ford, Terry Frost, Rory Mallinson, Wendy Waldron, Jack Reynolds, Steve Pendleton, Neyle Morrow, Rodd Redwing, Tito Renaldo, Dick Rich, David Sharpe, Tom Steele, Ferris Taylor, Anthony Sydes. Two U.S. marshals try to help Indians by clearing a tribe of rustling cattle and proving that two men were murdered by arrows fired by whites. Satisfactory theatrical feature composed of the 1951 \"Indian Bureau Story\" and \"Indian Pony Express\" episodes of \"The Adventures of Wild Bill Hickok\" (1951\u201358).\n\n**4492** _ **Trail of the Axe**_ **** American, 1922. 54 min. D: Ernest C. Warde. SC: Ridgwell Cullun. With Dustin Farnum, Natalie Kingston, George Fisher, Joseph J. Dowling. A lumber mill operator is at odds with his alcoholic brother over the woman they both love and to get revenge when he loses her the sibling tries to get workers to set fire to the business. Intense silent melodrama mainly of interest to Dustin Farnum fans.\n\n**4493** _ **Trail of the Hawk**_ **** Continental Pictures, 1935. 55 min. D: Edward Dmytryk. SC: Griffin Jay. With Yancey (Bruce) Lane, Betty Jordan, Dickie Jones, Lafe McKee, Henry Hall, Rollo Dix, Don Orlando, Marty Joyce, Edward Foster, Budd Buster, George Morrell, Barney Beasley, Ed Carey, Zandra (dog). A cowpoke seeks the identity of his real father and gets on a job on a ranch plagued by a cattle rustler known as \"The Hawk.\" Edward Dmytryk made his directorial debut with this tacky low budget affair, also called _**The Hawk**_ ; reissued in 1937 by Jay Dee Jay Productions. In 1949 it was re-edited with new footage featuring Ramblin' Tommy Scott and his group, including Frankie Scott, Sandra Scott, Eddy Williams and Gaines Blevins, and re-released by Tommy Scott Productions. Allegedly based on James Oliver Curwood's story \"The Coyote.\"\n\n**4494** _ **The Trail of the Lonesome Pine**_ **** Paramount, 1936. 99 min. Color. D: Henry Hathaway. SC: Harvey Thew and Horace McCoy. With Sylvia Sidney, Fred MacMurray, Henry Fonda, Fred Stone, Nigel Bruce, Beulah Bondi, Robert Barratt, George \"Spanky\" McFarland, Fuzzy Knight, Otto Fries, Samuel S. Hinds, Alan Baxter, Margaret Armstrong, Ricca Allen, Fern Emmett, Richard Carle, Henry (Brandon) Kleinbach, Philip Barker, Robert Kortman, Charlotte Wynters, Frank Rice, Frank McGlynn, Sr., James Burke, Clara Blandick, Irving Bacon, George Ernest, Charles Middleton, Russ Powell, Ed LeSaint, Lowell Drew, Hilda Vaughn, Norman Willis, Lee Phelps, John Larkin, Betty Farrington, Powell Clayton, Jack Curtis, Hank Bell, John Beck, Fred Burns, Jim Welch, Bud Geary, Bill McCormick, Jim Corey, Martin Beaumon, Tuffy (dog). The Falin's and the Tolliver's carry on a generations old blood feud in the mountains before a railroad surveyor and his crew arrive and he falls in love with a local girl who has been promised to her cousin. Delightful drama in beautiful Technicolor highlighted by top notch performances, especially Fred Stone and Beulah Bondi, plus Fuzzy Knight's renditions of \"When It's Twilight on the Trail\" and \"A Melody from the Sky.\" Based on John Fox, Jr.'s 1908 novel, this was the fourth screen version of the story, the first being released in 1914 by Broadway Picture Producing Company starring Dixie Compton, Richard Allen and Frank L. Dear, who also directed. Two years later Cecil B. DeMille helmed a second outing for Famous Players-Lasky headlining Charlotte Walker, Thomas Meighan, Theodore Roberts and Earle Fox, and in 1923 Paramount filmed it again starring Mary Miles Minter (her final film role), Antonio Moreno and Ernest Torrence, directed by Charles Maigne.\n\n**4495** _ **Trail of the Mounties**_ **** Screen Guild, 1947. 45 min. D: Howard Bretherton. SC: Elizabeth (Betty) Burbridge. With Russell Hayden, Jennifer Holt, Emmett Lynn, Harry Cording, Terry Frost, Zon Murray, Frank Lackteen, Britt Wood, Charles Bedell, Pedro Regas, Felice Richmond, Jack Tornek, Herman Hack, George Morrell, Tom Smith. Sent to a remote village to capture a comrade's killer, a Canadian Mounted Policeman finds the man he seeks is his twin brother, the leader of a gang of fur thieves. Russell Hayden is good in dual roles; this featurette is cheaply made but enjoyable.\n\n_**Trail of the Royal Mounted**_ see _**The Mystery Trooper**_\n\n**4496** _ **Trail of the Rustlers**_ **** Columbia, 1950. 55 min. D: Ray Nazarro. SC: Victor Arthur. With Charles Starrett, Smiley Burnette, Gail Davis, Eddie Cletro and His Roundup Boys, Tommy Ivo, Myron Healey, Don C. Harvey, Mira McKinney, Chuck Roberson, Gene Roth, Blackie Whiteford, Boyd \"Red\" Morgan, Herman Hack, Post Park. A woman and her two sons find a valley's secret water source and plot to run off other ranchers to get their spreads but end up being opposed by the Durango Kid. Fair series outing; Jock Mahoney does not appear (outside of stunt work) but the name of the villainous clan is Mahoney.\n\n**4497** _ **Trail of the Silver Spurs**_ **** Monogram, 1941. 58 min. D: S. Roy Luby. SC: Elmer Clifton. With Ray Corrigan, John King, Max Terhune, Dorothy Short, I. Stanford Jolley, George Chesebro, Milburn Morante, Eddie Dean, Kermit Maynard, Frank Ellis, Carl Mathews, Steve Clark, Chuck Baldra. The government sends three cowboys after a gold thief and they wind up in a deserted town where a man and his daughter are being harassed by a mysterious figure using \"ghost writing.\" Nice addition to \"The Range Busters\" series, enhanced by its mystery motif.\n\n**4498** _ **Trail of the Vigilantes**_ **** Universal, 1940. 75 min. D: Allan Dwan. SC: Harold Shumate. With Franchot Tone, Warren William, Broderick Crawford, Peggy Moran, Andy Devine, Mischa Auer, Porter Hall, Samuel S. Hinds, Charles Trowbridge, Paul Fix, Harry Cording, Max Wagner, Edmund MacDonald, Joe King, Frank Brownlee, Edmund Cobb, Ray Teal, Bob McKenzie, Duke York, Ralph Dunn, Earle Hodgins, Kermit Maynard, Lloyd Ingraham, Ted Adams, George Chandler, Heinie Conklin, Lew Kelly, Victor Potel, Hank Bell, George MacQuarrie, Jim Farley, Fred Walburn. An Eastern lawman is assigned to go West and bring in an outlaw gang. Pleasant blend of humor and action.\n\n**4499** _ **Trail of the Wild**_ **** American National Enterprises, 1974. 94 min. Color. With Gordon Eastman. An outdoorsman leads an expedition to the northern reaches of Canada to study the lives of the Eskimos. Well made documentary.\n\n**4500** _ **Trail of the Yukon**_ **** Monogram, 1949. 67 min. D: William X. Crowely (William Beaudine). SC: Oliver Drake. With Kirby Grant, Suzanne Dalbert, Bill Edwards, Iris Adrian, Dan Seymour, William Forrest, Anthony Warde, Maynard Holmes, Peter Mamakos, Jay Silverheels, Guy Beach, Stanley Andrews, Dick Elliott, Bill Kennedy, Alan Bridge, Harrison Hearne, Burt Wendland, Wally Walker, Roy Bucko, Chinook (dog). A gang of bank robbers are tracked into the Canadian wilderness by a Royal Mounted Policeman and his dog. Kirby Grant's initial Monogram series outing is overlong and lumbering; adapted from James Oliver Curwood's novel _The Gold Hunters._\n\n**4501** _ **Trail of Vengeance**_ **** Republic, 1937. 60 min. D: Sam Newfield. SC: George Plympton and Fred Myton. With Johnny Mack Brown, Iris Meredith, Warner Richmond, Earle Hodgins, Richard Cramer, Dick Curtis, Karl Hackett, Frank LaRue, Horace Murphy, Steve Clark, Budd Buster, Jack Kirk, Tex Palmer, Jim Corey, Herman Hack, Merrill McCormick, Horace B. Carpenter, Wally West, Clyde McClary, Ray Henderson. Dude Ramsey becomes involved in a range war and also finds he is the heir to a mine, his brother having been murdered by an outlaw gang leader. Well made action drama with Warner Richmond a most effective villain.\n\n**4502** _ **Trail Riders**_ **** Monogram, 1942. 55 min. D: Robert Tansey. SC: Frances Kavanaugh. With John King, David Sharpe, Max Terhune, Evelyn Finley, Forrest Taylor, Lynton Brent, Charles King, Kermit Maynard, John Curtis, Steve Clark, Kenne Duncan, Frank LaRue, Bud Osborne, Tex Palmer, Richard Cramer, Frank Ellis, Augie Gomez. After his sheriff son is killed trying to stop a bank robbery, a marshal sends for the Range Busters to help him round up the gang. Only average entry in the popular series.\n\n**4503** _ **Trail Street**_ **** RKO Radio, 1947. 84 min. D: Ray Enright. SC: Norman Houston and Gene Lewis. With Randolph Scott, Anne Jeffreys, Robert Ryan, George \"Gabby\" Hayes, Madge Meredith, Steve Brodie, Billy House, Virginia Sale, Harry Woods, Phil Warren, Harry Harvey, Jason Robards, Elena Warren, Betty Hill, Larry McGrath, Warren Jackson, Billy Vincent, Glen McCarthy, Ernie Adams, Kit Guard, Al Murphy, Lew Harvey, Roy Butler, Frank Austin, Carl Webster, Jessie Arnold, Si Jenks, Donald Kerr, Stanley Andrews, Sarah Padden, Frank McGlynn, Jr. Sam Lufkin. Marshal Bat Masterson teams with a land agent to battle cattle rustlers plaguing a Kansas town. Pretty good action melodrama.\n\n**4504** _ **Trail to Gunsight**_ **** Universal, 1944. 58 min. D: Vernon Keays. SC: Bennett Cohen. With Eddie Dew, Fuzzy Knight, Maris Wrixon, Lyle Talbot, Ray Whitley and His Bar-6 Cowboys, Buzzy Henry, Marie Austin, Sarah Padden, Glenn Strange, Ray Bennett, Charles Morton, Forrest Taylor, Terry Frost, Jack Clifford, Henry Wills. When outlaws murder a boy's father a lawman takes him home only to find the gang raiding the family ranch. Passable Eddie Dew vehicle with the slick Universal look; at its best when Ray Whitley sings \"Old Nevada Trail.\"\n\n**4505** _ **The Trail to Hope Rose**_ **** The Hallmark Channel, 2004. 88 min. Color. D: David S. Cass, Sr. SC: Kevin Cutts. With Lou Diamond Phillips, Ernest Borgnine, Lee Majors, Richard Tyson, Marina Black, Warren Stevens, Jonathan Murphy, David Shackelford, Casey Sander, Paul Hewitt, Buck Taylor, Tracey Walter, Spencer Redford, Tom Everett, Jenny Sullivan, Marty Papazian, Dennis Fitzgerald, Sara De Berry, Jacleen Haber, Dave McQuade, John Dybdahi, Julie Mikhaele Clark, Tom Mesmer, Betsy McIntyre, Osa Danam, Brian Brow. A half-breed ex-convict, now a miner, saves a woman beaten by her husband and then helps an old rancher fight for his spread against his mine owner boss. Acceptable TV Western.\n\n**4506** _ **Trail to Laredo**_ **** Columbia, 1948. 54 min. D: Ray Nazarro. SC: Barry Shipman. With Charles Starrett, Smiley Burnette, Jim Bannon, Virginia Maxey, Tommy Ivo, Hugh Prosser, The Cass County Boys (Jerry Scoggins, Bert Dodson, Fred Martin), George Chesebro, John Merton, Bob Cason, Robert Wilke, Ted Mapes, Ethan Laidlaw, Mira McKinney, Bob Reeves. Forced to become a fugitive by his crooked partner who is in cahoots with a saloon owner smuggling gold, a stage line operator is helped by the Durango Kid. Action filled series episode.\n\n**4507** _ **Trail to Mexico**_ **** Monogram, 1946. 57 min. D-SC: Oliver Drake. With Jimmy Wakely, Lee \"Lasses\" White, Dolores Castelli, Julian Rivero, Dora Del Rio, Terry Frost, Forrest Matthews, Brad Slaven, Alex Montoya, Jonathan McCall, Juan Duval, Arthur Smith, The Saddle Pals, The Guadalajara Trio, Jack Rivers, Jesse Ashlock. A crooning investigator goes south of the border to find out who is behind a series of mining outfit robberies. The music is good but the plot is only fair in this average Jimmy Wakely opus.\n\n**4508** _ **Trail to San Antone**_ **** Republic, 1947. 67 min. D: John English. SC: Luci Ward and Jack Natteford. With Gene Autry, Peggy Stewart, Sterling Holloway, William Henry, The Cass County Boys (Jerry Scoggins, Bert Dodson, Fred Martin), Tristram Coffin, John Duncan, Dorothy Vaughn, Edward Keane, Ralph Peters, Frankie Marvin, Cacuts Mack. A horse trainer tries to help an injured jockey while he also fights a wild stallion out to steal his horse herd. Routine Gene Autry vehicle.\n\n**4509** _ **Trail to Vengeance**_ **** Universal, 1945. 54 min. D: Wallace Fox. SC: Bob Williams. With Kirby Grant, Poni (Jane) Adams, Fuzzy Knight, Tom Fadden, John Kelly, Frank Jaquet, Stanley Andrews, Walter Baldwin, Roy Brent, Pierce Lyden, Dan White, Beatrice Gray, William Sundholm, Corey Loftin. While investigating his brother's murder, a cowboy learns a dishonest banker is about to foreclose the mortgage on his property. Standard Kirby Grant Universal series film.\n**4510** _ **Trailin'**_ **** Fox, 1921. 58 min. D-SC: Lynn Reynolds. With Tom Mix, Eva Novak, Bert Sprotte, James Gordon, Sidney Jordan, Carol Holloway, J. Farrell MacDonald, Duke R. Lee, William De Vaull, Al Fremont, Bert Hadley, Harry Dunkinson, Jay Morley, Cecil Van Auker. A man vows revenge for the murder of his father not knowing the killer is his real parent since he was abducted as an infant. Complicated silent screen adaptation of Max Brand's novel, remade as _**A Holy Terror**_ (q.v.).\n\n_**Trailin' Double Trouble**_ see _**Trailing Double Trouble**_\n\n**4511** _ **Trailin' Trouble**_ **** Universal, 1930. 60 min. D: Arthur Rosson. SC: Arthur Rosson and Harold Tarshis. With Hoot Gibson, Margaret Quimby, Pete Morrison, Olive Young, William McCall, Robert Perry, Ben Corbett, Bud Osborne. A ranch worker in love with the boss' daughter plans to herd horses to market as a rival orders him robbed so he will be discredited in the girl's eyes. Pleasant Hoot Gibson feature, not too involved but with plenty of fun for his fans.\n\n**4512** _ **Trailin' Trouble**_ **** Grand National, 1937. 60 min. D: Arthur Rosson. SC: Philip Graham White. With Ken Maynard, Lona Andre, Roger Williams, Vince Barnett, Grace Woods, Fred Burns, Phil Dunham, Ed Cassidy, Horace B. Carpenter, Marin Sais, Tex Palmer. Due to a case of mistaken identity, a cowboy has to round up an outlaw gang in order to prove his innocence. Well done low budget Ken Maynard affair with a nice balance of comedy.\n\n**4513** _ **Trailin' West**_ **** Warner Bros., 1936. 59 min. D: Noel Smith. SC: Anthony Coldeway. With Dick Foran, Paula Stone, Addison Richards, Robert Barrat, Joseph Crehan, Gordon (Bill) Elliott, Fred Lawrence, Eddie Schubert, Henry Otho, Stuart Holmes, Milton Kibbee, Carlyle Moore, Jr., Jim Thorpe, Edwin Stanley, Bud Osborne, Glenn Strange, Gene Alsace, Lee \"Lasses\" White, Tom Wilson. Sam Rice, Frank Prince, Cliff Saum, Baldy Belomont. During the Civil War a Secret Service agent is assigned to track an outlaw band in the West. Plentiful action highlights this Dick Foran outing with two songs.\n\n**4514** _ **Trailing Danger**_ **** Monogram, 1947. 58 min. D: Lambert Hillyer. SC: J. Benton Cheney. With Johnny Mack Brown, Raymond Hatton, Marshall Reed, Peggy Wynne, Bonnie Jean Hartley, Steve Darrell, Eddie Parker, Pat Desmond, Bud Osborne, Ernie Adams, I. Stanford Jolley, Artie Ortego, Cactus Mack, Dee Cooper, Wally West, Ray Jones. An outlaw plans to get revenge on the stage line superintendent who sent him to jail but is opposed by a U.S. marshal. A meager plot hurts this Johnny Mack Brown-Raymond Hatton teaming.\n\n**4515** _ **Trailing Double Trouble**_ **** Monogram, 1940. 56 min. D: S. Roy Luby. SC: Oliver Drake. With Ray Corrigan, John King, Max Terhune, Lita Conway, Roy Barcroft, Kenne Duncan, Tom London, William Kellogg, Carl Mathews, Forrest Taylor, Nancy Louise King, Jimmy Wakely and His Rough Riders (Johnny Bond, Dick Reinhart), Texas Rex Felker, Richard Cramer, Frank Ellis. The Range Busters gets involved in trying to find out who killed a rancher and kidnapped his sister. Very good second series entry with plenty of action and humor; also called _**Trailin' Double Trouble**_.\n\n**4516** _ **Trailing North**_ **** Monogram, 1933. 60 min. D: J.P. McCarthy SC: John Morgan. With Bob Steele, Doris Hill, Arthur Rankin, George Hayes, Dick Dickinson, Fred Burns, Norman Fensler. A criminal on probation is enlisted by the rangers to locate the killer of his lawman mentor and he heads to Canada seeking a woman who holds the clue to the murder. Mediocre Bob Steele feature.\n\n**Poster for** _**Trailing North**_ **(Monogram, 1933).**\n\n**  \n**\n\n**4517** _ **Trailing the Killer**_ **** B.F. Ziedman, 1932. 64 min. D: Herman C. Raymaker. SC: Jackson Richards. With Francis McDonald, Heinie Conklin, Jose De La Cruz, Pedro Regas, Caesar (dog). A wolf dog is falsely accused of killing its master, the deed actually committed by a mountain lion. Well done action drama, filmed in Canada and made in a semi-documentary style; also titled _**Call of the Wilderness**_.\n\n**4518** _ **Trailing Trouble**_ **** Universal, 1930. 57 min. D-SC: Arthur Rosson. With Hoot Gibson, Margaret Quimby, William McCall, Pete Morrison, Bob Perry, Olive Young, Art Acord, William Dyer. Two cowboys vie for the affections of their ranch boss's daughter while one of them takes part in a horse theft operation to ruin his rival's reputation with the girl. Star Hoot Gibson produced this pleasant outing, Art Acord's only sound film.\n\n**4519** _ **Trail's End**_ **** Beaumont, 1933. 57 min. D: Al Herman. SC: Jack Jevne. With Conway Tearle, Claudia Dell, Baby Charlotte Barry, Fred Kohler, Ernest (Ernie) Adams, Pat Harmon, Victor Potel, Gaylord (Steve) Pendleton, Stanley Blystone, Jack Duffy. Framed and sent to prison, a man gets out vowing revenge on his enemies and ends up the sheriff of a community harassed by his old gang. Despite low grade production values, this Conway Tearle vehicle is pretty good.\n\n**4520** _ **Trail's End**_ **** Monogram, 1949. 57 min. D: Lambert Hillyer. SC: J. Benton Cheney. With Johnny Mack Brown, Max Terhune, Kay Morley, Myron Healey, Douglas Evans, Zon Murray, George Chesebro, Keith Richards, William Norton Bailey, Carol Henry, Boyd Stockman, Eddie Majors. After finding gold on a rancher's land, a crook tries to buy it and when the man refuses to sell he finds himself accused of murder but a lawman believes in his innocence. Mediocre Johnny Mack Brown film; the title is sadly prophetic for the series Western at the time.\n\n**4521** _ **Trails of Peril**_ **** Big 4, 1930. 55 min. D-SC: Alvin J. Neitz (Alan James). With Wally Wales, Virginia Brown Faire, Jack Perrin, Frank Ellis, Lew Meehan, Joe Rickson, Buck Connors, Bobby Dunn, Pete Morrison, Hank Bell. Mistaken for a outlaw, a cowboy decides to capture the bad man for the reward but the hoodlum gets the same ideal. Bottom of the barrel poverty row feature.\n\n**4522** _ **Trails of the Wild**_ **** Ambassador, 1935. 61 min. D: Sam Newfield. SC: Joseph O'Donnell. With Kermit Maynard, Billie Seward, Monte Blue, Fuzzy Knight, Matthew Betz, Theodore Von Eltz, Frank Rice, Robert Frazer, Wheeler Oakman, Roger Williams, Charles Delaney, John Elliott, Dick Curtis, William Desmond, Ted Mapes, Eddie Phillips, Frank McCarroll, Richard Botiller, Ed Cassidy, Herman Hack, Clyde McClary, Artie Ortego. A member of the Royal Canadian Mounted Police is hunting the man who murdered his pal. Surprisingly poor adaptation of James Oliver Curwood's _Caryl of the Mountains_ , filmed under that title in 1914, starring Tom Santschi and Kathlyn Williams, and again in 1936 (q.v.).\n\n**4523** _ **The Train Robbers**_ **** Warner Bros., 1973. 92 min. Color. D-SC: Burt Kennedy. With John Wayne, Ann-Margret, Rod Taylor, Ben Johnson, Chris(topher) George, Ricardo Montalban, Bobby Vinton, Jerry Gatlin. A cowboy and his pals team with a beautiful widow in trying to recover gold stolen from her late husband. Okay feature filmed in Mexico but not one of John Wayne's best.\n\n**4524** _ **Train to Tombstone**_ **** Lippert, 1950. 60 min. D-SC: William Berke. With Don Barry, Robert Lowery, Tom Neal, Wally Vernon, Judith Allen, Nan Leslie, Minna Phillips, Barbara Stanley, Claude Stroud, Bill Kennedy, Jack Perrin. Passengers on a Tombstone bound railroad train are robbed by outlaws and then attacked by marauding Indians. _**Stagecoach**_ -on-a-train provides some excitement despite its low budget.\n\n**4525** _ **The Traitor**_ **** Puritan, 1936. 58 min. D: Sam Newfield. SC: Joseph O'Donnell. With Tim McCoy, Frances Grant, Karl Hackett, Dick Curtis, Jack Rockwell, Wally Wales, Pedro Regas, Frank Melton, Richard Botiller, Edmund Cobb, Tina Menard, George Chesebro, Slim Whitaker, Soledad Jiminez, J. Frank Glendon, Frank McCarroll, Wally West, Frank Ellis, Jimmy Aubrey, Oscar Gahan, Julian Rivero, Jack Kirk, Art Dillard, Al Taylor, Ray Henderson, Buck Morgan, Jack King. A Texas Ranger pretends to get thrown out of the service so he can masquerade as an outlaw and join a gang hiding across the border in Mexico. The time honored plot gets pretty thin in this ragged low budget effort.\n\n**4526** _ **La Trampa Mortal**_ (The Deadly Trap) **** Besne, 1962. 80 min. D: Zacarias Gomez Urquiza. With Luis Aguilar, Flor Silvestre, Jaime Fernandez, Rosario Galvez, Emma Roldan, Cuco Sanchez. Fifteen years after a thief murdered his father and stole his ranch, a man returns, dons a mask, and seeks revenge. Another in the long line of masked Mexican heroes, but still a nice offering.\n\n**4527** _ **The Tramplers**_ **** Embassy, 1966. 90 min. Color. D: Albert Band (Alfredo Antonini) and Mario Sequi. SC: Alfredo Antonini and Ugo Liberatore. With Joseph Cotten, Gordon Scott, James Mitchum, Ilaria Occhini, Franco Nero, Emma Vannoni, Georges Lycan, Mariel Franklin, Aldo Cecconi, Franco Balducci, Claudio Gora, Romano Puppo, Dario Michaelis, Ivan Scratuglia, Carla Calo. Following the Civil War a Confederate veteran comes home to the terrible aftermath of the conflict and finds his stern-willed father wants to keep the crusade alive. One of the better 1960s foreign oaters with a good plot (based on Will Cook's novel _Guns of North Texas_ ), cast and plenty of action; made in Argentina and issued in Italy in 1965 as _**Gli Uomini dal Passo Pesante**_ (The Man of the Heavy Step).\n\n_**Top:**_ **John Wayne and Ann-Margret in** _**The Train Robbers**_ **(Warner Bros., 1973).** _**Bottom:**_ **Advertisement for** _**The Tramplers**_ **(Embassy, 1966).**\n\n**  \n**\n\n_**Transcontinental Express**_ see _**Rock Island Trail**_\n\n**4528** _ **The Trap**_ **** Paramount, 1959. 84 min. Color. D: Norman Panama. SC: Richard Alan Simmons and Norman Panama. With Richard Widmark, Lee J. Cobb, Tina Louise, Earl Holliman, Carl Benton Reid, Lorne Greene, Peter Baldwin, Chuck Wassil, Richard Shannon, Carl Milletaire, Louis Quinn, Wayne Heffley, James Bell, Karl Lukas, Walter Coy, Roger Creed, John Indrisano, Russell Saunders, Mike Mahoney, Berel Firestone. A mobster and his gang take over a remote Western community as his lawyer must confront the town sheriff and deputy, his estranged father and drunken brother. Dull going in his modern-day Western.\n\n**4529** _ **The Trap**_ **** Continental, 1968. 106 min. Color. D: Sidney Hayers. SC: David Osborn. With Oliver Reed, Rita Tushingham, Rex Sevenoaks, Barbara Chilcott, Linda Goranson, Blain Fairman, Walter Marsh, Jo Golland. A Canadian trapper reluctantly takes a deaf-mute girl in a wife auction and her loyalty to him eventually saves his life. Well made Canadian drama with fine scenic values in addition to a good plot.\n\n**4530** _ **Trap on Cougar Mountain**_ **** Sun International, 1972. 94 min. Color. D-SC: Keith Larsen. With Eric Larsen, Keith Larsen, Karen Steele, Alvin Keeswood, Randy Burt, Lawrence J. Rink. A boy is befriended by a cougar and together they struggle to survive in a mountain wilderness. More than adequate family adventure fare; songs by Gene Merlino.\n\n**4531** _ **Trapped**_ **** Columbia, 1937. 55 min. D: Leon Barsha. SC: John Rathmell. With Charles Starrett, Peggy Stratford, Robert Middlemass, Allan Sears, Ted Oliver, Lew Meehan, Ed Peil, Sr., Jack Rockwell, Ed LeSaint, Francis Sayles, Art Mix, Richard Botiller, Blackjack Ward. When his brother is murdered a cowboy believes a neighboring rancher committed the crime to get his sibling's land but he cannot prove it because the suspect is a helpless invalid. An intriguing mystery element highlights this well done and entertaining Charles Starrett effort.\n\n**4532** _ **Trapped in Tia Juana**_ **** Mayfair, 1932. 60 min. D: Wallace Fox. SC: Bernard McConville, Wallace Fox and Carlos F. Borcosque. With Edwina Booth, Duncan Renaldo, Dot Farley, Joseph W. (Joe) Girard, Manuel Paris, Frank Lanning, Henry Roquemore, Arthur Thalasso, Monte Vandergrift, Ed Peil, Sr. After being separated as children, two brothers grow up to become a soldier and a bandit with both falling for the same beautiful woman. Producer Fanchon Royer spared every expense in this threadbare, barely tolerable comedy-drama based on a story by genre favorite Rex Lease.\n\n**4533** _ **The Traveling Saleslady**_ **** Columbia, 1950. 75 min. D: Charles F. Reisner. SC: Howard Dinsdal. With Joan Davis, Andy Devine, Adele Jergens, Joseph Sawyer, Dean Riesner, John Cason, Chief Thundercloud, Harry Hayden, Charles Halton, Minerva Urecal, Eddy Waller, Teddy Infur, Robert Cherry, William Newell, Harry Woods, Ethan Laidlaw, Harry Tyler, Alan Bridge, Gertrude Charre, Emmett Lynn, Stanley Andrews, George Yowlachie, Bill Wilkerson, Nick Thompson, George McDonald, Fred Aldrich, Louis Mason, Jessie Arnold, Robert Wilke. A woman sets out with her boyfriend to sell her dad's soap so his business will survive and the two head West where they get involved with crooks and an Indian uprising. Stilted comedy filled with wheezy gags; not even Joan Davis and Andy Devine can do much to save this weak feature.\n\n**4534** _ **Treachery Rides the Range**_ **** Warner Bros., 1936. 56 min. D: Frank McDonald. SC: William Jacobs. With Dick Foran, Paula Stone, Monte Blue, Craig Reynolds, Carlyle Moore, Jr., Henry Otho, Jim Thorpe, Milton Kibbee, Bud Osborne, Monte Montague, Don Barclay, Gene Alsace, Richard Botiller, Iron Eyes Cody, William Desmond, Frank McCarroll, Frank Ellis, Artie Ortego, Nick Copeland, Frank Bruno. Crooks try to defraud Plains Indians who threaten to go on the warpath but a cowboy plans to bring about continued peace. Top notch Dick Foran vehicle with a good balance between action and songs.\n\n**4535** _ **Treason**_ **** Columbia, 1933. 57 min. D: George B. Seitz. SC: Gordon Battle. With Buck Jones, Shirley Grey, Robert Ellis, Ed LeSaint, Frank Lackteen, Edwin Stanley, Art Mix, Frank Ellis, T.C. Jacks, Charles Brinley, Charles Hill Mailes, Ivar McFadden. In 1870 an Army scout tries to infiltrate a group of Confederate sympathizers led by a woman who wants to get lands unjustly taken from her in Kansas. High grade Buck Jones drama.\n\n**4536** _ **The Treasure of Lost Canyon**_ **** Universal-International, 1951. 82 min. Color. D: Ted Tetzlaff. SC: Brainerd Duffield and Emerson Crocker. With William Powell, Julia (Julie) Adams, Rosemary De Camp, Henry Hull, Charles Drake, Tommy Ivo, Chubby Johnson, John Doucette, Marvin Press, Frank Wilcox, Griff Barnett, Jack Perrin, Virginia Mullen, Philo McCullough, Paul \"Tiny\" Newlan, George Taylor, Jimmy Ogg, Ed Hinkle, Hugh Prosser. A boy adopted by a middle-aged couple accidentally finds a hidden cache that nearly brings tragedy to everyone it touches. Fairly good screen adaptation of Robert Louis Stevenson's _The Treasure of Franchard_ with good work by William Powell as Doc, an old prospector.\n\n**4537** _ **Treasure of Matecumbe**_ **** Buena Vista, 1976. 117 min. Color. D: Vincent McEveety. SC: Don Tait. With Peter Ustinov, Peter Foxworth, Joan Hackett, Vic Morrow, Jane Wyatt, Johnny Duran, Billy \"Pop\" Attmore, Dub Taylor, Don Knight, Virginia Vincent, Dick Van Patten, Mills Watson, Val De Vargas, Robert Doqui. In the Florida Everglades, two boys, using a treasure map, search for hidden riches with the help of three others while a bad guy is on their trail. Average Walt Disney feature.\n\n**4538** _ **The Treasure of Pancho Villa**_ **** RKO Radio, 1955. 96 min. Color. D: George Sherman. SC: Niven Busch. With Rory Calhoun, Shelley Winters, Gilbert Roland, Joseph Calleia, Fanny Schiller, Tony Carvajal, Pasquel Pena, Carlos Mosquiz. An American mercenary working for Pancho Villa plans the robbery of gold from a government train but the loot is stolen before it reaches the revolutionary. Standard action effort.\n\n_**The Treasure of Pancho Villa**_ (1966) see _**The Vengeance of Pancho Villa**_\n\n**4539** _ **Treasure of Ruby Hills**_ **** Allied Artists, 1955. 71 min. D: Frank McDonald. SC: Tom Hubbard and Fred Eggers. With Zachary Scott, Carole Mathews, Dick Foran, Barton MacLane, Lola Albright, Lee Van Cleef, Raymond Hatton, Gordon Jones, Steve Darrell, Rick Vallin, Charles Frederick, Stanley Andrews, James Alexander, Glenn Strange, John Cason, Carl Mathews, Ray Jones. Crooked cattlemen fight for control of range land with a rancher trying to stop them and their greed. Pretty fair feature with good work by Zachary Scott as the hero.\n\n**4540** _ **The Treasure of Silver Lake**_ **** Columbia, 1965. 82 min. D: Harald Reinl. SC: Harald G. Petersson. With Lex Barker, Pierre Brice, Herbert Lom, Gotz George, Karin Dor, Ralf Wolter, Eddi Arent, Marianne Hoppe, Mirko Boman, Jan Sid (Sima Kanicijevic), Jozo Kovacevic, Shobodan Dimitrijevic, Branko Spoljar, Milivoj Stojanovic, Velemir Hill, Ilija Ivezic, Sime Jargarinac, Antun Nails, Vladimir Medar. Frontiersman Old Shatterhand and his Indian blood brother Winnetou try to stop the pillage of Indian lands by crooks looking for secreted treasure. The initial big screen revival of the works of Karl May by producer Horst Wendlant is a beautifully done and enjoyable production released in West Germany in 1962 as _**Der Schatz im Silbersee**_ (The Treasure of Silver Lake) by Constantin-Filmverleih at 111 minutes and remade as an animated feature in East Germany in 1989 as _**Die Spur fuhrt zum Silbersee**_ (The Trail Leads to the Silver Sea).\n\n**4541** _ **Treasure of Tayopa**_ **** Reina Productions, 1974. 85 min. Color. D: Bob Cawley. SC: Robert Mason and Philip Michel. With Gilbert Roland, Rena Winters, Phil Trapani, Bob Corrigan, Frank Hernandez, Andrew Farnsworth, Robert \"Spanky\" Spangler, Randy Hill, Ken McConnell, Jose Contreras, Sagario Pacheo, Rick Roark. A woman leads an expedition into the Mexican wilderness looking for fabulous treasure but one of the party is a madman set on killing the others. Poor drama that will disappoint Gilbert Roland fans since he appears only at the beginning and end as the film's host.\n\n**4542** _ **The Treasure of the Aztecs**_ **** Gloria Film, 1965. 102 min. Color. D: Robert Siodmak. SC: Ladislas Fodor, R.A. Stemmle and Georg Marischka. With Lex Barker, Gerard Barray, Rik Battaglia, Michele Girardon, Ralf Wolter, Alessandra Panaro, Teresa Lorca, Fausto Tozzi, Gustavo Rojo, Hans Nielsen, Kelo Henderson, Jean-Roger Caussimon, Friedrich von Ledebur, Jeff Corey, Antun Nalis, Djordje Nenadovic, Mirko Kujacic, Milivoje Popovic-Mavid, Branimir Tori Jankovic, Rolf Rolphs, Nada Radovic, Petar Buntic, Willy Egger. In 1864 Mexico both the followers of Benito Juarez and Emperor Maxmillian search for lost Aztec treasure in order to use the riches for their causes. Pretty exciting West German film hurt by being cut to 90 minutes and dubbed for U.S. TV. German title: _**Der Schatz der Aztecan**_ (The Treasure of the Aztecs) and also called _**Mercenaries of the Rio Grande**_ ; followed by _**Pyramid of the Sun God**_ (q.v.) the same year.\n\n**4543** _ **The Treasure of the Sierra Madre**_ **** Warner Bros., 1948. 126 min. D-SC: John Huston. With Humphrey Bogart, Walter Huston, Tim Holt, Bruce Bennett, Barton MacLane, Alfonso Bedoya, Martin Garralaga, Jack Holt, John Huston, A. Soto Rangel, Manuel Donde, Jose Torvay, Margarito Luna, Jacqueline Dalya, Bobby Blake, Spencer Chan, Julian Rivero, Harry Vejar, Pat Flaherty, Clifton Young, Ralph Dunn, Guillermo Calleo, Roberto Canedo, Ernesto Escoto, Ignacio Villalbajo, Ann Sheridan, David Sharpe. Three men head into the mountains of Mexico looking for gold but one of them is turned into a madman by greed. A bit overlong, but still near-classic screen version of B. Traven's novel with fine photography (by Ted McCord) and some magnificent character work, specifically Walter Huston's old prospector, Barton MacLane as the big-mouth McCormick, and Alfonso Bedoya's maniac, Gold Hat. A must see.\n\n**4544** _ **Tres Hombres Malos**_ (Three Bad Men) **** Clasa-Mohme, 1949. 91 min. D-SC: Fernando Mendez. With Luis Aguilar, Carlos Lopez Moctezuma, Raul de Anda, Gloria Alonso, Manuel Noriega, Dagoberto Rodriguez, Enriqueta Lavat, Miguel Angel Ferriz, Gilberto Gonzalez, Jose L. Murillo. Three outlaws kidnap the mayor's little girl and take her to a mountain cabin but become enthralled with the child and plan to return her despite being hunted by a posse. Well done Mexican Western from producer Raul de Anda.\n\n**4545** _ **Trespasses**_ **** Shapiro Entertainment, 1987. 100 min. Color. D: Loren Bivens and Adam Roarke. SC: Loren Vivans, Lou Diamond Phillips and Jo Carol Pierce. With Robert Kuhn, Van Brooks, Ben Johnson, Mary Pillot, Adam Roarke, Lou Diamond Phillips, Deborah Neumann, Thom Meyer, Marina Rice, KaRan Reed, George Sledge, Lou Perry, John Henry Faulk. After being brutally raped, a young woman falls for a cattleman and both become the target of revenge by her husband. Only fair modern-day mystery in a Texas town setting.\n\n**4546** _ **The Trial of Billy Jack**_ **** Warner Bros., 1974. 170 min. Color. D: Frank Laughlin. SC: Frank Christian and Teresa Christian. With Delores Taylor, Tom Laughlin, Victor Izay, Teresa Laughlin, Riley Hill, Sparky Watt, Russell Lane, William Wellman, Jr., Michelle Wilson, Geo Anna Sosa, Lynn Baker, Guy Greymountain, Sacheen Littlefeather, Michael Bolland, Jack Stanley, Sandra Ego, Trinidad Hopkins, Marianne Hill, Jason Clark, Johnny West, Buffalo Horse, Dennis O'Flaherty, Bong Soo Han, Michael J. Sigezone, Kathy Cronkite, Alexandra Nicholson, Rolling Thunder. Released from prison, an Indian rights activist returns to the mountains to search for the meaning of life. Third film in the \"Billy Jack\" series and a pretentious bore.\n\n**4547** _ **Trianganlos Vivor o Muertos**_ **** Filmadora Chapultepec, 1974. 84 min. Color. D-SC: Ruben Galindo. With Rodolfo de Anda, Pedro Armendariz, Jr., Raquel Olmedo, Roberto Guzman, Marco Antonio Campos, Marcelo Villami, Angelines Fernandez, Federico Falcon, Oscar Sar., Gustavo del Castillo. When an outlaw and a beautiful woman bank robber join forces they are chased by lawmen and bounty hunters. Well done Mexican Western.\n\n**4548** _ **Tribute to a Bad Man**_ **** Metro-Goldwyn-Mayer, 1956. 95 min. Color. D: Robert Wise. SC: Michael Blankfort. With James Cagney, Stephen McNally, Irene Papas, Don Dubbins, Vic Morrow, James Griffith, Onslow Stevens, James Bell, Royal Dano, Jeanette Nolan, Chubby Johnson, Lee Van Cleef, Peter Chong, James McCallion, Tony Hughes, Roy Engel, Bud Osborne, John Halloran, Tom London, Dennis Moore, Buddy Roosevelt. A drifter saves a wealthy horse rancher from bushwhackers and soon learns the man is ruthless regarding his property and his woman, with whom the stranger falls in love. Highly competent Western with a powerful performance by James Cagney as the rancher.\n\n_**Tricked**_ see _**Bandits of El Dorado**_\n\n**4549** _ **Trigger Fingers**_ **** Victory, 1939. 53 min. D: Sam Newfield. SC: Basil Dickey. With Tim McCoy, Jill Martin, Joyce Bryant, Ben Corbett, Kenne Duncan, John Elliott, Ralph Peters, Ted Adams, Bud McTaggart, Forrest Taylor, Carleton Young, Carl Mathews, Budd Buster, Wally West, Herman Back, Bob Terry, Chick Hannon, Tex Palmer. A lawman masquerades as a gypsy so he can get the goods on an outlaw band. Slow moving, final low grade entry in the \"Lightning Bill Carson\" series from producer Sam Katzman.\n\n**4550** _ **Trigger Fingers**_ **** Monogram, 1946. D: Lambert Hillyer. SC: Frank H. Young. With Johnny Mack Brown, Raymond Hatton, Jennifer Holt, Riley Hill, Ed Cassidy, Ted Adams, Steve Clark, Eddie Parker, Cactus Mack, George Morrell, Pierce Lyden, Ray Jones, Frank McCarroll. A marshal tries to help a hot-head who shoots a crook during a card game and is framed for murder by the dead man's gang. Average Johnny Mack Brown-Raymond Hatton series entry.\n\n**4551** _ **Trigger Jr.**_ **** Republic, 1950. 68 min. Color. D: William Witney. SC: Gerald Geraghty. With Roy Rogers, Dale Evans, Pat Brady, Gordon Jones, Foy Willing and The Riders of the Purple Sage, Grant Withers, Peter Miles, George Cleveland, Frank Fenton, I. Stanford Jolley, Stanley Andrews, Jack Ingram, Tom Steele, DaleVan Sickel, The Raymor Lehr Circus. A rodeo show owner finds out a range protection service is really fleecing local ranchers. Good Roy Rogers vehicle.\n\n**4552** _ **Trigger Law**_ **** Monogram, 1944. 56 min. D: Vernon Keays. SC: Victor Hammond. With Bob Steele, Hoot Gibson, Beatrice Gray, Ralph Lewis, Ed Cassidy, Jack Ingram, George Eldredge, Pierce Lyden, Lane Chandler, Bud Osborne, George Morrell. Two men search for the killer of the father of one of them, the victim having been the manager of a stagecoach line. Slow moving, low budget post \"Trail Blazers\" affair.\n\n_**Trigger Man**_ see _**Billy the Kid's Fighting Pals**_\n\n**4553** _ **Trigger Pals**_ **** Grand National, 1939. 56 min. D: Sam Newfield. SC: George Plympton. With Art Jarrett, Lee Powell, Al St. John, Dorothy Fay, Ted Adams, Nina Guilbert, Ernie Adams, Earl Douglas, Stanley Blystone, Frank LaRue, Ethan Allen, Dirk Thane, Robert Walker, Wally West, Carl Mathews. A cowboy not only finds part of a cattle herd he has been guarding is stolen but that his female boss plans to turn her property into a dude ranch. Passable comedy-music-action vehicle for crooner Art Jarrett.\n\n**4554** _ **Trigger Smith**_ **** Monogram, 1939. 51 min. D: Alan James. SC: Robert Emmett (Tansey). With Jack Randall, Joyce Bryant, Frank Yaconelli, Ed Cassidy, Bobby Clark, Warner Richmond, Dave O'Brien, Forrest Taylor, Earl Douglas, Sherry Tansey, Jim Corey, Reed Howes, Bud Osborne, Dennis Moore, Horace B. Carpenter, Milton Kibbee, Mary Thompson, Frank LaRue, Chick Hannon, Archie Ricks, Denver Dixon. After outlaws murder his brother during a holdup, a cowboy, at the request of his marshal father, sets out to bring them to justice. Average Jack Randall outing.\n\n**4555** _ **Trigger Tom**_ **** Reliable, 1935. 57 min. D: Henri Samuels (Harry S. Webb). SC: Tom Gibson. With Tom Tyler, Al St. John, Bernadene Hayes, William Gould, Jack Evans, John Elliott, Bud Osborne, Wally Wales, Lloyd Ingraham, Jack Hendricks, Budd Buster, Milburn Morante, Art Dillard, George Morrell, Francis Walker, Barney Beasley, Tex Palmer, Rube Dalroy, George Hazel. A cattle buyer and his pal get mixed up with an outlaw gang when one of the leaders poses as a deputy sheriff. Low grade, somewhat hard to follow Tom Tyler film with the asset of having Al St. John as his sidekick.\n\n**4556** _ **Trigger Trail**_ **** Universal, 1944. 59 min. D: Lewis D. Collins. SC: Ed Ear Repp and Patricia Harper. With Rod Cameron, Fuzzy Knight, Eddie Dew, Vivian Austin, Ray Whitley, Lane Chandler, George Eldredge, Buzzy Henry, Davison Clark, Michael Vallon, Richard Alexander, Jack Rockwell, Budd Buster, Bud Osborne, Ray Jones, Jack Ingram, Artie Ortego, Ray Whitley's Bar-6 Cowboys. Returning home from law school, a man discovers a crook is trying to steal land from local ranchers before it becomes a legal territory. Well produced, sturdy Rod Cameron series feature.\n\n**4557** _ **Trigger Tricks**_ **** Universal, 1930. 60 min. D-SC: B. Reeves Eason. With Hoot Gibson, Sally Eilers, Robert Homans, Jack Richardson, Monte Montague, Neal Hart, Walter Perry, Max Asher. Out to avenge the murder of his brother, a cowboy intervenes in a dispute between a cattle rancher and a sheepherder. Lighthearted Hoot Gibson film, topical at the time of its release because of the much publicized Gibson-Sally Eilers romance.\n\n**4558** _ **The Trigger Trio**_ **** Republic, 1937. 55 min. D: William Witney. SC: Joseph Poland and Oliver Drake. With Ray Corrigan, Max Terhune, Ralph Byrd, Sandra Corday, Hal Taliaferro, Robert Warwick, Cornelius Keefe, Sammy McKim, Jack Ingram, Willie Fung, Harry Semuels, Henry Hall, Bob Burns, Fred Burns, Jerry Frank, Ted Billings, Buck (dog). A rancher, trying to prevent authorities from finding out his cattle have hoof-and-mouth disease, kills a range inspector. Exciting \"Three Mesquiteers\" adventure (Ralph Byrd substitutes for injured Robert Livingston) that marked the feature directorial debut of William Witney.\n\n**4559** _ **Triggerman**_ **** Monogram, 1948. 58 min. D: Howard Bretherton. SC: Ronald Davidson. With Johnny Mack Brown, Raymond Hatton, Virginia Carroll, Bill Kennedy, Marshall Reed, Forrest Mathews, Bob Woodward, Dee Cooper, Frank Ellis, Herman Hack, George Morrell, Jack Evans, Ray Jones, Foxy Callahan, Carol Henry, Al Haskell, Jack Tornek, Kansas Moehring. A Well Fargo agent goes to work for a woman rancher and deduces a crooked real estate agent is after her land because gold is hidden on it. Better-than-average Johnny Mack Brown-Raymond Hatton feature, due mainly to its interesting script.\n\n**4560** _ **Triggerman**_ **** Grindstone Entertainment Group, 2009. 97 min. Color. D: Terence Hill and Giulo Base. SC: Jess Hill. With Terence Hill, Paul Sorvino, Ornelia Muti, Micah Alverti, Linus Huffman, Christina July Kim, Darrian Chavez, Fabrizio Bucci, Gianni Biasetti, Christopher Hagen. A gambler gunfighter must stand up to a cheating outlaw in order to win a big poker tournament. Fair Italian TV movie; sequel to _**Doc West**_ (q.v.).\n\n_**Trinity**_ see _**Jesse and Lester**_\n\n**4561** _ **Trinity and Bambino**_ **** Central Film, 1995. 90 min. Color. D: E.B. Clucher (Enzo Barboni). SC: Marco Barboni. With Heath Kizzier, Keith Neubert, Yvonne De Bark, Fanny Cadeo, Ronald Nitschke, Siegfried Rauch, Renato D'Amore, Eduardo MacGregor, Jack Taylor, Riccardo Pizzuti, Blaki, Jorge Oscar Bosso Cuello, Renato Scarpa, Jose Lifante, Ana Coriano Perez, Paco Catala, Tony Lima. The sons of two notorious cowpokes meet in a small town and join forces to clean up its lawless element. Okay attempt to emulate the Spaghetti Western comedies of the two previous decades; made in Italy and released there as _**Trinita e Bambino...e Adesso Tocca a Noi**_ (Trinity and Bambino...And It is Now Up to Us).\n\n**4562** _ **Trinity and Sartana Are Coming**_ **** Metheus Film, 1972. 102 min. Color. D: Mario Siciliano. SC: Adriano Bolzoni. With Robert Widmark (Alberto Dell'Acqua), Harry Baird, Beatrice Pellh, Stelio Candelli, Dan May (Dante Maggio), Alan Abbott (Ezio Marano), Lars Bloch, Enzo Andronico, Carla Mancini, Nino Nini, Daniela Giordano, Romano Puppo, Claudio Ruffini, Fortunato Arena, Gilberto Galimberti, Dino Cassio, Franko Ukmar, Pietro Torrisi, Enzo Maggio, Nello Pazzafini, Pietro Torrisi, Ricccardo Paetrazzi, Osiride Pevarello. Two outlaws carry out a series of bank robberies but they can never keep the loot since one of them is always giving it away. The so-called humor misses the mark in this Italian production issued there as _**Trinita e Sartana:**_ _**Figli Di...**_ (Trinity and Sartana: Children of...).\n\n_**Trinity Is Back Again**_ see _**A Genius, Two Friends and an Idiot**_\n\n_**Trinity Is My Name**_ see _**Trinity Is Still My Name**_\n\n**4563** _ **Trinity Is Still My Name**_ **** Avco-Embassy, 1973. 117 min. Color. D-SC: E.B. Clucher (Enzo Barboni). With Terence Hill, Bud Spencer, Harry Carey, Jr., Jessica Dublin, Yanti Sommer, Enzo Tarascio, Pupo De Luca. Two bungling outlaw brothers, now on the right side of justice, try to right all the wrongs they can find. Funny follow-up to _**They Call Me Trinity**_ (q.v.), this Italian feature, made as _**Continuavano a Chiamarlo Trinita**_ (Continue to Call Me Trinity), offers Harry Carey, Jr., a good role as the father.\n\n_**Trinity Sees Red**_ see _**Revenge of Trinity**_\n\n**4564** _ **Triple Justice**_ **** RKO Radio, 1940. 66 min. D: David Howard. SC: Morton Grant and Arthur V. Jones. With George O'Brien, Virginia Sale, Peggy Shannon, Harry Woods, Paul Fix, LeRoy Mason, Glenn Strange, Bud McTaggart, Bob McKenzie, Wilfred Lucas, Herman Nolan, John Judd, Lloyd Ingraham, Lew Meehan, Steve Pendleton, Hank Worden, Fern Emmett, George Mendoza, Henry Roquemore, Paul Everton, Walter Patterson, Jean Del Val, Henrique Valdez, Elenda Lindeman, Clothidle Lindeman, Bertha Lindeman. A rancher, on the way to a friend's wedding, meets a trio of men who robbed the local bank and he is blamed for the crime. George O'Brien's last series film is a fast paced and exciting one, a fine finale to his \"B\" Western career.\n\n**4565** _ **Triumphs of a Man Called Horse**_ **** Jensen Farley Pictures, 1982. 90 min. Color. D: John Hough. SC: Ken Blackwell and Carlos Aured. With Richard Harris, Michael Beck, Ana De Sade, Vaughn Armstrong, Anne Seymour, Buck Taylor, Simon Andreu, Lautaco Murua, Roger Cudney, Jerry Gatlin, John Chandler, Jacqueline Evans. Following the murder of his white father, a Sioux half-breed tries to protect his people from the gold seekers swarming into their sacred Black Hills. Picturesque but tepid third entry in the trilogy preceded by _**A Man Called Horse**_ and _**Return of a Man Called Horse**_ (qq.v.); best thing about it is Rita Coolidge singing the title theme, \"He's Comin' Back.\"\n\n_**The Trooper**_ see _**The Fighting Trooper**_\n\n**4566** _ **Trooper Hook**_ **** United Artists, 1957. 82 min. D-SC: Charles Marquis Warren. With Joel McCrea, Barbara Stanwyck, Earl Holliman, Edward Andrews, John Dehner, Susan Kohner, Royal Dano, Terry Lawrence, Celia Lovsky, Rodolfo Acosta, Stanley Adams, Pat O'Moore, Jeanne Bates, Rush Williams, Dick Shannon, Sheb Wooley, Cyril Delevanti, D.J. Thompson. A trooper falls in love with a woman who has been rescued from the Apaches but is scorned because she bore the tribe's chief a son. Sturdy drama with Tex Ritter performing the title song.\n\n**4567** _ **Troopers Three**_. Tiffany, 1930. 80 min. D: Norman Taurog. SC: John F. (Jack) Natteford. With Rex Lease, Dorothy Gulliver, Roscoe Karns, Slim Summerville, Tom London, Joseph Girard, Walter Perry. Three starving ham actors mistakenly join the cavalry and one of them falls in love with his drill sergeant's daughter and saves the man's life. Mediocre early talkie set in the West; also available in an excruciating 22 minute silent version.\n\n**4568** _ **Trouble at Melody Mesa**_ **** Astor, 1949. 64 min. D: W. Merle Connell. SC: Ned Dandy. With Brad King, Carl Shrum, Lorraine (Miller) Michie, I. Stanford Jolley, Walt Shrum, Alta Lee, Jimmie Shrum, Carl Sepulveda, Stacey Alexander, Robert \"Pappy\" Hoag, Ace Dehne, Shorty (Bob) Woodward, Don Weston, Roy Butler, Sue Gamboa, John Blackburn, Paula Blackburn, Rusty Cline, Lefty Walker, Jack Gress, Frank Bertoldi. After getting jobs on a ranch where the owner mysteriously died, Western band musicians help the local sheriff since they suspect foul play. Cheap musical oater featuring the Shrum family.\n\n**4569** _ **Trouble at Midnight**_ **** Universal, 1938. 68 min. D: Ford Beebe. SC: Maurice Geraghty and Ford Beebe. With Noah Beery, Jr., Catherine (Kay) Hughes, Larry Blake, Bernadene Hayes, Louis Mason, Earl Dwire, Charles Halton, Frank Melton, George Humbert, Edward Hearn, Harlan Briggs, Henry Hunter, Harry C. Bradley, Virginia Sale, Ernie Adams. A young man tries to combat a gang using modern methods such as freight trucks to rustle cattle from ranches near a remote town. Highly efficient and entertaining \"B\" effort, the first to initiate the plot so often used in later genre outings.\n\n**4570** _ **Trouble Busters**_ **** Majestic, 1933. 55 min. D: Lewis D. Collins. SC: Oliver Drake. With Jack Hoxie, Lane Chandler, Kay Edwards, Harry Todd, Ben Corbett, William T. Burt, Roger Williams, Charles \"Slim\" Whitaker, Irving Bacon, Henry Roquemore, Olin Francis, Bob Fleming, Bart Carre, Jack Kirk, Chuck Baldra, Jack Jones, Bob Roper. A cowboy tries to help a woman about to be cheated out of her oil rich land by a crook. Pleasant Jack Hoxie vehicle with enough action and humor to satisfy his fans.\n\n**4571** _ **Trouble in High Timber Country**_ **** ABC-TV, 1980. 100 min. Color. D: Vincent Sherman. SC: Jeb Rosebrook. With Eddie Albert, Joan Goodfellow, Martin Kove, James Sloyan, Robin Dearden, Belinda J. Montgomery, Kevin Brophy, Steve Doubet, Scott Yeager, James Sikking, Bettye Ackerman, Richard Sanders, Brion James, Jimmy Mair, John Quade, Lyle Cannon, Steve Eastin, Michael J. Fox, Hugh Gillin, John Lupton, Sheldon Biggs, Brian Patrick Clarke, Ian Wolfe, Mel Fletcher. A large corporation will stop at nothing to absorb a family's lumber and mining business. Fairly good modern-day TV movie.\n\n**4572** _ **Trouble in Sundown**_ **** RKO Radio, 1939. 60 min. D: David Howard. SC: Oliver Drake, Dorrell McGowan and Stuart McGowan. With George O'Brien, Rosalind Keith, Ray Whitley, Chill Wills, Ward Bond, Cy Kendall, Howard Hickman, Monte Montague, John Dilson, Otto Yamaoka, Ken Card, Slim Whitaker, Bob Burns, Lafe McKee, Earl Dwire, The Phelps Brothers, Jack Perrin, Murdock MacQuarrie, Lloyd Ingraham, Tom London, Ted Mapes. A cowboy attempts to come to the rescue of his girlfriend's father, a banker falsely accused of robbing his own establishment and murdering the night watchman. Exceedingly good George O'Brien film with a solid script.\n\n**4573** _ **Trouble in Texas**_ **** Grand National, 1937. 64 min. D: Robert North Bradbury. SC: Robert Emmett (Tansey). With Tex Ritter, Rita (Hayworth) Cansino, Horace Murphy, Earl Dwire, Yakima Canutt, Charles King, Dick Palmer, Tom Cooper, Hal Price, Fred Parker, Chick Hannon, Oral Zumwalt, Foxy Callahan, Henry Knight, Bob Crosby, Jack Smith, Shorty Miller, Milburn Morante, George Morrell, Jack C. Smith, Glenn Strange, The Texas Tornados. When his brother is murdered, a rodeo performer teams with a female undercover agent to track an outlaw gang working the circuit. Pleasingly action packed, and tuneful, Tex Ritter vehicle.\n\n**4574** _ **Trouble on the Trail**_ **** Allied Artists, 1954. 54 min. D: Frank McDonald. SC: William Raynor. With Guy Madison, Andy Devine, Robert Livingston, Carole Mathews, Richard Alexander, Sam Flint, William (Merrill) McCormick, Ralph Sanford, Fred Kohler, Jr., Tom Monroe, Larry Hudson, Peter McGabe, Bill Coontz, George Sherwood. Wild Bill Hickok and his deputy, Jingle P. Jones, take on a gigantic blacksmith brutalizing local ranchers in order to get their land and then try to capture a gunman masquerading as a magician. Acceptable big screen feature made up of \"Blacksmith Story\" and \"Medicine Show,\" two 1952 segments of \"The Adventures of Wild Bill Hickok\" (1951\u201358).\n\n**4575** _ **Truce**_ **** Anthem Pictures, 2005. 106 min. Color. D-SC: Matthew Marconi. With Buck Taylor, Samantha Droke, Michaela Lange, George Kennedy, Brad Johnson, Harvey Jordan, Scarlett McAlister, Eileen Indelicato, Tommy Guy, Cliff Chamberlain, Henry Brown, Tony Dodds, Lora Fredericksen, Amy Rolland, Steve J. Warner, Barry Tubb, Robert Koftinow, Christina Canepa. A cattle rancher struggles to keep his land while fulfilling the promise he made to his late daughter to give his teenage granddaughter a stable home. Well done and enjoyable modern-day Western drama.\n\n**4576** _ **True Grit**_ **** Paramount, 1969. 128 min. Color. D: Henry Hathaway. SC: Marguerite Roberts. With John Wayne, Glen Campbell, Kim Darby, Jeremy Slate, Robert Duvall, Dennis Hopper, Alfred Ryder, Strother Martin, Jeff Corey, Ron Soble, John Fiedler, James Westerfield, John Doucette, Donald Woods, Edith Atwater, Carlos Rivas, Isabel Boniface, H.W. Gim, John Pickard, Elizabeth Harrower, Ken Renard, Jay Ripley, Kenneth Becker, Myron Healey, Hank Worden, Guy Wilkerson, Boyd \"Red\" Morgan, Robin Morse. After outlaws murder her father, a teenage girl convinces a one-eyed, hard drinking rogue lawman to help her hunt down the killers. Outstanding genre effort that won John Wayne a much deserved Academy Award as Rooster Cogburn. Followed by _**Rooster Cogburn**_ (q.v.) and remade in 1978 and 2010 (qq.v).\n\n**4577** _ **True Grit**_ **** ABC-TV\/Paramount, 1978. 100 min. Color. D: Richard T. Heffron. SC: Sandor Stern. With Warren Oates, Lisa Pelikan, Lee Meriwether, James Stephens, Jeff Osterhage, Lee Harcourt Montgomery, Ramon Bieri, Jack Fletcher, Parley Baer, Lee DeBroux, Fred Cook, Fredmond Gleeson. A young girl tries to make a drunken lawman tow the straight and narrow while they track down the lawless. Pale TV movie based on Charles Portis' characters.\n\n**4578** _ **True Grit**_ **** Paramount, 2010. 100 min. Color. D-SC: Ethan Coen and Joel Coen. With Jeff Bridges, Hailee Steinfeld, Matt Damon, Josh Brolin, Barry Pepper, Dakin Matthews, Jarlath Conroy, Paul Rae, Domhnall Gleeson, Elizabeth Marvel, Roy Lee Jones, Ed Corbin, Leon Russom, Bruce Green, Candyce Hinkle, Peter Leung, Don Pirl, Joe Stevens, David Lipman, Jake Walker, Orlando Storm Smart, Ty Mitchell, Nicholas Sadler, Scott Sowers, Jonathan Joss, Maggie A. Goodman, Brandon Sanderson, Ruben Nakai Campana, Brian Brown, Scott Flick, Marcello Murphy, Ted Ferguson, Cody Jones. A teenager enlists the help of a hardened lawman in capturing her father's killer and they are joined by a Texas Ranger who is also after the fugitive. Well photographed, box office success remake of the 1969 (q.v.) classic.\n\n**4579** _ **The True Story of Jesse James**_ **** 20th Century\u2013Fox, 1957. 92 min. Color. D: Nicholas Ray. SC: Walter Newman. With Robert Wagner, Jeffrey Hunter, Hope Lange, Agnes Moorehead, Alan Hale, Alan Baxter, John Carradine, Rachel Stephens, Barney Phillips, Biff Elliot, Frank Overton, Marian Seldes, Barry Atwater, Chubby Johnson, Frank Gorshin, Carl Thayler, John Doucette, Edmund Cobb, Carleton Young, Gene Roth, Ken Clark, Tom Greenway, Robert Adler, Alexander Campbell, J. Frederick Albeck, Clancy Cooper, Sumner Williams, Joe Di Reda, James Stone, Mike Steen, Mark Hickman, Kay E. Kuter, Bing Russell, Ray Bennett, Aaron Saxon, Fay Roope, Michael Ross, Clegg Hoyt, Adam Marshall, Tom Pittman, Jason Johnson, Anthony Ray, Owen McGiveney, Adam Marshall, Sally Corner, Kelly Thordsen, Paul Wexler, Jason Wingreen, Paul Weber, Hy Anzell, Ken Christy, Harry Carter. When the James gang fails in its bank holdup at Northfield, Minnesota, the story of Jesse and Frank is told by those who knew them. Psychological approach to the James brothers works fairly well but one gets tired of all the flashbacks.\n\n_**The True Story of...the Cowboy**_ see _**The Cowboy**_\n\n**4580** _ **The Trumpet Blows**_ **** Paramount, 1934. 72 min. D: Stephen Roberts. SC: Bartlett Cormack and Wallace Smith. With George Raft, Adolphe Menjou, Frances Drake, Sidney Toler, Edward Ellis, Nydia Westman, Douglas Wood, Lillian Elliott, Katherine DeMille, Francis McDonald, Morgan Wallace, Gertrude Norman, Aleth \"Speed\" Hansen, Howard Brooks, E. Alyn Warren, Joyce Compton, Charles Stevens, Hooper Atchley, Alan Bridge, Mischa Auer. The younger brother of a Mexican bandit wants to be a bullfighter but he falls for his sibling's dancer girlfriend. Unbelievably poor melodrama with Adolphe Menjou as a Pancho Villa-type.\n\n**4581** _ **The Trusted Outlaw**_ **** Republic, 1937. 60 min. D: Robert North Bradbury. SC: George Plympton and Fred Myton. With Bob Steele, Lois January, Joan Barclay, Charles King, Earl Dwire, Richard Cramer, Hal Price, Budd Buster, Frank Ball, Oscar Gahan, George Morrell, Chick Hannon, Sherry Tansey, Clyde McClary, Jack Rockwell, Wally West, Jack C. Smith, Al Taylor, Fred Parker, Ray Henderson. The only surviving member of a family of outlaws tries to stay on the side of the law but crooks are out to make him their patsy. Sturdy Bob Steele action drama.\n\n_**Trusting Is Good...Shooting Is Better**_ see _**Dead for a Dollar**_\n\n**4582** _ **The Truth**_ **** Wrather Corporation, 1956. 75 min. Color. D: Oscar Rudolph and Earl Bellamy. SC: Wells Root, Thomas Seller, Charles Larson and Robert Leslie Bellem. With Clayton Moore, Jay Silverheels, Allen Pinson, Wayne Burson, Jim Bannon, Claire Carleton, Slim Pickens, Dennis Moore, Hank Worden, Parley Baer, Victor Sen Yung, Judy Dan, Joseph Vitale, Lee Roberts, John Berardino, Mickey Simpson, Tudor Owen, Florence Lake, Brad Jackson, Ron Hagerthy, Ewing Mitchell, Pat Lawless. The Lone Ranger and Tonto try to help an innocent man, stop an old lady from inciting an Indian massacre and assist a Chinese laundryman fight discrimination in a small town. Well done telefeature culled from \"The Lone Ranger\" (ABC-TV, 1949\u201357) episodes \"The Banker's Son,\" \"The Law and Miss Aggie\" and \"Letter Bride.\"\n\n**4583** _ **Tu Vida Contra Mi Vida**_ (My Life Against Your Life) **** Producciones Potusi\/Peliculas Mexicanas, 1979. 95 min. Color. D: Alfred Martinez. With Juan Gallardo, Rosenda Bernal, Luis de Alba, Pedro Weber, Carlos Lopez Moctezuma, Mario Cid. A rancher and a gangster vie for the hand of a beautiful woman. Okay modern-day Mexican Western.\n\n**4584** _ **Tucson Raiders**_ **** Republic, 1944. 55 min. D: Spencer Gordon Bennet. SC: Anthony Coldeway. With Wild Bill Elliott, George \"Gabby\" Hayes, Bobby Blake, Alice Fleming, Peggy Stewart, LeRoy Mason, Ruth Lee, Stanley Andrews, John Whitney, Bud Geary, Karl Hackett, Tom Steele, Tom Chatterton, Ed Cassidy, Fred Graham, Frank McCarroll, Marshall Reed, Edward Howard, Charles Sullivan, Bud Wolfe, Tommy Coats; Kenne Duncan, Tom London, Jack Kirk (voices). Two corrupt men, a banker and governor, try to get control of a territory but are thwarted by Red Ryder, Gabby and Little Beaver, but not until after Red is falsely accused of murder. A mediocre plot, but lots of action in this \"Red Ryder\" adventure, the first in the long running Republic series.\n\n**4585** _ **Tulsa**_ **** Eagle-Lion\/Path\u00e9 Industries, 1949. 90 min. D: Stuart Heisler. SC: Frank S. Nugent and Curtis Kenyon. With Susan Hayward, Robert Preston, Pedro Armendariz, Lloyd Gough, Chill Wills, Edward (Ed) Begley, Jimmy Conlin, Roland Jack, Harry Shannon, Lola Albright, Lane Chandler, Charles Meredith, Pierre Watkin, Thomas Browne Henry, Billy Wilkerson, Charles D. Brown, John Dehner, Fred Graham, Chief Yowlachie, Chester Conklin, Selmer Jackson, Frank Hagney, William Norton Bailey, Iron Eyes Cody, Tom Dugan, Bill Hickman, Franklyn Farnum, Brick Sullivan, George Magrill, Creighton Hale, Sam Harris, Frank Mills, Cyril Ring, Dick Wessel, George Barrows, Yvonne Doughty, Sayre Dearing, Mike Donovan, James Conaty, Allegra Varron, Charles Sherlock, John Holland, Jack Low, Nolan Leary, Roger Moore, Frank Mills, Kenner G. Kemp, Harold Miller, Carl M. Leviness, George Meader, Wilbur Mack, David McMahon, Renny McEvoy, Bill Hickman. A beautiful woman becomes a wealthy oil wildcatter but her greed for riches almost costs her the man she loves. Entertaining melodrama that moves along at a good clip.\n\n**4586** _ **The Tulsa Kid**_ **** Republic, 1940. 57 min. D: George Sherman. SC: Oliver Drake and Anthony Coldeway. With Don \"Red\" Barry, Noah Beery, Luana Walters, David Durand, George Douglas, Ethan Laidlaw, Stanley Blystone, John Elliott, Jack Kirk, Fred \"Snowflake\" Toones, Charles Murphy, Art Dillard, Cactus Mack, Jimmy Wakely and His Rough Riders (Johnny Bond, Dick Reinhart). A young rancher almost loses his spread in a dispute with a famous gunfighter, who is really his father. Exciting Don Barry film with Noah Beery stealing the show as a likable gunman. Remake of _**Guns for Hire**_ (q.v.).\n\n**4587** _ **Tumbledown Ranch in Arizona**_ **** Monogram, 1941. 60 min. D: S. Roy Luby. SC: Milton Raison. With Ray Corrigan, John King, Max Terhune, Sheila Darcy, Marian Kerby, James Craven, Quen Ramsey, John Elliott, Jack Holmes, Steve Clark, Carl Mathews, Tex Palmer, Tex Cooper, Frank Ellis, Frank McCarroll, Chick Hannon, Sam Bernard, Rex Felker, Rudy Sooter, Oscar Gahan, Tom Smith, Lew Meehan, Herman Hack, Denver Dixon, Bud McClure, Jack Evans, The University of Arizona Glee Club. A college student suffers a fall and reverts back to a former time when his ancestor was one of the Range Busters opposed to a crooked politician and his saloon keeper henchman. Fast moving entry in \"The Range Busters\" series with a good music.\n\n**4588** _ **Tumbleweed**_ **** Universal-International, 1953. 79 min. Color. D: Nathan Juran. SC: John Meredyth Lucas. With Audie Murphy, Lori Nelson, Chill Wills, K.T. Stevens, Russell Johnson, Madge Meredith, Roy Roberts, I. Stanford Jolley, Lee Van Cleef, Ralph Moody, Ross Elliott, Lyle Talbot, King Donovan, Harry Harvey, Eugene Iglesias, Phil Chambers, Edmund Cobb, Gregg Barton, Ezelle Pule, Roy Butler, Lee Roberts, Clem Fuller, Jennings Miles, Belle Mitchell, Emile Avery, Jack Tornek, Felipe Turich. When Indians attack a wagon train and massacre its passengers, a guard hides two women, goes to the chief to negotiate peace and is later blamed for the killings. Despite its premise, this is a rather tame Audie Murphy feature.\n\n**4589** _ **Tumbleweed Trail**_ **** Producers Releasing Corporation, 1942. 57 min. D: Peter Stewart (Sam Newfield). SC: Fred Myton. With Bill \"Cowboy Rambler\" Boyd, Art Davis, Lee Powell, Marjorie Manners, Jack Rockwell, Charles King, Karl Hackett, George Chesebro, Maxine Leslie, Frank Hagney, Reed Howes, Curley Dresden, George Morrell, Art Dillard, Steve Clark, Dan White, Augie Gomez, Tex Palmer, Bert Dillard, Augie Gomez, Jack Evans, Jack Montgomery. Three Texas lawmen are after the man who murdered their pal and they trace him to a town run by a corrupt peace keeper. Barely passable entry in the \"Frontier Marshals\" series.\n\n**4590** _ **Tumbleweed Trail**_ **** Producers Releasing Corporation, 1946. 59 min. D: Robert Emmett Tansey. SC: Frances Kavanaugh. With Eddie Dean, Roscoe Ates, Shirley Patterson, Bob Duncan, Johnny McGovern, Ted Adams, Jack O'Shea, Kermit Maynard, William Fawcett, Carl Mathews, Lee Roberts, Frank Ellis, The Sunshine Boys (Eddie Wallace, J.D. Sumner, M.H. Richman, Freddie Daniel). A singing cowboy comes to the aid of a pretty girl whose ranch is sought by a gang of cattle rustlers. Fair Eddie Dean film, greatly helped by the title tune and lovely Shirley Patterson.\n\n**4591** _ **Tumbleweeds**_ **** United Artists, 1925. 81 min. D: King Baggott. SC: C. Gardner Sullivan. With William S. Hart, Barbara Bedford, Lucien Littlefield, J. Gordon Russell, Richard R. Neill, Jack Murphy, Lillian Leighton, Gertrude Claire, George F. Marion, Captain T.E. Duncan, James Gordon, Fred Gamble, Turner Savage, Monte Collins. A drover meets and falls in love with a woman and they decide to stake a claim in the Oklahoma Territory when it is opened to settlement. One of the truly great silent Westerns, with its well staged land rush sequence. Reissued in 1939 by Astor with music and sound effects along with an eight minute spoken prolog by William S. Hart.\n\n**4592** _ **Tumbling Tumbleweeds**_ **** Republic, 1935. 54 min. D: Joseph Kane. SC: Ford Beebe. With Gene Autry, Smiley Burnette, Lucille Brown, Norma Taylor, George Hayes, Jack Rockwell, George Chesebro, Frankie Marvin, Charles King, Slim Whitaker, Edward Hearn, Tom London, Cornelius Keefe, Cliff Lyons, Tracy Layne, Bud McClure, George Morrell, Oscar Gahan, Henry Hall, Bart Carre, Iris Meredith, Horace B. Carpenter, Joe Girard, Leonard Slye (Roy Rogers), Tom Smith. A medicine show entertainer returns home to find out who killed his father after his best friend is accused of the crime. Gene Autry's first starring feature film is a pleasant, fast paced affair.\n\n_**Tundra**_ see _**Arctic Fury**_\n\n**4593** _ **20 Mule Team**_ **** Metro-Goldwyn-Mayer, 1940. 84 min. D: Richard Thorpe. SC: Cyril Hume, Edward E. Paramore and Richard Maibaum. With Wallace Beery, Leo Carrillo, Marjorie Rambeau, Anne Baxter, Douglas Fowley, Noah Beery, Jr., Berton Churchill, Arthur Hohl, Clem Bevans, Charles Halton, Minor Watson, Oscar O'Shea, Ivan Miller, Lew Kelly, Lloyd Ingraham, Sam Appel, Eddy Waller, John Beck, Katherine Kentworthy, Mitchell Lewis, Hank Bell, Eddie Borden, Jim Mason, Bob Perry, Larry McGrath, Ed Brady, George Guhl, Joseph E. Bernard. In Death Valley a crook tries blackmail a miner into revealing the location of his valuable borax strike and at the same time romances the innocent daughter of a female tavern keeper. Fans of Wallace Beery and Leo Carrillo will enjoy them as borax digging partners.\n\n**4594** _ **Twenty Paces to Death**_ **** IFISA, 1970. 80 min. Color. D: Ted Mulligan (Manuel Esteba). SC: Ignacio F. Iquino and Giuseppe Rosati. With Dean Reed, Albert Farley (Alberto Farnese), Patty Shepard, Luis Induni, Maria Pia Conte, Mary (Marta) May, Tony Chandler, Cesar Ojinaga, Alejandro Ulloa, Gustavo Re, Marta Flores, Antonio Rojo, Indio Gonzalez, Angel Lombarte, Elena Pironti, Jose Ignacio Abadal, Alberto Severi. Adopted by a rancher, an Indian boy grows up to fall in love with the man's daughter who is wanted by a politician out to ruin her father. Poorly executed interracial love story and double cross plot, this Italian-Spanish co-production was made as _**Veinte Pasos para la Muerte**_ (Twenty Paces to Death).\n\n**4595** _ **Twice a Judas**_ **** Hispamex, 1969. 92 min. Color. D: Nando Cicero. SC: Jaime Jesus Belcazar. With Klaus Kinski, Antonio Sabato, Cristina Galbo, Pepe (Jose) Calvo, Emma Baron, Milo Quesada, Franco Leo, Linda Sini, Narciso Ianez Menta, Franco Beltramme, Damian Rabal, Maite Matalonga, Claudia Rivelli, Carlos Ronda, Gastone Pesucci, Giancarlo Pulone, Gaetano Scala, Jose Palomo, Ettore Bruson, Nino Nini, Antonietta Fiorito, Giuseppe Sciacqua, Sergio De Vcchi, Ettore Broschi, Walter Barnes (voice). Suffering from amnesia, a man opposes a corrupt land baron who forces poor Mexican to work his spread, although the bad guy may be his brother. Hard to follow, slow paced Italian feature filmed as _**Due Volte Giuda**_ (Two Times Judas) and also titled _**They Were Called Graveyard**_.\n\n**4596** _ **The Twilight Avengers**_ **** P.A.C.\/Caravel Film, 1970. 89 min. Color. D-SC: Al Albert (Alberto Albertini). With Tony Kendall (Luciano Stella), Peter Thorris, Alberto Dell'Acqua (Robert Widmark), Ida Medda, Albert Farley (Alberto Farnese), Helen Parker, Spartaco Conversi, Attilio Dottesio. A ruthless man takes over a Mexican village with a traveling circus arriving and its members try to get an old soldier and his men to help them free the locals. Mediocre, violent Italian feature issued there as _**I Vendicatori Dell'Ave Maria**_ (The Avengers of Ave Maria).\n\n**4597** _ **Twilight in the Sierras**_ **** Republic, 1950. 67 min. Color. D: William Witney. SC: Sloan Nibley. With Roy Rogers, Dale Evans, Estelita Rodriguez, Pat Brady, Russ Vincent, George Meeker, Fred Kohler, Jr., Edward Keane, House Peters, Jr., Pierce Lyden, Foy Willing and The Riders of the Purple Sage, Joe Carro, William Lester, Bob Burns, Robert Wilke. Parole officer Roy Rogers, assigned to a ranch employing prisoners, gets mixed up with crooks making counterfeit money and is falsely accused of murdering one of the gang. Fairly well done Roy Rogers entry, but not too interesting.\n\n**4598** _ **Twilight on the Prairie**_ **** Universal, 1944. 62 min. D: Jean Yarbrough. SC: Clyde Bruckman. With Johnny Downs, Vivian Austin, Eddie Quillan, Connie Haines, Leon Errol, Jack Teagarden and His Orchestra, Milburn Stone, Jimmie Dodd, Olin Howland, Perc Launders, Dennis Moore, Ralph Peters, Glenn Strange, Al Sloey, Foy Willing and The Riders of the Purple Sage, The Eight Buckaroos. On their way to Hollywood to break into the movies, members of a cowboy band get stranded on a ranch and agree to work there through harvest time. Plenty of songs fill this typically glossy Universal program feature.\n\n**4599** _ **Twilight on the Rio Grande**_ **** Republic, 1947. 71 min. D: Frank McDonald. SC: Dorrell McGowan and Stuart McGowan. With Gene Autry, Adele Mara, Sterling Holloway, Bob Steele, George J. Lewis, Charles Evans, Martin Garralaga, Howard Negley, Nacho Galindo, Tex Terry, The Cass County Boys (Jerry Scoggins, Bert Dodson, Fred Martin), Frankie Marvin, Bob Burns, George Magrill, Enrique Acosta, Barry Norton, Gil Perkins, Nina Campana, Kenne Duncan, Tom London, Alberto Morin, Keith Richards, Jack O'Shea, Bud Osborne, Frank McCarroll, Robert Wilke, Alex Montoya, Connie Henard. Gene Autry becomes involved with a gang of jewel thieves and a beautiful knife thrower. Not even a fine cast can save this slow moving Gene Autry vehicle which plays better in its 54 minute truncated TV version.\n\n**4600** _ **Twilight on the Trail**_ **** Paramount, 1941. 58 min. D: Howard Bretherton. SC: J. Benton Cheney, Ellen Corby and Cecile Kramer. With William Boyd, Andy Clyde, Brad King, Wanda McKay, Jack Rockwell, Norman Willis, Robert Kent, Tom London, Robert Kortman, Frank Austin, Clem Fuller, Frank Ellis, Bud Osborne, John Powers, The Jimmy Wakely Trio (Jimmy Wakely, Johnny Bond, Dick Reinhart), Hal Taliaferro, Kermit Maynard, Jim Corey. In order to round up a gang of rustlers, Hopalong Cassidy masquerades as a dandified Englishman. Just an average outing in the long running series, co-written by actress Ellen Corby.\n\n**4601** _ **Twinkle in God's Eye**_ **** Republic, 1955. 75 min. D: George Blair. SC: P.J. Wolfson. With Mickey Rooney, Coleen Gray, Hugh O'Brian, Don Barry, Touch (Michael) Connors, Joey Forman, Jil Jarmyn, Kem Dibbs, Tony Garcen, Raymond Hatton, Ruta Lee, Clem Bevans, Tony Garcen, Stanley Andrews, Joseph Crehan, Emmett Lynn, Dick Elliott, Frank Lackteen, Paul McGuire, Harry Harvey, Peter Mamakos, Larry J. Blake, Frank Kreig, Vicki Raaf, Renate Hoy, Shirley Whitney, Dan White, Dick Winslow, Rodd Redwing, Charles Williams, Harry Tyler, James Rawley, Milton Newberger, Sig Frohlich. In a tough Western town, the new parson tries to spread the world of the Lord via humor. A pleasant enough little feature but a bit sad to contemplate considering the cinematic stature of Mickey Rooney, who produced it, just a decade before.\n\n**4602** _ **Twisted Rails**_ **** Imperial, 1934. 51 min. D: Albert (Al) Herman. SC: L.V. Jefferson. With Jack Donovan, Alice Dahl, Philo McCullough, Donald Keith, Victor Potel, Buddy Shaw, Donald Mack, Henry Roquemore, Pat Harmon, Tom London, Bob McKenzie, Lawrence Underwood, Ada Belle Driver, Bill Patton, Gene \"Fatty\" Laymon, Elyn Glyn. Following the shooting of an informant, a railroad passenger offers to help a inspector find the killer and his gang who are after a gold shipment. Sparse action adventure made on poverty row.\n\n**4603** _ **Twisted Trails**_ **** Aywon, 1924. 45 min. D: Tom Mix. SC: Edwin Ray Coffin. With Tom Mix, Bessie Eyton, Eugenie Besserer, Al W. Wilson, Will Machin, Pat Chrisman, Sid Jordan, George Clark, Frank LeRoy, Olcott Byrnes. Two corrupt lawmen try to blame their cattle rustling operation on a ranch foreman in love with a girl wanted by a gambler. There is lots of action but a hard-to-follow plot in this Tom Mix feature made up of his 1916 Selig three reeler of the same title and padded with footage from some of his other short films.\n\n_**Two Against All**_ see _**Terrible Sheriff**_\n\n_**Two Brothers in Trinity**_ see _**Jesse and Lester**_\n\n_**Two Fisted Agent**_ see _**Bonanza Town**_\n\n**4604** _ **Two Fisted Justice**_ **** Arrow, 1924. 50 min. D: Dick Hatton. SC: Bennett Cohen. With Dick Hatton, Marilyn Mills, Morris Foster, Arthur Morrison, Star (horse). A cowpoke seeking revenge for the murder of his doctor brother ends up falling in love with the killer's wife. Moribund poverty row silent effort from producer Ben Wilson, directed by star Dick Hatton.\n\n**4605** _ **Two-Fisted Justice**_ **** Monogram, 1931. D-SC: G.A. Durlam. With Tom Tyler, Barbara Weeks, Bobby Nelson, Yakima Canutt, John Elliott, G.D. Wood (Gordon DeMain), Kit Guard, William Walling, Si Jenks, Pedro Regas, Carl Deloro, Joe Mills, Bob Fleming, Al Haskell, Tom Smith, Jack Low, F.R. Smith. President Lincoln sends a scout to the frontier to protect settlers and he uncovers an outlaw gang. Average Tom Tyler outing.\n\n**4606** _ **Two Fisted Justice**_ **** Monogram, 1943. 61 min. D: Robert Tansey. SC: William L. Nolte. With John King, David Sharpe, Max Terhune, Gwen Gaze, Joel Davis, John Elliott, Charles King, George Chesebro, Frank Ellis, Cecil Weston, Hal Price, Carl Mathews, Lynton Brent, Kermit Maynard, Richard Cramer, Tex Palmer, John Curtis, Augie Gomez, Denver Dixon, Milburn Morante, Jack Evans, Rose Plummer. The Range Busters arrive in a town to bring an outlaw gang to justice and after a run-in with the leader of the band they are made the community's lawmen. Only a fair entry in the popular series.\n\n**4607** _ **Two Fisted Law**_ **** Columbia, 1932. 64 min. D: D. Ross Lederman. SC: Kurt Kempler. With Tim McCoy, Alice Day, Wheeler Oakman, John Wayne, Wallace MacDonald, Tully Marshall, Richard Alexander, Walter Brennan, Merrill McCormick, Bud Osborne, Arthur Thalasso, Jack Hendricks, Hank Bell, Jack Evans, Rube Dalroy. Cheated out of his ranch a man makes a gold strike and returns home to try and save the property of a young woman being threatened by the same crook who stole his spread. Generally good Tim McCoy vehicle, based on a William Colt MacDonald story, with fine photography by Benjamin Kline. John Wayne has a small role as one of McCoy's loyal ranch hands.\n\n**4608** _ **Two-Fisted Rangers**_ **** Columbia, 1940. 62 min. D: Joseph H. Lewis. SC: Fred Myton. With Charles Starrett, Iris Meredith, The Sons of the Pioneers (Bob Nolan, Tim Spencer, Lloyd Perryman, Pat Brady, Hugh Farr, Karl Farr), Bill Cody, Jr., Hal Taliaferro, Kenneth MacDonald, Dick Curtis, Ethan Laidlaw, Bob Woodward, James Craig, Francis Walker. A cowboy plans to bring in the land baron responsible for the murder of his sheriff brother. Pretty good Charles Starrett feature.\n\n**4609** _ **Two-Fisted Sheriff**_ **** Columbia, 1937. 60 min. D: Leon Barsha. SC: Paul Perez. With Charles Starrett, Barbara Weeks, Bruce Lane, Ed Peil, Sr., Alan Sears, Walter Downing, Ernie Adams, Claire McDowell, Frank Ellis, Robert Walker, George Chesebro, Art Mix, Alan Bridge, Richard Botiller, George Morrell, Merrill McCormick, Edmund Cobb, Tex Cooper, Richard Cramer, Richard Alexander, Maston Williams, Ethan Laidlaw, Steve Clark, Wally West, Fred Burns, Blackie Whiteford, Charles Brinley, Art Dillard, Al Haskell, Ray Jones, Jack Evans, Blackjack Ward, Fred Parker, Hank Bell. A lawman loses his job when his pal is accused of killing his girl's father and is allowed to escape and he tries to find the real killer and clear his friend. Okay Charles Starrett vehicle enhanced by a superb supporting cast.\n\n**4610** _ **Two-Fisted Stranger**_ **** Columbia, 1945. 51 min. D: Ray Nazarro. SC: Robert Lee Johnson. With Charles Starrett, Smiley Burnette, Doris Houck, Zeke Clements, Lane Chandler, Ted Mapes, George Chesebro, Jack Rockwell, Herman Hack, I. Stanford Jolley, Edmund Cobb, Davison Clark, Maudie Prickett, Nolan Leary, Frank Ellis, Frank O'Connor, Charles Murray, Jr., Matty Roubert, Herman Hack, Tommy Coats. Outlaws try to force miners off their claims but find opposition from the Durango Kid. Short, but not much of an effort in the popular series. British title: _**High Stakes**_.\n\n**4611** _ **Two Flags West**_ **** 20th Century\u2013Fox, 1950. 92 min. D: Robert Wise. SC: Frank S. Nugent and Casey Robinson. With Joseph Cotten, Linda Darnell, Jeff Chandler, Cornel Wilde, Dale Robertson, Jay C. Flippen, Noah Beery, Jr., Harry Von Zell, John Sands, Arthur Hunnictt, Jack Lee, Robert Adler, Harry Carter, Ferris Taylor, Sally Corner, Everett Glass, Marjorie Bennett, Lee MacGregor, Roy Gordon, Aurora Castillo, Stanley Andrews, Don Garner. A group of Confederate prisoners agree to fight Indians in the West in order to get out of jail but the commander of the fort they are assigned hates all rebels. Fair drama buoyed by a fine cast.\n\n_**The Two from Rio Bravo**_ see _**Bullets Don't Argue**_\n\n_**Two Gangsters in the Wild West**_ see _**Two Mafiamen in the Far West**_\n\n**4612** _ **Two-Gun Justice**_ **** Monogram, 1938. 58 min. D: Alan James. SC: Fred Myton. With Tim McCoy, Betty Compson, Joan Barclay, John Merton, Lane Chandler, Alan Bridge, Tony Paton, Allan Cavan, Harry Strang, Earl Dwire, Enid Parrish, Olin Francis, Curley Dresden, Jack Ingram. A lawman pretends to be a Mexican bandit, \"The Vulture,\" so he can round up the notorious Kane gang. Cheap, quick on action and shouldered with an obtrusive canned music score, this Tim McCoy series film boasts two leading ladies, silent star Betty Compson and Joan Barclay.\n\n**4613** _ **Two Gun Lady**_ **** Associated Film Releasing, 1956. 75 min. D: Richard Bartlett. SC: Norman Jolley. With Peggie Castle, William Talman, Marie Windsor, Earle Lyon, Robert Lowery, Joe Besser, Ian MacDonald, Barbara Turner, Norman Jolley, Susan Long, Kit Carson, Arvo Jjala, Karl Hansen, Dave Tomack, Sid Lopez, Gregory Moffet, Ben Cameron, Kermit Maynard, Jack Ingram. A gun toting young woman teams with a lawman to hunt the men who murdered her father. Low budget action feature for fans of Peggie Castle and Marie Windsor.\n\n**4614** _ **The Two-Gun Man**_ **** Tiffany, 1931. 60 min. D: Phil Rosen. SC: John (Jack) Natteford. With Ken Maynard, Lucille Powers, Nita Martin, Charles King, Lafe McKee, Tom London, Murdock MacQuarrie, Walter Perry, Will Stanton, William Jackie, Ethan Allen, Jim Corey, Blackjack Ward, Buck Bucko, Roy Bucko. A cowboy opposes crooked cattlemen who are behind the rustling of area herds. Fast moving Ken Maynard oater.\n\n**4615** _ **Two-Gun Man from Harlem**_ **** Sack Amusement Enterprises, 1938. 65 min. D: Richard C. Kahn. SC: Fred Myton. With Herbert Jeffrey (Herb Jeffries), Margaret Whitten, Clarence Brooks, Mantan Moreland, Tom Southern, Mae Turner, Spencer Williams, Jr., Jesse Lee Brooks, Stymie Beard, Rose Lee Lincoln, Paul Blackman, Faithful Mary, The Four Tones (Lucius Brooks, Leon Buck, Ira Hardin, Rudolph Hunter), The Four Cats and a Fiddle, John Thomas. After being falsely accused of murder, a cowboy goes to Harlem where he pretends to be a minister turned gangster to expose the real killer. More gangster plotted than Western, this very low budget black cast musical feature was scripted by \"B\" film veteran Fred Myton; the first of a quartet of features headlining crooner Herb Jeffries, although he was not billed under that name in any of them.\n\n**4616** _ **Two Gun Marshal**_ **** Allied Artists, 1953. 54 min. D: Frank McDonald. SC: William Raynor and Maurice Tombragel. With Guy Madison, Andy Devine, Raymond Hatton, Carole Mathews, Richard Tyler, Frankie Darro, Danny Mummert, Gregory Marshall, Noralee Norman, Minerva Urecal, Sara Haden, Michael Vallon, Pamela Duncan, Francis McDonald, Ray Hyke, Wes Hudman. Wild Bill Hickok persuades his partner Jingles to masquerade as the wife of a peddler so they can find out who robbed a Wells Fargo stage and the two then go up against a family bent on not letting a son get an education. Acceptable patchwork theatrical feature made up of the \"Papa Antonelli\" and \"The Slocum Family,\" two 1951 episodes of \"The Adventures of Wild Bill Hickok\" (1951\u201358).\n\n**4617** _ **Two-Gun Sheriff**_ **** Republic, 1941. 54 min. D: George Sherman. SC: Doris Schroeder. With Don \"Red\" Barry, Lynn Merrick, Lupita Tovar, Fred Kohler, Jr., Jay Novello, Marin Sais, Fred \"Snowflake\" Toones, Milton Kibbee, Dirk Thane, Archie Hall, Charles Thomas, Lee Shumway, John Merton, Carleton Young, Slim Whitaker, John James, Stanley Price, Jack O'Shea, Forrest Taylor, Herman Hack, Frank Ellis, Curley Dresden, Buck Moulton, Bud McClure, Tex Parker, Herman Nolan, George Plues, Al Taylor, Pascale Perry, Rose Plummer. An outlaw is drafted by a gang leader to take the place of a sheriff but the two men turn out to be brothers and the fugitive double crosses the bandits. A complicated plot and plenty of action highlight this Don Barry film, with a fine music score by Cy Feuer.\n\n**4618** _ **The Two Gun Teacher**_ **** Allied Artists, 1954. 54 min. D: Frank McDonald. SC: William Raynor. With Guy Madison, Andy Devine, Rand Brooks, Tom Tyler, Murray Alper, Emory Parnell, Anne Carroll, Don C. Harvey, Steve Pendleton, Charles Stevens, Monte Montague, Neyle Morrow, Bob Woodward, Theodora Lynch, Sujata, Rory Mallinson, Peter Votrian, Isa Ashdown, Jim Flowers. A young woman dressed as a man helps save Wild Bill Hickok when he is ambushed by gun runners and Bill and Jingles are assisted by a student as they investigate crooks after an underground spring. Still another okay big screen feature made of two 1952 segments of \"The Adventures of Wild Bill Hickok\" (1951\u201358), \"Mexican Gun Running Story\" and \"School Teacher Story.\"\n\n**4619** _ **Two-Gun Troubador**_ **** Spectrum, 1939. 58 min. D: Raymond K. Johnson. SC: Richard L. Bare and Phil Dunham. With Fred Scott, Claire Rochelle, Harry Harvey, John Merton, Buddy Lenhart, Carl Mathews, Buddy Kelly, Harry Harvey, Jr., Gene Howard, Frank Ellis, William Woods, Jack Ingram, Bud Osborne, John Ward, Cactus Mack. Years after his uncle killed his father over property, a man returns to claim his inheritance and in doing so takes on the guise of a masked avenger. Pleasant Fred Scott vehicle with some good tunes.\n\n_**Two Gunmen**_ see _**Two Violent Men**_\n\n**4620** _ **Two Guns and a Badge**_ **** Allied Artists, 1954. 69 min. D: Lewis D. Collins. SC: Dan Ullman. With Wayne Morris, Beverly Garland, Morris Anrkum, Roy Barcroft, William Phipps, Damian O'Flynn, I. Stanford Jolley, Robert Wilke, Chuck Courtney, Henry Rowland, Lyle Talbot, William Fawcett, Mike Ragan (Holly Bane), Stanley Price, Ted Mapes. An ex-convict is mistaken for a deputy sheriff when he arrives in town and soon finds himself up against a corrupt rancher while falling in love with the man's daughter. Considered the final \"B\" series Western, this little film is a fit finale to a grand genre.\n\n**4621** _ **Two Guys from Texas**_ **** Warner Bros., 1948. 86 min. Color. D: David Butler. SC: I.A.L. Diamond and Allen Boretz. With Dennis Morgan, Jack Carson, Dorothy Malone, Penny Edwards, Forrest Tucker, Fred Clark, Gerald Mohr, John Alvin, Andrew Tombes, Monte Blue, The Philharmonic Trio, Richard Alexander, Fred Kelsey, Lane Chandler, Brandon Hurst, Jack Mower, Clifton Young, Lily Christine, Philo McCullough, Louis Mason, Fred Santley, Jack Baxley, Peggy Gordon, Charles Marsh, Joy Barlow, Cleatus Caldwell, Eileen Howe, William Steele, Tom Wells, Petra Silva, Mel Blanc (voice). Two vaudeville entertainers find themselves stranded on a Texas ranch where they fight crooks and meet two pretty girls. Okay reworking of _**The Cowboy from Brooklyn**_ (q.v.).\n\n_**Two Idiots at Fort Alamo**_ see _**The Two Sergeants of General Custer**_\n\n**4622** _ **Two in Revolt**_ **** RKO Radio, 1936. 65 min. D: Glenn Tryon. SC: Frank Howard Clark, Ferdinand Reyher and Jerry Hutchinson. With John Arledge, Louise Latimer, Moroni Olsen, Emmett Vogan, Harry Jans, Murray Alper, Willie Best, Max Wagner, Ethan Laidlaw, Clem Bevans, Erville Alderson, Billy Bletcher, Horace Murphy, Lightning (dog), Warrior (horse). A horse and a dog team to help a man in his battle with outlaws. Fair juvenile matinee fare centered around horse racing.\n\n**4623** _ **Two Mafiamen in the Far West**_ **** FIDA\/Epoca Film, 1964. Color. D: Giorgio Simonelli. SC: Marcello Ciorciolini and Giorgio Simonelli. With Francho Franchi, Ciccio Ingrassia, Helene Chanel, Fernando Sancho. Aroldo Tieri, Anna Casares, Aldo Giuffre, Adriano Micantoni, Luis Pena, Felix De Fauce, Alfredo Rizzo, Giovanni Vari. Two prisoners escape and travel to a Texas town where an outlaw has murdered his cousins for their ranch with the two convicts dead ringers for the dead men. Not much here for genre fans in this goofy Franco and Ciccio feature made as _**Due Mafiosi nel Far West**_ (Two Mafiosi in the Far West) and also called _**Two Gangsters in the Wild West**_.\n\n**4624** _ **Two Mules for Sister Sara**_ **** Universal, 1970. 105 min. Color. D: Don Siegel. SC: Albert Maltz. With Shirley MacLaine, Clint Eastwood, Armando Silvestre, Manolo Fabregas, John Kelly, Enrique Lucero, Jose Chavez, Alberto Morin, David Estuardo, Ada Carrasco, Poncho Cordoba, Pedro Galvan, Jose Angel Espinosa, Aurora Munoz, Xavier Marc, Hortensia Santovena, Rosa Furman, Jose Torvay, Mararito Luna, Javier Masse. A mercenary fighting for the cause of Juarez leads a free-wheeling, anti-rebel nun across the Mexican desert. Uneven but rather fun teaming of Clint Eastwood and Shirley MacLaine, based on a story by Budd Boetticher.\n\n**4625** _ **Two R-R-Ringos from Texas**_ **** Circus Film\/Tono Film, 1967. 94 min. Color. D: Frank Martin (Marino Girolami). SC: Amedeo Sollazzo and Roberto Gianviti. With Franco Franchi, Cicio Ingrassia, Ennio Girolami, Silvio Bagolini, Gloria Paul, Livio Lorenzon, Ignazio Balsamo, Helene Chanel, Gina Mascetti, Enzo Andronico, Rossell Bergamonti, Walter Marchetti, Mirella Pamphili, Lucio Fulci, Osiride Pevarello, Adriano Uriani, Guilielmo Bogliani, Maurizio Merli, Stefano Sibaldi. Led by a talking horse, two lamebrain Civil War soldiers try to get behind enemy lines to find a treasure. Silly Franchi and Ciccio Italian Western \"comedy\" made as _**Due RRRingos nel Texas**_ (Two Ringos from Texas).\n\n**4626** _ **Two Rode Together**_ **** Columbia, 1961. 109 min. Color. D: John Ford. SC: Frank S. Nugent. With James Stewart, Richard Widmark, Shirley Jones, Linda Cristal, Andy Devine, John McIntire, Paul Birch, Willis Bouchey, Henry Brandon, Harry Carey, Jr., Ken Curtis, Olive Carey, Chet Douglas, Annelle Hayes, David Kent, Anna Lee, Jeanette Nolan, Edward Brophy, John Qualen, Ford Rainey, Woody Strode, O.Z. Whitehead, Cliff Lyons, Mae Marsh, Frank Baker, Ruth Clifford, Ted Knight, Major Sam Harris, Jack Pennick, Chuck Roberson, Dan Borgaze, Bill Henry, Chuck Hayward, Big John Hamilton, Ted Mapes, Regina Carrol, Ed Sweeney, Robert Kenneally, Eunice Grey. A lawman and a cavalry lieutenant form an uneasy alliance in a mission to negotiate the return of settlers kidnapped by Comanches. Generally good John Ford film highlighted by a number of fine performances, especially Mae Marsh as the captive who explains whey she does not want to go back to her people.\n\n**4627** _ **The Two Sergeants of General Custer**_ **** FIDA\/Balcazar, 1965. 97 min. Color. D-SC: Giorgio Simonelli. With Franco Franchi, Ciccio Ingrassia, Fernando Sancho, Margaret Lee, Arolo Tieri, Moira Orfei, Riccardo Garone, Ernesto Calindri. During the Civil War two lunkheaded Union spies attempt to make it behind enemy lines but get involved with a female Confederate agent. Typically inane Italian \"comedy\" from the team of Franco and Ciccio made as _**Due Sergenti del General Custer**_ (Two Sergeants of General Custer) and also called _**Two Idiots of Fort Alamo**_.\n\n**4628** _ **Two Sons of Ringo**_ **** Flora Film\/Variety Film, 1966. 105 min. Color. D: Giorgio Simonelli. SC: Amedeo Sollazo, Roberto Gianviti, Marcello Ciorciolini and Dino Verde. With Franco Franchi, Ciccio Ingrassia, Gloria Paul, George Hilton, Pedro Sanchez, Mimmo Palmara, Umberto D'Orsi, Orchidea de Santis, Ivano Staccioli Fulvia Franco, Enzo Andronico, Armando Carini, Fortunato Arena, Giovanni Ivan Scratuglia, Guido Lollobrigida (Lee Burton), Nino Terzo, Fulvio Mingnazzi, Calogero Azzaretto, Gilliano Sbarra. Two fake sharpshooters enlist a bounty hunter to help them as they pretend to be the heirs of the late Ringo to inherit a fortune. The laughs are sparse in this Italian spoof headlining Franco and Ciccio; made as _**I Due Figli di Ringo**_ (The Two Sons of Ringo).\n\n**4629** _ **Two Violent Men**_ **** P.E.A.\/Arturo Gonzales, 1965. 94 min. Color. D: Anthony Greepy (Primo Zeglio). SC: Jesus Navarro and Primo Zeglio. With Alan Scott, George (Jorge) Martin, Susy Andersen, Mary Badmayer, Andrew Scott (Andrea Scotti), Pauline Baards, Sylvia Solar, Mike Brendell, Frank Brana, Jose Jaspe, Luis Induni, Aldo Sambrell. A lawman and the friend he is assigned to arrest on a murder charge team to stop the siege of a ranch by outlaws. Better than average Italian oater with a fairly literate script, released there as _**I Due Violent**_ (Two Violent Men), in Spain as _**Los Rurales de Texas**_ and in England as _**Two Gunmen**_.\n\n**4630** _ **Uccisore Nero**_ (Black Killer) **** Florida Cinemtaograpfica, 1971. 85 min. Color. D: Lucky Moore (Carlo Croccolo). SC: Charlie Foster (Carlo Croccolo) and Luigi Angelo. With Klaus Kinski, Fred Robsahm, Antonio Cantafora, Marina Mulligan, Paul Craine, Tiziana Dini, Ted Jones, Jerry Ross, Dan May (Dante Maggio), Claudio Trionfi, Robert Danish, Dick Foster (Mimmo Maggio), Carlo Croccolo. A vicious gang of Mexican brothers and a crooked judge rule the town of Tombstone but when a beautiful widow is physically abused by them she teams with a bounty hunter and a lawyer to destroy the bad men. Effective but very violent Spaghetti Western made in Italy.\n\n**4631** _ **The Ugly Ones**_ **** United Artists, 1968. 96 min. Color. D: Eugenio Martin. SC: Jose G. Maesso and Eugenio Martin. With Richard Wyler, Tomas Milian, Mario Brega, Hugo Blanco, Glenn Foster, Ella Karin, Manolo Zarzo, Lola Gaos, Ricardo Canales, Frank Brana, Luis Barboo. A young woman helps an outlaw escape from a bounty hunter only to learn he has become a killer due to his hard and roving life as a bandit. Very violent Spanish Western, based on the novel _Bounty Killer_ by Marvin H. Albert, issued there in 1966 as _**El Precio de un Hombre**_ (The Price of a Man); also called _**Bounty Killer**_.\n\n**4632** _ **El Ultimo Chinaco**_ (The Ultimate Trick) **** Clasa-Mohme, 1948. 91 min. D: Raul de Anda. SC: Raul de Anda and Carlos Gaytan. With Luis Aguilar, Katy Jurado, Irma Torres, Carlos Lopez Moctezuma, Miguel Arenas, Marga Lopez, Arturo Soto Rangel, Victor Parra, Jose Pardave, Lupe Inclan, Luis G. Barrierro. A mysterious figure, who loves a mine owner's daughter, takes from the rich and gives to the poor but is framed on a murder charge when someone commits a crime dressed like him. Another Mexican masked hero is played by Luis Aguilar in this okay feature from producer-director-writer Raul de Anda.\n\n**4633** _ **Ulzana's Raid**_ **** Universal, 1972. 103 min. Color. D: Robert Aldrich. SC: Alan Sharp. With Burt Lancaster, Bruce Davison, Jorge Luke, Richard Jaeckel, Joaquin Martinez, Lloyd Bochner, Karl Swenson, Douglas Walton, Dran Hamilton, John Pearce, Gladys Holland, Margaret Fairchild, Aimee Eccles, Richard Bull, Otto Reichow, Dean Smith, Larry Randles. Three men, an aging Indian fighter, a young lieutenant and a Native American scout hunt a raiding party that has been terrorizing area citizens. Fair melodrama, well acted but too violent.\n\n**4634** _ **Uncle Sam Magoo**_ **** U.P.A., 1969. 55 min. Color. D: Abe Levitow. SC: Larry Markes and Sam Rosen. With Jim Backus, Lennie Weinrib, Pattie Gilbert, Bob Holt, Barney Phillips, Dave Shelley, Sid Grossfeld, John Himes, Bill Clayton (voices). Mr. Magoo takes on the guise of Uncle Sam and looks back at the nation's history, including his being national heroes like Paul Revere and Davy Crockett. Fans of Mr. Magoo will get a kick out of this animated feature.\n\n**4635** _ **Unconquered**_ **** Paramount, 1947. 146 min. Color. D: Cecil B. DeMille. SC: Charles Bennett, Frederick M. Frank and Jesse Lasky, Jr. With Gary Cooper, Paulette Goddard, Howard Da Silva, Boris Karloff, Cecil Kellaway, Ward Bond, Katherine DeMille, Henry Wilcoxon, C. Aubrey Smith, Victor Varconi, Virginia Grey, Porter Hall, Mike Mazurki, Richard Gaines, Virginia Campbell, Gavin Muir, Alan Napier, Nan Sutherland, Marc Lawrence, Jane Nigh, Robert Warwick, Lloyd Bridges, Oliver Thorndike, Russ Conklin, John Mylong, George Kirby, Leonard Carey, Frank R. Wilcox, Davison Clark, Griff Barnett, Raymond Hatton, Julia Faye, Paul E. Burns, Mary Field, Clarence Muse, Matthew Boulton, Chief Thundercloud, Jack Pennick, Lex Barker, Charles Middleton, Dorothy Adams, Al Ferguson, Ethel Wales, Robert Kortman, Francis McDonald, Claire DuBrey, Christopher Clark, Iron Eyes Cody, Edgar Dearing, Earle Hodgins, Ray Teal, Frank Hagney, Chuck Hamilton, Erville Alderson, Belle Mitchell, Ottola Nesmith, Inez Palange, Bill Murray, Tiny Jones, Noble Johnson, Anna Lehr, Rose Higgins, Gertrude Valerie. In the 1760s a frontier Virginia militiaman opposes a sleazy trader who is selling guns to Chief Pontiac's followers while they both vie for the same beautiful woman, a bond servant. There is not much in the way of real history here but this Cecil B. DeMille production is colorful and exciting.\n\n**4636** _ **Unconquered Bandit**_ **** Reliable, 1935. 57 min. D: Harry S. Webb. SC: Rose Gordon and Lou C. Borden. With Tom Tyler, Lillian Gilmore, Charles \"Slim\" Whitaker, William Gould, John Elliott, Earl Dwire, Joe De La Cruz, George Chesebro, Richard Alexander, Lew Meehan, George Hazel, Wally Wales, Ben Corbett, Colin Chase. When his dad is murdered by a gang secretly led by a policeman, a cowboy plans to use the cop's pretty niece to get revenge for the crime. Better than average Tom Tyler vehicle for Reliable.\n\n**4637** _ **Undead or Alive:**_ _**A Zombedy**_ **** Image Entertainment\/Lionsgate, 2007. 92 min. Color. D-SC: Glasgow Phillips. With Chris Kattan, James Denton, Cristin Michele, Navi Rawatt, Chloe Russell, Mia Stallard, Brian Posehn, T. Jay O'Brien, Chris Coppola, Matt Besser, Todd Anderson, Lew Alexander, Christopher Allen Nelson, Ben Zeller, Michelle Greathouse, Gino Corgnale, Patricia Greer, Michael Patrick Metzdorff, Jeffrey Dashnaw, Elizabeth Slagsvol. A soldier fugitive and a jilted cowboy rob a lawman not knowing the area is filled with zombies thanks to a curse put on it by Geronimo. Okay horror Western comedy done direct to video.\n\n**4638** _ **The Undefeated**_ **** 20th Century\u2013Fox, 1969. 118 min. Color. D: Andrew V. McLaglen. SC: James Lee Barrett. With John Wayne, Rock Hudson, Tony Aguilar, Roman Gabriel, Marian McCargo, Lee Meriwether, Merlin Olsen, Melissa Newman, Bruce Cabot, Michael Vincent, Ben Johnson, Edward Faulkner, Harry Carey, Jr., Paul Fix, Royal Dano, Richard Mulligan, Carlos Rivas, John Agar, Guy Raymond, Don Collier, Big John Hamilton, Dub Taylor, Henry Beckman, Victor Junco, Robert Donner, Pedro Armendariz, Jr., James Dobson, Rudy Diaz, Richard Angarola, James McEachin, Gregg Palmer, Juan Garcia, Kiel Martin, Bob Gravage, Chuck Roberson. Veterans of the Civil War, both Union and Confederate, form an uneasy alliance as they head for Mexico to start new lives. Not one of the Duke's better features but still entertaining with a fine supporting cast, especially Dub Taylor as the cantankerous chuck wagon cook.\n\n**4639** _ **Under a Texas Moon**_ **** Warner Bros., 1930. 82 min. Color. D: Michael Curtiz. SC: Gordon Rigby. With Frank Fay, Raquel Torres, Myrna Loy, Armida, Noah Beery, George E. Stone, George Cooper, Fred Kohler, Betty Boyd, Charles Sellon, Jack Curtis, Sam Appel, Tully Marshall, Mona Maris, Francisco Maran, Tom Dix, Jerry Barrett, Inez Gomez, Edythe Kramera, Bruce Covington. A dashing Mexican adventurer and his pal romance two pretty women at a ranch where they plan to get the reward for capturing outlaws rustling the owner's cattle. Creaky and badly dated musical Western, but the title song endures.\n\n**4640** _ **Under Arizona Skies**_ **** Monogram, 1946. 60 min. D: Lambert Hillyer. SC: J. Benton Cheney. With Johnny Mack Brown, Raymond Hatton, Reno Blair, Riley Hill, Tristram Coffin, Reed Howes, Ted Adams, Ray Bennett, Frank LaRue, Steve Clark, Jack Rockwell, Bud Geary, Ted Mapes, Dusty Rhodes, Kermit Maynard, Smith Ballew, Leonard St. Leo, Lynton Brent, Ray Jones, The Sons of the Sage. A rancher leading an outlaw gang is out to get a rival's spread but two cowboys come to the rescue. Pretty fair Johnny Mack Brown\u2013Raymond Hatton vehicle with Smith Ballew along for a couple of tunes.\n\n_**Under Arrest**_ see _**Blazing Across the Pecos**_\n\n**4641** _ **Under California Stars**_ **** Republic, 1948. 70 min. Color. D: William Witney. SC: Sloan Sibley and Paul Gagelin. With Roy Rogers, Jane Frazee, Andy Devine, Bob Nolan and The Sons of the Pioneers (Doye O'Dell, Lloyd Perryman, Pat Brady, Hugh Farr, Karl Farr), Michael Chapin, Wade Crosby, George Lloyd, House Peters, Jr., Steve Clark, Joseph Carro, Paul Powers, John Wald. Bad guys steal Roy Rogers' horse Trigger and demand a $100,000 ransom. Colorful Roy Rogers action feature badly cut for TV showings at 54 minutes; the star reprises the Johnny Marvin song \"Dust,\" which he sang in his first starring film, _**Under Western Stars**_ (q.v.).\n\n**4642** _ **Under Colorado Skies**_ **** Republic, 1947. 65 min. Color. D: R.G. Springsteen. SC: Louise Rousseau. With Monte Hale, Adrian Booth, Foy Willing and The Riders of the Purple Sage, Paul Hurst, William Haade, John Alvin, LeRoy Mason, Edmund Cobb, Tom London, Steve Darrell, Gene Evans, Ted Adams, Steve Raines, Hank Patterson. From the knowledge he received at medical school, a cowboy is able to track down a gang of outlaws. Okay Monte Hale effort.\n\n**4643** _ **Under Fiesta Stars**_ **** Republic, 1941. 64 min. D: Frank McDonald. SC: Karl Brown and Eliot Gibbons. With Gene Autry, Smiley Burnette, Carol Hughes, Frank Darien, Joe Straugh, Jr., Pauline Drake, Ivan Miller, Sam Flint, John Merton, Jack Kirk, Curley Dresden, Hal Taliaferro, Frankie Marvin, Pascale Perry, Elias Gamboa, Inez Palange. Gene Autry manages a mine but half-interest is controlled by a woman who wants to sell the property, not knowing she is being bilked by two corrupt lawyers. There is not much of a fiesta in this slow moving Gene Autry vehicle.\n\n**4644** _ **Under Mexicali Stars**_ **** Republic, 1950. 67 min. D: George Blair. SC: Bob Williams. With Rex Allen, Buddy Ebsen, Dorothy Patrick, Roy Barcroft, Percy Helton, Walter Coy, Steve Darrell, Alberto Morin, Ray Walker, Frank Ferguson, Stanley Andrews, Robert Bice. A Treasury agent, who is also a cowboy, searches for a counterfeit operation and discovers it is using helicopters to smuggle gold. Exciting Rex Allen feature.\n\n**4645** _ **Under Montana Skies**_ **** Tiffany, 1930. 60 min. D: Richard Thorpe. SC: Bennett Cohen and James A. Aubrey. With Kenneth Harlan, Dorothy Gulliver, Slim Summerville, Nita Martan, Christian J. Frank, Harry Todd, Ethel Wales, Lafe McKee, Charles King, Slim Whitaker, Frank Ellis, Si Jenks, Bob Reeves, Barney Beasley, Tom Bay, George Blues, Hank Bell. A cowboy falls in love with a theatrical troupe actress and helps save her show but a cattle rustler he sent to jail is released and robs the box office. Antiquated early sound Western musical comedy.\n\n**4646** _ **Under Nevada Skies**_ **** Republic, 1946. 69 min. D: Frank McDonald. SC: PaulGangelin and J. Benton Cheney. With Roy Rogers, George \"Gabby\" Hayes, Dale Evans, Bob Nolan and The Sons of the Pioneers (Lloyd Perryman, Pat Brady, Shug Fisher, Hugh Farr, Karl Farr), Douglass Dumbrille, Tristram Coffin, Leyland Hodgson, Rudolph Anders, LeRoy Mason, George J. Lewis, Steve Darrell, Frank Marlowe, Broderick O'Farrell, George Magrill, Iron Eyes Cody, Eddie Parker. Radio singer Roy Rogers tries to find out who killed his friend, the manager of a nightclub in a Western town. A mystery element and the use of an A-bomb component in the plot greatly helps this fast moving Roy Rogers vehicle, which is somewhat hurt by losing 15 minutes for TV.\n\n**4647** _ **Under Strange Flags**_ **** Crescent, 1937. 61 min. D: I.V. Willat. SC: Mary Ireland. With Tom Keene, Lana (Luana) Walters, Budd Buster, Maurice Black, Roy D'Arcy, Paul Sutton, Paul Barrett, Donald Reed, Jane Wolfe. Americans mining silver in Mexico find their shipments being hijacked by Pancho Villa and his followers. Fair entry in the Crescent historical series starring Tom Keene.\n\n**4648** _ **Under Texas Skies**_ **** Syndicate, 1930. 60 min. D: J.P. McGowan. SC: G.A. Durlam. With Bob Custer, Natalie Kingston, Bill Cody, Tom London, Lane Chandler, Bob Roper, William McCall, Ted Adams, Joseph Marba. When a female rancher tries to sell her horses to the government, an agent claims one of her wranglers is working with Mexican revolutionaries. Slow moving Bob Custer affair.\n\n**4649** _ **Under Texas Skies**_ **** Republic, 1940. 57 min. D: George Sherman. SC: Anthony Coldeway and Betty Burbridge. With Robert Livingston, Bob Steele, Rufe Davis, Lois Ranson, Henry Brandon, Wade Boteler, Rex Lease, Yakima Canutt, Jack Ingram, Earle Hodgins, Walter Tetley, Burr Caruth, Curley Dresden, Jack Kirk, Ted Mapes, Vester Pegg, Forrest Taylor, Bob Burns, Donald Kerr, Fred Burns, Charles King, Frank Ellis, Jim Corey, John Beach, Matty Roubert, Kenneth Terrell, Chuck Baldra, Chick Hannon, Augie Gomez, Herman Nowlin, Franklyn Farnum, Bob Card, Herman Hack, Al Haskell, Pascale Perry, Silver Tip Baker, Lew Morphy. Tucson is accused of murdering Stony's lawman father and the Three Mesquiteers are estranged until Stony comes to believe in his pal's innocence. Fast paced series entry that introduced Bob Steele and Rufe Davis in the roles of Tucson and Lullaby.\n\n**4650** _ **Under the Pampas Moon**_ **** Fox, 1935. 78 min. D: James Tinling. SC: Ernest Pascal and Bradley King. With Warner Baxter, Ketti Gallian, J. Carrol Naish, John Miljan, Armida, Ann Codee, Jack LaRue, George Irving, Rita (Hayworth) Cansino, Veloz and Yolanda, Tito Guizar, Chris-Pin Martin, Max Wagner, Philip Cooper, Sam Appel, Arthur Stone, George Lewis, Paul Porcasi, Lona Andre, Martin Garralaga, Tommy Coats, Frank Cordell, Joe Rickson, Catherine Cotter, Charles Stevens, Pedro Regas, Fred Malatesta, Juan Ortiz, Joe Dominguez, Nick Thompson, Manuel Perez, Soledad Jiminez, John Eberts. In Argentina a gaucho's horse is stolen by crooks who want to enter it in a race in Buenos Aires and the cowboy heads to the big city to get his steed back. Mediocre attempt to transfer Warner Baxter's characterization of a Cisco Kid-type to the Pampas; Rita Hayworth's first film.\n\n**4651** _ **Under the Tonto Rim**_ **** Paramount, 1928. 60 min. D: Herman C. Raymaker. SC: J. Walter Ruben. With Richard Arlen, Mary Brian, Alfred Allen, Jack Luden, Harry T. Morey, William Franey, Harry Todd, Bruce Gordon, Jack Byron. A cowboy is blackmailed by a murderer but helped by the miner who loves his sister. Pretty good adaptation of the Zane Grey novel makes this silent film worth a look.\n\n**4652** _ **Under the Tonto Rim**_ **** Paramount, 1933. 63 min. D: Henry Hathaway. SC: Jack Cunningham and Gerald Geraghty. With Stuart Erwin, Verna Hillie, Raymond Hatton, Fred Kohler, Fuzzy Knight, John Lodge, George Barbier, Patricia Farley, Edwin J. Brady, Marion Burdell, Allan Garcia. A cowpoke who operates on the slow side ends up capturing a killer and wins the love of his boss' daughter. First sound version of the Zane Grey work is not quite as good as the silent outing but is still entertaining, especially for Stuart Erwin fans.\n\n**4653** _ **Under the Tonto Rim**_ **** RKO Radio, 1947. 61 min. D: Lew Landers. SC: Norman Houston. With Tim Holt, Nan Leslie, Richard Martin, Richard Powers (Tom Keene), Carol Forman, Tony Barrett, Harry Harvey, Jason Robards, Robert Clarke, Jay Norris, Lex Barker, Steve Savage, Bud Osborne. A mysterious outlaw gang is trailed by a cowboy determined to capture them. Although it bears little resemblance to the Zane Grey book, this is a fast paced effort with fine photography by Anent Hunt.\n\n**4654** _ **Under Western Skies**_ **** Universal, 1945. 57 min. D: Jean Yarbrough. SC: Stanley Roberts and Clyde Bruckman. With Martha O'Driscoll, Noah Beery, Jr., Leo Carrillo, Leon Errol, Irving Bacon, Ian Keith, Jennifer Holt, Edna May Wonacott, Earle Hodgins, Shaw and Lee, Dorothy Granger, Jack Rice, Gladys Blake, George Lloyd, Claire Whitney, Frank Lackteen, Jack Ingram, Patsy O'Bryne, Nan Leslie, Eddy Waller, Perc Launders, Donald Kerr, Donald Jackson, Charles Sherlock. The denizens of a Western town oppose the staging of a traveling show, especially when the leading lady gets involved with the local school teacher and a masked bandit. Entertaining oater musical program feature.\n\n**4655** _ **Under Western Stars**_ **** Republic, 1938. 65 min. D: Joseph Kane. SC: Dorrell McGowan, Stuart McGowan and Betty Burbridge. With Roy Rogers, Smiley Burnette, Carol Hughes, The Maple City Four, Guy Usher, Earl Dwire, Dick Elliott, Jack Rockwell, Frankie Marvin, Earle Hodgins, Jack Ingram, Kenneth Harlan, Tom Chatterton, Alden Chase, Brandon Beach, Slim Whitaker, Jean Fowler, Jack Kirk, Fred Burns, Tex Cooper, Curley Dresden, Bill Wolfe. A cowboy is drafted into running for Congress when the incumbent proves to be the pawn of a large company overcharging ranchers for water during a drought. Roy Rogers' first starring feature is a good one and in it he sings \"That Pioneer Mother of Mine\" and Johnny Marvin's Academy Award nominated \"Dust.\"\n\n**4656** _ **Undercover Man**_ **** Republic, 1936. 56 min. D: Albert Ray. SC: Andrew Bennison. With Johnny Mack Brown, Suzanne Kaaren, Ted Adams, Frank Darien, Lloyd Ingraham, Horace Murphy, Dick Moorehead, Ed Cassidy, Margaret Mann, Frank Ball, George Morrell, Jim Corey, Art Dillard, Ray Henderson. A Wells Fargo agent saves a woman and gold during a holdup and the local bar owner, the leader of an outlaw gang, plots revenge. Johnny Mack Brown's initial Republic series oater is pleasantly paced.\n\n**4657** _ **Undercover Man**_ **** United Artists, 1942. 68 min. D: Lesley Selander. SC: J. Benton Cheney. With William Boyd, Andy Clyde, Jay Kirby, Antonio Moreno, Nora Lane, Chris-Pin Martin, Esther Estrella, John Vosper, Ea Puig, Alan Baldwin, Jack Rockwell, Pierce Lyden, Martin Garralaga, Earle Hodgins, Frank Ellis, Ted Wells, Joe Dominguez, Tony Roux, Cliff Parkinson, Frank Ellis, Ben Corbett, George Sowards, Lem Sowards. Hopalong Cassidy is falsely accused of robbery and sets out to get the real culprits, a gang operating below the Mexican border. The first United Artists \"Hoppy\" release is a bit on the slow side and plays better in its edited 54 minute TV version.\n\n**4658** _ **Undercover Men**_ **** British Dominion, 1935. 60 min. D: Sam (Newfield) Neufeld. SC: Murison Dunn. With Charles Starrett, Adrienne Dore, Kenneth (Kenne) Duncan, Wheeler Oakman, Eric Clavering, Phil Brandon, Austin Morin, Grace Webster, Gilmore Young, Elliott Lorraine, Wilbur Freeman, Farnham Barter, Muriel Deane. A Mounted Policeman working undercover pursues of an outlaw in the wilds of Canada. Canadian made feature of interest to Charles Starrett fans since it pre-dates his Columbia years; based on a story by co-star Kenne Duncan.\n\n**4659** _ **Undercover Woman**_ **** Republic, 1946. 56 min. D: Thomas Carr. SC: Jerry Sackheim and Sherman Lowe. With Stephanie Bachelor, Robert Livingston, Richard Fraser, Betty Blythe, Isabel Withers, Helen Heigh, Edythe Elliott, John Dehner, Elaine Lange, Tom London, Larry J. Blake. When a murder is committed at a Western dude ranch, a female detective and the local sheriff try to solve the crime. Pleasing Republic program feature.\n\n**4660** _ **Underground Rustlers**_ **** Monogram, 1941. 59 min. D: S. Roy Luby. SC: Bud Tuttle, Elizabeth Beecher and John Vlados. With Ray Corrigan, John King, Max Terhune, Gwen Gaze, Robert Blair, Forrest Taylor, Bud Osborne, Steve Clark, Tom London, Carl Mathews, John Elliott, Richard Cramer, Tex Palmer, Ed Peil, Sr., Tex Cooper, Frank McCarroll, Rudy Sooter, Buck Connors, Post Park, Roy Bucko, Milburn Morante, Denver Dixon, George Morrell, Tex Phelps. Three cowboys are assigned to capture a gang of gold smugglers. Fair entry in \"The Range Busters\" series that belongs to the wonderful Max Terhune who disguises himself as a suspender salesman.\n\n**4661** _ **Unexpected Guest**_ **** United Artists, 1947. 61 min. D: George Archainbaud. SC: Ande Lamb. With William Boyd, Andy Clyde, Rand Brooks, Una O'Connor, Pamela Tate, Ian Wolfe, John Parrish, Robert B. Williams, Earle Hodgins, Ned (Nedrick) Young, Joel Friedkin, William Ruhl. Hopalong Cassidy tries to find out who is behind a series of mysterious incidents, including the framing of Lucky on a murder charge, when California inherits a portion of his late cousin's ranch. Acceptable later edition in the \"Hopalong Cassidy\" series helped by its mystery plot.\n\n**Spanish lobby card for** _**Unexpected Guest**_ **(United Artists, 1947); insert picturing Robert B. Williams, Patricia Tate, Una O'Connor and Andy Clyde.**\n\n**  \n**\n\n**4662** _ **The Unforgiven**_ **** United Artists, 1960. 120 min. Color. D: John Huston. SC: Ben Maddow. With Burt Lancaster, Audrey Hepburn, Audie Murphy, John Saxon, Charles Bickford, Lillian Gish, Albert Salmi, Joseph Wiseman, June Walker, Kipp Hamilton, Arnold Merritt, Carlos Rivas, Doug McClure. In a remote area the locals resent an Indian girl who has been raised as white and she is threatened by them when the Kiowa tribe goes on the warpath. Overwrought melodrama that should have been better but any film with Lillian Gish is worth watching.\n\n**4663** _ **Unforgiven**_ **** Warner Bros., 1992. 131 min. Color. D: Clint Eastwood. SC: David Webb Peoples. With Clint Eastwood, Gene Hackman, Morgan Freeman, Richard Harris, Jaimz Woolvett, Saul Rubinek, Frances Fisher, Anna Thomson, David Mucci, Rob Campbell, Anthony James, Tara Dawn Frederick, Beverley Elliott, Lisa Repo-Martell, Josie Smith, Shane Meier, Aline Levasseur, Cherrilene Cardinal, Robert Koons, Ron White, Mina E. Mina, Henry Kope, Jermey Ratchford, John Pyser-Ferguson, Jefferson Mappin, Walter Marsh, Garner Butler, Larry Reese, Blair Haynes, Frank C. Turner, Sam Karas, Lochlyn Munro, Ben Cardinal, Philip Hayes, Michael Charrois, Bill Davidson, Larry Joshua, George Orrison, Gregory Goossen, Michael Maurer, Paul McLean, James Herman. An aging ex-gunfighter enlists the help of an old friend in joining a young gunman in hunting the cowboy who brutalized a prostitute. Overlong but huge moneymaking Western that reaped a number of awards.\n\n**4664** _ **The Unholy Four**_ **** B.R.C.\/Atlas, 1970. 95 min. Color. D: E.B. Clucher (Enzo Barboni). SC: Franco Rossetti and Mario di Nardo. With Leonard Mann, Woody Strode, Peter Martell, Helmuth Schneider, Lucas Montefiori (George Eastman), Evelyn Stewart (Ida Galli), Alain Naya, Dino Strano, Andrew Ray (Andrea Aureli), Enzo Fiermonte, Romano Puppo, Fortunato Arena, Billa Salvatore, Emilio Messina, Osiride Pevarello, Giusepe Lauricella, Lucio Rosato, Remo Capitani, Remo De Angelis, Roberto Dell'Acqua, Claudio Scharchilli. Three men escape from prison with one of them an amnesiac American gunman hired to kill a man who may be his father. Good Spaghetti Western made in Italy as _**Ciakmull, L'Uomo della Vendetta**_ (Chuck Moll, the Vendetta Man) and also known as _**Chuck Moll**_.\n\n**4665** _ **Union Pacific**_ **** Paramount, 1939. 133 min. D: Cecil B. DeMille. SC: Walter DeLeon, C. Gardner Sullivan and Jesse Lasky, Jr. With Barbara Stanwyck, Joel McCrea, Robert Preston, Akim Tamiroff, Lynne Overman, Brian Donlevy, Robert Barrat, Anthony Quinn, Stanley Ridges, Henry Kolker, Francis McDonald, Willard Robertson, Harold Goodwin, Evelyn Keyes, Richard Lane, William Haade, Regis Toomey, Lon Chaney, Jr., J.M. Kerrigan, Fuzzy Knight, Harry Woods, Joseph Crehan, Julia Faye, Sheila Darcy, Joseph Sawyer, Stanley Andrews, Earl Askam, John Marston, Byron Foulger, Selmer Jackson, Morgan Wallace, John Merton, Ed Peil, Sr., Russell Hicks, May Beatty, Ernie Adams, William J. Worthington, Guy Usher, James McNamara, Gus Glassmire, Stanley Andrews, Paul Everton, Jack Pennick, John Marston, Iron Eyes Cody, Lew Short, Tom Burke. A troubleshooter for the Union Pacific Railroad romances an engineer's daughter and tries to combat sabotage to the line in its competition with the Central Pacific for the completion of the first transcontinental tracks. Sprawling, and typically entertaining, Cecil B. DeMille epic.\n\n**4666** _ **Unknown Ranger**_ **** Aywon, 1920. 45 min. With Rex Ray, Marie Newall, Ben Hall. Texas Rangers are on the trail of a gang smuggling opium across the Mexican border. Fairly interesting silent outing, not only for its plot but because the villain escapes at the end.\n\n**4667** _ **Unknown Ranger**_ **** Columbia, 1936. 58 min. D: Spencer Gordon Bennet. SC: Nate Gatzert. With Bob Allen, Martha Tibbetts, Harry Woods, Hal Taliaferro, Buzzy Henry, Edward Hearn, Robert Kortman, Lew Meehan, Bob McKenzie, Art Mix, Oscar Gahan, Rudy Sooter, Robert Hoag, Francis Walker, Ray Henderson, Al Taylor, Buck Moulton, Merrill McCormick, Tex Palmer, Allan Cavan, Cactus Mack, Bud McClure, Bob Card, Henry Hall, Art Dillard, Jack King, Horace B. Carpenter, Eva McKenzie, Bud Jamison, Rube Dalroy. Rustlers plan to use a wild stallion to steal a rancher's horse herd but a cowhand, actually a ranger, gets wind of the plan. Bob Allen's first series film is a good one and in it Hal Taliaferro (Wally Wales) sings a novelty tune.\n\n**4668** _ **Unknown Valley**_ **** Columbia, 1933. 69 min. D-SC: Lambert Hillyer. With Buck Jones, Cecilia Parker, Bert Black, Carlotta Warwick, Arthur Wanzer, Wade Boteler, Frank McGlynn, Charles Thurston, Ward Bond, Gaylord (Steve) Pendleton, Alf James, Frank Ellis. While searching for his father, an ex\u2013Army scout becomes lost in the desert and is rescued by a young woman belonging to a strange religious sect. Out-of-the-ordinary \"B\" Western that relies far more on its well written script than usual genre action; a very good film.\n\n**4669** _ **Unknown Wilderness**_ **** American National Enterprises, 1973. 94 min. Color. D: Austin Green. SC: Roger Davis and Austin Green. Two teenage boys learn to survive in the mountain wilderness of Montana and Wyoming as they search for the legendary treasure of Frenchy Latrek. Cheaply made but eye-pleasing docudrama.\n\n**4670** _ **The Unsinkable Molly Brown**_ **** Metro-Goldwyn-Mayer, 1964. 128 Color. D: Charles Walters. SC: Helen Deutsch. With Debbie Reynolds, Harve Presnell, Ed Begley, Jack Kruschen, Hermoine Baddeley, Martita Hunt, Vassili Lambrinos, Fred Essler, Harvey Lembeck, Kathryn Card, Hayden Rorke, Harry Holcombe, Amy Douglas, George Mitchell, Vaughn Taylor, Anthony Eustrel, Audrey Christie, Lauren Gilbert. In the late 1880s an orphan marries a miner and they strike it rich but are snubbed in Denver but go to Europe where they become the toast of society. Zesty musical with Meredith Willson-Richard Morris' score.\n\n**4671** _ **Untamed**_ **** Paramount, 1940. 83 min. Color. D: George Archainbaud. SC: Frederick Hazlett Brennan and Frank Butler. With Ray Milland, Patricia Morison, Akim Tamiroff, William Frawley, Jane Darwell, J.M. Kerrigan, J. Farrell MacDonald, Ester Dale, Eily Malyon, Faye Helm, Clem Bevans, Sybil Harris, Roscoe Ates, Gertrude W. Hoffman, Charles Waldron, Darryl Hickman, Charlene Wyatt, Bahe Denetdell, Donna Jean Lester, Byron Foulger, Helen Brown, Guy Wilkerson, Charles Stevens, Brenda Fowler, Ann Doran, Pauline Haddon, Dorothy Adams, Betsy Ross Clarke. An alcoholic New York City surgeon goes to the north woods to recover and decides to stay after falling in love with his guide's pretty wife. Well made melodrama enhanced by scenic locales and good use of Technicolor.\n\n**4672** _ **Untamed**_ **** 20th Century\u2013Fox, 1955. 111 min. Color. D: Henry King. SC: Talbot Jennings, Frank Fenton and Michael Blankfort. With Tyrone Power, Susan Hayward, Richard Egan, John Justin, Agnes Moorehead, Rita Moreno, Hope Emerson, Brad Dexter, Henry O'Neill, Paul Thompson, Alexander D. Havemann, Louis Mercier, Emmett Smith, Jack Macy, Bobby Diamond, Gary Diamond, Brian Corcoran, Kevin Corcoran, Eleanor Audley, Cecil Weston, Forrest Burns, Leonard Carey. In South Africa a Dutchman saves a wagon train of settlers from a Zulu attack and falls in love with the wife of a man killed in the skirmish. Entertaining melodrama with beautiful South African scenery although the plot tends more toward Susan Hayward's character than Tyrone Power's, a man who wants to set up a Dutch state.\n\n**4673** _ **The Untamed Breed**_ **** Columbia, 1948. 79 min. Color. D: Charles Lamont. SC: Tom Reed. With Sonny Tufts, Barbara Britton, George \"Gabby\" Hayes, Barbara Britton, William Bishop, George E. Stone, Joseph Sawyer, Gordon Jones, James Kirkwood, Harry Tyler, Virginia Brissac, Reed Howes, Russell Simpson, Syd Saylor, Dick Elliott, Frank Hagney, Kernan Cripps, Paul E. Burns, Louis Mason, Symona Boniface, Richard Gordon, Phil Schumacher. Hoping to improve their herds, ranchers along the Pecos River in Texas go along with a cattleman's plan to purchase a Brahma bull. Passable but standard drama.\n\n**4674** _ **Untamed Frontier**_ **** Universal-International, 1952. 75 min. Color. D: Hugo Fregonese. SC: Gerald Drayson Adams, John Bagni and Gwen Bagni. With Joseph Cotten, Shelley Winters, Scott Brady, Suzan Ball, Minor Watson, Katherine Emery, Antonio Moreno, Douglas Spencer, John Alexander, Richard Garland, Lee Van Cleef, Robert Anderson, Fess Parker, Edmund Cobb, Jose Torvay, Ray Bennett, Beatrice Gray, Alex Montoya, Forrest Taylor, Clem Fuller, Brick Sullivan, Connie Vera, Bob Burns, Joe Dominguez, Emile Avery, Forrest Burns, John Davidson, Carl Andre, Carlos Albert, Monte Montague, Lalo Rios, Bob Reeves, Mike Lally, Leo J. McMahon, Jennings Miles, David Janssen, Henry A. Escalante, Henry Orozco, Tom Smith, Denver Dixon, Mary Bayless. A wealthy cattle baron uses any means he can to stop settlers from taking free government land which he wants for his herds. Okay drama carried along by good direction and performances.\n\n**4675** _ **Untamed Heiress**_ **** Republic, 1954. 70 min. D: Charles Lamont. SC: Barry Shipman. With Judy Canova, Donald Barry, Taylor Holmes, George Cleveland, Chick Chandler, Jack Kruschen, Hugh Sanders, Douglas Fowley, William Haade, Ellen Corby, James Flavin, Tweeny Canova, Dick Wessel. A millionaire hires two talent agents to locate a woman who once gave him money and they find out she has died and her daughter is in an orphanage. Typical Judy Canova comedy that will appeal to her fans.\n\n**4676** _ **Until They Get Me**_ **** Triangle, 1917. 58 min. D: Frank Borgaze. SC: Kenneth B. Clark. With Pauline Starke, Jack Curtis, Joe King, Wilbur Higby, Anna Dodge, Walter Perry. A Mountie arrests a man for murder but he escapes and years later a young woman the policeman sets free from a life of drudgery helps him find the escapee and together they prove his innocence. Competent silent drama without a lot of action.\n\n_**Up Like a Shot!**_ see _**Blazing Stewardesses**_\n\n**4677** _ **Up River**_ **** Arcade, 1979. 90 min. Color. D: Carl Kitt. SC: Carol Hummel. With Morgan Stevens, Jeff Corey, Dale Wilson, John \"Bear\" Curtis, David Crowley, Mikal Dughi, Deborah Au Luce, Cindy Jensen, Robert George, Norm Weatherby, Lance Garrett, Ronnie Lester. A homesteader's wife is raped and murdered by a land baron who also burns his house and the sodbuster plots revenge. Poor, low grade production.\n\n**4678** _ **Up the MacGregors!**_ **** Columbia, 1968. 98 min. Color. D: Frank Garfield (Franco Giraldi). SC: Fernand Lion (Fernando Di Leo), Vincent Eagle (Enzo Dell'Aquila), Paul Levy (Paolo Levi), Jose Marie Rodriguez and Franco Giraldi. With David Bailey, Agatha Flory, Alberto Dell'Acqua (Robert Widmark), Leo Anchoriz, Roberto Camardiel, Cole Kitosh, Nick Anderson, Paul Carter, Julio Perez Tabernero, Hugo Blanco, Saturnino Cerra, George Rigaud, Roy Bossier, Victor Israel, Ann Casares, Francesco Tensi, Jesus Guzman, King Black, Antonio Vico, Enlena Montoya, Tito Garcia, Anne-Marie Noe, Margaret Horowitz, Margaret Merritt, Kathleen Parker, Anna Maria Mendoza. The seven MacGregor brothers are after an outlaw gang that stole all their families' possessions. Pleasant sequel to _**Seven Guns for the MacGregors**_ (q.v.) and, like its predecessor, less violent and more amusing than most of its ilk. Released in Italy in 1966 as _**Sette Donne per I MacGregor**_ (Seven Women for the MacGregors) by Produzione D.S.\/Jolly\/Talia Film.\n\n**4679** _ **Uphill All the Way**_ **** New World, 1985. 91 min. Color. D-SC: Frank Q. Dobbs. With Roy Clark, Mel Tillis, Burl Ives, Glen Campbell, Trish Van Devere, Richard Paul, Elaine Joyce, Jacque Lynn Colton, Frank Gorshin, Sheb Wooley, Burton Gilliam, Burt Reynolds, Gilard Sartain, Rockne Tarkington, Christopher Weeks, Pedro Gonzalez Gonzalez, Burt Reynolds, Danny Kwan, Jim Lau, Jo Perkins, David Logan Rankin, Noah Davison, Blue Deckert, Paul Menzel, Chet Warner, Tommy Collins, Ed Geldart. In 1916 Texas two con men are mistaken for notorious bank robbers and chased across the desert by a posse. Stars Roy Clark and Mel Tillis served as executive producers on this weak Western comedy in which Burt Reynolds appears unbilled as a gambler.\n\n**4680** _ **Uranium Boom**_ **** Columbia, 1956. 67 min. D: William Castle. SC: George W. Greene. With Dennis Morgan, Patricia Medina, William Talman, Tina Carver, Philip Van Zandt, Bill Henry, Gregg Barton, Mel Curtis, Henry Rowland, S. John Launer, Michael Bryant, Frank Wilcox, Ralph Sanford, Carlyle Mitchell, Nick Tell. A former lumberjack teams with a mining engineer to locate uranium in Colorado only to marry his partner's ex-fiancee, causing a rift between the two men. Mediocre program feature from producer Sam Katzman.\n\n**4681** _ **Utah**_ **** Republic, 1945. 78 min. D: John English. SC: Jack Townley. With Roy Rogers, George \"Gabby\" Hayes, Dale Evans, Peggy Stewart, Beverly Lloyd, Grant Withers, Hal Taliaferro, Bob Nolan and The Sons of the Pioneers (Tim Spencer, Ken Carson, Shug Fisher, Hugh Farr, Karl Farr), Jack Rutherford, Emmett Vogan, Ed Cassidy, Vivien Oakland, Jill Browning, Ralph Colby, Forrest Taylor, Horace B. Carpenter. Roy and Gabby try to stop an out of work show girl from selling the ranch she inherited to sheep herders. Fairly entertaining but shackled with a dull finale production number.\n\n**4682** _ **Utah Blaine**_ **** Columbia, 1957. 75min. D: Fred F. Sears. SC: Robert E. Kent and James B. Gordon. With Rory Calhoun, Susan Cummings, Max Baer, Angela Stevens, Paul Langton, George Keymas, Ray Teal, Gene Roth, Terry Frost, Dennis Moore, Jack Ingram, Steve Darrell, Norman Frederic (Dean Fredericks), Ken Christy. A gunman helps a rancher who is being harassed by marauders out to control the territory. Rory Calhoun fans will go for this better than average Sam Katzman production.\n\n**4683** _ **The Utah Kid**_ **** Tiffany, 1930. 60 min. D: Richard Thorpe. SC: Frank Howard Clark. With Rex Lease, Dorothy Sebastian, Tom Santschi, Mary Carr, Walter Miller, Lafe McKee, Boris Karloff, Bud Osborne, Art Mix, Wally Wales, Buffalo Bill, Jr., Fred Burns, Bud McClure, Al Taylor, Bob Burns, Chuck Baldra, Bob Card, Blackie Whiteford, Ralph Bucko, Roy Bucko. An outlaw returns to his gang's hideout to find a cohort trying to take advantage of a pretty school teacher who has wandered into their den and he defends and later marries her, although she is loved by the local sheriff. Rather interesting Rex Lease early talkie; Boris Karloff has a minor role as a gang member.\n\n**4684** _ **The Utah Kid**_ **** Monogram, 1944. 54 min. D: Vernon Keays. SC: Victor Hammond. With Bob Steele, Hoot Gibson, Beatrice Gray, Evelyn Eaton, Ralph Lewis, Mauritz Hugo, Jameson Shade, Mike G. Letz, Dan White, Bud Osborne, George Morrell, Al Ferguson, Lew Meehan, Earle Hodgins, Herman Hack, Jack Evans. A U.S. marshal and his new deputy investigate a gang that always wins the events on a rodeo circuit. An abundance of rodeo stock footage does not help this rather rag-tag dual bill effort; a remake of _**The Man from Utah**_ (q.v.).\n\n**4685** _ **Utah Trail**_ **** Grand National, 1937. 57 min. D: Albert (Al) Herman. SC: Edmund Kelso. With Tex Ritter, Adele Pearce (Pamela Blake), Dave O'Brien, Horace Murphy, Snub Pollard, Karl Hackett, Charles King, Ed Cassidy, Bud Osborne, Lynton Brent, Rudy Sooter and Tex Ritter's Tornados, Oscar Gahan, Ray Jones, Denver Dixon, George Morrell, Horace B. Carpenter, Herman Hack, Chick Hannon. An investigator is hired by the railroad to locate a stolen cattle train and find out who is sabotaging the company's property. Tex Ritter's final Grand National release is a ragged affair with average tunes. British title: _**Trail to Utah**_.\n\n**4686** _ **Utah Wagon Train**_ **** Republic, 1951. 67 min. D: Philip Ford. SC: John K. Butler. With Rex Allen, Penny Edwards, Buddy Ebsen, Roy Barcroft, Sarah Padden, Grant Withers, Arthur Space, Edwin Rand, Robert Karnes, William Holmes, Stanley Andrews, Frank Jenks, Forrest Taylor. A rancher gets himself made trail boss of a modern-day wagon train party searching for the route West used a century before by their ancestors, in order to find out who murdered his uncle, the group's original guide. Very fine Rex Allen feature with a good plot.\n\n**4687** _ **Vacation Days**_ **** Monogram, 1947. 68 min. D: Arthur Dreifuss. SC: Hal Collins. With Freddie Stewart, June Preisser, Frankie Darro, Warren Mills, Noel Neill, Milton Kibbee, Belle Mitchell, John Hart, Hugh Prosser, Terry Frost, Edythe Elliott, Claire James, Forrest Taylor, Spade Cooley Band, Jerry Wald and His Orchestra. A group of students, after graduation, go to a Western ranch inherited by their teacher and get involved with an outlaw gang. Average outing in the \"Teen Agers\" series, also called _**Freddie Goes West**_.\n\n_**The Valdez Horses**_ see _**Chino**_\n\n**4688** _ **Valdez Is Coming**_ **** United Artists, 1971. 90 min. Color. D: Edwin Sherin. SC: Roland Kibbee and David Rayfiel. With Burt Lancaster, Susan Clark, Jon Cypher, Barton Heyman, Richard Jordan, Frank Silvera, Hector Elizondo, Phil Brown, Ralph Brown, Roberta Haynes, Jose Garcia, Michael Hinn, Joaquin Parra, Rudy Ugland, Vic Albert, Allan Russell, Juan Fernandez, Tony Eppers, Nick Cravat, Raul Castro, Jose Morales, Mario Sanz. A gunman forced into a shootout is hunted by a posse and he plans to turn the tables on his trackers. Average drama filmed in Spain, based on Elmore Leonard's book.\n\n_**Valentine Lazan, El Ratero de las Pobres**_ see _**El Ratero de las Pobres**_\n\n**4689** _ **Valerie**_ **** United Artists, 1957. 84 min. D: Gerd Oswald. SC: Leonard Heideman and Emmett Murphy. With Sterling Hayden, Anita Ekberg, Anthony Steel, Peter Walker, John Wengraf, Iphigenie Castiglioni, Robert Adler, Gage Clarke, Jerry Barclay, Tom McKee, Stanley Adams, John Dierkes, Malcolm Atterbury, Darryl Duran, Norman Leavitt, Sydney Smith, Juney Ellis. A man's wife is wounded and her parents murdered with the trial for the crimes becoming a conflicting affair until the woman agrees to testify. A different kind of oater although it hardly seems worth the effort.\n\n**4690** _ **The Valiant Hombre**_ **** United Artists, 1948. 61 min. D: Wallace Fox. SC: Adele Buffington. With Duncan Renaldo, Leo Carrillo, Barbara Billingsley, John Litel, John James, Lee \"Lasses\" White, Stanley Andrews, Guy Beach, Gene Roth, Terry Frost, Frank Ellis, Ralph Peters, Herman Hack, George DeNormand, Ed Peil, Sr., Hank Bell, Bert Dillard, George Morrell, Rube Dalroy, Daisy (dog). The Cisco Kid and Pancho try to locate a mining engineer who disappeared after making a big strike. Good entry in \"The Cisco Kid\" series.\n\n_**El Valle de los Desaparecidos**_ see _**The Lone Rider**_ (1960)\n\n**4691** _ **Valley of Fear**_ **** Monogram, 1947. 54 min. D: Lambert Hillyer. SC: J. Benton Cheney. With Johnny Mack Brown, Raymond Hatton, Christine McIntyre, Ed Cassidy, Tristram Coffin, Ted Adams, Steve Darrell, Pierce Lyden, Eddie Parker, Gary Garrett, Cactus Mack, Robert O'Byrne, Ed Peil, Sr., Budd Buster. A cowpoke returns home to find his uncle dead and himself accused of taking money the deceased embezzled. Standard, but entertaining Johnny Mack Brown feature with a good mystery element.\n\n**4692** _ **Valley of Fire**_ **** Columbia, 1951. 70 min. D: John English. SC: Earle Snell. With Gene Autry, Pat Buttram, Gail Davis, Russell Hayden, Christine Larsen, Harry Lauter, Terry Frost, Riley Hill, Barbara Stanley, Duke York, Bud Osborne, Teddy Infur, Victor Sen Yung, Gregg Barton, Sandy Sanders, Fred Sherman, James Magill, Frankie Marvin, Pat O'Malley, Wade Crosby, William Fawcett, Syd Saylor, John Miller, Marjorie Liszt. In the town of Quantz Creek in the 1850s sheriff Gene Autry runs off a crooked gambler and his cohorts and the bad man vows revenge by trying to hijack a wagon train bringing brides to the community. Pretty good Gene Autry vehicle in which he sings \"On Top of Old Smoky.\"\n\n**4693** _ **The Valley of Gwangi**_ **** Warner Bros.-Seven Arts, 1969. 95 min. Color. D: James O'Connolly. SC: William E. Bast. With James Franciscus, Gila Golan, Richard Carlson, Laurence Naismith, Curtis Arden, Freda Jackson, Gustavo Rojo, Dennis Kibane, Marion De Barrros, Jose Burgos. In Mexico a cowboy finds a hidden valley and with his men is able to lasso a huge dinosaur they try to exhibit in a small circus. Basically dull combination of Western and sci-fi genres with fine special effects by Ray Harryhausen; Gwangi is a very likable monster.\n\n**4694** _ **Valley of Hate**_ **** Russell Productions, 1924. 63 min. D: Russell Allen. SC: George Hively. With Raymond McKee, Helen Ferguson, Earl Metcalfe, Wilfred Lucas, Ralph Yearsley, Helen Lynch, Frank Whitson. After inheriting property in a remote mountain area, a rich man goes there, is mistaken for a revenue agent and falls in love with a young woman, a moonshiner's ward. The scenery is the best part of this somewhat overwrought melodrama.\n\n**4695** _ **Valley of Hunted Men**_ **** Republic, 1942. 60 min. D: John English. SC: Albert DeMond and Morton Grant. With Bob Steele, Tom Tyler, Jimmie Dodd, Anna Marie Stewart, Edward Van Sloan, Roland Varno, Edythe Elliott, Arno Frey, Richard French, Kenne Duncan, Jack Kirk, Budd Buster, Hal Price, Rand Brooks, Billy Benedict, George Neise, Robert Stevenson, Duke Aldon, Charles Flynn, Mickey Rentschler, Hank Worden, Oscar \"Dutch\" Hendrian, Charles Flynn, Louis Adlon, Arvon Dale, John Frazer, Kermit Maynard, Jack O'Shea, Tex Terry, Bob Card, Charles Graham, Henry Morris, James Mitchell, Rose Plummer. Three cowboys are trailing an escaped Nazi posing as the nephew of a scientist working on a formula for making rubber from culebra plants. Topical outing in \"The Three Mesquiteers\" series, not very dated and still entertaining.\n\n**4696** _ **Valley of Terror**_ **** Ambassador, 1937. 59 min. D: Al Herman. SC: Stanley Roberts. With Kermit Maynard, Harley Wood, John Merton, Jack Ingram, Dick Curtis, Roger Williams, Frank McCarroll, Hank Bell, Hal Price, Slim Whitaker, George Morrell, Blackie Whiteford, Herman Hack, Jack Casey, Jack Evans, Tex Cooper, Ray Henderson. A crook is after mineral deposits on a woman's ranch and he has her boyfriend framed on a rustling charge to get him out of the way. Standard, but action laced, Kermit Maynard vehicle from producer Maurice Conn.\n\n**4697** _ **Valley of the Giants**_ **** Warner Bros., 1938. 79 min. Color. D: William Keighley. SC: Seton I. Miller and Michael Fessler. With Wayne Morris, Claire Trevor, Frank McHugh, Alan Hale, Donald Crisp, Charles Bickford, Jack LaRue, John Litel, Dick Purcell, El Brendel, Russell Simpson, Cy Kendall, Harry Cording, Wade Boteler, Helen McKellar, Addison Richards, Jerry Colonna, Trevor Bardette, Pierre Watkin, Don Barclay, Herbert Rawlinson, Stuart Holmes, Frank Darien, Clem Bevans, George Chandler, Sidney Bracy, Jack Mower, Al Herman, Fred Burton, Cliff Saum, Ben Hendricks, Sonny Bupp, Spencer Charters, Nat Carr, Eddy Chandler, William Pawley, Lee Shumway, Paul Panzer, Henry Otho, Lew Harvey, Bob Perry, Bob Stevenson, Don Turner, Tom Wilson, Sol Gorss. A lumberman, with the help of a saloon girl, fights a rival who wants to take timber and destroy the north woods in the process. Highly entertaining melodrama based on Peter B. Kyne's novel, first filmed in 1919 by Artcraft-Paramount with Wallace Reid and remade by Warner Bros. in 1927 starring Milton Sills. A fourth version was done as _**The Big Trees**_ (q.v.).\n\n**4698** _ **Valley of the Lawless**_ **** Supreme, 1936. 59 min. D-SC: Robert North Bradbury. With Johnny Mack Brown, Joyce Compton, George Hayes, Frank Hagney, Denny Meadows (Dennis Moore), Bobby Nelson, Charles King, Jack Rockwell, Frank Ball, Forrest Taylor, Blackie Whiteford, Horace Murphy, Steve Clark, Ed Cassidy, Bob McKenzie, George Morrell, Jack Kirk, Jack Evans, Milburn Morante, Francis Walker, Fred Parker, Anita Carmago, Clyde McClary, Bud Pope, Buck Morgan, Tex Phelps, Rube Dalroy. To retrieve a gold map for which his grandfather was killed, a cowboy must penetrate an area used as a refuge by outlaws. Fast paced Johnny Mack Brown effort; well written.\n\n**4699** _ **Valley of the Redwoods**_ **** 20th Century\u2013Fox\/Associated Producers, 1960. 63 min. D: William Witney. SC: Gene Corman. With John Hudson, Lynn Bernay, Ed Nelson, Michael Forest, Robert Shayne, John Brinkley, Bruno Ve Soto, Hal Torey, Chris Miller. Three people, two men and a woman, carry out a payroll robbery and try to escape through a forest area but their plan is foiled when one of them is injured. Compact and pleasing little action drama.\n\n**4700** _ **Valley of the Sun**_ **** RKO Radio, 1942. 79 min. D: George Marshall. SC: Horace McCoy. With Lucille Ball, James Craig, Sir Cedric Hardwicke, Dean Jagger, Peter Whitney, Billy Gilbert, Tom Tyler, Antonio Moreno, George Cleveland, Hank Bell, Richard Fiske, Don Terry, Chris Willow Bill, Fern Emmett, Al St. John, Harry Lamont, Al Ferguson, Chester Conklin, Ed Brady, Lloyd Ingraham, Frank Coleman, Francis McDonald, Harry Hayden, Bud Osborne, Steve Clemento, Chester Clute, Tom London, George Melford, Carleton Young, Robert Kortman, Stanley Andrews, Lloyd Ingraham, Ethan Laidlaw, John Cason, Pat Moriarty, George Lloyd, Iron Eyes Cody, Jay Silverheels. A government agent pretends to be a renegade scout to get the goods on a crooked Indian agent in the Arizona Territory. The cast is the best thing about this slow moving melodrama; Tom Tyler superbly plays Geronimo.\n\n**4701** _ **Valley of Vanishing Men**_ **** Columbia, 1942. 15 Chapters. D: Spencer Gordon Bennet. SC: Harry Fraser, George Grey and Lewis Clay. With Bill Elliott, Slim Summerville, Carmen Morales, Kenneth MacDonald, Jack Ingram, George Chesebro, John Shay, Tom London, Arno Frey, Julian Rivero, Roy Barcroft, I. Stanford Jolley, Ted Mapes, Lane Chandler, Ernie Adams, Michael Vallon, Robert Fiske, Davison Clark, Lane Bradford, Chief Thundercloud, Blackie Whiteford, Frank Ellis, Forrest Taylor, Horace B. Carpenter, Kenne Duncan, Karl Hackett, Thornton Edwards, Art Dillard, Tex Palmer, Kit Guard, Iron Eyes Cody, Hank Bell, Sherry Tansey, Roy Bucko, Rose Plummer. A cowboy and his pal search for the man's missing prospector father and they learn he and others are held as slaves in a mine operated by an outlaw and a renegade European general who plan to overthrow the Juarez government in Mexico. Bill Elliott's final Columbia assignment is a cheap, dull cliffhanger.\n\n**4702** _ **Valley of Vengeance**_ **** Producers Releasing Corporation, 1944. 57 min. D: Sam Newfield. SC: Joseph O'Donnell. With Buster Crabbe, Al St. John, Evelyn Finley, Glenn Strange, Donald Mayo, Charles King, John Merton, Lynton Brent, Jack Ingram, Bud Osborne, Nora Bush, Steve Clark, David Polonsky, Budd Buster, Ben Corbett, Artie Ortego, John Cason, Tex Cooper, Wally West, George Morrell, Herman Hack, Buck Bucko, Pascale Perry, Morgan Flowers, Ray Henderson, Merrill McCormick, Tom Smith, Hank Bell, Ed Cassidy, Jimmy Aubrey, Jess Cavin, Dan White, Curley Dresden, Carl Mathews. Years after their parents were murdered in a wagon train massacre, Billy Carson and Fuzzy Q. Jones stumble onto the gang responsible for the killings in a small town. Standard \"Billy Carson\" affair with a strong supporting cast. British title: _**Vengeance**_.\n\n**4703** _ **Valley of Wanted Men**_ **** Conn Pictures, 1935. 63 min. D: Alan James. SC: Barry Barringer and Forrest Barnes. With Frankie Darro, Grant Withers, Drue Layton, (Le)Roy Mason, Paul Fix, Russell Hopton, Walter Miller, Fred \"Snowflake\" Toones, Alan Bridge, William Gould, Jack Rockwell, Frank Rice, Slim Whitaker. A young man tries to bring peace to his home, a vale populated by outlaws with prices on their heads. Fast paced low budget dual bill feature.\n\n**4704** _ **The Vanishing American**_ **** Paramount, 1926. 110 min. D: George B. Seitz. SC: Ethel Doherty and Lucien Hubbard. With Richard Dix, Lois Wilson, Noah Beery, Malcolm MacGregor, Nocki, Shannon Day, Charles Crockett, Bert Woodruff, Bernard Siegel, Guy Oliver, Joe Ryan, Charles Stevens, Bruce Gordon, Richard Howard, John Webb Dillon. After fighting heroically in World War I, an Indian soldier returns home only to find the land barren and his people being cheated by corrupt government bureaucrats. A silent film classic and one of the first dramas to show the harsh treatment of Native Americans.\n\n**4705** _ **The Vanishing American**_ **** Republic, 1955. 90 min. D: Joseph Kane. SC: Alan LeMay. With Scott Brady, Audrey Totter, Forrest Tucker, Gene Lockhart, Jim Davis, John Dierkes, Gloria Castillo, Julian Rivero, Lee Van Cleef, George Keymas, Charles Stevens, Jay Silverheels, James Millican, Glenn Strange, Francis McDonald, Hank Worden, Fred Graham. When crooks attempt to steal land belonging to the Navajo Indians, a brave tries to stop them. Republic's remake of the silent classic is only average, due mainly to budget restrictions.\n\n**4706** _ **The Vanishing Frontier**_ **** Paramount, 1932. 65 min. D: Phil Rosen. SC: Stuart Anthony. With Johnny Mack Brown, Evalyn Knapp, ZaSu Pitts, J. Farrell MacDonald, Raymond Hatton, Wallace MacDonald, Ben Alexander, George Irving, Joyzelle Joyner, Deacon McDaniels. In old California an American tries to help officials in stopping military abuse. Johnny Mack Brown followers will go for this action outing, his last pre-series Western.\n\n**4707** _ **The Vanishing Land**_ **** Gold Key, 1975. 91 min. Color. D: John Elmore. With John Elmore. Photographer\/guide John Elmore goes on a journey through Alaska chronicling its wilderness and people. Top flight documentary.\n\n**4708** _ **The Vanishing Legion**_ **** Mascot, 1931. 12 Chapters. D: B. Reeves Eason. SC: Wyndham Gittens, Ford Beebe and Helmer Bergman. With Harry Carey, Edwina Booth, Rex (horse), Frankie Darro, Philo McCullough, William Desmond, Joe Bonomo, Edward Hearn, Al Taylor, Lafe McKee, Dick Hatton, Pete Morrison, Dick Dickinson, Robert Kortman, Paul Weigel, Frank Brownlee, Yakima Canutt, Tom Dugan, Robert Walker, Oliver Fuller Golden (Carey), Charles \"Rube\" Schaeffer, Bill Wolff, Boris Karloff (voice). The mysterious \"Voice\" and his gang frame a man on a murder charge as they try to take over an oil company. Fun Mascot cliffhanger with a good cast and lots of action; also issued in a feature version. An unseen Boris Karloff is \"The Voice.\"\n\n**4709** _ **The Vanishing Outpost**_ **** Western Adventure, 1951. 56 min. D: Ron Ormond. SC: Alexander White. With Lash LaRue, Al St. John, Riley Hill, Archie Twitchell, Lee Morgan, Ted Adams, Bud Osborne, Marshall Reed, Jack Ingram, Tom London, John Cason, Zon Murray, Clarke Stevens, Ray Broome, Cliff Taylor, Sharon Hall, Sue Hussey, Johnny Paul, Bob Duncan, Sandy Sanders, Riley Hill. A Pinkerton agent enlists the assistance of Lash and Fuzzy in stopping a notorious outlaw gang. Rag-tag Lash LaRue vehicle hurt by excessive use of footage from previous series features such as _**Mark of the Lash**_ , _**Outlaw Country**_ , _**Son of a Badman**_ and _**Son of Billy the Kid**_ (qq.v.).\n\n_**Vanishing Pioneer**_ see _**Rocky Mountain Mystery**_\n\n**4710** _ **The Vanishing Prairie**_ **** Buena Vista, 1954. 75 min. Color. D: James Algar. SC: James Algar, Winston Hibler and Ted Sears. With Winston Hibler (narrator). Life on the American prairie is presented, focusing on animals such as the buffalo, antelope, big-horn sheep, coyote and the prairie dog. Top notch Walt Disney documentary for the entire family.\n\n**4711** _ **The Vanishing Riders**_ **** Spectrum, 1935. 58 min. D: Robert Hill. SC: Oliver Drake. With Bill Cody, Ethel Jackson, Bill Cody, Jr., Wally Wales, Budd Buster, Milburn Morante, Donald Reed, Francis Walker, Roger Williams, Bert Young, Buck Morgan, Colin Chase, Oscar Gahan, Bud Pope, Barney Beasley. A cowpoke and his young pal set out to round up a rustling gang harassing area ranchers. Average oater of note only for its teaming of the Bill Cody's, father and son, who wear skeleton outfits to scare the bad guys.\n\n**4712** _ **The Vanishing Westerner**_ **** Republic, 1950. 60 min. D: Philip Ford. SC: Bob Williams. With Monte Hale, Aline Towne, Paul Hurst, Roy Barcroft, Arthur Space, Richard Anderson, William Phipps, Don Haggerty, Dick Curtis, Rand Brooks, Edmund Cobb, Harold Goodwin, Bob Reeves, Art Dillard, Bob Burns, Cactus Mack, Dale Van Sickel, Dudley Rose. Two wanted cowpokes are sent by a sheriff to a rancher to get jobs, but the lawman is really a killer who plans to use them as a front for his robbery activities and then have them murdered. A complicated scenario and plenty of action keep this Monte Hale outing moving at a fast clip.\n\n**Poster for** _**The Vanishing Westerner**_ **(Republic, 1950).**\n\n**  \n**\n\n**4713** _ **Vanishing Wilderness**_ **** Pacific International, 1974. 90 min. Color. D: Arthur R. Dubs and Heinz Seilmann. With Rex Allen (narrator), Arthur R. Dubs, Heinz Seilmann, Fred Truslow. The flora and fauna of North American, from Florida to Alaska, are chronicled in this theatrical documentary. Recommended for fans of this type of fare; Rex Allen sings the title song.\n\n**4714** _ **The Vanquished**_ **** Paramount, 1953. 84 min. Color. D: Edward Ludwig. SC: Winston Miller, Frank Moss and Lewis R. Foster. With John Payne, Jan Sterling, Coleen Gray, Lyle Bettger, Willard Parker, Roy Gordon, John Dierkes, Charles Evans, Ellen Corby, Ernestine Barrier, Russell Gage, Leslie Kimmell, Voltaire Perkins, Sam Flint, Louis Jean Heydt, Freeman Morse, Richard Shannon, Karen Sharpe, Howard Joslin, Llewellyn Johnson, John Halloran, Harry Cody, William Berry, Major Sam Harris, Jack Hill, Richard Beedle, Richard Bartell, Brad Mora. Following the Civil War an ex\u2013Confederate returns home and works undercover to get the goods on corrupt local officials. Although well produced by the Pine-Thomas unit, this feature has a plot used too often to make it very interesting.\n\n_**Vendetta**_ see _**Pancho Villa**_ (1972)\n\n_**Vendetta de Zorro**_ see _**The Lone Rider**_ (1960)\n**4715** _ **Las Vengadoras Enmascaradas**_ (The Masked Female Avengers). Estudios American, 1963. 87 min. D: Federico Curiel. SC: Federico Curiel and Alfredo Ruanova. With Kitty de Hoyos, Dacia Gonzalez, Dagoberto Rodriguez, Jorge Russek, Eric del Castillo, Santanon, Pancho Cordoba. When a silver shipment robbery results in the murder of a woman's fiance, she and her sister plan to round up the outlaws. Fast paced Mexican Western, a sequel to _**Las Hermanas X**_ (q.v.).\n\n**4716** _ **Venganza Apache**_ (Apache Vengeance) **** Alameda Films, 1960. 78 min. D: Fernando Mendez. SC: Alfredo Salazar. With Rafael Baledon, Mauricio Garces, Abel Salazar, David Reynoso, Dacia Gonzalez, Carlos Nieto, Angel Di Stefani, Carlos Suarez, Guillermo Carter, Consuelo Oviedo, Ignacio Peon, Alfonso Torres, Eduardo Bonada, Armando Gutierrez. Three brothers rescue a woman captured by the Apaches and try to stop a white man from selling guns to the tribe. Pretty good Mexican Western produced by co-star Abel Salazar.\n\n**4717** _ **Venganza del Charro Negro**_ (Vengeance of the Black Cowboy) **** Clasa-Mohme, 1942. 85 min. D-SC: Raul de Anda. With Raul de Anda, Irma Rosado, Carlos Lopez Moctezuma, Amanda de Llano, Miguel Angel Ferriz, Joaquin Busquets, Agustin Isunge, Tito Junco, Armando Soto la Marina, Consuel Quiroz, Salvador Quiroz. While investigating the murder of a pregnant girl, the Black Cowboy comes across the suicide of another young woman, sees a connection between the two incidents and tries to solve the puzzle. Well done Mexican Western mystery made by producer-director-writer-star Raul de Anda.\n\n**4718** _ **La Venganza del Lobo Negro**_ (The Vengeance of the Black Wolf) **** Telecine, 1981. 90 min. Color. D: Rafael Romero Marchent. SC: Joaquin Romero Hernandez and Rafael Romero Marchent. With Fernando Allende, Maria Silva, Fernando Sancho, Christian Bach, Alvaro de Luna, Carlos Ballesteros, Esperanza Roy, Lola Forner, Barta Barry, Frank Brana, Julian Ugarte, Alfonso del Real, Carmen Roldan, Tomas Zori, Eduardo Calvo, Alejandro de Enciso, Arturo Alegro, Jose Maria Caffarel. A masked avenger, The Black Wolf, assists the people of Monterrey when his bitter enemy, an Army Colonel, over taxes them and takes their land. Fair sequel to _**El Lobo Negro**_ (q.v.), also called _**Duelo a Muerte**_ (Duel to Death).\n\n_**Vengeance**_ (1935) see _**Range Warfare**_\n\n_**Vengeance**_ (1944) see _**Valley of Vengeance**_\n\n**4719** _ **Vengeance**_ **** Crown International, 1965. 80 min. D: Dene Hilyard. SC: Alex Sharp and Ed Erwin. With William Thourlby, Melora Conway, Owen Pavitt, Donald Cook, Ed Cook, Byrd Holland, John Bliss, James Cavanaugh, Tiger Joe Marsh. A Confederate prisoner released by the Yankees after the Civil War kills one of the men who murdered his brother and finds himself hunted by the man's fiancee and family. Average low budget melodrama.\n\n**4720** _ **Vengeance**_ **** Metro-Goldwyn-Mayer\/EMI\/Cinevision Films, 1971. 100 min. Color. D: Anthony Dawson (Antonio Marghereti). SC: Antonio Marghereti and Renato Savino. With Richard Harrison, Claudio Camaso, Alan Collins (Luciano Pignozzi), Sheyla Rosin, Lee Burton, Werner Pochath, Paolo Gozlino, Alberto Dell'Acqua (Robert Widmark), Pedro Sanchez. When his friend is mutilated and murdered by an outlaw gang, a cowboy plans to take revenge. Exceedingly brutal Spaghetti Western issued in Italy in 1968 as _**Joko, Invoco Dio...e**_ _**Muori**_ (Call to Your God...and Die).\n\n_**Vengeance in the Saddle**_ see _**Bullets and Saddles**_\n\n_**Vengeance Is a Colt**_ see _**Return of Django**_\n\n_**Vengeance Is Mine**_ see _**A Bullet for Sandoval**_\n\n_**Vengeance of a Gunfighter**_ see _**To Hell You Preach**_\n\n**4721** _ **The Vengeance of Pancho Villa**_ **** Lacy International Films, 1966. 83 min. Color. D: Jose M. Elorrieta. SC: Gonzalo Asensio Rey and Ricardo Vazquez. With John Ericson, James Philbrook, Gustavo Rojo, Mara Cruz, Nuria Torray, Ricardo Palacios, Pastor Serrador. Believing soldiers murdered his parents, a man agrees to help Pancho Villa in stealing gold from the government. Violent and action laced Spanish production issued there as _**Los 7 de Pancho Villa**_ (The 7 of Pancho Villa) and also called _**The Treasure of Pancho Villa**_.\n\n**4722** _ **Vengeance of Rannah**_ **** Reliable, 1936. 59 min. D: Franklin Shamray (Bernard B. Ray). SC: Joseph O'Donnell. With Bob Custer, Rin Tin Tin, Jr., Victoria Vinton, John Elliott, Roger Williams, George Chesebro, Eddie Phillips, Ed Cassidy, Wally West, Snub Pollard, Oscar Gahan, Jimmy Aubrey, William McCall, Jack Hendricks, Allen Greer, Jack Evans, Clyde McClary, Denver Dixon. An insurance detective investigates a stage payroll robbery and finds the murdered driver with the dead man's dog holding the clue to his killing. Tacky affair supposedly based on a work by James Oliver Curwood.\n\n**4723** _ **Vengeance of the West**_ **** Columbia, 1942. 60 min. D: Lambert Hillyer. SC: Luci Ward. With Bill Elliott, Tex Ritter, Adele Mara, Frank Mitchell, Dick Curtis, Richard Fiske, Ted Mapes, Eva Puig, Jose Tortosa, Guy Wilkerson, Edmund Cobb, Eddie Laughton, Stanley Brown, John Tyrrell, Steve Clark, George Morrell, Al Haskell, Tom Smith, Jim Corey, Jack Tornek. After his family is massacred, rancher Joaquin Murietta begins raiding gold shipments and eventually teams with a ranger who has been sent to capture him and together they try to stop the gang responsible for the murders. Final pairing of Bill Elliott and Tex Ritter treads very lightly on history but Tex sings \"Along the Trail Somewhere\" and \"Only Yesterday.\" A remake of _**The Avenger**_ (1931) [q.v.].\n\n**4724** _ **Vengeance Trail**_ **** Filme Cinematografica, 1972. 98 min. Color. D: William Redford (Pasquale Squittieri). SC: Monica (Venturini) Felt and William Redford. With Leonard Mann, Ivan Rassimov, Elizabeth Eversfield, Klaus Kinski, Steffen Zacharias, Enzo Fiermonte, Salvatore Billa, Teodoro Corra, Yotanka, Giorgio Dolfin, Isabella Gjidotti, Stefano Oppedisano, Gianfranco Tamborra, Pietro Torrisi. Seeking revenge for the murder of his parents by Indians, a man kidnaps a tribal maiden, planning to sell her but ends up falling in love with the girl, and tries to save her when she is abducted by outlaws. This Italian film, released there as _**La Vendetta e un Piatto Che Si Serve Freddo**_ (Vengeance Is a Plate Served Cold), is long on action and violence.\n\n**4725** _ **Vengeance Valley**_ **** Metro-Goldwyn-Mayer, 1951. 83 min. Color. D: Richard Thorpe. SC: Irving Ravetch. With Burt Lancaster, Joanne Dru, Robert Walker, Sally Forrest, John Ireland, Carleton Carpenter, Ray Collins, Ted De Corsia, Hugh O'Brian, Will Wright, Grace Mills, James Hayward, James Harrison, Stanley Andrews, Glenn Strange, Paul E. Burns, Robert E. Griffin, Harvey B. Dunne, John McKee, Tom Fadden, Monte Montague, Al Ferguson, Roy Butler, Margaret Bert, Norman Leavitt, Dan White, Robert Wilke, Louis Nicoletti. A ranch foreman tries to protect his no-good foster brother until he fathers an illegitimate child and tries to palm hit off as his, thus leading to a showdown between the two men. Burt Lancaster's first Western is a stout affair and good viewing.\n\n**4726** _ **Vengeance Vow**_ **** Wrather Corporation, 1956. 75 min. Color. D: Earl Bellamy. SC: Doane Hoag, De Vallon Scott and Thomas Seller. With Clayton Moore, Jay Silverheels, Allen Pinson, Wayne Burson, Jim Bannon, Francis McDonald, James J. Griffith, Ewing Mitchell, Maurice Jara, Joel Ashley, Jerry Brown, Walt LaRue, Mauritz Hugo, Harry Strang, Don C. Harvey, Margaret Aldrich, Gary Murray, Eugenia Paul, Baynes Barron, Robert Homans. An escaped convict plans to murder the Lone Ranger and Tonto as they also try to help an ex-prisoner and stop an Indian uprising. Action filled TV movie from the episodes \"Courage of Tonto,\" \"A Message for Abe\" and \"Two Against Two\" from \"The Adventures of the Lone Ranger\" (ABC-TV, 1949\u201357).\n\n**4727** _ **Vera Cruz**_ **** United Artists, 1954. 94 min. Color. D: Robert Aldrich. SC: Roland Kibbee and James R. Webb. With Gary Cooper, Burt Lancaster, Denise Darcel, Cesar Romero, Sarita Montiel, George Macready, Ernest Borgnine, Henry Brandon, Charles (Bronson) Buchinsky, Morris Ankrum, James McCallion, Jack Lambert, Jack Elam, James Seay, Archie Savage, Charles Horvath, Juan Garcia. In 1866 Mexico two mercenaries agree to accompany a countess to Vera Cruz since she is carrying a gold shipment for Emperor Maximilian's forces. Nicely photographed (by Ernest Laszlo) but mundane melodrama.\n\n**4728** _ **Via Pony Express**_ **** Majestic, 1933. 60 min. D: Lewis D. Collins. SC: Lewis D. Collins and Oliver Drake. With Jack Hoxie, Marceline Day, Lane Chandler, Doris Hill, Julian Rivero, Charles K. French, Matthew Betz, Joe Girard, Ben Corbett, Yakima Canutt, Charles Le Moyne, Slim Whitaker, Bob Burns, Chuck Baldra, Frank Ellis, Olin Francis, Bud Pope. A pony express rider tries to help a woman whose land grant is coveted by outlaws. Poor Jack Hoxie series outing with the star overshadowed by Lane Chandler.\n\n**4729** _ **The Vigilante**_ **** Columbia, 1947. 15 Chapters. D: Wallace Fox. SC: George H. Plympton, Lewis Clay and Arthur Hoerl. With Ralph Byrd, Ramsay Ames, Lyle Talbot, George Offerman, Jr., Robert Barron, Hugh Prosser, Jack Ingram, Eddie Parker, George Chesebro, Edmund Cobb, Terry Frost, Frank Ellis, Frank Merlo, Bill Brauer, John Fostine, Ted Mapes, Al Ferguson, Bud Osborne, Wallace Fox, Lane Bradford, Emmett Lynn, Rusty Wescoatt, Pierce Lyden, Jack Chefe, Bob Duncan, Baynes Barron, Al Wyatt, Tex Palmer, George DeNormand, Ted Adams, Kermit Maynard, Knox Manning (narrator). A Western movie star, who also works as an undercover agent, goes to a dude ranch where a gang is after a mysterious string of pearls. The plot is outlandish but this is a fun cliffhanger, based on the _Action Comics_ character; reworked as _**Roar of the Iron Horse**_ (q.v.).\n\n**4730** _ **Vigilante Hideout**_ **** Republic, 1950. 60 min. D: Fred C. Brannon. SC: Richard Wormser. With Allan \"Rocky\" Lane, Eddy Waller, Virginia Herrick, Roy Barcroft, Cliff Clark, Don Haggerty, Paul Campbell, Guy Teague, Art Dillard, Chick Hannon, Bob Woodward. Ranchers plagued by a rash of cattle thefts call in a range detective to stop the rustlers. Fairly exciting outing in Allan Lane's \"Famous Westerns\" series.\n\n**4731** _ **Vigilante Terror**_ **** Allied Artists, 1953. 70 min. D: Lewis D. Collins. SC: Sid Theil. With Bill Elliott, Mary Ellen Kay, Myron Healey, Fuzzy Knight, I. Stanford Jolley, Henry Rowland, George Wallace, Zon Murray, Richard Avonde, Michael Colgan, Denver Pyle, Robert Bray, Al Haskell, John James, Stanley Price, Lee Roberts, Ted Mapes, Ray Jones. Masked vigilantes carry off a successful gold hijacking and put the blame on a storekeeper but a cowboy comes to his defense. The plot is nothing new but good acting and production values makes this Bill Elliott film a pleasing one.\n\n**4732** _ **The Vigilantes Are Coming**_ **** Republic, 1936. 12 Chapters. D: Mack V. Wright and Ray Taylor. SC: John Rathmell, Maurice Geraghty and Winston Miller. With Robert Livingston, Kay Hughes, Guinn Williams, Raymond Hatton, Fred Kohler, Robert Warwick, William Farnum, Robert Kortman, John Merton, Ray Corrigan, Lloyd Ingraham, William Desmond, Yakima Canutt, Tracy Layne, Bud Pope, Steve Clemente, Bud Osborne, John O'Brien, Henry Hall, Philip Armetta, Stanley Blystone, Joe De La Cruz, Fred Burns, Frankie Marvin, Wally West, Wes Warner, Ken Cooper, Frank Ellis, Jerome Ward, Al Taylor, Herman Hack, Jack Ingram, Jack Kirk, Pascale Perry, Jack Kinney, Bob Jamison, Len Ward, Vinegar Roan, Lloyd Saunders, John Slater, Tommy Coats, Sam Garrett. In 1840 a man returns to his California home to find the ranch has been taken over by a general who is using Cossacks to help him in exploiting the land's gold and forming an empire. Outstanding Republic serial with a very fine cast; based on the Rudolph Valentino feature _**The Eagle**_ (United Artists, 1925), set in Russia.\n\n**4733** _ **Vigilantes of Boomtown**_ **** Republic, 1947. 56 min. D: R.G. Springsteen. SC: Earle Snell. With Allan Lane, Bobby Blake, Martha Wentworth, John Dehner, Roscoe Karns, Roy Barcroft, Peggy Stewart, George Chesebro, Ted Adams, George Lloyd, Earle Hodgins, George Turner, Harlan Briggs, Budd Buster, Jack O'Shea, Tom Steele, Eddie Lou Simms, Bobby Barker, Pascale Perry, Herman Howlin. In 1897 in Carson City factions oppose the sanctioning of the heavyweight championship boxing bout between James J. Corbett and Bob Fitzsimmons with Red Ryder getting into the fracas to keep the peace. Weak \"Red Ryder\" series effort that does not even bother to re-stage the famous fight which was also the subject of _**City of Badmen**_ (q.v.).\n\n**4734** _ **Vigilantes of Dodge City**_ **** Republic, 1944. 54 min. D: Wallace Grissell. SC: Norman S. Hall and Anthony Coldeway. With Wild Bill Elliott, Bobby Blake, Alice Fleming, Linda Stirling, LeRoy Mason, Tom London, Hal Taliaferro, Kenne Duncan, Bud Geary, Stephen Barclay, Robert Wilkie, Stanley Andrews, Horace B. Carpenter, Post Park, Dale Van Sickel, Roy Bucko. In Dodge City a gang of outlaws try to destroy the Duchess's freight line but Red Ryder and Little Beaver come to her rescue. Fast paced and pleasant outing in the popular series based on Fred Harman's comic strip character.\n\n**4735** _ **The Vigilantes Return**_ **** Universal, 1947. 67 min. Color. D: Ray Taylor. SC: Roy Chanslor. With Jon Hall, Margaret Lindsay, Andy Devine, Robert Wilcox, Paula Drew, Jonathan Hale, Arthur Hohl, Wallace Scott, Joan (Shawlee) Fulton, Lane Chandler, George Chandler, Jack Lambert, Robert Wilke, Monte Montague, John Hart. A federal marshal is assigned to a lawless region where crooks try to implicate him in a murder. Average program feature enhanced by Cinecolor.\n\n**4736** _ **The Vigilantes Ride**_ **** Columbia, 1944. 55 min. D: William Berke. SC: Ed Earl Repp. With Russell Hayden, Dub Taylor, Bob Wills and The Texas Playboys, Shirley Patterson, Tristram Coffin, Jack Rockwell, Robert Kortman, Richard Botiller, Jack Kirk, Stanley Brown, Blackie Whiteford, John Cason, Herman Hack, Tex Cooper, Barney Beasley, Jack Evans, Silver Tip Baker, Rube Dalroy, Matty Roubert. Leaving the rangers when his brother is killed, a man pretends to be a bandit in order to infiltrate and bring in the gang. Russell Hayden fans will like this action feature complimented by Bob Wills and his musical gang.\n\n**4737** _ **Villa**_ **** 20th Century\u2013Fox, 1958. 72 min. Color. D: James B. Clark. SC: Louis Vittes. With Brian Keith, Cesar Romero, Margia Dean, Rodolfo Hoyos, Carlos Muzquiz, Mario Navarro, Ben Wright, Elisa Loti, Enrique Lucero, Rosenda Monteras, Felix Gonzales, Jose Espinoza, Alberto Gutierrez, Jorge Trevino, Lee Morgan, Jose Lopez, Jose Trowe, Jorge Russek. To find a better life for himself and his people, Pancho Villa turns to banditry and begins harassing Mexican soldiers. Mediocre accounting of the famous bandit's early years although Rodolfo Hoyos is good in the title role.\n\n**4738** _ **Villa Rides!**_ **** Paramount, 1968. 125 min. Color. D: Buzz Kulik. SC: Robert Towne and Sam Peckinpah. With Yul Brynner, Robert Mitchum, Charles Bronson, Grazia Buccella, Robert Viharo, Frank Wolff, Herbert Lom, Alexander Knox, Diana Lorys, Robert Carricart, Fernando Rey, Regina de Julian, Andres Monreal, Antonio Padilla Ruiz, John Ireland, Jill Ireland, Jose Maria Prada. An American gun runner, captured by the forces of Pancho Villa, joins the revolutionary in his assault against the Mexican government. There is more mayhem than melodrama in his over-wrought European Western imitation.\n\n**4739** _ **The Villain**_ **** Columbia, 1979. 93 min. Color. D: Hal Needham. SC: Robert O. Kane. With Kirk Douglas, Ann-Margret, Arnold Schwarzenegger, Paul Lynde, Ruth Buzzi, Jack Elam, Strother Martin, Robert Tessier, Foster Brooks, Mel Tillis, Jan Eddy, Laura Lizer Sommers, Ray Bickel, Mel Todd, Jim Anderson, Dick Dickinson, Ron Duffy, Earl Smith. An outlaw tries to kidnap a beautiful young woman who has a muscle bound protector. Fast moving, but absurd, Western comedy.\n\n**4740** _ **The Violent Men**_ **** Columbia, 1955. 96 min. Color. D: Rudolph Mate. SC: Harry Kleiner. With Glenn Ford, Barbara Stanwyck, Edward G. Robinson, Dianne Foster, Brian Keith, May Wynn, Warner Anderson, Basil Ruysdael, Lita Milan, Richard Jaeckel, James Westerfield, Jack Kelly, Willis Bouchey, Harry Shannon, Peter Hanson, Don C. Harvey, Robo Bechi, Carl Andre, James Anderson, Katharine Warren, Thomas Browne Henry, Frank Ferguson, Raymond Greenleaf, Edmund Cobb, William \"Bill\" Phipps, Ethan Laidlaw, Kenneth Patterson. A Civil War veteran opposes a ruthless land baron who is trying to take over a fertile valley to expand his vast holdings. Another psychological Western from the 1950s, although this one has a bit more action than others of its ilk.\n\n**4741** _ **The Violent Ones**_ **** Feature Film Corporation of America, 1967. 96 min. Color. D: Fernando Lamas. SC: Doug Wilson and Charles Davis. With Fernando Lamas, Aldo Ray, Tommy Sands, David Carradine, Lisa Gaye, Melinda Marx. The sheriff of a New Mexico town has trouble controlling the Mexican population when three men are apprehended as suspects in the rape and murder of a young girl. Low grade modern-day Western melodrama.\n\n**4742** _ **Virginia City**_ **** Warner Bros., 1940. 121 min. D: Michael Curtiz. SC: Robert Buckner. With Errol Flynn, Miriam Hopkins, Randolph Scott, Humphrey Bogart, Frank McHugh, Alan Hale, Guinn Williams, John Litel, Douglass Dumbrille, Moroni Olsen, Russell Hicks, Dickie Jones, Frank Wilcox, Russell Simpson, Victor Kilian, Charles Middleton, Monte Montague, George Regas, Paul Fix, Thurston Hall, Charles Trowbridge, Howard Hickman, Brandon Tynan, Charles Halton, Ward Bond, Sam McDaniel, Harry Cording, Trevor Bardette, Tom Dugan, Spencer Charters, George Reeves, Lane Chandler, Reed Howes, Wilfred Lucas, Robert Homans, James Farley, DeWolfe (William) Hopper, Walter Miller, Henry Hall, Eddie Parker, Bud Osborne, Sam McDaniel, Roy Gordon, George Guhl, Philip Morris, Georgia Simmons, Wedgwood Newell, Davison Clark, Edward Keane, Claire Du Brey, Tom Dugan, Spencer Charters, Paul Fix, Max Hoffman, Jr., Normal Willis, Shirley Mills, Albert Russell. Escaping from a Southern prison camp, a Union soldier is sent to Virginia City to stop the shipment of gold to the Confederacy and falls for a pretty saloon singer who is a rebel spy. Pretty fair action drama based on historical fact.\n\n**4743** _ **The Virginian**_ **** Preferred Pictures, 1923. 60 min. D: Tom Forman. SC: Hope Loring and Louis D. Lighton. With Kenneth Harlan, Florence Vidor, Russell Simpson, Pat O'Malley, Raymond Hatton, Milton Ross, Sam Allen, Bert Hadley, Fred Gambold. A cowboy is forced to hang his best pal for stealing cattle and in doing so loses the affection of the girl he loves. Very good, and much underrated, second screen version (first filmed in 1914 by Paramount with Dustin Farnum) of Owen Wister's famous novel; well worth seeing.\n\n**4744** _ **The Virginian**_ **** Paramount, 1929. 90 min. D: Victor Fleming. SC: Edward E. Paramore, Jr. and Howard Estabrook. With Gary Cooper, Walter Huston, Richard Arlen, Mary Brian, Eugene Pallette, Chester Conklin, E.H. Calvert, Helen Ware, Victor Potel, Tex Young, Charles Stevens, Jack Pennick, George Chandler, Ernie Adams, Fred Burns, Randolph Scott. A cowboy fights with a bad man over a saloon girl and the crook later has the cowpoke's pal rustle cattle, causing him to be hanged by his friend. Flavorful early sound adaptation of the Owen Wister play and book that still holds up today; Richard Arlen is especially good as the tragic Steve and Mary Brian is the fetching leading lady.\n\n**4745** _ **The Virginian**_ **** Paramount, 1946. 87 min. Color. D: Stuart Gilmore. SC: Frances Goodrich and Albert Hackett. With Joel McCrea, Brian Donlevy, Sonny Tufts, Barbara Britton, Fay Bainter, Tom Tully, Henry O'Neill, Bill Edwards, William Frawley, Paul Guifoyle, Marc Lawrence, Vince Barnett, Stanley Andrews, Martin Garralaga, Willard Robertson, Ann Carter, Nana Bryant, James Burke, Esther Howard, Minerva Urecal, Josephine Whittel, Dorothy Ann Seese, Syd Saylor, Frances Morris, Paul Hurst, Dick Curtis, Robert Kortman, Al Ferguson, Wally West, Hank Bell, Teddy Infuhr, Harry Hayden, Arthur Loft, Edgar Dearing, Joseph Crehan, Buzz Henry, Betty Farrington, Perc Launders, Paul Hurst, Blackie Whiteford, Harry Lamont, Freddie Chapman, Larry Lawson, Tom Dillon, Jimmie Dundee, Al Murphy. A cowboy falls for a pretty girl but loses her when he hangs his pal after he joins a bad man in rustling cattle. A good cast and Technicolor cannot bring this fourth screen version of the novel above average. Owen Wister's 1902 work also spawned two television series, \"The Virginian\" (NBC-TV, 1962\u201370) and its sequel, \"The Men from Shiloh\" (NBC-TV, 1970\u201371).\n\n**4746** _ **The Virginian**_ **** Turner Network Television (TNT), 2000. 95 min. Color. D: Bill Pullman. SC: Larry Gross. With Bill Pullman, Diane Lane, John Savage, Harris Yulin, Colm Feore, James Drury, Gary Farmer, William MacDonald, Brent Strait, Sheila Moore, Darcy Beisher, Philip Granger, Dennis Weaver, Dawn Greenhalgh, Norman Edge, James Rattai, Mark Anderson, Maureen Rooney, Bill Merasty, Tom Glass, Joe Dodds, Stephen Hair, Marty Anthony, Keith Frey, Maesa Pullman, Arnold Lawson, Larry Austin, Tom Morris, Ben Tone, Dillinger Steele, Iloe Flewelling, Christopher Benson. Betrayed by his best friend, a cowpoke stands to lose the woman he loves if he brings in his pal for cattle rustling. A somewhat reworked version of the Owen Wister novel but still enjoyable.\n\n**4747** _ **Viva Cisco Kid**_ **** 20th Century\u2013Fox, 1940. 70 min. D: Norman Foster. SC: Samuel G. Engel and Hal Long. With Cesar Romero, Jean Rogers, Chris-Pin Martin, Minor Watson, Stanley Fields, Nigel de Brulier, Harold Goodwin, Francis Ford, Charles Judels, Harrison Greene, LeRoy Mason, Tom London, Jim Mason, Hank Worden, Eddy Waller, Ray Teal, Bud Osborne, Paul Sutton, Mantan Moreland, Paul Kruger, Willie Fung, Frank Darrien, Jacqueline Dalya, Margaret Martin, Inez Palange. The Cisco Kid and his pal Gordito are on the trail of an outlaw gang wanted for robbery. Well made and entertaining \"Cisco Kid\" series film.\n\n_**Viva! Django**_ see _**Man Called Django**_\n\n_**Viva la Revolucion**_ see _**Blood and Guns**_\n\n**4748** _ **Viva Maria!**_ **** United Artists, 1965. 114 min. Color. D: Louis Malle. SC: Louis Malle and Jean-Claude Carriere. With Brigitte Bardot, Jeanne Moreau, George Hamilton, Claudio Brook, Paulette Dubost, Gregor von Rezzori, Poldo Benani, Carlos Lopez Moctezuma, Luis Rizo. Two beautiful entertainers team with an Irish rebel to assist the revolutionary cause in Mexico. There is a pleasant blend of action and comedy in this French production.\n\n**4749** _ **Viva Max!**_ **** Commonwealth United, 1969. 96 min. Color. D: Jerry Paris. SC: Elliott Baker. With Peter Ustinov, Pamela Tiffin, Jonathan Winters, John Astin, Keenan Wynn, Harry Morgan, Alice Ghostley, Kenneth Mars, Morgan Guilford, Bill McCutcheon. A Mexican general, with a small band of confederates, tries to retake the Alamo 163 years after the initial siege. Forced comedy has its amusing moments thanks to an excellent cast.\n\n**4750** _ **Viva Villa!**_ **** Metro-Goldwyn-Mayer, 1934. 115 min. D: Jack Conway. SC: Ben Hecht. With Wallace Beery, Fay Wray, Leo Carrillo, Donald Cook, Stuart Erwin, George E. Stone, Joseph Schildkraut, Henry B. Walthall, Katherine De Mille, David Durand, Phillip Cooper, Frank Puglia, John Merkel, Charles Stevens, Steve Clemente, Pedro Regas, Carlos De Valdez, George Regas, Harry Cording, Nigel De Brulier, Charles Requa, Tom Ricketts, Clarence Wilson, James Martin, Anita Gordiana, Francis McDonald, Harry Semels, Julian Rivero, Dan Dix, Mischa Auer, Belle Mitchell, John Davidson, Brandon Hurst, Leonard Mudie, Herbert Prior, Emile Chautard, Henry Armetta, Hector Sarno, Ralph (Francis X., Jr.) Bushman, Shirley Chambers, Clarence Wilson, Adrian Rosley, Francis McDonald, Steve Clemente, Gino Corrado, Belle Mitchell, Nigel De Brulier, George Irving, Nick De Ruiz, Arthur Thalasso, Leo White, Claire DuBrey, Anita Gordiano, Sam Godfrey. Pancho Villa rises from petty bandit to revolutionary leader in Mexico. Overlong but passable biopic with a delightfully hammy performance by Wallace Beery in the title role.\n\n**4751** _ **Viva Zapata!**_ **** 20th Century\u2013Fox, 1952. 113 min. D: Elia Kazan. SC: John Steinbeck. With Marlon Brando, Jean Peters, Anthony Quinn, Joseph Wiseman, Arnold Moss, Alan Reed, Margo, Lou Gilbert, Harold Gordon, Mildred Dunnock, Frank Silvera, Nkina Varela, Florenz Ames, Fay Roope, Will Kuluva, Bernie Gozier, Frank De Kova, Pedro Regas, Richard Garrick, Ross Bagdasarian, Leonard George, Abner Biberman, Philip Van Zandt, Henry Silva, Guy Thomajan, George J. Lewis, Peter Mamakos, Ric Roman, Nestor Paiva, Robert Filmer, Julia Montoya, Salvador Baquez. A peasant, helped by his brother, rises above his surroundings and leads a revolution against the government of Mexico. Pretty fair big screen biography of Emiliano Zapata that will satisfy Marlon Brando fans.\n\n_**La Volpe**_ see _**Zorro the Fox**_\n\n**4752** _ **La Vuela del Charro Negro**_ (The Flight of the Black Cowboy) **** Clasa-Mohme, 1941. 90 min. D-SC: Raul de Anda. With Carmen Conde, Raul de Anda, Fernando Fernandez, Jose Torvay, Max Langler, Gilberto Gonzalez, Armando Soto la Marina, Agustin Isunza, Antonio R. Frausto. While visiting his late wife's resting place the Black Cowboy learns a series of grave robberies have taken place in the area and he traces them to a mad scientist who is using corpses in trying to learn the secret of life. Horror Western fans will like this early Mexican excursion into the genre by producer-director-writer-star Raul de Anda.\n\n**4753** _ **The Wackiest Wagon Train in the West**_ **** Topar Films, 1976. 86 min. Color. D: Jack Arnold, Earl Bellamy, Bruce Bilson and Oscar Rudolph. SC: Ron Friedman, Howard Ostroff, Brad Radnitz, Elroy Schwartz and Sherwood Schwartz. With Bob Denver, Forrest Tucker, Jeannine Riley, Lori Saunders, Ivor Francis, Lynn Wood, Bill Cort, Eddie Little Sky, Ernesto Esparza III, Donald Barry, Buck Young, Jim (James) Gammon, Dennis Fimple, John Quade, Dick Peabody, Taylor Lacher, James Jeter. Leading settlers West, a wagon master and his lame brained assistant get lost in the wilderness. Uninteresting Western rehash of \"Gilligan's Island\" (CBS-TV, 1964\u201367) made from episodes of \"Dusty's Trail\" (Syndicated, 1973\u201374).\n\n**4754** _ **Waco**_ **** Monogram, 1952. 68 min. D: Lewis D. Collins. SC: Dan Ullman. With Bill Elliott, Pamela Blake, Rand Brooks, I. Stanford Jolley, Richard Avonde, Stanley Andrews, Paul Pierce, Lane Bradford, Pierce Lyden, Terry Frost, Michael Whalen, Stanley Price, Ray Bennett, House Peters, Jr., Ray Jones, Ed Cassidy, Russ Whiteman, John Hart, Rory Mallinson, Ted Mapes, Richard Paxton. After killing a dishonest gambler in a fair fight, a cowboy is forced to run when he escapes after being denied a proper trial. Sturdy Bill Elliott film with an entertaining and literate script.\n\n**4755** _ **Waco**_ **** Paramount, 1966. 85 min. Color. D: R.G. Springsteen. SC: Steve Fisher. With Howard Keel, Jane Russell, Brian Donlevy, Terry Moore, Wendell Corey, John Agar, Richard Arlen, John Smith, Gene Evans, Ben Cooper, Tracy Olsen, DeForrest Kelley, Anne Seymour, Robert Lowery, Willard Parker, Jeff Richards, Fuzzy Knight, Reg Parton, Read Morgan, Dan White, Barbara Latell, Russ McCubbin, King Johnson. A former gunslinger is hired to bring peace to a Wyoming community and finds his ex-girl is married to the local preacher. An excellent cast lends credence to this otherwise mediocre horse opera from producer A.C. Lyles.\n\n**4756** _ **The Wagon Master**_ **** Universal, 1929. 70 min. D: Harry Joe Brown. SC: Marion Jackson and Leslie Mason. With Ken Maynard, Edith Roberts, Frederick Dana, Tom Santschi, Al Ferguson, Jack Hanlon, Bobby Dunn, Frank Rice, Whitehorse. A cowboy takes command of a wagon train after crooks murder its master who opposed their boss' food monopoly. This Ken Maynard part-talkie is a pleasant affair, proving the star adapted well to the sound medium.\n\n**4757** _ **Wagon Tracks**_ **** Paramount-Artcraft, 1919. 66 min. D: Lambert Hillyer. SC: C. Gardner Sullivan. With William S. Hart, Jane Novak, Robert McKim, Lloyd Bacon, Leo Pierson, Bertholde Sprotte, Charles Arling, Billy Hamilton. When is brother is shot for supposedly trying to molest a young woman, a wagon master tries to get the truth as he leads settlers, including the girl and the men who did the killing, to their new homes. Well made and entertaining William S. Hart silent melodrama with an austere denouement.\n\n**4758** _ **Wagon Tracks West**_ **** Republic, 1943. 55 min. D: Howard Bretherton. SC: William Lively. With Wild Bill Elliott, George \"Gabby\" Hayes, Anne Jeffreys, Tom Tyler, Rick Vallin, Robert Frazer, Roy Barcroft, Bryant Washburn, Charles Miller, Tom London, Cliff Lyons, Jack Rockwell, Kenne Duncan, Minerva Urecal, Hal Price, Frank Ellis, Hank Bell, Bill Nestell, Jack Ingram, Jack O'Shea, Ray Jones, Curley Dresden, Frank McCarroll, Marshall Reed, Ben Corbett, Jack Montgomery, Tom Steele, Roy Butler, J.W. Cody, Dick Rush, Bob Burns, Pascale Perry, Tom Steele, Bill Hazlett. A crooked Indian agent tries to cheat a tribe of its lands and a cowboy attempts to help them but meets opposition from a medicine man. A superb supporting cast is one of the many plus factors in this good \"Wild Bill Elliott\" series outing.\n\n**4759** _ **Wagon Trail**_ **** Ajax, 1935. 55 min. D: Harry Fraser. SC: Monroe Talbot. With Harry Carey, Gertrude Messinger, Edward Norris, Earl Dwire, Roger Williams, John Elliott, Chief Thundercloud, Chuck Morrison, Lew Meehan, Francis Walker, Allen Greer, Silver Tip Baker, Richard Botiller. A lawman attempts to help his gambler son when he is falsely accused of murder. A bit slow but otherwise entertaining Harry Carey oater.\n\n**4760** _ **Wagon Train**_ **** RKO Radio, 1940. 59 min. D: Edward Killy. SC: Morton Grant. With Tim Holt, Martha O'Driscoll, Ray Whitley, Emmett Lynn, Bud McTaggart, Cliff Clark, Ellen Lowe, Wade Crosby, Ethan Laidlaw, Monte Montague, Carl Stockdale, Glenn Strange, Bruce Dane. The new owner of a wagon train is the target of a trading post operator who killed his father and wants to buy out the franchise. Tim Holt's first series film is a fine sagebrush yarn and a good start to his RKO tenure.\n\n**4761** _ **Wagon Train**_ **** Columbia, 1952. 61 min. D: George Archainbaud. SC: Gerald Geraghty. With Gene Autry, Pat Buttram, Gail Davis, Dick Jones, Gordon Jones, Harry Harvey, Henry Rowland, George J. Lewis, The Cass County Boys (Jerry Scoggins, Bert Dodson, Fred Martin), Gregg Barton, Pierce Lyden, Carlo Tircoli, Syd Saylor, Sandy Sanders. Special investigator Gene Autry joins a medicine show while looking for money taken in an Army payroll holdup. Not one of Gene's better outings although he does reprise \"Back in the Saddle Again.\"\n\n**4762** _ **Wagon Wheels**_ **** Paramount, 194. 57 min. D: Charles Barton. SC: Jack Cunningham, Charles Logan and Carl Buss. With Randolph Scott, Gail Patrick, Billy Lee, Monte Blue, Raymond Hatton, Jan Duggan, Leila Bennett, Olin Howlin, J.P. McGowan, James Marcus, Helen Hunt, Julian Madison, James N. \"Pop\" Kenton, Alfred Delcambre, John Marston, Sam McDaniel, Michael Visaroff, Howard Wilson, Colin Tapley, E. Alyn Warren, Pauline Moore, Earl Conert and The Singing Guardsmen, Clara Lou (Ann) Sheridan, Lew Meehan, Leila Bennett, Fern Emmett, Harold Goodwin, Eldred Tidbury. Three scouts lead settlers on a wagon train to Oregon while a half-breed tries to stop them in order to keep his fur trade. Entertaining remake of Zane Grey's _**Fighting Caravans**_ (q.v.) with stock footage from that feature.\n\n**4763** _ **Wagon Wheels Westward**_ **** Republic, 1945. 55 min. D: R.G. Springsteen. SC: Earle Snell. With Wild Bill Elliott, Bobby Blake, Alice Fleming, Linda Stirling, Roy Barcroft, Emmett Lynn, Jay Kirby, Dick Curtis, George J. Lewis, Bud Geary, Tom London, Kenne Duncan, George Chesebro, Tom Chatterton, Frank Ellis, Bob McKenzie, Jack Kirk, Jack Sparks, Cactus Mack, Tommy Coats, Pascale Perry, Frances Gladwin, Lucille Byron. When crooks try to sabotage the Duchess' stage line in an isolated area, Red Ryder and Little Beaver come to her defense. Well written \"Red Ryder\" segment.\n\n**4764** _ **Wagonmaster**_ **** RKO Radio, 1950. 86 min. D: John Ford. SC: Frank S. Nugent and Patrick Ford. With Ben Johnson, Joanne Dru, Harry Carey, Jr., Ward Bond, Charles Kemper, Alan Mowbray, Jane Darwell, Ruth Clifford, Russell Simpson, Kathleen O'Malley, James Arness, Fred Libby, Hank Worden, Mickey Simpson, Francis Ford, Cliff Lyons, Don Summers, Movita Castenada, Jim Thorpe, Chuck Hayward, The Sons of the Pioneers (Ken Curtis, Lloyd Perryman, Shug Fisher, Tommy Doss, Hugh Farr, Karl Farr). Two cowboys join a Mormon wagon train traveling across the plains, fighting Indians, outlaws and the harsh elements. Somber, leisurely paced melodrama, not one of John Ford's best but still worth viewing; it spawned \"Wagon Train\" (NBC-TV, 1957\u201362; ABC-TV, 1962\u201365) with Ward Bond repeating the wagon master role until his death in 1960.\n\n**4765** _ **Wagons East**_ **** TriStar, 1994. 107 min. Color. D: Peter Markle. SC: Mathew Carlson. With John Candy, Richard Lewis, John C. McGinley, Ellen Greene, Robert Picardo, Ed Lauter, Rodney A. Grant, William Sanderson, Melinda Culea, Russell Means, Charles Rocket, Ingrid Nuernberg, Tony Pierce, Robin McKey, Abe Benrubi, Marvin McIntyre, Jill Boyd, Chad Hamilton, Thomas F. Duffy, David Dunard, Stuart Grant, Thomas F. Duffy, Joel McKinnon Miller, Deerek Senft, Lochlyn Munro, Ethan Phillips, Gailard Sartain, Marti Wells, Rick Damazio, Bud Davis, Bill Daydodge, Randy Hall, William Tucker, Patrick Thomas O'Brien. When they fail at homesteading, a group of settlers hire an old, incompetent cowboy to lead them back to their original homes in the East. Tepid comedy dedicated to the memory of John Candy, who played the wagon master.\n\n**4766** _ **Wagons West**_ **** Monogram, 1952. 73 min. D: Ford Beebe. SC: Dan Ullman. With Rod Cameron, Peggie Castle, Noah Beery, Jr., Michael Chapin, Henry Brandon, Sara Haden, Frank Ferguson, Anne Kimball, Wheaton Chambers, Riley Hill, I. Stanford Jolley, Glenn Strange, Harry Strang, John Parrish, Charles Stevens, Almira Sessions, Harry Tyler, Effie Laird. While leading a wagon train from Missouri across the plains, a man discovers some of his passengers are selling guns to the Indians. Cheaply made but entertaining action effort utilizing stock footage from _**Fort Osage**_ (q.v.).\n\n**4767** _ **Wagons Westward**_ **** Republic, 1940. 70 min. D: Lew Landers. SC: Joseph M. March and Harrison Jacobs. With Chester Morris, Anita Louise, Buck Jones, George \"Gabby\" Hayes, Guinn Williams, Ona Munson, Douglas Fowley, John Gallaudet, Virginia Brissac, Trevor Bardette, Selmer Jackson, Charles Stevens, James Conlin, Richard Cramer, Edmund Cobb, Horace B. Carpenter, Art Dillard, Tex Cooper, Tom Smith, Joe McGuinn, Bill Woolf, The Hull Twins. Siblings grow up on opposite sides of the law with the crooked one protected by a dishonest sheriff but when he is arrested his honest brother takes his place so he can round up an outlaw gang. A well made, adult Western with Chester Morris quite good in dual roles, although it is sad to see Buck Jones and Silver on the wrong side of the law.\n\n**4768** _ **Walk Like a Dragon**_ **** Paramount, 1960. 95 min. D: James Clavell. SC: James Clavell and Daniel Mainwaring. With Jack Lord, Nobu McCarthy, Mel Torme, James Shigeta, Josephine Hutchinson, Rodolfo Acosta, Benson Fong, Michael Pate, Don Kennedy, Donald Barry, Natalie Trundy, Lilyan Chauvin. After saving a Chinese girl from a prostitution ring, a man takes her to his Western hometown where she is snubbed by the locals. Passable melodrama dealing with racial prejudice.\n\n**4769** _ **Walk Tall**_ **** 20th Century\u2013Fox, 1960. 60 min. Color. D: Maury Dexter. SC: Stephen Kandel. With Willard Parker, Kent Taylor, Joyce Meadows, Russ Bender, Ron Soble, Alberto Monte, Bill Mims, Felix Locher, Dave De Paul. A renegade military man and his three cohorts murder women and children of the Shoshone Indian tribe with an Army captain assigned to bring them to justice. Despite its compact running time, this dual bill item is a trite affair.\n\n**4770** _ **Walk the Proud Land**_ **** Universal-International, 1956. 88 min. Color. D: Jesse Hibbs. SC: Gil Doud and Jack Sher. With Audie Murphy, Anne Bancroft, Patricia Crowley, Charles Drake, Tommy Rail, Jay Silverheels, Robert Warwick, Victor Milan, Anthony Caruso, Morris Ankrum, Addison Richards, Francis McDonald, John Pickard, Ed Hinton, Maurice Jara, George Keymas, Frank Chase, Eugene Iglesias, Mary Carrizosa. An Indian agent, wanting to bring peace to warring whites and Apaches, tries to capture Geronimo. Pretty good Audie Murphy action feature.\n\n**4771** _ **The Walking Hills**_ **** Columbia, 1949. 78 min. D: John Sturges. SC: Alan LeMay. With Randolph Soctt, Ella Raines, William Bishop, Edgar Buchanan, Arthur Kennedy, John Ireland, Jerome Courtland, Russell Collins, Charles Stevens, Houseley Stevenson, Reed Howes, Josh White, Ralph Dunn, Frank Yaconelli, Frank Marlo, John McKee, Jack Parker. A wanted killer, along with six men and a woman, rides into Death Valley in search of a wagon train carrying a fortune in gold lost in a sand storm years before. Underrated and very fine melodrama; well worth seeing.\n\n**4772** _ **Walking Thunder**_ **** Koan, 1997. 95 min. Color. D: Craig Clyde. SC: Craig Clyde and James Hennessy. With James Read, John Denver, David Tom, Klara Irene Miracle, Christopher Neame, Chief Ted Thin Elk, Kevin Conners, Billy Oscar, Don Shanks, Robert DoQui, Kasey Clyde, David Kirk Chambers, Carolyn Hurlburt, Duane Stephens, Wayne Brennan, Joseph Kelly Lockinland, Roy J. Cohoe, John Aspiras, Ivan N. Long, Bart the Bear, Brian Keith (narrator). Stranded in the Rocky Mountains, a boy and his family learn the ways of the wild from a mountain man, an Indian and a huge bear. Pleasant PG-rated family fare.\n\n**4773** _ **Wall Street Cowboy**_ **** Republic, 1939. 66 min. D: Joseph Kane. SC: Gerald Geraghty and Norman S. Hall. With Roy Rogers, George \"Gabby\" Hayes, Raymond Hatton, Ann Baldwin, Pierre Watkin, Craig Reynolds, Louisiana Lou, Ivan Miller, Reginald Barlow, Adrian Morris, Jack Roper, Jack Ingram, Hugh Sothern, Paul Fix, George Chesebro, Ted Mapes, Fred Burns, George (Montgomery) Letz. A cowboy tries to obtain money in the East to pay off the mortgage on his ranch since he suspects it contains a rich gold deposit. Despite its title and lack of continuing action, this Roy Rogers offering is pretty fair viewing.\n\n**4774** _ **The Walloping Kid**_ **** Awyon, 1926. 67 min. D-SC: Robert J. Horner. With Kit Carson (Boris Bullock), Dorothy Ward, Pauline Curley, Frank Whitson, Jack Richardson, Al Kaufman, Jack Herrick. When his father's ranch is threatened by outlaws, a boxer gives up the ring and infiltrates the rustling gang in order to bring them to justice. Tacky silent outing from Robert J. Horner Productions.\n\n**4775** _ **Wanda Nevada**_ **** United Artists, 1979. 107 min. Color. D: Peter Fonda. SC: Dennis Hackin. With Peter Fonda, Brooke Shields, Luke Askew, Fiona Lewis, Ted Markland, Severn Darden, Paul Fix, Henry Fonda, Larry Golden, John Demos, Bert Williams, Danny Zapien, Riley Hill. After a gambler wins a young girl in a poker game the two head for a sacred Indian burial ground to search for gold. There is not much here to recommend this 1950s-era oater except for Henry Fonda in a cameo as a grizzled prospector.\n\n**4776** _ **The Wanderer of the Wasteland**_ **** Paramount, 1935. 66 min. D: Otho Lovering. SC: Stuart Anthony. With Dean Jagger, Gail Patrick, Edward Ellis, Benny Baker, Larry \"Buster\" Crabbe, Trixie Friganza, Monte Blue, Raymond Hatton, Fuzzy Knight, Charles Waldron, Anna Q. Nilsson, Stanley Andrews, Pat O'Malley, Glenn (Leif) Erickson, Jim Thorpe, Tammany Young, Kenneth Harlan, Al St. John, Bud Osborne, Irving Bacon, Philo McCullough, Frank Lackteen, Bruce Mitchell, Alfred Delcambre, Marina Shubert, Chester Gan, Bob Burns, Hal Price, William Welsh, Jules Cowles, Lew Kelly, Brady Kline, Clarence L. Sherwood, Eddie Sturgis, Marian Mansfield, Maxine Reiner. A man who mistakenly thinks he murdered his brother during an argument heads into the desert where he meets and falls in love with a pretty girl after joining an outlaw gang preying on prospectors. Okay remake of the Zane Grey story first filmed by Paramount in 1924 in two-color Technicolor with Jack Holt, Billie Dove, Noah Beery and Kathlyn Willliams.\n\n**4777** _ **Wanderer of the Wasteland**_ **** RKO Radio, 1945. 67 min. D: Edward Killy and Wallace Grissell. SC: Norman Houston. With James Warren, Richard Martin, Audrey Long, Robert Barrat, Robert Clarke, Harry Woods, Minerva Urecal, Harry D. Brown, Tommy Cook, Harry McKim, Jason Robards, Dick Elliott, Myrna Dell, Sammy Blum, Budd Buster, Ethan Laidlaw, Sam Lufkin, Gordon Jones, Tanis Chandler, Nan Leslie, Allan Lee, Fred Aldrich, Larry Wheat, Cecil Stewart, Sam Shack, Lou Palfy. After his father is murdered a man grows up vowing to get revenge on the killer and he spends years tracking him down. Only fair third screen adaptation of Zane Grey's work, although leading lady Audrey Long is a knockout; James Warren's first series Western.\n\n**4778** _ **Wanderers of the West**_ **** Monogram, 1941. 58 min. D: Robert Hill. SC: Robert Emmett (Tansey). With Tom Keene, Betty Miles, Sugar Dawn, Arkansas Slim Andrews, Tom Seidel, Stanley Price, Gene Alsace, Tom London, Fred Hoose, James Sheridan (Sherry Tansey). A cowboy, on the trail of his father's killer, befriends a man not knowing he is the one he seeks. A somewhat complicated plot is about the only highlight of this otherwise average outing.\n\n**4779** _ **Wanted**_ **** Documento, 1969. 104 min. Color. D: Calvin J. Padget (Giorgio Ferroni). SC: Fernando Di Leo and Augusto Finocchi. With Giuliano Gemma, Teresa Gimpera, Serge Marquand, German Cobos, Daniele Vargas, Gia Sandri, Nello Pazzafini. Falsely branded an outlaw by a rustler, a lawman tries to clear his name by arresting the real culprit. Okay Spaghetti Western remake of _**Adios Gringo**_ (q.v.), which also starred Giuliano Gemma.\n\n**4780** _ **Wanted by the Law**_ **** Sunset, 1924. 51 min. D-SC: Robert North Bradbury. With J.B. Warner, Dorothy Woods, Jay Morley, Bill (William) McCall, Frank Rice, Tom Lingham, Jay Hunt, Billie Bennett, Ralph McCullough, Jacob Waldermyer. A cowboy takes the blame for a killing committed by his weakling brother and heads to Montana where he is accused of shooting his girl's uncle after the man's gold claim map is stolen. Action filled, low budget silent effort from producer Anthony J. Xydias.\n\n_**Wanted by the Law**_ (1944) see _**Dead or Alive**_\n\n**4781** _ **Wanted\u2014Dead or Alive**_ **** Monogram, 1951. 59 min. D: Thomas Carr. SC: Clint Johnson. With Whip Wilson, Jim Bannon, Fuzzy Knight, Christine McIntyre, Leonard Penn, Lane Bradford, Zon Murray, Marshall Reed, Stanley Price, Kenne Duncan, George DeNormand, William Fawcett, John Cason, Jack O'Shea, Ray Jones. A U.S. marshal and his two buddies are after a gang that captures and murders wanted men for the reward money. Better than average Whip Wilson series entry.\n\n**4782** _ **Wanted:**_ _**The Sundance Woman**_ **** ABC-TV\/20th Century\u2013Fox, 1976. 100 min. Color. D: Lee Phillips. SC: Richard Fielder. With Katharine Ross, Steve Forrest, Stella Stevens, Michael Constantine, Katherine Helmond, Hector Elizondo, Hector Elias, Warren Berlinger, Jorge Cervera, Lucille Benson. After the deaths of Butch Cassidy and the Sundance Kid, Etta Place finds herself a wanted woman and forms an uneasy alliance with Pancho Villa. Telefilm has Katharine Ross repeat the Etta Place role from _**Butch Cassidy and the Sundance Kid**_ (q.v.) but that is the only appeal in this mediocre outing.\n\n_**Wanted Woman**_ see _**Jessi's Girls**_\n\n**4783** _ **War Arrow**_ **** Universal-International, 1953. 78 min. Color. D: George Sherman. SC: John Michael Hayes. With Maureen O'Hara, Jeff Chandler, Suzan Ball, John McIntire, Charles Drake, Dennis Weaver, Noah Beery, Jr., Henry Brandon, Steve Wyman, Jim Bannon, Jay Silverheels, Brad Jackson, Lane Fuller, Bill Ward, Dee Carroll, Roy Whatley, Darla Ridgeway. When the Kiowa tribe threatens to engulf their Seminole neighbors, the U.S. cavalry sends a soldier to help the latter fight their traditional enemies. Although the plot of Indians vs. Indians is a bit different, the film is only average.\n\n**4784** _ **War Drums**_ **** United Artists, 1957. 75 min. Color. D: Reginald LeBorg. SC: Gerald Drayson Adams. With Lex Barker, Joan Taylor, Ben Johnson, Larry Chance, Richard Cutting, James Parnell, John Pickard, John Colicos, Tom Monroe, Jil Jarmyn, Jeanne Carmen, Mauritz Hugo, Ward Ellis, Fred Sherman, Paul Fierro, Alex Montoya, Stuart Whitman, Barbara Perry, Boyd \"Red\" Morgan. At the beginning of the Civil War, Apaches go on the warpath when gold seekers invade their lands and a cavalry officer is assigned to stop the trouble. Fairly exciting, well directed action feature.\n\n**4785** _ **War of the Range**_ **** Freuler\/Monarch, 1933. 59 min. D: J.P. McGowan. SC: Oliver Drake. With Tom Tyler, Caryl Lincoln, Charles K. French, Theodore (Ted) Adams, Lane Chandler, William Malan, Wesley Giraud, Fred Burns, Charles (Slim) Whitaker, Billy Franey, Lafe McKee, Frank Ellis. A cowboy sides with nesters against his rancher father, who at the behest of crooks, wants to start a range war in to fence off his spread. Static outdoor adventure from producer John R. Freuler. Also called _**War on the Range**_.\n\n**4786** _ **War of the Wildcats**_ **** Republic, 1943. 102 min. D: Albert S. Rogell. SC: Ethel Hill and Eleanore Griffin. With John Wayne, Martha Scott, Albert Dekker, George \"Gabby\" Hayes, Marjorie Rambeau, Dale Evans, Grant Withers, Sidney Blackmer, Paul Fix, Cecil Cunningham, Irving Bacon, Byron Foulger, Anne O'Neal, Richard Graham, Robert Warwick, Stanley Andrews, Will Wright, Harry Shannon, Emmett Vogan, Charles Arnt, Edward Gargan, Harry Woods, Tom London, Dick Rich, Slim Whitaker, LeRoy Mason, Lane Chandler, Arthur Loft, Bud Geary, Kenne Duncan, Hooper Atchley, Wade Crosby, George Chandler, Curley Dresden, Jack Kirk, Roy Barcroft, Yakima Canutt, Shirley Jean Rickert, Linda Scott, Bob Reeves, Jess Cavin, Pat Logan, Charles Agnew, Linda Brent, Rhonda Fleming, Fred Graham. A cowboy helps an Oklahoma Indian tribe in 1906 when a corrupt oil man wants to drill on their lands but keep the profits for himself. Brawling action melodrama that packs a lot of entertainment and is enhanced by a fine cast. Original title: _**In Old Oklahoma**_.\n\n_**War on the Range**_ see _**War of the Range**_\n\n**4787** _ **War Paint**_ **** United Artists, 1953. 90 min. Color. D: Lesley Selander. SC: Richard Alan Simmons and Martin Berkeley. With Robert Stack, Joan Taylor, Charles McGraw, Peter Graves, Keith Larsen, William Pullen, Richard Cutting, Douglas Kennedy, Walter Reed, Charles Nolte, James Farrell, Paul Richards, John Doucette, Robert Wilke. A madman tries to prevent the delivery of a peace treaty with Indians, first by murdering a commissioner and then leading a cavalry unit into an ambush. Pretty fair action drama.\n\n**4788** _ **War Party**_ **** 20th Century\u2013Fox, 1965. 72 min. D: Lesley Selander. SC: George Williams and William Marks. With Michael T. Mikler, Donald Barry, Davey Davison, Laurie Mack, Dennis Robertson, Charles Horvath, Guy Wilkerson, Michael Carr, Fred Krone. A rescue party tries reach an Army patrol under attack by warring Comanches. Standard, but effective, program feature.\n\n**4789** _ **War Party**_ **** TriStar, 1988. 99 min. Color. D: Franc Roddman. SC: Spencer Eastman. With Billy Wirth, Kevin Dillon, Tim Sampson, Jimmy Ray Weeks, Kevyn Major Howard, E. Emmet Walsh, Bill McKinney, Cameron Thor, Jerry Hardin. When a gun with bullets is brought to a modern-day re-enactment of a battle between the cavalry and Blackfoot Indians, a real fight results. A good premise goes awry in this pretentious \"social\" drama.\n\n**4790** _ **The War Wagon**_ **** Universal, 1967. 101 min. Color. D: Burt Kennedy. SC: Clair Huffaker. With John Wayne, Kirk Douglas, Howard Keel, Robert Walker, Keenan Wynn, Bruce Cabot, Valora Noland, Gene Evans, Joanna Barnes, Bruce Dern, Terry Wilson, Don Collier, Sheb Wooley, Ann McCrea, Emilio Fernandez, Frank McGrath, Chuck Roberson, Boyd \"Red\" Morgan, Hal Needham, Marco Antonio Arzate, Perla Walter. Framed for a crime he did not commit, a man is released from prison and joins forces with an ex-employee of the crook who stole his gold rich lands, and together they plot to steal his armor-plated war wagon. Action filled, light hearted and amusing drama which will delight fans of John Wayne and Kirk Douglas.\n\n**4791** _ **Warlock**_ **** 20th Century\u2013Fox, 1959. 122 min. Color. D: Edward Dmytryk. SC: Robert Alan Arthur. With Richard Widmark, Henry Fonda, Anthony Quinn, Dorothy Malone, Dolores Michaels, Wallace Ford, Tom Drake, Richard Arlen, DeForrest Kelley, Regis Toomey, Vaughn Taylor, Don Beddoe, Whit Bissell, J. Anthony Hughes, Donald Barry, Frank Gorshin, Ian MacDonald, Robert Osterloh, Mickey Simpson, James Philbrook, Robert Adler, Sam Gross, Ann Doran, Bartlett Robinson. A lawman tries to bring peace to a frontier town with the help of a gambler and a member of the gang he has to combat. Although no classic, this feature is entertaining and will appeal to fans of its stars.\n\n**4792** _ **Warpath**_ **** Paramount, 1951. 95 min. Color. D: Byron Haskin. SC: Frank Gruber. With Edmond O'Brien, Dean Jagger, Forrest Tucker, Polly Bergen, Harry Carey, Jr., James Millican, Wallace Ford, Paul Fix, Louis Jean Heydt, Paul Lees, Walter Sande, Charles Dayton, Robert Bray, Douglas Spencer, James Burke, Chief Yowlachie, Monte Blue, Frank Ferguson, Cliff Clark, Charles Stevens, Paul Burns, John Hart, John Mansfield. A man attempts to track the three outlaws who killed the woman he loved and gets involved in an Indian attack. A superb cast helps this otherwise standard melodrama.\n\n**4793** _ **The Warrior's Way**_ **** Rogue, 2010. 100 min. Color. D-SC: Sngmoo Lee. With Dong-gun Jang, Kate Bosworth, Geoffrey Rush, Danny Huston, Tony Cox, Lung Ti, Analin Rudd, Markus Hamilton, Rod Lousich, Matt Gillanders, Christina Asher, Jed Brophy, Carl Bland, Ian Harcourt, Tony Wyeth, Ryan Richards, Nic Sampson, Ashley Jones, Phil Grieve, Eddie Campbell, Ross Duncan, Makoto Murata, Chontelle Melgren, Cath Harkins, Neill Rea, Ken Smith, Youngmin Cho, Michael Deane, Ken McColl, David Austin, Bret Crozier. A Chinese swordsman takes an infant he is supposed to kill and flees to the American West where he opens a laundry, is attracted to a townswoman and forced to take up arms against outlaws who threaten her. Average East meets West action feature.\n\n_**Washington Cowboy**_ see _**Rovin' Tumbleweeds**_\n\n**4794** _ **Waterhole #3**_. Paramount, 1967. 100 min. Color. D: William Graham. SC: Joseph T. Steck and R.R. Young. With James Coburn, Carroll O'Connor, Margaret Blye, Claude Akins, Joan Blondell, James Whitmore, Timothy Carey, Bruce Dern, Harry Davis, Roy Jenson, Robert Cornthwaite, Jim Boles, Steve Whitaker, Ted Markland, Robert Crosse, Buzz Henry. A trio of Confederates rob the Union Army of a fortune in gold and bury it at a deserted waterhole with a gambler getting the map showing the location of the loot. Takeoff on genre films provides some amusing moments.\n\n**4795** _ **Way of a Gaucho**_ **** 20th Century\u2013Fox, 1952. 91 in. D: Jacques Tourneur. SC: Philip Dunne. With Gene Tierney, Rory Calhoun, Richard Boone, Hugh Marlowe, Everett Sloane, Enrique Chaico, Roland Dumas, Lidia Campos, John Henchley, Douglas Poole, Mario Abdah, John Paris. In 1875 a young gaucho and the girl he loves tries to find happiness together on the wild Argentine pampas. Turgid drama.\n\n**4796** _ **The Way of the West**_ **** First Division\/Superior, 1934. 52 min. D: Robert Emmett (Tansey). SC: Larry Berringer and Al Lane (Robert Emmett Tansey). With The American Rough Riders, Wally Wales, Myrla Bratton, William Desmond, \"Little\" Bobbie Nelson, Fred Baker, Jim Sheridan (Sherry Tansey), Art Mix, Billy Patton, Tex Jones, Harry Berry, Jimmy Aubrey, Helen Gibson, Gene Layman, Tiny Skelton. A federal investigator tries to stop cattlemen from using an outlaw to harass sheepherders leasing government land. This rawboned sagebrush quickie was Wally Wales' last starring effort and is recommended for his fans.\n\n_**Way of the West**_ (2011) see _**The Mountie**_\n\n**4797** _ **Way Out West**_ **** Metro-Goldwyn-Mayer, 1930. 80 min. D: Fred Niblo. SC: Byron Morgan, Alfred Block, Joe Farnham and Ralph Spence. With William Haines, Leila Hyams, Polly Moran, Francis X. Bushman, Jr., Cliff Edwards, Vera Marsh, Charles Middleton, Jack Pennick, Buddy Roosevelt, Jay Wilsey (Buffalo Bill, Jr.), Chief Yowlachie, Ann Dvorak, J.W. Cody. A sideshow barker cheats some cowboys and is forced to work off his debt on a ranch where he falls for the boss' daughter but is hated by her suitor. Badly dated William Haines comedy-drama; it is hard to understand the popularity of the star's brand of smart-aleck farce.\n\n**4798** _ **Way Out West**_ **** Metro-Goldwyn-Mayer, 1936. 66 min. D: James W. Horne. SC: Charles Rogers, Felix Adler and James Parrott. With Stan Laurel, Oliver Hardy, Sharon Lynne, James Finlayson, Rosina Lawrence, Stanley Fields, Jim Mason, James C. Morton, Frank Mills, Dave Pepper, Vivien Oakland, Harry Bernard, Mary Gordon, May Wallace, The Avalon Boys (Chill Wills, Art Green, Walter Trask, Don Brookins), Jack Hill, Sam Lufkin, Tex Driscoll, Flora Finch, Fred \"Snowflake\" Toones, Bobby Dunn, John Ince, Fritzi Brunette, Frank Montgomery, Fred Cady, Eddie Borden, Helen Holmes, Lester Dorr, Ham Kinsey, Art Mix, Ben Corbett, Buffalo Bill, Jr., Bill Woolf, Cy Clocum, Dinah (mule). Two bunglers arrive in a Western town to give their late partner's daughter the deed to a gold mine but a saloon owner tries pass off his pretty partner as the deserving girl. Laurel and Hardy comedy classic that is a delight from start to finish; also available in a colorized version.\n\n**4799** _ **The Way to Gold**_ **** 20th Century\u2013Fox, 1957. 94 min. D: Robert D. Webb. SC: Wendell Mayes. With Jeffrey Hunter, Sheree North, Barry Sullivan, Walter Brennan, Neville Brand, Jacques Aubuchon, Ruth Donnelly, Tom Pittman, Philip Ahn, Geraldo Mandia, Ted Edwards, Alan Jeffrey, Ken Scott, Jonathan Hole, Frank Mazzola, Blossom Rock (Marie Blake), Robert Adler, Edwin Jerome, Norman Leavitt, George Robotham, Rachel Stephens, Renny McEvoy, Jim Hayward, Mack Chandler, Harry Carter, Frank Eyman, Lou Hochstatter. A convict, who has fallen in love with a waitress, seeks hidden gold but finds himself at odds with the law and a vicious family. Well done modern-day chase thriller.\n\n**4800** _ **The Way West**_ **** United Artists, 1967. 122 min. Color. D: Andrew V. McLaglen. SC: Ben Maddow and Mitch Lindemann. With Kirk Douglas, Robert Mitchum, Richard Widmark, Lola Albright, Michael Witney, Sally Field, Katherine Justice, Stubby Kaye, William Lundigan, Paul Lukather, Roy Barcroft, Jack Elam, Patric Knowles, Ken Murray, John Mitchum, Nick Cravat, Harry Carey, Jr., Roy Glenn, Anne Barton, Eve McVeagh, Peggy Stewart, Stefan Arngrim, Hal Lynch, Timothy Scott, Gary Morris, Eddie Little Sky, Michael Keep, Clarke Gordon, Mitchell Schollars, Jack Coffer, Everett Creach, Jim Burk, Gary McLarty, Paul Wexler. An aging trail scout helps a widowed senator lead a wagon train across the northwest in the 1840s. Although this drama contains all the ingredients for being good fare it is sadly vapid.\n\n_**Welcome Stranger**_ see _**Across the Sierras**_\n\n**4801** _ **Welcome to Blood City**_ **** EMI, 1977. 96 min. Color. D: Peter Sasdy. SC: Stephen Scheck and Michael Winder. With Jack Palance, Keir Dullea, Samantha Eggar, Barry Morse, Hollis McLaren, Chris Wiggins, Allan Royale, Ken James, Henry Ramer, John Evans, Larry Reynolds, Al Bernardo, Larry Benedict, Chuck Shamata, Gary Reineke, Jack Creley, Steve Pernie, Alan Crofoot, Calvin Butler, Mina E. Mina, Lloyd White. Several people are tested by a research group to see how well they can survive in the environment of the Old West. A silly premise makes for an anemic, disappointing movie\u2014watch _**Westworld**_ (q.v.) instead. Also called _**Blood City**_.\n\n**4802** _ **Welcome to Hard Times**_ **** Metro-Goldwyn-Mayer, 1967. 103 min. Color. D-SC: Burt Kennedy. With Henry Fonda, Janice Rule, Aldo Ray, Keenan Wynn, Janis Paige, John Anderson, Warren Oates, Lon Chaney, Edgar Buchanan, Fay Spain, Denver Pyle, Michael Shea, Arlene Golonka, Royal Dano, Alan Baxter, Paul Birch, Don Ferrone, Paul Fix, Elisha Cook, Kalen Liu, Ann McCrea, Bob Terhune, Ron Burke. A sadistic killer terrorizes and destroys a frontier town with the survivors trying to rebuild their community. A dark, violent, well done melodrama with some interesting performances from its top notch cast, including saloon owner Lon Chaney, victim Paul Birch and Aldo Ray as the terrifying Big Bad Man from Bodie. British title: _**Killer on a Horse**_.\n\n**4803** _ **Wells Fargo**_ **** Paramount, 1937. 116 min. D: Frank Lloyd. SC: Paul Schoefield, Gerald Geraghty and Frederick Jackson. With Joel McCrea, Frances Dee, Bob Burns, Lloyd Nolan, Johnny Mack Brown, Henry O'Neill, Porter Hall, Robert Cummings, Ralph Morgan, Mary Nash, Barlowe Borland, Stanley Fields, Lane Chandler, Clarence Kolb, Jack Clark, Frank McGlynn, Granville Bates, Peggy Stewart, Bernard Siegel, Harry Davenport, Henry Brandon, Lane Chandler, Frank Conroy, Edward Earle, Lucien Littlefield, David Durand, Erville Alderson, I. Stanford Jolley, Syd Saylor, Ernie Adams, Willie Fung, Hank Bell, Paul Newlan, Brandon Tynan, Scotty Beckett, Dorothy Tennant, Jerry Tucker, Louis Natheaux, Jimmy Butler, Jane Dewey, Hal K. Dawson, Sheila Darcy, Spencer Charters, Robert Emmett O'Connor, Archie Twitchell, Herbert Heywood, Helen Dickson, Francis Sayles, Lee Shumway, George Ovey, Lowell Drew, Bert Lindley, Sidney D'Albrook, Bob McKenzie, Paul Kruger, Edgar \"Blue\" Washington, Ernie Adams, Richard Cramer, Merrill McCormick, Bruce Mitchell, Jack Baxley, Julian Rivero, Frank Austin, Harry Hayden, Bert Moorhouse, Sherry Hall, Ferdinand Munier, Fern Emmett, James Burtis, Gwen Kenyon, Philip Kieffer, Eddie Dunn, Ruth Warren, Ed LeSaint, Fritzi Brunette, Alphonse Martell, Buddy Roosevelt, Wade Boteler, Del Henderson, Chester Gan, Harry Woods, Al Ferguson, Jack Perrin, Hal Taliaferro, D'Arcy Corrigan, Arthur Aylesworth, Lee Phelps, Franklin Parker, Sam Ash, Monte Vandergrift, Ben Hendricks, Edward Keane, Kathryn Sheldon, Monte Montague, Edwin Brady, Barney Furey, George Guhl, Darby Jones, Harry Semels, Forbes Murray, Cyril Ring, Eric Mayne, Nell Craig, Ethel Clayton, Ann Evers, Dorothy Stevens, Cy Schindell, Frank Mills, Gus Glassmire, Earl Gunn, Lester Dorr, David Newall, Gertrude Astor, Harry Strang, Piertro Sosso, Jack Curtis, Oscar Rudolph. During the Civil War, a northern Wells Fargo employee believes his wife has betrayed him to her former suitor, a Confederate officer. Epic scale Western details the history of the Wells Fargo but overall is mediocre; some versions run 94 minutes.\n\n**4804** _ **Wells Fargo Gunmaster**_ **** Republic, 1951. 60 min. D: Philip Ford. SC: M. Coates Webster. With Allan \"Rocky\" Lane, Mary Ellen Kay, Chubby Johnson, Michael Chapin, Roy Barcroft, Walter Reed, Stuart Randall, William Bakewell, George Meeker, Anne O'Neal, James Craven, Jack Perrin, Forrest Taylor, Lee Roberts. When a series of robberies plague the Wells Fargo, a special investigator is called in to halt them and round up the gang responsible. Typically well made Allan Lane \"Famous Westerns\" vehicle.\n\n**4805** _ **The Werewolf**_ **** Columbia, 1956. 79 min. D: Fred F. Sears. SC: Robert Kent and James B. Gordon. With Don Megowan, Joyce Holden, Steven Ritch, Eleanore Tanin, Kim Charney, Harry Lauter, Larry J. Blake, Ken Christy, James Gavin, S. John Launer, George M. Lynn, George Cisar, Marjorie Stapp, Jean Charney, Jean Harvey, Don C. Harvey, Charles Horvath, Ford Stevens, Fred F. Sears (narrator). The denizens of a remote Western town are tormented by a man who has been turned into a werewolf by two mad scientists. Low budget, compact horror thriller that delivers the goods.\n\n_**The West Is Still Wild**_ see _**Mule Feathers**_\n\n**4806** _ **West of Abilene**_ **** Columbia, 1940. 57 min. D: Ralph Cedar. SC: Paul Franklin. With Charles Starrett, Marjorie Cooley, Bruce Bennett, The Sons of the Pioneers (Bob Nolan, Tim Spencer, Lloyd Perryman, Pat Brady, Hugh Farr, Karl Farr), William Pawley, Don Beddoe, George Cleveland, Forrest Taylor, William Kellogg, Francis Walker, Eddie Laughton, Vester Pegg, Bud Osborne, Frank Ellis, John Tyrrell, Frank Austin, Frank LaRue, Milton Kibbee, George Ovey, Carl Sepulveda, Slim Hazel. Settlers who bought property from an irrigation company find themselves up against land grabbers who plan to re-sell the area for a big profit. Pretty fair Charles Starrett action effort.\n\n**4807** _ **West of Broadway**_ **** Metro-Goldwyn-Mayer, 1931. 71 min D: Harry Beaumont. SC: Gene Markey. With John Gilbert, El Brendel, Madge Evans, Lois Morgan, Ralph Bellamy, Gwen Lee, John Miljan, Ralph Conroy, Hedda Hopper, Ruth Rennick, Willie Fung, Kermit Maynard, Buddy Roosevelt, Bob Reeves. Badly wounded in the war and rejected by his fiancee, a wealthy man marries a young woman while drunk and thinking she is only after his money he goes to his Arizona ranch but she follows, hoping to win his love. This downer added several nails in the coffin of John Gilbert's once illustrious screen career.\n\n**4808** _ **West of Carson City**_ **** Universal, 1940. 57 min. D: Ray Taylor. SC: Milton Raison, Sherman Lowe and Jack Bernhard. With Johnny Mack Brown, Bob Baker, Fuzzy Knight, Peggy Moran Harry Woods, Robert Homans, Roy Barcroft, Ted Wells, Charles King, Frank Mitchell, Al Hall, Edmund Cobb, Jack Roper, Jack Shannon, Ernie Adams, Kermit Maynard, Donald Kerr, The Notables Quartet, Dick Carter, Alan Bridge, Victor Potel. A rancher returns home to find a gold strike has been made in a nearby town and crooks have taken over the area. Entertaining and well made Johnny Mack Brown film.\n\n**4809** _ **West of Cheyenne**_ **** Syndicate, 1931. 56 min. D: Harry S. Webb. SC: Bennett Cohen and Oliver Drake. With Tom Tyler, Josephine Hall, Harry Woods, Ben Corbett, Robert Walker, Fern Emmett, Lafe McKee, Murdock MacQuarrie, Henry Roquemore, Lew Meehan, Slim Whitaker, Frank Ellis, Tex Palmer. A father and son framed for a crime set out to bring in the real culprits. Minor league Tom Tyler vehicle.\n\n**4810** _ **West of Cheyenne**_ **** Columbia, 1938. 53 min. D: Sam Nelson. SC: Ed Earl Repp. With Charles Starrett, Iris Meredith, The Sons of the Pioneers (Bob Nolan, Lloyd Perryman, Pat Brady, Hugh Farr, Karl Farr), Dick Curtis, Ed LeSaint, Edmund Cobb, Art Mix, John Tyrrell, Ernie Adams, Jack Rockwell, Tex Cooper, Frank Ellis, George Chesebro, Ed Peil, Sr., Richard Botiller, Al Haskell, Blackie Whiteford, Horace B. Carpenter, Barney Beasley. The new owner of a ranch finds the place nearly deserted due to mysterious raids by an outlaw gang and he and his pals try to capture them. Fine Charles Starrett feature.\n\n**4811** _ **West of Cimarron**_ **** Republic, 1942. 56 min. D: Lester Orleback. SC: Albert DeMond and Don Ryan. With Bob Steele, Tom Tyler, Rufe Davis, Lois Collier, James Bush, Guy Usher, Roy Barcroft, Budd Buster, Hugh Prosser, Cordell Hickman, John James, Bud Geary, Stanley Blystone, Mickey Rentschler, Eddie Dean, Nick Stewart, James Gillette, Tommy Coats, Sonny Bupp. In Texas after the Civil War three cowboys come across Union soldiers harassing the locals and try to stop them. Fairly exciting \"Three Mesquiteers\" entry that moves well but contains a rather dull plot.\n\n**4812** _ **West of Dodge City**_ **** Columbia, 1947. 57 min. D: Ray Nazarro. SC: Bert Horswell. With Charles Starrett, Smiley Burnette, Nancy Saunders, Fred F. Sears, Glenn Stuart, I. Stanford Jolley, George Chesebro, Robert Wilke, Nolan Leary, Steve Clark, Zon Murray, Marshall Reed, Tom Chatterton, Bud Osborne, Maudie Prickett, Mustard and Gravy (Frank Rice and Ernest L. Stokes), Jim Diehl. A crook trying to set up a phony power plant scheme murders a rancher for his land and wants the dead man's daughter to sell but the Durango Kid investigates the crime. Fair entry in the long running series that produced a sequel, _**Bonanza Town**_ (q.v.).\n\n**4813** _ **West of El Dorado**_ **** Monogram, 1949. 58 min. D: Ray Taylor. SC: Adele Buffington. With Johnny Mack Brown, Max Terhune, Reno Browne, Teddy Infuhr, Milburn Morante, Marshall Reed, William Norton Bailey, Terry Frost, Bud Osborne, Kenne Duncan, Bill Potter, Bob Woodward, Boyd Stockman, Artie Ortego. The kid brother of a murdered outlaw knows the location of the stolen money he hid and his cohorts are out to get it. Good plot, but this Johnny Mack Brown vehicle shows signs of wear despite a fine cast.\n\n**4814** _ **West of Nevada**_ **** Colony, 1936. 59 min. D: Robert Hill. SC: Rock Hawkey (Robert Hill). With Rex Bell, Joan Barclay, Al St. John, Steve Clark, Georgia O'Dell, Richard Botiller, Frank McCarroll, Forrest Taylor, Bob Woodward. A senator sends his son and a pal to investigate the thefts of gold from a mine on an Indian reservation. Passable Rex Bell film enhanced by the comedy of Al St. John as the hero's buddy, especially the scenes where he romances the ranch cook. Reissued by Grand National.\n\n**4815** _ **West of Pinto Basin**_ **** Monogram, 1940. 60 min. D: S. Roy Luby. SC: Earle Snell. With Ray Corrigan, John King, Max Terhune, Gwen Gaze, Tristram Coffin, Bud Osborne, George Chesebro, Jack Perrin, Carl Mathews, Dirk Thane, Phil Dunham, Richard Cramer, Jerry Smith, Budd Buster, Johnny Luther, Bud McClure. The Range Busters are after a gang whose holdups are causing money and supplies not to reach a dam construction site. Another fast paced episode in the popular series.\n\n**4816** _ **West of Rainbow's End**_ **** Monogram, 1938. 57 min. D: Alan James. SC: Stanley Roberts and Gennard Rea. With Tim McCoy, Kathleen Elliot, Walter McGrail, Frank LaRue, George Chang, Marry Carr, Ed Coxen, George Cooper, Robert Kortman, Jimmy Aubrey, Reed Howes, Ray Jones, Sherry Tansey, Slim Whitaker, Herman Hack, Ernie Adams, Hank Bell, Denver Dixon, Silver Tip Baker. When his foster father is murdered while investigating a series of train robberies, an ex-ranger comes out of retirement to capture the killers. Pretty good Tim McCoy vehicle.\n\n**4817** _ **West of Santa Fe**_ **** Columbia, 1938. 60 min. D: Sam Nelson. SC: Bennett Cohen. With Charles Starrett, Iris Meredith, Dick Curtis, Robert Fiske, The Sons of the Pioneers (Bob Nolan, Lloyd Perryman, Pat Brady, Hugh Farr, Karl Farr), LeRoy Mason, Edmund Cobb, Hank Bell, Richard Botiller, Edward Hearn, Ed LeSaint, Buck Connors, Bud Osborne, Clem Horton. Rustlers murder a rancher with a U.S. marshal arriving to help the dead man's daughter and capture the killers. Entertaining Charles Starrett outing.\n\n**4818** _ **West of Sonora**_ **** Columbia, 1948. 55 min. D: Ray Nazarro. SC: Barry Shipman. With Charles Starrett, Smiley Burnette, Steve Darrell, George Chesebro, Anita Castle, The Sunshine Boys (Eddie Wallace, Freddie Daniel, M.H. Richman, J.D. Sumner), Hal Taliaferro, Robert Wilke, Emmett Lynn, Lynn Farr, Lloyd Ingraham, Blackie Whiteford. A sheriff asks a pal to becomes his deputy and find a notorious outlaw and his gang who kidnapped the bad man's little granddaughter from a stage. Pretty good \"Durango Kid\" segment enhanced by a well written, mystery laden script.\n\n**4819** _ **West of Texas**_ **** Producers Releasing Corporation, 1943. 61 min. D-SC: Oliver Drake. With Dave O'Brien, Jim Newill, Guy Wilkerson, Frances Gladwin, Marilyn Hare, Henry Hall, Robert Barron, Tom London, Jack Rockwell, Jack Ingram, Charles King, Roy Butler, Art Fowler, Chuck Morrison, Victor Cox, Wally West, Carl Mathews, Chick Hannon, Hank Bell, Curley Dresden, Jack Tornek, Matty Roubert, Herman Hack, Carol Henry, George Morrell, Jack Evans, Rube Dalroy. Outlaws want a man's property and frame him on a murder charge but his daughter convinces a trio of rangers to help his cause. One of the better entries in \"The Texas Rangers\" series but still on the slow side. Reissued by Eagle-Lion in 1947 as _**Shootin' Irons**_ (40 minutes).\n\n**4820** _ **West of the Alamo**_ **** Monogram, 1946. 58 min. D: Oliver Drake. SC: Louise Rousseau. With Jimmy Wakely, Lee \"Lasses\" White, Iris Clive, Ray Whitley, Jack Ingram, Earl Cantrell, Betty Lou Head, Budd Buster, Eddie Majors, Billy Dix, Arthur Smith, Ted French, Ray Jones, Steven Keys, Marshall Reed, Artie Ortego, Roy Butler, George Turner, Jack Rivers, Jesse Ashlock, Billy Hamilton. A ranger works undercover to find who is behind a series of crimes. Sluggish Jimmy Wakely vehicle with songs.\n\n_**West of the Badlands**_ see _**The Border Legion**_ (1940)\n\n**4821** _ **West of the Brazos**_ **** Lippert, 1950. 59 min. D: Thomas Carr. SC: Ron Ormond and Maurice Tombragel. With James Ellison, Russell Hayden, Raymond Hatton, Fuzzy Knight, Betty (Julie) Adams, Tom Tyler, Stanley Price, Dennis Moore, George J. Lewis, John Cason, Bud Osborne, George Chesebro, Gene Roth, Jimmie Martin, Stephen Carr, Judith Webster, Cliff Taylor. Two cowboys try to stop a crook out to fleece a man of his oil rich land. Cheap but fast moving \"Irish Cowboys\" entry with the plot twist of having hero Russell Hayden being deaf throughout the film. TV title: _**Rangeland Empire**_.\n\n**4822** _ **West of the Divide**_ **** Monogram, 1934. 55 min. D-SC: Robert North Bradbury. With John Wayne, Virginia Brown Faire, Lloyd Whitlock, George Hayes, Billy O'Brien, Yakima Canutt, Lafe McKee, Blackie Whiteford, Earl Dwire, Dick Dickinson, Tex Palmer, Artie Ortego, Horace B. Carpenter, Wally Wales, Hal Price, Archie Ricks, Phillip Kieffer. A cowboy, finding the crook who years before murdered his parents and kidnapped his baby brother, pretends to be an outlaw to infiltrate the man's gang. Well paced \"Lone Star\" oater with a fine performance by Lloyd Whitlock as the dastardly Gentry.\n\n**4823** _ **West of the Law**_ **** Monogram, 1942. 60 min. D: Howard Bretherton. SC: Jess Bowers (Adele Buffington). With Buck Jones, Tim McCoy, Raymond Hatton, Harry Woods, Evelyn Cooke, Milt Moranti (Milburn Morante), Roy Barcroft, Bud McTaggart, George DeNormand, Jack Daley, Bud Osborne, Lynton Brent, Al Ferguson, Tom London, Eddie Parker, Artie Ortego, Warren Jackson, Tex Palmer, Foxy Callahan, Horace B. Carpenter, Augie Gomez, Chick Hannon. Three marshals work incognito in a town harassed by outlaw attacks on its newspaper and uncover a gold smuggling operation. The final entry in the popular \"Rough Riders\" series is a good one with an entertaining plot and lots of fast action.\n\n**4824** _ **West of the Pecos**_ **** RKO Radio, 1934. 68 min. D: Phil Rosen. SC: Milton Krims and John Twist. With Richard Dix, Martha Sleeper, Samuel S. Hinds, Fred Kohler, Louise Beavers, Maria Alba, Sleep 'n Eat (Willie Best), Pedro Regas, Russell Simpson, Irving Bacon, Maurice Black, G. Pat Collins, George Cooper. A cowboy fights lawlessness in post\u2013Civil War Texas. Well mounted adaptation of the Zane Grey novel; remade in 1945 (q.v.).\n\n**4825** _ **West of the Pecos**_ **** RKO Radio, 1945. 66 min. D: Edward Killy. SC: Norman Houston. With Robert Mitchum, Barbara Hale, Richard Martin, Thurston Hall, Rita Corday, Russell Hopton, Bill Williams, Bruce Edwards, Harry Woods, Perc Launders, Bryant Washburn, Philip Morris, Martin Garralaga, Sammy Blum, Robert Anderson, Italia De Nublia, Carmen Granada, Ariel Sherry, Virginia Wave, Henry Wills, Ethan Laidlaw, Jack Gargan, Allan Lee, Larry Wheat. When outlaws hold up a stagecoach carrying a rich meat packer and his pretty daughter, two cowpokes plan to capture the bad men. Well produced oater that sent Bob Mitchum on to bigger budget films.\n\n_**West of the Rockies**_ see _**Call of the Rockies**_\n\n**4826** _ **West of the Rio Grande**_ **** Monogram, 1944. 57 min. D: Lambert Hillyer. SC: Betty Burbridge. With Johnny Mack Brown, Raymond Hatton, Christine McIntyre, Dennis Moore, Lloyd Ingraham, Kenneth MacDonald, Frank LaRue, Art Fowler, Hugh Prosser, Edmund Cobb, Steve Clark, Jack Rockwell, Hal Price, John Merton, George Morrell, Bud Osborne, Al Ferguson, Robert Kortman, Pierce Lyden, Lynton Brent, Chick Hannon, Tommy Coats, Post Park, Foxy Callahan. Lawmen pose as a gunslinger and a teacher to work in a town where the head politician uses a gang to force citizens to give up their voting rights. Pretty good \"Nevada Jack McKenzie\" series offering.\n\n**4827** _ **West of Tombstone**_ **** Columbia, 1942. 59 min. D: Howard Bretherton. SC: Maurice Geraghty. With Charles Starrett, Russell Hayden, Cliff Edwards, Marcella Martin, Gordon DeMain, Clancy Cooper, Jack Kirk, Budd Buster, Tom London, Francis Walker, Ray Jones, Eddie Laughton, Lloyd Bridges, Art Mix, Alan Bridge, Steve Clark, Ernie Adams, George Morrell, Horace B. Carpenter, George Sherwood, Chuck Morrison, Rick Anderson. After a stagecoach robbery the citizens of a town believe Billy the Kid is still alive and doubting the sheriff opens his grave to find it empty. Action filled entry in the series teaming Charles Starrett and Russell Hayden.\n\n**4828** _ **West of Wyoming**_ **** Monogram, 1950. 57 min. D: Wallace Fox. SC: Adele Buffington. With Johnny Mack Brown, Gail Davis, Myron Healey, Dennis Moore, Stanley Andrews, Milburn Morante, Mary Gordon, Carl Mathews, Paul Cramer, John Merton, Holly Bane, Steve Clark, Frank McCarroll, Bud Osborne. A lawman is after an outlaw gang trying to keep settlers out of a newly opened territory. The plot is okay but this Johnny Mack Brown vehicle is a rather tired effort.\n\n**4829** _ **West to Glory**_ **** Producers Releasing Corporation, 1947. 61 min. D: Ray Taylor. SC: Elmer Clifton and Robert Churchill. With Eddie Dean, Roscoe Ates, Dolores Castle, Gregg Barton, Jimmie Martin, The Sunshine Boys (Eddie Wallace, Freddie Daniel, M.H. Richman, J.D. Sumner), Zon Murray, Alex Montoya, Ted French, Carl Mathews. Two cowpokes go after a couple of crooks who have stolen a man's gold and are also after his famous diamond. A dull Eddie Dean outing with an unfunny dream sequence where Soapy Jones (Roscoe Ates) imagines he is hero Dean. Film is partially saved by the star singing a trio of pleasant songs, including \"Cry, Cry, Cry.\"\n\n**4830** _ **Westbound**_ **** Warner Bros., 1959. 72 min. Color. D: Budd Boetticher. SC: Berne Giler. With Randolph Scott, Virginia Mayo, Karen Steele, Michael Pate, Andrew Duggan, Michael Dante, Wally Brown, John Day, Walter Barnes, Fred Sherman, Mack Williams, Ed Prentiss, Rory Mallinson, Rudi Dana, Tom Monroe, Jack Perrin, Buddy Roosevelt, Charles Morton, John Epper, Gary Epper, Kermit Maynard, Mary Boss, William A. Green, Jack E. Henderson, Felice Richmond, Creighton Hale, Gertrude Keeler, Walter Reed, Jack C. Williams, Gerald Roberts, John Hudkins. During the Civil War a Union officer is assigned to start a stage line to ship gold from California to help the Northern cause but he opposed by a rich rancher whose beautiful wife was once his girlfriend. Very well made and entertaining Randolph Scott feature.\n\n**4831** _ **Westbound Limited**_ **** Universal, 1937. 66 min. D: Ford Beebe. SC: Maurice Geraghty. With Lyle Talbot, Polly Rowles, Henry Brandon, Frank Reicher, Henry Hunter, William Lundigan, William Royale, Tom Steele, Charles Murphy, Monte Vandegrift, J.P. McGowan. Falsely accused of a crime and sent to prison, a railroad agent escapes to prove his innocence. Dandy \"B\" program feature.\n\n**4832** _ **Westbound Mail**_ **** Columbia, 1937. 54 min. D: Folmer Blangsted. SC: Frances Guihan. With Charles Starrett, Rosalind Keith, Edward Keane, Arthur Stone, Ben Welden, Alan Bridge, George Chesebro, Art Mix, Jack Rockwell, Edward Hearn, Ed LeSaint, Ed Peil, Sr., Francis Walker, Lew Meehan, Bill Patton, Fred Parker. An FBI agent masquerades as a mule skinner to help a woman whose property is sought by a miner who thinks his gold vein may extend onto her land. Top notch, well written Charles Starrett vehicle.\n\n**4833** _ **Westbound Stage**_ **** Monogram, 1939. 56 min. D: Spencer Gordon Bennet. SC: Robert Emmett (Tansey). With Tex Ritter, Muriel Evans, Reed Howes, Kenne Duncan, Nelson McDowell, Nolan Willis, Steve Clark, Tom London, Frank Ellis, Chick Hannon, Frank LaRue, Chester Gan, Hank Bell, Phil Dunham, Sherry Tansey, Wally West. After an outlaw gang massacres an Army patrol that included his cousin, a cowboy takes the job as guard of a stage carrying a gold shipment in order to round up the bandits. Pretty sturdy Tex Ritter outing with the star singing \"It's All Over Now\" and \"Trail to New Mexico.\"\n\n**4834** _ **Western Caravans**_ **** Columbia, 1939. 58 min. D: Sam Nelson. SC: Bennett Cohen. With Charles Starrett, Iris Meredith, The Sons of the Pioneers (Bob Nolan, Tim Spencer, Lloyd Perryman, Pat Brady, Hugh Farr, Karl Farr), Russell Simpson, Hal Taliaferro, Dick Curtis, Hank Bell, Sammy McKim, Edmund Cobb, Ethan Laidlaw, Glenn Strange, Edward Hearn, Steve Clark, Herman Hack, Charles Brinley, Sam Garrett, John Rand, Jack Montgomery. Rustlers attempt to cause range warfare between ranchers and incoming settlers with the local sheriff trying to maintain peace. Okay Charles Starrett action film.\n\n**4835** _ **The Western Code**_ **** Columbia, 1932. 60 min. D: J.P. McCarthy. SC: Milton Krims. With Tim McCoy, Nora Lane, Wheeler Oakman, Mathew Betz, Dwight Frye, Mischa Auer, Gordon DeMain, Bud Osborne, Emilio Fernandez, Chuck Baldra, Cactus Mack, Jack Kirk, Steve Clark, Artie Ortego, Hal Price. A cowboy tries to help a woman whose stepfather has stolen her ranch and plans to marry her and murder her brother. Pretty entertaining Tim McCoy opus of interest to horror film fans since Dwight Frye is cast as a pal of the hero who is framed by the villains.\n\n**4836** _ **Western Courage**_ **** Rayart, 1927. 43 min. D: Ben Wilson. SC: Leslie Curtis. With Dick Hatton, Elsa Benham, Robert Walker, Al Ferguson, Ed La Niece, George Kesterson (Art Mix), Cliff Lyons, Blackjack Ward. A cowboy loves a woman attracted to a dishonest man who plans to rustle her father's cattle and rob a bank. There is non-stop action in this compact silent feature from Ben Wilson Productions.\n\n**4837** _ **Western Courage**_ **** Columbia, 1936. 61 min. D: Spencer Gordon Bennet. SC: Nate Gatzert. With Ken Maynard, Geneva Mitchell, Charles K. French, Betty Blythe, Cornelius Keefe, Ward Bond, E.H. Calvert, Renee Whitney, Dick Curtis, Wally Wales, Bob Reeves, Bud McClure, Bart Carre, Wally West, Jack King, Buck Bucko, Roy Bucko, Arkansas Johnny. The foreman of a dude ranch falls for a guest, a spoiled rich girl who is being courted by a no-good and later held for ransom by outlaws. The story is silly but this Ken Maynard outing is fun anyway.\n\n**4838** _ **Western Cyclone**_ **** Producers Releasing Corporation, 1943. 62 min. D: Sam Newfield. SC: Patricia Harper. With Buster Crabbe, Al St. John, Marjorie Manners, Karl Hackett, Milton Kibbee, Glenn Strange, Charles King, Hal Price, Kermit Maynard, Frank Ellis, Frank McCarroll, Artie Ortego, Herman Hack, Al Haskell, Jack Ingram, Steve Clark, Lane Bradford, Lou Fulton, Charles Murray, Jr., Bert Dillard, Wally West, Hank Bell, Jack Evans, George Hazel, Barney Beasley, Jack Tornek, Robert Hill, Jimmy Aubrey, Art Dillard, Rube Dalroy, Victor Cox, Morgan Flowers, George Morrell, Lew Morphy. A woman, the leader of an outlaw gang, has herself kidnapped and then blames Billy Carson in an attempt to discredit the lawman. Passable, but low grade, entry in the PRC \"Billy Carson\" series; reissued by Eagle-Lion in 1947 in a 39-minute version called _**Frontier Fighters**_.\n\n**4839** _ **Western Frontier**_ **** Columbia, 1935. 56 min. D: Al Herman. SC: Nate Gatzert. With Ken Maynard, Lucille Browne, Nora Lane, Robert Henry, Otis Harlan, Frank Yaconelli, Harold Goodwin, Frank Hagney, Gordon Griffith, James Marcus, Tom Harris, Nelson McDowell, Frank Ellis, Art Mix, Slim Whitaker, William Gould, Dick Curtis, Budd Buster, Herman Hack, Horace B. Carpenter, Oscar Gahan. Using a medicine show as a front, a lawman arrives in a community trying to track an outlaw gang and learns his long lost sister is its leader. Ken Maynard wrote the story for this film, his first for Columbia, and overall it is quite good.\n\n**4840** _ **Western Gold**_ **** Principal\/20th Century\u2013Fox, 1937. 60 min. D: Howard Bretherton. SC: Forrest Barnes. With Smith Ballew, Heather Angel, LeRoy Mason, Ben Alexander, Howard Hickman, Alan Bridge, Bud Osborne, Victor Potel, Otis Harlan, Frank McGlynn, Horace Murphy, Tom London, Steve Clark, Paul Fix, Lew Kelly, Wesley Giraud, Ben Corbett. A Union officer is sent West by President Lincoln to find out why gold shipments are not reaching the East. Smith Ballew's first series vehicle, based on a Harold Bell Wright book, is a good one and nicely showcases the star's fine singing voice in patriotic and traditional songs.\n\n**4841** _ **Western Heritage**_ **** RKO Radio, 1948. 61 min. D: Wallace Grissell. SC: Norman Houston. With Tim Holt, Richard Martin, Nan Leslie, Lois Andrews, Tony Barrett, Richard Powers (Tom Keene), Harry Woods, Walter Reed, Jason Robards, Robert Bray, Perc Launders, Emmett Lynn, Monte Montague, Dick Rush, Bud Osborne, Chick Hannon, Rudy Sooter, Rita Lynn, Ralph Bucko. A cowpoke finds himself in double trouble, not only with outlaws but with the saloon girl he loves. Average Tim Holt effort.\n\n**4842** _ **Western Jamboree**_ **** Republic, 1938. 56 min. D: Ralph Staub. SC: Gerald Geraghty. With Gene Autry, Smiley Burnette, Jean Rouveral, Esther Muir, Frank Darien, Joe Frisco, Kermit Maynard, Jack Perrin, Jack Ingram, Margaret Armstrong, Harry Holman, Edward Raquello, Ray Teal, Frank Ellis, Eddie Dean, Davison Clark, Bentley Hewett, Kermit Maynard, George Wolcott. Gene Autry and his pals fix up a ranch for a woman so she will think her father, a broke prospector, is the owner while crooks want the land for its valuable helium deposits. Slight Gene Autry opus.\n\n**4843** _ **Western Justice**_ **** Supreme, 1935. 56 min. D-SC: Robert North Bradbury. With Bob Steele, Renee Bordon, Julian Rivero, Lafe McKee, Perry Murdock, Arthur Loft, Jack Cowell, Vane Calvert, Earl Dwire, Perry Murdock, Carmen LaRoux, Archie Ricks. A cowboy tries to help ranchers being forced off their spread by an outlaw gang. Exciting Bob Steele vehicle with one of the bad guys flayed alive, \u00e0 la _**The Black Cat**_ (Universal, 1934), plus a pleasing mystery element.\n\n**4844** _ **Western Mail**_ **** Monogram, 1942. 55 min. D: Robert Emmett Tansey. SC: Robert Emmett (Tansey) and Frances Kavanaugh. With Tom Keene, Jean Trent, Frank Yaconelli, LeRoy Mason, Glenn Strange, Fred Kohler, Jr., James Sheridan (Sherry Tansey), Gene Alsace, Karl Hackett, Tex Palmer. A U.S. marshal works undercover to find out who is behind an outlaw gang and he helps one of its members with an alibi so he can infiltrate their activities. Fair Tom Keene movie.\n\n**4845** _ **Western Pacific Agent**_ **** Lippert, 1950. 64 min. D: Sam Newfield. SC: Milton Raison. With Kent Taylor, Sheila Ryan, Robert Lowery, Mickey Knox, Morris Carnovsky, Sid Melton, Frank Richards, Dick Elliott, Anthony Jochim, Lee Phelps, Ted Jacques, Vera Marshe, Carla Martin, Margia Dean, Gloria Gray. After a railroad detective is murdered during a train robbery, a fellow agent hunts for the killer. Fairly good \"B\" program feature set in the modern-day West.\n\n**4846** _ **Western Racketeers**_ **** Aywon, 1935. 48 min. D: Robert J. Horner. SC: James P. Hogan. With Bill Cody, Edna Aslin, Wally Wales, George Chesebro, Richard Cramer, Bud Osborne, Frank Clark, Robert Sands, Tom Dwaine, Ben Corbett, Billy Franey, Gilbert \"Pee Wee\" Holmes, Wally West, Blackjack Ward, Budd Buster, Chuck Baldra, Johnny Luther, Gene Alsace, Jack Evans. A rancher is murdered when he tries to get the law to stop a gang from charging a toll on herds going to market and another cattleman investigates. Sluggish, arid Bill Cody vehicle from producer Robert J. Horner; its only asset is Brydon Baker's photography.\n\n**4847** _ **Western Renegades**_ **** Monogram, 1949. 58 min. D: Wallace Fox. SC: Adele Buffington. With Johnny Mack Brown, Max Terhune, Jane Adams, Riley Hill, Steve Clark, Marshall Bradford, Hugh Prosser, Marshall Reed, Constance Worth, James H. Harrison, Terry Frost, William Ruhl, Myron Healey, Milburn Morante, John Merton, Dee cooper, Chuck Roberson, Bill Potter, Lane Bradford. A U.S. marshal attempts to puzzle out the events surrounding the murder of a banker. A complicated plot somewhat helps this later Johnny Mack Brown effort, a remake of _**Down Texas Way**_ (q.v.).\n\n_**Western Terror**_ see _**Buzzy and the Phantom Pinto**_\n\n**4848** _ **Western Trails**_ **** Universal, 1938. 57 min. D: George Waggner. SC: Norton S. Parker. With Bob Baker, Marjorie Reynolds, Carlyle Moore, Jr., John Ridgely, Franco Casarro, Jack Rockwell, Bob Burns, Jack Kirk, Jimmy Phillips, Murdock MacQuarrie, Jack Ingram, Hank Worden, Forrest Taylor, Tex Palmer, Herman Hack, Oscar Gahan, Jack Montgomery. An undercover agent wants to put a stop to a vicious outlaw gang. Okay singing oater in the Bob Baker series; a loose remake of _**The Dawn Trail**_ (q.v.).\n\n**4849** _ **Western Union**_ **** 20th Century\u2013Fox, 1941. 93 min. Color. D: Fritz Lang. SC: Robert Carson. With Randolph Scott, Robert Young, Dean Jagger, Virginia Gilmore, John Carradine, Slim Summerville, Chill Wills, Barton MacLane, Russell Hicks, Victor Kilian, Minor Watson, George Chandler, Chief Big Tree, Chief Thundercloud, Dick Rich, Addison Richards, Irving Bacon, Harry Strang, Reed Howes, Tom London, Steve O'Brien, Cliff Clark, Charles Middleton, Arthur Aylesworth, Paul Burns, Francis Ford, Eddy Waller, Frank Ellis, Iron Eyes Cody, Russ Clark, Frank Ellis, James Flavin, Ralph Dunn, Frank Mills, Kermit Maynard, Herman Howlin, Tom Forman, Sid Jordan, George Plues, Hank Bell, Robert Clarke, Cecil Kellogg, Frank McGrath, J.W. Cody, John Epper, Merlyn Nelson. A former outlaw joins the crew stringing Western Union telegraph lines and becomes romantically involved with the sister of the project's chief engineer but has a rival in a fellow worker, a Harvard graduate. Well made but undistinguished historical feature.\n\n**4850** _ **The Westerner**_ **** Columbia, 1934. 58 min. D: David Selman. SC: Harold Shumate. With Tim McCoy, Marion Shilling, Joseph (Sawyer) Sauers, Hooper Atchley, Ed LeSaint, John Dilson, Eddie (Edmund) Cobb, Albert J. Smith, Harry Todd, Bud Osborne, Slim Whitaker, Merrill McCormick, Art Mix, Lafe McKee, Steve Clark, Hank Bell. A rancher learns his foreman and the local sheriff are in cahoots rustling cattle as they try to pin a murder charge on him. Complicated plot twists detract from this otherwise fine Tim McCoy outing which has a well staged pseudo-execution sequence.\n\n**4851** _ **The Westerner**_ **** United Artists, 1940. 99 min. D: William Wyler. SC: Jo Swerling and Niven Busch. With Gary Cooper, Walter Brennan, Doris Davenport, Fred Stone, Paul Hurst, Chill Wills, Charles Halton, Forrest Tucker, Tom Tyler, Arthur Aylesworth, Lupita Tovar, Julian Rivero, Lillian Bond, Dana Andrews, Roger Gray, Jack Pennick, Art Mix, Helen Foster, Trevor Bardette, Connie Leon, Charles Coleman, Lew Kelly, Heinie Conklin, Lucien Littlefield, Corbet Morris, Stanley Andrews, Henry Roquemore, Hank Bell, Dan Borgaze, William (Bill) Steele, Blackjack Ward, Jim Corey, Buck Moulton, Ted Wells, Joe De La Cruz, Frank Cordell, C.E. Anderson, Philip Connor, Bob Fleming, William Gillis, Buck Connors, Gertrude Bennett, Marie Layton, Bill Bauman, Aleth Hansen, Phil Tead, Miriam Sherwin, Annabelle Rousseau. A drifter falls in love with a homesteader's daughter and when her father is murdered by Judge Roy Bean's men he goes gunning for the maverick lawman. Not much history here but entertaining, especially Walter Brennan as Bean. There is a Colorized version.\n\n**4852** _ **Westward Bound**_ **** Syndicate, 1931. 60 min. D: Harry S. Webb. SC: Carl Krusada. With Buffalo Bill, Jr., Allene Ray, Buddy Roosevelt, Wally Wales, Ben Corbett, Yakima Canutt, Fern Emmett, Tom London, Robert Walker, Pete Morrison, Henry Roquemore, Frank Ellis, Bob Roper. A playboy is sent West by his father in hopes the wide open spaces will reform him and the job gets done when he comes to the rescue of a woman whose ranch is sought by crooks. Boring and tacky, one of the all-time worst \"B\" Westerns.\n\n**4853** _ **Westward Bound**_ **** Monogram, 1944. 59 min. D: Robert Emmett Tansey. SC: Frances Kavanaugh. With Ken Maynard, Hoot Gibson, Bob Steele, Betty Miles, John Bridges, Harry Woods, Karl Hackett, Weldon Heyburn, Hal Price, Roy Brent, Frank Ellis, Curley Dresden, Dan White, Al Ferguson, Horace B. Carpenter, Charles Murray, Jr., Chick Hannon, Denver Dixon, Foxy Callahan A trio of marshals help ranchers who are forced off their lands by an outlaw gang led by a government official who wants the area for its resale value once Montana becomes a state. Very entertaining \"Trail Blazers\" series entry with all three stars appearing to good advantage.\n\n**4854** _ **Westward Ho!**_ **** Republic, 1935. 55 min. D: Robert North Bradbury. SC: Lindsley Parsons, Robert Emmett (Tansey) and Harry Friedman. With John Wayne, Sheila Mannors, Frank McGlynn, Jr., James Farley, Jack Curtis, Bradley Metcalfe, Jr., Dickie Jones, Mary MacLaren, Yakima Canutt, Hank Bell, Glenn Strange, The Singing Riders, Lloyd Ingraham, Frank Ellis, Earl Dwire, Fred Burns, Jack Kirk, Tex Palmer, Henry Hall, Edward Hearn, Herman Hack, Eddie Parker, Charles Brinley, Chuck Baldra, Fred Parker, Al Taylor, Silver Tip Baker, Charles Sargent, Ray Henderson. A cowboy, separated from his young brother years before when outlaws murdered their parents, goes West with settlers and becomes friends with a man working as a gang's spy, not knowing he is his sibling. Very entertaining and action packed production from producer Paul Malvern; John Wayne's first Republic Western.\n\n**4855** _ **Westward Ho**_ **** Republic, 1942. 56 min. D: John English. SC: Morton Grant and Doris Schroeder. With Bob Steele, Tom Tyler, Rufe Davis, Evelyn Brent, Donald Curtis, Lois Collier, Emmett Lynn, John James, Jack Kirk, Kenne Duncan, Tom Seidel, Milton Kibbee, Edmund Cobb, Monte Montague, Al Taylor, Bud Osborne, Horace B. Carpenter, John Cason, Jack O'Shea, Ray Jones, Tex Palmer, Jack Montgomery, Curley Dresden, Budd Buster, Jayne Hazard, Roy Bucko. A gang of desperadoes, led by a woman banker, is trailed by the Three Mesquiteers. Good entry in the popular series bolstered by a fine cast. TV title: _**Riders for Justice**_.\n\n**4856** _ **Westward Ho the Wagons**_ **** Buena Vista, 1956. 90 min. Color. D: William Beaudine. SC: Tom Blackburn. With Fess Parker, Kathleen Crowley, George Reeves, Jeff York, David Stollery, Sebastian Cabot, Doreen Tracey, Barbara Woodell, John War Eagle, Cubby O'Brien, Tommy Cole, Leslie Bradley, Morgan Woodward, Iron Eyes Cody, Anthony Numkena, Karen Pendleton, Jane Liddell, Jon Locke. A wagon train filled with settlers moves West under the leadership of a likable head scout and along the way has many adventures. Leisurely paced Disney drama laced with songs.\n\n**4857** _ **Westward the Women**_ **** Metro-Goldwyn-Mayer, 1952. 112 min. D: William A. Wellman. SC: Charles Schnee. With Robert Taylor, Denise Darcel, Hope Emerson, Marilyn Erskine, Julie Bishop, John McIntire, Beverly Dennis, Lenore Lonegran, Henry Nakamura, Renata Vanni, Frankie Darro, John Cason, Michael Conrad, Henry Wills, Ted Adams, Roy Jenson, Chubby Johnson, Evelyn Finley. A wagon train with female passengers heads to California led by a wagon master who not only has to contend with them, but also Indians, outlaws and the elements. Frank Capra wrote the story for this rather lighthearted drama but the overall results are mild.\n\n**4858** _ **The Westward Trail**_ **** Eagle-Lion, 1948. 56 min. D: Ray Taylor. SC: Robert Alan Miller. With Eddie Dean, Roscoe Ates, Phyllis Blanchard, Eileen Harden, Andy Parker and The Plainsmen, Steve Drake, Bob Duncan, Carl Mathews, Lee Morgan, Bob Woodward, Budd Buster, Slim Whitaker, Frank Ellis. A U.S. marshal works incognito to help a woman whose ranch is sought by crooks. Poor Eddie Dean musical Western.\n\n**4859** _ **Westworld**_ **** Metro-Goldwyn-Mayer, 1973. 89 min. Color. D-SC: Michael Crichton. With Yul Brynner, Richard Benjamin, James Brolin, Norman Bartold, Alan Oppenheimer, Victoria Shaw, Dick Van Patten, Linda Scott, Steve Franken, Michael Mikler, Terry Wilson, Majel Barrett, Anna Randall, Julie Marcus, Sharyn Wynters, Anne Bellamy, Chris Holter, Charles Seel, Wade Crosby, Lin Henson, Nora Marlow, Lauren Gilbert, Howard Platt, Jared Martin. Two businessmen take a vacation in a fantasy world where the old West of the 1880s is recreated but a robot gunman goes haywire and begins killing the tourists. Interesting sci-fi effort that spawned a less satisfying sequel, _**Futureworld**_ (American International, 1976), in which Yul Brynner briefly reprises his gunman-robot role.\n\n**4860** _ **Wetbacks**_ **** Gibraltar Motion Picture Distributors, 1956. 87 min. D: Hank McCune. SC: Pete LaRoche. With Lloyd Bridges, Nancy Gates, Barton MacLane, John Hoyt, Harold Peary, Nacho Galindo, Robert Keys, David Colmans, Jose Gonzales Gonzales, Louis Jean Heydt, Richard Powers (Tom Keene), I. Stanford Jolley, Gene Roth, Wally Cassell, Roy Gordon, Scott Douglas, Maury Dexter, Joe Dominguez, Salvador Baquez. The U.S. Border Patrol tries to stop the influx of Mexicans being illegally smuggled across the U.S. border for cheap labor as a boat owner is forced by two thugs to carry out this activity. Low grade program feature partially filmed in Mexico.\n\n**4861** _ **Wheels of Destiny**_ **** Universal, 1934. 64 min. D: Alan James. SC: Nate Gatzert. With Ken Maynard, Dorothy Dix, Philo McCullough, Fred McKay, Fred Sale, Jr., Buffalo Bill, Jr., Jack Rockwell, Frank Rice, Nelson McDowell, William Gould, Ed Coxen, Merrill McCormick, Slim Whitaker, Hank Bell, Bob Burns, Artie Ortego, Wally Wales, Helen Gibson, Jack Evans, Bud McClure, Fred Burns, Chief Big Tree, Marin Sais, Chuck Baldra, Blackjack Ward, Bobby Dunn, Al Taylor, Roy Bucko. A cowboy leads a wagon train of settlers across Mt. Whitney's great salt flats and along the way battles outlaws, Indians and the harsh environment. Top notch Ken Maynard film which he also produced and wrote the music.\n\n**4862** _ **When a Man Rides Alone**_ **** Monarch, 1933. 60 min. D: J.P. McGowan. SC: Oliver Drake. With Tom Tyler, Adele Lacy, Alan Bridge, Bob Burns, Frank Ball, Alma Chester, Duke R. Lee, Barney Furey, Lee Cordova, Jack Rockwell, Bud Osborne, Jack Kirk, Herman Hack, J.P. McGowan, Jack Evans, Ed Burns, Horace B. Carpenter, Lillian Chay. A mysterious bandit robs a gold shipment and then gives it to settlers who have invested in a bogus mine. Average Tom Tyler outing with low grade production values.\n\n**4863** _ **When a Man Sees Red**_ **** Universal, 1934. 60 min. D-SC: Alan James. With Buck Jones, Peggy Campbell, Dorothy Revier, LeRoy Mason, Syd Saylor, Libby Taylor, Charles K. French, Jack Rockwell, Frank LaRue, Robert Kortman, Frank Ellis, Horace B. Carpenter, Horace Murphy, Horsea Stedman, William Steele, Silver Tip Baker, Charles Murphy, Bill Patton. A ranch foreman must contend with a strong willed new female boss as well as a cattle rustling gang. Buck Jones' second Universal feature is only average.\n\n**4864** _ **When a Man's a Man**_ **** Associated First National, 1924. 55 min. D: Edward F. Cline. SC: Walter Anthony and Harry Carr. With John Bowers, Marguerite De La Motte, Robert Frazer, June Marlowe, Forrest Robinson, Elizabeth Rhodes, Fred Stanton, George Hackathorne, Edward Hearn, John Fox, Jr., Arthur Hoyt, Ray Thompson, Charles Mailes. When his girl refuses a marriage proposal, a man goes West where he is mistaken for a cattle rustler and must prove his innocence. Fairly good silent adaptation of Harold Bell Wright's novel.\n\n**4865** _ **When a Man's a Man**_ **** Fox, 1935. 70 min. D: Edward F. Cline. SC: Frank M. Dazey and Agnes Christine Johnston. With George O'Brien, Dorothy Wilson, Paul Kelly, Harry Woods, Jimmy Butler, Richard Carlyle, Edgar Norton, Clarence Wilson, Frank Ellis, Tracy Layne, Dick Hunter, Ken Cooper, Richard Botiller, Slim Whitaker, Rose Plummer, Stanley Blystone, Jack Montgomery, Charles Brinley, Roy Bucko, Ralph Bucko, Lester Dorr, Sid Jordan, Arthur Thalasso. In a small town, a cowboy comes to the assistance of a woman ranch owner whose water supply has been taken over by outlaws. Well acted, entertaining George O'Brien remake of the 1924 feature, also directed by Edward F. Cline. Done a third time as _**Massacre River**_ (q.v.). TV title: _**Saga of the West**_.\n\n**4866** _ **When Lightning Strikes**_ **** Regal, 1934. 60 min. D: Harry Revier and Burton L. King. SC: George Morgan and J.P. McGowan. With Lightning The Wonder Dog, Francis X. Bushman, Jr., Alice Dahl, J.P. McGowan, Tom London, Blackie Whiteford, William Desmond, Marin Sais, Murdock MacQuarrie, Bart Carre, Richard Botiller, Shaggy (dog). When a crooks sends to thugs to steal his timber lease, a man entrusts the documents to his loyal dog who buries them in the woods. Lightning's cinema debut is a crudely filmed, photographed and edited juvenile oriented dual bill item.\n\n**4867** _ **When the Boys Meet the Girls.**_ Metro-Goldwyn-Mayer, 1965. 110 min. Color. D: Alvin Ganzer. SC: Robert E. Kent. With Connie Francis, Harve Presnell, Herman's Hermits, Liberace, Louis Armstrong, Sam the Sham and The Pharoahs, Sue Anne Langdon, Fred Clark, Frank Faylen, Joby Baker, David and Reese, Hortense Petra, Stanley Adams, Romo Vincent, Susan Holloway, Russ Collins, William T. Quinn. As their Nevada night club is about to go bankrupt, the owners turn it into a cabaret with spectacular success while a rich guy tries to avoid a showgirl's breach of promise suit. Meager third screen version of _**Girl Crazy,**_ filmed previously in 1932 and 1943 (qq.v.), is mainly a showcase for contemporary popular performers; produced by Sam Katzman.\n\n**4868** _ **When the Daltons Rode**_ **** Universal, 1940. 74 min. D: George Marshall. SC: Harold Shumate. With Randolph Scott, Kay Francis, Brian Donlevy, Broderick Crawford, George Bancroft, Stuart Erwin, Andy Devine, Frank Albertson, Mary Gordon, Harry Stephens, Edgar Dearing, Quen Ramsey, Dorothy Granger, Bob McKenzie, Fay McKenzie, June Wilkins, Walter Soderling, Edgar Buchanan, Sally Payne, Mary Ainslee, Erville Alderson, Lafe McKee, Jim Pierce, Alan Bridge, Lloyd Ingraham, Walter Long, James Flavin, Jack Clifford, Robert Kortman, Ethan Laidlaw, Harry Cording, Henry Roquemore, Tom London, Ian MacLaren, Stanley Blystone, Edwin Brady, George Guhl, Tom Chatterton, Eddie Parker, Russ Powell, Joe King, Don Rowan, James Morton, Charles McMurphy, Kernan Cripps, Jack Baxley, Duke York, Pat West. Four brothers are forced to become outlaws and they befriend a lawyer who falls in love with a woman one of them plans to marry. Slick Universal feature with more action than plot.\n\n**4869** _ **When the Legends Die**_ **** 20th Century\u2013Fox, 1972. 104 min. Color. D: Stuart Millar. SC: Robert Dozier. With Richard Widmark, Frederic Forrest, Luana Anders, Vito Scotti, Herbert Nelson, John War Eagle, John Gruber, Gary Walberg, Jack Mullaney, Malcolm Curley, Roy Engel, Rex Holman, Tillman Box. An aging rodeo circuit rider befriends an orphaned Ute Indian who eventually grows tired of his kind of life. Rather interesting story with an excellent performance by Richard Widmark as the rodeo rider.\n\n**Richard Widmark and Fredric Forrest in** _**When the Legends Die**_ **(20th Century** **\u2013** **Fox, 1972).**\n\n**  \n**\n\n**4870** _ **When the North Wind Blows**_ **** Sunn Classic Pictures, 1974. 100 min. Color. D-SC: Stewart Raffill. With Henry Brandon, Dan Haggerty, Fernando Celis, Rex Holman, Dale Ishimoto, Sander Johnson, Herbert Nelson, Henry Olek, Jack Ong, Jan Smithers. In the early 1900s two rare Siberian tigers attack near an Alaskan village with an old-time hunter and a group of youths hunting them and when one of the boys is accidentally wounded by the trapper he flees into the wilderness where he is befriended by a tigress and her cubs. Pleasant, easy to take family fare with scenic locales.\n\n**Dan Haggerty in** _**When the North Wind Blows**_ **(Sunn Classic Pictures, 1974).**\n\n**  \n**\n\n**4871** _ **When the Redskins Rode**_ **** Columbia, 1951. 77 min. Color. D: Lew Landers. SC: Robert E. Kent. With Jon Hall, Mary Castle, James Seay, John Ridgely, Sherry Moreland, Pedro De Cordova, John Dehner, Lewis L. Russell, William Bakewell, Gregory Gay, Rusty Wescoatt, Milton Kibbee, Rick Vallin, Steve Pendleton, Charles Horvath, J.W. Cody, Jack Chefe, Jessie Arnold, Harold Miller. In 1753 Virginia's Governor Dinwiddie and George Washington try to get the local Indian tribes to ally with the British against the French but a pretty spy attempts to convince the a chief's son to do the opposite. Standard historical fare set in the pre\u2013Revolutionary War period; produced by Sam Katzman.\n\n_**When the West Was Young**_ see _**Heritage of the Desert**_ (1932)\n\n**4872** _ **Where the Buffalo Roam**_ **** Monogram, 1938. 60 min. D: Al Herman. SC: Robert Emmett (Tansey). With Tex Ritter, Dorothy Short, Snub Pollard, Horace Murphy, John Merton, Richard Alexander, Karl Hackett, Dave O'Brien, Louise Massey and Her Westerners, Bob Terry, Charles King, Blackie Whiteford, Denver Dixon, Curt Massey, Ernie Adams, Hank Worden. A cowboy is hunting the murderers of his mother, an outlaw gang killing buffalo and trying to wreck a stage line. A complicated plot is somewhat offset by good action and music in this Tex Ritter feature.\n\n**4873** _ **Where the Hell's the Gold?**_ **** CBS-TV, 1988. 91 min. Color. D-SC: Burt Kennedy. With Willie Nelson, Delta Burke, Jack Elam, Alfonso Arau, Gregory Sierra, Michael Wren, Gerald McRaney, Jamie Lyn Bauer, Annabelle Gurwitch, John David Garfield, Sheila Foster, Deborah Carpenter, Pamela Moore, Adan Sanchez. Two adventurers after gold are talked into taking a train filled with dynamite through hostile territory and get pursued by Indians, outlaws, banking agents and Mexican soldiers. Pretty fair TV movie re-titled _**Dynamite and Gold**_ for video.\n\n**4874** _ **Where the Lillies Bloom**_ **** United Artists, 1974. 96 min. D: William A. Graham. SC: Earl Hamner. With Julie Gholson, Jan Smithers, Matthew Burill, Helen Harmon, Harry Dean Stanton, Sudie Bond, Rance Howard, Tom Spratley, Helen Bragdon, Alice Beardsley. In the mountain region of the South, youngsters try to keep their parents' deaths a secret so they will not be separated. Well made family film, produced in North Carolina.\n\n**4875** _ **Where the North Begins**_ **** Warner Bros., 1923. 60 min. D: Chester M. Franklin. SC: Raymond L. Schrock. With Rin Tin Tin, Walter McGrail, Claire Adams, Fred Huntley, Pat Hartigan, Myrtle Owen, Charles Stevens. After being raised by wolves, a German Shepherd is adopted by a fur trapper and later saves the man's life when he is attacked by thieves. Good initial Rin Tin Tin feature, well photographed and directed; the dog star's owner-trainer, Lee Duncan, co-wrote the story.\n\n**4876** _ **Where the North Begins**_ **** Screen Guild, 1947. 42 min. D: Howard Bretherton. SC: Betty Burbridge and Les Swabacker. With Russell Hayden, Jennifer Holt, Tristram Coffin, Denver Pyle, Stephen Barclay, Keith Richards, Anthony Warde, Frank Hagney, Artie Ortego, J.W. Cody, Chris Willow Bird, Billy Wilkerson, Lew Morphy, Charley Cypert, Billy Cypert. A member of the Royal Canadian Mounted Police battles a gang of outlaws in a remote town while romancing a pretty girl. This low budget short feature moves along quickly.\n\n**4877** _ **Where the North Holds Sway**_ **** Rayart, 1927. 56 min. D: Bennett Cohen. SC: Evan Merritt Post. With Jack Perrin, Pauline Curley, Billy Lamoreaux (Buzz Barton), Lew Meehan, Hal Walters, Starlight (horse). A Mountie leaves the service to seek his brother's killer and after being injured finds refuge with a gambler and his wife, not knowing the husband is the murderer. Picturesque quickie production mainly of interest because of its star and his beautiful steed; directed by genre writer Bennett Cohen.\n\n**4878** _ **Where the Red Fern Grows**_ **** Howco International\/Westamerica, 1974. 97 min. Color. D: Norman Tokar. SC: Douglas Stewart and Eleanor Lamb. With James Whitmore, Beverly Garland, Jack Ging, Lonny Chapman, Stewart Petersen, Jill Clark, Jeanna Wilson. In 1930s Oklahoma a boy raises two redbone hounds and trains them for two years to be champion coon hunters but during the big contest, held in the rugged Cherokee countryside, the canines are called off the hunt to rescue their master from a mountain lion. Pleasing family film based on Wilson Rawls' book with soundtrack songs by the Osmond Brothers and Andy Williams; followed by a sequel, _**Where the Red Fern Grows: Part Two**_ (q.v.), in 1992 and remade in 2003 (q.v.).\n\n**4879** _ **Where the Red Fern Grows**_ **** Doty-Dayton Releasing, 2003. 86 min. Color. D: Lyman Dayton and Sam Pillsbury. SC: Douglas Stewart, Eleanor Lamb, Lyman Dayton and Sam Pillsbury. With Joseph Ashton, Dave Matthews, Renee Faia, Mac Davis, Kris Kristofferson, Ned Beatty, Dabney Coleman, Gary Anson, Orvel Baldridge, Robert Bauman, Andrew Dickison, Stuart Dickison, Julia Downs, Tess Downs, Kevin Gourd, Zach Hamilton, Steve Sandalis, Stan Randolph, Eric Starkey, Charles Seal, Lindsey Labadie. A rural boy saves money to fulfill his dream of owing two hound pups, which he plans to raise to be coon hunters. Fine remake of the 1974 version (q.v.).\n\n**4880** _ **Where the Red Fern Grows: Part Two**_ **** UAV Entertainment, 1992. 93 min. Color. D: Jim McCullough (Jr.). SC: Samuel Bradford. With Wilford Brimley, Doug McKeon, Chad McQueen, Lisa Whelchel, Adam Faraizi, Karen Carlson, Devin Payne, Jessie Turner, Tom Bertino, Patricia Meeks, Maggie McCullough, Daniel Glover, Philip Dale, Sherry Spurrier, Cindy Garrett, Ace Williamson, Kim Bond, Rachel Bond. When a disturbed war veteran returns to his backwoods Louisiana home he is given two pups to raise by his grandfather. Average video follow-up to _**Where the Red Fern Grows**_ (q.v.); also called _**Where the Red Fern Grows 2: The Homecoming**_.\n\n_**Where the Red Fern Grows 2: The Homecoming**_ see _**Where the Red Fern Grows: Part Two**_\n\n**4881** _ **Where the Rivers Flow North**_ **** Caledonia Pictures, 1993. 106 min. Color. D: Jay Craven. SC: Don Bredes and Jay Craven. With Michael J. Fox, Rip Torn, Tantoo Cardinal, Treat Williams, Bill Raymond, Mark Margolis, George Woodard, Yusef Bulos, John Griesemer, Jeri Lynn Cohen, Amy Wright, Rusty De Wees, Dennis Mientka, John Rothman, Sam Lloyd, Sr., Burt Porter, Tony Washburn. In 1927 Vermont a logger and his Indian girlfriend are at odds over a dam construction company's offer to buy his land, which will be flooded. Fine adaptation of Howard Frank Mosher's novel; nice scenery.\n\n**4882** _ **Where the West Begins**_ **** Monogram, 1938. 54 min. D: J.P. McGowan. SC: Stanley Roberts and Gennaro Rea. With Jack Randall, Luana Walters, Fuzzy Knight, Richard Alexander, Budd Buster, Arthur Houseman, Ralph Peters, Ray Whitley, The Phelps Brothers, Kit Guard, Ken Card. A crook wants a woman's ranch and he has a cowboy thrown in jail to stop him from persuading her not to sell. Pleasant Jack Randall vehicle.\n\n**4883** _ **Where Trails Divide**_ **** Monogram, 1937. 60 min. D: Robert North Bradbury. SC: Robert Emmett (Tansey). With Tom Keene, Eleanor Stewart, Warner Richmond, David Sharpe, Lorraine Randall, Charles K. French, Steve Clark, Hal Price, Richard Cramer, James Sheridan (Sherry Tansey), Bud Osborne, Horace B. Carpenter, Jim Mason, Forrest Taylor, Oscar Gahan, Wally West. An express company lawyer is made the sheriff of a town to stop an outlaw gang. Austere Tom Keene film; well done.\n\n**4884** _ **Where Trails End**_ **** Monogram, 1942. 55 min. D: Robert Emmett Tansey. SC: Robert Emmett (Tansey) and Frances Kavanaugh. With Tom Keene, Joan Curtis, Frank Yaconelli, Charles King, Donald Stewart, Steve Clark, William Vaughn, Horace B. Carpenter, Nick Moro, Gene Alsace, Fred Hoose, James Sheridan (Sherry Tansey), Tex Palmer, Chick Hannon, Tom Seidel. When settlers are driven from their homes by a mysterious outlaw gang's terrorism, a U.S. marshal is sent to investigate and finds that enemy agents are after recently discovered tungsten. Average oater with a prophetic title as it was Tom Keene's final series film.\n\n**4885** _ **The Whirlwind**_ **** Columbia, 1933. 60 min. D: D. Ross Lederman. SC: Stuart Anthony. With Tim McCoy, Alice Dahl, Pat O'Malley, Matthew Betz, J. Carrol Naish, Joseph Girard, Lloyd Whitcomb, William McCall, Stella Adams, Theodore (Ted) Lorch, Hank Bell, Mary Gordon, Joe Dominguez. A cowboy returns home with his two pals to find his father and friends have been turned against him by a dishonest lawman. Fair Tim McCoy feature with rodeo and boxing sequences in addition to the usual action.\n\n**4886** _ **Whirlwind**_ **** Columbia, 1952. 70 min. D: John English. SC: Norman S. Hall. With Gene Autry, Smiley Burnette, Gail Davis, Harry Lauter, Thurston Hall, Dick Curtis, Harry Harvey, Kenne Dunan, Tommy Ivo, Gregg Barton, Al Wyatt, Gary Goodwin, Pat O'Malley, Bud Osborne, Boyd Stockman, Frankie Marvin, Stan Jones. Postal inspector Gene Autry is after a gang led by a crooked rancher. Pleasing Gene Autry vehicle.\n\n**4887** _ **Whirlwind Horseman**_ **** Grand National, 1938. 60 min. D: Robert Hill. SC: George Plympton. With Ken Maynard, Joan Barclay, Dave O'Brien, Kenny Dix, Roger Williams, Kenneth Harlan, Robert Frazer, Walter Shumway, Budd Buster, Lew Meehan, Joseph Girard, Bill Griffith, Glenn Strange, Wally West, Bud Osborne, Oscar Gahan, Carl Mathews, George Morrell, Jim Corey, Clyde McClary. After their pal disappears two men search for him and find a group of ranchers being menaced by outlaws. Low budget but entertaining Ken Maynard film with a good sprinkling of comedy.\n\n**4888** _ **Whirlwind Raiders**_ **** Columbia, 1948. 54 min. D: Vernon Keays. SC: Norman S. Hall. With Charles Starrett, Smiley Burnette, Nancy Saunders, Fred F. Sears, Little Brown Jug (Don Kay Reynolds), Doye O'Dell and His Radio Rangers, Jack Ingram, Philip Morris, Patrick Hurst, Eddie Parker, Lynn Farr, Arthur Loft, Maudie Prickett, Frank LaRue, Russell Meeker, Herman Hack. The Durango Kid is on the trail of the corrupt State Police, a group of outlaws hiding behind badges after the disbandment of the Texas Rangers. \"Durango Kid\" fans will like this action filled series entry. British title: _**State Police**_.\n\n_**Whirlwind Rider**_ see _**Ranger of the Law**_\n\n**4889** _ **The Whispering Skull**_ **** Producers Releasing Corporation, 1944. 56 min. D: Elmer Clifton. SC: Harry Fraser. With Tex Ritter, Dave O'Brien, Guy Wilkerson, Denny Burke, I. Stanford Jolley, Henry Hall, George Morrell, Ed Cassidy, Robert Kortman, Wen Wright, Frank Ellis, Jimmy Aubrey, Ray Henderson. Three lawmen are after a masked killer known as \"The Whispering Skull\" who is searching for a cache of diamonds. Slow moving and cheaply made installment of \"The Texas Rangers\" series that is lacking in atmosphere. Tex Ritter sings \"In Case You Change Your Mind\" and \"It's Never Too Late.\"\n\n**4890** _ **Whispering Smith**_ **** Paramount, 1948. 88 min. Color. D: Leslie Fenton. SC: Frank Butler and Karl Lamb. With Alan Ladd, Robert Preston, Brenda Marshall, Donald Crisp, William Demarest, Fay Holden, Murvyn Vye, Frank Faylen, John Eldredge, Robert Wood, J. Farrell MacDonald, Don Barclay, Will Wright, Eddy Waller, Gary Gray, Robert Kortman, Ashley Cowan, Ray Teal, Jimmy Dundee. A special agent is called in by the railroad to investigate a series of robberies and he discovers a friend is involved with the outlaws. Well done drama for Alan Ladd and Robert Preston fans. It was the basis for the series of the same name on NBC-TV in 1961 starring Audie Murphy and Guy Mitchell.\n\n**4891** _ **Whispering Smith Speaks**_ **** Fox, 1935. 65 min. D: David Howard. SC: Dan Jarrett and Don Swift. With George O'Brien, Irene Ware, Kenneth Thomson, Maude Allen, Spencer Charters, Victor Potel, Edward Keane, Frank Sheridan, William V. Mong, Maurice Cass, Si Jenks, John Ince, Dick Rush, J.P. McGowan, Bess Flowers. While learning the railroad business, a man finds out tungsten has been discovered on a farm owned by the woman he loves and he tries to dissuade her from selling it to a crook. Top notch George O'Brien vehicle based on the character created by Frank H. Spearman.\n\n**4892** _ **Whistlin' Dan**_ **** Tiffany, 1932. 60 min. D: Phil Rosen. SC: Stuart Anthony. With Ken Maynard, Joyzelle Joyner, George Renavent, Harlan E. Knight, Don Terry, Jack Rockwell, Lew Meehan, Bud McClure, Merrill McCormick, Wally Wales, Jessie Arnold, Frank Ellis, Hank Bell, Jim Corey. When outlaw Serge Karloff kidnaps and murders his pal, cowboy Whistlin' Dan joins his gang to get revenge on the gunman. Quite interesting Ken Maynard vehicle; leisurely paced. Remade as _**Along the Rio Grande**_ (q.v.).\n\n**4893** _ **Whistling Bullets**_ **** Ambassador, 1937. 58 min. D: John English. SC: Joseph O'Donnell. With Kermit Maynard, Harlene (Harley) Wood, Jack Ingram, Maston Williams, Bruce Mitchell, Karl Hackett, Sherry Tansey, Cliff Parkinson, Herman Hack, William McCall, Buck Moulton. Two Texas Rangers are on the trail of a gang of bond thieves. Good production values and direction heighten the affect of this Kermit Maynard film, supposedly based on the works of James Oliver Curwood.\n\n**4894** _ **Whistling Hills**_ **** Monogram, 1951. D: Derwin Abrahams. SC: Fred Myton. With Johnny Mack Brown, Jimmy (James) Ellison, Pamela Duncan, Noel Neill, I. Stanford Jolley, Lee Roberts, Marshall Reed, Lane Bradford, Bud Osborne, Pierce Lyden, Frank Ellis, Ray Jones, Merrill McCormick, Carl Mathews. A masked outlaw, who uses a mysterious whistle to signal his gang, is hunted by a sheriff and the cowboy who agrees to help him. A bit better than some of the later Johnny Mack Brown features.\n\n**4895** _ **The White Buffalo**_ **** United Artists, 1977. 97 min. D: J. Lee Thompson. SC: Richard Sale. With Charles Bronson, Will Sampson, Jack Warden, Kim Novak, Clint Walker, Stuart Whitman, Slim Pickens, John Carradine, Cara Williams, Shay Duffin, Douglas V. Fowley, Cliff Pellow, Ed Lauter, Martin Kove, Scott Walker, Ed Bakey, Richard Gilliland, David Roy Chandler, Philip Montgomery, Linda Moon Redfearn, Chief Tug Smith, Douglas Hume, Cliff Carnell, Ron Thompson, Eve Brent, Joe Roman, Bert Williams, Dan Vadis, Christopher Cary, Larry Martindale, Scott Bryson, Will Walker, Gregg White, Hal Southern, Harold Hensley. Wild Bill Hickok and Crazy Horse form an uneasy alliance in an effort to kill a legendary white buffalo. Strange, almost supernatural like, Western with dark, murky photography; mainly for Charles Bronson fans. TV title: _**Hunt to Kill**_.\n\n**4896** _ **White Comanche**_ **** International Producers Corporation, 1969. 90 min. Color. D: Gilbert Lee Kay (Jose Birz). SC: Robert Holt and Frank Gruber. With Joseph Cotten, William Shatner, Rossana Yani, Perla Christal, Barta Barry, Victor Israel, Luis Prendes, Vidal Molina. In the town of Rio Hondo, a sheriff tries to stop two feuding half-breed brothers from killing each other along with fighting a corrupt town boss. Made in Italy as _**Comanche Blanco**_ (White Comanche), this is one of the better 1960s European oaters, enhanced by good work by Joseph Cotten and William Shatner, the latter in dual roles.\n\n**4897** _ **The White Dawn**_ **** Paramount, 1974. 109 min. Color. D: Philip Kaufman. SC: James Houston and Tom Rickman. With Warren Oates, Timothy Bottoms, Lou Gossett, Simonie Kopapik, Joanasie Salomnie, Pilitak, Munamee Sake. During an 1896 whaling expedition three men get lost in the Arctic and are saved by Eskimos who they later use to their advantage. Overlong, but nicely photographed, melodrama.\n\n**4898** _ **White Eagle**_ **** Columbia, 1932. 67 min. D: Lambert Hillyer. SC: Fred Myton. With Buck Jones, Barbara Weeks, Ward Bond, Robert Ellis, Jason Robards, Russell Simpson, Frank Campeau, Robert Kortman, Robert Elliott, Jim Thorpe, Frank Hagney, Jimmy House, Clarence Geldert, Alf James. An Indian brave, who is really white, tries to protect his tribe from dishonest men wanting to steal their horses. Sturdy Buck Jones film that was remade as a serial (q.v.) in 1941 with the star again essaying the title role.\n\n**4899** _ **White Eagle**_ **** Columbia, 1941. 15 Chapters. D: James W. Horne. SC: Arch Heath, Morgan B. Cox, John Cutting and Lawrence Taylor. With Buck Jones, Raymond Hatton, Dorothy Fay, James Craven, Chief Yowlachie, Jack Ingram, Charles King, John Merton, Roy Barcroft, Edward Hearn, Al Ferguson, J. Paul Jones, Edward Cecil, Chick Hannon, Bob Woodward, Horace B. Carpenter, Steve Clark, Merrill McCormick, Yakima Canutt, Kit Guard, Constantine Romanoff, Harry Tenbrook, Ed Peil, Sr., Hank Bell, Lloyd Whitlock, Eddie Featherston, George Chesebro, Kenne Duncan, Bud Osborne, Edmund Cobb, Richard Cramer, Jack O'Shea, Robert Elliott, George Larkin, Jack Richardson, Charles Hamilton, Richard Ellis. An Indian pony express rider opposes a gang of renegades who dress like tribesmen to rob stagecoaches. Slim serial remake of the 1932 (q.v.) Buck Jones vehicle, but still worth viewing, especially for his fans.\n\n**4900** _ **White Fang**_ **** 20th Century\u2013Fox, 1936. 70 min. D: David Butler. SC: Hal Long and Sam Duncan. With Michael Whalen, Jean Muir, Charles Winninger, Slim Summerville, Jane Darwell, Thomas Beck, John Carradine, George Cucount, Joe Herrick, Edward Thorpe, Steve Clemento, Marie Chorre, Jack Curtis, Ken Evans, Robert St. Angelo, Nick De Ruiz, Desmond Gallagher, Walter James, Jack Stoney, Joe Brown, Ward Bond, William Wagner, Francis McDonald, Herbert Heywood. A woman promises to marry a man if he will guide her brother to a gold mine during the harsh Yukon winter. Average follow-up to _**Call of the Wild**_ (q.v.); based on the Jack London novel.\n\n**4901** _ **White Fang**_ **** Titanus, 1972. 97 min. Color. D: Lucio Fulci. SC: Guy Elmes, Piero Regnoli, Guillaume Roux, Roberto Gianviti, Thom Keyes and Peter Welbeck (Harry Alan Towers). With Franco Nero, Virna Lisi, Fernando Rey, Rik Battaglia, John Steiner, Missaele, Daniel Martin, Raimund Harmstorf, Daniele Dublino, Carole Andre, John Bartha, Luigi Antonio Guerra, Carla Mancini, Maurice Poli, Renzo Prevarello. A beautiful husky dog helps his master, a sourdough in the Klondike, in fighting both crooks and bad men when he strikes gold. Very well done French-Italian-Spanish co-production of the Jack London work. Italian title _**Zanna Bianca**_ (White Fang).\n\n**4902** _ **White Fang**_ **** Buena Vista, 1991. 107 min. Color. D: Randal Kleiser. SC: Jeanne Rosenberg, Nick Thiel and David Fallon. With Ethan Hawke, Klaus Maria Brandauer, Seymour Cassel, Susan Hogan, James Remar, Bill Moseley, Clint B. Youngreen, Pius Savage, Aaron Hotch, Charles Jimmie, Sr., Clifford Fossman, Irving Sogge, Tom Fallon, Dick Mackey, Suzanne Kent, Robert C. Hoelen, George Rogers, Michael David Lally, Raymond R. Menaker, David Fallon, Michael A. Hagen, Diane Benson, Rob Kyker, Tom Yewell, John Beers, Van Clifton, Jim Moore, Marliese Schneider, Jed (dog), Bart (bear). Going to Alaska in search of gold, a young prospector meets a boy and a wolf dog, who has been mistreated by his owner, as they struggle to survive claim jumpers and the harsh environment. Nicely done third screen version of Jack London's novel.\n\n**4903** _ **White Fang and the Gold Diggers**_ **** First Line Films, 1974. 86 min. Color. D: Alfonso Brescia. SC: Giuseppe Maggi and Piero Regnoli. With Robert Wood, Pedro Sanchez (Ignazio Spalla), Robert Hundar (Claudio Undari), Gabriella Lepori, Franco Lantieri, Paolo Lena, Jean-Pierre Clarain, Franco Calogero, Renato Malavasi, Andrea Fantasia, Amedeo Timpani, Giovanni (Nello) Pazzafini, Eolo Capritti, Benito Pacifico. Going with his little son and their dog to take over a mine he has inherited in Northern Canada, a man finds he is faced with an enemy who wants both his property and mail order bride. Mediocre \"White Fang\" imitation (the dog is called \"Whiskey\" in its original version), made in Italy as _**La Spacconata**_ (The Bluff) and cut by 12 minutes for U.S. release.\n\n**4904** _ **White Fang and the Hunter**_ **** First Line Films, 1975. 87 min. Color. D: Alfonso Brescia. SC: Giulio Berruti and Giuseppe Maggi and Franco Pietroletti. With Robert Wood, Ignazio Spalla, Malisa Longo, Robert Hundar (Claudio Undari), Massimo De Cecco, Franco Lantieri, Linda Sini, Amedeo Timpani, Giovanni Ukmar, Jean-Pierre Clarain, Guido Mariotti, Bruno Arie. Two hunters try to help a young woman save her homestead coveted by crooks. Sorry Italian production released there as _**Zanna Bianca e il Caccatore Solitario**_ (White Fang and the Solitary Hunter), which has little to do with White Fang and even less with Jack London's work.\n\n**4905** _ **White Fang to the Rescue**_ **** Paneuropean Production Pictures, 1974. 97 min. Color. D: Tonino Ricci. SC: Giovanni Simonelli and Sandro Continenza. With Maurizio Merli, Henry Silva, Renzo Palmer, Gisela Hahn, Benny Reeves (Benito Stefanelli), Donald O'Brien, Luciano Rossi, Manfred Freyberger, Sergio Smacchi, Marco Stefanelli, Cesar di Vito, Attilo Dottesio, Simone Santo, Natalie Siddi, Luciano Bonanni, Riccardo Pizzuti, Matteo Zoffoli. In the Yukon a wolf dog and a trapper team to take in the two men who murdered the canine's master. Average outdoor adventure highlighted by a brutal battle between the title character and a huge bear; made in Italy as _**Zanna Bianca alla Riscossa**_ (White Fang to the Rescue).\n\n**4906** _ **White Fang 2: The Myth of the White Wolf**_ **** Buena Vista, 1994. 106 min. Color. D: Ken Olin. SC: David Fallon. With Scott Bairstow, Alfred Molina, Geoffrey Lewis, Charmaine Craig, Victoria Racimo, Paul Coeur, Anthony Michael Ruivivar, Al Harrington, Matthew Cowles, Woodrow W. Morrison, Reynold Russ, Nathan Young, Charles Natkong, Sr., Edward Davis, Byron Chief Moon, Tom Heaton, Trace Yeomans, Thomas Kitchkeesic, Ethan Hawke. During the Alaskan Gold Rush a prospector, who is in love with an Indian princess, and his wolf-dog lead a tribe of starving Indians to a caribou herd. Filmed in British Columbia and Colorado, this Disney family films is only average except for the beautiful scenery.\n\n**4907** _ **White Feather**_ **** 20th Century\u2013Fox, 1955. 100 min. Color. D: Robert Webb. SC: Delmer Daves and Leo Townsend. With Robert Wagner, John Lund, Debra Paget, Jeffrey Hunter, Eduard Franz, Noah Beery, Jr., Virginia Leith, Emile Meyer, Hugh O'Brian, Milburn Stone, Iron Eyes Cody. A prospector, in love with an Indian maiden, tries to help the government in getting her tribe to move to a reservation. Standard melodrama with good performances.\n\n**4908** _ **White Fury**_ **** American National Enterprises, 1969. 100 min. Color. D-SC: Arthur R. Dubs. With Arthur R. Dubs (narrator). Documentary about the wilderness made up of three short films: _**Baja Big Horn**_ , _**High Desert**_ and _**White Fury**_. Pleasant sequel to _**Alaskan Safari**_ (q.v.).\n\n**4909** _ **White Gold**_ **** Producers Distributing Corporation, 1927. 65 min. D: William K. Howard. SC: Marion Orth, Garrett Fort and Tay Garnett. With Jetta Goudal, Kenneth Thomson, George Bancroft, George Nichols, Robert Perry, Clyde Cook. A pretty Mexican dancer marries a sheep raiser and moves with him to a remote ranch where she is distrusted by his father and lusted after by the hired hand. Highly regarded silent melodrama; well worth watching.\n\n_**White Justice**_ see _**Bloody Trail**_\n\n**4910** _ **White Oak**_ **** Paramount-Artcraft, 1921. 70 min. D: Lambert Hillyer. SC: Bennet Musson. With William S. Hart, Vola Vale, Alexander Gaden, Robert Walker, Bertholde Sprotte, Helen Holly, Chief Standing Bear, Red Wing. A gambler seeks revenge on the crook who took advantage of his younger sister. Typically dramatic William S. Hart silent feature based on his original story; he also produced.\n\n**4911** _ **The White Outlaw**_ **** Universal, 1925. 50 min. D: Clifford Smith. SC: Isadore Berstein. With Jack Hoxie, Marceline Day, William Welsh, Duke R. Lee, Floyd Shackelford, Charles Brinley, Scout (horse). A cowboy befriends and trains a wild stallion who is later accused of stealing area horses. Action packed, pleasing Jack Hoxie silent outing.\n\n**4912** _ **The White Outlaw**_ **** Exhibitors Film Corporation, 1929. 50 min. D: Robert J. Horner. SC: Bob McKenzie. With Art Acord, Vivian May, Lew Meehan, Bill Patton, Al Hoxie, Dick Nores, Betty Carter, Howard Davies, Walter Maly, Slim Mathews. An outlaw gets the blame for a robbery he did not commit and sets out to track down the real culprits. Silent Art Acord offering made at the end of his career; cheap and short on action but still entertaining. Written by character actor Bob McKenzie.\n\n**Art Acord, the star of** _**The White Outlaw**_ **(Exhibitors Film Corporation, 1929).**\n\n**  \n**\n\n**4913** _ **The White Squaw**_ **** Columbia, '956. 73 min. D: Ray Nazarro. SC: Lee Savage, Jr. With David Brian, May Wynn, William Bishop, Nancy Hale, William Leslie, Myron Healey, Robert C. Ross, Frank De Kova, George Keymas, Roy Roberts, Grant Withers, Wally Vernon, Paul Birch, Dennis Moore, Neyle Morrow, Guy Teague. A rancher plots revenge when the government informs him he did not properly file a land claim and his place is going to be used for an Indian reservation. Mediocre program feature from producer Wallace MacDonald.\n\n_**White Stallion**_ see _**The Harmony Trail**_\n\n**4914** _ **White Water Sam**_ **** Manson International, 1979. 87 min. Color. D-SC: Keith Larsen. With Keith Larsen, Lorne Greene (narrator). A man faces danger from the elements and Indians in the great Northwest. Another outdoor action feature from producer-director-writer-star Keith Larsen and it should please nature fans; originally called _**Run or Burn**_.\n\n**4915** _ **White Wilderness**_ **** Buena Vista, 1958. 73 min. Color. D-SC: James Algar. With Winston Hibler (narrator). Life in the Arctic is presented both in the warm and frigid months, dealing with animal life in the region. Another top notch Disney family documentary.\n\n**4916** _ **White Wolves: A Cry in the Wild II**_ **** Concorde, 1993. 90 min. Color. D-SC: Catherine Cyran. With Matt McCoy, David Moscow, Mark-Paul Gosselaar, Ami Dolenz, Amy O'Neill, Marc Riffon, Eric Drachman. Five teens trek through the mountains on a pleasure trip until they run into danger and a mystical white wolf. Pretty good family adventure film sequel to _**A Cry in the Wild**_ (q.v.) from producer Julie Corman.\n\n**4917** _ **White Wolves II: Legend of the Wild**_ **** Concorde\/New Horizon, 1995. 87 min. Color. D: Terence H. Winkless. SC: Dylan Kelsey Hadley. With Elizabeth Berkley, Ele Keats, Jeremy London, Corin Nemec, Emie Reyes, Jr., Justin Whalin. A naturalist, accompanied by troubled teens, tries to save the wolf population in a remote area as they encounter the legendary white wolf. Producer Julie Corman's third adventure outing, this one only fair.\n\n**4918** _ **White Wolves III: Cry of the White Wolf**_ **** Concorde, 2000. 87 min. Color. D-SC: Victoria Muspratt. With Mick Cain, Mercedes McNab, Rodney A. Grant, Robin Clarke, Tracy Brooks Swope, Margaret Howell, Frederick Dawson, David Campbell. Stranded in the Canadian wilds when their plane crashes, a trio of teenagers survive with the help of a white wolf. Poor rehash of previous \"White Wolves\" adventures.\n\n**4919** _ **Who Killed Johnny R?**_ **** CCC Filmkunst\/Tilma Films, 1966. 91 min. Color. D: Jose Luis Madrid. SC: Ladislaus Fodor and Paul Jarrico. With Lex Barker, Joachim Fuschsberger, Marianne Koch, Ralf Wolter, Barbara Bold, Sieghardt Rupp, Carlos Otero. A hunted Arizona outlaw is thought to be dead until a gun salesman is mistaken for him and almost lynched. Pretty fair Spanish-West German co-production called _**5000 Dollar fur der Kopf von Jonny R**_ ($5,000 for the Head of Johnny R) in West Germany and _**La Balada de Johnny Ringo**_ (The Ballad of Johnny Ringo) in Spain; also known as _**Kill Johnny R**_.\n\n_**Who Killed the Mysterious Mr. Foster?**_ see _**Sam Hill: Who Killed the Mysterious Mr. Foster?**_\n\n_**Who Killed Waring?**_ see _**Blazing the Western Trail**_\n**4920** _ **Why Kill Again?**_ **** Balcazar, 1967. 92 min. Color. D: Jose Antonio de la Loma. SC: Glen Vincent Davis (Vincenzo Musolino). With Anthony Steffen, Evelyn Stewart (Ida Galli), Aldo Berti, Hugo Blanco, Gemmo Cuervo, Pepe Calvo, Jose Torres, Franco Pesce. Swearing vengeance on the men who crippled him, an Army deserter causes a feud between two families. Another in the long string of violent oaters from Spain, also called _**Blood at Sundown**_ and _**Stop the Slayings**_.\n\n**4921** _ **Wichita**_ **** Allied Artists, 1955. 81 min. Color. D: Jacques Tourneur. SC: Daniel B. Ullman. With Joel McCrea, Vera Miles, Lloyd Bridges, Wallace Ford, Edgar Buchanan, Peter Graves, Keith Larsen, Carl Benton Reid, John Smith, Walter Coy, Walter Sande, Robert Wilke, Rayford Barnes, Jack Elam, Mae Clarke, I. Stanford Jolley, Kermit Maynard, Gene Wesson. In 1874 Wyatt Earp agrees to become sheriff of Wichita and combat the lawless forces operating there. Strong oater with fine work by Joel McCrea as Wyatt Earp; Tex Ritter sings the title song.\n\n**4922** _ **The Wicked Die Slow**_ **** Cannon, 1968. 75 min. Color. D: William K. Hennigar. SC: Gary Allen and Jeff Kanew. With Gary Allen, Steve Rivard, Jeff Kanew, Susannah Campbell, Yolanda Signorelli, Richard Palenske, Helen Srewart, Samantha Worthington, Racine. Searching for the renegade Indians who raped and murdered his girl friend, a stranger arrives in a remote town and helps a vagabond and his daughter attacked by outlaws. Tacky, soft core Western filmed in New Jersey.\n\n_**The Wicked, Wicked West**_ see _**Painted Angels**_\n\n**4923** _ **Wide Open Town**_ **** Paramount, 1941. 79 min. D: Lesley Selander. SC: Harrison Jacobs and J. Benton Cheney. With William Boyd, Andy Clyde, Russell Hayden, Evelyn Brent, Victor Jory, Morris Ankrum, Bernice Kay (Cara Williams), Kenneth Harlan, Roy Barcroft, Glenn Strange, Ed Cassidy, Jack Rockwell, Robert Kortman, George Cleveland, Charles Stevens, Frank Darien, Wen Wright, Lee Shumway, Chuck Morrison, Ethan Laidlaw, Ed Brady, Hank Bell. Hoppy, California and Lucky ride into a town looking for stolen Bar 20 cattle and learn a lady saloon owner and another crook are trying to get rid of the mayor-newspaperman. Very good \"Hopalong Cassidy\" feature with nice production values, fine performances and excellent photography by Sherman A. Rose.\n\n**4924** _ **The Wild and the Innocent**_ **** Universal-International, 1959. 85 min. Color. D: Jack Sher. SC: Sy Gomberg. With Audie Murphy, Joanne Dru, Gilbert Roland, Sandra Dee, Jim Backus, Peter Breck, Strother Martin, George Mitchell, Wesley Marie Tackett, Betty Harford, Mel Leonard, Lillian Adams, William Fawcett, Val Benedict. A peace loving trapper becomes involved with a runaway girl and during a town festival is forced to defend her in a gunfight. Well made and entertaining drama.\n\n**4925** _ **Wild and Woolly**_ **** 20th Century\u2013Fox, 1937. 64 min. D: Alfred Werker. SC: Lynn Root and Frank Fenton. With Jane Withers, Walter Brennan, Pauline Moore, Carl \"Alfalfa\" Switzer, Jackie Searl, Berton Churchill, Douglas Fowley, Robert Wilcox, Douglas Scott, Lon Chaney, Jr., Frank Melton, Syd Saylor, John Beck, Joseph E. Bernard, Sidney Fields, Fred Kelsey, Roger Gray, Eddy Waller, Josephine Drimmer, Alice Armand, Sidney Jarvis, Romaine Callender, Russ Clark, Vester Pegg, Alex Palasthy, Erville Alderson. During an annual Pioneer Day celebration, crooks use a feud between two families as a blind for their plans to rob the bank. Too much Jane Withers and not enough action hamper this comedy Western.\n\n**4926** _ **Wild and Wooly**_ **** Artcraft-Paramount, 1917. 70 min. D: John Emerson. SC: Anita Loos. With Douglas Fairbanks, Eileen Percy, Sam De Grasse, Monte Blue, Walter Bytell, J.W. Jones, Forest Seabury, Joseph Singleton, Tom Wilson, Charles Stevens. A man goes West and finds quite a different picture from the one he encountered in books. Fun silent Douglas Fairbanks feature based on a story by Horace B. Carpenter.\n\n**4927** _ **Wild and Wooly**_ **** ABC-TV, 1978. 100 min. Color. D: Philip Leacock. SC: Earl W. Wallace. With Chris DeLisle, Susan Bigelow, Elyssa Davalos, Doug McClure, Ross Martin, Vic Morrow, David Doyle, Paul Burke, Jessica Walter, Sherry Bain, Kenneth Tobey, Robert Wilke, Charles Siebert, Med Florey, Joan Crosby. Three beautiful women escape from an Arizona territorial prison and try to prevent an assassination attempt against President Theodore Roosevelt. Poor TV movie.\n\n**4928** _ **Wild Beauty**_ **** Universal, 1927. 60 min. D: Henry MacRae. SC: Edward Meagher and Tom Reed. With Rex (horse), June Marlowe, Hugh Allan, Scott Seaton, Hayes Robinson, William Bailey, J. Gordon Russell, Jack Pratt, Valerie (horse). A soldier brings home from the war a horse suffering from shellshock, planning to use her to help his girl's father win a big race but the man's enemies capture a wild stallion to run as opposition. Okay silent action feature.\n\n**4929** _ **Wild Beauty**_ **** Universal, 1946. 59 min. D: Wallace Fox. SC: Adele Buffington. With Don Porter, Lois Collier, Jacqueline De Wit, Robert Wilcox, George Cleveland, Buzz Henry, Dick Curtis, Eva Puig, Pierce Lyden, Roy Brent, Isabel Withers, Hank Patterson, Wild Beauty (horse). A school teacher from the East goes to work on an Arizona Indian reservation where she befriends a boy and tries to help him save a wild horse herd. Pretty fair family oriented program film.\n\n**4930** _ **Wild Bill**_ **** United Artists, 1995. 98 min. Color. D-SC: Walter Hill. With Jeff Bridges, Ellen Barkin, John Hurt, Diane Lane, Keith Carradine, David Arquette, Christina Applegate, Bruce Dern, James Gammon, Marjoe Gortner, James Remar, Karen Huie, Steve Reevis, Robert Knott, Pato Huffman, Patrick Gorman, Lee de Broux, Stoney Jackson, Robert Peters, Steven Chambers, Jimmy Medearis, Jason Ronard, Dennis Hayden, Teresa Gilmore, John Dennis Johnston, Boots Southerland, James Michael Taylor, Loyd Catlett, Janel Moloney, Ted Markland, Monty Stuart, Merritt Ohnka, Dennis Deveaugh, Jim Wilkey, Raleigh Wilson, Charles Gunning, Chris Doyle, Virgil Frye, Lauren Abels, Ritt Henn, Lise Hibolt, Charles Seybert, Luana Anders, Roland Nip, Mike Watson, Thomas Wilson Brown, Robert Kieth, Linda Harrison, Patricia M. Peters, Anthony De Longis, Bill Bolender, Alisa Christensen, Patricia Pretzinger, Peter Jason, Joseph Crozier, Mikey LeBeau, Jaime Elysse, Jaime Marsh, Burton Gilliam, Del Roy, Steve Brasfield, Juddson Keith Linn, Trisha Munford, Andre Alexsen. Aging Wild Bill Hickok comes to Deadwood where he again encounters Calamity Jane and is hunted by vengeful Jack McCall. Psychological biopic of the famed lawman often told in confusing flashbacks.\n\n**4931** _ **Wild Bill Hickok**_ **** Paramount-Artcraft, 1923. 70 min. D: Clifford S. Smith. SC: J.G. Hawk. With William S. Hart, Ethel Grey Terry, Kathleen O'Connor, James Farley, Jack Gardner, Carl Gerard, William Dyer, Bert Sprotte, Leo Willis, Nada Carle, Herschel Mayall. Following the Civil War, gunman Bill Hickok heads to Dodge City to become a gambler but ends up the town's sheriff and opposed to an outlaw gang leader. Star William S. Hart wrote the story for this sentimental screen biography; not one of his better efforts.\n\n**4932** _ **Wild Bill Hickok Rides**_ **** Warner Bros., 1942. 81 min. D: Ray Enright. SC: Charles Grayston, Paul Gerald Smith and Raymond Schrock. With Constance Bennett, Bruce Cabot, Warren William, Walter Catlett, Betty Brewer, Ward Bond, Russell Simpson, Frank Wilcox, Howard Da Silva, Trevor Bardette, Lillian Yarbo, Lucia Carroll, Faye Emerson, Julie Bishop, Elliott Sullivan, Richard Botiller, Ray Teal, J. Farrell MacDonald, Cliff Clark, Hobart Bosworth, Frank Mayo, Stuart Holmes, Charles Middleton, Francis McDonald, Alan Bridge, William Gould, Karl Hackett, Joseph Crehan, Dorothy Vaughan, Harry Woods, Frank M. Thomas, Ferris Taylor, Arthur Loft, Hank Mann, Forrest Taylor, Chief Thundercloud, Fred Kelsey, Harry Cording, Bud Osborne, Eddy Waller, Jack Mower, Charles K. French, Davison Clark, Sammy McKim, Francis Sayles, Herbert Heywood, Paul E. Burns, Walter Soderling, Howard Mitchell, Robert Homans, Sarah Padden, Frank Pharr, Bud Jamison, Jack \"Tiny\" Lipson, Pat McVeigh, Cliff Saum, Robert Strange, Lew Harvey, Georgia Caine, Albert Russell, Victor Zimmerman, Mary Thomas, John Maxwell. Wild Bill Hickok tries to stop a ruthless man from setting up his own Western empire. Despite a good cast and production values this is a pedestrian effort.\n\n**4933** _ **Wild Brian Kent**_ **** 20th Century\u2013Fox, 1936. 60 min. D: Howard Bretherton. SC: Earle Snell, Don Swift and Don Gruen. With Ralph Bellamhy, Mae Clarke, Helen Lowell, Stanley Andrews, Lew Kelly, Eddy Chandler, Richard Alexander, Jack Duffy. A self-centered playboy learns to become a man when he helps a woman save her ranch from a gang of crooks. Adequate adaptation of Harold Bell Wright's _The Re-creation of Brian Kent_ , first filmed under that title by Principal in 1925 with Kenneth Harlan, Helene Chadwick, Mary Carr, ZaSu Pitts and Rosemary Theby.\n\n**4934** _ **The Wild Bunch**_ **** Warner Bros.\u20137 Arts, 1969. 140 min. Color. D: Sam Peckinpah. SC: Walon Green and Sam Peckinpah. With William Holden, Ernest Borgnine, Robert Ryan, Edmond O'Brien, Warren Oates, Jaime Sanchez, Ben Johnson, Emilio Fernandez, Strother Martin, L.Q. Jones, Albert Dekker, Bo Hopkins, Dub Taylor, Jorge Russek, Alfonso Arau, Aurora Clavel, Elsa Cardenas, Fernando Wagner, Paul Harper, Constance White, Lilia Richards. A trail tired outlaw gang is forced to agree to rob a gun supply train for an enemy of Pancho Villa, resulting in a massacre. Exceedingly bloody and violent horse opera which has its main appeal to Sam Peckinpah fans; well done. The film was cut to 123 minutes for TV showings and is also available in 134-, 142- and 144-minute versions.\n\n**4935** _ **Wild Country**_ **** Producers Releasing Corporation, 1947. 59 min. D: Ray Taylor. SC: Arthur Orloff. With Eddie Dean, Roscoe Ates, Peggy Wynne, Douglas Fowley, I. Stanford Jolley, Steve Clark, Henry Hall, Lee Roberts, Forrest Matthews, William Fawcett, Richard Cramer, The Sunshine Boys (Eddie Wallace, J.D. Sumner, M.H. Richman, Freddie Daniel), Charles Jordan, Carl Mathews. Two marshals are after an escaped convict who killed the lawman who sent him to jail and took over his ranch. Standard Eddie Dean opus in which the title song (which he co-wrote) is better than the picture.\n\n**4936** _ **The Wild Country**_ **** Buena Vista, 1971. 100 min. Color. D: Robert Totten. SC: Calvin Clements, Jr. and Ralph Moody. With Steve Forrest, Vera Miles, Jack Elam, Ronny Howard, Frank DeKova, Morgan Woodward, Clint Howard, Dub Taylor, Woodrow Chambliss, Karl Swenson, Mills Watson. The story of a Pittsburgh, Pennsylvania, family's journey to Wyoming in the 1880s. Pretty fair Disney film for the family trade.\n\n**4937** _ **The Wild Dakotas**_ **** Associated Film, 1956. 75 min. D: Sam Newfield. SC: Thomas W. Blackburn. With Bill Williams, Coleen Gray, Jim Davis, John Litel, Dick Jones, Lisa Montell, John Miljan, I. Stanford Jolley, Wally Brown, Bill Henry, Iron Eyes Cody, Bill Dix. A trail guide is at odds with a corrupt wagon train leader who wants to settle in a valley belonging to Indians who will fight for their land. A good cast can do little to help this sub-standard feature.\n\n**4938** _ **The Wild Frontier**_ **** Republic, 1947. 59 min. D: Philip Ford. SC: Albert DeMond. With Allan \"Rocky\" Lane, Jack Holt, Eddy Waller, Pierre Watkin, John James, Roy Barcroft, Wheaton Chambers, Tom London, Sam Flint, Budd Buster, Ted Mapes, Bob Burns, Art Dillard, Bud McClure. A lawman helps an old sheriff and his sons fight a gang led by the town's corrupt saddle shop owner. First film in which Allan Lane was billed as \"Rocky\" and a good start to his \"Famous Westerns\" series, buoyed by Jack Holt as the villain.\n\n**4939** _ **Wild Fury**_ **** Ambassador Releasing, 1975. 90 min. Color. SC: Richard Wiles. Three men journey across the Alaskan wilderness in quest of a killer bear. Okay documentary.\n\n**4940** _ **Wild Geese Calling**_ **** 20th Century\u2013Fox, 1941. 77 min. D: John Brahm. SC: Horace McCoy. With Henry Fonda, Joan Bennett, Warren William, Ona Munson, Barton MacLane, Russell Simpson, Iris Adrian, James Morton, Paul Sutton, Mary Field, Stanley Andrews, Jody Gilbert, Robert Emmett Keane, Michael (Adrian) Morris, George Watts, Charles Middleton, Paul E. Burns, Jack Pennick, Nestor Paiva, George Melford, Tom London, Alan Bridge, Lee Phelps, Captain Anderson, Joe Bernard. A lumberjack with wanderlust is helped by a saloon woman as they battle crooks in Oregon and Alaska in the 1890s. One of those pictures that seems to care more about budget and detail than plot.\n\n_**Top:**_ **William Holden in** _**The Wild Bunch**_ **(Warner Bros.** **\u2013** **7 Arts, 1969).** _**Bottom:**_ **Warren William in** _**Wild Geese Calling**_ **(20th Century** **\u2013** **Fox, 1941).**\n\n**  \n**\n\n**4941** _ **Wild Gold**_ **** Fox, 1934. 75 min. D: George Marshall. SC: Dudley Nichols and Lamar Trotti. With John Boles, Claire Trevor, Harry Green, Roger Imhof, Ruth Gillette, Monroe Owsley, Edward Gargan, Suzanne Kaaren, Blanca Vischer, Elsie Larson, Gloria Roy, Winifred Shaw, Myrla Bratton. During the California Gold Rush an engineer loses his job because of his infatuation with a saloon girl whose husband arrives with an old prospector's claim. Fair melodrama with good work by the two leads.\n\n**4942** _ **Wild Grizzly**_ **** Monarch, 2000. 96 min. Color. D: Sean McNamara. SC: Jeff Phillips and Sean McNamara. With Riley Smith, Michele Greene, Fred Dryer, Courtney Peldon, Steve Reevis, Daniel Baldwin, Brendan O'Brien, John O'Hurley, Ron Rogge, Valerie Bickford, Chris Doyle, Tarren Noel Wilson, Patrick Ecclesine, Carlos Sanchez, Sean McNamara, Teresa Best. A teenager, whose policeman father was killed in the line of duty, moves with his mother to a remote town and tries to save a grizzly bear from being destroyed. Appealing TV movie.\n\n**4943** _ **Wild Heritage**_ **** Universal-International, 1958. 78 min. Color. D: Charles Haas. SC: Paul King and Joseph Stone. With Will Rogers, Jr., Maureen O'Sullivan, Troy Donahue, Gigi Perreau, Paul Birch, George Winslow, Casey Tibbs, Judi Meredith, Rod McKuen, Gary Gray, Jeanette Nolan, John Berardino, Phil Harvey, Lawrence Dobkin, Stephen Ellsworth, Ingrid Goude, Christopher Dark, Guy Wilkerson. Two families find their lives become intertwined as they migrate to the West. Sentimental stuff, but well made.\n\n**4944** _ **Wild Horse**_ **** Allied, 1931. 77 min. D: Richard Thorpe and Sidney Algier. SC: Jack Natteford. With Hoot Gibson, Alberta Vaughn, Stepin Fetchit, Edmund Cobb, Skeeter Bill Robbins, Neal Hart, George Bunny, Ed Peil, Sr., Joe Rickson, Glenn Strange, Slim Whitaker, Pete Morrison, Fred Gilman, Frank Ellis, Ben Corbett, Hank Bell, Tom Smith, Silver Tip Baker. A dishonest rodeo bronco buster murders a cowboy, who with his partner captured a wild stallion, and the pal is falsely blamed for the crime. A bit rawboned but an ingratiating Hoot Gibson vehicle, cut to 57 minutes for TV; reissued as _**Silver Devil**_.\n\n**4945** _ **Wild Horse Ambush**_ **** Republic, 1952. 54 min. D: Fred C. Brannon. SC: William Lively. With Michael Chapin, Eilene Janssen, James Bell, Richard Avonde, Roy Barcroft, Julian Rivero, Movita, Drake Smith, Scott Lee, Alex Montoya, John Daheim, Ted Cooper, Wayne Burson. Two youngsters help the law in tracking a counterfeiting operation. Mediocre juvenile fare; the final entry in the \"Rough Ridin' Kids\" series.\n\n**4946** _ **Wild Horse Canyon**_ **** Goodwill, 1925. 50 min. D: Ben Wilson. With Yakima Canutt, Helene Rosson, Edward (Ed) Cedil, Jay (Slim) Talbot, Boy (horse), Lad (dog). While looking for the murderer of his father, a cowboy tames a wild horse who the killer, a ranch foreman, plans to blame for thefts he plans to commit. Rather standard, but fast paced, Yakima Canutt (he co-produced with Ben Wilson) silent effort; reissued by Hollywood Film Enterprises.\n\n**4947** _ **Wild Horse Canyon**_ **** Monogram, 1938. 50 min. D: Robert Hill. SC: Robert Emmett (Tansey). With Jack Randall, Dorothy Short, Frank Yaconelli, Dennis Moore, Warner Richmond, Ed Cassidy, Walter Long, Charles King, Earl Douglas, Sherry Tansey. A cowpoke, who is looking for his brother's killer, and his pal happen on a area where a rancher and his daughter are having their horses rustled by a mysterious gang. Pretty good Jack Randall vehicle.\n\n**4948** _ **Wild Horse Hank**_ **** Film Consortium of Canada, 1979. 94 min. Color. D: Eric Till. SC: James Lee Barrett. With Linda Blair, Richard Crenna, Michael Wincott, Al Waxman, Pace Bradford, Helen Hughes, Lloyd Berry, Stephen E. Miller, Richard Fitzpatrick, James D. Morris, Michael J. Reynolds, Barbara Gordon, Gordie Tapp, Hardee Lineham. A college student works to keep horses from being butchered for dog food and tries to move a herd north so they can escape capture. Filmed in Canada, this is a scenic and fairly entertaining adventure drama.\n\n**4949** _ **Wild Horse Mesa**_ **** Paramount, 1925. 95 min. D: George B. Seitz. SC: Lucien Hubbard. With Jack Holt, Noah Beery, Billie Dove, Douglas Fairbanks, Jr., George Magrill, Edith Yorke, Bernard Siegel, Margaret Morris, Eugene Pallette, Gary Cooper, Tom Tyler. A trail rider tries to stop ranchers from capturing wild horses by the use of a barbed wire corral. Beautifully photographed (by Bert Glennon) and well acted adaptation of Zane Grey's novel.\n\n**4950** _ **Wild Horse Mesa**_ **** Paramount, 1932. 61 min. D: Henry Hathaway. SC: Harold Shumate and Frank Howard Clark. With Randolph Scott, Sally Blane, Fred Kohler, Lucille LaVerne, James Bush, Charles Grapewin, Jim Thorpe, George Hayes, Buddy Roosevelt, E.H. Calvert, Ted Adams, Jack Pennick. A horse trainer objects to roundup methods using barbed wire which may injure and kill the animals. Okay program feature with liberal use of footage from the 1925 version (q.v.).\n\n**4951** _ **Wild Horse Mesa**_ **** RKO Radio, 1947. 60 min. D: Wallace Grissell. SC: Norman Houston. With Tim Holt, Richard Martin, Nan Leslie, Richard Powers (Tom Keene), Jason Robards, Tony Barrett, Harry Woods, William Gould, Robert Bray, Richard Foote, Frank Yaconelli, John Elliott. Three cowpokes corral a herd of wild horses to sell but crooks try to take the animals for themselves. Well produced Tim Holt vehicle that bears little resemblance to the Zane Grey work.\n\n**4952** _ **Wild Horse Phantom**_ **** Producers Releasing Corporation, 1944. 56 min. D: Sam Newfield. SC: George Milton (Milton Raison and George Wallace Sayre). With Buster Crabbe, Al St. John, Elaine Morey, Hal Price, Kermit Maynard, Budd Buster, Frank Ellis, Frank McCarroll, Robert Meredith, John Elliott, Bob (John) Cason, Slim Whitaker, Reed Howes, Bud Osborne, Steve Clark, George Morrell, Herman Hack, Curley Dresden, Ed Peil, Sr., Jimmy Aubrey, Hank Bell, Jack Tornek. Billy Carson and Fuzzy Q. Jones work out a plan with a prison warden to let a convict escape so they can trail him to where he hid loot stolen from a bank job. Average \"Billy Carson\" series entry somewhat buoyed by horror trappings, including a haunted mine, a bogus ghost and the title prop from _**The Devil Bat**_ (PRC, 1941).\n\n**4953** _ **Wild Horse Range**_ **** Monogram, 1940. 58 min. D: Raymond K. Johnson. SC: Carl Krusada. With Jack Randall, Phyllis Ruth, Frank Yaconelli, Charles King, Tom London, Marin Sais, Ralph Hoopes, Forrest Taylor, George Chesebro, Carl Mathews, Ted Adams, Steve Clark, Tex Palmer. A cowboy gets on the trail of a gang of rustlers. Jack Randall fans will like this pleasing feature.\n\n**4954** _ **Wild Horse Rodeo**_ **** Republic, 1937. 55 min. D: George Sherman. SC: Betty Burbridge and Oliver Drake With Robert Livingston, Ray Corrigan, Max Terhune, June Martel, Walter Miller, Edmund Cobb, William Gould, Jack Ingram, Henry Isabell, Art Dillard, Ralph Robinson, Fred \"Snowflake\" Toones, Dick Weston (Roy Rogers), Jack Kirk, Kermit Maynard, Frank Ellis, Bob Burns, Bob Card, Jerry Frank, Charles Murphy, June Gittelson, Harry Willingham, Duke Green. The Three Mesquiteers capture a wild horse for a small rodeo's chief attraction and crooks try to steal the prize animal. Action filled entry in the long running series.\n\n**4955** _ **Wild Horse Roundup**_ **** Ambassador, 1936. 57 min. D: Alan James. SC: Joseph O'Donnell. With Kermit Maynard, Betty Lloyd (Beth Marion), Dickie Jones, Budd Buster, John Merton, Frank Hagney, Roger Williams, Dick Curtis, Jack Ingram, Nelson McDowell, Frank McCarroll, Bud Pope. A cowboy and his pals help a woman rancher being forced off her property by the mysterious \"Night Riders,\" a gang led by a man trying to buy all the nearby land needed for a railroad right-of-way. Pretty fair Kermit Maynard vehicle for producer Maurice Conn; supposedly based on a James Oliver Curwood story.\n\n**4956** _ **Wild Horse Rustlers**_ **** Producers Releasing Corporation, 1943. 58 min. D: Sam Newfield. SC: Joseph O'Donnell and Steve Braxton. With Robert Livingston, Al St. John, Linda Leighton, Lane Chandler, Stanley Price, Frank Ellis, Karl Hackett, Jimmy Aubrey, Ben Corbett, Curley Dresden, Artie Ortego, Kansas Moehring, Silver Harr. The Lone Rider discovers a Nazi spy has taken his twin brother's place as the foreman of a ranch in order to sabotage the government's horse procurement program. The plot of Axis-out-West adds a bit of zest to this otherwise routine \"Lone Rider\" adventure.\n\n**4957** _ **Wild Horse Stampede**_ **** Monogram, 1943. 59 min. D: Alan James. SC: Elizabeth Beecher. With Ken Maynard, Hoot Gibson, Betty Miles, Ian Keith, Bob Baker, I. Stanford Jolley, Forrest Taylor, Glenn Strange, Si Jenks, Donald Stewart, John Bridges, Reed Howes, Kenneth Harlan, Tom London, Tex Palmer, Kenne Duncan, Bob McKenzie, Chick Hannon, George Sowards, Augie Gomez. Two former lawmen come to the aid of a U.S. marshal trying to track a gang raiding railroad supply trains. The two stars are colorful and help pull together this minor opener for \"The Trail Blazers\" series but Bob Baker is poor as the federal man.\n\n**4958** _ **Wild Horse Valley**_ **** Metropolitan, 1940. 55 min. D: Ira Webb. SC: Carl Krusada. With Bob Steele, Phyllis Adair, George Chesebro, Ted Adams, Lafe McKee, Buzz Barton, Jimmy Aubrey, Bud Osborne, Tex Phelps, Robert Walker, Denver Dixon, Pirate (horse). Outlaws steal a prize Arabian stallion and a cowboy tries to get him back. Very low grade Bob Steele outing for producer Harry S. Webb.\n\n**4959** _ **Wild Horses**_ **** Satori Entertainment, 1984. 88 min. Color. D: Derek Morton. SC: Kevin Wilson. With Keith Aberdein, John Bach, Robyn Gibbes, Kevin Wilson, Kathy Rawlins, Helena Wilson, Tom Peata, Marshall Napier, Bruno Laurence. A free spirited cowboy attempts to make his way in the world by capturing and selling wild mustangs. Pleasant independent modern-day Western.\n\n**4960** _ **Wild Horses**_ **** CBS-TV, 1984. 104 min. Color. D: Dick Lowry. SC: Roderick Taylor and Daniel Vining. With Kenny Rogers, Pam Dawber, Ben Johnson, Richard Farnsworth, David Andrews, Richard Masur, Richard Hamilton, Ritch Brinkley, Karen Carson, Buck Taylor, Kelly Yunkerman, Cathy Worthington, R.W. Hampton, Brian Rogers, Jamie Fleenor, Dawn Holder, Roddy Salazar, Beckie Hinton, Charles H. Hunt, Dave Lowry, Jay H. Zirbel. A faded rodeo star goes on a wild horse roundup and helps a woman expose a corrupt agent's plan for the animals. There is nothing special in this made-for-TV oater.\n\n**4961** _ **Wild Mustang**_ **** Ajax, 1935. 62 min. D: Harry Fraser. SC: Weston Edwards. With Harry Carey, Barbara Fritchie, Del Gordon, Kathryn Johns, Robert Kortman, George Chesebro, Roger Williams, Chuck Morrison, Richard Botiller, George Morrell, Milburn Morante, Francis Walker, Budd Buster Phil Dunham, Sonny (horse). When outlaws brand his son so he will be forced to join them, an old time lawman takes up his badge to round up the thugs. Poor production values hurt this otherwise pleasant Harry Carey film with Robert Kortman excellent as the wicked gang leader.\n\n**4962** _ **The Wild North**_ **** Metro-Goldwyn-Mayer, 1952. 97 min. Color. D: Andrew Marton. SC: Frank Fenton. With Stewart Granger, Wendell Corey, Cyd Charisse, Morgan Farley, Howard Petrie, Houseley Stevenson, Lewis Martin, John War Eagle, Ray Teal, Clancy Cooper, J.M. Kerrigan, Henry Corden, Robert Stephenson, G. Pat Collins, Russ Conklin, Brad Morrow, Emile Meyer, Henri Letondal, Holmes Herbert, Cliff Taylor, Rex Lease, James Dime. A trapper falsely accused of murder falls in love with an Indian maiden as he is tracked in the north country by a Mountie. Idaho locales and color photography (by Robert Surtees) are the main assets of this otherwise pedestrian drama. Alternate title: _**The Big North**_.\n\n**4963** _ **The Wild Pony**_ **** Wonder Works, 1983. 87 min. Color. D: Kevin Sullivan. SC: Eda Lehman and Kevin Sullivan. With Marilyn Lightstone, Art Hindle, Josh Byrne, Kelsey McLeod, Paul Jolicoeur, Jack Ackroyd, Tommy Banks, Murrat Ord, Bob Collins, Mark Key, Philip Clark, Jack Goth, Ron Tucker, Roberta Meili. After accidentally killing a man in a fight, a Colorado rancher marries his widow and later permits her son to keep a wild pony. Minor, but enjoyable family fare.\n\n**4964** _ **Wild Prairie**_ **** American National Enterprises, 1975. 92 min. Color. With Larry Jones. Explorer Larry Jones roams the territory of the Southwestern United States observing its terrain and wildlife. Well done documentary.\n\n_**A Wild Ride**_ see _**Ride a Wild Stud**_\n\n**4965** _ **Wild Rovers**_ **** Metro-Goldwyn-Mayer, 1971. 106 min. Color. D-SC: Blake Edwards. With William Holden, Ryan O'Neal, Karl Malden, Lynn Carlin, Tom Skerritt, Joe Don Baker, James Olson, Leora Dana, Moses Gunn, Victor French, Rachel Roberts, Charles Gray, Sam Gilman, William Bryant, Jack Garner, Caitlin Wyles, Mary Jackson, William Lucking, Ed Bakey, Ted Gehring, Alan Carney, Ed Long, Lee De Broux, Bennie Dobbins, Boyd \"Red\" Morgan, Bob Beck, Geoffrey Edwards, Studs Tanney, Hal Lynch, Dick Crockett, Bruno VeSota. Two bored cattle drovers pull a bank robbery as a lark and after a wild spree find themselves being relentlessly hunted by a posse. This Western tries hard to be a classic but merely defeats its purpose and ends up slightly better than mediocre, although it is not without interest.\n\n**William Holden and Ryan O'Neal in** _**Wild Rovers**_ **(Metro-Goldwyn-Mayer, 1971).**\n\n**  \n**\n\n**4966** _ **Wild Stallion**_ **** Monogram, 1952. 72 min. Color. D: Lewis D. Collins. SC: Dan Ullman. With Ben Johnson, Edgar Buchanan, Martha Hyer, Hayden Rorke, Hugh Beaumont, Orley Indgren, Don Haggerty, Susan Odin, I. Stanford Jolley, Barbara Woodell, John Halloran. A cavalry lieutenant recalls his life in a military school and how another man helped him settle into the service. Flashback drama which is a sad waste of a good cast.\n\n**4967** _ **Wild Stampede**_ **** Producciones Raul de Anda, 1962. 77 min. Color. D-SC: Raul de Anda. With Luis Aguilar, Christiane Martel, Augustin de Anda, Jose Elias Moreno, Armando Soto la Marina \"Chicote,\" Jose Eudardo Perez, Yerye Beirute, Jose Chavez, Guillermo Alvarez Bianchi. A herd of wild horses is fought over by an outlaw gang and a band of revolutionaries. Action filled Mexican production first issued in that country in 1959 as _**Estampida**_ (Stampede).\n\n**4968** _ **Wild Times**_ **** Metromedia, 1980. 195 min. Color. D: Richard Compton. SC: Don Balluck. With Sam Elliott, Ben Johnson, Bruce Boxleitner, Penny Peyser, Timothy Scott, Cameron Mitchell, Gene Evans, Harry Carey, Jr., Leif Erickson, L.Q. Jones, Buck Taylor, Pat Hingle, Dennis Hopper, Trish Stewart, Chuck Hayward, William Smith, Geno Silva, R.L. Tolbert, Ben Zeller, Jose Masengale, Douglas Doran, George Stokes, Chris Noel Hanks, Arthur Wagner, Marianne Marks, Vernon Weddle, George B. Nason, Bob Tzudiker, Bill Hicks, Kenny Call. A Wild West show star attraction finds himself being hunted by the husband of the woman he once loved. Well acted and produced TV outing, based on Brian Garfield's novel.\n\n**4969** _ **Wild West**_ **** Producers Releasing Corporation, 1946. 75 min. Color. D: Robert Emmett Tansey. SC: Frances Kavanaugh. With Eddie Dean, Roscoe Ates, Al \"Lash\" LaRue, Robert \"Buzzy\" Henry, Louise Currie, Jean Carlin, Sarah Padden, Lee Bennett, Terry Frost, Warner Richmond, Lee Roberts, Chief Yowlachie, Bob Duncan, Frank Pharr, Matty Roubert, John Bridges, Al Ferguson, Bud Osborne. A singing cowboy tries to stop outlaws from stirring up local Indians against the building of telegraph lines. Eddie Dean's final Cinecolor outing is a fast moving and well done feature; reissued in 1948 in black and white, minus 15 minutes, as a new film entitled _**Prairie Outlaws**_ (q.v.).\n\n**4970** _ **The Wild West**_ **** Trans America Film Distributors, 1977. 100 min. Part Color. With Charles Bronson, Clint Eastwood, Steve McQueen, John Wayne, Ernest Borgnine, William Boyd, Raymond Burr, Lee Van Cleef, Broderick Crawford, John Derek, Angie Dickinson, Bill Elliott, Henry Fonda, Glenn Ford, William Holden, Rita Hayworth, Ben Johnson, Lash LaRue, Fred MacMurray, Ken Maynard, Joel McCrea, Tim McCoy, Robert Mitchum, Leonard Nimoy, Maureen O'Hara, Gregory Peck, Roy Rogers, Mickey Rooney, Randolph Scott, Barbara Stanwyck, Robert Vaughn, Ray Owens (narrator). A compilation feature made up of clips of Westerns from the past, issued briefly in the U.S. as well as Australia and New Zealand.\n\n**4971** _ **Wild West Days**_ **** Universal, 1937. 13 Chapters. D: Cliff Smith and Ford Beebe. SC: Wyndham Gittens, Norman S. Hall and Ray Trampe. With Johnny Mack Brown, Lynn Gilbert, Frank McGlynn, Jr., Walter Miller, Russell Simpson, Frank Yaconelli, Robert Kortman, George Shelley, Bob McClung, Bud Osborne, Lafe McKee, Iron Eyes Cody, Francis McDonald, Charles Stevens, Joe Girard, Sidney Bracey, Alan Bridge, Ed LeSaint, Bruce Mitchell, Frank Ellis, Chief Thundercloud, Jack Clifford, Hank Bell, William Royle, Mike Morita, Chief Thunderbird. Three frontiersmen help a woman and her brother, who has been framed on a murder charge, whose ranch is being raided by outlaws trying to find gold. Fast moving cliffhanger but an exaggerated version of W.R. Burnett's novel _Saint Johnson_ , filmed previously by Universal as _**Law and Order**_ (q.v.).\n\n**4972** _ **Wild West Whoopee**_ **** Cosmos, 1931. 57 min. D-SC: Robert J. Horner. With Jack Perrin, Josephine Hill, Buzz Barton, Fred Church, John Ince, George Chesebro, Horace B. Carpenter, Henry Roquemore, Ben Corbett, John Ince, Charles Austin, Walter Patterson, Starlight (horse). A rodeo rider tries to save a pretty girl from the attentions of a bad man. Really poor, with a plethora of stock rodeo footage.\n\n**4973** _ **The Wild Westerners**_ **** Columbia, 1962. 70 min. Color. D: Oscar Rudolph. SC: Gerald Drayson Adams. With James Philbrook, Nancy Kovack, Guy Mitchell, Duane Eddy, Hugh Sanders, Elizabeth MacRae, Marshall Reed, Nestor Paiva, Harry Lauter, Bob Steele, Ilse Burkert, Terry Frost, Don C. Harvey, Francis Osborne, Tim Sullivan, Pierce Lyden, Joe McGuinn, Charles Horvath, Henry Wills, Dan White. A marshal and his new bride try to carry gold across the desert for the Northern cause during the Civil War but find themselves at odds with Indians and a renegade lawman and his cohorts. Genre fans will like the cast better than the story.\n\n**Hugh Sanders, James Philbrook, Bob Steele and Duane Eddy in** _**The Wild Westerners**_ **(Columbia, 1962).**\n\n**  \n**\n\n**4974** _ **Wild Wild West**_ **** Warner Bros., 1999. 106 min. Color. D: Barry Sonnenfeld. SC: S.S. Wilson, Brent Maddock, Jeffrey Price and Peter S. Seaman. With Will Smith, Kevin Kline, Kenneth Branagh, Salma Hayek, M. Emmet Walsh, Ted Levine, Frederique van der Wal, Musetta Vander, Sofia Eng, Ling Bai, Garcelle Beauvais, Mike McGaughy, Jerry Wills, Rodney A. Grant, Buck Taylor, E.J. Callahan, Debra Christofferson, Carlos \"Gary\" Cervantes, Jerry Potter (Clayton Moore), Michael Sims, Scott Sandler, James Lashly, Dean Rader-Duval, Christian Aubert, Orestes Matacena, Ian Abercrombie, Ismael \"East\" Carlo, Bob Rumnock. President Grant enlists a Civil War hero and a U.S. marshal to capture a Southern sympathizer out to assassinate him. Slow moving Western comedy based on the popular series of the same name (CBS-TV, 1965\u201370); a box office bust.\n\n**4975** _ **The Wild Wild West Revisited**_ **** CBS-TV, 1979. 100 min. Color. D: Burt Kennedy. SC: William Bowers. With Robert Conrad, Ross Martin, Paul Williams, Harry Morgan, Rene Auberjonois, Jo Ann Harris, Trisha Noble, Alberto Morin, Skip Homeier, Joyce Jameson, Robert Shields, Lorene Yarnell, Jeff McKay, Susan Blu, Paula Ustinov, Wilford A. Brimley, Ted Hartley, Jacqueline Hyde, John Wheeler, Mike Wagner, Jeff Redford. Two government agents investigate a plot in which clones are being used to replace European royalty. Fun, tongue-in-cheek telefeature recreation of the long running \"The Wild Wild West\" (CBS-TV, 1965\u201370), followed by _**More Wild Wild West**_ (q.v.).\n\n**4976** _ **Wild Women**_ **** ABC-TV, 1970. 74 min. Color. D: Don Taylor. SC: Lou Morheim and Richard Carr. With Hugh O'Brian, Anne Francis, Marilyn Maxwell, Marie Windsor, Sherry Jackson, Robert F. Simon, Richard Kelton, Cynthia Hull, Pepe Callahan, Ed Call, Chuck Hicks, Jim Boles, Pedro Regas, Troy Melton. The U.S. government orders the Army Corps of Engineers to secretly map the Texas-Mexican border in the mid\u20131840s in case of war and five women convicts are recruited as a blind for the operation. Mediocre TV Western feature.\n\n**4977** _ **The Wild Women of Chastity Gulch**_ **** ABC-TV, 1982. 100 min. Color. D: Philip Leacock. SC: Earl W. Wallace. With Priscilla Barnes, Joan Collins, Donny Osmond, Lee Horsley, Howard Duff, Lisa Whelchel, Phyllis Davis, Pamela Bellwood, Jeannette Nolan, Morgan Brittany, Susan Kellerman. When their men go off to the Civil War, the women of a Mississippi town, both respectable and otherwise, join forces to fight a Yankee raiding party. Fair TV made comedy.\n\n**4978** _ **The Wildcat**_ **** Aywon, 1926. 50 min. D: Harry Fraser. SC: David M. Findlay. With Gordon Clifford, Charlotte Moore, Irwin Renard, Frank Bond, Hooper Phillips, Arthur Milleton. Going West to a ranch to train for a boxing match, a man finds a cache of diamonds stolen in an express holdup and hides them from the thief in order to capture him. Fairly picturesque and action laced poverty row silent feature.\n\n**4979** _ **Wildcat**_ **** Paramount, 1942. 73 min. D: Frank McDonald. SC: Maxwell Shane and Richard Murphy. With Richard Arlen, Arline Judge, Larry \"Buster\" Crabbe, William Frawley, Arthur Hunnicutt, Elisha Cook, Jr., Ralph Sanford, Alec Craig, John Dilson, Will Wright, Jessica Newcombe, Billy Benedict, Tom Kennedy, Sam Flint, Pierre Watkin, Dick Elliott, William Hall, Billy Nelson, Johnny Fisher, Fred Sherman, Edward Keane, Cy Schindell. A wildcatter and his pals drill for oil but find their operations being sabotaged by a rival. This William H. Pine-William C. Thomas production packs a lot of action.\n\n**4980** _ **Wildcat of Tucson**_ **** Columbia, 1941. 55 min. D: Lambert Hillyer. SC: Fred Myton. With Bill Elliott, Dub Taylor, Evelyn Young, Stanley Brown, Kenneth MacDonald, Ben Taggart, Edmund Cobb, George Lloyd, Sammy Stein, Francis Walker, Robert Winkler, Forrest Taylor, George Chesebro, Dorothy Andre, Bert Young, Newt Kirby, John Daheim, Murdock MacQuarrie, Jim Corey, Steve Clark, Bob Burns, Archie Ricks, Art Dillard, Ray Jones. Wild Bill Hickok and his brother help settlers cheated out of their lands by a speculator hooked up with a corrupt judge. Fairly entry in the \"Wild Bill Hickok\" series.\n\n**4981** _ **Wildcat Saunders**_ **** Atlantic, 1936. 60 min. D: Harry Fraser. SC: Monroe Talbot. With Jack Perrin, Blanche Mehaffey, William Gould, Fred \"Snowflake\" Toones, Ed Cassidy, Tom London, Roger Williams, Earl Dwire, Jim Corey, Bud Osborne, J.P. McGowan, Oscar Gahan, Tex Palmer, Ray Henderson. A boxer goes West and gets mixed up with a gang of outlaws. Passable outing in Jack Perrin's series for producer William Berke.\n\n**Lobby card for** _**Wildcat Saunders**_ **(Atlantic, 1936) picturing Jack Perrin, Tom London, Roger Williams and Fred \"Snowflake\" Toones.**\n\n**  \n**\n\n**4982** _ **Wildcat Trooper**_ **** Ambassador, 1936. 60 min. D: Elmer Clifton. SC: Joseph O'Donnell. With Kermit Maynard, Lois Wilde, Hobart Bosworth, Fuzzy Knight, Yakima Canutt, Eddie Phillips, John Merton, Frank Hagney, Roger Williams, Dick Curtis, Theodore Lorch, Hal Price, Jim Thorpe, Ben Hendricks, Jr., Wally West, Ray Henderson, Art Dillard, Tex Phelps. A member of the Royal Canadian Mounted Police is on the trail of an outlaw gang masterminded by a corrupt doctor. Better than average Kermit Maynard north woods affair, mainly thanks to Hobart Bosworth's good natured hamming as the villain. British title: _**Wild Cat**_.\n\n**4983** _ **The Wildcatter**_ **** Universal, 1937. 58 min. D: Lewis D. Collins. SC: Charles A. Logue. With Scott Colton, Jean Rogers, Jack Smart, Suzanne Kaaren, Russell Hicks, Ward Bond, Wallis Clark, Jack Powell, William Gould, Monte Montague, Henry Hall, Hattie McDaniel, James Farley, Tom Herbert, Donald Kerr, Frank Marlowe, Bob McKenzie, Jack Cheatham, Ruth Fallows, John Leeds, Frank H. Hammond, Jimmy Lucas, Jack Mack, George Ovey, Charles Murphy, Art Yeoman. Two pals leave their roadside caf\u00e9-gas station business and head to Texas to drill for oil. Okay Universal dual bill item.\n\n**4984** _ **Wilderness Calling**_ **** Aaro Films, 1969. 102 min. Color. D-SC: Paul O. Hansen. With Art Mercier (narrator). A young man follows the call of the wild from the Dakota prairies through Alaska and British Columbia to the Bering Sea. Interesting documentary shot on location.\n\n_**Wilderness Family, Part Two**_ see _**The Further Adventures of the Wilderness Family**_\n\n**4985** _ **Wilderness Journey**_ **** Gold Key Entertainment, 1970. 92 min. Color. With Tony Tucker Williams, Jimmy Kane. A Native American youth searches the Alaskan wilds for his father who he fears was injured in an accident. Well made semi-documentary with lots of beautiful scenery.\n\n**4986** _ **Wilderness Mail**_ **** Ambassador, 1935. 55 min. D: Forrest Sheldon. SC: Bennett Cohen and Robert Dillon. With Kermit Maynard, Doris Brook, Fred Kohler, Paul Hurst, Dick Curtis, Syd Saylor, Nelson McDowell, Roger Williams, Kernan Cripps, Merrill McCormick, Julian Rivero, Ray Henderson, George Morrell. Assigned to bring in the mail, a Mounted Policeman learns his twin brother is in the clutches of crooks. Scenic shots of heavy snow and dogsled action lend zest to this Kermit Maynard outing in which he has dual roles.\n\n**4987** _ **Wildfire**_ **** Screen Guild\/Action Pictures, 1945. 57 min. Color. D: Robert Tansey. SC: W.H. Tuttle. With Bob Steele, Sterling Holloway, John Miljan, Eddie Dean, William Farnum, Virginia Maples, Sarah Padden, Al Ferguson, Wee Willie Davis, Rocky Camron (Gene Alsace), Francis Ford, Frank Ellis, Hal Price, Wildfire (horse). After saving a wild horse from being shot by ranchers who think he is stealing their herds, two cowboys find themselves up against crooked traders. Pretty fair equestrian feature enhanced by Cinecolor; Eddie Dean sings \"On the Banks of the Sunny San Juan\" which he co-wrote with Glenn Strange. Also called _**Wildfire, the Story of a Horse**_\n\n_**Wildfire, the Story of a Horse**_ see _**Wildfire**_\n\n_**Will James' Sand**_ see _**Sand**_\n\n**4988** _ **Will Penny**_ **** Paramount, 1968. 106 min. Color. D-SC: Tom Gries. With Charlton Heston, Joan Hackett, Donald Pleasence, Lee Majors, Ben Johnson, Bruce Dern, Slim Pickens, Clifton James, Anthony Zerbe, Roy Jenson, J.D. Spradlin, Quentin Dean, William Schallert, Lydia Clarke, Matt Clark, Luke Askew, Anthony Costello, Chanin Hale, Stephen Edwards, Gene Rutherford, Jan Francis. Looking for work following a cattle drive, a veteran cowboy finds himself at odds with vicious rawhiders who later torture him when he is hired to be a line rider. Highly atmospheric account of the cowboy's lone existence with an especially poignant relationship between the title character and a widow; a very good film.\n\n**4989** _ **Winchester '73**_ **** Universal-International, 1950. 92 min. Color. D: Anthony Mann. SC: Robert L. Richards and Borden Chase. With James Stewart, Shelley Winters, Dan Duryea, Stephen McNally, Millard Mitchell, Charles Drake, John McIntire, Will Geer, Jay C. Flippen, Rock Hudson, John Alexander, Steve Brodie, James Millican, Abner Biberman, Tony Curtis, James Best, Gregg Martell, Frank Chase, Chuck Roberson, Carol Henry, Ray Teal, John Doucette, Chief Yowlachie, Edmund Cobb, Ethan Laidlaw, Jennings Miles, Guy Wilkerson, Gregg Martell, Virginia Mullens, Steve Darrell, Frank Conlan, Ray Bennett, Forrest Taylor, Bud Osborne, John War Eagle, Bob Anderson, Larry Olsen, Bonnie Kay Eddy. In 1873 Dodge City a cowboy wins a prize Winchester rifle in a contest only to have it stolen by the man who murdered his father. Very entertaining class \"A\" Western.\n\n**4990** _ **Winchester '73**_ **** NBC-TV\/Universal, 1967. 97 min. Color. D: Herschel Daugherty. SC: Stephen Kandel and Richard L. Adams. With Tom Tryon, John Saxon, Dan Duryea, John Drew Barrymore, Joan Blondell, John Dehner, Barbara Luna, John Doucette, David Pritchard, Paul Fix, John Hoyt, Jack Lambert, Jan Arvan, Robert Bice, Ned Romero, George Keymas. An ex-convict returns home and steals a valuable rife from his marshal brother who tries to get it back. Poor TV movie reworking of the 1950 (q.v.) near classic; Dan Duryea appears in both versions but in different roles.\n\n**4991** _ **The Wind**_ **** Metro-Goldwyn-Mayer, 1928. 88 min. D: Victor Seastrom. SC: Frances Marion. With Lillian Gish, Lars Hanson, Montagu Love, Dorothy Cumming, Edward Earle, William Orlamond, Carmencita Johnson, Laon Ramon, Billy Kent Schaefer. A young woman comes West to live with her cousin and family but when he becomes too fond of her she marries a rancher on the rebound only to be raped by an acquaintance when her husband is away on a roundup. Austere, spell binding silent classic (despite a tacked on happy ending) with Lillian Gish giving a magnificent performance as the bride nearly driven to madness by murder and incessant wind.\n\n**4992** _ **Wind River**_ **** Lionsgate Films, 1998. 98 min. Color. D: Tom Shell. SC: Elizabeth Hansen. With Blake Heron, A. Martinez, Russell Means, Wes Studi, Devon Gummersall, Karen Allen, Tim Griffin, Wayne Brennan, Brandon Baker, Pat Gordon, Rachel Hales, Dustin McQuay, Peter Looney, Everett Lightfoot, Cynthia Pyn Francisco, Ericke Willie, Falene Nemeth, Kiana Chournos, Payton Mackey, Maria Jejias, Alanzo Cody, Peter Yellow John, Joe Wandell, Peter Khoury, Tom Shell, Dalin Christiansen, Roy J. Cohoe, Patricking Shining Elk, Corrine Troester, Estella Namahoe, Dave Jensen, Kevin McNiven. When a Shoshone chief's wife dreams a white boy saves their daughter, a young man is abducted and must later chose between the tribe and his family. A somewhat interesting look at Shoshone Indian culture.\n\n**4993** _ **Winds Across the Everglades**_ **** Warner Bros., 1958. 93 min. Color. D: Nicholas Ray. SC: Bud Schulberg. With Burl Ives, Christopher Plummer, Gypsy Rose Lee, George Voskovec, Tony Galento, Howard I. Smith, Emmett Kelly, Pat Henning, Chana Eden, Curt Conway, Peter Falk, Fred Grossinger, Sammy Renick, Touch Brown, Frank Rothe, MacKinlay Kantor. At the turn of the 20th century a Florida game warden tries to protect animal life in the Everglades from the encroachment of civilization. Fairly entertaining drama if a bit oddly cast.\n\n**4994** _ **The Winds of Autumn**_ **** Howco International, 1976. 106 min. Color. D: Charles B. Pieerce. SC: Earl E. Smith. With Jack Elam, Jeanette Nolan, Andrew Prine, Dub Taylor, Charles B. \"Chuck\" Pierce, Jr., Earl E. Smith, Belinda Palmer, Jimmy Clem, Charles B. Pierce. In 1884 a Quaker boy travels across the Montana grasslands to take revenge on the brutal family who murdered his parents. Over long, but well photographed, frontier drama.\n\n**4995** _ **Winds of the Wasteland**_ **** Republic, 1936. 54 min. D: Mack V. Wright. SC: Joseph Poland. With John Wayne, Phyllis Fraser, Lane Chandler, Yakima Canutt, Douglas Cosgrove, Sam Flint, Lew Kelly, Robert Kortman, Lloyd Ingraham, Ed Cassidy, Merrill McCormick, Art Mix, Bud McClure, Jack Ingram, Charles Locher (Jon Hall), Joe Yrigoyen, Chris Franke, Jack Rockwell, Bob Burns, Horace B. Carpenter, Tracy Layne, Clyde McClary. Two pals buy and fix up an old stagecoach and plan to race a rival operation run by a crook for a valuable government mail contract. Fine John Wayne vehicle; well directed, full of action and well worth watching. Colorized as _**Stagecoach Run**_.\n\n_**The Wind's Fierce**_ see _**Revenge of Trinity**_\n\n**4996** _ **The Windwalker**_ **** Pacific International, 1980. 108 min. Color. D: Keith Merrill. SC: Ray Goldrup. With Trevor Howard, Nick Ramus, James Remar, Serene Hedin, Dusty Iron Wing McCrea, Silvana Gallardo, Billy Drago, Rudy Diaz. An old Indian chief returns to life to save his tribe from this twin son who was stolen at birth by a rival clan. Mystical drama of interest since it deals with Native Americans prior to contact with whites and was filmed in the Cheyenne and Crow languages with subtitles for theatrical showings. Co-produced by Arthur R. Dubs.\n\n**4997** _ **Wings of Adventure**_ **** Tiffany, 1930. 55 min. D: Richard Thorpe. SC: Harry Fraser. With Rex Lease, Armida, Clyde Cook, Fred Malatesta, Nadja, Eddie Boland, Charles K. French, Nick De Ruiz, Bruce Covington, Chris-Pin Martin, Steve Clemente. Two fliers crash land in Mexico and are captured by a bandit leader out to take over the government with the pilot falling for a pretty prisoner although he and his pal are framed for a robbery and sentenced to be shot. Very bad modern-day trifle that helped wreck Rex Lease's starring career.\n\n**4998** _ **Wings of an Eagle**_ **** Martin Green, 1976. 90 min. Color. With Ed Durden. The story of a rare California Golden Eagle, from her life in the nest through adulthood, as told by wild bird trainer Ed Durden. Well done documentary.\n\n**4999** _ **Wings of Chance**_ **** Universal-International, 1961. 76 min. Color. D: Edward Dew. SC: Patrick Whyte. With James Brown, Frances Rafferty, Richard Tretter, Patrick Whyte, Larry Trahan, Brian Burke, Len Crowther. A pilot is forced down in the Canadian wilderness by the carelessness of his co-pilot who is jealous of his attention to the pretty girl they both love. Standard drama made in Canada and directed by former genre star Eddie Dew.\n\n**5000** _ **Wings of the Hawk**_ **** Universal-International, 1953. 81 min. Color. D: Budd Boetticher. SC: James E. Mosier. With Van Heflin, Julia (Julie) Adams, Abbe Lane, Noah Beery, Jr., Rodolfo Acosta, George Dolenz, Pedro Gonzalez Gonzalez, Antonio Moreno, Paul Fierro, Mario Siletti, Rico Alaniz, John Daheim, Ricardo Alba, Nancy Westbrook. While working in Mexico, a mining engineer becomes involved with a pretty bandit queen and her efforts to overthrow the government. Originally issued in 3-D, this action drama is fairly entertaining.\n\n_**Wings Over Wyoming**_ see _**Hollywood Cowboy**_\n\n**5001** _ **Winners of the West**_ **** Universal, 1940. 13 Chapters. D: Ford Beebe and Ray Taylor. SC: George H. Plympton, Basil Dickey and Charles R. Condon. With Dick Foran, Anne Nagel, James Craig, Tom Fadden, Harry Woods, Charles Stevens, Trevor Bardette, Chief Yowlachie, Edward Keane, William Desmond, Edmund Cobb, Roy Barcroft, Chuck Morrison, Edgar Edwards, Ed Cassidy, Slim Whitaker, Alan Bridge, Hank Worden, Henry Hall, Jim Farley, Earl Douglas, Jim Pierce, Bud Osborne, Robert Kortman, Grace Cunard, Horace B. Carpenter, Tom London, Harry Tenbrook, Iron Eyes Cody, Frank Ellis, Jim Corey, Fred Graham, Eddie Parker, Cliff Lyons, Kenneth Terrell, Gene Alsace, Rose Plummer, Bud McClure, Dick Rush, Tex Palmer, Jack Voglin, George Plues, Viola Vonn, James Blaine, Evelyn Selbie, Robert Long, George Magrill, Paul Reed, Bill Hunter, Charles Sherlock, Charles Brunner. A railroad line president's assistant tries to stop a land baron who wants to keep the transcontinental rails from crossing his domain. Lighting fast Universal cliffhanger marred by excessive stock footage.\n\n**5002** _ **Winners of the Wilderness**_ **** Metro-Goldwyn-Mayer, 1927. 68 min. D: W.S. Van Dyke. SC: Josephine Chippo and Marian Ainslee. With Tim McCoy, Joan Crawford, Edward Connelly, Roy D'Arcy, Louise Lorraine, Edward Hearn, Tom O'Brien,Will R. Walling, Frank Currier, Lionel Belmore, Chief Big Tree, Jean Arthur. During the French and Indian War a British colonel captured by the enemy is helped to escape by the commandant's daughter. Picturesque silent historical drama highlighted by General Braddock's defeat at the hands of the French and Indians.\n\n**5003** _ **Winnetou and Shatterhand in the Valley of Death**_ **** CCC Filmkunst\/Super International\/Jadran-Film, 1968. 90 min. Color. D: Harald Reinl. SC: Herbert Reinecker. With Lex Barker, Pierre Brice, Karin Dor, Ralf Wolter, Eddi Arent, Rik Battaglia, Wojo Govedrizu, Clarke Reynolds, Vladimir Medar, Branco Spoliak, Kurt Waitzmann, Heinz Welzel, Vladimir Leib, Llija Lvezic, Jan Sid, Ivo Kristof, Nikola Gec, Vladimir Bacic, Sime Jagarinac, Dusko Ercegovi, Rajko Zakarija, Drago Sosa. Frontiersman Old Shatterhand and his Apache blood brother Winnetou help the daughter of a fort commander accused of stealing a fortune in gold but end up being left to die in the desert. The last Lex Barker-Pierre Brice teaming in the series based on Karl May's works does not have the big budget of its predecessors but still provides good entertainment. West German title: _**Winnetou und Shatterhand im Tal der Toten**_ (Winnetou and Shatterhand in the Valley of Death); U.S. title: _**In the Valley of Death**_.\n\n_**Winnetou I**_ see _**Apache Gold**_\n\n_**Winnetou II**_ see _**Last of the Renegades**_\n\n_**Winnetou III**_ see _**The Desperado Trail**_\n\n_**Winnetou the Warrior**_ see _**Apache Gold**_\n\n**5004** _ **The Winning of Barbara Worth**_ **** United Artists, 1926. 89 min. D: Henry King. SC: Frances Marion and Rupert Hughes. With Ronald Colman, Vilma Banky, Gary Cooper, Charles Lane, Paul McAllister, E.J. Ratcliffe, Clyde Cook, Erwin Connelly, Ed Brady, Sammy Blum, Fred Esmelton, William (Bill) Patton, Clarence Wilson, Glynn Waiters, Carmencita Johnson, Henry Wells, Margaret Wells. The foster son of a land baron falls in love with a rancher's adopted daughter while building a dam and finds he has a rival in the cattleman's foreman. Top notch drama that brought Gary Cooper to stardom; highlighted by a flood sequence beautifully photographed by George Banner and Gregg Toland.\n\n**5005** _ **Winning of the West**_ **** Columbia, 1953. 57 min. D: George Archainbaud. SC: Norman S. Hall. With Gene Autry, Smiley Burnette, Gail Davis, Richard Crane, Robert Livingston, House Peters, Jr., Gregg Barton, Ewing Mitchell, Rodd Redwing, George Chesebro, Frank Jaquet, Charles Delaney, Charles Soldani, Eddie Parker, Terry Frost, James Kirkwood, Boyd \"Red\" Morgan, Bob Woodward. A ranger loses his job for refusing to shoot his brother, a member of an outlaw gang terrorizing local miners and ranchers. Fast paced but somewhat tepid Gene Autry vehicle.\n\n_**Winning the West**_ see _**The Light of the Western Stars**_ (1930)\n\n**5006** _ **Winter Kill**_ **** ABC-TV\/Metro-Goldwyn-Mayer, 1974. 100 min. Color. D: Jud Taylor. SC: Joseph Michael Hayes. With Andy Griffith, John Larch, Tim O'Connor, Lawrence Pressman, Eugene Roche, Charles Tyner, Joyce Van Patten, Sheree North, John Calvin, Louise Latham, Robert F. Simon, Elayne Heilveil, Nick Nolte, Ruth McDevitt, Walter Brooke, David Frankham, Wes Stern, Vaughn Taylor, Devra Korwin. The sheriff of a Western ski resort community is baffled by a series of murders in which the killer leaves clues in spray paint. Interesting TV Western mystery that failed to sell as a continuing series; made by Andy Griffith Enterprises.\n\n**5007** _ **Winterhawk**_ **** Howco International, 1975. 90 min. Color. D-SC: Charles B. Pierce. With Leif Erickson, Michael Dante, Dawn Wells, Woody Strode, Denver Pyle, Arthur Hunnicutt, Elisha Cook, Jr., L.Q. Jones, Charles B. Pierce, Jr., Sacheen Littlefeather, Dennis Fimple, Seamon Glass. An Indian chief comes to a white settlement for smallpox serum, is treated badly and in revenge abducts a woman and her small brother. Fairly exciting Western made on a limited budget.\n\n_**Wishbone Cutter**_ see _**The Shadow of Chikara**_\n\n**5008** _ **The Wistful Widow of Willow Gap**_ **** Universal-International, 1947. 78 min. D: Charles Barton. SC: Robert Lees, Frederic I. Renaldo and John Grant. With Bud Abbott, Lou Costello, Marjorie Main, Aubrey Young, George Cleveland, Gordon Jones, William Ching, Peter Thompson, Olin Howlin, Bill Clauson, Billy O'Leary, Pamela Wells, Jimmie Bates, Paul Dunn, Diane Florentine, Rex Lease, Glenn Strange, Dewey Robinson, Edmund Cobb, Wade Crosby, Murray Leonard, Emmett Lynn, Iris Adrian, Lee \"Lasses\" White, George J. Lewis, Charles King, Jack Shutta, Harry Evans, Mickey Simpson, Frank Marlow, Ethan Laidlaw. Two salesmen arrive in a Montana community where one of them is falsely accused of shooting the town drunk and a crooked lawyer gets him off by using a state law which says he has to support the deceased's widow and pay off her debts. A good plot, Abbott and Costello and Marjorie Main in the title role make this an amusing genre spoof.\n\n_**With Buffalo Bill on the U.P. Trail**_ see _**Buffalo Bill on the U.P. Trail**_\n\n_**With Custer at the Little Big Horn**_ see _**General Custer at the Little Big Horn**_\n\n_**With Daniel Boone Thru the Wilderness**_ see _**Daniel Boone Thru the Wilderness**_\n\n_**With Davy Crockett at the Fall of the Alamo**_ see _**Davy Crockett at the Fall of the Alamo**_\n\n_**With General Custer at the Little Big Horn**_ see _**General Custer at the Little Big Horn**_\n\n_**With Sitting Bull at the Spirit Lake Massacre**_ see _**Sitting Bull at the Spirit Lake Massacre**_\n\n**5009** _ **Without Honors**_ **** Artclass, 1932. 62 min. D: William Nigh. SC: Harry P. (Fraser) Crist. With Harry Carey, Mae Busch, Gibson Gowland, George Hayes, Lafe McKee, Mary Jane Irving, Tom London, Ed Brady, Jack Richardson, Maston Williams, Jim Corey, Blackjack Ward, Bud McClure, Lee Sage, Buck Bucko, Roy Bucko. A man returns home to find his brother has been murdered and to capture the culprit he enlists with the rangers. Shaggy production values detract from this otherwise okay Harry Carey film.\n\n_**Without Risk**_ see _**Pecos River**_\n\n**5010** _ **Wolf Blood**_ **** Lee-Bradford (Artlee), 1925. 68 min. D: George Chesebro and Bruce Mitchell. SC: Bennett Cohen. With George Chesebro, Marguerite Clayton, Ray Hanford, Roy Watson, Milburn Morante, Frank Clark, Jack Cosgrove. A logging foreman is left for dead by a rival and a surgeon, who loves the daughter of the company's boss, gives him a transfusion of she-wolf's blood and when he survives the locales think he has become a lycanthrope. Scenic locales help this fairly interesting pseudo-horror silent feature starring the great George Chesebro, who co-directed.\n\n**5011** _ **Wolf Call**_ **** Monogram, 1939. 60 min. D: George Waggner. SC: Joseph West (George Waggner). With John Carroll, Polly Ann Young, Movita, George Cleveland, Wheeler Oakman, Guy Usher, Holmes Herbert, Peter George Lynn, John Sheehan, Charles Irwin, Roger Williams, Pat O'Malley. While inspecting his father's Western radium mine, a playboy discovers a gang of crooks are trying to steal the property. Paul Malvern produced his average dual bill item based on a Jack London story.\n\n**5012** _ **Wolf Dog**_ **** 20th Century\u2013Fox, 1958. 69 min. D: Sam Newfield. SC: Louis Stevens. With Jim Davis, Allison Hayes, Tony Brown, Austin Willis, Don Garrard, Juan Root, Lloyd Chester, Jay MacDonald, R. Braithwaite. An ex-convict moves with his family to remote area of Canada but finds a neighbor wants their land for himself. Filmed in Canada, this drama is passable entertainment.\n\n**5013** _ **The Wolf Hunters**_ **** Monogram, 1950. 70 min. D: Oscar (Budd) Boetticher. SC: W. Scott Darling. With Kirby Grant, Jan Clayton, Helen Parrish, Edward Norris, Ted Hecht, Charles Lang, Luther Crockett, Elizabeth Root. A Canadian Mounted Policeman, on the trail of murderous fur thieves, uncovers a plot concerning a lost gold mine. Allegedly based on the James Oliver Curwood novel, this minor outing bears little resemblance to it but is still enjoyable.\n\n**5014** _ **Wolf Riders**_ **** Reliable, 1935. 56 min. D: Harry S. Webb. SC: Carl Krusada. With Jack Perrin, Lillian Gilmore, Lafe McKee, Nancy Deshon, William Gould, George Chesebro, Earl Dwire, Budd Buster, Slim Whitaker, Frank Ellis, Robert Walker, George Morrell, Blackie Whiteford. An Indian agency inspector tries to protect a local tribe from a ruthless gang of fur thieves. Mediocre production values and a strung out plot hamper this Jack Perrin vehicle.\n\n**5015** _ **Wolf Song**_ **** Paramount, 1929. 80 min. D: Victor Fleming. SC: John Farrow and Keene Thompson. With Gary Cooper, Lupe Velez, Louis Wolheim, Constantine Romanoff, Michael Vavitch, Ann Brody, Russ Columbo, Augustina Lopez, George Regas, Leona Lane. The daughter of a Spanish don marries a backwoods Kentucky trapper and they live in a mountain settlement but he gets wanderlust and she returns home although they still long for each other. Location filming in the California Sierra mountains is the chief highlight of this early talkie.\n\n**5016** _ **Wolf Tracks**_ **** Sunset, 1923. 40 min. D: Robert North Bradbury. SC: William Lester. With Jack Hoxie, Andree Tourneur, Jim Welsh, Tom Lingham, William Lester, Marin Sais. Mistaken for an outlaw called \"The Wolf,\" a cowpoke tries to capture the villain not only to save himself but to protect a woman whose father has left her a mine the crook is trying to steal. Fast moving silent Jack Hoxie outing\u2014a good film.\n\n**5017** _ **Wolf Heart's Revenge**_ **** Aywon, 1925. 55 min. D: Charles L. Seeling. With Wolf Heart (dog), Guinn Williams, Kathleen Collins, Captain Bingham, Larry Fischer, Helen Walton, John Williams. After committing a murder, a ranch foreman tries to place the blame on an innocent cowboy. Good action entry in Guinn \"Big Boy\" Williams' silent series for producer-director Charles L. Seeling, with dog star Wolf Heart.\n\n**5018** _ **Wolves of the Range**_ **** Producers Releasing Corporation, 1943. 60 min. D: Sam Newfield. SC: Joseph O'Donnell. With Robert Livingston, Al St. John, Frances Gladwin, I. Stanford Jolley, Karl Hackett, Ed Cassidy, Jack Ingram, Kenne Duncan, Budd Buster, Bob Hill, Slim Whitaker, Jack Holmes, Bob Hill, John Elliott, Milton Kibbee, Lester Dorr, Reed Howes, Roy Brent, Wally West, Art Dillard, Bert Dillard, Jimmy Aubrey, Augie Gomez, Morgan Flowers, Al Haskell, Ray Jones, Cactus Mack, Jack Tornek, Tom Smith, Rose Plummer, Art Fowler, Chick Hannon, George Morrell, Foxy Callahan, Murdock MacQuarrie, Roy Bucko, Lew Morphy. While helping ranchers being forced off their land because it is needed for a government irrigation project, the Lone Rider becomes a victim of amnesia. A pleasant segment in the popular \"Lone Rider\" series.\n\n**5019** _ **Woman Obsessed**_ **** 20th Century\u2013Fox, 1959. 102 min. Color. D: Henry Hathaway. SC: Sidney Boehm. With Susan Hayward, Stephen Boyd, Arthur Franz, Dennis Holmes, Ken Scott, Theodore Bikel, James Philbrook, Florence MacMichael, Jack Raine, Barbara Nichols, Mary Carroll, Fred Graham, Mike Wally. Following the accidental death of her husband a woman struggles to make a living on her remote Saskatchewan ranch and when she remarries her young son resents his new stepfather. Well made but not overly interesting drama.\n\n**5020** _ **Woman of the North Country**_ **** Republic, 1952. 90 min. Color. D: Joseph Kane. SC: Norman Reilly Raine. With Rod Cameron, Ruth Hussey, John Agar, Gale Storm, J. Carrol Naish, Jim Davis, Jay C. Flippen, Taylor Holmes, Barry Kelley, Grant Withers, Howard Petrie, Hank Worden, Virginia Brissac, Stanley Andrews, Dub Taylor, Richard Alexander, Ray Bennett, Stephen Bekassy. A mining engineer finds himself opposed by a ruthless woman rival who will stop at nothing to destroy him. Good production with a fine script and performances.\n\n**5021** _ **Woman of the Town**_ **** United Artists, 1943. 89 min. D: George Archainbaud. SC: Aeneas MacKenzie. With Claire Trevor, Albert Dekker, Barry Sullivan, Henry Hull, Marion Martin, Porter Hall, Percy Kilbride, Beryl Wallace, George Cleveland, Arthur Hohl, Clem Bevans, Russell Hicks, Herbert Rawlinson, Dorothy Granger, Dewey Robinson, Hal Taliaferro, Wade Crosby, Glenn Strange, Claire Whitney, Russell Simpson, Frances Morris, Teddi Sherman, Marlene Mains, Charles Foy, Tom London, Eula Gray. Sheriff Bat Masterson is forced to choose between his job and the love of saloon woman Dora Hand. Underrated Harry Sherman production with fine work by Claire Trevor and Albert Dekker in the leading roles, ably supported by a good cast.\n\n**5022** _ **The Woman They Almost Lynched**_ **** Republic, 1953. 90 min. D: Allan Dwan. SC: Steve Fisher. With Brian Donlevy, Joan Leslie, John Lund, Audrey Totter, Jim Davis, Ben Cooper, James Brown, Ellen Corby, Reed Hadley, Virginia Christine, Richard Simmons, Gordon Jones, Nina Varela, Frank Ferguson, Ann Savage, Richard Crane, Ted Ryan, James Kirkwood, Fern Hall, Minerva Urecal, Marilyn Lindsey, Nacho Galindo, Post Park, Lee Roberts, Tom McDonough, Carl Pitti, Joe Yrigoyen, Jimmy Hawkins, Paul Livermore, Hal Baylor. After inheriting a saloon in a town run by crooks, a city girl becomes a bandit and is almost hung. Despite its exploitation title, this is pretty good with several interesting performances, especially Brian Donlevy and Audrey Totter as William Clarke Quantrill and his wife Kate.\n\n**5023** _ **Wonder of It All**_ **** Pacific International, 1986. 95 min. D: Arthur R. Dubs. SC: James T. Flocker. With Les Biegel (narrator). Wild life from around the world is shown, including various animals in North America. An outstanding documentary for the entire family.\n\n**5024** _ **The Wonderful Country**_ **** United Artists, 1959. 96 min. Color. D: Robert Parrish. SC: Robert Ardrey. With Robert Mitchum, Julie London, Gary Merrill, Pedro Armendariz, Jack Oakie, Albert Dekker, Charles McGraw, Satchel Paige, Victor Mendoza, Tom Lea, Jay Novello, Mike Kellin, Max Slaten, Joe Haworth, Chester Hayes, Chuck Roberson, Anthony Caruso, Claudio Brook, Judy Marsh, Mike Luna. A gun runner working for a Mexican revolutionary is sent to the U.S. for arms and gets involved with a pretty woman, outlaws and rampaging Indians. Sprawling adaptation of Tom Lea's novel provides good entertainment.\n\n**5025** _ **The World in His Arms**_ **** Universal-International, 1952. 104 min. Color. D: Raoul Walsh. SC: Borden Chase. With Gregory Peck, Ann Blyth, Anthony Quinn, John McIntire, Carl Esmond, Andrea King, Eugenie Leontovich, Hans Conreid, Rhys Williams, Sig Rumann, Gregory Gay, Bill Radovich, Bryan Forbes, Henry Kulky, Wee Willie Davis, Tudor Owen, Leo Mostovoy, Syl Lamont, Eve Whitney, Millicent Patrick, Dick Rich, George Scanlan, Gregg Barton, Frank Chase, Carl Harbaugh, Gregg Martell, Paul Newlan, Carl Andre, Suzan Ball. In 1850 San Francisco, a sea captain, who runs poached pelts out of Alaska, falls for a Russian princess but they are endangered by an evil prince. Stolid adaptation of Rex Beach's sprawling novel.\n\n**5026** _ **Wrangler**_ **** Samuel Goldwn Company, 1989. 92 min. Color. D: Ian Barry. SC: John Sexton. With Jeff Fahey, Tushka Bergen, Steve Vidler, Richard Moir, Shane Briant, Frederick Parslow, Cornelia Frances, Michael Winchester, Sandy Gore, Drew Forsythe, Robert Davis, Andrew Sharp, Kevin Healy, Owen Weingott, Colin Taylor, Conor McDermottroe, James Steele, Mic Conway, Fiona Stewart, Derek Mendl, Laurie Moran, Peter Collingwood, Paul Maclay, Barry McMahon, Suzette Williams, Jacqueline Kott, Robert Alexander. After the death of her rancher father, an Australian woman faces losing the place to a creditor while being romanced by two men, an entrepreneur and a cattleman. Nicely made Down Under Western originally called _**Outback**_.\n\n**5027** _ **Wrangler's Roost**_ **** Monogram, 1941. 57 min. D: S. Roy Luby. SC: John Vlahos and Robert Finkle. With Ray Corrigan, John King, Max Terhune, Gwen Gaze, Forrest Taylor, George Chesebro, Frank Ellis, Walter Shumway, Jack Holmes, Frank McCarroll, Carl Mathews, Hank Bell, Tex Palmer, Jim Corey, Al Haskell, Ray Jones, Horace B. Carpenter, Tex Cooper, Herman Hack, Chick Hannon, Jack Evans, Buck Moulton, Roy Bucko, Bob Card, Emma Tansey, Herman Hack, George Morrell, Silvertip Baker. The Range Busters are asked to investigate a series of stage holdups and they come to suspect an old time outlaw who they believe is masquerading as a deacon. Passable entry in the popular series but nothing special.\n\n**5028** _ **The Wrath of God**_ **** Metro-Goldwyn-Mayer, 1971. 111 min. Color. D-SC: Ralph Nelson. With Robert Mitchum, Rita Hayworth, Frank Langella, Victor Buono, John Calicos, Ken Hutcheson, Paula Pritchett, Gregory Sierra, Frank Ramirez, Enrique Lucero, Jorge Russek, Chano Urueta, Jose Luis Parades, Aurora Clavel, Victor Eberg, Pancho Cordova, Guillermo Hermandez, Ralph Nelson. In the late 1920s a loose living priest and his two pals help a Mexican revolutionary when a remote village is threatened by government forces. Not overly good tongue-in-cheek drama although Robert Mitchum is a delight as the priest.\n\n**5029** _ **Wyatt Earp**_ **** Warner Bros., 1994. 191 min. Color. D: Lawrence Kasdan. SC: Dan Gordon and Lawrence Kasdan. With Kevin Costner, Dennis Quaid, Gene Hackman, David Andrews, Linden Ashby, Jeff Fahey, Joanna Going, Mark Harmon, Michael Madsen, Catherine O'Hara, Bill Pullman, Isabella Rossellini, Tom Sizemore, JoBeth Williams, Mare Winningham, James Gammon, Rex Linn, Randle Mell, Adam Baldwin, Annabeth Gish, Lewis Smith, Ian Bohen, Betty Buckley, Alison Elliott, Todd Allen, Mackenzie Astin, James Caviezel, Karen Grassle, John Denis Johnston, Tea Leoni, Martin Kove, Jack Kehler, Kirk Fox, Norman Howell, Boots Southerland, Scotty Augare, Gabriel Folse, Kris Kamm, John Lawlor, Monty Stuart, Hugh Ross, Michael McGrady, Mary Jo Niedzielski, Darwin Mitchell, Heath Kizzier, Clark Sanchez, Giorgio E. Tripoli, Scott Rasmussen, Matt Langseth, David Doty, Steven G. Tyler, Billy Streater, David L. Stone, Jake Walker, Matt O'Toole, Dick Beach, Sarge McGraw, Owen Roizman, John Furlong, Adam Taylor, Michael Huddleston, John Doe, Matt Beck, Gary Dueer, Karen Schwartz. Lawman Wyatt Earp becomes a heartless killer in the name of justice as he and his brothers go up against the Clanton clan and other desperadoes. Effective, but somewhat lumbering, biopic of the famed peacemaker that failed at the box office; also available in an 212 minute extended version.\n\n**5030** _ **Wyatt Earp: Return to Tombstone**_ **** CBS-TV, 1993. 100 min. Color. D: Paul Landres and Frank McDonald. SC: Dan Ullman and Rob Word. With Hugh O'Brian, Bruce Boxleitner, Paul Brinegar, Harry Carey, Jr., Bo Hopkins, Alex Hyde-White, Martin Kove, Don Meredith, Jay Underwood, Tori Prince, Douglas Fowley, John Anderson, Steve Brodie, Bob Steele, Trevor Bardette, Lloyd Corrigan, George Wallace, Gregg Palmer, Nancy Hale, William Phipps, Stacy Harris, Rayford Barnes, Norman Alden, Ralph Reed, Ray Boyle, William Tannen. Former lawman Wyatt Earp returns to Tombstone where he is reunited with people from his past and helps to keep the town peaceful. Good nostalgic romp made up of colorized scenes from old episodes of \"The Life and Legend of Wyatt Earp\" (ABC-TV, 1955\u201361) interpolated with new footage.\n\n**5031** _ **Wyatt Earp's Revenge**_ **** Hybrid Productions, 2012. 93 min. Color. D: Michael Feifer. SC: Darren B. Shepherd. With Val Kilmer, Shawn Roberts, Matt Dallas, Daniel Booko, Scott Whyte, Steven Graham, Levi Fiehler, Trace Adkins, Diana DeGarmo, Brian Groh, Martin Santander, Wilson Bethel, Peter Sherayko, Lyle Kanouse, Rob Daly, Mason Cook, Caia Coley, Miracle Laurie, Andrew Hawkes, Kaitlyn Black, Wes Brown, Jonathan Erickson Eisley. After his girl is gunned down by a vicious gang, Wyatt Earp forms a posse that includes Bat Masterson, Doc Holiday, Bill Tilghman and Charlie Bassett to get revenge. Substandard video Western.\n\n**5032** _ **Wyoming**_ **** Metro-Goldwyn-Mayer, 1940. 88 min. D: Richard Thorpe. SC: Jack Jevne and Hugh Butler. With Wallace Beery, Leo Carrillo, Ann Rutherford, Lee Bowman, Joseph Calleia, Bobs Watson, Paul Kelly, Marjorie Main, Henry Travers, Addison Richards, Stanley Fields, William Tannen, Clem Bevans, Donald MacBride, Russell Simpson, Dick Curtis, Chill Wills, Richard Alexander, Chief Thundercloud, Glenn Lucas, Francis McDonald, Edgar Dearing, Glenn Strange, Ted Adams, Lee Phelps, Howard Mitchell, Richard Alexander, Ethel Wales, Richard Botiller, Frank Ellis, Archie Butler, Betty Jean Nichols. After the Civil War, a Missouri outlaw and his pal go West where they get involved with an earthy female blacksmith and end up on the right side of the law. This initial teaming of Wallace Beery and Marjorie Main, along with Leo Carrillo as the sidekick, makes for good entertainment.\n\n**5033** _ **Wyoming**_ **** Republic, 1947. 84 min. D: Joseph Kane. SC: Lawrence Hazard and Gerald Geraghty. With William Elliott, Vera Ralston, John Carroll, George \"Gabby\" Hayes, Albert Dekker, Virginia Grey, Maria Ouspenskaya, Grant Withers, Harry Woods, Minna Gombell, Dick Curtis, Roy Barcroft, Trevor Bardette, Paul Harvey, Louise Kane, Linda Green, Tom London, George Chesebro, Jack O'Shea, Charles Middleton, Eddy Waller, Olin Howlin, Glenn Strange, Charles King, Eddie Acuff, Marshall Reed, Rex Lease, Charles Morton, Tex Terry, Dale Fink, Ed Peil, Sr., Roque Ybarra, James Archuletta, David Williams, Lee Shumway, Ben Johnson. A Wyoming land baron has nesters encroaching on his range and when his foreman quits in their defense he also finds his college educated daughter deserting his cause. Grand scale (for Republic) drama with a great cast and lots of action; Yakima Canutt did the second unit work.\n\n**5034** _ **The Wyoming Bandit**_ **** Republic, 1949. 60 min. D: Philip Ford. SC: M. Coates Webster. With Allan \"Rocky\" Lane, Eddy Waller, Trevor Bardette, Victor Kilian, Rand Brooks, Reed Hadley, Harold Goodwin, Lane Bradford, Robert Wilke, John Hamilton, Edmund Cobb, William Haade. An outlaw teams with a lawman to get those responsible for the murder of his son. Trevor Bardette as the good-bad man Wyoming Dan steals the show in this above average \"Famous Westerns\" segment.\n\n**5035** _ **Wyoming Hurricane**_ **** Columbia, 1944. 58 min. D: William Berke. SC: Fred Myton. With Russell Hayden, Dub Taylor, Bob Wills and The Texas Playboys, Alma Carroll, Tristram Coffin, Joel Friedkin, Paul Sutton, Benny Petti, Robert Kortman, Hal Price, Steve Clark, Hank Worden, Tom Steele. A corrupt caf\u00e9 operator murders the local lawman and the blame is placed on the marshal's daughter's boyfriend. Lesser Russell Hayden vehicle with fine villainy by Tristram Coffin.\n\n_**The Wyoming Kid**_ see _**Cheyenne**_ (1947)\n\n**5036** _ **Wyoming Mail**_ **** Universal-International, 1950. 87 min. Color. D: Reginald LeBorg. SC: Harry Essex and Leonard Lee. With Stephen McNally, Alexis Smith, Howard Da Silva, Ed Begley, Dan Riss, Roy Roberts, Whit Bissell, Armando Silvestre, James Arness, Richard Jaeckel, Frankie Darro, Felipe Turich, Richard Egan, Gene Evans, Frank Fenton, Emerson Treacy, Chick Chandler, Frank Richards, John Cason. A former boxer is hired as an undercover agent for the railroad and after infiltrating a gang in Wyoming Territory he falls in love with its female member. The story is a bit farfetched but otherwise the film is okay.\n\n**5037** _ **Wyoming Outlaw**_ **** Republic, 1939. 56 min. D: George Sherman. SC: Jack Natteford and Betty Burbridge. With John Wayne, Ray Corrigan, Raymond Hatton, Donald Barry, Adele Pearce (Pamela Blake), LeRoy Mason, Charles Middleton, Elmo Lincoln, Katharine Kentworthy, Jack Ingram, David Sharpe, Jack Kenney, Yakima Canutt, Dave O'Brien, Curley Dresden, Tommy Coats, Ralph Peters, Jack Kirk, Al Taylor, Bud McTaggart, Frankie Marvin, Allan Cavan, John Hiestand, Jack Rockwell, Bob Burns, John Beach, George De Normand, Budd Buster, Ed Payson. The Three Mesquiteers are at odds with crooked politicians who are cheating small ranchers in the Dust Bowl and they are forced to hunt down a young man who broke the law in self defense. Exceedingly well done \"Three Mesquiteers\" series outing with a finale that predates _**High Sierra**_ (Warner Bros., 1941) by two years; Don Barry gives an outstanding performance as the hunted youth.\n\n**5038** _ **Wyoming Renegades**_ **** Columbia, 1955. 73 min. D: Fred F. Sears. SC: David Lang. With Phil(ip) Carey, Martha Hyer, Gene Evans, William Bishop, Douglas Kennedy, Roy Roberts, Don Beddoe, Aaron Spelling, George Keymas, Harry Harvey, Mel Welles, Henry Rowland, Boyd Stockman, Guy Teague, Bob Woodward, Don C. Harvey, John Cason, Don Carlos. Released from prison a man finds his past causes people to dislike him but he gets succor from the girl he loves. Standard program feature from producer Wallace MacDonald.\n\n**5039** _ **Wyoming Roundup**_ **** Monogram, 1952. 53 min. D: Thomas Carr. SC: Dan Ullman. With Whip Wilson, Phyllis Coates, Tommy Farrell, Henry Rowland, House Peters, Jr., Lyle Talbot, I. Stanford Jolley, Dick Emory, Robert Wilke, Stanley Price, Frank Jaquet, Herman Hack, Artie Ortego, Roy Bucko. After stopping a gunfight, two cowpokes are made the law in a town where someone is hiring gunmen to run out rival ranchers. Average Whip Wilson outing.\n\n**5040** _ **Wyoming Whirlwind**_ **** Willis Kent, 1932. 55 min. D: Armand L. Schaefer. SC: Wallace MacDonald. With Lane Chandler, Adele Tracy, Harry Todd, Alan Bridge, Yakima Canutt, Lois Bridge, Bob Roper, Harry Semels, Hank Bell, Jack Rockwell, Fred Burns, Lafe McKee, Jack Kirk, Frank Ellis, Bud Pope, Al Taylor, Silver Tip Baker, Raven (horse). The Lone Wolf, a wanted highwayman, is really the son of a rancher murdered years before and he is out to capture the killer, the foreman who inherited the spread. Cheaply made, strung out, low grade Lane Chandler vehicle. TV title: _**Roaring Rider**_.\n\n**5041** _ **Wyoming Wildcat**_ **** Republic, 1941. 56 min. D: George Sherman. SC: Bennett Cohen and Anthony Coldeway. With Don \"Red\" Barry, Julie Duncan, Syd Saylor, Frank M. Thomas, Edmund Cobb, Ed Brady, Richard Botiller, Ed Cassidy, George Sherwood, Ethan Laidlaw, Al Haskell, Frank Ellis, Curley Dresden, Art Dillard, Kermit Maynard, Cactus Mack, Frank O'Connor, Fred Burns. A wanted outlaw gets a job as a guard on a Wells Fargo stagecoach and is hunted by the law after a holdup. Average Don Barry outing.\n\n**5042** _ **Yankee Don**_ **** Capitol, 1931. 60 min. D: Noel Madison. SC: Frances Jackson. With Richard Talmadge, Lupita Tovar, Julian Rivero, Sam Appel, Gayne Whitman, Alma Reat, Victor Stanford. A Bowery desperado heads West and helps a Spanish don whose ranch is threatened by outlaws. Action laced poverty row feature produced by star Richard Talmadge.\n\n**5043** _ **Yankee Fakir**_ **** Republic, 1947. 71 min. D: W. Lee Wilder. SC: Richard S. Conway. With Douglas Fowley, Joan Woodbury, Clem Bevans, Ransom Sherman, Frank Reicher, Marc Lawrence, Walter Soderling, Eula Guy, Forrest Taylor, Elinor Appleton, Peter Michael, Elspeth Dudgeon, Ernie Adams, Tommy Bernard, Ed Peil, Sr., Marin Sais, Larry Steers, Charles Williams, Tex Terry, Rose Plummer, Edmund Cobb, Franklyn Farnum, Bud Osborne, Herman Hack, Stanley Blystone, Ben Corbett, Jack O'Shea, Victor Potel, Stanley Price, Marshall Reed, Tom Smith, Tex Palmer, John Ince, Buster Brodie. While working in the West a salesman falls in love with a border patrolman's daughter and when the lawman is murdered he tries to find the killer. Adequate mystery dual bill item set in the modern West.\n\n**5044** _ **Yaqui Drums**_ **** Allied Artists, 1957. 70 min. D: Jean Yarborough. SC: Jo Pagano and D.D. Beauchamp. With Rod Cameron, Mary Castle, J. Carrol Naish, Robert Hutton, Roy Roberts, Keith Richards, Denver Pyle, Ray Walker, Donald Kerr, John Merrick, Paul Fierro, G. Pat Collins. A rancher fighting a corrupt saloon proprietor is helped by a Mexican outlaw gang thwarted in a stagecoach holdup attempt. Standard program film enhanced by the performances of Rod Cameron and J. Carrol Naish.\n\n**5045** _ **The Yearling**_ **** Metro-Goldwyn-Mayer, 1946. 134 min. Color. D: Clarence Brown. SC: Paul Osborn. With Gregory Peck, Jane Wyman, Claude Jarman, Jr., Chill Wills, Clem Bevans, Margaret Wycherly, Henry Travers, Forrest Tucker, Don Gift, Dan White, Matt Willis, George Mann, Arthur Hohl, June Lockhart, June Wells, Jeff York, Chick York, Houseley Stevenson, Jane Green, Victor Kilian, Robert Porterfield, John Eldredge. A Florida farm boy becomes attached to a fawn his father must destroy. Award winning classic family film; well worth seeing.\n\n**Gregory Peck and Claude Jarman, Jr., in** _**The Yearling**_ **(Metro-Goldwyn-Mayer, 1946).**\n\n**  \n**\n\n**5046** _ **Yellow Dust**_ **** RKO Radio, 1936. 68 min. D: Wallace Fox. SC: Cyril Hume and John Twist. With Richard Dix, Leila Hyams, Moroni Olsen, Jessie Ralph, Andy Clyde, Onslow Stevens, Victor Potel, Ethan Laidlaw, Art Mix, Ted Oliver. A miner falls in love with a saloon girl and tries to win her from her boss, even after he is falsely accused of robbing a stage. Surprisingly poor Richard Dix feature.\n\n**5047** _ **Yellow Hair and the Fortress of Gold**_ **** Crown International, 1984. 102 min. Color. D: Matt Cimber. SC: Matt Cimber and John Kershaw. With Laurene Landon, Ken Roberson, John Ghaffari, Luis Lorenzo, Claudia Gravi, Aldo Sambrell, Eduardo Fajardo, Ramiro Oliveros, Suzannah Woodride, Tony Tarruella, Concha Marquez Piqua, Daniel Martin, Mario De Abros, Joaquin Lopez, Alfonso Delgado. A beautiful blonde half-breed teams with The Pecos Kid to try and get an Aztec treasure coveted by a corrupt military man and a brutal Indian tribe. Silly, sadistic, overlong adventure yarn.\n\n**5048** _ **Yellow Haired Kid**_ **** Monogram, 1952. 56 min. D: Frank McDonald. SC: Dwight V. Babcock and Maurice Tombragel. With Guy Madison, Andy Devine, David Bruce, Marcia Mae Jones, Alan Hale, Jr., Tom Tyler, Renie Riano, Riley Hill, Tommy Ivo, Emory Parnell, Bill Phipps, Wade Crosby, Tom Hubbard, John Carpenter, Alice Rolph. Marshals Wild Bill Hickok and Jingles P. Jones pursue an outlaw with yellow hair and take on corrupt townsmen who helped a gunfighter escape from jail. Okay theatrical compilation of \"Johnny Deuce\" and \"Yellow Haired Kid,\" 1951 episodes of \"The Adventures of Wild Bill Hickok\" (1951\u201358).\n\n**5049** _ **The Yellow Mountain**_ **** Universal-International, 1954. 78 min. Color. D: Jesse Hibbs. SC: George Zuckerman and Russell Hughes. With Lex Barker, Mala Powers, Howard Duff, William Demarest, John McIntire, Leo Gordon, Hal K. Dawson, Dayton Lummis, William Fawcett, James Parnell, Denver Pyle, Kermit Maynard, Frank Ellis, Jack Ingram, Paul McGuire, Carl Andre, Matty Fain, Paul Bryar, Joe Bailey, John Roy, Mel Ford. Two men vie for the same beautiful woman as well as a valuable gold claim. Fair \"B plus\" feature from producer Ross Hunter.\n\n**5050** _ **Yellow Rose of Texas**_ **** Republic, 1944. 69 min. D: Joseph Kane. SC: Jack Townley. With Roy Rogers, Dale Evans, Bob Nolan and The Sons of the Pioneers (Tim Spencer, Ken Carson, Shug Fisher, Hugh Farr, Karl Farr), George Cleveland, Harry Shannon, Grant Withers, William Haade, Weldon Heyburn, Hal Taliaferro, Tom London, Richard Botiller, Janet Martin, Robert Wilke, Jack O'Shea, Rex Lease, Emmett Vogan, John Dilson, Don Kay Reynolds, William Desmond, Chester Conklin, Fred \"Snowflake\" Toones, Horace B. Carpenter. On the run for falsely being accused of robbing a stagecoach, a man is sought by his daughter, a showboat entertainer, and an insurance investigator masquerading as a showman to see if the woman knows the whereabouts of her father. Dandy Roy Rogers outing with a good story and plenty of music in a showboat setting.\n\n**5051** _ **Yellow Sky**_ **** 20th Century\u2013Fox, 1948. 95 min. D: William A. Wellman. SC: Lamar Trotti. With Gregory Peck, Anne Baxter, Richard Widmark, Robert Arthur, John Russell, Henry (Harry) Morgan, James Barton, Charles Kemper, Robert Adler, Victor Kilian, Paul Hurst, William Gould, Norman Leavitt, Chief Yowlachie, Eula Guy. After robbing a bank, six outlaws ride into an Arizona ghost town inhabited only by a man and his granddaughter and trouble erupts with the old man hides their stolen loot. Highly entertaining drama from the novel by W.R. Burnett.\n\n_**The Yellow Streak**_ see _**Both Barrels Blazing**_\n\n**5052** _ **The Yellow Tomahawk**_ **** United Artists, 1954. 82 min. D: Lesley Selander. SC: Richard Alan Simmons. With Rory Calhoun, Peggie Castle, Noah Beery, Jr., Warner Anderson, Peter Graves, Lee Van Cleef, Rita Moreno, Walter Reed, Dan Riss, Adam Williams, Ned Glass, James Best, Robert Bray, Patrick Joseph Sexton. In order to prevent an attack on white settlers, an Indian guide plans hand-to-hand combat with his tribe's chief. Okay program film but nothing special.\n\n**5053** _ **Yellowneck**_ **** Republic, 1955. 83 min. Color. D: R. John Hugh. SC: Nat S. Linden. With Lin McCarthy, Stephen Courtleigh, Barry Kroeger, Harold Gordon, Bill Mason, Jose Billie, Roy Osceola, Al Tamez. Five escaped Confederate prisoners make their way through the Florida Everglades in an effort to stay alive and reach the safety of the ocean. Surprisingly atmospheric and entertaining melodrama.\n\n**5054** _ **Yellowstone**_ **** Universal, 1936. 63 min. D: Arthur Lubin. SC: Jefferson Parker, Stuart Palmer and Houston Branch. With Henry Hunter, Judith Barrett, Ralph Morgan, Alan Hale, Andy Devine, Monroe Owsley, Michael Loring, Paul Fix, Rollo Lloyd, Paul Harvey, Raymond Hatton, Diana Gibson, Mary Gordon, Claud Allister, Guy Kingsford, Russell Wade, Ed LeSaint, Dora Clement, Maxwell Sholes, Flo Wicks. A man searches for the bank loot his father supposedly hid two decades before and when an ex-convict is found murdered he becomes a suspect. Fairly good dual bill item with nice mystery elements, co-written by the famous detective novelist Stuart Palmer.\n\n**5055** _ **Yellowstone Kelly**_ **** Warner Bros., 1959. 91 min. Color. D: Gordon Douglas. SC: Burt Kennedy. With Clint Walker, Edward (Edd) Byrnes, John Russell, Ray Danton, Andra Martin, Claude Akins, Rhodes Reason, Gary Vinson, Warren Oates, Nesdon Booth, Harry Shannon, Buff Brady, Chief Yowlachie, Foster Hood, Clyde Howdy, Vince St. Cyr. A fur trapper finds himself in the middle of Indian warfare after whites take a pretty maiden as their prisoner. Clint Walker handles the title role of this speedy feature in good form.\n\n**5056** _ **Yodelin' Kid from Pine Ridge**_ **** Republic, 1937. 61 min. D: Joseph Kane. SC: Dorrell McGowan, Stuart McGowan and Jack Natteford. With Gene Autry, Smiley Burnette, Betty Bronson, LeRoy Mason, Charles Middleton, Russell Simpson, The Tennessee Ramblers, Jack Dougherty, Guy Wilkerson, Frankie Marvin, Henry Hall, Fred \"Snowflake\" Toones, Bud Osborne, Jack Kirk, Bob Burns, Al Taylor, George Morrell, Lew Meehan, Jim Corey, Jack Ingram, Art Dillard, Art Mix, Oscar Gahan, Herman Hack, Bill Nestell, Jack Evans, Charles Brinley, Jack Montgomery, Tom Smith. After being called a traitor by his father and joining a traveling Wild West show, Gene Autry returns home to the Turpentine Pine Forest of Florida and Georgia to try and stop a feud between cattle ranchers and turpentine makers. Fine Gene Autry vehicle with lots of action, an interesting story, good photography and location shooting and fine musical interludes. In this one Smiley Burnette is Colonel Millhouse, the carnival chief, and not his usual Frog character.\n\n**5057** _ **You Know My Name**_ **** Turner Network Television (TNT), 1999. 94 min. Color. D-SC: John Kent Harrison. With Sam Elliott, Arliss Howard, Carolyn McCormick, James Gammon, R. Lee Ermey, James Parks, Sheila McCarthy, Nataalia Rey, Jonathan Young, Perla Batalla, Johann Benet, Marilyn Norry, Andy Maton, Walter Olkewicz, Michelle Malmberg, David Lereaney, Alexander Pollick, Alex Diakun, David Barrett, Mel Crumb, Muse Watson, Chris Nelson Norris. After life as a cowboy and lawman, Bill Tilghman becomes a filmmaker attempting to showcase the real West. Sam Elliott gives an outstanding performance as Bill Tilghman in this well made TV movie.\n\n**5058** _ **Young and Free**_ **** Manson International, 1978. 90 min. Color. D-SC: Keith Larsen. With Erik Larsen, Ivy Angustain, Keith Larsen, Carrol McCall. A boy grows to manhood learning the ways of the wilderness and how to survive. Keith Larsen strikes again in this outdoor drama starring his son; for fans of lots of scenery.\n\n**5059** _ **Young Bill Hickok**_ **** Republic, 1940. 59 min. D: Joseph Kane. SC: Olive Cooper and Norton S. Parker. With Roy Rogers, George \"Gabby\" Hayes, Jacqueline Wells (Julie Bishop), John Miljan, Sally Payne, Monte Blue, Hal Taliaferro, Ethel Wales, Jack Ingram, Iron Eyes Cody, Dick Elliott, Slim Whitaker, Jack Kirk, Hank Bell, Henry Wills, William Desmond, John Elliott, Jack Rockwell, Bill Wolfe. Pony Express rider Bill Hickok is out to stop a foreign agent using a gang of marauders to annex part of California during the Civil War. Good, action packed early entry in Roy Roger's Republic series.\n\n**5060** _ **Young Billy Young**_ **** United Artists, 1969. 89 min. Color. D-SC: Burt Kennedy. With Robert Mitchum, Angie Dickinson, Robert Walker (Jr.), David Carradine, Jack Kelly, John Anderson, Deana Martin, Paul Fix, Willis Bouchey, Parley Baer, Bob Anderson, Rodolfo Acosta, Christopher Mitchum. A man arrives in a New Mexico town and becomes its sheriff to bring in the killer of his son. Good finale showdown adds some life to this leisurely paced Western.\n\n**5061** _ **Young Blood**_ **** Monogram, 1932. 60 min. D: Phil Rosen. SC: Wellyn Totman. With Bob Steele, Helen Foster, Charles King, Naomi Judge, Art Mix, Si Jenks, Earl Dwire, Henry Roquemore, Hank Bell, Harry Semels, Lafe McKee, Perry Murdock, Henry Hall, Fern Emmett, Horace B. Carpenter, Tex Palmer, Bud McClure, Blackjack Ward, Roy Bucko. A cowboy who steals from crooks to help the oppressed gets involved with a foreign actress. This Bob Steele vehicle has a plot that is hard to believe; below average.\n\n**5062** _ **Young Buffalo Bill**_ **** Republic, 1940. 59 min. D: Joseph Kane. SC: Harrison Jacobs, Robert Yost and Gerald Geraghty. With Roy Rogers, George \"Gabby\" Hayes, Pauline Moore, Hugh Sothern, Trevor Bardette, Chief Thundercloud, Julian Rivero, Gaylord (Steve) Pendleton, Wade Boteler, George Chesebro, Hank Bell, William Kellogg, Jack O'Shea, Iron Eyes Cody, Anna Demetrio, Estrelita Zarco. Youthful Bill Cody helps both settlers and Indians about to be defrauded by Spanish land grant claimants. Typically action filled Roy Rogers \"historical\" saga.\n\n**5063** _ **The Young Country**_ **** ABC-TV\/Universal, 1970. 73 min. Color. D-SC: Roy Huggins. With Roger Davis, Joan Hackett, Walter Brennan, Peter Deuel, Wally Cox, Skip Young, Steve Sandor, Robert Driscoll Miller, Richard Van Fleet, Elliott Street, Barbara Gates, Luis Delgado, Thomas Ballin. A gambler suddenly turns honest when he finds stolen bank money but when he tries to return it no one will claim the loot. Fair genre comedy made for television.\n\n**5064** _ **Young Daniel Boone**_ **** Monogram, 1950. 71 min. D: Reginald LeBorg. SC: Clint Johnson and Reginald LeBorg. With David Bruce, Kristine Miller, Damian O'Flynn, Don Beddoe, Mary Treen, John Mylong, William Roy, Stanley Logan, Richard Foote. Daniel Boone rescues the survivors of an Indian attack and learns a French spy is responsible for the uprising. This dual bill item tries hard but the lack of a sufficient budget is evident.\n\n**5065** _ **Young Fury**_ **** Paramount, 1965. 79 min. Color. D: Christian Nyby. SC: Steve Fisher. With Rory Calhoun, Virginia Mayo, Lon Chaney, Richard Arlen, William Bendix, John Agar, Preston Pierce, Linda Foster, Robert Biheller, Jody McCrea, Merry Anders, Rex Bell, Jr., Joan Huntington, Reg Parton, Marc Covell, Jay Ripley, Kevin O'Neal, Dal Jenkins, Fred Alexander, Jerry Summers, William Wellman, Jr., Steve Condit, Dave Dunlop, Bill Clark, William J. Vincent, Jesse Wayne, Robert Miles, Eddie Hice, Fred Krone, Joe Finnegan, Kent Hays, Jorge Moreno. A gang of young hellions take over a frontier town with the leader learning his mother is a saloon hostess and his ex-gunman father is being chased by the Dalton gang. Mediocre production, one of the least satisfying of the mid\u20131960s A.C. Lyles films, despite a good cast.\n\n**5066** _ **The Young Guns**_ **** Allied Artists, 1956. 84 min. D: Albert Band. SC: Louis Garfinkle. With Russ Tamblyn, Gloria Talbott, Perry Lopez, Scott Marlowe, Wright King, Walter Coy, Chubby Johnson, Myron Healey, James Goodwin, Rayford Barnes, I. Stanford Jolley, Emory Parnell, Dabbs Greer, Earle Hodgins, Ray Teal, Tom London, Kim Charney, Robert Bice, Ken Miller. The son of a famous gunman tries to lead a peaceful live in a Wyoming town but his father's reputation makes it difficult for him. Average outing with Guy Mitchell singing the title tune.\n\n**5067** _ **Young Guns**_ **** 20th Century\u2013Fox, 1988. 107 min. Color. D: Christopher Cain. SC: John Fusco. With Emilio Estevez, Kiefer Sutherland, Lou Diamond Phillips, Charlie Sheen, Casey Siemaszko, Terence Stamp, Jack Palance, Terry O'Quinn, Sharon Thomas, Geoffrey Blake, Alice Carter, Brian Keith, Tom Callaway, Patrick Wayne, Lisa Banes, Sam Gauny, Cody Palance, Gadeek, Victor Izay, Allen Robert Keller, Craig M. Erickson, Jeremy H. Lepard, Daniel Kamin, Richela Renkun, Pat Lee, Gary Kanin, Forrest Broadley, Alan Tobin, Joey Hanks, Loyd Lee Brown, Elena Parres. When their rancher benefactor is murdered, a group of young men become deputies led by Billy the Kid but when he guns down the killer they become the hunted. Big money making, good youth oriented Western, followed by _**Young Guns II**_ (q.v.).\n\n**5068** _ **Young Guns of Texas**_ **** 20th Century\u2013Fox, 1963. 78 min. Color. D: Maury Dexter. SC: Harry Cross. With James Mitchum, Alana Ladd, Jody McCrea, Chill Wills, Gary Conway, Barbara Mansell, Robert Lowery, Troy Melton, Fred Krone, Alex Sharp, Robert Hinkle, Will Wills. Two men, a soldier searching for stolen Army gold and a father trying to find his eloping daughter, join forces when caught in an Indian attack. Fine outing with a good script and the gimmick of having for its leads the children of famous stars.\n\n**5069** _ **Young Guns II**_ **** 20th Century\u2013Fox, 1990. 104 min. Color. D: Geoff Murphy. SC: John Fusco. With Emilio Estevez, Kiefer Sutherland, Lou Diamond Phillips, Christian Slater, William Petersen, Alan Ruck, R.D. Call, James Coburn, Balthazar Getty, Jack Kehoe, Robert Knepper, Tom Kurlander, Viggo Mortensen, Leon Rippy, Tracey Walter, Brad Whitford, Scott Wilson, Jenny Wright, John Hammil, William Fisher, Carlotta Garcia, Joy Bouton, Albert Trujillo, Alina Arenal, John Alderson, Lee de Broux, David Paul Needles, Joey Joe Amlin, Chief Buddy Redbow, Jerry Gardner, Mark Bustamante, Nicholas Sean Gomez, Stephan Kraus, Tony Frank, Don Simpson, Michael Eiland, Jon Bon Jovi, Richard Schiff, Frank Fierro, Jr., Rene L. Moreno, Alexis Alexander, Ginger Lynn Allen, Boots Southerland, Bo Gray. After Billy the Kid helps two pals escape from jail the trio head to Mexico but a former friend of Bonney's is hired to kill him. Action laced successful box office follow-up to _**Young Guns**_ (q.v.).\n\n**5070** _ **Young Jesse James**_ **** 20th Century\u2013Fox, 1960. 73 min. D: William F. Claxton. SC: Orville H. Hampton and Jerry Sackheim. With Ray Stricklyn, Willard Parker, Merry Anders, Robert Dix, Emile Meyer, Jacklyn O'Donnell, Rayford Barnes, Rex Holman, Bob Palmer, Sheila Bromley, Johnny O'Neill, Leslie Bradley, Norman Leavitt, Lee Kendall. During the Civil War, Jesse and Frank James join Quantrill's Raiders and later become outlaws when Yankees hang their father. Still another retelling of the James Brothers saga, but this one is a bit tattered although it contains good performances by Willard Parker as Cole Younger and Emile Meyer as William Clarke Quantrill.\n\n**5071** _ **The Young Land**_ **** Columbia, 1959. 89 min. Color. D: Ted Tetzlaff. SC: Norman Shannon Hall. With Pat(rick) Wayne, Yvonne Craig, Dennis Hopper, Dan O'Herlihy, Roberto de la Madrid, Cliff Ketchum, Ken Curtis, Pedro Gonzalez Gonzalez, Edward Sweeney, Miguel Camacho, Cliff Lyons, Randy Sparks, Mario Arteaga, Charles Heard, Carlos Romero, Tom Tiner, Jose Quijada. Trouble brews in the Republic of Texas when a citizen is to be tried for the murder of a Mexican. Well modulated and entertaining effort.\n\n**5072** _ **Young Mr. Lincoln**_ **** 20th Century\u2013Fox, 1939. 101 min. D: John Ford. SC: Lamarr Trotti. With Henry Fonda, Alice Brady, Marjorie Weaver, Arleen Whelan, Eddie Collins, Pauline Moore, Richard Cromwell, Eddie Quillan, Ward Bond, Donald Meek, Spencer Charters, Judith Dickens, Milburn Stone, Cliff Clark, Robert Lowery, Charles Tannen, Francis Ford, Fred Kohler, Jr., Kay Linaker, Russell Simpson, Charles Halton, Edwin Maxwell, Robert Homans, Jack Kelly, Dickie Jones, Harry Tyler, Louis Mason, Jack Pennick, Steven Randall, Clarence Wilson, Elizabeth Jones. In frontier Illinois novice attorney Abraham Lincoln agrees to act as defense council for two backwoods youths accused of murder. Top notch study of Lincoln's early life; a near classic.\n\n**5073** _ **Young Pioneers**_ **** ABC-TV, 1976. 96 min. Color. D: Michael O'Herlihy. SC: Blanche Hanalis. With Roger Kern, Linda Purl, Robert Hays, Shelly Juttner, Robert Donner, Frank Marth, Brendan Dillon, Charles Tyner, Jonathan Kidd, Arnold Soboloff, Bernice Smith, Janis Famison, Dennis Fimple. A teenage married couple leave their family home in Iowa and head West to settle in the rugged Dakotas of the 1870s. Fairly good TV movie adaptation of the works of Rose Wilder Lane.\n\n**5074** _ **Young Pioneers' Christmas**_ **** ABC-TV, 1976. 100 min. Color. D: Michael O'Herlihy. SC: Blanche Hanalis. With Rogert Kern, Linda Purl, Robert Hays, Kay Kimler, Robert Donner, Britt Leach, Arnold Soboloff, Brendan Dillon, Rand Bridges, Brian Melrose, Sherri Wagner. A young couple try to overcome the grief of losing their infant and bring a happy Christmas to other settlers in the Dakota Territory. Okay sequel to _**Young Pioneers**_ (q.v.), neither of which made it as a TV series.\n\n**5075** _ **The Younger Brothers**_ **** Warner Bros., 1949. 77 min. Color. D: Edwin L. Marin. SC: Edna Anhalt. With Wayne Morris, Janis Paige, Bruce Bennett, Geraldine Brooks, Robert Hutton, Alan Hale, Fred Clark, James Brown, Monte Blue, Tom Tyler, William Forrest, Ian Wolfe, Emmett Lynn, Hank Mann, Gene Roth, Paul Panzer, Syd Saylor, Lee Morgan, Creighton Hale, Kermit Maynard, Artie Ortego, Joan Blair, Phil McCullough, Jack Watt, J.G. MacMahon, Charles Sherlock, Dick Gordon, George Sherwood, Ben Corbett, Kansas Moehring. While awaiting a pardon from the governor, the Younger Brothers are forced into lawlessness when the youngest sibling kills in self defense. Good production values and cast overcome a mundane script.\n\n**5076** _ **You're Fired**_ **** Goodwill, 1925. 50 min. D: Paul Hurst. SC: William Lester. With Bill Bailey, Alma Rayford, Robert (Bob) McKenzie, Theodore Lorch, Sam Bloom, Velma Watkins, Foyce Brown, Victor Allen. When his sister tries to civilize a rancher by sending a group of dudes to his spread, he pretends to be a hired hand and gets fired but comes to their rescue when they are kidnapped by outlaws. Pleasant silent tongue-in-cheek poverty row feature.\n\n**5077** _ **Yukon Flight**_ **** Monogram, 1939. 58 min. D: Ralph Staub. SC: Edward Halperin. With James Newill, Louise Stanley, Warren Hull, Dave O'Brien, William Pawley, Karl Hackett, Jack Clifford, Roy Barcroft, Bob Terry, Earl Douglas, George Humbert, Ernie Adams, Jack Rutherford, Eddie Fetherston. Mounted Policeman Renfrew asks a former cohort to help him in stopping an outlaw gang carrying gold out of Canada via airplanes. Entertaining effort in the popular \"Renfrew of the Royal Mounted\" series. Also called _**Renfrew of the Royal Mounted in Yukon Flight**_.\n\n**5078** _ **Yukon Gold**_ **** Monogram, 1952. 62 min. D: Frank McDonald. SC: William Raynor. With Kirby Grant, Martha Hyer, Harry Lauter, Philip Van Zandt, Frances Charles, Mauritz Hugo, James Parnell, Sam Flint, I. Stanford Jolley, Hal Gerard, Roy Gordon, Ward Blackburn, Chinook (dog). A Royal Canadian Mounted Police officer arrives in a rugged mining camp searching for a killer and meets a pretty gambler. Average outing in Kirby Grant's series for Monogram supposedly based on James Oliver Curwood's _The Gold Hunters_.\n\n**5079** _ **Yukon Manhunt**_ **** Monogram, 1951. 61 min. D: Frank McDonald. SC: William Raynor. With Kirby Grant, Gail Davis, Margaret Field, Rand Brooks, Nelson Leigh, John Doucette, Paul McGuire, Dick Barron, Dennis Moore, Chinook (dog). A Mounted Policeman and his faithful husky dog try to find who is behind a series of payroll messenger robberies. Standard entry in Kirby Grant's Canadian Mounted series.\n\n**The Yukon Patrol** see _**King of the Royal Mounted**_ (1940)\n\n**5080** _ **Yukon Safari**_ **** American National Enterprises, 1976. 95 min. Color. Adventurers explore the Yukon from the Arctic south and look at its land and people. Very well done documentary.\n\n**5081** _ **Yukon Vengeance**_ **** Allied Artists, 1954. 68 min. D: William Beaudine. SC: William Raynor. With Kirby Grant, Mary Ellen Kay, Monte Hale, Henry Kulky, Carol Thurston, Marshall Bradford, Parke MacGregor, Fred Gabourie, Billy Wilkerson, Chinook (dog). A Mountie and his dog go to remote Bear Creek to investigate the robbery and murder of three mail carriers. Fair outing in the Kirby Grant series with the added treat of Monte Hale in a villainous role.\n\n**5082** _ **Yuma**_ **** ABC-TV, 1971. 73 min. Color. D: Ted Post. SC: Charles Wallace. With Clint Walker, Barry Sullivan, Kathryn Hays, Edgar Buchanan, Morgan Woodward, Peter Mark Richman, John Kerr, Robert Phillips, Miguel Alejandro, Neil Russell, Bruce Glover. The marshal of a rough town faces opposition from crooked officials as well as the brother of a prisoner. More than passable made-for-TV oater.\n\n**5083** _ **Zachariah**_ **** Cinerama Releasing Corporation, 1971. 93 min. Color. D: George Englund. SC: Joe Massot. With John Rubenstein, Pat Quinn, Don Johnson, Elvin Jones, Country Joe and The Fish, Doug Kershaw, William Challee, Robert Ball, Dick Van Patten, The James Gang, White Lightnin', The New York Ensemble. Two gunfighter friends go separate ways with one giving up his shooting irons for a life of peace but eventually they are forced into a showdown. Rock musical Western not likely to appeal to serious fans.\n\n**5084** _ **Zandy's Bride**_ **** Warner Bros., 1974. 116 min. Color. D: Jan Troell. SC: Marc Norman. With Gene Hackman, Liv Ullman, Eileen Heckart, Harry Dean Stanton, Joe Santos, Frank Cady, Sam Bottoms, Susan Tyrell, Bob Simpson, Fabian Gregory Cordova, Don Wilbanks, Vivian Gordon, Alf Kjellin. A rough hewn frontiersman wants a mail order bride in order to have children but the Swedish woman he gets is strong willed and he fears she may be too old for child bearing. Somewhat sluggish effort from the director and star (Liv Ullman) of _**The Emigrants**_ (Warner Bros., 1971) and _**The New Land**_ (q.v.); Frank Cady is quite good as Gene Hackman's hateful father.\n\n**5085** _ **Zorro**_ **** Allied Artists, 1976. 100 min. Color. D: Duccio Tessari. SC: Duccio Tessari and Giorgio Arlorio. With Alain Delon, Stanley Baker, Adriana Asti, Giacomo Rossi Stuart, Ottavia Piccolo, Moustache, Enzo Cerusico, Giampiero Albertini, Marino Mase, Rajka Jurcec, Yvan Chiffre. In nineteenth century Latin America the foppish Don Diego takes on the guise of the masked hero Zorro to avenge a friend's murder. Fairly entertaining French-Italian co-production about the famous screen hero although it does contain a talking dog; originally ran 124 minutes when released in Europe by Titanus in 1975.\n\n**Alain Delon and Stanley Baker in** _**Zorro**_ **(Allied Artists, 1976).**\n\n**  \n**\n\n_**Zorro Against Maciste**_ see _**Samson and the Slave Queen**_\n\n_**Zorro and the Comanches**_ see _**Zorro, Rider of Vengeance**_\n\n**5086** _ **Zorro and the Mystery of Don Cabrillo**_ **** Buena Vista, 1959. 73 min. D: Hollingsworth Morse. SC: Lowell S. Hawley and Bob Wehling. With Guy Williams, Annette Funicello, Henry Calvin, Gene Sheldon, Carlos Rivas, Arthur Space, Don Diamond, George J. Lewis, Wendell Holmes, Greigh Phillips, Perry Stanton, Edward Colmans, Bud Osborne. Zorro tries to help a young woman who has come from Spain to be reunited with her rancher father who seems not to exist. Well done \"The Adventures of Zorro\" (ABC-TV, 1957\u201359) compilation made up of the 1959 \"The Brooch\" and \"The Missing Father\" episodes and issued as a feature on tape.\n\n**5087** _ **Zorro and the Three Musketeers**_ **** Golden Era, 1963. 99 min. Color. D: Luigi Capuano. SC: Roberto Gravity and Italo De Tuddo. With Gordon Scott, Maria Grazia Spina, Jose Greci, Giacomo Rossi Stuart, Livio Lorenzon, Franco Fantasia, Nazzareno Zamperla, Roberto Risso, Mario Pisu, Gianni Rizzo, Nerio Bernardi, Amina Pirani Maggi. In the 17th century, Zorro joins forces with the Three Musketeers to rescue a princess kidnapped by a Spanish emissary. Made in Italy by Starlight as _**Zorro e i Tre Moschiettieri**_ (Zorro and the Three Musketeers), running 101 minutes and issued to TV by American International as _**Mask of the Musketeers**_ ; for diehard Zorro fans only!\n\n**5088** _ **Zorro at the Court of England**_ **** Romana Film, 1969. 92 min. Color. D: Franco Montemurro. SC: Arpad DeRiso and Franco Montemurro. With Spyros Focas, Anna Maria Guglielmotti, Daniele Vargas, Franco Ressel, Dada Gallotti, Massimo Carocci, Barbara Carroll, Franco Fantasia, Carole Wells, Angela De Leo, Mirella Pamphili, Liana Del Blazo, Spartaco Conversi. Zorro attempts to expose the machinations of a corrupt Central American governor and save a beautiful girl from his clutches. Average \"Zorro\" rehash made in Italy as _**Zorro alla Corte d'Inghilterra**_ (Zorro at the Court of England).\n\n**5089** _ **El Zorro Blanco**_ (The White Fox) **** Producciones Filmicas Agrasanchez S.A.\/Solis Hermanos S.A., 1978. 90 min. Color. D: Jose Luis Urquieta. SC: Adolfo Martinez Solares. With Juan Miranda, Hilda Aguirre, Carlos Agosti, Freddy Fernandez \"Pichi,\" Luis A. Elizondo, Rene Agrasanchez, Rebeca Sixton, Ernesto Solis, Josefina Sosa, Arlette Pacheco. A masked man saves his fiancee from her evil stepfather who wants her out of the way so he can have the family estate. Pretty fair modern-day \"Zorro\" film made in Mexico.\n\n**5090** _ **El Zorro de Jalisco**_ (The Fox of Jalisco) **** FICSA\/Pegaso Film, 1941. 68 min. D-SC: Jose Benavides, Jr. With Pedro Armendariz, Consuelo de Alba, Emilio Fernandez, Lucha Reyes, Augustin Isunza, Tito Junco, Alfonso Bedoya, Manuel Pozos, Miguel Indan, Manuel Donde, Julio Ahuet. An attorney returns home and takes on the guise of Zorro to combat the outlaw leader who murdered his girl's father. The first Mexican \"Zorro\" feature set in modern times is a good one.\n\n**5091** _ **El Zorro Escarlata**_ (The Scarlet Fox) **** Casa-Mohme, 1957. 74 min. D: Rafael Baledon. With Luis Aguilar, Irma Dorantes, Jaime Fernandez, Fernando Fernandez, Pascual Garcia Pena, Jose Eduardo Perez, Fanny Schiller, Emma Roldan. A mild mannered man becomes the Scarlet Fox to save his lady love from a revived corpse. Scary Mexican horror Western, followed by _**El Regreso del Monstruo**_ (The Return of the Monster) and _**El Zorro Vengador**_ (The Avenging Fox) (qq.v.).\n\n**5092** _ **Zorro in the Court of Spain**_ **** Starlight, 1962. 94 min. D: Luigi Capuano. SC: Nino Scolaro and Arpad DeRiso. With George Ardisson, Alberto Lupo, Nadia Marlowa, Tullio Altamura, Carlo Tamberlini, Gianni Rizzo, Adreina Paul, Maria Letizia, Franco Fantasia, Maria Grazia Spina, Nerio Bernardi, Carlo Cano, Livio Lorenzon, Gloria Parri, Nazzareno Zamperia, Pasquale De Filipp, Antonio Gradoli, Ugo Sasso, Amedeo Trilli. Zorro tries to help the queen of Spain by rescuing her daughter who has been kidnapped by her evil brother-in-law. So-so Italian production made as _**Zorro alla Corte di Spagna**_ (Zorro in the Court of Spain) and shown in the U.S. as _**The Masked Conqueror**_ by American International.\n\n**5093** _ **El Zorro Justiciero**_ (The Severe Zorro) **** Copercines\/Italian International Film, 1969. 78 min. Color D: Rafael Romero Merchant. SC: Rafael Romero Merchant, Fulvio Lucisano and Fernando Marco. With Martin Moore, Simone Blondell, Fabio Testi, Antonio Gradoli, Frank Brana, Luis Induni, Ana Maria Saijar, Eduardo Baldi, Emilio Rodriguez, Piero Lulli, Carlos Romero Merchant, Riccardo Garrone. Zorro steals the money made from a tyrant's auctioning his father's rancho and uses it to buy back the property. Passable French-Italian \"Zorro\" co-production.\n\n**5094** _ **Zorro, Marquis of Navarra**_. Romana Film, 1971. 91 min. Color. D: Jean Monty (Franco Montemurro). SC: Piero Pierotti and Francesco (Franco) Montemurro. With Nadir Moretti, Maria Luisa Longo, Daniele Vargas, Loris Gizzi, Renato Montalbano, Dada Gallotti, Ugo Adinolfi, Mimmo Poli, Fortunato Arena, Gisella Arden, Ignazio Balsamo, Rosy De Leo. Zorro allies himself with the exiled Spanish king in trying to help the oppressed people of his country against their French conquerors. Like _**Zorro in the Court of Spain**_ (q.v .) this average dubbed costumer is included here because of the Zorro character; made in Italy as _**Zorro, Marchese di Navarra**_ (Zorro, Marquis of Navarre).\n\n_**Zorro Nella Valle dei Fantasm**_ see _**The Lone Rider**_ (1960)\n\n**5095** _ **Zorro, Rider of Vengeance**_ **** D.C. Films\/Hispamer, 1971. 92 min. Color. D: Luigi Capuano and Jose Luis Merino. SC: Jose Luis Merino and Maria del Carmen Martinez Roman. With Charles (Carlos) Quiney, Malisa Longo, Maria Mahor, Arturo Dominici, Anna Farra, Fernando Hilbeck, Jose Cardenas, Enrique Avila, Pasquale Basile. Trying to stop a corrupt adventurer from obtaining a priceless diamond, Zorro finds himself against a female Pinkerton agent hired to stop him. Fair Italian-Spanish co-production filmed as _**Zorro il Cavaliere della Vendetta**_ (Zorro, the Cavalier of Vengeance) and also called _**Zorro and the Comanches.**_\n\n**5096** _ **Zorro Rides Again**_ **** Republic, 1937. 12 Chapters. D: William Witney and John English. SC: Barry Shipman, John Rathmell, Franklyn Adreon, Ronald Davidson and Morgan B. Cox. With John Carroll, Helen Christina, Reed Howes, Duncan Renaldo, Richard Alexander, Noah Beery, Nigel de Brulier, Robert Kortman, Jack Ingram, Roger Williams, Tony Martelli, Edmund Cobb, Mona Rico, Tom London, Harry Strang, Jerry Frank, Paul Lopez, George Mari, Yakima Canutt, Frank Ellis, Al Haskell, Dirk Thane, Lane Chandler, Murdock MacQuarrie, Chris-Pin Martin, Frank McCarroll, Frankie Marvin, Jack Kirk, Ray Teal, Merrill McCormick, Rosa Turich, Art Felix, Josef Swickard, Forrest Burns, Jason Robards, Jack Hendricks, Frank Leyva, Hector Sarno, Al Taylor, Duke Taylor, Bob Jamison. Zorro comes to the rescue of a family whose railroad is sought by the ruthless El Lobo and his henchmen. This action packed serial is a real treat; also issued in a 69 minute feature version in 1938 and 1959 by Republic.\n\n**5097** _ **Zorro the Avenger**_ **** Buena Vista, 1959. 90 min. D: Charles Barton. SC: Lowell S. Hawley and Bob Wehling. With Guy Williams, Charles Korvin, Henry Calvin, Gene Sheldon, George J. Lewis, Jay Novello, Ralph Clanton, Henry Rowland, Michael Pate, Jonathan Hole. In Old California, masked nemesis Zorro opposes a rogue trying to control the countryside and make it his empire. Made up of several episodes of \"The Adventures of Zorro\" (ABC-TV, 1957\u201359), this exciting theatrical feature was originally intended for countries that did not run the TV series.\n\n_**Zorro the Avenger**_ (1962) see _**Shadow of Zorro**_\n\n**5098** _ **Zorro the Dominator**_ **** Rosa Films, S.A., 1971. 89 min. Color. D: Jose Luis Merino. SC: Jose Luis Merino, Jose Luis Damiani, Enzo Gicca, Maria del Carmen Martinez Roman and Mario Merino. With Charles (Carlos) Quiney, Lea Nani, Vidal Molina, Pasquale Basile, Antonio Jiminez Escribano, Alex Marco, Juan Cortes, Pasquale Simeoli, Santiago Rivero, Jose Jaspe, Luis Marin, A.G. Scribano. Returning home from school to find his homeland under the thumb of a cruel alcalde, a nobleman becomes Zorro to lead the people in revolt. Still another rehash of the \"Zorro\" story, this one an Italian-Spanish co-production originally called _**El Zorro de Monterrey**_ (The Fox of Monterey).\n\n**5099** _ **Zorro the Fox**_ **** Magic Films, 1968. 89 min. Color. D: Guido Zurli. SC: Guido Leoni and Ambrogio Molteni. With George Ardisson, Consalvo Dell'Arti, Jack Stuart (Giacomo Rossi Stuart), Pedro Sanchez, Evaristo Maran, Femi Benussi, Spartaco Battisti, Artemio Antonini, Gustavo D'Arpe, Grazia Fei, Gippo Leone, Aldo Marianeci, Riccardo Pizzuti, Gianni Pulone, Juan Valejo. Zorro helps Mexican villagers to rise up in revolt against their despotic governor oppressor. Acceptable Spanish \"Zorro\" feature issued there as _**La Espada del Zorro**_ (The Sword of Zorro) and in Italy as _**La Volpe**_ (The Fox).\n\n**5100** _ **Zorro, the Gay Blade**_ **** 20th Century\u2013Fox, 1981. 93 min. Color. D: Peter Madak. SC: Hal Dresner. With George Hamilton, Lauren Hutton, Brenda Vacarro, Ron Liebman, Donovan Scott, James Booth, Helen Burns, Clive Revill, Carolyn Seymour, Eduardo Noriega, Jorge Russek, Eduardo Alcaraz, Carlos Bravo, Robert Dumont, Jorge Bolio, Dick Balduzzi, Ana Elisa Perez, Pilar Pellicar, Frank Welker (narrator). When an evil tyrant tries to oppress the people in frontier California, the two sons of a nobleman attempt to stop him but when one is sidelined by an injury he is replaced by his gay brother. Comic take on \"Zorro,\" not likely to appeal to the character's fans.\n\n**5101** _ **Zorro the Rebel**_ **** Romana Film, 1966. 95 min. D: Pierro Pierotti. SC: Pierro Pierotti and Gianfranco Clerici. With Howard Ross, Dina De Santis, Charles Borromel, Arturo Dominici, Gabriella Andreini, Ted Carter, Rosy De Leo, Nello Pazzafini. Zorro helps a beautiful woman forced to marry the son of the local governor against her will. This Italian production, issued there as _**Zorro il Ribelle**_ (Zorro the Rebel), is not one of the better \"Zorro\" efforts.\n\n**5102** _ **El Zorro Vengador**_ (The Avenging Fox) **** Alameda Film, 1962. 87 min. D: Zacarias Gomez Urquiza. With Luis Aguilar, Maria Eugenia San Martin, Jaime Fernandez, Arturo Martinez, Fernando Soto, Victorio Blanco, Guillermo Hernandez, Carlos Leon, Cuco Sanchez, Pascual Garcia Pena, Jesus Gomez, Emilio Garibay, Jose Eduardo Perez, Arturo Soto Rangel, Salvador Terroba. A mysterious brotherhood is after a woman's gold mine and her fiancee, the masked Zorro Escarlata, tries to defeat them. Okay third and final feature in the Mexican \"El Zorro Escarlata\" (The Scarlet Fox) series, preceded by _**El Zorro Escarlata**_ (The Scarlet Fox) and _**El Regreso del Monstruo**_ (The Return of the Monster) (qq.v.).\n\n_**Zorro vs. the Teenage Monster**_ see _**El Regreso del Monstruo**_ (The Return of the Monster)\n\n**5103** _ **Zorro's Black Whip**_ **** Republic, 1944. 12 Chapters. D: Spencer Gordon Bennet. and Wallace Grissell. SC: Basil Dickey, Jesse Duffy, Grant Nelson and Joseph Poland. With George J. Lewis, Linda Stirling, Lucien Littlefield, Francis McDonald, Hal Taliaferro, John Merton, John Hamilton, Tom Chatterton, Tom London, Jack Kirk, Jay Kirby, Si Jenks, Stanley Price, Tom Steele, Duke Green, Dale Van Sickel, Cliff Lyons, Roy Brent, Bill Yrigoyen, Forrest Taylor, Fred Graham, Marshall Reed, Augie Gomez, Carl Sepulveda, Horace B. Carpenter, Herman Hack, Carey Loftin, Cliff Parkinson, Kenneth Terrell, Duke Taylor, Jack O'Shea, Nolan Leary, Robert Wilke, Post Park, Vinegar Roan. After her newspaper editor brother is murdered by elements opposing statehood and promoting lawlessness, a woman not only takes over his job but also his guise of the Black Whip, the masked leader of vigilantes opposing the outlaws. Fast moving and enjoyable cliffhanger with pretty Linda Stirling as a female Zorro; re-released in 1957.\n\n**5104** _ **Zorro's Fighting Legion**_ **** Republic, 1939. 12 Chapters. D: William Witney and John English. SC: Ronald Davidson, Franklyn Adreon, Morgan Cox, Sol Shor and Barney A. Sareckey. With Reed Hadley, Sheila Darcy, William Corson, Leander de Cordova, Edmund Cobb, C. Montague Shaw, John Merton, Budd Buster, Carleton Young, Guy D'Ennery, Paul Marion, Joe Molina, James Pierce, Helen Mitchell, Curley Dresden, Charles King, Al Taylor, Charles B. Murphy, Billy Bletcher, Joe De La Cruz, Jason Robards, Theodore Lorch, Jack O'Shea, Jerome (Blackjack) Ward, Augie Gomez, Cactus Mack, Bud Geary, George Plues, Jack Carrington, Victor Cox, John Wallace, Bert Dillard, Kenneth Terrell, Wylie Grant, Carl Sepulveda, Yakima Canutt, Ernest Saracino, Reed Howes, Joe McGuinn, Bill Yrigoyen, Gordon Clark, Frank Ellis, Joe Yrigoyen, Ted Mapes, Henry Wills, Millard McGowan, Jimmy Fawcett, Martin Faust, Eddie Cherkose, Charles Murphy, Max Mark, Buel Bryant, Norman Lane, Ralph Faulkner, Alan Gregg, Clayton Moore, Bert Dillard, Bob Mabesa, Barry Hays, Jerry Frank. Three crooks try to block gold shipments to the government of Mexican president Benito Juarez and one of them pretends to be Don Del Oro to make the Indians think their deity has come to life but nobleman Don Diego becomes the masked avenger Zorro to thwart the troublemakers. A top notch serial, one of the best cliffhangers of the sound era; reissued in 1958.\n\n**5105** _ **El Zurdo**_ (The Left Hander) **** Radeant Films, 1965. 90 min. D: Arturo Martinez. SC: Marco Aurello Galindo and Carlos Gaytan. With Rodolfo de Anda, Ofelia Montesco, Francisco \"Charron\" Avitia, German Robles, Irma Serrano, Andres Soler, Noe Murayama, Pepito Velazquez. A young man is befriend by a gambler not knowing he is the one who killed his father. Solid Mexican Western drama.\n\n### Appendix 1: The Cowboys (and Cowgirls) and Their Horses\n\nAlthough they are not often included in the cast of the individual entries of this book, the horses of the cowboy stars were often big attractions and as much the star of a film as humans. Sometimes movies were built around the talents of the star's horse, as in the cases of Tom Mix's Tony, Ken Maynard's Tarzan and Roy Rogers' Trigger. The following list does not include every cinema star's movie mount but it does contain the main horse stars of the Western cinema.\n\n**Rex Allen** \u2014Koko\n\n**Gene Autry** \u2014Champion, Champion Jr.\n\n**Smith Ballew** \u2014Sheik\n\n**William Boyd** \u2014Topper\n\n**Harry Carey** \u2014Sonny\n\n**Lane Chandler** \u2014Raven\n\n**Bill Cody** \u2014Chico\n\n**Eddie Dean** \u2014Copper, Flash, White Cloud\n\n**Dale Evans** \u2014Buttermilk\n\n**William S. Hart** \u2014Fritz\n\n**Al Hoxie** \u2014Sunflash\n\n**Jack Hoxie** \u2014Dynamite, Scout\n\n**Buck Jones** \u2014Silver\n\n**Tom Keene** \u2014Flash, Prince\n\n**Allan Lane** \u2014Black Jack\n\n**Ken Maynard** \u2014Tarzan\n\n**Kermit Maynard** \u2014Rocky\n\n**Tom Mix** \u2014Tony, Tony Jr.\n\n**Dorothy Page** \u2014Snowy\n\n**Jack Perrin** \u2014Starlight\n\n**Jack Randall** \u2014Rusty\n\n**Tex Ritter** \u2014White Flash\n\n**Roy Rogers** \u2014Trigger\n\n**Reb Russell** \u2014Rebel\n\n**Fred Scott** \u2014White King\n\n**Charles Starrett** \u2014Raider\n\n**Fred Thompson** \u2014Silver King\n\n**John Wayne** \u2014Duke\n\n### Appendix 2: Screen Names\n\nOften people working in motion pictures used more than one name. Since this book contains thousands of credits the following will help with the confusion resulting from pseudonyms. Listed are the most commonly used screen name followed by alternate(s) and spelling differences.\n\n**Jane Adams** \u2014Poni Adams\n\n**Julie Adams** \u2014Betty Adams, Julia Adams\n\n**Richard Alexander** \u2014Dick Alexander\n\n**Gene Alsace** \u2014Rocky Camron, Buck Coburn\n\n**Morris Ankrum** \u2014Stephen Morris\n\n**John Archer** \u2014Ralph Bowman\n\n**Arkansas Slim Andrews** \u2014Lloyd Andrews\n\n**Anthony Ascott** \u2014Giuliano Carnimeo\n\n**Jimmy Aubrey** \u2014Jimmie Aubrey\n\n**Vivian Austin** \u2014Vivian Coe\n\n**William Norton Bailey** \u2014Bill Bailey\n\n**Silvertip Baker** \u2014Silver Tip Baker\n\n**Holly Bane** \u2014Michael Ragan, Mike Ragan\n\n**Joan Barclay** \u2014Germaine Greer\n\n**Don Barry** \u2014Don \"Red\" Barry, Donald Barry\n\n**Bruce Bennett** \u2014Herman Brix\n\n**Ray Bennett** \u2014Raphael Bennett\n\n**William Berke** \u2014Lester Williams\n\n**Willie Best** \u2014Sleep 'n Eat\n\n**Julie Bishop** \u2014Diane Duval, Jacqueline Wells\n\n**Pamela Blake** \u2014Adele Pearce\n\n**Robert Blake** \u2014Bobby Blake\n\n**Adrian Booth** \u2014Lorna Gray\n\n**Richard Botiller** \u2014Dick Botiller\n\n**Carlos Bravo** \u2014Charley Bravo, Charlie Bravo, Charly Bravo\n\n**Alan Bridge** \u2014Al Bridge\n\n**Sheila Bromley** \u2014Sheila LeGay, Sheila Mannors\n\n**Charles Bronson** \u2014Charles Buchinsky\n\n**Jean Brooks** \u2014Jeanne Kelly, Jeannie Kelly\n\n**Reno Browne** \u2014Reno Blair\n\n**Buffalo Bill, Jr.** \u2014Jay Wilsey\n\n**Adele Buffington** \u2014Jess Bowers\n\n**Boris Bullock** \u2014William Barrymore, Kit Carson\n\n**Francis X. Bushman, Jr.** \u2014Ralph Bushman\n\n**Jean Carmen** \u2014Julia Thayer\n\n**John Carpenter** \u2014Josh Carpenter, John Forbes\n\n**John Cason** \u2014Bob Cason, Chuck Cason, John L. Cason\n\n**Ed Cassidy** \u2014Edward Cassidy\n\n**Lon Chaney, Jr.** \u2014Creighton Chaney\n\n**Alden Chase** \u2014Guy Chase, Stephen Chase\n\n**Jan Clayton** \u2014Jane Clayton\n\n**Elmer Clifton** \u2014Elmer S. Pond\n\n**Junior Coghlan** \u2014Frank Coghlan\n\n**Lewis C. Collins** \u2014Cullen Lewis\n\n**Dorothy Comingore** \u2014Linda Winters\n\n**Michael Connors** \u2014Mike Connors, Touch Connors\n\n**Ray Corrigan** \u2014Ray Bernard\n\n**Bob Custer** \u2014Raymond Glenn\n\n**Art Davis** \u2014Larry Mason\n\n**Laraine Day** \u2014Lorraine Hayes, Laraine Johnson\n\n**Frank De Kova** \u2014Frank DeKova\n\n**Gordon DeMain** \u2014Gordon DeMaine, Bud Wood, G.D. Wood, Gordon Wood, G.D. Woods, Gordon D. Woods\n\n**Dick Dickinson** \u2014Peter Palmer\n\n**Denver Dixon** \u2014Victor Adamson, Al Mix, Art Mix\n\n**G.A. Durlam** \u2014George Arthur Durlam\n\n**Cliff Edwards** \u2014Ukulele Ike\n\n**Bill Elliott** \u2014Gordon Elliott, Wild Bill Elliott, William Elliott\n\n**Harry Fraser** \u2014Harry C. Crist, Weston Edwards, Harry O. Jones\n\n**Dean Fredericks** \u2014Norman Frederic\n\n**Rad Fulton** \u2014James Westmoreland\n\n**Guiliano Gemma** \u2014Montgomery Wood\n\n**Joseph Girard** \u2014Joe Girard, Joseph W. Girard\n\n**Chip Gorman** \u2014Andrea Giordana\n\n**Kirby Grant** \u2014Robert Stanton\n\n**James Griffith** \u2014James J. Griffith\n\n**Brett Halsey** \u2014Montgomery Ford\n\n**Arch Hall** \u2014Arch Hall, Sr., Archie Hall\n\n**Jon Hall** \u2014Charles Locher\n\n**Aleth Hansen** \u2014Aleth \"Speed\" Hansen, Speed Hansen, Aleth Hanson\n\n**George Hayes** \u2014George F. Hayes, George \"Gabby\" Hayes\n\n**Rita Hayworth** \u2014Rita Cansino\n\n**Ray Henderson** \u2014Jack Hendricks\n\n**Buzz Henry** \u2014Buzzy Henry, Robert Henry, Robert \"Buzz\" Henry\n\n**Riley Hill** \u2014Roy Harris\n\n**Robert Hill** \u2014Rock Hawkey\n\n**J. Merrill Holmes** \u2014Jack Holmes\n\n**Pee Wee Holmes** \u2014Gilbert Holmes\n\n**Olin Howlin** \u2014Olin Howland\n\n**Cris Huerta** \u2014Chris Huerta\n\n**Robert Hundar** \u2014Claudio Undari\n\n**Alan James** \u2014Alvin J. Neitz\n\n**Frank Jaquet** \u2014Frank Jacquet\n\n**Jennifer Jones** \u2014Phyllis Isley\n\n**Tom Keene** \u2014George Duryea, Dick Powers, Richard Powers\n\n**Robert Kellard** \u2014Robert Stevens\n\n**Charles King** \u2014Charles King, Jr., Charles L. King\n\n**Robert Kortman** \u2014Bob Kortman\n\n**Ed LeSaint** \u2014Edward J. LeSaint\n\n**Tom London** \u2014Leonard Clapham\n\n**Theodore Lorch** \u2014Ted Lorch\n\n**S. Roy Luby** \u2014Roy Claire\n\n**Jack Luden** \u2014John Luden\n\n**Cactus Mack** \u2014Taylor Curtis McPeters\n\n**Jock Mahoney** \u2014Jack Mahoney, Jock O'Mahoney, Jacques O'Mahoney\n\n**Daniel Mainwaring** \u2014Geoffrey Homes\n\n**Beth Marion** \u2014Betty Lloyd\n\n**Chris-Pin Martin** \u2014King Martin\n\n**Carl Mathews** \u2014Duke Mathews\n\n**William McCall** \u2014Bill McCall\n\n**Merrill McCormick** \u2014Merrill McCormack\n\n**Robert McKenzie** \u2014Bob McKenzie\n\n**Stephen McNally** \u2014Horace McNally\n\n**Murdock MacQuarrie** \u2014Murdock McQuarrie\n\n**Bud McTaggert** \u2014Malcolm McTaggert\n\n**Blanche Mehaffey** \u2014Janet Morgan\n\n**Lynn Merrick** \u2014Marilyn Merrick\n\n**John Merton** \u2014Mert Lavarre, Morton Lavarre\n\n**George Milton** \u2014Milton Raison & George Wallace Sayre\n\n**Art Mix** \u2014George Kesterson, Art Smith\n\n**George Montgomery** \u2014George Letz\n\n**Dennis Moore** \u2014Dennis Meadows, Denny Meadows, Smoky Moore\n\n**Alberto Morin** \u2014Albert Morin\n\n**Jack Natteford** \u2014John F. Natteford\n\n**Bud Nelson** \u2014James T. \"Bud\" Nelson\n\n**Bill Nestell** \u2014William Nestell\n\n**Sam Newfield** \u2014Sam Neufeld, Sherman Scott, Peter Stewart\n\n**James Newill** \u2014Jim Newill\n\n**Dave O'Brien** \u2014David Barclay, Tex O'Brien\n\n**Artie Ortego** \u2014Art Ardigan\n\n**Calvin Jackson Paget** \u2014Giorgio Ferroni\n\n**Robert Paige** \u2014David Carlyle, David Newell\n\n**Gregg Palmer** \u2014Palmer Lee\n\n**Eddie Parker** \u2014Edwin Parker\n\n**Post Park** \u2014Post Parks\n\n**Shirley Patterson** \u2014Shawn Smith\n\n**John Payne** \u2014Jack Payne\n\n**Ed Peil, Sr.** \u2014Ed Peil, Edward Peil, Ed Piel, Sr.\n\n**Jack Perrin** \u2014Jack Gable, Richard Terry\n\n**William \"Bill\" Phillips** \u2014Bill Phillips, William Phillips\n\n**Rose Plummer** \u2014Rose Plumer\n\n**Hal Price** \u2014Harry Price, Harry F. Price\n\n**Bernard B. Ray** \u2014B.B. Ray, Ray Bernard, Franklin Shamray\n\n**Rhodes Reason** \u2014Bart Roberts\n\n**Duncan Renaldo** \u2014Renault Duncan\n\n**Rin Tin Tin** \u2014Rin-Tin-Tin\n\n**Elisabeth Risdon** \u2014Elizabeth Risdon\n\n**Lynne Roberts** \u2014Mary Hart, Lynn Roberts\n\n**Mark Roberts** \u2014Robert Scott\n\n**Roy Rogers** \u2014Len Slye, Leonard Slye, Dick Weston\n\n**Henry Roquemore** \u2014Henry Rocquemore\n\n**Gene Roth** \u2014Eugene Roth, Gene Stutenroth\n\n**Al St. John** \u2014Al \"Fuzzy\" St. John, Fuzzy St. John\n\n**Joseph Sawyer** \u2014Joseph Sauers, Joe Sawyer\n\n**Bud Spencer** \u2014Carlo Pedersoli\n\n**Harry Dean Stanton** \u2014Dean Stanton\n\n**Alan Steel** \u2014Sergio Ciani\n\n**Anthony Steffan** \u2014Antonio De Teffe\n\n**Evelyn Stewart** \u2014Ida Galli\n\n**Robert Emmett Tansey** \u2014Robert Emmett, Al Lane, Robert Tansey\n\n**Sherry Tansey** \u2014James Sheridan\n\n**Dub Taylor** \u2014Cannonball Taylor\n\n**Kenneth Terrell** \u2014Ken Terrell\n\n**Daniel B. Ullman** \u2014Dan Ullman\n\n**Virginia Vale** \u2014Dorothy Howe\n\n**Gian Maria Volonte** \u2014John Wells\n\n**George Waggner** \u2014Joseph West\n\n**Wally Wales** \u2014Hal Taliaferro, Walt Williams\n\n**Blackjack Ward** \u2014Jack Ward, Jerome Ward\n\n**Harry S. Webb** \u2014Harry Samuels, Henri Samuels\n\n**Slim Whitaker** \u2014Charles Whitaker\n\n**Robert Wilke** \u2014Bob Wilke, Robert J. Wilke\n\n**Bill Wilkerson** \u2014Billy Wilkerson\n\n**Guinn Williams** \u2014Big Boy Williams, Guinn \"Big Boy\" Williams\n\n**Norman Willis** \u2014Jack Norman\n\n**Robert Woods** \u2014Robert Wood\n\n**Hank Worden** \u2014Heber Snow\n\n**Victor Sen Yung** \u2014Victor Sen Young, Sen Yung\n\n**Carleton Young** \u2014Gordon Roberts\n\n### Selected Bibliography\n\n#### _Books_\n\nAdams, Les, and Buck Rainey. _The Shoot-Em-Ups_. New Rochelle: NY: Arlington House, 1978.\n\nAlvarez, Max Joseph. _Index to Motion Pictures Reviewed by Variety, 1907\u20131980_. Metuchen, NJ: Scarecrow, 1982.\n\nAros, Andrew. _An Actor Guide to the Talkies 1965\u201374._ Metuchen, NJ: Scarecrow, 1977.\n\n_____. _A Title Guide to the Talkies 1964\u201374._ Metuchen, NJ: Scarecrow, 1977.\n\nBrooks, Tim, and Earle Marsh. _The Complete Directory to Prime Time Network TV Shows, 1946\u2013Present_. New York: Ballantine, 1979.\n\nCraddock, James M., ed. _Video Source Book_. Detroit: Gale, 1998.\n\n_Die Deutschen Filme_ (German Pictures). Langenbeckstrasse, West Germany: Export-Union die Deutschen Filmindustries, 1963\u201372.\n\nDimmitt, Richard. _An Actor Guide to the Talkies_. Metuchen, NJ: Scarecrow, 1967.\n\n_____. _A Title Guide to the Talkies._ Metuchen, NJ: Scarecrow, 1965.\n\n_Film Daily Yearbook of Motion Pictures._ 1920\u201370.\n\nFitzgerald, Michael B. _Universal Pictures_. New Rochelle, NY: Arlington House, 1977.\n\nFuente, Maria Isabel de la, ed. _Indice Bibliografico del Cine Mexicano 1930\u201365._ Mexico: Talleves de Editorial America, 1967.\n\nGreen, Douglas B. _Singing in the Saddle: The History of the Singing Cowboy_. Nashville: Country Music Foundation\/Vanderbilt University Press, 2002.\n\nGriffis, Ken. _Hear My Song: The Story of the Celebrated Sons of the Pioneers_. Los Angeles: John Edwards Memorial Foundation, 1977.\n\nHardy, Phil. _The Western_. New York: William Morrow, 1983.\n\n_Italian Production._ Rome, Italy: Unitalia Film Organization, 1963\u201372.\n\nKisch, John, and Edward Mapp. _A Separate Cinema: 50 Years of Black Cast Posters_. New York: Noonday, 1992.\n\nKoszarski, Diane Kaiser. _The Complete Films of William S. Hart: A Pictorial Record_. New York: Dover, 1980.\n\nLahue, Kalton C. _Continued Next Week_. Norman: University of Oklahoma Press, 1964.\n\nLimbacher, James L. _Feature Films on 8mm and 16mm._ New York: Bowker, 1977.\n\nMagers, Boyd, Bob Nareau and Bobby Copeland. _Best of the Badmen._ Madison, NC: Empire, 2005.\n\nMaltin, Leonard. _The Disney Films._ New York: Crown, 1973.\n\n_____. _Leonard Maltin's 2008 Movie Guide._ New York: Signet, 2007.\n\nMarrill, Alvin H. _Movies Made for Television: The Telefeature and the Mini-Series 1964\u20131984_. New York: Zeotrope, 1984.\n\nMartin, Len D. _The Allied Artists Checklist: The Feature Films and Short Subjects, 1947\u20131978_. Jefferson, NC: McFarland, 1993.\n\n_____. _The Columbia Checklist: The Feature Films, Serials, Cartoons and Short Subjects of Columbia Pictures Corporation, 1922\u20131988_. Jefferson, NC: McFarland, 1991.\n\n_____. _The Republic Pictures Checklist: Features, Serials, Cartoons, Short Subjects and Training Films of Republic Pictures Corporation, 1935\u20131959_. Jefferson, NC: McFarland, 1998.\n\nMathis, Jack. _Republic Confidential, Volume 2: The Players_. Barrington, IL: Jack Mathis Advertising, 1992.\n\nMcKinney, Grange B. _Art Acord and the Movies_. Raleigh, NC: Wyatt Classics, 2000.\n\nMichael, Paul, ed. _The American Movies Reference Book_. Englewood Cliffs, NJ: Prentice-Hall, 1969.\n\nMiller, Don. _B Movies_. New York: Curtis, 1973.\n\n_____. _Hollywood Corral_. New York: Popular Library, 1976.\n\nParish, James Robert, and Michael R. Pitts. _The Great Western Pictures._ Metuchen, NJ: Scarecrow, 1976.\n\n_____. _The Great Western Pictures II_. Metuchen, NJ: Scarecrow, 1988.\n\nPetzel, Michael. _Das Grosse Album der Karl-May-Filme_ (two volumes). Berlin: Schwarzkopf and Schwarzkopf Verlag, 2003\u201304.\n\nPitts, Michael R. _Poverty Row Studios, 1929\u20131940_. Jefferson, NC: McFarland, 1997.\n\n_____. _Western Film Series of the Sound Era_. Jefferson, NC: McFarland, 2009.\n\nQuinlan, David. _Quinlan's Film Directors: The Ultimate Guide to Directors of the Big Screen_. London: B.T. Batsford, 1999.\n\nRainey, Buck. _The Fabulous Holts_. Nashville, TN: Western Film Collector, 1976.\n\n______. _Saddle Aces of the Cinema_. San Diego, CA: A.S. Barnes, 1980.\n\n______. _Serials and Series: A World Filmography 1912\u20131956._ Jefferson, NC: McFarland, 1999.\n\n______. _The Shoot-Em-Ups Ride Again: A Supplement to Shoot-Em-Ups_. Metuchen, NJ: Scarecrow, 1990.\n\nRutherford, John A., and Richard B. Smith III. _More Cowboy Shooting Stars_. Madison, NC: Empire, 1992.\n\nScaramazza, Paul A. _Ten Years in Paradise_. Arlington, VA: Pleasant, 1974.\n\nSpeed, F. Maurice. _Film Review_. New York: A.S. Barnes, 1966\u201373.\n\nStanley, John. _John Stanley's Creature Features Movie Guide Strikes Again_. Pacifica, CA: Creatures at Large, 1994.\n\nTurner, George E., and Michael H. Price. _Forgotten Horrors: The Definitive Edition_. Baltimore: Midnight Marquee, 1999.\n\nTurner, Steve, and Edgar M. Wyatt. _Saddle Gals: A Filmography of Female Players in B-Westerns of the Sound Era!_ Madison, NC: Empire, 1995.\n\nTuska, Jon. _The Vanishing Legion: A History of Mascot Pictures 1927\u20131935._ Jefferson, NC: McFarland, 1982.\n\n_TV Feature Film Sourcebook_. New York: Broadcasting Information Bureau, 1978.\n\nWeiss, Ken, and Ed Goodgold. _To Be Continued_. New York: Crown, 1972.\n\nWeisser, Thomas. _Spaghetti Westerns\u2014the Good, the Bad and the Violent: 558 Eurowesterns and Their Personnel, 1961\u20131977_. Jefferson, NC: McFarland, 1992.\n\nWeldon, Michael J. _The Psychotronic Encyclopedia of Film._ New York: Ballantine, 1983.\n\n_____. _The Psychotronic Video Guide_. New York: St. Martin's Griffin, 1996.\n\nWhite, Raymond E. _The King of the Cowboys, Queen of the West: Roy Rogers and Dale Evans_. Madison: University of Wisconsin Press\/Popular, 2005.\n\nWilt, David. _The Mexican Filmography, 1916 Through 2001._ Jefferson, NC: McFarland, 2004.\n\n#### _Periodicals_\n\n_Film Collectors Registry_ (Knoxville, TN)\n\n_Film Fan Monthly_ (Teaneck, NJ)\n\n_Filmograph_ (Orlean, VA)\n\n_Films in Review_ (New York)\n\n_The Films of Yesteryear_ (Waynesville, NC)\n\n_Focus on Film_ (London, England)\n\n_Screen Facts_ (Kew Gardens, NY)\n\n_Screen Thrills Illustrated_ (Philadelphia, PA)\n\n_Under Western Skies_ (Waynesville, NC)\n\n_Variety_ (New York)\n\n_Views and Reviews_ (Milwaukee, WI)\n\n_Western Clippings_ (Albuquerque, NM)\n\n_Western Film Collector_ (Nashville, TN)\n\n_Western Revue_ (Maitland, FL)\n\n_Westerns All'Italiana_ (Keyport, NJ)\n\n_Wild West Stars_ (Knoxville, TN)\n\n_Wildest Westerns_ (Philadelphia, PA)\n\n_Wrangler's Roost_ (Bristol, England)\n\n_Yesterday's Saturdays_ (Lubbock, TX)\n\n#### _Websites_\n\nAmerican Film Institute (www.afi.com)\n\nInternet Movie Database (www.imdb.com)\n\nThe Spaghetti Western Database (www.spaghetti-western.net)\n\n### Index to Entry Numbers\n\nAaker, Lee 128, 693, 1078, 1082, 1921, 3221, 3411, 3512, 4227\n\nAbbott, Bud 2452, 3425, 3530, 5008\n\nAbbott, Charles 1332\n\nAbbott, John 4361\n\nAbel, Walter 1989\n\nAbraham, F. Murray 998\n\nAbrahams, Derwin 459, 481, 871, 1158, 1309, 1453, 1568, 1797, 2665, 2841, 3167, 3277, 3385, 3464, 3616, 3651, 3761, 3976, 4053, 4154, 4222, 4278, 4894\n\nAce the Wonder Dog 1811\n\nAcker, Jean 504\n\nAcker, Sharon 1770, 1784, 1834, 2738\n\nAckerman, Bettye 4571\n\nAcord, Art 141, 547, 754, 916, 1294, 1999, 2450, 4091, 4518, 4912\n\nAcosta, Rodolfo 116, 226, 759, 1161, 1371, 1372, 1441, 1921, 1936, 1945, 2227, 2322, 2382, 2498, 2598, 2901, 3031, 3054, 3149, 3185, 3391, 3406, 3518, 3641, 3690, 3731, 3755, 4035, 4661, 4566, 4768, 5000, 5060\n\nAcquanetta 2328\n\nAcuff, Eddie 191, 201, 203, 223, 283, 287, 354, 442, 541, 642, 956, 986, 1247, 1360, 1378, 1647, 1740, 1811, 1836, 2839, 3071, 3408, 3414, 3564, 3615, 3692, 3750, 3794, 3846, 3975, 4015, 4222, 4312, 4425, 5033\n\nAcuff, Roy 870, 3975\n\nAdair, Phyllis 341, 1725, 2166, 3457, 4958\n\nAdams, Abigail 795\n\nAdams, Brooke 966, 985\n\nAdams, Carol 201, 3490\n\nAdams, Casey 1150, 2444, 3376; _see also_ Showalter, Max\n\nAdams, Claire 2794, 4875\n\nAdams, Dorothy 312, 524, 550, 659, 1361, 1402, 1427, 1719, 1884, 2046, 2677, 2986, 3446, 3817, 4635, 4671\n\nAdams, Edie 1243\n\nAdams, Ernie 64, 69, 140, 201, 230, 244, 299, 306, 510, 582, 606, 792, 810, 866, 946, 960, 1128, 1148, 1193, 1270, 1329, 1340, 1370, 1438, 1446, 1462, 1494, 1520, 1540, 1545, 1573, 1586, 1604, 1626, 1681, 1682, 1683, 1758, 1811, 1932, 1933, 2016, 2125, 2165, 2255, 2272, 2274, 2283, 2291, 2356, 2398, 2403, 2547, 2552, 2572, 2595, 2637, 2647, 2652, 2660, 2773, 2886, 2908, 2923, 2967, 3003, 3016, 3082, 3089, 3100, 3139, 3173, 3194, 3224, 3229, 3259, 3275, 3344, 3351, 3383, 3414, 3429, 3446, 3450, 3492, 3504, 3526, 3540, 3550, 3557, 3562, 3565, 3589, 3609, 3636, 3679, 3744, 4011, 4019, 4089, 4103, 4104, 4130, 4139, 4315, 4367, 4382, 4419, 4503, 4514, 4519, 4553, 4569, 4609, 4665, 4701, 4744, 4803, 4808, 4816, 4827, 4872, 5043, 5077\n\nAdams, Gerald Drayson 257, 528, 733, 1184, 1188, 1364, 1485, 1490, 1499, 1666, 1670, 2161, 2878, 3132, 4244, 4376, 4674, 4784, 4973\n\nAdams, Jane 775, 1568, 1680, 1718, 2271, 2291, 2949, 3656, 3684, 3759, 4509, 4847\n\nAdams, Julie (Betty\/Julia) 290, 794, 896, 936, 1272, 1697, 1936, 1939, 2199, 2292, 2544, 2596, 2899, 3960, 4120, 4412, 4479, 4536, 4820, 5000\n\nAdams, Kathryn 135, 582, 3291\n\nAdams, Nick 168, 1482, 2241, 2695, 4148\n\nAdams, Richard L. 4900\n\nAdams, Stanley 1266, 1411, 2779, 2829, 2902, 3671, 4566, 4689, 4867\n\nAdams, Ted 2, 14, 70, 139, 140, 152, 187, 262, 286, 306, 332, 338, 342, 343, 345, 431, 478, 534, 542, 557, 688, 711, 773, 783, 792, 892, 899, 917, 924, 951, 956, 1041, 1051, 1052, 1193, 1211, 1260, 1297, 1317, 1327, 1373, 1439, 1443, 1445, 1461, 1503, 1512, 1529, 1557, 1667, 1684, 1690, 1712, 1724, 1732, 1756, 1758, 1823, 1824, 1853, 1888, 1895, 1931, 1950, 1956, 2012, 2085, 2111, 2134, 2233, 2250, 2251, 2255, 2263, 2271, 2290, 2298, 2360, 2399, 2407, 2412, 2451, 2645, 2734, 2762, 2804, 2946, 2965, 2979, 2999, 3002, 3018, 3027, 3085, 3098, 3102, 3157, 3209, 3231, 3257, 3326, 3446, 3448, 3450, 3452, 3477, 3485, 3592, 3629, 3657, 3680, 3696, 3802, 3804, 3901, 3944, 3946, 3953, 3967, 3977, 4011, 4022, 4028, 4110, 4114, 4144, 4178, 4185, 4220, 4324, 4367, 4372, 4391, 4448, 4485, 4498, 4549, 4550, 4553, 4590, 4640, 4641, 4648, 4656, 4691, 4709, 4729, 4733, 4785, 4857, 4950, 4953, 4958, 5032\n\nAdamson, Al 388, 1278, 1352, 1763, 2038\n\nAdamson, Ewart 2792\n\nAdamson, Victor 4, 156, 752, 2358, 2364, 2766, 2881, 3266, 3292, 3293, 3501, 3587, 3598; _see also_ Dixon, Denver\n\nAddy, Wesley 1422\n\nAdiman, Charles 247, 1036, 1107, 1943, 2917, 3306, 3740, 4249\n\nAdler, Bob (Robert) 237, 759, 1480, 1482, 1649, 1773, 1842, 2486, 3150, 3333, 3392, 3518, 3909, 3972, 4233, 4579, 4611, 4791, 5051\n\nAdler, Felix 870, 4798\n\nAdler, Jay 913, 2563, 3671, 3787\n\nAdler, Luther 4236\n\nAdler, Robert 482, 502, 519, 1995, 2902, 2941, 4333, 4411, 4689\n\nAdoree, Rene 2651, 4413\n\nAdorf, Mario 106, 1004, 1160, 2230, 2501, 2615, 3955, 4194\n\nAdren, Elaine 1810\n\nAdreon, Franklin 28, 37, 637, 951, 1849, 2035, 2132, 2399, 2403, 2564, 4011, 4033, 5096, 5104\n\nAdrian, Iris 46, 121, 329, 596, 664, 1577, 2159, 2646, 3020, 3739, 3925, 4467, 4500, 4940, 5008\n\nAdrian, Jane 3980\n\nAgar, John 65, 319, 690, 741, 1377, 1395, 1452, 2050, 2436, 3417, 4098, 4123, 4240, 4638, 4755, 5020, 5065\n\nAgee, James 1249\n\nAguilar, Antonio (Tony) 232, 3867, 4638\n\nAguilar, Luis 124, 171, 591, 635, 699, 746, 821, 839, 908, 1104, 1190, 1476, 1515, 2041, 2063, 2574, 2579, 2630, 2707, 2916, 3343, 3347, 3868, 3873, 3986, 3988, 4039, 4418, 4526, 4544, 4632, 4967, 5091, 5102\n\nAguilar, Rolando 124, 635\n\nAgutter, Jenny 736\n\nAherne, Brian 572, 2064\n\nAhn, Philip 1487, 2147, 2154, 2901, 3554, 4279, 4799\n\nAinslee, Mary 4868\n\nAitken, Spottiswoode 2816, 3108\n\nAkins, Claude 349, 363, 580, 1112, 1372, 1500, 1631, 1695, 1985, 2043, 2046, 2389, 2431, 2510, 2563, 3221, 3391, 3422, 3517, 4426, 4794, 5055\n\nAlaimo, Marc 2094, 2810, 3591\n\nAlaniz, Ric (Rico) 604, 654, 803, 828, 1161, 2066, 2497, 2809, 3861, 4111, 4469, 5000\n\nAlba, Maria 1850, 4824\n\nAlberghetti, Anna Maria 1182, 2187\n\nAlberni, Luis 327, 1591, 2344, 2527, 2838, 3710\n\nAlbert, Al _see_ Albertini, Alberto\n\nAlbert, Eddie 1098, 1175, 2853, 4571\n\nAlbert, Edward 587\n\nAlbert, Marvin H. 1183, 3612, 4631\n\nAlbertini, Alberto 2614, 4596\n\nAlbertson, Frank 3126, 4868\n\nAlbertson, Jack 2389, 3854, 4417\n\nAlbertson, Mabel 1773, 2717\n\nAlbright, Hardie 662, 1622, 2641, 4917\n\nAlbright, Lola 125, 2926, 3058, 3787, 3764, 3909, 3539, 4585, 4800\n\nAlbright, Wally 826, 881, 1227, 2648, 2879, 3584\n\nAlcaide, Chris 895, 979, 1188, 1743, 2068, 2105, 2178, 2517, 2955, 2996, 3974\n\nAlden, Norman 463, 1631, 2796, 4479, 5030\n\nAlderman, John 2155, 2460, 2671\n\nAlderson, Charles 1062\n\nAlderson, Erville 64, 155, 197, 648, 832, 975, 1302, 1378, 1594, 1796, 1827, 1847, 1853, 1884, 1925, 2031, 2159, 2181, 2208, 2516, 2543, 2936, 3195, 3245, 3276, 3711, 3750, 4089, 4231, 4442, 4622, 4635, 4803, 4868, 4925\n\nAlderson, John 1170, 3839\n\nAldon, Mari 1111\n\nAldrich, Charles W. 1733\n\nAldrich, Robert 100, 1220, 1422, 1437, 2237, 4633, 4727\n\nAldrich, Roma 1449\n\nAldridge, Kay 956, 3846\n\nAletter, Frank 3635, 4417\n\nAlexander, Ben 960, 1827, 2550, 4706, 4840\n\nAlexander, Betty 950\n\nAlexander, Jane 598, 1694, 4345\n\nAlexander, John 2902, 4674, 4989\n\nAlexander, Katharine 2923, 4215\n\nAlexander, Richard (Dick) 9, 53, 210, 318, 468, 475, 476, 638, 648, 727, 777, 853, 862, 877, 889, 922, 939, 958, 990, 1018, 1084, 1153, 1255, 1283, 1302, 1363, 1377, 1458, 1749, 1843, 1894, 1952, 1974, 1996, 2035, 2086, 2150, 2159, 2422, 2482, 2526, 2577, 2660, 2684, 2733, 2743, 2765, 2805, 2836, 2866, 2991, 2908, 2927, 2975, 3006, 3066, 3126, 3229, 3230, 3276, 3357, 3365, 3390, 3450, 3457, 3478, 3508, 3511, 3548, 3567, 3600, 3630, 3684, 3709, 3772, 3815, 3877, 3886, 3890, 3949, 4006, 4057, 4073, 4078, 4187, 4193, 4207, 4289, 4428, 4556, 4574, 4607, 4609, 4621, 4636, 4872, 4882, 4933, 5020, 5032, 5096\n\nAlgar, James 2330, 2384, 4710, 4915\n\nAlgier, Sidney 4944\n\nAli, Muhammad 372\n\nAllan, Jed 2772\n\nAllbritton, Louise 1136\n\nAllen, Alfred 1659, 4651\n\nAllen, Ariane 1491\n\nAllen, Arthur 3276\n\nAllen, Barbara Jo 2636; _see also_ Vague, Vera\n\nAllen, Barbara 997, 2863\n\nAllen, Bob (Robert) 1328, 2254, 2274, 3270, 3279, 3303, 3401, 3525, 4667\n\nAllen, Debra 786\n\nAllen, Elizabeth 720\n\nAllen, Eric 3963\n\nAllen, Ethan 56, 441, 3920, 4156, 4238, 4553, 4614\n\nAllen, Fred 306, 541, 1432, 1557, 2732, 3048, 3327, 3665\n\nAllen, Harry 4306\n\nAllen, Irving 3958\n\nAllen, Judith 424, 579, 1574, 1959, 2012, 4322, 4403, 4524\n\nAllen, Lewis 1225, 1676, 2555\n\nAllen, Mark 1502\n\nAllen, Maude Pierce 37\n\nAllen, Maude 879, 3012, 3764, 4891\n\nAllen, Patrick 1875\n\nAllen, Rex 134, 453, 469, 703, 796, 1142, 1383, 1888, 2006, 2200, 2330, 2632, 2880, 2882, 3091, 3234, 3328, 3342, 3342, 3578, 3758, 3803, 3892, 4056, 4381, 4488, 4644, 4686, 4713\n\nAllen, Sian Barbara 346\n\nAllende, Fernando 4718\n\nAllgood, Sara 2543, 3862\n\nAllison, Jean 192, 528, 1096, 1784\n\nAllister, Claud 5054\n\nAllwyn, Astrid 1566, 3362\n\nAllyson, June 1564\n\nAlmada, Fernando 2791\n\nAlmada, Fernando 837\n\nAlper, Murray 274, 433, 1092, 1143, 2688, 2964, 4618, 4622\n\nAlsace, Gene 37, 131, 138, 150, 387, 611, 1156, 1196, 1604, 1626, 1730, 1740, 2164, 2254, 2428, 2689, 2864, 2991, 2992, 3028, 3107, 3237, 3267, 3270, 3491, 3503, 3587, 3588, 4226, 4513, 4534, 4778, 4844, 4846, 4884, 5001; _see also_ Camron, Rocky; Coburn, Buck\n\nAltman, Robert 554, 2625\n\nAlvarado, Don 653, 1037, 2792, 3526, 3609, 3610\n\nAlvarez, Sofia 4438\n\nAlvin, John 415, 718, 1029, 2096, 2447, 3204, 3571, 3692, 4621, 4641\n\nAlyn, Kirk 624, 868, 1416, 2545, 2995, 3120, 3722, 3738\n\nAmateau, Rod 584, 4195\n\nAmber, Audrey 1737\n\nAmeche, Don 3245\n\nAmendola, Mario 275, 1582, 1643, 1653\n\nAmendola, Tony 756, 2341, 2604\n\nThe American Rough Riders 4796\n\nAmes, Ed 953\n\nAmes, Florenz 2118, 2563, 4751\n\nAmes, Judith 2926, 3411\n\nAmes, Leon 73, 530, 677, 766, 2552, 2599, 3895, 4426, 4445\n\nAmes, Rachel 1709\n\nAmes, Ramsay 272, 1520, 4729\n\nAmos, John 417\n\nAmy, George 1622\n\nAnders, Laurie 2601\n\nAnders, Luana 1243, 1590, 1624, 2666, 4412, 4869, 4930\n\nAnders, Merry 937, 1353, 1502, 3208, 4417, 5065, 5070\n\nAnders, Rudolph 1236, 3084, 4646\n\nAndersen, Elga 374\n\nAndersen, Susy 4629\n\nAnderson, Barbara 417\n\nAnderson, Carol 3612\n\nAnderson, Dusty 3923\n\nAnderson, Eddie \"Rochester\" 541\n\nAnderson, G.M. \"Bronco Billy\" 486, 3996, 4359\n\nAnderson, Herbert 1884, 2797\n\nAnderson, James 65, 170, 221, 320, 1164, 1184, 1481, 1635, 2200, 2373, 2578, 3295, 3644, 3779, 4740\n\nAnderson, John 93, 512, 980, 1034, 1133, 1354, 1533, 1631, 1765, 1832, 2429, 2506, 2674, 2738, 3070, 3435, 3737, 3987, 4802, 5030\n\nAnderson, Judith 1478, 2508, 3195\n\nAnderson, Mary 3053\n\nAnderson, Melissa Sue 2376\n\nAnderson, Michael, Jr. 965, 1575, 2501, 3840, 4035, 4192\n\nAnderson, Paul 2895\n\nAnderson, Richard 16, 1234, 1697, 1924, 2488, 3439, 4712\n\nAnderson, Rick 3159, 3453, 3550, 4393, 4827\n\nAnderson, Robert 539, 1253, 1481, 1502, 1703, 2084, 2292, 2314, 2955, 3090, 4227, 4235, 4674, 4825, 5060\n\nAnderson, Warner 191, 1161, 2226, 2302, 2919, 3518, 3702, 4740, 5052\n\nAndersson, Bibi 1183\n\nAndes, Keith 3096, 3754, 4069\n\nAndre, Carl 700, 798, 801, 935, 2544, 2565, 2677, 4168, 4232, 4390, 4674, 4740, 5025, 5049\n\nAndre, E.J. 1170, 2162, 2695, 3806\n\nAndre, Gaby 1185\n\nAndre, Lona 429, 917, 1558, 2732, 3126, 4512, 4650\n\nAndress, Ursula 1422, 3336\n\nAndreu, Simon 202, 2439, 4348, 4565\n\nAndrews, Arkansas Slim 138, 511, 555, 876, 880, 925, 1156, 1196, 1604, 2408, 3028, 3107, 3235, 3237, 3407, 3491, 3503, 3588, 3989, 4226, 4778\n\nAndrews, Dana 279, 648, 804, 2050, 2145, 2471, 3005, 3966, 4148, 4224, 4362, 4472, 4851\n\nAndrews, Dell 2947\n\nAndrews, Edward 1289, 1998, 2156, 2519, 2989, 4259, 4417, 4566\n\nAndrews, Lois 3646, 4841\n\nAndrews, Robert D. 2515\n\nAndrews, Robert Hardy 293, 2109, 2593\n\nAndrews, Stanley 9, 28, 42, 191, 200, 292, 514, 516, 534, 602, 637, 644, 664, 775, 832, 939, 956, 972, 990, 1136, 1153, 1216, 1247, 1268, 1390, 1451, 1585, 1883, 1941, 1977, 2085, 2132, 2145, 2184, 2221, 2399, 2416, 2512, 2515, 2543, 2586, 2652, 2678, 2711, 2733, 2774, 2781, 2830, 2839, 2939, 3005, 3020, 3032, 3150, 3163, 3216, 3394, 3499, 3544, 3563, 3667, 3685, 3750, 3834, 3848, 3890, 3928, 3972, 4057, 4060, 4061, 4099, 4123, 4167, 4309, 4310, 4400, 4467, 4500, 4503, 4509, 4533, 4539, 4551, 4584, 4601, 4611, 4645, 4665, 4686, 4690, 4700, 4725, 4734, 4745, 4754, 4776, 4786, 4828, 4933, 4940, 5020\n\nAndrews, Tod 1769, 4336\n\nThe Andrews Sisters 2688\n\nAngel, Heather 414, 952, 2213, 4840\n\nAngeli, Pier 3352\n\nAnglim, Sally 4408\n\nAnhalt, Edna 3386, 3862, 5075\n\nAnhalt, Edward 1943, 2028\n\nAnkers, Evelyn 2218, 2830, 3096, 4075, 4283\n\nAnkrum, Morris 65, 100, 128, 207, 212, 231, 248, 459, 462, 528, 541, 681, 715, 733, 798, 1071, 1092, 1138, 1151, 1164, 1403, 1407, 1452, 1481, 1526, 1734, 1792, 1847, 1866, 1977, 2152, 2187, 2353, 2381, 2504, 2691, 2748, 2889, 2955, 3116, 3213, 3222, 3339, 3425, 3540, 3750, 3848, 3851, 3897, 3906, 4060, 4244, 4257, 4258, 4364, 4376, 4620, 4727, 4770, 4923; _see also_ Morris, Stephen\n\nAnnakin, Ken 627, 1846, 3767\n\nAnn-Margret 4101, 4523, 4739\n\nAnsara, Michael 86, 247, 267, 455, 503, 808, 978, 1666, 1739, 2202, 2292, 2400, 2444, 2486, 2663, 2919, 2924, 3096, 3152, 3840, 4234, 4287, 4376\n\nAnthony, Joseph 3244\n\nAnthony, Mike 2053\n\nAnthony, Stuart 145, 429, 440, 471, 1043, 1058, 1153, 1227, 1329, 1457, 2239, 2628, 2774, 3269, 3675, 3817, 3880, 3971, 4395, 4706, 4776, 4885, 4892\n\nAnthony, Tony 394, 817, 1536, 4157, 4158, 4662\n\nAnthony, Walter 4152\n\nAnton, Amerigo 1582, 2015\n\nAntonelli, Laura 2510\n\nAntonini, Alfredo _see_ Band, Albert\n\nAntrim, Harry 484, 1093, 2302, 2370\n\nApfel, Oscar 619, 1827, 3243, 3657, 3695, 4071, 4091, 4215, 4281\n\nAppel, Sam 1143, 1865, 2646, 3245, 3609, 4593, 4639, 4650, 5042\n\nAppleby, Dorothy 2823, 4100\n\nAranda, Angel 574, 1841\n\nArau, Alfonso 3739, 4351, 4873, 4934\n\nArbuckle, Roscoe \"Fatty\" 3624\n\nArbus, Allan 1219, 1624\n\nArchainbaud, George 46, 104, 250, 309, 409, 456, 472, 950, 990, 1097, 1254, 1255, 1380, 1606, 1934, 1935, 2081, 2138, 2217, 2577, 2736, 2805, 2885, 2894, 3006, 3682, 3877, 3896, 4147, 4304, 4661, 4671, 4757, 5005, 5021\n\nArcher John 109, 293, 329, 798, 1026, 1876, 3576, 3702; _see also_ Bowman, Ralph\n\nArcher, Anne 1924, 2201, 2587, 2795\n\nArdell, Alice 3634, 4030\n\nArden, Eve 914, 2208\n\nArden, Gisela 4268, 5094\n\nArdigan, Art 2660, 4084\n\nArdisson, Giorgio (George) 1114, 1617, 2614, 5092, 5099\n\nArdrey, Robert 2163\n\nArent, Eddi 2219, 4540, 5003\n\nArgento, Dario 1358, 3604, 4444\n\nThe Arizona Wranglers 4142\n\nArkin, Alan 87, 1828\n\nArledge, John 129, 4622\n\nArlen, Ghia 1419\n\nArlen, Richard 47, 114, 309, 373, 381, 411, 486, 548, 557, 630, 633, 686, 690, 825, 1364, 1406, 1616, 1685, 1871, 1940, 2050, 2084, 2267, 2352, 2528, 2641, 2654, 2716, 2717, 3338, 3394, 3710, 3764, 3817, 3818, 3874, 3891, 4422, 4472, 4651, 4744, 4755, 4791, 4979, 5065\n\nArmendariz, Pedro 50, 450, 600, 669, 706, 1395, 1471, 1994, 2061, 2382, 3197, 3988, 4360, 4414, 4464, 4547, 4585, 5024, 5090\n\nArmendariz, Pedro, Jr. 87, 168, 232, 741, 756, 1007, 1515, 1615, 1729, 1731, 1786, 2341, 2488, 2604, 2876, 4040, 4176, 4177, 4209, 4453, 4638\n\nArmetta, Henry 1972, 2637, 4750\n\nArmida 204, 432, 451, 1516, 3603, 4055, 4639, 4650, 4997\n\nArms, Russell 1339, 2386, 3209, 3975, 4036, 4097, 4466\n\nArmstrong, Louis 4867\n\nArmstrong, R.G. 474, 838, 1206, 1212, 1441, 1527, 1641, 1809, 1834, 1987, 2076, 2077, 2229, 2333, 2429, 2627, 2723, 2813, 3055, 3192, 3317, 3435, 3645, 3798, 3941, 4253, 4295\n\nArmstrong, R.L. 1590, 2077, 2666\n\nArmstrong, Robert 278, 895, 1471, 2081, 2552, 3020, 3061, 3383, 3630, 3750, 3964, 4036\n\nArmstrong, Todd 2912, 3737, 4380, 4431\n\nArnall, Red, and The Western Aces 379\n\nArnaz, Desi, Jr. 346\n\nArness, James 45, 53, 277, 691, 1348, 1692, 1750, 1752, 1753, 1754, 1755, 1844, 1921, 1936, 1946, 2396, 2487, 2575, 3324, 3862, 4129, 4328, 4764, 5036\n\nArngrim, Allison 1958\n\nArnold, Danny 3299\n\nArnold, Eddy 1285, 1901\n\nArnold, Edward 94, 318, 2159, 2344, 4215\n\nArnold, Elliott 71, 2141\n\nArnold, Jack 474, 2511, 2550, 2813, 3337, 4240, 4753\n\nArnold, Jessie 1298, 1449, 1599, 1601, 1781, 1974, 2159, 2261, 2296, 3042, 3565, 3678, 4129, 4191, 4503, 4533, 4871, 4892\n\nArnold, Marion 2481\n\nArnold, Phil 30\n\nArnt, Charles 1645, 2605, 2659, 3409, 3669, 4786\n\nArquette, David 998, 4930\n\nArquette, Rosanna 3911, 4008\n\nArruza, Carlos 43\n\nArthur, Art 1830, 2839, 3499\n\nArthur, Carol 385\n\nArthur, Jean 129, 145, 2163, 3126, 3808, 3896, 4245, 4407, 5002\n\nArthur, Johnny 2012, 2027, 4225\n\nArthur, Louise 2687\n\nArthur, Robert Alan 4791\n\nArthur, Robert 1071, 1649, 2790, 5051\n\nArthur, Victor 2361, 3493, 4496\n\nArvan, Jan 1466, 1709, 2818, 3871, 4990\n\nAscott, Anthony 1030, 1342, 1351, 1717, 1801, 1954, 1955, 2354, 2675; _see also_ Carmineo, Guilliano\n\nAsh, Sam 65, 286, 324, 1134, 1836, 2125, 2153, 2815, 2923, 3414, 3561, 3669, 4121, 4803\n\nAshby, Joel 524\n\nAshcroft, Virginia 108\n\nAshe, Dick 4474\n\nAshe, Warren 1032\n\nAsher, Max 629, 3442, 4557\n\nAshley, Elizabeth 1642, 3252, 3253, 4102\n\nAshley, Joel 3299, 4726\n\nAshley, John 1948, 3963\n\nAshton, Tara 1352\n\nAskam, Earl 1222, 1805, 2353, 3111, 3126, 3177, 3325, 3903, 4485, 4665\n\nAskew, Luke 909, 1429, 1641, 2000, 2155, 2498, 2793, 3055, 3146, 3203, 4045, 4345, 4775, 4988\n\nAskin, Leon 611, 1731\n\nAslin, Edna 156, 1723, 3073, 3474, 4293, 4846\n\nAsner, Edward 1212, 3686, 3950\n\nAssante, Armand 393\n\nAstaire, Fred 2990\n\nAstar, Ben 1405\n\nAsther, Nils 46\n\nAstin, John 536, 1243, 1778, 4749\n\nAstin, Patty Duke 1778, 3723; _see also_ Duke, Patty\n\nAstor, Gertrude 170, 271, 546, 642, 1222, 2158, 2368, 2385, 2451, 2561, 2677, 2730, 2832, 2871, 3339, 3695, 4803\n\nAstor, Mary 514, 1131, 1290, 2181\n\nAtcher, Bob 1758\n\nAtcher, Leota 1758\n\nAtchley, Hooper 37, 153, 366, 432, 500, 767, 1162, 1172, 1307, 1519, 1592, 1677, 1827, 1925, 1974, 2135, 2389, 2701, 2737, 2764, 2785, 2838, 2872, 2889, 2947, 3126, 3173, 3549, 3675, 3681, 3710, 3745, 4067, 4190, 4580, 4786, 4850\n\nAtes, Roscoe 203, 331, 364, 381, 642, 711, 723, 795, 811, 876, 1144, 1157, 1588, 1626, 1803, 1888, 1923, 3241, 3254, 3257, 3360, 3454, 3564, 3640, 3745, 3799, 3815, 3875, 4131, 4163, 4349, 4372, 4383, 4437, 4466, 4590, 4671, 4829, 4858, 4935, 4969\n\nAthens, Vi 875, 3666\n\nAtherton, William 7, 1429, 1652, 4180\n\nAtkins, Thomas 1865\n\nAtterbury, Malcolm 212, 679, 932, 937, 1220, 1441, 1482, 1837, 2400, 2565, 3295, 3363, 3517, 4151, 4689\n\nAtwater, Barry 528, 1782, 2518, 3387\n\nAtwater, Edith 4576\n\nAtwater, Gladys 79, 1645, 2996\n\nAtwill, Lionel 410\n\nAuberjonois, Rene 234, 333, 2446, 2625, 2694, 2769, 4975\n\nAubert, Lenore 3154\n\nAubrey, Anne 1846\n\nAubrey, Jimmy (Jimmie) 6, 70, 151, 206, 382, 414, 422, 431, 428, 443, 452, 468, 479, 619, 642, 772, 773, 846, 856, 889, 946, 1001, 1017, 1090, 1155, 1260, 1271, 1312, 1338, 1458, 1459, 1473, 1486, 1505, 1542, 1674, 1798, 1894, 1965, 2040, 2067, 2107, 2151, 2172, 2177, 2251, 2274, 2276, 2284, 2355, 2434, 2589, 2640, 2645, 2690, 2719, 2734, 2742, 2847, 2991, 2953, 2997, 2999, 3016, 3038, 3073, 3086, 3097, 3098, 3102, 3169, 3226, 3278, 3348, 3446, 3480, 3489, 3528, 3545, 3653, 3801, 3825, 3884, 3888, 3939, 3942, 3942, 3946, 3952, 3977, 4108, 4132, 4143, 4193, 4269, 4274, 4363, 4393, 4394, 4460, 4480, 4490, 4525, 4702, 4722, 4296, 4816, 4838, 4889, 4952, 4956, 4958, 5018\n\nAuchubon, Jacques 1341, 1673, 3773, 4799\n\nAudley, Eleanor 3755, 4074, 4673\n\nAudran, Stephane 1204\n\nAuer, John 176\n\nAuer, Mischa 1084, 1361, 1521, 2212, 498, 4580, 4750, 4835\n\nAugust, Adele 101\n\nAumont, Tina 532; _see also_ Marquand, Tina\n\nAured, Carlos 4565\n\nAustin, Charlotte 3058\n\nAustin, Edward R. 1013\n\nAustin, Frank 152, 1247, 1444, 1687, 1792, 2159, 2744, 2975, 3260, 3450, 3498, 3549, 3750, 4072, 4257, 4503, 4600, 4803, 4806\n\nAustin, Gene 2147, 2722, 4031\n\nAustin, Lois 1146, 1256, 1438, 1600, 2805\n\nAustin, Marie 475, 3194, 4504\n\nAustin, Pamela 1243\n\nAustin, Vivian 475, 4556, 4598; _see also_ Coe, Vivian\n\nAustin, William 3361\n\nAutry, Gene 104, 183, 250, 283, 304, 323, 325, 389, 409, 410, 424, 437, 615, 662, 797, 818, 858, 864, 880, 1143, 1512, 1523, 1574, 1606, 1728, 1821, 1890, 1915, 1916, 1980, 1982, 1991, 2217, 2233, 2386, 2529, 2632, 2635, 2636, 2648, 2701, 2711, 2737, 2805, 2850, 2872, 2874, 2885, 2894, 3006, 2075, 3163, 3187, 3254, 3329, 3408, 3433, 3434, 3447, 3481, 3490, 3510, 3603, 3625, 3626, 3669, 3681, 3682, 3846, 3866, 3890, 3919, 3922, 3926, 3929, 4036, 4052, 4082, 4127, 4166, 4199, 4285, 4508, 4592, 4599, 4643, 4692, 4757, 4842, 4886, 5005, 5056\n\nAux, Victor 4243\n\nAvakian, Aram 2157\n\nAvalon, Frankie 43, 1741\n\nThe Avalon Boys 4798\n\nAvalos, Luis 34, 35, 36, 2769\n\nAvedis, Hikmet 4295\n\nAverback, Hy 1631, 2788, 2800\n\nAverill, Anthony 3362\n\nAvery, Emile 489, 695, 1938, 2544, 3640, 4588, 4674\n\nAvery, Phyllis 2613\n\nAvery, Tol 539, 2372, 3741\n\nAvery, Val 947, 2497, 2779, 3350\n\nAvonde, Richard 997, 2863, 2960, 3724, 3759, 3803, 3980, 4731, 4754, 4955\n\nAxton, Hoyt 1066, 2865, 3973\n\nAyars, Ann 112\n\nAylesworth, Arthur 147, 203, 514, 626, 1165, 1458, 1594, 1622, 2031, 2129, 2552, 2837, 2864, 3126, 3384, 3609, 3699, 3711, 3912, 4061, 4803, 4849\n\nAyres, Lew 654, 1827, 2789\n\nAyres, Mitchell 2688\n\nBabcock, Barbara 980, 1126, 1262, 1832\n\nBabcock, Dwight 3724, 5048\n\nBacall, Lauren 3847\n\nBach, Vivi 574\n\nBachelor, Stephanie 4033, 4083, 4659\n\nBackus, Georgia 105, 832, 2583\n\nBackus, Jim 22, 771, 1963, 2749, 4634, 4924\n\nBackus, Lionel 130, 493, 3173, 4389\n\nBacon, Irving 25, 49, 56, 74, 145, 170, 312, 367, 500, 642, 866, 932, 1048, 1070, 1092, 1152, 1214, 1256, 1300, 1402, 1405, 1564, 1591, 1599, 1638, 1831, 1847, 1923, 1928, 1932, 1947, 2083, 2124, 2393, 2782, 2864, 3126, 3153, 3374, 3409, 3431, 3571, 3608, 3636, 3669, 3695, 4036, 4061, 4072, 4088, 4284, 4309, 4336, 4494, 4570, 4654, 4758, 4776, 4786, 4824, 4849\n\nBacon, Lloyd 874, 1436, 1601, 1645, 2864, 3899\n\nBaddeley, Hermoine 26, 4670\n\nBadger, Clarence 197, 3281\n\nBadham, John 1583, 2015\n\nBaer, Buddy 324, 1838, 2066, 2601, 2895\n\nBaer, John 146, 2664, 3500\n\nBaer, Max 549, 2895, 3951, 4682\n\nBaer, Max (Jr.) 4431\n\nBaer, Parley 26, 807, 980, 1151, 1226, 1946, 1985, 2206, 2990, 3333, 3422, 3577, 3950, 4577, 4582, 5060\n\nBaffert, Al 1597, 3043, 4141\n\nBagdasarian, Ross 4373\n\nBaggott, King 1848, 3322, 3766, 4591\n\nBagni, Gwen 2241, 4674\n\nBagni, John 1264, 1427, 1836, 2132, 2252, 2716, 4674\n\nBagni, Owen 2252\n\nBailey, Bill 850, 1255, 1295, 1487, 2182, 5076; _see also_ Bailey, William Norton\n\nBailey, Buck 1279, 2988, 3720\n\nBailey, Carmen 1159, 3245, 3653\n\nBailey, Dick 2577\n\nBailey, Raymond 3376, 4240\n\nBailey, Rex 1259, 2834\n\nBailey, Richard 73, 1046, 1492, 2939, 3467\n\nBailey, Sherwood 327, 3732\n\nBailey, William Norton 42, 318, 354, 492, 680, 783, 1093, 1256, 1307, 1427, 1702, 1713, 2147, 2361, 2391, 3408, 3563, 3908, 3995, 4232, 4357, 4385, 4520, 4585, 4813; _see also_ Bailey, Bill\n\nBain, Sherry 3137, 3148, 4927\n\nBainbridge, Phyllis 855\n\nBainter, Fay 1029, 1140, 4745\n\nBaker, Art 2618, 2919, 3163, 3902, 4049, 4057\n\nBaker, Benny (Ben) 1153, 2056, 3009, 3609, 3611, 3737, 4776\n\nBaker, Betty 240\n\nBaker, Bob 69, 199, 353, 460, 739, 847, 1080, 1556, 1662, 1929, 2235, 2736, 2861, 2866, 2948, 2994, 3089, 3160, 3452, 3924, 4808, 4848, 4957\n\nBaker, Brydon 846, 3038, 3292, 4424, 4846\n\nBaker, Carroll 313, 651, 720, 1560, 1945\n\nBaker, Diane 216, 947, 3683\n\nBaker, Elliott 4749\n\nBaker, Fay 1256\n\nBaker, Graham 1420\n\nBaker, Jody 26, 4867\n\nBaker, Joe Don 1739, 2069, 3793, 4669\n\nBaker, Kenny 1792\n\nBaker, Roy Ward 1995, 3915\n\nBaker, Silver Tip (Silvertip) 297, 337, 458, 1057, 1192, 1389, 1440, 1455, 1459, 1474, 1486, 1489, 1494, 1687, 1751, 2548, 2590, 2701, 2764, 2826, 2850, 2954, 3092, 3098, 3116, 3153, 3165, 3270, 3506, 3574, 3680, 4004, 4135, 4181, 4389, 4406, 4649, 4736, 4760, 4816, 4854, 4863, 4944, 5027, 5040\n\nBaker, Stanley 636, 5085\n\nBaker, Tom 2199\n\nBakewell, William 654, 967, 970, 1639, 2122, 2128, 4804, 4871\n\nBalaban, Burt 1875\n\nBalcazar, Alfonso 769, 1197, 1355, 1359, 1419, 2562, 2846, 2905, 3715, 4332\n\nBalcazar, Jaime J. 2535\n\nBaldanello, Gianfranco 4012, 4340, 4342\n\nBaldassarre, Raf 61, 133, 296, 394, 1163, 1508, 1643, 1699, 2643, 2952, 3117, 3346, 3405, 3717, 3874, 3796, 4003, 4157, 4158, 4662\n\nBalderson, John 4257\n\nBaldi, Ferdinando 175, 394, 817, 1393, 1536, 1794, 3532, 4288\n\nBalding, Rebecca 419\n\nBaldini, Renato 84\n\nBaldra, Chuck 243, 327, 442, 777, 797, 974, 1009, 1298, 1312, 1334, 1335, 1432, 1681, 1860, 1895, 1916, 1928, 1952, 2152, 2174, 2220, 2263, 2264, 2269, 2298, 2999, 2300, 2303, 2522, 2542, 2778, 2785, 2786, 2872, 2954, 2956, 2975, 2998, 2036, 3074, 3111, 3160, 3163, 3165, 3188, 3210, 3254, 3268, 3325, 3300, 3360, 3427, 3443, 3491, 3552, 3564, 3574, 3728, 4110, 4248, 4406, 4480, 4483, 4497, 4570, 4649, 4683, 4728, 4835, 4846, 4854, 4861\n\nBaldwin, Alec 45\n\nBaldwin, Ann 2086, 3254, 4773\n\nBaldwin, Bill 2013, 2627, 3040\n\nBaldwin, Daniel 2769, 3800, 4942\n\nBaldwin, Peter 4435, 3528\n\nBaldwin, Walter 1, 49, 129, 720, 961, 1082, 1516, 2515, 2868, 3214, 3383, 3410, 3440, 3614, 4062, 4231, 4257, 4336, 4509\n\nBaledon, Rafael 569, 1376, 1857, 2405, 3145, 3190, 3360, 3765, 3867, 4176, 4216, 4716, 5091\n\nBalenda, Carla 2960, 2091\n\nBalin, Ina 704, 808, 815, 1077\n\nBall, Frank 140, 156, 297, 311, 423, 448, 495, 792, 848, 1057, 1193, 1281, 1301, 1414, 1474, 1494, 1503, 1683, 1969, 2110, 2220, 2274, 2298, 2531, 2584, 2785, 3044, 3188, 3242, 3270, 3275, 3303, 3332, 3492, 3581, 3742, 4125, 4189, 4317, 4460, 4581, 4656, 4698, 4862\n\nBall, George 2285, 3581\n\nBall, Lucille 1257, 4700\n\nBall, Suzan 733, 4674, 4783, 5025\n\nBallantine, Carl 3806\n\nBallew, Smith 659, 1158, 1512, 1802, 1956, 3022, 3029, 3289, 3305, 3584, 4278, 4640, 4840\n\nBalsam, Martin 751, 1610, 1908, 2373\n\nBancroft, Anne 2192, 3221, 3367, 4770\n\nBancroft, George 2831, 3139, 3240, 4071, 4100, 4286, 4868, 4909\n\nBand, Albert 1617, 3813, 4527, 5066; _see also_ Antonini, Alfredo\n\nBand, Charles 3039\n\nBanderas, Antonio 87, 2334, 2604\n\nBane, Holly 318, 377, 492, 557, 667, 858, 951, 1073, 1264, 1486, 1543, 1616, 1747, 1837, 1935, 2035, 2678, 2779, 2806, 2825, 2899, 3002, 3178, 3219, 3247, 3312, 3356, 3394, 3460, 3473, 3556, 3943, 4028, 4073, 4140, 4828\n\nBanks, Emily 1703, 3127\n\nBanks, Monty 4254\n\nBanky, Vilma 5004\n\nBannen, Ian 348, 1062\n\nBanner, Jill 3959, 4662\n\nBanner, Priscilla 1495\n\nBanning, Leslie 365, 3358, 4111\n\nBannon, Jim 368, 649, 819, 951, 1326, 1463, 1635, 2018, 2040, 2294, 2373, 2515, 2776, 2910, 3339, 3355, 3458, 3493, 3576, 3586, 3864, 4105, 4310, 4335, 4506, 4582, 4726, 4783\n\nBaragery, John 3040, 4232\n\nBarbeau, Adrienne 11, 1549\n\nBarber, Bobby 2020, 2163, 2452, 3040, 3524\n\nBarbier, George 1847, 3374, 4203, 4652\n\nBarboni, Enzo 568, 2553, 2614\n\nBarboo, Luis 61, 132, 296, 1508, 1530, 1581, 1776, 2390, 2612, 2900, 3286, 3515, 4182, 4631\n\nBarclay, Don 210, 448, 1436, 1925, 2288, 2371, 2864, 2948, 4534, 4697, 4890\n\nBarclay, Joan 230, 344, 345, 1253, 1284, 1576, 2360, 2640, 2979, 3083, 3105, 3194, 3679, 3924, 3944, 4324, 4581, 4612, 4184, 4887; _see also_ Greer, Geraine\n\nBarclay, Stephen 1134, 1380, 2175, 3176, 4734, 4876\n\nBarcroft, Roy 51, 67, 134, 146, 199, 231, 233, 236, 237, 286, 289, 309, 339, 365, 415, 453, 464, 634, 644, 661, 665, 710, 712, 726, 731, 779, 784, 793, 836, 892, 930, 933, 1008, 1048, 1063, 1072, 1073, 1074, 1129, 1130, 1141, 1142, 1238, 1246, 1247, 1254, 1264, 1345, 1367, 1400, 1431, 1454, 1469, 1543, 1618, 1666, 1668, 1721, 1727, 1767, 1809, 1843, 1860, 1870, 1873, 1918, 1923, 1934, 1964, 1971, 2022, 2032, 2035, 2135, 2170, 2176, 2179, 2193, 2198, 2234, 2310, 2357, 2368, 2411, 2427, 2530, 2542, 2545, 2564, 2565, 2592, 2593, 2594, 2597, 2600, 2607, 2648, 2669, 2678, 2685, 2727, 2804, 2825, 2838, 2842, 2853, 2856, 2857, 2873, 2878, 2880, 2882, 2909, 2910, 2927, 2931, 2942, 2971, 2995, 3024, 3025, 3088, 3104, 3115, 3128, 3132, 3151, 3178, 3220, 3228, 3254, 3271, 3354, 3356, 3436, 3450, 3453, 3461, 3475, 3483, 3567, 3578, 3600, 3649, 3652, 3679, 3685, 3691, 3707, 3711, 3724, 3803, 3826, 3827, 3828, 3851, 3898, 3899, 3921, 3933, 3938, 4011, 4033, 4048, 4056, 4073, 4075, 4083, 4096, 4110, 4127, 4152, 4153, 4181, 4183, 4197, 4202, 4206, 4212, 4253, 4357, 4287, 4336, 4361, 4400, 4425, 4463, 4487, 4515, 4620, 4644, 4686, 4701, 4712, 4730, 4733, 4759, 4763, 4786, 4800, 4804, 4808, 4811, 4823, 4899, 4923, 4938, 4955, 5001, 5033, 5077\n\nBardette, Trevor 25, 75, 103, 112, 203, 236, 318, 357, 392, 461, 519, 832, 863, 960, 1032, 1082, 1136, 1138, 1150, 1216, 1236, 1367, 1400, 1404, 1490, 1523, 1585, 1676, 1782, 1843, 1888, 1978, 2006, 2048, 2233, 2289, 2344, 2416, 2482, 2489, 2511, 2544, 2555, 2592, 2594, 2684, 2790, 2864, 3003, 3023, 3066, 3219, 3247, 3328, 3300, 3337, 3358, 3596, 3636, 3676, 3691, 3696, 3711, 3750, 3828, 3839, 3975, 4121, 4183, 4228, 4310, 4358, 4390, 4697, 4742, 4767, 4851, 4932, 5001, 5030, 5033, 5034, 5062\n\nBardot, Brigitte 236, 3807, 4748\n\nBare, Bobby 1112\n\nBare, Richard L. 3386, 3839, 4619\n\nBari, Lynn 2145, 2543, 3384, 3941\n\nBarkeley, Lynne 4931\n\nBarker, Jess 939, 2096, 3061, 3772\n\nBarker, Lex 106, 260, 1033, 1069, 1188, 1761, 2219, 2511, 2559, 2883, 3125, 3198, 3383, 4390, 4540, 4542, 4635, 4653, 4784, 4919, 5003, 5049\n\nBarker, Reginald 249, 1634, 2892\n\nBarkin, Ellen 4930\n\nBarkley, Lucille 146, 1433\n\nBarlow, Reginald 1855, 2213, 2269, 2786, 2990, 2923, 3308, 3626, 3874, 4395, 4773\n\nBarnes, Binnie 248, 1175, 1458, 1974, 2213, 4215\n\nBarnes, Forrest 4703, 4840\n\nBarnes, Jane 1455\n\nBarnes, Joanna 3193, 4790\n\nBarnes, Priscilla 4295, 4977\n\nBarnes, Rayford 508, 580, 1063, 1402, 1485, 1673, 1734, 1921, 1951, 2023, 2034, 2426, 3328, 3415, 3816, 4111, 4163, 4339, 4921, 5030, 5070\n\nBarnes, Walter 84, 106, 316, 526, 594, 769, 1185, 1663, 1761, 1880, 2490, 2675, 2926, 3246, 3517, 4595, 4830\n\nBarnett, Griff 129, 677, 1024, 1136, 1187, 1467, 1480, 1504, 1706, 2159, 2399, 2403, 2652, 2971, 3053, 3709, 3805, 3862, 3989, 4127, 4536, 4635\n\nBarnett, Vince 318, 425, 456, 664, 1041, 1509, 1854, 1862, 2149, 2386, 2554, 2711, 2993, 3084, 3211, 3426, 3866, 3920, 4080, 4127, 4203, 4383, 4409, 4512, 4745\n\nBaron, Lita 439, 524, 2038, 3337\n\nBarr, Byron 854, 141\n\nBarrat, Robert 63, 79, 196, 200, 207, 251, 254, 638, 757, 797, 857, 930, 1039, 1111, 1136, 1247, 1390, 1565, 1588, 1855, 2109, 2213, 2516, 2552, 2610, 2689, 2837, 3344, 3384, 3471, 3473, 3544, 3692, 3750, 4232, 4284, 4494, 4513, 4665\n\nBarray, Gerard 3198\n\nBarrett, Claudia 1074, 2804, 2875, 3652, 3791\n\nBarrett, Curt, and The Trailsmen 1158, 1497, 1529, 1725, 2175, 3231, 3802, 4023, 4327\n\nBarrett, Edith 2158\n\nBarrett, James Lee 182, 237, 3138, 3205, 3632, 3816, 3990, 4102, 4638, 4948\n\nBarrett, Judith 5054\n\nBarrett, Louise 3059\n\nBarrett, Tony 1736, 4653, 4841, 4951\n\nBarrie, Judith 1869, 2192, 5054\n\nBarrie, Mona 973, 2646, 3131\n\nBarrier, Edgar 25, 842, 2815, 3571, 3897, 4120\n\nBarringer, Barry 1177, 2832, 3308, 4703\n\nBarrington, Phyllis 1154\n\nBarriscale, Bessie 3609, 3766\n\nBarron, Baynes 74, 604, 841, 1014, 1188, 2482, 3751, 4726, 4729\n\nBarron, Robert (Bob) 210, 219, 475, 477, 481, 658, 726, 1506, 1738, 1749, 2546, 2638, 2994, 3344, 3390, 4017, 4053, 4072, 4078, 4081, 4135, 4272, 4729, 4819\n\nBarron, Wally 4438\n\nBarrows, George 541, 698, 2609, 2644, 2963, 4420, 4585\n\nBarry, Charlene 3413\n\nBarry, Charlotte 4519\n\nBarry, Don \"Red\" (Donald\/Don) 37, 71, 107, 114, 154, 237, 286, 388, 449, 466, 474, 559, 610, 644, 665, 771, 830, 925, 936, 986, 987, 995, 1042, 1106, 1397, 1406, 1442, 1467, 1472, 1548, 1668, 1713, 1940, 1942, 1959, 2033, 2038, 2069, 2082, 2087, 2267, 2357, 2545, 2559, 2632, 2667, 2909, 2911, 2935, 2966, 2967, 3074, 3128, 3312, 3338, 3527, 3675, 3788, 3806, 3989, 4085, 4106, 4185, 4318, 4467, 4472, 4524, 4586, 4601, 4617, 4675, 4753, 4768, 4788, 4791, 5037, 5041\n\nBarry, Gene 1086, 1412, 1500, 2647, 2963, 3093, 3215, 4420\n\nBarry, Patricia 379; _see also_ White, Patricia\n\nBarry, Wesley 433, 451, 2647, 2963, 3093, 3215, 4420\n\nBarrymore, Diana 1444\n\nBarrymore, Drew 194\n\nBarrymore, Ethel 2013, 3757\n\nBarrymore, John 4061\n\nBarrymore, John Drew (John, Jr.) 1225, 1226, 1876, 3201, 4344, 4990\n\nBarrymore, Lionel 198, 254, 1187, 2344, 2416, 4174, 4257\n\nBarrymore, William 3293; _see also_ Bullock, Boris; Carson, Kit\n\nBarsha, Leon 2907, 4531, 4609\n\nBartel, Paul 2483\n\nBarthelmess, Richard 2181, 2381, 2610, 3743, 4072\n\nBartlett, Bennie 27, 777, 1103, 1509, 2163, 4309\n\nBartlett, Hal 1151, 2757, 3020\n\nBartlett, Juanita 2788, 2810\n\nBartlett, Richard 1866, 2436, 2676, 3906, 3960, 4613\n\nBartlett, Sy 313\n\nBartlett, William 630\n\nBarton, Buzz 108, 646, 924, 1284, 1301, 1323, 1379, 1468, 1712, 1950, 1969, 2107, 2172, 2256, 2394, 2420, 2533, 2647, 2745, 3044, 3077, 3098, 3099, 3153, 3302, 3455, 3497, 3591, 3601, 3664, 3898, 3983, 4104, 4459, 4877, 4958, 4972\n\nBarton, Charles 471, 1394, 2774, 2937, 3573, 3859, 4395, 4762, 5008, 5097\n\nBarton, Finis 2726, 3763, 4117\n\nBarton, Gregg 49, 104, 189, 209, 290, 389, 390, 433, 819, 997, 1111, 1164, 1415, 2256, 2394, 2420, 2521, 2525, 2618, 2620, 2678, 2711, 2842, 2986, 3287, 3298, 3509, 3682, 3751, 3892, 4296, 4469, 4478, 4588, 4680, 4692, 4757, 4829, 4886, 5005, 5025; _see also_ Lee, Palmer\n\nBarton, James 1601, 2661, 2749, 3199, 3817, 5051\n\nBarton, Joan 2419, 3599, 4147\n\nBarty, Billy 2982\n\nBarwood, Hal 4180\n\nBascom, Texas Rose 2301, 3976\n\nBasehart, Richard 487, 645, 709, 1383, 2141, 3218, 3683, 3726\n\nBashful Brother Oswald (Pete Kirby) 3975\n\nBasie, Count 385\n\nBasinger, Kim 2696\n\nBasquette, Lina 153, 1178, 1781, 1848, 2702, 3610\n\nBastien, Yvonne 1484, 3725\n\nBateman, Kent 2168, 3580\n\nBates, Alan 3883\n\nBates, Barbara 111, 2941, 3684, 3757\n\nBates, Florence 278, 1949, 3534, 3692, 3696\n\nBates, Granville 874, 1378, 1594, 1622, 3126, 4803\n\nBates, Jeanne 399, 3751, 4191, 4566\n\nBattaglia, Rik 1069, 1171, 1525, 1653, 2316, 2569, 2883, 3198, 3829, 4342, 4380, 4542, 4901, 5003\n\nBattista, Lloyd 394, 741, 817, 1536, 4157\n\nBattle, John Tucker 2502, 3839\n\nBauer, Belinda 4433\n\nBauer, Michelle 397\n\nBaugh, Slingin' Sammy 2135\n\nBava, Mario 3543, 3628, 37225\n\nBavier, Frances 290, 1936\n\nBaxley, Jack 64, 158, 866, 1554, 1687, 1925, 2106, 2701, 3457, 3695, 3750, 3801, 4026, 4465, 4621, 4803, 4868\n\nBaxter, Alan 203, 741, 1248, 2162, 3009, 3154, 3711, 4494, 4579, 4801\n\nBaxter, Anne 748, 2899, 2941, 3972, 4073, 4161, 4237, 4373, 4411, 4593, 5051\n\nBaxter, Warner 142, 755, 1972, 3179, 3245, 3384, 3595, 4093, 4650\n\nBay, Tom 294, 327, 376, 785, 1294, 1335, 1432, 1796, 2231, 2404, 3047, 3321, 4442\n\nBaylor, Hal 190, 267, 580, 1220, 2906, 2959, 3535, 4426, 5022\n\nBaylor, Steve 808\n\nBazlen, Brigid 1945\n\nBazzoni, Luigi 532, 2441\n\nBeach, Guy 2482, 2502, 3128, 3972, 4500, 4690\n\nBeach, John 42, 242, 410, 459, 461, 670, 757, 1469, 1558, 1816, 1860, 1877, 1889, 1916, 1933, 1977, 2303, 2648, 2665, 2701, 2824, 2826, 2998, 3049, 3325, 3657, 4322, 4372, 4485, 4649, 5037\n\nBeach, Rex 176, 254, 1378, 2652, 4070, 4071, 4072, 4073, 5025\n\nBeach, Richard 323, 1751\n\nBeach, Scott 1656\n\nBeal, Frank 3985, 4203\n\nBeal, John 432, 944, 4253\n\nBean, Orson 1126\n\nBearclaw, Cody 1085, 1527, 2859\n\nBeard, Matthew \"Stymie\" 279, 3374, 4615\n\nBeasley, Barney 344, 369, 382, 411, 438, 443, 752, 1301, 1320, 1323, 1455, 1687, 1813, 1927, 2203, 2249, 2268, 2358, 2407, 2470, 2640, 2764, 2821, 2822, 2915, 2954, 2999, 3073, 3314, 3528, 3653, 3658, 3884, 3888, 3927, 3946, 3994, 4493, 4645, 4711, 4736, 4810, 4838\n\nBeatty, Ned 902, 2982, 4169, 4879\n\nBeatty, Robert 4065\n\nBeatty, Warren 2625\n\nBeauchamp, D.D. 53, 277, 953, 1082, 1383, 1747, 2038, 2252, 2506, 2544, 2565, 2611, 2754, 3234, 3534, 3699, 3839, 3996, 4258, 5044\n\nBeaudine, William 339, 470, 488, 1509, 1600, 2040, 2183, 3576, 4253, 4467, 4500, 4856, 5081\n\nBeaumont, Charles 3783\n\nBeaumont, Harry 4807\n\nBeaumont, Hugh 562, 632, 863, 2225, 2664, 2797, 2838, 3000, 4966\n\nBeavers, Louise 248, 279, 796, 2016, 3870, 4057, 4190, 4257, 4824\n\nBeck, Ethel 1674\n\nBeck, John 628, 1200, 1241, 1932, 2129, 2305, 2389, 2421, 3055, 3860, 3878, 4071, 4494, 4593, 4925\n\nBeck, Michael 1944, 4565\n\nBeck, Thomas 4900\n\nBeckett, Scotty 986, 1599, 2871, 4803\n\nBeckman, Henry 1014, 4113, 4638\n\nBeddoe, Don 277, 304, 324, 354, 409, 577, 602, 664, 857, 2005, 2066, 2153, 2547, 2549, 3295, 3535, 3578, 3839, 3859, 4238, 4286, 4314, 4469, 4791, 4806, 5038, 5064\n\nBedelia, Bonnie 4149\n\nBedford, Barbara 1058, 1564, 1577, 1599, 2181, 2211, 2654, 3771, 4070, 4591\n\nBedoya, Alfonso 50, 91, 313, 450, 604, 706, 2549, 3411, 4163, 4168, 4175, 4252, 4543\n\nBeebe, Ford 696, 780, 850, 936, 1328, 1444, 1488, 1959, 1993, 2073, 2139, 2212, 2254, 2367, 2522, 2607, 2730, 2860, 2861, 2927, 2994, 3065, 3082, 3173, 3312, 3401, 3450, 3452, 3508, 3720, 3945, 4006, 4117, 4569, 4592, 4708, 4766, 4851, 44971, 5001\n\nBeebe, Marjorie 1304, 2456, 2933\n\nBeecher, Elizabeth 573, 872, 878, 1025, 1519, 1798, 2166, 2378, 2411, 3613, 3616, 3666, 3889, 3914, 4221, 4261, 4669, 4957\n\nBeecher, Janet 2552, 2586, 2638, 3046\n\nBeery, Noah 37, 200, 248, 327, 665, 834, 1089, 1100, 1154, 1340, 1526, 1572, 1620, 1854, 1970, 2352, 2554, 2585, 2648, 2667, 2717, 2966, 2994, 3029, 3111, 3470, 3914, 4070, 4092, 4203, 4257, 4403, 4442, 4482, 4586, 4639, 4704, 4776, 4949, 5096\n\nBeery, Noah (Jr.) 196, 356, 378, 596, 657, 666, 771, 939, 969, 1026, 1136, 1179, 1238, 1274, 1341, 1444, 1741, 1832, 1863, 1922, 1985, 1988, 2059, 2065, 2225, 2353, 2585, 2994, 3156, 3323, 3438, 3450, 3655, 3728, 3734, 3783, 3860, 4065, 4142, 4310, 3382, 4569, 4593, 4611, 4654, 4766, 4783, 4907, 5000\n\nBeery, Wallace 145, 191, 198, 200, 248, 318, 331, 1634, 2020, 2211, 2238, 2516, 2646, 3139, 4120, 4593, 4750, 5032\n\nBegley, Ed 1346, 1769, 2161, 2416, 3672, 3878, 4129, 4585, 4670, 5036\n\nBegley, Ed, Jr. 1590, 3855\n\nBeir, Fred 450, 830, 1397\n\nBel Geddes, Barbara 402\n\nBelafonte, Harry 540\n\nBelasco, David 1570, 1571\n\nBelden, Charles 272, 1140, 1520, 2577, 3877, 4047\n\nBelford, Christine 2092, 2960, 3136\n\nBelgard, Andre 242\n\nBelgard, Arnold 596, 3156\n\nBelion, Edward 541\n\nBell, Hank 64, 67, 159, 206, 210, 240, 286, 306, 318, 327, 331, 338, 382, 411, 422, 429, 437, 458, 459, 460, 494, 511, 516, 623, 634, 666, 682, 712, 721, 728, 729, 739, 752, 779, 799, 814, 818, 832, 856, 872, 960, 1017, 1090, 1097, 1128, 1143, 1177, 1311, 1329, 1334, 1345, 1360, 1362, 1367, 1394, 1432, 1440, 1454, 1456, 1462, 1473, 1486, 1490, 1537, 1570, 1571, 1579, 1592, 1626, 1638, 1646, 1677, 1730, 1798, 1818, 1855, 1928, 1932, 1952, 1975, 2032, 2150, 2166, 2174, 2231, 2247, 2249, 2266, 2275, 2281, 2283, 2286, 2303, 2386, 2399, 2411, 2422, 2482, 2517, 2522, 2527, 2530, 2547, 2554, 2573, 2633, 2660, 2678, 2718, 2722, 2736, 2763, 2764, 2767, 2778, 2799, 2821, 2822, 2847, 2861, 2867, 2878, 2907, 2942, 2947, 2951, 2975, 2976, 2988, 2993, 2995, 3003, 3005, 3049, 3074, 3087, 3100, 3126, 3128, 3132, 3139, 3162, 3171, 3188, 3228, 3232, 3254, 3259, 3260, 3280, 3312, 3322, 3329, 3300, 3353, 3370, 3390, 3424, 3452, 3479, 3506, 3520, 3548, 3550, 3555, 3561, 3568, 3574, 3592, 3613, 3615, 3650, 3656, 3658, 3680, 3684, 3694, 3797, 3859, 3888, 3889, 3898, 3927, 3969, 3989, 4000, 4042, 4048, 4051, 4067, 4077, 4096, 4103, 4108, 4112, 4138, 4197, 4231, 4238, 4271, 4286, 4306, 4309, 4315, 4327, 4367, 4376, 4385, 4389, 4396, 4402, 4410, 4458, 4463, 4466, 4484, 4486, 4494, 4498, 4521, 4593, 4607, 4609, 4645, 4690, 4696, 4700, 4701, 4702, 4745, 4759, 4803, 4816, 4817, 4819, 4833, 4834, 4838, 4849, 4850, 4851, 4854, 4861, 4885, 4892, 4899, 4923, 4944, 4952, 4971, 5027, 5040, 5059, 5061, 5062\n\nBell, James 146, 357, 544, 933, 977, 1092, 2226, 2302, 2308, 2431, 2719, 2929, 3149, 3318, 3436, 3500, 3621, 3750, 4160, 4168, 4300, 4398, 4435, 4439, 4528, 4548, 4955\n\nBell, Marjorie 1929; _see also_ Champion, Marge\n\nBell, Rex 264, 458, 891, 973, 1102, 1323, 1333, 1440, 1469, 1712, 1965, 2248, 2416, 2640, 2661, 3239, 3664, 3954, 4143, 4456, 4459, 4460, 4814\n\nBell, Rex, Jr. 4098, 5065\n\nBellah, Ben 3979\n\nBellah, James Warner 819, 2561, 3776, 4387\n\nBellamy, Earl 41, 190, 377, 841, 1077, 1081, 1727, 1985, 2074, 2289, 2609, 2693, 2842, 2910, 3751, 3769, 3778, 4109, 4228, 4361, 4469, 4478, 4479, 4582, 4726, 4753\n\nBellamy, Madge 1614, 2005, 2841\n\nBellamy, Patsy 2264\n\nBellamy, Ralph 702, 2638, 3180, 4807, 4933\n\nBellaver, Harry 4099\n\nBellem, Robert Leslie 698, 2074, 2609, 2842, 3751, 4582\n\nBellini, Cal 1578, 2265, 2373, 2376, 2700, 3747\n\nBelmont, Virginia 850, 951, 1105, 1573, 2858, 3002, 3157, 3277, 3877\n\nBelmore, Lionel 2213, 3266, 3332, 5002\n\nBelson, Jerry 1243\n\nBeltran, Robert 2746\n\nBelushi, John 1590\n\nBenadaret, Bea 3134\n\nBenchley, Robert 3544\n\nBender, Joel 3398\n\nBender, Russ 211, 2813, 3193, 4769\n\nBendix, Lorraine 373, 2267\n\nBendix, William 2267, 4168, 5065\n\nBenedek, Laslo 2144\n\nBenedict, Billy 37, 488, 1915, 1976, 2032, 2240, 2342, 2552, 2722, 3005, 3162, 3468, 3544, 4096, 4423, 4695, 4799; _see also_ Benedict, William\n\nBenedict, Brooks 1563\n\nBenedict, Richard 149, 405, 2842, 3637\n\nBenedict, William 417, 2635, 4345; _see also_ Benedict, Billy\n\nBenet, Joile 3501\n\nBengell, Norma 218, 1811\n\nBenham, Elsie 2007, 4063, 4836\n\nBening, Annette 2920\n\nBenjamin, Richard 3500, 4390, 4859\n\nBennet, Spencer Gordon 13, 131, 180, 291, 486, 503, 610, 634, 644, 684, 778, 779, 876, 1280, 1474, 1711, 1716, 1804, 1835, 1861, 2024, 2125, 2192, 2274, 2427, 2673, 2852, 2869, 3068, 3088, 3270, 3279, 3303, 3365, 3461, 3491, 3509, 3525, 3582, 4001, 4011, 4584, 4667, 4701, 4833, 4837, 5103\n\nBennett, Bruce 386, 482, 718, 955, 1365, 1449, 1640, 1871, 2225, 2462, 2547, 3559, 3902, 4368, 4373, 4543, 4806, 5075; _see also_ Brix, Herman\n\nBennett, Constance 2815, 3912, 4932\n\nBennett, Enid 197, 1060\n\nBennett, Harriet 3589\n\nBennett, Joan 4284, 4940\n\nBennett, Lee 49, 658, 795, 835, 933, 1157, 2233, 3599, 4017, 4131, 4357, 4437, 4969\n\nBennett, Leila 4203, 4762\n\nBennett, Marjorie 339, 831, 1422, 2308, 2519, 2813, 3411, 3839, 3902, 4387, 4611\n\nBennett, Ray (Raphael) 73, 105, 234, 258, 354, 368, 427, 615, 664, 683, 695, 922, 928, 936, 972, 1001, 1017, 1099, 1138, 1155, 1167, 1257, 1362, 1394, 1463, 1497, 1513, 1520, 1541, 1560, 1627, 1645, 1687, 1693, 1797, 1823, 1830, 1870, 2081, 2124, 2152, 2399, 2512, 2543, 2547, 2595, 2633, 2667, 2761, 2839, 2872, 3150, 3167, 3187, 3229, 3267, 3288, 3310, 3335, 3340, 3348, 3370, 3385, 3423, 3500, 3511, 3557, 3596, 3651, 4072, 4080, 4313, 4315, 4322, 4401, 4408, 4504, 4579, 4640, 4674, 4754, 4989, 5020\n\nBennison, Andrew 1080, 2298, 4656\n\nBennison, Leslie 3700\n\nBenny, Jack 541\n\nBenson, Robby 2056\n\nBentley, Dick 1710\n\nBentley, Irene 1457\n\nBentley, John 3915\n\nBenton, Dean 1785, 4385\n\nBenton, Robert 192\n\nBenvenuti, Nino 4182\n\nBerardino, John 841, 2074, 2289, 3150, 3221, 3788, 4582, 4943\n\nBerenger, Tom 585, 2055, 2206, 3654\n\nBeresford, Harry 2147\n\nBergei, Oscar 2929\n\nBergen, Candace 348, 1951, 3987, 4460\n\nBergen, Connie (Constance) 310, 1883\n\nBergen, Polly 14, 128, 1234, 4792\n\nBerger, Burt 3060\n\nBerger, Santa 1575, 2501\n\nBerger, William 568, 1207, 1275, 1966, 2097, 2197, 2814, 3516, 3660, 3716, 4012, 4277, 4444\n\nBergerac, Jacques 4384\n\nBergh, Jerry 1900, 2743\n\nBerke, William 205, 229, 449, 786, 961, 1041, 1105, 1145, 1240, 1298, 1449, 1550, 1713, 1758, 1760, 1959, 2269, 2398, 2601, 2922, 3001, 3042, 3062, 3351, 3466, 3467, 3505, 3507, 3565, 3593, 3674, 3894, 4304, 4465, 4524, 4736, 4981, 5035; _see also_ Williams, Lester\n\nBerkeley, Busby 1564, 1649\n\nBerkeley, Martin 1757, 2768, 3337, 3698, 4240, 4787\n\nBerkeley, Xander 3775, 3809\n\nBerkes, John 497, 880\n\nBerle, Milton 1243, 2585\n\nBerlin, Abby 1140\n\nBerlinger, George 2434\n\nBerlinger, Warren 4782\n\nBernard, Ray 2742, 3926; _see also_ Ray, Bernard B.\n\nBernardi, Herschel 2513\n\nBernay, Lynn 4699\n\nBernds, Edward 1103, 1235, 1285, 1597, 1696, 3200, 4141, 4412\n\nBerne, Josef 1144\n\nBerringer, Barry 4796\n\nBerry, Ken 1660\n\nBertram, William 4084\n\nBertrand, Rafael 181, 1885, 2001, 3180, 3190\n\nBesser, Joe 3134, 3875, 4613\n\nBesserer, Eugenie 3743, 4442, 4603\n\nBessie, Alvah 2835\n\nBest, James 105, 258, 363, 374, 671, 749, 759, 789, 803, 807, 1346, 2084, 2202, 2314, 3208, 3221, 3430, 3641, 3643, 3734, 3768, 3779, 3816, 4338, 4989, 5052\n\nBest, Willie 95, 1594, 1884, 1967, 2081, 2159, 3334, 3886, 4043, 4622, 4824\n\nBestar, Barbara 2564, 2762\n\nBeswick, Martine 571\n\nBethman, Sabine 2535\n\nBethune, Ivy 2324\n\nBettger, Lyle 1039, 1082, 1164, 1225, 1273, 1645, 1701, 1702, 1741, 2050, 2400, 2779, 3387, 3856, 4472, 4714\n\nBetz, Carl 490, 759, 966, 1995, 3150\n\nBetz, Matthew 1320, 1440, 1592, 2257, 3880, 4522, 4728, 4835, 4885\n\nBevan, Billy 1571, 1588, 1880, 2344\n\nBevans, Clem 318, 489, 655, 967, 1041, 1127, 1392, 1487, 1567, 1577, 1597, 1599, 1622, 1774, 2081, 2095, 2386, 2549, 2864, 3020, 3046, 3087, 3344, 3409, 3510, 3892, 4121, 4163, 4168, 4231, 4252, 4286, 4456, 4593, 4601, 4622, 4671, 4697, 5021, 5032, 5043, 5045\n\nThe Beverly Hill Billies 323, 3038, 3589, 4279\n\nBey, Turhan 1438\n\nBeymer, Richard 2052\n\nBiachi, Mario 1275\n\nBiachi, Pablo 1580\n\nBiberman, Abner 1671, 2128, 3684, 4751, 4989\n\nBice, Robert 42, 228, 236, 284, 433, 895, 1053, 1263, 1338, 1428, 1608, 1691, 1726, 1866, 1936, 2022, 2068, 2591, 2616, 2805, 2894, 3068, 3334, 3750, 3936, 4124, 4213, 4645, 4990, 5066\n\nBickford, Charles 313, 317, 497, 1187, 1420, 1567, 1850, 1883, 2226, 3126, 3202, 3450, 3609, 4093, 4121, 4395, 4662, 4697\n\nBiehn, Michael 4453\n\nBierce, Ambrose 2912\n\nBig Tree, Chief 328, 514, 917, 952, 1084, 1093, 1165, 1300, 1570, 1571, 1605, 1805, 1862, 1889, 1949, 2004, 2121, 2212, 2660, 3015, 3111, 3314, 3814, 3926, 4214, 4298, 4849, 4861, 5002\n\nBikel, Theodore 2728, 5019\n\nBill, Tony 1372, 3959\n\nBillings, Ted 2016, 2826, 3650, 4558\n\nBillingsley, Barbara 1996, 4690\n\nBillingsley, Jennifer 538\n\nBing, Herman 246, 626, 3606, 4254\n\nBinns, Edward 913, 1842, 2468, 3127\n\nBinyon, Claude 129, 1256, 2829\n\nBirch, Paul 117, 681, 869, 939, 1274, 1357, 1719, 1720, 2043, 2561, 2565, 3223, 3423, 3744, 3897, 4237, 4626, 4802, 4913, 4943\n\nBishop, Jenifer 1278, 2039\n\nBishop, Joey 3777, 4287\n\nBishop, Julie 320, 1041, 2218, 2835, 4857, 4932; _see also_ Wells, Jacqueline\n\nBishop, William 25, 357, 835, 895, 1665, 2929, 2996, 3090, 3222, 3340, 4310, 4399, 4461, 4468, 4673, 4771, 4913, 5038\n\nBissell, Whit 170, 932, 1092, 1133, 1341, 1354, 1948, 1985, 2052, 2076, 2215, 2497, 2518, 2800, 2813, 3185, 3305, 3318, 3750, 4064, 4234, 4362, 4791\n\nBisset, Jacqueline 2348\n\nBixby, Bill 97, 121, 247, 2000, 3422\n\nBlack, Clint 2623\n\nBlack, Karen 1694\n\nBlack, Maurice 614, 3886, 4647, 4824\n\nBlack, Ralph 142\n\nBlackburn, Tom 685, 691, 801, 857, 967, 970, 2052, 2576, 3285, 3500, 3712, 3848, 3864, 4856, 4937\n\nBlackman, Honor 3807, 3990\n\nBlackman, Joan 1608\n\nBlackmer, Sidney 197, 552, 1143, 1187, 1824, 2270, 3696, 4119, 4786\n\nBlackwell, Carlyle 1438\n\nBlackwell, Carlyle, Jr. 633\n\nBlackwood, Hope 150\n\nBlaine, James 1297, 1367, 2526, 2861, 2927, 3409, 3450, 5001\n\nBlaine, Vivian 2815, 3039\n\nBlair, Betsy 1766\n\nBlair, George 1053, 2021, 2357, 2669, 3611, 3892, 4381, 4601, 4644\n\nBlair, June 2426\n\nBlair, Linda 4948\n\nBlair, Patricia 953, 1219\n\nBlair, Reno 1443, 1529, 2255, 3231, 4640; _see also_ Browne, Reno\n\nBlake, Amanda 685, 1754, 3741, 4129\n\nBlake, Bobby 115, 607, 731, 793, 827, 930, 1646, 1918, 1920, 2233, 2427, 2594, 2597, 2600, 2632, 3080, 3379, 3649, 3822, 3824, 4111, 4181, 4543, 4584, 4733, 4734, 4763; _see also_ Blake, Robert\n\nBlake, Gladys 3569, 4654\n\nBlake, Larry 1092, 1877, 1971, 2911, 3020, 3283, 3780, 4569, 4601, 4805\n\nBlake, Madge 3311, 3777\n\nBlake, Marie 4036, 4799\n\nBlake, Mary 780\n\nBlake, Oliver 53, 798, 1161\n\nBlake, Pamela 23, 449, 940, 1543, 1713, 2889, 3593, 4754; _see also_ Pearce, Adele\n\nBlake, Robert 2931, 2935, 3693, 4249, 4373; _see also_ Blake, Bobby\n\nBlakely, Ronee 1081, 3813\n\nBlakeney, Olive 2815\n\nBlanc, Erika 1117, 1351, 1653, 1954, 4150\n\nBlanc, Mel 4621\n\nBlanchard, Mari 367, 1082, 2626, 3000, 3234, 3376, 4111\n\nBlanchard, Phyllis 4858\n\nBlanchard, Susan 2788\n\nBlanche, Herbert 600\n\nBlanco, Hugo 175, 3715, 4631, 4678, 4920\n\nBlandick, Clara 642, 1165, 1450, 1961, 2831, 4494\n\nBlane, Sally 1317, 1847, 1854, 2388, 4950\n\nBlangsted, Folmer 2886, 4832\n\nBlankfort, Henry 519, 2149, 3925, 4672\n\nBlankfort, Michael 3127, 4286, 4548\n\nBlasco, Richard 275, 1700\n\nBleifer, John 1224, 1458, 1567, 1811, 1963, 2586, 2836, 4215\n\nBletcher, Billy 413, 479, 500, 541, 597, 606, 910, 1084, 1173, 1541, 1587, 1883, 2086, 2399, 2403, 2635, 2647, 3194, 3409, 4308, 4622, 5104\n\nBlinn, Holbrook 197\n\nBliss, Lela 1509, 2782\n\nBliss, Sally 3651, 4221\n\nBlocker, Dan 371, 771, 3991\n\nBlocker, Dirk 426, 512, 1065\n\nBlondell, Joan 22, 2158, 3422, 4133, 4211, 4794, 4990\n\nBlondell, Simone 99, 999, 1116, 1418, 2505, 3727, 5093\n\nBloom, Claire 2984\n\nBloom, Harold Jack 128, 1694, 1784, 1786, 1834, 2751\n\nBloom, John 1783\n\nBloom, Lindsay 1655, 4295\n\nBloom, Verna 1880, 1893\n\nBlore, Eric 1257\n\nBlossom, Roberts 3206\n\nBlue, Ben 1883\n\nBlue, Monte 23, 100, 160, 201, 288, 471, 718, 798, 935, 986, 1043, 1127, 1236, 1392, 1462, 1532, 1597, 1626, 1638, 1647, 1774, 1805, 1883, 2009, 2064, 2135, 2149, 2226, 2232, 2279, 2536, 2603, 2677, 2733, 2774, 2830, 2831, 2835, 2940, 3271, 3433, 3450, 3603, 3608, 3692, 3902, 3978, 4022, 4049, 4061, 4092, 4115, 4199, 4274, 4312, 4357, 4392, 4395, 4403, 4446, 4491, 4522, 4534, 4621, 4762, 4776, 4792, 4926, 5059, 5075\n\nBlye, Margaret 1908, 4794\n\nBlystone, John G. 3985\n\nBlystone, Stanley 63, 183, 474, 597, 599, 602, 606, 664, 776, 892, 898, 1073, 1159, 1213, 1228, 1246, 1264, 1315, 1322, 1323, 1491, 1496, 1543, 1605, 1810, 1927, 1974, 1983, 2014, 2017, 2226, 2249, 2292, 2302, 2386, 2396, 2403, 2473, 2517, 2552, 2687, 2716, 2759, 3020, 3035, 3051, 3066, 3075, 3142, 3151, 3179, 3221, 3243, 3262, 3325, 3401, 3497, 3541, 3611, 3646, 3664, 3702, 3890, 3921, 3942, 3943, 4086, 4132, 4152, 4190, 4199, 4278, 4365, 4372, 4419, 4488, 4519, 4553, 4586, 4732, 4188, 4865, 4868, 5043\n\nBlyth, Ann 3310, 3607, 5025\n\nBlythe, Betty 241, 973, 1925, 2436, 2816, 4659, 4837\n\nBoardman, Eleanor 1639, 4093\n\nBoardman, True 3014, 3425\n\nBoardman, Virginia True 495, 1474, 4275\n\nBoccardo, Delia 3033\n\nBochner, Lloyd 2739, 4661, 4633\n\nBodeen, De Witt 2705\n\nBoehm, Andre 678\n\nBoehm, Sidney 482, 497, 3221, 3612, 3721, 3861, 4233, 5019\n\nBoetticher, Budd 527, 539, 806, 1026, 1936, 2544, 3430, 3768, 3788, 4429, 4626, 4930, 5000, 5013\n\nBogarde, Dirk 636, 3915\n\nBogart, Humphrey 1907, 2864, 3071, 4543, 4742\n\nBohnen, Roman 602, 2088\n\nBohr, Jose 3582\n\nBois, Curt 410\n\nBoland, Mary 2790, 3634\n\nBolder, Cal 2034\n\nBoles, Jim 317, 2162, 2750, 3950, 4794, 4976\n\nBoles, John 2646, 3529, 3609, 3817, 4029, 4941\n\nBoleslawski, Richard 2923, 4359\n\nBolger, Ray 1792\n\nBolling, Tiffany 2140\n\nBonanova, Fortunio 207, 828, 1290, 1471, 2021, 2586, 3676, 4384\n\nBond, Johnny 155, 582, 587, 726, 872, 1028, 1187, 1449, 1456, 1821, 2378, 2424, 2595, 2866, 2873, 3227, 3707, 3675, 4024, 4208, 4081, 4135, 4170, 4222, 4261, 4318, 4515, 4586, 4600\n\nBond, Lilian 329, 1586, 4093, 4851\n\nBond, Raymond 2361, 3386, 4062\n\nBond, Rudy 1782\n\nBond, Sudie 4874\n\nBond, Tommy 1509, 2937, 3886\n\nBond, Ward 53, 328, 406, 541, 648, 684, 757, 824, 894, 932, 1127, 1165, 1302, 1324, 1328, 1395, 1457, 1458, 1471, 1640, 1679, 1757, 1766, 1831, 1848, 1921, 2048, 2073, 2145, 2288, 2422, 2502, 2691, 2718, 2864, 2919, 3043, 3096, 3384, 3711, 3752, 3817, 3912, 3921, 4187, 4231, 4360, 4572, 4635, 4668, 4742, 4764, 4837, 4898, 4900, 4932, 4983, 5072\n\nBondaduce, Danny 216\n\nBondi, Beulah 251, 1478, 2416, 3817, 4058, 4473, 4494\n\nBonet, Nai 4040\n\nBoniface, Symona 2515, 4673\n\nBonner, William 1278\n\nBonomo, Joe 264, 851, 1602, 2192, 2428, 3082, 3869, 4708\n\nBoone, Randy 190, 563, 1922, 3734\n\nBoone, Richard 41, 43, 319, 759, 1589, 1784, 184, 1908, 2079, 2494, 2565, 2738, 2739, 2917, 3333, 3392, 3518, 3559, 3740, 3847, 3861, 4123, 4235, 4252, 4387, 4795\n\nBooth, Adrian 68, 516, 605, 1490, 1918, 2184, 2193, 2357, 2537, 2851, 2935, 3132, 3567, 3728, 4075, 4641; _see also_ Gray, Lorna\n\nBooth, Delores 2881, 3501\n\nBooth, Edwina 2212, 4532, 4708\n\nBooth, James 1755, 1846, 2488, 5100\n\nBooth, Karin 212, 659, 895, 3769, 4461\n\nBooth, Nesdon 678, 1235, 1689, 5055\n\nBoothe, Powers 1244, 4453\n\nBorden, Eddie 824, 1449, 1450, 2163, 3414, 3619, 4593, 4798\n\nBorden, Eugene 324, 930, 1263, 1427, 2066, 2102, 2586, 3609, 3682, 3890\n\nBorden, Olive 4352\n\nBorden, Renee 646, 1312, 1949, 3487, 4843\n\nBorg, Sven Hugo 588, 1018, 2016, 2279, 2815, 3702, 4000\n\nBorg, Veda Ann 43, 160, 1451, 1925, 2080, 2256, 2590, 2635, 2665, 2748, 3066, 3431, 4384\n\nBorgaze, Dan 720, 1161, 1395, 1937, 3752, 3950, 4626\n\nBorgaze, Frank 3766, 4676\n\nBorgese, Sol 20, 537\n\nBorgnine, Ernest 7, 193, 209, 484, 570, 744, 1098, 1220, 1776, 2048, 2065, 2187, 2442, 2568, 2982, 3350, 3402, 3636, 3687, 4163, 4479, 4405, 4727, 4934, 4970\n\nBorland, Barlowe 1682, 4803\n\nBorland, Carroll 3738\n\nBorsche, Dieter 2615\n\nBosley, Tom 238\n\nBostock, Evelyn 879\n\nBosworth, Anne 696\n\nBosworth, Hobart 441, 547, 660, 1431, 2133, 2212, 2232, 3589, 3912, 4336, 4932, 4982\n\nBoteler, Wade 203, 337, 626, 811, 986, 1227, 1392, 1512, 1591, 1847, 1884, 1947, 2020, 2159, 2516, 2636, 2722, 2864, 2923, 3011, 3425, 3895, 3927, 4059, 4061, 4358, 4419, 4649, 4668, 4697, 4803, 5062\n\nBotiller, Richard (Dick) 130, 205, 414, 431, 614, 729, 753, 776, 799, 818, 1099, 1128, 1311, 1312, 1416, 1594, 1626, 1678, 1687, 1758, 1815, 1965, 2073, 2086, 2355, 2366, 2425, 2470, 2540, 2552, 2558, 2607, 2648, 2660, 2730, 2821, 2828, 2886, 2947, 2954, 3003, 3100, 3105, 3245, 3268, 3357, 3385, 3508, 3525, 3550, 3561, 3658, 3880, 3974, 3994, 4000, 4042, 4052, 4104, 4156, 4315, 4389, 4480, 4522, 4531, 4534, 4609, 4736, 4760, 4810, 4814, 4817, 4865, 4866, 4932, 4961, 5033, 5041, 5050\n\nBottoms, Sam 526, 2950, 3734, 5084\n\nBottoms, Timothy 1874, 2406, 3961, 4897\n\nBouchey, Willis 259, 720, 1161, 1266, 1356, 1665, 1937, 2046, 2202, 2234, 2486, 2496, 2561, 2813, 3096, 3387, 3392, 3776, 3815, 3837, 4073, 4210, 4211, 4329, 4626, 4740, 5060\n\nBoulton, Matthew 4115, 4180, 4635\n\nBoutell, Genee 1303, 2264, 3272, 3292\n\nBowden, Dorris 1165\n\nBowen, Harry 3665\n\nBower, Bertha Muzzy 2130\n\nBowers, Jess 131, 289, 973, 1146, 1389, 1546, 1554, 1716, 2457, 2761, 2969, 3229, 3460, 3480, 3804, 3938, 4153, 4299, 4823; _see also_ Buffington, Adele\n\nBowers, John 1984, 2702, 3957, 4864\n\nBowers, William 53, 354, 1225, 1487, 1663, 1704, 2087, 2246, 2694, 3439, 3815, 3860, 4069, 4211, 4975\n\nBowie, David 1745\n\nBowman, Lee 1599, 5032\n\nBowman, Ralph 1367, 2998; _see also_ Archer, John\n\nBoxleitner, Bruce 147, 1750, 1946, 3324\n\nBoxleitner, Bruce 7, 1958, 2092, 2093, 2094, 2487, 4968, 5030\n\nBoyce, Helen 1\n\nBoyd, Betty 1678\n\nBoyd, Beverly 1546\n\nBoyd, Bill (Cowboy Rambler) 70, 3164, 3232, 3592, 4302, 4589\n\nBoyd, Jimmy 3215, 3754\n\nBoyd, Rick 3, 20, 99, 1030, 1114, 1305, 1351, 1896, 1954, 1955, 2030, 3059, 3628, 3659, 3955\n\nBoyd, Stephen 502, 1776, 2509, 3807, 4438, 5019\n\nBoyd, William (Bill) 241, 242, 243, 446, 459, 461, 477, 579, 621, 660, 670, 800, 950, 990, 1097, 1138, 1202, 1254, 1255, 1368, 1380, 1416, 1469, 1685, 1816, 1824, 1870, 1880, 1889, 1931, 1932, 1933, 1934, 1935, 1977, 1978, 2370, 2311, 2451, 2480, 2577, 2632, 2637, 2736, 2826, 2972, 3011, 3049, 3116, 3177, 3267, 3354, 3458, 3479, 3657, 3704, 3761, 3851, 3877, 3898, 3928, 4112, 4135, 4139, 4147, 4304, 4322, 4364, 4367, 4485, 4600, 4657, 4661, 4923, 4970\n\nBoyd, William (Stage) 4071\n\nBoyland, Malcom Stuart 46, 355, 1368, 1567, 3330\n\nBoyle, Peter 2100\n\nBrabin, Charles 578, 1639\n\nBracey, Sidney 174, 2245, 4215, 4697, 4971\n\nBrackett, Leigh 1212, 1596, 3608, 3517, 3527\n\nBradbury, James, Jr. 295, 755, 1972, 2207, 4412\n\nBradbury, Robert North 54, 276, 297, 311, 411, 454, 495, 510, 689, 848, 946, 954, 968, 973, 1054, 1191, 1387, 1389, 1489, 1494, 1495, 1587, 1812, 1872, 1900, 2102, 2110, 2220, 2281, 2285, 2297, 2300, 2479, 2524, 2548, 2812, 3242, 3275, 3443, 3451, 3456, 3459, 3463, 3476, 3597, 3913, 3968, 4004, 4009, 4125, 4130, 4189, 4290, 4317, 4455, 4482, 4489, 4573, 4698, 4780, 4822, 4843, 4854, 4883, 5016\n\nBradford, Lane 28, 116, 134, 228, 284, 318, 364, 711, 784, 867, 914, 922, 994, 997, 1017, 1022, 1050, 1074, 1112, 1130, 1164, 1264, 1298, 1326, 1403, 1415, 1439, 1443, 1445, 1555, 1673, 1675, 1715, 1803, 1888, 2022, 2059, 2083, 2085, 2161, 2264, 2402, 2409, 2413, 2448, 2485, 2512, 2597, 2669, 2798, 2821, 2839, 2863, 2974, 2997, 3090, 3103, 3167, 3178, 3222, 3232, 3271, 3295, 3388, 3423, 3448, 3464, 3586, 3608, 3691, 3724, 3751, 3779, 3799, 3804, 3816, 3828, 3837, 3839, 3856, 3901, 3998, 4047, 4048, 4053, 4073, 4095, 4105, 4183, 4222, 4273, 4292, 4296, 4301, 4409, 4466, 4469, 4488, 4581, 4701, 4729, 4754, 4781, 4838, 4847, 4894, 5034\n\nBradford, Marshall 791, 842, 1844, 1996, 2448, 2616, 2804, 2899, 3839, 4266, 4329, 4847, 5081\n\nBradley, Curley 1980\n\nBradley, Grace 3554, 3609, 3870\n\nBradley, Harry C. 301, 1564, 1829, 2108, 2150, 2393, 3409, 4569\n\nBradley, Leslie 1452, 2143, 3663, 3781, 4856, 5070\n\nBradley, Truman 2208, 2423, 3837, 4062\n\nBradna, Olympe 1884\n\nBrady, Alice 5072\n\nBrady, Buff 313, 567, 940, 1603, 3578, 5055\n\nBrady, Edwin (Ed) 37, 158, 332, 442, 460, 825, 1002, 1058, 1392, 1475, 1494, 1626, 1925, 1974, 2012, 2147, 2207, 2281, 2599, 2645, 2733, 2756, 2777, 2862, 2864, 2943, 3162, 3258, 3275, 3456, 3609, 3675, 3846, 4050, 4059, 4093, 4187, 4215, 4281, 4284, 4308, 4382, 4612, 4700, 4807, 4868, 4923, 5004, 5009, 5041\n\nBrady, Pat 110, 284, 287, 386, 576, 615, 799, 1141, 1192, 1246, 1522, 1603, 1818, 1836, 1914, 1918, 1964, 2124, 2272, 2330, 2514, 2530, 2540, 2727, 2806, 2828, 2893, 2973, 2975, 2983, 3100, 3300, 3449, 3483, 3530, 3585, 3600, 3904, 4018, 4037, 4042, 4043, 4051, 4077, 4083, 4156, 4202, 4206, 4314, 4315, 4401, 4410, 4593, 4597, 4608, 4641, 4646, 4806, 4180, 4817, 4834\n\nBrady, Patti 25, 2138, 4115\n\nBrady, Scott 74, 132, 373, 399, 527, 595, 1352, 1406, 1487, 2048, 2084, 2229, 2287, 2624, 2672, 2678, 3066, 3338, 3367, 4098, 4141, 4337, 4674, 4705\n\nBrahm, John 4940\n\nBrana, Frank 18, 61, 416, 568, 1115, 1275, 1508, 1581, 2354, 2387, 2390, 2570, 2612, 3117, 3175, 3286, 3515, 3756, 4243, 4629, 4631, 4718, 5093\n\nBranch, Houston 1811, 2150, 3863\n\nBrand, Max 1070, 1082, 1084, 1433, 1907, 2725, 4510\n\nBrand, Neville 190, 247, 594, 700, 1007, 1075, 1481, 1666, 1672, 1899, 2237, 2389, 2395, 2431, 2462, 2544, 2672, 2919, 3203, 3287, 3318, 3376, 3735, 4361, 4368, 4435, 4799\n\nBrandes, Elaine 2413\n\nBrando, Jocelyn 4252\n\nBrando, Marlon 119, 2666, 2901, 4751\n\nBrandon, Henry 201, 677, 1527, 1847, 1953, 2375, 2599, 2663, 2836, 2827, 3020, 3140, 3269, 3741, 3752, 3817, 4061, 4494, 4626, 4649, 4727, 4766, 4783, 4803, 4851, 4870\n\nBrandon, Michael 3747\n\nBrands, X 512, 1238, 1720, 2748, 2868, 3712, 4361\n\nBrandt, Carolyn 404\n\nBrannon, Fred C. 28, 146, 228, 656, 784, 951, 1073, 1130, 1454, 1543, 1721, 2022, 2035, 2804, 3088, 3614, 3652, 3685, 4011, 4730, 4945\n\nBrasselle, Keefe 287, 2013, 4376\n\nBratton, Marla 4424, 4796, 4941\n\nBraun, Pinkas 358, 769\n\nBraveheart (dog) 1047, 1674\n\nBravo, Carlos (Charly) 416, 651, 736, 751, 1661, 1951, 2509, 2903, 3717, 4012, 4471, 5100\n\nBraxton, Steve 2407, 2411, 2734, 2965, 2999, 4956; _see also_ Robins, Sam\n\nBray, Robert 149, 402, 534, 1164, 1268, 1640, 1688, 1736, 1988, 2260, 2485, 2512, 2601, 3000, 3383, 3646, 3768, 3964, 4107, 4204, 4731, 4792, 4841, 4951, 5052\n\nBrazzi, Rossano 1163\n\nBreakston, George 2031\n\nBrecher, Egon 2064, 4061, 4442\n\nBrecher, Irving 1577\n\nBreck, Peter 523, 1575, 2157, 4924\n\nBreese, Edmund 578, 1340, 1605\n\nBrega, Mario 553, 1016, 1217, 1381, 1525, 1612, 1643, 2656, 2723, 2814, 3381, 4631\n\nBrendel, El 328, 1588, 2239, 4697, 4807\n\nBrennan, Andrew (Andy) 3690, 3838, 4470\n\nBrennan, Ruth 402, 612, 1852\n\nBrennan, Walter 65, 170, 193, 246, 293, 402, 516, 834, 866, 914, 930, 1164, 1263, 1307, 1423, 1426, 1848, 1945, 2249, 2254, 2440, 2718, 2832, 2837, 2989, 2990, 3038, 3173, 3185, 3323, 3392, 3517, 3655, 3838, 3852, 3880, 3921, 3963, 4211, 4212, 4284, 4294, 4359, 4411, 4607, 4799, 4851, 4925, 5063\n\nBrenon, Herbert 1572\n\nBrent, Eve 1412, 2576, 3663, 4426, 4895\n\nBrent, Evelyn 1600, 1917, 1932, 2288, 3231, 3562, 3896, 4027, 4178, 4855, 4923\n\nBrent, George 1252, 1588, 1594, 2367, 2678, 3310, 3899\n\nBrent, Linda 289, 1025, 2179, 3425, 4786\n\nBrent, Lynton 37, 201, 304, 452, 634, 868, 986, 1158, 1296, 1389, 1447, 1465, 1513, 1520, 1529, 1574, 1716, 1797, 2124, 2407, 2411, 2457, 2514, 2590, 2737, 2740, 2874, 2957, 3001, 3052, 3109, 3229, 3232, 3264, 3280, 3300, 3447, 3452, 3480, 3589, 3825, 3938, 4024, 4051, 4153, 4206, 4261, 4279, 4289, 4299, 4502, 4606, 4640, 4685, 4702, 4823, 4826\n\nBrent, Romney 3871\n\nBrent, Roy 37, 76, 155, 204, 383, 430, 665, 683, 775, 1028, 1155, 1749, 1894, 2286, 2546, 2590, 2737, 2740, 2874, 2957, 3001, 3052, 3109, 3229, 3232, 3264, 3300, 3447, 3452, 3480, 3589, 3825, 3938, 4024, 4051, 4153, 4206, 4261, 4279, 4289, 4299, 4502, 4606, 4640, 4685, 4702, 4823, 4826\n\nBrent, Timothy 4246, 4366\n\nBrescia, Alfonso 4903, 4904\n\nBressart, Felix 1105\n\nBretherton, Howard 243, 289, 299, 464, 621, 665, 973, 1146, 1202, 1472, 1554, 1687, 1824, 1873, 1931, 1977, 2012, 2131, 2280, 2545, 2761, 2798, 2967, 2972, 3115, 3357, 3453, 3475, 3480, 3484, 3693, 3708, 3764, 3851, 4367, 4495, 4559, 4600, 4759, 4823, 4827, 4840, 4876, 4933\n\nBrewer, Betty 3624, 4932\n\nBrewer, Ilene 3453\n\nBrewer, Teresa 4349\n\nBrewster, Diane 371, 2139, 2871, 3200\n\nBrian, David 75, 378, 972, 1408, 1481, 1945, 1996, 3066, 3283, 3438, 4080, 4428, 4913\n\nBrian, Mary 596, 1685, 2352, 2918, 4651, 4744\n\nBriant, Shane 5026\n\nBrice, Monte 3916\n\nBrice, Pierre 84, 106, 1069, 1366, 2883, 3125, 3246, 3689, 4268, 4380, 4540, 5003\n\nBricker, George 42, 125, 2101, 2830, 4468\n\nBridge, Alan (Al) 205, 210, 271, 283, 362, 386, 391, 410, 429, 461, 481, 602, 612, 621, 623, 723, 799, 869, 873, 960, 1089, 1128, 1136, 1236, 1271, 1297, 1333, 1340, 1367, 1378, 1414, 1440, 1480, 1488, 1496, 1586, 1626, 1756, 1852, 1925, 1971, 2006, 2066, 2114, 2159, 2189, 2200, 2273, 2299, 2391, 2415, 2540, 2546, 2573, 2636, 2652, 2722, 2727, 2737, 2785, 2822, 2825, 2851, 2861, 2907, 2954, 3049, 3051, 3109, 3209, 3291, 3485, 3540, 3544, 3563, 3711, 3884, 3902, 3916, 4007, 4049, 4067, 4082, 4156, 4203, 4261, 4365, 4398, 4403, 4468, 4484, 4485, 4500, 4533, 4580, 4609, 4612, 4703, 4808, 4827, 4832, 4840, 4868, 4932, 4940, 4971, 5001, 5040\n\nBridge, Loie 3927\n\nBridges, Beau 2468, 3319\n\nBridges, James 119\n\nBridges, Jeff 192, 1828, 3252, 4578, 4930\n\nBridges, Lloyd 1, 117, 648, 759, 801, 1758, 1877, 2204, 2372, 2633, 2827, 3042, 3244, 3247, 3310, 3432, 3466, 3707, 3629, 3645, 3666, 3859, 3878, 4000, 4236, 4635, 4827, 4860, 4921\n\nBriggs, Harlan 1458, 1480, 2722, 2727, 3386, 4075, 4257, 4569, 4733\n\nBrill, Patti 1253, 1573, 2775\n\nBrimley, A. Wilford 333, 463, 900, 1021, 1219, 1611, 2206, 2930, 3577, 3620, 4880, 4975\n\nBrinckman, Nancy 3670\n\nBrinegar, Paul 1, 678, 704, 833, 972, 1423, 1487, 1500, 1880, 1987, 2623, 3234, 5030\n\nBrinley, Charles 270, 609, 855, 975, 1128, 1172, 1228, 1307, 1328, 1432, 1469, 1626, 1855, 1983, 2232, 2249, 2300, 2558, 2660, 2886, 2947, 2975, 2998, 3259, 3289, 3322, 3658, 3880, 4067, 4077, 4138, 4315, 4376, 4535, 4609, 4834, 4854, 4865, 4911, 5056\n\nBrissac, Virginia 200, 203, 562, 757, 939, 1084, 1136, 2031, 2184, 2878, 3195, 3355, 4336, 4359, 4673, 4767, 5020\n\nBristow, Gwen 2066\n\nBrito, Phil 4086\n\nBritt, Elton 323, 2176\n\nBritton, Barbara 49, 229, 1706, 1960, 2386, 3222, 3436, 3761, 4073, 4673, 4745\n\nBritton, Florence 3880\n\nBritton, Milt 3499\n\nBrix, Herman 1805, 2165, 2399; _see also_ Bennett, Bruce\n\nBrocco, Peter 55, 277, 371, 585, 895, 1166, 1225, 1721, 4151, 4229\n\nBrochero, Eduardo 118\n\nBrodel, Mary 1148\n\nBroder, Jack 1775\n\nBroderick, Helen 4121\n\nBrodie, Don 808, 1658, 1852, 1942, 2528, 3185, 3953, 4382\n\nBrodie, Kevin 2705\n\nBrodie, Steve 149, 214, 534, 569, 700, 786, 1263, 1668, 1736, 2618, 2919, 3093, 3383, 3611, 3646, 3863, 4132, 4204, 4386, 4467, 4503, 4989, 5040\n\nBrodney, Oscar 127, 807, 977, 1433, 1487, 3741, 4073, 4123\n\nBrolin, James 195, 860, 3034, 4134, 4858\n\nBrolin, Josh 4578\n\nBromberg, J. Edward 1953, 1960, 2031, 2586, 3374, 3684, 4257\n\nBromfield, John 356, 749, 1451, 1478, 1790, 3213\n\nBromley, Sheila 977, 1013, 1963, 2995, 4074, 5070; _see also_ Le Gay, Sheila; Mannors, Sheila\n\nBronson, Betty 5056\n\nBronson, Charles 463, 508, 563, 709, 738, 1014, 1161, 1422, 1442, 1731, 1734, 2065, 2497, 2631, 2898, 3336, 3640, 3857, 4343, 4387, 4738, 4895, 4979; _see also_ Buchinsky, Charles\n\nBronson, Lillian 934, 1256, 1428, 2162, 2990, 3608, 4064\n\nBrook, Claudio 1204, 1911, 2056, 3369, 4748, 5024\n\nBrook, Doris 428, 2392, 3073, 4986\n\nBrooke, Hillary 2412, 2415, 2790, 3544, 3951, 4074\n\nBrooke, Walter 4336\n\nBrooks, Foster 4739\n\nBrooks, Geraldine 563, 2051, 5075\n\nBrooks, Jean 422, 1430, 2149; _see also_ Kelly, Jeanne\n\nBrooks, Joan 4031\n\nBrooks, Leslie 3001\n\nBrooks, Louise 1222, 2998\n\nBrooks, Mel 385\n\nBrooks, Norma 390\n\nBrooks, Phyllis 3904\n\nBrooks, Rand 274, 470, 472, 693, 700, 749, 806, 841, 880, 950, 990, 1097, 1255, 1380, 1715, 1823, 1935, 2319, 2512, 2577, 2609, 2622, 2680, 2693, 2837, 3149, 3365, 3512, 3873, 3928, 3989, 4109, 4147, 4183, 4618, 4661, 4695, 4712, 4754, 5034, 5079\n\nBrooks, Richard 348, 2198, 3180\n\nBroome, Ray 4408, 4708\n\nThe Broome Brothers 4085\n\nBrophy, Edward 2756, 3351, 4626\n\nBrophy, Kevin 2443\n\nBrosnan, Pierre 3775\n\nBrower, Otto 441, 767, 898, 1088, 1187, 1300, 1307, 1519, 1592, 1781, 2352, 2388, 2737, 2947, 3075, 3710, 3745, 4067\n\nBrown, Barry 192, 506, 1641\n\nBrown, Blair 702, 2930\n\nBrown, Charles D. 3429, 3711, 4585\n\nBrown, Clarence 2013, 2211, 5045\n\nBrown, Dottye 4028\n\nBrown, Ewing 3808, 4010\n\nBrown, Harry Joe 1315, 2226, 2472, 2575, 2699, 2913, 3035, 3631, 3770, 4020, 4038, 4756\n\nBrown, Helen 42, 127, 1247\n\nBrown, James 17, 516, 693, 700, 1247, 1356, 1392, 1490, 1658, 1670, 1689, 2504, 2677, 3152, 4080, 4390, 4472, 4999, 5022, 5075\n\nBrown, Jim 1208, 2113, 2903, 3518, 4224\n\nBrown, Joe E. 710, 2458, 3859, 4029, 4254\n\nBrown, Johnny Mack 114, 135, 187, 199, 244, 297, 331, 380, 423, 428, 471, 476, 477, 486, 499, 582, 643, 726, 739, 783, 791, 848, 897, 899, 997, 1028, 1052, 1080, 1158, 1241, 1297, 1325, 1340, 1362, 1367, 1443, 1447, 1503, 1529, 1540, 1546, 1639, 1687, 1732, 1724, 1797, 1868, 2169, 2171, 2180, 2250, 2255, 2259, 2271, 2273, 2280, 2282, 2298, 2306, 2378, 2424, 2457, 2512, 2526, 2539, 2601, 2632, 2679, 2682, 2761, 2861, 2863, 2873, 2926, 2949, 2969, 2988, 3002, 2052, 3142, 3220, 3227, 3229, 3231, 3261, 3264, 3291, 3365, 3425, 3452, 3581, 3650, 3804, 3823, 3848, 3889, 3901, 3938, 3943, 4006, 4103, 4118, 4153, 4155, 4261, 4292, 4299, 4301, 4501, 4514, 4520, 4550, 4559, 4640, 4656, 4691, 4698, 4706, 4803, 4808, 4813, 4826, 4828, 4827, 4894, 4971\n\nBrown, Karl 633, 852, 1827, 4643\n\nBrown (Browne), Lucille 264, 493, 725, 897, 2121, 2207, 2212, 3242, 4282, 4317, 4592, 4839\n\nBrown, Peter 172, 190, 1840, 1905, 4361, 4417\n\nBrown, Phil 3096, 3735, 4688\n\nBrown, Stanley 129, 386, 1100, 1298, 1449, 1579, 1768, 2269, 2547, 2973, 2983, 3003, 3109, 3449, 3505, 3707, 3520, 3565, 4238, 4391, 4723, 4736, 4980\n\nBrown, Tom 919, 1431, 1754, 1957, 2693, 2748, 2922, 3211\n\nBrown, Vanessa 318, 1293, 1427\n\nBrown, Wally 1256, 1573, 2314, 4830, 4937\n\nBrowne, Kathie 378, 1922, 3438\n\nBrowne, Reg 4010\n\nBrowne, Reno 14, 1279, 1800, 3262, 3331, 3460, 3802, 4813; _see also_ Blair, Reno\n\nBrowne, Roscoe Lee 883\n\nBrownlee, Frank 107, 129, 154, 1052, 1974, 2033, 2249, 2367, 2458, 2514, 2571, 2667, 3450, 3600, 3805, 3989, 4027, 4103, 4271, 4358, 4454, 4498, 4708\n\nBruce, David 642, 2620, 3538, 3684, 3711, 4421, 5048, 5064\n\nBruce, Ed 1147, 2190, 2650\n\nBruce, Kate 3743\n\nBruce, Nigel 1949, 4494\n\nBruce, Virginia 200, 2344, 2918\n\nBruckman, Clyde 4598, 4654\n\nBruckner, William 4184\n\nBrummell, Beau 4354\n\nBrunette, Fritzie 3650, 4798, 4803\n\nBrunetti, Argentina 115, 119, 519, 602, 1182, 1214, 1265, 1293, 3608, 3690, 3783, 3857, 4098, 4233, 4373\n\nBryan, Jane 714\n\nBryant, Buel 2547, 2975, 3393, 4156, 4313, 5104\n\nBryant, Jan 871, 893, 1373, 3804, 3901\n\nBryant, Joyce 13, 1327, 4549, 4554\n\nBryant, Nana 642, 1246, 2919, 2936, 2938, 4745\n\nBryant, Theona 2659\n\nBryant, William 93, 212, 370, 523, 741, 1007, 1770, 1832, 1922, 2333, 2488, 2559, 2616, 2698, 3152, 3422, 3428, 3962, 4969\n\nBryar, Claudia 192, 3055\n\nBryar, Paul 125, 620, 632, 691, 1263, 1719, 1945, 2482, 2665, 3093, 3208, 3563, 3864, 3912, 4086, 5049\n\nBrynner, Yul 20, 2002, 2141, 2497, 3391, 4859\n\nBuccella, Grazia 4738\n\nBuccholz, Horst 2497\n\nBuchanan, Edgar 25, 129, 292, 329, 552, 692, 748, 808, 835, 972, 979, 1070, 1082, 1093, 1096, 1205, 1364, 1421, 1640, 1727, 2139, 2436, 2482, 2513, 2515, 2598, 2626, 2989, 3219, 3290, 3310, 3355, 3435, 3622, 3686, 3750, 3808, 3815, 3891, 3906, 3959, 3991, 4074, 4286, 4456, 4470, 4771, 4802, 4921, 4966, 5082\n\nBuchinksy, Charles 100, 3333, 3500, 4727; _see also_ Bronson, Charles\n\nBuchs, Julio 570, 1287\n\nBuck (dog) 626, 630, 844, 1261\n\nBuckley, Betty 5029\n\nBuckley, Kay 3233, 4099, 4468\n\nBuckner, Robert 1127, 1441, 1594, 2462, 2864, 3387, 3471, 3711, 4742\n\nBuckner, Teddy 1422\n\nBucko, Buck 49, 287, 295, 379, 395, 446, 564, 607, 683, 1128, 1288, 1677, 1928, 2121, 2524, 2558, 2635, 2699, 3555, 3969, 3989, 4089, 4204, 4614, 4702, 4837, 5009\n\nBucko, Ralph 338, 461, 464, 800, 1017, 1090, 1474, 1546, 1687, 1730, 1907, 1971, 2015, 2263, 2274, 2288, 2419, 2457, 2527, 2915, 3015, 3188, 3263, 3300, 3478, 3589, 4011, 4153, 4199, 4208, 4304, 4364, 4365, 4683, 4865\n\nBucko, Roy 49, 183, 241, 338, 411, 493, 634, 785, 800, 853, 935, 951, 975, 984, 1017, 1059, 1090, 1128, 1130, 1269, 1288, 1296, 1474, 1503, 1513, 1546, 1627, 1677, 1687, 1730, 1824, 1920, 1928, 1934, 1935, 2014, 2111, 2250, 2288, 2368, 2404, 2419, 2421, 2448, 2527, 2558, 2589, 2847, 2908, 2915, 2957, 2978, 2986, 3015, 3188, 3241, 3263, 3314, 3353, 3458, 3478, 3480, 3525, 3552, 3555, 3589, 3673, 3693, 3834, 3969, 3989, 4000, 4011, 4047, 4089, 4099, 4108, 4135, 4153, 4190, 4204, 4208, 4289, 4304, 4364, 4365, 4402, 4483, 4614, 4660, 4683, 4701, 4734, 4837, 4841, 4855, 4861, 4865, 5009, 5018, 5027, 5037, 5039, 5061\n\nBuell, Jed 4270\n\nBuetel, Jack 293, 1762, 2038, 2713, 2943, 3608\n\nBuffalo Bill, Jr. 180, 240, 305, 410, 461, 722, 1008, 1303, 1394, 1861, 1869, 1907, 2274, 2297, 2358, 2660, 2745, 2766, 3075, 3087, 3153, 3188, 3242, 3270, 3272, 3292, 3462, 3486, 3501, 3555, 3658, 4282, 4317, 4376, 4406, 4683, 4798, 4852, 4861; _see also_ Wilsey, Jay\n\nBuffington, Adele 152, 204, 470, 857, 893, 1158, 1362, 1432, 1557, 1744, 1796, 1865, 2040, 2573, 3000, 3153, 3262, 3802, 3943, 4275, 4690, 4813, 4828, 4847, 4929; _see also_ Bowers, Jess\n\nBugner, Joe 3819\n\nBujold, Genevieve 96, 1109, 3313\n\nBuka, Donald 2789\n\nBull, Richard 1880, 1943, 2305, 2374, 2377, 4113, 4637\n\nBullock, Boris 12, 1651, 2264, 3292; _see also_ Barrymore, William; Carson, Kit\n\nBullock, Walter 1601\n\nBuono, Victor 421, 1422, 2694, 5028\n\nBupp, Tommy 130, 714, 1883, 1900, 2523, 2954, 3293, 3548, 3549, 4279\n\nBurbridge, Betty 51, 295, 473, 712, 758, 797, 810, 957, 1449, 1493, 1512, 1595, 1839, 1860, 1918, 1975, 1981, 2086, 2275, 2428, 2537, 2702, 2786, 2802, 2866, 2928, 2935, 2968, 3027, 3163, 3194, 3325, 3394, 3453, 3454, 3565, 3626, 3708, 3709, 4024, 4082, 4372, 4391, 4480, 4495, 4649, 4655, 4826, 4876, 4954, 5037\n\nBurgess, Dorothy 1972, 2158, 2180, 2423, 3658\n\nBurgess, Helen 3126\n\nBurke, Billie 3776\n\nBurke, Caroline 2734\n\nBurke, Delta 4873\n\nBurke, Denny 4889\n\nBurke, James 53, 166, 602, 626, 757, 832, 1039, 1883, 2147, 2416, 3285, 3409, 3499, 3634, 4425, 4442, 4470, 3394, 4745, 4792\n\nBurke, Kathleen 2774, 3573, 4203\n\nBurke, Paul 4927\n\nBurke, Samson (Sammy) 1955\n\nBurke, Walter 1140, 1945, 2990, 3127, 4661, 4210\n\nBurlinson, Tom 2538, 3396, 3858\n\nBurnett, Don 1485\n\nBurnett, W.R. 282, 2250, 2252, 3692, 3777, 4971\n\nBurnette, Smiley 9, 183, 234, 283, 291, 299, 323, 337, 399, 410, 418, 424, 447, 465, 542, 615, 624, 662, 779, 797, 818, 880, 922, 1046, 1059, 1143, 1213, 1309, 1345, 1404, 1460, 1497, 1512, 1574, 1595, 1606, 1725, 1728, 1806, 1815, 1818, 1821, 1827, 1915, 1916, 1938, 1964, 1983, 2068, 2103, 2105, 2124, 2175, 2176, 2178, 2179, 2189, 2217, 2262, 2361, 2317, 2529, 2632, 2636, 2648, 2701, 2737, 2850, 2872, 2874, 2885, 2894, 2939, 2991, 3006, 3063, 3075, 3095, 3163, 3167, 3168, 3176, 3187, 3209, 3228, 3233, 3253, 3329, 3358, 3408, 3433, 3434, 3464, 3490, 3919, 3922, 3926, 3940, 3974, 3978, 4044, 4052, 4053, 4082, 4127, 4154, 4166, 4167, 4199, 4272, 4296, 4496, 4506, 4592, 4610, 4643, 4655, 4812, 4818, 4842, 4886, 5005, 5056\n\nBurns, Bob (Bazooka) 278, 3409, 4803\n\nBurns, Bob 37, 183, 231, 337, 413, 437, 440, 493, 511, 522, 566, 610, 615, 631, 682, 731, 753, 777, 827, 848, 930, 975, 1254, 1282, 1298, 1307, 1323, 1329, 1345, 1474, 1489, 1491, 1513, 1542, 1553, 1574, 1626, 1678, 1679, 1728, 1740, 1781, 1796, 1862, 1898, 1920, 2151, 2152, 2165, 2226, 2247, 2257, 2285, 2299, 2300, 2311, 2353, 2363, 2432, 2480, 2514, 2530, 2549, 2599, 2642, 2673, 2701, 2722, 2736, 2765, 2786, 2915, 2967, 2968, 2975, 3027, 3035, 3036, 3075, 3080, 3101, 3111, 3126, 3164, 3165, 3170, 3173, 3187, 3094, 3260, 3263, 3270, 3300, 3314, 3427, 3434, 3452, 3478, 3486, 3504, 3549, 3551, 3561, 3564, 3614, 3626, 3649, 3664, 3669, 3674, 3680, 3742, 3926, 4009, 4022, 4048, 4096, 4104, 4181, 4190, 4225, 4248, 4297, 4304, 4320, 4347, 4358, 4372, 4396, 4423, 4454, 4558, 4572, 4597, 4599, 4649, 4674, 4683, 4712, 4728, 4759, 4776, 4848, 4861, 4862, 4938, 4954, 4980, 4995, 5056\n\nBurns, David 289\n\nBurns, Ed 3658\n\nBurns, Forrest 3523, 4673, 4674, 5096\n\nBurns, Fred 37, 108, 143, 144, 153, 201, 327, 337, 440, 442, 458, 479, 493, 496, 511, 611, 623, 634, 722, 790, 923, 986, 1038, 1055, 1089, 1138, 1143, 1162, 1172, 1307, 1335, 1369, 1426, 1432, 1462, 1474, 1512, 1558, 1651, 1751, 1781, 1799, 1814, 1818, 1854, 1916, 1928, 1964, 1973, 1976, 1978, 1980, 1982, 2032, 2154, 2163, 2192, 2247, 2257, 2263, 2275, 2300, 2398, 2399, 2403, 2514, 2530, 2600, 2603, 2642, 2660, 2699, 2703, 2764, 2778, 2786, 2826, 2850, 2860, 2886, 2908, 2968, 2975, 2998, 3048, 3075, 3105, 3163, 3173, 3194, 3228, 3269, 3289, 3291, 3322, 3427, 3491, 3520, 3529, 3550, 3574, 3582, 3626, 3665, 3675, 3797, 3805, 3827, 3904, 3922, 4037, 4059, 4082, 4127, 4193, 4199, 4202, 4206, 4208, 4238, 4308, 4314, 4321, 4364, 4401, 4410, 4460, 4494, 4512, 4516, 4558, 4609, 4649, 4655, 4683, 4732, 4744, 4773, 4785, 4854, 4861, 5040, 5041\n\nBurns, Marion 974, 1605, 2862, 3036\n\nBurns, Michael 3712, 4661\n\nBurns, Paul E. 25, 64, 100, 127, 248, 279, 282, 292, 357, 514, 757, 930, 1136, 1140, 1248, 1364, 1433, 1481, 1741, 1719, 1831, 1941, 2208, 2308, 2463, 2482, 2485, 2543, 2641, 2677, 2727, 2790, 3005, 3020, 3150, 3185, 3344, 3355, 3362, 3384, 3630, 3669, 3702, 3846, 3891, 3976, 4005, 4058, 4098, 4099, 4151, 4198, 4362, 4419, 4470, 4635, 4673, 4725, 4792, 4849, 4932, 4940\n\nBurr, Raymond 462, 501, 786, 843, 1633, 1936, 2486, 2502, 2789, 3054, 3762, 3858, 4132, 4392, 4970\n\nBurrud, Bill(y) 865, 1630, 2686\n\nBurson, Wayne 698, 841, 2074, 2609, 2693, 2842, 2910, 3751, 4228, 4478, 4582, 4726, 4955\n\nBurstyn, Ellen 3872\n\nBurt, Benny 842, 1658, 1802, 2013, 4069\n\nBurt, Charlene 305\n\nBurt, William 3482, 4570\n\nBurtis, James 145, 147, 1043, 2223, 4142, 4284, 4289, 4803\n\nBurton, Frederick 328, 514, 633, 755, 1427, 1432, 1577, 1767, 2516, 3895, 3898, 3899, 4257\n\nBurton, Lee 85, 268, 532, 920, 1117, 1163, 2441, 3604, 3628, 4348, 4720; _see also_ Lollobrigida, Guido\n\nBurton, Richard 1963\n\nBurton, Robert 501, 520, 843, 966, 1078, 1129, 1782, 1892, 1995, 2065, 2074, 2308, 2557, 2842, 3297, 3363, 3542, 3751, 3791, 3954, 4235, 4244, 4478\n\nBusch, Mae 1260, 3442, 5009\n\nBusch, Niven 654, 1111, 2691, 3195, 4538, 4851\n\nBusey, Gary 245, 1107, 1549, 1864, 1930, 2498, 2721\n\nBush, Billy Green 628, 822, 909, 2000, 2490, 2685, 4449\n\nBush, James 159, 258, 642, 1576, 1964, 2124, 2515, 2502, 2618, 2990, 3667, 4184, 4811, 4950\n\nBush, Johnny 266\n\nBushelman, John 521\n\nBushman, Francis X. 104, 1178, 3899\n\nBushman, Francis X., Jr. (Ralph) 668, 924, 1925, 2192, 2344, 4750, 4797, 4866\n\nBuster, Budd 67, 137, 140, 205, 214, 244, 258, 291, 297, 336, 338, 340, 341, 342, 345, 382, 384, 395, 422, 427, 448, 464, 479, 495, 573, 597, 615, 634, 687, 689, 711, 726, 728, 739, 753, 775, 781, 792, 797, 856, 872, 917, 929, 956, 1028, 1045, 1047, 1051, 1137, 1145, 1167, 1193, 1281, 1282, 1296, 1327, 1332, 1345, 1370, 1448, 1459, 1464, 1473, 1491, 1497, 1503, 1504, 1531, 1553, 1626, 1674, 1681, 1683, 1732, 1738, 1749, 1758, 1797, 1798, 1803, 1812, 1821, 1859, 1873, 1898, 1918, 2032, 2132, 2151, 2177, 2251, 2256, 2257, 2263, 2286, 2290, 2315, 2363, 2386, 2412, 2413, 2420, 2425, 2427, 2481, 2552, 2572, 2594, 2640, 2652, 2712, 2759, 2822, 2847, 2873, 2879, 2954, 2965, 2974, 2999, 3029, 3044, 3062, 3097, 3098, 3155, 3176, 3209, 3227, 3228, 3238, 3288, 3302, 3388, 3413, 3475, 3478, 3497, 3500, 3553, 3568, 3574, 3584, 3656, 3666, 3678, 3684, 3728, 3744, 3771, 3799, 3801, 3824, 3825, 3866, 3884, 3913, 3924, 3939, 3940, 3942, 4022, 4026, 4030, 4081, 4110, 4144, 4152, 4206, 4269, 4272, 4273, 4298, 4303, 4307, 4319, 4382, 4391, 4393, 4402, 4409, 4458, 4489, 4490, 4493, 4501, 4549, 4555, 4556, 4581, 4647, 4691, 4702, 4711, 4733, 4777, 4811, 4815, 4820, 4827, 4839, 4846, 4855, 4858, 4882, 4887, 4938, 4952, 4955, 4961, 5014, 5018, 5037, 5104\n\nButch and Buddy 2163, 2526\n\nButkus, Dick 751\n\nButler, Champ 2713\n\nButler, David 597, 819, 3692, 4621, 4900\n\nButler, Dean 2374, 2375, 2377\n\nButler, Frank 602, 2811, 3276, 4148, 4364, 4671, 4890\n\nButler, Hugo 4058\n\nButler, John K. 74, 299, 395, 605, 1134, 1141, 1164, 1458, 1849, 1873, 2357, 2534, 2727, 2938, 3176, 3228, 3510, 3563, 3578, 3904, 4197, 4213, 4470, 4686\n\nButler, Lois 1876\n\nButler, Robert 1942, 2156, 3739\n\nButler, Roy 368, 542, 599, 1093, 1272, 1298, 1345, 1360, 1422, 1456, 1461, 1497, 1523, 1687, 1915, 1991, 2123, 2161, 2169, 2175, 2294, 2435, 2530, 2798, 2873, 3002, 3234, 3357, 3370, 3388, 3670, 3702, 3804, 3866, 4081, 4114, 4154, 4301, 4378, 4503, 4568, 4588, 4725, 4759, 4819, 4820\n\nButterworth, Charles 2344\n\nButtons, Red 509, 4101\n\nButtram, Pat 104, 250, 304, 389, 409, 1510, 1523, 1890, 2632, 2711, 2805, 3447, 3661, 4166, 4285, 4692, 4757\n\nBuzzanca, Lando 1382, 3301\n\nBuzzell, Edward 1577, 4739\n\nBuzzi, Ruth 122\n\nByington, Spring 160, 1093, 3022\n\nByrd, Ralph 431, 1947, 2223, 2586, 2831, 3339, 4115, 4256, 4383, 4558, 4729\n\nByrnes, Edd 98, 3059, 3309, 3878, 5055\n\nByrnes, Jim 487, 996, 1707, 1754, 1946, 2092, 2093, 2188, 2487, 2658, 3282, 3661, 3798\n\nByron, Arthur 3179\n\nByron, Carol 190\n\nByron, Jean 2046\n\nByron, Marion 197, 510, 4029, 4254\n\nByron, Walter 892, 1013, 1464\n\nCaan, James 96, 816, 1212, 1575, 2059, 2379\n\nCabanne, Christy 187, 975, 2122, 2223, 2528, 2603, 2717, 2940, 3562, 3908, 4419\n\nThe Cabin Kids 3625\n\nCabot, Bruce 89, 200, 293, 319, 373, 674, 741, 808, 1127, 1257, 1361, 1490, 1706, 2213, 2267, 2452, 2626, 3096, 3561, 3567, 3745, 3821, 3899, 3872, 4472, 4638, 4790, 4932\n\nCabot, Sebastian 371, 1150, 2052, 4267, 4856\n\nCabot, Susan 258, 1184, 1402, 1747, 3423, 4450\n\nCactus Mack _see_ Mack, Cactus\n\nCady, Frank 1828, 1989, 2668, 3783, 4435, 4798, 5084\n\nCaffey, Michael 1086, 1770, 3878\n\nCagney, James 132, 1436, 2096, 2864, 3636, 4548\n\nCagney, Jeanne 4439, 4472\n\nCahn, Edward L. 1356, 1377, 1466, 1502, 1670, 1689, 1709, 2249, 2818, 2868\n\nCaiano, Mario 19, 1305, 2390, 3514, 3515\n\nCaillou, Alan 528, 3283\n\nCain, Ace 945, 2003, 2263, 3413, 3528, 3939, 4307, 4448\n\nCain, James M. 587, 4121\n\nCaine, Georgia 271, 1127, 2064, 2940, 3490, 3711, 4932\n\nCairns, Sally 856, 2134\n\nCaldwell, Hank 2301\n\nCalhern, Louis 159, 1093, 2064, 3319\n\nCalhoun, Alice 4260\n\nCalhoun, Rory 111, 114, 195, 373, 572, 972, 1129, 1423, 1675, 1892, 2618, 2663, 2710, 2739, 2815, 3150, 3191, 3287, 3337, 3386, 3432, 3535, 3583, 3676, 3698, 3849, 3909, 4073, 4411, 4538, 4682, 4795, 5052, 5065\n\nCallahan, Foxy 49, 146, 235, 289, 334, 345, 369, 459, 564, 610, 634, 872, 1146, 1155, 1296, 1309, 1370, 1505, 1546, 1920, 2003, 2276, 2286, 2421, 2514, 3264, 3728, 3826, 4458, 4559, 4573, 4823, 4826, 4853, 5018\n\nCallahan, Margaret 2223\n\nThe Callahan Brothers 4081\n\nCallam, Alex 925, 3084, 4401\n\nCallan, Michael 674, 1133, 2498, 4335\n\nCalleia, Joseph 43, 200, 497, 1420, 2005, 2064, 2351, 2382, 2722, 3023, 3556, 4538, 5032\n\nCallejo, Cecilia 758, 3353, 3684\n\nCalvert, E.H. 441, 826, 1576, 2732, 2918, 4744, 4837, 4950\n\nCalvert, John 1593, 2296, 3385\n\nCalvert, Vane 77, 1489, 2102, 3239, 3968, 4843\n\nCalvet, Corinne 114, 1263, 2720, 3134, 3150, 3201\n\nCalvin, Henry 524, 3871, 5086, 5098\n\nCalvo, Armando 61, 1185, 1351, 1954, 2390, 3117, 3514, 3515, 3516\n\nCamardiel, Roberto 18, 19, 220, 296, 316, 694, 1030, 1115, 1381, 205, 2532, 2904, 3346, 4003, 4679\n\nCambre, Del 126\n\nCamden, Joan 4148\n\nCameron, Jeff 1418, 4444\n\nCameron, Rod 282, 303, 475, 486, 516, 565, 574, 691, 762, 934, 1028, 1392, 1403, 1450, 1456, 1675, 1695, 1852, 2039, 2081, 2199, 2437, 2831, 2851, 2884, 3032, 3046, 3112, 3132, 3276, 3357, 3365, 3436, 3437, 3478, 3499, 3534, 3684, 3690, 3705, 3848, 4060, 4074, 4099, 4112, 4118, 4171, 4380, 4556, 4766, 5020, 5044\n\nCamp, Joe 1808\n\nCampana, Nina 129, 354, 630, 642, 2012, 2288, 2948, 3071, 3603, 4115, 4201, 4599\n\nCampanella, Frank 1879\n\nCampanella, Joseph 1655, 2327, 2663\n\nCampbell, Alexander 3234, 4300, 4579\n\nCampbell, Colin 4070\n\nCampbell, Glenn 4576, 4679\n\nCampbell, Kate 814, 1537, 1557\n\nCampbell, Paul 9, 379, 542, 1059, 1460, 2189, 3063, 3168, 3940, 3976, 4154, 4730\n\nCampbell, Peggy 311, 4138, 4863\n\nCampbell, R. Wright 1357, 1671, 3199\n\nCampbell, Virginia 4635\n\nCampbell, William 189, 1234, 2462, 2565, 2676, 2754, 3740, 3821\n\nCampeau, Frank 351, 460, 626, 653, 735, 1222, 1241, 1300, 1572, 1817, 1931, 1972, 2771, 2133, 2180, 2207, 2536, 2712, 3016, 3561, 3971, 4352, 4898\n\nCampillo, Anita 2548\n\nCampo, Dell 1089, 3245\n\nCampos, Rafael 119, 1770, 2351, 2859, 3378, 3731, 4255, 4457\n\nCampos, Victor 2598, 3378, 3731, 4255, 4457\n\nCamron, Rocky 632, 1224, 1330, 1789, 2957, 3599, 4017, 4032, 4987; _see also_ Alsace, Gene; Coburn, Buck\n\nCanada, Roy 1597\n\nCanary, David 2047, 3146\n\nCandido, Candy 874, 3134\n\nCandy and Coco _see_ Hall, Russell \"Candy\"; Heimel, Otto \"Coco\"\n\nCandy, John 4765\n\nCane, Charles 73, 277, 516, 599, 977, 1236, 1313, 1487, 1502, 1664, 1676, 2416, 2813, 2815, 3066, 3690\n\nCannell, Stephen J. 2800, 3747\n\nCannon, J.D. 640, 849, 1832, 2305, 2437, 3687\n\nCanova, Judy 611, 710, 1923, 2308, 2856, 3916, 4675\n\nCanova, Tweeny 2308, 4675\n\nCanova, Zeke 1974\n\nCantinflas (Mario Moreno) 3144\n\nCanty, Marietta 4072\n\nCanutt, Joe 1265, 2696\n\nCanutt, Tap 883, 2301, 4163\n\nCanutt, Yakima 28, 201, 240, 262, 411, 414, 498, 499, 609, 634, 646, 666, 667, 721, 722, 753, 886, 926, 930, 951, 974, 1008, 1044, 1087, 1088, 1333, 1334, 1340, 1467, 1513, 1553, 1557, 1558, 1647, 1730, 1751, 1822, 1845, 1873, 1898, 1952, 2007, 2082, 2086, 2124, 2129, 2212, 2247, 2297, 2300, 2301, 2359, 2367, 2399, 2432, 2434, 2479, 2523, 2548, 2660, 2766, 2778, 2802, 2844, 2857, 2867, 2945, 2954, 2998, 3015, 3026, 3027, 3036, 3111, 3165, 3172, 3176, 3188, 3256, 3259, 3269, 3451, 3456, 3462, 3476, 347, 3482, 3487, 3488, 3517, 3527, 3549, 3572, 3617, 3665, 3680, 3708, 3709, 3745, 3805, 3820, 3852, 4007, 4018, 4033, 4100, 4125, 4248, 4282, 4317, 4573, 4605, 4649, 4708, 4728, 4732, 4786, 4822, 4852, 4854, 4899, 4946, 4982, 4995, 5037, 5040, 5096, 5104\n\nCapitani, Giorgio 3659\n\nCapitani, Remo 3\n\nCapshaw, Kate 2721, 3205\n\nCapuano, Luigi 24, 2499, 5092, 5095\n\nCapucine 2829, 3336\n\nCapulina 1223\n\nCard, Bob 183, 337, 442, 493, 631, 773, 853, 1320, 1474, 1488, 1679, 1751, 1827, 1860, 1969, 1993, 2399, 2421, 2432, 2802, 2947, 2968, 3075, 3087, 3109, 3135, 3173, 3491, 3528, 3650, 3834, 4135, 4190, 4226, 4389, 4649, 4667, 4683, 4695, 4954, 5027\n\nCard, Ken 437, 1270, 1679, 2032, 3937, 4572, 4882\n\nCard, Virginia 2412\n\nCardenas, Elsa 168, 938, 1393, 1560, 4223, 4934\n\nCardi, Pat 86\n\nCardinale, Claudia 2326, 2898, 3180\n\nCardona, Rene 181, 1190, 1223, 1729, 2063, 2345, 2346, 2574, 2709, 3124, 3189, 3832, 4438\n\nCardona, Rene, Jr. 1223, 1739, 2345, 3189, 4338\n\nCardos, John \"Bud\" 1010, 1278, 1352, 2140, 3638, 3732, 4041\n\nCardwell, James 134, 648, 1141, 1790, 3563, 3691, 4239\n\nCarewe, Edwin 4071\n\nCarey, Ed 156, 1687, 2298, 2356, 3601, 4493\n\nCarey, Harry 8, 246, 298, 432, 434, 445, 546, 688, 944, 1089, 1187, 1550, 2203, 2212, 2223, 2249, 2288, 2554, 2794, 2799, 2940, 3153, 3166, 3179, 3323, 3653, 3750, 3817, 4072, 4145, 4215, 4359, 4360, 4403, 4708, 4961, 5009\n\nCarey, Harry, Jr. 71, 195, 223, 237, 319, 339, 402, 594, 677, 697, 720, 832, 1015, 1238, 1441, 1535, 1692, 1701, 2087, 2236, 2443, 2553, 2818, 2897, 2911, 2938, 3195, 3223, 3283, 3323, 3517, 3522, 3536, 3593, 3635, 3690, 3752, 3791, 3798, 3814, 3816, 3897, 3990, 4223, 4224, 4360, 4453, 4563, 4626, 4638, 4764, 4792, 4800, 4968, 5030\n\nCarey, Macdonald 523, 692, 807, 842, 1640, 1775, 2557, 2924, 4151, 4168\n\nCarey, Mary Jane 458\n\nCarey, Michele 93, 1106, 1212, 2333, 3739\n\nCarey, Olive (Golden) 43, 339, 434, 1249, 1702, 2756, 2797, 3096, 3640, 4626, 4708\n\nCarey, Philip (Phil) 190, 363, 597, 685, 843, 1644, 1695, 1778, 1844, 2504, 2617, 2768, 2955, 3397, 3749, 4080, 4338, 4361, 4457, 4472, 5038\n\nCarey, Timothy 1697, 2241, 2748, 2901, 3518, 3755, 4431, 4794\n\nCarle, Richard 145, 147, 1153, 1917, 2232, 2689, 2774, 3695, 4254, 4309, 4494\n\nCarleton, Claire 207, 841, 1096, 1293, 1402, 1450, 1647, 1693, 1923, 2482, 2659, 3436, 3720, 4327, 4582\n\nCarleton, George 2597, 2806, 2838, 3178, 3499, 3939, 4257\n\nCarlin, George 4169\n\nCarlin, Jean 658, 1542, 3942, 4026, 4969\n\nCarlin, Lynn 4969\n\nCarlisle, Mary 2877, 3626\n\nCarlson, June 1803, 2026, 3202\n\nCarlson, Richard 1423, 1947, 2112, 2187, 3676, 3768, 4693\n\nCarlyle, Patrick (Pat) 616, 2003, 2101\n\nCarlyle, Richard 1570, 2005, 3210, 3665, 4865\n\nCarmel, Roger G. 71\n\nCarmen, Jean 140, 468, 892, 1979, 3977; _see also_ Thayer, Julia\n\nCarmen, Jeanne 4368, 4784\n\nCarmichael, Hoagy 648, 4428\n\nCarmineo, Guiliano 568; _see also_ Ascott, Anthony\n\nCarney, Alan 26, 1573, 2829, 4969\n\nCarnovsky, Morris 4845\n\nCaro, Alicia 705, 750, 3765\n\nCarol, Sue 1970, 2422\n\nCaron, Leslie 2494\n\nCarpenter, Carleton 3954, 4725\n\nCarpenter, Frank (Red) 1957, 2959\n\nCarpenter, Horace B. 108, 140, 141, 201, 235, 244, 297, 323, 338, 344, 411, 422, 448, 510, 624, 634, 665, 682, 731, 779, 792, 793, 813, 836, 973, 986, 1127, 1137, 1172, 1258, 1307, 1322, 1345, 1362, 1367, 1374, 1447, 1474, 1486, 1505, 1513, 1546, 1553, 1574, 1626, 1646, 1677, 1678, 1681, 1683, 1687, 1751, 1818, 1904, 1974, 1983, 2003, 2072, 2119, 2121, 2129, 2186, 2220, 2257, 2280, 2298, 2299, 2362, 2391, 2392, 2403, 2410, 2412, 2424, 2427, 2522, 2529, 2534, 2600, 2673, 2701, 2826, 2850, 2877, 2909, 2928, 2951, 2966, 2968, 2978, 3015, 3029, 3036, 3044, 3051, 3086, 3092, 3159, 3176, 3177, 3188, 3254, 3256, 3259, 3264, 3266, 3270, 3332, 3390, 3408, 3451, 3459, 3465, 3474, 3555, 3581, 3589, 3592, 3601, 3636, 3650, 3657, 3675, 3704, 3707, 3709, 3805, 3820, 3826, 3834, 3844, 3893, 3894, 3968, 3969, 3970, 4000, 4032, 4037, 4077, 4128, 4138, 4152, 4155, 4181, 4205, 4208, 4238, 4299, 4303, 4315, 4358, 4391, 4427, 4483, 4501, 4512, 4554, 4592, 4667, 4681, 4685, 4701, 4734, 4767, 4810, 4822, 4823, 4827, 4839, 4853, 4855, 4862, 4863, 4883, 4884, 4899, 4926, 4972, 4995, 5001, 5027, 5050, 5061, 5103\n\nCarpenter, John (Johnny) 23, 213, 433, 444, 680, 692, 795, 807, 914, 1184, 1215, 1330, 1957, 2109, 2252, 2301, 2396, 2761, 2841, 2959, 3310, 3337, 3344, 3599, 3707, 4010, 4017, 4028, 4154, 4409, 4487, 5048; _see also_ Forbes, John\n\nCarpenter, Ken 4066\n\nCarpenter, Virginia 554, 2425, 2977, 3588\n\nCarpi, Tito 991, 1030, 1286, 1351, 1954, 1955, 2049, 2016, 2354, 3059, 3352, 3372, 3403, 4366\n\nCarr, June 1461, 3999\n\nCarr, Marian 1551, 2834\n\nCarr, Mary 1306, 1320, 1388, 1434, 1678, 2654, 2928, 4683, 4816\n\nCarr, Michael 830, 1809\n\nCarr, Nat 3308, 4697\n\nCarr, Richard 1832, 2518, 2990, 4976\n\nCarr, Stephen 896, 2596, 2970, 4820\n\nCarr, Thomas 51, 235, 274, 349, 671, 712, 783, 794, 896, 940, 984, 1063, 1215, 1272, 1314, 1415, 1748, 1939, 2035, 2512, 2596, 2928, 2970, 3259, 3298, 3326, 3524, 3613, 3707, 4028, 4124, 4234, 4462, 4491, 4659, 4781, 4821, 5039\n\nCarr, Trem 824\n\nCarradine, David 59, 532, 590, 1500, 1610, 1832, 1879, 2154, 2155, 2236, 2429, 2443, 2488, 2627, 2657, 2671, 2987, 4223, 4741, 5060\n\nCarradine, John 46, 248, 315, 339, 514, 523, 595, 720, 952, 1144, 1165, 1352, 1458, 1510, 1610, 1864, 1871, 2031, 2048, 2561, 2627, 2646, 2838, 2929, 3179, 3186, 3245, 3847, 3857, 3904, 4100, 4160, 4337, 4392, 4442, 4579, 4849, 4895, 4900\n\nCarradine, Keith 58, 884, 1220, 1583, 1694, 1864, 2090, 2154, 2236, 2443, 2625, 2987, 4930\n\nCarradine, Robert 883, 1705, 2443\n\nCarre, Bart (Bartlett) 262, 276, 458, 616, 721, 729, 753, 1303, 1310, 1334, 1687, 2364, 2744, 2799, 2954, 3268, 3288, 3292, 3490, 4321, 4570, 4837, 4866\n\nCarrera, Barbara 2429, 2619\n\nCarreras, Michael 3726\n\nCarricart, Robert 373, 401, 1133, 1734, 2174, 2587, 4738\n\nCarrillo, Leo 79, 158, 254, 957, 1444, 1471, 1516, 1521, 1562, 1568, 1571, 1572, 2180, 2638, 2688, 3031, 3450, 3540, 3720, 3912, 4419, 4593, 4654, 4690, 4750, 5032\n\nCarrol, Regina 388, 1278, 2039, 4626\n\nCarroll, Alma 3042, 3894, 4465, 5035\n\nCarroll, Anne 4618\n\nCarroll, John 91, 176, 277, 1026, 1247, 1290, 1577, 1865, 2878, 3096, 3530, 3610, 4212, 5011, 5033\n\nCarroll, Leo G. 4240\n\nCarroll, Madeleine 2831\n\nCarroll, Pat 536, 3753\n\nCarroll, Sidney 317\n\nCarroll, Virginia 207, 389, 893, 1273, 1836, 2233, 2607, 2869, 3002, 3074, 3159, 3232, 3481, 4074, 4151, 4256, 4559\n\nCarson, Jack 482, 753, 1084, 1658, 2108, 2496, 3316, 3846, 4621\n\nCarson, Ken 67, 286, 868, 1105, 1134, 1767, 1918, 1980, 2368, 2530, 2534, 2806, 3694, 4013, 4016, 4197, 4681, 5050\n\nCarson, Kit 759, 1259, 3496, 4774; _see also_ Barrymore, William; Bullock, Boris\n\nCarson, Robert 1070\n\nCarson, Sunset 51, 235, 263, 286, 465, 624, 712, 779, 984, 1003, 1215, 1321, 1345, 2928, 3326, 3521, 3524, 3613, 3707, 3820, 4196\n\nCarsten, Peter 85, 2724\n\nCarter, Ann 4745\n\nCarter, Ben 1792, 3431\n\nCarter, Carlene 2623\n\nCarter, Dick 3110, 4808\n\nCarter, Harry 519, 520, 759, 1480, 1601, 1995, 2902, 3150, 3333, 3377, 3909, 3972, 4411, 4579, 4611\n\nCarter, Helena 562, 1408, 3056, 3534\n\nCarter, Janis 1762, 3702\n\nCarter, Julie 4112\n\nCarter, Lynda 412\n\nCarter, Mrs. Leslie 3573\n\nCarter, Ted 19, 1342, 2441, 5101; _see also_ Pazzafini, Nello\n\nCartwright, Angela 2157\n\nCartwright, Veronica 1590, 4064\n\nCaruso, Anthony 209, 320, 489, 681, 766, 1053, 1077, 1161, 1264, 1372, 1402, 2005, 2043, 2295, 2324, 2328, 2504, 2663, 2831, 2871, 3024, 3054, 3705, 3862, 4258, 4416, 4770, 5024\n\nCaruth, Burr 634, 813, 865, 1504, 1553, 1751, 1916, 2153, 2786, 2998, 3074, 3325, 3424, 3549, 3574, 3805, 4649\n\nCarver, Louise 328\n\nCarver, Lynne 899, 1158, 1362, 1974, 2280, 2514, 4202\n\nCarver, Tina 4680\n\nCasas, Antonio 1419, 2570, 2655, 3030, 3118, 3175, 3380, 3420, 3997, 4182, 4325\n\nCase, Kathleen 2068, 2217, 3754, 4079\n\nCasell, Nancy 917\n\nCasey, Bernie 1739\n\nCasey, Claude 4085\n\nCasey, Sue 329, 3009, 3959\n\nCash, Johnny 971, 1694, 2190, 4102\n\nCash, June Carter 2190, 4102\n\nCason, John (Bob) 76, 107, 205, 325, 365, 368, 392, 427, 481, 489, 494, 695, 794, 807, 896, 994, 1024, 1073, 1130, 1145, 1272, 1285, 1296, 1330, 1362, 1363, 1404, 1459, 1505, 1540, 1542, 1672, 1687, 1782, 1789, 1806, 1894, 1939, 2065, 2105, 2171, 2176, 2189, 2192, 2233, 2287, 2289, 2302, 2363, 2397, 2451, 2582, 2590, 2596, 2609, 2847, 2974, 2991, 2992, 3155, 3167, 3168, 3169, 3230, 3232, 3262, 3312, 3328, 3342, 3344, 3453, 3460, 3511, 3648, 3652, 3718, 3724, 3801, 3805, 3940, 3982, 3995, 4001, 4032, 4053, 4078, 4099, 4108, 4141, 4167, 4196, 4228, 4310, 4390, 4408, 4461, 4467, 4506, 4533, 4539, 4700, 4702, 4709, 4736, 4781, 4820, 4855, 4857, 4952, 5036, 5038\n\nCaspary, Vera 2160\n\nCass, Maurice 3669, 4075, 4208, 4891\n\nThe Cass County Boys 104, 409, 542, 2103, 2189, 2894, 3481, 3563, 3669, 3929, 4506, 4508, 4757\n\nCassavetes, John 3671\n\nCassell, Wally 127, 191, 2252, 2372, 2851, 3247, 4860\n\nCassidy, Ed (Edward) 8, 37, 67, 76, 77, 137, 143, 150, 179, 277, 286, 365, 380, 415, 423, 425, 436, 460, 461, 479, 494, 495, 544, 557, 564, 573, 582, 665, 670, 683, 689, 725, 790, 793, 797, 810, 836, 848, 878, 886, 897, 939, 956, 984, 1001, 1008, 1072, 1074, 1080, 1090, 1138, 1215, 1229, 1241, 1242, 1279, 1284, 1332, 1348, 1367, 1465, 1468, 1505, 1512, 1550, 1674, 1760, 1812, 1873, 1898, 1900, 1920, 1925, 1965, 2035, 2124, 2135, 2152, 2248, 2269, 2298, 2529, 2546, 2589, 2622, 2640, 2647, 2690, 2701, 2759, 2761, 2798, 2816, 2863, 2893, 2931, 2948, 2979, 3029, 3048, 3062, 3084, 3097, 3115, 3155, 3194, 3220, 3226, 3289, 3291, 3325, 3332, 3452, 3483, 3490, 3491, 3497, 3525, 3332, 3452, 3483, 3490, 3491, 3497, 3525, 3526, 3546, 3548, 3666, 3679, 3694, 3703, 3706, 3708, 3728, 3820, 3898, 3899, 3907, 3989, 4011, 4022, 4078, 4108, 4110, 4127, 4128, 4181, 4189, 4197, 4202, 4279, 4363, 4409, 4427, 4448, 4488, 4489, 4512, 4522, 4550, 4554, 4584, 4656, 4681, 4685, 4691, 4698, 4702, 4722, 4754, 4889, 4923, 4947, 4981, 4995, 5001, 5018, 5041\n\nCassidy, Jack 771\n\nCassidy, Shaun 2897\n\nCassidy, Ted 586, 2489\n\nCastel, Lou 571, 2621, 2724, 3366\n\nCastellari, Enzo G. 98, 296, 751, 2049, 2097, 2016, 2900, 4246\n\nCastelli, Dolores 3406\n\nCastelnuovo, Nino 1358, 3406\n\nCastillo, Gloria 4247, 4705\n\nCastle, Anita 4818\n\nCastle, Don 320, 2837, 2936, 4118, 4171, 4456\n\nCastle, Mary 1747, 2234, 2292, 3168, 4285, 4468, 4829, 4871, 5044\n\nCastle, Peggy 857, 1849, 2038, 2870, 2996, 3213, 3998, 4232, 4613, 4766, 5052\n\nCastle, William 82, 259, 692, 828, 1188, 1405, 1691, 2037, 2150, 2287, 2620, 4680\n\nCatching, Bill 2488, 2525, 2924, 3422\n\nCatlett, Walter 203, 967, 1434, 1818, 1923, 4932\n\nCattrall, Kim 2658, 2800\n\nCaudell, Lane 560\n\nCaufield, Joan 548, 679, 3141, 3338\n\nCavan, Allan 410, 780, 813, 1228, 1626, 1827, 1898, 1979, 2212, 2257, 2399, 2785, 2802, 2879, 3194, 4612, 4667, 5037\n\nCavan, Taylor 154, 2033, 3893\n\nCavanagh, Paul 1591, 2009, 2287, 2664, 3362, 4093, 4139\n\nCavanaugh, Hobart 292, 874, 2106, 2020, 2081, 2170, 3711, 3846, 4096\n\nCavendar, Glen 203, 1378, 2777\n\nCavens, Fred 510, 2664\n\nCavin, Jess 183, 230, 395, 459, 494, 607, 793, 1328, 1338, 1389, 1469, 1546, 1553, 1969, 2673, 2864, 2953, 3126, 3177, 3280, 3550, 3673, 3827, 4463, 4702, 4786\n\nCawthorn, Joseph 2756\n\nCecil, Ed (Edward) 684, 1008, 1049, 1280, 1895, 2547, 3461, 4899, 4946\n\nCecil, Nora 467, 2245, 3750, 4100\n\nCedar, Jon 978, 1014\n\nCedar, Ralph 4806\n\nCeli, Adolfo 1019\n\nCelli, Teresa 439\n\nCesana, Renzo 604, 2583\n\nChabot, Amadee 168\n\nChadbourne (Chatburn), Jean 2756, 3038\n\nChadwick, Cyril 2005\n\nChadwick, Helene 626, 1436, 4933\n\nChaffey, Don 701, 3418\n\nChaliapin, Feodor, Jr. 553\n\nChallee, William 329, 339, 1071, 1641, 2023, 2555, 2565, 2609, 2693, 2818, 2902, 2910, 3133, 3671, 3750, 4136, 5083\n\nChamberlain, Howard 91, 245, 1877, 4212\n\nChamberlain, Richard 4387\n\nChambers, Wheaton 37, 251, 299, 366, 464, 939, 1041, 1215, 1253, 1573, 1690, 2125, 2292, 2597, 2665, 2775, 2871, 2893, 2966, 3061, 3132, 3165, 3750, 4011, 4021, 4045, 4047, 4058, 4106, 4110, 4231, 4461, 4766, 4938\n\nChambliss, Woodrow 1098, 1250, 1624, 3320, 4936\n\nChampion, John 530, 2714, 3032, 3849, 4118, 4325\n\nChampion, Marge 771; _see also_ Bell, Marjorie\n\nChan, Jackie 3809\n\nChance, Larry 260, 1396, 1485, 2319, 4784\n\nChandler, Chick 914, 1256, 1915, 2748, 2815, 3429, 4675, 5036\n\nChandler, David 105, 2202\n\nChandler, Eddy 541, 602, 660, 874, 1057, 1378, 1599, 2108, 2302, 3711, 3851, 4089, 4697, 4933\n\nChandler, George 16, 47, 101, 114, 129, 548, 552, 866, 1392, 1742, 1836, 1907, 2031, 2084, 2267, 2352, 2357, 2534, 2635, 2652, 2911, 3005, 3020, 3234, 3608, 3669, 3846, 3921, 4033, 4231, 4254, 4422, 4498, 4697, 4735, 4744, 4786, 4849\n\nChandler, Helen 3618\n\nChandler, Janet 877, 926, 1605, 3617\n\nChandler, Jeff 257, 519, 1151, 1428, 1645, 2025, 2550, 3096, 3133, 4073, 4244, 4384, 4611, 4783\n\nChandler, John Davis 255, 507, 1122, 1610, 1987, 2501, 2950, 3387, 3435\n\nChandler, Lane 47, 49, 64, 65, 73, 101, 262, 282, 300, 301, 450, 597, 602, 680, 700, 721, 813, 857, 1008, 1089, 1187, 1340, 1344, 1347, 1348, 1633, 1725, 1730, 1805, 1806, 1816, 1827, 1859, 1947, 1952, 1965, 1989, 2040, 2163, 2165, 2208, 2248, 2256, 2274, 2276, 2299, 2303, 2370, 2381, 2392, 2400, 2485, 2511, 2528, 2552, 2638, 2677, 2722, 2818, 2822, 2828, 2831, 2839, 2861, 2866, 2939, 2956, 2983, 3020, 3111, 3126, 3142, 3168, 3195, 3200, 3255, 3303, 3323, 3365, 3383, 3478, 3481, 3499, 3624, 3630, 3648, 3675, 3678, 3680, 3692, 3696, 3702, 3711, 3772, 3849, 3893, 3897, 3937, 4057, 4059, 4141, 4143, 4184, 4227, 4232, 4238, 4261, 4272, 4321, 4336, 4390, 4552, 4556, 4570, 4585, 4610, 4612, 4621, 4648, 4701, 4728, 4735, 4742, 4785, 4786, 4803, 4956, 4995, 5040, 5096\n\nChandler, Tanis 1105, 4777\n\nChanel, Helene 4268, 4623, 4625\n\nChaney, Creighton 2192, 3745, 4007; _see also_ Chaney, Lon, Jr.\n\nChaney, Lon 2816, 3412\n\nChaney, Lon, Jr. 49, 66, 114, 210, 260, 332, 373, 489, 548, 584, 725, 939, 955, 1266, 1278, 1444, 1458, 1877, 1989, 1996, 2031, 2050, 2267, 2444, 2646, 2676, 2830, 2831, 2874, 2919, 2994, 3040, 3054, 3057, 3341, 3450, 3906, 3919, 4080, 4159, 4229, 4472, 4665, 4802, 4925, 5065; _see also_ Chaney, Creighton\n\nChanning, Carol 1348\n\nChanning, Ruth 2961\n\nChanslor, Roy 611, 939, 1964, 2652, 4735\n\nChapin, Billy 4259\n\nChapin, Martha 2363\n\nChapin, Michael 146, 544, 933, 4013, 4080, 4641, 4766, 4804, 4945\n\nChaplin, Charles 1598\n\nChaplin, Charles, Jr. 1261\n\nChaplin, Geraldine 554\n\nChaplin, Sydney 1966, 3096, 3199\n\nChapman, Freddie 793, 1646, 1920, 3822, 4745\n\nChapman, Helen 472, 2953\n\nChapman, Janet 1820\n\nChapman, Lonny 426, 838, 947, 1943, 3635, 3645, 4113, 4878\n\nChapman, Marguerite 835, 2084, 3344\n\nCharisse, Cyd 1290, 1702, 2144, 2583, 4962\n\nCharlita 339, 516, 2517, 2747, 3440, 4043, 4470\n\nCharney, Kim 170, 482, 1569, 1735, 1945, 2521, 3200, 4805, 5066\n\nCharters, Spencer 200, 203, 1127, 1165, 1390, 1567, 2108, 2159, 2288, 2471, 2654, 3711, 3899, 3922, 4254, 4284, 4336, 4358, 4697, 4742, 4803, 4891, 5072\n\nChase, Alden 343, 773, 879, 1167, 2409, 2413, 2790, 3173, 3448, 3581, 3946, 4655; _see also_ Chase, Stephen\n\nChase, Barrie 3754\n\nChase, Borden 189, 190, 290, 1263, 1710, 2416, 2506, 2565, 2677, 2797, 3323, 3415, 4989, 5025\n\nChase, Charley 2136\n\nChase, Chevy 4351\n\nChase, Colin 1283, 2422, 3014, 3495, 4307, 4636, 4711\n\nChase, Frank 2565, 3294, 3415, 3718, 3768, 4073, 4770, 4989, 5025\n\nChase, Guy 1464\n\nChase, Howard 2529\n\nChase, Stephen 277, 691, 957, 1216, 1439, 1645, 1816, 1866, 2825, 2880, 3234; _see also_ Chase, Alden\n\nChasen, Dave 145\n\nChatterton, Tom 51, 144, 480, 602, 731, 779, 793, 827, 853, 1042, 1127, 1247, 1256, 1680, 1805, 1815, 1918, 1925, 2035, 2296, 2427, 2552, 2592, 2597, 2600, 2864, 2944, 2971, 2994, 3142, 3178, 3230, 3626, 3699, 3708, 3824, 4006, 4178, 4483, 4584, 4655, 4763, 4812, 4868, 5103\n\nChaudet, Louis 4063, 4260\n\nChautard, Emile 613, 4750\n\nChavez, Jose 167, 181, 837, 980, 1104, 1476, 1731, 1911, 2062, 2383, 2803, 3008, 3180, 3190, 3197, 3867, 3986, 4626\n\nChaykin, Maury 942, 114, 2090\n\nThe Checkerboard Band 4052\n\nCheney, J. Benton 187, 391, 459, 605, 869, 871, 1138, 1308, 1529, 1724, 1767, 1868, 1935, 1977, 2124, 2167, 2169, 2255, 2282, 2398, 2530, 2546, 2736, 2944, 2972, 2978, 2988, 3050, 3095, 3115, 3116, 3157, 3231, 3385, 3479, 3569, 3600, 3651, 3720, 3805, 3823, 3901, 3904, 3908, 3914, 3923, 4023, 4028, 4135, 4327, 4378, 4514, 4520, 4600, 4646, 4657, 4923\n\nChesebro, George 25, 28, 42, 157, 207, 214, 323, 332, 336, 338, 341, 343, 364, 368, 369, 382, 418, 427, 438, 443, 461, 473, 479, 489, 516, 592, 623, 658, 668, 682, 695, 711, 712, 724, 728, 780, 793, 794, 856, 862, 872, 889, 896, 917, 922, 926, 939, 945, 984, 1017, 1022, 1025, 1059, 1073, 1090, 1099, 1127, 1128, 1136, 1155, 1221, 1260, 1271, 1299, 1301, 1309, 1312, 1337, 1339, 1439, 1445, 1461, 1463, 1473, 1480, 1488, 1490, 1505, 1531, 1543, 1579, 1626, 1627, 1658, 1667, 1721, 1725, 1768, 1889, 1920, 1939, 1983, 2033, 2068, 2103, 2104, 2107, 2119, 2172, 2175, 2176, 2522, 2540, 2547, 2558, 2584, 2595, 2596, 2597, 2600, 2633, 2635, 2647, 2660, 2678, 2684, 2730, 2737, 2742, 2786, 2804, 2822, 2861, 2864, 2886, 2968, 2975, 2978, 2991, 2997, 3025, 3026, 3067, 3073, 3088, 3101, 3102, 3107, 3109, 3111, 3172, 3194, 3226, 3236, 3267, 3271, 3329, 3348, 3358, 3388, 3446, 3448, 3452, 3464, 3520, 3545, 3546, 3549, 3559, 3586, 3592, 3615, 3616, 3617, 3668, 3684, 3685, 3707, 3709, 3728, 3799, 3820, 3888, 3916, 3940, 3952, 3977, 3978, 4011, 4019, 4028, 4053, 4059, 4082, 4097, 4108, 4110, 4128, 4156, 4167, 4181, 4272, 4273, 4280, 4296, 4305, 4310, 4314, 4327, 4393, 4401, 4402, 4408, 4448, 4454, 4485, 4487, 4488, 4497, 4506, 4520, 4525, 4589, 4592, 4606, 4609, 4610, 4636, 4701, 4722, 4729, 4733, 4773, 4810, 4812, 4815, 4818, 4820, 4832, 4846, 4899, 4958, 4961, 4972, 4980, 5005, 5010, 5014, 5027, 5033, 5062\n\nCheshire, Harry V. 30, 134, 357, 516, 1158, 1234, 1247, 1319, 1348, 1842, 2839, 3367, 3481, 3698, 3929, 3975, 4083, 4086, 4381\n\nChester, Alma 877, 2886, 4190, 4862\n\nChevret, Lita 3697\n\nCheyenne Bill (Harry William McKenzie) 4406\n\nChiari, Walter 4263\n\nChickie and Buck 870\n\nChiles, Linden 1985, 4287\n\nChing, William 277, 1238, 2691, 2851, 3852, 4212, 4232, 5008\n\nChinook (dog) 620, 1259, 2834, 2840, 3980, 5078, 5079, 5081\n\nChissell, Noble \"Kid\" 23, 315, 930, 1616, 2552, 2710, 4013\n\nChomsky, Marvin 1277, 2490, 2706, 3686\n\nChristensen, Carol 1431\n\nChristie, Audrey 223\n\nChristie, Julie 2625\n\nChristina, Helen 5096\n\nChristine, Virginia 339, 679, 1371, 1422, 1877, 2052, 2784, 2910, 3080, 3224, 4062, 5022\n\nChristopher, Jordan 3391\n\nChristopher, Kay 4048\n\nChristy, Dorothy 868, 930, 1360, 3075, 3615, 3866\n\nChristy, Eileen 3066\n\nChristy, Jan 1934\n\nChristy, Ken 283, 377, 1482, 2641, 4579, 4682, 4805\n\nChristy, Vic 696\n\nChurch, Fred 108, 438, 1379, 3455, 3983, 4094, 4972\n\nChurchill, Berton 327, 866, 1457, 2732, 4100, 4593, 4925\n\nChurchill, Marguerite 328, 3470\n\nCiannelli, Eduardo 421, 602, 1617, 2489, 2614\n\nCilento, Diane 1908\n\nCimber, Matt 587, 5047\n\nCimino, Michael 1833\n\nCisar, George 339, 550, 2902, 4805\n\nCivirani, Osvaldo 991, 3372, 3830\n\nClair, Jany 3543\n\nClair, Rene 361\n\nClaire, Roy 753; _see also_ Luby, S. Roy\n\nClancy, Ellen 3172; _see also_ Shaw, Janet\n\nClapham, Leonard 457, 3065; _see also_ London, Tom\n\nClark, Bobby 304, 1668, 2885, 3003, 3299, 3510, 3521, 3890, 4036, 4041, 4554\n\nClark, Candy 3577\n\nClark, Cliff 49, 248, 324, 472, 659, 895, 1048, 1073, 1236, 1253, 1255, 1378, 1395, 1704, 3000, 3151, 3667, 3891, 3954, 4057, 4179, 4257, 4470, 4730, 4761, 4792, 4849, 4932, 5072\n\nClark, Colbert 1340, 2212, 2367\n\nClark, Cottonseed 161, 3976\n\nClark, Dane 252, 1029, 1398, 2611, 2627, 2981, 4257, 4392\n\nClark, Davison (Davidson) 179, 279, 352, 514, 812, 875, 878, 939, 1023, 1145, 1308, 1420, 1758, 2269, 2416, 2646, 2831, 2869, 3126, 3165, 3269, 3323, 3374, 3505, 3678, 3698, 3750, 3914, 3937, 4257, 4364, 4556, 4610, 4635, 4701, 4742, 4842, 4932\n\nClark, Dick 3732\n\nClark, Frank Howard 326, 1291, 1799, 2990, 4320, 4475, 4622, 4683, 4950\n\nClark, Frank 438, 443, 1306, 1820, 3073, 3272, 3551, 3655, 3797, 4070, 4084, 4846, 5010\n\nClark, Fred 1201, 1480, 3386, 4621, 4867, 5075\n\nClark, Harvey 478, 480, 674, 1095, 2258, 3049, 3934, 4061, 4359\n\nClark, James B. 86, 2728, 2902, 3663, 3863, 4747\n\nClark, Judy 710, 1073, 2419, 4051\n\nClark, Ken 1842, 2241, 2462, 2510, 3185, 3543, 3725, 4579\n\nClark, Mamo 2717\n\nClark, Montgomery 1120\n\nClark, Roy 4679\n\nClark, Russ 602, 2208, 4214, 4849\n\nClark, Steve 2, 28, 30, 54, 65, 70, 140, 150, 151, 228, 307, 334, 336, 340, 343, 345, 364, 369, 391, 414, 418, 422, 423, 427, 431, 573, 668, 682, 683, 687, 711, 728, 731, 818, 850, 867, 871, 872, 875, 899, 928, 945, 973, 1025, 1051, 1073, 1090, 1100, 1127, 1145, 1156, 1158, 1192, 1193, 1281, 1308, 1325, 1328, 1339, 1362, 1433, 1447, 1455, 1486, 1505, 1529, 1540, 1543, 1555, 1626, 1681, 1684, 1687, 1721, 1732, 1797, 1798, 1800, 1803, 1859, 1934, 1938, 2026, 2107, 2111, 2119, 2151, 2166, 2169, 2171, 2172, 2177, 2184, 2189, 2220, 2233, 2251, 2254, 2259, 2280, 2282, 2284, 2286, 2293, 2294, 2299, 2306, 2366, 2398, 2402, 2408, 2409, 2414, 2415, 2425, 2448, 2457, 2517, 2558, 2590, 2601, 2633, 2679, 2690, 2737, 2743, 2762, 2798, 2805, 2816, 2821, 2822, 2858, 2907, 2949, 2965, 2973, 2975, 2978, 2991, 3042, 3044, 3052, 3081, 3085, 3098, 3100, 3132, 3155, 3157, 3159, 3167, 3173, 3257, 3262, 3264, 3265, 3277, 3278, 3329, 3370, 3385, 3401, 3443, 3448, 3492, 3507, 3550, 3565, 3568, 3569, 3587, 3597, 3648, 3651, 3666, 3668, 3804, 3880, 3890, 3907, 3942, 3943, 3945, 3977, 4000, 4017, 4021, 4024, 4042, 4053, 4089, 4095, 4097, 4108, 4153, 4155, 4188, 4206, 4273, 4297, 4319, 4382, 4391, 4393, 4394, 4410, 4448, 4460, 4466, 4497, 4501, 4502, 4550, 4587, 4589, 4609, 4640, 4641, 4660, 4698, 4702, 4723, 4182, 4814, 4826, 4827, 4828, 4833, 4834, 4835, 4838, 4840, 4847, 4850, 4884, 4899, 4935, 4952, 4953, 4980, 5035\n\nClark, Susan 121, 831, 3855, 3991, 4249, 4688\n\nClark, Wallis 63, 1388, 1869, 2016, 2610, 2726, 3692, 4456, 4983\n\nClark, Matt 883, 909, 1220, 1641, 1828, 2028, 2094, 2113, 2156, 2229, 2335, 2348, 2488, 2685, 2950, 3055, 3136, 3205, 4045, 4345\n\nClarke, Angela 1704, 2013, 2705, 3722\n\nClarke, Gage 1482, 4689\n\nClarke, Gary 190, 507, 563, 1341, 1922, 2631, 4172, 4453\n\nClarke, Mae 317, 632, 1071, 1368, 1684, 1827, 1936, 1996, 2672, 4921, 4933\n\nClarke, Robert 652, 786, 1105, 3119, 3383, 3473, 4204, 4386, 4653, 4777, 4849\n\nClarke, Robin 1019, 2374\n\nClauser, Al, and His Oklahoma Outlaws 3603\n\nClaxton, William F. 417, 1261, 2267, 3211, 4098, 4111, 4264, 5070\n\nClay, Lewis 778, 4278, 4701, 4729\n\nClay, Nolan 4113\n\nClayton, Ethel 3414, 3766, 4803\n\nClayton, Jan (Jane) 1978, 2385, 3851, 3937, 4208, 5013\n\nClayton, Marguerite 5010\n\nCleese, John 81, 3911\n\nClemens, William 1253, 2127\n\nClement, Clay 63, 1622, 1827\n\nClemente (Clemento), Steve 764, 890, 956, 1324, 1387, 1489, 1730, 1889, 2012, 2359, 2367, 2741, 3240, 3266, 4280, 4700, 4732, 4750, 4900, 4997\n\nClements, Calvin 1086, 1346, 4936\n\nClements, Curley, and His Rodeo Rangers 3940\n\nClements, Marjorie 1001, 2884\n\nClements, Stanley 1096, 1409, 1942, 2205, 3559\n\nClements, Zeke 781, 4610\n\nCletro, Eddie, and His Roundup Boys 4496\n\nCleveland, George 46, 49, 386, 411, 642, 664, 895, 930, 1364, 1398, 1433, 1480, 1556, 1916, 2088, 2150, 2153, 2399, 2530, 2548, 2778, 2823, 2909, 2963, 2968, 2993, 3083, 3089, 3111, 3126, 3132, 3203, 3215, 3471, 3511, 3526, 3610, 3690, 3772, 4072, 4125, 4199, 4551, 4675, 4700, 4806, 4923, 4929, 5008, 5011, 5021, 5050\n\nCliff, John 185, 293, 304, 1092, 1225, 1433, 1676, 1748, 2037, 2260, 2287, 2339, 2511, 2609, 2693, 2868, 2910, 3234, 3787, 4069, 4361\n\nClifford, Gordon 3036, 4978\n\nClifford, Jack 135, 214, 231, 307, 602, 642, 648, 682, 686, 799, 951, 1153, 1488, 1526, 1592, 1627, 1792, 1883, 2129, 2334, 2388, 2435, 2522, 2712, 2884, 2907, 2994, 3126, 3220, 3401, 3450, 3569, 3582, 3684, 3772, 3953, 4050, 4193, 4454, 4504, 4868, 4971, 5077\n\nClifford, Ruth 69, 2179, 2635, 4626, 4764\n\nClift, Montgomery 2661, 3323\n\nClifton, Dorinda 2577\n\nClifton, Elmer 206, 395, 479, 494, 606, 726, 892, 917, 926, 987, 1001, 1028, 1299, 1448, 1456, 1506, 1738, 1749, 2106, 2278, 2363, 2589, 2873, 2953, 2974, 3026, 3097, 3209, 3227, 3238, 3280, 3331, 3390, 3617, 3887, 3952, 4026, 4078, 4152, 4185, 4196, 4220, 4363, 4497, 4829, 4889, 4982\n\nCline, Edward F. 879, 1177, 1336, 2688, 2727, 4864, 4865\n\nClive, E.E. 1378, 3022\n\nClive, Iris 2435, 3357, 4026, 4820\n\nClooney, Rosemary 3316, 3958\n\nClose, Glenn 2934\n\nClose, John 277, 1259, 3333, 4073, 4141, 4329\n\nClucher, E.B. 3349, 4334, 4561, 4563, 4664; _see also_ Barboni, Enzo\n\nClute, Chester 277, 648, 710, 796, 1070, 1127, 1144, 1392, 3917, 4072, 4085, 4700\n\nClutesi, George 2091, 2340, 2752, 2807, 4068, 4443\n\nClyde, Andy 2, 95, 152, 196, 241, 254, 318, 446, 459, 472, 661, 715, 800, 893, 990, 1097, 1138, 1254, 1255, 1279, 1380, 1416, 1800, 1934, 1935, 1977, 1978, 2311, 2451, 2480, 2577, 2736, 2979, 2972, 3116, 3128, 3262, 3458, 3460, 3479, 3542, 3761, 3802, 3877, 3900, 3928, 4023, 4135, 4147, 4188, 4304, 4327, 4364, 4378, 4600, 4657, 4661, 4811, 4826, 4923, 5046\n\nClyde, June 500\n\nCoates, Phyllis 399, 643, 649, 678, 1216, 1268, 1711, 1715, 2448, 2539, 2593, 2776, 2863, 2970, 3746, 4095, 4462, 5039\n\nCoats, Tommy 64, 544, 712, 984, 1046, 1057, 1128, 1333, 1470, 1618, 1725, 1815, 1860, 1862, 2022, 2035, 2128, 2132, 2262, 2297, 2304, 2403, 2421, 2479, 2514, 2558, 2850, 2928, 2968, 2998, 3088, 3167, 3242, 3256, 3271, 3303, 3483, 3552, 3574, 3691, 3805, 3820, 3989, 4011, 4048, 4106, 4181, 4190, 4272, 4395, 4584, 4610, 4661, 4811, 4826, 4923, 5046\n\nCobb, Edmund 15, 101, 130, 134, 153, 180, 183, 204, 234, 264, 284, 295, 305, 380, 410, 415, 486, 500, 511, 520, 543, 557, 564, 597, 609, 610, 611, 623, 624, 637, 664, 667, 682, 695, 712, 174, 729, 764, 775, 780, 799, 807, 814, 834, 854, 914, 922, 928, 956, 957, 958, 984, 1008, 1009, 1028, 1073, 1100, 1145, 1164, 1200, 1215, 1221, 1256, 1264, 1360, 1373, 1426, 1439, 1447, 1449, 1474, 1488, 1497, 1513, 1524, 1546, 1565, 1568, 1579, 1594, 1600, 1614, 1626, 1627, 1678, 1680, 1704, 1723, 1727, 1758, 1768, 1811, 1821, 1863, 1868, 1871, 1888, 1950, 1967, 1993, 2016, 2022, 2035, 2050, 2109, 2119, 2161, 2169, 2193, 2256, 2259, 2262, 2272, 2275, 2280, 2283, 2308, 2355, 2366, 2398, 2399, 2425, 2428, 2482, 2526, 2533, 2534, 2546, 2552, 2564, 2600, 2628, 2633, 2635, 2638, 2652, 2660, 2679, 2730, 2737, 2756, 2761, 2821, 2828, 2870, 2873, 2884, 2899, 2902, 2931, 2967, 2969, 2975, 2994, 3109, 3142, 3170, 3171, 3173, 3178, 3221, 3222, 3224, 3229, 3293, 3310, 3322, 3326, 3340, 3351, 3357, 3365, 3370, 3394, 3424, 3427, 3442, 3449, 3450, 3453, 3462, 3464, 3472, 3505, 3508, 3524, 3534, 3535, 3552, 3555, 3563, 3616, 3650, 3651, 3653, 3655, 3656, 3658, 3678, 3684, 3691, 3707, 3708, 3711, 3744, 3828, 3859, 3869, 3892, 3894, 3926, 3938, 3967, 3969, 4000, 4013, 4024, 4042, 4048, 4077, 4082, 4127, 4241, 4153, 4156, 4181, 4189, 4197, 4278, 4286, 4298, 4299, 4315, 4318, 4410, 4420, 4470, 4480, 4498, 4525, 4579, 4588, 4609, 4610, 4641, 4674, 4712, 4723, 4729, 4740, 4767, 4808, 4817, 4826, 4834, 4850, 4855, 4899, 4944, 4954, 4980, 4989, 5008, 5034, 5041, 5043, 5096, 5104\n\nCobb, Jerry 2612; _see also_ Cobos, German\n\nCobb, Julie 4134\n\nCobb, Lee J. 507, 549, 563, 831, 980, 1293, 1341, 1945, 2305, 2451, 2488, 2489, 2556, 2560, 2631, 3542, 4236, 45328\n\nCobos, German 3403, 3756, 4779; _see also_ Cobb, Jerry\n\nCoburn, Buck 1687; _see also_ Alsace, Gene; Camron, Rocky\n\nCoburn, Charles 713, 1487, 1649, 1924, 2497, 2501, 2519, 3297, 4358\n\nCoburn, James 348, 1152, 1171, 1248, 2141, 2196, 2623, 3055, 3430, 4311, 4794, 5069\n\nCoburn, Walt 186, 3658, 3838\n\nCoby, Fred 1092, 1132, 2515, 2544, 3154, 3741, 3744\n\nCoch, Ed 3500, 3509, 3769\n\nCochran, Steve 185, 935, 1005, 2370, 3200, 3285, 3811\n\nCodee, Ann 4650\n\nCody, Bill 384, 438, 443, 929, 1191, 1311, 1446, 1537, 1820, 2173, 2268, 2290, 2608, 2681, 2862, 2976, 3302, 3939, 4100, 4306, 4307, 4648, 4711, 4846\n\nCody, Bill, Jr. 199, 1080, 1084, 1446, 2927, 2976, 3232, 3302, 3597, 4608, 4711\n\nCody, Iron Eyes 49, 53, 101, 104, 115, 165, 327, 328, 352, 363, 402, 488, 519, 612, 642, 716, 771, 807, 819, 851, 864, 892, 917, 973, 1209, 1273, 1317, 1328, 1340, 1398, 1403, 1490, 1565, 1623, 1626, 1644, 1669, 1671, 1805, 1842, 1988, 2014, 2121, 2134, 2225, 2233, 2452, 2508, 2552, 2618, 2678, 2779, 2798, 2825, 2862, 2879, 2889, 2927, 2971, 2993, 3003, 3020, 3067, 3087, 3096, 3111, 3142, 3172, 3203, 3241, 3318, 3339, 3425, 3433, 3442, 3482, 3650, 3673, 3695, 3722, 3921, 3991, 4005, 4306, 4448, 4534, 4585, 4635, 4646, 4665, 4700, 4701, 4849, 4856, 4907, 4937, 4971, 5001, 5059, 5062\n\nCody, Lew 2843, 4254\n\nCoe, David Allan 2190, 4102\n\nCoe, Peter 166, 502, 673, 1844, 2902, 3054, 3572, 3849\n\nCoe, Vivian 37; _see also_ Austin, Vivian\n\nCoen, Franklin 71, 733, 1423, 2143\n\nCoffee, Lenore 4093\n\nCoffin, Tristram 131, 138, 283, 544, 605, 749, 759, 876, 880, 893, 972, 973, 1059, 1072, 1100, 1247, 1389, 1520, 1529, 1775, 1964, 2035, 2105, 2119, 2234, 2252, 2293, 2334, 2602, 2624, 2663, 2840, 2842, 2869, 2875, 2910, 2993, 3159, 3202, 3261, 3407, 3460, 3524, 3550, 3578, 3759, 3848, 3929, 3974, 4086, 4222, 4465, 4508, 4640, 4646, 4691, 4736, 4815, 4876, 5035\n\nCoghlan, Junior 1162, 2212, 3537\n\nCohen, Bennett 77, 230, 240, 447, 811, 1042, 1254, 1453, 1462, 1358, 1935, 2122, 2257, 2275, 2296, 2394, 2428, 2526, 2634, 2699, 2828, 2858, 2909, 3035, 3102, 3288, 3327, 3449, 3458, 3484, 3562, 3679, 3707, 3770, 3820, 3907, 3945, 3952, 4020, 4038, 4042, 4207, 4219, 4297, 4397, 4410, 4504, 4604, 4645, 4809, 4817, 4834, 4877, 4986, 5041\n\nCohen, Emma 751, 918, 2326\n\nCohen, Larry 378, 1066, 1208, 3391, 3438, 3840\n\nCohen, Sammy 3081, 3531\n\nColbert, Claudette 420, 1165, 1256, 4300\n\nColby, Marion 3999\n\nColdeway, Anthony 634, 779, 1023, 2600, 3157, 4318, 4513, 4584, 4586, 4649, 4734, 5041\n\nCole, Dennis 247, 808, 3152\n\nCole, Michael 744\n\nCole, Nat (King) 674, 3120, 4221\n\nCole, Royal K. 352, 1543, 1711, 2022, 3547, 4001, 4278\n\nCole, Slim 2192, 4289\n\nColeman, C.C., Jr. 780, 1128, 2983, 4077\n\nColeman, Charles 389, 1618, 2649, 3895, 4023, 4851\n\nColeman, Claudia 844, 1436, 2923, 4007\n\nColeman, Dabney 348, 3736, 4879\n\nColeman, Don 328, 331\n\nColeman, Ruth 1810\n\nColes, Mildred 187, 1072, 2592, 2857, 3711, 4021\n\nColizzi, Giuseppi 3, 421, 1581\n\nColla, Richard A. 2627\n\nCollier, Lois 1513, 3084, 3230, 3708, 4811, 4855, 4929\n\nCollier, William, Jr 747, 4413\n\nCollings, Anne 3778\n\nCollins, Alan 85, 376, 3174, 3660, 3716, 4720\n\nCollins, Cora Sue 2732, 2756\n\nCollins, Eddie 1165, 3374, 5072\n\nCollins, Gary 3737\n\nCollins, Joan 502, 1625, 4977\n\nCollins, Kathleen 173, 434, 445, 923, 954, 1203, 5017\n\nCollins, Lewis D. 2, 493, 643, 649, 716, 791, 867, 997, 1268, 1326, 1627, 1678, 1715, 2085, 2271, 2294, 2378, 2248, 2523, 2539, 2591, 2680, 2776, 2863, 2866, 2884, 3224, 3227, 3586, 3630, 3744, 4095, 4105, 4261, 4292, 4301, 4556, 4570, 4620, 4728, 4731, 4754, 4966, 4983; _see also_ Lewis, Cullen\n\nCollins, Monte (Monty) 541, 949, 1563, 1626, 1906, 2163, 3171, 4591\n\nCollins, Ray 214, 248, 491, 642, 803, 2143, 3334, 3606, 4074, 4300, 4725\n\nCollins, Richard 209, 1077, 1205, 2143\n\nCollins, Russell 193, 644, 2192, 4771, 4867\n\nCollins, Topsy 4340\n\nCollinson, Peter 2509\n\nCollyer, June 1176\n\nColman, Booth 324, 3739, 4329\n\nColman, Ronald 5004\n\nColmans, Edward 913, 2066, 2400, 2504, 2619, 3705, 3941, 5086\n\nColon, Miriam 62, 119, 528, 1027, 2417, 2631, 2901, 4169\n\nColona, Jerry 4697\n\nColt, Dennis 99, 999, 116, 1418, 3727\n\nColt, Lee 2826, 3657; _see also_ Cobb, Lee J.\n\nColter, Jessi 4102\n\nColton, John 2245\n\nColton, Scott 2245, 4983\n\nColumbo, Russ 4281, 5015\n\nColvin, John 3602\n\nCombs, Jeffrey 78\n\nComer, Anjanette 119, 1731, 2793, 4169\n\nComingore, Dorothy _see_ Winters, Linda\n\nCompson, Betty 886, 1587, 3139, 4071\n\nCompton, Joyce 1307, 1569, 1616, 2843, 3069, 3362, 3904, 4057, 4580, 4698\n\nCondon, Charles 777, 1491\n\nConklin, Chester 37, 117, 271, 317, 516, 621, 642, 739, 1257, 1394, 1603, 1853, 1974, 2032, 2025, 2782, 3171, 3475, 3600, 3679, 3916, 4005, 4023, 4028, 4037, 4083, 4585, 4700, 4744, 5050\n\nConklin, Heinie 73, 246, 642, 765, 1256, 1364, 1378, 1458, 1925, 2268, 2386, 2792, 3409, 3451, 3636, 3975, 4023, 4073, 4517, 4851\n\nConlin, Jimmy 145, 1392, 2722, 2922, 3490, 3593, 3606, 4585, 4767\n\nConn, Maurice 2832, 3083, 4027, 4427, 4696, 4955\n\nConnelly, Christopher 702, 1808, 1986, 2188\n\nConnelly, Joe 1266\n\nConners, Barry 1518\n\nConnery, Sean 3807\n\nConnor, Whitfield 759, 3741\n\nConnors, Buck 54, 186, 975, 1058, 1276, 1488, 1626, 1813, 1850, 1993, 2232, 2285, 2404, 2633, 2816, 3076, 3126, 4403, 4521, 4660, 4817, 4851\n\nConnors, Chuck 40, 238, 313, 523, 657, 1062, 1500, 1533, 1892, 2016, 2887, 2897, 3030, 3183, 3422, 3438, 4122, 4210, 4451\n\nConnors, Michael (Mike\/Touch) 938, 1357, 1377, 2021, 2870, 4101, 4601\n\nConrad, Eugene 1509, 3925\n\nConrad, Joseph 1249\n\nConrad, Michael (Mike) 229, 3274, 4361, 4857\n\nConrad, Mikel 127, 664, 711, 2515, 3095, 3698, 4049\n\nConrad, Robert 232, 2188, 2379, 2694, 4975\n\nConrad, William 640, 861, 1420, 1946, 2046, 2416, 2487, 2519, 3421\n\nConreid, Hans 536, 970, 2163, 2789, 4291, 5025\n\nConroy, Frank 626, 1457, 3005, 4803\n\nConsidine, John 554, 4434\n\nConstantin, Michael 4782\n\nConte, John 113, 1225, 1276, 2555, 3964, 4079, 4173\n\nConte, Richard 317, 1019, 1225, 1293, 1831, 3222, 4335\n\nConte, Steve 680, 1510, 1606, 1713\n\nConti, Albert 863\n\nConverse, Frank 1943\n\nConverse, Peggy 979, 1161, 4338\n\nConversi, Spartaco 61, 68\n\nConvy, Bert 1719\n\nConway, Curt 1948, 2002, 2488, 4993\n\nConway, Gary 5068\n\nConway, Jack 420, 1925, 2344, 4750\n\nConway, James L. 419, 1133, 1986, 2215, 2337\n\nConway, Kevin 1379\n\nConway, Kevin 3206, 4169\n\nConway, Lita 2132, 3668, 4515\n\nConway, Morgan 314, 644, 2016\n\nConway, Pat 515, 1533\n\nConway, Russ 354, 599, 1399, 1403, 1734, 2462, 2941, 4232\n\nConway, Tim 121, 122\n\nConway, Tom 198, 1253, 3530\n\nCoogan, Jackie 594, 1917, 2644, 2960, 3185, 3806, 3951\n\nCoogan, Richard 4362\n\nCook, Clyde 246, 1178, 2149, 4909, 4997, 5004\n\nCook, Donald 633, 826, 4750\n\nCook, Elisha, Jr. 401, 961, 981, 1161, 1208, 1220, 1631, 1641, 1989, 2431, 2901, 2963, 3055, 4390, 4428, 4449, 4802, 4979, 5007\n\nCook, Fiedler 317, 3687, 4345\n\nCook, Joe 145\n\nCook, Tommy 37, 258, 644, 964, 2672, 2797, 3608, 4013, 4777\n\nCook, Will 4527\n\nCooke, Evelyn 4823\n\nCookson, Peter 3744\n\nCooley, Marjorie 4806\n\nCooley, Spade 444, 710, 973, 1240, 1915, 2106, 2165, 2451, 2514, 2599, 2830, 2978, 3504, 3569, 3673, 3772, 3887, 3925, 4051, 4085, 4305, 4405, 4687\n\nCoolidge, Rita 3055, 4565\n\nCoon, Gene L. 2059, 2550, 2813, 3223\n\nCoontz, Bill 555, 559, 1733, 1871, 2301, 2426, 3225, 3422, 4010, 4574\n\nCooper, Ben 148, 1182, 1273, 1696, 1852, 2048, 2187, 2911, 2938, 2981, 3066, 3223, 3299, 3338, 4210, 4755, 5022\n\nCooper, Clancy 602, 718, 928, 934, 995, 1029, 1032, 1111, 1449, 2504, 2543, 2555, 3505, 3507, 3821, 3909, 4191, 4225, 4422, 4579, 4827, 4962\n\nCooper, Dee 280, 492, 711, 728, 893, 1461, 1520, 1540, 1555, 1724, 2006, 2280, 2710, 2946, 3103, 3231, 3262, 3460, 3670, 3802, 3820, 3901, 4097, 4295, 4514, 4559, 4847\n\nCooper, Gary 64, 406, 866, 935, 1111, 1434, 1772, 1877, 1961, 2013, 2222, 2385, 2556, 2773, 2831, 2918, 2923, 3126, 4071, 4080, 4281, 4335, 4635, 4727, 4744, 4851, 4949, 5004, 5015\n\nCooper, George 402, 1388, 1570, 3087, 3456, 3621, 4121, 4639, 4816, 4824\n\nCooper, Georgia 2546\n\nCooper, Inez 456, 1564, 2765, 2824, 3502, 3530\n\nCooper, Jackie 2392, 2638, 3374, 3577\n\nCooper, James Fenimore 1031, 1032, 1033, 1034, 2009, 2211, 2212, 2213, 2214, 2216, 2218, 3056, 3107, 3154\n\nCooper, Jeanne 1575, 2544, 2695, 3340, 3803, 4330\n\nCooper, Jeff 1183, 1648, 3882\n\nCooper, Ken 147, 723, 818, 880, 917, 1073, 1089, 1728, 1915, 3126, 3329, 3482, 3486, 3625, 3919, 4007, 4294, 4732, 4865\n\nCooper, Melville 1361\n\nCooper, Olive 228, 325, 442, 615, 880, 1143, 1647, 1827, 1888, 1964, 1976, 2124, 2942, 3564, 3827, 3929, 4016, 5059\n\nCooper, Ted 146, 4955\n\nCooper, Tex 14, 49, 307, 344, 369, 479, 683, 715, 800, 956, 960, 984, 1017, 1046, 1097, 1456, 1459, 1474, 1587, 1693, 1768, 1789, 1792, 1859, 1935, 2016, 2111, 2119, 2251, 2263, 2399, 3457, 2540, 2547, 2552, 2635, 2636, 2660, 2821, 2864, 2886, 2947, 2960, 2969, 2999, 3088, 3132, 3159, 3209, 3303, 3386, 3390, 3393, 3456, 3479, 3550, 3599, 3650, 3673, 3824, 4023, 4108, 4197, 4226, 4409, 4465, 4587, 4655, 4660, 4696, 4702, 4736, 4767, 4810, 5027\n\nCoote, Robert 196, 3281\n\nCopas, Lloyd \"Cowboy\" 4085\n\nCopeland, Nick 881, 1831, 2790, 4534\n\nCoppola, Christopher 1705\n\nCoral, Tito 1591\n\nCorbett, Ben (Benny) 14, 69, 130, 139, 205, 240, 270, 335, 351, 387, 414, 443, 458, 468, 496, 511, 552, 688, 700, 714, 718, 753, 754, 767, 773, 778, 798, 801, 813, 814, 851, 862, 889, 935, 958, 973, 1009, 1099, 1146, 1221, 1222, 1229, 1269, 1276, 1309, 1311, 1327, 1334, 1380, 1384, 1486, 1517, 1554, 1595, 1633, 1677, 1678, 1686, 1730, 1796, 1800, 1816, 1837, 1859, 1928, 1933, 1934, 1935, 1975, 1976, 1993, 2012, 2014, 2104, 2169, 2192, 2232, 2247, 2256, 2258, 2259, 2266, 2304, 2359, 2360, 2392, 2434, 2493, 2515, 2523, 2590, 2599, 2642, 2679, 2689, 2733, 2799, 2956, 2979, 2992, 3026, 3048, 3052, 3078, 3115, 3162, 3195, 3216, 3229, 3264, 3272, 3321, 3427, 3486, 3487, 3495, 3571, 3631, 3650, 3657, 3665, 3674, 3679, 3699, 3797, 3850, 3876, 3902, 3946, 3949, 3969, 4026, 4031, 4080, 4144, 4165, 4179, 4190, 4201, 4248, 4280, 4308, 4321, 4322, 4389, 4423, 4484, 4511, 4570, 4636, 4657, 4702, 4728, 4759, 4798, 4809, 4840, 4846, 4852, 4944, 4956, 4972, 5043, 5075\n\nCorbett, Glenn 319, 741, 2265, 3816, 4344\n\nCorbin, Barry 822, 900, 2060\n\nCorbin, Virginia Lee 3850\n\nCorbucci, Bruno 1382, 1419, 1643, 1793, 4366\n\nCorbucci, Sergio 225, 820, 1113, 1160, 1382, 1617, 1643, 1653, 1841, 2049, 2614, 2643, 2655, 2758, 3301, 3513\n\nCorby, Ellen 60, 329, 1422, 1704, 1982, 2795, 2797, 3744, 3808, 4111, 4171, 4210, 4600, 4675, 4714, 5022\n\nCorcoran, Brian 4672\n\nCorcoran, Donna 1757\n\nCorcoran, Kelly 4344\n\nCorcoran, Kevin 2887, 3731, 4417, 4673\n\nCord, Alex 1623, 2162, 2656, 4101\n\nCorday, Mara 972, 977, 1164, 1428, 2511, 2565, 2748, 3211, 3287, 4240\n\nCorday, Rita 1573, 4825\n\nCorday, Sandra 4558\n\nCordell, Chase 4474\n\nCordell, Frank 64, 145, 1187, 1842, 1876, 2525, 3040, 3126, 3722, 3891, 4005, 4168, 4284, 4309, 4395, 4435, 4650, 4851\n\nCordero, Joaquin 167, 732, 1187\n\nCording, Harry 42, 144, 207, 210, 329, 503, 582, 602, 825, 895, 950, 951, 952, 1084, 1307, 1380, 1472, 1627, 1820, 2128, 2132, 2150, 2159, 2213, 2545, 2577, 2599, 2716, 2805, 2828, 2983, 2994, 3012, 3216, 3291, 3323, 3351, 3425, 3453, 3618, 3692, 3702, 3711, 4057, 4072, 4096, 4121, 4215, 4257, 4294, 4314, 4442, 4495, 4498, 4697, 4742, 4750, 4868, 4932\n\nCordova, Fred 425, 2607, 2830, 3111, 3426, 3562\n\nCordova, Margarita 512\n\nCorey, Jeff 239, 585, 602, 675, 1163, 1935, 2159, 2373, 2423, 2716, 2719, 2781, 2789, 2830, 2919, 2947, 2986, 3198, 3247, 3318, 3567, 3837, 3921, 4057, 4257, 4542, 4576, 4677\n\nCorey, Jim 56, 153, 297, 306, 327, 337, 351, 423, 461, 525, 582, 586, 600, 631, 754, 767, 773, 800, 810, 814, 818, 823, 851, 899, 958, 1008, 1042, 1145, 1202, 1241, 1283, 1302, 1324, 1335, 1367, 1426, 1432, 1469, 1512, 1595, 1614, 1728, 1732, 1759, 1796, 1798, 1834, 1837, 1859, 1928, 1932, 1973, 1974, 1976, 1980, 1983, 1993, 2007, 2033, 2164, 2167, 2180, 2220, 2222, 2232, 2249, 2250, 2273, 2274, 2299, 2315, 2403, 2472, 2527, 2533, 2600, 2633, 2635, 2699, 2703, 2742, 2743, 2763, 2786, 2816, 2909, 2951, 2968, 2975, 3042, 3048, 3049, 2065, 3074, 3087, 3109, 3135, 3161, 3165, 3171, 3172, 3194, 3260, 3270, 3290, 3303, 3314, 3322, 3393, 3427, 3452, 3456, 3525, 3550, 3551, 3564, 4597, 3621, 3625, 3650, 3709, 3742, 3817, 3827, 3834, 3851, 3898, 3937, 3949, 3969, 3994, 4059, 4082, 4377, 4386, 4458, 4494, 4501, 4554, 4600, 4614, 4649, 4656, 4723, 4851, 4887, 4892, 4980, 4981, 5001, 5009, 5027, 5056\n\nCorey, Professor Irwin 815\n\nCorey, Wendell 53, 402, 523, 548, 1478, 1680, 2351, 2555, 3244, 3338, 4755, 4962\n\nCormack, Bartlett 4071, 4580\n\nCorman, Gene 4699\n\nCorman, Julie 4916\n\nCorman, Roger 117, 1357, 1743, 2870\n\nCornell, Ann 1593\n\nCornthwaite, Robert 981, 996, 2058, 2583, 3439, 4160, 4794\n\nCorrado, Gino 277, 2517, 2586, 2790, 3036, 3300, 3384, 3610, 3770, 4020, 4917, 4413, 4750\n\nCorrell, Mady 2688, 2873, 4304\n\nCorri, Adrienne 40\n\nCorrigan, D'Arcy 2147, 2249, 3245, 4803\n\nCorrigan, Lloyd 1176, 1638, 1871, 2124, 2225, 2368, 2720, 2789, 2830, 3374, 3864, 4005, 4016, 4079, 4115, 4171, 4257, 5050\n\nCorrigan, Ray 101, 151, 369, 422, 573, 631, 810, 872, 1129, 1475, 1553, 1751, 1860, 1898, 2114, 2166, 2171, 2786, 2802, 2998, 3015, 3027, 3075, 3914, 3258, 3259, 3325, 3351, 3454, 3482, 3549, 3568, 2668, 3709, 4365, 4372, 4393, 4458, 4488, 4497, 4515, 4558, 4587, 4660, 4732, 4815, 4954, 5027, 5037\n\nCorson, William 5104\n\nCort, Bud 4045\n\nCortes, Carlos 750\n\nCortes, Nancy 72\n\nCortez, Ricardo 614, 1436, 1831, 2682, 3139, 3596\n\nCorthell, Herbert 3361, 3590\n\nCosby, Bill 2503\n\nCosgrove, Douglas 4995\n\nCossart, Ernest 3022\n\nCosta, Mario 268, 553\n\nCostello, Don 64, 1646, 2163, 2208, 2597, 2736, 2815, 3431, 4184, 4304\n\nCostello, Lou 2452, 3425, 3530, 5008\n\nCostello, Maurice 2160, 3626\n\nCostello, Wally 2636\n\nCoster Nicholas 849, 1219\n\nCostner, Kevin 942, 2920, 3911, 5029\n\nCotner, Carl 2635\n\nCotten, Joseph 482, 515, 919, 1187, 1644, 1765, 1833, 1841, 2237, 4426, 4527, 4611, 4674, 4896\n\nCotter, Catherine 2976, 3101, 4189, 4307, 4650\n\nCotton, Carolina 104, 409, 869, 1285, 1901, 2978, 3619, 3923, 3975, 3976, 4023, 4114, 4305, 4327\n\nCoulouris, George 602, 2174, 4057\n\nCounselman, William 1457, 2208, 2423, 2843, 3429\n\nCountry Joe and The Fish 5083\n\nCounts, Eleanor 430, 870, 875, 2081\n\nCourtland, Jerome 373, 895, 2515, 3023, 3635, 3702, 4310, 4457, 4471\n\nCourtnay, Chuck 339, 479, 857, 883, 1212, 3527, 3712, 4129, 4247, 4361, 4620\n\nCourtwright, William 4200\n\nCousin Emmy 4221\n\nCowan, Jerome 935, 1565, 2635, 3624, 3904\n\nCowell, Jack 2640, 4460, 4843\n\nCowles, Jules 5, 200, 898, 1322, 1591, 1863, 3360, 3609, 4239, 4257, 4776\n\nCowling, Bruce 73, 258, 1093, 1665, 2620, 3014\n\nCox, Buddy 3303, 3546\n\nCox, Morgan B. 1008, 1444, 3003, 3224, 3227, 3540, 4899, 5096, 5104\n\nCox, Ronny 80, 845\n\nCox, Victor 1, 234, 383, 427, 573, 858, 1309, 1345, 1456, 1505, 1506, 1980, 2189, 2250, 2288, 2363, 2457, 2673, 2740, 2759, 2786, 2858, 3086, 3160, 3269, 3303, 3344, 3353, 4082, 4108, 4112, 4225, 4226, 4231, 4819, 4848, 5104\n\nCox, Virginia 534\n\nCox, Wally 771, 4064, 5063\n\nCoxen, Ed (Edward) 15, 682, 780, 1148, 1301, 1545, 1677, 2119, 2121, 3109, 3375, 3456, 3876, 3969, 4042, 4071, 4275, 4315, 4395, 4816, 4861\n\nCoy, Walter 801, 1356, 1670, 1697, 1720, 2052, 2485, 3030, 3096, 3672, 3752, 4079, 4329, 4528, 4921, 5066\n\nCoyle, Ellen 1542, 2997\n\nCoyote, Peter 558, 4433\n\nCrabbe, Buster 145, 147, 148, 212, 338, 340, 344, 345, 382, 427, 486, 683, 797, 815, 1043, 1090, 1153, 1155, 1296, 1394, 1459, 1473, 1486, 1505, 1531, 1542, 1666, 1709, 1894, 2111, 2218, 2251, 2295, 2363, 2554, 2632, 2734, 2774, 2847, 2852, 2974, 2997, 3155, 3169, 3348, 3648, 3801, 3825, 4108, 4273, 4402, 4403, 4442, 4702, 4776, 4838, 4952, 4979\n\nCraig, Alec 432, 2423, 2835, 2838, 2877, 3431, 4257, 4979\n\nCraig, Carolyn 111, 1482, 1560, 1742\n\nCraig, Catherine 49, 1214\n\nCraig, Daniel 884\n\nCraig, James 132, 315, 471, 491, 1166, 1406, 1407, 1421, 1526, 1940, 2153, 2205, 2543, 2557, 2611, 2750, 2838, 2839, 2889, 3003, 3069, 3134, 3177, 3746, 3839, 4238, 4395, 4608, 4700, 5001\n\nCraig, Michael 636, 1014, 4471\n\nCraig, Nell 747, 811, 1591, 2147, 4803\n\nCraig, Yvonne 5071\n\nCrain, Jeanne 759, 1274, 1741, 2565, 3754\n\nCramer, Marc 27, 1827\n\nCramer, Richard (Dick) 138, 151, 251, 299, 338, 342, 344, 422, 436, 847, 1127, 1228, 1282, 1284, 1384, 1426, 1455, 1604, 1626, 1732, 1859, 1970, 1979, 2067, 2151, 2172, 2249, 2266, 2283, 2303, 2342, 2356, 2411, 2415, 2826, 2837, 2990, 2956, 3011, 3016, 3073, 3081, 3086, 3101, 3102, 3115, 3135, 3278, 3279, 3322, 3332, 3413, 3442, 3489, 3526, 3531, 3546, 3561, 3568, 3591, 3679, 3701, 3703, 3949, 4017, 4030, 4072, 4254, 4323, 4382, 4393, 4489, 4501, 4515, 4581, 4606, 4609, 4660, 4767, 4803, 4815, 4846, 4883, 4899, 4935\n\nCrandall, Suzi 2582, 4132\n\nCrane, Frank Hall 2608, 2742, 2766\n\nCrane, Fred 1516\n\nCrane, Richard 841, 1096, 1225, 1676, 2310, 2549, 3084, 3458, 3751, 4339, 4400, 5005, 5022\n\nCrane, Stephen 1249, 3305, 3306\n\nCravat, Nick 970, 3736, 4688, 4800\n\nCraven, Frank 246, 2016, 2159\n\nCraven, James 984, 1072, 2885, 3690, 3824, 4076, 4147, 4587, 4804, 4899\n\nCraven, John 842, 1434, 2378\n\nCrawford, Broderick 207, 210, 1274, 2112, 2204, 2226, 2638, 2715, 2830, 3338, 3912, 4312, 4325, 4498, 4868, 4970\n\nCrawford, Joan 2048, 2682, 4413, 5002\n\nCrawford, John 28, 122, 221, 259, 419, 828, 951, 1081, 1183, 1543, 2022, 2077, 2487, 2549, 3593, 2840, 2880, 3285, 3298, 3387, 3751, 3759, 4001, 4033, 4124\n\nCrawford, Johnny 1212, 1500, 1990, 2093\n\nCrawford, Kathryn 823, 2130, 2699, 3770\n\nCregar, Laird 1949\n\nCrehan, Joseph 25, 134, 191, 207, 337, 714, 790, 1127, 1224, 1427, 1436, 1490, 1512, 1532, 1588, 1740, 1767, 2074, 2101, 2174, 2638, 2778, 2806, 3128, 3312, 3344, 3414, 3630, 3690, 3711, 3750, 3902, 4086, 4183, 4286, 4312, 4468, 4513, 4601, 4665, 4745, 4932\n\nCrenna, Richard 253, 508, 590, 675, 1062, 1926, 2509, 3333, 3840, 4948\n\nThe Crew Chiefs 4222\n\nCrews, Laura Hope 1361, 3362\n\nCrichton, Michael 4859\n\nCripps, Kernan 357, 602, 642, 865, 1253, 1573, 1627, 1716, 1898, 2014, 2233, 2638, 2649, 3234, 3374, 3574, 3903, 3912, 3938, 4053, 4138, 4673, 4868, 3986\n\nCrisp, Donald 1131, 1151, 2064, 2525, 2864, 3247, 3671, 4064, 4697, 4890\n\nCrist, Harry P. 434, 1712, 2799, 3664, 4459, 5009; _see also_ Fraser, Harry\n\nCristal, Linda 43, 804, 821, 1289, 2209, 2598, 3868, 4626\n\nCristal, Perla 742, 3785, 4896\n\nCrockett, Dick 417, 1235, 1597, 3032, 4069, 4462, 4969\n\nCrockett, Luther 418, 801, 1601, 1956, 5013\n\nCromwell, John 4281\n\nCromwell, Richard 5072\n\nCronyn, Hume 4330\n\nCrosby, Bing 53, 3409, 3544, 4005, 4101\n\nCrosby, Bob 3925\n\nCrosby, Cathy Lee 3620\n\nCrosby, Dennis 3777\n\nCrosby, Gary 2076\n\nCrosby, Lindsay 315, 3712, 3777\n\nCrosby, Mary 4102\n\nCrosby, Philip 3777\n\nCrosby, Wade 68, 129, 731, 1247, 1526, 1811, 1974, 1982, 2077, 2882, 3020, 3344, 3414, 3608, 3611, 3613, 3759, 3870, 4057, 4168, 4185, 4336, 4425, 4641, 4692, 4761, 4786, 4859, 5008, 5021\n\nCrosland, Alan 653, 2610\n\nCrosland, Alan, Jr. 1485, 2754\n\nCross, Dennis 487, 1206, 2750, 2989\n\nCross, Peter 18, 400\n\nCrothers, Scatman 526, 3847\n\nCrowe, Russell 3206\n\nCrowley, Kathleen 91, 3090, 3211, 3781, 3853, 3909, 4252, 4856\n\nCrowley, Patricia (Pat) 3316, 4770\n\nCruise, Tom 1262\n\nCruze, James 852, 3139, 4215\n\nCrystal, Billy 760, 761\n\nCukor, George 1842\n\nCulp, Robert 672, 1642, 1776, 3223\n\nCulver, Howard 375, 513, 678\n\nCummings, Dwight 864, 2386, 3682, 4166, 4398\n\nCummings, Irving 279, 755, 1907, 1972\n\nCummings, Irving, Jr. 2208, 2423, 3429\n\nCummings, Robert (Bob) 145, 1043, 1830, 4101, 4284, 4803\n\nCummings, Susan 2521, 3762, 4451, 4682\n\nCummins, Dwight 864\n\nCummins, Peggy 1649\n\nCunard, Grace 916, 1345, 1426, 1646, 1863, 3650, 5001\n\nCuneo, Lester 2253\n\nCunningham, Cecil 880, 2790, 4786\n\nCunningham, Jack 852, 1369, 1426, 2232, 2554, 3442, 3655, 3880, 4203, 4271, 4289, 4403, 4442, 4652, 4762\n\nCuriel, Federico 168, 181, 1885, 2001, 2227, 4209, 4715\n\nCurley, Pauline 1055, 3496, 4397, 4877\n\nCurrie, Finlay 636, 2079\n\nCurrie, Louise 336, 342, 1174, 1416, 1693, 3100, 4127, 4969\n\nCurrie, Sondra 2039, 3527\n\nCurrier, Frank 5002\n\nCurtis, Alan 103, 939, 1224, 1438, 2223, 3351\n\nCurtis, Billy 1880, 2971, 4270, 4372\n\nCurtis, Bob 492, 1321, 1680, 2293, 4196\n\nCurtis, Carolyn (Clarene) 2645\n\nCurtis, Clarence 2107\n\nCurtis, Clarissa 3028\n\nCurtis, Dan 2229, 3749\n\nCurtis, Dick 1, 15, 135, 244, 332, 344, 386, 410, 423, 607, 623, 682, 776, 799, 854, 878, 897, 936, 952, 972, 1503, 1544, 1563, 1704, 1732, 1925, 2016, 2020, 2272, 2291, 2306, 2370, 2371, 2660, 2690, 2762, 2832, 2886, 2907, 2975, 2983, 2986, 3003, 3042, 3083, 3109, 3220, 3289, 3290, 3305, 3351, 3449, 3467, 3520, 3547, 3567, 3608, 3624, 3744, 3828, 3859, 3869, 3918, 4013, 4023, 4042, 4077, 4078, 4135, 4156, 4238, 4310, 4314, 4364, 4410, 4456, 4501, 4525, 4608, 4696, 4712, 4723, 4745, 4763, 4810, 4817, 4834, 4837, 4839, 4886, 4929, 4955, 4982, 4986, 5032, 5033\n\nCurtis, Donald (Don) 191, 777, 1768, 1974, 2269, 2797, 3629, 3792, 4000, 4118, 4226, 4312, 4391, 4456, 4465, 4855\n\nCurtis, Jack 975, 1002, 1436, 2129, 2300, 3260, 3656, 3985, 4316, 4494, 4639, 4676, 4803, 4854, 4900\n\nCurtis, Joan 4884\n\nCurtis, Ken 43, 618, 720, 822, 869, 1130, 1240, 1431, 1561, 1754, 1937, 1945, 2337, 2419, 2668, 2896, 2897, 2991, 3410, 3468, 3522, 3752, 3923, 4023, 4114, 4327, 4378, 4626, 5071\n\nCurtis, Tony 2084, 3295, 4989, 3295, 4989\n\nCurtiz, Michael 489, 808, 1127, 1594, 1773, 3186, 3537, 3711, 4637, 4742\n\nCurwood, James Oliver 184, 185, 265, 663, 776, 973, 1337, 1585, 2088, 2279, 2816, 3083, 3538, 3907, 3980, 4260, 4274, 4427, 4482, 4493, 4500, 4522, 4722, 4955, 5013, 5078\n\nCusack, Cyril 1262, 3757\n\nCusack, John 2015\n\nCushman, Ralph 2392\n\nCuster, Bob 77, 136, 785, 855, 1173, 1759, 1813, 2231, 2275, 2283, 2567, 2584, 3047, 3210, 3465, 3474, 3706, 3742, 4009, 4648, 4722\n\nCutting, John 1008, 4899\n\nCutting, Richard 759, 1161, 1635, 1691, 1709, 1937, 2209, 2246, 2252, 2287, 2544, 3223, 3415, 3439, 3769, 3849, 3856, 4244, 4784, 4787\n\nCypher, Jon 2099, 2793, 4688\n\nCyphers, Charles 463, 2375\nDade, Frances 3092, 3263\n\nDaheim, John 796, 927, 2899, 2917, 3816, 3890, 4011, 4076, 4210, 4955, 4980, 5000; _see also_ Day, John\n\nDahl, Alice 889, 1009, 4377, 4866, 4885\n\nDahl, Arlene 73, 1996, 2174, 2986, 4057, 4602\n\nDailey, Dan 966, 4411, 4419\n\nDaily, Mary 1768\n\nDalbert, Suzanne 4590\n\nD'Albrook, Sidney 4121, 4803\n\nDalbes, Alberto 918, 2903, 3030\n\nDale, Esther 2829, 2871, 3972, 4212, 4671\n\nDale, Jim 663, 1942\n\nDale, Suzan 3407\n\nDale, Virginia 541, 1811, 2108, 3922, 4171\n\nDaley, Cass 3316, 3499\n\nDaley, Jack 131, 213, 289, 471, 489, 664, 798, 973, 1146, 1546, 2147, 2163, 3202, 3411, 3938, 4395, 4823\n\nDallesandro, Joe 4195\n\nDallimore, Maurice 2157\n\nDalmas, Herbert 2218, 2827, 3025, 3673\n\nDalroy, Rube 15, 369, 382, 392, 414, 428, 431, 973, 1017, 1202, 1296, 1328, 1332, 1389, 1447, 1469, 1505, 1529, 1553, 1894, 1935, 2171, 2259, 2263, 2398, 2408, 2552, 2740, 2764, 2969, 3042, 3171, 3260, 3280, 3446, 3528, 3742, 3801, 3913, 3938, 3946, 3977, 4053, 4108, 4378, 4459, 4480, 4555, 4607, 4667, 4690, 4698, 4736, 4819, 4838\n\nDalton, Abby 789, 3127\n\nDalton, Audrey 486, 1161, 2426\n\nDalton, Dorothy 110\n\nDalton, Timothy 80\n\nDaly, James 1358\n\nDaly, Matt 87\n\nDaly, Tim 3312\n\nDalya, Jacqueline 1519, 2160, 4543, 4747\n\nD'Amato, Joe 999\n\nDamiani, Damiano 571, 1436\n\nDamita, Lili 1090, 1300, 1436\n\nDamler, John 1664, 1670, 2202\n\nDamon, Mark 61, 1648, 2312, 3366, 3513, 4578\n\nDamon, Matt 62, 1534, 1611\n\nDana, Leora 4370, 4969\n\nDana, Mark 182\n\nDana, Rod 2614; _see also_ Mark, Robert\n\nD'Andrea, Tom 3902\n\nDandridge, Ruby 642, 1914, 2160\n\nDane, Karl 331, 2682\n\nDane, Patricia 2838, 3530\n\nDaniell, Henry 808\n\nDaniels, Bebe 1854, 3529, 3895\n\nDaniels, Bette 3691\n\nDaniels, Hank 1982\n\nDaniels, Harold 964, 1137, 1256, 1865, 1904, 2867, 3425, 4485\n\nDaniels, Mark 4257\n\nDaniels, William 849\n\nDanner, Blythe 849, 1828, 2468, 3860\n\nDanning, Sybil 1589\n\nDano, Royal 60, 190, 290, 507, 594, 748, 909, 947, 980, 1015, 1133, 1263, 1641, 1676, 1727, 1778, 1946, 2048, 2185, 2229, 2550, 2556, 2897, 2950, 3149, 3305, 3317, 3641, 3671, 3731, 3780, 3950, 4259, 4333, 4548, 4638, 4802\n\nDanson, Ted 860\n\nDante, Michael 109, 148, 1399, 2322, 4830\n\nDantine, Helmut 517, 1236, 2830\n\nDanton, Ray 102, 733, 1963, 3196, 3683, 4073, 5055\n\nDarby, Ken 3354, 3851, 4112\n\nDarby, Kim 657, 4345, 4576\n\nDarcel, Denise 4383, 4727, 4857\n\nD'Arcy, Ann 2523, 3650\n\nD'Arcy, Roy 1440, 1517, 2523, 2961, 4647, 5002\n\nDarcy, Sheila 1925, 4052, 4450, 4587, 4665, 4803, 5104\n\nDardern, Severn 1893, 2199, 2934\n\nDare, Helena 778\n\nDare, Margaret 3281\n\nDarien, Frank 129, 160, 191, 282, 642, 670, 747, 1526, 1847, 1953, 1961, 2020, 2135, 2943, 3750, 4121, 4231, 4643, 4656, 4697, 4747, 4842, 4923\n\nDarin, Bobby 1703, 2598\n\nDa Rita, Joe 388, 502, 1422, 2964\n\nDark, Christopher 979, 1766, 1945, 2046, 3739, 4255, 4943\n\nDarling, Sally 1730\n\nDarling, W. Scott 1053, 1455, 1592, 1600, 2133, 3135, 3912, 5013\n\nDarmour, Larry 2056\n\nDarnell, Linda 373, 514, 552, 932, 2586, 4611\n\nDarr, Vondell 457\n\nDarrell, Steve 28, 65, 134, 389, 392, 667, 858, 979, 1213, 1313, 1460, 1487, 1531, 1543, 1695, 1836, 2043, 2068, 2287, 2363, 2462, 2806, 2893, 2942, 2980, 3050, 3063, 3149, 3157, 3185, 3337, 3447, 3464, 3585, 3614, 3690, 4002, 4322, 4240, 4273, 4333, 4356, 4425, 4514, 4539, 4641, 4645, 4646, 4682, 4691, 4818, 4989\n\nDarren, James 1719\n\nDarrin, Diana 521, 2381, 3754\n\nDarro, Frankie 721, 1089, 1626, 2301, 2367, 3075, 3215, 4036, 4120, 4320, 4616, 4687, 4703, 4708, 4857, 5036\n\nDarrow, Henry 2623, 2793\n\nDarrow, John 4089\n\nDarvas, Lili 748\n\nDarwell, Jane 1300, 1829, 2031, 2638, 2718, 3005, 3245, 3310, 3334, 3342, 3609, 4212, 4360, 4671, 4764, 4900\n\nDa Silva, Howard 203, 439, 2779, 2889, 2984, 4635, 5036\n\nDaugherty, Herschel 2351, 2739, 3223, 4990\n\nDavalos, Dick 1014, 2442\n\nDavalos, Ellysa 122\n\nDavenport, Doris 4851\n\nDavenport, Harry 866, 1247, 1594, 1620, 1622, 1811, 1953, 2016, 2064, 2543, 3005, 4115, 4803\n\nDavenport, Nigel 701, 2925\n\nDaves, Delmer 209, 519, 859, 1161, 1772, 2065, 2241, 3071, 3392, 4064, 4370, 4907\n\nDavi, Jana 1669, 1720\n\nDavid, Thayer 1170, 1828, 2373\n\nDavidson, John 3132\n\nDavidson, Max 958, 2254, 2458, 3126, 3545, 3581\n\nDavidson, Ronald 14, 37, 365, 637, 850, 871, 956, 1073, 2128, 2135, 2399, 2403, 2564, 3261, 3556, 4559, 5096, 5104\n\nDavidson, William B. 874, 1105, 1622, 1638, 1847, 1884, 1982, 2159, 2238, 2245, 2722, 3128, 3896, 4016, 4257\n\nDavies, Marion 2923\n\nDavies, Richard 3540, 4170\n\nDavila, Luis 1197, 2562, 2621, 3030\n\nDavis, Altovise 2140\n\nDavis, Art 38, 70, 773, 1728, 1969, 2420, 3018, 3077, 3164, 3232, 3592, 3603, 3681, 3944, 4082, 4302, 4303, 4589\n\nDavis, Bette 1588, 2064, 3071\n\nDavis, Chet 999, 3364\n\nDavis, Dennis 1601\n\nDavis, Frank 489, 4080\n\nDavis, Gail 53, 492, 858, 1264, 1606, 1991, 2264, 2885, 2894, 3000, 3006, 3890, 3943, 4036, 4044, 4285, 4496, 4692, 4757, 4828, 4886, 5005, 5079\n\nDavis, Jim 116, 192, 208, 319, 324, 482, 516, 612, 659, 691, 816, 1036, 1182, 1212, 1247, 1352, 1365, 1406, 1451, 1466, 1502, 1842, 1852, 1922, 1924, 2034, 2066, 2187, 2205, 2234, 2265, 2372, 2484, 2624, 2685, 2818, 2851, 2906, 2938, 2963, 3211, 3225, 3335, 3367, 3436, 3527, 3608, 3643, 3728, 3852, 3890, 4257, 4337, 4357, 4428, 4469, 4479, 4705, 4937, 5012, 5020, 5022\n\nDavis, Jimmie 928, 1449, 2265, 3505, 4086, 4170\n\nDavis, Joan 1658, 4533\n\nDavis, Karl 116, 2557, 2622, 3542, 4428\n\nDavis, Lisa 937, 1481, 1485, 1499\n\nDavis, Nancy 2013\n\nDavis, Ossie 3688, 3736\n\nDavis, Owen, Jr. 1853, 2469\n\nDavis, Roger 2437, 4345, 5063\n\nDavis, Rufe 777, 1504, 1513, 2421, 2971, 3025, 3084, 3165, 3230, 3673, 4166, 4483, 4649, 4811, 4855\n\nDavis, Sammy, Jr. 2379, 4479\n\nDavis, Shirley 3178\n\nDavis, Wee Willie (William) 1380, 1427, 3319, 4005, 4987, 5025\n\nDavison, Bruce 2738, 4633\n\nDavison, Davey 4788\n\nDavison, Robert W. 596\n\nDaw, Evelyn 3029\n\nDaw, Marjorie 2844\n\nDawber, Pam 4960\n\nDawn, Isabel 1571, 2158, 3916\n\nDawn, Norman 126, 2932\n\nDawn, Sugar 150, 1196, 2418, 3028, 3503, 4778\n\nDawson, Anthony M. 85, 1198; _see also_ Margheriti, Antonio\n\nDawson, Anthony 1004, 1016, 3336\n\nDawson, Hal K. 678, 935, 1248, 1428, 2302, 3234, 3499, 4225, 4435, 4803, 5049\n\nDax, Donna 1792, 1981, 3801\n\nDay, Alice 4607\n\nDay, Dennis 541\n\nDay, Doris 223, 597, 3675\n\nDay, John 209, 759, 1399, 2026, 2544, 3768, 3966, 4123, 4249, 4830; _see also_ Daheim, John\n\nDay, Laraine 198\n\nDay, Lynda 640, 741; _see also_ George, Lynda Day\n\nDay, Marceline 254, 1306, 1322, 1440, 1592, 3135, 4248, 4728, 4911\n\nDay, Robert 3205, 3959\n\nDay-Lewis, Daniel 2216, 4331\n\nDayton, Lyman D. 177, 216, 3315, 4879\n\nDeacon, Richard 1026, 2308, 2829, 3185, 3223\n\nDe Aguillon, Pedro 217\n\nDe Alcaniz, Luana 1468\n\nDean, Eddie 151, 364, 658, 711, 795, 1143, 1144, 1297, 1513, 1604, 1789, 1803, 1870, 1889, 2082, 2124, 2152, 2270, 2300, 2353, 2385, 2403, 2408, 2632, 2867, 2971, 3025, 3232, 3257, 3267, 3354, 3588, 3599, 3704, 3799, 3851, 3866, 4017, 4106, 4112, 4131, 4199, 4437, 4466, 4497, 4590, 4811, 4829, 4842, 4858, 4935, 4969, 4987\n\nDean, James 1560\n\nDean, Jean 4183\n\nDean, Margia 74, 229, 251, 1261, 1451, 1616, 1960, 2205, 2436, 2644, 3312, 3377, 3511, 4111, 4225, 4737, 4845\n\nDean, Max 400, 2015\n\nDean, Priscilla 2146\n\nDe Anda, Raul 124, 215, 583, 706, 707, 708, 821, 1189, 1615, 1886, 1911, 1912, 2346, 2708, 2709, 3197, 3347, 3868, 4039, 4176, 4177, 4415, 4544, 4632, 4717, 4752, 4967\n\nDe Anda, Rodolfo 52, 164, 583, 699, 707, 1189, 1615, 1886, 1911, 1912, 1994, 2346, 2574, 2708, 2709, 4176, 4177, 4415, 4547, 5105\n\nDeane, Hazel 1295, 1651, 3760\n\nDeane, Shirley 3163\n\nDearing, Edgar 248, 437, 541, 664, 1134, 1440, 1228, 1257, 1577, 1892, 2105, 2361, 2638, 2813, 3063, 3126, 3233, 3493, 3544, 3606, 3609, 3702, 3890, 4240, 4635, 4745, 4868, 5032\n\nDe Borba, Dorothy 2639\n\nDe Brulier, Edgar 614, 1511, 1831, 2349, 3561, 3695, 4747, 4750, 5096\n\nDe Camp, Rosemary 2575, 4536\n\nDe Carlo, Yvonne 132, 354, 388, 450, 599, 1632, 1450, 1487, 1940, 2267, 2587, 2626, 3054, 3287, 3534, 3684, 3696, 3741, 3849, 3891, 4450\n\nThe De Castro Sisters 2991\n\nde Cordoba, Pedro 807, 957, 964, 1820, 1981, 2064, 2270, 2552, 2586, 3245, 3267, 3561, 3562, 3596, 3609, 3692, 3859, 4871\n\nDe Cordova, Arturo 4175\n\nde Cordova, Frederick 803, 1487\n\nDe Corsia, Ted 402, 1354, 1664, 1702, 2295, 2563, 2672, 2779, 2789, 2818, 2868, 2986, 3208, 3440, 3722, 3856, 4725\n\nDee, Frances 686, 1420, 1757, 4802\n\nDee, Ruby 540\n\nDee, Sandra 965, 4924\n\nDeem, Mickey (Miles) 99, 999, 1116, 1418, 3727\n\nDe Fore, Don 3247, 4229\n\nDe Grasse, Sam 2603, 4926\n\nDe Haven, Gloria 239, 2982\n\nde Havilland, Olivia 1127, 1594, 3186, 3711, 4336\n\nDehner, John 42, 100, 111, 234, 604, 608, 639, 671, 727, 861, 895, 977, 1050, 1106, 1188, 1199, 1274, 1285, 1404, 1660, 1765, 1926, 1938, 1941, 1946, 2008, 2068, 2088, 2314, 2511, 2556, 2784, 2935, 3150, 3404, 3468, 3871, 3991, 4060, 4210, 4232, 4259, 4296, 4310, 4461, 4566, 4585, 4659, 4733, 4871, 4991\n\nde Hoyos, Kitty 181, 1885, 2001, 2042, 4715\n\nDein, Edward 913, 3787\n\nDein, Mildred 3787\n\nDekker, Albert 549, 602, 1247, 1392, 1478, 1480, 1925, 1954, 2081, 2109, 3276, 3684, 4333, 4786, 5021, 5024, 5033\n\nDe Kova, Frank 111, 166, 324, 884, 981, 1161, 2025, 2047, 2332, 2400, 2444, 2525, 3096, 3143, 3363, 3640, 4069, 4148, 4338, 4751, 4912, 4936\n\nDe La Cruz, Joe (Jose) 37, 592, 625, 1468, 1850, 2249, 2290, 2801, 2861, 3036, 3245, 3894, 4201, 4517, 4636, 4732, 4851, 5104\n\nde la Loma, Antonio 416\n\nde la Loma, Jose 769, 1355\n\nDe La Motte, Marguerite 2585, 2994, 3797, 4864\n\nDelaney, Charles 484, 1337, 1762, 2084, 2434, 3644, 3764, 4522, 5005\n\nDelaney, Dana 4453\n\nDe Lay, Melville 2276\n\nDe Leon, Walter 3409, 3499, 3634, 4665\n\nDelevanti, Cyril 939, 1669, 2488, 3432, 4566\n\nDell, Claudia 425, 1083, 1544, 4519\n\nDell, Gabriel 488\n\nDell, Myrna 157, 584, 1105, 1478, 1487, 1736, 2205, 2482, 2749, 3226, 3611, 3621, 4777\n\nDell'Acqua, Alberto 175, 421, 1305, 2016, 2569, 2656, 3033, 3785, 3955, 4288, 4597, 4678, 4720; _see also_ Widmark, Robert\n\nDell'Acqua, Roberto 92, 1764, 1801, 3037, 3382, 3819, 4664\n\nDelon, Alain 3336, 4287, 5058\n\nDel Ray, Pilar 86, 3861\n\nDel Rio, Dolores 720, 1371, 1471, 1572, 2516\n\nDel Rio, Dora 4507\n\ndel Rosario, Rosa 428\n\nDel Ruth, Roy 248\n\nDe Luise, Dom 81, 385, 1243\n\nDeMain, Gordon 592, 710, 873, 891, 1172, 1333, 1337, 1470, 1586, 1814, 1925, 2297, 2479, 2764, 2777, 3001, 3015, 3239, 3485, 4009, 4067, 4405, 4827; _see also_ Wood, Gordon D. (G.D.)\n\nDemarest, William 64, 264, 318, 1234, 1265, 1695, 1712, 2782, 3295, 3431, 4890, 5049\n\nDe Mario, Donna 110; _see also_ Martell, Donna\n\nDe Martino, Alberto 1117, 1858, 4263\n\nde Metz, Danielle 1185\n\nDemichelli, Tullio 1772\n\nDeMille, Cecil B. 504, 1570, 1571, 2831, 3126, 3609, 4005, 4091, 4092, 4093, 4494, 4635, 4665\n\nDeMille, Katherine 362, 614, 626, 1153, 1973, 2518, 3245, 4580, 4635, 4750\n\nDeMille, William 4174\n\nDeming, Norman 3003, 3449, 4238\n\nDeMond, Albert 291, 453, 779, 1074, 1504, 1513, 2125, 2823, 2971, 3178, 3475, 3483, 3673, 4487, 4695, 4811, 4938\n\nDemongoet, Mylene 3915\n\nDempster, Carol 3743\n\nDenger, Frank 1761, 3246\n\nDenison, Leslie 503\n\nDennehy, Brian 585, 998, 2783, 3396, 3911\n\nD'Ennery, Guy 853, 2147, 2586, 2607, 3165, 5104\n\nDenning, Richard 259, 550, 1140, 1532, 1691, 1774, 2831, 2870, 3741, 4284\n\nDennis, Nick 1422, 1610, 1727, 2005, 3376\n\nDennison, Jo 304\n\nDenny, Reginald 674, 1407, 2009\n\nDeNormand, George 637, 795, 1136, 1279, 1516, 1546, 2108, 2132, 2163, 2271, 2280, 2403, 2511, 2539, 2636, 2775, 2863, 2949, 2970, 2988, 3015, 3388, 3720, 3834, 3943, 4047, 4057, 4105, 4127, 4409, 4690, 4729, 4781, 4823\n\nDent, Vernon 418, 878, 958, 1136, 1474, 1792, 2419, 2975, 3355, 3506, 3569, 4678, 3695, 3750, 3914, 4023, 4294, 4327\n\nDenver, Bob 4733\n\nDenver, John 4772\n\nDepp, Johnny 992\n\nDerek, Bo 21\n\nDerek, John 1482, 1875, 2226, 2616, 2938, 3636, 4970\n\nDern, Bruce 62, 883, 996, 1780, 1791, 1997, 3146, 3687, 4211, 4790, 4794, 4930, 4988\n\nDerr, E.B. 1032, 3288, 3300\n\nDerwin, Hal 1240\n\nDe Sales, Francis 1248, 2023, 3397, 3449\n\nDe Santis, Joe 86, 407, 539, 2198, 3152, 3180, 4259\n\nDeShon, Nancy 3884, 4455, 5014\n\nDesmond, Mary Jo 2192\n\nDesmond, William (Bill) 137, 303, 398, 438, 468, 582, 642, 689, 726, 764, 786, 846, 862, 926, 1145, 1269, 1345, 1436, 1444, 1446, 1450, 1545, 1614, 1812, 1863, 2192, 2367, 2424, 2552, 2595, 2756, 2774, 2862, 2884, 3073, 3082, 3153, 3232, 3293, 3322, 3344, 3385, 3555, 3617, 3650, 3655, 3744, 3889, 4022, 4025, 4165\n\nDe Stefani, Joseph 242, 3254\n\nDe Teffe, Antonio 1118, 2214, 2814, 3180; _see also_ Steffen, Anthony\n\nDe Toth, Andre 484, 664, 981, 1208, 1989, 2004, 2549, 3247, 3500, 4080, 4163, 4390\n\nDe Valdez, Carlos 414, 1167, 2064, 2385, 3173, 3561, 4750\n\nDevane, William 2625, 4434\n\nDeverall, Helen 395, 477\n\nDevi, Kamala 523, 1533\n\nDevine, Andy 210, 223, 274, 287, 433, 541, 648, 1246, 1247, 1264, 1361, 1438, 1444, 1450, 1490, 1522, 1541, 1618, 1945, 2184, 2249, 2528, 2561, 2602, 2622, 2641, 2652, 2678, 2716, 2717, 2782, 2789, 2806, 2830, 2878, 2893, 2980, 2989, 2990, 3093, 3305, 3540, 3759, 3936, 3958, 4083, 4100, 4392, 4419, 4420, 4439, 4491, 4498, 4533, 4574, 4616, 4618, 4626, 4641, 4735, 4868, 5048, 5054\n\nDeVito, Danny 1590, 3735\n\nDevlin, Joe 349, 1214, 1224, 1361, 1925, 2864, 3367, 4258, 4336\n\nDevon, Richard 209, 212, 679, 808, 1709, 2676, 3746, 4361\n\nDevry, Elaine 727\n\nDew, Eddie (Edward) 210, 299, 927, 1174, 1308, 2163, 2716, 2748, 2884, 3115, 3327, 3357, 3478, 3504, 3741, 3805, 3938, 4329, 4419, 4504, 4556, 4999\n\nDewhurst, Colleen 883\n\nDe Wilde, Brandon 1948, 2668, 2797, 3808, 4255\n\nDe Wit, Jacqueline 2308, 2688, 2782, 4929\n\nDe Witt, Jack 260, 288, 638, 1132, 1665, 2508, 2551, 2926, 3369, 3379, 3571, 3931\n\nDe Wolf, Karen 1579, 3675, 3859\n\nDexter, Al 3120\n\nDexter, Alan 759, 2664, 3009\n\nDexter, Anthony 652, 3045\n\nDexter, Brad 482, 2002, 2056, 2141, 2240, 2497, 2871, 4339, 4672\n\nDexter, Elliott 4092\n\nDexter, Maury 1343, 2952, 3193, 4769, 4860, 5068\n\nDexter, Rosemarie 1108, 1381\n\nDiamond, Bobby 3748, 3909, 4673\n\nDiamond, Don 462, 3225, 5086\n\nDiamond, Leo 1564\n\nDiaz, Rudy 237, 831, 1081, 1133, 1372, 2487, 2489, 2906, 4134, 4638, 4996\n\nDibbs, Ken 955\n\nDi Caprio, Leonardo 3206\n\nDiCenzio, George 860\n\nDick, Douglas 1371, 1499, 2829, 2871, 3305, 4136\n\nDickerson, Beach 3732\n\nDickerson, Dudley 271, 2158, 3156\n\nDickey, Basil 28, 492, 535, 725, 764, 951, 956, 1287, 1367, 1614, 1680, 1863, 2035, 2125, 2248, 2293, 2743, 2927, 2933, 2979, 3067, 3081, 3087, 3088, 3277, 3450, 3555, 3650, 4011, 4144, 4549, 5001, 5103\n\nDickinson, Angie 375, 1014, 1692, 1871, 2148, 2185, 2563, 2897, 3376, 3517, 3640, 3688, 3838, 4258, 4259, 4970, 5060\n\nDickinson, Dick 242, 353, 785, 939, 1057, 1089, 1263, 1306, 1320, 1333, 1362, 1438, 1440, 1470, 1489, 1494, 1540, 1872, 1927, 1929, 2268, 2281, 2365, 2367, 2457, 2524, 2608, 2745, 3082, 3112, 3160, 3275, 3500, 3898, 4004, 4155, 4290, 4299, 4516, 4708, 4739, 4822\n\nDickson, Gloria 1820, 2127\n\nDiege, Samuel 2133, 3426, 3920\n\nDiehl, Jim 1304, 1806, 2176, 2309, 2397, 2991, 3277, 4053, 4154, 4278, 4812\n\nDiehl, John 558, 1267\n\nDierkes, John 43, 399, 550, 808, 1063, 1182, 1274, 1735, 1766, 1772, 1852, 2065, 2691, 2859, 2901, 2929, 3054, 3066, 3218, 3221, 3294, 3305, 3808, 3891, 3897, 4428, 4689, 4705, 4714\n\nDieterle, William 1187, 3318, 4257\n\nDietrich, Marlene 1084, 1361, 3255, 4072\n\nDigges, Dudley 2610\n\nDi Leo, Fernando 302, 537, 1580, 1793, 2054, 2438, 2758, 3380, 3659, 3785, 4262, 4678, 4779\n\nDillard, Art 28, 37, 70, 183, 244, 299, 334, 335, 340, 341, 344, 395, 442, 666, 683, 687, 729, 776, 800, 827, 848, 933, 987, 1017, 1023, 1052, 1073, 1090, 1100, 1130, 1138, 1241, 1307, 1332, 1462, 1472, 1473, 1474, 1531, 1543, 1544, 1553, 1595, 1756, 1818, 1860, 1862, 1870, 1894, 1916, 1974, 1983, 2022, 2177, 2220, 2251, 2262, 2285, 2288, 2298, 2299, 2310, 2403, 2409, 2414, 2421, 2514, 2547, 2564, 2593, 2594, 2652, 2673, 2730, 2759, 2785, 2802, 2816, 2827, 2867, 2886, 2894, 2956, 2965, 2968, 2975, 2999, 3027, 2074, 3100, 3107, 3109, 3111, 3164, 3242, 3269, 3279, 3280, 3308, 3348, 3356, 3444, 3454, 3483, 3497, 3525, 3548, 3581, 3649, 3673, 3675, 3724, 3801, 3803, 3825, 3827, 3866, 3884, 3994, 4011, 4059, 4303, 4400, 4410, 4458, 4480, 4525, 4555, 4586, 4589, 4609, 4656, 4667, 4701, 4712, 4730, 4767, 4838, 4938, 4954, 4980, 4982, 5018, 5041, 5056\n\nDillard, Bert 345, 369, 382, 422, 481, 729, 798, 974, 1046, 1090, 1332, 1459, 1531, 1542, 1971, 2163, 2251, 2276, 2298, 2363, 2399, 2421, 2734, 2957, 2965, 2975, 2999, 3243, 3279, 3673, 3825, 3968, 4078, 4317, 4357, 4458, 4589, 4690, 4838, 5018, 5104\n\nDillaway, Don 1462, 1721, 4240\n\nDillman, Bradford 1036, 2162, 2340, 3127\n\nDillon, John Francis 1570\n\nDillon, John Webb 755, 1102, 1972, 4704\n\nDillow, Dickie 3822\n\nDilson, John 15, 37, 144, 307, 552, 927, 943, 1884, 2124, 2132, 2153, 2159, 2342, 3111, 3216, 3362, 4096, 4121, 4199, 4225, 4401, 4572, 4850, 4979, 5050\n\nDinehart, Alan 844, 2131\n\nDingle, Charles 318, 1187, 4057, 4257\n\nThe Dinning Sisters 4179, 4327, 4378\n\nDinsdale, Howard 914, 3772, 4533\n\nDivine (Harris Glenn Milstead) 2483\n\nDix, Billy 318, 453, 632, 1463, 2106, 2511, 2986, 3238, 3331, 3607, 3887, 3940, 4272, 4820, 4937\n\nDix, Dorothy 1162, 1728, 2777, 4201, 4861\n\nDix, Richard 79, 159, 210, 549, 715, 747, 826, 2081, 2552, 3362, 3624, 4456, 4704, 4824, 5046\n\nDix, Robert 595, 1010, 1352, 1412, 2381, 2426, 4041, 4339, 5070\n\nDixon, Denver 4, 77, 289, 290, 334, 335, 344, 422, 446, 473, 492, 573, 597, 621, 774, 872, 946, 973, 1254, 1282, 1303, 1324, 1352, 1416, 1542, 1545, 1576, 1604, 1728, 1763, 1859, 1935, 1977, 2249, 2259, 2271, 2276, 2300, 2302, 2358, 2364, 2424, 2457, 2527, 2572, 2640, 2647, 2687, 2872, 2881, 2957, 2969, 3042, 3081, 3095, 3098, 3102, 3108, 3264, 3266, 3292, 3293, 3446, 3447, 3480, 3495, 3501, 3512, 3556, 3587, 3589, 3597, 4112, 4128, 4231, 4324, 4357, 4385, 4391, 4458, 4554, 4587, 4606, 4660, 4674, 4685, 4722, 4816, 4853, 4872, 4958; _see also_ Adamson, Victor\n\nDixon, Joan 1050, 1726, 1941, 3119\n\nDixon, Lee 89\n\nDmytryk, Edward 71, 520, 3807, 4493, 4791\n\nDobkin, Lawrence 208, 861, 1433, 2054, 2143, 2347, 3225, 3333, 3964, 4264, 4329, 4440, 4943\n\nDobson, James 1434, 4141, 4234\n\nDobson, Kevin 2934\n\nDodd, Claire 2610\n\nDodd, Jimmie 42, 395, 1601, 2250, 2485, 2720, 3475, 3593, 3708, 3805, 3921, 4409, 4598, 4695\n\nDoherty, Ethel 3573\n\nDoherty, Shannon 2374, 2377\n\nDolenz, George 3630, 5000\n\nDomergue, Faith 603, 1184, 1238, 1645, 3705\n\nDominguez, Joe 91, 198, 325, 439, 461, 1478, 1532, 1570, 1571, 1601, 1757, 2066, 2144, 2409, 2527, 2556, 2608, 2648, 2677, 2936, 2977, 2986, 3015, 3036, 3255, 3275, 3421, 3440, 3459, 3561, 3692, 3954, 3994, 3998, 4047, 4055, 4099, 4168, 4309, 4650, 4657, 4674, 4860, 4885\n\nDominici, Arturo 2390, 5095, 5101\n\nDonahue, Elinor 3921\n\nDonahue, Troy 1112, 2325, 4943\n\nDonaldson, Ted 3334\n\nDonan, Stanley 3780\n\nDonath, Ludwig 3355\n\nDonati, Sergio 220, 316, 751, 2898, 2904\n\nDoniger, Walter 65, 1078, 1735\n\nDonlan, James 270, 3011\n\nDonlevy, Brian 63, 132, 246, 332, 514, 648, 859, 1084, 1235, 1638, 1830, 1940, 2031, 2084, 3436, 3958, 4057, 4665, 4745, 4755, 4868, 5022\n\nDonnell, Jeff 869, 1735, 1901, 2496, 2617, 2942, 3342, 3621, 4023, 4107, 4327, 4378\n\nDonnelly, Ruth 1247, 1829, 1982, 2302, 2722, 3241, 3624, 3757, 4073, 4799\n\nDonner, Richard 4122\n\nDonner, Robert 348, 741, 1212, 1880, 1998, 2196, 2560, 2598, 2623, 2706, 2752, 2911, 3527, 3990, 4224, 4264, 4638, 5073, 5074\n\nDonohoe, Amanda 1930\n\nDonovan, Jack 1310, 4602\n\nDonovan, King 520, 859, 1640, 1772, 2008, 2372, 2543, 2664, 3339, 4588\n\nDoqui, Robert 4537\n\nDor, Karin 2214, 2219, 4540, 5003\n\nDoran, Ann 318, 447, 599, 671, 979, 1893, 2487, 2782, 3014, 3294, 3486, 3520, 3576, 3839, 4329, 4450, 4671, 4791\n\nDoran, Mary 2756, 3070, 4205\n\nDore, Adrienne 1566, 4658\n\nDore, Nadine 2268\n\nDoret, Nica 1017\n\nDorian, Angela 744, 4255\n\nDorn, Dolores 484\n\nDorn, Philip 1236, 1313\n\nDornys, Judith 2317\n\nDorr, Lester 602, 757, 776, 854, 943, 1348, 1904, 1906, 1969, 2649, 2804, 3083, 3195, 3339, 3374, 3424, 3426, 3470, 3563, 4798, 4803, 4865, 5018\n\nDors, Diana 1776\n\nD'Orsay, Fifi 1566\n\nDorsey, Tommy 1564\n\nDortort, David 320, 2485, 2486\n\nDoss, Tommy 2330, 3522, 4764\n\nDoucet, Catherine 1175, 1256\n\nDoucette, John 75, 234, 318, 320, 456, 482, 519, 562, 691, 759, 860, 1053, 1082, 1133, 1263, 1273, 1274, 1415, 1427, 1541, 1551, 1606, 1714, 1877, 2009, 2059, 2202, 2295, 2431, 2482, 2624, 2779, 2906, 2911, 2919, 3090, 3213, 3255, 3335, 3386, 3535, 3608, 3696, 3781, 3783, 3864, 3921, 4035, 4132, 4381, 4388, 4470, 4536, 4576, 4579, 4787, 4989, 4990, 5078\n\nDouglas, Diana 1989\n\nDouglas, Don (Donald) 1008, 1810, 2031, 2923, 4231\n\nDouglas, Earl 892, 943, 1148, 1299, 1716, 2712, 3361, 3407, 3477, 3503, 3561, 3953, 4553, 4554, 4947, 5001, 5077\n\nDouglas, George 354, 853, 1915, 2086, 2122, 2421, 2802, 2936, 3027, 3478, 3857, 4586\n\nDouglas, Gordon 255, 3120, 700, 744, 1136, 1289, 1399, 1573, 1596, 1640, 2005, 2780, 2781, 2919, 4101, 4328, 5055\n\nDouglas, Kirk 65, 324, 329, 1152, 1694, 1702, 1989, 2237, 2240, 2430, 2538, 2565, 3146, 3735, 4330, 4739, 4800\n\nDouglas, Linda 4241, 4486\n\nDouglas, Mary 3204\n\nDouglas, Melvyn 22, 95, 1948, 3750\n\nDouglas, Robert 252, 3718\n\nDouglas, Warren 1150, 1259, 2018, 2795, 2834, 2835, 2840, 3320, 3376, 4086\n\nDourif, Brad 1068, 1652, 1833\n\nDove, Billie 2352, 2422, 3985, 4776, 4949\n\nDover, Nancy 747\n\nDowling, Doris 3644\n\nDown, Lesley Ann 3735\n\nDowning, Barry 3077\n\nDowning, Maryan 1037\n\nDowning, Rex 499\n\nDowns, Cathy 2096, 2618, 2718, 2870, 3032, 3848\n\nDowns, Johnny 147, 236, 803, 1888, 3085, 4598\n\nDoyle, David 831, 2598, 4927\n\nDoyle, Maxine 710, 810, 1345, 2995, 3228, 3526, 3625, 3694, 4018, 4378\n\nDozier, Robert 4869\n\nDrago, Billy 959, 1401\n\nDrago, Harry Sinclair _see_ Lomax, Bliss\n\nDrake, Charles 807, 1423, 1676, 1747, 2396, 2813, 3853, 4536, 4770, 4783, 4989\n\nDrake, Claudia 446, 864, 1254, 1529, 1988, 2291, 2419, 2834, 3351, 3379\n\nDrake, Dona 1136, 1142, 3998\n\nDrake, Frances 4580\n\nDrake, Oliver 14, 144, 262, 267, 336, 338, 424, 430, 437, 477, 492, 721, 729, 818, 886, 929, 1003, 1150, 1154, 1311, 1321, 1338, 1553, 1678, 1679, 1686, 1730, 1751, 1822, 1898, 1952, 2082, 2106, 2247, 2293, 2303, 2407, 2435, 2484, 2687, 2850, 2951, 2959, 3025, 3045, 3099, 3187, 3194, 3216, 3227, 3232, 3238, 3353, 3419, 3457, 3482, 3549, 3556, 3582, 3625, 3665, 3690, 3681, 3742, 3859, 3939, 4026, 4081, 4196, 4307, 4321, 4337, 4490, 4500, 4507, 4515, 4558, 4570, 4572, 4586, 4711, 4728, 4785, 4809, 4819, 4820, 4862, 4954\n\nDrake, Peggy 2128\n\nDrake, Tom 1947, 2050, 2396, 2676, 3338, 3701, 3740, 4791\n\nDrayton, Noel 2379\n\nDreifuss, Arthur 4687\n\nDresden, Curley 15, 37, 38, 70, 139, 154, 183, 201, 299, 334, 335, 340, 342, 343, 344, 345, 366, 382, 410, 442, 452, 459, 479, 631, 665, 666, 682, 683, 925, 1023, 1024, 1042, 1090, 1332, 1338, 1473, 1504, 1558, 1626, 1627, 1682, 1767, 1768, 1785, 1859, 1860, 1977, 1980, 2027, 2032, 2086, 2111, 2132, 2197, 2276, 2299, 2399, 2407, 2408, 2409, 2412, 2413, 2414, 2415, 2546, 2635, 2667, 2701, 2737, 2786, 2802, 2886, 2909, 2968, 2969, 2975, 2994, 2998, 3015, 3027, 3029, 3164, 3165, 3914, 3226, 3232, 3259, 3325, 3390, 3408, 3434, 3475, 3478, 3553, 3574, 3591, 3592, 3603, 3708, 3709, 3805, 3825, 3989, 4000, 4052, 4059, 4206, 4318, 4483, 4589, 4612, 4617, 4643, 4649, 4655, 4702, 4759, 4786, 4819, 4853, 4855, 4952, 4956, 5037, 5041, 5104\n\nDresser, Louise 686\n\nDrew, Ann 3321\n\nDrew, Ellen 251, 541, 961, 1532, 1640, 2515, 2549, 2981, 3046, 4129, 4312\n\nDrew, Lowell 1682, 2425, 4803\n\nDrew, Paula 4735\n\nDrexel, Nancy 2281, 2524, 3608, 3048, 3486, 4290\n\nDriscoll, Bobby 309\n\nDriscoll, Tex 622, 1560, 2756, 2940, 3126, 3709, 4798\n\nDriver, Ada Belle 2231, 2257, 2584, 3427, 4602\n\nDru, Joanne 1151, 1775, 2351, 3323, 3392, 3814, 3861, 4060, 4764, 4924\n\nDrummond, Jane 1170\n\nDrury, James 55, 190, 507, 509, 563, 1086, 1341, 1500, 1608, 1840, 2241, 2437, 2462, 2631, 3435, 3941, 4253, 4746\n\nDruxman, Michael B. 730\n\nDubbins, Don 1703, 3641, 4548\n\nDubov, Paul 117, 1412, 1877, 2830\n\nDuBrey, Claire 283, 288, 514, 930, 1132, 1238, 1452, 2031, 2357, 2368, 3245, 3411, 4052, 4326, 4635, 4742, 4750\n\nDubs, Arthur R. 48, 1479, 2698, 4713, 4908, 4996, 5023\n\nDudgeon, Elspeth 2482\n\nDudley, Florence 3482\n\nDuel, Geoffrey 741, 3723\n\nDuel, Peter 55, 641, 3723, 5063\n\nDuff, Howard 377, 524, 599, 2161, 3310, 3865, 4977, 5049\n\nDuff, Warren 1436, 1594, 1829, 2187, 2864\n\nDuffy, Albert 1519\n\nDuffy, Jack 4317, 4519, 4933\n\nDuffy, Jesse 465, 2035, 2125, 3067, 3088, 4011, 5103\n\nDuffy, Patrick 2463\n\nDugan, Jan 1153, 2722, 3179, 4762\n\nDugan, Mary 4222\n\nDugan, Michael 1078, 1234, 1274, 2578, 3814, 4291, 4360\n\nDugay, Yvette 681, 749, 1129, 1866\n\nDuggan, Andrew 253, 267, 502, 1026, 1129, 1575\n\nDuggan, Tom 466, 633, 3082, 3695, 4487, 4585, 4708, 4742\n\nDuke, Patty 2461; _see also_ Astin, Patty Duke\n\nDullea, Keir 2333, 2500, 4801\n\nDumbrille, Douglass 115, 633, 652, 939, 967, 1228, 1247, 1254, 1362, 1416, 2080, 2128, 2221, 2301, 2480, 2756, 2923, 3131, 3425, 3481, 3544, 3624, 3626, 3655, 3728, 3954, 3963, 4005, 4646, 4742\n\nDumke, Ralph 842, 1775, 2517, 2664, 3234, 3696, 4338\n\nDumont, Margaret 4197\n\nDuna, Steffi 1562, 1865, 2270, 3538\n\nDunaway, Faye 1121, 2373, 2859\n\nDuncan, Arletta 1301, 1489\n\nDuncan, Bob 436, 658, 758, 795, 1326, 1363, 2271, 2601, 2687, 2789, 2841, 2946, 3045, 3088, 3257, 3744, 3999, 4026, 4081, 4363, 4590, 4709, 4729, 4858, 4969\n\nDuncan, Julie 70, 573, 878, 1475, 2999, 4302, 4318, 4323, 5041\n\nDuncan, Kenne 14, 107, 139, 179, 213, 286, 291, 334, 338, 340, 342, 344, 382, 383, 389, 430, 597, 607, 644, 724, 726, 777, 784, 792, 827, 836, 893, 930, 956, 969, 987, 1008, 1166, 1196, 1360, 1370, 1377, 1445, 1461, 1464, 1472, 1626, 1684, 1767, 1868, 1873, 1890, 1918, 1982, 1991, 2040, 2107, 2111, 2135, 2172, 2176, 2179, 2251, 2282, 2286, 2293, 2301, 2408, 2411, 2537, 2542, 2545, 2600, 2667, 2673, 2711, 2712, 2727, 2754, 2776, 2797, 2828, 2863, 2868, 2894, 2928, 2965, 2967, 2977, 2995, 2999, 3006, 3088, 3098, 3327, 3399, 3404, 3447, 3503, 3524, 3556, 3585, 3587, 3613, 3677, 3694, 3707, 3708, 3802, 3822, 3824, 3826, 3890, 3929, 3946, 3953, 3989, 4016, 4036, 4118, 4181, 4183, 4185, 4212, 4302, 4303, 4310, 4313, 4319, 4420, 4487, 4490, 4502, 4515, 4549, 4584, 4599, 4658, 4695, 4701, 4734, 4759, 4763, 4781, 4786, 4813, 4833, 4855, 4886, 4899, 4957, 5018\n\nDuncan, Mary 3595\n\nDuncan, Pamela 1664, 2294, 3788, 4616, 4894\n\nDuncan, Renault 1132; _see also_ Renaldo, Duncan\n\nDuncan, Slim 418, 922, 1460, 3552, 4296\n\nDuncan, Tommy 4044\n\nDuncan, William 242, 1394, 1469, 1933, 2253, 2270, 2774, 3263, 4312, 4367, 4395\n\nDunham, Phil 8, 602, 687, 774, 777, 1284, 1322, 1345, 1470, 1550, 1760, 1965, 2548, 2777, 3062, 3153, 3239, 3494, 3601, 4143, 4220, 4512, 4619, 4815, 4833, 4961\n\nDunlap, Scott R. 298\n\nDunn, Bobby 511, 646, 785, 2458, 2767, 3035, 3047, 3153, 3555, 3631, 4050, 4521, 4756, 4798, 4861\n\nDunn, Eddie 272, 318, 544, 617, 648, 757, 1029, 1136, 1450, 1814, 1853, 1960, 2167, 2344, 2357, 2543, 2649, 3465, 3609, 3684, 4006, 4193, 4803\n\nDunn, Emma 866, 874, 2385, 4281\n\nDunn, James 1827, 2809, 3941\n\nDunn, Josephine 295\n\nDunn, Mary 4362\n\nDunn, Ralph 64, 337, 514, 602, 863, 961, 1029, 1080, 1247, 1313, 1458, 1600, 2159, 2403, 2790, 3384, 3772, 3921, 4170, 4212, 4222, 4498, 4543, 4771, 4849\n\nDunne, Harvey B. 2899, 4725\n\nDunne, Irene 1883, 2782\n\nDunne, Philip 2213, 2216, 4795\n\nDunne, Stephen 325, 2088, 2261\n\nDunnock, Mildred 2462, 4751\n\nDupuis, Art 158, 927, 1140, 1433, 1565, 2213, 2586, 2646, 2782\n\nDuran, Edna 1159\n\nDurand, David 3895, 4007, 4586, 4750, 4803\n\nDurante, Jimmy 2635\n\nDurbin, Deanna 642\n\nDurfee, Minta 3588\n\nDurkin, James 825, 1829, 4051\n\nDurkin, Junior 1685, 3710\n\nDurlam, G.A. (George Arthur) 5, 6, 300, 422, 772, 785, 917, 1626, 1970, 2355, 2434, 2517, 2681, 2764, 2862, 3038, 3051, 3307, 3465, 4605, 4648\n\nDurning, Charles 2094\n\nDuryea, Dan 42, 64, 354, 486, 1428, 1809, 1891, 1985, 2578, 2797, 3234, 3896, 3933, 3964, 4161, 4223, 4989, 4990\n\nDuryea, George 270, 1178, 1701, 3041, 4413; _see also_ Keene, Tom and Powers, Richard\n\nDuryea, Peter 486, 2613, 4223\n\nDusay, Marj 768\n\nDusenberry, Ann 1081\n\nDuval, Diane 1863; _see also_ Bishop, Julie and Wells, Jacqueline\n\nDuval, Maria 232, 2383, 3347\n\nDuvall, Robert 525, 1534, 1641, 2044, 2305, 2433, 2920, 4576\n\nDuvall, Shelly 554, 2625\n\nDvorak, Ann 1, 1360, 1829, 2610, 3377, 3757, 4797\n\nDwan, Allan 91, 277, 681, 1458, 2678, 2836, 3054, 3367, 3536, 3897, 4212, 4258, 4413, 4498, 5022\n\nDwan, Dorothy 619, 1315, 1636\n\nDwire, Earl 54, 56, 137, 143, 297, 311, 411, 468, 471, 668, 687, 689, 848, 897, 946, 974, 1047, 1127, 1137, 1177, 1191, 1221, 1323, 1440, 1470, 1491, 1494, 1503, 1550, 1574, 1595, 1626, 1674, 1683, 1812, 1900, 1927, 1965, 2073, 2110, 2127, 2129, 2146, 2203, 2249, 2268, 2281, 2283, 2297, 2300, 2356, 2479, 2524, 2529, 2548, 2608, 2660, 2733, 2740, 2766, 2785, 2816, 2850, 2862, 2872, 2968, 3036, 3062, 3101, 3914, 3256, 3329, 3443, 3451, 3456, 3459, 3476, 3489, 3545, 3597, 3664, 3680, 3703, 3949, 3968, 4004, 4022, 4059, 4125, 4130, 4189, 4290, 4423, 4448, 4455, 4482, 4569, 4572, 4573, 4581, 4612, 4636, 4655, 4760, 4822, 4843, 4854, 4981, 5014, 5061\n\nDyer, William (Bill) 772, 1049, 1677, 1927, 3451, 3680, 4290, 4518, 4931\n\nDylan, Bob 3055\n\nDyneley, Peter 709\n\nDysart, Richard 849, 2925, 3019\n\nEagles, James 1860, 2610, 3016, 3573, 4205, 4442\n\nEarl, Elizabeth 3538\n\nEarle, Edward 47, 94, 459, 464, 610, 710, 853, 1487, 1774, 1792, 1980, 1981, 2016, 2124, 2150, 2292, 2552, 2660, 2664, 2737, 2899, 3078, 3086, 3222, 3401, 3454, 3534, 3695, 4163, 4310, 4803, 4991\n\nEason, B. Reeves 410, 630, 834, 1221, 2101, 2164, 2212, 2283, 2660, 2701, 2737, 2765, 2824, 3075, 3161, 3172, 3329, 3511, 3551, 4084, 4207, 4275, 4557, 4708\n\nEast, Carlos 168, 406, 988\n\nEast, Jeff 2148\n\nEastman, George 220, 281, 627, 1117, 1119, 1794, 2097, 2724, 3735, 4366, 4999, 4664\n\nEastman, Gordon 3733\n\nEaston, Robert 1166, 1986, 2215, 3009, 3221, 4090, 4426\n\nEastwood, Clint 74, 526, 831, 1348, 1350, 1381, 1612, 1769, 1880, 2044, 2950, 3009, 3019, 4123, 4240, 4624, 4663, 4970\n\nEaton, Evelyn 4684\n\nEberhardt, Norma 2502\n\nEbsen, Buddy 965, 967, 970, 1485, 1571, 2108, 2437, 2500, 3316, 3578, 3892, 4137, 4381, 4644, 4686\n\nEburne, Maude 277, 442, 790, 973, 1904, 2534, 2701, 3132, 3454, 3558, 3634\n\nEby, Lois 2399\n\nEddy, Duane 3732, 4387, 4973\n\nEddy, Helen Jerome 1639, 2147\n\nEddy, Nelson 1571, 2344, 2756, 2790, 2836, 3009, 3605\n\nEdelman, Herb 1828, 3926\n\nEden, Barbara 1371, 3783\n\nEden, Chana 4993\n\nEdeson, Robert 398, 504, 2181, 2892, 3041, 3166, 3595\n\nEdmiston, James 977\n\nEdmunds, William 325\n\nEdward, Bill 1330, 3437, 4500, 4745\n\nEdwards, Alan 4052\n\nEdwards, Anthony 1209\n\nEdwards, Blake 2600, 3032, 4118, 4195, 4969\n\nEdwards, Bruce 433, 1105, 1400, 2294, 2680, 2863, 3151, 4420, 4825\n\nEdwards, Cliff \"Ukulele Ike\" 79, 179, 200, 205, 230, 1308, 1378, 1571, 2513, 2682, 3001, 3115, 3171, 3327, 3466, 3679, 4184, 4391, 4797, 4827\n\nEdwards, Edgar 1013, 3431, 5001\n\nEdwards, Elaine 2880\n\nEdwards, James 831, 1535, 2319, 3779\n\nEdwards, Julie 1352\n\nEdwards, Kay 4570\n\nEdwards, Mark 416\n\nEdwards, Neely 3912, 4170, 4215\n\nEdwards, Penny 656, 937, 1823, 1971, 2825, 3143, 3150, 3417, 4076, 4198, 4488, 4621, 4686\n\nEdwards, Ronnie Claire 4345\n\nEdwards, Sam 596, 1985, 3739\n\nEdwards, Sarah 605, 664, 1256, 1392, 1594, 1923, 2471, 2790, 3022, 3634, 4013, 4037, 4199\n\nEdwards, Snitz 2585\n\nEdwards, Thornton 1143, 1158, 1800, 2409, 2471, 3460, 3562, 3905, 4364, 4701\n\nEdwards, Vincent 1075, 1866, 1892, 3432\n\nEdwards, Weston 2203, 2766, 3653, 3949, 4961\n\nEgan, Richard 258, 982, 2084, 2462, 2613, 2663, 3386, 3781, 3840, 40069, 4217, 4333, 4343, 4672, 5036\n\nEggar, Samantha 971, 4801\n\nEggers, Fred 4392, 4539\n\nEgner, Red 3802\n\nThe Eight Buckaroos 4598\n\nEikenberry, Jill 585, 2934, 3633\n\nEilers, Sally 767, 835, 944, 1907, 2440, 3551, 4099, 4557\n\nEisenmann, Ike 239\n\nEisley, Anthony 4337\n\nEkberg, Anita 1422, 2664, 4689\n\nElam, Jack 122, 172, 258, 584, 681, 771, 808, 842, 965, 1106, 1147, 1150, 1205, 1263, 1346, 1422, 1623, 1665, 1702, 1776, 1807, 1808, 1876, 1877, 1942, 1946, 2065, 2066, 2156, 2185, 2228, 2237, 2474, 2525, 2565, 2684, 2691, 2744, 2784, 2795, 2797, 2897, 2898, 2989, 3040, 3055, 3141, 3255, 3283, 3290, 3320, 3423, 3440, 3527, 3661, 3662, 3715, 3860, 3882, 4210, 4211, 4388, 4411, 4727, 4739, 4800, 4873, 4921, 4936, 4994\n\nElcar, Dana 506, 1641, 1694, 2265, 3987\n\nElder, Ray 2171, 4017, 4053, 4155\n\nThe Elder Lovelies 4085\n\nEldredge, George 503, 589, 604, 642, 664, 1097, 1184, 1247, 1255, 1348, 1433, 1444, 1450, 1456, 1658, 1747, 1805, 2184, 2424, 2544, 2564, 2837, 2866, 2957, 2996, 3209, 3215, 3227, 3547, 3550, 3630, 3651, 3864, 3899, 4006, 4016, 4024, 4032, 4080, 4085, 4118, 4226, 4261, 4336, 4461, 4552, 4556\n\nEldredge, John 204, 433, 1356, 1431, 2502, 3652, 4890, 5045\n\nElizondo, Hector 3136, 4688, 4782\n\nElliot, Kathleen 3044, 4130\n\nElliot, Laura 3891, 4062\n\nElliott, Biff 982, 4579\n\nElliott, Bill (Gordon\/Wild Bill\/William) 15, 286, 307, 349, 424, 464, 575, 607, 634, 731, 793, 827, 1023, 1247, 1268, 1415, 1468, 1490, 1626, 1646, 1740, 1768, 1843, 1969, 1982, 2085, 2119, 2184, 2256, 2420, 2425, 2427, 2448, 2546, 2547, 2591, 2597, 2600, 2632, 2673, 2689, 2821, 2827, 2878, 2995, 3003, 3080, 3109, 3128, 3159, 3170, 3298, 3370, 3393, 3550, 3584, 3728, 3822, 3824, 3852, 4181, 4238, 4462, 4513, 4584, 4701, 4723, 4731, 4734, 4754, 4759, 4763, 4970, 4980, 5033\n\nElliott, Dick 1, 9, 49, 161, 869, 1175, 1182, 1247, 1285, 1398, 1458, 1516, 1877, 1923, 2529, 2556, 2635, 2678, 2909, 2940, 3020, 3179, 3255, 3454, 3567, 3611, 3917, 4199, 4212, 4327, 4386, 4439, 4500, 4601, 4673, 4845, 4979, 5059\n\nElliott, Edythe 575, 870, 1247, 1920, 2633, 4659, 4687, 4695\n\nElliott, Janis 4050\n\nElliott, John 77, 107, 134, 180, 214, 277, 311, 334, 344, 382, 452, 510, 564, 670, 683, 812, 825, 856, 877, 945, 1017, 1018, 1128, 1323, 1327, 1338, 1339, 1440, 1453, 1455, 1458, 1465, 1474, 1486, 1489, 1496, 1667, 1810, 1816, 1824, 1872, 2031, 2102, 2114, 2120, 2266, 2276, 2290, 2365, 2421, 2473, 2645, 2681, 2687, 2742, 2777, 2786, 2862, 2866, 2918, 2933, 2945, 2999, 3067, 3068, 3081, 3085, 3115, 3227, 3232, 3236, 3243, 3443, 3459, 3489, 3525, 3531, 3545, 3548, 3592, 3664, 3668, 3679, 3706, 3927, 3952, 3967, 3975, 4031, 4050, 4078, 4205, 4261, 4303, 4306, 4448, 4455, 4485, 4522, 4549, 4555, 4586, 4587, 4605, 4606, 4636, 4655, 4660, 4722, 4760, 4777, 4951, 4952, 5018, 5059\n\nElliott, Kathleen 4816\n\nElliott, Laura 1039\n\nElliott, Robert 653, 1097, 1378, 2639, 4898, 4899\n\nElliott, Ross 190, 507, 661, 778, 980, 1048, 1199, 1341, 1487, 1941, 2517, 4240, 4479, 4588\n\nElliott, Sam 558, 822, 1079, 1607, 1944, 1962, 2674, 3205, 3273, 3661, 3798, 4453, 4968, 5057\n\nElliott, Stephen 849, 2000, 4362\n\nEllis, Edward 2552, 4580, 4776\n\nEllis, Frank 5, 6, 14, 28, 49, 68, 70, 104, 135, 139, 151, 157, 174, 183, 205, 230, 244, 289, 304, 306, 327, 334, 340, 341, 345, 366, 369, 382, 383, 387, 413, 423, 427, 434, 460, 461, 476, 479, 511, 564, 582, 606, 607, 631, 643, 665, 682, 683, 711, 722, 731, 739, 778, 779, 818, 836, 850, 851, 856, 867, 872, 873, 877, 1001, 1003, 1023, 1025, 1028, 1057, 1080, 1090, 1146, 1172, 1191, 1228, 1229, 1271, 1288, 1302, 1309, 1320, 1322, 1324, 1333, 1340, 1367, 1445, 1448, 1459, 1461, 1473, 1474, 1488, 1503, 1531, 1543, 1544, 1556, 1563, 1574, 1626, 1635, 1677, 1781, 1687, 1732, 1855, 1859, 1894, 1907, 1932, 1933, 1959, 1969, 1974, 1976, 1980, 1983, 1991, 1993, 2068, 2073, 2114, 2167, 2170, 2177, 2226, 2235, 2250, 2256, 2276, 2281, 2282, 2288, 2300, 2355, 2363, 2371, 2392, 2396, 2399, 2403, 2408, 2409, 2410, 2412, 2413, 2414, 2415, 2420, 2471, 2526, 2527, 2540, 2549, 2573, 2589, 2599, 2660, 2673, 2678, 2699, 2722, 2734, 2737, 2786, 2822, 2847, 2866, 2885, 2886, 2990, 2907, 2927, 2945, 2951, 2953, 2957, 2965, 2974, 2975, 2977, 2995, 2997, 3006, 3025, 3029, 3048, 3074, 3075, 3078, 3086, 3087, 3092, 3100, 3126, 3153, 3155, 3162, 3164, 3165, 3172, 3187, 3194, 3210, 3222, 3226, 3232, 3242, 3259, 3260, 3291, 3322, 3353, 3388, 3426, 3452, 3482, 3525, 3528, 3539, 3545, 3547, 3548, 3568, 3574, 3586, 3587, 3597, 3615, 3619, 3626, 3651, 3658, 3704, 3797, 3801, 3802, 3827, 3848, 3880, 3884, 3900, 3944, 3994, 4000, 4001, 4011, 4032, 4037, 4059, 4082, 4096, 4097, 4103, 4112, 4124, 4127, 4131, 4135, 4156, 4163, 4197, 4206, 4207, 4208, 4220, 4245, 4248, 4273, 4294, 4297, 4302, 4304, 4309, 4319, 4323, 4324, 4327, 4363, 4364, 4377, 4396, 4405, 4458, 4460, 4480, 4484, 4490, 4497, 4502, 4515, 4521, 4525, 4534, 4535, 4559, 4587, 4590, 4600, 4606, 4609, 4610, 4617, 4619, 4645, 4649, 4657, 4668, 4690, 4701, 4728, 4729, 4732, 4759, 4763, 4785, 4806, 4809, 4833, 4838, 4839, 4842, 4849, 4852, 4853, 4854, 4858, 4863, 4865, 4889, 4892, 4894, 4944, 4952, 4954, 4956, 4971, 4987, 5001, 5014, 5027, 5033, 5040, 5041, 5049, 5096, 5104\n\nEllis, Mirko 553, 1119, 1135, 1793, 2532, 4580, 4776\n\nEllis, Robert 811, 958, 1002, 1058, 1306, 1329, 2307, 2471, 2552, 2573, 2702, 2889, 2908, 3096, 4214, 4309, 4377, 4535, 4898\n\nEllison, James 243, 254, 461, 621, 794, 896, 997, 1202, 1240, 1272, 1824, 1931, 1939, 1956, 2221, 2596, 2863, 3126, 4283, 4292, 4301, 4367, 4485, 4821, 4894\n\nEllsler, Effie 1153\n\nElorrieta, Jose Maria 2612, 4721; _see also_ Lacy, Joe\n\nEly, Ron 904, 1289, 1764, 2795\n\nEmerson, Faye 203, 1735, 4932\n\nEmerson, Hope 60, 277, 832, 2486, 3066, 4672, 4857\n\nEmerson, John 4926\n\nEmery, Gilbert 1591, 2064, 2128, 3538, 3874\n\nEmery, John 934, 1433, 2302\n\nEmery, Katherine 1866, 4674\n\nEmhardt, Robert 209, 1940, 2005, 2305, 2389, 2793, 4370\n\nEmmet, Michael 1692\n\nEmmett, Fern 230, 337, 411, 579, 642, 810, 939, 1080, 1308, 1414, 1444, 1458, 1579, 1586, 1638, 1712, 1834, 1847, 1974, 2032, 2167, 2170, 2391, 2998, 3242, 3276, 3362, 3451, 3482, 3485, 3487, 3675, 3817, 3859, 3888, 4037, 4185, 4269, 4317, 4484, 4494, 4564, 4762, 4803, 4809, 5061\n\nEmmett, Robert 616, 1587, 1900, 2418, 2542, 2572, 2640, 2647, 2785, 2993, 3016, 3036, 3235, 3237, 3456, 3476, 3503, 3588, 3597, 3599, 3905, 3913, 3937, 4017, 4022, 4236, 4424, 4554, 4778, 4796, 4833, 4844, 4854, 4872, 4883, 4884, 4947; _see also_ Tansey, Robert Emmett\n\nEmory, Richard 228, 784, 842, 1523, 1844, 2294, 2372, 3068, 3921, 4044, 5039\n\nEndore, Guy 2160\n\nEngel, Roy 60, 841, 842, 1451, 1837, 2140, 2199, 2246, 2747, 3583, 4141, 4373, 4548, 4869\n\nEngel, Samuel M. 2718, 3431, 3596, 4747\n\nEngle, Billy 3486\n\nEngle, Paul 1748, 2289, 2910, 3069\n\nEnglish, John 37, 137, 304, 389, 631, 777, 858, 864, 956, 995, 1023, 1134, 1504, 1523, 1805, 1890, 1895, 1991, 2132, 2135, 2179, 2233, 2386, 2399, 2403, 2546, 2711, 2995, 3084, 3228, 3230, 3308, 3447, 3481, 3510, 3694, 3890, 3893, 4036, 4166, 4409, 4508, 4681, 4692, 4695, 4855, 4886, 4893, 5096, 5104\n\nEnglish, Marla 1377, 3316\n\nEnright, Ray 49, 203, 835, 1364, 2084, 2638, 2677, 3383, 3538, 3912, 4029, 4049, 4072, 4254, 4476, 4932\n\nEpperson, Don 319, 595\n\nErdman, Richard 536, 3294, 3671, 3696\n\nErickson, Helen 848\n\nErickson, Leif (Glenn) 470, 749, 965, 1043, 1153, 1274, 2503, 2774, 2895, 3066, 3414, 3838, 3852, 4123, 4776, 4968, 5007\n\nEricson, John 193, 487, 935, 979, 1412, 2926, 3376, 3783, 4721\n\nErnest, George 1002, 4025, 4127, 4494\n\nErnest, Harry 1861\n\nErrol, Leon 2647, 2649, 410, 4598, 4654\n\nErskine, Chester 1430\n\nErskine, Laurie York 943\n\nErskine, Marilyn 4857\n\nErwin, Stuart 948, 1176, 1383, 1830, 4171, 4652, 4750, 4868\n\nEsmond, Carl 4384, 5025\n\nEssex, Harry 1011, 1092, 2503, 3287, 4035, 4060, 5036\n\nEssler, Fred 1348, 4099, 4670\n\nEssoe, Gabe 1098\n\nEstabrook, Howard 197, 681, 747, 3054, 4744\n\nEstelita _see_ Rodriguez, Estelita\n\nEstevez, Emilio 5067, 5069\n\nEstrada, Erik 2447, 3204\n\nEstrella, Esther 1143, 1974, 2353, 3164, 3165, 4364, 4657\n\nEthier, Alphonse 328, 480, 1252, 1972, 2249, 2637, 3700, 4139, 4208\n\nEvans, Charles 9, 359, 664, 801, 1478, 2103, 2225, 2720, 3066, 3154, 3295, 3701, 3702, 4099, 4599, 4714\n\nEvans, Dale 67, 110, 284, 286, 287, 868, 1134, 1141, 1603, 1836, 1914, 2368, 2534, 2632, 2727, 2935, 3024, 3236, 3585, 3694, 4013, 4016, 4043, 4197, 4213, 4551, 4597, 4646, 4681, 4786, 5050\n\nEvans, Douglas 134, 605, 759, 871, 899, 950, 1373, 1603, 1658, 1690, 1884, 2132, 2825, 3151, 3652, 4056, 4358, 4520\n\nEvans, Gene 45, 114, 221, 487, 502, 608, 681, 935, 1527, 1596, 1773, 2188, 2487, 2616, 2719, 2779, 2782, 2897, 3055, 3661, 3798, 3840, 3860, 3882, 4179, 4210, 4211, 4330, 4641, 4755, 4790, 4968, 5036, 5038\n\nEvans, Herbert 3766\n\nEvans, Jack 8, 234, 334, 337, 338, 341, 379, 382, 392, 411, 422, 431, 440, 443, 460, 461, 494, 573, 576, 616, 687, 729, 834, 889, 974, 1017, 1057, 1097, 1229, 1241, 1301, 1303, 1309, 1328, 1332, 1333, 1334, 1361, 1389, 1455, 1463, 1469, 1505, 1506, 1531, 1540, 1542, 1587, 1626, 1681, 1728, 1813, 1827, 1861, 1904, 1935, 1965, 2119, 2166, 2171, 2173, 2254, 2259, 2263, 2268, 2276, 2315, 2355, 2363, 2364, 2371, 2398, 2411, 2451, 2479, 2481, 2524, 2527, 2547, 2589, 2640, 2676, 2690, 2730, 2785, 2821, 2847, 2915, 2956, 2999, 3062, 3073, 3081, 3159, 3169, 3242, 3270, 3279, 3292, 3303, 3348, 3385, 3401, 3446, 3447, 3489, 3492, 3497, 3528, 3546, 3601, 3617, 3658, 3771, 3801, 3825, 3888, 3913, 3938, 3949, 3952, 3994, 4000, 4004, 4009, 4044, 4050, 4053, 4089, 4108, 4238, 4272, 4274, 4294, 4298, 4365, 4458, 4465, 4480, 4555, 4559, 4587, 4589, 4606, 4607, 4609, 4684, 4696, 4698, 4722, 4736, 4819, 4838, 4846, 4861, 4862, 5056\n\nEvans, Jacqueline 953, 955\n\nEvans, Joan 803, 2813\n\nEvans, Linda 1277, 1500, 2093, 2752, 4122, 4449\n\nEvans, Madge 4807\n\nEvans, Muriel 478, 480, 621, 1829, 2128, 2785, 3555, 3587, 3657, 3903, 3967, 4367, 4379, 4833\n\nEvans, Robert 1289\n\nEvanson, Edith 318, 2158, 3211, 3290, 3339, 3808, 3906, 4163\n\nEvelyn, Judith 1560\n\nEverett, Chad 2051, 2185, 3387\n\nEverett, Tom 44, 45, 333, 942, 2650, 4405\n\nEvers, Ann 1465, 1805, 3454, 4803\n\nEvers, Jason 766, 2506, 3740\n\nEverton, Paul 1679, 2256, 3162, 4121, 4257, 4564, 4665\n\nEwell, Tom 1042, 2452\n\nEyer, Richard 650, 1399, 1434, 3221\n\nEythe, William 3005, 4062\n\nEyton, Bessie 1817, 1984, 3308, 4070, 4603\n\nFabares, Shelley 3686\n\nFabian (Forte) 2829\n\nFabian, Francoise 1160\n\nFabray, Nanette 771\n\nFadden, Tom 207, 318, 602, 718, 935, 1084, 1093, 1106, 1166, 1175, 1205, 1236, 1371, 1444, 2083, 2163, 2293, 2423, 2516, 2835, 3195, 3630, 3817, 3921, 4184, 4233, 4470, 4725, 5001\n\nFago, Giovanni 1477\n\nFahey, Jeff 5026, 5029\n\nFahey, Myra 1248\n\nFain, Matty 478, 2315, 2482, 2496, 5049\n\nFair, Eleanor 326, 2799\n\nFairbanks, Douglas 1131, 1511, 2536, 2585, 2603, 4926\n\nFairbanks, Douglas, Jr. 4949\n\nFairbanks, Lucille 2149\n\nFairbanks, William 2520\n\nFairchild, Morgan 3317\n\nFaire, Virginia Brown 56, 511, 600, 2428, 2434, 3869, 4280, 4476, 4480, 4482, 4521, 4822\n\nFairfax, James 260, 2705\n\nFajardo, Eduardo 19, 61, 118, 202, 225, 1113, 1135, 1530, 1858, 1885, 2001, 2390, 2495, 2643, 3117, 3514, 3516, 3810, 4246, 4366, 5047\n\nFalk, Peter 4993\n\nFalkenburg, Jinx 2403, 4019\n\nFanning, Frank 3427, 3552, 4405\n\nFantasia, Franco 220, 553, 1794, 1954, 3405, 4012, 4369, 5087, 5088, 5092\n\nFarentino, James 3439\n\nFarley, Albert 1582, 4594, 4596\n\nFarley, Dot 145, 2304, 3194, 3657, 3694, 4152, 4532\n\nFarley, James (Jim) 231, 248, 611, 851, 1002, 1127, 1300, 1436, 1456, 1526, 1594, 1877, 1972, 1974, 1981, 2161, 1472, 2543, 2595, 2838, 2843, 3291, 3362, 3450, 3695, 3711, 3889, 3953, 4025, 4203, 4294, 4498, 4742, 4854, 4931, 4983, 5001\n\nFarley, Morgan 252, 2918, 2934, 4062, 4962\n\nFarmer, Donald 397\n\nFarmer, Frances 210, 1378, 3409, 3414\n\nFarmer, Mimsy 4964\n\nFarnsworth, Richard 816, 1068, 1170, 1183, 1539, 1650, 2335, 2950, 3136, 3323, 4040, 4287, 4449, 4960\n\nFarnum, Dustin 4091, 4492, 4743\n\nFarnum, Dustine 241\n\nFarnum, Franklyn 264, 300, 305, 438, 597, 623, 731, 797, 801, 917, 1008, 1023, 1247, 1254, 1299, 1446, 1449, 1455, 1545, 1647, 1658, 1762, 1928, 1931, 1950, 1969, 2152, 2226, 2302, 2355, 2418, 2584, 2678, 2843, 2862, 2878, 3126, 3153, 3234, 3270, 3591, 3597, 3666, 3888, 3899, 3902, 4127, 4163, 4252, 4289, 4367, 4585, 4649, 5043\n\nFarnum, William 37, 79, 297, 477, 493, 917, 964, 1028, 1149, 1154, 1202, 1340, 1369, 1444, 1504, 1574, 1585, 2003, 2110, 2145, 22007, 2208, 2352, 2399, 2416, 2422, 2423, 2638, 2648, 3011, 3046, 3153, 3187, 3240, 3469, 3593, 3626, 3709, 3832, 3889, 4052, 4070, 4071, 4072, 4257, 4488, 4732, 4987\n\nFarr, Felicia 1347, 1837, 2065, 2241, 3363, 4370\n\nFarr, Hugh 67, 110, 286, 287, 323, 386, 576, 611, 615, 623, 682, 799, 868, 1105, 1134, 1192, 1240, 1246, 1488, 1522, 1767, 1818, 1836, 1914, 1918, 1964, 2124, 2272, 2368, 2514, 2530, 2534, 2540, 2727, 2806, 2828, 2874, 2877, 2886, 2893, 2956, 2973, 2975, 2983, 3100, 3236, 3300, 3409, 3449, 3483, 3520, 3522, 3585, 3600, 3694, 3904, 4013, 4016, 4018, 4025, 4037, 4042, 4051, 4077, 4083, 4156, 4197, 4202, 4206, 4314, 4315, 4401, 4410, 4608, 4641, 4646, 4681, 4764, 4806, 4810, 4817, 4834, 5050\n\nFarr, Jamie 1225, 3422\n\nFarr, Karl 67, 110, 286, 287, 323, 386, 576, 611, 615, 623, 682, 799, 868, 1105, 1134, 1192, 1240, 1246, 1488, 1522, 1767, 1818, 1836, 1914, 1918, 1964, 2124, 2271, 2330, 2368, 2514, 2530, 2534, 2540, 2727, 2806, 2828, 2874, 2877, 2886, 2893, 2973, 2975, 2983, 3100, 3236, 3300, 3409, 3449, 3483, 3520, 3522, 3585, 3600, 3694, 3904, 4013, 4016, 4018, 4025, 4037, 4042, 4051, 4077, 4083, 4156, 4197, 4202, 4206, 4314, 4315, 4401, 4410, 4608, 4641, 4646, 4681, 4764, 4806, 4180, 4817, 4834, 5050\n\nFarrar, Stanley 211, 1248\n\nFarrare, Christine 2077\n\nFarrell, Charles 765\n\nFarrell, Conchata 1826, 2819\n\nFarrell, Glenda 115, 1829, 2150\n\nFarrell, Sharon 2229, 2429\n\nFarrell, Tommy 2, 791, 1711, 1713, 2798, 2970, 3547, 4001, 4136, 5039\n\nFarrow, John 572, 602, 832, 1921, 3131, 3362, 3440, 5015\n\nFaulk, John Henry 2468\n\nFaulkner, Edward 223, 741, 1998, 2381, 2626, 3527, 3739, 3806, 3816, 3990, 4412, 4638\n\nFaust, Martin 3245, 4106, 5104\n\nFaversham, Philip 2610\n\nFaversham, William 137, 3918\n\nFawcett, Charles 2227, 2883, 3730\n\nFawcett, Farrah 1609\n\nFawcett, George 1634, 3743, 4413\n\nFawcett, William 250, 364, 637, 649, 650, 680, 698, 711, 777, 932, 967, 972, 1063, 1157, 1555, 1608, 1673, 1711, 1747, 1890, 1940, 2034, 2085, 2109, 2161, 2287, 2308, 2370, 2448, 2591, 2609, 2652, 2680, 2856, 3103, 3204, 3222, 3509, 3547, 3586, 3608, 3637, 3769, 4070, 4080, 4095, 4124, 4131, 4232, 4278, 4428, 4437, 4590, 4620, 4692, 4781, 4924, 4935, 5049\n\nFay, Ann 3994\n\nFay, Dorothy 1464, 2278, 2821, 3160, 3235, 3237, 3590, 4019, 4152, 4186, 4553, 4899\n\nFay, Frank 4639\n\nFaye, Julia 602, 832, 2831, 4092, 4093, 4635, 4665\n\nFaye, Randall 496, 731, 1345, 1646, 2180, 4294\n\nFaylen, Frank 402, 602, 714, 832, 1201, 1702, 1810, 2395, 2485, 2781, 3053, 3156, 3316, 3792, 4867, 4890\n\nFazenda, Louise 1685\n\nFeatherston, Eddie 1008, 1134, 1228, 1560, 2505, 3063, 3485, 3953, 4899, 5077\n\nFehmiu, Bekim 1062\n\nFeist, Felix 260, 328, 2504\n\nFeld, Fritz 1422, 2319, 3500, 3859, 4225\n\nFelice, Lyle 1763\n\nFelix, Art 1045, 1805, 2003, 2263, 2356, 2371, 2399, 1403, 2589, 2640, 3413, 3458, 3884, 5096\n\nFelix, Maria 2063, 3988\n\nFelker, Texas Rex 4515, 4587\n\nFell, Norman 1660\n\nFellows, Edith 1821, 2247, 2937, 3442, 4127\n\nFellows, Rockcliffe 3360, 3658\n\nFelmy, Hansjorg 3113\n\nFelton, Verna 1704, 1741, 2513, 2789, 2837, 2871\n\nFenaday, Andrew J. 370, 741, 1922, 3422\n\nFender, Freddy 2650, 3813\n\nFenton, Earl 226, 1818, 3295, 3866, 4206\n\nFenton, Frank 27, 226, 552, 947, 1136, 1140, 1234, 1247, 1481, 1507, 1603, 1672, 1692, 2025, 2334, 2578, 2749, 2980, 3168, 3271, 3344, 3356, 3440, 3500, 3535, 3583, 3646, 3866, 3891, 3991, 4132, 4167, 4206, 4285, 4551, 4672, 4890, 4962, 5036\n\nFenton, Leslie 2516, 3339, 3168\n\nFerber, Edna 747, 748\n\nFerguson, Al 65, 76, 231, 303, 342, 390, 402, 602, 764, 813, 935, 1008, 1025, 1057, 1090, 1247, 1321, 1433, 1468, 1488, 1516, 1579, 1585, 1688, 1920, 1969, 2020, 2022, 2177, 2246, 2276, 2363, 2451, 2720, 2722, 2735, 2745, 2763, 2826, 2841, 2853, 2889, 2915, 2940, 2957, 2974, 3067, 3068, 3169, 3188, 3314, 3322, 3465, 3509, 3545, 3561, 3575, 3599, 3607, 3625, 3648, 3650, 3657, 3668, 3684, 3772, 3869, 4005, 4011, 4032, 4121, 4208, 4260, 4273, 4278, 4319, 4406, 4635, 4684, 4700, 4725, 4729, 4745, 4756, 4803, 4823, 4836, 4853, 4899, 4969\n\nFerguson, Frank 170, 290, 602, 648, 749, 759, 789, 1161, 1225, 1395, 1433, 1478, 1640, 1644, 1668, 1775, 1845, 2005, 2008, 2048, 2295, 2302, 2351, 2396, 2487, 2556, 2591, 2856, 2938, 3090, 3150, 3208, 3214, 3255, 2576, 3602, 3702, 4124, 4267, 4336, 4381, 4645, 4740, 4766, 4792, 4925, 5022\n\nFerguson, Helen 2070\n\nFernandez, Abel 114, 1026, 1409, 2241, 2575, 3518, 4136\n\nFernandez, Emilio 119, 517, 699, 706, 1190, 1994, 2167, 2860, 3055, 3391, 3406, 3873, 3988, 4193, 4464, 4790, 4835, 4934, 5090\n\nFernandez, Jaime 52, 217, 569, 571, 591, 600, 837, 980, 1386, 1661, 1731, 1856, 1912, 2014, 2579, 2611, 2708, 2791, 3008, 3343, 4416, 4526, 5091\n\nFernandez, Vicente 163, 1210, 1910, 2062\n\nFerraday, Lisa 604, 2095, 3255\n\nFerrer, Mel 3255\n\nFerroni, Giorgio 400; _see also_ Paget, Kelvin Jackson\n\nFerzetti, Gabriele 2898\n\nFetchit, Stephin 290, 4944\n\nFiedler, John 223, 1631, 1734, 1899, 2739, 4576\n\nField, Betty 831, 3817, 4058\n\nField, Charlotte 3177\n\nField, Margaret 3222, 5079\n\nField, Mary 874, 1311, 1423, 1947, 2163, 2342, 2668, 3053, 3415, 3791, 4358, 4483, 4635, 4940\n\nField, Sally 512, 1899, 4800\n\nField, Shirley Anne 2141\n\nField, Virginia 757, 1949\n\nFielding, Edward 210, 278\n\nFields, Stanley 441, 747, 1083, 1521, 1563, 1572, 1907, 2127, 2159, 2458, 2654, 2790, 2990, 3012, 3029, 3470, 3575, 3584, 4747, 4798, 4803, 4925, 5032\n\nFields, W.C. 2722\n\nFierro, Paul 1103, 2926, 3053, 3066, 3287, 3323, 3534, 3690, 4111, 4784, 5000, 5044\n\nFilmer, Robert 318, 695, 718, 798, 1215, 1309, 2400, 2815, 3093, 3095, 4398, 4751\n\nFimple, Dennis 1237, 1590, 1807, 1926, 2229, 2623, 2721, 2897, 3687, 3793, 4753, 5007\n\nFinch, Flora 4798\n\nFinch, Peter 1239, 3560\n\nFinch, Scott 2509, 3807\n\nFindlay, Ruth 1550, 1859, 2203, 3062\n\nFine, Budd 94, 1153, 2458, 3374, 3702, 3750, 4308\n\nFine, Larry 1422, 1597, 2964, 3569\n\nFink, Harry Julian 319, 594, 2501\n\nFink, Rita M. 319, 594\n\nFinkle, Robert 1475, 5027\n\nFinlayson, James 1618\n\nFinley, Evelyn 16, 138, 369, 864, 872, 1196, 1540, 1724, 2016, 2711, 3068, 3823, 4188, 4225, 4502, 4702, 4857\n\nFinley, George 18, 400, 1530\n\nFinley, Sylvia 3559\n\nFinn, Mickey 53, 1569, 1985, 2901, 340, 4266, 4435\n\nFinney, Charles G. 3783\n\nFinney, Edward 2134, 3905\n\nFirestone, Eddie 501, 1637, 2246, 2486, 2506, 3737, 4122\n\nFisher, Kai 574\n\nFisher, Shug 67, 286, 672, 673, 720, 919, 1105, 1240, 1561, 1733, 1762, 1836, 1918, 2368, 2534, 2561, 2727, 2893, 3236, 3468, 3512, 3585, 3661, 3694, 3776, 3816, 4013, 4016, 4083, 3114, 4426, 4646, 4681, 4764, 5050\n\nFisher, Steve 132, 373, 548, 1406, 1411, 1676, 1940, 2050, 2188, 2267, 2544, 3338, 3367, 3690, 4461, 4755, 5022, 5065\n\nFiske, Richard 15, 129, 2153, 2540, 2633, 2821, 2973, 3003, 3067, 3109, 3170, 4000, 4156, 4238, 4723\n\nFiske, Robert 69, 107, 258, 662, 679, 790, 799, 881, 928, 995, 1008, 1167, 1626, 2250, 2257, 2540, 2547, 2828, 2879, 3003, 3194, 3216, 3553, 4022, 4042, 4208, 4290, 4318, 4410, 4423, 4700, 4701, 4817\n\nFite, Buster, and His Six Saddle Tramps 3584\n\nFitzgerald, Barry 602, 3891\n\nFitzgerald, Ella 3425\n\nFitzroy, Emily 414, 1361, 1469\n\nFix, Paul 91, 174, 223, 243, 432, 563, 564, 594, 612, 930, 980, 1039, 1057, 1092, 1106, 1143, 1202, 1212, 1245, 1269, 1313, 1329, 1360, 1560, 1562, 1623, 1678, 1679, 1843, 1855, 1895, 1921, 1946, 2048, 2500, 2550, 2667, 2717, 2779, 2797, 2984, 3083, 3132, 3179, 3323, 3422, 3426, 3575, 3816, 3837, 3990, 3994, 4035, 4051, 4111, 4123, 4124, 4147, 4212, 4231, 4379, 4498, 4564, 4638, 4703, 4742, 4773, 4775, 4786, 4792, 4802, 4840, 4990, 5054, 5060\n\nFlaherty, Pat 89, 626, 1127, 1228, 1247, 1827, 1884, 2528, 4543\n\nFlanagan, Bud 579, 747, 2223, 3136, 3409, 3896; _see also_ O'Keefe, Dennis\n\nFlanagan, Fionnuala 1583, 2491, 2917\n\nFlanders, Ed 3686\n\nFlash (dog) 631\n\nFlavin, James 101, 279, 493, 757, 923, 2031, 2350, 2471, 2517, 2628, 2665, 2797, 2831, 2851, 3132, 3367, 3414, 3425, 3431, 3563, 3692, 3728, 4124, 4286, 4675, 4849, 4868\n\nFleischer, Richard 128, 226, 4065, 4333\n\nFleming, Alice 607, 731, 793, 827, 1646, 2427, 2597, 2600, 2995, 3080, 3693, 3822, 3824, 4181, 4584, 4734, 4763\n\nFleming, Bob 327, 1058, 2422, 3498, 3994, 4248, 4297, 4570, 4605, 4851\n\nFleming, Eric 913\n\nFleming, Rhonda 1, 53, 190, 577, 1201, 1673, 1702, 1995, 2225, 3140, 3339, 4258, 4349, 4786\n\nFleming, Susan 1588, 1854, 3260\n\nFleming, Victor 4744, 5015\n\nFletcher, Bill 55, 1354, 1843\n\nFletcher, Tex 3944\n\nFlint, Sam 28, 49, 67, 251, 277, 389, 475, 516, 602, 710, 716, 842, 857, 1254, 1404, 1420, 1433, 1516, 1541, 1600, 1806, 2081, 2084, 2235, 2299, 2300, 2368, 2419, 2432, 2534, 2539, 2691, 2727, 2785, 2815, 2840, 2878, 2963, 2969, 2970, 3095, 3167, 3215, 3329, 3383, 3541, 3553, 3567, 3794, 3839, 3893, 3923, 3929, 3975, 3978, 4023, 4024, 4051, 4057, 4153, 4166, 4222, 4225, 4409, 4421, 4574, 4643, 4714, 4938, 4979, 4995, 5078\n\nFlippen, Jay C. 290, 674, 1033, 1092, 1235, 1263, 1346, 1766, 1945, 2118, 2161, 2565, 2792, 2853, 3133, 3640, 2687, 3792, 3941, 4611, 4989, 5020\n\nFlippen, Lucy Lee 1590, 3158\n\nFlocker, James T. 57, 1559, 1656, 3758, 5023\n\nFlores, Iris 4047\n\nFlorey, Med 419, 1689, 2795, 3612, 4064, 4927\n\nFlori, Agata 1030\n\nFlorio, Aldo 1355\n\nFlower, George \"Buck\" 10, 37, 587, 655, 1085, 1479, 1777, 2047, 2698, 3961\n\nFlowers, Bess 597, 1256, 1361, 1564, 2302, 4891\n\nFlowers, Morgan 1, 244, 382, 1486, 1505, 2847, 2864, 3164, 4702, 4838, 5018\n\nFluellen, Joel 585, 1256, 1434, 1809, 3931, 3950, 4436\n\nFlynn, Emmett 2843\n\nFlynn, Errol 1127, 1185, 2677, 2835, 2572, 3692, 3711, 3902, 4336\n\nFlynn, Joe 1527\n\nFlynn, Maurice \"Lefty\" 1602, 2238\n\nFlynn, Sean 1185, 3786\n\nFocas, Spiros 1794\n\nFoch, Nina 1277, 1423\n\nFodor, Ladislas 2783, 3198, 4542, 4919\n\nFoley, Red 3107\n\nFonda, Bridget 2045\n\nFonda, Henry 317, 469, 727, 1165, 1346, 1395, 1471, 1945, 2031, 2379, 2718, 2723, 2898, 3005, 3320, 3374, 3622, 3993, 4061, 4064, 4330, 4435, 4494, 4661, 4775, 4791, 4802, 4940, 3970, 5072\n\nFonda, Jane 674, 816, 1219, 2876\n\nFonda, Peter 78, 1807, 1893, 2199, 3013, 4045, 4371, 4775\n\nFong, Benson 2154, 2155, 2815, 4768\n\nFons, Jorge 2056\n\nFontaine, Jacqueline 940, 2960, 3951\n\nFontaine, Joan 2552\n\nForan, Dick 42, 327, 387, 515, 611, 714, 874, 1041, 1099, 1214, 1221, 1395, 1740, 1820, 2164, 2689, 2722, 3071, 3172, 3276, 3425, 3450, 3540, 3865, 4025, 4223, 4513, 4534, 4539, 5001\n\nForbes, John 1957, 2959; _see also_ Carpenter, John (Johnny)\n\nForbes, Mary 2149\n\nForbes, Ralph 952\n\nForbes, Scott 3285, 3572\n\nFord, Constance 1913, 2008, 2198, 2555, 3838\n\nFord, Dorothy 1666, 2815, 3040, 3066, 4360\n\nFord, Francis 159, 196, 233, 264, 602, 764, 797, 916, 1070, 1083, 1165, 1246, 1264, 1395, 1454, 1532, 1570, 1571, 1591, 1614, 1677, 1863, 1993, 1999, 2128, 2208, 2292, 2349, 2471, 2516, 2527, 2601, 2703, 2718, 2966, 3005, 3126, 3132, 3179, 3355, 3442, 3471, 3596, 3691, 3692, 3766, 3814, 4020, 4038, 4100, 4116, 4284, 4336, 4358, 4360, 4425, 4470, 4491, 4747, 4764, 4849, 4987, 5072\n\nFord, Fritz 696, 2199, 3769\n\nFord, Glenn 22, 82, 455, 748, 859, 980, 1070, 1274, 1579, 1831, 1832, 2065, 2185, 2482, 2515, 2544, 2598, 3131, 3217, 3339, 3622, 3661, 3686, 3712, 3757, 3815, 3959, 3962, 4286, 4370, 4423, 4740, 4970\n\nFord, Grace 4274\n\nFord, Harrison 884, 1437, 1998, 2023, 2059, 4431\n\nFord, John 546, 720, 1165, 1395, 1471, 1937, 1945, 2004, 2070, 2223, 2561, 2718, 2940, 3179, 3522, 3752, 3776, 3814, 4100, 4145, 4352, 4360, 4626, 4764, 5072\n\nFord, Judith 781\n\nFord, Mary _see_ Summers, Colleen\n\nFord, Montgomery 4444; _see also_ Halsey, Brett\n\nFord, Paul 317, 2109, 2482, 2668\n\nFord, Peter 2598, 3183, 2686, 3959\n\nFord, Philip 91, 233, 415, 544, 605, 923, 1040, 1072, 1074, 2264, 2592, 2875, 2942, 3104, 3151, 3178, 3271, 3342, 3578, 3691, 4048, 4425, 4686, 4712, 4804, 4938, 5034\n\nFord, Ross 1029, 1439, 2261, 3614, 3902\n\nFord, Ruth 1100, 3550\n\nFord, Steven 767\n\nFord, Tennessee Ernie 2549\n\nFord, Wallace 282, 489, 835, 1082, 1347, 1478, 1635, 2046, 2302, 2486, 2525, 2543, 2624, 2768, 3244, 3335, 3576, 4111, 4388, 4791, 4792, 4921\n\nForde, Eugene 314, 844, 3971\n\nForeman, Carl 1877\n\nForest, Michael (Mike) 455, 838, 2695, 2846, 2903, 3364, 3878, 4699\n\nForman, Carol 786, 4653\n\nForman, Joey 4601\n\nForman, Tom 73, 387, 614, 647, 722, 1388, 2012, 2558, 3026, 3289, 4446, 4743, 4849\n\nForrest, Allan 948\n\nForrest, Fredric (Frederic) 598, 2433, 2666, 4869\n\nForrest, Helen 1537, 3665\n\nForrest, Sally 4725\n\nForrest, Steve 1034, 1371, 1754, 1770, 1842, 2215, 3620, 3683, 3755, 4782, 4936\n\nForrest, William 283, 339, 1247, 1395, 1407, 1937, 2899, 2901, 3040, 3219, 4069, 4076, 4086, 4336, 4469, 4500, 5075\n\nForrester, Kay (Cay) 383, 970, 2694, 4024\n\nForster, Robert 2058, 2752, 4113, 4122\n\nForsyth, Rosemary 3816, 4287\n\nForsythe, John 1234, 2835\n\nForsythe, Stephen 1968\n\nFort, Garrett 4909\n\nForte, Josef (Joe) 373, 1481, 1702, 2025, 2135, 2267, 3027, 3397, 3447, 3578, 3741, 4329\n\nFoster, Dianne 2095, 2797, 4362, 4740\n\nFoster, Helen 413, 2473, 5061\n\nFoster, Jodie 2623, 2906\n\nFoster, Lewis R. 47, 381, 642, 932, 1201, 1214, 2225, 3053, 3871, 4349, 4457, 4714\n\nFoster, Norman 515, 967, 970, 1532, 1572, 1990, 2322, 2757, 2809, 3214, 3429, 3954, 4747\n\nFoster, Preston 95, 159, 1792, 1829, 2138, 2252, 2519, 2601, 2684, 2719, 2831, 2940, 3247, 3624, 4357, 4398, 4399, 4450, 4468\n\nFoster, Ron 3345\n\nFoster, Susanna 1438\n\nFoulger, Byron 27, 104, 129, 170, 236, 248, 288, 293, 472, 550, 681, 698, 771, 895, 936, 961, 1071, 1096, 1174, 1392, 1568, 1573, 1664, 1734, 1862, 1934, 1960, 2081, 2139, 2144, 2469, 2521, 2652, 2717, 3066, 3093, 3128, 3312, 3344, 3377, 3435, 3490, 3536, 3608, 3637, 3685, 3720, 3865, 3897, 3904, 4073, 4168, 4221, 4225, 4330, 4358, 4665, 4671, 4786\n\nFoulk, Robert 101, 189, 671, 979, 1220, 1263, 2202, 2314, 2486, 2555, 3150, 3200, 3295, 3696, 3723, 3865, 3950, 4073, 4234\n\nFour Chicks and a Chuck 3923\n\nFowler, Art 155, 369, 1017, 1456, 1505, 1546, 2171, 2259, 2884, 2965, 3264, 3390, 4363, 4458, 4826, 5018\n\nFowler, Brenda 4100\n\nFowler, Gene 318, 332, 626, 2646\n\nFowler, Gene, Jr. 2725, 2929, 3857\n\nFowley, Douglas (Doug) 16, 67, 158, 161, 207, 208, 226, 241, 354, 524, 559, 632, 715, 800, 835, 1040, 1127, 1134, 1158, 1175, 1266, 1442, 1610, 1688, 1734, 1853, 1901, 1936, 2081, 2083, 2504, 2518, 2618, 2622, 2765, 2824, 2930, 3046, 3225, 3328, 3356, 3358, 3431, 3484, 3499, 3523, 3635, 3711, 3720, 3750, 3761, 3783, 4043, 4060, 4099, 4202, 4213, 4300, 4333, 4593, 4675, 4767, 4819, 4895, 4925, 4935, 5030, 5043\n\nFox, Bernard 319\n\nFox, Finis 445\n\nFox, Michael J. 4881\n\nFox, Michael 3509, 3637, 4417\n\nFox, Paddy 3246\n\nFox, Wallace 152, 204, 380, 575, 775, 957, 1279, 1516, 1546, 1682, 1693, 1718, 1744, 2291, 2425, 2647, 2679, 2764, 2949, 2969, 2988, 3051, 3153, 3176, 3478, 3656, 3900, 3943, 4024, 4509, 4532, 4690, 4729, 4847, 4929, 5046\n\nFoxe, Earl 1083\n\nFoxworth, Robert 2706, 2917, 3371, 4537\n\nFoy, Eddie, Jr. 1923, 4312\n\nFoy, Eddie III 2981\n\nFoy, Mary 4395\n\nFraker, William A. 2335, 2685\n\nFrancen, Victor 3692\n\nFranchi, Franco 1180, 1181, 1621, 3064, 4623, 4625, 4627, 4628\n\nFranciosa, Anthony 2506, 3518, 4102, 4345\n\nFrancis, Alec B. 3817, 4352\n\nFrancis, Anne 193, 239, 1892, 1998, 2692, 3030, 4976\n\nFrancis, Coleman 1695, 3741\n\nFrancis, Connie 4867\n\nFrancis, Diana 1569\n\nFrancis, Ivor 4417, 4753\n\nFrancis, Kay 4868\n\nFrancis, Noel 2315, 2726, 4138\n\nFrancis, Olin 262, 307, 467, 619, 753, 1057, 1227, 1455, 1785, 1869, 1916, 2003, 2151, 2257, 2364, 2366, 2388, 2391, 2547, 2802, 2864, 2990, 2954, 2998, 3003, 3027, 3110, 3278, 3325, 3449, 3461, 3588, 3609, 3618, 4226, 4280, 4570, 4612, 4728\n\nFrancis, Robert 4338\n\nFrancis, Stan 66\n\nFranciscus, James 587, 4693\n\nFrancks, Don 1168, 1349\n\nFranco, Jess 888, 2075, 3796\n\nFraney, Billy (William) 1432, 1489, 1553, 1602, 2766, 3048, 3360, 3424, 3631, 3994, 4125, 4651, 4785, 4846\n\nFrank, Christian 1781, 2726, 2773, 2790, 4057, 4203, 4645\n\nFrank, Harriet, Jr. 883, 1908, 1913, 1948, 3902, 4065\n\nFrank, Horst 358, 574, 1619, 1794, 2049, 2615, 2675, 3113\n\nFrank, Melvin 632, 1170, 2025, 3544, 4057\n\nFranken, Steve 2666, 4859\n\nFranklin, Chester M. 3774, 4875\n\nFranklin, Irene 2164\n\nFranklin, Joe 815\n\nFranklin, Pamela 4417\n\nFranklin, Paul 15, 386, 1145, 1297, 1326, 1768, 1810, 1916, 2540, 2973, 3001, 3370, 3408, 3466, 3764, 4077, 4156, 4401, 4405, 4806\n\nFranklin, Sidney 1187, 1825\n\nFranklin, Sidney, Jr. 1664\n\nFranz, Arthur 71, 3335, 3644, 5019\n\nFranz, Eduard 520, 580, 979, 1989, 2187, 2209, 4907\n\nFraser, Elizabeth 223, 632, 1888\n\nFraser, Harry 2, 8, 76, 494, 687, 1001, 1102, 1229, 1284, 1322, 1323, 1334, 1363, 1440, 1448, 1491, 1524, 1537, 1550, 1712, 1749, 1760, 1859, 1927, 2173, 2203, 2268, 2365, 2394, 2481, 2608, 2681, 2759, 2766, 2862, 2953, 3085, 3239, 3256, 3302, 3497, 3601, 3653, 3664, 3942, 3949, 4031, 4114, 4278, 4306, 4363, 4394, 4459, 4701, 4760, 4889, 4961, 4978, 4981, 4997; _see also_ Crist, Harry C.; Jones, Harry O.\n\nFraser, Phyllis 4995\n\nFraser, Richard 3794, 4659\n\nFraser, Ronald 1846, 4192\n\nFraser, Sally 2980\n\nFrawley, William 1360, 1883, 3255, 4671, 4745, 4979\n\nFrazee, Jane 309, 870, 1522, 1618, 2221, 2893, 4083, 4221, 4641\n\nFrazer, Robert 201, 300, 351, 777, 892, 956, 973, 995, 1307, 1337, 1416, 1504, 1716, 2259, 2284, 2315, 2660, 2745, 2991, 2909, 3025, 3052, 3241, 3480, 3665, 3903, 3944, 4153, 4482, 4522, 4759, 4864, 4887\n\nFreberg, Stan 632\n\nFrederic, Norman 2351, 2402, 4682; _see also_ Fredericks, Don\n\nFrederici, Blanche 331, 2207, 2554, 3766, 4403\n\nFrederick, Pauline 3245\n\nFredericks, Dean 1341, 3234, 3512, 3731; _see also_ Frederic, Norman\n\nFreed, Bert 330, 509, 1341, 1769, 2002, 2779, 3318, 4330\n\nFreeman, Howard 1, 602, 1564\n\nFreeman, Jerrold 463\n\nFreeman, Joan 3119, 3622\n\nFreeman, Kathleen 221, 966, 1263, 1610, 1899, 2500, 2668, 2829, 3058, 3066, 3622, 4210, 4211\n\nFreeman, Mona 497, 1150, 2161, 3542, 4168\n\nFreeman, Morgan 4663\n\nFrees, Paul 324\n\nFregonese, Hugo 105, 406, 1342, 2583, 2883, 3221, 3672, 3730, 4674\n\nFrench, Charles K. 310, 493, 686, 847, 894, 898, 916, 1045, 1083, 1110, 1812, 2130, 2220, 2470, 2523, 2573, 2816, 3075, 3087, 3322, 2626, 3903, 3934, 4379, 4480, 4489, 4728, 4785, 4837, 4863, 4883, 4932, 4997\n\nFrench, Ted 207, 870, 1158, 1803, 2175, 2259, 2266, 3257, 3484, 3651, 3666, 3678, 4081, 4820, 4829\n\nFrench, Valerie 1026, 1782, 3762, 3807\n\nFrench, Victor 709, 919, 1015, 1372, 2374, 2375, 2376, 2377, 3527, 4064, 4330, 4969\n\nFreuler, John R. 3983\n\nFrey, Arno 139, 2835, 4302, 4695, 4701\n\nFrey, Douglas 1233\n\nFriedkin, Joel 203, 205, 607, 925, 1100, 1255, 1449, 2068, 2361, 2971, 3095, 3230, 3669, 3678, 3914, 3989, 4147, 4191, 4661, 5035\n\nFriedlander, Louis 3322, 3650, 4142; _see also_ Landers, Lew\n\nFriedman, David F. 1233\n\nFriedman, Seymour 3762\n\nFriewald, Eric 698, 841, 2074, 2402, 2609, 2693, 2772, 3751, 4478\n\nFriganza, Trixie 4776\n\nFriml, Rudolf 2458, 2836, 3606\n\nFrisco, Joe 3434, 4842\n\nFritchie, Barbara 2232, 4385, 4961\n\nFrizzell, Lou 3218, 4113, 4134\n\nFrome, Milton 1243, 3040, 3426\n\nFrommer, Ben 3748\n\nFrost, Terry 23, 110, 231, 250, 251, 364, 428, 539, 597, 610, 658, 671, 711, 807, 913, 927, 940, 972, 977, 994, 997, 1158, 1263, 1268, 1279, 1447, 1513, 1520, 1529, 1555, 1711, 1715, 1748, 1797, 1803, 1823, 2074, 2085, 2109, 2273, 2293, 2504, 2680, 2687, 2798, 2857, 2884, 2899, 3970, 3068, 3103, 3509, 3648, 3751, 3804, 3901, 3999, 4047, 4073, 4095, 4097, 4163, 4252, 4278, 4292, 4301, 4437, 4466, 4478, 4491, 4495, 4504, 4682, 4687, 4690, 4692, 4729, 4754, 4813, 4847, 4969, 4973, 5005\n\nFryar, Hal 2964\n\nFrye, Dwight 3361, 3953, 4835\n\nFrye, Gil 4047\n\nFrye, Kathy 899\n\nFuchsberger, Joachim 2214, 4919\n\nFuest, Robert 1098\n\nFulci, Lucio 537, 697, 1424, 4625, 4901\n\nFuller, Barbara 759, 3567, 3728\n\nFuller, Clem 597, 1520, 1645, 1747, 1876, 3222, 3500, 3667, 3802, 4204, 4588, 4600, 4674\n\nFuller, Dolores 842, 2644, 3221\n\nFuller, Lance 117, 681, 981, 2096, 3762, 4244, 4783\n\nFuller, Robert 417, 597, 1133, 1510, 1985, 2613, 2623, 2714, 2738, 3391\n\nFuller, Samuel 251, 819, 1412, 1960, 1043, 1299, 2631, 3640\n\nFulton, Joan 2652, 3735; _see also_ Shawlee, Joan\n\nFulton, Lou 129, 139, 345, 1445, 1667, 2874, 4838\n\nFulton, Rad 1837, 2237\n\nFung, Willie 210, 460, 541, 621, 810, 1574, 1685, 1712, 1931, 2830, 3177, 3332, 3537, 3573, 3636, 3668, 3764, 3774, 4072, 4377, 4404, 4558, 4747, 4803, 4807\n\nFunicello, Annette 3941, 5086\n\nFurey, Barney 270, 314, 593, 1299, 2285, 2767, 2774, 3153, 3251, 3498, 4396, 4803, 4862\n\nFurey, Ed 3287\n\nFurie, Sidney J. 119\n\nFurio, Sonia 3197, 3347, 4176\n\nFurness, Betty 3360, 3745\n\nFurst, Stephen 3881\n\nFurstenberg, Ira 1011\n\nFurth, George 385, 586, 702, 3687\n\nFurthman, Jules 2943, 3517, 3934, 4061\n\nFutter, Walter 687\n\nGabel, Martin 4330\n\nGabel, Scilla 1120\n\nGable, Clark 16, 195, 420, 626, 632, 1507, 1925, 2018, 2416, 2661, 3011, 3695, 4233\n\nGable, Jack 2742; _see also_ Perrin, Jack\n\nGable, John Clark 195\n\nGabo, Louise 2742\n\nGabriel, John 4101\n\nGabriel, Lyn 1824\n\nGabriel, Roman 4638\n\nGahan, Oscar 6, 77, 140, 244, 334, 335, 338, 342, 344, 431, 428, 443, 666, 687, 725, 729, 810, 847, 946, 1017, 1474, 1488, 1491, 1503, 1556, 1574, 1576, 1674, 1683, 1732, 1756, 1813, 1859, 1861, 1898, 1900, 1929, 1969, 2256, 2257, 2306, 2355, 2408, 2432, 2481, 2522, 2540, 2690, 2740, 2786, 2821, 2822, 2954, 3073, 3272, 3278, 3303, 3332, 3456, 3546, 3553, 3581, 3589, 3591, 3597, 3601, 3827, 3884, 3907, 3913, 3918, 3946, 4000, 4082, 4104, 4256, 4298, 4303, 4365, 4525, 4581, 4587, 4592, 4667, 4685, 4711, 4722, 4839, 4848, 4883, 4887, 4981, 5056\n\nGaiano, Mario 1185\n\nGaines, Richard 1161, 1947, 4635\n\nGale, Joan 2660, 2961\n\nGale, June 1861, 2212, 3243, 3497, 4219\n\nGale, Roberta 54, 2742, 2812, 4269\n\nGalento, Tony 4993\n\nGalindo, Nacho 450, 462, 466, 520, 539, 934, 1212, 1214, 1364, 1520, 1757, 2021, 2066, 2416, 2486, 2677, 2901, 2938, 3344, 3530, 3671, 3752, 3852, 4049, 4212, 4388, 4599, 4860, 5022\n\nGallagher, Carole 1040, 1253, 1564\n\nGallagher, Ray 434, 2392, 4027\n\nGallagher, Skeets 826\n\nGallaher, Donald 774, 2979, 3946\n\nGallant, Kathleen 502\n\nGallardo, Juan 164, 168\n\nGallaudet, John 191, 1256, 2942, 3540, 4767\n\nGalli, Ida 18; _see also_ Stewart, Evelyn\n\nGalli, Rosina 1513\n\nGallian, Ketti 4650\n\nGalloway, Don 1341, 1703, 2896, 3283, 3439, 3612\n\nGam, Rita 2672, 3837, 3863\n\nGambon, Michael 2920\n\nGamet, Kenneth 25, 109, 638, 835, 1129, 1136, 1378, 1622, 1852, 1884, 1992, 2204, 2226, 2295, 2302, 2549, 2624, 3702, 3728, 4163, 4252\n\nGammon, James 822, 1250, 1267, 1998, 2059, 2488, 2508, 3661, 3911, 4169, 4753, 4930, 5029, 5057\n\nGan, Chester 145, 666, 844, 1153, 1594, 2147, 2647, 2722, 3028, 3291, 3322, 3695, 4241, 4242, 4776, 4803, 4833\n\nGangelin, Paul 309, 870, 939, 3585, 4002, 4036, 4641, 4646\n\nGannaway, Albert C. 208, 955, 1871, 2557, 3134, 3225\n\nGanzer, Gerry 3151\n\nGaras, Kaz 762\n\nGarces, John 2099\n\nGarces, Mauricio 217, 569, 3360, 4716\n\nGarcia, Allan 613, 1521, 1978, 4050, 4652\n\nGarcia, Joe 459, 573, 1805, 2821, 2993, 3003, 3693\n\nGarcia, Juan 745, 4233, 4250, 4638, 4727\n\nGarcia, Stella 2044, 2199\n\nGarcia, Tito 61, 1508\n\nGarde, Betty 2899\n\nGardiner, Reginald 1480\n\nGardner, Ava 2348, 2416\n\nGardner, Erle Stanley 1622\n\nGarfield, Frank 3785, 4678; _see also_ Giraldi, Franco\n\nGarfield, John 1378, 2064\n\nGarfield, John, Jr. 2489\n\nGargan, Edward 30, 246, 541, 944, 1175, 1253, 1285, 1567, 1577, 1627, 1827, 1831, 1883, 2837, 3156, 3669, 3694, 4086, 4171, 4284, 4786, 4941\n\nGargan, William 3295\n\nGarko, Gianni 202, 568, 1717, 1801, 1955, 1966, 2354, 3174, 3717; _see also_ Hudson, Gary\n\nGarland, Beverly 211, 349, 919, 1063, 1701, 1743, 3676, 4620, 4878\n\nGarland, Hank 3619\n\nGarland, Judy 1564, 1792\n\nGarland, Richard 258, 749, 803, 972, 1341, 1434, 2037, 2292, 2511, 3219, 4339, 4674\n\nGarner, James 53, 672, 1183, 1943, 2510, 2623, 2788, 2906, 3950, 4169, 4195, 4210, 4211\n\nGarner, Peggy Ann 312, 673, 2815\n\nGarnett, Tay 679, 696, 4426, 4909\n\nGaron, Pauline 863\n\nGarralaga, Martin 229, 272, 325, 352, 377, 425, 439, 447, 497, 602, 758, 907, 1132, 1138, 1293, 1311, 1420, 1518, 1520, 1740, 1974, 1981, 2064, 2066, 2109, 2179, 2184, 2252, 2263, 2270, 2287, 2288, 2290, 2314, 2430, 2502, 2550, 2646, 2943, 2948, 2986, 3003, 3128, 3407, 3476, 3502, 3610, 3690, 3854, 4000, 4022, 4047, 4055, 4096, 4128, 4213, 4543, 4599, 4650, 4657, 4745, 4825\n\nGarrett, Gary 1373, 2169, 4028, 4691\n\nGarrett, Grant 191, 248, 1917\n\nGarrett, Leif 370, 1589, 2113, 2598, 3070\n\nGarrett, Oliver H.P. 1187\n\nGarrett, Sam 1792, 2175, 2273, 3455, 4732\n\nGarrick, John 2458\n\nGarrick, Richard 2252, 3150, 3500, 4751\n\nGarrison, Sean 509\n\nGarrone, Sergio 1118, 2814\n\nGarroway, Dave 2437\n\nGarson, Greer 4148\n\nGarzcon, Gilbert 3347\n\nGasnier, Louis 2712\n\nGastaldi, Ernesto 1477, 1525, 1619, 1955, 2354, 2532, 2723, 4251\n\nGates, Harvey 2359, 2469, 2732, 2841\n\nGates, Larry 679, 1015, 1943, 2902, 4227\n\nGates, Nancy 482, 501, 711, 728, 806, 1697, 2496, 2620, 2775, 3294, 3586, 4160, 4860\n\nGateson, Marjorie 145, 1179, 1532, 1591, 2923, 3895\n\nGatlin, Jerry 319, 406, 417, 636, 860, 883, 909, 1170, 1245, 1354, 1765, 1924, 2331, 2780, 2982, 3019, 3602, 3840, 4523, 4565\n\nGatzert, Nate 180, 684, 1288, 1474, 1835, 1861, 1928, 2256, 2274, 2420, 3270, 3279, 3303, 3525, 3555, 3591, 3631, 3650, 4104, 4165, 4484, 4467, 4837, 4839, 4861\n\nGault, Slim 2026, 2722, 3891, 4085\n\nGavin, John 919, 1778, 3199\n\nGay(e), Gregory 2516, 4871, 5025\n\nGay, John 849, 1765, 3306, 3987, 3993\n\nGay, Nancy 2545, 2995, 3176\n\nGaye, Lisa 1164, 1485, 3871, 4741\n\nGaynor, Mitzi 1601, 4376\n\nGaze, Gwen 242, 3049, 4606, 4660, 4815, 5027\n\nGazzara, Ben 829\n\nGazzolo, Nando 1117, 1891\n\nGeary, Bud 37, 230, 231, 291, 395, 464, 602, 624, 634, 644, 665, 712, 779, 793, 880, 1023, 1345, 1360, 1439, 1504, 1513, 1646, 1814, 1873, 1907, 1915, 1920, 2027, 2124, 2125, 2132, 2163, 2170, 2175, 2176, 2427, 2537, 2545, 2546, 2597, 2600, 2660, 2673, 2831, 2889, 2928, 2967, 2971, 2995, 3001, 3080, 3084, 3115, 3151, 3176, 3179, 3230, 3693, 3707, 3708, 3822, 3824, 3826, 3893, 3970, 3972, 3989, 4185, 4394, 4409, 4463, 4487, 4494, 4584, 4640, 4734, 4763, 4786, 4811, 5104\n\nGeer, Ellen 2155\n\nGeer, Lennie 3759\n\nGeer, Will 237, 519, 807, 1770, 1926, 2028, 2109, 3687, 4989\n\nGeeson, Judy 3687\n\nGehrig, Lou 3289\n\nGehrig, Ted 122, 1413, 1998, 2335, 2685, 2859, 3687, 4969\n\nGeldert, Clarence 458, 2391, 2458, 2527, 2558, 3658, 4092, 4248, 4898\n\nGemma, Giuliano 18, 220, 976, 2438, 2532, 3175, 3955, 4182, 4277, 4779; _see also_ Wood, Montgomery\n\nGemora, Charles 3544\n\nGenest, Emile 322, 2808\n\nGentry, Rance 3234\n\nGentry, Roger 3248\n\nGeorge, Anthony 1714\n\nGeorge, Chief Dan 267, 941, 2373, 2950, 3795, 3962, 4068\n\nGeorge, Christopher 741, 978, 1212, 1654, 4523\n\nGeorge, George W. 111, 759, 1399, 2781, 3318, 3966\n\nGeorge, Gladys 2159, 3891\n\nGeorge, Gotz 84, 1761, 2507, 4540\n\nGeorge, Lynda Day 247, 978; _see also_ Day, Lynda\n\nGeorge, Sue 937\n\nGeorge, Susan 3961\n\nThe Georgia Crackers 1059, 1309\n\nGeraghty, Carmelita 921, 1335, 2238, 2642, 3078, 3582, 4308\n\nGeraghty, Gerald 67, 110, 143, 210, 236, 243, 409, 410, 667, 813, 858, 1142, 1438, 1444, 1618, 1758, 1836, 1870, 1890, 1914, 1934, 1973, 1980, 2006, 2119, 2648, 2701, 2711, 2885, 2894, 3091, 3111, 3132, 3236, 3269, 3328, 3505, 3608, 3724, 3761, 3803, 3880, 3890, 3912, 4198, 4202, 4203, 4488, 4551, 4652, 4757, 4773, 4802, 4842, 5033, 5062\n\nGeraghty, Maurice 112, 599, 934, 1889, 2272, 2462, 2672, 2733, 3310, 3559, 3898, 4450, 4569, 4732, 4827, 4851\n\nGerard, Hal 3980\n\nGeray, Steven 324, 1214, 1706, 2034, 2143, 4111\n\nGerber, Neva 609, 963, 1845, 2735, 4406\n\nGere, Richard 985\n\nGering, Marion 1961, 3609\n\nGerry, Alex 854, 3032, 3608, 4079\n\nGerry, Toni 2926\n\nGerstle, Frank 74, 1161, 2496, 3208, 3536\n\nGerwing, George 511, 3314, 4280, 4454\n\nGetty, Estelle 2810\n\nGhidra, Anthony (Antonio) 1119, 1902, 3121, 4262\n\nGhostley, Alice 4749\n\nGibbons, Eliot 1042, 1279, 1868, 4643\n\nGibson, Althea 1937\n\nGibson, Diana 5054\n\nGibson, Don 3768\n\nGibson, Helen 721, 896, 926, 1025, 1272, 1367, 2084, 2121, 2247, 2299, 2367, 2544, 2561, 2927, 4100, 4347, 4796, 4861\n\nGibson, Henry 1243, 2964\n\nGibson, Hoot 157, 383, 413, 543, 547, 600, 687, 739, 767, 823, 852, 873, 1284, 1322, 1455, 1517, 1781, 1817, 1897, 1937, 2130, 2223, 2286, 2388, 2440, 2541, 2573, 2590, 2601, 2632, 2703, 2957, 3015, 3072, 3153, 3161, 3243, 3497, 3551, 4032, 4067, 4084, 4145, 4205, 4219, 4316, 4511, 4518, 4552, 4557, 4684, 4853, 4944, 4957\n\nGibson, Julie 207, 488\n\nGibson, Mel 2623\n\nGibson, Mimi 170, 482, 1151, 2308, 2871, 3299\n\nGibson, Sonny 959\n\nGibson, Tom 668, 724, 856, 2172, 3601, 3630, 3744, 3918, 4555\n\nGifford, Frances 79, 459, 4456\n\nGilbert, Billy 1084, 2940, 4215, 4700\n\nGilbert, Bob 2106, 2713, 3045, 3238, 3331, 3887, 4026\n\nGilbert, Eugenia 186, 851, 1759, 4275\n\nGilbert, Florence 188\n\nGilbert, Helen 1020, 1585, 1950\n\nGilbert, Jody 49, 516, 586, 642, 1378, 1523, 1843, 1949, 2586, 3020, 3425, 3923, 3958, 4305, 4949\n\nGilbert, John 1851\n\nGilbert, Jonathan 2375, 2377\n\nGilbert, Melissa 2374, 2375, 2376, 2377\n\nGilchrist, Connie 112, 191, 332, 1263, 4129, 4411, 4412, 4417\n\nGiler, Bernie 1703, 1734, 3856, 4830\n\nGilkyson, Terry 3958, 4123\n\nGill, Vince 2623\n\nGillespie, Gina 1248\n\nGillespie, James 1042\n\nGillette, James 3471\n\nGillette, Ruth 1457, 3384, 4941\n\nGilliam, Burton 385, 1828, 2060, 2650, 3324, 3368, 4679\n\nGillingwater, Claude 825, 1634, 3179\n\nGillis, Ann 614, 2530, 3919\n\nGilman, Fred 94, 318, 851, 873, 1172, 2573, 3243, 3314, 3551, 3750, 4205, 4944\n\nGilmore, Douglas 1058\n\nGilmore, Lillian 3076, 5014\n\nGilmore, Lowell 804, 2416, 3718, 4636\n\nGilmore, Stuart 1762, 1941, 4241\n\nGilmore, Virginia 4184, 4745, 4849\n\nGilroy, Frank D. 1274, 1442, 1701\n\nGing, Jack 1695, 1769, 1880, 2695, 4878\n\nGinty, Robert 485\n\nGiordana, Andrea 1617, 2614, 4242; _see also_ Gorman, Chip\n\nGiorlami, Enzo 296, 565, 751, 1286, 3059\n\nGiorlami, Marino 296, 565, 4625\n\nGiraldi, Franco 2656; _see also_ Garfield, Frank\n\nGirard, Bernard 1341, 2379\n\nGirard, Joe (Joseph) 6, 327, 499, 851, 892, 1008, 1058, 1288, 1332, 1455, 1464, 1579, 1895, 2014, 2130, 2355, 2740, 2794, 2947, 2954, 2961, 3279, 3360, 3424, 3426, 3876, 3880, 3985, 4038, 4256, 4260, 4289, 4459, 4532, 4567, 4592, 4728, 4885, 4887, 4971\n\nGirardot, Ethienne 158\n\nGirdler, William 978, 1654\n\nGirotti, Mario 84, 1366, 2219; _see also_ Hill, Terence\n\nGish, Annabeth 5029\n\nGish, Lillian 1187, 4662, 4991\n\nGittens, Wyndham 1088, 1390, 2212, 2367, 2633, 3041, 4708, 4971\n\nGivot, George 2311\n\nGladstone, Marilyn 427\n\nGladwin, Frances 607, 683, 930, 1459, 2158, 4108, 4402, 4763, 4819, 5018\n\nGlas, Ursula (Uschi) 1761, 1764\n\nGlass, Everett 293, 1434, 1460, 1719, 1995, 2261, 3211, 4611\n\nGlass, Gaston 619, 4215, 4260\n\nGlass, Ned 307, 317, 632, 1579, 2013, 2025, 2119, 5052\n\nGlass, Seamon 2047, 5007\n\nGlaum, Louise 962, 1851, 3373\n\nGleason, James 1620, 1882, 2550, 2557, 2676, 2895, 4123\n\nGleason, Lucille 1134, 1572, 1620, 2147, 3409, 4225\n\nGleason, Pat 631, 1003, 3670, 4196\n\nGleason, Russell 1620, 4256\n\nGlecker, Robert 200, 1679, 3414, 4121\n\nGlendon, J. Frank 6, 451, 721, 753, 1678, 2073, 2129, 2247, 2371, 3075, 3322, 3681, 4321, 4525\n\nGlenn, Roy 1769, 3221, 4800\n\nGlenn, Scott 676, 1864, 2721, 3813, 3911\n\nGlennon, Bert 1970, 4050, 4949\n\nGlover, Bruce 1552, 2906, 3739, 4345, 5082\n\nGlover, Danny 561, 2433, 2623, 3911\n\nGoddard, Paulette 1392, 2831, 4464, 4635\n\nGodfrey, Arthur 3296\n\nGodfrey, Peter 252\n\nGodfrey, Renee 1144\n\nGodoy, Arturo 1620\n\nGoff, John 587, 655, 1085, 2047\n\nGolan, Gila 4693\n\nGoldbeck, Willis 2561, 3776\n\nGoldblum, Jeff 3911\n\nGoldman, William 586, 2623\n\nGoldsmith, Martin N. 671, 1402, 1697, 2996\n\nGoldstone, James 598, 2506, 3737\n\nGoldwyn, Samuel 866, 1436\n\nGolonka, Arlene 1769, 4802\n\nGombell, Minna 410, 1138, 2184, 3241, 3383, 5033\n\nGomberg, Sy 509, 4924\n\nGomez, Augie 37, 49, 147, 252, 255, 310, 335, 338, 340, 342, 344, 382, 410, 516, 613, 872, 956, 1073, 1296, 1426, 1470, 1472, 1565, 1798, 2166, 2251, 2302, 2403, 2407, 2408, 2411, 2412, 2413, 2415, 2421, 2549, 2600, 2701, 2734, 2847, 2878, 2999, 3015, 3088, 3132, 3448, 3456, 3574, 3589, 3677, 3918, 4032, 4402, 4502, 4589, 4606, 4649, 4823, 4957, 5018, 5103, 5104\n\nGomez, Thomas 91, 642, 939, 1201, 1438, 1444, 1499, 3143, 4133\n\nGonzales-Gonzales, Jose 1773, 4368, 4860\n\nGonzalez, Dacia 181, 1856, 1857, 1885, 1909, 2001, 2435, 3284, 4438, 4715, 4716\n\nGonzalez-Gonzalez, Pedro 26, 482, 741, 1692, 1773, 1940, 2483, 3411, 3517, 3815, 4148, 4679, 5000, 5071\n\nGoodfellow, Joan 4571\n\nGoodfriend, Pliny 3706\n\nGoodkind, Saul A. 2927\n\nGoodman, John 2015\n\nGoodrich, Frances 2756, 3605, 3780, 4745\n\nGoodrich, John 1368\n\nGoodwin, Aline 890, 1344, 3188, 3498\n\nGoodwin, Bill 1830, 3499\n\nGoodwin, Harold 415, 667, 1073, 1256, 1450, 1897, 2109, 2264, 2393, 2813, 3220, 3340, 3469, 3534, 3561, 3744, 3846, 3879, 3969, 4165, 4214, 4312, 4665, 4712, 4747, 4762, 4839, 5034\n\nGoodwin, Laurel 1575, 2267, 4098\n\nGoodwins, Leslie 1593, 2649, 2935\n\nGorcey, Bernard 488, 1103\n\nGorcey, David 488, 1103, 3163\n\nGorcey, Leo 488, 1103, 1810\n\nGordon, Alex 3365\n\nGordon, Bernard 915, 2292\n\nGordon, Bruce 913, 3445, 3651, 4704\n\nGordon, C. Henry 1518, 1858, 2145, 2552, 3384\n\nGordon, Don 641, 2252, 3404\n\nGordon, Gavin 1639, 2052, 2393, 2425, 3040, 3896\n\nGordon, Huntley 952, 1440, 2147, 3766\n\nGordon, Julia Swayne 1178, 1605, 1847\n\nGordon, Leo 111, 247, 371, 486, 548, 759, 1238, 1633, 1672, 1940, 2025, 2046, 2141, 2550, 2563, 2623, 2626, 2723, 2738, 2795, 2818, 3200, 3337, 3367, 3415, 3512, 3559, 3705, 3779, 3792, 4324, 4252, 4258, 4479, 5039\n\nGordon, Marjorie 945\n\nGordon, Mary 1041, 1395, 1800, 2523, 2599, 2907, 3216, 3551, 3916, 4115, 4294, 4798, 4828, 4868, 4885, 5054\n\nGordon, Michael 3757, 4287\n\nGordon, Muriel 2392\n\nGordon, Robert 358, 1510, 3294\n\nGordon, Rose 468, 889, 1271, 1312, 2177, 2742, 3495, 3884, 3888, 4480, 4636\n\nGordon, Roy 103, 304, 410, 681, 864, 1259, 1691, 1991, 2158, 2233, 2416, 2711, 2720, 3447, 3481, 3897, 4036, 4285, 4611, 4714, 4742, 4860, 5078\n\nGorman, Chip 1108, 2049; _see also_ Giordana, Andrea\n\nGorshin, Frank 3422, 4579, 4679, 4791\n\nGorss, Saul 934, 1130, 1140, 1378, 1820, 1963, 2026, 2910, 3234, 4336, 4697\n\nGortner, Marjoe 412, 1663, 4930\n\nGossett, Louis, Jr. 1209, 3395, 3860, 3950, 4897\n\nGottlieb, Franz 2615\n\nGoudal, Jetta 4909\n\nGoude, Ingrid 4909\n\nGough, Lloyd 354, 3255, 4057, 4249, 4585\n\nGould, Harold 2092, 2093\n\nGould, William 188, 203, 272, 310, 800, 877, 1045, 1047, 1308, 1378, 1474, 1626, 1677, 1785, 1823, 2067, 2124, 2365, 2403, 2452, 2526, 2737, 2939, 2994, 3092, 3101, 3194, 3200, 3270, 3361, 3452, 3528, 3692, 3969, 4023, 4072, 4193, 4215, 4219, 4269, 4286, 4305, 4484, 4555, 4636, 4703, 4839, 4861, 4932, 4951, 4954, 4981, 4983, 5014, 5051\n\nGowland, Gibson 2173, 2716, 2837, 4257, 5009\n\nGrable, Betty 271\n\nGraff, Wilton 1276, 4080\n\nGraham, Betty Jane 3616\n\nGraham, Fred 16, 28, 43, 148, 189, 212, 235, 309, 324, 415, 439, 552, 557, 637, 712, 775, 793, 796, 930, 935, 1127, 1234, 1313, 1395, 1561, 1635, 1646, 1733, 1823, 1937, 2083, 2198, 2424, 2600, 2673, 2727, 2790, 2829, 2880, 2893, 2935, 2967, 2996, 3000, 3066, 3080, 3136, 3255, 3326, 3367, 3411, 3517, 3542, 3606, 3696, 3704, 3707, 3741, 3750, 3788, 3791, 3814, 3893, 4002, 4011, 4069, 4333, 4388, 4425, 4463, 4584, 4585, 4705, 4786, 5001, 5019, 5103\n\nGraham, William A. 333, 1791, 1998, 2190, 2934, 3184, 4794, 4874\n\nGrahame, Gloria 370, 2853, 3422, 3432, 3621\n\nGrahame, Margot 159\n\nGranach, Alexander 2838\n\nGranada, Maria 316, 3726, 3997\n\nGrandin, Ethel 1999\n\nGrandstedt, Greta 1089, 2628\n\nGraneman, Eddy 917\n\nGranger, Dorothy 410, 951, 1071, 1306, 1925, 1974, 2159, 2163, 2597, 2790, 2830, 3020, 4058, 4197, 4654, 4868, 5021\n\nGranger, Farley 2509, 4334\n\nGranger, Stewart 84, 2197, 2829, 3246, 4962\n\nGrant, Cary 1947\n\nGrant, Frances 689, 2850, 3329, 4385, 4525\n\nGrant, James Edward 43, 89, 278, 612, 1921, 2241, 2626, 3567, 3815, 4210, 4212, 4373\n\nGrant, Kathryn 1719, 1735, 3363\n\nGrant, Kirby 620, 775, 1259, 1285, 1693, 1718, 1991, 2259, 2291, 2834, 2840, 3329, 3917, 3980, 4015, 4153, 4509, 5013, 5078, 5079, 5081; _see also_ Stanton, Robert\n\nGrant, Lawrence 2147, 4427\n\nGrant, Lee 4330\n\nGrant, Morton 69, 230, 241, 312, 1174, 1253, 2170, 3557, 3708, 4096, 4221, 4423, 4564, 4695, 4761, 4855\n\nGranville, Bonita 2335, 2400, 3772, 3895, 4025, 4171\n\nGrapewin, Charles 200, 1224, 1571, 1706, 1847, 3071, 3698, 4121, 4312, 4336, 4950\n\nGraser, Earle 2399\n\nGrassle, Karen 2376, 2377, 5029\n\nGrauman, Walter 4173\n\nGraves, Peter 223, 650, 1358, 1409, 3559, 3583, 3749, 4287, 4787, 4921, 5052\n\nGraves, Ralph 3743, 4372\n\nGraves, Robert 4275\n\nGray, Beatrice 4155, 4509, 4552, 4674, 4684\n\nGray, Billy 207, 1523, 2482, 2955, 3864, 3921, 4378\n\nGray, Charles 678, 2069, 3417\n\nGray, Coleen 105, 165, 375, 833, 1451, 1480, 3323, 3698, 4123, 4229, 4258, 4472, 4601, 4714, 4937\n\nGray, Dolores 2158\n\nGray, Gary 47, 292, 1688, 2605, 3014, 3214, 3383, 3576, 4222, 4266, 4890, 4943\n\nGray, Gilda 3666\n\nGray, Lawrence 2877\n\nGray, Linda 2094\n\nGray, Lorna 576, 930, 1008, 3325, 3483, 4156; _see also_ Booth, Adrian\n\nGray, Nadia 24, 4380\n\nGrayson, Charles 203\n\nGrayson, Donald 623, 682, 1128, 2886, 2975\n\nGrayson, Kathryn 2144, 3530\n\nGreaza, Walter 2789\n\nGreco, Jose 3183\n\nGreear, Geraine 1511, 2110, 3489; _see also_ Barclay, Joan\n\nGreen, Alfred E. 210, 1378, 1420, 3414, 3846, 3862, 3895\n\nGreen, Duke 712, 2128, 2170, 2399, 2524, 2966, 3561, 3704, 3826, 4954, 5103\n\nGreen, Harry 757, 2352, 4071, 4941\n\nGreen, Jane 191\n\nGreen, Joseph 251\n\nGreen, Mitzi 1176, 1563, 2452, 3710\n\nGreen, Nigel 40\n\nGreen, Walon 4934\n\nGreene, Angela 1236, 2122, 3066, 3849, 4074, 4115, 4412\n\nGreene, David 1539, 1609, 3313\n\nGreene, Harrison 160, 201, 414, 431, 1728, 1811, 1862, 2119, 2355, 2421, 2786, 2815, 3259, 3629, 3771, 3919, 4000, 4257, 4483, 4747\n\nGreene, Jaclynne 4120, 4160\n\nGreene, Joseph J. 2884, 3335\n\nGreene, Lorne 45, 1773, 1782, 2148, 2209, 2780, 3437, 4528, 4914\n\nGreenleaf, Raymond 42, 1936, 2226, 3150, 3757, 4074, 4300, 4373, 4411, 4740\n\nGreenstreet, Sydney 4336\n\nGreenway, Tom 2941, 3440, 3755, 4333, 4579\n\nGreenwood, Charlotte 2853\n\nGreer, Allen 140, 917, 1284, 1491, 1576, 1687, 2003, 2740, 3073, 3300, 3302, 3497, 3601, 3617, 3653, 3664, 4307, 4722, 4760\n\nGreer, Dabbs 349, 417, 727, 981, 1063, 1093, 1205, 1428, 2161, 2375, 2377, 2426, 2482, 3058, 3218, 3607, 3741, 3779, 3816, 3853, 5066\n\nGreer, Jane 1078, 3798, 4132, 4204\n\nGregg, Virginia 317, 1274, 1772, 1832, 3838, 4064\n\nGregory, James 1112, 1673, 2598, 3837, 4035\n\nGreig, Robert 4254\n\nGrey, Joel 554\n\nGrey, Nan 4215\n\nGrey, Shirley 301, 834, 2908, 3506, 4294, 4535\n\nGrey, Virginia 283, 1053, 1314, 1360, 1415, 1964, 2181, 2651, 2813, 3066, 3764, 3766, 3958, 4357, 4635, 5033\n\nGrey, Zane 145, 147, 441, 442, 471, 614, 1043, 1153, 1177, 1394, 1605, 1706, 1785, 1854, 1855, 1917, 2131, 2152, 2207, 2208, 2232, 2238, 2239, 2349, 2352, 2353, 2427, 2433, 2554, 2732, 2733, 2773, 2774, 2775, 3240, 3241, 3310, 3469, 3470, 3471, 3573, 3965, 4203, 4204, 4386, 4395, 4442, 4651, 4652, 4653, 4762, 4776, 4777, 4824, 4949, 4951\n\nGribbon, Eddie 271, 383, 644, 929, 1283, 1599, 1869, 1925, 2249, 2843, 2991, 2947, 3087, 3528, 4029, 4138, 4413\n\nGribbon, Harry 2458\n\nGrier, Roosevelt 1077, 4426\n\nGries, Tom 508, 584, 1569, 2058, 2713, 2903, 4988\n\nGriffies, Ethel 332\n\nGriffin, Merv 489, 685\n\nGriffin, Robert 252, 502, 519, 1482, 1742, 1963, 2287, 2486, 2555, 3058, 3849, 4725\n\nGriffin, Tod 482\n\nGriffith, Andy 1828, 3654, 3734, 3755, 5006\n\nGriffith, Charles B. 1377\n\nGriffith, D.W. 2603, 3245, 3743, 4174\n\nGriffith, Gordon 384, 2470, 2976, 4839\n\nGriffith, James 22, 42, 101, 105, 170, 317, 356, 489, 577, 659, 843, 964, 980, 1081, 1140, 1161, 1166, 1319, 1347, 1452, 1640, 1735, 1837, 1945, 1991, 2037, 2083, 2287, 2521, 2620, 2829, 3150, 3299, 3333, 3397, 3423, 3778, 3787, 3807, 4136, 4548, 4726\n\nGriffith, Julia 4317\n\nGriffith, Kay 853\n\nGriffith, Melanie 558, 4134\n\nGrimes, Gary 594, 909, 4065\n\nGrimes, Karolyn 1923\n\nGrinde, Nick 429, 1565, 4138\n\nGrissell, Wallace 836, 2600, 4734, 4841, 4951, 5103\n\nGrizzard, George 816, 2800\n\nGrodin, Charles 2631\n\nGrofe, Ferd, Jr. 982, 3183\n\nGruber, Frank 659, 1039, 1319, 1364, 1640, 1982, 2835, 3891, 4259, 4472, 4792, 4896\n\nThe Guadalajara Trio 2607, 3411, 3530, 4055, 4507\n\nGuard, Kit 47, 67, 352, 584, 592, 624, 666, 781, 832, 984, 1008, 1211, 1301, 1433, 1438, 1450, 1464, 1468, 1517, 1601, 1756, 1860, 1969, 2003, 2032, 2102, 2311, 2420, 2540, 2864, 3053, 3446, 3944, 4053, 4605, 4701, 4882, 4899\n\nThe Guardsmen Quartet 3046, 4103\n\nGuerrieri, Romolo 4251\n\nGuest, Inna 1604, 1667, 2836, 3938\n\nGuhl, George 541, 1127, 1591, 1595, 1647, 3695, 4593, 4742, 4803, 4868\n\nGuifoyle, Paul 100, 733, 944, 1228, 1855, 1967, 2288, 3744, 4745\n\nGuihan, Frances 351, 478, 480, 564, 865, 881, 1227, 1464, 2258, 2315, 3424, 3699, 4178, 4379, 4832\n\nGuilbert, Nina 4553\n\nGuillerman, John 1208, 4477\n\nGuinan, Texas 4116\n\nGuiol, Fred 3886\n\nGuizar, Tito 907, 1522, 2893, 3253, 4650\n\nGulagher, Clu 86, 702, 1705, 2092, 2674, 2721\n\nGulliver, Dorothy 917, 1299, 1320, 1969, 1975, 2192, 2420, 2951, 3082, 4567, 4645\n\nGunn, Moses 2265, 4969\n\nGurie, Sigrid 4358\n\nGuthrie, A.B., Jr. 324, 2095, 3808\n\nGuthrie, Jack 1453, 1803\n\nGwenn, Edmund 4328\n\nGwynne, Anne 199, 389, 620, 1224, 1444, 2123, 2428, 2638, 2861, 3032, 3425, 3540, 4247\n\nGwynne, Michael C. 1791, 4915\n\nGyton, Ed 3266\n\nGyton, George 3293\n\nHaade, William 73, 514, 566, 664, 715, 930, 956, 987, 1029, 1042, 1136, 1487, 1532, 1818, 1821, 1925, 1876, 1982, 2020, 2066, 2082, 2221, 2514, 2516, 2727, 2815, 2851, 2875, 2939, 3080, 3116, 3255, 3290, 3328, 3355, 3362, 3542, 3564, 3567, 3624, 3630, 3690, 3702, 3759, 3817, 3822, 4018, 4072, 4074, 4096, 4171, 4234, 4284, 4310, 4357, 4422, 4641, 4665, 4675, 5034, 5050\n\nHaas, Charles 3856, 4123, 4943\n\nHaas, Hugo 930, 1290, 1313, 1427, 2836\n\nHack, Herman 30, 70, 143, 146, 151, 154, 156, 187, 234, 262, 291, 303, 304, 334, 335, 344, 351, 382, 392, 411, 414, 431, 437, 440, 443, 446, 452, 461, 472, 479, 576, 597, 606, 634, 658, 667, 687, 795, 797, 848, 853, 858, 862, 867, 872, 929, 974, 1008, 1046, 1057, 1090, 1097, 1130, 1155, 1241, 1245, 1306, 1309, 1314, 1327, 1334, 1389, 1416, 1459, 1472, 1474, 1486, 1489, 1503, 1504, 1505, 1506, 1531, 1542, 1546, 1595, 1626, 1680, 1681, 1687, 1725, 1758, 1815, 1827, 1859, 1898, 1934, 1965, 2003, 2014, 2022, 2035, 2219, 2175, 2177, 2220, 2250, 2251, 2257, 2268, 2270, 2273, 2274, 2276, 2283, 2297, 2300, 2311, 2355, 2363, 2391, 2397, 2398, 2402, 2403, 2412, 2413, 2415, 2421, 2448, 2451, 2473, 2548, 2549, 2564, 2577, 2589, 2594, 2635, 2636, 2667, 2690, 2701, 2736, 2759, 2764, 2766, 2785, 2786, 2812, 2826, 2847, 2884, 2891, 2915, 2946, 2956, 2957, 2969, 2978, 2988, 2991, 2999, 3001, 3006, 3036, 3049, 3073, 3084, 3085, 3086, 3097, 3111, 3151, 3159, 3169, 3188, 3194, 3224, 3229, 3256, 3258, 3266, 3267, 3269, 3271, 3272, 3279, 3292, 3293, 3314, 3385, 3390, 3446, 3447, 3448, 3458, 3479, 3480, 3497, 3528, 3552, 3565, 3581, 3584, 3673, 3693, 3820, 3823, 3826, 3827, 3854, 3884, 3888, 3898, 3913, 3922, 3924, 3946, 3968, 3978, 4000, 4002, 4004, 4053, 4104, 4108, 4110, 4135, 4153, 4154, 4189, 4199, 4269, 4290, 4294, 4303, 4317, 4324, 4363, 4364, 4448, 4455, 4461, 4463, 4483, 4487, 4495, 4496, 4501, 4522, 4549, 4559, 4587, 4610, 4617, 4649, 4684, 4685, 4690, 4696, 4702, 4732, 4736, 4816, 4819, 4834, 4838, 4839, 4848, 4888, 4893, 4952, 5027, 5039, 5043, 5056, 5103\n\nHackathorne, George 300, 1369, 2434, 3465, 3817, 4864\n\nHackel, A.W. 1051, 1241, 1281, 1503, 1681, 2110, 2812, 3044, 3443, 3581, 4455\n\nHackett, Albert 2756, 3605, 3780, 4745\n\nHackett, Joan 2490, 4211, 4537, 4988, 5063\n\nHackett, Karl 6, 70, 140, 179, 201, 334, 342, 343, 345, 448, 461, 464, 476, 479, 494, 564, 610, 648, 689, 739, 779, 781, 792, 812, 1024, 1025, 1052, 1148, 1193, 1229, 1247, 1281, 1367, 1436, 1445, 1448, 1465, 1472, 1473, 1505, 1531, 1542, 1594, 1603, 1718, 1894, 1974, 2032, 2033, 2111, 2248, 2291, 2355, 2363, 2371, 2412, 2413, 2414, 2415, 2451, 2481, 2516, 2526, 2528, 2652, 2667, 2673, 2712, 2927, 2965, 2871, 2974, 2977, 3044, 3086, 3107, 3115, 3164, 3169, 3258, 3278, 3332, 3348, 3408, 3504, 3548, 3589, 3592, 3603, 3651, 3679, 3801, 3913, 3946, 3953, 4030, 4031, 4032, 4037, 4128, 4143, 4186, 4206, 4226, 4273, 4279, 4302, 4303, 4322, 4402, 4409, 4501, 4525, 4584, 4589, 4685, 4701, 4838, 4844, 4853, 4872, 4893, 4932, 4956, 5018, 5077\n\nHackman, Gene 1534, 1951, 3206, 4663, 5029, 5084\n\nHaddon, Pauline 139, 876\n\nHaden, Sara 191, 254, 312, 410, 2936, 2963, 3214, 3576, 3621, 4616, 4766\n\nHadley, Nancy 1466\n\nHadley, Reed 154, 935, 1247, 1361, 1616, 1626, 1762, 1960, 2083, 2221, 2372, 2528, 2543, 3032, 3312, 3377, 3394, 3473, 3511, 3540, 4057, 5022, 5034, 5104\n\nHagen, Jean 73, 128\n\nHagen, Kevin 417, 1527, 1748, 2185, 2374, 2375, 2377, 2519, 3445, 3518, 3816, 4040\n\nHagen, Ross 640, 1777, 2581, 2631\n\nHaggerty, Dan 29, 31, 32, 33, 657, 730, 1081, 1237, 1657, 2337, 2347, 2380, 2896, 4066, 4126, 4879\n\nHaggerty, Don 399, 472, 527, 632, 638, 678, 759, 867, 977, 990, 1039, 1199, 1214, 1255, 1644, 1688, 1697, 1775, 1985, 2109, 2795, 3091, 3201, 3646, 3791, 3877, 3928, 3950, 4048, 4074, 4076, 4140, 4300, 4712, 4730, 4966\n\nHaggerty, H.B. 3009\n\nHagman, Larry 3860\n\nHagney, Frank 64, 200, 331, 332, 382, 504, 578, 602, 634, 642, 824, 1300, 1434, 1492, 1553, 1616, 1674, 1702, 1772, 1774, 1790, 1861, 1904, 1928, 1974, 2226, 2238, 2273, 2276, 2407, 2409, 2410, 2413, 2415, 2502, 2516, 2549, 2635, 2638, 2756, 2837, 3020, 3066, 3082, 3276, 3322, 3348, 3427, 3456, 3537, 3607, 3696, 3702, 3750, 3772, 3900, 3912, 4057, 4093, 4099, 4163, 4168, 4257, 4302, 4362, 4383, 4396, 4419, 4423, 4428, 4478, 4585, 4589, 4635, 4673, 4698, 4839, 4876, 4898, 4955\n\nHahn, Gilesa 4334, 4905\n\nHaig, Douglas 3571\n\nHaig, Sid 55, 1220, 1343\n\nHaines, Connie 4598\n\nHaines, Donald 158\n\nHaines, William 4797\n\nHale, Alan 504, 718, 801, 844, 852, 1127, 1236, 1588, 1883, 2502, 2646, 3195, 3711, 4049, 4129, 4174, 4579, 4697, 4742, 5054, 5075\n\nHale, Alan (Jr.) 22, 60, 125, 328, 389, 567, 650, 1082, 1704, 1769, 1923, 1989, 2287, 2445, 2504, 2575, 3315, 3447, 3510, 3848, 3864, 3897, 4079, 4080, 4330, 4368, 5048\n\nHale, Barbara 548, 1253, 1265, 2204, 2396, 2871, 3768, 3792, 3960, 4041, 4825\n\nHale, Bill (William) 73, 101, 259, 597, 678, 850, 1348, 1690, 1868, 1703, 2059, 2264, 2339, 2502, 2517, 3233, 3261, 3890, 3982, 4223\n\nHale, Chanin 4988\n\nHale, Creighton 203, 844, 917, 1634, 1884, 2654, 2677, 3711, 3741, 4115, 4585, 4830, 5075\n\nHale, Georgia 1598, 2367\n\nHale, Jonathan 251, 362, 930, 1422, 2016, 2021, 2083, 2423, 3126, 3571, 3578, 3593, 3611, 3848, 3902, 4005, 4118, 4121, 4229, 4368, 4735\n\nHale, Louise Closser 948\n\nHale, Monte 68, 235, 309, 605, 607, 793, 1560, 1733, 1918, 2193, 2264, 2537, 2632, 2669, 2875, 2928, 2935, 2942, 3088, 3104, 3178, 3271, 3613, 3691, 4002, 4048, 4181, 4425, 4463, 4488, 4641, 5081\n\nHale, Nancy 4913, 5030\n\nHale, Richard 1, 214, 271, 644, 1434, 1996, 3054, 3096, 3316, 3433, 3690, 3739, 3777, 4080\n\nHale, Scott 3847\n\nHaley, Jack 4225\n\nHall, Alexander 1591\n\nHall, Arch (Archie) 340, 427, 1010, 1894, 2412, 2413, 2414, 2733, 2998, 3232, 3677, 4617\n\nHall, Arch, Jr. 1010\n\nHall, Charlie 2449\n\nHall, Edward 3801\n\nHall, Ellen 494, 624, 2293, 2480, 2969, 3229, 3264, 4394\n\nHall, Henry 76, 282, 386, 414, 427, 431, 472, 695, 739, 753, 810, 878, 899, 939, 1001, 1057, 1177, 1227, 1229, 1336, 1455, 1506, 1587, 1647, 1687, 1970, 2035, 2073, 2299, 2300, 2363, 2432, 2589, 2660, 2759, 2832, 2954, 2994, 3015, 3032, 3036, 3075, 2080, 3103, 3116, 3162, 3227, 3239, 3390, 3475, 3525, 3534, 3561, 3603, 3680, 3692, 3711, 4028, 4032, 4103, 4118, 4135, 4273, 4493, 4558, 4592, 4667, 4732, 4742, 4819, 4854, 4889, 4935, 4983, 5001, 5056, 5061\n\nHall, Huntz 488\n\nHall, James 948\n\nHall, Jon 503, 1041, 2145, 2218, 2652, 4735, 4871; _see also_ Locher, Charles\n\nHall, Lois 380, 716, 1256, 1460, 1938, 2798, 3556, 3780, 3958, 4292, 4296\n\nHall, Norman S. 37, 104, 304, 366, 464, 542, 610, 665, 836, 987, 995, 1023, 1443, 1472, 1523, 1805, 1991, 2132, 2135, 2189, 2264, 2545, 2668, 2673, 2678, 2805, 2966, 2967, 3006, 3326, 3611, 3691, 3693, 3822, 2826, 3989, 4183, 4185, 4285, 4409, 4463, 4734, 4773, 4886, 4888, 4971, 5005, 5071\n\nHall, Porter 129, 271, 960, 1070, 1762, 1853, 3046, 3071, 3140, 4498, 4635, 4802, 5021\n\nHall, Russell \"Candy\" 2722, 4031\n\nHall, Ruth 295, 1369, 2527, 3327, 4165\n\nHall, Thurston 229, 277, 362, 615, 664, 1127, 1638, 2027, 2717, 2805, 2936, 3510, 4016, 4023, 4085, 4092, 4107, 4222, 4291, 4742, 4825, 4886\n\nHall, Virginia 2882\n\nHalliday, Nan 2171\n\nHalligan, William 266, 863, 1360\n\nHalloran, John 49, 89, 214, 664, 1033, 1319, 2018, 2066, 2233, 2914, 3690, 4428, 4548, 4714, 4966\n\nHallyday, Johnny 1160\n\nHalop, Billy (William) 695\n\nHalperin, Edward 773, 942, 3953, 5077\n\nHalpin, Luke 2925\n\nHalsey, Brett 1421, 2544, 3628, 3964; _see also_ Ford, Montgomery\n\nHalton, Charles 514, 1127, 1434, 1594, 1974, 2031, 2064, 2159, 2691, 3362, 3916, 4072, 4360, 4456, 4533, 4569, 4593, 4742, 4851, 5072\n\nHalvey, Julian 915, 3030\n\nHamblen, Stuart 143, 665, 1360, 1980, 2125, 3128, 3728, 3989\n\nHamilton, Bernie 2738, 4661\n\nHamilton, Big John 237, 2626, 4626, 4638\n\nHamilton, Chuck 357, 979, 1136, 1227, 1364, 1433, 1601, 1626, 1773, 2396, 2452, 2511, 2544, 3126, 3422, 3702, 4044, 4053, 4073, 4635\n\nHamilton, Donna 1721\n\nHamilton, George 1913, 2560, 3138, 4431, 4748, 5100\n\nHamilton, John R. 42, 63, 1996\n\nHamilton, John 94, 214, 228, 233, 277, 284, 638, 895, 948, 969, 1005, 1072, 1216, 1247, 1319, 1427, 1490, 1638, 1918, 2006, 2017, 2022, 2149, 2153, 2264, 2593, 2669, 2804, 3841, 3088, 3104, 3231, 3355, 3383, 3637, 3724, 3750, 3822, 3828, 3931, 4123, 4179, 4241, 4257, 4336, 5034, 5103\n\nHamilton, Mahlon 772, 4326\n\nHamilton, Margaret 271, 2722, 3005, 3319\n\nHamilton, Murray 640\n\nHamilton, Neil 2135, 2381\n\nHammerstein, Oscar II 1883, 2854, 2855\n\nHammond, Hally 3118, 3380\n\nHammond, Victor 2590, 4055, 4552, 4684\n\nHamner, Earl 4874\n\nHampden, Walter 2831, 4148, 4336\n\nHampton, James 1251, 1808, 2076, 2490\n\nHampton, Orville H. 212, 375, 1261, 1451, 1709, 1959, 2205, 2644, 2868, 2910, 2960, 4357, 4368, 4469, 5070\n\nHampton, Royal 3272\n\nHampton, Ruth 2252\n\nHanin, Roger 3402\n\nHanna, Mark 1377, 1510, 1676, 2565, 4060\n\nHannah, Daryl 882\n\nHannalis, Blanche 1349, 2376, 5073\n\nHanneford, Grace 2841, 3319\n\nHanneford, Poodles 2841, 3319, 3692, 4080\n\nHannon, Chick 1, 37, 49, 138, 166, 307, 334, 335, 340, 345, 382, 391, 442, 465, 606, 719, 783, 793, 813, 853, 876, 946, 1073, 1130, 1146, 1304, 1327, 1370, 1468, 1486, 1529, 1540, 1587, 1604, 1687, 1725, 1789, 1969, 1974, 2022, 2171, 2174, 2258, 2286, 2365, 2403, 2413, 2418, 2421, 2457, 2514, 2542, 2572, 2589, 2600, 2635, 2636, 2740, 2762, 2875, 2880, 2927, 2999, 3028, 3052, 3107, 3224, 3264, 3278, 3303, 3407, 3456, 3552, 3564, 3587, 3673, 3709, 3728, 3803, 3826, 3834, 3913, 3924, 3938, 3945, 3974, 3989, 4056, 4078, 4081, 4104, 4110, 4128, 4130, 4153, 4458, 4549, 4554, 4573, 4581, 4587, 4649, 4685, 4730, 4819, 4823, 4826, 4833, 4841, 4853, 4884, 4899, 4957, 5018, 5027\n\nHansen, Aleth 323, 1136, 1455, 2451, 2599, 2953, 3038, 3478, 4580\n\nHansen, Eleanor 1367\n\nHansen, Juanita 2603\n\nHansen, Myrna 2565\n\nHansen, Peter 3722, 4373\n\nHansen, Valda 1641\n\nHanson, Lars 4991\n\nHanson, Lorna 1259\n\nHarden, Marcia Gay 2120\n\nHardie, Russell 2923, 3774\n\nHardin, Ty 195, 528, 915, 1163, 2228, 2240, 3324, 3730\n\nHarding, Ann 826, 1570, 4079\n\nHarding, Tex 481, 1059, 1453, 2189, 2296, 2978, 3385, 3651, 4305\n\nHardwicke, Sir Cedric 1947, 4700\n\nHardy, Oliver 1313, 2811, 4798\n\nHardy, Sam 3153, 4029\n\nHardy, Sophie 1069\n\nHardy, Thomas 763\n\nHare, Lumsden 1949, 2016, 2052, 2213, 2837, 2929, 3607\n\nHare, Marilyn 4819\n\nHargitay, Mickey 3831, 4355\n\nHarlan, Kenneth 230, 387, 956, 1028, 1297, 1751, 2119, 2149, 2278, 2286, 2927, 3170, 3177, 3267, 3361, 3695, 3704, 4025, 4185, 4208, 4485, 4645, 4655, 4743, 4776, 4887, 4923, 4933, 4957\n\nHarlan, Otis 2699, 2968, 3035, 3048, 3327, 3442, 4248, 4284, 4352, 4839, 4840\n\nHarlan, Rusell 3479\n\nHarmon, John 212, 349, 458, 615, 650, 842, 1455, 1924, 1936, 2018, 2511, 2772, 3234, 3637, 3794, 3817, 4073, 4287, 4373, 4376\n\nHarmon, Marie 1215, 1746, 2806, 4081\n\nHarmon, Mark 816, 900, 5029\n\nHarmon, Pat 30, 186, 262, 1269, 1426, 2207, 3676, 3994, 4519, 4602\n\nHarmonica Bill (William Russell) 9, 3063, 4154\n\nHarolde, Ralf 201, 2163, 3490\n\nHarper, Daryl 4031\n\nHarper, Patricia 369, 775, 1155, 1555, 3164, 3257, 3744, 4463, 4556, 4838\n\nHarper, Redd 4166\n\nHarper, Tex 3552, 3574\n\nHarr, Silver 130, 573, 924, 1023, 1459, 1542, 2590, 2594, 2673, 2915, 3153, 3348, 4011, 4956\n\nHarrell, Scotty 726, 1028, 2175, 2378, 2873, 3167, 3227, 3478, 4261\n\nHarrigan, William 129, 1430, 3458\n\nHarris, Brad 358, 2615, 3113, 3286, 4339\n\nHarris, Ed 120, 463\n\nHarris, Kay 1298, 3565\n\nHarris, Major Sam 597, 660, 1257, 1937, 2132, 2302, 2552, 2561, 2899, 3483, 3606, 3776, 4115, 4585, 4626, 4714\n\nHarris, Mitchell 1557\n\nHarris, Phil 1510\n\nHarris, Richard 2501, 2508, 2551, 3369, 3783, 4565, 4663\n\nHarris, Robert H. 114, 1641\n\nHarris, Rosemary 740\n\nHarris, Roy 151, 1361, 2273, 2641, 2861, 2994, 3291, 4323; _see also_ Hill, Riley\n\nHarris, Stacy 501, 671, 804, 1608, 1644, 1645, 2444, 3440, 5030\n\nHarris, Theresa 541, 1361, 4074\n\nHarris, Winifred 2790\n\nHarrison, James 3877, 3902, 4118\n\nHarrison, June 2169\n\nHarrison, Rex 1427\n\nHarrison, Richard 296, 1699, 1700, 1896, 2030, 2905, 3403, 4720\n\nHarrold, Kathryn 2807\n\nHarron, John 1594, 2792, 2864, 3172\n\nHarryhausen, Ray 4693\n\nHart, Christina 966\n\nHart, Dolores 3133\n\nHart, Dorothy 599, 1706, 3285\n\nHart, Gordon 387, 670, 714, 1740, 1916, 2164, 2529, 2998, 3452, 3626, 3761\n\nHart, Harvey 4122, 4137\n\nHart, John 66, 277, 743, 791, 867, 997, 1214, 1326, 1711, 2085, 2087, 2205, 2335, 2448, 2818, 2917, 3057, 3068, 3132, 3341, 3712, 4095, 4105, 4278, 4292, 4301, 4687, 4735, 4754, 4792\n\nHart, Maria 444, 680, 1330, 2485, 2960\n\nHart, Mary 337, 813, 1462, 1973, 3615, 3834, 4059; _see also_ Roberts, Lynne (Lynn)\n\nHart, Moss 1430\n\nHart, Neal 158, 464, 465, 607, 836, 1023, 1162, 1172, 1221, 1331, 1456, 2249, 2600, 2815, 3669, 3760, 3826, 4037, 4239, 4309, 4557\n\nHart, William S. 89, 248, 408, 962, 1110, 1843, 1851, 2753, 2892, 3373, 3412, 3697, 3879, 4087, 4088, 4276, 4375, 447, 4591, 4758, 4910, 4931\n\nHarte, Bret 608, 2469, 2940, 2941\n\nHartley, Mariette 255, 2498, 3435\n\nHartman, David 223, 2559\n\nHartman, Don 2013\n\nHartman, Edmund 1257, 3020\n\nHartman, Phil 4351\n\nHarvey, Don C. 101, 277, 377, 390, 698, 1103, 1129, 1130, 1199, 1263, 1330, 1408, 1568, 1691, 1697, 1711, 1720, 1721, 1901, 2065, 2609, 2693, 2804, 2840, 2885, 3068, 3168, 3511, 3779, 3995, 4285, 4496, 4618, 4726, 4740, 4805, 4973, 5038\n\nHarvey, Forrester 2741\n\nHarvey, Harry 127, 146, 149, 201, 214, 236, 282, 304, 352, 575, 599, 749, 774, 786, 796, 858, 1008, 1022, 1105, 1184, 1270, 1481, 1506, 1553, 1688, 1877, 1901, 1979, 2133, 2204, 2415, 2420, 2529, 2550, 2564, 2601, 2882, 2955, 3016, 3020, 3028, 3077, 3085, 3093, 3279, 3280, 3390, 3422, 3494, 3523, 3557, 3578, 3588, 3590, 3591, 3650, 3741, 3754, 3839, 3856, 3892, 3949, 3953, 4078, 4107, 4133, 4204, 4241, 4313, 4386, 4503, 4588, 4601, 4653, 4757, 4886, 5038\n\nHarvey, Harry, Jr. 2133, 2456, 2941, 3849, 3937, 4255, 4619\n\nHarvey, Jack 2221\n\nHarvey, Laurence 43, 2984\n\nHarvey, Paul 129, 597, 1134, 1256, 1591, 1818, 1836, 2357, 2530, 3071, 3126, 3431, 3609, 4057, 4058, 4381, 4033, 5054\n\nHasbrouck, Olive 454, 3631\n\nHaskell, Al 2, 234, 251, 327, 335, 337, 439, 459, 464, 555, 886, 914, 960, 973, 1128, 1143, 1301, 1461, 1586, 1626, 1971, 2086, 2111, 2166, 2264, 2298, 2421, 2540, 2607, 2648, 2737, 2826, 2956, 3015, 3063, 3073, 3169, 3173, 3272, 3300, 3353, 3448, 3486, 3508, 3528, 3556, 3655, 3827, 3927, 4050, 4163, 4238, 4294, 4318, 4389, 4559, 4605, 4609, 4649, 4723, 4731, 4838, 5018, 5027, 5041, 5096\n\nHaskell, Peter 2324\n\nHaskin, Byron 1039, 1347, 3891, 4792\n\nHasselhoff, David 887\n\nHassett, Marilyn 3795\n\nHatfield, Hurd 2314\n\nHathaway, Henry 482, 514, 1354, 1441, 1507, 1854, 1945, 2232, 2554, 2779, 2815, 2829, 3290, 3817, 3837, 4035, 4061, 4203, 4403, 4442, 4494, 4576, 4652, 4950, 5019\n\nHatton, Dick 1993, 4604, 4708, 4836\n\nHatton, Raymond 131, 147, 187, 200, 289, 362, 428, 783, 794, 834, 853, 857, 886, 896, 899, 940, 973, 1043, 1103, 1146, 1272, 1325, 1362, 1373, 1377, 1389, 1426, 1443, 1447, 1462, 1529, 1540, 1546, 1554, 1565, 1687, 1690, 1716, 1724, 1797, 1850, 1862, 1868, 1869, 1939, 2086, 2145, 2167, 2171, 2249, 2255, 2259, 2280, 2282, 2399, 2457, 2596, 2632, 2654, 2761, 2774, 2786, 2841, 2867, 2922, 2969, 3002, 3052, 2058, 3096, 3111, 3156, 3157, 3208, 3229, 3231, 3264, 3365, 3410, 3480, 3554, 3574, 3582, 3593, 3615, 3630, 3650, 3804, 3823, 3896, 3901, 3938, 3951, 4093, 4142, 4153, 4155, 4231, 4271, 4284, 4286, 4299, 4292, 4403, 4491, 4514, 4539, 4550, 4559, 4601, 4616, 4635, 4640, 4652, 4691, 4706, 4732, 4743, 4773, 4776, 4820, 4826, 4899, 5037, 5054\n\nHatton, Rondo 3005\n\nHauser, Dwight 2808\n\nHauser, Wings 178, 1539\n\nHaver, Phyllis 4352\n\nHavoc, June 4422\n\nHawke, Ethan 4902, 4906\n\nHawkey, Rock 877, 945, 4460, 4814; _see also_ Hill, Robert\n\nHawkins, Georgia 1138\n\nHawkins, Jack 3767, 3807\n\nHawkins, Jimmy 843, 2782, 3724, 3750, 5022\n\nHawks, Frank 2146\n\nHawks, Howard 246, 324, 2943, 3323, 3517, 3527\n\nHawks, J.G. 2651\n\nHawley, Monte 2249\n\nHawley, Wanda 3188, 4446\n\nHawn, Goldie 1170, 4180\n\nHaycox, Ernest 1, 4102\n\nHayden, Harry 207, 248, 271, 602, 757, 1136, 1175, 1256, 1438, 1458, 1588, 2208, 2226, 2288, 3046, 4533, 4700, 4745, 4803\n\nHayden, Russell 49, 103, 205, 242, 459, 794, 896, 1041, 1145, 1214, 1240, 1272, 1456, 1469, 1816, 1855, 1889, 1933, 1939, 1977, 1978, 2152, 2197, 2270, 2353, 2398, 2595, 2596, 2733, 2765, 2824, 2826, 3001, 3049, 3116, 3177, 3267, 3354, 3453, 3466, 3467, 3593, 3629, 3674, 3704, 3851, 3894, 3898, 4033, 4112, 4208, 4285, 4364, 4465, 4495, 4692, 4820, 4827, 4876, 4923, 5035\n\nHayden, Sterling 165, 751, 1039, 1364, 1664, 1844, 2008, 2048, 2083, 2187, 3849, 4227, 4250, 4267, 4428, 4461, 4689\n\nHaydn, Richard 26\n\nHaydock, Ron 404\n\nHaydon, Julie 811, 826, 3745, 4007\n\nHayers, Sidney 4529\n\nHayes, Allison 843, 1743, 2484, 2672, 4412, 5012\n\nHayes, Bernadene 2067, 2826, 3657, 3704, 4555, 4569\n\nHayes, George \"Gabby\" 49, 67, 143, 201, 214, 242, 243, 309, 411, 413, 434, 442, 461, 464, 493, 510, 621, 634, 659, 666, 790, 891, 960, 986, 1023, 1134, 1202, 1214, 1301, 1333, 1432, 1440, 1469, 1470, 1489, 1494, 1557, 1594, 1816, 1818, 1824, 1827, 1836, 1872, 1873, 1889, 1914, 1931, 1932, 1933, 1973, 1976, 1980, 1983, 2032, 2073, 2146, 2297, 2299, 2344, 2368, 2479, 2514, 2523, 2524, 2534, 2546, 2548, 2552, 2632, 2635, 2673, 2727, 2766, 2777, 2778, 2799, 2826, 2956, 2995, 3126, 3177, 3236, 3242, 3256, 3269, 3275, 3300, 3354, 3383, 3451, 3459, 3483, 3564, 3585, 3600, 2657, 3675, 3827, 3898, 3903, 3968, 4013, 4027, 4037, 4051, 4059, 4125, 4197, 4202, 4206, 4208, 4219, 4231, 4290, 4309, 4317, 4322, 4367, 4379, 4455, 4485, 4503, 4516, 4584, 4592, 4646, 4673, 4681, 4698, 4759, 4767, 4773, 4822, 4950, 5009, 5033, 5059, 5062\n\nHayes, Isaac 2849\n\nHayes, John Michael 2779, 2780, 4783\n\nHayes, Linda 2641, 4649, 3483, 4051\n\nHayes, Lorraine 1137, 2257; _see also_ Day, Laraine\n\nHayes, Margaret (Maggie) 1569, 1608, 1977\n\nHayes, Mary 3553\n\nHaymes, Dick 3740\n\nHaynes, Roberta 1293, 1672, 2768, 4688\n\nHays, Kathryn 509, 3422, 4344, 5082\n\nHays, Robert 608, 5073, 5074\n\nHayward, Chuck 313, 555, 920, 1073, 1238, 1354, 1412, 1937, 2044, 2141, 2335, 2379, 2485, 2564, 3602, 3640, 3690, 3752, 3776, 4035, 4626, 4764, 4968\n\nHayward, Jim 128, 324, 349, 1026, 1073, 1092, 1140, 1225, 1485, 2749, 3305, 3392, 3722, 4136, 4266, 4300, 4799\n\nHayward, Lillie 402, 527, 677, 1821, 2157, 2719, 2839, 3186, 3222, 3705, 4457\n\nHayward, Louis 744\n\nHayward, Susan 648, 1392, 1507, 2016, 2485, 3290, 3402, 4384, 4585, 4672\n\nHayworth, Rita 1898, 2646, 2879, 3300, 3353, 4335, 4573, 4650, 4970, 5028\n\nHazard, Jayne 4855\n\nHazard, Lawrence 930, 1247, 1526, 1847, 2020, 4072, 5033\n\nHaze, Jonathan 117, 1357, 1743, 2870\n\nHaze, Stan 239, 1442\n\nHazel, George 49, 334, 634, 2298, 2358, 2527, 2547, 2956, 3036, 3142, 3160, 3270, 3478, 3550, 3884, 3952, 4193, 4555, 4636, 4838\n\nHealey, Myron 14, 42, 111, 115, 418, 492, 681, 690, 698, 727, 766, 789, 791, 843, 1024, 1103, 1166, 1235, 1279, 1314, 1341, 1347, 1403, 1680, 1697, 1782, 1800, 1809, 1847, 1868, 1956, 2059, 2083, 2105, 2261, 2271, 2289, 2293, 2496, 2511, 2515, 2565, 2609, 2679, 2684, 2804, 2949, 2988, 3104, 3200, 3234, 3261, 3367, 3460, 3517, 3547, 3576, 3644, 3682, 3685, 3686, 3806, 3839, 3848, 3891, 3897, 3958, 3963, 3998, 4048, 4173, 4258, 4310, 4496, 4520, 4576, 4731, 4828, 4847, 4913, 5066\n\nHealy, Mary 3429, 4170\n\nHealy, Ted 2923, 3695\n\nHearn, Edward 37, 174, 180, 323, 480, 564, 684, 721, 767, 842, 853, 856, 954, 960, 968, 1008, 1095, 1312, 1340, 1626, 1983, 2020, 2105, 2129, 2212, 2299, 2388, 2504, 2516, 2552, 2660, 2722, 2730, 2737, 2756, 2790, 2889, 3011, 3119, 3150, 3171, 3241, 3329, 3401, 3424, 3541, 4009, 4071, 4080, 4082, 4121, 4179, 4232, 4308, 4315, 4321, 4569, 4592, 4667, 4708, 4817, 4832, 4834, 4854, 4864, 4899, 5002\n\nHeath, Ariel 366\n\nHeather, Jean 2233, 3335\n\nHeaton, Tom 237\n\nHecht, Ben 246, 2344, 4750\n\nHecht, Ted 207, 3502, 4075, 5013\n\nHeckart, Eileen 1842, 5084\n\nHedison, David 2093\n\nHedren, Tippi 3721\n\nHeffron, Richard T. 1962, 2746, 4577\n\nHeflin, Van 843, 1719, 3221, 3659, 3711, 3808, 4101, 4257, 4335, 4370, 4450, 5000\n\nHefner, Hugh 815\n\nHeggie, O.P. 197, 3179\n\nHeimel, Otto \"Coco\" 2147, 2722, 4031\n\nHeims, Jo 1675\n\nHeinz, Ray 458\n\nHeisler, Stuart 64, 580, 935, 2400, 4585\n\nHeller, Barbara 2426\n\nHeller, Lukas 1007, 2685\n\nHellman, Monte 736, 3428, 3845\n\nHellman, Sam 1458, 3374\n\nHelm, Faye 4671\n\nHelm, Frances 3404\n\nHelm, Levon 4356\n\nHelton, Percy 75, 586, 759, 832, 982, 1142, 1225, 1257, 1422, 1481, 1996, 2234, 2319, 2482, 2555, 3090, 3186, 3296, 3815, 4035, 4645\n\nHemingway, Mariel 1997, 4195\n\nHemmings, David 971\n\nHenaberry, Joseph 2311, 2536\n\nHenderson, Del 318, 1458, 1885, 2393, 2646, 3634, 4309, 4803\n\nHenderson, Douglas 937\n\nHenderson, Kelo 2234, 3918, 3671, 4542\n\nHenderson, Marcia 185, 650, 2749, 2754\n\nHenderson, Ray 8, 130, 334, 335, 338, 340, 341, 410, 422, 427, 431, 452, 945, 1001, 1057, 1306, 1447, 1459, 1486, 1503, 1506, 1542, 1626, 1813, 1894, 2273, 2274, 2278, 2285, 2300, 2399, 2524, 2589, 2596, 2822, 2847, 2865, 2974, 3029, 3062, 3086, 3097, 3116, 3165, 3210, 3279, 3289, 3332, 3444, 3446, 3448, 3703, 3801, 3825, 3888, 3946, 4303, 4363, 4365, 4437, 4501, 4525, 4581, 4656, 4667, 4696, 4702, 4854, 4889, 4981, 4982, 4986\n\nHendricks, Ben 611, 874, 1895, 2823, 3322, 3554, 4697, 4803\n\nHendricks, Ben, Jr. 776, 1570, 1634, 2254, 2832, 3126, 3308, 4982\n\nHendricks, Jack 668, 1057, 1214, 1448, 1455, 1463, 2177, 2300, 2582, 2858, 2954, 2956, 3157, 3162, 3268, 3478, 3589, 3679, 3884, 3994, 4294, 4301, 4466, 4555, 4607, 5096\n\nHendrix, Wanda 355, 2226, 2684, 2725, 4098, 3672, 3862\n\nHenner, Marilu 3654\n\nHenning, Pat 4993\n\nHenriksen, Lance 120, 992, 1708, 3158, 3206\n\nHenry, Bill (William) 43, 47, 158, 596, 637, 715, 979, 1022, 1040, 1212, 1697, 1727, 1748, 1809, 1937, 2074, 2149, 2402, 2561, 2564, 2593, 2616, 2620, 2875, 2923, 3356, 3724, 3751, 3776, 3950, 4127, 4223, 4368, 4400, 4478, 4508, 4626, 4680, 4937\n\nHenry, Buck 959, 2089\n\nHenry, Carol 14, 73, 152, 187, 282, 318, 643, 677, 685, 850, 871, 1033, 1403, 1680, 1684, 1690, 1713, 1724, 1727, 1800, 1868, 1959, 2271, 2248, 2684, 2798, 2949, 2988, 3262, 3277, 3423, 3500, 3552, 3586, 3707, 3802, 3820, 3823, 3839, 4118, 4153, 4357, 4520, 4559, 4819, 4989\n\nHenry, Charlotte 1587, 1827\n\nHenry, Gloria 25, 42, 94, 161, 598, 1285, 2261, 2361, 3255, 3447, 4166\n\nHenry, Hank 3777\n\nHenry, Louise 1228\n\nHenry, Mike 17, 2692, 3527, 4064\n\nHenry, O. (William Sydney Porter) 756, 2385\n\nHenry, Robert \"Buzz\/Buzzy\" 588, 589, 634, 710, 859, 1248, 1823, 1857, 1989, 2065, 2066, 2138, 2217, 2218, 2262, 2295, 2938, 3270, 3483, 3525, 3542, 3572, 3593, 3815, 3816, 4064, 4278, 4300, 4457, 4504, 4556, 4667, 4745, 4794, 4839, 4929, 4969\n\nHenry, Thomas Browne 277, 1140, 1696, 2205, 2252, 2502, 3200, 3741, 3857, 3931, 4585, 4740\n\nHensley, Harold 4895\n\nHepburn, Audrey 4662\n\nHepburn, Katharine 3244, 3602, 3750, 4662\n\nHerbert, F. Hugh 960, 2635, 4358\n\nHerbert, Holmes 844, 1256, 2009, 2064, 2991, 4049, 4170, 4962, 5011\n\nHerbert, Hugh 210, 271\n\nHerbert, Percy 651, 663, 2551\n\nHerchey, Barbara 2196\n\nHerman, Ace 3571\n\nHerman, Al 138, 206, 308, 384, 862, 1148, 1594, 1604, 1810, 1904, 2470, 2542, 2867, 2991, 2976, 3028, 3107, 3235, 3237, 3280, 3361, 3407, 3587, 3588, 3589, 3590, 4019, 4128, 4186, 4226, 4519, 4602, 4685, 4696, 4697, 4839, 4872\n\nHerman's Mountaineers 1822\n\nHern, Pepe 229, 462, 654, 1823, 2021, 2044, 2066, 3690, 4661\n\nHernandez, Juano 3776, 4129\n\nHerrick, Virginia 1461, 1956, 2679, 3547, 3900, 4730\n\nHerrmann, Edward 1021\n\nHershey, Barbara 1832, 2206, 3395\n\nHersholt, Jean 1110, 1131, 1851, 3636\n\nHerter, Gerard 20, 98, 316, 1477, 3309\n\nHervey, Irene 1084, 1177, 1785, 4359\n\nHessler, Gordon 903\n\nHessman, Howard 2056, 2100\n\nHeston, Charlton 166, 313, 627, 1265, 2196, 2501, 2696, 2700, 3140, 3184, 3722, 4373, 4453, 4988\n\nHeston, Fraser Clarke 2696, 2700\n\nHeston, John 4340, 4355\n\nHewston, Alfred 785, 1344, 1374, 1813, 2007, 3844, 4293\n\nHeyburn, Weldon 383, 465, 777, 779, 1023, 1025, 1453, 1518, 1574, 1977, 2733, 2831, 2995, 3568, 4057, 4135, 4336, 4483, 4853, 5050\n\nHeydt, Louis Jean 42, 207, 208, 1166, 1274, 1478, 1640, 2344, 2885, 3225, 3285, 3290, 3362, 3711, 4075, 4151, 4714, 4792, 4860\n\nHeyes, Douglas 347, 359, 1926, 2630, 3152, 3512\n\nHeyes, Herbert 634, 710, 870, 1023, 1214, 1265, 2124, 2838, 2967, 4257\n\nHeywood, Herbert 64, 147, 282, 514, 642, 1256, 2129, 2153, 2342, 2689, 2835, 2919, 3384, 3561, 3972, 4214, 4336, 4803, 4900\n\nHiatt, Ruth 1200, 3495, 4207\n\nHibbs, Jesse 367, 3234, 3415, 3423, 4073, 4770, 5049\n\nHibler, Winston 703, 2126, 2384, 4710, 4915\n\nHickman, Darryl 362, 491, 1535, 2008, 2020, 2838, 3069, 3411, 3870, 4060, 4671\n\nHickman, Dwayne 674\n\nHickman, Howard 279, 283, 566, 1328, 1953, 2086, 2108, 2158, 2516, 2838, 3557, 3874, 4572, 4840\n\nHicks, Catherine 1021\n\nHicks, Russell 160, 277, 1128, 1247, 1360, 1490, 1827, 1925, 2124, 2128, 2552, 2835, 2880, 3000, 3046, 3132, 3374, 3425, 3576, 3606, 3711, 3750, 3792, 3902, 3976, 4049, 4086, 4170, 4257, 4336, 4665, 4742, 4848, 4983, 5021\n\nHiggin, Howard 258, 1882, 3011\n\nHiggins, John C. 439, 524, 3143, 3213, 3781\n\nHightower, Slim 3752, 4204, 4284, 4395\n\nHill, Al 447, 621, 1136, 1762, 1904, 1925, 2547, 2838, 3020, 3049, 3534, 3544, 3692, 3764, 4132, 4163, 4254, 4257, 4367\n\nHill, Arthur 585, 2924, 2925\n\nHill, Craig 19, 92, 1163, 1385, 2941, 3861, 4079, 4243\n\nHill, Doris 262, 772, 891, 2681, 2915, 3275, 4020, 4038, 4051, 4067, 4321, 4516, 4728\n\nHill, George Roy 586\n\nHill, Josephine 108, 457, 711, 2104, 2362, 2428, 2533, 3065, 3251, 3869, 4972\n\nHill, Marianna 3345, 3723\n\nHill, Mary 2644\n\nHill, Ramsey 2601, 2879\n\nHill, Riley 14, 428, 649, 783, 1046, 1096, 1279, 1362, 1373, 1443, 1540, 1687, 1726, 2040, 2271, 2282, 2293, 2457, 2485, 2761, 2776, 2805, 2830, 2960, 3261, 3265, 3277, 3517, 3759, 3802, 3820, 3848, 3900, 3917, 3943, 4241, 4307, 4546, 4550, 4640, 4692, 4709, 4766, 4775, 4847, 5048; _see also_ Harris, Roy\n\nHill, Robert (Bob) 25, 382, 479, 723, 725, 811, 877, 929, 945, 1037, 1155, 1159, 1283, 1310, 1363, 1446, 1486, 1797, 1965, 2083, 2192, 2196, 2248, 2276, 2378, 2572, 2743, 2991, 3016, 3081, 3287, 3526, 3531, 3546, 3939, 4190, 4460, 4490, 4711, 4778, 4814, 4838, 4887, 4947, 5018; _see also_ Hawkey, Rock\n\nHill, Terence 3, 376, 421, 1123, 1525, 1581, 2474, 2477, 2553, 2723, 3349, 3400, 3532, 4560, 4568; _see also_ Girotti, Mario\n\nHill, Tommy 3935\n\nHill, Walter 525, 1244, 1534, 2443, 4930\n\nHiller, Arthur 2616, 2807, 4334, 4343\n\nHillerman, John 385, 1880, 2000, 2305, 3345\n\nHilliard, Harriet 4225\n\nHillie, Verna 2554, 2737, 4125, 4482, 4652\n\nHillyer, Lambert 270, 303, 307, 428, 613, 899, 1002, 1100, 1192, 1302, 1306, 1308, 1325, 1362, 1373, 1388, 1447, 1529, 1540, 1680, 1690, 1768, 1800, 1848, 2119, 2167, 2171, 2254, 2255, 2259, 2457, 2558, 2633, 2753, 2821, 2827, 2858, 2908, 2944, 3002, 3050, 3052, 3100, 3157, 3159, 3171, 3231, 3262, 3264, 3265, 3370, 3550, 3629, 3697, 3804, 3823, 3901, 3938, 4000, 4021, 4054, 4055, 4070, 4088, 4155, 4187, 4188, 4276, 4299, 4375, 4391, 4447, 4514, 4520, 4550, 4668, 4690, 4723, 4758, 4826, 4898, 4910, 4980\n\nHilton, George 98, 537, 570, 991, 1030, 1351, 1477, 1954, 2588, 2675, 3309, 3659, 4628\n\nHinds, Samuel S. 210, 1084, 1236, 1390, 1438, 2159, 2254, 3409, 3425, 3540, 3774, 3817, 3925, 4072, 4494, 4498, 4824\n\nHines, Gregory 713\n\nHingle, Pat 1755, 1769, 2002, 2738, 2779, 3206, 3645, 4968\n\nHinkle, Robert 1692, 1733, 2871, 2888, 2959, 5068\n\nHinn, Michael 486, 1669, 4688\n\nHinton, Ed 937, 1235, 1396, 2310, 2370, 2511, 3305, 3535, 3769, 3839, 4770\n\nHirliman, George A. 1090\n\nHiser, Joe 2106, 4196\n\nHittleman, Carl K. 339, 550, 1664, 2034, 2096, 3377, 3394\n\nHively, Jack 608, 2772, 2896, 4126\n\nHoag, Doane 698, 841, 842, 2289, 2910, 4726\n\nHobart, Rose 648\n\nHobbes, Halliwell 648, 360\n\nHodge, Patricia 4195\n\nHodges, Bob 4143\n\nHodgins, Earle 6, 42, 179, 199, 241, 248, 286, 349, 395, 446, 461, 476, 615, 631, 632, 710, 759, 800, 832, 929, 1028, 1097, 1127, 1144, 1254, 1274, 1297, 1345, 1367, 1380, 1415, 1416, 1438, 1444, 1456, 1553, 1626, 1635, 1693, 1728, 1697, 1810, 1859, 1873, 1889, 1903, 1916, 1934, 2124, 2235, 2250, 2258, 2304, 2306, 2416, 2424, 2480, 2545, 2552, 2561, 2577, 2668, 2727, 2776, 2850, 2857, 2871, 2872, 2878, 2928, 2931, 3020, 3036, 3049, 3053, 3080, 3177, 3194, 3227, 3258, 3259, 3267, 3278, 3327, 3344, 3377, 3379, 3388, 3458, 3750, 3828, 3859, 3866, 3877, 3919, 3967, 4058, 4072, 4086, 4199, 4221, 4261, 4307, 4322, 4390, 4463, 4498, 4501, 4635, 4649, 4654, 4655, 4661, 4684, 4733, 5066\n\nHodgson, Leyland 541, 3540, 4214, 4646\n\nHodiak, John 16, 73, 375, 828, 1792\n\nHoerl, Arthur 151, 444, 606, 1711, 2278, 3426, 3920, 4001, 4278, 4319, 4324, 4729\n\nHoey, Dennis 207, 1427, 2109, 3585\n\nHoey, Michael A. 4133\n\nHoffa, Portland 541\n\nHoffman, Charles 2659, 3663, 3754, 4073\n\nHoffman, Connie 388\n\nHoffman, Dustin 2373\n\nHoffman, Gertrude W. 602, 670, 2827, 4323, 4671\n\nHoffman, John 1956\n\nHoffman, Joseph 1184, 1906, 2396, 3234, 3370, 4232\n\nHoffman, Max, Jr. 1622, 3603, 4336, 4742\n\nHoffman, Otto 174, 246, 723, 747, 1328, 1796, 1831, 1848, 2349, 2471, 2552, 2722, 2940, 3327, 3361, 3679\n\nHogan, James 145, 147, 445, 1043, 3038, 4284, 4312, 4846\n\nHogan, Pat 166, 185, 733, 970, 1485, 1535, 1672, 1990, 2143, 2192, 2564, 2768, 2996, 3096, 3140, 3254, 3731, 3762, 3792, 3966, 4253\n\nHohl, Arthur 200, 1964, 2610, 3431, 3684, 4593, 5021, 5045\n\nHolbrook, Allen 473, 752, 1303, 2358, 3087, 3266, 3501, 4282\n\nHolbrook, Hal 182, 983, 2333\n\nHolcomb, Kathryn 1946, 2487\n\nHolden, Fay 648, 1740, 2936, 4890\n\nHolden, Gloria 112, 1127\n\nHolden, Harry 300, 772, 2423\n\nHolden, Jennifer 539\n\nHolden, Joyce 527, 4805\n\nHolden, Scott 3402\n\nHolden, William 71, 129, 1234, 1937, 2515, 3214, 3402, 4168, 4286, 4934, 4969, 4970\n\nHoldren, Judd 1593, 1963, 4074\n\nHole, William, Jr. 1421\n\nHoliday, Hope 3622\n\nHolland, John 277, 615, 2066, 2264, 2484, 2618, 3025, 3523, 3567, 4033, 4585\n\nHolliman, Earl 55, 520, 580, 640, 1077, 1092, 1560, 1702, 1754, 2240, 3244, 4035, 4528, 4566\n\nHolloway, Carol 3240, 4395, 4510\n\nHolloway, Sterling 271, 1020, 2096, 2923, 3022, 3563, 3669, 3929, 4507, 4599, 4987\n\nHolman, Harry 214, 826, 2031, 3092, 3805, 3889, 4257, 4842\n\nHolman, Rex 487, 2333, 2964, 2989, 3208, 4339, 4344, 4869, 4870, 5070\n\nHolmes, Ben 1967\n\nHolmes, Gilbert \"Pee Wee\" 454, 1058, 1276, 1301, 1333, 1369, 1779, 1869, 1975, 2268, 2493, 2524, 2699, 2703, 3072, 3558, 3655, 4084, 4846\n\nHolmes, Helen 398, 614, 1174, 2670, 2921, 4798\n\nHolmes, J. Merrill 231, 2232, 2279\n\nHolmes, Madeleine Taylor 1641, 2950\n\nHolmes, Phillips 145, 2918\n\nHolmes, Stuart 203, 251, 308, 578, 832, 1378, 1932, 2561, 2726, 2864, 3538, 4358, 4513, 4697, 4932\n\nHolmes, Taylor 832, 1166, 2486, 2624, 2938, 3061, 3436, 5020\n\nHolt, Jack 16, 149, 441, 516, 940, 1060, 1228, 1394, 1490, 2123, 2184, 2352, 2386, 2727, 2823, 2838, 3046, 3312, 3386, 3554, 3695, 4092, 4166, 4203, 4284, 4488, 4543, 4776, 4938, 4949\n\nHolt, Jennifer 303, 557, 726, 1028, 1339, 1456, 1687, 1738, 1803, 2378, 2424, 2457, 2595, 2687, 2761, 2866, 2873, 2957, 2991, 3103, 3227, 3228, 3265, 3357, 3478, 3799, 3889, 4017, 4097, 4135, 4261, 4437, 4466, 4495, 4550, 4654, 4876\n\nHolt, Mike 2027\n\nHolt, Tim 69, 149, 179, 230, 231, 456, 534, 812, 927, 1050, 1174, 1199, 1270, 1308, 1562, 1594, 1688, 1726, 1736, 1941, 1988, 2170, 2260, 2288, 2632, 2718, 2731, 3000, 3115, 3119, 3327, 3353, 3431, 3473, 3504, 3523, 3541, 3557, 3646, 3667, 3679, 3937, 4100, 4017, 4140, 4241, 4386, 4405, 4486, 4543, 4653, 4761, 4841, 4951\n\nHomans, Robert 56, 214, 303, 552, 698, 767, 823, 863, 1127, 1392, 1394, 1444, 1594, 1595, 1820, 1974, 2016, 2101, 2423, 2458, 2524, 2545, 2830, 2864, 2915, 3161, 3179, 3300, 3409, 3414, 3433, 3609, 3866, 3896, 3989, 4004, 4006, 4072, 4084, 4142, 4557, 4726, 4742, 4808, 4932, 5072\n\nHomeier, Skip 170, 312, 491, 567, 580, 806, 932, 972, 979, 1704, 2226, 2395, 3134, 3542, 3853, 4126, 4151, 4235, 4388, 4975\n\nHomes, Geoffrey 367, 562, 1063, 1201, 2225, 3150, 3621\n\nHooker, Hugh 23, 229, 1096, 1272, 1321, 1597, 2123, 4283, 4305\n\nHoose, Fred 150, 277, 1156, 1196, 2286, 2418, 2856, 2957, 3503, 3581, 3905, 3936, 4032, 4212, 4299, 4778, 4484\n\nThe Hoosier Hot Shots 161, 869, 1980, 2419, 2991, 3410, 3569, 3914, 3917, 3923, 3976, 4015, 4221, 4222, 4327, 4378\n\nHope, Bob 53, 1257, 2162, 3020, 3296, 3544, 4005\n\nHopkins, Anthony 2504\n\nHopkins, Bo 485, 730, 882, 909, 1147, 1539, 1607, 1944, 2000, 2229, 2488, 2560, 2685, 3013, 3146, 3577, 3643, 3961, 4045, 4934, 5030\n\nHopkins, Miriam 2941, 4742\n\nHopper, Dennis 1441, 1560, 1702, 2100, 2199, 2492, 4035, 4146, 4576, 4968, 5071\n\nHopper, E. Mason 318\n\nHopper, Hal 3512, 3807\n\nHopper, Hedda 4807\n\nHopper, Jerry 563, 2949, 2668, 2899, 3140, 3966\n\nHopper, William 1347, 3559, 3711, 3931, 3960, 4336, 4473, 4742\n\nHopton, Russell 1154, 1183, 2223, 2249, 2775, 2832, 3354, 3609, 4027, 4215, 4231, 4703, 4825\n\nHorn, Leonard 768, 2752\n\nHorn, Murray 3888\n\nHorne, James W. 1008, 3067, 4798, 4899\n\nHorne, Lena 1015\n\nHorne, Rick 2535\n\nHorne, Victoria 1982, 3744\n\nHorner, Harry 2518\n\nHorner, Robert J. 12, 108, 438, 443, 890, 1294, 2104, 3073, 3272, 3462, 3496, 3647, 3844, 4280, 4774, 4846, 4912, 4972\n\nHorse, Michael 177, 455, 2335\n\nHorsley, Lee 4977\n\nHorton, Clem 307, 682, 2540, 2821, 3550, 4817\n\nHorton, Robert 115, 128, 947, 3142, 3324, 3392\n\nHorvath, Charles 22, 418, 450, 603, 692, 733, 798, 932, 935, 953, 1077, 1130, 1151, 1735, 1720, 2050, 2105, 2143, 2504, 2550, 3058, 3096, 3149, 3232, 3547, 3741, 3853, 3978, 4244, 4390, 4727, 4788, 4805, 4871, 4973\n\nHosier, Lionel 960\n\nHossein, Robert 2898, 3604\n\nHouck, Doris 1815, 2175, 4167, 4610\n\nHouck, Joy, Jr. 3793, 3818\n\nHough, Emerson 4284\n\nHough, John 2051, 4565\n\nHouse, Billy 2748, 2960, 3702, 3891, 4503\n\nHouse, Newton 543, 4011\n\nHouseman, Arthur 332, 626, 1570, 1577, 4882\n\nHouser, Jerry 192\n\nHouser, Lionel 432\n\nHouston, George 452, 1464, 1947, 2407, 2408, 2409, 2410, 2411, 2412, 2413, 2414, 2415, 2965\n\nHouston, Norman 149, 241, 534, 549, 670, 687, 715, 1050, 1199, 1469, 1682, 1736, 1816, 1855, 1933, 1973, 1988, 2152, 2353, 2480, 2605, 3732, 3775, 3119, 3473, 3497, 3504, 3523, 3541, 4107, 4112, 4204, 4208, 4241, 4304, 4386, 4653, 4777, 4825, 4841\n\nHouston, Virginia 1136, 2805\n\nHoven, Adrian 4003\n\nHovey, Tim 2676, 3960\n\nHoward, Ann (Anne) 130, 891, 1333, 1489, 2303, 2364, 4455\n\nHoward, Boothe 2850, 3329, 4289\n\nHoward, Clint 730, 1245, 2664, 3320, 4936\n\nHoward, David 144, 437, 447, 566, 952, 1174, 1311, 1605, 1679, 1785, 1983, 2207, 2304, 2342, 2599, 2654, 2741, 2990, 3012, 3043, 3162, 3241, 3353, 3937, 3963, 4385, 4423, 4564, 4572, 4891\n\nHoward, Edward 204, 391, 481, 775, 1083, 1450, 2296, 2759, 3569, 3651, 3744, 4272, 4305, 4363, 4394, 4584\n\nHoward, Esther 271, 1843, 2020, 2147, 2159, 4284, 4745\n\nHoward, Jerry \"Curley\" 3569\n\nHoward, John 2516, 3438, 4312\n\nHoward, Ken 4149\n\nHoward, Leslie 3071, 3766\n\nHoward, Mary 332, 3471\n\nHoward, Moe 388, 1422, 1597, 2964, 3569\n\nHoward, Rance 959, 1245, 1262, 1549, 2325, 2442, 2657, 3320, 3800, 4441, 4874\n\nHoward, Ron 1262, 2662, 3847, 4065, 4936\n\nHoward, Ronald 40, 849, 1151, 1951, 4224\n\nHoward, Shemp 1361, 1597, 1810, 1906, 2688, 4170\n\nHoward, Susan 2793, 3878\n\nHoward, Trevor 4996\n\nHoward, William K. 2149, 4909\n\nHoward, Willie 3609\n\nHowell, Dorothy 4642\n\nHowell, Hoke 2140, 3816, 3843\n\nHowell, Ken 1971\n\nHowes, Reed 37, 70, 73, 183, 289, 334, 336, 340, 354, 390, 430, 494, 597, 665, 683, 724, 759, 773, 819, 853, 921, 938, 973, 974, 1001, 1024, 1026, 1063, 1136, 1214, 1327, 1340, 1370, 1438, 1445, 1475, 1556, 1627, 1709, 1735, 1774, 1862, 1929, 2082, 2124, 2226, 2276, 2302, 2350, 2365, 2386, 2399, 2412, 2413, 2418, 2421, 2504, 2549, 2844, 2953, 3018, 3036, 3085, 3089, 3115, 3226, 3232, 3262, 3327, 3477, 3534, 3666, 3667, 3702, 3708, 3728, 3759, 3787, 3863, 3900, 3914, 3944, 4023, 4052, 4099, 4144, 4163, 4199, 4252, 4261, 4318, 4324, 4408, 4409, 4458, 4482, 4554, 4589, 4640, 4673, 4742, 4775, 4816, 4833, 4849, 4952, 4957, 5018, 5104\n\nHowes, Sally Ann 1277\n\nHowland, Chris 2316\n\nHowland, Jobyna 3895\n\nHowland, Olin 200, 279, 930, 986, 1175, 1247, 1571, 1853, 2016, 3817, 4598; _see also_ Howlin, Olin\n\nHowlin, Herman 459, 853, 2457, 3693, 4733, 4849\n\nHowlin, Olin 89, 110, 207, 642, 1616, 1843, 1915, 1974, 2221, 2618, 2781, 3020, 3344, 3483, 3567, 3702, 3707, 3772, 3820, 3975, 4099, 4141, 4329, 4411, 4762, 5008, 5033; _see also_ Howland, Olin\n\nHoxie, Al 5, 141, 186, 3307, 3539, 3647\n\nHoxie, Jack 186, 188, 408, 454, 1054, 1387, 1495, 1592, 1678, 2247, 2359, 2951, 3463, 4359, 4446, 4570, 4728, 4911, 5016\n\nHoy, Robert 73, 247, 348, 417, 450, 1170, 1607, 1671, 1905, 1944, 2093, 2143, 2292, 2335, 2619, 2899, 2950, 3287, 4244\n\nHoyos, Rodolfo 82, 603, 654, 1293, 1347, 1757, 2066, 2486, 3285, 3387, 3762, 4111, 4250, 4368, 4469\n\nHoyt, Arthur 866, 1200, 4864\n\nHoyt, Harry O. 179, 1324, 1614, 2235, 2451, 2531, 3657, 3924, 4377\n\nHoyt, John 913, 1183, 1727, 1996, 1998, 2672, 3201, 3865, 4860, 4990\n\nHubbard, John 710, 749, 868, 1183, 1238, 1696, 1936, 2076, 2936, 4235\n\nHubbard, Lucien 4093, 4704, 4949\n\nHubbard, Thomas G. (Tom) 208, 1838, 1871, 2396, 2484, 3225, 3762, 4220, 4392, 4539, 5048\n\nHuber, Harold 1143, 1521, 2145, 2147, 2720, 2756, 3695\n\nHuddleston, David 192, 346, 385, 508, 1663, 1946, 2087, 2930, 3527, 3990, 4477\n\nHudson, Benny 2105\n\nHudson, Gary 719, 4251; _see also_ Garko, Gianni\n\nHudson, John 258, 749, 1409, 1702, 2575, 2578, 2672, 3897, 4699\n\nHudson, Rochelle 306, 523, 844, 1563, 2153, 3745\n\nHudson, Rock 185, 290, 1560, 1672, 1936, 2237, 2292, 2899, 3741, 3768, 3855, 4244, 4450, 4989\n\nHuerta, Cris (Chris) 2612, 2758, 3346, 3403, 3604, 3717, 3785, 3955, 4182, 4366, 4471, 4638\n\nHuffaker, Clair 738, 1062, 1371, 1372, 2903, 3149, 3518, 3791, 4790\n\nHuggins, Roy 1672, 1774, 2437, 4362, 5063\n\nHughes, Barnard 3218, 3282\n\nHughes, Carol 442, 1595, 2529, 3361, 4107, 4643\n\nHughes, Howard 2943\n\nHughes, J. Anthony (Tony) 53, 630, 659, 757, 1469, 1947, 2208, 2289, 2664, 4422, 4548, 4791\n\nHughes, Kathleen 972\n\nHughes, Kay 323, 1229, 1296, 1553, 3433, 3453, 3561, 4365, 4569, 4732\n\nHughes, Lloyd 1854\n\nHughes, Mary Beth 863, 1103, 1214, 1616, 1664, 2221, 2471, 3005, 3053, 3394, 3431, 3447, 3511, 3569, 4085, 4225, 4422\n\nHughes, Rupert 5004\n\nHughes, Russell 2192, 4179, 4390, 5049\n\nHugo, Mauritz 11, 391, 409, 481, 656, 933, 1022, 1073, 1348, 1439, 1480, 1603, 1664, 1697, 1726, 1920, 2105, 2189, 2564, 2590, 2969, 3119, 3356, 3541, 3637, 3636, 3667, 3744, 3787, 4109, 4234, 4339, 4411, 4684, 4726, 4784, 4078\n\nHull, Cynthia 4976\n\nHull, Henry 550, 798, 1029, 1214, 1995, 2031, 2096, 2226, 2563, 2929, 3186, 3374, 3377, 3384, 3511, 3821, 4390, 4536, 5021\n\nHull, Josephine 2161, 4809\n\nHull, Warren 892, 3434, 5077\n\nHullett, Otto 75\n\nHumberstone, H. Bruce 1480, 2471, 4252\n\nHumbert, George 613, 1785, 2279, 2720, 4569, 5077\n\nHume, Cyril 497, 4593, 5046\n\nHume, Phyllis 3581\n\nHumes, Fred 264, 592, 764, 1276, 2249, 3555, 3655, 4205\n\nHundar, Robert 918, 1242, 1305, 1619, 1902, 2904, 3346, 3420, 3660, 3784, 3796, 4003, 4903, 4904; _see also_ Undari, Claudio\n\nHunnicutt, Arthur 26, 114, 32, 439, 466, 487, 519, 674, 768, 966, 1092, 1111, 1212, 1298, 1449, 1478, 1758, 2187, 2269, 2482, 2485, 2706, 3042, 3053, 3305, 3402, 3505, 3507, 3565, 3837, 3899, 4065, 4069, 4129, 4179, 4235, 4411, 4417, 4479, 4611, 4979, 5007\n\nHunt, Eleanor 411, 2832\n\nHunt, Helen 734, 1997, 3106\n\nHunt, Jimmy 654, 1256, 2396, 3567, 3672, 4062\n\nHunt, Linda 2623\n\nHunt, Marsha 147, 1043, 2555, 3133\n\nHunt, Martita 4670\n\nHunter, Henry 1390, 1641, 4569, 4831, 5054\n\nHunter, Ian 332\n\nHunter, Jeffrey 742, 915, 1342, 1637, 1671, 2519, 3185, 3333, 3752, 3776, 3779, 3781, 4209, 4376, 4579, 4799, 4907\n\nHunter, Kim 2676\n\nHunter, Robert 324\n\nHunter, Ross 1658\n\nHunter, Tab 580, 1665, 1940, 2348, 2483, 4335, 4416, 4473\n\nHunter, Thomas 1891\n\nHunter, Virginia 1792, 2189, 3095, 3464, 3976\n\nHuntington, Joan 1924, 5065\n\nHuntington, Louise 1252\n\nHuntley, Chet 127\n\nHuppert, Isabelle 1833\n\nHurst, Brandon 1947, 3692, 4257, 4621, 4750\n\nHurst, Paul 89, 149, 196, 248, 327, 498, 522, 596, 605, 666, 930, 1020, 1038, 1179, 1521, 1573, 1688, 1799, 1831, 1982, 2016, 2182, 2264, 2359, 2458, 2472, 2669, 2699, 2726, 2815, 2823, 2875, 3005, 3046, 3104, 3128, 3178, 3271, 3321, 3561, 3691, 3774, 4002, 4048, 4184, 4413, 4470, 4641, 4712, 4745, 4851, 4986, 5051, 5076\n\nHurt, John 992, 1833, 4930\n\nHurwitz, Harry 815\n\nHussey, Ruth 2837, 3096, 4257\n\nHuston, Angelica 558, 2433, 3775\n\nHuston, John 2064, 2249, 2348, 2551, 2661, 3305, 4543, 4662\n\nHuston, Walter 197, 1187, 1478, 2249, 2943, 4543, 4744\n\nHutchins, Will 1705, 2623, 3203, 3845\n\nHutchinson, Josephine 1671, 2575, 2779, 4768\n\nHutchison, Charles 37, 1045, 2067, 2403, 3083, 3413\n\nHutton, Betty 94\n\nHutton, Dick 4063\n\nHutton, Jim 2501, 3723\n\nHutton, Lauren 4434, 5100\n\nHutton, Robert 2789, 2835, 2857, 3958, 4229, 5044, 5075\n\nHyams, Leila 3634, 4797, 5046\n\nHyde-White, Alex 5030\n\nHyer, Martha 259, 402, 982, 1439, 1688, 1963, 2080, 2143, 2795, 2895, 2939, 3337, 3685, 3856, 4035, 4386, 4966, 5038, 5078; _see also_ Julien, Martin\n\nHyke, Ray 49, 324, 659, 1214, 1395, 2515, 3323, 3333, 3814, 3891, 4616\n\nHyland, Diana 3737, 3973\n\nHyland, Frances 757, 1168, 1974, 1982\n\nHylands, Scott 1014\n\nHymer, Warren 1047, 1084, 2422, 2646, 2732, 3409, 3695\n\nHytten, Olaf 63, 287, 1257, 1512, 1947, 2213, 2552\n\nIce, Ada 1446, 3243\n\nIglesias, Eugene 604, 859, 1129, 1184, 1866, 1992, 2017, 2747, 3517, 4244, 4338, 4588, 4770\n\nIhnat, Steve 1924, 1943, 4162\n\nIlling, Peter 636, 1239\n\nImhoff, Peter 1165, 2159, 4257, 4359, 4941\n\nImhoff, Roger 1883, 2823, 3695\n\nInce, John 327, 343, 753, 777, 818, 927, 1083, 113, 1730, 1813, 1950, 2170, 2180, 2193, 2300, 2342, 2479, 2702, 2785, 2802, 2860, 3393, 3528, 3650, 3883, 4256, 4257, 4317, 4365, 4377, 4798, 4891, 4972, 5043\n\nInce, Ralph 1368, 1572, 2249, 2637, 4254\n\nInce, Thomas H. 248, 916, 1110, 1999\n\nInce, Thomas H., Jr. 2522, 2892\n\nInclan, Miguel 706, 1395, 1471, 1992\n\nIndrisano, John 183, 632\n\nInduni, Luis 61, 218, 220, 1120, 1508, 1801, 2214, 2326, 2354, 2499, 2570, 2817, 2905, 3286, 3346, 3420, 3717, 4003, 4325, 4332, 4594, 4629, 5093\n\nInescort, Frieda 1428\n\nInfante, Pedro 83, 1514\n\nInfante, Sonia 52, 1912, 2063, 2708\n\nInfuhr, Teddy 1523, 3095, 3552, 4533, 4692, 4745, 4812\n\nIngersoll, Felice 3261\n\nIngraham, Lloyd 37, 179, 199, 201, 230, 258, 297, 337, 383, 499, 579, 658, 790, 824, 834, 879, 927, 1084, 1162, 1174, 1177, 1222, 1241, 1336, 1447, 1450, 1453, 1503, 1544, 1545, 1679, 1681, 1682, 1687, 1827, 2032, 2159, 2207, 2248, 2296, 2299, 2342, 2356, 2432, 2529, 2599, 2635, 2682, 2722, 2832, 2861, 2971, 3043, 3052, 3084, 3142, 3162, 3242, 3264, 3269, 3329, 3362, 3426, 3443, 3456, 3557, 3581, 3728, 3834, 3880, 3889, 3920, 4006, 4031, 4071, 4072, 4081, 4103, 4143, 4170, 4200, 4257, 4297, 4317, 4327, 4385, 4405, 4419, 4427, 4460, 4489, 4490, 4498, 4555, 4564, 4572, 4593, 4656, 4700, 4732, 4818, 4826, 4854, 4868, 4995\n\nIngram, Jack 38, 70, 129, 143, 150, 151, 155, 205, 206, 229, 235, 258, 291, 303, 379, 410, 430, 475, 599, 631, 648, 666, 692, 749, 777, 778, 781, 797, 857, 986, 1008, 1051, 1059, 1073, 1090, 1130, 1137, 1148, 1155, 1193, 1229, 1247, 1268, 1282, 1357, 1362, 1384, 1400, 1407, 1433, 1447, 1448, 1450, 1456, 1459, 1464, 1468, 1473, 1487, 1490, 1540, 1554, 1555, 1558, 1576, 1627, 1647, 1738, 1749, 1751, 1756, 1823, 1827, 1916, 1964, 1969, 2109, 2119, 2135, 2256, 2278, 2282, 2284, 2399, 2407, 2408, 2413, 2418, 2420, 2424, 2432, 2452, 2514, 2546, 2552, 2565, 2589, 2635, 2648, 2673, 2687, 2701, 2778, 2786, 2802, 2847, 2866, 2874, 2909, 2948, 2953, 2968, 3018, 3052, 3067, 3068, 3077, 3097, 3103, 3165, 3187, 3224, 3227, 3228, 3230, 3254, 3279, 3454, 3475, 3494, 3509, 3525, 2547, 3556, 3592, 3616, 3626, 3651, 3675, 3708, 3744, 3820, 3827, 3834, 3848, 3862, 3894, 3995, 4000, 4010, 4051, 4053, 4059, 4104, 4123, 4144, 4155, 4166, 4167, 4185, 4188, 4191, 4206, 4278, 4283, 4286, 4402, 4551, 4552, 4556, 4612, 4613, 4619, 4649, 4654, 4655, 4682, 4696, 4701, 4702, 4709, 4729, 4732, 4759, 4773, 4819, 4820, 4848, 4842, 4848, 4888, 4893, 4899, 4954, 4955, 4995, 5018, 5037, 5049, 5056, 5059, 5096\n\nIngram, Rex 1238, 2059\n\nIngrassia, Ciccio 1180, 1181, 1621, 3064, 4623, 4625, 4627, 4628\n\nIreland, Jill 738, 1442, 3560, 4738\n\nIreland, John 61, 132, 418, 508, 584, 694, 845, 991, 1136, 1406, 1702, 1743, 1775, 1793, 1960, 2372, 2663, 2718, 3117, 3318, 3323, 3377, 3621, 3639, 4057, 4060, 4242, 4725, 4738, 4771\n\nIrish, Tom 1921, 3779\n\nIrons, Jeremy 120\n\nIronside, Michael 996\n\nIrving, Amy 81, 2023\n\nIrving, George 432, 1638, 1827, 2128, 2288, 2646, 2790, 2852, 2940, 4215, 4352, 4650, 4706, 4750\n\nIrving, Margaret 2940, 3695\n\nIrving, Richard 509, 4033\n\nIrwin, Boyd 1982, 2122, 2827\n\nIrwin, Charles 1407, 1427, 1809, 2677, 3276, 3821, 4214, 5011\n\nIsley, Phyllis 2786; _see also_ Jones, Jennifer\n\nIstabelita 1132\n\nIvarson, Dana 2109\n\nIvers, Robert 679, 4472\n\nIves, Burl 216, 313, 981, 1649, 2627, 3862, 3972, 4132, 4679, 4993\n\nIvins, Perry 277, 626, 1427, 1762, 2669, 3325, 3339, 3745, 3975, 4168\n\nIvo, Tommy 42, 1285, 1890, 1938, 2176, 2922, 2942, 3619, 3978, 4013, 4015, 4496, 4506, 4536, 4886, 5048\n\nJaccard, Celia 3771\n\nJaccard, Jacques 609, 722, 1044, 1045, 1845, 2007, 2945, 3078, 3488, 3771\n\nJaccard, Joan 722\n\nJackman, Fred 355, 1087, 2136, 2811\n\nJackman, Hugh 2854\n\nJackson, Anne 1106\n\nJackson, Danny 271\n\nJackson, Ethel 3939, 4711\n\nJackson, Eugene 1728, 1827, 2432\n\nJackson, Felix 1082, 1084\n\nJackson, Freda 4693\n\nJackson, Marion 1492, 2472, 3321, 4404, 4756\n\nJackson, Sammy 1273, 2795\n\nJackson, Selmer 514, 930, 1247, 1721, 2017, 2288, 2516, 3358, 3362, 3630, 3711, 3817, 3822, 4052, 4121, 4336, 4585, 4665, 4767\n\nJackson, Sherry 838, 854, 2370, 2561, 4976\n\nJackson, Thomas 379, 1567\n\nJackson, Warren 49, 631, 835, 973, 1716, 1906, 2163, 2649, 2677, 3383, 4023, 4049, 4986, 4503, 4823\n\nJacobs, Harrison 242, 254, 461, 790, 1469, 1816, 1855, 1932, 1978, 2270, 3049, 3354, 3704, 4767, 4923, 5062\n\nJacobs, William 2689, 4025, 4534\n\nJacobsson, Ulla 1341\n\nJacquet, Frank _see_ Jaquet, Frank\n\nJaeckel, Richard 101, 741, 788, 858, 978, 1371, 1422, 1578, 1654, 1704, 2188, 3055, 3320, 3964, 4370, 4472, 4633, 4740, 5036\n\nJaffe, Sam 1631, 1731, 3687\n\nJagger, Dean 46, 193, 514, 980, 1039, 1346, 1770, 1917, 2889, 3186, 3195, 3290, 3337, 3862, 3962, 3964, 4700, 4776, 4792, 4848\n\nJagger, Mick 2770\n\nJames, Alan 623, 814, 1162, 1367, 1677, 1928, 2121, 2165, 2886, 2391, 3015, 3092, 3969, 4163, 4219, 4454, 4484, 4554, 4612, 4703, 4816, 4861, 4863, 4955, 4957; _see also_ Neitz, Alvin J.\n\nJames, Alf 1302, 3575, 3919, 4294, 4377, 4668, 4898\n\nJames, Ann 250\n\nJames, Anthony 909, 1828, 1880, 3282, 3688, 4295, 4663\n\nJames, Brion 1068, 2093, 2309, 3519, 3911, 4571\n\nJames, Claire 3670, 4687\n\nJames, Clifton 1435, 2002, 2100, 2417, 3252, 3643, 4988\n\nJames, Gladden 4257\n\nJames, Jesse, Jr. 2036\n\nJames, John 465, 642, 925, 1487, 1646, 1680, 1873, 1920, 2179, 2435, 2546, 2944, 3265, 3357, 3457, 3475, 3484, 3670, 3708, 3989, 3999, 4028, 4409, 4462, 4617, 4690, 4731, 4811, 4855, 4938\n\nJames, Polly 3222, 3340\n\nJames, Sheila 3780\n\nJames, Sidney 663, 663, 3821\n\nJames, Walter 37, 130, 917, 1850, 2399, 2410, 2850, 4900\n\nJames, Will 2392, 3971, 3972, 3973\n\nJameson, Jerry 256, 538, 628, 860, 1750, 1753, 1755, 1879, 2000\n\nJameson, Joyce 2950, 3641, 4975\n\nJamison, Bud 1474, 1861, 2634, 2815, 4667, 4932\n\nJanis, Conrad 1170\n\nJanis, Elsie 4093\n\nJanney, William 747, 1932, 2137, 4215\n\nJanssen, David 733, 2488, 3106, 3856, 4674\n\nJanssen, Eilene 472, 544, 933, 1235, 2632, 3355, 4945\n\nJanuary, Lois 130, 244, 431, 847, 2355, 2356, 2381, 2690, 3332, 3546, 3581, 3952, 4581\n\nJaquet, Frank 291, 303, 615, 624, 712, 793, 957, 1360, 1564, 1974, 1981, 2123, 2928, 2949, 2988, 3178, 3230, 3447, 3707, 3834, 3893, 3943, 4121, 4257, 4428, 4463, 4509, 5005, 5039\n\nJara, Maurice 1161, 1560, 2066, 2400, 2768, 3024, 4335, 4338, 4726, 4770\n\nJarman, Claude, Jr. 1637, 1774, 1996, 2986, 3522, 3621, 5045\n\nJarmyn, Jil 2308, 2557, 4601, 4784\n\nJarrett, Art 4553\n\nJarrett, Dan 447, 633, 879, 1785, 1802, 1904, 2654, 2990, 3043, 3289, 3584, 3764, 4385, 4891\n\nJason, Leigh 2158\n\nJason, Peter 3527\n\nJason, Rick 982, 3863\n\nJason, Will 2088, 2543\n\nJay, Griffin 2641, 4419, 4496\n\nJefferson, L.V. 467, 1303, 2358, 2364, 3188, 3266, 3292, 4274, 4602\n\nJeffreys, Anne 338, 464, 634, 710, 1023, 1105, 1873, 2546, 2673, 2775, 2995, 3383, 4503, 4759\n\nJeffries, Herb 529, 713, 1787, 1788, 4615\n\nJeffries, Lang 1186\n\nJeffries, Lionel 1846\n\nJenkins, Allen 1084, 1579, 1812, 2856, 3916\n\nJenkins, Butch 491\n\nJenkins, Polly, and Her Plowboys 2529\n\nJenks, Frank 235, 1103, 1256, 1361, 1434, 1458, 1847, 2223, 2959, 3063, 3892, 4686\n\nJenks, Si 145, 311, 414, 431, 881, 1043, 1128, 1165, 1175, 1177, 1187, 1301, 1333, 1360, 140, 1480, 1489, 1496, 1513, 1537, 1567, 1585, 1647, 2027, 2254, 2303, 2349, 2451, 2470, 2482, 2517, 2534, 2764, 2856, 2862, 2886, 2923, 2928, 2940, 2987, 3038, 3269, 3289, 3290, 3292, 3434, 3443, 3451, 3508, 3561, 3692, 3916, 4004, 4011, 4016, 4256, 4257, 4290, 4358, 4483, 4503, 4605, 4645, 4891, 4957, 5061, 5103\n\nJennings, Al 42, 2167, 2182, 2864, 4022\n\nJennings, De Witt 328, 1340, 1897, 2610, 3895, 4093\n\nJennings, Maxine 2223\n\nJennings, Talbot 16, 2837, 4061, 4672\n\nJennings, Waylon 2439, 2490, 2623, 2770, 4102\n\nJens, Salome 490\n\nJenson, Roy 114, 232, 319, 373, 508, 538, 539, 953, 1170, 1250, 1354, 1371, 2058, 2093, 2155, 2348, 2664, 2700, 2829, 3009, 3152, 3320, 3338, 3430, 3993, 4098, 4449, 4794, 4857, 4988\n\nJergens, Adele 352, 700, 1240, 2261, 2436, 2959, 2996, 4148, 4179, 4533\n\nJerome, Jerry 1247, 2357, 3534, 3599, 3902, 4208, 4357\n\nThe Jesters 210, 481, 878, 1981, 3385\n\nJewell, Isabel 214, 282, 1161, 2837\n\nJeyne, Jack 1545, 4396, 4519, 5032\n\nJiminez, Soledad 91, 142, 204, 331, 354, 414, 606, 929, 934, 1390, 1972, 2248, 2831, 2864, 3081, 3384, 3561, 3595, 3768, 4055, 4281, 4525, 4650\n\nJochim, Anthony 324, 1319, 1341, 2043, 2339, 3234, 4123, 4845\n\nJodorowsky, Alexandro 1218\n\nJohns, Glynis 60, 3767, 4192\n\nJohns, Mervyn 4192\n\nJohnson, Ben 214, 348, 464, 508, 563, 720, 741, 959, 1395, 1396, 1398, 1623, 1769, 2069, 2100, 2501, 2721, 2853, 2865, 2901, 2982, 3283, 3299, 3320, 3522, 3632, 3661, 3798, 3814, 3960, 3990, 4180, 4253, 4360, 4452, 4523, 4545, 4638, 4764, 4784, 4934, 4960, 4966, 4968, 4970, 4988, 5033\n\nJohnson, Brad 11, 34, 35, 36, 2205, 2591, 2960, 3930, 4575\n\nJohnson, Chubby 1, 115, 185, 290, 597, 681, 819, 1263, 1274, 1343, 1347, 1400, 1408, 1433, 1714, 1747, 2252, 2804, 2996, 3219, 3536, 3572, 3688, 3783, 3954, 4211, 4258, 4536, 4548, 4579, 4804, 4857, 5066\n\nJohnson, Don 2265, 5083\n\nJohnson, June 1504, 1751, 2421\n\nJohnson, Kay 4071\n\nJohnson, Lamont 676, 1694\n\nJohnson, Laraine 429; _see also_ Day, Laraine; Johnson, Lorraine\n\nJohnson, Laura 3324\n\nJohnson, Linda 233, 1797, 4185\n\nJohnson, Lorraine 144, 3012; _see also_ Day, Laraine; Johnson, Laraine\n\nJohnson, Melodie 831, 3152, 3439\n\nJohnson, Noble 1165, 1462, 1490, 1805, 2064, 2359, 2825, 3126, 3128, 3269, 3814, 3859, 4635\n\nJohnson, Nunnally 64, 1371, 2031, 3179\n\nJohnson, Rafer 3776, 4041\n\nJohnson, Raymond K. 622, 724, 774, 856, 1260, 1979, 2107, 2172, 2284, 3098, 3446, 3494, 4619, 4953\n\nJohnson, Rita 60, 1256, 2719, 4398\n\nJohnson, Robert Lee 575, 1100, 1561, 3393, 3550, 4238, 4610\n\nJohnson, Russell 212, 803, 1112, 2252, 2575, 3255, 3423, 3676, 3768, 4120, 4148, 4588\n\nJohnson, Tor 2128, 3696\n\nJohnson, Van 482, 2013, 3175, 3861\n\nJohnston, Johnny 2543\n\nJohnstone, William 3500\n\nJolley, I. Stanford 28, 53, 76, 129, 131, 150, 189, 228, 251, 323, 352, 382, 418, 428, 452, 486, 494, 597, 599, 612, 649, 680, 711, 759, 794, 807, 842, 914, 928, 951, 973, 979, 997, 1059, 1063, 1073, 1130, 1272, 1296, 1325, 1363, 1403, 1415, 1448, 1449, 1456, 1481, 1505, 1506, 1543, 1553, 1664, 1680, 1715, 1724, 1747, 1748, 1766, 1800, 1860, 1939, 1957, 1963, 1980, 2008, 2083, 2085, 2096, 2111, 2169, 2292, 2294, 2310, 2363, 2381, 2426, 2448, 2512, 2521, 2530, 2564, 2591, 2659, 2759, 2765, 2776, 2824, 2863, 2871, 2902, 2953, 2965, 2978, 3067, 3068, 3419, 3157, 3164, 3169, 3185, 3202, 3222, 3255, 3280, 3295, 3298, 3305, 3377, 3390, 3511, 3576, 3586, 3588, 3676, 3779, 3780, 3806, 3862, 3901, 3942, 3989, 3998, 3999, 4081, 4095, 4108, 4118, 4141, 4220, 4266, 4273, 4278, 4285, 4301, 4339, 4408, 4439, 4462, 4490, 4497, 4514, 4551, 4568, 4588, 4610, 4620, 4701, 4731, 4754, 4766, 4803, 4812, 4860, 4889, 4894, 4921, 4935, 4937, 4957, 4966, 5039, 5066, 4078\n\nJolley, Norman 2943, 3915, 3902, 4613\n\nJones, Allan 2510, 3245, 3606, 3772, 4098\n\nJones, Arthur V. 69, 1297, 2765, 2824, 3162, 3557, 4096, 4564\n\nJones, Buck 131, 174, 289, 351, 429, 440, 478, 480, 496, 606, 613, 865, 894, 973, 975, 1002, 1058, 1146, 1222, 1302, 1324, 1329, 1384, 1388, 1389, 1554, 1614, 1716, 1810, 1848, 1906, 2014, 2070, 2258, 2278, 2315, 2404, 2558, 2628, 2632, 2642, 2908, 2961, 2892, 3087, 3260, 3322, 3424, 3450, 3480, 3486, 3555, 3575, 3699, 3797, 3903, 3967, 4054, 4138, 4152, 4178, 4187, 4201, 4308, 4377, 4379, 4410, 4535, 4668, 4767, 4823, 4863, 4898, 4899\n\nJones, Carolyn 1832, 1945, 1963, 2240, 4362\n\nJones, Darby 602\n\nJones, Dick 1091, 1408, 1705, 2217, 3365, 3572, 4036, 4166, 4757, 4937\n\nJones, Dickie 37, 460, 514, 952, 1084, 1469, 1626, 1906, 1947, 2165, 2399, 2689, 2885, 2943, 3342, 3361, 3967, 4493, 4742, 4854, 4955, 5072\n\nJones, Gordon 357, 1521, 1823, 1853, 2626, 2825, 2936, 2955, 3023, 3839, 3966, 4033, 4076, 4198, 4312, 4488, 4539, 4551, 4673, 4757, 4777, 5008, 5022\n\nJones, Grace 4146\n\nJones, Grover 825, 1176, 1685, 2352, 3817\n\nJones, Harmon 577, 650, 759, 977, 3909\n\nJones, Harry O. 510, 1470, 1489, 1494, 3275; _see also_ Fraser, Harry\n\nJones, Henry 586, 608, 771, 965, 1106, 2738, 3950, 3991, 4133, 4210, 4211, 4370\n\nJones, James Earl 1652\n\nJones, Jennifer 1187; _see also_ Isley, Phyllis\n\nJones, L.Q. 109, 190, 221, 239, 506, 748, 1091, 1371, 1769, 1951, 2015, 2162, 2328, 2429, 2501, 2604, 2627, 2706, 3055, 3324, 3435, 3661, 3853, 4122, 4133, 4253, 4433, 4934, 4968, 5007\n\nJones, Marcia Mae 5048\n\nJones, Morgan 4136\n\nJones, Ray 75, 155, 205, 234, 283, 366, 382, 390, 427, 428, 492, 510, 597, 606, 649, 658, 671, 683, 783, 834, 850, 857, 858, 867, 872, 956, 973, 1272, 1274, 1296, 1298, 1307, 1308, 1309, 1334, 1373, 1390, 1403, 1447, 1475, 1486, 1488, 1513, 1540, 1626, 1635, 1680, 1684, 1730, 1740, 1797, 1939, 2073, 2166, 2257, 2258, 2271, 2296, 2398, 2407, 2523, 2539, 2547, 2687, 2759, 2821, 2884, 2895, 2947, 2948, 2949, 2970, 2975, 2988, 2995, 3012, 3098, 3109, 3111, 3159, 3169, 3171, 3231, 3255, 3279, 3353, 3460, 3478, 3509, 3525, 3565, 3658, 3674, 2696, 3823, 3827, 3851, 3884, 3897, 3942, 3945, 4000, 4026, 4028, 4047, 4103, 4124, 4238, 4261, 4283, 4305, 4327, 4437, 4461, 4466, 4514, 4539, 4550, 4556, 4559, 4609, 4640, 4685, 4731, 4754, 4759, 4781, 4816, 4820, 4827, 4855, 4894, 4980, 5018, 5027\n\nJones, Shirley 727, 2853, 4626\n\nJones, Stan 2200, 3244, 3447, 3512, 4253, 4886\n\nJones, Tex 4424\n\nJones, Tommy Lee 1611, 2433, 2662, 4356\n\nThe Jones Boys 323\n\nJons, Beverly 667, 1516, 3484\n\nJordan, Betty 4493\n\nJordan, Bobby 488, 3759\n\nJordan, Dorothy 3752\n\nJordan, Judy 315, 1510\n\nJordan, Louis 2449\n\nJordan, Marsha 3248, 4218\n\nJordan, Richard 709, 2305, 3147, 3602, 4688\n\nJordan, Sid 243, 1817, 1904, 1931, 2342, 2541, 3956, 4385, 4423, 4510, 4603, 4849, 4865\n\nJordan, Ted 418, 584, 1754, 2601, 3862\n\nThe Jordanaires 559, 1371, 4133\n\nJory, Victor 241, 459, 549, 638, 654, 692, 715, 720, 800, 1127, 1319, 1364, 1934, 2081, 2151, 2234, 2311, 2353, 2489, 2544, 2552, 2663, 2700, 3281, 3437, 3479, 3538, 3971, 4049, 4136, 4214, 4428, 4456, 4470, 4923\n\nJoseph, Jackie 727\n\nJoslin, Howard 3201, 3891, 4309, 4714\n\nJoslyn, Allyn 536, 1274\n\nJostyn, Jay 523, 2462\n\nJowitt, Anthony 3695\n\nJoy, Leatrice 3335\n\nJoyce, Jean 2968, 2993, 3461\n\nJoyner, Jozelle 4020, 4706, 4892\n\nJudd, John 2171, 4299, 4322, 4564\n\nJudd, Naomi 3519\n\nJudels, Charles 323, 653, 1982, 3126, 3128, 3695, 4747\n\nJudge, Arline 1563, 4018, 4225, 4979\n\nJudge, Naomi 4271, 5061\n\nJulia, Raul 45, 2627\n\nJulien, Martin 3602; _see also_ Hyer, Martha\n\nJulien, Max 4346\n\nJunco, Victor 226, 1911, 3781, 4638, 5090\n\nJurado, Katy 166, 209, 520, 1150, 1877, 2518, 2901, 3055, 3690, 3973, 4133, 4632\n\nJuran, Nathan 1164, 1608, 1747, 2174, 2252, 4588\n\nJustice, James Robertson 636\n\nJustice, Katherine 1354, 4800\n\nJustin, John 4672\n\nKaaren, Suzanne 3086, 4656, 4941, 4983\n\nKadison, Ellis 673\n\nKadler, Karen 2143\n\nKahn, Gordon 868, 2368, 2838, 4016\n\nKahn, Madeline 385\n\nKahn, Richard C. 529, 588, 589, 1788, 4615\n\nKallen, Kitty 3754\n\nKallman, Dick 1838\n\nKanaly, Steve 2723, 4180\n\nKandel, Stephen 641, 1452, 2496, 4769, 4990\n\nKane, Eddie 67, 286, 961, 1174, 1836, 1906, 2534, 2877, 3626, 4225, 4254, 4377\n\nKane, Helen 948\n\nKane, Jimmy 300, 696, 4985\n\nKane, Joseph 143, 201, 337, 424, 442, 516, 612, 667, 790, 810, 868, 930, 986, 1182, 1360, 1462, 1490, 1553, 1574, 1595, 1647, 1714, 1728, 1751, 1767, 1818, 1822, 1852, 1964, 1973, 1976, 1980, 1982, 1983, 2032, 2066, 2124, 2129, 2182, 2184, 2234, 2295, 2299, 2432, 2514, 2529, 2530, 2624, 2636, 2778, 2850, 2851, 2872, 2874, 2878, 3128, 3132, 3187, 3269, 3330, 3433, 3436, 3483, 3498, 3542, 3564, 3567, 3600, 3615, 3625, 3675, 3681, 3728, 3827, 3834, 3904, 3963, 4016, 4018, 4037, 4051, 4059, 4074, 4082, 4202, 4206, 4388, 4428, 4592, 4655, 4705, 4773, 5020, 5033, 5050, 5056, 5059, 5062\n\nKane, Louise 5033\n\nKane, Marjorie \"Babe\" 451, 1634\n\nKantor, Hal 2895, 3296\n\nKantor, MacKinlay 1756, 4991\n\nKaren, James 2377, 2958\n\nKaris, Vassili 92\n\nKarlan, Richard 3864, 3980\n\nKarlen, John 3961\n\nKarloff, Boris 2211, 2606, 4635, 4683, 4708\n\nKarlson, Phil 25, 312, 362, 1719, 3560, 4310, 4338, 4399, 4431\n\nKarnes, Robert 704, 1356, 1888, 2930, 3576, 3768, 4074, 4111, 4686\n\nKarns, Roscoe 3499, 4567, 4733\n\nKarr, Mabel 918\n\nKarras, Alex 385, 1786\n\nKasdan, Lawrence 3911, 5029\n\nKashfi, Anna 859\n\nKasznar, Kurt 3440\n\nKatch, Kurt 3684\n\nKatt, William 585\n\nKatzin, Lee H. 1832, 1922, 2924, 3203, 3734\n\nKatzman, Sam 259, 535, 604, 1273, 2017, 2287, 2456, 2620, 2743, 2933, 3081, 3509, 4549, 4680, 4682, 4867, 4871\n\nKaufman, Millard 193\n\nKaufman, Philip 1641, 4897\n\nKavanaugh, Frances 150, 157, 383, 658, 795, 1156, 1157, 1196, 1224, 1330, 1585, 1789, 2286, 3503, 3598, 3670, 4017, 4032, 4081, 4131, 4502, 4590, 4844, 4853, 4884, 4969\n\nKay Bernice 4923; _see also_ Williams, Cara\n\nKay, Beatrice 4429\n\nKay, Gilbert Lee 4896; _see also_ Briz, Jose\n\nKay, Mary Ellen 453, 559, 796, 1048, 1400, 2693, 3892, 4167, 4381, 4392, 4478, 4731, 4804, 5081\n\nKaye, Stubby 674, 771, 4800\n\nKazan (dog) 2024\n\nKazan, Elia 3750, 4751\n\nKazan, Lainie 2483\n\nKeach, James 816, 2156, 2335, 2443\n\nKeach, Stacy 587, 1121, 2023, 2348, 2406, 2443, 3519\n\nKeane, Edward 251, 277, 429, 437, 442, 464, 610, 611, 614, 865, 1023, 1384, 1436, 1462, 1785, 1843, 1883, 1906, 2150, 2756, 2815, 2893, 2935, 3479, 3585, 3669, 4336, 4508, 4597, 4742, 4803, 4832, 4891, 4979, 5001\n\nKeane, Robert Emmett 337, 653, 1380, 1427, 1454, 1853, 1888, 2762, 3236, 4213, 4425, 4940\n\nKeating, Fred 1090\n\nKeating, Larry 664, 1757, 1995\n\nKeaton, Buster 1585, 2790, 4413\n\nKeats, Stephen 182\n\nKeays, Vernon 391, 2175, 2595, 3410, 3569, 3914, 4170, 4504, 4552, 4684, 4888\n\nKeckley, Jane 1751, 1979, 2350, 2399, 2422, 2552, 2635, 2802, 3011, 3126, 3549, 3949, 4459\n\nKeefe, Cornelius 323, 2874, 3673, 4100, 4558, 4592, 4837\n\nKeel, Howard 94, 132, 597, 632, 1078, 2513, 3009, 3338, 3440, 3607, 3780, 4291, 4755, 4790\n\nKeen, Chuck 696, 766, 4426\n\nKeene, Tom 150, 258, 306, 723, 811, 898, 1043, 1153, 1156, 1167, 1196, 1557, 1576, 1587, 2257, 2418, 2632, 2879, 2895, 3016, 3048, 3288, 3300, 3360, 3503, 3597, 3665, 3745, 4007, 4190, 4203, 4488, 4647, 4778, 4844, 4883, 4884; _see also_ Duryea, George; Powers, Richard\n\nKeene, Valley 4010\n\nKeiffer, Phil 3027, 3096, 3385, 4803, 4822\n\nKeighley, William 1588, 3572, 4697\n\nKeitel, Harvey 426, 554, 1204, 1745\n\nKeith, Brian 66, 563, 740, 849, 935, 1005, 1399, 1500, 1765, 1838, 2045, 2700, 2779, 3203, 3223, 3283, 3640, 3731, 3739, 3753, 3863, 3990, 4253, 4417, 4437, 4740, 4772, 5067\n\nKeith, Byron 368, 1631\n\nKeith, Donald 4602\n\nKeith, Ian 157, 328, 464, 875, 1188, 1634, 2546, 2841, 3080, 3923, 4017, 4185, 4654, 4957\n\nKeith, Robert 497, 500, 748, 860, 1083, 1084, 1092, 1161, 3149, 4335\n\nKeith, Rosalind 2131, 4572, 4832\n\nKellard, Robert 1298, 2132, 3165, 4278, 4378; _see also_ Stevens, Robert\n\nKellaway, Cecil 26, 2649, 3186\n\nKeller, Harry 236, 365, 979, 1048, 1216, 1400, 1535, 1701, 2310, 2593, 3091, 3199, 3328, 3608, 3724, 3791, 3933, 4119, 4400\n\nKellerman, Sally 3772, 4977\n\nKelley, Barry 539, 654, 1319, 1640, 1653, 1702, 1714, 1963, 2252, 3518, 3921, 4234, 5020\n\nKelley, DeForrest 114, 563, 1696, 2246, 3437, 4259, 4472, 4755, 4791\n\nKellogg, Bruce 248, 1032, 1600\n\nKellogg, Cecil 3555, 4142, 4309, 4849\n\nKellogg, John 207, 472, 506, 1314, 2084, 2138, 3222, 3255, 3563, 3882, 3928, 4132, 4349\n\nKellogg, Ray 116, 1561, 3612\n\nKellogg, William 283, 1915, 2909, 3393, 4415, 4806, 5062\n\nKelly, Brian 1653\n\nKelly, Carol 4267\n\nKelly, Claire 209, 3982\n\nKelly, Emmett 4993\n\nKelly, Gene 727\n\nKelly, Grace 1877, 2013\n\nKelly, Jack 762, 803, 1500, 1747, 2252, 2788, 2789, 3340, 4120, 4338, 4740, 5060, 5072\n\nKelly, Jeanne 210, 1297, 2526, 3450, 4006; _see also_ Brooks, Jean\n\nKelly, Jim 4224\n\nKelly, Lew 158, 442, 927, 1394, 1591, 1595, 1647, 1883, 1925, 1961, 1974, 2012, 2208, 2304, 2471, 2529, 2554, 2992, 3012, 3540, 3609, 3675, 3846, 4372, 4498, 4593, 4776, 4840, 4851, 4937, 4995\n\nKelly, Nancy 1458, 2031\n\nKelly, Patsy 866, 1974\n\nKelly, Paul 844, 1433, 1566, 1747, 1947, 2012, 2530, 3104, 3692, 4069, 4075, 4080, 4865, 5032\n\nKelly, Tommy 1247\n\nKelsey, Fred 798, 935, 1008, 1240, 1831, 2400, 2502, 2642, 2644, 2649, 2794, 2835, 2980, 3215, 3615, 3681, 3692, 3902, 4115, 4336, 4621, 4925, 4932\n\nKelso, Edmund 2740, 2910, 2927, 2968, 3491, 3589, 3870, 4186, 4279, 4685\n\nKelton, Pert 95, 3408\n\nKemmer, Ed 2576, 3865\n\nKemmerling, Warren 727, 1689, 2598, 2631, 2760\n\nKemp, Matty 38, 710, 2278\n\nKemper, Charles 282, 612, 1136, 1480, 1706, 2138, 2781, 4058, 4129, 4411, 4764, 5051\n\nKemper, Doris 2871, 3386, 4233\n\nKempler, Charles 3506\n\nKempler, Kurt 326, 4248, 4607\n\nKendall, Cy 332, 758, 1270, 1570, 1571, 1573, 1594, 1599, 1925, 2128, 2163, 2164, 2432, 2552, 2957, 3162, 3289, 3429, 3564, 3695, 3744, 3794, 3899, 4121, 4231, 4572, 4697\n\nKendall, Tony 358, 531, 1114, 3113, 4596\n\nKennedy, Arthur 290, 718, 720, 980, 1884, 2485, 2656, 2747, 2752, 2779, 3255, 3295, 3318, 4336, 4771\n\nKennedy, Bill 2, 282, 444, 449, 649, 659, 1236, 1959, 2282, 2776, 2851, 3002, 3630, 3802, 3823, 3892, 4140, 4500, 4524, 4559\n\nKennedy, Burt 45, 639, 806, 1062, 1106, 1147, 1399, 1610, 1692, 1776, 1946, 2082, 2500, 2694, 2897, 3391, 3430, 3622, 3788, 3840, 3860, 3933, 4133, 4210, 4211, 4235, 4523, 4790, 4802, 4873, 4975, 5055, 5060\n\nKennedy, Daun 3630, 3684\n\nKennedy, Douglas 632, 1319, 1665, 1696, 1847, 1992, 2017, 2202, 2241, 2395, 2402, 2426, 2617, 2831, 3271, 3931, 4048, 4049, 4115, 4148, 4310, 4787\n\nKennedy, Edgar 879, 899, 1627, 1829, 1830, 1974, 3561, 3695, 3745, 4007\n\nKennedy, George 237, 563, 594, 903, 1036, 1106, 1610, 1739, 1695, 1707, 2094, 2210, 2381, 2430, 3816, 4035, 4353, 4575\n\nKennedy, Leon Issac 2429\n\nKennedy, Madge 3134\n\nKennedy, Merna 814, 1517\n\nKennedy, Tom 449, 486, 686, 898, 1593, 2554, 2649, 2688, 3020, 3541, 4085, 4225, 4979\n\nKenney (Kenny), Jack 464, 1670, 1956, 2020, 2128, 2158, 2302, 2561, 2730, 2847, 2864, 2872, 3164, 3329, 3500, 4073, 4257, 4317, 4435, 4461, 4469, 4532, 5037\n\nKent, Barbara 1432, 2811\n\nKent, Crauford 952, 2990, 3626\n\nKent, Dorothea 615, 943, 2124\n\nKent, Gary 3638, 3732\n\nKent, Robert E. 402, 503, 604, 1273, 1405, 1509, 1573, 2037, 3056, 3208, 3334, 4682, 4805, 4867, 4871\n\nKent, Robert 844, 1392, 2131, 2835, 3088, 3298, 4106, 4199, 4600\n\nKent, Willis 130, 1686, 1952, 2303, 4321\n\nKenton, Erle C. 1228, 2830\n\nKentworthy, Katherine 4593, 5037\n\nKenyon, Charles 1884, 3071, 3080, 3537, 3879\n\nKenyon, Curtis 4585\n\nKenyon, Doris 270, 578, 1912\n\nKenyon, Ethel 496\n\nKenyon, Gwen 1981, 3499, 4803\n\nKenyon, May 795\n\nKerby, Marian 4587\n\nKercheval, Ken 598\n\nKern, James V. 4115\n\nKern, Roger 5073, 5074\n\nKerr, Deborah 4192\n\nKerr, Donald 290, 597, 863, 1364, 1904, 2161, 2288, 2396, 2452, 2853, 2899, 3234, 3362, 3802, 3925, 4023, 4073, 4086, 4225, 4503, 4649, 4654, 4808, 4983, 5044\n\nKerrigan, J. Warren 852, 1570\n\nKerrigan, J.M. 246, 254, 309, 1827, 2108, 2393, 2705, 3179, 3241, 3414, 3909, 4665, 4671, 4962\n\nKerry, Norman 254, 3078\n\nKerschner, Irvin 3369\n\nKershaw, Doug 989, 5083\n\nKesterson, George 4; _see also_ Mix, Art\n\nKetchum, Hal 2623\n\nKeyes, Evelyn 307, 1070, 2705, 3355, 4665\n\nKeyes, Evon 3521\n\nKeyes, Stephen 1003, 1298, 2106, 2171, 2713, 2866, 4196, 4820\n\nKeymas, George 101, 116, 148, 356, 449, 789, 1481, 1714, 1748, 1809, 2096, 2437, 2624, 2930, 3221, 3705, 4141, 4338, 4388, 4682, 4705, 4913, 4990, 5038\n\nKeys, Peggy 3288, 3456\n\nKeys, Robert 484, 3404, 3690, 3998, 4860\n\nKibbee, Guy 200, 826, 869, 1395, 1564, 1853, 2344, 2419, 2991, 3334, 3923\n\nKibbee, Milton 15, 203, 343, 345, 354, 366, 382, 387, 470, 611, 695, 714, 964, 1099, 1127, 1221, 1436, 1594, 1740, 1920, 1974, 2027, 2082, 2164, 2408, 2411, 2471, 2689, 2835, 2998, 3232, 3374, 3534, 3538, 3547, 3576, 4053, 4083, 4257, 4357, 4513, 4534, 4554, 4617, 4687, 4806, 4838, 4855, 4871, 5018\n\nKibbee, Roland 119, 507, 4688, 4727\n\nKidder, Margot 487, 1926, 2623, 3842\n\nKidman, Nicole 1262\n\nKiefer, Warren 2228; _see also_ Ricci, Luciano\n\nKiel, Richard 247, 2183\n\nKilbride, Percy 4058, 5021\n\nKilburn, Terence 2919\n\nKiley, Richard 1435, 1752, 2487\n\nKilian, Victor 229, 248, 278, 798, 1236, 1960, 2031, 2127, 2586, 2839, 2875, 2937, 3005, 3053, 3245, 3374, 3377, 3384, 3511, 3711, 3852, 3972, 4742, 4849, 5034, 5045, 5051\n\nKilly, Edward 69, 231, 812, 927, 2170, 2775, 3504, 3557, 4096, 4767, 4777, 4825\n\nKilmer, Val 993, 2662, 4453, 5031\n\nKilpatrick, Lincoln 2619, 4041\n\nKimball, Anne 2980, 4766\n\nKimble, Lawrence 176, 283, 3096\n\nKimbrough, John 2423, 4184\n\nKincaid, Aron 3183\n\nKing, Andrea 2583, 5025\n\nKing, Billy 1816, 1933, 3177, 4322\n\nKing, Brad 2972, 3479, 3692, 3761, 4568, 4600\n\nKing, Brett 2037\n\nKing, Charles 56, 70, 76, 107, 151, 153, 157, 206, 210, 289, 295, 300, 309, 334, 335, 340, 342, 343, 344, 351, 352, 382, 383, 430, 452, 464, 468, 479, 481, 494, 495, 500, 582, 610, 634, 658, 683, 724, 731, 779, 795, 834, 846, 878, 886, 891, 897, 946, 975, 1001, 1008, 1018, 1025, 1042, 1052, 1070, 1090, 1137, 1148, 1172, 1229, 1269, 1271, 1282, 1296, 1403, 1307, 1322, 1335, 1337, 1338, 1363, 1389, 1447, 1448, 1459, 1462, 1465, 1468, 1486, 1503, 1505, 1506, 1517, 1537, 1542, 1543, 1546, 1554, 1587, 1595, 1682, 1713, 1716, 1728, 1738, 1789, 1792, 1798, 1812, 1827, 1894, 1895, 1900, 1927, 1965, 1969, 2014, 2035, 2110, 2111, 2124, 2166, 2171, 2189, 2205, 2220, 2250, 2251, 2256, 2266, 2273, 2280, 2283, 2285, 2291, 2299, 2306, 2356, 2365, 2391, 2399, 2407, 2409, 2410, 2413, 2418, 2420, 2432, 2469, 2522, 2542, 2546, 2572, 2573, 2589, 2600, 2632, 2645, 2660, 2740, 2742, 2745, 2759, 2761, 2832, 2847, 2858, 2860, 2861, 2890, 2891, 2909, 2927, 2951, 2953, 2957, 2961, 2965, 2969, 2974, 2977, 2999, 3015, 3029, 3067, 3077, 3081, 3086, 3087, 3097, 3101, 3111, 3115, 3135, 3142, 3155, 3164, 3165, 3173, 3226, 3232, 3263, 3280, 3308, 3329, 3332, 3390, 3401, 3446, 3461, 3475, 3476, 3480, 3484, 3492, 3531, 3550, 3555, 3589, 3590, 3648, 3703, 3709, 3801, 3825, 3884, 3888, 3913, 3926, 3967, 4007, 4011, 4019, 4030, 4031, 4032, 4052, 4078, 4106, 4118, 4128, 4153, 4165, 4186, 4189, 4201, 4238, 4278, 4279, 4299, 4303, 4363, 4382, 4394, 4402, 4489, 4502, 4573, 4581, 4589, 4592, 4606, 4614, 4645, 4649, 4685, 4698, 4702, 4808, 4819, 4872, 4884, 4899, 4947, 4953, 5008, 5033, 5061, 5104\n\nKing, Claude 2628, 2790, 4367\n\nKing, Dennis, Jr. 3367\n\nKing, Henry 502, 1704, 2031, 3245, 3934, 4672, 5004\n\nKing, Jack 684, 729, 773, 975, 1084, 1474, 1716, 2549, 2915, 3135, 3268, 3270, 3279, 3303, 3486, 3525, 3658, 4294, 4364, 4525, 4667, 4837\n\nKing, John 1310, 1475, 1528, 3258, 3351\n\nKing, John (Dusty) 151, 422, 1310, 3568, 3668, 4215, 4319, 4343, 4393, 4458, 4497, 4502, 4515, 4606, 4660, 4787, 4815, 5027\n\nKing, Joseph 1588, 1820, 2164, 2689, 4336\n\nKing, Louis 440, 1058, 1329, 1433, 1649, 2349, 2370, 2404, 2611, 2642, 2705, 3099, 3150, 3558, 3698, 3797, 3972, 4025, 4398\n\nKing, Pee Wee 1362, 3493, 3619\n\nKing, Sonny 3777\n\nKing, Walter Woolf 482, 1577\n\nKing, Wright 671, 2058, 5066\n\nKing, Zalman 947, 1998, 4661\n\nThe King's Men 2081, 2152, 2270, 2526, 3354, 3624, 3851, 4112\n\nKingsford, Guy 1407, 3056, 4106, 4319, 5054\n\nKingsford, Walter 1430, 2064, 2756\n\nKingston, Natalie 4492, 4648\n\nKingston, Winifred 4091\n\nThe Kingston Trio 2339\n\nKinney, Jack 597, 1070, 2408\n\nKinski, Klaus 85, 268, 571, 1305, 1381, 1525, 1643, 1896, 1955, 1966, 2219, 2230, 2817, 3174, 3382, 3403, 3659, 3841, 4332, 4434, 4595, 4630, 4724\n\nKinski, Nastassia 763\n\nKinsky, Leonid 642, 1591, 1923, 2158, 3409\n\nKirby, Bruce 195\n\nKirby, Bruno 463, 760\n\nKirby, Jay 446, 800, 827, 984, 1934, 2125, 2311, 2451, 2600, 2857, 2944, 3050, 3569, 3822, 4002, 4188, 4657, 4763, 5103\n\nKirby, Luke 58\n\nKirby, Pete _see_ Bashful Brother Oswald\n\nKirk, Jack 6, 28, 37, 51, 89, 140, 154, 201, 235, 243, 291, 295, 297, 299, 325, 337, 414, 415, 442, 452, 460, 465, 607, 610, 611, 624, 644, 662, 665, 666, 684, 714, 721, 731, 777, 779, 793, 797, 810, 827, 834, 836, 853, 862, 868, 886, 927, 951, 1009, 1023, 1024, 1046, 1137, 1177, 1288, 1306, 1307, 1335, 1345, 1389, 1432, 1449, 1462, 1488, 1490, 1504, 1512, 1545, 1556, 1557, 1574, 1592, 1595, 1662, 1678, 1681, 1683, 1725, 1740, 1751, 1758, 1767, 1785, 1813, 1853, 1860, 1873, 1895, 1898, 1915, 1918, 1920, 1928, 1929, 1952, 1964, 1974, 1976, 1983, 1993, 2033, 2082, 2121, 2124, 2125, 2129, 2235, 2251, 2254, 2262, 2278, 2296, 2298, 2299, 2300, 2303, 2368, 2391, 2398, 2399, 2403, 2404, 2408, 2411, 2421, 2427, 2452, 2514, 2523, 2527, 2545, 2548, 2597, 2600, 2605, 2635, 2673, 2689, 2737, 2778, 2785, 2799, 2802, 2850, 2857, 2909, 2931, 2948, 2951, 2954, 2967, 2968, 2995, 2998, 3025, 3027, 3048, 3080, 3084, 3088, 3089, 3109, 3111, 3160, 3163, 3176, 3194, 3228, 3259, 3268, 3300, 3408, 3427, 3454, 3444, 3480, 3482, 3483, 3492, 3549, 3552, 3574, 3600, 3603, 3615, 3617, 3625, 3626, 3665, 3673, 3693, 3708, 3744, 3820, 3822, 3824, 3825, 3826, 3827, 3834, 3866, 3893, 3904, 3919, 3922, 3924, 4004, 4011, 4022, 4051, 4082, 4181, 4187, 4197, 4206, 4238, 4248, 4269, 4273, 4294, 4305, 4318, 4372, 4389, 4396, 4454, 4460, 4463, 4465, 4483, 4487, 4501, 4525, 4570, 4584, 4586, 4643, 4649, 4655, 4695, 4698, 4732, 4736, 4763, 4827, 4835, 4848, 4854, 4855, 4862, 4954, 5037, 5040, 5056, 5059, 5096, 5103\n\nKirk, Phyllis 2005, 2046, 4390\n\nKirk, Tommy 2887, 3731\n\nKirke, Donald 1426, 1869, 1935, 2850, 2966, 3259, 3289, 3424, 3851, 3967, 4201\n\nKirkland, David 3455, 3983\n\nKirkland, Jack 4215\n\nKirkland, Muriel 4442\n\nKirkwood, James 23, 277, 1136, 1427, 1907, 2159, 2226, 2549, 2726, 2781, 2782, 3054, 3241, 3302, 3335, 3702, 4071, 4099, 4174, 4257, 4673, 5005, 5022\n\nKirkwood, Ray 3939\n\nKitosch, Cole 175, 4678\n\nKjellin, Alf 205, 2629, 5084\n\nKleeb, Helen 913, 1434, 1765\n\nKlein, Philip 3470\n\nKleiner, Harry 1244, 1341, 1963, 2079, 3333, 4740\n\nKlick, Roland 1004\n\nKlimovsky, Leon 694, 1286, 3286\n\nKline, Benjamin 875, 878, 928, 2269, 2367, 3467, 3666, 3678, 4195, 4607\n\nKline, Herbert 1293, 4229\n\nKline, Kevin 3911, 4974\n\nKnaggs, Skelton 1811, 3020\n\nKnapp, Evelyn 1802, 1983, 3289, 3537, 4706\n\nKnapp, Robert 1720, 2065, 2644, 2981, 3294, 3404, 4451\n\nKnight, Fuzzy 30, 103, 112, 135, 145, 155, 199, 204, 210, 303, 460, 475, 476, 477, 486, 514, 582, 649, 726, 739, 775, 794, 847, 863, 868, 896, 1028, 1285, 1297, 1307, 1438, 1450, 1456, 1597, 1627, 1693, 1718, 1888, 1917, 1923, 1939, 1940, 2085, 2232, 2235, 2250, 2273, 2291, 2294, 2378, 2424, 2526, 2595, 2596, 2607, 2722, 2749, 2776, 3798, 2856, 2866, 2873, 2877, 2884, 2923, 2927, 3126, 3220, 3227, 3255, 3291, 3357, 3452, 3478, 3511, 3526, 3576, 3656, 3772, 3817, 3889, 3924, 3925, 3951, 4006, 4022, 4027, 4061, 4095, 4103, 4105, 4203, 4225, 4333, 4442, 4462, 4494, 4504, 4509, 4652, 4665, 4731, 4755, 4776, 4781, 4808, 4820, 4882, 4982\n\nKnight, Harlan 3555, 4442, 4892\n\nKnight, Sandra 4266\n\nKnight, Shirley 1435, 1609, 1963\n\nKnight, Ted 1341, 4339, 4626\n\nKnopf, Christopher 506, 1220, 1837, 2706, 3146, 4234\n\nKnopf, Edwin H. 2352, 3710\n\nKnotts, Don 121, 122, 1942, 2710, 3296, 3806\n\nKnowles, Patric 741, 3201, 3941, 4800\n\nKnox, Alexander 2549, 3807, 4738\n\nKnox, Elyse 362, 2688, 3827\n\nKnox, Mickey 4845\n\nKnox, Mona 4400\n\nKnox, Patricia 452, 1360, 1363, 1531, 2022, 2158, 3155, 3917, 4490\n\nKnudsen, Peggy 482, 832, 841, 2161, 4115\n\nKobe, Gail 1748\n\nKoch, Howard W. 466, 1396\n\nKoch, Marianne 769, 1350, 2230, 3125, 4194, 4919\n\nKohler, Fred 145, 309, 337, 429, 660, 1288, 1300, 1384, 1390, 1426, 1436, 1490, 1591, 1785, 1824, 1850, 1928, 2181, 2232, 2352, 2366, 2523, 3012, 3126, 3166, 3179, 3443, 3469, 4116, 4142, 4289, 4309, 4413, 4519, 4639, 4652, 4824, 4950, 4986\n\nKohler, Fred, Jr. 53, 251, 470, 477, 634, 800, 915, 955, 1073, 1516, 1843, 1907, 2304, 2386, 2423, 2552, 2778, 3062, 3215, 3230, 3261, 3305, 3414, 3554, 3936, 4076, 4267, 4315, 4339, 4448, 4467, 4574, 4597, 4617, 4844, 5072\n\nKohner, Susan 2241, 4566\n\nKolb, Clarence 1594, 2288, 4803\n\nKolker, Henry 866, 3046, 4665\n\nKolkmar, Lee 3895, 3954, 4254\n\nKomai, Tetsu 434, 2147, 2852\n\nKorda, Maria 1273\n\nKorff, Arnold 2639\n\nKorman, Harvey 385\n\nKornman, Mary 633, 1057, 3968\n\nKortman, Robert (Bob) 37, 64, 144, 159, 179, 230, 293, 295, 351, 366, 427, 446, 479, 496, 497, 573, 582, 610, 624, 747, 825, 832, 834, 894, 987, 1002, 1024, 1087, 1110, 1257, 1297, 1302, 1306, 1364, 1390, 1392, 1416, 1448, 1453, 1470, 1475, 1553, 1592, 1738, 1851, 1861, 1870, 2014, 2033, 2121, 2175, 2212, 2250, 2258, 2273, 2278, 2288, 2300, 2311, 2367, 2394, 2399, 2415, 2432, 2451, 2469, 2530, 2589, 2660, 2733, 2737, 2753, 2799, 2861, 2864, 2967, 3016, 3020, 3029, 3092, 3097, 3239, 3270, 3279, 3291, 3339, 3353, 3354, 3475, 3486, 3601, 3609, 3616, 3666, 3699, 3817, 3967, 3969, 4025, 4104, 4108, 4112, 4135, 4167, 4168, 4185, 4203, 4219, 4271, 4322, 4405, 4423, 4484, 4494, 4635, 4667, 4700, 4708, 4732, 4745, 4816, 4826, 4863, 4868, 4889, 4890, 4923, 4961, 4971, 5001, 5035, 5096\n\nKorvin, Charles 3701, 5097\n\nKosleck, Martin 1627\n\nKoslo, Paul 822, 965, 2044, 2093, 3602, 3661, 3739, 3747\n\nKotto, Yaphet 1064, 1354, 2503\n\nKovack, Nancy 1644, 2964, 4973\n\nKovacs, Ernie 2829\n\nKove, Martin 1657, 1780, 1905, 3789, 4571, 4895, 5030\n\nKowalski, Bernard 370, 2488\n\nKrafft, John 147\n\nKramer, Frank 20, 1589, 1966, 2053, 3381, 3660; _see also_ Parolini, Gianfranco\n\nKramer, Stanley 2859\n\nKramer, Wright 3851\n\nKrasna, Norman 1361\n\nKrasny, Paul 2045\n\nKress, Harold 115\n\nKreuger, Kurt 1236\n\nKrims, Milton 1388, 2672, 4258, 4824, 4835\n\nKristofferson, Kris 97, 517, 1109, 1375, 1833, 2190, 2199, 2417, 2439, 2658, 3017, 3034, 3055, 3984, 4102, 4477, 4879\n\nKroeger, Berry 260, 1319, 5053\n\nKrone, Fred 830, 1397, 2348, 4420, 4788, 5065, 5068\n\nKruger, Otto 254, 1187, 2187\n\nKruger, Paul 520, 798, 2147, 3179, 3222, 3431, 3954, 4057, 4312, 4336, 4747, 4803\n\nKrusada, Carl 305, 468, 592, 1211, 1271, 1282, 1312, 1344, 1602, 2107, 2177, 2742, 2745, 2822, 3018, 3073, 3079, 3098, 3446, 3477, 3487, 3495, 3528, 3703, 3869, 3888, 4269, 4298, 4852, 4953, 4958, 5014\n\nKruschen, Jack 211, 661, 1660, 1985, 2626, 2896, 3791, 4670, 4675\n\nKuhn, Mickey 519, 2064, 2192, 3323, 3552\n\nKulik, Buzz 3106, 4738\n\nKulky, Henry 234, 700, 1697, 1741, 2040, 2103, 2622, 3150, 3338, 3865, 5025, 5081\n\nKull, Edward A. 2571\n\nKulp, Nancy 843, 2795, 3808, 3839\n\nKwan, Nancy 2627\n\nKyne, Peter B. 1583, 1850, 3046, 4359, 4697\n\nLaborteaux, Matthew 2375\n\nLacher, Taylor 177, 216, 1081, 2752, 3712\n\nLackteen, Frank 324, 486, 642, 764, 811, 864, 934, 1053, 1364, 1377, 1444, 1450, 1519, 1554, 1562, 1656, 1626, 1804, 1811, 1863, 1991, 2064, 2086, 2173, 2181, 2192, 2315, 2471, 2639, 2688, 2731, 2825, 2834, 2931, 3068, 3139, 3365, 3609, 3655, 3744, 3916, 3995, 4112, 4306, 4495, 4535, 4601, 4654, 4776\n\nLacy, Adele 4862\n\nLacy, Joe 1484; _see also_ Elorietta, Jose M.\n\nLadd, Alan 209, 320, 471, 497, 1741, 1947, 2005, 2353, 2902, 3186, 3318, 3718, 4284, 4890\n\nLadd, Alana 1741, 3808, 5068\n\nLadd, David 320, 3186, 3663, 3808\n\nLadd, Diane 1124, 2488\n\nLaemmle, Edward 2180, 4289\n\nLahr, Bert 3606, 3754\n\nLaidlaw, Ethan 53, 204, 214, 314, 391, 437, 542, 582, 602, 631, 718, 797, 832, 880, 939, 1070, 1136, 1191, 1247, 1320, 1328, 1364, 1392, 1428, 1458, 1472, 1519, 1563, 1614, 1626, 1640, 1679, 1702, 1768, 1827, 1916, 1938, 1967, 1971, 2031, 2175, 2176, 2208, 2213, 2250, 2273, 2292, 2296, 2378, 2424, 2425, 2471, 2544, 2552, 2595, 2638, 2641, 2684, 2802, 2866, 2889, 2907, 2943, 2994, 3041, 3093, 3150, 3153, 3156, 3221, 3222, 3234, 3245, 3316, 3344, 3357, 3384, 3408, 3453, 3471, 3478, 3505, 3552, 3616, 3619, 3656, 3886, 3916, 3937, 3940, 4006, 4022, 4053, 4077, 4096, 4106, 4132, 4286, 4372, 4397, 4419, 4442, 4506, 4533, 4586, 4608, 4609, 4622, 4700, 4740, 4761, 4777, 4825, 4834, 4868, 4923, 4989, 5008, 5041, 5046\n\nLaine, Frankie 406, 577, 1702, 2565, 4148, 4370\n\nLaird, Effie 291, 4766\n\nLait, Jack, Jr. 1024, 2599, 4304\n\nLake, Florence 73, 349, 698, 1063, 1268, 2512, 2842, 3247, 4100, 4582\n\nLake, Stuart N. 1457, 1458\n\nLake, Veronica 4175\n\nLa Lanne, Jack 2694\n\nLally, Mike 3776, 4073\n\nLamarr, Hedy 420, 832\n\nLamas, Fernando 190, 2903, 3125, 3152, 3606, 3701, 4741\n\nLamas, Lorenzo 3389\n\nLamb, Eleanor 41, 3778, 4878, 4879\n\nLamb, Gil 26, 978, 1266, 3499, 4267\n\nLamb, Harold 3126\n\nLamb, Karl 4890\n\nLambe, Ande 1935, 2687, 2744, 3357, 3478, 4147, 4283, 4661\n\nLambert, Jack 1, 53, 170, 189, 290, 318, 516, 650, 934, 981, 1239, 1422, 1431, 1792, 1945, 2678, 2825, 3128, 3296, 3534, 3636, 3757, 4129, 4727, 4735, 4990\n\nLambert, Lucille 2398\n\nLamont, Charles 661, 914, 1450, 2308, 3411, 3540, 3684, 4673, 4675\n\nLa Mont, Connie 721\n\nLamont, Marten 3342\n\nLamour, Dorothy 1883, 3499, 3544, 4061\n\nL'Amour, Louis 1921, 2112, 2509, 3661, 3812, 4223\n\nLampert, Zohra 3149\n\nLampkin, Charles 3445\n\nLancaster, Burt 100, 554, 676, 1702, 2095, 2305, 3180, 3244, 3736, 4662, 4725, 4727\n\nLancaster, Iris 3494, 4482\n\nLanchester, Elsa 1433, 2756, 2836\n\nLand, Geoffrey 41, 388, 1278, 2039\n\nLandau, David 1854, 1961\n\nLandau, Martin 1765, 2155, 2779, 4109, 4471\n\nLandau, Richard 466, 1397, 1635\n\nLanders, Lew 30, 125, 183, 196, 352, 432, 604, 652, 870, 944, 969, 1012, 1032, 1199, 1562, 2261, 3490, 3637, 3794, 3922, 4107, 4221, 4386, 4653, 4767, 4871; _see also_ Friedlander, Louis\n\nLandis, Carole 886, 3362, 4372\n\nLandis, Cullen 556, 968, 2940\n\nLandis, James 1010, 2426\n\nLandis, John 4351\n\nLandon, Hal 3547\n\nLandon, Leslie 2374, 2375, 2377\n\nLandon, Michael 2339, 2374, 2375, 2376, 2377, 3437\n\nLandon, Michael, Jr. 417, 2459, 2464, 2465, 2466\n\nLandres, Paul 1452, 1616, 1838, 2202, 2426, 2521, 2659, 2926, 3997, 4085, 5030\n\nLane, Abbe 82, 5000\n\nLane, Al 2481, 4796; _see also_ Tansey, Robert Emmett\n\nLane, Allan (Rocky) 228, 233, 236, 286, 365, 415, 528, 656, 667, 784, 836, 854, 956, 1022, 1040, 1048, 1072, 1074, 1216, 1400, 1439, 1454, 1535, 1721, 1837, 1920, 2128, 2132, 2288, 2310, 2592, 2593, 2594, 2632, 2762, 2804, 2857, 2931, 2935, 3149, 3151, 3356, 3614, 3649, 3676, 3685, 3724, 3826, 3828, 3893, 4110, 4183, 4254, 4400, 4463, 4487, 4488, 4730, 4733, 4804, 4938, 5034\n\nLane, Carol 141\n\nLane, Charles 462, 541, 1638, 1899, 1915, 3147, 3425, 4312, 5004\n\nLane, Diane 676, 734, 2433, 4930\n\nLane, Jocelyn 1985, 2174, 4412\n\nLane, Lola 549, 2451\n\nLane, Mike 1838, 2379, 2619, 4133\n\nLane, Nora 461, 670, 710, 755, 1528, 1821, 1933, 2472, 2947, 3946, 4313, 4320, 4657, 4835, 4839\n\nLane, Priscilla 879, 3899\n\nLane, Richard 410, 863, 1240, 1959, 1967, 2717, 2940, 3426, 3429, 3471, 3596, 3929, 4225, 4665\n\nLane, Rose Wilder 5073\n\nLane, Rosemary 710, 2864, 3914\n\nLane, Rusty 1482, 2052, 3294, 3661\n\nLane, Vicky 758\n\nLane, Yancey (Bruce) 4493\n\nLanfield, Sidney 4132\n\nLang, Charles 539, 620, 1026, 3817, 4170, 5013\n\nLang, David 75, 101, 550, 1481, 1892, 2555, 2617, 2768, 2838, 2955, 3090, 3762, 5038\n\nLang, Fritz 3255, 3374, 4849\n\nLang, Howard 243, 621, 2067, 2147\n\nLang, Melvin 496\n\nLang, Otto 1485\n\nLang, Richard 2700\n\nLangan, Glenn 741, 1774, 2009, 2959, 3499\n\nLangdon, Sue Anne 727, 1807, 3622, 4867\n\nLange, Carl 904, 1069, 2214\n\nLange, Hope 4579\n\nLange, Johnny 1408\n\nLangella, Frank 2587, 5028\n\nLangford, Frances 1041, 1573, 3022\n\nLangton, Paul 22, 1422, 1526, 1569, 2018, 4682\n\nLanning, Frank 1320, 2249, 2367, 2394, 2777, 2978, 3082, 3619, 4326, 4532\n\nLansbury, Angela 1792, 2303\n\nLansing, Joi 315\n\nLansing, Robert 1245, 2350, 3438\n\nLansing, Sherry 3527\n\nLa Planche, Rosemary 1253, 1392, 1573, 3156\n\nLapp, Richard 4429\n\nLarch, John 349, 641, 1441, 1631, 1671, 1945, 2581, 2550, 3199, 3676, 3712, 3788, 4119, 5006\n\nLa Reno, Dick 556, 890, 1044, 1374, 2182, 2945, 3609, 4091\n\nLarkin, George 4899\n\nLa Rocque, Rod 504, 1865\n\nLa Roux, Carmen 1037, 1057, 1586, 3245, 4004, 4128, 4843\n\nLa Roy, Rita 437, 1907\n\nLarsen, Eric 3642, 4530, 5058\n\nLarsen, Ham 1479, 2698\n\nLarsen, Keith 116, 165, 733, 1407, 1485, 1866, 2202, 3317, 3642, 3998, 4530, 4787, 4914, 4921, 5058\n\nLarson, Bobby 3466, 3565, 3678\n\nLarson, Christine 282, 893, 1325, 1868, 2944, 3050, 3908, 4692\n\nLarson, Glen A. 55\n\nLarson, Jack 3342, 4124\n\nLaRue, Frank 138, 214, 307, 335, 391, 514, 774, 781, 792, 841, 850, 899, 925, 946, 1090, 1148, 1192, 1304, 1309, 1324, 1332, 1333, 1367, 1373, 1388, 1443, 1447, 1449, 1453, 1464, 1529, 1540, 1604, 1690, 1716, 1725, 1769, 1797, 1834, 1895, 1979, 1982, 2012, 2151, 2172, 2197, 2296, 2306, 2315, 2350, 2356, 2420, 2457, 2645, 2647, 2667, 2690, 2858, 2910, 2927, 2968, 2978, 2991, 2998, 3087, 3157, 3171, 3187, 3231, 3258, 3260, 3265, 3329, 3393, 3448, 3452, 3477, 3557, 3565, 3587, 3651, 3666, 3674, 3823, 3901, 3926, 3977, 4000, 4019, 4021, 4030, 4053, 4127, 4186, 4187, 4302, 4377, 4379, 4501, 4502, 4553, 4554, 4640, 4806, 4833, 4863, 4888\n\nLaRue, Jack 79, 930, 1973, 2286, 3120, 3436, 3544, 3562, 4256, 4442, 4650, 4697\n\nLaRue, Lash 368, 436, 728, 940, 994, 1224, 1339, 1461, 1463, 1555, 1783, 2123, 2266, 2582, 2632, 2946, 3017, 3103, 3388, 3995, 3999, 4017, 4097, 4102, 4408, 4709, 4969, 4970\n\nLaRue, Walt, 2693, 3019, 3088, 4228, 4726, 4816\n\nLasko, Edward 521\n\nLasky, Jesse, Jr. 2831, 2889, 3909, 4635, 4665\n\nLassie (dog) 2183, 2772\n\nLatell, Lyle 53, 170, 2543, 2902, 3335\n\nLatham, Louise 182, 1346, 2738, 4180, 4264, 5006\n\nLatimer, Jonathan 832, 3131, 3339\n\nLatimer, Louise 4622\n\nLatimer, Ross 4357\n\nLatimore, Frank 1484, 3796\n\nLatt, Jack, Jr. 2667\n\nLaughlin, Tom 330, 2335, 2619, 4546\n\nLaughton, Charles 3634\n\nLaughton, Eddie 15, 307, 386, 575, 576, 1298, 1758, 1768, 2016, 2269, 2425, 2547, 2633, 2907, 2973, 3678, 4000, 4191, 4314, 4391, 4401, 4723, 4806, 4827\n\nLaundres, Perc 602, 1364, 1392, 1433, 1627, 3534, 4598, 4654, 4745, 4825, 4841\n\nLauner, S. John 1963, 4680, 4805\n\nLaurel, Stan 2151, 4798\n\nLauren, Rod 1675\n\nLaurenz, John 23, 110, 352, 444, 786, 1981, 4204\n\nLaurie, Piper 972, 2664, 3966\n\nLauter, Ed 192, 532, 1014, 1583, 2498, 3218, 3775, 4433, 4895\n\nLauter, Harry 101, 104, 170, 208, 228, 255, 489, 508, 559, 562, 637, 830, 1103, 1314, 1397, 1406, 1415, 1454, 1485, 1608, 1697, 1890, 1996, 2103, 2601, 2693, 2805, 2871, 2959, 3006, 3149, 3178, 3225, 3397, 3839, 3853, 3892, 3959, 4228, 4259, 4368, 4381, 4420, 4462, 4469, 4478, 4692, 4805, 4886, 4973, 5078\n\nLaven, Arnold 1533, 1575, 3612, 3688\n\nLaVerne, Lucille 1639, 2239, 4950\n\nLaverre, Morton _see_ Merton, John\n\nLavi, Dahlia 675, 2883\n\nLaw, John Philip 1016, 2199\n\nLaw, Mildred 2296\n\nLawford, Peter 1564, 2079, 3777\n\nLawliss, Joe 3983\n\nLawlor, Anderson 723, 1221\n\nLawrence, Barbara 128, 2037, 2043, 2550, 2563, 2853\n\nLawrence, Delphi 2185\n\nLawrence, Edna 1167, 2399\n\nLawrence, Florence 1781, 3766\n\nLawrence, Jody 652, 4109\n\nLawrence, Mady 1821, 2363, 2847, 3097, 4078\n\nLawrence, Marc 514, 596, 615, 884, 915, 1134, 1358, 1360, 2051, 3005, 3561, 3730, 3817, 4635, 4745, 5043\n\nLawrence, Muriel 277\n\nLawrence, Peter Lee 1287, 1381, 1508, 2570, 3117\n\nLawrence, Rosina 4798\n\nLawson, Kate Drain 2781, 3567\n\nLawson, Linda 109, 3993\n\nLawson, Priscilla 332, 1571, 1860, 3087, 4215\n\nLawson, Wilfred 63\n\nLayne, Tracy 130, 243, 323, 729, 818, 1491, 1687, 1728, 2129, 2299, 2432, 2523, 2636, 3048, 3482, 3919, 4365, 4592, 4732, 4865, 4995\n\nLayton, Drue 4703\n\nLea, Tom 5024\n\nLeachman, Cloris 586, 2463\n\nLeacock, Philip 965, 4927, 4977\n\nLeahy, Agnes Brand 686, 1300, 2393, 4071\n\nLearned, Michael 1752\n\nLeary, Nolan 94, 324, 389, 391, 418, 607, 698, 779, 864, 987, 1247, 1478, 1497, 1523, 1601, 1641, 1725, 1793, 1815, 1892, 2022, 2175, 2176, 2189, 2427, 2609, 2676, 2935, 2944, 2967, 2978, 2991, 3069, 3464, 3481, 3552, 3556, 3651, 3666, 3669, 3707, 3741, 3750, 3826, 4167, 4232, 4327, 4585, 4610, 4812, 5103\n\nLease, Rex 6, 154, 189, 284, 286, 334, 342, 343, 597, 602, 624, 648, 731, 775, 779, 781, 784, 796, 832, 854, 862, 868, 914, 917, 925, 926, 930, 956, 984, 995, 1024, 1051, 1082, 1271, 1299, 1345, 1360, 1439, 1450, 1490, 1545, 1665, 1797, 1798, 1836, 1859, 1877, 1888, 1915, 1964, 1974, 1975, 1980, 2124, 2125, 2165, 2184, 2355, 2403, 2416, 2421, 2427, 2428, 2452, 2482, 2504, 2522, 2638, 2652, 2678, 2778, 2806, 2853, 2909, 2928, 2970, 2971, 2977, 3026, 3068, 3074, 3088, 3128, 3132, 3224, 3232, 3234, 3254, 3295, 3440, 3448, 2489, 3548, 3611, 3613, 3617, 3656, 3673, 3707, 3708, 3744, 3803, 3805, 3826, 3866, 3869, 3889, 3907, 3927, 4052, 4076, 4081, 4181, 4206, 4261, 4458, 4470, 4483, 4532, 4567, 4649, 4683, 4962, 4997, 5008, 5033, 5050\n\nLeavitt, Norman 501, 584, 1482, 1676, 1775, 1792, 2652, 2691, 2711, 3512, 3857, 4111, 4247, 4689, 4725, 4799, 5051, 5070\n\nLebeau, Madeline 1722\n\nLebedeff, Ivan 604, 1591\n\nLeBorg, Reginald 27, 937, 1635, 4784, 5036, 5064\n\nLederer, Francis 4212\n\nLederman, D. Ross 15, 201, 496, 958, 1227, 1320, 2628, 2689, 3082, 3216, 3260, 3486, 3506, 3658, 3880, 4294, 4308, 4401, 4607, 4885\n\nLedger, Heath 518, 2771\n\nLee, Alta 4220, 4568\n\nLee, Anna 292, 1395, 1937, 2561, 2800, 4626\n\nLee, Billy 145, 1980, 2027, 2778, 4395, 4762\n\nLee, Brandon 2155\n\nLee, Chen 1305, 4366\n\nLee, Christopher 1776\n\nLee, Dorothy 1563, 2877, 3529, 3886\n\nLee, Duke R. 1009, 1045, 1333, 1369, 1426, 1813, 2070, 2071, 2403, 2554, 2701, 2998, 3126, 3179, 3272, 3709, 3742, 4100, 4145, 4219, 4862, 4882, 4911\n\nLee, Glen 988\n\nLee, Gwen 1440, 4807\n\nLee, Gypsy Rose 278, 2989, 4993\n\nLee, Johnny 809\n\nLee, Laura 4421\n\nLee, Lila 2393\n\nLee, Margaret 1120, 1181, 4627\n\nLee, Mary 183, 662, 868, 1512, 2635, 3254, 3434, 3490, 3922, 4016, 4052\n\nLee, Palmer 749, 803, 3340; _see also_ Palmer, Gregg\n\nLee, Pinky 1971, 3024, 4043\n\nLee, Ruta 567, 1105, 1675, 3777, 4601\n\nLee, Stan 1937\n\nLeeds, Andrea 4027\n\nLeeds, Herbert I. 158, 757, 3384, 3431, 3596\n\nLeeds, Peter 3672\n\nLe Fieur, Jimmy, and His Saddle Pals 1465, 4082\n\nLeGault, Lance 2029, 3106\n\nLeGay, Sheila 617, 647; _see also_ Bromley, Sheila; Mannors, Sheila\n\nLeiber, Fritz 287, 758, 950, 1093, 1827\n\nLeigh, Barbara 474, 2069\n\nLeigh, Janet 2013, 2112, 2559, 2751\n\nLeigh, Nelson 56, 1347, 1434, 1702, 1773, 2018, 2037, 2381, 2622, 2963, 4069, 4333, 5078\n\nLeighton, Lillian 880, 1291, 1329, 1659, 2527, 4591, 4956\n\nLeighton, Melinda 777\n\nLeipnitz, Harald 1292, 3246, 4380\n\nLeith, Virginia 4907\n\nLelouch, Claude 96\n\nLeMat, Paul 1021\n\nLeMay, Alan 718, 1706, 3201, 3572, 3692, 4705, 4771\n\nLembbeck, Harvey 819, 4670\n\nLemmon, Jack 859\n\nLeMoyne, Charles 351, 865, 1200, 2258, 2315, 2975, 3087, 4178\n\nLenz, Kay 1642, 1707\n\nLenz, Rick 1784, 1834, 2738, 2739, 3739, 3740, 3847\n\nLenzi, Umberto 61, 3117, 3689, 4268\n\nLeonard, Barbara 2639\n\nLeonard, Elmore 1064, 1879, 2044\n\nLeonard, Jack 2578, 4222\n\nLeonard, Robert Z. 1571, 2790\n\nLeonard, Sheldon 2009, 2150, 3236, 4257, 4422\n\nLeone, Sergio 1171, 1350, 1381, 1612, 2898\n\nLeontovich, Eugenie 5025\n\nLerner, Alan Jay 3009\n\nLerner, Irving 915\n\nLerner, Michael 563, 3747\n\nLeRoy, Mervyn 182, 4148\n\nLeroy, Philippe 3033\n\nLeSaint, Ed (Edward) 144, 576, 623, 682, 686, 780, 975, 1002, 1083, 1228, 1328, 1436, 1457, 1458, 1869, 2031, 2073, 2239, 2272, 2540, 2547, 4730, 2886, 2908, 2927, 3276, 3409, 3508, 3520, 3634, 4077, 4156, 4284, 4289, 4314, 4359, 4377, 4385, 4494, 4531, 4803, 4817, 4832, 4971, 5054\n\nLescoulie, Jack 2867\n\nLeslie, Joan 1844, 1852, 2066, 2549, 2838, 3964, 4470, 5022\n\nLeslie, Kay 4303\n\nLeslie, Maxine 1473, 2407, 2993, 3461, 3825, 4589\n\nLeslie, Nan 1736, 1988, 2006, 2659, 3104, 3510, 4204, 4524, 4653, 4654, 4777, 4841, 4951\n\nLeslie, William 539, 859, 1937, 3397, 3746, 3792, 4244, 4913\n\nLesser, Sol 2012, 2654\n\nLester, Buddy 3777\n\nLester, Mark 412, 585, 3735\n\nLester, Vicky 2408\n\nLester, William 188, 1495, 1602, 1779, 2493, 4597, 5016, 5076\n\nL'Estrange, Dick 2493\n\nLettieri, Al 1007, 4471\n\nLettieri, Louis 550, 922, 2693\n\nLetz, George 337, 1462, 1595, 2399, 2552, 2802, 2968, 3027, 3194, 3615, 3834, 3926, 4082, 4773; _see also_ Montgomery, George\n\nLevering, Joseph 1969, 4104\n\nLevien, Sonya 866, 1165, 2853\n\nLevin, Henry 1075, 1499, 2431, 2515, 4376\n\nLevine, Nat 1089\n\nLevinson, Richard 2627\n\nLevitt, Gene 55\n\nLevy, Melvin 3561, 4478\n\nLewis, Cullen 1057; _see also_ Collins, Lewis D.\n\nLewis, Diana 1577\n\nLewis, Edgar 2493, 2573, 4297\n\nLewis, Fiona 4775\n\nLewis, Forrest 101, 528, 1234, 1672, 2292, 2550, 3149, 3950, 4073, 4120, 4227\n\nLewis, Geoffrey 192, 280, 526, 909, 1068, 1663, 1752, 1880, 1926, 2623, 2723, 2784, 3350, 3369, 3772, 3798, 3842, 4449, 4906\n\nLewis, George J. 28, 42, 234, 272, 320, 325, 352, 366, 295, 450, 497, 642, 778, 794, 808, 857, 893, 896, 936, 956, 967, 1024, 1092, 1116, 1272, 1520, 1543, 1741, 1939, 1990, 2005, 2082, 2103, 2123, 2179, 2207, 2289, 2334, 2552, 2596, 2858, 2966, 2972, 3088, 3112, 3222, 3236, 3356, 3433, 3453, 3512, 3540, 3718, 3808, 3823, 3848, 3871, 3908, 3912, 4043, 4047, 4054, 4055, 4234, 4299, 4314, 4599, 4646, 4650, 4751, 4757, 4763, 4820, 5008, 5086, 5097, 5103\n\nLewis, Jack 358, 2123, 2748, 2949, 3776\n\nLewis, Jarma 2578, 3535, 3780\n\nLewis, Jerry 2720, 3040\n\nLewis, Joseph H. 386, 460, 477, 847, 1078, 1766, 2235, 2302, 2547, 3393, 3792, 3924, 4267, 4608\n\nLewis, Louise 2289, 3609, 4228\n\nLewis, Louise 2910\n\nLewis, Mitchell 270, 467, 1577, 1599, 1792, 1996, 2144, 2344, 2416, 2838, 3530, 4121, 4593\n\nLewis, Ralph 468, 2590, 2954, 3695, 3744, 3924, 3994, 4205, 4219, 4269, 4550, 4684\n\nLewis, River 4028\n\nLewis, Robert Q. 523, 3422\n\nLewis, Ronald 3560\n\nLewis, Sheldon 308, 556, 684, 1344, 1677, 4280, 4454\n\nLewis, Texas Jim 199, 662, 2262, 2744, 3042, 4154\n\nLewis, Vance 4157, 4158, 4662; _see also_ Vanzi, Luigi\n\nLewis, Vera 203, 1127, 1622, 3410, 4115, 4336\n\nLibby, Fred 282, 659, 1313, 2718, 3776, 3814, 4360, 4764\n\nLiberace 4867\n\nLiberatore, Ugo 2656, 4527\n\nThe Light Crust Doughboys 323, 2850\n\nLightfoot, Gordon 1791\n\nLightning (dog) 619, 1820, 2571, 3361, 4866\n\nLigon, Tom 3009\n\nLime, Yvonne 3244\n\nLinaker, Kay 541, 1165, 2638, 5072\n\nLincoln, Caryl 924, 2167, 2531, 3210, 4475, 4785\n\nLincoln, Elmo 214, 352, 410, 797, 1449, 3385, 3593, 4423, 5027\n\nLinden, Eric 3561\n\nLinden, Judith 2119\n\nLinden, Tove 753, 3771\n\nLindfors, Viveca 1766, 3636\n\nLindon, Lionel 1166\n\nLindsay, Margaret 46, 482, 1426, 1436, 1594, 4735\n\nLine, Helga 736, 1185, 1801, 1968, 4348\n\nLingham, Tom 454, 954, 968, 1054, 1387, 1495, 3463, 3932, 4125, 4317, 4780, 5016\n\nLink, John F. 618\n\nLink, William 3687\n\nLinn, Rex 120, 417, 485, 900, 1500, 2015, 2721, 2865, 5029\n\nLinow, Ivan 1972, 3896\n\nLipman, William R. 191, 248, 4312\n\nLippert, Robert O. 1616, 2221\n\nLipson, Jack \"Tiny\" 535, 1831, 3946, 4932\n\nLipton, Peggy 2695\n\nLipton, Robert 406, 1589, 4249\n\nLisi, Verna 697, 4901\n\nLitel, John 804, 939, 1026, 1127, 1487, 1594, 1658, 1675, 1830, 1892, 2018, 2095, 2638, 2678, 3779, 2841, 3684, 3692, 3711, 3931, 4035, 4300, 4310, 4336, 4690, 4697, 4742, 4937\n\nLittle Brown Jug 867, 1326, 4167, 4888; _see also_ Reynolds, Don Kay\n\nLittle Sky, Dawn 748, 1183, 4253\n\nLittle Sky, Eddie 116, 508, 748, 1183, 1842, 2058, 2508, 2668, 2930, 3009, 3180, 3641, 3777, 3783, 4136, 4451, 4753, 4800; _see also_ Little, Eddie\n\nLittle, Ann 916, 1999, 2359, 4088, 4092, 4446\n\nLittle, Cleavon 385\n\nLittle, Eddie 3404, 3415; _see also_ Little Sky, Eddie\n\nLittlefeather, Sacheen 2047, 3842, 4546, 5007\n\nLittlefield, Lucien 207, 283, 471, 868, 1634, 1638, 1883, 1982, 2027, 2357, 2368, 3240, 3606, 3634, 4213, 4395, 4591, 4803, 4851, 5103\n\nLittlefield, Ralph 64, 979, 2334, 3750, 3975, 4114, 4222\n\nLively, Bob 917\n\nLively, William 139, 155, 343, 475, 481, 796, 933, 956, 984, 1018, 1304, 1327, 1445, 1543, 1597, 1693, 1718, 1941, 2022, 2128, 2135, 2409, 2595, 2884, 3085, 3265, 3677, 4302, 4466, 4486, 4759, 4945\n\nLivingston, Robert (Bob) 286, 291, 309, 388, 414, 631, 810, 853, 886, 930, 1017, 1134, 1504, 1618, 1751, 1822, 1860, 1862, 1898, 2086, 2179, 2260, 2276, 2403, 2421, 2711, 2731, 2805, 2867, 2895, 2968, 2993, 3025, 3111, 3120, 3165, 3176, 3194, 3226, 3259, 3447, 3454, 3482, 3549, 3574, 3667, 3673, 4359, 4365, 4483, 4558, 4574, 4649, 4659, 4732, 4954, 4956, 5005, 5018\n\nLivingstone, Mary 541\n\nLizzani, Carlo 1891, 3366\n\nLloyd, Alma 4025\n\nLloyd, Betty 4955\n\nLloyd, Beverly 4681\n\nLloyd, Christopher 585, 1590, 2156, 2335, 3772\n\nLloyd, Doris 3558, 3766\n\nLloyd, Frank 1947, 2159, 2181, 4803\n\nLloyd, George 42, 134, 201, 228, 489, 602, 642, 1040, 1130, 1161, 1247, 1285, 1438, 1454, 1512, 1823, 1914, 1974, 1976, 2006, 2176, 2549, 2815, 2864, 2877, 2923, 2942, 3005, 3323, 3393, 3630, 3652, 4015, 4048, 4215, 4222, 4641, 4654, 4700, 4733, 4980\n\nLloyd, Jimmy 3481\n\nLloyd, Kathleen 2666\n\nLloyd, Norman 599, 4058\n\nLloyd, Rollo 246, 1368, 4061, 5054\n\nLobo the Wonder Dog 4396\n\nLocher, Charles 2730, 4995; _see also_ Hall, Jon\n\nLocher, Felix 378, 603, 1343, 4384, 4769\n\nLocke, Sondra 526, 2950, 3793\n\nLockhart, Gene 115, 332, 1427, 1532, 2835, 4705\n\nLockhart, June 587, 657, 2161, 2183, 5045\n\nLockhart, Kathleen 2552\n\nLockwood, Alyn 213\n\nLockwood, Gary 1346, 2447, 3204\n\nLockwood, Margaret 4214\n\nLoden, Barbara 1251\n\nLodge, John 4652\n\nLoff, Jeanette 1335\n\nLoft, Arthur 64, 158, 183, 203, 552, 666, 853, 986, 1392, 1444, 2016, 2102, 2131, 2368, 2419, 2534, 2719, 2815, 2821, 2943, 3038, 3176, 3187, 3289, 3408, 3410, 3452, 3544, 3824, 3874, 3904, 4051, 4059, 4318, 4336, 4745, 4786, 4843, 4888, 4932\n\nLoftin, Carey 28, 637, 1130, 2125, 3574, 4509, 5103\n\nLogan, Helen 2471, 4214\n\nLogan, James 3607\n\nLogan, Joshua 3009\n\nLogan, Robert 10, 39, 1479, 2091, 2698, 3282, 3835, 3981\n\nLogan, Sidney 1505\n\nLoggia, Robert 679, 2809, 3747, 3941\n\nLogue, Charles 765, 824, 2651, 2991, 3361, 4983\n\nLollobrigida, Gina 202\n\nLollobrigida, Guido 4628; _see also_ Burton, Lee\n\nLom, Herbert 4540, 4738\n\nLomax, Bliss 2311, 3761\n\nLombard, Carole 142, 1880, 1961\n\nLommel, Uli 770\n\nLomond, Britt 3871, 4457\n\nLondon, Babe 2020\n\nLondon, Jack 46, 626, 627, 824, 904, 1625, 2016, 2830, 3202, 3594, 3879, 4229, 4900, 4901, 4902, 4904, 5011\n\nLondon, Julie 1151, 2556, 3386, 3671, 5024\n\nLondon, Tom 15, 51, 63, 68, 79, 104, 153, 154, 180, 201, 206, 230, 244, 240, 246, 286, 291, 306, 332, 389, 409, 413, 447, 492, 573, 592, 606, 610, 621, 644, 665, 712, 764, 778, 779, 793, 827, 836, 846, 853, 863, 930, 960, 979, 984, 1008, 1129, 1134, 1146, 1159, 1174, 1196, 1264, 1254, 1260, 1280, 1308, 1310, 1344, 1345, 1360, 1370, 1389, 1454, 1470, 1475, 1488, 1512, 1519, 1554, 1557, 1558, 1591, 1592, 1602, 1608, 1614, 1626, 1646, 1662, 1728, 1799, 1861, 1872, 1873, 1877, 1890, 1927, 1950, 1964, 1969, 2035, 2073, 2107, 2125, 2170, 2180, 2193, 2203, 2208, 2254, 2274, 2299, 2399, 2412, 2415, 2421, 2423, 2426, 2470, 2481, 2527, 2534, 2537, 2542, 2554, 2582, 2592, 2597, 2600, 2635, 2641, 2648, 2673, 2701, 2727, 2730, 2737, 2742, 2799, 2802, 2837, 2852, 2875, 2885, 2886, 2889, 2890, 2895, 2928, 2935, 2951, 2968, 2995, 3002, 3005, 3006, 3016, 3025, 3046, 3080, 3085, 3086, 3087, 3088, 3105, 3163, 3173, 3213, 3245, 3251, 3263, 3269, 3303, 3312, 3316, 3327, 3348, 3355, 3401, 3408, 3446, 3447, 3454, 3480, 3483, 3490, 3497, 3503, 3507, 3524, 3525, 3528, 3537, 3554, 3555, 3557, 3587, 3590, 3596, 3613, 3614, 3649, 3676, 3681, 3691, 3693, 3698, 3708, 3709, 3757, 3764, 3805, 3815, 3820, 3824, 3826, 3846, 3893, 3904, 3907, 3938, 3949, 3952, 4000, 4011, 4018, 4019, 4037, 4059, 4082, 4096, 4104, 4110, 4127, 4135, 4139, 4141, 4153, 4197, 4203, 4208, 4234, 4240, 4261, 4274, 4280, 4423, 4424, 4448, 4461, 4463, 4486, 4487, 4490, 4515, 4548, 4576, 4572, 4584, 4592, 4599, 4600, 4602, 4614, 4641, 4648, 4659, 4660, 4700, 4701, 4709, 4734, 4747, 4759, 4763, 4778, 4786, 4819, 4823, 3827, 4833, 4840, 4849, 4852, 4866, 4868, 4938, 4940, 4953, 4957, 4981, 5001, 5009, 5021, 5033, 5050, 5066, 5096, 5103; _see also_ Clapham, Leonard\n\nLong, Audrey 30, 691, 1992\n\nLong, Jack 2531, 2584, 2608, 2915, 3742\n\nLong, Lotus 3611\n\nLong, Richard 1481, 2084, 3718, 4255\n\nLong, Walter 242, 270, 414, 834, 1153, 1317, 1370, 1576, 1929, 2572, 2603, 2756, 2826, 3016, 3743, 3895, 3905, 3949, 4215, 4868, 4947\n\nLongstreet, Stephen 1348, 3902, 4115\n\nLoo, Richard 1600, 2016, 2154, 2911, 3761\n\nLoos, Anita 3695\n\nLoos, Mary 1175, 4111\n\nLopez, Perry 237, 1161, 1371, 1834, 2066, 2400, 2626, 3283, 5066\n\nLorch, Theodore (Ted) 8, 411, 626, 725, 1057, 1440, 1470, 1474, 1489, 1712, 1895, 1927, 2032, 2119, 2367, 2456, 2785, 2933, 3300, 3325, 3531, 3601, 3631, 3653, 3927, 4121, 4254, 4289, 4459, 4476, 4885, 4982, 5076, 5104\n\nLord, Del 3916\n\nLord, Jack 1773, 2556, 3439, 3916, 4768\n\nLord, Marjorie 432, 1142, 2605, 3298, 4419\n\nLord, Robert 826\n\nLoren, Sophia 1842\n\nLorenzon, Livio 3, 175, 1612, 2194, 3532, 4288, 4625, 5087, 5092\n\nLorimer, Louise 1354, 1487, 2622, 3006\n\nLoring, Ann 3561\n\nLoring, Lynn 370\n\nLoring, Michael 5054\n\nLoring, Teala 134, 3502\n\nLormer, Jon 419, 523, 808, 1441, 1663, 1695, 2437, 3602\n\nLorraine, Louise 300, 1779, 2703, 2763, 5002\n\nLorys, Diana 202, 376, 738, 1117, 1536, 1710, 3405, 3715, 3796, 4325, 4738\n\nLosch, Tilly 1187\n\nLoudermilk, Romaine 3235\n\nLoughery, Jackie 2664, 3040\n\nLouise, Anita 981, 1639, 1773, 2719, 3362, 4767\n\nLouise, Tina 1610, 4528\n\nLouisiana Lou 4773\n\nLove, Bessie 675\n\nLove, Courtney 4146\n\nLove, June 846\n\nLove, Lucretia 394\n\nLove, Montagu 1430, 1865, 1949, 2064, 2158, 2586, 2831, 2837, 3506, 4215, 4257, 4991\n\nLove, Suzanna 770\n\nLovejoy, Frank 354, 700, 789\n\nLovering, Joseph 2461\n\nLovering, Otho 1153, 1468, 2256, 2420, 2943, 3077, 3105, 3591, 4776\n\nLovsky, Celia 1188, 1427, 1428, 4300, 4566\n\nLow, Jack 2, 49, 307, 647, 867, 975, 1038, 2281, 2547, 2644, 2973, 3185, 3221, 3486, 4585, 4605\n\nLowe, Edmund 1842, 1972, 2149, 3134\n\nLowe, Edward T. 4422\n\nLowe, Ellen 395, 465, 3254, 3673, 4761\n\nLowe, Rob 1429\n\nLowe, Sherman 135, 352, 582, 892, 1046, 1102, 1491, 2250, 2273, 2283, 2378, 2607, 2636, 2874, 3142, 3220, 3547, 4659, 4808\n\nLowell, Helen 1883, 4933\n\nLowery, Robert 223, 449, 618, 857, 936, 961, 973, 1020, 1165, 1266, 1585, 1713, 1959, 2050, 2308, 2586, 2626, 3045, 3431, 3846, 4229, 4524, 4613, 4755, 4845, 5068, 5072\n\nLowry, Dick 2092, 2093, 2094, 2236, 4960\n\nLowry, Morton 1949\n\nLoy, Barbara 2499\n\nLoy, Dina 3346\n\nLoy, Myrna 197, 1634, 2207, 3319, 3582, 3745, 4639\n\nLu, Lisa 3445\n\nLubin, Arthur 1348, 3425, 5054\n\nLuby, S. Roy 130, 151, 369, 422, 448, 872, 897, 1052, 1475, 2166, 2366, 2622, 2954, 3258, 3268, 3332, 3568, 3581, 3668, 4323, 4393, 4458, 4497, 4515, 4587, 4660, 4815, 5027; _see also_ Claire, Roy\n\nLucas, John Meredyth 3871, 4588\n\nLucas, Nick 1828, 1972\n\nLucas, Wilfred 142, 144, 387, 510, 611, 921, 1127, 1221, 1436, 1594, 2164, 2342, 2473, 2599, 2756, 2923, 3092, 3172, 3179, 3216, 3220, 3469, 3711, 3895, 4142, 4254, 4564, 4694, 4742\n\nLucidi, Maurizio 2011, 2724\n\nLucking, William (Bill) 2155, 2498, 2700, 2859, 2962, 3369, 4969\n\nLuddy, Barbara 467, 1814\n\nLuden, Jack 465, 2131, 2222, 3077, 3105, 3591, 3613, 4104, 4214, 4651\n\nLudwig, Edward 381, 1247, 1675, 3701, 4714\n\nLudwig, William 491, 1673, 2853, 2936, 4800\n\nLuez, Laurette 219\n\nLufkin, Sam 49, 1024, 4053, 4503, 4777, 4798\n\nLugosi, Bela 1031\n\nLukather, Paul 71, 1151, 1942, 4800\n\nLukats, Nick 471, 1972\n\nLuke, Jorge 2780, 3369, 3402, 4633\n\nLuke, Keye 2154, 2155, 2830\n\nLukschy, Wolfgang 403, 1350, 1366\n\nLulli, Folco 1653, 3871\n\nLulli, Piero 19, 296, 416, 553, 565, 976, 1115, 1305, 1342, 1351, 1358, 1385, 1393, 1580, 1653, 1717, 1954, 2354, 2723, 3117, 3515, 3725, 3796, 5093\n\nLulu Belle, and Scotty 3834\n\nLumet, Sidney 2468\n\nLummis, Dayton 977, 1441, 3856, 4073, 5049\n\nLuna, Barbara 1346, 1510, 1527, 1770, 2500, 4990\n\nLund, Art 3203\n\nLund, John 257, 527, 733, 930, 1357, 2720, 4907, 5022\n\nLund, Lucille 881, 1334, 3268, 3526, 4427\n\nLundigan, William 112, 1127, 1811, 1884, 1995, 2838, 4831\n\nLupino, Ida 190, 1277, 1521, 2069, 2482\n\nLupo, Alberto 551, 1098, 1117, 5092\n\nLupo, Michele 220, 1242, 2532, 3064, 3819\n\nLupton, John 113, 982, 1091, 1151, 1234, 1637, 1669, 2034, 2563, 3324, 4571\n\nLuther, Johnny 357, 459, 1298, 1307, 1380, 1935, 2269, 2270, 2763, 2956, 3081, 3092, 3177, 3241, 3589, 3617, 4112, 4815, 4846\n\nLydecker, Howard 1827\n\nLyden, Pierce 27, 51, 187, 204, 325, 366, 379, 390, 395, 433, 446, 597, 599, 610, 643, 644, 664, 710, 712, 775, 778, 854, 899, 956, 994, 995, 1023, 1247, 1254, 1345, 1447, 1451, 1472, 1487, 1520, 1529, 1668, 1711, 1892, 2085, 2109, 2170, 2289, 2294, 2480, 2539, 2609, 2678, 2736, 2776, 2967, 2980, 3002, 3068, 3231, 3236, 3277, 3298, 3323, 3458, 3509, 3547, 3585, 3649, 3694, 3744, 3802, 3908, 3940, 3945, 4011, 4028, 4036, 4053, 4095, 4121, 4228, 4234, 4292, 4301, 4304, 4509, 4550, 4552, 4597, 4657, 4729, 4754, 4757, 4826, 4894, 4929, 4973\n\nLydon, James (Jimmy) 1015, 1063, 2784, 2851, 3070, 3219, 3739\n\nLyles, A.C. 114, 132, 373, 548, 1406, 1940, 2050, 2188, 2267, 4098, 4755\n\nLyman, Abe 3041\n\nLynch, Helen 1524, 1638, 2108\n\nLynch, Ken 109, 2557, 3641, 3791\n\nLynde, Paul 4739\n\nLyndon, Barre 2381\n\nLynley, Carol 838, 2237, 2351\n\nLynn, Betty 1535, 1671, 1773\n\nLynn, Diana 2095, 2720, 3131, 4473\n\nLynn, Emmett 69, 115, 129, 213, 293, 383, 481, 516, 632, 658, 665, 758, 786, 827, 867, 870, 895, 987, 995, 1053, 1264, 1270, 1309, 1326, 1459, 1505, 1601, 1618, 1620, 2035, 2066, 2175, 2179, 2184, 2189, 2286, 2416, 2419, 2537, 2775, 2834, 2856, 2966, 3305, 3344, 3390, 3540, 3557, 3586, 3599, 3649, 3794, 3801, 3954, 3958, 4000, 4017, 4072, 4106, 4167, 4185, 4221, 4378, 4495, 4533, 4601, 4729, 4761, 4763, 4818, 4841, 4855, 5008, 5075\n\nLynn, Jeffrey 874\n\nLynn, Rita 786\n\nLynne, Sharon 4798\n\nLyon, Ben 622\n\nLyon, Earl 2436, 3906, 4613\n\nLyon, Francis D. 1238, 1637, 1742, 2871, 4452\n\nLyon, Sue 1425\n\nLyons, Cliff 12, 141, 290, 323, 498, 617, 646, 647, 722, 785, 800, 855, 927, 930, 960, 1080, 1200, 1202, 1294, 1313, 1344, 1450, 1614, 1677, 1845, 1928, 1937, 1969, 2207, 2231, 2250, 2299, 2367, 2404, 2567, 2660, 2737, 2763, 2799, 2826, 2860, 2961, 3088, 3241, 3314, 3322, 2254, 3522, 3539, 3555, 3650, 2655, 3752, 3776, 3788, 3814, 3969, 4293, 4406, 4484, 4592, 4626, 4759, 4764, 4836, 5001, 5071, 5103\n\nLyons, Collette 4372\n\nLyons, Robert F. 3837\nMacArthur, Charles 246\n\nMacArthur, James 1769, 2351, 2695, 3422, 4064\n\nMacBride, Donald 1562, 2837, 4291, 5032\n\nMacDonald, Edmund 357, 514, 1084, 1519, 1818, 4286, 4419, 4422, 4498\n\nMacDonald, Grace 4179\n\nMacDonald, Ian 100, 406, 801, 1029, 1182, 1364, 1866, 1877, 2048, 2436, 2515, 2676, 2677, 2789, 3195, 3247, 3310, 3722, 3761, 3906, 3909, 4057, 4111, 4135, 4310, 4336, 4381, 4428, 4470, 4613, 4791\n\nMacDonald, J. Farrell 159, 271, 282, 813, 847, 934, 940, 958, 960, 1319, 1322, 1528, 1570, 1601, 1854, 1939, 1972, 2004, 2152, 2261, 2279, 2353, 2403, 2718, 2815, 2832, 2940, 3011, 3032, 3162, 3479, 3537, 3554, 3874, 3956, 3972, 4089, 4093, 4112, 4142, 4214, 4304, 4352, 4467, 4510, 4706, 4890\n\nMacDonald, Jeanette 1571, 2458, 2756, 2790, 3606, 3695\n\nMacDonald, Katherine 3412\n\nMacDonald, Kenneth 65, 125, 282, 352, 456, 458, 576, 664, 692, 718, 875, 934, 979, 1048, 1192, 1247, 1255, 1274, 1413, 1442, 1467, 1480, 1516, 1579, 1691, 1709, 1768, 1843, 1941, 2025, 2260, 2310, 2457, 2521, 2544, 2593, 2678, 2731, 2983, 3003, 3067, 3068, 3132, 3149, 3165, 3176, 3234, 3383, 3466, 3565, 3608, 3685, 3724, 3769, 3779, 3794, 3856, 3938, 4000, 4060, 4077, 4107, 4123, 4140, 4179, 4222, 4238, 4314, 4327, 4486, 4608, 4701, 4826, 4980\n\nMacDonald, Marie 271\n\nMacDonald, Wallace 295, 496, 834, 1335, 1481, 1720, 1848, 2137, 2617, 2955, 3075, 3260, 3397, 3506, 3762, 4280, 4294, 4468, 4607, 4706, 4913, 5038, 5040\n\nMacDonald, William Colt 886, 2263, 3153, 3482, 4365, 4607\n\nMacDouglas, John 400, 537, 1519, 1737, 1902, 2535, 3420, 4369; _see also_ Addobbati, Giuseppe\n\nMacDuff, Tyler 484, 489, 580, 1482, 2486, 2842, 4478\n\nMacFadden, Hamilton 1426, 3429, 3470, 3846\n\nMacGraw, Ali 1753\n\nMacGregor, Casey 318, 542, 2565\n\nMacGregor, Katherine 2375\n\nMacGregor, Lee 293, 1762, 1941, 4411, 4470, 4611\n\nMacGregor, Sean 1527\n\nMachiavelli, Nicoletta 1794, 1891, 2656, 2758, 2814\n\nMacht, Stephen 2700, 2807, 3930\n\nMack, Betty 1414, 1496, 1760, 1813, 2203, 2275, 2517, 2954, 3018, 3051, 3302, 3742, 3771, 4448\n\nMack, Cactus 107, 180, 286, 299, 318, 324, 387, 544, 566, 695, 777, 917, 936, 1311, 1647, 1772, 1818, 1842, 1873, 1920, 1927, 2006, 2022, 2082, 2151, 2169, 2288, 2403, 2457, 2529, 2534, 2599, 2689, 2785, 2802, 2880, 2909, 2966, 2969, 2971, 2995, 3087, 3088, 3165, 3316, 3230, 3231, 3270, 3277, 3278, 3434, 3447, 3494, 3591, 3613, 3617, 3649, 3675, 3679, 3804, 3805, 3901, 3922, 4011, 4110, 4185, 4188, 4202, 4365, 4386, 4409, 4423, 4483, 4508, 4550, 4586, 4619, 4667, 4691, 4712, 4763, 4835, 5018, 5041, 5104\n\nMack, Helen 613, 1269\n\nMackaill, Dorothy 1634, 2654\n\nMacKellar, Helen 960, 1143, 1504, 1647, 2837, 3403, 4185, 4357\n\nMacKenzie, Joyce 519, 3234, 4411\n\nMacLaine, Shirley 3815, 4624\n\nMacLane, Barton 91, 132, 189, 229, 548, 562, 718, 857, 1166, 1175, 1428, 1436, 1452, 1569, 1588, 1594, 1709, 1762, 1849, 2028, 2021, 2066, 2083, 2101, 2205, 2232, 2267, 2393, 2554, 2635, 2748, 2750, 2818, 3234, 3622, 3865, 3902, 3906, 4018, 4121, 4373, 4403, 4442, 4472, 4539, 4543, 4849, 4860, 4940\n\nMacLaren, Ian 825, 2213\n\nMacLaren, Mary 626, 899, 1270, 1338, 1447, 2129, 2306, 2552, 2571, 2761, 2785, 3165, 3303, 3574, 3664, 3938, 4106, 4206, 4854\n\nMacLean, Alistair 508\n\nMacLeod, Gavin 1998, 2506, 3282\n\nMacMahon, Aline 748, 1829, 2525, 3895\n\nMacMurray, Fred 170, 462, 632, 979, 1248, 1256, 1265, 1392, 1608, 1671, 2691, 2782, 2929, 3276, 3972, 4309, 4494, 4970\n\nMacnee, Patrick 1500\n\nMacQuarrie, George 626, 1883, 2147, 3126, 3561, 4498\n\nMacQuarrie, Murdock 153, 434, 447, 514, 797, 886, 898, 958, 1018, 1307, 1312, 1332, 1554, 1556, 1574, 1662, 1759, 1929, 2020, 2153, 2159, 2177, 2399, 2451, 2458, 2523, 2526, 2529, 2764, 2785, 2822, 2838, 2889, 2908, 2954, 3089, 3098, 3101, 3160, 3179, 3194, 3370, 3427, 3549, 3665, 3709, 3851, 3884, 3977, 4007, 4022, 4121, 4138, 4143, 4190, 4201, 4269, 4391, 4419, 4459, 4572, 4614, 4809, 4848, 4866, 4980, 5018, 5096\n\nMacRae, Gordon 2853, 3386\n\nMacRae, Henry 1993, 2130, 3655, 4928\n\nMacRae, Leslie 1278\n\nMacready, George 835, 1136, 1714, 2781, 3134, 4162, 4388\n\nMaddow, Ben 2515\n\nMadison, Guy 24, 238, 269, 274, 433, 577, 700, 819, 1166, 1355, 1541, 1722, 2192, 2316, 2602, 2618, 2622, 2883, 2980, 3059, 3093, 3363, 3372, 3403, 3759, 3936, 4342, 4420, 4439, 4491, 4574, 4616, 4618, 5048\n\nMadison, Mae 327, 4254\n\nMadison, Noel 5042\n\nMadrid, Jose Luis 4919\n\nMaffei, Mario 3516\n\nMagrill, George 271, 418, 453, 602, 956, 1049, 1519, 1601, 1804, 2223, 2367, 2516, 2652, 2727, 2790, 2861, 2928, 3075, 3112, 3414, 3454, 3534, 3650, 3707, 3750, 3877, 4057, 4099, 4289, 4488, 4585, 4599, 4646, 4692, 4949, 5001\n\nMahan, Larry 1924, 2490, 4432\n\nMaharis, George 1075, 2174\n\nMahin, John Lee 1937, 2245, 2756, 2829\n\nMahoney, Jack 1711, 1806, 2068, 2105, 2996, 3063, 3619, 3973, 3975; _see also_ Mahoney, Jock; O'Mahoney, Jock\n\nMahoney, Jock 234, 237, 379, 603, 695, 977, 2043, 2209, 2676, 2991, 3510, 3960, 4328, 4496; _see also_ Mahoney, Jack; O'Mahoney, Jock\n\nMahoney, Maggie 377, 3960\n\nMailes, Charles Hill 2585, 3985, 4404, 4535, 4864\n\nMain, Marjorie 191, 317, 960, 1434, 1526, 1925, 2013, 2020, 2756, 3411, 3607, 3817, 4257, 5008, 5032\n\nMainwaring, Daniel 830, 4769; _see also_ Homes, Geoffrey\n\nMajors, Eddie 2944, 3457, 4520, 4820\n\nMajors, Lee 1840, 1879, 4505, 4988\n\nMako 2155\n\nMala, Ray 630, 1565, 1567, 1626, 1805, 2717, 2831\n\nMalatesta, Fred 159, 173, 1328, 2064, 2586, 2794, 3276, 3771, 4650, 4997\n\nMalcolm, Robert 88, 392, 402, 658, 2400, 2482, 2543\n\nMalden, Karl 26, 407, 720, 1704, 1772, 1945, 2779, 2901, 4969\n\nMaley, Peggy 1757\n\nMallinson, Rory 100, 207, 503, 612, 664, 691, 749, 857, 1029, 1040, 1213, 1400, 1635, 1844, 2037, 2096, 2109, 2122, 2139, 2178, 2221, 2246, 2287, 2504, 2678, 2853, 2899, 3178, 3339, 3510, 3578, 3685, 3746, 3769, 3839, 3848, 3849, 4048, 4074, 4080, 4357, 4491, 4618, 4754, 4830\n\nMallory, Boots 3153\n\nMalone, Dorothy 170, 584, 798, 1357, 2018, 2237, 2252, 2395, 2781, 3096, 3199, 3667, 4049, 4232, 4259, 4621, 4791\n\nMalone, Molly 546, 1602, 2533, 4143\n\nMalone, Nancy 2560\n\nMaloney, Leo 735, 2541, 3065, 3251\n\nMalvern, Paul 1334, 1965, 2479, 3242, 3451, 3680, 3870, 4282, 4854, 5011\n\nMalyon, Eily 160, 1378, 1588, 1949\n\nMamakos, Peter 1341, 1396, 1499, 1936, 2578, 3213, 3752, 3890, 4266, 4500, 4601, 4751\n\nMamoulian, Rouben 1521, 1883, 2586\n\nMancini, Carla 92, 99, 1135, 1764, 4901\n\nMancuso, Nick 2340, 2696, 2807\n\nMander, Miles 112\n\nMankiewicz, Joseph L. 4330\n\nMann, Anthony 290, 439, 748, 1093, 1478, 2192, 2525, 2556, 2751, 4435, 4989\n\nMann, Daniel 1946, 3402\n\nMann, E.B. 2954, 3268\n\nMann, Hank 73, 142, 271, 621, 975, 1364, 1426, 1458, 2240, 3040, 3053, 3486, 3971, 4005, 4099, 4152, 4932, 5075\n\nMann, Larry 119, 838, 1365, 2859, 3739, 4064\n\nMann, Leonard 1393, 4664, 4724\n\nMann, Margaret 824, 2285, 3907, 4656\n\nManners, David 1827\n\nManners, Marjorie 382, 2965, 4319, 4589, 4838\n\nManni, Ettore 3513, 4162\n\nManning, Bruce 2066, 4074\n\nManning, Hope 2874; _see also_ Manning, Irene\n\nManning, Irene 3971; _see also_ Manning, Hope\n\nManning, Knox 3547, 4729\n\nManning, Monroe 2183, 2379\n\nManning, Robert 1280, 3927\n\nMannors, Sheila 873, 1052, 2173, 2300, 3173, 4297, 4306, 4854; _see also_ Bromley, Sheila; LeGay, Sheila\n\nMansfield, Jayne 3821\n\nMantee, Paul 401, 978\n\nMantegna, Joe 4351\n\nMantooth, Randolph 506\n\nManwaring, Daniel 789\n\nMany Treaties, Chief 354, 552, 557, 956, 1032, 2133, 2218, 2286, 2288, 2552, 2948, 3107, 3650, 3750, 4188\n\nMapes, Ted 143, 207, 234, 289, 352, 357, 391, 410, 442, 489, 634, 666, 827, 858, 878, 928, 951, 1001, 1017, 1059, 1072, 1158, 1213, 1228, 1247, 1263, 1362, 1447, 1449, 1480, 1486, 1506, 1513, 1520, 1626, 1635, 1704, 1805, 1915, 2035, 2119, 2132, 2166, 2197, 2259, 2285, 2403, 2514, 2521, 2561, 2727, 2797, 2867, 2875, 2907, 2942, 2943, 2978, 3042, 3052, 3088, 3159, 3233, 3269, 3283, 3330, 3385, 3453, 3464, 3507, 3520, 3552, 3564, 3574, 3629, 3651, 3763, 3876, 3892, 4011, 4117, 4166, 4188, 4191, 4272, 4305, 4323, 4372, 4393, 4427, 4458, 4465, 4649, 4701, 4723, 4729, 473, 4754, 4773, 4938, 5104\n\nMaple, Christine 323 3549\n\nThe Maple City Four 1594, 2872, 4655\n\nMaples, Virginia 4987\n\nMara, Adele 91, 176, 286, 375, 612, 842, 2806, 3563, 3567, 3859, 4599, 4723\n\nMarch, Fredric 1908, 2013\n\nMarch, Sally 143\n\nMarchand, Corinne 2532\n\nMarchent, Joaquin Luis Romero 281, 888, 1508, 1699, 2075, 2904, 3784, 3796, 4012\n\nMarchent, Rafael Romeo 1508, 2387, 2570, 3515, 3717, 5093\n\nMarcovicci, Andrea 3007\n\nMarcus, James 445, 810, 1300, 1639, 1928, 1972, 2004, 2121, 2129, 2173, 2432, 2608, 2923, 3179, 3609, 4165, 4281, 4316, 4482, 4762, 4839\n\nMarcuse, Theodore 2576, 4417\n\nMargheriti, Antonio 85; _see also_ Dawson, Anthony\n\nMargo 1441, 3561, 4751\n\nMargolin, Janet 2779, 4345\\\n\nMargolin, Stuart 985, 1998\n\nMarihugh, Tammy 4387\n\nMarin, Cheech 756, 1065\n\nMarin, Edwin L. 1, 638, 659, 801, 1408, 1599, 1853, 3285, 4179, 4231, 5075\n\nMarin, Gloria 888\n\nMarion, Beth 180, 297, 1241, 1384, 1464, 1474, 3077, 3081, 3531, 3903, 4489\n\nMarion, Charles R. 111, 2444, 3576\n\nMarion, Frances 3766, 4991, 5004\n\nMarion, George F. 1492, 3573, 4591\n\nMarion, Paul 439, 654, 853, 1407, 1973, 2143, 3233, 3849, 5104\n\nMaris, Mona 176, 3595, 3766, 4054, 4639\n\nMarizanno, Silvio 407, 1251\n\nMark, Michael 3414\n\nMark, Robert 1582, 2115; _see also_ Dana, Rod\n\nMarkey, Enid 962\n\nMarkham, Monte 1739, 1943, 3345\n\nMarkland, Ted 1765, 1893, 2056, 2198, 3345, 4775, 4794, 4930\n\nMarlen, Gloria 436\n\nMarley, John 674, 2056, 2113, 2510, 2696\n\nMarlo, Frank 588, 589, 680, 2122, 2262, 3502, 3656, 4771\n\nMarlo, Steve 1922\n\nMarlow, Rex 1010\n\nMarlow, Scott 3367, 5066\n\nMarlowe, Don 2871\n\nMarlowe, Frank 23, 82, 252, 489, 584, 1235, 1772, 2048, 2426, 3929, 4069, 4646, 4983, 5008\n\nMarlowe, Hugh 375, 562, 1507, 2445, 3290, 4120, 4795\n\nMarlowe, Jo Ann 2537, 3413, 4864, 4928\n\nMarlowe, June 765, 2394, 2792, 3413, 4864, 4928\n\nMarlowe, Kathy 1353\n\nMarquand, Serge 3604\n\nMarquand, Tina 4287; _see also_ Aumont, Tina\n\nMarques, Maria Elena 16\n\nMarquette, James (Jacques) 4247\n\nMarquis, Margaret 495, 670, 2220\n\nMarr, Eddie 4061\n\nMars, Kenneth 122, 586, 3753, 4749\n\nMarsh, Anthony 2998\n\nMarsh, Caren 2759\n\nMarsh, Mae 279, 863, 1165, 1313, 1395, 1704, 1831, 2718, 3150, 3752, 3776, 4360, 4626\n\nMarsh, Tiger Joe 4719\n\nMarshal, Alan 981, 1947\n\nMarshall, Brenda 1884, 2009, 4890\n\nMarshall, Connie 3682\n\nMarshall, E.G. 520, 2486\n\nMarshall, Gary 1243\n\nMarshall, George 22, 1082, 1084, 1257, 1392, 1735, 1945, 2482, 2646, 2782, 3096, 3316, 3355, 3499, 3722, 3754, 3815, 4286, 4428, 4700, 4868, 4941\n\nMarshall, Herbert 1187\n\nMarshall, Penny 3731\n\nMarshall, Tully 328, 410, 514, 1300, 1577, 2018, 2146, 2212, 2610, 4092, 4607, 4639\n\nMarshall, William 278, 1378, 2623, 3711\n\nMarshe, Vera 325\n\nMartan, Nita 4645\n\nMartel, Christiane 2063, 4967\n\nMartel, Jeanne 2456, 2933, 3703\n\nMartel, June 145, 3709, 4954\n\nMartell, Donna 1890, 4252; _see also_ De Mario, Donna\n\nMartell, Gregg 1092, 2620, 3862, 4457, 4989\n\nMartell, Peter 1393, 1580, 2724, 2904, 3515, 3727, 4342, 4664, 5025\n\nMartin, Al 4, 2285, 3526, 4103, 4485\n\nMartin, Ana 3387\n\nMartin, Andra 5055\n\nMartin, Andrew 1526\n\nMartin, Chris-Pin 64, 79, 134, 159, 198, 271, 282, 337, 402, 414, 424, 447, 593, 613, 755, 757, 1311, 1458, 1462, 1471, 1519, 1521, 1562, 1563, 1829, 1967, 1972, 2122, 2161, 2180, 2181, 2385, 2471, 2527, 2552, 2586, 2644, 2878, 2951, 3005, 3112, 3384, 3394, 3431, 3436, 3511, 3562, 3596, 3690, 4050, 4100, 4256, 4284, 4456, 4650, 4657, 4747, 4997, 5096\n\nMartin, D'Urville 474, 2331, 4040\n\nMartin, Dan 2214, 2507, 2656\n\nMartin, Daniel 202, 751, 1350, 1700, 2817, 3786, 4332, 4900, 5047\n\nMartin, Dean 237, 1354, 1422, 2720, 3040, 3612, 3777, 3855, 3990, 4035, 4287\n\nMartin, Deanna 4164, 5060\n\nMartin, Dewey 323, 2084, 3731, 3778\n\nMartin, Diana 1385, 4243\n\nMartin, Dick 2895\n\nMartin, Don 161, 501, 2395, 4141, 4160\n\nMartin, Eugenio (Gene) 202, 1186, 3030, 4631\n\nMartin, George 376, 769, 1385, 1484, 1737, 2817, 3028, 3118, 3309, 3380, 3405, 3715, 4243, 4332, 4629\n\nMartin, Janet 286, 1767, 5050\n\nMartin, Jared 4859\n\nMartin, Jill 1805, 4549\n\nMartin, Jimmie 794, 896, 940, 1272, 1463, 1939, 2123, 2596, 2765, 3942, 3945, 4021, 4155, 4408, 4821, 4829\n\nMartin, Jose Manuel 133, 202, 2330, 570, 571, 574, 918, 1355, 1393, 1519, 1581, 1699, 2532, 2656, 2903, 3118, 3380, 3726, 3782, 4243\n\nMartin, Lewis 166, 212, 389, 1166, 2234, 3140, 3211, 4123, 4962\n\nMartin, Marcella 4827\n\nMartin, Marion 420, 934, 2159, 2856, 4383, 5021\n\nMartin, Nora Lou 3889\n\nMartin, Pamela Sue 902, 1663\n\nMartin, Pepper 93, 594, 2379, 2990, 3748\n\nMartin, Richard 27, 149, 456, 534, 1050, 1199, 1421, 1688, 1726, 1736, 1941, 1988, 2260, 2605, 2731, 2775, 3119, 3222, 3441, 3473, 3523, 3646, 3667, 4107, 4140, 4241, 4386, 4486, 4653, 4777, 4825, 4841, 4951\n\nMartin, Ross 1533, 1958, 2513, 2694, 4927, 4975\n\nMartin, Sobey 3829\n\nMartin, Steve 4351\n\nMartin, Strother 221, 371, 375, 586, 833, 1005, 1161, 1245, 1642, 1776, 1937, 2002, 2561, 2626, 2779, 2807, 3136, 3602, 3816, 3853, 4035, 4134, 4576, 4739, 4924, 4934\n\nMartin, Todd 192, 242, 2056\n\nMartin, Tony 3213\n\nMartinelli, Arthur 3288\n\nMartinelli, Elsa 281, 1989\n\nMartinez, A. 713, 883, 2045, 3842, 4126\n\nMartinez, Joaquin 506, 1375, 1752, 2020, 2028, 4633\n\nMartini, Nino 1521\n\nMartino, Sergio 133, 2569\n\nMartinson, Leslie H. 363, 2157\n\nMarton, Andrew 40, 1757, 4962\n\nMarvin, Frankie 69, 104, 183, 304, 323, 389, 410, 424, 497, 797, 818, 858, 880, 1143, 1360, 1512, 1574, 1595, 1728, 1821, 1827, 1983, 1991, 2217, 2233, 2399, 2529, 2635, 2668, 2701, 2786, 2850, 2867, 2872, 2874, 2885, 2894, 3006, 3015, 3075, 3111, 3163, 3187, 3194, 3316, 3254, 3329, 3408, 3433, 3510, 3563, 3574, 3603, 3624, 2675, 3681, 3866, 3890, 3919, 3922, 3929, 4024, 4036, 4081, 4082, 4127, 4508, 4592, 4599, 4655, 4692, 4732, 4886, 5037, 5056, 5096\n\nMarvin, Johnny 2269, 2425, 3550, 4641, 4655\n\nMarvin, Lee 193, 674, 808, 1014, 1184, 1220, 1642, 1672, 1774, 2561, 2631, 2685, 3009, 3096, 3136, 3180, 3221, 3768, 4162\n\nMarx, Chico 1577\n\nMarx, Groucho 1577\n\nMarx, Harpo 1577\n\nMarx, Melinda 4741\n\nMarx, Neyle 3074\n\nMasak, Ron 4429\n\nMason, Bill 905, 5053\n\nMason, James 202\n\nMason, Jim (James) 144, 145, 243, 338, 344, 440, 621, 622, 660, 682, 684, 686, 823, 1162, 1177, 1328, 1563, 1679, 1812, 1889, 1980, 2111, 2207, 2323, 2245, 2304, 2391, 2404, 1440, 2549, 2699, 2737, 3011, 3012, 3087, 3126, 3153, 3187, 3353, 3359, 3408, 3538, 3606, 3702, 4104, 4203, 4297, 4442, 4593, 4747, 4798, 4883\n\nMason, Larry _see_ Davis, Art\n\nMason, LeRoy 15, 68, 107, 110, 230, 233, 366, 383, 447, 605, 610, 626, 644, 818, 827, 1177, 1311, 1337, 1345, 1519, 1522, 1553, 1558, 1595, 1767, 1836, 1860, 1873, 1918, 2012, 2020, 2035, 2125, 2192, 2307, 2349, 2427, 2481, 2545, 2600, 2608, 2649, 2673, 2727, 2786, 2790, 2832, 2838, 2948, 2967, 2995, 3003, 3015, 3016, 3088, 3228, 3242, 3258, 3269, 3326, 3408, 3471, 3557, 3574, 3625, 3669, 3693, 3694, 3709, 3846, 3889, 3905, 3924, 3929, 3937, 3971, 4016, 4181, 4184, 4306, 4317, 4564, 4584, 4642, 4646, 4703, 4734, 4747, 4786, 4817, 4840, 4844, 4863, 5037, 5056\n\nMason, Lesley 4020, 4028, 4756\n\nMason, Louis 1599, 2020, 2482, 2781, 3374, 4257, 4621, 4673\n\nMason, Mary 723\n\nMason, Shirley 1043\n\nMason, Sydney 104, 698, 841, 2693, 2910, 3222\n\nMassen, Osa 2016\n\nMassey, Curt 4872\n\nMassey, Darla 2005\n\nMassey, Ilona 2836, 3132\n\nMassey, Louise 4872\n\nMassey, Raymond 252, 664, 1945, 2489, 3711, 3779, 4179\n\nMasters, Howard 70, 289, 340, 343, 1554, 2409\n\nMasur, Richard 1833, 2671, 4433, 4960\n\nMate, Rudolph 497, 1265, 2664, 3295, 3861, 4373, 4740\n\nMateos, Julian 675, 1841, 3391, 4807\n\nMathers, Jimmy 2500\n\nMathes, Marissa 3433\n\nMatheson, Richard 3749\n\nMatheson, Tim 122, 1899, 2188, 2389, 2447, 3203, 3204, 3643\n\nMathews, Carl 139, 151, 206, 234, 236, 338, 343, 364, 369, 380, 395, 422, 425, 557, 573, 683, 726, 728, 758, 773, 774, 776, 856, 896, 917, 1071, 1272, 1282, 1299, 1327, 1338, 1339, 1370, 1445, 1448, 1459, 1464, 1475, 1576, 1626, 1667, 1690, 1718, 1744, 1756, 1800, 1803, 1818, 1859, 1894, 1939, 1979, 2107, 2114, 2119, 2151, 2172, 2251, 2266, 2278, 2363, 2365, 2398, 2407, 2409, 2424, 2435, 2448, 2457, 2481, 2596, 2634, 2679, 2690, 2759, 2826, 2835, 2863, 2891, 2949, 3002, 3050, 3073, 3085, 3097, 3098, 3157, 3164, 3169, 3232, 3258, 3261, 3262, 3278, 3280, 3308, 3353, 3385, 3388, 3446, 3448, 3494, 3546, 3568, 3589, 3601, 3613, 3656, 3668, 3670, 3677, 3799, 3825, 3876, 3918, 3943, 3944, 3949, 3951, 4021, 4024, 4030, 4097, 4131, 4144, 4303, 4319, 4323, 4324, 4393, 4458, 4461, 4497, 4534, 4549, 4553, 4587, 4590, 4606, 4619, 4660, 4702, 4815, 4819, 4828, 4829, 4830, 4838, 4858, 4887, 4894, 4935, 4953, 5027\n\nMathews, Carole 391, 759, 2189, 2618, 2978, 3811, 3857, 3914, 4062, 4221, 4339, 4539, 4574, 4616\n\nMathews, George 1702, 1842, 2204, 2241, 3185\n\nMathews, June 752\n\nMathews, Kerwin 255\n\nMatthau, Walter 1989, 2095, 2430, 3415\n\nMatthews, Carmen 702, 3687\n\nMatthews, Forrest 263, 368, 1003, 1321, 3331, 4026, 4197, 4507, 4559, 4935\n\nMatthews, Lester 1265, 1405, 1512, 2677, 2837, 3701, 4214, 4252\n\nMattison, Frank S. 294, 556\n\nMattoli, Mario 1382\n\nMattox, Martha 314, 546, 1200, 1426, 1796, 1863\n\nMature, Victor 733, 1238, 1480, 2192, 2718\n\nMauldaur, Diana 2627\n\nMauldin, Bill 3305\n\nMaunder, Wayne 2154, 2322\n\nMaurer, Norman 2964\n\nMaurey, Nicole 2025\n\nMauri, Roberto 3716\n\nMaxey, Paul 277, 354, 599, 914, 1427, 2226, 2665, 2829, 3046, 3377, 3534, 3857, 4049, 4162, 4252\n\nMaxey, Virginia 4506\n\nMaxwell, Edwin 514, 1165, 1566, 2145, 2790, 3126, 3431, 5072\n\nMaxwell, John 46, 154, 2048, 2161, 2197, 2398, 2609, 2620, 2693, 3020, 3856, 4329, 4932\n\nMaxwell, Lois 2088\n\nMaxwell, Marilyn 132, 2789, 4098, 4976\n\nMay, Ann 4404\n\nMay, Karl 106, 1069, 1761, 2219, 2316, 4540, 5003\n\nMay, Vivian 4912\n\nMayberry, May 2567\n\nMayer, Gerald 1996\n\nMayer, Ken 2616, 2659, 4064\n\nMayer, Ray 159, 2223, 2578, 2864, 3153\n\nMayes, Wendell 1441, 1772, 3402, 4113, 4799\n\nMaynard, Ken 56, 153, 157, 180, 295, 315, 327, 381, 425, 500, 684, 814, 1018, 1021, 1025, 1038, 1162, 1200, 1269, 1288, 1291, 1325, 1370, 1474, 1651, 1677, 1789, 1796, 1799, 1835, 1839, 1861, 1928, 1983, 2121, 2164, 2286, 2365, 2391, 2472, 2527, 2632, 2699, 2737, 3035, 3085, 3092, 3135, 3263, 3321, 3427, 3631, 3770, 3949, 3969, 4020, 4038, 4165, 4207, 4297, 4454, 4484, 4512, 4614, 4756, 4839, 4853, 4861, 4887, 4892, 4957, 4970\n\nMaynard, Kermit 68, 70, 76, 101, 129, 135, 151, 154, 189, 206, 210, 214, 234, 279, 289, 299, 332, 368, 377, 382, 395, 430, 476, 494, 542, 552, 582, 597, 659, 671, 695, 726, 739, 773, 776, 797, 819, 922, 1017, 1090, 1093, 1155, 1162, 1200, 1274, 1296, 1319, 1332, 1337, 1362, 1363, 1364, 1377, 1400, 1408, 1444, 1459, 1463, 1473, 1480, 1490, 1491, 1497, 1505, 1579, 1601, 1626, 1695, 1711, 1743, 1749, 1782, 1844, 1862, 1895, 1915, 1971, 2033, 2135, 2176, 2187, 2217, 2250, 2251, 2255, 2271, 2288, 2302, 2365, 2398, 2400, 2482, 2486, 2521, 2526, 2549, 2589, 2618, 2667, 2731, 2734, 2765, 2782, 2802, 2818, 2829, 2831, 2832, 2839, 2847, 2871, 2889, 2895, 2902, 2951, 2960, 2978, 3006, 3067, 3068, 3082, 3083, 3090, 3097, 3142, 3155, 3164, 3169, 3220, 3221, 3226, 3229, 3234, 3255, 3258, 3262, 3299, 3308, 3351, 3363, 3374, 3388, 3447, 3452, 3480, 3484, 3505, 3552, 3553, 3568, 3629, 3708, 3728, 3815, 3825, 3848, 3851, 3866, 3900, 3904, 3914, 4027, 4044, 4053, 4057, 4078, 4103, 4108, 4123, 4131, 4135, 4154, 4184, 4273, 4299, 4323, 4357, 4402, 4427, 4488, 4497, 4498, 4502, 4522, 4590, 4600, 4606, 4613, 4640, 4695, 4696, 4729, 4807, 4808, 4837, 4842, 4849, 4893, 4921, 4952, 4954, 4955, 4982, 4986, 5041, 5049, 5075\n\nMaynard, Mary 3388\n\nMayo, Archie 3071\n\nMayo, Donald 4702\n\nMayo, Frank 56, 144, 203, 579, 1043, 1378, 2064, 2861, 2864, 3078, 3263, 3538, 3695, 3711, 4336, 4932\n\nMayo, Virginia 65, 320, 798, 1399, 1406, 1633, 1795, 2005, 2106, 3185, 3344, 3365\n\nMazurki, Mike 26, 53, 696, 720, 804, 930, 967, 970, 1422, 1986, 3156, 3344, 3364, 4635\n\nMcAvoy, May 1673\n\nMcBain, Diane 363, 1112, 1133, 1963, 2896, 3315\n\nMcCall, William (Bill) 8, 186, 687, 752, 975, 1455, 1474, 1495, 1751, 1813, 1965, 1983, 2203, 2257, 2263, 2290, 2299, 2358, 2366, 2690, 2730, 2777, 2850, 2976, 3260, 3486, 3623, 3650, 3707, 3994, 4511, 4518, 4648, 4722, 4780, 4885, 4893\n\nMcCalla, Irish 1353\n\nMcCallister, Lon 312\n\nMcCallum, David 3560\n\nMcCambridge, Mercedes 748, 1560, 1996, 2048, 3638, 3661\n\nMcCann, Chuck 815\n\nMcCarey, Leo 3634\n\nMcCarey, Ray 863, 4205\n\nMcCargo, Marian 4638\n\nMcCarroll, Frank 27, 64, 143, 286, 311, 318, 379, 382, 392, 464, 492, 499, 542, 557, 573, 599, 624, 634, 656, 664, 665, 683, 695, 776, 781, 813, 827, 836, 853, 856, 858, 935, 956, 1080, 1145, 1156, 1279, 1334, 1360, 1362, 1433, 1447, 1472, 1475, 1486, 1505, 1738, 1718, 1744, 1815, 1818, 1873, 1894, 2026, 2111, 2164, 2166, 2178, 2250, 2258, 2264, 2293, 2435, 2457, 2546, 2884, 2944, 2953, 2967, 2975, 2988, 2995, 2999, 3063, 3115, 3142, 3164, 3165, 3172, 3220, 3228, 3230, 3358, 3471, 3493, 3504, 3564, 3565, 3615, 3648, 3651, 3670, 3679, 3744, 3801, 3822, 3824, 3834, 3859, 3893, 3902, 4006, 4016, 4027, 4059, 4108, 4184, 4364, 4419, 4455, 4522, 4525, 4534, 4550, 4584, 4587, 4599, 4660, 4696, 4759, 4828, 4838, 4952, 4955, 5027, 4096\n\nMcCarthy, John P. (J.P.) 688, 758, 891, 1414, 1586, 1813, 2167, 2263, 2290, 2473, 2531, 2590, 2860, 3229, 3444, 3485, 4022, 4193, 4516, 4835\n\nMcCarthy, Kevin 3, 317, 554, 941, 1499, 2661, 2897, 4160\n\nMcCarthy, Lin 1248, 5053\n\nMcCarthy, Nobu 4768\n\nMcCarty, Patti 1090, 1338, 1486, 1506, 1749, 2974, 2997, 3171, 3648, 4273\n\nMcCauley, Wilbur 729, 2948, 3598\n\nMcClary, Clyde 6, 448, 473, 495, 752, 773, 1301, 1303, 1455, 1503, 1683, 1687, 2003, 2298, 2355, 2364, 2812, 3062, 3081, 3086, 3266, 3268, 3272, 3292, 3293, 3434, 3446, 3476, 3501, 3528, 3589, 3617, 3913, 3946, 4294, 4298, 4324, 4385, 4424, 4489, 4501, 4522, 4581, 4698, 4722, 4887, 4995\n\nMcClory, Sean 237, 720, 982, 1735, 2555, 2576, 3131, 3621, 4329\n\nMcClure, Bud 295, 337, 411, 414, 431, 500, 684, 814, 853, 889, 1009, 1162, 1269, 1288, 1335, 1432, 1474, 1488, 1626, 1677, 1687, 1730, 1835, 1839, 1861, 1928, 1932, 1993, 2014, 2121, 2298, 2371, 2391, 2421, 2523, 2527, 2548, 3558, 2682, 2699, 2701, 2703, 2786, 2915, 2954, 2998, 3018, 3029, 3049, 3092, 3263, 3269, 3401, 3424, 3427, 3524, 3555, 3658, 3709, 3742, 3969, 4165, 4190, 4207, 4248, 4294, 4297, 4313, 4454, 4458, 4587, 4592, 4617, 4467, 4683, 4815, 4837, 4861, 4892, 4894, 4938, 4995, 5001, 5009, 5061\n\nMcClure, Doug 190, 247, 507, 996, 1292, 1341, 1500, 2623, 3816, 4662, 4927\n\nMcClure, Greg 1603, 4383\n\nMcClure, M'liss 2922\n\nMcConnell, Keith 453, 508\n\nMcConnell, Marilyn 2589\n\nMcConville, Bernard 437, 715, 726, 2129, 2432, 2529, 2872, 2972, 3454, 4532\n\nMcCormack, Maureen 3141\n\nMcCormack, Pat 2671\n\nMcCormack, Patty 60\n\nMcCormick, Merrill 38, 126, 129, 234, 289, 304, 335, 392, 413, 422, 424, 446, 473, 555, 616, 682, 725, 785, 810, 834, 873, 893, 946, 973, 1009, 1042, 1073, 1097, 1162, 1172, 1211, 1258, 1297, 1300, 1307, 1322, 1329, 1330, 1332, 1374, 1394, 1490, 1497, 1556, 1586, 1587, 1680, 1693, 1732, 1859, 1870, 1973, 1974, 2081, 2086, 2121, 2186, 2207, 2263, 2290, 2311, 2353, 2361, 2391, 2408, 2410, 2414, 2420, 2451, 2552, 2554, 2573, 2635, 2648, 2763, 2777, 2844, 2874, 2907, 2949, 2968, 2993, 3078, 3108, 3115, 3163, 3170, 3179, 3194, 3226, 3259, 3260, 3266, 3289, 3300, 3426, 3461, 3474, 3565, 3582, 3595, 3609, 3825, 3889, 3894, 3920, 3943, 3989, 3992, 4000, 4042, 4051, 4053, 4054, 4112, 4127, 4256, 4293, 4454, 4483, 4501, 4574, 4607, 4609, 4667, 4702, 4803, 4850, 4861, 4892, 4899, 4986, 4995, 5096\n\nMcCoy, Charlie 3378\n\nMcCoy, Horace 527, 1247, 2485, 3219, 3542, 4286, 4494, 4700, 4940\n\nMcCoy, Tim 6, 131, 139, 289, 301, 431, 564, 773, 781, 834, 852, 958, 1146, 1227, 1306, 1307, 1320, 1327, 1328, 1389, 1445, 1544, 1554, 1667, 1716, 1993, 2074, 2254, 2355, 2360, 2371, 2522, 2632, 2915, 2947, 2977, 2979, 3086, 3365, 3401, 3448, 3489, 3506, 3508, 3548, 3640, 3658, 3850, 3880, 3946, 4089, 4143, 4294, 4303, 4313, 4324, 4525, 4549, 4607, 4612, 4816, 4823, 4835, 4850, 4885, 5002\n\nMcCoy, Tony 2748\n\nMcCrea, Jody 521, 901, 1347, 2267, 2748, 5065, 5068\n\nMcCrea, Joel 246, 450, 677, 678, 798, 901, 1347, 1402, 1420, 1433, 1628, 1738, 1697, 1742, 2871, 2986, 3247, 3435, 3672, 3696, 3745, 4049, 4129, 4160, 4234, 4566, 4665, 4745, 4803, 4921, 4970\n\nMcCulley, Johnston 1138, 2587, 2683, 2969, 4055\n\nMcCullough, Philo 271, 290, 496, 592, 600, 718, 972, 1084, 1093, 1095, 1310, 1458, 1631, 1687, 1712, 1863, 1996, 2147, 2299, 2302, 2344, 2392, 2677, 2737, 2891, 3082, 3126, 3222, 3340, 3495, 3528, 4054, 4084, 4118, 4129, 4179, 4207, 4322, 4389, 4536, 4602, 4621, 4708, 4776, 4861, 5075\n\nMcCune, Hank 4860\n\nMcDaniel, Etta 79, 150, 662, 1576, 1638, 1827, 2299, 2432, 3179, 3969\n\nMcDaniel, Hattie 413, 1256, 1605, 2923, 4336, 4983\n\nMcDaniel, Sam 203, 496, 1427, 1574, 1947, 1963, 2349, 2923, 3409, 3899, 4336, 4742, 4762\n\nMcDermott, Hugh 651, 2305\n\nMcDevitt, Ruth 1266, 3806, 5006\n\nMcDonald, Francis 196, 204, 241, 317, 324, 415, 465, 534, 549, 597, 599, 612, 620, 662, 666, 731, 836, 950, 1050, 1097, 1103, 1182, 1187, 1402, 1403, 1487, 1523, 1567, 1646, 1679, 1735, 1872, 1927, 1970, 2027, 2043, 2064, 2081, 2181, 2234, 2480, 2641, 2727, 2736, 2747, 2816, 2831, 2842, 2923, 3020, 3032, 3058, 3126, 3151, 3179, 3222, 3254, 3255, 3287, 3318, 3510, 3512, 3561, 3585, 3611, 3646, 3669, 3676, 3690, 3702, 3849, 3995, 4002, 4055, 4075, 4099, 4147, 4162, 4252, 4271, 4284, 4290, 4300, 4304, 4362, 4388, 4477, 4517, 4580, 4616, 4635, 4665, 4700, 4705, 4726, 4750, 4770, 4900, 4932, 4971, 4978, 5032, 5103\n\nMcDonald, Frank 47, 67, 103, 160, 286, 325, 433, 1013, 1512, 1688, 1696, 2534, 2576, 2602, 2622, 2727, 2840, 2980, 3093, 3236, 3434, 3864, 3929, 3936, 3980, 3998, 4013, 4197, 4225, 4285, 4392, 4422, 4439, 4534, 4539, 4574, 4599, 4616, 4618, 4643, 4646, 5030, 5048, 5078, 5079\n\nMcDowall, Roddy 26, 1240, 1354, 2348, 2719, 3571\n\nMcDowell, Claire 369, 834, 1883, 2585, 4121, 4609\n\nMcDowell, Malcolm 4195\n\nMcDowell, Nelson 331, 424, 428, 468, 593, 814, 974, 1052, 1271, 1280, 1284, 1312, 1334, 1337, 1492, 1687, 1730, 1822, 1928, 2007, 2177, 2211, 2212, 2249, 2275, 2358, 2403, 2427, 2478, 2552, 2608, 2832, 2869, 2945, 3072, 3092, 3102, 3126, 3153, 3374, 3433, 3446, 3461, 3528, 3587, 3609, 3650, 3655, 3709, 3742, 3886, 3888, 4203, 4269, 4298, 4308, 4833, 4839, 4861, 4955, 4986\n\nMcEachin, James 540, 4638\n\nMcEnery, Red River Dave _see_ Red River Dave\n\nMcEntire, Reba 558, 1500\n\nMcEveety, Bernard 122, 267, 523, 672, 2447, 2487, 2906, 3422, 3620\n\nMcEveety, Vincent 378, 919, 1346, 1754, 2188, 4537\n\nMcFarland, Spanky 172, 868, 4494\n\nMcGann, William 79, 1444, 1884, 1974, 3046, 4456\n\nMcGaugh, Wilbur 498, 593, 609, 1993\n\nMcGavin, Darren 557, 1644, 1942, 3686\n\nMcGee, Vonetta 1643, 4346\n\nMcGhee, Gloria 3865\n\nMcGiver, John 2305, 2895, 3687\n\nMcGlynn, Frank 420, 1384, 1570, 1571, 1827, 1847, 1903, 1974, 2213, 2399, 2823, 2961, 3126, 3874, 4178, 4494, 4668, 4803, 4840\n\nMcGlynn, Frank, Jr. 145, 243, 467, 917, 1931, 2131, 2300, 2554, 2923, 3179, 3470, 3555, 4503, 4854, 4971\n\nMcGoohan, Patrick 1525\n\nMcGowan, Dorrell 309, 322, 818, 1134, 1574, 1728, 1842, 1980, 2027, 2129, 3229, 3433, 3626, 2694, 3852, 3919, 3921, 3982, 4052, 4127, 4573, 4599, 5056\n\nMcGowan, George 640, 2498, 2990, 3641, 3795\n\nMcGowan, J.P. 5, 136, 243, 300, 397, 429, 461, 617, 647, 772, 785, 855, 924, 1009, 1076, 1162, 1221, 1312, 1316, 1367, 1591, 1626, 1728, 1813, 1822, 1898, 1950, 2231, 2303, 2567, 2584, 2670, 2763, 2767, 2777, 2921, 2956, 3047, 3172, 3179, 3210, 3307, 3321, 3322, 3424, 3465, 3474, 3539, 3553, 3561, 3650, 3742, 3763, 3770, 3850, 3876, 3903, 3983, 3994, 4009, 4117, 4293, 4365, 4397, 4648, 4655, 4762, 4785, 4862, 4866, 4882, 4891, 4981\n\nMcGowan, Robert 1455, 1509\n\nMcGowan, Stuart 309, 322, 818, 1134, 1574, 1728, 1842, 1980, 2027, 2129, 3329, 3433, 3626, 3694, 3852, 3919, 3921, 3982, 4052, 4127, 4572, 4599, 4655, 5056\n\nMcGrail, Walter 289, 336, 338, 626, 774, 1023, 1037, 1979, 2207, 2208, 2212, 2422, 2682, 2722, 2891, 3126, 3480, 3537, 3558, 4100, 4205, 4816, 4875\n\nMcGrath, Frank 16, 318, 626, 1395, 1703, 1921, 2185, 3241, 3440, 3471, 3752, 3806, 3814, 4057, 4184, 4435, 4790, 4849\n\nMcGrath, Larry 140, 200, 660, 935, 1105, 2084, 4503, 4593\n\nMcGraw, Charles 402, 439, 748, 1086, 1769, 2043, 2893, 3671, 4249, 4390, 5024\n\nMcGuinn, Joe 283, 336, 575, 712, 925, 1100, 1502, 1974, 2017, 2599, 3028, 3111, 3159, 3434, 3466, 3550, 3674, 3857, 3859, 4056, 4185, 4197, 4313, 4391, 4767, 4973, 5104\n\nMcGuire, Don 193, 2046, 3296\n\nMcGuire, Dorothy 632, 1434, 1539, 2887\n\nMcGuire, John 287, 439, 1228, 1730, 2954, 3179, 3534, 4166\n\nMcGuire, Kathryn 314, 556, 968, 2440, 3108, 4094\n\nMcGuire, Mary 1105\n\nMcGuire, Paul 418, 1676, 1695, 2511, 2899, 3233, 3234, 3363, 3762, 3769, 3864, 3998, 4073, 4136, 4601, 5049, 5079\n\nMcHattie, Stephen 58, 829, 1021, 1534\n\nMcHugh, Frank 4248, 4417, 4468, 4697, 4742\n\nMcHugh, Kitty 4205\n\nMcHugh, Matt 246, 844, 1862, 2159, 2239, 3087, 3276, 3386, 3684\n\nMcIntire, John 73, 100, 189, 354, 696, 1263, 1423, 1697, 1936, 2095, 2292, 2352, 2664, 2784, 3152, 3310, 3534, 3602, 3612, 3672, 3791, 4073, 4160, 4435, 4626, 4783, 4857, 4989, 5025, 5049\n\nMcIntire, Reba 2623\n\nMcIntire, Tim 3662, 3816\n\nMcIntosh, Burr 543, 1602\n\nMcIntyre, Christine 430, 791, 973, 1389, 1447, 1529, 1690, 1716, 2169, 3052, 3278, 3480, 3568, 4153, 4691, 4781, 4826\n\nMcIntyre, Hal 3914\n\nMcKay, Doreen 2802, 3027\n\nMcKay, George 307, 1228, 2150, 3544, 4225\n\nMcKay, Scott 1187\n\nMcKay, Wanda 278, 1032, 1600, 2251, 2414, 3107, 3224, 3592, 3629\n\nMcKaye, Fred 2121, 3969, 4600, 4861\n\nMcKee, Lafe 56, 144, 262, 311, 327, 351, 411, 413, 473, 496, 722, 729, 772, 856, 889, 898, 917, 923, 924, 1037, 1049, 1057, 1172, 1200, 1203, 1228, 1301, 1304, 1307, 1310, 1320, 1329, 1337, 1340, 1384, 1446, 1455, 1474, 1494, 1517, 1545, 1587, 1592, 1677, 1687, 1835, 1839, 1845, 1863, 1928, 1952, 1967, 1993, 2014, 2024, 2101, 1220, 2121, 2151, 2177, 2203, 2220, 2263, 2274, 2275, 2283, 2355, 2364, 2394, 2403, 2404, 2432, 2434, 2456, 2527, 2531, 2533, 2548, 2567, 2584, 2634, 2642, 2660, 2730, 2733, 2737, 2740, 2742, 2743, 2745, 2763, 2826, 2832, 2927, 2933, 3015, 3048, 3051, 3087, 3102, 3109, 3135, 3242, 3259, 3263, 3270, 3279, 3289, 3292, 3303, 3308, 3321, 3401, 3427, 3451, 3452, 3486, 3495, 3501, 3506, 3508, 3528, 3555, 3591, 3650, 3703, 3706, 3711, 3797, 3880, 3886, 3888, 3924, 3927, 3949, 4006, 4042, 4067, 4104, 4219, 4248, 4271, 4274, 4282, 4308, 4379, 4385, 4396, 4454, 4480, 4484, 4489, 4493, 4572, 4614, 4645, 4683, 4708, 4785, 4809, 4822, 4843, 4850, 4868, 4958, 4971, 5009, 5014, 5040, 5061\n\nMcKee, Raymond 4694\n\nMcKenzie, Bob (Robert) 54, 159, 337, 431, 468, 566, 621, 642, 687, 747, 775, 795, 814, 818, 865, 1043, 1084, 1187, 1228, 1288, 1446, 1587, 1667, 1751, 1789, 1816, 1824, 1974, 2163, 2177, 2358, 2385, 2469, 2478, 2522, 2667, 2722, 2730, 2756, 3086, 3105, 3153, 3266, 3300, 3303, 3325, 3327, 3362, 3374, 3409, 3424, 3442, 3486, 3489, 3599, 3679, 3704, 3771, 3797, 3834, 3866, 3913, 3924, 3967, 3989, 4027, 4072, 4130, 4138, 4189, 4226, 4231, 4304, 4389, 4498, 4564, 4602, 4667, 4698, 4763, 4803, 4868, 4912, 4957, 4983, 5076\n\nMcKenzie, Ella 2730, 3022\n\nMcKenzie, Eva 460, 1751, 2358, 2522, 2730, 2802, 3087, 3105, 3270, 3279, 3303, 3424, 3456, 4389, 4667\n\nMcKenzie, Fay 473, 880, 1018, 1143, 1556, 1821, 1915, 2552, 3424, 3866, 3925, 4396, 4868\n\nMcKim, Harry 886, 2673, 2775, 4285, 4777\n\nMcKim, Robert 408, 1110, 1851, 2585, 3373, 3879, 4482, 4758\n\nMcKim, Sammy 631, 1626, 1751, 1822, 1898, 2399, 2786, 2802, 2872, 3015, 3325, 3574, 3626, 4319, 4558, 4834, 4932\n\nMcKinney, Bill 35, 55, 508, 526, 761, 1583, 2069, 2348, 2950, 3847, 4345, 4789\n\nMcKinney, Florine 2378\n\nMcKinney, Mira (Myra) 199, 387, 1136, 1257, 1823, 2084, 2226, 3163, 3500, 3613, 3711, 4496, 4506\n\nMcKinney, Nina Mae 832, 2432\n\nMcKuen, Rod 4943\n\nMcLaglen, Andrew V. 223, 237, 239, 594, 741, 1395, 1427, 1431, 1692, 2196, 2344, 2381, 2575, 2626, 2843, 2911, 3283, 3522, 3740, 3798, 3816, 3990, 4800\n\nMcLaglen, Victor 720, 1518, 2147, 2652, 3814, 4638\n\nMcLean, David 2140, 2779, 3875\n\nMcLeod, Catherine 1247, 1792, 2878, 2938, 3397\n\nMcLeod, Norman 53, 2020, 3020\n\nMcLeod, Victor 476, 582, 2250, 2607, 2716\n\nMcLerie, Allyn Ann 597, 883, 2028, 2498, 2685\n\nMcLiam, John 348, 628, 909, 1899, 2333, 2666, 2685, 3855\n\nMcMahon, Ed 587\n\nMcMahon, Horace 2635, 4300, 4422\n\nMcMahon, Leo J. 446, 461, 1816, 1889, 1932, 2494, 2733, 3177, 3354, 3657, 3877, 3891, 4005, 4132, 4140, 4322, 4485, 4674\n\nMcManus, George 2040\n\nMcManus, Sharon 491, 2013\n\nMcMillan, Kenneth 465, 3007\n\nMcMurtry, Larry 805, 998, 1948, 2055, 2433, 2468, 4169\n\nMcMyler, Pamela 2911\n\nMcNair, Barbara 4064\n\nMcNally, Stephen 105, 572, 1092, 1184, 1289, 1792, 1837, 1849, 2511, 2563, 2752, 3365, 4069, 4119, 4120, 4548, 5036\n\nMcNamara, Tom 898\n\nMcNear, Howard 1164, 1234, 1608, 1842\n\nMcQuade, Robert 1368\n\nMcQueen, Butterfly 1187, 1360\n\nMcQueen, Steve 2069, 2497, 4449\n\nMcTaggart, Bud 338, 812, 995, 1504, 3327, 3557, 3679, 3944, 4549, 4564, 4761, 4823, 5037\n\nMcVeagh, Eve 1877, 3363, 3865, 4229, 4800\n\nMcVey, Patrick 2838, 3095, 4136, 4267, 4336\n\nMcVey, Paul 844, 1165, 2132, 2425, 3179, 3808, 4100\n\nMcVey, Tyler 2426, 3705, 3860\n\nMcWade, Robert 633, 748, 1436, 1594, 2923\n\nMeacham, Anne 507\n\nMeadows, Denny 974, 1047, 1301, 1760, 2432, 3322, 4469; _see also_ Moore, Dennis\n\nMeadows, Herb 843, 2400, 3340, 4160\n\nMeadows, Jayne 760, 761\n\nMeadows, Joyce 1377, 1452, 4769\n\nMeans, Russell 4765, 4992\n\nMedford, Don 1851\n\nMedina, Patricia 269, 550, 1188, 1427, 3131, 3701, 4151, 4426, 4680\n\nMeehan, Elizabeth 1567, 1572, 1811, 2836\n\nMeehan, Lew 15, 140, 188, 231, 261, 335, 413, 423, 429, 467, 566, 592, 611, 688, 881, 886, 889, 898, 1055, 1057, 1137, 1203, 1271, 1281, 1284, 1306, 1312, 1316, 1324, 1329, 1334, 1344, 1370, 1474, 1518, 1626, 1681, 1687, 1712, 1732, 1759, 1834, 1860, 2014, 2071, 2177, 2203, 2248, 2249, 2250, 2254, 2299, 2306, 2362, 2366, 2388, 2391, 2403, 2415, 2520, 2533, 2558, 2634, 2639, 2690, 2736, 2737, 2742, 2954, 3029, 3041, 3073, 3092, 3135, 3163, 3173, 3259, 3260, 3266, 3279, 3302, 3307, 3332, 3374, 3449, 3492, 3495, 3546, 3548, 3550, 3673, 3880, 3884, 3888, 3937, 4000, 4007, 4082, 4138, 4207, 4282, 4298, 4308, 4367, 4382, 4407, 4521, 4531, 4564, 4587, 4636, 4665, 4684, 4698, 4760, 4762, 4809, 4835, 4877, 4887, 4892, 5056\n\nMeek, Donald 246, 248, 2031, 2722, 2889, 3374, 4100, 4072\n\nMeeker, George 110, 1040, 1522, 1914, 2108, 2841, 3005, 3271, 3615, 3640, 3922, 4016, 4076, 4225, 4597, 4804\n\nMeeker, Ralph 2047, 2751, 2793\n\nMegowan, Don 385, 970, 1637, 1692, 1838, 2025, 2103, 2302, 2484, 2676, 3749, 3982, 4805\n\nMehaffey, Blanche 1191, 2702, 2745, 2822, 3465, 3876, 4193, 4316; _see also_ Morgan, Janet\n\nMeighan, Thomas 1825, 4494\n\nMeins, Gus 614, 1619, 3584\n\nMejia, Miguel Aceves 745\n\nMelford, George 271, 279, 413, 514, 541, 759, 873, 1172, 1458, 1831, 2423, 2722, 3557, 4446, 4700, 4940\n\nMell, Marisa 220, 2230\n\nMelton, Frank 642, 710, 881, 1576, 3374, 4569, 4925\n\nMelton, Sid 53, 1240, 2426, 3377, 4357, 4467, 4845\n\nMeltzer, Lewis 65\n\nMenard, Tina 592, 654, 729, 743, 1235, 1560, 2066, 2556, 2777, 4212, 4525\n\nMendoza, Victor Manuel 83, 858, 1507, 5024\n\nMenez, Fernando 3360, 4716\n\nMenjou, Adolphe 16, 4428, 4580\n\nMenzie, William Cameron 1166, 1187\n\nMercer, Johnny 3780\n\nMercier, Louis 2718, 2756\n\nMercier, Michele 627, 3604\n\nMeredith, Burgess 317, 2087, 2389, 2489, 2619, 4123, 4330\n\nMeredith, Charles 42, 329, 685, 1266, 1676, 2400, 3208, 3702, 4585\n\nMeredith, Don 237, 5030\n\nMeredith, Iris 386, 623, 682, 799, 881, 1503, 2111, 2272, 2306, 2540, 2547, 2975, 2983, 3003, 3280, 3393, 3449, 3525, 4000, 4042, 4077, 4238, 4314, 4315, 4401, 4410, 4501, 4592, 4608, 4810, 4817, 4834\n\nMeredith, Judi 2676, 3223, 3990, 4943\n\nMeredith, Madge 1733, 4503, 4588\n\nMerivale, Philip 2158\n\nMeriwether, Lee 536, 4638\n\nMerkel, Una 1084, 2095, 2543, 4417\n\nMerli, Maurizio 2569, 4625, 4905\n\nMerlin, Jan 789, 977, 1734, 1837, 3061\n\nMerlo, Frank 4729\n\nMerman, Ethel 94\n\nMerrick, Doris 1330; _see also_ Merrick, Lynn\n\nMerrick, George 926, 1299, 2667, 3026, 3617\n\nMerrick, Lynn 107, 154, 665, 925, 987, 995, 1024, 1042, 1472, 2033, 2028, 2966, 3220, 3989, 4106, 4617; _see also_ Merrick, Doris\n\nMerrill, Dina 3645, 4192\n\nMerrill, Gary 356, 947, 1922, 2185, 2668, 3422, 5024\n\nMerrill, Keith 1628, 1629, 4374, 4996\n\nMerrill, Lou 1188, 1405, 2831\n\nMerton, Ivy 1839\n\nMerton, John 6, 64, 70, 140, 152, 154, 229, 231, 234, 235, 243, 335, 336, 345, 369, 387, 392, 409, 414, 428, 431, 449, 452, 573, 621, 712, 728, 774, 792, 853, 872, 897, 922, 952, 1046, 1090, 1146, 1167, 1202, 1214, 1279, 1332, 1362, 1370, 1445, 1451, 1461, 1473, 1486, 1491, 1497, 1513, 1520, 1526, 1540, 1573, 1597, 1600, 1683, 1716, 1751, 1756, 1762, 1792, 1797, 1800, 1815, 1931, 1959, 1979, 2111, 2151, 2165, 2166, 2171, 2197, 2251, 2257, 2274, 2286, 2370, 2371, 2398, 2399, 2421, 2539, 2602, 2635, 2660, 2667, 2734, 2736, 2885, 2946, 3086, 3164, 3202, 3231, 3235, 3259, 3279, 3322, 3323, 3355, 3393, 3454, 3460, 3553, 3615, 3648, 3679, 3682, 3804, 3825, 3890, 4031, 4057, 4155, 4256, 4365, 4372, 4411, 4420, 4465, 4483, 4486, 4506, 4612, 4617, 4619, 4643, 4665, 4696, 4702, 4732, 4826, 4828, 4847, 4872, 4899, 4955, 4982, 5103, 5104\n\nMessinger, Gertrude 8, 384, 1872, 2303, 2636, 3443, 3459, 3653, 4760\n\nMetcalfe, Bradley, Jr. 2129, 4854\n\nMetcalfe, Helen 4694\n\nThe Metzetti Brothers 444\n\nMeyer, Emile 211, 548, 680, 1164, 1289, 1608, 1692, 1940, 2139, 2325, 2563, 2624, 3287, 3808, 3897, 4160, 4223, 4233, 4431, 4441, 4907, 4962, 5070\n\nMeyer, Torben 271, 2016, 2458\n\nMeyler, Fintan 3857\n\nMichael, Gertrude 562\n\nMichaelangeli, Marcella 133\n\nMichaels, Dolores 1289, 2902, 4791\n\nMiddlemass, Robert 143, 200, 317, 679, 1596, 1599, 1740, 2149, 4061, 4121, 4531\n\nMiddleton, Charles 63, 203, 270, 279, 352, 514, 886, 1222, 1367, 1436, 1599, 1904, 1931, 2031, 2184, 2232, 2288, 2393, 2610, 2660, 2741, 2841, 2864, 3245, 3276, 3609, 3711, 3750, 3846, 3895, 4025, 4089, 4132, 4135, 4201, 4203, 4309, 4456, 4494, 4635, 4742, 4797, 4849, 4940, 5032, 5037, 5056\n\nMiddleton, Ray 94, 1565, 1834, 1953, 2066, 2158, 2160, 3542\n\nMiddleton, Robert 316, 679, 727, 979, 1434, 1837, 2246, 2431, 2462, 2587, 3185, 3337\n\nMifune, Toshiro 3336\n\nMikels, Ted V. 4172\n\nMikler, Michael T. 1696, 2631, 3055, 4789\n\nMilan, Frank 1904, 3027, 3276, 3584\n\nMilan, Lita 1666, 2314, 2750, 3421, 4740\n\nMiles, Art 642, 801, 930, 1332, 1526, 1579, 1805, 1925, 2641, 3067, 3370, 3450, 3546, 3884, 3912, 4072\n\nMiles, Betty 1156, 1506, 2276, 2418, 3370, 3491, 3503, 4032, 4778, 4853, 4957\n\nMiles, Joanna 900\n\nMiles, Lillian 776, 2877\n\nMiles, Peter 612, 1256, 1830, 2910, 3319, 4062, 4551\n\nMiles, Sarah 2023\n\nMiles, Sylvia 2198, 2560\n\nMiles, Vera 640, 672, 700, 2561, 2674, 2906, 3752, 4417, 4921, 4936\n\nMilestone, Lewis 2079, 3319\n\nMilhauser, Bertram 3538\n\nMilian, Tomas 225, 316, 396, 820, 1019, 1424, 2198, 2495, 3639, 4631\n\nMilius, John 1534, 2028, 2348\n\nMiljan, John 116, 145, 437, 863, 2402, 2705, 2790, 2823, 2827, 2864, 3126, 3414, 3636, 3722, 4118, 4312, 4650, 4807, 4937, 4987, 5059\n\nMillan, Victor 101, 1560, 2504, 2711, 3421, 4267, 4770\n\nMilland, Gloria 1287, 1699, 1793, 2562, 3784, 4369\n\nMilland, Ray 370, 562, 602, 832, 2502, 3536\n\nMillar, Stuart 3602, 4869\n\nMiller, Alice Duer 3606\n\nMiller, Ann 1579, 2144, 2635\n\nMiller, Arthur 2635\n\nMiller, Bodil 3741\n\nMiller, Charles 366, 886, 956, 1718, 1811, 1873, 2016, 2158, 3084, 3176, 3915, 3228, 3656, 4051, 4409, 4759\n\nMiller, Christine 1876\n\nMiller, Colleen 1423, 1696, 2550, 3295\n\nMiller, David 332, 2430\n\nMiller, Denny (Dennis) 540, 1485, 1840, 2210\n\nMiller, Dick 117, 1743, 2870, 4431\n\nMiller, Earl J. 4159\n\nMiller, Ernest 767, 1939\n\nMiller, Eve 126, 329, 542, 2083\n\nMiller, Flourney E. (F.E.) 529, 1787, 1788\n\nMiller, Ivan 630, 757, 886, 1467, 1532, 1599, 2158, 2514, 2529, 2838, 2872, 3561, 3846, 4593, 4643, 4773\n\nMiller, John \"Skins\" 145, 282, 1855, 2127, 3695, 4364\n\nMiller, Kristine 1852, 3069, 3759, 4388, 5064\n\nMiller, Lorraine 76, 299, 427, 1296, 1448, 2435, 3475, 3499, 4363, 4568\n\nMiller, Mark 743\n\nMiller, Marvin 1630\n\nMiller, Mirta 1526\n\nMiller, Patsy Ruth 3201\n\nMiller, Peggy 3195\n\nMiller, Roger 2474\n\nMiller, Seton I. 1436, 2422, 2664, 3882, 4697\n\nMiller, Walter 432, 478, 566, 1043, 1337, 1474, 1512, 1537, 1544, 1614, 1618, 1626, 1677, 1804, 1824, 1916, 1952, 2014, 2212, 2304, 2394, 3270, 3322, 3486, 3555, 3575, 3582, 3650, 3969, 4142, 4683, 4703, 4742, 4954, 4971\n\nMiller, Winston 381, 484, 489, 662, 664, 1265, 1821, 2004, 2225, 2514, 2633, 2718, 3171, 3344, 3434, 3572, 3629, 3636, 4018, 4132, 4259, 4714, 4732\n\nMillett, Arthur 543, 851, 958, 1320, 1474, 2274, 2297, 2371, 2670, 2777, 2921, 3135, 3179, 3279, 3303, 3361, 4125, 4357, 4705\n\nMillican, James 30, 42, 304, 562, 664, 691, 733, 857, 936, 972, 1093, 1240, 1319, 1392, 1487, 1616, 1640, 1665, 1704, 2066, 2515, 2525, 2938, 3290, 3337, 3394, 3500, 3511, 3909, 4075, 4080, 4162, 4461, 4792, 4803, 4849, 4989\n\nMills, Frank 2637, 4073, 4123, 4135, 4585, 4798\n\nMills, Haley 40\n\nMills, John 40, 744, 2859, 3915\n\nMills, Juliet 3283\n\nMills, Marilyn 4604\n\nMills, Mort 567, 967, 1225, 1676, 1696, 2008, 2550, 2964, 3208, 3387, 3415, 3987\n\nMills, Shirley 4742\n\nThe Mills Brothers 710, 870, 2923\n\nMilne, Peter 2651\n\nMilner, Martin 1702, 2204, 3096, 4080, 4361\n\nMilo, Sandra 238, 991\n\nThe Milo Twins 3589\n\nMilton, A.L. 559\n\nMilton, Dave 3571\n\nMilton, George 345, 427, 1894, 3348, 3592, 4273, 4952; _see also_ Raison, Milton; Sayre, George W.\n\nMimieux, Yvette 370, 3282, 3406, 3981\n\nMims, William 221, 509, 1372, 1703, 2322, 2430, 2813, 3009\n\nMinardos, Nico 641, 980\n\nMineo, Sal 720, 947, 1206, 1560, 4161, 4457\n\nMiner, Allen H. 371, 1551, 3421\n\nMinnelli, Vincent 1913\n\nMinter, Mary Miles 4494\n\nMintz, Eli 3186\n\nMiranda, Juan 2242, 2243, 2244\n\nMiranda, Soledad 2903\n\nMirisch, Walter 1866\n\nMiroslava 2961, 4160\n\nMitchell, Belle 2402, 2586, 3054, 3695, 4588, 4635, 4687, 4750\n\nMitchell, Bruce 159, 242, 824, 1332, 1336, 1491, 1544, 1870, 2434, 2850, 3075, 3076, 3116, 3177, 3294, 3461, 3888, 3898, 4776, 4803, 4893, 4971, 5010\n\nMitchell, Cameron 30, 60, 540, 1507, 1770, 1908, 2093, 2194, 2486, 2549, 2655, 2941, 3143, 3150, 3203, 3428, 4148, 4233, 4259, 4295, 4968\n\nMitchell, Don 2631\n\nMitchell, Ewing 381, 1936, 2565, 4080, 4582, 4726, 5005\n\nMitchell, Frank 22, 575, 2425, 2827, 3159, 3407, 3550, 4723, 4808\n\nMitchell, Geneva 684, 1328, 1566, 4837\n\nMitchell, Gordon 99, 991, 999, 1163, 1305, 1582, 1955, 3517, 3532, 3727, 4355\n\nMitchell, Grant 2064, 2790\n\nMitchell, Guy 3316, 4349, 4890, 4973\n\nMitchell, Helen 5104\n\nMitchell, Howard 2652, 2838, 4129, 5030\n\nMitchell, James 439, 464, 607, 798, 1093, 2853, 3061, 4129\n\nMitchell, Laurie 1696, 1837, 3871\n\nMitchell, Millard 1704, 2751, 4989\n\nMitchell, Rhea 2892, 4129, 4309\n\nMitchell, Thomas 552, 1082, 1877, 2943, 3902, 4100\n\nMitchum, Christopher (Chris) 319, 329, 941, 2196, 3527, 4453, 5060\n\nMitchum, James (Jim) 114, 1617, 2199, 2614, 4527, 5068\n\nMitchum, John 237, 315, 405, 508, 679, 741, 789, 1212, 1770, 1880, 2950, 3009, 3068, 3154, 4361, 4800\n\nMitchum, Robert 226, 241, 299, 402, 446, 800, 992, 1212, 1254, 1354, 1573, 1610, 1913, 1934, 2311, 2424, 2485, 2563, 2775, 3195, 3214, 3319, 3458, 3535, 4192, 4453, 4473, 4738, 4800, 4825, 4970, 5024, 5028, 5060\n\nMitic, Gojko 106, 737, 1069, 2219, 2833, 4034\n\nMix, Art 4, 15, 37, 144, 199, 205, 264, 307, 323, 413, 434, 511, 564, 623, 682, 774, 780, 834, 853, 856, 862, 926, 973, 975, 1043, 1099, 1100, 1145, 1162, 1172, 1221, 1228, 1320, 1324, 1335, 1474, 1489, 1496, 1545, 1626, 1758, 1768, 1835, 1928, 1974, 1993, 2032, 2137, 2197, 2249, 2272, 2283, 2303, 2311, 2398, 2410, 2424, 2434, 2478, 2633, 2635, 2642, 2736, 2737, 2778, 2821, 2886, 2907, 2915, 2969, 2975, 2999, 3003, 3026, 3042, 3087, 3105, 3153, 3159, 3172, 3173, 3177, 3188, 3293, 3314, 3322, 3483, 3486, 3505, 3506, 3520, 3525, 3564, 3598, 3600, 3626, 3650, 3674, 3680, 3834, 3859, 3880, 3894, 3924, 3944, 4009, 4037, 4042, 4077, 4089, 4156, 4165, 4206, 4219, 4225, 4238, 4286, 4377, 4409, 4410, 4465, 4531, 4535, 4609, 4667, 4683, 4796, 4798, 4810, 4827, 4832, 4836, 4839, 4850, 4851, 4995, 5046, 5056, 5061; _see also_ Kesterson, George\n\nMix, Ruth 917, 1323, 1712, 3497, 3664, 4459\n\nMix, Tom 314, 735, 1083, 1369, 1426, 1636, 1869, 1984, 2071, 2238, 2253, 2422, 2541, 2660, 2726, 3240, 3442, 3469, 3655, 3956, 3985, 4289, 4510, 3603\n\nMobley, Mary Ann 2322\n\nMobley, Roger 808, 3875\n\nMoehring, Kansas 289, 597, 607, 872, 973, 1017, 1023, 1040, 1146, 1389, 1472, 1473, 1546, 1687, 1820, 1873, 1935, 2171, 2545, 2969, 3052, 3176, 3228, 3229, 3264, 3673, 3707, 4559, 4956, 5075\n\nMoffatt, Donald 1064, 1086, 1607, 1641, 1944, 2168, 3855\n\nMoffett, Barbara 3327\n\nMoffitt, John C. 3247, 3414\n\nMohica, Vic (Victor) 608, 1034, 1538, 2017, 2376, 2487, 3855\n\nMohner, Carl 2194, 2562, 4340\n\nMohr, Gerald 550, 1184, 1830, 2124, 3974, 4621\n\nMoll, Richard 513\n\nMollinson, Henry 3763\n\nMong, William V. 328, 1306, 1307, 1685, 2213, 2610, 3012, 3940, 3880, 4089, 4891\n\nMonroe, Marilyn 2661, 3535, 4411\n\nMonroe, Tom 3652, 3705, 3848, 3950, 4441, 4574, 4830\n\nMonroe, Vaughn 3447, 3921, 4470\n\nMontague, Monte 63, 69, 107, 144, 214, 234, 442, 450, 534, 597, 614, 764, 811, 851, 925, 927, 1308, 1433, 1574, 1688, 1740, 1925, 1936, 1993, 2035, 2130, 2133, 2184, 2200, 2288, 2342, 2403, 2434, 2514, 2544, 2599, 2961, 3015, 3027, 3078, 3084, 3132, 3210, 3216, 3222, 3230, 3322, 3353, 3454, 3575, 3584, 3600, 3646, 3650, 3667, 3728, 3760, 3922, 4025, 4127, 4132, 4142, 4206, 4294, 4390, 4405, 4423, 4534, 4572, 4618, 4674, 4725, 4735, 4742, 4761, 4803, 4841, 4855, 4983\n\nMontalban, Carlos 907\n\nMontalban, Ricardo 16, 113, 407, 439, 720, 907, 1062, 1072, 2045, 2144, 2583, 2487, 2563, 4523\n\nMontana, Bull 1045\n\nMontana, Montie 132, 458, 753, 846, 1141, 1686, 1948, 2561, 3458, 3665, 3977, 4225\n\nMontana, Patsy 797\n\nMontell, Lisa 1343, 2402, 2445, 2809, 4451, 4937\n\nMontenegro, Conchita 755, 1518\n\nMonteros, Rosenda 2497, 3730, 4737\n\nMontes, Elisa 175, 651, 2715, 2952, 3391, 3782, 4288\n\nMontez, Maria 476, 3112\n\nMontgomery, Belinda 506, 2389, 4137, 4571\n\nMontgomery, Elizabeth 182, 280, 2706\n\nMontgomery, George 212, 259, 282, 371, 650, 757, 863, 895, 934, 969, 1405, 1665, 1805, 1940, 2009, 2017, 2139, 2202, 2208, 2395, 2521, 2620, 2952, 3056, 3058, 3471, 3559, 3721, 3769, 4164, 4310, 4469; _see also_ Letz, George\n\nMontgomery, Jack 183, 442, 460, 597, 798, 847, 853, 1042, 1254, 1503, 1505, 1558, 1626, 1673, 1718, 1932, 1935, 1992, 2014, 2235, 2270, 2403, 2414, 2635, 2785, 2947, 2961, 3195, 3323, 3325, 3327, 3348, 3600, 3649, 3924, 4000, 4006, 4309, 4357, 4367, 4589, 4759, 4834, 4848, 4855, 4865, 5056\n\nMontgomery, Lee H. 1277, 4577\n\nMontgomery, Peggy 5, 136, 1049, 1294\n\nMontgomery, Ray 236, 562, 803, 4049, 4450\n\nMonti, Carlotta 688, 1009\n\nMontiel, Sarita 3640, 4727\n\nMontiel, Silvia 376\n\nMontoya, Alex 101, 119, 272, 325, 604, 828, 935, 1234, 1520, 1601, 1805, 2006, 2233, 2878, 2986, 3054, 3502, 3562, 3690, 3998, 4087, 4111, 4168, 4376, 4469, 4507, 4599, 4674, 4784, 4829\n\nMoody, Ralph 803, 1265, 2198, 2339, 2402, 2575, 3058, 3219, 3318, 3363, 3744, 3768, 4087, 4099, 4148, 4588, 4936\n\nMoore, Alvy 1091, 2087, 2156, 2377, 3069, 3748\n\nMoore, Archie 508\n\nMoore, Candy 2795, 4452\n\nMoore, Carlyle, Jr. 144, 1099, 2948, 2992, 4513, 4534, 4848\n\nMoore, Clayton 28, 68, 101, 234, 250, 356, 555, 698, 841, 864, 922, 1050, 1142, 1264, 1454, 1516, 1543, 1711, 1806, 1836, 2035, 2074, 2083, 2145, 2289, 2334, 2400, 2402, 2592, 2605, 2609, 2684, 2693, 2805, 2842, 2910, 2966, 3054, 3132, 3222, 3481, 3751, 3828, 4001, 4036, 4044, 4228, 4439, 4478, 4582, 4726, 4974, 5104\n\nMoore, Cleo 1199, 3211, 3523\n\nMoore, Colleen 3957\n\nMoore, Constance 460, 1982, 2717\n\nMoore, Demi 3039\n\nMoore, Dennis 2, 13, 131, 152, 23, 289, 334, 369, 380, 390, 573, 642, 643, 698, 794, 795, 872, 874, 896, 973, 1073, 1157, 1260, 1398, 1442, 1447, 1456, 1522, 1633, 1691, 1702, 1744, 1800, 1956, 2123, 2166, 2284, 2408, 2410, 2414, 2485, 2511, 2539, 2596, 2762, 2842, 2866, 2899, 2965, 2993, 2999, 3025, 3090, 3116, 3219, 3224, 3230, 3255, 3237, 3238, 3265, 3447, 3480, 3556, 3574, 3592, 3681, 3900, 3903, 4081, 4261, 4437, 4548, 4554, 4582, 4598, 4682, 4821, 4526, 4913, 5079; _see also_ Meadows, Denny\n\nMoore, Dickie 778, 4093\n\nMoore, Grace 2790\n\nMoore, Ida 664, 1257, 1923, 3383, 3478\n\nMoore, Joanna 2519, 3415, 3960\n\nMoore, Juanita 3950, 4346\n\nMoore, Kieron 915, 3997\n\nMoore, Martin 5093\n\nMoore, Matt 625, 1996, 3040, 3267, 3704, 4139\n\nMoore, Michael 548, 1245, 1273, 3140, 3891\n\nMoore, Norma 1226, 1701, 4119\n\nMoore, Owen 1880\n\nMoore, Pauline 158, 160, 666, 790, 986, 2133, 4074, 4483, 4762, 4925, 5062, 5072\n\nMoore, Roger 1497, 1564, 1596, 1601, 4585\n\nMoore, Terry 373, 671, 4343, 4755\n\nMoore, Tom 3561\n\nMoore, Victor 948, 3499\n\nMoore, Vin 3650\n\nMoorehead, Agnes 381, 1945, 3040, 4132, 4349, 4579, 4672\n\nMoorehead, Jean 1601, 1720, 3960\n\nMoorhead, Natalie 653, 1816\n\nMoorhouse, Bert 67, 2775, 4197, 4803\n\nMorales, Carmen 4701\n\nMoran, Betty 1467, 3468\n\nMoran, Dolores 842, 3897\n\nMoran, Frank 291, 626, 922, 1179, 1228, 1361, 2288, 3939, 4099\n\nMoran, Patsy 876, 1604, 1918, 2163, 4021\n\nMoran, Peggy 2124, 4498, 4808\n\nMoran, Polly 3325, 4797\n\nMorante, Milburn 2, 6, 151, 244, 297, 334, 380, 384, 422, 556, 588, 687, 773, 810, 871, 899, 917, 926, 973, 1017, 1146, 1158, 1299, 1307, 1325, 1337, 1389, 1542, 1546, 1553, 1554, 1587, 1595, 1681, 1712, 1800, 1859, 1868, 2003, 2111, 2271, 2279, 2408, 2410, 2411, 2412, 2451, 2457, 2743, 2858, 2874, 2933, 2949, 2965, 2969, 2988, 2999, 3002, 3018, 3026, 3062, 3073, 3099, 3101, 3159, 3187, 3259, 3265, 3277, 3278, 3302, 3308, 3348, 3454, 3480, 3484, 3489, 3548, 3598, 3603, 3617, 3844, 3913, 3938, 3943, 3949, 3952, 4153, 4188, 4189, 4324, 4364, 4497, 4555, 4573, 4606, 4660, 4698, 4711, 4813, 4823, 4828, 4847, 4960, 5010\n\nMoray, Yvonne 4270\n\nMore, Kenneth 3821\n\nMoreau, Jeanne 2685, 4748\n\nMoreland, Betsy Jones 981\n\nMoreland, Mantan 809, 1464, 1600, 3461, 3870, 4615, 4747\n\nMoreno, Antonio 442, 935, 2232, 2482, 2583, 3595, 3610, 3618, 3672, 3718, 3752, 4174, 4494, 4657, 4674, 4700, 5000\n\nMoreno, Hilda 2247\n\nMoreno, Mario _see_ Cantinflas\n\nMoreno, Rita 113, 685, 1033, 1407, 1507, 1784, 3781, 4229, 4672, 5052\n\nMoreno, Rosita 3710\n\nMoretti, Nadir 5094\n\nMorey, Elaine 2273, 4952\n\nMorey, Harry T. 4651\n\nMorgan, Boyd \"Red\" 43, 486, 685, 727, 819, 932, 937, 1007, 1010, 1048, 1354, 1356, 1364, 1502, 1665, 1709, 1945, 2178, 2339, 2379, 2471, 2768, 2868, 2906, 3340, 3365, 3430, 3500, 3527, 3559, 3619, 3712, 3891, 3958, 3978, 4040, 4252, 4496, 4576, 5005\n\nMorgan, Buck 438, 929, 945, 1334, 1503, 1860, 1965, 2200, 2263, 2274, 2278, 2391, 2470, 2701, 2743, 2826, 2947, 3242, 3303, 3495, 3528, 3545, 3574, 3913, 3939, 3942, 3946, 3949, 3974, 4298, 4307, 4330, 4390, 4400, 4428, 4451, 4525, 4698, 4711, 4784, 4790, 4969\n\nMorgan, Claudia 4121\n\nMorgan, Dennis 209, 685, 718, 1691, 3285, 3538, 4621, 4680\n\nMorgan, Frank 420, 948, 1853, 2756\n\nMorgan, Gene 1228, 1512, 1567, 3582\n\nMorgan, George 174, 924, 1008, 1046, 1088, 1813, 1950, 2852, 3322, 3876, 4866\n\nMorgan, George 3722\n\nMorgan, Harry (Henry) 115, 121, 122, 128, 189, 277, 290, 482, 748, 1263, 1415, 1526, 1784, 1834, 1877, 1945, 2087, 2188, 2694, 2738, 2739, 2889, 2917, 3739, 3740, 3847, 3852, 3860, 3902, 4123, 4210, 4211, 4470, 4749, 4975, 5051\n\nMorgan, Helen 1430\n\nMorgan, Ira 3700\n\nMorgan, Janet 438, 862, 2956; _see also_ Mehaffey, Blanche\n\nMorgan, Kewpie 1149\n\nMorgan, Lee 364, 368, 390, 435, 597, 610, 728, 951, 955, 1184, 1339, 1463, 1890, 2209, 2227, 2504, 2511, 3222, 3233, 3388, 3493, 3521, 3586, 3799, 3863, 4097, 4709, 4737, 4858, 5075\n\nMorgan, Nancy 2474, 2477\n\nMorgan, Ralph 1532, 1593, 1627, 1823, 2018, 2149, 2233, 2552, 2936, 4803, 5054\n\nMorgan, Read 148, 373, 508, 1406, 1490, 2623, 3387, 4755\n\nMorgan, Robert (Bob) 71, 313, 1772, 2488, 3849\n\nMorgan, Stafford 655, 2671\n\nMorgan, William 283, 880, 1811, 1821, 1915, 3866, 4127, 4199\n\nMorheim, Lou 1951, 4976\n\nMoriarty, Michael 887, 1609, 3019\n\nMoriarty, Pat 73, 129, 1588\n\nMorin, Alberto (Albert) 190, 654, 727, 741, 934, 1313, 1427, 1570, 1571, 1704, 1936, 2144, 2504, 2544, 2583, 2619, 2972, 2983, 3096, 3522, 4147, 4257, 4599, 4624, 4644, 4975\n\nMorin, Gloria 2075\n\nMorison, Patricia 3276, 3394, 3596, 3624\n\nMorita, Miki 448\n\nMorita, Pat 1243, 3806\n\nMorley, Jay 276, 556, 954, 968, 2764, 3932, 4510\n\nMorley, Karen 470\n\nMorley, Kay 783, 2944, 3945, 4520\n\nMorley, Robert 3821\n\nMoro, Nick 150, 4884\n\nMorphy, Lew 76, 344, 634, 729, 922, 1017, 1059, 1389, 1416, 1506, 2264, 2276, 2701, 2736, 2999, 3385, 3552, 3574, 3693, 3801, 3938, 4110, 4135, 4153, 4649, 4838, 5018\n\nMorrell, George 49, 67, 187, 201, 244, 297, 307, 334, 337, 340, 341, 342, 344, 345, 369, 392, 425, 431, 468, 525, 564, 573, 588, 589, 592, 658, 682, 683, 753, 773, 774, 793, 810, 848, 862, 891, 917, 926, 945, 973, 974, 1017, 1052, 1059, 1146, 1172, 1241, 1254, 1260, 1296, 1299, 1301, 1333, 1337, 1378, 1448, 1459, 1469, 1486, 1503, 1506, 1529, 1530, 1540, 1542, 1546, 1556, 1574, 1593, 1626, 1680, 1683, 1687, 1693, 1728, 1756, 1768, 1789, 1859, 1894, 1898, 1900, 1965, 2114, 2119, 2151, 2171, 2248, 2249, 2251, 2259, 2263, 2271, 2274, 2276, 2280, 2285, 2298, 2299, 2311, 2355, 2408, 2409, 2410, 2415, 2451, 2457, 2470, 2478, 2479, 2481, 2482, 2552, 2590, 2634, 2640, 2690, 2711, 2736, 2743, 2759, 2766, 2821, 2822, 2826, 2850, 2858, 2933, 2965, 2974, 2975, 2979, 2993, 2998, 2999, 3026, 3036, 3042, 3077, 3085, 3086, 3109, 3115, 3154, 3159, 3164, 3169, 3170, 3171, 3177, 3209, 3226, 3231, 3232, 3259, 3264, 3267, 3268, 3270, 3280, 3289, 3329, 3480, 3520, 3546, 3552, 3581, 3589, 3601, 3603, 3617, 3624, 3629, 3656, 3684, 3704, 3709, 3810, 3851, 3884, 3888, 3894, 3898, 3901, 3944, 3946, 3949, 4019, 4026, 4042, 4044, 4108, 4112, 4143, 4144, 4153, 4238, 4272, 4273, 4286, 4303, 4304, 4314, 4324, 4364, 4378, 4389, 4396, 4402, 4427, 4455, 4458, 4465, 4493, 4495, 4550, 4552, 4555, 4559, 4573, 4581, 4589, 4592, 4609, 4656, 4684, 4685, 4690, 4696, 4698, 4702, 4723, 4819, 4826, 4827, 4828, 4838, 4887, 4889, 4960, 4986, 5014, 5018, 5027, 5056\n\nMorricone, Ennio 315, 574, 1700, 2758, 3955\n\nMorris, Adrian 757, 2471, 3071, 3153, 3374, 3384, 4773, 4940\n\nMorris, Chester 1430, 1567, 4358, 4767\n\nMorris, Frances 323, 602, 1482, 1661, 1730, 2480, 2705, 3022, 3026, 3293, 3485, 3839, 4170, 4745, 5021\n\nMorris, Gregg 378\n\nMorris, Kirk 3532\n\nMorris, Margaret 1045, 3927\n\nMorris, Stephen 461, 1899, 1932, 2826, 3657, 4485; _see also_ Ankrum, Morris\n\nMorris, Wayne 125, 559, 584, 1029, 1053, 1063, 1314, 2101, 2164, 2436, 2591, 3500, 3864, 4099, 4124, 4468, 4620, 4697, 5075\n\nMorrison, Chuck 8, 135, 210, 459, 777, 1323, 1550, 1604, 1712, 2258, 2273, 2641, 2821, 2852, 2971, 3025, 3116, 3142, 3230, 3235, 3237, 3393, 3434, 3453, 3489, 3540, 3626, 3653, 3664, 3922, 4006, 4364, 4760, 4819, 4827, 4923, 4960, 5001\n\nMorrison, Joe 1917\n\nMorrison, Pete 294, 305, 328, 851, 1172, 1975, 2192, 2682, 2951, 3079, 3108, 3353, 3462, 3487, 4084, 4511, 4518, 4521, 4708, 4852, 4944\n\nMorrow, Brad 1923, 3392, 3751, 4962, 3392, 3751, 4962\n\nMorrow, Jeff 833, 1347, 1353, 3040, 3861\n\nMorrow, Jo 1809, 2339\n\nMorrow, Neyle 325, 758, 950, 1412, 1606, 3112, 3222, 3640, 4075, 4364, 4491, 4618, 4913\n\nMorrow, Susan 381, 637, 3722\n\nMorrow, Vic 748, 2510, 3149, 4548, 4927\n\nMorrow, William 541\n\nMorse, Barry 1241, 2148, 4801\n\nMorse, Hollingsworth 4086\n\nMorse, Terry 288, 1132\n\nMortensen, Viggo 120, 1867, 5069\n\nMortimer, Edmund 1847, 3166\n\nMorton, Charles 64, 170, 654, 731, 975, 1247, 1345, 1873, 2035, 2480, 2722, 2967, 3128, 3534, 3849, 4005, 4212, 4504, 4830, 4868, 5033\n\nMorton, Danny 933, 1246, 1718, 3630, 3744\n\nMorton, James C. 145, 535, 606, 1974, 2160, 2756, 2923, 3187, 3374, 4798, 4840\n\nMoss, Arnold 439, 3201, 4751\n\nMoss, Stewart 256, 3437\n\nMostel, Zero 1631\n\nMoulton, Buck 135, 159, 180, 410, 454, 1100, 1302, 1557, 1768, 1861, 1993, 2110, 2785, 2975, 3220, 3270, 4083, 4190, 4289, 4617, 4667, 4851, 4893, 5027\n\nMovita 101, 2731, 3610, 4945, 5011\n\nMowbray, Alan 863, 2385, 2718, 3594, 3606, 4764\n\nMower, Jack 203, 387, 522, 564, 597, 714, 722, 914, 921, 1029, 1088, 1099, 1221, 1236, 1288, 1328, 1340, 1594, 2121, 2127, 2247, 2396, 2428, 2544, 2660, 2677, 2864, 3078, 3386, 3401, 3538, 3692, 3711, 3745, 3776, 3781, 3869, 3952, 4080, 4336, 4621, 4697\n\nMowery, Helen 9, 1309, 2105, 3257\n\nMoxey, John Llewellyn 487, 1786\n\nMudie, Leonard 4750\n\nMuir, Esther 1925, 2288, 4842\n\nMuir, Gavin 602, 3684, 4635\n\nMuir, Jean 1562, 2940\n\nMuldaur, Diana 2911, 2924\n\nMulford, Clarence H. 1705, 3177\n\nMulhall, Jack 800, 844, 917, 1360, 1392, 1916, 1964, 2081, 2223, 2765, 2824, 2968, 2979, 3668, 3764, 3912, 3952, 4174\n\nMullaney, Jack 4412, 4869\n\nMulligan, Richard 2373, 3138, 4638\n\nMulligan, Robert 4113\n\nMuni, Paul 1949, 2064\n\nMunier, Ferdinand 414, 2245, 2837, 3542, 4257, 4803\n\nMunro, Matt 3721\n\nMunson, Ona 930, 1964, 2160, 4767, 4940\n\nMurdock, George 506, 1266, 1809, 2336, 4223, 4346\n\nMurdock, Perry 311, 411, 448, 510, 647, 689, 855, 891, 1037, 1301, 1489, 1813, 1927, 2101, 2268, 2281, 2473, 2524, 2548, 2763, 2764, 2777, 2785, 3036, 3256, 4050, 4293, 4843, 5061\n\nMurphy, Audie 109, 148, 567, 671, 803, 1164, 1184, 1413, 1696, 1727, 1735, 1747, 1837, 2084, 2109, 2797, 2813, 3149, 3208, 3305, 3415, 3423, 3791, 3853, 3862, 3933, 4325, 4429, 4588, 4662, 4770, 4890, 4924\n\nMurphy, Ben 55, 512, 740, 2437, 4345\n\nMurphy, Donald 2620\n\nMurphy, Edna 2136, 4200\n\nMurphy, George 439, 2013\n\nMurphy, Horace 54, 131, 137, 201, 244, 297, 337, 420, 448, 495, 792, 810, 886, 892, 897, 1080, 1137, 1148, 1193, 1241, 1317, 1465, 1503, 1553, 1558, 1647, 1674, 1681, 1683, 1925, 2110, 2163, 2220, 2285, 2288, 2298, 2334, 2342, 2356, 2478, 2552, 2635, 2654, 2740, 2802, 2861, 2864, 2927, 3044, 3087, 3258, 3270, 3332, 3457, 3476, 3492, 3581, 3589, 2590, 3591, 3626, 3675, 3913, 4017, 4019, 4081, 4128, 4130, 4152, 4186, 4189, 4279, 4382, 4427, 4460, 4501, 4573, 4622, 4656, 4685, 4698, 4840, 4863, 4872\n\nMurphy, Mary 2069, 2502, 2624, 3931\n\nMurphy, Maurice 2651, 3179\n\nMurphy, Ralph 3335\n\nMurphy, Richard 107, 183, 520, 925, 2033, 4979\n\nMurray, Charles 2654\n\nMurray, Charles, Jr. 157, 383, 1023, 2957, 2965, 4032, 4610, 4838, 4853\n\nMurray, Don 838, 1441, 1946, 1998, 2076, 2112, 2902, 3127, 4333\n\nMurray, Forbes 107, 634, 644, 869, 880, 886, 990, 1380, 1433, 1487, 1934, 1936, 2033, 2158, 2161, 2311, 2399, 2790, 2815, 2899, 3171, 3434, 3541, 3549, 3599, 3673, 3707, 3877, 4057, 4077, 4409, 4803\n\nMurray, Gary 1235\n\nMurray, James 1970, 3952\n\nMurray, Jan 237, 982\n\nMurray, Ken 2601, 4800\n\nMurray, Zon 65, 349, 402, 418, 453, 597, 664, 698, 783, 899, 935, 1142, 1215, 1255, 1309, 1408, 1542, 1616, 1618, 1680, 1699, 1711, 1748, 1873, 2109, 2178, 2255, 2262, 2289, 2400, 2435, 2448, 2609, 2684, 2804, 2858, 2863, 2882, 2894, 2963, 2970, 3054, 3063, 3091, 3094, 3150, 3238, 3365, 3509, 3864, 3936, 3995, 4001, 4026, 4272, 4495, 4520, 4709, 4731, 4829\n\nMurtin, Jane 4121\n\nMusante, Tony 2643\n\nMuse, Clarence 105, 279, 540, 952, 1329, 1361, 2610, 3192, 4635\n\nMustard and Gravy (Frank Rice and Ernest L. Stokes) 234, 1285, 2397, 4812\n\nMustin, Burt 26, 597, 674, 1631, 1913, 2225, 2989, 2990, 3909, 4429\n\nMyers, Carmel 2158\n\nMyers, Harry 1084, 2606\n\nMylong, John 94, 4635, 5064\n\nMyton, Fred 340, 344, 366, 448, 774, 1051, 1503, 1531, 1634, 1681, 1715, 1756, 1787, 2111, 2137, 2151, 2276, 2412, 2690, 2847, 2999, 3100, 3109, 3155, 3159, 3169, 3170, 3393, 3467, 3546, 3590, 3801, 3944, 4108, 4270, 4314, 4402, 4501, 4581, 4589, 4608, 4612, 4615, 4894, 4898, 4980, 5035\n\nNader, George 1423, 2752, 3000, 3652, 3754\n\nNagel, Anne 1256, 1740, 2716, 2722, 3540, 4103\n\nNagel, Conrad 2651\n\nNagel, Don 3234\n\nNaish, J. Carrol 16, 94, 191, 432, 638, 826, 919, 1039, 1685, 2020, 2144, 2187, 2239, 2583, 2921, 3219, 3245, 3299, 3436, 3522, 3561, 3719, 3880, 3931, 4058, 4395, 4650, 4885, 5020, 5044\n\nNaismith, Laurence 3560, 3915, 4693\n\nNamath, Joe 2228\n\nNance, Anthony 4208\n\nNapier, Alan 16, 1290, 4635\n\nNapier, Charles 1771, 2962, 3282\n\nNardini, Tom 40, 674\n\nNarizzano, Silvio 407\n\nNash, George 411, 1470, 1489, 1494, 3239, 3275\n\nNash, Mary 1599, 4803\n\nNash, Noreen 1560, 1564, 2402, 3334, 3541, 4058, 4140\n\nNassour, Edward 269\n\nNatteford, John Francis (Jack) 153, 214, 337, 354, 377, 451, 493, 613, 619, 677, 767, 813, 873, 907, 986, 1037, 1172, 1335, 1595, 1781, 1822, 1862, 1869, 2086, 2112, 2212, 2279, 2432, 2726, 2909, 3111, 3289, 3383, 3439, 3443, 3549, 3603, 3615, 3646, 3656, 3834, 4059, 4067, 4228, 4365, 4487, 4508, 4567, 4944, 5037, 5056\n\nNatwick, Mildred 2144, 3814, 4360\n\nNavarro, Nieves 316, 1717, 2438, 3118, 3380\n\nNazarro, Cliff 615, 1105, 2765, 3601, 3918\n\nNazarro, Ray 42, 111, 161, 356, 379, 392, 542, 695, 869, 895, 922, 1046, 1213, 1404, 1460, 1497, 1665, 1725, 1815, 1892, 1901, 1992, 2068, 2083, 2103, 2176, 2178, 2189, 2262, 2395, 2397, 2419, 2684, 2939, 2978, 2991, 3023, 3090, 3094, 3209, 3358, 3397, 3552, 3619, 3917, 3933, 3946, 3975, 4015, 4023, 4044, 4060, 4167, 4272, 4296, 4305, 4327, 4378, 4461, 4468, 4496, 4506, 4610, 4812, 4818, 4913\n\nNeal, Ella 812\n\nNeal, Frances 812\n\nNeal, Patricia 1948, 3285\n\nNeal, Tom 103, 620, 940, 1635, 1959, 2123, 2150, 2936, 3312, 4524\n\nNearing, Margaret 1271\n\nNedell, Bernard 49, 1070, 2835, 3276\n\nNeedham, Hal 909, 2911, 4224, 4738, 4790\n\nNeeson, Liam 3775\n\nNegley, Howard 277, 638, 801, 1487, 1691, 2452, 2544, 2669, 3053, 3335, 3705, 3722, 3808, 3839, 3891, 3972\n\nNegulesco, Jean 1029\n\nNeill, James 2918\n\nNeill, Noel 2, 28, 2022, 2301, 2680, 2991, 3156, 3995, 4687, 4894\n\nNeill, Roy William 174\n\nNeilson, James 26, 1535, 2797, 3387\n\nNeise, George N. 1402, 4234, 4451, 4695\n\nNeitz, Alvin J. 188, 261, 467, 511, 646, 891, 1344, 2520, 2957, 3188, 3314, 3760, 4280, 4521; _see also_ James, Alan\n\nNelson, Barry 768, 1348, 3530\n\nNelson, Bobby 264, 423, 862, 873, 917, 926, 958, 1503, 1545, 1681, 2121, 3048, 3332, 3551, 3617, 4282, 4317, 4379, 4396, 4698, 4796\n\nNelson, David 981\n\nNelson, Ed 1096, 2519\n\nNelson, Evelyn 1054, 1387\n\nNelson, Frank 1322, 2370\n\nNelson, Gary 490, 2674, 3712\n\nNelson, Gene 2853, 3193\n\nNelson, Jack 443, 629, 1173, 1759, 3294\n\nNelson, James T. \"Bud\" 39, 481, 2296\n\nNelson, Kris 2989\n\nNelson, Lloyd 569, 3019\n\nNelson, Lori 290, 1082, 2672, 2981, 3040, 4588\n\nNelson, Ozzie 1183, 4170, 4225\n\nNelson, Ralph 3987, 5028\n\nNelson, Rick 2989, 3517\n\nNelson, Ruth 3750\n\nNelson, Sam 179, 576, 682, 799, 1626, 2153, 2272, 2540, 2828, 2973, 2975, 3003, 3109, 3170, 3520, 3679, 4042, 4156, 4315, 4410, 4475, 4810, 4817, 4834\n\nNelson, Tracy 2657\n\nNelson, Willie 97, 245, 1195, 1219, 2060, 2190, 2439, 2490, 2897, 3017, 3317, 4873\n\nNeri, Rosalba 88, 133, 1163, 1197, 1217, 1648, 2054, 2441, 2532, 3715, 4342\n\nNero, Franco 175, 537, 697, 751, 820, 1011, 1113, 1135, 2097, 2643, 4288, 4527\n\nNestell, William (Bill) 37, 183, 289, 446, 476, 549, 634, 647, 886, 950, 1009, 1146, 1253, 1315, 1335, 1345, 1389, 1416, 1432, 1453, 1546, 1932, 1973, 1977, 2152, 2250, 2257, 2311, 2451, 2480, 2552, 2673, 2736, 2745, 2786, 2802, 2826, 2872, 2972, 3025, 3048, 3230, 3269, 3393, 3558, 3574, 3673, 3707, 3761, 3805, 3834, 3989, 4103, 4106, 4127, 4190, 4357, 4364, 4483, 4759, 5056\n\nNettleton, Lois 563, 1106, 1610, 1924, 2500\n\nNeufeld, Sigmund 2444\n\nNeuman, Sam 555, 1144, 1956\n\nNeumann, Dorothy 2871\n\nNeumann, Harry 413\n\nNeumann, Kurt 116, 207, 677, 1033, 1071, 1175, 1866, 2109, 2672, 2726\n\nNeville, John T. 258, 387, 975, 1167, 1221, 1576, 2014, 2961, 3288, 3300, 3489\n\nNeville, Marjean 1725\n\nNewell, David 4057\n\nNewell, William 323, 1234, 1877, 2240, 2668, 2919, 2943, 3414, 3698, 4533\n\nNewfield, Sam 6, 140, 244, 382, 423, 427, 431, 452, 499, 564, 579, 682, 773, 776, 781, 792, 1017, 1018, 1051, 1090, 1137, 1155, 1193, 1281, 1296, 1304, 1317, 1327, 1365, 1370, 1451, 1459, 1464, 1473, 1486, 1503, 1505, 1531, 1542, 1544, 1681, 1732, 1756, 1787, 1894, 2151, 2205, 2306, 2355, 2356, 2360, 2371, 2407, 2408, 2409, 2410, 2411, 2412, 2413, 2414, 2415, 2444, 2634, 2690, 2724, 2832, 2847, 2960, 2965, 2974, 2979, 2997, 2999, 3044, 3057, 3086, 3155, 3169, 3226, 3278, 3341, 3348, 3492, 3548, 3549, 3648, 3801, 3944, 3946, 3951, 4030, 4108, 4143, 4144, 4270, 4273, 4324, 4357, 4368, 4382, 4402, 4427, 4501, 4522, 4525, 4549, 4553, 4658, 4702, 4838, 4845, 4937, 4952, 4956, 5012\n\nNewill, James (Jim) 206, 430, 479, 494, 892, 943, 1317, 1338, 1738, 1749, 2712, 2891, 2953, 3097, 3361, 3390, 3953, 4078, 4490, 4819, 5077\n\nNewlan, Paul 211, 288, 801, 967, 1100, 1145, 1480, 1792, 1908, 2431, 2452, 2782, 3136, 3150, 3222, 3276, 3535, 3544, 4057, 4072, 4179, 4536, 4803, 5025\n\nNewland, John 4033\n\nNewley, Anthony 4102\n\nNewman, Barry 2348, 2793\n\nNewman, Hank, and The Georgia Crackers 1309, 4053\n\nNewman, Joseph M. 1402, 1697, 2143, 2838, 2941, 3143, 3333, 4387\n\nNewman, Paul 554, 586, 1948, 2314, 2984, 3993\n\nNewman, Samuel 2037\n\nNewman, Scott 508\n\nNewman, Walter 674\n\nNewmar, Julie 2489, 2627, 2848, 2849, 3780\n\nNewton, Mary 207, 1059, 2068, 2189, 2577, 3040, 3751, 4015, 4162, 4469\n\nNewton, Theodore 1434, 1829, 3676\n\nNewton, Wayne 1206\n\nNeymeyer, Fred 2799, 3579\n\nNibley, Sloan 287, 664, 1141, 1264, 1485, 1522, 1603, 1940, 1971, 2806, 2893, 4076, 4983, 4212, 4338, 4641\n\nNiblo, Fred 2585, 4797\n\nNichols, Barbara 2118, 3535\n\nNichols, Dudley 159, 324, 1471, 1773, 2843, 3290, 3392, 3558, 4100, 4435, 4941\n\nNichols, George, Jr. 2552\n\nNicholson, Jack 426, 521, 1590, 2666, 3428, 3845\n\nNicol, Alex 972, 1633, 1710, 2252, 2396, 2525, 3340, 3420, 3726, 4450\n\nNicolai, Bruno 1530\n\nNielsen, Hans 574, 1359, 3125, 3198, 4542\n\nNielsen, Leslie 978, 1425, 1703, 3127, 3815\n\nNieto, Jose 2112, 2952, 3336, 3726, 3730, 3739\n\nNigh, Jane 456, 1319, 1403, 2922, 3523, 3576, 4635\n\nNigh, William 272, 434, 1335, 2257, 2823, 2877, 3502, 3610, 4047, 5009\n\nNilsson, Anna Q. 1638, 1811, 3479, 3780, 4070, 4336, 4447, 4776\n\nNimoy, Leonard 675, 2882, 4329, 4970\n\nNissen, Greta 2349\n\nThe Nitty Gritty Dirt Band 3009\n\nNiven, David 246, 3022, 3606\n\nNixon, Allan 116 2644, 2960\n\nNixon, Ione 3331\n\nNixon, Marion 2181, 3469\n\nNolan, Bob 67, 110, 286, 287, 323, 386, 576, 611, 615, 623, 682, 799, 868, 1105, 1134, 1192, 1246, 1488, 1522, 1767, 1818, 1836, 1914, 1964, 1983, 2124, 2272, 2368, 2514, 2530, 2534, 2540, 2727, 2730, 2806, 2828, 2874, 2877, 2886, 2893, 2973, 2975, 2983, 3100, 3236, 3330, 3409, 3449, 3483, 3520, 3585, 3600, 3694, 3904, 4013, 4016, 4018, 4025, 4037, 4042, 4051, 4077, 4083, 4156, 4197, 4202, 4206, 4314, 4315, 4401, 4410, 4608, 4641, 4646, 4681, 4806, 4810, 4817, 4834, 5018, 5050\n\nNolan, Danisue 3974\n\nNolan, Jeanette 55, 181, 1735, 1766, 1774, 2302, 2561, 2738, 2784, 2865, 3672, 3757, 3792, 4548, 4626, 4943, 4977, 4994\n\nNolan, Jim 149, 228, 1002, 1736, 4002, 4210\n\nNolan, Kathy 1071, 2008\n\nNolan, Lloyd 112, 4803\n\nNolte, Nick 1244, 3643, 5006\n\nNolte, William L. 310, 2266, 2470, 2531, 3742, 4186, 4606\n\nNoonan, Tommy 1833, 1960\n\nNorden, Eric 116, 449, 678, 833, 3211, 4111\n\nNoriega, Eduardo 269, 1201, 1214, 1265, 1483, 2209, 2227, 3131, 3781, 5100\n\nNoriega, Manuel 83, 706\n\nNorman, Lucille 664\n\nNorris, Chuck 285, 2429, 3881\n\nNorris, Edward 183, 389, 1458, 2095, 2544, 2731, 2756, 3925, 4212, 4760, 5013\n\nNorris, Mike 285\n\nNorth, Edmund H. 859, 1082, 1265, 2941, 3185\n\nNorth, Jay 2659\n\nNorth, Sheree 2305, 3847, 4799, 5006\n\nNorth, Ted 2586, 3005\n\nNorton, Barry 4599\n\nNorton, Edgar 4865\n\nNorton, Jack 248, 1360, 2081, 3156, 3609, 3634, 4072, 4257\n\nNorton, William 1951, 3688, 3736, 2773, 3961\n\nNorvo, Red 4291\n\nNosler, Lloyd 1496, 2517, 3927, 4007\n\nNosseck, Max 3379\n\nNostro, Nick 1232\n\nThe Notables 135, 2250, 4808\n\nNova, Lou 867, 1996, 3305\n\nNovack, Shelly 1998, 2627, 4249\n\nNovak, Eva 836, 932, 1420, 1843, 2238, 2561, 3078, 3414, 3417, 3563, 3776, 3956, 4232, 4276, 4463, 4510\n\nNovak, Jane 1478, 1550, 4375, 4758\n\nNovak, Kim 1631, 4895\n\nNovarro, Ramon 1842, 2245, 2986\n\nNovello, Jay 201, 283, 642, 685, 790, 1499, 1627, 1647, 2005, 2021, 2530, 3152, 3564, 3827, 4617, 5024, 5097\n\nNowlin, Herman 411, 1762, 3027, 3241, 3451, 4463\n\nNugent, Carol 2485, 3750, 4488\n\nNugent, Eddie 2877, 3526\n\nNugent, Elliott 2725\n\nNugent, Frank S. 1395, 1719, 1985, 3752, 3814, 4233, 4338, 4343, 4360, 4585, 4611, 4626, 4764\n\nNugent, Judy 266, 1142, 2805\n\nNusciak, Loredana 1113, 4251\n\nNuyen, France 2911\n\nNyby, Christian, II 3274\n\nNyby, Christian 3941\n\nNye, Carroll 1002, 2458, 2915, 4038, 4483\n\nNye, G. Raymond 327, 1149, 1574, 1781, 2207, 2549, 2573, 3694, 3997\n\nNye, William 413\n\nOakie, Jack 1176, 2839, 4309, 4550, 5024\n\nOakland, Simon 709, 1220, 1951, 3127, 3223\n\nOakland, Vivien 3240, 4254, 4681, 4798\n\nOakman, Wheeler 6, 184, 413, 776, 781, 834, 1227, 1298, 1446, 1544, 1851, 1979, 1983, 1984, 2165, 2403, 2522, 2633, 2730, 2923, 3038, 3075, 3506, 3507, 3548, 3551, 3616, 3658, 3674, 3880, 3979, 4027, 4070, 4089, 4187, 4191, 4284, 4294, 4427, 4522, 4607, 4658, 4835, 5011\n\nOates, Warren 255, 426, 517, 736, 2100, 2500, 2501, 3252, 3391, 3435, 3622, 3816, 3845, 3854, 3962, 3991, 4330, 4577, 4802, 4897, 4934, 5055\n\nOber, Philip 520, 2728\n\nOberon, Merle 866\n\nObon, Rafael 560, 583, 591, 2041, 4216\n\nObon, Ramon 168, 217, 750, 1190, 1376, 2242, 2244, 2383, 2579, 2803, 3566, 3765\n\nO'Brian, Hugh 40, 53, 185, 257, 304, 500, 520, 544, 692, 749, 1164, 1289, 1500, 1752, 2292, 2372, 3222, 3377, 3718, 3768, 3847, 4120, 4601, 4725, 4907, 4976, 5030\n\nO'Brien, Billy 2428, 2869, 4822\n\nO'Brien, Clay 594, 768, 2490, 2906\n\nO'Brien, Dave (Tex) 206, 334, 345, 430, 479, 494, 535, 588, 589, 773, 876, 892, 943, 1001, 1071, 1159, 1229, 1317, 1327, 1338, 1363, 1370, 1389, 1448, 1464, 1506, 1667, 1738, 1716, 1749, 2107, 2134, 2251, 2278, 2356, 2572, 2589, 2712, 2786, 2953, 2979, 3085, 3097, 3202, 3280, 3390, 3477, 3590, 3825, 3920, 3953, 4019, 4078, 4128, 4186, 4303, 4324, 4363, 4490, 4554, 4685, 4819, 4872, 4887, 4889, 5037, 5077\n\nO'Brien, Donald (Donal) 697, 1424, 2030, 2097, 2569, 3639, 4905\n\nO'Brien, Edmond 320, 857, 1038, 1998, 3339, 3518, 3891, 4792, 4934\n\nO'Brien, George 144, 437, 447, 566, 720, 879, 952, 1177, 1251, 1311, 1378, 1395, 1457, 1605, 1679, 1785, 1904, 1907, 2004, 2207, 2239, 2304, 2342, 2349, 2422, 2599, 2632, 2741, 2890, 3012, 3043, 3160, 3316, 3241, 3353, 3470, 3558, 3618, 3814, 3965, 4096, 4350, 4385, 4423, 4564, 4572, 4865, 4891\n\nO'Brien, Joan 43, 808, 1518, 3933\n\nO'Brien, Margaret 191, 1842\n\nO'Brien, Pat 874, 1368, 2066, 2989\n\nO'Brien, Pat J. 242, 582, 1138, 1363, 1801, 2961, 3029, 3555\n\nO'Brien, Peter 1619\n\nO'Brien, Virginia 1792\n\nO'Connell, Arthur 748, 2556, 3185, 3422, 3783, 3840, 4330, 4387\n\nO'Connell, William 3009\n\nO'Connolly, James 4693\n\nO'Connor, Carroll 1015, 2157, 2430, 2613, 3416, 4794\n\nO'Connor, Donald 914\n\nO'Connor, Frank 304, 337, 392, 557, 582, 695, 793, 867, 875, 922, 951, 979, 1073, 1136, 1153, 1254, 1378, 1679, 1762, 1873, 1895, 1918, 1920, 1929, 1970, 1974, 2006, 2022, 2131, 2161, 2193, 2302, 2304, 2386, 2427, 2549, 2552, 2594, 2664, 2711, 2802, 2978, 3194, 3316, 3359, 3383, 3434, 3454, 3524, 3552, 3565, 3586, 3634, 3640, 3649, 3666, 3669, 3702, 3709, 3728, 3893, 3898, 3974, 4110, 4127, 4181, 4204, 4327, 4423, 4461, 4468, 4610, 5041\n\nO'Connor, Glynnis 740, 2113\n\nO'Connor, Robert Emmett 248, 491, 1526, 1792, 2020, 2889, 3043, 4257, 4803\n\nO'Connor, Tim 5006\n\nO'Connor, Una 3606, 4661\n\nO'Connor, William 729, 1154, 3739\n\nO'Day, Dawn 1685, 3469; _see also_ Shirley, Anne\n\nO'Day, Molly 2263, 2290, 3817, 3952\n\nO'Day, Nell 135, 151, 476, 582, 1297, 2250, 2273, 2526, 2607, 3067, 3115, 3142, 3220, 3291, 3390, 3965, 4006, 4103, 4409\n\nO'Day, Peggy 3844\n\nO'Dea, John 693\n\nO'Dell, Doye 67, 1475, 1522, 1836, 2193, 2537, 3107, 3995, 4641, 4888\n\nO'Dell, Georgia 311, 4814\n\nO'Donnell, Cathy 1033, 2525\n\nO'Donnell, Joseph 38, 139, 334, 335, 342, 431, 436, 452, 564, 683, 711, 956, 1017, 1332, 1370, 1459, 1544, 1667, 1895, 2128, 2133, 2165, 2171, 2280, 2355, 2360, 2410, 2413, 2415, 2512, 2776, 2863, 3084, 3086, 3168, 3226, 3388, 3448, 3548, 3630, 3946, 4030, 4105, 4144, 4313, 4322, 4427, 4522, 4525, 4702, 4722, 4893, 4955, 4956, 4982, 5018\n\nO'Donnell, Spec 1564, 1594, 1848, 2652\n\nO'Driscoll, Martha 939, 1144, 4654, 4761\n\nOehman, Rita 1679\n\nOfferman, George, Jr. 1467, 1564, 2947, 3305, 3444, 4729\n\nO'Flynn, Damian 116, 955, 1151, 1263, 1638, 1696, 1762, 1871, 2899, 3104, 3481, 3669, 4620, 4064\n\nO'Flynn, Paddy 1324, 3745\n\nOgg, Sammy 2018, 2757, 4536\n\nOgle, Charles 852, 4092, 4446\n\nO'Hanlon, George 685, 2370\n\nO'Hara, Jim 720, 1005, 4064\n\nO'Hara, Kareen 98\n\nO'Hara, Maureen 319, 552, 807, 1005, 1427, 2079, 2626, 3283, 3320, 3340, 3522, 4064, 4783, 4970\n\nO'Hara, Quinn 1266\n\nO'Hara, Shirley 223, 288, 1253, 2381\n\nO'Hearn, Eileen 1100, 4391\n\nO'Herlihy, Dan 239, 2009, 2447, 2902, 2903, 3204, 5071\n\nO'Herlihy, Michael 1958, 2695, 3070, 3962, 5073, 5074\n\nOhmart, Carol 466, 4136\n\nO'Keefe, Dennis 200, 1150, 1201, 1847, 2108, 5053; _see also_ Flanagan, Bud\n\nO'Kelly, Don 1466\n\nThe Oklahoma Wranglers (Guy, Vic and Skeeter Willis) 1285, 1901\n\nOland, Warner 3469, 3469\n\nOld, John M. _see_ Bava, Mario\n\nOliver, Edna May 747, 826, 1165\n\nOliver, Gordon 3594, 4132\n\nOliver, Guy 686, 852, 1176, 1961, 2351, 2773, 2918, 4092, 4446, 4704\n\nOliver, Susan 1226, 1734, 2506\n\nOliver, Ted 159, 1925, 2020, 2147, 2516, 2790, 2837, 3558, 4121, 4531, 5046\n\nOlmos, Edward James 222, 998\n\nOlmstead, Nelson 586\n\nO'Loughlin, Gerald 3959\n\nOlsen, Merlin 2911, 3990, 4638\n\nOlsen, Moroni 63, 95, 362, 491, 514, 552, 1134, 1594, 2416, 3711, 4184, 4214, 4622, 4742, 5046\n\nOlsen, Ole 1317\n\nOlsen, Tracy 2050, 3338, 4755\n\nOlson, James 849, 2819, 4969\n\nOlson, Nancy 489, 638, 3962\n\nO'Mahoney, Jock 392, 778, 858, 1136, 1309, 1460, 1938, 2080, 2361, 2781, 3358, 3547, 3702, 4053, 4154, 4222, 4296, 4310; _see also_ Mahoney, Jack; Mahoney, Jock\n\nO'Malley, David 1660\n\nO'Malley, J. Pat 727\n\nO'Malley, Pat 73, 304, 377, 379, 389, 453, 695, 844, 858, 1028, 1320, 1458, 1638, 1835, 1925, 2105, 2146, 2273, 2367, 2515, 2654, 2660, 2711, 2885, 2894, 3025, 3447, 3538, 3574, 3594, 3846, 3890, 3916, 3950, 4187, 4228, 4257, 4692, 4743, 4776, 4885, 4886, 5011\n\nO'Moore, Patrick 375, 399, 678, 833, 3417, 4566\n\nO'Neal, Ann 94, 248, 354, 472, 718, 1136, 1925, 1974, 2482, 2782, 3989, 4786, 4804\n\nO'Neal, Charles 1601, 2183, 2677, 3383\n\nO'Neal, Patrick 71, 1208, 4172\n\nO'Neal, Ron 2619\n\nO'Neal, Ryan 4343, 4969\n\nO'Neil, Nance 747\n\nO'Neill, Ella 1367, 1614, 1863, 3087, 3322, 3650\n\nO'Neill, Henry 191, 248, 332, 1564, 1594, 2064, 2610, 3711, 3741, 4672, 4745\n\nO'Neill, Jennifer 3527\n\nOpatoshu, David 748, 1015\n\nOrbison, Roy 1273\n\nOrlandi, Felice 2443, 3686\n\nOrlando, Don 521, 2003, 3015, 3027, 3597, 3640, 4493\n\nOrlebeck, Lester 1513, 2971, 3025, 3111, 3165, 3673, 3805\n\nOrloff, Arthur E. 544, 728, 784, 1074, 1216, 2200, 2669, 3328, 4056, 4381, 4935\n\nOrmond, Ron 368, 794, 896, 940, 1272, 1461, 1939, 2123, 2596, 2748, 2946, 2960, 3312, 3995, 3999, 4085, 4196, 4408, 4709\n\nOrr, Gertrude 630\n\nOrtego, Artie 6, 49, 131, 198, 206, 297, 303, 325, 334, 387, 411, 413, 414, 431, 499, 714, 718, 776, 798, 801, 834, 872, 897, 899, 917, 973, 1057, 1090, 1099, 1146, 1221, 1299, 1320, 1327, 1337, 1459, 1461, 1473, 1496, 1505, 1520, 1529, 1544, 1546, 1554, 1563, 1614, 1626, 1627, 1718, 1724, 1820, 1835, 1933, 1981, 2026, 2108, 2121, 2131, 2164, 2177, 2249, 2255, 2257, 2259, 2291, 2297, 2302, 2366, 2548, 2558, 2664, 2764, 2766, 2777, 2822, 2832, 2858, 2862, 2915, 2946, 2957, 2969, 2988, 3026, 3111, 3115, 3157, 3231, 3242, 3256, 3268, 3279, 3322, 3444, 3485, 3506, 3548, 3613, 3617, 3650, 3658, 3889, 3938, 3943, 3946, 3951, 4009, 4026, 4027, 4094, 4125, 4153, 4204, 4248, 4274, 4290, 4317, 4367, 4445, 4482, 4490, 4514, 4522, 4534, 4556, 4702, 4813, 4822, 4823, 4861, 4876, 4956, 5039, 5075\n\nOrth, Frank 420, 552, 1099, 1480, 1599, 2158, 2164, 2649, 3172, 3431, 4231, 4336\n\nOrth, Marion 3595, 4909\n\nOrtolani, Riz 3420\n\nOsborne, Bud 13, 14, 25, 28, 63, 108, 152, 154, 179, 203, 206, 214, 231, 264, 289, 297, 307, 327, 364, 368, 387, 390, 395, 401, 418, 424, 427, 428, 430, 436, 444, 449, 453, 457, 473, 479, 488, 489, 495, 511, 562, 564, 575, 582, 607, 614, 617, 625, 626, 644, 647, 650, 658, 665, 691, 712, 714, 731, 739, 752, 758, 777, 783, 785, 796, 811, 841, 855, 858, 867, 871, 872, 876, 886, 894, 896, 899, 940, 946, 958, 973, 986, 1001, 1008, 1009, 1018, 1046, 1070, 1073, 1090, 1099, 1100, 1102, 1228, 1260, 1268, 1272, 1274, 1296, 1297, 1308, 1324, 1328, 1362, 1363, 1369, 1377, 1389, 1408, 1426, 1460, 1461, 1463, 1468, 1474, 1488, 1490, 1504, 1516, 1553, 1558, 1573, 1579, 1614, 1673, 1680, 1684, 1687, 1738, 1711, 1724, 1740, 1772, 1789, 1796, 1798, 1800, 1812, 1845, 1861, 1869, 1894, 1939, 1969, 1974, 1981, 1988, 1993, 2074, 2164, 2172, 2175, 2179, 2182, 2204, 2231, 2232, 2233, 2256, 2259, 2273, 2274, 2282, 2283, 2287, 2288, 2293, 2299, 2301, 2342, 2358, 2388, 2399, 2421, 2512, 2524, 2533, 2572, 2573, 2584, 2590, 2595, 2596, 2628, 2633, 2639, 2647, 2689, 2759, 2761, 2776, 2786, 2802, 2804, 2841, 2909, 2915, 2928, 2944, 2947, 2949, 2953, 2957, 2960, 2971, 2974, 2988, 2991, 2992, 2997, 2998, 3001, 3016, 3025, 3026, 3032, 3068, 3101, 3102, 3105, 3110, 3126, 3132, 3135, 3162, 3169, 3173, 3176, 3209, 3316, 3220, 3226, 3236, 3251, 3254, 3264, 3270, 3277, 3280, 3303, 3314, 3322, 3371, 3388, 3427, 3443, 3447, 3450, 3454, 3464, 3475, 3477, 3480, 3494, 3495, 3501, 3504, 3506, 3508, 3545, 3547, 3556, 3557, 3565, 3574, 3575, 3584, 3591, 3626, 3648, 3650, 3651, 3655, 3656, 3664, 3684, 3709, 3728, 3801, 3802, 3844, 3876, 3902, 3904, 3908, 3938, 3940, 3942, 3945, 3949, 3994, 3999, 4001, 4019, 4021, 4024, 4025, 4032, 4037, 4063, 4072, 4089, 4095, 4118, 4132, 4141, 4142, 4153, 4154, 4155, 4186, 4188, 4197, 4207, 4248, 4261, 4273, 4289, 4293, 4294, 4299, 4308, 4363, 4394, 4408, 4423, 4483, 4502, 4513, 4548, 4552, 4554, 4555, 4556, 4594, 4600, 4607, 4619, 4653, 4660, 4683, 4684, 4685, 4692, 4700, 4709, 4729, 4732, 4742, 4747, 4776, 4806, 4812, 4813, 4815, 4817, 4821, 4823, 4826, 4828, 4835, 4840, 4841, 4846, 4850, 4855, 4862, 4883, 4886, 4887, 4894, 4899, 4932, 4952, 4958, 4969, 4981, 5001, 5043, 5056, 5086\n\nOsbourne, Kent 595, 1352\n\nOscar and Elmer (Ed Platt and Lou Fulton) 1751, 2874, 3015\n\nO'Shea, Jack 37, 51, 183, 201, 235, 283, 323, 390, 460, 465, 488, 516, 610, 624, 658, 665, 731, 779, 810, 827, 868, 886, 925, 930, 956, 1042, 1143, 1264, 1345, 1373, 1415, 1438, 1450, 1461, 1462, 1486, 1504, 1543, 1567, 1647, 1730, 1751, 1767, 1805, 1969, 1974, 1980, 1982, 2032, 2066, 2122, 2124, 2184, 2218, 2264, 2266, 2288, 2304, 2355, 2410, 2479, 2530, 2545, 2546, 2597, 2600, 2624, 2893, 2946, 2966, 2967, 2968, 2974, 2995, 2997, 3000, 3012, 3066, 3074, 3077, 3088, 3128, 3132, 3162, 3259, 3322, 3353, 3454, 3475, 3524, 3586, 3599, 3613, 3690, 3693, 3707, 3728, 3820, 3826, 3828, 3890, 3904, 3933, 3944, 3989, 4011, 4016, 4018, 4037, 4051, 4052, 4096, 4124, 4131, 4206, 4409, 4463, 4590, 4579, 4617, 4695, 4733, 4759, 4781, 4855, 4899, 5033, 5043, 5056, 5062, 5103, 5104\n\nO'Shea, Michael 2106\n\nO'Shea, Milo 3070\n\nO'Shea, Oscar 1480, 2716, 3471, 3772, 3912, 4115, 4593\n\nOsmond, Cliff 122, 608, 1660, 2045, 2859, 3223, 4361\n\nOsmond, Donny 4977\n\nOsmond, Marie 1958\n\nThe Osmond Brothers 4878\n\nOssana, Diana 805, 988, 2055, 4169\n\nOsterhage, Jeffrey 959, 2333, 3661, 3798, 4577\n\nOsterloh, Robert 1071, 1136, 1166, 1402, 2046, 2048, 2563, 2789, 3023, 3779, 4123, 4791\n\nO'Sullivan, Maureen 3558, 4235, 4943\n\nOsuna, Gloria 1286, 3117\n\nOswald, Gerd 501, 1206, 1482, 2486, 4689\n\nOtho, Henry 387, 1221, 2012, 2164, 2403, 2648, 2998, 3172, 3177, 3899, 4513, 4534, 4697\n\nO'Toole, Annette 2090\n\nO'Toole, Peter 3729\n\nOuspenskaya, Maria 5033\n\nOverman, Lynne 1392, 3414, 3899, 4061, 4665\n\nOverton, Frank 3149, 4579\n\nOwen, Beverly 567\n\nOwen, Garry 630, 4057\n\nOwen, Granville 1099\n\nOwen, Michael 1024\n\nOwen, Reginald 626, 3095, 3316, 3558, 3606\n\nOwen, Tudor 165, 185, 841, 1433, 1466, 1945, 2074, 2289, 2677, 2829, 2870, 4582, 5025\n\nOwens, Patricia 373, 567, 1695, 2246, 4333\n\nOwsley, Monroe 1591, 4941, 5054\n\nThe Pacemakers 3434\n\nPacheco, Jose, and His Continental Orchestra 4022\n\nPadden, Sarah 368, 930, 1100, 1175, 1390, 1463, 1540, 1573, 1639, 1744, 1821, 2016, 2251, 2344, 2421, 2552, 2597, 3669, 2971, 3247, 3261, 3264, 3457, 3480, 4017, 4503, 4686, 4932, 4969, 4987\n\nPadget, Alfred 2603\n\nPadilla, Manuel 373, 919, 2508\n\nPadilla, Robert 182, 1411, 1946\n\nPadilla, Ruben 43\n\nPadjan, Jack 1751, 2004, 3015\n\nPagano, Ernest 1450\n\nPagano, Jo 25, 2311, 5044\n\nPage, Bradley 210, 1324, 2128, 2940, 4187\n\nPage, Dorothy 3426, 3920\n\nPage, Gale 1820\n\nPage, Geraldine 1921, 2077\n\nPage, Joy 828, 4457\n\nPaget, Debra 1676, 2198, 2462, 3536, 3779, 4907\n\nPaget, Kevin Jackson 400, 1410, 4779; _see also_ Ferroni, Giorgio\n\nPaget, Robert 1163\n\nPaige, Janis 718, 2810, 2802, 5075\n\nPaige, Robert 642, 714, 1444, 1827, 3334, 4069\n\nPaige, Satchel 5024\n\nPaiva, Nestor 53, 67, 214, 420, 603, 804, 1063, 1423, 1466, 1567, 1735, 2034, 2128, 2314, 2815, 3020, 3247, 3414, 3544, 3562, 3711, 3871, 4056, 4058, 4069, 4240, 4392, 4419, 4751, 4940, 4973\n\nPal, George 3783\n\nPalance, Cody 5067\n\nPalance, Jack 166, 532, 558, 709, 760, 761, 770, 1075, 1583, 1589, 2011, 2089, 2143, 2229, 2431, 2629, 2643, 2685, 2859, 3180, 3808, 4246, 4801, 5067\n\nPallette, Eugene 441, 1176, 1392, 1685, 1982, 2081, 2586, 3429, 3710, 3899, 4744, 4949\n\nPalmara, Mimmo 1062, 4628; _see also_ Palmer, Dick\n\nPalmer, Betsy 4435\n\nPalmer, Dick 2053, 2441, 3033, 3352, 4262; _see also_ Palmara, Mimmo\n\nPalmer, Don 991\n\nPalmer, Gregg 22, 319, 741, 808, 1356, 1578, 1670, 1942, 2720, 3208, 3404, 3663, 3748, 3816, 3847, 3962, 4244, 4638, 5030; _see also_ Lee, Palmer\n\nPalmer, Maria 4212\n\nPalmer, Patricia 3623, 3697\n\nPalmer, Peter 2322\n\nPalmer, Shirley 3932, 3994\n\nPalmer, Stuart 5054\n\nPalmer, Tex 70, 140, 150, 244, 335, 343, 344, 369, 382, 423, 443, 481, 542, 611, 724, 790, 853, 889, 891, 946, 974, 1017, 1051, 1057, 1162, 1241, 1281, 1297, 1301, 1322, 1327, 1333, 1370, 1389, 1468, 1486, 1494, 1587, 1604, 1662, 1681, 1683, 1732, 1798, 1803, 1812, 1898, 1969, 2003, 2035, 2107, 2114, 2129, 2151, 2166, 2172, 2203, 2220, 2256, 2263, 2266, 2268, 2271, 2274, 2285, 2299, 2300, 2306, 2365, 2407, 2411, 2412, 2418, 2420, 2473, 2479, 2548, 2572, 2590, 2740, 2763, 2740, 2763, 2785, 2847, 2948, 2951, 2956, 2957, 2965, 3018, 3027, 3036, 3062, 3073, 3077, 3089, 3107, 3153, 3160, 3232, 3237, 3242, 3256, 3264, 3270, 3272, 3278, 3279, 3291, 3303, 3332, 3359, 3388, 3443, 3446, 3448, 3451, 3476, 3568, 3581, 3591, 3592, 3597, 3653, 3668, 3884, 3888, 3894, 3913, 3968, 4067, 4104, 4112, 4125, 4128, 4130, 4294, 4303, 4305, 4317, 4319, 4393, 4437, 4458, 4482, 4501, 4502, 4512, 4549, 4555, 4587, 4589, 4606, 4660, 4701, 4729, 4809, 4822, 4823, 4844, 4848, 4854, 4855, 4884, 4953, 4957, 4981, 5001, 5027, 5043, 5061\n\nThe Pals of the Golden West 477, 3254, 3626, 3889, 4096\n\nPaluzzi, Luciana 1393, 3152\n\nPanama, Norman 632, 2025, 3544, 4528\n\nPangborn, Franklin 1361, 4170\n\nPanzer, Paul 203, 688, 1804, 1884, 2164, 2349, 2606, 2681, 3538, 3766, 4115, 4697, 5075\n\nParamore, Edward E., Jr. 441, 1300, 2552, 2864, 2918, 3573, 3710, 4358, 4456, 4593, 4744\n\nParfrey, Woodrow 526, 2788, 2859, 2950, 3687, 3688, 3737, 4345\n\nParis, Jerry 1243, 1433, 2813, 4749\n\nPark, Post 37, 235, 379, 442, 453, 464, 607, 664, 1458, 1672, 1774, 2006, 2027, 2397, 2399, 2403, 3002, 3088, 3107, 3150, 3484, 3814, 3820, 4011, 4233, 4363, 4483, 4496, 4600, 4734, 5022, 5103\n\nParker, Andy, and The Plainsmen 364, 711, 869, 1803, 3799, 4327, 4437, 4466, 4858\n\nParker, Carol 1155\n\nParker, Cecilia 1470, 1518, 1677, 1904, 1928, 2558, 2654, 2741, 2756, 2923, 2936, 3239, 3241, 3451, 3584, 4454, 4484\n\nParker, Eddie (Ed\/Edwin) 27, 28, 42, 250, 287, 352, 364, 458, 848, 951, 970, 979, 984, 987, 1073, 1080, 1263, 1362, 1412, 1447, 1543, 1545, 1574, 1587, 1803, 1806, 1936, 2035, 2105, 2125, 2282, 2297, 2396, 2403, 2424, 2457, 2479, 2544, 2648, 2711, 2727, 2766, 2785, 2837, 3088, 3120, 3151, 3220, 3222, 3234, 3242, 3254, 3261, 3363, 3415, 3464, 3799, 3890, 3901, 3902, 4011, 4053, 4073, 4125, 4155, 4240, 4278, 4336, 4409, 4419, 4427, 4482, 4514, 4550, 4646, 4691, 4729, 4742, 4823, 4854, 4868, 4888, 5001, 5005\n\nParker, Eleanor 1234, 1913, 2118, 2575\n\nParker, Fess 53, 484, 768, 953, 967, 970, 1773, 2025, 2351, 2887, 3973, 4080, 4329, 4390, 4674, 4856\n\nParker, Fred 6, 129, 156, 262, 337, 438, 443, 824, 848, 974, 1057, 1178, 1228, 1301, 1333, 1455, 1468, 1474, 1576, 1687, 2256, 2300, 2358, 2785, 3036, 3073, 3159, 3266, 3288, 3293, 3332, 3359, 3528, 3545, 3709, 3949, 4000, 4238, 4315, 4427, 4573, 4609, 4698, 4832, 4854\n\nParker, Jean 114, 254, 1565, 1704, 2152, 2302, 3045, 3593, 3774, 4309, 4349, 4470\n\nParker, Netta 3666\n\nParker, Norton S. 231, 460, 812, 847, 927, 1308, 2235, 2948, 3524, 3937, 4364, 4848, 5059\n\nParker, Willard 105, 229, 599, 1099, 1635, 2426, 2748, 3344, 3355, 3701, 4714, 4755, 4769, 5070\n\nParkhill, Forbes 2812\n\nParkins, Barbara 2265\n\nParkinson, Cliff 230, 395, 446, 461, 658, 793, 836, 1934, 1978, 2250, 2298, 2451, 2826, 3088, 3247, 3289, 3458, 3693, 3704, 3709, 4188, 4322, 4657, 4893, 5103\n\nParks, Eddie 318, 584, 1214, 2161, 3128, 3958\n\nParks, Gordon 4346\n\nParks, Larry 1032, 2827, 3355\n\nParks, Michael 169, 333, 2196, 2657, 3378, 4161\n\nParks, Nanette 4305\n\nParnell, Emory 53, 214, 259, 271, 277, 630, 1407, 1601, 1923, 2159, 2292, 2344, 2452, 2556, 2618, 2831, 2856, 2937, 2943, 3040, 3567, 3611, 3803, 4015, 4171, 4231, 4488, 4618, 5048, 5066\n\nParolini, Gianfranco 20, 1966, 2053\n\nParrish, Helen 328, 1974, 2994, 3209, 4206, 5013\n\nParrish, John 472, 990, 1420, 3447, 3682, 4662\n\nParrish, Leslie 3959\n\nParrish, Robert 3671, 3696, 4471, 5024\n\nParrott, Charles 2136\n\nParrott, James 4798\n\nParry, Harvey 286, 385, 1092, 2026, 2859, 3009, 3891\n\nParsons, Estelle 1663\n\nParsons, Harriet 2782\n\nParsons, Lindsley 137, 1057, 1465, 2300, 2548, 3036, 3133, 3242, 3256, 3589, 3680, 4482\n\nParsons, Milton 318, 654, 1567, 1638, 2204, 4942, 3593, 3875\n\nParton, Reg 111, 114, 132, 189, 373, 585, 1406, 1423, 1892, 1940, 2050, 2192, 2267, 2511, 3338, 3950, 4755, 5065\n\nPartos, Frank 1854\n\nPascal, Ernest 648, 1251, 2016, 2207, 4650\n\nPasco, Richard 3933\n\nPataki, Michael 256, 538, 628, 1879\n\nPate, Michael 22, 350, 378, 603, 639, 913, 1644, 1921, 2302, 2492, 2501, 2626, 2695, 2871, 3363, 3387, 3438, 3777, 3792, 4234, 4768, 4830, 5097\n\nPaton, Stuart 1975, 2702, 2745, 2794, 3876, 4396\n\nPaton, Tony 2132, 2830, 3870, 4612\n\nPatric, Jason 1534\n\nPatrick, Butch 1206\n\nPatrick, Dorothy 491, 1050, 2955, 3061, 3541, 3724, 4392, 4644\n\nPatrick, Gail 2138, 2732, 3128, 3362, 4442, 4776\n\nPatrick, Lee 432, 944, 1136, 1266, 3783, 3917, 4227\n\nPatten, Luana 1913, 2043, 2052, 2381, 3838, 4337, 4387\n\nPatterson, Elizabeth 826, 2422, 2929\n\nPatterson, Hank 1, 42, 282, 287, 604, 637, 784, 979, 1040, 1215, 1235, 1704, 1709, 1883, 1992, 2018, 2022, 2400, 2426, 2806, 2813, 2857, 2942, 3061, 3074, 3093, 3132, 3310, 3344, 3377, 3423, 3447, 3563, 3744, 3892, 3936, 4060, 4083, 4141, 4240, 4267, 4357, 4642, 4929\n\nPatterson, John 471, 1394, 1987\n\nPatterson, Neva 3950\n\nPatterson, Shirley 364, 482, 1157, 1792, 2269, 2827, 3466, 3505, 3507, 4131, 4299, 4590, 4736; _see also_ Smith, Shawn\n\nPatton, Bill 8, 156, 261, 262, 270, 308, 431, 493, 862, 923, 975, 1055, 1258, 1329, 1334, 1374, 1455, 1545, 1626, 1687, 1730, 1928, 1993, 2074, 2186, 2263, 2274, 2378, 2599, 2954, 3026, 3099, 3164, 3173, 3194, 3289, 3293, 3650, 3697, 3888, 3970, 4138, 4165, 4256, 4373, 4602, 4796, 4832, 4863, 4912, 5004\n\nPatton, Virginia 357, 648\n\nPaul, Gloria 1382, 4625, 4628\n\nPaul, Oscar 2501, 2503, 3755, 3757\n\nPaull, Morgan 280, 594, 2196\n\nPavan, Marisa 919, 1161\n\nPavone, Rita 3532\n\nPawley, Edward (Ed) 1679, 3538, 3600, 3615\n\nPawley, William 1070, 1378, 2864, 3163, 3374, 3558, 3953, 4312, 4697, 4806, 5077\n\nPawnee Bill, Jr. 12; _see also_ Wells, Ted\n\nPaxton, Bill 4453\n\nPayne, John 196, 381, 1201, 1214, 2127, 3053, 3234, 3299, 3542, 3705, 3897, 4258, 4337, 4714\n\nPayne, Laurence 3915\n\nPayne, Sally 323, 2032, 2159, 2514, 2529, 2778, 3330, 3600, 3827, 4868, 5059\n\nPayson, Ed 3450, 5037\n\nPayton, Barbara 1635, 2919, 3637\n\nPazzafini, Nello 2532, 3023, 4562, 4779, 4903, 5101\n\nPearce, Adele 4685, 4037; _see also_ Blake, Pamela\n\nPearce, Alice 2157\n\nPearson, Jesse 22\n\nPeary, Harold (Hal) 4417, 4860\n\nPeck, Gregory 313, 346, 502, 1187, 1704, 1945, 2489, 2876, 2919, 3837, 4113, 4970, 5026, 4045\n\nPeckinpah, Sam 221, 517, 736, 1005, 1225, 1575, 2069, 2162, 2501, 3055, 3435, 4738, 4934\n\nPeeples, Samuel A. 22, 2322\n\nPegg, Vester 471, 522, 546, 790, 975, 1394, 2722, 3179, 3827, 4100, 4145, 4245, 4352, 4395, 4649, 4806, 4925\n\nPeil, Ed, Sr. 174, 341, 411, 442, 452, 497, 499, 644, 682, 683, 767, 777, 789, 781, 799, 801, 810, 824, 834, 881, 952, 1008, 1083, 1088, 1145, 1291, 1301, 1370, 1389, 1475, 1486, 1517, 1553, 1627, 1636, 1751, 1859, 1972, 2004, 2032, 2082, 2084, 2111, 2124, 2233, 2342, 2349, 2388, 2411, 2412, 2413, 2414, 2514, 2540, 2548, 2599, 2802, 2850, 2909, 2943, 3015, 3075, 3138, 3194, 3316, 3289, 3330, 3401, 3424, 3448, 3452, 3482, 3520, 3565, 3648, 3669, 3711, 3801, 3859, 3893, 3912, 3920, 3924, 3949, 4077, 4186, 4206, 4303, 4308, 4236, 4378, 4410, 4458, 4531, 4609, 4660, 4665, 4690, 4691, 4810, 4832, 4899, 4944, 4952, 5033, 5043\n\nPellicar, Pilar 980, 2754, 3901, 5100\n\nPembroke, George 842, 976, 956, 1658, 1838, 2981, 3839, 3857\n\nPena, Pasquale Garcia 50, 232, 269, 591, 705, 707, 759, 2041, 2579, 2707, 3124, 3765, 4177, 4209, 4418, 4464, 4538, 5091, 5102\n\nPendleton, Gaylord (Steve) 392, 433, 544, 1336, 1635, 1640, 1713, 2349, 2599, 2641, 2895, 2899, 3093, 3262, 3379, 3386, 3414, 3522, 3586, 4036, 4198, 4491, 4519, 4564, 4618, 4668, 4871, 5062\n\nPendleton, Nat 1020, 1252, 1563, 1839, 2207, 2790, 2837, 4254\n\nPenn, Arthur 2314, 2373, 2666\n\nPenn, Christopher 3019\n\nPenn, Leo 3686, 3723\n\nPenn, Leonard 250, 850, 990, 1161, 1259, 1568, 1571, 1713, 1935, 2260, 2602, 2944, 2960, 3050, 3262, 3900, 3943, 4043, 4105, 4781\n\nPennell, Larry 1265, 1366, 2058, 3779\n\nPenner, Joe 1967\n\nPennick, Jack 43, 1153, 1165, 1313, 1395, 1591, 1839, 1847, 1937, 2160, 2192, 2561, 2701, 2718, 3179, 3359, 3522, 3567, 3606, 3752, 3776, 3814, 4100, 4360, 4626, 4635, 4665, 4744, 4797, 4851, 4940, 4950, 5072\n\nPennington, Ann 4318\n\nPenny, Hank 392, 1460, 1815, 2419\n\nPeppard, George 506, 641, 1226, 1913, 1945, 2911, 3612\n\nPepper, Barbara 642, 797, 2940, 3374, 3681, 4272\n\nPepper, Jack 3890\n\nPeralta, Gregorio 4150\n\nPercy, Eileen 3536, 3072, 4926\n\nPerez, Paul 1367, 1634, 2907, 3971, 4609, 4650\n\nPerkins, Anthony 1434, 2348, 2431, 2468, 4435\n\nPerkins, Gil 759, 1140, 1395, 2035, 4011, 4247, 4599\n\nPerkins, Millie 3428, 3845\n\nPerkins, Voltaire 1265, 3567, 3701, 4714\n\nPerreau, Gigi 1256, 2782, 4943\n\nPerreau, Janine 2783, 3339\n\nPerrin, Jack 108, 145, 229, 305, 336, 457, 516, 592, 593, 597, 776, 891, 934, 1047, 1196, 1200, 1214, 1490, 1586, 1626, 1674, 1760, 1839, 1959, 2084, 2104, 2172, 2205, 2302, 2362, 2422, 2533, 2670, 2677, 2745, 2790, 2822, 2921, 3015, 3018, 3914, 3299, 3303, 3328, 3340, 3444, 3450, 3487, 3779, 3804, 3844, 3869, 4004, 4016, 4252, 4280, 4284, 4298, 4312, 4329, 4397, 4521, 4524, 4536, 4572, 4803, 4804, 4815, 4842, 4877, 4972, 5014; _see also_ Gable, Jack; Terry, Richard\n\nPerrin, Nat 3609\n\nPerrin, Victor (Vic) 3500, 4343\n\nPerrine, Valerie 426, 1219\n\nPerry, Bob 1, 642\n\nPerry, Eleanor 2560\n\nPerry, Frank 1121, 3252\n\nPerry, Joan 1488, 1835, 2730\n\nPerry, Linda 611, 2164\n\nPerry, Luke 2055\n\nPerry, Luke 3130\n\nPerry, Pasquale 335, 338, 340, 382, 414, 442, 621, 776, 777, 853, 872, 984, 986, 1162, 1246, 1288, 1345, 1456, 1488, 1489, 1677, 1684, 1728, 1920, 1931, 1952, 2032, 2035, 2121, 2249, 2276, 2294, 2299, 2300, 2408, 2410, 2411, 2600, 2737, 2850, 2867, 3015, 3165, 3276, 3230, 3267, 3448, 3549, 3574, 3603, 3649, 3675, 3805, 3983, 4011, 4037, 4083, 4206, 4394, 4483, 4617, 4643, 4649, 4702, 4732, 4733, 4763\n\nPerry, Roger 673, 1832\n\nPerryman, Lloyd 110, 287, 386, 576, 615, 623, 682, 799, 1192, 1240, 1246, 1818, 1836, 1964, 2124, 2272, 2330, 2514, 2530, 2540, 2727, 2806, 2828, 2886, 2893, 2973, 2975, 2983, 3100, 3270, 3303, 3330, 3449, 3483, 3520, 3522, 3585, 3600, 3706, 3904, 4108, 4037, 4042, 4051, 4077, 4156, 4202, 4206, 4314, 4315, 4401, 4410, 4608, 4646, 4764, 4806, 4810, 4817, 4834\n\nPerschy, Maria 4237\n\nPersoff, Nehemiah 81, 209, 808, 919, 947, 981\n\nPeters, Brock 3, 2501, 2653\n\nPeters, House 2885, 4139\n\nPeters, House, Jr. 28, 371, 380, 433, 456, 664, 841, 850, 858, 933, 951, 1072, 1268, 1487, 1523, 1724, 2074, 2085, 2370, 2521, 2539, 2622, 2856, 2857, 2878, 2885, 2946, 2988, 2996, 3132, 3187, 3305, 3356, 3518, 3611, 3828, 3999, 4076, 4266, 5357, 4641, 4754, 5005, 5039\n\nPeters, Jean 100, 520, 2372, 4751\n\nPeters, Kelly Jean 1007, 3136\n\nPeters, Ralph 15, 129, 211, 283, 304, 343, 648, 797, 1018, 1082, 1192, 1370, 1558, 1664, 1925, 1944, 2124, 2407, 2572, 2789, 3815, 2869, 2968, 2977, 3109, 3425, 3448, 3626, 3709, 3859, 3944, 3958, 4006, 4152, 4199, 4225, 4286, 4508, 4549, 4598, 4690, 4882, 5037\n\nPeters, Susan 3711\n\nPeters, Werner 358\n\nPetersen, Paul 979, 2059, 3991, 4431\n\nPetersen, Stewart 41, 3141, 3778, 4878\n\nPeterson, Dorothy 648, 2937, 3429\n\nPetersson, Harald G. 106, 1069, 2219, 4380\n\nPetit, Pascale 1342\n\nPetracca, Joseph 3185, 3186\n\nPetrie, Daniel 1663, 1834, 3854\n\nPetrie, George 1809, 1948\n\nPetrie, Howard 450, 484, 677, 1257, 2046, 2624, 3143, 3219, 3376, 3572, 3780, 4428, 4435, 4962, 5020\n\nPetro, Hortense 4455\n\nPetroni, Guilio 396, 1016, 3955\n\nPettet, Joanna 407, 902, 3106\n\nPettyjohn, Angelique 1832\n\nPevney, Joseph 185, 1428, 2161, 2795, 3133\n\nPeyser, John 1425, 2595, 3133\n\nPflug, Jo Ann 675, 3749\n\nPhelps, Lee 64, 248, 274, 478, 480, 602, 842, 1073, 1093, 1105, 1134, 1187, 1378, 1392, 1436, 1526, 1568, 1599, 1626, 1638, 1680, 1785, 1870, 1888, 1925, 2315, 2830, 2856, 2941, 3022, 3288, 3323, 3447, 3480, 3486, 3561, 3606, 3697, 3202, 3969, 4086, 4178, 4225, 4257, 4431, 4494, 4803, 4845, 4940, 5032\n\nPhelps, Tex 297, 300, 327, 337, 411, 431, 606, 800, 974, 1328, 1556, 1816, 1855, 1859, 1898, 1993, 2027, 2129, 2239, 2270, 2297, 2404, 2408, 2432, 2479, 2526, 2529, 2548, 2766, 2785, 2908, 3081, 3160, 3242, 3256, 3272, 3314, 3374, 3479, 3486, 3548, 3582, 3680, 3704, 3709, 3889, 3944, 3968, 4226, 4317, 4364, 4427, 4445, 4660, 4698, 4958, 4982\n\nThe Phelps Brothers 427, 1679, 3353, 4572, 4882\n\nPhilbrook, James 218, 1441, 3997, 4721, 4791, 4973\n\nThe Philharmonic Trio 4621\n\nPhillipp, Harald 1761, 3246\n\nPhillips, Barney 2265, 3641\n\nPhillips, Doris 4009\n\nPhillips, Ed (Eddie\/Edward) 77, 327, 338, 414, 431, 898, 928, 1389, 1554, 1758, 2014, 2279, 3083, 3967, 4106, 4323, 4379, 4522, 4722\n\nPhillips, Jean 2972, 4419\n\nPhillips, Lou Diamond 90, 931, 2406, 2463, 4505, 4545, 5067, 5069\n\nPhillips, Michelle 2199\n\nPhillips, William (Bill) 42, 149, 211, 282, 318, 439, 524, 555, 562, 691, 1092, 1093, 1274, 1347, 1409, 1551, 1658, 1792, 1877, 2037, 2198, 2287, 2515, 2565, 2748, 2834, 2919, 2986, 3305, 3404, 3724, 3750, 4057, 4111, 4302, 4461, 4740\n\nPhipps, William (Bill) 690, 1071, 1072, 1235, 1403, 1703, 1985, 1989, 1998, 3219, 3305, 3441, 3608, 3966, 4132, 4264, 4620, 4712, 5030, 4058\n\nPhoenix, River 3883\n\nPiaget, Paul 1417, 1858, 3784, 3796\n\nPiazza, Ben 1772, 2807\n\nPicerni, Paul 484, 1408, 1485, 2174, 3397, 3672, 3736\n\nPichel, Irving 1827, 1883, 1949, 2064, 3702, 3814\n\nPickard, John 170, 211, 349, 367, 375, 524, 562, 700, 704, 741, 833, 1314, 1409, 1640, 1689, 1704, 1772, 1844, 1922, 2087, 2096, 2292, 2372, 2400, 2617, 2693, 2851, 2981, 3417, 3607, 3769, 3779, 3840, 3849, 3978, 4099, 4478, 4486, 4576, 4770, 4784\n\nThe Pickard Family 3291\n\nPickens, Slim 121, 239, 453, 489, 743, 796, 841, 883, 1062, 1077, 1086, 1142, 1206, 1238, 1245, 1575, 1637, 1663, 1666, 1742, 1808, 1899, 1924, 2006, 2187, 2200, 2322, 2324, 2501, 2880, 2882, 2901, 2938, 3055, 3091, 3142, 3252, 3328, 3572, 3612, 3661, 3687, 3705, 3731, 3793, 3803, 3815, 4056, 4068, 4101, 4151, 4217, 4387, 449, 4457, 4582, 4895, 4988\n\nPickford, Jack 1431, 2381\n\nPickford, Mary 1511, 3245, 3766\n\nPickrell, June 3001, 4304\n\nPidgeon, Walter 322, 960, 1571\n\nThe Pied Pipers 1901, 3410\n\nPierce, Charles B. 172, 1623, 1807, 3662, 4994, 5007\n\nPierce, Charles B., Jr. (Chuck) 1807, 4994, 4007\n\nPierce, James (Jim) 15, 138, 318, 680, 1519, 1591, 1604, 1996, 3235, 3237, 3375, 4129, 4312, 4427, 4868, 5001, 5104\n\nPierce, Maggie 679, 1273\n\nPierce, Preston 5065\n\nPierce, Sam 473, 3038, 3266\n\nPierce, Webb 559\n\nPierlot, Francis 832, 1175\n\nPierotti, Piero 403, 5094, 5101\n\nPierson, Carl L. 2785, 3036, 3926\n\nPiggot, Tempe 2554\n\nPinal, Silvia 1731\n\nPine, Robert 122, 267, 1727, 2059, 2772, 2906\n\nPine, William H. 47, 3636, 4979\n\nPink, Sidney 742, 4237\n\nPinsent, Gordon 2148, 3872\n\nPinson, Allen 698, 841, 1036, 2074, 2192, 2289, 2609, 2693, 2842, 2910, 3751, 4228, 4478, 4582\n\nPitt, Brad 169\n\nPitti, Carl 229, 1673, 1765, 1880, 2292\n\nPittman, Montgomery 2676, 3960\n\nPitts, ZaSu 1039, 1083, 2458, 3537, 3634, 4706, 4933\n\nPizor, William F. 3647\n\nPlatt, Edward 567, 1719, 2209, 2513, 2926, 3185, 3363\n\nPlatt, Louise 4061, 4100\n\nPlatt, Marc 2853, 3780\n\nPleasence, Donald 1765, 1828, 3987, 4988\n\nPleshette, Suzanne 26, 1112, 2087, 2779, 4210\n\nPlowman, Melinda 339, 3006\n\nPlowright, Hilda 3049\n\nPlues, George 414, 549, 800, 810, 956, 1272, 1309, 1367, 1595, 1728, 1932, 1993, 2132, 2250, 2270, 2399, 2404, 2420, 2730, 2786, 2826, 2884, 2927, 3005, 3027, 3087, 3160, 3187, 3220, 3279, 3289, 3424, 2478, 3525, 3549, 3834, 4104, 4204, 4261, 4365, 4479, 4617, 4645, 4849, 4865, 5001, 5018, 5104\n\nPlummer, Christopher 4993\n\nPlummer (Plumer), Rose 37, 327, 464, 479, 493, 494, 573, 634, 683, 695, 793, 810, 961, 1090, 1187, 1296, 1298, 1307, 1681, 1749, 2016, 2111, 2280, 2281, 2299, 2363, 2552, 2590, 2847, 2954, 2957, 2999, 3062, 3080, 3322, 3391, 3427, 3626, 3707, 3834, 3927, 3989, 4005, 4108, 4190, 4226, 4365, 4389, 4490, 4606, 4617, 4695, 4701, 5001, 5043\n\nPlympton, George 140, 244, 342, 390, 397, 423, 689, 897, 1088, 1137, 1193, 1281, 1367, 1503, 1505, 1614, 1681, 1683, 1711, 1779, 1863, 1965, 1993, 2192, 2218, 2306, 2645, 2915, 2927, 2977, 3044, 3068, 3087, 3173, 3260, 3278, 3322, 3332, 3450, 3509, 3547, 3650, 3977, 4001, 4030, 4142, 4324, 4326, 4382, 4501, 4553, 4581, 4729, 4887, 5001\n\nPoe, Edgar Allan 2024\n\nPoe, James 2240\n\nPoff, Lon 3694, 4284\n\nPogue, Ken 900, 1650, 2015, 2055, 3872\n\nPoitier, Sidney 540, 1183\n\nPoland, Joseph 364, 643, 682, 956, 997, 1268, 2128, 2132, 2135, 2271, 2299, 2794, 2874, 2891, 3003, 3259, 3681, 3903, 4095, 4292, 4301, 4558, 4995, 5103\n\nPollack, Sydney 1219, 2028\n\nPollard, Bud 2449\n\nPollard, Michael J. 1107, 1424, 2326\n\nPollard, Snub 137, 187, 271, 422, 718, 1256, 1285, 1465, 1812, 1900, 2111, 2177, 2280, 2386, 2481, 2645, 2712, 3476, 3569, 3589, 3706, 3851, 3907, 3913, 3928, 3953, 4019, 4128, 4129, 4279, 4304, 4685, 4722, 4872\n\nPollexfen, Jack 652, 1353\n\nPolo, Eddie 1028, 2250, 2994, 4006\n\nPolonsky, Abraham 4249\n\nPond, Elmer 3331; _see also_ Clifton, Elmer\n\nPoole, Roy 3993\n\nPop, Iggy 992\n\nPope, Bud 240, 438, 458, 725, 729, 862, 1301, 1587, 1730, 2003, 2129, 2280, 2299, 2432, 2456, 2522, 2542, 2640, 2743, 2933, 2947, 2954, 3018, 3268, 3303, 3486, 3498, 3531, 3589, 3949, 4007, 4256, 4282, 4307, 4317, 4406, 4459, 4698, 4711, 4728, 4732, 4955, 5040\n\nPorcasi, Paul 442, 1865, 3606, 4650\n\nPorter, Don 895, 2720, 3722, 4929\n\nPorter, Dorothy 161\n\nPorter, Ed 2954, 3268\n\nPorter, Gene Stratton 1431\n\nPorter, Jean 596, 1512, 1821, 1915, 3315, 3694\n\nPorter, Lew 773, 2407, 2408, 2411\n\nPost, Charles A. 3927\n\nPost, Ted 506, 1769, 2339, 3061, 4102, 5082\n\nPotel, Victor 145, 310, 451, 625, 642, 739, 862, 1360, 1446, 1564, 1567, 1570, 1571, 1588, 1785, 1831, 1836, 2159, 2203, 2344, 2689, 2890, 2891, 3048, 3247, 3410, 3431, 3626, 3634, 4025, 4093, 4121, 4357, 4358, 4389, 4498, 4519, 4602, 4744, 4808, 4840, 4891, 5043, 5046\n\nPotter, H.C. 866\n\nPotts, Annie 860\n\nPotts, Cliff 280, 1250, 229, 2780, 2917, 3993\n\nPowell, Dick 874, 2705, 3499, 4069\n\nPowell, Dinny 4472\n\nPowell, Jane 3780\n\nPowell, Lee 70, 1394, 2399, 2415, 3164, 3232, 3371, 3592, 4302, 4553, 4589\n\nPowell, Russ 159, 246, 328, 626, 866, 1898, 2199, 2249, 2288, 2741, 3171, 3374, 3609, 3709, 4000, 4215, 4442, 4494, 4868\n\nPowell, William 2013, 2773, 4536\n\nPower, Tyrone (Jr.) 514, 2031, 2586, 2664, 1832, 3143, 3290, 3374, 4671\n\nPower, Tyrone (Sr.) 327, 504, 4275\n\nPowers, Hunt 999\n\nPowers, Jill 1617, 2614\n\nPowers, Lucille 4289, 4614\n\nPowers, Mala 3219, 3608, 3863, 4141, 4059\n\nPowers, Richard 402, 534, 1073, 1103, 1627, 1988, 2362, 2963, 3383, 4140, 4285, 4386, 4653, 4841, 4860, 4951; _see also_ Duryea, George; Keene, Tom\n\nPowers, Stefanie 1786, 2379, 2498, 2626, 2793, 3840, 4101\n\nPowers, Tom 89, 91, 1039, 1775, 1936, 2226, 2591, 2781, 4062, 4132, 4252\n\nPrada, Jose Maria 3030, 4738\n\nPrather, Joan 1098\n\nPrather, Lee 471, 576, 1228, 2119, 2153, 2973, 3003, 3520, 4257, 4313, 4315\n\nPratt, Judson 720, 1112, 1701, 1937, 3776, 4255\n\nPratt, Purnell B. 797, 813, 1394, 1883, 3896\n\nPreisser, June 4687\n\nPreminger, Otto 3535\n\nPrentiss, Paula 3007\n\nPrescott, Guy 1402, 3200, 3219, 3849, 4234\n\nPresle, Micheline 2326\n\nPresley, Elvis 704, 1371, 2462, 4123, 4412\n\nPresnell, Harve 1575, 3009, 4670, 4867\n\nPreston, John 846, 4424\n\nPreston, Kelly 730\n\nPreston, Lew, and His Ranch Hands 3171\n\nPreston, Robert 293, 402, 740, 1249, 1945, 2069, 2159, 2192, 2725, 2831, 3772, 4585, 4665, 4890\n\nPreston, Wayde 2441, 2510, 3716, 4444\n\nPrevost, Francoise 2049\n\nPrevost, Marie 622, 660\n\nPrice, Hal 13, 70, 131, 140, 150, 277, 334, 336, 341, 369, 395, 510, 573, 631, 660, 666, 683, 689, 781, 826, 880, 897, 925, 946, 1052, 1228, 1338, 1363, 1389, 1439, 1445, 1473, 1474, 1486, 1494, 1504, 1604, 1789, 1835, 1847, 1859, 1904, 1915, 1916, 1980, 2068, 2251, 2259, 2407, 2408, 2410, 2427, 2529, 2542, 2600, 2634, 2647, 2673, 2730, 2786, 2802, 2856, 2937, 2957, 2969, 2993, 3046, 3052, 3088, 3105, 3159, 3163, 3187, 3230, 3232, 3264, 3275, 3303, 3385, 3424, 3427, 3451, 3453, 3492, 3565, 3569, 3614, 3648, 3680, 3761, 3820, 3825, 3866, 3893, 3922, 4032, 4104, 4130, 4155, 4181, 4188, 4189, 4197, 4313, 4393, 4573, 4581, 4606, 4695, 4696, 4759, 4776, 4822, 4826, 4835, 4838, 4853, 4883, 4952, 4982, 4987, 5035\n\nPrice, Kate 3797\n\nPrice, Nancy 1950\n\nPrice, Sherwood 2519, 3746\n\nPrice, Stanley 2, 51, 151, 251, 341, 349, 352, 369, 380, 643, 716, 794, 896, 936, 940, 997, 1020, 1063, 1156, 1196, 1268, 1272, 1324, 1338, 1415, 1444, 1447, 1453, 1591, 1604, 1616, 1627, 1815, 1843, 1844, 1890, 1919, 1939, 1960, 2096, 2128, 2262, 2294, 2349, 2418, 2512, 2539, 2591, 2596, 2660, 2680, 2798, 2863, 2909, 2966, 3068, 3229, 3264, 3267, 3298, 3326, 3426, 3511, 3599, 3828, 3920, 4011, 4095, 4105, 4110, 4124, 4299, 4301, 4312, 4462, 4467, 4617, 4620, 4731, 4754, 4778, 4781, 4821, 4956, 5039, 5103\n\nPrice, Vincent 251, 514, 1949, 2019, 2692\n\nPrickett, Maudie 304, 798, 864, 1225, 1309, 1595, 2339, 2397, 2452, 2482, 2563, 2677, 3063, 3090, 4015, 4610, 4812, 4888\n\nPrimus, Barry 1826\n\nPrince, William 526, 2482, 2793, 3762\n\nPrincipal, Victoria 2348\n\nPrine, Andrew 22, 237, 741, 1133, 1654, 1840, 2215, 2265, 2906, 4090, 4287, 4344, 4994\n\nPringle, Aileen 4336\n\nPrival, Lucien 1877, 2132, 2135\n\nProsser, Hugh 9, 290, 446, 477, 643, 775, 1360, 1768, 2171, 2396, 2451, 2680, 2802, 2949, 3088, 3167, 3264, 3458, 3547, 3866, 3940, 4024, 4506, 4536, 4687, 4729, 4811, 4826, 4847\n\nProuty, Jed 826, 1579, 4309\n\nProvost, Jon 2183\n\nProwse, Juliet 3752\n\nPrud'homme, Cameron 3244\n\nPryor, Richard 17, 385, 758\n\nPryor, Roger 2534\n\nPuglia, Frank 332, 580, 798, 1182, 1290, 1347, 1521, 1562, 1973, 2066, 2586, 3046, 3276, 3998, 4061, 4062, 4115, 4231\n\nPuig, Eva 289, 758, 2423, 2831, 3128, 3431, 3530, 3692, 3530, 3692, 4312, 4657, 4723, 4929\n\nPullman, Bill 4746, 5029\n\nPully, B.S. 2815\n\nPurcell, Dick 1964, 1974, 2149, 2790, 2884, 4422, 4697\n\nPurcell, Gertrude 1084, 1974\n\nPurcell, Lee 1107, 2092, 2100\n\nPurdom, Edmund 376, 1858, 2230\n\nPurl, Linda 1874, 2056, 2930, 5073, 5074\n\nPyle, Denver 29, 31, 32, 33, 237, 489, 490, 554, 594, 643, 671, 720, 1050, 1199, 1214, 1268, 1314, 1354, 1402, 1533, 1606, 1608, 1660, 1668, 1727, 1747, 1808, 1843, 1890, 1899, 1913, 1937, 1957, 1985, 2048, 2074, 2139, 2314, 2319, 2337, 2347, 2396, 2431, 2500, 2512, 2515, 2555, 2561, 2576, 2592, 2749, 2856, 2875, 2896, 3066, 3219, 3298, 3310, 3423, 3510, 3614, 3622, 3636, 3751, 3792, 3816, 3860, 3921, 3990, 4252, 4343, 4461, 4462, 4731, 4802, 4876, 5007, 5044, 5049\n\nQuade, John 192, 676, 1578, 1583, 1607, 1880, 1926, 1944, 2196, 2446, 2474, 2810, 2950, 3070, 3252, 4477, 4753\n\nQuaid, Dennis 44, 2443, 5029\n\nQuaid, Randy 518, 2229, 2443, 2666, 3192, 4169, 4374\n\nQualen, John 26, 75, 170, 200, 277, 317, 320, 1346, 1471, 1837, 2046, 2543, 2561, 2829, 3752, 3783, 3817, 4035, 4121, 4284, 4626\n\nQuarry, Robert 3843\n\nQuartero, Nena (Nina) 153, 929, 1329, 2163, 2315, 2527, 2639, 2943, 3078, 4365\n\nQuayle, Anthony 2489\n\nQuick, Eldon 2810\n\nQuigley, Charles 864, 2649, 3665\n\nQuigley, Rita 1947, 3429\n\nQuigley, Robert 1677, 2167, 3658, 3850\n\nQuillan, Eddie 22, 47, 63, 486, 961, 1361, 1563, 1696, 2688, 4598, 5072\n\nQuillan, Marie 721, 1952, 2636, 3665, 3926\n\nQuimby, Margaret 4511, 4518\n\nQuiney, Charles (Carlos) 5095, 5098\n\nQuinlan, Kathleen 4195\n\nQuinlivan, Charles 3787\n\nQuinn, Anthony 362, 406, 552, 602, 1011, 1372, 1731, 1842, 2240, 2518, 3005, 3126, 3421, 3440, 3536, 3729, 3768, 3781, 4312, 4336, 4665, 4751, 4791, 5026\n\nQuinn, Freddy 3829\n\nQuinn, Pat 3837, 5083\n\nQuinn, Tom 428, 710, 893, 950, 1105, 1362, 1540, 1792, 1974, 2171, 2280, 2652, 2761, 4155\n\nQuintana, Rosita 601, 746, 2630\n\nRaaf, Vicki 329, 1427, 2504, 2899, 3637, 4601\n\nRabock, Al 894\n\nRacimo, Victoria 1539, 2058, 2700, 2899, 3637, 4601\n\nRackin, Martin 320, 1111, 1937, 2780, 2829\n\nRae, Peggy 3783\n\nRafferty, Chips 350, 1239, 2079, 3004, 4192\n\nRafferty, Frances 27, 191, 248, 1564, 3576, 4999\n\nRaffetto, Michael 3112\n\nRaffill, Stewart 10, 39, 3835, 4870\n\nRafkin, Alan 3806\n\nRaft, George 2815, 4061, 4580\n\nRagan, Mike (Michael) 166, 349, 637, 933, 1452, 1541, 1727, 2205, 2252, 2555, 2693, 2988, 3219, 3423, 3998, 4080, 4109, 4228, 4232, 4241, 4285, 4361, 4478, 4620; _see also_ Bane, Holly\n\nRagland, Rags 1564\n\nRaine, Jack 5019\n\nRaine, Norman Reilly 1588, 2815, 5020\n\nRaines, Ella 3436, 3921, 4231, 4771\n\nRaines, Steve 678, 698, 1463, 2488, 2695, 2748, 3751, 3828, 3995, 4478\n\nRainey, Ford 209, 838, 1371, 1660, 1727, 2051, 2141, 2784, 4370, 4626\n\nRains, Claude 1594, 2064\n\nRaison, Milton 1919, 2712, 2880, 2882, 3825, 4062, 4075, 4462, 4587, 4808, 4845; _see also_ Milton, George\n\nRall, Tommy 3754, 3780\n\nRalli, Giovanna 641, 2643\n\nRalph, Jessie 1165, 2108, 2159, 3695, 5046\n\nRalston, Esther 4442\n\nRalston, Vera 277, 930, 1291, 1313, 1714, 2066, 3066, 3128, 4074, 4212, 4428, 5033\n\nRambeau, Marjorie 1831, 3684, 3704, 4593, 4786\n\nRambo, Dack 2810\n\nRamsey, Quen 3162, 4587, 4868\n\nRamsey, Ward 3149, 3791\n\nRamus, Nick 1962, 2746, 3203, 4007, 4996\n\nRand, Sally 323, 504\n\nRandall, Anne 3686, 4429\n\nRandall, Jack 13, 724, 856, 1159, 1682, 2107, 2165, 2572, 2647, 2869, 2993, 3102, 3446, 3456, 4130, 4554, 4882, 4946, 4953\n\nRandall, Lorraine 2826, 4883\n\nRandall, Meg 2202\n\nRandall, Monica 61, 376, 1355, 2905, 3030, 3064, 3301, 3309, 3336\n\nRandall, Rebel 1001\n\nRandall, Stuart 128, 146, 284, 562, 584, 652, 733, 1199, 1263, 1466, 1676, 1762, 1775, 1866, 1913, 2172, 2544, 2564, 3040, 3140, 3143, 3149, 3255, 3441, 3614, 3640, 3652, 4060, 4123, 4223, 4300, 4338, 4450, 4804\n\nRandall, Tony 2087, 3783\n\nRandell, Ron 2317, 2555, 2664, 3213\n\nRandle, Karen 867, 3684\n\nRandolph, Amanda 1842\n\nRandolph, Anders 622\n\nRandolph, Isabel 277, 433, 869, 1285, 3425, 3434, 4400\n\nRandolph, Jane 1380\n\nRandolph, John 3727, 4330\n\nRandolph, Lillian 290\n\nRandolph, Oscar 698, 2074\n\nRandom, Bob 563, 919, 2695, 3737, 4429\n\nThe Range Ranglers 926\n\nThe Range Riders 2956\n\nRankin, Arthur 757, 1239, 1306, 4271, 4377, 4516\n\nRansom, Lois 3075, 3348\n\nRapf, Maurice 200\n\nRassimov, Ivan 4724\n\nRasumny, Mikail 2144, 3112\n\nRathbone, Basil 2586\n\nRathmell, John 576, 1317, 1553, 2660, 3012, 3075, 3258, 3482, 4019, 4059, 4128, 4531, 4732, 5096\n\nRattray, Heather 10, 2698, 3835\n\nRaven, John 3696\n\nRavetch, Irving 883, 1908, 1948, 2986, 4065, 4725\n\nRawlins, John 149, 1398, 2618, 2641, 2716, 2994, 3583, 3761, 3811\n\nRawlins, Monte 38, 1931\n\nRawlinson, Herbert 135, 201, 472, 516, 800, 956, 987, 1319, 1361, 1416, 1490, 1523, 1588, 2101, 2124, 2132, 2135, 2451, 2480, 2595, 2600, 2866, 3067, 3458, 3877, 3899, 3928, 4103, 4147, 4162, 4697, 5021\n\nRay, Albert 625, 1080, 2801, 4656\n\nRay, Aldo 1795, 2379, 2563, 3778, 4741, 4802\n\nRay, Allene 1804, 1993, 4852\n\nRay, Bernard B. 555, 557, 668, 889, 1241, 1903, 3545, 4274; _see also_ Shamray, Franklin\n\nRay, Charles 1061, 2159, 4257\n\nRay, Fred Olen 78, 1777, 3738, 3843\n\nRay, Joey 842\n\nRay, Michael 4435\n\nRay, Nicholas 2048, 2485, 3636, 3729, 4579, 4993\n\nRay, Rex 4666\n\nRay, Vivian 785, 3047\n\nRay, Wade 2022\n\nRaye, Martha 3409\n\nRayford, Alma 522, 593, 1038, 1799, 2182, 2358, 2945, 3598, 3970, 5076\n\nRaymaker, Herman C. 2792, 4517, 4651\n\nRaymond, Gene 1352\n\nRaymond, Guy 223, 237, 2023, 4638\n\nRaymond, Paula 695, 1093, 1352, 1691, 1996, 4291\n\nRaynor, Bill (William) 433, 1259, 2602, 2622, 2840, 3093, 3759, 3936, 3998, 4420, 4439, 4574, 4616, 4618, 5078, 5079, 5081\n\nRaynor, Grace 4250\n\nRea, Peggy 2087\n\nReagan, Ronald 198, 681, 874, 2225, 2252, 3711, 4115, 4258\n\nReason, Rex 211, 528, 2143, 2659, 3287, 3294, 3663, 3718, 3966\n\nReason, Rhodes 1071, 5055\n\nRebane, Bill 655, 3250\n\nRed Elk, Lois 315, 2010, 2045, 2906, 4373\n\nRed River Dave (McEnery) 4221\n\nThe Red River Valley Boys 155, 1456, 2595, 2866, 4024\n\nRedd, Joyce 219\n\nRedfield, William 1183\n\nRedford, Robert 586, 1219, 2028, 4249\n\nRedgrave, Lynn 1135\n\nRedmond, Liam 26\n\nRedwing, Rodd 103, 114, 555, 681, 704, 828, 1371, 1711, 1842, 1844, 2021, 2050, 2176, 2204, 2233, 2372, 3056, 3255, 3339, 3468, 3777, 3807, 3916, 4001, 4032, 4491, 4601, 5005\n\nReed, Alan 1265, 2143, 2815, 3339, 4751\n\nReed, Barbara 835, 1020\n\nReed, Carol 1372\n\nReed, Dean 20, 1580, 4694\n\nReed, Donald 929, 2248, 2527, 2642, 3245, 3361, 3939, 4281, 4647, 4711\n\nReed, Donna 112, 189, 1265, 1526, 1672, 1774, 4338, 4362\n\nReed, Ione 12, 1076, 1316, 2636\n\nReed, Marshall 2, 89, 198, 235, 366, 415, 418, 464, 492, 612, 643, 649, 695, 711, 728, 850, 854, 857, 867, 936, 951, 994, 1023, 1040, 1158, 1325, 1339, 1403, 1454, 1490, 1506, 1529, 1540, 1543, 1684, 1687, 1711, 1726, 1765, 1797, 1868, 1920, 1982, 2022, 2066, 2085, 2169, 2178, 2179, 2259, 2271, 2280, 2282, 2294, 2319, 2448, 2485, 2582, 2600, 2673, 3679, 2680, 2762, 2776, 2798, 2851, 2863, 2893, 2949, 2988, 3002, 3050, 3052, 3104, 3157, 3231, 3264, 3277, 3356, 3441, 3460, 3509, 3556, 3607, 3652, 3690, 3728, 3744, 3804, 3892, 3900, 3943, 4001, 4021, 4028, 4085, 4097, 4105, 4118, 4188, 4292, 4296, 4299, 4301, 4400, 4421, 4439, 4466, 4514, 4559, 4584, 4709, 4759, 4781, 4812, 4813, 4820, 4847, 4894, 4973, 5033, 5043, 5102\n\nReed, Oliver 1642, 1951, 3395, 4529\n\nReed, Philip 229, 964, 969, 2147, 2213, 3112, 4227\n\nReed, Ralph 2980, 3333\n\nReed, Tom 185, 4072, 4139, 4673, 4928\n\nReed, Walter 381, 720, 830, 1050, 1201, 1265, 1397, 1936, 1937, 1945, 2202, 2295, 2544, 2911, 3053, 3383, 3682, 3768, 3776, 3788, 3960, 4241, 4339, 4349, 4429, 4464, 4787, 4804, 4830, 4841, 5030, 5052\n\nRees, Lanny 605, 842, 1914, 2255\n\nReese, Tom 401, 1371, 2249, 4161, 4223\n\nReeve, Christopher 359, 360, 361\n\nReeves, Bob 28, 235, 300, 424, 544, 637, 646, 987, 1076, 1246, 1316, 1444, 1456, 1474, 1492, 1497, 1543, 1563, 1725, 1767, 1861, 1974, 1993, 2032, 2250, 2264, 2424, 2434, 2478, 2628, 2722, 2786, 2806, 2856, 2968, 3135, 3260, 3270, 3344, 3401, 3442, 3498, 3575, 3649, 3707, 3827, 3884, 3969, 4917, 4294, 4320, 4386, 4419, 4506, 4645, 4712, 4786, 4807, 4837\n\nReeves, Del 3688\n\nReeves, George 241, 446, 549, 562, 800, 1934, 2311, 3255, 4062, 4383, 4752, 4856\n\nReeves, Jim 2117\n\nReeves, Richard 339, 550, 664, 1082, 1140, 1268, 1702, 1748, 3045, 3066, 3512, 3644, 4461\n\nReeves, Steve 2441\n\nRegan, Jayne 592, 2495, 3888, 4269, 4298\n\nRegas, George 200, 264, 614, 952, 1337, 1601, 1605, 1802, 2315, 2434, 2702, 2831, 3300, 3308, 3465, 3561, 4742, 4750, 5015\n\nRegas, Pedro 602, 1372, 1570, 1571, 1880, 2144, 3561, 4055, 4495, 4517, 4525, 4605, 4650, 4750, 4824, 4976\n\nRegehr, Duncan 333\n\nReicher, Frank 134, 309, 667, 1914, 2064, 2727, 4052, 4215, 4831, 5043\n\nReichow, Otto 139, 2815, 3902, 4633\n\nReid, Carl Benton 520, 819, 977, 1225, 1234, 1347, 1992, 2209, 2241, 2899, 4073, 4074, 4079, 4257, 4528, 4921\n\nReid, Dorothy (Davenport) 3610\n\nReid, Elliott 3862\n\nReid, Wallace 4697\n\nReid, Wallace, Jr. 2132, 2831, 2943\n\nReilly, Hugh 744, 2183\n\nReinhart, Dick 582, 1821, 2424, 2858, 2944, 3142, 3484, 3675, 4021, 4028, 4135, 4318, 4515, 4586, 4600\n\nReinl, Harald 106, 904, 1069, 1292, 2214, 2219, 2789, 4540, 5003\n\nReis, Irving 2789\n\nRemar, James 4902, 4930, 4996\n\nRemick, Lee 1765, 4333\n\nRemington, Colt 899\n\nRemsen, Bert 181, 463, 554, 2623, 2625\n\nRenaldo, Duncan 203, 288, 446, 654, 758, 853, 886, 957, 1143, 1512, 1516, 1568, 1767, 1862, 1981, 2086, 2128, 2135, 2403, 2876, 2972, 3015, 3300, 3574, 3610, 3615, 3693, 3720, 3826, 4052, 4055, 4061, 4532, 4690, 5096\n\nRenavent, Georges 1427, 3529, 3899, 4892\n\nRennie, James 197, 2570, 2181\n\nRennie, Michael 1922, 3422, 3781\n\nRenoir, Jean 4058\n\nRentschler, Mickey 493\n\nRepp, Ed Earl 623, 682, 695, 922, 1099, 1309, 1497, 1725, 1726, 1736, 1815, 2105, 2197, 2397, 2398, 2886, 2975, 3167, 3172, 3291, 3441, 3667, 3674, 3894, 3938, 4140, 4154, 4272, 4305, 4437, 4556, 4736, 4810\n\nRepp, Stafford 2563, 4123\n\nThe Republic Rhythm Riders 453, 796, 2200, 2880, 2882, 4056\n\nRettig, Tommy 170, 2241, 3221, 3535\n\nReuben, J. Walter 200\n\nRevere, Anne 174, 197, 625, 865, 1202, 1324, 2488\n\nRevier, Dorothy 174, 197, 625, 865, 1202, 1324, 2527, 4377, 4863\n\nRevier, Harry 4866\n\nRevill, Clive 5100\n\nRex (horse) 1087, 2133, 2136, 1237, 2283, 2811, 4708, 4928\n\nRey, Fernando 820, 2174, 2316, 2758, 3175, 3391, 3400, 3654, 3726, 3997, 4471, 4738, 4901\n\nRey, Rosa 4022\n\nReyes, Chuy 1240\n\nReynolds, Burt 378, 713, 1251, 1780, 2055, 2560, 2758, 2903, 3641, 3688, 4679\n\nReynolds, Clarke 1720, 2507, 3849, 3997\n\nReynolds, Craig 1528, 1595, 2515, 2775, 4534, 4773\n\nReynolds, Debbie 1945, 3755, 4670\n\nReynolds, Don Kay (Little Brown Jug) 304, 3319, 3586, 3978, 4013, 5050\n\nReynolds, Gene 312, 3711, 4395\n\nReynolds, Lake 3272\n\nReynolds, Lynn 543, 2071, 3240, 3469, 3956, 4316, 4510\n\nReynolds, Marjorie 207, 353, 927, 1174, 1662, 1830, 2235, 2279, 2572, 2992, 3089, 3156, 3564, 3949, 4279, 4423, 4848\n\nReynolds, Sheldon 3125\n\nReynolds, Vera 2404\n\nReynolds, William 257, 749, 1112, 1247, 1747, 2664, 3222\n\nRhoades, Barbara 3806, 3878, 4350\n\nRhodes, Betty Jane 69, 147\n\nRhodes, Donnelly 912, 1703\n\nRhodes, Grandon 638, 895, 1201, 2261, 2502, 2815, 2868, 2894, 4136, 4168, 4300\n\nRhue, Madlyn 1809, 4161\n\nRiano, Rene 642, 2145, 2530, 2563, 4336, 5058\n\nRicci, Tonino 1648\n\nRice, Florence 715, 2102, 4121\n\nRice, Frank 429, 440, 454, 825, 954, 968, 1054, 1315, 1324, 1387, 1388, 1432, 1848, 2121, 2232, 2297, 2702, 2741, 2774, 3035, 3092, 3099, 3153, 3322, 3465, 3558, 3631, 3634, 3797, 3850, 3994, 4020, 4038, 4093, 4138, 4269, 4484, 4494, 4522, 4703, 4756, 4780\n\nRice, Miriam 443\n\nRice, Patricia 3900\n\nRice, Rosalinda 3548\n\nRich, David Lowell 512, 966, 3127, 3282, 4361\n\nRich, Dick 100, 279, 514, 562, 1314, 1614, 1810, 1884, 2124, 2471, 2502, 2941, 2996, 3005, 3530, 3780, 4491, 4786, 4849, 5025\n\nRich, Gloria 1860, 2872, 2968, 3454\n\nRich, Irene 89, 504, 1395, 3202\n\nRich, Lillian 2584\n\nRichards, Addison 129, 196, 210, 248, 254, 286, 301, 465, 524, 715, 880, 1032, 1202, 1274, 1436, 1466, 1481, 1502, 1532, 1588, 1742, 1812, 1917, 2202, 2393, 2516, 2528, 2638, 2716, 2722, 2837, 2929, 3071, 3223, 3355, 3363, 3483, 3630, 3646, 3676, 3711, 3827, 4025, 4215, 4286, 4336, 4513, 4697, 4770, 4849, 5032\n\nRichards, Ann 214\n\nRichards, Dick 909, 1021\n\nRichards, Frank 129, 612, 864, 967, 1235, 1773, 1782, 2518, 2986, 3069, 3644, 3722, 4043, 4141, 4467, 5036\n\nRichards, Gordon 3607\n\nRichards, Grant 1932, 2868\n\nRichards, Jeff 466, 1078, 2575, 2578, 3780, 4755\n\nRichards, Keith 47, 74, 170, 211, 1214, 1264, 1392, 1502, 1522, 1836, 2022, 2035, 2453, 2825, 3298, 3751, 3761, 3802, 4033, 4076, 4304, 4520, 4599, 4976, 5044\n\nRichards, Paul 375, 499, 1421, 2990, 4232, 4787\n\nRichardson, Jack 12, 286, 298, 457, 891, 893, 1178, 1667, 1677, 2173, 2662, 2450, 2531, 3051, 3496, 3599, 3979, 4193, 4447, 4557, 4774, 4899, 5009\n\nRichardson, Jay 1777\n\nRichardson, Tony 426, 2770\n\nRichman, Mark Peter 417, 980, 1434, 2627, 5082\n\nRichmond, Kane 362, 688, 3384, 3471, 3876, 4396\n\nRichmond, Ted 749\n\nRichmond, Warner 331, 795, 848, 1137, 1317, 1604, 1789, 1824, 1827, 2306, 2349, 2785, 2927, 2957, 3028, 3075, 3163, 3235, 3237, 3243, 3407, 3426, 3456, 3920, 3926, 3949, 3968, 4022, 4130, 4501, 4554, 4883, 4946, 4969\n\nRickert, Shirley Jane 2766, 4786\n\nRicketts, Tom 952, 1591, 1614, 1865, 3322, 4750\n\nRicks, Archie 297, 351, 422, 438, 493, 496, 543, 872, 974, 1301, 1678, 1683, 1900, 1993, 2012, 2014, 2440, 2548, 2647, 2703, 2915, 3092, 3135, 3160, 3239, 3260, 3303, 3446, 3486, 3680, 3850, 3880, 3968, 4050, 4554, 4822, 4843, 4980\n\nRickson, Joe 243, 968, 1269, 1360, 1932, 2198, 2422, 3469, 4100, 4367, 4521, 4650, 4944\n\nThe Riders of the Purple Sage _see_ Willing, Foy, and The Riders of the Purple Sage\n\nRidgely, John 42, 439, 718, 874, 1390, 1403, 1884, 2225, 2835, 2941, 3538, 3672, 4336, 4381, 4848, 4871\n\nRidges, Stanley 648, 1658, 3898, 4168, 4326, 4665\n\nRidgeway, Fritzie 1850\n\nRieger, Manfred 574\n\nRiesner, Charles 4523\n\nRiesner, Dean 2519, 2644, 2922, 3951, 4161, 4523\n\nRigaud, George (Jorge) 220, 751, 2390, 3125, 3516, 3730, 3785, 4150, 4182, 4678\n\nRiley, Elaine 1097, 1253, 1255, 1573, 1890, 2310, 3040, 3441, 3928, 4147\n\nRiley, Jeannine 4172, 4753\n\nRilla, Walter 976\n\nRin Tin Tin (dog) 765, 1013, 2367, 2394, 2792, 4476, 4875\n\nRin Tin Tin, Jr. (dog) 668, 1260, 1950, 2283, 2284, 3907, 3952, 4274, 4722\n\nRin Tin Tin III (dog) 3379\n\nRing, Cyril 447, 1438, 1591, 1897, 1903, 1915, 2552, 3022, 3772, 4299, 4585, 4803\n\nRingwald, Molly 3007\n\nRio, Joanne 3509\n\nRiordan, Marjorie 4047\n\nRisdon, Elisabeth 874, 1141, 1888, 1947, 1971, 2013, 2649, 3585, 3862, 4231\n\nRiss, Dan 212, 3741, 4136, 4376, 5036, 5052\n\nRitch, Steven 259, 828, 841, 2074, 2617, 3509, 3512, 3769, 4805\n\nRitchie, Clint 41, 71, 506, 1081, 3137\n\nRitt, Martin 1908, 1948, 2984\n\nRitter, John 3739\n\nRitter, Tex 101, 137, 138, 155, 575, 726, 861, 876, 1001, 1028, 1100, 1148, 1229, 1363, 1444, 1465, 1506, 1604, 1812, 1877, 1900, 2119, 2378, 2424, 2425, 2542, 2589, 2595, 2601, 2632, 2740, 2827, 2866, 2873, 3028, 3107, 3159, 3227, 3235, 3237, 3407, 3461, 3476, 3491, 3550, 3587, 2588, 3589, 3590, 3913, 4019, 4022, 4128, 4186, 4226, 4261, 4279, 4363, 4566, 4573, 4685, 4723, 4833, 4872, 4889, 4921\n\nRitter, Thelma 3755\n\nThe Ritz Brothers 388\n\nRiva, Michael 1232\n\nRivas, Carlos 269, 938, 1033, 1483, 1510, 4576, 4638, 4662, 5086\n\nRivas, Jorge 3527, 3987\n\nRivas, Maria 2405\n\nRivelli, Luisa 268, 316\n\nRivero, Jorge 1729, 1994\n\nRivero, Julian 139, 159, 283, 306, 342, 347, 456, 520, 868, 877, 1018, 1143, 1159, 1191, 1512, 1562, 1591, 1767, 1859, 1865, 1907, 2144, 2164, 2196, 2247, 2298, 2409, 2414, 2416, 2470, 2473, 2502, 2524, 2644, 2799, 2878, 2943, 2948, 2991, 2999, 3083, 3320, 3406, 3413, 3448, 3492, 3530, 3562, 3681, 3803, 4004, 4025, 4049, 4052, 4168, 4252, 4310, 4388, 4507, 4525, 4573, 4701, 4728, 4750, 4803, 4843, 4851, 4945, 4986, 5042, 5062\n\nRivers, Jack 2435, 4507, 4820\n\nRivers, Johnny 1705\n\nRiviere, George 2655\n\nRivkin, Allen 2013, 3542, 4428\n\nRizzo, Gianni 20, 1966, 3381, 3639, 3660, 5087, 5092\n\nRoach, Bert 653, 1187, 1253, 1588, 1591, 1848, 2537, 3695, 4257\n\nRoach, Hal 355, 1087, 2136, 2639, 3156\n\nRoach, Hal, Jr. 596, 1179, 3156\n\nRoach, Joe 1280, 1863, 2024\n\nRoach, Margaret 3446\n\nRoadman, Betty 3385, 3574\n\nRoan, Vinegar 414, 1751, 2399, 3015, 3241, 3558, 3574, 4732, 5103\n\nRoarke, Adam 3732, 4545\n\nRobards, Jason 214, 826, 1105, 1736, 1938, 2146, 2552, 3194, 3267, 3383, 3481, 3511, 3561, 3695, 3899, 3975, 4002, 4044, 4386, 4475, 4503, 4653, 4777, 4841, 4898, 4951, 5096, 5104\n\nRobards, Jason, Jr. 221, 317, 816, 1943, 2335, 2898, 3055\n\nRobbins, Gale 1748, 3200\n\nRobbins, Marty 208, 219, 559, 1733, 1772, 3225\n\nRobbins, Skeeter Bill 413, 873, 1172, 1322, 1517, 1781, 2388, 2573, 4944\n\nRober, Richard 2549, 2960, 3053, 3722, 3862\n\nRoberson, Chuck 16, 22, 43, 49, 229, 313, 319, 373, 594, 654, 677, 720, 741, 857, 858, 883, 953, 1313, 1400, 1412, 1665, 1800, 1892, 1921, 1945, 1960, 2022, 2035, 2118, 2225, 2264, 2361, 2556, 2626, 2700, 2779, 2797, 3128, 3295, 3493, 3517, 3522, 3527, 3622, 3736, 3752, 3776, 3788, 3973, 4035, 4118, 4428, 4496, 4626, 4638, 4790, 4847, 4989, 5024\n\nRoberts, Adele 1046, 1497, 3552, 4378\n\nRoberts, Alice 3702\n\nRoberts, Bart 4244\n\nRoberts, Beatrice 1256, 1438, 3043, 3111\n\nRoberts, Beverly 1588\n\nRoberts, Edith 4756\n\nRoberts, Eric 2442, 3192\n\nRoberts, Gordon 3477; _see also_ Young, Carleton\n\nRoberts, Lee 2, 259, 263, 383, 390, 643, 650, 658, 716, 791, 841, 854, 857, 864, 936, 940, 1003, 1025, 1073, 1074, 1111, 1214, 1321, 1326, 1403, 1408, 1461, 1555, 1585, 1702, 1711, 1713, 1790, 2022, 2083, 2085, 2255, 2266, 2271, 2294, 2370, 2400, 2448, 2502, 2539, 2543, 2565, 2582, 2679, 2776, 2839, 2946, 2957, 3068, 3460, 3509, 3511, 3599, 3744, 3848, 4044, 4073, 4085, 4095, 4131, 4301, 4408, 4437, 4582, 4588, 4590, 4731, 4804, 4894, 4935, 4969, 5022\n\nRoberts, Lynne (Lynn) 309, 381, 389, 620, 631, 1199, 1246, 1822, 2208, 2357, 2399, 3431, 3471, 3563, 3596, 3669, 3929, 4033, 4425\n\nRoberts, Marguerite 1354, 1925, 3750, 3837, 4576\n\nRoberts, Mark _see_ Scott, Robert\n\nRoberts, Pernell 506, 702, 1064, 1425, 1879, 3430, 3815, 3878\n\nRoberts, Rachel 4969\n\nRoberts, Roy 189, 260, 462, 749, 895, 972, 1347, 1427, 1480, 2118, 2187, 2396, 2504, 2616, 2718, 2955, 3023, 3690, 3702, 3728, 3862, 3972, 4099, 4338, 4344, 4588, 4913, 5036, 5038, 5044\n\nRoberts, Stanley 781, 797, 1860, 2802, 3168, 3325, 4372, 4654, 4696, 4816\n\nRoberts, Stephen 4580\n\nRoberts, Theodore 4092, 4494\n\nRoberts, William 2335, 2497, 3146, 3336, 4257\n\nRobertson, Bob _see_ Leone, Sergio\n\nRobertson, Cliff 1641, 2077\n\nRobertson, Dale 401, 659, 759, 932, 977, 1092, 1319, 1499, 1601, 1695, 1838, 2229, 2267, 2513, 2865, 2941, 3392, 3737, 3909, 3931, 4136, 4611\n\nRobertson, Willard 64, 514, 755, 1029, 1252, 1480, 1518, 1829, 1855, 2031, 2207, 2213, 2641, 2722, 2831, 2877, 2923, 3005, 3043, 3267, 3355, 3442, 3554, 3895, 4286, 4289, 4359, 4665, 4745\n\nRobins, Sam 119, 2251, 3267\n\nRobinson, Ann 1666, 1668, 3738\n\nRobinson, Casey 572, 3359, 4611\n\nRobinson, Charles 3816\n\nRobinson, Chris 2445, 3838\n\nRobinson, Dewey 46, 271, 800, 935, 1097, 1214, 1290, 1591, 1843, 1996, 2158, 2249, 2261, 2652, 3276, 3414, 3534, 3951, 3953, 4115, 4147, 4225, 4257, 4357, 4422, 4467, 5008, 5021\n\nRobinson, Dick 530, 3580, 4445\n\nRobinson, Edward G. 246, 720, 2489, 2984, 3895, 4740\n\nRobinson, Frances 1080, 1390, 2973, 3452\n\nRobinson, Rad 2152, 2353, 3267, 3354, 3851, 4112\n\nRobinson, Ruth 277, 514, 853, 1143, 1974, 2086, 3584, 4257, 4318\n\nRobles, German 5105\n\nRobson, Mark 1622, 3621\n\nRobson, May 4284, 4312\n\nRobyns, William 1839, 2393, 3092\n\nRoc, Patricia 648\n\nRocco, Alex 538, 1828\n\nRochelle, Claire 423, 589, 774, 1211, 1222, 1732, 2107, 2414, 2790, 2815, 2821, 3018, 3477, 3492, 4619\n\nRockwell, Jack 37, 51, 54, 56, 153, 201, 230, 244, 289, 295, 299, 303, 309, 323, 331, 353, 366, 414, 431, 459, 464, 481, 493, 495, 500, 564, 576, 582, 623, 648, 666, 715, 793, 827, 869, 875, 914, 925, 939, 956, 960, 986, 995, 1158, 1192, 1269, 1288, 1308, 1362, 1373, 1416, 1444, 1447, 1449, 1453, 1467, 1488, 1529, 1626, 1627, 1662, 1677, 1728, 1730, 1749, 1835, 1839, 1861, 1870, 1928, 1983, 2032, 2074, 2119, 2121, 2252, 2259, 2272, 2273, 2285, 2296, 2297, 2299, 2355, 2366, 2371, 2391, 2420, 2472, 2478, 2479, 2480, 2514, 2522, 2523, 2540, 2546, 2558, 2660, 2730, 2733, 2737, 2766, 2812, 2951, 2954, 2975, 2978, 2981, 2994, 2995, 3003, 3077, 3080, 3092, 3100, 3135, 3142, 3160, 3163, 3173, 3224, 3228, 3259, 3263, 3279, 3291, 3303, 3322, 3327, 3330, 3348, 3354, 3393, 3450, 3458, 3476, 3508, 3525, 3540, 3548, 3552, 3555, 3575, 3581, 3591, 3613, 3615, 3616, 3650, 3704, 3733, 3761, 3805, 3827, 3834, 3894, 3898, 3919, 3924, 3969, 4082, 4112, 4130, 4135, 4155, 4156, 4165, 4181, 4185, 4188, 4189, 4206, 4207, 4208, 4297, 4322, 4391, 4456, 4459, 4484, 4525, 4531, 4556, 4581, 4589, 4592, 4610, 4640, 4655, 4657, 4698, 4703, 4736, 4759, 4810, 4819, 4826, 4832, 4848, 4861, 4862, 4863, 4892, 4923, 5037, 5040, 5059\n\nRockwell, Norman 4101\n\nRockwell, Robert 2772\n\nRodann, Ziva 1412, 2440\n\nThe Rodeo Revelers 4196\n\nRodgers, Jimmie 2381\n\nRodrigues, Percy 2800\n\nRodriguez, Dagoberto 171, 181, 705, 708, 745, 750, 821, 1189, 1386, 1856, 2001, 2345, 2708, 3189, 3832, 3868, 4176, 4028, 4438, 4544, 4715\n\nRodriguez, Estelita 67, 612, 1522, 1603, 1971, 2034, 2878, 2893, 3024, 3517, 4056, 4198, 4213, 4597\n\nRodriguez, Ismael 269, 955, 3988\n\nRoeca, Sam 1871, 2484, 2963, 3315, 3225, 3864, 4236, 4420\n\nRogell, Albert S. 660, 754, 1492, 1830, 2839, 2322, 3442, 3964, 4404, 4786\n\nRogers, Charles \"Buddy\" 3045\n\nRogers, Ginger 660, 1348, 1658, 4254\n\nRogers, Jean 514, 824, 1831, 4142, 4747, 4983\n\nRogers, Jimmy 596, 1179, 1254, 1416, 2480, 2736, 3156, 3458, 4304\n\nRogers, John 46, 2147\n\nRogers, Kenny 1500, 2092, 2093, 2094, 3519, 4960\n\nRogers, Mildred 4307\n\nRogers, Rita 1411, 1893, 3855\n\nRogers, Roy 53, 110, 143, 160, 284, 286, 287, 323, 337, 442, 611, 666, 790, 813, 868, 986, 1134, 1141, 1246, 1264, 1462, 1488, 1527, 1603, 1618, 1767, 1818, 1823, 1836, 1914, 1964, 1971, 1973, 2027, 2032, 2124, 2490, 2514, 2530, 2534, 2727, 2730, 2778, 2806, 2825, 2874, 2877, 2886, 2893, 2935, 3024, 3236, 3269, 3330, 3409, 3483, 3564, 3585, 3600, 3615, 3675, 3694, 3827, 3834, 3904, 4005, 4013, 4016, 4018, 4025, 4037, 4043, 4051, 4059, 4076, 4083, 4197, 4198, 4202, 4206, 4213, 4488, 4551, 4592, 4597, 4641, 4646, 4681, 4773, 4970, 5050, 5059, 5062; _see also_ Weston, Dick\n\nRogers, Roy, Jr. 132\n\nRogers, Ruth 2542, 2802, 3898, 4312\n\nRogers, Smokey 1090, 3569\n\nRogers, Wayne 1575, 3136\n\nRogers, Will, Jr. 489, 4943\n\nRohm, Maria 627\n\nRojo, Gustavo 570, 742, 1208, 2174, 2242, 2883, 3198, 4237, 4542, 4693, 4721\n\nRoland, Gilbert 98, 115, 226, 245, 272, 296, 720, 1175, 1478, 1572, 1741, 2049, 2064, 2122, 2209, 2583, 3587, 2639, 3112, 3276, 3406, 3502, 3562, 3645, 3659, 3661, 3715, 4047, 4373, 4395, 4464, 4538, 4541, 4924\n\nRoland, Jurgen 3113\n\nRolfe, Sam 768, 1786, 2751, 3096, 3345, 4345\n\nRollins, David 328\n\nRoman, Antonio 3725\n\nRoman, Lawrence 979, 2511, 2899, 3336\n\nRoman, Leticia 1366, 1596\n\nRoman, Ric 509, 1182, 2074, 2204, 2416, 2779, 3751, 3803, 3958, 4043, 4080, 4428, 4751\n\nRoman, Ruth 252, 406, 482, 801, 935, 978, 1263, 1633, 1789, 3299, 3661, 3882\n\nRomand, Gina 2708\n\nRomay, Lina 2504\n\nRomberg, Sigmund 2790\n\nRome, Sydney 4182\n\nRomero, Carlos 3180, 4255, 4335, 5071\n\nRomero, Cesar 271, 523, 757, 1458, 1519, 2471, 2483, 2563, 3178, 3384, 3431, 3596, 4041, 4426, 4727, 4747\n\nRomero, Eddie 690\n\nRomero, Ned 23, 1034, 1527, 1607, 1769, 1944, 1962, 2215, 2746, 3070, 4230, 4249, 4990\n\nRooney, Mickey 771, 1206, 1243, 1500, 1564, 2021, 2496, 2721, 2725, 2726, 2936, 2979, 4601, 4970\n\nRooney, Pat 3051\n\nRooney, Teddy 3791\n\nRoope, Fay 356, 632, 700, 1992, 2395, 3185, 3768, 4751\n\nRoosevelt, Buddy 1, 49, 107, 214, 343, 473, 541, 597, 752, 801, 832, 935, 1377, 1426, 1504, 1936, 2082, 2084, 2125, 2135, 2161, 2252, 2292, 2364, 2403, 2415, 2516, 2552, 2561, 2664, 2853, 2874, 2885, 2923, 3153, 3222, 3243, 3266, 3305, 3500, 3750, 3839, 4057, 4100, 4129, 4232, 4321, 4377, 4407, 4420, 4548, 4797, 4803, 4807, 4830, 4852, 4950\n\nRoot, Wells 198, 414, 841, 2552, 2609, 4139, 4175, 4257, 4287, 4582\n\nRoper, Bob 2517, 3492, 4570, 4648, 4852, 5040\n\nRopes, Bradford 134, 1512, 1767, 2530, 3254, 3342, 3490, 4221\n\nRoquemore, Henry 108, 258, 305, 510, 721, 747, 812, 891, 1153, 1280, 1301, 1562, 1564, 1586, 1599, 1756, 1827, 1925, 2104, 2170, 2288, 2424, 2471, 2481, 2745, 2756, 2764, 2774, 2889, 2921, 3047, 3153, 3227, 3242, 3243, 3634, 3926, 4004, 4031, 4257, 4290, 4312, 4317, 4532, 4564, 4570, 4602, 4809, 4851, 4852, 4868, 4972, 5061\n\nRorke, Hayden 1161, 1996, 4064, 4966\n\nRory, Rosanna 1838\n\nRoscoe, Alan 625, 723, 1200, 1839, 1848, 2211, 4316\n\nRose, Reginald 2556\n\nRosemond, Clifford 4121\n\nRosen, Phil 56, 153, 500, 633, 1517, 1981, 2146, 2573, 3135, 3202, 3263, 3554, 4297, 4706, 4824, 5061\n\nRosenberg, Stuart 2721, 3136\n\nRosenbloom, Maxie 1618, 2101, 3951\n\nRosener, George 666, 1317, 3043, 3827, 4274\n\nRosenwald, Francis 990, 3335, 4171\n\nRosing, Bodil 1827, 2130\n\nRosmer, Milton 3874\n\nRoss, Betsy King 1340, 3075, 3965\n\nRoss, Earl 689\n\nRoss, Howard 2495, 2509, 4348, 5101\n\nRoss, Katharine 586, 822, 1607, 1944, 3317, 3577, 3816, 4249, 4782\n\nRoss, Sarah 3659\n\nRoss, Shirley 3695\n\nRossen, Robert 4335\n\nRossetti, Franco 1108, 1113, 3513, 3532, 4288, 4664\n\nRossi, Rex 287, 2759\n\nRossitto, Angelo 229, 251, 288, 2619, 2644\n\nRosson, Arthur 425, 923, 1869, 2222, 2440, 2703, 4511, 4512, 4518\n\nRosson, Helene 4546\n\nRoth, Gene 28, 251, 303, 409, 597, 648, 794, 842, 1268, 1285, 1490, 1543, 1593, 1695, 1711, 1920, 1945, 2017, 2022, 2035, 2122, 2184, 2659, 2678, 3211, 3644, 3828, 3897, 4057, 4109, 4496, 4579, 4682, 4690, 4821, 4860, 4075; _see also_ Stutenroth, Gene\n\nRoth, Martha 2611, 3360\n\nRothwell, Robert 508, 1942, 2198, 3527\n\nRoubert, Matty 37, 395, 695, 922, 1505, 1513, 1565, 1725, 1815, 2397, 2419, 2421, 3074, 3599, 3834, 4023, 4131, 4272, 4483, 4610, 4649, 4736, 4819, 4969\n\nRoundtree, Richard 195, 701, 2962\n\nRourke, Mickey 2224\n\nRouse, Russell 1274, 4384\n\nRousseau, Louise 1296, 1486, 1725, 2419, 2435, 2665, 2687, 2991, 3178, 3457, 4642, 4820\n\nRouverol, Jean 243, 2288, 4842\n\nRoux, Tony 64, 1513, 1978, 2270, 2385, 2416, 3502, 4212, 4657\n\nRovere, Gina 1581\n\nRowan, Dan 2895\n\nRowe, Eileen 4001\n\nRowe, Prentiss 1170, 1660, 2443, 3981\n\nRowland, Henry 284, 320, 562, 652, 1668, 1672, 2074, 2143, 2511, 3084, 3298, 3852, 4073, 4462, 4620, 4680, 4731, 4757, 5038, 5039, 5097\n\nRowland, Roy 491, 1673, 1710, 2507, 2575, 2691, 2986\n\nRowland, Steve 1673, 1710, 3906\n\nRowlands, Gena 2430\n\nRowles, Polly 4082, 4831\n\nRoy, Gloria 1458, 2471, 4941\n\nThe Roy Rogers Riders 1971, 3024, 4043\n\nRoyal, Charles Francis 297, 792, 799, 848, 1732, 1993, 2356, 2547, 2821, 2872, 2983, 3078, 3520, 4238, 4314, 4465\n\nRoyce, Frosty 369, 461, 798, 1235, 1456, 2022, 2119, 2525, 2867, 2941, 3159, 3500, 3507\n\nRoyer, Fanchon 4532\n\nRoyle, Selena 497, 1792\n\nRoyle, William 144, 1311, 1367, 1462, 1576, 1805, 1862, 1904, 2471, 2528, 2552, 2648, 2712, 3126, 3325, 3353, 4831, 4971\n\nRub, Christian 3895\n\nRubel, James L. 2633, 3171\n\nRuben, J. Walter 2222, 4651\n\nRubin, Benny 1317, 2287, 2620, 2682, 3806\n\nRubinek, Saul 4663\n\nRudie, Evelyn 3367\n\nRudley, Herbert 502, 1837, 2025, 3287, 4457\n\nRudolph, Alan 554\n\nRudolph, Oscar 841, 2289, 2693, 2842, 2910, 3126, 4478, 4582, 4753, 4803, 4973\n\nRuffini, Claudio 88, 220, 568, 1030, 1305, 1643, 2030, 2569, 2814, 3382, 3819, 4246, 4262, 4562\n\nRuggles, Charles 1570, 3048, 3247, 3634\n\nRuggles, Wesley 129, 747\n\nRuhl, William 210, 274, 282, 492, 784, 871, 987, 1442, 1512, 1513, 1800, 2255, 2830, 2867, 3157, 3540, 3571, 3802, 4021, 4049, 4215, 4318, 4661, 4847\n\nRuick, Barbara 115\n\nRule, Janice 71, 1086, 1671, 2002, 2100\n\nRumann, Sig 439, 661, 2575, 5025\n\nRush, Barbara 1364, 1908, 2143, 2188, 3201, 4244\n\nRush, Dick 301, 369, 865, 1388, 1436, 1540, 1627, 1725, 1925, 2067, 2074, 2158, 2171, 2232, 2249, 2344, 2722, 2730, 2861, 3089, 3414, 3424, 3431, 3550, 3555, 3679, 3709, 3750, 4072, 4167, 4403, 4442, 4759, 4841, 4891, 5001\n\nRush, Richard 3732\n\nRussek, Jorge 517, 583, 908, 988, 1223, 1476, 1614, 1731, 1887, 1943, 1994, 2574, 2709, 3055, 3217, 3369, 3832, 3987, 4176, 4715, 4737, 4934, 5028\n\nRussell, Bing 678, 720, 903, 1608, 1702, 1937, 1945, 2059, 2497, 3417, 3439, 3517, 4240, 4579\n\nRussell, Gail 89, 3788, 3875\n\nRussell, Jane 1428, 2050, 2678, 2943, 3020, 4005, 4233, 4755\n\nRussell, John 55, 114, 548, 641, 937, 1402, 1406, 1433, 1487, 1852, 1940, 2031, 2187, 2549, 2563, 2856, 2950, 3019, 3517, 3672, 3963, 5051, 5055\n\nRussell, Kurt 1734, 2447, 2995, 3203, 3204, 3740, 4453\n\nRussell, Mary 323, 3482, 3549, 3907\n\nRussell, Reb 130, 458, 729, 1334, 1336, 2366, 2523, 2954, 3268\n\nRussell, Tony 275\n\nRussell, William D. 293\n\nRussell, William 3934\n\nRussell, Zella 180\n\nRusso, James 525\n\nRussoff, Lou 117, 2870\n\nRust, Richard 71, 806, 1411, 2339, 2598\n\nRuth, Mary 4019\n\nRuth, Phyllis 4953\n\nRutherford, Ann 210, 818, 2299, 2432, 2636, 2922, 2936, 3187, 3926, 5032\n\nRutherford, Jack 873, 1450, 1594, 2074, 3448, 3461, 3544, 3603, 4367, 5077\n\nRutherford, John 1367, 1824, 1933, 2826, 3288, 3588, 4681\n\nRuysdael, Basil 519, 798, 970, 1876, 1937, 2065, 3285, 4740\n\nRyan, Don 4811\n\nRyan, Fran 743, 1539, 1754, 2443, 3019, 3739\n\nRyan, Frank 642\n\nRyan, Irene 3411\n\nRyan, Joe 3114, 4377\n\nRyan, Kelly 2963\n\nRyan, Mitchell 1880, 1924, 1951, 2685\n\nRyan, Robert 193, 293, 639, 915, 1936, 1943, 1963, 1995, 2305, 2656, 2751, 2831, 3180, 3185, 3383, 4233, 4312, 4503, 4934\n\nRyan, Sheila 864, 1270, 1519, 1597, 2423, 2711, 2894, 3006, 4086, 4845\n\nRyan, Tim 488, 1268, 1392, 1600, 2208, 2591, 3569, 4118, 4170\n\nRydell, Mark 883\n\nRyder, Alfred 2002, 2559, 3223, 4576\n\nRyerson, Florence 1853, 2108\n\nRyland, Cecilia 1927\n\nRyno, William (Bill) 1303, 1316, 1817, 2801, 4063, 4070, 4245\n\nSabato, Antonio 1793, 2900, 4595\n\nSabu 2021, 4417\n\nSackheim, Jerry 803, 3669, 4659, 5070\n\nSackheim, William 252, 450, 2616\n\nSacks, Michael 4810\n\nThe Saddle Pals 2435, 4028, 4507\n\nSagal, Boris 181, 1036, 1734, 1899, 2930, 3643\n\nThe Sagebrush Serenaders 2537\n\nSaint, Eva Marie 1946, 2487, 4113\n\nSt. Clair, Arthur 70, 475, 2134, 3511, 3794\n\nSt. Clair, Malcolm 948, 2682\n\nSaint James, Susan 55, 762, 1081, 2559, 3723, 3747\n\nSt. John, Al 70, 107, 154, 243, 300, 334, 336, 338, 340, 341, 342, 343, 344, 345, 368, 382, 427, 436, 452, 630, 683, 940, 994, 1017, 1090, 1155, 1296, 1304, 1339, 1461, 1463, 1464, 1473, 1486, 1505, 1531, 1542, 1756, 1894, 1932, 2033, 2111, 2123, 2151, 2167, 2251, 2263, 2266, 2268, 2276, 2306, 2363, 2408, 2409, 2410, 2411, 2412, 2413, 2414, 2415, 2582, 2632, 2634, 2667, 2690, 2712, 2734, 2847, 2860, 2869, 2943, 2946, 2965, 2974, 2997, 2999, 3011, 3101, 3103, 3155, 3164, 3169, 3226, 3278, 3348, 3388, 3451, 3459, 3546, 3648, 3801, 3825, 3913, 3995, 3999, 4009, 4030, 4097, 4106, 4108, 4273, 4318, 4402, 4408, 4485, 4553, 4555, 4700, 4702, 4709, 4776, 4814, 4838, 4952, 4956, 5018\n\nSt. John, Betta 2287, 2747\n\nSt. John, Jill 762\n\nSt. Polis, John 447, 461, 621, 653, 2121, 2152, 3086, 3574, 3657, 4271, 4367\n\nSais, Marin 67, 276, 286, 334, 753, 867, 1008, 1192, 1303, 1326, 1389, 1459, 1635, 1861, 2125, 2363, 2420, 2847, 2954, 3077, 3105, 3169, 3292, 3361, 3461, 3463, 3586, 3673, 3709, 3866, 3953, 4110, 4273, 4512, 4617, 4861, 4866, 4953, 5016, 5043\n\nSakall, S.Z. 2013, 2677, 2692, 4179\n\nSalazar, Abel 888, 1857, 2075, 3124, 3369, 4716\n\nSale, Charles \"Chic\" 2637\n\nSale, Richard 1175, 2836, 4075, 4411, 4895\n\nSale, Virginia 214, 391, 642, 961, 1178, 1361, 1378, 1599, 1947, 3965, 4221, 4336, 4503, 4564, 4569\n\nSalisbury, Monroe 4091\n\nSalkow, Sidney 401, 1644, 1666, 1667, 1668, 2008, 2017, 2444, 3056, 3208, 3409, 3559, 3741, 3931\n\nSalmi, Albert 502, 1062, 1277, 1425, 1943, 2154, 2305, 2324, 2631, 2793, 2984, 3854, 3990, 4217, 4361, 4662\n\nSalt, Waldo 3214\n\nSambrell, Aldo 133, 202, 220, 571, 701, 1186, 1197, 1699, 1700, 1710, 1776, 1841, 1968, 2439, 2441, 2509, 2570, 2612, 2656, 2758, 2898, 2904, 3125, 3725, 3997, 4277, 4325, 4471, 4629, 5047\n\nSampson, Will 554, 1349, 2746, 2950, 3345, 4122, 4895\n\nSamuels, Henri 3101, 3703; _see also_ Webb, Harry S.\n\nSamuels, Raymond 77, 1271, 3706, 3907; _see also_ Ray, Bernard B.; Webb, Harry S.\n\nSan Juan, Olga 271\n\nSan Martin, Conrado 1968\n\nSanchez, Pedro 20, 98, 400, 1621, 2049, 3059, 3372, 3381, 3660, 4628, 4720, 4903, 5099\n\nSancho, Fernando 61, 316, 416, 769, 1180, 1181, 1197, 1359, 1699, 1722, 1793, 1966, 2387, 2532, 2562, 2655, 2905, 3118, 3380, 3532, 3782, 3784, 3785, 3786, 4012, 4243, 4251, 4623, 4627, 4718\n\nSande, Walter 64, 100, 129, 193, 638, 650, 934, 1015, 1151, 1184, 1408, 1645, 1666, 2008, 2052, 2109, 2240, 2334, 2486, 2624, 2818, 2868, 2996, 3150, 3208, 3290, 3318, 3510, 3925, 4300, 4419, 4792, 4921\n\nSanders, George 63\n\nSanders, Hugh 65, 109, 759, 1293, 1665, 1735, 1992, 2187, 2204, 2684, 2919, 3061, 3090, 3897, 4179, 4390, 4461, 4675, 4973\n\nSanders, Sandy 250, 304, 368, 389, 858, 1326, 1461, 1463, 1890, 1991, 2339, 2386, 2609, 2620, 2693, 2711, 2805, 2910, 2946, 3209, 3447, 3510, 3890, 3974, 3976, 3995, 4001, 4036, 4228, 4285, 4692, 4709, 4757\n\nSanderson, William 3930, 4765\n\nSandow (dog) 629\n\nSandrich, Mark 541\n\nSands, Gaston 569; _see also_ Santos, Gaston\n\nSands, Johnny 1247, 2618, 4611\n\nSands, Tommy 4741\n\nSanford, Isabel 4041\n\nSanford, Ralph 47, 60, 377, 662, 842, 858, 1398, 1415, 1434, 2027, 2641, 2727, 2815, 2929, 3414, 3583, 3702, 3849, 3929, 4057, 4080, 4574, 4680, 4979\n\nSanford, Tiny 1300, 1307, 1566, 1781, 3081, 3239, 3529, 4067\n\nSanforth, Clifford 38\n\nSangster, Jimmy 2819, 3726\n\nSantell, Alfred 142, 2016, 3595\n\nSanti, Giancarlo 1619\n\nSantley, Fred 2013, 4621\n\nSantley, Joseph 615, 710, 1143, 2635\n\nSanto 3713, 3714\n\nSantoni, Rene 1739, 3152\n\nSantos, Gaston 217, 1376, 2383, 2803, 3145, 3873, 4216; _see also_ Sands, Gaston\n\nSantschi, Tom 298, 1972, 3082, 3308, 3537, 4070, 4071, 4352, 4476, 4522, 4756\n\nSarafin, Richard C. 2551, 2560\n\nSarandon, Susan 2468\n\nSarecky, Barney A. 557, 1088, 2132, 3088, 5104\n\nSargent, Joseph 1435, 4169\n\nSargent, Lewis 629, 2531\n\nSaris, Marilyn 2295\n\nSarracino, Ernest 37\n\nSarrazin, Michael 1703, 2059, 2506, 2613, 3993\n\nSasdy, Peter 4801\n\nSaunders, Gloria 274, 417, 2840, 3936\n\nSaunders, Linda 2576\n\nSaunders, Lori 4753\n\nSaunders, Nancy 152, 418, 1461, 2262, 2946, 3167, 3940, 4053, 4812, 4888\n\nSavage, Ann 2150, 2197, 3351, 3674, 3720, 5022\n\nSavage, Brad 121\n\nSavage, John 192, 676, 4746\n\nSavage, Les, Jr. 3397, 4913\n\nSavage, Paul 417, 919, 965, 2490, 2784, 4428\n\nSavalas, George 190, 1251\n\nSavalas, Telly 225, 2174, 2589, 3030, 3297, 3736, 4471\n\nSavona, Leopoldo 118, 2312\n\nSawaya, George 830, 1098, 1397, 3306\n\nSawyer, Joseph (Joe) 47, 159, 442, 549, 693, 914, 960, 1041, 1143, 1458, 1516, 1820, 1945, 1992, 2088, 2127, 2159, 2208, 2223, 2471, 2635, 2689, 2922, 2943, 3071, 3156, 3173, 3224, 3333, 3401, 3500, 3512, 3711, 3925, 4107, 4132, 4184, 4244, 4336, 4533, 4665, 4673, 4850\n\nSaxon, John 119, 1015, 1219, 1998, 2044, 2900, 3133, 3149, 4662, 4990\n\nSaxson, Glenn 2499\n\nSayer, Diane 2500\n\nSayers, Jo Ann 2353\n\nSayers, Loretta 1002, 1329\n\nSayles, Francis 2399, 2552, 2802, 2827, 3003, 3126, 3194, 4106, 4170, 4289, 4391, 4531, 4803, 4932\n\nSaylor, Syd 76, 129, 161, 286, 318, 441, 471, 499, 552, 626, 686, 776, 1177, 1238, 1256, 1300, 1394, 1438, 1532, 1591, 1732, 1806, 1812, 1925, 1961, 2108, 2163, 2189, 2208, 2288, 2344, 2351, 2423, 2471, 2642, 2652, 2711, 2737, 2759, 2774, 2778, 2815, 2885, 2991, 3020, 3340, 3355, 3409, 3584, 3764, 3839, 3866, 3880, 3942, 3954, 4121, 4184, 4199, 4236, 4257, 4362, 4365, 4394, 4673, 4692, 4745, 4757, 4803, 4863, 4925, 4986, 5041, 5075\n\nSayre, George Wallace 46, 776, 1473, 3825, 4027; _see also_ Milton, George\n\nSayre, Joel 95\n\nScala, Gia 3415\n\nScannell, Frank 23, 2302, 2622, 3234, 4162\n\nScardon, Paul 2545\n\nSchaefer, Natalie 632\n\nSchaefer, Robert E. 698, 841, 2402, 2609, 2693, 3751, 4478\n\nSchafer, Armand L. 262, 721, 1333, 1340, 1952, 2247, 2367, 2660, 2737, 2951, 3075, 3680, 4271, 5040\n\nSchaff, Monroe 606\n\nSchallert, William 277, 981, 1427, 1434, 1607, 1743, 1943, 1944, 2400, 2430, 2550, 2616, 3221, 3287, 3608, 3688, 3966, 4329, 4988\n\nSchary, Dore 2013\n\nSchayer, Richard 109, 600, 1992, 2395, 2639, 4310, 4461\n\nScheider, Roy 3910\n\nSchell, Maria 748, 1772\n\nSchiaffino, Rosanna 2509\n\nSchildkraut, Joseph 1360, 1490, 2836, 2878, 3046, 3128, 3276, 4750\n\nSchilling, Gus 710, 1361, 1923, 2649, 3636\n\nSchnabel, Stefan 2261\n\nSchnee, Charles 1478, 3223, 4857\n\nSchneider, John 4102\n\nSchoener, Inge 24, 553\n\nSchofield, Paul 2070, 4205\n\nSchreiber, Avery 3753\n\nSchrock, Raymond L. 600, 964, 1785, 1897, 4932\n\nSchroeder, Doris 154, 235, 243, 566, 621, 950, 984, 1020, 1202, 1380, 1504, 1824, 1931, 2033, 2082, 2342, 2667, 2867, 3074, 3115, 3162, 3928, 4106, 4318, 4367, 4617, 4855\n\nSchubert, Eddie 4025, 4513\n\nSchubert, Karen 820\n\nSchulberg, Budd 4993\n\nSchunzel, Reinhold 3128\n\nSchuster, Harold 1150, 2018, 3376\n\nSchwarzenegger, Arnold 4739\n\nScott Robert 352, 869, 3167\n\nScott, Adrian 3046\n\nScott, Alan 4629\n\nScott, Andrew 400, 553, 991, 1699, 1966, 1968, 3372, 4629; _see also_ Scotti, Andrea\n\nScott, Brenda 2051, 2059, 4344\n\nScott, DeVallon 356, 828, 2624, 4338, 4726\n\nScott, Douglas 4925\n\nScott, Ernest 156, 752, 3293, 3501\n\nScott, Ewing 127, 1790, 1810, 1904, 1906\n\nScott, Fred 1304, 1979, 2151, 2223, 2632, 2634, 2690, 3278, 3494, 3546, 3579, 3601, 3918, 4030, 4405, 4619\n\nScott, George C. 1772, 2859, 3218, 4344\n\nScott, Gordon 553, 4527, 5087\n\nScott, Jacqueline 1015, 1346\n\nScott, Ken 502, 1289, 1441, 3755, 5019\n\nScott, Lester, Jr. 2852\n\nScott, Lizabeth 3318, 3897\n\nScott, Martha 1947, 4786\n\nScott, Mary 104, 2266\n\nScott, Pippa 507, 3752\n\nScott, Randolph 1, 214, 278, 279, 484, 539, 638, 659, 664, 774, 801, 806, 835, 1026, 1070, 1136, 1319, 1408, 1458, 1706, 1774, 1854, 1883, 1917, 2031, 2213, 2232, 2302, 2504, 2549, 2554, 2781, 3219, 3383, 3430, 3435, 3500, 3573, 3702, 3788, 3792, 3839, 4072, 4163, 4179, 4203, 4214, 4232, 4235, 4252, 4284, 4390, 4403, 4442, 4503, 4741, 4744, 4762, 4771, 4830, 4849, 4868, 4950, 4970\n\nScott, Sherman 338, 340, 341, 344, 345, 2111, 2251, 3825; _see also_ Newfield, Sam\n\nScott, Susan 20, 1717, 2354; _see also_ Navarro, Nieves\n\nScott, Zachary 226, 801, 2754, 3757, 3849, 4049, 4058, 4115, 4178, 4539\n\nScotti, Andrea 374, 2883, 3174, 3689, 3810, 4348; _see also_ Scott, Andrew\n\nScotti, Vito 563, 374, 3518\n\nScotto, Aubrey 3022\n\nScourby, Alexander 1560, 3340\n\nSeaforth, Susan 603, 1696\n\nSearle, Jackie 2722, 3020, 4925\n\nSears, Allan 682, 1328, 1384, 2074, 2603, 3168, 3401, 3926, 4201, 4531, 4609\n\nSears, Fred F. 9, 25, 75, 101, 212, 234, 392, 418, 922, 1059, 1404, 1460, 1481, 1806, 1901, 1938, 2103, 2105, 2176, 2178, 2262, 2361, 2397, 2419, 2482, 2515, 2617, 2768, 2955, 2996, 3063, 3094, 3168, 3233, 3358, 3493, 3619, 3973, 3975, 3978, 4015, 4044, 4053, 4296, 4682, 4805, 4888, 5038\n\nSeastrom, Victor 4991\n\nSeaton, George 3855\n\nSeay, James 223, 503, 652, 1405, 1434, 1666, 1915, 1919, 2017, 2831, 2867, 3310, 3425, 3483, 3998, 4330, 4336, 4419, 4727, 4871\n\nSebastian, Dorothy 143, 2082, 2682, 3615, 4683\n\nSeberg, Jean 2488, 3009\n\nSecchi, Antonio 3033\n\nSedan, Rolfe 1566, 2664, 3606, 3634\n\nSeddon, Margaret 1178, 4357\n\nSedgwick, Edward 1897, 4057\n\nSedgwick, Eileen 949\n\nSedgwick, Josie 4004\n\nSeegar, Miriam 975\n\nSeel, Charles 1937, 2561, 2829, 2911, 3776, 4344, 4859\n\nSeeling, Charles 173, 923, 1203, 2914, 3623, 5017\n\nSegal, George 1170, 2002\n\nSegal, Vivienne 4029\n\nSeidel, Tom 150, 1254, 1546, 2688, 3503, 4778, 4855, 4884\n\nSeiler, Lewis 1457, 1636, 1820, 2238\n\nSeiter, William A. 63, 278, 462, 1563, 1847, 4214\n\nSeitz, George B. 1324, 2145, 2213, 2334, 2936, 3095, 4377, 4535, 4704, 4949\n\nSeitz, George B., Jr. 2334\n\nSekely, Steve 4175\n\nSelander, Lesley 132, 165, 230, 241, 242, 255, 282, 446, 465, 480, 524, 549, 624, 670, 691, 715, 800, 830, 857, 934, 1050, 1138, 1222, 1345, 1397, 1403, 1406, 1407, 1409, 1416, 1469, 1646, 1726, 1736, 1816, 1855, 1870, 1933, 1988, 2080, 2152, 2193, 2260, 2315, 2353, 2402, 2451, 2480, 2605, 2731, 2733, 2935, 2981, 3000, 3032, 3049, 3080, 3116, 3119, 3177, 3222, 3267, 3327, 3334, 3354, 3404, 3424, 3441, 3458, 3473, 3479, 3523, 3541, 3563, 3624, 3646, 3667, 3669, 3699, 3704, 3822, 3826, 3848, 3849, 3898, 3967, 4112, 4118, 4135, 4140, 4171, 4208, 4232, 4325, 4364, 4405, 4451, 4472, 4486, 4487, 4657, 4787, 4788, 4923, 5052\n\nSelby, Sarah 1247, 2005, 4223\n\nSeldes, Marian 2351, 4579\n\nSelf, William 2594, 3323\n\nSelk, George 759, 1669, 1741, 3500\n\nSelleck, Tom 900, 2236, 3212, 3632, 3661, 3798\n\nSeller, Tom (Thomas) 2693, 2842, 2910, 3751, 4228, 4582, 4726\n\nSellon, Charles 600, 660, 1154, 1176, 2730, 3011, 3427, 4639\n\nSelman, David 881, 1328, 1488, 2074, 3173, 3401, 3508, 3763, 4089, 4322, 4850\n\nSelzer, Milton 317, 4345\n\nSelznick, David O. 306, 1187\n\nSemels, Harry 197, 642, 1162, 1440, 1518, 1521, 1557, 1925, 2016, 2137, 2145, 2993, 3401, 3609, 3626, 3631, 4054, 4138, 4290, 4377, 4558, 4750, 4803, 5040, 5061\n\nSepulveda, Carl 94, 135, 205, 210, 291, 318, 366, 369, 464, 624, 632, 773, 872, 986, 1444, 1473, 2166, 2169, 2273, 2378, 2403, 2408, 2435, 2600, 2633, 3050, 3088, 3128, 3227, 3232, 3238, 3348, 3385, 3449, 3534, 3651, 3669, 3691, 3708, 3826, 4011, 4024, 4026, 4077, 4103, 4465, 4568, 4806, 5103, 5104\n\nSepulveda, Lee 2007\n\nSequi, Mario 4527\n\nSerato, Massimo 1717, 1722, 2354, 2689, 4268\n\nSerna, Pepe 195, 222, 3827, 3911\n\nSernas, Jacques 1410\n\nService, Robert W. 4072\n\nSessions, Almira 389, 642, 1256, 1257, 1852, 2423, 2677, 2856, 2875, 2989, 3005, 4058, 4766\n\nSeven, Johnny 1703, 1741, 2760\n\nSevilla, Carmen 416, 1661\n\nSeward, Billie 499, 2074, 2254, 2522, 3362, 3401, 3508, 4522\n\nSeyffertitz, Gustav von 1511\n\nSeymour, Anne 903, 1828, 1913, 3959, 4098, 4123, 4343, 4565, 4755\n\nSeymour, Clarine 3743\n\nSeymour, Dan 1249, 2150, 2902, 3255, 3692, 4500\n\nSeymour, Jane 182\n\nShaff, Monroe 1906, 2278, 2992, 4152\n\nShahan, Rocky 399, 833, 1033, 2022, 2066, 2289, 3091, 3417, 3586, 4011, 4141\n\nShamray, Franklin 3528, 4722; _see also_ Ray, Bernard B.\n\nShane, Maxwell 47, 961, 4362, 4422, 4979\n\nShane, Sara 2118\n\nShanks, Don 31, 32, 33, 388, 740, 1147, 1660, 2039, 2215, 2337, 2347, 3580, 4066\n\nShannon, Ethel 543\n\nShannon, Frank 42, 1202, 1883, 2975, 3179, 3291, 3374, 4309\n\nShannon, Harry 170, 514, 648, 858, 914, 1182, 1228, 1704, 1811, 1849, 1877, 1964, 1974, 2018, 2083, 2431, 2557, 2578, 2815, 2839, 2941, 3061, 3091, 3234, 3383, 3646, 3921, 4018, 4233, 4585, 4740, 4786, 5050, 5055\n\nShannon, Peggy 4564\n\nShannon, Richard 166, 248, 678, 2025, 3140, 3417, 3432, 4435, 4528, 4566, 4714\n\nSharif, Omar 1867, 2489\n\nSharky, Sailor 467, 1295, 1651\n\nSharp, Alex 1790, 2267, 3337, 3572, 3768, 4069, 5068\n\nSharpe, David 28, 37, 69, 179, 287, 388, 637, 698, 749, 795, 856, 886, 951, 1047, 1137, 1167, 1491, 1516, 1522, 1550, 1568, 1664, 1798, 1965, 2022, 2125, 2128, 2132, 2256, 2257, 2348, 2403, 2420, 2544, 2602, 2634, 2716, 2802, 3230, 3327, 3340, 3452, 3454, 3626, 3679, 3706, 3834, 3905, 3936, 3937, 3944, 4057, 4213, 4319, 4372, 4391, 4450, 4491, 4606, 4883, 5037\n\nSharpe, Karen 2563, 4714\n\nShatner, William 247, 1098, 2140, 2168, 2819, 2984, 3106, 4896\n\nShaughnessy, Mickey 1205, 1719, 1773, 1945, 2204, 2829, 3815\n\nShaver, Helen 1791, 3017\n\nShaw, Anabel 170\n\nShaw, C. Montague 1519, 3482, 5104\n\nShaw, Janet 155, 206\n\nShaw, Reta 60\n\nShaw, Robert 4471\n\nShaw, Susan Damante 39, 1479, 2698\n\nShaw, Victoria 71, 1205, 4859\n\nShaw, Winifred 4941\n\nShawlee, Joan 734, 1450, 2911, 3991; _see also_ Fulton, Joan\n\nShawn, Dick 1243\n\nShayer, Richard 895, 969, 1083, 1356, 1665, 1670, 2009, 2444\n\nShayne, Robert 274, 292, 915, 993, 1063, 1199, 1992, 2261, 2386, 2593, 3441, 3692, 4417, 4699\n\nShayne, Tamara 2836, 3112\n\nShea, Gloria 1037, 1288, 2852, 3969\n\nShea, Michael 3416, 4802\n\nShean, Al 3698\n\nShear, Barry 1007\n\nSheehan, John 602, 1136, 1252, 1925, 2127, 2649, 2943, 4099, 4357, 4358, 5011\n\nSheen, Charlie 5067\n\nSheen, Martin 1204, 1705, 2324, 3218\n\nSheffield, Johnny 2471\n\nSheffield, Reginald 1949, 3051, 3762\n\nSheiner, David 2506, 3737\n\nSheldon, Barbara 2479\n\nSheldon, Forrest 295, 457, 1200, 1215, 1651, 1839, 2275, 2404, 2428, 2533, 3869, 3907, 4308, 4986\n\nSheldon, Gene 5086, 5097\n\nSheldon, James 507\n\nSheldon, John 1509\n\nSheldon, Julie 4144\n\nSheldon, Kathryn 131, 602, 1458, 1599, 1601, 1967, 2158, 2161, 2245, 3245, 3846, 4208, 4803\n\nSheldon, Norman 435\n\nSheldon, Patti 878\n\nSheldon, Sidney 94, 3040\n\nShelton, Marla 283, 3087\n\nShepard, Elaine 1332, 2274, 3926\n\nShepard, Patty 2326, 2509, 4150, 4594\n\nShepard, Sam 62, 169, 985, 1611, 3192, 3883, 4169\n\nShepperd, John _see_ Strudwick, Shepperd\n\nSher, Jack 4770, 4924\n\nSheridan, Ann 874, 1266, 1917, 3308, 3573, 3902, 4227, 4543, 4762\n\nSheridan, Dan 577, 605, 789, 1913, 1938, 2139, 2430, 4253\n\nSheridan, Frank 824, 1436, 2774, 3401, 3695, 4891\n\nSheridan, Gail 1889, 1932, 3126\n\nSheridan, James 150, 422, 773, 846, 917, 974, 1159, 2409, 2418, 2481, 3456, 3503, 3946, 4424, 4778, 4796, 4844, 4883; _see also_ Tansey, Sherry\n\nSherman, Evelyn 4020\n\nSherman, George 107, 154, 257, 318, 354, 450, 599, 733, 797, 804, 807, 843, 853, 886, 925, 953, 972, 1024, 1042, 1383, 1558, 1782, 1837, 1860, 2033, 2082, 2086, 2209, 2218, 2396, 2421, 2648, 2567, 2786, 2802, 2909, 2968, 2998, 3027, 3074, 3194, 3310, 3325, 3344, 3363, 3408, 3454, 3534, 3574, 3626, 3690, 3709, 3973, 3989, 4052, 4106, 4250, 4318, 4372, 4450, 4483, 4538, 4586, 4617, 4649, 4783, 4954, 5037, 5041\n\nSherman, Harry 2081, 2152, 3046, 5021\n\nSherman, Lois 800\n\nSherman, Ransom 5043\n\nSherman, Samuel M. 388\n\nSherman, Vincent 1963, 2416, 3755, 4571\n\nSherry, J. Barney 249, 916\n\nSherwood, Choti 340\n\nSherwood, Gale (Dale) 452, 3232, 3571\n\nSherwood, George 447, 664, 774, 827, 1236, 1996, 2485, 2534, 2994, 2998, 3171, 3684, 3702, 4127, 4574, 4827, 5041, 5075\n\nSherwood, John 3287\n\nShields, Arthur 105, 1165, 1383, 2118, 3535, 3814\n\nShields, Brooke 4775\n\nShigeta, James 4768\n\nShilling, Marion 687, 1687, 1965, 2470, 2573, 3322, 3528, 3601, 4138, 4190, 4389, 4850\n\nShipman, Barry 9, 161, 392, 418, 661, 777, 922, 1285, 1404, 1460, 1467, 1714, 1805, 1849, 1901, 1938, 2068, 2103, 2105, 2176, 2178, 2234, 2308, 2399, 2403, 2421, 2480, 2684, 2939, 3063, 3084, 3165, 3230, 3233, 3464, 3574, 3619, 3917, 3940, 3973, 3975, 3976, 2978, 4015, 4151, 4167, 4222, 4296, 4283, 4506, 4675, 4818, 5096\n\nShipman, Gwynne 258, 1932, 4485\n\nShipman, Nell 184, 1659, 3992\n\nShipman, Nina 2929\n\nShirley, Anne _see_ O'Day, Dawn\n\nShockley, Marion 2764\n\nShoemaker, Ann 3594\n\nSholem, Lee 3340, 3865, 4120\n\nShor, Sol 28, 951, 1543, 2022, 2035, 2132, 2403, 4011, 4033, 5104\n\nShore, Dinah 278\n\nShore, Roberta 507, 1341\n\nShores, Lynn 1576, 3300\n\nShort, Dorothy 575, 588, 773, 1445, 1816, 2410, 3085, 3142, 3920, 4497, 4872, 4947\n\nShort, Gertrude 3411\n\nShort, Martin 4351\n\nShrum, Cal 206, 2311, 2457, 2953, 3280, 3507, 3588, 4220, 4391, 4568\n\nShrum, Walt 410, 1046, 2165, 2872, 4220, 4568\n\nShubert, Nancy 3680\n\nShue, Elizabeth 393\n\nShuford, Andy 328, 1191, 1537, 1586, 1639, 1813, 2173, 2268, 2608, 2681, 2862, 3444, 4306\n\nShumate, Harold 1, 301, 898, 1228, 1392, 1762, 1854, 2081, 2153, 2372, 2554, 2638, 2719, 3046, 3596, 3624, 3672, 3745, 4007, 4054, 4089, 4498, 4850, 4950\n\nShumway, Lee 154, 351, 489, 514, 557, 582, 597, 853, 973, 995, 1008, 1024, 1320, 1360, 1389, 1468, 1519, 1526, 1550, 1785, 1904, 1906, 1971, 2014, 2033, 2035, 2238, 2256, 2274, 2315, 2394, 2420, 2529, 2667, 2794, 2802, 2928, 2943, 2971, 2975, 3012, 3048, 3087, 3115, 3165, 3194, 3289, 3384, 3414, 3561, 3585, 3626, 3690, 3710, 4027, 4061, 4076, 4127, 4395, 4617, 4697, 4803, 4923, 5027, 5033\n\nShumway, Walter 927, 1328, 1537, 1814, 2799, 2951, 3099, 3375, 3851, 3944, 4887, 4027\n\nShyer, Melville 2523\n\nSidener, Delores 3063\n\nSidney, George 94, 1792\n\nSidney, Sylvia 3981, 4494\n\nSiegel, Don 831, 841, 1015, 1184, 1205, 1371, 3847, 4161, 4624\n\nSierra, Gregory 672, 909, 1067, 1926, 2093, 2627, 4873, 5028\n\nSills, Milton 578, 4070, 4697\n\nSilva, Henry 92, 370, 502, 1891, 2025, 2246, 2483, 2503, 2568, 3127, 3406, 4235, 4751, 4905\n\nSilva, Maria 2115, 2387, 3717, 3796, 4263, 4718\n\nSilvera, Frank 119, 749, 1293, 1842, 1908, 4113, 4229, 4688, 4751\n\nSilverheels, Jay 53, 257, 356, 503, 519, 698, 841, 864, 922, 1164, 1256, 1266, 1423, 1480, 1535, 1762, 1990, 2017, 2074, 2176, 2178, 2204, 2233, 2289, 2334, 2400, 2402, 2482, 2560, 2609, 2620, 2693, 2768, 2835, 2836, 2842, 2906, 2910, 3056, 3154, 3318, 3397, 3698, 3712, 3718, 3751, 3916, 3917, 3962, 4228, 4478, 4500, 4582, 4700, 4705, 4726, 4770, 4783\n\nSilvers, Phil 2163\n\nSilverstein, Elliott 674\n\nSilvestre, Armando 105, 255, 669, 953, 1077, 1104, 1383, 1523, 1866, 2141, 2583, 3217, 3714, 3736, 3973, 4229, 4416, 4624, 5036\n\nSilvestre, Flor 591, 745, 2041, 2579, 2916, 3988, 4418, 4526\n\nSimcox, Tom 1985, 2906, 3816\n\nSimmons, Georgia 3846, 4742\n\nSimmons, Jean 313, 572, 2162, 3612\n\nSimmons, Michael 4053\n\nSimmons, Richard (Dick) 1078, 1842, 2132, 2183, 2564, 3777, 4057, 5022\n\nSimmons, Richard Alan 1899, 2118, 2389, 2981, 4362, 4528, 4787, 5052\n\nSimon, Robert F. 356, 732, 1348, 1427, 1719, 2322, 2561, 2809, 3779, 4976, 5006\n\nSimon, S. Sylvan 191, 2108, 2482, 3530\n\nSimone, Lisa 1561\n\nSimonelli, Giorgio 1180, 1181, 4623, 4627, 4628\n\nSimonelli, Giovanni 218, 1801, 2562, 3715, 3786, 4905\n\nSimpson, Ivan 2790\n\nSimpson, Mickey 91, 104, 664, 841, 1092, 1166, 1247, 1313, 1395, 1560, 1631, 1702, 1809, 2074, 2158, 2289, 2400, 2718, 3534, 3547, 3607, 3682, 3696, 3814, 4072, 4124, 4212, 4232, 4419, 4582, 4764, 4791, 5008\n\nSimpson, Russell 16, 49, 64, 159, 191, 203, 271, 309, 446, 488, 501, 514, 520, 607, 620, 622, 800, 835, 1020, 1080, 1165, 1247, 1434, 1457, 1487, 1532, 1541, 1570, 1571, 1594, 1639, 1820, 1847, 1848, 1937, 2081, 2187, 2208, 2249, 2416, 2422, 2423, 2486, 2718, 2853, 2986, 3245, 3486, 3506, 3672, 3695, 3711, 3780, 3859, 3895, 4072, 4183, 4231, 4233, 4357, 4304, 4357, 4435, 4673, 4697, 4742, 4743, 4764, 4824, 4834, 4898, 4932, 4940, 4971, 5021, 5032, 5073\n\nSinatra, Frank 1106, 1422, 2046, 2144, 3777\n\nSinatra, Frank, Jr. 2437\n\nSinbad 713\n\nSinclair, Diane 1302, 3655\n\nSinclair, Mary 166\n\nSinclair, Ronald 2937\n\nSinclair, Upton 4331\n\nSinger, Alexander 651, 2437\n\nThe Singing Buckaroos 3918\n\nThe Singing Constables 1895\n\nThe Singing Cowboys 3706\n\nThe Singing Riders 4854\n\nSingleton, Penny 1579\n\nSiodmak, Curt 1438\n\nSiodmak, Robert 3198, 4542\n\nSirk, Douglas 4227, 4244\n\nSitka, Emil 271, 288, 661, 1285, 2066, 2964, 3066, 4073, 4296, 4428\n\nSkala, Lilia 1826\n\nSkelton, Red 4057, 4291\n\nSkelton, Tiny 3555, 4389, 4424\n\nSkerritt, Tom 1098, 1878, 2188, 3138, 3872, 4311, 4969\n\nSkinner, Edna 3754\n\nSlate, Jeremy 2671, 4035, 4576\n\nSlater, Helen 760\n\nSlattery, Richard X. 122, 1112, 4431\n\nSlatzer, Robert F. 315, 2719\n\nSlaven, Brad 436, 728, 2266, 2687, 3257, 3388, 3484, 4026, 4097, 4466, 4507\n\nSlavin, George 759, 1766, 2781, 3318, 3966\n\nSleep 'n Eat _see_ Best, Willie\n\nSleeper, Martha 3409, 4205, 4824\n\nSlezak, Walter 3684\n\nSloane, Everett 1913, 2616, 4795\n\nSloane, Paul H. 1532, 2393, 4284\n\nSloman, Edward 686, 825, 1685\n\nSmall, Louise 2634, 3303\n\nSmalley, Phillips 445, 1882\n\nSmart, Jack 4983\n\nSmart, Ralph 350\n\nSmith, Albert J. (Al) 295, 397, 434, 496, 754, 780, 1058, 1848, 1928, 2628, 3173, 3797, 3850, 4215, 4248, 4377, 4475, 4850\n\nSmith, Alexis 692, 2677, 3692, 4049, 4115, 5036\n\nSmith, Art 91, 2608\n\nSmith, Arthur 850, 2435, 2687, 2858, 3050, 3457, 4026, 4221, 4507, 4820\n\nSmith, C. Aubrey 3766, 4635\n\nSmith, Carl 208, 559\n\nSmith, Charles Martin 909, 2265, 2783, 3055, 4065, 4169\n\nSmith, Charles 2052, 3694\n\nSmith, Clifford (Cliff) 186, 1110, 1291, 3072, 3462, 4282, 4911, 4931, 4971\n\nSmith, Constance 3333\n\nSmith, Dean 720, 2348, 2490, 2706, 3527, 3993, 4090, 4633\n\nSmith, Drake 680\n\nSmith, Hal 3411, 4131, 4399\n\nSmith, Jack C. 183, 566, 725, 1281, 1304, 1444, 1464, 1465, 1574, 1732, 1812, 1859, 1900, 1925, 2306, 2420, 2740, 2778, 2927, 2979, 3044, 3107, 3525, 3624, 3912, 3913, 3944, 4103, 4104, 4135, 4226, 4419, 4573, 4581\n\nSmith, John 1434, 1482, 1551, 1922, 2076, 2295, 3213, 3299, 3779, 4451, 4755, 4921\n\nSmith, Kate 1792, 1847\n\nSmith, Kent 209, 804, 1015, 1112\n\nSmith, Lois 4148\n\nSmith, Martin Cruz 2807\n\nSmith, Noel 387, 611, 685, 714, 765, 1740, 2307, 3979, 4513\n\nSmith, Paul 22\n\nSmith, Queenie 2618\n\nSmith, Tom 143, 337, 344, 369, 422, 606, 731, 810, 930, 960, 1070, 1145, 1214, 1327, 1488, 1586, 1859, 1873, 1973, 2119, 2249, 2355, 2403, 2636, 2673, 2759, 2861, 2867, 2927, 2947, 3025, 3116, 3165, 3194, 3241, 3267, 3447, 3465, 3649, 3834, 3888, 4030, 4231, 4386, 4463, 4483, 4490, 4491, 4495, 4587, 4592, 4605, 4702, 4723, 4767, 4944, 5018, 5043, 5056\n\nSmith, Wallace 3896\n\nSmith, Will 4974\n\nSmith, William 190, 474, 1007, 1437, 1527, 1840, 2623, 2989, 3843, 4066, 4361, 4968\n\nSmith and Dale (Joe Smith, Charles Dale) 2815\n\nSmithee, Allen 882, 1015, 1251\n\nSmits, Jimmy 756, 2876\n\nThe Smoky Mountain Boys 3975\n\nSmyrner, Anne 302\n\nSnell, Earle 51, 56, 68, 500, 579, 668, 793, 827, 853, 1052, 1059, 1071, 1241, 1269, 1920, 2012, 2114, 2131, 2223, 2867, 3080, 3156, 3263, 3358, 3504, 3568, 3574, 3581, 3649, 3668, 3764, 3824, 4044, 4110, 4138, 4181, 4201, 4393, 4454, 4692, 4733, 4763, 4815, 4933\n\nSnipes, Wesley 1498\n\nSnodgrass, Carrie 224, 3018\n\nSnow, Heber 3476, 3913, 4279; _see also_ Worden, Hank\n\nSnowflake _see_ Toones, Fred \"Snowflake\"\n\nSoble, Ron 965, 1670, 2488, 2631, 2760, 4576, 4769\n\nSoderling, Walter 395, 714, 1378, 2125, 2937, 2967, 3220, 3371, 3711, 4121, 4868, 4932, 5043\n\nSofaer, Abraham 741, 1347\n\nSojin 1178\n\nSokoloff, Vladimir 251, 748, 2064, 2497, 3414, 4061\n\nSolar, Silvia 1530, 2507, 4629\n\nSoldani, Charles 103, 157, 278, 519, 956, 973, 1238, 1427, 1815, 1894, 1938, 2534, 2678, 3107, 3499, 3611, 5005\n\nSoler, Julian 601, 3008\n\nSollima, Sergio 316, 3639\n\nSommer, Elke 84\n\nSommerfield, Helga 358\n\nSommers, Julie 1644\n\nSondergaard, Gale 2064, 2385, 2586, 3112, 3369\n\nThe Sons of the Pioneers 67, 110, 296, 287, 323, 386, 576, 611, 615, 623, 682, 799, 868, 1105, 1134, 1192, 1240, 1246, 1488, 1522, 1767, 1775, 1818, 1836, 1914, 1918, 1964, 2124, 2272, 2330, 2368, 2514, 2530, 2534, 2540, 2632, 2727, 2730, 2806, 2828, 2874, 2877, 2886, 2893, 2973, 2975, 2983, 3100, 3236, 3409, 3449, 3483, 3520, 3522, 3585, 3600, 3694, 3904, 4013, 4016, 4018, 4025, 4037, 4042, 4051, 4077, 4083, 4156, 4197, 4202, 4206, 4314, 4315, 4401, 4410, 4608, 4641, 4646, 4681, 4764, 4806, 4810, 4817, 4834, 5050\n\nThe Sons of the Sage 4640\n\nSoo, Jack 2559\n\nSooter, Rudy 323, 337, 1455, 1474, 1898, 2279, 2690, 3270, 3303, 3408, 3452, 3590, 3706, 3913, 4019, 4023, 4365, 4393, 4587, 4660, 4667, 4685, 4841\n\nSorel, George 2586, 2836\n\nSorel, Louise 2587\n\nSorvino, Paul 4560\n\nSothern, Ann 1599, 4029\n\nSothern, Hugh 203, 437, 2124, 2342, 2516, 2837, 2864, 4257, 4336, 4773, 5062\n\nSoule, Olan 497, 980, 1837\n\nSouthern, Eve 1300, 1511\n\nSouthern, Hal 4895\n\nSouthwood, Charles 1030, 1351, 1954, 2628\n\nSouthworth, Tommy 3235, 3237\n\nSowards, George 188, 459, 472, 712, 800, 896, 973, 1245, 1380, 1416, 1517, 1939, 2014, 2270, 2280, 2388, 2421, 2539, 2589, 2596, 2701, 2825, 2915, 2970, 2988, 3108, 3116, 3153, 3424, 3478, 3709, 4007, 4044, 4112, 4147, 4205, 4657, 4957\n\nSowards, Jack B. 1036, 1081\n\nSowards, Lem 459, 2388, 3153, 3307, 4205, 4657\n\nSpaak, Agnes 1580\n\nSpaak, Catherine 4224\n\nSpace, Arthur 185, 191, 256, 491, 637, 981, 1134, 1214, 1268, 1427, 1526, 1914, 2217, 2482, 2502, 2544, 2804, 3020, 3219, 3530, 3649, 3806, 3837, 3902, 4057, 4073, 4223, 4257, 4450, 4686, 4712, 5086\n\nSpacek, Sissy 1611, 4169\n\nSpain, Fay 363, 4802\n\nSparks, Ned 3766\n\nSparv, Camilla 2489\n\nSpelling, Aaron 2902, 4376, 5038\n\nSpellman, Martin 1260, 2284, 3709\n\nSpencer, Bud 3, 376, 421, 551, 1242, 1358, 1581, 2011, 3297, 3819, 4334, 4444\n\nSpencer, Douglas 63, 789, 2095, 2502, 2518, 3040, 3221, 3339, 3535, 3671, 3808, 3966, 3972, 4674, 4792\n\nSpencer, Norman 3243\n\nSpencer, Shelley 2311\n\nSpencer, Tim 67, 110, 286, 287, 323, 386, 611, 623, 868, 1105, 1134, 1192, 1246, 1488, 1522, 1767, 1818, 1836, 1914, 1964, 2124, 2272, 2368, 2514, 2530, 2534, 2540, 2727, 2730, 2806, 2828, 2874, 2877, 2893, 2973, 2983, 3100, 3236, 3330, 3409, 3449, 3483, 3600, 3694, 3904, 4013, 4016, 4018, 4025, 4037, 4042, 4051, 4077, 4156, 4197, 4202, 4206, 4314, 4315, 4401, 4410, 4608, 4641, 4681, 4806, 4810, 4834, 5050\n\nSperling, Milton 651, 3384\n\nSpielberg, Stephen 4180\n\nSpilsbury, Klinton 2335\n\nThe Sportsmen Quartette 2451\n\nSpradlin, G.D. 1944, 1951, 2685, 2865, 2930, 3687, 4988\n\nSpriggins, Ace 1813\n\nSpriggins, Deuce 869, 1090, 2978, 3569, 3649, 3923, 4023, 4327\n\nSpringsteen, R.G. 68, 114, 134, 373, 567, 607, 789, 793, 827, 854, 1022, 1439, 1695, 1809, 1843, 1888, 1918, 1920, 1923, 1940, 2050, 2139, 2537, 2594, 2597, 2762, 2856, 2931, 3066, 3101, 3338, 3356, 3824, 3828, 3853, 3921, 4002, 4110, 4181, 4183, 4223, 4470, 4642, 4733, 4755, 4763\n\nStack, Robert 210, 762, 828, 1633, 2638, 2725, 3723, 4787\n\nStacy, James 2924, 3146\n\nThe Stafford Sisters 1595, 2872\n\nStahl, John M. 1427\n\nStaley, Joan 1670, 1727\n\nStallings, Lawrence 2516, 2837, 3684, 3814, 4360\n\nStallone, Frank 4453\n\nStalmaster, Hal 2052\n\nStalnaker, Charles 915, 2174\n\nStamp, Terence 407, 5067\n\nStander, Lionel 421, 1982, 2588, 2898, 4246\n\nStanding Bear, Chief 926, 1323, 2245, 2660, 3664, 3710, 4306, 4910\n\nStanding, Sir Guy 3022\n\nStanding, Wyndham 331, 3750\n\nStanley, Edwin 160, 337, 552, 4513, 4535\n\nStanley, Forrest 2972, 3442\n\nStanley, Helene 228, 970\n\nStanley, Kim 2559\n\nStanley, Louise 724, 1193, 1681, 1682, 1756, 2165, 2172, 2298, 2927, 3098, 3476, 3913, 4382, 5077\n\nStanley, Paul 563, 838, 4361\n\nStanley, Phyllis 4227\n\nStanton, Dean 980, 1250, 1945, 1998, 3186, 3404, 4431, 4451; _see also_ Stanton, Harry Dean\n\nStanton, Harry Dean 947, 998, 2666, 3055, 3252, 3428, 4874, 5048; _see also_ Stanton, Dean\n\nStanton, Robert 566, 2304; _see also_ Grant, Kirby\n\nStanwyck, Barbara 95, 406, 602, 681, 1412, 1478, 1738, 2624, 2646, 2691, 4566, 4665, 4740, 4970\n\nStapleton, Joan 1094\n\nStapp, Marjorie 392, 1263, 1671, 2176, 3676, 3839, 4805\n\nStarke, Pauline 4676\n\nStarlight (horse) 593, 1047, 1674, 1760, 3709, 3314, 3844, 4219, 4877, 4972\n\nStarling, Pat 263, 710, 1003, 1321, 3238, 3694, 3925, 4196\n\nStarr, Barbara 3266\n\nStarr, Ringo 394\n\nStarr, Ronald 3435\n\nStarr, Sally 3041\n\nStarrett, Charles 9, 205, 379, 386, 391, 392, 418, 481, 543, 576, 623, 682, 695, 780, 799, 870, 875, 878, 881, 928, 1046, 1059, 1145, 1192, 1213, 1298, 1309, 1404, 1449, 1453, 1460, 1488, 1497, 1725, 1758, 1806, 1815, 1938, 2068, 2103, 2105, 2175, 2176, 2178, 2189, 2262, 2269, 2272, 2296, 2361, 2397, 2540, 2632, 2633, 2730, 2828, 2886, 2907, 2939, 2973, 2975, 2978, 2983, 3001, 3042, 3063, 3094, 3167, 3168, 3171, 3209, 3233, 3358, 3385, 3449, 3453, 3464, 3466, 3493, 3505, 3507, 3520, 3552, 3565, 3616, 3619, 3629, 3651, 3666, 3678, 3763, 3940, 3978, 4042, 4044, 4053, 4077, 4117, 4154, 4156, 4167, 4191, 4272, 4290, 4296, 4305, 4314, 4315, 4391, 4401, 4410, 4496, 4506, 4531, 4608, 4609, 4610, 4658, 4806, 4810, 4812, 4827, 4832, 4834, 4888\n\nStarrett, Jack 901, 2671, 3961, 4149\n\nStaub, Ralph 739, 843, 3163, 4842, 5077\n\nStayer, Frank R. 1579, 3772\n\nSteagall, Red 959\n\nSteckler, Ray Dennis 404\n\nStedman, Myrtle 4025\n\nSteel, Alan 503, 1275, 3689\n\nSteel, Anthony 2219, 4689\n\nSteele, Bill 1369, 1614, 3692\n\nSteele, Bob 54, 76, 140, 157, 233, 334, 335, 336, 395, 486, 493, 495, 510, 562, 567, 667, 677, 689, 718, 777, 792, 803, 808, 896, 968, 1025, 1026, 1037, 1051, 1137, 1164, 1182, 1193, 1211, 1281, 1282, 1408, 1422, 1489, 1494, 1504, 1513, 1631, 1647, 1671, 1681, 1683, 1769, 1809, 1814, 1837, 1872, 2102, 2110, 2167, 2205, 2220, 2281, 2285, 2356, 2370, 2421, 2524, 2590, 2626, 2632, 2645, 2759, 2763, 2764, 2777, 2812, 2813, 2841, 2860, 2895, 2938, 2957, 2971, 3108, 3025, 3040, 3044, 3230, 3275, 3332, 3365, 3443, 3459, 3475, 3477, 3485, 3492, 3517, 3524, 3527, 3608, 3673, 3690, 3708, 3724, 3728, 3805, 3816, 3824, 3890, 3832, 3933, 3942, 3950, 3968, 3977, 3990, 4004, 4032, 4049, 4050, 4073, 4189, 4223, 4290, 4293, 4382, 4394, 4409, 4445, 4472, 4483, 4489, 4516, 4552, 4581, 4599, 4649, 4684, 4695, 4811, 4843, 4853, 4855, 4958, 4973, 4987, 5030, 5061\n\nSteele, Hamilton 3292\n\nSteele, Karen 1026, 3430, 3642, 4530, 4830\n\nSteele, Marjorie 1249, 4467\n\nSteele, Tom 28, 291, 299, 309, 385, 631, 637, 665, 731, 779, 927, 956, 1040, 1073, 1345, 1367, 1454, 1543, 1576, 1721, 1873, 1920, 1971, 1980, 2022, 2026, 2035, 2074, 2125, 2128, 2424, 2427, 2564, 2600, 2673, 2942, 2966, 2994, 2995, 3088, 3230, 3271, 3322, 3367, 3482, 3608, 3693, 3751, 3856, 3892, 3893, 3936, 4011, 4016, 4259, 4319, 4487, 4551, 4584, 4733, 4759, 5035, 5103\n\nSteele, William 801, 1276, 1897, 2121, 2245, 2422, 2943, 3601, 3852, 4621, 4851, 4863\n\nSteenburgen, Mary 1590\n\nSteers, Larry 3076, 3504, 3669\n\nStefani, Joseph 3953\n\nSteffen, Anthony 118, 133, 1286, 1508, 1530, 2390, 2505, 3514, 3782, 4920; _see also_ De Teffe, Antonio\n\nStegani, Giorgio 18, 302; _see also_ Finley, George\n\nStehli, Edgar 1161, 2813\n\nSteiger, Rod 676, 1066, 1171, 2065, 2148, 2853, 3640\n\nStein, Paul S. 2458\n\nStein, Sammy 3550, 3866, 4980\n\nSteinbeck, John 3319, 3320, 4751\n\nSteiner, John 396, 697, 2569\n\nSteiner, Max 1563\n\nSteiner, William 3065\n\nStelling, William 437\n\nStephens, Harvey 22, 1394, 2864, 2926, 3133, 3561, 4112, 4284, 4312, 4376, 4456, 4868\n\nStephens, Laraine 1413, 3274\n\nStephens, Marvin 3429\n\nStephenson, Henry 2160\n\nStephenson, James 874, 1820\n\nSterling, Jan 2563, 3140, 3954, 4069, 4714\n\nSterling, Robert 803, 1519, 3621\n\nSterling, Tisha 831, 2059, 3152\n\nStern, Daniel 760, 761\n\nStevens, Andrew 978, 1014, 2930, 3843\n\nStevens, Angela 377, 1971, 2017, 2105, 2664, 2960, 4682\n\nStevens, Brinke 4351\n\nStevens, Charles 6, 73, 159, 198, 204, 235, 282, 328, 414, 428, 446, 472, 557, 612, 613, 616, 626, 825, 844, 864, 1080, 1162, 1234, 1390, 1458, 1511, 1518, 1562, 1605, 1854, 1991, 2066, 2145, 2240, 2536, 2552, 2564, 2585, 2586, 2590, 2693, 2718, 2741, 2851, 2868, 2927, 2994, 3040, 3095, 3126, 3353, 3383, 3440, 3550, 3586, 3608, 3609, 3690, 3692, 3728, 3974, 3999, 4054, 4055, 4228, 4280, 4408, 4456, 4478, 4580, 4618, 4650, 4704, 4705, 4744, 4750, 4766, 4767, 4771, 4792, 4875, 4923, 4926, 4971, 5001\n\nStevens, Clarke 2, 940, 1506, 4087\n\nStevens, Craig 539, 1166, 1188, 2161, 2627\n\nStevens, George 95, 1367, 1560, 3808\n\nStevens, Grace 4293\n\nStevens, Inger 1346, 1354, 1760, 3641, 4431\n\nStevens, Jean 1453, 3385\n\nStevens, K.T. 4588\n\nStevens, Lois 450\n\nStevens, Louis 442, 578, 749, 790, 1365, 1618, 1668, 1936, 4309, 5012\n\nStevens, Mark 1669, 1742, 1748, 2108, 3698\n\nStevens, Onslow 648, 700, 1261, 1535, 1579, 1605, 1767, 1863, 1890, 1964, 3696, 4206, 4329, 4338, 4367, 4548, 5046\n\nStevens, Paul 1477, 2265\n\nStevens, Robert 2397, 2419, 3042, 3067, 4327; _see also_ Kellard, Robert\n\nStevens, Rose Ann 1145\n\nStevens, Stella 22, 221, 702, 768, 1840, 1926, 2442, 2810, 3217, 4471, 4782\n\nStevens, Warren 1188, 1727, 2511, 2557, 2813, 3333, 3559, 3811, 4109, 4505\n\nStevenson, Hayden 2367\n\nStevenson, Houseley 599, 692, 718, 930, 1420, 1487, 2605, 2856, 3247, 3757, 3862, 4771, 4962, 5045\n\nStevenson, James 3538\n\nStevenson, Margot 1622\n\nStevenson, Robert Louis 25, 296, 4536\n\nStevenson, Robert 1261, 2052, 2887, 4962\n\nStevenson, Venetia 981, 3791\n\nStewart, Anna Marie 4695\n\nStewart, Anna 4046\n\nStewart, Charlotte 727\n\nStewart, Donald 157, 4884, 4957\n\nStewart, Douglas 41, 3778\n\nStewart, Elaine 1078, 1238, 1875, 2797, 3954\n\nStewart, Eleanor 137, 1370, 1638, 1683, 1812, 2647, 2736, 3016, 3116, 3279, 3479, 3591, 3706, 3899, 4104, 4883\n\nStewart, Evelyn 18, 400, 3786, 4664, 4920; _see also_ Galli, Ida\n\nStewart, James 81, 290, 519, 720, 727, 1084, 1346, 1945, 2525, 2561, 2751, 2797, 3283, 3606, 3816, 3847, 4626\n\nStewart, Nicodemus 930, 4059\n\nStewart, Patrick 2120\n\nStewart, Paul 348\n\nStewart, Peggy 51, 93, 235, 368, 412, 607, 657, 731, 778, 779, 827, 984, 994, 1059, 1133, 1326, 1345, 1463, 1689, 1905, 2085, 2597, 2680, 2928, 3088, 3326, 3649, 3822, 3824, 3893, 3936, 4011, 4110, 4278, 4508, 4584, 4681, 4733, 4800, 4803\n\nStewart, Peter 70, 139, 334, 335, 336, 1445, 1667, 2977, 3164, 3232, 3448, 3592, 3677, 4302, 4303, 4313, 4589; _see also_ Newfield, Sam\n\nStewart, Redd 3619\n\nStewart, Roy 556, 811, 814, 954, 1269, 1300, 1524, 1634, 1972, 2207, 2422, 2741, 3618, 3655\n\nStickland, Mabel 458, 3617\n\nStiers, David Ogden 45, 2001\n\nStirling, Linda 712, 930, 2035, 3524, 3693, 3707, 3820, 3826, 4463, 4734, 4763, 5103\n\nStockdale, Carl 69, 231, 258, 847, 894, 1095, 1228, 1270, 1802, 1925, 2014, 2153, 2207, 2245, 2257, 2258, 2304, 2399, 2599, 2850, 2943, 2961, 3003, 3109, 3289, 3371, 3575, 3575, 3695, 3846, 3924, 4096, 4257, 4314, 4385, 4401, 4761\n\nStockman, Boyd 14, 42, 116, 214, 304, 389, 492, 783, 850, 893, 1356, 1442, 1502, 1523, 1665, 1670, 1690, 1724, 1772, 1868, 1890, 1991, 2296, 2426, 2525, 2765, 2798, 2805, 2944, 3002, 3050, 3157, 3277, 3430, 3447, 3510, 3762, 3890, 4095, 4099, 4118, 4204, 4327, 4520, 4813, 4886, 5038\n\nStockwell, Dean 677, 1671, 2094, 3813, 4129\n\nStockwell, Guy 86, 1510, 3127, 4369\n\nStokey, Mike 3381\n\nStoll, Gunther 3381\n\nStoloff, Benjamin 1083, 2843\n\nStone, Carol 1431\n\nStone, Cliffie 4021\n\nStone, Fred 4494, 4851\n\nStone, George E. 53, 520, 715, 747, 1436, 1457, 1676, 2070, 2208, 2423, 2689, 2831, 3409, 3865, 4422, 4639, 4673, 4750\n\nStone, Harold J. 509, 2917, 3853, 4119, 4333\n\nStone, Jeff 2405\n\nStone, John 1636, 2222, 2238, 2773, 4352\n\nStone, Lewis 200, 2013, 2816, 2936, 4358\n\nStone, Milburn 166, 274, 497, 599, 606, 790, 892, 939, 1024, 1151, 1317, 1627, 1647, 1754, 2652, 3074, 3630, 3722, 3861, 3980, 4365, 4598, 4907, 5072\n\nStone, Paula 1931, 4513, 4534\n\nStone, Sharon 3206\n\nStonehouse, Ruth 397, 2606\n\nStoney, Jack 3234, 3544, 4057, 4132, 4240, 4900\n\nStorch, Larry 1631, 1669, 1986, 2437\n\nStorey, June 410, 662, 797, 1512, 1916, 1980, 2701, 3254, 4023, 4052\n\nStorm, Gale 42, 914, 1175, 2032, 2109, 2514, 3330, 3673, 4118, 4310\n\nStorm, Jerome (Jerry) 851, 1060, 3239, 4475\n\nStorm, Rafael 3634\n\nStorm, Wayne 2859\n\nStossel, Ludwig 4257\n\nStout, Archie 1932, 4282\n\nStowe, Madeleine 194, 2216\n\nStrafford, Peggy 4531\n\nStraight, Clarence 796, 1264, 1603, 1627, 1720, 1772, 1963, 2899, 3104, 3803\n\nStrait, George 3191\n\nStrang, Harry 170, 698, 757, 798, 844, 863, 1024, 1029, 1378, 1512, 1756, 1862, 1884, 2125, 2145, 2424, 2471, 2552, 2637, 2638, 2652, 2790, 2815, 2842, 2867, 2884, 2902, 3086, 3179, 3194, 3270, 3384, 3520, 3540, 3585, 3711, 3902, 3912, 3928, 4059, 4257, 4336, 4469, 4483, 4612, 4726, 4766, 4803, 4849, 5096\n\nStrange, Glenn 13, 135, 137, 144, 155, 180, 189, 204, 210, 230, 231, 272, 338, 353, 369, 387, 422, 440, 458, 460, 470, 573, 597, 605, 606, 611, 623, 632, 642, 683, 684, 714, 807, 812, 824, 856, 876, 926, 946, 960, 986, 1025, 1070, 1092, 1097, 1099, 1146, 1156, 1174, 1234, 1247, 1254, 1270, 1274, 1307, 1311, 1320, 1332, 1389, 1390, 1416, 1474, 1475, 1487, 1488, 1490, 1517, 1556, 1587, 1645, 1662, 1682, 1714, 1740, 1754, 1756, 1766, 1781, 1785, 1789, 1798, 1895, 1929, 1978, 2066, 2081, 2111, 2114, 2164, 2172, 2234, 2235, 2240, 2263, 2270, 2283, 2292, 2300, 2334, 2378, 2385, 2403, 2408, 2418, 2481, 2485, 2573, 2628, 2647, 2678, 2689, 2703, 2733, 2785, 2802, 2869, 2993, 2999, 3016, 3028, 3087, 3089, 3102, 3156, 3159, 3160, 3177, 3200, 3232, 3260, 3267, 3305, 3323, 3357, 3390, 3407, 3427, 3450, 3511, 3542, 3586, 3592, 3600, 3615, 3673, 3693, 3750, 3880, 3893, 3924, 3927, 3949, 3994, 4022, 4032, 4057, 4072, 4096, 4103, 4125, 4130, 4131, 4142, 4184, 4187, 4201, 4202, 4208, 4212, 4256, 4291, 4294, 4323, 4364, 4377, 4504, 4513, 4539, 4564, 4573, 4598, 4702, 4705, 4725, 4761, 4766, 4834, 4838, 4844, 4854, 4887, 4923, 4944, 4957, 4987, 5008, 5021, 5032, 5033\n\nStrange, Robert 135, 1042, 1264, 1436, 2132, 2482, 3564, 3908, 4051, 4932\n\nStrasberg, Susan 3398\n\nStratas, Teresa 639\n\nStratford, Dean 531, 3727, 3841\n\nStratton, Chet 2059\n\nStratton, Gil 1564\n\nStrauch, Joe, Jr. 283, 291, 1821, 1915, 4673\n\nStrauss, Peter 3184, 3340, 3987\n\nStrauss, Robert 1406, 4098\n\nStrauss, Theodore 602\n\nStrickland, Amzie 2563, 2695\n\nStricklyn, Ray 148, 2241, 3133, 5070\n\nStritch, Elaine 4373\n\nStrock, Herbert L. 528, 3445, 4172\n\nStrode, Woody 372, 421, 509, 1499, 1510, 2097, 2140, 2228, 2447, 2483, 2561, 2898, 3147, 3180, 3204, 3206, 3402, 3748, 3776, 3807, 4626, 4664, 5007\n\nStroud, Don 223, 965, 2044, 2519, 3735, 3991\n\nStrudwick, Sheppard 279, 2109, 3319\n\nStuart, Giacomo Rossi 220, 1305, 1359, 1617, 1700, 2614, 5085, 5087, 5099; _see also_ Stuart, Jack\n\nStuart, Jack 1358, 1519; _see also_ Stuart, Giacomo Rossi\n\nStuart, Mary 659, 4399\n\nStuart, Mel 740\n\nStuart, Nick 1037, 1746, 3526, 4190\n\nStuart, Randy 1427, 2521, 4123\n\nStubbs, Harry 2159, 3126, 3922, 4215\n\nStudi, Wes 36, 37, 805, 942, 1534, 2216, 2402, 2657, 4169\n\nSturges, John 189, 193, 292, 654, 738, 1234, 1702, 1765, 1843, 2013, 2044, 2240, 2246, 2497, 3777, 4771\n\nSturgess, Olive 3365\n\nSturgess, Preston 271\n\nStutenroth, Eugene (Gene) 204, 325, 2594, 2857, 3750; _see also_ Roth, Gene\n\nSuarez, Jose 175, 1393, 2683, 3297, 4288\n\nSullivan, Barry 207, 548, 640, 1150, 1412, 1996, 2151, 2624, 2986, 3055, 3791, 4098, 4224, 4249, 4300, 4344, 4799, 5021, 5082\n\nSullivan, Billy 2913, 4377\n\nSullivan, C. Gardner 1999, 2831, 2892, 4591, 4665, 4758\n\nSullivan, Don 913, 1561\n\nSullivan, Francis L. 3131, 3701\n\nSullivan, Jean 1236\n\nSullivan, Ruth 1280, 2024\n\nSully, Frank 550, 1071, 1225, 1361, 1451, 1688, 1883, 2205, 2287, 2521, 2549, 2805, 2834, 3168, 3305, 3355, 3374, 3897, 4053, 4073, 4079, 4086, 4227, 4378\n\nSummers, Ann 179, 1308\n\nSummers, Colleen 2435, 4024\n\nSummers, Hope 1354, 3806, 4064\n\nSummers, Jerry 319, 1343, 1631, 2267, 2381, 2426, 3193, 3737, 5065\n\nSummers, Neil 120, 194, 1733, 2057, 2092, 2328, 2348, 2474, 2723, 3836, 4475\n\nSummerville, Slim 1599, 1853, 2031, 2130, 2180, 3914, 4071, 4221, 4316, 4567, 4645, 4701, 4849, 4900\n\nSunderberg, Clinton 94, 317, 1945, 2144\n\nThe Sunshine Boys 695, 1157, 2068, 2105, 3168, 3209, 3257, 3976, 4015, 4131, 4590, 4818, 4829, 4935\n\nThe Sunshine Girls 2435, 4024\n\nSutherland, Donald 941\n\nSutherland, Kiefer 882, 5067, 5069\n\nSutherlin, Wayne 487, 909, 1641, 4345\n\nSutton, Grady 486, 1106, 1179, 1616, 1631, 2131, 2163, 2221, 3022, 3128, 3991, 4023, 4138, 4210, 4412\n\nSutton, John 639, 1949, 2213, 3701, 4214\n\nSutton, Kay 1967, 2304, 2528\n\nSutton, Paul 64, 242, 757, 1562, 1974, 1978, 2031, 2208, 2586, 2828, 2831, 3100, 3431, 3466, 3894, 4184, 4456, 4647, 4747, 4940, 5035\n\nSvenson, Bo 506, 719, 1899, 2568, 3981\n\nSwackhamer, F.W. 1067, 1068, 2503, 3371\n\nSwain, Mack 1598\n\nSwarthout, Gladys 3609\n\nSwarthout, Miles Hood 3847\n\nSweet, Blanche 3896, 4174, 4326\n\nSwenson, Karl 515, 1371, 1663, 1772, 1840, 1943, 1945, 1963, 2430, 2501, 2519, 2813, 2829, 2902, 4035, 4633, 4936\n\nSwerling, Jo 4851\n\nSwickard, Josef 480, 668, 817, 1602, 2003, 2394, 2648, 3018, 3079, 3699, 4020, 5096\n\nSwift, Don 633, 4385, 4891, 4933\n\nSwift, Susan 740\n\nSwit, Loretta 2188\n\nSwitzer, Carl \"Alfalfa\" 277, 1103, 1509, 3342, 4473, 4925\n\nSwofford, Ken 919, 1755, 1998, 2093, 2906, 3747\n\nSykes, Brenda 3950\n\nSykes, Eric 3807\n\nSyms, Sylvia 1075\n\nTabori, Kristoffer 2058\n\nTaeger, Ralph 1922, 4098\n\nTaggart, Ben 710, 870, 1627, 2633, 2678, 2994, 3561, 4006, 4072, 4980\n\nThe Tailor Maids 870\n\nTait, Don 122, 672, 2911, 4537\n\nTakei, George 2848, 2849\n\nTalbot, Helen 286, 610, 644, 836, 1134, 2125, 2427, 2967, 3210, 3694, 4016, 4487\n\nTalbot, Lyle 2, 449, 630, 716, 791, 940, 1063, 1074, 1240, 1253, 1597, 1693, 1711, 1829, 2085, 2146, 2294, 2539, 2622, 2644, 2665, 2680, 2824, 2863, 2960, 3209, 3936, 4001, 4013, 4124, 4292, 4301, 4383, 4504, 4588, 4620, 4729, 4831, 5039\n\nTalbot, Monroe 8, 1550, 1760, 2481, 4760, 4981\n\nTalbot, Nita 2677, 2895, 4217\n\nTalbot, Slim 935, 1560, 1772, 2561, 4284, 4546\n\nTalbott, Gloria 53, 148, 433, 678, 1053, 1245, 2834, 2868, 2871, 2929, 5066\n\nTaliaferro, Hal 69, 79, 201, 353, 402, 442, 459, 516, 575, 576, 612, 666, 715, 801, 868, 878, 960, 1416, 1456, 1468, 1490, 1626, 1647, 1662, 1683, 1815, 1818, 1822, 1934, 1964, 1982, 2032, 2068, 2128, 2273, 2274, 2311, 2378, 2399, 2480, 2530, 2552, 2828, 2907, 2983, 3003, 3015, 3077, 3088, 3105, 3111, 3128, 3247, 3279, 3323, 3330, 3461, 3474, 3483, 3520, 3525, 3550, 3600, 3603, 3626, 3675, 3692, 3728, 3744, 3827, 3904, 4018, 4037, 4042, 4059, 4081, 4104, 4156, 4318, 4410, 4456, 4558, 4600, 4608, 4643, 4667, 4734, 4803, 4818, 4834, 5021, 5050, 5059, 5103; _see also_ Wales, Wally; Williams, Walt\n\nTalmadge, Norma 1572, 1825\n\nTalmadge, Richard 357, 444, 1957, 2026, 3075, 5042\n\nTalman, William 223, 2042, 2109, 3069, 3966, 4613, 4680\n\nTalton, Alix (Alice) 1225, 3271\n\nTamberlani, Carlo 1185, 1966, 2316, 5092\n\nTamblyn, Russ 692, 748, 1274, 1278, 1945, 2198, 2199, 2575, 3780, 3997, 5066\n\nTamiroff, Akim 642, 1290, 1631, 1883, 2756, 2831, 3344, 3414, 4061, 4312, 4665\n\nTanchuck, Nat 66, 555, 1956, 3057\n\nTandy, Jessica 2351\n\nTani, Yoko 3729\n\nTanin, Eleanore 4805\n\nTannen, Charles 384, 759, 863, 1165, 1649, 1995, 2031, 3333, 3374, 3384, 4109, 4184, 5072\n\nTannen, William (Bill) 94, 211, 293, 377, 1216, 1347, 1375, 1487, 1644, 2107, 2037, 2287, 2482, 2731, 2789, 2790, 2818, 2838, 3208, 3473, 3530, 3541, 3702, 3931, 4057, 4121, 4198, 4461, 5030, 5032\n\nTansey, Emma 305, 2151, 2299, 3332, 5027\n\nTansey, Robert Emmett 13, 23, 138, 150, 157, 383, 658, 680, 795, 876, 1025, 1137, 1155, 1156, 1196, 1224, 1304, 1330, 1585, 1604, 1789, 1798, 2263, 2418, 2481, 2957, 3101, 3503, 4032, 4131, 4319, 4502, 4590, 4606, 4844, 4853, 4884, 4969, 4987; _see also_ Emmett, Robert; Lane, Al\n\nTansey, Sherry 130, 140, 297, 335, 336, 341, 343, 414, 423, 431, 499, 752, 946, 1156, 1281, 1445, 1455, 1503, 1587, 1604, 1681, 1682, 1728, 1752, 1756, 1789, 1812, 1965, 1969, 2151, 2263, 2299, 2300, 2306, 2360, 2542, 2572, 2640, 2647, 2690, 2740, 2785, 2923, 2977, 2993, 3028, 3036, 3044, 3073, 3085, 3086, 3101, 3107, 3237, 3278, 3332, 3407, 3443, 3491, 3528, 3545, 3588, 3591, 3898, 3907, 3913, 3944, 3946, 4030, 4128, 4130, 4226, 4324, 4382, 4554, 4581, 4701, 4816, 4833, 4893, 4947; _see also_ Sheridan, James\n\nTapley, Colin 129, 4762\n\nTashlin, Frank 3020, 4005\n\nTashman, Lilyan 1430\n\nTate, Kevin 3783\n\nTate, Lincoln 3727\n\nTate, Pamela 4662\n\nTate, Patricia 950\n\nTaurog, Norman 1564, 3040, 3409, 4412, 4567\n\nTayback, Vic 1220\n\nTaylor, Al 37, 137, 174, 306, 326, 327, 340, 341, 366, 395, 424, 500, 615, 631, 634, 684, 777, 810, 813, 814, 818, 853, 951, 956, 1023, 1073, 1088, 1187, 1276, 1306, 1307, 1320, 1474, 1504, 1512, 1513, 1537, 1557, 1558, 1574, 1586, 1595, 1751, 1859, 1862, 2014, 2032, 2132, 2247, 2250, 2268, 2274, 2276, 2283, 2299, 2403, 2411, 2412, 2514, 2546, 2600, 2648, 2701, 2745, 2802, 2867, 2890, 2915, 2966, 2971, 2995, 3052, 3082, 3084, 3135, 3163, 3164, 3210, 3228, 3230, 3259, 3260, 3270, 3279, 3303, 3325, 3424, 3452, 3486, 3524, 3525, 3548, 3564, 3665, 3708, 3825, 3827, 3834, 4106, 4248, 4294, 4409, 4483, 4525, 4581, 4617, 4667, 4683, 4708, 4732, 4854, 4855, 4861, 5037, 5040, 5056, 5096, 5104\n\nTaylor, Buck 46, 86, 266, 676, 822, 884, 1068, 1147, 1754, 1840, 2029, 2087, 2657, 2810, 3141, 3184, 3661, 4122, 4434, 4453, 4505, 4565, 4575, 4960, 4968, 4974\n\nTaylor, Cliff 896, 940, 994, 1272, 1461, 1463, 1939, 2123, 2582, 2596, 2960, 3999, 4408, 4821, 4962\n\nTaylor, Delores 330\n\nTaylor, Don 73, 1358, 1564, 1642, 1926, 2568, 2793, 3726, 3773, 3991, 4976\n\nTaylor, Dub 14, 15, 26, 28, 266, 307, 391, 481, 484, 700, 822, 850, 879, 871, 875, 878, 928, 966, 1015, 1274, 1453, 1500, 1680, 1684, 1765, 1768, 1828, 1913, 1926, 2069, 2119, 2197, 2293, 2296, 2398, 2501, 2403, 2508, 2547, 2623, 2632, 2721, 2821, 2858, 2897, 2909, 2944, 2978, 3050, 3055, 3109, 3141, 3170, 3265, 3277, 3370, 3393, 3416, 3467, 3484, 3500, 3556, 3616, 3651, 3666, 3674, 3678, 3687, 3806, 3840, 3894, 3991, 4021, 4064, 4191, 4210, 4232, 4305, 4329, 4361, 4465, 4537, 4638, 4736, 4934, 4936, 4980, 4994, 5020, 5035\n\nTaylor, Duke 286, 637, 1073, 2022, 2035, 2128, 2399, 2403, 2421, 3015, 3088, 3111, 4011, 5096, 5103\n\nTaylor, Elizabeth 632, 1560, 3138\n\nTaylor, Eric 796, 1823, 2825, 3024, 4043\n\nTaylor, Estelle 747, 4058\n\nTaylor, Ferris 739, 914, 1367, 1390, 1458, 1490, 1601, 1704, 2537, 2552, 2649, 2701, 3162, 3234, 3254, 3490, 3709, 3861, 4491, 4611, 4932\n\nTaylor, Forrest 25, 49, 68, 137, 139, 151, 235, 297, 303, 311, 341, 349, 352, 353, 380, 391, 453, 475, 489, 542, 573, 575, 597, 606, 642, 658, 682, 716, 724, 731, 773, 777, 784, 786, 795, 835, 848, 867, 880, 928, 972, 1022, 1025, 1041, 1051, 1052, 1145, 1157, 1192, 1193, 1281, 1298, 1311, 1326, 1327, 1330, 1364, 1420, 1445, 1465, 1487, 1497, 1556, 1600, 1604, 1682, 1747, 1812, 1860, 1915, 1929, 1956, 1964, 1974, 2006, 2082, 2107, 2124, 2128, 2133, 2166, 2197, 2256, 2278, 2294, 2296, 2360, 2392, 2403, 2415, 2425, 2456, 2469, 2572, 2601, 2622, 2640, 2660, 2673, 2677, 2690, 2740, 2744, 2762, 2781, 2798, 2806, 2812, 2927, 2933, 2948, 2966, 2967, 2979, 3016, 3025, 3067, 3077, 3081, 3089, 3132, 3160, 3168, 3234, 3264, 3280, 3332, 3351, 3361, 3407, 3443, 3449, 3451, 3456, 3461, 3483, 3490, 3491, 3520, 3526, 3531, 3546, 3569, 3581, 3599, 3616, 3626, 3649, 3652, 3677, 3678, 3750, 3799, 3904, 3914, 3922, 3974, 4016, 4018, 4032, 4049, 4056, 4072, 4110, 4114, 4121, 4130, 4136, 4143, 4154, 4191, 4202, 4279, 4302, 4305, 4324, 4327, 4409, 4460, 4483, 4489, 4502, 4504, 4515, 4549, 4554, 4617, 4649, 4660, 4674, 4681, 4686, 4687, 4698, 4701, 4804, 4806, 4814, 4848, 4883, 4953, 4957, 4980, 5027, 5043, 5103\n\nTaylor, Jack 742, 915, 1318, 4561\n\nTaylor, Joan 117, 1319, 1409, 3607, 3722, 4784, 4787\n\nTaylor, Jud 1251, 3007, 5006\n\nTaylor, Kent 46, 521, 939, 1343, 1396, 1451, 1551, 2008, 2267, 2732, 3993, 3245, 4203, 4456, 4769, 4845\n\nTaylor, Libby 1427, 1947, 3634, 3711, 4863\n\nTaylor, Norma 4592\n\nTaylor, Ray 135, 199, 264, 364, 436, 475, 476, 478, 582, 679, 711, 726, 728, 764, 865, 893, 939, 994, 1167, 1297, 1337, 1339, 1367, 1463, 1465, 1555, 1614, 1627, 1724, 1802, 1803, 1863, 1868, 2014, 2250, 2266, 2273, 2282, 2424, 2526, 2582, 2652, 2740, 2915, 2946, 2961, 3015, 3029, 3087, 3103, 3142, 3220, 3224, 3257, 3261, 3288, 3289, 3291, 3388, 3394, 3452, 3555, 3630, 3799, 3802, 3903, 3995, 3999, 4097, 4103, 4178, 4201, 4279, 4365, 4379, 4437, 4466, 4732, 4735, 4808, 4813, 4829, 4858, 4935, 5001\n\nTaylor, Rex 1805\n\nTaylor, Robert 73, 332, 1093, 1773, 1922, 2051, 2198, 2246, 2575, 3387, 3440, 3671, 3730, 3744, 4121, 4857\n\nTaylor, Rod 1007, 1560, 2903, 2962, 3152, 4461, 4523\n\nTaylor, Vaughn 221, 378, 859, 1026, 1748, 2442, 3133, 3180, 3806, 4670, 4791, 5006\n\nTazil, Zara 929, 2290, 2976, 3302\n\nTead, Phil 1259, 1261, 4851\n\nTeagarden, Jack 4598\n\nTeague, Guy 28, 260, 418, 489, 1073, 1560, 2103, 2122, 2302, 2564, 2955, 3493, 3852, 4162, 4730, 4913, 5038\n\nTeal, Ray 37, 64, 65, 73, 75, 112, 203, 248, 332, 354, 357, 580, 648, 679, 685, 715, 741, 891, 969, 1026, 1029, 1111, 1134, 1247, 1364, 1408, 1480, 1599, 1640, 1719, 1735, 1770, 1774, 1792, 1913, 1925, 1989, 2088, 2109, 2370, 2486, 2511, 2515, 2652, 2790, 2836, 2837, 2838, 2871, 2901, 2973, 3090, 3149, 3156, 3170, 3195, 3219, 3247, 3339, 3437, 3636, 3671, 3750, 3757, 4079, 4168, 4223, 4234, 4257, 4336, 4483, 4498, 4635, 4682, 4747, 4842, 4890, 4932, 4962, 4989, 5066, 5096\n\nTearle, Conway 1045, 1634, 2067, 2147, 3771\n\nTedrow, Irene 903, 3671, 3705, 4387\n\nTellegen, Lou 4352\n\nTemple, Shirley 1395, 4214, 4442\n\nTenbrook, Harry 1, 739, 1084, 1214, 1297, 1395, 1426, 1456, 1553, 1805, 1863, 1898, 1974, 2207, 2416, 2599, 2670, 2756, 2861, 3220, 3279, 3289, 3549, 3555, 3650, 3702, 4073, 4100, 4103, 4271, 4299, 4385, 4396, 4899, 5001\n\nThe Tennessee Ramblers 3433, 3491, 4191, 5056\n\nTerhune, Bob (Robert) 953, 2093, 2700, 3324, 3911, 3973, 4802\n\nTerhune, Max 68, 151, 323, 369, 422, 424, 631, 810, 870, 872, 1475, 1553, 1560, 1724, 1751, 1789, 1798, 1822, 1860, 1868, 1898, 2114, 2166, 2282, 2552, 2802, 2968, 3194, 3259, 3261, 3290, 3325, 3433, 3454, 3482, 3549, 3568, 3668, 3823, 3826, 4085, 4220, 4319, 4323, 4372, 4393, 4458, 4497, 4502, 4515, 4520, 4558, 4587, 4606, 4660, 4813, 4815, 4847, 5027\n\nTerrell, Kenneth (Ken) 28, 37, 415, 464, 779, 853, 880, 951, 1164, 1522, 1543, 1573, 1618, 1627, 1981, 2006, 2022, 2035, 2125, 2128, 2132, 2545, 2597, 2600, 2867, 2966, 3024, 3132, 3185, 3221, 3230, 3563, 4011, 4024, 4488, 4649, 5001, 5103, 5104\n\nTerry, Al 263, 577, 2484, 2529, 4196\n\nTerry, Alice 1634\n\nTerry, Bob 535, 606, 773, 943, 1148, 1370, 2360, 2365, 2481, 2891, 2979, 3361, 3590, 3699, 3946, 3949, 3953, 3977, 4019, 4031, 4128, 4153, 4186, 4324, 4549, 5077\n\nTerry, Don 451, 2716, 4892\n\nTerry, Ethel Grey 4931\n\nTerry, Gordon 1871\n\nTerry, Philip 2521, 2676, 2831, 3046\n\nTerry, Richard 881, 2024; _see also_ Perrin, Jack\n\nTerry, Ruth 615, 1767, 1818, 2530, 3120, 3976\n\nTerry, Sheila 1796, 1895, 2297, 2766, 3575\n\nTerry, Tex 51, 67, 110, 235, 304, 476, 516, 796, 856, 1215, 1490, 1823, 1836, 1862, 2035, 2066, 2082, 2534, 2564, 2878, 2885, 2928, 2929, 2966, 3006, 3111, 3128, 3132, 3291, 3326, 3524, 3542, 3615, 3693, 3929, 4011, 4129, 4197, 4199, 4212, 4400, 4428, 4469, 4599, 5033, 5043\n\nTessari, Duccio 1135, 1350, 3118, 3380, 3785, 4182, 4277, 5085\n\nTessier, Robert 508, 1527, 2215, 4739\n\nTesti, Fabio 736, 1424, 2898, 5093\n\nTetley, Walter 3163, 3316, 4649\n\nTetzel, Joan 1187\n\nTetzlaff, Ted 4536, 4071\n\nTevos, Herbert 2644\n\nTewksbury, Peter 3753, 4123\n\nTex Ritter's Tornadoes 1900, 3913, 4685\n\nThe Texas Rangers 161, 739, 2273, 2861, 3220, 3291, 4006\n\nThe Texas Troubadours 1903\n\nThalasso, Arthur 459, 543, 729, 1470, 1929, 2014, 2355, 2473, 2815, 4385, 4532, 4607, 4750, 4865\n\nThane, Dirk 2414, 2701, 2998, 3303, 3448, 3771, 4553, 4617, 4815\n\nThatcher, Torin 639\n\nThaxter, Phyllis 402, 1408, 3750, 4080\n\nThayer, Julia 1751, 3015; _see also_ Carmen, Jean\n\nThayer, Lorna 1431, 2485, 3963, 4292\n\nThayer, Otis B. 3114, 3472, 4481\n\nThayer, Tiffany 1089\n\nTheby, Rosemary 1570, 2238, 4933\n\nTheiss, Ursula 82, 226\n\nThew, Harvey F. 2753, 3895, 4029, 4494\n\nThinnes, Roy 370, 701\n\nThomas, B.J. 2056\n\nThomas, Buckwheat 793\n\nThomas, Frank M. 112, 160, 514, 944, 1532, 1638, 2223, 2516, 2940, 3675, 3866, 4202, 4206, 4932, 5041\n\nThomas, Jameson 1013\n\nThomas, Lyn 854, 1452, 2669, 2818, 3328\n\nThomas, Peter 2214\n\nThomas, Ralph 636\n\nThomas, Richard 3306\n\nThomas, William C. 47, 2003, 3636, 4062, 4980\n\nThompson, Al 472, 1392, 1626, 3481, 4468\n\nThompson, Carlos 2227\n\nThompson, Fred 1492, 2031, 4200, 4404\n\nThompson, Hilarie 1864, 3740\n\nThompson, J. Lee 2141, 2489, 4895\n\nThompson, Ken 616\n\nThompson, Kenne 686, 723, 1300, 5015\n\nThompson, Lotus 3767\n\nThompson, Marshall 191, 1093, 4079, 4129\n\nThompson, Nick 352, 452, 664, 942, 1256, 1570, 1571, 2144, 2192, 2660, 3067, 3074, 3351, 3609, 3916, 4047, 4139, 4533, 4650\n\nThompson, Peter 1404, 1483, 3493, 3702, 5008\n\nThompson, Slim 1213\n\nThompson, Thomas 679\n\nThomson, Kenneth 1931, 1983, 4891, 4909\n\nThordsen, Kelly 1071, 3723, 3816, 4579\n\nThorne, William 764, 1335, 2181, 2268, 2774, 3555\n\nThornton, Billy Bob 44, 62, 992, 4045, 4453\n\nThorpe, Jerry 980, 2154, 2389\n\nThorpe, Jim 138, 159, 264, 303, 471, 642, 682, 776, 846, 1464, 1853, 2014, 1247, 2542, 2552, 2664, 2726, 2927, 2957, 3087, 3170, 3322, 3544, 3650, 3886, 4215, 4513, 4776, 4950, 4982\n\nThorpe, Richard 112, 198, 317, 451, 1049, 1178, 1290, 1493, 2013, 2185, 2394, 4245, 4593, 4645, 4683, 4725, 4944, 4997, 5032\n\nThorson, Russell 1608, 1669, 1709, 1769, 4013\n\nThreatt, Elizabeth 324\n\nThe Three Stooges 388, 1422, 1597, 2964, 3569\n\nThring, Frank 2492\n\nThunderbird, Chief 1863, 1993, 2245, 2831, 4214, 4971\n\nThundercloud, Chief 73, 95, 214, 264, 352, 379, 414, 552, 555, 618, 801, 844, 917, 926, 956, 969, 1253, 1323, 1367, 1384, 1532, 1626, 1687, 1762, 1949, 1956, 1991, 2134, 2286, 2288, 2399, 2403, 2552, 2717, 2815, 2957, 2994, 3088, 3116, 3126, 3154, 3224, 3245, 3351, 3361, 3433, 3482, 3512, 3599, 3650, 3653, 3664, 3702, 3859, 3886, 3905, 3926, 4032, 4411, 4523, 4635, 4701, 4760, 4849, 4932, 4971, 5032, 5062\n\nThurman, Bill (Billy) 1807, 3013, 3911, 4180\n\nThurston, Carol 127, 828, 1364, 2233, 3853, 5081\n\nTibbets, Martha 3270, 4667\n\nTibbett, Lawrence 2790\n\nTibbs, Casey 469, 508, 527, 768, 883, 2069, 4387, 4419, 4452, 4943\n\nTichy, Gerard 820, 1519, 1722, 2905, 3125, 3715, 4325\n\nTidyman, Ernest 1880\n\nTierney, Gene 279, 1949, 3374, 3757, 4795\n\nTierney, Lawrence 214, 293, 584, 915\n\nTiffin, Pamela 1011, 1765, 4749\n\nTilbury, Zeffie 3827\n\nTillis, Mel 4679, 4739\n\nTillotson, Johnny 628\n\nTilton, Charlene 455\n\nTilton, Martha 4170\n\nTin Tan 1223; _see also_ Valdes, German\n\nTinling, James 2208, 2239, 2423, 3471, 4184, 4650\n\nTinti, Gabrielle 641, 3372\n\nTobey, Kenneth 330, 509, 950, 967, 970, 1413, 1637, 1702, 1704, 2720, 3218, 3219, 3290, 3941, 4431, 4927\n\nTobias, George 567, 2583, 3290, 3538\n\nTobin, Genevieve 3071\n\nTodd, Ann 203, 514, 1084, 1622, 1920\n\nTodd, Dick 1512\n\nTodd, Harry 496, 543, 851, 1002, 1306, 1307, 1315, 1320, 1678, 1975, 2130, 2254, 2275, 2428, 2908, 3869, 3957, 3996, 4038, 4187, 4308, 4377, 4570, 4645, 4651, 4850, 5040\n\nTodd, James 1319, 1487, 3470\n\nTodd, Lisa 1098, 1106\n\nTodd, Mabel 866, 1144\n\nTodd, Richard 1846\n\nTodd, Thelma 2146, 2773\n\nTokar, Norman 121, 322, 3731, 4417, 4878\n\nTolan, Michael 1866, 1943, 3722\n\nTolar, Gene 1280, 2024\n\nToler, Sidney 626, 1594, 1855, 2270, 2610, 2733, 2923, 4359, 4580\n\nTombes, Andrew 214, 277, 642, 844, 1134, 1143, 1450, 1935, 2208, 2856, 3499, 3694, 3925, 4286, 4621\n\nTombragel, Maurice 274, 794, 896, 940, 1272, 1396, 1461, 1627, 1701, 1939, 2294, 2539, 2596, 2602, 2622, 2641, 2716, 2798, 2980, 3093, 3540, 3936, 3941, 4255, 4383, 4439, 4491, 4616, 4821, 5048\n\nTomlin, Pinky 3914\n\nTompkins, Joan 182\n\nTone, Franchot 4498\n\nToney, Jim 670, 2315, 2432, 3409\n\nTonge, Philip 664, 1485, 3040, 3061, 3411, 4473\n\nToomey, Regis 129, 257, 932, 961, 1164, 1199, 1392, 1433, 1633, 1741, 1773, 2016, 2237, 2351, 2768, 3597, 2831, 2837, 3224, 3952, 3998, 4132, 4136, 4257, 4336, 4450, 4461, 4665, 4791\n\nToone, Geoffrey 2052\n\nToones, Fred \"Snowflake\" 8, 107, 154, 183, 248, 287, 1024, 1047, 1345, 1380, 1455, 1467, 1512, 1595, 1638, 1751, 1760, 1798, 1802, 1805, 1859, 1873, 1950, 2166, 2299, 2432, 2648, 2667, 2850, 2909, 3022, 3232, 3259, 3325, 3413, 3414, 3434, 3454, 3626, 3899, 3919, 3927, 4318, 4586, 4617, 4703, 4798, 4954, 4981, 5050, 5056\n\nTorme, Mel 2168, 4768\n\nTorn, Rip 97, 838, 1244, 1375, 1826, 3017, 4134, 4881\n\nTorne, Regina 168\n\nTornek, Jack 15, 30, 234, 334, 378, 341, 343, 379, 422, 494, 731, 872, 1071, 1146, 1296, 1301, 1309, 1459, 1486, 1809, 1898, 2276, 2298, 2457, 2547, 2673, 2894, 2999, 3080, 3171, 3280, 3303, 3348, 3478, 3552, 3801, 3938, 4078, 4136, 4363, 4383, 4386, 4458, 4495, 4559, 4588, 4723, 4819, 4838, 4952, 5018\n\nTorrance, David 3537\n\nTorrence, Ernest 852, 1300, 1854, 3139, 4494\n\nTorres, Jose 61, 98, 396, 106, 1114, 1955, 3639, 4920\n\nTorres, Liz 2694, 3138\n\nTorres, Raquel 4639\n\nTorrey, Roger 2377, 3133, 3737, 4472\n\nTorvay, Jose 83, 124, 226, 439, 462, 1190, 1141, 1471, 1911, 2100, 2237, 2382, 2725, 3144, 4039, 4148, 4543, 4624, 4674, 4752\n\nTotman, Wellyn 891, 1301, 1333, 1440, 1496, 1537, 1586, 1872, 2473, 3459, 4007, 4193, 4306, 5061\n\nTotten, Robert 919, 959, 1015, 3141, 3320, 3416, 3661, 4936\n\nTotter, Audrey 122, 2557, 2617, 4705, 5022\n\nTourneur, Andree 5016\n\nTourneur, Jacques 648, 1485, 1633, 2211, 4129, 4160, 4795, 4921\n\nTourneur, Maurice 2211\n\nTovar, Lupita 440, 1311, 4052, 4617, 4851, 5042\n\nTowers, Constance 1937, 3776\n\nThe Town Criers 4023\n\nTowne, Aline 3614, 3751, 4712\n\nTowne, Robert 4738\n\nTowne, Rosella 874, 3574\n\nTownes, Harry 1832, 2229, 3712, 4450\n\nTownley, Jack 174, 286, 389, 1619, 1916, 1923, 2223, 2233, 2649, 2727, 2856, 3481, 3886, 4681, 5050\n\nTownsend, Bud 1874\n\nTownsend, Leo 4907\n\nTozzi, Fausto 1287, 3198, 4542\n\nTrace, Al, and His Silly Symphonists 1453, 3651\n\nTracey, Emerson 1408, 3636, 5036\n\nTracy, Adele 5040\n\nTracy, Lee 4215\n\nTracy, Spencer 193, 420, 520, 1945, 2837, 3695, 3750\n\nTrahey, Madelyn 2714, 3045\n\nTrainor, Leonard 327, 454, 3920, 4271\n\nTraven, B. 4543\n\nTravers, Bill 1183\n\nTravers, Henry 198, 1183, 3095, 5032, 5045\n\nTravers, Victor 2978, 3914, 4053\n\nTravis, June 2101\n\nTravis, Merle 303, 922, 1497, 2419, 2884, 3552\n\nTravis, Randy 996, 2442, 2979, 3843, 4311\n\nTravis, Richard 2644, 2922, 3053, 3450\n\nTravolta, John 1098\n\nTreacher, Arthur 4750\n\nTree, Dorothy 1884, 4359\n\nTreen, Mary 1361, 1588, 1638, 1668, 1767, 4221, 5064\n\nTremayne, Les 3838\n\nTrenker, Luis 2078\n\nTrent, Jean 3912, 3925, 4844\n\nTrent, Philip 2971\n\nTrevor, Claire 63, 293, 462, 960, 1070, 2239, 2349, 2565, 4100, 4162, 4941, 5021\n\nTrevor, Hugh 3099\n\nTribuck, Augie 57\n\nTrintignant, Jean-Louis 1643\n\nTritt, Travis 2439, 3519\n\nTrivers, Barry 158, 3538\n\nTroell, Jan 5084\n\nTrosper, Guy 82, 1093, 1966, 2575, 2901\n\nTrotti, Lamar 279, 514, 844, 1165, 1949, 2019, 3005, 3245, 4941, 5051, 5072\n\nTrowbridge, Charles 279, 362, 584, 715, 1953, 1961, 2135, 2652, 2717, 3020, 3750, 4257, 4498, 4742\n\nTrueman, Paula 2950, 3009\n\nTruex, Ernest 60, 1903\n\nTrumbo, Christopher 2010, 2752\n\nTrumbo, Dalton 1831, 2010, 2237, 2430\n\nTryon, Glenn 2259, 2288, 4622\n\nTryon, Tom 1535, 1575, 1701, 4079, 4119, 4373, 4990\n\nTubb, Ernest 1298, 1903, 3509\n\nTuchock, Wanda 2385\n\nTuchock, Wanda 2815\n\nTucker, Forrest 55, 255, 312, 516, 562, 612, 741, 835, 1033, 1364, 1402, 1569, 1706, 1748, 1843, 1986, 2066, 2184, 2235, 2678, 2851, 3132, 3140, 3211, 3219, 3355, 3436, 3567, 3690, 3859, 4111, 4373, 4434, 4621, 4705, 4753, 4792, 4851, 5045\n\nTucker, Mary 990\n\nTucker, Richard 445, 1570, 1571, 1907, 2891, 2923, 4284\n\nTudor, Pamela 2904, 3405, 3716\n\nTufts, Sonny 3045, 3637, 4472, 4673, 4745\n\nTully, Tom 165, 497, 831, 2835, 3214, 3392, 4291, 4450, 4745\n\nTupper, Tristram 2146, 2149\n\nTurich, Felipe 272, 288, 450, 654, 934, 2066, 2409, 2901, 2977, 3255, 3562, 3848, 3999, 4047, 4362, 4588, 5036\n\nTurich, Rosa 28, 306, 488, 654, 853, 934, 2066, 2086, 2109, 2549, 2552, 2878, 3054, 3091, 3565, 3276, 3502, 3610, 3999, 4047, 4128, 5096\n\nTurkel, Joseph 93, 271\n\nTurley, Jack 1695\n\nTurner, Barbara 1077, 3987, 4613\n\nTurner, Florence 3485\n\nTurner, George 4011, 4733\n\nTurner, Lana 1925\n\nTurner, Martin 1556, 2533, 3969\n\nTurpin, Ben 2283\n\nTushingham, Rita 4529\n\nTuttle, B.R. (Burl) 752, 2766, 4004\n\nTuttle, Frank 1176, 2918\n\nTuttle, Lurene 1830, 2559, 2706\n\nTuttle, W.C. 1720\n\nTuttle, Wesley 2269, 2595, 2866, 3238, 3457, 3507, 4024, 4026, 4272\n\nTwain, Mark 258, 292\n\nTweed, Shannon 2446, 3627\n\nTwelvetrees, Helen 1906\n\nTwist, John 95, 314, 329, 798, 935, 1112, 1408, 2223, 2288, 2504, 2940, 3099, 3362, 4475, 4824, 5046\n\nTwitchell, Archie 201, 484, 542, 940, 1395, 1461, 1532, 1622, 1959, 1971, 2831, 3171, 3414, 4061, 4284, 4405, 4408, 4709, 4803\n\nTyke, Johnny 752\n\nTyler, Beverly 257, 749, 3023\n\nTyler, Harry 42, 145, 271, 514, 552, 979, 1577, 1843, 1974, 2031, 2068, 2302, 2452, 2756, 2782, 3344, 3441, 3702, 4171, 4300, 4329, 4601, 4673, 4766, 5072\n\nTyler, T. Texas 1938\n\nTyler, Tom 203, 214, 264, 293, 395, 402, 459, 468, 475, 535, 617, 647, 715, 718, 725, 764, 777, 794, 857, 889, 896, 940, 1009, 1165, 1175, 1271, 1272, 1283, 1312, 1414, 1458, 1496, 1513, 1586, 1600, 1640, 1927, 1939, 1960, 2177, 2223, 2353, 2370, 2456, 2482, 2517, 2531, 2596, 2605, 2632, 2742, 2743, 2767, 2802, 2933, 2960, 2970, 2971, 3015, 3081, 3082, 3084, 3101, 3153, 3230, 3323, 3383, 3444, 3473, 3475, 3479, 3489, 3495, 3523, 3528, 3537, 3541, 3545, 3692, 3703, 3708, 3805, 3814, 3884, 3888, 3914, 3927, 4085, 4100, 4269, 4312, 4320, 4409, 4480, 4488, 4555, 4605, 4618, 4636, 4695, 4700, 4759, 4785, 4809, 4811, 4821, 4851, 4855, 4862, 4949, 5048, 5075\n\nTyne, George 3319, 4249\n\nTyner, Charles 181, 192, 727, 883, 1220, 2028, 2305, 2685, 2950, 3070, 3993, 4113, 5006, 5073\n\nTyrell, Ann 3779, 4227\n\nTyrrell, John 386, 391, 575, 623, 870, 875, 878, 1192, 1228, 1298, 1453, 1725, 1768, 2153, 2175, 2189, 2269, 2296, 2425, 2547, 2633, 2978, 3003, 3171, 3520, 3552, 3565, 3569, 3678, 3859, 3894, 3914, 4023, 4042, 4053, 4167, 4238, 4391, 4401, 4723, 4806, 4810, 5084\n\nTyrrell, Susan 96, 605, 3138, 3837\n\nUgarde, Julian 2387, 4150, 4718\n\nUgland, Rudy 709, 2234, 4688\n\nUllman, Daniel B. (Dan) 170, 211, 650, 691, 716, 1103, 1248, 1314, 1347, 1403, 1415, 1597, 1608, 1866, 2083, 2085, 2202, 2448, 2591, 2679, 2680, 2871, 2964, 2970, 3312, 3779, 3900, 4085, 4124, 4412, 4620, 3754, 4766, 4921, 4966, 5030, 5039\n\nUllman, Elwood 3916\n\nUllman, Liv 2787, 5084\n\nUlmer, Edgar G. 2747; _see also_ Warner, John\n\nUlmer, Shirley 4389\n\nUlric, Lenore 2836\n\nUltra Violet 911\n\nUndari, Claudio _see_ Hundar, Robert\n\nUre, Mary 915\n\nUrecal, Minerva 110, 134, 160, 317, 488, 602, 863, 1084, 1256, 1464, 1480, 1974, 2150, 2158, 2502, 2592, 2688, 2856, 2942, 3236, 3505, 3669, 2677, 3783, 3929, 4037, 4183, 4285, 4336, 4523, 4616, 4745, 4759, 4777, 5022\n\nUrich, Robert 2309, 2433, 2452, 2796\n\nUris, Leon 1702\n\nUrueta, Chano 217, 517, 591, 750, 1515, 1729, 1731, 1857, 2041, 2579, 3566, 4208, 5028\n\nUsher, Guy 144, 179, 205, 231, 283, 424, 943, 1647, 1884, 1974, 2074, 2082, 2119, 2158, 2451, 2514, 2756, 2886, 2909, 3042, 3202, 3353, 3490, 3615, 3626, 3912, 4061, 4419, 4423, 4655, 4665, 4811, 5011\n\nUsher, Madeleine 2117\n\nUstinov, Peter 4192, 4537, 4749\n\nUys, Jamie 1846\n\nVaccaro, Brenda 4264, 5100\n\nVadis, Dan 526, 594, 1410, 1519, 1880, 2671, 3113, 3736, 4162, 4895\n\nVague, Vera 870, 1573, 2672, 4086; _see also_ Allen, Barbara Jo\n\nVail, Lester 1961\n\nValdes, German 1190; _see also_ Tin Tan\n\nValdez, Carlos 4178, 4418\n\nVale, Virginia 566, 2342, 2599, 3162, 3557, 4096\n\nVale, Vola 3879, 3934, 4910\n\nValentine, Karen 965, 1578, 1942\n\nValentine, Paul 4062\n\nValentino, Rudolph 4732\n\nValerie, Joan 3530\n\nValerii, Tonino 976, 1385, 2723, 3175, 3297, 4243\n\nValkis, Helen 387, 714, 2872\n\nVallee, Rudy 271, 3411\n\nValli, Virginia 4139\n\nVallin, Rick 170, 208, 577, 778, 807, 1235, 1314, 1451, 1919, 2133, 2218, 2511, 2591, 2602, 2616, 2748, 2836, 2842, 3067, 3068, 3208, 3225, 3475, 3509, 3523, 3547, 3980, 4001, 4124, 4141, 4462, 4539, 4759, 4871\n\nVallon, Michael 206, 430, 477, 842, 1018, 1456, 1573, 1664, 1749, 2365, 2378, 2424, 2595, 2866, 2873, 2884, 3227, 3298, 3889, 3936, 3982, 4364, 4420, 4462, 4556, 4616, 4701\n\nVallone, Raf 641, 1694, 2779\n\nVan Cleef, Lee 20, 128, 165, 208, 255, 302, 316, 378, 502, 972, 976, 1016, 1063, 1208, 1381, 1589, 1612, 1619, 1664, 1702, 1757, 1877, 1945, 2018, 2043, 2113, 2234, 2292, 2431, 2498, 2502, 2561, 2768, 3040, 3149, 3211, 3225, 3234, 3337, 3381, 3430, 3438, 3542, 3660, 4150, 4224, 4252, 4435, 4539, 4548, 4588, 4705, 5052\n\nVandergrift, Monty 3850, 4201, 4532, 4803, 4831\n\nVanders, Warren 734, 1942, 2780, 3175, 3402, 3437, 3602, 2612\n\nVan Devere, Trish 4679\n\nVan Dien, Casper 7\n\nVan Doren, Mamie 466, 3754, 3829, 4123\n\nVan Dyke, Barry 3947\n\nVan Dyke, Jerry 2616\n\nVan Dyke, W.S. 2245, 2756, 3606, 3695, 4121, 5002\n\nVan Eyck, Peter 3295, 3807\n\nVan Fleet, Jo 1206, 1702, 2118\n\nVan Horn, Buddy 348, 2044\n\nVan Nutter, Rick 1198\n\nVan Patten, Dick 1107, 1834, 2044, 4537, 4859, 5083\n\nVan Patten, Jimmy 740, 1942\n\nVan Patten, Joyce 3990, 5006\n\nVan Patten, Vincent 506, 738\n\nVan Pelt, John 897, 3482, 3919\n\nVan Rooten, Luis 1485\n\nVan Sickel, Dale 28, 125, 287, 637, 667, 671, 951, 997, 1072, 1073, 1543, 1573, 1603, 1713, 2022, 2026, 2035, 2050, 2125, 2132, 2233, 2357, 2427, 2564, 2664, 2834, 2857, 3088, 3234, 3356, 3365, 3374, 3614, 3741, 3752, 3791, 3839, 3853, 4011, 4400, 4462, 4472, 4551, 4712, 4734, 5103\n\nVan Sloan, Edward 159, 3475\n\nVan Tuyle, Bert 3992\n\nVan Zandt, Philip 100, 602, 652, 832, 1032, 1991, 2193, 2838, 2978, 3066, 3431, 3534, 4839, 4680, 4751, 5078\n\nVanzi, Luigi _see_ Vance, Lewis\n\nVarconi, Victor 653, 930, 3112, 3126, 4635\n\nVarden, Norma 1257\n\nVarela, Yolanda 3868\n\nVarga, Billy 2859\n\nVargas, Daniele 3372, 3604, 4162, 4268, 4348, 4779, 5088, 5094\n\nVargas, Eleanora 19\n\nVarno, Roland 4357\n\nVarsi, Diane 1441\n\nVaughn, Alberta 958, 2177, 3256, 4944\n\nVaughn, Dorothy 203, 874, 1579, 2638, 3441, 3750, 4015, 4086, 4508, 4932\n\nVaughn, Robert 1064, 1195, 1608, 1849, 2497, 3349, 4970\n\nVe Sota, Bruno 1743, 2870, 3234, 4699, 4969\n\nVega, Isela 45, 245, 517, 2057, 3144\n\nVelasquez, Lorena 3189, 4415, 4438\n\nVelasquez, Teresa 1223, 2345, 3197, 3343\n\nVelez, Josephine 3485\n\nVelez, Lupe 1511, 2245, 2649, 4139, 5015\n\nVeloz and Yolanda 4650\n\nVenable, Evelyn 1469, 1855, 2471, 2823\n\nVenturini, Edward D. 1978, 2385\n\nVenuta, Benay 94, 3411\n\nVerdon, Gwen 2664\n\nVerdugo, Elena 325, 688, 1213, 1523, 2591, 3056, 3980\n\nVergara, Luis Enrique 168, 4209\n\nVerneuil, Henri 1731\n\nVernon, Bobby 2393\n\nVernon, Dorothy 1301, 1469, 1831, 1859, 1916, 2249, 2451, 2482, 2567, 2727, 3169, 3288, 4254, 4274, 4308\n\nVernon, Glenn 277, 1105\n\nVernon, Howard 3796\n\nVernon, John 66, 247, 450, 644, 2444, 2911, 2950, 3661, 3984, 4249\n\nVernon, Wally 366, 449, 610, 1472, 1481, 1713, 1959, 2545, 2967, 3120, 3341, 3893, 4085, 4524, 4913\n\nVerrin, Raquell 2409\n\nVersini, Marie 106, 565, 4380\n\nVetri, Victoria 2141; _see also_ Dorian, Angela\n\nVickers, Martha 964, 1421\n\nVickers, Sunny 3493\n\nVickers, Yvette 1948, 3676\n\nVictor, Henry 504, 3874\n\nVidal, Gore 373\n\nVidor, Charles 159, 1070, 2013, 3594\n\nVidor, Florence 4743\n\nVidor, King 331, 3957\n\nVietrel, Salka 1029\n\nVigran, Herbert 982, 1742, 1808, 2205, 2856, 3134, 4210\n\nVillarias, Carlos 606, 613, 1468, 3610, 4128\n\nVillegas, Lucio 1311, 2353, 2586, 2878, 3112, 3353, 4067, 4364\n\nVincent, Billy \"Sailor\" 49, 597, 664, 1235, 1364, 1378, 2504, 2677, 3338, 3383, 3538, 3692, 3707, 4049, 4073, 4503\n\nVincent, Jan-Michael 232, 348, 1036, 1237, 3795\n\nVincent, June 161, 642, 796, 2659, 4015\n\nVincent, Romo 2267, 4867\n\nVincent, Russ 110, 2233, 3154, 4597\n\nVincent, Victoria 729, 4537\n\nVincent, Virginia 2760\n\nVincenzoni, Luciano 751, 1016, 1171, 1381, 1612, 2643\n\nVinson, Gary 1266, 5055\n\nVint, Alan 222, 280, 2373, 2629\n\nVint, Jesse 280, 413, 1583, 1834\n\nVinton, Arthur 3558\n\nVinton, Bobby 319, 4523\n\nVinton, Victoria 77, 3918, 4722\n\nVirgo, Peter 2549, 3696\n\nVisaroff, Michael 153, 930, 1132, 1521, 2121, 2836, 4762\n\nVitale, Joseph (Joe) 53, 1033, 1257, 2289, 2720, 3020, 4162, 4582\n\nVittes, Louis 2929, 3857, 4737\n\nVivyan, John 3445\n\nVlahos, John 1475, 3568, 3668, 4458, 4660, 5027\n\nVogan, Emmett 67, 210, 534, 644, 710, 793, 836, 874, 1040, 1141, 1221, 1360, 1490, 1820, 1920, 1947, 1953, 2124, 2159, 2218, 2395, 2838, 2852, 3024, 3150, 3224, 3328, 3334, 3408, 3409, 3481, 3711, 3772, 3794, 3846, 3976, 4016, 4048, 4127, 4257, 4622, 4681, 4786, 5050\n\nVogeding, Frederick 246, 866, 2646, 4357\n\nVogel, Mitch 4295\n\nVogel, Virgil 1064, 2265\n\nVohraus, Bernard 1953, 2160, 4358\n\nVohrer, Alfred 84, 1366, 4380\n\nVohs, Joan 1405, 1409\n\nVoight, Jon 829, 1943, 2201, 3395\n\nVoland, Herbert 3806\n\nVolanti, Vicki 1352\n\nVolkie, Ralph 741, 808, 1422, 2485, 2626, 3958, 4057\n\nVolonte, Gian Maria 571, 1350, 1381\n\nVon Bricken, William 2434, 2647, 2745\n\nVon Eltz, Theodore 142, 1638, 2811, 3766, 4522\n\nVon Seyffertitz, Gustav 1511\n\nVon Sternberg, Josef 1187\n\nVon Sydow, Max 2787, 3406\n\nVon Zell, Harry 4005, 4611\n\nVonn, Viola 324, 3220, 5001\n\nVoskovec, George 245, 502, 2005, 4993\n\nVosper, John 277, 365, 3750, 4657\n\nVye, Murvyn 367, 1569, 3535, 4890\n\nWade, Russell 230, 1308, 1378, 3115, 3327, 3351, 3679, 4188, 4231, 5054\n\nWade, Stuart 4240, 4247\n\nWadsworth, Henry 2923\n\nWagenheim, Charles 377, 489, 645, 1140, 1444, 2543, 2666, 2902, 2910, 3112, 3224, 3534, 3684, 3751, 3912, 4429, 4469\n\nWaggner, George 349, 353, 879, 1313, 1438, 1485, 1556, 1662, 1706, 1929, 2004, 2521, 2948, 3058, 3089, 3160, 4848, 5011; _see also_ West, Joseph\n\nWagner, Max 234, 271, 324, 381, 432, 475, 541, 642, 1214, 1364, 1433, 1588, 2208, 2660, 2852, 2902, 3012, 3222, 3234, 3319, 3359, 3362, 3431, 3757, 3972, 4428, 4498, 4622, 4650\n\nWagner, Robert 520, 1676, 2559, 3909, 4579, 4907\n\nWagner, Wende 1739, 3518\n\nWainwright, James 512, 2044\n\nWaite, Ralph 709, 2100, 2305, 2498, 3313\n\nWaizman, Max 37, 777, 1504, 1716, 1974, 2552, 3230\n\nWakely, Jimmy 14, 165, 392, 582, 726, 850, 871, 878, 928, 1028, 1680, 1684, 1821, 2293, 2378, 2424, 2435, 2601, 2687, 2858, 2873, 2944, 3050, 3142, 3227, 3238, 3265, 3277, 3457, 3484, 3556, 3565, 3616, 3666, 3670, 3675, 3678, 3906, 3908, 3945, 4021, 4024, 4026, 4028, 4081, 4135, 4170, 4191, 4219, 4221, 4261, 4318, 4507, 4515, 4586, 4600, 4820\n\nWalburn, Raymond 960, 1070, 1378, 1601, 1883, 2344, 3128, 3848, 4073\n\nWalcott, Gregory 212, 1133, 2044, 2553, 3069, 3203, 3333, 4148, 4180, 4300, 4388\n\nWald, Jerry 4222, 4687\n\nWaldis, Otto 439, 2518, 3298, 3829\n\nWaldo, Janet 231, 2170, 2909, 3905\n\nWaldron, Charles 4357, 4776\n\nWaldron, Wendy 2988, 4491\n\nWales, Ethel 243, 459, 852, 1257, 1306, 1322, 1462, 1870, 1973, 1982, 2146, 2223, 2480, 2573, 4635, 4645, 5032, 5059\n\nWales, Wally 77, 180, 240, 297, 438, 511, 592, 646, 862, 945, 1009, 1049, 1178, 1299, 1333, 1334, 1379, 1493, 1760, 1835, 1861, 1928, 2177, 2246, 2248, 2283, 2392, 2470, 2478, 2479, 2660, 2737, 2852, 3062, 3075, 3153, 3268, 3314, 3427, 3455, 3525, 3650, 3658, 3680, 3884, 3939, 3969, 3983, 4245, 4484, 4521, 4525, 4555, 4636, 4683, 4711, 4796, 4822, 4837, 4846, 4852, 4861, 4892; _see also_ Taliaferro, Hal; Williams, Walt\n\nWalken, Christopher 1833, 3842\n\nWalker, Cheryl 3410, 3805\n\nWalker, Cindy 1467, 3434\n\nWalker, Clint 216, 487, 1399, 1500, 1596, 1786, 2692, 2595, 3030, 3688, 3749, 3981, 4895, 5055, 5082\n\nWalker, Francis 8, 297, 386, 391, 631, 725, 945, 1192, 1241, 1283, 1288, 1384, 1491, 1626, 1687, 1859, 1900, 2067, 2119, 2203, 2256, 2258, 2263, 2274, 2300, 2355, 2403, 2425, 2432, 2547, 2802, 2821, 3001, 3062, 3081, 3100, 3109, 3171, 3279, 3302, 3371, 3385, 3393, 3424, 3453, 3466, 3489, 3497, 3550, 3591, 3664, 3924, 4238, 4314, 4401, 4480, 4555, 4608, 4667, 4698, 4711, 4760, 4806, 4827, 4832, 4960, 4980\n\nWalker, Hal 3544\n\nWalker, Johnnie 1570, 3979\n\nWalker, Nancy 1564\n\nWalker, Nella 2147\n\nWalker, Ray 354, 1901, 1919, 2008, 2790, 3104, 3298, 3563, 3951, 4086, 4253, 4644, 5055\n\nWalker, Robert 6, 240, 443, 461, 468, 511, 576, 646, 668, 773, 814, 848, 894, 917, 963, 1211, 1271, 1299, 1315, 1323, 1554, 1626, 1751, 1760, 1813, 1904, 2014, 2024, 2104, 2177, 2233, 2391, 2428, 2531, 2742, 2961, 3018, 3062, 3079, 3092, 3102, 3188, 3385, 3447, 3458, 3487, 3589, 3617, 3680, 3742, 3750, 3869, 3888, 3952, 4023, 4165, 4269, 4298, 4305, 4379, 4553, 4609, 4708, 4725, 4790, 4809, 4836, 4854, 4910, 4958, 5014\n\nWalker, Robert (Jr.) 1864, 3732, 5060\n\nWalker, Robert G. 693, 951, 3512\n\nWalker, Scott 3604, 4895\n\nWalker, Terry 335, 2633, 2891, 4226\n\nWalker, William (Bill) 319, 1427, 2121, 3415, 3698, 3701, 4361\n\nWallace, Beryl 3597, 4202, 4456, 5021\n\nWallace, George 128, 324, 450, 1082, 1164, 1919, 2292, 2396, 2485, 2549, 2565, 3219, 3754, 3933, 3950, 4123, 4124, 4287, 4731, 5030\n\nWallace, Helen 1151, 1265, 2226, 2594, 3417, 3929, 4074, 4151, 4376\n\nWallace, Irving 580, 1499, 4069\n\nWallace, Jerry 2050\n\nWallace, Morgan 337, 614, 1591, 1925, 2722, 3561, 3965, 4364, 4385, 4423, 4580, 4665\n\nWallace, Regina 3214, 4222\n\nWallach, Eli 3, 1135, 1612, 1945, 2489, 2497\n\nWaller, Eddy 1, 25, 63, 161, 200, 203, 228, 231, 233, 236, 354, 365, 415, 514, 612, 615, 631, 662, 667, 757, 784, 854, 914, 930, 979, 1022, 1040, 1074, 1216, 1263, 1367, 1428, 1439, 1454, 1478, 1599, 1626, 1721, 1768, 1925, 1982, 1992, 2031, 2081, 2153, 2163, 2226, 2288, 2310, 2342, 2423, 2482, 2525, 2528, 2565, 2592, 2593, 2618, 2652, 2684, 2717, 2762, 2786, 2857, 3090, 3128, 3150, 3151, 3195, 3224, 3355, 3356, 3367, 3384, 3409, 3534, 3540, 3613, 3615, 3652, 3656, 3685, 3692, 3711, 3724, 3750, 3828, 3859, 3903, 3904, 3912, 3933, 3937, 4000, 4112, 4121, 4166, 4181, 4183, 4184, 4231, 4318, 4400, 4533, 4593, 4654, 4730, 4747, 4849, 4890, 4925, 4932, 4938, 5033, 5034\n\nWallerstein, Herb 3981\n\nWalling, William 300, 765, 1095, 1636, 2004, 3011, 3260, 4605, 5002\n\nWallis, Shani 3249\n\nWalsh, Bill 66, 3739\n\nWalsh, George 2147, 3101, 3526, 4275\n\nWalsh, M. Emmett 1879, 2000, 2100, 4195, 4789, 4974\n\nWalsh, Raoul 65, 328, 718, 798, 960, 1111, 1112, 1672, 1972, 2118, 2147, 2292, 2835, 3195, 3718, 3821, 3902, 4115, 4233, 4336, 5026\n\nWalston, Ray 3009, 3324\n\nWalter, Jessica 3854, 4927\n\nWalters, Charles 4291, 4670\n\nWalters, Glen 1254, 1935, 4058\n\nWalters, Luana 6, 15, 131, 205, 287, 1192, 1227, 1260, 1333, 2114, 2284, 2425, 2648, 3258, 3393, 3424, 3540, 4405, 4586, 4647, 4882\n\nWalthall, Henry B. 254, 1827, 2146, 2223, 2654, 3245, 3427, 3994, 4174, 4750\n\nWalthall, Pat 1221\n\nWalton, Douglas 196, 599, 2033, 2837, 4633\n\nWanamaker, Sam 675, 2372\n\nWang, George 396, 1207, 1305, 1385, 2030, 4012, 4243\n\nWanzer, Arthur 4668\n\nWar Eagle, John 94, 563, 1150, 1372, 1605, 1645, 2178, 2204, 2525, 2931, 3243, 3682, 4338, 4411, 4450, 4457, 4856, 4869, 4962, 4989\n\nWarburton, John 3049\n\nWard, Amelita 3510\n\nWard, Blackjack (Jack) 15, 174, 295, 307, 327, 331, 496, 800, 814, 1162, 1128, 1269, 1320, 1323, 1328, 1329, 1367, 1469, 1474, 1545, 1677, 1756, 1796, 1839, 1928, 1933, 1952, 2014, 2152, 2272, 2299, 2358, 2391, 2403, 2404, 2524, 2526, 2527, 2584, 2730, 2864, 2890, 2915, 2951, 2956, 2968, 3029, 3087, 3092, 3109, 3135, 3250, 3263, 3354, 3424, 3427, 3466, 3582, 3597, 3658, 3664, 3680, 3709, 3834, 3859, 3969, 4010, 4190, 4297, 4308, 4315, 4318, 4442, 4531, 4614, 4732, 4836, 4846, 4851, 4861, 5009, 5061, 5104\n\nWard, Dorothy 4774\n\nWard, Fred 280\n\nWard, John 1491, 1748\n\nWard, Luci 205, 214, 307, 354, 377, 631, 714, 875, 1298, 2164, 2269, 2325, 2998, 3029, 3224, 3325, 3383, 3439, 3507, 3678, 3709, 4191, 4508, 4723\n\nWard, Rachel 2055\n\nWard, Rollo 2343\n\nWard, Skip 1908, 2155\n\nWarde, Anthony 288, 739, 758, 951, 1132, 1627, 2122, 2125, 2128, 2824, 2861, 3458, 3490, 4419, 4500, 4876\n\nWarde, Harlan 22, 1985\n\nWarden, Jack 346, 1583, 2560, 4895\n\nWare, Helen 1368, 1961, 2890, 4744, 4891\n\nWare, Mary 1935\n\nWarhol, Andy 770\n\nWarner, David 221, 1064, 2807\n\nWarner, H.B. 1214, 1571, 1843, 1983, 2344, 2790, 3609\n\nWarner, Hansel 481, 684, 3111\n\nWarner, J.B. 276, 326, 4780\n\nWarner, John 4389; _see also_ Ulmer, Edgar G.\n\nWarner, Wes 1088, 1728, 1751, 3681, 3919, 4321, 4732\n\nWarren, Bruce 1154, 1859, 2891, 3086, 3126\n\nWarren, Charles Marquis 166, 375, 399, 678, 704, 833, 980, 1844, 2372, 2851, 2919, 3140, 3417, 4080, 4259, 4566\n\nWarren, Dale 2330\n\nWarren, E. Allyn 200, 1300, 1570, 1571, 4580, 4762\n\nWarren, Gloria 288\n\nWarren, James 214, 786, 1105, 1564, 4204, 4257, 4777\n\nWarren, Jennifer 239, 4134\n\nWarren, Jerry 569\n\nWarren, Katherine 4740\n\nWarren, Kathleen 1481\n\nWarren, Lesley Ann 965, 3191\n\nWarren, Phil 786, 951, 4503\n\nWarren, Ruth 757, 1848, 2020, 2138, 2239, 2684, 3150, 3386, 4803\n\nWarrick, Ruth 1631, 3398\n\nWarwick, Robert 733, 776, 1032, 1340, 1480, 1566, 1688, 1907, 1931, 1980, 2064, 2153, 2272, 2583, 2664, 2790, 2843, 3054, 3112, 3422, 3839, 3895, 3897, 4179, 4215, 4257, 4427, 4446, 4558, 4635, 4732, 4770, 4786\n\nWarwick, Virginia 4, 3108\n\nWashburn, Beverly 2400, 2887, 3808\n\nWashburn, Bryant 665, 824, 878, 1025, 2003, 2132, 2286, 2775, 2940, 3805, 3922, 4100, 4214, 4379, 4759, 4825\n\nWashington, Edgar \"Blue\" 1796, 2121, 2472, 2699, 3035, 3126, 3665, 3969, 4803\n\nWaters, John 2773\n\nWaters, Ozie 875, 2175, 2736, 2939, 3094, 3167, 3678, 4167, 4272\n\nWaterston, Sam 1204, 1833, 3252\n\nWatkin, Pierre 161, 277, 614, 637, 693, 930, 1214, 1247, 1460, 1532, 1821, 1971, 2016, 2032, 2158, 2308, 2624, 2778, 2937, 2988, 3128, 3334, 3342, 3499, 3512, 3567, 3929, 4024, 4162, 4198, 4400, 4585, 4697, 4773, 4938, 4979\n\nWatkins, Linda 1518\n\nWatson, Bobs 3964, 5032\n\nWatson, Delmer 95, 1426, 2159, 2422, 2722, 2975, 3886, 4442\n\nWatson, Joseph K. 714\n\nWatson, Mills 1107, 1755, 2000, 2229, 2389, 2437, 3282, 4537, 4936\n\nWatson, Minor 491, 1249, 1253, 1870, 2385, 3046, 3276, 3609, 4057, 4121, 4252, 4336, 4593, 4674, 4747, 4849\n\nWatt, Harry 461, 1239, 1932, 2270, 2826, 2867, 3004\n\nWatt, Nate 1467, 1889, 3177, 3657, 4485\n\nWatts, Charles 109, 320, 489, 1560, 2402, 2813, 2902, 2981, 3411, 3909, 4232\n\nWatts, George 112\n\nWatts, Twinkle 610, 644, 836, 2545, 2967, 3826, 3893, 4463, 4487\n\nWayne, Aissa 43, 808, 2626\n\nWayne, Carole 1626\n\nWayne, David 2749, 3138\n\nWayne, John Ethan 45, 319, 2568, 3527, 3748\n\nWayne, John 43, 63, 319, 327, 328, 411, 471, 594, 741, 808, 824, 883, 930, 960, 974, 1057, 1212, 1313, 1360, 1636, 1692, 1796, 1921, 1937, 1945, 1974, 2129, 2158, 2160, 2163, 2297, 2299, 2300, 2432, 2479, 2527, 2548, 2561, 2626, 2632, 2766, 2785, 2786, 2802, 2829, 2998, 3027, 3036, 3242, 3256, 3260, 3323, 3324, 3325, 3427, 3451, 3517, 3522, 3527, 3602, 3618, 3680, 3709, 3752, 3814, 3817, 3847, 3994, 4035, 4072, 4100, 4125, 4231, 4248, 4294, 4317, 4358, 4360, 4372, 4482, 4523, 4576, 4607, 4638, 4786, 4790, 4822, 4854, 4995, 50037\n\nWayne, Patrick 33, 43, 267, 319, 523, 720, 808, 1062, 1245, 1510, 2626, 3522, 3654, 3752, 3816, 4295, 5067, 5071\n\nWayne, Walt 435\n\nWeathers, Carl 1014\n\nWeaver, Dennis 733, 803, 1183, 1277, 1877, 1936, 2010, 2252, 2292, 2510, 2544, 2627, 2664, 2768, 2925, 3222, 3779, 4252, 4783\n\nWeaver, Doodles 315, 1452, 2500, 2710, 3150, 3340, 3622, 3692, 3925, 4417\n\nWeaver, Frank 2027\n\nWeaver, June 2027\n\nWeaver, Leon 2027, 2386, 3481\n\nWeaver, Loretta 160, 1862, 2027\n\nWeaver, Marjorie 614, 757, 1627, 3846, 5072\n\nWeaver, Ned 2902, 3415, 4333\n\nThe Weaver Brothers and Elviry 160, 2027\n\nWebb, Harry S. 240, 305, 457, 468, 592, 593, 1282, 1312, 1602, 2177, 2428, 2533, 2645, 2822, 3018, 3079, 3089, 3102, 3477, 3487, 3495, 3869, 3977, 4269, 4488, 4809, 4852, 4958, 5014; _see also_ Semuels, Henri\n\nWebb, Ira 1211, 2096, 2123, 3995, 3999, 4958\n\nWebb, Jack 1784\n\nWebb, James R. 100, 201, 313, 329, 700, 720, 1731, 1945, 2005, 2032, 2141, 2677, 2778, 4049, 4051, 4727\n\nWebb, Richard 79, 356, 664, 673, 843, 1111, 2066, 2710, 2768, 3090, 3254, 4362, 4472\n\nWebb, Robert D. 1741, 2019, 2462, 3185, 4799, 4907\n\nWebber, Robert 517\n\nWebster, M. Coates 365, 656, 854, 1048, 1105, 1429, 1721, 1915, 2150, 2310, 2593, 2762, 2804, 3614, 3685, 4013, 4037, 4400, 4804, 5034\n\nWebster, Mary 4172, 4435\n\nWeeks, Barbara 1388, 2886, 2907, 3658, 4187, 4605, 4898\n\nWeeks, Ranny 1822\n\nWeidler, Virginia 1431, 1599, 1853, 2936, 2940\n\nWeis, Don 2013\n\nWeisberg, Brenda 1105, 2138\n\nWeiss, Louis 2244\n\nWelch, Niles 434, 814, 834, 844, 1222, 2014, 2391, 2628, 2660, 2732, 3194, 3241, 3508, 3895, 3926, 4138, 4187\n\nWelch, Raquel 237, 1776, 2340, 2903\n\nWelden, Ben 1175, 1290, 1871, 2693, 2805, 3447, 3874, 4121, 4467, 4832\n\nWeldon, Joan 489, 819, 979, 1742, 3500, 4162, 4239\n\nWeldon, Marion 792, 1051, 2151\n\nWelker, Frank 5100\n\nWeller, Peter 585\n\nWelles, Halstead 1772, 2162, 4229, 4370, 4371, 4431\n\nWelles, Mel 1188, 2617, 3364, 5038\n\nWelles, Orson 396, 587, 1187, 2550\n\nWellman, William, Jr. 1696, 1937, 5065\n\nWellman, William A. 16, 552, 626, 826, 1738, 2013, 3005, 3561, 4473, 4857, 5051\n\nWells, Alan 2842, 3376\n\nWells, Carole 1266, 4387\n\nWells, Dawn 5007\n\nWells, Jacqueline 183, 764, 2086, 3269, 4089, 5059; _see also_ Bishop, Julie\n\nWells, Ted 459, 465, 1977, 2403, 2493, 2600, 2972, 3073, 4154, 4188, 4657, 4808, 4851\n\nWelsh, William 689, 1432, 4190, 4776, 4911\n\nWelter, Ariadna 2227, 3217\n\nWendell, Howard 977\n\nWendkos, Paul 641, 1248, 1739, 2051, 2925\n\nWengraf, John 277, 1499, 4689\n\nWentworth, Martha 1920, 2594, 2597, 2931, 3649, 4110, 4733\n\nWenzel, Art 3390, 3507, 4052, 4221\n\nWerker, Alfred L. 170, 645, 1092, 1252, 1518, 2207, 2226, 3112, 3299, 4200, 4362, 4925\n\nWerle, Barbara 704, 1703, 2379, 4412\n\nWertmuller, Lina 281\n\nWescoatt, Rusty 363, 503, 671, 700, 778, 857, 1164, 3234, 3547, 3856, 3875, 3982, 4099, 4240, 4729, 4871\n\nWessel, Richard (Dick) 442, 602, 638, 930, 1042, 1071, 1378, 1460, 1647, 1805, 1923, 1983, 2124, 2292, 3330, 3534, 3600, 3625, 3904, 4057, 4206, 4291, 4329, 4336, 4585, 4675\n\nWesson, Dick 207, 283, 700, 2504\n\nWest, Adam 1523, 2576, 2770, 2964, 3346\n\nWest, Art, and His Sunset Riders 2991\n\nWest, Billy 1102\n\nWest, Dean 1353\n\nWest, Dottie 172\n\nWest, Jessamyn 1434, 1435\n\nWest, Joseph 353, 1556, 1662, 1929, 2869, 3089, 3160, 3202, 5011; _see also_ Waggner, George\n\nWest, Mae 1348, 1591, 2147, 2722\n\nWest, Martin 1431\n\nWest, Martin 3987\n\nWest, Victor 449, 1713\n\nWest, Wally 76, 77, 201, 286, 334, 335, 338, 340, 341, 342, 343, 344, 369, 383, 410, 414, 422, 425, 427, 431, 479, 481, 494, 499, 573, 682, 683, 711, 777, 792, 848, 868, 876, 892, 893, 945, 1017, 1018, 1024, 1025, 1057, 1146, 1155, 1156, 1281, 1296, 1327, 1400, 1447, 1459, 1486, 1504, 1505, 1506, 1520, 1529, 1542, 1546, 1553, 1756, 1827, 1898, 1931, 1939, 1983, 2074, 2251, 2264, 2269, 2274, 2276, 2286, 2290, 2302, 2306, 2355, 2360, 2392, 2407, 2408, 2409, 2410, 2411, 2412, 2415, 2534, 2589, 2596, 2635, 2743, 2847, 2891, 2956, 2971, 2976, 2979, 2997, 3052, 3068, 3073, 3075, 3085, 3086, 3101, 3103, 3169, 3194, 3226, 3232, 3332, 3385, 3390, 3482, 3489, 3508, 3531, 3545, 3547, 3548, 3581, 3601, 3648, 3677, 3703, 3801, 3946, 4001, 4030, 4047, 4097, 4108, 4144, 4193, 4324, 4365, 4396, 4489, 4490, 4501, 4514, 4525, 4549, 4553, 4581, 4609, 4702, 4722, 4732, 4745, 4819, 4833, 4838, 4846, 4883, 4887, 4982, 5018\n\nWestcott, Helen 260, 700, 748, 981, 1665, 1957, 3392, 3757, 4079, 4389\n\nWesterfield, James 407, 859, 988, 1026, 1697, 1769, 1773, 1884, 1947, 2486, 2506, 2563, 3133, 3186, 3737, 3962, 4035, 4362, 4419, 4576, 4740\n\nWestern, Johnny 937, 1396\n\nWestley, Helen 2160\n\nWestman, Nydia 198, 4580\n\nWeston, Brad 255, 1942, 2631, 3612, 3731, 4101\n\nWeston, Cecil 279, 514, 1176, 1974, 2818, 3111, 3179, 3245, 4606, 4672\n\nWeston, Dick 2872, 4954; _see also_ Rogers, Roy\n\nWeston, Don 1090, 1749, 2858, 3050, 3097, 3280, 3484, 4220, 4221, 4394, 4568\n\nWeston, Doris 739\n\nWeston, Garnett 1647, 2774\n\nWeston, Jack 1036\n\nWexler, Paul 981, 1161, 2095, 2659, 3909, 4300, 4579, 4809\n\nWhalen, Michael 53, 1240, 2123, 2959, 3870, 3906, 399, 4383, 4467, 4754, 4900\n\nWheeler, Bert 1563, 3529, 3886\n\nWhelan, Arleen 208, 1364, 3053, 3225, 3247, 3690, 4184, 5072\n\nWhelan, Tim 214, 1563, 3219, 4300\n\nWhipper, Leigh 1818, 2158, 3005, 3564\n\nWhitaker, Charles \"Slim\" 5, 69, 130, 131, 151, 174, 306, 323, 327, 332, 335, 340, 345, 382, 414, 427, 430, 431, 458, 461, 479, 500, 511, 564, 566, 582, 592, 721, 725, 753, 773, 797, 812, 184, 873, 889, 927, 975, 1017, 1045, 1049, 1058, 1070, 1076, 1155, 1162, 1241, 1253, 1271, 1283, 1308, 1311, 1315, 1329, 1333, 1334, 1340, 1367, 1369, 1384, 1432, 1464, 1468, 1472, 1474, 1488, 1493, 1544, 1557, 1586, 1626, 1712, 1730, 1732, 1738, 1756, 1768, 1796, 1822, 1839, 1849, 1906, 1907, 1928, 1952, 1969, 1974, 1975, 2111, 2179, 2203, 2247, 2254, 2256, 2258, 2266, 2273, 2274, 2283, 2300, 2342, 2355, 2360, 2378, 2392, 2399, 2403, 2408, 2414, 2420, 2456, 2523, 2527, 2530, 2531, 2552, 2573, 2567, 2584, 2595, 2599, 2634, 2635, 2660, 2701, 2734, 2743, 2786, 2799, 2860, 2866, 2869, 2886, 2915, 2921, 2933, 2951, 2956, 2974, 2997, 2998, 3048, 3052, 3075, 3077, 3087, 3101, 3103, 3105, 3160, 3162, 3172, 3173, 3220, 3226, 3227, 3232, 3254, 3268, 3288, 3289, 3303, 3314, 3388, 3427, 3444, 3450, 3452, 3454, 3489, 3497, 3525, 3528, 3545, 3546, 3548, 3553, 3555, 3581, 3590, 3591, 3624, 3650, 3653, 3658, 3665, 3668, 3673, 3680, 3703, 3706, 3797, 3834, 3880, 3884, 3888, 3889, 3904, 3907, 3937, 3967, 3969, 3994, 4022, 4050, 4052, 4104, 4190, 4200, 4205, 4207, 4245, 4248, 4269, 4278, 4321, 4324, 4410, 4484, 4490, 4525, 4570, 4572, 4592, 4617, 4636, 4645, 4655, 4696, 4703, 4728, 4785, 4786, 4809, 4816, 4839, 4850, 4858, 4861, 4865, 4944, 4952, 5001, 5014, 5018, 5059\n\nWhite Eagle, Chief 1227\n\nWhite, Alexander 4408, 4709\n\nWhite, Carol 3990\n\nWhite, Dan 37, 49, 82, 114, 157, 303, 309, 357, 383, 446, 452, 470, 479, 481, 486, 1017, 1111, 1166, 1187, 1214, 1254, 1327, 1338, 1348, 1363, 1389, 1420, 1447, 1459, 1461, 1468, 1633, 1672, 1693, 1704, 1714, 1718, 1724, 1738, 1789, 1936, 1980, 1995, 2027, 2034, 2066, 2198, 2256, 2412, 2413, 2418, 2511, 2595, 2782, 2946, 2953, 2969, 2980, 2999, 3046, 3068, 3163, 3165, 3186, 3200, 3219, 3244, 3290, 3305, 3318, 3323, 3338, 3348, 3385, 3386, 3542, 3615, 3616, 3692, 3750, 3814, 3825, 3834, 3902, 3909, 3922, 3963, 4049, 4144, 4179, 4196, 4244, 4490, 4509, 4589, 4601, 4684, 4702, 4725, 4755, 4853, 4973, 5045\n\nWhite, Jacqueline 654, 3383, 3473\n\nWhite, Jesse 536, 632, 1742, 4452\n\nWhite, Josh 4771\n\nWhite, Lee \"Lasses\" 46, 49, 231, 718, 812, 927, 1174, 1175, 1214, 1600, 1618, 1981, 1988, 2150, 2170, 2435, 2665, 2687, 2744, 2867, 2943, 3238, 3331, 3457, 3504, 3624, 3626, 3670, 3937, 3945, 4024, 4026, 4028, 4081, 4283, 4405, 4507, 4513, 4690, 4820, 5008\n\nWhite, Leo 1884, 3551, 4750\n\nWhite, Patricia 379, 3481, 3917; _see also_ Barry, Patricia\n\nWhite, Philip 425\n\nWhite, Ruth 1769\n\nWhiteford, Blackie 234, 271, 300, 311, 379, 392, 468, 493, 510, 682, 687, 775, 834, 851, 858, 891, 924, 1009, 1037, 1097, 1146, 1172, 1213, 1228, 1297, 1309, 1320, 1322, 1332, 1489, 1544, 1545, 1626, 1725, 1822, 1973, 1974, 1975, 2067, 2177, 2189, 2197, 2220, 2233, 2268, 2272, 2285, 2397, 2399, 2471, 2524, 2531, 2547, 2608, 2722, 2730, 2764, 2777, 2822, 2861, 2886, 2973, 2975, 2992, 3026, 3063, 3168, 3173, 3209, 3362, 3453, 3465, 3497, 3507, 3524, 3528, 3552, 3581, 3592, 3597, 3674, 3702, 3742, 3888, 3939, 3994, 4009, 4044, 4047, 4053, 4103, 4191, 4298, 4308, 4314, 4315, 4396, 4401, 4410, 4465, 4496, 4609, 4683, 4696, 4698, 4701, 4736, 4745, 4810, 4818, 4822, 4866, 4872, 5014\n\nWhitehead, Joe 832, 1853, 2020, 2278, 2645, 3431, 4135\n\nWhitehead, O.Z. 935, 1256, 1937, 2561, 3696, 4626\n\nWhitely, Crane 324, 602, 956, 1247, 1843, 3195, 3318, 3728\n\nWhiteman, Russ 51, 643, 1487, 1715, 2591, 2680, 3614, 4439, 4754\n\nWhitfield, Smoki 1256, 2544, 2899, 3779\n\nWhitley, Ray 69, 231, 303, 437, 475, 492, 812, 927, 1174, 1270, 1560, 1679, 1680, 1900, 1932, 2740, 2884, 2886, 2944, 3012, 3216, 3289, 3353, 3357, 3478, 3504, 3557, 3636, 3937, 4504, 4556, 4572, 4761, 4820, 4882\n\nWhitlock, Lloyd 1102, 1470, 2158, 2479, 2651, 2702, 2979, 3166, 3179, 3442, 4199, 4822, 4899\n\nWhitman, Gayne 37, 2790, 2792, 2877, 3234, 4357, 4309, 5042\n\nWhitman, Stuart 587, 651, 808, 1006, 1578, 1784, 2544, 2897, 3054, 3518, 3635, 3788, 3812, 3897, 4333, 4784, 4895\n\nWhitmore, James 16, 744, 1739, 1962, 2013, 2192, 2853, 2986, 4255, 4329, 4794, 4878\n\nWhitmore, James, Jr. 2443\n\nWhitney, Claire 739, 871, 1454, 1797, 2857, 3556, 3571, 3889, 4654, 5021\n\nWhitney, Grace Lee 2744\n\nWhitney, Peter 221, 356, 539, 648, 1633, 1645, 2192, 2518, 2836, 3530, 4700\n\nWhitney, Shirley 3509\n\nWhitson, Frank 1291, 3996, 4088, 4320, 4694, 4774\n\nWhorf, Richard 1658\n\nWhythe, Patrick 1407, 4999\n\nWiard, William 3747, 4449\n\nWickes, Mary 748, 1082, 3186\n\nWidener, Jimmie 3258\n\nWidmark, Richard 43, 71, 189, 520, 720, 1015, 1507, 1945, 2188, 2241, 2246, 2671, 2897, 3333, 4528, 4626, 4791, 4869, 5051\n\nWidmark, Robert 1764, 4012, 4342, 4562; _see also_ Dell'Acqua, Alberto\n\nThe Wiere Brothers 1767\n\nWiggins, Chris 359, 361, 845, 1349, 2126, 4801\n\nWilbanks, Don 509, 1250, 2990, 4109, 5084\n\nWilbur, Crane 1089, 2127, 2370, 3334\n\nWilcox, Art, and His Arizona Rangers 138, 3237\n\nWilcox, Frank 94, 203, 277, 356, 661, 1348, 1762, 1849, 1884, 2025, 2109, 2521, 2544, 2605, 2664, 2731, 3140, 3222, 3538, 3711, 3792, 4049, 4336, 4349, 4376, 4486, 4536, 4635, 4741, 4932\n\nWilcox, Larry 2229\n\nWilcox, Robert 2108, 4735, 4925, 4929\n\nWilcoxon, Henry 41, 159, 2213, 2551, 3141, 4635\n\nWilde, Cornel 604, 1205, 3054, 4611\n\nWilde, Lois 535, 668, 946, 1933, 3919, 4143, 4982\n\nWilde, Lyn 3828\n\nWilder, Gene 1437\n\nWilder, Laura Ingalls 2376\n\nWilder, Myles 3787\n\nWilder, Robert 313\n\nWilder, W. Lee 5043\n\nWiley, Jan 758, 973, 1450, 2259, 4393, 4458\n\nWilke, Robert (Bob\/Robert J.) 9, 37, 166, 189, 214, 235, 286, 291, 309, 418, 465, 610, 624, 650, 667, 685, 727, 731, 779, 813, 836, 857, 868, 922, 939, 951, 985, 1072, 1077, 1215, 1263, 1268, 1345, 1460, 1543, 1675, 1692, 1694, 1726, 1765, 1844, 1873, 1877, 1941, 1980, 2022, 2027, 2125, 2178, 2183, 2262, 2395, 2400, 2427, 2445, 2497, 2534, 2552, 2556, 2600, 2652, 2711, 2797, 2935, 2939, 2995, 3000, 3088, 3119, 3150, 3287, 3295, 3397, 3534, 3541, 3552, 3613, 3667, 3693, 3707, 3712, 3808, 3820, 3822, 3826, 3849, 3940, 3966, 3973, 4057, 4148, 4183, 4197, 4217, 4463, 4487, 4506, 4533, 4597, 4599, 4620, 4725, 4734, 4735, 4787, 4812, 4818, 4921, 4927, 5034, 5039, 5050, 5103\n\nWilkerson, Billy (Bill) 409, 488, 519, 1421, 1438, 1449, 1565, 1579, 2133, 2226, 3507, 3563, 3567, 3682, 3917, 4214, 4533, 4585, 4876, 5081\n\nWilkerson, Guy 1, 65, 206, 324, 373, 479, 489, 494, 602, 859, 1026, 1166, 1229, 1263, 1338, 1363, 1427, 1448, 1480, 1506, 1532, 1594, 1631, 1640, 1738, 1749, 1772, 2065, 2226, 2334, 2544, 2552, 2556, 2589, 2652, 2685, 2734, 2813, 2899, 2953, 2980, 3053, 3097, 3132, 3280, 3305, 3335, 3386, 3390, 3630, 3702, 3744, 3750, 4078, 4099, 4162, 4330, 4363, 4490, 4576, 4723, 4788, 4819, 4889, 4943, 4989, 5056\n\nWilkerson, Joy 315\n\nWilkins, June 3102\n\nWilkinson, June 4354\n\nWillard, Fred 3984\n\nWillat, Irvin 2469, 2651, 2879, 4647\n\nWilles, Jean 348, 727, 843, 1689, 2118, 2620, 3637, 4005, 4333\n\nWilliam, Warren 129, 4498, 4932, 4940\n\nWilliams, Bill 101, 524, 612, 1319, 1640, 1765, 1766, 2225, 2267, 2868, 2922, 2963, 3058, 3315, 3527, 3608, 3739, 3960, 4005, 4119, 4141, 4412, 4825, 4937\n\nWilliams, Billy Dee 3371\n\nWilliams, Bob (Robert) 27, 233, 291, 352, 391, 415, 465, 607, 624, 995, 1022, 1040, 1072, 1182, 1454, 1725, 1873, 2008, 2291, 2368, 2400, 2427, 2592, 2597, 2857, 2875, 3104, 3176, 3271, 3483, 3552, 3669, 3676, 3828, 3892, 4023, 4062, 4099, 4425, 4509, 4644, 4712\n\nWilliams, Cara 4895\n\nWilliams, Chilli 1433, 1509, 1573\n\nWilliams, Clara 249, 1851\n\nWilliams, Curly, and His Georgia Peach Pickers 3464\n\nWilliams, Elmo 116, 861, 4236\n\nWilliams, Esther 632, 1291, 4291\n\nWilliams, Grant 2426, 3337, 4119\n\nWilliams, Guinn \"Big Boy\" 42, 43, 79, 173, 196, 197, 200, 207, 278, 310, 332, 355, 516, 548, 578, 659, 808, 868, 869, 870, 877, 923, 945, 1070, 1203, 1228, 1353, 1639, 1767, 1774, 1854, 1871, 1982, 1900, 1913, 2138, 2263, 2470, 2518, 2549, 2554, 2617, 2717, 2775, 2823, 2914, 2963, 2991, 3153, 3410, 3450, 3572, 3623, 3711, 3899, 3914, 3916, 3923, 3975, 3976, 4060, 4089, 4132, 4221, 4327, 4339, 4378, 4389, 4732, 4741, 4767, 5017\n\nWilliams, Guy 2192, 2544, 2664, 3779, 3856, 3871, 4227, 5086, 5097\n\nWilliams, JoBeth 3632, 5029\n\nWilliams, John 1942\n\nWilliams, Kathlyn 4070, 4522, 4776\n\nWilliams, Lester 1047, 4448; _see also_ Berke, William\n\nWilliams, Maston 434, 631, 688, 767, 1340, 1517, 1822, 1860, 2399, 2628, 2907, 2956, 2992, 3015, 3187, 3449, 3549, 4442, 4609, 4893, 5009\n\nWilliams, Paul 4975\n\nWilliams, Rhys 577, 612, 1071, 1093, 1274, 1319, 2095, 2575, 2672, 3367, 3852, 4035, 5025\n\nWilliams, Roger 8, 54, 77, 468, 499, 535, 631, 684, 725, 774, 776, 781, 793, 810, 1047, 1052, 1281, 1283, 1284, 1323, 1455, 1464, 1545, 1550, 1664, 1687, 1732, 1859, 1860, 2248, 2290, 2298, 2355, 2403, 2456, 2470, 2640, 2701, 2743, 2802, 2812, 2891, 2933, 3062, 3083, 3101, 3268, 3302, 3303, 3325, 3408, 3482, 3489, 3497, 3546, 3548, 3552, 3653, 3664, 3706, 3907, 3918, 3939, 3949, 4027, 4143, 4307, 4448, 4512, 4522, 4570, 4696, 4711, 4722, 4760, 4955, 4960, 4981, 4982, 4986, 5011, 5096\n\nWilliams, Rush 832, 913, 1441, 2400, 3222, 3340, 3417, 3572, 3787, 4566\n\nWilliams, Spencer, Jr. 529, 1787, 1788, 4615\n\nWilliams, Tex 466, 710, 1046, 1090, 1459, 2978, 3169, 3569, 4305\n\nWilliams, Treat 1375, 4881\n\nWilliams, Tudor 3695\n\nWilliams, Walt 3888; _see also_ Taliaferro, Hal; Wales, Wally\n\nWilliamson, Fred 17, 474, 2057, 2331, 4040, 4224\n\nWilling, Foy, and The Riders of the Purple Sage 68, 284, 605, 875, 878, 1141, 1264, 1603, 1618, 1823, 2193, 2825, 2935, 3670, 3914, 4076, 4198, 4213, 4221, 4291, 4378, 4425, 4488, 4551, 4597, 4598, 4642\n\nWilling, William 3486\n\nWillingham, Calder 2373, 2901\n\nWillingham, Herman 464\n\nWillingham, Mary 148, 567, 1727\n\nWillingham, Noble 760, 761, 2092, 4045\n\nWillingham, Willard 148, 567, 1413, 1727, 3140, 3310, 3722, 4005\n\nWillis, Bruce 4195\n\nWillis, Leo 1162, 1851, 4375, 4931\n\nWillis, Matt 1319, 1974, 3001, 3923, 5045\n\nWillis, Norman 179, 199, 233, 279, 307, 1253, 1444, 1513, 1526, 1758, 1815, 1981, 2124, 2132, 2342, 2481, 2869, 2973, 2975, 3001, 3128, 3461, 3587, 3692, 3764, 3902, 4494, 4600, 4741\n\nThe Willis Brothers _see_ The Oklahoma Wranglers\n\nWillock, Dave 26, 1220, 3576, 4372\n\nWills, Bob, and His Texas Playboys 391, 1579, 2197, 2296, 2398, 2635, 3410, 3467, 3894, 4226, 4465, 4736, 5035\n\nWills, Chill 43, 63, 144, 198, 243, 248, 332, 420, 527, 621, 677, 1005, 1256, 1441, 1560, 1596, 1671, 1673, 1733, 1792, 1876, 1925, 2096, 2304, 2381, 2386, 2544, 2626, 2839, 2851, 2889, 2989, 2990, 3055, 3137, 3316, 3310, 3411, 3436, 3522, 3567, 3622, 3663, 3740, 4428, 4572, 4585, 4588, 4798, 4849, 4851, 5032, 5045, 5068\n\nWills, Henry 16, 209, 241, 299, 307, 309, 334, 367, 436, 446, 459, 465, 486, 488, 552, 607, 733, 779, 800, 836, 853, 883, 977, 1082, 1097, 1362, 1423, 1568, 1646, 1702, 1747, 2086, 2093, 2234, 2246, 2335, 2342, 2480, 2619, 2700, 2736, 2778, 2779, 2859, 2884, 2901, 2928, 2971, 3088, 3116, 3128, 3149, 3219, 3269, 3337, 3338, 3340, 3387, 3415, 3478, 3600, 3613, 3636, 3671, 3693, 3707, 3718, 3722, 3752, 3808, 3816, 3840, 3853, 3904, 3933, 4010, 4016, 4035, 4040, 4051, 4073, 4120, 4155, 4188, 4202, 4487, 4504, 4825, 4857, 4973, 5059, 5104\n\nWills, Walter 886, 1626, 1929, 2403, 2802, 3709, 4031\n\nWilsey, Jay 2409, 2413, 2415, 3109, 3279, 3501, 4271, 4797; _see also_ Buffalo Bill, Jr.\n\nWilson, Al 3076\n\nWilson, Ben 963, 2735, 2844, 3488, 3797, 4063, 4406, 4604, 4836, 4546\n\nWilson, Charles 379, 3554, 3904\n\nWilson, Clarence 1080, 1165, 1369, 2635, 2654, 2732, 2923, 3634, 3965, 4215, 4284, 4750, 4865, 5004, 5072\n\nWilson, Don 541, 1379, 3455, 3624, 3983\n\nWilson, Dooley 3053\n\nWilson, Dorothy 2637, 3745, 4865\n\nWilson, Elizabeth 692\n\nWilson, Harry 200, 597, 1842, 2291, 2416, 3234, 4057, 4073, 4329\n\nWilson, Lewis 2150\n\nWilson, Lois 852, 2249, 3442, 4704\n\nWilson, Margery 3373\n\nWilson, Marie 2720\n\nWilson, Michael 446, 800, 1416\n\nWilson, Richard 2563\n\nWilson, Scott 4577\n\nWilson, Terry 966, 1106, 2192, 2198, 2506, 2738, 2906, 3096, 3127, 3218, 3740, 3752, 3780, 3806, 4210, 4790\n\nWilson, Whip 2, 152, 649, 716, 892, 1279, 1585, 1715, 1744, 1800, 2294, 2680, 2776, 2798, 2970, 3262, 3460, 3802, 3900, 3908, 4095, 4105, 4781, 5039\n\nWindom, William 512, 679, 1943, 3753\n\nWindsor, Marie 271, 484, 594, 710, 934, 979, 1140, 1313, 1433, 1610, 1828, 1843, 2372, 2437, 2500, 2709, 2911, 2960, 3045, 3852, 3906, 4210, 4236, 4613, 4976\n\nWindust, Bretaigne 1249\n\nWindust, Penelope 628\n\nWing, Toby 3139\n\nWinkler, Robert 3157, 4980\n\nWinner, Michael 709, 2305\n\nWinninger, Charles 278, 1084, 1300, 1685, 2163, 3066, 4900\n\nWinslow, George 4943\n\nWinters, Gloria 1214\n\nWinters, Jonathan 2694, 4749\n\nWinters, Linda 2828, 3109\n\nWinters, Roland 1600, 1996, 3285, 3864\n\nWinters, Sally 785, 855, 1433, 2231, 2567, 2763, 2767, 3047, 3323, 3474, 3718, 3736, 4293, 4674\n\nWinters, Shelley 1372\n\nWinwood, Estelle 2661, 3776\n\nWisbar, Frank 3154, 3511\n\nWisberg, Audrey 652\n\nWise, Robert 402, 4548, 4611\n\nWiseman, Lulu Belle 3834\n\nWiseman, Scotty 3834\n\nWisemann, Joseph 2305, 4662, 4751\n\nWister, Owen 4743, 4744, 4745, 4746\n\nWithers, Grant 91, 112, 147, 277, 284, 656, 930, 1216, 1313, 1395, 1490, 1591, 1706, 1843, 1849, 1906, 1982, 2006, 2066, 2163, 2184, 2234, 2310, 2481, 2607, 2649, 2718, 2806, 2838, 2856, 2878, 3132, 3522, 3636, 3728, 3941, 4033, 4076, 4257, 4274, 4551, 4681, 4686, 4703, 4786, 4913, 5020, 5033, 5050\n\nWithers, Isabel 2259, 3447, 3938, 4659, 4929\n\nWithers, Jane 158, 1560, 3846, 4925\n\nWitherspoon, Cora 1430\n\nWitherspoon, Reese 3395\n\nWitney, William 37, 109, 110, 148, 284, 453, 796, 1141, 1142, 1246, 1264, 1413, 1522, 1603, 1618, 1805, 1823, 1836, 1862, 1914, 1971, 2006, 2128, 2132, 2135, 2200, 2399, 2403, 2445, 2806, 2825, 2880, 2882, 2893, 2938, 2966, 3015, 3024, 3437, 3585, 3705, 3803, 4043, 4056, 4076, 4083, 4151, 4213, 4488, 4551, 4558, 4597, 4641, 4699, 5050\n\nWolfe, Bill 155, 382, 395, 414, 431, 800, 1070, 1952, 2032, 2276, 2315, 2421, 2529, 2552, 2722, 2786, 2998, 3052, 3142, 3707, 3709, 3989, 4357, 4463, 4655, 4708, 5059, 5096, 5104\n\nWolfe, Bud 28, 880, 951, 2022, 2035, 2125, 2171, 2357, 2969, 3741, 4011, 4240, 4584\n\nWolfe, Ian 63, 91, 414, 602, 798, 832, 1949, 2577, 3195, 3780, 3902, 4571, 5075\n\nWolff, Frank 1286, 1581, 1643, 1955, 2116, 2898, 2900, 3514, 4158, 4738\n\nWolheim, Louis 2896, 5015\n\nWolter, Ralf 106, 1069, 1761, 2615, 2883, 3198, 4540, 4542, 4919, 5003\n\nWong, Anna May 3729\n\nWood, Britt 290, 448, 450, 459, 489, 718, 994, 1145, 1870, 2152, 2252, 2582, 3116, 3267, 3385, 3386, 3481, 3704, 3851, 4085, 4110, 4141, 4162, 4278, 4495\n\nWood, Douglas 1594, 1827, 1925, 3126, 3179, 4580\n\nWood, Edward D., Jr. 2300\n\nWood, Gordon D. (G.D.) 1414, 1440, 1496, 1927, 2133, 2473, 2681, 2862, 3927, 4605 _see also_ DeMain, Gordon\n\nWood, Harley (Harlene) 1283, 2248, 2285, 4696, 4893\n\nWood, Ken 394, 694, 2216, 3513, 3660\n\nWood, Lana 1623, 2076, 2899, 2990, 3752\n\nWood, Montgomery 18, 400, 1410, 3118, 3380; _see also_ Gemma, Guiliano\n\nWood, Natalie 580, 2782, 2899, 3752\n\nWood, Robert 1359, 4903, 4904; _see also_ Woods, Robert\n\nWood, Sam 73, 3276\n\nWoodbury, Joan 564, 626, 1070, 1202, 1362, 1577, 2371, 2469, 2841, 3431, 3859, 4022, 4206, 5043\n\nWoodell, Barbara 170, 251, 577, 649, 1240, 1403, 1635, 1713, 1919, 1959, 1960, 2680, 3333, 3377, 3857, 4856, 4966\n\nWoodell, Woody, and His Riding Rangers 2687\n\nWoods, Clarise 752\n\nWoods, Craig 89, 1398, 3052, 3229\n\nWoods, Donald 288, 470, 964, 1436, 1855, 2649, 3379, 4576\n\nWoods, Dorothy 4780\n\nWoods, Harry 159, 293, 410, 464, 476, 566, 624, 626, 642, 726, 798, 813, 824, 847, 960, 973, 986, 1028, 1146, 1247, 1362, 1392, 1458, 1488, 1490, 1546, 1796, 1802, 1835, 1843, 1852, 1861, 1969, 1973, 1975, 1988, 2020, 2164, 2187, 2208, 2223, 2249, 2260, 2299, 2404, 2416, 2595, 2605, 2642, 2718, 2767, 2775, 2969, 3029, 3041, 3087, 3126, 3255, 3259, 3260, 3269, 3303, 3480, 3561, 3591, 3600, 3609, 3650, 3814, 3815, 3848, 3886, 3893, 3902, 3924, 4047, 4049, 4057, 4072, 4104, 4142, 4200, 4204, 4231, 4384, 4297, 4308, 4386, 4503, 4533, 4564, 4665, 4667, 4777, 4786, 4803, 4808, 4823, 4825, 4841, 4853, 4865, 4932, 4951, 5001, 5039\n\nWoods, Robert 281, 374, 694, 1217, 1519, 2724, 3272, 3785; _see also_ Wood, Robert\n\nWoods, Walter 3139, 4215\n\nWoodward, Bob 9, 104, 111, 183, 187, 201, 335, 377, 389, 409, 436, 442, 492, 611, 649, 728, 850, 871, 893, 925, 960, 994, 1325, 1332, 1345, 1367, 1373, 1442, 1464, 1513, 1670, 1684, 1740, 1724, 1803, 1868, 1990, 1916, 1974, 2068, 2082, 2171, 2217, 2259, 2282, 2402, 2778, 2805, 2857, 2858, 2868, 2885, 2949, 2988, 2992, 3002, 3050, 3103, 3230, 3261, 3265, 3277, 3388, 3444, 3553, 3556, 3619, 3673, 3693, 3802, 3823, 3908, 4021, 4037, 4059, 4083, 4097, 4099, 4199, 4202, 4208, 4238, 4437, 4559, 4568, 4608, 4618, 4730, 4813, 4814, 4858, 4899, 5005, 5038, 5082\n\nWoodward, Edward 701\n\nWoodward, Frances 3458\n\nWoodward, Joanne 316, 843\n\nWoodward, Morgan 959, 1015, 1091, 1346, 1637, 1675, 1727, 1755, 2188, 2739, 2906, 3203, 3415, 3645, 3961, 4856\n\nWoodworth, Marjorie 1179\n\nWoody, Jack 664, 2204, 3500, 4080, 4162, 4390\n\nWooley, Sheb 489, 562, 685, 1111, 1560, 1844, 1877, 2048, 2372, 2485, 2565, 2871, 2950, 3417, 3517, 3572, 3607, 3754, 3911, 3954, 4267, 4470, 4566, 4790\n\nWoolf, Bill 464, 793, 2482, 4767, 4798\n\nWoolley, Monty 1571\n\nWoolsey, Robert 1563, 3529, 3886\n\nWorden, Hank 43, 89, 115, 319, 324, 369, 370, 459, 550, 577, 594, 644, 739, 741, 777, 880, 914, 1150, 1187, 1313, 1395, 1412, 1416, 1453, 1465, 1512, 1553, 1556, 1770, 1843, 1900, 1937, 1989, 2016, 2163, 2208, 2291, 2325, 2357, 2480, 2626, 2690, 2802, 2837, 2861, 2897, 2901, 2938, 3126, 3150, 3157, 3162, 3211, 3258, 3310, 3323, 3362, 3452, 3504, 3527, 3542, 3589, 3590, 3748, 3750, 3752, 3776, 3924, 4074, 4152, 4168, 4179, 4186, 4261, 4360, 4423, 4441, 4469, 4564, 4582, 4695, 4705, 4747, 4764, 4848, 4872, 5001, 5020, 5035; _see also_ Snow, Heber\n\nWorlock, Fredric 1949, 2218, 2837, 3095\n\nWormser, Richard 656, 1400, 1762, 2938, 3066, 3128, 3151, 3416, 3652, 4229, 4730\n\nWorth, Barbara 1276, 1337, 3161\n\nWorth, Constance 928, 2150, 3678, 4847\n\nWorth, David 2843, 3022, 3413\n\nWorth, Harry 37, 243, 323, 865, 927, 986, 1925, 1933, 2020, 2082, 2355, 2385, 2586, 3083, 3475, 4257\n\nWrather, Jack 4171\n\nWray, Fay 441, 653, 825, 2843, 4281, 4750\n\nWray, John 200, 1436, 2393, 251, 4061\n\nWright, Ben 2046, 2399, 2616, 3437, 4333, 4737\n\nWright, Harold Bell 633, 2618, 2654, 3764, 3817, 3818, 4840, 4864, 4933\n\nWright, Helen 4084\n\nWright, Howard 1691, 2217, 2325, 2339, 2899, 3221, 3769, 4151, 4441\n\nWright, Mack V. 136, 322, 397, 818, 1626, 1796, 1898, 2527, 2567, 2921, 3256, 3259, 3482, 3459, 3603, 3919, 3994, 4732, 4995\n\nWright, Tenny 326, 4248\n\nWright, Teresa 604, 654, 842, 3195, 4473\n\nWright, Wen 242, 459, 607, 1001, 1138, 1687, 1977, 2589, 3116, 3177, 3267, 3467, 3898, 4208, 4889, 4923\n\nWright, Will 53, 68, 317, 357, 602, 935, 1005, 1144, 1649, 1719, 1925, 2008, 2048, 2052, 2074, 2226, 2482, 2668, 2705, 3200, 3221, 3344, 3535, 3544, 3684, 3728, 3859, 4198, 4225, 4233, 4357, 4411, 4725, 4786, 4890, 4979\n\nWright, William 1144, 3611, 3674\n\nWrixon, Maris 2027, 3711, 4037, 4199, 4504\n\nWyatt, Al 418, 539, 1003, 1183, 1668, 1832, 1938, 2025, 2521, 2664, 3168, 3294, 3559, 3792, 3931, 3978, 4469, 4729, 4886\n\nWyatt, Charlene 461\n\nWyatt, Charlotte 145\n\nWyatt, Jane 549, 638, 1567, 1953, 2081, 4537\n\nWycherly, Margaret 5045\n\nWyenn, Than 3871\n\nWyler, Richard 3286, 4631\n\nWyler, William 313, 1434, 1779, 1850, 4139, 4851\n\nWyman, Jane 203, 718, 5045\n\nWymore, Patrice 329, 2504, 3572, 3663\n\nWynant, H.M. 378, 523, 1026, 2229, 2616, 2926, 3438, 3640, 3686, 4119, 4457\n\nWynn, Keenan 92, 94, 640, 657, 1078, 1098, 1206, 1653, 1784, 2013, 2046, 2324, 2447, 2578, 2563, 2749, 2795, 2838, 2898, 3033, 3203, 3204, 3944, 3962, 4098, 4101, 4291, 4749, 4802\n\nWynn, May 4338, 4740, 4913\n\nWynn, Tracy Keenan 3203\n\nWynne, Peggy 4935\n\nWynorski, Jim 1777\n\nWynter, Dana 3712\n\nWynters, Charlotte 633, 1428, 2014, 3354, 4208, 4494\n\nXochitl 1250\n\nXydias, Anthony J. 968, 1524, 1859, 4780\n\nYaconelli, Frank 13, 150, 272, 384, 724, 1156, 1159, 1175, 1299, 1344, 1389, 1774, 1835, 2418, 2470, 2478, 2586, 2782, 3035, 3102, 3502, 3503, 3561, 3601, 3770, 4047, 4165, 4365, 4554, 4771, 4839, 4844, 4884, 4947, 4951, 4953, 4971\n\nYamaoka, Otto 3409, 4572\n\nYanni, Rosanna 225, 4896\n\nYarbo, Lillian 1084, 1638, 2471, 3374, 4932\n\nYarbrough, Jean 2452, 2989, 4086, 4598, 4654, 5044\n\nYarnell, Sally 3607\n\nYarzi, Rosita 275\n\nYates, George Worthing(ton) 2225, 2399, 4349\n\nYerby, Frank 1427\n\nYoakam, Dwight 3013, 4045, 4356\n\nYordan, Philip 202, 406, 502, 513, 651, 981, 1166, 1289, 2048, 2192, 2525\n\nYork, Dick 859, 4335\n\nYork, Duke 620, 1627, 2020, 2665, 2840, 302, 3583, 3848, 3890, 3978, 3980, 4057, 4062, 4072, 4118, 4257, 4285, 4365, 4498, 4692, 4868, 5045\n\nYork, Francine 641, 2657, 4412\n\nYork, Jeff 2052, 2887, 3020, 3339, 3450, 3731, 4212, 4856\n\nYost, Dorothy 864, 1847, 2386, 3682, 3972, 4166\n\nYost, Robert 145, 147, 471, 644, 667, 1043, 1153, 1394, 2995, 3084, 4395, 4409, 5062\n\nYoung, Alan 216\n\nYoung, Carleton 37, 129, 210, 293, 335, 336, 341, 342, 344, 349, 353, 482, 720, 777, 810, 876, 1140, 1211, 1370, 1523, 1594, 1606, 1662, 1667, 1756, 1860, 1929, 1937, 1945, 2128, 2144, 2204, 2403, 2561, 2645, 2667, 2872, 2909, 2948, 2994, 3018, 3028, 3160, 3165, 3289, 3318, 3377, 3624, 3640, 3690, 3776, 3977, 4051, 4226, 4286, 4393, 4470, 4549, 4617, 4700, 5104; _see also_ Roberts, Gordon\n\nYoung, Clara Kimball 1469, 1889, 2850, 3624, 4367\n\nYoung, Clarence Upson 49, 196, 1562, 2288, 2830, 3502, 4006\n\nYoung, Clifton 284, 402, 599, 2432, 3685, 4543, 4621\n\nYoung, Evelyn 3170, 4980\n\nYoung, Faron 955, 1871, 3225\n\nYoung, Frank H. 428, 1373, 1540, 1687, 1797, 3052, 3264, 4021, 4155, 4550\n\nYoung, Gig 128, 517, 2482, 2919, 3958, 4336\n\nYoung, Lon 629\n\nYoung, Loretta 64, 2159, 3214, 3245\n\nYoung, Nedrick (Ned) 439, 599, 1097, 2005, 3500, 4080, 4267, 4662\n\nYoung, Polly Ann 447, 894, 1895, 2548, 2712, 2915, 5011\n\nYoung, Robert 1361, 1762, 2837, 3344, 4849\n\nYoung, Roland 3634, 4083\n\nYoung, Skip 5063\n\nYoung, Tammany 4776\n\nYoung, Terence 3336\n\nYoung, Tony 704, 1809, 2510, 4223\n\nYoungman, Henny 815\n\nYowlachie, Chief 94, 488, 555, 648, 691, 716, 864, 973, 1164, 1175, 1214, 1449, 1711, 1762, 1842, 1991, 2133, 2225, 2416, 2552, 2705, 2720, 2831, 2931, 2971, 3013, 3020, 3056, 3154, 3321, 3323, 3425, 3607, 3673, 3710, 3916, 3932, 4001, 4792, 4797, 4899, 4969, 4989, 5001, 5051, 5055\n\nYrigoyen, Bill 37, 710, 2399, 2403, 3088, 5104\n\nYrigoyen, Bob 3669\n\nYrigoyen, Joe 28, 37, 453, 637, 710, 883, 1543, 1668, 2339, 2399, 2403, 2514, 2529, 2635, 2886, 2975, 3015, 3088, 3563, 3816, 4995, 5022, 5103, 5104\n\nYule, Joe 332, 2020, 2040, 2790\n\nYulin, Harris 80, 2023, 2229, 2800, 3282, 4746\n\nYung, Victor Sen 1600, 1658, 2066, 2154, 3320, 3437, 3676, 4411, 4582, 4692\n\nYurka, Blanche 1478, 4058, 4384\n\nYuro, Robert 3439, 3806\n\nZacharias, Alfred 232\n\nZacharias, Steffan 3, 2510, 2553, 4334, 4724\n\nZadora, Pia 587\n\nZahler, Lee 388\n\nZalewska, Halina 1198\n\nZamperla, Nazzareno 19, 3785, 5087, 5092\n\nZandra (dog) 3038\n\nZanuck, Darryl F. 1949\n\nZapien, Danny 2563, 2700, 3060, 3193, 3345, 4775\n\nZappa, Frank 3638\n\nZaremba, John 3363, 3739\n\nZaremba, Paul 928, 3385, 3739\n\nZeglio, Primo 3346, 4629\n\nZeller, Ben 474, 585, 1066, 3712, 3855, 4346, 4637, 4968\n\nZerbe, Anthony 740, 1946, 1987, 2348, 4149, 4988\n\nZeta-Jones, Catherine 2341, 2604\n\nZieff, Howard 1828\n\nZimbalist, Efrem, Jr. 177, 3406\n\nZimmerman, Victor 412, 1884, 4336, 4932\n\nZinneman, Anna 2030\n\nZinneman, Fred 1877, 2853, 4192\n\nZsigmond, Vilmos 2623\n\nZucco, George 2790\n\nZuckerman, George 439, 972, 3423, 5049\n\nZuckert, William (Bill) 1631, 1769, 1899, 1986, 2319, 3739\n\nZugsmith, Albert 2550, 3287, 3337, 4123\n\nZuniga, Frank 1479\n\nZurakowska, Dianik 238, 1289, 3515\n\nZurli, Guido 4012, 5099\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nTHE FREE PRESS  \nA Division of Simon & Schuster Inc.  \n1230 Avenue of the Americas  \nNew York, NY 10020\n\nCopyright \u00a9 1999 by James W. Stigler and James Hiebert  \nAll rights reserved,  \nincluding the right of reproduction  \nin whole or in part in any form.\n\nTHE FREE PRESS and colophon are  \ntrademarks of Simon & Schuster Inc.\n\nLibrary of Congress Cataloging-in-Publication Data\n\nStigler, James W.  \nThe teaching gap : best ideas from the world's teachers for improvingeducation in the classroom \/ James W. Stigler and James Hiebert  \np. cm.  \nIncludes bibliographical references and index.  \n1. Mathematics\u2014Study and teaching\u2014United States.  \n2. Mathematics\u2014Study and teaching\u2014Germany.3. Mathematics\u2014Study  \nand teaching\u2014Japan.4. Comparative education.I. Hiebert, James.  \nII. Title.  \nQA13.S77 1999  \n510\u2032.71\u2014dc21 99-27270\n\nCIP\n\neISBN-13: 978-1-4165-8638-8  \neISBN-10: 1-4165-8638-5\n\nVisit us on the World Wide Web:  \n<http:\/\/www.SimonandSchuster.com>\nThis book is\n\ndedicated to the memory of\n\nAlbert Shanker\n\n1928-1997\n\n## Contents\n\nPreface\n\nChapter 1: The Teaching Gap\n\nChapter 2: Methods for Studying Teaching in Germany, Japan, and the United States\n\nChapter 3: Images of Teaching\n\nChapter 4: Refining the Images\n\nChapter 5: Teaching Is a System\n\nChapter 6: Teaching Is a Cultural Activity\n\nChapter 7: Beyond Reform: Japan's Approach to theImprovement of Classroom Teaching\n\nChapter 8: Setting the Stage for Continuous Improvement\n\nChapter 9: The Steady Work of Improving Teaching\n\nChapter 10: The True Profession of Teaching\n\nNotes\n\n## Preface\n\nTHIS BOOK IS ABOUT teaching and how to improve it. It is not another attempt to bash teachers or blame them for the ills that beset America's schools. It is also not another set of recommendations that tell teachers how to teach. It is, instead, a tribute to the importance of teaching, and to the key role that teachers must play in its improvement. School learning will not improve markedly unless we give teachers the opportunity and the support they need to advance their craft by increasing the effectiveness of the methods they use.\n\nOur viewpoint arises from a collaboration that started more than five years ago. At that time, the Third International Mathematics and Science Study (TIMSS) was well into the planning stages. This study, the latest in a series of international studies stretching back more than thirty years, compared mathematics and science achievement among students in forty-one nations. TIMSS was the most carefully designed international study of achievement ever conducted. One component of TIMSS was a video study that compared the teaching of eighth-grade mathematics in Germany, Japan, and the United States. The video study, on which we collaborated, marked the first time ever that national samples of teachers had been videotaped teaching in their classrooms. For the first time, we could see what teaching actually looks like on a national scale, and we could do this for three countries.\n\nFiguring out how to analyze and summarize these videos was challenging. But it also was a breathtaking experience. We often are blind to the most familiar aspects of our everyday environment, and teaching turns out to be one of these aspects. Looking across cultures is one of the best ways to see beyond the blinders and sharpen our view of ourselves. As we looked again and again at the tapes we collected, we were struck by the homogeneity of teaching methods within each culture, compared with the marked differences in methods across cultures.\n\nReaders who are parents will know that there are differences among American teachers; they might even have fought to move their child from one teacher's class into another teacher's class. Our point is that these differences, which appear so large within our culture, are dwarfed by the gap in general methods of teaching that exist across cultures. We are not talking about a gap in teachers' competence but about a gap in teaching methods. These cross-cultural differences in methods are instructive because they allow us to see ourselves in new ways.\n\nBut the teaching gap we describe refers to more than cross-cultural teaching differences. It refers to the difference between the kinds of teaching needed to achieve the educational dreams of the American people and the kind of teaching found in most American schools. Although many of the American teachers we observed were highly competent at implementing American teaching methods, the methods themselves were severely limited.\n\nThe teaching gap becomes even more significant when one realizes that while other countries are continually improving their teaching approaches, the United States has no system for improving. The United States is always reforming but not always improving. The most alarming aspect of classroom teaching in the United States is not how we are teaching now but that we have no mechanism for getting better. Without such a mechanism, the teaching gap will continue to grow.\n\n\u2022 \u2022 \u2022\n\nThis book started out as a description of teaching in different cultures based on the data we collected in the video study. As we wrote the book, however, these differences in teaching methods turned out to be only part of the story. Equally important are the general truths we came to understand about teaching and the implications of these truths for the improvement of classroom teaching. Thus, although this book was initially intended as a report of the TIMSS video study, it quickly became much more than that. We do describe mathematics teaching in Germany, Japan, and the United States. But we also examine current reform efforts in the United States, and based on what we learned about teaching and about learning to teach, we propose a new plan for improving classroom teaching in the United States. Because the video study focused on eighth-grade mathematics, most of the classroom examples we present are from eighth-grade mathematics classrooms. The points we make go well beyond mathematics, however\u2014and certainly well beyond eighth grade. Mathematics teachers might find the book especially interesting, but our intention became to write a book that would be of interest to teachers in all subjects at all levels.\n\nTeachers are not the only audience for this book. We have written it for school administrators, policymakers, politicians, and parents. Although teachers hold the key, they teach in a system that currently works against improvement. Unless other important players get involved, our country cannot implement a program that allows teachers to improve teaching. This would be unfortunate, not because it would miss an opportunity but because it would miss the only opportunity. The system must support teachers to improve teaching, because teachers are the key to closing the gap.\n\n\u2022 \u2022 \u2022\n\nThis book, and the study from which it grew, could not have been completed without the help of many people.\n\nThe TIMSS video study was funded by a contract from the National Center for Education Statistics (NCES), U.S. Department of Education, to WESTAT, Inc. The views expressed in this book, however, should in no way be construed to be those of NCES or of the U.S. Department of Education.\n\nWe are grateful to Emerson Elliot, past commissioner of NCES, and to Pascal Forgione, the current commissioner, whose support and enthusiasm went well beyond the financial. Lois Peak, the NCES program officer who oversaw the study, worked tirelessly to help us through the intricacies of a government project. Without her unwavering belief in the importance of this work, the study would never have been done. And we thank Nancy Caldwell of WESTAT for her constant and dependable help throughout the contract.\n\nThe TIMSS video study could not have been done without the help of our international collaborators. Juergen Baumert and Rainer Lehmann (in Germany) and Toshio Sawada (National Institute of Educational Research, Tokyo) managed the data-collection process and helped us to understand teaching in their countries.\n\nClea Fernandez, of Teachers College\/Columbia University, played a major role in the early planning and conceptualization of the project; Scott Rankin trained the videographers for the study; Takako Kawanaka, Steffen Knoll, and Ana Serrano led our coding-development efforts; Patrick Gonzales managed the transcription and translation process and contributed to our analyses of classroom discourse; Eric Derghezarian, Fumiko Ichioka, and Nicole Kersting worked endless hours in the analysis of videotapes; Gundula Huber and Alyne Delaney handled the digitizing of the tapes; and Ken Mendoza wrote the software that enabled us to manage the huge quantity of video collected in the study. We want to recognize these individuals for the important role they played in the project.\n\nA number of consultants advised us at key points. These included Nicolas Hobar, Christine Keitel, Magdalene Lampert, Gilah Leder, Shin-Ying Lee, Johanna Neubrand, Michael Neubrand, Yukari Okamoto, Hidetada Shimizu, Yoshinori Shimizu, Kenneth Travers, and Diana Wearne. We are grateful for their input.\n\nWe could never adequately acknowledge the tireless contribution of our mathematics content group, led by Alfred Manaster. Other members of this group included Phillip Emig, Wallace Etterbeek, and Barbara Wells. Their work greatly enhanced the findings of the study.\n\nThe writing of this book was supported in part by a generous grant from the Albert Shanker Institute. We want to express our deep appreciation to the late Albert Shanker for his early support of our efforts, and to Eugenia Kemble and Alice Gill, who followed through on the project after his death.\n\nMany people read drafts of the book at various stages; their suggestions and criticisms have improved the book greatly. These readers include Gail Burrill, Tom Carpenter, Megan Franke, Philip Jackson, Jennifer Jacobs, Jeremy Kilpatrick, Paul Kimmelman, Kevin Miller, Leigh Peake, Sheila Sconiers, Nanette Seago, Lisle Staley, William Stanley, Harold Stevenson, and Karen Stigler. Special thanks go to Ronald Gallimore, our most constant critic, whose good humor, encouragement, insight, and wisdom kept us going. Of course, none of these people are responsible for any flaws in the book (except for maybe Ron).\n\nWe also want to thank Susan Arellano, formerly of the Free Press, who believed in the book and sold the idea to the Free Press; and Philip Rappaport, our editor at the Free Press, whose clear vision and timely input helped us write a book that could speak to a real audience.\n\nFinally, we thank our families, whose support and tolerance made this book possible: Karen Stigler, Sam Stigler, Thomas Stigler, and Charlie Stigler; and Diana Wearne.\n\nWe welcome your comments on The Teaching Gap. Please visit our Web site at www.lessonlab.com\/teaching-gap.\n\n## CHAPTER 1\n\n## The Teaching Gap\n\nCONDITIONS FOR IMPROVING education in the United States are more favorable today than they have been in a generation. Both politicians and the public recognize that education needs to be improved. Bad news from international comparisons of student achievement is no longer seen as esoteric by the American public; these days it is on the front page and a linchpin of many politicians' stump speeches. In our increasingly global economy, citizens see direct evidence that America's future will depend on the education of its workforce, and they are determined to compete. Education has become a high priority among the electorate.\n\nBut the real reason for optimism is that all this attention to education is not just rhetoric. We are witnessing a tidal wave of educational reform that appears to gain momentum with each passing year. Virtually every state in the nation is working to develop high standards for what students should learn in school, along with means for assessing students' progress. In a field where fads have ruled, we are seeing something new: a growing commitment to the idea that clear and shared goals for student learning must provide a foundation on which to improve education and achievement. Without clear goals, we cannot succeed, for we cannot know in which direction to move.\n\nYet it is equally important to recognize that standards and assessments, though necessary, are not enough. What must be done now is to find ways of providing students with the learning opportunities they need to reach the new standards. Making higher standards a reality for students will require more than just the status quo inside our nation's classrooms; curriculum, assessments, and\u2014above all\u2014teaching must improve dramatically. In our view, teaching is the next frontier in the continuing struggle to improve schools. Standards set the course, and assessments provide the benchmarks, but it is teaching that must be improved to push us along the path to success.\n\nOur contention that standards alone are not enough is shared by many politicians and school reformers, and they stand ready to help. President Clinton has successfully pushed through legislation that will pour millions of dollars into reducing class size in elementary schools nationwide. Many states are actively considering making vouchers and school choice a central part of their educational systems. And many school districts are embarking on additional initiatives, such as creating charter schools, outfitting schools with new technologies, and sanctioning new forms of school management.\n\nWe believe that these highly visible efforts, though well intentioned, miss the mark, because they leave out the one ingredient most likely to make a difference in students' learning: the quality of teaching. Reducing the class size from thirty to twenty certainly will make teachers happier. But if teachers continue to use the same methods they used with larger classes, learning opportunities for students will change little. Similarly, implementing a voucher system might increase competition among schools and spur their desire to improve. But desire alone does not provide teachers with the knowledge they need to implement more effective methods. Class size reductions, vouchers, and most other popular efforts to improve schools will end in disappointment if they do not fundamentally improve what happens inside classrooms.\n\nWe are not the only ones to decry this lack of attention to the improvement of teaching. Jerome Bruner, an elder statesman in educational psychology, made the same point in his 1996 book, The Culture of Education:\n\nIt is somewhat surprising and discouraging how little attention has been paid to the intimate nature of teaching and school learning in the debates on education that have raged over the past decade. These debates have been so focused on performance and standards that they have mostly overlooked the means by which teachers and pupils alike go about their business in real-life classrooms\u2014how teachers teach and how pupils learn.\n\nOur goal in writing this book is to convince our readers that improving the quality of teaching must be front and center in efforts to improve students' learning. Teaching is the one process in the educational system that is designed specifically to facilitate students' learning. Of course, there are many other factors that influence learning in a significant way, such as students' home and social life, and the resources of the school and community. We do not want to minimize the importance of these for the well-being of children. But much of what our society expects children to learn, they learn at school, and teaching is the activity most clearly responsible for learning. Robert Slavin, long a leading educational researcher, made a similar observation in a recent article:\n\nThe problem, I would argue, is that reforms so often debated in the media, in the White House, in Congress, and in statehouses across the country do not touch on the changes needed to fundamentally reform America's schools.... These reforms ignore a basic truth. Student achievement cannot change unless America's teachers use markedly more effective instructional methods.\n\nWhat makes this argument compelling is that not only is teaching essential, it is a process we can do something about. Overemphasizing the importance of nonschool factors that often are, frustratingly, beyond the reach of public policy can become an excuse for not trying to improve. Teaching lies within the control of teachers. It is something we can study and improve.\n\n## The Learning Gap  \nand the Need to Improve\n\nGood questions to ask at this point are \"Why is it so important to improve teaching?\" and \"How do we know that improvement is needed? Maybe we are doing fine.\" Surprising as it may seem, there is considerable controversy about the answers to these questions. Influential educators and writers disagree. One answer is simply that there is always room for improvement; no matter how well our students are doing now, it would be foolish not to try to improve.\n\nThe truth, as we see it, however, is that the situation in the United States demands improvement, not just because improvement is possible but because it is needed. Our students are being shortchanged. They could be learning much more and much more deeply than they are learning now. In the most recent National Assessment of Educational Progress, a periodic thermometer of students' learning, only 38 percent of America's eighth-graders could figure out a 15 percent tip on the cost of a typical meal, even when given five choices from which they could select the correct answer. Is this good enough?\n\nBeyond the surveys of our own country's students, there are a number of sobering international reports. Several years ago, one of us coauthored a book called The Learning Gap. That book presented a study of schooling and achievement in Japan, Taiwan, China, and the United States. The findings were cause for concern: As early as fifth grade, U.S. students lagged far behind their counterparts in the other countries. On a test of mathematics achievement, for example, the highest-scoring classroom in the U.S. sample did not perform as well as the lowest-scoring classroom in the Japanese sample.\n\nInterest in international studies has grown since publication of The Learning Gap, heightened recently by release of the results of the Third International Mathematics and Science Study (TIMSS). As the name implies, this was the third in a series of international studies. The first was conducted in the 1960s and the second in the early 1980s. In both of these studies, U.S. students performed quite poorly compared with their peers in most Asian and many European countries. But neither of these two earlier studies came close to matching the size and quality of the TIMSS, by far the most comprehensive and methodologically sophisticated cross-national comparison of achievement ever completed. TIMSS investigated mathematics and science achievement among fourth-, eighth-, and twelfth-grade students in forty-one nations.\n\nThe results from TIMSS have garnered a great deal of media interest and have caught the attention of politicians, policymakers, and the general public. The results are dramatic, and they do not paint a flattering picture of American education. For example, in eighth-grade mathematics, twenty of the forty-one nations scored significantly higher, on average, than the United States, while only seven nations scored significantly lower than the United States. The seven nations scoring lower than the United States were Lithuania, Cyprus, Portugal, Iran, Kuwait, Colombia, and South Africa. Nations scoring significantly higher than the United States included Singapore, Korea, Japan, Canada, France, Australia, Hungary, and Ireland.\n\nOf course, the results of large international studies are always open to question. So much differs across cultures and educational systems, it is hard to know where to find the most meaningful comparisons. Are the samples comparable? Do we even have the same goals for education across cultures? Although the answers to these questions are important for interpreting the differences, the gap in achievement between U.S. students and those in other countries is simply too wide to be dismissed on methodological grounds. U.S. education is in need of improvement.\n\n## Beyond the Learning Gap\n\nAmericans increasingly are aware of this learning gap and are seeking ways to address it. The international comparisons grab the front-page headlines, and officials try to infer recommendations from how one country performs compared with the performance of another. Policymakers carefully study, state by state, scores on the most recent National Assessment of Educational Progress, as if one could divine a strategy, from the scores, for improving performance. Scores of all local schools are printed in the newspaper, and school boards and parents discuss why students in some schools score much lower than others.\n\nAs important as it is to know how well students are learning, examinations of achievement scores alone can never reveal how the scores might be improved. We also need information on the classroom processes\u2014on teaching\u2014that are contributing to the scores. Unfortunately, many policymakers have ignored this fact, making decisions about the future of education without even the most rudimentary information about what is happening in classrooms. In 1995, faced with low reading and mathematics performance on the National Assessment of Educational Progress, California's superintendent of public instruction formed two task forces, one for mathematics and one for reading, to study the situation and propose solutions. California, after all, was highly respected for its Curriculum Frameworks that guide reading and mathematics instruction in the state. The Frameworks provided a comprehensive outline for what students should learn and guidelines for appropriate instructional methods. If the Frameworks were so good, why was achievement so low?\n\nIn meetings of California's mathematics task force, the discussion often turned to the Frameworks. Were the teaching methods or curricular emphases recommended in the Mathematics Framework perhaps to blame for students' low achievement? A debate ensued among members of the task force, a debate that has been reflected more broadly in public debate around the country between proponents of \"reform\" teaching and those in favor of more \"traditional\" teaching methods. Some believed that the Frameworks were not working and should be changed; others believed that the state should stay the course. But often lost in the discussion was a key fact: the state of California had collected no data on the extent to which the Frameworks had been implemented in the state's classrooms. This did not stop the state, however, from undertaking a revision of its Mathematics Framework. But on what basis could the Framework be revised? Without knowing what teachers were doing, how could the effectiveness of the Framework be determined?\n\nWe do not mean to single out California; no state that we know of regularly collects and uses data directly related to instructional processes in the classroom. Policymakers adopt a program, then wait to see if student achievement scores will rise. If scores do not go up\u2014and this is most often what happens, especially in the short run\u2014they begin hearing complaints that the policy isn't working. Momentum builds, experts meet, and soon there is a new recommendation, then a change of course, often in the opposite direction. Significantly, this whole process goes on without ever collecting data on whether or not the original program was even implemented in classrooms\u2014or, if implemented, how effective it was in promoting student learning. If we wish to make wise decisions, we need to know what is going on in typical classrooms.\n\nFortunately, the same TIMSS that generated a new wave of concern about students' achievement also collected a wealth of information about educational factors that might help us understand the different levels of performance in different countries. TIMSS researchers analyzed textbooks; asked administrators, teachers, and students about their beliefs and practices; and videotaped teachers teaching typical lessons. The TIMSS video study of teaching, which forms the basis for this book, is especially significant because it provides a penetrating and unparalleled look into classrooms in three different countries. For the first time, we had a full video record of a representative sample of U.S. classrooms. More than that, we had the same kind of information from Germany and Japan. We could now compare more than achievement scores. We could examine similarities and differences in the instructional methods that lay behind these scores.\n\n## A Unique Opportunity\n\nThe data collected in the TIMSS video study allow us to answer questions that we could not answer previously yet are crucial for the formation of education policy in the years to come. What are the instructional methods that most teachers currently use? Are the highly publicized reform recommendations being implemented in the classrooms of the United States? Are there alternative ways of teaching in other cultures, or is mathematics teaching pretty much the same everywhere? As was pointed out earlier, a major obstacle in our efforts to improve education is the dearth of information about what is happening in our nation's classrooms. Video provides us with a unique way of gathering the information we need to examine our current practices and then improve them.\n\nVideo data, such as that collected in TIMSS, also help us discover new ideas about teaching. If alternative ways of teaching exist, video will capture them, even when they lie completely outside our society's current theories of teaching and learning. And because the new ideas are illustrated through actual classroom teaching, they can have immediate practical significance for teachers. Video information can shake up the way we think and let us take a fresh look at classrooms.\n\n## What We Have Learned  \nfrom the Video Study\n\nAs we look back over what we have learned from the TIMSS video study, several things stand out. We foreshadow these things here because they form the basis for the book you are reading.\n\nTeaching, Not Teachers, Is the Critical Factor\n\nAmericans focus on the competence of teachers. They decry the quality of applicants for teaching positions and criticize the talent of the current teaching corps. But we come away with a different conclusion: Although variability in competence is certainly visible in the videos we collected, such differences are dwarfed by the differences in teaching methods that we see across cultures. (In Chapters 2, 3, and 4 we present our analyses of teaching and describe what teaching looks like in each country.)\n\nWe have watched many examples of good teachers employing limited methods that, no matter how competently they are executed, could not lead to high levels of student achievement. Although there are teachers using extraordinary methods in all cultures, the extraordinary is not what defines most students' classroom experiences. Students' day-to-day experiences are mainly determined by the methods most commonly used by teachers within a culture. Cross-cultural differences in these commonly used methods are what we have termed the \"teaching gap.\"\n\nWhat we can see clearly is that American mathematics teaching is extremely limited, focused for the most part on a very narrow band of procedural skills. Whether students are in rows working individually or sitting in groups, whether they have access to the latest technology or are working only with paper and pencil, they spend most of their time acquiring isolated skills through repeated practice. Japanese teaching is distinguished not so much by the competence of the teachers as by the images it provides of what it can look like to teach mathematics in a deeper way, teaching for conceptual understanding. Students in Japanese classrooms spend as much time solving challenging problems and discussing mathematical concepts as they do practicing skills.\n\nTeaching Is a Cultural Activity\n\nTo put it simply, we were amazed at how much teaching varied across cultures and how little it varied within cultures. When we started, we believed there would be great variability in teaching methods within the United States. Political battles between advocates of, among other teaching techniques, phonics and whole language, and basic skills and conceptual understanding, would lead most Americans to assume that there are many different paths that teachers can follow. But these differences paled when we looked across countries from a cross-cultural, comparative perspective. Although we saw variation in the U.S. videos we collected, comparing them with videos from Germany and Japan allowed us to see something we could not see before: a distinctly American way of teaching, which differs markedly from the German way and from the Japanese way.\n\nTeaching is a cultural activity. We learn how to teach indirectly, through years of participation in classroom life, and we are largely unaware of some of the most widespread attributes of teaching in our own culture. (In Chapters 5 and 6 we pull together what we have learned about teaching and argue that if we are going to improve teaching, we must appreciate its cultural character.) The fact that teaching is a cultural activity explains why teaching has been so resistant to change. But recognizing the cultural nature of teaching gives us new insights into what we need to do if we wish to improve it.\n\nA Gap in Methods for Improving Teaching\n\nFinally, we have learned a great deal from the video study about the results of efforts to improve teaching in the United States. Earlier in this chapter we pointed to the dearth of information about the effects that educational policies have in the classroom. The videos provide us with this kind of information, and it is quite striking. Although most U.S. teachers report trying to improve their teaching with current reform recommendations in mind, the videos show little evidence that change is occurring. Furthermore, when teachers do change their practice, it is often in only superficial ways.\n\nThis will not surprise those who have worked in the field of teacher professional development. The problem of how to improve teaching on a wide scale is one that has been seriously underestimated by policymakers, reformers, and the public in this country. The American approach has been to write and distribute reform documents and ask teachers to implement the recommendations contained in such documents. Those who have worked on this problem understand that this approach simply does not work. The teaching profession does not have enough knowledge about what constitutes effective teaching, and teachers don't have a means of successfully sharing such knowledge with one another.\n\nTo really improve teaching we must invest far more than we do now in generating and sharing knowledge about teaching. This is another sort of teaching gap. Compared with other countries, the United States clearly lacks a system for developing professional knowledge and for giving teachers the opportunity to learn about teaching. American teachers, compared with those in Japan, for example, have no means of contributing to the gradual improvement of teaching methods or of improving their own skills. American teachers are left alone, an action sometimes justified on grounds of freedom, independence, and professionalism. This is not good enough if we want excellent schools in the next century. (In Chapters 7, 8, and 9 we discuss the problem of how to improve teaching, and offer a proposal to make improving teaching the focus of our efforts to close the achievement gap.)\n\nWe opened this chapter by describing the opportunities that exist at present for improving education. In this positive environment, the challenge that awaits our nation is to find a way to improve classroom teaching so that our educational goals can be realized.\n\n## CHAPTER 2\n\n## Methods for Studying  \nTeaching in Germany,  \nJapan, and the United  \nStates\n\nTHE STORY OF the TIMSS video study begins in 1993, when, for the first time in history, plans were made to videotape an international sample of eighth-grade mathematics teachers. Each teacher would be videotaped teaching one lesson in his or her own classroom.\n\nImagine a videographer, Ron Kelly, traveling around the United States for seven months, loaded with equipment, taping in a different school each day. At the same time, Andrea Lindenthal was driving around her native Germany, filming a different mathematics lesson each day, and Tadayuki Miyashiro was doing the same thing in Japan. They all were collecting data for the Third International Mathematics and Science Study.\n\n## Birth of the TIMSS Video Study\n\nPlanning for TIMSS was largely funded by the U.S. government, through the National Center for Education Statistics and the National Science Foundation. Long before any data were collected, teams of researchers were meeting and designing the tests and questionnaires that would be administered in forty-one countries. Most officials anticipated that American students would not fare well in the cross-national comparisons. The previous international comparisons had provided ample warning. But many of those involved in planning this study now understood that just worrying about low achievement scores would not help improve the scores. This time, officials wanted to be ready to discuss the reasons for low levels of student achievement and to suggest ways to improve achievement. They consulted widely with experts in mathematics and science education, looking for new ways of studying the processes that lead to student learning.\n\nFrom early on, there was a great deal of interest in collecting information on classroom instruction. Yet there was little agreement on how to do this. Previous large-scale international work had relied on questionnaires to collect information, a relatively inexpensive procedure. Perhaps teachers could be sent a questionnaire and asked to describe the methods they used. The problem was that it would be hard to interpret their responses. It is difficult to know how accurately teachers describe their methods and what they mean by the words they use. For example, if a teacher says she does \"problem solving\" (currently a popular phrase) with her students, what, exactly, does she do? Different teachers use the same words to mean different things.\n\nVideotaping teachers teaching in their own classrooms would solve this problem, but it was a radical idea. Video has been around for some time, and researchers have used it to study teaching. But no one had used video to collect a national sample of anything, certainly not teaching. The logistical problems alone were daunting. But the opportunity to peer into the classrooms of various countries was too exciting to pass up.\n\nSo was born the video study. Because the goal was to find out more about some of the things that might account for students' performance, the planners wanted to videotape teaching in a variety of countries. But which ones? Videotaping in all forty-one countries was impossible, for both logistical and financial reasons. Three countries were chosen: the United States, Japan, and Germany. Japan was an obvious choice because it has always scored near the top in international comparisons of mathematics achievement. Germany, though it had not participated previously in the large international studies, was also considered an important comparison country, because Germany, like Japan, is a major economic competitor of the United States. The stage was set.\n\n## Goals of the TIMSS  \nVideo Study\n\nAlthough the challenges of using video on a large scale are considerable, the major goals of the TIMSS video study were simple and straightforward:\n\n  1. To learn how eighth-grade mathematics is taught in the United States.\n  2. To learn how eighth-grade mathematics is taught in two comparison countries, Germany and Japan.\n  3. To learn something about the way American teachers view reform and whether they are implementing teaching reforms in their classrooms.\n\nThese goals point us toward the first step we must take to improve education in the United States. They focus attention on classroom teaching and on collecting basic information on the instructional methods we currently are using. They allow us to answer fundamental questions that until now we could only guess at: \"How do most American teachers teach?\" and \"How does this compare with how their peers in other countries teach?\"\n\n## Research Methods:  \nThe Nuts and Bolts\n\nWe started this chapter with three videographers, one in each country, traveling from school to school. Let us go back to describe in more detail how the video study was conducted.\n\nOnce inside the classroom, the videographers collected two main types of data: a videotape of the lesson and a questionnaire response from the teacher. They also collected supplementary materials, such as copies of textbook pages or worksheets, that were helpful for understanding the lesson. Each classroom was videotaped for one complete lesson on a date convenient for the teacher.\n\nBecause there are a number of ways in which the integrity of a study like this could be undermined, it is important to understand a few key features of the study. Having confidence in the findings depends on knowing how we dealt with the following issues.\n\nWhich Classrooms Should Be Videotaped?\n\nThe sample of the study is crucial: If we were to produce a national-level portrait of eighth-grade mathematics instruction, we needed to be sure that the sample of teachers was representative of eighth-grade mathematics teachers. Fortunately, the TIMSS sampling plan was highly sophisticated, and the video sample was constructed to be a random subsample of the full TIMSS sample.\n\nSelecting classroom lessons was not easy. Schools were selected first, then teachers, and then classes. Often we were asked by the school to substitute one teacher for another, or by the teacher to videotape a different class period than the one that had been sampled. We allowed no substitutions of either sort, because to do so would have introduced bias into the study.\n\nThe final sample was a \"national probability sample.\" This technical term means that every eighth-grade math teacher in the country and each of the teacher's classes had an equal chance of being selected for the study. This was true in all three of the countries studied. The final video study sample included 231 eighth-grade mathematics classrooms: 100 in Germany. 50 in Japan, and 81 in the United States. Although the number of lessons was not large, because they were sampled randomly we could be sure that they approximated, in their totality, the mathematics instruction to which students in the three countries were exposed. In short, we had a sample that met the highest standards in statistical methodology.\n\nWill the Camera Change What the Teacher Does?\n\nMany people wonder whether the videotapes show what teachers normally do when the camera is not present. Teachers knew, after all, that the videographer was coming. Surely they would try to prepare in some way. They might even go so far as to design a special lesson just for the videotaping. Even if this did not happen, the presence of the camera might affect a teacher's behavior in subtle, unconscious ways. This was a serious concern, and we did not take it lightly.\n\nWe explained the goal of the study to all the teachers and asked them to teach the same lesson they would have taught if the camera had not been there. Most teachers do not want to bias a research study, but some might inadvertently do so to please or impress the researchers. We told them we wanted to see a typical lesson, the one they had originally scheduled for that day.\n\nTo check on typicality, we asked teachers to describe in the questionnaire what they did in the same class the previous day and what they were planning to do the next day. This helped us to determine whether the teacher had taught a special, standalone lesson for the camera or whether, as we hoped, the taped lesson fit within an ongoing sequence. Teachers' responses indicate that few lessons were stand-alone, and we believe most were quite typical.\n\nFinally, we used common sense in deciding the kinds of indicators that might be susceptible to bias and took this into account when interpreting the results. It seems likely, for example, that students were on their best behavior in front of the camera, so we believe the videotapes do not show the normal frequency with which teachers must discipline students. On the other hand, it seems unlikely that teachers asked completely different kinds of questions while being videotaped than they did when the camera was not present. Some behaviors, such as the routines of classroom discourse, are so highly socialized that they are almost automatic and are difficult to change.\n\nTurning Videos into Information\n\nAfter we collected the videos, the difficult task of analysis began. Meaning is not contained in the videos; it must be constructed by the researcher through a difficult and painstaking process. We began this process in May 1994 with a set of nine trial tapes from each country. A team of six code developers\u2014two from Germany, two from Japan, and two from the United States\u2014and several mathematics educators spent the summer at UCLA watching and discussing the contents of the tapes. Our goal was to understand how teachers construct and implement lessons in each country, and to develop a common language for describing the lessons.\n\nThe process was a straightforward one: we would watch a tape (subtitled in English), discuss it, and then watch another. Anyone could stop the tape at any time. The discussion was so vigorous that it often would take a day or more to get through a single lesson. There were disagreements in the group about the contents of the tapes, and especially about how to describe them. But gradually, as we worked our way through the twenty-seven tapes, we began to develop a common view of the nature of teaching in the three countries. More than that, we began to develop a coding system to compare teaching across the three countries.\n\nMany people who are unfamiliar with behavioral research do not understand how it is possible to code video objectively. They assume that there is only one thing to do with video: watch it. But there is a great deal more one can do. It is possible to move beyond individual impressions and identify features of the events portrayed in a video objectively, so that anyone who watches can agree. These objective judgments can be used to quantify the events on the video so that one can know how frequently different categories of activities occur.\n\nThe process begins with a discovery that is turned into a hypothesis. For example, we might notice that German teachers develop concepts more fully than do U.S. teachers, or that Japanese teachers ask more open-ended questions than do German and U.S. teachers. We then propose this \"discovery\" as a hypothesis. The next step is to write a definition that will communicate to other coders what \"counts as\" developing a concept or as asking an open-ended question. Anyone who has engaged in this process knows that it is not easy to write such a definition. But it can be done. For example, how do you even know something is a question? Would something like \"Do you think you could open the door?\" count as a question just because it ends with a question mark? Clearly it should not, because in American culture such a statement is a request, not a question.\n\nThe test of how successful we were in developing objective codes was the degree to which independent coders made the same judgment when viewing the same segment of video. The convention in behavioral research is to accept as reliable only those codes on which independent coders make the same decision at least 80 percent of the time. All the codes we discuss in this book met this criterion.\n\nThe process of turning videos into information yielded two kinds of products: impressions or images of teaching in each country, and quantified results that indicate how often specific features of teaching occur in each country. The images are vivid and powerful. One picture, it is said, is worth a thousand words; one video may be worth millions. On the other hand, the images produced by video can be too powerful, because they can focus attention on one striking example, even when the example is not typical. Coded data help correct these errors and may themselves lead to interesting discoveries. So both kinds of information are crucial for learning about teaching across cultures.\n\nWe now turn to the results of the study, and we begin with the first kind of information\u2014images of teaching in Germany, Japan, and the United States.\n\n## CHAPTER 3\n\n## Images of Teaching\n\nIN THE FALL of 1994, after several months of watching tapes, the project staff met to present some preliminary impressions and interpretations. We invited distinguished researchers and educators from Germany, Japan, and the United States to attend, and we listened intently to what they had to say. We were ready for a fresh perspective. It came late on the last day of the meeting. One of the participants, a professor of mathematics education, had been relatively silent throughout the day. We asked him if he had any observations he would like to share.\n\n\"Actually,\" he began, \"I believe I can summarize the main differences among the teaching styles of the three countries.\" Everyone perked up at this, and here is what he had to say: \"In Japanese lessons, there is the mathematics on one hand, and the students on the other. The students engage with the mathematics, and the teacher mediates the relationship between the two. In Germany, there is the mathematics as well, but the teacher owns the mathematics and parcels it out to students as he sees fit, giving facts and explanations at just the right time. In U.S. lessons, there are the students and there is the teacher. I have trouble finding the mathematics; I just see interactions between students and teachers.\"\n\nMany of those present were somewhat dumbfounded by this description. How grossly oversimplified it seemed! It also was harshly critical of the American style of teaching and was disturbing to hear. But the image stayed with us, and as we watched the tapes over and over we began to understand that our colleague had captured an important aspect of what we saw in the tapes from all three countries. Although perhaps oversimplified, our colleague's description helped us to see features that might otherwise have been hidden within the noise and complexity of classroom life. Simplified descriptions provide an important starting point for understanding complex activities, provided we are open to revising, tempering, or even discarding them when they outlive their usefulness.\n\nWe begin our journey into the TIMSS videos with our own simplified descriptions of teaching in each country. We then present a typical lesson from each country. Because we realize that the typical lesson could never describe every teacher, we conclude the chapter by reporting some ways in which lessons might vary from the typical. Our goal is to help create accurate and rich visual images of teaching in each country.\n\n## Preliminary Descriptions  \nof Teaching\n\nOur impression is that teachers in Germany are in charge of the mathematics and that the mathematics is quite advanced, at least procedurally. In many lessons, teachers lead students through a development of procedures for solving general classes of problems. There is concern for technique, where technique includes both the rationale that underlies the procedure and the precision with which the procedure is executed. A good motto for German teaching would be \"developing advanced procedures.\"\n\nIn Japan, teachers appear to take a less active role, allowing their students to invent their own procedures for solving problems. And these problems are quite demanding, both procedurally and conceptually. Teachers, however, carefully design and orchestrate lessons so that students are likely to use procedures that have been developed recently in class. An appropriate motto for Japanese teaching would be \"structured problem solving.\"\n\nIn the United States, content is not totally absent, as was portrayed by our colleague, but the level is less advanced and requires much less mathematical reasoning than in the other two countries. Teachers present definitions of terms and demonstrate procedures for solving specific problems. Students are then asked to memorize the definitions and practice the procedures. In the United States, the motto is \"learning terms and practicing procedures.\"\n\nWhat do the mottoes \"developing advanced procedures,\" \"structured problem solving,\" and \"learning terms and practicing procedures\" look like in actual classrooms? In the following sections we describe three actual lessons selected to typify the lessons sampled from each country.\n\n## Portraits of Eighth-Grade  \nMathematics Lessons\n\nThe Classrooms\n\nEven though the videotaped classrooms are located thousands of miles apart and in different cultures, they look much the same. Rows of students' desks, posters on the walls, the teacher's desk and a chalkboard in front\u2014all provide few clues about the country from which the video comes. Students filing into class, individually and in pairs, jostling and joking and laughing, create a remarkably similar atmosphere in each country.\n\nBut there are differences. Although German and American students often dress alike\u2014casually, in denim pants and T-shirts or sweatshirts\u2014Japanese students usually dress in school uniforms: special jackets for boys, blouses and skirts for girls. There are fewer students in the German and U.S. classrooms than in the Japanese classrooms. The national average for eighth-grade class size in each country is twenty-five in Germany and the United States, and thirty-seven in Japan.\n\nThe typical lessons we describe in this chapter are from classrooms found in somewhat different school situations. In Germany, eighth-graders attend different schools based on their academic achievements and aspirations. The classroom we describe is located in a Realschule, the middle track of the three-tiered German school system. Most students attending a Realschule will not go on to university, but many expect to enroll in a technical or vocational college. The classroom in Japan is located in a small-city public school with no special distinctions. Japan has no system for tracking students in elementary and middle school. All eighth-graders take the same mathematics course, so the classroom we describe contains students of mixed achievement levels. The classroom in the United States is located in a large public school in the suburbs of a sprawling metropolitan area. The school offers only one mathematics course in eighth grade, so, as in Japan, the classroom contains students of mixed achievement levels.\n\nWhat about the lessons themselves? Again, there are some interesting similarities and some striking differences. The lessons we describe are about the same length, forty-five to fifty minutes. Some of the time in each lesson is devoted to teacher presentation, some to class discussion, and some to student work. But if the mottoes suggested earlier mean anything, there must be some significant differences in what happens during these activities and how they are arranged. To really understand the differences in classroom teaching, one needs to look carefully at the details of typical lessons, because this is where the teaching gap is revealed. The teaching gap is not an abstract idea concocted by ivory tower researchers; the teaching gap is a set of real differences in the teaching methods used every day in typical classrooms. These differences that accumulate over time and across the country are bound to affect what and how students learn.\n\nStudying the details of lesson design is important but not always easy. The following table might be helpful because it gives an overview of the three lessons we describe and it includes our observations about what features of the lessons typify teaching in each country. To appreciate the sometimes subtle but profound differences in teaching, however, one must study the lessons themselves.\n\nA German Lesson: Developing  \nAdvanced Procedures\n\nWhen the bell rings, Mr. Eisner, the teacher, greets the students: \"Good morning.\" The students respond, \"Good morning,\" and Mr. Eisner says, \"Okay, let's start right away, as usual, with our homework.\" As students pull out their worksheets, Mr. Eisner checks attendance by glancing around the room and recording the students who are absent.\n\nChecking Homework. Mr. Eisner then calls on students, one at a time, to give the answer to the next problem on the worksheet. After each response, he looks up to see if anyone disagrees. If so, he asks for other responses and endorses one of these as correct, or explains why the error might have been made and gives the correct answer. The first eleven problems are quite straightforward. They require finding the measure of the third angle in a triangle, given the first two, as in the drawing below. Students must simply add the measures of the two angles and subtract from 180 to find the measure of the missing angle.\n\nBut the next problems are more challenging. One presents the drawing below and asks students to find the angles labeled with capital letters. Students apparently had more trouble with these at home, and there is disagreement about the answers.\n\nMr. Eisner asks a volunteer to come to the board to explain the solution. As the student works, Mr. Eisner corrects errors, elaborates on the descriptions provided by the students, and makes sure students are using correct mathematical language. The discussion regarding the problem shown above begins as follows.\n\nThe rest of this problem, and the remaining homework problems (twenty-two in all), are checked in a similar way. It is now fourteen minutes into the lesson.\n\nPresenting the Topic for the Day. Mr. Eisner presents the new problem that will define today's lesson.\n\nAfter the construction, which takes about five minutes, Mr. Eisner asks the students to \"mark on the edge of the circle five arbitrary points and call them C1, C2,... C5\" and then \"connect with point A and point B so five triangles emerge.\" A sample drawing now looks like this:\n\nMr. Eisner asks the students to measure, with their protractors, all five angles at the five C's. After a few minutes, the students begin reporting that all five angles are the same size; they all measure 90 degrees. Mr. Eisner pretends to be startled by this result and tries to get students to share his surprise. He then notes that an ancient Greek mathematician (Thales) found that all angles drawn inside a semicircle like this will measure 90 degrees. Now the stage is set, and Mr. Eisner presents the real challenge: \"We did check it, but we also want to prove it, of course.... Prove that it really has to be that way and can't be any other way.\"\n\nHere we see the advanced nature of the mathematics in German classrooms: proving the Law of Thales is a challenging task for eighth-graders.\n\nWorking Through the Proof. As is common in Germany following the presentation of a challenging problem. Mr. Eisner neither leaves the students alone to complete the task by themselves nor demonstrates a quick method that students are supposed to imitate. Using the drawing below, he leads the class through a careful development of the proof.\n\nTwo triangles, AMC and BMC, have radii for two of their sides, so they are isosceles triangles. This means that each triangle has a pair of equal angles. By locating these angles in the drawing, it is possible to see that angle C comprises one angle equal to angle A and one angle equal to angle B. The sum of the three angles (A, B, and C) is 180 degrees, because they make up the large triangle ABC. Because angle C must be the same as angle A plus angle B, the measure of angle C must be exactly half of the 180 degrees, or 90 degrees. Mr. Eisner leads students through this proof step by step, asking students to respond to short-answer questions along the way.\n\nReviewing the Topic. It is now thirty-five minutes into the lesson, and Mr. Eisner hands out two pages summarizing the Law of Thales and its history. He selects students to read aloud small sections of the handout and asks if there are questions.\n\nAssigning Homework. The lesson concludes with the assignment of homework. The problems require finding the measure of missing angles. Some involve the Law of Thales, some a review of previous work. Mr. Eisner asks if there are any questions about the problems, then says, \"Okay. You can start with that until the bell.\" One minute later, forty-five minutes after the lesson began (about average for a German lesson), the bell rings.\n\nA Japanese Lesson: Structured Problem Solving\n\nFrom an American point of view, the Japanese lesson begins in a rather striking way. At the signal from the student monitor, all the students stand and bow, in unison, to the teacher. The teacher bows in return, and the lesson is officially under way.\n\nReviewing Yesterday's Lesson. After the customary exchange of bows, the students sit down and engage in a bit of joking with Mr. Yoshida, the teacher, about the video camera. Mr. Yoshida begins the lesson by reviewing the conclusion of the previous day's lesson. He notes that they had been working with \"the relationship between parallel lines\" and had ended by doing some problems. \"Do you remember what they were?\" he asks. There is no response, so he asks the students to get out the worksheet and look again at the first problem, which asked them to find the measure of the angle marked with an \"x\" in the drawing below. \"We hurried through this problem,\" he says, \"and were not able to summarize it well.\" Several solution methods had been presented, but briefly, and Mr. Yoshida asks the students to look again at the problem and finish it \"with the method you think is easiest.... If you can give an explanation, that would be terrific.\"\n\nAfter two minutes, he asks the students to present what they have found. Three different methods are presented by students, all based on drawing an additional line segment. In some cases, a triangle is formed and students then use what they already know about measures of triangles to find the measure of angle x. After each presentation, Mr. Yoshida asks how many students used that procedure. He concludes this ten-minute segment by summarizing each of the three methods, pointing out the usefulness of drawing additional lines when finding the measures of some angles.\n\nPresenting the Problem for the Day. The lesson continues with the problem for the day.\n\nWorking on the Problem Individually. As in the German lesson, the teacher presents a problem to the students that is mathematically challenging for eighth-graders. What is different is that Mr. Yoshida now asks the students to work out the solutions on their own rather than leading the class in developing the solution. Of course, students already have learned some methods that will help them get started.\n\nFor the next ten minutes, the students work individually on constructing a problem for their peers to solve and on making sure they can solve it themselves. Mr. Yoshida circulates around the room, answering questions and giving hints. He appears increasingly concerned that students are not making more progress and finally says, \"Well, it seems it was a little hard. I made a mistake. There are many of you that are in trouble.... Get in your groups, and from the problems you have made, pick a problem you and the others think is challenging, and group leaders please bring them up. Please check if the problem really can be solved and then bring it up.\"\n\nWorking on the Problem in Groups. As the students rearrange their desks, they move around the room and joke with one another about how hard the problems are that they have constructed. The noise gradually subsides, and after about two minutes, the leaders of the groups begin bringing up their problems for Mr. Yoshida to diagram on the chalkboard. After Mr. Yoshida records one problem for each of the six groups (see the diagrams below), he says, \"These are the problems. We don't know whether we can solve them or not until we try.... It seems impossible to do all of them in this lesson, so we'll think about them a little next time, too. Please hurry and copy the six problems.\"\n\nAs students copy the problems, Mr. Yoshida walks around the room, observing students' work and commenting periodically about how difficult the students have made some of the problems. It soon becomes apparent that the students are trying to solve the problems as they copy them. Mr. Yoshida might have intended this, and he certainly does not discourage it. The students are still sitting in their groups; some are working together as a group, some are working in pairs, and some are working individually. After about ten minutes, Mr. Yoshida asks how many students have solved each of the problems. He then continues to circulate around the room, mostly observing as students continue to work.\n\nStudents have been working at their seats, either individually or in small groups, for about twenty minutes now, an unusually long segment of seatwork for a Japanese lesson, and they will continue in this way for another nine minutes. As the students continue to work, they occasionally raise their hands and ask Mr. Yoshida to look at what they have done. It is apparent that some students become excited at finding solutions to the problems that Mr. Yoshida identified as being especially difficult. Mr. Yoshida studies their solutions but refuses to comment on whether they are correct.\n\nSummarizing the Main Point. The period is almost over, and Mr. Yoshida interrupts to say, \"I know this is bothersome, but I want to know the present situation.\" He then asks how many students have solved each problem. He concludes the lesson by observing, \"There are a lot of people who are using triangles. That's okay, but there are three types of auxiliary lines. Sometimes there are easier methods of solving these problems using other types of auxiliary lines. We will check these in the next period.\" His brief summary is more compressed than is typical for a Japanese lesson. The bell has already rung, forty-nine minutes after the lesson began (near average for a Japanese lesson), and the students now push their chairs back, stand, and bow.\n\nA U.S. Lesson: Learning Terms  \nand Practicing Procedures\n\nWarming Up. The video begins with Mr. Jones, the teacher, conducting a \"warm-up\" activity. He points to the top left-hand drawing on the chalkboard (shown below).\n\nChecking Homework. After five minutes of this quick-paced review, Mr. Jones asks the students to \"get out the worksheet I gave earlier in the week and make sure we understand complementary, supplementary, and angle measurements.\" The class goes over the worksheet in a similar way: Mr. Jones asks students for answers and continues questioning them until they give the correct answer. The class checks thirty-six problems on the worksheet during six minutes of question-and-answer interaction.\n\nDemonstrating Procedures. Reviewing previous work by checking homework is reminiscent of the German lesson. But the next activity is quite different from both the German and the Japanese lessons. Rather than presenting a topic or problem for the day, Mr. Jones distributes a worksheet that contains problems that, he notes, are \"just like the warm-up.\" At the top of the worksheet is a sample problem with the solution and a suggested method shown. Mr. Jones takes a minute to go over this with the students and then asks if there are any questions. There are none, and the students begin working independently.\n\nPracticing the Procedures. The worksheet contains forty problems, and the students spend the next eleven minutes working on them. The problems, like the homework and the warm-up, emphasize terms and procedures\u2014in this case, finding the measures of complementary, supplementary, and vertical angles. Mr. Jones circulates around the room, answering questions and giving hints.\n\nThe lesson clearly has taken a different turn from those in Germany and Japan. The mathematics is quite simple compared with that found in the previous two lessons. But more than that, the teaching method is different. In the German lesson, the teacher led the students through the development of some advanced mathematical procedures. In the U.S. lesson, the development is limited to a quick demonstration. As in the Japanese lesson, students in the U.S. classroom spend the heart of the lesson working on assigned problems. But American students are asked to practice the demonstrated procedures on many simple problems rather than to develop procedures for solving a few challenging problems.\n\nAs Mr. Jones walks around the room, he begins receiving questions about problems 37 and 38. Apparently believing he should intervene when students are struggling or become confused, Mr. Jones goes to the chalkboard and works these two problems with the whole class. He begins with problem 38: \"Write an equation that represents the sentence: The product of 12 and a number K is 192.\" Mr. Jones writes \"12K\" on the board and asks students what to write next. One student says \"Equal sign,\" and Mr. Jones completes the equation: \"12K = 192.\" It might strike the reader as curious that this task has nothing to do with the day's lesson (calculating the measures of angles), but some American curriculum materials include review of earlier topics in later problem sets. In fact, it is not uncommon to find this kind of topic switch during U.S. lessons.\n\nThe discussion then turns to problem 37: \"Angle QRS has the same measure as its supplement. Find m < QRS.\" Mr. Jones shows that the answer must be 90 degrees. Even these two problems, which were perceived to be the most difficult on the worksheet, are quite simple compared with those encountered by the German and Japanese students.\n\nDemonstrating More Procedures. Mr. Jones gives the students two more minutes to finish the worksheet and then asks them to get out the worksheet they completed the previous Friday, after a quiz. One of the problems asked students to measure the interior angles of a hexagon (shown below) and compute the total. Mr. Jones asks if everyone got an answer close to 720 degrees. He then proceeds to the second part of the problem.\n\n(It must be noted here that, based on what students have studied to this point, there is no way they can know the answer to Mr. Jones's opening question, or that the number of angles is the crucial fact in finding the sum of the angle measures. But the nature and tone of teachers' questions often give away the answer, and a number of students apparently picked up on these cues and answered the questions appropriately.)\n\nBy giving the students this formula, Mr. Jones has just taken a problem that could have been challenging for the students (at a level similar to that in Germany and Japan) and changed it into a routine problem for which they must simply follow a rule. One of the features that make this lesson typical of teaching in the United States is just this: stating rules, rather than developing procedures, and thereby turning mathematics into a matter of following rules and practicing procedures.\n\nReviewing Procedures and Definitions. After using the formula to calculate the sum of the interior angles in a triangle, Mr. Jones makes several announcements about upcoming activities and future quizzes and tests. He then conducts a quick oral review with the class on the meaning of terms such as complementary, supplementary, obtuse angle, and acute angle. A few minutes remain, and Mr. Jones tells the students to use the time \"to finish up any of this, and ask me questions.\" The lesson ends with a bell, forty-eight minutes after it began (about average for the United States).\n\n## Variations on a Theme\n\nAfter watching these lessons, and many others like them, we developed the images of teaching, complete with mottoes, that we sketched at the beginning of this chapter. But we noted then that these images are, in many ways, too simplistic. Why? Because there is a range of lessons in each country. Many lessons look much like the ones we have described, and it is from these that the simple images were formed. But some lessons look quite different. We can form richer images of teaching by seeing the full spectrum of lessons.\n\nGerman Variations: More Practice  \nand More Student Participation\n\nThe spectrum of lessons in Germany moves out from the center in two different directions. Some lessons focus more on practicing skills already learned than does the lesson taught by Mr. Eisner; other lessons include more student exploration of concepts and procedures.\n\nPracticing skills is illustrated by a lesson on solving linear equations. The lesson begins with the teacher reporting that the students' performance on a recent test was not very good and suggesting that more practice is needed. The teacher asks two students to come to the chalkboard, and he dictates a problem for each of them. The first problem is (2x \u2212 3)\/3 \u2212 (3x + 4)\/4 \u2212 \u22129\/20 \u2212 (4x \u2212 3)\/5. The rest of the students are expected to watch and correct errors as the two students work through the problems at the chalkboard. The teacher carefully monitors the step-by-step procedures of both students, often asking questions and correcting errors. After the two students finish, the teacher asks if there are questions and calls two more students to the front, dictating two new problems. The entire lesson proceeds in the same way. Some aspects of the lesson are quite typical: students working on complex procedures at the chalkboard with the teacher monitoring progress; and the teacher orchestrating a whole-class question-and-answer discussion of the solutions. But the emphasis on skills already learned, without the development of a new concept, is different from most lessons.\n\nThe second kind of variation\u2014more student participation in developing the mathematics\u2014is illustrated by the following lesson. The teacher begins by reviewing the main point from the previous lesson: special polygons, like squares and equilateral triangles, are defined by special relationships and special properties of their sides, angles, and so on. The teacher then distributes cardboard models of a variety of special polygons and asks the students to find all of the special properties they can\u2014sizes and relationships of sides and angles, and axes of symmetry. Students work together in small groups, and after about fifteen minutes group representatives come to the front, one at a time, and fill in the cells of a large chart, answering \"yes\" or \"no\" to statements such as \"sides are equal,\" \"angles are equal,\" and \"diagonals are axes of symmetry\" with respect to their polygon. One polygon, the kite, creates considerable disagreement among the students. The teacher demonstrates again how to check special properties and asks everyone to reexamine the kite.\n\nAs we put together the typical features with the variations, our image of German mathematics teaching as \"developing advanced procedures\" still seems appropriate. In Mr. Eisner's lesson, the procedures were methods for proving the Law of Thales, a powerful and rich theorem in geometry. The two variations can be interpreted as complements to the theme. In one case, the emphasis shifts from the initial development of procedures to proficiency of execution. In the second case, the teacher allows the students to participate more directly in the development of the procedures, at least for a short time. But the teacher is still in control, carefully constraining the task to ensure certain outcomes. Both kinds of variation support the image of the knowledgeable teacher leading students through the development of advanced mathematical procedures.\n\nJapanese Variations: Teacher Telling  \nand Students Memorizing\n\nAt first glance, the variations we find in Japanese lessons appear to conflict sharply with the lesson presented by Mr. Yoshida. It is not even clear that the typical lessons and the variations fit along the same spectrum. We see teachers lecturing about a topic or telling students how to solve a problem or asking students to memorize properties or facts through repeated recitation. It is especially interesting that when these activities occur, they often are put together, in the same lesson, with students' solving problems and sharing solution methods. From an American point of view, this looks rather odd.\n\nIn one lesson, the teacher begins by identifying the new topic that will continue for several weeks\u2014the analysis of polygons. For the next thirty-five minutes, he lectures. He talks about historical discoveries, about the prevalence of these figures in the real world, and about the fact that this topic is more closely related to the previous one\u2014linear functions\u2014than students might think, because both search for relationships. He concludes by posting the goal for mathematics: \"To learn to think logically while searching for new properties and relationships.\" He asks students to repeat this goal several times and memorize it. The teacher then says that the first task will be to study relationships among angles, and he draws a large X on the board. He asks students to draw a similar X on their papers and to use a protractor to investigate the relationships among the angles. \"Write down things you notice,\" he says. After five minutes the students share what they've found, including that the angles opposite each other are equal. The teacher wonders aloud whether this would always be true. Could it be proved? He then, with large hints, helps students discover a proof.\n\nIn another lesson, the teacher begins by reviewing three properties of parallelograms that they have developed thus far, such as \"opposite sides are of equal length.\" He posts the three statements, and the students spend fifteen minutes reciting them. The class recites them together, then a student stands and recites them alone, then they spend one minute reciting them quietly to themselves, and then they repeat the process. The teacher concludes this activity by saying that students need to remember the properties because they will need them. He then presents the picture shown below and says, \"If ABCD is a parallelogram and BE equals DF, prove that AE equals CF.\" The proof, not surprisingly, uses the properties just recited, and the students are asked to develop and share several different proofs.\n\nAt first glance, these two lessons look quite different from the one taught by Mr. Yoshida. But after viewing them several times, we can see how they complement the image of \"structured problem solving.\" When students are asked to solve challenging problems, teachers often build scaffolds to help them. The scaffolds come in many forms. Sometimes they are the outcomes of previous lessons, reviewed by the teacher (as in Mr. Yoshida's lesson). Sometimes they are in the form of information provided through lectures, and sometimes in the form of mental tools provided through memorization. What is constant is that challenging problems are selected and scaffolds are provided so that students can, at the least, begin developing methods for solution.\n\nNot every lesson, however, fits neatly along this spectrum. In one lesson, the teacher says that today they will be talking about the solutions and graphs for simultaneous linear equations. She then leads the students through a twenty-five-minute review of graphing linear equations, showing them how they can translate any form to y = mx + b and can then graph the equation easily. She then shows that when two equations are graphed on the same axes, they often intersect in one point, and this point is special\u2014it has coordinates that satisfy both equations. Throughout the entire discussion, she asks students short-answer questions and accepts their responses, but she essentially tells the students what they need to know and how to solve the problems.\n\nThis lesson shows that \"structured problem solving\" does not capture the full range of Japanese instruction. Indeed, it seems that the teaching method in this lesson is more like the methods typically used in Germany than the methods typically used in Japan. If nothing else, the lesson reminds us that not all teachers within the same country use the same methods.\n\nU.S. Variations: More Review and More Student Participation\n\nAs in Germany, the spectrum of lessons in the United States moves out from the most common in two different directions: even more review than in Mr. Jones's lesson on one side, and more student participation on the other.\n\nIn one lesson, the teacher announces that students will have time to review for the upcoming chapter test. Apparently referring to the suggestions in the teacher's manual, the teacher says, \"What they said is that I shouldn't review with you; you should do it in your groups.\" The students begin reviewing. Although their desks are placed together in groups of four, most students work through the textbook review page on their own, raising their hands when they have questions. The teacher circulates around the room for the full period, answering questions and briefly tutoring individual students who need help.\n\nAnother lesson, although also review, contrasts sharply in teaching method. The teacher hands back the previous day's quiz as each student enters the room. The teacher asks the students to get into their groups and compare responses, check mistakes they made, and decide to present one problem to the class that was hard for them. During most of the lesson, group representatives, in turn, present their selections on the chalkboard and lead a class discussion about methods of solution. One problem is to solve the following systems of equations: y = 2x \u2212 9; x + 2y = 2. Another problem is to factor 8x2 + 8. Apparently, the teacher intends for students to present and discuss their methods, but she often jumps in to correct or explain or cut off discussion in order to move to the next problem. At the end of the lesson, the teacher reviews the slopeintercept form for linear equations and how it can be used to construct graphs, and then assigns twenty-five problems from the textbook for homework.\n\nBoth kinds of variations, in different ways, are consistent with our simplified U.S. image of \"learning terms and practicing procedures.\" The first kind of variation\u2014additional review\u2014simply reinforces the theme. The second kind of variation expands the image. Although the goal of the second lesson is similar to the goal of the others\u2014learn terms and practice procedures\u2014the classroom activities in which the students are engaged look quite different. Working with other students to analyze problems, presenting problems to the class, describing one's own method for solving them, and asking questions of peers\u2014all these give students a more active role than they had in either the review lesson or Mr. Jones's classroom.\n\nLessons in which students participate in this way might show the effects of the current reform efforts. There are, however, very few of these lessons, and when they do occur, the variations from the theme appear in the form of activities, not the substance. Students are seen working in small groups or engaging in a discussion about solution methods, but the mathematics is simple compared with that encountered by their German and Japanese peers, and the work and discussion are mostly about memorizing definitions for terms and following rules and procedures.\n\n## Can We Trust the Images?\n\nPortraits of individual lessons are useful when they create images of teaching that represent the way teaching actually looks. But they can be dangerous if they misrepresent the situation. It is wise to be skeptical when deciding whether the description of a few lessons creates a fair image of teaching in each country. This is especially true when dealing with activities as complex as classroom teaching.\n\nOf course, we presented these typical lessons because we believe that they are useful. They capture quite well the images of teaching that we formed while watching the tapes and discussing them with our colleagues, and we believe they are a fair representation of teaching in each country. But as we stated at the end of Chapter 2, both impressionistic reports and coded data are necessary for learning about teaching. By examining the coded data from the lessons, readers can check the claims we made about what is typical in each country. The coded data also help to refine these images of teaching. In the next chapter, we present this information and then come to some fundamental conclusions about the nature of teaching, regardless of what it looks like or where you find it.\n\n## CHAPTER 4\n\n## Refining the Images\n\nONE OF THE ADVANTAGES of comparing activities across cultures is that we can see things we might never have noticed had we looked only within our own culture. This was illustrated one day early in the study, as we sat with our colleagues watching a U.S. lesson. The teacher in the video was standing at the chalkboard, in the midst of demonstrating a procedure, when a voice came over the public address system: \"May I have your attention, please. All students riding in bus thirty-one, you will meet your bus in the rear of the school today, not in the front of the school. Teachers please take note of this and remind your students.\"\n\nA Japanese member of our team reached over and pushed STOP on the VCR. \"What was that?\" he asked. \"Oh, nothing,\" we replied as we pushed the PLAY button. \"Wait,\" protested our Japanese colleague. \"What do you mean, nothing?\" As we patiently tried to explain that it was just a P.A. announcement, he became more and more incredulous. Were we implying that it was normal to interrupt a lesson? How could that ever happen? Such interruptions would never happen in Japan, he said, because they would ruin the flow of the lesson. As he went on, we began to wonder whether this interruption was more significant than we had thought.\n\nBut wait. Before we rush to interpret these interruptions, how sure are we that the countries really differ significantly on this score? The brief interaction with our Japanese colleague demonstrates the way images like those developed in the previous chapter can begin to form. But it also shows how easy it would be to build images that turn out to be invalid. How often are U.S. lessons really interrupted, and how infrequent are such interruptions in Japan? Fortunately, we do not have to rely on images alone to guide our understanding of how classrooms look in different cultures.\n\nIn this chapter we look across all the lessons and ask whether these images hold up for the full sample. How often do teachers just state concepts or procedures rather than develop them? In what percentage of lessons do students just practice procedures versus doing creative mathematical work? How different is the mathematical content presented by teachers in the three countries? By using the coded data we can answer questions like these and refine our images of teaching in each country.\n\n## Mathematics in the Classroom\n\nOne of the most striking impressions when watching the videotapes is that students in the United States encounter a different kind of mathematics from that encountered by their peers in Germany and Japan. The content appears to be less advanced and is presented in a more piecemeal and prescriptive way. We wanted to test these impressions, because the level and nature of the content to which students are exposed set boundaries on students' learning opportunities. If the content is rich and challenging, it is more likely that students have the opportunity to learn more mathematics and to learn it more deeply. If the content is fragmented and ordinary, students have less chance of learning important mathematics. In this section, we look at three indicators of content: its level of difficulty, how extensively it is developed, and how coherently it is presented.\n\nLevel of Content\n\nIt is very difficult to say how advanced or difficult particular mathematics content is from the students' point of view, because that depends on how well students have been prepared to deal with the topic, how it is presented, what is expected of them, and so on. But we can tell how advanced a topic is, on the international yardstick, by finding where it is placed in mathematics curricula around the world. As part of the Third International Mathematics and Science Study (TIMSS), William Schmidt and his colleagues at Michigan State University conducted an analysis of the grade level at which the majority of the forty-one TIMSS countries gave the most concentrated attention to each mathematical topic.\n\nThe topics treated in the eighth-grade videotaped lessons were matched against this scale. The United States lagged significantly behind Germany and Japan. By international standards, the mathematical content of the U.S. lessons was, on average, at a mid-seventh-grade level, whereas German and Japanese lessons were at the high eighth- and beginning ninth-grade levels, respectively. This means that most eighth-graders in the United States study topics that students in many other countries encounter a year earlier.\n\nDespite the obvious importance of this finding, it cannot by itself explain the relatively poor achievement of American students. German eighth-graders were more than one year ahead of U.S. students in the level of content they were studying, yet their performance on the achievement test was not significantly higher than that of the American eighth-graders. Challenging content alone does not lead to high achievement. The same content can be taught deeply or superficially. Students learning to solve algebraic equations might be asked to grapple with such deep mathematical concepts as variables, functions, and equivalence. On the other hand, they might simply be taught to mechanically follow the steps for solving equations. Students' learning of algebra will differ depending on how the content is taught.\n\nNature of Content\n\nOne of the reasons we dubbed American teaching \"learning terms and practicing procedures\" is that lessons in the United States seemed to place greater emphasis on definitions of terms and less emphasis on underlying rationale. When we counted the number of definitions presented in all lessons, we found that there were about twice as many in the United States as in Germany or Japan.\n\nOf course, there is nothing wrong with presenting definitions in mathematics; in fact, definitions are necessary. Knowing what the terms mean is crucial for communicating about mathematics. What matters most, however, is what one does with definitions. If students simply learn definitions to increase their mathematical vocabulary, they are just scratching the mathematical surface. If students use definitions to explore the deeper properties and relationships in mathematics, then they really are doing mathematics.\n\nStudents in Mr. Jones's class, the lesson we portrayed as typical of the United States, were learning definitions of terms such as \"supplementary angles\" and \"complementary angles\" (two angles are supplementary if they form a straight line; that is, if their measures add up to 180 degrees). Problems in Mr. Jones's lesson involved finding, for example, the supplement of a 70-degree angle. Definitions were the beginning and the end of the mathematics. In contrast, the problem in one of the Japanese lessons asked students to look for relationships among the angles formed when drawing an X. By using the definition of supplementary angles, a proof was developed to show that vertical angles are always equal. This treatment of mathematics goes much beyond learning a definition.\n\nOne way to tell how deeply the mathematics is being developed is to look at the kind of reasoning that was required. In Mr. Eisner's lesson, the task required deductive reasoning, a hallmark of mathematical thinking. One begins with a statement that is accepted as true and builds a logical chain of observations to reach a conclusion that is, necessarily, true. In mathematics, deductive reasoning is often found in proofs. As it turns out, there were no mathematical proofs in U.S. lessons. In contrast, there were proofs in 53 percent of Japanese lessons and 10 percent of German lessons. Whatever students in the United States were doing with the definitions, they clearly were not using them to develop proofs of mathematical relationships.\n\nContent Elaboration\n\nMost mathematics lessons include the presentation of concepts, either by the teacher or by the students. We used the label \"concepts\" broadly to apply to all instances in which information was presented by explaining an idea, by demonstrating an idea with an example, or simply by stating the information. We were interested in whether concepts were just stated or whether they were developed. As was described in Chapter 3, Mr. Jones stated the formula for the sum of the angles in a polygon, and Mr. Eisner developed the Law of Thales. How common is this difference among countries?\n\nTo answer this question, we first divided the lessons into topic segments. Topics were defined by the TIMSS curriculum analysis referred to earlier: items such as linear measurement, ratio and proportion, division of decimals, and so on. Then we analyzed whether the concepts within each topic were developed or just stated. We defined \"developed\" quite generously to include cases in which the concept was explained or illustrated, even with a few sentences or a brief example. We found that one-fifth of the topics in U.S. lessons contained developed concepts, while four-fifths contained only stated concepts. As is shown in Figure 4.1, this distribution was nearly reversed in Germany and Japan. These data add more weight to the impression that students in Germany and Japan have richer opportunities to learn the meanings behind the formulas and procedures they are acquiring.\n\nContent Coherence\n\nOne observation we made in Chapter 3, almost as an aside, was that the mathematics seemed to be more fragmented in the U.S. lesson, as evidenced by a curious shift of topics. Students in Mr. Jones's class spent most of their time considering the measures of angles, but one problem asked them to \"Write an equation that represents the sentence: The product of 12 and a number K is 192.\" The problem had nothing to do with the main lesson topic. Was this an aberration? Does it matter?\n\nBy itself, the event is not very significant. But it raises the question of lesson coherence\u2014the connectedness or relatedness of the mathematics across the lesson. And coherence is significant. Imagine the lesson as a story. Well-formed stories consist of a sequence of events that fit together to reach the final conclusion. Ill-formed stories are scattered sets of events that don't seem to connect. As readers know, well-formed stories are easier to comprehend than ill-formed stories. And well-formed stories are like coherent lessons. They offer the students greater opportunities to make sense of what is going on.\n\nFigure 4.1. Average percentage of topics in eighth-grade mathematics lessons that contained concepts that were DEVELOPED or STATED.  \nSource: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study. Videotape Classroom Study, 1994-95.\n\nThreats to Coherence. One way to measure coherence is to look for threats to coherence, features of lessons that make it difficult to design and sustain a smoothly developing story. Threats include things like switching topics frequently, or as we noted at the beginning of this chapter, being interrupted by outside intrusions. We found that U.S. lessons contained significantly more topics than did Japanese lessons, and significantly more switches from topic to topic than both German and Japanese lessons. This may mean that the curriculum materials in the United States are trying to cover more ground than the materials in the other countries but, as we found in the previous section, are covering it less deeply.\n\nAs for interruptions, we did, in fact, find that U.S. lessons were interrupted more frequently than lessons in Germany and Japan. The interruptions came from things like announcements over the public address system and visitors who entered the room to request something, like the lunch count. As claimed by our Japanese colleague, such interruptions never occurred during the Japanese lessons. But they did occur in 13 percent of the German lessons and 31 percent of the American lessons.\n\nMaking Connections. Threats to coherence tell only part of the story. Coherence is achieved through weaving together ideas and activities. One way to help students notice how ideas are related is to explicitly point out the connections among them. For example, several minutes into a German lesson, the teacher said, \"Next is a step you really need to pay close attention to because we're dealing here with different numbers from those we dealt with yesterday.\" The teacher went on to elaborate the differences and at the same time point out connections between the previous day's work and the current day's activities. Connections can also be made between ideas within lessons. Halfway through a U.S. lesson on solving sets of linear equations, the teacher asked the students to consider an alternative procedure and related it to a point made early in the lesson: \"What might have been the other choice? What would have been the logical choice that just had the two terms? This is what Hugh was talking about at the beginning, where you have just a square term and a constant.\"\n\nWe found that although the majority of teachers in all countries made explicit links from one lesson to another, only the Japanese teachers routinely linked together the parts of a lesson. In fact, 96 percent of Japanese lessons contained explicit statements by the teacher connecting one part of the lesson with another, whereas only 40 percent of German and U.S. lessons contained such statements.\n\nOther judgments about coherence, such as the flow of mathematical connections, are quite subtle and require a good deal of mathematical sophistication. So we asked a group of four university mathematics teachers (we'll call them the Math Group) to analyze the lessons. We asked them to devise a means of systematically describing the mathematical content of each lesson along dimensions that they thought were relevant for student learning. What they found is both interesting and important.\n\nThe Math Group worked from written descriptions of the lessons that contained information on how the lesson was organized, what activities were used, what tasks were presented, the solution strategies that were presented by the teacher and the students, and so on. One advantage of using these written charts rather than the videotapes was that country-specific references, such as coin systems or students' names, could be disguised so that the Math Group did not know which lesson was from which country. To reduce the immensity of their task, we gave them a subset of lessons to analyze: fifteen geometry and fifteen algebra lessons, randomly chosen from each country, for a total of ninety lessons.\n\nAs they studied the charts, the Math Group began mapping out the mathematical ideas in different parts of each lesson and how they were related. For example, in Mr. Yoshida's lesson (Chapter 3), students first reviewed finding the measures of angles by drawing auxiliary lines. Then they were asked to construct their own problems that could be solved using auxiliary lines. This segment was related to the previous one in a number of ways: (1) it was similar to the first segment with respect to the basic mathematical ideas; (2) it was dependent on the first segment procedurally\u2014students could apply the methods they used in the first segment to begin creating and solving their problems; and (3) it extended the first segment procedurally and conceptually by increasing the complexity of the problems.\n\nIn Mr. Jones's lesson, students first reviewed complementary, supplementary, and vertical angles. Then they spent most of the lesson solving problems that were similar. \"Just like the warm-up,\" said Mr. Jones. \"All... are done the same way.\" Although most segments of the lesson were related, because the problems were similar, in terms of content there were fewer mathematical relationships from one segment to the next. For example, there was no increase in mathematical complexity from the beginning of the lesson to the end.\n\nThe Math Group captured these kinds of differences by mapping out all of the mathematical relationships between segments of lessons. They reported their results in two ways. First, they counted the number of lessons in which all segments were connected through at least one appropriate mathematical relationship. Using our story analogy, these were lessons that told a single story. Of the thirty lessons analyzed from each country, 45 percent of the U.S. lessons, 76 percent of the German lessons, and 92 percent of the Japanese lessons fit this criterion.\n\nThen the Math Group calculated a summary score for each lesson to indicate the degree to which the parts of the lesson were interrelated. By this measure, German lessons scored four times as high as U.S. lessons. Japanese lessons scored six times as high as U.S. lessons.\n\nFigure 4.2. Percentage of lessons rated as having low, medium, and high quality of mathematical content.  \nSource: U.S. Department of Education, National Center for Education Statistics, Third international Mathematics and Science Study, Videotape Classroom Study. 1994-95.\n\nOverall Quality of Content. The Math Group conducted one final analysis. They assessed the overall quality of the mathematics in each lesson with regard to its potential for helping students understand important mathematics. This subjective judgment was, of course, related to coherence, but it also took into account the two aspects of mathematics we considered earlier\u2014the level of challenge and how the content was developed. The Math Group sorted the lessons into three quality categories: low, medium, and high. Remember, they did not know which lessons came from which countries. The results are shown in Figure 4.2. In the judgment of these experienced mathematicians and mathematics teachers, American students were at a clear disadvantage in their opportunities to learn, at least as indicated by the mathematics content of their lessons.\n\nSummary\n\nAn initial indication of the learning opportunities for students in a mathematics classroom is the nature and level of mathematics that is on the playing field. What is in the lesson, substance-wise, for the students to use to construct mathematical knowledge? We have learned that, in this regard, U.S. students are at a disadvantage. They encounter mathematics that is at a lower level, is somewhat more superficial, and is not as fully or coherently developed as the mathematics encountered by their German and Japanese peers. These findings add to the impressions formed in Chapter 3. Indeed, when the coded data are examined, differences in the content of the lessons appear to be even larger than when individual lessons are compared. U.S. students encounter less-challenging mathematics, and because it is presented in a less-coherent way, they must work harder to make sense of it than their peers in Germany and Japan. Still, this is not the whole story.\n\n## Engaging Students  \nin Mathematics\n\nDifferences among the lessons of Messrs. Eisner, Yoshida, and Jones seemed to lie in more than just the content. The way in which students were asked to engage in mathematics seemed to be different. In the German and U.S. lessons, students participated mostly by giving brief responses to the teacher's specific questions. Although the level of mathematics was higher in Germany, in both countries the teacher did most of the mathematical work In Japan, the reverse seemed to be true. The typical Japanese lesson invited students to do more of the mathematical work. How prevalent are these characteristics of lessons in each country?\n\nLesson Organization\n\nThe way a lesson is organized provides a shell, or context, within which the teacher engages students in learning the subject. All the teachers tended to divide their lessons into periods of classwork and seatwork. Classwork is when the teacher is working with all the students and, usually, orchestrating the discussion. Activities include learning a new concept, reviewing a previously learned concept or procedure, solving a problem together, or sharing solution methods for problems that have been solved. Seatwork is the time when students work individually or in small groups on assigned tasks. Talk is mostly private\u2014teacher-student or student-student.\n\nTeachers in all three countries spent more time in classwork than in seatwork. In Japan and the United States 60 percent of the time was spent in classwork; in Germany, it was 70 percent. Although the overall percentage of time spent in classwork was similar, shifts within the lesson from classwork to seatwork and vice versa were considerably more frequent in Japan than in the other two countries. As a consequence, the duration of segments defined by classroom organization tended to be shorter in Japan than in the other countries.\n\nThe full meaning of lesson organization becomes clear when we examine what happens during classwork and seatwork. Who does the work, and what kind of work is being done?\n\nWho Does the Work?\n\nMany educators agree that learning opportunities are enhanced when students do most of the mathematics work during the lesson. But just looking at whether things are being done during classwork or seatwork does not tell us enough. The teacher often is doing the work during classwork but might orchestrate the discussion so that students are required to do more of the work; students often are doing the work during seatwork but might be assigned tasks for which the teacher already has done the mental work. To measure more accurately who is doing the mathematical work, the Math Group looked at who controlled the solution method to a problem, the teacher or the students.\n\nFor example, Mr. Jones showed students the formula for finding the sum of the interior angles of a polygon: sum = 180\u00b0 \u00d7 (number of sides \u2212 2). He then asked students to calculate the sums of various polygons. Students were to execute the formula; Mr. Jones controlled the method. The same problem could have been presented in a much different way. The teacher might have asked students to measure the sums of interior angles of various polygons using a protractor, and then try to find some patterns that would help them compute the sums more quickly. Students could have been given responsibility to work out various solution methods\u2014including, perhaps, a general formula. Then students would have controlled the solution method.\n\nThe Math Group found that the tasks were predominantly student-controlled in 9 percent of the American lessons, 19 percent of the German lessons, and 40 percent of the Japanese lessons. Our impression is confirmed\u2014students in the Japanese lessons did more of the mathematical work than students in the other countries.\n\nWhat Kind of Work Is Expected?\n\nOne way to measure the kind of work students do is to apply the idea of student-controlled tasks and to look at whether students are asked to develop multiple solution methods for problems. Mathematical problems can have one right answer, but usually there are many ways to find the answer. For example, the problem in Mr. Jones's lesson\u2014finding the sum of the measures of the interior angles of a polygon\u2014can be solved by (1) measuring the angles of the particular polygon with a protractor and adding them up, or (2) measuring the angles of several polygons, looking for patterns, and predicting the answer, or (3) using the results of (2) to invent a formula. The fact that there are usually many ways to find an answer is important because it is generally agreed that richer, more conceptual learning opportunities are available if students are encouraged to examine the relative advantages of different methods, and this is a place where students can participate in doing mathematics.\n\nFigure 4.3. (a) Percentage of lessons that included student-presented alternative solution methods; (b) average number of student-presented alternative solution methods presented per lesson.  \nSource: U.S. Department of Education, National Center for Education Statistics, Third International Mathematics and Science Study, Videotape Classroom Study, 1994-95.\n\nAs was hinted at by the lessons described in Chapter 3, Japanese lessons included significantly more student-presented alternative solution methods than did German or American lessons (see Figure 4.3).\n\nNotice that even in Japan, however, fewer than half the lessons contained student-presented alternative solutions. Does this mean that we have overestimated the kind of creative work Japanese students are asked to do, or does it mean simply that some lessons do not provide class time for students to present their work? One way to check this is to look at the mathematics in which students were engaged during seatwork activity and the kind of thinking it required.\n\nThe thinking required during seatwork fell into three categories: practice routine procedures, apply concepts or procedures in new situations, and invent something new or analyze situations in new ways. The first category is well known and was prevalent in Mr. Jones's class. He demonstrated, for example, how to identify and calculate supplementary angles and then assigned a number of practice exercises. The second category consists of situations in which students are encouraged to use a particular idea to solve a problem but it is not immediately obvious how this might be done. The third category contains tasks in which students must invent something or in which they were asked to think about and analyze a mathematical situation in a new way. In Mr. Yoshida's class, students were asked to invent new problems that could be solved using what they knew about angle measures.\n\nAs is shown in Figure 4.4, Japanese students spent about equal amounts of time practicing routine procedures and inventing something new, whereas German and American students spent almost all their time practicing routine procedures.\n\nSummary\n\nWe have learned that students in Germany and the United States learn mathematics by following the teacher's lead. In Germany, this often takes the form of responding to specific questions from the teacher as the whole class develops a mathematical procedure. In the United States, this often takes the form of following the teacher's directions by practicing a procedure during seatwork.\n\nFigure 4.4. Average percentage of seatwork time spent in three kinds of tasks.  \nSource: U.S. Department of Education, National Center for Education Statistics. Third International Mathematics and Science Study, Videotape Classroom Study. 1994-95.\n\nAlthough Mr. Yoshida's lesson might make it tempting to say that the reverse is true in Japan, the data do not support this. A more accurate picture is that, on average, there is a balance in Japan. The mathematical work is shared by the teacher and the students. Students sometimes, but not always, do creative mathematical work by inventing new methods and presenting them to the class. At other times, teachers control the mathematics\u2014lecturing, demonstrating, asking students to memorize, and so on.\n\nUsing the tools of both subjective impression and objective codes, we have now built a picture of what eighth-grade mathematics teaching looks like in Germany, Japan, and the United States. We began with some simple impressions of teaching in each country and then checked these impressions against national samples. We zeroed in on specific features of teaching\u2014in research parlance, \"indicators\"\u2014that might influence students' learning. Now let us examine what these indicators can tell us about teaching.\n\n## CHAPTER 5\n\n## Teaching Is a System\n\nALTHOUGH VIDEOTAPES are a rich source of information, they provide only glimpses of the full activity of teaching. We have found images of teaching in each country, and we have constructed indicators that measure the features of classroom lessons in each country. These images and indicators provide only partial views of teaching, however. It is as if we are seeing the peaks of mountain ranges poking above the surface of the water. The videotapes provide views of these mountaintop islands, but still hidden, underneath the surface, are the mountain ranges.\n\nWe discovered that mountain ranges lay beneath the surface as we asked ourselves why the indicators revealed certain differences among the countries. Consider the following simple indicator. Many mathematics teachers in the United States use an overhead projector, whereas almost all teachers in Japan prefer the chalkboard. Some would say this is a trivial difference and not worth worrying about. But when we look more closely at this superficial difference we see that it points to a deeper, more significant difference in the way teaching is conducted.\n\nWhen we look again at teachers using overhead projectors and chalkboards, we begin to see that teachers in the two countries do not just use different visual devices, they use them in different ways. Most teachers in the United States use visual devices to focus students' attention. They use both overhead projectors and chalkboards to display information in written or graphic form while they are describing it orally. As they finish each part of their oral presentation, they often erase that part of the written material and move to the next item. Whether they use overhead projectors or chalkboards, they use these visual aids to keep students' attention directed toward the information of the moment. This observation is not a new revelation. Many preservice teacher-training programs offer advice on using overhead projectors in just this way. Readers who have participated in such teacher training might remember being told to cover up all the items on the transparency except the one being presented, then to move the cover down to the next item, and so on. When finished presenting the last item, the teacher is told to turn off the projector so as to reclaim students' attention.\n\nJapanese teachers use visual aids for a very different purpose: to provide a record of the problems and solution methods and principles that are discussed during the lesson. The first item of information in the lesson is placed at the far left of the chalkboard; the next item, whether presented by a student or the teacher, is written next to it; and so on. The record builds, left to right, as the lesson proceeds. Many Japanese teachers finish the lesson with a full chalkboard, showing a complete record of the lesson.\n\nThe fact that U.S. teachers frequently use overhead projectors and Japanese teachers use only chalkboards indicates much more than a whimsical preference in visual aids. Given how these aids are employed in each culture, we can now see that Japanese teachers would not use overhead projectors, whereas U.S. teachers would use either one but probably would find overhead projectors more effective. Visual aids function very differently in these two different systems of teaching.\n\nAnd here is the significant truth about teaching that this simple-seeming indicator reveals: teaching is a system. It is not a loose mixture of individual features thrown together by the teacher. It works more like a machine, with the parts operating together and reinforcing one another, driving the vehicle forward. In the U.S. machine, or system, there is a slot for a visual aid that helps focus students' attention. The overhead projector serves this purpose as well as, or better than, the chalkboard, so it is easy to see why many teachers have shifted to the overhead. In the Japanese system, there is no such slot. Instead, there is a slot for presenting a cumulative record of the day's lesson. The overhead projector does not function in this way, so Japanese teachers do not use it; they continue to use the chalkboard.\n\nIf teaching is a system, then each feature, by itself, doesn't say much about the kind of teaching that is going on. What is important is how the features fit together to form a whole. How does one feature connect with the next one? How does an activity near the end of the lesson link back with one at the beginning? This is a very different way to think about teaching. It means that individual features make sense only in terms of how they relate with others that surround them. It means that most individual features, by themselves, are not good or bad. Their value depends on how they connect with others and fit into the lesson.\n\nOne lesson we described briefly in Chapter 3 began with pure memorization. The teacher asked students to recite three properties they had learned already about parallelograms, such as \"opposite sides are parallel and of equal length.\" Individual students stood and recited them and were corrected by the teacher. The class recited them together. Students rehearsed them silently and then again recited them aloud. This continued for fifteen minutes. We wondered why the teacher was drilling the students so intensely. Some might see this as poor teaching\u2014or at least as unnecessary.\n\nThen the teacher told the students that they would be working on a problem for which they might find the properties useful. He presented a theorem about parallelograms and asked the students to prove it. As it turned out, the three properties they just had memorized were the key pieces they needed in order to work out a proof. Most students were reasonably successful. Now the memorization feature of the lesson took on new meaning for us. It fit well with the surrounding features and with the goal of the lesson.\n\n## Patterns of Teaching  \nin Three Countries\n\nReturning to our metaphor, we now became interested in discovering the nature of the mountain ranges that lay beneath the surface. How could we describe the systems of teaching in each country? We stepped back and watched the tapes again, looking for how the different pieces of the lesson fit together. The fact that we were watching lessons from different countries was invaluable.\n\nOur perceptions of classroom lessons changed, depending on the context in which we viewed them. When we watched only U.S. lessons, we tended to notice the differences among them. Some teachers demonstrated a procedure by working through several examples in a lecture form, others by asking students to fill in steps, and still others by passing out reference sheets and talking through the worked examples with the students. Some teachers presented rather long demonstrations and then gave students the rest of the time to work on the assignment, whereas others gave a brief demonstration, asked students to work on a few similar problems, checked their progress, gave another demonstration, assigned a few more problems, and so on. Some teachers asked students to work on the assigned problems in small groups; others asked students to work individually. Differences in how the demonstration was handled and how teachers structured seatwork were aspects of the lessons that stood out.\n\nWhen we watched a lesson from another country, we suddenly saw something different. Now we were struck by the similarity among the U.S. lessons and by how different they were from the other country's lesson. When we watched a Japanese lesson, for example, we noticed that the teacher presents a problem to the students without first demonstrating how to solve the problem. We realized that U.S. teachers almost never do this, and now we saw that a feature we hardly noticed before is perhaps one of the most important features of U.S. lessons\u2014that the teacher almost always demonstrates a procedure for solving problems before assigning them to students. This is the value of cross-cultural comparisons. They allow us to detect the underlying commonalities that define particular systems of teaching, commonalities that otherwise hide in the background.\n\nThrough this process, we began to see something that surprised us: The systems of teaching within each country look similar from lesson to lesson. At least, there are certain recurring features that typify many of the lessons within a country and distinguish the lessons among countries. These recurring features, or patterns, define different parts of a lesson and the way the parts are sequenced. They serve as a kind of shorthand for the common teaching approach in each country and metaphorically, begin to describe the nature of the mountain ranges underneath the visible islands.\n\nThe German Pattern\n\nGerman lessons usually unfold through a sequence of four activities.\n\n  * Reviewing previous material. This can take several forms, including reviewing homework or reminding students what they have accomplished up to this point.\n  * Presenting the topic and the problems for the day. In Mr. Eisner's lesson, the topic was the measure of angles, and the problem was to prove that all angles inscribed in a semicircle measured 90 degrees.\n  * Developing the procedures to solve the problem. Mr. Eisner directed the development (in this case, of a proof) from the chalkboard. In some lessons, students are asked to work at the chalkboard, taking suggestions from other students and the teacher. When students work at the chalkboard, the teacher retains control of the development, even when positioned at the back of the room.\n  * Practicing. Usually, this is handled through the assignment of seatwork, which can become homework if not finished. The problems often are similar to those that have been worked during the classwork portion of the lesson.\n\nMany lessons move through each activity only once, in sequence, but some lessons cycle through the second and third activities twice or even three times.\n\nThe Japanese Pattern\n\nJapanese lessons often follow a sequence of five activities.\n\n  * Reviewing the previous lesson. The review is conducted by a brief teacher lecture, or by the teacher's leading a discussion, or by the students' reciting the main points. Frequently, the day's lesson builds directly on the previous day's lesson, perhaps by using the methods that were developed on the previous day to solve the current day's problem. In the lesson from Chapter 3, Mr. Yoshida asked students to present again, in more detail, the previous day's methods with the expectation that the students would put these methods to use in the current day's lesson.\n  * Presenting the problem for the day. Usually, there is one key problem that sets the stage for most of the work during the lesson.\n  * Students working individually or in groups. This almost always follows the presentation of the problem and lasts anywhere from one to twenty minutes, often five to ten minutes. Students rarely work in small groups to solve problems until they have worked first by themselves.\n  * Discussing solution methods. After the students have worked on the problem, one or more solution methods are presented and discussed. Often, the teacher asks one or more students to share what they have found. Teachers often select students to share (rather than taking volunteers) based on the methods they have seen students develop as they circulated around the room. Sometimes, teachers themselves present methods they have seen students using or new methods they want students to learn. When students present methods, the teacher often summarizes and elaborates.\n  * Highlighting and summarizing the major points. Usually at the end of the lesson, and sometimes during the lesson, the teacher presents a brief lecture on the main point(s) of the lesson. Mr. Yoshida summarized the main point after the opening ten-minute review and again very briefly at the end of the lesson.\n\nActivities two through five can be cycled through several times in one lesson, but usually not more than twice. When a second problem is presented, it often is much like the first, and students are expected to practice the method(s) presented for solving the first problem.\n\nThe U.S. Pattern\n\nThe pattern of eighth-grade mathematics instruction in the United States shares some elements with the German pattern, but it devotes more time to practicing definitions and procedures and less time to developing the technical details and rationale of procedures. Four activities characterize U.S. lessons.\n\n  * Reviewing previous material. The lesson begins by checking homework or engaging in a warm-up activity. Mr. Jones conducted a warm-up activity and then checked homework, an opening that is quite common.\n  * Demonstrating how to solve problems for the day. After homework is checked, the teacher introduces new material, or reviews previous material, by presenting a few sample problems and demonstrating how to solve them. Often the teacher engages the students in a step-by-step demonstration by asking short-answer questions along the way.\n  * Practicing. Seatwork is assigned, and students are asked to complete problems similar to those for which the solution method was demonstrated. Seatwork usually is done individually, although sometimes students work in small groups to compare answers and help one another.\n  * Correcting seatwork and assigning homework. Near the end of the lesson, some of the seatwork problems are checked and, occasionally, some additional problems are worked out together. Homework, with more practice problems, is then assigned. Usually, some time is allowed during the lesson for students to begin the homework.\n\nVersions of activities two through four can be cycled through several times. In Mr. Jones's class, there were several demonstrations of definitions and procedures sandwiched around seatwork and checking homework and seatwork.\n\n## Comparing the  \nLesson Patterns\n\nThe three patterns share some basic features: the class reviewing previous material, the teacher presenting problems for the day, and students solving problems at their desks. Apparently, there is some international agreement about the importance of these activities. On closer inspection, however, it becomes clear that these activities play different roles. For example, presenting a problem in Germany sets the stage for a rather long development of a solution procedure, a whole-class activity, guided by the teacher. In Japan, presenting a problem sets the stage for students to work, individually or in groups, on developing solution procedures. In the United States, presenting a problem is the context for demonstrating a procedure and sets the stage for students practicing the procedure. The fact that similar lesson activities can function differently is not surprising, because the activities are embedded in different systems.\n\nThere also are differences in the core activities of lessons. For example, at the heart of the German pattern is an activity in which the teacher leads the students through the development of a mathematical procedure. This is not found in either the American or the Japanese pattern. As another example, the Japanese pattern includes an activity in which the teacher summarizes the major point of the lesson. This can be done quite quickly and with little fanfare, but it seems to be a stable part of the lesson design. Taken together, the differences in kinds of activities and in the roles that similar activities can play within the different systems generate large differences across systems of teaching in different cultures.\n\n## The Origins of  \nLesson Patterns\n\nMany people are not surprised to learn that Japanese lessons can be described by a simple, common pattern. After all, Japan is a country with a relatively homogeneous population and a highly centralized education system. Japanese teachers might be expected to teach in similar ways. But the United States seems to be a different case. How is it possible, in a country as diverse and decentralized as our own, to find a national pattern that can characterize teaching? Could a single method of teaching really have developed in this country? This possibility is not as far-fetched as it sounds, especially to those who study education. In fact, the American pattern we have described is consistent with a general method of teaching that has been prevalent in the United States for some time, not only in mathematics and not only in the eighth grade.\n\nThe most compelling question is: \"Where do these patterns come from, and what accounts for their apparent stability over time?\" Of course, patterns that we observe in the classroom come from teachers' heads; it is teachers, after all, who plan and implement the lessons we observe. Indeed, the national patterns of teaching that we have observed must arise out of a knowledge base that is widely shared by teachers within each culture. But where does this shared knowledge come from? One possibility is that it is imparted to teachers in teacher-training programs. Another possibility is that the knowledge is cultural, passed on from generation to generation through human interactions. We contend, as do other educational researchers, that although teachers learn some things about teaching from their formal training, mostly they learn from simple cultural participation. After all, teachers spend at least thirteen years in classrooms, as students, before they even enter a teacher-preparation program. We will explore this idea further in the next chapter.\n\n## CHAPTER 6\n\n## Teaching Is  \na Cultural Activity\n\nFOR MANY PEOPLE, family dinners are everyday events. They participate in these events without realizing all the aspects that are taken for granted. Everyone comes to the table and begins eating at about the same time. Menus are not distributed. Instead, the food is brought to the table in serving dishes and everyone eats the same things. The food is then parceled out by passing the serving dishes around the table, with everyone dishing up his or her own portion. Adults often help children with this task. Conversation usually is open, with no set agenda. Comments from everyone are welcome, with children and adults participating as conversational partners.\n\nFamily dinner is a cultural activity. Cultural activities are represented in cultural scripts, generalized knowledge about an event that resides in the heads of participants. These scripts guide behavior and also tell participants what to expect. Within a culture, these scripts are widely shared, and therefore they are hard to see. Family dinner is such a familiar activity that it sounds strange to point out all its customary features. We rarely think about how it might be different from what it is. On the other hand, we certainly would notice if a feature were violated; we'd be surprised, for example, to be offered a menu at a family dinner, or to be presented with a check at the end of the meal.\n\nCultural scripts are learned implicitly, through observation and participation, and not by deliberate study. This differentiates cultural activities from other activities. Take, for example, the activity of learning to use a computer. For older Americans, learning to use the computer is usually not a cultural activity. We learned to use the computer by consciously working on our skills\u2014by reading manuals, taking notes, getting help from experts, and practicing. Using computers is an interesting example because it is rapidly becoming a cultural activity. Children, for example, learn naturally, by hanging around their older siblings. But there still are those for whom it has the distinctly noncultural traits of intentionally, deliberately, and self-consciously working through the activity.\n\nTeaching, in our view, is a cultural activity. It is more like participating in family dinners than like learning to use the computer. This might be surprising, because teaching is rarely thought of in this way. As we noted earlier, some people think that teaching is an innate skill, something you are born with. Others think that teachers learn to teach by enrolling in college teacher-training programs. We believe that neither is the best description. Teaching, like other cultural activities, is learned through informal participation over long periods of time. It is something one learns to do more by growing up in a culture than by studying it formally.\n\nAlthough most people have not studied to be teachers, most people have been students. People within a culture share a mental picture of what teaching is like. We call this mental picture a script. The script is, in fact, a mental version of the teaching patterns we identified in Chapter 5. The difference is that the patterns were observable in the videotapes; scripts are mental models of these patterns. We believe that the scripts provide an explanation for why the lessons within a country followed distinctive patterns: the lessons were designed and taught by teachers who share the same scripts.\n\nIt is not hard to see where the scripts come from or why they are widely shared. A cultural script for teaching begins forming early, sometimes even before children get to school. Playing school is a favorite preschool game. As children move through twelve years and more of school, they form scripts for teaching. All of us probably could enter a classroom tomorrow and act like a teacher, because we all share this cultural script. In fact, one of the reasons classrooms run as smoothly as they do is that students and teachers have the same script in their heads: they know what to expect and what roles to play.\n\n## Implications of Teaching  \nas a Cultural Activity\n\nWe have already made the point that teaching is a complex system, and we have pointed out some implications of this fact. To say that teaching is a cultural activity reveals an additional truth about teaching: Cultural activities, such as teaching, are not invented full-blown but rather evolve over long periods of time in ways that are consistent with the stable web of beliefs and assumptions that are part of the culture. The scripts for teaching in each country appear to rest on a relatively small and tacit set of core beliefs about the nature of the subject, about how students learn, and about the role that a teacher should play in the classroom. These beliefs, often implicit, serve to maintain the stability of cultural systems over time. Just as we have pointed out that features of teaching need to be understood in terms of the underlying systems in which they are embedded, so, too, these systems of teaching, because they are cultural, must be understood in relation to the cultural beliefs and assumptions that surround them.\n\nLet's return to the example of the chalkboard versus the overhead projector. Recall that many teachers in the United States have replaced the chalkboard with the overhead projector, whereas Japanese teachers have not. In Chapter 5 we explained this difference in terms of the different instructional systems in which the visual aids are used. In U.S. classrooms visual aids function to guide and control students' attention. Seen in this light, the overhead projector is preferred because it gives teachers even more control over what students are attending to. Within the Japanese system of teaching, visual aids serve a different function. They are not used to control attention but to provide a cumulative record of the lesson's activities and their results. Japanese teachers do not use the overhead projector because it is not possible to fit the cumulative record on an overhead transparency.\n\nTo dig deeper we must ask why Japanese teachers want a cumulative record of the lesson to be available to students and why U.S. teachers want to control students' attention. To answer these questions we need to situate these two systems of teaching in the context of cultural beliefs about how students learn and about the role the teacher can play in this process.\n\n## Cultural Beliefs About  \nTeaching and Learning:  \nJapan and the United States\n\nAs we pursue deeper comparisons of teaching, we focus on Japan and the United States because this comparison is the most dramatic, and therefore illustrates well the role that beliefs can play in generating and maintaining cultural scripts for teaching.\n\nNature of Mathematics\n\nThe typical U.S. lesson is consistent with the belief that school mathematics is a set of procedures. Although teachers might understand that other things must be added to these procedures to get the complete definition of mathematics, many behave as if mathematics is a subject whose use for students, in the end, is as a set of procedures for solving problems.\n\nIn our study, teachers were asked what \"main thing\" they wanted students to learn from the lesson. Sixty-one percent of U.S. teachers described skills they wanted their students to learn. They wanted the students to be able to perform a procedure, solve a particular kind of problem, and so on.\n\nMany U.S. teachers also seem to believe that learning terms and practicing skills is not very exciting. We have watched them trying to jazz up the lesson and increase students' interest in nonmathematical ways: by being entertaining, by interrupting the lesson to talk about other things (last night's local rock concert, for example), or by setting the mathematics problem in a real-life or intriguing context\u2014for example, measuring the circumference of a basketball. Teachers act as if student interest will be generated only by diversions outside of mathematics.\n\nJapanese lessons appear to be generated by different beliefs about the subject. Teachers act as if mathematics is a set of relationships between concepts, facts, and procedures. These relationships are revealed by developing solution methods to problems, studying the methods, working toward increasingly efficient methods, and talking explicitly about the relationships of interest.\n\nOn the same questionnaire, 73 percent of Japanese teachers said that the main thing they wanted their students to learn from the lesson was to think about things in a new way, such as to see new relationships between mathematical ideas.\n\nJapanese teachers also act as if mathematics is inherently interesting and students will be interested in exploring it by developing new methods for solving problems. They seem less concerned about motivating the topics in nonmathematical ways.\n\nNature of Learning\n\nIf one believes that mathematics is mostly a set of procedures and the goal is to help students become proficient executors of the procedures, as many U.S. teachers seem to, then it would be understandable to believe that mathematics is learned best by mastering the material incrementally, piece by piece. This view of skill learning has a long history in the United States. Learning procedures occurs by practicing them many times, with later exercises being slightly more difficult than earlier ones. Practice should be relatively error-free, with high levels of success at each point. Confusion and frustration, in this traditional American view, should be minimized; they are signs that earlier material was not mastered. The more exercises, the more smoothly learning will proceed.\n\nSuppose students are studying how to add and subtract fractions with unlike denominators, such as 2\/3 + 4\/7. The U.S. beliefs about learning described above would dictate that students should first master adding fractions with like denominators, such as 1\/5 + 2\/5, then be shown how to add simple fractions with unlike denominators, such as \u00bd + \u00bc, being warned about the common error of adding the denominators (to minimize this error), and later practice more difficult problems, such as 2\/3 + 4\/7.\n\nJapanese teachers appear to hold a different set of beliefs about learning and probably would plan a different kind of lesson for adding fractions. One can infer that Japanese teachers believe students learn best by first struggling to solve mathematics problems, then participating in discussions about how to solve them, and then hearing about the pros and cons of different methods and the relationships between them. Frustration and confusion are taken to be a natural part of the process, because each person must struggle with a situation or problem first in order to make sense of the information he or she hears later. Constructing connections between methods and problems is thought to require time to explore and invent, to make mistakes, to reflect, and to receive the needed information at an appropriate time.\n\nWhat kind of lesson on adding and subtracting fractions with unlike denominators would these beliefs generate? A teacher's manual in a popular Japanese textbook series gives us a clue. It alerts teachers that the error students are most likely to make is to add the denominators. Students will learn to understand the process more fully, says the manual, if they are allowed to make this mistake and then examine the consequences. Some suggestions are given for how to help students reflect on the inconsistencies they will encounter if they add, for example, \u00bd and \u00bc, and get 2\/6. Teachers are to begin the lesson with a problem like this and then compare the different methods for solution that students develop. Obviously, struggling and making mistakes and then seeing why they are mistakes are believed to be essential parts of the learning process in Japan.\n\nRole of the Teacher\n\nGiven the differences between the United States and Japan in the apparent beliefs about the subject and learning, it is not surprising that marked differences can be inferred regarding beliefs about the role of the teacher. U.S. teachers appear to feel responsible for shaping the task into pieces that are manageable for most students, providing all the information needed to complete the task and assigning plenty of practice. Providing sufficient information means, in many cases, demonstrating how to complete a task just like those assigned for practice. Teachers act as if confusion and frustration are signs that they have not done their job. When they notice confusion, they quickly assist students by providing whatever information it takes to get the students back on track.\n\nWe saw the following sequence of events over and over. Teachers assign students seatwork problems and circulate around the room, tutoring and monitoring students' progress. Several students ask, in quick succession, about the same problem. Teachers interrupt the class and say, for example, \"Number twenty-three may be a little confusing. Remember to put all the x-terms on one side of the equation and all the y-terms on the other, and then solve for y. That should give the answer.\" In Mr. Jones's lesson (presented in Chapter 3), these problems were numbers 37 and 38, and as soon as he sensed that the students had reached them during their seatwork and were struggling, he stepped in to show the solutions. Teachers in the United States try hard to reduce confusion by presenting full information about how to solve problems.\n\nU.S. teachers also take responsibility for keeping students engaged and attending. Given their beliefs about the nature of mathematics and how it is learned, moment-by-moment attention is crucial. If students are watching the teacher demonstrate a procedure, they need to attend to each step. If their attention wanders, they will be lost when they try to execute the procedure on their own. Now we have a deeper explanation for the frequent use of the overhead projector by U.S. teachers. The projector's capability of focusing attention fits well with the teachers' beliefs about teaching mathematics.\n\nIn addition to the use of overhead projectors, U.S. teachers use a variety of other techniques to hold students' attention. They pump up students' interest by increasing the pace of the activities, by praising students for their work and behavior, by the cuteness or real-lifeness of tasks, and by their own power of persuasion through their enthusiasm, humor, and \"coolness.\"\n\nJapanese teachers apparently believe they are responsible for different aspects of classroom activity. They often choose a challenging problem to begin the lesson, and they help students understand and represent the problem so they can begin working on a solution. While students are working, the teachers monitor their solution methods so they can organize the follow-up discussion when students share solutions. They also encourage students to keep struggling in the face of difficulty, sometimes offering hints to support students' progress. Rarely would teachers show students how to solve the problem midway through the lesson.\n\nJapanese teachers lead class discussions, asking questions about the solution methods presented, pointing out important features of students' methods, and presenting methods themselves. Because they seem to believe that learning mathematics means constructing relationships between facts, procedures, and ideas, they try to create a visual record of these different methods as the lesson proceeds. Apparently, it is not as important for students to attend at each moment of the lesson as it is for them to be able to go back and think again about earlier events, and to see connections between the different parts of the lesson. Now we understand why Japanese teachers prefer the chalkboard to the overhead projector. Indeed, now we see, in a deeper way, why they cannot use the projector.\n\nIndividual Differences\n\nAs a consequence of their implicit beliefs about the subject, learning, and the teacher s role, all teachers appear to hold a set of beliefs about individual differences among students. Many U.S. teachers believe that individual differences are an obstacle to effective teaching. Meeting each student's needs means, ideally, diagnosing each student's level of performance and providing different instruction for different levels. This is not easy to do in a large class. As the range of differences increases, the difficulties of teaching increase. In simple terms, this is an obvious reason for tracking students into separate classes by ability or past performance. It is also the reason for reform efforts directed toward reducing class size. This belief says that the tutoring situation is best, academically, because instruction can be tailored specifically for each student or small group of students.\n\nJapanese teachers view individual differences as a natural characteristic of a group. They view differences in the mathematics class as a resource for both students and teachers. Individual differences are beneficial for the class because they produce a range of ideas and solution methods that provide the material for students' discussion and reflection. The variety of alternative methods allows students to compare them and construct connections among them. It is believed that all students benefit from the variety of ideas generated by their peers. In addition, tailoring instruction to specific students is seen as unfairly limiting and as prejudging what students are capable of learning; all students should have the opportunity to learn the same material.\n\nFor the Japanese teacher, the differences within a group are beneficial because they allow a teacher to plan a lesson more completely. Japanese teachers plan lessons by using the information that they and other teachers have previously recorded about students' likely responses to particular problems and questions. If the group is sufficiently large, they can be quite sure that these same responses will be given by these students. They can then plan the nature of the discussion that is likely to occur. The range of responses also provides the vehicle teachers use to meet the needs of different students. It is expected that different students will understand different methods and will think about the material at different levels of sophistication. Not all students will be prepared to learn the same things from each lesson, and the different methods that are shared allow each student to learn some things.\n\nSanctity of the Lesson\n\nAnother set of beliefs pertains to the significance of the classroom lesson. Lessons, of course, are the most common form of teaching. Classroom teaching, as it is known around the world, plays out through daily lessons. Students' lives in most schools are organized around the series of 45- to 60-minute periods that they move through in the course of a day. But different beliefs about teaching lead to treating lessons in quite different ways.\n\nIn Japan, classroom lessons hold a privileged place in the activities of the school. It would be exaggerating only a little to say they are sacred. They are treated much as we treat lectures in university courses or religious services in church. A great deal of attention is given to their development. They are planned as complete experiences\u2014as stories with a beginning, a middle, and an end. Their meaning is found in the connections between the parts. If you stay for only the beginning, or leave before the end, you miss the point. If lessons like this are going to succeed, they must be coherent. The pieces must relate to one another in clear ways. And they must flow along, free from interruptions and unrelated activities. It is clear why Japanese lessons we videotaped were never interrupted from the outside, not by P.A. announcements, not by lunch-count monitors, not by anyone.\n\nIt is quite easy to see how Japanese beliefs about mathematics, learning, and the role of the teacher lead to treating lessons in this way. In this belief system, mathematics is made up of relationships between ideas, facts, and procedures. To understand these relationships, students must analyze mathematical problems and different methods that can be used to solve them. They must struggle with problems first in order to make sense of later discussions about how to solve them and to understand the summary comments made by the teacher. So the lesson must tell a tightly connected, coherent story; the teacher must build a visible record of the pieces as they unfold so connections can be drawn between them; and the lesson cannot be sidetracked or broken by interruptions.\n\nIn the United States, lessons are treated differently. This is not surprising given the different beliefs about mathematics, learning, and the teacher. The activities within a lesson are more modular, with fewer connections between them. Practice time might be devoted to the procedures demonstrated on the current day, on the previous day, or during the previous week. Because learning procedures is believed to depend largely on practicing them, temporary interruptions, like outside intrusions or unrelated activities, do not ruin the lesson. They might be annoying, but they just reduce the number of practice exercises for that day. It might not be surprising, then, that we found that almost one-third of the U.S. lessons were interrupted in some way.\n\n## Changing Cultural Activities\n\nCultural activities are highly stable over time, and they are not easily changed. This is true for two reasons. First, cultural activities are systems, and systems\u2014especially complex ones, such as teaching\u2014can be very difficult to change. The second reason is that cultural activities are embedded in a wider culture, often in ways not readily apparent to members of the culture. If we want to improve teaching, both its systemic and its cultural aspects must be recognized and addressed.\n\nTeaching systems, like other complex systems, are composed of elements that interact and reinforce one another; the whole is greater than the sum of the parts. An immediate implication of this fact is that it will be difficult, if not impossible, to improve teaching by changing individual elements or features. In a system, all the features reinforce each other. If one feature is changed, the system will rush to \"repair the damage,\" perhaps by modifying the new feature so it functions the way the old one did. If all teachers in the United States started using the chalkboard rather than the overhead projector, teaching would not change much. The chalkboard simply would be used to fill the visual-aids slot in their system and therefore would be used just as the overhead projector was\u2014to catch and hold students' attention.\n\nThis point is missed in many popular attempts to reform teaching in the United States. These reforms start with indicators, like the ones we presented in Chapter 4, and try to improve teaching by influencing the level of the indicator. For example, having found that Japanese and German students encounter more advanced mathematics, reformers might propose that we present more challenging content in our schools. Or because Japanese teachers switch back and forth between classwork and seatwork more often than American teachers do, they might propose lessons with shorter classwork and seatwork segments. German and Japanese students do proofs, so perhaps we should include proofs in our lessons. Educational reforms in this country often have been driven by an effort to change our performance on quantifiable indicators like these.\n\nBut because teaching is a complex system, these attempts to change it generally don't work. It has now been documented in several studies that teachers asked to change features of their teaching often modify the features to fit within their pre-existing system instead of changing the system itself. The system assimilates individual changes and swallows them up. Thus, although surface features appear to change, the fundamental nature of the instruction does not. When this happens, anticipated improvements in student learning fail to materialize and everyone wonders why.\n\nA well-known example comes from the New Math reforms of the 1960s. A major thrust of these reforms was changing the textbooks. Because most mathematics teachers rely quite heavily on the textbook, one might think that changing the textbook would change teaching. In 1975, after the changes had had time to take effect, the National Advisory Committee on Mathematical Education commissioned a study of school mathematics instruction. The study concluded that in elementary schools, \"Teachers are essentially teaching the same way they were taught in school. Almost none of the concepts, methods, or big ideas of modern mathematics have appeared.\" Even textbooks can get swamped by the system.\n\nA more recent and personal illustration of the stability of systems of teaching occurred when one of us was working with a group of American teachers studying videotapes of Japanese mathematics instruction. After viewing the Japanese lessons, a fourth-grade teacher decided to shift from his traditional approach to a more problem-solving approach such as we had seen on the videotapes. Instead of asking short-answer questions as he regularly did, he began his next lesson by presenting a problem and asking students to spend ten minutes working on a solution. Although the teacher changed his behavior to correspond with the actions of the teacher in the videotape, the students, not having seen the video or reflected upon their own participation, failed to respond as the students on the tape did. They played their traditional roles. They waited to be shown how to solve the problem. The lesson did not succeed. The students are part of the system.\n\nSystems of teaching are much more than the things the teacher does. They include the physical setting of the classroom; the goals of the teacher; the materials, including textbooks and district or state objectives; the roles played by the students; the way the school day is scheduled; and other factors that influence how teachers teach. Changing any one of these individual features is unlikely to have the intended effect.\n\nTrying to improve teaching by changing individual features usually makes little difference, positive or negative. But it can backfire and leave things worse than before. When one or two features are changed, and the system tries to run as before, it can operate in a disabled state. Geoffrey Saxe and his colleagues at UCLA found that when elementary school teachers were asked to teach fractions by implementing an innovative curriculum, some did so with higher student achievement than a comparison traditional program and some did so with lower student achievement. The difference was that the successful teachers were provided with information and assistance by the project staff that, in our words, helped them improve their system. The less-successful teachers did not receive such assistance and tried to operate their conventional system with the new curriculum. This was not a good fit and did not promote students' learning. The point here is that trying to improve by changing individual features is not just ineffective; it is downright risky.\n\nBombarding teachers with waves of ineffective reforms can have another downside: Teachers can grow weary. They are asked over and over to change the way they do x, y, or z. Even when they try to accommodate the reformers and adopt a new feature or two, nothing much happens. They do not notice much improvement in students' learning. Although it might feel to teachers that they are changing, the basic system is running essentially as it did before. Always changing, and yet staying the same, is a discouraging state of affairs. It can lead to a defeatist kind of cynicism. \"Not another reform,\" says the veteran teacher; \"I'll just wait this one out.\" Quick fixes that focus on changing individual features leave behind a skeptical teaching corps.\n\nThe fact that teaching is cultural only further complicates and impedes efforts to change it. The widely shared cultural beliefs and expectations that underlie teaching are so fully integrated into teachers' worldviews that they fail to see them as mutable. The more widely shared a belief is, the less likely it is to be questioned\u2014or even noticed. This tends to naturalize the most common aspects of teaching to the point that teachers fail to see alternatives to what they are doing in the classroom, thinking, \"This is just the way things are.\" Even if someone wanted to change, things that seem this natural are perceived as unchangeable. It is no wonder that the way we teach has not changed much for many years.\n\nIs it impossible to change? We don't think so. But we must be sure that our efforts to improve are appropriate for changing cultural activities. If teaching were a noncultural activity, we could try to improve it simply by providing better information in teachers' manuals, or asking experts to demonstrate better techniques, or distributing written recommendations on more effective teaching methods. Note that this is exactly what we have been doing. We have been acting as if teaching is a noncultural activity.\n\nIf we took seriously the notion that teaching is a cultural activity, we would begin the improvement process by becoming more aware of the cultural scripts teachers are using. This requires comparing scripts, seeing that other scripts are possible, and noticing things about our own scripts that we had never seen before. Becoming more aware of the scripts we use helps us see that they come from choices we make. The choices might be understandable, but still they are choices, and once we are aware of them, other choices can be made.\n\nImproving cultural scripts for teaching is a dramatically different approach from improving the skills of individual teachers, but it is the approach called for if teaching is a cultural activity. No matter how good teachers are, they will be only as effective as the script they are using. To improve teaching over the long run, we must improve the script.\n\nOf course, knowing what must be done and actually doing it are two very different things, especially when it comes to complex, culturally embedded activities. Once again, we can learn something important by contrasting our own situation with that of others. In the next chapter, we look at how Japan has dealt with the challenge of improving teaching.\n\n## CHAPTER 7\n\n## Beyond Reform:  \nJapan's Approach  \nto the Improvement  \nof Classroom Teaching\n\nWE HAVE LEARNED that teaching is not a simple skill but rather a complex cultural activity that is highly determined by beliefs and habits that work partly outside the realm of consciousness. That teaching is largely a cultural activity helps to explain why, in the face of constant reform, so little has actually changed inside U.S. classrooms. The cultural nature of teaching might also help to explain why teaching per se has rarely been the direct focus of efforts to reform education. Teaching is so constant within our own culture that we fail even to imagine how it might be changed, much less believe that it should be changed.\n\nOn the other hand, our cross-cultural investigations have also revealed a different, yet equally important, fact about teaching: Although highly constant within a culture, variations in teaching methods across cultures are significant. This means that teaching might be an even more important influence on student learning than some studies have suggested. When studies are conducted within cultures, they might underestimate the effects of teaching because they probably are comparing methods that do not differ greatly from one another. The substantive differences in teaching that we see across cultures suggests that very different ways of teaching can be designed and implemented, and that these substantive changes might have large effects on students' learning. This bolsters our belief that efforts to directly improve classroom processes can lead to significant gains in students' learning.\n\nIn this chapter we briefly discuss the way reformers have sought to improve teaching in the United States, and we use the TIMSS videos to assess how successful these efforts have been. Then we briefly examine Japan's very different approach to the improvement of classroom teaching. Japan's record of high student achievement, together with its contrasting methods of teaching, entreats us to examine how Japan goes about improving its practice.\n\n## Reform in the United States:  \nEvidence from the Classroom\n\nAlthough most popular U.S. reform efforts have avoided a direct focus on teaching, there are some notable exceptions. One of these has been in the domain of mathematics, where the National Council of Teachers of Mathematics (NCTM) has made a strong effort to improve classroom mathematics teaching. The NCTM believes that teaching is of major importance in trying to explain students' learning of mathematics and has led a far-reaching campaign to move teachers in the direction of more effective practices.\n\nThe strategy employed by NCTM in this crusade exemplifies a common approach to reform in the United States. Experts are convened to review the research and experience of the profession and to formulate recommendations for change. These recommendations then are put in written documents that are widely disseminated. One of these documents, the NCTM's Professional Standards for Teaching Mathematics, is quite explicit in its vision of how teaching needs to change in order to raise the level of students' learning. The changes envisioned by NCTM are substantial.\n\nInvestigating the success of these NCTM efforts, and of similar efforts, was a major goal of the TIMSS video study. The findings are quite interesting. First of all, the good news: U.S. teachers appear to be highly aware of reforms advocated by NCTM and other organizations. We asked all of the videotaped teachers, on a questionnaire, to rate their awareness of current ideas regarding the best ways to teach mathematics. Almost all (95 percent) of the U.S. teachers sampled said they were \"somewhat aware\" or \"very aware\" of such ideas, with most claiming to have read documents published by NCTM or similar documents (such as California's Mathematics Framework). Not only are teachers aware of the reforms, the majority claimed to be implementing the reforms in their classrooms. When asked whether or not they implement reforms in their classrooms, and whether or not we would find evidence of such in the videotape we collected from their rooms, 70 percent of U.S. teachers we asked responded in the affirmative. The teachers even pointed us to specific places in the videos where we could see examples of their implementation of reform.\n\nBut this is where the good news ends. When we looked at the videos, we found little evidence of reform, at least as intended by those who had proposed the reforms. Looking at the situation as a whole, one might even argue that Japanese lessons better exemplify current U.S. reform ideas than do U.S. lessons. Japanese lessons, for example, emphasized student thinking and problem solving, multiple solution methods, and the kinds of discourse described in U.S. reform documents to a greater extent than U.S. lessons did. And this is not the worst of it. When we examined the places in the video that teachers referred to as examples of reform, we saw a disturbing confirmation of the suspicion we voiced in Chapter 6\u2014that reform teaching, as interpreted by some teachers, might actually be worse than what they were doing previously in their classrooms.\n\nOne teacher, for example, pointed to her use of calculators as an instance of reform in her classroom. True, NCTM recommends that calculators be introduced early in the curriculum, because, among other reasons, they can save computing time so students can focus their attention on problem solving and conceptual understanding. But this was not the way calculators were being used in this particular teacher's classroom. Midway through the solution of a simple problem, the class needed the answer to the problem 1 \u2212 4. \"Take out your calculators,\" the teacher said. \"Now, follow along with me. Push the one. Push the minus sign. Push the four. Now push the equals sign. What do you get?\" The calculator, in this case, was a diversion, and accomplished little on behalf of students' mathematical understanding.\n\nExamples like this one point to a problem in the U.S. approach to reform. Teachers can misinterpret reform and change surface features\u2014for example, they include more group work; use more manipulatives, calculators, and real-world problem scenarios; or include writing in the lesson\u2014but fail to alter their basic approach to teaching mathematics.\n\nAlbert Shanker, the late leader of the American Federation of Teachers, clearly anticipated this possibility. He often told a story of a trip he had made years earlier with his wife, Eadie. They were touring a housing project, a section that housed recently arrived Jews from African and Arabic countries. Shanker told the story this way:\n\nAs we were touring this housing project, we were told that most of these people had lived in tents or in very primitive housing and that most of them had not eaten on tables. There was this concerted effort to convince them to use tables. As we went through the development, our guides said, \"Let's visit one of these families; let's take a look at an apartment.\" And they knocked at a door and said, \"We have Mr. and Mrs. Shanker here from New York; can they come in?\" We walked in, and there was a family from Yemen, and they were eating from the table. But the table was upside down with the top on the floor and the legs standing up.\n\nShanker understood that teaching, like eating, is a cultural activity and that it is governed by powerful forces that function largely outside of conscious awareness, forces that change slowly over time\u2014if they change at all. Like the Yemeni immigrants, teachers can misinterpret the intentions of reformers, engaging in practices that verge on the bizarre. As we noted earlier, reform documents that focus teachers' attention on features of \"good teaching\" in the absence of supporting contexts might actually divert attention away from the more important goals of student learning. They may inadvertently cause teachers to substitute the means for the ends\u2014to define success in terms of specific features or activities instead of long-term improvements in learning. To the extent that this occurs, the best-laid plans of reformers will backfire. Far from being benign or simply ignored, reform recommendations might even worsen the quality of instruction. We are not the only researchers to document this phenomenon. But the TIMSS videotapes reveal that the problem is national in scope.\n\nAt first glance, this misapplication of reform seems quite discouraging: Given the energy that we as a nation have invested in reform, it is shocking to realize how little penetration there has been into the classroom. But on the other hand, given the cultural, systemic nature of teaching described earlier in this book, it would be surprising if reform were successful\u2014at least the type of reform that has been most widely practiced in the United States. Features of teaching recommended in documents such as the NCTM standards are easily misinterpreted, lacking reference to either the system of teaching in which they are embedded or the wider cultural beliefs that support them. This helps to explain how teachers could be aware of the NCTM's reform recommendations, and even think they were implementing them in the videotaped lesson, when, in fact, they often were aligning their teaching with the recommendations in only superficial ways.\n\nDisseminating models of effective teaching through static documents might work if teaching were a noncultural activity. If teachers learned to teach by studying books and memorizing techniques, written recommendations might have their intended effect. But everything we have learned indicates that teaching is a cultural activity, and consequently the writing and dissemination of reform documents is an unrealistic way to improve education.\n\nWhat are the alternatives? We were drawn initially to Japan's system of improvement as one possibility, because of Japanese students' high levels of achievement and because Japanese teaching methods provide such a clear contrast to our own. But we found another reason to examine Japan's system of improvement more closely: It is built on the idea that teaching is a complex, cultural activity. Just as we saw the U.S. method of teaching in a new way by comparing it with methods in other countries, so too can we use other systems of improvement to envision the possibilities in our own country more clearly.\n\n## Lesson Study: Japan's  \nAlternative to Reform\n\nDespite years of reform, research suggests that classroom teaching has changed little in the United States. In Japan, by contrast, teaching practices appear to have changed markedly over the past fifty years. What accounts for this difference? Japan, too, has sought to reform its educational practices. But the assumptions about how reform must work, and the mechanisms established to enact reform, are quite distinct from those in the United States. Whereas U.S. educators have sought major changes over relatively short time periods\u2014indeed, the very word reform connotes sudden and wholesale change\u2014Japanese educators have instituted a system that leads to gradual, incremental improvements in teaching over time. The system includes clear learning goals for students, a shared curriculum, the support of administrators, and the hard work of teachers striving to make gradual improvements in their practice.\n\nJapan has given teachers themselves primary responsibility for the improvement of classroom practice. Kounaikenshuu is the word used to describe the continuous process of school-based professional development that Japanese teachers engage in once they begin their teaching careers. In the United States, teachers are assumed to be competent once they have completed their teacher-training programs. Japan makes no such assumption. Participation in school-based professional development groups is considered part of the teacher's job in Japan. These groups play a dual role: not only do they provide a context in which teachers are mentored and trained, they also provide a laboratory for the development and testing of new teaching techniques.\n\nVirtually every elementary and middle school in Japan is engaged in kounaikenshuu. Run by teachers, kounaikenshuu consists of a diverse set of activities that together constitute a comprehensive process of school improvement. Teachers work together in grade-level groups, in subject-matter groups (for example, math or language arts), and in special committees (the technology committee, for example). The activities of these various groups are coordinated by a school-improvement plan that sets the goals and focus for each year's efforts. A significant percentage of teachers also engage in district-wide groups that meet in the evenings, generally on a monthly basis. Teachers spend a considerable amount of time each month on kounaikenshuu.\n\nLesson Study\n\nOne of the most common components of kounaikenshuu is lesson study (jugyou kenkyuu). In lesson study, groups of teachers meet regularly over long periods of time (ranging from several months to a year) to work on the design, implementation, testing, and improvement of one or several \"research lessons\" (kenkyuu jugyou). By all indications, lesson study is extremely popular and highly valued by Japanese teachers, especially at the elementary school level. It is the linchpin of the improvement process. One elementary school teacher interviewed by Catherine Lewis and Ineko Tsuchida, scholars of Japanese education, remarked, \"You won't find a school without research lessons.\"\n\nThe premise behind lesson study is simple: If you want to improve teaching, the most effective place to do so is in the context of a classroom lesson. If you start with lessons, the problem of how to apply research findings in the classroom disappears. The improvements are devised within the classroom in the first place. The challenge now becomes that of identifying the kinds of changes that will improve student learning in the classroom and, once the changes are identified, of sharing this knowledge with other teachers who face similar problems, or share similar goals, in the classroom.\n\nVery little has been written in English about the process of lesson study. We base our account on research conducted by Makoto Yoshida, by Catherine Lewis and Ineko Tsuchida, and by N. Ken Shimahara, as well as on informal discussions with teachers and teacher educators throughout Japan. Most of our specific examples will be drawn from the work of Yoshida.\n\nIn Tsuta Elementary School, the Hiroshima school studied by Yoshida, the lesson-study process was divided into trimesters. During the first trimester, teachers met first in several schoolwide meetings to determine the theme or focus of the year's kounaikenshuu efforts. Once a general theme was chosen, teachers continued meeting in grade-level groups to begin developing the goal that would guide their lesson study for the year. Because Tsuta Elementary School was a small school, the teachers divided into three groups: one each for lower, middle, and upper grades. Larger schools might divide into separate groups for each grade level, with a goal of keeping the sizes of the groups at approximately five to seven teachers.\n\nIn the second trimester, the teacher groups started developing \"research lessons.\" The group that Yoshida studied focused on first-graders' understanding of subtraction with borrowing. They met weekly, on Thursday afternoons, for approximately three to four hours. The principal and the head teacher of the school generally sat in with the group in an ex officio capacity.\n\nSteps in the Lesson Study Process\n\nAlthough the form of lesson study varies throughout Japan, we can describe the steps that seem to typify the process.\n\nStep 1: Defining the Problem. Lesson study is, fundamentally, a problem-solving process. The first step, therefore, is to define the problem that will motivate and direct the work of the lesson-study group. The problem can start out as a general one (for example, to awaken students' interest in mathematics) or it can be more specific (for example, to improve students' understanding of how to add fractions with unlike denominators). The group will then shape and focus the problem until it can be addressed by a specific classroom lesson.\n\nUsually the problem teachers choose is one they have identified from their own practice, something that has posed particular challenges for their own students. Sometimes, however, the problem is posed from above, perhaps by educational policymakers seeking teachers' input on problems identified as national priorities. The National Ministry of Education will ask a general question\u2014how can we help students become independent learners, for example\u2014and invite a sample of schools throughout the country to study the question in the context of lesson study and report their findings. At other times, the administrative authorities issue recommendations that teachers are expected to implement. This combination of top-down and bottom-up planning is a unique feature of the educational policy environment in Japan, and it provides a direct connection between classroom teachers and national education officials.\n\nStep 2: Planning the Lesson. Once a learning goal has been chosen, teachers begin meeting to plan the lesson. Although one teacher will ultimately teach the lesson as part of the process, the lesson itself is seen by all involved as a group product. Often the teachers will start their planning by looking at books and articles produced by other teachers who have studied a similar problem. According to one Japanese book on how to prepare a research lesson, the useful research lesson should be designed with a hypothesis in mind: some idea to be tested and worked out within the context of classroom practice. The goal is not only to produce an effective lesson but also to understand why and how the lesson works to promote understanding among students. The initial plan that the group produces is often presented at a schoolwide faculty meeting in order to solicit criticism. Based on such feedback, a revision is produced, ready for implementation. This initial planning process can take as long as several months.\n\nStep 3: Teaching the Lesson. A date is set to teach the lesson. One teacher will teach the lesson, but everyone in the group will participate fully in the preparation. The night before, the group might stay late at school, preparing materials and engaging in a dress rehearsal, complete with role playing. On the day of the lesson, the other teachers in the group leave their classrooms to observe the lesson being taught. (Teachers leave their classrooms without adult supervision. Two students, appointed to serve as class monitors, are left in charge of the class.) The teachers stand or sit in the back as the lesson begins, but when students are asked to work at their desks, the teacher-observers walk around, observing and taking careful notes on what students are doing as the lesson progresses. Sometimes the lesson is videotaped as well, for later analysis and discussion.\n\nStep 4: Evaluating the Lesson and Reflecting on Its Effect. The group generally stays after school to meet on the day the lesson has been taught. Usually, the teacher who taught the lesson is allowed to speak first, outlining in his or her own view how the lesson worked and what the major problems were. Then other members of the group speak, usually critically, about the parts of the lesson they saw as problematic. The focus is on the lesson, not on the teacher who taught the lesson; the lesson, after all, is a group product, and all members of the group feel responsible for the outcome of their plan. They are, in effect, critiquing themselves. This is important, because it shifts the focus from a personal evaluation to a self-improvement activity.\n\nStep 5: Revising the Lesson. Based on their observations and reflections, teachers in the lesson-study group revise the lesson. They might change the materials, the activities, the problems posed, the questions asked, or all these things. They often will base their changes on specific misunderstandings evidenced by students as the lesson progressed.\n\nStep 6: Teaching the Revised Lesson. Once the revised lesson is ready, the lesson is taught again to a different class. Sometimes it is taught by the same teacher who taught the lesson the first time, but often it is taught by another member of the group. One difference is that this time all members of the school faculty are invited to attend the research lesson. This is quite dramatic in a large school, where there may be more faculty crowded into the classroom than there are students in the class.\n\nStep 7: Evaluating and Reflecting, Again. This time, it is common for all members of the school faculty to participate in a long meeting. Sometimes an outside expert will be invited to attend as well. As before, the teacher who taught the lesson is allowed to speak first, discussing what the group was trying to accomplish, her or his own assessment of how successful the lesson was, and what parts of the lesson still need rethinking. Observers then critique the lesson and suggest changes. Not only is the lesson discussed with respect to what these students learned and understood, but also with respect to more general issues raised by the hypotheses that guided the design of the research lesson. What about teaching and learning, more generally, was learned from the lesson and its implementation?\n\nStep 8: Sharing the Results. All this work has focused on a single lesson. But because Japan is a country with national education goals and curricular guidelines, what this group of teachers has learned will have immediate relevance for other Japanese teachers trying to teach the same concepts at the same grade level. Indeed, the teachers in one lesson-study group see the sharing of their findings as a significant part of the lesson-study process. This sharing can be done in several ways. One is to write a report, and most lesson-study groups do produce a report that tells the story of their group's work. Often these reports are published in book form, even if only for the school's teacher resource room. They are read by the faculty and the principal, and might be forwarded to educational authorities at the prefectural level if judged to be interesting enough. If a university professor happens to have collaborated with the group, the report might be written for a wider audience and published by a commercial publisher.\n\nAnother method of sharing the results of a research lesson is to invite teachers from other schools to observe the teaching of the final version of the lesson. The school in Hiroshima that Yoshida observed hosted a \"lesson fair\" at the end of the school year and invited teachers from schools throughout the region to observe the research lessons they had produced in various subject areas. This is a festive occasion, and it is considered an important part of teachers' professional development. This also is one of the primary ways that teachers can learn about innovations that are being tried at other schools.\n\nWhat Teachers Talk About:  \nAn Example from Simple Subtraction\n\nWe have outlined the general lesson-study process. But what do teachers actually do during their meetings? In this section we again draw on the work of Makoto Yoshida, who spent a year with one lesson-study group in Hiroshima. It is one thing to say that teachers \"planned a lesson,\" quite another to witness the kind of detailed planning that goes on inside these groups. Yoshida brings this across clearly in his description of how a group of lower primary teachers planned the introductory lesson for a first-grade unit on simple subtraction with the minuend larger than ten.\n\nConsistent with the general cultural script for lessons in Japan, the basic flow of the lesson was determined at one of the early meetings of the group. The lesson would start with a problem. Students would work on the problem, then present their various solution methods to the whole class. The teacher would then lead a discussion of the methods students had invented, and conclude the lesson by summarizing the concept that students were intended to understand. But this general flow was only the beginning of what the teachers needed to decide. According to Yoshida, the teachers engaged in detailed discussions of the following topics over the weeks spent planning the lesson:\n\n  * The problem with which the lesson would begin, including such details as the exact wording and numbers to be used.\n  * The materials students would be given to use in trying to solve the problem.\n  * The anticipated solutions, thoughts, and responses that students might develop as they struggled with the problem.\n  * The kinds of questions that could be asked to promote student thinking during the lesson, and the kinds of guidance that could be given to students who showed one or another type of misconception in their thinking.\n  * How to use the space on the chalkboard (Japanese teachers believe that organizing the chalkboard is a key ingredient to organizing students' thinking and understanding).\n  * How to apportion the fixed time of the lesson\u2014about forty minutes\u2014to different parts of the lesson.\n  * How to handle individual differences in level of mathematical preparation among the students.\n  * How to end the lesson\u2014considered a key moment in which students' understanding can be advanced.\n\nDuring one of the early sessions, the Japanese teachers decided to use this problem at the beginning of the lesson:\n\n______________ (Student's name) collected _______ ginkgo leaves. Then he\/she drew ______ pictures of his\/her family on the leaves, one member on each leaf. How many leaves did not have pictures?\n\nThe teachers agreed that it would be good to use one of the students' names in the problem, though they hadn't determined which student. Also still unresolved was the question of what numbers to use in the problem. This question led to a great deal of discussion. Ms. Tsukuda, the teacher who had proposed the problem, began with the following comments, as related by Yoshida:\n\nNot long ago, the Vice-Principal (Ms. Furumoto) showed me several textbooks. All of those textbooks used 12 and 9 (i.e., 12 \u2212 9 =) and 13 and 9. What most of the textbooks said was, they started out by introducing the Subtraction-Addition Method (Genkahoo). In the case of 13 \u2212 9, first subtract the nine from ten (10 12 \u2212 9 12 = 1), then add what is left over in the Is position (which is 3) to the number (1 + 3 = 4). I thought if you narrow it down like that (introducing subtraction with borrowing by teaching the Subtraction-Addition Method), it's not very interesting. So on Saturday I suggested using 15 \u2212 8, or 15 \u2212 7. I thought that these are a little harder than 12 \u2212 9 and 13 \u2212 9. Using these numbers will bring out a lot more ideas or ways to solve the problem. But after reading a lot of different books on the subject, because kids can conceptualize in their heads about up to the number 6 at this age, I thought we should go with numbers like 11 \u2212 6.\n\nThe teachers agreed that the choice of numbers would influence which strategies the students would try when solving the problem. But they had other concerns as well. For example, one teacher wanted to use 12 12 \u2212 7, because one of her students, who was a low achiever, happened to have seven family members. Everyone agreed that this was a good idea. They also liked the number 12 because, since none of the students had fewer than three people in their families, subtracting the number of family members from 12 would involve decomposing ten, which was, of course, the point of the lesson. They briefly considered the number 13 instead of 12, but decided against it, as shown in the following dialogue.\n\nOnce the numbers were agreed on, they wondered how they could make it seem natural that students should start with the number 12. They decided to begin the problem by asking students to select their twelve favorites from among the leaves they had collected. These would be the leaves students would use to work on the problem.\n\nFrom there the discussion turned to the different strategies that students might be expected to generate. The teachers consulted some of the teachers' manuals and found five common ways of solving simple subtraction problems with borrowing. Each method was labeled with technical terminology that seemed quite familiar to the teachers (though not to the researcher). Ms. Furumoto, who was the vice principal and who had been in the school for only one year, named a particular student who appeared to use only the subtraction-addition method. She said this concerned her, because she found it difficult to move students from this particular method into using more sophisticated methods.\n\nAnd so the discussion continued, for weeks, touching on all the topics listed above at a level of detail similar to that manifested in this example. This kind of planning is decidedly intellectual in nature; these teachers are thinking deeply about the options available to them and the way the experiences they structure in their classrooms will facilitate students' understanding of mathematics. There is real excitement as this process unfolds, an excitement that is obvious to those who observe the weekly meetings of a lesson-study group.\n\n## Reflections on  \nLesson Study\n\nFrom the U.S. perspective, it is difficult to believe that a process as narrowly focused as lesson study could really be the driving force behind Japan's educational success. 12 \"One year on a single lesson? We could never do that here,\" mused one of our colleagues. \"It would take forever to make any significant improvements in teaching.\" Yet perhaps that is one of our problems: by being in a hurry and taking the short-term view, we undermine the kinds of gradual long-term improvements that add up to real change.\n\nOn reflection, we can identify a number of interesting aspects of lesson study that might contribute to its success. All these aspects appear to be consistent with what we know about changing complex, cultural activities. Yet, significantly, they differ markedly from most opportunities that U.S. teachers have to improve teaching. It is worth examining some of these features.\n\nLesson Study Is Based on a Long-term  \nContinuous Improvement Model\n\nWe have made this point already, but it is worth making again: Lesson study is a process of improvement that is expected to produce small, incremental improvements in teaching over long periods of time. It is emphatically not a reformlike process.\n\nThe lesson-study process, as conducted in Japan, thus respects the fact that teaching is a cultural activity. Ronald Gallimore, who has written extensively about these issues, says, \"Cultural activities are historically evolved solutions to adaptive challenges. They were constructed over time through collaborative human effort to achieve a stable daily routine. Changes in cultural activity are made slowly, gradually, and are built on existing routines.\" Because teaching is a cultural activity, it will not change quickly or drastically.\n\nLesson Study Maintains a Constant  \nFocus on Student Learning\n\nThe lesson-study process has an unrelenting focus on student learning. All efforts to improve lessons are evaluated with respect to clearly specified learning goals, and revisions are always justified with respect to student thinking and learning.\n\nAlthough this feature might seem obvious and trivial, it is not. Reforms in the United States often are tied to particular theories of teaching or to educational fads instead of to specific learning outcomes. Because of this, success often is measured by the degree to which teachers implement recommended practices. Someone is marked as a good teacher because he or she uses cooperative groups or concrete manipulatives, instead of on the basis of his or her students' successful learning.\n\nLesson Study Focuses on the Direct  \nImprovement of Teaching in Context\n\nBy attending to teaching as it occurs, lesson study respects teaching's complex and systemic nature, and so generates knowledge that is immediately usable. This is in marked distinction to teacher-development programs in the United States, which seek to take knowledge gained in one context (for example, knowledge produced by educational researchers) and translate it into the messy and complex world of the classroom. As useful as educational research might be, it is notoriously difficult to bridge the gap separating researchers and practitioners. Japanese teachers function both as teachers and researchers, making it unnecessary to translate one into the other.\n\nWhat keeps lesson study relevant to the improvement of classroom teaching is its focus on the lesson as the unit to be analyzed and improved. Some might see this focus as trivial: Are there not other, more important, issues that could organize the teachers' inquiry? But in fact, focusing on a particular lesson turns out to be a useful way of simplifying the work of the group while still preserving the complexity that characterizes life in classrooms. The challenge of choosing units for study that retain the important elements of the system one is trying to understand is a classic problem in the research literature, often referred to as the problem of ecological validity. If units that are selected do not have ecological validity, research results often cannot be generalized to actual real-world situations. Lessons, we believe, do have ecological validity. Even a single lesson retains the key complexities\u2014curriculum, student characteristics, materials, and physical environment, among other things\u2014that must be taken into account as we try to improve classroom learning.\n\nThe decision to focus on lessons is especially appropriate in Japan. Because Japan has a centralized educational system and a national curriculum, division of the content into lessons is done in a similar way for all teachers of a given grade level and subject. This means that knowledge developed about a specific eighth-grade mathematics lesson or sequence of lessons, for example, is highly sharable with teachers all over Japan who must teach the same lessons. Reports published by lesson-study groups describing their work and its consequences have an instant audience among their colleagues throughout Japan. Many such reports, in fact, can be purchased in neighborhood bookstores.\n\nLesson Study Is Collaborative\n\nBy working in groups to improve instruction, teachers are able to develop a shared language for describing and analyzing classroom teaching, and to teach each other about teaching.\n\nThe often-described isolation of U.S. teachers has greatly hindered our discussions about teaching and hence our ability to improve it. U.S. teachers rarely have the opportunity to observe other teachers in action and are rarely observed by other teachers. For whatever reason, teaching in the United States is considered a private, not a public, activity. The consequences of this isolation are severe. Teachers might agree in discussion, for example, that \"problem solving\" should be a central focus of the mathematics classroom. But in practice, different teachers might have completely different understandings of what \"problem solving\" entails. The term is the same, but the referent of the term is private and varies from person to person.\n\nAll of this might sound abstract and academic, but it is not. Several years ago one of us was invited to a school by the principal to observe one of the school's star math teachers. As we walked into her room, we saw that the third-grade children were in groups, and the teacher was working with one of the groups. \"Imagine,\" said the teacher, \"that eight kitten ears were visible over the top of a fence, and a whole bunch of kitten feet were visible below the fence. How many kittens are behind the fence?\" Within ten seconds every one of the children's hands shot up. One child, after being called on by the teacher, responded, \"Four.\" to which the teacher responded, \"Correct.\" The process was repeated twelve more times with different problems.\n\nAt this point the teacher walked over and said, \"Isn't it amazing the kinds of problems these kids are solving?\" We were stunned. Why did she call this problem solving? Can a problem that is solvable within ten seconds really be considered a problem? How could she possibly define problem solving with respect to the problem but without reference to the students' level of knowledge and skill relevant to the problem? Clearly, we had no shared understanding of what problem solving is all about.\n\nAnother important benefit of the collaborative nature of lesson study is that it provides a benchmarking process that teachers can use to gauge their own skills. Collaboration includes continuing interactions about effective teaching methods plus observations of one another's classrooms. These activities help teachers reflect on their own practice and identify things that can be improved. As researcher Catherine Lewis found, teacher collaboration can create a profound motivation to improve. A young teacher she interviewed recalled that after watching a lesson by her fellow first-grade teacher, she burst into tears: \"I felt so sorry for my own students. I thought their lives would have been so much better if they'd been in the other teacher's class.\"\n\nAt the same time, the collaborative nature of lesson study balances the self-critiquing of individual teachers with the idea that improved teaching is a joint process, not the province or responsibility of any individual. This idea is embodied in the fact that when Japanese teachers plan a lesson collaboratively, they treat the result as a joint product whose ownership is shared by all in the group. When one teacher teaches the lesson and the others observe, problems that emerge are generally attributed to the lesson as designed by the group, not to the teacher who implemented the lesson. It thus becomes possible for teachers to be critical without offending their colleague. The discussion can focus more pointedly and deeply on the merits and deficiencies of the lesson, and on the process of revising and improving it. This leads us to our final point.\n\nTeachers Who Participate in Lesson Study  \nSee Themselves as Contributing to the Development  \nof Knowledge About Teaching as Well as to Their  \nOwn Professional Development\n\nTeachers in Japan see themselves as developing the profession as well as themselves. Few U.S. teachers would feel this way. When U.S. teachers go to workshops and training seminars, they go to learn about a new activity or technique; most wouldn't conceive it possible that they might be making a contribution to the knowledge base of the teaching profession. The reason they feel this way is that, given our current system, they are right\u2014they are not making such a contribution. In the U.S. system, it is researchers who are supposed to discover and recommend new teaching practices. Teachers are supposed to implement these practices in their classrooms, but alas, they usually fail to do so, much to the chagrin and disappointment of the educational research community. This predictably sets the stage for talk of how the teaching profession attracts a less-than-able subset of the U.S. workforce, and how, apparently, U.S. teachers are just not smart enough to do what researchers tell them to do.\n\nBut there is another possibility: Perhaps what teachers are told by researchers to do makes little sense in the context of an actual classroom. Researchers might be very smart. But they do not have access to the same information that teachers have as they confront real students in the context of real lessons with real learning goals. For researchers to improve teaching, they must guess at many of the things that are readily perceivable by teachers. And they probably guess wrong a good deal of the time.\n\nJapan has succeeded in developing a system that not only develops teachers but also develops knowledge about teaching that is relevant to classrooms and sharable among the members of the teaching profession. Not only do lesson-study groups operate in individual schools, but the process of designing and critiquing research lessons is an integral part of the larger professional activity of both teachers and researchers. Professional conferences include sessions in which participants observe research lessons in local schools and then return to the conference meeting center for panel discussions of the lessons. Although some education conferences in the United States include field trips to special demonstration schools for a small number of interested participants, the kind of broad-based intensive examination of individual lessons common in Japan is almost unknown in this country.\n\nThrough the process of improving lessons and sharing with colleagues the knowledge they acquire, something remarkable happens to teachers: They begin viewing themselves as true professionals. They see themselves as contributing to the knowledge base that defines the profession. And they see this as an integral part of what it means to be a teacher. As one Japanese teacher said, when asked why she invests so much effort in trying to improve lessons, \"Why do we do research lessons? I don't think there are any laws. But if we didn't do research lessons, we wouldn't be teachers.\"\n\n## Conclusions\n\nThrough the gradual improvement of individual lessons, and through the knowledge developed and shared during this process, the Japanese system enables the steady improvement of teachers and teaching. In Japan, educators can look back over the past fifty years and believe that teaching has improved. In the United States, we cannot do this. We can see fashions and trends, ups and downs. But we cannot see the kind of gradual improvement that marks true professions.\n\nIt is clear that we need a research-and-development system for the steady, continuous improvement of teaching; such a system does not exist today. We must move beyond models of reform in which we try to replace one teaching method with another by distributing the written recommendations of experts. Like the Yemeni immigrants in Shanker's story, we get teachers to use our tables, but they often turn those tables upside down. Instead, we must take the first step toward building a system that will, over time, lead to improvement of teaching and learning in the American classroom. We need new ideas for teaching, ideas such as those provided by the videos from Japan and Germany. But instead of copying these ideas we must feed them into our own research-and-development system for the improvement of classroom teaching. And we must empower teachers to be the leaders in this process. In the next chapter we lay out a concrete proposal for building such a system.\n\n## CHAPTER 8\n\n## Setting the Stage for  \nContinuous Improvement\n\nALTHOUGH THE HIGH achievement of Japanese students has been a favorite media topic, Japan's system for improving teaching has attracted little interest in the United States. Perhaps this is because it is a program with little fanfare. Americans like to think big. Something as simple and straightforward as lesson study just isn't dramatic enough to capture the imagination of educational reformers in the United States. \"Wouldn't it take forever,\" a colleague asks, \"to improve teaching one lesson at a time?\" \"Our kids need help now, not ten years from now,\" says a politician. \"We need major restructuring, not modest improvements.\" Yet despite our desperate rush to reform, the evidence shows that little has changed inside U.S. classrooms. By trying to accomplish too much, we have sacrificed opportunities for small, cumulative improvements.\n\nJapan, in contrast, has attached great value to small improvements. Like the tortoise, Japan has pinned its hopes on steady progress, not on momentous leaps forward. There is nothing magical about the modest changes Japanese teachers devise in Is it possible to build such a system here in the United States? Our answer is yes, and the time to build such a system is now. A system of gradual improvement like that found in Japan depends on clear standards for what students should learn and means of assessing progress toward meeting the standards. These have been lacking in the United States up to now, but this is changing rapidly. Most states have set new, high standards for what students must learn. The problem now is to find ways to meet these standards.\n\nWe believe that the key to meeting the new standards is to improve teaching. The same curricula and teaching methods we have been using will no longer suffice. In this chapter and the next we propose a system of gradual improvement in teaching over time. We believe that if teaching can be improved\u2014not just in special locations but inside the average classroom\u2014student learning will follow. Even though we have seen that teaching, as a cultural system, can be difficult to change, the case of Japan shows that change is possible.\n\nThe core of our proposal is to establish something like Japan's lesson-study system here in the United States. Many readers will no doubt be skeptical that a program developed in another culture can be instituted here. Certainly, we do not expect the system that evolves here to be identical to the one that has evolved in Japan. Our system differs from the Japanese system in critical ways; not least among them is the lack of a national curriculum. Because no system of gradual improvement has ever been tried in this country on a national scale before, much will have to be tested and refined over time. Our goal is simply to convince the reader that something like lesson study deserves to be tested seriously in the United States. It is our hypothesis that if our educational system can find a way to use lesson study for building professional knowledge of teaching, teaching and learning will improve.\n\nImprovement will not happen by itself. It will require designing and building a research-and-development system that explicitly targets steady, gradual improvement of teaching and learning. Like the Japanese, we must be able to look back ten or twenty years from now and clearly see that we have improved\u2014gradually but continuously. We cannot make such a statement today. What kind of system will allow us do so in the future?\n\n## Six Principles for Gradual,  \nMeasurable Improvement\n\nWe propose six principles that must be taken seriously by anyone attempting to improve teaching. It is not coincidental that these principles are largely congruent with the key features of lesson study enumerated in Chapter 7. We note here, and again later, that lesson study is one process that is fully consistent with these principles. Indeed, that is why we are optimistic about the benefits that will accrue if lesson study can become part of the U.S. educational system.\n\nPrinciple #1: Expect Improvement to Be  \nContinual, Gradual, and Incremental\n\nBecause teaching is a system that is deeply embedded in the surrounding culture of schools, any changes will come in small steps, not in dramatic leaps. History supports this claim: In spite of waves of reform that have called for sudden, major shifts, teaching has always evolved like other complex, culturally embedded activities\u2014slowly and incrementally. Dramatic and fundamental changes can occur, even at the core, but these will result from accumulating small changes over time.\n\nThis means that we must take a long-term view when we design initiatives for improving teaching. We must reset our expectations, cultivated over a century of school reform, and anticipate slow and steady improvement, not momentous change. Efforts to reform teaching overnight, or even over a few years, are unlikely to have their intended effect. In addition, we must learn to value small improvements. Small improvements often are derided as \"too little, too late\" and are killed off before they have a chance to build into something significant. Teachers must be allowed and encouraged to invent small changes in the system of teaching and then to keep track of these changes so they can be accumulated and shared.\n\nPrinciple #2: Maintain a Constant  \nFocus on Student Learning Goals\n\nThe goal of teaching is students' learning. The goal of improving teaching is improving students' learning. Too often, however, reformers forget the goals of reform, gauging their success by changes in the particular forms of teaching. Bruce Joyce and colleagues, who have been involved in school reform for some time, have noted: \"The centrality of student learning becomes lost as the details of program implementation become ends in themselves.\" The question of whether and how these changes are improving students' learning in the teacher's classroom gets lost in the sheer effort to change.\n\nImproving complex systems, such as teaching, requires a relentless focus on the bottom-line goals\u2014in this case, students' learning\u2014and a commitment to evaluate changes with respect to these goals. Such a focus appears to be a necessary component of any successful school improvement program. Listen again to Bruce Joyce and his colleagues: \"In all reported cases of school improvement initiatives in which substantial student learning occurred, school staff kept students' interests as learners central throughout the planning, implementation, and assessment phases. We did not find a single case in the literature where student learning increased but had not been a central goal.\"\n\nPrinciple #3: Focus on Teaching,  \nNot Teachers\n\nA number of recent efforts to improve classroom instruction have targeted the competency of teachers. The National Board for Professional Teaching Standards, for example, has instituted a voluntary certification process to help raise the standards by which teachers are certified. Many states have instituted alternative certification programs designed to recruit high-achieving individuals without formal teacher training into the profession. Although we applaud the desire to raise standards for certification and to increase the pool of talented teachers, we believe that long-term improvement in teaching will depend more on the development of effective methods for teaching than on the identification and recruitment of talented individuals into the profession.\n\nIn biological evolution we know that it is the gene that is subject to natural selection, not the individual organism. Individuals are short-lived by evolutionary standards; they don't last long enough to undergo the slow process of evolutionary change. Genes, in contrast, persist over hundreds and thousands of generations. Teaching, similarly, persists, while teachers come and go. If we are to achieve long-term improvements in classroom teaching and learning, we must shift our focus from teachers to teaching. As we noted earlier, teachers follow scripts that they acquire as members of their culture, and their effectiveness depends on the scripts they use. Recruiting highly qualified teachers will not result in steady improvement as long as they continue to use the same scripts. It is the scripts that must be improved.\n\nPrinciple #4: Make Improvements in Context\n\nGoals give us the yardstick against which improvement can be measured. But how can we identify and develop ways to improve teaching? Because teaching is complex, improvements in teaching will be most successful if they are developed in the classrooms where teachers teach and students learn. Teaching is a system built from all the elements of the local context: teacher, students, curriculum, and so on. Improving the system requires taking all of these elements into account. What works in one classroom might or might not work in another classroom. Ideas for improvement that come from afar\u2014including, for example, what we've learned from Japanese lessons\u2014will need to be tested and adapted to our own local classrooms if they are to have any chance of success. Teaching is unlikely to improve through researchers' developing innovations in one place and then prescribing them for everyone. Innovations can spread around the country, but only by trying them out and adjusting them again and again as they encounter different kinds of classrooms.\n\nEducators have become increasingly aware of the importance of context in understanding and facilitating learning, but the arguments have been applied more often to students' learning than to teachers' learning. Teaching, given its systemic, cultural nature, is especially sensitive to context\u2014a good reason to take advantage of the very contexts in which teachers function every day. Teachers learning in the classrooms and schools in which they teach is an idea that has been proposed for some time, yet most teachers have not realized this opportunity. It certainly contrasts with many traditional methods of teacher development (for example, weekend workshops, university courses) in which teachers are expected to learn something new, disconnected from their context, then hope it works when they take it back to their classrooms.\n\nPrinciple #5: Make Improvement  \nthe Work of Teachers\n\nOne way to ensure that improvements can be developed in context is to entrust change to those engaged in the activity\u2014classroom teachers. Improving something as complex and culturally embedded as teaching requires the efforts of all the players, including students, parents, and politicians. But teachers must be the primary driving force behind change. They are best positioned to understand the problems that students face and to generate possible solutions. In fact, almost all successful attempts to improve teaching have involved teachers working together to improve students' learning.\n\nA second reason for making improvement the work of teachers is that there are so many teachers, especially when compared with the number of educational researchers. If we can find a way to marshal the efforts and experiences of our 2.5 million classroom teachers, the potential is far greater than anything that could be achieved by a few thousand researchers.\n\nTeachers should be engaged in improvement because they are the only ones who can ensure that students' learning improves. They are the gatekeepers of the classrooms in which teaching and learning take place. As we pointed out earlier, the classroom is the final common channel through which all efforts to improve school learning must flow. We cannot not work with teachers. They are, necessarily, the solution to the problem of improving teaching.\n\nPrinciple #6: Build a System That Can  \nLearn from Its Own Experience\n\nEach day, vast numbers of U.S. teachers solve problems, try new approaches, and develop their own knowledge of what works and what doesn't work in their own classrooms. Yet we have no way to harvest what even the most brilliant teachers have learned, no way to share that knowledge and use it to advance the professional knowledge base of teaching. U.S. teachers work alone, for the most part, and when they retire, all that they have learned is lost to the profession. Each new generation of teachers must start from scratch, finding its own way.\n\nIf efforts to improve schools are going to add up to more than just a temporary fix, it is necessary to find a way to accumulate knowledge about teaching and to share this knowledge with new practitioners entering the teaching profession. In the long run, of course, we need to change the teaching scripts that govern classroom practice. Scripts themselves might be the most effective means of storing professional knowledge. But scripts will not be changed unless we have a knowledge base to support the change. We must build a system with a memory, in other words, one that provides a means of accumulating the experiences and insights of teachers. Without this, there is no way of getting better over time.\n\n## Initiatives for Change:  \nSetting the Stage\n\nThe six principles proposed above are consistent with what we have learned about teaching. But principles alone will not bring about change. We must formulate action plans, based on these principles, to guide the work of school improvement over time. Because the United States has no tradition of gradual improvement, we must test the plans, monitor their level of success, and use the information to refine them over time. To begin the process, we propose a program based squarely on the process of lesson study, a program that we believe, if implemented, will produce long-term steady improvements in students' learning.\n\nEstablishing the program we envision is a monumental task, not so much because it is costly or requires new resources but because it involves a change in school culture. Changes in cultures, as we have noted many times, do not happen overnight. It is not that schools in the United States cannot improve classroom practice. Indeed, the program we propose shares many features with a number of local school improvement efforts that have shown great success. The problem is that these successes have never reached far beyond the sites receiving special attention and assistance. They certainly have not touched the average school and the average teacher.\n\nSimply recommending lesson study as a useful process is not enough, because the process cannot succeed, on a wide scale, without a supporting context. In the remainder of this chapter, we propose three initiatives in the form of school district tasks that must accompany the introduction of lesson study. The initiatives are directed toward creating cultures in schools and districts that would support any long-term teacher-driven improvement process, including lesson study. For each initiative, we describe the roles that must be played by the primary stakeholders\u2014school boards, superintendents, principals, teachers, and parents.\n\nRemember, our current system for improving teaching, just like the system of classroom teaching, is a cultural activity. As such, it will change gradually, just as teaching itself does. To succeed, we must be able to start small but also be able to expand. Whatever research-and-development system we construct to improve teaching over time must be capable of working even for a single district, for that is where such change will inevitably start. But there also must be a clear means of growing the system into one that can serve the nation as a whole.\n\nInitiative #1: Build Consensus  \nfor Continuous Improvement\n\nWho, in a school district, is in the best position to start the process of improving teaching? There is no single answer, because each district is different. Teachers, of course, are essential for the success of any effort. But it is rare for teachers to have the opportunity and time to lead such an effort. There must be involvement from the very top, especially from the school board. Principals will have difficulty starting such a process by themselves unless they have the unqualified support of their superintendents. Superintendents are a logical choice to lead the change, and they definitely will play a major role. But typically, they don't last long enough in their positions to follow through over long periods of time. School boards, though historically ineffective in leading reform efforts, have the power to make a real difference if they so choose. To be effective, school boards must build political consensus and public support for long-term improvement and then commit resources for many years to the task of improving students' learning.\n\nBuilding consensus around a process of slow, gradual improvement will not be easy. Traditionally, Americans have been more willing to accept dramatic failures than to applaud, or even appreciate, small successes. It seems as if we have an all-or-nothing culture: We want results fast or we are not interested. Gradual change, although it is the working model in Japan, has not been an option for many Americans.\n\nIn order to sell a model of gradual improvement to all stakeholders, outcome measures must be developed that are sophisticated enough to detect small changes in student learning and to differentiate such changes from random fluctuations. As students' learning gradually improves, it will be easier for school boards to strengthen the district's resolve to work hard for small improvements. It is important for school board members to remember that alternative paths, driven by impatience, have not worked well. The evidence suggests that efforts to produce rapid, wholesale change have failed to improve schools in virtually every case. It's time to start a realistic program of improvement.\n\nInitiative #2: Set Clear Learning Goals  \nfor Students and Align Assessments with These Goals\n\nWe hear a lot these days about standards. Much of the discussion, at least when politicians are involved, refers to how high or low standards are, and to the importance of raising them. But standards involve a great deal more than simply high expectations. Standards contain the goals that we value most. And clear goals for what we want students to learn are essential. Without them, it is not possible to set a course toward improvement. Without clear goals, we also cannot know whether changes represent improvement or... just change.\n\nNot only must there be clear goals, but if teachers are going to collaborate to find ways of improving instruction, the goals must be widely accepted. Most high-achieving countries have national goals for student learning. Japan, for example, has an explicit set of mathematics-learning goals for all of its students, grade by grade. When a Japanese teacher group works on improving lessons, it is not working alone. Many teacher groups throughout Japan are working on lessons with the same goals in mind. This means they can easily share the products of their efforts.\n\nIn the United States, we do not have national learning goals, and in some cases, we do not even share a set of learning goals within a district. We tend to rationalize this situation by referring to local and individual needs of different groups of students in different parts of the country and even in different schools and classrooms. Although local contexts might differ in some respects, we must realize that a set of shared learning and curricular goals is a minimum requirement for teachers to collaborate effectively. This does not mean that we must enforce, or even adopt, a set of national goals, but it does mean that we must, at least, develop shared goals at the district level.\n\nFortunately, there is a growing consensus around the country regarding the importance of developing clear learning and curricular goals for all students. As we noted in Chapter 1, the standards set by various professional organizations and the standards being developed within individual states have moved this activity to the top of the educational agenda. The National Science Board, an independent national science advisory panel established by the U.S. Congress in 1950, recently issued a statement urging \"all stakeholders in our vast grass-roots system of K-12 education to develop a nation-wide consensus for a common core of knowledge and competency in mathematics and science.\" In the short run, it will be difficult to develop a national consensus on learning goals, but a district-wide consensus surely can be achieved.\n\nSome thought must be given to the way in which learning goals are stated. Goals can emphasize skills (for example, \"students should be able to add two three-digit numbers\"), or they can emphasize conceptual understanding (\"students should understand how the place value nature of numbers allows different methods of adding to work equally well\"). They can be short-term or long-term. They also can be at various levels of generality. For example, they might refer to \"being able to perform basic arithmetic operations on whole numbers\" or \"being able to add single-digit numbers whose sum is less than ten.\" There is no one correct way to define learning goals. What is important, first, is that goals capture the kind of learning that we most value and, second, that they are clear enough to link directly to the lessons, units, and grade levels that make up a curriculum.\n\nWho should develop shared goals for student learning? Because such goals should be district-wide at least, superintendents must take a leadership role. But the development of shared goals will require participation of parents, principals, and especially teachers. Districts can take advantage of the best work that has already been done at the state and national levels in setting clear goals pitched at an appropriate level of specificity.\n\nThe process of attaching meaning to the learning goals is a long one requiring a great deal of work by teachers. A learning goal is not easily defined and not easily summarized in a written document. The true meaning of learning goals becomes apparent only as teachers link them to assessments and weave them into a coherent curriculum and use them to guide their teaching decisions.\n\nAssessments must be aligned with goals. Often, assessments are linked to goals for purposes of accountability, but assessments play an essential role in the improvement process. Even with clear goals, teachers cannot improve their practice unless they have access to a steady flow of information about the effectiveness of their teaching. As teachers engage in continual assessment of their students' learning, they will gradually develop understandings of how students learn from classroom instruction, and they will begin to perceive direct links between the goals they set, their own teaching, and their students' learning. With this information, teachers are in a good position to work together to improve their lessons.\n\nInitiative #3: Restructure Schools  \nas Places Where Teachers Can Learn\n\nA little-recognized truth in educational reform is that every recommendation for improving teaching requires teachers to learn. This is not surprising. Changing one's practice in any professional field requires examining the old and new practices, making the appropriate modifications, and learning to carry out new practices effectively. It would be silly to expect teachers to simply execute improved teaching methods without providing them with opportunities to develop these methods and learn how to use them.\n\nIf we expect teachers to play a major role in improving instruction, as they must, then we need to provide an environment in which they can do this work. Unfortunately, the vast majority of American schools are not suited to this purpose. Teachers work alone, for the most part, and have little time to interact, much less collaborate. They arrive at school shortly before the students do and leave shortly after the students leave. While at school, teachers spend most of their time teaching.\n\nThis may sound perfectly fine to many people. Aren't teachers paid to teach? Aren't teachers already trained in college and certified to teach? Why should they be doing anything else in school? Yet other professions that involve complex skills, such as law or medicine, expect their members to become more skilled over time and provide opportunities for learning new skills. This is true even for professions that often are assumed to require less training. Albert Shanker noted that the new process for building cars introduced by Saturn provides ninety-two hours of reeducation per year for each employee. Shanker said, \"It is ironic that a bunch of people whose business is building cars understand so well the importance of educating their employees, whereas people in education seem to assume that teachers and other school staff will be able to step right into a new way of doing things with little or no help. If it takes... 92 hours a year per employee to make a better automobile, it will take that and more to make better schools.\"\n\nBut the common American view of teaching does not include learning to teach while teaching. We typically expect that most teachers will be doing pretty much the same things in their classrooms when they are veteran teachers as when they begin teaching; in fact, some people believe that beginning teachers might be more effective, entering the profession with fresh ideas. And given the lack of learning opportunities for teachers, they might be right.\n\nImproving teaching is not something that can be left to refresher courses in the evenings or during the summer in university classrooms. Improving teaching must be done at school, in classrooms, and it must be seen by teachers, parents, and administrators as a substantial and important part of the teacher's workweek. Schools must be places where teachers, as well as students, can learn.\n\nChanging schools to support teachers' learning requires changing the culture of schools. We have emphasized culture as an important idea for understanding why teaching looks the way it does and why it is resistant to change. The reader could get the impression that it is impossible to change teaching because there are so many cultural factors that keep it in place. It is crucial, at this point, to distinguish between the wider American culture and the culture of schools. They are related, but they are not the same. What is needed to improve teaching is a change in school culture, and this is possible. Witness the individual schools and districts around the country that are changing.\n\nA requirement for beginning the change process is finding time during the workweek for teachers to collaborate. In our opinion, two hours per week is a reasonable goal, at least initially. Time is a precious commodity in teachers' schedules, and finding time for teachers to work together presents a challenge to the typical school district. Many people assume that the cost associated with doing so would be prohibitive. But there are low-cost solutions that can work and that are working in a number of school districts across the country. Some of these solutions are based on the fact that nonteaching personnel now account for more than half of the U.S. education workforce, a percentage that has increased steadily since World War II and that far exceeds the proportion of nonteaching personnel in most other countries. If funding can be shifted into teaching, and away from nonteaching personnel, it is possible to free teachers to devote significant amounts of time each week to the improvement of teaching.\n\nIn a recent book, Linda Darling-Hammond presents one analysis of how this can be accomplished. Darling-Hammond analyzed the way staff time is allocated in typical high schools and found that only 33 percent of the time was dedicated to teaching. By eliminating bureaucratic staff and adding teachers, some schools have been able to reduce class size and create up to ten hours per week for each teacher to invest in individual and collaborative efforts to improve teaching. And all of this has been accomplished without an increase in the overall cost of education. Darling-Hammond cites as an example the case of New York City's Community District 2. In this district, professional development has been targeted as the highest priority, and the school has been restructured to provide time for teacher collaboration and even cross-school observations by teachers. Student achievement has gone up.\n\nSome districts have been able to implement such changes in personnel quite quickly. For others this might be a long-term solution, as staff retirees are replaced by teachers and as teachers' jobs are redefined. For these latter districts, a more appealing short-term solution might be to review the current workday schedule along with all the in-service opportunities for teachers and to restructure these into regular collaborative work time. Principals and superintendents will need to take the lead in reorganizing schedules and in-service activities.\n\nThis much can be achieved without additional resources. The changes require only redirecting funds already spent operating our schools. Now imagine what would happen if we were to take the millions of dollars spent every year to reform American education, much of which has little affect on classroom practice, and use it to provide the time and resources teachers need to improve teaching. We would have plenty of funds to support a vigorous, highly focused research-and-development system. The evidence suggests that this would be a wise change in how we spend reform money: increased spending on teacher education has a greater positive effect on student learning than increased spending on other school variables.\n\nIt is becoming clear that the district is the unit that can restructure most successfully. As was noted earlier, the district is small enough that it should be possible to achieve consensus on students' learning goals and to implement a common curriculum. At the same time, the district level is large enough to allow substantive restructuring in terms of funding and staff allocations. The district office is usually the site of budgetary control. The district also is the level at which superintendents and school boards can exert strong leadership.\n\nA final reason that restructuring should be done at the district level is that a district-wide program provides teachers with a critical but often overlooked opportunity for professional growth. Teachers must have the opportunity to enlarge their horizons beyond their own classrooms and their own schools. We are asking them to undertake a significant task: to improve the quality of teaching as part of a nationwide effort. To do more than improve teaching in their own classrooms, to raise the standard of good teaching within the profession\u2014this demands that teachers work together, sharing what they learn in their classrooms to help one another learn even more. It demands that they assume responsibility for building the profession's knowledge base. Working with teachers from their own and other schools will allow them to develop this critical new professional perspective.\n\n## Summary\n\nIf these first three initiatives can be implemented, the stage will be ready for long-term, steady improvement. These are not initiatives that can be launched by a single teacher inside a single classroom, but they can be set in motion by a single school district. They will be sustained only with strong leadership at the highest levels of the community and broad participation by the school board, superintendent, principals, teachers, and parents.\n\nEven implementing these critical three initiatives, however, will not, in and of itself, produce gains in student achievement. For this we must make substantive changes inside classrooms, at the place where student learning occurs. Once teachers have time during their workweek to devote to the improvement of teaching, what should they do with this time? We turn to this question in the next chapter.\n\n## CHAPTER 9\n\n## The Steady Work of  \nImproving Teaching\n\nONCE A DISTRICT has laid out an organizational structure for school improvement, with time for teachers to work in collaborative groups, what should teachers do with their newfound time? Many reformers who thought increased planning time, by itself, would lead to improvements in teaching have found that it does not. Indeed, teachers who are told simply to collaborate often find that they are not sure what they are supposed to do, or how such collaborations can help them to improve their teaching. One school district that restructured to allow teachers time to collaborate found within months that teachers were complaining about the time they were supposed to spend meeting together. \"Let's just go home early,\" said one of the teachers, \"and use the time at home to prepare for tomorrow's lessons.\"\n\nThis is not teachers' fault; it merely demonstrates the fact that although most people now agree that teachers need opportunities for professional development, there is a dearth of knowledge about the process by which teachers actually learn to improve their practice. We are attracted to the Japanese notion of lesson study because it lays out a clear model for teacher learning and a clear set of principles or hypotheses about how teachers learn. Lesson study embodies a set of concrete steps that teachers can take, over time, to improve teaching. These steps may need to be modified to work in the United States. But we believe it is better to start with an explicit model, even if it needs revising, than with no model at all.\n\nWhich is why, in this chapter, we propose lesson study, American-style, as the foundation of our efforts to improve teaching in this country. We describe our vision of how lesson study can work for U.S. teachers. Lesson study, as it works in Japan, is fully consistent with what we know about how to improve complex cultural activities like teaching, and it works, simultaneously, toward improving both teaching and teachers' knowledge and skills. This means it could lead to the kind of continuing improvement in learning that American students deserve. The process is not magical or sudden, and it will encounter some significant challenges. It will be, above all, steady work. Our task is to explore through planning, implementation, and reflection how lesson study can grow and adapt to become a functioning research-and-development system for the improvement of teaching in the United States.\n\n## Can Lesson Study Work  \nin the United States?\n\nBefore we rush to implement a lesson-study program for the United States, we pause to question whether such a plan could work in the United States at all. There are several reasons to suppose that it could. For one, the features of lesson study, as practiced in Japan, are very similar to those reported by American researchers as characterizing successful experimental teacher-development programs. Reports of American school-based improvement initiatives that include such features\u2014for example, setting goals for students' learning, working together to improve practice, attending to the curriculum and students' thinking\u2014indicate that teaching can be improved by programs that resemble lesson study. Lesson study also is consistent with another, gradually expanding, movement that is often referred to by the phrase \"teacher-as-researcher.\" One of the goals of this movement is to encourage teachers to engage in research, thereby creating in the teacher a temperament oriented to inquiry and a disposition toward investigating one's own practice. Such a disposition is at the heart of the lesson-study process.\n\nBut such evidence is weak, at best. What works on a small scale in experimental settings almost never scales up, especially in education. Many teachers in the United States do not even prepare lesson plans, at least not around student learning goals. Why would they suddenly be willing to engage in lesson study? For lesson study to be a viable means of improving teaching nationwide, it must be able to flourish and grow within the current educational landscape. For this to happen, two tests must be passed. First, lesson study must meet the needs of teachers. Teachers must be motivated to engage in lesson study. They must be willing to try it, and once it is started, they must find it relevant and useful to the problems they face each day in the classroom. Only then will they continue to participate. Second, lesson study must fit within the current political and policy contexts that surround American education; in other words, it must meet the needs of the U.S. education system. Teachers are under great pressure to perform, and the stakes keep getting higher. Lesson study must meet the needs that teachers have to meet these pressures.\n\nMeeting Teachers' Needs\n\nTeaching is a difficult and demanding job. Teachers are isolated from their colleagues and rarely have the opportunity to participate in professional life outside the classroom. They are pressed by administrators and reformers to take on new responsibilities, to teach in new ways, and to show better results. But they are given few resources to meet these demands. Lesson study is not just another activity that teachers must add to the list of expectations, it is a way for teachers to deal with these expectations. Lesson study is a comprehensive program that can provide teachers with opportunities for practice-based professional development that, until now, they have been denied.\n\nWith its detailed analysis of practice and its frequent observations of other teachers, lesson study provides benchmarks against which teachers can measure their own practice and compare it with that of their colleagues. These comparisons, more than any external rewards, can create in teachers a strong desire to improve their own practice. As teachers watch other teachers, it is possible for them to imagine new possibilities for their own teaching. Lesson study provides a concrete means of trying out these possibilities in a nonthreatening context with the help of colleagues. This personal motivation is, in the end, the kind of demand that will produce improved teaching.\n\nSome may question whether, even if they become motivated to participate in lesson study, teachers have the capabilities to handle its demands. In our view, lesson study is not the kind of process in which teachers must first develop a list of capabilities and then begin to design improved lessons. Lesson study is, in fact, the ideal context in which teachers develop deeper and broader capabilities. This is what we mean when we say that lesson study is a form of teacher development as well as a program for improving teaching.\n\nSuppose, for example, that a group of fifth-grade teachers wants to improve an introductory lesson on decimal fractions. As part of the process, teachers would engage in research. They might consult other texts, invite special consultants for this aspect of their work, or meet with other teachers in the school or district. They might also study how students think about decimal fractions, and try to use this knowledge to predict how students would respond to various instructional alternatives. Through all of these activities, teachers would be increasing their own knowledge of decimal fractions, and doing so in a way directly relevant to the improvement of their ability to teach the subject. In many ways, teachers' learning more about mathematics in the context of lesson study is preferable to taking another mathematics course at a neighboring college, because the teachers have a concrete and compelling reason for learning more about this topic. Much of what they learn can be connected to other aspects of teaching the topic\u2014the tasks selected for students, the kinds of discussions that will be beneficial, and so on.\n\nLesson Study in the U.S. Context\n\nIf we assume that lesson study can meet teachers' needs, we still must examine how it fits within the U.S. education context. To succeed, lesson study must connect both to the curriculum of the school and to the current policy context in which teachers work.\n\nCurriculum. In Japan, where there is a national curriculum, it makes sense to spend sustained periods of time perfecting a small set of lessons. Because all teachers teach the same curriculum, knowledge generated by one lesson-study group is usable by everyone who teaches at the same grade level. But the United States does not have a national curriculum.\n\nAlthough the absence of a national curriculum will shape the nature of lesson study in the United States, it will not prevent lesson study from succeeding. The key is that the group of teachers who undertake lesson study must work from a shared curriculum. We have suggested that at the least, this must be done within the school district. As districts that use the same curriculum connect, teachers can share their results across district boundaries. In this sense, the same opportunity offered by Japan's national curriculum is available, though in a more limited fashion, to U.S. teachers.\n\nStandards, Assessment, and Accountability. In today's policy context, curriculum is increasingly aligned with content standards and assessments. Teachers, more and more, are working within contexts in which they are accountable for students' performance on assessments. Lesson study is unique in its potential connection to local standards and assessment. Because improvements are curriculum-based, lesson study allows teachers to devote their time to improvements that align with their local standards and for which they are held accountable.\n\nThe traditional researchers' method for testing improvements\u2014running an experiment that pits the new lessons against the old\u2014is hardly possible in a teacher's ongoing instructional program. But if lesson study is a district-wide activity, teachers can compare their own results with those obtained by colleagues elsewhere in the district or in other districts. The most important results pertain to the quality of students' thinking and learning during the lesson. Provided appropriate assessment systems are in place, feedback from different classrooms can be used to discard the new lessons or to revise them and make them even better. Repeating the lessons with other students, in other contexts, is a quality-control process that works slowly, but it is a powerful process that assures teachers that improvements will occur gradually and continuously.\n\nConnecting Policy with Classroom Practice. Finally, lesson study has the potential to solve what has been identified as a major problem in U.S. education, and that is the gap that exists between educational policymakers and classroom practice. Teachers are caught in a persistent dilemma: Although they frequently receive advice and recommendations on how to change their teaching, and they know that some of these changes would probably benefit their students, they also lack the learning opportunities needed to study the recommendations, decide which changes would be meaningful, and learn how to implement them. This leads teachers to devalue suggestions proposed by outsiders such as researchers or policymakers, because they fail to see them as relevant to their everyday classroom practice. And those who suggest the changes seldom get continuing feedback from teachers.\n\nAn example of the teacher's dilemma has been prompted by today's reform recommendations in mathematics and science. These recommendations ask teachers to teach in a more adventurous, ambitious way. Rather than demonstrating procedures for solving problems and then giving students worksheets of problems to practice, they are asked to present challenging problems to students and encourage students to develop their own methods of solution. Then they are to engage students in a thoughtful discussion of the alternative procedures, analyzing the pros and cons of each. When well executed, this method of teaching has been shown to be quite effective. But unless one knows what to expect from students, it is a scary way to teach. Success depends on making many split-second decisions about which student suggestions to follow up on and which to ignore. What is learned by students during the lesson seems to depend on whether students hit upon the solution methods that make for good class discussions. Teachers can feel that they have lost control of the lesson, but they are told to \"embrace the uncertainty,\" because this is what better teaching is like.\n\nLesson study offers an alternative that will appeal to the teachers caught in this difficult position. Lesson study shifts the key for effective teaching from on-the-fly decision making during the lesson to careful investigation and planning before the lesson. Through the lesson-study process, teachers can collect information about how students are likely to respond to the challenging problems, and they can plan which responses to introduce into the discussion in which order. They can thereby orchestrate a rich discussion without the debilitating uncertainty with which they previously had to deal. Of course, some decisions still will need to be made extemporaneously, but good advance planning shoulders much of the burden. Indeed, planning takes on new value as a premier teaching skill. And participation in lesson study allows teachers to improve their planning skills over time.\n\nThus, what starts as a vague and impractical suggestion from educational experts gets transformed, through lesson study, into an improvement in classroom practice. By communicating the results of this improvement, teachers, researchers, and policymakers all increase their power to contribute to the improvement of education.\n\n## Establishing the  \nLesson-Study Process\n\nThe fact that lesson study could provide the key ingredient for improving students' learning, district by district, across the United States does not mean it will be easy to implement. Even if the initiatives presented in the previous chapter are undertaken to provide a supportive context for continuous improvement, and even if the participants endorse the potential of lesson study, principals and teachers still face many challenges.\n\nLeadership for Lesson Study\n\nOne of the biggest problems schools will face is that there are few leaders among its teachers for launching this process. Very few teachers have experienced this kind of professional development. Most teachers, as we have noted, work alone, in isolation from their colleagues. Those who do collaborate with other teachers generally do everything except work on the improvement of classroom lessons.\n\nIn the absence of experienced teacher leaders, principals must take an active role in introducing lesson study. The principal must become personally and directly involved in beginning the process and establishing it as a permanent part of the school program. The principal's involvement is one way to signal that improving teaching is the most critical part of the school's development. The principal must work closely with teachers and must nurture teacher leaders who are willing to devote considerable time and energy to the lesson-study process. Even when teachers step up to take more and more responsibility, the principal must remain active in maintaining the school's long-term commitment to the process. This means that the principal needs to understand and believe in the six principles that we proposed in the previous chapter. The principal must expect improvement to be gradual and continual, which will not be easy. The principal will need the support of other principals in the district as he or she rethinks current in-service practices and finds creative ways to institutionalize the structures and support necessary for this process to become a new way of doing business.\n\nOf course, building a culture and tradition of lesson study in schools cannot depend solely on the principal. Teachers must begin to assume more and more leadership and responsibility. There are two leverage points in the U.S. educational system for encouraging this to happen. One is preservice education. Currently there are no teacher-preparation programs, of which we are aware, that engage students in a collaborative lesson-study experience. Thus, lesson study is a new concept for teachers entering the profession. If undergraduate methods courses were restructured to introduce students to collaboratively planning and testing lessons, new teachers would be ready to assume leadership roles more quickly. Collaboratively planning lessons means something very different from the traditional American exercise of writing lesson plans (an exercise with which most preservice teachers are too familiar). As was described in Chapter 7, lesson study includes setting clear learning goals, selecting and sequencing tasks by anticipating students' responses (and doing research, if necessary, to identify these responses), and planning the class discussions to build on students' responses and highlight the major points. These are challenging activities, but many undergraduate methods courses would benefit from taking them seriously.\n\nAs we noted earlier, preservice education, no matter how effective, cannot by itself produce continuous improvement. Continuous improvement requires ongoing opportunities to learn and improve while teaching. A second leverage point for jump-starting the lesson-study process is located in the field, where teaching occurs. As a program of school-based lesson study is getting started, it might help to bring in outside consultants to provide initial momentum and guidance. Even in Japan, where there is a fifty-year tradition of lesson study, many study groups work with an outside consultant. Such arrangements can provide additional focus and new ideas for the group. It must be remembered, however, that eventually teachers must lead the process and consultants must be just that\u2014consultants. This means that there needs to be a strategic plan for preparing and supporting teachers to become leaders of lesson-study groups.\n\nDistricts might begin the process of lesson study by inviting two teachers from each school to work intensively with a consultant who has the knowledge and skills to develop school leaders for lesson study. These teacher leaders, over time, would start lesson-study groups within their own schools but maintain their relations with leaders in other schools. Each school is different, and we cannot predict where the leaders will come from in any school. But because improving students' learning must be the number-one concern of each school's principal, the principal must assign priority to developing leadership for the improvement of teaching.\n\nMaking It Work\n\nLesson study is, at its core, a teacher activity. Teachers must make it work. True, it is impossible for teachers to initiate and sustain a vigorous program of lesson study without the active support of the school board, superintendent, principal, and parents. But the success of the activity ultimately depends on teachers.\n\nThe actual process of lesson study might take a variety of forms, depending on the mission and goal of the school, the learning goals set for students, the teachers' interests, and so on. However, some general guidelines can be suggested to keep the activity focused on the primary goal of improving classroom lessons to increase students' learning.\n\nWe already have identified time as an essential requirement. For teacher groups to make measurable progress in their efforts to improve lessons, they need two hours per week of uninterrupted study. This must be a priority in school scheduling.\n\nGroups can be formed on the basis of shared interests, shared problems, common curriculum expectations, or other criteria that make sense for planning common lessons. The school's sixth-grade mathematics teachers might form a group to design several lessons on introducing ratio and proportion. Or interested teachers of grades seven and eight might form a group to plan a sequence of lessons on the democratic election process. All the groups should have explicit goals that are consistent with the mission of the school. Groups of three to five members are ideal. Schools might announce the study groups for the year, perhaps following a planning phase. Teachers might sign up for a group of their choice or might be asked to serve on a group by the principal or a teacher committee.\n\nDeciding on the goals for lesson-study groups is an important first step. Many factors might come into play during this planning stage, but one important factor always should be teachers' judgments of the problems that impede students' learning. The goal of lesson study is to improve students' learning, so it makes sense to begin by addressing those topics and issues that are most in need of improvement. The problems students face might be formidable, but they are not random. For example, most fourth-grade teachers would report problems in their students' understanding of fractions. When teachers within a district are working with a shared curriculum, they will find commonality in the problems they face for given topics and given grade levels. Identifying such problems is an ideal way to start the lesson-study process.\n\nOnce they have identified a common problem, the group needs to select a particular lesson in which that problem can be addressed. Usually this will be a lesson they have taught before but with which they are not satisfied. The group must then spend time in the beginning clarifying in detail what the learning goal for the lesson is and how they will know that they have achieved it. They also should try to understand how the goal of the lesson fits into district and state standards and assessment systems. Neither the problem nor the learning goal needs to be overly ambitious. The aim is to produce small but solid improvements in one or two lessons with modest goals. Japanese teachers, who have engaged in this process for many years, often redesign only two or three lessons over the year.\n\nOnce a lesson has been identified, the group can develop an overall work plan for the year. The plan for the year should allocate initial time for finding out more about how students learn the concepts in question, determining what others recommend for teaching these concepts effectively, and learning more about the concepts themselves. The balance of the plan should then include time for designing improved lessons, trying them out with students, collecting information regarding their success, revising the lessons accordingly, and repeating the cycle. The plan should also include time for the group to write a report of their work to be shared with colleagues.\n\nNotice that part of the annual activity is researching and discussing others' recommendations for effective teaching of particular topics. This activity can resolve a long-standing problem for American educational reform. As we noted earlier, simply distributing written documents compiled by researchers and curriculum developers has been notoriously unsuccessful for improving practice. It is not that the written documents are deficient; they might contain excellent ideas. The problem is that the American system makes no provision for teachers to digest these recommendations and translate them into practice. In our opinion, lesson study is an ideal process for gradually working through new recommendations and giving them life in the classroom.\n\n## Building an Infrastructure  \nfor Sharing Professional  \nKnowledge\n\nIf we implement lesson study, we will have started the process of continuous improvement. The next challenge is to move from improving teaching practice in individual classrooms and individual schools to improving teaching across the country. The key questions are: \"How do teachers, collectively, improve the common standard of teaching? How do they build a professional knowledge base for teaching?\" The answers lie in finding ways for teachers to share what they are learning in their individual study groups. Large-scale improvements, over time, in the quality of classroom instruction can result only if teachers can effectively communicate their discoveries to their colleagues, both present and future.\n\nFinding effective ways to share knowledge about teaching is not a simple task. Previous approaches to this problem, such as codifying principles about teaching in reports and policy documents, have led to disappointing results. As we found in the TIMSS video study, teachers rarely interpret such documents in the ways intended by the authors, and have difficulty adapting the ideas into the realities of classroom life. On the other hand, approaches that stress the sharing of specific tips and techniques often lead to superficial changes that are disconnected from students' learning. In the next section we argue that one of the lasting benefits of lesson study is that the knowledge that results from lesson study is uniquely sharable.\n\nTheories Linked with Examples\n\nAs was noted in the previous chapter, Japanese lesson-study groups have several ways of communicating the results of their work. The participating teachers teach some of their lessons publicly, inviting other teachers to watch. They also write case reports that describe the sequence of plans, outcomes, and revisions that their group has gone through. These case reports are then accumulated into larger books or stored in archives for easy access by other teachers.\n\nThe knowledge contained in these reports is quite different from what one finds in U.S. books about teaching. It is not made up of principles devoid of specific examples or examples without principles. It is theories linked with examples. This kind of knowledge is notable in several respects. First, theoretical insights are always linked with specific referents in the classroom. When a lesson-study group reports, for example, that one of its hypotheses has been supported, it is never outside the context of a specific lesson with specific goals, materials, students, and so on, all of which would be described in the report. At the same time, specific suggestions of how to teach are always justified by the teachers' theoretical analysis of what they have been investigating in their lesson-study group. This blend of theory and example gives other teachers the information they need to relate the group's work to their own classrooms and to think through what might be different in a different context.\n\nThe Japanese lesson-study groups' case reports also contain information about students' responses to the lessons the group has taught. We mention this not only because such information turns out to be quite powerful in the context of teacher planning activities but also because it highlights an interesting kind of research that has been well exploited by Japanese educators. It is worth noting that much of American educational psychology has focused on predicting individual responses to various materials and situations but has not had a great deal of success. The Japanese have taken a different approach: Although it is not easy to predict what any particular student will do in a given situation, it is quite easy to predict what a group of forty students will do. This is simply the result of the statistical principle of aggregation. Japanese teachers have ready access to information of the form \"When presented with problem A, 60 percent of students will use Strategy One, 20 percent Strategy Two, 15 percent Strategy Three, and 5 percent some other strategy.\"\n\nThe fact that lessons are the unit of study is also important for the value of the case reports. The lesson is a unit that has ecological validity for teachers. Lessons are the smallest unit that maintains the complex and systemic properties of teaching. Quality instruction cannot be described with a list of features: there is no feature (be it real-world problems, concrete representations, or any other) that is always beneficial. Which features are the right ones depends on the context, and the context is the lesson in which the features are embedded. Lessons also are linked directly with the curriculum that teachers are using to guide instruction. This linkage, more than anything else, makes the results of lesson study of immediate interest to other teachers using the same curriculum. It is why lesson study is so powerful in Japan, where there exist specific national goals for each course of study, and it is why we believe that if lesson study is to succeed in the United States, it will be implemented within school districts that have adopted a comprehensive and common curriculum.\n\nAccumulating Knowledge About Teaching\n\nThe kind of knowledge produced by lesson-study groups is quite different from that traditionally produced by education researchers. Traditional research results are first validated in studies, then communicated in research journals to educators, who must then figure out how they apply to the classroom. The kind of knowledge produced by lesson study is accumulated in a different way. Because lesson study is carried out in classrooms, the problem of applying the findings to classrooms disappears. The application is direct and obvious. But we cannot immediately know how generalizable lesson-study results are across different teachers, schools, and children.\n\nThe fact that lessons linked with curricula are sharable, however, gives us a means of assessing the generality of findings. A first test concerns the ease with which lessons can be shared when they are passed through the filter of language. If teachers can describe lessons to other teachers in sufficient detail so that the other teachers can actually use the lessons, we can be fairly certain that both groups understand the essential characteristics of the lesson. Lessons that work across diverse groups of teachers can be said to generalize simply because they can be replicated.\n\nThe Role of Technology\n\nThe process of accumulating knowledge about teaching will be greatly enhanced by technology. The most useful information that can be shared about teaching includes examples of classroom lessons linked to evolving theoretical understandings of teaching. Instead of describing these lesson examples with reports such as those that Japanese teachers write, imagine if we could store them in large digital libraries that could link together video, audio, images of student work, and commentary by researchers and others into a single integrated database.\n\nIt is now possible to store video examples and related data on video servers that can be accessed over the internet and thus be physically located anywhere in the world. Curriculum developers, for example, could establish large archives of lessons that are organized around the specific structures of their curricula. Teacher groups could work on perfecting lessons, then post the results of their study, including a complete video record, on these large servers. Other teacher groups could access these archives in order to inform their own efforts to improve teaching. They could study archived lessons and actually collaborate interactively over the internet with the teachers who produced the lessons. The resulting discussions could be directly linked with concrete video examples, and over time such discussions would lead to development of a shared language for describing teaching\u2014the way it is and the way it could be.\n\nWe want to emphasize that these distributed databases would not be just collections of lessons that teachers could download and use. In fact, many such libraries of model lessons are already being developed. What we are proposing is far more complex and far more powerful. The goal of studying these lessons would not be to copy them but, instead, to use them as examples from which new theories of teaching can be constructed. For, in our view, the professional teacher is not someone who simply copies what others have done but is, rather, one who reflects on and improves on what others have done, working to understand the basis of these improvements.\n\nBuilding a national infrastructure for accumulating and sharing knowledge about teaching is something that should be done by the federal government in collaboration with curriculum developers and textbook publishers. When a plane crashes, the investigation is not left to local authorities. The reason for this is clear: The public has too much at stake to not allow everyone to learn from every crash. When problems are discovered, they are widely shared, together with possible solutions, so as to prevent more disasters from happening. The same attention must be paid to knowledge about teaching. Students are too precious to be guinea pigs for teachers forced to learn by themselves on the job. Creating knowledge that can be used, and then sharing that knowledge with each new generation of teachers, should be a high priority of our national education authorities.\n\n## Conclusion\n\nIn a book on teaching, it might be surprising that we have not recommended a particular way to teach. Instead, we have proposed the establishment of a national research-and-development system for the improvement of classroom teaching. Improving teaching, as we have said, is a cultural change and thus must happen in small steps. In the same way, building the kind of system we have proposed requires that we change the ways that teachers learn, which is also a cultural activity. This, too, will change slowly and in small steps. But that is not a bad thing. The system we have proposed is both modular and scalable. It can happen one district at a time and can evolve over many years. The important thing is that the system we are proposing is one that we desperately need if we are to look back a hundred years from now and recognize a history of gradual progress in improving teaching.\n\nAlthough constructing this system will be difficult, it should cause some current problems with educational reform to diminish. Take, for example, the highly charged political battles that are now being fought in California and elsewhere about the kind of mathematics instruction our children should be exposed to in school. Our earlier analyses of how policy affects practice reveal that regardless of who wins these battles, there is a good chance that little of any substance will change inside classrooms. But more to the point, the system we are proposing leaves the question of how best to teach up to the system itself to resolve, over time, through careful experimentation and study. Public debate is the only way to decide on the goals that will guide our educational system. But once the goals are decided, the best way we can help improve our children's education is not to continue taking sides on these explosive issues but to fight to institute some mechanism for improvement.\n\nProgress will be slow. For the sake of the children we can hasten the start toward better teaching by taking the first step toward the process of continual improvement. But that quick step is only the beginning of a long journey.\n\n## CHAPTER 10\n\n## The True Profession  \nof Teaching\n\nTHE SYSTEM FOR improving teaching that we have just presented challenges school districts, school boards, superintendents, principals, and parents to undertake a new, long-term process of improvement, but it places primary responsibility for the process squarely on teachers. The success of the system depends on teachers' initiative, creativity, and professional commitment. Some would question whether teachers are up to the task, whether they can be handed the job of improving their own teaching. There is, in our society, a widespread lack of confidence in teachers. When students' achievement scores are below expectations, and when stories of students' failures fill the media, teachers often are blamed for the problems. More than that, they are ignored when we look for solutions. Rather than turning to teachers for leadership and guidance, we look to business executives, to political leaders, or to so-called educational experts for the answers. Given this lack of trust, it is no wonder that, as we noted in Chapter 1, the current reforms leave teaching almost completely out of the equation.\n\nThe lack of confidence in teachers is not limited to public and political communities. Even educators display a certain skepticism of teachers' inclination or ability to improve teaching. Over the years, curriculum developers often have tried to create \"teacher-proof\" curricula\u2014content that is to be presented to students in such a straightforward way that it could not be distorted by incompetent teachers. There is also a long-standing degree of distrust between administrators and teachers, illustrated by the fact that principals usually observe teachers only when it is time to evaluate them. Teachers, in turn, take a very suspicious view of being observed. This mistrust ruins one of teachers' richest learning opportunities\u2014the opportunity to observe the practice of others and be observed yourself.\n\n## A Popular Solution:  \nProfessionalize Teachers\n\nThe perception that teachers are not up to the task of improving teaching and solving the country's educational problems is often captured in one short phrase: \"Teachers are not professionals.\" To combat this attack, some defenders have launched a counteroffensive. Teachers, they say, are unfairly blamed for students' failures. It is not the teachers' fault that students perform poorly or learn less than we expect. Poverty, demographic changes, an erosion of traditional values, and a breakdown of supportive families are among the real causes, yet society unfairly targets teachers because teaching has not been granted the status of other professions, like medicine or law or business. Society thinks that anyone can be a teacher, that little expertise is required. Because teachers are not fully appreciated for what they do, they are vulnerable to public attacks. To solve this problem, say their defenders, society should demand that teachers be given higher status and be treated as real professionals.\n\nThere are many ideas about how to turn teachers into high-status professionals: increased pay, increased certification requirements, more accountability, career ladders, peer review, training teachers as researchers, and encouraging teachers themselves to set the standards for entrance into their profession. Not all of these stratagems are proposed by teachers' advocates, but they do have one thing in common: They presume that attributing to teachers the characteristics common to professionals in other fields will bring higher status and respect.\n\nWe believe, however, that attacking the problem simply by arbitrarily assigning professional characteristics to teachers mistakes the trappings for the profession. In fact, a profession is created not by certificates and censures but by the existence of a substantive body of professional knowledge, as well as a mechanism for improving it, and by the genuine desire of the profession's members to improve their practice.\n\n## Redefining the Problem\n\nIf the desire to improve is a key to build a true profession, then what is standing in the way? Some would say the answer is obvious\u2014teachers do not have this desire, or at least they have not shown it. After all, teachers have not changed the way they teach for almost a century. They continue teaching in traditional ways despite regular waves of educational reform. But before condemning teachers, consider what we have learned about teaching in the previous chapters. We can now see that most of our nation's problems with teaching arise from the script for teaching that has evolved in this country and from the absence of a mechanism for changing it.\n\nThe TIMSS videotapes show most American teachers teaching in much the same way\u2014true to the American script. Many factors\u2014students, parents, administrators, school structure, and more\u2014maintain this cultural script, keep it in place, and discourage teachers from questioning it. The perception that they are unable to improve is premature at best; it is wrongheaded and destructive at worst.\n\nSuppose teachers really do wish to improve how they teach. What can they do? Under normal circumstances, they can become better at using the teaching script they already have learned. But without some new ways of examining teaching, they are unlikely to identify alternatives that could improve students' learning even more. The environments in which most teachers work have been structured in ways that actually work against the kind of sustained collaboration that we have suggested is needed for significant and steady improvement. So instead of assuming that teachers do not want to improve, we take a different view. We believe that the real problem lies in the teaching, not the teachers, and in the absence of resources available to help teachers improve how they teach.\n\n## How Did We Get to This Point?\n\nWhy is it that we in the United States have for so long accepted a script for teaching that might be inadequate, and why is there no system for improvement? An interesting turn of events in the early 1900s provides some insights into the problem. John Dewey, noted educator and philosopher, had shaped the laboratory school at the University of Chicago into a hotbed of educational improvement. This one school became a microcosm of the system we outlined in Chapters 8 and 9. Teachers and researchers, through collaborative planning and experimenting, developed knowledge of effective classroom practice and fed this knowledge back into the system. The lines between teachers and researchers were blurred; all were engaged in learning about teaching and how to improve it in the context of real classrooms.\n\nBut then things changed. Dewey left Chicago, and Charles Judd, Dewey's replacement, proposed a new method of improvement, one designed to bring the prestige and rigor of science to education. Central to Judd's approach was a distinction between scientists or researchers, who develop the best ideas, and teachers, who apply them. Judd was not alone. Many experts, including researchers such as Edward Thorndike, were impressed with the advances in science and wished to make education more scientific. The way to do this, they said, was to divide the labor: Researchers would discover the best methods and teachers would implement them in classrooms.\n\nOver time, the views of Judd and others won the day. Research became a specially designated activity, distinct from teaching, and researchers became differentiated from classroom teachers. They even located themselves in different places\u2014researchers moved to universities, and teachers stayed in schools.\n\nThe same distinction between research and practice continues today, perhaps even more strongly. A large gulf separates researchers and classroom teachers. Researchers work on better ways to teach and then hope that their findings will be applied by classroom teachers. We have pointed out in earlier chapters that this process has had little effect on standard teaching practice.\n\nBecause of the high status usually assigned to acquiring knowledge and the low status assigned to applying it, this distinction strongly reinforces the low professional status of teachers. And this is a distinction created and sustained within the educational community!\n\nWhat is most damaging about the distinction, however, is that it prevents our country from implementing a system, sustained by teachers, for improving teaching. By hanging on to the research\/teaching distinction, the educational world has robbed teachers of the opportunity to participate in the development of new knowledge about teaching. The U.S. educational establishment has been unable to envision a system that gives teachers the freedom and the responsibility to acquire and apply the knowledge needed to improve teaching over the long run. What is tragic about this is that there are no real alternatives. Teachers must be at the heart of the solution. Not only are they the gatekeepers for all improvement efforts, they are also in the best position to acquire the knowledge that is needed. They are, after all, the only ones who can improve teaching. Proposals that do not recognize this basic truth cannot succeed.\n\nClearly, blaming teachers for not improving teaching is unfair. It is unfair because, in fact, teachers have been encouraged, simply by virtue of their membership in our culture, to teach the way they do. It is unfair because they have been provided no system in which they might spend time and energy studying and improving teaching. But solving these problems does not require high-pitched rhetoric and protestations by defenders of teachers, claiming for them higher professional status. In fact, solving the problem of improving teaching requires shifting the focus from teachers to teaching. Instead of worrying about professionalizing teachers, we must think about what is required to professionalize teaching.\n\n## A Lasting Solution:  \nProfessionalize Teaching\n\nAs we noted in Chapter 8, a common American view of the classroom says that good teaching is an individual trait. Change the teacher, and the quality of teaching changes. Some teachers are effective and some are not, so the way to improve teaching is to recruit better teachers. But this perspective ignores a central truth about teaching: If the method is limited, students' learning will be limited no matter how talented the teacher. Teachers are only as good as the methods of teaching they use.\n\nBecause of America's focus on teachers, we tend to look to individual innovators for signs of improvement. Because the usual methods of teaching are recognized as uninspiring, we look to individuals who have figured out clever ways around these standard methods. We shine spotlights on the unusual, the variations from standard practice. These heroic individual teachers stand out from the crowd and are applauded by reformers for innovation and creativity. They are given special awards. The standard, routine way of teaching is treated as just routine.\n\nCelebrating individual innovations is fine, but individual innovations will never improve teaching in the average classroom. They cannot do so because they do not change standard practice. And if we hope to improve the practice of the profession, it is the standard, common practice that must improve. Sporadic end runs around the standard methods are not the answer; what is required is a steady, continuing effort to gradually improve the standard ways in which we teach.\n\nIn true professions, standard practices hold the wisdom of the profession. It is when the standard practices of teaching are improved that a profession begins to emerge. Some might worry that trying to improve standard practice limits teachers' individual creativity and spontaneity. But listen, again, to Albert Shanker as he testified before the U.S. House of Representatives' Committee on Economic and Educational Opportunities: \"Doctors don't try to figure out a new technique or procedure for every patient who comes into their office; they begin by using the standard techniques and procedures based on the experience of many doctors over the years. Nobody considers this a way of doctor-proofing medicine, although they do have a name for the failure to use standard practices\u2014it's malpractice. The standard practices that all doctors (and other professionals) use contain the wisdom of the profession.\"\n\nAmerica's script of teaching, the repository of its standard practices, has remained unexamined for too long. Teaching has not improved in this country because educators and parents have not provided a way for teachers to improve the script they use. It is time to begin building a research-and-development system like the one outlined in Chapters 8 and 9 to enable teachers to study teaching and to begin the long, steady process of improving the standard practice of the profession.\n\nPerhaps it is ironic that professional status for teachers will come only when the focus shifts away from teachers and onto teaching. But that is what is required. When teachers have a way to act on their desire to improve, when they can point to increases in students' learning over time, and when they can disseminate into standard practice the improvements in teaching that are responsible, teaching will be on its way to becoming a true profession. And when teaching becomes a profession, teachers will inherit the professional badges that come from being members.\n\n## A Story from the Past\n\nOur belief that teaching can become a true profession in the United States is bolstered by the fact that it has, at times, been a profession. No, not across the entire country and not for all time, but in some places and for brief periods. When teachers have seized unusual opportunities and taken on the task of improving teaching in their school or district, remarkable things have happened. A compelling and instructive example is related by William Johnson. At the turn of the century, the Baltimore city schools were in trouble. Physical conditions were deplorable, and the instructional methods used by many teachers were unenlightened, to say the least. Joseph Mayer Rice, an informed observer of America's schools, published a series of articles in 1892 that focused on the conditions of schools in America's cities. The first article was titled \"Evils in Baltimore.\"\n\nBut a group of Baltimore teachers, interested in improving classroom practices, convinced the Baltimore city public schools teachers' association to lobby the city school commissioners for time to meet and work together to improve teaching and learning in their classrooms. They wanted one afternoon each month to meet with other teachers. Teachers around the city who taught the same grade met to discuss methods of teaching that would facilitate improved student learning. Discussions centered on common problems of classroom practice. Wishing to share their insights and progress more widely, the city teachers began publishing a regular newsletter. The Expert Pedagogue, which was filled with suggestions for improved teaching.\n\nAfter several years of teacher meetings and newsletter publication, the district made an administrative decision that was to have a fatal impact on the burgeoning profession of teaching in Baltimore. In 1900, a new superintendent was hired. Eager to bring new, progressive ideas into the district, the superintendent reorganized the form of in-service education. The monthly meetings of teachers were replaced with presentations by educational experts. The presentations, given weekly and after hours, were mostly demonstrations of model lessons developed by the \"experts.\" Having no forum for continuing collaboration, the teachers' newsletter soon disappeared; arising in its place was the Maryland Educational Journal, produced by experts to promote progressive practices. Teachers were back in their role of trying to implement the recommendations of others. Business had returned to normal. William Johnson summarized this story by noting the irony: Teachers in Baltimore had begun the long journey to improve classroom practice and to professionalize teaching. The new superintendent entered with the same goals, but, not recognizing the central role that teachers must play in this process, he squelched the very movement he was trying to build. For a brief moment, however, teaching in the Baltimore schools had been on its way to becoming a true profession.\n\nThis story of what happened in Baltimore, along with the stories of what is happening now in a number of schools and districts scattered around the country, shows that building a profession of teaching in the United States is not an impossible dream. But it will take commitment\u2014and a vision.\n\n## A Vision of the Future\n\nThe star teachers of the twentieth century have been those who broke away from the crowd and created different and unusual methods of teaching. They distinguished themselves by being different, by leaving the standard practice behind. They gained fame by rising above the routine and showing the effectiveness of alternative forms of teaching. Although these efforts won the applause of educational critics, they did not have much effect on standard practice.\n\nThe star teachers of the twenty-first century will be those who work together to infuse the best ideas into standard practice. They will be teachers who collaborate to build a system that has the goal of improving students' learning in the \"average\" classroom, who work to gradually improve standard classroom practices. In a true profession, the wisdom of the profession's members finds its way into the most common methods. The best that we know becomes the standard way of doing something. The star teachers of the twenty-first century will be teachers who work every day to improve teaching\u2014not only their own but that of the whole profession.\n\n## Notes\n\nPREFACE\n\n  1. Husen, T. (1967). International study of achievement in mathematics. New York: Wiley; McKnight, C. C.; Crosswhite, F. J.; Dossey, J. A.; Kifer, E.; Swafford, J. O.; Travers, K. J.; and Cooney, T. J. (1987). The underachieving curriculum: Assessing U.S. school mathematics from an international perspective. Champaign, Ill.: Stipes.\n\nCHAPTER 1 : THE TEACHING GAP\n\n  1. On October 27, 1998, President Clinton signed into law the Labor, Health, and Human Services bill, which provided $1.2 billion to help local school districts hire and pay the salaries and benefits of more than 30,000 additional teachers. The funds are a down payment on the president's plan to hire 100,000 new teachers over seven years to reduce class size in grades one to three to a national average of eighteen. For details on the Class Size Initiative, see http:\/\/www.ed.gov\/PressReleases\/10-1998\/class.html.\n  2. Ample evidence exists to show that most reform efforts rarely change fundamentally what happens inside classrooms. For example: Cuban, L. (1993). How teachers taught: Constancy and change in American classrooms, 1890-1990, 2nd ed. New York: Teachers College Press; Educational Evaluation and Policy Analysis (1990). Vol. 12, No. 3 [special issue]; Griffin, G. A. (1995). Influences of shared decision making on school and classroom activity: Conversations with five teachers. Elementary School Journal 96, 29-45; Stake, R., and Easley, J. (eds.). (1978). Case studies in science education. Urbana, Ill.: University of Illinois; Tyack, D., and Cuban, L. (1995). Tinkering toward Utopia: A century of public school reform. Cambridge, Mass.: Harvard University Press.\n  3. Bruner, J. (1996). The culture of education. Cambridge, Mass.: Harvard University Press. P. 86.\n  4. Berliner, D. C., and Biddle, B. J. (1995). The manufactured crisis: Myths, frauds, and the attack on America's public schools. New York: Addison Wesley; Reynolds, A. J., and Walberg, H. J. (1992). A process model of mathematics achievement and attitude. Journal for Research in Mathematics Education 23, 306-28.\n  5. Slavin, R. E. (1996). Reforming state and federal policies to support adoption of proven practice. Education Researcher 25 (9), 4-5 (p. 4).\n  6. Berliner, D. C., and Biddle, B. J. (1995). The manufactured crisis: Myths, frauds, and the attack on America's public schools. New York: Addison Wesley; Hirsch, E. D. Jr. (1996). The schools we need: And why we don't have them. New York: Doubleday.\n  7. Wearne, D., and Kouba, V. L. (in press). Rational numbers. In E. A. Silver and P. A. Kenney (eds.), Results from the seventh mathematics assessment of the National Assessment of Educational Progress. Reston, Va.: National Council of Teachers of Mathematics. The relatively poor preparation of students in mathematics and science has real consequences. On September 24, 1998, the U.S. House of Representatives approved a bill allowing an additional 142,500 foreign skilled workers to enter the country, thus exempting them from normal immigration quotas. The bill was designed to meet the needs of high-technology industries that are unable to find adequate skilled workers among the U.S. population.\n  8. Stevenson, H. W., and Stigler, J. W. (1992). The learning gap: Why our schools are failing and what we can learn from Japanese and Chinese education. New York: Simon and Schuster.\n  9. Husen, T. (1967). International study of achievement in mathematics. New York: Wiley; McKnight, C. C.; Crosswhite, F. J.; Dossey, J. A.; Kifer. E.; Swafford, J. O.; Travers, K. J.; and Cooney, T. J. (1987). The underachieving curriculum: Assessing U.S. school mathematics from an international perspective. Champaign, Ill.: Stipes.\n  10. National Center for Education Statistics. (1996). Pursuing excellence: Initial findings from the Third International Mathematics and Science Study. Washington, D.C.: U.S. Department of Education.\n  11. California Department of Education. (1987). English\u2014Language Arts framework for California public schools. Sacramento: California Department of Education. (1992). Mathematics framework for California public schools. Sacramento: California Department of Education.\n  12. See, for example, an exchange in the New York Times, August 11, 1997, p. A19 (\"Creative Math or Just Fuzzy Math\"), and responding letters to the editor, August 17, 1997, p. E14.\n\nCHAPTER 2: METHODS FOR  \nSTUDYING TEACHING IN GERMANY,  \nJAPAN, AND THE UNITED STATES\n\n  1. Readers interested in more detailed descriptions of the full TIMSS study should consult http:\/\/nces.ed.gove\/timsspublist.html .\n  2. A full description of the methods used in conducting the video study is beyond the scope of this book. Interested readers should consult Stigler, J. W.; Gonzales, P.; Kawanaka, T.; Knoll, S.; and Serrano, A. ( 1999). The TIMSS videotape classroom study: Methods and findings from an exploratory research project on eighth grade mathematics instruction in Germany, Japan, and the United States. Washington, D.C.: National Center for Education Statistics (www.ed.gov\/NCES).\n  3. The original intent was to videotape one hundred classrooms in each country. The no-substitution rule eliminated nineteen classrooms in the United States, and the Japanese collaborators decided that fifty classrooms would be sufficient in their country because of its relatively small size and homogeneity.\n\nCHAPTER 3: IMAGES OF TEACHING\n\n  1. All names, in this lesson and all the others reported in this chapter, are fictional; however, the correct gender is retained.\n\nCHAPTER 4: REFINING THE IMAGES\n\n  1. Schmidt, W. H.; McKnight, C. C.; and Raizen, S. A. (1996). A splintered vision: An investigation of U.S. science and mathematics education. Boston: Kluwer Academic Publishers.\n  2. All cross-national differences reported in this chapter are supported by thorough statistical analyses. Interested readers should consult Stigler, J. W.; Gonzales, P.; Kawanaka, T.; Knoll, S.; and Serrano, A. (1999). The TIMSS videotape classroom study: Methods and findings from an exploratory research project on eighth grade mathematics instruction in Germany, Japan, and the United States. Washington, D.C.: National Center for Education Statistics (www.ed.gov\/NCES).\n  3. National Center for Education Statistics. (1996). Pursuing excellence: A study of U.S. eighth-grade mathematics and science teaching, learning, curriculum, and achievement in international context. Washington, D.C.: U.S. Department of Education.\n  4. Several studies have shown that students identify and remember the major points of more coherent lessons better than those of less coherent lessons [Fernandez, C.; Yoshida, M.; and Stigler, J. W. (1992). Learning mathematics from classroom instruction: On relating lessons to pupils' interpretations. Journal of the Learning Sciences 2(4), 333-65. Yoshida, M.; Fernandez, C.; and Stigler, J. W. (1993). Japanese and American students' differential recognition memory for teachers' statements during a mathematics lesson. Journal of Educational Psychology 85, 610-17].\n  5. This conclusion matches precisely the one arrived at by looking directly at the curricula commonly used in each country [Schmidt, W. H.; McKnight, C. C.; and Raizen, S. A. (1996). A splintered vision: An investigation of U.S. science and mathematics education. Boston: Kluwer Academic Publishers].\n  6. The Math Group was led by Professor Alfred Manaster of the University of California at San Diego. Other members of the group were Wallace Etterbeek, Phillip Emig, and Barbara Wells. For additional analyses conducted by the Math Group, see Manaster, A. B. (1998). Some characteristics of eighth grade mathematics classes in the TIMSS videotape study. American Mathematical Monthly, 105, 793-805.\n  7. See, for example, Doyle, W. ( 1983). Academic work. Review of Educational Research 53, 159-99; Doyle, W. (1988). Work in mathematics classes: The context of students' thinking during instruction. Educational Psychologist 23, 167-80; Schoenfeld, A. H. (1985). Mathematical problem solving. Orlando, Fla.: Academic Press.\n  8. The importance of engaging in creative, inventive work has long been recognized as crucial for developing deep understanding [Piaget, J. (1973). To understand is to invent. New York: Grossman; Resnick, L. B. (1980). The role of invention in the development of mathematical competence. In R. H. Kluwe and H. Spada (eds.), Developmental models of thinking. New York: Academic Press, pp. 213-44] and the benefits of engaging students in analyzing multiple solution methods to mathematical problems are detailed in Hiebert, J., et al. (1997). Making sense: Teaching and learning mathematics with understanding. Portsmouth, N.H.: Heinemann.\n\nCHAPTER 5: TEACHING IS A SYSTEM\n\n  1. Teachers in 57 percent of U.S. lessons used the overhead projector, in 67 percent, the chalkboard. In Japan, the percentages were 6 and 100, respectively. See Stigler, J. W; Gonzales, P.; Kawanaka, T.; Knoll, S.; and Serrano, A. (1999). The TIMSS videotape classroom study: Methods and findings from an exploratory research project on eighth grade mathematics instruction in Germany, Japan, and the United States. Washington, D.C.: National Center for Education Statistics (www.ed.gov\/NCES).\n  2. Japanese teachers also used something we called \"posters\": prepared statements of solution methods or principles that they attached to the chalkboard at critical points during the lesson to label students' work (which they apparently anticipated) or to summarize major ideas. Posters are fully consistent with a system in which visual aids provide records of instructional tasks, solution methods, and major ideas.\n  3. See, for example, Cuban, L. (1993). How teachers taught: Constancy and change in American classrooms. 1890-1990, 2nd ed. New York: Teachers College Press; Fey, J. (1979). Mathematics teaching today: Perspectives from three national surveys. Mathematics Teacher 72, 490-504; Hoetker, J., and Ahlbrand, W. P., Jr. (1969). The persistence of the recitation. American Educational Research Journal 6. 145-67; Sirotnik, K. A. (1983). What you see is what you get\u2014consistency, persistency, and mediocrity in classrooms. Harvard Educational Review 53, 16-31.\n  4. Lortie, D. C. (1975). Schoolteacher: A sociological study. Chicago: University of Chicago Press; Wideen, M.; Mayer-Smith, J.; and Moon, B. (1998). A critical analysis of the research on learning to teach: Making the case for an ecological perspective on inquiry. Review of Educational Research 68, 130-78; Nemser, S. F. (1983). Learning to teach. In L. Shulman & G. Sykes (eds.), Handbook of teaching and policy. New York: Longman. Pp. 150-70.\n\nCHAPTER 6; TEACHING IS A CULTURAL ACTIVITY\n\n  1. Ronald Gallimore makes many of these same points in \"Classrooms are just another cultural activity.\" In D. L. Speece and B. K. Keogh (eds.). (1996). Research on classroom ecologies: Implications for inclusion of children with learning disabilities. Mahwah, N.J.: Erlbaum. Pp. 229-50. The origins of these ideas can be traced to earlier writings, such as Cazden, C; John, V; and Hymes, D. (eds.). (1972). Functions of language in the classroom. New York: Teachers College Press.\n  2. The same categories of core beliefs have been suggested by other researchers. See, for example, Griffin, S., and Case, R. (1997). Rethinking the primary school math curriculum: An approach based on cognitive science. Issues in Education 3(1), 1-49; Thompson, A. G. (1992). Teachers' beliefs and conceptions: A synthesis of research. In D. A. Grouws (ed.), Handbook of research on mathematics teaching and learning. New York: Macmillan. Pp. 127-46.\n  3. There is a strong American tradition in behaviorist psychology, a psychology that addresses most directly issues of skill learning. Behaviorism, or connectionism, was developed most fully by E. L. Thorndike in the early 1900s and elaborated in different ways by B. F. Skinner and R. M. Gagne.\n  4. The psychology of learning that underlies this approach is familiar in the United States but is not the psychology that has taken hold in everyday teaching in the United States. See, for example, the writings of J. Dewey and J. Piaget and numerous recent works that have elaborated these ideas.\n  5. Kvoshiyo shidosho: Shogakko sansu 5 nen (Teacher's guidebook: Elementary mathematics 5th grade). ( 1991 ). Tokyo: Gakkotosho.\n  6. One item on the questionnaire given to U.S. eighth-grade mathematics teachers in the TIMSS sample asked them to select, among sixteen choices, those that limited their effectiveness in the classroom. The second most frequent choice, just behind lack of student interest, was the range of abilities among students in the same class (selected by 45 percent of the respondents). See also a survey of its members by the American Federation of Teachers, reported in the Spring 1996 (Vol. 20, No. 1) issue of American Educator, pp. 18-21.\n  7. See the following article for an analysis of how the variety of student responses in a Japanese classroom benefits the whole class: Hatano. G., and Inagaki, K. (1991). Sharing cognition through collective comprehension activity. In L. B. Resnick, J. M. Levine, and S. D. Teasley (eds.), Perspectives on socially shared cognition. Washington, D.C.: APA. Pp. 331-48.\n  8. Sasaki, Akira. (1997). Jugyo kenkyu no kadai to jissen (Issues and implementation of lesson study). Tokyo: Kioiku Kaihatsu Kenkyujo.\n  9. A common reform approach in the United States is to try to reform mathematics instruction by adding more features, such as concrete materials or problem solving. Many experts think that the indicators provide the road map we need to improve. See, for example, Stedman. L. C. (1997). International achievement differences: An assessment of a new perspective. Educational Researcher 26 (3), 4-15. After reviewing a few teaching indicators from an earlier international study, Stedman concludes: \"These descriptions of our pedagogical weaknesses are compelling and provide clear directions for reforming our teaching and curricula\" (p. 11 ).\n  10. Cohen, D. (1996). Standards-based school reform: Policy, practice, and performance. In H. F. Ladd (ed.). Holding schools accountable: Performance-based reform in education. Washington, D.C.: Brookings Institution; Guthrie, J. W. (ed.). (1990). Educational Evaluation and Policy Analysis 12 (3), special issue.\n  11. Bosse, M. J. (1995). The NCTM Standards in light of the new math movement: A warning! Journal of Mathematical Behavior 14, 177-201; DeVault, M. V, and Weaver. J. F. (1970). Forces and issues related to curriculum and instruction, K-6. In P. S. Jones (ed.), A history of mathematics education in the United States and Canada: Thirty-second yearbook. Washington, D.C.: National Council of Teachers of Mathematics. Pp. 93-152: Osborne, A. R., and Crosswhite, F. J. (1970). Forces and issues related to curriculum and instruction, 7-12. In Jones, A history of mathematics education in the United States and Canada. Pp. 155-297.\n  12. Freeman, D. J., and Porter, A. C. (1989). Do textbooks dictate the content of mathematics instruction in elementary schools? American Educational Research Journal 26, 403-21; Stodolsky, S. (1988). The subject matters: Classroom activity in math and social studies. Chicago: University of Chicago Press.\n  13. Conference Board of the Mathematical Sciences. (1975). Overview and analysis of school mathematics, K-12. Washington, D.C.: Conference Board of the Mathematical Sciences. The quote was taken from p. 77.\n  14. Leinhardt. G. (1993). On teaching. In R. Glaser (ed.), Advances in instructional psychology, Vol. 4. Hillsdale, N.J.: Erlbaum. Pp. 1-54. Even the most visible and well-intentioned reforms can be misinterpreted as single-feature changes: McLeod, D. B.; Stake, R. E.; Schappelle, B.; Mellissinos, M.; and Gierl, M. J. (1996). Setting the Standards: NCTM's role in the reform of mathematics education. In S. A. Raizen and E. D. Britton (eds.). Bold ventures, Vol. 3: Case studies of U.S. innovations in science and mathematics education. Dordrecht: Kluwer. Pp. 13-132.\n  15. Saxe, G. B.; Gearhart, M.; and Dawson, V. (1996). When can educational reforms make a difference? The influence of curriculum and teacher professional development programs on children's understanding fractions. Unpublished paper.\n\nCHAPTER 7: BEYOND REFORM: JAPAN'S  \nAPPROACH TO THE IMPROVEMENT  \nOF CLASSROOM TEACHING\n\n  1. Most efforts that have focused on improving teaching in a systematic way have been relatively small-scale, experimental efforts that have not gained national prominence outside the research community. With respect to mathematics teaching, see, for example. Fennema, E.; Carpenter, T. P.; and Peterson, P. L. (1989). Learning mathematics with understanding: Cognitively guided instruction. In J. E. Brophy (ed.), Advances in research on teaching, Vol. 1. Greenwich, Conn.: JAI Press, pp. 195-221; and Schifter, D., and Fosnot, C. T. (1993). Reconstructing mathematics education: Stories of teachers meeting the challenge of reform. New York: Teachers College Press. The effort of the National Council of Teachers of Mathematics that we describe here is somewhat unique in its high visibility, at least within the educational community.\n  2. National Council of Teachers of Mathematics. (1991). Professional standards for teaching mathematics. Reston, Va.: National Council of Teachers of Mathematics.\n  3. Albert Shanker told this story at a Pew Forum meeting in Jackson Hole, Wyoming, in July 1996. It was reprinted in a special issue of the American Educator, Spring\/Summer 1997, Vol. 21, Nos. 1 and 2, p. 37.\n  4. Educational Evaluation and Policy Analysis. Vol. 12, No. 3. (1990); Saxe, G. B.; Gearhart, M.; and Dawson, V. (1996). When can educational reforms make a difference? The influence of curriculum and teacher professional development programs on children s understanding of fractions. Unpublished paper; McLeod, D. B.; Stake, R. E.; Schappelle, B.; Mellissinos, M.; and Gierl, M. J. (1996). Setting the Standards: NCTM's role in the reform of mathematics education. In S. A. Raizen and E. D. Britton (eds.). Bold ventures. Vol. 3: Case studies of U.S. innovations in science and mathematics education. Dordrecht: Kluwer. Pp. 13-132.\n  5. Lewis, C., and Tsuchida, I. (1997). Planned educational change in Japan: The shift to student-centered elementary science. Journal of Educational Policy 12, 313-31.\n  6. Shimahara, N. K. (1997). Educational reforms in Japan and the United States: Implications for civic education. Paper presented at the annual meeting of the Educational Research Association of Singapore, Singapore.\n  7. Fewer high schools appear to engage in formalized kounaikenshuu: teacher development tends to be more idiosyncratic and to vary more from school to school. This is probably due, in part, to the departmental specialization of high schools and to the pressures imposed by entrance exam preparations [see Yoshida, M. (1999). Lesson study: An ethnographic investigation of school-based teacher development in Japan. Doctoral dissertation, University of Chicago.]\n  8. Takemura, S., and Shimizu, K. (1993). Goals and strategies for science teaching as perceived by elementary teachers in Japan and the United States. Peabody Journal of Education 68 (4), 23-33; Shimahara, N. K., and Sakai, A. (1995). Learning to teach in two cultures: Japan and the United States. New York: Garland.\n  9. Lewis, C., and Tsuchida, I. ( 1997). Planned educational change in Japan: The shift to student-centered elementary science. Journal of Educational Policy 12, 313-31.\n  10. Lewis, C., and Tsuchida, I. (1997). Planned educational change in Japan: The shift to student-centered elementary science. Journal of Educational Policy 12, 313-31; Lewis, C., and Tsuchida, I. (1998). A lesson is like a swiftly flowing river: How research lessons improve Japanese education. American Educator, 22 (4), 12-17, 50-52; Shimahara, N. K. (1998). The Japanese model of professional development: Teaching as a craft. Teaching and Teacher Education, 14, 451-62; Shimahara, N. K., and Sakai, A. (1995). Learning to teach in two cultures: Japan and the United States. New York: Garland; Yoshida, M. (1999). Lesson study: An ethnographic investigation of school-based teacher development in Japan. Doctoral dissertation, University of Chicago. Special issue of the Peabody Journal of Education 1993, Vol. 68, No. 4.\n  11. Orihara, Kazuo (ed.). ( 1993). Shogakko: Kenkyu Jugyo no susume kata mikata (Elementary school: Implementing and observing research lessons). Tokyo: Bunkyo-shoin.\n  12. Descriptions of the common professional development opportunities for American teachers can be found in Cohen, D. K., and Hill, H. C. (1998). Instructional policy and classroom performance: The mathematics reform in California. Ann Arbor: University of Michigan; Lubeck, S. Teachers and the teaching profession in the United States. In The education system in the United States: Case study findings, draft vol. Ann Arbor: University of Michigan Center for Human Growth and Development. Pp. 241-318; Weiss, I. (1994). A profile of science and mathematics education in the United States: 1993. Chapel Hill, N.C.: Horizon Research, Inc. There are, however, some local schools and districts where teachers have been given much richer opportunities to improve teaching. See, for example. Cognition and Technology Group at Vanderbilt. (1997). The Jasper project: Lessons in curriculum, instruction, assessment, and professional development. Mahwah, N.J.: Erlbaum; Elmore, R. F.; Peterson, P. L.; and McCarthey, S. J. (1996). Restructuring in the classroom: Teaching, learning, and school organization. San Francisco: Jossey-Bass; Franke, M. L.; Carpenter, T. P.; Fennema, E.; Ansell, E.; and Behrend J. (1998). Understanding teachers' self-sustaining, generative change in the context of professional development. Teaching and Teacher Education 14(1), 67-80; Stein, M. K.; Silver, E. A; and Smith, M. S. (1998). Mathematics reform and teacher development: A community of practice perspective. In J. Greeno and S. Goldman (eds.), Thinking practices in mathematics and science learning. Mahwah, N.J.: Erlbaum; and Swafford, J. O.; Jones, G. A.; and Thornton, C. A. (1997). Increased knowledge in geometry and instructional practice. Journal for Research in Mathematics Education 28, 467-83. It is interesting that the activities described in these reports share many features with Japanese lesson study. It also is the case that successful teaching-improvement programs in other countries contain some of these elements. See, for example. Paine. L., and Ma, L. (1994). Teachers working together: A dialogue on organizational and cultural perspectives of Chinese teachers. International Journal of Educational Research 19 (8), 675-98.\n  13. Gallimore, R. (1996). Classrooms are just another cultural activity. In D. L. Speece and B. K. Keogh (eds.), Research on classroom ecologies: Implications for inclusion of children with learning disabilities. Mahwah, N.J.: Erlbaum. Pp. 229-50. Quote taken from p. 232.\n  14. Lortie, D. C. (1975). Schooltcacher: A sociological study. Chicago: University of Chicago Press.\n  15. Lewis, C. C. (December 1997). Improving Japanese science education: How \"research lessons\" build teachers, schools, and a national curriculum. Paper presented at the Conference on Mathematics and Elementary Science Education, Berlin, Germany. P. 13.\n  16. Lewis, C. C. (December 1997). Improving Japanese science education: How \"research lessons\" build teachers, schools, and a national curriculum. Paper presented at the Conference on Mathematics and Elementary Science Education, Berlin, Germany.\n  17. Lewis. C. C. (December 1997). Improving Japanese science education: How \"research lessons\" build teachers schools, and a national curriculum. Paper presented at the Conference on Mathematics and Elementary Science Education, Berlin, Germany. P. 3.\n\nCHAPTER 8: SETTING THE STAGE  \nFOR CONTINUOUS IMPROVEMENT\n\n  1. Tyack, D., and Cuban, L. (1995). Tinkering toward utopia: A century of public school reform. Cambridge, Mass.: Harvard University Press.\n  2. Gallimore, R. ( 1996). Classrooms are just another cultural activity. In D. L. Speece and B. K. Keogh (eds.), Research on classroom ecologies: Implications for inclusion of children with learning disabilities. Mahwah, N.J.: Erlbaum. Pp. 229-50.\n  3. Joyce, B.; Wolf, J.; and Calhoun, E. (1993). The self-renewing school. Alexandria, Va.: Association for Supervision and Curriculum Development. Quote taken from p. 20.\n  4. Joyce, B.; Wolf, J.; and Calhoun, E. ( 1993). The self-renewing school. Alexandria, Va.: Association for Supervision and Curriculum Development. Quote taken from p. 19.\n  5. Carnegie Forum on Education and the Economy. (1986). A nation prepared: Teachers for the twenty-first century. Washington, D.C.: Carnegie Forum on Education and the Economy; Holmes Group. (1986). Tomorrow's teachers: A report of the Holmes Group. East Lansing, Mich.: Holmes Group; National Board for Professional Teaching Standards. (1996). Guide to National Board certification. Princeton, N.J.: Educational Testing Service.\n  6. Anderson, J. R.; Reder. L. M.; and Simon, H. A. (1996). Situated learning and education. Educational Researcher 25 (4), 5-11; Greeno, J. G. ( 1997). On claims that answer the wrong questions. Educational Researcher 26 (1), 5-17; Lave. J. (1988). Cognition in practice. New York: Cambridge University Press; Marsick, V J. (1998). Transformative learning from experience in the knowledge era. Daedalus. 127 (4), 119-36.\n  7. Eisner, E. W. (1979). The educational imagination: On the design and evaluation of school programs. New York: Macmillan; Sarason, S. B. (1983). Schooling in America: Scapegoat and salvation. New York: Free Press; Schaefer, R. J. (1967). The school as a center of inquiry. New York: Harper and Row; Tharp, R. G., and Gallimore. R. ( 1988). Rousing minds to life: Teaching, learning, and schooling in social context. Cambridge, England: Cambridge University Press; Cohen, D. K., and Hill, H. C. (1998). Instructional policy and classroom performance: The mathematics reform in California. Ann Arbor: University of Michigan; L\u00fcbeck, S. (1996). Teachers and the teaching profession in the United States. In The education system in the United States: Case study findings (draft volume). Ann Arbor: University of Michigan Center for Human Growth and Development. Pp. 241-318.\n  8. Numerous research reports have turned this claim into one of the most well-supported conclusions in the literature on educational improvement. See. for example, an early study by J. W. Little ( 1982). Norms of collegiality and experimentation: Workplace conditions of school success. American Educational Research Journal 19. 325-40; and a recent review by L. Darling-Hammond (1998). Teachers and teaching: Testing policy hypotheses from a National Commission report. Educational Researcher 27 (1), 5-15.\n  9. Elmore, R. F.; Peterson, P. L.; and McCarthey, S. J. (1996). Restructuring in the classroom: Teaching, learning, and school organization. San Francisco: Jossey-Bass; Stein, M. K.; Silver, E. A.; and Smith, M. S. (1998). Mathematics reform and teacher development: A community of practice perspective. In J. Greeno and S. Goldman (eds.), Thinking practices in mathematics and science learning. Mahwah, N.J.: Erlbaum; Tharp, R. G., and Gallimore, R. (1988). Rousing minds to life: Teaching, learning, and schooling in social context. Cambridge, England: Cambridge University Press.\n  10. Elmore, R. (1996). Getting to scale with good educational practice. Harvard Educational Review 66, 1-26: Cohen, D. K., and Hill, H. C. (1998). Instructional policy and classroom performance: The mathematics reform in California. Ann Arbor: University of Michigan; Lubeck, S. (1996). Teachers and the teaching profession in the United States. In The education system in the United States: Case study findings (draft volume). Ann Arbor: University of Michigan Center for Human Growth and Development. Pp.241-318.\n  11. Glass, T. E. (1992). The 1992 Study of the American School Superintendence: America's education leaders in a time of reform. Arlington, Va: American Association of School Administrators.\n  12. School boards are painted as ineffective by at least one recent study [Hess, F. (1998). Spinning wheels. Washington, D.C.: Brookings Institution]. But the survey of principals conducted as part of TIMSS showed that, in many districts, school boards have the final say in making many educational decisions and therefore are positioned to take an active leadership role.\n  13. Cuban, L. (1993). How teachers taught: Constancy and change in American classrooms, 1890-1990, 2nd ed. New York: Teachers College Press; Elmore, R. F., and McLaughlin, M. W. (1988). Steady work: Policy, practice, and the reform of American education. Santa Monica, Calif: Rand Corporation; Tyack, D., and Cuban, L. (1995). Tinkering toward utopia: A century of public school reform. Cambridge, Mass.: Harvard University Press.\n  14. National Science Board. (1998). Failing our children: Implications of the Third Mathematics and Science Study. Available on-line at http:\/\/www.nsf.gov\/nsb\/documents.\n  15. Cohen. D. K., and Barnes, C. A. (1993). Pedagogy and policy. In D. K. Cohen, M. W. McLaughlin, and J. E. Talbert (eds.). Teaching for understanding: Challenges for policy and practice. San Francisco: Jossey-Bass. Pp. 207-39.\n  16. Shanker, A. ( 1993). Where we stand: Ninety-two hours. New York Times, 24 January. Reprinted in American Educator 21 (1. 2), 1997, pp. 33-34.\n  17. Goldenberg, C. N., and Sullivan. J. (1994). Making change happen in a language-minority school: A search for coherence (EPR #13). Washington, D.C.: Center for Applied Linguistics; Elmore. R. F; Peterson, P. L.; and McCarthey, S. J. (1996). Restructuring in the classroom: Teaching, learning, and school organization. San Francisco: Jossey-Bass; Stein, M. K.; Silver, E. A.; and Smith, M. S. (1998). Mathematics reform and teacher development: A community of practice perspective. In J. Greeno and S. Goldman (eds.), Thinking practices in mathematics and science learning. Mahwah. N.J.: Erlbaum.\n  18. One of the major findings from the case studies of TIMSS was the difference in the weekly work schedule of teachers in the United States, Germany, and Japan. U.S. teachers say that one of the biggest constraints on improving practice is the lack of professional time outside of their classrooms. Indeed. U.S. teachers have less scheduled time to meet and plan for instruction than their foreign colleagues. Increasing the time for this purpose is seen by some federal policymakers as one of the primary implications of these cross-cultural data [Trying to beat the clock: Uses of professional time in three countries. (1998). Washington, D.C.: U.S. Department of Education, Office of Policy and Planning].\n  19. Organization for Economic Cooperation and Development. (1995). Education at a glance: OECD indicators. Paris: Organization for Economic Cooperation and Development.\n  20. Darling-Hammond. L. (1997). The right to learn. San Francisco: Jossey-Bass.\n  21. A more in-depth analysis of alternative ways of restructuring teachers' time can be found in Trying to beat the clock: Uses of teacher professional time in three countries. (1998). Washington, D.C.: U.S. Department of Education, Office of Policy and Planning.\n  22. Greenwald, R.; Hedges, L. V.; and Laine, R. D. (1996). The effect of school resources on student achievement. Review of Educational Research 66, 361-96.\n\nCHAPTER 9: THE STEADY WORK  \nOF IMPROVING TEACHING\n\n  1. In their 1996 book. Restructuring in the classroom: Teaching, learning, and school organization (San Francisco: Jossey-Bass), R. Elmore, P. Peterson, and S. McCarthey report that some reform-minded schools used teachers' time in a way that improved classroom practice and other schools did not. Other studies also have shown that what teachers do with their professional development time makes a big difference [see, for example. Cohen, D. K., and Hill, H. C. (1998). Instructional policy and classroom performance: The mathematics reform in California. Ann Arbor: University of Michigan; and Griffin. G. A. (1995). Influences of shared decision making on school and classroom activity: Conversations with five teachers. Elementary School Journal 96, 29-45].\n  2. The phrase \"steady work\" was used by R. Elmore and M. McLaughlin in their 1988 book, Steady work: Policy, practice, and the reform of American education. We have in mind much the same meaning as they did\u2014improving education is a long-term venture that requires continuing small improvements.\n  3. Brown, C. A.; Smith, M. S.; and Stein, M. K. (April 1996). Linking teacher support to enhanced classroom instruction. Paper presented at the annual meeting of the American Educational Research Association. New York; Cognition and Technology Group at Vanderbilt. (1997). The Jasper project: Lessons in curriculum, instruction, assessment, and professional development. Mahwah, N.J.: Erlbaum; Cohen, D. K.; McLaughlin, M. W.; and Talbert, J. E. (eds.), (1993). Teaching for understanding: Challenges for policy and practice. San Francisco: Jossey-Bass; Darling-Hammond, L. (1998). Teachers and teaching: Testing policy hypotheses from a national commission report. Educational Researcher 27 ( 1 ), 5-15; Elmore, R. E.; Peterson, P. L.; and McCarthey, S. J. (1996). Restructuring in the classroom: Teaching, learning, and school organization. San Francisco: Jossey-Bass; Fennema, E.; Carpenter, T. P.; Franke, M. L.; Levi, L.; Jacobs, V R.; and Empson, S. B. (1996). A longitudinal study of learning to use children's thinking in mathematics instruction. Journal for Research in Mathematics Education 27, 403-34; Franke, M. L.; Carpenter, T. P.; Fennema, E.; Ansell, E.; and Behrend, J. (1998). Understanding teachers' self-sustaining, generative change in the context of professional development. Teaching and Teacher Education 14 (1), 67-80; Little. J. W. (1982). Norms of collegiality and experimentation: Workplace conditions of school success. American Educational Research Journal 19, 325-40; Little, .J. W. (1993). Teachers' professional development in a climate of educational reform. Educational Evaluation and Policy Analysis 15, 129-51; Schifter, D., and Fosnot, C. T. (1993). Reconstructing mathematics education: Stories of teachers meeting the challenge of reform. New York: Teachers College Press; Tharp, R. G., and Gallimore, R. ( 1988). Rousing minds to life: Teaching, learning, and schooling in social context. Cambridge. England: Cambridge University Press.\n  4. Burnaford, G.; Fischer. J; and Hobson, D. (eds.). (1996). Teachers doing research. Mahwah, N.J.: Erlbaum; Cochran-Smith, M., and Lytle, S. L. (1990). Research on teaching and teacher research: The issues that divide. Educational Researcher 19 (2), 2-11; Cochran-Smith, M., and Lytle, S. L. ( 1993). Inside\/outside: Teacher research and knowledge. New York: Teachers College Press; Franke, M. L.; Carpenter, T. P.; Fennema, E.; Ansell, E.; and Behrend, J. (1998). Understanding teachers' self-sustaining, generative change in the context of professional development. Teaching and Teacher Education 14(1), 67-80. Hollingsworth, S., and Sockett, H. (eds.). (1994). Teacher research and educational reform: 93rd Yearbook of the Society for the Study of Education, Part I. Chicago: University of Chicago Press; Richardson, V (1990). Significant and worthwhile change in teaching practice. Educational Researcher 19 (7), 10-18; Richardson, V. (1994). Conducting research on practice. Educational Researcher 23 (5), 5-10; Wagner, J. (1997). The unavoidable intervention of educational research: A framework for reconsidering researcher-practitioner cooperation. Educational Researcher 26 (7), 13-22.\n  5. Jackson, P. W. ( 1968). Life in classrooms. New York: Holt, Rinehart, and Winston; Richardson, V. (1994). Conducting research on practice. Educational Researcher 23 (5), 5-10; Shavelson, R. J., and Stern, P. ( 1981 ). Research on teachers' pedagogical thoughts, judgments, decisions, and behavior. Review of Educational Research, 51, 455-98.\n  6. See the recommendations presented in NCTM's Professional standards for teaching mathematics (Reston, Va.: NCTM, 1991) and those in other recent reform-minded documents, such as Ball, D. L. (1993). With an eye on the mathematical horizon: Dilemmas of teaching elementary school mathematics. Elementary School Journal 93, 373-97; Hiebert, J.; Carpenter, T. P.; Fennema, E.; Fuson, K.; Wearne, D.; Murray, H.; Olivier, A.; and Human, P. (1997). Making sense: Teaching and learning mathematics with understanding. Portsmouth, N.H.; Heinemann; and Lampert, M. (1985). How do teachers manage to teach? Harvard Educational Review 55, 178-94.\n  7. For reviews of the empirical evidence regarding the effectiveness of this kind of teaching, see Hiebert, J. (1999). Relationships between research and the NCTM Standards. Journal for Research in Mathematics Education 30, 3-19; and Hiebert, J.; Carpenter, T. P.; Fennema, E.; Fuson, K.; Wearne, D.; Murray, H.; Olivier, A.; and Human, P. (1997). Making sense: Teaching and learning mathematics with understanding. Portsmouth, N.H.; Heinemann. It should be noted that the kind of instruction described in these reports shares many features with the Japanese script for teaching we described in chapters 3 through 6.\n  8. Ball, D. L. (1993). With an eye on the mathematical horizon: Dilemmas of teaching elementary school mathematics. Elementary School Journal 93, 373-97; Ball, D. L. (1996). Teacher learning and the mathematics reforms: What we think we know and what we need to learn. Phi Delta Kappan 77, 500-508; Cohen, D. K. (1988). Teaching practice (Issue Paper 88-3). East Lansing, Mich.: Michigan State University, National Center for Research on Teacher Education; Lampert, M. (1985). How do teachers manage to teach? Harvard Educational Review 55, 178-94; McDonald, J. P. (1992). Teaching: Making sense of an uncertain craft. New York: Teachers College Press; Schifter. D. (1996). A constructivist perspective on teaching and learning mathematics. Phi Delta Kappan 77. 492-99.\n\nCHAPTER 10: THE TRUE PROFESSION  \nOF TEACHING\n\n  1. Feiman-Nemser, S., and Floden. R. (1986). The cultures of teaching. In M. C. Wittrock (ed.), Handbook of research on teaching. New York: Macmillan. Pp. 505-26.\n  2. Apple, M. W. (1986). Teachers and text. New York: Routledge and Kegan Paul.\n  3. See De Vault and Weaver [DeVault, M. V, and Weaver, J. F. (1970). Forces and issues related to curriculum and instruction, K-6. In P. S. Jones (ed.). A history of mathematics education in the United States and Canada: Thirty-second yearbook. Washington, D.C.: National Council of Teachers of Mathematics] and Osborne and Crosswhite [Osborne, A. R., and Crosswhite, F. J. (1970). Forces and issues related to curriculum and instruction, 7-12. In Jones, A history of mathematics education in the United States and Canada] for a history of the development of mathematics curricula in the United States. An alternative way in which curricula might be developed if we trusted teachers to use them for improving students' learning and for improving their own teaching is presented in Ball, D. L., and Cohen, D. K. (1996). Reform by the book: What is or might be the role of curriculum materials in teacher learning and instructional reform? Educational Researcher 25 (9), 6-8, 14.\n  4. In their 1983 article, L. Darling-Hammond, A. Wise, and S. Pease describe the way in which efforts to analyze the quality of instruction turn into evaluations of teachers by administrators (Teacher evaluation in the organizational context: A review of the literature. Review of Educational Research 53, 285-328). The responses of teachers to the kind of distrust signaled by this tendency are described in Griffin, G. A. (1995). Influences of shared decision making on school and classroom activity: Conversations with five teachers. Elementary School Journal 96, 29-45; Jackson, P. W. ( 1968). Life in classrooms. New York: Holt, Rinehart, and Winston; and McDonald, J. P. (1992). Teaching: Making sense of an uncertain craft. New York: Teachers College Press.\n  5. Berliner, D. C., and Biddle, B. J. (1995). The manufactured crisis: Myths, frauds, and the attack on America's public schools. New York: Addison Wesley.\n  6. Carnegie Forum on Education and the Economy. (1986). A nation prepared: Teachers for the twenty-first century. Washington, D.C.: Carnegie Forum on Education and the Economy; Holmes Group. (1986). Tomorrow's teachers: A report of the Holmes Group. East Lansing, Mich.: Holmes Group; Kerchner, C. T., and Caufman, K. D. (1995). Lurching toward professionalism: The saga of teacher unionism. Elementary School Journal 96, 107-22; Romberg, T. A. (1988). Can teachers be professionals? In D. A. Grouws and T. J. Cooney (eds.), Effective mathematics teaching. Reston, Va.: National Council of Teachers of Mathematics. Pp. 224-44.\n  7. Cuban, L. (1993). How teachers taught: Constancy and change in American classrooms, 1890-1990, 2nd ed. New York: Teachers College Press; Hoetker, J., and Ahlbrand, W. (1969). The persistence of the recitation. American Educational Research Journal 6, 145-67; Tyack, D., and Cuban, L. (1995). Tinkering toward utopia: A century of public school reform. Cambridge, Mass.: Harvard University Press.\n  8. Cohen, D. K., and Hill, H. C. (1998). Instructional policy and classroom performance: The mathematics reform in California. Ann Arbor: University of Michigan; Lortie, D. C. (1975). Schoolteacher: A sociological study. Chicago: University of Chicago Press; Lubeck, S. (1996). Teachers and the teaching profession in the United States. In The education system in the United States: Case study findings (draft volume). Ann Arbor, Mich.: University of Michigan Center for Human Growth and Development. Pp. 241-318; Weiss, I. ( 1994). A profile of science and mathematics education in the United States: 1993. Chapel Hill, N.C.: Horizon Research, Inc.\n  9. A more complete version of this story can be found in Cremin, L. A. (1964). The transformation of the school: Progressivism in American education, 1876-1957. New York: Vintage Books; Johnson, W. R. (1987). Empowering practitioners: Holmes, Carnegie, and lessons of history. History of Education Quarterly 27, 221-40; Lagemann, E. C. (1989). The plural worlds of educational research. History of Education Research 29, 185-214; Lagemann, E. C. (1996). Contested terrain: A history of education research in the United States, 1890-1990. Educational Researcher 26 (9), 5-17; Tanner, L. N. (1997). Dewev's laboratory school: Lessons for today. New York: Teachers College Press; Urban, W. J. (1990). Historical studies of teacher education. In W. R. Houston (ed.), Handbook of research on teacher education. New York: Macmillan, pp. 59-71. Other stories of the separation can also be told. See, for example, Linda Darling-Hammond's discussion of other educational figures who contributed to the separation [Darling-Hammond, L. (1997). The right to learn. San Francisco: Jossey-Bass].\n  10. The problem has not gone unnoticed. There are countermovements that try to renegotiate the roles between teacher and researcher, such as the contemporary notion in the research community of \"teacher-as-researcher.\" But none of the countermovements have altered fundamentally the traditional division of labor on a wide scale.\n  11. The October 1995 testimony is reprinted in the American Educator 21 (1, 2), 1997, pp. 35-36.\n  12. Johnson, W. R. (1996). History and educational reform. History of Education Quarterly 36, 478-82.\n  13. Goldenberg, C. N., and Sullivan, J. ( 1994). Making change happen in a language-minority school: A search for coherence (EPR #13). Washington, D.C.: Center for Applied Linguistics; Elmore, R. E.; Peterson, P. L.; and McCarthey, S. J. (1996). Restructuring in the classroom: Teaching, learning, and school organization. San Francisco: Jossey-Bass; Stein, M. K.; Silver, E. A.; and Smith, M. S. ( 1998). Mathematics reform and teacher development: A community of practice perspective. In J. Greeno and S. Goldman (eds.), Thinking practices in mathematics and science learning. Mahwah, N.J.; Erlbaum.\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":"\n\nACRYLIC  \nINNOVATION\n\nStyles +  \ntechniques  \nfeaturing 64   \nvisionary  \nartists\n\n**NANCYREYNER** \u2022 Author of _Acrylic Revolution_\n\nwww.artistsnetwork.com\n**Acrylic Innovation**. Copyright \u00a9 2010 by Nancy Reyner. Manufactured in China. All rights reserved. No part of this book may be reproduced in any form or by any electronic or mechanical means including information storage and retrieval systems without permission in writing from the publisher, except by a reviewer who may quote brief passages in a review. Published by North Light Books, an imprint of F+W Media, Inc., 4700 East Galbraith Road, Cincinnati, Ohio, 45236. (800) 289-0963. First edition.\n\nOther fine North Light Books are available from your local bookstore, art supply store or online. Also visit our website at www.fwmedia.com.\n\n14 13 12 11 10 5 4 3 2 1\n\nDISTRIBUTED IN CANADA BY FRASER DIRECT  \n100 Armstrong Avenue  \nGeorgetown, ON, Canada L7G 5S4  \nTel: (905) 877-4411\n\nDISTRIBUTED IN THE U.K. AND EUROPE BY DAVID & CHARLES  \nBrunel House, Newton Abbot, Devon, TQ12 4PU, England  \nTel: (+44) 1626 323200,Fax: (+44) 1626 323319  \nEmail: postmaster@davidandcharles.co.uk\n\nDISTRIBUTED IN AUSTRALIA BY CAPRICORN LINK  \nP.O. Box 704, S. Windsor NSW, 2756 Australia  \nTel: (02) 4577-3555\n\n**Library of Congress Cataloging in Publication Data**\n\nReyner, Nancy.\n\nAcrylic innovation : styles and techniques featuring 64 visionary artists\/ Nancy Reyner. -- 1st ed.\n\np. cm.\n\nISBN 978-1-60061-864-2 (alk. paper)\n\n1. Acrylic painting--Technique. I. Title. II. Title: Styles and techniques featuring 64 visionary artists. III. Title: Styles and techniques featuring sixty four visionary artists.\n\nND1535.R48 2010\n\n751.4'26--dc22\n\n2010018580\n\nEdited by Kelly C. Messerly, Designed by Clare Finney, Production coordinated by Mark Griffin\n**About the Author**\n\nAuthor photo by Reynaldo Villalobos.\n\nBorn and raised on the East Coast in the United States, Nancy Reyner received a BFA from the Rhode Island School of Design and an MFA from Columbia University. Creating costumes and sets for theater and film to coordinating public arts programs for the state of New York, Reyner has had an expansive career in the arts, all of which inform her work. She now lives in Santa Fe, New Mexico, has been a painter for more than thirty years and exhibits and teaches both nationally and internationally. Her previous book, _Acrylic Revolution_ , contains over one hundred acrylic painting techniques, and continues to be a best seller. Please visit her Web site, www.NancyReyner.com, for current work, her painting blog, workshops and gallery representation.\n\n**Artwork on Cover and Back Cover:**\n\nGary Denmark, _Jongleur 2_ , pictured again\n\nPat Forbes, (detail) _Neutrino Blossoms_ , pictured in full\n\nCatherine Mackey, (detail) _Harrison Street Scooter_ , pictured in full\n\nBeth Ames Swartz, (detail) _The Fire and the Rose: But Heard, Half-Heard, in the Stillness_ , pictured in full\n\nJim Waid, _Blue Shimmy_ , pictured again\n\nInes Kramer, _Rooftop Eden_ , pictured again\n\nMartha Kennedy, _Avocado on Orange_ , pictured again\n\nRobin Sierra, _Aquamarine_ , pictured again\n\nRen\u00e9e Phillips, _Field of Dreams_ , pictured again\n\n**Art**\n\nNancy Reyner, _Koi and River_ , acrylic on canvas, 60\" \u00d7 46\" (152cm \u00d7 117cm)\n\n**Art**\n\nNancy Reyner, _Ocean Elixer_ , acrylic and gold leaf on panel 20\" \u00d7 20\" (51cm \u00d7 51cm)\n**Metric Conversion Chart**  \n---  \n|  |   \n**To convert** | **to** | **multiply by**  \nInches | Centimeters | 2.54   \nCentimeters | Inches | 0.4   \nFeet | Centimeters | 30.5   \nCentimeters | Feet | 0.03   \nYards | Meters | 0.9   \nMeters | Yards | 1.1\n\n**Acknowledgments**\n\nSpecial thanks to those who helped connect me with the extraordinary artists included in this book: George Alexander, Bruce Cody, Bill Hinsvark, Katherine Chang Liu, Michael Matassa, Keith Morant and Renee Phillips. Many thanks to Reynaldo Villalobos for photography. Thank you to Bonnie Teitelbaum, Tekla Johnson and Eve Munson for inspiration and assistance. Thank you Phillip Cohen and to my wonderful family and friends for support. I thank my art professors Phyllis Bramson, Jane Wilson and the late Art Wood for their wisdom and encouragement. I cherish my dance teacher, Lori Brody, and all my cool dancer friends in Brody's Ballerinas for making life wonderful. Thank you to Mark and Barbara Golden, and to the staff at Golden Artist Colors for all their wonderful inventions, products and support, with special thanks to their technical department. I have enjoyed immensely working with the F+W and North Light Books staff, with extra thanks to Jamie Markle and Kelly Messerly.\n**Dedication**\n\nThis book is dedicated to my son, Jacob Cooper Cohen, who has been and continues to be a great inspiration for me.\nTABLE OF CONTENTS\n\nIntroduction: How to Use this Book\n\nEssential Acrylic Tips\n\nSECTION 1\n\n**The Realistic Window**\n\nEpic Narrative\n\nPhotorealism\n\nPortrait\n\nStill Life\n\nLandscape\n\nSECTION 2\n\n**Abstracting the Window**\n\nFigure as Form\n\nReductive\n\nFlat Space\n\nRomanticized\n\nExaggeration\n\nSECTION 3\n\n**Changing Perspectives**\n\nInside & Outside\n\nMultiple Perspectives\n\nComposite Arrangements\n\nCollage\n\n**Sherry Loehr**\n\n_Tulips and Oranges_\n\nAcrylic on board\n\n24\" \u00d7 36\" (61cm \u00d7 91cm)\n\n**Gary Denmark**\n\n_Jongleur 2_\n\nAcrylic and collage on board\n\n48\" \u00d7 48\" (122cm \u00d7 122cm)\n\nSECTION 4\n\n**Engaging the Picture Plane**\n\nMapping: Planes & Fragments\n\nConfronting the Viewer\n\nIsolated Imagery\n\nEmbellished Form\n\nHard Edge Graphic\n\nPattern Fields\n\nSECTION 5\n\n**Spatial Push-Pull**\n\nMarking Space\n\nFree-Floating Organics\n\nText, Line & Image\n\nGeometry & Space\n\nSECTION 6\n\n**Minimal**\n\nSheens & Texture\n\nIlluminated\n\nColor Field\n\nA Singular Note\n\nVeiling Forms\n\nContributing Artists\n\n**Bruce Cody**\n\n_Capitol City Sunlight_\n\nAcrylic on canvas\n\n16\" \u00d7 30\" (41cm \u00d7 76cm)\n\n**Hamish Allan**\n\n_House and Hills_\n\nAcrylic on canvas\n\n24\" \u00d7 24\" (61cm \u00d7 61cm)\nINTRODUCTION:  \nHOW TO USE TH IS BOOK\n\n**Nancy Reyner**\n\n_Essence_ (detail)\n\nAcrylic and gold leaf on panel\n\n32\" \u00d7 26\" (81cm \u00d7 66cm)\n\nNeed an idea? Want new motivation? Desiring new tips and techniques? Just flip through the twenty-nine styles in this book ranging from photorealism to minimal color field and everything in between. Read how other artists have taken the journey to create unique and personal art, offering advice and gems from their own experience and encouraging artists to invent, not replicate. An artist's process can be viewed both mentally (the idea) and physically (the technique). A visionary artist develops and combines both to create a personal vision and unique art. This book is meant to inspire artists to find their own personal voice.\n\nThroughout history the discovery of new art products spawned the invention of new art styles, ideas and techniques. The creation of oil paint was a catalyst for new styles of blending and realism that were previously unheard of for the egg tempera painters of that time. The invention of storing oil paint in tubes added portability for painters, taking them out of the studio and paving the way for plein air and onsite painting. Photography, digital imagery and computers have also made their mark.\n\nAcrylic paint, first developed in the 1930s, continued to improve in quality and durability, and eventually established itself as one of the most dependable, archival and permanent artist materials offered. Available to artists now for over sixty years, it is currently helping to push the envelope of contemporary vision. Acrylic is often used as a substitute for oil paint or watercolor. But the real gold mine for the artist is in allowing the medium freedom to do what it does best. New paint requires a new way of thinking. Acrylic has a wide range of possibilities, uses and techniques. My previous book, Acrylic Revolution, demonstrates this variety with over one hundred acrylic painting techniques in an easy resource format. More and more artists are choosing to paint in acrylic, and this growing trend is reflected in the new contemporary art styles emerging on the scene. This book contains incredible artwork from sixty-four acrylic artists worldwide, all contemporary, using acrylic in part or completely in their work, who are taking advantage of acrylic's potential to create unique genres or styles. This book addresses methods and techniques for finding your own personal voice in this medium.\n\nTo view a painting is to experience the idea of space. In this book I take a fresh stance by grouping styles in terms of their perceived spatial qualities, even inventing some of the style names. Notably, many artists in this book have work represented simultaneously in several style categories. A style is merely a tool, meant to facilitate the expression of an idea, and therefore most artists' work will not fit neatly into any specific category. A style is a tool, not a label.\n\nTo create an original idea, we can use the wide variety of visual stimuli around us every day, in advertisements and media, our relationships and experiences, objects, nature and architecture. All these can offer artistic references. Instead of replicating, art can be fresh and original when we add our own individuality, rearrange images, work with unusual combinations, and let materials and mediums influence us.\n\nESSENTIAL ACRYLIC TIPS\n\n**\u2022 Acrylic is available in a wide variety of paints and products.** It is helpful to understand the different categories of products to select the most appropriate one for the technique at hand. In general, acrylic paint color has two components: pigment (for color) and acrylic binder (also known as polymer). Usually, the binder is also called a medium, and, when thickeners are added, it's a gel. The fluid binder without thickeners is combined with pigment to make fluid paints. Binder with thickener added to pigment creates the heavy bodied paints. A small amount of whiting agent added to the binder creates matte products such as matte mediums, matte gels and matte varnishes. Larger quantities of whiting or opaque components make pastes. Adding materials such as pumice, glass beads, metal and fiber can make unusual gels. Any of these products can be used as a ground to change the painting surface, when applied as a first layer to customize and change the surface absorbency, texture or color.\n\n**\u2022 Acrylic products are compatible with each other.** Any acrylic product can be combined together while wet to create new mixtures and hybrids. Gels, mediums and pastes can all be mixed together. Keep in mind that the characteristics of one particular paint or product will change when you mix it with another. This can work to your advantage; for example, mixing a gloss medium with a matte medium will produce an excellent semigloss. However, some specialty products, such as Clear Tar Gel, which makes the paint tar-like and stringy, will lose their special abilities when too much of another paint or product is added. You can also layer acrylic products and paints in any order in your painting process. Gels can go on top of mediums and mediums on gels. Diluted paint works on undiluted paint and vice versa. Glossy can go on top of matte, then on satin and on glossy again.\n\n**\u2022 Acrylic can be used in combination with other mediums.** Most other mediums such as oil, gouache, watercolor, pastel or casein can be applied on top of acrylic grounds or acrylic underpaintings. In general, though, avoid using acrylic on top of other mediums, or combining other mediums together with acrylic in wet form. For example, watercolor can be used to wonderful effect on top of acrylic that is already dry, but mixing the two together wet is less effective.\n\n**\u2022 A few acrylic products are best used alone.**\n\n1. Use acrylic gesso as a primer and not as a white paint. In thick applications, acrylic gesso may crack.\n\n2. Use a removable archival varnish as a final protective coat. Even though it can be used in the middle layers of a painting, it is meant to be removable for cleaning purposes and as a final coat.\n\n3. Additives such as retarder, thinners and flow release need to be used in correct proportions. It is always a good idea to read product labels for correct use and application information.\n\n**\u2022 Acrylic is a glue.** Any acrylic paint or product can function very well as a glue. You can add collage elements and mixed media to acrylic while it's wet and they will adhere nicely.\n\n**\u2022 Stir gently and don't shake.** The small amount of soap remaining in the acrylic products from processing may create bubbles with vigorous stirring or handling. To stir a batch of paint, use a palette knife instead of a stiff bristle brush. After stirring mediums for use in pours, let the bubbles settle out overnight before pouring.\n\n**\u2022 In general, mix paint with a knife.** Brushes are meant to hold paint, while knives more easily release the paint. Mix a batch of paint with a knife for a clean, homogenized mixture. Use a brush to mix paint in a hodgepodge manner, for a painting technique called \"dirty brushing.\"\n\n**\u2022 As you paint, keep brushes in water** until you can clean them well with soap. Soap removes paint from the brush more effectively than water alone. If a brush in use is left out of water too long, the acrylic will dry on it, making it difficult to remove later. To make cleaning easier, slightly dampen the brush bristles with water before dipping them into the acrylic paint. Damp bristles reduce the gripping power of acrylic, making the paint easier to remove later.\n\n**\u2022 There are two choices for thinning acrylic paint:** water or acrylic medium. Adding water will break down the acrylic binder in the paint, creating a thinner paint that appears like watercolor, resulting in a matte finish. The more water you add, the more the paint will be affected by the absorbency of the painting surface. On the other hand, adding acrylic mediums, gels or pastes, while minimizing the addition of water, will maintain the acrylic's properties, offering a rich, glossy appearance. Either choice is effective, depending on your desired result.\n\n**\u2022 Acrylic shrinks in volume while drying.** As acrylic dries, the paint layer will shrink by about a third in volume. This is more noticeable when using thick pastes and gels. Generally, apply a paint layer that is a little thicker than what you want at completion.\n\n**\u2022 Acrylic appears lighter in color when wet.** The acrylic binder is naturally white when wet, but dries clear and glossy. Therefore while painting with acrylic, the color will appear lighter when wet but goes to its natural color when dry. The more mediums and gels you add to a paint color, the greater the difference between this lighter version when wet and its natural color when dry. Paint color added to pastes, however, will not change in color very much between wet and dry. That is because pastes are white when wet, remaining white when dry.\n\n**\u2022 Acrylic is naturally glossy.** Matte products such as matte mediums, matte gels and matte varnishes have matting agent (a fine white powder) added to them to create their characteristic appearance. This matting agent means that matte products slightly cloud and lighten colors underneath. The thicker the application of a matte product, the more this cloudy and lightening effect will be apparent. Acrylic products labeled semigloss or satin have matting agent in them as well, just less in proportion to a matte product. Also, colored paints when dry will vary naturally in sheen, according to the pigment.\n\n**\u2022 Acrylic has a two-part drying process.** The first part of the acrylic drying process, known as \"dry to the touch,\" means the top layer of the paint skin has dried due to the evaporation of the water in the paint, enabling layering. The second part of the drying process involves the curing of the polymer or acrylic in the paint, which takes several days to several weeks to \"lock down,\" or fully cure. The actual curing time is dependent on the layer's thickness and environmental factors. During this curing time, it is important to allow air to flow around it. Avoid tightly wrapping the painting, storing the artwork in a closed environment or exposing it to extreme temperatures during the curing phase.\n\n**\u2022 Be aware of the \"tacky phase.\"** While the paint is still wet, it is very malleable. You can scrape it, wipe it off and rework it with ease. As you continue to work into this wet paint, however, it is already beginning to dry. Once it dries to the touch, it has a wonderfully resistant surface on which you can overlay new paints without disturbing what is underneath. It is between the wet stage and this \"dry to the touch\" stage when problems can occur. Between the wet and dry stages, the acrylic gets tacky, and continued working over this tacky area can create unwanted effects such as streaking or pulling as the paint sticks to your brush. Initially, acrylic paint glides smoothly and easily, but as it reaches this tacky phase, it will begin to pull and feel difficult to manipulate. At this point, stop painting in that area, and move on to a drier area of the painting. If you need to keep working in the tacky area, use a blow-dryer for a minute to dry it quickly, then resume painting. Avoid blow-drying very thick layers. Also, be aware that the paint on your brush will dry quickly. Make a habit of rinsing frequently to keep the paint from getting tacky on your brush.\n\n**\u2022 Do not freeze acrylic.** Oil painters sometimes freeze the excess paint on their palettes to prolong the life of the paint. Avoid this with acrylic. Acrylic contains a certain amount of antifreeze, but after a few freeze\/thaw cycles, the paint will no longer be stable. Acrylic paintings, when fully cured, will be fine if exposed to cold. However, fine art should not be exposed to extreme temperatures. To extend the drying time of wet paint, consider using Golden's slow drying OPEN Acrylics.\n**Section 1**\n\nTHE  \nREALISTIC  \nWINDOW\n\n**Jason de Graaf**\n\n_Untitled (Self Portrait)_\n\nAcrylic on canvas\n\n30\" \u00d7 30\" (76cm \u00d7 76cm)\n\nRealism, convincing portrayals of the physical world, presents imagery that has fascinated artists and viewers for centuries, and continues in popularity today. Still life, landscape and portraiture are all just as potent today as they were in the past. This section includes five realistic styles, each using similar principles. These principles, popularized in the Renaissance, include singular or fixed perspective, the use of a horizon line, modeling of light and shadow to create volume, and the illusion of receding space, all of which turn the canvas into a viewing window. Inspired by evolving technologies in photography, digital imagery and new pigment development, realism has expanded to include hyperrealism, along with new techniques and exciting new color palettes. This section investigates both traditional and new realistic styles with an emphasis on keeping them personal and contemporary.\n\nEPIC NARRATIVE\n\nAn epic narrative style is characterized by multiple figures in a landscape\u2014it's the \"big scene\" that has been used historically to convey weighty narratives such as religious stories and myths.\n\nPaul Pletka's _Tears of the Lord_ is a new vision on the epic narrative. This striking piece uses a bold-colored, flat backdrop and a vibrant color palette to set off multiple figures, along with incredible attention to detail.\n\nPletka works in large formats for greater impact and has centered his work on his desire to explore the ritual side of Native American culture, to bring forth the spiritual side in a respectful way, and to consider this an interpretation from his own cultural point of view.\n\nPletka has painted for many years, working through his own personal desire to invent something original. Each painting is heavily researched to use authentic and accurate material. But for Pletka, making something beautiful is the most important aspect; therefore, some elements may be slightly compromised for aesthetics.\n\n**The Artist's Process**\n\nTwo key motivations for Pletka are scale and a new technical challenge. He creates many thumbnail sketches until a cogent image emerges, then he moves directly to the canvas, drawing first with vine charcoal to transfer his energy and excitement into the work. He then overpaints the line work with Raw Sienna. Even with the initial drawing in place, Pletka will rearrange and redesign to accommodate a piece's evolution. He often obliterates a figure he painstakingly worked and repaints it in a new location, sometimes several times. To maintain the integrity of the canvas's surface, he sands or scrubs off an area, instead of overpainting. By working in acrylic, Pletka is able to maintain a sense of immediacy while working with multiple glaze layers on one painting at a time.\n\n**Other Art in This Style**\n\n\u2022 Contemporary artist Mark Tansey, Modern artist George Bellows\n\n\u2022 Renaissance painters Sandro Botticelli, Masaccio, Leonardo da Vinci, Michelangelo Buonarroti, Titian, Tintoretto and Raphael\n\n\u2022 Historic religious and mythic paintings from French Classicism and Baroque periods\n\n\u2022 Fantastic imagery from Hieronymus Bosch\n\n\u2022 Muralist Diego Rivera\n\n **TIPS**  \n **from the Artist**\n\nGreat art contains the artist's originality, which then shows through the work. Be willing to destroy elements in your work to push the piece further. Your work has to be you, it has to come from your core and be honest. It can't come from someone else and be successful. There are so many permutations available in this world that it's always possible to come up with a unique vision.\n\nFor Pletka, inspiration comes from many trips across the Western United States, visiting museums, national parks and wildlife refuges, reading books and attending cultural events.\n\n**Paul Pletka**\n\n_Tears of the Lord_\n\nAcrylic on linen canvas\n\n96\" \u00d7 72\" (244cm \u00d7 183cm)\n\n\u00a9 copyright Paul Pletka, 2005\n\nCourtesy of the Museum of the American\n\nWest, Autry National Center, Los Angeles\n\nVARIATIONS ON EPIC NARRATIVE\n\n**Paul Pletka**\n\n_The Revelation_\n\nAcrylic on linen canvas\n\n44\" \u00d7 66\" (112cm \u00d7 168cm)\n\n\u00a9 copyright Paul Pletka, 2008\n\nPrivate collection\n\n**Variation 1: Action and Narrative**  \nA wide variety of strong colors and distorted proportions adds visual tension and movement to emphasize action and narrative.\n\n**Daniel Barkley**\n\n_Far Shore_\n\nAcrylic on canvas\n\n52\" \u00d7 114\" (132cm \u00d7 290cm)\n\n**Variation 2: Emphasize Mood**  \nA limited palette with muted tones creates mystery and mood. The simplified and elegant background keeps the focus on the figures and narrative.\n\n**More Variations**\n\n\u2022 Use your own personal stories to paint scenes depicting memorable moments. Take liberties and change, rearrange and re-create to paint your ideal version. Paint yourself into new scenes you desire or fear. Using childhood photographs, replace figures or backgrounds with new ones. Change color palettes.\n\n\u2022 Experiment with arrangements of multiple figures. Cut out figures, using a photocopier to reduce and enlarge for a variety of sizes. Place and arrange on various backgrounds from photographs, drawings or other paintings. Backgrounds can be as simple as a color field and as complex in detail as a photographic landscape.\n\n PREPARING A TONAL BASE\n\nEpic narrative paintings combine multiple figures in a landscape, coordinating many forms in one image. These compositions rely heavily on tonal areas (shapes of light and dark) to create movement and focus.\n\nStarting the painting with a tonal drawing gives you a jumpstart to the composition process. This technique transforms a drawing into tonal shapes, then seals and prepares it for subsequent paint layers.\n\n**Materials**\n\n**\u2022 Drawing Materials:** A variety of water-soluble and waterproof drawing materials (i.e., charcoal, Cont\u00e9, graphite, water-soluble pencils), smudge stick, chamois, erasers\n\n**\u2022 Surface:** Any primed surface\n\n**\u2022 Painting Tools:** Painting knife with stepped handle, paintbrush\n\n**\u2022 Other:** Acrylic polymer medium (matte), water\n\n**1. Create a Line Drawing**  \nMake a drawing outlining the general areas to be painted, using a variety of drawing materials, both water-soluble and waterproof. Make sure to have at least one water-soluble item. Test it first on a scrap surface to see if the drawn line will blend when brushed over with water. Experiment by using a variety of line qualities; change your grip, vary the angle onto the surface, vary pressure applied, and smear lines using a smudge stick, eraser or chamois.\n\n**2. Create Tonal Areas Using Water and Medium**  \nDip a brush in water and\/or acrylic medium. Apply the brush over the drawn lines, softly moving tones out along the surface to create softer edges and tonal areas. When applied over waterproof materials, such as graphite pencil, the line will not create tones. By using both waterproof and water-soluble materials in your drawing, you get more variations with this step. If you need to get some white areas back, selectively overpaint with gesso. In addition, experiment to see the differences using medium versus water. This step alone may suffice for sealing. If some lines, however, did not get painted with water or medium and are still water-soluble, continue to step 3 to seal the drawing completely.\n\n**3. Add a Waterproof Coating**  \nLiberally pour an acrylic medium over the entire surface. If working on a large area, do this step in smaller sections. Using a painting knife with a stepped handle, gently spread an even coat of the acrylic medium over the drawing. Avoid scraping the surface by raising the knife about 1\/8-inch (3mm) from the surface with the acrylic rolling out from under it. Keep the knife at a slight angle so it isn't parallel to the surface. If the knife keeps scraping the surface, apply more acrylic medium.\n\nPHOTOREALISM\n\nPhotorealism relies on sharp detail, with super-real images incorporating photographic characteristics. Daniel Smith, photorealist wildlife painter, considers it a compliment when someone comments that his work is as good as a photo. Yet Smith goes even further. Making his paintings believably realistic while still able to stand alone as good works of art takes daring and creativity beyond copying a photograph. Too much emphasis on the story, however, can feel contrived, like an illustration. Even with his intent for portraying reality truthfully, Smith will stretch things a bit to enhance aesthetics when needed. Photographic images contain inherent limitations from the camera's depth of field, including distortion, compressed images and mixed focal ranges\u2014blurring certain areas while leaving others crisp.\n\nIn _Zero Tolerance_ , Smith transforms the work from photographic generality by adding his own compositional design, ideas and imagery, along with simplifying and editing detail. Here the image is enhanced by the addition of foreground dust, which reduces background detail and places the focus on the animals and action. The lionesses were photographed on one trip, while the elephant was from another. In addition to using multiple references, Smith researches animal behavior to get correct wildlife appearances and habitats. Smith has always been intrigued by wildlife, and has been painting this imagery since childhood.\n\n**The Artist's Process**\n\nSmith works on masonite boards, and rolls out his gesso primer to add a pebbly texture (good for rock textures). Drawing and then underpainting gives Smith his jumpstart, whereupon many thin layers of paint are applied, along with drybrushing techniques. Keeping brushstrokes and texture to a minimum, the final surface is smooth and flat, reducing distraction from the exacting detail.\n\n **TIPS  \nfrom the Artist **\n\nThere's a difference between technical skill and turning it into a true work of art. Composition, a strong focal point, editing, balance, aesthetics and the right amount of negative space are all essential. There's a lifetime of experience that goes into it.\n\nSmith lives near Yellowstone National Park where moose, elk, bear and fox are frequently visible. He also travels to places like Alaska, Africa, Glacier National Park and the Canadian Rockies to sketch and photograph. Firsthand experience is key for Smith, adding adventurous stories and personal relationships with the animals in his paintings.\n\n**Other Art in This Style**\n\n\u2022 Wildlife painters Bob Kuhn, Robert Bate-man and Carl Rungius\n\n\u2022 Urban landscape painter Max Ferguson\n\n\u2022 Photorealist Richard Estes\n\n\u2022 The Dutch Masters\n\n**Daniel Smith**\n\n_Zero Tolerance_\n\nAcrylic on Masonite\n\n55\" \u00d7 42\" (140cm \u00d7 107cm)\n\nVARIATIONS ON PHOTOREALISM\n\n**Tom Palmore**\n\n_Mugsy the Jungle Boy_\n\nAcrylic and oil on canvas\n\n46\" \u00d7 72\" (117cm \u00d7 183cm)\n\n**Variation 1: Unexpected Juxtapositions**  \nUnexpected juxtapositions add humor, making this realism more personal. A trademark for Palmore, stylistic plays between background and object creates surprises. Palmore's tip: Develop your own techniques so you don't paint like someone else.\n\n**Jason de Graaf**\n\n_Vesalius Skeleton_\n\nAcrylic on canvas\n\n24\" \u00d7 36\" (61cm \u00d7 91cm)\n\n**Variation 2: The Real and the Surreal**  \nClose-cropping, a hint of narrative, and an intent to make paintings more intriguing than photographs all add dramatic impact and a sense of the surreal. De Graaf consciously tries to paint things that haven't been done before.\n\n**More Variations**\n\n\u2022 Our best work has roots in our everyday lives and past experiences. Spend time photographing your home, personal spaces and favorite places, friends and activities you like. Experiment by cutting and reassembling your photos to create new combinations.\n\n\u2022 In a corner of a room, create your own stage set using fabric and add friends willing to pose with costumes and objects. Photograph this setup using different lighting and use it for a painting.\n\n  TRANSFERRING DIGITAL IMAGERY\n\nPhotorealism demands exact painting techniques using extreme detail, perfect proportions and scale. Many realist painters enjoy the challenge of painting with minimal references and mostly imagination. Other painters take advantage of photographic processes in their paintings. This technique prints a color image directly onto an acrylic skin for use as an underpainting or collage element.\n\n**Tip**\n\nInstead of using gloss acrylic gel for a skin, try other acrylic products such as colored paint, matte gels or pastes to get an opaque or translucent skin. Consider using unusual papers such as aluminium foil.\n\n**Materials**\n\n\u2022 **Acrylic Products:** Any clear glossy acrylic gel, mineral spirits-based acrylic spray fixative\n\n\u2022 **Surface:** A nonstick surface that will release acrylic (freezer paper\u2014not wax paper, HDPE plastic, garbage bags or cheap plastic wrap\u2014test first)\n\n\u2022 **Painting Tools:** A squeegee tool (piece of cardboard or plasterer knife), sponge applicator, masking tape, paper\n\n\u2022 **Other:** Clear digital coating product or ground for inkjet printing (i.e., inkAID or Golden's Digital Ground for Non-Porous Surfaces), ink-jet printer (with a flat feed, i.e., paper should enter in a different place than it exits)\n\n**1. Make an Acrylic Skin**  \nOn a removable surface where acrylic will not stick, squeeze out a generous amount of gloss acrylic gel. Using a squeegee tool, spread the gel in an even layer about \u00bc-inch (6mm) thick. (Acrylic will shrink by 30 percent in volume. A thicker application will keep the dried skin from tearing.) Let this dry until the gel turns clear.\n\n**2. Coat and Prepare the Skin for Printing**  \nWhile the skin is still on its surface, use a sponge applicator to apply a thin layer of digital ground. Let it dry for about an hour, or quick dry for a minute with a blow-dryer. Peel the skin off the surface and cut the skin to a size smaller than a sheet of regular-sized paper. Leaving about \u00bd-inch (12mm) or more as a bottom margin, secure the skin onto a piece of paper with masking tape, along the bottom and on only one side where the printer grips the paper.\n\n**3. Print the Image Onto the Prepared Skin**  \nSelect an image (here, I'm using a photograph taken by Bruce Cody). If using an all-in-one printer, then place the image in the scanner. Otherwise, send the image through a computer, or use an attached scanner. Set the printer preferences for high quality and to print on gloss. Place the paper with taped skin into the printer's paper feed and print.\n\n**4. Secure With Fixative**  \nHere the image is printed onto the clear glossy skin. Fix the image by using a mineral spirits-based acrylic spray fixative. If desired, you can paint on the back of the skin to change transparent areas to opaque, and\/or paint a background. Glue the skin onto a painting surface using gloss acrylic gel.\n\nPORTRAIT\n\nPortraiture has been a popular style for many artists and audiences throughout the ages. For portrait painter Lea Bradovich, continuity between humanity and nature is a key concept, visible in her portrayals using fashion and headgear to create fanciful portraits. Inspired while looking at a photograph of a giant bee head, Bradovich integrated botany with Renaissance style portraiture in the painting shown opposite, adding irony, freshness and an element of surprise. Note the honeycombed collar, bee-head hat, and mouth located where a bee's mandibles would be in _Queen Bee's Regalia II_ . According to Bradovich, \"We humans seem to fashion everything we find into headgear. And headgear is symbolic.\"\n\nPainted in the style of Italian Master Bronzino (Agnolo de Cosimo), _Queen Bee's Regalia II_ follows his traditional, refined and aristocratic flavor. A \"queen bee\" in human society would be cool, aloof and alluring, and here, appropriately paired with Bronzino's painting style, it creates an interesting dialogue between old and new.\n\n**The Artist's Process**\n\nIf an idea amuses Bradovich and makes her smile, she will investigate it further, even if some ideas seem goofy or silly at first. Bradovich starts by collecting reference materials including historic costumes, old masters portraits, fashion magazines and science photographs. These are maintained in boxes, files and on her computer so that she can leaf through for ideas, allowing for cross-fertilization.\n\nInitial drawings give Bradovich a jumpstart, continuing by painting on panel with acrylic gouache, leaving room for accidental things to happen. On occasion she will work from models. Serendipity keeps the work changing, thus avoiding a paint-by-number appearance. Using washes, dry brush, and any technique that works for her at the time, she sometimes builds up areas with small overlapping brushstrokes.\n\n **TIPS  \nfrom the Artist **\n\nWhile mixing paint simultaneously consider the brush strokes you will be painting with that color. This creates a mindfulness and preparatory gestural memory.\n\nDramatic studio lighting plays off multiple portraits in Lea Bradovich's studio.\n\n**Other Art in This Style**\n\n\u2022 Renaissance masters Agnolo Bronzino and his teacher Jacopo da Pontormo\n\n\u2022 Portrait photographers Annie Leibovitz, Dorothea Lange and Cindy Sherman\n\n\u2022 Psychological aspects and technical mastery of contemporary figurative realists Michael Grimaldi, Lucian Freud, Eric Fischl, Philip Pearlstein, Avigdor Arikha, John Currin, Alice Neel, K\u00e4the Kollwitz, Chuck Close, Gerhard Richter and Bernardo Torrens\n\n\u2022 Photographic portraits by Nan Goldin, Cindy Sherman and William Wegman\n\n\u2022 Lifelike sculptures of Duane Hanson.\n\n**Lea Bradovich**\n\n_Queen Bee's Regalia II_\n\nAcrylic gouache on panel\n\n24\" \u00d7 18\" (61cm \u00d7 46cm)\n\nCollection of Renee Goshin\n\nVARIATIONS ON THE PORTRAIT\n\n**More Variations**\n\n\u2022 Compare painting a portrait from life versus photographs by getting a friend or model to pose. Paint them from life in several sittings, then photograph them to work from the image as reference.\n\n\u2022 Use your most available model\u2014yourself\u2014and paint a self-portrait using a mirror, photographs or memory. Paint self-portraits at intervals in your life to create a series.\n\n\u2022 Research and interview to find out about the person you are painting. What do you like about this person? Translate these personality traits into the work through unusual cropping, varying color palettes, placement in unusual backgrounds, and adding costume and prop elements.\n\n\u2022 Stage your own settings using costumes and props like Cindy Sherman's self-portrait series.\n\n**Daniel Smith**\n\n_Colors of Kenya\u2014_\n\n_Warrior_\n\nAcrylic on panel\n\n14\" \u00d7 28\" (36cm \u00d7 71cm)\n\n**Variation 1: Add Personality With Clothing and Objects**  \nCostume and personal objects add a hint of narrative, personality and culture.\n\n**Daniel Barkley**\n\n_Petit nuage a l'horizon_\n\nAcrylic on canvas\n\n54\" \u00d7 76\" (137cm \u00d7 193cm)\n\n**Variation 2: Enhance Mood**  \nA soft, muted limited palette, simplified background and unusual elongated format adds mood and a contemporary feel to this portrait.\n\n FIXING A LINEAR UNDERDRAWING\n\nPreparing a preliminary detailed line drawing offers a jumpstart to any painting, especially when it's drawn directly on the painting surface. Starting with the flexibility of charcoal, where lines can be easily erased and changed, this technique changes the charcoal drawing into waterproof graphite, making it smudgeproof for subsequent overpainting.\n\n**Tip**\n\nIf your drawing surface is too slick and the charcoal will not easily adhere, apply a thin amount of matte medium onto the surface, let dry, then continue drawing.\n\n**1. Create a Charcoal Drawing**  \nA semi-slick surface is best, particularly a surface primed with acrylic gesso. Draw onto the primed surface with soft vine charcoal. Erase and rework as necessary, using your fingers, eraser, chamois, smudge sticks and rags. Keep working the drawing until it is exactly what you want. Here is a finished portrait underdrawing using vine charcoal.\n\n**Materials**\n\n**\u2022 Drawing Materials:** Vine charcoal, soft graphite pencil, chamois or soft rag **\u2022 Surface:** A primed surface\n\n**2. Retrace With Graphite**  \nUsing a graphite pencil, retrace the important lines with medium pressure directly over the vine charcoal lines.\n\n**3. Wipe off the Charcoal**  \nWipe away the vine charcoal using a chamois or a rag to reveal the graphite lines underneath. Graphite will not be affected by any subsequent overpainting with acrylic, producing an underdrawing ready for painting.\n\nSTILL LIFE\n\nStill life depicts subjects such as flowers, shells, inanimate objects, fruit and skulls\u2014just about any object with the exception of people and animals. This style generally involves a grouping of objects, set off by a closely distanced background. Still life was popularized in the seventeenth century, and still holds fascination for us by implying humanity's symbolic presence, adding perspective on culture, lifestyles and attitudes of the times.\n\n**The Artist's Process**\n\nTraditional still life offers Sherry Loehr the opportunity to evoke a quiet and contemplative mood. For her, simplicity is essential. She chooses all of one type of fruit, for example, or clusters several items together to create one large shape. Loehr stages the objects in a classic still life format, then photographs them, leaving background areas open for interpretation. She uses photographs only to get an initial compositional sketch, then lets go of this reference to allow plenty of room for imagination. \"I can't think of anything more boring than copying a photo because you know what it will be like and the best it can be is as good as the photo,\" states Loehr.\n\nLoehr adds paint layers, continually working and reworking areas to expose varying underlayers. She loosely paints background areas using spontaneous patterns and textural areas in quiet tones. Her simplified backgrounds contrast with tightly detailed objects, keeping Loehr's work fresh and contemporary. Loehr was inspired early on by instructor Katherine Chang Liu (featured). Liu, an abstract painter, encouraged Loehr to find her own voice.\n\n **TIPS  \nfrom the Artist **\n\nLook beyond your normal vision for inspiration and explore a wide variety of work and resources that are not necessarily similar to your own style. Go beyond art museums and art books, and look at science museums, fashion, commercials, your house and local environment.\n\n**Loehr At Work in Her Studio**  \nLoehr's \"love-hate\" relationship with acrylic comes from the struggle necessary to achieve controlled edges and blending, while enjoying freedom and playfulness in the loosely painted areas.\n\n**Other Art in This Style**\n\n\u2022 Seventeenth century painters from the Netherlands, especially Dutch flower painting and trompe l'oeil\n\n\u2022 American modernists such as Raphaelle Peale and his group of early American still life artists\n\n\u2022 American trompe l'oeil painters John Haberle, William Michael Harnett and John Frederick Peto\n\n\u2022 Moody works of masters Francisco Goya, Gustave Courbet and Eug\u00e8ne Delacroix\n\n\u2022 Contemporary artist Janet Fish\n\n**Sherry Loehr**\n\n_Tulips and Oranges_\n\nAcrylic on board\n\n24\" \u00d7 36\" (61cm \u00d7 91cm)\n\nVARIATIONS ON STILL LIFE\n\n**Sherry Loehr**\n\n_Koi Chi_\n\nAcrylic on board\n\n36\" \u00d7 24\" (91cm \u00d7 61cm)\n\n**Variation 1: Using Living Subject Matter**  \nInstead of using the traditional inanimate objects for a still life, Loehr paints live swimming fish. Visible here is Loehr's trademark style contrasting realistically detailed forms with looser experimental backgrounds.\n\n**More Variations**\n\n\u2022 Change your painting formula. Make a list of all the habits and processes you use. Then make another list that contains their opposites. Break up your routine by trying one of the opposites. For instance, if you always start with a drawing or undersketch, switch to starting spontaneously with paint. If you always work from photographic references, try working without them. Vary your usual subject matter. Paint on different surfaces. Use a new medium.\n\n\u2022 Change the balance of elements between the background and foreground. Enlarge objects while minimizing the background, or expand the background and minimize the objects.\n\n**Mary L. Parkes**\n\n_Angelic Apples_\n\nAcrylic and oil on linen\n\n30\" \u00d7 40\" (76cm \u00d7 102cm)\n\n**Variation 2: Add Decorative Flourishes**  \nThe addition of decorative flourishes and geometric patterns turns this still life into a delightful fantasy.\n\n MIXING CLEAN BRIGHT REDS\n\nLoehr loves red; it plays a key role in her color palette and composition. Loehr enjoys employing its full range from a cool magenta-purple red to a warm orangey red. Here are two ways to keep reds looking their brightest with acrylic paint.\n\n**Overlay a Glaze**  \nSqueeze a warm bright red such as Cadmium Red and a cool bright red like Quinacridone Magenta onto your palette. Paint out an area of red by using either the Cadmium Red straight out of the tube or mix a small amount of white with Quinacridone Magenta (use about 5 percent white if you're using an opaque white like Titanium White. With a transparent white like Zinc White, use 20 percent). This will bring out the color intensity of the modern color. Do not add white to the mineral color Cadmium Red or it will lose its intensity and get chalky. Overpaint the red with a thin glaze made from a 1:1 mixture of untinted Quinacridone Magenta and gloss acrylic medium. Apply thinly using a soft flat brush or rag.\n\n**Pair the Red With a Complement**  \nTo make the red appear even brighter, paint a contrasting color next to it, such as a muted red, a green (which is red's complement) or a dark color. A muted red can be mixed using red, a touch of black (or green) and some white. Here a muted green is applied in varying values next to red to make it appear brighter.\n\n**Tip**\n\nAvoid buying paint from manufacturers that add white to their modern pigmented colors like Quinacridones and Phthalos. Start with an unadulterated paint (one that comes out dark from the tube) to get good color mixing control.\n\nThe ideas above can be performed with the other two primaries, yellow and blue. For yellow, use Hansa Yellow Medium, Cadmium Yellow Medium or Hansa Yellow Light. For blue, use Ultramarine Blue or Phthalo Blue (Green Shade). Phthalo Blue is the only color in this grouping of yellows and blues that would intensify with white. For the other colors, use them straight out of the tube for the brightest effect.\n\nLANDSCAPE\n\nLandscape painting depicts natural scenery such as mountains, trees and rivers, and still holds a powerful influence and inspiration for artists and viewers.\n\nDennis Culver, living in the southwestern United States, enjoys the local vistas, especially midday, when colors are quiet with no shadows. Not wanting his paintings to merely re-create nature, Culver always starts with a vision from his imagination. By painting directly outdoors and avoiding any photographic references, he distances himself from photographic reality. In the painting shown on Dennis Culver, Culver has added subtle geometry infused with triangles and native symbols, most visible in the foreground rocks, encouraging the viewer to move from scenery toward ideas.\n\n**The Artist's Process**\n\nIn _Arroyo Logic_ , Culver started with several earth tones on a toothbrush, spritzing a variety of dots along the bottom half of the painting. He then drew lines connecting the dots to create irregular geometric patterns which formed the initial inspiration and unusual composition in the foreground. Much of Culver's starting processes come from playing with materials. This play keeps his work spontaneous and varied. He prefers acrylic for its nontoxic, fast drying qualities, and vast potential in attaining varied sheens and surface treatments. Additionally, he enjoys the freedom of layering acrylic with no rules, as opposed to oil with its \"fat over lean\" restrictions.\n\n **TIPS  \nfrom the Artist **\n\nFamiliarize yourself with all the traditional mediums from watercolor to egg tempera. Experiment. Don't leave out acrylic because it's less traditional\u2014you'll be missing a whole lot of fun and surprising results.\n\n**Dennis Culver at Work in His Studio**  \nCulver's work isn't about being realistic, and always contains something surprising.\n\n**Other Art in This Style**\n\n\u2022 Minimized detail and grand panoramas of imaginary landscapes in early Chinese landscape paintings\n\n\u2022 Large scale and vast epic views of Frederic Edwin Church, Albert Bierstadt and other Hudson River School artists\n\n\u2022 Idealized landscapes of the Italian Renaissance painters like Titian\n\n**Dennis Culver**\n\n_Arroyo Logic_\n\nAcrylic on canvas\n\n56\" \u00d7 62\" (142cm \u00d7 157cm)\n\nVARIATIONS ON THE LANDSCAPE\n\n**Bruce Cody**\n\n_Capitol City Sunlight_\n\nAcrylic on canvas\n\n16\" \u00d7 30\" (41cm \u00d7 76cm)\n\n**Variation 1: Urban Landscapes**  \nFor Bruce Cody, art is about people and architecture, and can be seen as a social symbol. Cody uses inanimate elements such as buildings, cars, signage and telephone transmission poles to transform the objects into layers of space, atmosphere and time of day.\n\n**Phil Garrett**\n\n_Jones Gap\/Red Rocks_\n\nAcrylic on linen\n\n42\" \u00d7 60\" (102cm \u00d7 152cm)\n\nCollection of Palmetto Bank,\n\nGreenville, SC\n\n**Variation 2: Close Cropping**  \nHere Garrett's use of close framing strengthens the composition, creating an intimate and unusual view.\n\n**More Variations**\n\n\u2022 Use multiple references to create a new scene. One photograph can be used for its color palette, another for composition, and other photographs that contain natural forms such as trees, clouds, animals, etc. for added elements. Combine a sky from one photograph with ground from another.\n\n\u2022 Paint the same landscape three ways: Outdoors, from a photograph and from memory, then compare the results. Paint the same scene several ways in varying weather, or light during different times of day (see Claude Monet's Haystacks paintings).\n\n\u2022 Paint a landscape using unexpected colors, such as green for the sky and blue for the ground.\n\n  THE INVISIBLE GRID\n\nSmaller images or sketches (often called maquettes) make great starts for large paintings. This is especially helpful for landscapes. When traveling outdoors, it's easy and convenient to create many quick, small, on-the-spot pieces. Gridding provides a quick way to enlarge these later in the studio. Traditional grids using charcoal or graphite lines are difficult to remove later. This gridding technique, however, leaves behind no marks.\n\n**Tip**\n\nIf your painting surface has no sides, place push pins close along the edges. For large-scale works, use an additional color for main gridding lines to help navigate while transferring.\n\n**Materials**\n\n**\u2022 Paints:** Any acrylic paints\n\n**\u2022 Surface:** Any support (optimally with sides)\n\n**\u2022 Painting Tools:** A brush\n\n**\u2022 Other:** A small image you want to transfer and enlarge, cotton twine, pushpins, hammer, pencil, ruler, two waterproof markers differing in color, mylar or other clear plastic, scissors, tape, a small piece of cardboard, water\n\n**1. Prepare the Maquette**  \nBegin with a small image you want to transfer to a larger scale. Protect it by taping it onto a piece of cardboard and covering it with a taped piece of clear mylar or plastic.\n\n**2. Grid the Image to be Transferred**  \nUsing a waterproof marker, draw onto the plastic by tracing only essential lines and shapes. Keep detail to a minimum, focusing on large compositional lines (pictured in red). Change marker color for the grid lines (here using black), and create a checkerboard grid. Slip white paper between the plastic and the original drawing to isolate the lines and minimize unnecessary detail.\n\n**3. Grid the Painting Surface With String**  \nUsing a painting surface with sides, hammer pushpins at all points along the sides angled slightly upward. Take one end of a ball of string (the same color as the grid lines) and knot it onto one of the pushpins closest to a corner. Wind the string along the pushpins, creating all the vertical and horizontal lines, winding a bit around each pushpin as you go. Secure the string to the last pushpin with a knot. The string will be slightly raised from the surface if pushpins are angled correctly. Add water to a light colored paint and brush apply to re-create the same compositional lines from the maquette onto the painting surface. Focus on one grid square at a time. Allow the brush to go under the string, continuing lines that travel between grid squares. Instead of erasing to correct a wrong line, try changing color.\n\n**4. Remove Pins and String**  \nWhen all the compositional lines are painted, pull out the pushpins along with the string grid. The end result is a transferred image with no visible grid remaining.\n**Section 2**\n\nABSTRACTING  \nTHE WINDOW\n\n**Tom Palmore**\n\n_Phil With His Favorite Chicken_\n\nAcrylic on canvas\n\n72\" \u00d7 60\" (183cm \u00d7 152cm)\n\nAn abstract painting engages the viewer in a different way than realism. Abstraction minimizes detail and recognizable imagery, moving the emphasis toward aesthetic elements such as line, shape, perspective, form, texture and space. This section explores styles that still use some realistic principles such as approaching the canvas like a viewing \"window,\" and maintaining a sense of background, foreground and fixed perspective. Imagery here combines both realistic and abstract elements directing the eye toward more subtle aesthetics, often resulting in an unexpected twist.\n\nFIGURE AS FORM\n\nAs humans, we have a natural and immediate response to body imagery in art, and its use can ground a work in space and create a connection with viewers.\n\nFigures have always been a focus in Jylian Gustlin's work. She creates characters and sets them in moody and unfamiliar landscapes. By reducing detail in the figure and facial features, Gustlin also obscures the figure's gender and race. In this way, she hopes to move the focus of her work away from particular individuals and toward the idea of being human. Most communication, Gustlin muses, comes from body language or gesture, which carry the meaning of her work.\n\n**The Artist's Process**\n\nGustlin's fascination with and study of science and math, especially theories on the Fibonacci sequence, are visible in the way she divides her backgrounds and proportions the body. Metaphysical ideas, including stillness in motion, also play a key role in her work.\n\nGustlin starts with a wood panel, gesso and many layers of plaster and creates an etched relief using chisels. She continues to build multiple layers with a variety of materials: oil, acrylic, charcoal, wax, gold leaf, pastel and graphite. In the painting process, she makes as many mistakes as she can for as long as she can until the painting is finished. She eventually adds a final thin coat of resin to intensify the colors and emphasize texture. A creative block is rare for her, as she leaves her process open-ended, never deciding where she's going\u2014and, states Gust-lin, \"You can't get lost if you have no destination.\" When frustrated, she likes to load a brush with paint and throw it at a surface until something comes up. Inspirational artists for Gustlin include Nathan Oliveira, Richard Diebenkorn and the brooding, haunting qualities of Odd Nerdrum's portraits.\n\n **TIPS  \nfrom the Artist **\n\nMake art as much as possible to find your voice. Play as much as possible; don't be too serious. Combine everything you learn like a soup\u2014play, relax and paint. Mistakes are a way to bring the image into focus and to find your way to the fin-ish. If it doesn't work, it's not finished.\n\nAn expert in computer science, Gustlin finds parallel painting techniques to those found in Photoshop's layering tool. She even created her own computerized software which takes her scanned images of drawings, textures and photos and randomly combines them to make her abundant reference material.\n\n**Other Art in This Style**\n\nInventive combinations of figure and abstraction, including the Bay Area Figurative Movement, Francis Bacon, Aristide Maillol, Willem de Kooning's Woman series, Philip Guston and Jean Dubuffet.\n\n**Jylian Gustlin**\n\n_Bivium 18_\n\nAcrylic and mixed media (plaster, oil stick, pencil, resin) on panel\n\n48\" \u00d7 48\" (122cm \u00d7 122cm)\n\nCorporate collection\n\nVARIATIONS OF THE FIGURE AS FORM\n\n**Joey Fauerso**\n\n_Mouth to Mouth (1)_\n\nWatercolor and acrylic on paper\n\n48\" \u00d7 46\" (122cm \u00d7 117cm)\n\nPrivate collection\n\n**Variation 1: Isolation**  \nEmphasize or isolate parts of the body. Here an openmouthed portrait echoes the open mouth shape silhouetting or framing it.\n\n**More Variations**\n\n\u2022 Draw or photograph figures in various positions. Cut them out and place them on a variety of backgrounds\u2014solid or multicolored, smooth or textural, patterned or simple, abstract or realistic. Keep playing until ideas emerge that you like.\n\n\u2022 Trace outlines of figures. Working only with these generalized lines, change, expand, eliminate and add lines to transform the figure into something else.\n\n**Dennis Culver**\n\n_Politician_\n\nCharcoal and acrylic on paper\n\n44\" \u00d7 30\" (112cm \u00d7 76cm)\n\n**Variation 2: Transformation**  \nAllow figural shapes to transform and transmute. Starting with a charcoal drawing of textures, and combining with acrylic gesso, this playful investigation of materials formed into a central figure while creating a range of grays.\n\n  WORKING WITH MULTIPLE TEXTURES\n\nGustlin's work features seductive surfaces employing a variety of textures. Her technique uses multiple layers of plaster, chiseled to create etched relief lines and contrasting texture. Here is a variation using acrylic molding paste.\n\n**Materials**\n\n**\u2022 Surface:** A rigid and sturdy primed painting surface such as gessoed wood or panel\n\n**\u2022 Painting Tools:** A large plaster knife or other applicator; an array of texture tools such as combs, rakes, sticks and fingers\n\n**\u2022 Other:** Acrylic molding paste\n\n**1. Etch an Outline Into Wet Paste**  \nUsing a primed rigid support, generously apply acrylic molding paste all over the panel using a palette knife or wide plaster knife and smooth it as best as you can. Keep the layer about \u00bc-inch (6mm) thick or more. Using a tool that will easily create a line, such as a stick or the tip of the painting knife, draw the outline of a central form into the wet paste.\n\n**2. Create Varying Levels**  \nThe painting now has two distinct areas created from the etched lines in the previous step: inside the form and its background. Select one of the areas to be raised up higher than the other. Apply another layer of paste in that area in any level you prefer. The varying heights help to emphasize form and background. Continue to the next step while the paste is still wet.\n\n**3. Contrast Textures**  \nSelect one of the areas to add a texture. If the paste has started to dry in that area, reapply a fresh layer of paste. Apply combs and other tools to the wet paste, scrape and draw to get a variety of lines and patterns. Allowing one area untouched to remain smooth contrasts nicely with the textural area just created. Alternatively, in the smooth area, create a different texture by adding more paste, using different tools, or embedding beads or other small objects. Allow the surface to fully dry before applying paint.\n\n**Tip**\n\nTry adding color to the paste. Use different colored layers, sanding back to reveal the various colors. Sanding is also a good way to reduce texture. If sanding, always use waterproof sandpaper and water to eliminate exposure to toxic dust.\n\nREDUCTIVE\n\nOur eyes absorb an infinite amount of detail as we perceive reality. A reductive style reveals what the artist finds essential. Distillating unnecessary detail places the focus on bold shapes and the artist's personal vision.\n\nFrank Webster portrays postindustrial landscapes\u2014 the modern city\u2014combining minimalism and realism and revealing the connection between nature and technology. His editing process, essential to a reductive style, is driven by intuition and taste used for clarity and impact. Webster succeeds in carrying a hint of narrative with a clean elegance and a feel of minimal essentials.\n\n**The Artist's Process**\n\nInspiration comes from reading science fiction books, including the dystopian works of J.G. Ballard, with a great sense \"of the not too distant future.\" Trips to shoot reference photos, architectural research, even finding great compositions in films with epic cinematography, all add to his collective resources. Webster sketches and works digitally to get to the essence of the image. Multiple glazing layers add intensity to his simplified areas. Some of Webster's favorite artists include Tom Morandi, Dan Graham, Robert Smithson and photographer Robert Adams.\n\n**Other Art in This Style**\n\n\u2022 Reductive process: Compare Piet Mondrian's well-known grid abstractions with his early landscapes.\n\n\u2022 Reductive urban and industrial: Charles Sheeler and Ralston Crawford.\n\n\u2022 Contemporary and modern artists: Ed Ruscha, Andy Warhol, Roy Lichtenstein, Philip Guston, Jacob Lawrence, John Marin, Wolf Kahn, David True and Milton Avery\n\n **TIPS  \nfrom the Artist **\n\nAn editing process is idiosyncratic.\n\nWebster noticed postindustrial changes growing up in the midwestern United States. Charles Sheeler and his modernist contemporaries painted romanticized, optimistic versions of modernity and technology. Now, with hindsight and contemporary eyes, Webster adds a new outlook on the subject, with an intention toward subtlety.\n\n**Frank Webster**\n\n_High Rise_\n\nAcrylic on canvas\n\n86\" \u00d7 65\" (218cm \u00d7 165cm)\n\nVARIATIONS ON REDUCTIVE IMAGERY\n\n**More Variations**\n\n\u2022 Take photographs of places, people or events you like. Paint over areas with opaque paint to eliminate detail. See how much you can eliminate and still portray what you think is important.\n\n\u2022 Transform color images to black and white using photography and\/or computers to see them differently.\n\n\u2022 Play with colored cutout shapes, arranging and collaging onto larger colored paper, as an exercise in working with bold areas and no detail.\n\n**James Strom-botne**\n\n_Escalator Girl_\n\nAcrylic on canvas\n\n48\" \u00d7 60\" (122cm \u00d7 152cm)\n\nCourtesy of Handsel\n\nGallery, Santa Fe, NM\n\n**Variation 1: Contrast Flat Space With Perspective**  \nSimplified forms combine with linear perspective to create an interesting visual tension between flat and deep space.\n\n**Frank Webster**\n\n_Plastic Bags_\n\nAcrylic on canvas\n\n60\" \u00d7 80\" (152cm \u00d7 203cm)\n\n**Variation 2: Seek a Different Point of View**  \nSingling out an atypical view creates an immediate attraction in this piece.\n\n  ELEGANT CURVED EDGES\n\nA reductive style often makes use of simplified lines and edges. Dave Yust, whose work is shown, uses this method to get refined hard edges with curved designs.\n\nFor this technique, use masking tape that's \u00bc-inch (6mm) because it curves easily with minimal crimping.\n\n**Materials**\n\n**\u2022 Paints:** Any acrylic paint colors\n\n**\u2022 Surface:** Any primed painting surface\n\n**\u2022 Painting Tools:** Any soft painting brush\n\n**\u2022 Other:** \u00bc-inch (6mm) masking tape, any clear or non-colored acrylic medium or gel (glossy or matte), optional hair blow-dryer to quicken drying time\n\n**1. Tape the Area**  \nUsing a surface that is primed and\/or pre-painted with a background, plan where you wish to have hard edge painted lines or forms. Apply \u00bc-inch (6mm)-thick masking tape in a design or bordering a shape. Pull the tape firmly with one hand while pressing it into place with the other to keep the tape lying as smooth and flat as possible. Press along the edges to secure. Here a pre-painted background using a gradation of green and pink will remain visible in the final outcome only where tape is applied.\n\n**2. Seal the Tape Edge**  \nReduce paint seepage under the taped edges by applying a clear non-colored acrylic medium or gel with a brush along the edge of the tape. Quickly dry with a hair blow-dryer for a minute and continue to the next step.\n\n**3. Apply Paint Color**  \nApply paint color to either or both sides of the tape depending on your preferences, feeling free to paint over the tape when needed. Immediately remove the tape after painting. If the tape is left on the surface while the paint dries for a prolonged period, it may be difficult to remove later without pulling up the paint along with it. To reduce problems with tape removal, roll or peel it back onto itself rather than pulling upward.\n\n**Finished Example**\n\nFLAT SPACE\n\nSince a painting surface is naturally flat, one challenge for artists is to construct the illusion of space. In the twentieth century, a new trend emerged, rejecting the deep illusionistic space from the Renaissance. Instead, the emphasis was on flat space. The challenge changed to maintaining the integrity of a painting's image when flat to avoid looking like wallpaper. Detail and complexity still inform the work, but emphasis is placed on the painted surface and perspective is minimized. A master of this style, James Strombotne's sophisticated works employ deceivingly simplified areas and forms, clearly visible in his painting _Circus Elephant_ shown.\n\n**The Artist's Process**\n\nStrombotne makes it a point to draw every day, and believes this is an essential tool to keep his work fresh, while paint quality and personal experience are also key. Strombotne says, \"The craft of painting what you see is as rich and compelling as the use of the paint.\" Painting is more than just an illustration telling a story. Using a minimal palette helps Strombotne add simplicity with the appearance of effortlessness. He doesn't want his pieces to look like a lot of work, but instead to carry a sense of freedom and naturalness. Too reductive is boring, so the challenge for him is letting it be reductive but still compelling. After three to six layers of paint, he draws over the layers with charcoal, then paints over the drawing, sometimes covering the lines up entirely. An acrylic painter for over thirty years, he favors this medium over oil, because it maintains its luminosity and the feeling of naturalness.\n\nHe finds Georgia O'Keeffe's watercolors fresh, interesting and accomplished. He also likes artists such as Rembrandt, Francisco Goya, Henri Matisse and Pierre Bonnard.\n\n **TIPS  \nfrom the Artist **\n\nEach painting expresses your own personal vision when you express how you see the subject. You don't need to fill in every square inch of canvas. Being selective comes with maturity.\n\nStrombotne notes that artists in general seem to gravitate towards more reductive and simplified work as they get older. They are more apt to \"cut to the chase,\" as they mature.\n\n**James Strombotne**\n\n_Circus Elephant_\n\nAcrylic on canvas\n\n40\" \u00d7 36\" (102cm \u00d7 91cm)\n\nCourtesy of Handsel Gallery,\n\nSanta Fe, NM\n\n**Other Art in This Style**\n\n\u2022 Pop artist Tom Wesselmann, and the comic book style of Roy Lichtenstein\n\n\u2022 Comic book artists\n\n\u2022 Strong patterns and simplified flat space of Henri Matisse, Milton Avery and Joan Mir\u00f3\n\n\u2022 A wide range of flat space imagery from American modernists like Marsden Hartley and Stuart Davis. Also the work of Fernand L\u00e9ger and Anna (Grandma) Moses\n\n\u2022 Great masters from the Florentine School and Byzantine period\n\nVARIATIONS ON FLAT SPACE\n\n**Barbara Moody**\n\n_Chinois_\n\nAcrylic on canvas\n\n30\" \u00d7 40\" (76cm \u00d7 102cm)\n\nPrivate collection\n\n**Variation 1: Contrast Detail with Space**  \nThe contrast between exact detail with the shallow space adds a compelling quality to this still life.\n\n**More Variations**\n\n\u2022 Any techniques that bring attention to the actual surface of the painting, such as texture or sheen, will create an interesting visual tension when combined with the illusion of spatial depth. Experiment with gels and pastes to create texture before or after applying a painted image.\n\n\u2022 Reduce or eliminate perspective techniques. Take out horizon lines or other lines that angle into the picture. Increase the use of drawn lines and repeated geometric patterns. Play with the idea of spatial depth by enlarging the size of forms that appear far back, and reducing the size of objects and forms that are closer (nearer to the bottom of the picture).\n\n\u2022 Paint a realistic scene minimizing the color shifts or gradations that are normally used to create the illusion of light and volume. Instead, paint some areas using flat, evenly applied color.\n\n**James Strombotne** _Summer_\n\nAcrylic on canvas\n\n56\" \u00d7 68\" (142cm \u00d7 173cm)\n\nCourtesy of Handsel Gallery, Santa Fe, NM\n\n**Variation 2: Reduce Detail and Minimize Color**  \nReducing detail and minimizing the color palette places an emphasis on the artist's intent.\n\n  EDGE CONSCIOUSNESS\n\nPaintings using a flat or shallow space still depend on some illusion of depth within the image. Overlapping forms give the illusion of spatial depth, created through the use of edge relationships. A variety of edges from hard to soft, as well as distinct overlapping adds space and a clean look to the work.\n\n**Tip**\n\nRework and refine edges in a painting with a small brush and work with the colors both inside the forms as well as outside or background.\n\n**Undefined Edges Create Ambiguous Space**  \nHere a variety of forms are painted on a peach background. The edges of these forms are all somewhat ambiguous\u2014they are neither hard nor soft and do not clearly indicate whether one form is in front or in back of another. Notice how the space feels and compare this to the next image.\n\n**Define Edges to Enhance Space**  \nA wider variety of edges are created by intentionally making some edges cleaner, crisper and harder while others are softened and blended. Now it's easier to tell which form is in front of which, and the space feels more expansive.\n\n**Eliminate Edge \"Kissing\"**  \nThis image illustrates a common problem of shape \"kissing.\" On the left there are three shapes all sharing common edges. This makes it difficult to tell which form is in front of another. On the right the same set of three shapes overlap one another, thereby increasing the illusion of space.\n\nROMANTICIZED\n\nRomanticism originated toward the end of the eighteenth century in Europe, using images of untamed nature and eliciting emotions such as loneliness and a dreamlike state of mind. The use of soft edges, soft focus, hazy atmospheric qualities and muted palettes add to the surreal beauty often found in these works. Artists can use these techniques in conjunction with contemporary subject matter to evoke a nostalgic or emotional response. This is visible in the work of McCreery Jordan, whose images take on mysterious and haunting tones. In _Companeros_ , a horse and raven are engaged in some form of secret communication, the ring in the bird's beak underscoring the romantic flavor.\n\n**The Artist's Process**\n\nJordan's inspiration is time. Layers and veiling of time and history are echoed in her surface \"patinas,\" which she creates using antiquing, sanding and texturing, then repeating these processes in many layers. Sometimes she'll find and use old doors with their own history. If an object looks too new, she'll sand and stain it, texturizing and antiquing the surface for a few days while allowing the image to emerge. Collage and the use of photographic images are integral to her work and process. Jordan stays connected with her content on a deep emotional level, encouraging this to come through in the finished work.\n\nJordan finds inspiration in the work of Joseph Campbell, the super-realistic still lifes of William Harnett and Jean-Baptiste-Sim\u00e9on Chardin; the loose painting style and philosophy of contemporary artist Richard Schmid, and the textural imagery of Randall Lagro.\n\n **TIPS  \nfrom the Artist **\n\nThe ability to work from your subconscious will allow your vision to come through. This requires many years to develop the necessary skills and tools: including drawing, mistakes, lessons, workshops and miles of canvas.\n\nMcCreery Jordan in her studio\/gallery space with new work. Jordan uses acrylic for its fast drying capabilities, which allows multiple glaze layers, and the ability to work thickly without cracking.\n\n**Other Art in This Style**\n\n\u2022 Emotional dreamlike imagery of Surrealists such as Andr\u00e9 Breton, Marc Chagall, Salvador Dali, Ren\u00e9 Magritte and Dorothea Tanning\n\n\u2022 Moody work and color palettes of Paul Gauguin and Albert Pinkham Ryder\n\n\u2022 Emotional content of Expressionists such as Edvard Munch\n\n\u2022 Masters of Romanticism such as Joseph Mallord William Turner and Caspar David Friedrich\n\n\u2022 John Constable's romantic interpretations of the English landscape\n\n\u2022 Frida Kahlo, Emil Nolde\n\n**McCreery Jordan**\n\n_Companeros_\n\nAcrylic on wood\n\n48\" \u00d7 36\" (122cm \u00d7 91cm)\n\nVARIATIONS ON ROMANTICIZED IMAGERY\n\n**Ren\u00e9e Phillips**\n\n_Field of Dreams_\n\nAcrylic and oil on canvas with powdered pigments\n\n12\" \u00d7 12\" (30cm \u00d7 30cm)\n\n**Variation 1: Soft Colors and Harmonious Flow**  \nRomantic soft colors are used here along with a soothing directional flow moving toward a distant dreamy horizon.\n\n**More Variations**\n\n\u2022 Make a list of things (colors, images, stories, events, adjectives) you find romantic, emotional or beautiful. Select a few of these to assemble together in one image.\n\n\u2022 Mix a palette of colors that feels romantic to you, and use this to create a painting.\n\n\u2022 Make a series of images you find romantic, each using a different context or subject matter including portrait, landscape, still life and abstraction.\n\n**Beth Ames Swartz**\n\n_The Thirteenth Moon: Only the Mindless Waters Remain_\n\nAcrylic and mixed media on canvas\n\n60\" \u00d7 72\" (152cm \u00d7 183cm)\n\n**Variation 2: Limited Palette**  \nUsing a limited color palette with muted tones, Swartz infuses a spiritual and mysterious quality to this moonlit scene.\n\n  GLAZED EDGE ENHANCEMENT\n\nThis technique of vignette glazing can be used to add a romantic or mysterious feeling to a painting. By staining and darkening the outside edges of an image, the space feels more intimate. It can also add a feeling of containment in areas where the image appears to spill out of its frame.\n\nAvoid extending the glaze color too far beyond the edge into the image or it can become distracting.\n\n**Materials**\n\n**\u2022 Paints:** Any acrylic paint (suggested colors: black, and a variety of earth tones in brown, green, rust and ochre)\n\n**\u2022 Surface:** Any painting close to finish\n\n**\u2022 Painting Tools:** Lint-free rag and\/or painting brushes; color mixing palette\n\n**\u2022 Other:** A slow drying clear acrylic medium in gloss or matte\n\n**1. Select a Painting**  \nSelect a painting close to completion that will benefit by adding a romantic mood or more spatial containment. This painting's sky area feels too expansive in comparison to the dark and moodily painted ground. By selectively applying the vignette glazing technique in only the top sky area, the top and bottom halves will be more integrated, the whole image will feel more intimate and contained, and its romantic mood will be enhanced.\n\n**2. Make a Dirty Mix Glaze**  \nPlace several earth tone colors along with slow drying acrylic medium separately on a palette. Use at least four or five distinctly different color choices (pictured here are Sap Green, Burnt Sienna, Payne's Gray, Raw Umber and Carbon Black).\n\nDip a brush or rag into the medium, picking up about \u00bc teaspoon of it, then directly dip into a small amount of paint color. Both medium and paint will be on your rag or brush simultaneously but not mixed together, which is called a dirty mix glaze.\n\n**3. Glaze Along the Edges**  \nApply this dirty mix along a portion of the outside edge using small circular motions. Keep working in a small area while the paint is still wet. Switch to a clean brush or dry rag to remove some of the excess glaze so the color subtly blends into the image. Repeat with more glaze and keep cleaning the brush or use a new dry rag in areas as needed. As you finish one area, move along the same edge using different combinations of color every 1 (25mm) to 2 (51mm) inches. Avoid using the same color for too long or applying it opaquely. Instead, use a variety of color and transparency to subtly enhance the image.\n\n**Finished Example**  \nCompare this to the original image in step 1. Here, the colored edging in the upper half creates a cozy feel to the sky and enhances its mood.\n\nEXAGGERATION\n\nEmphasizing aesthetic elements such as line, shape, proportions, color perspective, texture and space bring individuality and personality to a painting. Martha Kennedy defines her work as \"bold compositions in mouthwatering color,\" and, sure enough, her color is juicy enough to taste. By exaggerating color, Kennedy enables the viewer to better respond with emotion. Kennedy wants her paintings to give people pleasure, to enable them to escape into another world. \"Colors help you feel better, like sharing a smile,\" she says. Kennedy strives for simplicity. When elements are simplified, exaggeration can then come into play to enhance those aesthetic elements the artist prefers, such as color.\n\n**The Artist's Process**\n\nTo get bold areas of exaggerated color, Kennedy works from photographs to assist in simplifying compositional areas. However, her color selections come from her imagination. She chooses colors that play against each other to pop, push, pull and create space. Once she's applied color to an area, she alters it, going lighter to darker, warmer to cooler, or brighter to duller as she creates volume.\n\nOil paint's slow drying properties allow for subtle blending that creates soft shifts in color, making oil Kennedy's primary medium. To replicate this effect, Kennedy has been experimenting with the new slow drying, or extended, acrylics. She has found ways to further extend her working time by using mediums, gels and thinners to resoften dried areas. The reworked surface creates a different type of texture, adding a new quality that she finds desirable and interesting.\n\n**Other Art in This Style**\n\n\u2022 Exaggeration\/simplification of Fernand L\u00e9ger, Milton Avery, Georgia O'Keeffe, Ed Mell, Amedeo Modigliani and Fernando Botero\n\n\u2022 The bold color of Vincent Van Gogh\n\n **TIPS  \nfrom the Artist **\n\nTemporarily switching mediums (i.e., from oil to acrylic) can add a new twist to your work and process. Stick to one medium long enough, though, to go deeper into an idea. Avoid being seduced or distracted by switching mediums too often. For some artists, however, switching to a new medium can add a sense of play, experimentation and new inspiration.\n\nKennedy tapes reference images to the top of her easel while she works. References include an original composition sketch and ink-jet prints of her photographs. Paints and painting tools are located on easy-to-reach dollies. Natural light from windows and skylights are a boost for color decisions.\n\n**Martha Kennedy**\n\n_Road Through Golden Hills_\n\nAcrylic on panel\n\n16\" x 16\" (41cm x 41cm)\n\nCollection of Debra and Chuck Blitzer,\n\nLas Vegas, NV\n\nVARIATIONS OF EXAGGERAT ION\n\n**Martha Kennedy**\n\n_Avocado on Orange_\n\nAcrylic on panel\n\n24\" x 24\" (61cm x 61cm)\n\n**Variation 1: Enlarge and Zoom In**  \nHere Kennedy changes her subject matter from landscape to still life. Simplicity is achieved by selecting and enlarging one object, close cropping and keeping the background minimal. The large, simplified shapes are then painted with exaggerated color.\n\n**More Variations**\n\n\u2022 Try temporarily switching to a medium you haven't used before, such as pastel, watercolor, oil or gouache, and see how this changes your work.\n\n\u2022 Create a few paintings with the same image, but in varying sizes, going from very small to very large to compare the impact.\n\n\u2022 Create a painting from your head that eliminates all recognizable imagery, perhaps using geometric shapes and other nonrealistic forms, and modulate each color shape to give it volume.\n\n**Jakki Kouffman**\n\n_Red Rock, Near Creek_\n\nAcrylic on canvas\n\n40\" x 30\" (102cm x 76cm)\n\nCollection of New Mexico Arts, Art in Public Places,\n\nSan Miguel County Courthouse Annex\n\n**Variation 2: Exaggerate Brushwork**  \nExaggerated textural brushstrokes add a sensual feel to the surface. In addition, the edges between forms are exaggerated, made crisper like stain glass, emphasizing line and pattern.\n\n  REWORKING SLOW DRY ACRYLICS\n\nSlow drying acrylics have a longer open time, which means they stay wet longer than normal acrylic paints. Plus, when the slow-dry paint layers become dry to the touch, you can still rework them for a day or two by using any of the slow drying mediums, gels, thinners and\/or water. These products can be added directly into fresh paint color, or applied straight over a dried layer to rework it. However, once a paint layer has been drying for more than a day or two, it may not be possible to revive it, and then it's best to apply new layers of fresh paint. There are a variety of extended paints. The best results will be obtained using those that stay wet more than twenty-four hours, like Golden's OPEN Acrylics.\n\n**Materials**\n\n**\u2022 Paints:** Slow drying acrylic paint\n\n**\u2022 Surface:** Any painting surface\n\n**\u2022 Painting Tools:** Brush, palette knife\n\n**\u2022 Other:** Mixing palette, slow drying acrylic medium, slow drying acrylic gel, slow drying acrylic thinner, water\n\n**1. Apply slow drying acrylic paint**  \nUsing two slow drying acrylic paint colors, paint two shapes which touch, creating a hard edge. Let dry for several hours.\n\n**2. Resoften dried areas**  \nWith a brush, apply a slow drying acrylic medium, gel, thinner or water over the dried colors near the area where the two colors touch. Let this remain on the dried paint layer for a few minutes to resoften the paint.\n\n**3. Blend**  \nWith a brush, rework the softened color layers to create a smooth transition or softer edge between the two shapes.\n**Section 3**\n\nCHANGING  \nPERSPECTIVES\n\n**Catherine Mackey**\n\n_Pier With Cruise Liner_\n\nAcrylic and mixed media on wood panels\n\n65\" \u00d7 48\" (165cm \u00d7 122cm)\n\nFor centuries a painting was seen as a \"viewing window.\" Most images included a background, foreground, horizon line and fixed perspective. Cubism emerged in the early twentieth century, offering an alternative to the way an image was perceived. Using multiple perspectives in the same image, the fixed viewing point was eliminated, adding a new sense of movement and an interactive approach to viewing art. Cubist masters Pablo Picasso and Georges Braque spearheaded these concepts, while contemporary artists such as David Hockney continue to push them even further. New concepts are continually emerging based on how we really see. In real life, our eyes constantly move, darting around from close-up detail to the distant horizon. We remember an event or scene using a combination of snapshot views taken with our eyes from multiple perspectives. Multiple viewing points in a painting can often feel more real than a conventional photograph taken by a camera. On the other hand, going against our expectations of visual reality can feel surprising, upsetting or jolting to one not familiar with this new way of seeing in contemporary art. The styles in this section reflect the artists' preference to work with these new concepts and to involve the viewer in perceiving more actively.\n\nINSIDE & OUTSIDE\n\nAn image that combines indoor spaces with outdoor environments can convey a sense of spaciousness, freedom and movement. This journey through both natural and human-made environments often employs a broad spectrum of light and shadow effects ranging from natural light sources to incandescent. Hyperrealist painter Gerard Boersma takes us on a magic carpet ride with his paintings through the world's street corners, shops and stores to underground subways. Emphasizing the theme of modern isolation and society, Boersma's work hints at subtle narratives where solitary figures are placed in mundane activities, suggesting the feeling of being alone even in a crowd.\n\n**The Artist's Process**\n\nWorking from photographs and the computer to adjust color and composition, Boersma draws the image onto his painting panel in pencil and continues with paint, making changes when needed. Since he concentrates on one painting at a time, acrylic's fast drying qualities are an advantage, allowing Boersma to layer quickly. Clarity and realistic detail are important to Boersma, who wants his images to engage the viewer in a virtual experience. Artists inspiring to him include Edvard Munch, Johannes Vermeer, Max Ginsburg, John Currin and his great uncle, also a painter, Jopie Huis-man. Super-realist favorites include Ralph Goings and Richard Estes.\n\n **TIPS  \nfrom the Artist **\n\nPractice! Paint what intrigues you. Outliers: The Story of Success by Malcolm Gladwell reports it takes ten thousand hours of time and practice to make something successful.\n\nGerard Boersma deliberately keeps a clean and organized studio, which he notes helps him paint good realism.\n\n**Other Art in This Style**\n\n\u2022 American scenes focusing on loneliness in a modern city: Edward Hopper\n\n\u2022 Scenes using strong lighting contrasts: Masters such as Caravaggio and Rembrandt\n\n**Gerard Boersma**\n\n_The Smoker (self portrait)_\n\nAcrylic on Masonite\n\n37\" \u00d7 28\" (94cm \u00d7 71cm)\n\nPrivate collection\n\nVARIATIONS ON INSIDE & OUTSIDE\n\n**Paul Sarkisian**\n\n_Untitled (El Paso)_\n\nAcrylic on canvas\n\n14' \u00d7 21' (4.3m \u00d7 6.4m)\n\nPhoto by Eric Swanson\n\n**Variation 1: Blur Boundaries**  \nThis trompe l'oeil painting depicts a storefront fa\u00e7ade with incredible lifelike detail, and is even more startling when viewed in person. Here interior and exterior mingle. The artist's use of a black-and-white palette changes the viewing experience from an immediate association with it as a painting to a more conceptual one.\n\n**Hamish Allan**\n\n_House and Hills_\n\nAcrylic on canvas\n\n24\" \u00d7 24\" (61cm \u00d7 61cm)\n\nCollection of the artist\n\n**Variation 2: Simplicity and Repetition**  \nAreas are simplified, creating a bold abstracted landscape. Here the eye travels back and forth between the outdoor and indoor spaces, assisted by the repetition of green used in both places.\n\n**More Variations**\n\n\u2022 Experiment with small collages using interior and exterior images to find unique ways of combining the two.\n\n\u2022 Go out on a photography mission to find shots that naturally combine interior and exterior, like storefront windows that reflect the street while revealing the store's interior.\n\n\u2022 Journal about your emotional response on how interior and exterior spaces differ in feel, and paint that.\n\n  OBTAINING A RANGE OF VALUES\n\nPainting a convincing image that portrays both indoor and outdoor spaces requires a broad range of light and dark values. One surefire way to obtain a variety of these tones in a painting is to pre-mix a full range of colors ahead of time, offering maximum potential. Here is my favorite full palette to achieve this.\n\n**\u2022 Select Optimal Colors for Maximum Color Mixing.** You need a full palette to obtain a full range of colors. Full palettes contain both a warm and a cool for each of the three primary colors, plus white. Adding other colors such as black, green, orange and violet add convenience, but are not necessary.\n\nAlong the outer rim of a large palette, apply paints in an arc, allowing ample space in the middle for mixing color and placing mediums, if desired. Squeeze out a generous amount of each of these suggested colors plus Titanium White:\n\n1. Choose either Hansa Yellow Light or Cadmium Yellow Light for a cool yellow.\n\n2. Use either Hansa Yellow Medium or Cadmium Yellow Medium for a warm yellow.\n\n3. Select from: Cadmium Red Medium or Light, Naphthol Red Medium or Light, Pyrrole Red or Pyrrole Red Light for a warm red.\n\n4. Quinacridone Magenta, a cool red that mixes a clean violet.\n\n5. Ultramarine Blue, a warm blue.\n\n6. Phthalo Blue (Green shade), a cool blue.\n\n**\u2022 Add a Tinted Swatch Near Each Color.** Modern colors look very dark on the palette, while mineral colors appear bright. It can be deceiving to see the colors only in their thick form or mass-tone. Adding white, which turns the colors into a tint, offers different results depending on whether the pigment is modern or mineral.\n\nNext to each color, add a small amount of white in a separate mixture, giving you a reminder of the true potential of each color. In the tinted form, modern colors get brighter, while mineral colors get chalky and muted.\n\n**\u2022 Pre-Mix Lights.** Most paint colors unmixed out of the tube have a middle to mid-dark value range; therefore, a palette using only paint directly from the tubes will produce a dark painting. Pre-mixing a few light values before starting to paint helps solve this problem. Make a 20:1 mixture of Titanium White with red. If the resultant mixture appears too pink, add more white. Make a second light mixture using the same ratio of white mixed with yellow, and a third using white and blue.\n\n**\u2022 Pre-Mix Darks.** There's nothing wrong about using black straight from the tube. Darks are further enhanced, however, by the addition of pre-mixed custom blacks. Make a mixed black combining a modern red and blue (i.e., Quinacridone Magenta with Phthalo Blue) and a small amount of any yellow. Test your mixed black by adding a small amount of white in a separate mixture to see its tinted version of gray. Make one or two more mixtures using the same three colors, but vary the combination ratios to make rich dark browns and cooler or warmer grays.\n\nMULTIPLE PERSPECTIVES\n\nCombining multiple perspectives in the same image allows a variety of viewing points, movement and an invitation for multiple interpretations. This idea has been used by many artists in the twentieth century, most notably with Cubism. Ines Kramer's imaginary cityscapes with a kaleidoscope of planes offers unexpected associations and an overall surprising image. Born in Caracas, Venezuela, Kramer has moved and traveled extensively. Through this rich variety of surroundings Kramer developed a habit of synthesis, absorbing images from all sources and arranging them in new and different ways.\n\n**The Artist's Process**\n\nKramer calls herself a \"hunter-gatherer of images,\" and she goes on several photographing trips each year, later reassembling these images into landscapes of her imagination. Her current interest in urban landscape takes her to large cities, where she captures objects close up such as windows, doors, architecture, rooftop gardens and city trees. Placing close-up shots next to far shots in these reconstructed cityscapes suggests a live city experience with its jostling, overstimulating space.\n\nKramer takes photographs with the intent of capturing a shape or object, not taking a great photograph. This way, she doesn't mind cutting them up or overpainting the photos later. After manipulating them on the computer, Kramer prints out the images, cuts them up and collages them onto a painting surface, which she overpaints with acrylic paint. Some images are completely buried with layers of opaque paint, while others are still visible through layers of transparent glazes. Kramer also adds watercolor, colored pencil, lead pencil and found imagery. Working in layers, Kramer has found the fast drying nature of acrylic works best. Acrylic is a glue as well as a painting medium, making it a good choice for collage, without harming the photographs as oil would.\n\n **TIPS  \nfrom the Artist **\n\nMake sure to have a designated space devoted exclusively to your art\u2014even if it's just a TV cart (my very first \"studio\")\u2014so that you can work on or just look at and think about your work every day. The easiest way to become inspired and stay inspired is to do the work every day. Avoid teachers whose students' work looks just like the teacher's\u2014that teacher will never help you find your own vision.\n\nInes Kramer in her Santa Fe studio. Her large size paintings can contain over twenty-five photographic images. The collaged photographs give her a jumpstart and add inspiration as she continues to change and shift the work.\n\n**Other Art in This Style**\n\nFor multiple perspectives, check out Cubist artists such as Pablo Picasso, Georges Braque and Paul Klee. David Hockney works with a unique contemporary version of Cubism especially visible in his photo collages. See also Wassily Kandinsky, David Salle, James Rosenquist, Jos\u00e9 Clemente Orozco and Richard Diebenkorn.\n\n**Ines Kramer**\n\n_Rooftop Eden_\n\nAcrylic and collage on panel\n\n32\" \u00d7 36\" (81cm \u00d7 91cm)\n\nVARIATIONS OF MULTI PLE PERSPECTIVE S\n\n**Catherine Mackey**\n\n_Whiz Burgers_\n\nAcrylic and mixed\n\nmedia on wood panel\n\n24\" \u00d7 53\" (61cm \u00d7 135cm)\n\n**Variation 1: Combine Separate Panels**  \nJoining separate painted panels contrasts varying perspectives in this assemblage painting.\n\n**Mary Morrison**\n\n_Grasslands_\n\nAssembled acrylic painted fragments on panel\n\n36\" \u00d7 45\" (91cm \u00d7 114cm)\n\nCollection ofThe Herman Memorial Wellness Center, Houston, TX\n\n**Variation 2: Merge Different Images**  \nCombine two completely different images. Here a landscape and abstract designs are spliced into geometric shapes and reassembled into a completely new image.\n\n**More Variations**\n\n\u2022 Pick one particular topic that interests you. Gather all the images of these you can from varying sources: magazines, photocopies from books, your own photos and drawings; and collage these together in unusual patterns and arrangements.\n\n\u2022 Find and cut out pieces of color with no recognizable imagery (i.e., paint swatches from home decorating stores, cut pieces of ads and product labels). Arrange these and collage them to create an abstract or color field, with or without the addition of painted areas.\n\n\u2022 Exaggerate viewpoints: Pick a subject you like: people, landscapes, skies, pets and animals, food, clothes, etc. Photograph it several times from different viewpoints. Collage these by overlapping images.\n\n\u2022 Create a painting that has no horizon line, thus eliminating a finite point of view.\n\n  THE POWER OF HORIZONTALS\n\nHorizontal lines in a painting are easily interpreted as a horizon, and can add a grounded feeling to the work. Many abstract styles add a horizontal line to the image to ease visual tension, while others eliminate it to move the viewer into another reality. Here are some examples to illustrate this idea.\n\n**Tip**\n\nA horizontal line is a grounding tool, not an essential element in a work. It can be used or avoided depending on your intent for the image.\n\n**Evoking a Landscape**  \nThis painting contains very few elements. With only three colors, the composition consists of a gradation of two colors in the top area and a simple horizontal band of color for the bottom. Note how easy it is to interpret this as a landscape, even with nontraditional sky and ground colors. It is our sense of being human, our relationship with the planet and its continual pull of gravity that defines the essence of this viewing experience.\n\n**No Horizon Line Increases Visual Tension**  \nThis painting has no obvious horizontal lines, giving it a floating quality with a visual tension that can offer a new and vital viewing experience.\n\n**Adding a Horizon Line Grounds the Viewer's Perspective**  \nHere the same painting is changed completely with the addition of a simple horizontal line. There is a new feeling of comfort and ease in the viewing.\n\nCOMPOSITE ARRANGEMENTS\n\nThis style, also called assemblage, is a form of artistic construction where multiple individual images and surfaces are combined to create a new, but segmented larger whole. Catherine Mackey relies on her experiences as an interior architect to create an unusual body of work using this process. In a definite reaction to her former career's pristine and controlled nature, her paintings add elements of dirt and grit. Collisions of color and fragments in her urban environment catch her eye as she notes inadvertent beauty, searching for urban decay on walls that carry a visual history of stories about people.\n\n**The Artist's Process**\n\nTen to thirty bits of found wood, reworked and often repainted, can always be found hanging around Mackey's studio. She uses these to play and arrange. Allowing for accidental associations and chance, her process carries an unpredictable timing. Texture is important, and she'll often use building materials like ceramic tiles or Sheetrock compound, pressing other objects into it. She uses acrylic for most of her painting purposes. Her techniques are varied, and include stenciling and thick stippling for adding text in relief. Colored areas are sanded, scratched and washed down. Scrubbing creates an accelerated look of aging. When pieces come together in a way she likes, she connects them by constructing a backing frame, then continues painting them together as one image.\n\nArtists who inspire Mackey include: Robert Rauschenberg and the complete freedom he used to work with anything, repeat imagery, and reuse things in other works; Frank Auerbach's thick layers using lots of subtraction and peeling back; Anselm Kiefer's monolithic early work, using massive scale and architectural composition with miles of field; famed graffiti artist Jean-Michel Basquiat; David Ireland's unusual installations.\n\n **TIPS  \nfrom the Artist **\n\nMackey works mostly from photographs in her studio. Yet even with plenty of photographs on hand, there are times when she still needs to energize her work and\/or attitude, and ventures out with her camera exploring for more. This will inevitably reveal something new. She recommends one of her daily rituals: spending ten minutes every night looking through art books she might not have picked up for a while. Falling asleep while thinking of new ideas keeps her creative energy alive.\n\nCatherine Mackey surrounded by a variety of assemblage panels and materials in her California studio.\n\nHer term \"architectural palimpsests\" aptly describes her interest in graffiti, signs and wall writing. Ripped posters stuck onto each other year after year build up with a patina she can't resist. She peels these treasures off their original surface, soaking them in a tray of water to become unique collage fodder.\n\n**Other Art in This Style**\n\n\u2022 Assemblage artists: Robert Rauschenberg, John Baldessari, Louise Nevelson, Mike Kelly, Jennifer Bartlett\n\n\u2022 Cubist masters: Georges Braque and Pablo Picasso\n\n\u2022 Fluxus Movement artists\n\n\u2022 Marcel Duchamp's readymades, Kurt Schwitters, Man Ray, Joseph Cornell\n\n**Catherine Mackey**\n\n_Tony's Journey_\n\nMixed media on wood panels\n\n24\" \u00d7 238\" (61cm \u00d7 605cm)\n\nVARIATIONS OF COMPOSITE ARRANGEMENTS\n\n**William Dunlap**\n\n_Landscape & Variable: Landscape Askew _\n\nPolymer paint on canvas (with wood, slate, leather, mixed media)\n\n41\" \u00d7 180\" (104cm \u00d7 457cm)\n\nCollection of Washington Convention Center, Washington D.C.\n\n**Variation 1: Use a Variety of Materials**  \nWilliam Dunlap uses materials both found and fashioned, often confusing viewers over what is made or found. The implied narrative is made exciting and contemporary by Dunlap's arrangement, angled placement and use of materials.\n\n**More Variations**\n\n\u2022 Assemble a collection of objects and images that you find or make. Do not permanently adhere the elements together in the beginning to allow for maximum play and flexibility. As you find arrangements you like, photograph them for later use as reference, while continuing to rearrange.\n\n\u2022 Dig out old paintings and drawings that are no longer of interest to you. Cut them into new shapes and pieces. Rearrange them on a new surface or over another painting.\n\n**Don Quade**\n\n_Winter Lotus_\n\nAcrylic, oil stick, graphite and paper collage on wood panel\n\n48\" \u00d7 48\" (122cm \u00d7 122cm)\n\nPrivate collection\n\n**Variation 2: Assemble Non-Related Sections**  \nThree distinct sections offer a playful dialogue about the artist's interest in nature. During a winter visit to a botanical garden, Don Quade noted flowers trapped in ice. The combination of abstraction and strong divisions turned his experience into a metaphor for space and time. The top sections refer to clouds and sky, while the bottom represents rain.\n\n  ANTIQUE EFFECTS\n\nWhen Mackey needs to make something look old, she likes to work with acrylic washes, alternating with subtractive scrubbing. Here is a technique to get similar results, especially helpful with found objects and collage items that look too new.\n\n**Tip**\n\nBefore starting, hold your surface up to the light, and notice the sheen. If some areas are matte and some are glossy, you'll have a greater variety of effects. Optionally, you can prepare the surface by applying matte and semigloss mediums or gels separately in several places on the surface. Let dry before starting. To emphasize the stained effect even more, reapply semigloss acrylic gel in between wash layers, letting it dry before adding any more washes.\n\n**Materials**\n\n**\u2022 Paints:** Any acrylic paint colors in earthy tones such as black, brown, olive green and rust (suggested colors: Burnt Sienna, Quinacridone Nickel Azo Gold, Viridian Green Hue, Transparent Red Iron Oxide, Green Gold, Bone Black)\n\n**\u2022 Surface:** A primed surface with a semigloss or matte sheen\n\n**\u2022 Painting Tools:** Any brush\n\n**\u2022 Other:** Dish scrubbers or sandpaper, water, spray bottle, paper towel or rag\n\n**1. Apply Acrylic Washes**  \nWet the painting surface with enough water to form small puddles. Select a few colors and dilute heavily with water. Apply all the colors randomly over the wet surface. Lightly spray more water over the applied paint and entire surface. Allow the paint to puddle up and move around on its own. Let this dry on a level surface.\n\n**2. Scrub the Paint**  \nIf the dried stains from the previous step are too noticeable, reduce them with this step. Select a dish scrubber or any abrasive tool such as waterproof sandpaper. Apply water to the surface, then remove some of the paint using your scrubber, applying medium pressure in a variety of scrubbing directions. Wipe the excess paint off with a paper towel or rag, and let dry.\n\n**3. Repeat Steps**  \nIf the stains are too subtle, you can repeat steps 1 and 2, layering over the previous one, until you've obtained the desired results.\n\n**A New Use for Old Water**\n\nInstead of diluting a fresh batch of paint for the washes in step 1, use some leftover dirty paint water from your studio.\n\nCOLLAGE\n\nThe term collage comes from the French word _coller_ which means \"to stick.\" A collage incorporates multiple images applied to a surface, usually overlapping one another to partially reveal and partially cover. A whole new image emerges, offering multiple perspectives.\n\nCollage artist Nancy Scheinman takes her paintings to another level by making most of her unusual collage materials. For instance, she washes hand-embossed sheets of copper with acid to create patinas that she nails onto wood panel, which she paints or prints. The unusual formats and ways she combines these materials add a unique quality to her work.\n\nScheinman presents and assembles new stories incorporating themes such as finding balance in life or the myth of a purer past. Her signature motifs include towers, borders, woman protagonists, birds, leaping dogs and red circles\u2014her personal symbol\u2014which she presents in dreamlike landscape. She presents her stories in snippets, allowing the audience to bring their own experience and interpretation to the work. Using the framework of a modernist geometric grid, borders visually contain the multiple stories within each piece, which are slowly revealed to the viewer.\n\n**The Artist's Process**\n\nScheinman freely applies layers of paint and materials, and, just as freely, takes them off. Like fabric in a quilt, she cuts and layers sheets of copper paint and patterns, and collages them together with nails, tacks and antique materials, literally connecting the past with the present. She is inspired by the late contemporary artist Hollis Sigler, whose work contains an admirable strength in dealing with personal narrative; as well as Italian Baroque painter Artemisia Gentileschi, a bold artist who dared to represent historical scenes at a time when it was not considered appropriate for women artists. Gentileschi was the first female painter to become a member of the famed art academy in Florence.\n\n **TIPS  \nfrom the Artist **\n\nA well-known art instructor, Scheinman uses her theme \"Nine Who Dared,\" based on nine artists who dared to face artistic criticism, as inspiration for students to bravely create more personal work.\n\nNancy Scheinman at work in her Maryland studio. A visual storyteller, her use of personal narratives and internalized images from extensive travels are transformed into collective myths and pieced together as part of a universal, spiritual and domestic experience.\n\n**Other Art in This Style**\n\n\u2022 Contemporary artists David Salle, David Hockney, Jim Dine and Gilbert and George\n\n\u2022 Collage work of modernists Pablo Picasso, Georges Braque, Henri Matisse, Joseph Cornell, Man Ray, Robert Motherwell and Peter Blake\n\n\u2022 Merz pictures of collage artist Kurt Schwitters\n\n**Nancy Scheinman**\n\n_Repeated by a Bird\u2014Melody_\n\nAcrylic; canvas; antique tin; brass; etched and\n\ninked vinyl; hand-embossed printed, painted\n\nand patinated copper on wood panel\n\n21\" \u00d7 21\" (53cm \u00d7 53cm)\n\nVARIATIONS OF COLLAGE\n\n**Cate Goedert**\n\n_Repose_\n\nAcrylic and mixed media on canvas panel\n\n16\" \u00d7 8\" (41cm \u00d7 20cm)\n\n**Variation 1: Merge Photography with Paint**  \nThe use of transfer techniques allows an interplay between photography and paint. A photographer with a theater background, Cate Goedert plays with dramatic lighting and atypical formats. A copyright-free photograph in the top portion of the painting is kept transparent and transferred onto a painted background. The bottom portion of the painting contains a photograph that Goedert shot, as well as paint stenciling, crackle paste and textured surfaces. (See for the digital transfer technique.)\n\n**Darlene McElroy**\n\n_Ode to the Masters_\n\nAcrylic on panel with image transfer, collage and gold leaf\n\n39\" \u00d7 39\" (99cm \u00d7 99cm)\n\n**Variation 2: Unusual Formats**  \nOld covered doors are used here, adding an interesting shaped surface resembling a house or shrine. The image is a tribute to Diego Vel\u00e1zquez, painted in the ornate style of his time.\n\n**More Variations**\n\n\u2022 Find printed images to cut out for a collage, but cut them with excess borders to create shapes that differ from the images themselves.\n\n\u2022 Set up a still life using one of your paintings as the backdrop and assembling real objects in front. Photograph the setup to use as a reference for a whole new painting.\n\n\u2022 Collect and save the numerous paper items you receive daily in junk mail, wrappers, product labels, etc. Color, cut, mix and match them to create something different.\n\nSMOOTHING RAISED EDGES\n\nA collage usually combines paint with other materials. Paper, fabric, magazine images and other printed imagery often use inks and dyes that are not lightfast or permanent. Thicker materials like cardboard or heavy fabric can create a raised edge and present another challenge for artists who prefer the edges to transition smoothly into the painting. A smoother surface can be obtained by applying an acrylic gel over the collaged materials. Fading can be reduced by using a gel with UV protection.\n\n**Tip**\n\nThe gel adds a nice handmade texture to the final surface. Generally, a thick gloss gel will suffice. Use a matte gel if you prefer a slightly cloudy, veiled layer that will somewhat obscure the underlying painting. If you desire a glassy smooth surfboard-like finish, you can add a layer of pourable acrylic. See for additional pouring tips.\n\n**Materials**\n\n**\u2022 Surface:** Any collage acrylic painting that is finished or in-process on any surface\n\n**\u2022 Painting Tools:** Palette knife or other wide, flat application tool\n\n**\u2022 Other:** A thick gloss acrylic gel with UV protection (if unavailable use two separate products\u2014a gel and a UV product)\n\n**1. Preparation**  \nSelect a painting with texture or relief edges that needs smoothing. Using a knife with a stepped handle so that your hand will not drag through the gel while applying, load up the back of the knife with a generous amount of the gel.\n\n**2. Spread the Gel**  \nSpread the gel evenly over the entire surface. While applying the gel, keep the knife lifted slightly off the surface, at least by \u00bc inch (6mm), so the knife never actually touches the surface. This will allow the layer to be thick enough to cover all the texture and relief. If your edges are thicker than \u00bc inch (6mm), apply the layer more thickly, or apply a second layer after the first has dried. Additionally, you can add more objects or images into the gel while wet and they will adhere. The gel will turn clear when dry.\n\n**3. Add UV Protection**  \nIf the gel used in the previous step does not have UV protection, then apply another layer using a UV product such as Golden's Archival Varnish. Follow the label's instructions, diluting if required, and brush or spray accordingly. The more coats of a UV product you apply, the longer the colors will remain intact and resist fading.\n**Section 4**\n\nENGAGING  \nTHE  \nPICTURE  \nPLANE\n\n**Jim Waid**\n\n_Blue Shimmy_\n\nAcrylic on canvas\n\n78\" \u00d7 96\" (198cm \u00d7 244cm)\n\nCollection of Kevin Osborn, Tucson, AZ\n\nImages that create the illusion of coming forward confront the viewer, or break the illusion of pictorial depth, and attempt to visually engage us on the front surface of the canvas. This is a significant change from the Renaissance \"window\" where attention to the front picture plane was avoided. The styles in this section contain works which are sometimes bold, blatant and appear to meet us head-on. Shock imagery quickly comes to mind as one way to create this effect. Highly textured surfaces, flatly painted forms and shallow space can also bring our attention to the front picture plane. This section offers a variety of styles and approaches to engage the picture plane, using a range of aesthetic aspects.\n\nMAPPING: PLANES & FRAGMENTS\n\nMapping involves a schematic display that is readily seen in maps, science book diagrams and graphs. A painting that uses mapping can create a sequence in the way it's viewed, as in comic book strips, scrolls and altered books.\n\nDannielle Tegeder makes use of this new approach, inspired primarily by the architectural blueprints and technological sketches she has been exposed to since childhood. Her use of this style, along with mechanical references and forms, creates abstract fields that evoke postmodern visions of the future. The spaces can feel disorienting with no recognizable horizon line, floating architectural fragments and a precarious balance of objects.\n\n**The Artist's Process**\n\nTegeder sands layers of acrylic molding paste on canvas over panel to create a smooth drawing surface, where she then draws with pencil. She then layers paint, transfer skins and gels together to create the final textured surface. For her, adding handmade elements and imperfections offer a humanizing element and balance the hard edges and analytical motifs. Using an intuitive process, Tegeder keeps the end result unplanned, allowing each step to respond to the one before. The sense of space created in Tegeder's paintings shifted when she started working with installations and sculpture. _Inropataciland_ shows her earlier work using a flatter space, compared with _Deconia_ which shows a more expanded space using angles and loosely worked areas.\n\n**Other Art in This Style**\n\n\u2022 Contemporary artists: Squeak Carnwath, Ed Ruscha, Joyce Kozloff, Newton and Helen Mayer Harrison, Julie Mehretu, Vernon Fisher, \u00d6yvind Fahlstr\u00f6m, Jean-Michel Basquiat, Pat Steir's early work, William Wiley\n\n\u2022 Egyptian hieroglyphics\n\n\u2022 The Map as Art by Katharine Harmon, a book on contemporary artists using this style\n\n **TIPS  \nfrom the Artist **\n\nEmerging artists often overdose on art magazines. Look at things not directly related to your work, including unfamiliar art styles, music, books and poetry. Follow things that interest you, even non-art interests, which can still inspire. Don't buy into other people's stuff.\n\nDannielle Tegeder in her New York studio. Tegeder is inspired by cities and charts organizing information about people, like transportation routes, population charts and aviation maps. Organizing color and form is integral to Tegeder, and she allows some type of narrative to come through and read like a map.\n\n**Dannielle Tegeder**\n\n_Inropataciland: White Winter_\n\n_City with Dot lower Tunnel_\n\n_Routes, Love Circle Production Expulsion Center, with five station Route, Twin Station Igloo Housing, with New Central Hospital and Lace Grid Station, multiple Box Square Hotel Housing and 27 circle; Storage from Above Ground and Below Ground; and Sideways color coding Grid, with Triangle Training Center, 2005_\n\nAcrylic, ink, dye, pencil, design marker and gouache on paper\n\n55\" \u00d7 79\" (140cm \u00d7 201cm)\n\nImages courtesy of Priska C.\n\nJuschka Fine Art, New York,\n\nNew York\n\n**Dannielle Tegeder**\n\n_Deconia: Thermal Atomic Rate at Red Midnight 2007_\n\nAcrylic, ink, colored pencil, graphite, gouache and pastel on paper\n\n55\" \u00d7 79\" (140cm \u00d7 201cm)\n\nImages courtesy of Priska C.\n\nJuschka Fine Art, New York,\n\nNew York\n\nVARIATIONS ON MAPPING\n\n**Catherine Mackey**\n\n_Harrison Street Scooter_\n\nAcrylic and mixed media on wood panel\n\n48\" \u00d7 45\" (122cm \u00d7 114cm)\n\n**Variation 1: Strong Divisions**  \nThis composition presents imagery within strong geometric divisions, presenting a flat diagrammatic space.\n\n**Keith Morant**\n\n_Airs & Graces II _\n\nAcrylic and oil pastel on canvas\n\n18\" \u00d7 24\" (46cm \u00d7 61cm)\n\nBryce Gallery\n\n**Variation 2: No Horizon Line**  \nWith no horizon to ground the viewer, Keith Morant creates a dialogue between the free-floating forms by harmonizing, contrasting, and using structural and directional forces.\n\n**More Variations**\n\n\u2022 Secure several maps and diagrams onto a surface. Extend elements from one diagram into another, exploring how to integrate the aspects and create a flow.\n\n\u2022 Collect diagrams and graphs of all kinds. Select your favorite and replicate its format, replacing the information formerly displayed with personal imagery. Find multiple ways of expressing the same concept (pictures, text, lines, arrows, products, shapes, etc). Books on tarot will often show special \"layouts\" for readings that can be used as inspiration for compositional structures.\n\n\u2022 Instead of writing entries in a journal or diary, find a new way of presenting events. For instance, draw or list images from your day by displaying them differently, e.g., chronologically, by priority, like a family tree, circularly, etc. Use text, arrows and pictures to get your point across.\n\n  SEDUCTIVELY SMOOTH SURFACES\n\nTegeder's paintings end up with a multi-textured finish, but she still likes to start with a very smooth surface, facilitating detail and line work. Here is a technique simulating her surface preparation.\n\n**Tip**\n\nTo lend a warmer white color to the paste, add a small amount of Transparent Red Iron Oxide (about two\u2013four drops per eight ounces of paste) before applying the paste to the surface. Other colors can be added in more quantity to the paste to create a colored ground.\n\n**Materials**\n\n**\u2022 Surface:** Any rigid surface\n\n**\u2022 Painting Tools:** A large flat plaster knife or other application tool\n\n**\u2022 Other:** Acrylic molding paste, waterproof sandpaper, paper towels or rags, water\n\n**1. Apply Acrylic Paste**  \nUsing a large flat plaster knife or other application tool, apply acrylic molding paste onto your surface. Use a larger tool for a larger surface. Apply a generous amount of paste and smooth it the best you can. Let it dry at least twelve hours or until it's no longer cool to the touch.\n\n**2. Wet Sand the Dried Paste**  \nUsing waterproof sandpaper (usually black in color, and sold at home improvement stores) wet-sand the paste surface by always keeping water between the sandpaper and the surface. Dip the sandpaper into water or use a brush or spray bottle to apply water separately to the surface. Using moderate pressure, sand in circular motions, continuing to add water at intervals into your sanding area. Wipe excess paint off frequently with a rag. Work the entire surface until smooth.\n\n**3. Repeat as Desired**  \nRepeat with a second layer of paste, letting it dry before sanding. Keep repeating as desired. The surface will become unusually smooth with a dense feel. Drawing and taping are easily accomplished on this surface.\n\nCONFRONTING THE VIEWER\n\nContent in images that confront the viewer head-on can be organized in two different ways: aesthetic or psychological confrontation. Confrontation using aesthetic concepts was spearheaded by German abstract expressionist Hans Hofmann, an influential artist and teacher, who used color relationships to enhance his \"push-pull\" spatial theory. Blocks of color with no recognizable imagery can create the illusion of an assertively forward or recessive backward movement.\n\nPsychological confrontation uses imagery and associations that confront our societal, political, religious or moral values. This latter mode is easily seen in works by Frances Ferdinands, who uses her training in conceptual art to convey ideas on social and environmental concerns of postmodern society. In _Becoming One_, a package of Wonder bread becomes an offering in the lap of an ancient Buddha, addressing issues on global consumption.\n\nFerdinands wants the ideas in her works to take precedence. She uses humor to diffuse confrontation, which she incorporates into her work by mixing contemporary and historic sources with surprising juxtapositions.\n\n**The Artist's Process**\n\nFerdinands' paintings start from personal experiences she transforms to the universal. She allows the process to take whatever time it needs. Ideas often take years to percolate, and sometimes work needs to be put aside for her to return to later. She allows her process full freedom, adding the risk that some pieces can and will get destroyed or overworked. Multiple glaze applications create depth and richness to her color and surfaces.\n\n **TIPS  \nfrom the Artist **\n\nEveryone's life is sufficiently interesting to provide fodder for unique work. You don't have to look \"out there.\" Capture your vision and combine it with technical mastery. Great art has to touch the soul. The harshest critic is yourself.\n\nFor inspiration, Ferdinands researches art history. Artists that inspire her include: Edward Hopper, Jim Dine, Jasper Johns, Robert Rauschenberg, as well as pop artists, conceptual art and surrealist movements.\n\n**Other Art in This Style**\n\n\u2022 Confrontational works of Hans Hofmann, Damien Hirst, Mike Kelly, Paul McCarthy, Otto Dix, Eric Fischl, John Heartfield, Barbara Kruger\n\n\u2022 Realist Impressionist painter \u00c9douard Manet created controversial work for his time in the mid-nineteenth century\n\n\u2022 Pop artists like Andy Warhol\n\n**Frances Ferdinands**\n\n_Becoming One_\n\nAcrylic on canvas\n\n30\" \u00d7 30\" (76cm \u00d7 76cm)\n\nVARIATIONS ON CONFRONTING THE VIEWER\n\n**Daniel Barkley**\n\n_Cage 1_\n\nAcrylic on canvas\n\n42\" \u00d7 42\" (107cm \u00d7 107cm)\n\n**Variation 1: A Direct Gaze**  \nA Japanese fencing mask only slightly obscures the portrait's direct gaze. The mouth is hidden by leather, adding an intimidating quality.\n\n**Grant Wiggins**\n\n_Where Is Gibarian?_\n\nAcrylic on canvas\n\n21\" \u00d7 16\" (53cm \u00d7 41cm)\n\nCollection of Laurence Blake, Palmdale, CA\n\n**Variation 2: Shockingly Bright Color**  \nFluorescent orange paint vibrates, layers collide and colors bounce off each other. Grant Wiggins likes his work to clash and uses advertising and corporate logos for the basis of his inspiration.\n\n**Darlene McElroy**\n\n_Ma\u00eetresse sans Visage_\n\nAcrylic and mixed media on panel\n\n24\" \u00d7 24\" (61cm \u00d7 61cm)\n\n**Variation 3: Add Surprise**  \nAn unexpected image is created by removing something expected, giving this work a disturbing quality. Darlene McElroy appropriates some imagery from historic painting, while adding her own. Paint and image transfers are used over stenciling.\n\n  PAINTING A LUMINOUS BACKGROUND\n\nFerdinands often uses backgrounds that are simple yet luminous, creating a sensual contrast for her realistically painted imagery. This technique takes a simple yellow background and adds luminosity and a halo effect to frame or accent any centrally painted form.\n\n**Materials**\n\n**\u2022 Paints:** Heavy Body or Fluid Acrylic colors: Naples Yellow Hue, Indian Yellow Hue, Quinacridone\/Nickel Azo Gold, Transparent Yellow Iron Oxide, Nickel Azo Yellow\n\n**\u2022 Surface:** Any primed surface\n\n**\u2022 Painting Tools:** Brushes, mixing knife, rag\n\n**\u2022 Other:** Acrylic slow drying or glazing gloss medium\n\n**1. Paint a Colored Background**  \nBrush-apply an opaque color, such as Naples Yellow Hue, over a primed surface, creating a solid, evenly applied undercoat. Apply a second coat if it appears streaky.\n\n**2. Apply Transparent Color**  \nGenerously brush-apply some uncolored slow drying medium or acrylic glazing medium gloss in the center of the painting, covering an area in excess of any central imagery you plan to paint later. Mix a transparent glazing color using medium to color in a 4:1 ratio. Using a brush or rag, apply the glazing color, starting at the corners and outer edges, where the strongest color will be. Continue spreading the glaze, moving towards the center. As you continue to spread the color toward the center, allow the color to dissipate. When the remaining colored glaze meets the wet medium in the center, the color will blend easily. Let the glaze dry fully.\n\nRepeat this step using a variety of glazes. The example here uses Indian Yellow Hue for the first glaze, Quinacridone\/Nickel Azo Gold for the second and Transparent Yellow Iron Oxide for the third.\n\n**3. Integrate With an Overall Glaze**  \nThe background now has a luminous quality from the several glaze layers. The central area, however, appears like a hole since color was withheld there. To integrate the center, a final glazing color is applied with Nickel Azo Yellow. But this time, brush-apply the glaze over the entire image, including the center. For added luminosity, apply another glaze using Interference Gold or Iridescent Gold.\n\n**Finished Example**\n\nISOLATED IMAGERY\n\nWhile complex or overcrowded compositions have their place, a bold use of simplicity can allow a deeper insight into an idea. By isolating a particular form, fragment or area, the composition takes on a graphic quality.\n\nFormer figurative and portrait painter Joey Fauerso offers a great example of isolated imagery in the work pictured opposite. Isolating the figure from its environment allows it to be seen in a different way. When she expanded her medium to include installation and animation, the figure continued to remain her primary focus. Fauerso films people in various activities, and from this creates work that includes her films, stills and paintings from the stills. Fauerso sees animation as \"moving painting\" and her installations offer \"epic paintings in time.\"\n\n**The Artist's Process**\n\nFauerso warms up by working on fifty or sixty sheets of paper all the same size that are soaked then stretched. She experiments on these sheets, trying out new techniques, materials, surfaces and mediums. What she learns becomes part of her new series. She protects the backgrounds by coating the paper with matte medium. Projecting the stills onto the paper, she then traces the forms using thin washes of color. She masks the background around the form with blue tape and seals the edges with matte medium to ensure the background stays clean during painting. Fauerso then applies paint onto the wet paper using brushes, mops and giant assembled brushes to get big pools of color. Working subtractively, she wipes out areas of paint, then continues working to get the visual impact she wants.\n\nInspiration for Fauerso include artists Max Beckmann and Alice Neel, studying Eastern philosophy, reading favorite writers and poets such as Mark Strand, and listening to the music of Bach and Arthur Russell.\n\n **TIPS  \nfrom the Artist **\n\nMore experimenting produces more discoveries. Avoid only working on finished pieces where you are invested in the outcome. Allow mistakes, since these can often lead to surprising techniques.\n\nJoey Fauerso in her Texas studio. Her intent to convey the inside and outside of things, representing our physical and metaphysical boundaries, is aptly expressed in her work. She strives for strong graphic impact, contrasting form against the white page.\n\n**Other Art in This Style**\n\n\u2022 Andy Warhol and other pop artists\n\n\u2022 Photorealist Chuck Close\n\n\u2022 Susan Rothenberg, Jean Dubuffet, Leon Golub\n\n**Joey Fauerso**\n\n_Wide Open Wide_\n\n_(installation detail)_\n\nWatercolor and acrylic on paper\n\n335 paintings, each 8.5\" \u00d7 11\"\n\n(22cm \u00d7 28cm)\n\n**Joey Fauerso**\n\n_Get Naked (Detail)_\n\nOil and acrylic\n\non paper\n\n8.5\" \u00d7 11\"\n\n(22cm \u00d7 28cm)\n\nVARIATIONS ON ISOLATED IMAGERY\n\n**Frances Ferdinands**\n\n_Seven-Day Wonder_\n\nAcrylic on canvas, mounted on panel\n\n24\" \u00d7 36\" (61cm \u00d7 91cm)\n\n**Variation 1: Repetition**  \nA graphic quality is created by the use of bright, bold background colors, and repeating views of the same object.\n\n**Olga Seem**\n\n_Duality (11)_\n\nAcrylic on paper on canvas\n\n18\" \u00d7 18\" (46cm \u00d7 46cm)\n\nCourtesy Couturier Gallery, Los Angeles, CA\n\nand Davis & Cline Gallery, Ashland, OR\n\n**Variation 2: Fragment**  \nThe unconventional divisions in this painting present separate fragmented sections of the same plant. Olga Seem researches unusual or extinct botanical flora. Her paintings incorporate material from reference guides containing various illustrations using dissections and close-up views.\n\n**More Variations**\n\n\u2022 Find photographs or other images you like. Pick out one object or form in each image, and isolate it by painting over the background or cutting it out and placing it on a piece of colored or white paper.\n\n\u2022 Starting with a picture you like, cut out the strongest forms, for example an animal, figure or flower. Recut forms further into smaller fragments or pieces, (i.e., figure is separated into arms, legs, hair, etc.), and experiment by placing the pieces separately on different backgrounds or combine some together on one common background. Take the original picture with its cut-out, and place it over other images to \"fill\" the hole with something else.\n\n\u2022 Place strips of white cardboard over paintings to recrop and reframe images to view them in a new way.\n\n  MASKING A SIMPLIFIED BACKGROUND\n\nThis technique is a great way to simplify a busy background, and isolate a singular form for emphasis or focus.\n\n**1. Select a Painting**  \nFind a painting with a busy background that's in need of an emphasized focus area.\n\n**2. Cut a Mask**  \nApply contact paper or masking paper over your painting in excess of the focus area. (Pictured here is shelf paper found in home improvement stores.) Select a shape or section to single out as a main focus. Using a permanent marker, draw its outline onto the plastic. Transfer the plastic to an appropriate cutting surface, cut the outline with a craft knife, then situate the cut shape over the focus area once again.\n\n**3. Overpaint the Background**  \nSelect your choice of paint application (pictured here is a Preval sprayer found in home improvement stores), your preferred color and preferred transparency (here I am using fluid Iridescent Translucent Pearl mixed with some water to ease spraying). Keep the color fairly transparent at first by adding more medium or water if necessary. More control can be obtained by starting transparent and repeating layers until desired coverage is achieved. Overpaint the entire painting. Remove the remaining contact paper.\n\n**Finished Example With a Simplified Background**  \nSelecting a dark color for use in the background will increase the brightness of the focus area, while a bright or light color will create more of a silhouette.\n\n**Materials**\n\n**\u2022 Paints:** Any acrylic paint\n\n**\u2022 Surface:** A painting in-process with a busy background\n\n**\u2022 Painting Tools:** Your choice of paint applicator such as a brush, sponge or commercial spray bottle (like Preval) or household water sprayer\n\n**\u2022 Other:** Clear or transparent contact paper or masking paper, scissors or cutting knife, permanent marker\n\nEMBELLISHED FORM\n\nEnhancing an image by using decorative elements such as gems, gold leaf and painted detail adds a unique visual flavor. Depending on how the artist handles the embellishments, emotions and qualities such as playful, delicate, disturbing or disruptive can emerge. Jim Barsness combines pattern with pictorial space. Working on canvases covered in rice paper, his elaborate detail brings to mind an enlarged antique illuminated manuscript. Finished painted canvases are even left unstretched to be hung like a tapestry. Ancient meets contemporary with a touch of humor.\n\n**The Artist's Process**\n\nBarsness makes a point of seeking new subject matter that he knows little about. Looking outside himself for the unknown, Barsness puts in hours of research for each project. This process allows him to fall in love with different images and absorb them while actively engaging his own imagination.\n\nBarsness has created a cast of characters that he reuses while continually inventing new ones. He works with acrylic like oil, using traditional glazing techniques, and layering to develop a depth in the character field. Other techniques he frequently uses include distressing the paint by sanding and using bright analogous glazes to enhance color (i.e., red gets an Alizarin glaze then Magenta\u2014see COLOR ENHANCEMENT). Barsness also works with different materials such as glitter, silicone molds and casts from craft jewels, interference paints and flocking. Borders are hand-drawn.\n\nBarsness is inspired by a wide range of art including graffiti; the Rubin Museum of Art's collection of Tibetan thangka paintings with hand-painted fabric borders; Northern Renaissance artists such as Pieter Bruegel the Elder, who painted huge crowd scenes; Early Netherlandish painter Hieronymus Bosch; Persian miniatures; and the stylization of works from Cambodia and Thailand.\n\n **TIPS  \nfrom the Artist **\n\nStart with what moves you visually, not to duplicate but to get you started. Instead of copying work you admire, try to emulate the way that artist thinks visually. Be prepared to make and embrace mistakes and find little moments that work. Find a path from all of the experimentation and mistakes and find your own territory. It takes a long time.\n\nJim Barsness in his studio. Barsness is intrigued by stylization and how it operates in various cultures. For him, pattern is a metaphor for social and cultural contexts, and these designs become a language.\n\n**Other Art in This Style**\n\n\u2022 Gustav Klimt, particularly work from his Golden Phase; other artists from the Symbolist and Art Nouveau movements\n\n\u2022 Dot paintings in Indigenous Australian art\n\n\u2022 Ancient Italian mosaics; illuminated manuscripts from the Middle Ages\n\n**Jim Barsness**\n\n_Hanuman's Rescue_\n\nAcrylic on canvas\n\n90\" \u00d7 66\" (229cm \u00d7 168cm)\n\nVARIATIONS ON THE EMBELLISHED FORM\n\n**Diana Ingalls**\n\n_Ties that Bind_\n\nAcrylic on canvas with pencil and paper\n\n20\" \u00d7 16\" (51cm \u00d7 41cm)\n\n**Variation 1: Combine Symbols With Realism**  \nA pattern overlay infuses this image with symbols, transforming a portrait into archetype. Graphic elements combine with realism to create a complex image and rich surface.\n\n**More Variations**\n\n\u2022 Collect patterns and designs from books, images, advertisements, labels, fabric and wallpaper. Cut out, trace or transfer these to your painting in select areas or overall. Allow pattern designs to be opaque, covering the painting underneath, or transparent, allowing the underpainting to be visible but partially obscured.\n\n\u2022 Make your own patterns by starting with one thing: a symbol, letter, number or abstract shape. Photocopy, transfer or redraw this one item onto a larger surface. Use the same form repeatedly, attached to itself in a chain, turned, flipped and chopped in various ways and combinations with itself until a pattern is formed that goes beyond the initial singular element.\n\n**Darlene McElroy**\n\n_Ingres Redux 2_\n\nAcrylic and mixed media on panel\n\n8\" \u00d7 8\" (20cm \u00d7 20cm)\n\n**Variation 2: Mix Contemporary With Traditional**  \nA formal portrait is applied as an image transfer over a vintage scrapbook from the 1800s, then painted and collaged to create a contemporary tribute to French Neoclassical painter Jean Auguste Dominique Ingres.\n\n  JEWEL-LIKE STIPPLING\n\nThe painting surfaces in Barsness's work contain multiple techniques such as glazing, sanding, layering and casting. One of his techniques creates a jewel-like surface by applying paint color in small dot patterns using bottle applicators.\n\n**Materials**\n\n\u2022 **Paints:** Any acrylic paint color\n\n**\u2022 Surface:** Any primed surface\n\n**\u2022 Painting Tools:** Plastic squeeze bottle with small application nozzle, brush, palette knife, mixing palette\n\n**\u2022 Other:** Acrylic gel or medium\n\n**1. Prepare the Materials**  \nSelect a paint color and brush-apply the paint onto your surface to create a background.\n\nWhile the background dries, on a palette, combine this same paint color with acrylic medium or gel in a 1:5 ratio. If you use a thick gel, the stippling will hold higher peaks and stronger dot shapes, while fluid mediums create a softer texture. Using a knife, scoop this mixture into a plastic squeeze bottle or applicator found in hobby and art stores. Here, I combined Golden Fluid Cobalt Teal with Golden Soft Gel Gloss and placed the mixture into a squeeze bottle.\n\n**2. Apply Dot Patterning**  \nSqueeze the paint from the bottle onto the painting surface, creating small dots. After all the dots are applied, let this dry for several hours or overnight. Make sure the dots are dry to the touch before any subsequent overpainting.\n\n**Finished Example**  \nThe raised dot pattern creates a jewel-like texture. Experiment with different sized dots, varying patterns and colors to embellish painted areas.\n\n**Tip**\n\nPastry bags with icing tips can also be used to create varying dot patterns. For a different effect, try squeezing clear gel with no added color from a squeeze bottle directly onto a plain surface to create an underlying jewel-like ground, then apply thin paint color diluted with water on top.\n\nHARD EDGE GRAPHIC\n\nThe phrase \"hard edge graphic\"  \nimmediately brings to mind geometric abstraction, popular in California in the 1960s, with roots going back to the late nineteenth century and later artists such as Piet Mondrian. Images in this genre use precision and clarity with crisp delineations between colors, readily visible in the work of Kasarian Dane.\n\nDane is concerned with the presence of a painting, its physical flat surface, and the canvas as an object in space. Instead of a spatial depth, Dane goes for a substance depth. He works on 1\/8-inch (3mm) thick aluminum sheeting, braced to extend out about 1 inch (25mm) into the space from the wall when hanging for exhibition, which makes it appear to float (see _Untitled_). The relationship of the exhibited paintings with each other and the space around them are key. The thin aluminum projected outward creates a presence that differs from a canvas with edges sitting flat against the wall.\n\n**The Artist's Process**\n\nTo generate ideas, Dane works with the computer, experiments on paper, and collects bits of color from found ads, and other things he finds. His aluminum surfaces are first sanded, degreased and primed. He then divides his surface into horizontal and vertical areas. From there the process is allowed to flow, not planned too far ahead.\n\nDane has experimented with all types of paint applications, like knives and rollers, to get the surface exactly the way he wants. He prefers brush-applying matte acrylics to give a smooth sensual quality to the paint with a hint of brushstroke. He paints, repaints, widens areas, changes colors, and allows his process to be visible in the final piece by allowing remnants of the changes to remain, carrying the history of his process. Perfection from the hard edged forms contrasts with subtle irregular elements. The irregularity comes from: paint drips and other remnants of Dane's process which are allowed to remain visible along the thin aluminum sides; the light brushstroke texture reveals a \"hand at work;\" and a surface texture is created from underlying layers where forms were moved and repainted.\n\n **TIPS  \nfrom the Artist **\n\nOften art schools add pressure to form your work around trends. Don't try to fig-ure out what's hot but do the work that keeps you coming back. Look at a lot of art galleries and museums and soak in everything you can. Try out a lot of stuff. Anything that comes to mind\u2014just try it. Make work that gets you excited to go into your studio day after day.\n\nKasarian Dane in his New York studio.\n\n**Kasarian Dane**\n\n_Untitled, 2009_\n\nAcrylic and Flashe on aluminum\n\n16\" \u00d7 24\" (41cm \u00d7 61cm)\n\n**Other Art in This Style**\n\n\u2022 Piet Mondrian and other artists from the De Stijl or Neo-Plasticism art movement\n\n\u2022 Color Field painters such as Kenneth Noland, Joseph Albers, Ellsworth Kelly, Kasimir Malevich, and shaped canvases of Frank Stella\n\n\u2022 Futurist painters such as Joseph Stella\n\n\u2022 Frederick Hammersley, Al Held, Abraham Gelbart, Keith Haring\n\n\u2022 Minimalist Brice Marden\n\n\u2022 Stained glass artists\n\n\u2022 Contemporary artist Ed Mell paints realistic landscapes and florals using hard edge.\n\nVARIATIONS OF HARD EDGE GRAPHIC\n\n**Dave Yust**\n\n_Chromaxiologic\u2014Inclusion VI with catenary_\n\n_Ogee curve_\n\nAcrylic on canvas (on 38 piece laminated wood stretcher frame)\n\n54\" \u00d7 84\" (137cm \u00d7 213cm)\n\n**Variation 1: Unusual Canvas Shape**  \nThis elliptically shaped painting conveys a relationship between inside painted forms and the outer shaped and constructed canvas.\n\n**Harry Doolittle**\n\n_Gentle Pandemonium_\n\nAcrylic on board with glass and aluminum leaf\n\n43\" \u00d7 32\" (109cm \u00d7 81cm)\n\n**Variation 2: Cut Away Layers**  \nThis painting employs an unusual construction. Canvas board is cut away in a variety of shapes. The board is then placed over metal and painted glass. White areas in the image are silver metal. The glass is seen in the areas that are gray, gold, red and blue. Other colors are painted acrylic.\n\n**Anne Seidman**\n\n_Untitled_\n\nWatermedia on wood\n\n20\" \u00d7 20\" (51cm \u00d7 51cm)\n\n**Variation 3: Contrast Hard Edge With Organic**  \nA variety of high key and neutral colored forms contrast with a textural organic background. The shapes are packed into a set and unified to form a larger shape.\n\n  SOFTENING HARD EDGES\n\nTaping is a great way to get a clean hard edge between two paint colors. Taping, however, can appear mechanical, which can enhance your painting or work against it, depending on the results you want to achieve. By taping one color and hand-painting its adjacent color, the mechanical effects are softened, adding a handmade look to the edge.\n\n**Tip**\n\nTo reduce seepage under the tape and ensure a very clean edge, you can apply a clear gloss acrylic gel or medium before applying the first paint color, along the tape's painting edge to seal any gaps between the tape and painting surface. This is especially helpful if your paint is thin or watery. Apply your paint color immediately over the gel\/medium seal once it is dry to the touch.\n\n**Materials**\n\n**\u2022 Paints:** Any two acrylic paint colors\n\n**\u2022 Surface:** A primed painting surface\n\n**\u2022 Painting Tools:** Soft flat brush\n\n**\u2022 Other:** Masking tape or clear tape\n\n**1. Tape and Paint the First Color**  \nOn a primed surface, apply tape, lining up its edge where the two colors will meet. Press the tape along the painting edge firmly.\n\nPaint the first color, allowing the color to go over the tape. Immediately remove the tape after painting. If left on too long, the tape may be difficult to remove later. Let the paint dry.\n\n**2. Hand Apply the Second Color**  \nSelect a second acrylic paint color and load it onto a soft flat brush. Place the brush at one end and press gently, gliding alongside the previous line. Make sure you breathe while painting to relax your hand and help it move in a straight path. If your brush goes over the edge, correct it later with the first color. Go back and forth alternating with both colors until the edge looks clean.\n\nPATTERN FIELDS\n\nRepeated forms or patterns in an overall \"field\" can produce varying spatial effects, depending on the compositional design and treatment. A flat, tight pattern can evoke a shallow space, yet the illusion of depth can be enhanced by overlapping, contrasting, movement and the treatment of negative space. Patterns are tricky elements to use in paintings. Symmetrical patterning risks producing a wallpaper effect\u2014timidly hanging in our periphery and easily ignored. However, patterns organized into asymmetrical areas and combined with spatial effects can become vibrant and seductive.\n\nJim Waid's sophisticated use of aesthetics, especially contrast, transforms flat decorative patterns into a painting that presents the experience of three-dimensional space. Textural areas contrast with smoothly painted ones, reflective sheens contrast with matte, transparent colors play off opaque, and patterns mingle with solidly painted areas. Waid feels the point of his art is to create a sense of \"presence\" in and on the canvas. He strives for convincing space so the image becomes real, not illusion. His paintings are not illustrations, but rather enactments of the world around him.\n\n**The Artist's Process**\n\nEarly in his painting career, Waid made a commitment to find something original. Along the way he passed through several distinct phases, invented many processes and techniques, even creating his own tools. Waid calls his process the \"guerrilla school of technique\"\u2014using whatever it takes to get the image he wants. He rarely starts a piece knowing what it will look like and lets it develop during the act of painting. The physicality of the paint plays an important role. He works wet in wet, adding colored paint directly into wet gel already on the surface, mixing it all directly on the canvas. He never uses mixing palettes. Super large kitchen spatulas are a favorite tool, along with sponges, rags, scrapers, sticks, brooms, spray guns and paint rollers.\n\n **TIPS  \nfrom the Artist **\n\nUse inspiration from the masters and others who inspire you, and combine this with ambition to create the best. Let go of academic dogma. Think with the paint, not about the paint.\n\nJim Waid working large in his Arizona studio. Inspired by his desert surroundings, Waid keeps his forms and patterns hovering at the edge of recognition, and notes \"The Sonoran Desert has the craziest landscapes. If the gods can use every kind of twisty thing in the world, why can't I?\"\n\n**Other Art in This Style**\n\n\u2022 Wassily Kandinsky\n\n\u2022 Modern artists Henri Matisse and Joseph Stella\n\n\u2022 Postmodern artist David Hockney\n\n\u2022 Decorative patterning in Gothic and early Renaissance art frescoes with Italian masters such as Giotto de Bondone (1267\u20131337)\n\n**Jim Waid**\n\n_Evening Notes_\n\nAcrylic on canvas\n\n79\" \u00d7 61\" (201cm \u00d7 155cm)\n\nVARIATIONS ON PATTERN FIELDS\n\n**Pat Forbes**\n\n_Neutrino Blossoms_\n\nAcrylic and paper on panel\n\nThree panels, each 16\" \u00d7 16\" (41cm \u00d7 41cm)\n\n**Variation 1: Combine Pattern and Sheen**  \nCombining decorative papers with acrylic in a unique layering process produces elegant and rich imagery with a touch of whimsy. Pat Forbes creates unusual surfaces using reflective paints to add a shimmering magical sheen.\n\n**Thaneeya McArdle**\n\n_A Salutary Encounter_\n\nAcrylic on wood panel\n\n8\" \u00d7 10\" (20cm \u00d7 25cm)\n\nPrivate collection\n\n**Variation 2: Highly Detailed Design**  \nThaneeya McArdle uses meditative states and tribal sensibilities to create highly detailed designs conveying energy and flow of spirit and matter. Complementary pairs and intensely bold colors add an optical vibration.\n\n**More Variations**\n\n\u2022 Add decorative papers and patterned objects such as mosaics or fabric to a painting. Compare the difference between adding patterns along a border surrounding an image and adding patterns directly into the image.\n\n\u2022 Start with patterned papers secured onto a surface as background. Use opaque paint to overpaint and eliminate some patterned areas, and transparent paint to emphasize or change others.\n\n\u2022 Paint a realistic still life using several patterned fabrics in your setup.\n\n  GHOST IMAGERY: REVERSE PAINTING\n\nJim Waid has a continual desire to invent. One of his earliest processes was to paint on the back of canvas. The paint partially seeped through to the front creating a new \"ghost image,\" offering surprising new patterns and inspiration for the painting's continued emergence.\n\n**Tip**\n\nPretest the canvas by dripping some water onto its surface. If the droplets hold their form, the fabric needs special washing to remove any water-repellent coatings. Wash the canvas in a washing machine using warm water, adding one tablespoon each of Synthrapol and soda ash. Once any coatings are removed, place the dry canvas over protective plastic on the floor. Either side of the canvas is fine to use; however, whatever side faces up in this step will be the back of the painting later.\n\n**Materials**\n\n**\u2022 Paints:** Any acrylic paint colors\n\n**\u2022 Surface:** Untreated, unstretched medium or heavy weight raw canvas or linen\n\n**\u2022 Painting Tools:** A variety of painting brushes, water\n\n**\u2022 Other:** Any liquid soap or Golden Acrylic Flow Release\n\n**1. Apply Soap to the Reverse Side of the Canvas**  \nDilute liquid soap (a more refined version is Golden Acrylic Flow Release) with water using a 1:4 ratio, and generously apply onto the canvas using a large brush, or other applicator, to the entire surface. Apply enough to wet through to the other side of the canvas. Lift the canvas and remove any excess puddling from underneath or move the canvas to a dry area.\n\n**2. Apply Paint Color**  \nWhile the canvas is still wet, apply paint diluted with water. Start with a 1:1 ratio of paint to water, then experiment with the amount of water to create a variety of dilutions and color intensities. Add enough water so the paint can soak through the canvas threads. Lift the painted canvas and remove excess paint from underneath or move to a dry area. Allow to dry flat.\n\n**3. Flip the Canvas to the Reverse Side**  \nFlip the canvas over to see the results on the reverse side. The colors seep through in a variety of ways, but more subtle and softer, creating an interesting new appearance. Use this side as your painting's new front and first layer. If the front and back are identical, then try this again on a thicker piece of canvas, or use less water in the paint colors.\n**Section 5**\n\nSPATIAL  \nPUSH-PULL\n\n**Gary Denmark**\n\n_Depth Charge_\n\nAcrylic on canvas\n\n36\" \u00d7 36\" (91cm \u00d7 91cm)\n\nRenowned teacher and artist Hans Hofmann coined the phrase push-pull to describe a visual play of pictorial space\u2014the sensation of the eye being pushed and pulled into the illusory depths of the painting's space. Here lies a signature of contemporary art, containing both directional movements, a push toward the illusory depths and a pull back toward the front picture plane, and a continued tug of war between the two. This section offers examples of nonobjective abstract imagery that make use of this effect.\n\nMARKING SPACE\n\nThis style has roots in the Abstract Expressionist and Surrealist movements, which focused on the subconscious and spontaneous coming through the work. Here marks carry expression, and are also key to creating the spatial qualities. Jackson Pollock's drip paintings are one example of how an artist collaborates with the material to express a gesture, creating a signature in the paint. An ambiguous shape, form or mark can appear as a Rorschach test, prodding us to find something recognizable and yet also existing as just a mark. Leah Dunaway's paintings have an emphasis on mark making and architectural space. She uses linear elements such as straight lines and corners, delineating architectural spaces. These contrast with organic marks such as squiggles and loops to create an evocative pictorial space, a \"vibratory push-pull.\" Her paintings often involve a sheet of Plexiglas, bolted and overlayed onto the work, adding an industrial component and drama by casting shadows and enhancing the sense of space. Her work keeps evolving, each new series emerging from the previous one.\n\n**The Artist's Process**\n\nGrowing up with an architect father, Dunaway learned to read blueprints and to this day is attracted to architectural forms and spaces, which she continually adds to her subconscious \"file.\" Patterns, shadows and the play of light through forms continually interest her. Architectural elements appear naturally in her work, since her subconscious plays a main role in her process. Dunaway works on several canvases at the same time, usually working flat as opposed to on an easel, encourages drips, and creates most of her marks using charcoal and oil stick. Her agenda is to \"engage in serious play\" like a child, with no expectations. During her painting process she continually turns the painting around in different ways, waiting a long time to declare it finished. Artists who inspire Dunaway include Hans Hofmann, Jackson Pollock, Richard Diebenkorn and former art instructor Katherine Chang Liu (featured).\n\n **TIPS  \nfrom the Artist **\n\nContinual growth will keep your work fresh. Continue to modify, change and expand what your work is about. Your whole life will feed into it. Everything you do goes into the same giant soup pot to make a chili.\n\nDunaway often finds inspiration in familiar surroundings such as hardware stores, where she can find small sample bottles of house paint, in decorator colors both opaque and muted, for experimenting cheaply at home.\n\n**Other Art in This Style**\n\n\u2022 German and Abstract Expressionists such as Jackson Pollock (check out his early work in addition to the drip paintings), Mark Tobey, Joan Mir\u00f3, Wassily Kandinsky, Willem de Kooning, Paul Klee, Henri Michaux, L\u00e1szl\u00f3 Moholy-Nagy\n\n\u2022 Contemporary artists: Sam Scott, Agnes Martin, Vernon Fisher's \"blackboard\" paintings, Joan Mitchell\n\n\u2022 The calligraphic-style scribble paintings of Cy Twombly\n\n**Leah Dunaway**\n\n_Fusion 2_\n\nAcrylic on canvas\n\n56\" \u00d7 46\" (142cm \u00d7 117cm)\n\nVARIATIONS ON MARKING SPACE\n\n**Beth Ames Swartz**\n\n_The Fire and the Rose: But Heard, Half-Heard, in the Stillness_\n\nAcrylic on canvas\n\n54\" \u00d7 66\" (137cm \u00d7 167cm)\n\n**Variation 1: Contrast Marks With Background**  \nSmall vertical strokes repeat to create a background field. This emphasizes the larger lines which are curvy, bold and gestural.\n\n**Katherine Chang Liu**\n\n_Drawing #1_\n\nAcrylic and mixed media on paper\n\n11\u00bd\" \u00d7 12\u00bd\" (29cm \u00d7 32cm)\n\nCourtesy Jenkins Johnson Gallery, San Francisco and New York\n\n**Variation 2: Vary Marks**  \nA limited palette of colors contrasts with the wide variety of marks, including text, hand drawn lines, graphs and brushstrokes.\n\n**More Variations**\n\n\u2022 Experiment making a variety of marks using different tools. Vary how you hold the tool, the type and consistency of paint you use, and the surfaces. Keep these experiments as reference for later ideas.\n\n\u2022 Start with one mark on a blank canvas. Take a moment to consider what this one mark feels like. Make another mark that responds to the first, either by being similar or opposite, or keeping some qualities while changing others. Continue in this dialogue making, more marks, each one responding to the last, and noting how each new mark affects the whole image.\n\n  HOMEMADE PAINTING TOOLS\n\nJim Waid, featured in the previous chapter, likes to invent his own painting tools to obtain the marks he prefers. Here are some ideas to expand your application options.\n\n**Tip**\n\nPriming your cardboard before or after you cut the comb from it will make it more durable. Applying a coat of gloss acrylic medium over the background color will give cleaner marks and ease for the subtractive part in step 3.\n\n**Materials**\n\n**\u2022 Paints:** Any acrylic paint color different than the surface color\n\n**\u2022 Surface:** Any primed surface pre-painted with color\n\n**\u2022 Other:** Gloss acrylic gel, painting knife, cardboard, cutting knife\n\n**1. Cut Out a Cardboard Comb**  \nUsing thick cardboard and a sharp cutting knife, cut small triangles or wedge shapes into the cardboard, leaving a margin at the top. Vary each triangle in size and length to obtain a less predictable comb effect with the paint.\n\nHere is a finished hand-cut cardboard comb, along with a surface pre-painted with brush-applied Pyrrole Orange.\n\n**2. Apply a Gel Layer**  \nSelect a color that contrasts with the background color. Using a knife, mix it with thick gloss gel in a 1:1 ratio. Mixing gloss gel into the paint slows the drying, allowing more working time. Generously apply the color mixture over the background. Here the blue color is made with Primary Cyan, Titanium White and Soft Gel Gloss that is then applied over the Pyrrole Orange background.\n\n**3. Create Subtractive Marks by Combing**  \nComb through the wet blue paint layer to \"subtract\" the blue while creating lines in the underlying orange color. Smooth out some areas with a painting knife, then repeat subtractive combing, moving in a different direction. Repeat as much as desired.\n\n**Other Homemade Tool Ideas**  \nJoin commercial brushes together to make an oversized brush for large strokes and marks. Consider using non-art tools for unusual marks such as kitchen spatulas and other household objects, scrapers, squeegees, combs, mops and whisk brooms. Browse through home improvement, decoration and discount stores for objects, imagining what type of mark they would make, and experiment.\n\nFREE-FLOATING ORGANICS\n\nForms that are abstract or nonobjective can appear to float in space when there is no distinguishing horizon line or sense of ground. Barbara Moody's painting pictured opposite gives a feeling of being underwater or suspended in air. It is easy to see from the organic shapes in the painting that nature is an inspiration for Moody, and no surprise that she lives in a house filled with specimen jars from her incredible self-made nature collection.\n\n**The Artist's Process**\n\nStarting flat on the floor, Moody makes marks on her surface, then applies overlapping layers of paint. The marks she makes determine the content from there. Moody does not rely on reference material on hand, working mostly out of her head. She admits that years of traditional figurative painting instruction and many years of painting experience have made this viable. However, because she is accomplished at drawing and painting realistically, it would be too easy for her to copy reference material, so by keeping them out of sight she is able to make her actions more her own. She relies on accidents that can then be more deliberately worked.\n\nMoody pushes herself into unknown places by using asymmetrical composition, or colors she finds unattractive. She mixes colors in big plastic cups in large quantities, helpful for big paintings, and never uses primary colors right out of the tube. She likes to make \"nameless\" colors, such as celery, peach or apricot, ones that are not as easy to identify as red or blue. Each layer has areas that are softened, quieted down by glazing, and edited, while a few select items are enhanced. Every day she sketches, and at night she goes through her journal and makes notes, often writing about conceptual ideas to get to the essence of the mood or feeling\u2014not so much the subject. After completing a series, Moody moves on, fueling her sense of discovery and her desire to create something that has never been seen before. She feels her work has to come from within and not from an external source, representing who she is and containing a type of authenticity.\n\n **TIPS  \nfrom the Artist **\n\nWork flat or on the floor instead of at an easel. Pour, scrape, use odd tools to add spontaneity. Respond to happy accidents. Feel as if you are back in kindergarten but with wise intent. Allow yourself to discriminate, edit, refine, eliminate and enhance.\n\nBarbara Moody in her Massachusetts studio. Moody is inspired by young cutting-edge painters because they push the boundaries of painting today and have fewer preconceived habits from modernist dogma.\n\n**Other Art in This Style**\n\nWorks by Joan Mir\u00f3, Wassily Kandinsky, Marc Chagall, Ross Bleckner, Joan Snyder, Lee Krasner and Roberto Matta\n\n**Barbara Moody**\n\n_Ecosystem_\n\nAcrylic on canvas\n\n60\" \u00d7 48\" (152cm \u00d7 122cm)\n\nVARIATIONS OF FREE-FLOATING ORGANICS\n\n**Lisa Ferguson**\n\n_Escaping the chaos to a peaceful haven_\n\nAcrylic and mixed media on canvas\n\n80\" \u00d7 60\" (203cm \u00d7 152cm)\n\n**Variation 1: Minimal Color Palette**  \nUsing a minimal color palette places the focus on the fluid flowing forms.\n\n**Sam Scott**\n\nBlues for Charlie Parker\n\nAcrylic on canvas\n\n80\" \u00d7 66\" (203cm \u00d7 168cm)\n\n**Variation 2: No Horizon Line**  \nWith no recognizable horizon line to evoke a landscape, organic forms are energized in space.\n\n**Patti Brady**\n\n_Blue Bonkers_\n\nAcrylic on paper (with transfers and digital images)\n\n27\" \u00d7 27\" (69cm \u00d7 69cm)\n\n**Variation 3: Variety of Patterns and Forms**  \nA variety of patterning and forms are created through collage, stencils and ink-jet printed acrylic skins.\n\n  EDGE GHOSTING\n\nGary Denmark, whose work is shown on Gary Denmark and Gary Denmark, uses this printmaking technique in his paintings to soften some of the edges of his painted forms. It's a great way to integrate hard edged geometric forms with soft, organic areas and shapes.\n\n**Tip**\n\nIf you don't clean the plastic from step 2, you can save it and reuse it in a collage by cutting up the plastic and regluing it into another painting. Another recycling idea: Apply clear acrylic gel over the dry shape left on the plastic. Let dry. Peel off as a skin and use in a collage.\n\n**Materials**\n\n**\u2022 Paints:** Any acrylic paint color\n\n**\u2022 Surface:** A primed painting surface\n\n**\u2022 Painting Tools:** A variety of brushes\n\n**\u2022 Other:** A clear plastic sheet\n\n**1. Paint a Shape**  \nOn a primed or pre-painted surface, brush-apply a shape. This background is painted with a light green color made from Titanium White, Turquoise Phthalo and Permanent Green, while the shape uses a pink made of Titanium White and Quinacridone Red.\n\n**2. Blot the Wet Paint**  \nWhile the painted shape is wet, place a sheet of plastic over it and rub gently to pick up some excess color. This will create an imprint or light textural pattern.\n\n**3. Remove the Plastic**  \nCarefully remove the plastic by lifting straight up. Place the plastic down again slightly off register to the original shape to create a double image along the edge. Rub to transfer the excess color onto the surface. The plastic can be cleaned and reused by wiping with a damp cloth.\n\n**Finished Example**  \nOriginally a hard edge shape, it is now softened with the addition of a ghost or double of itself slightly overlapping.\n\nTEXT, LINE & IMAGE\n\nUsing text in an image can create a peculiar visual play, and an enriched viewing experience. Text can be seen as symbol, sound or as a graphic, and can enhance any narrative qualities. Former graphic designer Debi Pendell learned to see areas of text as shape, color and value, and to respond to letters and words abstractly. Pendell views everything as a symbol: letters, words, trees, brushstrokes, line, color and texture, and combines these to create a readable space. At first she paints a believable reality, and then she purposely \"spoils\" it to change the way we view it, like placing a wrong value or odd size somewhere, or peppering her image with letters to bring attention to the front surface.\n\n**The Artist's Process**\n\nPendell starts by building up an initial background using collaged papers, often adding painted elements. Some imagery, like the trees in _Proposed Possibility_ , are her own photos turned into gel skin transfers. Pendell creates a faux encaustic (wax) appearance by applying a variety of acrylic gels ranging from very thick to very thin, both gloss and matte. The multiple layers of gel add a depth and a real physical space, underscoring the illusion of space she creates with her landscape format. The matte gels add a nice encaustic look, but often obscure the text and imagery too much, so instead she works with Golden High Solid Gel Gloss to get the thickness, followed by several final coats of matte medium on top. This adds control over how much veiling the matte acrylic will produce. To ensure that letters are read as abstract marks, Pendell purposely spaces them apart enough to avoid having them come together as words. Pendell finds inspiration in artists such as Henri Matisse, Robert Rauschenberg, Sylvia Plimack Man-gold, Giorgio Morandi, Sean Scully and Louise Bourgeois.\n\n **TIPS  \nfrom the Artist **\n\nIn the process of painting, many emotions can come up, like frustration, anger and fear. If you hold back, the work shows repression and the desire to be polite, creating kitsch. Instead, welcome all these emotions, including joy, allowing things to flow. All you have inside will then pour out and the censoring stops. Don't be afraid of the struggle.\n\nDebi Pendell in her Massachusetts studio..\n\n**Other Art in This Style**\n\n\u2022 Contemporary artists: Ed Rusha, Jenny Holzer, Julian Schnabel, Robert Indiana\n\n\u2022 Asian woodcuts, especially from Buddhist texts, include poetry and writing seamlessly into the imagery\n\n\u2022 Joseph Kosuth, Gino Severini, Barbara Kruger, Larry Rivers\n\n**Debi Pendell**\n\n_Proposed Possibility_\n\nAcrylic, collage and mixed media on canvas\n\n48\" \u00d7 48\" (122cm \u00d7 122cm)\n\nCollection of Caron and Scott Palladino,\n\nSaratoga Springs, NY\n\nVARIATIONS OF TEXT, LINE & IMAGE\n\n**Teresa Stanley**\n\n_Not At All_\n\nAcrylic and mixed media on wood panel\n\n36\" \u00d7 48\" (91cm \u00d7 122cm)\n\n**Variation 1: Enhance Painted Forms With Linear Elements**  \nA physical three-dimensional quality is sustained by Teresa Stanley's addition of collaged paint skins and paint. Lighting grid plans with technical references were silkscreened into this image, creating a dialogue between art and science.\n\n**Bill Dunlap**\n\n_Unholy Trinity_\n\nPolymer paint on canvas\n\n49\" \u00d7 37\" (124cm \u00d7 94cm)\n\nCollection of the artist\n\n**Variation 2: Use Readable Text for Impact**  \nText here is kept readable as word with attention to placement and shape creating an immediate association and impact.\n\n**More Variations**\n\n\u2022 Experiment with balancing lines, text and imagery by changing the amount of each in a piece. Try almost all text with barely any imagery, and the reverse, mostly imagery with scant text. Change your process by starting with text then adding imagery, or the reverse.\n\n\u2022 Explore the spacing between letters to find the amount of space needed to form words, and the distance needed for them to read as design or graphic elements.\n\n\u2022 Use writing you find inspirational, from books, poetry or other sources, whether personal or found, and find ways of incorporating sections of the writing into your work.\n\n  ACCIDENTAL STENCILING\n\nStencils are used here in an unorthodox way to create accidental marks. When used conventionally, stencils can sometimes offer imagery that is too specific to integrate, with hard edged forms and a mechanical look. This technique shows a way to use stencils that avoid the commercial cutout appearance, adding softer abstract elements by using smaller portions of the stencil design.\n\n**Materials**\n\n**\u2022 Paints:** Any acrylic paint colors\n\n**\u2022 Surface:** Any primed painting surface\n\n**\u2022 Painting Tools:** Painting knife with stepped handle\n\n**\u2022 Other:** Heavy or very thick acrylic gloss gel, a plastic stencil, paper towel\n\n**1. Prepare the Materials**  \nSelect a paint color and stencil design. Tape the stencil onto a painting support with or without a background color. Make a mixture of color and acrylic gel in a 1:1 ratio. Adding thick gloss gel slows the drying, eases application and reduces leakage underneath the stencil.\n\n**2. Apply Color to the Stencil**  \nLoad a painting knife along the back side, with plenty of the paint\/gel mixture. Begin by depositing the paint onto the plastic part of the stencil, not too close to the open cutout portion of the design. Press down hard with the flat side of the knife, and swipe across the stencil, allowing mistakes, missed areas and skimpy paint in some places. By pressing hard, only small portions of the design get transferred, creating unusual abstract \"hand-applied\" design elements.\n\n**3. Remove the Stencil**  \nCarefully lift the stencil off to reveal fragments of the design. Repeat using different stencils and new colors to get a buildup of interesting abstract marks.\n\nGEOMETRY & SPACE\n\nComposing with geometric shapes offers a contemporary way to create space. Katherine Chang Liu creates a delicate balance between geometric and organic shapes; color contrasts; and combinations of drawing, collage and paint. Liu's paintings result in a balance between chaotic and controlled emotional content. Her work focuses on our daily sensory overload with visual, literary, musical and verbal information. Of her work, Liu states she maps \"this emotional state with autobiographical marks and a reference to the urban environment in which I live and work.\"\n\n**The Artist's Process**\n\nLiu gets her inspiration from words, through poetry, news articles\u2014anything that triggers a visual possibility. She continually creates lists of words, allowing time for ideas to simmer. Around these ideas, Liu makes twenty or more thumbnail sketches for general design, overall shapes and color atmosphere. Liu also collects items, which she readily collages into her work. Working in a series, she still only focuses on one painting at a time. Each painting in the series is a comment on the last one. Surfaces are built up using a variety of materials, including acrylic, charcoal, fabric, wood, direct drawing and fragmented paint areas. Working for that true image, Lui remains unattached and continues adding and subtracting paint and layers. Sometimes this fails, but Liu doesn't try to please others; she just needs to please herself. The work needs to reflect back to her like a self-portrait or mirror. If the work comes too easily to her technically, she doesn't trust it and changes it. Questions arise for her such as: How do you cover something and still reveal it? Is it just sitting there on the surface, or worse, jumping off the page? Would slightly veiling it make it more integrated and add more mystery? Something too obvious might be improved if veiled and allowed to be discovered at a second glance. How to hold the weight of importance? How much is allowed to sit on the surface is the artist's choice.\n\n **TIPS  \nfrom the Artist **\n\nEvery painting we make is a self-portrait.\n\nLiu frequently curates art exhibitions and acts as juror to many competitions, keeping her current on the art scene. Artists that inspire Liu include Robert Irwin, Robert Rauschenberg, Anselm Kiefer and Joseph Cornell.\n\n**Other Art in This Style**\n\n\u2022 Hans Hofmann, Fernand L\u00e9ger, Mark Rothko and other Color Field artists\n\n\u2022 Antonio Tapies, Eva Hesse, Piet Mondrian, Jean H\u00e9lion, Wassily Kandinsky, Jasper Johns, Franz Kline, Robert Mangold\n\n\u2022 Auguste Herbin, who developed a movement emphasizing color through geometry\n\n**Katherine Chang Liu**\n\n_Night Walk_\n\nAcrylic and mixed media on panel\n\n30\" \u00d7 30\" (76cm \u00d7 76cm)\n\nCourtesy LewAllen Contemporary Gallery,\n\nSanta Fe, NM\n\nVARIATIONS OF GEOMETRY & SPACE\n\n**Sandra Duran Wilson**\n\n_The Other Side_\n\nAcrylic, gold leaf and collage on panel\n\n40\" \u00d7 48\" (102cm \u00d7 122cm)\n\nPrivate collection of Naresh and Kavita Nakra\n\n**Variation 1: Experimental Techniques and Surfaces**  \nThe combination of techniques and surfaces with geometry creates a unique space. Painting on old hollow core doors cut down to size, Sandra Duran Wilson first applies gold leaf to the entire surface. Heat and viscosity are used to push the medium around to create effects. Monoprints and wood block prints are cut in strong shapes then collaged, while glazing shifts colors. The circles are recurring symbols for the artist, representing overlapping dimensions of time.\n\n**Gary Denmark**\n\n_Jongleur 2_\n\nAcrylic on canvas\n\n48\" \u00d7 48\" (122cm \u00d7 122cm)\n\n**Variation 2: Contrast Geometric With Organic Shapes**  \nThe bold use of geometric shapes contrasts well with gradations of color, fluid spaces and organic forms.\n\n**More Variations**\n\n\u2022 Collect a variety of dispensable items such as smooth and textured papers and old paintings you no longer care for. Cut out different geometric shapes in a variety of sizes and in a range of colors. Arrange these on different painting surfaces to see how they change the painting. Rearrange the items to see how the spaces between the shapes change the idea of space in the painting.\n\n\u2022 Find a painting you admire. Overlay tracing paper and trace the large shapes created by elements in the painting. Create a new painting using these large shapes. Try rearranging them, overlapping them differently and adding your own imagery.\n\n  AVOIDING ATTACHMENT\n\nLiu does not get too attached to any part of the painting while working. Favored elements frequently present problems by inhibiting continued progress. Liu often uses materials that are precious and has ways of avoiding this attachment \"trap.\" Try this project below to test your unattachment skills.\n\n**Materials**\n\n**\u2022 Paints:** A variety of opaque and transparent acrylic paint colors\n\n**\u2022 Surface:** Any sturdy surface\n\n**\u2022 Painting Tools:** Favorite painting brushes, knives, rags, a mixing palette\n\n**\u2022 Other:** Any collage item you like (paper, photographs, fabric, objects, drawings), acrylic gloss gel\n\n**1. Glue the Collage Item**  \nUsing a surface that is much bigger than your collage item (could be a new surface or a painting already started), find a placement for the collage item with ample space around it for adding paint later. Glue the item securely in place using an acrylic gel. Here lies the big trick to avoiding attachment syndrome\u2014making a commitment by permanently gluing the object to the surface.\n\n**2. Mix Up a Variety of Opaque and Transparent Colors**  \nFor opacity, choose mineral (inorganic) pigments such as Cadmiums and Titanium White. Transparent pigments can be made opaque by using them thick, straight out of the container or by adding white. For transparent colors, choose modern (organic) pigments such as Phthalos and Quinacridones. Opaque pigments can be made transparent by adding large amounts of gloss mediums, or by applying thinly. Pre-mix a few colors that match the background of your painting and also of the collaged item. Match colors 10 percent lighter than desired, since acrylic appears lighter when wet.\n\n**3. Apply Varying Degrees of Coverage**  \nApply pre-mixed colors over the collage item and onto the painting surface. Obtain various degrees of coverage by heavily applying the paint using a knife in some areas, and by applying lighter applications with a brush or rag in others. Scrape off paint using a knife in some places to add even more variety.\n\n**Tip**\n\nIf your collage item is thin, like tissue paper, a thick acrylic (gel) offers a better choice for a glue than a thin acrylic (medium) and will reduce wrinkling. Use a matte gel if the glue will seep through the collage material, otherwise a gloss gel is best.\n**Section 6**\n\nMINIMAL\n\n**Paul Sarkisian**\n\n_Untitled (black mark\/amber\/vertical3)_\n\nResin epoxy on wood\n\n12'\u00d7 12' (366cm \u00d7 366cm)\n\nPhoto by Eric Swanson\n\nMinimalism presents exciting challenges for both the artist and the viewer. By limiting or eliminating representative imagery, the artist works solely with aesthetic aspects to create work that attracts. A painter's aesthetic tool box consists of color, line, shape, plane, space, surface and light. These tools in themselves carry associative and narrative connections. In this section, styles use a minimum of aesthetic aspects to get maximum impact, usually evoking a meditative mood or an altered state or reality, allowing viewers to lose themselves in the purely visual.\n\nSHEENS & TEXTURE\n\nUsing sheens and texture places the focus on the painting's surface and can create a seductive tactile experience. _Canyon and Poem_ combines metal leaf with acrylic paint. This combination presents an interesting challenge for me, as the metal leaf changes qualities like a chameleon. As available light shifts throughout the day, the metallic changes from a dark value to light, and from bright to dull. Realizing that busy imagery and bold color can overpower the reflective qualities of the metal leaf, I researched historic Asian landscapes and their expert use of minimal forms and color. A play with opposites adds a sense of space in the painting, with areas containing both high and low refractive qualities, smooth and textured areas, and warm and cool glazing on the metal leaf.\n\n**The Artist's Process**\n\nExperimentation is my main painting tool. I keep playing around until I discover something new that takes me over. I allow my techniques, processes and materials to change according to the needs of each series, allowing the idea to best come through the work. Inventing the way I paint is just as exciting to me as what I paint. For _Canyon and Poem_ , metal leaf is applied onto smooth panels then overpainted with acrylic. I get inspired by artists such as Anselm Kiefer, Wassily Kandinsky, Damien Hirst, and not surprisingly, many of the artists I interviewed for this book. Other ideas come from a continual searching for things I find beautiful, whether natural or man-made: fabrics, butterfly wings, cityscapes, mountains, skies, oceans and people. I draw from models once a week to keep my drawing skills alive, and continually enjoy visiting artists' studios, galleries and museums.\n\n**Other Art in This Style**\n\n\u2022 Chinese landscape paintings from about AD 900\u20131600\n\n\u2022 Illuminated manuscripts from the Middle Ages into the Renaissance period\n\n\u2022 Gustav Klimt's (1862\u20131918) use of metal leaf as backgrounds for sensuously painted figures\n\n **TIPS  \nfrom the Artist **\n\nDon't take action on every idea that comes to you. Keep a journal to write down ideas as they come, allowing them to percolate until the right one really fuels you. Then take action. Inspired action tells you the next step. Time, experimenting and experiences bring artists to their personal vision. The inspiration for our unique vision is already in us, making it difficult for anyone to copy what we do successfully. Do what comes naturally, but make the time to create and the bravery to make things you might not have seen before. Workshops and classes are helpful, but take long periods or enough time in between to work in the studio and let the new tools and techniques simmer into your own stew.\n\nNancy Reyner in her Santa Fe studio at work on her metal leaf paintings.\n\n**Nancy Reyner**\n\n_Canyon and Poem_\n\nAcrylic and gold leaf on panel\n\n48\" \u00d7 36\" (122cm \u00d7 91cm)\n\nPrivate collection\n\nVARIATIONS OF SHEENS & TEXTURE\n\n**Bette Ridgeway**\n\n_It Was a Rainy Red Autumn_\n\nAcrylic with resin on aluminum\n\n32\" \u00d7 24\" (81cm \u00d7 61cm)\n\n**Variation 1: Create a Glasslike Surface with Resin**  \nClear layers of resin are applied in between colored layers, generating a glasslike surface with embedded textural effects. An aluminum surface is allowed to reflect through multiple transparent layers.\n\n**Reynaldo Villalobos**\n\n_Redemption 1_\n\nAcrylic on panel\n\n60\" \u00d7 48\" (152cm \u00d7 122cm)\n\n**Variation 2: Textured, Monochromatic Palette**  \nA monochromatic palette places focus on textures and sheens. The texture here is actually only visual, with no relief or physical texture. The effect is created by allowing paint to seep under previously taped areas, creating a thin but dramatic variation in darks and lights.\n\n**More Variations**\n\n\u2022 Create a textured ground or surface under metal leaf by first applying thick acrylic pastes or gels and letting it dry before applying the leaf. The leaf will then emphasize this underlying texture.\n\n\u2022 Create a collage using materials with a variety of sheens and textures.\n\n  CONTRASTING METALLICS\n\nMetal leaf is an appealing material to use. There is a noticeable difference in the refraction between metal leaf and metallic paints. By contrasting them together in the same painting an interesting result can be obtained.\n\n**Tip**\n\nUse imitation gold leaf with a waterbased adhesive to avoid tarnishing. When using paint color, avoid reducing the reflective quality of the metal leaf by using modern (organic) pigmented paints like Phthalos and Quinacridones instead of the mineral (inorganic) paints like Cadmiums, which are opaque. Avoid matte or satin acrylic products as they will reduce the leaf's reflective qualities.\n\n**Materials**\n\n**\u2022 Paints:** A variety of metallic acrylic paints (copper, gold, bronze, silver)\n\n**\u2022 Surface:** Any painting support\n\n**\u2022 Painting Tools:** Soft flat or foam brush, painting knife\n\n**\u2022 Other:** Metal leaf, leafing adhesive, mixing palette, gloss acrylic gel, mineral spirits-based acrylic varnish in spray or jar\n\n**1. Seal the Metal Leaf**  \nApply metal leaf on a support using leaf adhesive and following directions on the product's label. Overpaint or spray the leaf with a mineral spirits-based acrylic and let dry. This coating will ensure good adhesion for any subsequent acrylic paint layers.\n\n**2. Create a Variety of Metallic Mixtures**  \nUsing a variety of metallic acrylic paints (i.e., copper, bronze, gold, silver, etc.), mix them together in various combinations to create new metallic colors. As an option, add clear gloss acrylic gels into the paint to add texture.\n\n**3. Overpaint the Metal Leaf**  \nApply the various metallic paint mixtures over selected areas of the metallic leaf surface. Use a brush or knife, and apply thinly for a smooth application or thickly to create texture. Avoid covering the leaf surface completely so the leaf can be seen along with the paint. Metallic paint will result in a satin finish as compared to the leaf's highly reflective sheen.  \nAs an alternative in this step, substitute thinly applied transparent glazes of color to subtly shift the metallic leaf instead of overpainting with the metallic paints.\n\nILLUMINATED\n\nIllumination or the representation of light in a painting can be conveyed both figuratively or literally. Light has always played an essential role for painters. Chiaroscuro, developed in the Renaissance, uses paint to create dramatic lights and dark shadows indicating a strong light source and adding depth and mystery to the image. Contemporary artists such as Dan Flavin create art using actual light as the medium itself. Stained glass, photography and light projections are all art forms that use actual or physical light sources in the final art.\n\nPaul Sarkisian's art career spans an astonishing range. Sarkisian continually invents and reinvents to create works that allow us to question our perception of reality. _Untitled (bronze\/ green42)_ is from a series of three hundred works by Sarkisian and uses contoured surfaces and pearlescent pigment to create a surface that color-shifts from different viewing angles and in different light. Sarkisian sees them as fields of energy, greatly affected by their placement in space.\n\n**The Artist's Process**\n\nSarkisian's favorite tool is his creative thoughts. He thinks of how to make his idea in its purest form, resulting in a wide range of mediums, styles and processes. Works are created using tools such as airbrush, trowel and brush and in modes such as collage, photography, photorealism and minimalist abstraction, with mediums including acrylic, oil, asphalt, tar, gouache, gypsum, resin and foam. What he does with the technology he chooses is very important to him. In general, Sarkisian works on one piece at a time within a series.\n\n **TIPS  \nfrom the Artist **\n\nLots of people are afraid to use their talents, and are intimidated by the art scene and superstars. If you have talent, why not use it?\n\nPaul Sarkisian in his studio, in front of one of his paintings.\n\n**Other Art in This Style**\n\nWorks using real light sources by artists such as Dan Flavin, Olafur Eliasson, James Turrell, Waltraut Cooper, Aleksandra Stratimirovic and Austine Wood Comarow.\n\n**Paul Sarkisian**\n\n_Untitled (bronze\/green42)_\n\nPolymer resin and automotive enamel on wood\n\n6'5\" \u00d7 11'8\" (196cm \u00d7 336cm)\n\nPhoto by Eric Swanson\n\n**Detail**\n\nVARIATIONS ON ILLUMINATION\n\n**Sandy Keller**\n\n_Lapis_\n\nAcrylic on panel\n\n24\" \u00d7 36\" (61cm \u00d7 91cm)\n\n**Variation 1: A Rich Value Range in a Single Color**  \nThe use of a monochromatic palette places emphasis on the rich value range and feeling of illumination from within the painting's depths.\n\n**More Variations**\n\n\u2022 Find different lighting situations and paint, photograph or draw these to capture their mood and feeling.\n\n\u2022 Notice shadows and patterns created from the sun and other light sources and use these for painting inspirations.\n\n\u2022 Create a still life using one light source. Paint the dark cast shadows and highlighted areas to capture the feel of light on the objects.\n\n**Bette Ridgeway**\n\n_Yellow Moon_\n\nAcrylic with resin on aluminum\n\n30\" \u00d7 24\" (76cm \u00d7 61cm)\n\n**Variation 2: Colorful Layers Over an Aluminum Surface**  \nResin layers over an aluminum surface creates refraction and illumination. The image itself, although minimal in forms, parallels this light idea by subtly referring to a moonlit landscape.\n\n  ILLUMINATED POURING\n\nThis technique takes advantage of two categories of reflective paints: Iridescent and Interference. Both can be used singly or in combinations to create new, unusual, contemporary illuminated effects. This technique uses Iridescent in the background and Interference in a transparent overpour. Alternatively, the addition of color to this technique will create a vast array of possibilities. Some helpful tips and definitions before starting:\n\n**\u2022 Rigid surface:** Pouring is easiest when done over a rigid, sturdy support such as panel or wood.\n\n**\u2022 Reflective Mixtures:** Adding any Iridescent or Interference to a clear medium or gel will make a reflective mixture. Adding a small amount of black to interference enhances its color. If adding color, use small amounts to keep the color transparent.\n\n**\u2022 Pourable Mediums:** Any acrylic medium that will dry clear and glossy and readily pours from its container can be used in a pour technique. Th in the medium with up to 20 percent water or use specialty pouring products like Golden's GAC 800 or commercial resins.\n\n**Materials**\n\n**\u2022 Paints:** Several Iridescent and Interference paints (optional addition: any transparent modern paint color)\n\n**\u2022 Surface:** Any rigid or sturdy paint surface\n\n**\u2022 Painting Tools:** Any brush or other painting application tool, plastic mixing container and stirrer, wide flat spreading tool or knife, mixing palette\n\n**\u2022 Other:** A pourable acrylic medium, household spray bottle, isopropyl alcohol, water\n\n**1. Paint the Background with Reflective Paint**  \nUsing a brush or knife, apply any Iridescent and\/or Interference paint in any combination to create a reflective background. Here Iridescent Silver (fine) is knife-applied. Let the background dry fully.\n\n**2. Add Black Washes**  \nWith a brush, add enough clean water to the entire surface to create puddles. On a separate palette, mix black acrylic paint with water in a 1:10 ratio. Apply this black wash to the surface puddles. Let dry naturally, and the black washes will form small irregular shapes and patterns. These dark shapes help enhance the pour you will do in Step 3.\n\n**3. Pour a Transparent Reflective Mixture**  \nCreate a reflective mixture by adding any reflective paint to a pourable medium in a 1:10 ratio to maintain transparency. (Pictured here is 9 parts Clear Tar Gel with 1 part water and 5 drops each of Interference Blue, Interference Violet and Interference Red.) Lightly stir, then pour the mixture over the entire surface. Mist lightly with isopropyl alcohol. Let dry flat. Repeat Steps 2 and 3 as much as desired, letting each layer dry before adding another.\n\n**Finished Example**\n\nCOLOR FIELD\n\nEmerging out of New York City in the 1940s and 1950s, Color Field paintings generally consist of large unbroken areas of pure color, often creating an emotional or expressive field, with minimal attention to brushstroke, texture and forms. Aleta Pippin's desire to experiment and work with color as her primary tool is readily visible in her work, and her choice of working minimally allows an internal viewing.\n\n**The Artist's Process**\n\nPippin does not like to over-plan. She usually starts by pouring a color onto the surface, setting up spontaneous events or color combinations and continuing until she finds something beautiful. She then decides to enhance, add or change areas to bring visual depth and light to the color. Pippin chooses application techniques that minimize the brush mark. She finds the color richer when applied with a knife. Applied thinly, the knife allows the paint to overlay without affecting the bottom layers, easing a building up of paint layers. Pippin continually revisits and reevaluates what she is doing and why she is doing it. She is not afraid to take a different direction and try something new, realizing each new path has a learning curve, requiring a sense of bravery to allow the feeling of being a beginner again. Early in her career Pippin was inspired by Wassily Kandinsky, Willem de Kooning and Georgia O'Keeffe. Now she is more influenced by artists such as Gustav Klimt and Helen Frankenthaler. Additional inspiration comes from listening to music from Pink Floyd to George Gershwin.\n\n **TIPS  \nfrom the Artist **\n\nBelieve in what you're doing. Be focused and don't listen to what others say. Artists are entrepreneurs\u2014don't buy into the starving artist myth. Learn basic business skills to get started.\n\nAleta Pippin at work in her Santa Fe studio.\n\n**Other Art in This Style**\n\n\u2022 Pioneering Color Field painters: Mark Rothko, Helen Frankenthaler, Clyfford Still, Robert Motherwell, Barnett Newman, Hans Hofmann, Ad Reinhardt, Sam Francis, Anselm Kiefer\n\n\u2022 Pat Steir's waterfall paintings\n\n**Aleta Pippin**\n\n_The Fabric of Life #4_\n\nAcrylic on Yupo paper mounted on panel\n\n8\" \u00d7 8\" (20cm \u00d7 20cm)\n\nCollection of Patti and Bill Nash\n\nVARIATIONS ON COLOR FIELD\n\n**Robin Sierra**\n\n_Aquamarine_\n\nAcrylic on canvas\n\n25\" \u00d7 27\" (64cm \u00d7 69cm)\n\nCollection of Anton Lengmueller\n\n**Variation 1: Minimize Forms**  \nThis strong unified color field contains a range of color in softly integrated formless areas. The image verges on becoming recognizable as water or reflections.\n\n**Sandy Keller**\n\n_Time Lapse_\n\nAcrylic on panel\n\n48\" \u00d7 36\" (122cm \u00d7 91cm)\n\n**Variation 2: Flowing Areas**  \nA bold use of flowing forms creates movement, light and pattern within a strong field of color. \"It is always about color and creating that magic moment when two or more colors vibrate against each other. That compelling force draws the viewer in, perhaps evoking a memory of a transient moment once seen,\" says Keller.\n\n**More Variations**\n\n\u2022 Collect pieces of pure color (without recognizable forms to distract attraction) by ripping pieces of magazine pictures and collecting chips from paint stores and swatches of fabric. Remake these colors in paint form and build them up in paint layers to enhance the color's richness.\n\n\u2022 Minimize the forms from an image you like by only painting its colors and shapes, and avoiding any detail. Then remove your original reference from view to continue painting. Merge and blend shapes into each other by softening edges to create an overall field of color.\n\n  COLOR ENHANCEMENT\n\nWhen color is applied in several layers, the pigment takes on a richer quality. With multiple layers, light is able to refract in more complex ways, enhancing color and surface quality. Here is a technique using three layers to enhance color quality without changing hue.\n\n**Tip**\n\nThis technique works best when each step uses very transparent layers. For the pour mixture in step 3 it is important to use very little paint color and\/or keep application thin.\n\n**Materials**\n\n**\u2022 Paints:** Any acrylic color in several variants of its hue\n\n**\u2022 Surface:** Any rigid absorbent surface\n\n**\u2022 Painting Tools:** Soft flat blending brush, mixing palette\n\n**\u2022 Other:** Acrylic Glazing Medium or other slow drying medium, household spray bottle, mixing cup, mixing stick, palette knife, isopropyl alcohol, water\n\n**1. Apply Color as a Wash**  \nStart with an absorbent painting surface, or make it absorbent by applying an absorbent product on first as a ground. A surface merely primed with gesso will not be as effective. Products that make good absorbent grounds will dry matte with a discernible tooth or texture, such as acrylic pastes, matte gels or gels with added materials like pumice. Alternatively, you can adhere watercolor paper to your painting surface.\n\nOnce you've attained an absorbent surface, select a color and heavily dilute the acrylic paint with water in a 1:5 ratio. Apply this color wash using a soft, wide, flat brush over the entire surface. Here Cadmium Red Medium Hue, properly diluted with water, is applied over a panel prepared with Golden's Acrylic Ground for Pastel. Let the surface dry. Repeat this step if more color is desired using less water in the wash mixtures.\n\n**2. Apply the Color as a Glaze**  \nUsing the same color from the prior step, or a variant of the same hue, combine color to gloss medium in a 1:10 ratio without adding any water. Optimally, use a slow-dry medium for this mixture to make glazing easier. Here, Primary Magenta is mixed with Acrylic Glazing Liquid and evenly brush-applied over the dried surface from the previous step. Let dry.\n\n**3. Pour the Color**  \nRaise the painting surface off your work table with jars or other objects for support and place on plastic. Add the same color as before, or a variant of the same hue into a pourable gloss medium in a 1:20 ratio. Optionally, add up to 20 percent water if the medium is thick. Here Pyrrole Red is mixed with GAC 800 (no water is needed with this Golden product). Pour this mixture in a puddle in the center of the surface. Gently spread the mixture outward toward the corners and edges with a wide knife. Immediately spray the surface lightly with isopropyl alcohol in a household sprayer to remove any bubbles. Keep level while drying.\n\nA SINGULAR NOTE\n\nA dramatic visual impact can be obtained by presenting one strong form against a minimal field of color. This is readily seen in Jim Alford's painting _Confluence_. A hard edged circle placed against a soft background allows the illusion of floating. Jim Alford aligns with Minimalist philosophies, in particular with the difference between gazing at a painting and staring at it in a conventional Western way. Alford's intent is to offer a meditative experience for the viewer. Interestingly, Alford has his own mind-altering experience in the painting process. Alford typically paints using an airbrush. As he works on layer after layer, he progressively steps back with the sprayer farther and farther from the work, in a rhythmic movement similar to Tai Chi, often placing Alford in a meditative state as well.\n\n**The Artist's Process**\n\nAlford's work spans an impressive range from super-realistic to minimal abstraction. This diversity mandates an equal range in techniques and processes. Alford often begins his paintings on the computer, experimenting with colors and shapes, sometimes constructing an image with combinations of several photographs he has taken. Once he finds an image he likes, he takes it to his studio, transferring the idea into paint. Acrylic is his choice for spraying, which is less toxic than oil. Alford finds his inspiration in nature and landscape, walking along California beaches looking at the sun reflecting in the ocean, and in horizon lines and figures silhouetted against dark skies. Artists inspiring to him include Larry Rivers, Richard Dieben-korn, Mark Rothko, James Terrell, and contemporary landscape painters such as Robert Workman, Peter Nisbet and Doug West.\n\n **TIPS  \nfrom the Artist **\n\nEvery artist has to find his voice and it takes a long time. Graduate school is one way to assist this process, especially by selecting teachers who allow freedom.\n\nJim Alford's studio view offers a continual reminder of nature and landscape.\n\n**Other Art in This Style**\n\nWorks by Morris Louis, Barnett Newman's zip paintings, Yves Klein, Agnes Martin and Susan Rothenberg\n\n**Jim Alford**\n\n_Confluence_\n\nAcrylic on canvas\n\n60\" x 48\" (152cm x 122cm)\n\nVARIATIONS ON A SINGULAR NOTE\n\n**Steve Stone**\n\n_Nol_\n\nAcrylic on canvas\n\n60\" \u00d7 36\" (152cm \u00d7 91cm)\n\nPrivate collection\n\n**Variation 1: A Subtle, Singular Form**  \nA textural field supports a subtle, singular emerging form, a cross, in the upper right through the use of acrylic pastes and gels and a variety of mark-making tools.\n\n**Grant Wiggins**\n\n_Spaceloop Two_\n\nAcrylic on canvas\n\n30\" \u00d7 30\" (76cm \u00d7 76cm)\n\nCollection of the artist\n\n**Variation 2: A Bold Singular Form**  \nRacing stripes are graphically illuminated with fluorescent paint and emphasized on a solid black background.\n\n**More Variations**\n\n\u2022 Select some favorite forms including geometric, organic, realistic and abstract. Draw or cut them out of colored paper. Arrange each one on different surfaces as backgrounds using paintings, tiles, fabric, wallpaper, etc. Use these as a model for creating a painting from scratch.\n\n\u2022 Experiment with materials and techniques to create an overall background or field of color containing some accidental marks or forms. Select one of these accidental forms and make it more visible with painting techniques such as increasing the color hue or intensity, making it lighter or brighter, sharpening the edges and\/or blurring or softening everything around it.\n\n  TEXTURE WITH MESH TAPE\n\nA subtly textured background can add visual appeal to a minimal abstraction. The grid pattern of dry wall mesh tape creates an interesting effect when used in this two-layer technique using paste and washes.\n\n**Tip**\n\nEliminate step 3 by substituting thick acrylic paint color for the paste in step 2.\n\n**Materials**\n\n**\u2022 Paints:** Several acrylic paint colors\n\n**\u2022 Surface:** A primed or pre-painted surface\n\n**\u2022 Painting Tools:** Any brush, painting knife or other applicator tool\n\n**\u2022 Other:** Drywall self-adhesive mesh tape, absorbent acrylic paste, scissors\n\n**1. Attach Drywall Tape**  \nOn a dry surface, pre-painted with a background color, apply strips of drywall mesh tape. Press firmly to stick the tape to the surface. The background color pictured here is a mixture of Titanium White and Primary Blue.\n\n**2. Apply Paste**  \nGenerously load an acrylic paste that is absorbent or dries matte to the back of a painting knife (Light Molding Paste is shown here). Apply the paste over the tape using medium pressure. For variety, allow some areas of the tape to remain without any paste. Remove the tape and allow paste to dry at least a few hours or overnight.\n\n**3. Overpaint With Washes**  \nDilute each of several acrylic paint colors with water in a 1:5 ratio to create several wash colors. You can also pre-wet the dry paste surface with water in some areas to increase a bleed effect. Brush-apply the diluted colored washes separately over the paste, varying colors in areas. Blot with paper towel in some areas to lighten. Pictured here are washes using Phthalo Blue, Carbon Black, Dioxazine Purple, Anthraquinone Blue, Phthalo Turquoise, Quinacridone Burnt Orange and Nickel Azo Yellow.\n\n**Finished Example**\n\nVEILING FORMS\n\nA veil is something that hides yet reveals. Veiling in painting happens when a transparent or translucent layer is applied over another layer, partially hiding what's underneath. Veiling can soften a focus, shape or edge. It can create a misty atmospheric space or add a surface quality similar to encaustic or wax. Veiling adds a sense of mystery in the act of partially hiding things. Bonnie Teitelbaum's paintings use multiple layers, building one on top of another, veiling forms and color areas to create fluid fields. Teitelbaum feels veiling adds more dimension, simulating nature, while flat applications of color feel more inorganic. Her paintings often evoke distilled ecosystems.\n\n**The Artist's Process**\n\nMeditative processing is a key part of Teitelbaum's painting process. She starts her day painting early in the morning to take advantage of feeling centered. Teitelbaum sees painting as a collaboration between herself and the chemistry of the paint, noting that she needs to put in a certain amount of paint and time to get the image to a certain point. First she experiments to see what happens, often discovering new and unusual forms. These accidental discoveries then get developed further with more paint. She frequently applies gels over the painting in stages between layers. Gels are mixed with color, then poured on top of other layers. While the gels are still wet, she adds more color and manipulates the work.\n\n **TIPS  \nfrom the Artist **\n\nIt's good to discover new tools. Abstraction relies on experimentation and discovery. Education is also important, learning about the characteristics of paint and materials.\n\nTeitelbaum gets inspiration out in nature, offering her peace of mind, and which she intends to convey through her work. Teitelbaum is inspired by James Abbott McNeil Whistler's reduction of elements to distill a landscape. She is also inspired by Vincent Van Gogh for movement, brushwork and color.\n\n**Other Art in This Style**\n\n\u2022 The painted forms of Joseph Mallord William Turner (1775\u20131851) that dissolve into atmosphere\n\n\u2022 The evocative, moody works of James Abbott McNeill Whistler's (1834\u20131903) Nocturnes series\n\n\u2022 Abstract Expressionist painters Morris Louis and Helen Frankenthaler\n\n**Bonnie Teitelbaum**\n\n_Tidal Flow_\n\nAcrylic on panel\n\n24\" \u00d7 24\" (61cm \u00d7 61cm)\n\nVARIATIONS ON VEILING FORMS\n\n**Doug Trump**\n\n_Breezeway_\n\nMixed media on polymesh\n\n21\" \u00d7 33\" (53cm \u00d7 84cm)\n\n**Variations: Neutral Colors and Obscured Forms**  \nThese paintings by Doug Trump represent excellent examples of veiling. Trump uses neutral colors plus obscured forms to create a sense of mystery, allusiveness and space. For Trump a painting is a demonstration of its history. He works it until it breathes on its own.\n\n**Doug Trump**\n\n_Tiptoe_\n\nMixed media on paper\n\n32\" \u00d7 40\" (81cm \u00d7 102cm)\n\n**More Variations**\n\n\u2022 While creating a painting, place pieces of tissue paper over areas without attaching them just to see how veiling will change its appearance. When you find a place where veiling improves the work, apply the tissue permanently using acrylic as glue, or apply layers of matte gel, or transparent applications of colored paint.\n\n\u2022 Dry-brushing or using thin layers of paint can also produce translucent layers of color that can act as veils. Try finding several ways of applying a layer of color on top of another without being so opaque that it completely hides the underlayer.\n\n  EMBEDDING PAINTED FORMS\n\nAn intriguing spatial depth can be created when paint is embedded in transparent layers of acrylic. You can create some interesting options and effects with this technique.\n\n**Tips**\n\n\u2022 Use a matte gel for this technique to create an encaustic or waxy look.\n\n\u2022 Gloss will give the most clarity, while matte or satin will give a frosted or veiled appearance. Try varying the types of gel each time you repeat Step 1 for a new layer.\n\n**Materials**\n\n**\u2022 Paints:** A variety of acrylic paint colors\n\n**\u2022 Surface:** A nonstick surface for acrylic such as freezer paper, HDPE plastic or garbage bags\n\n**\u2022 Painting Tools:** Palette knife or spatula, brush\n\n**\u2022 Other:** Acrylic gel (matte, satin or gloss)\n\n**1. Apply Gel to a NonStick Surface**  \nSelect an acrylic gel. Using a wide spatula, apply gel onto a nonstick surface in a thick layer, at least 1\/8-inch (3mm) thick. Work with the knife to get the surface fairly smooth.\n\nIt's your choice to continue to the next step while the gel is still wet, or to wait until it's dry. Each option will produce different results. Experiment both ways to find your preference.\n\n**2. Apply Paint Color**  \nSelect several acrylic paint colors using a range of hues and transparencies. Add water to each of the paint colors. Start with a 1:1 ratio, then create variety by adding more water to increase the transparency, and less to increase the opacity. Apply paint to the top surface of the wet or dry gel using a brush or knife. If desired, spray water over the paint before it dries to get softer edges and blurred forms. Let dry on a level surface.\n\n**3. See the Reverse Side**  \nThe gel should dry for at least a day or more until the white of the acrylic has turned clear. Peel the gel off the surface. Note its reverse side is smoother and the painted forms appear embedded. This \"skin,\" or unattached piece of painted gel, can now be glued onto a background with either side facing up.\n\n**Alternative for Step 3**  \nBefore securing the skin onto a background, repeat the first two steps to create more skin layers. Combine skins one on top of the other, attaching them by gluing with more gel.\nCONTRIBUTING ARTISTS\n\n**Jim Alford**\n\nwww.jimalford.com\n\nNew Mexico\n\n**Hamish Allan**\n\nwww.hamishallan.co.nz\n\nChristchurch, New Zealand\n\n**Daniel Barkley**\n\nwww.danielbarkley.com\n\nMontreal, Canada\n\n**James Barsness**\n\nwww.georgeadamsgallery.com\n\nwww.cclarkgallery.com\n\nGeorgia\n\n**Gerard Boersma**\n\nwww.gerardboersma.nl\n\nThe Netherlands\n\n**Lea Bradovich**\n\nwww.leabradovich.com\n\nwww.nuartgallery.com\n\nNew Mexico\n\n**Patti Brady**\n\nwww.pattibrady.com\n\nSouth Carolina\n\n**Bruce Cody**\n\nwww.brucecody.com\n\nNew Mexico\n\n**Dennis Culver**\n\nwww.doculver.com\n\nNew Mexico\n\n**Kasarian Dane**\n\nwww.kasariandane.com\n\nNew York\n\n**Jason de Graaf**\n\njasondegraaf.blogspot.com\n\nQuebec, Canada\n\n**Gary Denmark**\n\nwww.garydenmark.com\n\nNew Mexico\n\n**Harry Doolittle**\n\nwww.harrycdoolittle.com\n\nNew York\n\n**Leah Dunaway**\n\nwww.leahdunaway.com\n\nTexas\n\n**William Dunlap**\n\nwww.williamdunlap.com\n\nVirginia\n\n**Joey Fauerso**\n\nwww.joeyfauerso.com\n\nTexas\n\n**Frances Ferdinands**\n\nwww.francesferdinands.com\n\nToronto, Canada\n\n**Lisa Ferguson**\n\nwww.lisaferguson.com\n\nNoosa, Australia\n\n**Pat Forbes**\n\nwww.patriciaforbesart.com\n\nNew Mexico\n\n**Phil Garrett**\n\nwww.philgarrett.com\n\nSouth Carolina\n\n**Cate Goedert**\n\nwww.categoedert.com\n\nNew Mexico\n\n**Jylian Gustlin**\n\nwww.jyliangustlin.com\n\nCalifornia\n\n**Diana Ingalls Leyba**\n\nwww.dianaingallsleyba.com\n\nNew Mexico\n\n**McCreery Jordan**\n\nwww.mccreeryjordan.com\n\nNew Mexico\n\n**Sandy Keller**\n\nwww.sandykellerart.com\n\nNew Mexico\n\n**Martha Kennedy**\n\nwww.marthakennedy.com\n\nNew Mexico\n\n**Jakki Kouffman**\n\nwww.jakkikouffman.com\n\nNew Mexico\n\n**Ines Kramer**\n\nwww.ineskramer.com\n\nNew Mexico\n\n**Katherine Chang Liu**\n\nwww.lewallencontemporary.com\n\nwww.jenkinsjohnsongallery.com\n\nCalifornia\n\n**Sherry Loehr**\n\nwww.sherryloehr.com\n\nCalifornia\n\n**Catherine Mackey**\n\nwww.catherinemackey.com\n\nCalifornia\n\n**Thaneeya McArdle**\n\nwww.thaneeya.com\n\nFlorida\n\n**Darlene McElroy**\n\nwww.darleneoliviamcelroy.com\n\nNew Mexico\n\n**Barbara Moody**\n\nwww.barbaramoody.com\n\nMassachusetts\n\n**Keith Morant**\n\nwww.keithmorant.com\n\nChristchurch, New Zealand\n\n**Mary Morrison**\n\nwww.marymorrison.info\n\nColorado\n\n**Tom Palmore**\n\nwww.flyingpaintbrush.com\n\nOklahoma\n\n**Mary L. Parkes**\n\nwww.marylparkes.com\n\nNew Mexico\n\n**Debi Pendell**\n\nwww.debipendell.com\n\nMassachusetts\n\n**Ren\u00e9e Phillips**\n\nwww.ReneePhillipsArt.com\n\nNew York\n\n**Aleta Pippin**\n\nwww.aletapippin.com\n\nNew Mexico\n\n**Paul Pletka**\n\nwww.rivayaresgallery.com\n\nNew Mexico\n\n**Don Quade**\n\nwww.donquade.com\n\nColorado\n\n**Nancy Reyner**\n\nwww.nancyreyner.com\n\nNew Mexico\n\n**Bette Ridgeway**\n\nwww.ridgewaystudio.com\n\nNew Mexico\n\n**Paul Sarkisian**\n\nNew Mexico\n\n**Nancy Scheinman**\n\nwww.scheinman.com\n\nMaryland\n\n**Sam Scott**\n\nwww.samscottart.com\n\nNew Mexico\n\n**Olga Seem**\n\nolgaseem@earthlink.net\n\nCalifornia\n\n**Anne Seidman**\n\nwww.anneseidman.com\n\nPennsylvania\n\n**Robin Sierra**\n\nwww.rsierra.net\n\nNew Mexico\n\n**Daniel Smith**\n\nwww.danielsmithwildlife.com\n\nMontana\n\n**Teresa Stanley**\n\nwww.teresastanley.com\n\nCalifornia\n\n**Steve Stone**\n\nwww.stevestoneart.com\n\nArizona\n\n**James Strombotne**\n\nwww.strombotnestudio.com\n\nCalifornia\n\n**Beth Ames Swartz**\n\nwww.bethamesswartz.com\n\nArizona\n\n**Dannielle Tegeder**\n\nwww.dannielletegeder.com\n\nNew York\n\n**Bonnie Teitelbaum**\n\nwww.bonnieteitelbaum.com\n\nNew Mexico\n\n**Doug Trump**\n\nwww.dougtrump.com\n\nVermont\n\n**Reynaldo Villalobos**\n\nreynaldovillalobos@earthlink.net\n\nCalifornia\n\n**Jim Waid**\n\nwww.jimwaidart.com\n\nArizona\n\n**Frank Webster**\n\nwww.fwebster.com\n\nNew York\n\n**Grant Wiggins**\n\nwww.wiggz.com\n\nArizona\n\n**Sandra Duran Wilson**\n\nwww.sandraduranwilson.com\n\nNew Mexico\n\n**Dave Yust**\n\ndavyust@lamar.colostate.edu\n\nColorado\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n# Milk Proteins\n\n## From Expression to Food\n\nSecond edition\n\nHarjinder Singh\n\nMike Boland\n\nAbby Thompson\n\nRiddet Institute, Massey University, Palmerston North, New Zealand\n\n# Table of Contents\n\nCover\n\nTitle page\n\nFood Science and Technology International Series\n\nCopyright\n\nList of Contributors\n\nPreface to the Second Edition\n\nPreface to the First Edition\n\nChapter 1: The World Supply of Food and the Role of Dairy Protein\n\nAbstract\n\nIntroduction\n\nHunger and the need for food\n\nThe dietary essential amino acids in proteins\n\nIdentifying the countries deficient in dietary essential amino acids\n\nDemographic changes, aging populations, and the need for quality protein and essential amino acids\n\nGlobal trade in proteins, the long-term prospects, with a focus on dairy foods\n\nConclusions\n\nChapter 2: Milk: An Overview\n\nAbstract\n\nIntroduction\n\nEvolution of mammals and lactation\n\nUtilization of milk\n\nComposition of milk\n\nMilk constituents\n\nSummary\n\nChapter 3: The Comparative Genomics of Monotremes, Marsupials, and Pinnipeds: Models to Examine the Functions of Milk Proteins\n\nAbstract\n\nIntroduction\n\nThe echidna (Tachyglossus aculeatus)\n\nThe tammar wallaby (Macropus eugenii)\n\nA role for milk in the control of mammary function\n\nThe fur seal\n\nNew player in milk bioactives; MicroRNA\n\nConclusions\n\nChapter 4: Significance, Origin, and Function of Bovine Milk Proteins: The Biological Implications of Manipulation or Modification\n\nAbstract\n\nIntroduction\n\nOrigins of milk proteins\n\nConstraints and opportunities for evolution or manipulation of bovine milk proteins\n\nConclusion\n\nChapter 5: Post-translational Modifications of Caseins\n\nAbstract\n\nIntroduction\n\nThe caseins\n\nCaseins from other species\n\nConclusions\n\nChapter 6: Casein Micelle Structure and Stability\n\nAbstract\n\nIntroduction\n\nCasein primary structure and interactions\n\nCasein micelle properties\n\nModels of casein micelle structure\n\nConcluding remarks\n\nChapter 7: Structure and Stability of Whey Proteins\n\nAbstract\n\nIntroduction\n\nBovine \u03b2-Lactoglobulin\n\n\u03b1-Lactalbumin\n\nSerum albumin\n\nImmunoglobulins\n\nLactoferrin\n\nConcluding remarks\n\nAcknowledgments\n\nChapter 8: Effects of High-pressure Processing on Structure and Interactions of Milk Proteins\n\nAbstract\n\nIntroduction\n\nHigh-pressure-induced changes in caseins\n\nEffects of high pressure on interactions of milk proteins involving whey proteins\n\nConcluding remarks\n\nAcknowledgment\n\nChapter 9: The Whey Proteins in Milk: Thermal Denaturation, Physical Interactions, and Effects on the Functional Properties of Milk\n\nAbstract\n\nIntroduction\n\nThe casein micelle\n\nThe heat treatment of milk\n\nRelationships between denaturation\/interactions of the whey proteins in heated milk and the functional properties of milk\n\nConclusion\n\nChapter 10: Effects of Drying on Milk Proteins\n\nAbstract\n\nIntroduction\n\nProperties of spray-dried milk products\n\nPrinciples of spray drying\n\nProcess improvement\n\nDrying of proteins\n\nConclusions\n\nChapter 11: Changes in Milk Proteins during Storage of Dry Powders\n\nAbstract\n\nIntroduction\n\nThe formation of maillard and pre-maillard compounds\n\nFormation of isopeptide bonds\n\nAmino acids other than lysine\n\nImplications for nutritional value of milk proteins\n\nProduct-specific storage trials\n\nConclusions\n\nChapter 12: Interactions and Functionality of Milk Proteins in Food Emulsions\n\nAbstract\n\nIntroduction\n\nAdsorption of Milk Proteins During the Formation of Emulsions\n\nStability of Milk Protein-Based Emulsions\n\nHeat-Induced Changes in Milk Protein-Based Emulsions\n\nPressure-Induced Changes in Milk Protein-Based Emulsions\n\nMilk Protein Hydrolysates and Oil-In-Water Emulsions\n\nLactoferrin-Based Oil-In-Water Emulsions\n\nLipid Oxidation in Milk Protein-Based Emulsions\n\nBehavior of Milk Protein-Stabilized Emulsions Under Physiological Conditions\n\nConclusions\n\nChapter 13: Milk Protein\u2013Polysaccharide Interactions\n\nAbstract\n\nIntroduction\n\nMixing behavior of biopolymers\n\nPhase diagram\n\nNature of interactions in protein\u2013polysaccharide systems\n\nMilk protein\u2013polysaccharide interactions in the aqueous phase\n\nMilk protein\u2013polysaccharide interactions at the interface\n\nRheological properties and microstructures of protein\u2013polysaccharide systems\n\nConcluding remarks\n\nChapter 14: Interactions between Milk Proteins and Micronutrients\n\nAbstract\n\nIntroduction\n\nInteractions Between native Milk Proteins and Micronutrients\n\nInteractions between process-modified milk proteins and micronutrients\n\nConclusions\n\nChapter 15: Model Food Systems and Protein Functionality\n\nAbstract\n\nIntroduction\n\nProtein functionality in foods\n\nRole of interactions in determining food characteristics\n\nProcessing effects\n\nUses of model food systems\n\nApplications of model food systems\n\nUse of model food systems for other food components\n\nLimitations\n\nConclusions\n\nChapter 16: Sensory Properties of Dairy Proteins\n\nAbstract\n\nIntroduction\n\nSensory analysis\n\nWhey proteins\n\nMilk proteins\n\nCaseins and hydrolysates\n\nFlavor binding\n\nConclusions\n\nAcknowledgment\n\nChapter 17: Milk Protein Gels\n\nAbstract\n\nIntroduction\n\nRennet-induced gels\n\nAcid-induced milk gels\n\nWhey protein gels\n\nConclusions\n\nAcknowledgment\n\nChapter 18: Milk Proteins\u2014A Cornucopia for Developing Functional Foods\n\nAbstract\n\nIntroduction\n\nFunctional foods\n\nMilk proteins as a source of amino acids\u2014specialized nutritionals\n\nMilk proteins as a source of amino acids\u2014specific physiological roles\n\nMilk proteins as a source of amino acids\u2014role in providing calories and in promoting satiety\n\nMilk proteins as a source of bioactive peptides\n\nConclusions\n\nChapter 19: Milk Proteins and Human Health\n\nAbstract\n\nIntroduction\n\nMilk proteins, metabolic health, and type 2 diabetes\n\nMilk proteins, obesity, and weight control\n\nMilk proteins and bone health\n\nConclusions\n\nChapter 20: Milk Proteins: Digestion and Absorption in the Gastrointestinal Tract\n\nAbstract\n\nIntroduction\n\nDigestion of milk proteins\n\nMilk protein hydrolysis in the intestinal lumen\n\nPeptides released during digestion\n\nImpact of processing on milk protein digestion and absorption\n\nConclusions\n\nChapter 21: Milk Proteins: The Future\n\nAbstract\n\nIntroduction\n\nGlobal issues for food\n\nConsumer demands and trends for food and ingredients\n\nNew technologies and their possible effect on milk protein ingredients and products\n\nConclusions\n\nIndex\n\nFood Science and Technology: International Series\n\n# Food Science and Technology International Series\n\nSeries Editor\n\nSteve L. Taylor\n\nUniversity of Nebraska\u2014Lincoln, USA\n\nAdvisory Board\n\nKen Buckle\n\nThe University of New South Wales, Australia\n\nMary Ellen Camire\n\nUniversity of Maine, USA\n\nRoger Clemens\n\nUniversity of Southern California, USA\n\nHildegarde Heymann\n\nUniversity of California\u2014Davis, USA\n\nRobert Hutkins\n\nUniversity of Nebraska\u2014Lincoln, USA\n\nRon S. Jackson\n\nQuebec, Canada\n\nHuub Lelieveld\n\nBilthoven, The Netherlands\n\nDaryl B. Lund\n\nUniversity of Wisconsin, USA\n\nConnie Weaver\n\nPurdue University, USA\n\nRon Wrolstad\n\nOregon State University, USA\n\nA complete list of books in this series appears at the end of this volume.\n\n# Copyright\n\nAcademic Press is an imprint of Elsevier\n\n32 Jamestown Road, London NW1 7BY, UK\n\n225 Wyman Street, Waltham, MA 02451, USA\n\n525 B Street, Suite 1800, San Diego, CA 92101-4495, USA\n\nCopyright \u00a9 2014, 2009 Elsevier Inc. All rights reserved\n\nNo part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher\n\nPermissions may be sought directly from Elsevier's Science & Technology Rights\n\nDepartment in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: permissions@elsevier.com. Alternatively, visit the Science and Technology Books website at www.elsevierdirect.com\/rights for further information\n\nNotice\n\nNo responsibility is assumed by the publisher for any injury and\/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made\n\nBritish Library Cataloguing-in-Publication Data\n\nA catalogue record for this book is available from the British Library\n\nLibrary of Congress Cataloging-in-Publication Data\n\nA catalog record for this book is available from the Library of Congress\n\nISBN : 978-0-12-405171-3\n\nFor information on all Academic Press publications visit our website at elsevierdirect.com\n\nTypeset by Thomson\n\nPrinted and bound in United States of America\n\n14 15 16 17 10 9 8 7 6 5 4 3 2 1\n\n# List of Contributors\n\nSkelte G. Anema\n\nFonterra Research and Development Centre, Palmerston North, New Zealand\n\nS.D. Berry\n\nSchool of Population Health, University of Auckland, Auckland, New Zealand\n\nSwathi Bisana\n\nSchool of Medicine, Deakin University, Geelong, Victoria, Australia\n\nMike J. Boland\n\nRiddet Institute, Massey University, Palmerston North, New Zealand\n\nSrikanta Chatterjee\n\nMassey University, Palmerston North, New Zealand\n\nRod Collins\n\nSchool of Medicine, Deakin University, Geelong, Victoria, Australia\n\nTh\u00e9r\u00e8se Considine\n\nFonterra Research and Development Centre, Palmerston North, New Zealand\n\nM. Digby\n\nDepartment of Zoology and CRC for Innovative Dairy Products, University of Melbourne, Victoria, Australia\n\nM.A. Drake\n\nDepartment of Food Science, Southeast Dairy Foods Research Center, North Carolina State University, Raleigh, North Carolina, USA\n\nDidier Dupont\n\nUMR 1253 INRA, Rennes, France\n\nPatrick J.B. Edwards\n\nCentre for Structural Biology, Institute of Fundamental Sciences, Massey University, Palmerston North, New Zealand\n\nJohn Flanagan\n\nRiddet Institute, Massey University, Palmerston North, New Zealand; Naturex S.A., Avignon Cedex, France\n\nP.F. Fox\n\nSchool of Food and Nutritional Sciences, University College, Cork, Ireland\n\nKelvin K.T. Goh\n\nInstitute of Food, Nutrition and Human Health, Massey University, Palmerston North, New Zealand\n\nW. James Harper\n\nDepartment of Food Science and Technology, The Ohio State University, Columbus, Ohio, USA\n\nKerianne Higgs\n\nFonterra Research and Development Centre, Palmerston North, New Zealand\n\nJohn W. Holland\n\nInstitute for Molecular Bioscience, The University of Queensland, Australia\n\nDavid S. Horne\n\nFormerly Hannah Research Institute, Ayr, Scotland, UK\n\nThom Huppertz\n\nNIZO Food Research BV, Ede, The Netherlands\n\nGeoffrey B. Jameson\n\nCentre for Structural Biology, Institute of Fundamental Sciences, Massey University, Palmerston North, New Zealand\n\nAmit Kumar\n\nSchool of Medicine, Deakin University, Geelong, Victoria, Australia\n\nJoly Kwek\n\nSchool of Medicine, Deakin University, Geelong, Victoria, Australia\n\nChristophe Lefevre\n\nSchool of Medicine, Deakin University, Geelong, Victoria, Australia\n\nSimon M. Loveday\n\nRiddet Institute, Massey University, Palmerston North, New Zealand\n\nJohn A. Lucey\n\nDepartment of Food Science, University of Wisconsin-Madison, Madison, Wisconsin, USA\n\nRobin A. McGregor\n\nHuman Nutrition Unit, Institute for Innovation in Biotechnology, School of Biological Sciences, University of Auckland, Auckland, New Zealand\n\nK. Menzies\n\nDepartment of Zoology and CRC for Innovative Dairy Products, University of Melbourne, Victoria, Australia\n\nR.E. Miracle\n\nDepartment of Food Science, Southeast Dairy Foods Research Center, North Carolina State University, Raleigh, North Carolina, USA\n\nVengama Modepalli\n\nSchool of Medicine, Deakin University, Geelong, Victoria, Australia\n\nPaul J. Moughan\n\nRiddet Institute, Massey University, Palmerston North, New Zealand\n\nKevin R. Nicholas\n\nSchool of Medicine, Deakin University, Geelong, Victoria, Australia; Victorian Bioinformatics Consortium, Monash University, Clayton, Victoria, Australia\n\nJ.A. O'Mahony\n\nSchool of Food and Nutritional Sciences, University College, Cork, Ireland\n\nHasmukh A. Patel\n\nDairy Science Department, South Dakota State University, South Dakota, USA\n\nSally D. Poppitt\n\nHuman Nutrition Unit, Institute for Innovation in Biotechnology, School of Biological Sciences, University of Auckland, Auckland, New Zealand; Riddet Institute, Massey University, Palmerston North, New Zealand\n\nAnwesha Sarkar\n\nNestec Ltd., Vevey, Switzerland\n\nArnab Sarkar\n\nSynlait Milk Ltd., Rakaia, New Zealand\n\nPierre Schuck\n\nINRA, UMR 1253, STLO, Rennes, France\n\nJulie A. Sharp\n\nSchool of Medicine, Deakin University, Geelong, Victoria, Australia\n\nP.A. Sheehy\n\nCentre for Advanced Technologies in Animal Genetics and Reproduction (REPROGEN), Faculty of Veterinary Science, The University of Sydney, Camden, New South Wales, Australia\n\nHarjinder Singh\n\nRiddet Institute, Massey University, Palmerston North, New Zealand\n\nR.G. Snell\n\nSchool of Biological Sciences, University of Auckland, Auckland, New Zealand\n\nDaniel Tome\n\nUMR 914 INRA, Paris, France\n\nStephen Wanyonyi\n\nSchool of Medicine, Deakin University, Geelong, Victoria, Australia\n\nAshalyn Watt\n\nSchool of Medicine, Deakin University, Geelong, Victoria, Australia\n\nP. Williamson\n\nCentre for Advanced Technologies in Animal Genetics and Reproduction (REPROGEN), Faculty of Veterinary Science, The University of Sydney, Camden, New South Wales, Australia\n\nJ.M. Wright\n\nDepartment of Food Science, Southeast Dairy Foods Research Center, North Carolina State University, Raleigh, North Carolina, USA\n\nP.C. Wynn\n\nCentre for Advanced Technologies in Animal Genetics and Reproduction (REPROGEN), Faculty of Veterinary Science, The University of Sydney, Camden, New South Wales, Australia\n\nAiqian Ye\n\nRiddet Institute, Massey University, Palmerston North, New Zealand\n\n# Preface to the Second Edition\n\nIt is now five years since the first edition of Milk Proteins: From Expression to Food was published. In that time there have been considerable advances in the topics covered in the first edition. There is new awareness both of the importance of proteins in nutrition globally and of a burgeoning demand for animal-derived protein on a global scale, coupled with a recognition of the higher resource demands for producing proteins from animals in comparison with plant proteins. New knowledge is emerging on the roles of milk proteins in the prevention of chronic age-related conditions such as adverse metabolic health and type 2 diabetes, muscle wasting and sarcopenia, atherosclerosis, hypertension, and cardiovascular disease risk, as well as bone health and osteoporosis. Understanding the digestion of proteins and how processing can influence digestion kinetics is another emerging area, supported by the emergent science of peptidomics. Because of these changing research interests, three new chapters have been added to this volume, and several other chapters have been extensively expanded and rewritten. All chapters have been considerably revised and updated. Because not all of the original authors were available, some new authors came aboard to rewrite and update the chapters.\n\nLike the first edition, this book provides a comprehensive overview of the biology and chemistry of milk proteins and is intended for academics, researchers, students, and industry personnel interested in milk proteins. We would like to express our sincere appreciation to all the authors, in particular the new authors, for sharing their knowledge and expertise. Their cooperation and timely delivery of manuscripts made our task as editors a pleasure. Finally, we would like to thank the excellent staff at Elsevier, particularly Nancy Maragioglio, Carrie Bolger, and Caroline Johnson for their efforts in making this volume a reality.\n\nHarjinder Singh\n\nAbby Thompson\n\nMike Boland\n\nFebruary 2014.\n\n# Preface to the First Edition\n\nProteins are vital ingredients for the food industry because they provide all the essential amino acids needed for human health combined with a wide range of dynamic functional properties, such as the capacity to form network structures and stabilize emulsions and foams. The proteins of milk have excellent functional properties and nutritional value, and some have distinctive physiological properties, which are widely exploited in the food industry. Milk proteins have been the subject of intensive research during the last 50 years in an effort to unravel their molecular structures and interactions, relationship between structure and functional attributes, interactions of proteins during processing and, more recently, their physiological functions.\n\nRecent studies on the interactions of milk proteins in complex food systems are leading to a new understanding of the nature of these interactions and their impact on food quality. The knowledge has resulted in the development of several specialized milk protein ingredients tailored to meet specific needs of the food industry. Currently, there is a growing demand by the food industry for milk protein ingredients for specialist high-value applications such as functional foods. In the future, application of novel analytical approaches (genomics, proteomics, nanotechnology) to milk proteins and food materials will provide further understanding of molecular structures and interactions to enable the dairy industry to produce highly functional and healthy protein ingredients for specific applications.\n\nSeveral books have been published about milk and milk proteins\u2014so why another one? Most of the earlier books have addressed different specialist aspects of dairy science and technology. The primary theme of this book is to present a view along the dairy food chain\u2014starting at the cow (and its mammalian relatives) and finishing with nutritional aspects affecting the consumer, dipping into important current research topics along the way. The molecular structures and interactions of milk proteins under various processing environments are covered most prominently. More importantly, the book also contains a considerable amount of material from dairy industry-based or industry-funded research. Thus, it provides fresh perspectives on milk proteins, from an advanced dairy industry point of view.\n\nThe editors particularly thank Fonterra Research and Development Centre for making available time for staff members to contribute their chapters, and for making available hitherto unpublished material. This book is designed to provide an update and call for attention, for industry and academic researchers alike, to important and relevant milk protein science in areas that have the potential to advance the dairy industry.\n\nThe overall theme covered in this book was piloted at a meeting organized by the Riddet Institute and Fonterra Research and Development Centre in February 2006, with invited presentations from a number of experts in the relevant fields from Australasia, the USA and the UK. This meeting was particularly successful, with a large number of international delegates attending from a broad range of disciplines. This confirmed the growing interest of milk protein scientists in looking beyond the boundaries of their immediate topic area to gain an understanding of how the whole food chain fits together. Such an understanding can help elucidate mechanisms and processes, identify novel research opportunities, and provide additional applications for new developments.\n\nThis book includes chapters covering many of the topics addressed at the meeting, as well as some new subjects that we felt were important in order to provide a more complete picture of the journey from expression to food. We would like to thank both the contributors who have been involved from the meeting in 2006 and those who have come on board more recently.\n\nWe have chosen to start the book with a comprehensive overview of the biology and chemistry of milk, to set the stage and give a broad underpinning of the later chapters for readers not familiar with this field. Attention is then turned to the biology, and particularly the molecular biology, of lactation, looking first at some \"extreme\" mammals\u2014the tammar wallaby, which can express two different milk compositions at the same time, and the fur seal, which produces an extremely concentrated milk\u2014to give an idea of the range of biology of milk production. The book thereafter focuses on bovine milk, with mention of the milk from other domestic species as appropriate. This starts with an update on the genomics of bovine milk proteins, and is followed by an overview on post-translational modifications, which completes our view of the biology of milk protein production.\n\nThe structural chemistry of milk proteins, including the latest model of casein micelle and molecular structures of whey proteins, is covered in detail. The behavior of milk proteins under a variety of processing regimes, including ultra-high-pressure, functional systems, drying and storage of powders, is dealt with in a series of chapters. These chapters address our current state of knowledge about existing and emerging processes for the production of milk protein-based food ingredients.\n\nAttention is then turned to the behaviors of milk proteins in real and model food systems, and finally to consumer aspects\u2014the sensory and nutritional\/functional food aspects of milk proteins. A wrap-up chapter gives a view on likely issues of future importance for milk proteins, including the emerging area of nutrigenomics.\n\nAs with any volume written by a large number of contributors, this contains a variety of styles of presentation. We have made no attempt to homogenize the authors' styles, but have provided guidance on chapter content to make for best possible continuity.\n\nA volume of this kind always requires a large amount of work by a large number of people. We would like particularly to thank all the contributing authors for their efforts and their expeditious preparation of manuscripts that allowed for the timely publication of this book. We are pleased to acknowledge Claire Woodhall for assisting with the technical editing, and the staff at Elsevier for producing this book.\n\nAbby Thompson\n\nMike Boland\n\nHarjinder Singh\nChapter 1\n\n# The World Supply of Food and the Role of Dairy Protein\n\nSrikanta Chatterjee*\n\nArnab Sarkar**\n\nMike J. Boland***\n\n* Massey University, Palmerston North, New Zealand  \n** Synlait Milk Ltd., Rakaia, New Zealand  \n*** Riddet Institute, Palmerston North, New Zealand\n\n## Abstract\n\nWorld hunger continues to be a major problem. The main focus of those concerned with world hunger is on the availability of adequate calories for all, but this overshadows another problem: the availability of enough protein and enough dietary essential amino acids. We estimate that about a quarter of the world's population is getting barely enough protein; a particular issue is that of the essential amino acid lysine, which is deficient in cereal proteins, the biggest source of protein nutrition. Milk protein is an important source of dietary protein and is particularly rich in lysine and branched-chain amino acids. It accounts for 10% of all the global protein supply and provides the third highest supply after cereals and meat, but it is more important than the amounts would suggest because of its rich supply of essential amino acids and high digestibility (in contrast to cereal protein) and because of its acceptability to vegetarians. Global trade in dairy is still small (about 8% of dairy production is traded) but growing. Supplementation of cereal protein with milk protein has a potentially important role to play in balancing world protein nutrition.\n\n## Keywords\n\nProtein nutrition, milk protein, dairy trade, dairy production\n\nOutline\n\nIntroduction 2\n\nHunger and the need for food 3\n\nHunger-reduction Targets 3\n\nWorld Hunger and Undernutrition Status 4\n\nGlobal Hunger Index 4\n\nProtein and Its Composition and Bioavailability 6\n\nProtein Composition 6\n\nBioavailability 6\n\nWorld Protein Supply and Its Regional Distribution 7\n\nVegetable and Animal Protein Sources with a Focus on Dairy Foods 8\n\nGrowing Global Demand for Animal Proteins and Implications 8\n\nThe dietary essential amino acids in proteins 9\n\nIdentifying the countries deficient in dietary essential amino acids 9\n\nProtein and Dietary Essential Amino Acid Contents of Food Items 10\n\nDemographic changes, aging populations, and the need for quality protein and essential amino acids 10\n\nProtein Nutritional Needs of the Elderly 12\n\nRole of Essential Amino Acids in Nutrition of the Elderly 13\n\nGlobal trade in proteins, the long-term prospects, with a focus on dairy foods 14\n\nThe Global Dairy Food Scene: An Overview 14\n\nConclusions 16\n\n## Introduction\n\nAs one of the basic necessities of life, the availability, quality, and affordability of food are of concern to individuals and nations alike. The Green Revolution started in the late 1960s and involved the introduction of some new, high-yielding seed varieties, better use of irrigation facilities, and fertilizers. Following these innovations, the world enjoyed several decades of relative stability in the price of basic items of food, including food grains. The period from the early 1970s to 1990 saw world output of food grains and oilseeds rise steadily by an average of 2.2% a year, with periodic fluctuations. With the exception of parts of the African continent, the global rate of growth of food crops exceeded that of the world population, leading to an increase in their per capita availability and to relatively stable prices. Indeed, in 2000 world food prices in real terms were at their lowest for one hundred years (Trostle, 2008).\n\nSince the early 1990s, however, the global rate of growth for grain and oilseed production has declined to 1.3% a year and is projected to decline further to around 1.2% over the next decade. This and a few other adverse factors have contributed to the rapid rise in the world market prices for major food products since early 2006, recording an increase of around 60% in just two years to early 2008. This upward trend has moderated somewhat since then, and the inflation-adjusted food price index stood at 140.3 in June 2013, having risen from its 2002\u20132004 base of 100.0. The most recent figures indicate that the Food and Agriculture Organization (FAO) food price index (FFPI) averaged 210 points in February 2013, remaining virtually stable since November 2012. The index of cereal prices, at 246 points, was 5.4% higher in June 2013 than it was a year earlier. The index of cereal prices was 5 points (2.07%) higher, and the index of dairy prices, after a sharp decline (22%) in the year to June 2012, rose 38% by June 2013 (FAO, 2013a). The general climate of rising retail prices of food items globally has raised the specter of another global food crisis, especially in the poorer countries, where the drive for food security has suffered a major jolt.\n\nSide by side with this worsening situation with respect to the availabilities and prices of the major cereals, there has been another recent development involving food consumption patterns in a number of countries. With growing affluence, tastes change and consumers shift to more varied diets, which usually include larger proportions of noncereal items. Over the last few decades, several poorer countries, among them the two most populous ones, China and India, have experienced rapid growth and transformation in their economies. This economic growth has helped to lift several million people out of extreme poverty and to make many others more affluent, bringing in its trail significant changes in food consumption. One of the more noticeable changes has been a decline in the consumption of cereals and a corresponding increase in the consumption of animal protein. These developments have important policy implications for the global food economy. At the same time, hunger still afflicts a large number of people globally and there is a need for policies to resolve or mitigate this.\n\nIn light of these ongoing changes, this chapter seeks to examine several interrelated issues relating to the evolving world food situation. In particular, it investigates the issue of nutrition and the role proteins and their constituent amino acids play in it. It starts by looking at the issue of hunger, its measurement, global incidence, and mitigation targets. It next discusses the issue of nutrition, its global and regional perspectives, and the role of animal and vegetable proteins. The evolving global demographic trends, with a rapidly increasing elderly population that has special nutritional needs, call for policies to deal with the issue of nutrition for the aged and the role of proteins. Since the production and consumption of protein foods in different countries do not always match, significant international trade in protein products exists. This is briefly discussed to identify the major players in the global protein markets. The chapter concludes with observations on policy implications. In examining the various dimensions of proteins, this chapter focuses on the evolving role of dairy proteins and its implications for future policy.\n\n## Hunger and the need for food\n\nEvery day, millions of people around the globe do not get enough food to eat and remain hungry. Hunger has been referred to as \"the uneasy or painful sensation caused by a lack of food\" and \"the recurrent and involuntary lack of access to food\" (Anderson, 1990). There is no assurance that these hungry people will get the minimum required quantity of food on a daily basis. This unpredictability about where the next meal will come from is called food insecurity. The FAO of the United Nations defines food insecurity as \"a situation that exists when people lack secure access to sufficient amounts of safe and nutritious food for normal growth and development and an active and healthy life\" (FAO, 2000).\n\nAccording to this definition, people are hungry if they do not get enough energy supply from food (fewer than about 1800 kilocalories a day), or if the food they consume is not of sufficiently high quality (i.e., does not contain essential nutrients). Hunger is usually understood to refer to the discomfort associated with lack of food (von Grebmer et al., 2012).\n\n### Hunger-Reduction Targets\n\nHaving thus defined hunger as an operational concept, we need to measure it and to track how it changes over time. This is probably better achieved if a target or targets are set, and the observed incidences of hunger are measured against those targets to ascertain whether the observed trends indicate an improvement or a deterioration.\n\nThe FAO currently monitors two main hunger-reduction targets: the World Food Summit target and Goal 1 of the Millennium Development Goals.\n\n\u2022 During the World Food Summit in Rome (1996), world leaders made the commitment to decrease the number of undernourished people to around 425 million by 2015 (considering 850 million undernourished people as the baseline during the period 1990\u20131992) (FAO, 2011).\n\n\u2022 At the 2000 Millennium Summit in New York, this objective was reiterated when the eight Millennium Development Goals were introduced. The first goal pertains directly to hunger, which is the FAO's fundamental global concern. It aims to reduce the number of people suffering from hunger between 1990 and 2015.\n\nGoal 1 of the Millennium Development Goals calls for a reduction by half of the proportion of people suffering from hunger between 1990 and 2015. Rather than setting a definite number to be reached, this hunger objective therefore depends on the size of the future world population (FAO, 2011; United Nations 2010).\n\n#### World Hunger and Undernutrition Status\n\nAccording to the most recent FAO report, the total number of undernourished people in the world was estimated to be 1023 million in 2009, and it was projected to decrease by 9.6% to 925 million in 2010. The largest numbers of undernourished people live in the developing countries. Two-thirds live in just seven countries (Bangladesh, China, the Democratic Republic of the Congo, Ethiopia, India, Indonesia, and Pakistan), and over 40% live in China and India alone (FAO, WFP, and IFAD, 2012).The territory with the world's largest number of undernourished people continues to be Asia and the Pacific, with an estimated 578 million (Fig. 1.1).\n\nFigure 1.1 Undernourishment in 2012, by region (millions). Data from FAO, WFP, and IFAD (2012).\n\n#### Global Hunger Index\n\nThe International Food Policy Research Institute (IFPRI) has introduced the Global Hunger Index (GHI) tool to measure and track global hunger. The index combines three hunger indicators: (1) the number of undernourished as a proportion of the total population; (2) the proportion of underweight children under the age of 5; and (3) the mortality rate of children under the age of 5. The three indicators are assigned equal weights. On a 100-point scale, the higher the value of the index, the worse is the incidence of hunger, implying that a score of 0 indicates no hunger and a score of 100 indicates the worst possible hunger. Both of these extremes are, obviously, just notional, and are not observed in practice. Different hunger scenarios are defined with the help of the GHI. An index value less than 4.9 indicates \"low hunger\"; values of 5 to 9.9 \"moderate hunger\"; 10 to 19.9 \"serious hunger\"; 20 to 29.9 \"alarming hunger\"; and values in excess of 30 \"extremely alarming hunger.\"\n\nThe 2010 GHI showed some improvement over the 1990 value, falling from 19.8 to 15.1. The index fell by 14% in Sub-Saharan Africa; about 25% in South Asia; 33% in the Near East and North Africa; 40% in Southeast Asia; and 43% in Latin America and the Caribbean compared with the 1990 score (von Grebmer et al., 2012). Figure 1.2 graphs these values for the different regions.\n\nFigure 1.2 Global and regional trends of the Global Hunger Index: Contribution of components in 1990, 1996, 2001, and 2012. Reproduced with permission from the International Food Policy Research Institute www.ifpri.org. The publication from which this figure originates can be found online at <http:\/\/www.ifpri.org\/sites\/default\/files\/publications\/ghi12.pdf>.\n\nTHE IMPORTANCE OF PROTEIN IN WORLD NUTRITION\n\nMost reports on hunger and undernutrition focus primarily on calories, and rightly so. If a person does not get enough calories, his or her well-being will be compromised. However, calories are a necessary, but not sufficient, condition for good nutrition. Many micronutrients, such as vitamins and minerals, are also an important part of the diet, but they are not the focus of this discussion.\n\nA further aspect of undernutrition involves people not getting enough protein, and particularly a sufficient amount of the dietary essential amino acids. This aspect of undernutrition, as well as the role of dairy protein in meeting these needs, is the major concern of the rest of this chapter. Table 1.1 lists the recommended daily intake of protein and of the dietary essential amino acids. Adequate intake of total protein and of all the essential amino acids is essential to maintaining health.\n\nTable 1.1\n\nRecommended Daily Intake for Adults of Protein and Dietary Essential Amino Acids\n\nDietary item | Recommended daily intake (mg\/kg body weight)  \n---|---  \nProtein | 800  \nHistidine | 10  \nIsoleucine | 20  \nLeucine | 39  \nLysine | 30  \nMethionine + cystine | 15  \nPhenylalanine + tyrosine | 25  \nThreonine | 15  \nTryptophan | 4  \nValine | 26\n\nData from WHO (2007).\n\n### Protein and Its Composition and Bioavailability\n\nThe assessment of protein nutrition is more complex than that for calories because proteins vary widely in terms of their composition and bioavailability.\n\n#### Protein Composition\n\nAll proteins are composed of linear chains of amino acids, and each species of protein has its own defined amino acid sequence, which is determined by the genetics of the producing organism. Thus, the amino acid composition, and by implication the amount of each essential amino acid in a given protein, are defined. In practice, most food protein sources contain a complex mixture of proteins. Nevertheless, the overall composition can be determined empirically and is generally quite constant. This has enabled the composition of almost all the major food protein sources in terms of essential amino acids to be determined, and thus intakes of dietary essential amino acids can be estimated from knowledge of the types and amounts of food protein in a diet. In practice, it turns out that intake of most essential amino acids in most diets is adequate, provided total protein intake is adequate. The exception to this rule is the dietary essential amino acid lysine, which is discussed in detail later in this chapter.\n\n#### Bioavailability\n\nAdequate protein supply is one aspect of protein nutrition. A further important aspect is bioavailability: getting the amino acids from the food structures in the gastrointestinal tract to the cells that need them throughout the body. In the adult gastrointestinal tract, proteins must be broken down to very small oligopeptides (at most di- or tripeptides) in order to be taken up, and to single amino acids in order to enter most metabolic pathways. Thus it is necessary for the protein in foods to be both accessible to digestive enzymes and to be broken down by the digestive enzymes in the stomach and small intestine in particular. Furthermore, it is necessary that the broken down protein is able to be taken up into the bloodstream, where it can be redistributed to the tissues that need it. The efficiency of digestion of most of the common food protein sources has been determined, using a range of different methods. Past methodology has largely been based on so-called fecal digestibility. This method is now known to be flawed, particularly with respect to foods with poor digestibility; however it has been widely used and is the only method for which literature values are available for most common foods. For a full discussion of protein quality and nutritional requirements, the reader is referred to a recent FAO report, Protein Quality Evaluation in Human Nutrition (FAO, 2013b).\n\nBy using digestibility values, dietary intake can be converted to an estimated uptake into the body for these proteins, noting that these values are derived from a flawed methodology and so, particularly in the case of plant-derived proteins, represent an upper limit of their true bioavailability. The digestibility of a range of dietary proteins is given in Figure 1.3.\n\nFigure 1.3 True protein digestibility of common food proteins. Data from FAO (1970) and WHO\/FAO (1991).\n\nAnimal-derived proteins generally have good bioavailability and content of dietary essential amino acids, but many plant proteins are deficient in one or more dietary essential amino acids, and many are not efficiently digested and therefore the constituent amino acids are not highly bioavailable. Most Western countries are characterized by a high protein diet with a strong emphasis on animal-derived protein, so protein nutrition is not generally a problem (although there may be some issues with protein nutrition among the elderly). Most developing countries are very dependent on plant protein as the main dietary source, and that protein may be inadequate, due to poor digestibility and poor amino acid balance, particularly in the case of lysine.\n\n### World Protein Supply and Its Regional Distribution\n\nInformation on the amount of protein available per capita and by protein source in each country is available from the FAOSTAT database (<http:\/\/faostat.fao.org\/>). In Figure 1.4, we present the average protein availability for individual countries, along with their total population. These statistics are presented as the total number of people who reside in countries with corresponding average protein availability for each incremental 5 g band. The figure is striking in having two clear peaks, one in the range 55\u201360 g protein per capita per day, and the second in the band 85\u201390 g protein per capita per day, with a considerable tail to the right. The first of these peaks is of some concern: It represents more than a quarter of the world's population, and it is simple to calculate that with a standard body weight of 70 kg; based on the dietary recommendations for protein requirements, a person will need 56 g of protein a day to stay healthy. The band at 55\u201360 includes India, Indonesia, Bangladesh, and the Democratic People's Republic of Korea (North Korea). This band is of concern because, although the average availability figure is just above the minimum requirement, disparities of income and situation in these countries will mean that a large proportion of these people are not getting enough protein. Furthermore, these figures are just for total protein supply, with no correction for bioavailability. When the dietary pattern is corrected for digestibility of the main protein components of the diet (from FAOSTAT), the situation is more serious, with almost 1.8 billion people getting, on average, less than 55 g of protein per day based on FAOSTAT figures for 2009.\n\nFigure 1.4 Protein intake by population on a country average basis. Each bar represents the total population of countries with average per capita daily protein intake in bands of 5 g\/capita\/day. Protein figures for 2007 from FAOSTAT September 2012; population data from United Nations (2009).\n\n### Vegetable and Animal Protein Sources with a Focus on Dairy Foods\n\nThere is considerable debate over the merits of vegetarianism and over eating only vegetable-origin foods in consideration of global sustainability. It is often estimated that production of 1 kg of animal-origin food requires 10 kg of plant-origin food, leading to the simplistic assumption that 10 times as many people could be fed off the same resources if everyone was vegetarian. For a full discussion of the subject, the reader is referred to Fairlie (2010). In the case of dairy products and eggs, the situation is somewhat better than it is for meat, because the animal can continue producing throughout its adult lifetime. This leads to conversion ratios of about 4:1. In fact, the argument is much more complex, partly because of the role of animals in subsistence agriculture, largely eating food waste or processing residue, or grazing and browsing plant species that are not suitable for human consumption, and partly because of the niche many animals occupy in developed agricultural economies, either grazing pastures intensively, or being farmed on land not suitable for arable cultivation (e.g., see Elferink et al., 2008). In this context, we have calculated that in the Canterbury Plains in New Zealand, the main wheat-growing area, the yield per hectare per annum of protein from the wheat crop, processed to the form of white flour and its consequent baked products, is somewhat less than the yield of protein from milk that is produced over the same period in the same area. Thus, the efficiencies of production need to be considered in the context of what is the target of that production (there is no doubt that wheat produces the greater number of calories). Nevertheless, it must be recognized that the changing protein consumption patterns, involving more animal-based products, have significant implications for global land-use patterns, agriculture, agri-food industries, cereal prices, and the environment.\n\n### Growing Global Demand for Animal Proteins and Implications\n\nThe demand for animal protein foods is expected to increase to about double the present consumption by 2050, driven by population growth and by the emerging middle classes in developing countries (FAO, 2006). As people get more money, one of the first priorities is better food, and this usually means animal protein foods. This phenomenon was first described by Bennett (1941), who related comparative studies of the consumption of staple foods leading to what has come to be known as Bennett's Law: the empirical generalization that there is an inverse relationship between the percentage of total calories derived from cereals and other staple foods and per capita income. This principle has since become generalized to mean a move away from carbohydrate-based foods to protein-based foods.\n\nA simple extrapolation from past increases in animal production indicates that we should be able to meet this demand if past rates of increase can be sustained (Boland et al., 2013). However, past increases have been based on bringing in new land for farm production, increases in efficiency through breeding gains, better livestock management and nutrition, and other factors revolving around the Green Revolution. Most of these options are either reaching their limits or entering a phase of diminishing returns. The carbon footprint of livestock production is a further constraint, although the good news is that as animal production has intensified, the carbon footprint has massively decreased. For example, Capper et al. (2009) have calculated that the carbon footprint for milk in the United States in 2007 was just 37% of that for the same milk in 1944. Nonetheless, past increases will not continue ad infinitum, and new ways of sustainably meeting the increasing demand are needed.\n\n## The dietary essential amino acids in proteins\n\nAlthough there are nine dietary essential amino acids, it is rare for a diet with adequate overall protein intake to be deficient in most of them. The exception is lysine.\n\nLysine may be an issue for two reasons: The first is that many staple protein sources, particularly the cereals, are deficient in lysine. The second is that lysine is chemically unstable under heating and undergoes a range of reactions when food is heated. The most important of these reactions is the Maillard reaction, in which the side chains of the lysine residues in the protein cross-react with sugar molecules to produce glycosyl lysine side chains that are indigestible and thus no longer bioavailable. This reaction can occur under mild heating conditions, and under more extreme conditions it is responsible for much of the browning of food that occurs during cooking. Another reaction of importance for dairy products is the reaction with phosphoserine, leading to the formation of lysinoalanine, which is not bioavailable. This problem is specific to casein-containing products (mainly milk powders and caseinate), because of its high phosphoserine content (see Chapter 11 for a detailed discussion of this reaction).\n\n## Identifying the countries deficient in dietary essential amino acids\n\nIn an attempt to obtain an understanding of the dietary availability of the essential amino acids, countries with low intakes of protein were analyzed to determine the dietary essential amino acid content of the mix of protein sources for that country (from FAOSTAT), corrected for digestibility for each protein source. Because literature values were unavailable for some minor protein sources, a sensitivity test was performed, changing the digestibility figure from 1.0 to 0.8 for plant proteins and 1.0 to 0.9 for animal proteins in these cases. Because this change did not make a noticeable difference to overall lysine bioavailability for the countries in question, the method was considered to be robust. The countries found to be lysine deficient are given in Table 1.2, together with information about the main dietary protein sources. Of these countries, only Liberia was found to be deficient for any of the other essential amino acids (leucine and isoleucine in this case).\n\nTable 1.2\n\nLysine-Deficient Countries\n\nCountry | Lysine (% RDI) | Animal protein in diet (%) | Main protein source(s) | Main protein source (%)  \n---|---|---|---|---  \nEritrea | 63.1 | 11 | Other cereals* | 48  \nLiberia | 71.6 | 14 | Rice | 43  \nMozambique | 75.1 | 11 | Maize  \nWheat | 28  \n15  \nGuinea-Bissau | 79.4 | 18 | Rice | 39  \nZambia | 79.9 | 16 | Maize | 57  \nHaiti | 87.5 | 19 | Rice  \nWheat | 19  \n16  \nTogo | 89.0 | 12 | Maize | 30  \nEthiopia | 92.7 | 10 | Other cereals* | 34  \nTajikistan | 98.0 | 21 | Wheat | 61  \nBangladesh | 98.7 | 16 | Rice | 61  \nZimbabwe | 98.8 | 19 | Maize | 46  \nYemen | 99.1 | 22 | wheat | 45\n\n* Other cereals include sorghum, barley, oats, and rye.\n\nThe countries that are lysine deficient show a clear pattern of low levels of consumption of animal protein and strong dependence on cereals for their protein.\n\n### Protein and Dietary Essential Amino Acid Contents of Food Items\n\nThe amino acid composition, particularly the lysine content, of proteins is of particular concern for countries that tend to be protein deficient. Thus maintenance of an adequate intake of lysine, especially in populations with a high dependence on cereals, requires attention. Figure 1.5 indicates the levels of bioavailable lysine in a range of common dietary protein sources. Meat is clearly the best source of lysine but may not be a suitable dietary component for many because of cost and cultural restrictions. Dairy protein is also an excellent source of lysine. Inclusion of supplementary dairy protein in the diet may offer an effective solution that is acceptable to vegetarians, price notwithstanding.\n\nFigure 1.5 Bioavailable lysine content in a range of food protein in g\/100 g protein, corrected for digestibility.\n\n## Demographic changes, aging populations, and the need for quality protein and essential amino acids\n\nRecent global demographic trends indicate a steady increase in the number of people aged 60 years and over. The projection is for this population to more than triple from 600 million in 2000 to over 2 billion in 2050 (United Nations, 2009). As a consequence, in the more developed world, the fastest growing section of the population is that of adults aged 80 years or over. This clearly presents unique challenges for health care, diets, and nutrition, as well as for certain age-specific clinical conditions.\n\nBoth the number and the proportion of older persons are growing in virtually all countries, and these trends are likely to continue worldwide. For example, in 2009, Japan had the highest percentage of the population aged 60 or over, at 29.7%, followed by Italy at 26.4%, whereas in Qatar it was only 1.9%. It is expected that the proportion of the population aged 60 or over will be 22% in 2050 compared to 11% in 2009. Figure 1.6 shows the predicted percentage of the elderly population in 2050 in major continents. By 2050, it is projected that there will be more than 1.4 billion elderly people in Asia alone.\n\nFigure 1.6 Percentage of elderly population in 2050. Sarkar (2012).\n\nAging is a continuous, ongoing, and progressive process of damage accumulation. It is associated with reduction in muscle mass and function, and reduced physical activity. The loss of muscle mass with aging is known as sarcopenia. With the aging of the population globally, the prevalence of sarcopenia is likely to increase. Sarcopenia is accelerated by inadequate diet, mainly due to lack of quality protein in optimal quantity and lack of essential amino acids. The issue of the nutritional needs of the growing aging population in terms of the role of dietary protein and essential amino acids with particular reference to sarcopenia is described in more detail in the following sections.\n\nOverall, a strong case can be made that an aging population will require a substantially increased intake of protein and of essential amino acids (particularly leucine), a demand that milk proteins are particularly well suited to meet.\n\n### Protein Nutritional Needs of the Elderly\n\nThe aging process is characterized by changes in body composition, with a progressive loss of muscle and bone mass, strength, and metabolic function. The loss of muscle with aging is the result of a chronic imbalance between muscle protein synthesis and breakdown. There are many causes of sarcopenia, and an understanding of the complex mechanism is evolving. This degenerative loss of skeletal muscle occurs at a rate of 3 to 8% per decade after the age of 30 and accelerates with advancing age; chronic muscle loss is estimated to affect 30% of people older than 60 years and 50% of those older than 80 years (Katsanos et al., 2006; Paddon-Jones et al. 2008). With the aging of the population, the prevalence of sarcopenia and the resulting burden of disability are likely to increase. Strategies to prevent sarcopenia are, therefore, of considerable importance, and there is a need for public awareness, as simple health strategies can be effective.\n\nResearchers have identified two measures that can play a role in fighting against sarcopenia: diet and exercise. However, in the case of many elderly individuals, the ability to perform exercise is compromised due to disease and disability. In this case, daily high-quality protein intake can be helpful to slow down or prevent muscle protein loss. Different protein sources have been found to stimulate muscle protein synthesis in varying degrees. The most important factor is the amount of essential amino acids in the protein, in particular, leucine. Differences in digestibility and bioavailability of certain protein-rich foods may also influence muscle protein synthesis (Paddon-Jones et al., 2008).\n\nCurrently, there is no agreement on whether dietary protein needs change with advancing age. For adults the recommended dietary allowance for protein is 0.8 g protein per kg body weight per day (WHO, 2007). The report of the FAO\/WHO\/UNU expert consultation, published in 2007, recommends that the essential amino acid requirement for elderly people should be the same as for adults, as the current acceptable methodologies are not appropriate to make a separate set of essential amino acid values for elderly people (WHO, 2007). A more recent FAO-sponsored expert consultation has failed to resolve this issue, with one group maintaining that \"the data based on the currently acceptable methodologies... are inadequate to make a separate recommendation for dietary IAA requirements in elderly people\" (Pillai & Kurpad, 2012), while another group has advised that:\n\ndietary protein intake, and the resulting increased availability of plasma amino acids, stimulates muscle protein synthesis. If all other variables are controlled, increased muscle protein synthesis leads to improved muscle mass, strength, and function over time. Increased muscle mass, strength, and function are related to improved health outcomes in older individuals. Since adverse effects of reasonable increases in protein intake above the recommended dietary allowance (RDA) of 0.8 g protein\/kg\/day have not been reported, it is reasonable to conclude that the optimal protein intake for an older individual is greater than the RDA (Wolfe, 2012).\n\nSome studies suggest that an intake of 1.0\u20131.5 g protein per kg body weight per day or about 15\u201320% of total caloric intake is essential to preserve proper nitrogen balance in the healthy elderly instead of the recommended RDA value (Morais et al., 2006; Wolfe et al., 2008).\n\n### Role of Essential Amino Acids in Nutrition of the Elderly\n\nEssential amino acids are mainly responsible for the stimulation of muscle protein anabolism in the aged (Volpi et al., 2003). It is considered that 15 g of essential amino acids taken as bolus is required for maximum stimulation of muscle protein synthesis (Wolfe, 2002). This indicates that quality of protein is very important in the diet of the elderly.\n\nPreliminary data from a recent randomized controlled trial indicate that it is more important to ingest a sufficient amount of high-quality protein (25\u201330 g) with each meal rather than one large bolus, because more than 30 g in a single meal may not further stimulate muscle protein synthesis (Symons et al., 2009). It is also recognized in recent studies that intake of whey protein brings beneficial effects to muscle protein anabolism in the elderly. Furthermore, ingestion of intact whey protein has been found to provide a greater anabolic benefit than ingestion of the equivalent essential amino acids alone. Thus, whey protein may be more than just a simple source of essential amino acids with respect to providing a stimulus for enhancing muscle protein anabolism in the elderly (Katsanos et al., 2008). For a fuller discussion of the function of whey proteins and other milk proteins in human health, the reader is referred to Chapter 19 of this volume.\n\nThere is a general agreement that the essential amino acid leucine increases protein anabolism and decreases protein breakdown (Paddon-Jones and Rasmussen, 2009). Leucine-rich food sources include legumes such as soybeans and cowpea, and animal products such as beef, fish, and particularly dairy proteins (whey protein). It is reported that amino acid supplements without adequate leucine do not stimulate protein synthesis (Rieu et al., 2007; Hayes and Cribb 2008). Leucine has recently been acknowledged to be especially important as a signaling molecule and a building block for muscle. Rat studies show that leucine can directly stimulate muscle protein synthesis through increasing mRNA translation (Anthony et al., 2000). Insulin and leucine are anabolic stimuli for muscle, and both share a common pathway of action via activation of a kinase known as mTOR. mTOR is the main regulator of cell growth and acts by phosphorylating target proteins involved in mRNA translation. Because insulin sensitivity decreases with age, one possible mechanism by which amino acids (mainly leucine) might improve muscle mass is by providing another anabolic stimulus to activate the mTOR-controlled pathway (Gaffney-Stomberg et al., 2009; Casperson et al. 2012).\n\nNo differences exist in protein balance in the elderly relative to the young following administration of either 30 g of beef protein or 15 g of essential amino acids as a bolus (Paddon-Jones et al., 2004). However, when 6.7 g of a mix of the dietary essential amino acids is given, the overall protein synthetic response is reduced in the elderly relative to the young (Katsanos et al., 2005). This anabolic resistance has been attributed to a decrease in leucine sensitivity and may be overcome by increasing the proportion of this amino acid in the diet. For example, when a 6.7 g bolus of dietary essential amino acids enriched with leucine (46% leucine compared to the 26% normally found in whey protein) was given to the elderly individual, protein synthesis was fully restored (Katsanos et al., 2006).\n\n## Global trade in proteins, the long-term prospects, with a focus on dairy foods\n\nGlobal food consumption patterns have been changing in recent decades in several significant ways. Among them is the noticeable and continuing shift in favor of proteins, especially animal proteins. Global consumption of protein is forecast to grow by 96% over the three decades from 1990 to 2020 (von der Heyde, 2012). The growth is largely due to the rising incomes in the developing world, particularly in some of the more populous countries such as China, Brazil, and, to a lesser extent, India. Over the decade since 2000, however, global protein demand has been driven by increased consumption in other countries and geographic areas too. For example, demand in the African continent has increased by around 70%, in Southeast Asia by 49% and in Central America by 29% (von der Heyde, 2012). Between 1999 and 2011, world protein trade grew by 74% (von der Heyde, 2012). Since only a few countries currently have surplus protein to export, the projected increase in its demand is likely to pose serious challenges to these countries and to the world in general.\n\nIn overall world protein nutrition, milk products, representing about 10% of all protein consumption, are the third most important source of protein after cereals (40%) and meat (18%) (data for 2009 from FAOSTAT). When the low levels of lysine in cereals are taken into account (about one-third of that in dairy products), it is clear that milk protein plays a very important nutritional role in the world today.\n\n### The Global Dairy Food Scene: An Overview\n\nMilk and other dairy products have always been among the major everyday food items in human consumption in many cultures. It is a particularly useful food for the large, and possibly growing, number of vegetarians around the world. Its value for both infants and the elderly is easily recognized. Apart from its consumption in liquid form, there are many other ways in which milk is transformed and consumed. Innovations keep occurring to make new milk-based products available in the market.\n\nWith growing world population and changing food habits, the production and consumption of milk and other milk-based products have also been rising over time. Over the five decades since 1961, world milk production more than doubled from 344 million tons to 703 million tons in 2009 (FAOSTAT) and 749 million tons in 2011 (IDF, 2012).\n\nThe pattern of regional distribution of the production and consumption of milk reveals that, as of 2009, Asia's share is the highest\u2014with 36% of global production and 38% of consumption; Europe comes next, with 31% and 30%, respectively, followed by North America with a balanced 13% of both production and consumption, and South America, again with a balanced 8% of production and consumption. The only region with a significant exportable surplus is Oceania, which produces 3.7% and consumes 1.5% of the global totals.\n\nIt is important to note that much of the milk produced is consumed in the country (or economic bloc, in the case of the EU) where it is produced, and that just over 8% of dairy production is involved in international trade (2011 figures; IDF, 2012). In this context, it is noted that about 80% of whole milk powder, 55% of skim milk powder, and only about 10% of cheese is traded internationally (IDF, 2012). It is of interest to note, too, that trade in cheese within the EU-27, for example, was five times the volume exported, which in turn was only 8% of production. In 2012, the major exporters of cheese were the EU-27, the United States, New Zealand, and Australia, and the main importers were Russia, Japan, the EU-27, Mexico, and Korea (USDA, 2012). There is, evidently, some intraindustry trade in cheese. Given its variety and established regional specialties, this is not difficult to understand.\n\nThe six major exporters account for 80% of world dairy trade in cow's milk (IDF, 2012). They were, in 2011, New Zealand (26%), the EU-27 (26%), the United States (12%), Australia (8%), Argentina, and Belarus (4% each). Of course, all of the exported 'milk' consists of processed products, of which the main ones that contain protein (in order of importance) are whole milk powder, skim milk powder and cheese. Different countries dominate the export markets of the different dairy products (Table 1.3).\n\nTable 1.3\n\nVolume of Major Dairy Exports for the Six Main Exporters, 2011\n\n| Whole milk powder  \n(000 tons) | Skim milk powder  \n(000 tons) | Cheese  \n(000 tons) | Total protein  \n(000 tons)a  \n---|---|---|---|---  \nNew Zealand | 1080.8 | 349.8 | 245.9 | 454.1  \nEU-27 | 390.0 | 517.6 | 680.2 | 448.7  \nUSA | 21.6 | 435.7 | 224.3 | 214.0  \nAustralia | 143.2 | 165.9 | 207.0 | 145.6  \nArgentina | 199.1 | 18.4 | 58.8 | 70.9  \nBelarus | 26.7 | 55.2 | 122.1 | 56.5\n\na Excludes other products such as casein, whey products, and liquid and condensed milks. Values are based on protein of: WMP 25%; SMP 35%; and cheese 25%. The cheese value is based on the bulk of traded cheese being cheddar. Values were obtained from the Canadian Dairy Commission dairy ingredient profiles on www.milkingredients.ca and values near the lower end of each range used.\n\nSource: IDF, 2012.\n\nOne notable feature of our discussion is perhaps the absence of the poorer developing countries among the major exporters and importers of such processed high-value milk products as butter and cheese. The possible explanation for this may be that the consumption of these products is income-sensitive; they are consumed in noticeable quantities only when income has reached a certain level.\n\nThis presumption is further confirmed when one examines the trade patterns in respect of milk powders, which are usually reconstituted for consumption as liquid milk\u2014the demand for which is likely to be less income-sensitive. Among the major importers of whole milk powder are China, Algeria, Brazil, Indonesia, and the Philippines, while the major exporters are New Zealand, Australia, and the EU-27. The situation is very similar in the market for skim milk powder. The major importing countries are Mexico, China, Indonesia, and the Philippines, and the major exporters are New Zealand, Australia, the United States, and the EU-27.\n\nWhile China features as a milk importer, India, perhaps surprisingly, is neither a major importer nor a major exporter of dairy products, although it has the largest bovine herd in the world. With the introduction around the mid-1960s of a system of dairy cooperatives under the umbrella of the National Dairy Development Board (NDDB), India's dairy industry has achieved a remarkable transition. Set up in 1965, the NDDB oversaw the dairy cooperatives collecting the often-small marketable surplus milk from the small herds scattered around the villages and supplying the growing market for milk in the urban areas. This linking of the milk producers with the markets\u2014both of which are scattered in locations and are large in numbers\u2014generated a five-fold growth in India's milk production in three decades from the late 1960s, as domestic consumption of milk also rose steadily (Chatterjee, 1990; Brown 2009). This transformation is all the more remarkable in that India's dairy industry is built almost entirely on crop residues\u2014wheat or rice straw, corn stalks, vegetable residues, and grass gathered from roadside\u2014a rather different protein production model.\n\nAlthough the consumption of dairy products is projected to grow as the standards of living improve in the developing world, some new developments, so far mainly in the more affluent countries, have also been creating additional demand for certain types of foods referred to as 'specialty foods.' These include functional foods, defined as 'food and drink products making a specific health claim,' organic foods, and genetically modified (GM) foods. While international trade in specialty foods is still relatively small and confined to a few countries, evidence suggests that it is growing rapidly (Chatterjee, 2012). Dairy products feature prominently among both functional foods and organic foods currently traded internationally; other animal protein products less so. The market for these products is likely to grow over time as rising affluence spreads globally. Resources including land devoted to mostly export- oriented organic farming, for example, have also been growing, particularly in the developing countries of Asia, Africa, and Latin America.\n\n## Conclusions\n\nWorld hunger continues to be a major problem. Hunger has several dimensions, notably the need first and foremost for adequate intake of calories. A close second is the need for adequate intake of protein and of dietary essential amino acids. Protein nutrition is more complex than calories because all proteins are not equal: Nutritional value depends on the type of protein and on how it has been treated prior to consumption, as much as the amount of protein itself. Milk protein is a very high quality protein, with a good supply of the dietary essential amino acids and high bioavailability. It can therefore be used to supplement poorer plant-derived proteins, such as cereal protein, to greatly improve the nutritional value of the combination. Milk production is growing globally, and the amounts of dairy products (and implicitly milk protein) traded internationally are also growing. Milk protein already accounts for 10% of the global food protein supply and makes a disproportionate contribution to global protein nutrition, based on its bioavailability and desirable composition. 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Amino acid requirements in children and the elderly population .  _British Journal of Nutrition_.  2012 ;108 : S44 \u2013 S49 .\n\nRieu I ,  Balage M ,  Sornet C ,  Debras E ,  Ripes S ,  Rochon-Bonhomme C ,  Pouyet C ,  Grizard J ,  Dardevet D .  Increased availability of leucine with leucine-rich whey proteins improves postprandial muscle protein synthesis in aging rats .  _Nutrition_.  2007 ;23 : 323 \u2013 331 .\n\nSarkar A .  _Global protein nutrition: essential amino acids availability_ .  New Zealand :  Thesis, Massey University ;  2012 .\n\nSymons TB ,  Sheffield-Moore M ,  Wolfe RR ,  Paddon-Jones D .  A moderate serving of high-quality protein maximally stimulates skeletal muscle protein synthesis in young and elderly subjects .  _Journal of the American Dietetic Association_.  2009 ;109 : 1582 \u2013 1586 .\n\nTrostle R .  _Global agricultural supply and demand: Factors contributing to the recent increase in food commodity prices. A Report from the Economic Research Service_ .  United States Department of Agriculture ;  2008 .\n\nUnited Nations, 2009. World population prospects: The 2008 revision. Population Division of the Department of Economic and Social Affairs of the United Nations Secretariat. <http:\/\/esa.un.org\/unpp>, downloaded January 30, 2011.\n\nUnited Nations, 2010. Millennium Development Goals. <http:\/\/www.un.org\/millenniumgoals\/poverty.shtml> Retrieved September 6, 2011.\n\nUSDA .  _Dairy: World markets and trade_ .  United States Department of Agriculture, Foreign Agricultural Service ;  2012, December .\n\nVolpi E ,  Kobayashi H ,  Sheffield-Moore M ,  Mittendorfer B ,  Wolfe RR .  Essential amino acids are primarily responsible for the amino acid stimulation of muscle protein anabolism in healthy elderly adults .  _American Journal of Clinical Nutrition_.  2003 ;78 : 250 \u2013 258 .\n\nvon der Heyde, C., 2012. JBS Pilgrims Presentation, 2012 NOPA Industry Forum. <http:\/\/www.nopa.org\/content\/newsroom\/2012\/Charles%20von%20der%20Heyde_JBS%20Pilgrims%20Presentation_2012%20NOPA%20Industry%20Forum.pdf>, retrieved 6 January 2013.\n\nvon Grebmer, K., Ringler, C., Rosegrant, M.W., Olofinbiyi, T., Wiesmann, D., Fritschel, H., Badiane, O., et al., 2012. The challenge of hunger: Ensuring sustainable food security under land, water, and energy stresses. Deutsche Welthungerhilfe; International Food Policy Research Institute; Concern Worldwide. Bonn, Germany; Washington, DC; Dublin, Ireland.\n\nWHO\/FAO .  _Protein quality evaluation: Report of Joint FAO\/WHO Expert Consultation_ .  Rome :  FAO ;  1991 .\n\nWHO .  _Protein and amino acid requirements in human nutrition: Report of a Joint FAO\/WHO\/UNU Expert Consultation_ .  Geneva, Switzerland :  World Health Organization ;  2007 .\n\nWolfe RR .  Regulation of muscle protein by amino acids .  _Journal of Nutrition_.  2002 ;132 : 3219S \u2013 3224S .\n\nWolfe RR .  The role of dietary protein in optimizing muscle mass, function and health outcomes in older individuals .  _British Journal of Nutrition_.  2012 ;108 : S88 \u2013 S93 .\n\nWolfe RR ,  Miller SL ,  Miller KB .  Optimal protein intake in the elderly .  _Clinical Nutrition_.  2008 ;27 : 675 \u2013 684 . \nChapter 2\n\n# Milk: An Overview\n\nJ.A. O'Mahony\n\nP.F. Fox School of Food and Nutritional Sciences, University College, Cork, Ireland\n\n## Abstract\n\nMilk is the characterizing excretion of mammals, of which there are about 4500 species, produced to meet the complete nutritional, and some defensive and other physiological, requirements of the neonate of the species. The milk of all species is basically similar, but there are significant species-specific differences. In addition to supplying all the nutritional requirements of the neonate, many of the minor constituents of milk serve protective roles (e.g., oligosaccharides, immunoglobulins, metal-binding proteins, and enzymes). Milk is an aqueous solution (milk serum) of lactose, inorganic and organic salts, and numerous compounds at trace levels, in which are dispersed colloidal particles of three size ranges: whey proteins dissolved at the molecular level, caseins dispersed as large (50\u2013500 nm) colloidal aggregates (micelles), and lipids emulsified as large (1\u201320 \u03bcm) globules.\n\nThe colloidal stability of milk, especially of the casein micelles, is very important from nutritional and technological viewpoints. The micelles are destabilized and aggregate or gel following limited proteolysis or acidification to \u223cpH 4.6. In vivo, aggregation occurs in the stomach of the neonate, thereby slowing transit and improving digestibility. Technologically, destabilization of the micelles can be undesirable or can be exploited in the production of cheese and fermented milk products.\n\nHumans have used the milk of other species in their diets for about 8000 years, and a major industry has developed around the processing of milk of a few species, especially cattle, buffalo, sheep, and goats, for human foods. Milk processing, which exploits certain physicochemical properties of milk, is practiced worldwide, especially in Europe and North America. Milk is a very flexible raw material, from which a wide range of different products, including about 1000 varieties of cheese, are produced.\n\n## Keywords\n\nMammals, milk, evolution, proteins, lipids, lactose, casein micelle, whey proteins, biological functions, physicochemical properties\n\nOutline\n\nIntroduction 20\n\nEvolution of mammals and lactation 21\n\nClassification of Mammals 22\n\nClassification and Phylogenetic Relationships of the Principal Dairying Species 23\n\nEvolution of the Mammary Gland 24\n\nStructure of the Mammary Gland 24\n\nUtilization of milk 25\n\nComposition of milk 25\n\nMilk constituents 26\n\nCarbohydrates 26\n\nLactose 26\n\nModification of the Concentration of Lactose in Milk through Genetic Engineering 28\n\nNutritional Problems Associated with Lactose 29\n\nProduction and Utilization of Lactose 29\n\nOligosaccharides 30\n\nLipids 32\n\nFatty Acids 32\n\nDistribution of FAs in Triglycerides 34\n\nDegradation of Lipids 34\n\nMilk Lipids as an Emulsion 35\n\nMilk Proteins 37\n\nPreparation of Casein and Whey Proteins 39\n\nComparison of Key Properties of Casein and Whey Proteins 40\n\nHeterogeneity and Fractionation of Casein 41\n\nApplication of Gel Electrophoresis to the Study of Milk Proteins 41\n\nMicroheterogeneity of the Caseins 42\n\nVariability in the Degree of Phosphorylation 42\n\nGenetic Polymorphism 42\n\nDisulfide Bonding 43\n\nVariations in the Degree of glycosylation 43\n\nHydrolysis of the Caseins by Plasmin 43\n\nMolecular Properties of the Milk Proteins 43\n\nNomenclature of Milk Proteins 45\n\nWhey Proteins 45\n\nFractionation of Whey Proteins 46\n\nMajor Characteristics of Whey Proteins 46\n\nBlood Serum Albumin 48\n\nImmunoglobulins 48\n\nWhey Acidic Protein 49\n\nProteose Peptone 3 49\n\nMinor Proteins 50\n\nMetal-binding Proteins 50\n\n\u03b22-Microglobulin 50\n\nOsteopontin 50\n\nVitamin-binding Proteins 51\n\nAngiogenins 51\n\nKininogen 51\n\nGlycoproteins 51\n\nProteins in the Milk Fat Globule Membrane 52\n\nGrowth Factors 52\n\nIndigenous Milk Enzymes 52\n\nBiologically Active Cryptic Peptides 53\n\nNonprotein Nitrogen 53\n\nCasein Micelles 53\n\nStability of Casein Micelles 53\n\nMicelle Structure 54\n\nInterspecies Comparison of Milk Proteins 56\n\nMilk Salts 58\n\nVitamins 60\n\nWater 60\n\nPhysical Properties of Milk 61\n\nMicrobiology of Milk 61\n\nSummary 61\n\n## Introduction\n\nMilk is a fluid secreted by the female of all mammalian species, of which there are about 4500 extant species (about 80% of mammalian species are extinct), and it meets the complete nutritional requirements of the neonate. The principal requirements are for energy (supplied by lipids and lactose and, when in excess, by proteins), essential amino acids and amino groups for the biosynthesis of nonessential amino acids (supplied by proteins), essential fatty acids, vitamins, inorganic elements, and water. The nutritional requirements of the neonate depend on its maturity at birth, its growth rate, and its energy requirements, which in turn depend mainly on environmental temperature. Therefore, the gross composition of milk shows large interspecies differences, which reflect these requirements (see Fox and McSweeney, 1998; Fuquay et al., 2011; McSweeney and Fox 2013). Milk also serves a number of physiological functions, which are performed mainly by proteins and peptides, including immunoglobulins, enzymes, enzyme inhibitors, growth factors, hormones, and antibacterial agents.\n\nOf the 4500 species of mammal, the milk of only about 180 species has been analyzed, and of these, the data for only about 50 species are considered reliable (a sufficient number of properly taken samples, representative sampling, adequate coverage of the lactation period). Milk from the commercially important species\u2014cattle, goat, sheep, buffalo, yak, horse, camel, and pig\u2014are quite well characterized; human milk is also well characterized, as is that of experimental laboratory animals, especially rats and mice. Reviews on nonbovine milks include: general (Evans, 1959; Jenness and Sloan, 1970; Fuquay et al. 2011; and Medhammar et al., 2011), buffalo (Laxminarayana and Dastur, 1968), goat (Parkash and Jenness, 1968; Jenness 1980; Haemlein, 1980), sheep (Bencini and Pulina, 1997), sheep and goats (Jandal, 1996; IDF 1996, Park et al., 2007); camel (Rao et al., 1970; Farah 1993), horse (Doreau and Boulot, 1989; Solaroli et al., 1993; Doreau, 1994; Park et al. 2006), human (Atkinson and Lonnerdal, 1989; Jensen, 1989, ) and sow (Verstegen et al., 1998). The Handbooks edited by Park and Haemlein (2006, 2013) are a particularly useful source of information on the milk of nonbovine mammals; they include chapters on goat, sheep, buffalo, mare, camel, yak, reindeer, sow, llama, minor species (moose, musk ox, caribou, alpaca, ass, elk, seals, sea lion, and polar bear), and human.\n\nThe milk of certain domesticated animals and dairy products derived from them are major components of the human diet in many parts of the world. Domesticated goats, sheep, and cattle have been used for milk production since about 8000 BC (Kindstedt, 2012). Recorded milk production today is greater than 700 \u00d7 106 tonnes per annum, about 84% of which is bovine, 13% is from buffalo, and about 2% each is ovine and caprine, with small amounts produced from horses, donkeys, camels, yaks, and reindeer (IDF, 2010). Milk and dairy products are consumed throughout the world but are particularly important in Europe, the United States, Canada, Argentina, India, Australia, and New Zealand. The contribution of milk and dairy products to dietary intake varies widely for different regions of the world; for example, the kilocalories per day supplied by milk range from 12 in China to 436 in Ireland; and in the UK, milk and dairy products supply \u223c30% of dietary protein consumed by young children, \u223c27% of dietary lipids, and \u223c65% of calcium (Barker, 2003; Patton 2011). Overall, the contribution of milk and dairy products to macro- and micro-nutrient intakes in the human diet is substantial, as reflected by the 2005 Dietary Guidelines for Americans recommendations to consume three servings of low-fat or fat-free milk or equivalent milk products daily as part of a healthy diet and lifestyle (see Cifelli et al., 2011).\n\nThe chemistry and physicochemical properties of milk have been studied for about 200 years and are now understood in considerable detail as well as described in a voluminous literature. The objectives of this chapter are to provide a summary and overview of the evolution of mammals and lactation and of the principal constituents of milk, especially proteins, which are the subject of this book. Where possible, an interspecies comparison of milk and its constituents is made. Numerous textbooks and review articles are cited.\n\n## Evolution of mammals and lactation\n\nThe secretion of milk is one of the characteristic features of mammals that evolved from egg-laying, premammalian reptiles, Synapsids and Cynodonts. Cynodonts are believed to be the ancestors of all mammals, which evolved \u223c200 M years ago (at the end of the Triassic Period). The word 'mammal' is derived from mamma, Latin for breast. Initially, mammals were small shrew-like creatures, but they have evolved and diversified to occupy all niches on land, sea, and air. They range in size from a few grams (pigmy shrew) to 200 tonnes (blue whale). Their dominance occurred especially after the extinction of the dinosaurs, 60\u201370 M years ago, at the interface between the Cretaceous and Tertiary Periods (C\/T interface). Mammals have been successful because the young of most species are born alive (viviparous) and all are supplied with a specially designed food, milk, for the critical period after birth. No other class of animal is so pampered (for an interesting discussion on this point, see Peaker, 2002). Not surprisingly, the evolution of mammals is a popular subject; reviews include Crampton and Jenkins (1973), Kemp (1982), Lillegraven et al. (1987), Lillegraven (2004), Forsyth (2003), Benton (1999), Easteal (1999), Springer et al. (2004), and Oftedal (2013).\n\nMammals are distinguished from other classes of animals by four criteria:\n\n\u2022 They secrete milk to nourish their young.\n\n\u2022 They are endothermic; that is, they can control their body temperature.\n\n\u2022 They grow body hair or wool for insulation; even aquatic mammals have some hair.\n\n\u2022 They have different types of teeth (flat incisors, conical canines, and multicusped molars) which allow them to masticate different types of food.\n\nThe class Mammalia contains two subclasses, Prototheria and Theria (young born alive):\n\nPrototheria: These egg-laying mammals, known as monotremes because they have only one opening for the elimination of waste, mating, and egg laying, were the first mammals. Only five species of these mammals survive: the duck-billed platypus and four species of echidna (also called spiny anteaters), which are found only in Australia and New Guinea. Presumably, there were other species of monotreme that have become extinct.\n\nTheria: About 90 M years ago, the Theria split into two infraclasses, Metatheria and Eutheria. However, the fossil of a eutherian mammal, believed to be \u223c 125 M years old, was discovered recently in northeastern China; it was named Eomonia scansoria, meaning \"earliest eutherian mammal with specialised features for climbing\" (Ji et al., 2002, p. 816).\n\nMetatheria: In this class of mammals, usually called marsupials, there are about 330 species. The young are born alive (viviparous) but are very immature and develop in an abdominal pouch (marse = pouch, purse). Marsupials survive mainly in Australia and the surrounding islands (>200 species), with several species in South America, and one species, the Virginia opossum, in North America; there are none in Europe, Asia, or Africa.\n\nEutheria: The fetus of this mammal develops in utero where it receives nourishment from the maternal blood via a highly-specialized organ, the placenta (these are called placental mammals); \u223c95% of all mammals are eutherians.\n\n### Classification of Mammals\n\nThere are \u223c4500 species of mammal, which are classified into 20 orders (see MacDonald, 2004). It is estimated that about 80% of the species that have evolved over the last 200 M years are extinct. The classification and nomenclature of mammals commenced with the work of Carolus Linnaeus in 1758 and was based initially on morphological characteristics (see MacDonald, 2004). More soundly based classification is now possible based on DNA sequences (Murphy et al., 2001; Madsen et al., 2001; Springer et al. 2004) and on the primary sequence of certain proteins, for example, growth hormone and prolactin (Forsyth and Wallis, 2002). It should be possible to classify mammals based on the structure of milk proteins, especially caseins, which are fast-mutating proteins. Some preliminary work has been reported (Chanat et al., 1999; Goldman, 2002; Rijnkels 2002, Simpson and Nicholas, 2002; Peaker, 2002; Rijnkels, 2002; Kawasaki and Weiss, 2003; Kawasaki et al., 2004, ; Lefevre et al., 2009; Le Parc et al. 2010).\n\nAlthough the sequence of milk proteins from a sufficient number of species has not been determined to make a comprehensive classification scheme possible, considerable progress on the structure of milk proteins is being made (see Martin et al., 2003; 2011; 2013a) and is discussed further later in this chapter.\n\nA minor whey protein, whey acidic protein (WAP), has already been useful for tracing the relationships between mammalian families (Hajjoubi et al., 2006). To date, WAP has been found only in the milk of platypus, echidna, tammar wallaby, possum, mouse, rat, rabbit, camel, and pig. In humans and ruminants, the gene for WAP has been frame-shifted and occurs as a pseudogene that is not transcribed. The distribution of WAP suggests that the loss of a functional WAP gene occurred after the divergence of the pig and ruminantia lines but before the bovidae diverged from the other ruminants. Analysis of the milk from a wider range of species for WAP should be interesting.\n\n### Classification and Phylogenetic Relationships of the Principal Dairying Species\n\nAll of the principal, as well as many of the minor, dairying species belong to the Family Bovidae, a member of the Order Artiodactyla [even-toed ungalates (hoofed) mammals, i.e., cloven-hoofed]. A few minor dairying species (horse and ass) are members of Perissodactyla (odd-toed ungalates). The Bovidae evolved \u223c 18 M years ago; the earliest fossil attributed to the Bovidae is Eotragus, found in 18 M-year-old deposits in Pakistan. The Artiodactyla Order has three suborders: Ruminantia (ruminants, to which all major dairying species belong), Suidae (pigs and related species), and Tylopoda (camels, llama, alpaca, and guanaco).\n\nThe Ruminantia are classified into six families: Tragulidae (chevrotains), Moschidae (musk deer), Antilocapridae (pronghorns), Giraffidae (giraffes and okapi), Cervidae (deer; 43 species in 16 genera), and Bovidae (137\u2013138 species in 46\u201347 genera).\n\nThe Bovidae are divided into six subfamilies, of which the Bovinae is the most important. The Bovinae are divided into three Tribes, of which the Bovini are the most important from our viewpoint. The Bovini are classified into five genera: Bubalus (water buffalo), Bos (cattle), Pseudoryx (Saola), Syncerus (African (Cape) buffalo), and Bison (American and European; the European Bison is also called a wisent).\n\nThere are seven or eight species of Bos: B. primigenus (aurochs, the ancestor of domestic cattle, are extinct, with the last animal killed in Poland in 1627), B. javanicus (banteng), B. gaurus (gaur), B. frontalis (gayol), B. mutus (yak), B. sauvali (kouprey), B. taurus (European cattle), and B. indicus (Indian, humped zebu cattle). (B. taurus and B. indicus may be sub-species rather than species.) The phylogenetic relationships of the Bovini have been studied by molecular biology techniques (see Finlay, 2005).\n\nToday, there are about 1.3 \u00d7 109 cattle worldwide, of which there are two species, B. taurus, of European origin, and B. indicus, which originated in India. B. indicus (zebu) cattle also dominate in Africa, but apparently African zebu cattle have some B. taurus genes, probably as a result of cross-breeding many centuries ago. Zebu are less efficient producers of milk or meat than B. taurus but are more resistant to heat stress and various diseases and therefore dominate in tropical regions.\n\nSince cattle were domesticated \u223c 8000 years ago, they have been bred selectively, especially during the past 200 years (i.e., since Herd Books have been kept). These breeding practices have selected for various characteristics, for example, health, fertility, docility, milk or meat production, or both. Today, there are about 1000 breeds of cattle (Buchanan, 2011), including dairy, beef, or dual-purpose breeds. There are \u223c 200 M dairy cows classified into many (mainly local) breeds; Holstein-Friesian is the principal breed of dairy cow, representing \u223c 35% of the total (\u223c 70 M cows). Other important international dairy breeds are Brown Swiss (\u223c 4 M), Jersey (\u223c2 M), Ayrshire, Guernsey, and Red Dane.\n\nThere are about 170 M buffalo worldwide, of which there are two types, river and swamp, found mainly in Southeast Asia, India, and Egypt, with smaller numbers in Bulgaria, Italy, Brazil, and Australia. Buffalos are usually named after the area from which they come. Depending largely on the region, buffalo are used for milk, meat, work, or combinations of these.\n\n### Evolution of the Mammary Gland\n\nEvolution of the mammary gland is believed to have begun with the synapsids, \u223c 300 M years ago. These reptiles laid membrane-shelled eggs that lost water rapidly through evaporation and were kept moist by an aqueous oily secretion from sebaceous apocrine glands on the breast\/abdomen of the mother. These were somewhat like the brood patches on the breast of birds. The secretions are believed to have contained a range of bactericidal substances, such as oligosaccharides, lysozyme, lactoferrin, transferrin, immunoglobulins, and peroxidases, which protected the eggs against microbial infection. Presumably, the secretions were licked by the neonate from the mother's abdomen and served as a source of nutrients.\n\nConsidering its importance in the evolution of mammals, including humans, the evolution of the mammary gland and the origin of lactation have attracted considerable attention, including Darwin (1872). Reviews on the subject include Pond (1977), Hayssen and Blackburn (1985), Blackburn et al. (1989), Blackburn (1991; 1993), Hayssen (1993), Oftedal (2002a,b; ), and Lemay et al. (2009).\n\n### Structure of the Mammary Gland\n\nThe microscopic structure of the mammary gland of all species, monotremes, marsupials, and eutherians, is basically similar. The structure of the bovine gland from the cellular to the organ level is described by Fox and McSweeney (1998) and Oftedal (2013). The cells (mammocytes), whose structure is basically similar to that of all animal cells, are arranged as a monolayer in a pear-shaped organelle, called an alveolus. The alveoli are connected via a system of ducts to a cistern where the milk is stored until it is expressed from the gland, usually through a teat that is sealed by a sphincter muscle. There is little de novo synthesis in the mammary gland. Rather, the major constituents of milk are synthesized from molecules imported from the blood through the basal cell membrane. Within the mammocyte, mainly in the rough endoplasmic reticulum (RER), these molecules are polymerized to lactose, lipids, or proteins. The mammocytes are provided with a good blood supply through an extensive system of capillaries and are surrounded by contractile myoepithelial cells which, under the control of the hormones, oxytocin and prolactin, contract and express milk from the alveoli through the ducts and eventually from the gland. The hormonal control of mammary growth and function was described by Forsyth (1986).\n\nAlthough at the microscopic level, the mammary gland is essentially similar across species, the number and appearance of the gland are characteristic of the species. Monotremes have many glands on the abdomen; the glands do not end in a teat, and the milk is licked from the abdomen by the young. Marsupials have two or four glands, which end in a teat, within the pouch. On entering the pouch, the young attaches to a teat and remains attached during the period it spends permanently in the pouch. During this period, an older offspring may use another gland during its visits to the pouch. The two glands secrete milk of very different composition, designed for the specific requirements of the neonate; the composition changes markedly when the offspring leaves the pouch intermittently (Sharp et al., 2011). The mammary glands of eutherians are located external to the body cavity and end in a teat; their number varies from 2 (human, goat, sheep, horse, elephant, etc.), 4 (cattle), 12\u201314 (pig) to 24 (some insectivores). The glands are separate anatomically.\n\nThe external location of the mammary gland facilitates study of the biosynthesis of milk constituents by isotope dilution techniques, arteriovenous concentration differences, profusion of the severed gland, tissue slices, and cell homogenates (Larson and Smith, 1974\u20131979; Mepham, 1983).\n\n## Utilization of milk\n\nSince young mammals are born at a different stage of maturity and with different nutritional requirements, the milk of each species is designed to meet the requirements of the neonate of that species, that is, it is species-specific. Milk is intended to be consumed unchanged by the young suckling its mother. However, humans have consumed the milk of other species for at least 8000 years. Several species have been used for milk production, but today, cattle, especially Bos taurus, is the principal dairying species, accounting for \u223c84% of total milk production. The other important dairying species are buffalo (Bubalus bubalus) (13%) and goats and sheep (\u223c2% each); other species are significant in certain regions, or for certain purposes, for example, camel, yak, reindeer, horse, and donkey.\n\nMilk is often described as the most 'nearly perfect' food, and although this is true only for the young of the producing or closely-related species. The milk of all species is a nutrient-rich and well-balanced food (Kon, 1959; du Puis, 2002; Patton 2004; ). Many of the minor constituents of milk have biological properties, which will be described in the appropriate section (see also Korhonen, 2006); these minor constituents have been attracting considerable attention recently. However, milk is very susceptible to the growth of microorganisms that will cause spoilage if the milk is not stored properly. To counteract this problem, humans have developed a range of products that are more stable than milk; some of these date from 4000 BC and have evolved desirable epicurean characteristics, in addition to their nutritional value. Today, several thousand food products are produced from milk; these fall into the following principal groups: liquid\/beverage milk (40%), cheese (35%), milk powders (15%), concentrated milks (2%), fermented milk products (2%), butter (30%; some of which is produced from cream\/fat obtained as a by-product in the manufacture of other products), ice cream, infant formula, creams, protein-rich products, and lactose. Some of these groups are very diverse, for example, 1400 varieties of cheese have been listed.\n\n## Composition of milk\n\nMilk is a very complex fluid containing several hundred molecular species (several thousand if all triglycerides are counted individually). The principal constituents are water, lipids, sugar (lactose), and proteins. In addition, there are numerous minor constituents, mostly at trace levels\u2014for example, minerals, vitamins, hormones, enzymes, and miscellaneous compounds. The chemistry of these compounds is generally similar across species, but in many cases their structure differs in detail, reflecting evolutionary changes. The concentrations of the principal constituents vary widely among species: lipids, 2\u201355%; proteins, 1\u201320%; and lactose, 0\u201310%, reflecting mainly the energy requirements (lipids and lactose) and growth rate (mainly proteins) of the neonate. The concentrations of the minor constituents also vary widely (e.g., lysozyme and lactoferrin in equine, human, and bovine milks).\n\nWithin any species, the composition of milk varies among individual animals, between breeds, with the stage of lactation, feed, and health of the animal, along with many other factors. The fat content of bovine milk shows large inter-breed differences, and within any breed there is a wide range of fat and protein content for individual animals; similar differences occur in the milk of sheep, goat, and buffalo.\n\nReflecting mainly the nutritional and physiological requirements of the neonate, the composition of milk, and even the profile of constituents therein, change markedly during lactation. The changes are most marked during the first few days postpartum, especially in the immunoglobulin fraction of proteins. For marsupials, the milk changes from a high-carbohydrate (mainly oligosaccharides) to a high-fat secretion when the neonate begins to leave the pouch, a time that corresponds roughly to the birth of eutherians. The composition of milk remains relatively constant during mid-lactation but changes considerably in late lactation, reflecting the involution of the mammary gland tissue and the greater influx of blood constituents.\n\n## Milk constituents\n\nIn the following sections, the chemistry of milk carbohydrates, lipids, proteins, salts, and some minor constituents are described; where possible, interspecies comparisons are made.\n\n### Carbohydrates\n\n#### Lactose\n\nThe principal carbohydrate in the milk of most species is the reducing disaccharide, lactose, which is composed of galactose and glucose linked by a \u03b21-4 glycosidic bond. Its concentration varies from 0 to \u223c10% (Fox and McSweeney, 1998; McSweeney and Fox, 2009), and milk is the only known source of lactose. Research on lactose commenced with the work of Carl Scheele in about 1780; its chemistry and its important physicochemical properties have been described very thoroughly. The very extensive literature has been reviewed by Whittier (1925; 1944), Weisberg (1954), Zadow (1984; 1992), Fox (1985; 1997), Fox and McSweeney (1998), McSweeney and Fox (2009), Fox (2011), Paterson (2011), and Schuck (2011).\n\nThe concentration of lactose in milk is inversely related to the concentration of lipids and casein because lactose causes the influx of water into the mammocytes, thereby causing dilution of milk (Jenness and Sloan, 1970; Jenness and Holt 1987). The principal function of lactose and lipids in milk is as a source of energy; since lipids are 2.5 times as energy-dense as lactose, when a highly calorific milk is required, for example, by animals in a cold environment (e.g., marine mammals or hibernating bears, this is achieved by increasing the fat content of the milk. The inverse relationship between the concentrations of lactose and casein reflects the fact that the synthesis of lactose draws water into the Golgi vesicles, thereby diluting the concentration of casein (Jenness and Holt, 1987).\n\nLactose is synthesized in the epithelial mammary cells from two molecules of glucose absorbed from the blood. One molecule of glucose is phosphorylated and converted (epimerized) to galactose-P via the Leloir pathway, which is widespread in animal tissues and bacterial cells. Galactose-P is condensed with a second molecule of glucose through the action of a unique two-component enzyme, lactose synthetase. One component is UDP-galactosyl transferase (EC 2.4.1.22), which transfers galactose from UDP-galactose to any of several acceptor molecules in the biosynthesis of glycoproteins and glycolipids. The specificity of the transferase is controlled and modified by \u03b1-lactalbumin (\u03b1-La), one of the principal milk proteins, which reduces the Michaelis constant (KM) for glucose 1000-fold. In its presence, most of the galactose is transferred to glucose, with the synthesis of lactose. There is a positive correlation between the concentrations of lactose and \u03b1-La in milk; the milk of the California sea lion or the hooded seal lack both \u03b1-La and lactose.\n\nLactose serves two important functions in milk: It is a ready source of energy for the neonate (it provides 30% of the calories in bovine milk); and it is responsible for about 50% of the osmotic pressure of milk, which is isotonic with blood and hence is essentially constant. The synthesis of lactose draws water osmotically into the Golgi vesicles and hence affects the volume of milk and the concentration of casein, which is packaged in the Golgi vesicules.\n\nFor milk with a low level of lactose, the concentration of inorganic salts is high to maintain the osmotic pressure at the desired level; there is an inverse relationship between the concentrations of lactose and salts (ash) in milk (Jenness and Sloan, 1970). During mastitis or in late lactation, the integrity of the mammocyte cell membranes is damaged, and there is an influx of blood constituents into milk, the osmotic pressure increases, and to adjust this imbalance, the concentration of lactose is reduced. This relationship is expressed as the Koesler Number, % chloride \u00d7 100 \u00f7 % lactose, which is normally <2 and a value >3 is considered abnormal. Today, the Koesler Number is rarely used as a diagnostic indicator of mastitis, but the electrical conductivity of milk, which depends mainly on the milk salts and can be measured in-line during milking, is commonly used for this purpose.\n\nWhy milk contains lactose rather than some other sugar(s) is not clear. The presence of a disaccharide rather than a monosaccharide can be explained on the basis that twice as much (mass) disaccharide as monosaccharide can be accommodated for any particular incremental increase in osmotic pressure, which is fixed. Maltose, which consists of two molecules of glucose, would seem to be the obvious choice of disaccharide. Since energy is expended in converting glucose to galactose, some benefit must accrue from this conversion. A possible benefit is that galactose or derivatives thereof occurs in some physiologically important lipids and proteins and a galactose-containing sugar in milk provides the neonate with a ready supply of this important monosaccharide.\n\nThe properties of lactose are generally similar to those of other sugars, but it differs in some technologically important respects. Some important characteristics of lactose are:\n\n\u2022 Lactose is a reducing sugar; that is, it has a free, or potentially free, carbonyl group (an aldehyde group in the case of lactose).\n\n\u2022 Like other reducing sugars, lactose exists partially as an open-chain form with an aldehyde group that can form a hemiacetal and thus a ring structure. Formation of a hemiacetal creates a new chiral center (asymmetric carbon), which may exist as two isomers (enanthiomorphs), \u03b1 or \u03b2. By alternatively opening and forming the ring structure, the molecule can interchange between \u03b1 and \u03b2 isomers, a process referred to as mutarotation.\n\n\u2022 The \u03b1 and \u03b2 isomers of lactose have very different properties, the most important of which are specific rotation, [\u03b1]20 D (+89o and +35o for \u03b1 and \u03b2, respectively) and solubility (70 and 500 g\/L, for \u03b1 and \u03b2, respectively).\n\n\u2022 Like all reducing sugars, lactose can participate in the Maillard (nonenzymatic) browning reaction, resulting in the production of (off-) flavor compounds and brown polymers. The Maillard reaction contributes positively to the flavor and color of many foods (e.g., crust of bread, toast, and deep-fried products), but the effects in dairy products are negative and must be avoided.\n\n\u2022 Redox titration using alkaline CuSO4 (Fehling's solution) or chloramine-T is the principal standard method for the quantitative determination of lactose. It may also be determined by polarimetry, spectrophotometrically after reaction with phenol or anthrone in strongly acidic solution, enzymatically, or by high-performance liquid chromatography.\n\n\u2022 Among sugars, lactose, especially the \u03b1 enanthiomorph, has low solubility in water, but when in solution, it is difficult to crystallize and may cause problems in lactose-rich dairy products (e.g., skim milk powder and especially whey powder), unless precautions are taken to induce and control crystallization.\n\n\u2022 \u03b1 and \u03b2 lactose are soluble in water to the extent of about 70 and 500 g\/L, respectively, at 20 \u00b0C; at equilibrium, the ratio of \u03b1:\u03b2 is about 1:2, giving a total solubility of about 180 g\/L at 20 \u00b0C. The solubility of \u03b1 lactose is more temperature dependent than that of the \u03b2 isomer; the solubility-temperature curves intersect at \u223c94oC, making \u03b1 lactose more soluble than the \u03b2 anomer >94 \u00b0C. Hence, \u03b1-lactose is the form of lactose that crystallizes <94 \u00b0C and is the usual commercial form of lactose; \u03b2 lactose may be prepared by crystallization >94 \u00b0C\n\n\u2022 \u03b1 lactose crystallizes as a monohydrate, while \u03b2 lactose forms anhydrous crystals; thus, the yield of \u03b1 lactose is 5% higher than that of \u03b2 lactose.\n\n\u2022 When milk or whey is spray-dried, any lactose that has not been precrystallized forms an amorphous glass that is stable if the moisture content of the powder is maintained low. If it increases to >6%, however, the lactose crystallizes as \u03b1 hydrate, the crystals of which form interlocking masses and clumps that may render the powder unusable if very extensive; (i.e., inadequately crystallized powder is hygroscopic). The problem can be avoided by adequate crystallization of lactose before drying or by using effective packaging.\n\n\u2022 Interestingly, crystalline lactose has very low hygroscopicity and is used in icing sugar blends.\n\n\u2022 Among sugars, lactose has a low level of sweetness; it is only about 16% as sweet as sucrose at 1% in solution and hence has limited value as a sweetening agent, the principal application of sugars in foods. However, it is a useful bulking agent when excessive sweetness is undesirable.\n\n\u2022 Lactose is important in the manufacture of fermented dairy products where it serves as a carbon source for lactic acid bacteria that produce lactic acid.\n\n#### Modification of the Concentration of Lactose in Milk through Genetic Engineering\n\nBecause lactose is the least valuable constituent in milk, there is considerable interest in modifying the lactose content of milk, as it costs energy on the part of the animal to synthesize it. Therefore, it would be economically advantageous to reduce the lactose content of milk. In addition, lactose effectively controls the water content of milk, and most dairy processes require the removal of water. Hence, it would be advantageous to reduce the amount of water in milk by reducing the level of lactose. Since the concentration of lactose is controlled by the concentration of \u03b1-La in the secretory cells, the approach to changing the concentration of lactose involves altering the level of \u03b1-La by genetic engineering. However, if the level of lactose is reduced too much, the viscosity of the milk will be too high for easy expression from the mammary gland. It has been shown that the viscosity of mouse milk engineered to contain no lactose was so high that the pups were unable to suckle and died (Leaver and Law, 2003). Obviously, this problem could be overcome by reducing the level of lactose rather than eliminating it. Alternatively, it may be possible to modify the milk secretory mechanism to produce a more useful, or at least a less problematic sugar than lactose, for example, glucose, maltose, or lactulose (which is a laxative and a prebiotic). It might be possible to increase the concentration of salts in milk. As discussed below, most adult humans are unable to digest lactose. If the problems arising from high viscosity were resolved, lactose-free or -reduced milk would be nutritionally desirable. The possibility of engineering the mammary cell to secrete \u03b2-galactosidase into milk and to hydrolyze lactose in situ has been suggested.\n\nHowever, in some cases it would be advantageous to increase the lactose content of milk. The economic benefits of increasing the milk output of sows by increasing its lactose content were discussed by Wheeler (2003).\n\n#### Nutritional Problems Associated with Lactose\n\nMammals cannot absorb disaccharides from the small intestine, where they are hydrolyzed to monosaccharides, which are absorbed. Lactose is hydrolyzed by \u03b2-galactosidase (lactase), which is secreted by cells in the brush border of the small intestine. The young of most mammalian species secrete an adequate level of lactase, but as the animal ages, the secretion of lactase declines and eventually becomes inadequate to hydrolyze indigested lactose. The lactose then enters the large intestine where it causes an influx of water, resulting in diarrhea, and is metabolized by bacteria with the production of gas that causes cramps and flatulence. In humans, this condition may occur at 8\u201310 years of age and cause many individuals to exclude milk from the diet. The problems may be avoided by prehydrolyzing the lactose using exogenous \u03b2-galactosidase (see Mahoney, 1997; Shakeel-Ur-Rehman 2009). The frequency and intensity of lactose intolerance\/malabsorption vary widely among populations from \u223c100% in Southeast Asia to \u223c5% in northwest Europe (Mustapha et al., 1997; Ingram and Swallow 2009).\n\n#### Production and Utilization of Lactose\n\nPreviously, whey from cheese or casein production was considered a waste material that was fed to farm animals, irrigated on land, or disposed of in sewers. Environmental and economic considerations now dictate, however, that whey should be used more efficiently. The principal product lines produced from whey are whey powders (various), whey protein products produced by membrane technology, and lactose and its derivatives.\n\nLactose is prepared commercially by crystallization from concentrated whey or ultrafiltrate. The crystals are usually recovered by centrifugation; this process is essentially similar to that used for sucrose or other sugars. About 500,000 tonnes of crystalline lactose are produced annually (Paterson, 2009), compared to \u223c100 M tonnes of sucrose. Owing to its relatively low sweetness and low solubility, the applications of lactose are much more limited than those of sucrose or glucose. Its principal application is in the production of 'humanized' infant formula based on cow's milk (human milk contains \u223c7% lactose in comparison with \u223c4.6% in bovine milk). The lactose used may be in the form of a purified crystalline product or demineralized whey (for physiological reasons, it is necessary to reduce the concentration of inorganic salts in whey).\n\nLactose has a number of low-volume special applications in the food industry, for example, as a free-flowing or agglomerating agent, to accentuate\/enhance the flavor of some foods, to improve the functionality of shortenings, and as a diluent for pigments, flavors, or enzymes. It is widely used in the tabletting of drugs in the pharmaceutical industry where low hygroscopicity is a critical property.\n\nLactose can be converted to several more valuable food-grade derivatives, of which the most significant are glucose-galactose syrups (\u223c3 times as sweet as lactose; produced by hydrolysis by \u03b2-galactosidase), lactulose (galactose-fuctose; a prebiotic and laxative), lactitol (the alcohol of lactose), lactobionic acid (a sweet-tasting acid, which is a very rare property), tagatose, oligosaccharides (prebiotics), and fermentation products (ethanol, lactic, acetic, and propionic acids) (Playne and Crittenden, 2009; Ganzle, 2011a,b).\n\n#### Oligosaccharides\n\nIn addition to lactose, the milk of most, and probably all, species contains other free saccharides, mainly oligosaccharides (OSs), the concentration, proportions and types of which show large interspecies differences. The concentration of OSs is higher in colostrum than in milk. General reviews on the OSs in milk include Newburg and Newbauer (1995), Mehra and Kelly (2006), and Urashima et al. (2001; ; 2011).\n\nAlmost all of the OSs have lactose at the reducing end, they contain three to eight monosaccharides, they may be linear or branched, and contain either or both of two unusual monosaccharides, fucose (a 6-deoxyhexose) and N-acetylneuraminic acid. Fucose occurs quite widely in tissues of mammals and other animals where it serves a wide array of functions (Becker and Lowe, 2003). Its significance in the OSs in milk is not clear; perhaps it is to supply the neonate with preformed fucose.\n\nThe OSs are synthesized in the mammary gland, catalyzed by special transferases that transfer galactosyl, sialyl, N-acetylglucosaminyl, or fucosyl residues from nucleotide sugars to the core structures. These transferases are not affected by \u03b1-La and are probably similar to the transferases that catalyze the glycosylation of lipids and proteins.\n\nThe milk of all species examined contains OSs, but the concentration varies markedly. The highest levels are in the milk of monotremes, marsupials, marine mammals, humans, elephants, and bears. With the exception of humans and elephants, the milk of these species contains little or no lactose, and OSs are the principal carbohydrates.\n\nThe milk of the echidna contains mainly the trisaccharide, fucosyllactose, while that of the platypus contains mainly the tetrasaccharide, difucosyllactose. Among marsupials, the best studied is the Tammar wallaby; presumably, its lactation pattern and milk composition are typical of marsupials. A low level of lactose is produced at the start of lactation, but about 7 days after birth, a second galactosyltransferase appears and tri- to penta-saccharides are produced, which by \u223c180 days are the principal saccharides. During this period the saccharide content is high, \u223c50% of total solids, and the level of lipids is low (\u223c15% of total solids). At about 180 days, the carbohydrate decreases to a very low level and consists mainly of monosaccharides, while the level of lipids increases to \u226560% of total solids (Sharp et al., 2011).\n\nHuman milk contains \u223c130 OSs, at a total concentration of \u223c15 g\/L; these are considered to be important for neonatal brain development. Bear milk contains little lactose but a high level of total sugars (mainly OSs) \u2013 1.7 and 28.6 g\/kg, respectively (Oftedal et al. 1993; ). Elephant milk contains \u223c50 and 12 g\/kg of lactose and OSs, respectively, a few days postpartum, but as lactation progresses, the concentration of lactose decreases while that of OSs increases (e.g., 12 and 18 g\/kg, respectively), at 47 days (Osthoff et al., 2005). The milk of seals contains both lactose and OSs, but milks of the Californian sea lion, Northern fur seal, and Australian fur seal contain neither, probably because they contain no \u03b1-La (Urashima et al., 2001).\n\nBovine, ovine, caprine, and equine milk contain relatively low levels of OSs, which have been characterized (see Urashima et al. 2001; ; ). Caprine milk contains about 10 times as much OSs as bovine and ovine milk, and a process for their isolation by nanofiltration has been reported (Martinez-Ferez et al., 2006). Possible methods for producing OSs similar to those found in human milk, by fermentation or by transgenic animals or by recovering OSs from cow's milk whey or UF permeate were discussed by Mehra and Kelly (2006) and O'Mahony and Tuohy (2013).\n\nAs discussed earlier, OSs with bactericidal properties were probably the saccharides present in the mammary secretions of early mammals; the high level of OSs in the milk of monotremes and marsupials conforms with their secretion early in evolution. It is proposed that the primitive mammary glands of the first common ancestor of mammals produced lysozyme (a predecessor of \u03b1-La), and a number of glycosyltransferases but little or no \u03b1-La. This resulted in the production of a low level of lactose that was utilized in the synthesis of OSs and did not accumulate (Messer and Urashima 2002; Urashima et al. 2009). Initially, the OSs served mainly as bactericidal agents but later became a source of energy for the neonate. Both of these functions persist for monotremes, marsupials, and some eutherians such as bears, elephants, and marine mammals. However, most eutherians evolved to secrete predominantly lactose as an energy source, due to the synthesis of an increased level of \u03b1-La, while OSs continued to play a bactericidal role. Human and elephant milk, both of which contain high levels of lactose and OSs, seem to be anomalous. Work on the OSs of a wider range of species is needed to explain this situation.\n\nThe significance of OSs is not clear, but the following aspects may be significant: For any particular level of energy, they have a smaller impact on osmotic pressure than smaller saccharides, they are not hydrolyzed by \u03b2-galactosidase, and fucosidase or neuraminidase is not secreted in the intestine. Hence the OSs are not hydrolyzed and absorbed in the gastrointestinal tract, and they function as soluble fiber and prebiotics that affect the microflora of the large intestine. It is claimed that they prevent the adhesion of pathogenic bacteria in the intestine; galactose, and especially N-acetylneuraminic acid, are important for the synthesis of glycolipids and glycoproteins, which are vital for brain development. It has therefore been suggested that the OSs are important for brain development (see Kunz and Rudloff, 2006).\n\nIn addition to lactose and free OSs, the milk of all species examined contains small amounts of monosaccharides and some milk proteins, especially \u03ba-casein, are glycosylated, and there are low levels of highly glycosylated glycoproteins, especially mucins, and glycolipids in the milk fat globule membrane.\n\nThere is considerable interest in the development of OS-enriched ingredients from bovine milk (O'Mahony and Tuohy, 2013), primarily for infant formula applications. This interest has been spurred by the demonstrated bioactive functionality of these compounds in humans (Kunz and Rudlof, 2006).\n\n### Lipids\n\nLipids (commonly called oils or fats, which are liquid or solid, respectively, at ambient temperature) are those constituents of tissues, biological fluids, or foods that are soluble in an apolar solvent (e.g., diethyl ether, chloroform, or carbon tetrachloride). Historically, the fat of milk was regarded as its most valuable constituent, and until recently, milk was valued largely or totally on the basis of its fat content. This was due at least partially to S. M. Babcock and N. Gerber's development of rather simple methods for quantifying the fat content of milk in the 1890s, long before comparable fast and simple methods for proteins became available. Milk lipids are very complex chemically and exist as a rather unique emulsion. Milk lipids have been thoroughly studied and characterized (see Fox 1983; ; Fox and McSweeney 1998; ; and the references therein).\n\nThe level of fat in milk shows very large interspecies differences, ranging from \u223c2% to >50% (see Fox and McSweeney, 1998). The fat content of milk reflects the energy requirements of the neonate; the requirement is high in the milk of species that live in a cold environment or need to build up a layer of subcutaneous fat quickly (marine mammals).\n\nLipids are commonly divided into three classes:\n\n\u2022 Neutral lipids. These are esters of glycerol, and one, two, or three fatty acids for mono-, di-, and triglycerides, respectively. Neutral lipids are by far the dominant class of lipids in all foods and tissues, representing 98.5% of total milk lipids.\n\n\u2022 Polar lipids (a complex mixture of fatty acid esters of glycerol or sphingosine). These may contain phosphoric acid, a nitrogen-containing compound (choline, ethanolamine, or serine), or a sugar\/OS. Although present at low levels (\u223c1% of total milk lipids), the polar lipids play critical roles in milk and dairy products. They are very good natural emulsifiers and are concentrated in the milk fat globule membrane that maintains the milk lipids as discrete globules and ensures their physical and biochemical stability.\n\n\u2022 Miscellaneous lipids. This is a heterogeneous group of compounds that are unrelated chemically to each other or to neutral or polar lipids. This group includes cholesterol, carotenoids, and the fat-soluble vitamins, A, D, E, and K.\n\nThe carotenoids are important for two reasons: They are natural pigments (yellow, orange, red), and they are responsible for the color of butter and cheese. Some consumers prefer highly colored cheese, which is obtained by adding a carotenoid-containing extract from annatto beans. Some carotenoids are converted to vitamin A in the liver.\n\n#### Fatty Acids\n\nFatty acids (FAs) are carboxylic acids with the general formula R-COOH, where the alkyl group, R, is a hydrocarbon chain containing 3 to 25 carbons (total number of carbons, 4 to 26), which may be saturated or unsaturated (one to six double bonds), and is usually straight (normal), with small amounts of branched chain, hydroxyl, and keto (oxo) acids. The vast majority of FAs have an even number of carbon atoms because they are synthesized from, and elongated by adding, a 2-C compound, acetyl CoA, on each cycle of the multienzyme fatty acid synthetase (FAS). Although the hydroxy fatty acids are present at low levels, they are important in milk fat because upon heating they are converted to lactones, which give a desirable flavor to milk fat, which is considered the premium cooking fat. Although keto acids are also minor components, they are important flavor precursors since they are converted to highly-flavored methyl ketones.\n\nThe melting point (MP) of FAs increases progressively with molecular weight (MW), while solubility in water decreases. The MP decreases with the introduction of double bonds, and for unsaturated FAs, the MP of the cis isomer is lower than that of the trans isomer.\n\nMilk lipids are chemically similar to all other lipids but contain a wide range of FAs (up to 400 FAs have been reported in milk lipids, although most of these are present at trace levels). The milk lipids of ruminants are unique in that they are the only natural lipids that contain butyric (butanoic) acid (C4:0). They also contain substantial amounts of medium-chain FAs [hexanoic (C6:0), octanoic (C8:0), and decanoic (C10:0)], the only other sources of which are coconut and palm kernel oil. The short- and medium-chain FAs are water-soluble and volatile and have a strong aroma and taste.\n\nThe fatty acids in milk fat are obtained from three sources:\n\n\u2022 Butanoic acid is produced by reducing \u03b2-hydroxybutanoic acid, which is synthesized by bacteria in the rumen.\n\n\u2022 All hexanoic (C6:0) to tetradecanoic (C14:0) acids and 50% of hexadecanoic (C16:0) acid are synthesized in the mammary gland from acetyl CoA (CH3COSCoA). These FAs are released from the FAS by chain-length-specific thioesterases, the relative activities of which are responsible for interspecies differences in the proportions of medium-chain FAs. Decanoic acid (C10:0) and dodecanoic (C12:0) are major FAs in the milk fat of elephant, horse, donkey, zebra, tapir, rhinoceros, rabbit, and hare, but these fats contain very little or no butanoic acid (Glass et al., 1967; Glass and Jenness, 1971; Christie, 1995; Osthoff et al., 2005; MacGibbon and Taylor 2006). These species are nonruminant herbivores with a large secum, a feature that presumably is somehow responsible for the high levels of C10:0 and C12:0; some of the above species also practice coprophagy. All octadecanoic (C18:0) and 50% of hexadecanoic (C16:0) acids are obtained from dietary lipids.\n\nThe unsaturated FAs are synthesized as follows:\n\n\u2022 C18:1 is produced from C18:0 in the liver by \u0394-9 desaturase.\n\n\u2022 C18:2 is obtained from the diet; that is, it is an essential FA.\n\n\u2022 The other unstaturated FAs are produced from C18:2 by further desaturation and\/or elongation.\n\nRuminant milk fats contain low levels of polyunsaturated fatty acids (PUFAs) because PUFAs in the diet are hydrogenated by bacteria in the rumen. Biohydrogenation can be prevented by encapsulating dietary PUFAs or PUFA-rich sources in cross-linked protein or cross-linked crushed oilseeds. PUFA-enriched milk has improved spreadability and perceived improved nutritional qualities (Parodi, 2006; O'Brien and O'Connor 2011).\n\nIncomplete biohydrogenation by the rumen bacterium, Butyrivibrio fibrisolvens, results in the formation of conjugated linoleic acid (CLA; also called rumenic acid), which has potent anticarcinogenic properties. Eight isomers of CLA are possible, but cis-9, trans-11 is the most biologically active. The formation of CLA and its nutritional benefits have been the subject of considerable research during the past 15 years and has been reviewed by Bauman and Lock (2006), Parodi (2006), Bauman et al. (2011), and Mills et al. (2011).\n\n#### Distribution of FAs in Triglycerides\n\nAs well as the constituent FAs, the position of the FAs in triglycerides (TGs) affects their MP and rheological properties. For these reasons and to completely characterize the structure of TGs, the position of FAs in milk TGs has been determined. An index of the length of the FAs can be obtained by determining the acyl carbon number (ACN) of TGs, that is, the sum of the number of carbons in the three-component FAs, which can be done by gas chromatography (GC). Probably the first study on this aspect was done by Breckenridge and Kuksis (1967), who reported the ACN of the milk TGs from seven species. More recent work has been reviewed by Christie (1995), MacGibbon and Taylor (2006), and Taylor and MacGibbon (2011).\n\nThe complete structure of TGs can be determined by stereospecific analysis, the results of which for milk fat are described by Christie (1995) and MacGibbon and Taylor (2006). The most notable feature is the almost exclusive esterification of the short-chain FAs, C4:0 and C6:0, at the Sn3 position. Since many lipases are specific for the Sn3 position, these short-chain FAs (which are highly flavored\/off-flavored) are released rapidly, causing desirable\/undesirable changes in sensory properties.\n\n#### Degradation of Lipids\n\nFood lipids are susceptible to two forms of deterioration: lipid oxidation leading to oxidative rancidity and hydrolysis of lipids by lipases (lipolysis), leading to hydrolytic rancidity. Lipid oxidation involves a very complex set of chemical reactions that have been well characterized; the literature has been comprehensively reviewed by Richardson and Korycka-Dahl (1983) and O'Connor and O'Brien (2006; 2011).\n\nMilk contains an indigenous lipoprotein lipase (LPL) that is normally inactive because it is separated from the TG substrates by the milk fat globule membrane (MFGM), but if the membrane is damaged, lipolysis and hydrolytic rancidity ensue rapidly. When milk lipids are hydrolyzed by milk LPL, the short- and medium-chain FAs, which are esterified mainly at the Sn3 position, are released preferentially, and are major contributors to flavor, which may be desirable or undesirable, depending on the product. Hydrolytic rancidity caused by milk LPL is potentially a very serious problem in raw milk and in some dairy products. Lipolysis in milk has been reviewed comprehensively by Deeth and Fitz-Gerald (1983; ; 2006); and Deeth (2011).\n\nA low level of lipolysis is desirable in all types of cheese, especially in blue cheeses, in which the principal lipases are those secreted by the blue mold, Penicillium roqueforti. The free fatty acids (FFAs) are converted to alk-2-ones, the principal flavor compounds in blue cheeses. The characteristic piquant flavor of some cheeses, such as Pecorino Romano, is due to short- and medium-chain FAs that are released mainly by an added lipase, pregastric esterase. Other important derivatives of FAs are alk-2-ols (secondary alcohols), lactones, esters, and thioesters; these are important flavor compounds in cheese.\n\n#### Milk Lipids as an Emulsion\n\nLipids are insoluble in water or aqueous systems. When mixed, a lipid and water (or aqueous solvent) form distinct layers and a force, interfacial tension (\u03b3), exists between the layers. Lipids can be dispersed in water by vigorous agitation (homogenization), but when agitation ceases, the droplets of lipid coalesce quickly into a single mass (i.e., phase separation), driven by the need to reduce the interfacial area and, consequently, interfacial tension, \u03b3, to a minimum. If \u03b3 is reduced, the droplets of lipid will remain discrete, although they will rise to the surface (i.e., cream) owing to the lower density of lipids compared to water. Interfacial tension can be reduced by using a surface-active agent (emulsifier, detergent). Natural emulsifiers include proteins, phospholipids, mono- and diglycerides; there is a wide range of synthetic emulsifiers.\n\nIn milk, the lipids are dispersed in the milk serum (specific gravity, 1.036) as globules with a diameter in the range <1 to \u223c20 \u03bcm (mean 3\u20134 \u03bcm). The FAs (from the sources described above) and monoglycerides (from blood lipids) are synthesized to TGs in the rough endoplasmic reticulum (RER) at the basal region of the epithelial cells. The TGs form into globules within the RER and are released into the cell cytoplasm. The globules are stabilized by a complex layer of proteins and phospholipids, known as the MFGM. The inner layer of the MFGM is acquired within the epithelial cell as the fat globules, after release from the RER, move toward the apical membrane. The outer layer of the MFGM is the apical membrane of the secretory cell through which the lipid globules are pushed as they are expressed from the cell. Since the stability of the milk emulsion is critical in most dairy products, the structure and stability of the MFGM have been the subject of research for more than 100 years.\n\nSodium dodecyl sulfate (SDS)-PAGE indicates that there are eight main proteins in the MFGM butryophilin (BTN), xanthine oxidoreductase (XOR), adipophilin (ADPH), mucin 1 (MUC 1), mucin 15 (MUC 15), periodic acid Schiff glycoproteins (PAS) 6 and 7 and fatty acid-binding protein (FABP)], which have been isolated and characterized (see [Mather, 2000; Keenan and Mather 2006). However, SDS-PAGE followed by microcapillary HPLC-MS indicates 120 proteins, of which 23% are involved in protein trafficking, 23% in cell signaling, 21% in unknown functions, 11% in fat trafficking\/metabolism, 9% in transport, 7% in protein synthesis\/folding, 4% are immune proteins, and 2% are contaminating skim milk proteins (Reinhardt and Lippolis, 2006). Many of the 70 indigenous enzymes in milk are concentrated in the MFGM. The very extensive literature on the MFGM has been the subject of many reviews, including Keenan and Mather (2006) and Mather (2011), who present an up-to-date model of the MFGM (Fig. 2.1).\n\nFigure 2.1 Topology of the major bovine milk fat globule membrane proteins (from Mather, 2011).\n\nSome of the MFGM is shed during the aging of milk, especially if agitated, and forms vesicles (sometimes called microsomes) in the skim milk. The MFGM may be damaged by agitation, homogenization, whipping, or freezing, which may lead to hydrolytic rancidity and non-globular fat, potentially causing cream plug, oiling-off in coffee and tea, and poor wettability of milk powder (for a review see Evers, 2004). The MFGM is stripped from the fat globules by extensive agitation (usually of cream), a process referred to as 'churning'; the free fat coalesces and is kneaded (worked) to give a water-in-oil emulsion, butter. The MFGM partitions into the aqueous phase, referred to as buttermilk. The phospholipids in buttermilk give it good emulsifying properties, and there is commercial interest in using it as a food ingredient (Singh, 2006). Several approaches, many using membrane filtration technology, have been proposed and evaluated over the last 20 years or so for their suitability to fractionate buttermilk\/butter serum in the enrichment of MFGM material (see Zanabria Eyzaguirre and Corredig, 2011; O'Mahony and Tuohy 2013). Some of the polar lipids in the MFGM are reported to have desirable nutritional properties (see Spitsberg, 2005; Ward et al. 2006), but there are conflicting views (see Moss and Freed, 2003; Riccio 2004).\n\nPresumably, the fat globules in the milk of all species are stabilized by a membrane similar to that in bovine milk, but there is very little information on the MFGM in nonbovine milks. Buchheim et al. (1989) and Welsch et al. (1990) studied the glycoproteins in the MFGM of human, rhesus monkey, chimpanzee, dog, sheep, goat, cow, gray seal, camel, and alpaca by SDS-PAGE with periodic acid Schiff (PAS) staining, Western blotting, and lectin biochemistry. Large intra- and interspecies differences were found; and very highly glycosylated proteins were seen in the MFGM of primates, horse, donkey, camel, and dog. Long (0.5\u20131 \u03bcm) filamentous structures extend from the surface of the fat globules in equine and human milk; the filaments are composed of mucins (highly glycosylated proteins) that dissociate rapidly from the surface of bovine globules into the milk serum on cooling; they are also lost on heating human milk (e.g., at 80oC for 10 min; see Patton, 1999, and references therein). The filaments facilitate the adherence of fat globules to the intestinal epithelium and probably improve the digestion of fat. The mucins prevent bacterial adhesion and may protect mammary tissue against tumors (mammary tumors are very rare in cattle) (Patton, 1999). Why the filaments on bovine milk fat globules are lost much more easily than those in equine and human milk is not known; work in this area is warranted. Proteomic methodology is being applied to study the human MFGM and membranes of the mammary epithelial cells (for references see Reinhardt and Lippolis, 2006; Lopez and Menard 2011).\n\nThe fat globules in bovine milk form a cream layer due to the difference in specific gravity between the fat and aqueous phases, but the cream layer is readily dispersed by gentle agitation. The rate of creaming can be calculated from Stokes's equation:\n\nV = 2 r 2 ( \u03c1 1 \u2212 \u03c1 2 ) g \/ 9 \u03b7\n\nwhere V is the velocity of creaming, r is the radius of the fat globules, \u03c11 and \u03c12 are the specific gravity of the continuous and dispersed phases, respectively, g is acceleration due to gravity, and \u03b7 is the viscosity of the continuous phase. Based on the typical values for r, \u03c11, \u03c12, and \u03b7 for milk, one would expect a cream layer to form in milk in about 60 h, but, in fact, a cream layer forms in about 30 min. The faster than expected rate of creaming is due to the aggregation of fat globules, aided by an immunoglobulin M-type protein, called cryoglobulin, because it precipitates onto the fat globules when the milk is cooled. The clusters of globules behave as a unit with a large radius. Creaming can be prevented by homogenizing the milk, which reduces the size of the globules and denatures the cryoglobulins. The fat globules in buffalo, ovine, caprine, equine, and camel milk do not agglutinate because these milks lack cryoglobulins.\n\nPreviously, the creaming of milk was a very important attribute and was a popular research topic; the extensive literature has been the subject of several reviews, most recently by Huppertz and Kelly (2006). Traditionally, the fat was removed from milk by natural (gravity) creaming. Gravity creaming is still used to standardize the fat content for some cheese varieties (e.g., Parmigiano Reggiano), but the removal of fat from milk is now usually accomplished by centrifugal separation, in which g is replaced by \u03c92R, where \u03c9 is the centrifugal velocity in radians per second and R is the radius of the centrifuge bowl. Centrifugal separation is very efficient, essentially instantaneous, and continuous.\n\n### Milk Proteins\n\nThe properties of milk and most dairy products are affected more by the proteins they contain than by any other constituent. The milk proteins have many unique properties. Because of this and their technological importance, the milk proteins have been studied extensively and are probably the best characterized food protein system.\n\nResearch on milk proteins dates from the early nineteenth century. Pioneering work was reported by J. J. Berzelius in 1814, by H. Schubler in 1818 on the physicochemical status of milk proteins, and by H. Braconnot in 1830 who published the first paper in which the word 'casein' was used. A method for preparing protein from milk by acid precipitation was described in 1938 by J. G. Mulder, who coined the term protein ('primary' or 'of first rank'). The acid-precipitated protein was referred to as casein. Some early authors called acid-precipitated milk protein caseinogen, which was converted by rennet to casein, which coagulated in the presence of Ca2+. This situation is analogous to the conversion of fibrinogen in blood by thrombin to fibrin, which coagulates in the presence of Ca2+. About 70 years ago, the term casein was universally adopted as the English word for the pH 4.6-insoluble protein in milk. The method for acid (isoelectric) precipitation of casein was refined by O. Hammarsten in 1883\u20131885, and, consequently, isoelectric casein is frequently referred to as 'casein nach Hammarsten.' An improved method for isolating casein was published by L. L. van Slyke and J. C. Baker in 1918.\n\nAs early as 1846, J. E. Schlossberger reported the separation of casein into two fractions (see Woodward, 1976) and in 1880, A. Danilewsky and P. Radenhausen suggested that isoelectric casein is heterogeneous (see Hammarsten, 1883; Lindqvist 1963), but Hammarsten (1883) maintained that properly produced casein is homogeneous. Based on differential solubility in ethanol-HCl solutions, evidence began to emerge from the work of T. B. Osborne and A. J. Wakeman in 1918 and of K. Linderstr\u00f8m-Lang and collaborators during the period 1925\u20131929 that isoelectric casein is heterogeneous. Heterogeneity was confirmed by K. O. Pederson in 1936, using analytical ultracentrifugation, and by O. Mellander in 1939, using free-boundary electrophoresis (see McMeekin, 1970, for references to early literature).\n\nThe liquid whey remaining after isoelectric precipitation of casein from skim or whole milk is a dilute solution of proteins (whey or serum proteins; \u223c0.7% in bovine milk), lactose, organic and inorganic salts, vitamins, and several constituents at trace levels. By salting-out with MgSO4, the whey proteins were fractionated by J. Sebelein, in 1885, into soluble (albumin) and insoluble (globulin) fractions. According to McMeekin (1970), A. Wichmann, in 1899, crystallized a protein from the albumin fraction of whey by the addition of (NH4)2SO4 and acidification, a technique used to crystallize blood serum albumin and ovalbumin. Using the techniques available at that time, researchers found the whey proteins to be generally similar to the corresponding fractions of blood proteins and were considered to have passed directly from blood to milk. Consequently, the whey proteins attracted little research effort until the 1930s.\n\nIn addition to the caseins and whey proteins, milk contains two other groups of proteinaceous materials, proteose peptones (PPs) and nonprotein nitrogen (NPN), which S. J. Rowland recognized in 1938. Rowland observed that after heating milk at 95 \u00b0C for 10 min, the whey proteins co-precipitated with the caseins on acidification to pH 4.6. When the pH 4.6-soluble fraction of heated milk was made to 12% trichloroacetic acid (TCA), some nitrogenous compounds precipitated which were designated 'proteose peptone'; nitrogenous compounds which remained soluble in 12% TCA were designated nonprotein nitrogen (NPN). A modified version of Rowland's scheme is now used to quantify the principal nitrogenous groups in milk (Aschaffenburg and Drewry, 1959).\n\nThus, by 1938, the complexity of the milk protein system had been described, that is, caseins, lactalbumin, lactoglobulin, PPs, and NPN, which represent approximately 78, 12, 5, 2, and 3%, respectively, of the nitrogen in bovine milk. However, knowledge of the milk protein system was rudimentary and vague at this stage. Advancement of knowledge on the chemistry of milk proteins during the twentieth century can be followed through the progression of textbooks and reviews on dairy chemistry (see Fox 1982; ; ; Fox and McSweeney, 2003; O'Mahony and Fox 2013).\n\n#### Preparation of Casein and Whey Proteins\n\nThe protein fractions may be prepared from whole or skim milk, but skim milk is usually used since the fat is occluded in isoelectric casein and interferes with further characterization of the proteins. The fat is easily removed from milk by centrifugation (e.g., 3000 \u00d7 g for 30 min), and any remaining fat may be removed by washing the precipitated protein with ether. Isoelectric precipitation is the most widely used method for separating the casein and noncasein fractions of milk protein, but several other techniques are used in certain situations (see Fox, 2003; O'Mahony and Fox, 2013):\n\n\u2022 Isoelectric precipitation at\u223c4.6 at 20oC: The precipitate is recovered by filtration or low-speed centrifugation. Essentially similar methods are used to prepare casein on a laboratory or industrial scale.\n\n\u2022 Ultracentrifugation: In milk, the casein exists as large micelles that may be sedimented by centrifugation at 100,000 \u00d7 g for 1 h; the whey proteins are not sedimentable. The casein pellet can be redispersed in a suitable buffer as micelles with properties similar to those of natural micelles.\n\n\u2022 Salting-out methods: casein can be precipitated by any of several salts, usually by (NH4)2SO4 at 260 gL\u20131 or saturated NaCl. The immunoglobulins co-precipitate with the caseins.\n\n\u2022 Ultrafiltration and microfiltration: All the milk proteins are retained by small-pore, semipermeable membranes and separated from lactose and soluble salts. This process, ultrafiltration, is used widely for the industrial-scale production of whey protein concentrates (WPCs) and to a lesser extent for the production of total milk protein. Intermediate-pore membranes are used to separate casein micelles from whey proteins. In microfiltration (MF), using large-pore membranes (1.4 m), both the caseins and whey proteins are permeable, but >99.9% of bacteria and other large particles are retained; MF is used to produce extended shelf-life beverage milk or cheese milk or to remove lipoprotein particles from whey to improve the functionality of WPC.\n\n\u2022 Gel filtration: It is possible to separate the caseins from the whey proteins by permeation chromatography, but this method is not used industrially and rarely on a laboratory scale.\n\n\u2022 Precipitation by ethanol: The caseins are precipitated from milk by \u223c40% ethanol, while the whey proteins remain soluble. However, precipitation by ethanol is rarely used, either on a laboratory or an industrial scale, for the precipitation of casein.\n\n\u2022 Cryo-precipitation: Caseins, in a micellar form, may be destabilized and precipitated by freezing milk or, preferably, concentrated milk, at about \u201310 \u00b0C. Precipitation is caused by a decrease in pH and an increase in [Ca2+]; the precipitated micelles may be redispersed as micelles by heating to about 55oC. Alternatively, the cryo-precipitated casein may be recovered, washed, and dried; it has many interesting properties for food applications, but it is not produced commercially.\n\n\u2022 Rennet coagulation: The casein micelles are destabilized by specific, limited proteolysis and coagulate in the presence of Ca2+. The properties of rennet-coagulated casein are very different from those of isoelectric casein, and it is very suitable for certain food applications, for example, cheese analogues.\n\n\u2022 Caseinates: Isoelectric casein is insoluble in water, but it may be converted to water-soluble caseinates by dispersion in water and adjusting the pH to \u223c6.7 with alkali, usually NaOH, to yield sodium caseinate. KOH, NH4OH, or Ca(OH)2 give the corresponding caseinates which may be freeze-dried or spray-dried.\n\n#### Comparison of Key Properties of Casein and Whey Proteins\n\n\u2022 Solubility at pH 4.6. The caseins are, by definition, insoluble at pH 4.6, whereas the whey proteins are soluble under the ionic conditions of milk. The isoelectric precipitation of casein is exploited in the production of caseins and caseinates, fermented milk products, and acid-coagulated cheeses.\n\n\u2022 Coagulability following limited proteolysis. The caseins are coagulable following specific, limited proteolysis, whereas the whey proteins are not. This property of the caseins is exploited in the production of rennet-coagulated cheese (\u223c75% of all cheese) and rennet casein.\n\n\u2022 Heat stability. The caseins are very heat-stable. Milk at pH 6.7 may be heated at 100 \u00b0C for 24 h without coagulation and withstands heating at 140 \u00b0C for up to 20\u201325 min; aqueous solutions of sodium caseinate may be heated at 140 \u00b0C for several hours without apparent changes. The heat stability of the whey proteins is typical of globular proteins; they are denatured completely on heating at 90 \u00b0C for 10 min. The remarkably high heat stability of the caseins, which is probably due to their lack of typical stable secondary and tertiary structures, permits the production of heat-sterilized dairy products with relatively small physical changes.\n\n\u2022 Amino acid composition. The caseins contain high levels of proline (17% of all residues in \u03b2-casein), which explains their lack of \u03b1- and \u03b2-structures. The caseins are phosphorylated, while the principal whey proteins are not. Whole isoelectric casein contains approximately 0.8% phosphorus, but the degree of phosphorylation varies among the individual caseins. The phosphate is attached to the polypeptides as phosphomonoesters of serine: the presence of phosphate groups has major significance for the properties of the caseins, for example, molecular charge and related properties, such as hydration, solubility, and heat stability, and metal binding which affects their physicochemical, functional, and nutritional properties. Metal binding by casein is regarded as a biological function since it enables a high concentration of calcium phosphate to be carried in milk in a soluble form (to supply the requirements of the neonate). Otherwise, calcium phosphate would precipitate in and block the ducts of the mammary gland, leading to the death of the gland and perhaps of the animal.\n\n\u2022 Sulfur content. The caseins are low in sulfur (0.8%), while the whey proteins are relatively rich (1.7%). The sulfur in casein is mainly in methionine, with little cystine or cysteine; the principal caseins are devoid of the latter two amino acids. The whey proteins are relatively rich in cysteine and\/or cystine, which have major effects on the physicochemical properties of these proteins and of milk.\n\n\u2022 Site of biosynthesis. The caseins are synthesized in the mammary gland and are unique to this organ. Presumably, they are synthesized to meet the amino acid requirements of the neonate and as carriers of important metals required by the neonate. The principal whey proteins are also synthesized in, and are unique to, the mammary gland, but several minor proteins in milk are derived from blood, either by selective transport or due to leakage. Most of the whey proteins have a biological function.\n\n\u2022 Physical state in milk. The whey proteins exist in milk as monomers or as small quaternary structures, while the caseins exist as large aggregates, known as micelles, with a mass of \u223c108 Da and containing about 5000 molecules. The white color of milk is due largely to the scattering of light by the casein micelles. The structure, properties, and stability of the casein micelles are of major significance for the technological properties of milk and have been the subject of intensive research (see below).\n\n#### Heterogeneity and Fractionation of Casein\n\nHammarsten believed that isoelectric casein was a homogeneous protein, but during the early years of the twentieth century, T. B. Osborne and A. J. Wakeman, and especially K. Linderstr\u00f8m-Lang and collaborators, presented evidence that it was heterogeneous (see McMeekin, 1970). By extraction with ethanol-HCl mixtures, K. Linderstr\u00f8m-Lang and S. Kodoma obtained three major casein fractions, which contained about 1.0, 0.6, or 0.1% P, and several minor fractions. The heterogeneity of casein was confirmed by analytical ultracentrifugation and free boundary electrophoresis by Pedersen and Mellander, respectively (see McMeekin, 1970). Electrophoresis resolved isoelectric casein into three proteins, which were named a-, b- and k- in order of decreasing electrophoretic mobility and represented about 75, 22, and 3% of whole casein, respectively.\n\nFollowing the demonstration of its heterogeneity, several attempts were made to isolate the individual caseins. The first reasonably successful method was developed in 1944 by R. C. Warner, who exploited differences in the solubility of \u03b1- and \u03b2-caseins at pH 4.4 and 2 \u00b0C. A much more satisfactory fractionation method was developed in 1952 by N. J. Hipp and co-workers based on the differential solubility of \u03b1-, \u03b2-, and g-caseins in urea solutions at pH 4.9. This method was widely used for many years until the widespread application of ion-exchange (Visser et al., 1986) and reverse-phase chromatography (Visser et al., 1991; Bobe et al. 1998). Reviews describing the application of high-performance and fast protein-liquid chromatography for the analysis of milk and dairy products include Gonzalez-Llano et al. (1990), Strange et al. (1992), and Dupont et al. (2013).\n\nIn 1956, \u03b1-casein was resolved by D. F. Waugh and P. H. von Hippel into Ca-sensitive and Ca-insensitive proteins that were called \u03b1s\\- and \u03ba-caseins, respectively. \u03ba-Casein, which represents \u223c12% of total casein, is responsible for the formation and stabilization of casein micelles and affects many technologically important properties of the milk protein system. Numerous chemical methods were soon developed for the isolation of \u03ba-casein (see Fox, 2003; O'Mahony and Fox 2013). \u03b1s-Casein prepared by the method of Waugh and von Hippel contains two proteins, now called \u03b1s1\\- and \u03b1s2-caseins (Annan and Manson, 1969).\n\nChemical methods for fractionation of the caseins have now been largely superseded by ion-exchange chromatography, which gives superior results when urea and a reducing agent are used (see Strange et al., 1992; Imafidon et al. 1997).\n\n#### Application of Gel Electrophoresis to the Study of Milk Proteins\n\nZone electrophoresis on a solid medium, paper or cellulose acetate, was introduced in the 1940s. This technique gave good results with many protein systems, but the caseins, owing to a very strong tendency to associate hydrophobically, were resolved poorly on these media. Electrophoresis in starch gels (SGE) using discontinuous buffer systems was introduced to general protein chemistry by M. D. Poulik in 1957 and applied to the study of the caseins by R. G. Wake and R. L. Baldwin in 1961. The resolving power of SGE was far superior to that of any of its predecessors. When urea (7 M) and a reducing agent, usually, 2-mercaptoethanol, were incorporated into the starch gel, isoelectric casein was resolved into about 20 bands, most of which are due to the microheterogeneity of one or more of the caseins.\n\nElectrophoresis on polyacrylamide disc gels (PAGE) was introduced by L. Ornstein in 1964 and applied to the study of the caseins by R. F. Peterson in 1966. PAGE and SGE give similar results, but PAGE is far easier to use and has become the standard electrophoretic method for the analysis of caseins (and most other protein systems). Gel electrophoretic methods for the analysis of milk proteins have been reviewed by Swaisgood (1975), Strange et al. (1992), Tremblay et al. (2003), Chevalier et al. (2011a), and Dupont et al. (2013).\n\nSodium dodecyl sulfate (SDS)-PAGE, which resolves proteins mainly on the basis of molecular mass, is very effective for most proteins, but since the masses of the four caseins are quite similar, SDS-PAGE is not very effective. \u03b2-Casein, which has very high surface hydrophobicity, binds a disproportionately high amount of SDS and, consequently, has a higher electrophoretic mobility than \u03b1s1-casein, although it is a larger molecule. SDS-PAGE is very effective for the resolution of whey proteins and is the method of choice.\n\nThe study of proteins (proteomics) has evolved considerably in recent years to incorporate electrophoresis, chromatography, mass spectrometry, and immunoassays (O'Donnell et al., 2004; Manso et al., 2005; Chevalier 2011b; Roncada et al., 2012; Dupont et al. 2013).\n\n#### Microheterogeneity of the Caseins\n\n\u03b1s1-, \u03b1s2-, b-, and k-caseins represent approximately 38, 10, 35, and 12%, respectively, of whole bovine casein. However, SGE or PAGE indicates much greater heterogeneity due to small differences in one or more of the caseins, referred to as microheterogeneity, which arises from five factors which are described in the following sections.\n\n#### Variability in the Degree of Phosphorylation\n\nAll the caseins are phosphorylated but to a variable degree \u03b1s1-, 8 or 9P; \u03b1s2-, 10, 11, 12, or 13P; \u03b2-, 4 or 5P; \u03ba-, 1 or 2P per molecule). The number of phosphate residues is indicated thus: \u03b1s1-CN 8P, \u03b2-CN 5P, and so on. See Chapter 5 for more details.\n\n#### Genetic Polymorphism\n\nIn 1955, R. Aschaffenburg and J. Drewry discovered that \u03b2-lactoglobulin exists in two forms (variants, polymorphs) now called A and B, which differ by only two amino acids. The variant in the milk is genetically controlled, and the phenomenon is called genetic polymorphism. It was soon shown that all milk proteins exhibit genetic polymorphism, and at least 45 polymorphs have been detected by PAGE, which differentiates on the basis of charge, and therefore only polymorphs that differ in charge have been detected (Ng-Kwai-Hang, 2011). It is very likely that only a small proportion of the genetic polymorphs of milk proteins have been detected. The potential of peptide mapping of enzymatic hydrolysates by HPLC-MS has been assessed (Leonil et al., 1995). The genetic polymorph(s) present is indicated by a Latin letter as follows: \u03b2-CN A 5P, \u03b1s1-CN B 9P, \u03ba-CN A 1P, and so on. Genetic polymorphism also occurs in the milk of sheep, goat, buffalo, pig, and horse, and probably of all species.\n\nTechnologically important properties of milk (e.g., rennetability, heat stability, yield, and proportions of milk proteins) are affected by the genetic polymorphs of the milk proteins present, and work in this area is being expanded and refined (Jakob and Puhan, 1992). The extensive literature on the genetic polymorphism of milk proteins has been the subject of several reviews, including Ng-Kwai-Hang and Grosclaude (2003), Ng-Kwai-Hang (2011), and Martin et al. (2013b).\n\n#### Disulfide Bonding\n\n\u03b1s1\\- and \u03b2-caseins lack cysteine and cystine, but both \u03b1s2\\- and \u03ba-caseins contain two 1\/2 cystine residues, which occur as intermolecular disulfide bonds. \u03b1s2-Casein exists as a disulfide-linked dimer, while up to 10 \u03ba-casein molecules may be linked by disulfide bonds. Inclusion of a reducing agent (usually 2-mercaptoethanol) in SGE or PAGE gels is required for good resolution of \u03ba-casein; in its absence, \u03b1s2-casein appears as a dimer (originally called \u03b1s5-casein).\n\n#### Variations in the Degree of Glycosylation\n\n\u03ba-Casein is the only glycosylated casein; it contains galactose, N-acetylgalactosamine, and N-acetylneuraminic (sialic) acid, which occur as tri- or tetrasaccharides, the number of which varies from 0 to 4 per molecule of protein (i.e., a total of nine variants) attached to threonine residues. See Chapter 5 for more details.\n\n#### Hydrolysis of the Caseins by Plasmin\n\nMilk contains several indigenous proteinases, the principal of which is plasmin, a trypsin-like, serine-type proteinase from blood; it is highly specific for peptide bonds with a lysine or arginine at the P1 position. The preferred casein substrates are \u03b2- and \u03b1s2-; \u03b1s1\\- is also hydrolyzed, but \u03ba-casein is very resistant, as are the whey proteins. All the caseins contain several lysine and arginine residues, but only a few bonds are hydrolyzed rapidly. \u03b2-Casein is hydrolyzed rapidly at the bonds Lys28-Lys29, Lys105-His106, and Lys107-Glu108. The resulting C-terminal peptides are the \u03b3-caseins (\u03b31: \u03b2-CNf29\u2013209; \u03b32: \u03b2-CNf106\u2013209; \u03b33: \u03b2-CNf108\u2013209), while the N-terminal peptides are proteose peptones 5, 8slow and 8fast. The \u03b3-caseins, which represent \u223c3% of total caseins, are evident in PAGE gels. Other plasmin-produced peptides are probably present but are either too small to be readily detectable by PAGE or their concentrations are very low relative to the principal caseins.\n\nAlthough \u03b1s2-casein in solution is also quite susceptible to plasmin, \u03b1s2-derived peptides have not been identified in milk. \u03b1s1-Casein in solution is also hydrolyzed readily by plasmin; members of a minor casein fraction, \u03b3-casein, are N-terminal fragments of \u03b1s1-casein produced by plasmin (O'Flaherty, 1997).\n\n#### Molecular Properties of the Milk Proteins\n\nBoth the principal and many of the minor milk proteins have been very well characterized. The principal properties of the six milk-specific proteins are summarized in Table 2.1. A number of features warrant comment.\n\nTable 2.1\n\nProperties of the Principal Lactoproteins\n\n| Caseins | \u03b2-Lg | \u03b1-La  \n---|---|---|---  \nProperties | \u03b1s1 | \u03b1s2 | \u03b2- | \u03ba- |  |   \nMW | 23612 | 25228 | 23980 | 19005 | 18362 | 14174  \nResidues | 199 | 207 | 209 | 169 | 162 | 123  \nConc in milk (g\/L) | 12-15 | 3-4 | 9-11 | 2-4 | 3.0 | 0.7  \nPhosphate residues | 8-9 | 10-13 | 4-5 | 1-2 | 0 | 0  \n\u00bd Cystine | 0 | 2 | 0 | 2 | 5 | 8  \nSugars | 0 | 0 | 0 | Yes | 0 | 0  \nProlyl residue per molecule | 17 | 10 | 35 | 20 | 8 | 2  \nA280, 1% 1 cm | 10.1 | 11.1 | 4.6 | 9.6 | 9.4 | 20.1  \nSecondary structure | Low | Low | Low | Low | High | High  \nH\u03d5ave | 4.89 | 4.64 | 5.58 | 5.12 | 5.03 | 4.68  \npI | 4.96 | 5.27 | 5.20 | 5.54 | 5.2 | 4.2-4.5  \nPartial specific volume (ml\/g) | 0.728 | 0.720 | 0.741 | 0.734 | 0.751 | 0.735\n\nThe six principal lactoproteins are small molecules, a feature that contributes to their stability. The primary structure of the principal lactoproteins of several species is known, as are the substitutions in the principal genetic variants (Martin et al., 2013b).\n\nThe whey proteins are highly structured, but the four caseins lack stable secondary structures. Classical physical measurements indicate that the caseins are unstructured, but theoretical considerations indicate that rather than being unstructured, the caseins are very flexible molecules and have been referred to as rheomorphic (Holt and Sawyer, 1993; see also Horne, 2002; Farrell et al. 2006a). The inability of the caseins to form stable structures is due mainly to their high content of the structure-breaking amino acid, proline; \u03b2-casein is particularly rich in proline, with 35 of the 209 residues. The open, flexible structure of the caseins renders them very susceptible to proteolysis, which facilitates their natural function as a source of amino acids. In contrast, the native whey proteins, especially \u03b2-Lg, are quite resistant to proteolysis, and at least some are excreted in the feces of infants. This feature is important since most of the whey proteins play a non-nutritional function in the intestine, and, therefore, resistance to proteolysis is important.\n\nThe caseins are generally regarded as very hydrophobic proteins but, with the exception of \u03b2-casein, they are not exceptionally hydrophobic. Owing to their lack of stable secondary and tertiary structures, most of their hydrophobic residues are exposed. Consequently, they have a high surface hydrophobicity. One of the more notable features of the amino acid sequence of the caseins is that the hydrophobic and hydrophilic residues are not distributed uniformly, thereby giving the caseins a distinctly amphiphatic structure (Huppertz, 2013). This feature, coupled with their open flexible structure, gives the caseins good surface activity, as well as good foaming and emulsifying properties, making casein the functional protein of choice for many food applications. Owing to their hydrophobic sequences, the caseins have a propensity to yield bitter hydrolysates. Also, due to their open structure, the caseins have a high specific volume. Consequently, they form highly viscous solutions, which is a disadvantage in the production of caseinates. Owing to its high viscosity, it is not possible to spray-dry sodium caseinate solutions containing >20% protein, thereby increasing the cost of drying and resulting in low-bulk density powders.\n\nThe lack of stable tertiary structures means that the caseins are not denaturable sensu stricto. Therefore, they are extremely heat stable; sodium caseinate, at pH 7, can withstand heating at 140 \u00b0C for several hours without visible change. This very high heat stability makes it possible to produce heat-sterilized dairy products with very little change in physical appearance; other major food systems undergo major physical changes upon severe heating.\n\nThe caseins have a very strong tendency to associate due mainly to hydrophobic bonding. Even in sodium caseinate, the most soluble form of casein, the molecules form aggregates of 250\u2013500 kDa, that is, containing 10\u201320 molecules. This strong tendency to associate makes it difficult to fractionate the caseins, for which a dissociating agent (e.g., urea or SDS) is required. On the other hand, a tendency to associate is important for some functional applications and in the formation and stabilization of casein micelles. In contrast, the whey proteins are molecularly dispersed in solution.\n\nOwing to their high content of phosphate groups, which occur in clusters, \u03b1s1-, \u03b1s2-, and \u03b2-caseins have a strong tendency to bind metal ions, which in the case of milk are mainly Ca2+. This property has many major consequences; the most important from a technological viewpoint is that these three proteins, which represent approximately 85% of total caseins, are insoluble at Ca concentrations >\u223c6 mM at temperatures >20 \u00b0C. Since bovine milk contains \u223c30 mM Ca, one would expect the caseins to precipitate under the conditions prevailing in milk. However, \u03ba-casein, which contains only one organic phosphate group, binds Ca weakly and is soluble at all Ca concentrations found in dairy products. Furthermore, when mixed with the Ca-sensitive caseins, \u03ba-casein can stabilize and protect up to \u223c10 times its mass of the Ca-sensitive caseins by forming large colloidal particles called casein micelles. The micelles act as carriers of inorganic elements, especially Ca and P, but also Mg and Zn, and are, therefore, very important from a nutritional viewpoint. Through the formation of micelles, it is possible to solubilize much higher levels of Ca and PO4 than would otherwise be possible.\n\n#### Nomenclature of Milk Proteins\n\nDuring the period of greatest activity on the fractionation of casein (1950\u20131970), several casein (and whey protein) fractions were prepared that were either similar to proteins already isolated and named, or were artifacts of the isolation procedure. In order to standardize the nomenclature of the milk proteins, the American Dairy Science Association established a Nomenclature Committee in 1955, which has published seven reports, the most recent of which is by Farrell et al. (2004). In addition to standardizing the nomenclature of the milk proteins, the characteristics of the principal milk proteins are summarized in these articles.\n\n#### Whey Proteins\n\nAbout 20% of the total proteins of bovine milk are whey (serum) proteins. The total whey protein fraction is prepared by any of the methods described for the preparation by casein. That is, the proteins that are soluble at pH 4.6 or in saturated NaCl or after rennet-induced coagulation of the caseins, are permeable on microfiltration, or not sedimented by ultracentrifugation.\n\nThe proteins prepared by these methods differ somewhat: Acid whey contains PPs; immunoglobulins are co-precipitated with the caseins by saturated NaCl; rennet whey contains the macropeptides, produced from \u03ba-casein by rennet, plus small amounts of casein; and small casein micelles remain in the ultracentrifugal serum.\n\nOn a commercial scale, whey protein-rich products are prepared by:\n\n\u2022 ultrafiltration\/diafiltration of acid casein or rennet whey to remove varying amounts of lactose and spray-dried to produce whey protein concentrates (30\u201385% protein)\n\n\u2022 ion-exchange chromatography and spray-dried to yield whey protein isolate, containing \u223c95% protein,\n\n\u2022 demineralization by electrodialysis or ion exchange, thermal evaporation of water, and crystallization of lactose, and\n\n\u2022 thermal denaturation, removal of precipitated protein by filtration\/centrifugation and spray-drying, to yield lactalbumin, which has very low solubility and poor functionality.\n\n#### Fractionation of Whey Proteins\n\nIt was recognized early that acid whey contains two well-defined groups of proteins: lactalbumins, which are soluble in 50% saturated (NH4)2SO4 or saturated MgSO4; and lactoglobulins, which are salted-out under these conditions. The lactoglobulin fraction contains mainly immunoglobulins. The lactalbumin fraction contains two principal proteins, \u03b2-lactoglobulin and a-lactalbumin, and several minor proteins, including blood serum albumin and lactoferrin, which have been isolated by various procedures and crystallized (see Imafidon et al., 1997; Fox, 2003; O'Mahony and Fox 2013).\n\nThere is considerable interest in the production of the major and many minor whey proteins on a commercial scale for nutritional, nutraceutical, or functional applications. Several methods have been developed for the industrial-scale production of several whey proteins (see Mulvihill and Ennis, 2003).\n\n#### Major Characteristics of Whey Proteins\n\n##### \u03b2-Lactoglobulin\n\n\u03b2-Lactoglobulin (\u03b2-Lg) represents \u223c50% of the whey proteins, \u223c12% of the total proteins, in bovine milk. It is a typical globular protein that has been characterized very well. The extensive literature has been reviewed, among others, by Sawyer (2003; 2013); and Creamer et al. (2011).\n\n\u03b2-Lg is the principal whey protein in the milk of cattle, buffalo, sheep, and goat, although there are slight interspecies differences. Initially, it was considered that \u03b2-Lg occurs only in the milk of ruminants, but it is now known that a similar protein occurs in the milk of many other species, including the sow, mare, kangaroo, dolphin, and manatee. However, \u03b2-Lg does not occur in the milk of human, rat, mouse, guinea pig, camel, llama, or alpaca, in which \u03b1-La is the principal whey protein.\n\nBovine \u03b2-Lg consists of 162 amino acid residues per monomer, with an MW of \u223c18 kDa; its amino acid sequence and that of several other species have been established. Its isoelectric point is \u223cpH 5.2. It contains two intramolecular disulfide bonds and one mol of cysteine per monomer. The cysteine is especially important since it reacts, following thermal denaturation, with the intermolecular disulfide of \u03ba-casein and significantly affects the rennet coagulation and heat stability of milk. It is also responsible for the cooked flavor of heated milk. Some \u03b2-Lgs (e.g., porcine) lack a sulfydryl group. Twelve genetic variants of bovine \u03b2-Lg have been identified, the most common being A and B. Genetic polymorphism also occurs in \u03b2-Lg of other species.\n\n\u03b2-Lg is a highly structured protein: in the pH range 2\u20136, 10\u201315% of the molecule exists as \u03b1-helices, 43% as \u03b2-sheets, and 47% as unordered structures, including \u03b2-turns; the \u03b2-sheets occur in a \u03b2-barrel-type calyx. The molecule has a very compact globular structure. Each monomer exists almost as a sphere, measures about 3.6 nm in diameter, and exists as a dimer, MW \u223c36 kDa, in the pH range 5.5\u20137.5, as a monomer <pH 3.5 and >pH 7.5, and as a tetramer (MW, \u223c144 kDa) in the pH range 3.5\u20135.5. Porcine and other \u03b2-Lgs that lack a free thiol do not form dimers, a property that is probably not due to the absence of a thiol group.\n\n\u03b2-Lg is very resistant to proteolysis in its native state; this feature suggests that its primary function is not nutritional. It may have either or both of two biological roles:\n\n\u2022 It binds retinol (vitamin A) in a hydrophobic pocket, protects it from oxidation, and transports it through the stomach to the small intestine where the retinol is transferred to a retinol-binding protein, which has a similar structure to \u03b2-Lg. It is not clear how retinol is transferred from the core of the fat globules, where it occurs in milk, to \u03b2-Lg and why some species lack this protein. \u03b2-Lg can bind many hydrophobic molecules, and hence its ability to bind retinol may be incidental. \u03b2-Lg is a member of the lipocalin family, all of which have binding properties (Akerstrom et al., 2000).\n\n\u2022 Through its ability to bind fatty acids, \u03b2-Lg stimulates lipase activity, which may be its most important physiological function.\n\n\u03b2-Lg is the most allergenic protein in bovine milk for human infants, and there is interest in producing whey protein products free of \u03b2-Lg for use in infant formula. \u03b2-Lg has very good thermogelling properties and determines the gelation of whey protein concentrates (WPCs).\n\n##### \u03b1-Lactalbumin\n\nAbout 20% of the protein of bovine whey (3.5% of total milk protein) is \u03b1-lactalbumin (\u03b1-La), which is the principal protein in human milk. It is a small protein containing 123 amino acid residues, with a mass of \u223c14 kDa, which has been well characterized. The literature has been reviewed, among others, by McKenzie and White (1991) and Brew (2003; ; 2013).\n\n\u03b1-La contains four tryptophan residues per mole, giving it a specific absorbance at 280 nm of 20. It contains four intramolecular disulfide bonds per mole but no cysteine, phosphate, or carbohydrate. Its isoionic point is \u223cpH 4.8. The milk of Bos taurus breeds contains only one genetic variant of \u03b1-La, B, but Zebu cattle produce two variants, A and B. \u03b1-La has been isolated from the milk of cattle, sheep, goat, sow, human, buffalo, rat, guinea pig, horse, and many other species; there are minor interspecies differences in its composition and properties.\n\nThe primary structure of \u03b1-La is homologous with lysozyme. Out of a total of 123 amino acid residues in \u03b1-La, 54 are identical to corresponding residues in chicken egg white lysozyme and 23 others are structurally similar. \u03b1-La is a compact, highly structured globular protein. The tertiary structure of \u03b1-La is similar to that of lysozyme (McKenzie and White, 1991) and x-ray crystallography-based analysis of apo- and holo- bovine \u03b1-La have been published by Chrysina et al. (2000). In evolutionary terms, lysozyme is a very ancient protein; it is believed that \u03b1-La evolved from it through gene duplication (see Nitta and Sugai, 1989).\n\nAs discussed earlier, \u03b1-La is a component of lactose synthetase, the enzyme that catalyses the final step in the biosynthesis of lactose. There is a direct correlation between the concentrations of \u03b1-La and lactose in milk. The milk of the California sea lion or the hooded seal contains no \u03b1-La.\n\n\u03b1-La is a metalloprotein containing two Ca2+ per molecule in a pocket containing four Asp residues. The Ca-containing protein is the most heat-stable of the principal whey proteins, or, more correctly, the protein renatures following heat denaturation, which occurs at a relatively low temperature, as indicated by differential scanning calorimetry. When the pH is reduced to <\u223cpH 5, the Asp residues become protonated and lose their ability to bind Ca2+. The calcium-free protein is denatured at quite a low temperature and does not renature on cooling; this characteristic has been exploited to isolate \u03b1-La from whey. \u03b1-La has poor thermogelling properties. Most lysozymes do not bind a Ca2, but equine lysozyme is an exception (Nitta et al., 1987) and seems to be an intermediate in the evolution of lysozyme to \u03b1-La (see Nitta and Sugai, 1989).\n\n\u03b1-La is synthesized in the mammary gland, but a very low level is transferred, probably via leaky mammocyte junctions, into blood serum, in which the concentration of \u03b1-La increases during pregnancy or following administration of steroid hormones to male or female animals (Akers, 2000). The concentration of \u03b1-La in blood serum is a reliable, noninvasive indicator of mammary gland development and of the potential of an animal for milk production.\n\nA high MW form of \u03b1-La, isolated recently from human acid-precipitated casein, has anticarcinogenic activity; it was named HAMLET (human \u03b1-La made lethal to tumor cells). Bovine \u03b1-La can be converted to a form with similar activity, called BAMLET; it is the molten globular state formed from apo-\u03b1-La and cis\u03949-octadecenoic acid (see Svensson et al., 2000; Chatterton et al. 2006; and Pettersson et al., 2006). Both forms are reported to have comparable cytotoxic activity against three different cancer cell lines (Brinkmann et al., 2011).\n\n#### Blood Serum Albumin\n\nNormal bovine milk contains 0.1\u20130.4 gL\u20131 of blood serum albumin (BSA; 0.3\u20131.0% of total nitrogen), presumably as a result of leakage from blood; it has no known biological function in milk. BSA has been studied extensively; for reviews see Fox (2003) and O'Mahony and Fox (2013). Owing to its low concentration in milk, BSA has little effect on the physicochemical properties of WPC and whey protein isolate (WPI).\n\n#### Immunoglobulins\n\nMature bovine milk contains 0.6\u20131 g immunoglobulins (Ig)L\u20131 (\u223c3% of total nitrogen), but colostrum contains \u224810% (w\/v) Ig, the level of which decreases rapidly postpartum. IgG1 is the principal Ig in bovine, buffalo, caprine, or ovine milk, with lesser amounts of IgG2, IgA, and IgM; IgA is the principal Ig in human milk. The cow, sheep, and goat do not transfer Ig to the fetus in utero, and the neonate is born without Ig in its blood. Consequently, it is very susceptible to bacterial infection with a very high risk of mortality. The young of these species can absorb Ig from the intestine for several days after birth and thereby acquire passive immunity until they synthesize their own Ig, within a few weeks of birth. The human mother transfers Ig in utero, and the offspring is born with a broad spectrum of antibodies. Although the human baby cannot absorb Ig from the intestine, the ingestion of colostrum is still very important because its Igs prevent intestinal infection. Some species, such as the horse, transfer Ig both in utero and via colostrum (see Hurley, 2003; Marnila and Korhonen 2011; Hurley and Thiel, 2013).\n\nIt has been suggested that the neonate secretes chymosin rather than pepsin because chymosin is weakly proteolytic and does not inactivate Ig. Colostrum contains \u03b12-macroglobulin, which may inhibit proteinases in the GIT and protect Ig.\n\nThe modern dairy cow produces colostrum far in excess of what its calf requires; surplus colostrum is available for the recovery of Ig and other nutraceuticals (Pakkanen and Aalto, 1997). Some work has been done on hyperimmunizing cows against certain human pathogens, such as rota virus, for the production of antibody-rich milk for human consumption, especially by infants. Interest in this approach dates back to the 1950s, when L.M. Spolverini suggested using bovine colostrum in the diet of infants to confer protection against shared human and bovine diseases (Campbell and Petersen, 1959). Milk powder manufactured from 'immune milk' is commercially available in several markets (e.g., Stolle Milk Biologics Inc., Cincinnati, Ohio).\n\n#### Whey Acidic Protein\n\nWhey acidic protein (WAP) was first identified in the milk of mouse and has since also been found in the milk of rat, rabbit, pig, camel, wallaby, possum, echidna, and platypus. Since the milk of all of these species lacks \u03b2-Lg, it was thought that these proteins were mutually exclusive. However, porcine milk, which contains \u03b2-Lg, was recently found to contain WAP also (see Saidi et al., 2007). The MW of WAP is 14\u201330 kDa (the variation may be due to differences in glycosylation), and it contains two (in eutherians) or three (in monotremes and marsupials) 4-disulfide domains. Since human milk lacks \u03b2-Lg, it might be expected to contain WAP, but there are no reports to this effect. In humans and ruminants, the WAP gene is frame-shifted and is a pseudogene. WAP functions as a proteinase inhibitor, is involved in terminal differentiation in the mammary gland, and has antibacterial activity (for reviews see Simpson and Nicholas, 2002; Hajjoubi et al., 2006; Martin et al., 2011).\n\n#### Proteose Peptone 3\n\nThe proteose-peptone (PP) fraction of milk protein is a very complex mixture of peptides, most of which are produced by the action of indigenous plasmin (see above), but some are indigenous to milk. The fraction has been only partially characterized; the current status has been described by Fox (2003) and O'Mahony and Fox (2013). PP fractions 5, 8slow, and 8fast, have little or no technological significance; proteose peptone 3 (PP3) has several interesting technological functionalities.\n\nBovine PP3 is a heat-stable phosphoglycoprotein that was first identified in the PP (heat-stable, acid-soluble) fraction of milk. Unlike the other peptides in this fraction, PP3 is an indigenous milk protein, synthesized in the mammary gland. Bovine PP3 consists of 135 amino acid residues, with five phosphorylation and three glycosylation sites. When isolated from milk, the PP3 fraction contains at least three components of MW \u224828, 18, and 11 kDa; the largest of these is PP3, while the smaller components are fragments thereof generated by plasmin (see Girardet and Linden, 1996). PP3 is present mainly in acid whey, but some is also found in the MFGM. Girardet and Linden (1996) proposed changing the name to lactophorin or lactoglycophorin; it has also been referred to as the hydophobic fraction of proteose peptone.\n\nOwing to its strong surfactant properties (Campagna et al., 1998), PP3 can prevent contact between milk lipase and its substrates, thus preventing spontaneous lipolysis. Its emulsifying properties have also been evaluated in dairy products such as ice cream and recombined dairy cream (Vanderghem et al., 2007; Innocente et al. 2011). Although its amino acid composition suggests that PP3 is not a hydrophobic protein, it behaves hydrophobically, possibly owing to the formation of an amphiphilic \u03b1-helix, one side of which contains hydrophilic residues while the other side is hydrophobic. The biological role of PP3 is unknown.\n\n#### Minor Proteins\n\nMilk contains several proteins at very low or trace levels, many of which are biologically active (see Schrezenmeir et al., 2000; Wynn and Sheehy 2013). Some are regarded as highly significant and have attracted considerable attention as nutraceuticals. When ways of increasing the value of milk proteins are discussed, the focus is usually on these minor proteins, but they are, in fact, of little economic value to the overall dairy industry. They are found mainly in the whey, but some are also located in the fat globule membrane. Reviews on the minor proteins include Fox and Flynn (1992), Haggarty (2003), Fox and Kelly (2003), Wynn et al. (2011), O'Mahony and Fox (2013), and Wynn and Sheehy (2013).\n\n#### Metal-binding Proteins\n\nMilk contains several metal-binding proteins: the caseins (Ca, Mg, Zn), \u03b1-La (Ca), xanthine oxidase (Mo, Fe), alkaline phosphatase (Zn, Mg), lactoperoxidase (Fe), catalase (Fe), ceruloplasmin (Cu), glutathione peroxidase (Se), lactoferrin (Fe), and transferrin (Fe).\n\nLactoferrin (Lf), a non-hem iron-binding glycoprotein, is a member of a family of iron-binding proteins, which includes transferrin and ovotransferrin (conalbumin) (see Lonnerdal, 2003; Lonnerdal and Suzuki, 2013). It is present in several body fluids, including saliva, tears, sweat, and semen. Lf has several potential biological functions: It improves the bioavailability of Fe, is bacteriostatic (by sequestering Fe and making it unavailable to intestinal bacteria), and has antioxidant, antiviral, anti-inflammatory, immunomodulatory, and anticarcinogenic activity. Human milk contains a very high level of Lf (\u224820% of total N), and therefore there is interest in fortifying bovine milk-based infant formula with Lf. The pI of Lf is \u22489.0; that is, it is cationic at the pH of milk, whereas most milk proteins are anionic, and it can be isolated on an industrial scale by adsorption on a cation exchange resin. Hydrolysis of Lf by pepsin yields peptides called lactoferricins, which are more bacteriostatic than Lf, and their activity is independent of iron status. Milk also contains a low level of serum transferrin.\n\nMilk contains a copper-binding glycoprotein, ceruloplasmin, also known as ferroxidase (EC 1.16.3.1). Ceruloplasmin is an \u03b12-globulin with a MW of \u223c126 kDa; it binds six atoms of copper per molecule and may play a role in delivering essential copper to the neonate.\n\nGlutathione peroxidase (GTPase) is a Se-containing protein. It has been reported that milk contains GTPase and that it binds 30% of the total Se in milk (see Fox and Kelly, 2006b; O'Mahony et al., 2013). GTPase has no known enzymatic function in milk and the activity attributed to GTPase in milk may be due to sulfhydryl oxidase (Stagsted, 2006).\n\n#### \u03b22-Microglobulin\n\n\u03b22-Microglobulin, initially called lactollin, was first isolated from bovine acid-precipitated casein by M. L. Groves in 1963. Lactollin, reported to have an MW of 43 kDa, is a tetramer of \u03b22-microglobulin, which consists of 98 amino acids, with a calculated MW of 11,636 Da. \u03b22-Microglobulin, a component of the immune system, is probably produced by proteolysis of a larger protein, mainly within the mammary gland; it has no known significance in milk.\n\n#### Osteopontin\n\nOsteopontin (OPN) is a highly phosphorylated acidic glycoprotein, consisting of 261 amino acid residues with a calculated MW of 29,283 Da (total MW of the glycoprotein, \u224860,000 Da). OPN has 50 potential calcium-binding sites, about half of which are saturated under normal physiological concentrations of calcium and magnesium.\n\nOPN occurs in bone (it is one of the major non-collagenous proteins in bone), in many other normal and malignant tissues, in milk and urine, and can bind to many cell types. It is believed to have a diverse range of functions (Bayless et al., 1997), but its role in milk is not clear. A rapid method for the isolation of OPN from milk was reported by Azuma et al. (2006) who showed that it binds Lf, lactoperoxidase, and Igs and may serve as a carrier for these proteins. An acidic whey protein fraction, which contains OPN, reduces bone loss in ovariectomized rats (Kruger et al., 2006; Wynn et al., 2011; O'Mahony and Fox, 2013; Wynn and Sheehy 2013).\n\n#### Vitamin-binding Proteins\n\nMilk contains binding proteins for at least the following vitamins: retinol (vitamin A, i.e., \u03b2-lactoglobulin), biotin, folic acid, and cobalamine (vitamin B12). The precise role of these proteins is not clear, but they probably improve the absorption of vitamins from the intestine or act as antibacterial agents by rendering vitamins unavailable to bacteria. The concentration of these proteins varies during lactation, but the influence of other factors such as individuality, breed, and nutritional status is not known. The activity of these proteins is reduced or destroyed on heating at temperatures somewhat higher than high temperature short time (HTST) pasteurization (see Wynn et al., 2011; Wynn and Sheehy 2013).\n\n#### Angiogenins\n\nAngiogenins induce the growth of new blood vessels, that is, angiogenesis. They have high sequence homology with members of the RNase A superfamily of proteins and have RNase activity. Two angiogenins (ANG-1 and ANG-2) have been identified in bovine milk and blood serum; both strongly promote the growth of new blood vessels in a chicken membrane assay. The function(s) of the angiogenins in milk is unknown. They may be part of a repair system to protect either the mammary gland or the intestine of the neonate and\/or part of the host-defense system (see Wynn et al., 2011; Wynn and Sheehy 2013).\n\n#### Kininogen\n\nTwo kininogens have been identified in bovine milk: a high (88\u2013129 kDa, depending on the level of glycosylation) and a low (16\u201317 kDa) MW form. Bradykinin, a biologically active peptide containing nine amino acids which is released from the high MW kininogen by the action of the enzyme, kallikrein, has been detected in the mammary gland, and is secreted into milk, from which it has been isolated. Plasma kininogen is an inhibitor of thiol proteinases and has an important role in blood coagulation. Bradykinin affects smooth muscle contraction and reduces hypertension. The biological significance of bradykinin and kininogen in milk is unknown (see Wynn et al., 2011; Wynn and Sheehy 2013).\n\n#### Glycoproteins\n\nMany of the minor proteins discussed above are glycoproteins; in addition, several other minor glycoproteins have been found in milk and especially in colostrum, the functions of which have not been elucidated. One of the high MW glycoproteins in bovine milk is prosaposin, a neurotrophic factor that plays an important role in the development, repair, and maintenance of nervous tissue. It is a precursor of saposins A, B, C, and D, which have not been detected in milk. The physiological role of prosaposin in milk is not known, although saposin C, released by digestion, could be important for the growth and development of the young (Patton et al., 1997; Patton, 1999; Campana et al. 1999).\n\n#### Proteins in the Milk Fat Globule Membrane\n\nAbout 1% of the total protein in milk is in the milk fat globule membrane (MFGM). Most of the proteins are present at trace levels, including many of the indigenous enzymes in milk. The principal proteins in the MFGM include mucins, adipophilin, butyrophilin, and xanthine oxidoreductase (see section on milk lipids).\n\n#### Growth Factors\n\nMilk contains many peptide hormones, including epidermal growth factor, insulin, insulin-like growth factors 1 and 2, three human growth factors (\u03b11, \u03b12, and \u03b2), two mammary-derived growth factors (I and II), colony-stimulating factor, nerve growth factor, platelet-derived growth factor, and bombasin. It is not clear whether these factors play a role in the development of the neonate or in the development and functioning of the mammary gland, or both (see Fox and Flynn, 1992; Gauthier et al., 2006; Wynn and Sheehy 2013).\n\n#### Indigenous Milk Enzymes\n\nMilk contains about 70 indigenous enzymes, which are minor but very important members of the milk protein system (see ; O'Mahony et al., 2013). The enzymes originate from the secretory cells or the blood; many are concentrated in the MFGM and originate in the Golgi membranes of the cell or the cell cytoplasm, some of which becomes entrapped as crescents inside the encircling membrane during exocytosis. Plasmin and lipoprotein lipase are associated with the casein micelles, and several enzymes are present in the milk serum; many of the enzymes are derived from the MFGM that is shed as the milk ages.\n\nThe indigenous enzymes are significant for several reasons:\n\n\u2022 Deterioration of product quality: plasmin, lipoprotein lipase, acid phosphatase, xanthine oxidoreductase.\n\n\u2022 Bactericidal agents: lactoperoxidase and lysozyme.\n\n\u2022 Indices of the thermal history of milk: alkaline phosphatase, \u03b3-glutamyltransferase, or lactoperoxidase.\n\n\u2022 Indices of mastitic infection: catalase, acid phosphatase, and especially N-acetylglucosaminidase.\n\nThe concentration\/activity of indigenous enzymes in milk shows greater interspecies variability than any other constituent (e.g., 3000-fold greater concentration of lysozyme in equine and human milk than in bovine milk). Lactoperoxidase is a major enzyme in bovine milk but is absent from human milk. Human milk and the milk of a few other species contain bile salt-stimulated lipase, but the milk of most species lacks this enzyme. The principal lipase in milk is lipoprotein lipase, of which there is \u223c500 times as much in guinea pig milk as in rat milk. Bovine milk has a high level of xanthine oxidoreductase activity (XOR), but all other milks that have been studied have low XOR activity because the protein lacks Mo (XOR plays a major role in the excretion of fat globules from the mammocyte, but in this function it does not act as an enzyme). (see milk lipids section). The reasons for these interspecies differences are not known, but some of them may be significant.\n\n#### Biologically Active Cryptic Peptides\n\nOne of the most exciting recent developments in milk proteins is the discovery that all milk proteins contain sequences that have biological\/physiological activities when released by proteolysis. The best studied are phosphopeptides, angiotensin-converting enzyme inhibitory peptides, platelet-modifying peptides, opiate peptides, immunomodulating peptides, and the caseinomacropeptides which have many biological properties (see FitzGerald and Meisel, 2003; Korhonen, 2006; Korhonen and Pihlanto, 2006; Mills et al. 2011).\n\n#### Nonprotein Nitrogen\n\nThe nonprotein nitrogen (NPN) fraction of milk contains those nitrogenous compounds soluble in 12% TCA; it represents \u223c5% of total nitrogen (\u223c300 mgL\u20131). The principal components are urea, creatine, uric acid, and amino acids. Human milk contains a high level of taurine, which can be converted to cysteine and may be nutritionally important for infants. Urea, the concentration of which varies considerably, has a significant effect on the heat stability of milk (Walstra and Jenness, 1984). The amino acids present in the NPN fraction also support the growth of lactic acid bacteria.\n\n#### Casein Micelles\n\nIt has been known since the work of H. Schuler in 1818 that the casein in milk exists as large particles, now called casein micelles. The stability of the micelles is critically important for many of the technologically important properties of milk and consequently has been the focus of much research, especially since the discovery of \u03ba-casein in 1956 by Waugh and von Hipple (1956). Early views and research on casein micelles were reviewed by Fox and Brodkorb (2008), and a recent review is that of McMahon and Oommen (2013). Although views on the detailed structure of the casein micelle are divided, there is widespread, or unanimous, agreement on their general structure and properties.\n\nElectron microscopy shows that casein micelles are spherical with a diameter in the range 50\u2013500 nm (average \u223c120 nm) and a mass ranging from 106 to 3 \u00d7 109 Da (average \u223c108 Da) for bovine micelles. There are numerous small micelles, but these represent only a small proportion of the mass. There are 1014\u20131016 micelles mL\u20131 in milk, and they are roughly two micelle diameters (\u223c250 nm) apart. The dry matter of the micelles is \u223c94% protein and 6% low-molecular-mass species, referred to collectively as colloidal calcium phosphate (CCP), and consisting mainly of calcium phosphate with some magnesium and citrate and trace amounts of other species. The micelles bind \u223c2.0 g H2O g\u20131 protein. They scatter light, and the white color of milk is due largely to light scattering by the casein micelles. The white color is lost if the micelles are disrupted, by dissolving CCP with citrate, EDTA, or oxalate, by increasing pH, or by urea (>5 M) or ethanol (\u223c35% at 70 \u00b0C).\n\n#### Stability of Casein Micelles\n\nThe micelles are quite stable to the principal processes to which milk is normally subjected. They are very stable at high temperatures, and they withstand heating at 140 \u00b0C for 15\u201320 min at pH 6.7. Coagulation is caused by heat-induced changes, for example, a decrease in pH due to the pyrolysis of lactose to acids, dephosphorylation of casein, cleavage of the carbohydrate-rich moiety of \u03ba-casein, denaturation of the whey proteins and their precipitation on the casein micelles, and precipitation of soluble calcium phosphate on the micelles (see O'Connell and Fox, 2003).\n\nThe micelles are stable to compaction (e.g., they can be sedimented by ultracentrifugation and redispersed by mild agitation), to commercial homogenization, and to Ca2+ up to at least 200 mM at temperatures up to 50 \u00b0C. The effects of high pressure (up to 800 MPa) on the casein micelles in bovine, ovine, caprine, and buffalo milk have been studied; the size of the micelles increases up to 2\u2013300 MPa but decreases at higher pressure (see Huppertz et al., 2006; Huppertz and de Kruif 2007).\n\nAs the pH of milk is reduced, CCP dissolves and is fully soluble at \u2264pH 4.9. Acidification of cold (4 \u00b0C) milk to pH 4.6, followed by dialysis against bulk milk, is a convenient technique for altering the CCP content of milk. If acidified cold milk is readjusted to pH 6.7, the micelles re-form, provided that the pH had not been reduced below 5.5 (Lucey et al., 1996). This result seems to suggest that most of the CCP can be dissolved without destroying the structure of the micelles.\n\nSome proteinases, especially chymosin, catalyze a very specific hydrolysis of \u03ba-casein, as a result of which the casein coagulates in the presence of Ca2+ or other divalent ions. This is the key step in the manufacture of most cheese varieties. The proteinase preparations used for cheesemaking are called rennets.\n\nAt room temperature, the casein micelles are destabilized by \u223c40% ethanol at pH 6.7 or by lower concentrations if the pH is reduced (Horne, 2003a). However, if the system is heated to \u226570 \u00b0C, the precipitate redissolves, and the system becomes translucent. When the system is recooled, the white appearance of milk is restored, and a gel is formed if the ethanol\u2013milk mixture is held at 4 \u00b0C, especially if concentrated milk is used. If the ethanol is removed by evaporation, very large aggregates (average diameter, \u223c3000 nm) are formed. The dissociating effect of ethanol is promoted by increasing the pH (35% ethanol causes dissociation at 20 \u00b0C and pH 7.3) or by adding NaCl (Horne, 2003b). Methanol and acetone have an effect similar to ethanol, but propanol causes dissociation \u223c25 \u00b0C. The mechanism by which ethanol dissociates casein micelles has not been established, but it is not due to the solution of CCP, which is unchanged (O'Connell et al., 2003).\n\nThe micelles are also reversibly dissociated by urea (5 M) (McGann and Fox, 1974; Holt, 1998), SDS (Lefebvre-Cases et al., 1998), or by raising the pH to >9. Under these conditions, the CCP is not dissolved.\n\nThe micelles are destabilized by freezing (cryodestabilization) due to a decrease in pH and an increase in the Ca2+ in the unfrozen phase of milk; concentrated milk is very susceptibile to cryodestabilization (Moon et al. 1988; ). Cryodestabilized casein can be dispersed by warming the thawed milk to 55oC to give particles with micelle-like properties.\n\n#### Micelle Structure\n\nSince the beginning of the twentieth century, there has been speculation on how the casein particles (micelles) are stabilized (see Fox and Brodkorb, 2008), but no significant progress was possible until Waugh and von Hippel (1956) achieved the isolation and characterization of \u03ba-casein. The first attempt to describe the structure of the casein micelle was made by Waugh in 1958, and since then numerous models have been made and refined. Progress has been reviewed regularly; recent reviews include de Kruif and Holt (2003), Horne (2002; 2003a; 2003b; 2006), Farrell et al. (2006b), and McMahon and Oommen (2013). The principal features that any micelle model must meet are the following: \u03ba-casein, which represents \u223c12\u201315% of total casein, must be located so as to be able to stabilize the calcium-sensitive \u03b1s1-, \u03b1s2-, and \u03b2-caseins, which represent approximately 85% of total casein; chymosin and other rennets, which are relatively large molecules (MW\u223c 35 kDa), very rapidly and specifically hydrolyze most of the \u03ba-casein; when heated in the presence of whey proteins, \u03ba-casein, and \u03b2-lactoglobulin (MW \u223c36 kDa) interact to form a disulfide-linked complex that modifies the rennet and heat coagulation properties of the micelles. The arrangement that would best explain these features is a surface layer of \u03ba-casein surrounding the Ca-sensitive caseins, somewhat analogous to a lipid emulsion in which the triglycerides are surrounded by a thin layer of emulsifier. Most models of the casein micelle propose a surface location for \u03ba-casein, but some early models envisaged \u03ba-casein serving as nodes in the interior of the micelle.\n\nRemoval of CCP causes disintegration of the micelles into particles of MW \u223c106 Da, suggesting that the casein molecules are held together in the micelles by CCP. The properties of the CCP-free system are very different from those of normal milk (e.g., it is precipitated by relatively low levels of Ca2+, it is more stable to heat-induced coagulation, and it is not coagulable by rennets). Many of these properties can be restored, at least partially, by an increased concentration of calcium. However, CCP is not the only integrating factor, as indicated by the dissociating effect of urea, SDS, ethanol, or alkaline pH. At low temperatures, casein, especially \u03b2-casein, dissociates from the micelles.\n\nThere has been strong support for the view, first proposed by C. V. Morr in 1967, that the micelles are composed of submicelles (\u223c106 Da and 10\u201315 nm in diameter) linked together by CCP, giving a micelle with an open porous structure. On removing CCP (by acidification\/dialysis, EDTA, citrate, or oxalate) the micelles disintegrate. Disintegration may also be achieved by treatment with urea, SDS, 35% ethanol at 70 \u00b0C, or pH >9. These reagents do not solubilize CCP, suggesting that hydrophobic interactions and hydrogen bonds contribute to micelle structure. Much of the evidence for a submicellar structure relies on electron microscopy studies that appear to show variations in electron density, a raspberry-like structure, which was interpreted as indicating submicelles.\n\nViews on the proposed structure of the submicelles have evolved over the years (see McMahon and McManus, 1998; Dalgleish et al., 2004; Dalgleish, 2011; McMahon and Oommen 2008; ). Proposals have included the following: A rosette-type structure similar to that of a classical soap micelle, in which the polar regions of \u03b1s1-, \u03b1s2-, and \u03b2-caseins are oriented toward the outside of the submicelle to reduce electrostatic repulsion between neighboring charged groups; each submicelle was considered to be surrounded by a layer (coat) of \u03ba-casein, thus providing a \u03ba-casein coat for the entire micelle. Several authors have suggested that the submicelles are not covered completely by \u03ba-casein and that there are \u03ba-casein-rich, hydrophilic, and \u03ba-deficient, hydrophobic, regions on the surface of each submicelle. The latter aggregate via the hydrophobic patches such that the entire micelle has a \u03ba-casein-rich surface layer but also with some of the other caseins on the surface. In a popular version of this model, it was proposed that the hydrophilic C-terminal region of \u03ba-casein protrudes from the surface, forming a layer 5\u201310 nm thick and giving the micelles a hairy appearance. This hairy layer, functioning as an ionic brush, is considered to be responsible for micelle stability through major contributions to zeta potential (\u201320 mV) and steric stabilization. If the hairy layer is removed through specific hydrolysis of \u03ba-casein or collapsed (e.g., by ethanol), the colloidal stability of the micelles is destroyed, and they aggregate.\n\nA further variant of the subunit model envisages two main types of subunits\u2014one consisting of \u03b1s1-, \u03b1s2-, and \u03b2-caseins, which are present in the core of the micelle, and the other consisting of \u03b1s1\\- and \u03b1s2\\- and \u03ba-caseins, which forms a surface layer. It has also been proposed that \u03b2-casein associates to form thread-like structures with which \u03b1s1\\- and \u03b1s2-caseins associate hydrophobically to form the core of the micelle or submicelles which are surrounded by a layer of \u03ba-casein; CCP cements neighboring submicelles within the micelle. A recent study by Bouchoux et al. (2010), using SAXS analysis, supports a submicelle structure.\n\nAlthough the submicelle model of the casein micelle explains many of the principal features of, and physicochemical reactions undergone by, the micelles, and has been supported widely, it has never enjoyed unanimous support. Indeed, new electron microscopy techniques have cast doubts on the authenticity of submicelles. Using cryopreparation electron microscopy with stereo-imaging, McMahon and McManus (1998) found no evidence to support the submicellar model and concluded that if the micelles do consist of submicelles, these must be smaller than 2 nm or less densely packed than previously presumed. Like other forms of electron microscopy, field emission scanning electron microscopy showed that casein micelles have an irregular surface, but Dalgleish et al. (2004) concluded that the caseins form tubular structures rather than spherical submicelles. In principle, this model seems basically similar to earlier subunit models. McMahon and Oommen (2008) also found no evidence for a submicellar structure using high-resolution transmission electron microscopy.\n\nAlternative models have been proposed. Visser (1992) proposed that the micelles are spherical conglomerates of randomly aggregated casein molecules held together by amorphous calcium phosphate and hydrophobic bonds, with a surface layer of \u03ba-casein. Holt (1992) considered the casein micelle to be a tangled web of flexible casein molecules forming a gel-like structure in which microgranules of CCP are an integral feature and from the surface of which the C-terminal region of \u03ba-casein extends, forming a hairy layer. In what he referred to as the dual binding model, Horne (1998; ; 2003a; 2003b; 2006); described how casein molecules interact hydrophobically and through calcium phosphate nanoclusters to form micelles. These three models retain the key features of the submicellar model, that is, the cementing role of CCP and the predominantly surface location and micelle-stabilizing role of \u03ba-casein, and differ from it mainly with respect to the internal structure of the micelle.\n\n#### Interspecies Comparison of Milk Proteins\n\nThe protein content of milk varies widely depending on species, ranging from \u223c1% to \u223c20%. The protein content reflects the growth rate of the neonate of the species, that is, its requirements for essential amino acids. The milk of all species for which data are available contain two groups of protein, caseins and whey proteins, but the ratio of these varies widely. Both groups show genus- and even species-specific characteristics that presumably reflect some unique nutritional or physiological requirements of the neonate of the species. Interestingly, and perhaps significantly, of the milks that have been characterized, human and bovine milks are more or less at opposite ends of the spectrum. Among the general interspecies comparisons of milk proteins are Woodward (1976), Jenness (1973; ; 1982), Ginger and Grigor (1999), and Martin et al. (2003); reviews on milk proteins of individual species include buffalo (Addeo et al., 1977), goat (Trujillo et al. 1997; ), sheep (Amigo et al., 2000), camel (Kappeler et al., 1998; Ochirkhuyag et al., 1997), yak (Ochirkhuyag et al., 1997), horse (Ochirkhuyag et al., 2000; Park et al., 2006; Uniacke-Lowe et al., 2010; Uniacke-Lowe and Fox 2011), and sow (Gallagher et al., 1997).\n\nThere is considerably more and better information on the interspecies comparison of individual milk proteins than on overall milk composition, probably because only one sample of milk from one animal is sufficient to yield a particular protein for characterization. The two principal milk-specific whey proteins, \u03b1-La and \u03b2-Lg, from quite a wide range of species have been characterized and, in general, show a high degree of homology (see Brew 2003; ; Sawyer 2003; ). The caseins show much greater interspecies diversity, especially in the \u03b1-casein fraction; most of the species that have been studied contain a protein with an electrophoretic mobility similar to that of bovine \u03b2-casein (see Fig. 2.2), but the \u03b2-caseins that have been sequenced show a low level of homology (Martin et al. 2003; 2013a). Sheep's milk is used mainly for cheese production, with small amounts used for the production of fermented milks; hence the coagulation and gel-forming properties of ovine milk are particularly important. The \u03b1s1-casein of caprine milk is very heterogeneous. Not only do the properties of the variants differ, but the concentration of \u03b1s1-casein varies from 0 to 26% of total casein, and consequently, the total protein content varies considerably. This, in turn, has major effects on the rennet-induced coagulation properties of ovine and caprine milk as well as on the yield and quality of cheese produced therefrom (Amigo et al., 2000; Clark and Sherbon 2000a; 2000b; Martin et al., 2013a; 2013b).\n\nFigure 2.2 Urea polyacrylamide gel electrophoretogram of milk from 15 species. Lanes: 1, Bovine; 2, Camel; 3, Equine; 4, Asinine; 5, Human; 6, Rhinoceros; 7, Caprine; 8, Ovine; 9, Asian elephant; 10, African elephant; 11, Vervet monkey; 12, Macaque monkey; 13, Rat; 14, Canine; 15, Porcine (from Uniacke-Lowe, unpublished data).\n\nHuman \u03b2-casein occurs in multiphosphorylated forms (0\u20135 mol P per mol protein; see Atkinson and Lonnerdal, 1989), as does mare's \u03b2-casein (Ochirkhuyag et al., 2000; Girardet et al., 2006; Uniacke-Lowe et al. 2013). Considering the critical role played by \u03ba-casein, it would be expected that all casein systems contain this protein, but Ochirkhuyag et al. (2000) failed to identify \u03ba-casein in mare's milk and suggested that the micelle-stabilizing role was played by \u03b2-casein with zero or a low level of phosphorylation. More recent work has shown that equine milk does contain a low level of \u03ba-casein (Iametti et al., 2001; Egito et al., 2002; Uniacke-Lowe et al. 2013). Human \u03ba-casein is very highly glycosylated, containing 40\u201360% carbohydrate (compared with \u223c10% in bovine \u03ba-casein). The \u03b1s-casein fraction differs markedly between species; human milk probably lacks an \u03b1s-casein (Darragh and Lonnerdal, 2011), while the \u03b1-casein fractions in horse and donkey milk are very heterogeneous. The urea-PAGE gels of milk from 15 species are shown in Figure 2.2.\n\nThere are considerable interspecies differences in the minor proteins of milk. The milk of those species that have been studied in sufficient depth contain approximately the same profile of minor proteins, but there are very marked quantitative differences. Most of the minor proteins in milk have some biochemical or physiological function, and the quantitative interspecies differences presumably reflect the requirements of the neonate of the species. The greatest interspecies differences, in some cases 4000-fold, seem to occur in the indigenous enzymes (Fox and Kelly, 2006a,b).\n\nIn the milk of all species, the caseins exist as micelles (at least the milks appear white), but the properties of the micelles in the milk of only a few nonbovine species have been studied: caprine (Ono and Creamer, 1986; Ono et al., 1989); ovine (Ono et al., 1989); buffalo (Patel and Mistry, 1997); camel (Attia et al., 2000); mare (Welsch et al., 1988; Ono et al. 1989, Uniacke-Lowe, 2011); and human (Sood et al. 1997; ). The appearance and size of casein micelles in the milk of 19 species, guinea pig, rat, nutria (coypu), dog, cat, gray seal, rabbit, donkey, horse, alpaca, dromedary camel, cow, red deer, sheep, pig, water buffalo, goat, porpoise, and humans, were studied by Buchheim et al. (1989). The structures of all micelles appeared similar on electron microscopy, but there were large interspecies differences in size: Human micelles were smallest (64 nm) while those of the alpaca, goat, camel, and donkey were very large (300\u2013350 nm); the micelles in equine milk were as large as 700 nm (Uniacke-Lowe, 2011).\n\n#### Milk Salts\n\nWhen milk is heated at 500oC for \u223c5 h, an ash derived mainly from the inorganic salts of milk and representing \u223c0.7 %, w\/w, of bovine milk, remains. However, the elements in the ash are changed from their original form to oxides or carbonates, and the ash contains P and S derived from caseins, lipids, sugar phosphates, or high-energy phosphates. The organic salts, the most important of which is citrate, are oxidized and lost during ashing; some volatile metals, such as sodium, are partially lost. Thus, ash does not accurately represent the salts of milk. However, the principal inorganic and organic ions in milk can be determined directly by potentiometric, spectrophotometric, or other methods. The typical concentrations of the principal elements, the macroelements, are shown in Table 2.2; considerable variability occurs, due, in part, to poor analytical methods and\/or to samples from cows in very early or late lactation or suffering from mastitis.\n\nTable 2.2\n\nConcentration and Distribution of the Principal Milk Salts\n\nSpecies | Concentration (mg\/l) | % Soluble | Form | % Colloidal  \n---|---|---|---|---  \nSodium | 500 | 92 | Completely ionized | 8  \nPotassium | 1,450 | 92 | Completely ionized | 8  \nChloride | 1,200 | 100 | Completely ionized | -  \nSulfate | 100 | 100 | Completely ionized | -  \nPhosfate | 750 | 43 | 10% bound to Ca and Mg  \n54% H2PO4\u2212   \n36% HPO42\u2212 | 57  \nCitrate | 1,750 | 94 | 85% bound to Ca and Mg (undissociated)  \n15% Citr3\u2212 | 6  \nCalcium | 1,200 | 34 | 35% Ca2+   \n55% bound to citrate  \n10% bound to phosphate | 66  \nMagnesium | 130 | 67 | Probably similar to calcium | 33\n\nMilk also contains 20\u201325 elements at very low or trace levels. These microelements are very important from a nutritional viewpoint; many, notably Zn, Fe, Mo, Cu, Ca, Se, and Mg, are present in enzymes, many of which are concentrated in the MFGM; some microelements, such as Fe and Cu, are very potent lipid pro-oxidants. Although the salts are relatively minor constituents of milk, they are critically important for many technological and nutritional properties of milk.\n\nSome of the salts in milk are fully soluble, but others, especially calcium phosphate, exceed their solubility under the conditions in milk and occur partly in the colloidal state, associated with the casein micelles. These salts are referred to as colloidal calcium phosphate (CCP), although some magnesium, citrate, and traces of other elements are also present in the micelles. As discussed earlier, CCP plays a critical role in the structure and stability of the casein micelles. The typical distribution of the principal organic and inorganic ions between the soluble and colloidal phases of bovine milk is summarized in Table 2.2. The actual form of the principal species can be determined or calculated after making certain assumptions. Typical values are shown in the table.\n\nThe solubility and ionization status of many of the principal ionic species are interrelated, especially H+, Ca2+, PO43\u2013 and citrate3\u2013. These relationships have major effects on the stability of the caseinate system and, consequently, on the processing properties of milk. The status of various species in milk can be modified by adding certain salts to milk; for example, Ca2+ is reduced by adding PO43\u2013 or citrate3\u2013. Addition of CaCl2 affects the distribution and ionization status of calcium and phosphate and the pH of milk.\n\nThe precise nature and structure of CCP are uncertain (Choi et al., 2011). CCP is associated with the caseins, probably via the casein phosphate residues. It probably exists as nanocrystals, which include PO4 residues of casein. The simplest stoichiometry is Ca3(PO4)2, but spectroscopic data suggest that CaHPO4 is the most likely form.\n\nThe distribution of species between the soluble and colloidal phases is strongly affected by pH and temperature. As the pH is reduced, CCP dissolves and is completely soluble <Ca. pH 4.9; the reverse occurs when the pH is increased. The solubility of calcium phosphate decreases as the temperature is increased, and soluble calcium phosphate is transferred to the colloidal phase, with the release of H+ and a decrease in pH:\n\nCaHP O 4 \/ Ca ( H 2 P O 4 ) 2 \u2194 C a 3 ( P O 4 ) 2 + 3 H +\n\nThese changes are quite substantial but are at least partially reversible upon cooling. Since milk is supersaturated with calcium phosphate, concentration of milk by evaporation of water increases the degree of supersaturation and the transfer of soluble calcium phosphate to the colloidal state, with the concomitant release of H+. Dilution has the opposite effect.\n\nMilk salts equilibria are also shifted on freezing; as pure water freezes, the concentrations of solutes in unfrozen liquid are increased. Soluble calcium phosphate precipitates as Ca3(PO4)2, releasing H+ (the pH of the unfrozen liquid may decrease to 5.8). The crystallization of lactose as a monohydrate increases the degree of supersaturation by reducing the amount of solvent water.\n\nThere are substantial changes in the concentrations of the macroelements in milk during lactation, especially at the beginning and end of lactation and during mastitic infection. Changes in the concentration of some of the salts in milk, especially calcium phosphate and citrate, have major effects on the physicochemical properties of the casein system and on the processability of milk, especially rennet coagulability and related properties and heat stability.\n\nReviews on the chemistry of milk salts include Pyne (1962), Holt (1985; ; 2003), Holt et al. (1996; 1998), Holt and Carver (2012), Lucey and Fox (1993), Lucey and Horne (2009), and Choi et al. (2011). Reviews on the nutritional significance of milk salts include Flynn and Power (1985), Flynn and Cashman (1997), Cashman (2003a,b; ), and Hunt and Nielsen (2009).\n\n### Vitamins\n\nMilk contains all the vitamins in sufficient quantities to enable normal maintenance and growth of the neonate. Bovine milk is a very significant source of vitamins, especially biotin (B7), riboflavin (B2), and cobalamine (B12), in the adult human diet. For general information on the vitamins and for specific aspects in relation to milk and dairy products, including stability during processing and storage, the reader is referred to a set of articles in Roginski et al. (2003) and Fuquay et al. (2011) and those by Morrissey and Hill (2009) and Nohr and Biesalski (2009).\n\nIn addition to their nutritional significance, four vitamins are significant for other reasons: vitamin A (retinol) and carotenoids are responsible for the yellow-orange color of fat-containing products made from cow's milk; vitamin E (tocopherols) is a potent antioxidant; vitamin C (ascorbic acid) is an antioxidant or pro-oxidant, depending on its concentration; and vitamin B2 (riboflavin), which is greenish-yellow, is responsible for the color of whey or UF permeate, it co-crystallizes with lactose and is responsible for its yellowish color, which may be removed by recrystallization or bleached by oxidation, and acts as a photocatalyst in the development of light-oxidized flavor in milk, which is due to the oxidation of methionine.\n\n### Water\n\nWater is the principal constituent in the milk of most species. In addition to meeting the requirement of the neonate for water, the water in milk serves as solvent for milk salts, lactose, and proteins, and affects their properties and stability. It controls the rate of many reactions, for example, Maillard browning, lipid oxidation, enzyme activity, and microbial growth, thus affecting the stability of milk and milk products. The preservation of milk products is effected by reducing the water activity by dehydration or by adding sugar or NaCl. The stability and quality of most dairy products are strongly affected by their water content, and small differences can cause major instability problems.\n\nThe chemistry of water, its measurement, and its significance in foods in general and in dairy products, will not be discussed in this text. For that information, the reader is referred to Duckworth (1975), Rockland and Beuchat (1987), Fennema (1996), Roos (1997; 2011), Franks (2000), Chieh (2006), and Simatos et al. (2009; 2011).\n\n#### Physical Properties of Milk\n\nThe physical properties of milk are important from several points of view, including appearance, quality, analytical, and processing characteristics. A summary of the key physical properties of milk is provided in Table 2.3, and further information can be found in McCarthy and Singh (2009) and McCarthy (2011).\n\nTable 2.3\n\nSummary of the Principal Physical Properties of Milk\n\nPhysical property | Typical value  \n---|---  \npH1 | 6.5\u20136.7  \nTitratable acidity | 0.14\u20130.16%  \nDensity2 | 1030 kg m\u22123  \nSpecific heat capacity | 3880\u20134000 Jkg\u22121K\u22121  \nThermal conductivity2 | 0.548 Wm\u22121K\u22121  \nThermal diffusivity2 | 1.34 times 10-7m2s\u22121  \nSurface tension2 | 52 mNm\u22121  \nElectrical conductivity3 | 0.460 Sm\u22121  \nDielectric permittivity3 | 67.9  \nDielectric loss | 17.6  \nRefractive index2 | 1.3440\u20131.3485  \nFreezing point | \u22120.512 to \u22120.550 \u00b0C  \nOsmolality4 | 0.285 osmolkg\u22121\n\n1 Measured at 25 \u00b0C\n\n2 Measured at 20 \u00b0C\n\n3 Relative to value for free space\n\n4 Assuming freezing point of \u22120.53 \u00b0C\n\n#### Microbiology of Milk\n\nDue to its high water content, almost neutral pH, and gross chemical composition, milk is a good growth medium for many microorganisms. Milk from a healthy cow is secreted free of microorganisms, but it can quickly become contaminated with microorganisms of varied origin (e.g., air, human intervention, soil, bedding, milking equipment) (Hassan and Frank, 2011). The microorganisms typically found in milk can be divided into three broad groups\u2014animal pathogens and toxin producers, spoilage microorganisms, and microorganisms used to produce fermented products. Milk-handling conditions (e.g., temperature and duration of storage) and thermal processing parameters need to be designed to eliminate pathogenic microorganisms and reduce or control the growth of spoilage microorganisms in milk and dairy products made therefrom (Tamime, 2009).\n\n## Summary\n\nMilk is a very complex fluid. It contains several hundred molecular species, mostly at trace levels. Most of the microconstituents are derived from blood or mammary tissue, but most of the macroconstituents are synthesized in the mammary gland and are milk-specific. The constituents of milk are in true aqueous solution (e.g., lactose and most inorganic salts) or as a colloidal solution (proteins, which may be present as individual molecules or as large aggregates of several thousand molecules, called micelles) or as an emulsion (lipids). The macroconstituents can be readily fractionated and are used widely as food ingredients. The natural function of milk is to supply the neonate with its complete nutritional requirements for a period (sometimes several months) after birth and with many physiologically important molecules, including carrier proteins, protective proteins, and hormones.\n\nThe properties of milk lipids and proteins may be readily modified by biological, biochemical, chemical, or physical means and thus may be converted into novel dairy products. In this overview, the chemical and physicochemical properties of milk sugar (lactose), lipids, proteins, and inorganic salts were discussed. 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Milk proteins, including growth factors .  In:  McSweeney PLH ,  Fox PF ,  eds.  _Advanced Dairy Chemistry, Volume 1A: Proteins: Basic Aspects_ .  4th ed. New York :  Springer ;  2013 : 317 \u2013 335 .\n\nWynn PC ,  Morgan AJ ,  Sheehy PA ,  Minor proteins, bovine serum albumin, vitamin-binding proteins .  Fuquay JW ,  Fox PF ,  McSweeney PLH ,  eds.  _Encyclopedia of Dairy Sciences_ ,  Volume 3 .  2nd ed. Oxford :  Academic Press ;  2011 : 795 \u2013 800 .\n\nZadow JG .  Lactose: Properties and uses .  _J. Dairy Sci_.  1984 ;67 : 2654 \u2013 2679 .\n\nZadow JG .  _Lactose and Whey Processing_ .  London :  Elsevier Applied Science ;  1992 .\n\nZanabria Eyzaguirre R ,  Corredig M ,  Buttermilk and milk fat globule membrane fractions .  Fuquay JW ,  Fox PF ,  McSweeney PLH ,  eds.  _Encyclopedia of Dairy Sciences_ ,  Volume 3 .  2nd ed. Oxford :  Academic Press ;  2011 : 691 \u2013 697 . \nChapter 3\n\n# The Comparative Genomics of Monotremes, Marsupials, and Pinnipeds: Models to Examine the Functions of Milk Proteins\n\nJulie A. Sharp*\n\nAshalyn Watt*\n\nSwathi Bisana*\n\nVengama Modepalli*\n\nStephen Wanyonyi*\n\nAmit Kumar*\n\nJoly Kwek**\n\nRod Collins*\n\nChristophe Lef\u00e8vre*, ***\n\nKevin R. Nicholas*\n\n* School of Medicine, Deakin University, Geelong, Victoria, Australia  \n** Australian Regenerative Medicine Institute, Monash University, Clayton, Victoria, Australia  \n*** Victorian Bioinformatics Consortium, Monash University, Clayton, Victoria, Australia\n\n## Abstract\n\nMilk has many functions, ranging from the provision of factors crucial to the operation of the mammary gland and the development of the suckled young through to their protection from infection. Lactation evolved about 200 million years ago with the aplacental, egg-laying monotremes, but since that time there has been extensive adaptation to reproduction, including a large repertoire of lactation strategies. This chapter discusses three animal models with extreme adaptation to lactation, and examines the option of exploiting their comparative biology to identify milk protein bioactives that may have potential in functional foods or pharmaceuticals. The echidna (Tachyglossus aculeatus, a monotreme) has a fascinating combination of reptilian and mammalian characteristics. It retains a primitive component of reptilian reproduction in the form of laying shelled eggs, but it also has a prototherian lactation process. The tammar wallaby (Macropus eugenii, a marsupial) has adopted a reproductive strategy that includes a short gestation (26.5 days), birth of an immature young, and a relatively long lactation (300 days). The composition of milk changes progressively during the lactation cycle, and these changes in milk composition subsequently control development of the young. The tammar can also practice concurrent asynchronous lactation; the mother provides a concentrated milk for an older animal that is out of the pouch, and a dilute milk from an adjacent mammary gland for a newborn pouch young. The third study species, the Cape fur seal (Arctocephalus pusilluspusillus, a eutherian), has a lactation characterized by a repeated cycle of long at-sea foraging trips (up to 23 days) alternating with short suckling periods of 2\u20133 days ashore. Lactation almost ceases while the seal is offshore, but the mammary gland does not progress to involution and apoptosis.Technology platforms using genomics, proteomics, and bioinformatics have been used to exploit these models to identify milk bioactives. In addition, the availability of sequenced marsupial, dog, platypus, and bovine genomes permits rapid transfer of information to the cow to provide outcomes for the dairy industry.\n\n## Keywords\n\nExtreme lactation, milk composition, comparative genomics, bioactive, growth and development\n\nOutline\n\nIntroduction 76\n\nThe echidna (Tachyglossus aculeatus) 78\n\nMonotremes: An Animal Model to Discover Antimicrobials 79\n\nThe tammar wallaby (Macropus eugenii) 82\n\nMilk Protein Gene Expression; Naturally Occurring Gene Knockouts Used to Assess Protein Function 82\n\nChanges in Milk Composition Regulate Growth of the Tammar Pouch Young 84\n\nIdentification of Milk bioactives in Tammar Milk Using a Functional Genomics Platform 85\n\nTemporal Delivery of Domain-specific Bioactive Proteins in Wallaby Milk 85\n\nCathelicidins\u2014Domain-specific Delivery of Bioactivity 85\n\nWhey Four-disulfide Core-domain- containing Proteins WAP and WAP Four-disulfide Core Domain Protein 2 88\n\nWFDC2 88\n\nWAP 90\n\nS100A19\u2014A New Member of the S100 Family of Proteins 93\n\nA role for milk in the control of mammary function 95\n\nThe fur seal 96\n\nTranscriptomic Analysis of the Lactating Cape Fur Seal Mammary Gland during Suckling and Foraging 97\n\nAbsence of Local Apoptotic Milk Factors in Fur Seal Milk 99\n\nNew player in milk bioactives; microRNA 103\n\nConclusions 104\n\n## Introduction\n\nThe composition of milk includes all the factors required to provide appropriate nutrition for growth of the neonate. However, it is now clear that milk has many functions and also consists of bioactive molecules that play a central role in regulating developmental processes in the young and providing a protective function for both the suckled young and the mammary gland during milk production (Trott et al., 2003; Waite et al., 2005; Kwek et al., 2009b). Identifying these bioactives and their physiological function in eutherians can be difficult and requires extensive screening of milk components that may function to meet the demand for foods that improve wellbeing and options for prevention and treatment of disease.\n\nIn addition, human targets for milk bioactives are focused on specific and changing health needs during the progression from infancy to old age (Sharp et al., 2009). For example, the need to support growth and development in early life differs from the maintenance of muscle, eyes, and brain function in the elderly. Health issues such as obesity, diabetes, hypertension, and cancer increase with an aging population. Therefore, functional foods that improve both general nutrition and well-being, and also target prevention and treatment of health problems are very attractive. New animal models with unique reproductive strategies are now becoming increasingly relevant in the search for these factors. This review focuses on three animal models that provide new insights into how the lactation cycle is regulated, and when combined with technology platforms that include genomics, proteomics and bioinformatics can be exploited to identify milk proteins with bioactivity and subsequent application as nutraceuticals and pharmaceuticals.\n\nThe lactation cycle is common to all mammals, although monotremes, marsupials, and some pinnipeds have evolved a reproductive strategy that is very different from that of most eutherians. The evolution of lactation has led to new opportunities to better understand the functionality of milk. The first split in mammalian phylogeny resulted in the Prototheria approximately 200 million years ago and produced the monotreme lineages. Australia is the only continent on which the monotremes, platypus (Ornithorhynchus anatinus), and echidnas (Tachyglossus and Zaglosus genera) are still found. The Theria split into the Metatheria (Marsupialia) and Eutheria (Placentalia) lineages over 150 million years ago. These animal models have an extreme adaptation to lactation and have increased our understanding of the function of milk proteins by revealing mechanisms that are present but often not readily apparent in many eutherian mammals (Brennan et al., 2007; Sharp et al., 2009). Eutherians have a long gestation relative to their lactation period, and the composition of the milk doesn't change substantially. This contrasts with reproduction and lactation in the other two classes of mammals, the monotremes (such as platypus and echidna) and the marsupials (such as the tammar wallaby, Macropus eugenii).\n\nMonotremes do not have a placenta, and they lay eggs that are incubated outside the mother's body. This is followed by a prototherian lactation process, marking them as representatives of early mammals. Apart from a short weaning period (Morrow and Nicol, 2013), milk is the sole source of nutrition and protection for the hatchlings which are altricial and immunologically na\u00efve (Griffiths, 1978). Since much of the development of the monotreme young occurs in the nonsterile, ex-utero environment, the role of milk is of importance in the growth, development, and protection of the young from disease (Bisana et al., 2013). Reproduction in marsupials is characterized by a short gestation followed by a long lactation period, and all the major milk constituents change substantially and progressively during lactation (Tyndale-Biscoe and Janssens, 1988; Nicholas, 1988a,b). The tammar neonate is altricial and remains attached to the teat for the first 100 days of lactation, and it may be considered a fetus maintained in the pouch as opposed to the uterus. The conversion of milk to body mass is very similar to the conversion of precursors to body mass observed in the eutherian fetus in utero (Tyndale-Biscoe and Janssens, 1988). Therefore, examination of the milk provides a unique opportunity to identify specific molecules that play a primary role in regulating the development of the young. The lactation cycle of the Cape fur seal (Arctocephalus pusillus pusillus) consists of maternal foraging and infant nursing that are spatially and temporally separate (Bonner, 1984). Lactating mothers suckle offspring over a period of 4\u201312 months, and females alternate between short periods ashore suckling their young with longer periods of up to 3\u20134 weeks foraging at sea (Gentry and Holt, 1986). Fur seals provide new opportunities to examine a potential role for milk in protecting the mammary gland from cell death and infection while the lactating animal is foraging, and to examine a function for milk in the growth and protection of the pup that remains on shore.\n\nTransmitting passive immunity through milk is a key factor for survival of the suckled young. Understanding the relationship between the developmental delay of the immune system in the newborn marsupial and the role of the mammary gland in providing immunity for the young to survive in a pathogenic environment provides new opportunities to identify molecules with a role in innate immunity. The transmission of passive immunity is very important until the active immune system has matured in the neonate (Bailey et al., 2001; Butler, 1998; Sinkora et al., 2003; Brambell 1970). This process is defined by the reproduction strategies of mammals and includes the type of placenta and gestation period. The placental system has huge diversity both structurally and physiologically, and fetal\/maternal exchanges have evolved from a primitive form of yolk sac placenta in marsupials to well-developed chorioallantoic placenta in eutherians (Carter et al., 2004). Marsupials have a placenta, but the fetal and maternal interaction is very limited, with minimal transfer of immunity to the embryo. In contrast, in eutherians the embryo is implanted even before the germ layers are formed in the blastocyst and the interactions between the fetus and the mother are more developed (Tyndale-Biscoe and Janssens, 1988). This has extended to support of the fetus for longer periods in utero. This diversity among the mammals in acquiring passive immunity prior to birth has led to wide adaptations in postnatal life in order to survive. If the newborn is unable to acquire any immune protection prior to birth, the mammary gland plays a substantial role in providing the immune protection either through colostrum or mature milk. Therefore, it is likely that the mammary gland has evolved prior to the placenta as monotremes are aplacental (Goldman, 2002) and developed a mammary gland by modifying the epidermal gland on the ventral part of the abdomen to provide secretions to support the offspring (Daniel, 1993).\n\n## The echidna (Tachyglossus aculeatus)\n\nThe monotremes occupy an important position in the mammalian tree of evolution because of their egg-laying mode of reproduction, which is a transition between that of reptiles and therian mammals (Lef\u00e8vre et al., 2010). The extant monotremes, the platypus and echidna, are confined to Australia and New Guinea and show a mixture of specialized adaptations and plesiomorphic features (Bininda-Emonds et al., 2007; Nicol 2003). The short-beaked echidna (Tachyglossus aculeatus) is more widely distributed, occupying a diverse range of habitats from coastal areas to above the snow lines (Griffiths, 1978). The maternal care provided by the mother echidna from fertilization until weaning can be divided into four phases (Fig. 3.1). The 20\u201324 days of gestation forms phase 1 and ends when the egg is laid into the pouch. The mammary glands of the mother hypertrophy and form the margins of the rudimentary pouch during the last two weeks of gestation and shortly before laying the egg (Griffiths, 1978). The egg is incubated inside the pouch for about 10.5 days (phase 2). Following hatching, the young is carried in the pouch and is able to suckle continuously (phase 3). Unlike other mammals, the platypus does not have a defined nipple, and the young consumes milk from a milk pad (Griffiths et al., 1988). In phase 4, the young is evicted from the pouch and sequestered in a nursery burrow, where the mother returns to feed it at intervals of 5\u201310 days (Griffiths et al., 1988; Rismiller and McKelvey 2003). The duration of lactation is about 200 days and the young leave the burrow after 180 to 205 days. Key differences in maternal care during the first weeks of lactation have been observed between the geographical subspecies of echidnas. Weaning in the Tasmanian populations of echidnas is reported not to be abrupt as there is a period when the young echidnas begin exploratory foraging while their mothers are still lactating (Morrow and Nicol, 2013).\n\nFigure 3.1 Stages of echidna development.\n\nAlthough the composition of monotreme milk has not been extensively studied, some of the main components have been described, such as caseins (Lef\u00e8vre et al., 2009) and whey proteins including alpha-lactalbumin (Messer et al., 1997; Shaw et al. 1993), lysozyme (Guss et al., 1997), and whey acidic protein (WAP) (Sharp et al., 2007b). Platypus milk is reported to contain more solids than most eutherian milk, as well as having a high iron content (Griffiths et al., 1984; Grant 2007). The composition of milk secreted by the echidna mother changes over time. At the time of birth, the milk is dilute and contains 1.25% fat, 7.85% protein, and 2.85% carbohydrates and minerals. Mature milk is more concentrated, with 31.0% fat, 12.4% protein, and 2.8% carbohydrate (Augee et al., 2006). Near weaning, protein levels continue to increase, which is thought to be due to the need for keratin synthesis for hair and spines.\n\nIn addition to providing general nutrition, monotreme milk is a potential source to discover new antimicrobials. The protective properties of milk in a range of mammals have been described, including the antiparasitic activity of human milk lipase (Gillin et al., 1983), antifungal activity of bovine milk lactoferrin (Manzoni et al., 2012), and bacteriostatic activity of murine milk WAP (Iwamori et al., 2010). Egg laying, incubation, and hatching of monotreme eggs, and the subsequent development of the monotreme young occurs in nonsterile ex-utero environments, which requires that milk provide both the source of nutrition and antibacterial protection (Griffiths, 1978). It has often been speculated that during reproduction in monotremes, the protolacteal secretions enhanced survival of eggs and the young by virtue of their antimicrobial properties (Oftedal 2002; ; Blackburn et al., 1989; Hayssen and Blackburn 1985).\n\n### Monotremes: An Animal Model to Discover Antimicrobials\n\nA recent study using genomics and proteomics platforms identified a novel, monotreme-specific antibacterial milk protein (Bisana et al., 2013). Since the monotremes are protected species and access to their tissue samples is limited, a noninvasive approach for analysis of lactation was employed (Lef\u00e8vre et al., 2009). Echidna milk cells were harvested from milk collected during peak lactation, and cDNA sequencing identified several casein and whey protein genes expressed at high levels, indicating that monotreme milk cells harvested during lactation were enriched for mammary epithelial cells (Fig. 3.2). A novel, monotreme-specific transcript named EchAMP was identified during this analysis (Bisana et al., 2013). An exploration of EchAMP expression relative to GAPDH in different tissues of echidna using Reverse transcription polymerase chain reaction (RT-PCR) further confirmed a high level of expression in milk cells and a low level of expression in other tissues.\n\nFigure 3.2 Echidna milk cell transcriptome. Relative abundance of the EchAMP transcript in echidna milk cells. A novel sequence (EchAMP) was one of the most highly expressed transcripts.\n\nIn silico analyses indicated that the cognate protein of EchAMP would be hydrophilic and secretory in nature. This was confirmed through identification of peptides corresponding to EchAMP protein in a sample of echidna milk (Bisana et al., 2013). The presence of three consecutive sequences in the platypus genome showing homology to EchAMP cDNA with high percentage identity provided supporting evidence for the presence of an orthologous EchAMP gene in platypus, designated as PlatAMP. However, the expression of the PlatAMP gene in the mammary\/milk cells of the platypus still requires confirmation.\n\nWhile characterizing the functional attributes of EchAMP, in silico analyses indicated that EchAMP contained an overall significant alpha-helical structure, suggestive of antimicrobial activity (Boman, 2003; Lai and Gallo 2009). This was confirmed by predictions from the Antimicrobial Peptide Database (Wang and Wang, 2004). Similarly, the predicted post-translational modification of EchAMP, especially the presence of a mucin-type GalNAc O-glycosylation site, also suggested that EchAMP protein may carry antimicrobial activity. To confirm this hypothesis, a recombinant EchAMP protein was synthesized by transfecting HEK293T cells with a c-Flag pcDNA3-EchAMP construct. The protein was secreted into conditioned media (Bisana et al., 2013). The media was analyzed for antibacterial activity with Gram positive and Gram negative bacteria (Fig. 3.3). Significant antibacterial activity was observed against the pathogenic bacteria E. coli, S. enterica, P. aeruginosa, and Staphylococcus spp, which have been reported to be present in monotremes (McOrist and Smales, 1986; Munday et al., 1998). However, there was no antibacterial activity against the Gram positive commensal bacterium Enterococcus faecalis, which is reported to be one of the predominant harmless commensals found in the gastrointestinal tract in a diverse range of species, including humans, marsupials, and most other vertebrates (Fanaro et al., 2003; Yadav et al., 1972; Witte et al. 1999). Mastitis is the most common infection in the mammary gland, and Staphylococcus aureus, E. coli, and Streptococcus spp. have been frequently isolated in conditions of human and bovine mastitis (Barkema et al., 2009; Borm et al., 2006; Bradley and Green, 2001; Oeding 1952). Taken together, it appears that EchAMP, a novel monotreme-specific milk protein, is capable of conferring protection to the underdeveloped, immunologically na\u00efve young outside the uterus. The monotreme mammary gland lacks nipples, and the altricial young is therefore more likely  to ingest pathogens while suckling compared with other species. We propose that the evolution of nipples and development of offspring in utero in the placental mammals (Metatherian and Eutherian) could have led to the loss of selective pressure for the preservation of this gene. This could explain its subsequent disappearance in these species. For example, there is a precedence of the selective loss of genes involved in gastric function in the platypus, which diverged from the Therian lineage early during mammalian evolution. Notwithstanding the high conservation in vertebrates for more than 200 million years, genes encoding the gastric proteinases, hormone gastrin, both the subunits of the gastric H+\/K+-ATPase and the neurogenin-3 transcription factor have either been deleted or inactivated in the platypus genome, giving rise to physiological differences in digestion between monotremes and therians (Ordonez et al., 2008).\n\nFigure 3.3 Antibacterial assays illustrating bacteriostatic activity using (A) E. coli, (B) Salmonella enterica 43971, (C) Staphylococcus aureus 29213, (D) Staphylococcus aureus 25923, (E) Staphylococcus epidermidis, (F) Pseudomonas aeruginosa 27853, and (G) Enterococcus faecalis 10100. * indicates a statistically significant result (P<0.05). Each assay was performed in triplicate, and the experiments were repeated at least three times. Standard error bars are indicated.\n\n## The tammar wallaby (Macropus eugenii)\n\n### Milk Protein Gene Expression; Naturally Occurring Gene Knockouts Used to Assess Protein Function\n\nLactation in the tammar wallaby has been divided into phases that are defined by the composition of milk and apparent sucking pattern of the young (Nicholas et al. 1995; ) (Fig. 3.4). Phase 1 comprises a 26.5-day pregnancy followed by parturition, and the subsequent 200 days of phase 2 are characterized by lactogenesis and the secretion of small volumes of dilute milk high in complex carbohydrate and low in fat and protein. The pouch young (PY) remains attached to the teat for approximately the first 100 days (phase 2A), after which it relinquishes permanent attachment to the teat and sucks less frequently while remaining in the pouch (phase 2B). The onset of phase 3 of lactation (200\u2013330 days) is characterized by temporary exit from the pouch by the young, a large increase in milk production and a change in the composition of milk to include elevated levels of protein and lipid and low levels of carbohydrate. There is a progressive increase in protein concentration (Nicholas, 1988a,b) and protein production during lactation, and the composition of the proteins changes considerably (Nicholas, 1988a; Simpson et al., 2000; Trott et al. 2002). Interestingly, macropodids such as the tammar can practice concurrent asynchronous lactation; the mother provides a concentrated milk high in protein and fat for an older animal which is out of the pouch, and a dilute milk low in fat and protein but high in complex carbohydrate from an adjacent mammary gland for a newborn PY (Nicholas, 1988a,b) (Fig. 3.5). This phenomenon suggests that the mammary gland is controlled locally, and more recent studies (Wanyonyi et al., 2013) have shown that mammary extracellular matrix (ECM) together with endocrine stimuli regulates the expression of the major milk protein genes specific to each phase of lactation.\n\nFigure 3.4 The lactation cycle of the tammar wallaby. (a) Development of the pouch young from day 6 to day 220 of age. (b) The lactation cycle in the tammar has been divided into four phases characterized by changes in milk composition and the sucking pattern of the pouch young. (c) Expression of the major milk protein genes during the lactation cycle. The \u03b2-casein, \u03b1-casein, \u03b1-lactalbumin, and \u03b2-lactoglobulin genes are induced at parturition, and expression remains elevated for the entire lactation. The genes encoding ELP (early lactation protein), WAP (whey acidic protein), and the LLPs (late lactation proteins A and B) are expressed only for specific phases of the lactation cycle.\n\nFigure 3.5 Concurrent asynchronous lactation. The pouch has been retracted to expose the four mammary glands. A 6-day-old young is attached to a teat from a mammary gland secreting phase 2A milk. An older animal at approximately 275 days of age has vacated the pouch and sucks from the elongated teat which provides phase 3 milk from the enlarged mammary gland. The remaining two teats are from quiescent mammary glands.\n\nTo provide more detailed information on changes in milk protein during lactation in the tammar, and to confirm earlier studies describing changes in milk protein gene expression (Bird et al., 1994; Simpson et al., 2000; Trott et al. 2002), we developed a custom tammar mammary EST (expressed sequence tag) array printed with 10,000 cDNAs representing approximately 7000 genes expressed across the lactation cycle. This array was used to transcript profile the mammary gland at all the major stages of pregnancy and lactation. There were two temporally different patterns of milk protein gene expression during the lactation  cycle: One group of genes is induced to high levels around parturition and is expressed throughout lactation, and a second group of genes is expressed highly only during specific phases of lactation (Sharp et al., 2009; Simpson and Nicholas 2002). For example, the genes for the whey proteins \u03b2-lactoglobulin, trichosurin, and \u03b1-lactalbumin, and the \u03b2-casein and \u03b1-casein genes were induced coordinately around the time of parturition, and were expressed for the duration of lactation (Sharp et al., 2009). In contrast, the early lactation protein (ELP) gene was expressed at very high levels in phase 2A; the WAP gene was most highly expressed in phase 2B; and a gene coding for an outlier lipocalin protein referred to as late lactation protein A (LLP-A) was highly expressed in phase 3 (Sharp et al., 2009). A second LLP gene referred to as LLP-B was not on the microarray, but was first expressed after 200 days of lactation (Trott et al., 2005). Whereas some milk proteins may be specific to the tammar, in most cases orthologues of the proteins are found in other species (Simpson and Nicholas, 2002). As a result, the tammar is now emerging as a valuable model to provide a 'temporal gene knockout' which allows for a more accurate assessment of the role of the gene product in regulating either a specific stage of development of the young or mammary function during each phase of the lactation cycle.\n\n### Changes in Milk Composition Regulate Growth of the Tammar Pouch Young\n\nA study by Trott et al. (2003) examined the hypothesis that the sucking pattern of the PY controls the pattern of milk secretion in the tammar wallaby. To test this hypothesis, groups of 60-day-old PY were fostered at 2-week intervals onto one group of host mothers so that a constant sucking stimulus on the mammary gland was maintained for 56 days. This allowed the lactation stage to progress ahead of the age of the young. Analysis of the milk in fostered and control groups showed that the timing of changes in the concentration of protein and carbohydrate was unaffected by altering the sucking pattern. However, rate of growth and development of the foster PY was significantly increased relative to control PY. This most likely resulted from the foster PY ingesting more milk of higher energy content and milk with a composition that was approximately 50 days more advanced than normally consumed at their age (Trott et al., 2003; Waite et al., 2005; Kwek et al. 2009b). It was concluded from these studies that the lactating tammar wallaby regulates both milk composition and the rate of milk production, and these determine the rates of PY growth and development regardless of the age of the PY. The tammar therefore allows studies that correlate milk with a specific composition, particularly the milk proteins, with defined developmental stages of the suckled young.\n\nSubsequent experiments (Kwek et al., 2009b; Waite et al., 2005) extended the studies of Trott et al. (2003) and examined a role for milk in specific changes in the development of the gut of the tammar PY. Parietal cells positive for the alpha subunit of the H + \/K + -ATPase pump were present in both the forestomach and hindstomach of PY at 168 days of age. However, at 260 days of age parietal cells were seen in sections from the hindstomach but were no longer detectable in the forestomach (Kwek et al., 2009b). Experiments where a PY was fostered to a mother at a more advanced stage of lactation showed that this loss of parietal cells was accelerated. However, it is likely that colonization of the gut also has an important role in this process (Kwek et al., 2009b).\n\n### Identification of Milk bioactives in Tammar Milk Using a Functional Genomics Platform\n\nSome bioactives are required continuously for the development of the young, whereas others have a role at specific stages of lactation. Therefore, it is likely that their pattern of secretion can be matched to the developmental profile of the suckled young. A genomics platform may be used to identify the differentially expressed genes that code for a secreted protein (Sharp et al., 2009). This approach used the tammar microarray to identify the differentially expressed genes in the tammar mammary gland that code for secreted proteins. These proteins were synthesized in vitro and examined for activity in cell-based assays for immune modulation, inflammatory responses, growth, and differentiative effects (Sharp et al., 2009). At least 30 of these proteins had at least one of these bioactivities. In addition, some of the proteins showed the capacity to alter the development of three-dimensional structures of tammar mammary epithelial cells in a mammosphere culture model. This raises the possibility that these proteins may also contribute to growth and function of the mammary gland during lactation.\n\n### Temporal Delivery of Domain-specific Bioactive Proteins in Wallaby Milk\n\nApart from the major lactation phase markers, tELP, tWAP, tLLPB-A, and tLLP-B, the wallaby appears to deliver domain-specific bioactive peptides selectively throughout lactation so that specific parts of multifunctional genes are expressed only when they are required (Nicholas et al., 2012). As discussed earlier, a genomics approach has been used to identify differentially expressed genes that were later shown to perform an array of functions (Sharp et al., 2009). The genomics data were corroborated by a later study that used a variety of proteomics tools, including matrix-assisted laser desorption ionization mass spectrometry (MALDI MS) and de novo peptide sequencing to identify a mix of proteins associated with various functions, including antimicrobial defense, growth, and development in tammar milk (Joss et al., 2009). This study identified lactoferrin, complement B protein, haptocorrin, amphiphysin, and myeloid cathelicidin (among other proteins), all of which were delivered in milk at specific times during lactation. Lactoferrin and cathelicidin are well-known antimicrobial proteins, while haptocorrin has also been shown to have bacteriocidal activity against E. coli (Adkins and Lonnerdal, 2003). All of them were expressed in phase 2A and phase 2B (Joss et al., 2009). Amphiphysin, which plays a role in endocytosis (Butler et al., 1997), was found in milk produced from days 172\u2013250 of lactation, while complement B protein, which plays a role in pathogen opsonization (Peters et al., 1988) was detected in phase 2B. The temporal delivery of these proteins by the tammar mother may therefore be able to be exploited to assist in developing a better understanding of the function and role of the bioactives in milk for the development of the young and protection of the mammary gland.\n\n### Cathelicidins\u2014Domain-specific Delivery of Bioactivity\n\nCathelicidins are a well-studied family of antimicrobial proteins found in a wide variety of vertebrates (Boman, 2003; Hancock and Lehrer, 1998; Lehrer and Ganz, 1999; Zanetti et al., 1995; Rollins-Smith et al., 2002; Zasloff 2002). Together with many other cationic amphiphilic peptides collectively known as host-defense peptides (Hancock et al., 2006; Nijnik and Hancock 2009), cathelicidins represent a significant constituent of innate immunity in mammals. They display a remarkably broad spectrum of antipathogenic activity, with a single cathelicidin often showing activity against bacteria, fungi, and viruses (Ramanathan et al., 2002).\n\nThe generic primary structure of cathelicidins includes four exons (Fig. 3.6A; Gudmundsson and Agerberth, 1999; Gudmundsson et al., 1995; Zhao et al. 1995), the first three of which consist of the 22\u201335 amino acid N-terminal signal peptide and the cathelin-like region that covers amino acids 90\u2013115 (Yang et al., 2004). The fourth exon consists of the variable C-terminal antimicrobial domain preceded at its N-terminal by a species-specific proteinase cleavage site (Gudmundsson et al., 1996; Zanetti et al. 1991). All cathelicidins have a net positive charge of 2\u20139, and a high percentage of hydrophobic residues that confer the amphipathic properties upon which their antimicrobial activity is dependent (Hancock and Lehrer, 1998).\n\nFigure 3.6 Tammar MaeuCath1 structure and function. (A) Schematic diagram of the cathelicidin gene structure, including the signal-peptide, cathelin-like region and the variable C-terminal antimicrobial region (AMP). The molecular sizes of the respective regions of MaeuCath1 are indicated. (B) MaeuCath1 exon structure. MaeuCath1a includes all four exons, while MaeuCath1b includes exons 1 and 2 and the 3\u2032 untranslated region. Shaded boxes represent the translated sequence, while unshaded boxes indicate spliced-out exons. (C) An agarose gel showing differential expression of MaeuCath1a and MaeuCath1b genes during wallaby lactation. NP = not pregnant, P = day 16 pregnant, Early = phase 2A lactation, Mid = phase 2B lactation and Late = phase 3 lactation. Involution 1, 5, and 10 represents the number of days post-weaning. Samples in each phase were loaded onto the gel in order of increasing days of lactation. (D) Antibacterial activity of MaeuCath1-derived peptides, Con73 and Con218 directed to S. aureus. The bactericidal activity of the peptides was determined relative to the untreated control and is expressed as the relative bacterial survival (y-axis). Control cells were treated with BSA. (E) Cell proliferation of wallaby mammary epithelial cells (Wall-MEC) following treatment with MaeuCath1. Vector represents conditioned media collected after transfection with empty plasmid. FL MaeuCath1 and MaeuCath1b represent full length Maeucath1 and MaeuCath1b conditioned media, respectively. Control cells were treated with BSA.\n\nThe sequence heterogeneity of the antimicrobial domain between species has been proposed to be an adaptation to the extremely diverse pathogenic challenges for the various host animals. This point of view gains credence from the tendency in mammals to have more than one cathelicidin gene (Castiglioni et al., 1996; Daly et al., 2007; Wang et al. 2011b; Wu et al., 1999), each presumably targeting different microbes. There is increasing evidence that cathelicidins also play significant physiological roles in mammals (Yang et al., 2004). Cathelicidins have been shown to play chemotactic roles (Agerberth et al., 2000; Huang et al. 1997), neutralize endotoxins (Golec, 2007), enhance re-epithelialization of dermal wounds (Yang et al., 2004), and modulate angiogenesis (Li et al., 2000). The secretion of cathelicidins in milk (Joss et al., 2009) must therefore raise questions as to whether milk performs multiple roles in the young and perhaps the mammary gland of the mother.\n\nAlthough eight cathelicidin genes, MaeuCath1-8, have previously been reported in the wallaby (Daly et al., 2009), only MaeuCath1 has been investigated for its function in neonatal development. MaeuCath1 is expressed in the spleen, liver, skin, bone marrow, thymus, and gut of the PY (Daly et al., 2007; Lef\u00e8vre et al., 2007) and the adult mammary gland (Lef\u00e8vre et al., 2007; Khalil et al. 2008).\n\nA recent study showed that MaeuCath1 and MaeuCath7, both of which are secreted in wallaby milk, have antimicrobial activity in their variable C-terminal portion and the capacity to enhance mammary epithelial cell proliferation in the cathelin domain (Fig. 3.6E; Wanyonyi et al., 2011). Synthetic peptides Con73 and Con218, designed from the heterogeneous C-terminal portion of MaeuCath1 and MaeuCath7, respectively, were shown to have bactericidal activity against S. aureus, Pseudomonas aureginosa, Salmonella enterica and E. faecalis (Fig. 3.6D). This is consistent with the need to provide antimicrobial protection to the young before it develops its own immunity (Daly et al., 2007). However, the ability of MaeuCath peptides to kill S. aureus also points to a possible role in the protection of the lactating mammary gland against mastitis agents (Oikonomou et al., 2012).\n\nInitially, the bactericidal activity of MaeuCath peptides against E. faecalis was somewhat paradoxical, because E. faecalis is a commensal of the gut and its clearance would perturb the gut microfloral balance. However, upon examining the expression profile of MaeuCath1, it became evident that its delivery through milk is temporally regulated through alternate splicing (Fig. 3.6B,C), so that the antimicrobial domain (MaeuCath1a) is only expressed in early lactation when it is required for protection of the immune-compromised neonate (Daly et al., 2007) and at involution when the mammary gland of the mother would be vulnerable to potential mastitic infections (Sordillo et al., 1987; Williams and Patnode 1948). In this way, the mother provides broad-spectrum antimicrobial molecules through milk without disturbing the microfloral balance of the neonate's gut.\n\nDuring involution (when MaeuCath1a is highly expressed) a large number of immune cells converge in the mammary gland, similar to a wound-healing environment (Schedin et al., 2007; Findlay 1982). Accompanying the complex milieu of pro-inflammatory activities is an elevated risk of mastitic infections (Sordillo et al., 1987; Williams and Patnode 1948) and a possible release of endotoxins due to the bactericidal activity of immune cells. Interestingly, cathelicidins have been shown to play a role in the different levels of immune response, including regulation of immune genes (Yang et al., 2004), chemotaxis of immune cells (Ciornei et al., 2005), and neutralization of endotoxins (Bhunia et al., 2009). It is therefore conceivable that MaeuCath1a plays these roles in the involuting mammary gland.\n\nApart from the possible roles in immune response, MaeuCath1a may participate in the process of involution by directly inducing apoptosis of epithelial cells, as has been demonstrated for BMAP-28, a bovine cathelicidin. BMAP-28 induces apoptosis through the initiation of the mitochondrial permeability transition pore (MPT pore) and the consequent release of cytochrome c (Risso et al., 2002). Although the mechanism by which BMAP-28 initiates MPT is not well understood, this role is likely to link it to a number of signaling pathways related to mitochondria-mediated apoptosis. BMAP-28 has been shown to selectively kill transformed and proliferating lymphocytes relative to normal nonproliferating cells through MPT (Zanetti, 2004). If MaeuCath1a has pro-apoptotic properties similar to BMAP-28, it would be interesting to investigate for the possibility of it having the ability to selectively kill transformed cells and having a role in the inhibition of tumor progression.\n\nMaeuCath1b is expressed during peak lactation (Nicholas et al., 1997), and, in addition to playing a proliferative role, may induce a number of physiological processes in the mammary gland, similar to the porcine cathelicidin PR-39. In a wound-healing environment, PR-39 has been thought to induce cell proliferation via the expression of syndecans (Bernfield et al., 1992), a family of heparan sulfate proteoglycans known for regulating cell adhesion and blood coagulation, among other functions. The syndecans effect cell proliferation by sensitizing cells to growth factors, including the transforming growth factor-\u03b2 (TGF-\u03b2), vascular endothelial growth factor (VEGF) and fibroblast growth factor (FGF) (Bernfield et al., 1992; Zanetti 2004). The possibility of MaeuCath1b inducing cell proliferation through the syndecans would therefore link Maeucath1b to multiple TGF-\u03b2-regulated physiological processes, including angiogenesis, extracellular matrix neogenesis, and immune suppression (Ijichi et al., 2004). However, although this proposition is feasible, it is also somewhat controversial because in normal epithelial cells TGF-\u03b2 signals apoptosis instead of proliferation (Massague, 2000). Nevertheless, the possible role of Maeucath1b in mammary growth during peak lactation is worth further investigation.\n\n### Whey Four-disulfide Core-domain-containing Proteins WAP and WAP Four-disulfide Core Domain Protein 2\n\nWAP and WAP four-disulfide core domain protein 2 (WFDC2) are members of a larger family of whey four-disulfide core (WFDC) proteins (Fig. 3.7). The WFDC genes are composed of multiple four-disulfide core (4-DSC) domains that consist of a core of eight cysteine residues in a conserved arrangement that form intramolecular disulfide bonds (Campbell et al., 1984; Bingle 2011).\n\nFigure 3.7 The 4-disulfide domain structure of WFDC2, marsupial WAP, eutherian WAP, and monotreme WAP from platypus and echidna. Modified from Sharp et al. (2007b).\n\n### WFDC2\n\nWFDC2 is a secreted protein that consists of two 4-DSC domains. Tammar wallaby WFDC2 comprises two 4-DSC domains, which have previously been annotated domain III on the amino terminal end and domain II at the carboxyl terminal end (Sharp et al., 2007b). WFDC2 also shows a similar structure in monotremes and eutherians, and consists of domains III and II (Watt et al., 2012) (Fig. 3.7). This conservation of the WFDC2 domain structure between all species suggests that these proteins may have similar conserved functions.\n\nThe WFDC2 gene was first identified in epididymis and was later found expressed in the oral cavity (Bingle et al., 2002; Kirchhoff et al., 1991; Idoji et al. 2008). More recently, the WFDC2 gene has been identified in the mammary gland of the tammar wallaby during lactation (Sharp et al., 2007b). There is limited information on the biological activity of the WFDC proteins and their domains, with the exception of the secretory leukocyte peptidase inhibitor (Hiemstra, 2002; Zhu et al. 2002), Elafin (Tomee et al., 1997), and Eppin (Yenugu et al., 2004), all of which have been shown to function as antibacterial, anti-inflammatory, antiviral, and antiproteinase proteins (Moreau et al., 2008; Williams et al. 2006).\n\nThe WFDC2 gene expression profile in the tammar wallaby mammary gland during the lactation cycle showed relatively low WFDC2 gene expression in the nonpregnant wallaby mammary gland, but elevated expression in pregnancy (phase 1) and early lactation (phase 2A). The WFDC2 gene was downregulated in midlactation (phase 2B) and subsequently increased toward the end of late lactation (phase 3) and into involution (Watt et al., 2012) (Fig. 3.8).\n\nFigure 3.8 Expression profiles of WAP and WFDC2. RNA was extracted from tammar wallaby mammary tissue at the indicated stages of pregnancy and lactation, and RT-PCR was used to show the expression profile with GAPDH as a housekeeping gene. WFDC2 and WAP densitometry values were normalized against the GAPDH densitometry values to show relative gene expression. Modified from Watt et al. (2012).\n\nThe WFDC2 protein and domain II of the protein have been found to have antibacterial activity against S. enterica, P. aeruginosa, and S. aureus (Fig. 3.9). However, these proteins demonstrated no activity against E. faecalis (Watt et al., 2012). The WFDC2 domain III protein showed no significant antibacterial activity, clearly demonstrating that antibacterial activity correlated with domain II of WFDC2, and that this domain accounted for the strain-specific activity\n\nFigure 3.9 Antibacterial activity of WFDC2. Cultures of S. aureus and S. enterica with recombinant WFDC2 protein and WFDC2 domain II protein showed significant inhibition of growth of the bacteria. Recombinant WFDC2 DIII protein showed no significant inhibition of growth when compared to the empty vector (control, no protein) treatment, P<0.05. Bacteriostatic assays of Enterococcus faecalis with WFDC2 protein, WFDC2 domain II protein and WFDC2 domain III protein all showed no significant inhibition of growth of the bacteria when compared to the empty vector (control, no protein) (P>0.05). Modified from Watt et al. (2012).\n\nThe expression pattern of WFDC2 at specific times during the tammar reproductive strategy was similar to Maeucath1b, and relates to times when the function of the protein is required. For example, WFDC2 expression during pregnancy, early lactation (phase 2A), and in involution correlated with stages of the lactation cycle when the mammary gland has increased risk of infection (Basden et al., 1997; Daly et al., 2007; Old and Deane 2000). The most common infection in the mammary gland during lactation is mastitis, which is typically caused by bacterial infection from S. aureus, E coli, and Streptococcus spp. in human and bovine mastitis (Barkema et al., 2009; Borm et al., 2006; Bradley and Green 2001). Mastitis in the tammar wallaby is yet to be observed, which may be explained by the bioactive antimicrobial proteins expressed in the milk. Elevated expression of the WFDC2 gene in early lactation also suggests a role of immune protection to the developing PY when it does not have a fully functioning immune system (Basden et al., 1997).\n\nCommensal microbial flora in the gut of the tammar PY, predominately E. faecalis and E. coli (Yadav et al., 1972), was observed at 50 days of age (phase 2A). Interestingly, WFDC2 is expressed when microbial flora is present, and while WFDC2 antibacterial activity is targeted to S. aureus, S. enterica, and P. aeruginosa, it seems that WFDC2 may selectively contribute to the absence of problematic bacterial strains in the gut of the developing young while having no activity against the commensal E. faecalis.\n\n### WAP\n\nWAP is a secreted protein found only in milk. The WAP found in the milk of eutherians has two 4-DSC domains annotated DI-DIIA (Fig. 3.7) (Sharp et al., 2007b; Simpson and Nicholas, 2002). Not all species express WAP; in humans and in ruminants, WAP is a pseudogene due to a frameshift mutation (Clauss et al., 2002; Hajjoubi et al. 2006). In marsupials studied to date, WAP has three domains annotated DIII-DI-DIIB (De Leo et al., 2006; Demmer et al., 2001; Simpson et al. 2000). The monotreme WAP from platypus has three 4-DSC domains like marsupials, but with the annotation of DIII-DIIA-DIIB (Fig. 3.7). However, the monotreme echidna WAP comprises two 4-DSC domains DIII-DIIA (Sharp et al., 2007b). The differences between the number of domains present in different species and the domain annotation could suggest that WAP from different species may differ in its role and function.\n\nThe WAP gene-expression profile in the tammar wallaby mammary gland during the lactation cycle showed relatively low WAP gene expression in the nonpregnant wallaby mammary gland, which continued into pregnancy (phase 1) and early lactation (phase 2A). The WAP gene was upregulated in mid lactation (phase 2B) and then became downregulated in late lactation (phase 3) and into involution (Fig. 3.8).\n\nEarlier studies have shown that mouse WAP added to culture medium with mouse HC-11 cells (a mouse mammary epithelial cell line) is antiproliferative and may act by an autocrine or paracrine mechanism (Nukumi et al., 2004). This is consistent with reports showing that overexpression of WAP in transgenic mice inhibits development of the mammary gland and secretion of milk (Burdon et al., 1991). Studies in our laboratory have used in vitro models to examine the effect of tammar WAP on proliferation of an epithelial-enriched population of tammar mammary cells. In contrast to the inhibitory action of mouse WAP on proliferation of HC-11 cells, results show that tammar WAP or domain III added to culture media stimulates proliferation of primary wallaby mammary epithelial cells (Wall-MEC) and increases expression of the cell-cycle gene cyclin D1. Interestingly, a protein construct comprised of DI-II, which more closely resembles the 2-domain eutherian WAPs, did not have any effect on the proliferation of Wall-MEC (Topcic et al., 2009) (Fig. 3.10). Previous studies have shown that DNA synthesis in mammary tissue is higher in phase 2 than phase 3 (Nicholas, 1988b), which is consistent with a potential role of tammar WAP in the development of the mammary gland. This is further supported by our studies showing that tammar WAP stimulates DNA synthesis in primary cultures of mammary epithelial cells, most likely by upregulating the expression of cyclin genes known to be essential for development of the mammary gland (Fig. 3.10) (Topcic et al., 2009). Our analyses of the expression of G1-phase cyclins demonstrated that expression of tWAP and DIII in HC11 and Wall-MEC cells significantly upregulated the expression of cyclin D1 and CDK-4 genes compared to control HEK293 cells (Topcic et al., 2009). This increase in gene expression supports the argument that tammar WAP is a positive regulator of cell-cycle progression of mammary epithelial cells in culture by the regulation of cyclin D1 and CDK-4 expression. It is likely that evolution has adapted the structure and function of the protein and the expression pattern of the gene, which presumably relates to changes in the reproductive strategies of marsupials and eutherians.\n\nFigure 3.10 Proliferative activity of WAP. Proliferation assay using primary wallaby mammary epithelial cells. tWAP or the DIII significantly increased cell number after 72 and 96 hours in culture P<0.05 when compared to the empty vector control, whereas DI-II showed no significant increase in proliferation P>0.05. Expression of the cyclin D1 gene in Wall-MEC cells grown in the presence and absence of tammar WAP (day 3 post-treatment). Expression of the cyclin D1 and GAPDH genes was determined by RT-PCR analyses. The relative expression levels of the cyclin D1 gene in Wall-MEC was quantified by NIH. Image presented as a proportion of the housekeeping gene GAPDH. Modified from Topcic et al. (2009).\n\nStudies using a WAP gene knockout mouse have shown that mammary development is normal. However, the pups had limited development at the later stages of lactation (Triplett et al., 2005). As tammar WAP is a major milk protein secreted during the middle third of lactation (phase 2B) when the PY's diet comprises of only milk, it has been speculated that this protein plays a specific role in development of both the mammary gland and the suckled young at this time. It is likely that a putative function for WAP in mammary development is predominantly found only in marsupials and monotremes, and the activity has been lost in humans and ruminants due to the reduction in evolutionary pressure related to changes in the reproductive strategy of eutherians.\n\n### S100A19\u2014A New Member of the S100 Family of Proteins\n\nThe S100 proteins are calcium-binding proteins implicated in governing diverse intracellular and extracellular processes such as contraction, motility, cell growth, differentiation, cell-cycle progression, gene transcription, protein secretion, and antimicrobial function (Donato 1999; ; Donato et al., 2013). A new member of the S100 family has been identified, which is expressed in both gut and mammary gland tissue of the tammar wallaby (Kwek et al., 2013). Initially, S100A19 was identified from a subtracted cDNA library in an effort to identify genes that were differentially regulated between the tammar forestomach and hindstomach (Kwek et al., 2009a). This differential expression pattern was later confirmed by Northern blot analysis showing that S100A19 was expressed in the immature forestomach at a time when it displays a gastric phenotype, but expression is downregulated as the stomach matures into a fermentation sac (Fig. 3.11) (Kwek et al., 2013). This change in gut phenotype corresponds to a time when the diet of the young transitions from milk only to a diet that is supplemented with herbage (Kwek et al., 2009a).\n\nFigure 3.11 Expression analysis of S100-A19 gene during the tammar wallaby stomach development.   \n(A) Northern blot analysis of S100A19 gene expression in tissue isolated from the developing forestomach (F) and hindstomach (H) at days 140, 190, 230, 260, and adult tammar wallabies. (B) Northern blot analysis of S100A19 expression in tissue isolated from the developing forestomach at days 65, 142, 168, 180, 205, 260, and adult tammar wallabies. Analysis shows the presence of a single transcript (625 bp) that increases in expression from day 65 to day 180 and then declines at day 205. S100A19 is no longer expressed from day 230 of age onward.\n\nS100A19 expression in the tammar wallaby mammary gland was also found to be differentially regulated (Kwek et al., 2013). S100A19 is upregulated during pregnancy and involution, but not expressed during the milk production phase of lactation (Fig. 3.12). Comparison of the tammar wallaby S100A19 protein sequence with S100 protein sequences from eutherian, monotreme, and other marsupial species suggests that the marsupial S100A19 has two functional EF hand domains and an extended His tail. It also has close similarity to S100A7 and S100A15 proteins, which have been shown to have antibacterial activity (Kwek et al., 2013; Buchau et al., 2007; Glaser et al. 2005). Human S100A8, S100A9, and S100A12 have been shown to be expressed in the inflamed gastric mucosa of children infected with Helicobacter pylori, but absent in the normal gut, suggesting that these proteins may be involved in the host response at this site (Leach et al., 2008). The regulated differential expression patterns of S100A19 in the tammar wallaby suggest that S100A19 may play a role in gut development, a process that differs between metatherians and eutherians, and\/or possesses potential antibacterial properties to establish the correct flora and protect against bacterial infection in the immature forestomach. In the mammary gland, S100A19 may act to protect the tissue from infection at times of vulnerability during the lactation cycle.\n\nFigure 3.12 Expression analysis of S100A19 across the tammar wallaby lactation cycle.   \nQuantitative RT-PCR was used to determine levels of S100A19 gene expression at different phases of lactation. One of the milk proteins, \u03b2-lactoglobulin (BLG) gene expression, was also measured and used as the lactation marker gene. The pregnancy period (P), lactation period, and involution period (I) consist of 26, 350, and approximately 10 days, respectively. Data represent mean \u00b1 S.E.M. from three animals.\n\n## A role for milk in the control of mammary function\n\nThere is increasing evidence to suggest that milk plays an important role in regulating mammary epithelial function and survival, and this is particularly evident during involution (Brennan et al., 2007). Apoptosis was induced preferentially in the sealed teats of lactating mice (Li et al., 1997; Marti et al. 1997), while the litter suckled successfully on the remaining teats, which indicates that cell death is stimulated by an intramammary mechanism sensitive to milk accumulation (Quarrie et al., 1996). A protein known as the feedback inhibitor of lactation (FIL) has been suggested as a candidate and is secreted in the milk of the tammar (Hendry et al., 1998) and other species. It acts specifically through interaction with the apical surface of the mammary epithelial cell to reduce secretion (Wilde et al., 1995).\n\nMore recent studies using the tammar mammary explant culture model (Nicholas and Tyndale-Biscoe, 1985) to examine the process of involution have confirmed the likely role of milk, and particularly putative autocrine factors, for controlling mammary function during involution (Brennan et al., 2007). Mammary explants from pregnant tammars, cows, and mice were cultured for three days with insulin, cortisol, and prolactin to induce milk protein gene expression. To mimic involution, all hormones were subsequently removed from culture medium for 10 days to downregulate expression of the milk protein genes. Surprisingly, the explants retained the same level of response during a subsequent challenge with lactogenic hormones. Previous studies have shown that there is limited secretion of milk proteins from tammar mammary explants, but it is unlikely that milk constituents accumulated to elevated concentrations (Nicholas and Tyndale-Biscoe, 1985; McFadden, 1997). The maintenance of epithelial cell viability and hormone responsiveness in explants cultured in the absence of hormones is consistent with a more active mechanism, such as the accumulation of local factors in the milk, being the primary stimulus for apoptosis of mammary epithelial cells in the tammar wallaby mammary gland. This model permits the uncoupling of hormone- and milk-regulated involution. A primary outcome of these studies is evidence for the extraordinary capacity for survival and maintenance of hormone responsiveness by tammar mammary epithelial cells cultured in a chemically defined medium with no exogenous hormones and growth factors.\n\nIt is likely that milk components will play a major role in the shape of the lactation curve in dairy cattle following peak lactation at approximately 20 weeks of milk production. The lactation cycle in dairy cattle includes a period of increasing milk yield in early lactation followed by a steadily declining yield for the remainder of lactation. The amount of milk produced during lactation is determined by the peak yield and persistency of lactation. Milk production is largely a function of the number and activity of secretory cells in the udder that decline significantly between the time of peak yield and late lactation. Therefore, it follows that approaches to address the decline in milk yield and lactation persistency after peak lactation must involve changes to the frequency of apoptosis in mammary secretory cells. Endocrine treatment of cattle with bovine sommatotrophin increases milk yield, but in many cases persistency is not altered. Furthermore, although moderate heritability of persistency suggests that selection for this trait is possible, it is achieved at the cost of milk yield. Increased frequency of milking has been shown to increase milk yield, suggesting that the mammary gland has a local intrinsic resistance to regression. Identification of the milk components that impact significantly on mammary cell fate may provide new approaches for strategies to improve lactation persistency.\n\n## The fur seal\n\nThe three families of Pinnipeds (also known as fin-footed mammals) evolved from a carnivorous ancestor around 25 million years ago and diverged into the Phocids (true seals), Odobenids (walrus), and Otariids (sea lions, fur seals) during the middle Miocene (10 million years ago) (Fordyce, 2002). Each group adopted a different approach to lactation. Phocid seals evolved large body sizes to increase body reserves and reduce heat loss and risk of predation, enabling them to adopt a 'fasting strategy' during lactation (Oftedal et al., 1987). In contrast, the small ancestral otariid seals retained smaller body sizes and insulating fur, breeding at rockeries to gain proximity to local prey resources and adopting a 'foraging lactation' strategy (Bonner, 1984). The smaller size of otariid seals made it necessary to feed during lactation in order to replenish the body stores required to continue milk production. Reduced prey availability contributed to lengthening the lactation period (4 to >12 months), and otariid seals began exploring resources farther off shore, increasing the duration but reducing the frequency of foraging trips during lactation. The current-day otariid seals have adopted a lactation strategy that is characterized by alternation between periods of several days of copious milk production on shore and extended periods of maternal foraging at sea. Intersuckling intervals have been recorded in otariid seals of up to 23 days and are among the longest ever recorded for a mammal. The need to increase the duration of foraging trips due to distant foraging grounds during lactation has selected for an adapted otariid mammary gland, which remains functional despite sustained interruptions in suckling activity. This type of lactation is unique among mammals due to the extreme duration of intersuckling bouts and the rapid rate of energy transfer while suckling, and results in a slow growth rate of the otariid pups (Oftedal et al., 1987) (Fig. 3.13).\n\nFigure 3.13 The 'foraging' lactation strategy of the fur seal. The pregnant female arrives at shore to give birth and remains with the pup for approximately one week. For the remainder of lactation (10\u201312 months), females alternate trips to the sea with short trips ashore to suckle their pup.\n\nFur seal milk is considered among the most nutritious mammalian milk and is composed of 30\u201360% fat, 5\u201315% protein, and around 40% water (Oftedal and Iverson, 1995). The high-energy content of fur seal milk allows for a short and rapid period of energy transfer from mother to pup. Interestingly, fur seal milk is devoid of lactose (Dosako et al., 1983; Urashima et al. 2001b), an osmole that normally regulates the water content of milk (Oftedal and Iverson, 1995).\n\nStudies on milk production and mammary gland function in fur seals have shown that during a foraging bout, the lactating mammary gland produces less milk compared to that of the onshore lactating female, but does not progress to involution. For example, milk production in Antarctic fur seals (A. gazella) has been shown to continue while the female is foraging at sea, but at only 19% of the rate of production on land (Arnould and Boyd, 1995). During onshore lactation, the mammary alveoli of the Cape fur seal appear engorged with milk containing a large amount of lipid (Fig. 3.14a,b) (Sharp et al., 2006b). In contrast, during the mother's extended foraging trip, the alveoli appear less distended, epithelial cells surrounding the alveoli appear columnar, and the lipid component is decreased within the milk, consistent with a gland that is producing less milk. During natural weaning, in most mammals the alveoli fill with milk due to cessation of suckling, with a resulting decline in milk protein gene expression in the mammary epithelial cells that causes the epithelium to regress and enter involution (Li et al., 1997). This process is characterized by apoptotic cell loss and mammary gland remodeling (Lund et al., 1996; Metcalfe et al., 1999; Strange et al., 1992; Walker et al. 1989). Apoptosis associated with involution in the mammary gland of the foraging seal has been analyzed and found to be barely detectable. Even after extended periods when there is no sucking stimulus, the gland does not regress (Sharp et al, unpublished).\n\nFigure 3.14 Histological sections of the mammary gland from seals (a) lactating while nursing onshore and (b) lactating while foraging at sea. Sections are stained with hematoxylin and eosin. Immature alveoli in the pregnant gland are present in lipid (white) and milk protein (grey) and are indicated in the onshore and offshore mammary glands. Magnification x100. (c), \u03b2-casein expression during Cape fur seal lactation cycle. Analysis of expression using canine Affymetrix chips hybridized to cDNA probes generated from RNA from pregnant (placental gestation and nonlactating, n = 2); lactating on shore (n = 2) and lactating at sea (n = 1) (animals in embryonic diapause) Cape fur seals. (d) Cluster analysis of gene-expression profiles from the Cape fur seal mammary gland during different stages of lactation. A total of 1020 Cape fur seal mammary messenger RNA (mRNA) transcripts were identified with expression levels above an intensity of 250 in any sample type. Hierarchical clustering was conducted using euclidean distance. Pregnant and onshore lactating data represent an average of two animals. Offshore data represent a single sample.\n\nAs mentioned previously, one candidate factor involved in the regulatory mechanism of milk production is a milk protein termed FIL, which acts within hours to downregulate milk production (Peaker et al., 1998; Wilde et al. 1987). FIL is a heat-labile protein secreted by the mammary gland and present in the whey fraction of milk which, when exposed to mammary cells for an extended period, acts to block translation of milk protein transcripts and inhibiting secretion of milk proteins by binding a putative receptor in the apical surface of the epithelial cells (Blatchford et al., 1998). Fractionated fur seal milk has demonstrated a FIL-like activity at a similar level to that reported for other species (Cane et al., 2005). It is predicted that FIL may play a role in the downregulation of milk volume in the fur seal mammary gland during foraging to assist in the prevention of prolonged engorgement.\n\n### Transcriptomic Analysis of the Lactating Cape Fur Seal Mammary Gland during Suckling and Foraging\n\nThe transcript profile of the Cape fur seal (Arctocephalus pusillus pussillus) lactation cycle has been analyzed using Affymetrix canine arrays to examine the pattern of gene expression in mammary tissue during pregnancy, onshore lactation, and offshore lactation (Fig. 3.14c) (Sharp et al., 2006a). High-sequence conservation between the Cape fur seal and dog, 95% similarity at the DNA level (Sharp et al., 2006a), permits a significant detection rate of measurable hybridization signals between seal cDNA and the Affymetrix Canine microarray.\n\nCluster analysis of expression profiles from microarray data has revealed that the overall expression profile of lactating mammary gland of the foraging Cape fur seal is more closely related to the profile of pregnant nonlactating animals than the profile obtained from onshore lactating animals (Fig. 3.14d). This suggests that the interruption of lactation in foraging animals involves a major reprogramming of mammary gland expression. This analysis also shows that the acute immune response is upregulated in the mammary gland of the foraging seal, presumably to protect the mammary gland from infection during the extended foraging trip while milk remains in the gland (Sharp et al., 2007a). Gene-expression profiles associated with survival and preservation of tissue architecture were also either maintained or upregulated, preventing degradation of mammary tissue. These global gene-expression data suggest that the immune and acute phase responses observed in the mouse at involution (Clarkson and Watson, 2003), and mimicked in the mammary gland of the foraging fur seal, are independent of the apoptotic phase of involution.\n\nAnalysis of individual genes showed that the reduced rate of milk production was controlled at a transcriptional level by a reduction in milk protein gene-expression profiles (Sharp et al., 2006a). For example, expression of \u03b2-casein showed a significant downregulation of expression in the mammary gland of a foraging animal and correlated with decreased milk volume (Fig. 3.14c). This pattern of expression has been confirmed by RT-PCR and is also observed for other milk protein genes such as \u03b1S2-casein, \u03ba-casein, and \u03b2-lactoglobulin. The fur seal mammary gland therefore undergoes repeated cycling of high milk protein gene expression which correlates with high milk production while the pup is suckling, and low milk protein gene expression that correlates with low milk production in the absence of suckling. It has been postulated that the absence of lactose and subsequent production of milk with low water content may prevent extensive distension of the mammary gland during the long foraging trip (Reich and Arnould, 2007). However, this seems unlikely as fur seal milk is devoid of lactose during both onshore and offshore milk production and milk is not viscous. Instead, we hypothesize that the prevention of overdistension of alveoli in the fur seal mammary is achieved by preventing milk accumulation via downregulation of milk protein gene expression (Sharp et al., 2006a), which in turn lowers milk volume. This mechanism prevents the physiochemical signaling triggers that normally lead to involution upon milk accumulation.\n\n### Absence of Local Apoptotic Milk Factors in Fur Seal Milk\n\nFor other mammals, the accumulation of milk in the mammary gland due to cessation of sucking by the young allows factors present within milk to initiate involution, leading to mammary gland remodeling (Li et al., 1997). In vivo studies, such as teat sealing experiments in mice and goats, have demonstrated the importance of local factors controlling mammary gland involution at weaning (Li et al., 1997; Theil et al. 2005). The effects of local milk factors have also been observed in lactating goats, showing reduced cell number and increased apoptosis in individual glands with infrequent milk removal for several weeks; within those glands, apoptotic cells were concentrated in smaller regressing alveoli (Li et al., 1999). Effects due to alveoli distention were discounted as mammary epithelium was not duly compromised by stored milk (Wilde et al., 1999). Conversely, frequent milking of cows during early lactation has also shown an increase in milk production, which is proposed to be due to removal of these local factors (Wall and McFadden, 2007). The presence of local milk apoptotic factors in the mammary gland is therefore likely to act as a stimulus to initiate involution and also may cause a decline in lactation persistency during normal lactation.\n\nThe absence of sucking during the period of fur seal maternal foraging results in milk remaining within the gland. Apoptotic factors present in milk of other species would be predicted to cause involution. However, this does not occur, and fur seal mammary glands remain functional and alveoli remain intact, as demonstrated by apoptotic staining of mammary tissue by Apotag (Chemicon) (Sharp, Hornby, Nicholas, unpublished data). The absence of involution at this time suggests that the local trigger of involution that is present in milk of other species may be absent or limited in the milk of the fur seals. In addition, milk secretion in fur seals does not follow the pattern of secretion of most other mammals, where peak milk secretion tends to occur at the start of lactation. Studies on the amount of milk volume produced in fur seals suggests that the mother secretes larger volumes of milk as lactation progresses in order to meet the needs of her growing young. This observation may also suggest an absence of a local trigger of involution in fur seal milk.\n\nThe \u03b1-lactalbumin protein, which is required for lactose synthesis and is secreted in milk, has also been implicated in the process of apoptosis (Baltzer et al., 2004; Hakansson et al. 1999; ). The \u03b1-lactalbumin protein plays a central role in the mammary gland as the regulatory subunit of lactose synthase (Lonnerdal and Lien, 2003) and as a potential apoptotic milk factor (Baltzer et al., 2004). A multimeric form of \u03b1-lactalbumin (MAL) isolated from the casein fraction of human milk has previously been shown to localize to the nucleus and cause apoptosis in selected cell lines (Hakansson et al., 1999). Similarly, HAMLET (human \u03b1-lactalbumin made lethal to tumor cells) (Svensson et al., 2002), a complex of apo-\u03b1-lactalbumin combined with oleic acid, has been shown to localize to the nucleus, where it binds to histones, disrupts chromatin structure, and leads to cell death (Duringer et al., 2003). However, negatively charged \u03b1-lactalbumin can also bind to positively charged histones without the aid of oleic acid, causing aggregation (Permyakov et al., 2004). Native bovine \u03b1-lactalbumin has also been found to have an antiproliferative effect on human colon adenocarcinoma cell lines which were dose and time dependent (Sternhagen and Allen, 2001). Affymetrix analysis has shown that \u03b1-lactalbumin is expressed at extremely low levels in the fur seal mammary gland compared to expression levels in other species during lactation; however, the fur seal \u03b1-lactalbumin transcripts are not translated into secreted protein (Sharp et al., 2008). This finding supports previous observations that were unable to detect \u03b1-lactalbumin or lactose in fur seal milk (Dosako et al., 1983; Johnson et al., 1972; Messer et al., 1988; Schmidt et al., 1971; Urashima et al. 2001a). Analysis of \u03b1-lactalbumin function showed that \u03b1-lactalbumin had apoptotic effects on a variety of cell types. These apoptotic effects were also observed when seal mammary cells were exposed to \u03b1-lactalbumin, suggesting that the \u03b1-lactalbumin-induced apoptotic pathway is still functional in fur seal mammary cells (Sharp et al., 2008). These observations were also seen for mammary cells derived from other species such as the cow and mouse. It is speculated that the absence of \u03b1-lactalbumin in fur seal milk contributes to the lack of apoptosis in the mammary gland of lactating seals during foraging. Therefore, the pseudogenization of the \u03b1-lactalbumin gene has enabled the adoption of the current lactation strategy of the fur seals, facilitating avoidance of involution while undertaking long foraging bouts.\n\nIn addition to regulating the amount of milk produced by the mammary gland during foraging, transcriptome analysis has shown that the Cape fur seal mammary gland also upregulates a number of genes involved in maintaining cell structure during foraging compared to the level of expression during lactation on shore (Sharp et al., 2007a). This response involves a >2-fold upregulation of the Na-dependent transporter of taurine (TAUT), which is involved in maintaining cell volume (Christensen et al., 2005) and may act to control the viscosity of milk, and the ZO-1\u2013associated Y-box factor (ZONAB), which is involved in regulating cell density and cell proliferation (Balda et al., 2003).\n\nGlobal gene-expression analysis during the fur seal lactation cycle has also revealed that the foraging fur seal mammary gland fails to downregulate cell survival genes, a phenomenon that usually occurs during first stage of involution. For example, Ank3, Mcl1, Bcl2, and BclXL have been shown to be downregulated in the involuting mouse mammary gland (Clarkson et al., 2004). However, these genes showed no change in expression between the mammary glands of onshore lactating and at-sea foraging fur seals (Fig. 3.15).\n\nFigure 3.15 Expression of involution-related genes in the mammary gland during the fur seal lactation cycle. Genes were grouped according to ontological category. The mean intensity for each category was plotted. Analysis shows no changes in expression of genes involved in promotion of   apoptosis (P53, p21\/WAF1, Daxx, Bax) and   cell survival (Ank3, Mcl1, Bcl2, and BclXL) in the transition between onshore lactation (shore) and foraging (sea). However, induction of immune-related genes, such as   immunoglobulins (IgA, IgJ and IgC),   complements factors (C1qa, C1qc, C1r, C1s, and C3a) and   acute phase response (SAA3) is observed. Genes involved in maintenance of   cell structure (TAUT and ZONAB) are also shown to be upregulated in the foraging fur seal mammary gland (sea) compared to the onshore lactating gland (shore). There are also a set of genes,   sea-specific genes (unannotated hypothetical genes), which are specifically upregulated in the mammary gland of foraging ('sea') seals. Values shown on the y-axis represent average intensity values from canine Affymetrix analysis for each group. The x-axis shows mammary gland tissue type collected at different stages of Cape fur seal lactation. 'Pregnant' represents pregnant mammary gland (n = 2), 'Shore' are samples collected from the mammary gland of onshore-lactating Cape fur seals (n = 2), and 'Sea' is a single sample collected from the lactating mammary gland of a seal caught at sea more than 100 km from the nearest colony.\n\nPro-apoptotic genes, such as Bax, shown to be upregulated in the second phase of involution in mice (Clarkson et al., 2004; Lund et al. 1996), also showed no change in expression in the foraging fur seal (Fig. 3.15), suggesting apoptosis is also not initiated by this pathway. Another pro-apoptotic gene, Daxx, an activator of apoptosis in mice (Yang et al., 1997), was also examined and showed reduced expression in the mammary gland of seals at sea. This finding suggested that any apoptosis that may be mediated in the mammary gland by Daxx is actively prevented during foraging. The absence of expression of these pro-apoptotic genes indicates that apoptosis by these pathways is not initiated in the fur seal mammary gland during foraging, and is consistent with earlier observations showing a lack of apoptotic activity in intact mammary gland tissue from these animals (Sharp et al., 2006a).\n\nOther genes associated with this second stage of involution include downregulation of the tissue inhibitors of matrix metalloproteinases (TIMPs) TIMP-1 and TIMP-3 (Fata et al., 2001). Tissue remodeling during the second phase is largely dependent on matrix metalloproteinases (MMPs), which act to remodel the extracellular matrix and stroma within the gland. MMPs are maintained in the gland in an inactive state by TIMPs during the first phase of involution. The ratio of MMP to TIMP expression is critical for determining the two phases of involution (Talhouk et al., 1992). In order for the first phase of involution to maintain its reversibility, MMPs are not activated until day 3 of involution. TIMP expression is high in the first phase, but as involution progresses, TIMP levels decline, leading to activation of MMPs which then remodel the extracellular matrix and stroma (Lund et al., 1996). Upregulation of the TIMP-3 gene has been observed in microarray analysis of mice involution at 96 hours (Clarkson and Watson, 2003), and TIMP-3 deficient mice show an accelerated involution process, indicating that TIMP-3 is necessary to control MMP activity during the first stage of involution (Fata et al., 2001). During fur seal lactation on shore, TIMP-3 is expressed at high levels (5200 intensity units) and is maintained at a high level while at sea (3400 intensity units), suggesting no significant change in expression of TIMP-3. TIMP-1 is expressed at a lower level while lactation occurs on shore (164 intensity units), and expression is elevated 2.3 fold (371 intensity units) while out foraging. Expression of the MMP-2 and the serine proteinase urokinase-type plasminogen activator (uPA), which are normally low during lactation in mice, are strongly upregulated in parallel starting at day 4 after weaning, coinciding with start of the collapse of the lobulo-alveolar structures and the intensive tissue remodeling in involution (Lund et al., 1996). These genes are expressed at very low levels in the mammary gland of fur seals during lactation on shore and at sea (<100 intensity units), and upregulation of gene expression was absent in the mammary gland of the foraging fur seal. Maintenance of high levels of TIMP-3 and elevated expression of TIMP-1 combined with generally low levels of MMP-2 may help to maintain the stoichiometry between MMPs and TIMPs and prevent involution while on a foraging trip. Indeed, the characteristic upregulation of TIMP-3 expression during the typical first phase of involution was not observed in the foraging fur seal, suggesting that the pathway associated with TIMP and MMP expression is not initiated. This is consistent with the preservation of alveoli architecture observed in mammary tissue of foraging fur seals (Sharp et al., 2006a).\n\nWithin 72\u201396 hours post forced weaning in mice, a sharp rise in immunoglobulin gene expression has been observed (Clarkson and Watson, 2003). It is unknown what purpose these immunoglobulins have during involution. It has previously been suggested that the vast amount of apoptosis associated with involution leads to a localized immunoglobulin response. However, the fur seal also shows a very high level of immunoglobulin expression (IgA, IgJ, and IgC) during foraging at sea in the absence of apoptosis, suggesting that the immunoglobulins are not expressed as a result of apoptosis. Complement factor genes (C1qa, C1qc, C1r, C1s, and C3a) are also upregulated in the mammary gland of the foraging fur seal when compared to the mammary gland of the onshore fur seal. These genes have been observed to be upregulated in the mouse involution mammary gland and are proposed to mark monocyte and lymphocyte attraction to the site (Clarkson et al., 2004). It is unknown why these cells would be attracted to the foraging fur seal mammary gland, but it may be the result of an inflammatory response for prevention of mastitis during milk stasis. The acute phase response gene, serum amyloid A (SAA), was also seen to be upregulated in the mammary gland of the foraging fur seal (Fig. 3.15). SAA is induced during clinical mastitis in cows (Eckersall et al., 2006; Weber et al., 2006) and during involution (Clarkson et al., 2004). Expression of SAA in the fur seal at this time may act to protect the mammary gland from infection. The sea environment may provide adequate protection against bacterial infection for the teat, which is the usual path of mastitis infection.\n\nA number of other genes are specifically upregulated in the mammary gland of the foraging fur seal (Fig. 3.15). These include a number of unannotated hypothetical genes that may act as potential mediators for prevention of involution. These genes are yet to be studied; however, the expression of these genes due to other factors such as disease cannot as yet be ruled out.\n\nThis unique capacity to inhibit apoptosis associated with involution, to maintain mammary epithelial cells in a viable and hormone-responsive state, and to stimulate mammary gland growth to increase milk production during the lactation cycle provides new opportunities to further identify the mechanisms controlling these events. A thorough understanding of the genetic and\/or local factors in the residual milk regulating this process may have applications for improving lactation persistence in the dairy cow, either by improved breeding programs or by manipulation of milk factors.\n\n## New player in milk bioactives; MicroRNA\n\nMicroRNAs (miRNAs) are a class of 18\u201326 nucleotide small noncoding RNAs that regulate post-transcriptional inhibitory effects on gene expression. They are thought to provide a new level of regulation directed to fine-tuning expression of the genome in a tissue- and time-specific manner. MiRNAs show stage-specific expression (Pigati et al., 2010) and are established mediators of mammary gland development (Avril-Sassen et al., 2009). Recent studies have found miRNAs in high concentrations in milk (Chen et al., 2010); these miRNAs are most concentrated in milk exosomes (Zhou et al., 2012). The presence of miRNA in bovine milk provides new opportunities for use as biomarkers for milk quality (Chen et al., 2010) and potentially as markers of udder function. In addition to the cow, miRNAs have been found in milk from a variety of animals, including goat, pig, yak (Li et al., 2012; Gu et al., 2012; Bai et al. 2013), and human (Zhou et al., 2012). Milk miRNAs appear to be very stable and withstand harsh environmental conditions, including prolonged storage at room temperature, multiple freeze-thaw, RNAse digestion, and boiling (Chen et al., 2010; Bail et al. 2010). Studies with human breast milk have shown that miRNAs were able to withstand highly acidic conditions (pH 1) for 1 hour (Kosaka et al., 2010). An interesting report has shown that exogenous miRNAs consumed from plant foods may pass across the gut and directly influence gene expression in animals (Zhang et al., 2012). These observations suggest that miRNA consumed in the diet may have a putative role in mammary gland physiology, together with signaling potential for bioactive effects on growth and development of the young. Rapid advances in miRNA sequencing combined with the option of exploiting mammalian lactation diversity, provide a platform to address the role of miRNA during lactation.\n\n## Conclusions\n\nThis review has described three animals with extreme adaptation to lactation, and explored approaches to exploit their unique reproductive strategies to identify new proteins in milk with bioactivity that have the potential to regulate mammary function and growth of the suckled young. In contrast, the lactation cycle in most eutherian mammals is characterized by the initiation of lactation around parturition and the production of milk in which the individual components vary little during the entire period of lactation. For example, in the dairy cow, evolution and selection pressure has led to a lactation cycle that does not include significant changes in the secretory pattern of the milk proteins. Therefore, identification of new protein bioactive molecules in the dairy cow requires a broad screening program and will most likely identify molecules in milk that are secreted at low and unchanged levels during lactation. In contrast, in the tammar, echidna, platypus, and fur seal the temporal pattern of secretion of a milk bioactive that is correlated with either specific developmental phases of the suckled young or altered mammary function is more likely to indicate a cause-and-effect relationship, and these factors are more likely to have increased commercial value. Monotremes and marsupials are particularly interesting as their genomes have evolved under evolutionary pressure to protect immunologically na\u00efve young with broad spectrum antibiotics (Wang et al., 2011a). This presents a potential opportunity to discover new antimicrobial proteins. In addition, it may be possible to exploit the specificity of these naturally occurring antibacterial proteins in the dairy industry to help treat infections such as mastitis as an alternative to synthesizing new antibiotics. The increasing availability of sequenced genomes in public databases has underpinned increased interest in comparative genomics and bioinformatics, and our results using the echidna, tammar, and fur seal models have led to the identification of many bovine and human orthologues of proteins with bioactivity. Conceptually, it is very likely that the relevance and use of species with an extreme adaptation to reproduction or environmental pressure will become increasingly attractive for improved understanding of the genetic regulation of physiological processes.\n\n# References\n\nAdkins Y ,  Lonnerdal B .  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Snell*****\n\n* School of Population Health, University of Auckland, Auckland, New Zealand  \n** Centre for Advanced Technologies in Animal Genetics and Reproduction (REPROGEN), Faculty of Veterinary Science, The University of Sydney, NSW, Australia  \n*** Department of Zoology and CRC for Innovative Dairy Products, University of Melbourne, Victoria, Australia  \n**** Victorian Bioinformatics Consortium, Monash University, Victoria, Australia  \n***** School of Biological Sciences, University of Auckland, Auckland, New Zealand  \n****** School of Medicine, Deakin University, Geelong, Victoria, Australia\n\n## Abstract\n\nThe rapid advances in technology for both evaluating and understanding the structure of animal genomes and their functional significance have presented opportunities for scientists to more clearly understand the complexity of bovine milk proteins and the control of their expression. Used in conjunction with protein data, we can begin to understand the significance of naturally occurring genetic variation influencing milk protein production and composition, as well as how milk proteins are processed into the biologically active peptides that are resident within colostrum and milk. The challenge then remains to translate this information into useful food and therapeutic products, helping to underpin the commercial viability of the dairy industry. Recent advances in genetic technologies provide the potential for accelerated selection of existing traits. Of course, composition could be manipulated via genetically modified (GM) approaches, although consumer acceptance of GM foods is still a major issue. In this chapter, we present a review of the current status of bovine milk genomics and functional genomics and describe the roles, characteristics, and key bioactivities of the major bovine milk proteins and their encrypted peptides. The application of modern molecular analytical tools to the full spectrum of lactation strategies adopted by eutherians, marsupials, and monotremes to improve our understanding of the milk proteome is also discussed.\n\n## Keywords\n\nBovine milk proteins, milk genomics, bovine genome science, evolution or manipulation of bovine milk proteins\n\nOutline\n\nIntroduction 114\n\nMilk Genomics: A Contemporary Approach to Milk Composition 114\n\nAdvances in Bovine Genome Science 115\n\nDNA Sequence 116\n\nGenome Map\/Structure 116\n\nPolymorphism 117\n\nFunctional Genomics 117\n\nTransgenic Technologies for Milk Manipulation 118\n\nComparative Milk Genomics 119\n\nOrigins of milk proteins 120\n\nConstraints and opportunities for evolution or manipulation of bovine milk proteins 122\n\nThe Detailed Structure of Milk Proteins and Their Genes 124\n\nThe Function of Bovine Milk Proteins 125\n\nBioactive Peptides Sequestered within Milk Proteins 128\n\nExisting Variation in Bovine Milk Proteins and the Impact on Expression Function and Milk Quality: Experimental Modifications of Bovine Milk Proteins 129\n\nExperimental Modifications of Bovine Milk Proteins 130\n\nAdding Value to Milk through the Use of Milk Protein Genomics 132\n\nConclusion 133\n\n## Introduction\n\nIn addition to being the source of nutrition for the neonates of mammals, milk has long been considered a healthy source of dietary proteins and minerals for human consumption, and a global dairy industry has evolved to cater to that need. Global demand for dairy continues to grow, driven by the increasing world population, improving economic conditions, and transition toward westernization of diets. In addition, increased competition from other foods and beverages providing enhanced nutritive properties has driven an impetus for diversification of dairy-based foods, such as the manipulation of milk composition to remove saturated fats and the fortification of milk with specific fatty acids, minerals, and vitamins.\n\nAlthough the protein component of bovine milk is only \u22482\u20134% of the total weight, which typically is marginally less than both total fat and carbohydrate, the population of proteins present in milk is diverse and impacts on the physical properties of milk, the nutrition of a neonate, and the metabolic homeostasis and development of both the dam and the offspring. Approximately 80% of the protein present in milk consists of four structurally and functionally interrelated proteins called the caseins. These include \u03b1S1\\- and \u03b1S2-caseins as well as \u03b2\\- and \u03ba-caseins. Some of these proteins show some conservation of gene and protein sequence and structure, and there is similar evidence of relatedness between proteins observed in milks of different species. The remaining 20% of the milk protein population includes the major whey proteins \u03b2-lactoglobulin (\u03b2-Lg) and \u03b1-lactalbumin (\u03b1-La) as well as other constituent proteins, including immunoglobulins, serum proteins, milk fat globule proteins, transferrin, lactoferrin, \u03b22-microglobulin, an array of enzymes, and numerous proteolytic products.\n\nThere has been considerable effort to increase the supply of milk and the diversity of milk products, as well as to characterize the biological activities of milk components. This work has been accelerated by rapid advances in genetic and genomic methods, which have allowed (1) the utilization of comparative genomics to characterize the bioactivities in bovine milk and (2) the accurate selection of specific traits occurring in the natural population. More controversially, such technologies also allow the precise engineering of valuable traits tailored to potential market segments.\n\n### Milk Genomics: A Contemporary Approach to Milk Composition\n\nTrends in the consumer market are driving an increased focus on the health and nutritional functions of food products, and there is an increasing global demand for milk products. One major focus in dairy research is increasing the quantity and improving the quality of milk proteins, which are rich in nutritive and bioactive components. Production of milks with optimized quantities of specific milk proteins may therefore be particularly valuable to dairy processors and suppliers, enabling the provision of specialist milk to precise specifications. The completion of the bovine genome sequencing project has accelerated the ability to define milk protein composition, abundance, and function. In addition, the availability of the bovine genome sequence has facilitated the ability to discover naturally occurring genetic variation, which may influence milk protein quantity and composition.\n\nMilk has been a subject for nutrition and dairy science research for many years, but especially during the post-Second World War period, when it received significant support from the British government for its role as a major source of public nutrition. Increasingly, milk is the source of animal protein of choice worldwide because of its ease of handling and excellent reputation as a high-quality food source. Industrial processing of milk into dairy products has been based largely on the insights into milk chemistry that arose from seminal studies conducted during this period. These studies were concerned primarily with nutritional and processing properties of the major milk proteins. Recent advances in bovine genomics, including the sequencing of the bovine genome and the development of related bioinformatics tools, now extend the capacity via analysis of sequence data and gene expression to understand the properties of milk proteins at a more detailed level and, along with parallel advances in humans and other animals, to open a window on the evolutionary origins and specific benefits of milk. This new knowledge will also enable the discovery of animals with naturally occurring beneficial genetic variations. The study of milk genomics has been bolstered in recent years through the development of the Milk Genome Consortium in 2004 (http:\/\/milkgenomics.org), which endeavors to understand the biological processes underlying milk genomics, as well as the function of, and health traits conferred by, milk proteins.\n\n### Advances in Bovine Genome Science\n\nGenomics is the study of an organism's gene sequence, gene expression, and gene function. Genomic tools can be used to study population diversity, the understanding of which provides strategies for the genetic selection of desirable traits. Genome science has developed rapidly in the past decade as a result of a vast and rapid expansion of genome sequence data, initially arising from the human genome sequencing project, and the concomitant development of applied information technology (bioinformatics). In December 2003, only months after the completion of the human genome sequencing project, an international consortium formally declared the initiation of the bovine genome sequencing project. The human genome sequencing project had resulted in the establishment of significant infrastructure, expertise, and technological advances. Consequently, sequencing of the bovine genome progressed quickly and was completed in 2005, well ahead of the original schedule. Since that time, the development of high-throughput, 'next-generation' sequencing technologies has made whole genome sequencing accessible to many research groups. Genome analysis of individuals is well underway in humans and is being progressively applied in other species, including dairy cows, with a number of large-scale private and public bovine genome sequencing projects currently in progress. A major focus of these projects is the identification of naturally occurring genetic variation, which will facilitate opportunities for discovery of proteins or peptides with novel or improved function and will provide the ability to use advanced breeding technologies to optimize the composition of milk for specific product requirements.\n\n### DNA Sequence\n\nBovine genetics, with a focus on the identification of genetic variation underlying quantitative trait loci (QTL) or linked to important biological phenotypes, has been an active area of study for many years (Khatkar et al., 2004). Using classical genetic and molecular genetic approaches, researchers had mapped over 1000 bovine genes by the mid-1990s. A phase then followed that was based on the positional mapping of candidate genes and on the creation and sequencing of bovine expressed sequence tag (EST) libraries. Subsequently, the sequencing strategy employed for the bovine genome sequencing project combined the use of bacterial artificial chromosome sequences with a whole genome shotgun sequencing approach, using an individual Hereford cow as the source of DNA. The total sequencing effort provided approximately 7x coverage of the genome, representing about 2.7 billion base pairs (reviewed in Tellam, 2007). The completion of the bovine genome provided a foundation for developing bovine-specific post-genome research tools and applied industrial applications. For example, the Bovine Genome Database provides an online resource for the analysis and interpretation of the bovine genome (Childers, 2011). Further discussion of the tools and their application to the study of milk protein composition is included below.\n\n### Genome Map\/Structure\n\nThe bovine genome comprises 29 autosomes, plus the X and Y chromosomes. An international bovine bacterial artificial chromosome (BAC) map was produced to assist in assembling the genome, complemented by an integrated mapping strategy using radiation hybrid and existing physical mapping data. A virtual assembly was released in 2005, and the first de novo assembly was released in 2006. More complete genome builds followed (Zimin et al., 2009; Elsik et al., 2009). There are two current genome assemblies available: Btau_6.1 and UMD3.1. The UMD3.1 bovine genome was derived from a total of 36.82 million reads and 2.649 billion bp. Of these, 99% (2.64 billion bp) have been mapped to chromosomes. A total of 26,618 genes are predicted, coding for an estimated 42,806 proteins (<http:\/\/www.cbcb.umd.edu\/research\/bos_taurus_assembly.shtml>).\n\nComparative genetic maps have revealed significant similarity (synteny) between the bovine genome and genomes of other species (Elsik et al., 2009). This can be effectively viewed using the OXGRID display at <http:\/\/oxgrid.angis.org.au\/>, and as described by Elsik et al. (2009). The synteny between human and bovine genome organization is greater than that between the human and rodent genomes. Further, the mean length of syntenic segments between bovine and human is approximately twice as long as the average length of conserved syntenic segments between human and mouse. The sequencing of the bovine genome and subsequent comparative analysis allowed the discovery of 124 evolutionary breakpoint regions, the majority of which were cattle- or ruminant-specific. Thus, genome sequencing has greatly facilitated further understanding of the evolutionary biology of mammals and lactation.\n\n### Polymorphism\n\nUnderstanding the naturally occurring bovine genetic polymorphism was a primary goal of the genome sequencing project. An understanding of such variation underlying milk proteins provides an opportunity to capture the diversity in milk protein function. Such functional variation could affect the processing characteristics of the milk or change the nutritive properties of the resultant milk product. The Bovine HapMap Consortium resulted in the discovery of a large number of single-nucleotide polymorphisms (SNPs). This effort has been extended to re-sequence six breeds: Holstein, Angus, Jersey, Brahman, Limousin, and Norwegian Red, at low coverage, with the aim of identifying genome-wide SNPs (Tellam, 2007). In addition, mining of existing ESTs has contributed a putative 17,344 SNPs (Hawken et al., 2004).\n\nInitial analysis of dairy herds using a subset of SNPs revealed that the predominant cattle breed used internationally for milk production, Holstein Friesian, has low genetic diversity and may be considered to be a single population unit (Zenger et al., 2007). A subsequent report (Gibbs et al., 2009) found that Bos taurus had a large ancestral population size with an SNP diversity similar to that of humans. Selection for domestication has reduced the effective population size of the breed to small numbers. This has implications for animal breeding. The length of linkage disequilibrium blocks was found to be smaller than expected, but larger than that of humans (Gibbs et al., 2009).\n\nThe genes and genetic variations underlying many milk composition traits have subsequently been identified using the molecular tools that were developed as part of the genome sequencing projects. One of the most significant was the discovery of a variant in the bovine acylCoA:diacylglycerol acyltransferase (DGAT1) gene that affects the production and composition of both milk protein and milk fat (Grisart et al., 2002). This was the first time the causative mutation for a bovine QTL had been identified. Subsequently, a large number of genes and mutations have been identified for milk composition traits as diverse as fatty acid saturation (Mele et al., 2007; Moioli et al., 2007), milk fat color (Berry et al., 2009), and IgA content (Berry et al., 2013).\n\n### Functional Genomics\n\nThe bovine genome sequencing project has also led to improved gene annotation, facilitating the ability to study bovine gene expression and gene function in more detail than was previously possible.\n\nGene-expression tools allowing the study of mammary gland gene expression, and consequently lactation functional genomics, include (among others) microarray platforms and next-generation sequencing technologies. Microarray platforms have been used for many years to measure global gene expression in tissues of interest (Dalma-Weiszhausz et al., 2006; Hoheisel, 2006); to determine changes in gene expression in response to physiological stimuli (for example, Littlejohn et al., 2010); and to identify predicted milk polypeptides, which through subsequent validation and functional analysis may become targets for the development of new dairy ingredients or products. The use of microarrays is being gradually phased out by the decreasing cost of next-generation sequencing technologies and the concomitant development of the RNA sequencing (RNA-seq, or Whole Transcriptome Shotgun Sequencing) approach (Wang et al., 2009). This approach is essentially the application of next-generation sequencing technologies to RNA and enables the identification of transcript abundance, alternatively spliced isoforms and individual-specific sequence variants. RNA sequencing has been used effectively to conduct transcriptional profiling of milk (Wickramasinghe et al., 2012) and to identify differential expression of genes related to oligosaccharide metabolism within lactating mammary tissue, despite the abundance of transcripts from the major milk proteins (Wickramasinghe et al., 2011). It has also allowed identification of genetic variation associated with citrate content in milk (C\u00e1novas et al., 2013), as well as identification of miRNA molecules associated with lactation (Li et al., 2012). miRNAs are small nonprotein-coding RNAs that control expression via base pairing to specific transcripts, targeting them for degradation. They are part of a mechanism involving a variety of small RNAs for controlling expression following transcription.\n\nA complementary approach to gene-expression studies of the mammary gland is provided by milk proteomics (Smolenski et al., 2007; Rawson et al., 2012). Cow's milk comprises approximately 3.5% protein, 80% of which are caseins; the remaining proteins are predominantly the major whey proteins, but there are also a large number of low-abundant proteins, some of which have significant bioactivity or useful nutritional properties. This area has also benefited markedly from the bovine genome sequencing project, and the sequencing of comparative genomes, through the detailed annotation and cataloguing of genes, genetic variations, and proteins. It is now possible to identify variants of both major and low-abundant proteins in milk based on bovine protein sequences. Together with rapid advances in mass spectrometry, particularly improvements in sensitivity and quantification, and complemented by gene-expression data, the protein composition of milk can now be dissected with great accuracy and the variation can be captured using breeding techniques.\n\n#### Transgenic Technologies for Milk Manipulation\n\nAnimals produced using transgenic approaches are a potential source of increased milk production or specifically altered milk composition. The original transgenesis methodology applied to farm animals involved the random integration of a transgene at any location within the genome and was heralded by the production of cattle that expressed human lactoferrin (Krimpenfort et al., 1991). It was originally thought that the use of yeast and bacterial artificial chromosomes capable of conveying large gene constructs up to a Mb may be useful in supporting the overexpression of the 200 kb casein locus (with all four caseins), including the regulatory elements co-coordinating their mammary-specific expression (Zuelke, 1998). To date, there have been only a limited number of successes (Brophy et al., 2003; Laible et al., 2007) including examples where both \u03b2-casein and \u03ba-casein were over-expressed. Since that time, considerable improvements have been made in the methodology for generating transgenic animals (for reviews, see Garrels et al., 2012; Gaj et al., 2013). These methods utilize enzymatic approaches via RNA-guided DNA endonucleases (Zinc-fingers, transcription activator-like effector nucleases, TALENs) and clustered, regularly interspaced short palindromic repeats of Cas9 nuclease (CRISPR\/Cas9) that cause site-specific double-stranded breaks. These breaks subsequently stimulate errors in repair, which can then inactivate genes or facilitate homologous recombination of a sequence delivered along with the enzyme coding vectors or RNA. Through use of these methods combined with nuclear transfer or 'cloning,' the potential therefore exists to make any milk composition as long as the resulting trait is biologically possible and does not harm the animal in any way. This includes the potential for the complete removal of specific genes or the full 'humanization' of bovine milk composition (Wilmut et al., 1997). The major advantage of these improved approaches is that (depending on the target) they can be applied in such a way that leaves no exogenous sequence, potentially overcoming one of the public's concerns about genetic modification technologies. Another genetic modification approach, recently applied to the knockdown of bovine \u03b2-Lg (Jabed et al., 2012), is the use of small RNA molecules, specifically designed for a particular transcript, to target the endogenous transcripts, consequently inhibiting gene expression. (For a review of this technology, see Filipowicz et al., 2008; Valencia-Sanchez et al., 2006.)\n\n### Comparative Milk Genomics\n\nThe bovine genome sequencing initiative was not the only project to follow the human genome sequencing effort. As this initial project abated, international consortia formed to sequence the genomes of other organisms, including the chicken, dog, opossum, chimpanzee, macaque, and platypus. A comparison of the genome sequences of 29 eutherian species was reported in 2011, as a result of the effort by the 29 Mammals Genome Project (Lindblad-Toh et al., 2011). The diverse range of species included in the project was selected to maximize the total branch length of the evolutionary tree. The 29 Mammals Project has facilitated the ability to understand the origin and evolution of lactation strategies and milk protein composition, which vary considerably across different species. It also complements a growing body of work on whole genome expression studies in mammary tissue of different species, including cattle, humans, and mice. For example, Lemay et al. (2009) compared the genomes of five mammalian species and showed that the more divergent milk proteins were associated with nutritional and immunological functions, whereas the more conserved milk proteins were associated with milk secretory processes. Further insight into the evolution of milk synthesis and lactation has been provided by the completion of the kangaroo genome sequence in 2011 (Renfree et al., 2011).\n\nAligned with the goal to develop a deep understanding of milk proteins, a comparative approach to assess the genomic basis of diversity in milk composition between species has been developed (Menzies et al., 2009). Experimental models have been chosen on the basis of reproductive\/lactation strategy and have been cross-referenced to the dairy cow. The study is based on gene expression profiles and integration with genomic mapping information and dairy QTL analysis. Within this framework, a specific methodology has been applied to identify those genes that may code for milk proteins that are either inherently active in milk or could be developed as functional dairy products.\n\nOne example of comparative genomics to alter milk protein composition is the use of functional genomics to exploit animal models, with extreme adaptations to lactation as an alternative approach to identifying genes that regulate protein synthesis in the mammary gland. For example, milk protein is the only component of milk that is synthesized de novo in the mammary gland of the Cape fur seal (in comparison to other mammals, where both milk fat and milk carbohydrate components are synthesized, at least in part, de novo in the mammary gland). The protein content of this seal milk is more than double that of bovine milk (Cane et al., 2005). Understanding the molecular basis for such a lactation strategy may provide insight into mechanisms to increase the protein content of bovine milk. In another model organism, the Tammar wallaby, both milk protein concentration and milk protein yield are dramatically increased in the latter half of lactation (Nicholas et al., 1997). Changes in global gene expression comparing mammary tissues from the Tammar wallaby at early and late lactation using a cDNA array (see Chapter 3), and comparing pregnant and lactating seals and dairy cows using an Affymetrix microarray, indicate that folate metabolism, particularly the role of the folate receptor, may be a crucial regulatory point of milk protein synthesis in mammary epithelial cells (Menzies et al., 2009).\n\nFolate may influence milk protein production, either by producing a direct effect on the mammary gland or by reducing the competition for precursors between gluconeogenesis and methylneogenesis, thereby increasing the metabolic efficiency of the mammary gland. The importance of the folate receptor 1 (or \u03b1) for the cellular uptake of folate was established by analyzing renal folate handling in mice that had targeted gene knockouts of folate-binding proteins 1 and 2 (folbp1 and folbp2, equivalent to human and cow folr\u03b1 and folr\u03b2) (Birn, 2006). Molecular studies in human, monkey, and mouse cell lines have shown that the folate receptor \u03b1 mediates cellular uptake of folate and that folate receptor abundance is regulated by multiple mechanisms. Furthermore, the folate receptor population may play a crucial role in the capacity of mammary epithelial cells to respond and utilize circulating serum folate.\n\nThe folate requirements of lactating dairy cows have been extensively reviewed by Girard et al. (2005) and by Regalle et al. (2009). Lactation increases the demands both for methylated compounds (synthesis of milk choline, creatine, creatinine, and carnitine) and for methionine to support milk protein synthesis (Xue & Snoswell, 1985; Girard & Matte, 2005). The high demand for folate during lactation is demonstrated by the observation that total serum folates decrease by 40% across the lactation period in dairy cows (Girard et al., 1989).\n\nFolate supplement experiments in dairy cows have shown a positive response to milk production and milk protein yields (Girard & Matte, 1998; Girard et al., 2005; Graulet et al., 2007). This positive response was not consistent among all experimental cows in the first two studies and appeared to be dependent on cow lactation status (primiparous were nonresponsive, multiparous were responsive), stage of lactation, and serum vitamin B12 status. The more recent study by Graulet et al. (2007) including vitamin B12 supplementation suggested that the increases in milk and milk protein yields as a result of folic acid supplements were not dependent on the supply of vitamin B12 and were probably closely related to the role of folic acid in the DNA cycle. However, this does not diminish the importance of the role that folate metabolism plays in milk protein synthesis. This is an example of how the combined approach of comparative genomics and bioinformatics, plus species with extreme adaptation to lactation, may be utilized to identify key genes and cellular processes involved specifically in modifying milk composition and milk protein production.\n\n## Origins of milk proteins\n\nThe study of milk protein genes and encoded milk proteins of domesticated mammals and undomesticated mammals alike has made these data relevant to dairy science. The data highlight adaptive features and the diversity of milk protein genes over evolutionary time. Evolutionary evidence suggests that complex lactation preceded divergence of the extant mammalian lineages. Monotreme lactation appears to be the most ancestral form of lactation, whereas marsupials and eutherians developed divergent strategies. Study of the evolution of milk proteins helps determine the significance of maintaining certain proteins while changing or losing others. These differences may reflect divergence of lactation strategies and may provide clues as to why some proteins have become nonessential and are in the process of being lost by some eutherians.\n\nBioinformatic predictions of genes encoding secreted proteins upregulated during lactation in the mammary gland suggest that as many as 300 different proteins may be found in milk. However, caseins are the major proteins in milk, representing about 80% of total milk protein content. Expression of a number of casein genes has been confirmed in monotremes (platypus and echidna) and marsupials. Three evolutionarily related calcium-sensing casein-like genes and the functionally related \u03ba-casein were identified in monotremes. As in other mammals, a number of casein splice variants could be identified. Interestingly, all of these casein genes cluster tightly within a 100-kb region of the platypus genome, a physical linkage that has been observed in the casein locus of all mammalian genomes examined so far (Rijnkels, 2002). The respective organization of the casein locus is also highly conserved in all mammalian lineages. Casein genes occur in the order \u03b1S1-, \u03b2-, \u03b1S2-, and \u03ba-, where the \u03b2-casein gene is encoded in the opposite direction from the other genes. The close proximity of \u03b1S1\\- and \u03b2-casein genes in an inverted tail\u2212tail orientation and the relative orientation of the more distant \u03ba-casein genes are similar in all mammalian genome sequences available so far. This observation suggests that the synchronized tissue-specific expression of casein genes in the lactating mammary gland may be controlled, in part, by a common mechanism at this locus. However, genome sequence analysis has also revealed variation in the \u03b1S2-casein gene content of mammalian lineages. Marsupials possess only one copy of \u03b1-casein corresponding to \u03b1S1-casein (Lefevre et al., 2007; ). In the monotreme lineage, a recent duplication or a gene-conversion event involving \u03b2-casein occurred to produce an \u03b1-like casein gene at a genomic location similar to that of the eutherian \u03b1S2-casein. A similar but more ancestral duplication of \u03b2-casein occurred in eutherians to produce \u03b1S2-casein, and further duplications have also occurred since, in human and mouse lineages, to produce \u03b1S2a\\- and \u03b1S2b-casein genes.\n\n\u03b2-Lg is the major whey protein in ruminant milk and is present in milk from a variety of species. However, it is absent in rodents, such as the mouse; lagomorphs, such as the rabbit; and humans, although reports to the contrary exist. \u03b2-Lg in human milk probably arises either from spurious antibody cross-reactivity (Brignon et al., 1985) or from the presence of ingested bovine \u03b2-Lg in the mother's milk (Fukushima et al., 1997). Not all primate milks are devoid of \u03b2-Lg, as both the macaque and the baboon have the protein within their milk (Hall et al., 2001).\n\nBecause of the absence of \u03b2-Lg in human milk, the dairy industry has developed ways in which to convert the protein composition of bovine milk to something more similar to the protein composition of human milk or remove it altogether using transgenic approaches. Genetic variants have been observed in essentially all of the species from which the protein has been found. There are several genes encoding \u03b2-Lgs within the different species. Sequence analysis of species genomes suggests that some species such as the dog and the cat have three \u03b2-Lg genes, whereas others such as the horse and the donkey have two \u03b2-Lg genes, bovine, sheep, and goat have one \u03b2-Lg gene, and bovine has a pseudogene. Marsupials, monotremes, and some primates have one \u03b2-Lg gene. Comparison of \u03b2-Lg protein sequences shows high conservation of the protein during evolution. Bioactive peptides derived from \u03b2-Lg are currently being intensively studied. These peptides are active only once they are released from the precursor protein. Once released, it is suggested that these peptides play important roles in human health, including having antimicrobial, antihypertensive, and antioxidant activities (Hernandez-Ledesma et al., 2008; Saito et al., 2008). The variation of gene number and protein content in milk represents a specific evolution of \u03b2-Lg for each lineage. It has not yet been explained how a major protein in cow's milk such as \u03b2-Lg can be completely abolished in human, rat, or rabbit milk with little to no consequence to the offspring.\n\nWhey acidic protein (WAP) is a major whey protein that is found in the milk of numerous species such as the mouse (Hennighausen & Sippel, 1982), rat (Campbell et al., 1984), pig (Simpson et al., 1998), rabbit (Devinoy et al., 1988), camel (Beg et al., 1986), wallaby (Simpson et al., 2000), brushtail possum (Demmer et al., 2001), echidna, platypus (Sharp et al., 2007), and dog (Seki et al., 2012), but it is absent in the milk of cows, sheep, and humans (Hajjoubi et al., 2006). WAP pseudogenes have been identified in the bovine and the human. A nucleotide deletion at the end of the first exon is reported to have caused a truncation of the bovine WAP protein (Hajjoubi et al., 2006). This deletion is also found in ovine and caprine species. In comparison, it is the absence of an ATG initiation codon in human WAP that is suggested to be the cause of the pseudogene in this species. Interestingly, the polyadenylation signal AATAAA is still present in the ruminant sequence but not in the human sequence. WAP proteins from different species show moderate sequence similarity; however, they differ in the number of four-disulfide core (4-DSC) motifs between the species. Each specific 4-DSC domain is recognizable by sequence similarity, and it is possible to trace the origins of each domain during evolution (Fig. 4.1). Although not expressed, both bovine and human pseudogenes have similar gene structure to the eutherian WAPs, and it is suggested that these were once functional. As suggested in Chapter 2, the function of the WAP proteins appears to differ with the presence or absence of each of these domains. It is interesting to speculate why this protein appears to be dispensable in some species.\n\nFigure 4.1 Schematic representation of the origins of WAP gene structure. (a) Boxes represent exons within the WAP genes. Each 4-DSC domain is encoded by different exons as indicated. In eutherian WAP, exons 2 and 3 encode the two 4-DSC domains (DI and DIIA). In marsupial WAP, there is an additional exon, and exons 2, 3, and 4 encode the three 4-DSC domains (DIII, DI, and DIIB). In echidna WAP, exons 2 and 3 encode the two 4-DSC domains (DIII and DIIA). In platypus WAP, exons 2, 3, and 4 encode the three 4-DSC domains (DIII, DIIA, and DIIB). Comparison of sequence similarity suggests that DIIB of marsupial WAP and DIIB of platypus WAP are derived from similar exons and that DIIA of eutherian WAP and DIIA of monotreme WAP are derived from similar exons. (b) Schematic representation of stepwise evolutionary history of 4-DSC domains of WAP within the major lineages. The ancestral progenitor, represented by six exons (numbered) and four 4-DSC domains (DIII, DI, DIIA, DIIB), is shown, followed by events of exon loss leading to the evolution of the present-day WAPs in platypus, echidna, marsupial, and monotremes. Numbers represent the order of exons relative to the ancient WAP. Adapted from Sharp et al. (2007).\n\nThe apparently dispensable role of proteins such as \u03b2-Lg and WAP may be related to the diet of the young, which is being adequately compensated in more domesticated animals or may be linked to improved care and hygiene in domesticated life.\n\n## Constraints and opportunities for evolution or manipulation of bovine milk proteins\n\nThe gene and protein structures and sequences of the six major bovine milk proteins (the four caseins, \u03b2-Lg, and \u03b1-La) have been well characterized, although the precise secondary and tertiary structures of the caseins have been difficult to demonstrate experimentally, as they have proven to be challenging to investigate via x-ray crystallography. The casein phosphoproteins precipitate at pH 4.6, which is a significant characteristic that is utilized in cheese production, but, in milk, these proteins aggregate to form protein aggregates or micelles of \u224850\u2212300 nm in diameter. \u03ba-Casein stabilizes the surface of the protein micelles, with the other caseins forming the core, which is stabilized by the phosphorylation of seryl residues. The organic phosphate moieties in turn enable calcium binding, which also stabilizes the micelle structure and, hence, as well as being a significant source of dietary amino acids, the micelle aggregates also form a valuable source of calcium and phosphorus for neonatal nutrition and development. The casein micelle is considered in detail in Chapter 6.\n\nApart from the neonatal nutritional and industrial (dairy products) significance of bovine milk proteins, a wide range of biological activities is observed in the neonate as well as the dam. These activities are attributed to both the intact proteins and, more importantly, to their products of in vivo digestion (see the review by Meisel, 2005; Ricci-Cabello et al., 2012; Pepe et al., 2013). When considered together, bovine milk proteins, which provide neonatal nutrition, regulate physiological processes, coordinate neonatal development, and form the basis of an internationally significant agricultural commodity market, are globally important proteins. For this reason, their functional characteristics need to be considered when evaluating the potential for enhancement or modification of their structure or biosynthesis.\n\n### The Detailed Structure of Milk Proteins and Their Genes\n\nAn understanding of the detailed structure of milk proteins and their genes is critical in the consideration of the impact of modifying bovine milk proteins to increase productivity, to enhance milk product manufacturing efficiency, or to exploit inherent or proposed bioactivity. The physical and structural characteristics of bovine milk proteins have been extensively reviewed (Farrell et al., 2004) and are covered in Chapter 7 of this volume. The genomics coordinates and gene information given below are from the Btau_6.1 annotation (National Center for Biotechnology Information; www.ncbi.nlm.nih.gov\/).\n\nThe bovine casein genes are present as single-copy genes per chromosome and are clustered together on chromosome 6 in a 200-kb region at position q31\u221233 in the order \u03b1S1-, \u03b2-, \u03b1S2-, and \u03ba\\- (Threadgill & Womack, 1990). The bovine \u03b1S1-casein gene (CSN1S1 casein alpha s1 Gene ID: 282208) is 21,900 bp in length and contains 19 exons. The protein is the most abundant of the bovine caseins, present at 12\u221215 g\/L. There are up to eight reported genetic variants of this protein, with the B variant being the most common in Bos taurus species and comprising 199 amino acids with a molecular weight of 23.6 kDa. The \u03b1S1-casein protein may contain up to eight serine monophosphate residues, which cluster in a hydrophilic domain between amino acids 43 and 80 and, through modeling studies, are thought to be connected to a hydrophobic domain (amino acids 100\u2212199) by helical and sheet secondary structures. The less abundant bovine \u03b1S2-casein gene (CSN1S2 casein alpha-S2 Gene ID: 282209) is 18,479 bp in length and contains 18 exons, with significant exon duplication (Mercier & Vilotte, 1993). This gene codes for 207 amino acids, with a molecular weight of 25.23 kDa. This protein exhibits a number of phosphorylation states as well as four variants that have been observed in common bovine populations.\n\nIn contrast, the bovine \u03b2-casein gene (CSN2 casein beta Gene ID: 281099) is significantly less complex, containing nine exons in 8505 bp. The protein is present at 9\u221211 g\/L in modern dairy cattle, consists of 209 amino acids, and has a molecular weight of approximately 24 kDa. Up to 12 genetic variants have been observed and the protein exhibits a flexible structure that can adopt multiple conformations, although a small amount of secondary structure in the form of \u03b1 helices and \u03b2 sheets has been predicted. There are five phosphoserine residues that reside at the N-terminal domain of the protein. \u03b2-Casein can self-assemble into micelles in neutral and acidic environments at room temperature (Moitzi et al., 2008). The bovine \u03ba-casein gene (CSN3 casein kappa Gene ID: 281728) exhibits fewer structural similarities than the other casein genes, contains five exons, and is approximately 13,700 bp in length and not completely annotated on the Btau 6.1. build. It has diverged from the fibrinogen gene family but is more highly conserved between species than the other casein genes. Two common variants are observed in dairy cattle (although several others have been reported), with the B variant containing 169 amino acids with a molecular weight of 19 kDa. Up to six threonine residues may be glycosylated, and, similar to the other caseins, \u03ba-casein has a relatively high proline content that influences the predicted secondary structure of two sets of antiparallel \u03b2 sheet structures exhibiting hydrophobic side chains that may be important for micelle formation. As with the other caseins, the detailed structure of bovine \u03ba-casein is poorly characterized, although the N-terminal domain of amino acids 1\u221244 has been experimentally shown to contain a helical structure (Bansal et al., 2006).\n\nThe bovine \u03b1-La gene (LALBA lactalbumin, alpha Gene ID: 281894) contains four exons and is 2022 bp in length. It is situated on bovine chromosome 5 and exhibits significant interspecies homology in exon 4 (significant for its interaction with galactosyltransferase) and the 5' untranslated region. Two variants have been characterized for the protein, with the most common B variant exhibiting a molecular weight of 14.1 kDa and being present at 0.6\u22121.7 g\/L in skim milk. The 123 amino acids are organized into four helical domains and a small sheet structure that is dependent on pH and metal ion concentrations (Sawyer & Holt, 1993). Different glycosylation states have been observed, and structurally bovine \u03b1-La has some similarity to lysozyme proteins (Brew et al., 1970). This protein, like the caseins, also has calcium-binding properties, as described by Hiraoka et al. (1980).\n\nBovine \u03b2-Lg (PAEP progestagen-associated endometrial protein\/ \u03b2-lactoglobulin Gene ID: 280838) is located on chromosome 11, contains four exons within 4893 bp, and is well conserved between species (Folch et al., 1994). Structural similarities occur throughout the lipocalin protein family, and \u03b2-Lg has been shown to bind to a number of hydrophobic ligands, including palmitic acid (Kontopidis et al., 2004; Konuma et al., 2007). There are two common variants of this protein, A and B, with the protein itself containing 162 amino acids, with a molecular weight of 18.3 kDa. It exhibits a number of \u03b2 strands, which form a \u03b2-barrel structure that contains the hydrophobic-ligand-binding site, and a single \u03b1 helical domain is observed at the C-terminal end of the protein.\n\nThe secondary structures of the major bovine milk proteins are diverse, yet are clearly critical for the functions of the proteins, including micelle formation, calcium and phosphate binding, and other bioactivities. Although there may be opportunities for manipulation of structure and therefore function of these proteins, even modest changes to nucleotide and amino acid sequence may significantly alter protein and peptide functionality.\n\n#### The Function of Bovine Milk Proteins\n\nThe structural complexity of these milk proteins suggests that they serve many functions in the neonate, in addition to the mere provision of a source of amino acids for protein biosynthesis in the fast-developing neonate. Yet the range of functions is not likely to be as great as that in more altricial species such as the monotremes and marsupials, in which the milk constituents orchestrate almost the entire developmental process from the fetus through to attaining physiological independence (see Chapter 3). The contributions that individual proteins make to this developmental spectrum are poorly understood, although longitudinal studies on gene expression and related changes in milk composition will help unravel this relationship.\n\nThe protracted emphasis on selection of cows for milk volume has probably also influenced the balance of biological activities residing within these proteins, because contemporary dairy herds are larger in stature than their predecessors. Thus, developmental activities encoded within the protein fraction may have been altered to emphasize growth promotion rather than other key functions such as the storage of body energy reserves to support reproductive activity and even to promote maturation of the immune system.\n\nThe complexity increases further once these proteins are digested enzymatically or are hydrolyzed in the developing gastrointestinal tract and then absorbed to be transported to their site of action. The precise balance between the peptide component that is absorbed as functional entities and the peptide component that remains for further processing into short peptides as a source of dietary amino acids is not well understood. However, the need to conserve a constant supply of developmental molecules is probably more important than the need for protein synthetic substrate in sustaining the development of the neonate.\n\nThe observation that over 100 hormones and growth factors have been identified in colostrum or milk (Koldovsky, 1996) would suggest that the milk proteins may play important roles, working in concert with these established endocrine\/paracrine signals. Therefore, the sustained selection of animals for milk volume output will have altered the balance of bioactivities in colostrum and milk to account for this response. Thus, the balance of bioactive peptides may differ from those in species in which this selection pressure has been absent. From a commercial point of view, these milk constituents also play a role in determining the sensory properties of milk and its manufactured products. Of course, these same sensory factors probably also serve as attractants to the newborn as it seeks its nutrient supply from the udder.\n\nThe extraction of intact proteins of high commercial value from milk has been seen as a lucrative commercial opportunity by dairy farmers who have incomes tied to the ebb and flow of the commodity markets. The profitability of such ventures depends on the ease of extraction, the relative concentration of the protein in milk, and the value placed on the bioactivity. A range of these proteins is presented in Table 4.1.\n\nTable 4.1\n\nMajor Milk Proteins with Potential for Extraction on a Commercial Scale to Exploit Bioactivities\n\nProtein | Concentration in colostrum (g\/L) | Concentration in milk  \n(g\/L) | Molecular weight | Exploitable bioactivity  \n---|---|---|---|---  \nCaseins (\u03b1s1-, \u03b1s2-, \u03b2\\- and \u03ba-) | 26 | 28 | 14,000\u221222,000 | Ion carriers, bioactive peptide sources, immunomodulators  \n\u03b2-Lactoglobulin | 8.0 | 3.3 | 18,400 | Antioxidant, vitamin-A-binding protein, bioactive peptide source  \n\u03b1-Lactalbumin | 3.0 | 1.2 | 14,200 | Lactose synthesis and milk volume. Calcium binding, immunomodulator, bioactive peptide source  \nImmunoglobulins | 20\u2212150 | 0.5\u22121.0 | 150,000\u22121,000,000 | Specific immune protection, potential bioactive peptide source  \nGlycomacropeptide | 2.5 | 1.2 | 8,000 | Antimicrobial, bifidogenic, gastric modulator, anticlotting factor  \nLactoferrin | 1.5 | 0.1 | 80,000 | Antimicrobial, antioxidant, anti-inflammatory, anticarcinogenic, bioactive peptide source, immunomodulator  \nLactoperoxidase | 0.02 | 0.03 | 78,000 | Antimicrobial, immunopotentiator  \nLysozyme | 0.0004 | 0.0004 | 14,000 | Antimicrobial, immunopotentiator  \nSerum albumin | 1.3 | 0.3 | 66,300 | Bioactive peptide source\n\nAdapted from Korhonen & Pihlanto, 2007.\n\nBoth \u03b1-La and \u03b2-Lg have been isolated on a large scale using a range of chromatographic, ultra-high-pressure, and membrane separation techniques (Korhonen & Pihlanto, 2007). Both of these proteins perform important activities (Table 4.1) that provide the impetus for large-scale production. One of the most important clinical findings is the potential ability of these whey proteins to decrease the incidence of certain cancers (Pepe et al., 2013).\n\nMany of the highly expressed proteins clearly have important immunological functions. The passive immunity provided through the immunoglobulin complement of colostrum to the neonate is supplemented by a range of other bioactivities designed to countenance pathogen loads to which the newborn is exposed in its new environment. The activity of these proteins extends to the initiation of mucosal immunity in the gut and an optimal environment for beneficial microflora. Such proteins include immunoglobulins, many chemokines, mucin 1, lactoferrin, cathelicidin 1, lactoperoxidase, S100 calcium-binding proteins, complement proteins, and polymeric immunoglobulin receptor.\n\nIt is beyond the scope of this review to detail all of the bioactivities identified in milk. However, some specific proteins that are illustrative of principles found across the milk proteome are described below.\n\nLactoferrin has dual roles of scavenging iron and acting as a bacteriostat. It is found in exocrine secretions, including milk, tears, tubotympanum and nasal exudate, saliva, bronchial mucus, gastrointestinal fluids, cervicovaginal mucus, and seminal fluid, all of which require antibiotic activity. It is also found in neutrophils that are attracted to sites of infection where both iron sequestration and antibiotic activity assist in the healing process. In addition to these antibacterial effects, lactoferrin has significant commercial value because of its antiviral, antioxidant, antitoxigenic, antithrombotic, and, importantly, anticarcinogenic activities. It is important to note that lactoferrin is just one component of a mix of protective molecules in milk and other exocrine secretions, as lysozyme, beta defensins, and the surfactant proteins A and D (SP-A, SP-D), among others, also contribute to this important function.\n\nTypically, lactoferrin is found in concentrations in excess of 100 mg\/L in bovine colostrum and is extracted by conventional fractionation and chromatography. However, the expression of milk proteins in cereal grains such as rice, using recombinant DNA technology (Lonnerdal, 2006) is another possible way of providing these unique milk proteins in our diet. Similarly, recombinant human lactoferrin given orally has been suggested as a mechanism for the prevention of certain gastrointestinal infections in preterm infants (Sherman & Petrak, 2005). More recently, bovine lactoferrin has been tested as a therapeutic approach for diarrhea in children (Ochoa et al., 2013) and as a way to prevent fungal infections in low-birth-weight infants (Manzoni et al., 2012) and, using a recombinant form of human lactoferrin (Talactoferrin), sepsis in adults (Guntupalli et al., 2013). Incidence of diarrhea did not diminish, but longitudinal prevalence and severity were decreased. Oral delivery of bovine lactoferrin reduced the incidence of invasive fungal infections in preterm infants. Talactoferrin reduced mortality in patients with sepsis. There is little doubt that demand for the use of lactoferrin as a therapeutic agent will increase. Without the development of the ability to produce sufficient quantities in milk, transgenic plants or even recombinant bacterial production (Garcia-Montoya et al., 2013) may well supersede milk as a source of this and other unique bioactive milk proteins.\n\n#### Bioactive Peptides Sequestered within Milk Proteins\n\nMilk proteins contain latent biofunctional peptide sequences within their primary structures that exert physiological responses in vivo. They remain latent until released through the processes of enzymatic processing in the gastrointestinal tract. A large range of these peptides have been identified, including opioid, antimicrobial, immunomodulatory, mineral-transporting, growth-promoting, anticancer, and proteinase and angiotensin-converting enzyme (ACE) inhibitory peptides (Shah, 2000; Ferranti et al., 2004; Ricci-Cabello et al., 2012). In an evolutionary sense, the cow is targeting peptides to specific effector sites through the provision of such peptides to the neonate. Presumably, if the availability of metabolic substrates to support maximal biological response is inadequate, then the animal has the ability to limit the quantities of peptides released by suppressing the release of rate-limiting enzymes.\n\nThe biofunctional peptides currently most studied in food proteins appear to be those that inhibit ACE. This enzyme plays a central role in the regulation of blood pressure through production of the potent vasoconstrictor angiotensin (Ang) II and the degradation of the vasodilator bradykinin. Therefore food-sourced ACE inhibitory peptides may have the ability to lower blood pressure in vivo by limiting the vasoconstrictory effects of Ang II and by potentiating the vasodilatory effects of bradykinin (Murray & Fitzgerald, 2007; Miguel et al., 2009; Jiang et al., 2010).\n\nThe development of functional foods with antihypertensive properties provides an attractive and potentially commercially lucrative range of products.\n\nIn view of the extensive range of peptides derived from various precursor proteins that are involved in the regulation of blood pressure, it may be too simplistic to expect a functional food containing just one or two of these peptides to decrease suppressor activity in the long term. However, several products are currently being evaluated as beneficial functional foods\/food ingredients.\n\nOne success story in deriving a commercial product from milk protein tryptic digests is the casein phosphopeptides used in the chewing gum 'Recaldent' that is capable of remineralizing carious lesions in dental enamel (Cross et al., 2007). These peptides, containing the sequence \u2212Pse\u2212Pse\u2212Pse\u2212Glu\u2212Glu\u2212, where Pse is a phosphoseryl residue, stabilize calcium and phosphate ions in aqueous solution and make these essential nutrients bioavailable. Interestingly, these peptides are naturally formed during gastric digestion, making dairy products containing them nutritionally useful only in dental applications. Investigations of the chemistry of these intriguing peptides have shown that they can be altered to form casein phosphopeptide\u2212amorphous calcium phosphate, which in turn is capable of forming a calcium fluoride.\n\nRecent studies have indicated that there is benefit in incorporating dairy products into the diet with wide-reaching clinical effects (McGregor and Poppitt, 2013). Increasing consumption of dairy products, particularly dairy protein, can decrease the prevalence of metabolic disorders, including type 2 diabetes. One mechanism that has been proposed is milk protein effects on increasing insulin secretion. (See Chapter 19 for an expansion of this topic.)\n\nThe potential for the discovery of new peptides is high, and it would seem reasonable to suspect that careful screening of digestion from various segments of the gastrointestinal tract may prove to be a valuable approach for this discovery process.\n\n#### Existing Variation in Bovine Milk Proteins and the Impact on Expression Function and Milk Quality: Experimental Modifications of Bovine Milk Proteins\n\nAs described above, a number of genetic variants of the major bovine milk proteins have been observed and characterized. These variations range from minor amino acid substitutions to larger deletions. Many studies suggest that the presence or absence of a particular variant expressed within an individual may be associated with altered production or processing quality of milk (see the reviews of Jakob & Puhan 1992; Martin et al., 2002; Caroli et al., 2009). Many of these studies are based on statistical associations, but others have been well characterized to elucidate cause\u2212effect relationships. These naturally occurring variations in milk protein amino acid sequences serve as examples for the breadth of potential that modification of milk proteins may have on milk production and quality.\n\nThe relationship between genetic variants of bovine milk proteins and production characteristics has been extensively reviewed. In particular, variation in the \u03ba-casein and \u03b2-Lg genes has significant impact on milk composition (Hill, 1993; Hill et al., 1997). The characteristics observed to be associated with milk protein polymorphism include first lactation milk production, protein production and fat percentage in Californian Jersey cows (Hill, 1993; Ojala et al., 1997); milk and protein yield and fat content in Finnish Ayrshire cows (Ikonen et al., 2001); milk, protein, and fat yield in Holstein and Ayrshire cows (Lin et al., 1986); processing and manufacturing properties (Hill et al., 1997; Manderson et al., 1997; Creamer et al., 2004); the effect of nutritional regimens on milk composition (Mackle et al., 1999; Auldist et al., 2000); somatic cell count (Ng-Kwai-Hang et al., 1987); total solids and milk protein profile (McLean et al., 1984); and lifetime performance (Lin et al., 1989).\n\nYet, despite this body of evidence, milk protein genotype is only slowly being introduced into selection systems to enhance genetic gain for milk production characteristics. This is due to the concern over a potential reduction in genetic diversity, through reducing the pool of sires to select from, based on a few loci. This is now changing rapidly with the introduction and widespread use of DNA- or genomic selection-based methods (Kemper and Goddard, 2012). By using this approach, both positive and negative effects across the genome can be evaluated; therefore, at least in theory, enabling the selection of lines with altered protein characteristics without affecting overall performance or longer term genetic diversity.\n\nThe B variant of bovine \u03b1S1-casein is the most abundant variant in Western dairy populations, possibly because of indirect selection as other variants may in some way be associated with decreased \u03b1S1-casein synthesis, such as the A variant which may be subject to a potential decrease in translational efficiency because of observed polymorphism at a polyadenylation signal site (McKnight et al., 1989) or because of promoter polymorphisms that may alter transcriptional efficiency (Lum et al., 1997; Prinzenberg et al., 2003). Similarly, animals that are observed to have a lower bovine \u03b1S1-casein protein concentration in milk (heterozygous for the G allele) exhibit a lower casein content and a slower clot formation time, yet display a faster curd-firming time and a higher curd firmness (Mariani et al., 2001). In contrast to bovine \u03b1S1-casein, although there are four known variants of the bovine \u03b1S2-casein protein, few studies have shown any relationships between the genotype of this protein and milk production or quality traits.\n\nThe A1 and A2 variants of bovine \u03b2-casein are present in modern dairy populations at high frequencies, and the A2 variant has been shown in defined populations of Danish dairy cattle breeds to be associated with higher milk, fat, and protein yields when homozygous (Bech & Kristiansen, 1990). Some controversy surrounds the A2 protein genotype, as it has been suggested that human populations who consume milk containing higher concentrations of the A2 variant exhibit a lower incidence of cardiovascular disease, a lower incidence of type 1 diabetes, and decreased severity of symptoms of neurological diseases because the A2 variant does not liberate the opioid bioactive peptide casomorphin 7 upon proteolytic digestion (Laugesen & Elliott, 2003; Tailford et al., 2003; Bell et al., 2006; Kaminski et al., 2007). In a review of evidence, however, Truswell (2005) and, more recently, Clemens (2011) suggested that there was limited evidence to suggest that milk containing the A2 variant of \u03b2-casein had any significant human health advantage. Indeed, human trials suggested that there was no differential effect between dietary products containing either the A1 variant or the A2 variant on human plasma cholesterol concentrations (Venn et al., 2006) or cardiovascular health (Chin-Dusting et al., 2006).\n\nWith the significant structural role of bovine \u03ba-casein in the casein micelle, it is not surprising that there are only two common variants, A and B, in modern dairy populations, although up to nine other variants have been characterized in bovine species (Farrell et al., 2004). A number of studies have been conducted to investigate associations between \u03ba-casein protein variants and milk production and quality characteristics. Of all the genetic polymorphisms of dairy cattle affecting milk composition, the \u03ba-casein genotype is one of the more significant, with association differences identified between genotype and lifetime production (Lin et al., 1989), concentrations of individual milk proteins (McLean et al., 1984), protein yield and percentage (Tsiaras et al., 2005), as well as cheese production characteristics, including rennet clotting time, curd formation, and coagulation strength (Pagnacco & Caroli, 1987; Bittante et al., 2012).\n\nOf the 12 identified bovine \u03b2-Lg variants, only the A and B variants occur at high frequencies in dairy cow populations. The B variant has been associated with higher casein percentage (Lund\u00e9n et al., 1997; Berry et al., 2010), higher total solids (McLean et al., 1984), and higher milk yield, fat yield, fat percentage, and lactose yield (Tsiaras et al., 2005), whereas the A variant is associated with higher whey protein content and lower casein content (Auldist et al., 2000). This expression difference has been attributed to promoter variation affecting allele-specific transcription factor binding (Lum et al., 1997). The B variant association with the casein composition of milk therefore has implications for milk quality for cheese production (Boland & Hill, 2001). Of the three bovine \u03b1-La variants, the B variant is the most common in modern dairy populations, yet the A and B variants differ by a single amino acid residue. The role of this protein in lactose biosynthesis is such that there seem to be little genetic divergence and little function or associated differences observed between the common variants.\n\n#### Experimental Modifications of Bovine Milk Proteins\n\nThe experimental manipulation of patterns of expression or characteristics of bovine milk proteins has previously been reviewed (Bremel et al., 1989; Yom & Bremel, 1993; Clark, 1996). Both overexpression and impairment or 'knockout' of milk proteins have been conducted predominantly in rodent models, presumably because of the expense and technical challenges of conducting studies in bovine or other ruminant species. Although observations from experiments conducted in model species must be considered in context, valuable insight into the potential for future manipulation of bovine milk proteins in ruminant species has been gained.\n\nAlthough a number of experiments have been conducted to either overexpress or inhibit expression of milk proteins in mice and other rodents, only a few have further characterized the milk to assess the impact of manipulation of expression profiles on the quality of milk for dairy product manufacture. A transgenic mouse model overexpressing bovine \u03ba-casein (Guti\u00e9rrez-Ad\u00e1n et al., 1989) exhibited no changes in milk protein concentration, yet milk from these mice exhibited a smaller micelle size and a tendency toward stronger curd characteristics. Conversely, a mouse model in which the endogenous \u03ba-casein was suppressed exhibited a loss in micelle stability, resulting in casein precipitation in alveolar lumens, which prevented lactation (Shekar et al., 2006). These investigations highlight the significance of the role of \u03ba-casein in the tertiary structure of protein in milk regardless of species. In comparison, the manipulation of \u03b2-casein expression results in less dramatic consequences. The overexpression of bovine \u03b2-casein in a mouse model (Hitchin et al., 1996) had little observed effect on lactation, although the processing quality of milk was not assessed in this study. In another study, the inhibition of endogenous murine \u03b2-casein (Kumar et al., 1994) changed micelle size and reduced pup growth.\n\nSimilarly, in cloned transgenic cattle in which both bovine \u03b2-casein and \u03ba-casein were overexpressed (Brophy et al., 2003), milk levels of both proteins were increased, demonstrating that milk protein composition can be altered in large ruminants using recombinant gene technologies. Subsequent analysis revealed that the expression of additional copies of the casein genes resulted in a color change in the milk (Laible et al., 2007). The higher expression of \u03ba-casein resulted as expected in smaller casein micelles, which may explain the color change. The smaller micelles also had an effect on cheese making predicted to be because of a more rapid separation of the fat globules and therefore a lower rate of incorporation into the cheese. There was a slight change in amino acid composition of the cheese consistent with the change in \u03ba-casein content. The authors stated that the fat, lactose, and mineral contents of the transgenic milk were within normal ranges.\n\nThe functionality of caseins can be altered through enzymatic modification of the proteins, for example, through dephosphorylation. This results in increased pepsin hydrolysis, which in turn alters digestibility (Li-Chan & Nakai, 1989) and potentially the release of bioactive peptides. Importantly, none of the studies discussed identified defects or significant changes to mammary development or structure, suggesting that manipulation of milk proteins may not have undesirable effects on the mammary gland itself, compromising either the potential to lactate or animal health.\n\nThe whey proteins have also been investigated for manipulation to alter the processing characteristics of milk. Because of its role in the synthesis of lactose and the implications for manipulation of osmotic potential of milk, \u03b1-La has been a target for manipulation. Mice transgenic for expression of bovine \u03b1-La exhibited increased lactose concentration in milk and a slight increase in pup growth rate (Boston et al., 2001). Similarly, transgenic pigs overexpressing bovine \u03b1-La produced milk with lower total solids, higher milk yield in early lactation, higher lactose concentration, and increased litter growth rates (Noble et al., 2002). In contrast, a mouse model of \u03b1-La deficiency resulted in milk that was so viscous that pups were not able to suckle effectively (Stinnakre et al., 1994). Bovine \u03b1-La itself has also been modified to alter its characteristics, and various mutations have resulted in changes in affinity for galactosyltransferase, increased glucose binding (Grobler et al., 1994), and changes in molten globule conformation (Uchiyama et al., 1995).\n\nThe \u03b2-Lg protein has also been evaluated experimentally. Work describing the small RNA-mediated knockdown has been referred to earlier in this chapter, in the section entitled Transgenic Technologies for Milk Manipulation (Jabed et al., 2012). Overexpression in transgenic mice resulted in normal mammary physiology and therefore normal milk secretion and pup growth (Hyttinen et al., 1998; Guti\u00e9rrez-Ad\u00e1n et al., 1999). However, a similar study in which ovine \u03b2-Lg was overexpressed reported an increase in total protein content (Simons et al., 1987). Mutants of bovine \u03b2-Lg have also been studied, with variation in thermal stability (Cho et al., 1994), rate of secretion (Katakura et al., 1999), and rate of denaturation and digestion (Jayat et al., 2004) being reported.\n\n#### Adding Value to Milk through the Use of Milk Protein Genomics\n\nThe potential value of milk as a source of animal protein varying in characteristics that promote human health has been demonstrated by the range of bioactivities residing within the mammary protein phenome. The ease with which dairy products are distributed to consumers regardless of their socioeconomic status also makes this an ideal vehicle for the administration of therapeutics. The hyperimmune nature of colostrum, containing up to 40% by weight of immunoglobulin, suggests that immunization of cows against specific pathogens may provide a rich source of therapeutic antibodies. As human colostrum contains antibodies reactive to colonization factors 1 and 2 of enterotoxigenic Escherichia coli (Correa et al., 2006), thereby preventing diarrhea in infants, the production of similar antibodies in cows seemed to be a logical commercial target. Although progress in this area has been relatively slow, two recent studies suggest the potential of the production of antibodies in milk. Kramski et al. (2012) immunized cows with HIV-1 gp140 envelope (Env) oligomers and found production of a very high anti-gp 140 IgG titer in serum and colostrum. This may ultimately lead to a topical HIV-1 microbicide. Enterohemorrhagic Escherichia coli (EHEC) O157:H7 transferred from cattle to humans is an important cause of intestinal disease and hemolytic uremic syndrome, which is fatal in 5\u201310% of cases. Vaccination in cattle decreases bacterial shedding and can also induce antibodies in colostrum, which may protect calves from colonization (Rabinovitz et al., 2012).\n\nHowever, perhaps the greatest potential that milk proteins and peptides have for human health is through the addition of whey protein and casein peptides to foods to increase their functionality. Estimates of world whey protein production exceed 0.5 million tons, which provides a resource that should be used more effectively than by simply disposing of it as a liquid waste or as an animal feed. Considerable progress has been made in developing methods for separation and then purification of specific proteins from the whey fraction. Native immunoglobulins, lactoferrin, lactoperoxidase, \u03b1-La, and \u03b2-Lg have all been recovered in industrial quantities. The use of recombinant technologies may ultimately provide a cost-competitive alternative supply of these proteins.\n\nIdeally, a milk enriched in peptides promoting immune function, controlling blood pressure, acting as a bacteriostat, and minimizing oxidative stress and cancer risk, while at the same time relieving depression and preventing dental caries, would seem to have the makings of a highly valuable functional food. Combining this with an enrichment with n-3 fatty acids thought to increase insulin sensitivity and therefore prevent diabetes, together with certain milk carbohydrates capable of improving cognition, adds greatly to a product that already serves as a rich source of amino acids and energy to promote normal growth processes. Manipulation of these proteins in milk will inevitably occur in the factory and potentially in the cow. The challenge remains to turn this speculation into a commercial reality for the benefit of societies in both the developed and developing world.\n\n## Conclusion\n\nThe bovine milk proteins form the basis for a global industry in dairy products and as such have been intensively studied by both classical deconstructive observation and in dynamic whole animal systems utilizing postgenomic or functional genomics approaches. From these studies we have a greater understanding of the roles of these proteins in milk, their biological significance in the neonate, and the biochemical characteristics of bovine milk proteins that affect the efficiencies of manufacturing processes.\n\nThe use of bovine milk proteins and their peptides as bioactivities in clinical trials is an exciting new development. The potential of milk proteins, other than as a simple source of nutrition, is slowly being realized. This success will likely encourage further work in this area. The development of the milk-based nutraceutical business has been slowed by the requirements of some regulators to provide clinical evidence for claims of efficacy. This is not an unreasonable request, and developing a product that meets these requirements will ensure market share. The increasing world demand for high-quality dairy protein has perhaps delayed some research initiatives on product differentiation. However, research results indicate that considerable value can be gained from the protein components of milk\n\nThe bovine genome project has provided the underpinning knowledge necessary to develop tools for accelerated selection of superior animals in both production and specific milk composition. These tools are still in an early phase of development, but combined with accelerated breeding methods it is relatively straightforward and time efficient to generate specific herds for pharmaceutical, nutraceutical, or product requirements.\n\nThe further development of efficient methods of transgene integration or gene knockout in theory means that any compositional change could be made relatively quickly and cheaply. These approaches to modifying food are not yet accepted but may ultimately be desirable, depending on the value of the end product to human health and nutrition.\n\n# References\n\nAuldist MJ ,  Thomson NA ,  Mackle TR ,  Hill JP ,  Prosser CG .  Effects of pasture allowance on the yield and composition of milk from cows of different \u03b2-lactoglobulin phenotypes .  _Journal of Dairy Science_.  2000 ;83 : 2069 \u2013 2074 .\n\nBansal PS ,  Grieve PA ,  Marschke RJ ,  Daly NL ,  McGhie E ,  Craik DJ ,  Alewood PF .  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Transgenic modification of cow's milk for value-added processing .  _Reproduction, Fertility and Development_.  1998 ;10 : 671 \u2013 676 . \nChapter 5\n\n# Post-translational Modifications of Caseins\n\nJohn W. Holland*\n\nMike J. Boland**\n\n* Institute for Molecular Bioscience, The University of Queensland, Australia  \n** Riddet Institute, Massey University, Palmerston North, New Zealand\n\n## Abstract\n\nThe caseins exhibit a high degree of heterogeneity as a result of post-translational modifications (PTMs). Phosphorylation of the \u03b1s1-, \u03b1s2, and \u03b2-caseins and glycosylation of \u03ba-casein are the best known modifications and are critical for the formation and stability of casein micelles. \u03ba-Casein, in particular, has long been known to exhibit a high degree of variability in glycosylation, specifically in the caseinomacropeptide region, which is responsible for stabilization of the micelle. It is somewhat surprising to see so much variability in such important structural features. In recent years, the adoption of proteomic techniques has greatly enhanced our ability to unravel heterogeneity in proteins arising from complex and variable patterns of PTMs. In this chapter, a summary of our knowledge of the PTMs of caseins is attempted, with a particular emphasis on \u03ba-casein and the implications that variations in PTMs have for dairy processors.\n\n## Keywords\n\ncaseins, \u03ba-casein, post-translational modifications, phosphorylation, glycosylation, disulfide bonding, proteomic techniques\n\nOutline\n\nIntroduction 141\n\nThe caseins 142\n\n\u03b1s1-Casein 143\n\n\u03b1s2-Casein 144\n\n\u03b2-Casein 145\n\n\u03ba-Casein 145\n\nPhosphorylation 150\n\nGlycosylation 150\n\nDisulfide Bonding 152\n\nSources and Functional Significance of \u03ba-casein Heterogeneity 153\n\nSources of Heterogeneity 153\n\nFunctional Significance 156\n\nBiological Significance 160\n\nCaseins from other species 161\n\nConclusions 162\n\n## Introduction\n\nThe caseins are phosphoproteins and constitute about 80% of the protein in milk (Swaisgood, 2003; Farrell et al., 2004). They are assembled in a colloidal complex with calcium phosphate and small amounts of other minerals. Although obviously important for the provision of amino acids, calcium, and phosphorus for infant nutrition, the casein micelle structure is also critical in determining the physical properties of milk. The stability of the micelle, or its controlled destabilization in the case of cheese and yogurt manufacture, is of primary concern to the dairy industry.\n\nA number of reviews of micelle structure have been published in recent years (Rollema, 1992; Holt & Horne, 1996; Horne, 1998; Walstra, 1999; Horne, 2002). In simple terms, the micelle is a network of protein molecules held together by a combination of hydrophobic interactions between protein molecules and electrostatic interactions between phosphoserine-rich regions of the \u03b1\\- and \u03b2-caseins and micellar calcium phosphate. Whereas the internal structure is still the subject of debate (Horne, 1998; Walstra 1999; and Chapter 6 of this volume), there is general acceptance of the 'hairy micelle' concept, in which the hydrophilic C-terminal portion of \u03ba-casein extends from the surface, providing steric and electrostatic repulsion, which prevents micelle aggregation.\n\nA critical factor in micelle formation and stability is the presence of post-translational modifications (PTMs) such as phosphorylation on the \u03b1s1-, \u03b1s2, and \u03b2-caseins, and glycosylation on \u03ba-casein. PTMs on secreted proteins such as the caseins occur in the endoplasmic reticulum and\/or Golgi complex after synthesis of the polypeptide chain. As such, they are not encoded by the genes per se but may be dependent on protein (and hence gene) sequence motifs that are necessary. Of themselves, however, they are not sufficient for modification to take place. A number of other factors determine whether or not a PTM occurs, including expression of the genes encoding the enzymes necessary for the modification, the availability of their substrates, and the accessibility of the modification site on the protein, especially after folding. Therefore, although it may be possible to predict the theoretical sites of modification on proteins, determination of the actual sites and the degree to which they are modified requires considerable experimental characterization.\n\nAdvances in our understanding of complex systems such as the casein micelle are frequently preceded by advances in technology. In recent years, the development of proteomic technologies has greatly enhanced our ability to analyze milk proteins, particularly with respect to PTMs. Two-dimensional electrophoresis (2-DE), in particular, provides a high-resolution methodology for displaying the heterogeneity of the major milk proteins. As can be seen in Figure 5.1, genetic variants, phospho-variants, and glyco-variants of the caseins can be resolved on a single 2-D gel. Advances in mass spectrometry (MS) have enhanced our ability to analyze the proteins arrayed on 2-D gels. Therefore, it is possible not only to resolve many proteins and their isoforms but to characterize them, particularly with respect to the many PTMs that affect their electrophoretic mobility.\n\nFigure 5.1 2-D gel of bovine milk. Section of a 2-D gel highlighting the major protein forms found in bovine milk. Adapted from Holland et al. (2004).\n\nAs will be seen below, the four caseins are present in many diverse forms as a result of differential PTMs. The biological reasons behind the diversity are not clear. However, what is clear is that the PTMs of the caseins are critical for their function in micelle formation and stability. The first part of this chapter summarizes what is known about the PTMs of the \u03b1\\- and \u03b2-caseins. The second part focuses on \u03ba-casein, which has been the subject of recent proteomic studies, and includes an extended discussion on the functional significance of \u03ba-casein heterogeneity. For the main part of the chapter, the discussion is specific to the caseins of the cow, Bos taurus. A short review on the caseins of other species is included in a later section.\n\n## The caseins\n\nThe caseins are phosphoproteins that are generally well characterized in terms of their PTMs and have been the subject of a number of reviews (Mercier, 1981; Ginger & Grigor, 1999; Farrell et al., 2004). In fact, caseins have been used as model phosphoproteins in the development of proteomic techniques to examine global phosphorylation patterns (e.g., Cox et al., 2005; Kapkova et al., 2006; Sweet et al., 2006; Zhou et al., 2006; Wu et al., 2007). The multitude of techniques developed for the analysis of phosphorylation is beyond the scope of this chapter but has been well reviewed (Bodenmiller et al., 2007; Collins et al., 2007). Indeed, the topic of phosphoproteomics has become a field of study in its own right. The PTMs of the \u03b1s1-, \u03b1s2, and \u03b2-caseins are summarized in the sections below.\n\nAn important note concerns the numbering of the amino acids in the casein sequences described in subsequent sections. The numbering of residues is based on the Swiss-Prot database entry for the relevant protein (see Figs 5.2 and 5.3) and includes the signal peptide, which is normally removed during processing to generate the mature protein. This differs from the numbering in most of the dairy literature, in which the N-terminal amino acid of the mature protein is numbered 1.\n\nFigure 5.2 Amino acid sequences of the bovine \u03b1\\- and \u03b2-caseins. Swiss-Prot accession numbers, entry names, protein names, species, and sequences for the \u03b1\\- and \u03b2-caseins. The database sequences are for the B variant of \u03b1s1-casein, the A variant of \u03b1s2-casein, and the A2 variant of \u03b2-casein. Potential phosphorylation sites are shown in bold type, and those that have been experimentally confirmed are underlined. The signal peptides are shown in italics.\n\nFigure 5.3 Amino acid sequence of \u03ba-casein A (Swiss-Prot accession number P02668). The N-terminal signal sequence (1\u201321) is shown in italics. The arrow indicates the amino-terminus of the mature protein. Recognized sites of potential phosphorylation and glycosylation are indicated in bold, underlined text. Amino acid substitutions distinguishing the B variant are shown above the main sequence.\n\n### \u03b1s1-Casein\n\nThe predominant form of \u03b1s1-casein in bovine milk contains eight phosphate groups. The phosphates are attached to hydroxyamino acids occurring in the sequence motif Ser\/Thr\u2013Xxx\u2013Glu\/Asp\/pSer (Mercier, 1981). However, the vast majority of casein phosphorylation sites, including the eight sites in the major form of \u03b1s1-casein, occur in the more restricted Ser\u2013Xxx\u2013Glu\/pSer motif. Phosphorylation of threonine or of serine in the Ser\u2013Xxx\u2013Asp motif is relatively uncommon. A minor form of \u03b1s1-casein with nine phosphates, originally called \u03b1s0-casein, also occurs in bovine milk. It contains one extra phosphate on Ser56, which occurs in a Ser\u2013Xxx\u2013Asp motif (Manson et al., 1977). The reference protein for \u03b1s1-casein is \u03b1s1-CN B-8P, where B-8P signifies the B genetic variant with eight phosphates (Farrell et al., 2004). A number of genetic variants of \u03b1s1-casein have been described (Farrell et al., 2004). Only the less common forms, D and F, are likely to have altered phosphorylation profiles. Variant D has Ala68 substituted with Thr, which generates a phosphorylation site of the form Thr\u2013Xxx\u2013Glu. Phosphorylation of this residue was detected when the variant was first identified (Grosclaude et al., 1972a). Variant F has Ser81 substituted with Leu, which disrupts the serine cluster, Ser79\u2013Ile\u2013Ser\u2013Ser\u2013Ser\u2013Glu\u2013Glu85, and eliminates the secondary phosphorylation sites at Ser79 and Ser81. The amino acid sequence and the modifications of \u03b1s1-casein are shown in Figure 5.2.\n\n### \u03b1s2-Casein\n\nThe \u03b1s2-casein component of bovine milk is more varied than the \u03b1s1-casein component. It generally presents as a mixture of four phosphoforms with 10\u201313 phosphates. The reference protein for \u03b1s2-casein is \u03b1s2-CN A-11P (Farrell et al., 2004). The A variant has 12 serine residues in Ser\u2013Xxx\u2013Glu\/pSer motifs and four threonine residues in Thr\u2013Xxx\u2013Glu motifs (Fig. 5.2). Consequently, up to 16 phosphates are theoretically possible. Presumably, the 12 serine residues are the first to be phosphorylated. However, it is not known whether specific residues remain unphosphorylated in the different forms but, given the consistent appearance of the forms in milk, it is likely to be the case. Unfortunately, it is not possible to draw any conclusions with regard to phosphorylation site occupation until the individual phosphoforms are analyzed. This should be possible using gel-based proteomic techniques.\n\nOnly four genetic variants of \u03b1s2-casein have been described (Farrell et al., 2004), and, in each case, an altered phosphorylation profile would be expected. In variant B, Ser23 is changed to Phe as a result of a single-nucleotide substitution (Ibeagha-Awemu et al., 2007). This causes loss of a phosphorylation site in the first phosphoserine cluster. Variant C has three amino acid changes: Glu48 is changed to Gly, with loss of the phosphorylation site at Ser46; Ala62 is changed to Thr, creating a potential site with the motif Thr62\u2013Xxx\u2013Glu64; and Thr145 is changed to Ile, with loss of the potential site at Thr145. Variant D has a deletion of nine amino acids as a result of skipping exon VIII (Bouniol et al., 1993). This results in loss of the first three serines from the second phosphoserine cluster. A second PTM on \u03b1s2-casein is the formation of an intramolecular disulfide bond between the two cysteine residues in the protein (Rasmussen et al., 1994). The functional role of disulfide bonding is not clear at this stage, but it may contribute to micelle stability. (This is discussed further in the section on \u03ba-casein.)\n\n### \u03b2-Casein\n\nBovine \u03b2-casein is usually present as a single form with five phosphates, indicating that all five Ser\u2013Xxx\u2013Glu\/pSer sites in the sequence are constitutively phosphorylated. The reference protein is \u03b2-CN A2-5P (Farrell et al., 2004). Some 12 genetic variants of \u03b2-casein have been characterized, but only two variants appear to have altered phosphorylation profiles. Variant C has a Glu to Lys substitution at residue 52, which removes the phosphorylation site at Ser50. Variant D has a Ser to Lys substitution at residue 33, which removes the primary phosphorylation site at Ser33. Variation in \u03b2-casein arises primarily as a result of proteolysis rather than PTMs. The sequence and the phosphorylation sites of \u03b2-casein are summarized in Figure 5.2.\n\n### \u03ba-Casein\n\n\u03ba-Casein does not contain any phosphoserine clusters and probably plays little part in calcium binding. Its major feature is a variable degree of glycosylation. The keen interest in \u03ba-casein arises largely from its key role as a stabilizer of the micelle structure. In mice, the absence of \u03ba-casein causes a failure of lactation, as the lumina of the mammary gland become clogged with aggregated caseins (Shekar et al., 2006). \u03ba-Casein is usually thought of as having two distinct domains that are separated by the very specific Phe-Met (126\u2013127) bond cleaved by chymosin. The N-terminal para-\u03ba-casein is the hydrophobic part that remains in the micelle and contains the disulfide bonds. The C-terminal caseinomacropeptide is the part that extends into the solution, and it is removed to become part of the whey fraction during cheese making. It contains the glycosylation sites, which is why this peptide is often referred to as the glycomacropeptide, although strictly speaking this term pertains only to forms that are actually glycoslated. The PTMs of \u03ba-casein have been the subject of more recent research and are covered here in much greater detail than those of the other caseins.\n\nThe full amino acid sequence of bovine \u03ba-casein was first reported in 1973 (Mercier et al., 1973). The mature protein consists of a single chain of 169 amino acids (Fig. 5.3) and has a theoretical molecular weight of 18,974 Da and a theoretical pI of 5.93 (A variant). The amino terminal glutamine is cyclized to form a pyrrolidone glutamic acid residue. The bovine \u03ba-casein gene sequence was published in 1988 (Alexander et al., 1988). It consists of five exons spread over about 14 kilobases, with most of the protein-coding region located in exon 4. A cleavable amino terminal signal sequence of 21 amino acids directs secretion of the mature protein. A number of polymorphisms in the \u03ba-casein gene have been identified, resulting in one or more amino acid substitutions in the mature protein. The most common variants are the A and B variants, which differ by two amino acids in the caseinomacropeptide region (Asp169\/Thr157 in variant A and Ala169\/Ile157 in variant B). The genetic variants of \u03ba-casein are summarized in Table 5.1. A number of polymorphisms in the noncoding region have also been identified (Schild et al., 1994; Keating et al., 2007). Although these do not affect the amino acid sequence, they have the potential to affect expression levels. The full amino acid sequences of \u03ba-casein from 32 species are currently in the Swiss-Prot database, with another 112 entries covering subspecies and incomplete sequences (Table 5.2). The reference protein for \u03ba-casein is \u03ba-CN A-1P, Uni-prot P02668 (Farrell et al., 2004).\n\nTable 5.1\n\nGenetic Variants of Bovine \u03ba-Casein\n\nVarianta | Amino acid changes  \n(Relative to A variant) | Reference  \n---|---|---  \nA |  | (Grosclaude, Mahe, Mercier, Ribadeau-Dumas, 1972b)  \nB | Thr157 to Ile, Asp169 to Ala | (Mercier, Brignon, Ribadeau-Dumas, 1973)  \nB2 | Thr157 to Ile, Asp169 to Ala, Ile174 to Thr | (Gorodetskii, Kershulite, Korobko, 1983)  \nC | Arg118 to His, Thr157 to Ile, Asp169 to Ala | (Miranda, Anglade, Mahe, Erhardt, 1993)  \nE | Ser176 to Gly | (Schlieben, Erhardt, Senft, 1991)  \nF1 | Asp169 to Val | (Sulimova, Sokolova, Semikozova, Nguet, Berberov, 1992)  \nF2 | Arg31 to His | (Prinzenberg, Hiendleder, Ikonen, Erhardt, 1996)  \nG1 | Arg118 to Cys | (Prinzenberg et al., 1996)  \nG2 | Asp169 to Ala | (Sulimova, Badagueva Iu, Udina, 1996)  \nH | Thr156 to Ile | (Prinzenberg, Krause, Erhardt, 1999)  \nI | Ser125 to Ala | (Prinzenberg et al., 1999)  \nJ | Thr157 to Ile, Asp169 to Ala, Ser176 to Arg | (Mahe et al., 1999)  \nA(1) | Silent (Pro150, CCA to CCG) | (Prinzenberg et al., 1999)\n\na Other variants have been described but have not been confirmed or have been proven to be identical to those given.\n\nTable 5.2\n\n\u03ba-Casein Entries in the Swiss Protein Database (December 2013)\n\nEntry name | | Organism  \n---|---|---  \nQ6YLM6_AEPME | Fragment | Aepyceros melampus (Impala)  \nP79091_AILFU | Fragment | Ailurus fulgens (Lesser panda) (Red panda)  \nQ95MY6_ALCBU | Fragment | Alcelaphus buselaphus (Hartebeest)  \nQ8HXL0_9CETA | Fragment | Alces alces (Eurasian elk)  \nQ5C9H0_AMMLE | Fragment | Ammotragus lervia (Barbary sheep) (Aoudad)  \nQ95MY0_ANTMR | Fragment | Antidorcas marsupialis (Springbok)  \nQ7JFQ1_ANTAM | Fragment | Antilocapra americana (Pronghorn)  \nK7Z0L5_ANTCE | Fragment | Antilope cervicapra (Blackbuck)  \nCASK_BALPH | Fragment | Balaenoptera physalus (Finback whale) (Common rorqual)  \nQ95MY7_BEAHU | Fragment | Beatragus hunteri (Hunter's antelope) (Damaliscus hunteri)  \nP79093_BISBI | Fragment | Bison bison (American bison) (Bos bison)  \nB2Z897_BISBO | Fragment | Bison bonasus (European bison)  \nO18899_BOSGA | Fragment | Bos gaurus (Seladang) (Indian bison)  \nB6CXY4_BOSGF | Fragment | Bos gaurus frontalis (Domestic gayal) (Bos frontalis)  \nL8IIT8_BOSMU |  | Bos grunniens mutus  \nB6CXY5_BOSIN | Fragment | Bos indicus (Zebu)  \nP79097_BOSJA | Fragment | Bos javanicus (Wild banteng)  \nK9ZTJ1_9CETA | Fragment | Bos mutus (wild yak)  \nI6UFY2_BOSMU |  | Bos mutus grunniens (Wild yak) (Bos grunniens)  \nA3FJ56_BOVIN | Fragment | Bos taurus (Bovine)  \nO18903_BOSTR | Fragment | Boselaphus tragocamelus (Nilgai)  \nQ712N6_BUBBU |  | Bubalus bubalis (Domestic water buffalo)  \nK9ZS88_BUBDE | Fragment | Bubalus depressicornis (Lowland anoa) (Anoa depressicornis)  \nQ5C9G9_BUDTA | Fragment | Budorcas taxicolor (Golden takin)  \nL0P304_CAMBA |  | Camelus bactrianus (Bactrian camel)  \nL0P3Z7_CAMDR |  | Camelus dromedarius (Dromedary) (Arabian camel)  \nS9WI70_9CETA |  | Camelus ferus (Wild Bactrian camel)  \nF5CIN1_CAPMR | Fragment | Caperea marginata (Pigmy right whale) (Balaena marginata)  \nQ71EC2_CAPAE | Fragment | Capra aegagrus (Wild goat)  \nQ5C9G8_CAPFA | Fragment | Capra falconeri (Markhor)  \nCASK_CAPHI |  | Capra hircus (Goat)  \nQ7YRV1_CAPII | Fragment | Capra ibex ibex (Alpine ibex)  \nQ5C9G7_CAPNU | Fragment | Capra nubiana (Nubian ibex) (Capra ibex nubiana)  \nQ7YRU9_CAPSI | Fragment | Capra sibirica (Siberian ibex) (Capra ibex sibirica)  \nCASK_CAPCA | Fragment | Capreolus capreolus (Roe deer)  \nCASK_CAPCR |  | Capricornis crispus (Japanese serow) (Naemorhedus crispus)  \nCASK_CAPSU |  | Capricornis sumatrensis (Serow)  \nCASK_CAPSW |  | Capricornis swinhoei (Taiwan serow) (Naemorhedus swinhoei)  \nCASK_CAVPO |  | Cavia porcellus (Guinea pig)  \nQ6YLM4_CEPDO | Fragment | Cephalophus dorsalis (Bay duiker)  \nCASK_CEREL | Fragment | Cervus elaphus (Red deer)  \nCASK_CERNI |  | Cervus nippon (Sika deer)  \nQ95MY5_CONGN | Fragment | Connochaetes gnou (Black wildebeest)  \nQ6YLM2_DAMPY | Fragment | Damaliscus pygargus (Bontebok)  \nQ28354_9CETA | Fragment | Delphinidae gen. sp.  \nK7ZCW9_9CETA | Fragment | Dorcatragus megalotis (beira)  \nCASK_ELADA | Fragment | Elaphurus davidianus (Pere David's deer)  \nB1PLB8_EQUAS | Fragment | Equus asinus (Donkey)  \nF0V6V5_EQUAS |  | Equus asinus africanus  \nF6VKB4_HORSE |  | Equus caballus (Horse)  \nCASK_EQUGR | Fragment | Equus grevyi (Grevy's zebra)  \nB1PLB7_EQUZE | Fragment | Equus zebra (Mountain zebra)  \nF5CIN0_ESCGI | Fragment | Eschrichtius gibbosus (California gray whale) (Eschrichtius robustus)  \nK7YUI7_9CETA | Fragment | Gazella arabica (Arabian gazelle)  \nK7Z353_GAZDO | Fragment | Gazella dorcas (Dorcas gazelle)  \nK7ZHH2_GAZSU | Fragment | Gazella subgutturosa marica (sand gazelle)  \nCASK_GIRCA | Fragment | Giraffa camelopardalis (Giraffe)  \nF5CIM6_GRAGR | Fragment | Grampus griseus (Risso's dolphin) (Delphinus griseus)  \nQ074L9_HEMHY | Fragment | Hemitragus hylocrius (Nilgiri tahr)  \nQ074M0_HEMJA | Fragment | Hemitragus jayakari (Arabian tahr)  \nQ5C9G5_HEMJE | Fragment | Hemitragus jemlahicus (Himalayan tahr)  \nG5AYC4_HETGA |  | Heterocephalus glaber (Naked mole rat)  \nCASK_HIPAM | Fragment | Hippopotamus amphibius (Hippopotamus)  \nQ95MY8_HIPEQ | Fragment | Hippotragus equinus (Roan antelope)  \nQ6YLM3_HIPNI | Fragment | Hippotragus niger (Sable antelope)  \nCASK_HUMAN |  | Homo sapiens (Human)  \nT2MHA8_HYDVU | Fragment | Hydra vulgaris (Hydra) (Hydra attenuata)  \nA8IY17_INIGE | Fragment | Inia geoffrensis (Amazon dolphin)  \nQ95MY2_KOBLE | Fragment | Kobus leche (Lechwe)  \nCASK_LAMGU | Fragment | Lama guanicoe (Guanaco)  \nQ95MX9_LITWA | Fragment | Litocranius walleri (Gerenuk)  \nG7P5P0_MACFA |  | Macaca fascicularis (Crab-eating macaque) (Cynomolgus monkey)  \nG7MT61_MACMU |  | Macaca mulatta (Rhesus macaque)  \nK7Z0L8_MADSL | Fragment | Madoqua saltiana (Salt's dikdik)  \nCASK_MAZAM | Fragment | Mazama americana (Red brocket)  \nF5CIM8_MESPE | Fragment | Mesoplodon peruvianus (Peruvian beaked whale) (Pygmy beaked whale)  \nQ6YLM7_MOSMO | Fragment | Moschus moschiferus (Siberian musk deer) (Moschus sibiricus)  \nA8IY19_9CETA | Fragment | Moschus sp. JEG-2007  \nCASK_MUNRE | Fragment | Muntiacus reevesi (Reeve's muntjac) (Chinese muntjac)  \nCASK_MOUSE |  | Mus musculus (Mouse)  \nS7Q8N5_MYOBR |  | Myotis brandtii (Brandt's bat)  \nG8FPL3_NANDA | Fragment | Nanger dama (Dama gazelle) (Gazella dama)  \nQ5C9H1_NANGR | Fragment | Nanger granti (Grant's gazelle) (Gazella granti)  \nCASK_NEMGO |  | Nemorhaedus goral (Gray goral)  \nQ95MX5_NEOMO | Fragment | Neotragus moschatus (Suni)  \nCASK_ODOHE | Fragment | Odocoileus hemionus (Mule deer) (Black-tailed deer)  \nCASK_ODOVI | Fragment | Odocoileus virginianus virginianus (Virginia white-tailed deer)  \nQ6YLM8_OKAJO | Fragment | Okapia johnstoni (Okapi)  \nCASK_OREAM |  | Oreamnos americanus (Mountain goat)  \nQ95MX4_OREOR | Fragment | Oreotragus oreotragus (Klipspringer)  \nD0QJA9_ORNAN |  | Ornithorhynchus anatinus (Duckbill platypus)  \nCASK_RABIT |  | Oryctolagus cuniculus (Rabbit)  \nQ95MY9_ORYDA | Fragment | Oryx dammah (Scimitar-horned oryx)  \nQ95MZ0_ORYGA | Fragment | Oryx gazella (Gemsbok)  \nQ5C9G2_OVIMO | Fragment | Ovibos moschatus (Muskox)  \nD2SZ86_9CETA | Fragment | Ovis ammon collium  \nD2SZ87_9CETA | Fragment | Ovis ammon sevetzovi  \nCASK_SHEEP |  | Ovis aries (Sheep)  \nD2SZ88_OVICA | Fragment | Ovis canadensis canadensis  \nD2SZ94_OVICA | Fragment | Ovis canadensis nelsoni (desert bighorn sheep)  \nQ5C9G0_OVIDA | Fragment | Ovis dalli (Dall sheep)  \nD2SZA4_9CETA | Fragment | Ovis orientalis anatolica  \nD2SZA5_9CETA | Fragment | Ovis orientalis gmelini  \nD2SZB9_9CETA | Fragment | Ovis orientalis isphahanica  \nD2SZE2_OVIVI | Fragment | Ovis vignei (Urial sheep)  \nD2SZC5_OVIVI | Fragment | Ovis vignei arkal  \nD2SZD4_OVIVI | Fragment | Ovis vignei blanfordi  \nD2SZD7_OVIVI | Fragment | Ovis vignei bochariensis (Bukhara urial)  \nD2SZD9_OVIVI | Fragment | Ovis vignei cycloceros  \nD2SZE4_OVIVI | Fragment | Ovis vignei punjabensis  \nD2SZE7_OVIVI | Fragment | Ovis vignei vignei  \nQ5C9F9_PANHO | Fragment | Pantholops hodgsonii (Chiru)  \nQ95MY1_PELCP | Fragment | Pelea capreolus (Gray rhebok)  \nQ95MX3_PHIMO | Fragment | Philantomba monticola (Blue duiker) (Cephalophus monticola)  \nF5CIM7_PHOPH | Fragment | Phocoenoides phocoena (Harbor porpoise)  \nP79230_PHYCD | Fragment | Physeter catodon (Sperm whale) (Physeter macrocephalus)  \nK7YUI9_PROGT | Fragment | Procapra gutturosa (Mongolian gazelle)  \nQ5C9F8_PSENA | Fragment | Pseudois nayaur (Bharal)  \nQ5C9H3_PSENG | Fragment | Pseudoryx nghetinhensis (Saola)  \nL5K924_PTEAL |  | Pteropus alecto (Black flying fox)  \nCASK_RANTA | Fragment | Rangifer tarandus (Reindeer) (Cervus tarandus)  \nQ95MX8_RAPCA | Fragment | Raphicerus campestris (Steenbok)  \nQ95MX6_RAPME | Fragment | Raphicerus melanotis (Grysbok)  \nQ95MX7_RAPSH | Fragment | Raphicerus sharpei (Sharpe's grysbok)  \nCASK_RAT |  | Rattus norvegicus (Rat)  \nQ6YLM5_REDFU | Fragment | Redunca fulvorufula (Mountain reedbuck)  \nQ95MY3_REDRE | Fragment | Redunca redunca (Bohor reedbuck)  \nCASK_RUCDU | Fragment | Rucervus duvaucelii (Swamp deer) (Cervus duvaucelii)  \nQ074J7_RUPPY | Fragment | Rupicapra pyrenaica (Pyrenean chamois)  \nCASK_RUPRU |  | Rupicapra rupicapra (Chamois)  \nCASK_RUSUN | Fragment | Rusa unicolor (Sambar) (Cervus unicolor)  \nCASK_SAITA |  | Saiga tatarica (Saiga antelope)  \nCASK_PIG |  | Sus scrofa (Pig)  \nQ95MX2_SYLGR | Fragment | Sylvicapra grimmia (Grey duiker)  \nQ95MZ2_SYNCA | Fragment | Syncerus caffer (Cape buffalo)  \nO18901_SYNCA | Fragment | Syncerus caffer nanus (forest buffalo)  \nD0QJA7_9MAMM |  | Tachyglossus aculeatus (Australian echidna)  \nCASK_TAPIN | Fragment | Tapirus indicus (Asiatic tapir) (Malayan tapir)  \nF5CIM9_TASSH | Fragment | Tasmacetus shepherdi (Shepherd's beaked whale)  \nCASK_TAYTA | Fragment | Tayassu tajacu (Collared peccary) (Pecari tajacu)  \nQ5C9H2_TETQU | Fragment | Tetracerus quadricornis (Four-horned antelope) (Chousingha)  \nO18902_TRAIM | Fragment | Tragelaphus imberbis (Lesser kudu)  \nQ95MZ1_TRAOR | Fragment | Tragelaphus oryx (Eland) (Taurotragus oryx)  \nCASK_TRAJA | Fragment | Tragulus javanicus (Lesser Malay chevrotain) (Lesser mouse deer)  \nQ9XSD6_TRIVU |  | Trichosurus vulpecula (Brush-tailed possum)  \nCASK_UNCUN | Fragment | Uncia uncia (Snow leopard) (Panthera uncia)\n\n#### Phosphorylation\n\n\u03ba-Casein appears to be constitutively phosphorylated at Ser170 and only partially phosphorylated at Ser148 (Talbot & Waugh, 1970; Mercier et al., 1973). A minor tri-phosphorylated form has also been detected (Vreeman et al., 1986; Molle & Leonil, 1995). Other studies have managed to detect only mono-phosphorylated forms (Rasmussen et al., 1997; Riggs et al., 2001), although, in one of these (Riggs et al., 2001), the phosphorylation site appears to have been identified incorrectly. Phosphorylation has also been examined by MS of intact \u03ba-casein extracted from 2-D gels. Both mono- and di-phosphorylated forms were observed, and phosphorylation at Ser170 was confirmed by MS\/MS (Claverol et al., 2003). However, the electrophoretic mobility of some phospho-forms was not consistent with the MS analysis and probably reflected artifactual modification (e.g., deamidation) during purification. Using 2-DE with isoelectric focusing as the first dimension, phosphorylation variants in whole milk can be easily resolved because of the pI shifts caused by the acidic phosphate groups (Holland et al., 2004). The two main phosphorylation sites have been confirmed by tandem MS sequencing of peptic peptides released from protein forms separated by 2-DE. Tri-phosphorylated forms of both the A and B variants have been observed, and the third phosphorylation site has been identified as Thr166 (Holland et al., 2006). This site is consistent with the general observation of casein phosphorylation on the Ser\/Thr\u2013Xxx\u2013Glu\/pSer motif, with only relatively low levels of phosphorylation on threonine residues. A recent study has identified two minor phosphorylation sites at Thr166 \u2013 as above \u2013 and also at Ser187 (Hernandez-Hernandez et al., 2011).\n\n#### Glycosylation\n\nWhereas about 40% of \u03ba-casein has been estimated to be nonglycosylated, the remaining 60% has up to six glycans attached (Vreeman et al., 1986). The presence of sugars on \u03ba-casein was recognized as long ago as 1961 (Alais & Jolles, 1961), and during the 1970s and 1980s, a large number of studies were directed at elucidating the sugar composition, sequence, and sites of attachment to the protein. The major glycan is a tetrasaccharide composed of galactose (Gal), N-acetylgalactosamine (GalNAc), and sialic or neuraminic acid (NeuAc) of the form NeuAc\u03b1(2-3)Gal\u03b2(1-3)NeuAc\u03b1(2-6)]GalNAc, but monosaccharide (GalNAc), disaccharide (Gal\u03b2(1-3)GalNAc), and trisaccharide (NeuAc\u03b1(2-3)Gal\u03b2(1-3)GalNAc or Gal\u03b2(1-3)[NeuAc\u03b1(2-6)]GalNAc) are also found ([Fournet et al., 1975; ; van Halbeek et al., 1980; Fiat et al., 1988; Saito & Itoh, 1992). The relative amounts have been determined by high-performance liquid chromatography (HPLC) as 56.0% tetrasaccharide, 36.9% trisaccharide (18.4% linear and 18.5% branched), 6.3% disaccharide, and 0.8% monosaccharide (Saito & Itoh, 1992). It is not known whether the minor forms arise from incomplete synthesis of the tetrasaccharide in mammary epithelial cells or are products of degradation after synthesis and\/or secretion of \u03ba-casein into the lumen of the mammary gland.\n\nEstablishment of the attachment site(s) of the glycans has been more controversial. On the basis of Edman sequencing of short glycopeptides obtained by enzymatic digestion, Jolles et al. (1973) proposed Thr152 or Thr154 as the glycan attachment site. Kanamori et al. (1980) proposed Thr152, Thr154, and Thr156 (or Thr157) after analyzing a glycopeptide that was derived from \u03ba-casein and that contained three GalNAc residues. Work from the same laboratory on bovine \u03ba-casein from colostrum also indicated glycosylation at Thr152, Thr154, and Thr156 (Doi et al., 1980). Subsequently, using a different peptide fraction prepared from \u03ba-casein of normal milk, glycosylation at Thr154 and Ser162 was reported (Kanamori et al., 1981). Meanwhile, further work from Jolles's laboratory identified Thr152 as the glycan attachment site on \u03ba-casein from normal milk and Thr152 and Thr163 as the attachment sites on \u03ba-casein from colostrum (Fiat et al., 1981). Zevaco and Ribadeau-Dumas (1984) suggested that glycans could be attached to any of the previously identified sites (Thr152, Thr154, Thr156, Thr157, Ser162, or Thr163), but their published study contained no conclusive evidence for any site. All these studies were limited by the technology available at the time. In normal Edman sequencing, glycosylated serine or threonine residues are not detected, and their presence is inferred from a blank in the sequencing cycle where serine or threonine is expected (for a more detailed discussion, see Pisano et al., 1994). This is further complicated by the fact that serine and threonine are themselves low-yield amino acids. Thus, in assigning O-glycosylation sites, Edman sequencing data can easily be misinterpreted. Conclusive identification of glycosylation sites in \u03ba-casein was achieved using solid-phase Edman sequencing, which allows the direct detection of glycosylated serine and threonine residues (Pisano et al., 1994). Variable levels of glycosylation at Thr142, Thr152, Thr154, Thr157 (A variant only), Thr163, and Thr186 were detected, and no evidence of glycosylation at any serine residue was obtained. However, even this study did not give the full picture of \u03ba-casein glycosylation, as it could detect only average glycosylation site occupancy in a crude mixture of glycoforms.\n\nWe have shown that \u03ba-casein glycoforms can be separated by 2-DE (Holland et al., 2004). and the resolution obtained is shown in Figure 5.4. When the glycosylation site occupancy of individual glycoforms was investigated using tandem MS sequencing of chemically tagged peptides, an interesting pattern was observed. The mono-glycoform was glycosylated exclusively at Thr152, the di-glycoform was glycosylated at Thr152 and Thr163, and the tri-glycoform was glycosylated at Thr152, Thr154, and Thr163 (Holland et al., 2005). Further studies using enriched fractions of \u03ba-casein separated on 2-D gels showed up to six glycans on \u03ba-casein B where only five sites had been previously identified. Tandem MS analysis provided evidence for glycosylation at Thr166 on the tetra-glycoform (Holland et al., 2006). The remaining two glycosylation sites were not confirmed but were presumably on Thr142 and Thr186, as proposed earlier (Pisano et al., 1994). This pattern is illustrated in Figure 5.5. As Thr166 can be phosphorylated or glycosylated, there is potential for competition at this site. However, as both the tri-phosphate and tetra-glycoforms are very minor forms, it may be of little significance. Overall, the glycosylation of \u03ba-casein in the mammary epithelial cells appears to take place in a highly controlled manner; this suggests that it is a rather important process with considerable functional significance.\n\nFigure 5.4 Heterogeneity of \u03ba-casein in cow's milk. 2-D gel showing multiple forms of \u03ba-casein. The genetic variant, number of phosphate residues, and glycosylation state are indicated. Numbers in brackets indicate the extra negatively charged residues relative to \u03ba-casein B-1P. Adapted from Holland et al. (2004).\n\nFigure 5.5 Attachment of tetrasaccharide units to the four major glycoforms of \u03ba-casein. Schematic representation of \u03ba-casein highlighting the glycomacropeptide portion and the attachment sites of phosphate and sugar residues in the major glycoforms. ( ) GalNAc; ( ) Gal; (\u2666) NeuAc.\n\n#### Disulfide Bonding\n\n\u03ba-Casein purified from bovine milk occurs in both monomeric and oligomeric forms, with up to eight or more monomers linked by disulfide bonds (Swaisgood et al., 1964; Talbot & Waugh, 1970). A more recent study has shown that reduced and carboxymethylated \u03ba-casein can form large fibrillar structures, although these do not occur in milk (Farrell et al., 2003). The nature of the disulfide-linked complexes has been examined by a number of authors (Groves et al., 1992; ; Rasmussen et al., 1992; ; ; Farrell et al., 2003).\n\nThere are only two cysteine residues (Cys32 and Cys109) in bovine \u03ba-casein (Mercier et al., 1973), both occurring in the para-\u03ba-casein part of the molecule, and they appear to be randomly linked in disulfide bonds in oligomeric forms (Rasmussen et al., 1992). In monomeric \u03ba-casein, the two cysteines form an intramolecular disulfide bond (Rasmussen et al., 1994). As can be seen in Figure 5.6, disulfide-bonded monomers, dimers, and trimers can be resolved on 2-D gels of whole milk when reducing agents are omitted (Holland et al., 2008). It is not clear whether these higher-order complexes of \u03ba-casein have any importance in micellar structure, but it would be expected that they would be less likely to dissociate from the micelles. The cysteine residues are not well conserved across species, with human, porcine, and rodent \u03ba-caseins containing only a single cysteine, precluding the formation of disulfide-linked oligomers larger than dimers (Rasmussen et al., 1999; Bouguyon et al., 2006). However, the ability of \u03ba-casein to form disulfide-linked complexes with itself or with other proteins during heat treatment is relevant to dairy processing (see below). The combined PTMs of bovine \u03ba-casein are summarized in Figure 5.7.\n\nFigure 5.6 Distribution of disulfide-linked isoforms of \u03ba-casein on a non-reducing 2-D gel. Dimers, trimers, and tetramers of \u03ba-casein are labeled to show the participating monomeric forms. The dimers and trimers run as doublets, depending on the disulfide linkages. MALDI-TOF mass spectra show the disulfide-linked peptides obtained from tryptic digests of the homodimer and homotrimer of \u03ba-casein B:1P.\n\nFigure 5.7 Potential modifications on \u03ba-casein B. This schematic summarizes the potential PTMs on \u03ba-casein B. (P) Phosphorylation sites; ( ) GalNAc; ( ) Gal; (\u2666) NeuAc. Note that trisaccharides and disaccharides lacking one or both NeuAc residues also occur. The monomer has an intramolecular disulfide bond between Cys32 and Cys109.\n\n### Sources and Functional Significance of \u03ba-casein Heterogeneity\n\nAlthough the heterogeneity of \u03ba-casein has been recognized for many decades and the structural elements are now fairly well defined, the source of the heterogeneity, particularly in glycosylation, and its functional role(s) are not known. Early studies on the influence of the glycosylation of \u03ba-casein on its biological properties have been reviewed previously (Dziuba & Minikiewicz, 1996). Dziuba and Minikiewicz highlighted a number of studies addressing factors that could influence the degree of glycosylation and what influence glycosylation might have on micelle stability. The sections below cover some of those studies again and highlight more recent work related to the sources and functional significance of \u03ba-casein heterogeneity.\n\n#### Sources of Heterogeneity\n\nA large number of studies have examined the influence of milk protein polymorphism on milk composition and yield, and these have been extensively reviewed (e.g., Ng-Kwai-Hang, 1997; Martin et al., 2002; Heck et al., 2009). However, in many cases, the results have been inconsistent, which probably reflects the multifactorial nature of milk production. It is difficult to isolate the effects of a single protein polymorphism from those of the other major milk proteins, especially as there appears to be a substantial degree of co-ordination of their expression. There are also a number of environmental or cow-related factors such as feed type and lactational stage that are frequently interrelated, as they all vary with the seasonal changes in dairy farming. Studies on specific effects of \u03ba-casein variants have largely focused on the common A and B variants, and there appears to be a consensus that milk from B variant cows contains more fat, protein, casein, and \u03ba-casein than milk from A variant cows (Bovenhuis et al., 1992; Ng-Kwai-Hang, 1997; Bobe et al., 1999).\n\nStudies relating to the glycosylation status of \u03ba-casein are more limited. Robitaille et al. (1991a) identified a number of factors that appeared to contribute to variation in the NeuAc content of bovine \u03ba-casein. The NeuAc content was higher in cows with the \u03ba-casein AB phenotype than in cows with the AA phenotype. It decreased with increasing parity and varied over the course of lactation, dropping to a minimum at 2\u20133 months after calving before increasing over the next 9\u201310 months. Robitaille et al. (1991b) also examined the association between \u03ba-casein glycosylation and milk production\/composition. Although there appeared to be a statistically significant association between the NeuAc content of \u03ba-casein and milk yield, the most striking result of these investigations was the variability of NeuAc\/\u03ba-casein measurements (mean, 64 \u00b1 21 \u03bcg\/mg; range, 23\u2013166 \u03bcg\/mg), which suggests that other factors could have had a large impact on glycosylation or that the inherent variability in the assay masked any true associations. Limited 2-D gel analyses suggest that the pattern of glycosylation is far more consistent than these measurements indicate (Holland et al., 2004; ). It would be very informative to examine some of these supposed extremes in NeuAc and hence the glycosylation level of \u03ba-casein on 2-D gels.\n\nSignificant differences in the content of nonglycosylated \u03ba-casein in milk have been reported for cows of different \u03ba-casein genotypes (Lodes et al., 1996). Nonglycosylated \u03ba-casein levels were higher (as a percentage of total protein) in milk from cows with the B variant than in milk from cows with the A variant. The rarer variants, C and E, were generally associated with lower levels. However, as no measurements of glycosylated \u03ba-casein levels were reported, no effect of genetic variant on glycosylation can be inferred from this report.\n\nElectrophoretic and chromatographic techniques have been used to profile the caseinomacropeptide from cows of the AA and BB phenotypes (Coolbear et al., 1996). They found that the B variant macropeptide was more highly glycosylated than the A variant, with an increased content of both hexosamine (i.e., GalNAc) and sialic acid (i.e., NeuAc). After anion-exchange HPLC on a MonoQ column, the elution profile of the B variant contained more peaks, suggesting that an increased number of oligosaccharide chains were attached. These results were consistent with earlier studies, suggesting more extensive glycosylation of the B variant (Vreeman et al., 1986) compared with the A variant (Molle & Leonil, 1995), despite the fact that the A variant contains an extra (potential) glycosylation site (Pisano et al., 1994). From these and other results, Coolbear et al. (1996) suggested that there were generally consistent patterns of glycosylation for the genetic variants but that the overall extent of glycosylation could vary.\n\nVariations in \u03ba-casein glycosylation during lactation have been touched on above. Early studies indicated a higher degree of glycosylation of \u03ba-casein in colostrum than in mature milk as well as the presence of an additional sugar moiety, N-acetylglucosamine (GlcNAc) (Guerin et al., 1974; Fournet et al., 1975). Subsequently, a number of studies addressed the structure of the oligosaccharides attached to colostral \u03ba-casein and how they varied with time after parturition (Saito et al., 1981a; 1981b; ; van Halbeek et al., 1981; Fiat et al., 1988). As well as the structures already identified above in normal milk, the following structures have been reported: the acidic hexasaccharide, NeuAc\u03b1(2-3)Gal\u03b2(1-3)NeuAc\u03b1(2-3)Gal\u03b2(1-4)GlcNAc\u03b2(1-6)]GalNAc; the acidic pentasaccharide, NeuAc\u03b1(2-3)Gal\u03b2(1-3)[Gal\u03b2(1-4)GlcNAc\u03b2 (1-6)]GalNAc; the acidic tetrasaccharide, GlcNAc\u03b2(1-3)Gal\u03b2(1-3)[NeuAc\u03b1(2-6)]GalNAc; the neutral pentasaccharide Gal\u03b2(1-3)[Gal\u03b2(1-4){Fuc\u03b1(1-3)}GlcNAc\u03b2(1-6)]GalNAc; the neutral tetrasaccharide, Gal\u03b2(1-3)[Gal\u03b2(1-4)GlcNAc\u03b2(1-6)]GalNAc; and the neutral trisaccharide, Gal\u03b2(1-3)[GlcNAc\u03b2(1-6)]GalNAc. This extra complexity is already observable 15 min after parturition but decreases to normal over about 66 h ([Fiat et al., 1988). These results suggest changes in the expression profiles of the glycosyltransferases responsible for assembling the O-linked glycans on \u03ba-casein. The initial step of attachment of GalNAc to a threonine residue is catalyzed by UDP-GalNAc:polypeptide N-acetylgalactosaminyltransferase (GalNAcT) and, although not much is known about the bovine enzymes, there are at least 12 mammalian GalNAcTs that have been cloned and functionally expressed (Ten Hagen et al., 2003). In mice, the expression of several isoforms changes markedly during pregnancy and lactation (Young et al., 2003). It is likely that expression of the other required glycosyltransferases also varies, presumably under control of lactogenic hormones.\n\nOther seasonal factors related to climate, such as heat stress, drought, and nutrition (e.g., pasture versus fodder), can have an impact on milk production and composition. However, we are not aware of any specific studies on their effect on \u03ba-casein glycosylation.\n\n### Functional Significance\n\n\u03ba-Casein plays a key role in micelle stability by acting as a hairy layer that provides both steric and electrostatic repulsion between micelles, preventing aggregation. Glycosylation of \u03ba-casein increases both the size of the hydrophilic C-terminal 'hairs' and their charge, because of the bulk of the hydrophilic sugar residues with their hydration shells and the negative charge of the neuraminic acid groups, respectively. Theoretically, the higher the degree of glycosylation of \u03ba-casein, the greater its stabilizing ability should be. As such, it might be expected that the degree of glycosylation of \u03ba-casein would have a marked effect on both the size and the stability of the casein micelles. However, whereas the size of the casein micelles has been shown to be inversely related to their \u03ba-casein content (relative to total casein) by a number of authors (O'Connell & Fox, 2000), there is no clear correlation between micelle size and degree of glycosylation. Slattery (1978) found an apparent inverse relationship between the proportion of glycosylated \u03ba-casein and micelle size, but it did not apply to all of the size fractions isolated. In contrast, Dalgleish (1985; 1986) found that the proportions of glycosylated and nonglycosylated \u03ba-casein did not vary with micelle size. More recently, O'Connell and Fox (2000) showed an apparent increase in \u03ba-casein glycosylation with increasing micelle size. Some of these discrepancies are probably the result of the different experimental approaches adopted. Currently, there is no conclusive evidence for a distinct relationship between micelle size and \u03ba-casein glycosylation.\n\nAs stated above, micellar stability, or controlled destabilization in the case of cheese and yogurt manufacture, is of key importance in dairy manufacturing. A number of authors have looked for effects of \u03ba-casein heterogeneity on micellar stability and the processing properties of milk. Takeuchi et al. (1985) used ion-exchange chromatography to prepare nine fractions of \u03ba-casein A-1P that varied in the level of glycosylation. The ability of these subfractions to stabilize \u03b1s1-casein was shown to increase with increasing carbohydrate content. In cheese manufacture, the initial step is the chymosin- (rennet-) catalyzed cleavage of the Phe126\u2013Met127 bond in \u03ba-casein, resulting in release of the hydrophilic caseinomacropeptide from the micelle surface, which leads to micellar aggregation or clotting. Doi et al. (1979) examined the susceptibility to chymosin action of \u03ba-casein preparations with different degrees of glycosylation. They found that more highly glycosylated forms were less susceptible to hydrolysis not only by chymosin but also by other proteinases. Others have also found an inverse relationship between glycosylation and chymosin susceptibility with purified \u03ba-casein fractions (Addeo et al., 1984; Vreeman et al., 1986) and in model systems (Addeo et al., 1984; Leaver & Horne, 1996).\n\nIn milk, the relationship is not so clear. Chaplin and Green (1980) claimed that all \u03ba-casein molecules were hydrolyzed with equal efficiency, whereas van Hooydonk et al. (1984) found that the rate of chymosin-catalyzed hydrolysis decreased with increasing glycosylation. Again, differences in experimental approach were probably responsible for at least some of the apparent discrepancy. Rennet coagulation time (RCT), rate of curd firming, and curd firmness have been measured to assess the effect of \u03ba-casein glycosylation on the coagulation properties of milk (Robitaille et al., 1993). Whereas no effect on RCT was observed, the rate of curd firming decreased and the curd firmness increased at higher glycosylation levels. A more recent study of more than 2000 samples of milk from Simmental cows (Bonfatti, et al., 2013) observed that RCT decreased when total \u03ba-casein (as a proportion of total casein) and glycosylated-\u03ba-casein increased, whereas unglycosylated-\u03ba-casein exhibited a slightly unfavorable effect on the onset of the coagulation process. A slight decrease of RCT was also observed for milks with a high degree of glycosylation of \u03ba-casein, although this effect was less clear than that of glycosylated-\u03ba-casein. A favorable effect of \u03ba-casein, glycosylated \u03ba-casein, and degree of glycosylation on curd firmness was also detected. In another recent study, a detailed characterization of \u03ba-casein isoforms was conducted on milks with extremes of good or poor coagulation properties, selected from 892 samples of milk from Holstein-Friesian and Jersey cows (Jensen et al., 2012). Six \u03ba-casein isoforms varying in phosphorylation and glycosylation levels from each of the genetic variants of \u03ba-casein were separated and identified, along with an unmodified \u03ba-casein form at low abundance. Relative quantification showed that around 95% of total \u03ba-casein was phosphorylated with 1 or 2 phosphates attached, whereas approximately 35% of the identified \u03ba-casein was glycosylated with 1 to 3 tetrasaccharides. In a comparison of isoforms from individual samples, a very consistent \u03ba-casein isoform pattern was found, with only minor differences in relation to breed, \u03ba-casein genetic variant, and milk coagulation ability.\n\nDifferences in rennet coagulation properties have also been observed for genetic variants of \u03ba-casein. Shorter RCTs, higher curd-firming rates, and higher curd firmness have been reported for milk from cows with the BB variant than for milk from cows with the AA variant (Walsh et al., 1998, and references therein). These differences between A and B variant milks were maintained after heat treatments of up to 80 \u00b0C for 2 min, despite an overall deterioration of the coagulation properties at elevated temperatures (Choi & Ng-Kwai-Hang, 2003). Milks containing the rarer \u03ba-casein C variant form rennet gels even more slowly than the A or B variant milks, possibly because of the substitution of histidine for arginine at residue 118, which may affect chymosin binding (Smith et al., 1997). Similar results have been observed for the \u03ba-casein G variant, which has cysteine at residue 118 (Erhardt et al., 1997), and a similar explanation has been proposed (Smith et al., 1997).\n\nCoagulation of milk can also be induced by acid, as is the case in yogurt manufacture. Because this step does not involve \u03ba-casein cleavage, a lesser or different effect might be expected. There are fewer studies on the effect of the glycosylation of \u03ba-casein on acid coagulability. Cases et al. (2003) found that partial deglycosylation with neuraminidase had little effect on micellar surface charge and solvation but caused a decrease in acid gelation time, a higher rate of gel firming, and a higher final firmness.\n\nHeat treatment of milk can also destabilize the casein micelle structure. The heat-induced coagulation of milk is a very complex process that is affected by many parameters (O'Connell & Fox, 2003). A number of studies have examined the influence of genetic variants of \u03ba-casein on heat stability parameters, and it is generally accepted that B variant milks are more stable than A variant milks (FitzGerald & Hill, 1997). The reason may be related more to the effects on \u03ba-casein concentration and micelle size mentioned above than to the structural differences between the variants (Smith et al., 2002). Again, there are fewer studies related to the influence of the glycosylation of \u03ba-casein on heat stability. Using a model system composed of casein micelles in simulated milk ultrafiltrate, Minkiewicz et al. (1993) showed that enzymatic removal of neuraminic acid using neuraminidase caused a decrease in heat stability. However, Robitaille and Ayers (1995), using whole milk, could not find a significant effect of neuraminidase treatment on heat stability. When milk is heated above 65 \u00b0C, \u03b2-lactoglobulin denatures, exposing a previously buried sulfhydryl group that can participate in disulfide exchange reactions with other cysteine-containing proteins including \u03ba-casein. This interaction has been recognized for many decades (Sawyer, 1969) and has been the subject of numerous investigations and reviews over the years; a detailed analysis is beyond the scope of this review (for an extensive review, see Chapter 9 of this volume). Recent studies have addressed both the mechanism of formation (Guyomarc'h et al., 2003) and the impact on product quality (Vasbinder et al., 2003) of disulfide-linked complexes. Despite the vast amount of literature on this topic, there do not appear to be any studies that have addressed the impact of the variable glycosylation of \u03ba-casein on its ability to form disulfide-linked complexes either with itself or with \u03b2-lactoglobulin, although this is perhaps not surprising because the sulfhydryl amino acids are in the para-\u03ba-casein domain of the molecule and the glycosylation sites are in the caseinomacropeptide.\n\nHeat-induced changes in micelle structure are particularly relevant for ultra-high-temperature (UHT) milk production and storage. The extremes of heat treatment (of the order of 140\u2013145oC for 4\u201310 s) produce a number of changes in the milk, not least of which is the formation of \u03ba-casein\u2013 \u03b2-lactoglobulin complexes. On storage, UHT-treated milks show a variable tendency to form gels, and this phenomenon, known as age gelation, affects product shelf life (for a review, see Datta & Deeth, 2001). Again, despite extensive studies over many years on UHT processing and product performance, the influence of \u03ba-casein heterogeneity, particularly heterogeneity with respect to glycosylation, has not been addressed. From a theoretical perspective, higher initial levels of glycosylation may act to temper the deleterious effects of heat treatment through effects on micellar size, micellar stability, and formation of disulfide-linked complexes. The heat treatment itself may affect the glycosylation level at the surface of the micelle either indirectly, through loss of \u03ba-casein in complex formation with \u03b2-lactoglobulin, or directly, through degradation of glycosidic residues (van Hooydonk et al., 1987; as quoted in Dziuba & Minikiewicz, 1996). Subsequent changes in the glycosylation level during storage could be mediated by the action of heat-stable glycosidases originating from psychrotrophic bacteria present in the raw milk (Marin et al., 1984). Release of monosaccharides during the storage of UHT milk has been observed (Recio et al., 1998; Belloque et al., 2001). Thus, both the initial glycosylation level of the \u03ba-casein and the residual amount after UHT treatment may affect the storage properties of UHT-treated milk. As the actions of heat-resistant proteinases can contribute to the age gelation of UHT milk, the inhibitory effects of glycosylation on the activity of proteinases such as plasmin (Doi et al., 1979) may be important for prolonging shelf life. Unraveling specific effects will require the application of modern proteomic technologies for \u03ba-casein analysis (Claverol et al., 2003; Holland et al., 2004; ; ; O'Donnell et al., 2004). Through use of these technologies, it will be possible to elaborate the heterogeneous glycoforms of \u03ba-casein in raw milk, after pretreatment(s), after UHT processing, and during storage leading up to gelation. This will allow a definitive assessment of the functional significance of \u03ba-casein glycosylation.\n\nFunctional performance of the caseinomacropeptide is of importance both for use of the caseinomacropeptide as a food ingredient in its own right and for the properties it can impart to cheese whey products. Emulsifying performance has been found to vary with glycosylation (Kreuss et al., 2009a). Nonglycosylated caseinomacropeptide showed significantly better emulsifying properties than glycosylated caseinomacropeptide. While nonglycosylated caseinomacropeptide showed an emulsifying activity index of 150.7 g\/m2, glycosylated caseinomacropeptide achieved a value of only 98.5 g\/m2. Stability of emulsions was 1.4 times higher for nonglycosylated than for glycosylated caseinomacropeptide. Droplet size measurements and creaming studies showed a strong influence of pH on both fractions, with minimal emulsion stabilities at pH 4.1 (glycosylated caseinomacropeptide) and 4.9 (nonglycosylated). Investigation of the flocculation behavior and variations of the ionic strength indicated that the glycan side chains induce a combination of electrostatic, steric, and hydrophilic effects, preventing an ordered adsorption of glycosylated caseinomacropeptide molecules at the oil\/water interface, while nonglycosylated caseinomacropeptide builds a stable network at the surface.\n\nIn a further study, caseinomacropeptide fractions, both glycosylated and nonglycosylated, were studied in detail for their foaming properties (Kreuss et al., 2009b). The nonglycosylated caseinomacropeptide-stabilized foams showed significantly higher foam rigidity and stability than foams stabilized with glycosylated caseinomacropeptide, whereas both fractions yielded a high foaming ability with overruns of around 600%. The glycosylated caseinomacropeptide-stabilized foams, in particular, were considerably influenced by pH and showed reduced foaming properties above the pI, but superior properties at strong acidic pH, below the pI. This influence was less significant for nonglycosylated caseinomacropeptide. An increase in ionic strength did not appear to influence either fraction. The combination of electrical, steric, and hydrophilic barriers caused by the glycosylation of glycosylated caseinomacropeptide appears not to allow an ordered adsorption at the air\/water interface, whereas nonglycosylated caseinomacropeptide can build a stable network at the surface.\n\n#### Biological Significance\n\nOne aspect of \u03ba-casein heterogeneity that has not been considered is its influence on the biological properties of milk. This aspect has been reviewed extensively (Dziuba & Minikiewicz, 1996). There are three areas to consider. First, the effect of post-translational modification on digestibility and bioavailability has only received attention relatively recently. The effect of glycosylation on hydrolysis by chymosin has been discussed above and has been intensively studied because of its importance for cheese making. The digestion of the resultant glycosylated or nonglycosylated caseinomacropeptide by brush border membrane peptidases has been described by Boutrou et al. (2008). Their key finding was that the digestion of unglycosylated and glycosylated caseinomacropeptide through the action of exopeptidases was similar, but the activity of endopeptidases on glycosylated caseinomacropeptide was limited, certainly due to the attached O-glycosylations. Consequently, many more peptides were identified from the unglycosylated than from the glycosylated caseinomacropeptide. In addition, the glycosylation core, as well as the number of the attached glycosylated chains, modified the kinetics of digestion, the most heavily glycosylated forms being the slowest digested.\n\nSecond, there is the nutritional contribution of the carbohydrate residues in \u03ba-casein, particularly NeuAc. The importance of NeuAc and its roles in numerous biological functions have been reviewed (Schauer, 2000). NeuAc is commonly found as the terminal sugar residue on mammalian glycoproteins. Although mammals can synthesize NeuAc, the high levels in milk and especially colostrum may be related to a high demand for neonatal growth and development. The normal glycans on \u03ba-casein are part of a class known as the Thomsen-Friedenreich-related antigens (Dall'Olio & Chiricolo, 2001). The terminal NeuAc residues may play a key role in preventing colonization of the gut by pathogenic organisms by providing alternative binding sites that minimize binding to the normal gut epithelium.\n\nThe third area relates to the enormous interest in bioactive peptides derived from milk proteins (Clare & Swaisgood, 2000; Kilara & Panyam, 2003). Numerous in vitro activities have been ascribed to \u03ba-casein, its caseinomacropeptide or peptides derived from them (Dziuba & Minikiewicz, 1996; Brody, 2000). Some of these activities appear to be associated with particular forms of \u03ba-casein (Malkoski et al., 2001) and can be glycosylation-dependent (Li & Mine, 2004). Whether or not the same activities occur in vivo is not always clear because it requires both generation and absorption of the active component during digestion and this is not easy to detect. In vivo production of caseinomacropeptide is known to occur after milk ingestion (Ledoux et al., 1999, and references therein) and has been detected in the plasma of infants after the ingestion of milk (Chabance et al., 1995). Any naturally occurring bioactivity of caseinomacropeptide-derived peptides could be strongly influenced by the glycosylation status of \u03ba-casein either directly, by modifying the activity of the peptide, or indirectly, by affecting proteolysis of \u03ba-casein and hence release of the peptide.\n\n## Caseins from other species\n\nAlthough much of the focus has been on bovine milk, other species have not been neglected (Ginger & Grigor, 1999), as in recent years attention has focused on milk from alternative species as a source of nutrition. It is apparent that considerable variations in caseins and their PTMs occur between different species. In this section of the chapter, all positions reported are for the mature protein, that is, without the promoter region.\n\nThe \u03b2-casein of human milk exists as six different forms with 0 to 5 phosphates (Greenberg et al., 1984). In a recent study on human milk, the phosphorylation status of \u03b2-casein was investigated in human milk samples over time (Molinari et al., 2013). The phosphorylation of \u03b2-casein varied significantly between term and preterm milk. A longitudinal trend was observed among the term population, with the phosphorylation state of \u03b2-casein decreasing during lactation in seven of the eight mothers analyzed. In the preterm population, no change in the distribution of phosphorylated isoforms was observed over time in 10 of the 16 mothers, whereas in the other 6 mothers, the level of phosphorylation increased during lactation. A study on glycosylation in human milk revealed changing patterns of glycosylation of many of the whey proteins, but not for \u03ba-casein (Froehlich et al., 2010).\n\nEquine \u03b2-casein also shows variation in phosphorylation, with typically 3 to 7 phosphates on full-length \u03b2-casein (Girardet et al., 2006) and 1 to 7 phosphates on a low-molecular-weight form that arises from an internal deletion (Miclo et al., 2007). The phosphorylation sites have recently been reported for equine milk (Mateos et al., 2010). The isoform 4P was found to be phosphorylated on residues Ser9, Ser23, Ser24, and Ser25. Addition of phosphate groups on Ser18, Thr12, and Ser10 led to the formation of the isoforms 5P\u20137P, respectively. The results indicate that the in vivo phosphorylation of the equine \u03b2-casein follows a sequential path and is not random.\n\nOvine \u03b2-casein has also been reported to be variably phosphorylated, with 0 to 7 phosphates (Ferranti et al., 2001), although it is not clear where the seventh phosphorylation site is. Caprine \u03b2-casein appears to be more like bovine \u03b2-casein with the same five phosphorylation sites and an additional site on Thr27 (Neveu et al., 2002). A more recent study compared goats' milk from the indigenous Greek breed and from the international breeds Saanen and Alpine (Moatsou et al., 2008). This study identified a wide range of protein polymorphisms and phosphorylations. Phosphorylation of \u03b1s1-casein was 7P, 8P, and 9P, with lesser amounts of 6P and, rarely, 10P; phosphorylation of \u03b1s2-casein ranged from 6P to 11P; phosphorylation of \u03b2-casein was 5P, 6P, and 7P, with small amounts of lesser phosphorylation; and \u03ba-casein was phosphorylated with mostly 2P but some lesser amounts. A recent study using nanoscale liquid chromatography coupled with tandem electrospray mass spectrometry has identified the majority of the phosphorylation sites of the goat caseins (Olumee-Shabon & Boehmer, 2013). These are summarized in Table 5.3.\n\nTable 5.3\n\nPhosphorylation Positions for the Caprine Caseins\n\nCasein | No of phosphorylations | Phosphorylated residues  \n---|---|---  \n\u03b1s1 | 9 | S61, S63, S130 (6 others not detected)  \n\u03b1s2 | 10 | S23, S24, S25, S72, S73, S74, S77, S145, S147, S159  \n\u03b2 | 5 | S32, S33, S34, S37, S50\n\nData from Olumee-Shabon, Boehmer, 2013.\n\nA detailed proteomic analysis of water buffalo milk (D'Ambrosio et al., 2008) showed high consistency with the bovine counterpart proteins (residue numbers here based on the mature protein sequence). The study identified phosphopeptides from \u03b1sl-casein where phosphorylation occurred at Ser 41, 46, 48, 64, 66, 67, 68, and 75; \u03b1s2-casein phosphorylation occurred at the same sites of the bovine counterpart (Ser 129, 131, and 143); \u03b2-casein phosphorylation sites were consistent with bovine (Ser 15, 17, 18, 19, and 35); and the main and secondary sites of phosphorylation in buffalo \u03ba-casein were Ser 149 and Ser 127 as observed for the bovine protein. Diglycosylated forms of \u03ba-casein were identified, but the specific residues modified were not reported. However, there is no reason to believe they would be different from the bovine sites.\n\nA recent study of the donkey 'caseome' (Chianese et al., 2010) identified 11 components for \u03ba-casein, six phosphorylated components for \u03b2\\- and \u03b1s1-casein, and three main phosphorylated components for \u03b1s2-casein.\n\n## Conclusions\n\nPTMs such as phosphorylation, glycosylation, and perhaps disulfide bond formation play a critical role in casein micelle formation and stability. It seems somewhat surprising that so much variability occurs in these PTMs on the caseins. Whereas significant functional differences in milk properties have been consistently reported for milks with different genetic variants of the caseins and are well established, the effects reported for variable PTMs have been investigated only more recently, enabled by proteomic methods and supported by a range of highly sensitive mass spectrometry techniques. 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The various strands are drawn together step by step to develop the dual-binding model of casein micelle assembly and structure as a polymerization of a block copolymer system. The model is then used to predict the behavioral properties of the micelle in fluid milk, where, by considering the rheology of high concentration milks, the hard sphere colloidal approach is shown to be a special case limited to milk of normal concentration and pH. The necessity for a dual-binding approach is then forcefully demonstrated in its ability to provide full mechanistic explanations of observed behavior in the renneting, acid gelation, and alcohol-induced destabilization of skim milk. Bond mobility is identified as a crucial factor but also important is bond location and whether bonding can be extended beyond the protective shell of the micelle, its hairy layer. It is emphasized that the dual-binding model is only a tool and that the most important features are the interactions of the caseins with themselves and mineral calcium phosphate.\n\n## Keywords\n\nCasein physicochemical properties, casein micelle structure, micellar calcium phosphate, dual-binding model, casein micelle stability\n\nOutline\n\nIntroduction 169\n\nCasein primary structure and interactions 171\n\nCasein micelle properties 176\n\nModels of casein micelle structure 178\n\nThe Dual-binding Model for Micelle Assembly and Structure 178\n\nCalcium Phosphate Equilibria in the Dual-binding Model 180\n\nPredictions of Casein Micelle Properties in the Context of the Dual-binding Model 182\n\nSize and Appearance 182\n\nEffects of Urea, pH, Sequestrants, and Temperature 183\n\nThe Dual-binding Model and Micellar Interactions 184\n\nConcentrated Micellar Dispersions 186\n\nThe Dual-binding Model and Micellar Destabilization 189\n\nDual-binding Model and Rennet Curd Formation 189\n\nDual-binding Model and Ethanol Stability 191\n\nDual-binding Model and Acid Gel Formation 192\n\nConcluding remarks 195\n\n## Introduction\n\nThe caseins are a family of phosphoproteins found in the milks of all mammals. They exist in these milks generally as complex aggregates or micelles of the proteins and mineral calcium phosphate (Fox and Brodkorb, 2008). Because the caseins utilize the same calcium-sequestering mechanism to regulate the calcium phosphate concentration of their environment, they have recently been identified as members of a wider family of secretory calcium-binding phosphoproteins descended from a common ancestor gene (Kawasaki and Weiss, 2003; 2006). These secretory phosphoproteins include enamel matrix proteins, dentine, salivary proteins, bone extracellular matrix proteins, and the caseins, among others. All are descended from early primordial genes by duplication and divergence to serve their specialized adaptive functions. Their genes retain common functional and sequence features even after this extensive divergence. It is thought that primordial calcium-sensitive casein genes diverged from enamel matrix protein genes before the appearance of monotremes in the Jurassic era (Kawasaki and Weiss, 2003). More controversially, Kawasaki et al. (2011) have recently argued that all caseins, both calcium-sensitive and calcium-insensitive, that is, \u03ba-type caseins, evolved from the odontogenic ameloblast-associated gene (ODAM) via two different pathways; calcium-sensitive genes are postulated to originate directly from secretory calcium-binding phosphoprotein genes (SCPP), whereas the calcium-insensitive genes directly differentiated from follicular dendritic cell-secreted peptide gene (FDCSP), both SCPP and FDCSP having a common ancestor in ODAM. This ancestral line for \u03ba-casein genes is inconsistent with the hypothesis of Jolles et al. (1978), based on high levels of sequence identity, that \u03ba-casein was derived from \u03b3-fibrinogen, a blood coagulation factor.\n\nCasein allows milk to appear supersaturated with calcium phosphate. Essentially, it transports the mineral calcium phosphate safely through the mammary gland, which is essential for development of bones and teeth in the suckling infant, through the medium of the casein micelle. Locking up the calcium phosphate in this package is one aspect of the biological function of the micelle. Ensuring release of this same calcium phosphate in the gastric destination of the milk is a property of the micelle that is not generally given great consideration. More research effort has been put into trying to give mechanistic understanding to the technological behavior of the micelle, understanding necessary to achieve efficient conversion of milk into products such as cheese and yogurt, or the behavior of milk components in emulsions or reconstituted dairy products.\n\nMany of the physical and technological properties of the casein micelle (diffusion, viscosity, and light scattering) can be described by treating the casein micelles as colloidal hard spheres (Alexander et al., 2002). The initial stages, up to the onset of instability, in processes such as renneting, acid-induced gelation, and flocculation in the presence of ethanol can apparently be well described by allowing these micellar hard spheres to become adhesive, essentially, as described later, treating their \u03ba-casein outer layer as a salted polyelectrolyte brush (De Kruif and Zhulina, 1996; De Kruif and Holt, 2003). Beyond the critical point in all three processes, however, changes in micellar integrity and internal structure render the simple colloidal particle approach inadequate (Horne, 2003a,b; Choi et al., 2007; Ozcan-Yilsay et al., 2007).\n\nNeither is the adhesive sphere approach at all helpful in furthering our understanding of the processes of micellar assembly, the pathways to dissociation or the maintenance of micellar integrity. For that, we must turn to a structural model of the casein micelle, bearing in mind that, notwithstanding the inadequacy of the adhesive sphere approach, our micelle model also has to be adaptable enough to explain why that primitive approach has been so successful within its limitations.\n\nVarious models of casein micelle assembly and structure have appeared over the years and have been subjected to regular review and appraisal, most recently by Fox and Brodkorb (2008). The three principal contenders are: (1) the submicelle model of Slattery and Evard (Slattery and Evard, 1973; Slattery, 1977), subsequently elaborated by Schmidt (1980); (2) the nanocluster model of Holt (Holt 1992; 2004; De Kruif and Holt, 2003); and (3) the dual-binding model proposed by Horne (1998). There has been considerable debate in more recent years over the suitability and success of these models (Farrell et al., 2006; Horne, 2006; Dalgleish, 2011). Dalgleish (2011) repeated the arguments of Horne (1998) in dismissing the submicelle model and proposed minor but dubiously founded amendments to the dual-binding model, as did McMahon and Oommen (2008). De Kruif et al. (2012) have also described the latest incarnation of the nanocluster model in terms that are essentially those of a dual-binding model. The differences between these amendments and the original dual-binding model of Horne (1998) are small and subtle and are considered in detail later. It is not our intention to repeat previous arguments; rather we would point out that the dual-binding model has its basis in the interactions and chemistry of the caseins and calcium phosphate. As we demonstrate in the following, the model is only an aid to visualizing how the casein micelle responds in the production of dairy products such as yogurts or rennet curds, or in stability situations with ethanol or heat. The reality is in how the interactions and chemistry dictate the behavior of the caseins in those situations. We first summarize the physicochemical properties and interactions of the caseins and show how these lead naturally to the dual-binding picture.\n\n## Casein primary structure and interactions\n\nJust as casein micelles are aggregates of all of the casein proteins and micellar calcium phosphates, so does the dual-binding model involve the properties and interactions of all of the caseins. Central to this argument are those features of the proteins that are conserved across species and through millennia. The caseins were identified as members of the wider secretory calcium phosphate binding family by their possession of functional and sequence features common to that family (Kawasaki and Weiss, 2003; 2006). Among the conserved motifs is the SXE peptide (Ser\u2013Xaa\u2013Glu) where Xaa may be any amino acid. In the caseins, this peptide provides a recognition template for post-translational phosphorylation of the serine in the mammary gland by a casein kinase (Mercier, 1981). Moreover, in the caseins, the serine residues are often found clustered in groups of two, three, or four. Such clusters in the \u03b1S\\- and \u03b2-caseins are highly conserved (Martin et al., 2003), and their numbers attest to the significance of the calcium phosphate requirement for postnatal growth in mammals\u2014even more so when it is noted by reference to their sequences (Swaisgood, 2003) that the \u03b1S-caseins, for example, the \u03b1S1\\- and \u03b1S2-caseins of bovine milk, themselves possess two or more such clusters. From now on, discussion will be confined largely to the behavior of bovine caseins because it is for these that the largest body of research data is available. However, the extensions to the behavior in the micelles of other milks will be obvious.\n\nThese clusters of phosphoserine residues and the necessary glutamic acid residues templating their existence give rise to massive downward spikes in the hydrophobicity profiles of the bovine \u03b1S1\\- and \u03b2-caseins (Fig. 6.1). Associated with these are significantly high densities of negative charge at normal milk pH. There is a charge density of \u20139e within the span of residues 65\u201372 of \u03b1S1-casein and a further \u20136e along the sequence 48\u201353 of the same protein. A similarly high-charge density of \u20139e is found between residues 16\u201323 of \u03b2-casein, encompassing the phosphoserine cluster there. Similar high densities are found around the phosphoserine clusters of \u03b1S2-casein. All of these estimates of charge density assume a contribution of \u20131.5e from each phosphoserine residue at or near the natural pH of milk, 6.7.\n\nFigure 6.1 Hydrophobicity plots of (a) \u03b1S1-casein, (b) \u03b1S2-casein, and (c) \u03b2-casein calculated as a moving average (window n = 3) of amino acid hydrophobicities taken from the consensus scale used by Horne (1988). Asterisks denote centers of electrostatic repulsion arising from phosphoserine cluster motifs, the size indicating the number of negative charges associated with each, as listed in the text.\n\nAway from the phosphoserine clusters, the casein molecules are distinctly hydrophobic. This segregation of hydrophilic and hydrophobic residues confers on the caseins a definite amphipathic nature, which contributes to their ability to function successfully as stabilizers in oil-in-water emulsions. The topography of \u03b2-casein adsorbed at the oil\u2013water interface was probed by testing the accessibility of the reactive sites to the proteolytic enzyme trypsin (Leaver and Dalgleish, 1990). In aqueous solution, the reactive sites of \u03b2-casein were attacked randomly at no preferential rate. With \u03b2-casein-stabilized emulsions, however, the peptides released showed the lysines at positions 25 and 28 of the sequence to be readily accessible to trypsin, whereas all other possible attack sites were less so (Leaver and Dalgleish, 1990). These residues lie in the center of the highly charged, hydrophilic N-terminal region containing the four-phosphoserine cluster in \u03b2-casein. Measured by dynamic light scattering, a decrease of approximately 13 nm in hydrodynamic radius of the emulsion droplets also accompanied the scission of these peptides, indicating the extent to which they stretched out into the aqueous phase from the emulsion droplet surface (Dalgleish and Leaver, 1991). The remaining hydrophobic portion of the molecule was speculated to lie along the droplet surface, shielded from trypsin attack. Similar changes in hydrodynamic radius were observed when \u03b2-casein was adsorbed from aqueous buffers on to the surface of polystyrene latex particles, indicating a similar adsorption pattern (Dalgleish, 1990; Brooksbank et al., 1993), a pattern that was replicated at the air\u2013water interface, as observed by neutron reflectivity (Dickinson et al., 1993).\n\nThe combined experimental evidence was therefore consistent with the view that much of the hydrophobic end of the adsorbed \u03b2-casein was directly associated with the hydrophobic interface, with the hydrophilic N-terminal tail extending significantly out into the aqueous phase. Self-consistent-field calculations of the conformation of \u03b2-casein adsorbed at a planar hydrophobic interface confirmed this picture of a tail\u2013train structure, and also predicted a train\u2013loop\u2013train structure for adsorbed \u03b1S1-casein with anchor points at both ends of the molecule (Leermakers et al., 1996; Dickinson et al., 1997a,b). Schematic representations of these structures were drawn by Horne (1998) and were used in depicting assembly of the casein micelle via the dual-binding model. Perhaps rather than depicting the hydrophobic regions of the molecules as rectangular bars, it would have been more realistic to depict these regions as puckered, as in Figure 6.2, because not all amino acids therein are equally hydrophobic, as the profile plots of Figure 6.1 demonstrate.\n\nFigure 6.2 Schematic structures of \u03b1S1\\- and \u03b2-casein, based on self-consistent field calculations of the proteins adsorbed onto a hydrophobic interface, illustrating the train\u2013loop\u2013train structure of \u03b1S1-casein and the loop\u2013train structure of \u03b2-casein. The dashed circles give an idea of the range of the interaction potential components, the larger circle around the loops being the electrostatic repulsion arising from the negative charge centers thereon and the smaller circle being the regions of hydrophobic attraction in the train.\n\nSuch representations were also used by Horne (1998) to picture the aggregates produced by self-association of \u03b2-casein or \u03b1S1-casein (Fig. 6.3). Thus \u03b2-casein was envisaged as a hedgehog-like micelle subject to a monomer\/micelle equilibrium, a mechanism proposed by Payens and co-workers (Payens and van Markwijk, 1963; Payens et al., 1969). In this picture, the hydrophobic trains are buried inside, and the charged hydrophilic tails extend from the surface into solution. More recently, the self-association of \u03b2-casein has been revisited by De Kruif and collaborators (De Kruif and Grinberg, 2002; Mikheeva et al., 2003; O'Connell et al., 2003), who applied high-sensitivity differential scanning calorimetry, as well as static and dynamic light scattering techniques to the problem. These experiments again concluded that a micelle-like structure was adopted but, rather than being formed by the highly cooperative monomer\/micelle equilibrium suggested previously, instead the micellization took place as a series of consecutive additions of monomer to a growing micelle, as suggested in the shell model of Kegeles (1979). In a like fashion, with its hydrophobic chains at opposite ends of the molecule and a central section containing the highly charged phosphoserine clusters, \u03b1S1-casein self-associates to produce a worm-like chain polymer (Payens and Schmidt, 1966; Schmidt, 1970a,b).\n\nFigure 6.3 Diagrammatic representations of the polymeric structures generated when the hydrophobic chains of the caseins interact: (a) the worm-like chain of \u03b1S1-casein; and (b) the micelle of \u03b2-casein, where only two molecules have been included to simplify the diagram.\n\nThough driven by hydrophobic interactions, electrostatic repulsive interactions are also very important in the self-association of these caseins. In particular, note how the equilibrium structures adopted by their polymers place the centers of charge as far apart as possible while still permitting the self-association to take place. Compared with hydrophobic interaction, electrostatic repulsion is a long-range force, an important factor now being recognized in studies of protein\u2013protein interactions (Kegel and Van der Schoot, 2004; Piazza, 2004; Stradner et al., 2004), and certainly manifesting itself in these reactions of the caseins. These electrostatic interactions define the degree of polymerization and limit further growth. Thus, increasing the pH, which increases the protein charge, decreases the polymer size in both \u03b1S1-casein and \u03b2-casein solutions, whereas increasing the ionic strength, which decreases the range of the electrostatic repulsion component, allows the formation of larger polymers for both casein species (Payens et al., 1969; Schmidt, 1970a,b).\n\nThe importance of charge in controlling the extent of aggregation of the caseins cannot be stressed too highly. Precipitation of the caseins can be achieved by lowering the pH and titrating away a sufficient amount of the charge of the phosphoseryl and carboxyl groups to reach the isoelectric points of the proteins. \u03b1S2-Casein, \u03b1S1-casein, and \u03b2-casein are termed the calcium-sensitive caseins because they can be precipitated in the presence of ionic calcium, the order of sensitivity being as given (Swaisgood, 2003). The most extensive dataset is available for \u03b1S1-casein. Here the aggregation shows a lag phase with little change in molecular weight with time until a critical time, beyond which rapid aggregation occurs. Horne and Dalgleish (1980) demonstrated that the logarithm of this critical coagulation time was a linear function of Q2, where Q is the net negative charge of the protein. Thus Q is the algebraic sum of the negative and positive charges of the protein, reduced by twice the number of calcium ions bound to the protein, each calcium carrying two positive charges. Furthermore, this relationship held when changes in the net negative charge were produced by chemical modification, whether by conversion of positively charged lysine residues to neutral or negatively charged derivatives, or even by introduction of new negatively charged sites by iodination of tyrosine residues to the di-iodo form (Horne, 1983; Horne and Moir, 1984). Each of these modifications effectively increases the net negative charge of the protein, thereby reducing its propensity for calcium-induced precipitation and slowing down the rate of aggregation. However, once the protein charge is corrected for the measured extent of modification, the logarithm of the rate of precipitation has been shown to remain linear in Q2, all points lying on the same line as those obtained with the unmodified protein, all other reaction conditions being the same (Horne, 1983; Horne and Moir, 1984). The net charge of the protein therefore dominates its precipitation behavior. Farrell et al. (2006) have suggested that positively charged residues in the N-terminal hydrophobic chain of \u03b1S1-casein could participate in binding to the phosphate groups of phosphoseryl residues, but such +\/\u2013 bridging does nothing to reduce the net negative charge of the protein having already been accounted for in the algebraic summation leading to Q, which, as we have demonstrated, controls the level of aggregation in these proteins.\n\nIt is only when the local balance of electrostatic repulsion and hydrophobic attractive interaction is in favor of attraction that hydrophobic bonds are formed. Sequentially, the major centers of electrostatic repulsion, the phosphoserine clusters, are remote from the hydrophobic regions in the trains of Figure 6.2, though, depending on the adopted conformation, they may not be remote spatially. Figure 6.2 is a depiction of a possible conformation of each protein at a hydrophobic interface, the puckering displaying the tendency to form a multitude of weak, short-range bonds by the hydrophobic chain. Lowering the temperature weakens hydrophobic bonds, hence the tendency to find monomeric \u03b2-casein at low temperatures but a micellar aggregate at room temperature and above. The aggregation of \u03b2-casein induced by calcium also shows a marked temperature dependence, with no precipitation observed at 4 \u00b0C (Parker and Dalgleish, 1981). At higher temperature, these hydrophobic bonds are in general stronger, but, because they are relatively weak overall, statistically individual bonds are readily ruptured by the increased thermal energy available, leading to more mobile, labile interactions between molecules.\n\n## Casein micelle properties\n\nAlmost all of the casein proteins present in bovine milk expressed at 37\u00b0C are incorporated into the casein micelles, together with a high proportion of the available calcium and inorganic phosphate. The calcium and phosphate within the micelle form low-molecular-mass species collectively known variously as colloidal calcium phosphate, micellar calcium phosphate, and latterly calcium phosphate nanoclusters. The micelles are very open, highly hydrated structures, with typical hydration values of 2\u20133 g H2O\/g protein, depending on the method of measurement.\n\nElectron microscopy shows that casein micelles are generally spherical in shape, with diameters ranging from 50 to 500 nm (average \u2248 150 nm) and a molecular weight ranging from 106 to >109 Da (average \u2248108 Da) (Fox and Brodkorb, 2008). For a casein content of 2.5 g\/100 mL milk, there are some 1014\u20131016 micelles\/mL milk, which implies a relatively close packing with inter-surface separations less than one micelle diameter.\n\nMilk is white largely because the colloidal dimensions of the casein micelles are such that they scatter significant amounts of light, an effect compounded by their high number density. Scattering of shorter wavelength radiation (neutrons and x-rays) reveals the internal structure to be heterogeneous, with a correlation length for variations in scattering length density within the particle of approximately 18 nm. Stothart and Cebula (1982) have interpreted this scattering behavior as being due to a structure composed of closely packed spherical subunits of this diameter, a picture that mirrors the raspberry-like appearance in early electron micrographs of the casein micelle (Schmidt, 1982).\n\nMore recent electron microscopy studies (McMahon and McManus, 1998; McMahon and Oommen, 2008) have suggested that these well-defined structures are likely to be artifacts of the fixation process, although micrographs recently obtained by field emission scanning electron microscopy (SEM) show a complex surface structure of cylindrical or tubular, but not spherical, protrusions between 10 and 20 nm in diameter, extending from the surface of the micelle (Dalgleish et al., 2004). These samples were not metal coated, although they were of necessity subjected to a fixation and dehydration process, which might have introduced some collapse of more loosely bound protein on to a denser skeleton. In a careful study using cryo-transmission electron microscopy (cryo-TEM) and small angle x-ray scattering (SAXS), Marchin et al. (2007) have extended previous structural studies to elucidate greater detail of micellar fine structure. Their cryo-TEM pictures show small regions of high electron density, approximately 2.5 nm in diameter, uniformly distributed in a homogeneous web of protein, giving the micelles a granular aspect that diminishes when the pH is reduced from 6.7 to 5.2. Paralleling this change in appearance, the SAXS scattering profile loses its characteristic shoulder at high wave vector when the pH is reduced. This clearly demonstrates that the shoulder is directly linked to the presence of micellar calcium phosphate and its dissociation from the micelle on acidification to pH 5.2. The results also demonstrate that this micellar calcium phosphate is uniformly distributed through the micelle in small particles, approximately 2.5 nm in diameter, and that, after their removal, other forces\/interactions must contrive to maintain micellar structural integrity. Their loss on acidification does not result in micellar disruption. Trejo et al. (2011) have also used cryo-TEM to study internal micellar structure. By varying the angle of incidence of the electron beam onto the same micelle, they have constructed tomographic images that demonstrate the presence of water-filled channels and cavities within the micelle and map the size, number, and location of the calcium phosphate nanoclusters, significantly different from the seemingly homogeneous networks apparent in earlier cryo-TEM studies (Marchin et al., 2007; McMahon and Oommen, 2008). While these interpretations chime with observations on the internal accessibility of the micelles to enzymes and ready release of \u03b2-casein, a word of caution should be added. The analysis by Trejo et al. (2011) is essentially a paint-by-numbers exercise. If the electron density is less than x, color it black and call it a channel or cavity; if the density is greater than y, then color it purple and call it a calcium phosphate nanocluster. There is no chemical identification of the scattering species, so both the number and size of the nanoclusters are likely to be an overestimate and the same is true for the channels and cavities. However, any structural model of casein micelle assembly would have to lead to a structure that could reproduce both the cryo-TEM pictures and the field emission SEM pictures, allowing for possible changes in appearance brought about by techniques of sample preparation and the vagaries of any analysis approach.\n\nCasein micelle structure is not fixed, but dynamic. In various ways, it responds to changes in micellar environment, temperature, and pressure. Cooling milk on release from the udder at 37 \u00b0C to storage at refrigeration temperatures brings about significant solubilization of \u03b2-casein, some \u03ba-casein, and much lower amounts of \u03b1S1\\- and \u03b1S2-casein from the micelles (Dalgleish and Law, 1988). Raising the temperature back to 37 \u00b0C reverses the process. None of this movement of \u03b2-casein does anything to disrupt the internal structure of the micelle, as observed by cryo-TEM and SAXS (Marchin et al., 2007). Almost complete disruption of the micelles, manifested by a loss of their scattering power and removal of the white color of milk, can be achieved by addition of a strong calcium sequestrant such as ethylene diamine tetraacetic acid (EDTA) (Griffin et al., 1988), by addition of urea (McGann and Fox, 1974), by dialysis against a phosphate-free buffer (Holt et al., 1986), by increasing the pH, by exposure to high pressure (Huppertz et al., 2006), or by addition of ethanol at \u224870 \u00b0C (O'Connell et al., 2001). Significantly, the colloidal calcium phosphate can also be solubilized by lowering the pH but, as confirmed by Marchin et al. (2007), without substantial disruption of the micelle structure.\n\nFractionation of the casein micelles according to size can be realized by a stepwise centrifugation protocol. The proportions of \u03b1S1\\- and \u03b1S2-caseins remain constant with micelle size, but \u03ba-casein content increases inversely with that size (Donnelly et al., 1984; Dalgleish et al., 1989). For a solid sphere, the surface-to-volume ratio is inversely proportional to the radius of the sphere, and these results imply that \u03ba-casein resides on the micellar surface, where its content controls the micellar total surface area and hence the micelle size. A surface location for the \u03ba-casein component may also be inferred from the requirement that this protein be readily accessible for rapid and specific hydrolysis by chymosin and similar proteinases, a reaction that destabilizes the micelles and leads to clot formation, which is exploited in cheese manufacture. A surface location is also required to enable the \u03ba-casein to interact with \u03b2-lactoglobulin in milk to form a complex on heating, the formation of which modifies the rennet and acid coagulation properties of the micelles. It is evident that a principal requirement, which must be met by any micelle model, is that it should generate a surface location for \u03ba-casein.\n\n## Models of casein micelle structure\n\nCasein micelle structure and casein micelle models have been extensively reviewed (Schmidt, 1982; Walstra 1990; 1998; Holt, 1992; Rollema, 1992; Horne 1992; 1998; 2006; Fox, 2003; Farrell et al., 2006; Fox and Brodkorb, 2008). As mentioned previously, based on the biochemical and physical properties of the micelles and the casein proteins outlined above, three main models have been proposed: the submicelle model (Slattery and Evard, 1973; Schmidt, 1982; Walstra, 1998); the nanocluster model of Holt (Holt, 1992; De Kruif and Holt, 2003); and the dual-binding model (Horne, 1998; 2002).\n\nIn the first model, the casein micelles are composed of smaller proteinaceous subunits, the submicelles, linked together via colloidal calcium phosphate. In the second model, the nanoclusters of colloidal calcium phosphate are randomly distributed, cross-linking a three-dimensional web of casein molecules. Both of these models have been severely criticized (Farrell et al., 2006; Horne, 2006), and the dual-binding model arose first as an attempt to overcome their deficiencies. It is significant that the most recent description of the nanocluster model (De Kruif et al., 2012) adopts tacitly an essentially dual-binding approach, replacing the terminology 'hydrophobic interactions' with 'nonspecific interactions.' Since most groups now appear to be accepting it or subtle variants, we present first a summary of the dual-binding model as providing a rational mechanism for micelle assembly and structure, and demonstrate how this model may be exploited to explain various observations of micellar properties and behavior.\n\n### The Dual-Binding Model for Micelle Assembly and Structure\n\nThe description presented in this section largely follows that found in Horne (2002), with minor refinement highlighted.\n\nIn the dual-binding model, micellar assembly and growth take place by a polymerization process involving, as the name suggests, two distinct forms of bonding, namely, cross-linking through hydrophobic regions of the caseins and bridging across calcium phosphate nanoclusters. Central to the model is the concept that bond formation is facilitated, and hence micellar integrity and stability are maintained, by a local excess of hydrophobic attraction over electrostatic repulsion, bearing in mind the quite different ranges of these interaction components. The individual casein molecules behave and interact as they do in their self-association equilibria, as described previously.\n\nEach casein molecule effectively functions as a block copolymer, as detailed in Figure 6.2, with the hydrophobic region(s) offering the opportunity for a multitude of individual, weak, hydrophobic interactions. The hydrophilic regions of the casein molecules contain the phosphoserine cluster (or clusters), with the exception of \u03ba-casein, which has no such cluster, each offering multiple functionality for cross-linking. Thus, as we have seen, \u03b1S1-casein can polymerize (self-associate) through the hydrophobic blocks, giving the worm-like chain of Figure 6.3. Further growth is limited by the strong electrostatic repulsion of the hydrophilic regions, but, in the casein micelle situation, the negative charges of the phosphoserine clusters are neutralized by intercalating their phosphate groups into a facet of the calcium phosphate nanocluster. This has two very important implications for the micelle. First, by removal of a major electrostatic repulsion component, it increases the propensity for hydrophobic bonding upstream and downstream of the nanocluster link. It effectively permits and strengthens those bonds. Second, it allows for multiple protein binding to each nanocluster, permitting a different network to be built up. \u03b2-Casein, with only two blocks, a hydrophilic region containing its phosphoserine cluster and the hydrophobic C-terminal tail, can form polymer links into the network through both, allowing further chain extension through both. \u03b1S2-Casein is envisaged in this model as having two of each block, two (possibly three; see below) phosphoserine clusters and two hydrophobic regions. It is only a small fraction of the total bovine casein but, by being able to sustain growth through all its blocks, it is likely to be bound tightly into the network. \u03ba-Casein is the most important of the caseins in the dual-binding model of micellar assembly and structure. It can link into the growing chains through its hydrophobic N-terminal block, but its C-terminal block is hydrophilic and cannot sustain growth by linking hydrophobically to another casein molecule. Neither does \u03ba-casein possess a phosphoserine cluster; therefore it cannot extend the polymer cluster through a nanocluster link. Thus, chain and network growth are terminated wherever \u03ba-casein joins the chain. This leaves the network with an outer layer of \u03ba-casein, satisfying the prime requirement recognized earlier. This assembly process is in its essential requirements identical to the model proposed by McMahon and Oommen (2008). Their main contribution is to describe the structure resulting from the assembly processes as an interlocked lattice, which seems an excellent description of a complex structure.\n\nThe nanocluster bridging pathway through the phosphoserine clusters was the only pathway allowed in the original nanocluster micelle model of Holt (Holt, 1992; De Kruif and Holt, 2003), where around 50 casein molecules are considered to bind to each calcium phosphate nanocluster. Horne (2006) and Horne et al. (2007a) have argued on the basis of mineral content and stoichiometry that the functionality of these nanoclusters will be much lower and that they link to four to six phosphoserine clusters, which may not necessarily originate from different casein molecules. Some consideration also has to be given to what constitutes a phosphoserine cluster capable of linking into the calcium phosphate nanocluster. Aoki et al. (1992) suggested a minimum of three phosphoserine residues, but De Kruif and Holt (2003) argued that two might be sufficient. This would allow the phosphoserine pair at positions 46 and 48 of \u03b1S1-casein, or those at positions 129 and 131 of \u03b1S2-casein, to function as nanocluster linkage sites, particularly if the carboxyls of the neighboring glutamate residues acted as pseudo-phosphate groups. This would give \u03b1S1-casein two linkage sites and \u03b1S2-casein three linkage sites. This level of functionality in these caseins is absolutely essential to the Holt model to build the required three-dimensional network, as, without them, an \u03b1S2-casein molecule with only two linkage sites and with such a low percentage of the total casein would probably prove to be insufficient. Although they are not essential to the dual-binding model, these mini-clusters of pairs of phosphoserines may provide for a weaker bridging link to the calcium phosphate nanocluster, allowing a range of nanocluster bond strengths to prevail.\n\n### Calcium Phosphate Equilibria in the Dual-Binding Model\n\nFrom the mineral viewpoint, casein micelle assembly is a frustrated crystallization of calcium phosphate. Milk is supersaturated in calcium and phosphate and, were it not for the presence and intervention of the highly phosphorylated caseins, a precipitation of calcium phosphate and potentially painful calcification of the mammary gland duct system would occur. There is considerable controversy over the possible (crystalline) structure of the nanocluster. The inorganic components of colloidal calcium phosphate have a stoichiometry close to that of the mineral, hydroxyapatite, whereas, if the phosphates of the casein phosphoserines are included in the mix, the stoichiometry moves closer to that of the mineral, brushite. Other groups have argued that the nanoclusters are actually an amorphous calcium phosphate (Cross et al., 2005), a form found in many biological situations that may have so-called Posner's clusters (Ca9(PO4)6) as basic building blocks (Yin and Stott, 2003). Posner's cluster can readily be detected as a unit within the structure of apatites, but it is also recognizable with distortion in other phosphate minerals. Amorphous calcium phosphate may be the first solid phase to appear upon mixing calcium and phosphate-containing aqueous solutions at pH >7 and concentrations sufficiently high to produce an immediate precipitation (Dorozkhin, 2010), but, in the formation of casein micelles, this ignores the observation of Horne (1982) that the inclusion of phosphate in a solution of \u03b1S1-casein and calcium initiates precipitation at ion-product levels, which are orders of magnitude below those observed in the absence of casein. This would appear to preclude the phosphoserines binding to preformed calcium phosphate nanoparticles, as suggested by DeKruif et al. (2012) but would offer the potential for the phosphoserine clusters of the caseins to act as templates to initiate nanocluster growth, to moderate that growth by the cutting of growth at a particular facet and finally to terminate that growth at the final free facet. In this way, the overall total number of phosphoseryl clusters in a milliliter of milk controls the size and number of nanoclusters present in the milk (Horne et al., 2007a). This can happen only if all (or close to all) of the phosphoserine cluster motifs are involved in nanocluster stabilization. In the scenario developed by Horne et al. (2007a), the phosphoserine clusters are incorporated into the planes of phosphate groups forming the faces of a hexagonal bipyramid. Examination of the molecular model of the brushite crystal in the photograph shown in Figure 6.4 reveals how these planes transect the crystal. In the Golgi vesicle, nanocrystal growth and number are unlikely to be limited by the availability of the inorganic components but only by the number of capping phosphoserine motifs on the caseins. In turn, this must mean that the binding of a serine phosphate group into the nanocluster must present a bonding advantage thermodynamically over that of a free phosphate group going into a growing calcium phosphate crystal in an equilibrium situation. It is a more favorable outcome thermodynamically. For these reasons, the presence of unbound, free phosphoserine clusters on \u03b2-casein in freshly drawn milk at 37 \u00b0C, as envisaged by Dalgleish (2011) is unlikely.\n\nFigure 6.4 Molecular model of the crystal structure of the calcium phosphate mineral, brushite, courtesy of Beever's Miniature Models, Department of Chemistry, University of Edinburgh. Lines are drawn to aid the eye in discerning planes through the crystal containing phosphorus atoms. Such solids were used by Horne et al. (2007a) to calculate a molecular weight and size for the nanocluster.\n\nBut just as the calcium phosphate crystal in equilibrium in solution with free calcium and phosphate is subject to environmental constraints shifting that equilibrium, so too is the micellar nanocluster species. The two equilibria may exist in parallel, though, when micelles are present in normal milk conditions, the nanocluster form is the favored option. However, conditions may change where 'solution' crystal growth is favored; we try to explore these speculations in some of the following discussions. The possibility of shifting calcium phosphate equilibria is an aspect of casein micelle structural behavior that has not been considered previously but, as the nanoclusters are involved, may offer alternative avenues to explore in addressing problems of micellar behavior associated with the application of high pressure, or of the addition of ethanol, or indeed in lowering milk temperature, where in the first edition we reproduced the effects later envisaged by Dalgleish (2011).\n\n### Predictions of Casein Micelle Properties in the Context of the Dual-binding Model\n\n#### Size and Appearance\n\nA major failing of the earlier micelle models was their lack of a plausible mechanism for assembly, growth, and, more importantly, termination of growth. All such elements are in place in the dual-binding model. Furthermore, the product of the dual-binding model satisfactorily represents the appearance and scattering behavior of the native casein micelle. Network growth is envisaged as a random process, and its termination along any particular pathway depends on the serendipitous arrival of a \u03ba-casein molecule. Micelle size will therefore depend on the proportion of \u03ba-casein in the mix, but will also present a range of sizes dependent as it is on random events. The model also reproduces the heterogeneity in structure required by the x-ray and neutron scattering data. The dense calcium phosphate nanoclusters will be rather homogeneously distributed through the matrix and will give rise to the structures observed by cryo-TEM and inferred from SAXS (Marchin et al., 2007). Their size, as predicted by the stoichiometric analysis of Horne et al. (2007a), has been confirmed in terms of molecular weight by Choi et al. (2011) and length scale by De Kruif et al. (2012).\n\nUntil this point, we have been emphasizing the interactions involved in casein micelle assembly, but the final structure of the micelle is dictated by the kinetics of the aggregation processes. Simulations of particle aggregation reactions (Meakin, 1999) demonstrated that open, ramified structures result when weak repulsion exists between particles. Essentially every collision between particles creates a bond and chains, and clusters grow from their extremities. When reaction barriers are higher, more collisions between particles are required before a favorable encounter produces a bond. This extensive sampling of phase space produces denser, more closely packed aggregates. These computer simulations were confirmed experimentally (Weitz et al., 1991). There is no doubt that casein micelles are open, ramified, highly hydrated structures with extensive water-logged channels and cavities, most recently pictured in the cryo-TEM study of Trejo et al. (2011). The creation of such structures is dictated by the kinetics of the assembly process, implying that such reactions are rapid, occurring on almost every encounter, and demonstrate low or insignificant energy barriers to bond formation.\n\nThe dual-binding model presents the casein micelle as a dynamic, 'living' entity. The hydrophobic interactions are individually weak and capable of breaking and recombining on an almost continuous basis. To some extent, the molecular movements this allows will be restricted by the potentially stronger nanocluster linkages and the low probability of rupturing simultaneously all hydrophobic bonds involving any particular molecule. Molecular movement has several consequences, however. The micelle may have an outer layer of individual \u03ba-casein molecules when initially constructed, but conditions within the Golgi vesicle are suitable for disulfide bond formation; otherwise \u03b2-lactoglobulin and the other whey proteins would not fold properly. Movements within that outer 'hairy' layer may bring those \u03ba-caseins into proximity and allow their polymerization through disulfide bridging, the size of the polymer depending on when the chain closes into a loop, but giving rise to the polymeric \u03ba-casein entities observed on micellar dissociation.\n\nAnother consequence of the 'living' nature of these hydrophobic bonds is that the dehydration of the micelle required in the preparation of a sample for some forms of electron microscopy would also tend to be accompanied by the collapse of the more mobile, weaker, and less multitudinously bonded regions on to those more strongly cross-linked\u2014hence, perhaps giving rise to the raspberry-like (Schmidt, 1982) or tubular (Dalgleish et al., 2004) structures seen in some electron micrographs. Even the putative caps suggested for those tubules by Dalgleish et al. (2004) can be provided by the dual-binding model, as the disulfide bridging between the \u03ba-casein molecules would enhance the Velcro effect of the weak hydrophobic bonding of an individual molecule to many such molecules in the chain.\n\n#### Effects of Urea, pH, Sequestrants, and Temperature\n\nThe concept of a local excess of hydrophobic attraction over electrostatic repulsion, as well as permitting the visualization of micellar growth, successfully accommodates the response of the micelle to changes in pH, temperature, urea addition, or removal of calcium phosphate by sequestrants, all in accordance with experimental observations.\n\nUrea disrupts hydrophobic bonds, and high concentrations will bring about micellar disintegration. In some regions of the micelle, this may be only partial because the nanocluster cross-links through the phosphoserines remain, unaffected by this reagent. Micellar fragments in a range of sizes may be produced, some even as large as some of the original micelles, though perhaps more open and swollen from their own starting state before urea treatment. Extensive disruption does occur, however, as is observed by the loss of the white appearance of skim milk (McGann and Fox, 1974). The dual-binding model fully accounts for these observations.\n\nRemoval of calcium from the calcium phosphate nanocluster by sequestrant addition, whether EDTA, citrate, or oxalate, restores the negative charge of the hydrophilic region, if the pH is maintained at the native milk pH. This shifts the hydrophobic attraction\/electrostatic repulsion balance in favor of repulsion, and the micelle breaks up. Decreasing the milk pH solubilizes the colloidal calcium phosphate (Dalgleish and Law, 1989), but the negative charges associated with the cross-linking phosphoseryl groups are also titrated away. The strength of the hydrophobic bonds remains unaffected or may be enhanced if other carboxyl charges are also titrated away. The integrity of the micelles is maintained, but their scattering behavior and their appearance in cryo-TEM micrographs reflect the loss of the nanoclusters (Marchin et al., 2007; Moitzi et al., 2011). The loss of the nanocluster linkages also produces a smoothing and spreading of the micelles as seen by atomic force microscopy (AFM) on lowering the pH around an adsorbed micelle, as internal bonding becomes dominated by the multitudinous but ephemeral hydrophobic bonds (Ouanezar et al., 2012).\n\nIncreasing the pH may be expected to be the reverse of the dissolution process and to favor the formation of calcium phosphate species. Fox (2003) noted that raising the pH to >9.0 does not dissolve colloidal calcium phosphate but rather increases its level. However, increasing the milk pH to these levels does lead to dissociation of the micelles and creation of a translucent solution. Dialysing these high pH solutions against excess of the original milk restores the milk pH and produces milks with enhanced levels of colloidal calcium phosphate. It is argued (Ozcan et al., 2011) that this is incorporated into larger nanoclusters rather than increasing the number of them.\n\nDecreasing the temperature is known to decrease the strength of hydrophobic attraction and to shift the monomer\/micelle equilibrium in \u03b2-casein solutions toward the monomer side at temperatures below 15 \u00b0C (De Kruif and Grinberg, 2002; Aschi et al., 2009). Lowering the temperature of milk to refrigeration levels also brings about dissociation of a large fraction of the \u03b2-casein from the casein micelle (Dalgleish and Law, 1988), possibly some of which is not bound into the micellar matrix through its phosphoserine cluster. However, nature has invested considerable energy in creating the phosphoserine clusters and in preserving those clusters through eons of evolutionary development, not to have all of those clusters involved in their designated role. Moreover, Ca binding to caseins is known to decrease when temperature is lowered (Dalgleish and Parker, 1980). Hence some weakening of Ca-PSer bonds is possible when temperature is lowered, with consequent release of \u03b2-casein where the increased negative charge will contribute enormously to shifting the balance in binding energies without the necessity for postulating 'free' \u03b2-caseins at all temperatures as suggested by Dalgleish (2011). Raising the temperature back to its initial value reverses the process and the \u03b2-casein is reincorporated into the micelle.\n\nThere are also shifts in the soluble calcium phosphate equilibria in milk associated with temperature change. Ultrafiltration permeate is a clear, straw-yellow liquid when prepared at 4 \u00b0C, but it becomes turbid when heated to room temperature and above because of the precipitation of calcium phosphate. Even permeate collected at room temperature clouds on heating but reverts to clarity on cooling.\n\nHilgemann and Jenness (1951) noted that calcium phosphate also precipitates in milk. However, the calcium phosphate precipitate was only slowly re-solubilized (Jenness and Patton, 1959). Weakening the calcium phosphate 'solution' equilibrium would favor preservation of the nanoclusters, but anything that pushes that 'solution' equilibrium to the solid side could have an effect on the continuing existence of the nanoclusters. There are indications that heating milks in the temperature range 50\u201390 \u00b0C brings about increasing mobility in the micelle (Rollema and Branches, 1989), which would be in line with partial disruption. Thachepan et al. (2010) have also found that prolonged heating at 60\u00b0C at pH 7 for weeks in the case of micellar casein and days for \u03b2-casein\/CCP constructs, yields mesocrystals of hydroxyapatite and products of dissociated micelles. The behavior of casein micelles in this temperature range merits further scrutiny, particularly as so many processes in the dairy industry are conducted just in this range.\n\n### The Dual-Binding Model and Micellar Interactions\n\nThe ideas outlined earlier in this chapter allow us to schematically describe in Figure 6.5 how the casein micelle might appear as an interacting species at the various pH values indicated.\n\nFigure 6.5 Representations of casein micelle structures at various pH values as indicated. The pale chains indicate protein molecules, where they cross being a hydrophobic interaction junction, the depth of color indicating the intensity of attraction at that pH. The small black circles are the calcium phosphate nanoclusters that are solubilized when the pH is lowered. The outer circle is indicative of the range of steric repulsion generated between micelles and preventing interaction of the surface protein chains.\n\nInternally, at pH 6.7, the micellar matrix is closely interlinked through a combination of nanocluster bridging bonds (the small black circles) and hydrophobic interactions, occurring randomly along any selected polymer chain. The hydrophobic interactions at this pH (indicated as crossover points in the tangled protein network in the diagrams in Fig. 6.5) are many but relatively weak, being counterbalanced by the negative charges present on ionized carboxyl groups, dispersed along the chains and throughout the network. The micellar outer reaches are mainly \u03ba-casein molecules, which have terminated polymer extension and limited micellar growth in the dual-binding model. The negative charges from the ionized carboxyls and sialic acid groups on the \u03ba-casein macropeptides provide the electrostatic repulsion component in the inter-micellar interaction potential, which inhibits micellar aggregation. Its longer range, illustrated by the thickness of the shell around the micelle, prevents close approach of the hydrophobic regions buried beneath the shell and amply fulfills the requirements of a hard sphere model colloid at this pH, 6.7.\n\nAt the lower intermediate pH of 5.6 in this series of illustrations, the same shell continues to prevent close approach of the micelles. The pK values of the acidic groups giving rise to the negative charge are generally lower than 5.5 and have yet to be titrated away. Internally, however, most of the micellar calcium phosphate nanoclusters have been solubilized and the bridges between phosphoserine cluster motifs have been lost, weakening the overall network structure of the micelle. The bond strengths of hydrophobic interactions remain relatively weak, still being counterbalanced by ionized carboxyl groups dispersed through the micelle. The relatively weak bonding, however, allows for rapid interchange and restructuring of the micelle in this range of pH, smoothing out gross structural features apparent in AFM pictures at pH 6.7 (Ouazenar et al., 2012).\n\nBy pH 5.1, the surface charges are being titrated away; the shell depicted in Figure 6.5 is much thinner, and aggregation begins. Internally, the hydrophobic interactions are effectively being strengthened (indicated by a deepening of the color of the chains) because the counterbalancing electrostatic repulsions are also being removed from the equation, leading to reduced mobility within the micellar particles.\n\n### Concentrated Micellar Dispersions\n\nIn milk as produced from the cow at its natural pH of 6.7 and temperatures from ambient to blood heat, casein micelles closely follow the behavior of hard sphere colloids (De Kruif, 1998; Alexander et al., 2002). Justification for this assertion comes from studies utilizing light and neutron scattering to measure micelle size and polydispersity (Hansen et al., 1996), from sedimentation behavior (De Kruif, 1998), and from measurements of micellar voluminosity (De Kruif, 1998), diffusivity (De Kruif, 1992), and viscosity of micellar suspensions (Griffin et al., 1989).\n\nParalleling colloidal hard sphere behavior holds only for a limited range of concentrations, and, above a critical concentration, micellar suspensions show strong deviations from expected hard sphere behavior (Mezzenga et al., 2005). The viscosity continues to increase but at a slower rate than that expected for hard spheres. This is accompanied by a transition from Newtonian viscosity behavior at natural milk concentration to non-Newtonian viscoelastic behavior in the high concentration regime. More enlightening demonstrations of the departure from hard sphere behavior come from studies of the rheology of high concentration micellar suspensions produced by ultrafiltration (Karlsson et al., 2005), by evaporation to 45% total solids (Bienvenue et al., 2003), by centrifugal sedimentation and pelleting (Horne, 1998), and by osmotic compression (Bouchoux et al., 2010). In these instances, concentrated micellar suspensions are close packed and show a gel-like behavior, which can be interpreted with the assistance of the dual-binding model.\n\nKarlsson et al. (2005) concentrated skim milk by ultrafiltration to produce a micellar suspension with 19.5% casein and studied the effects of pH and ionic strength on its viscoelastic properties. Their suspensions exhibited Newtonian viscosities at very low (Brownian) and very high (hydrodynamic) shear rates, with shear thinning at intermediate shear rates and stresses. The concentration of the micelles by ultrafiltration forced the micelles to interact, jamming them together at this high-volume fraction and producing a honeycomb-like structure in freeze\u2013fracture electron micrographs. The elastic modulus of these gels decreased as the pH was lowered from the value achieved in the ultrafiltration retentate. Addition of NaCl at levels of 0.33 and 0.66 mol\/kg prior to ultrafiltration increased the elasticity of the gels but shifted their pHs to more acidic values. Thereafter in the salt-added systems, lowering the pH produced a decrease in elasticity that paralleled the untreated suspension behavior, the higher salt level giving the greater elasticity throughout. Karlsson et al. (2005) also measured the phase angle, the partitioning between viscous and elastic components in these gels, as the pH was reduced. In the no-added-salt system, they found the phase angle to increase through a maximum close to 45\u00b0 and thereafter decrease with decreasing pH. In the presence of added salt, the maximum in phase angle was again observed but shifted to much lower pH values: in the case of the higher salt level, to a pH value lower than that for acid gel formation in milk of normal concentration, and, in both cases, where elasticity had been observed to increase again in these salted concentrated suspensions.\n\nThe dual-binding model explains this behavior with reference to the schematic of the micellar interaction potential depicted in Figure 6.6. The increase in micellar concentration in the no-added-salt case forces the micelles together and into the secondary minimum generated by hydrophobic interactions. This is the source of the attractive interaction giving rise to the viscoelasticity observed. The micelles are also in a jammed structure, and their internal bonding contributes to the measured elasticity. On lowering the pH, the loss of the calcium phosphate nanocluster bridges weakens this structure, and the elasticity decreases, as observed. The bonding due to hydrophobic interactions is relatively weak, and the loss of the nanocluster bridges further contributes to the mobility in the gel, as evidenced by the observed increase in phase angle. Dropping the pH further titrates away carboxyl groups. However, it reduces the counterbalancing electrostatic component and thereby strengthens hydrophobic bonds in the matrix, reducing mobility and producing the subsequent drop in phase angle.\n\nFigure 6.6 Repulsive inter-micellar interaction potential with inner hydrophobic interaction minimum. The dashed line shows the effect of salt addition on the range of the electrostatic repulsion component.\n\nThe major effect of the addition of salt is to reduce the Debye\u2013Huckel parameter and shorten the range of the electrostatic repulsion between micelles. This makes it easier to enter the secondary minimum in the interaction potential and increases the gel elasticity, as observed, with more salt producing the stronger gel. Again, however, the calcium phosphate nanocluster bridges contribute stress-carrying bonds and their removal by lowering the pH leads to the observed decrease in the elasticity of the gel. Throughout this titration, the bonds in the system are relatively stronger than in the no-salt case\u2014the salt also contributes to decreasing the effectiveness of intra-micellar electrostatic repulsion\u2014and the phase angles are lower in comparison.\n\nKarlsson et al. (2005) suggested that a significant effect of the salt addition is to exchange bound calcium within the micelle for monovalent ions, which would imply no nanoclusters in the system to be solubilized on decreasing the pH and thereby voiding the above explanation for the decrease in elasticity with pH. Huppertz and Fox (2006) did indeed find increased levels of calcium in serum when 600 mM NaCl was added to a two-times-concentrated milk, but they found no increase in serum inorganic phosphate, suggesting that the increase in calcium came from displacement of casein-bound calcium rather than a salt-induced dissociation of the calcium phosphate nanoclusters, leaving these to be solubilized on acidification.\n\nThe evaporated milks produced by Bienvenue et al. (2003) had 45% total solids or were approximately concentrated from normal by a factor of four, rather than the eight times concentration of the micelles in the ultrafiltration retentates of Karlsson et al. (2005). The milks of Bienvenue et al. (2003) increased in viscosity on storage at 50 \u00b0C, with salt addition accelerating the increase. Such behavior is in line with the predictions of the dual-binding model outlined above. The collision rate increased by concentration will be further increased by raising the temperature, bringing about a higher frequency of micelles attempting to enter the secondary minimum. A higher success rate and flocculation due to more thermal energy will give rise to the observed increase in viscosity. The weak flocs can be disrupted by higher shear stresses, giving the observed shear-thinning behavior. The effect of salt, as above, would be to render it easier to enter the secondary minimum and promote the flocculation reaction.\n\nIn another study of the rheological behavior of concentrated micellar sytems, casein micelle pellets were produced by the centrifugation of skim milk at 19,000 g for 60 min, giving protein concentrations of approximately 20% (Horne, 1998). At high temperatures (40oC), this pellet flowed freely. Its viscosity was Newtonian, independent of shear rate or frequency. At low temperature (5 \u00b0C), however, this micellar suspension exhibited all the properties of a classical viscoelastic gel, with elastic moduli independent of frequency and phase angles less than 45\u00b0. At intermediate temperatures, there was a crossover between viscous and elastic behavior. The behavior here is dominated by that of the hydrophobic interactions. At low temperatures, the strength of these interactions is low. Both \u03b2-casein and \u03ba-casein are known to depart from the micelle under such conditions (Dalgleish and Law, 1988), but, in the close-packed conditions prevailing in the pellet, they are liable to migrate or link to neighboring micelles or to become entangled with proteins loosened from those micelles, leading to the gel-like behavior. As the temperature is increased, the strength of the hydrophobic interaction increases, but the ability to break bonds is also enhanced and more mobility is allowed. The strengthening of the bonding may also lead to a tightening up of the micelles, and their becoming more compact may allow the suspension to flow more freely.\n\nFinally, Bouchoux et al. (2010) combined the osmotic stress technique with SAXS to study the structural response of the casein micelle to an increase in concentration. Their SAXS results indicate that as the micelles are compressed, they lose water and shrink to a smaller volume, but this compression is nonaffine; that is, some soft parts of the micelle lose water and readily collapse, whereas other hard parts resist deformation and are pushed closer together. Bouchoux et al. (2010) argue that existing models of the casein micelle fail to reproduce this behavior and suggest a physical model based on hard regions that assume the structure of Voronoi tessellations, but without providing any mechanism for the creation of this structure. Moreover, the structure they believe corresponds to the behavior deduced from their SAXS spectra, with hard regions containing the nanoclusters and major water-filled channels and cavities, is nothing more than the structure deduced by Trejo et al. (2011) from their tomographic analysis of their cryo-TEM pictures. As we saw earlier in this chapter, such a structure can be developed within the dual-binding model, provided due consideration is given to the kinetics of protein aggregation and micellar assembly.\n\n### The Dual-Binding Model and Micellar Destabilization\n\nThe concept of the casein micelle electrosterically stabilized by a 'hairy layer' coat of \u03ba- casein appears to enjoy universal acceptance (Holt, 1975; Walstra, 1979; Holt and Horne, 1996).\n\nBecause the dual-binding model of the casein micelle naturally provides a surface location for \u03ba-casein in a growth-limiting role, it readily explains the destabilization of the casein micelle system on the proteolysis of \u03ba-casein by chymosin and the loss of the steric-stabilizing hairs. Such proteolysis also leads to a significant drop in the micellar zeta potential (Dalgleish, 1984), and consequent reduction in the electrostatic repulsion between micelles. Further confirmation of the importance of electrostatic repulsion in inter-micellar interactions is evinced by the necessary presence of ionic calcium to bring about\/promote the aggregation of the chymosin-treated micelles. Notwithstanding the importance of electrostatics, hydrophobic interactions also play an important part, as evidenced by the fact that fully renneted micelles show no signs of aggregation at low temperatures (<10 \u00b0C) (Dalgleish, 1983) or that rennet gels increase in elasticity as the incubation temperature is raised (Horne, 1998). Unraveling and separating the contributions of electrostatics and hydrophobic attraction still remain a major gap in our understanding of rennet coagulation.\n\nSimilarly, the action of ethanol in collapsing the hairs and inducing micellar aggregation is a major coup for the 'hairy micelle' model, translated into the adhesive sphere picture of De Kruif and Holt (2003). As the \u03ba-casein hairs are also negatively charged, their neutralization on acidifying milk would also remove a component of the stabilizing barrier and induce aggregation (De Kruif and Holt, 2003). Attractive as these scenarios are for explaining these three routes to micellar destabilization, in none of them does the adhesive sphere\/hairy micelle approach tell the whole story. To achieve this, the influence of reaction conditions on micellar integrity has to be considered, and it is here that the full power of the dual-binding model comes into play.\n\n### Dual-Binding Model and Rennet Curd Formation\n\nThe aggregation and gelation of casein micelles induced by the chymosin proteolysis of \u03ba-casein is the reaction that comes closest to the colloidal aggregation model, especially in strictly controlled laboratory studies, many of which maintained the pH of the milk at its natural value of 6.7. It is under such conditions that the casein micelle exhibits most closely the properties of a colloidal hard sphere and, importantly for the aggregation observed, where the internal integrity of the micelle is undisturbed. It seems that the internal binding through calcium phosphate nanocluster bridges limits extensive rearrangements and constrains excursions of hydrophobic regions so that only those close to the surface behind the barrier of the charged macropeptide can take part in micellar aggregation once that barrier is removed. This explains the success of the reaction schemes for the initial stages of aggregate growth based on the particle model (reviewed by Hyslop, 2003, and previously by Dalgleish, 1992). It also explains the success of the gel strength model described by Horne (1995; 1996), with the particles remaining largely unchanged through the gel formation process.\n\nHowever, cheeses are seldom manufactured at the natural pH of milk, because this is not the optimum pH for enzyme action (Dalgleish, 1992). Generally, some acidification of the milk is applied, and this modifies the internal integrity of the micelles, with consequent effects on the rennet coagulation and curd properties. Some of these properties, and their influence on the cheeses produced from such curds, have been studied by Choi et al. (2007; 2008), by varying the milk pH or by adding EDTA at a fixed milk pH of 6.0, the objective being to examine in detail the impact of removing micellar calcium phosphate. In all samples, the elastic modulus of the rennet gel passed through a maximum as the gel was formed, declining during longer reaction times. Rennet gels produced at pH 6.4 had the highest maximum, probably due to a lower electrostatic repulsion because only low levels of micellar calcium phosphate were solubilized at this pH. The maximum elasticity in the gelation profile thereafter decreased with the decreasing pH of the preparation from 6.4 to 5.4. This is explained in the dual-binding model by the decline in the number of nanocluster bridges within the micelle that contribute to the overall strength of the gel matrix. There was also a decrease in the maximum gel elasticity with an increase in the added concentration of EDTA, which removed nanocluster links by sequestration of calcium. In both cases, pH adjustment and EDTA addition, the decrease in the maximum elasticity in the gel curds was accompanied by an increase in the rheological loss tangent. The removal of the nanocluster bridges was permitting greater mobility in these gels; a weaker, more flexible network was being produced.\n\nThe microstructure of these rennet-induced gels was also examined, near the point of their maximum elasticity and again some 2\u201310 h later using fluorescence microscopy (Choi et al., 2007). When elasticity was at its maximum, the gels obtained at pH 6.4 manifested more branched, interconnected networks than those obtained at pH 5.4 where the strands\/clusters were larger with more obvious open regions between. In all cases, there was a decrease in apparent interconnectivity between strands in the gel microstructure during aging, which agreed with the decrease in elasticity beyond the maximum. It is apparent that gel strength is a function not only of the number and strength of potential bonds in a system but also of their spatial distribution. Here the loss of the nanocluster bridges weakens the network but also introduces into it greater mobility. Rearrangements occur at a rate governed by that mobility and apparently toward a more compact clustering of the casein proteins, which weakens the matrix structure. Predicting the relationship between matrix morphology, bond strength, and bond number is one of the challenges yet to be addressed in food materials science.\n\nThese changes in rennet gel matrix structure have a direct impact on the functional properties of the cheeses made from the gels. Choi et al. (2008) demonstrated that removal of micellar calcium phosphate contributed to a greater softening and ease of flow of these cheeses at higher temperature. However, they found that there was an optimum pH of preparation, below which the increasing influence of attractive hydrophobic interactions in the balance of forces reduced bond lability and inhibited curd stretching.\n\n### Dual-Binding Model and Ethanol Stability\n\nAlthough studies of the response of dilute suspensions of casein micelles to the addition of ethanol constituted one of the greater successes of the hairy micelle model, they also provided pointers to the failings of the adhesive sphere concept developed from that model (Horne, 2003a).\n\nDynamic light-scattering studies (Horne 1984; 1986; Horne and Davidson, 1986) demonstrated the collapse of the hairy layer and the consequent loss of the steric-stabilizing component with subcritical concentrations of ethanol, but attempts to measure layer thickness as a function of buffer pH or ionic calcium concentration were confounded by the initial observation that initial micelle size was a function of these parameters. Raising the pH or decreasing the ionic strength produced an increase in the hydrodynamic size of the micelle, presumably due to a loosening up of the micelle as either treatment increased the effective electrostatic repulsion between the hairs. More importantly for the ethanol-induced collapse of the hairs, this loosening of the structure extended deeper into the micelle, and greater apparent layer thickness was shown as the micelle structure was caused to expand. This behavior is accommodated within the dual-binding model as a manifestation of the control of internal binding by the balance of hydrophobic attraction and electrostatic repulsion and also by the presence of the calcium phosphate nanocluster bridges, which can be lost on acidification. The model allows for changes in the rigidity of the micelle structure, particularly in the surface layers, which would then control the apparent size of the micelle when the steric-stabilizing hairs are collapsed by the nonsolvent, ethanol.\n\nThe experiments described above confirm yet again the contribution of steric stabilization to micelle stability but relate to the behavior of the casein micelle in a highly dilute suspension in an artificial environment, devoid of inorganic phosphate. In milk, the results of the alcohol stability test, and particularly the behavior of the alcohol stability\/pH profile as a function of mineral content, demand consideration of a different mechanism of destabilization (Horne, 1987), but a mechanism wholly consistent with the dual-binding model, although initially proposed well in advance of that model.\n\nThe mechanism proposed by Horne (1987) suggests that ethanol has two competing effects on the micellar system: destabilization through loss of the hairy layer, and shifts in the calcium phosphate equilibria, first noted by Pierre (1985). The behavior of calcium phosphate and its colloidal form has been at the center of much discussion in this chapter. If ethanol promotes the precipitation of calcium phosphate external to the micelle, it would first of all reduce the concentration of free calcium, reduce the level of caseinate-bound calcium, and disrupt the binding through calcium phosphate nanoclusters. Moderate losses would increase the negative charge of the caseins and increase the thickness of the steric-stabilizing layer. The higher the alcohol level, the faster and more extensive would be the precipitation of calcium phosphate. The ensuing adjustment in protein charge and conformation, though relatively rapid, still requires a finite response time. Countering these changes are the effects of ethanol as a nonsolvent for the proteins, promoting cross-linking and collapse of the hairy layer. When the coagulation reaction occurs faster than the adjustment of charge and conformation resulting from shifts in the calcium phosphate equilibria or the extent of the latter is limited by insufficient ethanol, the aggregation reaction dominates and precipitation of micelles follows.\n\nThe origin of the sigmoidal ethanol stability\/pH profile can also be explained through the effect of pH on calcium phosphate precipitation. Increasing the pH brings about increased calcium phosphate precipitation, possibly further enhanced by the ethanol, which means that more ethanol is required to precipitate the protein, that is, to overcome the increased energy barrier being erected following the transfer of calcium phosphate from the nanocluster state. Conversely, decreasing the pH acts to diminish the influence of ethanol-induced precipitation of calcium phosphate by titrating away negative charge and reducing electrostatic repulsion between protein species. Other effects of milk serum composition, of forewarming the milk, and of modifying the milk concentration and ionic strength can all be explained in a similar fashion (Horne, 2002).\n\nFinally, what is effectively a competition between mineral precipitation and protein aggregation explains the anomalous destabilization of milk by trifluoroethanol (TFE) (Horne and Davidson, 1987) and the behavior of ethanol in milks at high temperature (\u224870 \u00b0C) (O'Connell et al., 2001). With TFE, it was found that, after passing through a critical range of TFE concentrations, which caused protein precipitation, higher levels gave rise to micellar dissociation and produced translucent suspensions. When ethanol\/milk mixtures were heated by O'Connell et al. (2001), these, too, became translucent as the micelles dissociated. In both instances in the mechanism proposed here, this behavior would be seen as the result of the calcium phosphate precipitation proceeding so fast or to such an extent that the micelle would disintegrate before the micellar aggregation reaction could occur, any protein aggregates remaining small and giving rise to insignificant turbidity.\n\n### Dual-Binding Model and Acid Gel Formation\n\nIt is in trying to describe milk acid gel formation in terms of molecular events that the dual-binding model is most useful. Superficially, the initial stages of acid-induced aggregation of casein micelles can be accommodated by the adhesive sphere model, where titration of micellar charge collapses the 'hairy layer' and allows aggregation to proceed (De Kruif and Holt, 2003), but closer study, particularly of the kinetics of gel formation using glucono-\u03b4-lactone (GDL) as acidulant, reveals anomalies (Horne, 2003b).\n\nFirst, at any given temperature, increasing the quantity of GDL used, and hence the rate of acidification of the milk, leads to stiffer gels. There is no mechanism for this in the adhesive sphere model, which has pH as the only variable, but, in the dual-binding model, we postulate that titration of the negative charges of the caseins will also affect internal bonds in the micelle and that reduction of electrostatic repulsion will deepen the attraction between the molecules. If the acidification is proceeding slowly, then this may allow equilibration and rearrangement into localized denser structures with few linkages between, giving rise to weaker gels. More rapid drops in pH may lock the protein into a more dispersed structure with greater density of possibly stronger strands, as was observed microscopically when similar trends in elasticity were detected in rennet gels (Choi et al., 2007).\n\nSecond, if the skim milk is heat-treated at 90 \u00b0C for 10 min, a process practiced by the dairy industry and known as forewarming, prior to acid-induced gelation by GDL at 40 \u00b0C, not only is the critical coagulation pH shifted to a higher pH value and a much stiffer gel produced but also a distinct step is introduced on to the gelation profile. This is not an artifact introduced by slippage in the rheometer. It is fully reproducible, and its presence is subject to the reaction conditions applied. It can be made to disappear with forewarmed milk, for example, by lowering the incubation temperature to 30 \u00b0C and below (D. S. Horne, unpublished observations). A similar step can be introduced into the profile of a non\u2013heat-treated milk by raising the incubation temperature to 40 \u00b0C or above and employing a high concentration of GDL. Such steps in the profiles were also seen following the acidification of milks in the presence of xanthan (Aichinger, 2005), where their magnitude and definition were dependent on the level of xanthan introduced. With heat-treated milks, the presence, magnitude, and definition of the stepped gelation profile change in response to the temperature and duration of the preheating (Fig. 6.7). The critical pH for gelation and the maximum in complex modulus are monotonic functions of the level of denaturation of the whey proteins brought about by the prior heat treatment (Fig. 6.8). Stepped gelation profiles like this are also seen during the formation of fermented milk gels, again with forewarmed milks, where the step can be removed by suitable modest additions of trisodium citrate (TSC) (Ozcan-Yilsay et al., 2007). The critical pH and the loss tangent remained unaffected by additions of TSC up to 10 mM, although the gel elasticity recorded at pH 4.6 was enhanced and the gelation profile lost its step and assumed a monotonic increase in G' with decreasing pH.\n\nFigure 6.7 Exemplary gelation profiles showing the complex modulus of the developing gel as a function of the measured pH for skim milks preheated at 90 \u00b0C for the durations indicated on each curve. All milks were gelled at 40 \u00b0C using 4% GDL as acidulant.\n\nFigure 6.8 Gelation pH and maximum value of the complex modulus as a function of the level of denaturation of \u03b2-lactoglobulin in milks preheated at temperatures ranging from 60 to 90 \u00b0C. Different symbols indicate different durations of heating. All milks were gelled at 40 \u00b0C using 4% GDL as acidulant.\n\nAll of the above observations can be reconciled within the following picture, which relies heavily on the precepts of the dual-binding model. Aggregation, and thereafter gelation, of the casein micelles on acidification begin only when the electrostatic repulsion component is reduced below a critical level, that is, when a critical pH is reached. The rate of aggregation is a function of a number of factors: the nature of the surface, the energy (or temperature) involved, and the collision rate. Modifying the surface by partially or wholly coating it with denatured whey proteins through formation of a \u03b2-lactoglobulin\/\u03ba-casein complex, raising the incubation temperature, and increasing the micellar concentration by imposing a phase separation through addition of incompatible polysaccharide will all increase the rate of aggregation and shift the critical pH. These are all operational in the above examples, and in all cases we are forcing the micelles to interact, probing further into the inner reaches of the interaction potential with greater frequency and accessing the hydrophobic minimum there. So far, this is no more than another adhesive hard sphere description, but, during acidification, the internal integrity of the casein micelles is also being compromised through the solubilization of the micellar calcium phosphate nanoclusters. Pushing the critical gelation pH to higher values means that more nanoclusters are still present in the aggregating micelles, which are in consequence more rigid than at lower pH. Because the hydrophobic interactions are counterbalanced by any remaining electrostatic repulsion, these hydrophobic bonds are weaker the higher the coagulation pH. Rearrangements and bond breakage are relatively rapid, as evidenced by the higher value of the phase angle at this point. As the pH continues to drop, the loss of the remaining nanoclusters allows more mobility in the gel, giving the increasing phase angle, but also permitting stronger strands to be formed in the rearrangements, and hence the greater elasticity in the gel. However, the elasticity is not increasing so rapidly that it cannot be overtaken by its diminution on the loss of the nanocluster bridges\u2014hence the maximum or inflexion point in G' with system pH. Eventually, the titration of the carboxyl groups wins out, and the increasing contribution from attractive hydrophobic interactions produces increasing elasticity in the gels. Compared with the gel produced in a reaction regime with a low critical gelation pH, this 'hi-pH' gel is anticipated to have a more uniform distribution of contributing strands, as has been observed in yogurt studies (Ozcan-Yilsay et al., 2007). When the critical pH is low, as for an unheated milk for example, the reactions described above relating to the behavior on the loss of the remaining calcium phosphate nanoclusters still occur but are now confined within the micellar particle, which remains unaggregated because the inter-micellar potential is still repulsive. Mobility is introduced and bonds are strengthened, but only within the micelle so that, when eventually aggregation and gelation do take place, fewer inter-cluster bonds are formed and a more open gel structure is obtained. Location and confinement are major factors in defining the final architecture and strength of the gels formed, and the dual-binding model helps us to understand what is going on.\n\n## Concluding remarks\n\nThis chapter has reviewed the physicochemical properties and interactions of the caseins and blended them into the construction of a mechanistic framework for a working model of the casein micelle, the dual-binding model. The various properties of the casein micelle have been rationalized in terms predicted by the model. It is stressed that the dual-binding model is only a framework and that the real emphasis, in explaining, for example, the behavior of the micelle in processing, should be placed on the effects of those conditions on the ongoing interactions in their totality. This is well demonstrated by the mechanism proposed for ethanol-induced destabilization, where the behavior of the calcium phosphate equilibria and consequent reappearance of electrostatic repulsion have to be set against the poor solvent properties of the proteins in ethanol.\n\nAs well as the interactions, however, due consideration must be given to the location of the effects, and time should also be allowed for these to occur. Examples of these effects are seen in the rapid kinetics of micellar assembly leading to open, ramified structures of varying textures and densities, and in the relaxation behavior of the labile hydrophobic bonds in the growing clusters in acidified milk just below the critical gelation pH leading to stronger gels. Just as when critical pH is lower, the same hydrophobic interchange and readjustment occur only within the confines of the micelle and are locked in as the pH continues to drop, eventually reaching the critical value that leads to aggregation but a weaker final gel.\n\nConstraints of space have limited the number of cases we have been able to consider. A major omission has been explaining the dissociation and reassociation behavior of the casein micelle under high pressure, but this aspect has recently been covered by Huppertz et al. (2006) in terms that can be easily aligned with the dual-binding model. Neither have we extensively mentioned the application of this model to explaining the behavior of nonbovine milks, although a recent study by Horne et al. (2007b) has demonstrated its usefulness in this context for marsupial milks. Our objective has not been to be exhaustive but rather to provide examples of how the model can be made to work in various situations. 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Differences between the association of genetic variants B, C and D of \u03b1S1-casein .  _Biochimica et Biophysica Acta_.  1970 ;221 : 140 \u2013 142 .\n\nSchmidt DG .  Colloidal aspects of the caseins .  _Netherlands Milk and Dairy Journal_.  1980 ;34 : 42 \u2013 64 .\n\nSchmidt DG .  Association of caseins and casein micelle structure .  In:  Fox PF ,  ed.  _Developments in Dairy Chemistry\u20141_ .  London :  Applied Science Publishers ;  1982 : 61 \u2013 86 .\n\nSlattery CW .  Model calculations of casein micelle size distribution .  _Biophysical Chemistry_.  1977 ;6 : 59 \u2013 64 .\n\nSlattery CW ,  Evard R .  A model for the formation and structure of casein micelles from subunits of variable composition .  _Biochimica et Biophysica Acta_.  1973 ;317 : 529 \u2013 538 .\n\nStradner A ,  Sedgwick H ,  Cardinaux F ,  Poon WCK ,  Egelhaaf SU ,  Schurtenberger P .  Equilibrium cluster formation in concentrated protein solutions and colloids .  _Nature_.  2004 ;432 : 492 \u2013 495 .\n\nStothart PH ,  Cebula DJ .  Small angle neutron scattering study of bovine casein micelles and sub-micelles .  _Journal of Molecular Biology_.  1982 ;160 : 391 \u2013 395 .\n\nSwaisgood HE .  Chemistry of the caseins .  In:  Fox PF ,  McSweeney PLH ,  eds.  _Advanced Dairy Chemistry\u20141. Proteins_ .  New York :  Kluwer Academic\/Plenum Publishers ;  2003 : 139 \u2013 201 .\n\nThachepan S ,  Li M ,  Mann S .  Mesoscale crystallization of calcium phosphate nanostructures in protein (casein) micelles .  _Nanoscale_.  2010 ;2 : 2400 \u2013 2405 .\n\nTrejo R ,  Dokland T ,  Jurat-Fuentes J ,  Harte F .  Cryo-transmission electron tomography of native casein micelles from bovine milk .  _Journal of Dairy Science_.  2011 ;94 : 5770 \u2013 5775 .\n\nWalstra P .  The voluminosity of bovine casein micelles and some of its implications .  _Journal of Dairy Research_.  1979 ;46 : 317 \u2013 323 .\n\nWalstra P .  On the stability of casein micelles .  _Journal of Dairy Science_.  1990 ;73 : 1965 \u2013 1979 .\n\nWalstra P .  Casein sub-micelles: do they exist? .  _International Dairy Journal_.  1998 ;9 : 189 \u2013 192 .\n\nWeitz DA ,  Lin MY ,  Lindsay HM .  Universality laws in coagulation .  _Chemometrics and Intelligent Laboratory Systems_.  1991 ;10 : 133 \u2013 140 .\n\nYin X ,  Stott MJ .  Biological calcium phosphates and Posner's cluster .  _Journal of Chemical Physics_.  2003 ;118 : 3717 \u2013 3723 . \nChapter 7\n\n# Structure and Stability of Whey Proteins\n\nPatrick J.B. Edwards\n\nGeoffrey B. Jameson Centre for Structural Biology, Institute of Fundamental Sciences, Massey University, Palmerston North, New Zealand\n\n## Abstract\n\nThe chemical and physical stability of the more common proteins of bovine and, where available, ovine, caprine, and equine whey (\u03b2-lactoglobulin, \u03b1-lactalbumin, serum albumin, immunoglobulins, and lactoferrin) is reviewed with regard to their molecular structures and dynamics. The behavior of the proteins separately and in combination with temperature, pressure, pH, denaturants (such as guanidinium chloride and urea), and stabilizers (such as fatty acids and metal ions) has been considered. The combination of high temperature and low pH in the hydrolysis and subsequent formation of fibrils constitutes a new body of study and knowledge since the first edition of this chapter in 2009. Particular emphasis has been placed on studies that have utilized x-ray, NMR, fluorescence, and circular dichroism techniques. Attention is directed to the role of cysteines and disulfide bridges with regard to chemical stability. Whereas there is considerable knowledge of structure\u2013function relationships of individual proteins, there is a dearth of three-dimensional structural knowledge of combinations of proteins, despite the clear importance of such knowledge to functionality, especially with regard to food processes.\n\n## Keywords\n\nWhey protein structure\n\nwhey protein stability\n\nbovine whey\n\nbovine \u03b2-lactoglobulin\n\nx-ray\n\nNMR\n\nfluorescence\n\ncircular dichroism techniques\n\nOutline\n\nIntroduction 202\n\nBovine \u03b2-lactoglobulin 203\n\nMolecular Structure of Bovine \u03b2-Lactoglobulin 203\n\nStructure of Bovine \u03b2-Lactoglobulin in Aqueous Solution 205\n\nThe Monomer\u2013Dimer Equilibrium of Bovine \u03b2-Lactoglobulin in Aqueous Solution 206\n\nStudies of Bovine \u03b2-Lactoglobulin by NMR at Neutral pH 207\n\nBovine \u03b2-Lactoglobulin Dynamics 208\n\nStructures of \u03b2-Lactoglobulins from other Species 209\n\nLigand Binding to \u03b2-Lactoglobulin 210\n\nEffect of Temperature on Bovine \u03b2-Lactoglobulin 213\n\nEffect of Pressure on Bovine \u03b2-Lactoglobulin 216\n\nEffect of Chemical Denaturants on Bovine \u03b2-Lactoglobulin 218\n\nFibrillar Formation from Bovine \u03b2-Lactoglobulin 219\n\n\u03b1-Lactalbumin 220\n\nMolecular Structure of Bovine \u03b1-Lactalbumin 221\n\nEffect of Temperature on Bovine \u03b1-Lactalbumin 222\n\nEffect of Pressure on Bovine \u03b1-Lactalbumin 222\n\nEffect of Denaturants on Bovine \u03b1-Lactalbumin 223\n\nSerum albumin 223\n\nStructure of Serum Albumins 225\n\nEffect of Temperature on Serum Albumins 225\n\nEffect of Pressure on Serum Albumins 226\n\nEffect of Chemical Denaturants on Serum Albumins 226\n\nImmunoglobulins 227\n\nStructure of Immunoglobulin G 227\n\nEffects of Temperature, Pressure, and Chemical Denaturants on Ig Structure and Stability 227\n\nLactoferrin 229\n\nBovine Lactoferrin Structure 229\n\nEffects of Temperature, Pressure, and Chemical Denaturants on Lactoferrin Structure and Stability 231\n\nConcluding remarks 231\n\nAcknowledgments 232\n\n## Introduction\n\nInformation regarding whey protein structure and stability has great potential to facilitate knowledge-based product design. Reviews that highlight the importance of knowledge of structure and stability, including the effects of pressure, temperature, and chemical denaturants, have been made by L\u00f3pez-Fandi\u00f1o (2006). In this chapter, we discuss the structures of the whey proteins shown in Table 7.1 under quiescent and destabilizing conditions (change in pH, temperature, and pressure and addition of chaotropes) and in the presence or absence of small-molecule ligands. A host of other proteins are found in the whey of milk, but only in trace amounts, including the hyperphosphorylated osteopontin, trefoil factors, and growth factors (TGF-\u03b2, IGF-I and IGF-II, EGF, HB-EGF), and the enzymes lactoperoxidase, superoxide dismutase, platelet-activating factor acetylhydrolase, and alkalinephosphatase (Chatterton et al., 2006; Chatterton et al., 2013). Particular emphasis is placed on information that has been obtained via high-resolution x-ray crystallographic and high-field nuclear magnetic resonance (NMR) studies. The results of these studies have been applied to many projects reported in the present volume (e.g., Chapters 8, , , and ). The focus of this chapter is directed toward properties of proteins in milk whey that impinge on functionality, rather than the intrinsic function and effects on organisms imbibing milk.\n\nTable 7.1\n\nTypical Protein Composition of Whey\n\nProtein | Proportion by mass | No. amino acids | Mol mass (\/ Da) | Iso-ionic point | Disulfide bond\/thiols | Comments  \n---|---|---|---|---|---|---  \n\u03b2-Lactoglobulin  \n(\u03b2-Lg) | 60% | 162 | 18 363a | 5.35 | 2\/1 | Two common variants, A and B  \n\u03b1-Lactalbumin  \n(\u03b1-La) | 20% | 123 | 14 178 | 4.80 | 3 | About 10% of molecules are glycosylated  \nBovine serum albumin (BSA) | 3% | 583 | 66 399 |  | 17\/1 | Also present in blood serum  \nImmunoglobulin G (IgG) | 10% | >500 | 161 000 (G1)b | Many isoforms |  | Passive transfer of immunities  \nLactoferrin (Lf) | <0.1% | 689 | 76 110 | 8.95 | 17 | Bacteriostatic role; glycoprotein\n\na Molar mass for the A variant.\n\nb G1 is the major immunoglobulin; two other classes, IgM and IgA, are present in much lower abundance.\n\nBased on Farrell et al., 2004.\n\n## Bovine \u03b2-Lactoglobulin\n\n\u03b2-Lactoglobulin (\u03b2-Lg) contains 162 amino acids and has a molecular weight of 18.3 kDa (Hambling et al., 1992). It is a member of the lipocalin (a contraction of the Greek lipos, meaning 'fat, grease' and calyx, meaning 'cup') family of proteins (Banaszak et al., 1994; Flower, 1996), so called because of their ability to bind small hydrophobic molecules into a hydrophobic cavity. This led to the proposal that \u03b2-Lg functions as a transport protein for retinoid species, such as vitamin A (Papiz et al., 1986).\n\n\u03b2-Lg is the most abundant whey protein in the milk of most mammals (\u223c10% of total protein or \u223c50% of whey protein), but has not been detected in the milk of humans, rodents, or lagomorphs. In the case of human milk, \u03b1-lactalbumin \u03b1-la (see below) is the dominant whey protein. Bovine \u03b2-Lg is by far the most commonly studied milk protein.\n\nThere are ten known genetic variants of bovine \u03b2-Lg. The most abundant variants are labeled \u03b2-Lg A and \u03b2-Lg B (Farrell et al., 2004) and differ by two amino acid substitutions, Asp64Gly and Val118Ala, respectively. The quaternary structure of the protein varies among monomers, dimers, or oligomers depending on the pH, temperature, and ionic strength, with the dimer being the prevalent form under physiological conditions (Kumosinski and Timasheff, 1966; McKenzie and Sawyer, 1967; Gottschalk et al., 2003). This variable state of association is likely to be the result of a delicate balance among hydrophobic, electrostatic, and hydrogen-bond interactions (Sakurai et al., 2001; Sakurai and Goto, 2002).\n\n### Molecular Structure of Bovine \u03b2-Lactoglobulin\n\n\u03b2-Lg was an early target of x-ray diffraction as newly applied at the Royal Institution to protein crystals. This was due to its high abundance and relatively easy purification from milk and to its propensity to form suitable crystals. In retrospect, this was a very ambitious project because \u03b2-Lg was not the easiest protein to analyze (Green et al., 1979) partly because of the multiple crystal forms. Nevertheless, this study established that the protein monomer was near spherical with a block of electron density with a rod-like structure across one face.\n\nThe next attempt (Creamer et al., 1983) to determine the structure was by calculation using sequence data and structural probabilities to estimate which portions of the amino acid sequence might form into the helices, strands, and sheets. The secondary structure of \u03b2-Lg was predicted to comprise \u223c15% \u03b1-helix, \u223c50% \u03b2-sheet, and 15\u201320% reverse turn (Creamer et al., 1983). It is interesting to note that many of the residues that are present in the extended structures of the native protein have been shown to have a nascent propensity to form \u03b1-helical structures in the presence of trifluoroethanol or amphiphiles (Hamada et al., 1995; Kuroda et al., 1996; Chamani et al., 2006).\n\nIn 1986, the first medium resolution structure of \u03b2-Lg was published (Papiz et al., 1986). Structural similarity to a seemingly different type of protein, plasma retinol-binding protein, has given rise to much speculation as to the role of \u03b2-Lg in bovine milk. Higher resolution structures subsequently revealed the now familiar eight-stranded \u03b2-barrel (calyx), flanked by a three-turn \u03b1-helix. A final ninth strand forms the greater part of the dimer interface at neutral pH (Papiz et al., 1986; Bewley et al., 1997; Brownlow et al., 1997). The \u03b2-barrel is formed by two \u03b2-sheets, where strands A to D form one sheet and E to H the other (with some participation from the A strand, facilitated by a 90o bend at Ser21). Two disulfide bonds link Cys66 on loop CD (which, as its name suggests, connects strands C and D) with Cys160 near the C terminus, and Cys106 on strand G with Cys119 on strand H, leaving Cys121 as a free, but unexposed thiol. The loops connecting the strands BC, DE, and FG are relatively short, whereas those at the open end of the barrel, AB, CD, EF, and GH, are longer and more flexible. These features are illustrated in Figure 7.1.\n\nFigure 7.1 Diagram of the dimeric structure of bovine \u03b2-Lg A looking down the two-fold axis. The coordinates are taken from the structure of \u03b2-Lg A in the trigonal Z lattice with 12-bromododecanoic acid bound (PDB code: 1bso). The strands that form the \u03b2 barrel are labeled A to H. The I strand, together with part of the AB loop, forms the dimer interface at neutral pH. The locations of the sites of difference between the A and B variants are also shown. The structure is rainbow colored, beginning with blue at the N-terminus and ending with red at the C-terminus. Ser21, which shows conformational flexibility, and the 12-bromododecanoate anion, are shown as spheres. Figure drawn with PyMOL (Delano, 2002).\n\nThe structures of the A and B variants are very similar. However, the Asp64Gly substitution results in the CD loop adopting differing conformations (Qin et al., 1999). The Val118Ala causes no detectable change to the structures, but the void created by substituting the bulky isopropyl substituent, with the smaller methyl group results in the hydrophobic core of the B variant being less well packed, and may account for the lower thermal stability of the B variant under some measurement conditions (Qin et al., 1999).\n\nVery careful titrimetric and thermodynamic measurements in the late 1950s (Tanford et al., 1959; Tanford and Nozaki, 1959) established the presence of a carboxylic acid residue with an anomalously high pK a value of 7.3. This was attributed to a pH-dependent conformational change, a conclusion that rationalized earlier measurements of pH-dependent sedimentation coefficients (Pedersen, 1936) and specific optical rotation data (Groves et al., 1951). Much later, x-ray structure analyses (Qin et al., 1998a) at pH values above and below this so-named Tanford transition established that, at pH 6.2, the EF loop is closed over the top of the barrel, burying Glu89 (the carboxylic acid with the anomalous pK a) inside the calyx. At pH 8.1 this loop is articulated away from the barrel such that the formerly buried glutamic acid becomes exposed in the carboxylate form (Qin et al., 1998a).\n\nAn early structure of bovine \u03b2-Lg crystallized in the presence of retinol appeared to show retinol bound externally to the protein (Monaco et al., 1987), apparently later confirmed by a body of fluorescence data (Dufour et al., 1994; Lange et al., 1998; Narayan and Berliner, 1998). Subsequent structural analyses have shown, however, that fatty acids, retinoid species (including vitamin A), and cholesterol (including vitamin D) all bind inside the calyx (Kontopidis et al., 2004). Induced circular dichroism (CD) measurements and NMR measurements confirm the x-ray crystallographic observations. Ligand binding will be discussed in more detail below, since it relates to the probable physiological function of \u03b2-Lg as well as to the stability of this molecule and current technological interest in the role of protein-ligand interactions in flavor perception (see Chapter 14). At this stage, there is no evidence for binding of fatty acids or retinoid species outside the calyx. Except for very bulky ligands (see below), ligands bind inside the calyx of \u03b2-Lg at pH \u223c7.\n\nAt about the same time as the Tanford transition and ligand-binding modes were elucidated by high-resolution x-ray crystallography (Qin et al., 1998a), NMR studies of \u03b2-Lg structure in low-pH (\u223c2.5), very low ionic strength (<10 mM) solutions, where the protein is monomeric, were initiated (Ragona et al., 1997; Fogolari et al., 1998; Kuwata et al., 1998; Uhr\u00ednov\u00e1 et al., 1998). These NMR studies, described below, have provided proof of persistence of tertiary structure down to pH 2 and have yielded a depth of insight into structural stability and protein dynamics that is not possible by standard x-ray crystallographic techniques.\n\n### Structure of Bovine \u03b2-Lactoglobulin in Aqueous Solution\n\nNMR spectroscopy is used to obtain protein structures in solution (Cavanagh et al., 1995). The technique is best suited to monomeric proteins with molecular weights <\u223c25 kDa and usually requires recombinant singly (15N) or doubly (15N \/13C) labeled material for molecules with molecular weights >\u223c8 kDa. Therefore, most NMR studies of bovine \u03b2-Lg have been at a pH of between 2 and 3, and very importantly at very low ionic strength, where the molecule is monomeric. Early studies by Molinari's group using wild-type B-variant protein (Fogolari et al., 1998) confirmed the presence of the eight-stranded \u03b2-barrel. However, the full structure (of the A variant) (Kuwata et al., 1999; Uhr\u00ednov\u00e1 et al., 2000) required the use of isotopically labeled recombinant material (Kim et al., 1997; Denton et al., 1998). As the full structure was determined by NMR techniques independently and near-simultaneously by two groups from Tokyo and from Edinburgh and Palmerston North, this has provided objective and very useful comparisons (Jameson et al., 2002a).\n\nThe solution structure was shown by both Kuwata et al. (1999) and Uhr\u00ednov\u00e1 et al. (2000) to have an overall similarity to that established earlier by x-ray crystallography at pH 6.2, despite the considerably lower pH (and concomitant increase in the protein's surface charge) necessary to obtain usable NMR spectra from monomeric bovine \u03b2-Lg. The EF loop is closed over the open end of the \u03b2-barrel at this pH, and the side chain of the Glu89 'latch' is buried, as in the x-ray structures. The biggest difference, when compared with the Z lattice x-ray structure at pH 6.2 (Qin et al., 1998a), is that the three-turn \u03b1-helix adopts a different position with respect to the \u03b2-barrel, possibly because of the pH-induced increase in positive charge on this part of the protein's surface (Uhr\u00ednov\u00e1 et al., 2000). In addition, the loss of the dimer interface at low pH removes restraints on the position of the helices, which in the dimer are in proximity but not direct contact (see Fig. 7.1), and on the conformation of the AB loop, which moves by up to 3.5 \u00c5.\n\nFurther differences were found at the N- and C-termini, but these can be ascribed to limitations imposed by the use of recombinant protein with a non-native N terminus for the NMR structure and possible crystal packing effects at the C terminus for the x-ray structure. In some crystal forms, much of the C-terminus from residue \u223c152 to 162 is not observed or is very poorly defined in electron density maps.\n\n#### The Monomer\u2013Dimer Equilibrium of Bovine \u03b2-Lactoglobulin in Aqueous Solution\n\nThe monomer-dimer equilibrium of bovine \u03b2-Lg has been widely studied over a long period of time by a host of techniques, including analytical ultracentrifugation (AUC), light scattering, small-angle x-ray scattering (SAXS), and isothermal dilution calorimetry (for a comprehensive and critical tabulation, see Table S1 in Supporting Information in Mercadante et al., 2012). What had not been measured until recently were the rate constants associated with the monomer-dimer equilibrium, despite the importance of the dimer dissociation step as the first step in the denaturation of bovine \u03b2-Lg and in subsequent chemical processes. Analytical centrifugation measurements confirmed by earlier values for the dimer dissociation equilibrium were constant over the pH range 2.5 to 7.5 at an essentially constant ionic strength provided by 100 mM NaCl. The equilibrium constants and rate constants, where obtainable by AUC, are summarized in Table 7.2. At pH 2.5\u20133.5 and 6.5\u20137.5, the equilibrium dissociation constants K D are \u223c10 \u03bcM. At pH around the isoelectric point, K D decreases markedly and falls outside the range of standard AUC detection. The dissociation rate constant is quite slow (\u223c0.008 s\u22121), and the association rate constant is well removed from the diffusion-controlled limit. This was attributed (Mercadante et al., 2012) to the considerable restructuring of the cloud of counter-ions on dimerization: At low pH, where \u03b2-Lg is strongly positively charged, there is substantial restructuring of counter-anions, and at pH >7, where it is mildly negatively charged, there is a restructuring of counter-cations as well as a sharp distance dependence for optimization of hydrophobic contacts.\n\nTable 7.2\n\nEquilibriuma and Rateb Constants Calculated for \u03b2Lg A and \u03b2Lg B Dimer Dissociation over the pH Range 2.5 to 7.5 in 100 mM NaCl from Global Fitsc of SE and SV Data\n\n| K D2-1 (\/\u03bcM) | k off (\/s\u22121) | k on (calc.) (\/M\u22121 s\u22121)  \n---|---|---|---  \n| \u03b2Lg A | \u03b2Lg B | \u03b2Lg A | \u03b2Lg B | \u03b2Lg A | \u03b2Lg B  \npH 2.5d | 15 \u00b1 3 | 8 \u00b1 3 | 0.008 \u00b1 0.005 | 0.009 \u00b1 0.005 | 540 | 1,100  \npH 3.5d | 4.0 \u00b1 1.5 | 1.4 \u00b1 1.5 | [0.007]f | [0.041]f | \\---f | \\---f  \npH 6.5e | 4.0 \u00b1 1.5 | 2.5 \u00b1 1.5 | Fast (> 0.1) | Fast (> 0.1) | >25,000 | >40,000  \npH 7.5e | 11 \u00b1 3 | 9 \u00b1 2 | Fast (> 0.1) | Fast (> 0.1) | >9,200 | >12,000\n\na Calculated from global fitting of sedimentation velocity (SV) and sedimentation equilibrium (SE) data.\n\nb Calculated from global fitting of SV data (SE data contain no kinetic signal).\n\nc Calculated error ranges represent the sensitivity of the values to changes in other fitting parameters.\n\nd Citrate buffer.\n\ne 3-(N-morpholino)propanesulfonic acid (MOPS) buffer.\n\nf Indicative value, as no error range could be determined; kon not calculated.\n\nReproduced from Mercadante, Melton et al., 2012.\n\nBello et al. (2011) also observed by isothermal dilution calorimetry that the dimer dissociation constant of variant B of \u03b2-Lg is smaller than that of variant A at temperatures up to 35 \u00b0C, consistent with analytical ultracentrifugation data (Mercadante et al., 2012). Indeed, the dissociation constant of 14.5(1) \u03bcM measured at pH 7.0 in 50 mM phosphate buffer and 100 mM NaCl is very similar to that determined by analytical ultracentrifugation measurements (Table 7.2).\n\n### Studies of Bovine \u03b2-Lactoglobulin by NMR at Neutral pH\n\nThe large size of the bovine \u03b2-Lg dimer at pH 7 is expected to cause some broadening of the peaks in its1H NMR spectrum due to slower molecular reorientation. However, this problem is exacerbated by chemical exchange broadening of peaks in the vicinity of the dimer interface by the dynamic equilibrium of molecules between the associated (dimeric) and unassociated (monomeric) states. These factors render the resulting spectra unsuitable for structure determination. Several methods have been employed to allow NMR studies at neutral pH. The most straightforward of these methods has been to use a nonruminant \u03b2-Lg that is intrinsically monomeric, yet with the same overall tertiary structure as the bovine protein, in this case equine \u03b2-Lg. (Kobayashi et al., 2000). Alternatively, the dimer interface may be disrupted by producing bovine \u03b2-Lg mutants with amino acid substitutions carefully chosen to disrupt the intermolecular interactions between either the I strands or the AB loops (Sakurai and Goto, 2002) (see Fig. 7.1).\n\nAn attempt to form dimeric equine \u03b2-Lg by producing a mutant with amino acid substitutions chosen to mimic those of the bovine protein at the interface was not successful (Kobayashi et al., 2002). This failure indicates that subtle features in \u03b2-Lg conformation remote from the interface have an impact on successful dimer formation. Indeed, reaction of the free thiol of Cys121 (located away from the interface in the H strand and covered by the main \u03b1-helix, Asp129\u2014Lys141; see Fig. 7.2) with 2-nitro-5-thiobenzoic acid produces a monomeric species with native structure at pH 2 and a monomeric but unfolded structure at pH 7 (Sakai et al., 2000). The configuration of the \u03b1-helix is known to change with pH (Uhr\u00ednov\u00e1 et al., 2000), and this may therefore also have an important influence on both the protein's stability and quaternary state.\n\nFigure 7.2 Ligand-binding sites on \u03b2-Lg as inferred from NMR measurements of binding of small (<12 atoms) ligands to \u03b2-Lg at acidic pH (Luebke et al., 2002). The binding site of 12-bromododecanoic acid is shown for reference (Qin et al., 1998a, b). (a) View into the calyx showing primary binding site of fatty acids at pH \u223c7 and of flavor components at \u223cpH 2, highlighted in yellow. (b) Secondary binding site for flavor components at pH 2 at N-terminal ends of strands A, B, C, and D, and the C-terminal strand, highlighted in pink. (c) Secondary binding site for flavor components at pH 2 adjacent to the three-turn helix and strand G, highlighted in cyan. To show more clearly the attachment of side chains to the main chain, loops and strands have not been smoothed. The pH-sensitive EF loop is colored in magenta. Figure drawn with PyMOL (Delano, 2002) using coordinates with PDB code 1bso.\n\nThe third approach to overcome the problems of the rate constants for the dissociation\/ reassociation equilibrium of the bovine \u03b2-Lg dimer being in the intermediate exchange regime has been to covalently bond two monomers via an Ala34Cys mutant (Sakurai and Goto, 2006). This variant was used to study the dynamics of the EF loop across the Tanford transition (see the following section) and to examine the nature of ligand binding to \u03b2-Lg (Konuma et al., 2007). Although no full structure determination was reported, the amide chemical shifts of the mutant were within 0.1 ppm of those from monomeric \u03b2-Lg (except for seven residues that encompassed the substitution site). This fact, combined with the similarity of the mutant and wild-type \u03b2-Lg CD spectra, indicated that the tertiary structures of the mutant and wild-type protein were similar.\n\n### Bovine \u03b2-Lactoglobulin Dynamics\n\nCrystallographic atomic displacement parameters, often loosely referred to as temperature factors or just B factors, describe the spread of an atom's electron density in space and can therefore be used to infer residue-specific mobility. However, for surface residues, the B factors of both main-chain and side-chain atoms are highly sensitive to intermolecular crystal-packing contacts. Moreover, except where data are available to ultra-high resolution (better than 1.0 \u00c5, which is not yet the case for any \u03b2-Lg), similarity restraints are imposed on B values of adjacent atoms and residues along the polypeptide chain to ensure stable refinement.\n\nNonetheless, in the case of isomorphous structures at similar resolution (where structures share the same average B value, the same space group, very similar unit cell parameters, and, hence, intermolecular contacts), or in regions where non-isomorphous structures lack intermolecular contacts, some meaning can be placed on differences observed in B values. These differences can be both within a particular structure or between structures determined, for example, at different pH or in the presence\/absence of added ligands. High B values, indicating apparent high mobility, can also arise from a distribution of slightly different yet immobile conformations or from errors in model building. For these reasons, it is advantageous to study dynamics of the protein (particularly those of the backbone) by NMR techniques, using uniformly15N-labeled protein.\n\nFlexibility on the nanosecond timescale can be inferred from low15N steady-state nuclear Overhauser effect values. As might be expected, mobile residues for \u03b2-Lg tend to have highly accessible surface areas (and such residues identified in NMR studies correlate in general with those that have relatively high B values; see Kuwata et al., 1999). Slower conformational exchange processes can be indicated by large values for the ratio of the T1 (spin-lattice) and T2 (spin-spin) relaxation times of15N nuclei. Such residues include Ser21 at the midpoint (kink) of the A strand, possibly caused by fluctuations of the barrel, and residues 61 and 66 at either end of the CD loop, consistent with a slow segmental or a hinging motion of this loop (Uhr\u00ednov\u00e1 et al., 2000).\n\nNMR measurements of the dynamics of the covalently bonded Ala34Cys dimer mutant have recently given complementary information regarding the structural changes associated with the Tanford transition established previously using x-ray crystallography (Qin et al., 1998a). The15N dynamics of the EF loop measuring either side of the transition indicates a three-step process. With increasing pH, the first event is a conformational change to the GH loop. This is followed first by the breaking of hydrogen bonds at the hinges of the EF loop and then by the articulation of the EF loop away from the calyx (Sakurai and Goto, 2006; Sakurai et al., 2009). The dynamic flexibility of the EF loop at pH \u223c2 is important because it means that at low pH neither ingress into nor egress from the hydrophobic pocket is kinetically prevented.\n\nRecent molecular dynamics simulations confirm what has been suspected from x-ray crystallographic studies that dimerization of bovine \u03b2-Lg is accompanied by an increase in molecular flexibility, an entropically favorable contribution to the free energy of dimer formation (Bello et al., 2011).\n\n### Structures of \u03b2-Lactoglobulins from Other Species\n\nEquine (horse) \u03b2-Lg, which shares 58% identity with bovine \u03b2-Lg, has been shown to be monomeric over a wide pH range (Kobayashi et al., 2000), whereas porcine (pig), which shares 63% identity with bovine \u03b2-Lg, is dimeric below pH 5 and monomeric at pH 5 and above (in contrast to bovine \u03b2-Lg) (Ugolini et al., 2001). At pH 7 both equine and porcine \u03b2-Lg are monomeric and therefore amenable to NMR study. Equine \u03b2-Lg has been extensively studied by NMR with regard to denaturation processes, especially at low pH (<4) where it assumes a molten globule conformation, (Yamamoto et al., 2011), but full structural characterization by either NMR or x-ray methods has yet to be published. However, the x-ray crystal structure at 2.0-\u00c5 resolution and at near-neutral pH of Gyuba \u03b2-Lg, a chimeric protein formed from bovine \u03b2-Lg secondary structure elements and equine \u03b2-Lg loops, revealed a bovine-like dimeric structure (Ohtomo et al., 2011). This is in contrast to an earlier chimera where replacing just the I strand and AB loop of equine \u03b2-Lg by the dimer-forming I-strand and AB loop of bovine \u03b2-Lg failed to elicit dimerization of the chimera (Kobayashi et al., 2002).\n\nThe x-ray crystal structure of porcine \u03b2-Lg at pH 3.2 clearly revealed a dimeric structure formed by domain swapping of N-terminal regions, a quaternary structure quite different from that observed for bovine \u03b2-Lg (Hoedemaeker et al., 2002). The EF loop adopts the closed conformation over the calyx, as found also for bovine \u03b2-Lg at acidic pH, consistent with the notion that this loop acts as a lid to the calyx. However, the porcine protein is much less conformationally stable at acidic pH than its bovine counterpart (Burova et al., 2002; Invernizzi et al., 2006), which has led to questioning of the role of \u03b2-Lg as a transporter of hydrophobic molecules through the acidic environment of the stomach (Burova et al., 2002). Despite 63% identity between bovine and porcine \u03b2-Lg, the RMS difference in C\u03b1 positions between these two structures is remarkably high at 2.8 \u00c5, although inspection of the two structures shows that the core \u03b2-barrel structure superimposes closely and that these differences are concentrated in the flexible loop regions, which comprise nearly a third of the structure.\n\nThe structure of rangiferine (reindeer) \u03b2-Lg at pH \u223c6.5 has been determined to 2.1 \u00c5 resolution (Oksanen et al., 2006). Both the monomeric tertiary structure and the dimeric quaternary structure are very similar to those of bovine \u03b2-Lg, which is not unexpected as polypeptide lengths are identical and sequence identity is greater than 94%. At pH \u223c6.5, the EF loop was observed to be in the closed position. There are few structural data on ovine (sheep) and caprine (goat) \u03b2-Lg, despite the commercial importance of their milk.\n\nAlthough \u03b2-Lg is absent from human milk, humans secrete two other lipocalins that share limited sequence identity (but close structural similarity) with bovine \u03b2-Lg. Tear lipocalin is the major protein in tears, and structural and functional studies indicate that it, like \u03b2-Lg, binds a broad range of hydrophobic molecules (Glasgow et al., 1995). Glycodelin, a heavily glycosylated lipocalin, is found in the human endometrium in early pregnancy where its function remains unclear. Kontopidis et al. (2004) have suggested that \u03b2-Lg arose by gene duplication of glycodelin and exists solely for nutritive value in milk. However, there appears to have been considerable selection pressure to retain not only the Glu89 in all sequences of \u03b2-Lgs found to date but also the pH-sensitive conformational switch for the EF loop of all \u03b2-Lgs (and indeed for tear lipocalin) that have been functionally characterized. Such preservation is not consistent with a solely nutritive role for \u03b2-Lg, although the origin of the \u03b2-Lg gene from glycodelin remains an intriguing possibility.\n\n### Ligand Binding to \u03b2-Lactoglobulin\n\n\u03b2-Lg has the ability to bind a large number of small molecules (for a review see Sawyer et al., 1998), although the location of the bound ligand continues to generate controversy (Dufour et al., 1994; Lange et al., 1998; Narayan and Berliner, 1998; Yang et al., 2008). However, in recent years, NMR techniques have been used to map possible ligand-binding sites on the protein through changes in chemical shift and relaxation times of protein residues.\n\nAnother technique that provides reliable evidence of binding to the protein is induced circular dichroism, where an achiral chromophore 'lights up' in the CD spectrum on being placed into a chiral environment on or in the protein. Creamer et al. (2000) applied this technique to bovine \u03b2-Lg to show unequivocally that retinol and fatty-acid binding (e.g., palmitic and cis-parinaric acids) is competitive. The binding of these and other chirally active ligands, including trans-parinaric acid and retinoic acid, were explored over a range of pH and with nonchromophoric fatty acid ligands to gain an understanding of the parameters surrounding the Tanford transition. More recently Zsila et al. have used induced circular dichroism to study binding of these (Zsila et al., 2002) and other ligands (Zsila, 2003; Zsila et al., 2005), including piperidine, to \u03b2-Lg and other lipocalins.\n\nLigand binding has also been extensively studied by fluorescence spectroscopy, in which, typically, the fluorescence from tryptophan residues is monitored for changes, which may be positive or negative. For bovine \u03b2-Lg, one tryptophan (Trp19) is buried and is part of the highly conserved lipocalin Gly-X-Trp motif at the beginning of strand A (see Fig. 7.2b), whereas the other (Trp61) is largely exposed and is part of the mobile CD loop. Fluorescence by both tryptophans is sensitive to small changes in positions of nearby quenchers, respectively, a nearby charged side-chain of Arg124 and the Cys66-Cys160 disulfide bond. NMR spectroscopy, induced CD, and x-ray results clearly indicate that interpretation of fluorescence measurements for evidence of ligand binding poses hazards.\n\nRagona et al. (2003) used a combination of electrostatic calculations, docking simulations, and NMR measurements to suggest that the pH-dependent conformational change of the EF loop triggered by the protonation of Glu89 is common to all \u03b2-Lgs and that ligand binding (of palmitic acid) is determined by the opening of this loop. In earlier work using13C-labeled palmitic acid, this group had shown that the ligand also undergoes conformational change with increasing pH (Ragona et al., 2000).\n\nRecent NMR studies of the binding of palmitic acid to the 'NMR friendly' Ala34Cys mutant dimer of bovine \u03b2-Lg have indicated that, although a rigid connection is made by the protein with the ligand at the bottom of the calyx, the interaction at the open end of the calyx is more dynamic (Konuma et al., 2007). These observations complement those of Ragona et al. (2003) and also the x-ray studies on binding of fatty acids (Qin et al., 1998b; Wu et al., 1999), which showed the carboxylate head group to be substantially less well ordered than the hydrophobic tail. The results of the study with the Ala34Cys mutant suggest that it is the plasticity of the D strand and the EF and GH loops that allows \u03b2-Lg to accommodate such a wide range of ligands (Konuma et al., 2007). With the exception of changes in conformations of the side chains of Phe105 and Met107, NMR and x-ray studies show that the core lipocalin structure remains invariant upon ligand binding.\n\nCrystallographic data clearly show that both fatty acids and retinoids bind in the calyx of bovine \u03b2-Lg (Qin et al., 1998b; Wu et al., 1999; Sawyer et al., 1998; Kontopidis et al., 2002). Recently, complexes of bovine \u03b2-Lg with a host of different ligands have been crystallographically characterized: dodecyl sulfate (separately bovine variants A and B) (Guti\u00e9rrez-Magdaleno et al., 2013; Loch et al., 2013), unsaturated C18 fatty acids, oleic acid, and linoleic acid (Loch et al., 2013a), saturated fatty acids, lauric (separately bovine variants A and B), myristic, stearic, palmitic, capric, caprylic and acid (Loch et al., 2011; ), trialkyl ammonium species, dodecylammonium (separately bovine variants A and B) (Loch et al., 2013b), and an alkyl-tailed rhodium(III) organometallic complex (Cherrier et al., 2013).\n\nAs a crystallization artifact, n-dodecyl-\u03b2-D-maltoside was found in the canonical binding pocket of a complex of bovine \u03b2-Lg with the D1\/Fab fragment of human IgE (immunoglobulin type E) (Niemi et al., 2008).\n\nAlthough all crystallographic studies of ligand binding have been under conditions of high ionic strength, congruence of these data with NMR and induced CD data collected under conditions of low ionic strength indicate that the x-ray results are not an artifact of ionic strength. The preservation of structure of \u03b2-Lg, in particular the hydrophobic cavity at conditions of near-zero ionic strength at the pI of the protein (\u223c5.3) (Adams et al., 2006) further demonstrates that the primary, and possibly only, ligand-binding site at \u223cpH 7 is inside the calyx.\n\nThe structural studies of Loch et al. (2011; 2012), and Guti\u00e9rrez-Magdaleno et al. (2013) have been complemented by measurements of the thermodynamics of ligand binding by isothermal titration calorimetry (see Table 7.3), with several key provisos: The structural studies were on crystals grown at very high ionic strength (1.34 M sodium citrate at pH 7.5), while ligand-binding thermodynamics were conducted at much lower ionic strength (\u223c50 mM Tris-HCl\/5 mM KOH) and where \u223c0.2 mM protein solution is typically titrated into \u223c20 \u03bcM ligand solutions at initial protein concentrations where the \u03b2-Lg will dissociate from dimer into monomer (see Table 7.2).\n\nTable 7.3\n\nBinding of C8 to C18 Species Bovine \u03b2-Lg\n\n| K(assoc)  \n\/M\u22121 x 105 | \u0394 G   \n\/kJ mol\u22121 | \u0394 H   \n\/kJ mol\u22121 | T \u0394 S   \n\/kJ mol\u22121  \n---|---|---|---|---  \nCaprylate (C8:0)a | 0.107(17) |  |  |   \nCaprate (C10:0)a | 0.060(5) |  |  |   \nLaurate (C12:0)b |  |  |  |   \n\u03b2-Lg A | 1.88 \u00b1 0.08 | \u221231.2 \u00b1 0.8 | 19.2 \u00b1 0.3 | 50.5 \u00b1 0.8  \n\u03b2-Lg B | 3.70 \u00b1 0.02 | \u221232.8 \u00b1 0.7 | 6.5 \u00b1 0.2 | 39. \u00b1 1  \nLauryl sulfateb |  |  |  |   \n\u03b2-Lg A | 1.07 \u00b1 0.02 | \u221229.2 \u00b1 0.5 | \u221210.54 \u00b10.08 | 18.7 \u00b1 0.4  \n\u03b2-Lg B | 5.79 \u00b1 0.06 | \u221266.5 \u00b1 0.7 | \u221217.54 \u00b10.04 | 15.94 \u00b1 0.04  \nLaurate (C12:0)c | 1.72 \u00b1 0.15 | \u221229.9 | \u221223.2 \u00b11.2 | 6.7  \nMyristate (C14:0)c | 7.8 \u00b1 1.6 | \u221233.5 | \u221218.7 \u00b1 0.9 | 14.9  \nPalmitate (C16:0)c | 20. \u00b1 4 | \u221236. | \u221211.6 \u00b1 0.3 | 82.1  \nOleate (C18:1) | 10 \u00b1 3 | \u221234.0 \u00b1 0.1 | \u221215 \u00b1 4 | 19 \u00b1 4  \nLinoleate (C18:2) | 9.0 \u00b1 0.8 | \u221233.2 \u00b1 0.1 | \u22128.8 \u00b1 1.9 | 25 \u00b1 2\n\na Loch et al. (2011). Fluorescence measurements on 20 \u03bcM \u03b2-Lg variant B in 50 mM Tris, pH 7.5 at room temperature.\n\nb Bello et al. (2011). Measurements performed at pH 7.0 in a 50 mM phosphate buffer, 100 mM NaCl, at 35 and 30 \u00b0C for laurate and dodecyl sulfate, respectively. Ligand (30 mM) titrated into \u03b2-Lg (\u223c13 \u03bcM as monomer). Values converted from reported values in kcal mol\u22121.\n\nc Loch et al. (2012). Measurements performed at pH 7.5 in a 50 mM Tris-HCl\/5 mM KOH buffer at 25 \u00b0C. Laurate (2 mM) and myristate (0.6 mM) were titrated into 50\u2013100 \u03bcM \u03b2-Lg (variant B). Palmitate (20 \u03bcM) was titrated into 0.2 mM \u03b2-Lg (variant B). Stearate (C18:0) gave a very small enthalpy of binding, and so its binding is not accessible by calorimetric methods. Heats of dilution were measured by titration beyond saturation of \u03b2-Lg by ligand, and so may be affected by enthalpy of dissociation of the \u03b2-Lg dimer.\n\nd Loch et al. (2013a). Measurements performed at pH 7.5 in a 50 mM Tris-HCl buffer at 25 \u00b0C. Bovine \u03b2-Lg (variant B) (0.25\u20130.40 nM) titrated into oleic acid (20 \u03bcM) or linoleic acid (25\u201330 \u03bcM). Heats of dilution were measured by titration beyond saturation of \u03b2-Lg by ligand.\n\nIsothermal titration calorimetric methods, when properly conducted, give not only binding equilibrium constants and stoichiometry of binding, but also the enthalpy of binding and, after conversion of equilibrium constant to Gibbs free energy, the entropy of binding.\n\nThat ligand binding is a complex process is revealed by the observation that binding of laurate (dodecanoate) and lauryl sulfate (dodecyl) to bovine \u03b2-Lg (variant A) at pH 7.5 is endothermic in phosphate buffer (proton ionization enthalpy of 3.6 kJ mol\u22121) and exothermic in Tris buffer (proton ionization enthalpy of 47.5 kJ mol\u22121) (Loch et al., 2013b). Furthermore, ligand binding may be accompanied by dissociation of the dimer (Bello et al., 2011; Guti\u00e9rrez-Magdaleno et al., 2013). The ligand-binding enthalpy with proton transfer effects removed (\u0394 H dissoc,b) is given by Baker and Murphy (1996)\n\n\u0394  H dissoc,obs = \u0394  n H * \u0394  H ion + \u0394  H dissoc,b\n\nwhere \u0394 H ion have been measured for a number of common buffers (Goldberg et al., 2002) and \u0394 n H is the number of protons transferred in ligand binding (for \u03b2-Lg and fatty acid ligands \u0394 n H \u223c0.25\u22120.6; see Loch et al., 2013a,b).\n\nLoch et al. (2011; ; 2013a,b) systematically determined ligand:\u03b2-Lg binding stoichiometries to be considerably less than the expected value of unity. Errors in measurement of \u03b2-Lg concentrations are probably the cause, since Bello et al. (2011), in work remarkably paralleled by that of on the binding of dodecyl sulfate and laurate (dodecanoate) to \u03b2-Lg variants A and B, report stoichiometries that are not significantly different from 1:1. Affinities for fatty acids and other ligands are consistently smaller (by more than an order of magnitude) when measured by isothermal titration calorimetry than by fluorescence methods.\n\nRemarkably, enthalpic and entropic contributions to the free energy of binding dodecanoate, dodecyl sulfate, and dodecyltrimethyl ammonium differ between variant A and B. Moreover, whereas binding of dodecanoate is primarily entropically driven, that of dodecyl sulfate has both enthalpic and entropic contributions to binding to \u03b2-Lg (Bello et al., 2011). Bello et al. (2012) report significantly stronger binding of dodecanoate and dodecyl sulfate to variant B than to variant A, in contrast to who report the opposite, but in the latter case the differences in affinity are not statistically significant.\n\nMeasurements of ligand-binding affinity, especially of amphipathic ligands such as fatty acids, by spectroscopic means are often compromised by light scattering from ligand micelles unless care is taken to reduce the ligand concentration below the critical micelle limit. The fatty acid ligand is typically dissolved first in ethanol before diluting into buffer, which leaves a residual \u223c0.1% trace in the ligand-protein mixture.\n\nVitamin D3 was reported to bind in half occupancy not only at the canonical site but also at an external site in the groove between the subunits comprising the canonical dimer of bovine \u03b2-Lg (Yang et al., 2008). Quite apart from unconvincing discontinuous electron density for the vitamin D3 in this external site (reconstructed from coordinates and structure factors deposited in the protein data bank with PDB Code 2gj5) (Berman et al., 2000), there are a host of uncomfortably close contacts involving the hydrophobic parts of vitamin D3 with polar (Asp137_O, Ser150_OG) or charged (Arg148) groups and the hydroxyl moiety eschews close contacts with polar residues. Although a number of hydrophobic contacts are reported, most of the residues proposed as being involved in hydrophobic contacts, Phe136, Ala139, Ile147, Ala142, Ile143, and Pro144, have their side chains more than 5.4 \u00c5 distant from the putative vitamin D3. Arg148 is noted as buried, but is only buried by the putative vitamin D3; otherwise it is well exposed to solvent. In brief, there remains no unequivocal crystallographic evidence for the binding of (largely) hydrophobic molecules to an external site on bovine \u03b2-Lg.\n\nAlthough ligands such as palmitic acid appear to be released at acid pH (Ragona et al., 2000), NMR evidence (based on perturbations of backbone chemical shifts) for the binding of the flavor compounds \u03b3-decalactone and \u03b2-ionone at pH 2 has been reported (Luebke et al., 2002; Tromelin and Guichard, 2006). There is, then, evidence for three binding sites to \u03b2-Lg: the canonical site inside the calyx, a second site involving perturbation of residues Trp19, Tyr20, Tyr42, Glu44, Gln59, Gln68, Leu156, Glu157, Glu158, and His161, and a third site involving perturbation of residues Tyr102, Leu104, and Asp129. These regions are illustrated in Figure 7.2.\n\nInitially it was thought that porcine \u03b2-Lg did not bind fatty acids (Frapin et al., 1993). However, NMR studies have shown that the pH for 50% uptake of ligand has shifted by nearly 4 pH units from \u223c5.8 for bovine \u03b2-Lg to 9.7 for porcine \u03b2-Lg, whereupon the EF loop undergoes a structural change analogous to that of its bovine counterpart (Ragona et al., 2003). Indeed, this loop with its absolutely key Glu residue (Glu89 in bovine \u03b2-Lg) is predicted to be pH-gated for all \u03b2-Lgs (Ragona et al., 2003).\n\n### Effect of Temperature on Bovine \u03b2-Lactoglobulin\n\nThe thermal properties of \u03b2-Lg variants are of considerable commercial relevance due to their role in the fouling of processing equipment as well as the functional qualities that can be imparted to dairy products by thermally induced \u03b2-Lg aggregation. Consequently, this aspect of the protein's behavior has been the focus of extensive experimental work.\n\nAt neutral pH the midpoint of the thermal unfolding transitions, as determined by differential scanning calorimetry (DSC), is \u223c70 \u00b0C (de Wit and Swinkels, 1980) whereupon the protein dimer dissociates and the constituent molecules begin to unfold. This reveals the free thiol of Cys121 (located at the C-terminal end of the H strand; see Fig. 7.1) and a patch of hydrophobic residues, leading to the possibility of both covalent and hydrophobic intermolecular association (Qi et al., 1995; Iametti et al., 1996). The ensuing disulfide interchange reactions lead to the formation of a variety of mixed disulfide-bonded polymeric species (Creamer et al., 2004). Genetically engineered mutants with an extra cysteine positioned to allow a third disulfide bond to be formed to Cys121 have been shown to both retard thermal denaturation by 8\u201310 \u00b0C and to resist heat-induced aggregation (Cho et al., 1994). In mixtures of bovine \u03b2-Lg, \u03b1-La, and BSA, or of \u03b2-Lg and one or other of \u03b1-La and BSA at pH 6.8 subjected to high temperatures, homo- and heteropolymeric disulfide-bridged species were observed (Havea et al., 2001). The formation of \u03b1-La \u2212 \u03b1-La disulfide links (\u03b1-La has no free cysteine; see below) is attributed to catalysis by BSA or \u03b2-Lg (Havea et al., 2001). At low pH where the protein is monomeric, denaturation is largely reversible at temperatures below 70 \u00b0C (Pace and Tanford, 1968; Alexander and Pace, 1971; Mills, 1976; Edwards et al., 2002). Heating above this temperature leads to the formation of large aggregates, but in contrast to the behavior at neutral pH, the species are predominantly noncovalently bonded (Schokker et al., 2000).\n\nThe precise denaturation process is complex and is influenced by factors such as pH, protein concentration, ionic environment, genetic variant, and presence of ligands. Lowering the pH (Kella and Kinsella 1988; Relkin et al. 1992), or adding calyx-bound ligands (Puyol et al., 1994; Considine et al., 2005a; Busti et al., 2006) can make the protein more resistant to thermal unfolding. The stability of the genetic variants (at pH 6.7) appears to decrease in the order C > A > B, with the A variant showing the least cooperative unfolding transition (Manderson et al., 1997). The protein's susceptibility to thermal denaturation at pH 6.7\u20138 is strongly concentration dependent up to about 6 mM, being most susceptible to unfolding at a concentration of \u223c1.4 mM (Qi et al., 1995). It is possible that at high protein concentration (\u223c6 mM) tertiary structure is lost directly from the native dimer state (Qi et al., 1995).\n\nThere is some evidence that the thermal unfolding occurs in more than one step. Kaminogawa et al. (1989) used antibody binding affinities to propose that thermal unfolding of variant A of \u03b2-Lg occurs in at least two stages, starting with conformational changes near the N terminus followed by changes in the region of the 3-turn \u03b1-helix. FTIR measurements by Casal et al. have also indicated a loss of helical content early in the denaturation process (using variant B of \u03b2-Lg in 50 mM phosphate buffer at pH 7) (Casal et al., 1988). Qi et al. used FTIR and CD measurements to propose that variant A of \u03b2-Lg forms a molten globule with reduced \u03b2 structure when heated above 65 \u00b0C in 30\u201360 mM NaCl at pH 6.5 (Qi et al., 1997). NMR studies, observing hydrogen\/deuterium (H-D) exchange of the backbone amide protons of \u03b2-Lg A at pH 2\u20133, have revealed a stable core comprising the FG and H strands, possibly stabilized by the Cys106\u2013Cys119 disulfide bond between strands F and G (Belloque and Smith, 1998; Edwards et al., 2002). The observation of significant secondary structure even at a temperature as high as 90 \u00b0C has been reported (Casal et al., 1988; Qi et al., 1997; Bhattacharjee et al., 2005).\n\n### Effect of Pressure on Bovine \u03b2-Lactoglobulin\n\nHigh-pressure treatment of food is of increasing commercial importance due to increasing consumer demand for products that have been subjected to minimal processing damage. Pressure treatment as part of the processing regime has greater potential to produce dairy products with improved functional and organoleptic properties than those produced by thermal treatment alone (Messens et al., 2003).\n\nOf the major whey proteins \u03b2-Lg is the most susceptible to pressure-induced change (Stapelfeldt et al., 1996; Patel et al., 2005). Presumably this is due to its relatively inefficient packing caused by the presence of the \u03b2-barrel, with its large solvent-exposed hydrophobic pocket and the lower number of disulfide bonds (two compared to four in, for example, the similar-sized \u03b1-lactalbumin). A reduction in the molar volume of bovine \u03b2-Lg has been detected at pressures as low as 10 MPa, possibly due to a contraction of the calyx (Vant et al., 2002). A number of studies have shown that \u03b2-Lg becomes more susceptible to enzymatic cleavage when exposed to pressure, possibly due to pressure-induced conformational change. The free cysteine has been shown to become exposed at between 50 and 100 MPa (Stapelfeldt et al., 1996; Tanaka and Kunugi,1996; M\u00f8ller et al., 1998). Exposure of the protein to pressures in excess of \u223c300 MPa causes irreversible changes to \u03b2-Lg tertiary and quaternary structure. A combination of CD and fluorescence spectroscopy of \u03b2-Lg at neutral pH exposed to pressures as high as 900 MPa indicated that pressure induces monomer formation with subsequent aggregation, but with only small irreversible effects on \u03b2-Lg tertiary structure (Iametti et al., 1997). However, more recent results from tryptic hydrolysis suggest that while exposure to pressures below 150 MPa has no detectable permanent effect on \u03b2-Lg A's conformation, pressures above 300 MPa lead to the detachment of strands D and G from the \u03b2-barrel together with the formation of disulfide-bonded oligomers (Knudsen et al., 2002).\n\nIn mixtures of bovine \u03b2-Lg with either \u03b1-La or BSA at pH 6.6 subjected to high pressures, intermolecular disulfide-bridged aggregates form only between \u03b2-Lg and itself. No \u03b2-Lg \u2212 \u03b1-La or \u03b2-Lg \u2212 BSA disulfide-bridged species are detected (Patel et al., 2005), in contrast to heat-treated mixtures where such species are observed (Havea et al., 2001).\n\nIn order to correlate the pressure-induced conformational changes with the protein's primary sequence, Belloque et al. have made NMR amide H-D exchange observations of \u03b2-Lg A and B following exposure of solutions at neutral pH to pressures of up to 400 MPa (Belloque et al., 2000). Little H-D exchange was reported at 100 MPa, which indicates that any conformational change that occurs is not increasing exposure of most amide protons to the solvent compared to their exposure in the native conformation at ambient pressure. A large increase in the extent of H-D substitution at 200 MPa and above indicated increased conformational flexibility, but the similarity of the spectra of control samples recorded in H2O rather than D2O before and after pressurization demonstrated that any pressure-induced conformational changes were largely reversible up to 400 MPa. The authors proposed that the A variant's structure was more sensitive to changes in pressure than that of the B variant and that the F, G, and H strands of the protein's \u03b2-barrel were the most resistant to conformational change, the latter conclusion paralleling the effects of temperature (Belloque and Smith 1998; Edwards et al., 2002).\n\nFTIR and SAXS experiments suggest that even at 1 GPa the unfolded state contains significant secondary structure (Panick et al., 1999). Combined application of pressure and heat has shown that changing the temperature over the range 5 to 37 \u00b0C has a negligible effect on the susceptibility of \u03b2-Lg to pressures up to 200 MPa (Skibsted et al., 2007). However, combined application of pressure and moderate temperature at 600 MPa\/50 \u00b0C (Yang et al., 2001) and 294 MPa\/ 62 \u00b0C (Aouzelleg et al., 2004) has indicated the formation of a molten globule with an \u03b1-helical structure on the basis of results obtained from CD spectroscopy.\n\nIt should be noted, therefore, that the potential for temperature increases induced by rapid pressurization of the sample needs to be considered when studying the effects of pressure on protein conformation and stability.\n\nEnzymatic proteolysis observations indicate that \u03b2-Lg is less susceptible to pressure- induced change at acidic pH (Dufour et al., 1995) than at neutral or basic pH, but this result may be confounded by pressure-induced changes in the proteinases, thermolysin and pepsin. Nevertheless, NMR measurements of monomeric \u03b2-Lg at pH 2 while under pressure of up to 200 MPa, have shown that the two \u03b2 sheets unfold independently to form two intermediates to an unfolded state that still appears to contain significant secondary structure (Kuwata et al., 2001).\n\nA three-step mechanism has been proposed for \u03b2-Lg denaturation at neutral pH and ambient temperature, which broadly encompasses the above observations: A pressure of 50 MPa causes partial collapse of the calyx (with concomitant reduction in ligand-binding capacity) together with exposure of Cys121. Increasing the pressure to 200 MPa causes further (partially reversible) disruption to the hydrophobic structure, together with a decrease in the molecular volume. Higher pressures cause irreversible aggregation reactions involving disulfide interchange reactions (Stapelfeldt and Skibsted, 1999; Considine et al., 2005b).\n\n### Effect of Chemical Denaturants on Bovine \u03b2-Lactoglobulin\n\nChemical denaturants are often used to unfold proteins and to characterize mechanisms and transition states of protein-folding processes. Commonly used denaturants include alcohols, particularly 2,2,2-trifluoroethanol (TFE), urea, and guanidinium chloride (GdmCl).\n\nTheoretical calculations predict a significantly higher amount of \u03b1-helical secondary structure than is actually observed in native \u03b2-Lg (Creamer et al., 1983; Nishikawa and Noguchi, 1991). That is, the native structure is the result of competition between \u03b1-helix favoring local interactions and \u03b2-sheet forming long-range interactions. However, addition of alcohols such as TFE can disturb this balance by weakening the hydrophobic interactions and strengthening the helical propensity of the peptide chain (Thomas and Dill, 1993). The ability to increase the \u03b1-helical content of bovine \u03b2-Lg by the addition of alcohols (ethanol, 1-propanol, 2-chloroethanol) was first demonstrated by Tanford using optical rotary dispersion measurements (Tanford et al., 1960). Contemporary studies tend to favor the use of TFE, where the \u03b2-Lg \u03b2-sheet-to-\u03b1-helix transition has been shown to be highly cooperative, occurring over the range \u223c15\u201320% v\/v of cosolvent (Shiraki et al., 1995; Hamada and Goto, 1997; Kuwata et al., 1998). The higher proportion of \u03b1-helical structures in the so-called TFE-state is found in the N-terminal half of the molecule (Kuwata et al., 1998). Magnetic relaxation dispersion measurements of the solvent nuclei have shown that this state is an open, solvent-permeated structure (unlike the collapsed state of a molten globule), and its formation is accompanied by a progressive swelling of the protein with increasing TFE concentration (Kumar et al., 2003). High-protein and TFE concentration (8% v\/w and 50% v\/v, respectively) can lead to fibrillar aggregation (see below) and gel formation of bovine \u03b2-Lg at both acid and neutral pH (Gosal et al., 2002).\n\nThe ability of urea to induce protein unfolding is thought to be via a combination of hydrogen-bond formation with the protein backbone and a reduction in the magnitude of the hydrophobic effect (Bennion and Daggett, 2003). Therefore, in contrast to TFE, both the helical propensity and the hydrophobic effect are reduced. Urea-induced unfolding of bovine \u03b2-Lg at acidic pH was first reported as a two-state process (Pace and Tanford, 1968). Subsequent NMR H-D exchange measurements of bovine \u03b2-Lg B at pH 2.1 also allowed the urea-induced unfolding to be well approximated as a two-state transition between folded protein and unfolded state via a cooperative unfolding of the \u03b2-barrel and the C-terminus of the major \u03b1-helix (Ragona et al., 1999). However, Dar et al. have provided evidence that urea also causes unfolding via an intermediate, albeit with structural properties between those of the native and unfolded states (Dar et al., 2007). Addition of anionic amphiphiles, sodium dodecyl sulfate (SDS), or palmitate, causes \u03b2-Lg to resist urea-induced unfolding due to binding inside the calyx (Creamer, 1995).\n\nGuanidinium chloride is often used as an alternative to urea in studies of protein stability. At the neutral or acidic pH of most stability studies, GdmCl will be fully dissociated. At low GdmCl concentration (<\u223c1 M), chloride ions screen the electrostatic repulsion between positively charged groups of the protein (Hagihara et al., 1993). The result is that the additional electrostatic interactions of GdmCl compared to the neutral urea molecule GdmCl have the potential to both stabilize and destabilize protein structure depending on the concentration of GdmCl (Hagihara et al., 1993). D'Alfonso et al. have compared denaturation of bovine \u03b2-Lg B with both GdmCl and urea between pH 2 and pH 8, as monitored by CD, UV differential absorption, and fluorescence measurements (D'Alfonso et al., 2002). Discrepancies between unfolding free energies obtained using the two denaturants could be reconciled if GdmCl denaturation was assumed to occur via an intermediate state. The secondary structure of this state is similar to that of the native protein, but with greater rigidity in the vicinity of the Trp residues consistent with the screening of electrostatic repulsion between charged residues (D'Alfonso et al., 2002). The GdmCl-induced unfolding intermediate of bovine \u03b2-Lg A at pH 2 has been reported to have increased \u03b1-helical structure (Dar et al., 2007).\n\nPorcine \u03b2-Lg has also been shown to unfold via an intermediate state on addition of GdmCl. The stability of the porcine protein was lower than that of its bovine counterpart, and the intermediate state was richer in \u03b1-helical structure. Most of the hydrophobic\u2013hydrophobic interactions of the buried core of the native state are conserved between bovine and porcine \u03b2-Lg. However, four pairwise interactions of the Phe105 side chain of bovine \u03b2-Lg are lost on the change to Leu in the porcine protein. This indicates that the presence of the aromatic residue may play an important role in the increased stability of the bovine protein (D'Alfonso et al. 2004). This residue is particularly resistant to H-D exchange in heated \u03b2-Lg solutions (Edwards et al., 2002).\n\n### Fibrillar Formation from Bovine \u03b2-Lactoglobulin\n\nAt high temperatures (>80 \u00b0C), low pH (typically pH \u223c2) and low ionic strength (<\u223c 20 mM), \u03b2-Lg self-assembles into fibrils (Aymard et al., 1999; Gosal et al., 2002; Arnaudov et al., 2003). These structures have some characteristics of classic \u03b2-amyloid fibrils, such as binding the fluorophores thioflavin-T and Congo red and showing x-ray diffraction consistent with a cross-\u03b2-sheet structure (Bromley et al., 2005). However, they differ substantially in that they are easily dissociated (Akkermans et al., 2008; Jordens et al., 2011). The fibrils have contour lengths in the order of \u03bcm (Aymard et al., 1999) and diameters in the range 4\u20138 nm (Gosal et al., 2004).\n\nResults from the combined use of AFM, PAGE, and in-situ FTIR suggest that the formation of heat-induced \u03b2-Lg fibrils involves its partial unfolding, self-hydrolysis, and self-assembly of some of the resulting species (Oboroceanu et al., 2010). It has been proposed that each fibril is typically composed of several protofilaments that grow to a length of \u223c0.5\u20131 \u03bcm before aligning and twisting together (Adamcik et al., 2010; Bolisetty et al., 2011; Adamcik and Mezzenga, 2012). Hydrolysis of \u03b2-Lg appears to play an important role in fibril formation (Akkermans et al., 2008), but it is likely to also be responsible for their disintegration if the temperature is raised to 120 \u00b0C for extended periods (Loveday et al., 2012).\n\nFibrils have been shown to contain peptide fragments in the range of 2\u20138 kDa, with sequences weighted toward \u03b2-Lg's termini (Akkermans et al., 2008), and those toward the N terminus being the more abundant (Dave et al., 2013). 2-D gel electrophoresis indicates the presence of disulfide-linked fragments, possibly involving the Cys66-Cys160 linkage found in the native protein (Dave et al., 2013). Recently, irradiation with microwaves followed by sample storage has been shown to be a particularly efficient method of fibril formation. Notably, the resulting fibrils contain not only peptides, but sequences consistent with intact \u03b2-Lg monomers (Hettiarachchi et al., 2012).\n\nAlthough fibril formation in food products has yet to be utilized, there is potential for their application as functional ingredients to control such properties as viscosity, gelation propensity, and emulsion and foam stability (Kroes-Nijboer et al., 2012; Nicolai and Durand, 2013).\n\n## \u03b1-Lactalbumin\n\n\u03b1-Lactalbumin (\u03b1-La) is a 123 amino acid, 14.2 kDa globular protein found in the milk of all mammals. The bovine protein binds Ca2+ with the holo form, being the more abundant form in milk. Within the Golgi apparatus of the mammary epithelial cell, \u03b1-La is the regulatory component of the lactose synthase complex (in which it combines with N-acetyl galactosamine synthase, now named \u03b2-1,4-galactosyltransferase-I); its role is to transfer galactose from UDP-galactose to glucose (Brew, 2003). The structure of human \u03b1-La in a 1:1 complex with \u03b2-1,4-galactosyltransferase-I has been determined (Ramakrishnan and Qasba, 2001; Ramakrishnan et al., 2001; Ramakrishnan et al., 2006), and a 2:1 structure has recently become available (Ramakrishnan, 2013, unpublished data).\n\nIn the absence of \u03b1-La and in the presence of a transition metal ion such as manganese(II), the catalytic domain of bovine \u03b2-1,4-galactosyltransferase-1 (residues 130\u2013402) transfers galactose (Gal) to N-acetylglucosamine (GlcNAc), which may be either free or linked to an oligosaccharide, generating a disaccharide unit, Gal-\u03b2-1,4-GlcNAc (N-acetyllactosamine). The calcium-ion binding site is remote from the active site of the \u03b1-La \u2212 \u03b2-1,4-galactosyltransferase-I complex (Brew, 2003). \u03b1-Lactalbumin has been studied extensively, largely due to its formation of a molten globule state under mild denaturing conditions (Dolgikh et al., 1981).\n\n### Molecular Structure of Bovine \u03b1-Lactalbumin\n\nThe tertiary structure of bovine \u03b1-La is typical of that of the protein from other mammalian species (Acharya et al., 1991; Calderone et al., 1996; Pike et al., 1996) and is similar to that of lysozyme, with which it shares significant homology. As illustrated in Figure 7.3a, \u03b1-La is made up of two lobes: the \u03b1-lobe contains residues 1\u201334 and 86\u2013123, and the smaller \u03b2-lobe spans residues 35\u201385. The \u03b1-lobe contains three \u03b1-helices (residues 5\u201311, 23\u201334, and 86\u201398) and two short 310-helices (residues 18\u201320 and 115\u2013118). A small, three-stranded \u03b2-sheet (residues 41\u201344, 47\u201350, and 55\u201356) and a short 310-helix (residues 77\u201380) make up the \u03b2-lobe (Calderone et al., 1996; Pike et al., 1996). The structure is stabilized by four disulfide bonds (Cys6\u2013Cys120 and Cys28\u2013Cys111 in the \u03b1-lobe, Cys60\u2013Cys77 in the \u03b2-sheet, and Cys73\u2013Cys90 tethering the two lobes together) (Brew, 2003). Unlike \u03b2-Lg, \u03b1-La has no free thiol. A calcium ion binds with a submicromolar dissociation constant at the so-called binding elbow formed by residues 79\u201388 located in a cleft between the two lobes, with the metal ion coordinated in a distorted pentagonal bipyramidal configuration by the side-chain carboxylate groups of Asp82, Asp87, and Asp88, the carbonyl oxygens of Lys79 and Asp84, and the oxygen atoms of two water molecules (Calderone et al., 1996; Pike et al., 1996).\n\nFigure 7.3 (a) Structure of bovine \u03b1-lactalbumin showing the Ca2+ binding site (PDB code: 1f6s). The peptide chain is rainbow colored, beginning at the N-terminus in blue and progressing to the C-terminus in red, in order to show the assembly of the subdomains. The Ca2+ ion is seven-coordinate. Loop 79\u201384 provides three ligands, two from main-chain carbonyl oxygen atoms of Lys79 and Asp84 and one from the side chain of Asp82. Coordination about the Ca2+ is completed by carboxylate oxygen atoms from Asp87 and Asp88 at the N-terminal end of the main 4-turn helix, and by two water molecules. The four disulfide bonds are shown in ball-and-stick representation (one in the helical domain is obscured, and the two linking the helical domain and calcium-ion binding loop to the \u03b2 domain are on the left half of the panel). (b) The lactose synthase complex formed from bovine \u03b1-lactalbumin (yellow) with \u03b2-1,4-galactosyltransferase (gray) (PDB code 2fyd). Several substrate molecules are observed, together with the cleaved nucleotide-sugar moiety (cyan sticks). The MnII ion is shown as a pink sphere and the Ca2+ ion as a gray sphere. For clarity, loop regions are given a smoothed representation. \u03b2-1,4-galactosyltransferase is not present in milk, and the lactose synthase complex is present only in the milk-producing cells of the alveoli. Figure drawn with PyMOL (Delano, 2002).\n\nThe structure of the apo form of bovine \u03b1-La is similar to that of the holo protein. The largest changes involve the movement of the Tyr103 side chain in the interlobe cleft with little change in the vicinity of the Ca2+ binding site (Chrysina et al., 2000). The salient features of the structure of the holo protein are depicted in Figure 7.3, together with its complex with \u03b2-1,4-galactosyltransferase-I with bound substrates and, interestingly, a trapped intermediate species.\n\nThe structures of \u03b1-La from several other species, including goat, baboon, human, guinea pig, and buffalo, have also been characterized by x-ray diffraction methods (Acharya et al., 1989; Acharya et al., 1991; Pike et al., 1996; Makabe et al., 2012; Calderone et al., 1996). Consistent with high-sequence identity, there are no significant differences among these structures, except for a flexible loop at residues 105\u2013110 implicated in formation of the lactose synthase complex (Acharya et al., 1989; Calderone et al., 1996; Pike et al., 1996). The recombinant goat protein, which has an added methionine at the N-terminus, is markedly less stable, by \u223c14 kJ mol\u22121, than the native protein, mostly the result of an increased rate of unfolding (but preserved rate of refolding) (Chaudhuri et al., 1999). Similar observations have been made on recombinant bovine \u03b1-La (Acharya et al., 1989). Thus, native protein functionality and stability should not in general be inferred from measurements of recombinant proteins heterologously expressed in bacterial systems (which generally add an N-terminal methionine residue).\n\n### Effect of Temperature on Bovine \u03b1-Lactalbumin\n\nIn general, holo \u03b1-La undergoes a thermal unfolding at a lower temperature than \u03b2-Lg (Ruegg et al., 1977). The role of bound calcium ions appears to confer stability to the tertiary structure: With less than equimolar amounts of bound calcium, the thermal unfolding transition is lowered substantially, decreasing to about 35 \u00b0C for the apo form (Relkin, 1996; Ishikawa et al., 1998). The presence of calcium ions also accelerates the rate of refolding of \u03b1-La by more than two orders of magnitude (Wehbi et al., 2005). The presence of calcium ions also aids in the refolding and formation of the correct disulfide linkages of denatured reduced protein (Wehbi et al., 2005). The structurally closely related, but functionally unrelated, enzyme lysozyme can be subdivided into two classes, a noncalcium-binding subclass, typified by egg-white lysozyme (Grobler et al., 1994; Steinrauf, 1998) and a calcium-binding subclass, including equine and echidna lysozyme (Tsuge et al., 1992; Guss et al., 1997).\n\nThe thermal denaturation behavior of bovine \u03b1-La from three different sources has been studied; significant differences were reported (McGuffey et al., 2005), thereby providing some resolution of the apparently discordant denaturation data from different groups. In the presence of \u03b2-Lg or BSA, each of which has an unpaired cysteine, \u03b2-Lg \u2212 \u03b1-La, \u03b1-La \u2212 BSA and even \u03b1-La \u2212 \u03b1-La oligomers form at high temperature. Since \u03b1-La (which lacks a free thiol) by itself fails under similar conditions to form disulfide-linked oligomers, intermolecular disulfide-sulfhydryl interchange reactions appear to play a role in forming \u03b1-La \u2212 \u03b1-La oligomers (Havea et al., 2001; Hong and Creamer, 2002).\n\n### Effect of Pressure on Bovine \u03b1-Lactalbumin\n\nAgain the absence of free thiol groups renders \u03b1-La intrinsically less susceptible to irreversible structural and functional change induced by high pressure. Reversible unfolding to a molten globule state begins at 200 MPa, and loss of native structure becomes irreversible beyond 400 MPa (McGuffey et al., 2005) (corresponding numbers for \u03b2-Lg are 50 and 150 MPa (Stapelfeldt and Skibsted, 1999; Considine et al., 2005b)). In the presence of calcium ions, denaturation pressure increases by 200 MPa for \u03b1-La (Dzwolak et al., 1999). Only in the presence of thiol reducers does oligomerization of \u03b1-La occur at high pressures (Jegouic et al., 1996).\n\n### Effect of Denaturants on Bovine \u03b1-Lactalbumin\n\nAt neutral pH, the calcium-depleted, or apo form of \u03b1-La, reversibly denatures to a variety of partially folded or molten globule states upon moderate heating (45 \u00b0C), or, at room temperature, by dissolving the protein in aqueous TFE (15% TFE) or by adding oleic acid (7.5 equivalents) (Svensson et al., 2000; de Laureto et al., 2002). Under these various conditions, the UV-CD spectra of apo-\u03b1-La are essentially identical to those of the most studied molten globule form of \u03b1-La, the A-state found at pH 2.0 (Kuwajima, 1996). At 4 \u00b0C and pH 8.3, proteolysis of apo-\u03b1-La by proteinase-K occurs slowly and nonspecifically, leading to small peptides only. On the other hand, at 37 \u00b0C, preferential cleavage by proteinase-K is observed at peptide bonds located in loop regions of the \u03b2-sheet subdomain of the \u03b2 domain of the protein (residues 35\u201385), creating peptides in which disulfide bridges link N-terminal residues 1\u201334 to C-terminal fragments, residues 54\u2013123 or 57\u2013123. Preferential cleavage at similar sites and similar disulfide-bridged fragments has also been observed for proteolysis of the molten globule states induced by TFE and oleic acid. Therefore, polypeptides formed from the molten globule A-state of \u03b1-La comprise a well-structured native-like conformation of the \u03b1-domain and a disordered conformation of the \u03b2-subdomain, residues 34\u201357 (de Laureto, et al., 2002).\n\nThe oleic acid treatment leads to a kinetically trapped folding variant of the protein, which can also bind calcium ions, called HAMLET (human \u03b1-lactalbumin made lethal to tumor cells, and its bovine analogue BAMLET), which has been shown to induce apoptosis in tumor cells (Svensson et al., 2000). Under conditions where thermal denaturation of \u03b1-La is reversible, thermal denaturation of HAMLET is irreversible, with respect to loss of its apoptotic effect on tumor cells (Fast et al., 2005).\n\nHuman and bovine \u03b1-La also weakly bind a second calcium ion, structurally characterized for human \u03b1-La (Chandra et al., 1998). In addition, zinc-ion binding to possibly structurally inequivalent sites has been characterized for human and bovine \u03b1-La by fluorescence spectroscopic techniques (Permyakov and Berliner, 2000). The binding of zinc ions to calcium-loaded \u03b1-La has been shown to destabilize the native structure to heat denaturation (Permyakov and Berliner, 2000). However, the weak binding of zinc (submillimolarity dissociation constant) means that this binding is probably not physiologically relevant.\n\n## Serum albumin\n\nSerum albumin (SA) is an approximately 580-residue protein found in both the blood serum and milk of all mammals and appears to function as a promiscuous transporter of hydrophobic molecules. However, as with many proteins, this transport role appears not to be the only physiological function for serum albumins. The structure of human serum albumin (HSA), described in more detail in the next subsection, is notable also for the number of disulfide bridges, 17 in total. There is one unpaired cysteine, Cys34, in HSA (highlighted in Figure 7.4), also Cys34 in BSA. This cysteine is part of a highly conserved QQCP(F\/Y) motif and is susceptible to various oxidations, including a two-electron oxidation to sulfenic acid (-SOH) (Carballal et al., 2007).\n\nFigure 7.4 Structure of human serum albumin (HSA) complexed with halothane (slate\/purple) partially occupying seven distinct sites and myristic acid (yellow\/red) occupying fully five distinct sites (PDB code: 1e7c). Domain IA (residues 5\u2013107) is shown in blue; domain IB (108\u2013196) in light blue; domain IIA (197\u2013297) in green; domain IIB (residues 297\u2013383) in light green; domain IIIA (residues 384\u2013497) in red; and domain IIIB (residues 498\u2013582) in light red. The single cysteine, Cys34, is arrowed (Bhattacharya et al., 2000). The 17 disulfide bonds, which tie together individual subdomains, are represented in stick format. Figure drawn with PyMOL (Delano, 2002).\n\nEvidence has been produced that, at least in blood serum, this cysteine is involved in HSA's role in the control of redox properties (Kawakami et al., 2006). A similar role can be postulated for bovine serum albumin (BSA) in blood serum, but in both human milk and bovine milk, this redox role has not been established (or even investigated). In terms of milk flavor and the flavor of milk products, control of redox states of milk components is obviously of importance. It appears also that in blood serum, where HSA is the major protein component, present at a concentration of 0.6 mM, HSA is the first line of defense against radicals, including reactive oxygen species and nitric oxide.\n\nIn vitro studies showing the reactivity of Cys34 in HSA have been complemented by in vivo studies showing that in primary nephrotic syndrome, Cys34 is oxidized to sulfonate, \u2013SO3- (Musante et al., 2006). Again, in milk, defenses against reactive oxygen species are essential to preserve milk quality, but HSA and BSA are at much lower concentrations in milk than in blood. HSA has also been shown to have esterase activity (Sakurai et al., 2004). Whether this is physiologically important in blood (or in milk) has not been established for either BSA or HSA. Finally, an active role for HSA in transport of fatty acids across membranes has been characterized (Cupp et al., 2004). It is in this process that serum albumins are introduced into mammalian milks.\n\n### Structure of Serum Albumins\n\nThe structure of HSA, with which BSA shares 75% sequence identity, has been well characterized for apo protein, as well as for a variety of complexes with a variety of long-chain fatty acids and other more compact hydrophobic molecules. The structure of HSA complexed with the anesthetic halothane (C2F3Cl2Br) and myristic acid (CH3(CH2)12COOH) is shown in Figure 7.4.\n\nThe structure of HSA comprises three structurally homologous domains, each of just under 200 residues, denoted I-III (Curry et al., 1998) and involving residues 5\u2013196, 197\u2013383, and 384\u2013582, respectively. Each domain has two subdomains, each of \u223c100 residues, denoted A and B. The structure lacks \u03b2-strands and is predominantly (68%) \u03b1-helical, with several lengthy loops connecting A and B subdomains. On ligand binding, there is substantial movement of the domains with respect to each other, but the tertiary structure of each domain undergoes only small changes (Curry et al., 1998). Medium- and long-chain fatty acids occupy five distinct sites (dissociation constants 0.05\u20131 \u03bcM) (Spector, 1975), one in domain I, a second between domains I and II, and the remaining three in domain III, as characterized by x-ray techniques for HSA. In the case of halothane binding, two sites are located in domain I and five in domain II.\n\nA comprehensive study of the binding of 17 distinct drugs to HSA, in the presence and absence of myristate, has been published (Ghuman et al., 2005). Whereas binding of steroids to BSA is influenced by binding of fatty acids, for HSA, there is much less influence (Watanabe and Sato, 1996). NMR titrations have shown that BSA, like HSA, binds five myristates; four of the five sites appear to be in structurally homologous sites to those identified crystallographically for HSA (Hamilton et al., 1984; 1991; Cistola et al., 1987; Simard et al., 2005). The x-ray structure of equine serum albumin (ESA) is very similar to that of HSA, consistent with \u223c75% sequence identity between these two proteins (Ho et al., 1993).\n\nThe structures of bovine, equine, and leporine (rabbit) SA have been recently published by two independent groups (Bujacz, 2012; Majorek et al., 2012; Sekula et al., 2013). HSA has also been heterologously expressed in rice and structurally characterized (He et al., 2011), finessing problems associated with availability and disease risks when sourced for clinical and cell-culture applications from human blood sources.\n\n### Effect of Temperature on Serum Albumins\n\nCareful differential scanning calorimetry (DSC) measurements have been made of defatted HSA and its binding to short- to medium-chain fatty acids. In the absence of fatty acids, a single sharp endotherm is observed, yielding a midpoint temperature for denaturation, T m, of 64.7 \u00b0C, consistent with a concerted unfolding. In the presence of fatty acids, the endotherm broadens, and there is a steady increase in T m as the chain length of the fatty acid increases from n-butanoate (T m = 77.6 \u00b0C) to n-octanoate (T m = 87.2 \u00b0C); T m for n-nonanoate and n-decanoate are very similar to that for n-octanoate. The short-chain fatty acids, formate, acetate, and n-propionate show evidence for inducing increased stability in HSA through binding of the fatty acids at secondary sites inaccessible to the longer-chain fatty acids. The concentration of fatty acid at which maximum stability of HSA is achieved decreases from more than 2900 mM for formate to less than 15 mM for a 30 mg\/mL (\u223c0.5 mM) solution of HSA at pH 7.0 (Shrake et al., 2006).\n\nFor the native protein, DSC data could be fitted to a two-state model with \u223c7 more or less equivalent binding sites for n-decanoate. This number may be contrasted with the value of 5 observed by x-ray and NMR methods for the binding of myristate (tetradecanoate acid) to HSA and also BSA (see above) (Simard et al., 2005).\n\n### Effect of Pressure on Serum Albumins\n\nBSA, despite the unpaired cysteine, is relatively stable to high pressures (800 MPa) (L\u00f3pez-Fandi\u00f1o, 2006). BSA undergoes substantial secondary structure changes, but unlike the case with \u03b2-Lg, these changes are reversible. It appears that the large number of disulfide bonds protects the hydrophobic core of the protein, including the largely buried Cys34 in subdomain IA, which is held together by three disulfide bonds (L\u00f3pez-Fandi\u00f1o, 2006). The effects of binding partners, such as fatty acids or other whey components, such as \u03b2-Lg, on structure and stability of BSA at high pressure remain uncharacterized.\n\nCombined pressure-temperature infrared studies of the amide vibrational modes of ESA have shown very recently that high pressure (400 MPa) can convert an intermolecular \u03b2-sheet aggregate formed by heating ESA to 60 \u00b0C at 0.1 MPa (i.e., ambient pressure) to a disordered structure, which reverts to the native structure upon release of pressure. The activation volume of \u223c+92 mLmol\u22121 and partial molar volume difference between native and heat-denatured states of (\u0394 V N\u2192HA = +32 mLmol\u22121) are consistent with the decreasing stability of the heat-denatured intermolecular \u03b2-sheet with increasing pressure (Okuno et al., 2007).\n\n### Effect of Chemical Denaturants on Serum Albumins\n\nThe denaturation of BSA and HSA by urea and GdmCl has been extensively studied (Lapanje and Skerjanc, 1974; Khan et al., 1987; Guo and Qu, 2006). In the presence of fatty acids and other molecules, especially molecules that bind to domain III (e.g., diazepam), denaturation of BSA by urea changes from a three-step process to a two-step process, indicating that the initial denaturation involves changes in tertiary and secondary structure of domain III (Ahmad and Qasim, 1995; Tayyab et al., 2000). For HSA, denaturation appears to be an intrinsically two-step process with fatty acids converting denaturation to a one-step process (Muzammil et al., 2000; Shrake et al., 2006). In both cases, the binding of ligands to BSA or HSA stabilizes the protein against urea-induced denaturation. There is evidence that the urea-induced denatured state and that induced by high pressure are similar, at least for HSA (Tanaka et al., 1997).\n\nIn the presence of cations, such as guanidinium and cetylpyridinium salts, denaturation of SA again features domain III as being the most susceptible to denaturation (Ahmad et al., 2005; Sun et al., 2005). Consistent with the high helical content of SA, perfluorinated alcohols and alcohols with bulky hydrophobic heads stabilize HSA (and presumably BSA) against denaturation by both urea and GdmCl (Kumar et al., 2005).\n\nThe stability of HSA in the presence of polyethylene glycols (PEGs) has also been examined by Farruggia et al. (1999); low-molecular-weight PEGs affect ionization of surface tyrosines, and high-molecular-weight PEGs lower the thermal transition temperatures.\n\n## Immunoglobulins\n\nThe immunoglobin proteins (Ig) form a diverse family whose members, when in milk, protect the gut mucosa against pathogenic microorganisms. In bovine milk, the predominant species of Ig proteins are members of the IgG subfamily, in particular the IgG1. Colostrum contains 40\u2013300 times the concentration of IgG proteins than does milk; its role is to confer passive immunity to the neonate while its own immune system is developing (Gapper et al., 2007).\n\nIgG proteins have multiple functions, including complement activation, bacterial opsonization (rendering bacterial cells susceptible to immune response), and agglutination. They inactivate bacteria by binding to specific sites on the bacterial surface. Given the significance attributed to bovine milk and milk products in human nutrition and health, it is important to note that there are significant differences in the levels of the various subfamilies of immunoglobins in milk from different species. Human colostrum and milk contain relatively low levels of the IgG subfamily compared to bovine milk; the opposite is the case with respect to the IgA subfamily. The properties and accurate quantitation of bovine Ig proteins have been reviewed in detail by Gapper et al. (2007).\n\n### Structure of Immunoglobulin G\n\nThe structure of IgG is illustrated schematically in Figure 7.5a and as revealed by x-ray techniques in Figure 7.5b. Both the heavy chain and light chain are predominantly \u03b2-sheet structures. Disulfide bridges link pairs of molecules, as well as the heavy chain to the light chain. The protein is generally glycosylated at a number of sites. However, the actual structure is much less tidy than the schematic Y-shaped figure; in particular, the disulfide bonds are at the base of the light chain\u2013heavy chain associations.\n\nFigure 7.5 (a) Schematic of the general structure of immunoglobulins. Reproduced from Gapper et al. (2007) with permission. The different classes are distinguished by the constant or Fc regions of the heavy and light chains. Reproduced from Gapper et al. (2007) with permission. (b) The x-ray structure of the human IgG1 molecule (PDB code 1hzh). The heavy chains are in blue and green, the light chains are in magenta and pink. The asparagine N-linked glycan is shown in stick representation. The lack of two-fold symmetry indicates the extreme flexibility of the domains with respect to one another and consequent sensitivity to crystal-packing effects. Disulfide bridges are shown as spheres. Each domain has a disulfide bridge joining the two sheets; additional disulfide bridges link the two heavy chains and the light chains to the heavy chains. The N-termini of each chain is at the top left and top right of the diagram; the C-termini of the heavy chains are at the base of the molecule. Structures (e.g., PDB code: 1wej, 3hfm) where antigens are bound at the light chain\u2013heavy chain interface indicate that binding of antigen occurs across the top of the molecule with little embedding of antigen between the domains, contrary to the mode implied by (a). Figure drawn with PyMOL (Delano, 2002).\n\nBovine \u03b2-Lg has long been recognized for eliciting strong immunological responses in selected susceptible individuals (Wal, 1998). The structure of bovine \u03b2-Lg in complex with a recombinant human IgE Fab fragment, that is, an IgE epitope of bovine \u03b2-Lg, has been reported (Niemi et al., 2007).\n\n### Effects of Temperature, Pressure, and Chemical Denaturants on Ig Structure and Stability\n\nThe response of bovine IgG (isoform not specified) to temperature and chemical denaturants, urea, and guanidinium hydrochloride has been reported (Ye et al., 2005). Thermal denaturation and thermal denaturation in the presence of denaturants was irreversible, producing via a series of steps an incompletely unfolded aggregate. Isothermal chemical denaturation produced, also by a series of steps, a completely unfolded random-coil state (Ye et al., 2005). The response of IgG to high pressure (200\u2013700 MPa) in the presence of the kosmotrope sucrose has also been reported (Zhang et al., 1998).\n\nA comparative study of the thermal denaturation of bovine IgG, IgA, and IgM has been published, with stability in the order just given (Mainer et al., 1997). As retention of immunological properties on standard milk-processing techniques was of interest, the activity of heat-treated protein was determined by an immunological assay using antibodies raised against these Ig proteins. The response of IgG to pulsed electric fields (and to heat) was reported (Li et al., 2005). Little change in secondary structure or in immunoactivity was reported for samples subjected to a pulsed electric field of \u223c41 kV\/cm.\n\n## Lactoferrin\n\nLactoferrin (Lf) is a monomeric, globular, Fe3+-binding glycoprotein comprising \u223c680 amino acids giving a molecular mass of \u223c80 kDa. It is a member of the transferrin family, but unlike the eponymous protein, to date there is no strong evidence that lactoferrin is involved in iron transport or metabolism under normal circumstances. Indeed, the protein is only lightly loaded with iron(III) in milk, allowing it to perform a major bacteriostatic role by sequestering iron(III) despite bacteria-producing iron(III)-sequestering agents (siderophores) that have affinities for iron(III) many orders of magnitude higher than that for lactoferrin. Lf is, however, a multifunctional protein, with evidence to suggest that it, and especially its N-terminal Arg-rich fragments, called lactoferricin(s), obtained by pepsin hydrolysis of the entire protein, have active antimicrobial activity as well as activity as antiviral and antiparasitic agents (Str\u00f8m et al., 2000; ).\n\nAccordingly, commercial applications utilizing bovine Lf and its partially digested peptides are appearing as nutraceuticals in infant formulas, health supplements, oral care products, and animal feeds. In addition, reputed antioxidant properties are being utilized in cosmetics (van Hooijdonk and Steijns, 2002). For a review on the remarkable properties of Lf and its commercial applications, see Brock (2002). The protein is synthesized in the mammary gland, but it is also found in other exocrine fluids besides milk. Bovine Lf is sometimes used as a supplement in bovine milk-based infant formulas (to offset the lower abundance relative to that found in human milk) and is potentially useful as a constituent in functional foods, with marked effects on bone-cell activity (Cornish et al., 2004).\n\n### Bovine Lactoferrin Structure\n\nBovine Lf has a tertiary structure (Moore et al., 1997) very similar to that of the Lfs of other species determined so far: human (Anderson et al., 1987; ; Haridas et al., 1995), buffalo (Karthikeyan et al., 1999), horse (Sharma et al., 1999), and camel (Khan et al., 2001). All Lfs and transferrins contain two lobes, which share internal homology (\u223c40% sequence identity between the N- and C- terminal lobes) and a common fold (see Fig. 7.6). The homology between lobes for a given species is less than that between corresponding lobes from different species, indicating that the gene duplication event is of ancient origin. Each lobe contains two \u03b1\/\u03b2 domains divided by a cleft that incorporates an iron-binding site. Huge structural changes accompany iron binding: Each lobe closes over the iron(III), encapsulating a synergistic bicarbonate anion. Remarkably, for camel lactoferrin, the N-terminal lobe appears to have sequence and iron-binding properties similar to those of other lactoferrins, whereas the C-terminal lobe resembles more closely in sequence and iron-binding properties of transferrins, hen ovo-transferrin in particular (Khan et al., 2001).\n\nFigure 7.6 Structure of lactoferrin. (a) Human apo-lactoferrin (PDB code: 1cb6), showing the domain structure. The N lobe (blue for subdomain N-IN and cyan for subdomain N-IC) is in the open conformation, the C-lobe (magenta for subdomain C-IN and pink for subdomain C-IC), despite no metal ion present, is in the closed conformation. The helix connecting the two lobes is shown in yellow. Metal-binding ligands are shown as spheres. (b) Holo-(FeIII)-bovine lactoferrin (PDB code: 1blf; the human analogue, 1b0l, is structurally very similar). The polypeptide is rainbow colored, blue at the N-terminus to red at the C-terminus, to highlight the manner in which the subdomains N-IN and C-IN are formed from residues \u223c1\u201390 and \u223c250\u2013320 (N-IN) and \u223c350\u2013440 and \u223c600\u2013680 (C-IN); for reference, these subdomains are shown in approximately the same orientation in frames (a) and (b). The iron-binding ligands, including the synergistic bicarbonate ion in (b), are shown in stick form, the iron atom as a red sphere and the cysteines as orange spheres (16 Cys forming 8 disulfides for human Lf and 34 Cys forming 17 disulfides for bovine Lf). Loops are not smoothed. Figure drawn with PyMOL (Delano 2002).\n\nNotwithstanding these structural changes, for the apo protein, at least for human Lf, there is little difference in free energy between open and closed forms (indeed, the structure of human apo-Lf has one lobe open, the other closed). This delicate balance is achieved by both open and closed conformations of the protein having, remarkably, the same surface area exposed to solvent. The same applies, but by a different structural mechanism, to open and closed forms of the N-terminal recombinantly produced half molecule of human lactoferrin (Jameson et al. 1998; ; 2002b;). Like \u03b2-Lg, holo-Lf undergoes a pH-dependent conformational change (in this case as low as pH \u223c3) that releases the bound ferric ion; the structurally related, but genetically and functionally distinct, serum transferrin releases its cargo at significantly higher pH (pH \u223c5.5) (Baker and Baker, 2004).\n\n### Effects of Temperature, Pressure, and Chemical Denaturants on Lactoferrin Structure and Stability\n\nAs a minor component of milk proteins, Lf seems largely to have escaped detailed study of its intrinsic response to temperature, pressure, and chemical denaturants or of its influence, if any, on stability of either itself, or other whey proteins, in the presence of other whey proteins. Calcium ions have been reported to bind to bovine Lf, probably to the sialic acid groups of the asparagine N-linked glycan, with micromolar dissociation constants; both apo- and holo forms of calcium-bound bovine Lf are more stable to heat and chemical denaturants (Rossi et al., 2002). The calcium-bound forms appear to reduce the Lf-induced release of lipopolysaccharide moieties from bacterial membranes (Rossi et al., 2002).\n\nThe heat stability of bovine Lf in isolation (S\u00e1nchez et al., 1992) and in association with the major bovine whey proteins, \u03b2-Lg (weak association), \u03b1-La (no detectable association), and BSA (weak 1:1 association) (Lampreave et al., 1990) has also been studied. The stability of bovine Lf is such that standard pasteurization conditions (but not ultra-high-temperature treatment) are likely to have little effect on structure and properties, especially immunological properties (Oria et al., 1993).\n\n## Concluding remarks\n\nThree-dimensional structure determines physiological function as well as the functionality of whey proteins in their varied applications, both in whole milk and in whey itself. In this review we have tried to look beyond individual proteins to see what structural features may be important to protein\u2013protein interactions under stresses of temperature, pressure, and chemical denaturants. In particular, we have attempted to uncover more recent results that challenge existing paradigms or offer new perspectives. In general, changes that are brought about as a consequence of pH, heat, pressure, chaotropes, and so on, shift equilibrium points (e.g., between monomer and dimer, native and unfolded states, etc.), and it is only when a new covalent bond is formed (or removed) that there is irreversible change.\n\nOne key interaction between like and unlike proteins arises from disulfide bond interchange, generally facilitated by a single cysteine residue. For this reason, we have focused, especially in the figures, on the observed locations of cysteines. However, detailed structural information, at a level comparable to that of individual partners, is generally lacking in intermolecular assemblies, even when only pairwise associations are formed and isolated.\n\nGiven sufficient time at elevated temperatures and low pH, proteins, in particular \u03b2-Lg, are observed to hydrolyze and reassemble in fibrillar species, for which signatures of extended \u03b2 sheet structures are seen, but the extreme chemical and thermal stability that characterizes true \u03b2-amyloid aggregates is absent.\n\nWhereas considerable attention has been directed toward understanding interactions among components of milk and of whey, relatively little attention has been given to date to the redox state and redox changes during storage and processing of milk proteins and their effects on milk processing and flavors of milk-derived products. This remark originates from recent studies on blood sera, where serum albumins play key roles in redox state and protection against reactive oxygen species. It is important, then, that these structures and functional states are characterized and understood, especially if milk processing, particularly by means of relatively new methods such as pressure treatment, results in refolded proteins that may have altered properties and functionality, such as processability and digestibility.\n\n## Acknowledgments\n\nThe authors would like to acknowledge illuminating discussions with Lindsay Sawyer over many years, and long-standing collaboration with Lawrie Creamer, facilitated by Mike Boland.\n\n# References\n\nAcharya KR ,  Ren JS ,  Stuart DI ,  Phillips DC ,  Fenna RE .  Crystal structure of human \u03b1-lactalbumin at 1.7 \u00c5 resolution .  _Journal of Molecular Biology_.  1991 ;221 (2) : 571 \u2013 581 .\n\nAcharya KR ,  Stuart DI ,  Walker NPC ,  Lewis M ,  Phillips DC .  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Patel*\n\nThom Huppertz**\n\n* Dairy Science Department, South Dakota State University, South Dakota, USA  \n** NIZO food research BV, Ede, the Netherlands\n\n## Abstract\n\nHigh-pressure processing (HPP) is a rapidly growing nonthermal preservation technology that can potentially be used to create novel protein structures by bringing about particular changes in the molecular structure of proteins and thus may give rise to innovative, new generation value-added food products with new properties that are not possible through conventional methods of protein modification. Compared to heat treatment, HPP has been reported to have some similar and some different effects on the structure of milk proteins. Knowledge of the effects of HPP on the structure and interactions of milk proteins, therefore, can be used to develop food products with novel textures.\n\nIn this chapter, we review recent literature reports on how HPP can affect the structure of casein micelles and individual casein fractions, as well as denaturation and aggregation of whey proteins and their interactions with caseins. The mechanisms and pathways underlying pressure-induced denaturation and aggregation of whey proteins and their interactions with casein micelles are explained in various systems, including model systems, commercial whey protein solutions, and skim milk.\n\n## Keywords\n\nHigh pressure\n\nhigh-pressure processing\n\nheat, milk proteins\n\ncasein\n\nwhey proteins\n\nprotein structure\n\ndenaturation\n\naggregation\n\nprotein interactions\n\nOutline\n\nIntroduction 244\n\nHigh-pressure-induced changes in caseins 244\n\nEffects on Individual Casein Fractions 244\n\nEffects on Casein Micelles 245\n\nEffects of HPP on Caseinates 247\n\nEffects of high pressure on interactions of milk proteins involving whey proteins 247\n\nDenaturation and Aggregation of Pure Whey Protein Fractions in Model Systems 248\n\n\u03b2-Lactoglobulin (\u03b2-LG) 248\n\n\u03b1-Lactalbumin (\u03b1-LA) 250\n\nBovine Serum Albumin (BSA) 251\n\nImmunoglobulins (Igs) and Lactoferrin (LF) 251\n\nMixtures of \u03b1-LA and \u03b2-LG 252\n\nMixtures of \u03b2-LG, \u03b1-LA, and BSA 252\n\nCommercial Whey Protein Solutions 252\n\nPressure-induced Gelation of Whey Proteins 253\n\nHPP-induced Changes in Milk 256\n\nDenaturation of Whey Proteins in the Milk System 257\n\nInteractions of Whey Proteins with Casein Micelles 258\n\nConcluding remarks 261\n\nAcknowledgment 261\n\n## Introduction\n\nProcessing treatments such as heat and high pressure are normally applied in the food industry, for the purpose of microbial destruction, shelf-life extension, or achievement of desired functionality in the final product (Jelen & Rattray, 1995; Singh, 1995). It is well known that the heat-induced interactions of milk proteins have a marked impact on the functionality of the final products and such interactions are of considerable commercial importance in the dairy and food industries. Therefore, considerable research efforts have been directed to studying the detailed pathways\/mechanisms of heat-induced functionality and the effects of heat treatments on milk proteins (denaturation, aggregation, and gelation of whey proteins) and protein\u2013protein interactions, including interactions of caseins and whey proteins, have been studied in great detail over five to six decades (e.g., Havea et al. 1998; ; ; ; Manderson et al., 1998; Schokker et al. 1999; ; Anema & Li, 2003a,b; Cho et al., 2003; Livney & Dalgleish, 2004; Patel et al. 2004; ; ; ), and the subject has often been reviewed (e.g., De la Fuente et al., 2002; O'Connell & Fox, 2003; Singh & Havea, 2003; Singh, 2004).\n\nMore than a century ago, pioneering work by Hite (1899) showed the potential of high-pressure processing (HPP) as a nonthermal (alternative) preservation process, and Bridgman (1914) demonstrated the effects of HPP on the denaturation and functional properties of egg proteins. The recent interest in the HPP of food materials as an alternative to or in addition to temperature treatment led to many fundamental studies on the behavior of proteins under HPP. It is also evident from the literature that, in many cases, heat treatment and high-pressure treatment have different effects on different milk proteins. This suggests that HPP has the potential for both preservation and modification of the structure of proteins, alteration to their functional properties, and the creation of value-added products (e.g., L\u00f3pez-Fandi\u00f1o et al., 1996; Garc\u00eda-Risco et al., 2000; ; Patel et al. 2004; ; ).\n\nToday, many high-pressure-treated products such as fresh fruit jams, jellies, juices, salad dressings, rice, cakes, and guacamole are commercially available (L\u00f3pez-Fandi\u00f1o, 2006a,b). As there is increased interest in the high-pressure (HP) treatment of dairy products, it is currently a major focus of investigation. This subject has also been extensively reviewed (e.g., Datta & Deeth, 1999; ; Farkas & Hoover, 2000; Boonyaratanakornkit et al., 2002; Huppertz et al., 2002; Lullien-Pellerin & Balny, 2002; Trujillo et al., 2002; Claeys et al., 2003; Huppertz et al., 2006a,b; L\u00f3pez-Fandi\u00f1o, 2006a,b; Considine et al., 2007b; Rastogi et al., 2007; Patel and Creamer, 2008; Patel et al., 2008).\n\nThis chapter focuses on the effects of HPP on structures of individual whey proteins and caseins and their interactions in various systems. Where possible, the discussion is extended to describe pathways and mechanisms of pressure-induced denaturation, aggregation, and interactions of milk proteins.\n\n## High-pressure-induced changes in caseins\n\n### Effects on Individual Casein Fractions\n\nResearchers have found very different effects of HPP on \u03b1s1-casein and \u03b2-casein fractions (Schmidt & Payens, 1972; Payens & Heremans, 1969). For \u03b1s1-casein, the intensity of light scattered at an angle of 90\u00b0 (I90\u00b0) under pressure decreased with increasing pressure in the range of 0.1\u2013300 MPa. Reductions in I90\u00b0 indicate a reduction in the size and molar mass of the \u03b1s1-casein aggregates present in the suspension. In contrast, a reduction in I90\u00b0 was observed for \u03b2-casein with increasing pressure up to 150 MPa at 20\u201325 \u00b0C, but further increase in pressure resulted in increase in I90\u00b0, similar to that of original suspension (Schmidt & Payens, 1972). However, such changes were completely reversible upon pressure release (Schmidt and Payens, 1972). On the other hand, when \u03b2-casein was pressurized at 5 \u00b0C rather than 20\u201325 \u00b0C, pressures up to 150 MPa had little effect on I90\u00b0, whereas higher pressures caused a progressive increase in I90\u00b0 (Payens & Heremans, 1969).\n\nDifferences in the behavior of \u03b1s1\\- and \u03b2-casein under pressure can readily be related to their differences in primary structure and self-association. The type of self-association displayed by \u03b1s1-casein is best described as consecutive self-association in the form of dimers, trimers, tetramers, and the like (Huppertz, 2013). In contrast, due to its highly amphipathic structure, with a hydrophilic and highly charged N-terminus and a hydrophobic C-terminus with only few charged amino acid residues, \u03b2-casein is prone to micellization above critical micellar concentration (CMC) (Huppertz, 2013). Similar minima for I90\u00b0 at 100\u2013200 MPa are also observed for a wide-range of surfactant molecules and thus cannot be related directly to specific properties of \u03b2-casein. It is more likely that HPP-induced changes in water structure contributed to the observed effects of HPP on micellization of \u03b2-casein. Moreover, at low pressures, micellization may be reduced due to reduced hydrophobic interactions or increased electrostatic repulsion as a result of enhanced dissociation under high pressure (Morild, 1981). At higher pressures, such effects are overruled and micellization is promoted. Comparatively smaller reductions in I90\u00b0 were observed for \u03b1s1-casein, indicating a shift of the association equilibrium toward particles of lower molecular mass and enhanced dissociation of side-groups of ionizable amino acid residues, which are more randomly distributed on \u03b1s1-casein and thus may have contributed to such phenomena (Huppertz, 2013). No studies have been reported on the effects of HPP on \u03b1s2-casein and \u03ba-casein.\n\n### Effects on Casein Micelles\n\nAs for individual caseins, the behavior of casein micelles under high pressure has also been studied in detail, largely by determination of light transmission, rather than I90\u00b0. Light transmission of skim milk or other suspensions containing micellar casein increased with increase in pressure, indicating disruption of casein micelles (Huppertz et al., 2006c; Huppertz & De Kruif, 2006; Orlien et al., 2006; Gebhart et al., 2005). At micellar casein concentrations comparable to those in skim milk, that is, 2.5\u20132.8%, light transmission increased with increase in pressure up to \u223c400 MPa, above which no further increase in light transmission was observed (Huppertz et al., 2006c; Orlien et al., 2006). The increase in the light transmission of skim milk or micellar casein suspensions at this pressure indicated significant disruption of casein micelles. The extent of micellar disruption under pressure also decreased with increase in pH (Huppertz & De Kruif, 2006), increase in concentration of micellar casein (Huppertz & De Kruif, 2006), or increase in temperature of pressurization (Orlien et al., 2006). Moreover, addition of calcium to milk prior to HPP also resulted in increased stability of casein micelles to HPP-induced disruption (Huppertz & De Kruif, 2006).\n\nWhen considering the factors governing the stability of casein micelles to dissociation, two primary factors can be considered: (1) the weak interactions (such as hydrophobic interactions, electrostatic interactions, and hydrogen bonding) which collectively facilitate the association of casein micelles; and (2) the calcium phosphate nanoclusters, whose stability is primarily governed by the solubility of calcium phosphate in the serum phase of the product. At atmospheric pressure, micellar disruption as a result of the disruption of the weak cohesive protein\u2013protein interactions can be achieved by the addition of urea or sodium dodecyl sulfate (SDS), whereas the addition of calcium-chelating agents such as citrate or ethylene diamine tetraacetic acid (EDTA) results in the disruption of micelles through solubilization of the micellar calcium present in the calcium phosphate nanoclusters. Considering this theory, association of \u03b1s1-casein is reduced, whereas that of \u03b2-casein is actually enhanced at pressures where extensive micellar disruption is observed. Therefore, explaining HPP-induced disruption of casein micelles solely on the basis of reduced interactions between caseins appears unrealistic. Contributions from solubilization of micellar calcium phosphate (MCP) arise due to the 'electrostrictive' effect; that is, water molecules arrange in a more compact structure around ions than salts (Stippl et al., 2005). As a result, ionization reactions and solubilization of minerals are promoted under pressure. Studies on the solubilization of MCP using NMR (Hubbard et al., 2002) or light scattering from casein nanogel particles (Huppertz & De Kruif, 2007a) have reported that MCP is progressively solubilized with increasing pressure. At a micellar casein concentration of 2.5%, full solubilization of MCP was achieved at \u223c400 MPa (Huppertz & De Kruif, 2007a), which coincides with the pressure at which the highest extent of micellar disruption under HPP was achieved (Huppertz et al., 2006c; Orlien et al., 2006). While the initial phase of HPP treatment results in disruption of casein micelles, prolonged holding of the samples under pressure can result in partial reversal in light transmission effects, indicating some reformation of particles from the micellar fragments (Huppertz et al., 2006c; Orlien et al., 2006). This was observed primarily at pressures in the range 200\u2013300 MPa, but was not observed at >350 MPa, where solubilization of MCP and micellar disruption were nearly complete. Furthermore, this process was more extensive when HPP was performed at a higher temperature (Orlien et al., 2006) and at lower pH (Huppertz & De Kruif, 2006). This process was not influenced by the presence of whey proteins (Huppertz & De Kruif, 2007a,b), suggesting that it is primarily driven by casein\u2013casein interactions.\n\nWhen the pressure is released, a subsequent reduction in light transmission is observed, indicating further reformation of casein particles. This reduction in light transmission coincided with the reversal in the solubility of MCP upon pressure release (Hubbard et al., 2002), to the extent that the distribution of calcium and inorganic phosphate between the micellar and serum phase of milk was identical to that of untreated samples (Regnault et al., 2006). However, when treatment was carried out at pressures >300 MPa, initial light transmission values were not restored, indicating that casein micelles were not reformed in their native state (Keenan et al., 2001; Huppertz et al., 2006c; ). Another factor to take into consideration is the pressure-release rate, which effectively controls the rate of reformation of casein micelles upon pressure release (Merel-Rausch et al., 2006).\n\nAs a result of the above-mentioned changes, the properties of casein micelles in high-pressure-treated milk can differ strongly from those in untreated milk, for example, in terms of particle size, turbidity and distribution. It has been reported that HPP treatment of milk for up to 10 min at 250 MPa causes a reduction in particle size, whereas treatment for >15 min at this pressure causes increases in particle size (Huppertz et al., 2004a). Treatment at pressures \u2264200 MPa showed little effect on size of casein micelle and turbidity, whereas in milk treated at >300 MPa, particle size and turbidity were reduced considerably (Gaucheron et al., 1997; Huppertz et al., 2004a,b; Regnault et al., 2004; Anema et al., 2005b; Anema, 2008). Such HP-induced changes were also dependent on pressurization temperature (Huppertz et al., 2004a; Anema et al., 2005b), pH (Huppertz et al., 2004a) and the solids content of the milk (Anema, 2008). However, such changes were independent of treatment time.\n\nThe increase in the particle size was accompanied by reductions in turbidity and light-scattering intensity suggesting that increase in average particle size was most likely due to the presence of a small proportion of large particles, as was also apparent from the electron micrographs of Knudsen and Skibsted (2009). The increase in particle size at 250 MPa was reversible on subsequent storage of the milk, with the greater reversibility observed upon storage at 20 \u00b0C than at 5 \u00b0C. In contrast, HPP-induced reductions in particle size were largely irreversible on subsequent storage (Huppertz et al., 2004a).\n\n### Effects of HPP on Caseinates\n\nCompared to milk, fewer reports are available on HPP-induced changes in caseinates. High-pressure treatment of caseinate suspensions at 200 MPa caused strong reductions in the turbidity (Lee et al., 1996; Anema et al., 1997). The reduction in turbidity was more extensive with increase in pH or when 5 or 10 mM CaCl2 was added, although the pH at which large reductions in turbidity occurred shifted to a higher pH with increasing amounts of added CaCl2. Unlike milk, the reductions in the turbidity of calcium caseinate suspensions were largely irreversible upon storage and also occurred when whey proteins were present in the calcium caseinate suspension (Lee et al., 1996; Anema et al., 1997).\n\n## Effects of high pressure on interactions of milk proteins involving whey proteins\n\nProteins are composed of amino acids, and the particular sequence of amino acids in a protein determines its structure (primary, secondary, tertiary, and quaternary structures), conformation, and properties. The native three-dimensional structure of a protein is maintained by a variety of noncovalent interactions (such as hydrogen bonding, electrostatic, van der Waals, and hydrophobic interactions) between amino acid residues within the polypeptide chain and between residues and solvent molecules (Singh, 1995). Three-dimensional structure has a very important role to play in the stability and functional properties of a protein.\n\nAccording to Le Chatelier-Braun's principle, under pressure, reactions with a negative volume change are enhanced and reactions with a positive volume change are suppressed (Buchheim & Prokopek, 1992; Johnston, 1995; Balci & Wilbey, 1999). Applications of high pressure cause proteins to lose their native three-dimensional structure and lead to denaturation and aggregation of whey proteins and\/or their interactions with each other (whey protein \u2212 whey protein interactions) or with the caseins (casein \u2212 whey protein interactions).\n\nIn contrast to heat treatments, where covalent bonds and noncovalent bonds are affected, it has been reported that HPP at room temperature (\u224820 \u00b0C) disrupts only relatively weak bonding such as intramolecular hydrophobic and electrostatic interactions (Balny & Masson, 1993; Silva & Weber, 1993), whereas hydrogen bonds are relatively insensitive to pressure, suggesting that high pressure affects the tertiary (three-dimensional configuration held together mainly by hydrophobic and ionic interactions) and quaternary (the spatial arrangement by noncovalent interactions into a multimeric protein) structures of globular proteins and has little effect on their secondary structure. There are views that covalent bonds are largely insensitive to pressure treatment at relatively low temperature (Hayakawa et al., 1996), which means that the primary protein structure (the amino acid sequence) will remain intact. This partly explains why HPP has been reported to have slightly different effects on protein structure compared with heat treatments.\n\nThe effects of high pressure on the denaturation, aggregation, and interactions of whey proteins have been studied extensively under various conditions and in different systems such as milk (e.g., Felipe et al., 1997; Law et al., 1998; Arias et al., 2000; Scollard et al., 2000; Huppertz et al., 2002; 2004a,b; Anema et al., 2005a; Patel et al., 2005), whey protein concentrate (WPC) (e.g., Van Camp et al., 1997a,b) and whey protein isolate (WPI) (Hinrichs et al., 1996a,b; Michel et al., 2001), and using pure whey proteins (Dumay et al. 1994; ; Funtenberger et al., 1995; Galazka et al., 1996a; Jegouic et al., 1996; Olsen et al., 1999). It has been reported that pressure-induced reactions of whey proteins lead to the unfolding of monomeric proteins, aggregation, and gelation (Van Camp & Huyghebaert, 1995a,b; Van Camp et al., 1997a,b; Balci & Wilbey, 1999; Tedford et al., 1999a,b; Huppertz et al., 2002; Fertsch et al., 2003), by reformation of intra- and intermolecular bonds within or between the molecules, linked by hydrophobic interactions and disulfide bridges (e.g., Cheftel, 1992; Masson, 1992; Hoover, 1993; Galazka et al., 1996a; Messens et al., 1997; Trujillo et al., 2002), depending on the type of protein, protein concentration, pH, ionic strength, applied pressure, pressurizing temperature and duration of the pressure treatment, and so on (Messens et al., 1997; Fertsch et al., 2003; Huppertz et al., 2004a). Also, different pressurizing temperatures may have different effects on the denaturation and aggregation of proteins (e.g., Huppertz et al., 2004a; Patel, 2007), because of the combined effects of pressure and temperature, which can have different effects on the interactions that maintain protein structures. Many studies related to the combined effects of pressure and temperature on milk proteins have been published (e.g., Tedford et al., 1999b; Huppertz et al., 2004a; Patel, 2007). However, the present review focuses mainly on the HPP- induced changes in milk proteins that occur at ambient temperature, unless specified otherwise.\n\nSelected reports on the effects of high pressure on individual whey protein fractions such as \u03b2-lactoglobulin (\u03b2-LG), \u03b1-lactalbumin (\u03b1-LA), bovine serum albumin (BSA), or their combinations in various systems are summarized in the following sections.\n\n### Denaturation and Aggregation of Pure Whey Protein Fractions in Model Systems\n\n#### \u03b2-Lactoglobulin (\u03b2-LG)\n\nWith increasing pressure, protein molecules undergo a sequence of conformational changes because of alterations in stabilizing interactions (Johnston et al., 1992). Different effects of high pressure on proteins have been observed when the samples are analyzed under high pressure (in situ analysis) and when analyzed after pressure release. Many reports have suggested that \u03b2-LG is the most sensitive of the major whey proteins to high pressure and that it dominates the pressure-induced denaturation, aggregation, and gelation of the whey protein system (Stapelfeldt et al., 1996; Van Camp et al., 1996; 1997a,b; Belloque et al., 2000; Patel et al. 2004; ; L\u00f3pez-Fandi\u00f1o, 2006a,b; Considine et al., 2005a,b; 2007b). Therefore, the majority of studies have concentrated on the effects of high pressure on \u03b2-LG in order to gain insight into the mechanisms of unfolding and aggregation that occur during pressurization or after pressure release. During the pressure release phase and after pressure treatment, new intermolecular interactions are formed, and the proteins may be newly structured (Fertsch et al., 2003). The majority of the reports included in the present review deal with the interactions of proteins after pressure release.\n\nAt relatively low pressures (50 MPa), analysis of thiol reactivity (M\u00f8ller et al., 1998; Stapelfeldt et al., 1999) and NMR studies (Tanaka & Kunugi, 1996) suggested the existence of a 'pre-denatured' state of \u03b2-LG. This pre-denatured form corresponded to a not-completely-unfolded structure, which preceded reversible denaturation. Belloque et al. (2000), using1H NMR, showed that the degree of deuterium exchange was very small at 100 MPa and that there were no variations in the resonances belonging to the strongly bonded 'core' of \u03b2-LG. These observations might suggest that the regions of \u03b2-LG affected by the pre-denatured state are likely to be different from the core, as the core was still tight and remained unaltered at 100 MPa. Whereas pressures ranging between 0 and 140 MPa did not affect \u03b2-sheets (Subirade et al., 1998), the reactivity of the free sulfhydryl group of \u03b2-LG increased with pressure up to 150 MPa (Tanaka et al., 1996a,b). These results suggested that, in spite of having a similar overall conformation, the architectures of \u03b2-LG before and after dynamic high pressure were stabilized by slightly different interactions (Subirade et al., 1998).\n\nThe pressure-induced denaturation of \u03b2-LG was believed to be a simple two-step mechanism until Jonas and Jonas (1994) reported pressure-induced pre-denaturation transitions and thus demonstrated that the pressure-induced denaturation of \u03b2-LG could be a stepwise process. A few years later, Stapelfeldt and Skibsted (1999) proposed a three-step denaturation process, and recently Considine et al. (2005b) published a three-stage model of the pressure denaturation of \u03b2-LG (Fig. 8.1; for a detailed description, also refer to Considine et al., 2007b). It has also been reported that addition of hydrophobic ligands such as all-trans-retinol, palmitic acid, SDS, and 8-anilino-1-naphthalenesulfonate (ANS) to \u03b2-LG solution before pressure treatment (see Fig. 8.1) affects the pathways of denaturation (Considine et al., 2005b, 2007b).\n\nFigure 8.1 Proposed three-stage model of the pressure denaturation of \u03b2-LG A and \u03b2-LG B with added ANS, retinol or SDS. Reproduced with the permission of [Considine et al. (2005b), copyright 2005 Journal of Agricultural and Food Chemistry.]\n\nThe overall major changes that occur to the structure of \u03b2-LG include monomerization of the dimeric state (Iametti et al., 1997), a decrease in \u03b1-helix and \u03b2-sheet content (Hayakawa et al., 1996; Panick et al., 1999), and irreversible changes involving the formation of intermolecular disulfide bonds (Funtenberger et al., 1997; Iametti et al., 1997; L\u00f3pez-Fandi\u00f1o et al., 1997; M\u00f8ller et al., 1998). In the pressure-induced mechanism proposed (Iametti et al., 1997), release of monomers represents one of the earliest events, whereas association of transiently modified monomers stabilizes the denatured forms of the protein. In addition, it has been reported that inter- and intramolecular reactions of sulfhydryl groups can occur (Tanaka et al., 1996a,b), leading to the formation of new disulfide bonds through sulfhydryl\u2013disulfide interchange reactions (Funtenberger et al., 1997) when samples are pressure treated at 450 MPa. At 800 MPa, most of the \u03b2-LG present in the system becomes involved in hydrophobic and disulfide-linked aggregates (Patel, 2007); this behavior is quite similar to the effects of heat treatment on \u03b2-LG (Havea et al. 1998; ; ).\n\nMoreover, it has been reported that factors, such as protein concentration (Dumay et al., 1994), pH, ionic strength, type, and molarity of the buffer used for preparation of the protein solutions (Funtenberger et al., 1995; Cheftel & Dumay, 1996), pressure intensity, pressurizing time and pressurizing temperature (Yang et al., 2001; Patel, 2007), and binding of hydrophobic ligands (Considine et al., 2005b, 2007a) and small molecules such as sucrose (Dumay et al., 1994) can affect the pressure-induced denaturation and aggregation of \u03b2-LG. It has been found that high-pressure-induced denaturation is partially reversible at lower (2.5%) protein concentration but that the denaturation is irreversible and that aggregation occurs at higher (5.0%) concentration (Dumay et al., 1994). The progressive formation of intermolecular disulfide-bonded dimers to hexamers or higher polymers of \u03b2-LG (pH 7.0) has been reported to be a function of the pressure level and of the buffer type and molarity (Funtenberger et al., 1995). It was suggested that high pressure induced the formation of intermolecular disulfide bonds, especially at neutral pH. When the combined effects of pressure, temperature, and time were evaluated, the pressure intensity was found to have major effects on the structure of \u03b2-LG (Aouzelleg et al., 2004), which is somewhat different from the finding that the combined effects of pressure intensity and temperature have the greatest effect on the denaturation of \u03b2-LG (Patel, 2007).\n\nDifferent aspects of the pressure-induced unfolding and aggregation of \u03b2-LG have also been reviewed in detail (see L\u00f3pez-Fandi\u00f1o, 2006b; Considine et al., 2007b), including the effects of high pressure on the functional properties of \u03b2-LG (L\u00f3pez-Fandi\u00f1o, 2006b).\n\n#### \u03b1-Lactalbumin (\u03b1-LA)\n\nSeveral studies reported that, compared with \u03b2-LG, \u03b1-LA is resistant to pressure denaturation (L\u00f3pez-Fandi\u00f1o et al., 1996; Tanaka & Kunugi, 1996; Scollard et al., 2000; Huppertz et al., 2004a; Patel et al. 2004; ). A comparison of pressure-induced changes of two major whey proteins, \u03b1-LA and \u03b2-LG, at neutral pH showed that the reversible unfolding to a molten globule state of \u03b1-LA begins at 200 MPa and loss of native structure becomes irreversible only beyond 400 MPa, as compared with 50 and 150 MPa, respectively, for \u03b2-LG (Tanaka & Kunugi, 1996; Tanaka et al., 1996c; Stapelfeldt & Skibsted, 1999; McGuffey et al., 2005). Various explanations have been provided for such differences in the stability of these two proteins, including the differences in their secondary structures (which lead to a higher effective hydrophobicity in \u03b2-LG) and\/or in the number of disulfide bonds (four in \u03b1-LA and two in \u03b2-LG) and also the Ca2+ binding sites (Tanaka & Kunugi, 1996). In fact, the binding of calcium is reported to remarkably stabilize \u03b1-LA to pressure, by a 200 MPa increase in the pressure value at which denaturation occurs (Dzwolak et al., 1999; Hosseini-nia et al., 2002). This observation is partly supported by the finding that there was a difference in \u0394 V values between apo- and holo-\u03b1-LA (Kobashigawa et al., 1999).\n\nFluorescent measurement of dansylated (prepared at atmospheric pressure) proteins, especially the energy transfer from the intrinsic tryptophan residue to the dansyl group, showed that the protein structure was deformed by pressure and that the energy transfer mechanisms of the two proteins were differently affected by high pressure, probably reflecting the degree of compactness of their pressure-perturbed structures (Tanaka et al., 1996c). It has been reported that \u03b1-LA is present in a molten globule state beyond 200 MPa and up to 400 MPa (Jonas, 2002) and that \u03b1-LA changes its conformation from the molten globule state to the unfolded state without volume changes (Kobashigawa et al., 1999). The volume of \u03b1-LA changes only at the transition from the native state to the molten globule state (Kobashigawa et al., 1999). Lasselle et al. (2003) reported that heat and high pressure had similar effects, supporting the view that the molten globule state is stabilized by hydrophobic interactions.\n\nIn samples of severely heated solutions of \u03b1-LA, dimers and larger aggregates of \u03b1-LA were formed (Lyster, 1970; Havea et al., 2001). However, no effects on monomeric \u03b1-LA were noticeable when pure \u03b1-LA was pressure treated at 800 MPa (Patel, 2007), except that some changes in the structure of \u03b1-LA were found when \u03b1-LA samples were pressure treated at 1000 MPa (Jegouic et al., 1996).\n\n#### Bovine Serum Albumin (BSA)\n\nBSA has been found to be quite resistant to pressure treatment up to 400 MPa (Hayakawa et al., 1992; L\u00f3pez-Fandi\u00f1o et al., 1996; Patel et al. 2004; ; ; L\u00f3pez-Fandi\u00f1o, 2006b). Several reports explaining the pressure stability of BSA are available. There are views that pressure-induced changes in the secondary structure of BSA are mainly reversible (Hosseini-nia et al., 2002) and that the greater stability of BSA is probably related to the fact that this molecule, through its 17 intramolecular disulfide bonds and the presence of several separate domains, has an extremely rigid structure (Hayakawa et al., 1992; L\u00f3pez-Fandi\u00f1o et al., 1996; L\u00f3pez-Fandi\u00f1o, 2006b). It is possible that the relatively high number of disulfide linkages in BSA may impede pressure-induced aggregation by protecting the hydrophobic core\/groups present inside the molecule from exposure to the solvent (Hosseini-nia et al., 2002).\n\nCeol\u00edn (2000) studied the hydrodynamic behavior of BSA, using a perturbed angular correlation technique, as a function of high pressure up to 410 MPa. It was reported that, at moderate pressure (\u2248150 MPa), the BSA molecule suffers structural modifications that produce an increase in the molecular volume and the rotational correlation time of the molecule. However, it may be possible that, unlike \u03b2-LG, the changes in the secondary structure of BSA are largely reversible (L\u00f3pez-Fandi\u00f1o, 2006b). However, processing at 800 MPa was reported to have a substantial effect on the secondary structure of BSA, and BSA was polymerized through disulfide bonding involving the free sulfhydryl residue (Galazka et al. 1996b; ; Patel, 2007).\n\n#### Immunoglobulins (Igs) and Lactoferrin (LF)\n\nBoth IgG and LF are more stable under pressure than under heat (Patel et al. 2005; ; Carroll et al., 2006; Palmano et al., 2006). This finding has great commercial significance for using HPP in the manufacture of nutritional products containing IgG and LF. It was reported that the pressure stability of IgG was better in colostrum solutions than in pure IgG solutions (Indyk et al., 2008), suggesting that some other colostrum milk components had protective effects on the denaturation of IgG. A study on the response of IgG to high pressure (200\u2013700 MPa) in the presence of the kosmotrope sucrose has been reported (Zhang et al., 1998). Brisson et al. (2007) studied the effects of iron saturation on the thermal aggregation of LF at neutral pH and found that iron saturation markedly increased the thermal stability of LF and decreased aggregation. Palmano et al. (2006) made a similar observation for pressure-treated iron-saturated LF solutions.\n\n#### Mixtures of \u03b1-LA and \u03b2-LG\n\n\u03b1-LA does not form aggregates when pressure treated alone at 800 MPa (Patel, 2007), but it forms high-molecular-weight disulfide-bonded oligomers at high pressure in the presence of thiol reducers (Jegouic et al., 1996). The oligomers of \u03b1-LA are stabilized mainly by non-native interchain disulfide bridges. As reported for heat-treated mixtures of \u03b1-LA and \u03b2-LG (e.g., Havea et al., 2001; Hong & Creamer, 2002), mixed aggregates of denatured \u03b1-LA and \u03b2-LG were readily formed in pressure-treated whey protein solutions (Jegouic et al., 1997). This observation supports the view that the presence of reactive thiol groups is a prerequisite for pressure-induced denaturation, aggregation, and oligomerization of \u03b1-LA (Jegouic et al., 1997; Grinberg & Haertl\u00e9, 2000).\n\nYet another possibility is that the interactions of \u03b1-LA and \u03b2-LG occur in a hydrophobic environment. However, it has been reported that, under pressure, the volume change of \u03b1-LA is much less (Lassalle et al., 2003) than that of \u03b2-LG (Royer, 2002) and therefore there is little possibility for such interactions to take place in a hydrophobic environment. This partly explains why \u03b1-LA retains most of its structure under high pressure. At very high pressure (e.g., 800 MPa), the irreversible denaturation of \u03b1-LA was much less than that of \u03b2-LG, which was assigned to the difference in the number of bonds stabilizing the structure of each protein (Hinrichs et al., 1996b; Messens et al., 1997).\n\n#### Mixtures of \u03b2-LG, \u03b1-LA, and BSA\n\nComparatively little information is available on the effects of high pressure on mixtures of \u03b2-LG, \u03b1-LA, and BSA in pure protein systems. Recently, Patel (2007) reported that, when mixtures of \u03b2-LG, \u03b1-LA, and BSA were pressure treated, a somewhat similar aggregation trend was observed to that reported for heat-treated mixtures (Gezimati et al. 1996; ; Havea et al., 2001). However, in the pressure-treated samples, it appeared that \u03b2-LG, being the most pressure-sensitive whey protein, formed early aggregates prior to the unfolding of either \u03b1-LA or BSA.\n\nPressure treatment of a ternary mixture of BSA, \u03b2-LG, and \u03b1-LA generated aggregates comprising a mixture of hydrophobically linked and disulfide-linked aggregates (Patel, 2007), whereas, when pure BSA solutions or combinations of BSA and other whey proteins (e.g., a binary mixture of BSA and \u03b1-LA) were pressure treated, almost all the aggregates were disulfide-linked and only a small proportion of the aggregates were hydrophobically linked (Patel, 2007). This may be due to structural differences in each of these proteins, and\/or the effects of high-pressure treatment on the structure of each of these proteins.\n\n### Commercial Whey Protein Solutions\n\nIn addition to the above studies using pure protein systems, several studies have been conducted on heat- and pressure-induced denaturation, aggregation, and gelation of whey proteins using commercial whey protein ingredients, such as WPC or WPI.\n\nPatel et al. (2004; 2005); reported that the sensitivities of each of the whey proteins to heat treatment (Ig > LF > BSA > \u03b2-LG B > \u03b2-LG A > \u03b1-LA) and pressure treatment (\u03b2-LG B > \u03b2-LG A > IgG > LF > BSA > \u03b1-LA) were considerably different. Also, high-pressure treatment generated a comparatively greater proportion of smaller aggregates than did heat treatment (Patel et al., 2004). These results confirmed the view that there are some similarities and some differences between the heat- and high-pressure-induced aggregation and gelation of whey proteins (Van Camp & Huyghebaert, 1995a,b; Van Camp et al., 1996; Dumay et al., 1998). It was concluded that the large internal hydrophobic cavity of \u03b2-LG may have been partially responsible for its sensitivity to high-pressure treatment. Conversely, \u03b1-LA responds to pressure by modifying its structure to be more molten globule and does not fully unfold at very high pressures (Patel et al., 2006).\n\nCharacterization of pressure-treated WPC solutions using 2D PAGE (Patel et al. 2004; ) suggested that HPP generated both hydrophobically bonded and disulfide-bonded aggregates consisting of all whey proteins, including \u03b2-LG, IgG, LF, BSA, and \u03b1-LA (Fig. 8.2), fairly similar to those reported by Havea et al. (1998) for heat-treated WPC solutions. Almost all of the \u03b2-LG was incorporated into the aggregates via disulfide bonds and to a lesser extent via hydrophobic interactions (Havea et al., 1998). However, when similar samples were pressure treated, the \u03b2-LG dimer was predominant (Patel et al. 2004; ). The detailed characterization and identification of the disulfide-linked aggregates formed in pressure-treated WPC solution are shown in Figure 8.3, which clearly shows that severe pressure treatment of WPC solutions generated disulfide-bonded dimer, trimer, tetramer, 1:1 complexes of \u03b2-LG:\u03b1-LA, and the like, as well as forming higher-molecular-weight disulfide-linked aggregates consisting of BSA, LF, Ig, \u03b2-LG, and \u03b1-LA.\n\nFigure 8.2 2D PAGE patterns of control and pressure-treated WPC solutions (12% w\/v). Native- and then non-reduced SDS-PAGE patterns of (A) the control and (B) a sample pressure treated for 20 min at 800 MPa. Similarly, SDS- and then reduced SDS-PAGE patterns of (C) the control and (D) a sample pressure treated for 20 min at 800 MPa. Gel strips marked as a\u2032 and a\u2033 represent the sample strip and the stained strip, respectively. X2 and X3 are dimer and trimer of \u03b2-LG, respectively, and X4, X5, and X6 are high-molecular-weight aggregates, which were caught up at the beginning of the resolving gel, caught up within the stacking gel and could not enter the gel, respectively. For a detailed description, refer to Patel et al. (2005). Reproduced with permission from [Patel et al. (2005), copyright 2005 Journal of Agricultural and Food Chemistry.]\n\nFigure 8.3 Detailed identification of the disulfide-linked aggregates and protein interactions on a 2D SDS- and then reduced SDS-PAGE pattern of a pressure-treated (800 MPa for 30 min) WPC solution.\n\nHinrichs et al. (1996b) determined orders of reaction of n = 2.0 for \u03b1-LA and n = 2.5 for \u03b2-LG in a WPI solution. These reaction rate constants were found to vary slightly at higher protein concentrations (Keim & Hinrichs, 2004). \u03b2-LG, \u03b1-LA, and BSA participate in pressure-induced aggregation and gelation through disulfide bonding. Moreover, it has been reported that the number of stabilizing disulfide bonds directly influences the texture properties of pressure-induced whey protein gels. At high-protein concentration (10%), intermolecular interactions and irreversible aggregation are favored (Wong & Heremans, 1988; Dumay et al., 1994). High-pressure treatment of concentrated (80\u2013160 g\/kg) \u03b2-LG isolate solutions (pH 7.0) prepared in water or various buffers induces \u03b2-LG gelation at low temperature (Zasypkin et al., 1996; Dumay et al., 1998). The decreasing solubility (in various dissociating media) of the protein constituents of pressure-induced gels as a function of storage time after pressure release suggests that the aggregation and gelation result from hydrophobic interactions and also disulfide bonds and that a progressive build-up of these interactions takes place after pressure release (Dumay et al., 1998).\n\n### Pressure-induced Gelation of Whey Proteins\n\nThe effects of protein concentration, intensity of pressure treatment, holding time, and pressurizing temperature on whey protein aggregation in WPC solutions have also been investigated (Patel, 2007). It was reported that the rate of aggregation of the whey proteins increased with an increase in the concentration of protein in the WPC solution and the pressurizing temperature. The combination of low-protein concentration, mild pressure treatment (200 MPa), and low pressurizing temperature (20 \u00b0C) led to minimal loss of native-like and SDS-monomeric \u03b2-LG, whereas the combination of high-protein concentration, severe pressure treatment (600 MPa), and higher pressurizing temperature (40 \u00b0C and higher) led to significant loss of both native-like and SDS-monomeric \u03b2-LG. The sensitivity of the pressure-resistant whey proteins, such as \u03b1-LA and BSA, to aggregation was significantly increased at pressurizing temperatures of 40 \u00b0C and higher. Self-supporting gels were formed when 8 or 12% (w\/v) WPC solutions were pressure treated at 600\u2013800 MPa and 20 \u00b0C. At protein concentrations sufficiently high for gel formation, WPC was found to produce pressure-induced gels in the pressure range 200\u2013400 MPa (Van Camp & Huyghebaert, 1995a,b; Van Camp et al., 1996). Also, the WPC gels produced by high pressure (400 MPa for 30 min) at protein concentrations ranging from 110 g\/L up to 183 g\/L differed significantly from heat-induced protein gels (80 \u00b0C for 30 min) with respect to gel strength and appearance (Van Camp & Huyghebaert, 1995a,b).\n\nAs discussed earlier, significant differences in protein denaturation and aggregation induced by heat compared with high pressure have been demonstrated (Heremans et al., 1997; Patel et al. 2004; ). This might suggest that the gels produced from whey proteins by high-pressure treatment may have different properties from those made by heat treatment. For example, HPP generated gels that had a more porous structure and lower firmness (Van Camp & Huyghebaert, 1995b; Zasypkin et al., 1996; Dumay et al., 1998), and that were weaker, less elastic, and more exudative than heat-induced gels (Cheftel & Dumay, 1996; Dumay et al., 1998). In contrast to heat-induced gels, pressure-induced gels of \u03b2-LG underwent mechanical and protein solubility changes when stored at 4 \u00b0C following pressure release, clearly indicating a time-dependent strengthening of protein\u2013protein interactions, probably because the primary aggregates of \u03b2-LG further aggregated during storage through hydrophobic interactions and disulfide bonds (Dumay et al., 1998).\n\nA recent study on characterization of protein\u2013protein interactions during pressure-induced gel formation using combinations of techniques such as transmission electron microscopy (TEM), size exclusion chromatography (SEC), and 1D and 2D PAGE (Patel et al., 2006) clearly showed a time-dependent loss of native whey proteins and a corresponding increase in non-native proteins and protein aggregates of different sizes. Using 1D PAGE (native, SDS, and SDSR PAGE) and 2D PAGE (native: SDS and SDS:SDSR PAGE), these aggregates were shown to be cross-linked by intermolecular disulfide bonds and by noncovalent interactions (e.g., hydrophobic bonds). These aggregates altered the viscosity and opacity of the samples.\n\nVarious possible hypotheses in support of pressure-induced gel formation have been discussed (Patel et al., 2006). It was proposed that at 800 MPa, the formation of a \u03b2-LG disulfide-bonded network precedes the formation of disulfide bonds between \u03b1-LA or BSA and \u03b2-LG to form multiprotein aggregates, possibly because the disulfide bonds of \u03b1-LA and BSA are less exposed than those of \u03b2-LG either during or after pressure treatment. It may be possible that intermolecular disulfide bond formation occurs at high pressure and that hydrophobic association becomes important after the high-pressure treatment, that is, a novel pathway of whey protein gel formation using high pressure. It was postulated that \u03b2-LG plays a major role in the aggregation and gel formation of WPC under pressure (Van Camp et al., 1997a,b; Patel et al., 2006), which suggested that the major whey protein component in WPC primarily determines its functional behavior under high pressure. However, some additional studies will be needed in model systems to confirm this hypothesis, as well as to deduce the role of other whey proteins (i.e., \u03b1-LA, BSA, and Ig) in gel formation. Similar to heat-induced gelation of whey proteins, it has been reported that factors such as protein concentration, applied pressure, holding time, and pressurizing temperature (Van Camp & Huyghebaert, 1995a; Walkenstr\u00f6m & Hermansson, 1997), pHs (Van Camp & Huyghebaert, 1995b; Arias et al., 2000), and calcium contents (Van Camp et al., 1997b) affect the aggregation behavior, pressure-induced functionality such as gel formation, and physical, rheological, and microstructural properties of whey proteins. Protein\u2013protein interactions are favored near the isoelectric point of the whey proteins, and neutral and alkaline pHs stimulate the formation of intermolecular disulfide bonds (Van Camp & Huyghebaert, 1995b). Pressure-induced \u03b2-LG denaturation increases considerably at alkaline pH and decreases at acidic pH (Arias et al., 2000). Longer pressure holding times improve the strength of the gel network, stimulating the formation of more intensive intermolecular interactions (Van Camp & Huyghebaert, 1995a). Further, it has been reported that the role of calcium in the aggregation and gelation of whey proteins under pressure may be explained in a similar manner to the heat-induced effects on whey proteins (Mulvihill & Kinsella, 1988; Kinsella & Whitehead, 1989; Van Camp et al., 1997b). A combination of pressure and higher pressurizing temperature (up to 70 \u00b0C) has been recommended for inactivating microbial spores, and therefore it is important to determine its effects on the proteins in food systems.\n\n### HPP-induced Changes in Milk\n\nStudies on the effects of high pressure on milk can be broadly grouped into several topics, including the effects of high pressure on casein micelle size and its dissociation, changes in the appearance of pressure-treated milks, denaturation of whey proteins and their interaction with the casein micelles in milk, and effects of high pressure on milk from various species. Some of these topics have been reviewed recently (Huppertz et al., 2002; 2006a,b; L\u00f3pez-Fandi\u00f1o, 2006a,b; Considine et al., 2007b), including the effects of HPP on technological properties including rennet coagulation and cheese-making properties, and acid coagulation properties. The main focus of this section is on the main aspects of pressure-induced denaturation of whey proteins and their interactions with casein in the milk system.\n\n#### Denaturation of Whey Proteins in the Milk System\n\nConsiderable differences in the sensitivities of the different proteins to heat (LF > Ig > BSA > \u03b2-LG > \u03b1-LA) and pressure (\u03b2-LG> LF > Ig > BSA > \u03b1-LA) have been reported (Patel et al., 2006), showing that \u03b2-LG is the most pressure sensitive among all the whey proteins. About 70\u201380% denaturation of \u03b2-LG occurs at 400 MPa (L\u00f3pez-Fandi\u00f1o et al. 1996; L\u00f3pez-Fandi\u00f1o & Olano, 1998; Arias et al., 2000; Garc\u00eda-Risco et al., 2000; Scollard et al., 2000). Relatively little further denaturation of \u03b2-LG occurs at 400\u2013800 MPa (Scollard et al., 2000). Compared with \u03b2-LG, \u03b1-LA is stable to pressures up to about 400\u2013500 MPa in the milk environment at ambient temperature (Hinrichs et al., 1996a,b; L\u00f3pez-Fandi\u00f1o et al., 1996; Felipe et al., 1997; Gaucheron et al., 1997; L\u00f3pez-Fandi\u00f1o & Olano, 1998; Arias et al., 2000; Garc\u00eda-Risco et al., 2000; Needs et al., 2000; Scollard et al., 2000; Huppertz et al. 2002; 2004b).\n\nDifferences in the pressure stabilities of \u03b1-LA and \u03b2-LG may be linked to the more rigid molecular structure of the former (L\u00f3pez-Fandi\u00f1o et al., 1996; Gaucheron et al., 1997), caused probably by differences in the secondary structure and in the number of disulfide bonds and Ca2+ binding sites. The pressure resistance of \u03b1-LA is partially caused by the different numbers of intramolecular disulfide bonds in the two proteins (Hinrichs et al., 1996a,b; Gaucheron et al., 1997) or by the lack of a free sulfhydryl group in \u03b1-LA (L\u00f3pez-Fandi\u00f1o et al., 1996; Funtenberger et al., 1997). It has also been reported that the molecular structure of \u03b1-LA is more stable than that of \u03b2-LG, and that oligomerization takes place only if, during unfolding, free sulfhydryl groups from other molecules are available (Hinrichs et al., 1996b; L\u00f3pez-Fandi\u00f1o et al., 1996; Gaucheron et al., 1997; Jegouic et al., 1997). This difference in pressure sensitivity can also be explained by the types of bonds stabilizing the conformational structures of \u03b2-LG and \u03b1-LA (Hinrichs et al., 1996b; Messens et al., 1997).\n\nBSA has also been found to be resistant to pressures up to 400 MPa in raw milk (Hinrichs et al., 1996b; L\u00f3pez-Fandi\u00f1o et al., 1996) or 600 MPa (Hayakawa et al., 1992). The high stability of BSA can be explained by the fact that BSA carries one sulfhydryl group and 17 disulfide bonds. The energy received under pressure treatment is too small to disrupt all the disulfide bonds and to change the molecular structure of BSA. IgG in caprine milk (Felipe et al., 1997) and bovine milk (Carroll et al., 2006; Patel et al., 2006) has been reported to be more resistant to pressure denaturation than to heat denaturation.\n\nVarious studies have reported different extents of denaturation of \u03b2-LG following high-pressure treatment at 600 MPa of pasteurized milk (Needs et al., 2000) or reconstituted skim milk powder (Gaucheron et al., 1997). This finding may be attributed to the level of denaturation caused by treatments before pressurization, which may influence the amount of denaturation measured afterward. The pressure intensity and the holding time have also been reported to affect the level of denaturation of whey proteins in milk (L\u00f3pez-Fandi\u00f1o & Olano, 1998; Huppertz et al., 2004a; Anema et al., 2005b; Hinrichs & Rademacher, 2005). The reaction order of pressure- induced denaturation of \u03b2-LG is 2.5 (Hinrichs et al., 1996b), indicating that the denaturation process is concentration dependent and that a lower initial concentration of native \u03b2-LG should reduce the extent of denaturation of \u03b2-LG under pressure. Also, \u03b2-LG and \u03b1-LA are reported to be comparatively more pressure resistant in whey than in milk, which may be attributed to the absence of casein micelles and colloidal calcium phosphate in whey (Huppertz et al., 2004b).\n\n#### Interactions of Whey Proteins with Casein Micelles\n\nFor reasons of functionality, one of the major reactions of interest in the heat-treated and pressure-treated milk systems is the interaction between the denatured whey proteins and the casein micelles. Unlike the heat-treated milk system, comparatively little information is available on the interactions of casein and whey proteins in high-pressure treated milk systems.\n\nOn high-pressure treatment of milk at 300\u2013600 MPa, \u03b2-LG may form small aggregates (Felipe et al., 1997) or may interact with the casein micelles (Needs et al., 2000; Scollard et al., 2000; Huppertz et al. 2004c). It was reported that, when mixtures of \u03ba-CN and \u03b2-LG were pressure treated at 400 MPa, the presence of \u03b2-LG reduced the susceptibility of \u03ba-CN to subsequent hydrolysis by chymosin, indicating interactions between the proteins (L\u00f3pez-Fandi\u00f1o et al., 1997). SDS-PAGE analysis of pressure-treated and untreated milks or solutions containing \u03ba-CN or \u03b2-LG or both in the presence or absence of denaturing agents showed evidence of the formation of aggregates linked by intermolecular disulfide bonds (L\u00f3pez-Fandi\u00f1o et al., 1997). Interestingly, \u03b1s2-casein (\u03b1s2-CN) occurs at the same concentration as \u03ba-CN and has one disulfide bond, but it has not normally been reported to interact with \u03b2-LG in milk systems heated at 85\u221290 \u00b0C. In contrast, Patel et al. (2006) reported that the effects of heat treatment and high-pressure treatment on the interactions of the caseins and whey proteins in milk were significantly different by demonstrating the formation of disulfide-linked complexes involving \u03b1s2-CN, \u03ba-CN, and whey proteins in heat- and pressure-treated milks. The results have been explained using modified 2D SDS- and then reduced SDS-PAGE and by proposing the possible reactions of the caseins and whey proteins in heat- and pressure-treated milk (Figs 8.4 and 8.5). The virtual absence of \u03b1s2-CN from the heat-induced aggregates formed at 85\u201390 \u00b0C in milk, as reported in previous studies, might be because \u03b1s2-CN is not a surface component of the micelle and therefore its disulfide bond(s) are inaccessible to the denatured \u03b2-LG. On the other hand, \u03ba-CN is on the surface of the micelles, and its disulfide bond(s) could be readily accessible to a thiol group of \u03b2-LG. Moreover, it has been reported that large quantities of very large aggregates that cannot enter the gel are present to a greater extent in heat-treated milk than in pressure-treated milk (Fig. 8.4; Patel et al., 2006), indicating that the sizes of the aggregates are comparatively smaller in pressure-treated milks than in heat-treated milks. Such differences can be attributed to different effects of heat treatment and pressure treatment on the structure of the proteins, which may ultimately lead to different textures of the final products.\n\nFigure 8.4 2D SDS- and then reduced SDS-PAGE patterns of the control sample (A) and samples heat treated at 72 \u00b0C for 15 s (B) and 140 \u00b0C for 5 s (C). Similarly, 2D PAGE patterns of samples pressure treated at 200 MPa for 30 min (D) and 800 MPa for 30 min (E). Gel strips marked as a\u2032 and a\u2033 represent the sample strip and the stained strip, respectively, and X4, X5, and X6 are high-molecular-weight aggregates, which were caught up at the beginning of the resolving gel, caught up within the stacking gel, and could not enter the gel, respectively. For a detailed description, refer to Patel et al. (2006). Reproduced in part with the permission of [Patel et al. (2006), copyright 2006 Journal of Agricultural and Food Chemistry.]\n\nFigure 8.5 Pictorial representation of the likely effects of medium (\u2248250 MPa) and high (>600 MPa) pressure treatment at \u224822 \u00b0C. The casein micelle swells at \u2248250 MPa and the \u03b2-LG unfolds and aggregates via disulfide bonds. \u03b2-LG forms disulfide-bonded dimers at lower pressure and probably aggregates with \u03ba-CN, but does not form larger \u03b2-LG aggregates. The proportion of \u03b1-LA that is included in the aggregates is less than that of \u03b2-LG because it does not readily unfold. At pressures >600 MPa, \u03b1s2-CN becomes available for thiol interchange reactions, assisted by the permeation of water into the micelle and the dissolution of the calcium phosphate. Also the \u03b2-LG molecules can polymerize into larger aggregates than dimers. Redrawn with the permission of [Patel et al. (2006), copyright 2006 Journal of Agricultural and Food Chemistry; and Considine et al. (2007b), copyright Elsevier.]\n\nUpon HPP treatment of milk serum depleted of casein micelles, no sedimentable whey proteins were observed despite high levels of whey protein denaturation, indicating that sedimentable whey proteins in HPP-treated milk are mostly associated with the casein micelles (Huppertz et al, 2004a). The level of denatured \u03b2-LG associated with the casein micelles increased with increase in pressure intensity, treatment time, and pressurization temperature (Huppertz et al., 2004a; Zobrist et al., 2005; Anema 2008; ). No effects of \u03b2-LG or the solids content of milk were observed (Anema, 2008). However, the association of \u03b2-LG with casein micelles increased with increase in pH of the milk before HP treatment (Huppertz et al., 2004a; Anema 2010), whereas the addition of KIO3 to milk prior to HPP treatment resulted in a lower level of denatured \u03b2-LG associated with casein micelles in HPP-treated milk, probably because the formation of disulfide bridges through thiol\u2013thiol interactions rather than thiol\u2013disulfide interchange reactions is favored in the oxidizing environment (Zobrist et al., 2005). In contrast to \u03b2-LG, most denatured \u03b1-LA was found in the serum phase of HPP-treated milk.\n\nThe 2D-PAGE results of Patel et al, (2006) also showed that pressure treatment of milk at 200 MPa (Fig. 8.4D) caused \u03b2-LG to form disulfide-bonded dimers and incorporated \u03b2-LG into aggregates, probably disulfide bonded to \u03ba-CN, suggesting that preferential reaction occurred at this pressure. The other whey proteins appeared to be less affected at 200 MPa. In contrast, pressure treatment at 800 MPa incorporated \u03b2-LG and most of the minor whey proteins (including Ig and LF), as well as \u03ba-CN and much of the \u03b1s2-CN, into large aggregates (Fig. 8.4E). However, only a proportion of the \u03b1-LA was denatured or incorporated into the large aggregates. Relatively lower degree of \u03b1-LA reactivity at high pressures is probably related to the relative stability of this protein compared with \u03b2-LG, as discussed earlier, and is based on the unusual pressure-dependent behavior of \u03b1-LA (Kuwajima et al., 1990; Kobashigawa et al., 1999; Lasalle et al., 2003). At higher pressures (>400 MPa), the polymerization of \u03b2-LG becomes the norm, and pressure-induced \u03b2-LG aggregation becomes similar to heat-induced \u03b2-LG aggregation (Fig. 8.5). The \u03b2-LG in WPC or in milk is not significantly modified by the other components; that is, \u03b2-LG dominates the denaturation and aggregation pathway during pressure (>400 MPa) treatment, as it has been shown to dominate the reaction at high-temperature heat treatments.\n\nAll these results show that the differences between the stabilities of the proteins and the accessibilities of the disulfide bonds of the proteins at high temperature or pressure affect the formation pathways that result in differences among the compositions of resultant aggregation or interaction products (including their sizes) that ultimately may affect product functionalities.\n\n## Concluding remarks\n\nIn the present chapter, we reviewed the mechanisms and pathways of pressure-induced denaturation and aggregation of whey proteins and their interactions with casein in various systems. It has been reported that compared to heat treatment, HPP has many different effects on denaturation, aggregation, and interactions of milk proteins. Such differences can be attributed to different effects of heat treatment and pressure treatment on the structure of the proteins. In view of this consideration, HPP has potential to improve rennet, acid coagulation, and many other functional properties of milk and milk products. HPP can be applied to develop dairy and food products with novel texture and unique functional properties that are inaccessible via conventional methods.\n\n## Acknowledgment\n\nThe authors are grateful to Lawrence Creamer and Harjinder Singh.\n\n# References\n\nAnema SG .  Effect of milk solids concentration on whey protein denaturation, particle size changes and solubilization of casein in high-pressure-treated skim milk .  _International Dairy Journal_.  2008 ;18 : 228 \u2013 235 .\n\nAnema SG .  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Ames, Iowa :  Blackwell Publishing ;  2008 : Chapter 21 .\n\nPayens TJ ,  Heremans K .  Effect of pressure on the temperature-dependent association of \u03b2-casein .  _Biopolymers_.  1969 ;8 : 335 \u2013 345 .\n\nRegnault S ,  Thiebaud M ,  Dumay E ,  Cheftel JC .  Pressurisation of raw skim milk and of a dispersion of phosphocaseinate at 9 \u00b0C or 20 \u00b0C: Effects on casein micelle size distribution .  _International Dairy Journal_.  2004 ;14 : 55 \u2013 68 .\n\nRegnault S ,  Dumay E ,  Cheftel JC .  Pressurisation of raw skim milk and of a dispersion of phosphocaseinate at 9 \u00b0C or 20 \u00b0C: Effects on the distribution of minerals and proteins between colloidal and soluble phases .  _Journal of Dairy Research_.  2006 ;73 : 91 \u2013 100 .\n\nRastogi NK ,  Raghavarao KSMS ,  Balasubramaniam VM ,  Niranjan K ,  Knorr D .  Opportunities and challenges in high pressure processing of foods .  _Critical Reviews in Food Science and Nutrition_.  2007 ;47 : 69 \u2013 112 .\n\nRoyer CA .  Revisiting volume changes in pressure-induced protein unfolding .  _Biochimica et Biophysica Acta_.  2002 ;1595 : 201 \u2013 209 .\n\nSchmidt DG ,  Payens TAJ .  The evaluation of positive and negative contributions to the second virial coefficient of some milk proteins .  _Journal of Colloid and Interface Science_.  1972 ;39 : 655 \u2013 662 .\n\nSchokker EP ,  Singh H ,  Pinder DN ,  Norris GE ,  Creamer LK .  Characterization of intermediates formed during heat-induced aggregation of \u03b2-lactoglobulin AB at neutral pH .  _International Dairy Journal_.  1999 ;9 : 791 \u2013 800 .\n\nSchokker EP ,  Singh H ,  Creamer LK .  Heat-induced aggregation of \u03b2-lactoglobulin A and B with \u03b1-lactalbumin .  _International Dairy Journal_.  2000 ;10 : 843 \u2013 853 .\n\nScollard PG ,  Beresford TP ,  Needs EC ,  Murphy PM ,  Kelly AL .  Plasmin activity, \u03b2-lactoglobulin denaturation and proteolysis in high pressure treated milk .  _International Dairy Journal_.  2000 ;10 : 835 \u2013 841 .\n\nSilva JL ,  Weber G .  Pressure stability of proteins .  _Annual Review of Physical Chemistry_.  1993 ;44 : 89 \u2013 113 .\n\nSingh H .  Heat-induced changes in casein, including interactions with whey proteins .  In:  Fox PF ,  ed.  _Heat-induced Changes in Milk_ .  Brussels :  International Dairy Federation ;  1995 : 86 \u2013 104 .\n\nSingh H .  Heat stability of milk .  _International Journal of Dairy Technology_.  2004 ;57 : 111 \u2013 119 .\n\nSingh H ,  Havea P .  Thermal denaturation, aggregation and gelation of whey proteins .  In:  Fox PF ,  McSweeney PLH ,  eds.  _Advanced Dairy Chemistry, Volume 1, Proteins_ .  New York :  Kluwer Academic\/Plenum Publishers ;  2003 : 1257 \u2013 1283 .\n\nStapelfeldt H ,  Skibsted LH .  Pressure denaturation and aggregation of \u03b2-lactoglobulin studied by intrinsic fluorescence depolarization, Rayleigh scattering, radiationless energy transfer and hydrophobic fluoroprobing .  _Journal of Dairy Research_.  1999 ;66 : 545 \u2013 558 .\n\nStapelfeldt H ,  Petersen PH ,  Kristiansen KR ,  Qvist KB ,  Skibsted LH .  Effect of high hydrostatic pressure on the enzymic hydrolysis of \u03b2-lactoglobulin B by trypsin, thermolysin and pepsin .  _Journal of Dairy Research_.  1996 ;63 : 111 \u2013 118 .\n\nStapelfeldt H ,  Olsen CE ,  Skibsted LH .  Spectrofluorometric characterization of \u03b2-lactoglobulin B covalently labeled with 2-(4'-maleimidylanilino) naphthalene-6-sulfonate .  _Journal of Agricultural and Food Chemistry_.  1999 ;47 : 3986 \u2013 3990 .\n\nStippl VM ,  Delgado A ,  Becker TM .  Ionization equilibria at high pressure .  _European Food Research and Technology_.  2005 ;221 : 151 \u2013 156 .\n\nSubirade M ,  Loupil F ,  Allain A-F ,  Paquin P .  Effect of dynamic high pressure on the secondary structure of \u03b2-lactoglobulin and on its conformational properties as determined by Fourier transform infrared spectroscopy .  _International Dairy Journal_.  1998 ;8 : 135 \u2013 140 .\n\nTanaka N ,  Kunugi S .  Effect of pressure on the deuterium exchange reaction of \u03b1-lactalbumin and \u03b2-lactoglobulin .  _International Journal of Biological Macromolecules_.  1996 ;18 : 33 \u2013 39 .\n\nTanaka N ,  Koyasu A ,  Kobayashi I ,  Kunugi S .  Pressure-induced change in proteins studied through chemical modifications .  _International Journal of Biological Macromolecules_.  1996 ;18 : 275 \u2013 280 .\n\nTanaka N ,  Nakajima K ,  Kunugi S .  The pressure-induced structural change of bovine \u03b1-lactalbumin as studied by a fluorescence hydrophobic probe .  _International Journal of Peptide and Protein Research_.  1996 ;48 : 259 \u2013 264 .\n\nTanaka N ,  Tsurui Y ,  Kobayashi I ,  Kunugi S .  Modification of the single unpaired sulfhydryl group of \u03b2-lactoglobulin under high pressure and the role of intermolecular S-S exchange in the pressure denaturation [single SH of \u03b2-lactoglobulin and pressure denaturation] .  _International Journal of Biological Macromolecules_.  1996 ;19 : 63 \u2013 68 .\n\nTedford L-A ,  Smith D ,  Schaschke CJ .  High pressure processing effects on the molecular structure of ovalbumin, lysozyme and \u03b2-lactoglobulin .  _Food Research International_.  1999 ;32 : 101 \u2013 106 .\n\nTedford L-A ,  Kelly SM ,  Price NC ,  Schaschke CJ .  Interactive effects of pressure, temperature and time on molecular structure of \u03b2-lactoglobulin .  _Journal of Food Science_.  1999 ;64 : 396 \u2013 399 .\n\nTrujillo AJ ,  Capellas M ,  Saldo J ,  Gervilla R ,  Guamis B .  Applications of high-hydrostatic pressure on milk and dairy products: a review .  _Innovative Food Science and Emerging Technologies_.  2002 ;3 : 295 \u2013 307 .\n\nVan Camp J ,  Huyghebaert A .  A comparative rheological study of heat and high pressure induced whey protein gels .  _Food Chemistry_.  1995 ;54 : 357 \u2013 364 .\n\nVan Camp J ,  Huyghebaert A .  High pressure-induced gel formation of a whey protein and haemoglobin protein concentrate .  _Lebensmittel-Wissenschaft und -Technologie_.  1995 ;28 : 111 \u2013 117 .\n\nVan Camp J ,  Feys G ,  Huyghebaert A .  High pressure induced gel formation of haemoglobin and whey proteins at elevated temperatures .  _Lebensmittel-Wissenschaft und -Technologie_.  1996 ;29 : 49 \u2013 57 .\n\nVan Camp J ,  Messens W ,  Cl\u00e9ment J ,  Huyghebaert A .  Influence of pH and sodium chloride on the high pressure-induced gel formation of a whey protein concentrate .  _Food Chemistry_.  1997 ;60 : 417 \u2013 424 .\n\nVan Camp J ,  Messens W ,  Cl\u00e9ment J ,  Huyghebaert A .  Influence of pH and calcium chloride on the high-pressure-induced aggregation of a whey protein concentrate .  _Journal of Agricultural and Food Chemistry_.  1997 ;45 : 1600 \u2013 1607 .\n\nWalkenstr\u00f6m P ,  Hermansson A-M .  High-pressure treated mixed gels of gelatin and whey proteins .  _Food Hydrocolloids_.  1997 ;11 : 195 \u2013 208 .\n\nWong PTT ,  Heremans K .  Pressure effects on protein secondary structure and hydrogen deuterium exchange in chymotrypsinogen: A Fourier transform infrared spectroscopic study .  _Biochimica et Biophysica Acta_.  1988 ;956 : 1 \u2013 9 .\n\nYang J ,  Dunker AK ,  Powers JR ,  Clark S ,  Swanson BG .  \u03b2-Lactoglobulin molten globule induced by high pressure .  _Journal of Agricultural and Food Chemistry_.  2001 ;49 : 3236 \u2013 3243 .\n\nZasypkin DV ,  Dumay E ,  Cheftel JC .  Pressure- and heat-induced gelation of mixed \u03b2-lactoglobulin\/xanthan solutions .  _Food Hydrocolloids_.  1996 ;10 : 203 \u2013 211 .\n\nZhang H ,  Deligeersang ,  Guo J ,  Mu Z ,  Zhang Y ,  Zhu H .  Denaturation of bovine milk IgG at high pressure and its stabilization .  _Shipin Kexue (Beijing)_.  1998 ;19 : 10 \u2013 12 .\n\nZobrist MR ,  Huppertz T ,  Uniacke T ,  Fox PF ,  Kelly AL .  High pressure-induced changes in rennet-coagulation properties of bovine milk .  _International Dairy Journal_.  2005 ;15 : 655 \u2013 662 . \nChapter 9\n\n# The Whey Proteins in Milk: Thermal Denaturation, Physical Interactions, and Effects on the Functional Properties of Milk\n\nSkelte G. Anema Fonterra Research and Development Centre, Palmerston North, New Zealand\n\n## Abstract\n\nThis chapter reviews the literature on the denaturation of the whey proteins in milk, examines their interaction with other milk protein components, and provides some examples on the relationships between denaturation\/interaction reactions of the whey proteins and the functional behavior of the milk in selected applications. Early studies on whey protein denaturation in milk were aimed at developing methods to assess denaturation levels, and determining the relationships between the denaturation of the whey proteins and the functional behavior of milk products in bakery and other applications. Subsequent studies were directed toward modeling the whey protein denaturation processes through kinetic and thermodynamic evaluations, and determining the role of various milk components on these denaturation processes. Although the denaturation of whey proteins is critical in modifying the functional behavior of dairy products, it has become increasingly apparent that a measure of the denaturation level alone is not in itself a good predictor of functional performance. Therefore further studies have investigated the interactions of the denatured whey proteins with other milk protein components, in particular the interactions between the denatured whey proteins and the casein micelles (including identification of the specific disulfide bonds involved in complex formation between the denatured \u03b2-lactoglobulin and \u03ba-casein). A limited number of recent studies have indicated that manipulation of the interactions of the denatured whey proteins with the other milk protein components may provide a significant tool for modifying or controlling the functional performance of milk protein products in some applications.\n\nOutline\n\nIntroduction 270\n\nThe casein micelle 270\n\nThe heat treatment of milk 273\n\nWhey Protein Denaturation 274\n\nAssessment of the Denaturation of Whey Proteins in Milk 274\n\nKinetic Evaluation and Modeling of Whey Protein Denaturation 276\n\nInteractions between Denatured Whey Proteins and \u03ba-Casein\/Casein Micelles 279\n\nInteractions between Denatured Whey Proteins and \u03ba-Casein in Model Systems 279\n\nInteractions between Denatured Whey Proteins and \u03ba-Casein\/Casein Micelles in Milk Systems 280\n\nDetermination of the Specific Disulfide Bonds Formed between \u03ba-Casein and \u03b2-Lactoglobulin 290\n\nRelationships between denaturation\/interactions of the whey proteins in heated milk and the functional properties of milk 295\n\nExamples of the Relationships between Whey Protein Denaturation and the Functional Properties of Milk 295\n\nExamples of the Relationships between the Level of Interactions of Whey Proteins with \u03ba-Casein\/Casein Micelles and the Functional Properties of Milk 297\n\nAcid-induced Aggregation\/Gelation of Heated Milk 298\n\nChymosin-induced Aggregation\/Gelation of Heated Milk 306\n\nExamples of the Effect of Denaturing Whey Proteins Separately from Casein Micelles on the Functional Properties of Milk 310\n\nConclusion 311\n\n## Introduction\n\nMilk is produced in the mammary gland of female mammals and is intended for the feeding of the neonate from birth to weaning. Milk is a highly nutritious, readily digested food, rich in protein, minerals, and energy in an aqueous solution. It also provides the neonate with many other essential compounds such as protective agents, hormones, and growth factors. Milk is a highly perishable fluid and was intended by nature to be consumed soon after production. However, humans have used milk and dairy-derived foods to supplement the diet for centuries, and dairy products are still a major food source. Because of the commercial and nutritional significance of dairy products, manufacturing processes to preserve the food value of milk long after its initial production have been developed.\n\nOver the last century, modern dairy milk processing has been transformed from an art into a science. Traditional products such as cheeses and yogurts combine centuries-old knowledge with modern science, technology, and processing techniques. In contrast, more recently developed products (such as spray-dried milk products, milk protein concentrates, and whey protein concentrates) have been based on modern technologies of the time. The majority of the milk processed is of bovine (cow) origin; however, significant quantities of buffalo, goat, and sheep milk are also manufactured into dairy products (Fox, 2003).\n\nIn milk, the lactose, some of the mineral components, and the native whey proteins are in true molecular solution. However, the casein and most of the calcium and phosphate are found in large macromolecular assemblies called casein micelles. The colloidal suspension of casein micelles in milk serum is a remarkably stable food protein system. Milk can be subjected to high temperatures and pressures, high shear, and variations in concentration without appreciable damage to the casein micelle system. Even the extreme action of drying milk to a powder does not significantly alter the milk system, as milk powders can be reconstituted to produce liquid milks that have many properties similar to those of the milk from which they were derived (Kelly et al., 2003; Nieuwenhuijse & van Boekel, 2003; O'Connell & Fox, 2003; Singh & Newstead, 1992).\n\n## The casein micelle\n\nIn order to understand and rationalize any changes to the properties and stability of milk, it is necessary to have some knowledge of the casein micelle structure. Despite extensive research efforts, the detailed structure and assembly of the casein micelle have not been unequivocally established. Several models have been proposed over the years, and these models have been progressively updated or modified as more information on the casein micelle has become available (Fig. 9.1; Dalgleish, 2011; Holt, 1992; Holt & Horne, 1996; Horne, 1998; Schmidt, 1982; Walstra, 1990; ). Although there is some agreement on some aspects of the various models, there is still considerable debate over the detailed structure of the casein micelle, as evidenced from the numerous recent papers and reviews devoted to this subject\u2014see Chapter 6 (Dalgleish, 2011; Dalgleish & Corredig, 2012; de Kruif et al., 2012; Farrell et al., 2006; Qi 2007; ; Trejo et al., 2011).\n\nFigure 9.1 Recent models of the casein micelle. (a): Original 'hairy' submicelle model of the casein micelle. Adapted with permission from Walstra & Jenness (1984). Copyright (1984) John Wiley & Sons. (b): modified hairy submicelle model of the casein micelle. Adapted with permission from Walstra (1999). Copyright (1999) Elsevier. (c): Nanocluster model of the casein micelle. Original depiction used with permission from Holt (1992). Copyright (1992) Elsevier. (d): The dual-binding model of the casein micelle. Adapted with permission from Horne (1998). Copyright (1998) Elsevier. (e) Dalgleish model of the casein micelle. Reproduced with permission from Dalgleish (2012). Copyright (2012) Annual Reviews.\n\nEvidence from early electron microscopy and light-scattering studies suggested that the casein micelle was assembled from smaller subunits and, as a consequence, submicelle models of the casein micelle structure were developed. In the later iterations of these submicelle models, the casein proteins were hydrophobically aggregated to form the submicelle units, and these submicelle units were linked by colloidal calcium phosphate (CCP) to form the casein micelle. The distribution of \u03ba-casein between submicelles was heterogeneous, and submicelles with high levels of \u03ba-casein were located at the micelle surface, whereas those with low levels of \u03ba-casein were in the interior, thus giving a surface location to \u03ba-casein that was consistent with experiments (Fig. 9.1a; Schmidt, 1982; Walstra, 1990).\n\nSubsequent experimental evidence did not support the existence of submicelles, and therefore the validity of the submicelle model of the casein micelles was questioned (Holt, 1992; Holt & Horne, 1996; Horne 1998; ; ;; Walstra, 1999). In particular, there was evidence showing that the CCP was uniformly distributed through the casein micelle, which precluded submicelles being linked by CCP to form the micelle. In addition, it was considered unlikely that there would be heterogeneous populations of casein submicelles with different levels of \u03ba-casein, or that assembly into casein micelles via calcium phosphate would occur only after the casein submicelles had been formed.\n\nIn addition, electron micrograph images of casein micelles using modern techniques did not display the internal substructure expected for casein submicelles, and it was considered that the appearance of submicelles in earlier micrographs were artifacts of the early preparation techniques for electron microscopy (Horne, 2006; McMahon & McManus, 1998). In an attempt to reconcile this new evidence, the submicelle model of the casein micelle was refined to change the role of CCP from that of linking the submicelles to a charge-neutralizing agent to allow for a uniform distribution of CCP, and the submicelles were now linked together via hydrophobic interactions (Fig. 9.1b; Walstra, 1999).\n\nHowever, new models for the casein micelle that do not rely on the formation of submicelles have been proposed (Dalgleish, 2011; de Kruif & Holt, 2003; Holt, 1992; Holt & Horne, 1996; Horne, 1998). Recent models include the nanocluster model (Fig. 9.1c; de Kruif & Holt, 2003; Holt, 1992; Holt & Horne, 1996) and the dual-binding model (Fig. 9.1d; Horne, 1998). There has been some convergence of these two models, as the combined results of detailed experiments on micelle structure using small-angle x-ray\/neutron scattering and static light scattering were only consistent with calculation if weak interactions (hydrophobic interactions, hydrogen bonding, ion pairing, etc.) were incorporated into the nanocluster model (de Kruif et al., 2012).\n\nThe dual-binding model proposed by Horne (1998) describes the types of interactions involved in the assembly of casein micelles and demonstrates that micelles with a consistent arrangement of casein proteins and CCP can be achieved. However, this dual-binding model does not give a detailed description of the surface or internal structure of the casein micelle. The most recent model of the casein micelle, proposed by Dalgleish (2011; Fig. 9.1e), has a relatively sparse hairy layer of \u03ba-casein on the surface, dense enough to stabilize against approach by other casein micelles or other large colloidal particles, but sufficiently diffuse to allow denatured whey proteins to interact with the para-\u03ba-casein region of \u03ba-casein or to allow \u03b2-casein to dissociate and reassociate with the micelles on cooling and subsequent warming. In addition, this model attempts to reconcile the structural arrangement of the interior of the micelles with its high hydration by giving a specific role to \u03b2-casein. It was proposed that some of the \u03b2-casein acts as a surfactant in stabilizing the hydrated internal channels of the micelle. This \u03b2-casein may be loosely bound to the micelles so that it dissociates and reassociates during cold and warm temperature cycling.\n\nDespite the emergence of new models without submicelles, there is still no universal agreement on casein micelle structure. There are proponents of submicelle models, and convincing arguments based on a structural biology perspective have been presented (Farrell et al., 2006; Qi, 2007). Even within the groups that support the models without submicelles, diverse views on the structural arrangements and relative importance of different types of bonding still exist (Dalgleish, 2011; Dalgleish & Corredig, 2012; de Kruif et al., 2012; Horne, 2006; McMahon & Oommen, 2008).\n\nAlthough various models of the casein micelles have been proposed, these have largely been derived from the same pool of research data and are therefore different depictions or interpretations of similar information. As a consequence, many of the salient features of the structure, assembly, and stability of the different models are similar (Dalgleish, 2011; Dalgleish & Corredig, 2012; de Kruif et al., 2012; Holt & Horne, 1996; Horne, 1998; Schmidt, 1982; Walstra 1990; ;). Hydrophobic interactions and CCP are important in maintaining micelle integrity. Therefore, micelle integrity can be modified or destroyed by disruption to hydrophobic interactions or by the dissolution of the CCP. In all recent models, \u03ba-casein has a preferential surface location, with the C-terminal region protruding from the surface layer as a flexible hair. These models of the casein micelles, with the surface layers of \u03ba-casein and an internal structure maintained by hydrophobic interactions and CCP, have been used to explain micelle stability and the destabilization by the enzymes in rennet, by acidification, or by the addition of alcohol (Holt & Horne, 1996; Horne 1998; ; Walstra, 1990). However, less studied and less well understood are the mechanisms responsible for the changes that occur to the casein micelles during the heating of milk, in particular the interactions with the denatured whey proteins, the heat-induced, pH-dependent dissociation of the casein (especially \u03ba-casein) from the micelles, and the eventual heat-induced coagulation of the casein micelles.\n\n## The heat treatment of milk\n\nThe effect of heat on the milk system is an important consideration in dairy chemistry, as a heat treatment is involved in the manufacture of almost all milk products. The heat treatment may range from thermization (about 65 \u00b0C for 15 s) to sterilization (about 120 \u00b0C for 10\u201320 min) or ultra-high-temperature (UHT) treatment (typically 138\u2013142 \u00b0C for several seconds). As the thermal history of milk influences the behavior of the milk in subsequent applications, the effects of heat on milk have been the subject of intensive, if somewhat intermittent, research and many reviews and books on the subject are available (IDF 1995; ; O'Connell & Fox, 2003; Singh & Creamer, 1992; Singh, 2004).\n\nWhen milk is heated, a number of competitive and often interdependent reactions occur; the importance of each reaction is determined by the heating conditions as well as by factors such as milk composition or concentration. When considering the protein components of milk, reactions of particular importance are whey protein denaturation, the interactions of denatured whey proteins with other proteins (including those of the casein micelles), and casein micelle dissociation. These three reaction processes can markedly modify the physicochemical properties of milk and may play a major role in determining the stability of milk and the functional performance of heated milk products.\n\n### Whey Protein Denaturation\n\nThe whey proteins are typical globular proteins with well-defined secondary and tertiary structures. The whey proteins (especially \u03b1-lactalbumin and \u03b2-lactoglobulin) retain their native conformations only within relatively limited temperature ranges. Exposing the whey proteins to extremes of temperature results in the denaturation and aggregation of the proteins; this process can be expressed using the simple reaction scheme as shown in Equation 9.1.\n\nFor protein species where the native protein is in the form of noncovalently linked oligomers (such as dimeric \u03b2-lactoglobulin), the first step in the denaturation process is the reversible dissociation of the oligomer into monomeric species (Equation 9.1a). The monomeric protein can then unfold, disrupting the native conformation (Equation 9.1b). In principle, this unfolding step is reversible; however, in complex mixtures such as milk, the unfolding process is accompanied by the exposure of reactive amino side-chain groups, which allows irreversible aggregation reactions to occur. The unfolded whey protein can undergo aggregation reactions with other (unfolded) whey proteins, with aggregates or with the casein micelles (represented by A in Equation 9.1c).\n\nAt a fundamental level, protein denaturation is often defined as any noncovalent change to the secondary or tertiary structure of the protein molecule (Equation 9.1b). From this denatured state, the protein can revert to its native state (refold) or interact with other components in the system (aggregate). Under this definition, \u03b1-lactalbumin is generally regarded as one of the most heat-labile whey proteins, whereas \u03b2-lactoglobulin is one of the most heat-stable whey proteins (Ruegg et al., 1977).\n\nHowever, for the dairy industry, it is the irreversible aggregation processes that largely determine the functional properties of dairy products. Hence, it is common practice to define whey protein denaturation as the formation of irreversibly denatured and aggregated whey proteins (Kelly et al., 2003; Sanderson, 1970b; Singh & Newstead, 1992); therefore this encompasses only the irreversible process shown in Equation 9.1c. Unless otherwise stated, the irreversible denaturation process is the definition used in this chapter. Using this definition for the denaturation of whey proteins in milk, the immunoglobulins are the most heat labile and \u03b1-lactalbumin is the most heat stable of the whey proteins, with \u03b2-lactoglobulin and bovine serum albumin being intermediate (Larson & Rolleri, 1955). In general, significant denaturation of the major whey proteins, \u03b1-lactalbumin and \u03b2-lactoglobulin, occurs only on heating milk at temperatures above about 70 \u00b0C.\n\n(PN)n\u21c6nPN\n\n (9.1a)\n\nPN\u21c6PU\n\n (9.1b)\n\nPU+A\u2192(P\u2212A)\n\n (9.1c)\n\n#### Assessment of the Denaturation of Whey Proteins in Milk\n\nA considerable amount of research has been directed toward determining and understanding the denaturation processes of the major whey proteins when milk is heated. In early studies, the casein and denatured whey proteins were precipitated by adjustment of the pH to the isoelectric point of the casein (about pH 4.6). The supernatant was analyzed from the unheated and heated milks for protein nitrogen, which gave estimates of initial native whey protein levels and levels following heat treatment (Rowland, 1933).\n\nA rapid method for determining the native whey protein levels was required for assessing milk powders for suitability in applications in the bakery industry (Harland & Ashworth, 1947), and also for categorizing milk powders based on the heat treatments received during the powder manufacturing process (Sanderson, 1970a,c). From these requirements, the whey protein nitrogen index (WPNI) method was developed.\n\nIn the WPNI method, the casein and the denatured whey proteins were precipitated and separated from the native whey proteins by saturating the milk with salt, and the supernatant containing the native whey proteins was analyzed for protein content. This was originally achieved by dilution and pH adjustment of the supernatant to produce a turbid solution, with the turbidity proportional to the level of native whey protein originally present (Harland & Ashworth, 1947; Kuramoto et al., 1959; Leighton, 1962). However, the WPNI method displayed considerable variability in the degree of turbidity developed for samples with similar levels of whey protein denaturation. To overcome this problem, Sanderson (1970b) combined a dye-binding method for determining the total protein content of milk with the original WPNI method, thus giving a more accurate and reliable method for determining the WPNI.\n\nThe original WPNI method or one of its variants is still the industry standard for determining the native whey protein levels of milk powder products. It is still widely used to classify milk powders according to the heat treatments received (Kelly et al., 2003; Singh & Newstead, 1992). However, a recent report indicates that Fourier transform near-infrared spectroscopy may have potential as a rapid method for determining the WPNI of milk powders in the dry state, eliminating the necessity of reconstitution, precipitation, and filtration (Patel et al., 2007).\n\nAlthough the WPNI method can give an estimate of the level of whey protein denaturation, research into the denaturation and interactions of the individual whey proteins requires more accurate separation and analysis procedures. There are numerous quantitative methods for separating and determining the level of the individual whey proteins in milk, and these methods can be used directly or adapted to determine the level of denaturation after defined heat treatments. The methods that have been used include polyacrylamide gel electrophoresis (PAGE; e.g., Anema & McKenna, 1996; Dannenberg & Kessler, 1988a; Hillier & Lyster, 1979; Kessler & Beyer, 1991), capillary electrophoresis (e.g., Butikofer et al., 2006; Fairise & Cayot, 1998), differential scanning calorimetry (e.g., Manji & Kakuda, 1987; Ruegg et al., 1977), high-performance liquid chromatography (HPLC; e.g., Kessler & Beyer, 1991), and various immune-based assays (e.g., Lyster, 1970). In recent years, miniaturized and\/or rapid automated methods have been adapted for determining whey protein denaturation and protein compositions, such as PAGE techniques on microfluidic chips (Anema, 2009; Wu et al., 2008) or optical biosensor-based assays (Dupont & Muller-Renaud, 2006; Indyk, 2009).\n\nIn general, good correlations have been observed when the various methods for determining whey protein denaturation have been compared (Anema & Lloyd, 1999; Anema, 2009; Indyk, 2009; Kessler & Beyer, 1991; Manji & Kakuda, 1987; Patel et al., 2007). These methods are more time consuming than the traditional WPNI methods and therefore cannot be used for routine analysis and classification of milk products. However, they have higher accuracy and reproducibility, and can be used to determine the denaturation behavior of the individual whey proteins. In addition, variations of the techniques or coupling to additional detection devices can provide further information on the interactions of the denatured whey proteins with other components in the milk (e.g., Donato & Guyomarc'h, 2009; Lowe et al., 2004; Patel et al., 2006; ).\n\n#### Kinetic Evaluation and Modeling of Whey Protein Denaturation\n\nEarly studies showed that the denaturation of whey proteins was a kinetic phenomenon, and was therefore dependent on both the temperature and duration of the heat treatment (Harland & Ashworth, 1945; Rowland, 1933). Although these early studies considered the whey protein components as a single entity, it was noted that the denaturation process did not follow a simple exponential law and was not a first-order (uni-molecular) process. In addition, there was a change in temperature dependence above about 80 \u00b0C, which was probably the first indication of the complex nature of the irreversible denaturation of the whey proteins in milk (Rowland, 1933).\n\nAlthough early studies on the effect of temperature and heating time on the denaturation of the individual whey proteins had been performed (e.g., Gough & Jenness, 1962; Harland & Ashworth, 1945), it was the kinetic study of Lyster (1970) over a wide temperature range (68\u2013155 \u00b0C) that conclusively demonstrated the complexity of the denaturation process of the whey proteins. Lyster (1970) found that the denaturation of \u03b1-lactalbumin appeared to follow first-order kinetics and that the denaturation of \u03b2-lactoglobulin was second-order. Arrhenius plots for both \u03b1-lactalbumin and \u03b2-lactoglobulin indicated that the irreversible denaturation reaction was not a simple process, as a change in temperature dependence was observed at about 80\u201390 \u00b0C for both \u03b1-lactalbumin and \u03b2-lactoglobulin (Fig. 9.2). The rate constants increased more rapidly with an increase in temperature in the low-temperature ranges than at higher temperatures.\n\nFigure 9.2 Effect of milk concentration on the Arrhenius plot for the thermal denaturation of \u03b2-lactoglobulin (a) and \u03b1-lactalbumin (b) over a 75\u2013100 \u00b0C temperature range.  , 9.6% total solids milk;  , 19.2% total solids milk;  , 28.8% total solids milk;  , 38.4% total solids milk. Part (a) was adapted with permission from Anema (2000). Copyright (2000) American Chemical Society. Part (b) was adapted with permission from Anema (2001). Copyright (2001) Blackwell Publishing.\n\nFurther studies confirmed the complexity of the denaturation process and provided relationships between compositional aspects and the rate of denaturation (Hillier & Lyster, 1979; Lyster, 1970; Manji & Kakuda, 1986). The kinetic and thermodynamic studies of Dannenberg and Kessler (1988a) provided insights into the possible mechanisms responsible for the complex temperature dependences of the denaturation of \u03b1-lactalbumin and \u03b2-lactoglobulin. Dannenberg and Kessler (1988a) found that, in milk, the denaturation of \u03b2-lactoglobulin had an order of about 1.5, which is now generally accepted, and that the denaturation of \u03b1-lactalbumin was pseudo first-order.\n\nFrom thermodynamic evaluations of the denaturation reactions of \u03b2-lactoglobulin and \u03b1-lactalbumin in the two temperature ranges (i.e., at temperatures above and below the marked change in temperature dependence for the denaturation reactions; Fig. 9.2), information on the possible rate-determining steps in the denaturation reactions was obtained. At temperatures below about 90 \u00b0C for \u03b2-lactoglobulin and 80 \u00b0C for \u03b1-lactalbumin, the high values for the activation energies and enthalpies indicated that a large number of bonds were disrupted, and the positive activation entropies indicated a lower state of order of the reaction products. These kinetic and thermodynamic parameters were interpreted as indicating that the unfolding (reversible denaturation) of the whey proteins was the rate-determining step in the lower temperature ranges.\n\nAt higher temperatures, above 80 \u00b0C for \u03b1-lactalbumin and above 90 \u00b0C for \u03b2-lactoglobulin, the considerably lower activation energies and enthalpies were typical of chemical reactions and the negative activation entropies indicated a higher state of order. These parameters suggested that chemical (aggregation) reactions were the rate-determining step in the higher temperature ranges. Subsequent studies have supported these interpretations in skim and whole milk under industrial processing conditions (Anema & McKenna, 1996; Oldfield et al., 1998a).\n\nThe denaturation reactions of both \u03b2-lactoglobulin and \u03b1-lactalbumin are enhanced when the pH of the milk is increased from the natural pH and are retarded when the pH is decreased (Law & Leaver, 2000). The denaturation of \u03b2-lactoglobulin was retarded when all components in the milk were concentrated, although the effect was less pronounced as the temperature was increased (Fig. 9.2a; Anema, 2000). In contrast, the denaturation of \u03b1-lactalbumin was hardly affected by milk concentration, with similar rates of denaturation at all milk concentrations regardless of the heating temperature (Fig. 9.2b; Anema, 2001).\n\nThe seemingly contrasting effects of milk concentration on the denaturation of \u03b1-lactalbumin and \u03b2-lactoglobulin have been explained through detailed studies on the effect of the concentrations of the individual components of milk on the denaturation reactions. Increasing the protein concentration of milk while maintaining essentially constant concentrations of nonprotein soluble components increased the rate of denaturation of both \u03b1-lactalbumin and \u03b2-lactoglobulin (Fig. 9.3; Anema et al., 2006; Law & Leaver, 1997), with a similar effect at all temperatures (Anema et al., 2006).\n\nFigure 9.3 Comparison of the effects of the concentrations of protein ( ,  ), nonprotein soluble components ( ,  ), lactose ( ,  ), and total solids ( ,  ) on the rate constants for the denaturation of \u03b2-lactoglobulin (a) and \u03b1-lactalbumin (b) at 80 \u00b0C (filled symbols) and 95 \u00b0C (open symbols). Reproduced with permission from Anema et al. (2006) Copyright (2006) American Chemical Society.\n\nIncreasing the concentration of nonprotein soluble components while maintaining constant protein concentrations retarded the denaturation of both \u03b2-lactoglobulin and \u03b1-lactalbumin. However, the effects on these two proteins were somewhat different (Fig. 9.3; Anema et al., 2006). For \u03b2-lactoglobulin, increasing the nonprotein soluble components caused a substantial retardation of denaturation in the lower temperature range, and this effect became less pronounced at higher temperatures. In contrast, the effect of increasing the nonprotein soluble components on \u03b1-lactalbumin denaturation was less pronounced than for \u03b2-lactoglobulin and was similar at all temperatures investigated (Fig. 9.3). The increase in lactose concentration, the major component of the nonprotein soluble components, explained much of the effect of increasing nonprotein soluble components. Clearly, however, other compositional factors such as pH and ionic components also have an effect (Fig. 9.3; Anema et al., 2006).\n\nFrom these results, it was possible to explain the effects of milk concentration on the denaturation of \u03b2-lactoglobulin and \u03b1-lactalbumin. For \u03b1-lactalbumin, on increasing the total solids concentration of the milk (both protein and nonprotein soluble components), the retardation of the reaction rate by increasing the nonprotein soluble components concentration was almost exactly offset by the increase in the denaturation rate on increasing the protein concentration. As this effect was similar at all temperatures, increasing total solids appeared to have no effect on the rate of denaturation of \u03b1-lactalbumin (Figs 9.2b and 9.3b ; Anema, 2001; ).\n\nFor \u03b2-lactoglobulin, the retardation in the rate of denaturation on increasing the concentration of the nonprotein soluble components was not completely offset by the increasing rate of denaturation on increasing the protein concentration. Therefore \u03b2-lactoglobulin denaturation was retarded by increasing the total solids concentration of the milk. However, the nonprotein soluble components were less effective in retarding the denaturation of \u03b2-lactoglobulin at higher temperatures and, as a consequence, the increase in total solids concentration appeared to have a smaller effect on the denaturation of \u03b2-lactoglobulin at the higher temperatures, and particularly above about 90 \u00b0C (Figs 9.2a and 9.3a ; Anema, 2000; Anema et al., 2006). The effects of the nonprotein soluble components concentration or the lactose concentration on the denaturation of \u03b2-lactoglobulin and \u03b1-lactalbumin have been discussed in terms of the preferential hydration theory (Anema, 2000; Anema et al., 2006).\n\n### Interactions between Denatured Whey Proteins and \u03ba-Casein\/Casein Micelles\n\nAn understanding of the denaturation reactions of the whey proteins provides information on the initial steps of a complex series of aggregation reactions that can occur when milk is heated. This aggregation process can involve other milk protein components and may involve numerous reaction pathways or interaction processes. Although the reactions of the denatured whey proteins with other milk protein components are important, these types of reactions are considerably more difficult to measure than the irreversible denaturation processes, particularly in a complex mixture of components such as is found in (skim) milk.\n\n#### Interactions between Denatured Whey Proteins and \u03ba-Casein in Model Systems\n\nOne of the major reactions of interest is the interaction between the denatured whey proteins and the casein micelles, particularly interactions of denatured \u03b2-lactoglobulin with \u03ba-casein at the micelle surface. Early studies on model systems indicated that there was an interaction between \u03b2-lactoglobulin and \u03ba-casein when these components were heated together (Long et al., 1963; Sawyer et al., 1963; Zittle et al., 1962). These conclusions were drawn from electrophoretic studies, which showed that the discrete bands assigned to \u03ba-casein and \u03b2-lactoglobulin observed in unheated solutions produced species of intermediate mobility when the solutions were heated together. Sedimentation velocity experiments also confirmed complex formation, as the \u03b2-lactoglobulin\u2013 \u03ba-casein complex formed on heating had markedly higher sedimentation coefficients than did the individual proteins when heated separately (Zittle et al., 1962).\n\nOnce interactions between \u03ba-casein and denatured \u03b2-lactoglobulin had been confirmed, subsequent investigations in heated model systems were aimed at determining the types of bonds involved in complex formation, the stoichiometry of the complexes formed, and the involvement of other whey proteins (particularly \u03b1-lactalbumin). It was shown that reducing agents dissociated the heat-induced complexes and that thiol-blocking agents prevented the formation of the complexes (Sawyer et al., 1963). These results supported earlier suggestions that the free thiol group of \u03b2-lactoglobulin was involved in the interactions (Trautman & Swanson, 1958; Zittle et al., 1962), and it was suggested that intermolecular disulfide bonds were formed between \u03ba-casein and denatured \u03b2-lactoglobulin (Sawyer et al., 1963). This has been corroborated by numerous subsequent studies (e.g., Grindrod & Nickerson, 1967; Purkayastha et al., 1967; Sawyer, 1969; Tessier et al., 1969).\n\nSome studies indicated that the heat-induced self-aggregation of \u03b2-lactoglobulin was limited when \u03ba-casein was present, which suggested that \u03ba-casein formed complexes with intermediate species of aggregated \u03b2-lactoglobulin (McKenzie et al., 1971; Sawyer, 1969). In contrast, other studies indicated that the aggregation of \u03b2-lactoglobulin was not a prerequisite for interaction with \u03ba-casein (Euber & Brunner, 1982). The reason for these apparently conflicting observations may have been resolved through the detailed study of Cho et al. (2003) in which many of the possible pathways involved in the aggregation of \u03b2-lactoglobulin with \u03ba-casein in heated model systems were elucidated. Cho et al. (2003) proposed that, when mixtures of \u03b2-lactoglobulin and \u03ba-casein were heated, the free thiol of \u03b2-lactoglobulin was exposed and this initiated a series of thiol\u2013disulfide exchange reactions of \u03b2-lactoglobulin with other denatured \u03b2-lactoglobulin molecules or with \u03ba-casein. The products formed ranged from 1:1 \u03b2-lactoglobulin\u2013 \u03ba-casein complexes to large heterogeneous aggregates, and the product mix was dependent on the ratio of \u03ba-casein to \u03b2-lactoglobulin. The aggregate species were held together by either or both disulfide bonds and hydrophobic interactions.\n\nAlthough there have been some indications of interactions between \u03b1-lactalbumin and \u03ba-casein on heating (Doi et al., 1983; Shalabi & Wheelock, 1976), others have reported that interactions between these proteins do not occur (Baer et al., 1976; Elfagm & Wheelock, 1978). It is now generally believed that interactions between \u03b1-lactalbumin and \u03ba-casein will occur only if \u03b2-lactoglobulin (or another whey protein with a free thiol) is present during heating, and this may require the initial formation of a \u03b2-lactoglobulin\u2013 \u03b1-lactalbumin complex, which subsequently interacts with \u03ba-casein (Baer et al., 1976; Elfagm & Wheelock, 1978).\n\nThere is considerable evidence to show that disulfide bonds are involved in the aggregates formed between the denatured whey proteins and \u03ba-casein; however, there are reports suggesting that noncovalent bonding may be important in these interactions, particularly in the early stages of heating and at lower heating temperatures (Haque et al., 1987; Haque & Kinsella, 1988; Hill, 1989; Sawyer, 1969). Other studies have shown that, although a substantial part of the denatured whey proteins in heated milk are involved in disulfide-bonded aggregates, a significant proportion can be recovered as monomeric protein under dissociating but nonreducing conditions, indicating that noncovalent interactions are also involved (Anema, 2000; Oldfield et al., 1998b). As Cho et al. (2003) have suggested, it is likely that both hydrophobic and disulfide interactions are important in the early stages of aggregate formation, with the interaction mechanism dependent on the composition of the system and the conditions of heating.\n\n#### Interactions between Denatured Whey Proteins and \u03ba-Casein\/Casein Micelles in Milk Systems\n\nMost of the early studies examining the heat-induced interactions between denatured whey proteins and \u03ba-casein involved model systems using purified proteins in buffer solutions. Milk is considerably more complex, with numerous protein species that could potentially interact upon heating. A number of the milk proteins have free thiol groups and\/or disulfide bonds. Although \u03b2-lactoglobulin is the major whey protein component, denatured \u03b1-lactalbumin and bovine serum albumin can also be involved in thiol\u2013disulfide exchange reactions and therefore can be incorporated in the aggregated products. For the caseins, both \u03ba-casein and \u03b1S2-casein have disulfide bonds; therefore, both could participate in thiol\u2013disulfide exchange reactions with denatured \u03b2-lactoglobulin or other denatured thiol-bearing whey proteins. As a consequence of this complexity, there are numerous potential thiol\u2013disulfide interaction pathways, as well as noncovalent interactions, and therefore the separation and analysis of the reaction products can be difficult.\n\nStudies on the interactions between the proteins in heated milk suggest that, despite the complexity of the system, the reactions between \u03b2-lactoglobulin and \u03ba-casein may be similar to those occurring in the model systems. In early electrophoretic studies on heated milk, it was noted that the bands corresponding to \u03b2-lactoglobulin disappeared, along with a reduction in the intensity of the bands corresponding to casein. This was accompanied by the formation of bands corresponding to new (heterogeneous) components (Slatter & van Winkle, 1952; Tobias et al., 1952). When thiol-blocking agents were added, the band pattern was comparable with that of the original skim milk, indicating that thiol\u2013disulfide exchange reactions were involved in the interaction mechanisms (Trautman & Swanson, 1958). Subsequent studies confirmed that an interaction between denatured \u03b2-lactoglobulin and \u03ba-casein on the casein micelles occurred on heating milk although, as expected, the other denatured whey proteins were also involved in the interactions (Corredig & Dalgleish, 1996a,b, ; Elfagm & Wheelock, 1978; Noh et al., 1989a,b; Oldfield et al., 1998b; Smits & van Brouwershaven, 1980; Snoeren & van der Spek, 1977).\n\nUnlike \u03ba-casein, \u03b1S2-casein does not readily interact with denatured whey proteins when milk is heated, although some interactions in UHT milks have been reported (Patel et al., 2006; Snoeren & van der Spek, 1977). This low reactivity may be due to the location of \u03b1S2-casein in the interior of the casein micelles, which makes it less accessible for interaction, whereas \u03ba-casein is located at the casein micelle surface and is therefore more accessible for interaction (Horne, 1998; Walstra, 1990). Interestingly, in pressure-treated skim milk, disulfide-bonded aggregates between \u03b1S2-casein and the denatured whey proteins are observed, suggesting that the disulfide bonds of \u03b1S2-casein may become accessible to thiol groups of the denatured whey proteins when the casein micelle structure is disrupted under pressure (Patel et al., 2006).\n\nThe degree of interaction of the denatured whey proteins with the casein micelles is dependent on many variables, including the time, temperature, and rate of heating, the milk and individual protein concentrations, the milk pH, and the concentration of the milk salts (Anema & Li, 2003a; Corredig & Dalgleish, 1996a,b; Oldfield et al., 2000; Oldfield, et al., 2005; Smits & van Brouwershaven, 1980). For example, when the temperature of milk is gradually increased above 70 \u00b0C, as in indirect heating systems, most of the denatured \u03b2-lactoglobulin and \u03b1-lactalbumin associates with the casein micelles, presumably as disulfide-bonded complexes with \u03ba-casein at the micelle surface (Corredig & Dalgleish, 1996a; Smits & van Brouwershaven, 1980). In contrast, when milk is heated rapidly, as in direct heating systems, only about half of the denatured \u03b2-lactoglobulin and \u03b1-lactalbumin associates with the casein micelles, with the rest remaining in the milk serum (Corredig & Dalgleish, 1996b; Oldfield, et al., 1998b; Singh & Creamer, 1991a).\n\nCorredig and Dalgleish (1999) suggested that, on heating milk, \u03b1-lactalbumin and \u03b2-lactoglobulin initially aggregate in the serum phase at a ratio dependent on the initial individual whey protein concentrations. These complexes subsequently associate with \u03ba-casein at the casein micelle surface on prolonged heating. However, Oldfield et al. (1998b) proposed that, under rapid heating rates, \u03b2-lactoglobulin forms aggregates in the serum before interacting with the casein micelles and this limits the level of association with the casein micelles, whereas, at slower heating rates, monomers or smaller aggregates of \u03b2-lactoglobulin may interact with the micelles and this may allow higher association with the casein micelles.\n\nThe pH of the milk at heating is important in determining the level of interaction between the denatured whey proteins and the casein micelles. When milk is heated at high temperatures (about 140 \u00b0C), the heat coagulation time\/pH profiles of most milks show increasing heat stability with increasing pH to a maximum at about pH 6.7, followed by decreasing stability to a minimum at about pH 6.9, and increasing stability again as the pH is increased further (Rose, 1961). Considerable research has been undertaken over decades in an attempt to explain this unusual pH-dependent heat stability of milk, and numerous factors are known to influence the heat stability behavior. Many review papers on the heat stability of milk are available (IDF, 1995; O'Connell & Fox, 2003; Singh & Creamer, 1992; Singh, 2004).\n\nThe results from the studies on the heat stability of milk have influenced the direction of future research on the effects of heat on milk, and in particular the interactions between denatured whey proteins and \u03ba-casein\/casein micelles. Therefore, it is appropriate to briefly review aspects of the pH-dependence of heat stability that are relevant to understanding the interactions between denatured whey proteins and \u03ba-casein\/casein micelles.\n\nElectron microscopic studies showed that when milk was heated at high temperatures (90\u2013140 \u00b0C) for long periods (30 min) at pH below 6.7, the denatured whey proteins complexed on to the micelle surfaces as filamentous appendages. However, when the milk was heated at higher pH, the denatured whey proteins were found in the serum phase as aggregated complexes (Creamer et al., 1978; Creamer & Matheson, 1980). These were the first indications that the pH at heating the milk may influence the interactions between the denatured whey proteins and the casein micelles.\n\nKudo (1980) showed that the amount of nonsedimentable protein in milk heated at pH 6.5 was lower than that in unheated milk; however, the level of nonsedimentable protein increased with the pH at heating so that, above pH 6.7, the level was markedly higher than in the unheated milk and increased with increasing pH. Kudo (1980) concluded that the denatured whey proteins co-sedimented with the casein micelles at low pH (about pH 6.5), whereas most of the denatured whey proteins along with some casein (particularly \u03ba-casein) was released from the casein micelles at pH above 6.8. It was also proposed that the transition from whey-protein-coated casein micelles to protein-depleted forms with changing pH at heating could explain the pH-dependence of the heat stability of milk at high temperatures.\n\nSingh and Fox (1985a,b; ; 1987a,b,c), in a series of extensive studies, showed that the dissociation of \u03ba-casein-rich protein on heating was dependent on the pH at heating. At pH below about 6.8, little dissociation of micellar \u03ba-casein occurred, whereas at higher pH, particularly above pH 6.9, high levels of \u03ba-casein dissociated from the micelles, with the level increasing proportionally with increased pH. The whey proteins, particularly \u03b2-lactoglobulin, played an important role in the heat-induced pH-dependent dissociation of \u03ba-casein (Singh & Fox, 1987b,c), as did mineral components such as calcium and phosphate (Singh & Fox, 1987a). The results from these studies have been used to develop detailed mechanisms for the pH-dependent heat stability of milk and concentrated milk systems (O'Connell & Fox, 2003; Singh & Creamer, 1992; Singh, 2004).\n\nInitially, it was reported that the dissociation of \u03ba-casein from the casein micelles only occurred when milk at high pH (above about pH 6.8) was heated at high temperatures, particularly 90 \u00b0C or above (Singh & Fox, 1985b). However, subsequent studies demonstrated that, at these pH values, the dissociation of \u03ba-casein occurred as soon as the temperature was raised above ambient, with the level of dissociated \u03ba-casein increasing proportionally with temperature up to 90 \u00b0C. In these studies, the dissociation of \u03b1S-casein (\u03b1S1-casein and \u03b1S2-casein combined), and \u03b2-casein showed unusual temperature dependence. Increasing levels of these caseins dissociated as the temperature was increased up to about 70 \u00b0C, with the levels then decreasing again at higher temperatures (Fig. 9.4; Anema & Klostermeyer, 1997; Anema, 1998).\n\nFigure 9.4 Effect of temperature and pH on the level of protein in the supernatants obtained from 10% total solids reconstituted skim milk samples heated for 30 min: \u03ba-casein (a); \u03b1s-casein (b); \u03b2-casein (c).  , pH 6.3;  , pH 6.5;  , pH 6.7;  , pH 6.9;  , pH 7.1. Adapted with permission from Anema & Klostermeyer (1997). Copyright (1997) American Chemical Society.\n\nThe unusual temperature dependence of \u03b1S-casein and \u03b2-casein was a consequence of the whey proteins, particularly \u03b2-lactoglobulin. When whey-protein-depleted milk was heated, the levels of \u03b1S-casein and \u03b2-casein dissociating from the casein micelles increased with increasing temperature up to 90 \u00b0C. When compared with heating standard milk, this indicated that higher levels of \u03b1S-casein and \u03b2-casein dissociated from the micelles in the whey-protein-depleted milks at temperatures above about 70 \u00b0C (Anema & Li, 2000).\n\nIt was postulated that all the caseins dissociated from the micelles on heating. On subsequent cooling, the dissociated \u03ba-casein stabilized the dissociated \u03b1S-casein and \u03b2-casein as small serum-phase aggregates if the heating temperature was below about 70 \u00b0C. However, above about 70 \u00b0C, \u03ba-casein associated with denatured whey proteins. It was already known that the complex formed between \u03ba-casein and denatured \u03b2-lactoglobulin was less effective at stabilizing \u03b1S-casein and \u03b2-casein in the presence of calcium ions than uncomplexed \u03ba-casein (Zittle et al., 1962). Therefore, this interaction may have prevented \u03ba-casein from stabilizing the other caseins, and they either reassociated with the casein micelles or formed larger aggregates on subsequent cooling (Anema & Li, 2000).\n\nEarly studies on the effect of the pH at heating on the interaction of denatured whey proteins with the casein micelles tended to use relatively large pH steps. In a model milk system containing casein micelles and \u03b2-lactoglobulin, about 80% of the denatured \u03b2-lactoglobulin associated with the casein micelles when the milk was heated at pH 5.8 or pH 6.3, whereas only about 20% associated with the casein micelles at pH 6.8 or pH 7.1 (Smits & van Brouwershaven, 1980).\n\nThe studies on the heat-induced, pH-dependent dissociation of \u03ba-casein from the casein micelles showed that this dissociation was accompanied by increases in the levels of denatured whey proteins remaining in the serum (Singh & Creamer, 1991b). This finding was confirmed by Anema and Klostermeyer (1997) and Oldfield et al. (2000), who reported that 80\u201390% of the denatured whey proteins associated with the casein micelles when milk was heated at pH below 6.7, whereas only about 20% of the denatured whey proteins was associated with the casein micelles at pH above 6.8. Corredig and Dalgleish (1996a) measured the ratio of \u03b2-lactoglobulin or \u03b1-lactalbumin to \u03ba-casein in the colloidal phase obtained from heated milk adjusted to pH 5.8, 6.2, or 6.8. Although the denatured whey proteins interacted with the casein micelles at a faster rate at lower pH and at higher temperatures, the ratios of denatured whey proteins to \u03ba-casein on the casein micelles were not markedly different under the different heating conditions.\n\nFurther studies demonstrated the extreme importance of pH on the association of denatured whey proteins (\u03b1-lactalbumin and \u03b2-lactoglobulin) with the casein micelles when milk was heated above 70 \u00b0C, particularly at pH 6.7 or below, where differences in association behavior could be measured at pH differences as small as 0.05 pH units (Anema & Li, 2003a,b; Vasbinder & de Kruif, 2003). From these studies, it was shown that about 80% of the denatured whey protein associated with the casein micelles at pH 6.5 and that this level of association decreased linearly as the pH at heating was increased, so that only about 30% was associated at pH 6.7. At higher pH (above pH 6.7), very low levels of denatured whey proteins associated with the casein micelles on heating milk (Fig. 9.5).\n\nFigure 9.5 Level of whey proteins associated with the casein micelles\/nonsedimentable whey proteins in skim milk samples that were heated at 90 \u00b0C for various times. The pH values of the milk samples prior to heating were:  , pH 6.5;  , pH 6.55;  , pH 6.6;  , pH 6.65;  , pH 6.7;  , pH 6.9;  , pH 7.1. Adapted with permission from Anema et al. (2004a). Copyright (2004) American Chemical Society.\n\nAlthough the heat-induced pH-dependent dissociation of \u03ba-casein from the casein micelles could explain the low levels of denatured whey proteins interacting with the casein micelles at pH above 6.8, it had been reported that very little \u03ba-casein dissociated from the casein micelles at pH below 6.8 (Nieuwenhuijse et al., 1991; Singh & Fox, 1985b; Singh, 2004). Therefore, it was initially unknown why small shifts in pH between pH 6.5 and pH 6.7 affected the association of denatured whey proteins with the casein micelles when milk was heated. The level of \u03ba-casein in the serum phase was low; therefore, it was initially believed that \u03ba-casein was not involved in this partition of the whey proteins between the serum and colloidal phases (Anema & Li, 2003a; Oldfield et al., 1998b; Vasbinder & de Kruif, 2003).\n\nSubsequent studies, however, showed that the heat-induced dissociation of \u03ba-casein was pH dependent from pH 6.5 to pH 7.1, with a linear increase in serum-phase \u03ba-casein as the pH was increased throughout the pH range from 6.5 to 7.1 (Fig. 9.6a), and that the level of serum-phase \u03ba-casein was correlated with the level of serum-phase denatured whey protein (Fig. 9.6b; Anema, 2007). A similar pH-dependent dissociation of \u03ba-casein and formation of serum-phase denatured whey proteins was observed when concentrated milks were heated at different pH (Anema, 2008b). The differences in the level of dissociated \u03ba-casein between the earlier and later studies may be related to the centrifuging conditions, which may have masked the effects at the lower pH, especially under conditions where the particles are less hydrated and more readily deposited (Anema, 2007; Parker et al., 2005; Rodriguez del Angel & Dalgleish, 2006).\n\nFigure 9.6 (a) Effect of the pH at heating on the level of nonsedimentable \u03ba-casein in milk.  , serum-phase \u03ba-casein in unheated milk;  , serum-phase \u03ba-casein in milk heated at 90 \u00b0C for 20 min;  , serum-phase \u03ba-casein in milk heated at 90 \u00b0C for 25 min;  , serum-phase \u03ba-casein in milk heated at 90 \u00b0C for 30 min. (b) Relationship between the serum-phase denatured whey protein and the level of serum-phase \u03ba-casein for the heated milk samples.  , milk heated at 90 \u00b0C for 20 min;  , milk heated at 90 \u00b0C for 25 min;  , milk heated at 90 \u00b0C for 30 min. Adapted with permission from Anema (2007). Copyright (2007) American Chemical Society.\n\nAlthough the level of \u03ba-casein in the serum phase at pH below 6.7 was relatively low (less than about 30% of the total \u03ba-casein), the ratio of denatured whey protein to \u03ba-casein was high and relatively constant (about 2.5 whey proteins to each monomeric \u03ba-casein) for the serum-phase proteins at all pH. In contrast, the ratio of denatured whey protein to \u03ba-casein was only about 1:1 for the whey protein associated with the casein micelles at pH 6.5, and this decreased to about 0.5:1 at pH 7.1 (Anema, 2007). Intensive studies on the soluble whey protein\u2013 \u03ba-casein complexes formed when milk was heated at the natural pH also showed that \u03ba-casein was intimately involved in the serum-phase aggregates, and that a high ratio of denatured whey proteins to \u03ba-casein was observed (Guyomarc'h et al., 2003). Electron micrographs of the serum-phase whey protein\u2013 \u03ba-casein aggregates indicated that these particles were roughly spherical, with a relatively uniform size of about 20\u201350 nm (Parker et al., 2005; Rodriguez del Angel & Dalgleish, 2006).\n\nThere is still some debate over the sequence of events in the interaction reactions between the denatured whey proteins and \u03ba-casein. Some reports suggest that \u03ba-casein dissociates from the micelles early in the heating process and that the denatured whey proteins subsequently interact with the \u03ba-casein either in the serum phase or on the micelles, with a preferential serum-phase reaction (Anema & Li, 2000; Anema 2007; 2008a;). This proposal was supported by the observations that the dissociation of \u03ba-casein is a rapid process and that significant dissociation of \u03ba-casein can occur at temperatures below those where the denaturation of whey proteins occurs (Anema & Klostermeyer, 1997). In addition, significant dissociation of \u03ba-casein occurs in systems that have been depleted of whey proteins (Anema & Li, 2000). The higher ratio of denatured whey protein to \u03ba-casein for the serum phase regardless of the pH at heating or the level of dissociated \u03ba-casein may also suggest a preferential serum-phase reaction between the denatured whey proteins and \u03ba-casein (Anema, 2007).\n\nHowever, other reports suggest that, on heating milk, the denatured whey proteins first interact with the casein micelles and that the whey protein\u2013 \u03ba-casein complexes subsequently dissociate from the casein micelles (Donato & Dalgleish, 2006; Donato et al., 2007b; Donato & Guyomarc'h, 2009; Parker et al., 2005). This proposal was supported by the observation that the addition of sodium caseinate to milk did not increase the level of serum-phase complexes between the denatured whey proteins and \u03ba-casein, which was interpreted as indicating that the complexes between the denatured whey proteins and \u03ba-casein were formed on the casein micelle surface regardless of the pH at heating (Parker et al., 2005).\n\nMilks with added \u03ba-casein were analyzed by size exclusion chromatography (Donato et al., 2007b). The difference profiles between the sera from unheated and heated milk with added \u03ba-casein produced a negative peak in the region of the native whey proteins and a positive peak in the region of the whey protein\u2013 \u03ba-casein aggregates. As the difference spectra was the same as for milks without added \u03ba-casein, this was also interpreted as indicating that the added \u03ba-casein was not involved in the formation of the serum-phase aggregates, and therefore the denatured whey proteins only interacted with micelle bound \u03ba-casein, with the complexes subsequently dissociating from the casein micelles (Donato et al., 2007b; Donato & Guyomarc'h, 2009).\n\nThe partial hydrolysis of \u03ba-casein by chymosin prevented dissociation of unhydrolyzed \u03ba-casein from the casein micelles. This in turn prevented the formation of serum-phase whey protein\u2013 \u03ba-casein complexes and therefore increased the level of denatured whey proteins associated with the casein micelles (Renan et al., 2007). This observation was also used as evidence to support the initial interaction of denatured whey proteins with the casein micelles, with the subsequent dissociation of whey protein\u2013 \u03ba-casein complexes as it was suggested that the unhydrolyzed \u03ba-casein should still be able to dissociate and interact with serum-phase denatured whey proteins if this was the preferential reaction pathway (Renan et al., 2007). However, this proposal did not take into account the polymeric nature of \u03ba-casein (Holland et al., 2008), and therefore the partial hydrolysis of a \u03ba-casein would substantially increase the hydrophobicity of the polymer even when it contained some unhydrolyzed \u03ba-casein. This increased hydrophobicity may account for the reduced dissociation of \u03ba-casein from the micelles and the increased interaction of denatured whey proteins with the casein micelles (Donato & Guyomarc'h, 2009).\n\nIt was also suggested that two mechanisms occur depending on the pH at heating. This suggestion was based on observations that the protein composition of the serum phase appeared to vary markedly depending on whether the milk was heated at pH above or below the natural pH of the milk (Donato & Dalgleish, 2006). However, other studies did not display a marked difference in composition of serum-phase proteins with pH (Anema, 2007).\n\nA detailed study was conducted aimed specifically at elucidating the sequence of events occurring when denatured whey proteins interacted with \u03ba-casein in heated milks (Anema, 2008a). It was shown that \u03ba-casein could dissociate from the casein micelles at temperatures that were below those where the whey proteins denatured (Fig. 9.7a). When heated at temperatures at which the whey proteins could denature, it was found that \u03ba-casein dissociated from the casein micelles in the early stages of heating and before significant levels of whey proteins were denatured. In addition, the maximum level of serum-phase \u03ba-casein was obtained when less than half the whey proteins were denatured. Once this maximum level of \u03ba-casein was dissociated, any additional denatured whey proteins formed on prolonged heating were predominantly found in the serum phase (Fig. 9.7b), indicating a preferential interaction of the denatured whey proteins with the serum-phase whey protein\u2013 \u03ba-casein complexes.\n\nFigure 9.7 (a) Level of serum-phase whey protein (filled symbols) and \u03ba-casein (open symbols) in milk samples heated for 15 min at different temperatures. (b) Level of denatured (filled symbols) and micelle-bound (open symbols) whey proteins in milk samples heated at 90 \u00b0C for different times. (c) Level of serum-phase \u03ba-casein in milk samples heated at 90 \u00b0C for different times. The skim milks were adjusted to pH 6.5 ( ,  ), pH 6.7 ( ,  ), and pH 6.9 ( ,  ) before heating. Adapted with permission from Anema (2008a). Copyright (2008) Cambridge University Press.\n\nWhen \u03ba-casein was added to the milk prior to heating, the denatured whey proteins preferentially interacted with the added serum-phase \u03ba-casein, regardless of the pH at heating (Table 9.1). Taken together, these results provide unequivocal evidence that \u03ba-casein dissociation from the micelles can precede the interaction of denatured whey proteins with the \u03ba-casein, and that denatured whey proteins will preferentially interact with serum-phase \u03ba-casein (Anema, 2008a).\n\nTable 9.1\n\nSerum-phase denatured whey proteins from skim milk with different levels of added \u03ba-casein that were adjusted to pH 6.5, 6.7, and 6.9 before heating at 90 \u00b0C for 15 min.\n\nAdded \u03ba-casein (%) | pH at heating | Serum-phase denatured whey protein  \n(% of total)  \n---|---|---  \n0 | 6.5 | 35 \u00b1 1a  \n0.1 | 6.5 | 67 \u00b1 2b  \n0.2 | 6.5 | 84 \u00b1 3c  \n|  |   \n0 | 6.7 | 71 \u00b1 2a  \n0.1 | 6.7 | 78 \u00b1 3b  \n0.2 | 6.7 | 92 \u00b1 3c  \n|  |   \n0 | 6.9 | 86 \u00b1 3a  \n0.1 | 6.9 | 91 \u00b1 3a,b  \n0.2 | 6.9 | 95 \u00b1 3b\n\nThe numbers represent the average and standard deviation of triplicate measurements.\n\nData at a given pH with the same letters are not significantly different from each other at P <0.05.\n\nReproduced with permission from Anema (2008a). Copyright (2008) Cambridge University Press.\n\nThe pH-dependent changes in the association of the denatured whey proteins with the casein micelles, and the dissociation of \u03ba-casein from the micelles can have an effect on some of the physical properties of the milk. A marked increase in casein micelle size was observed when high levels of denatured whey protein were associated with the colloidal phase, as is observed on heating milk at pH 6.5. This change in size was less pronounced as the pH at heating the milk was increased to pH 6.7, and a decrease in casein micelle size was observed when significant levels of \u03ba-casein were dissociated from the colloidal phase, as is seen on heating milk at pH above 6.7 (Fig. 9.8a; Anema & Li, 2003a,b). Similar changes in viscosity (Fig. 9.8b) and turbidity (Fig. 9.8c) with the pH at heating were also observed (Anema et al., 2004c).\n\nFigure 9.8 Effects of the pH at heating on the changes in the size of casein micelles (a), the viscosity of the milk (b), and the turbidity of the milk (c). The milk samples were heated at 90 \u00b0C for various times and the pH values of the milk samples prior to heating were:  , pH 6.5;  , pH 6.55;  , pH 6.6;  , pH 6.65;  , pH 6.7;  , pH 6.9;  , pH 7.1. Some of the particle size and viscosity results were adapted with permission from Anema et al. (2004c). Copyright (2004) Elsevier.\n\nThe difficulty in interpreting these changes in size, viscosity, and turbidity is determining (1) whether the association of the denatured whey proteins with the casein micelles is directly responsible for the change in size\/volume of the casein micelles by increasing the diameters of the individual particles as the proteins interact, or (2) whether there is some associated phenomenon, such as aggregation of the casein micelles, that is related to the level of whey protein or \u03ba-casein that is in the serum phase or associated with the casein micelles. The strong relationship between the level of whey protein associating with the colloidal phase and the size\/volume of the casein micelles, the observation that the size change plateaus on prolonged heating, and the relationships between the protein composition of the micelles and size, viscosity, and turbidity seem to suggest that the size changes are a direct consequence of the distribution of protein between the colloidal and serum phases, rather than an associated aggregation reaction (Anema & Li, 2003a,b; Anema et al., 2004c).\n\n#### Determination of the Specific Disulfide Bonds Formed between \u03ba-Casein and \u03b2-Lactoglobulin\n\nAlthough many types of bonding may be involved in the early stages of interactions between the denatured whey proteins and \u03ba-casein, there is clear evidence that disulfide bonds are involved in complex formation when model systems and milk are heated. Recent studies have focused on determining the specific thiol groups of \u03ba-casein and, in particular, \u03b2-lactoglobulin that are involved in the disulfide bonding between these two protein species when they are heated in model systems or milk (Creamer et al., 2004; Henry et al., 2002; Livney & Dalgleish, 2004; Lowe et al., 2004). Understanding the specific disulfide bonds involved in the interaction process may provide useful insights into the mechanisms for the denaturation and subsequent aggregation reactions of the whey proteins in milk.\n\nNative \u03b2-lactoglobulin has two disulfide bonds and one free thiol group at Cys121 (Qin et al., 1999), whereas \u03ba-casein is found as a heterogeneous polymeric protein cross-linked in a random manner by disulfide bonding via the two Cys groups in the monomer protein (Holland et al., 2008; Rasmussen et al., 1999). It was believed that the formation of disulfide bonds between the denatured \u03b2-lactoglobulin and other milk proteins, including \u03ba-casein, during heating first involved the dissociation of the \u03b2-lactoglobulin dimer to monomer species, followed by the unfolding of the native structure, exposing the buried side groups including the free thiol at Cys121. The exposure of this free Cys121 then initiated a series of intermolecular thiol\u2013disulfide exchange reactions with other denatured whey proteins and with \u03ba-casein on the casein micelle surface (Creamer et al., 2004; Hoffmann & van Mil, 1997; Iametti et al., 1996; Snoeren & van der Spek, 1977; Vasbinder & de Kruif, 2003; Verheul et al., 1998).\n\nStudies on pure \u03b2-lactoglobulin, however, indicated that in the early stages of the denaturation process, non-native monomeric \u03b2-lactoglobulin species are formed, which may be intermediates in the intermolecular aggregation processes. These non-native monomeric species are stable upon subsequent cooling and can be separated by alkaline PAGE techniques (Hong & Creamer, 2002; Manderson et al., 1998) or by gel permeation chromatography (Croguennec et al., 2003; ; Iametti et al., 1996). It was hypothesized that the non-native monomers were formed from intramolecular thiol\u2013disulfide exchange reactions between the free Cys121 of \u03b2-lactoglobulin and the Cys106\u2013Cys119 and\/or Cys66\u2013Cys160 disulfide bonds within the same \u03b2-lactoglobulin monomer (Croguennec et al. 2003; ; Hong & Creamer, 2002; Iametti et al., 1996; Manderson et al., 1998).\n\nWith the advent of sensitive mass spectrometric techniques, the identification of the Cys residues involved in disulfide-bonded protein species was possible. The strategy for identifying specific disulfide bonds involves a number of steps (Gillece Castro & Stults, 1996; Gorman et al., 2002; Kehoe et al., 2007; Lowe et al., 2004). For the sample under analysis, the protein species are first hydrolyzed under conditions where no further thiol\u2013disulfide bond exchange reactions are likely to occur. The peptides formed from this hydrolysis are separated, usually by reverse-phase HPLC, and the mass of individual peptides is determined by mass spectrometry (MS). The identification of individual peptides can be achieved by comparing the measured masses with those of expected peptides for the hydrolysis of the protein under study. For the disulfide-bonded peptides, usually other criteria also need to be satisfied, such as the peptides being present in nonreduced hydrolysates but absent in the reduced system.\n\nFurther confirmation can be gained by the use of tandem MS, where single molecular ions are isolated and analyzed in the first mass analyzer and then passed into a collision cell where fragmentation of the peptide is induced by collision with an inert gas collision-induced dissociation (CID)] and the fragments are characterized in the second mass analyzer. From the mass of the fragments, the sequence of the amino acids in the peptides can be determined, providing conclusive characterization of the peptides ([Gorman et al., 2002; Kehoe et al., 2007; Lowe et al., 2004).\n\nUsing these types of mass spectrometric techniques, researchers found a stable non-native \u03b2-lactoglobulin monomer with a free thiol group at position Cys119 rather than the natural position of Cys121 in heated \u03b2-lactoglobulin solutions (Croguennec et al., 2003). This finding confirmed that intramolecular thiol\u2013disulfide exchange within monomeric \u03b2-lactoglobulin could occur. It was suggested that this \u03b2-lactoglobulin with the free thiol at Cys119 may be the activated monomer that was proposed as the starting point for intermolecular aggregation reactions leading to large polymers. It was equally possible, however, that unfolded protein with a free thiol at the natural position of Cys121 was that activated monomer (Creamer et al., 2004; Croguennec et al. 2003; ;).\n\nA more recent investigation on the disulfide bonding patterns in heated \u03b2-lactoglobulin found that a significant proportion of Cys160 was in the reduced form after heating \u03b2\\- lactoglobulin in solution (Creamer et al., 2004; Kehoe et al., 2007), indicating that the Cys66\u2013Cys160 disulfide bond was broken during the early stages of heating. This may occur concurrently with the interchange of the free thiol from Cys121 to Cys119. It was suggested that a monomeric \u03b2-lactoglobulin species with a free thiol at Cys160 may be (one of) the reactive species involved in the intermolecular thiol\u2013disulfide bonding responsible for cross-linking in heat-induced whey protein aggregates because of its position near the C-terminal end of the protein.\n\nAttempts have been made to identify the specific Cys residues involved in disulfide bonds formed between \u03ba-casein and \u03b2-lactoglobulin when these proteins are heated together. Livney and Dalgleish (2004) compared masses of peptides from tryptic digests of heated \u03ba-casein\/\u03b2-lactoglobulin mixtures with theoretical values and concluded that Cys106\/119\/121 of \u03b2-lactoglobulin were involved in disulfide bonds with both Cys11 and Cys88 of \u03ba-casein. (Note that the hydrolysis pattern does not allow the separation of the three Cys106\/119\/121 residues of \u03b2-lactoglobulin unless CID is used for sequencing.) Although some peptides involving Cys66 and Cys160 of \u03b2-lactoglobulin and the two Cys residues of \u03ba-casein were also identified based on mass comparisons, the high abundance of disulfide-bonded peptides containing Cys106\/119\/121 led these authors to conclude that \u03b2-lactoglobulin with a free thiol at Cys119\/121 was the predominant species involved in intermolecular disulfide bonding. The potential disulfide-bonded species were characterized based on mass analysis alone; no confirmatory experiments were performed, such as comparing reduced with nonreduced systems to ensure that the proposed intermolecular peptides were disulfide bonded, or confirming sequences by CID\u2013MS to preclude misidentification of similarly massed peptides.\n\nIn a novel study, Lowe et al. (2004) used an activated monomeric \u03ba-casein with reduced thiol groups that were blocked with thionitrobenzoate (TNB). \u03b2-lactoglobulin was added to the mixture, and the system was heated under very mild conditions (60 \u00b0C). The TNB groups on the thiols of \u03ba-casein are good leaving groups and, when a reactive thiol from \u03b2-lactoglobulin is exposed, it is capable of interacting with the activated TNB groups on \u03ba-casein in a specific 1:1 oxidative reaction, forming a disulfide-bonded complex and releasing the TNB as a brightly colored compound. This approach allowed the formation of specific disulfide bonds between \u03ba-casein and \u03b2-lactoglobulin under mild heating conditions. Because of the chemical nature of the reaction, it limited further thiol\u2013disulfide exchange reactions, which allowed specific interactions between \u03b2-lactoglobulin and \u03ba-casein to be monitored during the early stages of the denaturation of \u03b2-lactoglobulin.\n\nThe interacted \u03b2-lactoglobulin\u2013 \u03ba-casein complexes were hydrolyzed with trypsin and separated by reverse-phase HPLC followed by MS. In addition, disulfide bonding was confirmed by comparing HPLC traces of nonreduced systems with those of reduced systems, and the identities of some peptides were confirmed by sequencing using CID\u2013MS. Although it was possible to identify disulfide bonds between Cys106\/119\/121 of \u03b2-lactoglobulin and Cys88 of \u03ba-casein, Cys160 of \u03b2-lactoglobulin was found to have formed disulfide bonds with both Cys11 and Cys88 of \u03ba-casein as major products (Fig. 9.9). This supported the earlier findings on pure \u03b2-lactoglobulin that intramolecular thiol\u2013disulfide exchange may precede the intermolecular reactions and that the non-native monomeric form of \u03b2-lactoglobulin species with a free thiol at Cys160 is likely to be one of the reactive monomer species that initiates intermolecular thiol\u2013disulfide exchange reactions (Creamer et al., 2004; Kehoe et al., 2007).\n\nFigure 9.9 Diagram indicating the identified peptides on the linear sequences of \u03ba-casein and \u03b2-lactoglobulin, and the intermolecular disulfide bonds formed between \u03ba-casein and \u03b2-lactoglobulin on heating model systems and milk. The horizontal box lines represent the protein sequence; the lines over the boxes represent the peptides; and S\u2013S indicates the presence of a disulfide bond. Arrows indicate the major proteolytic sites, the chymosin site for \u03ba-casein, and the rapid tryptic sites for \u03b2-lactoglobulin. Potential glycosylation and phosphorylation sites are indicated for \u03ba-casein. Reproduced with permission from Lowe et al. (2004). Copyright (2004) American Chemical Society.\n\nLowe et al. (2004), using the techniques developed for the model system, expanded the study to examine the specific disulfide bonds involved in aggregation between \u03b2-lactoglobulin and \u03ba-casein in heated milk systems. Interestingly, no disulfide bonds between Cys106\/119\/121 of \u03b2-lactoglobulin and either of the Cys residues of \u03ba-casein could be found, even though the disulfide bond between Cys88 of \u03ba-casein and Cys106\/119\/121 of \u03b2-lactoglobulin was readily identified in the model system. In the heated milk system, it was found that Cys160 of \u03b2-lactoglobulin formed disulfide bonds with both Cys88 and Cys11 of \u03ba-casein, as was found in the model system (Fig. 9.9). In independent studies, a similar disulfide bond between Cys160 of \u03b2-lactoglobulin and Cys88 of \u03ba-casein was identified in a heated model goat milk system consisting of isolated casein micelles and \u03b2-lactoglobulin suspended in milk ultrafiltrate, although it appears that no attempts were made to isolate and characterize other intermolecular disulfide bonds between these protein species (Henry et al., 2002).\n\nFrom these observations, it was concluded that, in the model system of \u03b2-lactoglobulin and activated \u03ba-casein, non-native monomeric \u03b2-lactoglobulin species with a free thiol at either Cys119 or Cys121 (but probably not Cys106) could be the reactive monomer involved in intermolecular thiol\u2013disulfide exchange reactions, as Cys119\/121 was involved in disulfide bonds with \u03ba-casein. However, further intramolecular thiol\u2013disulfide exchange reactions in \u03b2-lactoglobulin must precede or occur concurrently with the intermolecular reactions, as disulfide bonds between Cys160 of \u03b2-lactoglobulin and the two Cys residues of \u03ba-casein were also observed as major products in the model system (Fig. 9.9).\n\nIn the heated milk system, no peptides involving Cys106\/119\/121 and the two Cys residues of \u03ba-casein were isolated. Only peptides involving Cys160 and Cys66 with both Cys residues of \u03ba-casein were found (Fig. 9.9). As Cys160 and Cys66 are involved in a disulfide bond in native \u03b2-lactoglobulin, this indicates that intramolecular thiol\u2013disulfide exchange reactions in \u03b2-lactoglobulin precede the intermolecular thiol\u2013disulfide exchange reactions, and that a \u03b2-lactoglobulin (monomeric) species with a free thiol group at Cys160 may play a significant role in the interprotein disulfide bonding that occurs in heated milk or whey protein systems.\n\nThe differences in reaction products between the model systems and milk may be a consequence of factors such as the heating conditions, the nature of the reactions (oxidative interaction compared with thiol\u2013disulfide interchange reactions), and the fact that the \u03ba-casein in milk is found on the casein micelles, whereas in the model system it is not (Lowe et al., 2004). Because of the C-terminal location of Cys160 in \u03b2-lactoglobulin, when this Cys is in the free thiol form and not linked to Cys66, it may be able to productively react with the disulfide bonds of \u03ba-casein to give stable \u03ba-casein\u2013whey protein aggregates (Lowe et al., 2004).\n\n## Relationships between denaturation\/interactions of the whey proteins in heated milk and the functional properties of milk\n\nWhen milk is heated, numerous changes take place, including changes to the proteins, the milk salts (such as changes in the mineral equilibria between the colloidal and serum phases), and lactose. Many of the changes can involve more than one of the milk constituents (IDF, 1995). The changes can be irreversible or reversible to various extents, depending on the changes being monitored and the conditions of the heat treatment. Although the changes to the protein system are an important determinant of the functional properties of milk products, all other changes to the milk system should also be considered to obtain a full understanding of the relationships between heat treatments, interactions, and functional performance. However, there are limited examples of changes to components other than the proteins, and the functional behavior of milk products, and therefore this review, are restricted to some examples of the relationships between the changes in the milk protein system and the functional performance of the milk.\n\n### Examples of the Relationships between Whey Protein Denaturation and the Functional Properties of Milk\n\nIn the early days of milk powder manufacture, it was recognized that the level of whey protein denaturation could be used as an index for the extent of heat treatment the milk had received during the manufacture of the milk powders. The functional properties of the milk products were related to some extent to the heat treatment that the milk had received during processing and therefore the level of whey protein denaturation (Harland & Ashworth, 1947; Harland et al., 1952; Larson et al., 1951).\n\nEven as early as 1952, the concept of 'tailor-made' milk powders was discussed, where powders were processed to provide specific requirements, such as low-heat powders for beverage applications and cottage cheese manufacture, and high-heat powders for bakery applications (Harland et al., 1952). Although there were no standards of quality or processing at this time, it was recognized that the proper control of processing conditions, particularly preheating of the milk, was necessary to produce satisfactory products. Measurement of the level of whey protein denaturation could be used as an objective method for determining the suitability of milk (powder) products for particular commercial and functional applications.\n\nThe heat treatment of milk, whether in liquid milk applications or prior to drying for milk powder manufacture, remains one of the major processes for manipulating the functional properties of milk products. Products such as milk powders are still generally classified according to the heat treatments received during manufacture using one of the derivatives of the WPNI test (Kelly et al., 2003; Singh & Newstead, 1992). With the extensive research on the denaturation of the whey proteins and the ability to predict the denaturation levels after defined heat treatments, it would be envisaged that the level of denaturation of the whey proteins could be used as an indicator of the functional properties of milk products. In a broad sense, this is true. For example, certain heat classifications of milk powders will give improved functionality for particular applications over other classes of milk powders. Some of the general applications of different heat-classified milks and their functional uses, in particular for milk powder products, have been summarized in numerous publications (IDF 1995; ; Kelly et al., 2003; Singh & Newstead, 1992).\n\nHowever, an infinite range of temperature and heating time combinations are available to denature the whey proteins when milk is heated. As a consequence, specific correlations between the level of whey protein denaturation and the functional properties of milk across all possible heating conditions and milk sources do not exist. For example, the WPNI method was developed for assessing the suitability of milk powders for use in bakery applications; however, it was noted that a powder with a low WPNI did not always correspond to good baking qualities (Harland & Ashworth, 1947). Some of these variations are due to factors such as natural variations in the initial whey protein levels in the milk (Harland et al., 1955; Sanderson, 1970a). Others, however, are due to the methods of heat treatment during milk processing. As such, the WPNI or level of whey protein denaturation is, at best, a guide for the suitability of powders for specific applications, or an in-factory guide on processing conditions. Many manufacturers impose additional specifications on the milk powders to ensure suitability in their specific applications (Sanderson, 1970c; Singh & Creamer, 1991a).\n\nSome of the most detailed studies on the relationship between the functional performance of milk and the heat treatment conditions or whey protein denaturation levels have been reported for acid gel or yogurt systems. Parnell-Clunies et al. (1986) showed correlations between the level of whey protein denaturation and the firmness and apparent viscosity of yogurt, regardless of the method used to heat the milk [batch (85 \u00b0C), high-temperature short-time (98 \u00b0C), and ultra-high-temperature (140 \u00b0C) heating systems for different holding times]. However, other properties such as water-holding capacity\/syneresis were more dependent on the heating system used, and it was concluded that high levels of whey protein denaturation in milk were not necessarily associated with an improved water-holding capacity in yogurt.\n\nIn extensive studies, Dannenberg and Kessler (1988b,c) examined the relationship between the denaturation level of whey proteins in milk and the functional performance (firmness, flow properties, and syneresis) of the milk in set yogurt applications. There was a clear relationship between whey protein denaturation and yogurt firmness, with a higher firmness at higher levels of whey protein denaturation. Similar results were obtained for the flow properties of the yogurt. However, very high levels of whey protein denaturation appeared to be detrimental, with a decrease in the firmness and flow properties at denaturation levels above about 95% (Dannenberg & Kessler, 1988c).\n\nFor syneresis of the yogurt, a negative relationship between whey protein denaturation and the level of serum expelled from the yogurt was observed (Dannenberg & Kessler, 1988b). Despite the apparent correlations between denaturation and firmness, flow properties, or syneresis, there were significant variations at each denaturation level, indicating that the temperature of heating used to denature the whey proteins, rather than just the whey protein denaturation level, may be an important factor in determining the functional performance in acid gels.\n\nIn a study on reconstituted whole milk, McKenna and Anema (1993) also observed a positive correlation between the denaturation of the whey proteins in the milk and the firmness of the yogurt made from the milk, regardless of whether the heat treatment was performed before or after powder manufacture (Fig. 9.10a). However, when individual heating conditions were examined, it was also noted that excessive heat treatment of the milk could be detrimental to the firmness of the set yogurt (McKenna & Anema, 1993). A less clear relationship between syneresis and the level of whey protein denaturation was observed, with the level of syneresis appearing to have a greater dependence on the heating conditions (temperature, time, and before\/after reconstitution) than on the level of denaturation itself (Fig. 9.10b; McKenna & Anema, 1993), which supports the findings of Parnell-Clunies et al. (1986).\n\nFigure 9.10 Relationship between the level of whey protein denaturation in reconstituted whole milk and the firmness (a) and syneresis (b) of acid gels prepared from the heated milks. The milks were heated only before powder manufacture ( ), heated only after reconstitution ( ) or heated both before powder manufacture and after reconstitution ( ). Adapted with permission from the results of McKenna & Anema (1993). Copyright (1993) International Dairy Federation.\n\n### Examples of the Relationships between the Level of Interactions of Whey Proteins with \u03ba-Casein\/Casein Micelles and the Functional Properties of Milk\n\nA major limitation in using whey protein denaturation as an index of the functional properties of milk is that it does not consider the subsequent interaction reactions of the denatured whey proteins. These interactions will be dependent on the conditions of denaturation such as temperature and time as well as on the properties of the milk such as pH, concentration, and composition. It is more complex to investigate these aggregation reactions, as there are potentially numerous pathways, and there is great difficulty in isolating and characterizing the specific reaction products. However, in recent years some effort has been made in identifying the interaction reactions of the denatured whey proteins with other components in milk, and in some cases their effects on the functional properties of the milk.\n\n#### Acid-induced Aggregation\/Gelation of Heated Milk\n\nAnema et al. (2004a) showed that small changes in pH of milk from the natural value at the time of heating markedly affected the properties of acid gels prepared from these heated milks. During acidification of the heated pH-adjusted milks, the acid gelation profiles were progressively shifted to higher firmness as the pH at heating was increased (Fig. 9.11a), so that the final firmness of the acid gels (measured as the storage modulus, G\u2032, at pH 4.2) was almost doubled as the pH at heating was increased from pH 6.5 to pH 7.1 (Fig. 9.11). The effect was particularly pronounced in the milks that were heated for times sufficient to fully denature the whey proteins (Fig. 9.11b). This effect of small changes in the pH of the milk at the time of heating on the firmness of acid gels prepared from the heated milk has been independently confirmed (Lakemond & van Vliet, 2005; 2008b; Rodriguez del Angel & Dalgleish, 2006).\n\nFigure 9.11 (a) Changes in firmness (G\u2032) with time after the addition of glucono-\u03b4-lactone (GDL) for heated (90 \u00b0C\/30 min) skim milk samples. (b) Changes in the final firmness (final G\u2032) for acid gels prepared from milk samples heated for various times at 90 \u00b0C. The pH values of the milk samples prior to heating were  , pH 6.5;  , pH 6.55;  , pH 6.6;  , pH 6.65;  , pH 6.7;  , pH 6.9;  , pH 7.1. In all samples, the pH was readjusted back to pH 6.7 before addition of GDL, and this pH reduced to about pH 4.2 after 5.5 h. Reproduced with permission from Anema et al. (2004a). Copyright (2004) American Chemical Society.\n\nLarge strain deformation experiments were also conducted on the set gels. In these experiments, the strain was increased at a constant rate and the stress was monitored until the gel structure yielded and the stress decreased. The maximum in the strain versus stress curves was considered to be the point at which the gel structure broke (Fig. 9.12). The yield properties of the final set acid gels were affected by the pH at which the milk was heated. The yield stresses of the gels increased markedly, and the yield strains of the gels decreased slightly as the pH at which the milk was heated was increased (Fig. 9.12; Anema, 2008b; Lakemond & van Vliet, 2008b).\n\nFigure 9.12 Stress versus strain curves for acid gels prepared from heated (90 \u00b0C\/15 min) skim milk samples. The pH values of the milk samples prior to heating were  , pH 6.5;  , pH 6.6;  , pH 6.7;  , pH 6.9;  , pH 7.1. The maximum in the stress represents the breaking point of the gel, and the stress and strain at this point are considered to be the breaking stress and breaking strain, respectively.\n\nIn addition to influencing the final firmness of the acid gels, the pH at the heat treatment of the milk also influenced the pH at which the milk started gelling\/aggregating during acidification (Anema et al., 2004a; 2004b; Lakemond & van Vliet, 2005; 2008b; Rodriguez del Angel & Dalgleish, 2006; Vasbinder & de Kruif, 2003). On subsequent acidification of milk samples that were heated over a pH range from pH 6.5 to 7.1, samples heated at higher pH (pH 7.1) started gelling at significantly higher pH than samples heated at a lower pH. These effects were very dependent on the temperature at which the milks were acidified (Anema et al., 2004b).\n\nIn further detailed studies on the properties of the acid gels prepared from milks heated at different pH (Lakemond & van Vliet, 2008b), it was shown that the permeability coefficients of the gels increased as the pH of the milks at heating increased. In addition, adding thiol blocking agents to the milks during acid gelation did not affect the firmness of gels prepared from milks heated at low pH (about pH 6.20), but markedly reduced the firmness of the gels prepared from milks heated at higher pH (about pH 6.90). Based on the permeability, effects of thiol blocking agents and the small and large strain rheological results of the acid gels prepared from pH adjusted heated milks, it was concluded that acid gels from milks heated at low pH had finer stranded structure with a higher strand curvature and the gel contained fewer (intermolecular) disulfide bonds. These differences in gel properties accounted for the lower firmness of the gels prepared from milks heated at lower pH (Lakemond & van Vliet, 2008b).\n\nIn concentrated milk, a similar effect of milk pH at heating on acid gel firmness and yield stress was observed, with a marked increase in final gel firmness (Fig. 9.13a and b) and yield stress (Fig. 9.14a and b) as the pH of the milk at heating was increased from the natural pH, and a marked decrease in firmness when the pH of the milk at heating was decreased (Anema, 2008b). As the natural pH of milk decreases when milk is concentrated, the percentage change in firmness or yield stress of the gels from that obtained at the natural pH was plotted against the relative change in pH, and it was found that the firmness (Fig. 9.13c) or yield stresses (Fig. 9.14c) of the gels at all milk concentrations fell on a single curve. This indicates that changing the pH of milk at heating had the same effect on the final firmness or yield stresses of the acid gels at all milk concentrations (Anema, 2008b).\n\nFigure 9.13 (a) Changes in storage modulus, G\u2032, with time after GDL addition for heated (80 \u00b0C\/30 min) 10% TS skim milk samples.  , pH 6.48;  , pH 6.55;  , pH 6.59;  , pH 6.67;  , pH 6.90;  , pH 7.10. (b) Changes in storage modulus, G\u2032, with time after GDL addition for heated (80 \u00b0C\/30 min) 20% TS skim milk samples.  , pH 6.28;  , pH 6.39;  , pH 6.48;  , pH 6.75;  , pH 6.95. (c) Percentage change in final G\u2032 versus change in pH at heating from the natural pH of the milk.  ,  : 10% TS milk samples;  ,  : 15% TS milk samples;  ,  : 20% TS milk samples;  ,  : 25% TS milk samples. Open symbols: samples at the natural pH; filled symbols: samples that were adjusted in pH before heating. Reproduced with permission from Anema et al. (2008b). Copyright (2008) Elsevier.\n\nFigure 9.14 (a) Stress versus strain curves for acid gels prepared from heated (80 \u00b0C\/30 min) 10% TS skim milk samples.  , pH 6.48;  , pH 6.55;  , pH 6.59;  , pH 6.67;  , pH 6.90;  , pH 7.10. (b) Stress versus strain curves for acid gels prepared from heated (80 \u00b0C\/30 min) 20% TS skim milk samples.  , pH 6.28;  , pH 6.39;  , pH 6.48;  , pH 6.75;  , pH 6.95. (c) Percentage change in breaking stress versus change in pH at heating from the natural pH of the milk.  : 10% TS milk samples;  : 15% TS milk samples;  : 20% TS milk samples;  : 25% TS milk samples. Reproduced with permission from Anema et al. (2008b). Copyright (2008) Elsevier.\n\nWhen low levels of starch (up to 1% w\/w) were added to milks prior to heating and acidification, the pH effect on gel firmness was similar to that of milk without starch addition; however, the firmness of the gels increased as the starch level increased (Oh et al., 2007). The gelatinized starch absorbed the aqueous phase on the milk and consequently increased the density of the protein network in the acid gels, increasing the firmness in a similar fashion to increasing the concentration of the milk. However, when higher levels of starch were added to the milk (>1% w\/w), the pH effect diminished, although the gels were still firmer as the starch level increased (Oh et al., 2007). At these high levels of starch, the viscosity of the milk markedly increased after heating due to the gelatinization of the starch and the leaching of amylose into the continuous phase. It was proposed that the high viscosity of the continuous phase may affect the diffusion of protein components during heating and subsequent acidification. This in turn changed the network structure formed in the acid gel, diminishing the importance of serum-phase components (Oh et al., 2007).\n\nThe changes in acid gel firmness or yield stress on changing the pH at heating of the milk could not be related solely to the level of whey protein denaturation (Fig. 9.11a). Small changes in the pH of the milk before heating markedly affected the distribution of the denatured whey proteins and \u03ba-casein between the colloidal and serum phases in milk at its natural concentration (Figs 9.4 and 9.5 ; Anema & Klostermeyer, 1997; Anema & Li, 2003a,b; Lakemond & van Vliet, 2008a; Rodriguez del Angel & Dalgleish, 2006; Vasbinder & de Kruif, 2003) and similar effects were observed in concentrated milks (Anema, 2008b; Chandrapala et al., 2010).\n\nAlthough heating milk prior to acidification markedly increased the firmness of the acid gels, i.e., acid gels prepared from heated milks always had a considerably higher firmness than acid gels prepared from unheated milks (Dannenberg & Kessler, 1988c; Lucey et al., 1997; Lucey & Singh, 1998), the distribution of the denatured whey proteins and \u03ba-casein between the colloidal and serum phases also appeared to influence the firmness of the acid gels. When the final firmness of the acid gels was plotted against the level of nonsedimentable denatured whey proteins in the milk, the results for all pH values fell on a single curve for milk at its natural concentration (Fig. 9.11b). Similarly for concentrated milks, when the percentage change in final gel firmness (Fig. 9.15a) or yield stress (Fig. 9.15b) was plotted against the level of nonsedimentable denatured whey proteins in the milk, the results for all pH and milk concentrations fell onto a single curve (Anema et al., 2004a; Anema, 2008b).\n\nFigure 9.15 (a) Relationship between the change in final G\u2032 for acid skim milk gels and the level of nonsedimentable denatured whey protein in heated (80 \u00b0C\/30 min), pH-adjusted skim milk. (b) Relationship between the change in breaking stress for acid skim milk gels and the level of nonsedimentable denatured whey protein in heated (80 \u00b0C\/30 min), pH-adjusted skim milk.  : 10% TS milk samples;  : 15% TS milk samples;  : 20% TS milk samples;  : 25% TS milk samples. Reproduced with permission from Anema et al. (2008b). Copyright (2008) Elsevier.\n\nFrom these results it was concluded that, although the denatured whey proteins that associated with the micelles have a significant effect on the final firmness of the acid gels, those denatured whey proteins that remain in the serum appear to have a more dominant influence over the final firmness than those associated with the casein micelles (Anema et al., 2004a; Anema, 2008b). For samples where virtually all the whey proteins were denatured, the final gel strength for acid gels prepared from milks in which all the denatured whey proteins were in the serum phase was found to be essentially a factor of two higher than that for acid gels prepared from milks in which all the whey proteins were associated with the casein micelles (Fig. 9.16; Anema et al., 2004a; Anema, 2008b). This effect was observed over a wide range of milk concentrations (Anema, 2008b).\n\nFigure 9.16 Comparison between the final firmness (final G\u2032) and the level of denatured whey protein (open symbols) and the level of soluble denatured whey protein (filled symbols) for acid gels prepared from heated (90 \u00b0C\/30 min) skim milk samples. The pH values at heating of the milks were ( ,  ): pH 6.5; ( ,  ): pH 6.55; ( ,  ): pH 6.6; ( ,  ): pH 6.65; ( ,  ): pH 6.7; ( ,  ): pH 6.9; dotted ( ,  ): pH 7.1. Adapted with permission from Anema et al. (2004a). Copyright (2004) American Chemical Society.\n\nRodriguez del Angel and Dalgleish (2006) separated the nonsedimentable whey protein\u2013 \u03ba-casein aggregates from milks heated at different pH using size exclusion chromatography, and related the peak area of these aggregates to the firmness of the acid gels. They also concluded that the gel firmness appeared to be strongly dependent on the formation of soluble complexes in the milks, and that there appeared to be a linear relationship between the level of soluble aggregates in the heated milk and the final strength of the acid gels.\n\nBased on these results, a hypothesis on the roles of the nonsedimentable and micelle-bound denatured whey protein\u2013 \u03ba-casein aggregates has been developed (Anema et al., 2004a; 2004b; Anema, 2008b; Donato et al., 2007a; Lakemond & van Vliet, 2008a,b; Rodriguez del Angel & Dalgleish, 2006). The increased pH of gelation and the increased acid gel strength of heated milk when compared with unheated milk has been attributed to the incorporation of the whey proteins as well as casein (micelles) in the acid gel structure during the acidification of milk (Graveland-Bikker & Anema, 2003; Lucey et al., 1997; Lucey, 2002).\n\nIn milk, the casein is insoluble at its isoelectric point (about pH 4.6), whereas the native whey proteins remain soluble at all pH. Therefore, unheated milk starts aggregating when the milk pH approaches the isoelectric point of casein, and visible gelation is observed at about pH 4.9. However, for heated milk, the denatured whey proteins are insoluble at their isoelectric points (about pH 5.3 for \u03b2-lactoglobulin, the major whey protein). Therefore, on acidification of heated milk, the proteins will start aggregating at a much higher pH, closer to the isoelectric points of the whey proteins. As a consequence, the contribution of the denatured whey proteins to the acid gel structure and the firmness of the acid gels is markedly higher than that observed for unheated milk (Graveland-Bikker & Anema, 2003; Lucey et al., 1997).\n\nThe pH at heating the milk will produce casein micelle particles with markedly different compositions (Figs. 9.5 and 9.6 ; Anema & Li, 2003a; Anema, 2008b; Vasbinder & de Kruif, 2003). Therefore, on acidification of milks heated at different pH values, different aggregation and gelation behavior is observed. For the milks heated at high pH, the serum-phase denatured whey proteins\/\u03ba-casein complexes may aggregate separately and at a higher pH than the casein micelles. As the isoelectric point of these serum-phase protein components will be higher than that of the casein micelles, the pH at which aggregation occurs will be progressively shifted to higher pH as the heating pH and the concentration of the serum-phase denatured whey proteins\/\u03ba-casein are increased (Anema et al., 2004a; Guyomarc'h et al., 2009; Rodriguez del Angel & Dalgleish, 2006).\n\nIn addition, the dissociation of \u03ba-casein from the casein micelles may also contribute to the higher aggregation pH as the pH at heating is increased, particularly above pH 6.7. Lower levels of \u03ba-casein on the micelles will reduce the density of the surface hairy layer. This may cause the surface hairy layer to collapse at a higher pH, or this layer may have a reduced efficiency in stabilizing the casein micelles. Either effect will allow the \u03ba-casein-depleted micelles to aggregate at a pH that is markedly higher than that observed for the native casein micelles or for casein micelles in milk heated at a lower pH.\n\nIn a study where the serum-phase and colloidal-phase protein aggregates were labeled with different fluorescent dyes before remixing and acidification, it was not possible to identify separate aggregation stages of the different fractions at the early stages of gelation. In addition, the final gel had co-localized serum and colloidal-phase aggregates (Guyomarc'h et al., 2009). This suggests that there may not be a two-stage gelation process and that when gelation starts, both serum-phase and colloidal-phase components are involved in the aggregation process. However, it is still possible that either the serum phase denatures whey proteins\/\u03ba-casein complexes, or the \u03ba-casein-depleted micelles begin aggregating at higher pH and incorporate all phases in the gelling matrix.\n\nThe firmness of acid gels can be related to the number and properties of the contact points between the protein components in the acid gel (Lakemond & van Vliet, 2008b; Lucey et al., 1997; Mellema et al., 2002; van Vliet & Keetals, 1995). As the pH at heating of the milk is increased, the level of serum-phase denatured whey protein\u2013 \u03ba-casein complexes increases, and therefore there are a greater number of particles to aggregate during the subsequent acidification to form the acid gels. There is also the potential for the formation of a more complex acid gel structure when the milk is heated at high pH, where there are high levels of serum-phase denatured whey protein\u2013 \u03ba-casein complexes than when the milk is heated at low pH, where most of the denatured whey protein and \u03ba-casein are associated with the casein micelles. In the latter case, the acid gel process will probably involve only entire whey protein\u2013casein micelle complexes. Therefore, there may be fewer contact points in the acid gels formed from milk with the denatured whey proteins associated with the micelles than in those formed from milk with soluble denatured whey proteins; hence a gel with a lower firmness is observed (Anema et al., 2004a,b; Lakemond & van Vliet, 2008b).\n\nThe large-strain deformation properties also give some indication of the types of bonds involved in the acid gel network. As the pH at heating was increased, the breaking stress of the acid gels prepared from the heated milks was found to increase markedly. However, the breaking strain was virtually unchanged (Figs. 9.12 and 9.14 ; Anema, 2008b; Lakemond & van Vliet, 2008b). For a gel to break on increasing the strain, the strands within the gel network are first straightened and then stretched until the strands, or the bonds within the strands, rupture (Lakemond & van Vliet, 2008b; Mellema et al., 2002; vanVliet & Walstra, 1995). Therefore, the breaking strain is dependent on factors such as the degree of curvature of the strands, with a higher breaking strain when the strands have greater curvature. As the breaking strain of the acid gels only changed slightly with the pH at heating of the milk, despite the marked change in final firmness (Figs. 9.12 and 9.14), this indicates that the relative curvature of the individual strands within the gel network was similar for all acid gel samples.\n\nThe types of bonds involved in the acid gel network will have an influence on the breaking stress (Lakemond & van Vliet, 2008b; Mellema et al., 2002; van Vliet & Walstra, 1995; van Vliet, 1996). The breaking of strands containing covalent bonds requires a greater force than the breaking of strands held together by noncovalent bonds, as covalent bonds have higher bond energies. Therefore, a change in the number or distribution of covalent bonds within the gel network may explain the differences in breaking stress as the pH of the milk at heating was changed (Figs. 9.12 and 9.13). It seems unlikely that the difference in breaking stress can be due to a greater degree of disulfide bonding within the gelled sample; although continuing thiol\u2013disulfide exchange reactions may be occurring during acidification (Vasbinder et al., 2003), the physical number of disulfide bonds is unlikely to be markedly different between the samples.\n\nThe denatured whey proteins, along with some of the \u03ba-casein, are progressively transferred to the serum phase when the pH of the milk is increased before heating (Figs. 9.5 and 9.6). As these interactions involve disulfide bonding, this indicates that the interaction between the denatured whey proteins and \u03ba-casein is transferred from the colloidal phase (casein micelle) to the serum phase as the pH of the milk at heating is increased. On subsequent acidification, both nonsedimentable and colloidal-phase denatured whey proteins are incorporated in the acid gel structure. The nonsedimentable denatured whey protein\u2013 \u03ba-casein complexes can form strands that may be involved in interconnecting the colloidal particles. As the nonsedimentable aggregates are disulfide bonded, those samples heated at high pH and with high levels of nonsedimentable whey protein\u2013 \u03ba-casein aggregates will have a greater number of these strands interconnecting the residual casein micelles.\n\nIn contrast, the samples heated at lower pH will have the denatured whey proteins predominantly associated with the casein micelles and therefore fewer of the whey protein\u2013 \u03ba-casein aggregates interconnecting the colloidal particles. Therefore, the samples heated at higher pH may have a greater number of disulfide bonds interconnecting the colloidal particles and therefore a higher breaking stress, whereas for the samples heated at lower pH, most of the disulfide bonds are on the colloidal particles and fewer disulfide bonds interconnect the colloidal particles; this may explain the lower breaking stress (Figs. 9.12 and 9.14).\n\n#### Chymosin-Induced Aggregation\/Gelation of Heated Milk\n\nNote: The pH of milk has a marked effect on the action of chymosin in destabilizing the system (Walstra & Jenness, 1984; Walstra et al., 1999). In all studies where the pH of milk at heating is being discussed in relation to chymosin treatment, after the heat treatment the pH of the milks were readjusted back to the natural pH, so that pH effects on the enzyme activity were eliminated.\n\nChanging the pH of milk at heating changed the interactions between denatured whey proteins and the casein micelles; however, this did not appear to influence the gelation behavior of heated milk by chymosin to any great extent (Anema et al. 2007; ; Kethireddipalli et al. 2010; ;). The rate of release of glycomacropeptide (GMP) was similar in all milks regardless of whether they were unheated or heated, and the pH at heating had almost no effect on GMP release (Anema et al., 2007; Kethireddipalli et al., 2011; Vasbinder et al., 2003). When looking at gelation profiles by chymosin, all heated milks had very long gelation times and formed very weak gels compared with unheated milks (Fig. 9.17; Anema et al. 2007; ; Kethireddipalli et al., 2010).\n\nFigure 9.17 Changes in storage modulus (G\u2032) with time after addition of rennet to unheated milks (solid symbols) and skim milk samples heated at 90 \u00b0C for 30 min (open symbols). The pH values at heating of the milks were ( ,  ): pH 6.5; ( ,  ): pH 6.7; ( ,  ): pH 6.9; ( ,  ): pH 7.1. All samples were readjusted back to the natural pH (pH 6.67) before addition of rennet (40 \u03bcL of 1:3 diluted rennet per 1.3 mL of milk). Reproduced with permission from Anema et al. (2011). Copyright (2011) American Chemical Society.\n\nThese results indicated that there was a retardation in the chymosin-induced gelation regardless of whether the denatured whey proteins were associated with the caseins micelles (as observed on heating milk at low pH) or in the serum phase associated with \u03ba-casein that had dissociated from the casein micelles (as is observed on heating milk at high pH). The inhibition of the gelation process was therefore not due to steric or charge effects of the denatured whey proteins that are associated with \u03ba-casein at the casein micelle surface. The denatured whey proteins, whether as serum-phase complexes with \u03ba-casein or associated with \u03ba-casein at the casein micelle surface, interfere with the aggregation process and therefore increase the gelation time (Anema et al., 2007; Kethireddipalli et al., 2010).\n\nThis effect was confirmed by experiments in which serum and colloidal phases from unheated and heated milks were exchanged, as it was shown that denatured whey proteins, whether in the serum phase or associated with casein micelles, inhibited the gelation of milk by chymosin. However, this inhibition may be more complex, as it was also shown that heated casein micelles in the absence of whey proteins and nonprotein serum components from heated milks also inhibited gelation (Kethireddipalli et al., 2010). From these observations it was concluded that the inhibition of gelation by chymosin for heated milks was complex and may be due to the combined effects of heat on casein micelles, denatured whey proteins (both serum and colloidal phase), and nonprotein serum components (Kethireddipalli et al., 2010).\n\nGelation of pH adjusted, heated milks by chymosin did not reveal differences; however, this is not an objective measure of the destabilization of the system, as it requires the formation of an interconnected network structure, and it is possible for the casein micelles to be destabilized without forming a gel. By monitoring the particle size changes of milk during chymosin treatment, it was possible to examine the early stages of the aggregation process (Anema et al., 2011). For all milk samples, after adding chymosin, the size initially decreased slightly due to the cleavage of GMP from \u03ba-casein (lag phase), and then increased as the destabilized particles aggregated (aggregation phase; Fig. 9.18).\n\nFigure 9.18 Changes in particle size on the rennet treatment of (a) unheated skim milk, (b) skim milk heated at 90 \u00b0C for 2.5 min, and (c) skim milk heated at 90 \u00b0C for 30 min. Milk samples (20 \u03bcL) were diluted in Ca-imidazole buffer (1 mL) before the addition of rennet (10 \u03bcL of 1:120 diluted rennet). The pHs of the milk samples were ( ) 6.5, ( ) 6.6, ( ) 6.7, ( ) 6.9, and ( ) 7.1. (d) Effect of pH on the time taken for the particle size to increase by 50 nm from the initial size (T50) after the addition of rennet to the milk samples. The milk samples were unheated ( ) or heated at 90 \u00b0C for ( ) 1 min, ( ) 2.5 min, ( ) 5 min, ( ) 10 min, ( ) 15 min, or ( ) 30 min. Reproduced with permission from Anema et al. (2011). Copyright (2011) American Chemical Society.\n\nMilks heated at temperatures below those where whey proteins denatured (\u2264 60\u00b0C) had short lag phases and rapid aggregation phases; these were similar to those from unheated milks and were only slightly affected by the pH of the milk at heating. However, milks heated at higher temperatures (> 60 \u00b0C) and at low pH (e.g., pH 6.5) had extremely long lag phases and once destabilized, the particles aggregated at a slow rate. Increasing the pH at heating shortened the lag phase and increased the rate at which the particles aggregated. In fact, samples heated at pH 7.1 had lag phases and aggregation rates not dissimilar to those of unheated milks (Anema et al., 2011).\n\nSimilarly, for milks heated at 90 \u00b0C for different times, the lag phase decreased, and the rate of aggregation increased markedly as the pH of the milk at heating was increased from pH 6.5 to pH 7.1. In fact, the lag phases and aggregation rates for the milks heated at pH 7.1 were not substantially different from those from unheated milks (Fig. 9.18). The effect of pH was greater as the heating time was increased, predominantly as a result of an increased lag phase and decreased aggregation rate of the samples heated at lower pH when compared with those heated at a higher pH (Anema et al., 2011; Fig. 9.18).\n\nFor the milk samples heated at high temperatures, there were positive correlations between the aggregation time and the level of whey protein or \u03ba-casein that was associated with the casein micelles (Fig. 9.19). It was proposed that casein micelles that were substantially coated in denatured whey proteins and \u03ba-casein (obtained on heating milks at low pH) were more resistant to chymosin-induced aggregation than casein micelles with low levels of denatured whey proteins and \u03ba-casein on the casein micelle surface (as observed on heating milk at high pH; Anema et al., 2011). It was also proposed that, although casein micelles in heated milks aggregated on chymosin treatment, the denatured whey proteins, whether associated with the micelles or in the serum phase, inhibited the gelation of the milk by stabilizing the surface of aggregated particles and preventing or slowing the formation of an interconnected network structure (Anema et al., 2011).\n\nFigure 9.19 Relationship between the time taken for the particle size to increase by 50 nm and (a) the percentage of micelle-bound \u03b2-lactoglobulin, (b) the percentage of micelle-bound \u03b1-lactalbumin, (c) the percentage of micelle-bound \u03ba-casein (all data), and (d) the percentage of micelle-bound \u03ba-casein (selected data with high levels of denatured whey proteins). The milk samples were heated either at 90 \u00b0C for various times (open symbols) or at different temperatures for 30 min (filled symbols). The pHs of the samples were ( ,  ) 6.5, ( ) 6.55, ( ) 6.6, ( ) 6.65, ( ,  ) 6.7, ( ,  ) 6.9, and ( ,  ) 7.1. Reproduced with permission from Anema et al. (2011). Copyright (2011) American Chemical Society.\n\n### Examples of the Effect of Denaturing Whey Proteins Separately from Casein Micelles on the Functional Properties of Milk\n\nCompared with heated milks, different effects on functional properties can be observed when the whey proteins are denatured and aggregated separately from the casein micelles and then remixed. However, the effects of adding pre-denatured whey proteins to casein micelle suspensions or milk on the functional properties are dependent on the conditions under which the whey proteins are denatured. Lucey et al. (1998) showed that acid gels prepared from milk samples where the whey proteins were denatured in the presence of casein micelles had a markedly higher firmness than acid gels prepared from milk samples where the whey proteins in milk serum were pre-denatured and then added back to the casein micelle suspensions. In many cases, the samples with denatured whey proteins added back to the casein micelles produced acid gels with firmness similar to or only slightly higher than those prepared from unheated milks.\n\nIn another study, Schorsch et al. (2001) prepared model milk systems in which the whey proteins in a simulated milk serum were either heated in the presence of casein micelles or heated separately and added back to the casein micelles. Acid gels were prepared from these model milk systems. It was shown that the acid-induced gelation occurred at a higher pH and in a shorter time when the whey proteins were denatured separately from the casein micelles than when they were heated in the presence of casein micelles. However, the gels formed were weaker and more heterogeneous because of the particulate nature of the denatured whey proteins.\n\nIt was suggested that the large denatured whey protein aggregates, as formed when the whey proteins in milk serum were heated separately from the casein micelles, hinder the formation of a casein gel network when the milk is subsequently acidified, and that a weak acid gel with a heterogeneous structure results. When the whey proteins are heated in the presence of the casein micelles, the denatured whey proteins interact with the \u03ba-casein at the casein micelle surface. On subsequent acidification, the denatured whey protein\u2013casein micelle complexes aggregate to form a firmer acid gel with a more homogeneous structure (Schorsch et al., 2001). This proposal is supported by early studies, which showed that the aggregation of the denatured whey proteins, and in particular \u03b2-lactoglobulin, formed large aggregate species when heated in the absence of \u03ba-casein, whereas aggregation was limited when the whey proteins were heated in the presence of \u03ba-casein (McKenzie et al., 1971).\n\nIn contrast, if the whey proteins are denatured at relatively low protein concentrations, at low ionic strengths, and at a pH far from the isoelectric point (>pH 6.5), then soluble denatured whey protein polymers can be formed. The polymers are linear and can be induced to gel when salt is added or the pH is reduced (Britten & Giroux, 2001; Gustaw et al., 2006; ). When these whey protein polymers are added to heated skim milk, and the preparations are acidified, the acid gels formed had markedly higher firmness and water-holding capacities than those from the skim milk or milks heated with equivalent levels of native whey proteins (Britten & Giroux, 2001; Gustaw et al. 2006; ;). The firmness and water-holding capacities were markedly higher than those where the whey proteins were denatured in milk serum and then added back to casein micelle suspensions or milk before acidification (Lucey et al., 1998; Schorsch et al., 2001).\n\nThe isoelectric points of denatured whey protein complexes were chemically modified through succinylation or methylation (Morand et al., 2012b). These modified denatured whey protein complexes were added to whey protein-free milk suspensions to produce model heated milk systems, and the milks were subsequently acidified to form gels. The gelation pH of these milks increased markedly as the isoelectric point of the whey protein complexes increased, supporting the proposition that it is the higher isoelectric point of the whey proteins that causes heated milks to gel at markedly higher pH than unheated milks (Anema et al., 2004a; Lucey et al., 1997). Interestingly, the final firmness of the gels was not markedly affected by the pI of the complexes (Morand et al., 2012b).\n\nIn similar studies, the hydrophobicity of denatured whey protein complexes was altered by acylation of lysine amino acids using anhydrides of various carbon chain lengths (Morand et al., 2012a). These complexes were also added to whey protein-free milk suspensions, and the milks were acidified to form gels. Increasing the hydrophobicity of the whey protein complexes also increased the pH at which the milks gelled as well as the maximum firmness of the gels, although this maximum was not always at the same pH. Taken together, these results indicate that both the isoelectric properties and the hydrophobicity of the serum-phase complexes influence the acid gelation properties of heated milk systems (Morand et al., 2011; 2012a, b).\n\n## Conclusion\n\nA considerable amount of work has gone into understanding the irreversible denaturation reactions of the whey proteins in heated milk systems. These detailed studies have produced models that allow reasonably accurate prediction of the level of whey protein denaturation in milks under a wide range of heating conditions, even in milk samples with markedly modified concentrations and compositions. However, with a few exceptions, monitoring of the whey protein denaturation levels provides only a crude indication of the functionality of the milk system. As a consequence, more recent research efforts have focused on trying to understand the specific interaction reactions of the denatured whey proteins with other proteins in the milk system. Early indications suggest that these types of studies on the interactions of denatured whey proteins may provide greater insights into the functional properties of heated milk products than can be obtained by monitoring just whey protein denaturation levels. These initial studies on protein interactions have been conducted under relatively narrowly defined conditions (temperatures, heating times, pH, milk concentrations, and milk compositions). It is likely that changes to these variables will markedly influence the interaction behavior and will explain the changes in functional behavior when the heating conditions are changed (even though the whey protein denaturation levels may be similar). 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Electrophoretic separations on microfluidic chips .  _Journal of Chromatography A_.  2008 ;1184 : 542 \u2013 559 .\n\nZittle CA ,  Custer JH ,  Cerbulis J ,  Thompson MP .  \u03ba-Casein\u2013 \u03b2-lactoglobulin interaction in solution when heated .  _Journal of Dairy Science_.  1962 ;45 : 807 \u2013 810 . \nChapter 10\n\n# Effects of Drying on Milk Proteins\n\nPierre Schuck*\n\n* INRA, UMR 1253, STLO, Rennes, France\n\n## Abstract\n\nDehydration by spray drying is a valuable technique for water evaporation, using hot air to stabilize the majority of dairy ingredients. In view of the increased development of filtration processes, the dairy industry requires greater understanding of the effects of spray drying on the quality of protein powders. Several publications have reported that the proteins have an important role in the mechanisms of water transfer during drying and rehydration. The residence time of the droplet is so short that it is very difficult to study the mechanisms of the structural change in the protein without fundamental research into relationships with the process\/product interactions. Following an introduction to spray drying, this chapter on the effects of drying on milk proteins covers five areas: the world dairy powder situation, the properties of spray-dried milk products, the principles of spray drying, the drying of proteins, and the rehydration of dried protein powders.\n\n## Keywords\n\nspray-dried milk products\n\ndairy powders\n\nwhole milk powder\n\nskim milk powder\n\nwhey products\n\nOutline\n\nIntroduction 319\n\nWorld Dairy Powder Situations 321\n\nWhole Milk Powder 321\n\nSkim Milk Powder 322\n\nWhey Products, Casein, and Other Dairy Ingredients 323\n\nProperties of spray-dried milk products 323\n\nPrinciples of spray drying 324\n\nProcess improvement 328\n\nDrying of proteins 328\n\nExamples of Dairy Protein Concentrates and Powders 329\n\nResearch Approach Using Drying by Desorption 330\n\nDesorption Results 331\n\nIndustrial Implications 333\n\nIntroduction 333\n\nAvailability of Water 333\n\nWhey Proteins 335\n\nCaseins 335\n\nRehydration of Protein Powders 335\n\nConclusions 339\n\n## Introduction\n\nThe purpose of the dehydration of milk and whey is to stabilize these products for their storage and later use. Milk and whey powders are used mostly in animal feed. With changes in agricultural policies, such as the implementation of the quota system and the dissolution of the price support system in the European Union (EU), the dairy industry was forced to look for better uses for the dairy surplus and for the by-products of cheese (whey) produced from milk and buttermilk produced from cream. Studies on the reuse of protein fractions with their nutritional qualities and functional properties led us to believe that they could have several applications.\n\nMainly because of the emergence of filtration technology (microfiltration, ultrafiltration, nanofiltration, and reverse osmosis), in the past 30 years the dairy industry has developed new technological processes for extracting and purifying proteins (casein, caseinates, whey proteins, etc.) (Kjaergaard et al., 1987; Maubois, 1991), such as:\n\n\u2022 dairy proteins and whey concentrates (Le Gra\u00ebt & Maubois, 1979; Goud\u00e9dranche et al., 1980; Madsen & Bjerre, 1981; Maubois et al., 1987; Caron et al., 1997);\n\n\u2022 micellar casein concentrates (Fauquant et al., 1988; Schuck et al., 1994a);\n\n\u2022 micellar casein (MC) (Pierre et al., 1992; Schuck et al., 1994b);\n\n\u2022 whey concentrates, selectively demineralized concentrates (Jeantet et al., 1996); and\n\n\u2022 super-clean skim milk concentrates (Piot et al., 1987; Vincens & Tabard, 1988; Trouv\u00e9 et al., 1991; Schuck et al., 1994a)\n\nUsed as either nutritional or functional ingredients, most of these proteins are marketed in dehydrated form (Fig. 10.1). The application of different processing steps allows the production of a wide range of different dried and stable intermediate dairy products. Many new uses for these constituents have emerged with the manufacture of formula products, substitutes, and adapted raw materials.\n\nFigure 10.1 Fractionation of milk.\n\nThe most frequently used technique for the dehydration of dairy products is spray drying. It became popular in the dairy industry in the 1970s, but at that time there were few scientific or technical studies on spray drying and, in particular, there were none on the effects of spray drying parameters or the effects of the physicochemical composition and microbiology of the concentrates on powder quality. Manufacturers acquired expertise in milk drying and eventually in whey-drying processes through trial and error. Because of the variety and complexity of the mixes to be dried, a more rigorous method based on physicochemical and thermodynamic properties has become necessary. Greater understanding of the biochemical properties of milk products before drying, water transfer during spray drying, and the properties of powders and influencing factors is now essential in the production of milk powders. A lack of technical and economic information and of scientific methods prevents the manufacturer from optimizing the powder plant in terms of energy costs and powder quality.\n\nThe aim of this chapter is to provide a brief summary of the process of spray drying of dairy products and to review current knowledge on the properties of spray-dried milk products, the modeling and simulation of water-transfer processes (drying and rehydration) and dairy powders, spray drying equipment, and energy consumption.\n\n### World Dairy Powder Situations\n\nThe nature of dairy powders has changed over the last 20\u201325 years (CNIEL 1991; 2005). There has been a decrease in production, mainly of skim and fat-filled milk powders, although the production of whole milk powder and whey powder increased between 1986 and 2004. This increase was reflected in the types of whey and derived powders (protein concentrates) produced. Cheese production from cow's milk increased between 1986 and 2004, with a corresponding increase in whey production. Having fallen in 2004, the production of dry milk products did not recover significantly in the first months of 2005, the decline being due mainly to slower increase in milk supplies in many parts of the world.\n\nAccording to the International Dairy Federation (2012), production increased worldwide in 2011 for every dairy product, but growth was especially strong for butter and milk powder. In 2011 SMP production recovered strongly after a fall in 2010. In contrast to the previous year, stocks were fairly low in 2011 while sustained international demand stimulated production. World butter production continued to grow in 2011, reaching levels substantially above previous years owing to exceptional growth in output from the United States and New Zealand. The industrial output of cow's milk cheese and whole milk powder was broadly in line with levels observed in the preceding ten years.\n\n### Whole Milk Powder\n\nAccording to the International Dairy Federation (2012), world production of whole milk powder (WMP) is estimated at around 4.5 million tons. The two main players, China (+4.5%) and New Zealand (+5.6%), increased their production strongly, but the most impressive growth occurred in Argentina (+35.8%), where the high additional volumes of milk produced in 2011 were mainly processed into WMP. The structural decline in WMP production in Europe continued (\u20132.4%).\n\nWorld trade in WMP increased by 7%, up to 2.2 million tons. While more than half of this was channeled into Asia and the Middle East, market growth focused on Latin America and Africa, where key markets such as Algeria, Venezuela, Brazil, and Mexico all stepped up their imports significantly. New Zealand strengthened its position as the leading WMP supplier to the world market (49%), achieving another milestone and for the first time breaking through the barrier of one million tons of WMP exports. The geographical focus for New Zealand remained the Middle East and Asia. Exports to all major destinations in these regions expanded, including those to China, despite the fact that the enormous increase in WMP flow to that country came to an end in 2011. China alone absorbed about 28% of New Zealand's total WMP exports. Meanwhile, New Zealand also benefited from better export opportunities in Algeria. On the other hand, it lost share in a key market, Venezuela, because of renewed competition from within South America, in particular Argentina.\n\nArgentina experienced a most remarkable increase in export volume in 2011. With a massive additional 75,000 tons, Argentina was the largest contributor to world trade expansion after New Zealand. Spurred on by favorable production circumstances and high price levels on international markets, the country's dairy sector more than recovered from the previous year's export slump, reaching a level that remained not far from its historical height of five years before. Besides New Zealand and Argentina, of all the major exporters of WMP only Australia and the Philippines managed to increase export volumes in 2011. By contrast, exports from the EU dropped for the third consecutive year, as a result of lack of competiveness in the international markets. The fact that the EU lost ground to New Zealand, Argentina, and Australia in a number of its traditional key markets, such as Algeria, Nigeria, and the Middle East, is illustrative of this decline. Unlike its neighbor, Uruguay experienced a decline in WMP exports after years of expansion. This was apparently due to a priority switch in the export range in favor of product categories such as butter, cheese, and skim milk powder (SMP), which absorbed the additional milk available in 2011 (International Dairy Federation, 2012).\n\n### Skim Milk Powder\n\nWorld production of skim milk powder (SMP) is estimated at around 4 to 4.5 million tons. Stimulated by firm demand, the output of SMP increased in most parts of the world between 2010 and 2011. SMP had an increasing role over these two years in exports from the EU and the United States in that there is no longer a price gap between these two blocks and New Zealand. Since 2010, North American and European processors have been competitive throughout the year in the world SMP market, which is not the case for butter and WMP. This convergence in terms of price is corroborated by the presence of American and European companies in the SMP auction on the Global Dairy Trade platform (International Dairy Federation, 2012).\n\nWorld trade in SMP in 2011 reflected very dynamic development. Worldwide exports for this category soared to over 1.7 million tons (+19%). After a period of weaker interest, this was the fourth consecutive year that the world trade volume increased. All major exporters expanded their trade, benefiting from growth in demand in various regions. The main reason for this was both developments in Southeast Asia and increased trade to various key markets in the Middle East, South America, and Africa. Mexico remained the world's largest single SMP market, but the expanding export volume entering Mexico was of benefit mainly to American exporters, as they were facilitated by NAFTA arrangements, and this is why the United States now retains nearly 90% of that market. The SMP export market is rather concentrated on the supply side, since effectively 75% of world trade volume is supplied by only three exporters: the EU, the United States, and New Zealand. This situation was reaffirmed in 2011, when the EU reestablished its position as the leading exporter. International price developments in combination with internal dairy policy in recent years have contributed to the EU's competitiveness in milk protein in the world market.\n\nDestinations in Northern Africa remained very important to EU exporters, as did several key destinations in Asia, such as Indonesia and China. They also seized opportunities in India, South Korea, and Mexico, while on the other hand the 2010 surge in exports to the Russian market was reversed. Though not expanding to the extent of the EU, the United States remained by far the second largest global SMP supplier. As already mentioned, its exports are closely linked to trade with Mexico, which took an additional 59,000 tons of U.S. milk powder in 2011 and absorbed nearly 40% of total U.S. exports. Moreover, U.S. exporters developed substantial markets in Asia, which also allowed them to benefit from the growth in that region. Of the minor suppliers, Australia, Uruguay, and Ukraine all made significant progress, recovering from weak export results in 2010. At the same time, exports from Belarus and Argentina were down; for Belarus this was a reaffirmation of a downward trend since the previous year, closely related to reduced demand from Russia (International Dairy Federation, 2012).\n\n### Whey Products, Casein, and Other Dairy Ingredients\n\nIn 2011 casein production increased in most countries where statistics are available. Output in the EU was estimated at around 145,000 tons\u2014that is, 15,000 tons more than in 2010. Liquid whey production results primarily from the industrial production of cheese, which generates more than 80% of the total whey available, and secondarily from casein output. The major processors of whey are therefore located in Europe, North America, and the South Sea Islands, corresponding to the major cheese production areas. Compared to 2010, no significant changes occurred in 2011 in the United States; production remained fairly stable at 500,000 tons of whey powder and condensed whey, as well as 195,000 tons of whey protein concentrates and almost 30,000 tons of whey protein isolates. The production of whey powder within the EU is estimated at around 1.9 million tons. It increased slightly (+1.6%) in 2011 (International Dairy Federation, 2012).\n\nWorld trade in whey powder and whey products in 2011 continued its dynamic development, increasing to over 1.5 million tons (+11%). This trade remained dominated by exports from the EU and the United States, which represented two-thirds of total world trade. While exports from the former increased substantially, mainly on the wave of expanding export opportunities in China and other Asian destinations, exports from the latter stabilized as a result of reduced exports to neighboring markets and mixed results in Asia. China is by far the largest and the most dynamic market, absorbing 22% of world trade after a 36% growth.\n\nExcept for Switzerland and Australia, all other significant whey exporters increased their exports. In absolute terms, Argentina expanded the most, after the EU, mainly due to imports to Asia in general and China in particular. In relative terms, Belarus showed the highest export increase, which reflected recent investments in whey processing. As usual, most of its products went to Russia, ousting competitors from a contracting market (International Dairy Federation, 2012).\n\n## Properties of spray-dried milk products\n\nA dairy powder is characterized not only by its composition (proteins, carbohydrates, fats, minerals, and water) but also by its microbiological and physical properties (bulk and particle density, instant characteristics, flowability, floodability, hygroscopicity, degree of caking, whey protein nitrogen index, thermostability, insolubility index, dispersibility index, wettability index, sinkability index, free fat, occluded air, interstitial air, and particle size), which form the basic elements of quality specifications. There are well-defined test methods for their determination according to international standards (Pisecky, 1986; 1990; 1997; American Dairy Products Institute 1990; Masters, 1991). These characteristics depend on drying parameters (type of tower spray drier, nozzles\/wheels, pressure, agglomeration, and thermodynamic conditions of the air, such as temperature, relative humidity, and velocity), and the characteristics of the concentrate before spraying (composition\/physicochemical characteristics, viscosity, thermosensitivity, and availability of water). Several scientific papers have been published on the effects of technological parameters on these properties (Hall & Hedrick, 1966; De Vilder et al., 1979; Baldwin et al., 1980; Pisecky, 1980; 1981; 1986; Kessler 1981; Bloore & Boag, 1982; De Vilder, 1986; Tuohy, 1989; Ilari & Loisel, 1991; Masters, 1991; Mahaut et al., 2000).\n\nWater content, water dynamics, and water availability are among the most important biochemical properties of dairy powders (Fig. 10.2).\n\nFigure 10.2 Properties and qualities of powders.\n\nThe nutritional quality of dairy powders depends on the intensity of the thermal processing during the technological process. The thermal processing induces physicochemical changes that tend to decrease the availability of the nutrients (loss of vitamins, reduction of available lysine content, and whey protein denaturation) or to produce nutritional compounds such as lactulose (Straatsma et al., 1999a,b).\n\n## Principles of spray drying\n\nAccording to Pisecky (1997), spray drying is an industrial process for the dehydration of a liquid containing dissolved and\/or dispersed solids (e.g., dairy products) by transforming the liquid into a spray of small droplets and exposing these droplets to a flow of hot air. The very large surface area of the spray droplets causes evaporation of the water to take place very quickly, converting the droplets into dry powder particles.\n\nIndeed, when a wet droplet is exposed to hot dry gas, variations in the temperature and the partial pressure of the water vapor are established spontaneously between the droplet and the air:\n\n\u2022 Heat transfer from the air to the droplet occurs under the influence of the temperature variation.\n\n\u2022 Water transfer occurs in the opposite direction, explained by variation in the partial pressure of water vapor between the air and the droplet surface.\n\nAir is thus used both for fluid heating and as a carrier gas for the removal of water. The air enters the spray drier hot and dry and leaves wet and cool. Spray drying is a phenomenon of surface water evaporation maintained by the movement of capillary water from the interior to the surface of the droplet. As long as the average moisture is sufficient to feed the surface regularly, the evaporation rate is constant; if not, it decreases.\n\nThe drying kinetics are related to three factors.\n\n\u2022 Evaporation surface created by the diameter of the particles. Spraying increases the exchange surface: 1 L of liquid sprayed as particles of 100 \u03bcm diameter develops a surface area of 60 m2, whereas the surface area is only approximately 0.05 m2 for one sphere of the same volume.\n\n\u2022 Difference in the partial pressure of water vapor between the particle and the drying air. A decrease in the absolute humidity of the air and\/or an increase in the air temperature tend to increase the difference in the partial pressure of water vapor between the particle and the drying air.\n\n\u2022 Rate of water migration from the center of the particle toward its surface. This parameter is essential for the quality of dairy powders. Indeed, it is important that there is always water on the surface of the product so that the powder surface remains at the wet bulb temperature for as long as possible. The rate of water migration depends on the water diffusion coefficient, which varies according to the biochemical composition, water content, and droplet temperature. Calculation of this coefficient is therefore complex and the mathematical models suggested are not easily exploitable by the dairy industry.\n\nAccording to Masters (1991) and Pisecky (1997), the main components of the spray drier, as shown in Figure 10.3, are as follows:\n\n\u2022 A drying chamber (Fig. 10.3, point 7). The chamber can be horizontal (box drier), although in the dairy industry the chamber design is generally vertical with a conical or flat base.\n\n\u2022 An air disperser with a hot air supply system such as a main air filter, supply fan, air heater, and air disperser (Fig. 10.3, point 3). The air aspiration is performed through filters, the type depending on the local conditions and the nature of the product to be treated. The air can be heated in two different ways: by direct heating (gas) and\/or by indirect heating (vapor, gas, oil, or electricity). The air flow chamber can be in co-current, countercurrent, or mixed mode.\n\n\u2022 An atomizing device with a feed supply system such as a feed tank, feed pump, water tank, concentrate heater, and atomizing device. There are three types of atomizing device: rotary (wheel or disc), nozzle (pressure, pneumatic or sonic), and combined (rotary and pneumatic) (Fig. 10.3, point 3).\n\n\u2022 A powder recovery system. Separation of the dried product can be achieved by a primary discharge from the drying chamber followed by a secondary discharge from a particulate collector (using a cyclone, bag filter, or electrostatic precipitation), followed by total discharge from the particulate collector and finishing with final exhaust air cleaning in a wet scrubber and dry filter (Fig. 10.3, points 8 and 9).\n\nFigure 10.3 Multiple effect spray dryer.\n\nAccording to Sougnez (1983), Masters (1991), and Pisecky (1997), the simplest types of installation are single-stage systems with a very short residence time (20\u201360 s). Thus, there is no real balance between the relative humidity of the air and the moisture content of the powder. The outlet temperature of the air must therefore be higher, and the thermal efficiency of the single-stage spray drier is then reduced. This type of drying chamber was the standard equipment for drying milk in the 1960s. Space requirements were small and building costs were low. Generally, installations without any post-treatment system are suitable only for nonagglomerated powders that do not require cooling. If necessary, a pneumatic conveying system could be added to cool the powder while transporting the chamber fraction and the cyclone fraction to a single discharge point.\n\nThe two-stage drying system consists of limiting the spray drying process to a process with a longer residence time (several minutes) to provide a better thermodynamic balance. This involves a considerable reduction in the outlet air temperature, as well as an increase in the inlet air temperature. A second final drying stage is necessary to optimize the moisture content by using an integrated (static) fluid bed or an external (vibrating) fluid bed, the air temperatures of which are 15\u201325 \u00b0C lower than with a single-stage system to improve and\/or preserve the quality of the dairy powder (Fig. 10.3, points 11 and 14). Consequently, the surrounding air temperature at the critical drying stage and the particle temperature are also correspondingly lower, thus contributing to further economic improvement. The integrated fluid bed can be either circular (e.g., the Multi Stage Drier (MSD\u2122) chamber) or annular (e.g., the Compact Drier (CD) chamber). Two-stage drying has its limitations, but it can be applied to products such as skim milk, whole milk, precrystallized whey, caseinates, whey proteins, and derivatives. The moisture content of the powder leaving the first stage is limited by the thermoplasticity of the wet powder\u2014that is, by its stickiness in relation to the water activity and the glass transition temperature (Roos, 2002). The moisture content must be close to 7\u20138%, 9\u201310%, and 2\u20133% for skim\/whole milk, caseinate\/whey protein, and precrystallized whey powders, respectively. The two-stage drying techniques can be applied to the production of both nonagglomerated and agglomerated powders, but this technique is very suitable for the production of agglomerated powders, by separating the nonagglomerated particles from the agglomerates [i.e., collecting the cyclone fractions and reintroducing these fine fractions (called fines) into the wet zone around the atomizer of the chamber].\n\nThe three-stage drying systems, with an internal fluid bed as a second stage in combination with an external vibrating fluid bed as a third-stage drier, first appeared at the beginning of the 1980s and were called the Compact Drier Instantization (CDI) or MSD\u2122. Today, they dominate the dairy powder industry (Fig. 10.3). Three-stage systems combine all the advantages of extended two-stage drying, using spray drying as the primary stage, fluid bed drying of a static fluid as the second drying stage, and drying on an external vibrating fluid bed as the third drying stage. The final drying stage terminates with cooling to under the glass transition temperature. Evaporation performed at each stage can be optimized to achieve both gentle drying conditions and good thermal economy.\n\nThe compact dryer (CD) is suitable for producing both nonagglomerated and agglomerated powders of practically any kind of dried dairy product. It can also cope successfully with whey powders, fat-filled milk, and whey products as well as caseinates, both nonagglomerated and agglomerated. It has a fat content limit of about 50% fat in total solids. Powder quality and appearance are comparable to those of products from two-stage drying systems, but they have considerably better flowability and the process is more economical. In comparison with the CD, the MSD\u2122 can process an even wider range of products and can handle an even higher fat content. The main characteristics of MSD\u2122 powder are very good agglomeration and mechanical stability, low particle size fractions (below 125 \u03bcm), and very good flowability.\n\nOptimization of the process has allowed considerable improvement in the drying efficiency, and the quality of the product obtained is generally better. The various advantages are:\n\n\u2022 Improved thermal efficiency: significant reduction in the outlet air temperature, permitting an increase in the inlet air temperature;\n\n\u2022 Reduction in material obstruction: the capacity in one volume is two or three times higher than that for a traditional unit;\n\n\u2022 Considerable reduction in powder emission to the atmosphere: a reduction in the drying air flow and an increase in powder moisture content reduce the loss of fine particles in the outlet air;\n\n\u2022 Improved powder quality in relation to the agglomeration level, solubility, dispersibility, wettability, particle size, density, and so on\n\nThere are other examples of drying equipment such as the 'tall form drier,' the 'Filtermat\u00ae drier,' the 'Paraflash\u00ae drier,' and the 'Tixotherm\u00ae drier.' All these towers have characteristics related to the specific properties of the product being dried (e.g., high fat content, starch, maltodextrin, egg, and hygroscopic products).\n\n## Process improvement\n\nThis section shows the use of a thermohygrometric sensor, with some examples of such measurements (temperature, absolute humidity (AH), and relative humidity (RH), dry air flow rate, water activity), through calculation of mass and AH to prevent sticking in the drying chamber and to optimize powder moisture and water activity in relation to the RH of the outlet air.\n\nSchuck et al. (2005) demonstrated that a thermohygrometer can be used to avoid sticking and to optimize water content and water activity in dairy powders. These results demonsrate that the calculated AH is systematically higher than the measured AH because the calculated AH corresponds to the maximum theoretical value that can be reached. Calculation of AH by means of the mass balance is based on the hypothesis that the air circulating in the spray drier removes all the water from the concentrate. Thus, if the difference between the calculated and the measured AH of the outlet air is below 2 g of water kg-1 dry air (depending on the spray drier with regard to measurement accuracy), there is no problem of sticking in the spray dryer chambers, whatever the dairy concentrate used. On the other hand, sticking was observed in this study for differential AH above 2 g of water kg-1 dry air, corresponding to lower water removal and consequently to favorable sticking conditions. The operator can follow the AH and anticipate a variation in drying parameters according to the differences between the calculated and the measured AH.\n\nThe operator can also follow the relative humidity in the outlet air. To achieve a dairy powder with the same water activity and moisture content, the same RH in the outlet air using the previous equations according to each dairy product, whatever the spray drying conditions (inlet air temperature, RH and AH), must always be maintained.\n\nThe changes in RH and AH (resulting from variations in AH of inlet air, total solid content of concentrate, crystallization rate, outlet air temperature, etc.) can be rapidly observed in the outlet air using a thermohygrometer before such changes significantly affect powder moisture, water activity, and powder behavior with regard to sticking.\n\n## Drying of proteins\n\nThe native properties of milk components are substantially unaffected by moderate drying conditions. Depending on the preheating conditions, drier design, and temperature operation, the properties of spray-dried powder may vary significantly. An evaporating milk droplet in a spray drier in co-current air flow does not initially appreciably exceed the wet bulb temperature and can be held effectively at temperatures below 60 \u00b0C. As the falling temperature period is approached in the course of further evaporation, the temperature rises to a final value determined by the final temperature of the drying gas and the residence time in the drier. Under properly controlled spray drying conditions, the changes in milk protein structure and solubility are minor. Spray drying does not denature the whey protein significantly, and the levels of denatured whey protein in dairy powders are more or less equal to those of condensed milk and heated milk, which is substantially more denatured than during spray drying. An excellent example is in relation to the whey protein nitrogen index (WPNI). According to Pisecky (1997), the WPNI expresses the amount of undenatured whey protein (milligrams of whey protein nitrogen per gram of powder). It is a measure of the sum of heat treatments to which the milk has been subjected prior to evaporation and spray drying. The heat treatment of a concentrate, and subsequently of a powder, has only a negligible effect on the WPNI. The main operation to achieve the required value is the pasteurization process, that is, time\/temperature combination. However, many other factors influence the WPNI, including the total amount of whey protein and the overall composition of the processed milk as influenced by animal breed and seasonal variations. The individual design of the processing equipment, that is, the pasteurizer and holding tubes, is also very important. It is therefore difficult to predict the conditions required to achieve the required WPNI on a general basis. The main purpose of heat pretreatment is obviously to ensure the microbiological quality of dairy products.\n\nThe influence of the heat treatment on the denaturation of whey proteins to achieve the desired properties of the final products is just as important in milk powder production. SMP for cheese manufacture should have as much undenatured protein as possible; that is, it should be low heat-produced (WPNI >6), whereas, for bakery products, high heat-produced powder with high denaturation is required (WPNI <1.5). For ice cream, chocolate, and confectionery, medium heat-produced powder is required. According to Schuck et al. (1994a), the use of microfiltration (pore diameter, 1.4 \u03bcm), combined with low-heat treatment during vacuum evaporation, allows the production of a low low heat SMP with a WPNI close to 9 mg of whey protein nitrogen\/g of powder, bacterial count <1000 CFU\/g powder, solubility index >99.5%, dispersability index >98.5%, and wettability index <15 s. After water rehydration, such a powder has the same renneting time as the original raw milk and can be used as a reference powder for both industrial and scientific purposes.\n\nThe stability of protein powders during storage is critically affected by the moisture content and the storage temperature. More precisely, such stability is governed by the water activity (aw) and the glass transition (Tg) temperature. The aw should be close to 0.2 at 25 \u00b0C for optimal preservation, with an ideal moisture content determined by using the sorption isotherm of some dairy powders. For example, the corresponding moisture content for skim milk, whey, and protein powders must be close to 4%, 2\u20133%, and 6%, respectively. The optimal storage temperature must be below the Tg temperature, which is close to 40\u201350 \u00b0C at 0.2 aw.\n\n### Examples of Dairy Protein Concentrates and Powders\n\nTo illustrate this chapter through examples, micellar casein concentrate (MCC) was prepared by microfiltration and diafiltration (pore diameter, 0.1 \u03bcm) on an MFS 19 (Tetra Laval, \u00c5arhus, Denmark; 4.6 m2) at 50 \u00b0C, at 200 g\/kg total solids, according to Fauquant et al. (1988) and Pierre et al. (1992). The whey protein concentrate (WPC) was obtained by ultrafiltration and diafiltration of the microfiltrate (0.1 \u03bcm) using a DDS module (GEA, Soeborg, Denmark) with the plane membrane (10,000 g\/mol molecular weight cut-off, 9 m2, 50 \u00b0C) at 200 g.kg-1 total solids. The sodium, calcium, and potassium caseinate concentrates were reconstituted from sodium, calcium, and potassium caseinate powder at 190 g.kg-1 total solids. The microfiltration retentate (R4 MF) was obtained by microfiltration (0.1 \u03bcm) on an MFS 19 (Tetra Laval, \u00c5arhus, Denmark; 4.6 m2) at 50 \u00b0C. The volume reduction ratio was 4. The ultrafiltration retentate (R4 UF) was manufactured by ultrafiltration on a 2-S37 module with M1 membranes (Tech Sep, Rh\u00f4ne Poulenc, St Maurice de Beynost, France; 100,000 g\/mol nominal molecular weight cut-off, 6.8 m2) at 50 \u00b0C. The volume reduction ratio was 4. NaCl solution, CaCl2 solution, sodium phosphate solution at pH 7.1 (for MCC) or pH 6.6 (for WPC) and sodium citrate solution at pH 7.1 (for MCC) or pH 6.6 (for WPC) in 205 5 g.kg-1. Total solids were added to the MCC or the WPC to obtain a concentrate with 12% (w\/w) (NaCl, CaCl2, and sodium phosphate solution) and 30% (w\/w) (sodium citrate) of mineral salts\/total solids. After the addition of salt, the pH was adjusted to 7.1 (for MCC) or 6.6 (for WPC) with 1 N KOH (NaCl, CaCl2, and sodium phosphate solution) or 1 N HCl (sodium citrate) at 20 \u00b0C.\n\nSpray drying of the concentrates was performed at Bionov (Rennes, France) in a three-stage pilot plant spray drier (GEA, Niro Atomizer, St Quentin en Yvelines, France) according to Schuck et al. (1998a) and Bimbenet et al. (2002) to obtain a micellar casein powder (MCP) or a whey protein powder (WPP). The temperature of the concentrate before drying was 40 \u00b0C for MCC and 20 \u00b0C for WPC. The atomizer was equipped with a pressure nozzle (0.73 mm diameter orifice) and a four-slot core (0.51 mm nominal width), providing a 60\u00b0 spray angle. The evaporation capacity was 70\u2013120 kg.h-1 (depending on the inlet and outlet air temperatures and the air flow). The pressure in the nozzle was 16 MPa. The inlet temperature was 208 \u00b0C for WPC and 215 \u00b0C for MCC; the integrated fluid bed air temperature was 70 \u00b0C for MCC and WPC; and the outlet temperature was 80 \u00b0C for WPC and 70 \u00b0C for MCC. The inlet air humidity was controlled and adjusted by a dehumidifier (Munters, Sollentuna, Sweden). Two granulation states were obtained for each MCP or WPP, i.e., nongranulated (NG) and granulated (G) powders, by reintroduction of the fine particles after the cyclones at the top of the spray drier.\n\n#### Research Approach Using Drying by Desorption\n\nThe concentrates were dried in a water activity meter (Novasina RTD 200\/0, Pf\u00e4ffikon, Switzerland) at 20 \u00b0C (constant temperature). The concentrate (100 mg) was placed in a plastic support (area, 95 mm2) with a zeolite WE 291 dryer below (7 g) (Bayer, Puteaux, France). This method was used to simulate the conditions of spray drying by establishing a difference in vapor pressure equilibrium between the dairy concentrate and the drying air and to determine the water transfer from inside the dairy concentrate to the surface. The RH was measured in relation to time following water transfer from the dairy concentrate to the zeolite. The final slope of the absolute value of the decrease in RH (\u03b2) represented the ability to remove bound water from the solute at the end of the drying phase (Schuck et al. 1998b; 1999); the lower the \u03b2 slope, the greater the difficulty of removing water at the end of drying and the higher the bound water content.\n\n#### Desorption Results\n\nThe drying slopes (\u03b2) of the various dairy products tested are shown in Table 10.1. The dairy products, ranging from the highest to the lowest absolute value of the slope, were skim milk (\u03b2 = 0.90%.min-1), R4 UF (\u03b2 = 0.75%.min-1), R4 MF (\u03b2 = 0.70%.min-1), and MCC (\u03b2 = 0.34%.min-1). These results could be explained by a decrease in water diffusion through the dried product, that is, the final residue obtained at the end of drying, when the micellar casein concentration increased. Water transport was probably affected by the high micellar casein content of the sprayed droplet in the atomization tower, and similarly when the powder granule was dissolved in water. These results are in accordance with the results of Schuck et al. (1994a, b).\n\nTable 10.1\n\nDrying of Dairy Concentrates by Desorption\n\nDairy concentrate | \u03b2 slope (%.min-1)  \n---|---  \nSkim milk | 0.90  \nR4 UF | 0.75  \nR4 MF | 0.70  \nMCC | 0.34  \nWPC | 0.68  \nSodium caseinate | 0.64  \nPotassium caseinate | 0.65  \nCalcium caseinate | 0.51  \nWPC + NaCl | 0.24  \nWPC + CaCl2 | 0.46  \nWPC + Phosphate | 0.41  \nWPC + Citrate | 0.48  \nMCC + NaCl | 0.19  \nMCC + CaCl2 | 0.36  \nMCC + Phosphate | 0.49  \nMCC + Citrate | 0.45\n\nR: retentate; UF: ultrafiltration; MF: microfiltration; MCC: micellar casein concentrate; WPC: whey protein concentrate\n\nDrying of the caseinates showed that sodium caseinate and potassium caseinate dried more easily (\u03b2 = 0.64% and 0.65%\/min, respectively) than calcium caseinate (\u03b2 = 0.51%\/min) (Table 10.1). The limitation of water diffusion through the calcium caseinate may be explained by the structure of this colloidal dispersion. In calcium caseinate, the casein subunits are more aggregated because of calcium binding, whereas in sodium and potassium caseinates, the caseins are more soluble. The above results showed that the water bound in a micellar structure (e.g., MCC) was more strongly bound than that bound to the soluble caseins in sodium caseinate. The situation was intermediate for calcium caseinate. We assumed that these differences in water transfer during drying could be explained by the casein structure. Therefore the decrease in water inside the dairy concentrate led to a decrease in the water concentration on the surface of the concentrate in the water activity meter or on the surface of the droplet during spray drying and thus decreased the drying kinetics. These results were confirmed by the desorption drying of two different classes of proteins, that is, MCC (micellar structure), with a \u03b2 value of 0.34%\/min, and WPC (globular structure), with a \u03b2 value of 0.68%\/min (Table 10.1). These two different types of protein had the same protein content (89% of total solids) and the same water content before desorption drying, but not the same drying time or \u03b2 value. All these results show that the drying rate is dependent on the nature and structure of the casein. Water may be less available during the drying of a protein with a micellar structure than during the drying of a protein with a globular structure.\n\nAddition of minerals to WPC decreased the \u03b2 value, the smallest decrease being with citrate (29% reduction) and the greatest decrease being with NaCl (65% reduction) (Table 10.1). A decrease in the \u03b2 value means a lower rate of water transfer at the end of drying (Schuck et al., 1998b). The decrease in water transfer during the drying of modified WPC could be explained by the high hygroscopicity of the added mineral salts. This result suggested that the water is more closely bound to the mineral salts than to the whey proteins at the end of drying. The addition of minerals to WPC under the test conditions had little effect on the whey protein structure but probably had some effect on the increase in bound water in the modified WPC.\n\nAddition of NaCl to MCC decreased the \u03b2 value (0.19%\/min) (Table 10.1). Water in a casein system containing NaCl is more rotationally mobile than water in a casein model system without NaCl, with the same water activity (Curme et al., 1990). For high electrolyte concentrations, the amount of bound water decreases because of the suppression of the electrical double layer surrounding the protein molecule. This is directly related to the hydration of ions (Na+, Cl\u2013) and hence to the ability to separate the water molecules from the protein molecules (Robin et al., 1993). Water is less closely bound to micellar casein in the presence of NaCl (Cayot & Lorient, 1998). The decrease in water transfer at the end of drying can be explained by the hygroscopicity of NaCl. The water is more closely bound to NaCl than to the micellar caseins.\n\nAddition of CaCl2 to MCC increased the \u03b2 value (0.36%\/min) (Table 10.1). Moreover, addition of CaCl2 to milk decreases micellar solvation (Tarodo de la Fuente, 1985; Van Hooydonk et al., 1986; Jeurnink & de Kruif, 1995; Le Ray et al., 1998). First, the water inside the micellar structure in the case of MCC without addition of salt might be less available than the water inside the micellar structure of MCC with CaCl2. Second, the lower water content inside micellar structures with additional calcium might lead to an increase in water transfer during drying.\n\nAddition of phosphate ions to MCC increased the \u03b2 value (0.49%\/min) (Table 10.1). The increase in water transfer was explained by the partial solubilization of caseins (Le Ray et al., 1998), although an increase in casein micelle solvation and an increase in viscosity were observed. These results may be discussed in terms of the strength of water binding to caseins, either in a soluble form as in sodium caseinate (Schuck et al., 1998b) or in the micellar structure (MCC) as in the experiments reported here. The water bound to micellar caseins was probably less easy to remove than the water bound to soluble caseins. Partial solubilization of caseins improved water transfer during dehydration of phosphate solution + MCC.\n\nAddition of citrate solution to MCC at 30%\/total solids increased the \u03b2 value (0.45%\/min) (Table 10.1). Addition of citrate induces the release of large amounts of soluble casein from the micellar phase because of solubilization of colloidal calcium phosphate (Le Ray et al., 1998). Similar results occurred with the addition of phosphate ions. Solubilization of the micellar casein improved the water transfer during dehydration of citrate solution + MCC.\n\n### Industrial Implications\n\n#### Introduction\n\nSeveral studies (Masters, 1991; Pisecky, 1997) have reported that the moisture content can vary for the same outlet air temperature in the spray drier according to the product. For example, Pisecky (1997) reported that the moisture content is close to 4% and 5% for WMP and SMP, respectively, for an outlet air temperature close to 85 \u00b0C. On the other hand, to produce WMP and SMP at the same moisture content (4%, for example), the outlet air temperatures must be different (80 \u00b0C and 90 \u00b0C, respectively). All these differences can be explained by the effects of the biochemical composition of the ingredients on the availability of the water to be transferred from the droplet to the drying air.\n\n#### Availability of Water\n\nConcerning the availability of the water, Schuck et al. (1998a,b; 2009) have developed a new method (drying by desorption, using a thermohygrometer sensor) in order to determine major drying parameters of food components in relation to their interactions with water (bound and free water) and linked to water-transfer kinetics. Studies by Schuck et al. (1998b; 2009) and Zhu et al. (2011a,b) have shown that drying by desorption is an excellent tool to determine and optimize the major spray-drying parameters in relation to biochemical composition according to water availability and desorption behavior (calculation of extra energy, \u0394E). The experimental device proposed by the authors differs from spray-drying equipment in terms of duration of drying, drying temperature, surface\/volume ratio, and so on, because the concentrate is dried in a cup and not in a droplet. Computational tools have been developed to improve the method by taking these factors into account such as the 'Spray Drying Parameter Simulation and Determination Software' (SD2P\u00ae) registered under the following identification: IDDN.FR.001.480002.003.R.P.2005.000.30100. For reasons of calculation speed and reliability, this method has been computerized, and it can already be used in determining the parameters of spray drying for food products. Validation tests (>80 products) have indicated that this method could be applied to a wide range of food products and spray-dryer types.\n\nCombined with knowledge of the temperature, the software provides analysis of the desorption curve (measured relative humidity versus time), total solids, density, and specific heat capacity of the concentrate, air flow rates, water content, RH of the outlet air, current weather conditions, and cost per kWh. The percentage of drying in the integrated fluid allows determination of enthalpy H, T, RH (including \u0394E) for each inlet air, concentrate and powder flow rate, specific energy consumption, energy and mass balance, yield of the dryer, and cost (in \u20ac or in $) to remove 1 kg of water or to produce 1 kg of powder (summarized in Fig. 10.4). This figure is a representation of the software delivery:\n\n\u2022 air characteristics at the dryer (with or without integrated fluid bed) inlet and outlet (upper part)\n\n\u2022 flow, energy, and cost calculations (lower part) (Schuck et al., 2009)\n\nFigure 10.4 Parameters of spray drying calculated by SD2P\u00ae software.\n\nThus, the interests of the desorption curves lay in evaluatimg water transfer during spray drying of various dairy concentrates using thermodynamic and biochemical approaches. Whey protein concentrates and isolates (WPC35, WPC50, WPC70, WPI90) with or without heat denaturation, MC, sodium caseinate (NaCas), and milk with and without whey protein enrichment were dried in a three-stage pilot plant spray drier. When the concentrate temperature, air flow rate, concentrate flow rate, total solids content of the concentrate, inlet air temperature absolute humidity, inlet air temperature before and after heating, and outlet air temperature after drying are known, it is possible to determine the specific energy consumption (SEC)\u2014that is, the ratio of the energy consumed to the evaporation of 1 kg of water (measured in kJ.kg-1 water) (Schuck et al., 1998a; Bimbenet et al., 2002). Thus, if you spray dry only free water, the energy used in terms of SEC would be close to 2500 kJ.kg-1 water. If the concentrate amounts of bound water to free water increase, the SEC increases (up to 10,000 kJ.kg-1 water, for example). The significance of a very high SEC relates to the decreasing availability of the water, limiting water transfer, and thus increasing both the surface temperature of the droplet and the risk of protein denaturation of the powder.\n\n#### Whey Proteins\n\nThe results presented in Table 10.2 show that water transfer during spray drying decreased when the whey proteins were native proteins. For the same moisture content, the SEC for drying was higher when (a) the native whey protein content increased in WPC and in milk and (b) the whey proteins were heat denatured in WPC35. However, the SEC was lower when (c) the whey proteins were heat denatured in WPC50, WPC70, and WPI90. These results can be explained by the availability of the water (bound or unbound) in the concentrate in relation to the nature and content of the whey proteins.\n\nTable 10.2\n\nSpecific Energy Consumption at 4% Moisture Content for the Drying of Dairy Proteins\n\n| SM | SM + WPI | WPC | WPI | MC | NaCas  \n---|---|---|---|---|---|---  \nProtein content (%) | 34 | 50 | 35 |  | 50 | 70 | 90 |  | 90 | 90  \nHeat treatment (72 \u00b0C\/4 min) | N | N | N | Y | N | Y | N | Y | N | Y | N | N  \nSEC (\u00b13%) (kJ.kg-1 water) | 5900 | 6400 | 5950 | 7700 | 6800 | 6550 | 7050 | 6600 | 7200 | 6500 | 6900 | 5900\n\nY: heat treatment; N: no heat treatment; SM: skim milk; WPI: whey protein isolate; WPC: whey protein concentrate; MC: micellar casein; NaCas: sodium caseinate; SEC: specific energy consumption at 4% of moisture content.\n\n#### Caseins\n\nThe results presented in Table 10.2 also show that water transfer during spray drying decreased when the micellar casein content increased. For the same moisture content, the SEC for drying was higher when (a) the micellar casein content increased in MC compared with skim milk and (b) casein remained in a micellar state (as in MC) compared with a soluble state (e.g,. in NaCas). These results can be explained by the availability of the water in the concentrate in relation to the content and the structure of the caseins. Water is more available when the caseins are soluble than when they are in a micellar state.\n\nAll these results also show that water transfer depends on the relationship between the water and the protein components and that these components should be taken into account when optimizing spray drying parameters for proteins.\n\n#### Rehydration of Protein Powders\n\nMost food additives are prepared in powder form and need to be dissolved before use. Water interactions in dehydrated products and dissolution are thus important factors in food development and formulation (Hardy et al., 2002). Dissolution is an essential quality attribute of a dairy powder as a food ingredient (King, 1966). Many sensors and analytical methods such as the insolubility index (ISI, International Dairy Federation, 1988; American Dairy Products Institute, 1990), nuclear magnetic resonance (NMR) spectroscopy (Davenel et al., 1997), turbidity, viscosity, and particle size distribution (Gaiani et al., 2006) can now be used to study water transfer in dairy protein concentrates during rehydration. Using combinations of these methods, it is very easy to determine the different stages of the rehydration process, that is, wettability, swellability, sinkability, dispersibility, and solubility.\n\nThe ISI (in %) described by the IDF standard (International Dairy Federation, 1988; Schuck et al., 2012) for skim milk is the volume of sediment (for 50 mL) after rehydration (10 g of powder in 100 mL of distilled water, at 25 \u00b0C), mixing (for 90 s, at 4000 rev\/min) and centrifugation (for 300 s, at 160 g). With this method, the quantity of insoluble material (whether true or false) can be determined.\n\nNMR spectroscopy is a technique for determining the rate of solution, the time required for complete reconstitution of powders, and the transverse relaxation rate of reconstituted solutions. The method was first described by Davenel et al. (1997). A 40 mm diameter glass tube filled with 20 mL of water at 40 \u00b0C was placed in the gap of the magnet of a Minispec Bruker PC 10 NMR spectrometer operating at a resonance frequency of 10 MHz. A suitably designed funnel and an electric stirrer (glass spatula) were inserted into the tube. They showed that the solubilization rate was independent of the quantity of powder poured (up to 20 g powder\/100 mL water) and increased with the stirring rate. In subsequent experiments, the rotation rate of the stirrer was adjusted after starting at 1150 rev.min-1 for spray-dried powders, and 1 g of powder was poured into the water. The NMR measurements were generally continued until the solution was completely reconstituted, except if insoluble material was formed. Each decay curve was obtained by sampling a maximum of 845 spin echoes of a Carr-Purcell-Meiboom-Gill (CPMG) sequence every 20 s during the reconstitution period. Interpulse spacing between 180\u00b0 pulses was fixed at 2 ms to limit the diffusion effect caused by stirring. The NMR kinetics method was used in triplicate. The CPMG curves were well approximated by the sum of two exponential curves to determine the protons attributed to water protons in fast exchange with exchangeable protons of nondissolved powder particles, as well as the protons attributed to water protons and exchangeable protons in the reconstituted phase (Davenel et al., 1997). With this method, it is possible to differentiate between truly insoluble material and falsely insoluble material. The falsely insoluble material can be explained by the low water transfer during rehydration and not by denatured protein, which is truly insoluble (Schuck et al., 1994b).\n\nFor viscosity measurement, a rheometer can be used to obtain viscosity profiles. In the studies of Gaiani et al. (2005; 2006), the blades were placed at right angles to each other to provide good homogenization. Industrial dissolution processes usually include stirring at a constant speed, and the experiments were therefore designed to provide a constant shear rate (100 s\u20131). MCP was added to the rheometer cup manually. The aqueous phase used was distilled water at a volume of 18 mL. The powder was dispersed in the rheometer cup 50 s after starting the rheometer. Dissolution was highly dependent on temperature and concentration. The total nitrogen concentration employed to study these effects was about 5% (w\/v), and the temperature was about 24 \u00b0C.\n\nThe experiments to provide the turbidity profiles were carried out in a 2 L vessel equipped with a four-blade 45\u00b0 impeller rotating at 400 rev\/min. A double-walled jacket vessel maintained the temperature at 24 \u00b0C. The turbidity sensor was placed 3 cm below the surface of the water and was positioned through the vessel wall to avoid disturbance during stirring. Turbidity changes accompanying powder rehydration were followed using a turbidity meter. The apparatus used light in the near-infrared region (860 nm), the incident beam being reflected back at 180\u00b0 by any particle in suspension in the fluid to a sensitive electronic receptor (Gaiani et al., 2005).\n\nA laser light diffraction apparatus with a 5 mW He\u2013Ne laser operating at a wavelength of 632.8 nm can be used to record particle-size distributions. The particle-size distribution of dried particles was determined by using a dry powder feeder attachment, and the standard optical model presentation for particles dispersed in air was used. To measure the particle size distribution of micellar casein in concentrates, 0.5 mL of suspension was taken from the rheometer cup and introduced into 100 mL of prefiltered distilled water (membrane diameter, 0.22 \u03bcm) to reach the correct obscuration. The results obtained corresponded to average diameters calculated according to the Mie theory. The criterion selected was d(50), meaning that 50% of the particles had diameters lower than this criterion (midpoint of cumulative volume distribution) (Gaiani et al. 2005; 2006).\n\nBy using this combination of three methods, it was possible to follow the water transfer during rehydration and obtain the wetting time, determined using the first peak of increased viscosity and turbidity, and the swelling time, determined using the second peak of viscosity in relation to the increase in particle size. The rehydration time was then determined according to stabilization of the viscosity, turbidity, and particle-size values.\n\nThe results in Table 10.3 show that rehydration of MCP occurs in different stages: First wetting and swelling of the particles take place, followed by slow dispersion to reach a homogeneous fluid, in agreement with Gaiani et al. (2005; 2006). Using an NMR method, Davenel et al. (1997) also demonstrated two stages during MC rehydration, attributed to water absorption by powder and solubilization of particles (i.e., swelling and dispersion stages). They estimated the water uptake by the powder at around 5 g water\/g powder during the first 20 min of rehydration but could not identify a wetting stage with this method.\n\nTable 10.3\n\nReconstitution Period, Insolubility Index, and Rehydration Time of Dairy Protein Powders\n\nPowders | RP using NMR (min) | ISI using IDF Standard (mL) | WT (min) | ST (min) | DT + SolT (min) | RT (min)  \n---|---|---|---|---|---|---  \nMCP G | 22 | 14.5 | 1 | 2 | 804 | 807  \nMCP NG | 8 | 3.5 | 3 | 17 | 551 | 571  \nMCP + Carbohydrate, G | 18 | 5.0 | 1 | nm | nm | 116  \nMCP + Carbohydrate, NG | nm | nm | 2 | 0 | 95 | 97  \nMCP + NaCl, G | 9.5 | 0.9 | nm | nm | nm | nm  \nMCP + CaCl2, G | \u221e | 14.5 | nm | nm | nm | nm  \nMCP + SCS \/ SPS, G | 6 \/5 | <0.5 | nm | nm | nm | nm  \nWPP, G | 5 | <0.5 | 4 | 0 | 0 | 4  \nWPP, NG | 15 | <0.5 | 17 | 0 | 0 | 17\n\nMCP: micellar casein powder; G: granulate; NG: nongranulate; SCS: sodium citrate solution; SPS: sodium phosphate solution; WPP: whey protein powder; RP: reconstitution period; ISI: insolubility index; WT: wetting time; ST: swelling time; DT: dispersability time; SolT: solubility time; RT: rehydration time = WT + ST + DT + SolT; \u221e: infinite delay; nm: not measured.\n\nMCPs with a high ISI (14.5 mL) are generally considered to be poorly soluble powders in which rehydration of the micelle remains incomplete (Jost, 1993). Addition of NaCl to the MC concentrate before spray drying considerably reduced the ISI and reconstitution period (RP) values (ISI 0.9 mL; RP 9.5 min) (Table 10.3). It has been hypothesized that the significant decrease in the RP value is probably related to the hygroscopic strength of NaCl.\n\nAddition of sodium citrate solution (SCS) or sodium phosphate solution (SPS) resulted in fast solubilization, as shown by the very low RP value and by the lower than 0.5 mL ISI value (Table 10.3). The resulting solution consisted of changing casein micelles in sodium caseinate form, associated with the occurrence of a single proton population, characterized by NMR relaxation rates that were much lower than the relaxation rate measured with reconstituted MC. This could be attributed to a decrease in the amount of hydration water induced by the change in micelle structure. The transparency of the solution indicated the formation of soluble caseins related to greater quantities of calcium complexes.\n\nReconstitution of MCP in the presence of CaCl2 led to considerable changes in protein structure, associated with instability of the casein micelles, which began to precipitate just after mixing, as shown by the high ISI and the nonmeasurable RP value due to experimental delay. This precipitate probably resulted from aggregation of casein micelles or submicelles through a decreasingly negative charge on the protein by additional calcium binding, leading to a reduction in electrostatic repulsion (Dalgleish, 1982). In this case, the high ISI of these solutions was related to the presence of insoluble substances, whereas in the case of rehydration of MCP the high ISI represented only the low water transfer rate in the casein (Table 10.3).\n\nOn the other hand, the rehydration of whey powders was totally different (Table 10.3). As the wettability of whey powders is poor, the turbidity instability at the beginning of the profile could be due to lump formation going past the sensor, as reported by Freudig et al. (1999). For nongranulated (NG) WPI powder, the very long signal instability could be explained by a tendency for the lumps to be stuck together in a thick layer of wet particles, due to the small size of the particles (Kinsella, 1984). Powder swelling has not been reported for WPI powders, probably because globular protein powders bind less water than intact casein micelle powders (Kinsella, 1984; Robin et al., 1993). De Moor and Huyghebaert (1983) also reported that whey powders have a lower water-holding capacity than casein powder.\n\nAs expected, granulation had a positive effect on wetting. The wetting time was systematically better for granulated particles. This phenomenon is well known, as fast wetting is enhanced, with large particles forming large pores, high porosity, and small contact angles between the powder surface and the penetrating water (Pisecky, 1986; Freudig et al., 1999; Gaiani et al., 2005). A surprising influence of granulation on the rehydration time was observed. Depending on the nature of the protein, the granulation influence involved opposite effects. WPI rehydration was enhanced for granulated particles, whereas the rehydration time was shorter for nongranulated particles of MCP. This was unexpected and could be explained by the controlling stage rate. The controlling stage for whey proteins is wetting (Baldwin & Sanderson, 1973; Schubert, 1993). As granulation improves the wetting stage, the rehydration of whey powders is enhanced for granulated particles. In contrast, the controlling stage for casein proteins is dispersion. Indeed, even with a shorter wetting time, a granulated powder is slower to rehydrate than a nongranulated powder (Gaiani et al., 2005).\n\nThe above results are not compatible with those of other studies, in which it was generally accepted that a single particle size around 200 \u03bcm (Neff & Morris, 1968) or 400 \u03bcm (Freudig et al., 1999) represented optimal dispersibility and sinkability. In fact, the optimal particle size depends on the composition of the dairy powder. As shown in Table 10.3, if the industry wishes to optimize powder rehydration, it seems to be better to rehydrate granulated powders if the protein is whey and to rehydrate nongranulated powders if the protein is casein.\n\n## Conclusions\n\nThe aim of this chapter was to explain the process of dehydration, that is, spray drying, in order to understand the effects of spray drying on the quality of protein powders (micellar caseins and globular proteins) during drying and rehydration. We then demonstrated that the quality of these powders depends on the biochemical environment.\n\nIt is very important for the dairy industry to understand that enrichment of milk in micellar casein (by ultrafiltration or microfiltration) decreases water transfer during the drying and rehydration processes. Insolubility (International Dairy Federation, 1988) is related to the decrease in water transfer required for rehydration and not to thermal denaturation, and decrease in water transfer is related to the micellar structure. The destabilization of the micellar structure induced by the addition of phosphate or citrate solution to MC increases water transfer during drying and during rehydration. Water transfer in WPC or MC containing added carbohydrates is improved during rehydration. Addition of NaCl to MC decreases water transfer during drying but increases water transfer during rehydration, and thus is related to the hygroscopicity of the carbohydrate and the NaCl.\n\nThe industrial requirement for protein powders with specific properties is expanding. As powder is the easiest way to carry and store milk derivatives, an understanding of the rehydration behavior of a dairy powder will become more and more important in the future.\n\nMoreover, it is essential for both dairy powder producers and dairy powder users to have a method to evaluate the rehydration behavior of dairy powders. As demonstrated in several studies, the industry should take into account certain technological factors such as granulation and the incorporation mode, and also the nature of the protein being rehydrated, to optimize the rehydration of a dairy powder. In contrast to other studies, we found that improving the wetting stage by using granulated powders did not systematically improve total rehydration. Depending on the nature of the protein, it seems to be better to work with granulated (for whey) or nongranulated (for micellar casein) powders to obtain more rapid rehydration (Gaiani et al., 2007).\n\nIn conclusion, water transfer in dairy protein concentrates during dehydration and during rehydration depends on the aqueous environment, the nature of the mineral salts, and the structure of the dairy proteins (MC or WPC). The water\u2013protein interaction requires further study, to understand the effects of preheat treatment and spray drying on the functional properties of protein powders.\n\n# References\n\nAmerican Dairy Products Institute .  _Standards for Grades of Dry Milk Including Methods of Analysis_ .  Chicago, Illinois :  American Dairy Products Institute ;  1990 .\n\nBaldwin AJ ,  Sanderson WB .  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Caract\u00e9risation \u00e9nerg\u00e9tique d'une tour de s\u00e9chage par atomisation multiple effet .  _Industries Alimentaires et Agricoles_.  1998 ;115 : 9 \u2013 14 .\n\nSchuck P ,  Roignant M ,  Brul\u00e9 G ,  Davenel A ,  Famelart MH ,  Maubois JL .  Simulation of water transfer in spray drying .  _Drying Technology_.  1998 ;16 : 1371 \u2013 1393 .\n\nSchuck P ,  Briard V ,  M\u00e9jean S ,  Piot M ,  Famelart MH ,  Maubois JL .  Dehydration by desorption and by spray drying of dairy proteins: influence of the mineral environment .  _Drying Technology_.  1999 ;17 : 1347 \u2013 1357 .\n\nSchuck P ,  M\u00e9jean S ,  Dolivet A ,  Jeantet R .  Thermohygrometric sensor: a tool for optimizing the spray drying process .  _Innov. Food Sci. and Emerg. Technol_.  2005 ;6 : 45 \u2013 50 .\n\nSchuck P ,  Dolivet A ,  M\u00e9jean S ,  Zhu P ,  Blanchard E ,  Jeantet R .  Drying by desorption: A tool to determine spray drying parameters .  _Journal of Food Engineering_.  2009 ;94 : 199 \u2013 204 .\n\nSchuck P ,  Dolivet A ,  Jeantet R .  _Analytical Methods for Food and Dairy Powders_ .  Oxford :  John Wiley & Sons ;  2012 .\n\nSougnez M .  L'\u00e9volution du s\u00e9chage par atomisation .  _Chimie Magazine_.  1983 ;1 : 1 \u2013 4 .\n\nStraatsma J ,  Vanhouwelingen G ,  Steenbergen AE ,  Dejong P .  Spray drying of food products. 1. Simulation model .  _Journal of Food Engineering_.  1999 ;42 : 67 \u2013 72 .\n\nStraatsma J ,  Vanhouwelingen G ,  Steenbergen AE ,  Dejong P .  Spray drying of food products. 2. Prediction of insolubility index .  _Journal of Food Engineering_.  1999 ;42 : 73 \u2013 77 .\n\nTarodo de la Fuente B .  Solvation of casein in bovine milk .  _Journal of Dairy Science_.  1985 ;58 : 293 \u2013 300 .\n\nTrouv\u00e9 E ,  Maubois JL ,  Piot M ,  Madec MN ,  Fauquant J ,  Rouault A ,  Tabard J ,  Brinkman G .  R\u00e9tention de diff\u00e9rentes esp\u00e8ces microbiennes lors de l'\u00e9puration du lait par microfiltration en flux tangentiel .  _Lait_.  1991 ;71 : 1 \u2013 13 .\n\nTuohy JJ .  Some physical properties of milk powders .  _Irish Journal of Food Science and Technology_.  1989 ;13 : 141 \u2013 152 .\n\nVan Hooydonk ACM ,  Hagedoorn HG ,  Boerrigter IJ .  The effect of various cations on the renneting of milk .  _Netherlands Milk and Dairy Journal_.  1986 ;40 : 369 \u2013 390 .\n\nVincens D ,  Tabard J .  L'\u00e9limination des germes bact\u00e9riens sur membranes de microfiltration .  _Techniques Laiti\u00e8res_.  1988 ;1033 : 62 \u2013 64 .\n\nZhu P ,  Patel K ,  Lin S ,  M\u00e9jean S ,  Blanchard E ,  Chen XD ,  Schuck P ,  Jeantet R .  Simulating industrial spray drying operations using a reaction engineering approach and a modified desorption method .  _Drying Technology_.  2011 ;29 : 419 \u2013 428 .\n\nZhu P ,  M\u00e9jean S ,  Blanchard B ,  Jeantet R ,  Schuck P .  Prediction of dry mass glass transition temperature and the spray drying behaviour of a concentrate using a desorption method .  _Journal of Food Engineering_.  2011 ;105 : 460 \u2013 467 . \nChapter 11\n\n# Changes in Milk Proteins during Storage of Dry Powders\n\nKerianne Higgs*\n\nMike J. Boland**\n\n* Fonterra Research and Development Centre, Palmerston North, New Zealand  \n** Riddet Institute, Palmerston North, New Zealand\n\n## Abstract\n\nMilk proteins undergo chemical changes, even in dried powders. This chapter reviews the changes undergone by caseins and whey proteins in milk powders and in purified protein products. Maillard compounds are of particular importance in milk powders, milk protein concentrates, and whey protein concentrate products, where lactose is present. Caseins undergo isopeptide bond formation, as the result of dephosphorylation of phosphoserine residues and subsequent reactions of the dehydroalanine produced.\n\nWe discuss the nutritional significance of these changes. Both the Maillard reaction and the formation of isopeptides lead to loss of bioavailable lysine. This is not a problem in dairy proteins, which contain an excess of lysine, but it can affect the nutritional value of protein blends which may be limiting in lysine.\n\n## Keywords\n\nchanges in milk proteins\n\nstorage of dry powders\n\ndephosphorylation\n\nphosphoserine residues\n\ndehydroalanine\n\nOutline\n\nIntroduction 343\n\nThe formation of Maillard and pre-Maillard compounds 345\n\nMeasuring Lactulosyl Lysine Levels 346\n\nRates of Formation of Lactulosyl Lysine 346\n\nFormation of isopeptide bonds 349\n\nRates of Formation of Lysinoalanine 350\n\nAmino acids other than lysine 351\n\nImplications for nutritional value of milk proteins 352\n\nLysine 352\n\nSulfur Amino Acids 354\n\nOther Amino Acids 355\n\nProduct-specific storage trials 355\n\nConclusions 356\n\n## Introduction\n\nMilk is an unstable foodstuff, prone in particular to microbiological degradation, but also to long-term chemical change. A large part of the world's dairy production occurs in areas remote from the markets in which it is consumed, and production is often seasonal, requiring storage to smooth out supply. The production of milk powders and other dried milk protein-containing products has been the method of choice for over 100 years for the storage and shipping of milk over long distances and\/or times, as it confers stability and massively reduces weight and bulk.\n\nMilk powders were known to the Chinese and were described by Marco Polo. The production of milk powders was described by Nicolas Appert in the early nineteenth century, and commercial processes for the spray drying of milk were patented in the United States in 1872 and 1905. This opened the way for large-scale industrial production of milk powders throughout the twentieth century. Milk powder production is covered in Chapter 10.\n\nTable 11.1 summarizes the biggest exporters and importers of milk powders in 2011. In the same period, somewhere between 8 and 9 million tons of milk powder was produced globally (IDF, 2012).\n\nTable 11.1\n\nTop Six Exporters and Importers of Milk Powders in 2011 (000 tons)\n\nExport | New Zealand | European Union | United States | Australia | Argentina | Belarus  \n---|---|---|---|---|---|---  \nSMP | 350 | 518 | 436 | 166 | 18 | 55  \nWMP | 1081 | 390 | 22 | 143 | 199 | 27  \nTotal | 1431 | 908 | 458 | 309 | 217 | 82  \nImport | China | Algeria | Mexico | Indonesia | Singapore | Philippines  \n---|---|---|---|---|---|---  \nSMP | 130 | 125 | 191 | 128 | 61 | 111  \nWMP | 320 | 203 | 31 | 68 | 81 | 28  \nTotal | 450 | 328 | 222 | 195 | 142 | 139\n\nSMP, skim milk powder; WMP, whole milk powder\n\nSource: World Dairy Situation, International Dairy Federation, Brussels, 2012.\n\nIn addition to milk powders, dried dairy protein products that are traded on the world market, largely as food ingredients, include casein and caseinate, whey powders, whey protein concentrates, whey protein isolates, milk protein concentrates, milk protein isolates, and specialist nutritional powders and blends, which may also contain hydrolyzed dairy proteins.\n\nMilk powders are used primarily for making reconstituted and recombined milks and are usually sold to consumers in UHT format, although substantial amounts are used to make other dairy products such as yogurt and ice cream as well as being minor ingredients in a wide range of nondairy foods.\n\nThe main uses for dried milk protein products are nutritional, and products include infant formulas, medical foods, specialist foods for weight management, and foods for muscle building (where high protein and high levels of branched-chain amino acids are desirable). They are also used in non-nutritional applications, including desserts, confectionery, toppings, imitation cheeses, sauces and dressings, and coffee whiteners, where their functional properties are important.\n\nThese products, especially those from New Zealand and Australia, often have long storage times because of geographic distance to market and seasonal production. Research from our own and other laboratories has shown that a range of reactions can occur in the dry powders and that these are known to affect powder functionality and nutritional value. The quality of dried milk protein products deteriorates upon storage at ambient temperatures mainly because of two reactions: the Maillard reaction and isopeptide bond formation. There are also some other minor reactions, which are covered later in this chapter.\n\nThe standard abbreviations used throughout the world for many of the dried milk products are as follows: SMP = skim milk powder; WMP = whole milk powder; WPC = whey protein concentrate; WPI = whey protein isolate (usually >85% protein); MPC = milk protein concentrate; and MPI = milk protein isolate. Milk powders sold internationally must conform to standard protein levels and fat levels for WMP. A number following the abbreviation for the other products is the percentage of protein by weight in the dry powder. Caseins and caseinates are not usually abbreviated and are typically >90% protein by dry weight.\n\nA range of chemical reactions that modify proteins can occur in dried milk products, particularly at elevated temperatures. Both of the most important of these reactions involve lysyl side chains: they are the formation of Maillard and pre-Maillard compounds, in the presence of sugars, and the formation of isopeptide bonds, particularly in products containing phosphoseryl residues, such as casein.\n\n## The formation of maillard and pre-maillard compounds\n\nThe Maillard reaction occurs when lysine-containing proteins interact with reducing sugars. The first stable compound formed during the Maillard reaction is the Amadori product (so-called because it is the result of a class of reaction called the Amadori rearrangement), shown in Figure 11.1. These compounds block the \u025b-amino groups of lysine residues, reducing the bioavailability of that essential amino acid. This reaction is dependent on a reducing sugar, usually lactose, being present as the co-reactant (Erbersdobler, 1986).\n\nFigure 11.1 Formation of lactulosyl lysine.\n\nPure protein products such as casein and caseinate do not suffer significantly from this reaction, as not enough lactose is present. The reaction is particularly important in WMPs, SMPs, WPCs, and MPCs, and can occur to a limited extent in some MPIs and WPIs. The rate of reaction is critically dependent on the level of moisture (water activity) and the temperature as well as the lactose content.\n\nAdvanced Maillard reaction products are partly responsible for the development of aromas and color during food processing and preparation.\n\n### Measuring Lactulosyl Lysine Levels\n\nLactulosyl lysine formation is conveniently monitored by measuring furosine concentrations (Erbersdobler, 1987) and can be indirectly measured by monitoring the amount of available amine (Hern\u00e1ndez et al, 1991). Lactulosyl lysine in WPCs can be measured directly by mass spectrometry where the addition of one lactose molecule increases the molecular weight of the protein by 324 Da. Figure 11.2 shows the mass spectrum of a mixture of the native A and B variants of \u03b2-lactoglobulin and their mono-, di-, and trilactosylated derivatives. The results from all three methods have been shown to correlate well (Fig. 11.3).\n\nFigure 11.2 Mass spectrum of native and lactosylated \u03b2-lactoglobulin.\n\nFigure 11.3 (a) Correlation between available amine and furosine values for stored samples of a WPC56. (b) Average number of lactose molecules bound, determined by mass spectrometry, against furosine for a WPC56. K. Higgs, unpublished data.\n\n### Rates of Formation of Lactulosyl Lysine\n\nWe studied the rate of lactosylation for WPC products as dry powders. The rates were found to be dependent on the lactose concentration (a consequence of processing to reach desired protein levels), water activity (aW), and temperature (T). The trials lasted only a few months, and extrapolation beyond that timeframe cannot be done with confidence.\n\nDetailed kinetics developed using WPC80 for a range of aW and T values allowed the rates of lactosylation to be predicted for periods of up to 4 months.\n\nFor the kinetic evaluation, Figure 11.1 can be simplified to\n\nreactants \u27f6 k 1 lactulosyl lysine \u27f6 k 2 advanced Maillard products\n\nwhere k 1 and k 2 are the rate constants for the formation and degradation of lactulosyl lysine.\n\nThe rate equation for the formation of lactulosyl lysine is\n\nd L d t = k 1 [ reactants ] n 1 \u2212 k 2 [ L ] n 2\n\n (11.1)\n\nwhere L represents lactulosyl lysine and n 1 and n 2 are the rate orders for the formation and degradation of lactulosyl lysine respectively.\n\nBecause the formation of lactulosyl lysine in dairy powders uses only a fraction of the available reactants, the first reaction can be considered to be zero order, and the degradation of lactulosyl lysine is a first-order process. This simplifies the equation to\n\nd L d t = k 1 \u2212 k 2 [ L ]\n\n (11.2)\n\nRearrangement of this equation and solving for [L]t gives\n\n[ L ] t = k 1 k 2 \u2212 k 1 k 2 \u2212 [ L ] 0 ) exp ( \u2212 k 2 t )\n\n (11.3)\n\nFurosine is a hydrolysis product of lactulosyl lysine and gives a direct measure of its concentration (Fig. 11.3). Therefore, furosine can be substituted for lactulosyl lysine in the rate equations.\n\nThe samples used were individual samples that were removed from controlled storage at seven time points (2, 6, 12, 24, 40, 78, and 116 days). They were analyzed for furosine, and the values were plotted against time. The rates were determined using nonlinear regression with Sigma Plot 8.0. At 40 \u00b0C, the samples at time points after 40 days for water activities of 0.54 and 0.80 showed advanced Maillard browning and were not included in the regression analysis. R2 values for the regressions were between 0.95 and 0.99.\n\nThe lactosylation rate constants were greater at higher temperatures and at higher water activities. The rates were low at low water activity (aW = 0.33), with small increases with temperature; however, at higher water activities, the rates increased substantially with increasing temperature (Fig. 11.4).\n\nFigure 11.4 Rate constants for the formation of furosine in a WPC80.\n\nIt was observed for a number of WPC80 specifications that lactosylation appeared to stop when only 2.5 (average) or 3 lactose molecules had been bound per \u03b2-lactoglobulin molecule. This corresponded to about 20% of the total lysine being blocked. This condition was specific to the 80% protein products, which contained about 12% lactose. WPC56 products showed a much greater degree of lactosylation. However, much higher levels of lactosylation have reportedly been seen in overseas laboratories in WPC80 samples stored for long periods (W. J. Harper, Ohio State University, 2003, personal communication).\n\nSimilar kinetic work has been reported for SMPs stored at a range of temperatures (37, 50, and 60 \u00b0C), with a series of initial water activities (0.33, 0.43, 0.52, 0.69, 0.85, and 0.98) (Pereyra Gonzales et al., 2010). As with WPCs, there were significant increases in Maillard reaction rates with temperature, but they did not see an increase related to the powder's initial water activity.\n\n## Formation of isopeptide bonds\n\nIsopeptide bonds are formed largely by the breakdown of the phosphoseryl side chains that are present in products containing casein during processing or storage to form dehydroalanyl side chains (Fig.11.5) (Friedman, 1999). The side chains are reactive and will form cross-links, mainly with adjacent lysyl (but also with histidinyl or cysteinyl) side chains to form lysinoalanyl, histidinoalanyl, or lanthionyl isopeptides, respectively. The reaction during milk powder processing is considered to be an alkali-catalyzed reaction that is accelerated by heat treatment. This reaction is not known to be significant in whey products, which do not contain significant amounts of phosphoseryl residue.\n\nFigure 11.5 Formation of lysinoalanine.\n\nThe main isopeptide product in both acid and gastrointestinal digestion gives lysinoalanine, which renders the lysine nonbioavailable. Additional minor reactions form histidinoalanine and lanthionine on digestion. Although lanthionine is only a minor component, it is important because it renders cysteine partially nonbioavailable, and the sulfur amino acids are often nutritionally limiting in milk proteins. (Note: Lanthionine formation blocks the bioavailability of cysteine, which is not normally considered to be an essential amino acid, because it can be synthesized from methionine; however, methionine is itself a nutritionally limiting amino acid in casein.) Studies have indicated that, although lanthionine and histidinoalanine linkages are formed under the alkaline conditions encountered during processing, it is only lysinoalanine that is formed in the neutral conditions encountered in powders.\n\nLysinoalanine formation in \u03b2-casein has been found to be enhanced by pressure treatment under alkaline conditions (Schwarzenbolz & Henle, 2010). Still to be investigated is whether pressure treatment also affects the rate at which lysinoalanine is formed during subsequent storage.\n\nLysinoalanine is usually measured directly in protein hydrolysates as part of an extended amino acid analysis (Henle et al., 1991). In casein products, measurement of available amine is a useful and simpler, though less specific, alternative.\n\n### Rates of Formation of Lysinoalanine\n\nThe rates of formation of lysinoalanine in caseinates and proteinates (TMPates) have been investigated by W. Thresher (1996; 1997; personal communications) for a range of temperatures and water activities. The rate constants for lysinoalanine formation as a function of temperature are shown in Figure 11.6.\n\nFigure 11.6 Rate constants for lysinoalanine formation in caseinates stored at various temperatures. W. Thresher, 1996, personal communication.\n\nThe rate of lysinoalanine formation increased with increasing temperature and water activity for both caseinates and TMPates. Calcium TMPates and caseinates were found to have lower rates of lysinoalanine formation than potassium or sodium TMPates and caseinates.\n\n## Amino acids other than lysine\n\nCysteine, methionine, and tryptophan are other essential amino acids that can be rendered nonbioavailable by reacting during processing and\/or storage. Cysteine undergoes \u03b2-elimination to give dehydroalanine when treated with alkali. This can then react with a lysine residue to give lysinoalanine. Cysteine can also react with dehydroalanine to give lanthionine (Friedman, 1999). Chemically determined values for cysteine and lysine availability have been found to correlate well with rat protein efficiency ratios for heat- and alkali-treated caseinates (W. Thresher, 1996, personal communication).\n\nTryptophan residues are relatively stable during processing and storage. They are not easily oxidized and have been found to be relatively resistant to oxidizing lipids, alkali, quinines, and reducing sugars (Nielsen et al., 1985). Any losses are small and not significant when compared with the losses of other amino acids such as methionine and lysine. However, we have seen small amounts of oxidized tryptophan residues in digests of skim milk purchased from the supermarket (Fig. 11.7).\n\nFigure 11.7 Time-of-flight mass spectrometry of the M2+ ion from the tryptophan-containing \u03ba-casein peptide SPAQILQWQVLSNTVPAK. The chemical structures in the figure show the various levels of oxidation found.\n\nMethionine is relatively easily oxidized to the sulfoxide, but methionine in the sulfoxide form is still bioavailable (Nielsen et al., 1985).\n\nTable 11.2 indicates levels of key essential amino acids following either alkali treatment of casein or extensive lactosylation of WPC. Note that the losses of amino acids other than lysine are considerably lower than the losses of lysine. The conditions used for the casein are well beyond any normal exposure during processing or storage.\n\nTable 11.2\n\nAmino Acid Concentrations of a Control and an Alkali-treated Casein, and a Control and a Lactosylated WPC56\n\nAmino Acid | Concentration (mg\/g crude protein) | Concentration (mg\/g sample)  \n---|---|---  \n| Control casein | Alkali-treated casein a | Control WPC | Lactosylated WPC b  \nTryptophan | 13.4 | 12.1 (90%) | 8.4 | 8.1 (96%)  \nLysine | 91.0 | 60.8 (67%) | 50.3 | 44.5 (88%)  \nMethionine | 31.4 | 25.3 (80%) | 12.3 | 12.5 (102%)  \nCysteine | 4.4 | 3.3 (75%) | 14.9 | 14.8 (99%)\n\na Casein was heated for 4 h at 80 \u00b0C in 0.15 M NaOH (Nielsen et al., 1985).\n\nb WPC was heated at 40 \u00b0C, aW 0.75, for 100 h. The median number of lactosyl groups on \u03b2-lactoglobulin was 5 (K Higgs, unpublished results).\n\nSource: Nielsen et al., 1985; K Higgs, unpublished results.\n\n## Implications for nutritional value of milk proteins\n\nMilk protein is rich in essential amino acids, with many, including lysine, well exceeding recommended requirements (Table 11.3). Use of these proteins as the predominant nutritional source thus poses few problems if some of the lysine is nonbioavailable.\n\nTable 11.3\n\nEssential Amino Acid Content of Milk and Other Proteins\n\nEssential amino acid (AA) | Recommended requirementa (mg\/g protein) | Caseinate | TMP, MPI | WPC | Soy | Wheat  \n---|---|---|---|---|---|---  \nIsoleucine | 28 | 46 | 44 | 54 | 47 | 33  \nLeucine | 66 | 91 | 103 | 119 | 85 | 68  \nLysine | 58 | 77 | 81 | 94 | 63 | 27  \nSulfur AAb | 25 | 33 | 39 | 52 | 24 | 39  \nAromatic AA | 63 | 106 | 102 | 68 | 97 | 78  \nThreonine | 34 | 43 | 45 | 66 | 8 | 29  \nTryptophan | 11 | 12 | 14 | 20 | 11 | 11  \nValine | 35 | 57 | 57 | 51 | 49 | 43  \nHistidine | 19 | 29 | 27 | 21 | 29 | \u2013\n\na For 2- to 5-year-olds.\n\nb Includes cysteine, cystine, and methionine.\n\n### Lysine\n\nThe main concern during the storage of most nutritional proteins is the loss of lysine as a result of Maillard reactions or isopeptide bond formation.\n\nLactulosyl lysine renders lysine nonbioavailable. The protein efficiency ratio (PER) was found to be decreased in a WPC56, with an average of five lactulosyl lysine residues per protein molecule (Fig. 11.8). A more detailed study using skim milk diets with pigs confirmed that lactulosyl lysine was nonbioavailable (Rerat et al., 2002). That study also found a decrease in the digestibility of lysine, phenylalanine, valine, cystine, aspartic acid, glycine, and methionine residues. This decrease suggests that lactulosyl lysine residues hinder the release and therefore the utilization of adjacent amino acids.\n\nFigure 11.8 Plot of lactosylation level against PER. The PER value for ANRC casein was used for zero lactosylation. PER, protein efficiency ratio.\n\nLimited human studies have been conducted on Maillard reaction products that include lactulosyl lysine. One study looked at adolescent males on diets containing different levels of Maillard reaction products, which principally showed that the Maillard reaction reduces protein digestibility (Gilani et al., 2012).\n\nStudies on infant formula have shown that products with an increased amount of Maillard reaction products had decreased PER values. Those formulas with lower PER values also resulted in the rats having lower levels of plasma lysine.\n\nLysinoalanine is not a bioavailable source of lysine (Robbins et al., 1980; Friedman, 1999; Gilani et al. 2012). Alkali treatment of casein with 0.2 N NaOH at 80 \u00b0C for 1 h reduced the PER of casein from 3.09 to 0.02 for a diet containing 10% casein (Possompes et al., 1989; cited in Friedman, 1999).\n\nMilk proteins are unusually rich in lysine (Table 11.3), and can stand to lose a significant proportion of the lysine before it becomes limiting (around 50% in the case of whey proteins and 25% in the case of casein). The real concern arises when milk proteins, and particularly whey proteins, are being added to a mixture to provide a source of lysine supplementation. When this is the case, it is particularly important to ensure that the lysine content has not been compromised after long storage. Steps to ensure the integrity of the lysine content include keeping the product at temperatures less than 25 \u00b0C for most of the time and ensuring that the water activity in the product is kept low, preferably at or below 0.3. Experiments carried out in our laboratories to determine bioavailable lysine showed that loss of lysine upon storage was negligible at 30 \u00b0C, whereas losses were severe at 40 \u00b0C (Fig. 11.9).\n\nFigure 11.9 Loss of bioavailable lysine with time for SMP, WMP, and ALACEN 421 with a water activity (aW) of 0.3 during storage at (a) 30 \u00b0C and (b) 40 \u00b0C. The solid and dashed lines on the graphs show the levels at which lysine becomes limiting in WPC, SMP, and WMP, respectively. K. Higgs, unpublished results.\n\n### Sulfur Amino Acids\n\nModification of sulfur amino acids is possible through loss of cysteine via lanthionine formation and through oxidation of methionine to methionine sulfoxide. Whey proteins are relatively rich in sulfur amino acids, and casein has more than adequate quantities. It should be noted that these scores are relative to World Health Organization (WHO) requirements for children aged 2\u20135 years. The requirement for rat diets is higher, probably because of the rat's requirement for hair production (hair is rich in sulfur amino acids).\n\nLanthionine is a problem only for caseinates, and lanthionine formation is known to occur under the conditions used in the caseinating process (Aymard et al., 1978). The rates of lanthionine formation in dry powder have not been studied.\n\nOxidation of methionine to the sulfoxide does not alter its bioavailability per se. It has been claimed, however, that the presence of the sulfoxide side chain in intact proteins may hamper digestion of the protein, thus affecting its overall bioavailability (Anon., 1973).\n\n### Other Amino Acids\n\nOne essential amino acid known to be destroyed during some processes is tryptophan. Extensive investigations by Nielsen et al. (1985) on whey proteins, casein, and WMP showed that tryptophan remained relatively intact even when substantial lysine modification had occurred. Tryptophan analysis gave a result of 100% (within experimental error) and a bioavailability of >90% in milk powder that had been stored at 60 \u00b0C for 5 weeks, which resulted in a loss of 80% of available amine. This powder was considered to show 'advanced' Maillard browning.\n\n## Product-specific storage trials\n\nSamples of WPC80 powders that had earlier been shipped from New Zealand to the United States or Europe were obtained and analyzed for lactosylation levels. The powders were at the time less than 2 years old and thus were considered to be current stock. The powders had average bound lactose levels of 0.7 to 1.2. This correlates to between 87% and 95% of the lysine in the products being bioavailable (estimated by extrapolation from the remaining lysine in \u03b2-lactoglobulin). This compares well with freshly produced WPC80 powders, which had an average number of lactose bound of 0.6, or 96% of available lysine remaining (Fig. 11.10).\n\nFigure 11.10 Levels of lactosylation in market samples of a WPC80 compared with levels in freshly produced WPC80 in a subsequent season.\n\nWe carried out a two-year storage study on >80% protein powders to determine changes, if any, in nutritional properties, but these powders were kept at constant temperature and were not exposed to any of the temporal variations possible during shipping and storage in overseas warehouses.\n\nA 5% decrease in available amine was seen in MPC85 when stored at 20 \u00b0C for 2 years; this increased to 10% when the storage temperature was increased to 30 \u00b0C. A single MPI stored for 2 years gave consistent available amine results over the storage period.\n\nThe caseinates and caseins in the 2-year study showed no definitive trend in available amine values. Most values remained consistent over the 2 years. This was expected as a previous 3-month study showed that storage temperatures in excess of 30 \u00b0C were required for significant levels of lysinoalanine formation.\n\n## Conclusions\n\nMilk proteins do undergo change when dry powders are stored. Powders containing appreciable amounts of lactose (milk powders and WPCs) form pre-Maillard reaction products, rendering lysine nonbioavailable, whereas those containing casein undergo formation of isopeptide bonds, reducing the availability of lysine and sulfur amino acids. Because milk proteins are rich in lysine, loss of some lysine will not be a significant problem. However, when milk proteins are used as 'balancers' in formulations with other proteins that are poor in lysine, attention should be given to the storage history of the protein.\n\nBoth the Maillard reaction and the formation of isopeptide bonds are undesirable, and are best avoided by ensuring that shipping and storage temperatures do not exceed 30 \u00b0C for significant periods. This should not be a problem in temperate climates; however, in climates where high temperatures routinely occur, consideration should be given to storage in a cool store.\n\n# References\n\nAnon .  Nutritional implications of sulfur amino acid oxidation .  _Nutrition Reviews_.  1973 ;31 : 220 \u2013 221 .\n\nAymard C ,  Cuq JL ,  Cheftel JC .  Formation of lysino-alanine and lanthionine in various food proteins, heated at neutral or alkaline pH .  _Food Chemistry_.  1978 ;3 : 1 \u2013 5 .\n\nErbersdobler H .  Twenty years of furosine\u2014better knowledge about the biological significance of the Maillard reaction in food and nutrition .  In:  Fujimaki M ,  Namiki M ,  Kato H ,  eds.  _Amino-Carbonyl Reactions in Food and Biological Systems. Proceedings of the 3rd International Symposium on the Maillard Reaction_ .  Susono, Shizuoka, Japan, 1\u20135 July 1985 .  Amsterdam :  Elsevier ;  1986 : 481 \u2013 491 .\n\nErbersdobler H ,  Dehn B ,  Nangpal A ,  Reuter H .  Determination of furosine in heated milk as a measure of heat intensity during processing .  _Journal of Dairy Research_.  1987 ;54 : 147 \u2013 151 .\n\nFriedman M .  Chemistry, biochemistry, nutrition and microbiology of lysinoalanine, lanthionine, and histidinoalanine in food and other proteins .  _Journal of Agricultural and Food Chemistry_.  1999 ;47 : 1295 \u2013 1319 .\n\nGilani GS ,  Xiao CW ,  Cockell KA .  Impact of antinutritional factors in food proteins on the digestibility of protein and the bioavailability of amino acids and on protein quality .  _British J Nutrition_.  2012 ;108 : S315 \u2013 S332 .\n\nHenle T ,  Walter H ,  Krause I ,  Klostermeyer H .  Efficient determination of individual maillard compounds in heat-treated milk products by amino acid analysis .  _International Dairy Journal_.  1991 ;1 : 125 \u2013 135 .\n\nHern\u00e1ndez MJM ,  Domingo EB ,  Cama\u0148as RMV ,  Alvarez-Coque MCG .  Use of the \u03bf-phthalaldehyde and N-acetyl-L-cysteine in the evaluation of milk proteins .  _Journal of Dairy Science_.  1991 ;74 : 1779 \u2013 1785 .\n\nIDF, 2012. The world dairy situation, 2012. Bulletin of the International Dairy Federation, 458\/2012, Brussels.\n\nNielsen HK ,  de Weck D ,  Finot PA ,  Liardon R ,  Hurrell RF .  Stability of tryptophan during food processing and storage. 1. Comparative losses of tryptophan, lysine and methionine in different model systems .  _British Journal of Nutrition_.  1985 ;53 : 281 \u2013 292 .\n\nPereyra Gonzales AS ,  Naranjo GB ,  Leiva GE ,  Malec LS .  Maillard reaction kinetics in milk powder: Effect of water activity at mild temperatures .  _International Dairy Journal_.  2010 ;20 : 40 \u2013 45 .\n\nRerat A ,  Calmes R ,  Vaissade P ,  Finot P-A .  Nutritional and metabolic consequences of the early Maillard reaction of heat treated milk in the pig .  _Significance for man. European Journal of Nutrition_.  2002 ;41 : 1 \u2013 11 .\n\nRobbins KR ,  Baker DH ,  Finley JW .  Studies on the utilization of lysinoalanine and lanthionine .  _Journal of Nutrition_.  1980 ;110 : 907 \u2013 915 .\n\nSchwarzenbolz U ,  Henle T .  Non-enzymatic modifications of proteins under high-pressure treatment .  _High Pressure Research_.  2010 ;30 : 458 \u2013 465 . \nChapter 12\n\n# Interactions and Functionality of Milk Proteins in Food Emulsions\n\nHarjinder Singh\n\nAiqian Ye*\n\n* Riddet Institute, Massey University, Palmerston North, New Zealand\n\n## Abstract\n\nBecause of their high nutritional quality and versatile functional properties, milk proteins are widely used as ingredients in many manufactured food colloids, e.g., dairy desserts, nutritional beverages, ice cream, yogurt, spreads, confectionery, and baked goods. Milk proteins perform a wide range of key functions in prepared foods, including emulsification, thickening, gelling, and foaming. An important functionality of milk proteins in food colloids is their ability to facilitate the formation and stabilization of oil droplets in emulsions. The ability of milk protein products to adsorb at the oil\u2013water interface and to stabilize emulsions is influenced by the structures, flexibility, and aggregation state of the constituent proteins. This chapter deals mainly with the properties and functionalities of food emulsions formed with a range of milk protein products and how they are influenced by different environmental and processing conditions. Of particular importance are the effects of pH, calcium ions, and protein content and the influences of thermal and high-pressure processing. The chapter focuses on the structure and composition of adsorbed protein layers, competition between proteins, and the creaming and flocculation behaviors of emulsion droplets. Recent work on the influence of the emulsion structure, in particular of the adsorbed layers, on lipid oxidation and lipid digestion behavior is also reviewed briefly.\n\n## Keywords\n\nmilk proteins\n\nfood emulsions\n\nemulsifying abilities of milk proteins\n\noral behaviour\n\nOutline\n\nIntroduction 359\n\nAdsorption of milk proteins during the formation of emulsions 361\n\nStability of milk protein-based emulsions 366\n\nHeat-induced changes in milk protein-based emulsions 370\n\nPressure-induced changes in milk protein-based emulsions 372\n\nMilk protein hydrolysates and oil-in-water emulsions 373\n\nLactoferrin-based oil-in-water emulsions 374\n\nLipid oxidation in milk protein-based emulsions 376\n\nBehavior of milk protein-stabilized emulsions under physiological conditions 378\n\nConclusions 380\n\n## Introduction\n\nMilk proteins possess functional properties that provide desirable textural and other attributes to the final product. For this reason, they have found numerous applications in traditional dairy products and other foods. The functional properties of milk proteins, such as emulsification, thickening, gelling, flavor binding, and foaming, contribute to the sensory characteristics and the stability of the manufactured foods (Table 12.1). Several types of milk protein products (e.g., caseins and caseinates, whey protein concentrates (WPCs) and whey protein isolates (WPIs), milk protein concentrate (MPC) powders, and hydrolyzed proteins) are manufactured from milk by the dairy industry.\n\nTable 12.1\n\nFunctional Properties of Milk Proteins in Food Systems\n\nFunctional property | Food system  \n---|---  \nSolubility  \nEmulsification  \nFoaming  \nWater binding  \nHeat stability  \nGelation  \nAcid stability | Beverages  \nCoffee whiteners, cream liqueurs, salad dressings, desserts  \nWhipped toppings, shakes, mousses, cakes, meringues  \nBreads, meats, snack bars, custard, soups, sauces, cultured foods  \nUHT- and retort-processed beverages, soups and sauces, custard  \nMeats, curds, cheese, surimi, yogurt  \nAcid beverages, fermented drinks\n\nCaseinates are produced from skim milk by adding acid (hydrochloric acid or lactic acid) or microbial cultures to precipitate the casein from the whey at pH 4.6. The acid-precipitated casein can then be re-solubilized with alkali or an alkaline salt (using calcium, sodium, potassium, or magnesium hydroxide) to about pH 6.7 and spray dried to form caseinate. Caseinates have exceptional water-binding capacity, fat emulsification properties and whipping ability, and a bland flavor. Emulsion-type products, including coffee whiteners, whipped toppings, cream liqueurs, and low-fat spreads, are an important application of caseinates in the food industry. In recent years, the use of casein and caseinate in dietary preparations, nutritional products, and medical applications has increased; many of these preparations are also oil-in-water emulsions containing relatively small amounts of fat.\n\nWPC and WPI are concentrated forms of whey protein components. Ultrafiltration, diafiltration, and ion-exchange technology are used to concentrate and separate the protein from other components. The whey protein is then dried to obtain WPC or WPI, both of which are highly soluble in water, with protein levels ranging from 80% to 95%. Both WPC and WPI have a wide range of food applications and, because of their high protein content, can function as water-binding, gelling, emulsifying, and foaming agents. Processing treatments used in the manufacture of WPC and WPI may sometimes cause some protein denaturation, which tends to affect their functionality.\n\nMPCs are processed directly from skim milk by a combination of ultrafiltration\/diafiltration (Mulvihill, 1992). The protein content of MPCs can vary from 56% to 82%; the caseins are in a micellar form, similar to that found in milk, and the whey proteins are also in their native state. Recently, new MPC products in which the casein micelles have been dissociated to some extent by reducing the colloidal calcium content have been developed. MPCs have been used as ingredients in many food applications, such as milk extension in cheese, yogurt manufacture, and nutritional beverages.\n\nThe functionality of milk proteins in processed foods is determined by their molecular structures and interactions with other food components, such as fats, sugars, polysaccharides, salts, flavors, and aroma compounds. The type and strength of various interactions determine the structure, texture, rheology, sensory properties, and shelf life of manufactured food products. Much knowledge on the structure and properties of individual milk protein components has been gained, but less is known about interactions between different components that occur in a food system as a result of processing and formulation. Controlling these interactions is of key significance for development of novel products and processes as well as for improvement of conventional products and processes. In recent years, there has been an increased interest in the understanding of how the interactions of food components and food structure design influence the rates of nutrient digestion and bioavailability. This research is aimed at developing new foods that regulate calorie intake, provide increased satiety responses, provide controlled digestion, and\/or deliver bioactive molecules (Singh & Sarkar, 2011). Most of this research effort focuses on understanding the changes in physical and biochemical structures of food emulsions during digestion (Singh et al., 2009; McClements et al., 2009; Le R\u00e9v\u00e9rend et al., 2010; Golding & Wooster, 2010).\n\nThis chapter focuses on the emulsifying properties of milk proteins, as this functional property is very important in all the food applications of milk protein products. The adsorption behavior of different milk protein products at oil-in-water interfaces and the stability of the resulting emulsions in different environments are considered. Recent advances in understanding the behavior of milk protein-based emulsions under physiological conditions are discussed briefly.\n\n## Adsorption of Milk Proteins During the Formation of Emulsions\n\nEmulsions are composed of oil droplets (average range 0.5\u20135 \u03bcm diameter) enveloped by a continuous film of surfactant material that stabilizes the droplets. In the food industry, homogenization is widely used for finely dispersing oils in food products, and proteins are most commonly used as emulsifying agents. The state of the droplet size distribution after homogenization reflects the emulsifying capacity of the proteins, the energy input during formation, as well as the effects of various factors, such as pH, temperature, ionic strength, and ratio of the two phases, on the surface activity of the proteins (Walstra, 1993; Dickinson 1998a). In addition, the droplet size distribution influences markedly the properties of food emulsions, such as stability, viscosity, texture, and mouthfeel.\n\nDuring homogenization, the milk protein, in the form of individual molecules or protein aggregates, becomes rapidly adsorbed at the surface of the newly formed oil droplets. The amount of protein present at the interface per unit surface of dispersed phase is defined as the protein load, which is usually expressed as milligrams of protein per unit area of the dispersed phase (mg\/m2). The protein load determines the amount of protein required to make an emulsion with a desired oil volume and droplet size and is dependent on the concentration and type of protein as well as on the conditions used for emulsion formation. The factors that affect the protein load include protein concentration, volume of oil, energy input, state of protein aggregation, pH, ionic strength, temperature, and calcium ions (Dickinson & Stainsby, 1988; Walstra, 1993).\n\nThe properties of the adsorbed layers depend on the amounts and structures of the proteins present during homogenization. Proteins are amphipathic molecules containing both polar and nonpolar parts and orientate at the interface in such a way that a substantial proportion of the nonpolar amino acids remains in contact with the oil phase, and the polar groups are in contact with the aqueous phase (Dickinson 1992; 1998a). The main thermodynamic driving force for the adsorption of proteins is the removal of hydrophobic residues from the unfavorable environment of the bulk aqueous phase by displacement of structured water molecules from the close vicinity of the interface. An additional important driving force is the unfolding and reorganization of the native protein structure, which is due to interaction with the interface. By adsorbing at the interface, the protein reduces the free energy of the system and hence the interfacial tension. The effectiveness of any particular protein in lowering the surface tension depends on the number and type of contacts it makes with the interface (Dickinson et al., 1988a; Dickinson, 1999). A protein molecule that spreads out a lot, and thus has a substantial proportion of its nonpolar residues in contact with the surface, is also very effective in reducing the interfacial tension. Flexible proteins (caseins) with a higher proportion of nonpolar groups are more effective in reducing the interfacial tension than rigid proteins with fewer nonpolar groups (Dickinson & McClements, 1995). The order of surface activity that has been reported for the individual milk proteins is: \u03b2-casein > monodispersed casein micelles > serum albumin > \u03b1-lactalbumin > \u03b1s-caseins = \u03ba-casein > \u03b2-lactoglobulin (Mulvihill & Fox, 1989).\n\nOnce a protein is adsorbed at an interface, it undergoes unfolding and rearrangement to form a stabilizing adsorbed layer (Dickinson, 1992; Dalgleish 1996) and the extent of unfolding depends on the flexibility of the protein molecule, that is, on the strength of the forces maintaining the secondary and tertiary structures. Because the caseins have flexible structures, they unfold rapidly at the interface and may form extended layers up to about 10 nm thick (Dalgleish, 1990). Dalgleish (1999) suggested that casein molecules are stretched to their maximum extent when their overall surface coverage is less than about 1 mg\/m2. Conversely, the presence of excess casein increases the monolayer coverage to a maximum value of 3 mg\/m2, the parts of the molecules in contact with the interface adopt a more compact conformation, and the hydrophilic moieties protrude further from the interface.\n\nWhey proteins (such as \u03b2-lactoglobulin), which give adsorbed layers that are only about 2 nm thick, change conformation and unfold their structure to some extent at the surface (Dalgleish & Leaver, 1993; Mackie et al., 1993; Dalgleish, 1995; Dickinson & McClements, 1995; Dalgleish, 1996; Fang & Dalgleish, 1998). The adsorbed whey protein structure lies somewhere in between the native structure and the fully denatured state, which may have a native-like secondary structure and an unfolded tertiary structure (Dickinson, 1998a). Additionally, the partial unfolding of the globular whey protein structure following adsorption causes exposure of the reactive sulfhydryl group, leading to slow polymerization of the adsorbed protein in the aged layer via sulfydryl\u2013disulfide interchange (Dickinson & Matsumura, 1991; McClements et al., 1993).\n\nThe amount of protein adsorbed on the interface of an emulsion droplet suggests the state of the protein adsorbed at the interface. If the protein load is < 1mg\/m2, it suggests that the protein molecules are fully unfolded. If the protein load is 1\u20133 mg\/m2, a monolayer of globular proteins may be present, or unfolded molecules may be adsorbed in the conformation of trains, loops, and tails. Protein load values above 5 mg\/m2 suggest the adsorption of aggregates of proteins or multilayers of proteins. Some proteins of higher molecular weight may also give higher protein loads (Phillips, 1981; Hunt & Dalgleish, 1994; Dam et al., 1995; Srinivasan et al., 1996).\n\nExtensive studies on purified milk protein systems show that a disordered casein monomer may be regarded as a complex linear copolymer that adsorbs to give an entangled monolayer of flexible chains, with some sequences of segments in direct contact with the surface (trains) and other sequences of segments protruding into the aqueous phase (loops and tails) (Dickinson, 1998a). Based on various experimental studies and molecular modeling, the \u03b2-casein molecule has been shown to adsorb with an extensive hydrophobic region anchored directly at the surface and a hydrophilic region (40\u201350 residues at the N-terminus) protruding extensively into the aqueous phase. This is probably also the portion of the molecule that forms the hydrodynamically thick layer. In contrast to the dangling tail predicted for \u03b2-casein, a loop-like conformation has been predicted for \u03b1s1-casein, and it does not have such a pronounced inequality in the distribution of hydrophobic and hydrophilic residues in its primary structure. It has been suggested that \u03b1s1-casein adsorbs to the oil\u2013water interface via peptides toward the middle of its sequence, rather than the end as in \u03b2-casein, and it may be this that causes the protein to form a thinner adsorbed layer than \u03b2-casein (Dalgleish, 1996). The simple train\u2013loop\u2013tail model is not adequate to describe the molecular configuration of adsorbed \u03b2-lactoglobulin. A closely packed, dense, and rather thin (2\u20133 nm at neutral pH) layer of \u03b2-lactoglobulin is formed, which can be modeled as a dense two-dimensional assembly of highly interacting deformable particles.\n\nMilk proteins used by the food industry contain complex mixtures of proteins in various states of aggregation. The structures of the adsorbed layers formed with these complex mixtures are not understood in molecular detail. The most commonly used milk proteins, (i.e., sodium caseinate and whey proteins) show excellent emulsifying ability, and it is possible to make stable emulsions at a relatively low protein-to-oil ratio (about 1:60). In emulsions formed with sodium caseinate or whey proteins, the protein load increases with an increase in protein concentration until it reaches a plateau value of about 2.0\u20133.0 mg\/m2 (Singh, 2005) (Fig. 12.1). The emulsifying ability of 'aggregated' milk protein products, such as MPC and calcium caseinate, is much lower than that of whey protein or sodium caseinate; that is, much higher concentrations of protein are required to make stable emulsions, and larger droplets are formed under similar homogenization conditions. The surface protein concentration of emulsions formed with MPC is in the range 5\u201320 mg\/m2, depending on the protein concentration used in making the emulsions (Euston & Hirst, 1999). At low protein-to-oil ratios, protein aggregates are shared by adjacent droplets, resulting in bridging flocculation and consequently a marked increase in droplet size. In addition, the spreading of protein at the interface is limited in these emulsions, because the aggregates are held together by calcium bonds and\/or colloidal calcium phosphate. These bonds are unlikely to be affected during the emulsification process. The higher conformational stability of these aggregates will also contribute to their reduced emulsifying ability (Euston & Hirst, 1999; Srinivasan et al., 1999). New MPC products called calcium-depleted MPCs that show improved emulsifying ability have been developed recently (Ye, 2011). In these MPCs, the depletion of colloidal calcium results in disintegration of the casein micelle structure that consequently produces smaller casein particles. These smaller casein particles are adsorbed more efficiently at the droplet surface than the native casein micelles, resulting in more stable emulsions with smaller droplets. The surface protein concentration in emulsions formed with MPC with 80% calcium removal is very similar to that of sodium caseinate (Ye, 2011).\n\nFigure 12.1 Influence of protein concentration on average droplet size d 43 (left) and surface protein coverage (right) in emulsions (30% soya oil) made with sodium caseinate ( ), calcium caseinate ( ), WPC ( ), or MPC (\u2666). From Singh, 2005, reproduced with the permission of The Royal Society of Chemistry.\n\nThe composition of the interfacial layer is determined by the quantities and structures of the proteins present at the moment the emulsion is formed. If proteins are the only emulsifiers present, they will adsorb to the oil\u2013water interface, generally in proportion to their concentration in the aqueous phase (Dalgleish, 1997). However, certain mixtures of caseins show competition during adsorption at oil\u2013water interfaces and rapid exchanges between adsorbed and unadsorbed caseins after emulsion formation. Studies have demonstrated that \u03b2-casein, because of its greater surface activity, adsorbs in preference to \u03b1s1-casein in emulsions stabilized by mixtures of these proteins and that \u03b2-casein displaces \u03b1s1-casein rapidly from the droplet surface (Dickinson et al., 1988b). In binary mixtures containing \u03b2-lactoglobulin and \u03b1-lactalbumin, some limited competitive adsorption does occur, but little exchange between the adsorbed and unadsorbed protein occurs. The protein that arrives at the interface first during homogenization is the protein that predominates there afterward.\n\nIn contrast to model systems, no competitive adsorption has been observed in emulsions stabilized by more complex casein mixtures, such as sodium caseinate (Hunt & Dalgleish, 1994). Interestingly, this behavior appears to be related to the ratio of protein to oil in the emulsions (Euston et al., 1995; Srinivasan et al. 1999). Srinivasan et al. (1999) has shown that in sodium caseinate emulsions when the ratio of protein to oil is very low (about 1:60), \u03b2-casein is preferentially adsorbed at the droplet surface. However, when the total amount of protein is greatly in excess of the amount needed for full surface coverage, \u03b1s1-casein is adsorbed in preference to the other caseins. At all concentrations, \u03ba-casein from sodium caseinate appears to be less readily adsorbed (Fig. 12.2). The concentration dependence of the competitive adsorption of \u03b1s1-casein and \u03b2-casein in sodium caseinate emulsions may be a consequence of the different complexes that can be formed by caseins in solution (Rollema, 1992). The preferential adsorption of \u03b2-casein, because of its high surface activity, appears to exist only at low concentrations where caseins may exist as monomers. With increasing protein concentration, caseins aggregate to form various complexes (Lucey et al., 2000) and it is likely that \u03b2-casein loses its competitive ability because of its self-aggregation to form micelles or through the formation of complexes with other caseins. Therefore, the surface composition of emulsions formed using a relatively high sodium caseinate concentration is likely to be determined by the surface activities and flexibilities of the casein aggregates and complexes. Although extensive information on the surface activity and hydrophobicity of individual caseins is available, little is known about how these characteristics are modified when casein molecules undergo self-association under different environmental conditions.\n\nFigure 12.2 Surface concentrations of \u03b1s1-casein ( ), \u03b2-casein ( ) and \u03ba-casein ( ) in sodium caseinate emulsions (30% oil). From Srinivasan et al., 1999, reproduced with the permission of Elsevier Inc.\n\nWhen the casein is in the highly aggregated form of casein particles, as in calcium caseinate or MPC, very little competitive adsorption and protein exchange take place (Euston & Hirst, 1999; Srinivasan et al., 1999). In these systems, the average surface composition is probably determined by the adsorption of protein aggregates of fixed composition. For instance, calcium caseinate solution consists of large \u03b1s1-casein-rich aggregates, which appear to dominate the droplet surface after emulsification (Srinivasan et al., 1999). When WPCs or WPIs are used to make emulsions, there is no preferential adsorption between \u03b2-lactoglobulin and \u03b1-lactalbumin regardless of the protein-to-oil ratio in the emulsion (Euston et al., 1996; Ye & Singh, 2000; 2006a).\n\nThe aggregation state and the flexibility of protein molecules can be altered by changes in pH, addition of divalent cations, and various processing treatments prior to emulsification. These changes will inevitably influence the adsorption behavior of milk proteins at the oil\u2013water interface. For example, addition of CaCl2 at above a certain critical concentration to a sodium caseinate or whey protein solution before homogenization increases the droplet size, increases the surface protein coverage, and, in sodium caseinate emulsions, also affects the competition between different proteins (Ye & Singh, 2000; 2001). The proportions of \u03b2-lactoglobulin and \u03b1-lactalbumin at the droplet surface remain unaffected by the addition of CaCl2 to a whey protein solution prior to emulsification. In contrast, addition of CaCl2 to a sodium caseinate solution markedly enhances the adsorption of \u03b1s1-casein at the droplet surface, with a much smaller effect on \u03b2-casein adsorption. The effects of calcium on surface coverage and composition can be explained by the binding of the ions to the negatively charged amino acid residues on the protein. This reduces electrostatic repulsions between the protein molecules and increases the potential for intermolecular associations. Because of the presence of clusters of phosphoserine residues, the caseins have stronger affinity than the whey proteins to bind calcium. Consequently, the caseins (except \u03ba-casein) are precipitated by calcium, with \u03b1s1-casein being the most sensitive to aggregation and precipitation by calcium. In sodium caseinate emulsions, the increased surface coverage upon addition of calcium prior to emulsification is probably due to adsorption of casein aggregates onto the droplet surface (Ye & Singh, 2001). Greater \u03b1s1-casein adsorption reflects its stronger tendency to be aggregated by calcium ions in solution or at the interface.\n\nThe native whey proteins do not bind much calcium and are not precipitated in the presence of calcium (Baumy &; Brule, 1988a,b), although heat-denatured whey proteins are able to bind considerable amounts of calcium and undergo aggregation (Pappas & Rothwell, 1991). The increase in surface protein coverage suggests the formation of aggregates of whey proteins in the presence of calcium, which subsequently become adsorbed during emulsification (Ye & Singh, 2000). This has been attributed to a decrease in the denaturation temperature of the whey proteins in the presence of calcium.\n\nAll of these results confirm that, under a given set of homogenization conditions, the surface composition is largely dependent on the protein-to-oil ratio and the aggregation state of the proteins in solution. It appears that the structure of the interfacial layer in emulsions can be manipulated by controlling the protein concentration, the protein type, and the ionic environment. Because of their different interfacial structures, these droplets would be expected to exhibit different reactivities that could be exploited to develop new food textures. Further studies are required for an understanding of the relationship between droplet surface structures and the sensitivity of the droplets to different environments and processing conditions.\n\n## Stability of Milk Protein-Based Emulsions\n\nThe term emulsion stability refers to the ability of an emulsion to resist any alteration in its properties over the timescale of observation (McClements, 1999; Dickinson, 2003; McClements 2005;). An emulsion is thermodynamically unstable, as the free energy of mixing is positive because of the large interfacial area between the oil and the aqueous phase. Therefore, the kinetic stability\u2014that is, the time period for which the emulsion is stable\u2014is important (Damodaran, 1997; McClements, 1999; Dickinson, 2003; McClements 2005;). For instance, an emulsion can be considered to be 'stable' if the inevitable process of separation has been slowed to an extent that it is not of practical importance during the shelf life of the product. An emulsion may become unstable because of a number of different types of physical and chemical processes. Physical instability refers to the change in spatial arrangement or size distribution of emulsion droplets, such as creaming, flocculation, or coalescence, whereas chemical instability includes change in the composition of the emulsion droplet itself, such as oxidation and hydrolysis (McClements & Decker, 2000; McClements, 2005). Creaming is the movement of oil droplets, under gravity or in a centrifuge, to form a concentrated layer at the top of an oil-in-water emulsion sample, with no accompanying change in the droplet size distribution. Creaming is reversible, and the original uniform distribution of droplets can usually be obtained by gentle mixing. The creaming process can be explained by Stokes's Law (Hunter, 1986; McClements 2005):\n\n\u03bd s t o k e s = 2 \u03b3 2 \u03c1 1 \u2212 \u03c1 2 9 \u03b7\n\n (12.1)\n\nwhere \u03bd stokes = velocity of creaming, \u03b3 = emulsion droplet radius, \u03c1 1 and \u03c1 2 = density of the continuous phase, and the dispersed phase, respectively, and \u03b7 = shear viscosity of the continuous phase. The creaming rate can be reduced by lowering the radius, increasing the continuous phase viscosity, or decreasing the difference in density between the two phases. However, this law often fails to define the rate of creaming due to flocculation or coalescence.\n\nCoalescence, that is, an increase in droplet size by accretion, gradually results in separation of the oil and the aqueous phase and is always irreversible. Coalescence requires rupture of the stabilizing film at the oil\u2013water interface, but this occurs only when the layer of continuous phase between the droplets has thinned to a certain critical thickness (Dickinson & Stainsby, 1988; Britten & Giroux, 1991; Das & Kinsella, 1993; Walstra, 1993).\n\nFlocculation has been defined as the reversible aggregation mechanism that arises when droplets associate as a result of unbalanced attractive and repulsive forces (Dalgleish, 1997). Generally, two types of flocculation are distinguished: depletion flocculation and bridging flocculation (Dickinson, 2003). The type of mechanism prevailing depends upon the interaction between the interfacial layer and the emulsion droplets.\n\nBridging flocculation normally occurs when a high-molecular-weight biopolymer at a significantly low concentration adsorbs to two or more emulsion droplets, forming bridges (Dickinson & Pawlowsky, 1998; Dickinson, 1998b; 2003; McClements, 1999; 2005; Fellows &; Doherty 2006;). Depletion flocculation occurs as a result of the presence of unadsorbing biopolymer in the continuous phase, which can promote association of oil droplets by inducing an osmotic pressure gradient within the continuous phase surrounding the droplets (de Hek & Vrij, 1981; Dickinson, 1999; Tuinier & de Kruif, 1999; McClements, 2005).\n\nIf the added biopolymer is either unadsorbed or poorly adsorbed, it is squeezed out of the area between two approaching emulsion droplets. The concentration of biopolymer between the emulsion droplets becomes less than its overall solution concentration, resulting in osmotic imbalance. The net effect is that the particles are attracted to each other, resulting in flocculation. The attraction energy is determined by the concentration of the polymer, and the range of interaction depends on the radius of gyration of the polymer molecule. The bonds formed through the depletion flocculation mechanism are generally weak, flexible, and reversible.\n\nThe ability of proteins to stabilize emulsions is the most important criterion besides the emulsion formation in most food applications. The forces involved in stabilizing and destabilizing emulsions include van der Waals's attractive forces, electrostatic interactions, and steric factors. At pH values away from their isoelectric point, as proteins are electrically charged, there is an electrostatic repulsion, which prevents dispersed droplets from closely approaching one another. With the possible exception of highly charged proteins, a predominant contribution to emulsion stabilization by protein comes from the steric stabilization mechanism. Interactions between droplets stabilized by proteins may be influenced by the presence of certain ions, particularly calcium, as proteins are capable of binding ions.\n\nAs long as sufficient protein is present during homogenization to cover the oil droplets, emulsions stabilized by milk proteins are generally very stable to coalescence over prolonged storage. However, these emulsions are susceptible to different types of flocculation, which in turn leads to enhanced creaming or serum separation. At low protein-to-oil ratios, there is insufficient protein to fully cover the oil\u2013water interface during homogenization, and this results in bridging flocculation. Another consequence of insufficient protein is coalescence of droplets during or immediately after emulsion formation. Bridging flocculation is commonly observed in emulsions formed with aggregated milk protein products, such as calcium caseinate or MPC, in which the droplets are bridged by casein aggregates or micelles. Optimum stability can generally be attained at protein concentrations high enough to allow full saturation coverage at the oil\u2013water interface. However, at very high protein-to-oil ratios, the presence of excess, unadsorbed protein may lead to depletion flocculation in some emulsions. Both depletion flocculation and bridging flocculation cause an emulsion to cream more rapidly.\n\nDepletion flocculation has been observed in sodium-caseinate-based emulsions but not in emulsions formed with calcium caseinate, MPC, or whey proteins (Dickinson & Golding, 1997; Euston & Hirst, 1999; Srinivasan et al., 2001; Singh, 2005) (Fig. 12.3). In sodium-caseinate-based emulsions, it was shown that at a protein content of nearly 2.0 wt%, the emulsion droplets were protected from flocculation by a thick steric-stabilizing layer of sodium caseinate. The emulsion was stable against flocculation, coalescence, and creaming for several weeks. However, when the protein content was increased to above 3.0 wt%, unadsorbed protein gave rise to depletion flocculation. Because of this depletion flocculation, the effective diameter of the droplets increased, resulting in a marked decrease in creaming stability, with an increase in the caseinate concentration from 3 to 5 wt%. Further increasing the protein content to 6.0 wt% and above resulted in very high depletion flocculation, leading to a strong emulsion droplet network that was stable to creaming.\n\nFigure 12.3 Creaming stability and microstructure of emulsions made with sodium caseinate ( ) or WPC ( ) (30% oil). Scale bar represents 10 \u03bcm. From Singh, 2005, reproduced with the permission of The Royal Society of Chemistry.\n\nThe differences in the creaming stabilities of emulsions made with different kinds of milk protein products are largely related to depletion flocculation effects (Singh, 2005). The depletion interaction free energy (G DEP), of the order of a few kT, can be estimated using Equation 12.2 (Walstra, 1993):\n\n\u0394 G D E P = \u2212 2 \u03c0 \u03b3 2 \u220f \u03b3 d \u2212 2 \u03b3 m \/ 3\n\n (12.2)\n\nwhere \u03a0 is the osmotic pressure of the polymer solution, represented as a fluid of hard spheres of radius \u03b3 m, and \u03b3 d is the mean droplet radius. The osmotic pressure under ideal conditions is given by the following equation:\n\n\u220f = c R T \/ M\n\n (12.3)\n\nwhere R is the molar gas constant, T is the temperature, M is the molecular mass of the polymer, and c is the number concentration of the polymer.\n\nFor depletion flocculation to occur, the polymer has to have a fairly high M, so that the \u03b3 m is relatively large. However, at a given c, M is inversely proportional to \u03c0. Therefore, an increase in the polymer molecular mass will reduce the osmotic pressure driving the depletion interaction. Hence, at a given concentration, the depletion interaction free energy is low for a polymer of low molecular mass, increases with an increase in molecular mass until it reaches a maximum, and then decreases with a further increase in molecular mass. Similarly, a reduction in the polymer number concentration will reduce the osmotic pressure.\n\nAlthough the exact state of the casein molecules in concentrated sodium caseinate solutions is unknown, a sodium caseinate solution has been reported to have a radius of gyration of about 20\u201330 nm, as determined by static light scattering (Lucey et al., 2000). Depletion flocculation in sodium caseinate emulsions is likely caused by the presence of these casein aggregates formed from self-assembly of sodium caseinate in the aqueous phase of the emulsion at concentrations above 2 wt% (Dickinson & Golding, 1998). Theoretically, a casein aggregate with a radius of approximately 20 nm causes the strongest depletion flocculation of fine emulsion droplets (mean diameter \u223c0.4 \u03bcm), that is, corresponding to a size ratio of about 10:1 (Radford et al., 2004). The estimated optimum size of casein particles for inducing depletion flocculation is similar to the size of small casein aggregates actually found in sodium caseinate dispersions at low ionic strength (Lucey et al., 2000).\n\nEmulsions formed with whey proteins, MPC, and calcium caseinate do not show depletion flocculation, probably because there are no suitably sized protein particles at the required concentrations in the aqueous phase. The molecular size of whey proteins is less than the optimum, whereas the casein micelles in MPC are too large to induce depletion flocculation. Calcium caseinate consists of mixtures of casein aggregates of different sizes, but the concentration of aggregates capable of inducing depletion flocculation is probably too low. The extent of creaming in these emulsions is largely determined by the particle size of the droplets. Generally, in these emulsion systems, the creaming stability increases with increasing protein concentration up to a certain concentration and then remains almost constant (Euston & Hirst, 1999; Srinivasan et al., 2001). However, the creaming stability of emulsions formed with calcium caseinate or MPC at relatively high protein concentration tends to be higher than that of whey-protein-stabilized emulsions. This can be attributed to an increase in the droplet density as a result of the presence of a much thicker and denser adsorbed protein layer at the droplet surface.\n\nThe addition of moderate amounts of CaCl2 to emulsions containing excess sodium caseinate has been shown to eliminate depletion flocculation and to improve the creaming stability (Ye & Singh, 2001). This effect appears to be due to an increase in the average size of the casein aggregates in the aqueous phase, resulting in a large increase in the molecular mass of the caseins (Dickinson et al., 2001). In addition, there is a reduction in the concentration of unadsorbed caseinate. Both of these effects are expected to cause a substantial reduction in the concentration of small particles, which are assumed to be the main depleting species responsible for inducing reversible flocculation in the calcium-free systems. In contrast, depletion of calcium from MPC causes dissociation of casein micelles, resulting in much smaller casein aggregates that induce depletion flocculation in the MPC-stabilized emulsions (Ye, 2011). Presumably, the substantial reduction in osmotic pressure makes the magnitude of G DEP predicted from Equation 12.2 too small to cause depletion flocculation. Similarly, addition of NaCl at above a certain concentration reduces the extent of depletion flocculation of sodium caseinate emulsions and improves the creaming stability (Srinivasan et al., 2000). This effect is due to increased adsorption of protein at the droplet surface and hence a lower concentration of unadsorbed protein remaining in the solution. Decreasing the pH of emulsions formed with excess sodium caseinate also gradually eliminates depletion flocculation, through aggregation of adsorbed protein and a transfer of more protein to the droplet surface (Singh, 2005). Therefore, it seems to be possible to switch depletion flocculation off and on by controlling the concentration and the aggregation state of the casein molecules in the aqueous phase.\n\n## Heat-Induced Changes in Milk Protein-Based Emulsions\n\nFood emulsions are often heat treated at relatively high temperatures to provide a long shelf life to the product via microbial sterility. These heat treatments can cause denaturation and aggregation of adsorbed and unadsorbed proteins, resulting in aggregation or coalescence of droplets and gel formation. Emulsions formed with whey proteins at neutral pH are stable against heating when the ionic strength and\/or the concentration of protein in the emulsions are low. Addition of KCl at 100 mM or above has been shown to cause destabilization of whey protein emulsions, leading to gel formation (Hunt & Dalgleish, 1995).\n\nBoth the unadsorbed and the adsorbed proteins are necessary for the heat-induced aggregation of whey-protein-stabilized emulsions. Aggregation of emulsion droplets is more extensive and proceeds more rapidly as the concentration of protein in the emulsion is increased, whereas removal of unadsorbed protein from the emulsion decreases the rate of droplet aggregation (Euston et al., 2000). During heat treatment, the protein-covered droplet appears to interact more readily with the unadsorbed protein than with another emulsion droplet. This has been explained by assuming that the relative surface hydrophobicities of the emulsion droplet and the unadsorbed denatured whey proteins are different. Interaction of two emulsion droplets through their respective adsorbed protein layers will have a relatively low probability because the surface hydrophobicity is likely to be relatively low. When an emulsion droplet and an unadsorbed protein molecule aggregate, at least one of them (the denatured protein molecule) has a relatively high surface hydrophobicity, and this will increase the probability of interaction and aggregation (Euston et al., 2000).\n\nIn emulsions made with 3.0% WPI and 25% soya oil, the amount of adsorbed protein was shown to increase from 2.9 to 3.7 mg\/m2 within the first 10 min of heating at 75 \u00b0C, but further heating had no effect (Sliwinski et al., 2003). At 90 \u00b0C, the plateau value of about 4 mg\/m2 was reached within 5 min of heating. Studies on the effects of heating temperature in the range 50\u201390 \u00b0C on WPI emulsions (pH 7.0) (Monohan et al., 1996; Demetriades & McClements, 1998) show that droplet aggregation occurs on heating in the range 75\u201380 \u00b0C, which causes an increase in viscosity and a loss of creaming stability, but the degree of aggregation and the susceptibility to creaming decrease on heating at temperatures above 80 \u00b0C. It has been suggested that in the temperature range 75\u201380 \u00b0C, the whey protein molecules at the droplet surface are only partly unfolded and that not all of the hydrophobic amino acid residues are directed toward the oil phase. Consequently, the surface of the droplet is more hydrophobic, making it susceptible to droplet aggregation. At higher temperatures, the proteins become fully unfolded, with all of the hydrophobic residues being directed into the oil phase, which makes the droplets less prone to aggregation. The role of sulfhydryl\u2013disulfide interchange reactions in droplet aggregation is not clear. It has been suggested that disulfide-mediated interactions during heat treatment are not critical during the initial stages of aggregation, but they tend to strengthen the aggregates (Demetriades & McClements, 1998).\n\nDickinson and Parkinson (2004) and Parkinson and Dickinson (2004) reported that the addition of a very small proportion of caseinate (0.03\u20130.15% of the total emulsion) can stabilize a whey-protein-based emulsion against heat treatment. The magnitude of the effect is dependent on the type of casein, with the order of effectiveness being \u03b2-casein > sodium caseinate > \u03b1s1-casein. The stabilizing effect of the casein in these mixed milk protein systems is strongly synergistic. The casein polymer appears to be acting in a colloidal stabilizing capacity at a surface concentration very much lower than that at which it could be used as an emulsifying or stabilizing agent simply on its own. It has been suggested that adsorbed casein molecules keep the emulsion droplet surfaces sufficiently far apart to prevent the normal cross-linking processes that occur between whey-protein-coated droplets during heat-induced aggregation and gelation, because of the steric hindrance from the loops and tails of the disordered casein polymers (Parkinson & Dickinson, 2004).\n\nIn contrast to whey proteins, emulsions formed with sodium caseinate (2 wt% protein, 20% soya oil) are stable to heating at 90 \u00b0C for 30 min or 121 \u00b0C for 15 min, as determined by droplet size analysis (Hunt & Dalgleish, 1995; Srinivasan et al., 2002). However, the protein coverage and the adsorbed casein composition change upon heat treatment, indicating that interactions between unadsorbed caseinate molecules and caseinate at the droplet surface may occur during heating (Srinivasan et al., 2002). Analysis of adsorbed caseins isolated from emulsions heated at 121 \u00b0C for 15 min has shown that a substantial proportion of the adsorbed caseinate is polymerized to form high-molecular-weight aggregates (Srinivasan et al., 2002), held together through covalent bonds other than disulfide bonds. These covalent bonds appear to form mainly between caseinate molecules at the surface of the same droplet because of the higher local concentrations of casein molecules at the droplet surface. Interestingly, the adsorbed caseins also appear to undergo thermal degradation, resulting in the formation of low-molecular-weight species. Relatively high proportions of casein degradation products present at the droplet surface indicate that the adsorbed caseinate molecules are more susceptible to fragmentation during heating than those in solution and that these peptides remain adsorbed. This is probably due to different structures and conformations of the caseins at the droplet surface than of those in the solution.\n\nThe creaming stability of sodium caseinate emulsions has been found to improve upon heating, with the onset of depletion flocculation occurring at higher protein concentration than in unheated emulsions (Srinivasan et al., 2002). This can be attributed to a reduction in the number of unadsorbed caseinate molecules\/aggregates in the aqueous phase as a result of increased surface coverage and heat-induced polymerization and degradation of the casein molecules. The improvement in the creaming stability in heated emulsions at low protein concentrations may be attributed to an increase in droplet density because of the presence of greater amounts of polymerized protein at the droplet surface.\n\nThe surface protein composition of emulsion droplets may also change during heat treatment in emulsions formed with whey proteins. For WPI-stabilized emulsions, the amount of \u03b2-lactoglobulin at the droplet surface was found to increase during heat treatment, whereas the amount of adsorbed \u03b1-lactalbumin decreased markedly (Ye & Singh, 2006a; Ye, 2010). It seems that \u03b2-lactoglobulin displaces \u03b1-lactalbumin from the interface on heating at temperatures up to 90 \u00b0C, but the reason for this is not clear. Similar phenomena were observed in studies of exchanges of caseins and whey proteins at the interfaces of oil-in-water emulsion droplets (Dalgleish et al., 2002). It was found that, at temperatures above 40 \u00b0C, addition of WPI to the aqueous phase of caseinate-stabilized emulsions caused a displacement of adsorbed caseins. As the \u03b2-lactoglobulin and \u03b1-lactalbumin adsorbed, \u03b1s1\\- and \u03b2-caseins were desorbed, principally the \u03b1s1\\- caseins, whereas the \u03b1s2\\- and \u03ba-caseins were not displaced. The rate of the displacement or exchange reaction was temperature dependent, being almost undetectable at room temperature, but complete within 2 min at 80 \u00b0C. The displacement reaction was not affected by ionic strength; neither were any of the reactions apparently dependent on sulfhydryl exchange reactions (Dalgleish et al., 2002). However, no exchange of proteins occurred when an emulsion prepared with WPI was treated with caseinate and heat treated at 80 \u00b0C for 2 min (Brun & Dalgleish, 1999). This was surprising in view of the known interactions of whey proteins with \u03b1s2\\- and \u03ba-caseins, involving sulfhydryl\u2013disulfide interchange reactions.\n\n## Pressure-Induced Changes in Milk Protein-Based Emulsions\n\nThe effect of ultra-high pressure (100\u20131000 MPa) on the structures of milk proteins in aqueous solution has received considerable attention over the last few years (see Chapter 8). High pressure can disrupt the quaternary and tertiary structures of globular proteins with relatively little influence on their secondary structure. In addition, the proteinaceous colloidal aggregates (e.g., casein micelles), which are held together by ionic and hydrophobic interactions, can be dissociated by high-pressure treatment (Gaucheron et al., 1997; Huppertz et al. 2004). Whey proteins are sensitive to high-pressure treatments (L\u00f3pez-Fandin\u00f3 et al., 1996; Anema et al., 2005). Solution studies (Patel et al., 2005) of native \u03b2-lactoglobulin and whey protein products have shown that high-pressure treatment has a marked effect on the protein's conformation and consequently its aggregation behavior; the aggregation is more extensive at high protein concentrations (Patel et al., 2005). The formation of aggregates is most probably due to the generation of intermolecular disulfide bridges through sulfhydryl\u2013disulfide interchange reactions (Patel et al., 2006).\n\nIn model oil-in-water emulsions, high-pressure treatment has been shown to have no effect on the droplet size distribution or the emulsion viscosity of sodium-caseinate-based emulsions at pH 7 (Dumay et al., 1996). However, high-pressure treatment significantly induced flocculation of emulsion droplets and increased the emulsion viscosity of oil-in-water emulsions stabilized by \u03b2-lactoglobulin or WPC at neutral pH (Dumay et al., 1996; Dickinson & James, 1998). The unfolded unadsorbed whey proteins in the emulsion treated by high pressure appear to be the major contributor to the cross-linking or flocculation of emulsion droplets because greater emulsion flocculation was observed in emulsions with higher proportions of unadsorbed protein in the aqueous phase. As in the case of emulsions treated by heat processing, whey-protein-stabilized emulsions are more sensitive to pressure and temperature at pH values closer to the isoelectric point and at high ionic strength. In terms of the change in emulsion rheology, severe high-pressure treatment (800 MPa for 60 min) is equivalent to relatively mild thermal treatment (65 \u00b0C for 5 min) (Dickinson & James, 1998). In a concentrated emulsion formed with \u03b2-lactoglobulin (1% protein and 40% vol% n-tetradecane), an emulsion gel was produced following high-pressure treatment. When \u03b2-lactoglobulin or WPC solution was treated by high pressure before emulsion formation, the emulsions had larger droplet sizes than emulsions made with the native protein (Galazka et al., 1995). The results indicated a modification of protein structure, leading to the loss of emulsifying efficiency as a result of protein aggregation, despite an increase in surface hydrophobicity. After adsorption on the surface, the protein probably became partially unfolded at the interface, and subsequent pressure treatment caused no further conformational change. No studies on the behavior of emulsions formed with aggregated milk proteins, such as micellar casein, upon high-pressure treatment have been reported.\n\n## Milk Protein Hydrolysates and Oil-In-Water Emulsions\n\nMilk protein hydrolysates have been used extensively in infant and specialized adult nutritional formulations. Extensively hydrolyzed proteins are more easily digested and have substantially reduced immunological reactivities. These formulations are essentially multicomponent emulsion systems, and therefore the emulsifying properties of protein hydrolysates are important.\n\nThe flexibility and thus the availability of hydrophobic and hydrophilic segments within the protein chain can be improved by moderate enzymatic hydrolysis of globular proteins (e.g., whey proteins), thus improving the emulsifying properties of the protein. However, extensive hydrolysis (above 20% degree of hydrolysis), because of the production of many short peptides, has been found to be detrimental to the emulsifying and stabilizing properties of whey proteins (Singh & Dalgleish, 1998). The main form of instability in emulsions formed with highly hydrolyzed whey proteins is the coalescence that arises because of the inability of the predominantly short peptides to adequately stabilize the large oil surface generated during homogenization (Singh &; Dalgleish, 1998; Agboola et al., 1998ab). Nevertheless, it seems to be possible to make a fairly stable emulsion using highly hydrolyzed whey proteins at high peptide concentrations (protein-to-oil ratio about 1:1) and at low homogenization pressures as the sole emulsifier (Agboola et al., 1998a,b). Under these conditions, there is a sufficient amount of high-molecular-weight peptides (>5000 Da) in the emulsion to cover and stabilize the emulsion droplets.\n\nAddition of calcium or magnesium at above 20 mM has been shown to reduce the emulsion stability of emulsions formed with whey protein hydrolysates (Ramkumar et al., 2000). This instability arises mainly from the binding of calcium to the adsorbed peptides, leading to a reduction in the charge density at the droplet surface, which would reduce the inter-droplet repulsion and enhance the likelihood of droplet flocculation. The formation of calcium bridges between peptides present on two different emulsion droplets would also enhance flocculation.\n\nIn these emulsions, some very large droplets, apparently formed by coalescence, are also formed in the presence of calcium. This is likely to be due to the binding of calcium ions to the negatively charged peptides, causing aggregation of larger, more surface-active peptides. This situation would reduce the effective concentrations of emulsifying peptides available during emulsion formation.\n\nHeat treatment of emulsions stabilized by highly hydrolyzed whey proteins at 121 \u00b0C for 16 min results in destabilization of the emulsions, which appears to occur mainly via a coalescence mechanism (Agboola et al., 1998b). As the adsorbed peptide layers in these emulsions lack the cohesiveness of the parent proteins and have poor ability to provide steric or charge stabilization, increased collisions between the droplets during heating would cause droplet aggregation, leading to coalescence. It is also possible that desorption of some loosely adsorbed peptides occurs during heating, as indicated by the decrease in the amount of peptides associated with the oil surface after heating, which would also enhance coalescence.\n\n## Lactoferrin-Based Oil-In-Water Emulsions\n\nBovine milk contains low levels of lactoferrin, an iron-binding glycoprotein with about 700 amino acid residues and a molecular weight of about 80,000 Da (Baker & Baker, 2005). The polypeptide is folded into two globular lobes, representing its N- and C-terminal halves, commonly referred to as the N-lobe and the C-lobe. The surface of the lactoferrin molecule has several regions with high concentrations of positive charge, giving it a high isoelectric point (pI \u22489). This positive charge is one of the features that distinguishes lactoferrin from other milk proteins, such as \u03b2-lactoglobulin, which have isoelectric points in the range 4.5\u20135.5 and are negatively charged at neutral pH. This unique difference could allow the formation of oil-in-water emulsions containing cationic emulsion droplets, through adsorption of lactoferrin, over a wider pH range.\n\nYe and Singh (2006b) showed that, similar to other milk proteins (e.g., caseinate and \u03b2-lactoglobulin), lactoferrin adsorbs onto the interface of oil-in-water emulsion droplets and forms stable emulsions, but emulsion droplets with an overall positive surface charge are produced. In contrast to caseinate- and whey-protein-stabilized emulsions, the cationic emulsion droplets formed by lactoferrin are stable against a change in the pH from 7.0 to 3.0. For emulsions prepared under the same conditions (concentrations of oil and protein, pH, homogenization pressure), the droplet sizes in the lactoferrin emulsions are similar to those in \u03b2-lactoglobulin-stabilized emulsions, but the surface protein coverage (mg\/m2) of the emulsions made at pH 7.0 is higher in lactoferrin emulsions possibly because of its higher molecular weight.\n\nThe formation of a positively charged adsorbed layer in lactoferrin-stabilized emulsions over a wide pH range provides an opportunity for electrostatic interactions with other milk proteins that are mostly negatively charged around neutral pH. In aqueous solutions, lactoferrin tends to form a complex with \u03b2-lactoglobulin via electrostatic interactions (Wahlgren et al., 1993). Adsorption of such a complex onto the droplet surface during emulsion formation results in greater amounts of protein at the droplet surface and the formation of thick interfacial layers. It is interesting to note that oil-in-water emulsions formed using a binary mixture of \u03b2-lactoglobulin and lactoferrin are very stable, even though the overall charge (\u03b6-potential) of the emulsion droplets is close to zero. This suggests that steric repulsion plays an important role in this binary protein-stabilized emulsion.\n\nMultilayered emulsions can be produced by interactions of oppositely charged milk proteins, that is, lactoferrin and \u03b2-lactoglobulin or caseinate at neutral pH (Ye & Singh, 2007). A primary emulsion, containing either anionic droplets coated with \u03b2-lactoglobulin or cationic droplets coated with lactoferrin, can be produced. A secondary emulsion can then be made by mixing either \u03b2-lactoglobulin solution or lactoferrin solution with the primary emulsion (Ye & Singh, 2007). For example, when the emulsions formed with lactoferrin (1 wt%, pH 7.0) were diluted with aqueous phase containing a range of \u03b2-lactoglobulin concentrations, the adsorption of \u03b2-lactoglobulin increased considerably with an increase in the \u03b2-lactoglobulin concentration up to 0.42 wt%, with very little change above this concentration. This increase in \u03b2-lactoglobulin on the surface of emulsions formed with lactoferrin was further confirmed by the change in the \u03b6-potential. In the absence of \u03b2-lactoglobulin, the \u03b6-potential of the emulsion droplets was around +50 mV, because the lactoferrin used to stabilize the droplets has a net positive charge at pH 7.0. The \u03b6-potential became less positive, and eventually changed from positive to negative, as the \u03b2-lactoglobulin concentration in the emulsion was increased (Fig. 12.4).\n\nFigure 12.4 Influence of addition of \u03b2-lactoglobulin into emulsions formed with 1 wt% lactoferrin (30 wt% soya oil, pH 7.0) on the \u03b6-potential of the emulsion droplets. From Ye & Singh, 2007, reproduced with the permission of Springer.\n\nRecently, it was reported that the multilayered protein emulsions are more stable against various environmental condition such as Ca2+, high ionic strength, and heat treatment than the standard protein emulsions (Schmelz et al., 2011; Ye et al. 2012). For example, the addition of Ca2+ to the casein- or whey protein-stabilized emulsions at neutral pH causes droplet aggregation (Agboola & Dalgleish, 1995; Ye & Singh, 2000; Dickinson & Davies, 1999; Ye & Singh, 2001). Addition of lactoferrin to the caseinate or whey protein-stabilized emulsions results in the association of lactoferrin with adsorbed caseins or whey proteins via electrostatic interactions (Ye et al., 2012). This multisurface layer significantly reduces the calcium-induced destabilization of the emulsions, even when a small amount of lactoferrin is involved in the surface layer. Steric repulsion interactions produced by the large lactoferrin molecules on the surface were considered to contribute to this stabilizing effect (Ye et al., 2012).\n\n## Lipid Oxidation in Milk Protein-Based Emulsions\n\nIn addition to the physical properties of emulsions in foods, lipid oxidation is one of the major issues in food storage and consumption as it greatly influences the flavor, odor, and color of foods. Similar to the physical properties of emulsions, lipid oxidation in oil-in-water emulsions is influenced by the droplet size, interfacial characteristics of the lipid droplets, and the type of emulsifying agent (Dickinson & Stainsby, 1982; McClements & Decker, 2000). In addition to their remarkable emulsifying properties, both WPI and caseinate have been shown to inhibit the oxidative deterioration of unsaturated fatty acids, either as part of triacylglycerols or in free form (Hu et al., 2003; Djordjevic et al., 2004; Kiokias et al., 2006; Ries et al. 2010). WPI and caseinate therefore appear to be useful for the design of emulsions that serve as delivery systems for omega-3 fatty acids because of their dual functionality as emulsifiers and antioxidants (Singh et al., 2006). Such emulsions may be incorporated into real food emulsion systems, notably milk, yogurt, mayonnaise, ice cream, and cheese (Ye et al., 2009).\n\nRecently, the oxidation stability of the emulsions made with various milk protein products and linoleic acid was evaluated by determining the formation of lipid hydroperoxides and hexanal (Ries et al., 2010). The oxidative stability of both WPI- and sodium-caseinate-stabilized linoleic acid emulsions with smaller droplet size was greater than that of emulsions with larger droplet size. Other studies have reported contradictory results; some found greater lipid oxidation in emulsions with small droplets (Gohtani et al., 1999; Jacobsen et al., 2000; Lethuaut et al. 2002), whereas others found greater lipid oxidation for large droplets (Nakaya et al., 2005; Let et al. 2007).\n\nCaseinate appears to be a better antioxidant than WPI in the emulsions (Hu et al., 2003; Djordjevic et al., 2004; Kiokias et al., 2006; Ries et al. 2010). The inhibition of lipid oxidation by proteins in emulsions is considered to be mostly due to metal ion chelation and free radical scavenging (Benjelloun et al., 1991). In general, the specific antioxidative activity of caseinate appears to be due to its chelating capacity owing to its phosphoseryl groups (Baumy &; Brule 1988ab; Gaucheron et al., 1996; Bennett et al., 2000; Sugiarto et al. 2010), and that of whey protein appears to be due to its free radical scavenging activity as a result of free sulfhydryl groups (McClements & Decker 2000; Tong et al., 2000; Hu et al., 2003; Kiokias et al., 2007). As caseins do not possess a free sulfhydryl group, their free radical scavenging activity would be expected to be lower than that of whey proteins. On the other hand, whey proteins have a limited ability to chelate metal ions, due to lack of phosphoseryl groups. However, phosphoseryl groups and free sulfhydryl groups do not contribute solely to the total antioxidative capacity of the respective protein. It has been shown that the dephosphorylation of \u03b1s1\\- and \u03b2-casein only partially suppresses their antioxidative activity in a liposome system (Cervato et al., 1999) and that blocking of sulfhydryl groups of whey protein with N-ethylmaleimide in aqueous solution reduces its free radical scavenging activity by only 20% (Tong et al., 2000).\n\nRies et al. (2010) reported that the extent of lipid oxidation decreased with an increase in the protein concentration (Fig. 12.5). Furthermore, an increase in protein concentration led to a decrease in the difference in lipid hydroperoxide production between large- and small-droplet-sized emulsions. At high protein concentrations, the antioxidative effect of the protein in the emulsions appeared to offset the effects of emulsion droplet size and protein type. In addition to the physical barrier of the interfacial protein layer and the antioxidative effect of protein on the interface of emulsion droplets, unadsorbed protein in the continuous phase played an important role in the oxidative stability of emulsions.\n\nFigure 12.5 Lipid hydroperoxide concentration after 4 h (top) and hexanal concentration after 24 h (bottom) of storage at 50 \u00b0C in WPI-stabilized emulsions with average droplet sizes of 0.65 \u03bcm ( ) and 0.31 \u03bcm (\u02c6), and in sodium-caseinate-stabilized emulsions with average droplet sizes of 0.65 \u00b1 0.03 \u03bcm ( ) and 0.31 \u00b1 0.03 \u03bcm ( ). Each data point is the average of determination on two separate emulsions. Bars indicate standard errors. From, Ries et al., 2010 reproduced with the permission of Elsevier.\n\nThe experiments involving the replacement of the continuous phase of the emulsions with water or protein solutions showed that, compared with the control emulsion, the replacement of the continuous phase with water increased the production of lipid hydroperoxides. Replacement of the continuous phase with protein solution decreased the production of lipid hydroperoxides (Ries et al., 2010). In addition, the lipid hydroperoxide concentration was lower in the aqueous phase of emulsions containing caseinate than that containing WPI solution. Furthermore, it has been shown that the oxidative stability increased with increasing protein concentration in the continuous phase. This suggests that the antioxidative mechanism of protein at the interfacial region, such as binding trace metal ions from the lipid phase and free radical scavenging activity, may involve a dynamic exchange process with protein molecules from the continuous phase.\n\nThe antioxidative properties of the milk proteins are also influenced by processing and environmental conditions\u2014that is, heat treatment and change in the pH. When whey proteins that had been heated at temperatures higher than 80 \u00b0C were added to the fish oil emulsions, the oxidation stability of fish oil improved significantly compared to the control samples, as assessed by the lipid hydroperoxide formation and TBARS (Tong et al., 2000; Elias et al. 2007). It has been suggested that the increased oxidative stability could have been due to a greater exposure of free radical scavenging amino acid residues (e.g., tryptophan, tyrosine, phenylalanine, or methionine) to whey proteins and\/or greater interfacial contact of the protein because of increased hydrophobicity after heat treatment, both leading to improved effectiveness of the free radical scavenging process.\n\n## Behavior of Milk Protein-Stabilized Emulsions Under Physiological Conditions\n\nIn recent years, there has been considerable research on the physicochemical and structural changes in food emulsions during oral and gastrointestinal processing. These studies have focused mainly on understanding the role of emulsion structure on the lipolysis of emulsified triacylglycerols, with a view to developing emulsion systems with a controlled rate of lipid digestion and delivery of lipid soluble nutrients. Several studies have shown that the ability of lipases to digest emulsified oil droplets is affected by the composition of the interfacial layer and droplet size of emulsions (see reviews by Singh, 2011; Singh & Ye, 2013). Digestion involves complex mechanical, physicochemical, and physiological processes. Different parts of the human gastrointestinal tract, including the mouth, stomach, small intestine, and large intestine, work in a highly cooperative manner in the overall digestion and absorption of food. Here we provide a brief overview of the basic physicochemical and physiological processes that occur during digestion of milk protein-based emulsions.\n\nThe behavior of milk protein-based emulsions in the oral cavity is largely driven by the interactions of saliva with the adsorbed layer on emulsion droplets (van Aken et al., 2005; Vingerhoeds et al. 2008). Emulsions formed with WPI, sodium caseinate, or lysozyme showed flocculation of droplets when mixed with human saliva. This flocculation was considered to be driven by depletion, van der Waals's forces, and\/or electrostatic interactions between emulsion droplets and salivary mucins, and was largely dependent on the initial charge of the emulsion droplets (Vingerhoeds et al., 2005; Sarkar et al. 2009; Silletti et al., 2007a,b). For example, negatively charged protein-stabilized emulsions (i.e., \u03b2-lactoglobulin emulsions at neutral pH) did not interact with the artificial saliva because of strong repulsive forces between anionic mucin and the anionic \u03b2-lactoglobulin interfacial layer at neutral pH, but underwent depletion flocculation on the addition of higher concentrations of mucin (\u22651.0 wt%). In contrast, positively charged lactoferrin-stabilized emulsions interacted with mucin via electrostatic interaction and resulted in the formation of a secondary layer around the lactoferrin-stabilized droplets, with some bridging-type flocculation. These interactions could have an impact on sensorial and textural perceptions of food emulsions in vivo. For instance, Vingerhoeds et al. (2008) showed that positively charged lysozyme-stabilized emulsions, which underwent irreversible flocculation with saliva, were perceived to be dry and astringent in the mouth.\n\nThe biochemical conditions prevailing in the stomach have a major impact on the structure and stability of protein-based emulsions. The stomach has highly acidic pH (between 1 and 3 for a fasting stomach) and contains various minerals and both proteolytic and lipolytic enzymes. There is also some mechanical agitation because of peristalsis in the stomach (Ekmekcioglu, 2002; Kalantzi et al., 2006; Pal et al. 2007). As most protein-based emulsions are negatively charged at neutral pH, the decrease in the pH to below 2.0 causes substantial changes in the droplet charge, as well as some droplet aggregation around the isoelectric point. The action of pepsin is most critical as it causes major modifications of the adsorbed protein layers and the droplet characteristics, affecting the stability of the emulsion and the digestibility of its components. Because of their highly disordered structures, caseins undergo rapid hydrolysis by pepsin, but \u03b2-lactoglobulin shows some resistance to gastric digestion owing to its highly folded conformation (Reddy et al., 1988; Schmidt & van Markwijk, 1993). However, the rate of hydrolysis of \u03b2-lactoglobulin by pepsin increases when this protein is present as the adsorbed layer in an emulsion (Macierzanka et al., 2009; Sarkar et al. 2009), possibly due to change in the conformation of the \u03b2-lactoglobulin molecules upon adsorption at the oil\u2013water interface (Macierzanka et al., 2009). Surprisingly, adsorbed \u03b1-lactalbumin in oil-in-water emulsions appears to be more resistant to hydrolysis by pepsin, compared with native \u03b1-lactalbumin in solution (Nik et al., 2010). Casein and bovine serum albumin adsorbed at the surface of emulsion droplets have been shown to be readily hydrolyzed by pepsin after mixing with the gastric fluid (Li et al., 2012; Kenmogne-Domguia et al. 2013).\n\nBecause of the hydrolysis of interfacial protein by pepsin (Macierzanka et al., 2009; Sarkar et al., 2009; Nik et al., 2010; Li et al., 2012; Kenmogne-Domguia et al. 2013), the emulsions stabilized by milk proteins (such as WPI, sodium caseinate, \u03b2-lactoglobulin, or \u03b2-casein) undergo flocculation and coalescence of the droplets. In WPI-stabilized emulsions, the presence of excess unadsorbed protein appears to significantly improve the stability of the oil droplets during gastric digestion (Nik et al., 2010).\n\nThe human stomach also contains gastric lipase that is able to penetrate the adsorbed layer and act on the triglyceride core, preferentially cleaving at the sn-3 ester bonds of triglycerides. This lipolysis leads to the accumulation of protonated free fatty acids at the oil\u2013water interface, which competitively displace the proteins and peptides from the interface (Armand et al., 1994; 1996; Pafumi et al. 2002). However, the effects of gastric lipase on the stability of protein-stabilized emulsions have not been studied in any detail, partly because of the nonavailability of commercial gastric lipase for in vitro experiments. Several other materials, such as mucins and phospholipids, that are present in the stomach could alter the physicochemical properties of emulsions.\n\nThe small intestine is the main site for digestion and absorption, and it contains various salts, pancreatic enzymes, coenzymes, bile salts, and phospholipids. The pH of the partly digested\/modified emulsions entering the small intestine increases (to between 6 and 7), because of the secretion of sodium bicarbonate, which allows pancreatic enzymes to act efficiently (Bauer et al., 2005; Krondahl et al. 1997). The change in pH and ionic strength affect the stability of protein-stabilized oil droplets via electrostatic interactions. The pancreatic proteinases (i.e., trypsin and chymotrypsin) cause further hydrolysis of the adsorbed and unadsorbed proteins\/peptides, although the mechanisms of complex interactions of these proteinases with the adsorbed proteins\/peptides are not known. Pancreatic lipase adsorbs to the droplet interface, usually by complexation with co-lipase and\/or bile salts, and then cleaves triglycerides to form 2-monoglycerides and free fatty acids. Bile salts are highly surface-active and are able to displace any protein or peptide material remaining on the droplet surface. Partial or complete displacement of protein from the droplet surface after the introduction of bile salts into the simulated intestinal fluid for emulsions formed with caseins, WPI, and lactoferrin has been demonstrated (Maldonado-Valderrama et al., 2008; Torcello-Gomez et al., 2011; Hur et al., 2009; Mun et al., 2007; Sarkar et al., 2010; Klinkerson &; McClements 2010;). Whey proteins appeared to be more readily displaced than caseinates from the emulsion droplet surface (Mun et al., 2007). \u03b2-Lactoglobulin was rapidly displaced from the oil\u2013water interface compared to lactoferrin, possibly because of differences in droplet charges. Bile salts appeared to bind to the positively charged lactoferrin emulsion droplet, forming a mixed lactoferrin\/bile salt interfacial layer (Sarkar et al., 2010). The exposure of protein-stabilized emulsions to in vitro intestinal conditions has been shown to cause coalescence of some droplets initially (Golding & Wooster, 2010; Sarkar et al., 2010), but all aggregated\/flocculated droplets are broken down, resulting in uniform dispersions of small droplets. The negative charge imparted by adsorbing bile salts\/other surface-active molecules is thought to provide electrostatic repulsions between the droplets and to prevent their further aggregation.\n\n## Conclusions\n\nMilk proteins in soluble and dispersed forms have excellent surface-active and emulsion-stabilizing properties. Differences in the emulsifying abilities of milk proteins arise largely from the differences in structure, flexibility, state of aggregation, and composition of the proteins. These attributes of milk proteins (and hence their emulsifying abilities) are modified through various interactions occurring during the processing required to isolate the protein components, as well as during the manufacture of prepared foods. Emulsions with different surface compositions and structures can be made using different kinds of milk proteins; these emulsions exhibit different sensitivities to solution conditions, such as pH and ionic strength, and processing conditions, such as heat and high-pressure treatments. This could offer possibilities for the formation of emulsions with a range of functionalities for different food applications.\n\nMost of the research during the last 20 years has been performed on oil-in-water emulsions using purified or simple mixtures of caseins and whey proteins. A great deal of information is now available on the conformation of proteins at oil\u2013water interfaces, competitive exchange reactions between adsorbed and unadsorbed proteins, protein\u2013polysaccharide interactions, and factors controlling the rheology and stability of emulsions. In addition, some understanding of how processing conditions (heat treatments, high-pressure treatments) influence interfacial structures and emulsion properties has been achieved. There is much less understanding of the behavior of more complex mixtures of proteins in emulsions and the stability behavior of emulsions under processing environments commonly encountered in the food industry. In addition, there is a lack of understanding of the behavior of emulsions during oral processing in the mouth, as well as during digestion processes. It is critical to understand the oral behavior of emulsions, as common sensorial attributes (e.g., creaminess, smoothness) and the release of fat-soluble flavors are based on interfacial structures and rheological parameters. There is some evidence to show that the behavior of emulsions in the gastrointestinal tract is affected by their physicochemical properties, and that the properties of the interface modulate fat digestion and consequently influence the bioavailability of lipid nutrients. This is the emerging area of emulsion science which may contribute to the development of novel products with health and sensory attributes.\n\n# References\n\nAgboola SO ,  Dalgleish DG .  Calcium-induced destabilization of oil-in-water emulsions stabilized by caseinate or by \u03b2-lactoglobulin .  _Journal of Food Science_.  1995 ;60 : 399 \u2013 403 .\n\nAgboola SO ,  Singh H ,  Munro PA ,  Dalgleish DG ,  Singh AM .  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Influence of calcium chloride addition on the properties of emulsions stabilized by whey protein concentrate .  _Food Hydrocolloids_.  2000 ;14 : 337 \u2013 346 .\n\nYe A ,  Singh H .  Interfacial composition and stability of sodium caseinate emulsions as influenced by calcium ions .  _Food Hydrocolloids_.  2001 ;15 : 195 \u2013 207 .\n\nYe A ,  Singh H .  Heat stability of oil-in-water emulsions formed with intact or hydrolysed whey proteins: Influence of polysaccharides .  _Food Hydrocolloids_.  2006 ;20 : 269 \u2013 276 .\n\nYe A ,  Singh H .  Adsorption behaviour of lactoferrin in oil-in-water emulsions as influenced by interaction with \u03b2-lactoglobulin .  _Journal of Colloid and Interface Science_.  2006 ;295 : 249 \u2013 254 .\n\nYe A ,  Singh H .  Formation of multilayers at the interface of oil-in-water emulsion via interactions between lactoferrin and \u03b2-lactoglobulin .  _Food Biophysics_.  2007 ;2 : 125 \u2013 132 .\n\nYe A ,  Cui J ,  Taneja A ,  Zhu X ,  Singh H .  Evaluation of processed cheese fortified with fish oil emulsion .  _Food Research International_.  2009 ;42 : 1093 \u2013 1098 .\n\nYe A ,  Lo J ,  Singh H .  Formation of interfacial milk protein complexation to stabilize oil-in-water emulsions against calcium .  _Journal of Colloid and Interface Science_.  2012 ;378 : 184 \u2013 190 . \nChapter 13\n\n# Milk Protein\u2013Polysaccharide Interactions\n\nKelvin K.T. Goh*\n\nAnwesha Sarkar**\n\nHarjinder Singh***\n\n* Institute of Food, Nutrition and Human Health, Massey University, Palmerston North, New Zealand  \n** Nestec Ltd., Vevey, Switzerland  \n*** Riddet Institute, Massey University, Palmerston North, New Zealand\n\n## Abstract\n\nProteins and polysaccharides are common ingredients present in many food formulations. They are generally responsible for imparting key sensory attributes (e.g., textural attributes, controlled flavor release) and are capable of modifying phase stability in food colloidal systems. Their physicochemical properties depend not only on the molecular parameters of the individual biopolymers but also on the nature of interactions between the protein and polysaccharide molecules. This chapter provides an overview of the possible types and nature of interactions that can occur between protein and polysaccharide molecules in aqueous solutions and at interfaces. Extensive research carried out in this field over the last few decades outlining different milk protein polysaccharide interactions is summarized in tables. The last section attempts to categorize the different types of interactions and their impact on microstructures and rheological properties of the systems. The chapter concludes by stressing the importance of understanding these interactions, which potentially provide food scientists with the opportunities to modify or create novel food structures and functionalities.\n\n## Keywords\n\nMicrostructures\n\nbiopolymers\n\nphase separation\n\ndepletion flocculation\n\nthermodynamic incompatibility\n\nco-solubility\n\ncoacervates\n\nemulsion\n\nMaillard\n\ninterface\n\nOutline\n\nIntroduction 388\n\nMixing behavior of biopolymers 388\n\nPhase diagram 390\n\nNature of interactions in protein\u2013polysaccharide systems 392\n\nRepulsive Interactions 392\n\nAttractive Interactions 393\n\nCovalent Bonds 394\n\nMilk protein\u2013polysaccharide interactions in the aqueous phase 395\n\nMilk protein\u2013polysaccharide interactions at the interface 398\n\nRheological properties and microstructures of protein\u2013polysaccharide systems 401\n\nNoninteracting Protein\u2013Polysaccharide Mixtures 402\n\nNongelling Phase-separated System 403\n\nCasein Micelles and Galactomannans 403\n\nMilk Proteins and Xanthan 404\n\nGelling Phase-separated System 404\n\nWhey Protein and Galactomannans 405\n\nWhey Protein Isolate and Xanthan 405\n\n\u03b2-Lactoglobulin and Pectin 406\n\n\u03ba-Carrageenan and \u03b2-Lactoglobulin 406\n\nInteracting Protein\u2013Polysaccharide Mixtures 406\n\nNongelling Phase-separated System 407\n\n\u03b2-Lactoglobulin and Chitosan 407\n\nWhey Proteins and Exopolysaccharides 407\n\nWhey Proteins and Gum Arabic 407\n\nSodium Caseinate and Gum Arabic 408\n\nCasein Micelle and Pectin 408\n\nGelling Phase-separated System 409\n\nSodium Caseinate and Pectin 409\n\nCasein Micelles and Iota-carrageenan 409\n\nConcluding remarks 410\n\n## Introduction\n\nProteins and polysaccharides are broadly classified as biopolymers because of their large molecular structures. These macromolecules are known to possess important physicochemical roles, such as imparting thickening, stabilizing, gelling, and emulsifying properties in food products (Dickinson, 2003; Dickinson et al., 2003; Hemar et al., 2001a,b). The physicochemical properties of individual proteins and polysaccharides have been studied extensively in the last several decades. It is well established that the factors influencing the physicochemical properties of these macromolecules in solution include molar mass, molecular conformation, polydispersity, charged density, concentration, pH, ionic strength, temperature, solvent quality, and the nature of molecular (intra-\/inter-) interactions (de Kruif & Tuinier, 2001; Doublier et al., 2000; Tolstoguzov, 1997). In many food systems, their physical properties become more complex, as both proteins and polysaccharides are present (either naturally or added as ingredients) among the complex multicomponent mixtures. The overall stability and microstructure of these food systems depend not only on the physicochemical properties of proteins or polysaccharides alone, but also on the nature and strength of interaction between protein and polysaccharide (Dickinson, 1998b; Dickinson et al., 1998). This chapter reviews a number of studies carried out in the field of protein\u2013polysaccharide interactions, with a particular focus on milk proteins and a diverse range of polysaccharides in aqueous systems.\n\n## Mixing behavior of biopolymers\n\nWhen aqueous solutions of proteins and polysaccharides are mixed, one of four phenomena can arise: (1) co-solubility, (2) thermodynamic incompatibility, (3) depletion interaction, or (4) complex coacervation (Fig. 13.1) (Benichou et al., 2002; de Kruif & Tuinier, 2001; de Kruif et al., 2004; Dickinson, 2003; Martinez et al., 2005; Schmitt et al., 1998; Syrbe et al., 1998; Tolstoguzov, 1991; ; ). These phenomena can be explained as follows:\n\nFigure 13.1 Different types of interactions (co-solubility, thermodynamic incompatibility, depletion interaction, and complex coacervation) between protein and polysaccharide in aqueous solutions.\n\nCo-solubility refers to the creation of a stable homogeneous solution, that is, the generation of one phase in which the two macromolecular species either do not interact or exist as soluble complexes in the aqueous medium. When intermolecular attraction is absent, macromolecules are only co-soluble in dilute solutions where the entropy of mixing favors more randomness in the system (Tolstoguzov, 2003). To achieve co-solubility from a thermodynamic angle, the Gibbs free energy of mixing (\u0394Gmixing) given in Equation 13.1 must be negative. This means that the entropy of mixing should favorably exceed the enthalpy term. (Note: The highest level of entropy is achieved when the different kinds of molecules are randomly distributed throughout the system) (McClements, 2005a). The expression for Gibbs free energy accompanying mixing in standard conditions is given by\n\n\u0394Gmixing=\u0394Hmixing\u2212T\u0394mixing\n\n (13.1)\n\nwhere \u0394Gmixing, \u0394Hmixing, and T\u0394Smixing are the free energy, enthalpy (interaction energy), and entropy changes between the mixed and unmixed states, respectively.\n\nWhen the size of the molecules is small, as it is in the case of monomer sugars and hydrophilic amino acids, mixing the two species results in a co-soluble system. However, with increasing molecular weight and concentration of the polymers, the system tends to become less co-soluble due to thermodynamic incompatibility (Tolstoguzov, 1991; ). This is because the entropy of mixing of biopolymers is significantly lower than that of the monomers. The bulky size and rigid structure of biopolymer molecules decrease the entropy of mixing, resulting in a higher free energy. For a mixed biopolymer solution, the enthalpy\u2013entropy balance generally results in mutual exclusion of one biopolymer from the local vicinity of the other. This means that biopolymers in mixed solution show a preference to be surrounded by their own type; otherwise, consequently, their mixtures tend to separate into liquid phases, as described below (Grinberg & Tolstoguzov, 1972; ; Polyakov et al., 1997; Tolstoguzov, 1988; ; Tolstoguzov et al., 1985).\n\nThermodynamic incompatibility occurs when the two dissimilar noninteracting macromolecular species separate into two different phases as enthalpy of mixing exceeds the entropy difference (Benichou et al., 2002; Grinberg & Tolstoguzov, 1997; Schmitt et al., 1998; Tolstoguzov, 2002). The driving force to segregation is the enthalpic advantage of molecules being surrounded by others of the same type. For small molecules, this is normally outweighed by the entropic advantage of both species being free to move throughout the entire volume. However, for polymer solutions, where there are far fewer individual molecules, the entropy of mixing is much smaller, which can allow phase separation to occur. Of the two distinct immiscible aqueous phases formed, each phase is mainly loaded with only one biopolymer species, that is, a protein-rich phase and a polysaccharide-rich phase. Phase separation due to incompatibility can also occur if each biopolymer shows varying affinity toward the solvent (Piculell & Lindman, 1992; Tolstoguzov, 1991). In this case, solvent\u2013protein (or solvent\u2013polysaccharide) interactions are favored over protein\u2013polysaccharide interactions and solvent\u2013solvent interactions, leading to two phases, one enriched in protein and the other in polysaccharides (Doublier et al., 2000). Thermodynamic incompatibility can also arise within a mixture of polysaccharides or proteins. Some examples include polysaccharides with different structures; proteins of different classes like water-soluble albumins with salt-soluble globulins; native and denatured forms of the same protein, as well as aggregated and nonaggregated forms of the same protein (Tolstoguzov, 2002).\n\nThermodynamic incompatibility is highly dependent on pH and ionic strength, and is prevalent when protein and neutral polysaccharide are present or when both protein and polysaccharide carry the same negative charge at neutral pH (Doublier et al., 2000). Although thermodynamic incompatibility is prevalent in mixed-polymer systems, some of these systems do not achieve thermodynamic equilibrium within a limited timescale due to the presence of kinetic energy barriers. When the kinetic energy exceeds the thermal energy of the system, the molecules become 'trapped' in a metastable state (McClements, 2005a). Some examples of kinetic energy barriers include the formation of a gel network within an incompatible system or a highly viscous continuous phase that slows down the phase separation process. The choice of which phase to gel and the component used to promote gelation depends on the type of biopolymers used in the system. (Bryant & McClements, 2000a,b; Kim et al., 2006; Norton & Frith, 2001).\n\nDepletion flocculation usually involves spherical particles in the presence of macromolecules (Asakura & Oosawa, 1954; ; Bourriot et al., 1999a). Phase separation of particulate suspension is enhanced by the addition of a polymer. This phenomenon usually occurs in the colloidal dispersion in the presence of noninteracting polymers (e.g., polysaccharides in an emulsion, polysaccharides, and colloidal casein micelles). The higher osmotic pressure of the polymer molecules surrounding the colloidal particles (as compared to the interparticle region) causes an additional attractive force between particles leading to the flocculation of particles. The attractive force depends on the size, shape, and concentration of the polymer molecules and the colloidal particles (Hemar et al., 2001b). When colloidal particles approach each other, the excluded (or depleted) layer starts to overlap, allowing more space for the polymer molecules. The increase in volume causes the total entropy of the system to increase (i.e., free energy to decrease), which in turn encourages attraction between the colloidal particles (de Bont et al., 2002). In a mixed protein\u2013polysaccharide system containing casein micelles, phase separations are often attributed to depletion flocculation phenomena (Bourriot et al., 1999a; Tuinier & De Kruif, 1999; Tuinier et al., 2000). This is because of the large colloidal particle size of casein micelles and because increasing the concentration of polysaccharides results in greater attraction between the casein micelles (Doublier et al., 2000).\n\nComplex coacervation is the formation of electrostatic complexes between the protein and polysaccharide molecules, leading to a two-phase system. One phase has both the biopolymers in a complex matrix, while the other phase contains mainly the solvent water and is depleted in both biopolymers. Complex coacervation commonly occurs between oppositely charged biopolymers. Complex coacervation between oppositely charged proteins and polysaccharides was first reported by mixing gelatin and gum arabic in acetic acid solution (Tiebackx, 1911). The term coacervation was first introduced in 1929 to describe a process in which aqueous colloidal solutions separate into two liquid phases, one rich in colloid, that is, the coacervate, and the other containing little colloid (Bungenberg de Jong & Kruyt, 1929). If the two biopolymers are present in equal proportions by weight at a pH such that they carry net equal but opposite charges, the yield of coacervates will be at its maximum (Schmitt et al., 1998). The size and morphology of these structures may be exploited to bring about new functionalities and textural changes in processed foods.\n\n## Phase diagram\n\nMixing two aqueous solutions of proteins and polysaccharides may give rise to a one-phase or two-phase system depending on the solution composition and environmental conditions, as depicted in Figure 13.1 (Benichou et al., 2002; de Kruif & Tuinier, 2001; de Kruif et al., 2004; Dickinson, 2003; Martinez et al., 2005; Schmitt et al., 1998b; Syrbe et al., 1998; Tolstoguzov, 1991;1997).\n\nIn a one-phase system, protein and polysaccharide can exist either as individual molecules or as soluble complexes that are uniformly dispersed throughout the entire system. However, with increasing molecular weight and concentration of biopolymers, the system tends to become less co-soluble and to give rise to a two-phase system; that is, the system separates into two distinct phases that have different biopolymer concentrations.\n\nFor a system with relatively strong net repulsion between protein and polysaccharide in aqueous solution, the two biopolymers move into two different phases due to thermodynamic incompatibility. Two distinct immiscible aqueous phases are formed, and each of them is mainly loaded with only one-biopolymer species, that is, one phase protein-rich and the other phase polysaccharide-rich. A typical phase diagram for segregating the biopolymer system is shown in Figure 13.2 which has been explained by many researchers (Antonov et al., 1982; Bourriot et al., 1999a; Clark, 2000; Closs et al., 1999; Ercelebi & Ibanoglu, 2007; Grinberg & Tolstoguzov, 1972; ; Lundin et al., 2003; Polyakov, et al., 1980; Thaiudom & Goff, 2003; Tolstoguzov, 2003; Tolstoguzov et al., 1985). The phase diagram consists of a typical binodal curve (the solid line curve), which divides the single-phase miscible region (below the curve) from the two-phase immiscible region (shaded region). The binodal branches exhibit the points of limited co-solubility. The points of the binodal curve connected by the tie line represent the composition of the coexisting equilibrium phases. From the phase diagram, it is possible to determine the effective concentrations of biopolymers in the two phases and the concentrations at which maximal co-solubility of the biopolymers is achieved. In addition, it helps to establish which of the two biopolymers forms the continuous phase.\n\nFigure 13.2 A typical phase diagram showing a protein\u2013polysaccharide solution, with water as the solvent at a particular pH, temperature, and ionic strength. A sample of composition O (which was initially made with A% of protein and B% of polysaccharide) separates out into two bulk polymer-rich phases. The protein-enriched phase will have a composition C% protein, whereas the polysaccharide-enriched phase will have composition D% polysaccharide. The binodal (solid curve) separates the single-phase region from the two-phase domain (which can be obtained by direct observation of the phase separation in test tubes). The % protein in the polysaccharide phase will be negligible and vice versa. The tie-line can be calculated. The points on the tie-line have the same effective concentration of the phases at equilibrium even though their phase volume ratios differ. The ratio of DO\/OC represents the volume ratio of protein-rich phase C and polysaccharide-rich phase D, respectively, by inverse-lever rule. If O is shifted along the tie-line to O1, the new phase volume ratio will be DO1\/ O1C. The line obtained by joining the midpoints (+) of two or more tie-lines gives the rectilinear diameter. The coordinates of the critical point E (obtained from the intersection of the binodal to the rectilinear diameter) show the composition of a system separating into two phases of the same volume and composition, which means the separated-phase systems will have 50% protein and 50% polysaccharide in the same phase-volume ratio. Point F represents the separation threshold that is the minimum critical concentration required for the biopolymers to separate into two phases.\n\n## Nature of interactions in protein\u2013polysaccharide systems\n\nThe interactions responsible for complex formation between biopolymers (Fig. 13.1) can be classified as weak or strong, specific or nonspecific, attractive or repulsive (Dickinson, 1993). The overall interaction between protein and polysaccharides is the average of the following different intermolecular forces arising between the various segments and chains of the two biopolymers (Dickinson, 1998b; Schmitt et al., 1998).\n\n### Repulsive Interactions\n\nRepulsive interactions are always nonspecific and of transient duration. They usually arise from excluded volume effects and\/or electrostatic interactions and tend to be weak, except at very close range or very low ionic strength.\n\nThe excluded volume or steric exclusion effects are the nonspecific and transient interactions. This arises when proteins and polysaccharides are non-ionic and noninteracting. As a result, the volume surrounding one polymer molecule becomes unavailable to the other polymer molecule in the aqueous solution (Polyakov et al., 1997; Schmitt et al., 1998; Tolstoguzov 1991; ; ). Excluded volume effects exhibit mutual spatial restrictions and competition between the biopolymers for solution space; that is, there is a reduction in the mixing entropy of the system due to the reduction in the volume available for the biopolymer molecules to occupy.\n\nNet repulsive interactions, due to electrostatic effects, depend largely on the pH and ionic strength of the background electrolyte concentration. The electrostatic repulsive interactions are commonly found in mixtures of proteins and anionic polysaccharides under conditions where both the biopolymers carry the same net charge\u2014for example, pH is above the isoelectric point (pI) of the protein.\n\n### Attractive Interactions\n\nAttractive interactions between proteins and polysaccharides may be weak or strong and either specific or nonspecific. Nonspecific attractive interactions arise as a result of a multitude of weak interactions between groups on the biopolymers, such as electrostatic, van der Waals, hydrogen bonding, and hydrophobic interactions. Hydrogen bonding and hydrophobic interactions are actually collective interactions (e.g., electrostatic, van der Waals, and steric overlap), including some entropy effects (McClements, 2005a).\n\nElectrostatic interactions are the most important forces involved in the complex formation between proteins and ionic polysaccharides. These interactions between charged biopolymers lead to a decrease in the electrostatic free energy of the system. Moreover, the enthalpy contribution due to interactions of oppositely charged biopolymers and liberation of counter-ions along with water molecules often compensates for the loss of configurational entropy of mixing rigid biopolymers (Piculell & Lindman, 1992; Tolstoguzov, 1997). Strong electrostatic attractive interactions occur between positively charged proteins (pH < pI) and anionic polysaccharides, especially at low ionic strengths. Generally, two types of complexes are formed by electrostatic interactions (Schmitt et al., 1998; Tolstoguzov, 1997; ; ). Soluble complexes are obtained when opposite charges carried by the two biopolymers are not equal in number, whereas insoluble complexes result when the net charge on the complex is close to zero.\n\nVan der Waals forces are extremely weak electrical attractions arising because of temporary dipole interactions (Dickinson, 1998b; Sherony & Kintner, 1971; Stainsby, 1980). Basically, every atom has an electron cloud that can yield a temporary electric dipole. The dipole in one atom can induce a corresponding dipole in another atom. This is possible only if the atoms are close. However, if they are too close, repulsive forces between the adjacent negatively charged electron clouds may not allow these van der Waals attractions. Although these transient electrical attractive forces are very weak, they can influence macromolecular interactions, together with other noncovalent forces described above (Damodaran, 1997).\n\nHydrophobic bonding is an entropy-driven long-range interaction between nonpolar groups, and it is promoted by conformational and structural modifications of biopolymers, mostly by the unfolding of polymeric chains exposing hydrophobic groups. These kinds of interactions are promoted by an increase in temperature (Antonov et al., 1996a; Piculell & Lindman, 1992; Samant et al., 1993; Stainsby, 1980; Tolstoguzov, 1997).\n\nHydrogen bonding is a moderately strong bond that becomes relatively insignificant at high temperatures. These bonds are ionic in nature and refer to the interaction between hydrogen atoms attached to an electronegative atom (oxygen, sulfur) with another electronegative atom (e.g., sulfur of sulfate group), that is, \u2013O-H\u03b4+...\u03b4 \u2212O <. A classical example of hydrogen bonding has been shown in the complex coacervation of gelatin and pectin (Braudo & Antonov, 1993), which is obtained over a wide range of pH, including the isoelectric pH (4.8) of gelatin. Protein\u2013polysaccharide hydrogen bonding between gelatin\u2013pectin, gelatin\u2013alginate, and chitosan\u2013collagen has been well established by various studies over a wide range of pH values (Antonov et al., 1996a; Taravel & Domard, 1995).\n\n### Covalent Bonds\n\nCovalent bonds are very strong, specific, non-electrostatic, and permanent linkages. Two principal methods can be used to generate a covalent linkage between proteins and polysaccharides. The most commonly used method utilizes the chemical reaction between amino groups of proteins and carboxylic group of polysaccharides (the Maillard reaction) to give an amide covalent bond (Stainsby, 1980). Covalent bonds can also be generated enzymatically using the oxidoreductase family of enzymes (E.C. 1.XXX) that catalyze the oxidation of the phenolic group of tyrosine residues with carbohydrate groups containing phenolic residues, such as cereal arabinoxylans (Boeriu et al., 2004). Tyrosine-containing peptides have also been conjugated with ferulic acid (Oudgenoeg et al., 2001) and with whey proteins through the use of three different oxidoreductases (Faergemand et al., 1998).\n\nThe potential for cross-linking proteins and polysaccharides using transglutaminase has been suggested (E.C. 2.3.2.13) (Flanagan & Singh, 2006). Many polysaccharides contain residual protein, for example, gum arabic, guar gum, and locust bean gum (LBG), all containing low levels of protein. Gum arabic (approximately 2% protein, depending on source) consists of, among other subunits, a glycoprotein and an arabinogalactan protein. Provided the residual protein in these polysaccharides contains lysine and\/or glutamine residues, the treatment of protein and polysaccharide mixtures with transglutaminase could theoretically lead to the formation of heteropolymers (i.e., protein\u2013polysaccharide conjugates) in addition to homopolymers (cross-linked protein or cross-linked polysaccharide). Flanagan and Singh (2006) demonstrated that sodium caseinate\u2013gum arabic conjugates catalyzed by transglutaminase can be produced.\n\nThese kinds of interactions are generally very stable to pH and ionic strength. Because of their stable properties, this kind of bonding has been intentionally used to produce conjugated emulsifiers (Akhtar & Dickinson, 2003; ; Benichou et al., 2007; Dunlap & C\u00f4t\u00e9, 2005; Neirynck et al., 2004; Shepherd et al., 2000; Song et al., 2002). In most of these studies, the covalent conjugation between proteins and polysaccharides has been achieved through the Maillard reaction.\n\nApart from these major interactions, ion-bridging involving the binding of cations like Ca2+ may also contribute to protein\u2013polysaccharide interactions to some extent, although they do not have the predominant influence (Antonov et al., 1996a; Dickinson, 1998b; Stainsby, 1980). For example, firm sodium caseinate gels (G \u2032 > 100 Pa) were formed with pectin concentrations \u22650.6% at one particular degree of methylation (\u223c31%) and amidation (\u223c17%) in the presence of Ca2+ ions (1.8 mM) at pH \u223c3.6 (Matia-Merino et al., 2004).\n\n## Milk protein\u2013polysaccharide interactions in the aqueous phase\n\nMilk proteins and polysaccharides dissolved in the aqueous phase form a pseudoternary system of milk protein\u2013polysaccharide water. Various interactions in these systems could lead to complex formation or bulk-phase separation. Extensive studies have been carried out in areas of protein\u2013polysaccharide interactions, particularly using well-studied milk proteins and commercially available polysaccharides (Dickinson, 1998b). Table 13.1 and 13.2 show a compilation (non-exhaustive) of various milk proteins (casein and\/or whey proteins) and polysaccharide mixtures in aqueous systems, and the conditions in which different kinds of interactions occur. Following this section, we describe the microstructure and rheological properties of some of these systems.\n\nTable 13.1\n\nCasein\u2013Polysaccharide Interactions in Aqueous Systems\n\nSL. NO. | Casein\u2013polysaccharide aqueous systems | Conditions | Interactions | References  \n---|---|---|---|---  \n1. | Milk proteins (Casein micelle + Whey proteins) \\+ Pectin (High methoxyl-62.7% methylated) | 20 \u00b0C, pH 6.0\u201310.5, 0\u20130.5 M NaCl | Thermodynamic incompatibility | (Antonov et al., 1982)  \nMilk proteins (Casein micelle + Whey proteins) \\+ Gum arabic  \nMilk proteins (Casein micelle + Whey proteins) \\+ Arabinogalactan  \n2. | Casein micelle + Alginate | 25 \u00b0C, pH 7.2 | Thermodynamic incompatibility | (Suchkov et al., 1988; Suchkov et al., 1981)  \n3. | Casein micelle (2.5%) \\+ Pectin (Low methoxyl- 35%, High methoxyl- 73%, Low methoxyl amidated- 35% methylated & 20% amidated) (0.1\u20130.2%) | 60 \u00b0C, pH 6.7\/ 5.3 | pH 6.7: Depletion interaction Methoxylation affects interaction | (Maroziene & de Kruif, 2000)  \n4. | Casein micelle (0.8-4%) \\+ Galactomannans (Guar Gum, LBG) (0.09\u20130.3%) | 5\/20 \u00b0C,  \npH 6.8\/ 7.0, 0.08\/0.25 M NaCl, Sucrose  \n(10\u201340 wt%) | Depletion interaction Sucrose affects interaction | (Bourriot et al., 1999a; Schorsch et al., 1999)  \n5. | Casein micelle (1.0%) \\+ Carrageenan (\u03b9-, \u03ba-, \u03bb-forms) (0.12%) | 60\/ 50\/ 20 \u00b0C, pH 6.7\/ pH 7.0, 0.25 M NaCl\/ 0.05 M NaCl\u20130.01 M KCl | Depletion interaction  \n| (Bourriot et al., 1999c; Dalgleish & Morris, 1988; Langendorff et al., 1997; ; )  \n6. | Sodium caseinate (0.1\u20130.5%) \\+ Gum arabic (0.01\u20135%) | pH 2.0\u20137.0, 0.5 M NaCl, slow acidification with glucono-\u03b4-lactone | Soluble electrostatic complexation | (Ye, Flanagan, & Singh, 2006)  \n7. | Casein micelle (0.1%) \\+ Exopolysaccharide (5.0%) (Lactococcus lactis subsp. cremoris B40) | 25 \u00b0C, pH 6.6 | Depletion interaction | (Tuinier & De Kruif, 1999; Tuinier et al., 1999)  \n8. | Sodium caseinate + Maltodextrin  \n(2:1, 1:1, and 1:4) | 60 \u00b0C, 2\u20134 days | Covalent conjugate via Maillard reaction. No phase separation | (Morris et al., 2004; Shepherd et al., 2000)  \n9. | Casein (\u03b2-casein, \u03b1s-casein) \\+ Polysaccharide  \n(Dextran, Galactomannan)  \n1:1) | 60 \u00b0C, 24 hours | Covalent conjugate via Maillard reaction. No phase separation | (Dickinson & Semenova, 1992; Kato et al., 1992)  \n10. | Sodium caseinate (6.0%) \\+ Sodium alginate (1%) | 23 \u00b0C, pH 7.0, | Thermodynamic incompatibility | (Guido et al., 2002; Simeone et al., 2002)\n\nTable 13.2\n\nWhey Protein\u2013Polysaccharide Interactions in Aqueous Systems\n\nSL. NO. | Whey protein\u2013polysaccharide aqueous systems | Conditions | Interactions | References  \n---|---|---|---|---  \n1. | \u03b2 -lactoglobulin ( \u03b2 -lg) (0.5%) \\+ Chitosan (Degree of Deacetylation: 85%) (0\u20130.1%) | pH 3.0\u20137.0, 5 mM phosphate buffer | pH dependent \u03b2-lg-chitosan Soluble\/insoluble complex coacervation | (Guzey & McClements, 2006a, 2006b)  \n2. | Heat denatured whey protein isolate (HD-WPI) (8.0%) \\+ Pectin (28, 35, 40, 47, and 65% methylation) (0.1\u20131.5%) | 80 \u00b0C\/85 \u00b0C, pH 6.0\/ 7.0, 5.0\/10.0 mM CaCl2 | Thermodynamic incompatibility | (Beaulieu et al., 2001; Kim et al., 2006)  \n3. | \u03b2 -lg (12.0%) \\+ Alginate (0.1\u20131.0%) | 87 \u00b0 C\/30 \u00b0C, pH 7.0\/ (3.0\u20137.0),  \nHigh pressure | pH dependent \u03b2-lg-chitosan Soluble\/insoluble complex coacervation | (Dumay et al., 1999; Harnsilawat et al., 2006)  \n4. | \u03b2 -lg (0.05%) \\+ Pectin (Low methoxyl- 28.3\/42.6%, High methoxyl- 71.3\/73.4%) (0.0125%) | 4\u201340 \u00b0C\/ 25 \u00b0C\/87 \u00b0C, pH 4.0\u20137.5\/ 6.5, 0.11\/ 0.1\u20131.0 M NaCl, High pressure | pH, ionic strength, & temp. Complex coacervation. Precipitation for modified pectin. Methylation affects complexation | (Dumay et al., 1999; Girard et al., 2002; 2003a; 2004; Maude et al., 2003b; Kazmiersi et al., 2003; Wang & Qvist, 2000)  \n5. | Whey protein isolate (WPI) (5.0%) \\+ Galactomannans (LBG) (0\u20130.4%) | pH 5.0\u20137.0 | pH and concentration: Biphasic gel | (Tavares & Lopes da Silva, 2003)  \n6. | HD-WPI (8.5%) \\+ Xanthan gum (0\u20130.2%) | 25\u201390 \u00b0C\/ 75\u201380 \u00b0C, pH 7.0\/ 5.4, 0.2 M NaCl, High pressure treatment | Native WPI: Co-solubility.  \nHD-WPI: Thermodynamic incompatibility | (Bryant & McClements, 2000b; Li et al., 2006)  \n7. | WPI (4\u201312.5%) \\+ Xanthan gum (0.01\u20131.0%) | pH 5.5\/ 6.0\/ 6.5\/7.0, 0.1\/0.5 M NaCl, high-pressure treatment | Depletion interaction, pH-dependent electrostatic complexation | (Benichou et al., 2007; Bertrand & Turgeon, 2007; Hemar et al., 2001b; Laneuville et al., 2000; Zasypkin et al., 1996)  \n8. | Bovine serum albumin (BSA) + Sulfated polysaccharides (\u03b9-, \u03ba-carrageenan, dextran sulfate) (2.5:1 and 5:1) | pH 6.5\u20138.0, High-pressure treatment | Complex coacervation | (Galazka et al., 1996; ; )  \n9. | HD-WPI (10.0%) + \u03ba -Carageenan (0.5%) | 80 \u00b0C,  \npH 1.0\u201312.0 | Complex coacervation | (Mleko et al., 1997)  \n10. | \u03b2 -lg (0.5\u201310.0%) + \u03ba- Carageenan (1.0%)  \n(1:2, 5:1, and 10:1) | 45\u201380 \u00b0C, pH 7.0, 0.1 M NaCl\/ 0.01 M CaCl2 | Temp., pH, & concentration dependent phase separated bi-continuous gel formation | (Capron et al., 1999; Ould Eleya & Turgeon, 2000)  \n11. | \u03b2 -lg + Gum arabic  \n(2:1) | pH 3.6\u20135.0, 0.005\u201310.7 mM NaCl | Complex coacervation | (Sanchez et al., 2002; ; Sanchez & Renard, 2002; Schmitt et al. 1998; ; )  \n12. | WPI + \u03bb -Carrageenan  \n(1:1 to 150:1) | pH over a wide range, 0\u20130.1 M (NaCl\/ CaCl2) | Electrostatic complexation Precipitation | (Weinbreck et al., 2004a)  \n13. | WPI + Gum arabic  \n(2:1) | pH 4.0\u20137.0, 0\u20130.1 M NaCl | Complex coacervation Glassy state | (Weinbreck et al., 2003a; 2004b; 2004c)  \n14. | \u03b2 -lg + Carboxymethyl dextran  \n(1:1 and 7:2) | 4 \u00b0C\/ 25 \u00b0C, pH 5.5\/ 4.75 | \u03b2-lg-Carboxymethyl dextran covalent conjugate. No phase separation | (Hattori et al., 1994)  \n15. | WPI + Carboxymethyl potato starch  \n(2:1) | 24 \u00b0C, pH 7.0 | WPI-Carboxymethyl starch covalent conjugate | (Hattori et al., 1995)  \n16. | WPI + Exopolysaccharide (Lactococcus lactis subsp. cremoris B40)  \n(2:1) | 25 \u00b0C, pH over a wide range, 0\u20130.1 M (NaCl\/CaCl2), Heat treatment of WPI | Electrostatic complexation Precipitation HD-WPI: Depletion interaction | (de Kruif & Tuinier, 1999; Tuinier & de Kruif, 1999; Weinbreck et al., 2003b)  \n17. | \u03b2 -lg + Pullulan | 4 \u00b0C, 0.01 M NaCl | Depletion interaction | (Wang et al., 2001)  \n18. | \u03b2 -lg + Carboxymethyl cellulose  \n(CMC) | 60 \u00b0C, pH 2.5\u20137.0, 0.05\u20130.2 M | Insoluble electrostatic complex Sedimentation | (Hansen et al., 1974; Hidalgo & Hansen, 1969)  \n19. | WPI + Maltodextrin  \n(1:2 and 1:3) | 80 \u00b0C, 2 hours, 79% RH | Covalent conjugation No phase separation | (Akhtar & Dickinson, 2007)  \n20. | WPI\/ Whey protein concentrate (WPC) + Pectin  \n(4:1, 2:1, 1:1, and 1:2) | 60 \u00b0C, pH 7.0, 14 days, | Covalent conjugation No phase separation | (Mishra et al., 2001; Neirynck et al., 2004)\n\n## Milk protein\u2013polysaccharide interactions at the interface\n\nIn an emulsion system containing both milk protein and polysaccharide, protein generally forms the primary interfacial layer by directly adsorbing to the oil surface. The hydrophilic polysaccharide possibly forms a thick secondary steric-stabilizing layer on the outside of protein-adsorbed emulsion droplets, providing the protein\u2013polysaccharide interaction is sufficiently attractive (Dickinson, 1994). Generally, strong electrostatic interaction between the oppositely charged adsorbed protein and added polysaccharide leads to the formation of multilayered interfacial membranes stabilizing emulsion droplets (Dickinson & James, 2000; Guzey et al., 2004; G\u00fczey & McClements, 2006a,b; Hong & McClements, 2007; Laplante, et al., 2005; Moreau et al., 2003; Mun et al., 2006).\n\nTo date, much research has been done on protein\u2013polysaccharide interactions in emulsion systems under different conditions of temperature, pH, ionic strength, concentration of protein and polysaccharide, pressure treatment, and so on. In most of these cases, the presence of polysaccharides creates flocculation by bridging interactions at lower concentrations, followed by emulsion stabilization at sufficiently high concentrations to enable the saturation of the protein-adsorbed emulsion droplets. However, in systems where polysaccharides do not interact with the proteins, depletion flocculation can be expected. Protein\u2013polysaccharide interactions are sensitive to details of protein structure as well as to the charge density on the biopolymers. For example, partial denaturation of globular proteins can result in increased complexation with hydrocolloids at the interface as compared to that of native proteins in an aqueous solution at the same pH and ionic strength (Dickinson, 2003). Generally, strong electrostatic interaction between the adsorbed protein and the added polysaccharide leads to the formation of a stabilizing layer. At sufficiently high hydrocolloid concentrations, emulsion stability is increased by immobilizing the emulsion droplets in a gelled protein\u2013polysaccharide network. However, the same polysaccharide can induce irreversible bridging flocculation of the protein-coated emulsion droplets if an insufficient amount of polysaccharides is present for surface coverage.\n\nTable 13.3 summarizes some of the recent investigations on potential milk\u2013protein polysaccharide interactions in the emulsion system (non-exhaustive).\n\nTable 13.3\n\nMilk Protein\u2013Polysaccharide Interactions in Emulsion Systems\n\nSL. NO. | Milk protein\u2013polysaccharide in emulsion systems | Oil phase | Conclusion | References  \n---|---|---|---|---  \n1. | Bovine serum albumin (BSA) + Carrageenan  \n(\u03b9-and \u03ba\\- forms)  \n| n-Tetradecane  \n(20, 40 vol%) | Strong bridging flocculation, BSA-\u03ba-interaction weaker than BSA-\u03b9-carrageenan at same pH & ionic strength | (Dickinson & Pawlowsky, 1997; )  \n2. | Caseins (\u03b1s1-, \u03b2-casein and caseinate) + High-methoxy Pectin | Sunflower oil  \n(11, 40 vol%) | Steric stabilization at low pH, depletion flocculation at neutral pH, salt-induced destabilization for \u03b1s1-stabilized emulsion only | (Dickinson et al., 1998)  \n3. | \u03b2-lactoglobulin + Low-methoxy pectin | Soybean oil (20 vol%) | Gelation at high-pressure treatment (400 MPa): | (Dickinson & James, 2000)  \n4. | WPI (85% \u03b2-lg) + Low-methoxy pectin | Soybean oil (20 wt%) | Covalent conjugation, better emulsion stability at pH 5.5 | (Neirynck et al., 2004)  \n5. | WPI + Dextran sulfate | Medium-chain triglyceride oil, Silicone oil, Orange oil &  \nn-Tetradecane (20 vol%) | Covalent conjugation, long-term emulsion stability at low concentration, steric stabilization | (Akhtar & Dickinson, 2003)  \n6. | Casein + Maltodextrin | Soybean oil (30 wt%) | Covalent conjugation, better emulsion stability | (Shepherd et al., 2000)  \n7. | Hydrolyzed WPI + Hydrocolloid (Xanthan gum, Guar gum, \u03ba-Carrageenan) | Corn oil  \n(4wt%) | Coalescence rate: guar gum > xanthan gum > \u03ba-carrageenan, depletion interaction induced coalescence at high-pressure treatment | (Ye et al., 2004; Ye & Singh, 2006)  \n8. | WPC + Hydrocolloids (Xanthan gum, Polypropylene glycol alginate (PGA), Carrageenan) | Soybean oil  \n(20 wt%) | Droplet aggregation by depletion mechanism (xanthan gum) on heating. Better creaming stability with PGA-WPC complexation | (Euston et al., 2002)  \n9. | WPI + Maltodextrin | Medium-chain triglyceride oil & orange oil  \n(20 vol%) | Covalent conjugation, no creaming over 40 days experimental period | (Akhtar & Dickinson, 2007)  \n10. | WPC + CMC | Corn oil  \n(10 vol%) | WPC-CMC complex formation inducing bridging flocculation at pH5, 0.3 M NaCl | (Damianou & Kiosseoglou, 2006)  \n11. | Sodium caseinate + Pectin (High-methoxyl: 59% DE, Low-methoxyl: 32% DE) | Corn oil  \n(10 wt%) | Bridging flocculation\/depletion interaction depending on pH and pectin type | (Surh et al., 2006)  \n12. | WPI + Chitosan | Canola oil  \n(10 vol%) | Electrostatic interaction and stable emulsion at pH > 5, depletion flocculation at pH < 5 | (Laplante et al., 2005)  \n13. | \u03b2-lactoglobulin + Carrageenan | Corn oil  \n(5 wt%) | Stable emulsion on thermal processing at ionic strength (< 500 mM NaCl, < 2 mM CaCl2) | (Gu et al., 2005a, 2005b)  \n14. | \u03b2-lactoglobulin + Dextran | Sunflower oil  \n(10 wt%) | Depletion flocculation but stability obtained by sucrose (> 20%) addition | (Blijdenstein et al., 2004)  \n15. | Sodium caseinate + \u03ba-Carrageenan | Soybean oil  \n(30 wt%) | Electrostatic repulsion; depletion flocculation at 55\u00b0 C, pH 7.0 | (Singh et al., 2003)  \n16. | Sodium caseinate + Apple pectin | Corn oil  \n(25 vol%) | Covalent conjugation, better emulsion stability than that of gum arabic and commercial emulsifiers | (Al-Hakkak & Kavale, 2002)  \n17. | Sodium caseinate + Xanthan gum | Soybean oil  \n(30 wt%) | Depletion flocculation, creaming stability obtained by flocculated droplet network | (Hemar et al., 2001a)  \n18. | Sodium caseinate + Portulaca oleracea gum | Medium-chain triglycerides (5wt%) | Caseinate-portulaca gum electrostatic complex providing better emulsion stability | (Garti et al., 1999)  \n19. | \u03b2-lactoglobulin + Pectin (59% DE) | Corn oil  \n(10 wt%) | Stable emulsion at pH (3\u20134), 0.1 M NaCl | (Guzey et al., 2004)  \n20. | \u03b2-lactoglobulin + Pectin (59% DE) + Chitosan | Corn oil  \n(2.5 wt%) | Stable tertiary emulsion at pH (3\u20135), 0.1 M NaCl | G\u00fczey & McClements, 2006a, 2006b)  \n21. | \u03b2-lactoglobulin + Sodium alginate | Hydrogenated palm oil  \n(5 wt%) | Electrostatic attraction at pH (3\u20135), bridging flocculation at pH (6\u20137), improved emulsion stability with sonication | (Pongsawatmanit et al., 2006)  \n22. | Lysozymes + Dextran | Corn oil  \n(25.0\u2013vol%) | Covalent conjugation and improved emulsion stability | (Nakamura et al., 1991; Scaman et al., 2006)  \n23. | Lysozymes + Galactomannan | Corn oil  \n(25.0-vol%) | Covalent conjugation and cationic electrostatic repulsion providing emulsion stability | (Nakamura et al., 1994; Scaman et al., 2006)  \n24. | \u03b2-lactoglobulin + Dextran | Sunflower oil  \n(20.0-wt%) | Covalent conjugation providing better emulsion stability | (Dunlap & C\u00f4t\u00e9, 2005)  \n25. | Lysozymes + Chitosan | Corn oil  \n(25.0-vol%) | Covalent conjugate: Emulsion stability\u2013High-molecular-type chitosan > Low molecular type | (Song et al., 2002)\n\nThe interaction of polysaccharides with proteins is not limited to electrostatic interactions. Apart from the use of transglutaminase as cross-linkers between proteins, covalent conjugates formed via Maillard reactions between milk proteins and polysaccharides have gained much interest due to their improved emulsification abilities compared with the biopolymer alone. These conjugates are stable over a wide range of temperature, pH, and ionic strength. The conjugates with a high molecular weight possess both the properties of a hydrophobic protein being adsorbed to the surface of the oil droplet and the properties of a hydrophilic polysaccharide being highly hydrated by the aqueous phase. Although these conjugates possess both a hydrophobic and hydrophilic group and are effective surface active polymers, the presence of excess unreacted hydrocolloid may lead to depletion effects (Syrbe et al., 1998). Consequently, interfacial layers made up of different structures, thicknesses, compositions, and charges require knowledge of the functionality of different protein\u2013polysaccharide combinations to meet the structural demands, environmental challenges, and stability of food emulsions.\n\n## Rheological properties and microstructures of protein\u2013polysaccharide systems\n\nThe rheological properties of a solution containing only protein are expected to be different from those of a pure polysaccharide solution. Polysaccharide molecules generally have a greater effect in causing a significant increase in solution viscosity than proteins. This is because polysaccharide molecules are usually much larger and more extended (\u223c5.0 \u00d7 105 to 2.0 \u00d7 106 Da) than globular proteins (\u223c1.0 \u00d7 104 to 1.0 \u00d7 105 Da). Hence, polysaccharide molecules generally occupy larger hydrodynamic volumes that give rise to higher solution viscosity. The above assumes that intermolecular interactions are absent or negligible (e.g., in dilute solution). When intermolecular interactions are present among neighboring polymer molecules (i.e., polysaccharide\u2013polysaccharide or protein\u2013protein), the rheological properties of many systems are expected to change significantly. The changes in rheological properties may arise as a result of an increase in the size of particles (e.g., protein\u2013polysaccharide complexes) or when depletion interactions occur in the mixed system or if one or more polymer species form continuous network structures. The overall effect results in the formation of different microstructures. Schematic illustrations of some possible microstructures formed from mixtures of protein and polysaccharides under some specific conditions (e.g., pH, ionic strength, heat treatment, etc.) are shown in Figures 13.3a and 13.3b.\n\nFigure 13.3a Schematic diagrams of some possible microstructures formed between noninteracting protein\u2013polysaccharide mixtures. Circle (\u2022) represents protein; coil structure represents polysaccharide molecules. (a) Flocculated protein network is formed, with polysaccharide filling the space in the network; (b) polysaccharide molecules overlap and form continuous 'network,' with protein filling the space; (c) particulate protein gel network formed, with polysaccharide filling the space; (d) polysaccharide gel network formed, with protein filling the space; (e) bi-continuous network formed from protein and polysaccharide; (f) polysaccharide gels dispersed among weakly flocculated protein network; and (g) protein gels dispersed among entangled polysaccharide molecules.\n\nFigure 13.3b Schematic diagrams of some possible microstructures formed between interacting protein\u2013 polysaccharide mixtures. Circle (\u2022) represents protein; coil structure represents polysaccharide molecules. (a) Protein\u2013polysaccharide complexes formed; (b) protein interacting with gelling polysaccharides helices; (c) polysaccharide interacting with protein particulate gel network; and (d) polysaccharide gel helices interacting with protein particulate gel network.\n\nTo characterize the physicochemical properties of protein\u2013polysaccharide systems, various rheological techniques have been employed. Generally, if the mixtures are liquid-like, viscosity measurements using rotational viscometers are commonly used to obtain steady-state viscosity curves, yield stress, and the like. Other simpler methods include the use of a kinematic viscometer (e.g., the Ubbelodhe capillary viscometer) to obtain a single point relative viscosity measurement. If the samples are viscoelastic (e.g., gels), rheometers are widely used to obtain rheological data (e.g., loss and storage moduli obtained within the linear viscoelastic region) by performing small deformation oscillatory measurements. The rheological data yield information on the viscosity and viscoelastic properties of the mixed systems. Knowledge of the rheological properties of mixed protein\u2013polysaccharide systems is essential to gain insights into the nature of the interactions and the resulting microstructure of the system. A fundamental understanding of the interactions at the molecular and colloidal levels will provide a strong foundation in exploiting the physical functionality of such complex systems in different applications (e.g., microencapsulation technology, imparting specific sensory characteristics, time\/temperature\/pH\/ionic control-release, emulsion stability, etc.).\n\nIn the following sections, we provide various examples of mixed systems involving different milk proteins and polysaccharides. An attempt was made to classify these mixed systems into two broad categories (i.e., interacting and noninteracting). Under each of these headings, they are further grouped according to whether the systems form or do not form gels (i.e., gelling or nongelling). The discussion focuses mainly on the techniques used and the rheological properties of the systems.\n\n### Noninteracting Protein\u2013Polysaccharide Mixtures\n\nNoninteracting protein\u2013polysaccharide mixtures existing as one phase are rare, but may take place when the two different molecular species have good chemical resemblance in terms of hydrophilicity and conformation (Tolstoguzov, 1991; ). Many polymer mixtures are thermodynamically incompatible, and segregative interactions often occur in the absence of electrostatic interaction or in the presence of electrostatic repulsion (Neiser et al., 1998). Protein\u2013polysaccharide mixtures that commonly exist as two separate phases are the result of either thermodynamic incompatibility or depletion phenomenon (Doublier et al., 2000).\n\n### Nongelling Phase-separated System\n\nThe following are examples of noninteracting protein and polysaccharide mixtures. Both proteins and polysaccharides were mixed under conditions where the mixtures did not form gels. The rheological properties of these systems are discussed in relation to their interactions and the microstructures formed.\n\n#### Casein Micelles and Galactomannans\n\nA noninteracting protein\u2013polysaccharide mixture wherein phase separation occurs has been reported in the case of a mixed system consisting of micellar casein (3%) and guar gum (0.2%) at pH 7 (Bourriot et al., 1999b). The rheological properties showed a significant change in the flow and viscoelastic properties compared to the individual biopolymer system. With the mixed system, an increase in the apparent viscosity was reported. Furthermore, the mechanical spectra (elastic modulus G', viscous modulus G\") of the frequency sweeps showed slightly higher values of the moduli, which were less frequency dependent. The results suggested the formation of a weak network structure within the system due to the flocculation of casein micelles as the polysaccharide molecules were excluded from the protein phase. The appearance of a slightly thixotropic behavior indicated that the network can be easily broken under shear because the network formed by the micellar casein was weakly flocculated and reversible, presumably attributable to the depletion\u2013flocculation mechanism. The study also showed that the lower the intrinsic viscosity of the polysaccharide, the higher the concentration of the polysaccharide required before phase separation occurred (Bourriot et al., 1999b). An increase in the concentration of polysaccharide resulted in stronger flocculation of the casein micelles as the volume occupied and the osmotic pressure from the surrounding polysaccharides increased.\n\nSimilar thixotropic behavior has been reported for a ternary solution consisting of micellar casein\/LBG\/sucrose (Schorsch et al., 1999). The results from the ternary solution showed that at pH \u223c6.8, casein micelles and LBG were thermodynamically incompatible, behaving as a water-in-water emulsion. The presence of sucrose even at high concentration (40%) did not significantly improve the compatibility of the biopolymers (Schorsch et al., 1999).\n\n#### Milk Proteins and Xanthan\n\nAnother study investigated the interaction between xanthan gum (0\u20131% w\/w, a polysaccharide known to have 'weak gel' properties) and different types of milk proteins 5% w\/w, sodium caseinate (Na-CN), skim milk powder (SMP), whey protein isolate (WPI), and milk protein concentrate (MPC)] in an aqueous solution at neutral pH ([Hemar et al., 2001b). Depending on the xanthan gum concentrations and the protein type, the microstructures of the mixtures were different. In the case of xanthan mixtures with either MPC or SMP, depletion flocculation of casein micelles took place. The size of the depleted protein aggregates decreased with increasing xanthan concentration (microstructures resembled a particulate network). In the case of xanthan solutions containing either Na-CN or ultracentrifuged WPI, no phase separation occurred within the timescale of the experiment. This was attributed to the larger size of casein micelles (average diameter \u223c0.2 \u03bcm) compared to the nanometer-size scale of WPI and Na-CN (0.05 \u03bcm) (Lucey et al., 2000). However, the rheological behavior of the mixtures was found to be very similar to the rheological behavior of xanthan. The differences in microstructures of the mixtures observed by the confocal laser scanning microscope (CLSM) were not detected by viscosity measurements probably because the weakly flocculated proteins were easily re-dispersed by the shearing action of the viscometer during measurement.\n\n### Gelling Phase-separated System\n\nIn a system where two biopolymer species (e.g., proteins and polysaccharides) do not interact, gelation of one or more of the components in a thermodynamically incompatible system will cause competition between phase separation and gelation (Neiser et al., 1998). Gelation basically means the formation of a three-dimensional aggregated network structure, which is generally induced by heating, cooling, acidification, enzymatic treatments, high-pressure processing, and so on. Generally, heating enhances hydrophobic and covalent interactions. In the case of whey protein, unfolded proteins interact to give rise to aggregates (Boye et al., 1997; Kinsella, 1984). In mixed systems, the microstructure will depend on the rates of phase separation and gel formation (Tavares et al., 2005). The gel may appear homogeneous at a macroscopic level but heterogeneous at the microscopic level. However, the rheological properties of such gels depend on the concentration and arrangement of each species in the different phases. If the gelling species is in the continuous phase, the gel strength is higher than one in the dispersed phase where the network is disrupted (Neiser et al., 1998).\n\n#### Whey Protein and Galactomannans\n\nOne such study was based on a mixture of LBG (a nongelling neutral polysaccharide) and whey protein at neutral pH and pH 5 (close to the pI of whey proteins) (Tavares & Lopes da Silva, 2003). At neutral pH, it is known that whey protein forms clear fine-stranded gels (protein aggregation is hindered by electrostatic repulsion), while at lower pH (e.g., pH 5), an opaque course particulate gel is formed (Aguilera, 1995; Langton & Hermansson, 1992). Rheological measurements showed that whey protein isolate gel (13% w\/w) had a stronger and more elastic character at pH 5 than at pH 7 because of the thick particulate network formed (Bertrand & Turgeon, 2007; Stading et al., 1993). For the protein gels at pH 7, increasing LBG concentration (>0.25%) decreased both the onset temperature for gelation and the gelation time. The presence of LBG was also found to increase gel rigidity. The authors attributed this effect to a decrease in macromolecular mobility within the network in the presence of LBG, due to segregative interactions and the 'local' concentration of each polymer species. The LBG molecules acted as fillers in the continuous protein network. At pH 5, the elastic character of the particulate gel network was shown to decrease in the presence of LBG, especially at low protein concentration (5%). It was suggested that LBG chains hampered protein\u2013protein interactions and were detrimental to the protein gel development. However, at a higher protein concentration (13%) where sufficient particulate gel network was formed, LBG acted as fillers within the network, improving the gel strength.\n\nIn a subsequent study carried out using whey protein isolate and guar gum at pH 7, an increase in protein gel strength was found with a decreasing degree of branching of the galactomannans (Tavares et al., 2005). Like LBG, the guar gum was dispersed as droplets among the whey protein network at low concentration (0.2%). However, at higher gum concentration (0.6%), the dispersed droplets joined to form a continuous polysaccharide-rich phase. Despite the different microstructures observed, the linear viscoelastic profiles were rather similar, indicating that viscoelasticity was fairly insensitive to microstructural changes of this nature.\n\n#### Whey Protein Isolate and Xanthan\n\nA very similar trend was observed for whey protein and xanthan mixtures after heat treatment (Bertrand & Turgeon, 2007). The microstructures and rheological properties of the gels were highly dependent on pH and salt concentration. At pH 6.5, the presence of xanthan improved the elastic modulus of the WPI gel. This was attributed to segregative phase separation, where xanthan was dispersed among the protein gel network. However, upon lowering the pH to 5.5 (close to the pI of WPI), the addition of xanthan decreased the elastic modulus of the gel. It was suggested that the possible formation of WPI\u2013xanthan complexes decreased protein\u2013protein interactions, producing a weaker gel network.\n\n#### \u03b2-Lactoglobulin and Pectin\n\nA different type of network was formed in mixtures of \u03b2-lactoglobulin (8% w\/w) and low methoxyl (LM) pectin (0.85% w\/w) after thermal treatment at pH 6.8. The storage modulus of the mixed gel system was significantly lower than the protein gel alone. Microstructure observed by CLSM revealed phase separation, with \u03b2-lactoglobulin appearing as spherical colloidal particles distributed in a continuous pectin network (Donato et al., 2005). A similar type of protein depletion-induced phase separation was reported for a mixed system containing aggregated whey protein and an exopolysaccharide (EPS) from lactic acid bacteria (Tuinier et al., 2000).\n\n#### \u03ba-Carrageenan and \u03b2-Lactoglobulin\n\nIf two gelling species are present in a binary system, the mixed gels may form interpenetrating, coupled, or phase-separated networks (Morris, 1986). Interpenetrating networks are the result of two independent continuous networks formed throughout the gel, and only topological interactions exist between the networks. Coupled networks (ordered into junction zones like those of a polysaccharide gel) are formed when favorable interactions between the two molecular species exist. However, such systems involving protein\u2013polysaccharide interactions are uncommon (Rao, 1999).\n\nPhase-separated networks are formed when one polymer species is incompatible with the other, forming phase-separated regions within the network gel (Piculell & Lindman, 1992; Turgeon & Beaulieu, 2001). An example of a phase-separated gel can be found with \u03ba-carrageenan and \u03b2-lactoglobulin (Capron et al., 1999). The mixed polymer formed a weaker gel than the carrageenan gel alone when the protein was in its native state. Upon heating the mixture to 90 \u00b0C, holding for 30 min and then cooling to 20 \u00b0C, the gel rheology indicated the melting of \u03ba-carrageenan and the gelation of \u03b2-lactoglobulin above 65 \u00b0C. There was no aggregation of \u03ba-carrageenan with \u03b2-lactoglobulin upon heating. The gelation time of \u03b2-lactoglobulin was reduced in the presence of \u03ba-carrageenan, which was attributed to micro phase separation, which caused an increase in local concentration of the \u03b2-lactoglobulin (Capron et al., 1999). Upon cooling, the mixed gel system formed a phase-separated bicontinuous network (Ould Eleya & Turgeon, 2000).\n\n### Interacting Protein\u2013Polysaccharide Mixtures\n\nAnother phase separation phenomenon is the associative phase separation where associative interactions are present. Associative interactions between protein and polysaccharide can occur as a result of electrostatic interactions, hydrogen bonding, hydrophobic interactions, or poor solvent conditions (Antonov et al., 1996b; de Kruif et al., 2004; Doublier et al., 2000; Gao & Dubin, 1999). In some cases, complexes known as coacervates are formed via electrostatic interactions. Coacervates of protein\u2013polysaccharide can occur when the pH of the mixture is lower than the pI of the protein. At this pH, protein possesses a net positive charge while the polysaccharides still possess a negative charge. The result of the complexation is the formation of a solvent-rich phase and a coacervate-rich phase (Doublier et al., 2000; Ould Eleya & Turgeon, 2000). The rheological properties of milk\u2013protein polysaccharide complexes are related to the interaction between the complexes and the water molecules, which forms soluble (or liquid coacervate phase) or insoluble (or precipitate) complexes. The solubility of complexes is based on the energetic difference between biopolymer\u2013biopolymer and biopolymer\u2013solvent interactions (Damodaran, 1997). The main parameters that affect the solubility of biopolymer complexes are charge density, pH, ionic strength, and protein:polysaccharide (PP:PS) ratio (Schmitt et al., 1998). It has been suggested that a complex involving a strong polyelectrolyte will form a precipitate rather than a liquid coacervate. A number of protein\u2013polysaccharide systems where complex coacervations occur have been reviewed by several authors (de Kruif et al., 2004; Schmitt et al., 1998; Turgeon et al., 2003). The following section presents some examples of interacting polymers in mixed systems and the effect on rheological properties.\n\n### Nongelling Phase-separated System\n\n#### \u03b2-Lactoglobulin and Chitosan\n\nIt has been reported that protein solubility increases below its pI when it complexes with an anionic polysaccharide (Tolstogusov, 1986; Tolstoguzov et al., 1985). A recent study of the \u03b2-lactoglobulin-chitosan complex has shown that depending on the pH, the complex is either soluble or insoluble (Guzey & McClements, 2006a, 2006b). The interaction of soluble chitosan (MW = 15,000 Da, DD = 85%, 0\u20130.1 wt%, 5mM phosphate buffer) with \u03b2-lactoglobulin (0.5 wt% \u03b2-lg, 5mM phosphate buffer) in aqueous solutions studied at pH 3\u20137 showed that at pH 3, 4, and 5, the majority of the \u03b2-lactoglobulin\u2013chitosan complex in the solutions was soluble, but at pH 6 and 7 a significant fraction of the two biopolymers was insoluble.\n\n#### Whey Proteins and Exopolysaccharides\n\n'Soluble complexes' formed via electrostatic interactions were reported for EPS B40 (an exopolysaccharide from Lactococcus lactis subsp cremoris NIZO B40) and whey proteins (PP:PS = 2:1) under specific pH and ionic conditions (with no macroscopic phase separation) (Weinbreck et al., 2003b). Decreasing the pH of the mixtures increased further aggregation of the complexes, which led to phase separation. In addition, increasing the ionic strength of the solution caused a shift to lower pH value for the onset of complexation. In this study, complexation in this system led to a decrease in solution viscosity as intramolecular repulsion of the EPS was reduced in the presence of whey proteins. The decrease in viscosity was attributed to a reduction of the quantity of dispersed phase, that is, water present within the complexes. Consequently, it was suggested that dilute solution viscosity measurement (which is related to the size of complexes) could be used to determine the optimum conditions for complexation (Weinbreck et al., 2003b). A potential benefit of this complexation is that it protects the protein from loss of solubility due to aggregation during thermal or high-pressure treatments (Galazka et al., 1997; Imeson, 1977).\n\n#### Whey Proteins and Gum Arabic\n\nViscosity curves were obtained to evaluate the 'strength' of electrostatic interactions of whey protein\/gum arabic coacervates (Weinbreck & Wientjes, 2004). This study showed that the stronger the interaction, the greater the shear-thinning behavior and the slower the reformation of the complexes after shearing. The highly viscous coacervates at pH 4 were considered to be due to electrostatic interactions. At pH above the pI (without electrostatic interactions), the mixtures appeared to be more elastic than viscous.\n\n#### Sodium Caseinate and Gum Arabic\n\nIn contrast to whey proteins, sodium caseinate and gum arabic mixtures show some peculiar behavior (Ye et al., 2006) as no coacervation is observed in these systems. Below a certain pH (pH 5.4), electrostatic interactions between sodium caseinate and gum arabic leads to the formation of stable composite particles in the size range 100\u2013200 nm. Over a pH range of 3.2\u20135.4, the particle complexes were consistent in size and remained stable and soluble. This pH range could shift depending on the ratio of sodium caseinate to gum arabic and ionic strength in the mixtures. The sodium caseinate\/gum arabic particles associated to form larger particles, which resulted in phase separation when the pH was lower than 3.0. A mechanism for the formation of these particles based around the self-aggregation of casein and the electrostatic interaction between the aggregated particles of casein and gum arabic molecules has been proposed. As the pH of the mixture decreases below pH 5.4, the caseinate molecules tend toward small-scale aggregation, prior to large-scale aggregation and precipitation at pH values closer to their pI (pH 4.6). In this case, in the early stages of aggregation the gum arabic molecules may attach to the outside of these small-scale aggregates through electrostatic interactions between negatively charged gum arabic and exposed positive patches on the surface of the caseinate aggregates. The presence of hydrophilic gum arabic molecules on the outside of the caseinate aggregate may be enough to sterically stabilize these nanoparticles and consequently prevent self-aggregation. As the charge on these particles is quite low (\u223c15 mV at pH 4.0), steric stabilization is probably important. In a recent study, the formation of sodium casein\u2013gum arabic complexes was reported to occur at temperatures above 60 \u00b0C at a certain mass ratio of protein to gum arabic (e.g., 1:5) and pH (maximum complexation at pH 6.5) (Ye et al., 2012). Interestingly, the complex formation is reversible when the temperature is decreased to below 60 \u00b0C (although not in the case of pH 5.0). The temperature-dependent complexation between sodium caseinate and gum arabic was attributed to hydrophobic interactions between the two polymer molecules. These unique complexes can potentially be used to form interfacial layers of emulsion droplets that can be altered by temperature.\n\n#### Casein Micelle and Pectin\n\nProtein\u2013polysaccharide interactions have been shown to be pH-dependent, as in the case of pectin and casein micelles (Ambjerg & J\u00f8rgensen, 1991; Maroziene & de Kruif, 2000). At pH 6.7, pectin did not adsorb onto casein micelles. With sufficient pectin present (0.1\u20130.2%), phase separation occurred due to depletion interactions of the casein micelles (\u223c0.1%). However, adsorption of pectin onto the casein micelles did occur at pH 5.3. Viscosity measurements were employed to study the changes that occurred at different polymer concentrations. At low pectin concentrations (\u223c0.1%) and at pH 5.3, a maximum viscosity was reached that was attributed to bridging flocculation. Bridging among the casein particles was interpreted as having a larger effective volume. As pectin concentration increased (>0.1%), the casein micelles became fully covered, and interactions between casein particles were reduced. This caused a decline in viscosity, but the solution viscosity remained higher than the pure milk samples (without added pectin). The amount of pectin required for full coverage of the casein micelles differed depending on the type of pectin in the sequence high methoxyl (HM) < low methoxyl amidated (LMA) < LM pectin. Adding pectin at levels beyond the concentration required for full coverage led to phase separation due to depletion interactions. A further increase in pectin reduced the thickness of the casein depleted layer, as the viscosity of the continuous phase became very high and gelled polymer networks were formed (Maroziene & de Kruif, 2000). When the pH of the mixture was increased from 5.3 back to 6.7, desorption of pectin from the casein occurred, but over a much longer timescale (\u223c10\u201315 min) than the adsorption process (Maroziene & de Kruif, 2000).\n\n### Gelling Phase-separated System\n\n#### Sodium Caseinate and Pectin\n\nThe dynamic rheological properties of glucono-\u03b4-lactone (GDL) acidified protein gels (2% w\/v Na-CN) were studied in the presence of LMA pectin (0.01\u20131%w\/v) at pH\u223c4 (Matia-Merino et al., 2004). The presence of pectin (0.01\u20130.05% w\/v) decreased the storage modulus and increased the gelation time as pectin adsorped onto the casein particles. At pectin concentrations >0.08% w\/v, acid-induced gelation appeared to be completely inhibited over \u223c9 hr at 25 \u00b0C.\n\n#### Casein Micelles and Iota-Carrageenan\n\nIn the case of casein\u2013carrageenan mixed systems, the attractive interactions involved the negatively charged sulfated groups of the polysaccharides and the positive 'patches' between residues 97 and 112 of \u03ba-caseins (Snoeren, 1975), despite a pH above the pI isoelectric point and an overall net negative charge of the casein micelles. The interaction between \u03b9-carrageenan (0.5%) and skim milk (based on 3.3% protein) mixtures was studied above and below carrageenan's coil-helix transition temperature (Langendorff et al., 1999). Above this temperature, carrageenan did not adsorb to casein micelles, resulting in depletion flocculation, while below this temperature, attractive interactions between carrageenan and casein micelles occurred. The higher charge density of the double-helix form as compared to the coil conformation of carrageenan probably explained the stronger attractive interaction between casein micelles and carrageenan. The presence of casein micelles increased the gel strength (indicated by higher G' and G\") and the gelation temperature (from 39 \u00b0C to 47 \u00b0C) when the mixtures were heated to 65 \u00b0C and cooled down to 25 \u00b0C. Depending on the concentration of carrageenan, different types of gel network were deduced from the frequency sweep. At low carrageenan concentrations (<0.2%), one type of network was formed on cooling. This was probably due to the bridging of casein micelles by the adsorbed carrageenan helical chains. The network was much more thermally stable than the pure carrageenan gels. At carrageenan concentrations above 0.2%, in addition to the formation of a network as described above, a second network was formed, similar to that of carrageenan gel in the absence of proteins. This was attributed to interactions between carrageenan chains (Langendorff et al., 1999). For the different types of carrageenans, the amount required for full coverage increased from \u03ba< \u03b9< \u03bb (Langendorff et al., 1997) as the charge density of the polymer determined the strength of adsorption (Maroziene & de Kruif, 2000; Pereyra et al., 1997).\n\nFrom the above examples, it is clear that mixed protein and polysaccharide systems can provide very different physicochemical properties. The properties of these systems are the results of cumulative effects from the molecular parameters of the macromolecules (e.g., size, conformation, charged density, concentration, polysaccharide:protein ratio), the conditions the mixed systems are subjected to (e.g., pH, temperature, ionic strength), and the resulting interactions among the macromolecules (e.g., type of interaction, strength of interactions, gels, or aggregates). Understanding the physicochemical properties of these systems may help in the development of novel food structures with unique sensory properties and functionalities, including microencapsulation and controlled-release applications.\n\n## Concluding remarks\n\nProteins and polysaccharides are the two main structural entities in foods, and a great deal of work has been published on their interactions over the last few decades. The following highlights some of the current and future key areas of research in protein\u2013polysaccharide interactions.\n\nProtein\u2013polysaccharide interactions using plant proteins (e.g., soy and pea) and egg proteins are gaining increasing interest among researchers. Proteins from different sources differ in their amino acid sequence, tertiary structures, molar mass, and distribution of reactive groups. All of these features can contribute to different functionalities in different environmental conditions when these proteins interact with polysaccharides (Andrade et al., 2010; Elmer et al., 2011; Mession et al., 2012; Miquelim et al., 2010; Souza et al., 2013; Tran & Rousseau, 2013; Yin et al., 2012).\n\nResearch on novel polysaccharides from new sources will continue, as the structural complexity of these materials continues to intrigue researchers. Their interactions with proteins depend largely on their chemical composition, chain length, chain conformation, and reactive groups present (Ettelaie et al., 2012; Yadav et al., 2010; Yin et al., 2012).\n\nProtein\u2013polysaccharide conjugates formed by Maillard-type reactions with different protein and polysaccharide combinations continue to be effective in improving emulsion stability. The area of research is envisaged to continue, with better understanding of the functionality of new protein and polysaccharide sources (Kasran et al., 2013; Markman & Livney, 2012; O'Regan & Mulvihill, 2010; Xu et al., 2012; Zhang et al., 2012).\n\nProtein\u2013polysaccharide interactions based on different ratios of proteins to polysaccharides over a range of pH, temperature, and ionic strengths have been shown to produce complexes that differ in functionality. Thus, further exploration of the impact of environmental conditions is likely. There is much room for continued research due to wide permutation of proteins and polysaccharides (Liu et al., 2012; Paraskevopoulou et al., 2013; Ru et al., 2012; Souza et al., 2013; van Gruijthuijsen et al., 2012).\n\nModifications of existing polysaccharides and proteins by chemical, enzymatic, and physical techniques have also drawn interest among scientists. Among these techniques, physical modification techniques such as ultra-high pressure, ultrasonication, electric-pulsed, and microwave irradiation appear to be favored because of their environmentally friendly nature (Bigikocin et al., 2011; Ma & Wang, 2013; Panteloglou et al., 2010; Singh et al., 2012; Wong et al., 2010).\n\nParticles of nanometric size formed by a combination of proteins and polysaccharides through various interactions such as hydrogen, electrostatic, hydrophobic, and covalent bonds will continue to offer opportunities to create new structures with desired functionalities. For instance, such particles are used to form and stabilize emulsions and dispersed particles. The physicochemical properties of emulsions stabilized by nanoparticles are substantially different from protein-stabilized emulsions. This provides opportunities for texture modification and controlled delivery of nutrients (Dickinson 2012; ).\n\nIn recent years, excellent progress has been made on understanding the key variables and interactions that control the physical stability, rheology, and microstructure of protein\u2013 polysaccharide mixtures. Although milk proteins, particularly whey proteins, have formed the basis of the most widely studied protein\u2013polysaccharide systems, most work has dealt with relatively simple binary combinations of one protein and one polysaccharide. More complex systems, including ternary mixtures, still remain to be investigated in detail. At a practical level, it seems possible to manipulate these interactions and produce different microstructures by controlling the internal (pH, ionic strength, biopolymer ratio, molecular weight, and charge of the biopolymer) and external (temperature, pressure, and shear rate) parameters. However, there is a considerable challenge in understanding how different microstructures relate to the sensory and nutritional properties of food products, such as mouthfeel and nutrient release.\n\nThe formation of complexes through hydrogen, electrostatic, hydrophobic, and covalent interactions has been the subject of intensive studies. This is mainly because of the potential for better functionality of the composites compared with the protein or polysaccharide alone, for example, in terms of rheology, gelation, and interfacial properties. Protein\u2013 polysaccharide complexes can serve as texturing agents, encapsulating agents, fat replacers, and stabilizers of emulsions and other dispersed systems. However, information is still lacking on the detailed molecular structures of protein\u2013polysaccharide complexes, and describing the experimental observations within the known theoretical frameworks remains a challenge.\n\nIt is now becoming apparent that the modification of food structure through modulation of macromolecular interactions can also be used to control the release of nutrients and bioactive components during digestion, and to target where and how such components are released. The basic science underpinning these functions is largely unknown. New knowledge in this area will enable development of composite food systems and ingredients that are superior in nutritional value and textural characteristics.\n\n# References\n\nAguilera JM .  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Pressure- and heat-induced gelation of mixed \u03b2-lactoglobulin\/xanthan solutions .  _Food Hydrocolloids_.  1996 ;10 (2) : 203 \u2013 211 .\n\nZhang JB ,  Wu NN ,  Yang XQ ,  He XT ,  Wang LJ .  Improvement of emulsifying properties of Maillard reaction products from beta-conglycinin and dextran using controlled enzymatic hydrolysis .  _Food Hydrocolloids_.  2012 ;28 (2) : 301 \u2013 312 . \nChapter 14\n\n# Interactions between Milk Proteins and Micronutrients\n\nTh\u00e9r\u00e8se Considine*\n\nJohn Flanagan**\n\nSimon M. Loveday***\n\n* Fonterra Research and Development Centre, Palmerston North, New Zealand  \n** Previously: Riddet Institute, Massey University, Palmerston North, New Zealand; Currently: Naturex S.A., Avignon, France  \n*** Riddet Institute, Massey University, Palmerston North, New Zealand\n\n## Abstract\n\nMilk proteins can interact with micronutrients through a variety of mechanisms, with hydrophobic interactions being of particular importance. This chapter focuses on the interactions of milk proteins with a range of micronutrients, including vitamins, fatty acids, sugars, and minerals. Milk proteins can potentially be used as micronutrient carriers in foods, thereby increasing the nutritional benefit of milk and milk-based products.\n\nIt is widely known that the processing of milk proteins via heat or high pressure can result in the modification of protein structure, resulting in altered interactions between proteins and micronutrients. Interestingly, the presence of some micronutrients can retard the denaturation of some milk proteins. The addition of specific micronutrients may therefore be used as a processing tool to prevent denaturation of milk proteins under physical conditions that normally result in denaturation.\n\n## Keywords\n\nMilk proteins\n\n\u03b2-lactoglobulin\n\n\u03b1-lactalbumin\n\ncasein\n\nmicronutrient\n\nvitamin\n\nmineral\n\nfatty acid\n\nsugar\n\nsurfactant\n\nfood processing\n\nmicrostructures\n\nbiopolymers\n\nphase separation\n\ndepletion flocculation\n\nthermodynamic incompatibility\n\nco-solubility\n\ncoacervates\n\nemulsion\n\nMaillard\n\ninterface\n\nOutline\n\nIntroduction 421\n\nInteractions between native milk proteins and micronutrients 422\n\nVitamin A 422\n\nVitamin C 424\n\nVitamin D 424\n\nOther Vitamins 426\n\nMinerals 426\n\nFatty Acids 427\n\nSurfactants 430\n\nSugars and Polyols 431\n\nFlavors 433\n\nOther Micronutrients 434\n\nInteractions between process-modified milk proteins and micronutrients 435\n\nProtein Denaturation by Thermal and Pressure Treatments and Effect of Micronutrients 436\n\nProcessing Treatments Involving Ligands 437\n\nProcessing Treatments Involving Sugars or Polyols 438\n\nConclusions 441\n\n## Introduction\n\nThe existence of a three-dimensional, folded protein structure is dependent on several forces. These include hydrogen bonding, hydrophobic interactions, van der Waals forces, and electrostatic interactions. Some amino acid residues exhibit a hydrophobic character, and electrostatic forces are based on interactions between charged residues. Thus, the conformation of a protein is dependent on the presence of particular amino acids and the variation of residues within the primary structure. Although proteins may be in the native state, interactions through hydrophobic, electrostatic, van der Waals, and other forces can occur at exposed regions on the protein's surface or in cavities and pockets. It is through these mechanisms that interactions between milk proteins and various micronutrients such as vitamins, fatty acids, minerals, and surfactants can occur.\n\nProtein structures can be readily destabilized from their native state by relatively minor changes in the environmental conditions. Variations in pH, temperature, and pressure, for example, can all induce structural transitions in proteins. In some cases, the objective of processing is to induce changes in protein structure, for example, the heating of whey proteins to form gels. In other cases, however, changes in environmental conditions can elicit changes in protein structure that result in undesirable functional properties (e.g., loss of solubility or biological activity).\n\nIn addition to pH-, temperature- and pressure-induced changes in protein structure, the presence of micronutrients can affect how the protein structure reacts to variations in pH, temperature or pressure. By interacting with specific sites within the protein's three-dimensional structure, micronutrients can render a protein more, or less, susceptible to denaturation.\n\n## Interactions Between native Milk Proteins and Micronutrients\n\nMicronutrients such as vitamins, sugars, fatty acids, and minerals may interact with milk proteins through a variety of mechanisms. The main mechanism is through hydrophobic interaction, and the majority of studies of interactions between milk proteins and micronutrients focus on globular whey proteins, which have hydrophobic cavities and extensive secondary and tertiary structures. By contrast, the interactions between caseins and micronutrients are based mostly on electrostatic driving forces. Interactions between milk proteins and micronutrients can have appreciable effects on the physical and chemical properties of proteins during processing, and these effects are discussed in the next section.\n\n### Vitamin A\n\nVitamin A refers to a group of molecules structurally related to retinol, which consists of a \u03b2-ionone ring and an isoprenoid 'tail' containing four conjugated double bonds and a terminal hydroxyl group. These 'retinoids' may have carboxylic acid, aldehyde, or esterified fatty acid terminal groups, but the all-trans conjugated double-bond structure is needed for biological activity (Weiser and Somorjai, 1992).\n\nCis-trans isomerization, which leads to loss of biological activity, is catalyzed by heat, transition metal ions, free radicals, and light, especially at wavelengths below 500 nm (Loveday and Singh, 2008). In foods, retinoids are often dissolved in the fat matrix, where they are protected from the oxidizing action of atmospheric oxygen by vitamin E and other antioxidants (Ball, 1988). They may also be dispersed in various colloidal carrier systems (Loveday and Singh, 2008).\n\nMost lipocalin molecules such as \u03b2-lactoglobulin (\u03b2-LG) have clear biological roles as ligand carriers. Papiz et al. (1986) observed that the structure of \u03b2-LG was remarkably similar to the structure of retinol-binding protein (RBP). Nonetheless, no definite biological function has been attributed to \u03b2-LG (Creamer et al., 2011). One high-affinity retinoid binding site has been identified on \u03b2-LG, located inside the calyx, and there is some evidence for additional low-affinity binding on the outside of the molecule (Qin et al., 1998; Wu et al., 1999). The structural changes induced in different environments have been determined by x-ray crystallography (Qin et al., 1998). Further information about \u03b2-LG structure and binding can be found in Chapter 7.\n\nDifferent researchers have used a variety of methods to determine binding constants for the \u03b2-LG retinoid complexation, thus making comparison between studies difficult. For example, Muresan et al. (2001) compared fluorimetry with equilibrium dialysis and found that fluorimetry yielded higher binding affinities than equilibrium analysis. The pH, the genetic variant, and the source of the protein all contribute to the discrepancies in the literature. The range of retinoid binding constants reported for \u03b2-LG and the factors affecting their measurement are discussed in detail by Kontopidis et al. (2004).\n\nFree retinol is a rather unstable compound, especially in an aqueous environment, but its stability is greatly improved when bound to an RBP (Futterman and Heller, 1972). However, little endogenous retinol is found bound to \u03b2-LG when it is first purified, and the ligand most closely associated with the protein is palmitate (P\u00e9rez et al., 1989).\n\nIn 1972, Futterman and Heller used fluorescence measurements to deduce that bovine \u03b2-LG, like RBP, forms water-soluble complexes with retinol. Complexation with \u03b2-LG can make vitamin A more resistant to heat and UV light (Loveday and Singh, 2008). The \u03b2-LG-retinoid complex is slightly more resistant to tryptic hydrolysis than uncomplexed \u03b2-LG (Shimoyamada et al., 1996).\n\nDufour et al. (1991) monitored the binding of retinol, retinyl acetate, retinoic acid, and \u03b2-carotene to native, esterified, or alkylated \u03b2-LG by the quenching of tryptophan fluorescence. The retinoids were bound to native or modified \u03b2-LG in a 1:1 molar ratio with apparent dissociation constants in the range of 10\u20138 M, whereas the molar ratio was 1:2 (ligand:protein) for \u03b2-carotene. Chemical modification of \u03b2-LG by methods such as methylation, ethylation (Dufour and Haertle, 1990), or alkylation (Dufour et al., 1991) has been shown to enhance the binding affinity for retinol by opening up a second binding site. It may therefore be assumed that the partial change of the \u03b2-LG secondary structure produced by these treatments does not destroy the structure of the retinol-binding pocket. Similarly, moderate amounts of surfactants do not appear to affect the retinol binding site of \u03b2-LG (Sahihi et al., 2012; Taheri-Kafrani et al., 2008).\n\nVery few studies have been carried out with ligand binding to \u03b1-lactalbumin (\u03b1-LA), in comparison with the vast range of studies with \u03b2-LG. However, there is the potential for ligands to bind to \u03b1-LA. Puyol et al. (1991) studied the binding of retinol and palmitic acid in a whey protein mixture. From this study, \u03b1-LA was shown to bind retinol more strongly than \u03b2-LG, but a much lower percentage of palmitic acid was bound to \u03b1-LA in comparison to \u03b2-LG.\n\nFutterman and Heller (1972) showed that, as with \u03b2-LG, bovine serum albumin (BSA) forms a strong fluorescent water-soluble complex with retinol. They also postulated that, although no detectable retinol is bound to BSA in vivo, the possibility exists that this protein could serve as an auxiliary carrier if excess free retinol were introduced into the circulation. BSA can inhibit photo-catalyzed oxidation of retinol, but not retinoic acid (Shimoyamada et al., 1996). BSA is sometimes used in cell culture to reduce nonspecific binding of retinoids to plastic surfaces, but binding of retinoids to BSA can reduce their bioavailability (Klaassen et al., 1999).\n\nRaica et al. (1959) reported that retinol can also be bound to casein, and Mohan et al. (2013) found that a significant proportion of the vitamin A in commercial skim milk was bound to unmodified casein micelles. The insolubilization of the casein micelles in the acid casein and rennet casein forms promotes nutritional activity in retinol, which is not observed for the milk proteins in the native state (Adrian et al., 1984). Reassembled casein micelles have been shown to afford thermal protection to \u03b2-carotene (S\u00e1iz-Abajo et al., 2013).\n\n### Vitamin C\n\nFew studies have explored the interactions between vitamin C and milk proteins. A single binding site on BSA for ascorbic acid was recorded by Tukamoto et al. (1974). Oelrichs et al. (1984) investigated the interactions between ascorbate and BSA. They suggested an intrinsic association constant of 2600 M\u22121 at 20 \u00b0C. Dai-Dong et al. (1990) observed an increased stability of ascorbic acid in the presence of \u03b2-LG compared with that in pure water, and vitamin C was more thermoresistant when heated in the presence of \u03b2-LG. In contrast to these studies, Puyol et al. (1994) reported the lack of interaction of ascorbic acid with \u03b2-LG or indeed any of the other whey proteins. Puyol et al. (1994) suggested that the discrepancy between their work and that of Dai-Dong et al. (1990) may have been related to the methods used. Monitoring the reducing ability of ascorbic acid may not provide sufficient allowance for the effects of ascorbate losses through autoxidation. Puyol et al. (1994) also suggested that the antioxidant effect of reductive thiols in \u03b2-LG and serum albumin may have a protective effect.\n\n### Vitamin D\n\nThe affinity of \u03b2-LG for vitamin D2 is about 10-fold greater than that for vitamin A and other retinoids (Cho et al., 1994; Dufour and Haertle, 1990; Wang et al., 1997b). For vitamin D2, 2 mol\/mol of protein is bound, but only 1 mol of the various retinoids binds to \u03b2-LG (Cho et al., 1994; Dufour and Haertle, 1990; Wang et al., 1997a). Ergosterol and vitamin D3 will bind to \u03b2-LG at 2 mol\/mol, and the binding affinity appears to be significantly greater than for vitamin D2. Wang et al. (1997b) showed that both vitamin D and cholesterol can bind to \u03b2-LG. However, they reported that only one molecule of vitamin D or cholesterol would bind in the calyx, and so they suggested that the other molecule could be bound to an external site, as postulated by Monaco et al. (1987) and later confirmed using x-ray crystallography (Yang et al., 2008). Figure 14.1 illustrates the sites at which vitamin D molecules will bind to \u03b2-LG, as identified crystallographically. The external site is located in a hydrophobic cleft between the \u03b2-I strand and an \u03b1-helix, where hydrophobic ligands are stabilized by interactions with hydrophobic residues near the C-terminus (residues 136\u2013149) (Yang et al., 2008).\n\nFigure 14.1 Structure of \u03b2-LG (gray) complexed with two molecules of vitamin D3 (white). Source: Drawn using PyMOL (The PyMOL Molecular Graphics System, Version 1.3, Schr\u00f6dinger, LLC) and PDB entry 2GJ5.\n\nThe binding of retinal, vitamin D2, and retinyl palmitate by \u03b2-LG was studied by Wang et al. (1999). Analysis of competitive binding experiments with palmitate indicated that retinal and palmitate did not compete for the same site; however, vitamin D2 appeared to displace palmitate at higher concentrations. Retinoids and vitamin D2 were bound more tightly than palmitate.\n\nFogolari et al. (2000) demonstrated the importance of pH when binding ligands to \u03b2-LG, in some cases due to the effect of pH on monomer\u2013dimer\u2013oligomer equilibria (Mercadante et al., 2012). Forrest et al. (2005) reported on the interactions of vitamin D3 with \u03b2-LG A under a range of environmental conditions (i.e., pH and ionic strength). These results showed that binding depended greatly on the solution conditions. For example, at low pH, 2.5 (I = 0.15 M), the EF loop (gate) is closed, and thus vitamin D3 was probably weakly bound in the external hydrophobic surface. Upon lowering the ionic strength to 0.08 M, binding increased. It has been suggested that lowering the salt concentration allowed more surface binding (Aymard et al., 1996). A dissociation constant of 0.02\u20130.29 \u03bcM was reported for \u03b2-LG A, with apparent mole ratios of vitamin D3 bound per mole of \u03b2-LG A ranging from 0.51 to 2.04 (Forrest et al., 2005).\n\nForrest et al. (2005) also studied the binding of vitamin D3 to \u03b2-casein and reported on interactions under a range of environmental conditions. The binding constants of vitamin D3 to \u03b2-casein were dependent on pH and ionic strength. In agreement with the study of Lietaer et al. (1991), an increase in binding as a function of ionic strength was apparent at pH 6.6. This was attributed to enhanced hydrophobic interactions, creating more surface area for binding (Lietaer et al., 1991). Increased binding was associated with a weaker affinity, compared with lower ionic strength where binding was stronger. Although stronger interactions at low ionic strength were attributed to fewer protein interactions, Forrest et al. (2005) could not identify a reason for decreased binding at pH 8. A dissociation constant of 0.06\u20130.26 \u03bcM was reported for \u03b2-casein, with apparent mole ratios of vitamin D3 bound per mole of \u03b2-casein ranging from 1.16 to 2.05. Forrest et al. (2005) suggested that the rheomorphic nature of \u03b2-casein allowed the hydrophobic area to bind strongly with vitamin D3, in the most thermodynamically stable conformation. The hydrophobic interactions were aligned with the perturbation of phenylalanine and the quenching of tryptophan, both of which are located in the hydrophobic core.\n\nHaham et al. (2012) recently reported that encapsulating vitamin D in reassembled casein micelles improves its stability during thermal treatment or prolonged storage, while maintaining bioavailability. The concept of delivering hydrophobic compounds in casein micelles is claimed in a patent from this group (Livney and Dalgleish, 2007). The potential of \u03b2-LG as a carrier for vitamin D3 in fortified foods has recently been explored (Diarrassouba et al., 2013).\n\n### Other Vitamins\n\nMilk also contains an array of vitamin-binding proteins, including vitamin-B12-binding protein, folate-binding protein (FBP), vitamin-D-binding protein, and riboflavin-binding protein. These proteins occur at low concentrations but may play a significant role in the uptake of vitamins from the diet (Salter and Mowlem, 1983). FBPs are specifically involved in the uptake of folate in the intestine. They also reduce the availability of folate to bacteria in the gut and hence may have antibacterial properties (Ford, 1974). Raw bovine milk contains a riboflavin-binding protein (Kanno et al., 1991), and riboflavin bound to this milk protein has similar antioxidant activities to riboflavin bound to egg white riboflavin-binding protein (Toyosaki and Mineshita, 1988). This area, including the binding of trace elements, has been reviewed in detail by Vegarud et al. (2000). Subsequent to that review, Nixon et al. (2004) investigated the source of the cooperativity between FBP and folate, and their results suggested stoichiometric interactions.\n\n### Minerals\n\nThe abilities of certain milk proteins, in particular the caseins, to bind calcium are well known. The extent of binding of the caseins is directly related to the number of phosphoserine residues and thus follows the order \u03b1S2\\- > \u03b1S1\\- > \u03b2\\- > \u03ba-casein (see Chapter 6). Increased binding of calcium to the caseins results in reduced negative charges on the casein molecule, producing diminished electrostatic repulsion and consequently inducing precipitation.\n\nCaseins with high numbers of phosphoserine residues, such as \u03b1S1-casein B, \u03b1S1-casein C and the \u03b1S2-caseins, are insoluble in Ca2+ concentrations above about 4 mM (Singh and Flanagan, 2005). However, \u03b2-casein is soluble at high concentrations of Ca2+ (0.4 M) at temperatures below 18 \u00b0C, but very insoluble above 18 \u00b0C, even in the presence of low concentrations of Ca2+ (4 mM). \u03ba-Casein, with only one phosphoserine, binds little calcium and remains soluble in Ca2+ at all concentrations. Although \u03ba-casein does not bind calcium to any great extent, its ability to stabilize \u03b1S1-, \u03b1S2-, and \u03b2-caseins against precipitation by Ca2+ is well known and plays a large part in the stabilization of the casein micelle. This is discussed in more detail in Chapter 6.\n\nSugiarto et al. (2009) tested whether sodium caseinate and\/or whey protein isolate (WPI) could bind and solubilize iron (ferrous sulfate) for food fortification. Caseinate had more binding sites than WPI, and Fe was bound more strongly to caseinate, but caseinate was increasingly precipitated at >4 mM Fe. Caseinate-iron complexes with 2 mM Fe remained soluble as the pH was decreased from 7 to 5.5, where the solubility of WPI-iron complexes decreased with decreasing pH. Chelation of iron with milk proteins mitigated iron-catalyzed oxidation in emulsions, although some contribution from antioxidant amino acid side chains was also postulated (Sugiarto et al., 2010). It was recently discovered that prior depletion of calcium from milk proteins dramatically improves Fe binding, allowing much higher levels of Fe to be stabilized for food fortification (Das et al., 2013).\n\nIntact casein will bind zinc and calcium, but tryptic hydrolysates of \u03b1S1-, \u03b1S2-, \u03b2-, and \u03ba-caseins also display mineral-binding properties. Termed caseinophosphopeptides (CPPs), these peptides can bind and solubilize high concentrations of calcium because of their highly polar acidic domain. Calcium-binding CPPs can have an anticariogenic effect in that they inhibit caries lesions through recalcification of the dental enamel (FitzGerald, 1998). This effect has been exploited in CPP-fortified chewing gums (Recaldent and Trident brands) using ingredients developed by CSIRO Australia. CPPs have also been reported to improve the intestinal absorption of zinc, as studied using an isolated perfused rat intestinal loop system (Peres et al., 1998). The amount of iron bound to CPPs produced with the enzyme alcalase depends on the degree of hydrolysis as well as the temperature and pH during the binding reaction (Wang et al., 2011). Binding of iron to CPPs reduces iron-induced peroxidation in Caco-2 cells, suggesting that CPPs could help mitigate against the unintended side effects of iron fortification (Kibangou et al., 2008). Enzymatic hydrolysis of \u03b2-LG dramatically increases its iron-binding capacity, which may be due to improved contact between iron and aromatic amino acids (Zhou et al., 2012).\n\nLactoferrin has the ability to bind iron very strongly. In vivo, the ferric III form of iron is bound to lactoferrin (Anderson et al., 1989). Considerable interest has been expressed in supplementing bovine-milk-based infant formulas with lactoferrin, as bovine milk contains much lower levels of lactoferrin than human milk and lactoferrin, isolated from human milk, can bind two moles of iron per mole of protein (Bezwoda and Mansoor, 1986). Nagasko et al. (1993) reported that lactoferrin can bind iron at sites other than its chelate-binding sites, probably on the surface of the molecule. The thermal stability of lactoferrin-iron complexes is enhanced by soluble soybean polysaccharide, which was apparently due to enhanced electrostatic repulsion (Ueno et al., 2012). Other studies involving the interactions of minerals\/ions and milk proteins are listed in Table 14.1.\n\nTable 14.1\n\nInteraction of Milk Proteins and Minerals\n\nMilk protein | Mineral | Reference  \n---|---|---  \n\u03b2-LG A  \n\u03b2-LG A and B  \nCaseins and \u03b2-LG  \n\u03b1-LA | Chromium  \nLead  \nMercury  \nCopper | Divsalar et al. (2006)  \nDivsalar et al. (2005)  \nMata et al. (1997)  \nPermyakov et al. (1988)\n\n### Fatty Acids\n\nMost of the fatty acids present in milk are found as triglycerides, which form the fat globule. P\u00e9rez et al. (1992) proposed that ruminant \u03b2-LG, because of its activity to bind fatty acids, might play a role in the activity of pregastric lipases. P\u00e9rez et al. (1989) demonstrated that two types of lipids, namely, free fatty acids and triglycerides, bound to \u03b2-LG. The total amount of fatty acids extracted from \u03b2-LG was 0.71 mol per mol of monomer protein. The predominant saturated fatty acids found were palmitic (31\u201335%) and myristic (14\u201317%) acids, which when combined account for 66\u201375% of the total fatty acids bound to \u03b2-LG. The remaining fatty acids extracted from \u03b2-LG were unsaturated (<31% of the total fatty acids) and were mainly oleic (22\u201323%) and palmitoleic (4\u20135%) acids. Although \u03b2-LG is often associated with fatty acids in milk at physiological pH, Frapin et al. (1993) showed that \u03b2-LG isolated at acid pH contained minimal bound fatty acids, and delipidating had almost no effect on fatty acid binding affinities.\n\nAs with retinol, there also seems to be controversy regarding the binding location of fatty acids. Narayan and Berliner (1998) suggested that fatty acids bind at the 'external site' of \u03b2-LG. However, this conflicts with earlier studies by Puyol et al. (1991), which suggested competitive binding, and by Creamer (1995), which suggested an internal location as the primary binding site for fatty acids. Since then, several studies have shown ligands to bind in the internal cavity. Qin et al. (1998), using x-ray crystallography, showed 12-bromododecanoic acid binding inside the calyx, and Wu et al. (1999) revealed that palmitate binds in the central cavity (Fig. 14.2) in a manner similar to the binding of retinol to the related lipocalin, serum RBP. Ragona et al. (2000) provided further evidence for cavity binding of \u03b2-LG and palmitic acid, as did Zsila et al. (2002) using circular dichroism (CD) spectroscopy, electronic absorption spectroscopy, and electrospray ionization mass spectrometry (ESI-MS) with cis-parinaric acid.\n\nFigure 14.2 Diagram of the three-dimensional structure of \u03b2-LG that shows the relative positions of the five Cys residues, Lys60, Lys69, and the bound palmitate (Wu et al., 1999). The helix and the strands that constitute Sheets 1 and 2 are also labeled. Source: The diagram was drawn from the PDB file 1GXA using RASMOL Ver 2.6. Reproduced with the permission of Considine et al. (2005). Copyright 2005 Journal of Agricultural and Food Chemistry, American Chemical Society.\n\nKonuma et al. (2007) examined palmitic acid binding to a dimeric \u03b2-LG mutant A34C using heteronuclear nuclear magnetic resonance (NMR) spectroscopy. Their results suggested a 1:1 binding stoichiometry. They indicated that the protein conformation should be complementary, at least in part, to the ligand's structure if tight binding (dissociation constant of <10\u20137 M) is to occur. They further highlighted the role of the flexible loops above the barrel in ligand binding. Konuma et al. (2007) hypothesized that the barrel's entrance accommodates a variety of ligands, because of its plasticity, whereas the bottom of the cavity shows rigid and somewhat selective binding.\n\nThus, it has been established that \u03b2-LG strongly binds one mole of long-chain fatty acids (myristic, palmitic, stearic acid, etc.) per mole of monomeric protein (Dufour et al., 1994; Frapin et al., 1993). The relative binding strength for saturated fatty acids from C8 to C20 has been measured with a variety of methods, including isothermal titration calorimetry, fluorescence spectroscopy, mass spectrometry, and equilibrium partition analysis (Loch et al., 2012). There is remarkable agreement on both the order and the relative magnitude of association constants, and data clearly show increasing affinity with increasing chain length: C8 \u2248 C10 < C12 < C14 < C16. Binding with arachidonic acid (C20) is weaker than with palmitic acid (C16), and there is disagreement for stearic acid (C18), which is thought to bind with a similar strength to palmitic acid or more strongly (Loch et al., 2012). Frapin et al. (1993) reported binding constants for several monounsaturated fatty acids, and Bello et al. (2011) measured the binding of lauric acid (C10) and SDS to \u03b2-LG genetic variants A and B.\n\nFatty acid binding to \u03b2-LG is sensitive to changes in pH. Changes in binding constants are observed over the pH range 5.5\u20138.5 (P\u00e9rez and Calvo, 1995). This may be due to the electrostatic interactions; for example, as the pH increases, \u03b2-LG becomes negatively charged, thus making it less electrostatically inviting for a negatively charged fatty acid. The two lysine residues at the opening of \u03b2-LG's ligand-binding cavity, Lys60 and Lys69, are likely to play a significant role in ligand affinity. The inability of porcine \u03b2-LG to bind fatty acids may be due to the substitution of Lys69 by glutamate, as suggested by Frapin et al. (1993) and P\u00e9rez et al. (1993). Creamer (1995) also hypothesized that lysine was involved in the binding process, whereby at neutral pH the carboxylate group of the fatty acid salt bridged to the positively charged \u025b-amino group. Loch et al. (2012) reported x-ray crystallography data suggesting that fatty acid head groups could form hydrogen bonds with Glu62 and Lys69.\n\nPuyol et al. (1991) studied the competition between the binding of retinol and free fatty acids to \u03b2-LG. They observed that, when the ratio between the concentrations of the total fatty acids (as palmitic acid) and retinol is similar to that found in milk, the fatty acids compete with retinol for binding to \u03b2-LG. Using intrinsic fluorescence studies, Frapin et al. (1993) and Dufour et al. (1994) suggested that an external, independent fatty-acid-binding site on the \u03b2-LG\u2013retinol complex was in the groove between the \u03b1-helix and the \u03b2-sheets of the protein. Narayan and Berliner (1997) supported simultaneous binding of retinoids and fatty acids to \u03b2-LG. However, binding is more difficult to determine when several ligands are present.\n\nThe organic-anion-binding sites of BSA are composed of two parts: a pocket lined with nonpolar amino acid chains and a cationic group located at or near the surface of the pocket (Swaney and Klotz, 1970). Most of the information available on the mechanism of BSA binding has been obtained using organic dyes, anionic detergents, and fluorinated or spin-labeled derivatives. Free fatty acid binding involves hydrophobic interactions with the hydrocarbon chain and electrostatic interactions with the carboxylate anion of BSA (Spector et al., 1969). Andersson et al. (1971) suggested that the fatty-acid-binding sites are located in clefts between the globular regions of the albumin polypeptide. One tryptophan is located deep inside the globular structure, whereas the other is superficially located where it is fairly accessible to solvent (Futterman and Heller, 1972). Several of the strong fatty-acid-binding sites are located within 10 \u00c5 of the buried tryptophan residue (Spector, 1975). Spector et al. (1969) reported that palmitate and palmitoleate were bound more tightly than oleate, linoleate, stearate, or myristate and much more tightly than laurate. When a long-chain hydrocarbon did not contain a free carboxyl group (methyl palmitate, cetyl alcohol, and hexane), they were bound to a limited extent.\n\nAccording to P\u00e9rez et al. (1989), the amount of fatty acids found bound to BSA in milk was 4.8 mol per mol, and the predominant acids were oleic, palmitic, and stearic acids. Although the number of high-affinity binding sites and the values of apparent association constants for fatty acids to \u03b2-LG are lower than those for BSA (Anel et al., 1989), the molar concentration of \u03b2-LG in milk is much higher than that of BSA (about 30 times higher). Therefore, \u03b2-LG is considered to be the main fatty-acid-binding protein in ruminant whey (P\u00e9rez et al., 1989; ), whereas \u03b1-LA binds little or no palmitic acid (Barbana et al., 2006).\n\nBarbana et al. (2006) reported that bovine holo-\u03b1-LA neither contains bound fatty acids in vivo nor has the ability to bind them in vitro. Cawthern et al. (1997) observed the lack of binding of stearic acid with bovine holo-\u03b1-LA, using the fluorescent indicator acrylodated intestinal fatty-acid-binding (ADIFAB) protein. However, these results were in contrast to their spin resonance and intrinsic protein fluorescence results (Cawthern et al., 1997), which showed that stearic acid was bound to holo-\u03b1-LA with a dissociation constant of 10\u2013100 \u00d7 10\u22126 M. On the other hand, interactions of apo-\u03b1-LA with fatty acids have been reported by Barbana et al. (2006). Bovine apo-\u03b1-LA displayed apparent affinity binding constants of 4.6 \u00d7 106 and 5.4 \u00d7 105 M\u22121 for oleic acid and palmitic acid, respectively, using partition equilibrium, and fluorescence spectroscopy showed a binding constant of 3.3 \u00d7 106 M\u22121 for oleic acid. The small fluorescence changes observed for palmitic acid made it difficult to obtain a binding constant.\n\nComplexes of \u03b1-LA with oleic acid have strong cytotoxic effects on cancer cells, and the complex has been named HAMLET\/BAMLET (Human\/Bovine Alpha-lactalbumin Made LEthal to Tumor cells) (Brinkmann et al., 2011). Similar cytotoxic activity has recently been demonstrated with \u03b2-LG-oleate complexes (Li\u0161kov\u00e1 et al., 2011).\n\n### Surfactants\n\nSome interesting amphiphilic surfactants have been widely used to study ligand binding to \u03b2-LG, including cationic, anionic, and non-ionic surfactants. These ligands provide useful information regarding the structure of the molecule and the way in which it unfolds in response to chemical and physical stimuli. At low concentration, ionic surfactants are thought to bind in the hydrophobic calyx of \u03b2-LG, whereas they unfold the protein at higher concentration (Hansted et al., 2011).\n\nSodium dodecyl sulfate (SDS) is an anionic surfactant, and it binds strongly to a small number of sites on \u03b2-LG at low SDS concentration (Jones and Wilkinson, 1976; Lamiot et al., 1994). Creamer (1995) demonstrated that SDS had a profound effect on the equilibrium unfolding of bovine \u03b2-LG by maintaining \u03b2-LG in the native confirmation, despite high concentrations of urea. Busti et al. (2005) examined the interaction of alkyl sulfonates (AS) with \u03b2-LG and demonstrated one binding site per molecule. The efficiency of AS stabilizing native \u03b2-LG was related to the length of the hydrocarbon tail: Longer AS generally stabilized \u03b2-LG better against urea denaturation (Busti et al., 2005). AS also retarded thermal denaturation of \u03b2-LG to an extent that depended on the length of the alkyl group\u2014longer tails were again more effective (Busti et al., 2006). This mirrors the link between alkyl group length and binding affinity that occurs with fatty acids (see above).\n\nWaninge et al. (1998) showed a substantial increase in unfolding temperature of \u03b2-LG\u2013SDS complex at a molar ratio of 1:1. In contrast, a decrease in the unfolding temperature was observed with the addition of a cationic surfactant dodecyl trimethyl ammonium chloride (DTAC) at a similar ratio. Whereas DTAC was easily removed by dialysis, it was impossible to remove SDS by this method. Viseu et al. (2007) showed that DTAC disrupted the tertiary structure of \u03b2-LG, but also drove an increase in the amount of \u03b1-helical secondary structure, in contrast to total disruption of structure caused by guanidine-HCl.\n\nLu et al. (2006) showed that the anionic surfactant sodium perfluorooctanoate was a strong denaturant of \u03b2-LG. However, the denaturing ability of sodium perfluorooctanoate could be tempered with cationic surfactants, such as alkyl trimethyl ammonium bromide. Maulik et al. (1996) observed the binding of cetyl trimethyl ammonium bromide with \u03b2-LG and reported a two-stage interaction by first-order kinetics. Tetradecyl trimethyl ammonium bromide (TTAB) was also shown to interact with \u03b1-LA and cause protein unfolding below TTAB concentrations of 2 mM (Housaindokht et al., 2001).\n\nThe interactions of non-ionic sucrose esters with the casein micelle (Fontecha and Swaisgood, 1995) and \u03b2-casein (Clark et al., 1992) have also been studied. Creamer (1980) examined the effect of SDS on the \u03b2-casein self-association. The results indicated that SDS binds on an external site of \u03b2-casein, such that the hydrophobic tail of SDS becomes involved in the casein self-association. This is supported by the lack of displacement of 8-anilino-1-naphthalene sulfonate (ANS) by SDS. It was also postulated that SDS binds to sites in or on the protein such that the amino acid residues involved in the self-association reaction can interact more favorably with one another. At low concentrations, SDS is thought to bind to a limited number of sites. Despite the increase in the negative charge of the protein when low concentrations of SDS are added, the normal monomer\u2013polymer equilibrium moves predominantly to the polymer in solution, whereas at high concentrations of SDS, only protein monomers are present.\n\n### Sugars and Polyols\n\nSugars and polyols belong to a family of small hydrophilic molecules that stabilize proteins, and can be referred to as osmolytes, cosolvents, compatible solutes, or cosolutes (the last named will be used here). The effect of sugars and polyols on the unfolding and denaturation of proteins is generally attributed to preferential exclusion of these solutes from the protein surface, or, equivalently, 'preferential hydration' of the protein. Two main elements contribute to this effect (McClements, 2002; Timasheff, 1998), as illustrated in Figure 14.3. First, nonspecific steric exclusion arises from the fact that sugars and polyols are larger than water molecules. This effect is common across a wide range of cosolutes and depends more on the size of cosolute molecules than on their chemical nature (Ebel et al., 2000; R\u00f6sgen, 2007). Second, hydroxyl-rich polyols and sugars have a strong affinity for hydrogen-bonded water molecule networks and a strong phobia of protein nonpolar groups, so they are drawn to the aqueous environment in preference to the protein surface. This leads to the strengthening of hydrophobic interactions in cosolute solutions (Kamiyama et al., 1999; Timasheff, 1998).\n\nFigure 14.3 Protein\u2013cosolvent\u2013solvent interactions as a result of (a) steric interaction or (b) differential interaction effects. In (b), the exclusion of the cosolvent from the region surrounding the protein is clearly shown. Source: Taken from McClements (2002).\n\nA recent study comparing the effects of trehalose, maltose, and sucrose on the structure of water found that trehalose binds to a larger number of water molecules than do maltose or sucrose, thus affecting the structure of water to a greater extent (Lerbret et al., 2005). The effect of cosolutes on the surface tension at the protein\u2013solvent interface may also contribute to preferential exclusion, but these effects are very complex and somewhat controversial (Chanasattru et al., 2007c; Kaushik and Bhat, 1998; Lin and Timasheff, 1996).\n\nWater is able to get into the layer immediately adjacent to the protein, but cosolutes are sterically excluded; thus, a concentration gradient of the cosolvent molecules between the inner layer and the outer solution arises. This is a thermodynamically unfavorable situation because of the free energy that is required to maintain this concentration gradient. Subsequent movement of water molecules from the area surrounding the protein to outer parts leads to a dehydration of the protein molecule. This dehydration can result in tighter folding of the protein molecules.\n\nUnfolding of a compact protein structure increases the area of contact between the protein and the solvent, and reveals nonpolar groups that were buried in the interior in the native protein. For these reasons, preferential exclusion is greater in the denatured state than in the native state, and it follows that denaturation is accompanied by a net increase in the degree of preferential exclusion (i.e., a net increase in cosolute activity in the aqueous phase). This adds to the energetic cost of unfolding, so denaturation is inhibited (Timasheff, 1998).\n\nAn alternative explanation of the effect of sugars on the unfolding and denaturation of proteins has been put forward by Semenova et al. (2002), who proposed a direct hydrogen bonding between sugars and proteins, which results in additional hydration. However, this hypothesis does not fully explain the exclusion of sugars from the protein domain (Hammou et al., 1998; Timasheff, 2002).\n\nHigher concentrations of polyols and sugars increase viscosity substantially, possibly inhibiting diffusion-limited reactions such as protein aggregation (Kulmyrzaev et al., 2000). However, the more compact protein conformation induced by preferential exclusion may diffuse faster, overcoming viscosity effects (Rodr\u00edguez Ni\u00f1o and Rodr\u00edguez Patino, 2002). The stabilizing effect of polyols increases with more hydroxyl groups on the polyols (Politi and Harries, 2010; Romero et al., 2007; Tiwari and Bhat, 2006), and with lower pH (Singh et al., 2011) and lower temperature (Xie and Timasheff, 1997b).\n\nThe Timasheff research group has been dominant in research into the interactions of proteins and sugars, or cosolvents as they describe them (Timasheff, 1993). Xie and Timasheff (1997a) reported the exclusion of trehalose from the domain of ribonuclease A at low temperatures. However, at 52 \u00b0C there was preferential binding of trehalose in both the native and unfolded state (pH 5.5 and 2.8, respectively). Binding was stronger in the native state than the unfolded state, meaning that denaturation in the presence of trehalose was still accompanied by a net increase in preferential exclusion (or in other words a net decrease in binding), and therefore denaturation was inhibited. The same group conducted a lot of earlier research showing the exclusion of water from the domains of a range of globular proteins, in the presence of sucrose (Lee and Timasheff, 1981), lactose and glucose (Arakawa and Timasheff, 1982). In all cases, they argued that the exclusion of sugars from the protein domain made unfolding of the protein less thermodynamically favorable.\n\nThe Timasheff group has measured the preferential exclusion of various cosolutes from BSA and \u03b2-LG in a number of studies (Timasheff, 1998). In a rare study involving sugars and casein, Mora-Gutierrez and Farrell (2000) also proposed preferential exclusion of sugar molecules from the casein domain, resulting in preferential hydration of the caseins. The ability of sugars to alter the heat- and pressure-induced denaturation of milk proteins is discussed later in this chapter.\n\n### Flavors\n\nThe interaction of milk proteins and volatile flavor has been reviewed in detail by K\u00fchn et al. (2006), and the reader should refer to this review for more in-depth discussion of protein\u2013flavor interactions. However, this section covers the area briefly. A number of flavor compounds are known to bind to milk proteins. Despite this knowledge, there are large discrepancies in the binding data because of the use of different methodologies, which appears to be a common feature of determining binding constants.\n\n\u03b2-LG is known to interact with a variety of flavor compounds, including ionones (Jouenne and Crouzet, 2000; Jung and Ebeler, 2003), lactones (Guth and Fritzler, 2004; Sostmann and Guichard, 1998), alkanes (Mohammadzadeh et al., 1969), aldehydes (van Ruth et al., 2002), esters, and ketones (Guichard and Langourieux, 2000; Jouenne and Crouzet, 2000). In contrast, very few studies have explored flavor binding to \u03b1-LA, with an exception being the binding of aldehydes and methyl ketones (Franzen and Kinsella, 1974), and 2-nonanone and 2-nonanal (Jasinski and Kilara, 1985). BSA has been shown to bind alkanes (Mohammadzadeh et al., 1969), and studied interactions of 2-nonanone and BSA. Jasinski and Kilara (1985) compared the binding of 2-nonanone and nonanal to BSA.\n\nThe binding of flavors to caseins or sodium caseinate has also received some attention, including the binding of diacetyl (Reineccius and Coulter, 1969), vanillin (McNeill and Schmidt, 1993), \u03b2-ionone, n-hexanol, ethylhexanoate, and isomyl acetate (Voilley et al., 1991) to sodium caseinate.\n\nThe most extensively studied flavor compound is 2-nonanone. Thus the binding strengths of this flavor to the whey proteins can be compared. Although different authors have reported different affinity constants, the binding strength consistently follows the trend BSA >\u03b2-LG >\u03b1-LA. Table 14.2 illustrates the interactions between 2-nonanone and various milk proteins (K\u00fchn et al., 2006).\n\nTable 14.2\n\nBinding Data for the Interactions between 2-Nonanone and Milk Proteins (25 \u00b0C): n, Number of Binding Sites per Monomer; K, Intrinsic Binding Constant\n\nProtein | n | K (M\u20131) | Method | Reference  \n---|---|---|---|---  \nWPCa | 61 | 1,920,000 | Equilibrium dialysis | Jasinski & Kilara (1985)  \n| 0.2 | 53,000,000 | Fluorescence spectroscopy | Liu et al. (2005)  \nWPIb | 1 | 2059 | Headspace SPMEc | Zhu (2003)  \nsodium caseinate | 0.3 | 1858 | Headspace SPME | Zhu (2003)  \n\u03b2-LG | 1 | 2439 | Equilibrium dialysis | O'Neill & Kinsella (1987)  \n| 0.2  \n0.5 | 6250 (\u226440 ppm)  \n1667 (\u226545 ppm) | Static headspace analysis | Charles et al. (1996)  \n| 14 | 122 | Equilibrium dialysis | Jasinski & Kilara (1985)  \n\u03b1-LA | 33 | 11 | Equilibrium dialysis | Jasinski & Kilara (1985)  \nBSA | 5\u20136 | 1800 | Liquid\u2013liquid partitioning | Damodaran & Kinsella (1980)  \n| 15 | 14,100 | Equilibrium dialysis | Jasinski & Kilara (1985)  \n| 7 | 833 | PFG-NMRd spectroscopy | Jung et al. (2002)\n\na WPC: whey protein concentrate\n\nb WPI: whey protein isolate\n\nc SPME: solid phase microextraction\n\nd PFG-NMR: pulsed-field gradient NMR\n\nSource: Reproduced with the permission of K\u00fchn et al. (2006); copyright 2006 Journal of Food Science, Institute of Food Technologists.\n\n### Other Micronutrients\n\nSome of the milk proteins, most particularly the whey proteins, have been used as model proteins in studies involving a range of other micronutrients. The interaction of small heat-shock proteins, such as alpha-crystallin, prevents the precipitation of \u03b1-LA when in the molten globule state (Lindner et al., 1997). This finding was confirmed by Sreelakshmi and Sharma (2001), who found that the active site of alpha-cystallin by itself can maintain a significantly denatured and unfolded protein in soluble form. Zhang et al. (2005) reported on the chaperone-like activity of \u03b2\\- and \u03b1-caseins. \u03b2-Casein was able to suppress the thermal and chemical aggregation of insulin, lysozyme, and catalase. A similar chaperone-like effect is seen with \u03b2-LG, \u03b1-LA, and BSA (Kehoe and Foegeding, 2011).\n\nThe use of milk proteins as chaperones for drugs has also been studied. The interaction of chlorpromazine with \u03b2-LG and \u03b1s-casein affected the proteins in different ways. Far UV CD studies revealed that chlorpromazine increased the secondary structure of \u03b2-LG, whereas the structure of casein became further disordered (Bhattacharyya and Das, 2001). Divsalar et al. (2006) also reported on the interaction between genetic variants of \u03b2-LG and an anticancer component.\n\nA number of recent studies that examined the interactions between milk proteins and various bioactive compounds are listed in Table 14.3. Most studies were carried out with a view to using milk proteins as carrier molecules or particles for protecting and\/or delivering bioactives. This potential application was examined in several recent reviews (Abd El-Salam and El-Shibiny, 2012; Elzoghby et al., 2011; Livney, 2010). The number of studies on the binding of polyphenols by milk proteins is notable.\n\nTable 14.3\n\nRecent Studies of Milk Protein Interactions with Miscellaneous Biologically Active Compounds\n\nProtein | Active agent | Notes | Reference  \n---|---|---|---  \nPolyphenolic compounds  \n\u03b2-LG | Epigallocatechin | Binding study | Wu et al. (2011)  \n\u03b2-LG | Epigallocatechin-3-gallate | Binding study | Wu et al. (2013)  \n\u03b2-LG | Tea polyphenols | Binding study | Kanakis et al. (2011)  \nCasein | Quercitin | Chitosan-casein nanoparticles | Ha et al. (2013)  \nCasein | Polymethoxyflavones | Raman spectroscopy study of binding | He et al. (2013)  \nVarious milk proteins | Flavonoids | \u03b2-casein showed strongest interactions | Bohin et al. (2012)  \nVarious milk proteins | Polyphenols | Protein sequence influence on noncovalent binding | Nagy et al. (2012)  \nVarious milk proteins | Green tea catechins | Effect of milk proteins on bioaccessibility | Xie et al. (2013)  \nVarious bioactive compounds  \n---  \nCasein micelles | Curcumin | Bioactive component of turmeric | Benzaria et al. (2013)  \nRahimi et al. (2012)  \n\u03b2-LG | Curcumin | Binding and encapsulation study | Sneharani et al. (2010)  \n\u03b1\\- and \u03b2-caseins | Folic acid |  | Bourassa and Tajmir-Riahi (2012)  \nVarious milk proteins | Norbixin | Cheese coloring agent | Zhang and Zhong (2013a, b)  \nReassembled casein micelles | n-3 Polyunsaturated fatty acids |  | Zimet et al. (2011)  \n\u03b2-LG | Piperine (pepper alkaloid) | Binding study | Zsila et al. (2005)  \nPharmaceutical compounds  \n---  \n\u03b2-LG A | Bioactive peptides | Antihypertensive peptide | Roufik et al. (2006)  \n\u03b2-LG | Doxorubicin | Antibiotic | Agudelo et al. (2012)  \nCasein | Alfuzosin | Prostate cancer drug in genipin-cross-linked casein nanoparticles | Elzoghby et al. (2013)  \n\u03b2-casein nanoparticles | Paclitaxel, vinblastine, mitoxantrone | Antitumor drugs | Shapira et al. (2010)  \nLactoferrin | Gambogic acid | Antitumor compound | Zhang et al. (2013)\n\n## Interactions between process-modified milk proteins and micronutrients\n\nHeat has been used extensively in food processing for centuries and is a widely applied treatment in food production, primarily for the control of microbial populations. Fields of application include pasteurization under mild temperatures and sterilization under higher temperatures. However, heating may also affect texture as well as taste development and may result in flavor and color changes. The latter effects are often described as disadvantages of heat treatment. Changes in the organoleptic properties are generally a result of structural changes occurring within the constituents of the food, namely, the proteins, polysaccharides, or fats.\n\nAnother technology that is similar in its control of the microbial population of food products is high-pressure treatment. Foods are preserved with minor changes in texture, flavor, or color, in contrast to heat processing, and high pressure can be considered a cold preservation technology. High pressure is a long-used technique in Japan and has also become increasingly popular worldwide. However, high-pressure treatment may cause some conformational and structural changes to the individual constituents of the food, possibly resulting in altered functional and organoleptic properties.\n\nHeat and high-pressure treatment may both cause the denaturation of globular whey proteins such as \u03b2-LG; although there may be differences in the mechanisms behind the denaturation process, the general process appears to be similar. These processes are examined in detail in Chapter 9.\n\nThe denaturation of whey proteins during the heat treatment of milk, the interactions of the denatured whey proteins with other milk components, and the effect of these reactions on the physical and functional properties of milk products have been extensively reported and reviewed in great detail (O'Connell and Fox, 2003; Singh and Havea, 2003). Studies have shown that heat-induced aggregation and gelation occur along detailed pathways and are influenced by the types of proteins and forces present (disulfide bonding and hydrophobic interactions) (Abbasi and Dickinson, 2002; Havea et al., 2001; Schokker et al., 1999). The use of heat to induce self-assembly and co-assembly of milk proteins into micro\/nanoparticles is discussed in Loveday et al. (2012).\n\nThe effect of high pressure on whole milk and individual constituents has become a subject of much recent activity, particularly regarding the effect of pressure treatment on the physical and functional properties of milk products (Anema, 2010), and the pressure-induced changes to individual proteins (Anema, 2012). Interested readers are referred to the reviews of Huppertz et al. (2006) and Considine et al. (2007). The mechanistic effects of high-pressure processing and several other novel processing technologies were reviewed in the context of the industrial potential of these technologies in yogurt manufacture (Loveday et al., 2013). The use of high pressure in other dairy systems, such as whey or casein gels, has also been reviewed (Devi et al., 2013).\n\n### Protein Denaturation by Thermal and Pressure Treatments and Effect of Micronutrients\n\nThe caseins have not been suitable candidates for observing changes in protein denaturation due to their lack of defined secondary and tertiary structure. In contrast, the whey proteins have been studied widely as model globular proteins because of their well-defined secondary and tertiary structures, as outlined above.\n\nInteractions between whey proteins and other species induced by either heat treatment or pressure treatment may be divided into two separate classes: covalent interactions and noncovalent interactions. The most important covalent interaction involving whey proteins upon storage is their reaction with reducing sugars via the Maillard reaction to form discolored protein powders, which also have reduced solubilities and diminished nutritional properties. Noncovalent interactions can also occur; these too may lead to a loss of protein solubility after association of the proteins with polysaccharides, and these noncovalent interactions are primarily driven by reversible electrostatic interactions.\n\nIn the present work, the effects of noninteracting species on the unfolding and structural transitions of whey proteins are of specific interest. The marked increase in the thermal and conformational stability of globular proteins in aqueous media in the presence of sugars is well known and has been extensively studied.\n\n### Processing Treatments Involving Ligands\n\nSeveral studies have shown that ligands can retard the heat- or pressure-denaturation of \u03b2-LG, and the type of ligand has an impact on this process. For example, during the heat denaturation of \u03b2-LG, both SDS and palmitate stabilized the native structure of \u03b2-LG against heat-induced structural flexibility, subsequent unfolding, and denaturation up to approximately 70 \u00b0C, whereas both retinol and ANS provided very little stabilization (Considine et al., 2005). When a similar range of ligands were used during pressure denaturation, a similar effect was noted; that is, higher pressures were required to cause unfolding of \u03b2-LG when a ligand was present (Considine et al., 2005). These studies and the comparison study of heat and pressure using myristate and conjugated linoleic acid as ligands showed that \u03b2-LG unfolds slightly differently with respect to the type of treatment (Fig. 14.4) (Considine et al., 2007).\n\nFigure 14.4 Proposed three-stage model of the pressure denaturation of \u03b2-LG B, and \u03b2-LG B with added ANS, retinol, or SDS. Source: Reproduced with the permission of Considine et al. (2005). Copyright 2005 Journal of Agricultural and Food Chemistry.\n\nBarbiroli et al. (2011) have shown that endogenous ligands bound to \u03b2-LG (mostly palmitic and stearic acid) stabilize the tertiary structure against denaturation by urea or heat. They reported evidence that the binding of palmitic acid in the calyx enhanced the thermal stability of both the calyx region and the helix held against the outside of the beta barrel (the helix conceals the free thiol at Cys 121). They believed that fatty acid binding in the calyx made the whole structure 'tighter,' and inhibited the movement of the helix region and exposure of Cys 121, which is crucially involved in disulfide-bonded aggregation. In related work with synthetic ligands, Busti et al. (2006) reported that alkyl sulfonates (AS) with a chain length >10 increased the denaturation temperature of \u03b2-LG at pH 6.8 by up to 13 \u00b0C.\n\nHansted et al. (2011) conducted a detailed investigation of how surfactants affect thermally-induced unfolding and aggregation of \u03b2-LG, using homologous a series of cationic (alky trimethyl ammonium chlorides, xTAC), anionic (sodium alkyl sulfates, SxS), and non-ionic (alkyl maltopyranosides, xM) surfactants. SxS inhibited thermal unfolding and aggregation at concentrations well below the critical micelle concentration (CMC), indicating that surfactant monomers were responsible for the effect. xM also inhibited aggregation, though only above the CMC, and smaller xM promoted unfolding at such concentrations. xTAC strongly promoted aggregation at sub-CMC concentrations. The findings highlight the effect of surfactant charge on aggregation at pH 6.5: anionic SxS and non-ionic xM reduced aggregation, whereas cationic xTAC promoted aggregation. The authors also showed how the concentration of surfactants strongly modifies their effects, and they postulated surface interactions between \u03b2-LG and micelles of non-ionic or cationic surfactants (Hansted et al., 2011).\n\nCelej et al. (2005) compared the effects of the binding of two ANS derivatives, namely, 1,8-ANS and 2,6-ANS, on BSA thermostability. They reported that 1,8-ANS had a stronger effect on BSA thermal stability and that the binding parameters of the two ANS derivatives were quite different. This was thought to indicate that stereochemistry is an important factor in determining protein\u2013ligand interactions. Thus electrostatic interactions should also be considered along with hydrophobic interactions. The authors emphasized the importance of free ligand concentration rather than the ligand:protein mole ratio when determining protein stability.\n\nAs discussed earlier, the binding of retinol to casein is through hydrophobic interactions (Poiffait and Adrian, 1991). \u03b2-Casein is the most hydrophobic casein and has a highly charged N-terminal domain, containing an anionic phosphoserine cluster, that is clearly distinct from a very hydrophobic C-terminal domain (Swaisgood, 2003). Little work has been done on the ability of the caseins to bind retinol, although Poiffait and Adrian (1991) reported that casein plays an important role in stabilizing retinol over time or during heat treatment. However, information in this area is limited.\n\n### Processing Treatments Involving Sugars or Polyols\n\nThe effect of up to 70% w\/w glycerol or sorbitol on the properties and functionality of \u03b2-LG was examined in several studies by Chanasattru and co-workers. Sorbitol strongly increased the thermal denaturation temperature of \u03b2-LG at pH 7, whereas glycerol had a very minor effect (Chanasattru et al., 2007b). This translated to much stronger gels with glycerol when 10% \u03b2-LG solutions were heated to 90 \u00b0C for 70 min. Both polyols increased the complex modulus (G*) relative to controls, which was attributed to the strengthening of protein\u2013protein interactions, but the inhibitory effect of sorbitol on denaturation was thought to explain the low G* with this polyol. Later studies noted that glycerol decreases the surface tension at hexadecane\/water interfaces, whereas sorbitol slightly increases it (Chanasattru et al., 2007c). The authors proposed that glycerol could interact with nonpolar regions on the surface of proteins in a way that counterbalanced steric exclusion effects, leading to small net effects on denaturation temperature (Chanasattru et al., 2008).\n\nThis group also studied the effects of polyols in \u03b2-LG-stabilized emulsions (Chanasattru et al., 2007a). Glycerol and sorbitol improved emulsion stability against salt-induced flocculation to an approximately equal extent on a % w\/w basis. This effect was attributed partly to increased viscosity (especially for sorbitol) but also a predicted reduction in attractive van der Waals and hydrophobic interactions large enough to overcome a slight weakening of electrostatic repulsion. Similar studies on \u03b2-LG- and casein-stabilized emulsions were discussed by Dickinson (2010).\n\nThe effect of small mono- and polyhydroxy alcohols on \u03b2-LG thermal stability at pH 5.5 was examined in more detail by Romero et al. (2007), using a homologous series of 4-carbon alcohols with 1 to 4 hydroxyl groups. All alcohols destabilized \u03b2-LG, but to an extent that decreased as the number of hydroxyl groups increased. The authors proposed that 1-butanol was hydrophobic enough to interact with nonpolar regions on the surface of \u03b2-LG, whereas more hydroxylated (and therefore more hydrophilic) alcohols interacted preferentially with water instead of protein, and thereby had a less destabilizing effect. This theory aligns well with the proposal from Chanasattru et al. (2008) that glycerol (1,2,3-propanetriol) interacts with nonpolar regions on the surface of protein.\n\nBoye and Alli (2000) reported on the thermal denaturation of 1:1 mixtures of \u03b1-LA and \u03b2-LG in the presence of a range of sugars, using differential scanning calorimetry (DSC). Sugars protected against heat-induced denaturation, and the protection offered (i.e., size of the increase in the thermal transition temperature of \u03b2-LG) was in the order galactose = glucose > fructose = lactose > sucrose > sugar-free control. No significant effects of sugar were observed with apo-\u03b1-LA. Interestingly, an earlier study by the same authors solely on \u03b1-LA found an increase in the thermal transition temperature of both the apo and holo forms of \u03b1-LA when either 50% sucrose or 50% glucose was added (Boye et al., 1997). This increase was fully reversible in the holo form, but only partly reversible in the apo form. The thermal transition temperature of \u03b2-LG was found to be increased in the presence of sucrose, lactose, and glucose at 10\u201350% (Boye, Ismail, et al., 1996).\n\nJou and Harper (1996) found an increase in the DSC thermal transition temperature of whey protein concentrates following the addition of sugars, and the protection offered by the sugars was in the order maltose > trehalose > sucrose. Lactose was also found to provide some protection against heat-induced denaturation. Dierckx and Huyghebaert (2002) followed the heat-induced gelation of a WPI solution using DSC and small-amplitude oscillatory rheometry. They found that by adding increasing concentrations of sucrose or sorbitol, both the thermal transition temperature of the protein denaturation process and the gelation temperature were increased, with a linear relationship existing between the transition and gelation temperatures. They suggested that because of the differences in the gelation mechanisms observed at different pH values, sucrose and sorbitol affected protein\u2013protein interactions in gels through enhancement of hydrophobic interactions.\n\nKulmyrzaev et al. (2000) had previously conducted a study on the effect of sucrose on the thermal denaturation, gelation, and emulsion stabilization of WPI. They also observed increases in the thermal transition temperatures on the addition of increasing concentrations of sucrose, as well as improved gel formation and enhanced emulsification flocculation. They postulated that sucrose played different roles in a pre-denatured (improved heat stability) and a post-denatured (enhanced protein\u2013protein interactions) whey protein solution system.\n\nIn a study on the effects of different lactose concentrations (within a naturally occurring range) on the formation of whey protein microparticles, Spiegel (1999) put forward a two-stage process in the aggregation of whey proteins: Up to approximately 85 \u00b0C, the aggregation of whey proteins is limited by the slow unfolding of the individual proteins; above 100 \u00b0C aggregation is the rate-limiting step, as the rate of unfolding is high. Lactose (at 500 mM) was also found to increase the temperature of the denaturation of WPI at pH 9.0 by approximately 3 \u00b0C. However, the authors realized the effect that the Maillard reaction was having on these systems, a factor that some reports seem to ignore.\n\nBaier and McClements (2001) found that increased concentrations of sucrose (up to 40%) could increase the thermal stability of BSA. These systems had a higher gelation temperature and produced gels with a lower complex shear modulus. Similar effects were found in a subsequent study (Baier and McClements, 2003). A further study by the same group (Baier et al., 2004) showed that 40% glycerol increased the temperature of gelation of BSA, but no change in the temperature of denaturation of BSA with increasing concentration of glycerol was detected.\n\nSome early DSC work (Dumay et al., 1994) showed that the presence of 5% sucrose was enough to reduce the extent of \u03b2-LG unfolding by 22% following high-pressure treatment at 450 MPa for 15 min. In a later study, Dumay et al. (1998) found that adding sucrose to \u03b2-LG solutions prior to pressure-induced gelation resulted in gels with decreased pore size and strand thickness. They attributed this to a reduction in the number of protein\u2013protein interactions occurring under the influence of pressure.\n\nKeenan et al. (2001) reported that low concentrations of sucrose aided in the pressure-induced gel formation of a range of milk-protein-containing systems, but that gel formation was reduced at higher sucrose concentrations. In another group of studies, the pressure- induced gelation properties of skim milk powder were found to be improved by adding low concentrations of sucrose, glucose, or fructose, whereas high (45\u201350%) sugar concentrations inhibited gel formation (Abbasi and Dickinson, 2001).\n\nBoye et al. (1996) described how lactose, sucrose, and glucose increased the temperature of denaturation of BSA, with 50% glucose having a greater stabilizing effect than 50% sucrose. Wendorf et al. (2004) studied the ability of different proteins (ribonuclease A, BSA, and egg white lysozyme) to adsorb to a liquid\u2013solid interface in the presence of a range of sugars. They found that the ability of sugars to reduce protein adsorption followed the trend trisaccharides > disaccharides > 6-carbon polyols > monosaccharides, and this was explained by the stabilization of the protein in the native state in solution.\n\nOther studies have also shown the beneficial effects of sugars in protecting against denaturation induced by freeze drying, spray drying, and chemicals. At low temperatures, high concentrations of sugars cause a substantial increase in solution viscosities and can thus affect protein denaturation. Tang and Pikal (2005) reported that, by negating the thermal stabilizing effects of sucrose by adding denaturants, the increased stability of \u03b2-LG in the freeze drying process could be directly attributed to a viscosity effect. Murray and Liang (1999) explored the addition of sucrose, trehalose, lactose, and lactitol to whey protein concentrate solutions prior to spray drying and found that the foaming properties of the spray-dried powders were dramatically decreased when sugars were absent. Trehalose was particularly successful in retaining the original foaming properties of both whey protein concentrate and \u03b2-LG, but did not perform as well in spray-dried BSA powders (Murray and Liang, 1999).\n\n## Conclusions\n\nThe interaction of milk proteins with various micronutrients is primarily governed by the physicochemical properties of the proteins. The whey proteins, with extensive secondary and tertiary structures and significant hydrophobicity (albeit largely shielded in the native form), tend toward hydrophobic interactions with ligands. Preferential exclusion effects govern the interaction of sugars and polyols with proteins, thus affecting their denaturing properties in the presence of pressure or heat. Electrostatic interactions drive the association of minerals and proteins.\n\nIn the food industry, an increasing emphasis is being placed on foods that will have a physiologically functional benefit, in addition to the nutritional benefit of the food. This emphasis is being driven by consumers who are becoming increasingly more health aware and health responsible. The challenge for the food scientist is now to deliver the required physiologically functional activities into the final food product, while retaining product quality and shelf life. Knowledge of the interactions of these micronutrients with milk proteins, a major component in many food products, is necessary to achieve this aim. Relevant examples of this concept are detailed in patents concerning the delivery of micronutrients in complexes with \u03b2-LG (Swaisgood, 2001) or casein micelles (Livney and Dalgleish, 2007).\n\nWhile the concept of using milk proteins as nutrient carriers has been explored in a range of protein\u2013nutrient combinations, there is relatively little knowledge about how nutrient interactions can be used to manipulate the functionality of proteins during processing. Binding of ligands to certain whey proteins increases their resistance to thermal denaturation, and noninteracting solutes like sugars can also stabilize proteins against heat processing. Mineral binding to caseins affects their solubility, which has obvious implications for beverage products. A stronger knowledge of protein\u2013micronutrient interactions will enable the use of milk proteins as nutrient carriers and allow the use of micronutrients as processing aids; perhaps both objectives could even be achieved simultaneously.\n\n# References\n\nAbbasi S ,  Dickinson E .  Influence of sugars on high-pressure induced gelation of skim milk dispersions .  _Food Hydrocolloids_.  2001 ;15 : 315 \u2013 319 .\n\nAbbasi S ,  Dickinson E .  Influence of high-pressure treatment on gelation of skim milk powder + low methoxyl pectin dispersions .  _High Pressure Res._ 2002 ;22 : 643 \u2013 647 .\n\nAbd El-Salam MH ,  El-Shibiny S .  Formation and potential uses of milk proteins as nano delivery vehicles for nutraceuticals: A review .  _Int. J. Dairy Technol._ 2012 ;65 : 13 \u2013 21 .\n\nAdrian J ,  Frangne R ,  Rabache M .  Effect of milk-proteins on retinol efficiency .  _Sci. Aliments_.  1984 ;4 : 305 \u2013 308 .\n\nAgudelo D ,  Beauregard M ,  B\u00e9rub\u00e9 G ,  Tajmir-Riahi HA .  Antibiotic doxorubicin and its derivative bind milk \u03b2-lactoglobulin .  _J. Photochem. Photobiol._ 2012 ;B117 : 185 \u2013 192 .\n\nAnderson BF ,  Baker HM ,  Norris GE ,  Rice DW ,  Baker EN .  Structure of human lactoferrin: Crystallographic structure analysis and refinement at 2.8 \u00c5 resolution .  _J. Mol. Biol._ 1989 ;209 : 711 \u2013 734 .\n\nAndersson LO ,  Brandt J ,  Johansson S .  The use of trinitrobenzenesulfonic acid in studies on the binding of fatty acid anions to bovine serum albumin .  _Arch. Biochem. Biophys._ 1971 ;146 : 428 \u2013 440 .\n\nAnel A ,  Calvo M ,  Naval J ,  Iturralde M ,  Alava MA ,  Pineiro A .  Interaction of rat \u03b1-fetoprotein and albumin with polyunsaturated and other fatty acids: Determination of apparent association constants .  _FEBS Lett._ 1989 ;250 : 22 \u2013 24 .\n\nAnema SG .  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Induced chirality upon binding of cis-parinaric acid to bovine \u03b2-lactoglobulin: Spectroscopic characterization of the complex .  _FEBS Lett._ 2002 ;520 : 81 \u2013 87 . \nChapter 15\n\n# Model Food Systems and Protein Functionality\n\nW. James Harper Department of Food Science and Technology, The Ohio State University, Ohio, USA\n\n## Abstract\n\nFabricated foods generally comprise a mixture of components made up of lipids, proteins, simple and complex carbohydrates, emulsifiers, and salts, which are capable of interacting with each other and modifying the final characteristics of the food. Often, processing utilized in the manufacture of the food also modifies these interactions. Model food systems were first developed because of the disparity between laboratory functional tests for proteins and the functionality in the food.\n\nModel food systems today find utility for the functionality of many other food components, including starches, gums, and emulsifiers, as well as areas of continued interest (lipid oxidation, Maillard reaction, etc.). Therefore, model food systems provide a means of determining how the ingredients and the process alter the characteristics of the final product, as well as evaluating the sensitivity of the characteristics of the food to the different ingredients and processing steps. Model food systems are based on the formulation and processing of real foods, using laboratory and pilot plant facilities. Generally, ingredients that do not have a main effect on the final characteristics of the product are eliminated. One potential limitation is the use of processing equipment that does not scale up to commercial production.\n\nThe utilization of carefully selected statistical designs is essential to unravel the multiple interactions that do occur and to optimize food formulation and processing.\n\nA major limitation of model food systems is that they do not provide any information as to the mechanisms by which the ingredients and the process control the final characteristics of the product. Thus they have application to only the food under investigation. They do have a major role in food product development.\n\n## Keywords\n\nModel food systems, protein functionality, food characteristics, sodium caseinate, pasting characteristics, potato starch, milk proteins, statistical design\n\nOutline\n\nIntroduction 451\n\nProtein functionality in foods 453\n\nRole of interactions in determining food characteristics 453\n\nA Case Study 455\n\nEffect of Sodium Caseinate on the Pasting Characteristics of Different Starches 455\n\nEffect of Milk Proteins on the Pasting Characteristics of Potato Starch, with Emphasis on Peak Viscosity 455\n\nProcessing effects 459\n\nUses of model food systems 461\n\nInitial Steps to Developing Model Food Systems 461\n\nStatistical Design 461\n\nApplications of model food systems 462\n\nBakery Products 463\n\nDairy Analogues 464\n\nCoffee Whiteners 464\n\nWhipped Toppings 465\n\nSalad Dressings 465\n\nMeat Products 466\n\nUse of model food systems for other food components 467\n\nLimitations 467\n\nConclusions 467\n\n## Introduction\n\nThe utilization of proteins in food for nutritional and functional purposes goes back many centuries, but the relationship between structure and function has been given close attention only during the past 30\u201340 years (Owusu-Apenten, 2004). Numerous studies and many reviews have contributed to gaining an understanding of precisely how proteins act in a complex food system and how their structure and function are altered by the other ingredients in the food, its intrinsic properties, and its processing. These include Anfinsen (1972), Kinsella (1982), Nakai (1983), Mulvihill and Fox (1987), Mangino et al. (1994), Zayas (1996), Li-Chan (2004), Luyten et al. (2004), Owusu-Apenten (2004), Turgeon et al. (2007), Ghousch et al. (2008), Foegeding and Davis (2011), Singh (2011), and Dickinson (2012).\n\nThere are two broad ways of gaining knowledge of the structure and function of protein systems: (1) study of pure proteins in simple systems and (2) study of commercial proteins in the food systems in which they are used. These methods are entirely different (Luyten et al., 2004; Owusu-Apenten, 2004) and provide quite different information. Functionality tests can be very useful in obtaining reproducible functional properties, even though such tests cannot be used to predict the final characteristics in a real food system (de Wit, 1984; 1989; Harper, 1984; Owusu-Apenten, 2004). Some differences include the following.\n\n\u2022 In pure structure\/function studies, pure proteins are generally used and are used at concentrations much lower than those used in food systems (Owusu-Apenten, 2004).\n\n\u2022 In food systems, proteins are seldom pure and may actually involve complex mixtures of proteins from a given food source (such as milk proteins, egg proteins, and soy proteins) or proteins from multiple food sources (i.e., meat, soy, milk, and gluten) or proteins that have been selectively denatured to provide the desired functionality (Mangino et al., 1994).\n\n\u2022 In structure\/function studies, care is taken to avoid interactions with other components and to avoid modifying the secondary and tertiary structure during the experiments. The proteins are fully hydrated (Kinsella, 1982; Owusu-Apenten, 2004).\n\n\u2022 In food systems, the proteins are constantly exposed to other ingredients, which can modify the structure and hence function, as well as being modified by processes that often include pH, heat, and shear (Lee et al., 1992; Kilara, 1994). Competition for water can also modify functionality, as can changes in intrinsic properties (Zayas, 1996).\n\n\u2022 In structure\/function studies, outcome is generally measured for a specific and single response (Owusu-Apenten, 2004).\n\n\u2022 In food systems, the ingredients can influence product functionality at different points in the process, or functionality can be expressed in more than one outcome with respect to the characteristics of the food (de Wit, 1984; 1989; Harper, 1984).\n\nUnquestionably, proteins and other hydrocolloids are important and are required to give the food desirable characteristics. However, our knowledge remains incomplete today because we still cannot fully predict the characteristics of a formulated food on the basis of our knowledge of the structure and function of pure proteins or hydrocolloids under strictly controlled conditions (Kinsella, 1982; de Wit, 1984; 1989; Harper, 1984; Owusu-Apenten, 2004; Zayas, 1996).\n\nTesting for functionality using simplified systems has been reviewed in depth by numerous investigators, including Kinsella (1982), Kilara (1984), Modler and Jones (1987), Mulvihill and Fox (1987), Patel and Fry (1987), Hall (1996), Zayas (1996), and Owusu-Apenten (2004). The continued need to develop standardized testing for protein solubility, viscosity, water absorption, gelation, emulsification, and foaming properties has been emphasized by Mulvihill and Fox (1987), Patel and Fry (1987), German and Phillips (1994), Kilara (1994), Hall (1996), Zayas (1996), and Luyten et al. (2004).\n\n## Protein functionality in foods\n\nProteins used in foods include plant proteins (soy, wheat, rice, corn, and other plant sources), milk proteins (caseins, caseinates, whey proteins, and milk protein concentrates\u2014both caseins and whey proteins), egg proteins (egg white and egg yolk proteins), meat proteins, and fish proteins. Each type of protein exhibits different functional properties and has application in different types of food products (Inglett and Inglett, 1992; Kinsella, 1982; Lee et al., 1992; Kilara, 1994; Mangino et al., 1994; Owusu-Apenten, 2004).\n\nThe major functionalities of food proteins include solubility, emulsification, gelation and foaming, water binding, and heat stability. As shown in Table 15.1, different types of foods have different functional requirements and may require multiple functionalities.\n\nTable 15.1\n\nMultiple Functionalities in Selected Food Products\n\nFood Type | Multiple Functionalities  \n---|---  \nBeverages | Solubility, heat stability, pH stability, color  \nBaked goods | Emulsification, foaming, gelation  \nDairy analogues | Gelation, foaming, emulsification  \nEgg substitutes | Foaming, gelation  \nMeat emulsions | Emulsification, foaming, gelation, adhesion\/cohesion  \nSoups and sauces | Viscosity, emulsification, water adsorption  \nInfant formulae | Emulsification, heat stability  \nToppings | Foaming, emulsification  \nFrozen desserts | Foaming, gelation, emulsification\n\nSource: Adapted from Kinsella (1982), de Wit (1984), Kilara (1994) and Owusu-Apenten (2004).\n\nFactors that may modify the protein during processing, and hence its effect on the product characteristics, include heat, shear, salts, and other hydrocolloids (de Wit, 1984; Mangino et al., 1987; Yada, 2004).\n\n## Role of interactions in determining food characteristics\n\nInteractions between ingredients and modifications caused by processing are the primary reasons why the functionality of proteins and other colloids cannot be predicted in food systems (de Wit, 1984; 1989; Harper, 1984; Zayas, 1996; Owusu-Apenten, 2004; Yada, 2004).\n\nThe following diagram provides an overview of the potential interactions that can occur in a food product (adapted from Harper, 1984):\n\nEssentially, almost everything can modify the functionality of everything else. Salts are somewhat unique in that they do not in themselves affect product characteristics, but can act on proteins, surfactants, polysaccharides, and, to some extent, polar lipids to modify the functionality of each in the food system. The nature and extent of these interactions will be modified by pH, ionic strength, ingredient concentrations, and process-induced modifications. Some examples include the following.\n\n\u2022 Surfactants and proteins can interact competitively at the surface of an oil to modify the characteristics of the emulsion, such as stability, size distribution, and light scattering. The extent to which a given component will dominate the characteristics will depend on the relative concentrations (Fig. 15.1) and chemical natures of the surfactant and protein.\n\nFigure 15.1 Effect of emulsifier (mono- and diglycerides) on protein and phase separation of an oil-in-water emulsion (coffee whitener). Adapted from Harper (1984).\n\n\u2022 Starch is frequently used to provide texture in food products. However, the viscosity during processing and the final viscosity can be greatly altered by interactions with other components. The data in Figure 15.2 show that interactions that are observed with two-component systems do not always predict the effect of three- and four-ingredient interactions. In addition, 'who sees who first' can further modify other interactions and change product characteristics.\n\nFigure 15.2 Effect of components singly and in combination on the peak viscosity of potato starch (starch (S)): S + X = starch + xanthan gum; S + M = starch + mono- and diglycerides; S + F = starch + fructose; S + X + M = starch + xanthan gum + mono- and diglycerides; S + X + F = starch + xanthan gum + fructose; S + M + F = starch + mono- and diglycerides + fructose; S + X + M + F = starch + xanthan gum + mono- and diglycerides + fructose. Adapted from Harper (2006).\n\nRecent reviews of the use of simple model systems to obtain a better understanding of protein food ingredient interactions and the mechanisms involved include Dickinson (2011; 2012), Foegeding and Davis (2011), and Singh (2011).\n\n### A Case Study\n\nTo further expand on protein interactions, the following presents a case study of the effect of various milk proteins on the modification of the pasting characteristics of potato starch, including the effect of different types of milk proteins and of differences in the concentrations of protein and\/or starch.\n\nThe texture of formulated foods is strongly modified by interactions. This section gives results of a simplified model system that provides a better understanding of food formulations that involve protein\/food component interactions. The information provides an understanding of how milk proteins can modify the pasting properties of potato starch as investigated by Harper and Illingworth (unpublished data); Harper and Hemar (unpublished data); Doublier et al. (2001); and Bertolini et al. (2005). Starch is important in a number of food products in which the texture can be modified by protein\/starch interactions to change the textural properties of the food.\n\n#### Effect of Sodium Caseinate on the Pasting Characteristics of Different Starches\n\nThe Rapid Visco Analyzer (RVA) was used to evaluate the pasting properties of starch. Figure 15.3 shows a typical starch pasting curve for potato starch. Generally, an 8% starch paste was utilized, and the various characteristics are shown as a function of time. The two most important characteristics are peak height and final viscosity. Peak height shows the maximum viscosity during manufacture, and the final viscosity gives a measure of the final texture.\n\nFigure 15.3 Diagram of the steps in starch pasting, including time to initiate gelation. Peak viscosity, trough, final viscosity, breakdown, and setback\n\nThe addition of sodium caseinate to six different starches (corn, rice, wheat, potato, cassava, and waxy maize) influenced the peak (pasting) temperature, the time to reach peak viscosity, the peak viscosity, and the viscosity after cooling. Sodium caseinate had different effects on these parameters for the different starches.\n\nThe percent change in peak viscosity of the six sodium caseinate\/starch mixtures is shown in Figure 15.4. All starches, except potato starch, showed an increase in peak viscosity, whereas potato starch showed a dramatic decrease in peak viscosity. There was no statistically significant change in the final viscosity.\n\nFigure 15.4 Peak viscosity during pasting of six different starches\n\nThe differences in the pasting characteristics will be important with respect to both processing and the characteristics of different food products using starch and caseinate. Based on the results of the marked difference between the pasting characteristics of potato starch and those for all the other starches, attention was given only to potato starch for further studies.\n\n#### Effect of Milk Proteins on the Pasting Characteristics of Potato Starch, with Emphasis on Peak Viscosity\n\nProteins and polysaccharides, including starch, are frequently used together in food systems to import specific attributes to the final product. Hardacre et al., (2004) showed that sodium caseinate decreased the peak viscosity of potato starch at a starch to protein ratio of 1000:1. Different milk proteins were used to determine their effect on the pasting characteristics of potato starch. These included individual milk casein fractions, sodium caseinate with different concentrations of protein and starch, sodium caseinate, milk protein concentrate, and milk protein isolate.\n\n##### Individual Casein Fractions\n\nCasein was fractionated into \u03b1s\\- (\u03b1s1 \\+ \u03b1s2) and \u03b2-caseins, and these were in turn converted to their sodium and calcium salts. The peak viscosity, final viscosity, and pasting time of starch, starch + 1% calcium caseinate + starch, 1% calcium \u03b1s-caseinate starch + 1% calcium \u03b2-caseinate are shown in Table 15.2 and Figure 15.5 using standard starch concentrations (8%).\n\nTable 15.2\n\nRVA Results Showing the Effect of the Calcium Salts of Caseinate Fractions on the Gelation and Pasting of Potato Starch\n\n| Viscosity (cP) | Pasting time (s)  \n---|---|---  \n| Peak | Final |   \nStarch | 7315 | 2952 | 196.2  \nStarch + 1% calcium \u03b1s-caseinate | 4487 | 2975 | 232.2  \nStarch + 1% calcium \u00df-caseinate | 3806 | 2834 | 288  \nStarch + 1% calcium caseinate | 3467 | 2543 | 336\n\nFigure 15.5 Effect of caseinate fractions and sodium casein on the peak viscosity of potato starch.\n\nPasting time increased in the order of starch < starch + calcium caseinate < starch + \u03b2-caseinate < starch + sodium caseinate. As the starch concentration decreased, there was a loss in final viscosity, especially at a 2% starch concentration. This finding would indicate a definite lack in texture for food products using 2% or less of starch.\n\n##### Effect of the Concentration of Sodium Caseinate on the Pasting Characteristics of Potato Starch\n\nThe effect of concentration of sodium caseinate, from 0.1% to 2.5%, on the pasting characteristics of potato starch was evaluated. Data for time to reach the maximum viscosity, peak viscosity, breakdown, setback, and final viscosity are shown in Table 15.3. The time to reach maximum viscosity increased. Peak viscosity and setback decreased with concentration, where there was no significant change in setback or final viscosity. The data suggest that the addition of casein would put less stress on processing, through a decrease in peak viscosity, without a significant effect on the final viscosity\n\nTable 15.3\n\nRVA Results for the Effect of Casein Fractions on the Gelation and Pasting of Potato Starch\n\n| Viscosity (cP) | Pasting time (s)  \n---|---|---  \n| Peak | Final |   \nStarch | 7361 | 2968 | 199.8  \nStarch + 2.5% \u03b1s-casein | 5885 | 3106 | 228.0  \nStarch + 2.5% \u03b2-casein | 4359 | 2712 | 228.0  \nStarch + 2.5% sodium caseinate | 2113 | 2671 | 343.8  \nStarch + 5% \u03b1s-casein | 5676 | 3329 | 220.2  \nStarch + 5% \u03b2-casein | 4727 | 2947 | 211.8\n\n##### Effect of the Concentration of Starch and Milk Proteins on Potato Starch Pasting Characteristics\n\nThe concentration of starch for determining the effects of starch on pasting characteristics generally has been 6\u20138%. However, the use level in most food applications ranges from 2% to 4%. Therefore, from a practical viewpoint, effects on starch pasting characteristics in food use concentration would be useful.\n\nThe effect of protein concentration and starch concentration on the loss of peak viscosity, final viscosity, and phase separation on storage was determined for sodium caseinate, milk protein concentrate, and milk protein isolate. All of the different proteins gave similar results, and only the effect of milk protein isolate is presented in this investigation, for illustrative purposes.\n\nThe loss in peak viscosity increased as the starch content decreased, and the protein content increased as illustrated in Figure 15.6. The loss of peak viscosity increased to some degree as the starch content increased. The effect of protein concentration was very significant and linear, from 0.02% to 1%. However, once the protein concentration reached 1%, the loss was the same for higher protein concentration increases, ranging from 80% to nearly 100%.\n\nFigure 15.6 Effect of protein and starch concentration of peak viscosity of potato starch\n\nUnder standard test conditions with the RVA (8% starch), there was little to no loss in final viscosity with the addition of milk protein. However, as the starch content was decreased below 6%, the loss in final viscosity markedly increased, as shown in Figure 15.7. This would limit the advantage of using potato starch in foods containing milk protein, especially at starch concentrations below 4%.\n\nFigure 15.7 Effect of MPI concentrations on loss of final viscosity as a function of starch concentration.\n\n## Processing effects\n\nThe functionality of commercial food proteins and other hydrocolloids can be modified both during their production and during the processing of the food product itself. An overview of the conversion of a raw protein source to a functional food ingredient and the subsequent further processing during food manufacture is outlined in Figure 15.8.\n\nFigure 15.8 Steps in the manufacture of food proteins and the subsequent processes during food manufacture. From Owusu-Apenten (2004).\n\nDuring the production of commercial food proteins for use as food ingredients, the proteins may be exposed to a wide range of processing steps that can include thermal processes (pasteurization, sterilization), shear (pumping, mixing, homogenization), pressure (high-pressure processing, retorting), concentration (membrane processing, evaporation, drying), and precipitation (heat, acid, salts, solvents). Each of these steps will modify the functional properties of the protein and thus will affect the final characteristics of the food (Kinsella, 1982; de Wit, 1984; 1989; Harper, 1984; Dybing and Smith, 1991; Kilara, 1994; Zayas, 1996; Owusu-Apenten, 2004). Such processes can alter functionality in food through a number of different modifications of the protein, including changes in sulfydryl interactions, modification of secondary and quaternary structure, and shifts in the hydrophilic\/lipophilic balance (Kinsella, 1982). Subsequent processing during use of the protein as a functional ingredient in food will bring further changes in the system, especially those occurring in the presence of other interacting ingredients. Such changes in the characteristics of the food generally cannot be predicted; thus, there is a need for use of model food systems as an intermediate step in product development (Owusu-Apenten, 2004).\n\n## Uses of model food systems\n\nModel food systems can be used in a variety of ways (de Wit, 1984; Harper, 1984; Owusu-Apenten, 2004), including the following.\n\n\u2022 Determining the relative significance of the main effects of ingredients.\n\n\u2022 Studying factors in food that affect chemical and physical changes (Maillard reaction, lipid oxidation, etc.).\n\n\u2022 Evaluating the sensitivity of the food to alterations in formulation and processing.\n\n\u2022 Defining ingredient interactions.\n\n\u2022 Optimizing the formulation for robustness.\n\n\u2022 Determining critical steps in the processing of the product.\n\n\u2022 Determining interrelationships between ingredients and the process.\n\n\u2022 Tailor-making ingredients for a specific food application.\n\n\u2022 Evaluating and minimizing the sensitivity of product attributes to the formulation and the process.\n\nOwusu-Apenten (2004) stated that the advantages of model food systems over standard functionality tests included: (a) their ease of use, (b) the lack of a need for specialized equipment and methodologies, (c) the ability to aid in product optimization, and (d) the ability to test for multiple factors and interactions with respect to formulation and processing.\n\n### Initial Steps to Developing Model Food Systems\n\nThe approach to developing a model food system is the same, whether the ingredient being investigated is a protein, lipid, emulsifier, starch, or gum.\n\nThe development of a model food system begins by reviewing as many formulations as can be found and selecting those that are common to all formulations at a concentration that is at the central point of the various formulas (Harper, 1984). Next, a small-scale process for making the products is developed using processing steps and conditions as close to the commercial process as possible. When more than four or five ingredients are involved, it is often necessary to do a screening experiment to eliminate ingredients that do not have a main effect on important characteristics.\n\nEach different food will have different characteristics, which may include taste, color, and texture, that can be modified by the formulation and the process. Key attributes and methods for their evaluation need to be selected. Generally, the evaluation methods are different from those that are used in research (Owusu-Apenten, 2004).\n\n### Statistical Design\n\nStatistical design is an essential component of model food systems because of the information it provides on ingredient and processing interactions (Dziezak, 1990; Earle et al., 2001; Hanrahan and Lu, 2006). Most fabricated food products have from 5 to 25 variables when both the ingredients and the processing steps are taken into consideration. This makes full factorial designs, which would exceed several hundred experiments, an impractical choice. Thus fractional factorial screening designs are generally required. For most food products, the experimental design is a stepwise process, starting with screening experiments to minimize the variables that do not have major effects on the characteristics of the products. One of the most common screening designs is the Plackett-Burman, which can be used with up to 36 variables (Mullen and Ennis, 1985; Hanrahan and Lu, 2006). The screening experiments allow determination of the main effects that can be used in further fractional factorial designs; these designs will provide a better understanding of ingredient and process interactions and will generate response surfaces that give an understanding of the interactions (Hanrahan and Lu, 2006).\n\nIn developing a fractional factorial experimental design in model food systems, it is necessary to know: (a) the critical factors associated with the ingredients and the process, (b) the region of interest where the factor levels influencing the product characteristics are known, (c) that the factors vary continuously throughout the experimental range tested, (d) that a mathematical function relates the variable factors to the measured response, and (e) that the response defined by the function is a smooth curve.\n\nNumerous studies have used statistical design and response surface methodology to determine the effect of interactions on product characteristics and to optimize specific characteristics in a food (Dziezak, 1990).\n\nIn developing an experimental design, consultation with a statistician familiar with the factors that affect the outcomes of the specific design is needed to avoid common pitfalls. Among these pitfalls are the following: (a) incorrectly defined or specified critical factors, (b) too narrow or too broad a range of factors selected, so that the optimum cannot be defined, (c) lack of the use of good statistical practices, (d) too large a variation in the range of the factors utilized, introducing bias and error, (e) over-reliance on computer-generated results, and (f) failure to ensure that the results make good sense.\n\n## Applications of model food systems\n\nInitially, model food systems were applied to milk proteins to gain a better understanding of what was required to get desired characteristics in complex food products that could not be predicted from standard functionality tests. de Wit (1998) has stated: \"Information obtained from functional characterization tests in model systems is more suitable to explain retroactively protein behaviors in complex food systems than to predict functionality.\" What has been learned using milk proteins in model food systems has been shown to be equally applicable to other food proteins. In addition to understanding the protein being used, there is a need to know the functionality of other ingredients in the food, the probability of how they will interact and modify the function of the food protein, and the use of statistical design to gain the full potential of the model system approach.\n\nStudies of model food systems, used to assess their performance in foods, have included a large number of different types of foods, as shown in Table 15.4. These include bakery products, dairy products, dairy analogues, meat products, sauces and dressings, fermented foods, wine, and infant formulas.\n\nTable 15.4\n\nModel Food Systems Used to Assess Functionality in foodsa\n\nFirst-generation model foodsa | Additional examplesb  \n---|---  \nCakes (angel food, chocolate, yellow, pound) | Low-fat spreads  \nMeringues | Beer batters  \nBread | Beef patties  \nCoffee whitener | Gravies  \nHam, restructured meats | Meat emulsions  \nInfant formula | Cream  \nSalad dressing | Milk  \nSausage | Cheese  \nStarch pudding | Processed cheese  \nWhipped topping | Soups and sauces  \nIce cream | Surimi  \n| Wine  \n| Yogurt\n\na Adapted from Harper (1984) and de Wit (1984).\n\nb Adapted from Owusu-Apenten (2004).\n\nThe examples of the model food systems used to illustrate applications in this chapter are primarily from the first generation category. They include bakery products, dairy analogues, meat products, salad dressings, and sauces.\n\n### Bakery Products\n\nBread represents a system in which the methods of evaluation of ingredients have been standardized and covered by AACC-approved methods (AACC methods 10-9, 10-10 and 10-11). Details of the procedures and evaluation techniques have been given by various investigators (Lindblom, 1977; Pomeranz et al., 1984; Ranhortra et al., 1992; Fenn et al., 1994; Cauvain and Young, 2006). In general, the substitution or addition of other proteins (milk, whey proteins, etc.) leads to a loss of loaf volume (Harper et al., 1980; de Wit, 1984). Harper and Zadow (1984) found that heat treatments that prevented loss of loaf volume in bread made with milk powder were ineffective in preventing loss of loaf volume in bread made with whey protein concentrates.\n\nModel food systems have been used widely in cake systems including: pound cake (Lee, 1999), Madeira cake (de Wit, 1984), white cake (Harper et al., 1980), and angel food cake (Kissell and Bean, 1978).\n\nOf these, angel food cake has received the most attention (Lowe et al., 1969; DeVilbiss et al., 1974; Cunningham, 1976; Regenstein et al., 1978; Johnson and Zabik, 1981a,b; Ball and Winn, 1982; Froning et al., 1987; Froning, 1988; Martinez et al., 1995). The primary protein evaluated has been egg white, for which cake height and texture can be related to the individual egg white proteins (Johnson and Zabik, 1981a,b; Ball and Winn, 1982). Attempts to replace egg white with whey proteins have never been completely successful (DeVilbiss et al., 1974; Harper et al., 1980). Arunepanlop et al. (1996) were able to replace 25\u201350% of the egg white with whey protein and could achieve greater replacement by the addition of xanthan gum. Cake volume is essentially the same as it is with egg white, but the cakes collapse upon baking. This emphasizes the requirement for both foaming and gelation (Owusu-Apenten, 2004). This is due in part to the lower gelation temperature for foams made with egg white (Pernell et al., 2002) and in part to the shear-induced denaturation of egg white with mixing (DeVilbiss et al., 1974).\n\nOther proteins evaluated for angel food cake include blood plasma protein (Kahn et al., 1979; Raeker and Johnson, 1995) and dried beef plasma (Duxbury, 1988). Factors that should be considered in developing a model food system for bakery products are outlined briefly in Table 15.5.\n\nTable 15.5\n\nFactors Affecting Functionality of Protein in Bakery Products\n\nProduct type | Functional requirement of protein | Ingredient modifying functionality | Processing factors affecting functionality  \n---|---|---|---  \nBread | Dough formation, water binding, gelation, elasticity of dough | Protein source, polar lipids, oxidizing and reducing agents, other proteins with sulfhydryl groups | Mixing, method of bread making (sponge dough versus mechanical development)  \nCakes | Fat binding, foaming, gelation | Protein type and concentration, gums, fat, sugar concentration | Mixing speed and time, pre-emulsification\n\n### Dairy Analogues\n\nDairy analogues include coffee whiteners, whipped toppings, and processed cheese products.\n\n#### Coffee Whiteners\n\nCoffee whiteners, first developed in the 1950s, generally are protein-stabilized oil-in water emulsions, with vegetable oil as the dispersed phase. A model system developed by Harper and Raman (1979) and Harper et al. (1980) utilized caseinate, soy bean oil, carbohydrate, phosphate, emulsifier, and a gum (xanthan gum or carrageenan). The role of the ingredients has been reviewed by Knightly (1969) and Patel et al. (1992), and the process has been reviewed by Owusu-Apenten (2004). This is summarized in Table 15.6.\n\nTable 15.6\n\nFactors Affecting Functionality of Protein in Coffee Whiteners\n\nProduct type | Functional requirement of protein | Ingredient modifying functionality | Processing factors affecting functionality  \n---|---|---|---  \nCoffee whiteners | Emulsification, stability to the pH and temperature of coffee | Emulsifiers, gums, phosphate, calcium | Homogenization, pasteurization time and temperature, temperature and pH of coffee\n\nPatented processes include using milk protein retentate (Kosikowski and Jimenez- Florez 1987), reformed casein micelles (McKenna et al., 1992), phosphate-modified milk protein (Melachouris et al., 1994), and soy proteins (Melmychyn, 1973).\n\nAlternative proteins that have been suggested to replace caseinate include milk protein concentrate (Euston and Hirst, 2000), whey protein (Hlavacek et al., 1970; Gruetzmacher and Bradley, 1991; Euston and Hirst, 2000), wheat protein (Golde and Schmidt, 2005; Patil et al., 2006), soy protein (Hlavacek et al., 1970; Golde and Schmidt, 2005), peanut protein (Malundo et al., 1992), and cottonseed protein (Choi et al., 1982).\n\nCoffee whiteners are evaluated to ensure that they provide an emulsion with a small particle size to maximize whiteness, minimize astringency of the coffee by binding with the coffee tannins, maintain stability in hot coffee under acidic conditions, minimize feathering in the presence of hard water salts, and readily disperse in the coffee (Pearce and Harper, 1982; Tran and Einerson, 1987; Kneifel et al., 1992; Kelly et al., 1999).\n\nGolde and Schmidt (2005) compared coffee whiteners made from sodium caseinate, soy protein isolate, and wheat protein isolate, and found that they gave similar whiteness (L*) to the coffee. However, the liquid coffee whiteners made with wheat protein tended to separate upon storage, whereas the whiteners made with soy protein isolate tended to show feathering.\n\n#### Whipped Toppings\n\nMost commercial whipped toppings contain sodium caseinate as the protein of choice (Knightly, 1968). Other proteins used for whipped toppings include whey protein concentrate (Peltonen-Shalaby and Mangino, 1986; Liao and Mangino, 1987) and soy protein isolates (Kolar et al., 1979; Lah et al., 1980; Chow et al., 1988; Abdullah et al., 1993; Shurtleff et al, 1994).\n\nWhipped toppings are high-fat, foamed emulsions with about 40% total solids, and model food systems generally also contain sugars, gums, and small-molecular-weight emulsifiers (Knightly, 1968; Harper et al., 1980). A brief summary of key factors that affect the functionality of protein in whipped toppings is given in Table 15.7. The model system differs from whipping or foaming tests with respect to both compositions and much lower fat content (Owusu-Apenten, 2004). Min and Thomas (1977) found that calcium addition to a 15%-fat-containing whipped topping stabilized with sodium caseinate gave improved stability to the system. Peltonen-Shalaby and Mangino (1986) showed that pasteurization also improved the overrun of the topping. Liao and Mangino (1987) used whey proteins to make a model whipped topping and found a positive correlation between exposed hydrophobicity and overrun. Other factors that affect overrun and stability include the hardness of the fat, the type and percentage of emulsifier, and the equipment used for mixing (Harper, 1984).\n\nTable 15.7\n\nFactors Affecting Functionality of Protein in Whipped Toppings\n\nProduct type | Functional requirement of protein | Ingredient modifying functionality | Processing factors affecting functionality  \n---|---|---|---  \nWhipped toppings | Emulsification, whipping to \u2248200% overrun in the presence of high fat and high solids | Emulsifiers, gums, phosphate, calcium | Homogenization, pasteurization time, equipment used to produce the whipped product\n\n### Salad Dressings\n\nSalad dressings are high-fat emulsions, frequently stabilized by high shear in the presence of egg yolk as the primary emulsifier (Parker et al., 1995). Mayonnaise, a spoonable dressing, contains 75% oil by definition. Subsequently, starch pastes were used to make a spoonable dressing with about 40% oil. Today, the most common dressings are pourable, with a wide range of oil contents and are stabilized primarily by xanthan gum (Franco et al., 1995).\n\nModel food systems have been used to gain a better understanding of both ingredients (Smith, 1977; Paredes et al., 1988) and processing (Parker et al., 1995).\n\nSmith (1977), using a central composite statistical design, found that the coefficients of the regression analysis were larger for the interaction terms than for the main effect terms in pourable salad dressing with 40% oil and containing egg, vinegar, xanthan gum, and mustard powder. The order of addition was also found to be important to the viscous properties of the pourable salad dressing.\n\n### Meat Products\n\nModel meat products including beef, pork, lamb, poultry, or fish, have been utilized for recombined meats (ham, steaks, etc.) and meat macroemulsions that include bologna, sausages, liver sausages, frankfurters, and meat loaves.\n\nNon-meat proteins have been injected into beef and ham, together with water, followed by tumbling to maintain nutritionally equivalent protein levels, increase yield, and improve texture (Zayas, 1996; Yada, 2004; Szerman et al., 2007). Szerman et al. (2007) found whey protein isolates to be superior to vegetable proteins on the basis of flavor.\n\nMeat emulsions generally have size distributions between 0.1 and 50 \u03bcm, and many investigators suggest that they are three-dimensional gel networks with entrapped fat (Regenstein, 1989; Krishnan and Sharma, 1990; Xiong et al., 1992; Correia and Mittal, 1993; Barbut, 1995). However, most reviewers continue to classify them as meat emulsions (Gordon, 1969; Webb, 1974; Owusu-Apenten, 2004).\n\nThe factors that affect the functionality of proteins in these products include meat extraction temperature, emulsification temperature, shear during emulsification, fat melting point, pH, ionic strength, ratios of ingredients, salt, soluble protein concentration, and type of salt (salt, phosphates, citrates, etc.).\n\nAchievement of functionality has been determined by a number of different methods, including:\n\n\u2022 emulsification capacity (EC) (Swift et al., 1961; Swift and Sulzbacher, 1963)\n\n\u2022 emulsion activity (EA) (Acton and Saffle, 1972)\n\n\u2022 emulsion stability (ES) (Carpenter and Saffle, 1964; Townsend et al., 1968; Marshall et al., 1975)\n\nAlthough the tests for EC and ES for comminuted meat products are widely used, there does not appear to be much collaborative testing of the different methods (Owusu-Apenten, 2004). The type of protein affects the EC of meat emulsions, with T isolated muscle proteins giving different EC values. In general, the EC was in the order of myosin > actomycin > actin for different types of meat (Tsai et al., 1972; Galluzzo and Regenstein, 1978; Li-Chan et al., 1984). Substitution of meat protein by other protein in meat emulsions, as measured by large deformation rheological testing, showed that\n\n\u2022 gluten, soy protein isolate, or egg white increased the yield after the cooking of meat emulsions (Randall et al., 1976)\n\n\u2022 corn germ protein at 2% substitution reduced the shear force and reduced cooking losses (Mittal and Usborne, 1985)\n\n\u2022 partial substitution of meat with sodium caseinate, soy protein isolate, whey protein concentrate, or wheat germ protein all increased cook yield, increased protein level, and decreased fat in frankfurters, without affecting quality (Atughonu et al., 1988)\n\n\u2022 addition of bovine blood plasma to meat emulsion products improved emulsion stability and yield, and contents of protein, phenylalanine, and valine (Marquez et al., 1997)\n\n## Use of model food systems for other food components\n\nIn addition to evaluating the performance of proteins in food systems, a wide range of other applications have been utilized. During the past several years, more than 200 papers have been published on other uses of model food systems. A full review of such uses is outside the scope of this chapter. However, selected applications from studies over the past several years are cited both to present a basis for understanding the scope of the use of model food systems in the food industry and to provide a starting point for obtaining more detailed information.\n\nApplications include:\n\n\u2022 factors affecting flavor release in foods (Bylaite et al., 2005; Heinemann et al., 2005; Conde-Petit et al., 2006; Nongonierma et al., 2006; Seuvre et al., 2007)\n\n\u2022 factors affecting D values in food (Rodriguez et al., 2006)\n\n\u2022 lipid oxidation (Jaswir et al., 2004; Sakanaka and Tachibana, 2006; Wijeratne et al., 2006)\n\n\u2022 water migration in foods (Guignon et al., 2005; Doona and Moo, 2007)\n\n\u2022 Maillard reaction investigations (Severini, et al., 2003; Miao and Roos, 2005; Acevedo, 2006; Casal, 2006)\n\n\u2022 effects of high-pressure processing of food (Severini et al., 2003; Sila et al., 2007)\n\n## Limitations\n\nModel food systems can tell you 'what,' but they cannot tell you the mechanism(s) by which the effects occur. Frequently, the results with a model system cannot be scaled up to full commercial practice because of differences in equipment and processes. However, they do provide insight into directions to take to overcome scale-up problems. Generally, the results are valid only within the parameters that have been established. Optimization of a food system can sometimes be outside the limits of either the processing equipment or the functionality of a specific ingredient.\n\n## Conclusions\n\nHistorically, model food systems were used first to improve the functionality of milk proteins in food systems. Currently, there are very few publications on the use of milk proteins for these purposes, although it is known that a number of dairy food companies use model food systems in their product development programs. 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Properties of broiler myofibril gels containing emulsified lipids .  _Poultry Science_.  1992 ;71 (9) : 1548 \u2013 1555 .\n\nYada RY .  _Proteins in Food Processing_ .  Cambridge :  Woodhead Publishing ;  2004 .\n\nZayas JF .  _Functionality of Proteins in Food_ .  Berlin :  Springer-Verlag ;  1996 . \nChapter 16\n\n# Sensory Properties of Dairy Proteins\n\nM.A. Drake\n\nR.E. Miracle\n\nJ.M. Wright Department of Food Science, Southeast Dairy Foods Research Center, North Carolina State University, North Carolina, USA\n\n## Abstract\n\nProduction and applications of dairy proteins are increasing globally. The dairy protein category is a wide one, encompassing milk proteins and whey proteins, as well as many subcategories of these products such as caseins, hydrolysates, and serum or native whey proteins. Although functionality and nutrition continue to be key aspects of these products, flavor is a critical parameter that should not be overlooked. An array of sensory analysis techniques can be applied to measure flavor intensities and flavor variability and to determine flavor sources when applied in conjunction with analytical chemistry. This chapter addresses the current status and ongoing research on the sensory properties of dairy proteins.\n\n## Keywords\n\nSensory properties\n\ndairy proteins\n\nmilk proteins\n\nwhey proteins\n\nflavor performance\n\nflavor properties\n\nflavor variability\n\nOutline\n\nIntroduction 473\n\nSensory analysis 474\n\nWhey proteins 474\n\nMilk proteins 485\n\nCaseins and hydrolysates 488\n\nFlavor binding 489\n\nConclusions 489\n\nAcknowledgment 490\n\n## Introduction\n\nDairy proteins are valuable dried ingredients with a host of functional and nutritional properties (Foegeding et al., 2002; Miller, 2005; O'Connell & Flynn, 2007). Within the category of dairy proteins, there are a variety of ingredients, including whey proteins and milk proteins of various protein contents. Dried caseins and caseinates as well as serum proteins ('native' whey proteins or whey proteins separated from milk prior to the cheesemaking process) are also in this category. Dairy proteins (primarily dried) are used in an increasingly wide array of ingredient applications for functionality, but with the current consumer focus on health and nutrition, these ingredients are also used widely to enhance nutrition.\n\nMilk proteins play a crucial role in the flavor of all dairy foods. As part of the sensory experience, proteins provide mouthfeel, viscosity, and structure for dairy foods. Amino acids and peptides can elicit basic tastes but can also serve as the starting substrates for numerous volatile aroma-active compounds. Proteolysis and the subsequently released amino acids and peptides are the sources as well as the substrates for many desirable and undesirable flavors in cheeses and other fermented dairy products (Singh et al. 2003; 2005; Carunchia Whetstine et al., 2005a; Drake et al., 2007). Heat processing influences the flavor potential of proteins via denaturation and the release of sulfuric compounds and the typical eggy aroma of scalded milk. Denaturation can also make proteins more susceptible to breakdown and thus influences flavor and flavor development in this manner as well. Theoretically, pure undegraded protein should be flavorless. However, dairy proteins as food ingredients are not 100% protein. Fat, ash, carbohydrate, and other components are present in various amounts and also clearly influence the final flavor and flavor stability of dairy proteins.\n\nAs with all foods, flavor plays a large role in product acceptance and success. Dried ingredients certainly affect the quality of the final product (Caudle et al., 2005), and the sensory properties of these valuable ingredients should not be overlooked. Dried dairy proteins should ideally be bland or mild and dairy-like in flavor. Recent research has demonstrated that dairy proteins are not flavorless and display a wide array of flavor variability. Understanding and documenting the flavor of these proteins is the key to strategic research and marketing. This chapter addresses and reviews current research on the sensory properties of dairy proteins.\n\n## Sensory analysis\n\nSensory analysis is a scientific discipline that encompasses the depth and breadth of all properties of a food that are perceived by the human senses (Drake, 2007). As such, sensory properties are crucial for product success. Dairy foods continue to enjoy a positive flavor perception by the consumer (Drake & Gerard, 2003; Russell et al., 2006) and this competitive edge should ideally be maximized. This means that a complete understanding of flavor, flavor variability, sources of flavors, and consumer perception is mandatory.\n\nA wide array of sensory tests is available to objectively or subjectively measure the sensory properties of foods. These tests and their specific application to dairy products are covered in several textbooks and review articles (Lawless & Heymann, 1999; Singh et al., 2003; Drake, 2004; 2007; Meilgaard et al., 2007). Two basic categories exist: analytical tests and affective tests. Several types of tests exist within each category, and selection of the specific and appropriate test is dependent on the specific objective in mind. Analytical sensory tests are a group of sensory tests that are objective in nature and use trained or screened panelists. Some examples include descriptive analysis, discrimination tests, and threshold tests. Affective tests are subjective tests and comprise tests that use consumers in qualitative or quantitative measurements.\n\n## Whey proteins\n\nWhey proteins are recovered from membrane processing and concentration of the liquid whey stream resulting from cheesemaking. Thus, one source of flavor and flavor variability of whey proteins is the flavor of the liquid whey source. The flavor of liquid whey varies, not surprisingly, with the type of cheese (Tomaino et al., 2004; Gallardo Escamilla et al., 2005). The flavor (sensory perception and volatile components) of fresh liquid whey from thermophilic starters (pasta filata cheeses) differs from that from mesophilic starter cultures (Cheddar cheese) (Liaw et al., 2011). The flavor of whey from direct acid-set curd will deviate further (Table 16.1) (Campbell et al., 2011a,b). The addition of enzymes such as lipases will increase the free fatty acid content of the whey, and this will also influence flavor in the form of rancid, waxy, and\/or animal flavors, depending on the lipase and the milk source.\n\nTable 16.1\n\nFlavor Profiles of Fluid Whey Obtained from Mozzarella Cheese, Cheddar Cheese, or Acid Casein Manufacturea\n\nAttributeb | Cheddar | Mozzarella | Acid casein  \n---|---|---|---  \nHeated milkc | 41a | 20c | 27b  \nCaramelized milk | 21a | 4b | 4b  \nNatural yogurt | 4c | 29a | 5c  \nStale | 9b | 9b | 22a  \nRancid | 16c | 24b | 38a  \nOaty | 8a | 1b | 7a  \nDirty | 2b | 7b | 35a  \nAcid | 5b | 39a | 36a  \nSweet | 37a | 8b | 12b  \nBitter | 2b | 14a | 7ab  \nSalty | 6c | 12b | 18a\n\na Adapted from Gallardo Escamilla et al. (2005).\n\nb Attributes were scored on a scale from 0 to 100.\n\nc Means in a row followed by different letters are different (p < 0.05).\n\nWithin a single type of cheese, the flavor of the whey will vary depending on starter culture rotation and\/or other variables in the cheesemaking process. Carunchia Whetstine et al. (2003) documented tremendous variability in flavor and volatile compound profiles within and between two Cheddar cheese facilities with starter culture rotation. These results were further confirmed by Karagul-Yuceer et al. (2003a). Free fatty acid profiles and proteolysis were also distinct. Tomaino et al. (2004) documented that cold storage of liquid pasteurized whey increased lipid oxidation products and resulted in cardboard flavors. They speculated that lactic starter culture enzymes accelerated these storage-induced changes because the concentrations of lipid oxidation products were higher in fermented whey than in whey from direct acid coagulation. Further, differences in lipid oxidation products were observed in fresh whey from three different single mesophilic starter strains. Campbell et al. (2011a, b) confirmed the role of the starter culture in lipid oxidation as well as differences in oxidative contributions between strains of starter cultures. Liaw et al. (2011) also documented consistent differences in flavor initially and following storage between mesophilic starter cultures and thermophilic starter cultures, concurrent with distinct volatile compound profiles. Clearly, the initial raw product stream in whey protein processing displays tremendous flavor, flavor variability, and flavor precursors.\n\nIt is not unexpected then that finished dried protein concentrates and isolates also display flavor variability. Liquid whey is subjected to a host of processing techniques to concentrate and separate the whey protein. Pasteurization, membrane filtration, concentration, and spray drying are all steps that can induce the formation of flavor compounds. Although there is a general process of whey protein production, each facility is distinct, with facility-specific storage parameters and\/or time\/temperature profiles that further contribute variability to the finished product. In the case of whey protein from colored Cheddar cheese, a bleaching process with hydrogen or benzoyl peroxide is also involved. The process of oxidizing the whey stream to decolorize it will result in a host of possible flavors and flavor precursors. Recent published work has highlighted the impact of these specific processing steps (cheesemaking procedure, liquid storage, dry storage, bleaching, solids) on final whey protein flavor (Whitson et al., 2011; Wright et al., 2009; Croissant et al., 2009; Evans et al. 2009; 2010; Jervis et al., 2012; Campbell et al., 2013).\n\nRecently, application of defined sensory analysis in combination with instrumental volatile analysis has shed light on the sources of many dairy flavors and will ultimately aid in identifying methods to control flavor. This approach recently led to the identification of a method to enhance the nutty flavor in Cheddar cheese (Avsar et al., 2004; Carunchia Whetstine et al., 2006) and the specific identification of a cabbage off-flavor in whey protein isolate (Wright et al., 2006). The reader is referred to three recent reviews on the application of these techniques to control the flavor in dairy products (Singh et al., 2003; Drake, 2004; Drake et al., 2006). A host of defined flavors in whey proteins have been documented (Drake et al., 2003; Carunchia Whetstine et al., 2005b; Drake, 2006; Russell et al., 2006; Wright et al., 2006) (Table 16.2), and volatile compound flavor variability has also been documented (Morr & Ha, 1991; Mills, 1993; Quach et al., 1999; Carunchia Whetstine et al., 2005b; Wright et al., 2006). The many processing variables listed above undoubtedly contribute to these differences in flavor among fresh products. Figure 16.1 demonstrates the flavor variability in fresh (<1 month old) product collected from different manufacturers. Products 4, 5, 6, 8, 9, and 10 were manufactured from Mozzarella or white Cheddar cheese whey, whereas products 1, 2, 3, and 7 were manufactured from colored Cheddar cheese whey. These differences in flavor and volatile compounds also suggest that some flavors and volatile components are specifically formed from whey protein manufacturing bleaching processes. Certainly this area of research should be investigated in ongoing efforts to minimize whey protein flavor. The flavor intensities of many whey proteins are comparable with those of soy proteins (Russell et al., 2006); this is a crucial issue for competitive global marketing.\n\nTable 16.2\n\nFlavors Reported in Whey Proteins [Whey Protein Concentrate (WPC80) and Whey Protein Isolate (WPI)] by Sensory Analysisa\n\nTerm | Definition | Reference | Example\/Preparation  \n---|---|---|---  \nOverall aroma intensity | The overall orthonasal aroma impact |  | Evaluated as the lid is removed from the cupped sample  \nFlavors, tastes, feeling factors (evaluated in the mouth) |  |  |   \nSweet aromatic | Sweet aroma associated with dairy products |  | Vanilla cake mix or 20 ppm vanillin in milk  \nCooked\/milky | Aromatics associated with | cooked milk | Heating skim milk to 85 \u00b0C for 30 min  \nDoughy\/fatty\/  \npasta | Aromatics associated with canned biscuit dough and cooked pasta | (Z)-4-heptenal | 1 ppm (Z)-4-heptenal in water from canned biscuit dough, or cooked pasta water  \nFatty\/frying oil | Aromatics associated with old frying oil and lipid oxidation products | 2,4-decadienal | Old (stored) vegetable oil  \nMetallic\/meat serum | Aromatics associated with metals or with juices of raw or rare beef | Aromatics of fresh raw beef steak or ground beef or juices from seared beef steak |   \nCucumber | Aromatics associated with freshly sliced cucumber | (E)-2-nonenal | 1 ppm (E)-2-nonenal or freshly sliced cucumbers  \nBrothy | Aromatics associated with broth or boiled potatoes | Methional | 1 ppm methional in water or boiled potatoes  \nCabbage | Aromatics associated with medium-chain fatty acids and soaps | Dimethyl trisulfide | Boiled fresh cut cabbage, 10 ppb dimethyl trisulfide on filter paper in sniff jar  \nCardboard\/ wet paper | Aromatics associated with cardboard | Cardboard paper | Brown cardboard or brown paper bag cut into strips and soaked in water  \nAnimal\/wet dog | Aromatics associated with wet dog hair | Knox gelatin | One bag of gelatin (28 g) dissolved in two cups of distilled water  \nPasta water | Aromatics associated with water after pasta has been boiled in it |  | Pasta boiled in water for 30 min  \nSoapy | Aromatics associated with soap | Lauric acid | 1 ppm lauric acid or shaved bar soap  \nBitter | Basic taste associated with bitterness | Caffeine | Caffeine, 0.5% in waterb  \nAstringency | Drying tongue sensation | Alum | Alum, 1% in waterb\n\na Adapted from Drake et al. (2003), Karagul-Yuceer et al. (2003a), Carunchia Whetstine et al. (2005b), and Wright et al. (2006).\n\nb From Meilgaard et al. (2007).\n\nFigure 16.1 Principal component biplot of descriptive sensory analysis of WPC80. The WPC80s are represented by numbers. Underlined samples were evaluated by consumers.\n\nStorage of whey proteins is another source of flavor variability. The purported shelf life of whey protein concentrate (WPC80) and whey protein isolate (WPI) varies from 12 to 24 months, depending on the supplier. Volatile compound work on WPC80 subjected to accelerated storage conditions has demonstrated key volatile component changes with storage (Javidipour & Qian, 2007). Wright et al. (2009) went on to apply sensory and volatile compound analysis to demonstrate changes in flavor with storage time and instantization. Products are often agglomerated to enhance their functional properties. The agglomeration process (rewet or single pass) can include addition of lecithin to further increase wettability (instantization). Both of these processes (agglomeration and agglomeration with lecithin) may impact flavor and decrease shelf life. Figure 16.2 demonstrates the flavor changes with storage time of nonagglomerated, agglomerated, and instantized (agglomerated with lecithin) WPC80 from a single supplier, as documented by a trained sensory panel. The results suggest that agglomeration, especially agglomeration with lecithin, affects the storage stability of WPC80, with more rapid development of fatty, cucumber, and lipid oxidation types of off-flavors.\n\nFigure 16.2 Flavor changes during storage for 12 months at 21 \u00b0C of nonagglomerated, agglomerated, and instantized (agglomerated with lecithin) WPC80s. C, control nonagglomerated product; A, steam-agglomerated product; AL, product agglomerated with lecithin.\n\nSamples of agglomerated and nonagglomerated WPI and WPC80 were collected from suppliers whose products were previously noted to develop a cucumber off-flavor. The samples were stored at 21 \u00b0C and were monitored by descriptive sensory analysis (rehydrated to 10% solids w\/w) and by instrumental volatile analysis [headspace solid phase microextraction gas chromatography\u2013olfactometry (HS-SPME GC\u2013O) with gas chromatography\u2013mass spectrometry (GC\u2013MS)]. Samples for volatile analysis were prepared as previously described (Wright et al., 2006). Briefly, 20 g of each reconstituted whey protein, a stirring bar, and 1 g of NaCl were placed in a 40 mL amber glass SPME vial and sealed air tight with a Teflon\u2122-sided silicon septum (PTFE\/silicon) and a plastic cap (Supelco, Bellefonte, Pennsylvania, USA). Samples were heated to 40 \u00b0C and stirred for 30 min before the SPME fiber (three phase: 2 cm \u2013 50\/30 \u03bcm DVB\/Carboxed\u2122\/PDMS Stable Flex, Supelco) was exposed in the headspace at a depth of 3.8 cm for an additional 30 min prior to injection on to the gas chromatograph. The fiber was desorbed at 250 \u00b0C for 5 min in the injection port fitted with an SPME inlet at a depth of 7.6 cm. GC\u2013O analysis was performed using an HP 5890 series II gas chromatograph equipped with a flame ionization detector (FID), a sniffing port, and a splitless injector. For GC\u2013MS, samples were prepared analogously prior to injection on to the GC\u2013MS system by a CTC Analytics CombiPal Autosampler (Zwingen, Switzerland). The fiber was desorbed at 250 \u00b0C for 5 min in the injection port fitted with an SPME inlet at a depth of 50 mm. GC\u2013MS analysis was performed using an Agilent 6890N gas chromatograph with a 5973 inert MSD with a DB-5ms (20 m \u00d7 0.25 mm internal diameter x 0.25 \u03bcm film thickness) column. Each sample was analyzed in triplicate.\n\nThe sensory profiles confirm that agglomerated samples, particularly those with lecithin, are more prone to cucumber off-flavor (Figs. 16.3 and 16.4). A number of compounds with cucumber or fatty aromas were recorded by GC\u2013O in samples with and without cucumber flavor (Table 16.3). Unfortunately, these compounds were at or below MS detection limits by the extraction technique used. The aroma of a compound when it is isolated is not necessarily the same aroma or flavor that compound elicits when it is in a food matrix, which means that sensory analysis of the compound in the food matrix is recommended (Drake & Civille, 2003). The character of an aroma can also change with compound concentration (Drake & Civille, 2003). However, when suspect compounds were placed into WPC80 without cucumber flavor within their reported threshold range and were presented to trained panelists (n = 8), many of them elicited cucumber flavors (Table 16.4), suggesting that one or a combination of these compounds are responsible for this off-flavor that develops during the storage of whey proteins. In agreement with the previous example (Fig. 16.2), these compounds are also lipid oxidation compounds, again indicating that lipid oxidation is a major source of off-flavor development in these protein products.\n\nFigure 16.3 Flavor changes during storage for 18 months at 21 \u00b0C of nonagglomerated, agglomerated, and instantized (agglomerated with lecithin) WPC80s. L, lecithinated; C, control; S, steam agglomerated. Number indicates months of storage.\n\nFigure 16.4 Flavor changes during storage for 18 months at 21 \u00b0C of nonagglomerated, agglomerated, and instantized (agglomerated with lecithin) WPIs. L, lecithinated agglomerated; C, control. Number indicates months of storage.\n\nTable 16.3\n\nAroma-active 'Green' Compounds Identified by HS-SPME GC\u2013O from Stored Agglomerated and Nonagglomerated WPC80 and WPI\n\n|  | Post-peak intensityb |  |   \n---|---|---|---|---  \nCompound | Odor a | C8 | C10 | C12 | C14 | S8 | S10 | S12 | S14 | L8 | L10 | L12 | L14 | RI c | Method of ID d\n\nWPC80\n\nHexanal | Grassy\/green | 2.50 | 2.50 | 2.50 | 3.00 | 3.25 | 2.50 | 2.50 | 3.00 | 2.50 | 2.75 | 1.50 | 3.25 | 806 | RI, odor, MSe  \nE-2-nonenal | Carpet\/green | NDf | 2.00 | 1.50 | 2.25 | 3.00 | 2.00 | 3.00 | ND | ND | 2.75 | 2.00 | 3.00 | 1163 | RI, odor  \nE,Z-2,6-nonadienal | Cucumber\/herb | ND | 3.00 | 2.00 | ND | 3.50 | ND | 3.00 | ND | 2.00 | 3.00 | ND | 3.00 | 1168 | RI, odor  \nIsobutyl-methoxy-pyrazine | Green pepper | 3.00 | ND | ND | ND | 3.00 | ND | ND | ND | 3.00 | 2.50 | ND | ND | 1185 | RI, odor  \nE,Z-2,4-nonadienal | Carpet\/green | ND | ND | ND | ND | ND | ND | ND | 2.25 | 1.75 | ND | ND | ND | 1189 | RI, odor  \n6-Decenal | Cucumber\/rosy | 2.50 | ND | 2.50 | ND | 2.00 | ND | 1.50 | ND | ND | 2.00 | 1.50 | ND | 1205 | RI, odor\n\nWPI\n\n---  \n|  | Post-peak intensityb |  |   \n---|---|---|---|---  \nCompound | Odor a | C8 | C10 | C12 | C14 | L8 | L10 | L12 | L14 | RI c | Method of ID d  \nMethyl-2-butenol | Green\/hay | NDf | ND | ND | ND | 1.50 | ND | ND | ND | 777 | RI, odor  \nHexanal | Grassy\/green | ND | 2.25 | 1 | 2 | 2.00 | 2.00 | 1.5 | 2.5 | 806 | RI, odor, MSe  \nE-2-nonenal | Carpet\/green | ND | ND | ND | ND | ND | 1.75 | ND | ND | 1163 | RI, odor  \nE,Z-2,6-nonadienal | Cucumber\/herb | ND | ND | ND | ND | 1.75 | 1.75 | ND | 3.0 | 1168 | RI, odor  \nIsobutyl-methoxy-pyrazine | Green pepper | 3.25 | 2.50 | ND | ND | 5.00 | ND | ND | ND | 1185 | RI, odor  \nE,Z-2,4-nonadienal | Carpet\/green | ND | ND | ND | ND | 2.00 | ND | ND | ND | 1189 | RI, odor  \n6-Decenal | Cucumber\/rosy | 2.75 | ND | ND | ND | 3.50 | 1.50 | ND | ND | 1205 | RI, odor\n\nNote: C refers to nonagglomerated product, S to steam agglomerated product, and L to product agglomerated with added lecithin. The number following the treatment letter designation indicates storage time at 21\u00b0C in months.\n\na Odor description at the gas chromatograph sniffing port.\n\nb Mean post-peak intensities as determined by two experienced sniffers at the gas chromatograph sniffing port (van Ruth, 2001).\n\nc Retention index was calculated from GC\u2013O data on a DB-5 column.\n\nd Compounds were identified by comparison with authentic standards on the following criteria: retention index (RI) on a DB-5 column, odor property at the gas chromatograph sniffing port, and mass spectra in the electron impact mode. Positive identifications indicate that mass spectral data were compared with authentic standards.\n\ne MS = mass spectra.\n\nf ND = not detected.\n\nTable 16.4\n\nSensory Analysis of Model Whey Protein Systems for Cucumber Flavor\n\nCompound | Concentration in WPC80 Model (ppb) | Aromaa | Reported threshold (ppb)  \n---|---|---|---  \nE-2-nonenal | 0.4 | New carpet\/cucumber | 0.4  \nE-2-nonenal | 4.0 | Cucumber | -  \nE,Z-2,6-nonadienal | 0.14 | Cucumber | 0.14  \nE,Z-2,6-nonadienal | 1.4 | Cucumber\/fatty | -  \nE,Z-2,4-nonadienal | 0.158 | Fatty\/cucumber | Not reported  \nE,Z-2,4-nonadienal | 1.58 | Fatty\/cucumber | -  \nHexanal | 73.3 | Sweet chemical-like | 50  \n2-Pentyl furan | 7.97 | Cucumber\/fatty (very faint) | 6  \n2-Pentyl furan | 79.7 | Hay\/licorice | -  \n2-Ethyl 1 hexanol | 300 | Chemical\/cleaning agent | 300\n\nNote: WPC80 without discernible cucumber flavor was used as the base. WPC80 was rehydrated to 10% solids (w\/w) and then spiked with suspect compounds at their reported threshold concentration.\n\na Aroma as perceived by eight trained sensory panelists who were also experienced GC\u2013O sniffers. Panelists were provided with the list of descriptors used for GC\u2013O and were asked to select one or two descriptors that best described the aroma.\n\nRecent work has suggested that native whey proteins might provide a product with the functional and nutritional benefits of whey proteins with superior flavor properties (Evans et al. 2009; 2010; Campbell et al., 2013). Native whey proteins are simply whey proteins that are removed from fluid milk prior to the initiation of cheesemaking. In fact, serum or whey proteins can be removed from fluid milk, and the cheesemaking procedure can subsequently be initiated as normal with few or no effects on cheese yield. As the native whey proteins have not been subjected to the normal cheesemaking and whey protein processing procedures, their flavor profiles are remarkably bland and nearly free of flavor (Fig. 16.5). Significant economic challenges face the industrial scale-up of these products but, at a minimum, research with these products may reveal key information on minimizing the flavor of traditional whey proteins.\n\nFigure 16.5 Sensory profiles of rehydrated WPC34 from Cheddar whey and rehydrated native WPC34 (serum proteins).\n\nDairy products and, by association, dairy ingredients continue to enjoy a positive flavor image with consumers (Drake & Gerard, 2003; Russell et al., 2006). However, many consumers are unaware that whey proteins are dairy proteins; this is a sensory issue because consumer perception influences consumer liking. Drake (2006) found that consumers were generally less sure of their responses when asked to comment on the properties of specific protein types. The U.S. consumer was generally less informed about whey proteins and was more confident and aware of soy protein than were New Zealand consumers (Jones et al., 2007). In a follow-up study (Childs et al., 2007), focus groups with U.S. consumers confirmed that most U.S. consumers were unaware that whey proteins were dairy or milk proteins. Consumer education is a current challenge to the dairy protein industry.\n\nOne other issue pertinent to whey proteins and consumer acceptance is whether the flavor variability documented by trained panelists is detected by consumers and\/or whether it affects the quality of the finished product. Intuitively, the freshest and highest quality ingredients make the best finished product. However, research also indicates that consumers can discern differences in whey protein flavors and that these flavors carry through into ingredient applications (Drake, 2006). Figure 16.6 demonstrates this concept with protein beverages manufactured from different fresh WPC80s. The nature of the off-flavor and the ingredient application will also influence flavor carry-through (Drake, 2006). Some ingredient applications will be more tolerant than others of variability in the flavor of the ingredients. Childs et al. (2007) recently demonstrated flavor and texture\/mouthfeel differences between meal replacement beverages and bars made with whey proteins, soy proteins, or mixtures of whey and soy proteins. The ingredient applications were made using standard formulas to allow direct comparison of the influence of the different proteins. Trained panelists documented discernible flavor carry-through of whey and soy proteins in vanilla meal replacement beverages (Fig. 16.6). In contrast, no differences in flavor between bars made with whey or soy proteins were noted, although several differences in bar texture were impacted by the protein type (Figs. 16.7 and 16.8). Wright et al. (2009) and Evans et al. (2009; 2010); conducted consumer acceptance testing with whey proteins that were documented as distinct when profiled in 10% solution by trained panelists. For consumer testing, proteins were incorporated into clear acidic beverages. Consumers documented differences in flavor liking among beverages that differed only in protein source, confirming that (off) flavors from whey proteins carry through into ingredient applications.\n\nFigure 16.6 Trained panel flavor and mouthfeel profiles of vanilla meal replacement shakes made with whey protein, soy protein, or a mixture of whey protein and soy protein. * Indicates significant attributes (p <0.05). Adapted from Childs et al. (2007).\n\nFigure 16.7 Trained panel flavor profiles of peanut butter-flavored meal replacement bars made with whey protein, soy protein, or a mixture of whey protein and soy protein. No attribute differences were noted (p > 0.05). Adapted from Childs et al. (2007).\n\nFigure 16.8 Trained panel texture profiles of peanut butter-flavored meal replacement bars made with whey protein, soy protein, or a mixture of whey protein and soy protein. * Indicates significant attributes (p <0.05). Adapted from Childs et al. (2007).\n\n## Milk proteins\n\nMilk protein concentrates (MPCs) and isolates (MPIs) represent a newer category of dried dairy ingredients that are rapidly gaining in popularity. These products are manufactured by concentrating milk proteins (whey proteins and caseins) from fluid milk by membrane processing, followed by spray drying. Recent work in the primary author's laboratory has addressed the sensory properties of milk proteins across increasing protein concentration. MPCs with lower protein content (56, 70% protein dry weight) are characterized by fluid milk types of flavors: cooked\/milky, sweet aromatic, sweet taste, and cereal (Fig. 16.9). As the protein content is increased, the flavor profiles change, and MPC77, MPC80, and MPI are characterized by tortilla, brothy, cardboard, and animal flavors as well as higher astringency.\n\nFigure 16.9 Principal component biplot of trained panel flavor profiles of rehydrated MPC and MPI. Number indicates protein content.\n\nChanges in flavor with increasing protein content were also observed when the sensory properties of lower protein WPCs were compared with those of WPC80 and WPI. Increases in whey protein content also resulted in decreases in sweet aromatic and milky flavors. These changes in flavor are probably directly linked to changes in composition and different concentrations of resulting volatile components. A comparison of aroma-active volatile components isolated from WPC80\/WPI and whey powder revealed few differences (Mahajan et al., 2004; Carunchia Whetstine et al., 2005b). Differences in flavor are probably due to differences in the relative abundance of specific compounds. Similarly, volatile compound changes are evident in MPCs and MPIs as the protein content is increased. MPCs with higher final protein content have lower sulfur compound response as well as lower aldehyde levels when analyzed by HS-SPME techniques (Table 16.5). Changes appear to be due to changes in relative abundance rather than the evolution of new compounds.\n\nTable 16.5\n\nMean Relative Concentration (ppb) of Selected Volatile Components Extracted from the Headspace of Rehydrated (10% solids w\/w) Domestic and International MPCs with Various Protein Contents\n\nMPC Sample |  | Dimethyl sulfide | Propanal, 2-methyl- | Furan, 2-methyl- | Butanal, 3-methyl- | Butanoic acid, methyl ester | Hexanal | 2-Heptanone | Heptanal | Hexanoic acid, methyl ester | Pentanoic acid, 1-methyl ethyl ester | Benzaldehyde | Furan, 2-pentyl- | Octanal | 2-Nonanone | Nonanal | Octanoic acid, methyl ester | Decanal  \n---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---  \n| Protein (%)  \n1 | 56 | 0.66  \n(0.18) | 2.33  \n(0.06) | 0.22  \n(0.03) | 0.07  \n(0.10) | 0.22  \n(0.03) | 2.40  \n(0.14) | 0.33  \n(0.07) | 0.44  \n(0.13) | 0.15  \n(0.01) | 0.50  \n(0.07) | 0.32  \n(0.05) | 0.56  \n(0.02) | 0.17  \n(0.02) | 0.18  \n(0.02) | 1.12  \n(0.22) | 0.56  \n(0.05) | 0.10  \n(0.02)  \n2 | 56 | 0.45  \n(0.04) | 2.73  \n(0.42) | 0.21  \n(0.02) | 0.35  \n(0.03) | 0.43  \n(0.01) | 7.44  \n(0.13) | 0.70  \n(0.01) | 1.72  \n(0.11) | 1.11  \n(0.18) | 0.46  \n(0.01) | 1.49  \n(0.10) | 1.86  \n(0.01) | 0.36  \n(0.00) | 0.33  \n(0.01) | 1.43  \n(0.04) | 0.76  \n(0.01) | 0.13  \n(0.02)  \n3 | 56 | 0.53  \n(0.00) | 2.70  \n(0.69) | 0.15  \n(0.00) | 0.20  \n(0.02) | 0.26  \n(0.01) | 3.12  \n(0.06) | 0.74  \n(0.03) | 0.49  \n(0.02) | 0.06  \n(0.01) | 0.49  \n(0.02) | 0.61  \n(0.24) | 0.71  \n(0.00) | 0.18  \n(0.01) | 0.30  \n(0.02) | 1.00  \n(0.04) | 0.08  \n(0.00) | 0.19  \n(0.03)  \n4 | 56 | 0.26  \n(0.02) | 3.72  \n(0.09) | 0.11  \n(0.00) | 0.11  \n(0.01) | 0.18  \n(0.02) | 3.96  \n(0.08) | 0.59  \n(0.00) | 0.59  \n(0.11) | 0.07  \n(0.02) | 0.53  \n(0.01) | 0.77  \n(0.05) | 0.80  \n(0.01) | 0.26  \n(0.01) | 0.24  \n(0.02) | 1.07  \n(0.03) | 0.09  \n(0.01) | 0.18  \n(0.02)  \n5 | 70 | 0.23  \n(0.01) | 2.56  \n(0.44) | 0.11  \n(0.01) | 0.00  \n(0.00) | 0.24  \n(0.02) | 2.25  \n(0.04) | 0.29  \n(0.04) | 0.40  \n(0.04) | 0.04  \n(0.01) | 0.45  \n(0.02) | 0.95  \n(0.01) | 0.43  \n(0.01) | 0.15  \n(0.00) | 0.17  \n(0.00) | 1.29  \n(0.02) | 0.17  \n(0.01) | 0.10  \n(0.01)  \n6 | 70 | 0.39  \n(0.05) | 2.50  \n(0.16) | 0.08  \n(0.01) | 0.00  \n(0.00) | 0.31  \n(0.05) | 0.27  \n(0.04) | 0.36  \n(0.01) | 0.05  \n(0.01) | 0.12  \n(0.03) | 0.52  \n(0.01) | 0.26  \n(0.01) | 0.41  \n(0.03) | 0.08  \n(0.00) | 0.30  \n(0.01) | 0.18  \n(0.02) | 0.38  \n(0.01) | 0.16  \n(0.01)  \n7 | 70 | 0.18  \n(0.02) | 2.60  \n(0.44) | 0.09  \n(0.01) | 0.14  \n(0.06) | 0.29  \n(0.07) | 3.27  \n(0.42) | 0.39  \n(0.07) | 0.44  \n(0.04) | 0.30  \n(0.06) | 0.53  \n(0.07) | 1.03  \n(0.12) | 0.94  \n(0.06) | 0.20  \n(0.02) | 0.22  \n(0.02) | 1.37  \n(0.20) | 1.04  \n(0.03) | 0.15  \n(0.01)  \n8 | 70 | 0.34  \n(0.07) | 4.13  \n(0.45) | 0.06  \n(0.02) | 0.07  \n(0.01) | 0.30  \n(0.13) | 2.34  \n(0.31) | 0.20  \n(0.01) | 0.58  \n(0.19) | 0.15  \n(0.01) | 0.52  \n(0.00) | 0.23  \n(0.00) | 0.20  \n(0.03) | 0.16  \n(0.06) | 0.16  \n(0.02) | 1.28  \n(0.21) | 0.20  \n(0.01) | 0.18  \n(0.02)  \n9 | 70 | 0.00  \n(0.00) | 2.69  \n(0.04) | 0.29  \n(0.01) | 0.14  \n(0.01) | 0.28  \n(0.08) | 0.94  \n(0.03) | 0.15  \n(0.00) | 0.94  \n(0.05) | 0.03  \n(0.00) | 0.58  \n(0.02) | 1.61  \n(0.13) | 0.09  \n(0.03) | 0.18  \n(0.05) | 0.10  \n(0.03) | 0.80  \n(0.01) | 0.01  \n(0.01) | 0.07  \n(0.01)  \n10 | 70 | 0.11  \n(0.02) | 3.08  \n(0.40) | 0.04  \n(0.01) | 0.08  \n(0.01) | 0.84  \n(0.24) | 0.82  \n(0.21) | 0.39  \n(0.04) | 0.21  \n(0.02) | 1.21  \n(0.04) | 0.52  \n(0.01) | 0.19  \n(0.02) | 0.12  \n(0.04) | 0.06  \n(0.00) | 0.16  \n(0.01) | 0.17  \n(0.01) | 1.41  \n(0.06) | 0.06  \n(0.00)  \n11 | 80 | 0.01  \n(0.02) | 1.62  \n(1.00) | 0.01  \n(0.01) | 0.17  \n(0.09) | 0.15  \n(0.14) | 2.97  \n(0.29) | 0.72  \n(0.04) | 0.46  \n(0.08) | 0.04  \n(0.00) | 0.44  \n(0.05) | 2.08  \n(0.20) | 1.75  \n(0.08) | 0.17  \n(0.03) | 0.75  \n(0.04) | 0.92  \n(0.08) | 0.10  \n(0.00) | 0.24  \n(0.03)  \n12 | 80 | 0.00  \n(0.00) | 4.20  \n(0.30) | 0.02  \n(0.01) | 0.03  \n(0.05) | 0.19  \n(0.00) | 1.92  \n(0.19) | 0.30  \n(0.02) | 0.26  \n(0.13) | 0.03  \n(0.00) | 0.45  \n(0.04) | 1.83  \n(0.13) | 0.34  \n(0.04) | 0.11  \n(0.05) | 0.12  \n(0.01) | 0.33  \n(0.05) | 0.00  \n(0.00) | 0.13  \n(0.02)  \n13 | 80 | 0.00  \n(0.00) | 0.00  \n(0.00) | 0.03  \n(0.04) | 0.04  \n(0.06) | 0.14  \n(0.20) | 0.49  \n(0.69) | 0.82  \n(0.74) | 0.09  \n(0.13) | 0.19  \n(0.27) | 0.26  \n(0.37) | 0.00  \n(0.00) | 0.00  \n(0.00) | 0.00  \n(0.00) | 0.00  \n(0.00) | 0.00  \n(0.00) | 1.69  \n(1.61) | 2.39  \n(3.28)  \n14 | 80 | 0.06  \n(0.00) | 2.35  \n(0.04) | 0.02  \n(0.02) | 0.00  \n(0.00) | 0.20  \n(0.00) | 0.47  \n(0.00) | 0.21  \n(0.04) | 0.32  \n(0.05) | 0.03  \n(0.00) | 0.49  \n(0.06) | 0.46  \n(0.04) | 0.03  \n(0.01) | 0.04  \n(0.02) | 0.09  \n(0.00) | 0.32  \n(0.05) | 0.08  \n(0.02) | 0.08  \n(0.00)  \n56 | 0.48 | 2.87 | 0.17 | 0.18 | 0.27 | 4.23 | 0.59 | 0.81 | 0.35 | 0.50 | 0.80 | 0.98 | 0.24 | 0.26 | 1.16 | 0.37 | 0.15  \n70 | 0.21 | 2.93 | 0.11 | 0.07 | 0.38 | 1.65 | 0.30 | 0.44 | 0.31 | 0.52 | 0.71 | 0.37 | 0.14 | 0.19 | 0.85 | 0.54 | 0.12  \n80 | 0.02 | 2.04 | 0.02 | 0.06 | 0.17 | 1.46 | 0.51 | 0.28 | 0.07 | 0.41 | 1.09 | 0.53 | 0.08 | 0.24 | 0.39 | 0.47 | 0.71\n\nNote: All concentrations given as mean relative concentration and (standard deviation)\n\nNote: MPC volatile compound mean relative concentration (ppb); all concentrations given as mean relative concentration\n\n## Caseins and hydrolysates\n\nCaseins represent the primary protein constituent of milk; whey or serum proteins are the other fraction. Just as whey proteins comprise a large group of functional ingredients, so do caseins and caseinates. Caseins are traditionally produced by acid or rennet precipitation of casein, followed by spray or roller drying. Caseinate or soluble casein is produced when casein curd (usually acid precipitated) is treated with alkali at pH 6\u20137 and fully dissolved prior to spray drying (O'Connell & Flynn, 2007). Potassium, sodium, and calcium are commonly used counter-ions. Caseins display a unique set of functional properties, including solubility and heat stability, and are thus used for a host of ingredient applications. Caseins have relied on functionality for their success because a host of relatively intense and unpleasant flavors, including sulfur, animal, tortilla, musty, cardboard, burnt feathers, glue, and bitter taste, have been associated with them (Ramshaw & Dunstone, 1969; Walker & Manning, 1976; Drake et al., 2003; Karagul-Yuceer et al., 2003b) (Fig. 16.10). Micellar casein can be manufactured by membrane fractionation and spray drying of fluid milk and may represent a blander option. The net result is a more mildly flavored product that still displays some of the previously reported flavors.\n\nFigure 16.10 Trained panel flavor profiles of rehydrated caseins [10% solids (w\/w)]. Caseins 1 and 2 are rennet caseins, caseins 3 and 5 are acid caseins, and casein 4 is a sodium caseinate.\n\nDairy protein hydrolysates are another promising category of protein-derived ingredients with valuable functional and nutritional properties (Nnanna & Wu, 2007). Hydrolysis improves digestibility, and hydrolysates are used widely in infant formulas. Recent research has demonstrated that peptides with specific bioactive properties can also be generated, and certainly an array of functional properties such as solubility and heat stability can be altered via hydrolysis. Whey protein hydrolysates are commonly added to meal replacement bars to inhibit bar hardening with storage time (Childs et al., 2007). These products can be manufactured from casein, whey protein, or milk protein by enzymatic hydrolysis and are classified based on their degree of protein hydrolysis (molecular weight). Flavor is a significant challenge to increased usage of these products, particularly in beverages. Brothy and free fatty acid flavors and bitter tastes are distinct (Fig. 16.11), and intensities can vary with the degree of hydrolysis, the specific processing steps, the enzyme used, and the protein source (Leksrisompong et al., 2010). Because of their intense aromas and flavors, these products should be rehydrated to a lower solids concentration prior to sensory analysis (e.g., 5% w\/w compared with 10% w\/w for all other dairy proteins).\n\nFigure 16.11 Trained panel flavor profiles of commercial rehydrated whey protein hydrolysates (5% solids w\/w). pH 1 has a higher degree of hydrolysis than pH 2\u2013pH 6. pH 2\u2013pH 6 represent different enzymatic digestions.\n\n## Flavor binding\n\nAlthough somewhat beyond the scope of this chapter, it is important to note that, in addition to displaying and contributing flavors, dairy proteins can interact and bind with desirable flavors in foods and influence flavor in this manner as well. An excellent review on this subject has been published (Kuhn et al., 2006). Most of the research in this arena has been conducted with instrumental analysis (e.g., headspace analysis and calculation of binding constants), and very little research to relate these results directly back to sensory perception has been attempted. Future research should address this issue.\n\n## Conclusions\n\nApplications for dairy proteins continue to increase, and flavor will remain a crucial aspect. An abundance of research on the functional properties of dairy proteins exists, but there is still a relative dearth of information on the flavor of dairy proteins. Flavor sources, flavor formation during processing, and flavor carry-through and stability in ingredient applications are key areas for future research.\n\nThe flavor of dairy proteins and their flavor performance in ingredient applications will ultimately influence their widespread usage and competitiveness with other protein sources. Published research has only recently begun to reflect and emphasize the importance of this issue. The positive flavor image of dairy foods, combined with the numerous functional and nutritional benefits of dairy proteins, provides a powerful marketing juggernaut for these products, but specific flavor properties and the flavor variability of these proteins should not be overlooked in ongoing research.\n\n## Acknowledgment\n\nThe authors gratefully acknowledge Dairy Management, Inc. and the California Dairy Research Foundation for providing financial support.\n\n# References\n\nAvsar YK ,  Karagul-Yuceer Y ,  Drake MA ,  Singh T ,  Yoon Y ,  Cadwallader KR .  Characterization of nutty flavor in Cheddar cheese .  _Journal of Dairy Science_.  2004 ;87 : 1999 \u2013 2010 .\n\nCampbell RE ,  Miracle RE ,  Drake MA .  The effect of starter culture and annatto on the flavor and functionality of whey protein concentrate .  _Journal of Dairy Science_.  2011 ;94 : 1185 \u2013 1193 .\n\nCampbell RE ,  Miracle RE ,  Gerard PD ,  Drake MA .  Effects of starter culture and storage on the flavor of liquid whey .  _Journal of Food Science_.  2011 ;76 : S354 \u2013 361 .\n\nCampbell RE ,  Adams S ,  Drake MA ,  Barbano DM .  Effect of bleaching permeate from microfiltered skim milk on 80% serum protein concentrate .  _Journal of Dairy Science_.  2013 ;96 : 1387 \u2013 1400 .\n\nCarunchia Whetstine ME ,  Parker JD ,  Drake MA ,  Larick DK .  Determining flavor and flavor variability in commercially produced liquid Cheddar whey .  _Journal of Dairy Science_.  2003 ;86 : 439 \u2013 448 .\n\nCarunchia Whetstine ME ,  Cadwallader KR ,  Drake MA .  Characterization of aroma compounds responsible for the rosy\/floral flavor in Cheddar cheese .  _Journal of Agricultural and Food Chemistry_.  2005 ;53 : 3126 \u2013 3132 .\n\nCarunchia Whetstine ME ,  Croissant AE ,  Drake MA .  Characterization of WPC80 and WPI flavor .  _Journal of Dairy Science_.  2005 ;88 : 3826 \u2013 3829 .\n\nCarunchia Whetstine ME ,  Drake MA ,  Broadbent JR ,  McMahon DJ .  Enhanced nutty flavor formation in Cheddar cheese made with a \"malty\" Lactococcus lactis adjunct culture .  _Journal of Dairy Science_.  2006 ;89 : 3277 \u2013 3284 .\n\nCaudle AD ,  Yoon Y ,  Drake MA .  Influence of flavor variability in skim milk powder on consumer acceptability of ingredient applications .  _Journal of Food Science_.  2005 ;70 : S427 \u2013 S431 .\n\nChilds JL ,  Yates MD ,  Drake MA .  Sensory properties of meal replacement bars and beverages made from whey or soy proteins .  _Journal of Food Science_.  2007 ;72 : S425 \u2013 S434 .\n\nCroissant AE ,  Kang EJ ,  Campbell RE ,  Bastian E ,  Drake MA .  Impact of bleaching agent on the flavor and flavor chemistry of whey proteins .  _Journal of Dairy Science_.  2009 ;92 : 5917 \u2013 5927 .\n\nDrake MA .  Defining dairy flavors .  _Journal of Dairy Science_.  2004 ;87 : 777 \u2013 784 .\n\nDrake MA .  Flavor and flavor carry-through of whey proteins in beverages .  _The Wonders of Whey... Catch the Power. Proceedings of the 4th International Whey Conference_ .  Elmhurst, IL :  American Dairy Products Institute ;  2006 : 292 \u2013 300 .\n\nDrake MA .  Sensory analysis of dairy foods .  _Journal of Dairy Science_.  2007 ;90 : 4925 \u2013 4937 .\n\nDrake MA ,  Civille GV .  Flavor lexicons .  _Comprehensive Reviews in Food Science and Food Safety_.  2003 ;2 : 33 \u2013 40 .\n\nDrake MA ,  Gerard PD .  Consumer attitudes and acceptability of soy-fortified yogurts .  _Journal of Food Science_.  2003 ;68 : 1118 \u2013 1122 .\n\nDrake MA ,  Karagul-Yuceer Y ,  Cadwallader KR ,  Civille GV ,  Tong PS .  Determination of the sensory attributes of dried milk powders and dairy ingredients .  _Journal of Sensory Studies_.  2003 ;18 : 199 \u2013 216 .\n\nDrake MA ,  Miracle RE ,  Caudle AD ,  Cadwallader KR .  Relating sensory and instrumental analyses .  In:  Marsili R ,  ed.  _Sensory-directed Flavor Analysis_ .  Boca Raton, FL :  CRC Press, Taylor and Francis ;  2006 : 23 \u2013 55 .\n\nDrake SL ,  Carunchia Whetstine ME ,  Drake MA ,  Courtney P ,  Fligner K ,  Jenkins J ,  Pruitt C .  Sources of umami taste in Cheddar and Swiss cheeses .  _Journal of Food Science_.  2007 ;72 : S360 \u2013 S366 .\n\nEvans J ,  Zulewska J ,  Newbold M ,  Drake MA ,  Barbano DM .  Comparison of composition, sensory and volatile components of 34% whey protein and milk serum protein concentrates .  _Journal of Dairy Science_.  2009 ;92 : 4773 \u2013 4791 .\n\nEvans JP ,  Zulewska J ,  Newbold M ,  Drake MA ,  Barbano DM .  Comparison of composition and sensory properties of 80% whey protein and milk serum protein concentrates .  _Journal of Dairy Science_.  2010 ;93 : 1824 \u2013 1843 .\n\nFoegeding EA ,  Davis JP ,  Doucet D ,  McGuffey M .  Advances in modifying and understanding whey protein functionally .  _Trends in Food Science and Technology_.  2002 ;13 : 151 \u2013 159 .\n\nGallardo Escamilla FJ ,  Kelly AL ,  Delahunty CM .  Sensory characteristics and related volatile flavor compound profiles of different types of whey .  _Journal of Dairy Science_.  2005 ;88 : 2689 \u2013 2699 .\n\nJavidipour I ,  Qian M .  Volatile component change in whey protein concentrate during storage investigated by headspace solid-phase microextraction gas chromatography .  _Dairy Science and Technology_.  2008 ;88 : 95 \u2013 104 .\n\nJervis SM ,  Campbell RE ,  Wojciechowski K ,  Drake MA ,  Barbano DM .  Impact of bleaching whey on sensory and functional properties of 80% whey protein concentrate .  _Journal of Dairy Science_.  2012 ;95 : 2828 \u2013 2862 .\n\nJones VS ,  Drake MA ,  Harding R ,  Kuhn-Sherlock B .  Consumer perception of soy and dairy products: A cross-cultural study .  _Journal of Sensory Studies_.  2008 ;23 : 65 \u2013 79 .\n\nKaragul-Yuceer Y ,  Drake MA ,  Cadwallader KR .  Aroma active components of liquid Cheddar whey .  _Journal of Food Science_.  2003 ;68 : 1215 \u2013 1219 .\n\nKaragul-Yuceer Y ,  Vlahovich KN ,  Drake MA ,  Cadwallader KR .  Characteristic aroma components of rennet casein .  _Journal of Agricultural and Food Chemistry_.  2003 ;51 : 6797 \u2013 6801 .\n\nKuhn J ,  Considine T ,  Singh H .  Interactions of milk proteins and volatile flavor compounds: Implications in the development of protein foods .  _Journal of Food Science_.  2006 ;71 : R72 \u2013 R82 .\n\nLawless HT ,  Heymann H .  _Sensory Evaluation of Food: Principles and Practices_ .  1st ed. New York :  Chapman and Hall ;  1999 .\n\nLeksrisompong PP ,  Miracle RE ,  Drake MA .  Characterization of flavor of whey protein hydrolysates .  _Journal of Agricultural Food Chemistry_.  2010 ;58 : 6318 \u2013 6327 .\n\nLeksrisompong PP ,  Gerard PD ,  Lopetcharat K ,  Drake MA .  Bitter taste inhibiting agents for whey protein hydrolysate and whey protein hydrolysate beverages .  _Journal of Food Science_.  2012 ;77 : S282 \u2013 S287 .\n\nLiaw I ,  Miracle RE ,  Jervis SM ,  Listiyani M ,  Drake MA .  Comparison of the flavor chemistry and flavor stability of Mozzarella and Cheddar whey .  _Journal of Food Science_.  2011 ;76 : C1188 \u2013 C1194 .\n\nMahajan SS ,  Goddik L ,  Qian MC .  Aroma compounds in sweet whey powder .  _Journal of Dairy Science_.  2004 ;87 : 4057 \u2013 4063 .\n\nMeilgaard MM ,  Civille GV ,  Carr T .  _Sensory Evaluation Techniques_ .  4th ed. New York :  CRC Press ;  2007 .\n\nMiller G .  Healthy growth ahead for wellness drinks .  _Food Technology_.  2005 ;59 : 21 \u2013 26 .\n\nMills OE .  Flavour of whey protein concentrate .  _Food Flavours, Ingredients, and Composition. Proceedings of the 7th International Flavour Conference_ .  New York :  Elsevier ;  1993 : 139 \u2013 149 .\n\nMorr CV ,  Ha EYW .  Off-flavors in whey protein concentrates: A literature review .  _International Dairy Journal_.  1991 ;1 : 1 \u2013 11 .\n\nNnanna IA ,  Wu C .  Dairy protein hydrolysates .  In:  Hui YH ,  ed.  _Handbook of Food Products Manufacturing_ .  Hoboken, NJ :  John Wiley and Sons ;  2007 : 537 \u2013 556 .\n\nO'Connell JE ,  Flynn C .  The manufacture and applications of casein-derived ingredients .  In:  Hui YH ,  ed.  _Handbook of Food Products Manufacturing_ .  Hoboken, NJ :  John Wiley and Sons ;  2007 : 557 \u2013 591 .\n\nQuach ML ,  Chen XG ,  Stevenson RL .  Headspace samplings of whey protein concentrate solutions using solid phase microextraction .  _Food Research International_.  1999 ;31 : 371 \u2013 379 .\n\nRamshaw EH ,  Dunstone EA .  Volatile compounds associated with the off-flavor in stored casein .  _Journal of Dairy Research_.  1969 ;36 : 215 \u2013 223 .\n\nRussell TA ,  Drake MA ,  Gerard PD .  Sensory properties of whey and soy proteins .  _Journal of Food Science_.  2006 ;71 : S447 \u2013 S455 .\n\nSingh T ,  Drake MA ,  Cadwallader KR .  Flavor of Cheddar cheese: A chemical and sensory perspective .  _Comprehensive Reviews in Food Science and Food Safety_.  2003 ;2 : 139 \u2013 162 .\n\nSingh TK ,  Young ND ,  Drake MA ,  Cadwallader KR .  Production and sensory characterization of a bitter peptide from \u03b2 casein .  _Journal of Agricultural and Food Chemistry_.  2005 ;53 : 1185 \u2013 1189 .\n\nTomaino RM ,  Turner LG ,  Larick DL .  The effect of Lactococcus lactic starter cultures on the oxidative stability of liquid whey .  _Journal of Dairy Science_.  2004 ;87 : 300 \u2013 307 .\n\nvan Ruth S .  Methods for gas chromatography-olfactometry: A review .  _Biomolecular Engineering_.  2001 ;17 : 121 \u2013 128 .\n\nWalker NJ ,  Manning DJ .  Components of the musty off-flavor of stored dried lactic casein .  _New Zealand Journal of Dairy Science and Technology_.  1976 ;11 : 1 \u2013 9 .\n\nWhitson ME ,  Miracle RE ,  Bastian E ,  Drake MA .  Effect of liquid retentate storage on flavor of spray dried whey protein concentrate and isolate .  _Journal of Dairy Science_.  2011 ;94 : 3747 \u2013 3760 .\n\nWright JM ,  Carunchia Whetstine ME ,  Miracle RE ,  Drake MA .  Characterization of a cabbage off-flavor in whey protein isolate .  _Journal of Food Science_.  2006 ;71 : C91 \u2013 C96 .\n\nWright BJ ,  Zevchak SE ,  Wright JM ,  Drake MA .  Impact of agglomeration on flavor and flavor stability of whey proteins .  _Journal of Food Science_.  2009 ;74 : S17 \u2013 S29 . \nChapter 17\n\n# Milk Protein Gels\n\nJohn A. Lucey Department of Food Science, University of Wisconsin-Madison, USA\n\n## Abstract\n\nThe formation and properties of the main types of milk protein gels are described, that is, casein gels made with rennet or acid, heat-induced whey protein gels, and gels made by a combination of approaches. The impact of various factors on these gelation properties is discussed. Recent key advances are highlighted, including the use of high pressure, exopolysaccharides, and transglutaminase cross-linking of proteins, and new insights are given into the ubiquitous use of thermal processing to alter the texture of these gels.\n\n## Keywords\n\nmilk protein gels\n\ncasein gels\n\nheat-induced whey protein gels\n\nrennet-induced gels\n\nacid-induced milk gels\n\nmixed gels, casein micelles\n\nOutline\n\nIntroduction 494\n\nRennet-induced gels 494\n\nPrimary Phase of Rennet Coagulation 494\n\nSecondary Phase of Rennet Coagulation 495\n\nMonitoring Gelation 497\n\nRheological Properties of Rennet-induced Milk Gels 497\n\nSyneresis of Rennet-induced Milk Gels 498\n\nFactors Influencing the Texture of Rennet-induced Gels 500\n\nMilk Heat Treatment 501\n\nEnzymatic Cross-linking of Caseins 501\n\nHigh Hydrostatic Pressure 502\n\nStarch Addition 502\n\nAcid-induced milk gels 502\n\nImpact of Acid on Casein Micelles 502\n\nSome Factors Influencing the Texture of Yogurt Gels 503\n\nHomogenization and Fat Globule Surface Material 504\n\nHigh Hydrostatic Pressure 504\n\nEnzymatic Modification of Proteins 504\n\nHeat Treatment 505\n\nIncubation Temperature 508\n\nProduction of Exopolysaccharides 508\n\nWhey protein gels 509\n\nThermal Denaturation of Whey Proteins 510\n\nTypes and Properties of Whey Protein Gels 512\n\nOther Factors Influencing Properties of Whey Protein Gels 513\n\nCold Gelation of Whey Proteins 514\n\nEnzymatic Modification of Whey Protein for Gelation Purposes 515\n\nMixed Gels Made with Rennet and Acid 515\n\nConclusions 516\n\nAcknowledgment 516\n\n## Introduction\n\nGelation of the proteins in milk is the basis for the manufacture of cheese and fermented milk products. Various approaches can be used to destabilize the milk proteins, including heat (whey proteins), use of rennet enzyme (caseins), and acidification (caseins and denatured whey proteins). Combinations of these approaches can also be used to form dairy products, for example, the use of a low concentration of rennet in cottage cheese (or quarg), which is primarily a cultured product. Yogurt is a cultured product in which caseins and denatured whey proteins are responsible for the gelation properties. Milk protein gels are irreversible, in contrast to many polysaccharide gels that are thermoreversible. Milk gels are often classified as particle gels, although it is now recognized that they are not simple particle gels, as the internal structure of the casein particle plays an important role in their rheological properties (Horne 2001; ). The properties of milk protein gels have been reviewed (Green, 1980; de Kruif et al., 1995; Lucey 2002; van Vliet et al., 2004). The casein particles in rennet gels undergo rearrangement, fusion, and syneresis in the process of forming cheese curd. Thus they are inherently dynamic in nature, and the rearrangement processes involved have been studied (e.g., see the review by Dejmek & Walstra, 2004).\n\n## Rennet-induced gels\n\nCoagulation of milk by rennet probably occurred initially by accident, as warm milk was stored in sacks made from the stomachs of ruminant animals that contained some residual proteinase enzymes. Crude extracts, prepared from the fourth stomach of young calves (called rennets, which are a type of aspartic proteinase), have been used for cheesemaking for thousands of years. Pepsin is the predominant proteinase in adult mammals. Naturally produced calf chymosin (EC 3.4.23.4) may contain up to six molecular species, which have slight differences in their amino acid residues (Crabbe, 2004). Chymosin has been cloned into several genetically modified organisms to produce fermentation-derived chymosin, which is widely used in many countries around the world (Crabbe, 2004). Recently, Kappeler et al. (2006) expressed the gene for camel (Camelus dromedarius) chymosin in Aspergillus niger and produced camel chymosin by fermentation. The rennet coagulation of milk has been reviewed (Dalgleish 1987; ; Hyslop, 2003; Horne & Banks, 2004).\n\n### Primary Phase of Rennet Coagulation\n\nThe basic building blocks of rennet-induced gels are the casein micelles. Both \u03b1s\\- (\u03b1s1\\- and \u03b1s2-) and \u03b2-caseins are sensitive to precipitation by the Ca2+ in milk and are protected by association with \u03ba-casein, which is one reason for the formation of micelles. \u03ba-Casein molecules are thought to have a predominantly surface position on micelles (although some \u03ba-casein is also present in the interior of the micelle), where the hydrophilic C-terminal apparently acts as a 'hairy' layer providing steric stabilization and a barrier against association with other micelles (Walstra, 1990).\n\nThe two stages of the rennet coagulation of milk are shown in Figure 17.1. In the primary phase of rennet coagulation, the C-terminal part (residues 106\u2013169) of the \u03ba-casein molecule is hydrolyzed, and this hydrophilic peptide diffuses away from the micelle (called para-casein) into the serum phase. This macropeptide is called caseinomacropeptide (CMP) or, if it is highly glycosylated, glycomacropeptide (GMP). Most microbial coagulants, including those derived from Rhizomucor miehei, hydrolyze the same Phe105\u2013Met106 bond as chymosin; however, Cryphonectria parasitica hydrolyzes the Ser104\u2013Phe105 bond (Dr\u00f8hse & Foltmann, 1989). The proteolysis of other proteins in milk by chymosin occurs at a much slower rate (Crabbe, 2004).\n\nFigure 17.1 The two stages of the rennet coagulation of milk.\n\nThe enzymatic reaction in milk appears to obey first-order kinetics. The proteolysis of \u03ba-casein is usually described by standard Michaelis\u2013Menten kinetics, although Hyslop (2003) questioned whether this was truly appropriate. It should be noted that the primary phase and the secondary phase of clotting overlap as the aggregation begins before the enzymatic reaction is complete.\n\n### Secondary Phase of Rennet Coagulation\n\nThe stability of the casein micelles of milk is attributed to their net negative charge and to steric repulsion by the flexible macropeptide region of \u03ba-casein (the so-called hairs that extend out into the solution), calcium-induced interactions between protein molecules, hydrogen bonding, and electrostatic and hydrophobic interactions. The release of the CMP (or GMP), which diffuses away from the micelles, leads to a decrease in the zeta potential, by \u22485\u20137 mV (\u224850%), which reduces electrostatic repulsion between rennet-altered micelles. Removal of the 'hairs' results in a decrease in the hydrodynamic diameter by \u22485 nm and a loss of steric stabilization, and causes a slight minimum in the viscosity during the initial lag phase of renneting.\n\nVarious attempts have been made to model the aggregation reaction (see the review by Horne & Banks, 2004). The nature of the attractive forces during the aggregation of casein micelles is still not completely clear, although calcium bridges, van der Waals forces, and hydrophobic interactions may be involved. Destabilized micelles will aggregate only in the presence of free Ca2+. Rennet acts on casein at temperatures as low as 0 \u00b0C, but milk does not clot at temperatures below 15 \u00b0C, whereas aggregation is very rapid at high temperature (e.g., 55 \u00b0C).\n\nWhen milk is clotted under normal conditions of pH and protein content, the viscosity does not increase until the enzymatic phase is mostly complete, that is, at >60% of the (visual) rennet coagulation time. Coagulation does not occur until the enzymatic phase is at least \u224887% complete. Sandra et al. (2007) studied the rennet gelation process using diffusing wave spectroscopy, which allowed gelation to be monitored without the need for dilution. Sandra et al. (2007) suggested that partially renneted casein micelles do not begin to approach one another until the extent of breakdown of the \u03ba-casein hairs has reached about 70%; above this point, they interact increasingly strongly with an increase in the extent of proteolysis. This interaction initially restricts the diffusive motion of the particles rather than causing true aggregation. Only after more extensive removal of the protective \u03ba-casein hairs does true aggregation occur, with the appearance of a space-filling gel (as defined by rheology terms, such as having a loss tangent value <1). A micrograph of a rennet-induced skim milk gel is shown in Figure 17.2.\n\nFigure 17.2 A confocal laser scanning micrograph of a rennet gel made from skim milk. Protein is white; dark areas are water. Scale bar = 10 \u03bcm.\n\nSrinivasan and Lucey (2002) studied the impact of plasmin enzyme on the rennet coagulation of skim milk. They found that hydrolysis of \u03b1s\\- and \u03b2-caseins (as plasmin hardly degrades \u03ba-casein) accelerated the rennet coagulation time. Srinivasan and Lucey (2002) hypothesized that plasmin could have degraded non-\u03ba-casein 'hairs' present on the surface of micelles and that this could have reduced the repulsive barrier to aggregation of rennet-altered micelles such that aggregation could take place at a lower degree of \u03ba-casein hydrolysis. At low temperatures \u03b2-caseins protrude from the micelle surface due to a weakening of intramicellar hydrophobic interactions, and these proteins provide a substantial barrier to the aggregation of renneted micelles. This phenomenon could be involved in the slow rennet coagulation of milk at low temperatures (Walstra, 1990). There is also a reduction in the amount of calcium ions bound to caseins with a decrease in temperature (Walstra, 1990).\n\nCompletely hydrolyzed micelles initially form small linear chains, and these continue to aggregate to form clumps, clusters and eventually a system-spanning network that has a fractal-like appearance.\n\nLittle aggregation occurs at low temperatures (e.g., <15 \u00b0C), which is usually taken as an indication of the importance of hydrophobic interactions. It is more likely that, with decreasing temperature, the activation free energy for flocculation increases, presumably because of the presence of \u03b2-casein chains on the outside of the micelle (Walstra, 1993). There is an increase in the strength of rennet gels at lower temperatures (where hydrophobic interactions are weak), reflecting swelling of casein particles, which results in an increase in the contact area between aggregated particles and strands.\n\n### Monitoring Gelation\n\nThere have been several recent reviews of techniques to monitor milk gelation (Lucey, 2002; O'Callaghan et al., 2002; Klandar et al. 2007). The interest in monitoring gelation comes from the cheesemaker's desire to know the 'optimum' time to initiate cutting, as well as from the researcher's desire to better understand this complex process.\n\nTwo promising techniques for the study of milk gels are diffusing wave spectroscopy and ultrasonic spectroscopy (Alexander & Dalgleish, 2004; Dalgleish et al., 2006). These techniques could be used to complement existing approaches. For example, Wang et al. (2007) used both ultrasonic and (traditional) rheological methods to investigate the effects of milk pretreatment at ultra-high temperatures on the rennet gelation of a whey-protein-free casein solution. Wang et al. (2007) found that the ultrasonic velocity was able to measure the enzymatic hydrolysis and aggregation process, but was not as sensitive in detecting gel formation. In contrast, the oscillatory rheological method was not able to detect the enzymatic hydrolysis reaction, but was very suitable for characterizing the formation of a gel network.\n\n### Rheological Properties of Rennet-induced Milk Gels\n\nRennet-induced gels are viscoelastic and their rheological properties can be characterized using dynamic low-amplitude oscillatory rheology, which determines both the viscous component and the elastic component. These measurements should be performed in the linear viscoelastic range, where the deformation (strain) is proportional to the applied stress. Often, for rennet and acid gels, that means trying to operate at \u22643% strain, which can be difficult during the early stage of gel formation for many (of the popular) controlled stress rheometers (because of the very low torque resulting on the measuring geometry of the rheometer from such a weak gel). Some new techniques\/software can be used to reduce this problem in commercial rheometers (e.g., Lauger et al., 2002).\n\nParameters that can be determined include the elastic or storage modulus (G'), which is a measure of the energy stored per oscillation cycle; the viscous or loss modulus (G''), which is a measure of the energy dissipated as heat per cycle; and the loss tangent (tan \u03b4), which is the ratio of the viscous properties to the elastic properties (loss tangent = G''\/G'). The loss tangent is related to the relaxation of bonds in the gel during deformation and is a useful parameter.\n\nDuring gelation, there is a lag period before a measurable storage modulus value is obtained (this depends on the sensitivity of the rheometer to measure events close to the gelation point). The loss tangent decreases from \u226b1 to <1 at the gelation point and then attains a relatively constant value (about 0.35 for rennet gels). The dynamic moduli initially increase relatively rapidly and then, after a period of several hours, tend to plateau. In commercial practice, rennet-induced gels are cut once they have attained a certain firmness (usually assessed subjectively by the cheesemaker) or, more commonly, at a fixed time after rennet addition. The increase in the moduli after gelation probably reflects ongoing fusion of micelles, which results in an increase in the contact area between aggregated particles, and possibly the incorporation of additional particles into the gel network. Some micelles that have incomplete hydrolysis of their \u03ba-casein hairs could be trapped within the space-filling network at the point of network formation and they might later become attached to the matrix once their \u03ba-casein hairs get completely hydrolyzed. Mellema et al. (2002) reported that their analysis suggested that nearly all casein was incorporated in the rennet gel, at least very soon after network formation. Mellema et al. (2002) also considered that changes in the storage modulus and microstructure during aging could be explained in terms of (various types of) rearrangements of the gel network at various length scales.\n\nTypical plateau values for the storage modulus of rennet-induced gels (made from unconcentrated milk) range from 100 to 200 Pa. Both moduli have lower values at low frequencies, reflecting relaxation of more bonds when the time scale of the applied stress is longer. The loss tangent at low frequencies is an important indicator of rearrangements as this is approximately the same timescale as that over which rearrangement processes related to syneresis in rennet gels are estimated to occur (van Vliet et al., 1991).\n\nThe development of the complex or shear modulus as a function of time after rennet addition can be re-plotted against a reduced time t\/t g, where t g is the gelation time. Various individual renneted milk curves can be normalized against its complex or shear modulus value at a low multiple (two or three) of the t g. These various curves then collapse into a single or master curve because of the scaling behavior of the dynamics of the gel formation process (Horne 1995; ).\n\nVarious mathematical, empirical, and kinetic models have been applied to predict the development of gel firmness or shear moduli; their effectiveness in performing this function has been reviewed by Horne and Banks (2004).\n\n### Syneresis of Rennet-induced Milk Gels\n\nThe syneresis of rennet-induced gels has been reviewed (Walstra et al., 1985; Pearse & MacKinlay, 1989; Walstra 1993 van Vliet & Walstra, 1994; Dejmek & Walstra, 2004). Rennet-induced gels remain stable for several hours if left undisturbed. They rapidly synerese if disturbed by cutting or by wetting the gel surface. A rennet-induced gel may lose up to two-thirds of its volume (as whey) under quiescent conditions and more than 90% if external pressure is applied (Dejmek & Walstra, 2004). Cheesemaking can be viewed as a dehydration process, and syneresis is the crucial method by which most of the moisture is lost from curd particles. As syneresis is the main method that cheesemakers have to control cheese moisture, it is also the process that is most often manipulated; it also helps to facilitate differentiation between cheese varieties. Most of the water in milk gels is not chemically bound to proteins but rather is physically entrapped in the network structure (van Vliet & Walstra, 1994).\n\nBecause of the complexity of the syneresis process, researchers have often used thin gel slabs to monitor one-dimensional shrinkage (e.g., van Dijk & Walstra, 1986). The one-dimensional syneresis of rennet gels is related to the flow of liquid (whey) through the network (because liquid flows out of the gel concomitantly with gel shrinkage) and is governed by the equation of Darcy:\n\nv = B \u03b7 p x\n\nwhere v is the superficial flow velocity of the syneresing liquid, B is the permeability coefficient, \u03b7 is the viscosity of the liquid, p is the pressure acting on the liquid, and x is the distance over which the liquid must flow.\n\nIt is believed (e.g., Walstra, 1993) an internal (endogenous) pressure or driving force within rennet gels is responsible for the shrinkage of the gel once the initial gel is disturbed (presumably this overcomes the yield stress of the system). It has not been possible to measure this small endogenous pressure experimentally. Endogenous syneresis pressure (i.e., the pressure within the rennet gel causing the syneresis) is not constant. It increases initially as a function of time after renneting but decreases at longer times, presumably because of the fusion of para-casein micelles and a reduction in permeability of the contracting network. In practice, syneresis in curd particles occurs in three dimensions simultaneously and is much harder to study than the one-dimensional model.\n\nIn rennet-induced milk gels, the mechanism responsible for the gels' strong tendency to exhibit syneresis is related to the (extensive) rearrangements of the casein network that occur after gel formation. As acid-induced gels undergo much less rearrangement, they synerese less. The rearrangement process is accelerated, and is more extensive, at high temperatures and lower pH values (\u22655.1) (the loss tangent is also higher under these conditions). Aging of rennet-induced gels results in a coarsening (sometimes called 'microsyneresis') of the gel (i.e., as a result of more rearrangements), and an increase in the permeability and the fractal dimensionality takes place.\n\nRearrangements of casein particles into a more compact structure would increase the number of bonds and hence would decrease the total free energy of the system (Walstra, 1993). However, the casein particles are already part of the gel network, which must be deformed or broken locally to form new junctions. Breakage of the bonds in the strands requires a sufficiently low-yield stress if it is to be exceeded. In cheesemaking, conditions such as cutting, stirring, acid production, and the increase in the cooking temperature all encourage syneresis and the rearrangement processes that facilitate syneresis of the gel network. If the strands become too thick (e.g., because of a very high casein concentration), syneresis hardly occurs.\n\nOne-dimensional syneresis of rennet-induced skim milk gels was studied in gels with different thicknesses and at pH values of 6.4 and 6.0 using a laser displacement sensor (Lodaite et al., 2000). Syneresis was much faster at the lower pH, and the initial syneresis rate increased linearly with slab thickness.\n\nSeveral (mostly empirical) techniques have been used to estimate the syneresis of rennet gels, including shrinkage of gel slabs, determining the volume of whey expelled as a function of time, the dry matter content or density of curd particles, and low-resolution nuclear magnetic resonance (NMR) (Dejmek & Walstra, 2004). A recent development has been a light backscatter sensor, with a large field of view relative to curd size, for continuous online monitoring of coagulation and syneresis to help cheesemakers improve their control over the moisture content of the curd (Fagan et al., 2006).\n\n### Factors Influencing the Texture of Rennet-induced Gels\n\nMany factors influence the milk-clotting process and the consistency of rennet gels, including pH, temperature, casein content, ionic strength, enzyme concentration, calcium content, presence of homogenized fat globules, concentration of denatured whey proteins, and casein hydrolysis by proteinases such as plasmin. These factors have been reviewed many times (Dalgleish, 1987; 1993 Green & Grandison, 1993; Lomholt & Qvist, 1999; Hyslop, 2003; Horne & Banks, 2004) because of the importance of rennet gels for the cheese industry.\n\nThe effects of pH (5.19\u20136.21) and NaCl concentration (0, 1.75, and 3.5%) on the rheological and microstructural properties of rennet-induced casein gels made from ultrafiltered skim milk (19.8%, w\/w casein) were recently investigated (Karlsson et al., 2007a). Low pH and high NaCl concentration reduced the rate of development of the gel elasticity after coagulation. Strain at fracture and stress at fracture 48 h after coagulation showed maximum and minimum values at pH 5.8 and 5.29, respectively. The microstructure examined with confocal laser scanning microscopy was unaffected by the changes in pH and the concentrations of NaCl, probably because of the very high-volume fraction of caseins in this type of gel (Karlsson et al., 2007a). Rennet-induced coagulation of ultrafiltered skim milk (19.8%, w\/w casein) at pH 5.8 was studied and compared with coagulation of unconcentrated skim milk of the same pH (Karlsson et al., 2007b). At the same rennet concentration, coagulation occurred at a slower rate in ultrafiltered skim milk but started at a lower degree of \u03ba-casein hydrolysis, compared with the unconcentrated skim milk. Confocal laser scanning micrographs revealed that, during storage for up to 60 days (at 13 \u00b0C), the microstructure and the size of the protein strands of the ultrafiltered gel hardly changed, probably because of the high zero shear viscosity of the concentrated system (Karlsson et al., 2007b).\n\nPlant coagulants obtained from the flowers of Cynara sp. have been used to make rennet gels and cheeses (Esteves et al., 2001; ; ). These coagulants are less sensitive to changes in gelation temperature, they cause more casein rearrangements during gelation, and they have higher values for the storage modulus (at least during the initial stages of gelation), compared with gels made with chymosin (probably because of greater proteolysis of the caseins).\n\nThe properties of fermentation-produced camel chymosin have been compared with those of calf chymosin. Camel chymosin has a 70% higher clotting activity per mol on bovine milk than calf chymosin, but only 20% of its general proteolytic activity and, hence, has a seven-fold higher ratio of clotting to general proteolytic activity (Kappeler et al., 2006).\n\nChoi et al. (2007) demonstrated that the concentration of colloidal calcium phosphate (CCP) associated with the casein micelles had an important influence on the properties of rennet gels. Removal of some CCP from milk prior to gelation using calcium-chelating agents lowered the storage modulus of rennet gels because of the reduction in the amount of CCP cross-linking in the casein micelles. Reduction in the CCP content prior to rennet gelation resulted in gels with higher loss tangent values, indicating greater bond mobility. Choi et al. (2007) also studied the impact of preacidification of milk prior to gelation. They found that gels made at pH 6.4 had higher storage modulus values than gels made at pH 6.7, probably because of the reduction in electrostatic repulsion, whereas the CCP content only slightly decreased at this pH value. The storage modulus values were highest at pH 6.4 and decreased with decreasing pH from 6.4 to 5.4 because of the reduction in CCP cross-linking within the casein micelles (Choi et al., 2007).\n\n### Milk Heat Treatment\n\nIt is well known that severe heat treatment of milk at temperatures sufficiently high to denature the whey proteins results in an increased rennet coagulation time as well as weaker gels (Lucey, 1995). There are some reports that the interaction of denatured whey proteins with the \u03ba-casein inhibits the primary phase of rennet action on \u03ba-casein (to some extent). For example, Reddy and Kinsella (1990) reported that very high heat treatments decreased the initial velocity (V i) and GMP release. However, most studies have concluded that the secondary phase of the coagulation process is the step that is mainly inhibited by the presence of denatured whey proteins on the micelle surface. These denatured whey proteins probably sterically interfere with the (normal) aggregation of rennet-altered micelles (Lucey, 1995). Vasbinder et al. (2003) concluded that whey protein denaturation had only a small effect on rennet activity and that the release of GMP (or the formation of para-\u03ba-casein) was similar in heated and unheated milks. Anema et al. (2007) adjusted the pH of the milk prior to heat treatment, which allowed them to manipulate the distribution of denatured whey proteins and \u03ba-casein between the serum and micellar phases. They reported that the retardation in rennet gelation as a result of heat treatment was observed regardless of whether the denatured whey proteins were associated with the casein micelles or in the serum phase.\n\n### Enzymatic Cross-linking of Caseins\n\nTransglutaminase (TGase; EC 2.3.2.13) catalyzes covalent intermolecular protein cross-linking through an acyl-transfer reaction, between the \u03b3-carboxyamide group of a peptide-bound glutamine residue (acyl donor) and the primary amino group of an amine (acyl acceptor). The application of TGase in various types of dairy products has been reviewed (Jaros et al., 2006). In a system where caseins and whey proteins are available as substrates for TGase, such as milk, the caseins are preferentially cross-linked over native whey proteins (Han & Damodaran, 1996).\n\nLorenzen (2000) incubated preheated milk with TGase for various incubation times prior to rennet addition, and found that increasing TGase incubation times, as well as an increasing intensity of preheat treatment of the milk, resulted in increasing coagulation times up to the point of a complete absence of coagulation. Lorenzen (2000) attributed the reduced rennetability of preheated milk to a 'surface sealing' of the casein micelles with cross-linked \u03b2-lactoglobulin, leading to a steric inhibition of the release of the macropeptide from the surface of the casein micelle. O'Sullivan et al. (2002b) also attributed the loss of rennetability to the impact of TGase cross-linking on the primary enzymatic phase, that is, reduced rate of hydrolysis of \u03ba-casein. Huppertz and de Kruif (2007) criticized the analytical method used by O'Sullivan et al. (2002b) to study the hydrolysis reaction because they suggested that this method detects only the products of hydrolysis of non\u2013cross-linked milk; hydrolysis products of cross-linked \u03ba-casein would not be adequately detected because the macropeptide remains attached to the micelle. Huppertz and de Kruif (2007) suggested instead that TGase treatment affects mainly the secondary stage of rennet-induced coagulation. They suggested that this inhibition was due to the progressive cross-linking of the \u03ba-casein located on the surface of the casein micelles, which provided additional steric hindrance to the aggregation of renneted micelles.\n\n### High Hydrostatic Pressure\n\nHigh hydrostatic pressure influences various properties of milk, including a reduction in the size of the casein micelles, denaturation of \u03b2-lactoglobulin, and a reduction in the CCP content. Huppertz et al. (2005) studied the impact of milk heat treatment (90 \u00b0C for 10 min) and subsequent application of high-pressure treatment at pressures from 0 to 600 MPa. Heated unpressurized milk or heated milk treated for 0 min (immediate release of pressure) at 100 MPa was not coagulable by rennet. However, heated milk treated at 250\u2013600 MPa for 0\u201330 min had a rennet coagulation time equal to, or lower than, that of unheated unpressurized milk; the coagulation time decreased with increasing pressure and treatment time. The strength of the rennet-induced coagulum from heated milk treated at 250\u2013600 MPa for 30 min or 400 or 600 MPa for 0 min was considerably greater than that of the rennet-induced coagulum from unheated unpressurized milk. There was also an increase in the yield of cheese curd by \u224815%. Other studies on applying a combination of pressures and heat treatment for rennet gels have also been reported (Al-Nabulsi et al., 2012). The impact of high pressure and heat treatment on milk proteins has been reviewed (Considine et al., 2007).\n\n### Starch Addition\n\nStarch is sometimes added in cheesemaking in order to bind water and increase cheese yield. It is not a permitted ingredient in many cheese types, but where permitted they are used to replace more expensive milk proteins. Waxy corn or rice starch appear to be mostly retained in rennet curds, whereas modified tapioca starch appears to interfere with the rennet gelation process, producing very weak rennet gels (Brown et al., 2012). Losses of starch into the cheese whey would be a significant concern for whey producers due to greatly increased viscosity and increased likelihood of fouling during membrane filtration.\n\n## Acid-induced milk gels\n\n### Impact of Acid on Casein Micelles\n\nIn cultured products, such as yogurt, as the pH of milk is reduced, CCP is dissolved, and the internal casein micelle structure is altered because of the loss of CCP. Little casein dissociation occurs at the high temperatures (>40 \u00b0C) commonly used for yogurt manufacture. Aggregation of casein occurs as the isoelectric point (pH \u22484.6) is approached (Horne, 1999). Native casein micelles (in milk of normal pH) are stabilized by their negative charge and steric repulsion (Lucey & Singh, 2003). Lucey (2003) distinguished three (arbitrary) pH regions in the acidification of milk from pH 6.7 to 4.6. (a) pH from 6.7 to \u22486.0. The decrease in pH causes a reduction in the net negative charge on the casein micelles, thereby reducing electrostatic repulsion. As only a relatively small amount of CCP is dissolved above pH 6.0, the structural features of the micelles are relatively unchanged. (b) pH from \u22486.0 to \u22485.0. The decrease in pH causes a reduction in the net negative charge on the casein micelles, thereby reducing electrostatic repulsion. As the \u03ba-casein 'hairs' on the micelle surface are charged, these charged 'hairs' may shrink\/collapse as the pH decreases. The net result is a decrease in both electrostatic repulsion and steric stabilization. The CCP within the casein micelles is dissolved completely by pH \u22485.0. (c) pH \u22645.0. The net negative charge on the casein micelles declines with the approach of the isoelectric point, and there are increased +\/\u2013 charge interactions (and van der Waals forces). The reduction in electrostatic repulsion allows increased hydrophobic interactions (Horne, 1998; 2001). In unheated milk gels where acidification is the only coagulation method, gelation occurs at around pH 4.9; if acidification is performed at very high temperatures, a higher gelation pH is observed.\n\nOn acidification, casein particles aggregate as a result of (mainly) charge neutralization. With acidification, the overall charge density of the colloidal system changes and together with it the van der Waals attraction\/electrostatic repulsion balance (Mezzenga & Fischer, 2013). Acidification eventually leads to the formation of chains and clusters that are linked together to form a three-dimensional network (Kal\u00e1b et al., 1983). Acid casein gels can be formed from sodium caseinate (this ingredient is sometimes used as a yogurt stabilizer). Another approach to acid gel formation involves direct acidification of milk at a low temperature and subsequent warming. Glucono-\u03b4-lactone (GDL) is also used to acidify milk, but these acid-induced gels have different rheological and structural properties from gels produced by bacterial cultures (Lucey et al., 1998a; Renan et al., 2008).\n\nHydrophobic interactions are unlikely to play a direct role in the strength of acid gels as the storage modulus of acid gels increases with decreasing measurement temperature (Lucey, 2003). Cooling gels result in an increase in the storage modulus, probably as a result of the swelling of casein particles (caused by the weaker hydrophobic interactions) and an increase in the contact area between particles (Lucey, 2003). With increasing ionic strength, the charged groups on casein are screened, thereby weakening interactions between casein particles.\n\nMilk has been reversibly acidified by means of carbonation, injecting pressurized CO2 as the acidifying agent, in order to reduce the pH (usually done at low temperature). Neutralization is obtained by pressure release followed by degassing under vacuum. Upon CO2 treatment, the zeta potential and the size of the casein micelles were restored, although the total amount of CCP was not restored (Raouche et al., 2007). The rheological properties of acid gels (made using GDL) from CO2-treated milk were similar to those of acid gels from untreated milk (Raouche et al., 2007).\n\n### Some Factors Influencing the Texture of Yogurt Gels\n\nIt is well established that the way in which the milk is handled or prepared, including the processing conditions used in yogurt manufacture, greatly influences the gel texture, strength, and stability (Lucey & Singh, 1998; Walstra, 1998; Tamime and Robinson, 1999; Jaros & Rohm, 2003a, b). These factors include (a) fortification level and material(s) used in the mix, (b) stabilizer type and usage levels, (c) fat content and homogenization conditions, (d) milk heat treatment conditions, (e) starter culture (type, rate of acid development, and production of exopolysaccharides), (f) incubation temperature (which influences growth of starter cultures, gel aggregation, and bond strength), (g) pH at the breaking of the gel (stirred) and\/or the start of cooling (set), (h) cooling conditions (when cooling is started, rate of cooling), and (i) postmanufacture handling of the product, for example, vibration and temperature fluctuations (i.e., if the product is not maintained at \u22485 \u00b0C).\n\n### Homogenization and Fat Globule Surface Material\n\nThe fat globules in milk are surrounded by membrane proteins, and, unless homogenized, fat acts as an inert filler in milk gels. Cho et al. (1999) prepared fat globules with different surface materials and studied the effects of these surface materials on the rheological properties of acid milk gels. Gels containing fat globules stabilized by noninteracting materials ('structure breaker'), i.e., Tween and unheated whey protein concentrate (WPC) had low storage moduli compared with interacting surface materials ('structure promoter') (skim milk powder, sodium caseinate, and heated WPC).\n\nMilk for the manufacture of yogurt is normally homogenized (15\u201320 MPa) in order to increase the yogurt's consistency and to decrease whey separation during storage (Tamime & Robinson, 1999). High-pressure homogenization has a similar principle to conventional homogenization but works at significantly higher pressures (up to 400 MPa). Milk given a high-pressure (>200 MPa) treatment gave firmer yogurt gels than heat treated milk (90 \u00b0C for 90 s) and traditionally homogenized at 15 MPa (Serra et al., 2007). Presumably this effect reflects a combination of the creation of very small fat globules, whey protein denaturation, and possible modification to the CCP content (Huppertz & de Kruif, 2006; L\u00f3pez-Fandi\u00f1o, 2006). The combination of high-pressure homogenization and heat treatments has been explored to improve the properties of acid gels (Hernandez & Harte, 2008).\n\n### High Hydrostatic Pressure\n\nHigh hydrostatic-pressure treatment of milk enhances the mechanical properties of yogurt gels (Needs et al., 2000). The storage moduli of gels made from high-pressure-treated milk were considerably higher than those of gels made from heat-treated milk (85 \u00b0C for 20 min), although the yield stress and the yield strain were lower in the pressure-treated gel (Needs et al., 2000). The combined use of high thermal treatment and high hydrostatic pressure results in extensive whey protein denaturation and casein micelle disruption, respectively (Harte et al., 2003). The net effect of the combined high hydrostatic pressure and thermal treatments was an improvement in the yield stress of the yogurt and a reduction in syneresis (Harte et al., 2003). High-pressure treatment up to 600 MPa (for 20 min) improved the viscosity of stirred yogurt, which had similar rheological properties to yogurt made from milk heated at 90 \u00b0C for 30 min (Knudsen et al., 2006). The pH of milk at pressure treatment also influences acid gelation (Anema, 2010), with higher pH values producing firmer gels as a result of greater whey protein denaturation.\n\n### Enzymatic Modification of Proteins\n\nAcid-induced gelation of TGase-cross-linked casein resulted in increased gel firmness, lower permeability, finer protein networks, and improved whey drainage (Faergemand & Qvist, 1997; F\u00e6rgemand et al., 1999; Schorsch et al., 2000). Lauber et al. (2000) reported that even a very small amount of casein cross-linking, due to the action of TGase, is capable of inducing significant changes in yogurt texture (i.e., a large increase in gel strength). TGase treatment may reduce protein rearrangements during the gelation phase, thereby lessening the likelihood of (unwanted) whey separation (Ercili-Cura et al., 2013). A slightly slower acidification rate by the starter culture was observed in yogurts made from TGase-treated milk; possibly there was a reduction in availability of the low-molecular-weight peptides required by Streptococcus thermophilus as a result of the cross-linking reaction (Faergemand et al., 1999; Ozer et al. 2007). Cross-linking of caseins restored the sensory texture profile of a lower protein yogurt to be comparable with that of a higher protein yogurt, suggesting that TGase could be used instead of some of the milk solids currently used in yogurt fortification (Faergemand et al., 1999). Excessive protein cross-linking increased the gel firmness, but the yogurt became grittier than the control samples (Faergemand et al., 1999). TGase is capable of cross-linking caseins even under high pressure (Lauber et al., 2001). When TGase treatment was performed during high hydrostatic-pressure treatment, a markedly higher final storage modulus was observed in acid milk gels compared with gels with only pressure treatment or when a separate TGase treatment was performed (Anema et al., 2005). Anema et al. (2005) proposed that there is an increase in cross-linking of the whey proteins and an increase in cross-linking between the whey proteins and casein when TGase treatment is performed under high pressure. It appears that treatment of goat's milk with TGase is less effective than cow's milk, resulting in weaker acid gels (Ardelean et al., 2013).\n\n### Heat Treatment\n\nAcid gels formed from unheated milk are very weak, at least partly because the interparticle contact area is still dominated by the presence of the \u03ba-casein hairs (GMP), which have collapsed but are still present (Li & Dalgleish, 2006). The \u03ba-casein hairs are rich in hydroxylated amino acids, some of which are glycosylated, and also acidic and basic residues. Thus, the interface between the aggregating particles will tend to be highly hydrated, and attractive interactions will be partly offset by the hydrophilic tendency of the \u03ba-casein hairs (Li & Dalgleish, 2006).\n\nThere has been considerable recent research on the topic of how whey proteins influence yogurt texture. Native whey proteins in unheated milk are inert fillers in yogurt (Lucey et al., 1999). Added whey proteins alter yogurt gelation and texture as long as the mix is given a sufficiently high-heat treatment to denature the whey proteins and cause them to associate with the casein micelles (Lucey et al., 1999). Commercially, WPC is often used to increase the solids content of yogurt and to give improved viscosity and lower whey drainage. High-heat treatment causes considerable whey protein denaturation (e.g., 85 \u00b0C for 15 min results in >80% \u03b2-lactoglobulin denaturation). As a result, \u03b2-lactoglobulin becomes mostly attached to the \u03ba-casein of the casein micelles or forms soluble complexes (with serum casein), depending on the heating conditions (i.e., pH) (Lucey et al., 1998b).\n\nDenatured whey proteins (DWPs) attached to the surface of casein micelles during heating (i.e., bound DWP) are a critical factor in the increased stiffness of yogurt gels made from heated milk. DWPs cause micelles to aggregate at higher pH because of the higher isoelectric pH (\u22485.3) of the main whey protein, \u03b2-lactoglobulin, than that of caseins (Lucey et al., 1997; Guyomarc'h et al., 2003; Morand et al. 2012). An alternative view is that the DWPs associated with the micelles alter the hydrophobic interactions between heated micelles, which facilitates gelation at higher pH values (although there is greater electrostatic repulsion at higher pH) (Jean et al., 2006). More cross-linking of gels by bound DWP increases the gel strength. Soluble DWPs are not able to increase the gel stiffness of milk in which there are no bound DWP present. That is, the micelle surface does not contain any 'bound' DWP and can be created experimentally (Lucey et al., 1998b) (Fig. 17.3).\n\nFigure 17.3 Storage modulus as a function of time during the formation of acid-induced milk gels made from heated milk ( ), heated milk containing bound DWP (\u25cb), heated milk containing soluble DWP (\u25be), and unheated milk (\u25b5). Heat treatment was at 80 \u00b0C for 30 min, and acidification was at 30 \u00b0C with 1.3% GDL. Reproduced, with the permission of Cambridge University Press, from Lucey et al., 1998b.\n\nIn industrial practice, heating milk always creates some bound DWP, which allows soluble DWP to become attached to the micelles and to contribute to the gel strength. The pH at heating influences the association of DWP with casein micelles. At pH 6.5, most DWPs are associated with micelles (e.g., >70% for milk heated at 90 \u00b0C for 30 min). At higher pH (e.g., 7.0), fewer DWPs are associated with micelles as more \u03ba-casein dissociates from the micelles to interact with \u03b2-lactoglobulin during heating. The gel strength of acid gels made from milk heated at high pH is higher than that of acid gels made from milk heated at the natural pH of milk (Lucey et al., 1998b; Anema et al., 2004); this may not be valid for situations in which there is a lot of added whey protein. At high pH values, there is an increase in the concentration of CCP (additional cross-linking) in milk (McCann & Pyne, 1960), which could potentially increase the stiffness of acid gels made from high-pH milk. Increasing the pH of heat treatment of the milk from 6.5 to 7.0 should also alter protein unfolding and disulfide bond formation, involving \u03b2-lactoglobulin, as the pK value of its free thiol group is 9.35 (Kella & Kinsella, 1988a).\n\nThe creation of additional covalent disulfide bonds that involve whey protein and caseins should increase the strength of the yogurt gel. Regardless of the pH of the milk at heating, DWPs (i.e., those designated as 'soluble' and 'bound' at the pH of heat treatment) are insoluble at low pH and should associate with casein at the pH values involved in yogurt fermentation. As the pH decreases during fermentation, virtually all the residual soluble complexes become attached to caseins via the bound DWP. The rate of acidification and the gelation temperature may also influence how these complexes associate with the caseins during acidification. The extent of denaturation of the whey proteins is often determined by their loss of solubility at pH 4.6 (de Wit, 1981), so that all the DWP should precipitate as the pH approaches 4.6.\n\nThe addition of WPC to milk that was then given a high-heat treatment resulted in an increase in the pH of gelation, an increase in gel stiffness, and a reduction in fracture strain compared with gels made from heated milk without added WPC (Lucey et al., 1999). If WPC was added to heated milk and this mixture was not given any further heat treatment, the acid gels formed after acidification were weaker than those made from heated milk without WPC. This suggests that any added whey proteins must be denatured in order to reinforce the network, even when DWPs are already present in the milk. Schorsch et al. (2001) examined the effect of heating whey proteins in the presence or absence of casein micelles on the subsequent acid gelation properties of milk. The acid-induced gelation occurred at a higher pH (around pH 6.0) and in a shorter time when the whey proteins (concentration of 1 g whey protein\/kg) were denatured separately from the casein micelles than when the whey proteins were heated in the presence of the casein micelles. However, the gels formed were very weak, probably because of the formation of a weak network in which whey proteins entrapped caseins.\n\nVarious studies have shown some conflicting results about the relative importance of the soluble and bound DWP fractions to the texture of acid milk gels (Lucey et al., 1998b; Guyomarc'h et al., 2003; Anema et al. 2004). Differences in the proportions of soluble and bound DWP fractions in these studies could have contributed to these conflicting results. Guyomarc'h et al. (2003) had only a small proportion (10\u201315%) of \u03b2-lactoglobulin in the bound DWP fraction, whereas Lucey et al. (1998b) had around 80%. Guyomarc'h et al. (2003) suggested that differences in the quantitative amounts of aggregates (and the total amount of DWP) present in the systems, independently of whether or not they were soluble, could be the reason for some of the conflicting results reported by the different groups.\n\nIn gels made from heated milk, because of the high gelation pH, the gel goes through a period of solubilization of the CCP that is present within casein particles that are already part of the gel network (this event is responsible for the maximum in the loss tangent during gelation) (Lucey et al., 1997). This process loosens the interactions between caseins in the gel network, and the higher bond mobility in yogurt gels during this period has been associated with whey separation (Lucey, 2001). The rheological changes during the acid-induced gelation (with GDL) of unheated and heated milk at 30 \u00b0C are shown in Figure 17.4. Note the much shorter gelation time, the large increase in the storage modulus, and the maximum in the loss tangent (as indicated by the hatched region between the two arrows, region A) in the heated milk sample. As the low gelation pH (4.8) of the unheated milk gel occurs after most or all of the CCP is already solubilized, there is no maximum in the loss tangent in this type of gel. When acid-induced gelation of heated milk occurs rapidly at high temperature, a plateau in the storage modulus, which corresponds to the region where there is a maximum in the loss tangent, can be observed (Horne, 2001).\n\nFigure 17.4 Storage modulus (solid lines) and loss tangent (dashed lines) of acid gels made from heated milk ( ) and unheated milk (\u25cb). Heat treatment was at 80 \u00b0C for 30 min, and acidification was at 30 \u00b0C with 1.3% GDL. The area marked by the letter A indicates the region in which the loss tangent increases after gelation because of solubilization of CCP in casein particles that are already part of the gel network.\n\nBikker et al. (2000) reported that the addition of \u03b2-lactoglobulin variant B or variant C to the milk prior to heating and acidification caused a larger increase in the storage modulus of acid gels than the addition of \u03b2-lactoglobulin variant A.\n\nSoluble whey protein polymers have been used as ingredients for yogurt applications (Britten & Giroux, 2001). The use of whey protein polymers to standardize the protein content of milk increased the yogurt viscosity to about twice that obtained using skim milk powder at the same protein concentration. The water-holding capacity of yogurt standardized with whey protein polymers was considerably higher than that of yogurt standardized with skim milk powder (Britten & Giroux, 2001).\n\n### Incubation Temperature\n\nAlthough 42 \u00b0C is a commonly used fermentation temperature for yogurt, the use of slightly lower incubation temperatures (e.g., 40 \u00b0C) leads to slightly longer gelation times, but firmer and more viscous gels that are less prone to whey syneresis are formed (Lee & Lucey, 2004). At a lower incubation temperature, there is an increase in the size of the casein particles because of a reduction in hydrophobic interactions, which, in turn, leads to an increased contact area between the casein particles (Lee & Lucey, 2004); a similar trend occurs when the gels are cooled. A high incubation temperature also makes the gel network more prone to rearrangements (more flexible) during gelation, and these changes can lead to greater whey separation (Lucey, 2001; Mellema et al. 2002). Peng et al. (2010) investigated the effect of altering temperature immediately after gel formation. Cooling after gelation resulted in an increase in gel stiffness and greater inter-cluster strand formation, whereas heating of gels may promote intra-cluster fusion and breakage of strands between clusters.\n\n### Production of Exopolysaccharides\n\nSome yogurt starter cultures produce exopolysaccharides (EPS) during the fermentation process. They can be viewed as a naturally produced thickener. This EPS can be produced as a capsular layer around the bacterial cell or excreted into the medium to produce an effect sometimes called 'ropy' or 'stringy' (Hassan, 2008). Popular examples of ropy yogurt are Viili and L\u00e5ngfil from Scandinavia. Capsular EPS has little impact on yogurt gelation or texture. Ropy EPS can be either charged or uncharged. It is possible that charged EPS may associate electrostatically with the caseins, depending on the pH of the milk, whereas uncharged EPS may influence gelation via a depletion flocculation-type mechanism (Girard & Schaffer-Lequart, 2007). The molar mass, concentration of EPS, and the exact period during fermentation (before, during, or after gelation) when EPS is produced may all play a critical role in determining the impact of EPS on yogurt gels.\n\n## Whey protein gels\n\nWhey is usually obtained as a by-product of cheesemaking (although recent developments in membrane technology mean that, in future, whey will come not necessarily from a cheese vat but as native whey directly from milk prior to cheesemaking). Its composition, however, depends on the cheesemaking conditions; for example, acid whey derived from cottage cheese has different mineral (ash), lactic acid, and pH values from whey derived from rennet-coagulated cheeses such as Cheddar (Table 17.1). Whey products are widely used as food ingredients because of their excellent functional and nutritional properties. Various types of whey products are made commercially, ranging from dried whey to WPC (WPC has protein contents ranging from \u224835 to 80%) to whey protein isolate (WPI) (protein contents \u226590%) (Table 17.2). Membrane filtration, that is, ultrafiltration (UF) and diafiltration (DF), is used to concentrate the protein fraction before spray drying into WPC. Two different approaches are used to produce WPI: (a) membrane filtration (microfiltration, UF and DF) and (b) ion-exchange chromatography coupled with UF\/DF. These two approaches result in WPI with different protein profiles (Table 17.3; Wang & Lucey, 2003). Many serum proteins take part in heat-induced gelation, whereas GMP and proteose peptones do not (Walstra et al., 2005).\n\nTable 17.1\n\nComposition of Rennet and Acid Wheys\n\n| Average composition  \n---|---  \n| Rennet whey | Acid whey  \nTotal whey protein (g\/L) | 6.7 | 5.8  \n\u03b2-lactoglobulin (g\/L) | 3.5 | 3.5  \n\u03b1-lactalbumin (g\/L) | 1.3 | 1.3  \nSerum albumin (g\/L) | 0.1 | 0.1  \nImmunoglobulins (g\/L) | 0.4 | 0.4  \nProteose peptones (g\/L) | 0.2 | 0.2  \nGlycomacropeptide (g\/L) | 1.0 | \u2014  \nLactose (g\/L) | 5.0 | 4.4  \nLipid (g\/L) | 0.6 | 0.1  \nAsh (g\/L) | 0.5 | 0.6  \nNa (mg\/100 g) |  35 | 40  \nK (mg\/100 g) |  109 | 133  \nCa (mg\/100 g) |  22 | 86  \nMg (mg\/100 g) |  6 | 9  \nP (mg\/100 g) |  42 | 63\n\n(Adapted from Oakenfull et al., 1997)\n\nTable 17.2\n\nTypical Composition of Some Whey Powders (approximate, wet, or as-is basis)\n\nWhey ingredient | Moisture (%) | Fat (%) | Protein (%) | Lactose (%) | Ash (%)  \n---|---|---|---|---|---  \nSweet whey | 3\u20135 | 1.1\u20131.5 | 11\u201314.5 | 75 | 8\u201310  \nAcid whey | 3.5 | 0.5\u20131.5 | 11\u201313.5 | 70 | 10\u201312  \nWPC35 | 3\u20134.5 | 3\u20134.5 | 34\u201336 | 48\u201352 | 6.5\u20138  \nWPC80 | 3.5-4.5 | 6\u20138 | 80\u201382 | 4\u20138 | 3\u20134  \nWPI | 4\u20135 | <1.0 | 90\u201392 | <1.0 | 2.5\u20133.5\n\nTable 17.3\n\nApproximate Protein Composition of Whey Protein Isolates Made by Different Technologies\n\nProtein type | Membrane filtration | Ion-exchange chromatography  \n---|---|---  \n\u03b2-lactoglobulin | 48\u201355% | 60\u201373%  \n\u03b1-lactalbumin | 15\u201322% | 12\u201325%  \nBovine serum albumin and Immunoglobulins | 4\u20137% | 6\u201316%  \nGlycomacropeptide | 17\u201326% | 0.2\u20131.4%\n\n(Data from several sources, including Wang and Lucey, 2003)\n\nWhey proteins are globular proteins, and heating induces denaturation and aggregation. Gelation depends on many factors, especially pH. At sufficiently high protein levels (usually \u22656%, except for purified individual whey proteins), gelation occurs during heating or cooling. The formation and the properties of whey protein gels are influenced by many factors (Table 17.4). There have been several reviews of the aggregation\/gelation of globular proteins (Oakenfull, 1987; Clark 1992; ; ; Doi, 1993; Oakenfull et al., 1997; Gosal & Ross-Murphy, 2000). The thermal denaturation and gelation of whey proteins have been reviewed (Mulvihill & Kinsella, 1987; Mangino, 1992; Aguilera, 1995; Singh & Havea, 2003; Foegeding, 2006; Mezzenga & Fischer, 2013).\n\nTable 17.4\n\nFactors that Influence the Heat-induced Gelation Properties of Whey Proteins\n\npH\n\nProtein content\n\nIonic strength\n\nRate and temperature\/time of heating\n\nTypes and ratios of the serum proteins\n\nConcentration of divalent ions (e.g., Ca2+)\n\nConcentration of sugars\n\nConcentration of lipids, including phospholipids\n\nUpon denaturation, whey proteins unfold and expose their hydrophobic residues, which were initially buried within the folded structure to minimize contact with water. The protein molecules will then tend to aggregate, thereby minimizing the contact of the hydrophobic residues toward water (Mezzenga & Fischer, 2013).\n\n### Thermal Denaturation of Whey Proteins\n\nMany studies on the denaturation of whey proteins have been conducted (see the review by Mulvihill & Donovan, 1987), especially \u03b2-lactoglobulin, as this is the major whey protein and its behavior dominates the gelation behavior of whey protein products. Denaturation has been used to describe both the loss of native structure (conformational change) and the loss of solubility (e.g., at pH values close to the isoelectric point). At around neutral pH values, denaturation becomes irreversible above about 65 \u00b0C (Holt & Sawyer, 2003); with a decrease in the pH, the denaturation temperature increases (Kella & Kinsella, 1988b). Disulfide bond formation is favored as the pH is increased toward the pK value of the thiol group on \u03b2-lactoglobulin (9.35; Kella & Kinsella, 1988b).\n\nDenaturation, that is, conformational change, can be reversible and, for whey proteins, the cause of irreversibility is often the formation of new covalent (mostly disulfide) bonds. Various mechanisms for the thermal denaturation\/aggregation of \u03b2-lactoglobulin (at neutral pH) have been proposed, in which the basic steps are: (a) the dissociation of the dimer into monomers and a conformational change leading to the exposure of Cys121, which initiates sulfydryl\u2013disulfide interchange reactions, (b) an endothermic transition to a molten globule state, and (c) the unfolding of the protein and a second, high-temperature endothermic transition (Holt & Sawyer, 2003). The reactive monomers formed during the denaturation process initially form dimers and trimers via the thiol\u2013disulfide exchange reaction, and the conversion of dimer to trimer is considered to be the rate-limiting step in the aggregation process (Prabakaran & Damodaran, 1997). Patel et al. (2006) proposed that the following reactions occur when milk is heated at \u224885 \u00b0C. The major whey proteins (\u03b2-lactoglobulin and \u03b1-lactalbumin) alter their structures and the free cysteine (CysH121) of \u03b2-lactoglobulin initially reacts reversibly with the adjacent Cys106\u2013Cys119 disulfide bond to give a free CysH119, which, in turn, reacts with the Cys66\u2013Cys160 disulfide bond of the same or another \u03b2-lactoglobulin molecule to give a free CysH160. CysH160 is mobile and free to move because it is so close to the C-terminus of the molecule. Thus, it reacts with disulfide bonds in other proteins, allowing a chain reaction with other \u03b2-lactoglobulin or casein molecules to occur (Patel et al., 2006).\n\nA possible model of these reactions during the denaturation and aggregation of \u03b2-lactoglobulin is shown in Figure 17.5. In the presence of different types of whey proteins, various heteropolymers (e.g., \u03b2-lactoglobulin\u2013\u03b1-lactalbumin or \u03b2-lactoglobulin\u2013bovine serum albumin) are formed during heating (Havea et al., 2001).\n\nFigure 17.5 Model for the aggregation and formation of heat-induced \u03b2-lactoglobulin gels.\n\nDuring the heating of \u03b2-lactoglobulin, the loss of native structure occurs via both disulfide-linked aggregate formation and noncovalently linked aggregates (e.g., hydrophobic interactions) (McSwiney et al., 1994). When \u03b2-lactoglobulin was heated at 75 \u00b0C, gelation was not observed until most of the protein had aggregated (McSwiney et al., 1994). Pure \u03b1-lactalbumin is very heat stable (because it does not have a free thiol group), although it does undergo a reversible transition at 64 \u00b0C (Ruegg et al., 1977). In the presence of \u03b2-lactoglobulin, it undergoes irreversible aggregation via the thiol\u2013disulfide exchange reaction as well as other types of interactions (Elfagm & Wheelock, 1978).\n\nDuring the heating of \u03b2-lactoglobulin, most of the helical conformation is lost by about 65 \u00b0C; with increasing temperature there is progressive loss of \u03b2-sheet structure (Qi et al., 1997). However, in \u03b2-lactoglobulin, a considerable amount of secondary structure, particularly \u03b2-sheet, still remains intact even at 90 \u00b0C (Bhattacharjee et al., 2005). Aggregation of globular proteins starts when heat causes some unfolding of the molecule, which exposes reactive groups or sites (e.g., hydrophobic regions) that favor intermolecular interactions (Foegeding, 2006). If \u03b2-lactoglobulin is denatured at low temperatures (e.g., around 60 \u00b0C), the molecules partially unfold into the state of a molten globule, while higher denaturation temperatures trigger a complete unfolding and rapid aggregation (Mezzenga & Fischer, 2013).\n\nGupta et al. (1999), using Monte Carlo computer simulations, indicated that protein-like molecules need to only partially unfold before they are susceptible to aggregation. Aggregation ultimately results in gelation if the protein concentration and other gelling conditions are favorable. This aggregation process is governed by a balance between attractive hydrophobic and repulsive electrostatic interactions.\n\nFractal aggregation theory has been applied to the aggregation and formation of whey protein gels (Vreeker et al., 1992; Ikeda et al., 1999; Mezzenga & Fischer, 2013). Euston (2004) argued that theories of fractal aggregation are not necessarily a good representation of protein gel structure as they treat the aggregating protein as a rigid particle and ignore any structural changes that occur in the protein during denaturation and aggregation. This criticism could be particularly important for the gelation of globular proteins, such as \u03b2-lactoglobulin.\n\nA gel is formed when the extent of aggregation exceeds a critical level for the formation of a self-supporting network that is able to entrap the solvent.\n\n### Types and Properties of Whey Protein Gels\n\nDifferent types of gel networks can be formed by globular proteins, such as whey proteins. The network structure in a heat-induced globular protein gel is strongly dependent on the balance between attractive and repulsive forces among (partially) denatured protein molecules during the aggregation process. As whey proteins have isoelectric points (pI) in the vicinity of pH 5, they are negatively charged at neutral pH values. In whey protein solutions, the ionic strength is important as it regulates the amount of ions available for the screening of charged groups on the proteins. At neutral pH values and under low ionic strengths, there is intermolecular repulsion. Aggregation of denatured proteins proceeds via hydrophobic sites, and this leads to the formation of fine-stranded gels (with a transparent or translucent appearance and strands that are often 10\u201320 nm in thickness) (Stading & Hermansson, 1991).\n\nIntermolecular repulsion can be reduced by increasing the ionic strength or by adjusting the pH to be closer to the isoelectric point of the whey proteins (\u22485). Under gelation conditions of high ionic strength or pH values close to 5, whey proteins form opaque or particulate or turbid gels. The particles\/clusters in this type of gel are in the micron-size range. This type of gel structure has a poorer water-holding capacity than fine-stranded gels (Bottcher & Foegeding, 1994). Particulate gels break down rapidly during mastication to yield a homogeneous distribution of small particles, whereas fine-stranded gels break down into large, inhomogeneous particles with irregular shapes (Foegeding, 2006). A fine-stranded gel formed at neutral pH is rubbery and deformable to a large strain, with a small fracture stress (Stading & Hermansson, 1991). At acidic pH values, intermolecular disulfide bonding is unlikely to occur, and the fine-stranded networks formed at very low pH values (e.g., 3) are brittle. Particulate gels normally fracture at a small strain, but the stress required to reach the fracture strain is relatively large (Stading & Hermansson, 1991; Bottcher & Foegeding, 1994; Foegeding et al., 1995). After heat-induced gelation, cooling results in strengthening of the network because of hydrogen bond formation.\n\nHeat-induced \u03b2-lactoglobulin gels exhibit the characteristics of a 'strong' gel. That is, they have a low-frequency dependence on the storage modulus (the linear slope, n, of the plot of log frequency versus log storage modulus is <0.06) (Stading & Hermansson, 1990). The slope n is slightly higher for particulate gels than for fine-stranded gels (Stading & Hermansson, 1990).\n\nAt pH values around 2 and low ionic strengths, whey protein gels that have some similarities in structure to \u03b2-amyloid fibrils are formed (Gosal et al., 2004; Bolder et al. 2006). Fibrils are usually rigid, nonbranching and filamentous structures, around 8 nm (or larger) in width (for \u03b2-lactoglobulin) and often more than 1 \u03bcm long, that arise from linear aggregation of partly unfolded proteins (Gosal et al., 2004). \u03b1-lactalbumin and bovine serum albumin can also form fibrils during heating at pH 2 (Goers et al., 2002; Veerman et al. 2003). Extensive hydrolysis occurs at low pH, and aggregation occurs through \u03b2-strands and \u03b2-sheets creating fibrils (Mezzenga & Fischer, 2013).\n\n### Other Factors Influencing Properties of Whey Protein Gels\n\npH and ionic strength greatly impact the type of gel formed and its properties. The strength of whey protein gels increases with protein content. The minimum protein content needed for gelation depends on whether an individual whey protein (e.g., \u03b2-lactoglobulin) or a commercial mixture (e.g., WPC) is used, as well as the gelation conditions (e.g., pH, heat treatment, ionic strength). Pure solutions of \u03b2-lactoglobulin can form a self-supporting gel at 5% protein content when tested at pH 8.0 and a heat treatment of 90 \u00b0C for 15 min (100 mM Tris-HCl buffer) (Matsudomi et al., 1991).\n\nThe protein profile is important for whey gelation. For example, higher gelling whey products can be made by increasing the proportion of \u03b2-lactoglobulin and decreasing the proportion of GMP. As \u03b1-lactalbumin is a poorer gelling protein than \u03b2-lactoglobulin, increasing the proportion of \u03b2-lactoglobulin to \u03b1-lactalbumin also increases the gelation properties of whey products (Hines & Foegeding, 1993). Commercial whey products with a higher ratio of \u03b2-lactoglobulin to \u03b1-lactalbumin are available (e.g., WPI made by ion-exchange chromatography compared with WPI made by membrane filtration, or acid whey WPC; both have little or no GMP).\n\nSalts have a major effect on the type of whey protein gel formed as a result of heat treatment and its mechanical\/sensory properties. It is generally recognized that the addition of NaCl or CaCl2 to dialyzed samples of WPC or WPI results in an increase in gel strength. Above a level of 10\u201320 mM CaCl2 and 100\u2013200 mM NaCl, the gel firmness starts to decrease (Schmidt et al., 1979; Kuhn & Foegeding, 1991). Excessive calcium has been speculated to cause rapid protein aggregation, which limits protein unfolding and network formation (Mangino, 1992). Caussin et al. (2003) reported that the addition of calcium to whey proteins resulted in the formation of very large protein aggregates during heating. Most commercially available WPC products probably have calcium contents that are greater than that required for optimal gel strength (Mangino, 1992).\n\nThere is considerable variability in the thermal aggregation behavior of commercial whey products; some of these differences could be removed by dialysis of these samples to a common ionic strength (McPhail & Holt, 1999). The concentration of divalent cations is higher in WPC made from cheese whey than in WPC made from acid whey. These cations are not easily removed by dialysis, suggesting some binding with the whey proteins (Havea et al., 2002). Although acid whey starts with a higher calcium content than cheese whey (Table 17.1), it is presumably easier to remove these salts in the manufacture of acid whey WPC than in the manufacture of cheese whey WPC. Acid whey WPC is known as a superior heat-gelling product compared with cheese whey WPC (Veith & Reynolds, 2004). These differences could be due to the absence of GMP and the low calcium concentration in acid whey WPC. It has been reported that polyphosphates have been added to WPC to improve the gelling properties (Veith & Reynolds, 2004). There are various possible processing approaches to reduce the calcium\/mineral content of cheese whey WPC (e.g., electrodialysis, addition of chelating agents, low-pH UF\/DF) in order to improve its gelling properties.\n\nThe gelling time is also dependent on temperature, with the time required for gelation decreasing with increasing temperature, although, at very high temperatures, gelation may occur only during the subsequent cooling stage (Hillier & Cheeseman, 1979). Many reports show that, when all other factors are kept constant, the gel strength increases with increasing temperature (presumably reflecting greater unfolding and reactivity of the proteins) (Mulvihill & Kinsella, 1987). The presence of lipids and lactose impairs the gelation of whey proteins (Mulvihill & Kinsella, 1987). Sugars, such as lactose, are known to protect the protein against loss of solubility during heat treatment and increase the thermal denaturation temperature of whey proteins (de Wit, 1981; Jou & Harper, 1996). Possibly, lipids might interfere with the hydrophobic interactions that play a role in the aggregation of partly unfolded whey proteins during heat treatment.\n\n### Cold Gelation of Whey Proteins\n\nGels can also be produced using a two-step process that involves heat treatment at low ionic strength and\/or far from the isoelectric point, followed by an increase in ionic strength and\/or an adjustment in pH (Barbut & Foegeding, 1993; Britten & Giroux, 2001). These gels are labeled cold-set gels, as the initial heat treatment produces a polymerized solution, with gelation occurring during the subsequent cold-set conditions through screening of the repulsive forces. To obtain gels via the cold-set gelation method, it is necessary to prepare a heat-denatured solution, with a protein concentration below the critical gelation concentration. Gelation can then be induced at low temperatures by the addition of mono- or polyvalent cations (e.g., Ca2+).\n\nBritten and Giroux (2001) acidified whey protein polymers to pH 4.6 with GDL and formed opaque particulate gels. The storage modulus and the firmness of the gels were affected by the conditions used to prepare the protein polymers.\n\n### Enzymatic Modification of Whey Protein for Gelation Purposes\n\nExtensive hydrolysis of whey protein using proteinases results in gelation mainly via hydrophobic interactions, with hydrogen bonding and electrostatic interactions also playing a minor role (Otte et al., 1996; Doucet et al. 2003;).\n\nThe casein fractions in milk are more susceptible to TGase cross-linking than the globular whey protein fractions (Jaros et al., 2006). Some unfolding of \u03b2-lactoglobulin improves the extent of cross-linking with TGase (Faergemand et al., 1997; O'Sullivan et al. 2002a). Cold-set whey protein gels at low pH have been cross-linked with the TGase enzyme under either low pH or alkaline conditions (Eissa et al., 2004; Eissa & Khan, 2005). One approach involved two steps; first, cross-linking whey proteins with TGase at pH 8 and 50 \u00b0C; and second, cold-set acidifying the resulting solution using GDL (Eissa et al., 2004). During the first step, the whey proteins undergo enzyme-catalyzed \u025b-(\u03b3-glutamyl) lysine bond formation with a substantial increase in viscosity. Enzyme-cross-linked gels had significantly higher yield\/fracture stress and strain than cold-set gels prepared without TGase enzyme or conventional heat-set gels. In addition, the elastic modulus of the enzyme-catalyzed gel was higher than that of its nonenzyme-treated counterpart.\n\n### Mixed Gels Made with Rennet and Acid\n\nMilk coagulation can be induced by the combined action of acid and enzyme (i.e., mixed gels). The study of mixed milk coagulation has received very little attention when compared with rennet- or acid-induced coagulation (Roefs et al., 1990; Lucey et al. 2000; 2001; Tranchant et al., 2001). Cottage cheese is generally manufactured by acid coagulation of pasteurized skim milk, and a small concentration of rennet is sometimes added after the starter has been allowed to develop some acidity (i.e., at pH around 5.5) (Castillo et al., 2006). The use of rennet in combination with acid development initiates gelation at a high pH, and the gel can undergo a 'weakening' stage (as indicated by a decrease\/plateau of the storage modulus, a decrease in the light backscatter ratio, or an increase in the loss tangent). This weakening is more pronounced with unheated milk gels and where there have been very high levels of \u03ba-casein hydrolysis prior to acidification (Li & Dalgleish, 2006). This 'weakening' stage is related to rearrangements caused by CCP demineralization of the casein particles in the gel network because this CCP solubilization occurs after gelation (gelation is initiated at a high pH in mixed gels) (Lucey et al., 2000). The final storage modulus of mixed gels can be considerably higher than that of acid gels made without rennet. Mixed gels made from heated milk formed firmer gels, as they were cross-linked by denatured whey proteins and underwent fewer large-scale rearrangements (Lucey et al., 2000).\n\nThe rheological and microstructural properties of mixed gels are complex, and these properties can be adjusted by varying the rennet level or the acidification rate (Tranchant et al., 2001). The use of low rennet levels during the fermentation of milk resulted in a coarser acid gel network and higher syneresis (Aichinger et al., 2003). Micelle fusion was faster in gels with added rennet because of the removal of the \u03ba-casein hairs (Aichinger et al., 2003). Gastaldi et al. (2003) studied the acid-induced gelation of milk samples in which chymosin was used to vary the degree of \u03ba-casein hydrolysis prior to acidification (further chymosin activity during acidification was blocked using an inhibitor). The gelation pH increased and the gelation time decreased with an increasing degree of \u03ba-casein hydrolysis. Gels with much higher storage moduli were formed as a result of partial \u03ba-casein hydrolysis prior to gelation, although the loss tangent and the serum-holding capacity were lower (Gastaldi et al., 2003). Presumably, partial \u03ba-casein hydrolysis prior to acid gelation facilitated greater rearrangements\/fusion of casein, which was responsible for the increase in the storage modulus but also increased the serum separation (Lucey et al., 2001).\n\n## Conclusions\n\nThe formation and the physical properties of milk protein gels have been the subject of intense study because of the great economic impact of these gels on dairy products such as cheese, yogurt, and heat-set whey gels. There is a growing recognition that the internal structure of casein micelles plays an important role in the structural properties of rennet, acid, and mixed gels. These gels are dynamic in nature and undergo rearrangements. Technologists have recently studied the impact of high pressure and enzymatic cross-linking of these proteins to modify their functionality. The interaction between DWP and caseins has received a lot of attention; this interaction has been used to alter the texture of acid gels, although there is disagreement about the exact mechanism(s) involved. DWP polymers have been used for making cold-set gels, and they have interesting possible applications in various milk gels\/products. Fine-stranded whey proteins made at very low pH values have been shown to be similar in structure to amyloid fibrils. From an industrial viewpoint, these fine-stranded fibril types of gels might have some useful applications because they gel at low-protein concentrations.\n\n## Acknowledgment\n\nThis material is based on work supported by the National Institute of Food and Agriculture, United States Department of Agriculture, under project WIS01650.\n\n# References\n\nAichinger P-A ,  Michel M ,  Servais C ,  Dillmann M-L ,  Rouvet M ,  D'Amico N ,  Zink R ,  Klostermeyer H ,  Horne DS .  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General considerations and literature review .  _Netherlands Milk and Dairy Journal_.  1985 ;39 : 209 \u2013 246 .\n\nWalstra P ,  Wouters JTM ,  Geurts TJ .  _Dairy Science and Technology_ .  2nd ed. New York :  CRC Press ;  2005 .\n\nWang Q ,  Bulca S ,  Kulozik U .  A comparison of low-intensity ultrasound and oscillating rheology to assess the renneting properties of casein solutions after UHT heat pre-treatment .  _International Dairy Journal_.  2007 ;17 : 50 \u2013 58 .\n\nWang T ,  Lucey JA .  Use of multi-angle laser light scattering and size-exclusion chromatography to characterize the molecular weight and types of aggregates present in commercial whey protein products .  _Journal of Dairy Science_.  2003 ;86 : 3090 \u2013 3101 . \nChapter 18\n\n# Milk Proteins\u2014A Cornucopia for Developing Functional Foods\n\nPaul J. Moughan Riddet Institute, Massey University, Palmerston North, New Zealand\n\n## Abstract\n\nMilk proteins have a central role to play in the development of functional foods, foods that have targeted physiological effects in the body over and above the normal effects of food nutrients. Milk proteins contain high amounts of bioavailable amino acids, making them ideal ingredients for the manufacture of nutritionals, foods designed for specific nutritional purposes. Certain amino acids have specific physiological roles (e.g., tryptophan as a precursor of serotonin or leucine in the regulation of muscle metabolism), and some isolated milk proteins have particularly high concentrations of these amino acids, allowing foods to be developed to target physiological end points. Milk proteins, especially whey protein and glycomacropeptide, have an application in inducing satiety in humans, and the relatively low yield of adenosine triphosphate (ATP) per unit amino acid in comparison with glucose or fatty acids means that milk proteins are ideal ingredients for weight-loss foods. Finally, milk proteins are known to be a rich source of bioactive peptides, released in the gut naturally during digestion. These peptides have a plethora of physiological effects including effects locally at the gut level. The multiple nutritional and physiological properties of milk proteins and peptides in the context of functional foods are discussed.\n\n## Keywords\n\nMilk\n\nproteins\n\namino acids\n\nbioactive peptides\n\nfunctional foods\n\nOutline\n\nIntroduction 525\n\nFunctional foods 526\n\nMilk proteins as a source of amino acids\u2014specialized nutritionals 528\n\nMilk proteins as a source of amino acids\u2014specific physiological roles 531\n\nMilk proteins as a source of amino acids\u2014role in providing calories and in promoting satiety 533\n\nMilk proteins as a source of bioactive peptides 534\n\nConclusions 537\n\n## Introduction\n\nOver the last century, scientists have gradually come to better understand the complexity of foods. In addition to delivering nutrients, satisfying hunger, and providing pleasure, food components are now known to have a role in directly influencing physiological processes in the body. In particular, certain food components may assist in maintaining or promoting health and well-being and preventing the development of disease. These components may be non-nutrients (e.g., the antioxidant effect of polyphenols or the blood cholesterol-lowering effect of plant sterols) or nutrients (e.g., the effects of short-chain volatile fatty acids in modulating gut development and function). In fact, the multiple effects of some chemical compounds released from foods during digestion, traditionally considered to have a sole role in nourishment, call into question the very definition of 'a nutrient.'\n\nIt is also now widely appreciated that body growth and maintenance processes are a subtle interaction between nutrient supply and genetically regulated metabolism. The assimilation and metabolism of food compounds is subject to genetic and epigenetic control, which will vary among individuals and diverse populations, leading to important gene\u2013nutrient interactions. In turn, nutrients and food non-nutrients also greatly influence gene expression, with effects once again varying among individuals, and the complexity and the subtlety of nutrigenomics and nutrigenetics are really only beginning to be understood (Ferguson, 2011). Over the next 50 years, it is expected that great advances will be made in understanding how food influences gene expression, how genetic regulation influences the assimilation and utilization of nutrients, and how the individual's genome explains differences among individuals in their physiological and nutritional response to different foods under different conditions. Such understanding will pave the path toward personalized nutrition and personalized health foods and dietary\/exercise regimens.\n\nMilk is an excellent example of a food having both nutritional and non-nutritional physiological roles in the human diet. Milk and, in particular, milk proteins not only supply the body with amino acids necessary for the maintenance and growth of body protein, but during food manufacture and\/or food digestion give rise to a myriad of protein fragments and large and small peptides that have distinct biological functions (Ward and German, 2004). Certain amino acids (e.g., leucine, tryptophan) released during digestion have regulatory functions or act as precursors for the synthesis of key nonprotein metabolites. Such compounds are a rich source of bioactive components for the development of functional foods. Although these compounds are undoubtedly of major significance to the suckled infant, whereby milk should be regarded as a biological fluid specifically designed by nature for optimal growth and development, they are also probably of importance in the adult diet, where milk has long been an important constituent.\n\n## Functional foods\n\nA functional food has been defined by Diplock et al. (1999, p. 1) as follows:\n\nA food can be regarded as functional if it has beneficial effects on target functions in the body beyond nutritional effects in a way that is relevant to health and well-being and\/or the reduction of disease.\n\nIn a sense, and following this definition strictly, most if not all foods could be described as 'functional.' Perhaps, then, the definition needs to be expanded to include the notion that the 'beneficial effect on target functions' occurs at a meaningful level. Minor beneficial effects on target functions, which would be consequent upon the ingestion of many mainstream foods, may be relevant to health, well-being, and disease prevention in the context of a long-term balanced diet, but such foods are not widely regarded as functional foods.\n\nUnder the latter definition of a functional food, foods may be functional naturally (e.g., oily fish containing high amounts of omega 3 fatty acids) or may be rendered functional, usually by adding a bioactive component or by removing some component that is inhibiting bioactivity. Foods may also be enriched with a given bioactive component or components, through conventional animal or plant breeding practices, by genetic engineering or by manipulation of the feeding and nutrition of the plant or animal (Moughan, 2011).\n\nFunctional foods is a rapidly growing sector of the international food industry, with development spurred by a number of technical, social, and economic drivers. First, there is a high degree of awareness of the link between diet and health, which has been established largely through well-publicized epidemiological studies. This knowledge has been expanded by the more cogent 'proof of cause and effect' from human intervention studies. Well-educated consumers, aware of the importance of diet to health, are demanding healthy and functional foods and are prepared to pay a price premium. Escalating health care costs, a major concern to governments, encourage 'disease prevention rather than cure' community health strategies. Completion of the human genome project with the subsequent rapid accumulation of knowledge concerning single and multiple gene effects (heralding presymptomatic testing for disposition to particular conditions) is being coupled with a better understanding of intergenerational nutritional effects (Barker hypothesis) on predisposition to chronic disease states. This is likely to lead in the near future to personalized nutrition strategies and to further demand for specific functional foods tailored to the requirements of the individual (mass customization).\n\nIf the functional foods industry is to achieve its full potential, it will be critical that regulatory bodies have a clear, consistent, and rigorous approach to safety, labeling, and health claims issues, and that food manufacturers contract reputable science providers to independently establish 'proof of concept' around their products (Roberfroid, 2000). There will also need to be a considerable investment in research and development to clearly establish cause-and-effect relationships between food compounds and targeted physiological end points and recognized disease risk factors. Consumer confidence can quickly be eroded by conflicting messages received from the scientific community. This underlines the critical role of regulatory authorities in ensuring that sufficient credible information is available and is correctly analyzed using meta-analysis techniques in order to make sustainable health claims, both qualified and approved claims. The burden of proof for generic health claims and for claims on the efficacy of specific food products needs to be considerable, such that consumers can have high levels of confidence.\n\nFoods undoubtedly have a major role to play in preventing disease and ensuring health and vitality. However, if functional foods are to achieve their potential as part of an overall lifestyle stratagy toward healthfulness, then consumers must be guided by the highest quality information and distilled findings that have a strong likelihood of remaining substantiated over time. The food industry, science providers, and government bodies all have a responsibility to ensure that the functional foods movement is led by ethical and informed decision making. Although the Institute of Food Technologists' (IFT, USA) Expert Report on Functional Foods (2005) recognized the enormous potential for functional foods, it stated: \"But that is not to say that IFT believes that all foods on the market for which claims are being made are being properly represented based on science and proper regulatory policies. IFT does not support some claims on foods marketed today because they are not supported by today's science.\" Clearly there is a challenge! The IFT Expert Panel goes on to recommend basing structure\/function health claims on broad-based scientific criteria that address the underlying link between health and nutrition and meet the need for sound scientific substantiation supporting the structure\/function effect. The panel discusses principles around ensuring the safety of functional foods and has introduced the concept of GRAE (generally recognized as efficacious), analogous to GRAS (generally recognized as safe), to encourage public confidence in the labeling of functional foods.\n\nThe dairy industry is in an excellent position to take advantage of the trend toward more healthy diets and more healthy and functional foods. Because milk proteins are readily available sources of amino acids and give rise to many bioactive compounds, they have a central place in the development of both specialized nutritionals and functional foods.\n\n## Milk proteins as a source of amino acids\u2014specialized nutritionals\n\nMilk and milk proteins have long been regarded as a rich source of readily digestible and nutritionally available amino acids. However, early in vivo determinations of protein digestibility were based on fecal measurement, which is now known to be flawed. This is due to the significant degree of colonic microbial metabolism now known to occur, and the fact that the resulting amino acids are at best absorbed from the large intestine to only a very limited extent (Moughan, 2003). The preferred accurate method for determining amino acid digestibility is to determine unabsorbed amino acids at the end of the small bowel (terminal ileum). This can be achieved in humans using naso-intestinal intubation or through the cooperation of ileostomates. Alternatively, animal models can be used, with both rat and pig ileal digestibility assays being suitable (Moughan et al., 1994). Where a protein has undergone structural alteration due to processing or storage (especially at high temperatures), conventional digestibility measures are inappropriate for some amino acids, in particular the often first-limiting amino acid, lysine. A new lysine bioavailability assay has been developed that can be usefully applied to damaged proteins, based on the collection of ileal digesta and application of the guanidination reaction (Moughan, 2003). When digestibility determinations are based on the sampling of ileal digesta, it is important to recognize that digesta contain copious quantities of endogenous (of body origin) protein in addition to undigested protein. This endogenous protein component needs to be taken into account to yield 'true' rather than 'apparent' estimates of digestibility (Moughan et al., 1998; Moughan and Rutherfurd, 2012).\n\nThere are very few published data on true ileal protein digestibility as determined using human subjects. A comprehensive set of studies in humans demonstrated a high true ileal digestibility of protein in milk of around 95% (Gaudichon et al. 1994; 1995; 1996; Mah\u00e9 et al., 1995; 1996). Comparable values, using the same methodology, for soya and pea proteins were 91 and 89%, respectively. Sandstrom et al. (1986) gave soya- and meat-based diets to ileostomates and reported ileal digestibility coefficients for total nitrogen in the range 80\u201385%. The naso-intestinal intubation method with normal volunteers has also been used to determine digestibility coefficients for individual amino acids (Gaudichon et al., 2002). For cow's milk, true ileal digestibility ranged from 92% for glycine to 99% for tyrosine, whereas for soya bean, digestibility ranged from 89% for threonine to 97% for tyrosine. In our own laboratory, true ileal amino acid digestibility determined using ileostomates ranged from 98% for aspartate to 100% for cysteine in sodium caseinate; from 93% for threonine to 99% for cysteine in whey protein concentrate; from 95% for glycine to 99% for arginine in soya protein isolate; and from 91% for cysteine to 100% for arginine in soya protein concentrate (Moughan et al., 2005a). There are more comprehensive data on the true ileal digestibility of amino acids in various milk proteins, which have been obtained using animal models for digestion in humans (Rutherfurd and Moughan, 1997). Table 18.1 shows ileal digestibility data obtained using the laboratory rat for selected amino acids in a range of protein sources. These data confirm the very high digestibility of milk-derived proteins to the end of the ileum in simple-stomached mammals. The dietary essential amino acids are virtually completely digested.\n\nTable 18.1\n\nMean True Ileal Digestibility of Selected Amino Acids in a Range of Soya and Dairy Products\n\nAmino acid | Soya protein concentrate | Soya protein isolate | Lactic casein | Sodium caseinate | Whey protein concentrate | \u03b1-Lactalbumin | Milk protein concentrate  \n---|---|---|---|---|---|---|---  \nLysine | 97.3 | 98.5 | 98.8 | 98.0 | 98.2 | 94.7 | 98.3  \nMethionine | 95.3 | 100.0 | 100.0 | 99.6 | 100.0 | 99.2 | 100.0  \nCysteine | 86.9 | 95.3 | 99.2 | 93.0 | 99.6 | 96.1 | 97.8  \nIsoleucine | 96.4 | 96.8 | 94.8 | 90.6 | 98.1 | 95.4 | 94.9  \nLeucine | 95.7 | 95.3 | 99.1 | 97.6 | 99.1 | 96.1 | 98.9\n\nAdapted from Rutherfurd and Moughan (1997), with the permission of the publisher.\n\nAlmost all dairy proteins have been subjected to some type of processing during their manufacture. Given that milk products often contain the reducing sugar lactose, they are susceptible to damage to the amino acid lysine. A specific assay designed to allow an accurate determination of lysine bioavailability in processed foods (Moughan and Rutherfurd, 1996) has recently been applied to a range of commercially available milk protein-based products (Table 18.2). Once again, this underscores the high bioavailability of milk proteins and the limited amount of lysine damage incurred by proteins with modern controlled dairy processing. In contrast, when the same bioassay was applied to grain-based processed foods often containing high amounts of sucrose, including cereals for children, a substantial degree of lysine damage was found (Table 18.3) (Rutherfurd et al., 2006). Milk and milk-based products have an important role in complementing cereal foods and in supplying available lysine.\n\nTable 18.2\n\nTrue Ileal Reactive Lysine Digestibility (Bioavailability, %) and Digestible Total and Reactive Lysine Contents (g\/kg air-dry) for 12 Dairy Protein Sources\n\n| | Digestible lysine  \n---|---|---  \nProduct | Bioavailability | Total | Reactivea  \nWhole milk protein | 98.3 | 26.2 | 24.0  \nInfant formula A | 91.0 | 8.3 | 8.6  \nInfant formula B | 92.3 | 9.1 | 9.2  \nInfant formula C | 93.1 | 11.1 | 11.7  \nWhey protein concentrate | 98.5 | 79.9 | 77.5  \nUHT milk | 100.0 | 31.7 | 31.4  \nEvaporated milk | 96.7 | 23.4 | 20.5  \nWeight-gain formula | 99.0 | 24.4 | 24.1  \nSports formula | 98.0 | 20.4 | 19.1  \nElderly formula | 97.1 | 11.7 | 11.8  \nHydrolyzed lactose milk powder | 98.6 | 27.2 | 25.1  \nHigh-protein supplement | 99.9 | 14.3 | 14.3\n\na Bioavailable lysine; minimal difference between total lysine and reactive lysine denotes minimal Maillard damage.\n\nAdapted from Rutherfurd and Moughan (2005), with the permission of the publisher.\n\nTable 18.3\n\nTrue Ileal Digestible Total and Reactive Lysine Contents (g\/kg air-dry) in Selected Cereal-based Foods\n\n| Digestible lysine  \n---|---  \nCereal product | Total | Reactivea  \nWheat-based (shredded) | 1.3 | 0.8  \nCorn-based (flaked) | 0.4 | 0.2  \nRice-based (puffed) | 1.1 | 0.6  \nMixed cereal (rolled) | 3.2 | 1.9\n\na Bioavailable lysine; a difference between total lysine and reactive lysine denotes Maillard damage.\n\nAdapted from Rutherfurd et al. (2006), with the permission of the publisher.\n\nFigure 18.1 highlights the substantial differences that exist in the amounts of digestible amino acids supplied by plant proteins (e.g., soya) and milk proteins (e.g., \u03b1-lactalbumin). The often first-limiting amino acids (lysine and methionine plus cysteine) are found in much higher concentrations in the milk proteins. This makes them excellent sources of amino acids and very important dietary constituents in order to achieve a balanced dietary protein intake. Recently, an international consensus view has formed that the traditional protein digestibility corrected AA score (PDCAAS) method for describing dietary protein quality for humans should be replaced by a method based on true ileal amino acid digestibility and availability and not involving truncation of the score for individual food ingredients (DIAAS; digestible indispensable amino acid score). Calculated DIAAS values for whey protein isolate, milk protein concentrate, and whey protein concentrate are 1.25, 1.31, and 1.10, and values for soya protein isolate and pea protein concentrate are 0.99 and 0.93, further showing the significant complementary value of milk proteins.\n\nFigure 18.1 Digestible (true ileal) amino acid contents of a plant protein and a milk protein. Adapted from Rutherfurd and Moughan, 1997, with the permission of the publisher.\n\nBecause of their relatively high levels of nutritionally important amino acids, milk proteins are utilized efficiently by humans when given as a sole protein source. Tom\u00e9 and Bos (2000) reported net postprandial protein utilization values of 80 and 72% for milk protein and soya protein, respectively, measured over 8 h after the ingestion of standard meals by healthy human subjects.\n\nGiven the high bioavailability of amino acids in milk proteins and their abundant supply, it is hardly surprising that milk proteins are commonly used for the manufacture of so-called nutritionals\u2014foods designed for a specific nutritional purpose (e.g., formulas for infants, the elderly, and athletes).\n\nIn the future, with increasing human population growth and greater pressure on food supplies, milk proteins will likely play an ever more important role as protein 'balancers' in plant-based processed foods.\n\n## Milk proteins as a source of amino acids\u2014specific physiological roles\n\nAmino acids may have physiological roles that are unrelated to their direct involvement in protein synthesis. These include their roles as neurotransmitters (e.g., glutamate, aspartate, and glycine) and as precursors for the synthesis of other molecules involved in neuromuscular function (e.g., creatine and taurine) and in host defenses (e.g., glutathione and nitric oxide). Tryptophan is a precursor for the synthesis of serotonin, potentially impacting mood control (van de Poll et al., 2006) and appetite regulation (Fernstrom and Wurtman, 1972; Fernstrom and Fernstrom, 1995). The nitrogen-rich amino acid arginine leads to the production of nitric oxide (Wu and Morris, 1998), which is considered to have a significant role in cell signaling and the control of endothelial tone. Depending on its site of release, nitric oxide has several functions, including stimulation of the pituitary gland, vasodilation, neurotransmission and immune modulation. Another example of an amino acid with a specific metabolic role is the branched-chain amino acid leucine, which has a unique role in the regulation of muscle protein synthesis (Kimball and Jefferson, 2001; McNurlan, 2012). Interestingly, leucine stimulates protein synthesis directly in skeletal muscle but not in liver. The other branched-chain amino acids, isoleucine and valine, are less effective in stimulating muscle protein synthesis. Leucine supplementation has been shown to stimulate recovery of muscle protein synthesis during food restriction and after endurance exercise (Gautsch et al., 1998; Anthony et al., 2000). It has also been suggested that leucine has a potential regulatory role in glycemic control (Layman 2002; 2003).\n\nIt has been known for many years that glutamine, glutamate, and aspartate are preferred oxidative fuels for the gut, a highly metabolic organ. Consequently, many studies with humans and animals have been undertaken to investigate the effects on intestinal mucosal integrity, glutathione synthesis, and immune function, especially using dietary glutamine. This has led to debate as to whether glutamine should be regarded as a 'conditionally essential' dietary amino acid (Grimble, 1993). In the traumatized patient, dietary glutamine may be needed to maintain immune responsiveness and for maintenance of the mucosal barrier against bacterial action and endotoxins.\n\nThere are also amino acids with specific physiological functions that are not found in proteins (i.e., nonprotein amino acids). The classic example is taurine (\u03b2-aminoethanesulfonic acid), which is synthesized by the body from cysteine or methionine and is essential for the production of conjugated bile salts (taurocholic acid). Taurine is found in milk, normally in the free form. It is recognized that cow's milk has low concentrations of taurine relative to human milk, and so taurine is now often added to cow's milk-based infant formulas.\n\nThe above are examples of specific physiological functions of amino acids; there are many others. It is anticipated that over the next decade our understanding of the physiological roles of individual amino acids will increase, leading to opportunities to develop functional foods containing higher or lower amounts of certain amino acids. Summaries of current evidence for proven functional effects (clinical benefits) in humans consequent upon dietary supplementation with specific amino acids have been provided by van de Poll et al. (2006) and Jonker et al. (2012). Arginine has been widely used in supplemental nutrition for surgical patients and patients with burns, to modify the inflammatory response, to enhance organ perfusion, and to stimulate wound healing. However, the benefits of arginine supplementation are not uniformly proven and accepted. There is some evidence that taurine supplementation improves retinal development in premature babies receiving parenteral nutrition. Glutamine is one of the more extensively studied amino acids and has been used in the preparation of medical foods. There is evidence that glutamine supplementation may reduce infectious morbidity and the length of hospital stay in surgical patients. Phenylalanine-free preparations have application in phenylketonuria.\n\nEtzel (2004) highlighted an opportunity for the dairy industry, whereby a number of refined high-quality proteins are produced and marketed. The diverse amino acid compositions of these proteins can be exploited. The mixtures of proteins in milk and whey may be fractionated to give isolated proteins (\u03b1-lactalbumin, \u03b2-lactoglobulin, immunoglobulins, bovine serum albumin, the caseins, lactoferrin, lactoperoxidase, and the peptide glycomacropeptide that is cleaved from \u03ba-casein by chymosin) and blends of proteins with unique amino acid patterns. Etzel (2004) compared the amino acid compositions of several milk proteins with the composition of a theoretical 'average' protein calculated from the frequency of occurrence of each amino acid in 207 unrelated proteins of known sequence. Table 18.4 provides an abridged version of the Etzel (2004) dataset. Some interesting comparisons can be made. First, glycomacropeptide is completely devoid of cysteine, histidine, phenylalanine, tyrosine, and tryptophan. Cysteine content is relatively high in \u03b1-lactalbumin, whereas glutamine has a relatively high concentration in \u03b2-lactoglobulin, and glutamic acid is found at a high concentration in three of the dairy products. Leucine content is some twofold higher in \u03b2-lactoglobulin compared with the 'average' protein, and the threonine content of glycomacropeptide is extraordinarily high. As a group, the branched-chain amino acids in milk proteins are higher in concentration than the 'average' protein. It is clear from this type of comparison that milk-based food products with amino acid compositions targeting particular physiological end points can be developed.\n\nTable 18.4\n\nThe Amounts (% air-dry) of Selected Amino Acids in Various Milk Proteins and in an 'Average' Protein\n\nAmino acid | \u03b2-Lactoglobulin | \u03b1-Lactalbumin | Glycomacropeptide | Whey protein isolate | 'Average' proteina  \n---|---|---|---|---|---  \nCysteine | 2.8 | 5.8 | 0 | 1.7 | 2.6  \nGlutamine | 6.3 | 4.5 | 3.8 | 3.4 | 4.6  \nGlutamic acid | 11.3 | 7.3 | 15.5 | 15.4 | 7.3  \nHistidine | 1.5 | 2.9 | 0 | 1.7 | 2.6  \nIsoleucine | 6.2 | 6.4 | 11.9 | 4.7 | 4.8  \nLeucine | 13.6 | 10.4 | 1.7 | 11.8 | 7.8  \nValine | 5.4 | 4.2 | 8.9 | 4.7 | 6.2  \nPhenylalanine | 3.2 | 4.2 | 0 | 3.0 | 4.7  \nTryptophan | 2.0 | 5.3 | 0 | 1.3 | 1.9  \nTyrosine | 3.6 | 4.6 | 0 | 3.4 | 5.2  \nThreonine | 4.4 | 5.0 | 16.7 | 4.6 | 5.5\n\na Based on amino acid compositions of 207 unrelated sequenced proteins.\n\nAdapted from Etzel (2004), with the permission of the publisher.\n\n## Milk proteins as a source of amino acids\u2014role in providing calories and in promoting satiety\n\nIn addition to their role as a substrate for body protein synthesis and for the synthesis of various nonprotein nitrogenous compounds, amino acids may also be used as a source of dietary energy, and the interaction between dietary protein and energy has long been understood.\n\nIt is often overlooked, however, that different dietary macronutrients give rise, biologically, to quite different amounts of free energy (adenosine triphosphate\u2014ATP) per unit gross energy (bomb calorimeter). It is argued that 'a calorie is a calorie' regardless of the macronutrient giving rise to the energy. This is true, but what often fails to be appreciated is that the numbers of calories in a food deemed to be derived from the respective macronutrients are usually estimates, not absolute measures. Conversion factors such as the Atwater factors are applied to determine the amounts of macronutrients in a food, and 'available' energy is estimated. The point is that the conversion factors are not completely accurate, and thus neither are the estimated calories. This point has particular relevance in the case of amino acids.\n\nAtwater factors attempt to take into account the loss of energy due to incomplete absorption of the amino acid during digestion and the loss of energy in excreted urinary metabolites post catabolism. However, the net yield of ATP during the catabolism of the amino acid and the ATP cost of synthesizing urea are not accounted for. The capture of net energy as ATP for an amino acid is less efficient than for other nutrients such as glucose and fatty acids, with an accompanying higher dietary-induced thermogenesis compared with glucose and fatty acids. This has important implications for designing weight-loss diets. Foods containing high amounts of protein will provide less 'available energy' (i.e., ATP) per unit dry matter or gross energy, compared with foods high in available carbohydrate and\/or fat.\n\nDairy proteins are a highly versatile source of amino acids for the design of weight-loss foods, and more care should be taken in describing the calorific values, especially for functional foods designed for weight loss. Protein has a further advantage for the formulation of weight-loss foods, as it is now widely accepted that it is a satiating nutrient and is more effective than carbohydrate and fat in suppressing voluntary food intake independent of its calorific value. The role of dietary protein in the regulation of food intake and body weight in humans, and underlying mechanisms, has been the subject of recent reviews (Anderson and Moore, 2004; Westerterp-Plantenga and Lejeune, 2005). There is strong evidence that the protein content of a food is a determinant of short-term satiety and of how much food is subsequently eaten. The role of protein in the regulation of long-term food intake and body weight is less clear because of a paucity of relevant experimental observations.\n\nThe role of protein in regulating body weight, in comparison with other macronutrients, is considered to consist of several often-related but different aspects: satiety, thermogenesis, metabolic energy efficiency, and body composition. First, the highly satiating effect of protein has been observed both postprandially and postabsorptively. Second, and also as discussed, high-protein diets are associated with a high dietary-induced thermogenesis, which could be related to the satiety effect of proteins. Third, high-protein diets help maintain or increase fat-free body mass, and the maintenance of a higher lean mass is costly energetically (i.e., a higher resting energy expenditure), leading to a lower associated metabolic efficiency of energy utilization.\n\nOf particular interest to the dairy industry is the observation that protein source per se may be a factor influencing short-term satiety in humans. Whey protein has been identified as potentially more effective in promoting satiety (Vandewater and Vickers, 1996; Portman et al., 2000; Hall et al., 2003). A basis for differences in satiety related to source of protein may be found in amino acid composition (e.g., a high leucine content stimulating protein synthesis and altering body energetics), in bioactive peptides released from the protein during digestion (refer to the following section), in different kinetics of protein digestion, and, in the case of whey, in the presence of glycomacropeptide.\n\n## Milk proteins as a source of bioactive peptides\n\nMilk is known to contain proteins (e.g., lactoferrin, lactoperoxidase, immunoglobulins) and free peptides having specific non-nutritional physiological functions. These compounds are undoubtedly important in the case of the human infant and may also have a functional role in adults. Of potentially greater significance, however, are the many small (from 3 to 20 amino acids) bioactive peptides encrypted in food protein amino acid sequences and released during digestion. Bioactive peptides are specific protein fragments that influence body function. These peptides are inactive within the sequence of the parent protein and can be released during proteolysis or fermentation. Bioactive peptides may act as physiological modulators both locally in the gut and systemically. Most, if not all, proteins appear to contain bioactive sequences, although the majority of research to date has been conducted with milk proteins.\n\nAn opioid activity of peptides derived from partial enzymatic digestions of milk proteins and wheat gluten was reported in the literature as early as 1979 (Brantl et al., 1979; Zioudrou et al., 1979). Since then, a considerable body of research has been undertaken, and many different bioactive amino acid sequences have been discovered and physiological functions have been defined. The potential for bioactive peptides in the development of functional foods is great. It is now appreciated that bioactive peptides have a wide range of physiological effects, some of which are listed in Table 18.5. Specific bioactive peptides and protein hydrolysates can now be produced commercially, allowing for dietary supplementation and protein fortification. Casein-derived peptides are already in commercial use as food supplements (e.g., phosphopeptides) and as pharmaceuticals (Meisel, 1997).\n\nTable 18.5\n\nReported Effects of Natural Food-derived Peptides\n\nModulation of gastrointestinal motility\n\nStimulation of secretory processes\n\nMineral binding\n\nAntibacterial properties\n\nImmunomodulation\n\nAntithrombotic activity\n\nInhibition of angiotensin-converting enzyme (ACE) in the control of hypertension\n\nAnalgesic (pain relief) and other neuroactive effects\n\nThe remainder of this section focuses on the first two functions listed in Table 18.5 (i.e., gut function), as an example of the potential of food-derived peptides as natural bioactive peptides. Several studies have described the involvement of bioactive peptides (exorphins) in regulating the stomach-emptying rate, gastrointestinal motility, and gut secretory activity in mammals (Rutherfurd-Markwick and Moughan, 2005). A role for bioactive peptides in influencing gut function is not unexpected, as the effects may be mediated both directly and hormonally, involving receptor sites in the gut without the need for absorption and systemic uptake of the peptide.\n\nOur own quantitative studies within the Riddet Institute at Massey University highlight the potential importance of the net effect of food-derived peptides on overall gut metabolism. The gut is a highly metabolic organ, with changes in the rate of organ metabolism having significant implications for total body energetics, protein dynamics, and amino acid and other nutrient requirements. In our series of studies, terminal ileal digesta amino acid or nitrogen flow was determined as an indicator of overall protein dynamics in the upper digestive tract consequent upon the ingestion of a meal. This is reflective of the various cellular and dietary controls on the protein secretion and amino acid reabsorption processes. Endogenous amino acid flows (the net result of secretion and reabsorption) at the end of the ileum were determined following the provision of semisynthetic corn starch-based diets, differing in the source of dietary nitrogen (protein-free, synthetic amino acids, protein, hydrolyzed protein). A range of methods to determine endogenous (of body origin) as opposed to exogenous (diet origin) nitrogen were applied (Moughan et al., 1998).\n\nTable 18.6 gives results for endogenous lysine (a marker for total protein) flow at the end of the small bowel from a representative study from our series of experiments using the pig as a generalized mammalian model. The results clearly demonstrate that when amino acids were present in the gut (either directly from the hydrolyzed casein or after being released from the digestion of dietary zein, a corn-derived protein), endogenous protein loss at the end of the small bowel was significantly enhanced. The peptides led to an enhanced secretion of protein into the gut lumen and\/or a reduced reabsorption of endogenous amino acids. In any case, the loss of extra protein into the colon, whereupon the amino acids are not salvageable, represents a considerable loss of amino acids and is associated with a high metabolic energy cost. Further work has demonstrated that the quite dramatic effect (an almost 60% increase in endogenous flow for the hydrolyzed casein) of dietary peptides is dose dependent (Hodgkinson et al., 2000; Hodgkinson and Moughan, 2006).\n\nTable 18.6\n\nEndogenous Lysine Loss at the Terminal Ileum of the Growing Pig Given Protein-free, Synthetic Amino Acid, Hydrolyzed Casein- or Zein-based Diets\n\n| Diet  \n---|---  \n| PF | SAAa | EHCb | Znc | Significance  \nLysine loss (mg\/kg dry matter intake) | 252 | 284 | 448 | 389 | P < 0.05\n\nPF, protein-free; SAA, synthetic amino acid; EHC, hydrolyzed casein; Zn, zein.\n\na Devoid of lysine, with intravenous lysine infusion.\n\nb Digesta were centrifuged and ultrafiltered (10,000 Da molecular weight cut off).\n\nc Naturally devoid of lysine, with intravenous lysine infusion.\n\nAdapted from Butts et al. (1993), with the permission of the publisher.\n\nThese results, combined with those of several other similar studies, provide compelling evidence for a significant influence of dietary peptides on gut protein dynamics and overall metabolism. Little is known about how these effects are mediated or how the magnitude of the effect is influenced by the source of dietary protein. Claustre et al. (2002) have recently shown that casein and lactalbumin hydrolysates (but not egg or meat hydrolysate) greatly stimulate mucin secretion in rat jejunum. The casein hydrolysate-mediated effect was blocked by the administration of naloxone (an opioid antagonist), and \u03b2-casomorphin-7, an opioid peptide released from \u03b2-casein during digestion, also induced mucin secretion. The peptide effect was also inhibited by naloxone. It may be that the effects are more pronounced with milk proteins.\n\nThe cooperation of ileostomates has allowed our results obtained from animal studies to be confirmed in humans (Moughan et al., 2005b; refer to Table 18.7).\n\nTable 18.7\n\nEndogenous Ileal Lysine and Total Nitrogen Losses (\u03bcg\/g dry matter intake) in Adult Humans Receiving a Protein-free or Hydrolyzed Casein-based Diet\n\n| Diet  \n---|---  \n| PF | EHCa | Significance  \nLysine | 383 | 614 | P < 0.01  \nTotal nitrogen | 2061 | 4233 | P < 0.001\n\nPF, protein-free; EHC, hydrolyzed casein.\n\na Digesta were centrifuged and ultrafiltered (10,000 Da molecular weight cutoff).\n\nAdapted from Moughan et al. (2005b), with the permission of the publisher.\n\nBioactive peptides (and it would seem particularly those arising from the digestion of milk proteins) have been shown to have multiple physiological effects often at modest dietary intakes. As more is understood about these effects, it will become possible to develop novel functional foods. The potential to develop protein hydrolysates, peptide fractions, and commercially synthesized peptides, to target physiological end points associated with gut motility, digestion, energetics and satiety, is particularly promising.\n\n## Conclusions\n\nIn this chapter, a case has been made for the central place of dairy proteins in the development of functional foods and specialized nutritionals. Dairy proteins are a source of highly bioavailable amino acids and offer a diverse range of amino acid patterns and specific amino acid concentration ratios. Certain amino acids found in milk proteins at high concentration have direct physiological as well as nutritional functions in humans. Milk proteins may have antimicrobial and immunomodulatory functions, and also induce and maintain satiety. Additionally, amino acids have a relatively high dietary thermogenesis; thus dairy proteins are ideal for the formulation of specialized weight-loss foods. Almost every week, new information is reported concerning bioactive peptides, which are released in the gut during the natural digestion of milk proteins. Milk is indeed a veritable cornucopia for developing functional foods.\n\n# References\n\nAnderson GH ,  Moore SE .  Dietary proteins in the regulation of food intake and body weight in humans .  _J. Nutr_.  2004 ;134 : 974S \u2013 979S .\n\nAnthony JC ,  Yoshizawa F ,  Gautsch-Anthony T ,  Vary TC ,  Jefferson LS ,  Kimball SR .  Leucine stimulates translation initiation in skeletal muscle of postabsorptive rats via a rapamycin-sensitive pathway .  _J. Nutr_.  2000 ;130 : 2413 \u2013 2419 .\n\nBrantl VH ,  Teschemacher H ,  Blasig J ,  Henschen A ,  Lottspeich F .  Novel opioid peptides derived from casein (-casomorphins). I. 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Exogenous and endogenous nitrogen flow rates and level of protein hydrolysis in the human jejunum after [15N]-labeled milk and [15N]-labeled yogurt ingestion .  _Br. J. Nutr_.  1995 ;74 : 251 \u2013 260 .\n\nGaudichon C ,  Mah\u00e9 S ,  Luengo C ,  Laurent C ,  Meaugeais M ,  Krempf M ,  Tom\u00e9 D .  A 15N-leucine-dilution method to measure endogenous contribution to luminal nitrogen in the human upper jejunum .  _Eur. J. Clin. Nutr_.  1996 ;50 : 261 \u2013 268 .\n\nGaudichon C ,  Bos C ,  Morens C ,  Petzke KJ ,  Mariotti F ,  Everwand J ,  Benamouzig R ,  Dar\u00e9 S ,  Tom\u00e9 D ,  Metges CC .  Ileal losses of nitrogen and amino acids in humans and their importance to the assessment of amino acid requirements .  _Gastroenterology_.  2002 ;123 : 50 \u2013 59 .\n\nGautsch TA ,  Anthony JC ,  Kimball SR ,  Paul GL ,  Layman DK ,  Jefferson LS .  Availability of eIF-4E regulates skeletal muscle protein synthesis during recovery from exercise .  _Am. J. Physiol_.  1998 ;274 : C406 \u2013 C414 .\n\nGrimble GK .  Essential and conditionally-essential nutrients in clinical nutrition .  _Nutr. Res. Rev_.  1993 ;6 : 97 \u2013 119 .\n\nHall WL ,  Millward DJ ,  Long SJ ,  Morgan LM .  Casein and whey exert different effects on plasma amino acid profiles, gastrointestinal hormone secretion and appetite .  _Br. J. Nutr_.  2003 ;89 : 239 \u2013 248 .\n\nHodgkinson SM ,  Moughan PJ .  An effect of dietary protein content on endogenous ileal lysine flow in the growing rat .  _J. Sci. Food Agric_.  2006 ;87 : 233 \u2013 238 .\n\nHodgkinson SM ,  Moughan PJ ,  Reynolds GW ,  James KAC .  The effect of dietary peptide concentration on endogenous ileal amino acid loss in the growing pig .  _Br. J. Nutr_.  2000 ;83 : 421 \u2013 430 .\n\nInstitute of Food Technologists. 2005. Expert Report onFunctional Foods. Available at: www.ift.org.\n\nJonker R ,  Engelen MPKJ ,  Deutz NEP .  Role of specific dietary amino acids in clinical conditions .  _Br. J. Nutr_.  2012 ;108 : S139 \u2013 S148 .\n\nKimball SR ,  Jefferson LS .  Regulation of protein synthesis by branched-chain amino acids .  _Current Opinion in Clinical Nutrition and Metabolic Care_.  2001 ;4 : 39 \u2013 43 .\n\nLayman DK .  Role of leucine in protein metabolism during exercise and recovery .  _Canadian Journal of Applied Physiology_.  2002 ;27 : 592 \u2013 608 .\n\nLayman DK .  The role of leucine in weight loss diets and glucose homeostasis .  _J. Nutr_.  2003 ;133 : 261S \u2013 267S .\n\nMah\u00e9 S ,  Benamouzig R ,  Gaudichon C ,  Huneau JF ,  De Cruz I ,  Tom\u00e9 D .  Nitrogen movements in the upper jejunum lumen in humans fed low amounts of caseins or \u03b2-lactoglobulin .  _Gastroenterology and Clinical Biology_.  1995 ;19 : 20 \u2013 26 .\n\nMah\u00e9 S ,  Roos N ,  Benamouzig R ,  Davin L ,  Luengo C ,  Gagnon L ,  Gausser\u00e8s N ,  Rautureau J ,  Tom\u00e9 D .  Gastrojejunal kinetics and the digestion of [15N]\u03b2-lactoglobulin and casein in humans: The influence of the nature and quantity of the protein .  _Am. J. Clin. Nutr_.  1996 ;63 : 546 \u2013 552 .\n\nMcNurlan M .  New perspectives in the control of body protein metabolism .  _Br. J. Nutr_.  2012 ;108 : S94 \u2013 S104 .\n\nMeisel H .  Biochemical properties of regulatory peptides derived from milk proteins .  _Biopolymers_.  1997 ;43 : 119 \u2013 128 .\n\nMoughan PJ .  Amino acid availability: Aspects of chemical analysis and bioassay methodology .  _Nutr. Res. Rev_.  2003 ;16 : 127 \u2013 141 .\n\nMoughan PJ .  Functional foods .  In:  Moughan PJ ,  McCool P ,  eds.  _Floreat Scientia_ .  New Zealand :  Wairau Press ;  2011 : 218 \u2013 221 .\n\nMoughan PJ ,  Rutherfurd SM .  A new method for determining digestible reactive lysine in foods .  _J. Agric. Food Chem_.  1996 ;44 : 2202 \u2013 2209 .\n\nMoughan PJ ,  Rutherfurd SM .  Gut luminal endogenous protein: Implications for the determination of ileal amino acid digestibility in humans .  _Br. J. Nutr_.  2012 ;108 : S258 \u2013 S263 .\n\nMoughan PJ ,  Cranwell PD ,  Darragh AJ ,  Rowan AM ,  The domestic pig as a model for studying digestion in humans .  Souffrant WB ,  Hagemeister H ,  eds.  _Digestive Physiology in the Pig_ ,  Volume II .  Dummerstorf, Germany :  Forschungsinstitut fur die Biologie Landwirtschaftlicher Nutztiere (FBN) ;  1994 : 389 \u2013 396 .\n\nMoughan PJ ,  Souffrant WB ,  Hodgkinson SM .  Physiological approaches to determining gut endogenous amino acid flows in the mammal .  _Arch. Anim. Nutr_.  1998 ;51 : 237 \u2013 252 .\n\nMoughan PJ ,  Butts CA ,  van Wijk H ,  Rowan AM ,  Reynolds GW .  An acute ileal amino acid digestibility assay is a valid procedure for use in human ileostomates .  _J. Nutr_.  2005 ;135 : 404 \u2013 409 .\n\nMoughan PJ ,  Butts CA ,  Rowan AM ,  Deglaire A .  Dietary peptides increase endogenous amino acid losses from the gut in adults .  _Am. J. Clin. Nutr_.  2005 ;81 : 1359 \u2013 1365 .\n\nPortman R ,  Bakal A ,  Peikin SR .  Ingestion of a premeal beverage designed to release CCK reduces hunger and energy consumption in overweight females .  _Obes. Res_.  2000 ;8 (Suppl 1) : PB59 (Abstract) .\n\nRoberfroid MB .  Concepts and strategy of functional food science: The European perspective .  _Am. J. Clin. Nutr_.  2000 ;71 (Suppl) : 1660S \u2013 1664S .\n\nRutherfurd SM ,  Moughan PJ .  The digestible amino acid composition of several milk proteins: Application of a new bioassay .  _J. Dairy Sci_.  1997 ;81 : 909 \u2013 917 .\n\nRutherfurd SM ,  Moughan PJ .  Digestible reactive lysine in selected milk-based products .  _J. Dairy Sci_.  2005 ;88 : 40 \u2013 48 .\n\nRutherfurd-Markwick KJ ,  Moughan PJ .  Bioactive peptides derived from food .  _J. AOAC Int_.  2005 ;88 : 955 \u2013 966 .\n\nRutherfurd SM ,  Torbatinejad NM ,  Moughan PJ .  Available (ileal digestible reactive) lysine in selected cereal-based food products .  _J. Agric. Food Chem_.  2006 ;54 : 9453 \u2013 9457 .\n\nSandstrom B ,  Andersson H ,  Kivisto B ,  Sandberg AS .  Apparent small intestinal absorption of nitrogen and minerals from soy and meat-protein based diets .  _A study on ileostomy subjects. J. Nutr_.  1986 ;116 : 2209 \u2013 2218 .\n\nTom\u00e9 D ,  Bos C .  Dietary protein and nitrogen utilization .  _J. Nutr_.  2000 ;130 : 1868S \u2013 1873S .\n\nvan de Poll MCG ,  Luiking YC ,  Dejong CHC ,  Soeters PB .  Amino acids\u2014specific functions .  In:  Caballero B ,  Allen L ,  Prentice A ,  eds.  _Encyclopedia of Human Nutrition_ .  2nd ed. Amsterdam :  Elsevier ;  2006 : 92 \u2013 100 .\n\nVandewater K ,  Vickers Z .  Higher-protein foods produce greater sensory-specific satiety .  _Physiol. Behav_.  1996 ;59 : 579 \u2013 583 .\n\nWard RE ,  German JB .  Understanding milk's bioactive components: A goal for the genomics toolbox .  _J. Nutr_.  2004 ;134 : 962S \u2013 967S .\n\nWesterterp-Plantenga MS ,  Lejeune MPGM .  Protein intake and body-weight regulation .  _Appetite_.  2005 ;45 : 187 \u2013 190 .\n\nWu G ,  Morris SM .  Arginine metabolism: Nitric oxide and beyond .  _Biochem. J_.  1998 ;336 : 1 \u2013 17 .\n\nZioudrou C ,  Streaty RA ,  Klee WA .  Opioid peptides derived from food proteins, the exorphins .  _J. Biol. Chem_.  1979 ;254 : 2446 \u2013 2449 . \nChapter 19\n\n# Milk Proteins and Human Health\n\nRobin A. McGregor*,**\n\nSally D. Poppitt*,**,***\n\n* Human Nutrition Unit, University of Auckland, Auckland, New Zealand  \n** Institute for Innovation in Biotechnology, School of Biological Sciences, University of Auckland, Auckland, New Zealand  \n*** Riddet Institute, Palmerston North, New Zealand\n\n## Abstract\n\nCow's milk proteins and related bioactive peptides are purported to have a wide range of effects on human health across the life span. Casein and whey protein have both been proposed to play a role in the prevention of chronic age-related conditions such as adverse metabolic health and type 2 diabetes, muscle wasting and sarcopenia, atherosclerosis, hypertension and cardiovascular disease risk, as well as bone health and osteoporosis. Epidemiological studies have shown regular dairy consumption to be associated with decreased prevalence of cardiometabolic risk, an outcome of improved metabolic health, while intervention studies have shown milk proteins to promote postprandial insulin secretion and glycemic control, and under some conditions improve postprandial lipemia and hypertension. The branched-chain amino acids of dairy protein also promote muscle anabolism, important for the maintenance of muscle mass and mobility during aging and, in our current epidemic of obesity, for the maintenance of lean body mass during energy restriction and weight loss. Dietary protein is known to suppress satiety and food intake, and if consumed from a dairy source may support preferential loss of fat rather than lean mass during weight loss. Milk proteins may also be beneficial for bone health during aging. In addition, milk proteins are important for maternal and infant nutrition, with protein playing a major role in growth and development during early life. While public health recommendations for breastfeeding underpin nutrition in young infants, commercial infant formulas are required for mothers unable to breastfeed long term, and formulas containing cow's milk protein aim to optimize infant health. In this chapter we review the evidence for the potential health benefits of milk proteins, based on human clinical trials, with a particular focus on metabolic health.\n\n## Keywords\n\nMilk protein\n\nwhey protein\n\ncasein\n\nmetabolic health\n\ncardiovascular\n\nskeletal muscle\n\nbone\n\ninfant\n\nOutline\n\nIntroduction 541\n\nMilk proteins, metabolic health, and type 2 diabetes 542\n\nMilk proteins, obesity, and weight control 543\n\nMilk Proteins, Muscle Wasting, and Sarcopenia 545\n\nMilk Proteins and Heart Health 547\n\nAtherosclerosis 547\n\nBlood Pressure and Vascular Reactivity 547\n\nMilk proteins and bone health 548\n\nMilk Proteins and Infant Health 549\n\nConclusions 550\n\n## Introduction\n\nMilk provides a rich source of proteins that are not found in any other food source and that have a range of putative positive health outcomes in both adults and children. Epidemiology has pointed to a positive association between cow's milk and human health, the underpinning mechanisms of which are as yet not well understood (Elwood et al., 2008). Humans consume milk primarily from cows, but around the world communities consume milk from many other animals, including buffalo, yaks, goats, sheep, reindeer, and camels. Cow's milk protein consists of approximately 80% (w\/w) casein and 20% (w\/w) whey protein, both of which may elicit beneficial health outcomes, and which are the source of a wide range of peptides with potential bioactive properties. The processing of milk protein, which may be an important factor in digestion and absorption kinetics, may also have physiological effects and in turn potential health benefits (Morr and Ha, 1993; Dalgleish and Corredig, 2012).\n\nIn clinical studies, whey protein concentrate (WPC), micellar casein, or sodium caseinate have most commonly been investigated, but trials have also used milk protein concentrate (MPC), calcium caseinate, casein hydrolysate, whey hydrolysates, and whey protein isolate (WPI) as sources of cow's milk protein. The production of these products is described elsewhere in this volume (see in particular Chapters 2 and ).\n\nThe impact of milk proteins and their associated peptides on health outcomes across the life span is a growing area of research interest. Whey protein in particular has been a focus due to its properties of rapid absorption, serum AA profile, and insulinotropic effects among other properties of interest. In this chapter, we review the evidence (primarily from human clinical trials) that milk proteins may improve or prevent a range of age-related chronic health conditions, particularly those associated with metabolic health, including metabolic syndrome, type 2 diabetes (T2DM), atherosclerosis, and hypertension, as well as the role that they may play in the control of body weight and maintenance of lean body and\/or skeletal muscle mass during aging and weight loss. We also review recent evidence in support of bone health and maternal and infant nutrition.\n\n## Milk proteins, metabolic health, and type 2 diabetes\n\nOver the past two decades poor metabolic health has emerged as a major public health concern. It has been linked to a more sedentary lifestyle and poor diet, and is a forerunner of T2DM and adverse cardiovascular health. Metabolic syndrome has been defined to represent a cluster of commonly measured metabolic markers of adverse health, including abdominal obesity, hyperglycemia, dyslipidemia, and hypertension (Grundy et al., 2005; Alberti et al., 2009). These provide intermediary biomarkers of poor health that can be targeted for improvement. Individuals with metabolic syndrome are at higher risk of developing T2DM and also cardiovascular disease (CVD) (Alberti et al., 2009), and younger, lean individuals typically have a better metabolic profile than older, overweight men and women. This gradual impairment is commonly accompanied by physiological changes such as an excess accumulation of lipids in adipose tissue and subsequent overspill into liver, skeletal muscle, and other tissues. Adipose tissue provides a major lipid storage depot; however, excessive expansion of abdominal and possibly other adipose stores and consequent lipid overspill and infiltration into other tissues is linked to release of proinflammatory mediators (Despres and Lemieux, 2006). Blunting of CHO, fat and protein metabolism, insulin resistance, and impairment in endothelial function are other common consequences. Metabolically active lean body mass is important for good metabolic health, not least as a mediator for glucose regulation among many other roles. For example, skeletal muscle accounts for up to 75% of whole-body glucose uptake. In combination, progressive loss of skeletal muscle and excess accumulation of abdominal fat (recently termed 'sarcobesity' (Parr et al., 2013)) can severely impair metabolic health.\n\nEvidence from large epidemiological studies has shown that the consumption of dairy products may be associated with a lower risk of metabolic disorders and CVD (Elwood et al. 2008; Rice et al., 2011), which in turn has been attributed to milk protein and in particular to the whey protein component. Insulin is sensitive to both the composition and concentration of plasma AAs, and both whey protein and casein ingestion stimulate increased insulin secretion (Nilsson et al. 2004; 2007;). Ingestion of whey protein leads to more rapid secretion of insulin than micellar casein (Boirie et al., 1997), a consequence of its rapid absorption kinetics, and hence underpins better glucose control in metabolically impaired individuals (Pal and Radavelli-Bagatini, 2012). Evidence predominantly generated from single-meal postprandial studies shows that milk proteins may increase tissue glucose uptake and suppress post-meal blood glucose fluctuations (Claessens et al., 2008; ; Pal and Ellis, 2010a; Akhavan et al., 2010).\n\nThese effects have also been shown in patients with T2DM. In a postprandial study of T2DM patients, the addition of 18 g whey protein into a breakfast or lunch meal caused a marked increase in plasma insulin concentrations compared to an isoenergetic substitution of nondairy protein and carbohydrate (CHO). The higher insulin levels were associated with a greater suppression of postprandial blood glucose levels (Frid et al., 2005). Casein also has effects on blood glucose. In a group of overweight individuals with T2DM, casein hydrolysate (0.3 g\/kg) with leucine (0.1 g\/kg) decreased plasma glucose levels over 24 hours when consumed after breakfast, lunch, and dinner (Manders et al., 2006). Notably, however, the casein supplement had been enriched in this study with the branched-chain amino acid (BCAA) leucine. Conversely, ingestion of 40 g casein hydrolysate at breakfast, lunch, and dinner caused little improvement in 24-hour blood glucose levels in a group of patients with long-standing T2DM. The discrepancy in outcome has been attributed to impairment of the insulin-secreting pancreatic beta-cell in long-standing T2DM patients, impairing the response to the milk protein (Manders et al., 2009). This remains unconfirmed, with some preliminary evidence that free AAs may in fact reactivate the insulin secretory mechanism in this metabolically impaired T2DM patient group (van Loon et al., 2003). Obesity and metabolic syndrome are often accompanied by the development of nonalcoholic fatty liver disease (NAFLD), where excessive lipids accumulate in the liver, eventually leading to fibrosis and cirrhosis. Experimental evidence from rodents has shown whey protein to decrease lipid accumulation in the liver (Hamad et al., 2011), but there are as yet no clinical studies.\n\nThere have been few long-term clinical studies of milk proteins on hyperglycemia. In overweight and obese individuals with increased metabolic risk through impaired glucose tolerance (IGT), raised plasma triglyceride (TG) levels, low HDL-cholesterol levels, and\/or abdominal obesity, daily ingestion of a high-dose WPI supplement (54 g\/day) for 12 weeks successfully decreased fasting blood insulin levels and insulin resistance but unexpectedly did not in turn decrease fasting glucose concentrations (Pal et al., 2010a). Clearly, more long-term studies of insulin and glucose control are needed.\n\n## Milk proteins, obesity, and weight control\n\nThere is growing evidence that high-protein diets may be efficacious for weight loss and\/or longer-term weight-loss maintenance (Larsen et al., 2010), but the issue as to whether different protein sources may be more or less successful is not well understood. The primary mechanisms that underpin high-protein diets or protein-based supplements for weight loss are enhanced appetite control and suppression of food intake (Poppitt et al., 1998; Anderson and Moore, 2004; Paddon-Jones et al., 2008; Dove et al., 2009). Sparing of skeletal muscle in favor of adipose loss during weight reduction through use of a higher-protein (>20% of energy as protein), energy-restricted diet may also be important. This is of particular relevance to dairy protein, where the high BCAA content drives muscle protein synthesis, which will contribute to the maintenance of metabolically active lean body mass. High-protein diets suppressing hunger and food intake have been reported in many studies, although the majority of data have been from shorter-term, postprandial interventions. Gradually, evidence for longer-term suppression and weight loss is growing (Clifton et al., 2008; Larsen et al., 2010). Milk proteins are commonly used in meal replacement drinks, which provide a convenient way to undertake a low-energy, high-protein diet. Some preliminary data show that milk proteins may promote greater satiety than other protein sources, but it is not well substantiated (Anderson et al., 2004; Veldhorst et al., 2009; Baer et al., 2011). Whether whey protein or casein may have differential effects is also not well understood. In a recent study, ingestion of an isoenergetic bolus of skimmed milk containing both casein and whey protein was reported to decrease energy intake more than protein alone (Lorenzen et al., 2012). In contrast, other studies have reported whey protein to be more anorectic than casein (Hall et al., 2003), with \u03b1-lactalbumin proposed as the whey protein fraction responsible for the greatest suppression of intake in a study comparing whey, whey protein plus glycomacropeptide (GMP), casein, and soy (Veldhorst et al., 2009). In a recent study on overweight women, there was no difference in energy intake between four whey protein fractions comprising WPC, colostrum WPC, GMP, and \u03b2-lactoglobulin, despite promising rodent studies, although \u03b2-lactoglobulin induced greater fullness (Poppitt et al., 2013). GMP has long been purported to differentially suppress appetite, but there is little evidence to support this hypothesis (Veldhorst et al., 2009; Keogh et al., 2010).\n\nTo date, there have been few long-term studies investigating the role of milk proteins in weight loss, in the absence of other lifestyle interventions. A 6-month study of overweight and obese individuals reported that 56 g\/day WPC resulted in significant loss of body weight, fat mass, and waist circumference compared to a CHO control (Baer et al., 2011), while a shorter 12-week study found no effect of 54 g\/day of WPI (Pal et al., 2010a). Interestingly, fasting blood lipids and insulin levels improved in this study, supporting the hypothesis that whey protein may improve metabolic health even in the absence of weight loss (Pal et al., 2010a). Energy-restricted diets are a common way to successfully decrease body weight, at least in the short term, but characteristic of this is loss of both fat mass and lean mass. Loss of lean mass results in concomitant reduction in basal metabolic requirements, and a return to habitual dietary habits may then result in rapid weight regain. Middle-aged or older individuals with the age-related sarcopenia and obesity known as 'sarcobesity' (Parr et al., 2013) who lose weight through traditional energy-restricted diets may be even more susceptible to rapid weight regain due to already low muscle mass. Ingestion of whey protein as part of an energy-restricted diet has been proposed as a strategy to decrease fat mass while preserving lean mass. There is some data to support this proposal. In a study of obese individuals undertaking a severely energy-restricted diet of \u223c2 MJ\/day for 12 weeks, supplementation twice daily with a milk protein successfully led to greater weight loss, fat loss, and maintenance of lean muscle mass (Frestedt et al., 2008). Another similar study supplementing with a high-protein meal replacement comprising whey, soy, and free AAs led to greater loss of body fat compared to a low-protein meal replacement (Treyzon et al., 2008). Milk proteins may also be beneficial for maintaining a lower body weight after energy restriction and fat loss. In a study of whey protein and casein supplementation, there was significantly better weight loss maintenance over a 12-week period compared with a CHO control (Claessens et al., 2009a).\n\nWhether whey protein or casein supplements may be more effective for preservation of lean muscle mass during periods of weight loss induced by an energy-restricted diet was addressed in a recent trial (Adechian et al., 2012) where obese individuals underwent a 6-week energy-restricted diet with whey protein or casein supplementation. While all individuals showed weight loss after 6 weeks, there were no differential effects of whey protein or casein for weight loss, fat loss, or preservation of lean muscle mass. Intriguingly, assessment of whole body protein synthesis and whole body protein breakdown showed casein to cause greater inhibition of protein breakdown, while whey protein increased protein synthesis (Adechian et al., 2012). The implications are that casein supplementation may be optimal for preservation of skeletal muscle mass during energy restriction. Notably, however, no long-term trials have as yet compared milk proteins with other protein sources for long-term weight loss and maintenance.\n\n### Milk Proteins, Muscle Wasting, and Sarcopenia\n\nAdvancing age and a sedentary lifestyle is associated with a gradual decline in skeletal muscle mass, function, and strength, which in the extreme form is termed sarcopenia or muscle wasting. Loss of skeletal muscle mass or function of the muscle has major implications for quality of life since activities of daily living such as walking upstairs become difficult or are no longer possible. At the extreme end, patients with chronic or end-stage diseases, including cancer, heart failure, AIDS, and chronic obstructive lung disease and sepsis, are also often susceptible to muscle wasting (Tan and Fearon, 2008; Fearon et al., 2011).\n\nIn addition to these mobility issues, skeletal muscle also has a significant impact on metabolic health. As one of the major organs responsible for insulin-stimulated glucose uptake, loss of skeletal muscle mass is often associated with insulin resistance (Evans, 2010). In addition to the insulin stimulatory effect of milk protein, it is the high level of BCAAs in whey protein and casein that prevents loss of lean body mass through increased skeletal muscle protein synthesis and\/or decrease in breakdown (Adechian et al., 2012). There is evidence that the anabolic effect of milk protein is decreased during aging, which has been termed anabolic resistance (Volpi et al., 2000). Whether this is an anabolic resistance to dietary intake of BCAAs or simply a reflection of underutilization of the major muscle groups in older people as exercise levels decline is a matter of considerable debate. Whether milk proteins can prevent the development of anabolic resistance or overcome established anabolic resistance in older individuals is of great interest.\n\nBoth circulating insulin and AAs are important for the activation of muscle protein synthesis. Whey protein and casein are high-quality proteins based on both the protein digestibility corrected AA score (PDCAAS) (Boye et al., 2012) and the recently developed digestible indispensable AA score (DIAAS) (FAO, 2013), both of which take into account human AA requirements and protein digestibility. However, whey protein contains a greater amount of the BCAAs leucine, isoleucine, and valine than does casein. Of the BCAAs, leucine is thought to be the most potent activator of protein synthesis (Katsanos et al., 2006; van Loon, 2012), although a recent study has shown that high levels of nonleucine BCAAs can induce equivalent protein synthesis when given with a whey protein supplement (Churchward-Venne et al., 2012). Casein in turn contains several essential amino acids (EAAs), including histine, methionine, and phenylalanine in a greater amount than whey protein, and also contains a greater amount of the non-EAAs arginine, glutamic acid, proline, serine, and tyrosine (Hall et al., 2003). The EAAs have been demonstrated to be primarily responsible for the activation of muscle protein synthesis (Volpi et al., 2003), although they are not necessarily efficacious for inhibition of protein breakdown. As noted above, studies have demonstrated greater whole body protein synthesis following ingestion of whey protein versus casein, while protein catabolism was greater following casein supplementation (Boirie et al., 1997; Adechian et al., 2012). The faster digestion rate of whey protein compared with casein, which due to its micellar structure tends to clot in the stomach, has been commonly believed to lead to more rapid delivery of AAs into plasma following whey protein, and longer, more sustained delivery following casein ingestion. There is certainly a body of data showing greater muscle protein accretion over the 6 hours following whey protein supplementation compared to casein or casein hydrolysate (Pennings et al., 2011).\n\nConversely, a recent study of intrinsically labeled whey protein and casein, co-ingested as milk, showed the absorption and retention of AAs from whey protein and casein to be similar after 2 hours, with AAs derived from casein showing greater absorption and retention rates beyond 3 hours (Soop et al., 2012). Some discrepancies in outcome might be due to the differential effects of aging on the response to these two dairy proteins. It may be worthwhile to increase the typically low whey protein content of milk, which then theoretically would provide both an early (whey) and sustained (casein) stimulation of muscle protein synthesis and an inhibition of muscle protein breakdown (Reitelseder et al., 2011). However, it remains to be seen whether milk protein-enhanced muscle protein synthesis is effective for prevention of muscle wasting and promotion of muscle protein accretion in healthy aging or elderly populations, or in patients with chronic end-stage disease. Additional evidence is also required to confirm that this in turn results in functional improvements in strength and\/or mobility.\n\nRegular resistance-type exercise transiently activates muscle protein synthesis and over time can lead to increases in skeletal muscle mass. Milk protein supplements are popular among recreational gym users seeking to increase muscle mass, but there is also research interest in whether milk protein ingestion in conjunction with resistance exercise may be beneficial for individuals with sarcopenia. Whey protein and casein ingestion after resistance exercise have been shown to both cause comparable increases in net protein balance (Tipton et al., 2004) and myofibrillar protein synthesis rate (Reitelseder et al., 2011), while other data show that whey protein ingestion causes greater increase in the muscle protein synthesis rate during the early 3-h period (Tang et al., 2009). Longer-term studies to date have produced mixed findings. In obese postmenopausal women, WPI in combination with an energy-restricted diet plus exercise over 6 months led to 4% greater weight loss than in the control group, and notably greater loss of subcutaneous adipose tissue and greater increase in leg muscle mass (Mojtahedi et al., 2011). Conversely, in elderly men who undertook resistance training for 12 weeks, protein supplementation did not improve muscle hypertrophy (Verdijk et al., 2009). Although the effects on muscle anabolism and catabolism are clear, the evidence underpinning clinically significant gains in lean body remain equivocal (Cermak et al., 2013).\n\n### Milk Proteins and Heart Health\n\n#### Atherosclerosis\n\nAtherosclerosis is a common cause of myocardial infarction, stroke, and peripheral vascular disease. It represents the progressive damage to the vascular endothelium due to build-up of lipids, immune cell infiltration, and plaque formation, leading to impaired endothelial function and reduced blood flow. It has been suggested that milk protein may improve adverse circulating lipid levels, one of the primary metabolic risk factors for CVD. Most of these studies have focused on the effects on postprandial lipemia, based on the premise that rapid clearance of blood lipids after a meal decreases arterial exposure to these circulating lipids. One study in overweight postmenopausal women given a high-fat meal found WPI and sodium caseinate to both decrease circulating TG and TG:ApoB48 ratio compared with glucose ().\n\nAnother study of obesity compared the effect of different milk protein fractions, including WPI, whey protein hydrolysate, \u03b1-lactalbumin, and GMP, on postprandial lipemia after a high-fat meal, but found no significant differences (Holmer-Jensen et al., 2012). Whey protein has also been shown to suppress postprandial circulating TG, free fatty acids (FFA), and chylomicron-rich lipoprotein appearance in diabetic patients following a high-fat meal, compared to controls of casein, fish, or plant protein sources (Mortensen et al., 2009). Casein alone, on the other hand, failed to improve postprandial TG in T2DM patients (Brader et al., 2010). There have been few long-term clinical studies. A 3-month trial of a fermented whey product in individuals with metabolic syndrome found some improvements in metabolic markers, although confounding effects on body weight make this trial difficult to interpret (Gouni-Berthold et al., 2012).\n\n#### Blood Pressure and Vascular Reactivity\n\nObservational and clinical studies have shown that the consumption of dairy products is associated with decreased risk of hypertension (Soedamah-Muthu et al., 2012). Much work has focused on identifying and isolating the bioactive peptides that may be responsible. The discovery that milk-derived peptides inhibit angiotensin converting enzyme (ACE) activity, and hence alter vasoconstriction, vasodilation, and blood pressure (BP) in vitro, led to a plethora of animal and subsequently human trials. Despite promising evidence that lactokinins or caseinkinins can reduce BP in spontaneously hypertensive animals, these findings are yet to be confirmed.\n\nThe most well-studied milk protein-derived peptides are isoleucine-proline-proline (IPP) and valine-proline-proline (VPP), which are generated by the fermentation of milk (Saito, 2008; Boelsma and Kloek, 2009). IPP and VPP have been shown to be weak ACE inhibitors based on in vitro experiments (FitzGerald and Meisel, 2000). Several meta-analyses of clinical trials of IPP and VPP on BP have been published which suggest these milk-derived peptides may have antihypertensive effects in humans (Xu et al., 2008; Turpeinen et al., 2013; Cicero et al., 2013). For example, a recent meta-analysis of 19 clinical trials of a daily dose of <10 mg milk tripeptides on BP over the last 15 years reported that overall systolic BP was decreased \u20134.0 mmHg, and diastolic BP was decreased \u20131.9 mmHg (Turpeinen et al., 2013). Previous meta-analysis reports also reported that VPP and IPP supplementation resulted in significant decreases in systolic and diastolic BP, although suppression was greater in hypertensive individuals (Xu et al., 2008).\n\nNot all studies have been positive, however. A European trial of VPP and IPP supplementation in a workplace environment found no effect on BP (Engberink et al., 2008). In the few human trials that assessed changes in ACE activity, neither VPP nor IPP appears to result in detectable inhibition of in vivo ACE activity. Hence, the mechanism behind the BP lowering effects observed in many studies is not yet clear. A recent report from the European Food Safety Authority (EFSA), which assesses the scientific basis for health claims of nutraceutical food products, concluded there was currently insufficient evidence to substantiate the claim that consumption of the milk-derived peptides IPP or VPP help maintain normal BP (European Food Safety Authority, 2012). Efforts are ongoing to identify further milk-derived peptides with potential to improve cardiovascular health.\n\nThere is evidence that supplementation with intact milk proteins may also potentially exert beneficial effects on endothelial function and BP control. An acute trial showed that ingestion of 5 g whey protein extract increased brachial artery flow mediated dilation in older, overweight individuals (Ballard et al., 2012). This was not associated with an inhibition of ACE activity. Hence regulation of vascular reactivity\/endothelial function takes place by some other as yet unknown mechanism. In overweight individuals, whey protein hydrolysate or WPC (28 g\/day) decreased systolic and diastolic BP compared to baseline, but only in subjects with hypertension at the start of the trial, and was again not associated with any detectable changes in ACE activity. A more recent study found 12-week supplementation with WPI or sodium caseinate (54 g\/day) decreased BP compared to glucose supplementation (Pal and Ellis, 2010b). In summary, human intervention studies suggest that milk peptides or milk protein supplementation may improve hypertension, but this is primarily in individuals with increased risk. Moreover, the mechanism underlying these improvements is not well understood.\n\n## Milk proteins and bone health\n\nBone remodeling occurs across the life span in response to environmental changes such as diet, physical activity, and external loading. Nutrition during childhood and adolescence is one of the major factors that influences the development of bone mass and strength, alongside physical activity, endocrine status, and exposure to risk factors (Caroli et al., 2011). Calcium, protein, and vitamin D are essential nutrients for bone development and bone health maintenance (Pampaloni et al., 2011), and dairy foods such as milk and cheese are a rich source of both Ca and protein, arguably providing an optimal source of essential nutrients for bone health (Caroli et al., 2011). Aging is associated with decreased bone mineral density (BMD), which leads to increased susceptibility to fracture in the elderly.\n\nAlthough Ca derived from cow's milk has long been shown to ameliorate bone loss (Tang et al., 2007) and provides a more conservative and arguably safer route by which to increase dietary Ca than through supplementation with Ca salts per se (Reid et al., 2006; Bolland et al. 2008; 2010;), recent studies show that cow's milk protein may influence bone remodeling (Tsuji-Naito and Jack, 2012). In these studies, whey proteins have been shown to promote growth of bone cells through a number of mechanisms, including stimulation of osteoblast differentiation, activation of intracellular signaling molecules, increased alkaline phosphatase activity, and mineralization (Tsuji-Naito and Jack, 2012). It has long been known that milk protein supplementation increases serum insulin-like growth factor 1 (IGF-1). In turn it has been hypothesized that this may be a mechanism for protection of BMD. Supplementation with MPC over a period of 6 months successfully increased serum IGF-1 in elderly individuals with recent osteoporotic hip fracture, decreased mean rehabilitation ward stay to 33 days compared with 54 days in control patients, and 6 months after the intervention was completed continued to show attenuation of proximal femur BMD loss (Sch\u00fcrch et al., 1998). Conversely, in a longer 2-year study of elderly women, while a high-protein whey drink increased IGF-1 levels when measured after 1 and 2 years of supplementation, there were no effects on BMD outcomes (Zhu et al., 2011). Whether whey protein may enhance the effects of multinutrient supplementation for improving recovery from fractures in the elderly is of interest, but is yet to be established.\n\nThe effects of overweight, obesity, and weight loss have long been of concern to the field of bone remodeling and fracture risk. Energy-restricted diets for weight loss have long been reported to increase the risk of bone loss, possibly due to inadequate energy and nutrients for maintenance of BMD (Villareal et al., 2006). Interestingly, there is some evidence that bone health may be preserved when the diet is rich in dairy products (Josse et al., 2012). Which component of dairy may be responsible for this protective effect, however, is not clear.\n\n### Milk Proteins and Infant Health\n\nAppropriate early nutrition is essential for the growth and development of infants. The World Health Organization and other health bodies recommend breastfeeding infants up to 6 months of age and continuing breastfeeding with complementary foods up to 2 years of age. However, commercial infant formulas (IF) have been widely developed to provide appropriate nutrition should breastfeeding not be possible. Cow's milk proteins are often used as the main protein source in IF, the composition of which is governed by strict regulatory guidelines. Milk protein, in particular whey-modified protein, is used as a primary protein source. Cow's milk requires modification in order for it to be the basis of IF since it contains two to three times the level of total protein compared with human breast milk, and has a different protein composition. Whether high-protein IFs may exceed infant requirements is widely debated, with discussion focused on later development of obesity and associated noncommunicable diseases (NCDs) (Michaelsen et al., 2012), although systematic reviews examining associations between early feeding and later-life obesity or BMI are not conclusive (Owen et al., 2005a,b).\n\nCertainly there may be differences in infant body composition driven by feeding practices (Gale et al., 2012). In light of these issues, IFs with low protein levels close to that of human milk have been developed, and these lower protein content IFs remain an area of great interest and a current 'hot topic.' Differences in protein composition between human and cow's milk include the whey protein:casein ratio and the level of free AAs, which are high in human milk. Mature human milk contains \u223c60% whey protein, of which \u03b1-lactalbumin is the major component, and \u223c40% casein. It also has high levels of EAAs. Conversely, the ratio of whey protein:casein in cow's milk is 20:80, and it contains a major whey protein \u03b2-lactoglobulin that is absent in human breast milk. The primary objective in protein-modified IFs has been to achieve a low-protein composition with sufficient indispensable and conditionally indispensable AAs but with decreased nonessential AAs. Adequate protein nutrition in infants given low-protein IF (as demonstrated by comparable growth curves to infants fed standard IF) has been demonstrated in formulas enriched with components such as \u03b1-lactalbumin (Lien et al., 2004).\n\nIn low-birth-weight (LBW) infants, dairy protein provides a useful way to increase intake in order to improve growth and development. A systematic review of IF feeding showed that 3\u20134 g\/kg body weight per day of protein may successfully accelerate weight gain (Premji et al., 2006). Again there has been debate as to whether this accelerated growth may increase susceptibility to weight and adipose gain in later life, with data showing that it may be associated with school age obesity (Koletzko et al., 2009). Studies are ongoing to establish whether the benefits of increased protein intake in infancy may be offset in later adolescence and adulthood, or whether IF-fed infants have better long-term health in adult life. Outcomes remain equivocal, with prospective studies showing both IF and breast milk to be more beneficial in areas such as bone health in later adult life (Fewtrell et al., 2009; M\u00f8lgaard et al., 2011; Piril\u00e4 et al., 2011).\n\nAn area of protein composition that has been reviewed in detail is immune function. Breast milk contains a range of fractions, including immunoglobulins, lactoferrins, antibodies, macrophages, neutrophils, lymphocytes, and cytokines among others, which may aid immune development (Field, 2005). Because of the importance of this fact, IF commonly also contains factors that may help support immunity. In addition, the use of hydrolyzed proteins for suppression of allergy is of great interest. Hydrolyzed whey protein in IF has been shown to decrease the risk of atopic dermatitis, an inflammatory, chronic form of childhood eczema, after birth and up to 3 years of age. This beneficial effect has been attributed to the presence of immune components such as \u03b2-lactoglobulin, GMP, and lactoferrin (Osborn and Sinn, 2006). Also of possible relevance with respect to allergy is recent evidence from a probiotic supplementation study. Having initially supplemented mothers during pregnancy with two different probiotic strains, these were then given to their infants using a range of delivery formats, including cow's milk IF. This study showed Lactobacillus rhamnosus to decrease the risk of childhood eczema (Wickens et al., 2012). Whether the mother or the infant is the critical window for supplementation, and whether there may be synergies between specific strains of probiotics and the protein component of dairy, is not known.\n\n## Conclusions\n\nThere is an evolving body of evidence showing that milk proteins or milk-derived peptides may have significant functional properties for the prevention and\/or treatment of important chronic health conditions, although much of the science is as yet at an early stage. Clearly, there remains considerable scope for optimizing the processing of whey protein or casein in order to maximize positive health outcomes. The role that dairy protein may play in the prevention of adverse metabolic health and T2DM, atherosclerosis, hypertension and CVD risk, muscle wasting and sarcopenia, as well as osteoporosis, is gradually being elucidated, but research to identify specific protein components, optimum protein dose, synergies with other nutrients, and also the underpinning mechanisms of these positive effects on health are needed. Bovine milk proteins are also an important component of infant formulas and provide an essential alternative when breastfeeding cannot be achieved or maintained. However, further clinical intervention studies, particularly long-term well-controlled studies, are required to build the evidence base necessary to fully support the development of functional foods based on bovine milk proteins for the maintenance and improvement of human health.\n\n##### Disclosures\n\nSDP holds the Fonterra Chair in Human Nutrition at the University of Auckland. 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Formulas containing hydrolysed protein for prevention of allergy and food intolerance in infants. Cochrane. Database. Syst. Rev. CD003664.\n\nOwen CG ,  Martin RM ,  Whincup PH ,  Davey-Smith G ,  Gillman MW ,  Cook DG .  The effect of breastfeeding on mean body mass index throughout life: A quantitative review of published and unpublished observational evidence .  _Am. J. Clin. Nutr_.  2005 ;82 : 1298 \u2013 1307 .\n\nOwen CG ,  Martin RM ,  Whincup PH ,  Smith GD ,  Cook DG .  Effect of infant feeding on the risk of obesity across the life course: A quantitative review of published evidence .  _Pediatrics_.  2005 ;115 : 1367 \u2013 1377 .\n\nPaddon-Jones D ,  Westman E ,  Mattes RD ,  Wolfe RR ,  Astrup A ,  Westerterp-Plantenga M .  Protein, weight management, and satiety .  _Am. J. Clin. Nutr_.  2008 ;87 : 1558S \u2013 1561S .\n\nPal S ,  Ellis V .  The acute effects of four protein meals on insulin, glucose, appetite and energy intake in lean men .  _Br. J. 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Nutr_.  2011 ;2 : 396 \u2013 407 .\n\nSaito T .  Antihypertensive peptides derived from bovine casein and whey proteins .  _Adv. Exp. Med. Biol_.  2008 ;606 : 295 \u2013 317 .\n\nSch\u00fcrch MA ,  Rizzoli R ,  Slosman D ,  Vadas L ,  Vergnaud P ,  Bonjour JP .  Protein supplements increase serum insulin-like growth factor-I levels and attenuate proximal femur bone loss in patients with recent hip fracture. A randomized, double-blind, placebo-controlled trial .  _Ann. Intern. Med_.  1998 ;128 : 801 \u2013 809 .\n\nSheikholeslami Vatani D ,  Ahmadi Kani Golzar F .  Changes in antioxidant status and cardiovascular risk factors of overweight young men after six weeks supplementation of whey protein isolate and resistance training .  _Appetite_.  2012 ;59 : 673 \u2013 678 .\n\nSoedamah-Muthu SS ,  Verberne LD ,  Ding EL ,  Engberink MF ,  Geleijnse JM .  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Behav_.  2009 ;96 : 675 \u2013 682 .\n\nVeldhorst MAB ,  Nieuwenhuizen AG ,  Hochstenbach-Waelen A ,  Westerterp KR ,  Engelen MPKJ ,  Brummer R-JM ,  Deutz NEP ,  Westerterp-Plantenga MS .  A breakfast with alpha-lactalbumin, gelatin, or gelatin + TRP lowers energy intake at lunch compared with a breakfast with casein, soy, whey, or whey-GMP .  _Clin. Nutr_.  2009 ;28 : 147 \u2013 155 .\n\nVerdijk LB ,  Jonkers RAM ,  Gleeson BG ,  Beelen M ,  Meijer K ,  Savelberg HHCM ,  Wodzig WKWH ,  Dendale P ,  Van Loon LJC .  Protein supplementation before and after exercise does not further augment skeletal muscle hypertrophy after resistance training in elderly men .  _Am. J. Clin. Nutr_.  2009 ;89 : 608 \u2013 616 .\n\nVillareal DT ,  Fontana L ,  Weiss EP ,  Racette SB ,  Steger-May K ,  Schechtman KB ,  Klein S ,  Holloszy JO .  Bone mineral density response to caloric restriction-induced weight loss or exercise-induced weight loss: A randomized controlled trial .  _Arch. Intern. Med_.  2006 ;166 : 2502 \u2013 2510 .\n\nVolpi E ,  Kobayashi H ,  Sheffield-Moore M ,  Mittendorfer B ,  Wolfe RR .  Essential amino acids are primarily responsible for the amino acid stimulation of muscle protein anabolism in healthy elderly adults .  _Am. J. Clin. Nutr_.  2003 ;78 : 250 \u2013 258 .\n\nVolpi E ,  Mittendorfer B ,  Rasmussen BB ,  Wolfe RR .  The response of muscle protein anabolism to combined hyperaminoacidemia and glucose-induced hyperinsulinemia is impaired in the elderly .  _J. Clin. Endocrinol. Metab_.  2000 ;85 : 4481 \u2013 4490 .\n\nWickens K ,  Black P ,  Stanley TV ,  Mitchell E ,  Barthow C ,  Fitzharris P ,  Purdie G ,  Crane J .  A protective effect of Lactobacillus rhamnosus HN001 against eczema in the first 2 years of life persists to age 4 years .  _Clin. Exp. Allergy_.  2012 ;42 : 1071 \u2013 1079 .\n\nXu J-Y ,  Qin L-Q ,  Wang P-Y ,  Li W ,  Chang C .  Effect of milk tripeptides on blood pressure: A meta-analysis of randomized controlled trials .  _Nutrition_.  2008 ;24 : 933 \u2013 940 .\n\nZhu K ,  Meng X ,  Kerr DA ,  Devine A ,  Solah V ,  Binns CW ,  Prince RL .  The effects of a two-year randomized, controlled trial of whey protein supplementation on bone structure, IGF-1, and urinary calcium excretion in older postmenopausal women .  _J. Bone Miner. Res_.  2011 ;26 : 2298 \u2013 2306 . \nChapter 20\n\n# Milk Proteins: Digestion and Absorption in the Gastrointestinal Tract\n\nDidier Dupont*\n\nDaniel Tome**\n\n* UMR 1253 INRA, Rennes, France  \n** UMR 914 INRA, Paris, France\n\n## Abstract\n\nMilk proteins are present in numerous foods and are key players in the human diet. They are known to be highly digestible and excellent sources of indispensable amino acids. Milk proteins can be divided into two families: the caseins, which are proteins with a loose structure that are part of a supramolecular structure called the micelle; and the whey proteins, which exhibit a compact and well-characterized three-dimensional structure. These opposite structures give them opposite behaviors when entering the gastrointestinal tract. Whereas caseins are rapidly hydrolyzed by digestive enzymes, whey proteins show great resistance to hydrolysis by pepsin in the stomach. However, caseins can coagulate and be retained longer in the stomach, and for this reason have been referred to as 'slow proteins,' whereas whey proteins remain soluble and are rapidly transferred into the small intestine. Processing can significantly affect protein digestion and absorption through phenomena such as aggregation, denaturation, coagulation, and hydrolysis.\n\n## Keywords\n\nProtein\n\namino acids\n\nmilk\n\ndigestion\n\nabsorption\n\nhydrolysis\n\naggregation\n\ncoagulation\n\nprocessing\n\nOutline\n\nIntroduction 557\n\nDigestion of milk proteins 558\n\nMilk protein hydrolysis in the intestinal lumen 559\n\nCaseins 559\n\nWhey Proteins 559\n\nPeptides released during digestion 561\n\nImpact of processing on milk protein digestion and absorption 562\n\nHeat Treatment of Milk 562\n\nHomogenization of Milk 563\n\nPhysicochemical Modifications of Proteins 563\n\nCoagulation of Milk 563\n\nConclusions 566\n\n## Introduction\n\nDairy products are an important part of the diet in the industrialized world, especially in northern Europe and North America. In these regions, milk products contribute around 30% of the total dietary protein supply and represent about 65% of the intake of animal protein.\n\nThe protein content of cow's milk ranges from 32 to 35 g\/L. There are two major types of milk protein: the caseins (80%), which are represented by four distinct proteins (\u03b1s1, \u03b1s2, \u03b2, and \u03ba) and the whey proteins (20%), which are represented by proteins such as \u03b2-lactoglobulin and \u03b1-lactalbumin (see Chapter 2 for more details). These two families of proteins are opposite in terms of structure. Caseins exhibit a loose, highly flexible structure and are part of a supramolecular structure called the micelle, whereas whey proteins have a globular, well-defined three-dimensional (3D) structure. These structural differences between the two families markedly affect the behavior of these proteins in the gastrointestinal tract, particularly their susceptibility to hydrolysis by the digestive enzymes.\n\nThe nutritive value of milk protein is generally assessed via two components: nitrogen (indicative of total protein level, a marker of protein quantity) and essential amino acids (a marker of protein quality). The most common method for measuring protein retention is via nitrogen retention. In terms of protein quality, the nutritive value is related to the amino acid content and the bioavailability of these amino acids. The content of indispensable amino acids, that is, those that cannot be synthesized in the body and consequently must be supplied through the diet, is of particular concern.\n\n## Digestion of milk proteins\n\nIn the evaluation of the nutritive value of dietary proteins, nitrogen and individual amino acid digestibility, ileal and fecal digestibility, and apparent and true digestibility should be considered (Fuller & Tom\u00e9, 2005).\n\nThe true digestibility of milk protein, as measured in the ileum, averages 95%, which corresponds to one of the highest digestibilities for dietary proteins (Table 20.1) (AFSSA, 2007). The ileal digestibility of caseins has been estimated to be around 93% in pigs and 94% in humans; that of whey proteins appears to be even higher (97\u221298%) (Gilani & Sepehr, 2003; Lacroix et al., 2006; Rutherfurd & Moughan, 2003) but has never been precisely assessed in humans. Measurement of the true digestibility values of dietary nitrogen and amino acids in healthy human volunteers after the ingestion of milk indicated that ileal digestibility values for the individual amino acids ranged from 92% for serine to 99% for tyrosine, with an average amino acid digestibility of 95.3% (Gaudichon et al., 2002), that is, the same value as for nitrogen digestibility.\n\nTable 20.1\n\nFecal versus Ileal Digestibility (%) of Milk Proteins in Humans\n\n| Fecal | Ileal  \n---|---|---  \nProtein | True | Apparent | True | References  \nMilk protein | 96.6 | 91 | 95 | Bos et al., 2003; Gaudichon et al., 2002; Mah\u00e9 et al., 1994b  \nFermented milk | \u2014 | 90 | \u2014 |  Mah\u00e9 et al., 1994a  \nCasein | \u2014 | \u2014 | 94.1 |  Deglaire et al., 2009\n\nIn the context of digestion, although caseins exhibit a structure that makes them highly sensitive to hydrolysis by digestive enzymes, they are considered to be 'slow proteins' (Boirie et al., 1997) because they cause a slow postprandial release of amino acids in the plasma. This contrasts with whey proteins, which rapidly give rise to an intense peak of amino acids in the plasma. This property of caseins has been attributed to their ability to form a coagulum in the stomach through the joint action of acidic secretions and digestive enzymes. This coagulum of caseins is retained in the stomach longer than the whey proteins, which remain soluble and pass rapidly through the stomach and into the small intestine. These differences in gastric emptying lead to differences in the rate at which dietary amino acids enter the bloodstream (Lacroix et al., 2006; Mah\u00e9 et al., 1996). The longer retention of caseins in the stomach leads to a lower level, but a longer persistence, of amino acids in the plasma than is observed for amino acids from whey proteins (see Fig. 20.1). The type of protein can also specifically influence post-meal aminoacidemia. The chemical composition of whey protein is characterized by high leucine and isoleucine contents, and its ingestion is followed by a peripheral plasma elevation of these amino acids, which are known to be poorly oxidized in the liver. Similarly, the higher plasma proline concentration observed after the ingestion of casein is due to the higher proline content of this fraction (Lacroix et al., 2006).\n\nFigure 20.1 Mean (\u00b1 standard deviation) changes from baseline in serum total amino acid (AA), indispensable AA (IAA), branched-chain AA (BCAA), and dispensable AA (DAA) concentrations in subjects after the ingestion of total milk protein (TMP; n = 8), micellar casein (MC; n = 8), and milk soluble protein isolate (MSPI; n = 7). A significant effect of time (P = 0.0001) and a significant meal-by-time interaction (P = 0.01) were observed for all variables, as tested on the crude values using a mixed-model analysis of variance (ANOVA) with time as a repeated measure.*, **, *** Significantly different from baseline: *P = 0.05, **P = 0.01, ***P = 0.005. Lacroix et al., 2006.\n\n## Milk protein hydrolysis in the intestinal lumen\n\n### Caseins\n\nCaseins are extensively degraded during the gastric phase of digestion, an observation that is consistent with the fact that pepsin has a preference for mobile, loosely structured polypeptides (Dupont et al., 2010a). All in vitro studies on purified proteins have clearly demonstrated that caseins are hydrolyzed within the first minutes of pepsin hydrolysis, in adults as well as in infants (Fig. 20.2A).\n\nFigure 20.2 Evolution of residual immunoreactivity of \u03b2-lactoglobulin (A) and \u03b2-casein (B) during in vitro gastric (left) and duodenal (right) digestion using an infant model ( ) and an adult model ( ), as determined by inhibition enzyme-linked immunosorbent assay (data are the results of three independent determinations made in duplicate).\n\nMore recently, an animal trial conducted on mini-pigs fed skim milk and yogurt also showed a rapid and extensive hydrolysis of caseins during the first minutes of digestion, with intact caseins being detected for only 20 min after the meal intake (Barb\u00e9 et al., 2013). Similarly, in piglets fed milk-based infant formulas, caseins were shown to be more rapidly hydrolyzed than whey proteins, with only 23% of intact caseins being present in the stomach 30 min after ingestion (Bouzerzour et al., 2012).\n\nA recent in vivo study on human volunteers fed caseins showed extensive release in the jejunum of medium-sized peptides (750\u22121050 kDa) during the first 6 h after the meal intake. Most of the identified peptides originated from the two major caseins, that is, \u03b2-casein (61%) and \u03b1s1-casein (25%), and most contained two or more proline residues; the largest contained seven proline residues out of the 26 residues in its sequence (Boutrou et al., 2013). This finding is in agreement with the generally reported resistance of proline-containing peptides to gastric and pancreatic digestive enzymes (Agudelo et al., 2004; Vanhoof et al., 1995) and to epithelial proteinases (Bauchart et al., 2007).\n\n### Whey Proteins\n\nIn contrast to caseins, whey proteins, because of their globular structure, are known to be extremely resistant to proteolysis. This is particularly the case for \u03b2-lactoglobulin, which is not affected during gastric digestion, being virtually unaltered after 60 min of simulated digestion (Fig. 20.2B) (Dupont et al., 2010a; Mandalari et al., 2009b; Schmidt et al., 1995). It has also been shown that the molecular interaction of \u03b2-lactoglobulin with phosphatidylcholine from the gastric mucosa protects the protein from duodenal digestion by trypsin and chymotrypsin (Mandalari, 2009a). However, \u03b2-lactoglobulin has been found to be more sensitive to pepsinolysis when located at the interface of a lipid droplet than when in solution because of drastic conformational changes (Macierzanka et al., 2009).\n\nConflicting results have been published for the second major bovine whey protein, that is, \u03b1-lactalbumin. Whereas some researchers have found that \u03b1-lactalbumin is even more resistant to simulated digestion than \u03b2-lactoglobulin (Inglingstad et al., 2010a), others have found that \u03b1-lactalbumin is susceptible to hydrolysis in solution (Nik et al., 2010). In contrast to \u03b2-lactoglobulin, \u03b1-lactalbumin appears to be more resistant to digestion when located at the oil\u2212water interface than when in solution. The same protective effect of phospholipids from the gastric mucosa on the susceptibility of \u03b1-lactalbumin to hydrolysis by pancreatic enzymes has been described (Moreno et al., 2005).\n\nIn contrast to \u03b1-lactalbumin and \u03b2-lactoglobulin, lactoferrin has been shown to be extensively degraded during simulated gastric digestion (Furlund et al., 2013). Multiple sequence analysis of the identified peptides indicated a motif consisting of proline and neighboring hydrophobic residues that could restrict proteolytic processing. Further structure analysis showed that almost all proteolytic cleavage sites were located on the surface and mainly on the nonglycosylated half of lactoferrin.\n\n## Peptides released during digestion\n\nThe hydrolysis of milk proteins in the gastrointestinal tract will result in the production of myriad peptides (Jahan-Mihan et al., 2011), some of which have been shown to exert biological activities such as antihypertensive (Martinez-Maqueda et al., 2012), antiatherogenic (Ricci-Cabello et al., 2012), antimicrobial, and immunomodulatory activities (Agyei & Danquah, 2012). Mass spectrometry is the best tool for tracking peptides released during digestion, and the concept of nutritional peptidomics has recently been proposed (Panchaud et al., 2012).\n\nTo date, milk peptides have been identified by submitting food to simulated in vitro digestion (Dupont et al., 2010b; Kopf-Bolanz et al., 2012; Picariello et al., 2010). In the in vitro situation, identifying and quantifying the peptides in digested samples is rather easy because most of the proteins in the samples originate from the food itself. However, it is still questionable whether mimicking digestion with in vitro models perfectly reflects the physiological reality. Only a few in vivo studies have been conducted; detection of dietary peptides is made more difficult by the presence of endogenous proteins secreted in the different compartments of the gut. In a pioneer work, milk caseinomacropeptide, that is, \u03ba-casein (f106\u2212169), was detected in the jejunum of humans fed15N-labeled casein, whey protein, and yogurt (Ledoux et al., 1999). Similarly, caseinophosphopeptides were identified in the effluent collected from milk-fed humans (Meisel et al., 2003). More recently, Bauchart et al. (2007) identified 26 dietary peptides in the duodenum and jejunum of pigs fed beef meat or cooked trout fillets; the peptides were short (<2000 Da) and particularly rich in proline residues. In all these studies, only a limited number of targeted peptides were followed, and the food peptidome found in the lumen of the small intestine was not extensively characterized. In 2013, Boutrou et al. established the peptidome of jejunal effluents collected from milk protein-fed humans. Totals of 356 and 146 peptides were detected and sequenced in the jejunum following casein and whey protein ingestion, respectively. However, the subjects were fed pure protein fractions, and the possible effect of the food structure was not investigated. Indeed, the structure of the chyme, as affected by food structure, could limit or modify the accessibility of digestive enzymes to some cleavage sites.\n\nIn recent work, the impact of the structure of the dairy matrix on the number and nature of milk peptides released in the duodenum was investigated using cannulated mini-pigs fed dairy liquid or acid or rennet dairy gels of identical composition. The formation of peptides in vivo was followed by tandem mass spectrometry over a postprandial period of 5 h after ingestion of the dairy matrices by the mini-pigs. The effect of the meal structure was investigated at two levels: the microstructure, as modified by thermal treatment, and the macrostructure, as modified by milk gelation. More than 16,000 peptides were sequenced and unambiguously identified. The results obtained showed that the structure of the dairy products had only little influence on the location of the cleavage sites on the protein sequences (Barb\u00e9 et al., submitted). However, the structure markedly impacted the number of peptides identified, especially for the rennet dairy gels; about three times fewer peptides were detected than for the other matrices. This effect was attributed to greater extents of dilution by digestive secretions associated with longer gastric retentions for the rennet gels. Potential bioactive peptides were also produced over time, and their identification has increased our knowledge of peptides present in the lumen in vivo. Our results indicate that the structure of dairy matrices markedly affects the kinetics of milk protein digestion in vivo, more than the mechanism of proteolysis itself.\n\n## Impact of processing on milk protein digestion and absorption\n\nMilk proteins are introduced into the human diet as processed milk products. It is therefore critical to determine the impact of the major processing technologies on milk protein digestion.\n\n### Heat Treatment of Milk\n\nOne of the most common processes applied to milk is heat treatment to ensure product safety. Because of the structural differences already mentioned, heat treatment affects caseins and whey proteins quite differently. Heat treatment highly modifies the 3D structure of whey proteins, resulting in an 'opening' of the globular structure and making whey proteins more sensitive to the action of digestive enzymes, as demonstrated for \u03b2-lactoglobulin (Barb\u00e9 et al., 2013) and \u03b1-lactalbumin (Inglingstad et al., 2010b). In contrast, caseins, with their loose and highly flexible structure, are not strongly modified by heat treatment. Heat treatment at high temperature results in an increased resistance of the caseins to simulated digestion (Almaas et al., 2006; Dupont et al., 2010b), which has been attributed to the formation of thermally induced aggregates between caseins and between caseins and whey proteins.\n\n### Homogenization of Milk\n\nHomogenization of milk results in the disruption of the milk fat globule membranes. Lipids are present as smaller droplets that are stabilized by milk proteins covering the oil\u2212water interface. \u03b2-Lactoglobulin and \u03b2-casein have been shown to be more susceptible to pepsinolysis when they are adsorbed to an oil\u2212water interface than when they are in solution (Macierzanka et al., 2009; Sarkar et al., 2009). This has been attributed to the unfolding of the proteins at the droplet surface, which improves their accessibility to pepsin. It has been found that the rate of gastric digestion of \u03b2-casein is twice as fast as when adsorbed to the oil\u2212water interface than when in solution. In the small intestine, proteins are displaced from the interface by bile salts (Sarkar et al., 2010), making triglycerides more accessible to the pancreatic lipase.\n\n### Physicochemical Modifications of Proteins\n\nMilk protein modification with cross-linking enzymes such as transglutaminase (TG) has been used extensively to change the functionality of proteins and thereby to improve the textural quality and stability of protein-based food products. In dairy products, TG-induced cross-linking can increase the firmness and water-holding capacity of acid-induced gels in products with low solids and fat contents or to improve the stability of emulsions and foams. The effect of the TG-induced cross-linking of sodium caseinate on postprandial metabolic and appetite responses was recently investigated in 13 healthy individuals (Juvonen et al., 2012). The results indicated that enzymatically cross-linked sodium caseinate and native sodium caseinate had comparable metabolic responses in a liquid matrix, suggesting similar digestion and absorption rates and first pass metabolism despite the structural modification of the cross-linked sodium caseinate.\n\nThe hydrolysis of milk proteins has been widely used to reduce their allergenicity properties in infant nutrition. However, hydrolysis could also be considered to be a 'pre-digestion' of proteins, facilitating their digestion and absorption in the gastrointestinal tract. This was confirmed in a study on 10 elderly subjects who received either intact or hydrolyzed caseins (Koopman et al., 2009). The plasma amino acid concentrations increased extensively (25\u221250%) after ingestion of the hydrolyzed casein, compared with the intact casein (P < 0.01).\n\n### Coagulation of Milk\n\nMilk coagulation (liquid\/gel\/solid transition) is used extensively in the dairy industry, especially for yogurt and cheese manufacture, even though the mechanisms of milk clotting for these two types of product are quite different. Studies on the digestion of real dairy matrices (yogurt, cheese) are scarce, compared with studies on purified fractions of casein or whey protein. Gaudichon et al. (1994) showed, using mini-pigs, that the half gastric emptying time of the liquid phase was not different between milk and yogurt. However, the intestinal deliveries of both the liquid phase and the nitrogenous fraction of the chyme were more delayed in pigs fed yogurt than in pigs fed milk (Fig. 20.3). The kinetics of exogenous nitrogen delivery into the intestine were correlated with those of exogenous nitrogen absorption. These results suggest that milk proteins are rapidly absorbed after they reach the intestine and that gastric emptying is a major factor controlling the kinetics of milk nitrogen absorption.\n\nFigure 20.3 Remaining fraction of exogenous nitrogen in the stomach of mini-pigs after the ingestion of 500 mL of milk or 500 g of yogurt. The ingested milk and yogurt contained 17 and 18 g of nitrogen, respectively. Values are means \u00b1 standard error of the mean (SEM) for three or four pigs. No significant differences were found by ANOVA, P < 0.05.\n\nRychen et al. (2002) examined the postprandial portal absorption of15N in the growing pig after the ingestion of milk, yogurt, and heat-treated yogurt. Although the total portal absorption was similar between the three products, yogurt nitrogen was absorbed more slowly than milk nitrogen, with significant differences being observed after 30, 60, and 180 min. Heat-treated yogurt showed similar behavior to milk; it was hypothesized that heat treatment of the gel was responsible for destroying the natural body and viscosity of yogurt. These effects were therefore attributed to different emptying rates between milk, yogurt, and heat-treated yogurt.\n\nMore recently, a determination of the kinetics of milk protein digestion and amino acid absorption after the ingestion of liquid or gelled (acid and rennet gels) dairy matrices by six mini-pigs showed that the gelation of milk slowed down the outflow of the meal from the stomach, and the subsequent absorption of amino acids, and decreased their bioavailability in peripheral blood (Fig. 20.4) (Barb\u00e9 et al., 2013). The nature of the matrix seemed to affect the release of the gastrointestinal hormones involved in appetite regulation, with the gel matrices appearing to be potentially more satiating. It was also shown that two gels with the same composition and similar rheological and structural properties, but differing in their mode of coagulation (acidification\/renneting), exhibited different behaviors during digestion. Indeed, ingestion of the rennet gel resulted in lower levels of both proteins in the duodenum and lower levels of amino acids in the plasma, compared with ingestion of the acid gel. This was probably due to the formation of a coagulum with high stiffness after ingestion of the rennet gel, under the simultaneous action of the stomach acidity and the rennet, leading to a very long retention of the rennet matrix in the stomach (Barb\u00e9 et al., 2013). The plasma cholecystokinin and ghrelin concentrations suggested a potentially more satiating effect of the rennet gel than the acid gel.\n\nFigure 20.4 Plasma leucine concentration (\u03bcmol\/h) in mini-pigs over a 7-h period after the ingestion of liquid (L) and gel (G) matrices, from unheated (R) and heated (H) milk products. Values are means \u00b1 SEM calculated for four mini-pigs (n = 4). The data were analyzed using a mixed-model ANOVA. The time effect was significant (P < 0.001), and the lines at the bottom of the figure indicate a significant difference (P < 0.05) from baseline for each curve. The time-by-matrix interaction was significant (P < 0.001), and, at a given time, differences between matrices are indicated by different letters a and b (P < 0.05).\n\nStudies performed on cheese digestion are scarce. A recent study compared the kinetics of the matrix degradation of different cheeses in a gastrointestinal environment (Lamothe et al., 2012). The relationship between the physical characteristics of the cheeses (rheological properties, microstructure) and their digestion patterns was also studied. Rheological measurements and compositional and microstructural analyses were performed on mild cheddar, aged cheddar, light cheddar, and mozzarella cheeses. Mozzarella cheese showed the highest rate of matrix degradation. Aged cheddar cheese showed rapid degradation during the gastric phase, but was more resistant to the duodenal environment. Light cheddar showed the opposite behavior, being highly resistant to the gastric environment; however, it underwent extensive degradation at the end of the duodenal phase. The extent of matrix degradation for mild cheddar was similar to that for mozzarella in the gastric phase but was much lower than that for the other cheeses in the duodenal phase. The results suggest that degradation of the cheeses was driven mainly by their physical characteristics.\n\nThe production of parmigiano reggiano cheese is closely related to the nutritional quality of the final product; in particular, the high digestibility of its proteins is claimed to be proportional to the ripening stage of the cheese. The effect of cheese aging on the kinetics of protein digestion was recently investigated. Two different kinds of parmigiano reggiano, young (aged 15 months) and old (aged 30 months), were separately digested using an in vitro system that simulated digestive processes in the mouth, stomach, and small intestine (Bordoni et al., 2011). The results indicated that the digestion of cheeses with different aging times, although starting from different initial compositions, concluded in similar ways, in terms of free amino acids and small organic compounds, but evolved with different kinetics of hydrolysis and peptide formation, discriminating the young cheese from the old cheese.\n\n## Conclusions\n\nThe digestion and absorption of milk proteins have been extensively studied, and the mechanisms involved are well described. However, many of the studies have been performed either in vitro or with purified protein fractions, and more work is needed to better understand the disintegration of real dairy products in the human gastrointestinal tract. Nevertheless, it appears that casein and whey protein exhibit different behaviors in the gastrointestinal tract because of differences in their structure and physicochemical properties, and that processing has a significant impact on the kinetics of protein digestion by modifying the residence time of the products in the stomach. In the context of the nutritional properties of food, it appears that the micro- and macrostructures of a meal, resulting from technological processes used in the food industry, markedly affect the different steps of milk protein digestion. Thus, the design of food matrices at the technological level is of particular interest in the control of the delivery of nutrients, especially for specific subpopulations, such as the elderly or overweight people.\n\n# References\n\nAFSSA, 2007. Protein intake: Consumption, quality, requirements and recommendations. In: AFSSA., (Ed.), Maisons-Alfort. 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Dietary proteins as determinants of metabolic and physiologic functions of the gastrointestinal tract .  _Nutrients_.  2011 ;3 (5) : 574 \u2013 603 .\n\nJuvonen KR ,  Lille ME ,  Laaksonen DE ,  Mykkanen HM ,  Niskanen LK ,  Herzig KH ,  Poutanen KS ,  Karhunen LJ .  Crosslinking with transglutaminase does not change metabolic effects of sodium caseinate in model beverage in healthy young individuals .  _Nutrition Journal_.  2012 ;11, 35 .\n\nKoopman R ,  Crombach N ,  Gijsen AP ,  Walrand S ,  Fauquant J ,  Kies AK ,  Lemosquet S ,  Saris WHM ,  Boirie Y ,  Van Loon LJC .  Ingestion of a protein hydrolysate is accompanied by an accelerated in vivo digestion and absorption rate when compared with its intact protein .  _American Journal of Clinical Nutrition_.  2009 ;90 (1) : 106 \u2013 115 .\n\nKopf-Bolanz KA ,  Schwander F ,  Gijs M ,  Vergeres G ,  Portmann R ,  Egger L .  Validation of an in vitro digestive system for studying macronutrient decomposition in humans .  _Journal of Nutrition_.  2012 ;142 (2) : 245 \u2013 250 .\n\nLacroix M ,  Bos C ,  Lonil J ,  Airinei G ,  Luengo C ,  Dar S ,  Benamouzig R ,  Fouillet H ,  Fauquant J ,  Tom D ,  Gaudichon C .  Compared with casein or total milk protein, digestion of milk soluble proteins is too rapid to sustain the anabolic postprandial amino acid requirement .  _American Journal of Clinical Nutrition_.  2006 ;84 (5) : 1070 \u2013 1079 .\n\nLamothe S ,  Corbeil MM ,  Turgeon SL ,  Britten M .  Influence of cheese matrix on lipid digestion in a simulated gastro-intestinal environment .  _Food and Function_.  2012 ;3 (7) : 724 \u2013 731 .\n\nLedoux N ,  Mahe S ,  Dubarry M ,  Bourras M ,  Benamouzig R ,  Tome D .  Intraluminal immunoreactive caseinomacropeptide after milk protein ingestion in humans .  _Nahrung-Food_.  1999 ;43 (3) : 196 \u2013 200 .\n\nMacierzanka A ,  Sancho AI ,  Mills ENC ,  Rigby NM ,  Mackie AR .  Emulsification alters simulated gastrointestinal proteolysis of beta-casein and \u03b2-lactoglobulin .  _Soft Matter_.  2009 ;5 (3) : 538 \u2013 550 .\n\nMahe S ,  Gaudichon C ,  Roos N ,  Benamouzig R ,  Luengo C ,  Bouley C ,  Tome D .  N-15-labeled milk and yogurt digestion and absorption in the human jejunum .  _FASEB Journal_.  1994 ;8 (5) : A714 \u2013 A1714 .\n\nMahe S ,  Roos N ,  Benamouzig R ,  Sick H ,  Baglieri A ,  Huneau JF ,  Tome D .  True exogenous and endogenous nitrogen fractions in the human jejunum after ingestion of small amounts of N-15-labeled casein .  _Journal of Nutrition_.  1994 ;124 (4) : 548 \u2013 555 .\n\nMahe S ,  Roos N ,  Benamouzig R ,  Davin L ,  Luengo C ,  Gagnon L ,  Gausserges N ,  Rautureau J ,  Tome D .  Gastrojejunal kinetics and the digestion of N-15 \u03b2-lactoglobulin and casein in humans: The influence of the nature and quantity of the protein .  _American Journal of Clinical Nutrition_.  1996 ;63 (4) : 546 \u2013 552 .\n\nMandalari G ,  Mackie AM ,  Rigby NM ,  Wickham MS ,  Mills E .  Physiological phosphatidylcholine protects bovine \u03b2-lactoglobulin from simulated gastrointestinal proteolysis .  _Molecular Nutrition and Food Research_.  2009 ;53 : S131 \u2013 S139 .\n\nMandalari G ,  Mackie AM ,  Rigby NM ,  Wickham MSJ ,  Mills ENC .  Physiological phosphatidylcholine protects bovine \u03b2-lactoglobulin from simulated gastrointestinal proteolysis .  _Molecular Nutrition and Food Research_.  2009 ;53 : 131 \u2013 139 .\n\nMartinez-Maqueda D ,  Miralles B ,  Recio I ,  Hernandez-Ledesma B .  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Mass spectrometry for nutritional peptidomics: How to analyze food bioactives and their health effects .  _Journal of Proteomics_.  2012 ;75 (12) : 3546 \u2013 3559 .\n\nPicariello G ,  Ferranti P ,  Fierro O ,  Mamone G ,  Caira S ,  Di Luccia A ,  Monica S ,  Addeo F .  Peptides surviving the simulated gastrointestinal digestion of milk proteins: Biological and toxicological implications .  _Journal of Chromatography B \u2013 Analytical Technologies in the Biomedical and Life Sciences_.  2010 ;878 (3-4) : 295 \u2013 308 .\n\nRicci-Cabello I ,  Herrera MO ,  Artacho R .  Possible role of milk-derived bioactive peptides in the treatment and prevention of metabolic syndrome .  _Nutrition Reviews_.  2012 ;70 (4) : 241 \u2013 255 .\n\nRutherfurd SM ,  Moughan PJ .  The rat as a model animal for the growing pig in determining ileal amino acid digestibility in soya and milk proteins .  _Journal of Animal Physiology and Animal Nutrition_.  2003 ;87 (7\u20138) : 292 \u2013 300 .\n\nRychen G ,  Mpassi D ,  Jurjanz S ,  Mertes M ,  Lenoir-Wijnkoop I ,  Antoine JM ,  Laurent F .  N-15 as a marker to assess portal absorption of nitrogen from milk, yogurt and heat-treated yogurt in the growing pig .  _Journal of Dairy Research_.  2002 ;69 (1) : 95 \u2013 101 .\n\nSarkar A ,  Goh KKT ,  Singh RP ,  Singh H .  Behaviour of an oil-in-water emulsion stabilized by \u03b2-lactoglobulin in an in vitro gastric model .  _Food Hydrocolloids_.  2009 ;23 (6) : 1563 \u2013 1569 .\n\nSarkar A ,  Horne DS ,  Singh H .  Interactions of milk protein-stabilized oil-in-water emulsions with bile salts in a simulated upper intestinal model .  _Food Hydrocolloids_.  2010 ;24 (2\u20133) : 24 .\n\nSchmidt DG ,  Meijer R ,  Slangen CJ ,  Vanberesteijn ECH .  Raising the pH of the pepsin-catalyzed hydrolysis of bovine whey proteins increases the antigenicity of the hydrolysates .  _Clinical and Experimental Allergy_.  1995 ;25 (10) : 1007 \u2013 1017 .\n\nVanhoof G ,  Goossens F ,  Demeester I ,  Hendriks D ,  Scharpe S .  Proline motifs in peptides and their biological processing .  _FASEB Journal_.  1995 ;9 (9) : 736 \u2013 744 . \nChapter 21\n\n# Milk Proteins: The Future\n\nMike J. Boland*\n\n* Riddet Institute, Massey University, Palmerston North, New Zealand\n\n## Abstract\n\nThis final chapter contemplates future trends and their likely impact on the production and use of milk proteins. First, we consider global issues, including energy consumption, the global water economy, and specific issues for dairy relating to greenhouse gases. We then review current and emerging trends in consumer demands and how they might impact the market for milk proteins. Important factors are expected to be food safety and traceability, as well as an increasing concern for the effect of food on health and an increasing importance of personalized nutrition. Finally, we consider some emerging technologies and how they might affect the future of milk protein production and processing.\n\n## Keywords\n\nGlobal issues for food\n\nmilk and energy\n\ndairy methane production\n\nfood safety and traceability\n\nallergies to milk\n\nA2 milk\n\nbioactive peptides\n\ngenetic modification\n\nOutline\n\nIntroduction 571\n\nGlobal issues for food 571\n\nCompetition for Land Use 572\n\nMilk and Energy 572\n\nMilk and the Water Economy 573\n\nImplications of Dairy Methane Production 574\n\nConsumer demands and trends for food and ingredients 575\n\nFood Safety and Traceability 575\n\nFood and Health: Nutrigenomics and Personalized Nutrition 576\n\nNew technologies and their possible effect on milk protein ingredients and products 578\n\nGenetic Modification 578\n\nNovel Processing 579\n\nNew Analytical Methods 580\n\nMaterials Science and Nanotechnology 580\n\nConclusions 581\n\n## Introduction\n\nAs a wrap-up of our journey from expression to food, this chapter takes a look at the possible future of food, especially as it relates to milk proteins. Global macroenvironmental factors are considered first, and then we examine consumer demands and trends, and the likely impact of new technologies.\n\n## Global issues for food\n\nThe demand for high-quality protein for nutrition, especially for dairy protein, has been discussed in the first chapter of this volume. The Food and Agriculture Organization (FAO) has predicted a demand for dairy production by 2050 of 843 million tons\u2014more than double that of today. In order to achieve this figure sustainably, the industry will need to address a range of resource constraints. Global resource issues expected to have a major impact on future food production, including production of milk proteins, are:\n\n\u2022 competition for land use;\n\n\u2022 increasing cost of energy\u2014primarily because of the greenhouse gas implications of energy use, but also because of the rising cost of energy production;\n\n\u2022 the water economy; and\n\n\u2022 methane emissions from cows and the effect on global warming.\n\nThe position of animal production in a constrained world has been discussed in Chapter 1 of this volume and is covered in more depth by Steinfeld et al. (2006), FAO (2009), and Godfray et al. (2010).\n\n### Competition for Land Use\n\nAn important consideration related to competition for land use is the opportunity cost incurred in land used for milk production, including land used for growing crops to feed cows. If cows are fed using arable crops, the equation is simple: The opportunity cost is the value of the crop that has been fed to cows that could otherwise have been used to feed humans. In the case of pasture grazing, the consideration changes to one of the value of competing uses for land (e.g., Godfray et al., 2010). In some cases, when the land is unsuitable for arable farming or horticulture, for example, because the terrain is not suitable, the opportunity cost is low. In cases where the land is suitable for cropping, a comparison has to be made on the value of the crop in relation to the value of the equivalent milk production from that land. This comes down to the purpose of production. If the purpose is simply production of energy, a cereal crop will always come out best. The global need for protein nutrition was addressed in Chapter 1. If the purpose of farming is to produce protein, calculations based on product yields and composition suggest that under the right circumstances, the yield of bioavailable protein per hectare per annum can be at least as high for milk as it is for a cereal crop such as wheat once both are converted into a food format. Pastoral farming can be relatively efficient, partly because of the perennial nature and low maintenance costs of pasture and partly because the cow is doing the work of harvesting as well as re-fertilizing the pasture.\n\n### Milk and Energy\n\nMilk is energetically very expensive. Milk is an animal product: To produce it requires that the cow eat vegetable material that has already been produced in a nutritional format\u2014although not one necessarily edible by humans. However, milk is one of the most efficiently produced of the animal-produced foods, largely because the animal itself is not consumed. It has been estimated that production of 50 kg of milk protein in the United States requires 7 x 106 kcal of feed energy (i.e., 585 kJ\/kg), an energy efficiency of 30:1 (Pimentel and Pimentel, 1979). In contrast, the total energy input per kilogram for production of corn or soy protein in the United States calculates out to 58 kJ\/kg (calculated from data in Pimentel and Pimentel, 1979), one-tenth of the energy. These figures do not take into account the uptake of direct solar energy through photosynthesis as the crops grow, or the opportunity cost in energy for other use of the land that grows these products. It is generally accepted that for grain-fed beef, about 10 kg of grain is required to produce 1 kg of meat. Included in this calculation is the requirement to replace the animal and a loss of around 50% of carcass mass either as waste or by-products. It has been estimated that the energy efficiency for milk is 4:1 and the protein efficiency is 4.75:1 (Council for Agricultural Science and Technology, quoted in Fairlie, 2010). More recent calculated values of the efficiency of conversion of grain and forage into animal protein have been provided by Pimentel (2006), and selected values are given in Table 20.1. A significant difference between the inputs for grass-fed and grain-fed beef production is evident, and presumably a similar difference can be expected for dairy production, although the conversion factor presented is a hybrid.\n\nTable 20.1\n\nEnergy Efficiency of Conversion of Animal Feed to Animal Protein\n\nLivestock | Grain  \n(kg) | Forage  \n(kg) | Energy input \/ energy protein  \n---|---|---|---  \nLamb | 21 | 30 | 57:1  \nBeef cattle | 13 | 30 | 40:1  \nGrass-fed beef cattle | \\-- | 200 | 20:1  \nSwine | 5.9 | \\-- | 14:1  \nDairy (milk protein) | 0.7 | 1 | 14:1\n\nData are from Pimentel (2006).\n\nEnergy sensitivity in some markets, particularly in Europe, has resulted in the use of 'food miles'\u2014an inappropriately named descriptor of the carbon footprint (i.e., the energy cost) expended in producing, distributing, and consuming foods. The methods used in calculating carbon footprints have varied, sometimes leading to inappropriate comparisons. Calculations can be expected to become more accurate and more meaningful in the future as the methodology becomes standardized, but they are also open to misuse as nontariff barriers in some jurisdictions. Most food products are shipped by sea, and the greenhouse gas component of shipping is small compared with production costs, even when food is shipped long distances. For example, in shipping from New Zealand to the UK, the contribution of CO2 from the shipping was estimated at 125 kg CO2\/ton milk solids out of a total of 1422 kg CO2\/ton for the life-cycle footprint, which in turn compared favorably with the equivalent figure of 2921 kg CO2\/ton milk solids for the locally produced equivalent in the UK (Saunders et al., 2006). It is becoming increasingly recognized that sea freight is a minor part of the carbon footprint of a food product, typically 5\u201310%, and that low carbon footprints through more efficient production systems can more than offset this.\n\n### Milk and the Water Economy\n\nIncreasingly, international attention is being paid to the 'water economy' as water becomes a limiting resource in many regions due to the effects of population growth and of climate change. The amount of 'virtual water' in a product is the amount of water required to produce it throughout the production chain. The amount of virtual water in a range of products is given in Table 20.2.\n\nTable 20.2\n\nVirtual Water Content of Dairy and Related Products\n\nProduct | Virtual water  \n(m3\/ton) | Reference  \n---|---|---  \nMilk | 990 |  Hoekstra and Chapagain (2007)  \nMilk powder | 4,602 |  Hoekstra and Chapagain (2007)  \nMilk protein powders | 18,400 | Calculated from above  \nSoybeans | 1,789 |  Hoekstra and Chapagain (2007)  \nSoy protein | 5,400 | Calculated from above\n\nMost of the virtual water in these products arises from on-farm activities, with processing water a minor component. Hence, protein product values have been calculated here by simply adjusting for the amount of protein in the parent product, without adjusting for processing water or credit for the water value of any co-products. The key point is that, as with energy, the cost of water for producing milk-origin products is several-fold higher than for producing similar plant-origin products. This means that only countries that are water-rich can sustainably produce animal-based products for export. This fact will impact in future as water distribution changes with climate change, but also threatens production in some parts of the world where existing water use is unsustainable, such as parts of Australia where water offtake has led to saline ingress into soils (Anderies et al., 2006).\n\nThe use of water for dairy production must be considered in relation to the availability of water and competition for that water. In countries such as New Zealand and Brazil, there is an abundance of fresh water in many regions, and water not used for dairying is unlikely to be used for another more beneficial purpose; thus there is no real opportunity cost. As water availability becomes limiting in different parts of the world, due to the effects of population growth and climate change, the locus of dairy production may shift to water-rich regions. An interesting development is the emergence of 'seawater greenhouse' farming based on using seawater as the primary source of water, with various methods of production of fresh water to support plant growth. This method is already in operation in a number of countries for horticulture. Whether this practice will extend significantly to dairy production remains to be seen.\n\n### Implications of Dairy Methane Production\n\nMethane merits special mention as a greenhouse gas because emissions from cows contribute substantially to the global greenhouse gas load as a by-product of rumen digestion. Methane is recognized as a greenhouse gas and is rated as having a global warming potential 21 times that of the equivalent amount of carbon dioxide, based on a 100-year time scale. It has been estimated that, of all the greenhouse gases, methane is second in effect only to carbon dioxide and is responsible for around 10\u201315% of the present greenhouse gas effect in the atmosphere. Globally, ruminant livestock produce about 28% of methane emissions from human-related activities. A single adult cow is a relatively minor contributor, emitting only 80\u2013110 kg of methane, but, with about 100 million cattle in the United States alone and 1.2 billion large ruminants in the world, ruminants are one of the largest sources of methane. In the United States, cattle emit about 5.5 million tons of methane per annum into the atmosphere, accounting for 20% of U.S. methane emissions, with dairy cattle producing around one-quarter of the total (see www.epa.gov\/methane\/rlep\/faq.html). Although the greenhouse gas effect of methane is a matter of concern, to put it in perspective, it has been estimated that the methane emissions from the entire dairy herd in the United States in 2007 were 112 billion kg CO2 equivalent, less than half of the emissions calculated for the buffalo (American bison) herd in 1860 (228 billion kg CO2 equivalent) (Capper, 2011). It is further recognized that increasing efficiencies in farming have a substantial effect in reducing the carbon footprint of milk. In the United States, it has been estimated that the carbon footprint of milk was reduced by 63% between 1944 and 2007 (Capper et al., 2009). In New Zealand, a strong negative correlation with greenhouse gas emissions has been shown for both kg milk solids per hectare per year and kg milk solids per cow, indicating that more efficient production systems have a lower footprint in terms of their product (Judge et al., 2010).\n\nMost governments recognize the need to limit greenhouse gases, and international negotiations following the Kyoto Protocol were expected to impose penalties on greenhouse gas producers (carbon taxes). To date, governments have shied away from imposing carbon taxes on pastoral farming, and it remains to be seen whether this tax will ever be levied.\n\nNevertheless, methane generation represents both a source of pollution and a waste of energy, and research efforts to specifically target the removal of methanogenic organisms from the rumen are important for the future economic viability of the industry. Because methanogens are believed to have an important role in managing the hydrogen concentration in the rumen, it may be necessary to find or create a microorganism that can transfer hydrogen into a product other than methane.\n\n## Consumer demands and trends for food and ingredients\n\n### Food Safety and Traceability\n\nThroughout the world, awareness of foodborne disease has risen in response to the high-level publicity that such outbreaks receive. The toll exacted in human and economic terms is considerable. Notable dairy outbreaks in recent years include Salmonella in ice cream (United States, 1994: 224,000 cases of illness) and staphylococcal enterotoxin in milk (Japan, 2000: 15,000 cases). Contaminated soft cheeses and raw milk are often in the news. Most dairy products, processed to modern standards of hygiene, have an excellent safety record, but consumers are demanding increased surveillance and control of all foods, including dairy. The contamination of animal feed with dioxin in Belgium in 1999 highlighted the importance consumers place on the absence of toxic chemicals in their food. The deliberate adulteration of infant formula with melamine in China in 2008 resulted in the illnesses and deaths of infants. There will be no lessening in the demands on food producers to control risks and deliver assurances of safety. The increased costs from providing this assurance through effective process control will become the norm for dairy businesses in the future.\n\nIn recent times, increasing attention has been paid to traceability, so that any food safety issue can be quickly traced to its origin and other food from the same batch can be quickly quarantined. Traceability can also be important because of consumers' desire for products that are sustainably produced or have other connotations of quality (such as organically produced products). Traceability is usually managed through the labeling and tracking of products through manufacture and distribution, usually by means of labels on the packaging. This is usually well handled by most food manufacturers and distributors. There have, however, been attempts at 'false-flagging' products in the past, and this will no doubt continue.\n\nFor products containing milk proteins, it is often possible to obtain an internal check on the origin of the product: Dairy herds in different countries and regions tend to have a rather unique mixture of breeds and genetics. This is reflected in the distribution of polymorphisms of the proteins, which can be relatively simply analyzed using gel electrophoresis and\/or mass spectroscopy. Additional information about processing can be gained from mass spectroscopic analysis of postproduction changes in the chemistry of milk proteins (see Chapter 11).\n\nA recent concern has arisen because of deliberate adulteration of milk used for infant formula in China with melamine, leading to the deaths of babies in the summer of 2008 (Tyan et al., 2009). Melamine has a high nitrogen content (66% w\/w) and is relatively inexpensive, and when milk is paid for on the basis of nitrogen content as a proxy for protein content, melamine can be a cheap way to fraudulently boost the apparent protein content. An unfortunate side effect of the addition of melamine can be the presence of cyanuric acid, a common contaminant and hydrolysis product of melamine. Although neither melamine nor cyanuric acid on its own is particularly toxic, the combination of the two can result in the formation of melamine cyanurate crystals, which are very insoluble and tend to form in the kidneys. This can lead to kidney failure. Protein measurements using Fourier transform infrared spectroscopy (FTIR), now a common standard method for routine milk testing in factories, can detect low levels of melamine in dairy products (Balabin and Smirnoff, 2011), and melamine itself is relatively easy to test for in other ways (Tyan et al., 2009).\n\n### Food and Health: Nutrigenomics and Personalized Nutrition\n\nConsumers are being increasingly sensitized to the effects of diet on health (and on personal appearance). The success of diet clinics attests to this trend. The occurrence of (and attention being paid to) current high levels of obesity in affluent societies is spurring interest in diet at all levels of society, from individual to government. Food products on supermarket shelves are increasingly differentiated by the presence of (omega-3 fats, antioxidants) or absence of (fat-free, gluten-free) components believed to affect health.\n\nIndividuals can now obtain information about their own genetic profile, with respect to known genetic polymorphisms related to health and metabolism. Although early-entrant mail-order gene-sequencing companies such as Sciona and Genelex failed, it is now possible for anyone to have their DNA genotyped for under 300US$ (23andMe DNA Spit Kit). The combination of the availability of individual genetic data on an unprecedented scale with a detailed understanding of nutrition has led to the field of 'nutrigenomics', the study of the relationship between a person's genetic make up and their nutritional needs. While early studies in nutrigenomics suggested quick wins, with simple gene differences (single-nucleotide polymorphisms, or SNPs) indicating specific dietary effects, the situation has been found to be much more complex. The focus is now on understanding 'nutritional phenotypes,' which take into account not only SNPs, but other genetic differences such as copy number variants of genes (CNVs), as well as differences in expression of those genes that can be caused by a range of factors, including epigenetic effects, health status, and medications (van Ommen et al., 2010).\n\n'Personalized nutrition' is a nutritional response to differences between individuals\u2014whether from a nutrigenomics input or through other identified needs and preferences\u2014and attempts to balance an individual's diet to specific individual (nutritional phenotype) and situational needs. Nutrition today is not just about balance of macro- and micronutrients: A plethora of functional (bioactive) food components are also known to affect health in ways that extend far beyond the simple supply of nutrients. They can also be modifiers of nutrient uptake and usage, thus modifying the effect of nutritional balance as seen by the body's metabolism. The kinetics of nutrient uptake are just as important as overall absolute uptakes of nutrients. In the case of carbohydrates, this has translated into the 'glycemic index'\u2014an indicator of the rapidity of glucose uptake and thus the effect of a food on insulin production in the body. In the case of proteins, 'fast' and 'slow' proteins have been identified that exit the stomach either rapidly or slowly following ingestion (Boirie et al., 1997), and these may have significant effects on hormone levels and satiety. Personalized nutrition attempts to take this into account, to provide optimal customized nutrition for the individual.\n\nIn sophisticated markets today, there is increasing acceptance of the idea that nutrition has a profound effect on health and wellness; as individuals become more aware of their specific nutritional needs, the demand for personalized nutrition is set to increase.\n\nThe impact of all this on milk proteins has to date been minimal. However, three aspects are notable.\n\n\u2022 Allergies to milk. Such allergies, particularly in infants, have been attributed to \u03b2-lactoglobulin in cow's milk (although recent work has cast some doubt on this theory\u2014see Brix et al., 2003). This protein is not produced in human milk and is the dominant whey protein in bovine milk (see Chapter 2). Whey proteins are important nutritionally, as they are a valuable source of essential amino acids. \u03b2-Lactoglobulin is a particularly important source of branched-chain amino acids. So-called hypoallergenic products are therefore produced by hydrolyzing milk proteins, more particularly whey proteins, so that fragments are sufficiently small to be nonallergenic.\n\n\u2022 A2 milk: Some literature suggests a weak correlation between consumption of milk containing the \u03b2-casein A1 variant and some diseases, notably type 1 diabetes (Elliott at al., 1997) and ischemic heart disease (McLachlan, 2001). Further studies on diabetes have proved to be inconclusive (Beales et al., 2004), and the heart disease data do not stand up to scrutiny. Furthermore, other epidemiological data show that the A2 hypothesis does not hold up (Truswell, 2005). Nevertheless, a New Zealand-based company, the A2 Corporation, markets a niche milk product from cows that do not carry the A1 gene. The milk is sold mostly in Australia, where the company is now very careful not to make claims about any specific health benefits after having been prosecuted and fined $15,000 in Queensland in 2004 for making such claims. The product has since then been launched in New Zealand, where it has limited availability, and more recently in the United Kingdom. A report on A2 milk has been released by the European Food Safety Authority (De Noni et al., 2009), which concludes: \"Based on this review, EFSA concluded that a cause and effect relationship is not established between the dietary intake of BCM7, related peptides or their possible protein precursors and non-communicable diseases. Consequently, a formal EFSA risk assessment is not recommended.\"\n\n\u2022 Bioactive peptides: There is increasing evidence that some milk proteins, and more particularly peptides, have physiological functionality. Effects on cardiovascular health, immune modulation, health of bones and teeth, and anticancer effects have all been reported (see Chapters 18  ,). The validity of these effects and the efficacy of functional foods based on them remain to be fully proven, but in time may lead to new functional foods based on milk proteins and their products.\n\n## New technologies and their possible effect on milk protein ingredients and products\n\nA range of new technologies have the potential to affect dairy production and processing in the near future. They include gene technologies that could lead to new, different milk proteins, new kinds of processing that can produce novel milk protein materials and products containing them, and new analytical techniques that have the potential to improve processing and place ever more stringent requirements on product quality.\n\n### Genetic Modification\n\nMilk proteins have been genetically modified and expressed in nonbovine animals (e.g., Bleck et al., 1998) and in cows (Brophy et al., 2003). However, it seems unlikely that transgenic modification of milk proteins for functional or nutritional purposes will occur widely in the foreseeable future. There are several reasons for this (Creamer et al., 2002).\n\n\u2022 Consumer acceptance of genetically modified (GM) foods is still variable throughout the world, with most countries now having strict labeling requirements. Because milk is a liquid product handled in large volumes during processing, maintaining batch identity and keeping GM milk separate are more problematic than is the case with discrete products, although recent efforts with organic milk have proven that this separation is possible.\n\n\u2022 Milk is an animal product that is strongly targeted at the health of babies and young people. This area has been identified in consumer surveys as very sensitive (compared, for example, with the acceptability of GM fruit and vegetables), and milk will probably be one of the last foods for which genetic modification is acceptable to most consumers.\n\n\u2022 The cost of producing herds of GM cows will be very high, and developing herds will be very slow unless expensive cloning and embryo transfer methods are used. This is not justified by a small premium for improved nutrition or functionality arising from genetic modification.\n\n\u2022 More importantly, a switch to genetic modification will severely limit genetic gain because the gene pool will be restricted to the genetics of the donor animals for the original GM parents. This segregation from the global bovine gene pool will prevent, or severely limit, participation in the ongoing genetic improvement of the species, which is currently occurring at about 2% per annum.\n\nNotwithstanding these points, if a strong nutriceutical or pharmaceutical component were to be identified, enhanced expression through genetic modification would not be out of the question. However, much-touted 'gene-pharming' in dairy animals has not yet been notably successful commercially.\n\nThe announcement of development of a female calf that does not produce \u03b2-lactoglobulin in its milk is of great interest (Jabed et al., 2012). Because \u03b2-lactoglobulin does not occur in human milk, it is often regarded as an undesirable component in milk for infant formula, and it has been implicated in some milk allergies (see above). Hydrolysis of whey protein has been an approach to overcome this problem, but this approach incurs a processing cost and raises the osmolarity of the resulting formula, which is considered undesirable. Although the composition of milk from this experimental animal has been shown to be quite unusual, with no detectable \u03b2-lactoglobulin and elevated casein levels, this composition must be viewed with caution, as it was artificially produced from a female calf (induced by hormones) rather than from natural lactation. It remains to be seen if this animal is able to breed. The fact that this construct removes a protein from milk, rather than adding a foreign protein to it, may lead to a product that meets with less consumer resistance.\n\nIn contrast to milk proteins, competitive plant-origin proteins are well advanced in improvement, using genetic modification. GM soybeans are now predominant in world soybean crops, covering 69 million hectares in 2009, which makes up 77% of worldwide soybean production (www.gmo-compass.org\/eng\/agri_biotechnology\/gmo_planting\/342.genetically_modified_soybean_global_area_under_cultivation.html). Soybeans can be genetically modified to remove undesirable proteins such as trypsin inhibitor, soy hemagglutinin, and allergens (e.g., Friedman et al., 1991), while at the same time soy proteins can be modified to provide a more favorable nutritional balance of essential amino acids (Mandal and Mandal, 2000). The more efficient production of soy proteins in terms of energy and water, coupled with these improvements from genetic modification, means that soy proteins will increasingly outcompete dairy proteins as generic nutritional and functional food ingredients. Having said that, we can expect increasing demand for high-quality protein, so there is likely to be a continuing strong market pull for all forms of food protein.\n\n### Novel Processing\n\nHigh-pressure processing was originally developed in the late 19th century (Hite, 1899) but did not find application in food processing until the 1990s, when new materials enabled the development of production-scale processing equipment. High-pressure processing has been used commercially as an alternative method for preservation, particularly for acidic foods (Dunne and Kluter, 2001).\n\nWhen milk is subjected to high pressure, the casein micelle undergoes dramatic nonreversible changes, leading to a smaller micelle that is less opaque (Chapter 6). It has also been reported that high-pressure processing can alter the functionality of whey proteins (Patel et al., 2005). Whether these novel modified proteins will find application in new foods remains to be seen.\n\nA range of other 'nonthermal' processes for the preservation of food have been developed. These include pulsed electric field, irradiation, and cold plasma. Although some of these innovations have been successful in the processing of acidic foods and solid foods (in the case of cold plasma), their commercial application to dairy products is not an immediate prospect.\n\n### New Analytical Methods\n\nRecent years have seen a range of new and improved analytical methods that have potential to improve process control and tighten product specification. Particularly important are methods that can control product safety (especially microbiology), as well as nutritional and functional properties.\n\nOne weakness in product safety in the past has been the need to grow samples on Petri dishes to test for the presence of undesirable microbial species. This process is time consuming and laborious, and can identify issues with process and product only well after they have occurred. A range of novel microbial detection methods is emerging that may allow at-line detection of microbiological problems, or, conversely, provide early assurance of food safety. For example, use of flow cytometry was able to reduce times for measuring bacterial numbers from the 3 days required for the traditional plate count to 2 h (Flint et al., 2006). This advance could be particularly important for proteins manufactured using ultrafiltration, such as whey protein concentrates and milk protein concentrates, because the ultrafiltration step co-concentrates any microbial contaminants that may be present.\n\nModern electrospray mass spectrometric analysis has enhanced our ability to understand and control processing effects that can alter the nutritional value of milk proteins, particularly the loss of bioavailable lysine due to processing and storage effects (see Chapter 11). Similarly, once the relationship between functionality and protein chemistry is well understood, the same techniques will allow better management of functional properties. Novel inline and at-line methods are becoming possible through a range of techniques, including nuclear magnetic resonance, Fourier transform infrared spectroscopy, selected ion flow tube mass spectrometry, and surface plasmon resonance analysis. These methods can, in principle, measure attributes such as water activity inside packaging and can be used to predict the flavor characteristics of cheeses at maturation (for example, Langford et al., 2012).\n\n### Materials Science and Nanotechnology\n\nFood structure is important at all dimensional scales for the sensory properties of food (including texture, mouthfeel, and flavor release) and may have an important effect on nutrient release and bioavailability (Parada and Aguilera, 2007). Increasingly, attention is being paid to materials science approaches to understanding and potentially managing these effects. An example is the physics of soft materials being applied to food (Mezzenga et al., 2005; Ubbink et al., 2008).\n\nThe potential application of nanotechnology and nanoscience to food can be expected to become an important area. Much of the higher dimensional structure of food is a consequence of nanostructures. It is unlikely that nano robotics will be applied to food in the foreseeable future: Regardless of the considerable technical challenges, public acceptance can be expected to be a major barrier. Notwithstanding this attitude, nanotechnology is having a considerable impact on food science, in part through the use of new improved instrumentation that is becoming available to support nanotechnology research (Foegeding, 2006; Weiss et al., 2006). One of the important features of nanotechnology is the occurrence of self-assembling molecular superstructures (nanostructures). It turns out that foods naturally contain many such systems, examples being the actin-myosin complex in the muscle fibres of meat, starch granules in plant foods, and the casein micelle in milk. Whey proteins have been shown to form self-assembling systems under a variety of conditions\u2014as whey protein (a mixture), as \u03b2-lactoglobulin, and as \u03b1-lactalbumin. Early publications on these (Bolder et al., 2006; Graveland-Bikker and de Kruif, 2006) have been followed by a wide range of published work, recently reviewed by Loveday et al. (2012) and Jones and Mezzenga (2012), that shows how to control the generation and properties of such structures and their physical functional properties.\n\n## Conclusions\n\nThis brief chapter has provided a glimpse of some of the global issues and new technologies that may influence the future development and use of dairy protein products. Global trends such as rising energy costs, scarcity of water, and effect of greenhouse gases will increase the cost of production of milk proteins and will restrict land areas where they can be sustainably produced, especially to areas that are water-rich. Milk proteins are relatively expensive nutritional and functional food ingredients, and although they are nutritionally superior to plant origin competitors, they are likely to be increasingly restricted to niche applications as less expensive plant-based alternatives become more widely available.\n\n# References\n\nAnderies JM ,  Ryan P ,  Walker BH .  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Inhibiting, promoting, and preserving stability of functional protein fibrils .  _Soft Matter_.  2012 ;8 : 876 \u2013 895 .\n\nJudge A ,  Ledgard S ,  Smeaton D ,  Boyes M .  _Greenhouse gas emissions from Rotorua dairy farms_ .  Hamilton, New Zealand :  Report to MAF. AgResearch ;  2010 .\n\nLangford VS ,  Reed CJ ,  Milligan DB ,  McEwan MJ ,  Barringer SA ,  Harper WJ .  Headspace analysis of Italian and New Zealand Parmesan cheeses .  _Journal of Food Science_.  2012 ;77 : C719 \u2013 C726 .\n\nLoveday SM ,  Su J ,  Rao MA ,  Anema SG ,  Singh H .  Whey protein nanofibrils: The environment \u2212 morphology \u2212 functionality relationship in lyophilization, rehydration, and seeding .  _Journal of Agricultural and Food Chemistry_.  2012 ;60 : 5229 \u2013 5236 .\n\nMandal S ,  Mandal RK .  Seed storage proteins and approaches for improvement of their nutritional quality by genetic engineering .  _Current Science_.  2000 ;79 : 576 \u2013 589 .\n\nMcLachlan CN .  Beta-casein A1, ischaemic heart disease mortality, and other illnesses .  _Medical Hypotheses_.  2001 ;56 : 262 \u2013 272 .\n\nMezzenga R ,  Schurtenberger P ,  Burbridge A ,  Michel M .  Understanding foods as soft materials .  _Nature Materials_.  2005 ;4 : 729 \u2013 740 .\n\nParada J ,  Aguilera JM .  Food microstructure affects the bioavailability of several nutrients .  _Journal of Food Science_.  2007 ;72 : R21 \u2013 R32 .\n\nPatel HS ,  Singh H ,  Havea P ,  Considine T ,  Creamer LK .  Pressure-induced unfolding and aggregation of the proteins in whey protein concentrate solutions .  _Journal of Agricultural and Food Chemistry_.  2005 ;53 : 9590 \u2013 9601 .\n\nPimentel D .  Impacts of organic farming on the efficiency of energy use in agriculture .  _An Organic Center State of Science Review_.  2006 ; : 2013 : www.organiccenter.org\/science.pest.php?action=view&report_id=59 downloaded May 2013 .\n\nPimentel D ,  Pimentel M .  _Food, energy, and society_ .  New York :  Wiley ;  1979 .\n\nSaunders C ,  Barber A ,  Taylor G .  _Food Miles\u2014 Comparative Energy\/Emissions Performance of New Zealand's Agriculture Industry_ .  Lincoln, New Zealand :  Research Report No. 285. Lincoln University ;  2006 .\n\nSteinfeld H ,  Gerber P ,  Wassenaar T ,  Castel V ,  Rosales M ,  de Haan C .  _Livestock's long shadow: Environmental issues and options_ .  Rome :  Food and Agriculture Organization ;  2006 .\n\nTyan Y-C ,  Yang M-H ,  Jong S-B ,  Wang C-K ,  Shiea J .  Melamine contamination .  _Analytical and Bioanalytical Chemistry_.  2009 ;395 : 729 \u2013 735 .\n\nTruswell AS .  The A2 milk case: A critical review .  _European Journal of Clinical Nutrition_.  2005 ;59 : 623 \u2013 631 .\n\nUbbink J ,  Burbidge A ,  Mezzenga R .  Food structure and functionality: A soft matter perspective .  _Soft Matter_.  2008 ;4 : 1569 \u2013 1581 .\n\nvan Ommen B ,  Bouwman J ,  Dragsted LO ,  Drevon CA ,  Elliott R ,  de Groot P ,  Kaput J ,  Mathers JC ,  Muller M ,  Pepping F ,  Saito J ,  Scalbert A ,  Radonjic M ,  Rocca-Sera P ,  Travis A ,  Woperis S ,  Evelo CT .  Challenges of molecular nutrition research 6: the nutritional phenotype database to store, share and evaluate nutritional systems biology studies .  _Genes and Nutrition_.  2010 ;5 : 189 \u2013 203 .\n\nWeiss J ,  Takhistov P ,  McClements DJ .  Functional materials in food nanotechnology .  _Journal of Food Science_.  2006 ;71 : R107 \u2013 R116 . \n\n# Index\n\nA\n\nA2 milk, see A2 protein\n\nA2 protein, ,\n\nAllergenic effects, ,\n\nAlpha-cystallin,\n\nAcid gels\n\nbonds, types of, 305\u2013306\n\ncarbonation,\n\n\u03ba-casein, , ,\n\nCCP, ,\n\ncross-linking,\n\ndirect acidification,\n\nenzymatic modification of proteins,\n\nfat globule surface material,\n\nglucono-\u03b4-lactone (GDL), , , ,\n\nhomogenization,\n\nhydrolysis\n\n\u03ba-casein,\n\nwhey proteins,\n\nmicelles, effect of acidification, , 502\u2013503\n\nmicelle-whey protein association, , 302\u2013304\n\nmolecular interactions,\n\nnon sedimentable protein, , ,\n\npH, 192\u2013193, , , 305\u2013306, 502\u2013503,\n\npressure,\n\nprocessing variables, , ,\n\nprotein concentration, , , , 305\u2013306, ,\n\nrheological properties, 192\u2013193, , , ,\n\nsoluble aggregates and acid gel properties, , , 304\u2013305\n\nstorage modulus, , ,\n\nstructure, , , , , ,\n\nsyneresis, , ,\n\ntemperature\n\nacidification temperature,\n\nincubation temperature, ,\n\npH, , , 298\u2013299, 305\u2013306,\n\npressure,\n\npre-treatment, , , , 505\u2013508\n\nwhey protein denaturation, 296\u2013311, 504\u2013508\n\ntexture, 192\u2013193, , , ,\n\ntransglutaminase (TGase),\n\ntrisodium citrate (TSC),\n\nwhey protein denaturation, , 505\u2013508\n\nin the presence of micelles, , 297\u2013307\n\nseparate denaturation to micelles, 310\u2013311\n\nzeta potential,\n\nAcid precipitation, , 274\u2013275, ,\n\nAlkysulfonate (AL),\n\nAlveolus, see Mammary gland\n\nAmino acids\n\ncalorie provision, 533\u2013534\n\ncaseins,\n\ncomparison of proteins,\n\ncysteine, , , ,\n\ndietary essential,\n\ncountries deficient, identification of, 9\u201310\n\ndietary availability of,\n\nfood items, contents of,\n\nformation of lysinoalanine,\n\nlysine, see Lysine\n\nflavor,\n\nlanthionine, , ,\n\n\u03b1-lactalbumin,\n\n\u03b2-lactoglobulin variants,\n\nlysine, 352\u2013353, , , , , ,\n\nmethionine, , , ,\n\nnutrition, 528\u2013531\n\nphysiological roles, 531\u2013532\n\nproline,\n\nsatiety, 533\u2013534\n\ntaurine, , ,\n\ntryptophan, , , , 439\u2013440\n\nAngiogenins,\n\nAnimal protein sources,\n\ngrowing global demand, 8\u20139\n\nBennett's Law,\n\ncarbon footprint,\n\ngreen revolution,\n\nAtherosclerosis,\n\ncommon cause of,\n\nmilk protein fractions,\n\npostmenopausal women, overweight,\n\nwhey protein\n\nfunctions,\n\nvs. casein,\n\nB\n\nBakery products, 463\u2013464\n\ninclusion of milk proteins,\n\nBAMLET, ,\n\nBiologically active molecules, see Functional components; See also Functional foods Peptides\n\nBiopolymers, see Polysaccharide-protein systems\n\nBlood serum albumin, see Serum albumin\n\nBlood pressure, 547\u2013548\n\ndairy products consumption,\n\nintact milk proteins supplementation,\n\nlowering effects,\n\nmilk-derived peptides\n\nantihypertensive effects,\n\nACE inhibitory activity,\n\nisoleucine-proline-proline (IPP), ,\n\nmeta analysis,\n\nvaline-proline-proline (VPP), ,\n\nwhey protein hydrolysate,\n\nBone health, 548\u2013550\n\nbone mineral density (BMD),\n\nbone remodeling,\n\ncow's milk,\n\nCa derived from,\n\nenergy-restricted diets, weight loss,\n\nessential nutrients,\n\ninfant, see Infant health\n\ninsulin-like growth factor 1 (IGF-1),\n\nnutrition\n\nadolescence,\n\nchildhood,\n\nwhey protein,\n\nfunction,\n\nBonds, see Molecular interactions\n\nBovine classification, 22\u201323\n\nBovine serum albumin, see Serum albumin\n\nBradykinin, ,\n\nC\n\nCalcium-binding proteins, , , , , , , ,\n\nCalcium phosphate\n\nequilibria, 180\u2013181\n\nforms in milk,\n\ngeneral, 53\u201356,\n\nmicelles, role in, 55\u201356, , , , 270\u2013273\n\npH, effect on, , , 502\u2013503,\n\npressure, effect on, ,\n\nrennet gels,\n\nCape fur seal\n\napoptosis, protection from, 99\u2013103\n\ncDNA library,\n\nchanges in milk composition,\n\ncomparison with other seals, ,\n\nFIL protein,\n\ngene expression, 102\u2013103\n\ninvolution, 102\u2013103\n\n\u03b1-lactalbumin,\n\nlactation cycle,\n\nprotein content of milk,\n\nCarbohydrates, see Lactose; See also Oligosaccharides\n\nCarbon footprint,\n\nCarrageenans\n\nCasein\n\navailable amine,\n\ncalcium binding,\n\ncomparison with whey proteins,\n\nflavors, interactions with,\n\nfunctionality, 171\u2013173,\n\ngenes, ,\n\nglycosylation, , 150\u2013151\n\nheterogeneity, , See also \u03ba-casein\n\ncasein phosphoproteins,\n\nexperimental modifications, 130\u2013132\n\nfunction, 125\u2013127\n\nmicelle, see Casein micelle\n\nprecipitation,\n\nprotein characterization,\n\nvariants\n\nbovine \u03b1-Lg,\n\nbovine \u03b1s2-casein,\n\nbovine \u03b1s1-casein, B variant,\n\nbovine \u03b2-casein,\n\nbovine \u03b2-Lg,\n\nbovine \u03ba-casein,\n\n\u03ba-casein genes,\n\ngenetic,\n\n\u03b2-Lg genes,\n\nhistory of, 37\u201338\n\nhydrolysis, , , , , , ,\n\nintestinal lumen,\n\nin vitro studies,\n\nin vivo study, human,\n\nhydrophobicity, ,\n\ninterspecies comparison,\n\nisoelectric point, ,\n\nisopeptide bonds, , ,\n\nmanufacture, 37\u201338, ,\n\nmetal binding,\n\nmicroheterogeneity,\n\nmolecular interactions\n\ndisulfide bonds, , , , , , ,\n\nelectrostatic repulsion, , , , , , , ,\n\nhydrophobic bonds, , , , , 174\u2013175, , 182\u2013184, , , , , , , , ,\n\nproperties, , 43\u201345\n\nrehydration of powders, 337\u2013338\n\nretinol, interactions with, , ,\n\nresidual immunoreactivity, evolution of \u03b2-caesin, digestion of,\n\nsensory characteristics,\n\nstorage and processing changes, , , ,\n\nstructure, ,\n\nsugars, interactions with,\n\ntransglutaminase (TGase), , ,\n\nvariants, proportions of,\n\nvitamin D, interactions with, , see also \u03b1-casein; \u03b2-casein; Casein micelles; \u03b3-casein; \u03ba-casein\n\n\u03b1-Casein\n\naggregation, ,\n\ncalcium binding,\n\ncalcium sensitivity, , ,\n\n\u03ba-casein, interactions with, ,\n\nchaperone-like activity,\n\ndissociation,\n\ndisulfide bonding, ,\n\nemulsions, , , , ,\n\ngenes, ,\n\nhydrolysis, ,\n\nhydrophobicity,\n\nhydrophobic bonds, , , 173\u2013174, 178\u2013179\n\ninterspecies comparison, ,\n\nisolation, ,\n\n\u03b2-lactoglobulin, interactions with, , , ,\n\nlocation in the micelle,\n\nminerals, interactions with, ,\n\nphosphorylation, , , ,\n\npolymorphism,\n\npost-translational modification, ,\n\nproportion in milk,\n\nself-association, , 178\u2013179\n\nstability,\n\nstructure, , ,\n\nvariants, , , , ,\n\nwhey proteins, interactions with,\n\n\u03b2-Casein\n\nA2 protein, ,\n\ncalcium sensitivity, ,\n\nCape fur seal,\n\nchaperone-like activity,\n\ndissociation, , ,\n\nmolecular interactions\n\ndisulfide bonding,\n\nhydrophobic bonds, , , , 173\u2013174, , ,\n\nemulsions, 171\u2013173, , , , ,\n\ngenes, , ,\n\nhydrolysis,\n\nhydrophobicity, ,\n\ninteractions with \u03ba-casein,\n\ninteractions with minerals, , ,\n\ninteractions with surfactants,\n\ninteractions with vitamin D,\n\ninterspecies comparison, , ,\n\nisolation,\n\nphosphorylation, , , ,\n\npost-translational modifications, ,\n\nproportion in milk,\n\nself-association, 173\u2013175, 178\u2013179\n\nstability,\n\nstructure, ,\n\ntemperature changes, ,\n\nvariants, , , ,\n\n\u03b3-Casein\n\nhydrolysis,\n\nisolation,\n\n\u03ba-Casein\n\nacid gels, , ,\n\naggregation,\n\ndissociation\n\nchanges in physical properties of milk,\n\npH, effect of, 282\u2013285,\n\nstabilization of other caseins,\n\ntemperature, effect of, , ,\n\ndisulfide bonding, , , 152\u2013153\n\nemulsions, , ,\n\ngenes, , , ,\n\nglycosylation, , , , , , 150\u2013151, 155\u2013160\n\nheterogeneity\n\nenvironmental influences, ,\n\nhydrolysis, effect on,\n\nlactation phase,\n\nmicelle stability, effect on, ,\n\nbiological significance,\n\npolymorphism,\n\nrennet clotting time (RCT), effect on,\n\nvariants, ,\n\nhydrolysis, , , ,\n\nglycosylation, effect of, 157\u2013158\n\nkinetics,\n\nrennet, , , , ,\n\ninteractions with other whey proteins, ,\n\ninterspecies comparison, , ,\n\nisolation, ,\n\n\u03b2-lactoglobulin, interactions with, , , , 279\u2013293, 505\u2013507\n\ncomplexes, formation of,\n\npH, effect of, 282\u2013285\n\nspecific disulfide bonds, 290\u2013293\n\nstabilizing caseins against calcium ions,\n\nlocation in micelle, , , , , , , 270\u2013273,\n\nmacropeptides, , , , , ,\n\nminerals, interaction with, , ,\n\nmolecular interactions\n\ndisulfide bonds, , , , 152\u2013153, 182\u2013183, , 280\u2013281, 290\u2013293, 305\u2013306\n\nsteric effects, , , , , , , ,\n\nNeuAc, , , ,\n\nphosphorylation, , , ,\n\npost-translational modification,\n\npolymorphism, , ,\n\nproportion in milk,\n\nstability\n\nethanol, , ,\n\npH, 282\u2013285, ,\n\nthermal treatment, , , ,\n\nstructure, ,\n\nvariants, , , , ,\n\nCasein micelles, 53\u201358\n\nacid gels, 192\u2013193\n\naggregation, 495\u2013497, 497\u2013498\n\nbiological purpose, ,\n\ncalcium phosphate, , , 55\u201356, , , 180\u2013186, , , , 270\u2013273\n\ncasein aggregation\n\n\u03b1-casein aggregation, ,\n\n\u03b2-casein self association, 173\u2013174, , 178\u2013179\n\nforces involved, , , 182\u2013183\n\nionic concentration, effect of, 174\u2013175,\n\nmicellization, 173\u2013174\n\ntemperature, effect of,\n\n\u03b1-casein, location in the micelle,\n\n\u03ba-casein\n\ndissociation, , 282\u2013285, ,\n\ndistribution, ,\n\nglycosylation,\n\nhydrolysis, , , ,\n\nlocation in the micelle, , , , , , , 270\u2013273,\n\ncentrifugation, , ,\n\ncomposition\n\ngeneral, ,\n\npH, effect of, , , 184\u2013186, ,\n\nsize fractionation, effect of,\n\ntemperature, effect of, , , , , , ,\n\nconcentration, effect of, 186\u2013188\n\nenzymes, interactions with,\n\ninterspecies comparison,\n\nisoelectric point,\n\nisolation,\n\nmetal binding,\n\npara-\u03ba-casein, , , , , ,\n\npolysaccharides, interactions with\n\n\u03b9-carrageenan,\n\ngalactomannans,\n\nlocust bean gum,\n\npectin,\n\npost-translational modification,\n\nproperties, , 186\u2013188\n\nrennet gels, 189\u2013190, 494\u2013497\n\nrheology, 186\u2013188\n\nself-assembly,\n\nsize, , , , , 191\u2013192, , , ,\n\nsurfactants,\n\nstability\n\nalcohols, , , , , , 191\u2013192,\n\ncalcium,\n\n\u03ba-casein glycosylation,\n\ncompaction, ,\n\ncooling, , , , ,\n\nheating, , , , 282\u2013285\n\npH changes, , , , , , , 191\u2013192, , 282\u2013285, , , 502\u2013503\n\npressure, ,\n\nproteinases, , , ,\n\nsequestrants, , ,\n\ntrifluoroethanol (TFE),\n\nurea, , ,\n\nstructural models, dual-binding\n\nacid gels, 192\u2013193\n\nassembly, 178\u2013179, 182\u2013183\n\ncalcium phosphate equilibria, 180\u2013181\n\nethanol, 191\u2013192\n\nmicellar interactions, 184\u2013186, 186\u2013188\n\nmineral addition, ,\n\npH, , 184\u2013187, 190\u2013192\n\nrennet gels, 186\u2013190\n\nsequestrants, effect of,\n\nstructure, 182\u2013183,\n\ntemperature, effect of, ,\n\ntrifluoroethanol (TFE),\n\nurea, effect of,\n\nstructural models, general, 54\u201356, , , 270\u2013273\n\nsubmicelles, 55\u201356,\n\nsurface structure, , , , , ,\n\nSXE peptide,\n\nwhey proteins, interactions with\n\n\u03b1-lactalbumin,\n\n\u03b2-lactoglobulin, , , 280\u2013293, 305\u2013306, , see also \u03ba-casein; hydrolysis\n\nCaseinomacropeptide (CMP), see Macropeptide\n\nCaseinate\n\nacid gel stabilization, ,\n\nemulsions\n\ncalcium caseinate, , , ,\n\nsodium caseinate, , , 368\u2013369, , ,\n\nflavors, interactions with,\n\nfunctionality,\n\nhydrophobic bonding,\n\nlanthionine,\n\nmanufacture, , , ,\n\nnon-dairy food applications\n\ncoffee whitener,\n\nmeat products, 466\u2013467\n\nwhipped toppings,\n\npolysaccharides, interactions with\n\ngum Arabic, ,\n\npectin, ,\n\nxanthan gum,\n\nproperties, ,\n\nstability, ,\n\nstorage and processing changes, , ,\n\nCaseinophosphopeptides (CPPs),\n\nCCP, see Calcium phosphate\n\nCeruloplasmin,\n\nChaperone-like activity,\n\nChemical denaturants\n\n\u03b1-lactalbumin,\n\nLactoferrin,\n\n\u03b2-lactoglobulin, 218\u2013219\n\nserum albumin,\n\nsugars, effect of,\n\nChitosan\n\nChymosin, , , , , , , , , , see also Rennet\n\nCircular dichroism (CD), , , 216\u2013218, ,\n\nComplex coacervation, see Polysaccharide-protein systems, mixing behaviors\n\nCoagulation, 563\u2013566\n\ncheese,\n\naging effect,\n\ncheddar vs. mozzarella,\n\ndigestion studies,\n\nParmigiano Reggiano, production of,\n\nrheological measurements,\n\ndairy industry, use in,\n\ngel matrices,\n\nkinetics of\n\namino acids absorption,\n\nmilk protein digestion,\n\nmini-pigs stomach,\n\nexogenous nitrogen, remaining fraction of,\n\nplasma leucine concentration,\n\n15N, postprandial portal absorption of,\n\nreal dairy matrices, digestion of,\n\nrennet\n\ngel,\n\nmatrix,\n\nyogurt,\n\ndigestion studies,\n\nheat treated,\n\nColloidal calcium phosphate (CCP), see Calcium phosphates\n\nCommercial milk protein products, , ,\n\nConfocal laser scanning microscope (CLSM), ,\n\nCo-solubility, see Polysaccharide-protein systems, mixing behaviors\n\nCosts of milk production\n\nenergy,\n\nmethane, 574\u2013575\n\nwater, 573\u2013574\n\nCryoglobulin,\n\nCysteine, , , , , , , ,\n\nCysteine residues, , ,\n\nD\n\nDairy food, global\n\ncheese,\n\nexporters, major,\n\nproduction,\n\nconsumption of,\n\nfood consumption patterns, global,\n\nimportance of,\n\nimporters, major,\n\nIndia's dairy industry,\n\nmajor dairy exports, volume of,\n\nprotein consumption, global,\n\nprotein demand, global,\n\nspecialty foods,\n\nworld milk production,\n\nregional distribution, pattern of,\n\nworld protein trade,\n\nDairy product analogues\n\ncoffee whitener,\n\nwhipping products,\n\nDehydration, see Spray drying\n\nDenaturation\n\ndefinition, , see also beta-lactoglobulin\n\nDepletion interactions, see Polysaccharide-protein systems, mixing behaviors\n\nDifferential scanning calorimetry (DSC), , , , , ,\n\nDiffusing wave spectroscopy, ,\n\nDigestion, milk protein\n\namino acids digestibility,\n\ncaseins\n\ncoagulum,\n\nretention time,\n\nslow proteins,\n\nvs. whey protein,\n\nileal digestibility\n\ncaseins,\n\nvs. fecal digestibility,\n\nwhey,\n\npeptides released during digestion,\n\ntrue digestibility of,\n\nDried proteins, see Drying by desorption; See also Protein powders; Spray drying\n\nDrying by desorption\n\ncaseinates, ,\n\ncitrate addition,\n\ndrying slopes,\n\nion addition,\n\nprinciples,\n\nWPC, , see also Spray drying\n\nE\n\nEchidna (Tachyglossus aculeatus)\n\ndevelopment stages,\n\nextant monotreme,\n\nhabitats,\n\nmaternal care, phases of,\n\ngestation,\n\nincubation,\n\nlactation,\n\nweaning,\n\nmonotreme milk\n\ncomponents,\n\ncomposition,\n\nEdman sequencing,\n\nElderly, see also Sarcopenia; Atherosclerosis; Blood pressure\n\nessential amino acids,\n\nanabolic resistance,\n\nhigh-quality protein,\n\nleucine, , see also Leucine\n\nresponsible for,\n\nsupplements,\n\nwhey protein, see Whey protein\n\nprotein nutritional needs of, 12\u201313\n\naging process, characterized by,\n\nhigh-quality protein intake, daily,\n\nmuscle protein synthesis,\n\nprotein dietary allowance, recommended,\n\nsarcopenia, see Sarcopenia\n\nElectron microscopy, , 55\u201356, , 176\u2013178, , , , , , , ,\n\nElectrophoresis, 41\u201342, , ,\n\n2-dimensional (2-DE), , , , , , , ,\n\nnative-PAGE,\n\nSDS-PAGE, , , , ,\n\nElectrospray ionization mass spectrometry,\n\nEmulsifying characteristics of milk constituents, , ,\n\nEmulsions\n\ncasein emulsions\n\ncalcium caseinate, , , ,\n\n\u03b1-casein, , , , ,\n\n\u03b2-casein, 171\u2013173, , , , ,\n\n\u03ba-casein, , ,\n\nformation, ,\n\nsodium caseinate, , , 368\u2013369, , ,\n\nwhey emulsion, stabilization of,\n\ncharacteristics,\n\ncomparison of milk proteins,\n\ncompetitive adsorption, ,\n\ndroplet size, , ,\n\neffect of calcium addition, ,\n\neffect of polysaccharide addition,\n\nformation drivers,\n\nformation process, ,\n\ngel, emulsion,\n\nheat-induced changes, 370\u2013371\n\nhomogenization, ,\n\nhydrolysate-stabilised\n\nbenefits of hydrolysis,\n\ncauses of instability,\n\neffect of polysaccharide addition,\n\nwhey hydrolysates, 373\u2013374\n\ninstability\n\nbridging flocculation, ,\n\ncalcium-induced instability, 373\u2013374\n\ncoalescence, ,\n\ncreaming, , , ,\n\ndefinition,\n\ndepletion flocculation, , 368\u2013369, 371\u2013373\n\nlactoferrin-based emulsions, 374\u2013375\n\nmeat emulsions, 466\u2013467\n\nMPC emulsions, , , ,\n\nmultilayered emulsions,\n\nprocessing of proteins,\n\nprotein load, ,\n\nstability\n\ndefinition,\n\nheat, 370\u2013371\n\nions, , , , , 373\u2013374\n\npH, , ,\n\npressure,\n\nprotein concentration, ,\n\nprotein-oil ratios, 363\u2013364, ,\n\nsynergistic effects,\n\ntypes of instability,\n\nstructure,\n\nsurface activity, ,\n\nsurface tension,\n\nwhey protein emulsions\n\ncasein addition,\n\nformation, , ,\n\nhydrolysates, 373\u2013374\n\nion addition, , 373\u2013374\n\n\u03b1-lactalbumin, , , ,\n\n\u03b2-lactoglobulin, , , , , , ,\n\nmixed whey proteins, , , ,\n\nEnergy to produce milk,\n\nEnzymes, , , ,\n\nEvolutionary changes in proteins, ,\n\nExpressed sequence tag (EST), , ,\n\nExtreme adaptation to lactation, see Cape fur seal; See also Tammar wallaby\n\nF\n\nFAO, see Food and agriculture organization\n\nFood and agriculture organization, , see also Hunger\n\nfood insecurity, definition of,\n\nprotein quality evaluation in human nutrition,\n\nFatty acids\n\nconjugated linoleic acid (CLA), ,\n\nflavors,\n\nproperties,\n\nproteins, interactions with\n\nbinding sites, , , 428\u2013429,\n\ncompetitive binding, ,\n\n\u03b1-lactalbumin, , ,\n\n\u03b2-lactoglobulin, , , , , , , 428\u2013429\n\npH, effect of,\n\nserum albumin, , , , 429\u2013430\n\nstructure, 32\u201334\n\nsynthesis, ,\n\ntriglycerides, distribution in,\n\nFeedback inhibitor of lactation (FIL) protein, see Cape fur seal, Tammar wallaby\n\nFerroxidase, see Ceruloplasmin\n\nFIL, see Cape fur seal; See also Tammar wallaby\n\nFlavor, see Sensory characteristics\n\nFlavor binding, ,\n\nFlocculation, see Polysaccharide-protein systems, mixing behaviors\n\nFluorescence spectroscopy, , , , , ,\n\nFolate-binding proteins (FBPs),\n\nFolate\n\nlactation, regulation of, 119\u2013120\n\nlactation, requirements in,\n\nprotein synthesis, role in,\n\nreceptors,\n\nsupplementation during lactation,\n\nFood miles,\n\nFood safety, 575\u2013576,\n\nFood traceability, 575\u2013576\n\nFourier transform infrared (FTIR) spectroscopy, , , 274\u2013275\n\nFourier transform of Raman spectra,\n\nFreeze drying, protective effect of sugars, 440\u2013441\n\nFunctional characteristics\n\nACE inhibitors, 128\u2013129,\n\nangiogenesis,\n\nantimicrobial, , , , , , , , ,\n\nanticancer, , , , , ,\n\nanticariogenic, ,\n\nanti-hypertensive, 128\u2013129\n\nanti-inflammatory,\n\nantioxidant, , ,\n\nantiviral, , ,\n\nblood coagulation,\n\nbone-cell activity, ,\n\ncopper delivery,\n\ngastrointestinal motility, 535\u2013536\n\ngrowth promotion,\n\nimmunity, , , , , , see also Immunoglobulins\n\nintestinal absorption,\n\nmuscle contraction,\n\nlipase stimulation,\n\nsatiety, 533\u2013534\n\nsecretory processes,\n\nthermogenesis,\n\nvitamin-binding,\n\nFunctional components\n\namino acids\n\narginine, , ,\n\nglutamine,\n\nleucine, , , ,\n\ntaurine, , ,\n\ntryptophan, , ,\n\nbioactive peptides, , , 128\u2013129, 534\u2013536\n\ncaseinophosphopeptides (CPPs),\n\nexorphins,\n\nlactoferricin, ,\n\nmacropeptides, , ,\n\ndiscovery of,\n\nwhole proteins\n\nangiogenins,\n\nenzymes,\n\nglycoproteins, 50\u201351\n\nimmunoglobulins,\n\nkininogens,\n\n\u03b1-lactalbumin,\n\nlactoferrin, , , ,\n\nosteopontin (OPN), 50\u201351\n\nwhey proteins, , 49\u201351\n\nFunctional foods\n\nbackground, , , ,\n\nconsumer considerations,\n\ndefinition,\n\ndrivers, , 128\u2013129,\n\nexamples\n\n\"Anadis\" tablets to prevent diarrhoea,\n\nAnti-hypertensive dairy products, 128\u2013129\n\nPossible future product,\n\n\"Recaldent\" chewing gum to repair tooth enamel,\n\nmanipulation of milk composition, 130\u2013132\n\nregulatory considerations, 526\u2013527, see also Functional compounds; Nutrigenomics; Personalized nutrition\n\nFunctionality of protein systems, see Structure-function relationships\n\nFurosine, ,\n\nG\n\nGalactomannans\n\n\u03b2-Galactosidase,\n\nGels, see Acid-induced gels; See also Mixed gels; Polysaccharide-protein systems; Rennet-induced gels\n\nGene expression, see Cape fur seal; See also Post-translational modifications; Tammar wallaby\n\nGenetic modification\n\nbarriers to uptake, ,\n\neffects on milk characteristics, 130\u2013132\n\nmodification of mammals, ,\n\nmodification of other organisms, , ,\n\ntechniques for manipulating milk proteins,\n\nGenetic polymorphism, , , , ,\n\nCharacteristics influenced, ,\n\nhuman gene polymorphisms,\n\ntypes of milk protein polymorphism, , see also Variants\n\nGenomics\n\nbovine\n\ngenome sequencing project, ,\n\nDNA sequence,\n\nfunctional genomics, 117\u2013118\n\ngenome map,\n\npolymorphisms, ,\n\nCape fur seal, 102\u2013103\n\ncDNA library,\n\ndefinition, 114\u2013115\n\nExpressed sequence tag (EST), , ,\n\nFIL protein,\n\nfolate metabolism and milk production, 119\u2013120\n\ngene expression during lactation cycle, , , ,\n\n\u03b1-lactalbumin, , ,\n\n\u03b2-lactoglobulin, , , ,\n\nMammalian Genome Project,\n\nTammar wallaby\n\ngrowth regulation,\n\ngene expression, 102\u2013103\n\nwhey acidic protein (WAP), , , 90\u201393,\n\nGlobal hunger index\n\nglobal and regional trends of,\n\nhunger indicators,\n\nhunger scenarios, different,\n\nGlucono-\u03b4-lactone (GDL), , , , ,\n\nGlutathione peroxidase,\n\nGlycomacropeptide (GMP), see Macropeptide\n\nGlycoproteins, , , , , ,\n\nGlycosylation, , , , , , , 150\u2013151, 155\u2013160\n\nGrowth factors, ,\n\nGuar gum\n\nGum Arabic\n\nH\n\nHAMLET, ,\n\nHeterogeneity, see Casein; See also \u03ba-casein\n\nHigh pressure processing (HPP), see Pressure treatment\n\nHormones, , , ,\n\nHunger\n\ndefinition of,\n\nGHI, see Global hunger index\n\nprotein, importance of, 5\u20139, see also Protein bioavailability; Protein composition\n\nFAOSTAT database,\n\nprotein daily intake for adults, recommended,\n\nprotein intake, country population,\n\nworld protein supply,\n\nreduction targets,\n\ngoal 1, millennium development goals,\n\nworld food summit target,\n\nworld hunger and undernutrition status,\n\nundernourished people,\n\nundernourishment by region,\n\nHydrolysates\n\nbioactive hydrolysates\n\ncasein, ,\n\n\u03b1-lactalbumin,\n\nfunctional properties\n\nemulsification properties, 373\u2013374\n\nprevention of bar hardening,\n\nnutritional benefits, ,\n\nsensory characteristics, ,\n\nHydrolysis\n\n\u03b1-casein, ,\n\n\u03b2-casein,\n\n\u03ba-casein, , , , , , , , , ,\n\nchymosin, , , , , , , , ,\n\nlactoferrin, ,\n\nplasmin, , ,\n\nrennet, , 39\u201340, , 46\u201347, , , , , , 189\u2013190, ,\n\nHyper-immune milk, ,\n\nHypoallergenic milk products, ,\n\nI\n\nIFPRI, see International food policy research institute\n\nImmunoglobulins\n\nbioactive potential, ,\n\nconcentration in milk, ,\n\nhyper-immunization,\n\nImmunoglobulin G (IgG)\n\nbiological purpose,\n\nstability, , ,\n\nstructure,\n\ninterspecies variation, ,\n\nstability, ,\n\nvariants,\n\nInfant health\n\nbreastfeeding,\n\nbreast milk,\n\nfractionation,\n\ncommercial infant formulas (IF)\n\nhigh-protein content,\n\nhydrolyzed whey protein,\n\nlow-protein levels,\n\nobesity,\n\nprotein-modified,\n\nwhey-modified protein,\n\ncow's milk proteins,\n\nhuman's milk\n\nvs. cow's milk protein composition,\n\nlow-birth-weight (LBW) infants,\n\nInternational food policy research institute,\n\nIsopeptide bonds, , ,\n\nK\n\nKininogens,\n\nL\n\n\u03b1-Lactalbumin\n\nalpha-crystallin, interactions with,\n\napoptosis, protection against,\n\nBAMLET, , , ,\n\nbiological purpose, ,\n\ncasein micelles\/\u03ba-casein, interactions with, 279\u2013281,\n\nemulsions,\n\nfatty acids, interactions with, , ,\n\nflavors, interactions with,\n\nforms, ,\n\ngels, ,\n\ngenes, ,\n\nglycosylation,\n\nHAMLET,\n\nheat-shock proteins, interactions with,\n\nhydrolysates,\n\ninterspecies variation, , 47\u201348,\n\ninvolution, role in,\n\nisolation, ,\n\nlactoferrin, interactions with,\n\n\u03b2-lactoglobulin, interactions with, , , , , , , , , ,\n\nlactose synthesis,\n\nligand binding, ,\n\nlysozyme, similarity to, , , ,\n\nminerals, interactions with, ,\n\nmineral binding\n\nbinding site, ,\n\ngeneral, ,\n\nstability, effect on, ,\n\nmolecular interactions\n\ndisulfide bonding, , , , ,\n\nelectrostatic repulsion,\n\noverview, 47\u201348\n\nphospholipids, interactions with,\n\nproteolysis,\n\nproportion in milk, ,\n\nretinol, interactions with,\n\nserum albumin, interactions with, ,\n\nstability\n\nbound calcium, , , ,\n\nchemical denaturants,\n\ndenaturation kinetics, 276\u2013279\n\nions, , ,\n\nnon-protein soluble components,\n\npH, , ,\n\npressure, , 250\u2013253, ,\n\nprotein concentration,\n\nsugars, 438\u2013440\n\ntemperature, , , , 276\u2013279,\n\nstructure\n\nbovine, 220\u2013222\n\ngeneral, 47\u201348, ,\n\nnon-bovine,\n\nrecombinant,\n\nsurfactants, interactions with,\n\nsynthesis,\n\nvariants,\n\nLactase, see beta-galactosidase\n\nLactation\n\nbovine, ,\n\nCape fur seal, , , 96\u201397\n\neffect of folate supplementation,\n\ngene expression, , 102\u2013103,\n\ninterspecies comparison, 24\u201325, , , 90\u201397\n\noverview, ,\n\nTammar wallaby, , , , ,\n\nLactoferricin, ,\n\nLactoferrin\n\napplications,\n\nbioactivity, , 126\u2013127,\n\ncalcium-binding,\n\nconcentration in milk, ,\n\nemulsification properties, 374\u2013375\n\nhydrolysis, ,\n\ninteractions with other milk proteins\n\n\u03b1-lactalbumin,\n\n\u03b2-lactoglobulm, , 374\u2013375\n\nserum albumin,\n\ninterspecies comparison,\n\niron-binding, , , ,\n\nisolation,\n\nnon-bovine,\n\npH, ,\n\npI,\n\nstability, , ,\n\nstructure, ,\n\n\u03b2-Lactoglobulin\n\nallergenic effects, ,\n\nanalogy to WAP,\n\nbioactivity,\n\nbiological purpose, , ,\n\n\u03b1-casein, interactions with,\n\n\u03ba-casein, interactions with, , , , 279\u2013293, ,\n\nconcentration in milk,\n\ncysteine,\n\ndynamics,\n\ndisulfide bonding, , , , , 216\u2013217, , , , , , 290\u2013293,\n\nemulsions, , , , , ,\n\nfatty acids, interactions with, , , , , , 428\u2013429\n\nbinding site,\n\nflavors, interactions with, ,\n\ngels, ,\n\ngenes, ,\n\ninterspecies variation, , , , 121\u2013122, , ,\n\nisoelectric point,\n\nisoelectric pH,\n\nisolation, ,\n\n\u03b1-lactalbumin, interactions with, , , , , , , , ,\n\nlactoferrin, interactions with, , 374\u2013375\n\nlactose, interactions with,\n\nligand binding, , , , , , 210\u2013213, , ,\n\nbinding sites,\n\ncompetitive binding between retinol and fatty acids,\n\nlipase stimulation,\n\noverview, 46\u201347\n\nphospholipids, interactions with,\n\npolymorphism, , see also Variants\n\npolysaccharides, interactions with,\n\nrennet gels, ,\n\nretinol-binding protein, similarities to, ,\n\nSDS, interactions with, , 430\u2013431\n\nserum albumin interactions with, , , , ,\n\nstability\n\nchemical denaturants, 218\u2013219\n\ndenaturation process, 213\u2013216,\n\ndenaturation kinetics, 276\u2013279\n\nions, ,\n\nligand binding,\n\nnon-protein soluble component, concentration,\n\npH, , 213\u2013216, 218\u2013219,\n\npressure, 216\u2013217, , , , , ,\n\npressure and pH,\n\npressure and temperature, ,\n\nprotein concentration, 213\u2013216,\n\nproteolysis, , ,\n\nsugars, protective effect of, 438\u2013440\n\ntemperature, , 213\u2013216, , 276\u2013279, , 438\u2013440, ,\n\nstructure\n\nbovine, 203\u2013205\n\nchemical denaturants, effect on, 218\u2013219\n\ngeneral, , ,\n\nnon-bovine, , , ,\n\npH dependence, , , 203\u2013205, ,\n\nsugars, protective effect of\n\npressure,\n\nthermal, 438\u2013440\n\nsurfactants, interactions with, 430\u2013431\n\ntransglutaminase (TGas), interactions with,\n\nvariants, , , , ,\n\ndenaturation process,\n\ndifferences between, ,\n\ndistinguishing between variants,\n\ndynamics,\n\nNMR-friendlyAla34Cys mutant, ,\n\nother mutants,\n\npressure, 216\u2013217,\n\nSDS binding,\n\nstability, ,\n\nstructure,\n\nvitamins, interactions with\n\nbinding sites,\n\nvitamin A, , , , , 422\u2013424,\n\nvitamin C,\n\nLactollin, see beta-Microglobulin\n\nLactoperoxide,\n\nLactose, 26\u201328\n\napplications,\n\nbiological purpose,\n\nconcentration, , 28\u201329,\n\ncrystallization, , ,\n\ngenetic engineering to modify concentration, 28\u201329\n\ninterspecies comparison, , ,\n\nintolerance, ,\n\nKoesler Number,\n\n\u03b1-lactalbumin involvement, ,\n\n\u03b2-lactoglobulin, interactions with,\n\nmanufacture, ,\n\nosmotic pressure,\n\npreferential hydration theory,\n\nproperties, 27\u201328\n\nprotection against protein degradation, 439\u2013440\n\nproteolysis,\n\nstructure,\n\nsynthesis, ,\n\nwhey protein denaturation, stabilizing effect on, , see also Lactulosyl lysine; Maillard browning\n\nLactose synthetase,\n\nLactulosyl lysine\n\nbioavailability, 352\u2013353\n\ndetection,\n\nformation kinetics, 345\u2013349\n\nstorage trial, 348\u2013349\n\n\u03b2-lactoglobulin, , , , , 210\u2013213, , ,\n\ncasein,\n\nLeucine\n\nanabolic stimuli, muscle,\n\nimportance of,\n\nas signaling molecule,\n\nLight scattering techniques, , , , , ,\n\nLipids\n\nCLA, ,\n\nclasses,\n\nconcentration,\n\ndegradation, , 35\u201337\n\nform in milk, ,\n\ninterspecies comparison, , 32\u201334,\n\nphospholipid binding,\n\nprotein binding,\n\nPUFA,\n\nsynthesis, , , see also Fatty acids\n\nLipocalin,\n\nLipoprotein lipase (LPL), , , ,\n\nLocust bean gum (LBG)\n\nLysine, , 352\u2013353, , , , , ,\n\nbioavailablity of,\n\ndairy protein,\n\ndeficient countries,\n\nMaillard reaction,\n\nreaction conditions,\n\nmeat, best source of,\n\nLysoalanine, , ,\n\nLysozyme, ,\n\n\u03b1-lactalbumin, similarity to,\n\nM\n\nMaillard reactions, , , 345\u2013349, , , , , ,\n\nMacropeptide, , ,\n\nbioactivity, , ,\n\nformation, , , ,\n\ntypes of macropeptide,\n\nzeta potential, effect on,\n\nMammals\n\nclassification of, 22\u201323\n\nevolution of, 21\u201322, see also Mammary gland\n\nMammary gland\n\nevolution,\n\ninterspecies comparison, 24\u201325\n\nstructure,\n\nhormones,\n\nManipulation of bovine proteins genes, 130\u2013132\n\nMass spectrometry (MS), , , , 291\u2013292\n\nMastitis,\n\nMetabolic health, 542\u2013543\n\nbranched-chain amino acid (BCAA) leucine,\n\ncardiovascular disease (CVD),\n\ndairy products consumption,\n\nepidemiological studies,\n\nhyperglycemia,\n\nimpaired glucose tolerance (IGT),\n\nmilk proteins, long-term clinical studies,\n\ninsulin,\n\nmetabolically active lean body mass,\n\nimportance of,\n\nmetabolic syndrome\n\ndefinition,\n\nphysiological changes,\n\nproinflammatory mediators,\n\nT2DM, see Type 2 diabetes\n\nMeat products, inclusion of milk proteins, 466\u2013467\n\nMetal-binding proteins, see Mineral-binding proteins\n\nMethane production, 574\u2013575\n\nMicrobiological techniques,\n\n\u03b2-Microglobulin,\n\nMicronutrients,\n\nMicroRNAs (miRNAs)\n\nbovine milk,\n\nexogenous,\n\nfunction of,\n\nmammary gland development,\n\nmilk bioactives,\n\nmilk exosomes,\n\nphysical properties,\n\nsignificance,\n\nstage-specific expression,\n\nMilk\n\nbiological purpose, , , , , ,\n\ncommercial products,\n\ncomposition\n\nchanges during lactation, ,\n\ngeneral, , , , , , ,\n\nmanipulation of composition,\n\nratio of components, , see also Lactose; Lipids; Minerals and specific protein names\n\ncosts of production\n\nenergy, 572\u2013573\n\nmethane, 574\u2013575\n\nwater, 573\u2013574\n\nglobal production, ,\n\nheat treatment,\n\neffect on caseins,\n\nwhey protein, modified,\n\nhomogenization,\n\nmilk fat globule membranes, disruption of, , see also Milk fat globule membranes\n\npepsinolysis,\n\nphysicochemical modifications\n\nallergenicity, to reduce,\n\npre-digestion of proteins,\n\nprotein-based food products, stability of,\n\ntransglutaminase (TG), cross-linking enzyme,\n\nsecretion, see Lactation\n\nsynthesis, 24\u201325, see also Bovine; Lactation; Mammary gland; Non-bovine species\n\nMilk fat globule membrane (MFGM)\n\ndamage to,\n\nformation,\n\nglycolipids,\n\ninterspecies comparison, 35\u201337\n\nproperties,\n\nprotein components, , , ,\n\nstructure, ,\n\nMilk protein concentrates\n\ncommercial products,\n\nemulsions, , , ,\n\nfunctionality,\n\ninteractions with xanthan gum,\n\nmanufacture,\n\nnon-dairy food applications\n\nbakery products, 463\u2013464\n\ncoffee whitener,\n\nproduction, global,\n\nsensory characteristics, 483\u2013484\n\nstorage changes\n\navailable amine,\n\nlactulosyl lysine, 345\u2013349, 352\u2013353\n\nMaillard browning, 345\u2013349\n\nMinerals\n\nconcentration in milk, , 28\u201329,\n\ninteractions with proteins,\n\nmodification of product functionality, , ,\n\noverview,\n\ntypes of minerals in milk, , see also Casein micelles; Colloidal calcium phosphate; Mastitis\n\nMineral-binding proteins, , , , see also Casein; Casein micelle; Ceruloplasmin; Glutathione peroxidase; Lactoferrin; Transferrin\n\nMixed gels,\n\nModel food systems\n\nadvantages,\n\napplications,\n\ndevelopment\n\ninitial stages,\n\nstatistical design, 461\u2013462\n\nexample models\n\nbakery products, 463\u2013464\n\ncoffee whitener,\n\nmeat products, 466\u2013467\n\nwhipped toppings,\n\nlimitations,\n\nMolecular interactions\n\ndisulfide bonds\n\ncaseins, , , , , , ,\n\n\u03ba-casein, , , , 152\u2013153, 182\u2013183, , 280\u2013281, 290\u2013293, 305\u2013306\n\nimmunoglobulins,\n\n\u03b1-lactalbumin, , , , ,\n\n\u03b2-lactoglobulin, , , , , 216\u2013217, , , , , , , 290\u2013293, 305\u2013306, , 510\u2013512,\n\n\u03b2-lactoglobulin and \u03ba-casein, , , , , , 280\u2013281, 290\u2013293, 305\u2013306\n\nmixed milk proteins, , ,\n\nserum albumin, , , , , ,\n\nWAP,\n\nelectrostatic repulsion\n\ncaseins, , , , , , , ,\n\nemulsions, , ,\n\n\u03b1-lactalbumin,\n\n\u03b2-lactoglobulin, , ,\n\nmicelles, , , , , , , , , 191\u2013192, ,\n\nprotein-polysaccharide mixtures, , , ,\n\nwhey proteins and other species, , ,\n\ngeneral covalent bonds, , , , , , , ,\n\nhydrogen bonds\n\ngels, ,\n\n\u03b2-lactoglobulin, , ,\n\nmicelle structure, , ,\n\nprotein-polysaccharide mixtures,\n\nsugars,\n\nhydrophobic bonds\n\n\u03b1-casein, , , 173\u2013174, 178\u2013179\n\n\u03b2-casein, , , , 173\u2013174, , ,\n\ncaseins, , , , , 174\u2013175, , 182\u2013184, , , , , , , , ,\n\n\u03b2-lactoglobulin, , , , , , , , , 424\u2013426, , , ,\n\nmicelle structure, , , 182\u2013187, 189\u2013192, ,\n\nprotein-polysaccharide mixtures,\n\nwhey proteins, , , , , , , ,\n\nsteric effects\n\n\u03ba-casein, , , , , , ,\n\nemulsions, , , ,\n\nprotein-polysaccharide mixtures, , ,\n\nsugars,\n\nwhey proteins,\n\nvan der Waals' bonds, , , ,\n\nMucins, 35\u201337\n\nN\n\nNanotechnology,\n\nNational dairy development board,\n\nNDDB, see National dairy development board\n\nNative-PAGE, see Electrophoresis Non-bovine species\n\n\u03b1-caseins, ,\n\n\u03b2-caseins, ,\n\n\u03ba-caseins, , ,\n\nclassification of, 21\u201322\n\ncryoglobulin,\n\nenzymes,\n\ngenome,\n\nimmunoglobulins, ,\n\n\u03b1-lactalbumin, 47\u201348,\n\n\u03b2-lactoglobulin, , 121\u2013122, , ,\n\nlactose, , ,\n\nlipid classes, , 32\u201334, ,\n\nmilk fat globule membrane, 35\u201337\n\noligosaccharide levels, ,\n\nprotein comparison, 56\u201358\n\nwhey acidic protein (WAP), , , 90\u201393, , see also Cape fur seal; Mammary gland; Tammar wallaby\n\nNon-protein nitrogen,\n\nNuclear magnetic resonance (NMR)\n\ngels,\n\n\u03b2-lactoglobulin, 205\u2013208, , , 216\u2013218, , ,\n\npowders, ,\n\nserum albumin,\n\nNutraceuticals, see Functional foods\n\nNutrigenomics, 576\u2013578\n\nNutrigenetics, 576\u2013578\n\nNutritional characteristics\n\namino acids, 528\u2013534\n\ncalorie provision, 533\u2013534\n\nmanufacture of nutritional products, , see also Functional foods Nutrigenomics\n\nO\n\nObesity and weight control, 543\u2013548\n\nadipose loss,\n\nbranched-chain amino acids (BCAA),\n\nmuscle protein synthesis,\n\nenergy-restricted diets,\n\nwhey protein,\n\nheart health, see Atherosclerosis; See also Blood pressure\n\nhigh-protein diets,\n\nsuppressing hunger,\n\nhigh-protein meal replacement,\n\nmilk protein role,\n\nlong-term studies,\n\nlower body weight,\n\nmuscle wasting, see Sarcopenia\n\nsarcobesity,\n\nskimmed milk,\n\nisoenergetic bolus, ingestion of,\n\nwhey protein vs. casein,\n\nsupplements\n\ncasein,\n\nwhey protein,\n\nOligosaccharides\n\nbiological purpose, , ,\n\nconcentrations,\n\ninterspecies comparison, ,\n\nosmotic pressure,\n\nstructure,\n\nsynthesis,\n\nOsmotic pressure, ,\n\nOsteopontin (OPN),\n\nP\n\nparacasein, ,\n\nPectin, see Polysaccharide-protein systems\n\nPeptides, , , 128\u2013129, , see also Functional components\n\nPeptides released, digestion\n\nbioactive peptides,\n\nbiological activities,\n\n\u03ba-casein,\n\ndetection of,\n\ncaseinophosphopeptides,\n\nidentification of,\n\nin vitro digestion of food,\n\nin vivo peptide formation,\n\nin duodenum,\n\nin jejenum,\n\nmeal structure, effect of,\n\nmilk proteins hydrolysis,\n\nPepsin, , , , see also Chymosin; Rennet\n\nPersonalised nutrition, 576\u2013578\n\nPhospholipid binding,\n\nPhosphorylation, , ,\n\nPlasmin, , ,\n\nPolymorphism, , , , , , ,\n\nPolysaccharide-protein systems\n\ncasein micelles\n\n\u03b9-carrageenan,\n\ngalactomannans,\n\nlocust bean gum,\n\npectin,\n\ncaseinate\n\ngum Arabic, ,\n\npectin, ,\n\nxanthan gum,\n\nemulsions,\n\ngel formation,\n\ngelling phase-separated systems, 404\u2013406,\n\ninteracting mixtures, 406\u2013409\n\n\u03b2-lactoglobulin\n\n\u03ba-carrageenan,\n\nchitosan,\n\npectin,\n\nmixing behaviors\n\nco-solubility, 388\u2013389,\n\ncomplex coacervation, , , ,\n\ndepletion flocculation, , , , ,\n\ndepletion interaction, , , , ,\n\nthermodynamic incompatibility, , , ,\n\nmolecular interactions\n\nattractive, 393\u2013394\n\ncovalent,\n\nrepulsive, 392\u2013393\n\nMPC and xanthan gum,\n\nnon-gelling phase-separated systems, , 407\u2013408\n\nnon-interacting mixtures, 402\u2013406\n\nphase diagrams, 390\u2013391\n\nrheological measurements, ,\n\ntransglutaminase cross-linking,\n\nwhey proteins, mixed\n\nEPS,\n\ngalactomannans,\n\nguar gum,\n\ngum Arabic,\n\nxanthan gum, ,\n\nPost-translational modifications (PTMs)\n\nanalytical techniques,\n\n\u03b1-casein, ,\n\n\u03b2-casein, ,\n\n\u03ba-casein, , 150\u2013151\n\ndisulfide bonding, 152\u2013153\n\nglycosylation, , 150\u2013151, 155\u2013156\n\nimportance of,\n\noccurrence, ,\n\nphosphorylation, , , ,\n\nPressure treatment\n\nacid gels,\n\nbond disruption,\n\ncommercially pressure-treated products,\n\ncomparison with thermal, , , ,\n\ndenaturation of proteins caseins,\n\ncasein-whey interactions, , ,\n\neffect of processing variables,\n\ngeneral, 247\u2013248\n\nimmunoglobulins, ,\n\n\u03b1-lactalbumin, 250\u2013251,\n\nlactoferrin, ,\n\n\u03b2-lactoglobulin, , 248\u2013250, , ,\n\nmicelles, ,\n\nmilk systems, 256\u2013261\n\nserum albumin, ,\n\nwhey proteins, mixtures of, ,\n\nwhey protein products, commercial, 252\u2013253\n\nemulsions,\n\ngelation of whey proteins\n\ncomparison with thermal gels,\n\neffect of processing variables, ,\n\nformation, ,\n\n\u03b2-lactoglobulin, ,\n\nproperties,\n\nsugars,\n\nstorage,\n\nWPI,\n\nimmunoglobulins, ,\n\n\u03b1-lactalbumin, , 250\u2013253, ,\n\nlactoferrin,\n\n\u03b2-lactoglobulin\n\nconcentration,\n\ndenaturation, , 248\u2013250, , , ,\n\nenzymatic cleavage,\n\ngels,\n\ninteractions with \u03b1-casein,\n\ninteractions with \u03ba-casein,\n\ninteractions with whey proteins,\n\npH,\n\npresence of ligands, ,\n\npressure level, , , 248\u2013250,\n\npre-denatured state,\n\nprocessing variables,\n\nstructural changes, 216\u2013217, 248\u2013250\n\ntemperature, ,\n\nvariants,\n\nLe Chatelier-Braun's principle,\n\nmilk systems\n\nwhey protein denaturation,\n\nwhey proteins and casein micelles, 258\u2013261,\n\npreservation,\n\nprocess development, ,\n\npurpose, , , ,\n\nserum albumin, , , ,\n\nstructural modification, , , , ,\n\nanalytical techniques, 244\u2013247\n\ntemperature of pressure treatment,\n\nwhey protein products, commercial, 252\u2013253,\n\nProtein bioavailability\n\nanimal-derived proteins,\n\ncommon food proteins\n\ntrue protein digestibility,\n\nfecal digestibility,\n\nplant-derived proteins,\n\nprotein breakdown, gastrointestinal tract,\n\nprotein supply, adequate,\n\nProtein composition\n\ndietary essential amino acids,\n\nlinear chains amino acids, composed of,\n\nProtein powders\n\nagglomeration, 476\u2013479\n\napplications, 295\u2013296,\n\ncommercial products, 321\u2013323,\n\ndenaturation during spray drying, , , , ,\n\nfunctionality, effect of denaturation, 295\u2013296\n\nglobal production,\n\nhistory of drying proteins,\n\nphysical properties,\n\nrehydration\n\ncaseins, 337\u2013338\n\nions, effect of,\n\ngranulation,\n\nstages of water transfer, ,\n\ntechniques for monitoring rehydration, 335\u2013337\n\nwhey powders,\n\nsensory characteristics\n\ncasein and hydrolysates,\n\nflavor binding, ,\n\nmixed proteins, 483\u2013484\n\nwhey proteins, 474\u2013482\n\nstorage changes\n\ncysteine, , ,\n\nflavor changes, , 476\u2013479\n\nisopeptide bonds, , , ,\n\nlanthionine, , ,\n\nlysine, 352\u2013353, ,\n\nMaillard compounds, 345\u2013349, ,\n\nmethionine, , , ,\n\nnutritional consequences, , 352\u2013355\n\ntryptophan, , , see also Drying by desorption Spray drying; WPNI\n\nProteins, see \u03b1-casein; See also \u03b2-casein; \u03ba-casein; Immunoglobulins; \u03b1-lactalbumin; Lacto-ferrin; \u03b2-lactoglobulin; Serum albumin\n\nProtein quality, need for\n\naging,\n\nassociated with,\n\nsarcopenia,\n\nchallenges,\n\nelderly population, , see also Elderly\n\ndemographic trends,\n\npopulation percentage in 2050,\n\nglobal demographic trends,\n\nProteolysis, , , , , , , , , , , see also chymosin; rennet\n\nProteomics, , ,\n\nProteose peptones, , , \nR\n\nRecombinant DNA technology, ,\n\nRegulatory issues\n\ncarbon footprint,\n\nfunctional foods, 526\u2013527,\n\nfood safety, 575\u2013576\n\nfood traceability, 575\u2013576\n\ngenetic modification, 578\u2013579\n\nmethane production, 574\u2013575\n\nRennet, , 39\u201340, , 46\u201347, , , , , , 189\u2013190, , see also chymosin; rennet gels\n\nRennet clotting time (RCT),\n\nRennet gels\n\naggregation of micelles, , 495\u2013497\n\ncalcium phosphate, ,\n\ncross-linking of caseins,\n\nformation\n\n\u03ba-casein hydrolysis, , 494\u2013497, ,\n\nions, effect of,\n\nmodeling aggregation,\n\npH, effect of, ,\n\nprimary phase, 494\u2013495\n\nrheological changes, 497\u2013498\n\nsecondary phase, 495\u2013497\n\ntemperature, effect of, , ,\n\nzeta potential, changes in,\n\nhistory,\n\nions, ,\n\nmonitoring gelation,\n\npH, 189\u2013190, , ,\n\nplant coagulants,\n\npre-heat treatment, 505\u2013508\n\npressure,\n\nprotein concentration,\n\nrennet concentration, ,\n\nrennets, types of,\n\nrheological changes during formation, 497\u2013498\n\nrheological properties, 189\u2013190, 497\u2013499\n\nrheological parameters,\n\nrheological techniques,\n\nsequestrants, 189\u2013190\n\nstorage,\n\nstorage modulus, , ,\n\nstructure, , 497\u2013499,\n\nsyneresis\n\ncasein concentration,\n\nendogenous pressure,\n\nfree energy,\n\nfunctional purpose,\n\nmechanism,\n\nmicrosyneresis,\n\nmodeling,\n\nmonitoring,\n\npH,\n\ntemperature,\n\ntemperature, 499\u2013501\n\ntexture, ,\n\ntransglutaminase (TGase),\n\nwhey protein denaturation, ,\n\nRetinol-binding protein, , ,\n\nRetinol-protein interactions, see Vitamin A-protein interactions\n\nRheology, see Rennet gels\n\nS\n\nS100A19 proteins\n\nidentification of,\n\npeptide sequence,\n\nS100A19 gene,\n\ndifferential\n\nexpression,\n\nregulation,\n\nexpression analysis\n\nwallaby lactation cycle,\n\nwallaby stomach development,\n\nfunctions,\n\nvariants\n\nS100A8,\n\nS100A9,\n\nS100A12,\n\nSalts, see Minerals\n\nSarcopenia\n\naging effects,\n\ncasein,\n\nbranched-chain amino acid (BCAA),\n\ndigestion rate,\n\nessential amino acids (EAAs),\n\ncauses of,\n\nend-stage diseases,\n\nmilk protein,\n\nanabolic effect,\n\nmuscle protein\n\nbreakdown, inhibition of,\n\nsynthesis,\n\nactivation of,\n\nessential amino acids (EAAs),\n\npreventive strategies,\n\ndiet,\n\nexercise,\n\nresistance-type exercise, regular,\n\nlong-term studies,\n\nmilk protein supplements,\n\nskeletal muscle\n\ninsulin resistance,\n\nloss of function,\n\nloss of mass,\n\nmetabolic health, impact on,\n\nwhey protein,\n\nbranched-chain amino acid (BCAA),\n\ndigestion rate,\n\nSatiety, 533\u2013534\n\nSelf-assembly of protein structures, ,\n\nSensory analysis,\n\nSensory characteristics\n\ncasein,\n\nflavor binding, ,\n\nhydrolysates, ,\n\nmixed protein powders, 483\u2013484\n\nrennet gels,\n\ntexture, , 192\u2013193, , , ,\n\nwhey proteins, 474\u2013482\n\nagglomeration, effect of, 476\u2013479\n\nconsumer issues, 480\u2013482\n\nenzymes, effect of, ,\n\nflavor control,\n\nnative whey proteins,\n\nprocess variations, effect of,\n\nstorage, effect of, 474\u2013479\n\nwhey type, effect of,\n\nwhole milk,\n\nSerum albumin\n\nbiological purpose, ,\n\n\u03ba-casein, interactions with,\n\nconcentration in milk,\n\nfatty acids, interactions with, ,\n\nflavors, interactions with,\n\n\u03b1-lactalbumin, interactions with,\n\nlactoferrin, interactions with,\n\n\u03b2-lactoglobulin, interactions with, , ,\n\nminerals, interactions with,\n\ninterspecies comparison, ,\n\nstability\n\nchemical denaturants,\n\nligand binding, effect of, ,\n\nions,\n\npressure, , , ,\n\npressure and temperature,\n\nsugars, ,\n\ntemperature, , , ,\n\nstructure\n\nbovine,\n\ngeneral,\n\nnon-bovine,\n\nvitamins, interactions with, ,\n\nSmall-angle X-ray scattering (SAXS), 176\u2013178, ,\n\nSodium dodecyl sulfate (SDS)\n\n\u03b2-lactoglobulin\n\nbinding, , , 430\u2013431\n\nstabilizing in presence of urea,\n\ncasein micelle dissociation, ,\n\nSodium dodecyl sulfate polyacrylamide gel electrophoresis (SDS-PAGE), see Electrophoresis\n\nSoft materials science,\n\nSpray drying\n\nagglomeration,\n\ncaseins, ,\n\nCompact Drier Instantization (CDI),\n\ndenaturation, , , , ,\n\nindustrial implications,\n\nions,\n\noptimization,\n\nprinciples\n\ncomponents, 325\u2013326\n\nkinetics,\n\nsingle-stage system,\n\nthree-stage system,\n\ntwo-stage system,\n\nsugars, protective effects,\n\nwater transfer\n\ncaseins,\n\nwhey proteins, ,\n\nStorage\n\nbar hardening,\n\nchanges during storage\n\ncysteine, , ,\n\nisopeptide bonds, , , ,\n\nlanthionine, , ,\n\nlysine, 352\u2013353, ,\n\nMaillard compounds, 345\u2013349, ,\n\nmethionine, , , ,\n\ntryptophan, ,\n\nflavor changes, , 476\u2013479\n\nnutritional consequences, , 352\u2013355\n\npowder stability,\n\nStructure, techniques for assessing\n\ncircular dichroism (CD), , , , 216\u2013218, ,\n\nconfocal laser scanning microscope (CLSM), ,\n\ndifferential scanning calorimetry (DSC), , , , , ,\n\ndiffusing wave spectroscopy, ,\n\nelectron microscopy, , 55\u201356, , 176\u2013178, , , , , , , ,\n\nelectrospray ionization mass spectrometry,\n\nFourier transform infrared (FTIR)\n\nspectroscopy, , , 274\u2013275\n\nFourier transform of Raman spectra,\n\nfluorescence spectroscopy, , , , , ,\n\nlight scattering techniques, , , , , ,\n\nmass spectrometry (MS), , , , 291\u2013292\n\nnuclear magnetic resonance (NMR) gels,\n\n\u03b2-lactoglobulin, 205\u2013208, , , 216\u2013218, , ,\n\npowders, 336\u2013338\n\nserum albumin,\n\nsmall-angle X-ray scattering (SAXS), 176\u2013178, ,\n\nneutron small angle scattering,\n\nultrasonic spectroscopy,\n\nUV differential absorption, ,\n\nX-ray crystallography\n\nCaseins,\n\nIgG,\n\n\u03b1-lactalbumin,\n\n\u03b2-lactoglobulin, , , , , ,\n\nserum albumin,\n\nStructure of proteins, see \u03b1-casein; See also \u03b2-casein; \u03ba-casein; immunoglobulins; \u03b1-lactalbumin; lactoferrin; \u03b2-lactoglobulin\n\nStructure-function relationships\n\nhistory, 451\u2013452\n\nfunctionality, modification of\n\ningredients, , 453\u2013458\n\nprocessing, 359\u2013361, 459\u2013460\n\nsimple model systems vs complex real\n\nsystems, 451\u2013452\n\nprediction of function, 451\u2013453, ,\n\ntypes of functionality, ,\n\nSugar-protein interactions\n\ncasein,\n\ncomparison of different sugars, , , 439\u2013440,\n\ndenaturation, protective effect chemical denaturants,\n\nfoaming,\n\nfreeze drying, 440\u2013441\n\noverview, 431\u2013433\n\npressure, ,\n\npromotion,\n\nthermal treatments, 436\u2013440\n\ndirect bonding, ,\n\n\u03b1-lactalbumin, 438\u2013440\n\n\u03b2-lactoglobulin, 438\u2013441\n\nMaillard reactions,\n\nserum albumin,\n\nsteric exclusion effect, 431\u2013433\n\nthermodynamics,\n\nSurfactant-protein interactions, 430\u2013431,\n\nSyneresis, see Acid gel; See also Rennet gel\n\nT\n\nTammar wallaby, ,\n\nasynchronous lactation,\n\ncDNA library,\n\nchanges in milk composition, , 84\u201385\n\nchanges in gene expression,\n\ncomparative milk genomics, 119\u2013120\n\neffect of milk composition on growth of young,\n\nexpressed sequence tag (EST),\n\nFIL protein,\n\ngenomics,\n\ngrowth regulation, 23\u201324,\n\nlactation\n\npattern, ,\n\nstrategy,\n\noligosaccharide levels,\n\nwhey acidic protein (WAP), , , 90\u201393, see also Mammary gland\n\nThermal treatments\n\n\u03b1-casein and whey protein complexes,\n\n\u03ba-casein glycosylation,\n\nchanges in milk due to heat\n\ncaseins, , , , , ,\n\nmicelles, , , , , , 282\u2013285\n\nminor proteins, , ,\n\nmixed proteins, , , , 279\u2013289,\n\nnon-protein components, , ,\n\nwhey proteins, , , , 276\u2013277, 291\u2013292\n\nwhole milk, ,\n\nurea,\n\ncomparison with pressure treatments, , , ,\n\ndenaturation, effect on milk product functionality\n\nacid gel strength, heating whey proteins and caseins together, 303\u2013306\n\nacid gel strength, heating whey proteins and caseins separately, 310\u2013311\n\nhigh pH, heating at,\n\nlow pH, heating at,\n\nmicelle-bound \u03ba-casein, 304\u2013306\n\nnon-sedimentable \u03ba-casein, 304\u2013306\n\nyoghurt properties,\n\ndenaturation, whey proteins\n\nchanges in functional properties of milk products, 295\u2013311\n\ninteractions with \u03ba-casein in milk systems, 280\u2013289\n\ninteractions with \u03ba-casein in model systems, 279\u2013280\n\nkinetics of denaturation, 276\u2013279\n\nmeasurement of denaturation, 274\u2013275,\n\nwhey protein nitrogen index (WPNI), 274\u2013275, ,\n\ndirect heating,\n\nindirect heating,\n\nion binding, protective effect,\n\nkinetics of denaturation, 276\u2013277\n\n\u03b1-lactalbumin and \u03ba-casein complexes, ,\n\n\u03b2-lactoglobulin\n\ninteractions with \u03ba-casein in model systems, 279\u2013280\n\ninteractions with \u03ba-casein in milk systems, 280\u2013289, , 291\u2013292\n\ndenaturation, 213\u2013216, , , ,\n\nthermal stability,\n\nlactoferrin,\n\nmicelles, interactions with,\n\npH, effect on\n\ndissociation of \u03b1-casein,\n\ndissociation of \u03b2-casein,\n\ndissociation of \u03ba-casein, 282\u2013285,\n\nhigh temperature stability,\n\n\u03b2-lactoglobulin,\n\nmoderate temperature stability,\n\nnon-sedimentable protein, , ,\n\nsensory changes, , ,\n\nsugars, 436\u2013440\n\ntemperatures,\n\ntypes of thermal treatments,\n\npurpose, ,\n\nThermodynamic incompatibility, see Polysaccharide-protein systems, mixing behaviors\n\nTransferrin,\n\nTransgenic animals, ,\n\nTransgenic plants,\n\nTransglutaminase, (TGase), , , ,\n\nType 2 diabetes (T2DM)\n\ncasein supplement,\n\ninsulin, high levels of,\n\nmetabolically impaired,\n\noverweight,\n\npostprandial study,\n\nU\n\nUltrasonic spectroscopy,\n\nV\n\nVariants\n\n\u03b1-casein, , , , ,\n\n\u03b2-casein, , , ,\n\n\u03ba-casein, , , , ,\n\nexisting variation in genes,\n\n\u03b1-lactalbumin,\n\n\u03b2-lactoglobulin, , , , , , , see also Genetic polymorphism\n\nVegetable protein sources\n\nproduction efficiencies,\n\nvegetarianism,\n\nmerits of,\n\nVitamin-binding proteins, , 424\u2013426\n\nVitamin A-protein interactions\n\ncaseins,\n\ncompetitive binding, ,\n\ndifferent forms of vitamin A,\n\n\u03b1-lactalbumin,\n\n\u03b2-lactoglobulin, , 422\u2013424\n\nretinol-binding protein, ,\n\nserum albumin,\n\nstabilization of retinol,\n\nVitamin B2-protein interactions,\n\nVitamin C-protein interactions\n\n\u03b2-lactoglobulin,\n\nserum albumin,\n\nVitamin D-protein interactions\n\n\u03b2-casein,\n\n\u03b2-lactoglobulin, 424\u2013426\n\nVitamins, other,\n\nVitamins present in milk, ,\n\nW\n\nWAP four-disulfide core domain protein 2\n\nantibacterial acitivity, 89\u201391\n\ndomain structure,\n\nduring lactation,\n\nstructure,\n\nin tammar wallaby,\n\nWFDC2 gene,\n\nepididymis, identified in,\n\nexpression pattern,\n\nexpression profile, ,\n\nmammary gland, tammar wallaby,\n\nWater\n\ncomponent of milk,\n\nsyneresis, loss through,\n\nwater activity in powders, , , , ,\n\nwater economy, 573\u2013574\n\nwater transfer in powders, , 330\u2013332, , , 337\u2013338\n\nWhey four-disulfide core\n\nWAP, see Whey acidic protein\n\nWFDC2, see WAP four-disulfide core domain protein 2\n\nWhey proteins\n\nbioactivity, ,\n\n\u03ba-casein, interactions with, ,\n\ncommercial products\n\nagglomerated products,\n\napplications,\n\nmanufacture, , , ,\n\nshelf-life,\n\ntypes of whey products,\n\ncomparison with caseins,\n\nconsumer acceptance,\n\nconcentration in milk,\n\ndenaturation\n\nassessment, 274\u2013275\n\ndefinition,\n\nfunctionality, effect on, 295\u2013311\n\ninteractions with \u03ba-casein in milk systems, 280\u2013289, 505\u2013507\n\ninteractions with \u03ba-casein in model systems, 276\u2013280\n\nsugars, protective effect of, 438\u2013440\n\ntemperature, , , , 510\u2013512\n\nemulsions\n\ncasein addition,\n\nformation, , , ,\n\nhydrolysates, 373\u2013374\n\nion addition, ,\n\n\u03b1-lactalbumin, , ,\n\n\u03b2-lactoglobulin, , , , , , ,\n\nmixed whey proteins, , ,\n\ngenetic modification,\n\nhistory of,\n\nhydrolysates,\n\nhydrolysis, intestinal lumen\n\n\u03b1-lactalbumin,\n\nvs. \u03b2-lactalbumin,\n\nlactoferrin,\n\nmultiple sequence analysis,\n\npepsinolysis, sensitivity to,\n\nproteolysis, resistant to,\n\nfractions, , ,\n\nfunctionality,\n\ngels\n\ncold gelation, 514\u2013515\n\nenzymatic modification,\n\nformation variables,\n\nionic strength, 513\u2013514\n\nnon-protein compounds, effect of,\n\npH, , 512\u2013513\n\npolysaccharide gels,\n\nproperties of gels, 512\u2013513\n\nprotein concentration,\n\nstructure, 512\u2013513\n\ntemperature,\n\nthermal denaturation, 510\u2013512\n\ntypes of whey protein gels, 512\u2013513,\n\nisoelectric points,\n\nintact,\n\ninterspecies comparison, 56\u201357\n\n\u03b2-lactoglobulin and \u03ba-casein interactions, , , 279\u2013293, ,\n\nmineral binding,\n\nmuscle protein anabolism, elderly\n\nbeneficial effects,\n\nnative whey proteins, , ,\n\nnon-dairy food applications\n\nbakery products, 463\u2013464\n\ncoffee whitener,\n\nmeat products, 466\u2013467\n\nwhipped toppings,\n\npolysaccharides, interactions with\n\nEPS,\n\ngalactomannans,\n\nguar gum,\n\ngum Arabic,\n\nxanthan gum, ,\n\npreparation, ,\n\npressure treatments of commercial products, 252\u2013253\n\npressure-induced gels, 253\u2013256\n\nproperties,\n\nproteolysis,\n\nrehydration,\n\nself-assembly,\n\nsensory characteristics\n\nagglomeration, effect of, 476\u2013479\n\nconsumer issues,\n\nenzymes, effect of,\n\nflavor control,\n\nnative whey proteins,\n\nprocess variations, effect of,\n\nstorage, effect of, 474\u2013480\n\nwhey type, effect of,\n\nstorage changes\n\n\u03b2-lactoglobulin-lactose interactions,\n\nlactulosyl lysine, 346\u2013349, 352\u2013353\n\nMaillard reactions, 346\u2013349, ,\n\nstructure,\n\nsugars, interactions with, , 438\u2013440,\n\ntransglutaminase (TGase),\n\nyoghurt, roles in, , see also \u03b1-lactalbumin; \u03b2-lactoglobulin; immunoglobulins\n\nWhey acidic protein (WAP), , , , 90\u201393,\n\nWhey protein nitrogen index (WPNI), 274\u2013275, ,\n\nX\n\nXanthan gum\n\nXanthine oxidoreductase (XOR),\n\nX-ray crystallography\n\nCaseins,\n\nIgG,\n\n\u03b1-lactalbumin,\n\n\u03b2-lactoglobulin, , , 212\u2013213, , ,\n\nserum albumin,\n\nY\n\nYoghurt\n\nwhey protein denaturation, effect of,\n\nwhey proteins, role of, 505\u2013507,\n\nZ\n\nZeta potential, , , \n\n# Food Science and Technology: International Series\n\nAmerine, M.A., Pangborn, R.M., and Roessler, E.B., 1965. Principles of Sensory Evaluation of Food.\n\nGlicksman, M., 1970. Gum Technology in the Food Industry.\n\nJoslyn, M.A., 1970. Methods in Food Analysis, Second Ed.\n\nStumbo, C. R., 1973. Thermobacteriology in Food Processing, Second Ed.\n\nAltschul, A.M. (Ed.), New Protein Foods: Volume 1, Technology, Part A\u20141974. Volume 2, Technology, Part B\u20141976. Volume 3, Animal Protein Supplies, Part A\u20141978. Volume 4, Animal Protein Supplies, Part B\u20141981. Volume 5, Seed Storage Proteins\u20141985.\n\nGoldblith, S.A., Rey, L., and Rothmayr, W.W., 1975. Freeze Drying and Advanced Food Technology.\n\nBender, A.E., 1975. Food Processing and Nutrition.\n\nTroller, J.A., and Christian, J.H.B., 1978. Water Activity and Food.\n\nOsborne, D.R., and Voogt, P., 1978. The Analysis of Nutrients in Foods.\n\nLoncin, M., and Merson, R.L., 1979. Food Engineering: Principles and Selected Applications.\n\nVaughan, J. G. (Ed.), 1979. Food Microscopy.\n\nPollock, J. R. A. (Ed.), Brewing Science, Volume 1\u20141979. Volume 2\u20141980. Volume 3\u20141987.\n\nBauernfeind, J. C. (Ed.), 1981. Carotenoids as Colorants and Vitamin A Precursors: Technological and Nutritional Applications.\n\nMarkakis, P. (Ed.), 1982. Anthocyanins as Food Colors.\n\nStewart, G.G., and Amerine, M.A. (Eds.), 1982. Introduction to Food Science and Technology, Second Ed.\n\nIglesias, H.A., and Chirife, J., 1982. Handbook of Food Isotherms: Water Sorption Parameters for Food and Food Components.\n\nDennis, C. (Ed.), 1983. Post-Harvest Pathology of Fruits and Vegetables.\n\nBarnes, P.J. (Ed.), 1983. Lipids in Cereal Technology.\n\nPimentel, D., and Hall, C.W. (Eds.), 1984. Food and Energy Resources.\n\nRegenstein, J.M., and Regenstein, C.E., 1984. Food Protein Chemistry: An Introduction for Food Scientists.\n\nGacula Jr. M.C., and Singh, J., 1984. Statistical Methods in Food and Consumer Research.\n\nClydesdale, F.M., and Wiemer, K.L. (Eds.), 1985. Iron Fortification of Foods.\n\nDecareau, R.V., 1985. Microwaves in the Food Processing Industry.\n\nHerschdoerfer, S.M. (Ed.), Quality Control in the Food Industry, second edition. Volume 1\u20141985. Volume 2\u20141985. Volume 3\u20141986. Volume 4\u20141987.\n\nUrbain, W.M., 1986. Food Irradiation.\n\nBechtel, P.J., 1986. Muscle as Food.\n\nChan, H.W.-S., 1986. Autoxidation of Unsaturated Lipids.\n\nCunningham, F.E., and Cox, N.A. (Eds.), 1987. Microbiology of Poultry Meat Products.\n\nMcCorkle Jr. C.O., 1987. Economics of Food Processing in the United States.\n\nJaptiani, J., Chan Jr., H.T., and Sakai, W.S., 1987. Tropical Fruit Processing.\n\nSolms, J., Booth, D.A., Dangborn, R.M., and Raunhardt, O., 1987. Food Acceptance and Nutrition.\n\nMacrae, R., 1988. HPLC in Food Analysis, Second Ed.\n\nPearson, A.M., and Young, R.B., 1989. Muscle and Meat Biochemistry.\n\nPenfield, M.P., and Campbell, A.M., 1990. Experimental Food Science, Third Ed.\n\nBlankenship, L.C., 1991. Colonization Control of Human Bacterial Enteropathogens in Poultry.\n\nPomeranz, Y., 1991. Functional Properties of Food Components, Second Ed.\n\nWalter, R.H., 1991. The Chemistry and Technology of Pectin.\n\nStone, H., and Sidel, J.L., 1993. Sensory Evaluation Practices, Second Ed.\n\nShewfelt, R.L., and Prussia, S.E., 1993. Postharvest Handling: A Systems Approach.\n\nNagodawithana, T., and Reed, G., 1993. Enzymes in Food Processing, Third Ed.\n\nHoover, D.G., and Steenson, L.R., 1993. Bacteriocins.\n\nShibamoto, T., and Bjeldanes, L., 1993. Introduction to Food Toxicology.\n\nTroller, J.A., 1993. Sanitation in Food Processing, Second Ed.\n\nHafs, D., and Zimbelman, R.G., 1994. Low-fat Meats.\n\nPhillips, L.G., Whitehead, D.M., and Kinsella, J., 1994. Structure-Function Properties of Food Proteins.\n\nJensen, R.G., 1995. Handbook of Milk Composition.\n\nRoos, Y.H., 1995. Phase Transitions in Foods.\n\nWalter, R.H., 1997. Polysaccharide Dispersions.\n\nBarbosa-Canovas, G.V., Marcela Go'ngora-Nieto, M., Pothakamury, U.R., and Swanson, B.G., 1999. Preservation of Foods with Pulsed Electric Fields.\n\nJackson, R.S., 2002. Wine Tasting: A Professional Handbook.\n\nBourne, M.C., 2002. Food Texture and Viscosity: Concept and Measurement, Second Ed.\n\nCaballero, B., and Popkin, B.M. (Eds.), 2002. The Nutrition Transition: Diet and Disease in the Developing World.\n\nCliver, D.O., and Riemann, H.P. (Eds.), 2002. Foodborne Diseases, Second Ed.\n\nKohlmeier, M., 2003. Nutrient Metabolism.\n\nStone, H., and Sidel, J.L., 2004. Sensory Evaluation Practices, Third Ed.\n\nHan, J.H., 2005. Innovations in Food Packaging.\n\nSun, D.-W. (Ed.), 2005. Emerging Technologies for Food Processing.\n\nRiemann, H.P., and Cliver, D.O. (Eds.), 2006. Foodborne Infections and Intoxications, Third Ed.\n\nArvanitoyannis, I.S., 2008. Waste Management for the Food Industries.\n\nJackson, R.S., 2008. Wine Science: Principles and Applications, Third Ed.\n\nSun, D.-W. (Ed.), 2008. Computer Vision Technology for Food Quality Evaluation.\n\nDavid, K., and Thompson, P., (Eds.), 2008. What Can Nanotechnology Learn from Biotechnology?\n\nArendt, E.K., and Bello, F.D. (Eds.), 2008. Gluten-free Cereal Products and Beverages.\n\nBagchi, D. (Ed.), 2008. Nutraceutical and Functional Food Regulations in the United States and Around the World.\n\nSingh, R.P., and Heldman, D.R., 2008. Introduction to Food Engineering, Fourth Ed.\n\nBerk, Z., 2009. Food Process Engineering and Technology.\n\nThompson, A., Boland, M., and Singh, H. (Eds.), 2009. Milk Proteins: From Expression to Food.\n\nFlorkowski, W.J., Prussia, S.E., Shewfelt, R.L. and Brueckner, B. (Eds.), 2009. Postharvest Handling, Second Ed.\n\nGacula Jr., M., Singh, J., Bi, J., and Altan, S., 2009. Statistical Methods in Food and Consumer Research, Second Ed.\n\nShibamoto, T., and Bjeldanes, L., 2009. Introduction to Food Toxicology, Second Ed.\n\nBeMiller, J. and Whistler, R. (Eds.), 2009. Starch: Chemistry and Technology, Third Ed.\n\nJackson, R.S., 2009. Wine Tasting: A Professional Handbook, Second Ed.\n\nHeldman, D.R., 2011. Food Preservation Process Design.\n\nTiwari, B.K., Gowen, A. and McKenna, B. (Eds.), 2011. Pulse Foods: Processing, Quality and Nutraceutical Applications.\n\nCullen, PJ., Tiwari, B.K., and Valdramidis, V.P. (Eds.), 2012. Novel Thermal and Non-thermal Technologies for Fluid Foods.\n\nStone, H., Bleibaum, R., and Thomas, H., 2012. Sensory Evaluation Practices, Fourth Ed.\n\nKosseva, M.R. and Webb, C. (Eds.), 2013. Food Industry Wastes: Assessment and Recuperation of Commodities.\n\nMorris, J.G. and Potter, M.E. (Eds.), 2013. Foodborne Infections and Intoxications, Fourth Ed.\n\nBerk, Z., 2013. Food Processing Engineering and Technology, Second Ed.\n\nSingh, R.P., and Heldman, D.R., 2014. Introduction to Food Engineering, Fifth Ed.\n\nHan, J.H. (Ed.), 2014. Innovations in Food Packaging, Second Ed.\n\nMadsen, C., Crevel, R., Mills, C., and Taylor, S. (Eds.), 2014. Risk Management for Food Allergy\n\nMatthews, K.R., Sapers, G.M., and Gerba, C.P. (Eds.), 2014. The Produce Contamination Problem, Second Ed.\n\nBagchi, D. (Ed.), 2014. Nutraceutical and Functional Food Regulations in the United States and Around the World, Second Ed.\n\nJackson, R.S., 2014. Wine Science: Principles and Applications, Fourth Ed.\n\n# Table of Contents\n\n  1. Cover\n  2. Title page\n  3. Table of Contents\n  4. Food Science and Technology International Series\n  5. Copyright\n  6. List of Contributors\n  7. Preface to the Second Edition\n  8. Preface to the First Edition\n  9. Chapter 1: The World Supply of Food and the Role of Dairy Protein\n    1. Abstract\n    2. Introduction\n    3. Hunger and the need for food\n    4. The dietary essential amino acids in proteins\n    5. Identifying the countries deficient in dietary essential amino acids\n    6. Demographic changes, aging populations, and the need for quality protein and essential amino acids\n    7. Global trade in proteins, the long-term prospects, with a focus on dairy foods\n    8. Conclusions\n  10. Chapter 2: Milk: An Overview\n    1. Abstract\n    2. Introduction\n    3. Evolution of mammals and lactation\n    4. Utilization of milk\n    5. Composition of milk\n    6. Milk constituents\n    7. Summary\n  11. Chapter 3: The Comparative Genomics of Monotremes, Marsupials, and Pinnipeds: Models to Examine the Functions of Milk Proteins\n    1. Abstract\n    2. Introduction\n    3. The echidna (Tachyglossus aculeatus)\n    4. The tammar wallaby (Macropus eugenii)\n    5. A role for milk in the control of mammary function\n    6. The fur seal\n    7. New player in milk bioactives; MicroRNA\n    8. Conclusions\n  12. Chapter 4: Significance, Origin, and Function of Bovine Milk Proteins: The Biological Implications of Manipulation or Modification\n    1. Abstract\n    2. Introduction\n    3. Origins of milk proteins\n    4. Constraints and opportunities for evolution or manipulation of bovine milk proteins\n    5. Conclusion\n  13. Chapter 5: Post-translational Modifications of Caseins\n    1. Abstract\n    2. Introduction\n    3. The caseins\n    4. Caseins from other species\n    5. Conclusions\n  14. Chapter 6: Casein Micelle Structure and Stability\n    1. Abstract\n    2. Introduction\n    3. Casein primary structure and interactions\n    4. Casein micelle properties\n    5. Models of casein micelle structure\n    6. Concluding remarks\n  15. Chapter 7: Structure and Stability of Whey Proteins\n    1. Abstract\n    2. Introduction\n    3. Bovine \u03b2-Lactoglobulin\n    4. \u03b1-Lactalbumin\n    5. Serum albumin\n    6. Immunoglobulins\n    7. Lactoferrin\n    8. Concluding remarks\n    9. Acknowledgments\n  16. Chapter 8: Effects of High-pressure Processing on Structure and Interactions of Milk Proteins\n    1. Abstract\n    2. Introduction\n    3. High-pressure-induced changes in caseins\n    4. Effects of high pressure on interactions of milk proteins involving whey proteins\n    5. Concluding remarks\n    6. Acknowledgment\n  17. Chapter 9: The Whey Proteins in Milk: Thermal Denaturation, Physical Interactions, and Effects on the Functional Properties of Milk\n    1. Abstract\n    2. Introduction\n    3. The casein micelle\n    4. The heat treatment of milk\n    5. Relationships between denaturation\/interactions of the whey proteins in heated milk and the functional properties of milk\n    6. Conclusion\n  18. Chapter 10: Effects of Drying on Milk Proteins\n    1. Abstract\n    2. Introduction\n    3. Properties of spray-dried milk products\n    4. Principles of spray drying\n    5. Process improvement\n    6. Drying of proteins\n    7. Conclusions\n  19. Chapter 11: Changes in Milk Proteins during Storage of Dry Powders\n    1. Abstract\n    2. Introduction\n    3. The formation of maillard and pre-maillard compounds\n    4. Formation of isopeptide bonds\n    5. Amino acids other than lysine\n    6. Implications for nutritional value of milk proteins\n    7. Product-specific storage trials\n    8. Conclusions\n  20. Chapter 12: Interactions and Functionality of Milk Proteins in Food Emulsions\n    1. Abstract\n    2. Introduction\n    3. Adsorption of Milk Proteins During the Formation of Emulsions\n    4. Stability of Milk Protein-Based Emulsions\n    5. Heat-Induced Changes in Milk Protein-Based Emulsions\n    6. Pressure-Induced Changes in Milk Protein-Based Emulsions\n    7. Milk Protein Hydrolysates and Oil-In-Water Emulsions\n    8. Lactoferrin-Based Oil-In-Water Emulsions\n    9. Lipid Oxidation in Milk Protein-Based Emulsions\n    10. Behavior of Milk Protein-Stabilized Emulsions Under Physiological Conditions\n    11. Conclusions\n  21. Chapter 13: Milk Protein-Polysaccharide Interactions\n    1. Abstract\n    2. Introduction\n    3. Mixing behavior of biopolymers\n    4. Phase diagram\n    5. Nature of interactions in protein-polysaccharide systems\n    6. Milk protein-polysaccharide interactions in the aqueous phase\n    7. Milk protein-polysaccharide interactions at the interface\n    8. Rheological properties and microstructures of protein-polysaccharide systems\n    9. Concluding remarks\n  22. Chapter 14: Interactions between Milk Proteins and Micronutrients\n    1. Abstract\n    2. Introduction\n    3. Interactions Between native Milk Proteins and Micronutrients\n    4. Interactions between process-modified milk proteins and micronutrients\n    5. Conclusions\n  23. Chapter 15: Model Food Systems and Protein Functionality\n    1. Abstract\n    2. Introduction\n    3. Protein functionality in foods\n    4. Role of interactions in determining food characteristics\n    5. Processing effects\n    6. Uses of model food systems\n    7. Applications of model food systems\n    8. Use of model food systems for other food components\n    9. Limitations\n    10. Conclusions\n  24. Chapter 16: Sensory Properties of Dairy Proteins\n    1. Abstract\n    2. Introduction\n    3. Sensory analysis\n    4. Whey proteins\n    5. Milk proteins\n    6. Caseins and hydrolysates\n    7. Flavor binding\n    8. Conclusions\n    9. Acknowledgment\n  25. Chapter 17: Milk Protein Gels\n    1. Abstract\n    2. Introduction\n    3. Rennet-induced gels\n    4. Acid-induced milk gels\n    5. Whey protein gels\n    6. Conclusions\n    7. Acknowledgment\n  26. Chapter 18: Milk Proteins--A Cornucopia for Developing Functional Foods\n    1. Abstract\n    2. Introduction\n    3. Functional foods\n    4. Milk proteins as a source of amino acids--specialized nutritionals\n    5. Milk proteins as a source of amino acids--specific physiological roles\n    6. Milk proteins as a source of amino acids--role in providing calories and in promoting satiety\n    7. Milk proteins as a source of bioactive peptides\n    8. Conclusions\n  27. Chapter 19: Milk Proteins and Human Health\n    1. Abstract\n    2. Introduction\n    3. Milk proteins, metabolic health, and type 2 diabetes\n    4. Milk proteins, obesity, and weight control\n    5. Milk proteins and bone health\n    6. Conclusions\n  28. Chapter 20: Milk Proteins: Digestion and Absorption in the Gastrointestinal Tract\n    1. Abstract\n    2. Introduction\n    3. Digestion of milk proteins\n    4. Milk protein hydrolysis in the intestinal lumen\n    5. Peptides released during digestion\n    6. Impact of processing on milk protein digestion and absorption\n    7. Conclusions\n  29. Chapter 21: Milk Proteins: The Future\n    1. Abstract\n    2. Introduction\n    3. Global issues for food\n    4. Consumer demands and trends for food and ingredients\n    5. New technologies and their possible effect on milk protein ingredients and products\n    6. Conclusions\n  30. Index\n  31. Food Science and Technology: International Series\n\n## List of tables\n\n  1. Tables in Chapter 1\n    1. Table 1.1\n    2. Table 1.2\n    3. Table 1.3\n  2. Tables in Chapter 2\n    1. Table 2.1\n    2. Table 2.2\n    3. Table 2.3\n  3. Tables in Chapter 4\n    1. Table 4.1\n  4. Tables in Chapter 5\n    1. Table 5.1\n    2. Table 5.2\n    3. Table 5.3\n  5. Tables in Chapter 10\n    1. Table 10.1\n    2. Table 10.2\n    3. Table 10.3\n  6. Tables in Chapter 11\n    1. Table 11.1\n    2. Table 11.2\n    3. Table 11.3\n  7. Tables in Chapter 12\n    1. Table 12.1\n  8. Tables in Chapter 13\n    1. Table 13.1\n    2. Table 13.2\n    3. Table 13.3\n  9. Tables in Chapter 14\n    1. Table 14.1\n    2. Table 14.2\n    3. Table 14.3\n  10. Tables in Chapter 15\n    1. Table 15.1\n    2. Table 15.2\n    3. Table 15.3\n    4. Table 15.4\n    5. Table 15.5\n    6. Table 15.6\n    7. Table 15.7\n  11. Tables in Chapter 16\n    1. Table 16.1\n    2. Table 16.2\n    3. Table 16.3\n    4. Table 16.4\n    5. Table 16.5\n  12. Tables in Chapter 17\n    1. Table 17.1\n    2. Table 17.2\n    3. Table 17.3\n    4. Table 17.4\n  13. Tables in Chapter 20\n    1. Table 20.1\n  14. Tables in Chapter 21\n    1. Table 20.1\n    2. Table 20.2\n\n## List of figures\n\n  1. Figures in Chapter 1\n    1. Figure 1.1\n    2. Figure 1.2\n    3. Figure 1.3\n    4. Figure 1.4\n    5. Figure 1.5\n    6. Figure 1.6\n  2. Figures in Chapter 2\n    1. Figure 2.1\n    2. Figure 2.2\n  3. Figures in Chapter 3\n    1. Figure 3.1\n    2. Figure 3.2\n    3. Figure 3.3\n    4. Figure 3.4\n    5. Figure 3.5\n    6. Figure 3.6\n    7. Figure 3.7\n    8. Figure 3.8\n    9. Figure 3.9\n    10. Figure 3.10\n    11. Figure 3.11\n    12. Figure 3.12\n    13. Figure 3.13\n    14. Figure 3.14\n    15. Figure 3.15\n  4. Figures in Chapter 4\n    1. Figure 4.1\n  5. Figures in Chapter 5\n    1. Figure 5.1\n    2. Figure 5.2\n    3. Figure 5.3\n    4. Figure 5.4\n    5. Figure 5.5\n    6. Figure 5.6\n    7. Figure 5.7\n  6. Figures in Chapter 6\n    1. Figure 6.1\n    2. Figure 6.2\n    3. Figure 6.3\n    4. Figure 6.4\n    5. Figure 6.5\n    6. Figure 6.6\n    7. Figure 6.7\n    8. Figure 6.8\n  7. Figures in Chapter 7\n    1. Figure 7.1\n    2. Figure 7.2\n    3. Figure 7.3\n    4. Figure 7.4\n    5. Figure 7.5\n    6. Figure 7.6\n  8. Figures in Chapter 8\n    1. Figure 8.1\n    2. Figure 8.2\n    3. Figure 8.3\n    4. Figure 8.4\n    5. Figure 8.5\n  9. Figures in Chapter 9\n    1. Figure 9.1\n    2. Figure 9.2\n    3. Figure 9.3\n    4. Figure 9.4\n    5. Figure 9.5\n    6. Figure 9.6\n    7. Figure 9.7\n    8. Figure 9.8\n    9. Figure 9.9\n    10. Figure 9.10\n    11. Figure 9.11\n    12. Figure 9.12\n    13. Figure 9.13\n    14. Figure 9.14\n    15. Figure 9.15\n    16. Figure 9.16\n    17. Figure 9.17\n    18. Figure 9.18\n    19. Figure 9.19\n  10. Figures in Chapter 10\n    1. Figure 10.1\n    2. Figure 10.2\n    3. Figure 10.3\n    4. Figure 10.4\n  11. Figures in Chapter 11\n    1. Figure 11.1\n    2. Figure 11.2\n    3. Figure 11.3\n    4. Figure 11.4\n    5. Figure 11.5\n    6. Figure 11.6\n    7. Figure 11.7\n    8. Figure 11.8\n    9. Figure 11.9\n    10. Figure 11.10\n  12. Figures in Chapter 12\n    1. Figure 12.1\n    2. Figure 12.2\n    3. Figure 12.3\n    4. Figure 12.4\n    5. Figure 12.5\n  13. Figures in Chapter 13\n    1. Figure 13.1\n    2. Figure 13.2\n    3. Figure 13.3a\n    4. Figure 13.3b\n  14. Figures in Chapter 14\n    1. Figure 14.1\n    2. Figure 14.2\n    3. Figure 14.3\n    4. Figure 14.4\n  15. Figures in Chapter 15\n    1. Figure 15.1\n    2. Figure 15.2\n    3. Figure 15.3\n    4. Figure 15.4\n    5. Figure 15.5\n    6. Figure 15.6\n    7. Figure 15.7\n    8. Figure 15.8\n  16. Figures in Chapter 16\n    1. Figure 16.1\n    2. Figure 16.2\n    3. Figure 16.3\n    4. Figure 16.4\n    5. Figure 16.5\n    6. Figure 16.6\n    7. Figure 16.7\n    8. Figure 16.8\n    9. Figure 16.9\n    10. Figure 16.10\n    11. Figure 16.11\n  17. Figures in Chapter 17\n    1. Figure 17.1\n    2. Figure 17.2\n    3. Figure 17.3\n    4. Figure 17.4\n    5. Figure 17.5\n  18. Figures in Chapter 18\n    1. Figure 18.1\n  19. Figures in Chapter 20\n    1. Figure 20.1\n    2. Figure 20.2\n    3. Figure 20.3\n    4. Figure 20.4\n\n## Landmarks\n\n  1. Cover\n  2. Title page\n  3. Table of Contents\n\n  1. i\n  2. ii\n  3. iii\n  4. iv\n  5. ix\n  6. x\n  7. xi\n  8. xiii\n  9. xv\n  10. xvi\n  11. \n  12. \n  13. \n  14. \n  15. \n  16. \n  17. \n  18. \n  19. \n  20. \n  21. \n  22. \n  23. \n  24. \n  25. \n  26. \n  27. \n  28. \n  29. \n  30. \n  31. \n  32. \n  33. \n  34. \n  35. \n  36. \n  37. \n  38. \n  39. \n  40. \n  41. \n  42. \n  43. \n  44. \n  45. \n  46. \n  47. \n  48. \n  49. \n  50. \n  51. \n  52. \n  53. \n  54. \n  55. \n  56. \n  57. \n  58. \n  59. \n  60. \n  61. \n  62. \n  63. \n  64. \n  65. \n  66. \n  67. \n  68. \n  69. \n  70. \n  71. \n  72. \n  73. \n  74. \n  75. \n  76. \n  77. \n  78. \n  79. \n  80. \n  81. \n  82. \n  83. \n  84. \n  85. \n  86. \n  87. \n  88. \n  89. \n  90. \n  91. \n  92. \n  93. \n  94. \n  95. \n  96. \n  97. \n  98. \n  99. \n  100. \n  101. \n  102. \n  103. \n  104. \n  105. \n  106. \n  107. \n  108. \n  109. \n  110. \n  111. \n  112. \n  113. \n  114. \n  115. \n  116. \n  117. \n  118. \n  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563. \n  564. \n  565. \n  566. \n  567. \n  568. \n  569. \n  570. \n  571. \n  572. \n  573. \n  574. \n  575. \n  576. \n  577. \n  578. \n  579. \n  580. \n  581. \n  582. \n  583. \n  584. \n  585. \n  586. \n  587. \n  588. \n  589. \n  590. \n  591. \n  592. \n  593. \n  594. \n  595. \n  596. \n  597. \n  598. \n  599. \n  600. \n  601. \n  602. \n  603. \n  604. \n  605.\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n# A MARRIED WOMAN\n\nManju Kapur\n\nFor  \nmy daughter  \nAmba\n\n&\n\nIra\n\n# Contents\n\nTitle Page  \nDedication  \nChapter I  \nChapter II  \nChapter III  \nChapter IV  \nChapter V  \nChapter VI  \nChapter VII  \nChapter VIII  \nChapter IX  \nChapter X  \nAcknowledgements  \nAbout the Author  \nCopyright\n\nChapter I\n\nAstha was brought up properly, as befits woman, with large supplements of fear. One slip might find her alone, vulnerable and unprotected. The infinite ways in which she could be harmed were not specified, but Astha absorbed them through her skin, and ever after was drawn to the safe and secure.\n\nShe was her parents' only child. Her education, her character, her health, her marriage, these were their burdens. She was their future, their hope, and though she didn't want them to guard their precious treasure so carefully, they did, oh they did.\n\nHer mother often declared, 'When you are married, our responsibilities will be over. Do you know the shastras say if parents die without getting their daughter married, they will be condemned to perpetual rebirth?'\n\n'I don't believe in all that stuff,' said Astha, 'and I think, as an educated person, neither should you.'\n\nHer mother sighed her heavy soul-killing sigh. 'Who can escape their duty?' she asked, as she put in a steel almirah another spoon, sheet, sari, piece of jewellery towards the girl's future.\n\n*\n\nEvery day in her temple corner in the kitchen, she prayed for a good husband for her daughter.\n\n'You pray too,' she insisted as they stood before the shrine on the shelf, ordinarily hidden by curtains made from an old silk sari border, woven through with gold so pure that if the cloth was burnt, the metal would emerge in a little drop.\n\nAstha obediently closed her eyes to delicious images of a romantic, somewhat shadowy young man holding her in his strong manly embrace.\n\n'Are you praying?' asked the mother suspiciously.\n\n'Of course I'm praying,' replied the daughter indignantly, 'you never trust me.'\n\nTo prove her sincerity she fixed her gaze firmly on Krishna, Krishna the one so many had adored. He would send her marriage, love and happiness. She fingered the rope of tiny pearls around his image. On either side were miniature vases with fresh jasmine buds. There was also a picture of Astha's dead grandparents, a little silver bell and thali, two small silver lamps which were lit every evening, while a minuscule silver incense holder wobbled next to it. Whatever meal Astha's mother cooked was first offered to the gods before the family ate. She believed in the old ways.\n\n*\n\nWhile her father believed in the new. His daughter's future lay in her own hands, and these hands were to be strengthened by the number of books that passed through them. At least once a day he said to her, 'Why aren't you studying?'\n\nHow much studying could Astha do to satisfy the man? Through her school years she never found out.\n\n'Where is the maths work I asked you to do?' he would continue.\n\n'I haven't finished it yet.'\n\n'Show me whatever you have completed.'\n\nSums indifferently done were produced. The father tightened his lips. The girl felt afraid, but refused to show it. She looked down.\n\n'You worthless, ungrateful child. Do you know how much money I spend on your education?'\n\n'Don't then, don't spend anything,' she muttered, her own lips as tight as his.\n\nDriven by her insolence, carelessness and stupidity, he slapped her. Tears surfaced, but she wouldn't act sorry, would rather die than show how unloved and misunderstood she felt.\n\nHer mother looked on and said nothing. Later, 'Why don't you do the work he tells you to? You can't be drawing and painting all the time.'\n\n'So he hits me?' She didn't want her mother's interventions, she hated her as well as him.\n\n'It's his way of showing concern.'\n\nAstha looked away.\n\nThe mother sighed. The girl was good, only she got into these moods sometimes. And how much she fiddled with brush and pencil, no wonder her father got anxious, there was no future in art. If she did well in her exams, she could perhaps sit for the IAS, and find a good husband there. You met all kinds of people in the administrative services, and the girl was not bad looking. She must tell her to frown less. Frowns mislead people about one's inner nature.\n\n*\n\nThe girl's body was nurtured by walks that started every morning at five.\n\n'Get up, get up. Enough laziness.'\n\n'You will thank us later when you realise the value of exercise and fresh air.'\n\n'How can you waste the best part of the day? This is Brahmakaal, the hour of the gods.'\n\nSo Astha dragged her feet behind her parents' straight backs as they strode towards the dew and space of the India Gate lawns. Her parents arranged their walk so that they would be facing the East as the sun rose, showing their respect for the source of all life, while Astha, lagging behind, refused to participate in their daily satisfaction over the lightening sky, or the drama of the sun suddenly rising behind India Gate.\n\nWhen they came home they all did Pranayam together. Pranayam, in the patchy grass surrounded by a short straggly hedge outside their flat. Inhale through one nostril, pinch it, exhale through the other, pinch that, right left, left right, thirty times over, till the air in the lungs was purified and the spirit uplifted.\n\n*\n\nAt other times Astha's father took her for a stroll through the colony in the evenings. Away from her studies he was more amiable. He didn't want his daughter to be like himself, dissatisfied and wasted. You have so much potential, you draw, you paint, you read, you have a way with words, you do well academically, the maths is a little weak, but never mind, you must sit for the competitive exams. With a good job comes independence. When I was young, I had no one to guide me, I did not know the value of time, did not do well in my exams, had to take this job, thinking later I can do something else, but once you are stuck you are stuck.\n\nHere he grew silent and walked on moodily, while Astha linked her arm through his, feeling slightly sorry for him.\n\nAfter her father died and experience had drilled some sense of the world into her, Astha realised how emancipated he had been. At the time she felt flattered by his attention, but bored by his words.\n\nThe family counted their pennies carefully. Their late marriage, their daughter still to be settled, their lack of any security to fall back on, meant that their pleasures were planned with thrift firmly in the forefront. Once a month on a Sunday they went as a treat to the Bengali Market chaat shop. They gazed at the owner sitting on a narrow platform, cross-legged before his cash box, a small brass grille all around him. His dhoti kurta was spotless white, his cash box rested on a cloth equally spotless.\n\n'This man came from Pakistan, a refugee'' said the father.\n\n'Look at him now'' echoed the mother.\n\nAnd the shop grew glitzier every time they came, with marble floors added, mirrors expanding across the walls, extensions built at the back and sides. The tikkis and the papri did not remain the same either, but grew more and more expensive. What was in the tikki that made him charge one rupee per plate?\n\n'The potatoes he must be buying in bulk, so that is only one anna worth of potato, the stuffing is mostly dal, hardly any peas, a miserable half cashew, fried in vanaspati, not even good oil, let alone ghee; the chutney has no raisins, besides being watery, and what with the wages of the waiter and the cook, the whole thing must be costing him not more than... than...'\n\nFather, mathematician, closes his eyes to concentrate better on the price of the potato tikki.\n\n'Not more than eight annas, maximum''\n\n'Hundred per cent profit'' said the mother gloomily. 'How much does he sell in a day? Five hundred?'\n\nThey looked around the crowded restaurant, with its tables jammed together, and the large extended section behind the sweet counters. They looked at the people being served on the road. Yes, five hundred would not be wrong.\n\nThen they calculated eight annas multiplied by 500 made 250 rupees. And this was just on potato tikkis.\n\nWhat about the papri, the kulchas, the dahi barhas, the gol gappas in spicy water, the gol gappas in dahi and chutney, the kachoris with channa, the puri aloo, the channa bhatura, the newly introduced dosas, the dry savouries, sweets, and chips that he was cunningly displaying in glass cases, what about every one of those? How much money would he be taking home at the end of just one day? To top it all, he was uneducated.\n\n'I could make better tikkis at home'' offered the mother.\n\nAstha stared miserably at her plate of two small swollen tikkis, buried under sweet and sour chutney. Then she stared at the fat man behind the payment counter. He sat there with his paan-stained mouth, taking people's money, opening the lid of the cash box, calmly lifting the change katoris to add to the growing piles of ones, twos, fives, tens, hundreds.\n\n'Do you want anything else, beti?' asked the parents after they had eaten every crumb from their little steel plates.\n\n'No.'\n\n'Are you sure?'\n\n'Yes.'\n\n'Let's go home then.'\n\n'That was a nice outing, wasn't it?' they said to each other as they started the old Fiat and headed back to their flat on Pandara Road.\n\nOver these smaller worries, loomed the larger one, their unbuilt house, the place they would go to when they retired, the shelter that lay between them and nothingness. It was towards this end that they counted every paisa, weighed the pros and cons of every purchase with heavy anxiety. From time to time they drove to the outer parts of South Delhi with dealers to look at plots. Somehow, the places they wanted to live in were always outside their budget, the places they could afford seemed wild and uninhabitable.\n\n'There was a time when Defence Colony too was on the outskirts of Delhi'' pointed out the mother. 'Let us at least buy wherever we can''\n\n'No, Sita'' said the father irritably. 'How can we live so far away from everything?'\n\nHow far do we have to go before we can afford something, thought Astha, who was forced to come on these expeditions, she couldn't be left alone with the servant. She herself never intended to land in any house on any tiny plot they were looking at. Her husband would see to it.\n\nMeanwhile Delhi grew and grew, and plots they had once rejected as being too far, now became part of posh and expensive colonies, and not as far as they had once thought.\n\n*\n\nRetirement was coming nearer, the pressure to buy was growing, when in the early sixties, ministries started forming co-operative housing societies.\n\n'Thank goodness'' grumbled Astha's mother, 'at least the government will do for us, what we have not been able to do for ourselves''\n\n'It's one thing to form a co-operative housing society, another thing to get land allotted to it, and still another to build a house'' said the father, born pessimist. 'In what god-forsaken corner will they allocate land to a ministry as unimportant as relief and welfare, that too you have to see''\n\n'Arre, wherever, whatever, we have to build. Otherwise you plan that after retirement we live in your ancestral palace?'\n\nThe husband looked pained at his wife's coarseness.\n\n*\n\nThey continued to worry. When would their housing society have land assigned to it, how many more years for the father to retire, how many more working years for the mother, how long before they had to leave this government house in the centre of Delhi, so convenient?\n\nOnce the land was allotted, how much would it cost to build, how much did they have in fixed deposits, in their provident funds, how much could they borrow, how much interest would they have to pay? After discussing all this, they allowed themselves to dream a little.\n\n'I will have a special place for my books'' said the father, 'cupboards with glass to protect them from dust and silverfish''\n\n'I will have a big kitchen'' said the mother, 'with screen windows to keep flies out, and a stainless steel sink, not like this cement one which always looks filthy. I will have a long counter, so I don't have to unpack the mixi every time I need it. I will have a proper place to do puja, rather than a shelf''\n\n'We will have a study, maybe an extra bedroom for guests'' mused the father, and then they looked frightened at the money their dreams were going to cost. Maybe not a guest room, their voices trailed off.\n\nBy the time Astha was sixteen, she was well trained on a diet of mushy novels and thoughts of marriage. She was prey to inchoate longings, desired almost every boy she saw, then stood long hours before the mirror marvelling at her ugliness. Would she ever be happy? Would true love ever find her?\n\n*\n\nThen the day dawned, the day Astha saw Bunty. Bunty the beautiful, Bunty whose face never left her, Bunty whose slightest word, look and gesture she spent hours nursing to death.\n\nBunty's family lived in one of the bigger houses of the Pandara Road colony, a duplex with a large garden, and a roomy verandah. They were on visiting terms with Astha's parents, the younger sister was in her school. The boy was away in Kharakvasala in the Defence Academy and now home for the holidays.\n\nHe came over with his father. Oh, how he stood out. He had glossy black hair which he wore in a small puff over a high wide forehead. His eyes were like soft black velvet, set in pale sockets over the faint blush of his cheek. And just beneath that the bluish black shadow of an incipient beard, framing a red mouth. As she stared, steady, unwavering, he felt her gaze, looked up and smiled. His teeth were small, white and uneven, and as she lost herself in them, he raised his left eyebrow slightly. She shuddered, and weakly smiled back.\n\nThus began her torture.\n\nIf only she didn't see him so often, but Bunty was restless during his holidays. Boarding school, boarding college, as a result he knew few people in Delhi. He took to dropping in with his sister. There was the attraction of her devotion.\n\nDay and night the thought of him kept her insides churning; she was unable to eat, sleep, or study. Away from him her eyes felt dry and empty. Her ears only registered the sound of his voice. Her mind refused to take seriously anything that was not his face, his body, his feet, his hands, his clothes. She found temporary relief in sketching him, sketches that were invariably too bad to be mulled over.\n\nHours were spent in planning accidental meetings, how to bump into him in the colony, how to cross his father on his evening walk, how to fall into enough conversation to be invited over, how to borrow a book to prolong the stay, how to fall into a faint, how to die at his doorstep.\n\nOnce in Bunty's house she saw him pet his dog, who promptly put her paws in his lap, wagged her tail and salivated. At that moment she felt a keen shamed kinship with the animal.\n\nShe was too overwhelmed by her feelings to actually want to talk to him. To approach the site of all this wonder would be apostasy. To think that he would ever have anything to say to her was past crediting. Finally it was so unbearable, she had to tell someone.\n\n*\n\nGayatri, school friend, eventual confidante, decided that this affair needed managing.\n\n'What have you actually done?' she demanded.\n\n'Done?' quavered Astha, immediately feeling worse instead of better. 'Nothing''\n\n'You are such a ninny'' scolded Gayatri, 'invite him to a movie.'\n\n'How can I?'\n\n'How can you?' Gayatri stared at her. 'There is a Charlie Chaplin film at the National Stadium next Sunday morning. Ask him if he has seen it. Go on. Give him a chance.'\n\nEach day was now an exam, in which she failed daily. Gayatri was insistent. There had to be movement to the whole thing, otherwise she might as well not be in love. Astha was forced to admit the logic of this.\n\nThe day came when she stood tongue-tied before him, stammering out her request that the god come to a film with her and her friend.\n\n'Of course hell go, won't you, Bunty beta?' boomed his father.\n\n'Th-Thank-you, Uncle'' stammered Astha, not looking at Bunty.\n\nBunty seemed stiff and bored through the film. Gayatri chattered gaily in the interval, while Astha gritted her teeth and waited for the nightmare to end. Words rushed around in her head, words that would show how clever and interesting she was, but when she actually looked at him she could not speak. She wanted to never see Bunty again. She hated him. She wished his holidays would quickly end.\n\n*\n\nThey did, and Astha grew desperate. The point of getting up every morning had been the hope that she would be able to look at him, feed on a glance, a word, a smile. Now her rich inner world would become stale with nothing new to add to the store.\n\n'Suggest writing. You know, like pen pals'' said Gayatri.\n\n'No.' Suppose he laughed? Looked contemptuous?\n\n'What do you have to lose?'\n\n'Why should he write to me?'\n\n'Why not? He does drop by, and you also visit him.'\n\nAstha hesitated. 'That means very little'' she pronounced finally, thinking of those visits, the long pauses, she pulverised with emotion and Bunty shifting about in his seat, saying from time to time, 'So what's new?'\n\nGayatri pressed home her point.\n\n'He does talk to you, and objectively speaking, you're not bad looking. You have no figure, but your features are sharp, you have clear skin, and high cheekbones. If your hair was styled instead of pulled back, it would help, but still, it is thick and curly. You are on the short side, but tall men like short girls, that is one thing I have noticed, time and again.' (Gayatri herself was tall.)\n\n'I can't just walk up to him and say give me your address, I want to write to you.'\n\n'It's not anything so great you are asking. Once you write, he will write back.'\n\n'He may not.'\n\n'Then he is no gentleman'' said Gayatri severely.\n\nEventually Astha blurted out the request, shoving her diary and pen at him.\n\n*\n\nShe wrote, and he did reply, weeks later.\n\n'Who is this from?' asked her mother, holding the letter away from her.\n\n'How do I know?' demanded Astha.\n\nShe snatched the letter and tucked it into her school bag. It was from him, she knew it was. He had written.\n\nDear Astha,\n\nI received your letter a few weeks back. We do not really get time to write, we are very hard worked here.\n\nTomorrow, I am leaving for camp. There is much work to be done; we do a lot of studies on tactics and strategy of defence and attack. We leave early in the morning, first marching 20 miles, from where we will be transported another 80 miles. At the end of it all we will land in some remote village. After lunch, which we carry, we will 'dig-in' for the night to carry out a defence exercise. Digging trenches in the Deccan plateau isn't quite as easy as you might think. Each one takes 3 to 4 hours. We shall also have to climb Simhagarh, Shivaji's famous fortress, and incidentally the highest one. At night we shall ambush and patrol, the sole difference between this and a real war being that we shall fire blank rounds at each other instead of live ones.\n\nAnd so on. It was a soldier's letter, what else had she expected? If the reality of Bunty was a little flat after her image of him, her love could take it. She re-read it all day and the days to come, till she got his next.\n\nIt turned out Bunty liked corresponding. Through the year Astha heard about his friends, the war with Pakistan, Lal Bahadur Shashtri, his academic subjects, his service subjects, his feelings about the Indian Army in general, and cadets in particular.\n\nAnd Astha, Astha was witty, clever, chatty, all the things she could not be when he was in front of her. Her writing was laced with little drawings which he found ingenious and talented. She started flirting. Letters were safe.\n\nAs the correspondence established itself, so did the mother's suspicions.\n\n'Why is he writing so much to you?' she asked every time a letter came. By this time there were two people waiting for the post, Astha and her mother.\n\nIs it a crime?' Astha replied.\n\n'You are too young to be indulging in such goings-on.'\n\nShe made it sound so sordid. What words could Astha use to a woman who saw the world in terms of goings-on?\n\n'There is nothing going on'' she said, lying with great dignity. There was no need to explain the pulpiness of her heart, the wretched and permanent knot in her stomach. No doubt her mother would consider that a going-on too. How she wished she could really be gone, gone in the arms of Bunty, who would hold her close, whisper his love, confide that her letters had made him realise she was his soulmate, they would marry after he graduated, could she wait for him.\n\n'You have got your exams coming'' went on Astha's mother, staring hard and penetratingly at her daughter.\n\n'I know'' said the daughter, staring back just as hard.\n\nAstha's mother sniffed, a tight cold sniff.\n\nAstha paid no attention. She was living in a world of her own, waiting for the holidays to come, so that she could see Bunty. It would be different now, no awkwardness or shyness. They were closer, they had shared their thoughts and feelings. Hopefully they would kiss. Where and how? She imagined the places, grew lost in her fantasies.\n\nThe holidays came. The minute the mother knew that Bunty had come, she went to his house and from then on Bunty refused to have anything to do with Astha.\n\nFor a long time she didn't know what had happened, nor could she bear to find out. She lived in pain and anything that touched it was too much for her.\n\n*\n\nThe night before, on the phone, she had fixed to see him, this time she would not need Gayatri. She had spent many hours thinking about her hair, her clothes, should she wear casual or formal, new or old? How should she do her hair? Up or down? Loose or tied?\n\nDressed in her newest churidar kameez, tight around the hips, loose around the waist, Astha went to Bunty's house, at eleven o'clock as planned. His father met her at the door.\n\n'Bunty is not at home, beta'' he said politely, without asking her in, a slap in the face for Astha, standing awkwardly in her new churidar kameez, so tight around her hips, so loose around her waist.\n\n'When will he be back, Uncle?' she managed, dread making her voice heavy. Did Bunty's father hate her? Had Bunty said something to him? On the train home from the Defence Academy had he decided to loathe her instead of like her? _Was this his way of letting her know_?\n\n'I don't know, beta. It is his holidays, he has so many friends and relatives to see. You can phone him some time. Bye now.'\n\nThe door was closed before she was even down the steps. No seeing her off, no nothing.\n\nShe walked home, feeling sick. The year of writing to each other, he had said he wanted to see her, had he been lying, seeing how far she would reveal her feelings in those stupid letters before he showed them to his father? How could she have forgotten the little interest he had shown in her when he was actually in Delhi? He was amusing himself, that was why he had written, now when it was time to meet he intended to drop her. How Gayatri would laugh. Was there any way she could stop being friends with Gayatri right that minute? Dump her for ever, and never see her again?\n\nAstha had not been in the house ten minutes when Gayatri called. 'What happened?' she asked breathlessly, as though she had been the one waiting all these months to be kissed.\n\n'Oh nothing'' said Astha airily, through gritted teeth.\n\n'Nothing? What do you mean nothing?'\n\n'It's very sad. One of his uncles died, and he has to go to Bombay immediately with the whole family.'\n\n'But why didn't he tell you?'\n\n'There wasn't time to write.'\n\n'Odd'' said Gayatri after a pause. 'He might have met you for a few seconds alone. After all those letters.'\n\n'I'm telling you there wasn't time'' said Astha her voice rising.\n\n'Oh Asu, poor you.'\n\n'Not at all. I found I didn't like him so much when I actually saw him. He looked very silly. All he could say was \"So what's new\". One tends to build people up through letters.'\n\n'I suppose'' said Gayatri, sounding dissatisfied.\n\n*\n\nThe holidays passed. Astha suffered daily. Neither drawing nor reading could engage her. Her heart felt like lead, her mind like stone. She couldn't get Bunty on the phone, he was always out. Shyness, reticence, some shreds of self-esteem forbade her from persisting beyond politeness. No matter what had happened, he should also want to see her, if only to clear any misunderstanding. And so pride carried her through each miserable day.\n\n*\n\nA year later, when the pain was less, and college had made her feel more a woman of the world, she wrote, a light casual letter, 'What happened?'\n\nHe wrote back, 'I thought you knew. Your mother visited us the very night I arrived and told my father that I was distracting you from your studies. At the same time she asked him what my intentions were. My father thought it better if we had nothing to do with each other. Why create complications? I wish you well in life. Yours sincerely, Bunty''\n\nCan one die of shame twice? Astha did. How dare her mother interfere in her friendships? But then Bunty too had given in so easily, not bothered to find out how she felt, no word, no sign.\n\nWhere was the man whose arms were waiting to hold her? Till his arrival, she would walk alone, alone in college, through corridors of happy, independent, bustling girls, through classrooms devoted to the study of English Literature, alone in the colony through the dreary lanes between the houses.\n\nShe tried to put Bunty from her mind, though once or twice when girls huddled together, heads bent in the canteen, she brought out his name experimentally, to show she too had lived and knew what love was.\n\n'Yes, these boys\u2014'\n\n'Yes, there was someone, only last year\u2014'\n\n'Yes, he was handsome\u2014'\n\n'Oh, he doesn't study here. The Defence Academy at Kharakvasala.'\n\n'Yes, we still meet during the holidays, nothing special from my side. I thought it better not to have a long-distance relationship, you know how it is... '\n\nThe girls listened sceptically, how could they believe in the reality of one who was never seen hanging out at the back gate? Still, they teased her sometimes saying, 'Astha, tell us more about Bunty'' and Astha cursed her need to feel part of a group by making light of something that still tightened her chest with grief.\n\nFive years after its inception the housing society of the Ministry of Relief and Welfare was awarded a piece of land across the Jamuna.\n\nThe habitual gloom on the father's face became even more pronounced as he conveyed this news to his family. 'Other ministries, where the bureaucrats have pull, managed to get allotments in South Delhi. But what do we get? A site across the Jamuna, where there is no water, no electricity, no markets, no bus services, no amenities, no proper roads even.'\n\n'Never mind'' consoled his wife, concealing how bitter the blow was for herself as well, so much had depended on the promised piece of land. 'Once construction starts, things will change. Everything has to have a beginning. How much are the plots going for?'\n\n'How much can they go for?' replied the father. 'In the middle of the jungle with thugs, dacoits, and wild animals. 7-8,000 rupees.'\n\n'Size?'\n\n'225, 280, or 350 square yards.'\n\nTheir future home was going to be small and relatively cheap.\n\n*\n\nThe lots were chosen by draw. On the appointed day, the mother said to her daughter, 'I hope he draws a 350 plot, in the corner. There is very little difference in price.'\n\n'From?' asked, the daughter languidly. She had been paying insufficient attention to the family fortunes knowing that wherever they built a house, she would not be in it.\n\n'Don't you ever listen? After we are gone it will be yours.'\n\n'I don't want it.'\n\nThe mother trembled with annoyance. 'Don't you see our lives?' she hissed. 'Have you still not realised the value of land, that once you have it, there is always something solid to fall back on?'\n\nAstha looked at her mother, at the sallow skin with liver markings, at the carelessly dyed hair, black and white, at the hands gnarled from a lifetime of housework, the veins standing out on the backs, only fifty, despairing, shrivelled, and old. Her dream of a house was coming true in a way that made that dream for ever unrealisable. Her thoughts were now of 225, 280 or 350 square yards, of whether it would be near a park, or near the main road, near a market or bus stop, whether her husband would be happy there or not, because she of course could be happy anywhere.\n\nSlowly she took her mother's hands in her own. 'It will be all right, Mama'' she said beginning to rattle off what she had heard so often. 'Trans-Jamuna will grow, South Delhi too was once like this, it will be different once the new bridge is built. Just imagine, we will have our own house at last. A garden too.'\n\nThe mother looked at her daughter's young hands holding her own loose-skinned bony ones.\n\n'Yes darling, at last'' she sighed heavily.\n\n*\n\nThe father came home.\n\n'Well?' asked the mother, taking his briefcase.\n\n'280.'\n\nThere was silence as the family digested this. 280. They were going to live on 280 square yards. But still, that was more than 225, of which there were so many.\n\n'There were only four 350 ones.'\n\n'Only four? Then naturally we won't get one.'\n\nThe father paused before continuing, 'One of them the President drew.'\n\n'I'm sure it was rigged'' flashed the mother.\n\n'God knows. It seemed fair enough. It was done in front of us all.'\n\n'Of course it will seem fair. These people are not children.'\n\nThey consoled themselves over tea with reflections of the general perfidy of the world, and their own inabilities to succeed in the games that were demanded of life's players.\n\n*\n\nNow that they had their plot, they drove out in the direction of the wilderness to see it. Along with them was the President of the Ministry of Relief and Welfare Group Housing Society, needed to show the way.\n\nBehenji'' he turned around to address Astha's mother, sitting in the back seat with the reluctant daughter, 'in ten years time this area will be really built up. The future of Delhi is here. How far can Delhi spread south?'\n\n'It is a long way around'' murmured Astha's mother.\n\nThey were heading north, towards the Red Fort, beyond the ghats for burning the dead, towards Shahadra, across the old British-built double-storied bridge for cars and trains, the lone bridge across the river to east Delhi.\n\n'When the new bridge is completed the journey will be quicker, Behenji'' consoled the President. 'Twenty minutes and you will be in Connaught Place, heart of Delhi.'\n\nAstha's father drove without responding to any of these remarks. The privilege of owning a plot in this godforsaken place would come as a result of belonging to a ministry in which he had felt wasted all his life. The bitterness of this kept him silent.\n\nThe roads they were now passing were potholed and badly kept, establishing kinship with the dirt roads of villages. From time to time they caught sight of a brave house standing alone.\n\n'People are already constructing'' pointed out the President.\n\nThe road got narrower and bumpier, the trails of dust bloomed larger.\n\n'Nirman Vihar, Swasth Vihar, and there, Preet Vihar'' said the President, waving his hand at the bare expanses around him. A gloomy silence filled the car, they were too old and too young to regard with excitement this particular future laid out before them.\n\n'Here, turn here'' indicated the President. They left the narrowness of the main road for an indistinguishable little lane, bumped along, turned once, and there they were. The land was dry, dusty, bare, treeless and nondescript. Asman Vihar. Sky Colony.\n\nWhat had they imagined? Neat plots, lined with trees, and little lanes, waiting for owners to come and build houses? For 8,000 rupees? Were they crazy?\n\n'What's that?' asked Astha's father pointing to a village they could see in the distance.\n\n'Oh, that will go, we are dealing with them'' said the President. 'They think they have a right just because nobody has dislodged them so far. The land is vacant, so these villagers use it to farm. And the odd group of Gujjars roams around.'\n\n'Is it safe?' trembled Astha's mother.\n\n'The more people come, the safer it will be.'\n\n'And water, electricity connections?'\n\n'For water we have to dig our own tube wells. And they have promised a temporary line for electricity. It is only a matter of time when this will be like your Golf Links, Jor Bagh, or Defence Colony.'\n\nAfter this trip they did not talk about their dream home anymore. They heard stories of how, in one of the lonely houses there, dacoits had broken in at night, stolen everything, and injured the owner so much that he was in a state of semi-paralysis.\n\nWhen the future was taken out and aired they concentrated on the difference the new bridge would make, the changes in infrastructure that would come about once the area became more populated. When the prices went up, they could sell their plot and buy a little flat in south Delhi. They had to trust in God and wait, though with the father's retirement only six years away, the period they could wait was limited.\n\nNow that Astha was in college her mother focused anxiously on their primary parental obligation. Every Sunday she scanned the matrimonial pages meticulously, pencil in hand, circling ads. Later on she would show them to the father.\n\n'You have to take her to the studio to get nice photos taken. One full standing, one close-up of the face.'\n\n'She is only in second year, Sita, for heaven's sake. Let her finish her education at least.'\n\n'In the time it takes to finalise a match she will have graduated. Good boys are not to be found so easily.'\n\n'She has just turned eighteen. Let her be.'\n\n'You want her to turn out like us? Marrying in her thirties? And everybody wondering what is wrong?'\n\n'Let her settle down to a career, then we will see. I can't go around begging people to marry my daughter.'\n\n'There is a time for everything'' went on the mother. 'The girl is blossoming now. When the fruit is ripe it has to be picked. Later she might get into the wrong company and we will be left wringing our hands. If she marries at this age, she will have no problem adjusting. We too are not so young that we can afford to wait.'\n\nAstha, eavesdropping, wondered where this stream of logic would lead. She herself had tasted love and wanted nothing of an arranged marriage, but what did her father think?\n\nHer father refused to answer and refused to take Astha to the studio.\n\n*\n\nThe day the mother found a suitor, was a day when Astha came home from college, tired, stinking of sweat, her bag heavy on her shoulders, her red pointed ballerina shoes pinching her feet. All she wanted was to quickly bathe, get lunch out of the way and then lose herself in the Georgette Heyer she had borrowed from a friend.\n\nHer mother was sitting in the drawing room with a young man dressed in khaki.\n\n'Beti'' she called. 'Come here.'\n\n'Coming'' Astha shouted back, but she didn't like the tone of her mother's voice. She hid behind the curtain dividing the room and listened.\n\nMother: 'That was my daughter.'\n\nYoung man: 'Does she like sports?'\n\nMother: 'Very much.'\n\nDread filled Astha. Her mother was lying. She had somehow found a groom without a studio photo. Did her father know? She locked herself in the bathroom.\n\n'Astha.'\n\nNo answer.\n\nThe door rattled. 'Come out beta. Hurry up.'\n\n'Why?'\n\n'There is someone here to meet you.'\n\n'Who?'\n\n'Someone.'\n\n'First you tell me.'\n\n'Oho. A boy.'\n\n'Why are you so interested in a boy meeting me _now_?' asked Astha bitterly.\n\nBang, bang, bang \u2013 the wooden bathroom door shook against the onslaught of the mother's rage. Astha watched the towels hanging from the hooks shudder, she heard the tap next to the toilet dripping into the tin can below it.\n\n'I'm not coming'' she shouted.\n\n'You don't object to seeing boys otherwise. Suddenly you become so high and mighty, and refuse to even be polite to someone who has come all this way.' The mother dropped her voice to wheedle, 'Now come, what is the harm? It is just a meeting, nothing else.'\n\nAstha stared at a tiny crack in the old wood of the bathroom door. She was about to humiliate her mother in front of a stranger. She took a deep breath. 'I can't'' she whispered hopelessly, 'I can't meet anyone like this.'\n\nThe mother finally gave up, leaving Astha collapsed against the bathroom door, tears falling, crying, crying for Bunty, crying for the lack of love in her barren life, crying because she didn't want to see a dull stolid man in the drawing room who advertised for a wife and asked about sports.\n\nShe remained in the bathroom long after the suitor left. The bathroom represented her future; she had better start getting acquainted with it now.\n\nHours, years, later her mother banged irritably on the door, 'He has gone, fool that I was to try and arrange anything for you, you are just like your father, thinking everything is going to happen on its own. The food has gone stone cold, you can reheat it and clear up everything after you have finished.'\n\nOne month after this the boy appeared. In his final year of college, he was a bit older. They had been introduced by friends of friends at the University Coffee House. Like everybody they knew, they were missing classes in order to haunt these hunting grounds, their gaze sorting through the tables speculatively.\n\nAstha began to be included in groups that included him, at cinemas, restaurants, appointed meetings at the Coffee House, instead of casual ones. A boy was interested in her. With every visit he made to the back gate, her stock grew in college.\n\nShe began to lie at home about where she was going, and what she was doing. Most of the girls she knew who were seeing boys lied. It was the routine, self-protective thing. And how necessary, Astha had already seen.\n\nNow every evening Astha took a walk in the colony, announcing her intention of dropping in on old school friends on the way back. My head gets so tired with studying, she complained, I need a change.\n\nGo, beti, go. Take some exercise, and remember, walk briskly, swing your arms, and breathe deeply from the stomach, said the mother, glad that her daughter was at last beginning to understand the value of fresh air.\n\nAnd Astha would dash out of the colony, down the main road to the corner where Rohan was waiting in his old\n\nVauxhall. A quick check to make sure no one was looking, then she would jump in and they would roar off.\n\nIt was the first time in the old Vauxhall, side by side, in the front seat. The car was parked in a narrow empty lane, next to a Minister's house in the Lutyens part of New Delhi. Rohan had shut the windows, and locked the doors. It was late November and dark. The car smelled of Old Spice, Rohan's aftershave lotion, and the musty scent of ancient leather.\n\nAstha glanced at Rohan sideways. He was twisting the car keys in his long slender hands with the hairy fingers, tapping them restlessly on the steering wheel. Finally he turned and reached purposefully towards her.\n\n'Do you want to marry me?' she asked breathlessly, anxious to get this thing out of the way.\n\n'What?' he asked distractedly.\n\n'Marry me. Do you want to?'\n\nHe took his hands away and stared at her. 'Isn't it a bit early to decide that?' Astha felt she had offended him.\n\n'Well, you know, otherwise\u2014' she trailed off, trying to look as cute and disarming as possible.\n\n'Good God, is that what you are worried about? I'm not the type to let a girl get pregnant, what do you think I am?'\n\nAstha realised she was sounding very un-hep, but she couldn't help it. She had to know how safe she was. 'Marriage'' she persisted, 'you know\u2014' She inched a little further away from him.\n\n'Oh God, all right. We might get married. One day.'\n\nHis mouth closed on hers, his tongue was exploring, while Astha choked and wriggled frantically.\n\n'What's the matter?' he asked letting her go. 'Don't you like it?'\n\n'I don't know'' she mumbled non-committedly.\n\n'Well, let's find out.' Rohan was beginning to sound impatient.\n\nAstha had volunteered to go with him in the car. Her body had registered excitement when he had parked in the dark lane. When he bent close to her, she had been overcome with dread and longing. There was no going back. She offered her lips, trying not to shrink into the seat.\n\n'God you're so stiff'' said Rohan shaking her a little bit.\n\n'Sorry'' she mumbled.\n\n'This is not some kind of torture you know.'\n\nAstha didn't know what to say. Rohan let her go, also silent. Finally he said, 'Give me your hands.'\n\nShe held her fists tensely out. Slowly he moved his thumb around her wrist, stroking the closed hand open. He kissed the fingers, nails, palms, he felt the small hair on the back with his closed lips. Astha felt something flow inside her as she stared at his bent head. She had never been so aware of her body's separate life before. After a little more of this he dropped her at their secret corner.\n\nShe got out of the car reluctantly. Something hadn't happened. But then again, something had. On the whole the encounter left her anxious for more.\n\nWhen Rohan at last slid his tongue into Astha's mouth she was putty in his hands. Her neck, her ears, her throat, eyes, chin, lips, all had been explored, and this time her strongest feeling was gratitude.\n\n*\n\nThis was the start of numberless kisses in the car. The only problem was in finding a place sufficiently isolated.\n\nTo Astha all places looked the same, but no, muttered Rohan, his eyes scanning various deserted spots, his fingers kneading the palm of her hand, not that one, not there, that's not safe, while Astha burned with impatience. Finally before rolling up the windows he always put out his hand and locked the door from outside.\n\n'Why are you doing that?' Astha asked.\n\n'Precautions.'\n\n'Against what?'\n\n'Just.'\n\nAstha was not really interested. All she wanted was for him to start, so that the world could fall away, and she be lost. This is love, she told herself, no wonder they talk so much about it.\n\nOne evening, they had parked in their usual dark corner in a by-lane off Akbar Road. Rohan liked Akbar Road, he considered it among the safest places he could find.\n\nAstha had slid as far down the seat as she could go without dislocating her hip. Rohan was lying as much as he could on top of her without dislocating his own. Their eyes were closed, their breathing audible. Absorbed in what they were doing, they did not hear footsteps approaching, did not see faces pressed against the windows on either side, eyes peering down at them.\n\nThe car shook, hands banged violently on the glass, rattled the handles. Astha and Rohan jerked upright, the whole car was swarming over with threatening bodies trying to get in.\n\n'Oye, oye'' they shouted, leering and grimacing, furiously working the handles. Rohan frantically turned the key in the ignition, pressed the accelerator, the old Vauxhall shook and roared. The men fell off as he sped in reverse down the dark lane, lights off. They ran after the car for a while, then couldn't be seen anymore. Some distance down the main road, Rohan stopped.\n\n'Who were they? And why\u2014?' stammered Astha, shaking with fright.\n\nRohan took her hand. 'Some fools. I'm sorry'' he said.\n\nShe started to cry.\n\n'Calm down.'\n\n'Take me home.'\n\n'Calm down first. Look, nothing actually happened.'\n\nAstha felt worse and worse. Those men had wanted to attack. Suppose they had managed to break the car window, gang rape her because of her shameless behaviour in a public place, and beat up Rohan when he tried to intervene? And all the while her parents would be thinking she was breathing in fresh air. If her mother knew she would first kill her and then herself. Astha's tears grew copious and she began to choke in her dupatta, while Rohan took sly glances at his watch. 'Come on'' he said at last, 'it was bad, but now it is over. Don't cry, for heaven's sake. We won't go there again.'\n\n'Hoon'' sniffled Astha.\n\n'You are all right, so what exactly is the problem?'\n\nAstha only knew she felt terrible. Finally when Rohan dropped her off, she sensed eyes hidden in every bush, eyes that saw and condemned. She pulled her dupatta around her head, and hurried home trying to concentrate on the various lies she would have to tell as to why she was so late.\n\nThe day Astha's mother read her diary dawned cool and clear, beautiful like all winter days, with the flowers blooming in the garden, and the promise of basking for hours in the sun.\n\nShe was deep in a book when her mother called, 'Come here, I have something to show you.'\n\nReluctantly Astha marked her place, and went inside. When she saw her journal in her mother's hands, she wanted to instantly erase herself. There she was, with her skin ripped off, exposed in all her abandoned thoughts and deeds.\n\n'Is this you?' the mother's voice quavered, her grip like iron on Astha's arm.\n\nAstha shook her head nervously.\n\n'Then, what is it?'\n\nDesperate silence while she tried to think of something plausible.\n\n'Answer me'' screamed the mother in a whisper.\n\n'I\u2013I don't know'' stammered the daughter, 'I mean, I don't remember.'\n\nBut she did, of course. All her secret fantasies, the things she did with Rohan, painstaking details of the furtive, exciting moments in his car.\n\n'Well, look at it'' the mother waved the offending notebook in front of Astha's nose, an innocuous old brown paper covered thing with _St_ _Theresa's_ _Convent_ _School_ in a half moon on top. It had been hidden behind her college books, how had her mother discovered it? It looked awful in her hands, soaked in sin.\n\n'You have no right to read my diary'' she weakly muttered in self defence, averting her eyes.\n\nThe mother ignored this remark and continued leafing through it gingerly.\n\n'Here, what does this mean?'\n\nThe usual scene of passion. Astha went through puzzled motions of glancing, page turning, furrowed brow.\n\n'These are notes for a story I am writing'' she said, inspired at last.\n\nThe mother's body sagged as some of the tension went out. 'This is your imagination?'\n\n'Yes. Yes, it is my imagination.'\n\nThe mother was silent for a moment, then sighed heavily and held the tender young body of her innocent daughter close to her. 'My child is too sensible to do anything like this'' she whispered. The girl remained rigid, arms by her side.\n\nThey avoided each other for the rest of the day.\n\n*\n\nThere were three consequences to this.\n\nOne was that Astha stopped being able to write in her journal. She tried a few entries in an elaborate code, but an audience was now branded into every page, and she could inscribe nothing beyond a casual account of her day in college.\n\nThe second was that Astha's parents took an annoying interest in her reading matter. Her father began diligently to bring her books of moral and intellectual substance. 'You need a sense of your cultural background'' said the bureaucrat. 'Of what made this country great. Know your artistic heritage, since your interest lies there.'\n\nHer mother decided that the virtues of tradition needed to be made more explicit. 'Our shashtras teach us how to live. You will learn from the _Gita,_ __ the Vedas, the Upanishads.'\n\n'I can't read them'' protested Astha violently. 'My Hindi is not good enough, you know that. It is your fault for sending me to a convent school.'\n\n'Your father happened to think that a convent gave the best education. That doesn't mean we can't broaden your base now'' replied the mother. And she began getting Hindi books and magazines from her school library, so that Astha's Hindi could improve and the sacred texts of the Hindus be made available to her.\n\nThe third consequence was that the parents tightened their surveillance. Getting to meet Rohan proved more and more difficult.\n\n*\n\nShe didn't want to tell him of what she was going through. He was preoccupied with his final year exams and seemed to have less time for her.\n\n'Why do you keep phoning all the time?' he complained one evening when they met. 'I have to study.'\n\n'I don't phone you all the time. Once or twice a day to ask how's it going.'\n\n'It distracts me.'\n\n'If you don't want to talk to me, just say so. Don't look for excuses.' Astha's voice shook. Rohan sighed and put his arms around her. Astha snuggled into him and slid her hands under his shirt.\n\n'Baby, don't be unreasonable. A man has to do well. Get a scholarship. Go abroad.'\n\nThis was the first time he had talked of going abroad so definitely. Astha shifted herself out of his arms.\n\n'Hey, little one'' cooed Rohan, reaching out for her. 'What's the matter?'\n\n'Nothing'' she said forlornly.\n\n'Do you think I'm going to forget you?'\n\nAstha did think exactly that, but how could she admit it?\n\n'Let me just finish my exams, little one, and then\u2014'\n\nRohan's words helped bolster Astha's illusions, it was all right, she was still safe, their affair was going to end in marriage. But the cold feeling did not go away, though each time Rohan spoke, Astha clung to his assurances.\n\n*\n\nRohan did very well in his exams, and on that stepping stone began the process of his going away. Away to the West, where ordinary mortals cannot go, where even the words PPE and Oxford showed Astha how great the distance was between them.\n\n'Oxford'' she breathed in awe. Suddenly her life seemed very small.\n\nRohan looked nonchalant. 'Might as well follow in the family tradition, keep the old folks happy. My father is an Oxford man, you know.'\n\nNo, Astha had not known. 'How lucky you are, Rohan'' she said.\n\n'Well, my father is keen'' continued Rohan, his gaze centred on the middle distance.\n\n'When are you going?' she asked, and then hated herself for being in a situation where she was forced to prise answers from the man she had been so intimate with.\n\n'Soon.'\n\n'Doesn't it cost a lot of money?' asked Astha tentatively.\n\n'Lots'' said Rohan, lighting a cigarette and breathing the _smoke sexily out. 'But we are trying to manage something_.'\n\nAstha thought it might seem rude to ask for more information, and waited for Rohan to tell her. Rohan did not. He looked at her briefly and smiled. 'Come, let me drop you home'' he said, 'There are a lot of things I have to do.' He hadn't touched her once. As he turned the key in the ignition, Astha thought, he was going to Oxford, he had the resources, his father was an Oxford man. What was her father? A minor bureaucrat, who had never studied abroad, whose sole possession was 280 square yards in the wilderness beyond the Jamuna.\n\n'Wait, Rohan'' she said, 'I hardly get to see you nowadays, you are so busy, and it is still early.'\n\nRohan continued drumming his keys against the steering wheel.\n\n'How come you never mentioned your family traditions before?' Astha went on carefully, scratching and poking at the leather so hard, she could smell the car on her hands long afterwards.\n\n'Well, there was no point talking about things, until things got certain.'\n\n'Yes, but you might have told me that there was a possibility of your going away.'\n\n'You knew that'' said Rohan coldly, not looking at her. 'I never tried to hide anything. There is no need for me to hide.'\n\n'No, no, of course not. That is not what I meant'' floundered Astha. 'But you just mentioned once or twice, like people do, you know, about going abroad, and I didn't know... Why, your results are just out, and you must have been trying since last year if you are going so soon.'\n\n'Sending applications is not something to make a song and dance about. I would look very stupid afterwards if I got neither admission nor funding.'\n\nAstha felt hopeless. She sat in silence, next to this boy whom she had thought she knew. The hands that he had used on her body were now clenched around her heart, slowly squeezing, slowly hurting.\n\n'What about us?' she asked abruptly, drawing a deep breath.\n\n'We will see'' said Rohan briefly. He was waiting to take her home, waiting to get rid of her. He started the car, while Astha stared out of the window all the way to the edge of her colony.\n\n'Bye'' she said, closing the door carefully, feeling it would be her last time in the Vauxhall, which was just as well. Old cars were so ugly, so useless, so slow, why did anyone bother with them?\n\n'Bye'' said Rohan. 'Be seeing you'' he added, a remark which her dignity threw back silently, with all the coldness and contempt her falling to pieces self was capable of.\n\n*\n\n'Where have you been? Dinner has been getting cold'' came her mother's voice, as Astha tried her usual unobtrusive entrance.\n\n'Nowhere.'\n\n'Nowhere has a name or no?'\n\n'No'' said Astha, going to lock herself in the bathroom, free from voices, free from everything except the terrible things she was feeling, because Rohan didn't love her, Rohan had lied to her. Rohan was what her mother had been warning her about since she was old enough to be warned, and how pleased she would be to know she had been right all along.\n\n*\n\nIn the days to come Rohan neither called, nor sent the usual secret messages through her girlfriends.\n\nIt was over. Over. Over.\n\nAstha entered her third year with a desire to get her education over as quickly as possible. Every day was painful to her. She was constantly reminded of Rohan, in the Coffee House, at the back gate, at their secret corner of the road, every evening at home. As for old black cars, they made her physically sick.\n\nRohan went abroad and Astha enrolled in MA, bored and unenthusiastic. Three years of an Honours course in English Literature had given her all she wanted to know of the subject. Not for her did the excitement of intellectual discovery lie ahead, only more of the same. After two years were over she supposed she would drift into teaching, that is what women like her did. School or college, remained to be decided.\n\nAll through third year her classmates had been busy preparing for competitive exams, or like Rohan, applying for higher studies abroad. Those not in this category had married and disappeared, to be heard of occasionally, moving around with husband, and later baby, stamped with the marks of confirmed adulthood.\n\nNow in MA, her friends were few.\nChapter II\n\nAstha was in her final year when the proposal came. The MBA, foreign returned son of one of the bureaucrats who lived in the larger houses bordering Lodhi colony, had seen her and wanted to meet her. His father dropped in on them, and acquainted the parents with their good fortune.\n\n'I don't know him'' objected Astha when they told her the news.\n\n'He also doesn't know you, Madam'' said the mother crossly. 'That is why he wants to meet you.'\n\n'Give her the details, Sita'' reproved the father. 'This is a question of our girl's happiness. There is no hurry.'\n\n'With you retiring in one year, there is every hurry.'\n\n'That is no reason to marry.'\n\nThe mother fell into despairing silence. Retirement, father's uncertain health, finances in a meagre state, the bridge to the plot unbuilt and their dearest daughter still to be settled.\n\n'How many times can I meet him?' demanded Astha, a little excitement rising in her. Somebody found her desirable and had gone to lengths to find out who she was.\n\n'One, two times, what is the need for more?' said the mother. 'You cannot tell about a person before marriage, no matter how many times you meet him.'\n\nAstha sat silent, twiddling her thumbs, staring down at her flat feet in their bathroom slippers. Had she known Rohan? Not really, and the soiled feeling she now associated with that interlude came over her again.\n\n'Papa?' she quickly asked. 'You think this is a good thing?'\n\n'I'm not sure'' said the father uncertainly. 'The plus side is that he is the only son and both his sisters are married. The younger one, settled in the US, wanted to sponsor him, but he decided to return to his parents. He is twenty-six, five eleven, he works as an assistant manager in a bank.'\n\n'Clearly a good, family-minded boy'' said the mother complacently.\n\n'And Vadera's ministry was allotted land in South Delhi. They will be able to build on it, they won't have to wait for bridges and water and electricity connections, they won't have to worry about thugs or gypsies'' continued the father bitterly.\n\n'Isn't that a good thing for our daughter? She at least will have a decent home. God has heard my prayers'' added the mother piously.\n\n'Sita, are your prayers that the girl be married to a plot in Vasant Vihar? Why don't you go and do the pheras there?'\n\n'What's wrong, Papa? You don't like the family?'\n\n'I have heard things about Vadera.'\n\n'What things?'\n\n'He is in the commerce ministry. Nice place to be if you want to keep a certain standard of living, and licences are needed by every manufacturing unit, big or small, for anything they do.'\n\n'So?'\n\n'He travels abroad, gives his daughters big weddings, buys a car, a _new_ car every three or four years.'\n\nThis did indeed seem very bad; such high living had to have some dark reason behind it. 'How does he do it, I ask you?' went on the father. 'Must be taking bribes. Will you be happy in a house that doesn't share our values?'\n\n'Papa, you don't think it is a good idea, I won't meet the boy.'\n\nThe mother collapsed into rage. 'Everybody is corrupt, are they? Throw out nine tenths of the government, run the country yourself with your high blood pressure. Expect the whole nation to be like Gandhi. Send your daughter to an ashram, because we have neither the means nor the money to get her properly married.'\n\n'I will look after myself'' said Astha bravely.\n\nNo one paid attention.\n\n'Their sole interest is the girl, her looks, her education, her qualities. That is something'' said the harassed father.\n\n'It's more than something'' insisted the mother. 'How many people do you know like that?'\n\n'Big dowries are being offered for Hemant. He is known to be quite smart.'\n\n'Is that his name?' asked Astha.\n\n'Yes.'\n\nA nice name. Hemant and Astha. It had a certain ring to it.\n\n'Why aren't they going in for big dowries?' inquired the prospective bride.\n\n'The boy does not believe in dowry. Must be the foreign influence, couldn't be his father.'\n\nAstha felt an even greater interest in the boy. 'Let me meet him, Papa'' she declared. 'After all, the father came with the proposal, they must be thinking alike to a certain extent.'\n\n*\n\nHemant came to the house. The parents left them alone for half an hour. Astha was so nervous her palms were sweating. He had only gone by her face, she knew that was passable, but what about the rest of her? Should she tell him about Rohan, but what to tell? That though she had kissed a boy, her hymen was intact? That he had broken her heart but she hoped to find happiness in marriage? How could she say this to someone she didn't know? She looked up at Hemant and smiled tentatively, he smiled back, asked about her hobbies before continuing to talk about his experiences in the USA.\n\nA few weeks later the engagement between Astha and Hemant Vadera was decided upon. The wedding was to be held in June. By then Astha's MA exams would be over, Hemant's elder sister's children would have their holidays, and his younger sister would be able to come for a month from the States. With all this settled, Astha and Hemant began to date.\n\nThe day of Astha's wedding was like any other day in June. The heat rose and rose, dust gathered, and all activity struggled against a dull and heavy lassitude. In the morning Astha's mother brought her tea and gazed at her approvingly.\n\n'Today you are getting married and leaving for your new home'' she murmured, tears in her eyes, while relatives clustered and consoled, speaking of the necessity of this moment, the pain of a mother at parting, the joy of a mother at her duty successfully completed. These murmurs fluttered around Astha, who, restless and ill at ease, waited for the action to begin.\n\nOutside, the tent wallahs had started converting the central common ground into a wedding hall, enclosing it with shamianas, covering the stubbly parched grass with durries. In the afternoon the caterers came, putting up tandoors near the garage. Five hundred invitations had been sent out. Both the families lived in the same colony, worked for the same government, had lived for years in the same city. A huge guest list was unavoidable.\n\nIn the evening there was a havan, during which Astha's red and white wedding bangles, with the dangling chura, were put on. She sat before the fire, tossing the samagri in, feeling dazed and unreal.\n\nAfter this it was time to get ready. Her wedding sari, fresh from the ironers, was laid out. She contemplated the thick red and gold silk in which sweat and discomfort were guaranteed. Normally summer brides wore thin tissue woven with gold, but with the wedding costing so much, she had to wear the heavy bridal temple sari bought years ago at a special price by her father on a government tour to South India. Now Astha hoped Hemant would not find her dowdy or unfashionable. She hoped he would not mind that she had little jewellery, she hoped he would like her without beauty parlour bridal make-up, and that he would hate, like she did, those assembly line creations, pink and white, with black-lined eyes.\n\n*\n\nNight came, the barat arrived. Astha was called to garland her groom to the taped music of shehnais. At the auspicious hour they sat down next to each other under a small rickety pandal, with a fan trained on them. The hot air from the fan, the smoke from the fire, the sight of her father waiting to do the kanyadaan, the feel of her hand in the hand of her bridegroom, in a trance she realised this was the beginning of the life ordained for her.\n\nThe pundit was chanting. They were taking the seven steps around the fire, the steps that meant they were legally married. It was stifling, perhaps she was going to faint. Her cousins clustered around her, fussing with her jewellery, adjusting her palla and teasing the bridegroom.\n\nThis time tomorrow she would be in Kashmir, surrounded by mountains, trees, and lake, where waters rippled gently around gliding shikaras, where a book would not be her companion, but her husband bending tenderly over her, her companion for life.\n\nOn the plane to Srinagar Hemant held Astha's hand, while she looked shyly out of the window at the mountains they were flying over. A deep seed of happiness settled in the pit of her stomach, she was married, she didn't have to be the focus of her parents' anxieties any longer. She was now a homemaker in her own right, a grown woman, experiencing her first plane ride.\n\nThroughout the journey, Hemant kept touching Astha, a finger slid inside the sleeve of her blouse, a hand pressed on her knee.\n\nThe consummation took place in a houseboat on Dal Lake. Hemant undressed her slowly, gazing at her steadily, while Astha nervously looked at his stomach.\n\n'Now you undress me'' he said.\n\nShe shook her head modestly, wrapping the sheet around herself, tucking one edge in.\n\n'Come on'' he urged.\n\nShe bent her head still further and stared at his shoes planted next to her smooth, freshly waxed bare legs and pedicured feet, their mehndi patterns still a deep orange.\n\nHe took her hands and put them on his chest. She undid the buttons, and slid his shirt off. As he lifted his arms for her to remove his vest, the hair of his underarms sprang out at her, along with the smell of him. She pulled off the vest quickly and stopped. He drew her hands again to him, unzipping his pants himself in his impatience to guide her to the right spot.\n\n'There'' he whispered, jamming her hand into his underwear, swelling bulkily, 'it's yours. Do you like it?'\n\nAstha hardly knew. She snatched her hand away, and rolled face down on the pillow, while Hemant excitedly finished his own undressing.\n\nAfterwards they found a spot of marriage blood on the sheet. They both peered at it.\n\n'Did it hurt?' asked Hemant fondly.\n\n'A little'' confessed his wife.\n\nLater, in the privacy of the bathroom, Astha allowed herself to wonder whether she had been misled about the magnitude of the act.\n\n*\n\nSex. There was so much of it. The pain Astha felt between her legs was never quite absent. She could only thank God they never spent that much time actually doing it. Hemant attacked the whole thing with great urgency, gazing at her a little anxiously after each time, while she uncertainly smiled back to a look of satisfaction that came over his face.\n\nUnbidden thoughts of Rohan came. How slow his kisses had been, how infinitely long, how thorough. Then she scolded herself. Rohan had abandoned her, Hemant had married her, he valued her, he thought her pretty. The question of whose lovemaking she preferred did not arise.\n\nDuring the day they wandered around the tourist spots of Srinagar, hand in hand. People looked at the bangles on her wrists and smiled. She was a bride, and her grip on Hemant's hand grew more certain, and the blush on her face more conscious. Hemant's attention was constant. He took endless photographs and never let her read.\n\n*\n\nAstha wanted to record what she felt. This was her honeymoon, a one-time thing. She tried writing in her journal, but as usual the words didn't come. She tried sketching her surroundings, but the beauty was too overwhelming. She drew Hemant instead, his face, body, torso, arms, legs, it was the first time she had had so much nudity to work with. And when she was tired of art, she attempted writing.\n\nOne evening, sitting on the roof of the houseboat, drinking her evening tea, looking out on the lake, she wrote a poem about the sky, the shikaras, the sound of the birds, the sun behind the mountains reflected in the water. She wrote that she, the watcher, was part of that harmony, and it was fitting that her new life begin in beauty. As she put down her pen, tears filled her eyes.\n\nHer husband saw. 'What's the matter, darling?' he enquired, leading her downstairs to the bed, where they had made love last night, this morning and afternoon.\n\n'Nothing'' gulped Astha, her head laid out on the pillow. Hemant scooped her legs up, and lay down next to her.\n\n'I don't like to see my baby crying'' he said softly.\n\nAstha pressed her face into his shoulder, and laced her arms around his back.\n\n'Are you missing your mother?'\n\nShe started to laugh, the idea was so absurd. 'No, silly'' she said.\n\n'Then what? Tell me, I'm your husband'' urged Hemant. 'Tell me, wife.'\n\nAstha didn't know why she had been crying. Nothing in her present life seemed to justify tears. Finally she mumbled, a little sheepish, 'It hurts.'\n\n'Where?'\n\nAstha's hand vaguely danced over her middle. Hemant put his hand firmly between her legs. 'There?' he asked. She nodded.\n\n'Why didn't you tell me? You must tell me these things, I will never know otherwise. We are one, you know. Now promise.' He bent to kiss her.\n\nAstha responded more warmly than she had in her entire marriage. 'I didn't know what to say'' she went on whispering in the ears of her lawful wedded husband, her husband who would take care of all her hurts like he was taking care of this one.\n\n'Poor baby'' murmured Hemant, 'we won't make love till it stops hurting, all right?'\n\n*\n\n'Hemant?' asked Astha, a week after they were married.\n\n'Hum?' replied Hemant sleepily. Astha's head was on his shoulder, his arm was around her, and he had spread her hair across his chest.\n\n'Why didn't you marry an American?'\n\n'Why do you ask?'\n\n'Well, you were there a long time, you must have gone out with girls. Fallen in love, thought of marriage.'\n\n'Never.'\n\n'Never had women?'\n\nHemant side-stepped this. 'I was always sure I wanted to marry a girl from here.'\n\n'Me, you mean.'\n\nHemant's grip tightened around his wife, while Astha felt thrilled and wanted. 'Besides'' he said, playing dreamily with her hair, 'I had responsibilities to my parents. I am the only son, and I wanted someone who would fit in with our family life. American women are too demanding. Their men have to cater to all their whims and fancies.'\n\n'Is that true?' demanded Astha, visions of American women waited on hand and foot, basking in love, flashing through her mind.\n\n'You bet'' said Hemant with great certainty. 'Besides you can't be sure they won't be up to something.'\n\n'Like?'\n\n'Other men. It's not so unthinkable for them as it is for an Indian girl.'\n\nAstha fell silent. She was wondering if she liked this conversation. She turned to kiss him, but Hemant was not to be distracted. This was a topic he had considered deeply. 'I wanted an innocent, unspoilt, simple girl'' he went on.\n\nThere was a pause.\n\n'A virgin'' he elaborated.\n\n'Suppose I had not been one?' asked the wife carefully.\n\n'And the blood on the sheet, what was that? A mirage?'\n\n'Some women don't bleed, even though they have had no sex, you know'' said Astha. She had read this in a magazine.\n\n'Since that didn't happen in our case, why talk about it'' said Hemant.\n\nRohan's face bending over hers arose before Astha's eyes. Had she been a virgin? Unlike Hemant, she was not sure. She decided to forget the whole business, after all now she was definitely not one, and what was the point of thinking about the past?\n\n*\n\n'I see you are a writer', said Hemant, looking through her notebook, 'as well as an artist.'\n\n'Not really'' said Astha modestly, and waited for him to draw her out.\n\n'What are you doing now?' he asked.\n\nAstha showed him the paper on which she was writing a poem.\n\n'They say a picture's worth a thousand words'' he read, then looked up and frowned. 'But this is not a picture'' he objected.\n\n'I know'' said Astha, quickly. 'I was just looking for a way to start. Whenever I sketch the scene, it ends up looking like a post card, so I thought I'd try words instead.' She reached out to take the page, 'I'll work on it more and show you.'\n\n'No, no, let me read. Maybe I can help you. I used to read a lot when I was in college.'\n\n'Really?' asked Astha interestedly. 'What?'\n\n'Harold Robbins. Erie Stanley Gardner, Somerset Maugham, Agatha Christie, P. G. Wodehouse, all kinds of authors. I was quite a reader, you know.'\n\nAstha was silent, while Hemant's eyes quickly scanned the page. 'You certainly have a nice imagination'' he said, 'You put things well.'\n\nAstha looked pleased.\n\n'And for being so clever...'\n\nHe leaned towards her, and reached under her blouse. Astha pressed him close, and breathed my husband into his waiting ear.\n\n'My baby'' responded Hemant.\n\nAstha heard him with satisfaction. Her husband was going to encourage her writing. Maybe she could become a poetess as well as a painter. Her life was opening up before her in golden vistas.\n\n'Do you think there will be golden vistas in our life, darling?' she asked, taken with the sound of the words.\n\n'Of course, baby'' he replied. 'Golden like your body in the sunlight when it comes through the window touched by the water of this lake.'\n\n'Oh, Hemant'' laughed Astha, 'I didn't know you were a poet!'\n\nHemant looked modest. After they had kissed, fondled and not made love, Hemant told the bearer to take the drink tray upstairs to the roof.\n\nThey reclined on deck chairs facing the lake. The ice tinkled in Hemant's glass, bird sounds tinkled in their ears, water lapped around the boat. They were too high to see the sludge that had gathered around the houseboat, too high to notice the slight smell that came from the stagnant edge. Upon the roof, hand in hand, Astha's heart was as full of love as the lake was full of water.\n\nBack in Delhi, Astha submerged herself in the role of daughter-in-law and wife. The time spent in the kitchen experimenting with new dishes was time spent in the service of love and marriage. Hemant's clothes she treated with reverence, sliding each shirt in his drawers a quarter centimetre out from the one above so they were easily visible, darning all the tiny holes in his socks, arranging his pants on cloth-wrapped hangers so there would be no crease. With her mother-in-law she visited and shopped in the mornings, the memory of the night past, and the expectation of the night to come insulating her from any tedium she might otherwise have felt.\n\nEvery evening her father-in-law remarked, 'How nice it is to have a daughter in the house.'\n\nHemant looked as though it were all his doing, while Astha's mother-in-law sighed and talked of her absent daughters; Seema, so far away in America, and Sangeeta, well, now that Hemant was married, he and his wife were responsible for Sangeeta, whose troubles with her husband and in-laws were always hinted at rather than spelt out.\n\nAstha, proud that she was considered responsible enough to share the family problems invariably replied, 'Don't worry Mummy, she has us'' though she was seven years younger than Sangeeta, and had only seen her at the wedding.\n\nAfter they had had tea Hemant and Astha dropped in on her parents. 'I do not want them to feel they have lost a daughter'' Hemant insisted, as they walked through the colony to Astha's old house, while Astha thought how nice he was, and how lucky she.\n\n'Why do we have to drink tea twice every day?' she complained occasionally, for the pleasure of hearing Hemant say, 'And disappoint Mama and Papa, who are waiting? And when Mama makes snacks especially for us, no fears.'\n\n'Especially for you, you mean'' said Astha.\n\n'It is the same thing'' said Hemant drawing Astha's hand through his arm even more tightly.\n\nIn the kitchen, Astha's mother would hiss 'Happy?' and Astha would give the slightest non-committal nod, wanting to keep her happiness to herself. To share it or voice it might encourage its departure.\n\n*\n\nMeanwhile Hemant immersed himself in sex manuals. He hid them in his cupboard under rows of shirts.\n\n'Mummy might see'' Astha objected nervously. Her mother-in-law frequently visited their room, examined all the items, and straightened the covers on the bed.\n\n'So what?' laughed Hemant. 'We are married, what can anybody say?'\n\nThe number of sex manuals increased. All the books had graphic illustrations.\n\n'Why do you have to read these things?' Astha demanded for form's sake.\n\n'They are interesting. Look.' Hemant tried to show her, but Astha turned away her head, and Hemant did not persist. 'I will show you in other ways'' he murmured in her hair.\n\nAstha blushed and said nothing, too diffident to tell him that she had already noticed a change in his lovemaking, he was less in a hurry, and his focus had widened from the single point of her vagina.\n\nNew positions, timing the length of intercourse, variations on a theme. There seemed no end to what one could do with two bodies. At the suggestion of sexy clothes she balked.\n\n'What do you think I am? A whore?'\n\n'There is nothing to be ashamed of darling'' said Hemant caressing her. 'It is to increase married pleasure.'\n\nAstha looked at the lacy black thing he was offering her. 'What is it?' she asked suspiciously.\n\n'A teddy.'\n\n'So I am to be your teddy bear?'\n\nHemant was not interested in double meanings. 'I went to a lot of trouble to get it for you'' he said.\n\n'For me?'\n\n'Who else is the woman in my life?' asked Hemant, pushing her towards the bathroom. Thank God their room was slightly separated from her parents-in-law's bedroom, thought Astha, and they had a bathroom to themselves. Otherwise there was no way she could do these things. She locked the door and looked at herself in the mirror, clad from throat to ankle, neck to wrist. Diaphanous, lacy, and a soft pink she had all along thought this nightie made her look quite attractive. Slowly she took it off and looked at her body. She was in her hairless condition, the way Hemant liked her, with legs, arms, and underarms freshly waxed, shining smooth, with not an unsightly black stump in sight, only a series of pink bumps where the wax had pulled too hard and left its protest. She raised her arms and anxiously sniffed the wet place underneath. Hemant didn't like the smell of sweat, or vaginal fluids, he was a little squeamish in that respect, and she now washed and dusted herself with powder before turning her attention back to the thing. Single piece, lace and satin, slinky, with holes and slits, she could crumple it in one fist, its only stiffness the wires in the cups.\n\nShe put it on, and there from below her chin, a deep cleavage appeared with black laced mounds on either side, the dark nipples straining through black net hearts. She almost didn't recognise herself, with the sexual parts so emphasised. She raised her arms to take out the pins from her hair, watching as her breasts rose and thrust forwards, feeling an excitement that embarrassed her.\n\nAstha wrapped a dressing gown around herself, and slowly went into the bedroom locking the door quietly. Hemant was lying on the bed with the small bedside lamp on, his arms and chest shone brown and shapely. He kept his eyes on her, as she took off the dressing gown and walked self-consciously across to him, desire rising still higher, trying not to think of what she was wearing, what it was doing to him, to her. She sat next to him, and he grabbed her tightly encased body.\n\n'Sit on me'' he said hoarsely, pulling her on to him, twisting the little bit of lace aside.\n\nAstha sat on him, her breasts tight and forward, falling over him, over my husband, she thought, as they rocked together, while sensation took over, drowning thoughts even of husbands.\n\n*\n\nThe days passed. Astha had not imagined that sex could be such a master. Slightly ashamed, she kept hidden that she longed to dissolve herself in him, longed to be the sips of water he drank, longed to be the morsels of food he swallowed. The times he was away she was focused on one thing, the moment of their union. When he came through the door, she wanted to jump on him, tear his clothes off, thrust her nipples into his mouth, and have him charge his way through her. One with him, one with all that mattered.\n\nI haven't really lived, thought Astha, till now I did not know what life was all about.\n\nShe felt a woman of the world, the world that was covered with the film of her desire, and the fluids of their sex.\n\nA few months and dullness began to taint Astha's new life. What was she to do while waiting for Hemant to come home? Her in-laws were not demanding, for the housework they had help, and supervision, no matter how painstaking, still left her with enough free time to be restless in.\n\n'You need to work'' said her mother.\n\nThe teaching job she had never considered with interest loomed large. Now that she was married, Astha could see that its hours qualified it as the ideal job, a fact her mother was even now pointing out.\n\n'As a teacher you will earn some money, but you will only be out half the day so the home will not suffer.'\n\nAstha looked resentful. Her future suddenly seemed very pedestrian.\n\n*\n\nIt was some evenings later that Astha's mother brought up the subject with Hemant. 'She needs to be occupied, beta.'\n\n'Yes, Ma, I know'' said Hemant. 'I myself was thinking.'\n\n'What about your painting and writing?' asked her father. 'You can make use of these talents in journalism.'\n\nMother and husband expressed scepticism.\n\nAs they walked back through the colony to their own house, Hemant repeated, 'Journalists have to stay out late, they have very odd hours. We must see about a teaching job. You read quite a lot.'\n\n'I don't think that alone will equip me'' said Astha, briefly wondering whether all women were destined to be teachers or nothing.\n\nHemant laughed. 'You will probably know more than anyone'' he said.\n\n*\n\nWith the newly introduced 10+2 system, it was not difficult to get a job teaching elective subjects to classes eleven and twelve. In answer to the combined wishes of Astha's relatives one of her college teachers phoned with news of a vacancy at St Anthony's School, and if she was interested she should go and see the Principal, Mrs Dubey.\n\nAstha's in-laws approved. 'It is a good time pass.'\n\n'It's near enough. You won't have to spend much time on the road'' commented the mother.\n\nHer father merely said, 'It will do until you decide to develop yourself in other ways.'\n\nHer husband said, 'With a job you won't be so fidgety if I am a minute late.'\n\n'Oh, I am to work so you can do what you like?'\n\n'Who says I want to do what I like? It will benefit you to leave the house in the mornings. When the children come we will see whether to continue this.'\n\nAt the interview Mrs Dubey made it clear that a teacher at her school needed to show commitment to the institution, foster students' interests in extra-curricular activities, and make sure they did well in the tenth and twelfth board exams, the reputation of a school unfortunately depending on results. Astha agreed to everything and was hired. Later she thought that since the job fell into her lap, her destiny must be teaching.\n\nBeing a teacher meant the languor of her days was over. No longer did she have the luxury of leisurely brooding over her love, she had to get up early and go to work. She had exercises to correct, and lessons to prepare. She started a reading club, a writing club, a painting club, directed by the principal's suggestions and followed through with her encouragement. The peripheries of her world now stretched to include many schoolgirls. Life was shaping up nicely, with her mind and heart gainfully employed.\n\nHemant dropped her occasionally when she was getting late for morning assembly. Both families exclaimed at his devotion as a husband.\n\n*\n\nA day, as usual, with Hemant coming in late. Astha had been waiting the whole evening, and now took this opportunity to gaze at him, her soul in her eyes, the soul that she was waiting to hand over on a platter.\n\n'How are you, darling?' he asked, looking at her affectionately. 'How was your day in school?'\n\n'They have asked me to edit the school magazine'' she managed, but even those few words were difficult, so heavy was the passion weighing her down. Her tongue felt useless in her mouth, unless it was activated by his.\n\nHe sat down on the sofa, and Astha knelt to take off his shoes. She unlaced them, and pulled off his socks, gathering the day's dust in her lap. At that moment she loved Hemant so intensely, that every fetid, stale, sweaty smell that came from his foot was a further nail in the armour of her love.\n\n'How was your day?' she asked. 'Why are you so late? I have been waiting hours.'\n\n'The director called a meeting'' replied Hemant looking disgruntled.\n\n'At this time?'\n\n'What does he care? Slow, pompous, ass-licking fucker.'\n\n'What has happened now?'\n\n'The latest directives for distributing loans. Our target has been increased, and he is worried we might not make it. Then his head will be on the chopping block.'\n\nOh dear, this was not going to be a happy subject.\n\n'This percentage for cottage units, that for farmers, this for small scale units, that for backward classes, and without any security! No collateral, no third-party guarantor, because the government has to look good in the next election while we bear the losses. How can any bank function in this manner? This is what happens when you nationalise banks, constant meddling and interference.'\n\nHow long would it take for him to notice her? 'I kept thinking of you in school'' she started, but Hemant hadn't finished.\n\n'How are we encouraging any initiative, if these buggers get money for free? And how do you make sure someone is scheduled caste, for fuck's sake? Just a few months ago I had a branch officer complaining that the local bigwig was demanding a larger than usual cut for supplying the bank with certified scheduled caste people. He was falling short of his target and he had to give in. Bloody country, this is why we never progress. In America such interference would be unheard of.'\n\n'Well, this is India, dearest'' said Astha, not wanting Hemant to start on the subject of America versus India. 'This is the way things function. If you get angry, you will only harm your health. My father got blood pressure because he hated his job. Fire burns itself'' she added, a saying she had grown up with.\n\nHemant deflated. 'When I think of how my classmates are doing, how much money they are making \u2013 with an American MBA you can do anything, but there are no opportunities in this bloody country, none. Sometimes I wish I had never come back.'\n\n'Money isn't everything darling. Look, you have your family, me, our parents.'\n\n'Maybe we all emigrate, huh? Seema's husband keeps calling, he's willing to sponsor me.'\n\nLive abroad? 'Yes, let's go'' she said excitedly.\n\nHemant sighed, 'No, Az, I came for Papaji and Mummy, I have to stay. Papaji knows I am being wasted here, and he tries his best to make me happy, but still, what can he do about the job? This is not satisfying work, it is a clearing division, clear this loan, that loan, deal with union demands and government meddling, nothing is allowed to become efficient.'\n\nAstha's desire receded. She felt cold, dreary, and distanced from him. She had been waiting for him all day, thinking of their being together, but nothing of this was reciprocated. He was a criminal, destroying her anticipation, ruining her happiness.\n\nHer subservient position struck her. She had no business kneeling, taking off his shoes, pulling off his socks, feeling ecstatic about the smell of his feet.\n\n'What's the matter, darling?' said Hemant as her hands stopped moving. He reached out and ran his fingers through her hair. 'Leave my shoes, I'll do it.'\n\nHe got up, put them away, and catching her by the elbow sat her down next to him. Poor man, thought Astha softening, he must have had a hard day in the office, was that anything to mind? She must make his home a haven for him, not a place of recrimination.\n\n'So what were you saying about school?' he asked, passing his hand down her back, gently pressing the dividing line between her haunches.\n\nAstha sidled closer to him, and the pressure became a little firmer. 'I kept thinking of you'' she whispered. 'I missed you every minute.'\n\n'Baby'' he murmured, accepting this as his due. 'And school, how was that?'\n\n'Well, they have asked me to help with the school magazine, as I am the teacher for the senior elective English classes. And I thought, why not?'\n\n'Do they know you write?'\n\n'Of course not. Anybody with reasonable English is enough for this job. My class XI girls got really excited, they want to organise a creative writing competition. We can publish the best poems and stories, maybe even send them to the children's page in the newspaper.'\n\nHemant wasn't really listening. Astha stopped talking about creative writing as he got up to lock the door.\n\n'They are waiting'' objected Astha.\n\n'Just a quick one'' said Hemant.\n\n'They will know what we are doing'' said Astha, already imagining what was to come, even if it was a quick one.\n\n'Let them know. We are married.'\n\nAstha lay back, aware of every inch of her skin, aware of every thread she wore, now about to be dislodged. The day, with its petty vexations flowed away from her. This, what was going to happen, was the central thing in her life.\n\nThe last year of Astha's father's service drew to a close. They would have to leave their house soon. Hemant threw himself into their plans, politely suppressing his surprise at their unworldliness.\n\n'Az'' he said frequently to his wife, after visiting his in-laws, 'how come Papa didn't plan more for his retirement?'\n\n'He was planning'' said Astha hopelessly, 'in fact they were always planning.'\n\n'Then, what happened?'\n\n'They kept trying to buy, but it was always too expensive. Then this housing society thing came up and they were allotted land trans Jamuna. They thought once the bridge was built and prices went up, they could sell the plot and buy a small flat this side.'\n\nTo Astha now, this seemed like not very much planning.\n\n'As an investment, Az, this is not good strategy'' said Hemant, banker. 'The bridge is nowhere in sight, you can't depend on government promises.'\n\n'Well, I don't know, that is what they did'' said Astha pettishly.\n\n'They can still live on it, though. People are building, after all. Then when prices rise, their property will be worth even more.'\n\n'How can they? It's still so undeveloped it's not safe. No infrastructure, no nothing. You should see it, it's just a patch of mud. In one of the nearby colonies, the owner was alone in the house when dacoits broke in, stole everything, and beat him till he almost died. You want this to happen to them?' Astha's voice rose slightly.\n\n'Now, now, baby, don't get upset, of course they shouldn't go if it is not safe. We'll help them all we can.'\n\n'OK'' said the wife, feeling momentarily soothed, pushing away the knowledge that it is one thing to offer help, another to give it, still another to take it, and that her father was a very proud man.\n\n*\n\nAstha's father retired, and in six months they had to vacate the house in Lodhi colony. It had been central and inexpensive, the rent 10 per cent of the father's salary. Now they were thrown to the outside world. While the mother was at school, the father trudged around various colonies with property dealers. The private colonies near Lodhi Colony were all too posh, there was no question of trying Sundar Nagar, Golf Links, or even Defence Colony, where army officials had bought plots for a song not so many years before. Finally he found a small two-room apartment in Jangpura. Its advantage was a large terrace, its disadvantage that it was on the first floor.\n\n'You will have to go up and down everyday'' said Astha, 'are you sure your health can take it?'\n\n'Climbing stairs is good exercise'' said the father.\n\n'When you have high blood pressure?'\n\n'I will be all right.'\n\nWhat choice did they have? The flat was comparatively cheap, the location comparatively central. The landlord was kind, only demanding three months' rent in advance, not insisting on a company lease.\n\n*\n\nDismantling the house in which they had lived for fifteen years was not easy. They took the furniture the new flat could accommodate, the rest they sold. The father's books were put in boxes which were then placed so as to make two beds and a living-room divan. They would still be with him, that knowledge would have to replace the pleasure of seeing them every day. The bed linen, the small pieces of bric-a-brac that Astha's mother had stored through the years were given to the daughter. 'I don't need them anymore'' she said.\n\n'But these are brand new'' said Astha looking at the carefully preserved things, wrapped in soft, old saris. 'Why don't you use them?'\n\n'No, no, I do not need'' insisted the mother.\n\nHemant helped them to move. 'I don't like asking him to do so much for us, beti'' said the father.\n\n'He is your son-in-law, Papa. It is all right'' said Astha.\n\nAgain they had no choice.\n\n*\n\nIn the small flat, near the highway, noisy, confined, far from tree and grass, alone for half the day while his wife was at work, eating things he was not supposed to, the father wandered through his life, looking at what was left behind and what lay ahead, and decided there was no use living. Other people decide that with less success.\n\nThey had been in the new flat a little over a year when one evening after dinner he complained of a slight chest pain. That night he died in his sleep. Through the period of shock and mourning, Astha and her mother clung to each other.\n\n'It was the move'' the mother kept sobbing. 'He was never the same after he retired.'\n\n'He was a saint'' said the relatives. 'Never liking to trouble others.'\n\n'I kept telling him, do not strain, do not exert yourself, but no. He was never careful. And now he has left me and gone.'\n\n'You have me, Ma'' said Hemant.\n\n'Yes, Ma. We are all with you.'\n\nAs consolation to the widow, now all alone, the relatives said, thank God he saw his child settled, he will rest in peace.\n\n*\n\nIn the months that followed the father's death, the mother became listless and withdrawn. The evenings Astha spent with her she would desperately try and cheer her up.\n\n'You are still young, Ma, still working. Think of all the things you can do.'\n\n'You don't worry about me, beti'' said the mother dully.\n\n'You can travel, you can do social work, you can do something for the children of the poor, you always said you wanted to help other people. Now you can.'\n\n'Han, beti. You don't worry about me.'\n\n'But I do worry. Why don't you come and live with me?'\n\n'You live with your in-laws, and besides where is the room in these government flats.'\n\nThat was true enough.\n\nAstha tried to interest Hemant in the problem of her mother. He was a good son-in-law, everybody said so, his own parents in particular, closely echoed by the mother-in-law herself. If there was an illness he would call the doctor, if she needed money he would offer it, if she needed help in shifting he would provide it. But appeals beyond this irritated and annoyed him.\n\n*\n\nThen the mother met a swami. She informed her daughter of this casually.\n\n'A swami?' repeated Astha, puzzled. 'How did you meet him?'\n\n'One of the teachers in school took me. Often she has mentioned him, but when your father was alive I never felt the need for anything more in my life. Also he was suspicious of this kind of thing, your father always thought he knew best.'\n\n'With reason, Ma. Swamis are known to take advantage of women, especially widows'' said Astha.\n\nHer mother ignored this. 'Why don't you come?' she went on. 'He lectures on the _Gita_ at Gandhi Bhavan. He teaches you how to accept things, how to look inside yourself, how to deal with your wants and desires. There are lots of young girls there.'\n\n'I don't want to look inside myself'' said Astha.\n\n'Well, I am learning a lot from him. Through him I understand the _Gita,_ it is something I have wanted to do all my life.'\n\n'Really? How come I didn't know?'\n\n'Where was the time or place to say I want this or that?'\n\n'And now you have a swami? Is that what you wanted time for?'\n\nAstha's mother looked offended. 'Why don't you come and see before you start your criticising?'\n\n*\n\nThat evening Astha said to Hemant, 'Ma has found some swami. She wants me to go to him and look inside myself.'\n\n'Rubbish. These people just try and sound clever.'\n\n'That is what I said.'\n\n'Who is this man?'\n\n'I don't know.'\n\nHemant looked alert. 'One has to be careful around swamis'' he said. 'Thank God I am handling her money.'\n\n'I know'' said Astha, her wretchedness increasing. 'But what can I do?'\n\n'Somebody is putting ideas into her head. People think old women are easy targets.'\n\n'She doesn't listen.'\n\n'Don't worry sweetheart, once we have a child, she will forget all this nonsense. There will be a new interest in her life.'\n\nAstha smiled her agreement.\n\nLoving Hemant as she did, Astha longed to get pregnant. During sex she imagined his seed spurting into her womb; later she would gather his wet shrivelled penis, adoring it strong, thick and hot, or wet, limp and woebegone. 'I want to have your baby'' she would murmur.\n\n'You can't be so old fashioned'' remonstrated the progressive husband. 'This is like villagers, marry, impregnate wife, pack of children. No, no sweetheart, we need to be by ourselves. Time enough for these responsibilities later. With a young wife one can afford to wait.'\n\nAstha looked at him in admiration. Everything about him was so masculine, his decisiveness, his hairy blunt fingers, his tall heavy set figure, his muscled limbs, his moustache that tickled, the bitter tobacco taste from his tongue.\n\n'These ideas are all from America'' said his parents, refusing to see the value of bonding time for the young couple. They had married, now they should get on with it.\n\n*\n\nIt was two years before Hemant relented, two years before Astha could stop using birth control, two years before his seed found its home.\n\nAstha and Hemant drove to Jangpura on their weekly visit, full of the good news.\n\n'Ma'' said Hemant, 'You are going to be a nani.'\n\nTears filled the mother's eyes. 'If only he had been here'' she said.\n\nAstha thought of her father and felt sad. He had sent her forth, and then left, duty done.\n\n'Ma, this is a time to celebrate'' reproached Hemant.\n\n'Beta, you are right. May it be a boy, and carry your name for ever. A great son of a great father.'\n\nAstha thought her mother was overdoing it.\n\n'But Ma, I want a daughter'' said Hemant.\n\n'That's true, Ma'' repeated Astha, 'He wants a girl.'\n\n'In America there is no difference between boys and girls. How can this country get anywhere if we go on treating our women this way?'\n\nThere was no mistaking the admiration in both women's eyes.\n\n*\n\nAstha enjoyed every aspect of her pregnancy. As it advanced, she became more and more bucolic. Teaching was an effort, and she had no energy for any extra activity. At home she slept most of the time.\n\nHemant adored what was happening to her. 'My wife is becoming a woman before my very eyes'' he said passing his hands over her belly, large and full, over her breasts, certainly larger and fuller than they had ever been. 'I hope they remain like this'' he said holding them possessively.\n\n'What'll happen if they don't?'\n\n'Another baby, what else?'\n\n'You'll get tired earning for all these children you plan to produce.'\n\n'With you looking like this, never'' declared Hemant passionately. 'A real woman rather than a girl.'\n\nAstha had heard men were revolted by the way women looked when they were pregnant, but not Hemant. He loved touching her belly and breasts, her breasts especially, sucking on them experimentally, drawing a little milk when he sucked long enough.\n\n'It's very sweet'' he said with surprise.\n\n'It's called colustrum'' she informed him knowledgeably. 'It comes for the first three days, and is full of nutrients to prevent the baby from getting sick.'\n\nHemant smiled, 'How full of information my wife is'' he said. 'Where did you find that out from?'\n\n'Books.'\n\n'Our baby will be the best looked after baby there is'' said Hemant, caressing the taut stomach, gently stroking the raised belly-button, following the linea niger down to her pubic hair with his fingers, before inserting them into her vagina.\n\n*\n\nAnuradha. Born in March, after fifteen hours of labour at a private nursing home. Six pounds, eight ounces, nineteen inches. Long delicate nails, a head of thick black hair, pink, wrinkled, foetus like.\n\n'Oh'' chorused the new grandparents. 'Just like Hemant. Same nose and forehead.'\n\n'Such a straight little nose'' detailed Astha's mother, 'such big eyes. Handsome like her father. Girls who look like their fathers are lucky.'\n\nHemant leaned over the tiny baby and kissed her cone-like dome enthusiastically. Astha thought with amazement, he doesn't see through my mother's flattery, before tightening her own hold on the child.\n\n*\n\nThe first time Anuradha put her mouth to her mother's breast and started pulling, Astha was astonished. Hemant's own pullings were nothing in comparison, mild as the winter sun. Anuradha meant business. She tugged ferociously, and Astha's womb in response obligingly contracted, spurting out blood into the pad she wore.\n\nA month of wet before the blood ceased to come, before the womb had contracted all it was going to. A time of swollen aching breasts charged with milk that dribbled constantly, soaking the towel inside her nursing bra, staining her clothes, a time when she had to beg Hemant to drink from them to relieve the pressure.\n\nHemant always willingly obliged, putting a gentle mouth to the tight breast with its blue veins now clearly marked. 'It's very watery'' he said the first time, surprised once more.\n\n'Is it?' asked Astha, 'Let me see.' She cupped a hand under her nipple to catch a drop of the still-oozing milk, and tasted it. It was sweet and watery, bluish-white in colour. 'I guess we are used to cow's milk, which has more fat. That is meant for calves, this is meant for humans'' she explained pedantically, her new-found knowledge still burgeoning in her mind.\n\nAnuradha yawned in her sleep, and made wuffling baby sounds while both parents gazed at the little variations in her movements, with a joy that spilled into each other.\n\n*\n\n'Darling'' said Hemant one night.\n\n'What?' said Astha preoccupiedly. Anuradha was six months old now, and had just begun to sleep through the night. Astha was looking forward to sleeping through the night too, something she felt she had never appreciated before.\n\n'Where's the teddy?'\n\n'What on earth for?'\n\n'I wonder how it looks. It's been a long time since you tried it on.'\n\n'It's been a long time since I had a figure'' retorted Astha.\n\n'You have a figure'' said Hemant, gazing upon his wife's fullness appreciatively. 'Go on, try it'' he urged, pushing her stubborn form towards the bathroom.\n\n'No, no, I don't want to'' expostulated Astha.\n\n'Why? You think because we've had a baby, our life is over. I haven't touched you in months.'\n\n'I know, I'm sorry. Soon it'll stop hurting. And our life isn't over, if by that you mean sex, but it's not necessary to have sex with that thing on, is it? What'll happen to Anu's subconscious? She might grow up with a problem.'\n\n'Look at her. She's totally unconscious. How do you think half the country fucks? You think they have separate rooms?'\n\nAstha knew they didn't. She didn't like the leer on Hemant's face, but she could think of no more reasons for objecting. What could she say? That she was too old? She was twenty-five. That the early days of their marriage were over? They had been married three years. That Hemant should want her without her prancing around in a tight black cut-away garment? But she had worn it before, she had been turned on herself, wasn't she being rather prudish now? She threw a glance at the baby, maybe she was waking? But no, Anuradha slept peacefully, while her mother made her way slowly to the drawer where the teddy was hidden en route to the bathroom.\n\nShe pulled it on. Her breasts spilled over the top, and looked more voluptuous than they were. That was all very well, thought Astha, but the sight of her stomach bulging through the shiny stretchy lace see-through stuff, that sight was not pretty. Also she hadn't been so regular about her waxing, there was hair growing all over her limbs.\n\nThis'll put him off teddies for ever, thought Astha, surveying herself in the mirror, a little regretful that her body should have this deterrent effect. Finally she wrapped his dressing gown around her waist and emerged complaining, it's so tight, look darling it doesn't fit, I'll never be my old self again.\n\nHemant saw her point. The teddy was put away and never mentioned again.\n\n'Once we build our new house, we can start planning for our next child.'\n\n'Um'' said Astha absently, handing her husband the baby oil. Hemant poured a little into his palm and began carefully rubbing it on his daughter, her bath part of his Sunday morning ritual. He insisted on doing this, ideas about fatherhood are so antiquated in India.\n\n'I want to have my son soon'' declared Hemant, looking emotional and manly at the same time. 'I want to be as much a part of his life as Papaji is of mine.'\n\n'How do you know we will have a son?' asked Astha, feeling a little scared.\n\n'Of course we will have a son, and if we don't we needn't stop at two.'\n\nAstha silently took the oil bottle from him and closed it.\n\n'Is the water ready?'\n\nHis wife hastily tested the water in the bath-set crammed into a corner of the bedroom. 'Yes'' she said.\n\nThe father gently lowered his daughter into the water, while the mother stood ready with the shampoo, rubber toy, and soft towel.\n\nAfter the bath Astha called the servant to mop the floor and throw out the water while she hung the towel, disposed of the oil, comb, powder, toy, dirty diaper and night clothes. She then settled down to nurse the baby while Hemant went on discussing their house and their future.\n\n'Hemant?' said Astha after a while.\n\n'Yes?' replied Hemant engrossed in the soft feet and tiny legs of his child.\n\n'I thought these things didn't matter to you. What if we don't have a boy?'\n\n'Of course they don't matter to me. I was so pleased Anu was a girl. But that doesn't mean we should not try for a boy. I am the only son.'\n\n'It is not in our hands, at least not in mine. It is the man's chromosome that decides the sex, and with two sisters in your family, it may be a girl. I have read about these things.'\n\n'You are always reading'' said Hemant coldly.\n\n'I am sorry. Does it bother you?'\n\n'It fills your head with unnecessary ideas. Let us first not have a son and then we will see. Keep it simple. All right?'\n\nAstha looked dissatisfied but could think of nothing to say.\n\nIn the family she had married into Astha had ample opportunity to witness how the business of building a house and planning for retirement should be gone about. Papaji's ministry's housing society, Papaji's rank, Papaji's draw had achieved for them 633 square yards in Vasant Vihar. And for the next ten years the family watched in amazement, satisfaction, and smugness the rate at which their initial investment of twenty thousand rupees multiplied five hundred times over.\n\nAstha's marriage entitled her to the same emotions. This is what my parents hoped would happen to them, she thought wistfully every time the latest price of their plot was discussed, and it was discussed many times.\n\nVasant Vihar too was once wilderness, home to rabbits, peacocks and deer, but by the late seventies almost a third of it was under construction, a boom which the Vadera family now joined.\n\n*\n\nFor the plans Papaji contacted the chief architect of the New Delhi Municipal Corporation, who enjoyed the same secretary level status he did. A senior teacher of the Delhi School of Planning and Architecture was recommended, drawings were made, their relative living convenience minutely examined.\n\nThe house was going to be double storied, the ground floor for Hemant and Astha, the one above for Papaji and Mummy. Each floor would have a drawing-dining, kitchen, two bedrooms with attached baths, and a small study to double as a guest room. In the centre, overlooking a patch of lawn running on the side of the house, would be an open informal area where the family could congregate. There would be one large verandah beyond the drawing-dining, and small balconies outside the bedrooms. On the roof would be the servant's quarters.\n\n*\n\nA puja was done at the site, and the building started. Steel and cement could only be obtained on quotas, and construction on the house lasted nearly two years, despite Papaji's contacts, as thirty tons of rolled steel bars and thousands of bags of cement were released in dribs and drabs by the concerned ministry.\n\nPeriodically Hemant and Papaji would go shopping along with the contractor. To GB Road for cast iron and galvanised iron pipes, toilets, taps, stainless steel and ceramic sinks, wall tiles and marble chips; to Bhagirath Palace for mild steel conduit pipes, electrical cables for light and power, switch boxes, switches, fans, hinges, door locks and door handles; to Paharganj for wood and plywood; and last of all to Kotla for glass and paint, Snowcem for the outside, oil bound distemper for the inside, lime wash for the ceilings.\n\nTwo to three times a week Hemant visited the site, he was a junior officer, and he didn't have the pressures Papaji did. Sometimes Astha accompanied him, audience to Hemant's sense of himself as the child of fortune. 'This \u2013 this'' he said waving his hand at their plot, 'this is worth over a crore today.'\n\n'A crore?' breathed Astha. 'So much.' And she warmed with the pleasure of being part of a family that was in tune with the ways of the world. Now and for ever she would be looked after.\n\n*\n\nTo avoid death duties, the five Vaderas were registered as co-owners, with a letter of intention signed amongst them as to future rights. Hemant was to get the ground floor; Seema, who had contributed dollars towards the construction, was to get the first floor; and Sangeeta, who had contributed nothing, was to get the terrace, which allowed a built up area of 25 per cent. Should either of the sisters wish to sell they had to give their brother first offer.\n\nAfter the house was built, it was given on rent to an embassy, at over a hundred times the rate they paid for their Lodhi Colony government accommodation. Astha's mother listened to the details of the increase in the family finances with glistening eyes, sighing heavily, blessing her daughter, remembering her departed husband, a very simple man, with no sense of this world.\n\nThe two-year excitement and absorption of building a house over, Hemant began to get bored. On his way home from work he took to frequenting the club where, swimming, playing tennis or drinking, he met men like himself.\n\nThey were a new breed, these men. Their fathers had opted for the security and prestige of the civil services, but the sons wanted challenge and money. Educated abroad, their idealism had been exercised in their choice to return to India, now they wanted tangible returns for that sacrifice. Certainly Hemant did. He decided to try his hand at business.\n\n'Isn't it terribly risky?' asked Astha nervously. 'Business is full of bribes and corruption, headache and uncertainty.'\n\n' Az, this is the thinking of the past. Maybe a government job was all right for our parents, they wanted to serve their country after Independence. And perhaps it once was the place where you could make a difference, but no longer. The inertia, red tape and small-mindedness kills you. Now people sit on their asses and push files around all day. As an entrepreneur you can see the result of what you are doing. And it generates work.'\n\n'But we are comfortably off, you have a secure position, your work is not demanding. Even now, we spend so little time together, what will it be like with longer hours?'\n\n'I miss you too'' said Hemant absently. 'But I am not starting the business immediately. I can get a loan more easily if I am at the bank, and the company, my dear, will be registered in your name.'\n\n'And what will I be doing?' inquired Astha.\n\n'Making TVs.'\n\n'TVs? What market is there for TVs? All you get are rubbishy programmes, like _Krishi_ _Darshan,_ _Chitrahaar,_ and half an old black and white Hindi movie on Saturday with the other half on Sunday.'\n\n'You wait, Az, TVs are the thing of the future. In developed countries, TV has taken over the culture, and here too, when colour comes to India... ' He paused and, stirred by his vision of the future, put his arm around his wife.\n\nHis wife had less imagination. 'What will happen?' she asked.\n\n'Do you know how much profit margin there is on a colour TV?'\n\n'No, I don't know, and what's the point, there is no colour, even if we do make the sets.'\n\n'You wait and see.'\n\nWell, thought Astha, at least we have the security of the house if anything goes wrong.\n\n*\n\nHemant applied for a plot of land with the Uttar Pradesh Industrial Development Corporation, and he was allotted one in the ambitiously called 'Electronic City' of Noida, Sector 16. For this as yet undeveloped piece of property he had to pay nine lakhs in installments, with ten per cent down payment. His connections in his bank made applying for a loan easy, a few trips to Lucknow, and the loan was routed through the Noida branch. He registered his wife's unit as a small-scale industry, something that Papaji's position in the commerce ministry facilitated.\n\nAlong with other erstwhile factory owners, Hemant waited for Noida to develop, in the meantime hiring a factory. He made his parents board members, and started his unit with a thousand black and white TVs a month. They had the standard 20-inch screen, sold at 1,850 rupees a set, with a profit margin of 20 per cent.\n\nAll this took a year to accomplish. Hemant now left the house every morning at seven to first visit the factory, and then make the long drive to Parliament Street. 'My family comes first'' he would say, as he juggled factory, bank and home.\n\nAstha watched Hemant in his new avatar and felt moved by his grasp of the rules of getting on, by his ability to exploit situations rather than be defeated by them. Because he was her husband this meant that she too would not fall between the wide cracks of the world like her parents had done.\n\nSomewhere along the way Hemant's attitude to Astha changed. She told herself it was only slightly, but it oppressed her. Occasionally she tried addressing this directly.\n\n'Hemant, why is it that we never talk anymore?'\n\n'We talk all the time.'\n\n'About the business, the house, or Anuradha. Not about ourselves. Like we did before.'\n\n'Grow up, Az, one can't be courting for ever.'\n\n'Is it courting to be interested in the other person? Their feelings?'\n\n'Why are you so childish? I work hard all day, and when I come home I want to relax. If you are feeling something, tell me. I have no time for all these games.'\n\n'I want to be close to you, have a better relationship\u2014' faltered Astha, knowing she had lost the argument before she had been able to define its parameters.\n\n'There is nothing wrong with our relationship.'\n\n'Are you saying there is something wrong with me?'\n\n'You said it, not I.'\n\n'But I'm not happy, so how can you...' She bit back words that might seem to indicate some insensitivity on his part.\n\n'You think too much, that is the trouble.'\n\nAstha stared at him nonplussed. 'I love you'' she said lamely, but she meant something else.\n\n'I know, baby, I know'' said Hemant, drawing her to himself, caressing her. 'Maybe we should go out together more? Would you like that?'\n\n'What about Anu? I don't like leaving her with Mummy so much. She looks after her when I am at school as it is.'\n\n'We'll take her, you are the one with all the scruples. Come on, darling.' He slipped his hand under her sari, undid the first two hooks of her blouse and slid his hand over her breasts.\n\n'Poor little things'' he cooed, 'Have I been neglecting them?'\n\n'It's not that'' murmured Astha.\n\n'Cheer up, baby. Make it nice for me to be with you.'\n\nBaby. That is how he liked her. The look on his face became focused as he pulled her sari palla away and yanked at the rest of the hooks on her blouse, drawing it down from her shoulders and arms. Now he would bury his face in her breasts, pressing them against himself from either side, suck on her nipples, and they could both be babies together.\n\nShe found this soothing, and later scolded herself for being so demanding. Hemant was busy, Hemant was building their future, she had to be adjusting, that was what marriage was all about.\n\nWhen Anuradha was four, Papaji retired. The tenants left, the family moved into their Vasant Vihar house, and Astha conceived again.\n\n'God willing it will be a boy'' said her mother. 'I have asked swamiji's advice as to what offerings to make.'\n\n'Nonsense, Ma'' retorted Astha uneasily. 'These people are not like that.'\n\n'You are still such an innocent. What people say and what they do are two different things. Besides why is Hemant working so hard? For whom, if not his son?'\n\n'It doesn't matter to Hemant'' said Astha valiantly.\n\n'I hope for your sake you are right.'\n\n*\n\nA few nights later. Hemant laughing, 'Mummy is so sweet.'\n\nHemant often found the things his mother said or did sweet, so Astha paid not much attention. 'She is hiring a pundit to come every day and do some special pujas.'\n\n'Why?'\n\n'To ensure a grandson.'\n\n'But puja may not make a difference, it may still be another granddaughter'' objected Astha in alarm.\n\n'Don't worry, sweetheart, then we will try again, it's perfectly all right. Why do you get so tense for nothing?'\n\n'But Hem, I do not wish to go on trying and trying until we get a son. It's very difficult with the teaching as it is.'\n\n'Oh-ho, what is there in teaching? Hardly a serious job, you just go, talk to some children about poems and stories, organise a few clubs, and come back. If you do feel it is important, all the more reason not to mind if Mummy does some puja. Who knows it may yield good results?'\n\nBut Astha did get worked up, she couldn't help it. She tried to stay calm for the baby's sake, she took to meditation, she concentrated on peaceful thoughts. But she was not allowed to forget that everybody, her colleagues, her in-laws, her husband's friends' wives, her mother, the cook, the gardener and the part-time help all had an opinion about her baby's gender, and that almost universal opinion was that it would be a son and heir.\n\n'Baby, it's you they want to be a boy'' Astha would whisper sometimes, 'are you a boy or a girl? I'll love you no matter what'' and she soothed the foetus she imagined so troubled with her troubled hands.\n\n*\n\nWhen Astha's son was finally born she felt a gratitude as profound as it was shamed.\n\n'The family is complete at last'' said Astha's mother piously, feeling her own contribution.\n\nHemant's mother agreed, too happy in the birth of her grandson, carrier of the line, the seed, the name, to respond with her usual reserve to someone she increasingly felt was her social inferior.\n\nThe naming ceremony of the boy was carried out on a much grander scale than that of Anuradha's. Caterers were called, and they came early in the morning, setting up their fires in the narrow driveway. The priests arrived for an elaborate puja and havan. The letter taken out for the baby's name was 'h'. An auspicious sign, same letter as his father said everybody, and he was christened Himanshu.\n\nAstha was given gold jewellery and a new sari. Anuradha and the child's aunts were given gold necklaces. The newborn was given gold guineas.\n\nAstha was officially declared the mother of a son. Her status rose, and she pushed from her mind thoughts of what might have happened had she been unable to do her duty.\n\n*\n\nHimanshu was two months old when he raised his wobbly head from his mother's chest to smile at her, wet pink lips stretched over little toothless gums. Astha thought, he recognises me, and she smiled back, silently, across her chest, this human being and her connected. The baby, trying out the strength of his neck, began to laugh, which made Astha laugh too. Happiness flowed through her like a river, lapping at her mind. She never forgot this first exchange, it lived on in her memory, a link between a male and her that was joyous, simple, and unproblematic. So what if it was with her two-month son.\n\nAstha often looked at her family, husband, daughter, son. She had them all. She was fulfilled. Her in-laws frequently commented, 'Woman is earth'' and it is true she felt bounteous, her life one of giving and receiving, surrounded by plenty. Visitors to the house would say, 'A mother's love' and then trail off, words collapsing into significant silence, which in turn washed over Astha and made her feel that she had partaken of the archetypal experiences marked out for the female race. \nChapter III\n\nBetween Anuradha's birth and Himanshu's, Hemant changed from being an all-American father to being an all-Indian one.\n\nAfter he came home the last thing he wished to bother about was taking care of a child.\n\n'It's your job'' he said.\n\n'That's not what you thought when we had Anu'' replied his wife. 'I can't do everything myself. It's tiring.'\n\nIt was also boring, though this was not acknowledged.\n\n'It's woman's work'' said Hemant firmly. 'Hire somebody to help you, or quit your job.'\n\n'This is our son, the one you wanted so much. It's nice if we look after him together.'\n\n'Send him up to Mummy if you can't manage.'\n\nAstha was struck dumb. Were Mummy and he interchangeable?\n\n'And'' continued Hemant, 'my son is going to be very lucky for us.'\n\n'Oh Hemant, how?' asked Astha with an effort that wasn't noticed.\n\n'Wait and see.'\n\nHemant had invested in the future with his TV project, and was now about to witness the fruits of his foresight. Three months before the Asiad of 1982, the Minister for Information and Broadcasting declared that India would go colour: we have a certain dignity to uphold, an image to project. The games will be beamed internationally, conveying the pomp and splendour, hopes and aspirations of a developing nation. How can all this be done in black and white, when colour technology is prevalent worldwide? It is a question of marching with the times.\n\nThe Left protested: such a priority is elitist, false and a waste of precious foreign exchange. When the nation is still poor and backward, when electricity, water, roads, education and basic health care have yet to reach hundreds of villages, why should we develop a totally useless technology that will neither feed nor clothe?\n\nBut whether the technology was useless or not, whether it would help the nation or not, it was there to stay. Hemant now needed to travel to South East Asia; the indigenous black and white TV was possible to piece together locally but colour expertise was still not available in India. He resigned from the bank and security, to devote himself full-time to risk and money.\n\nHemant placed his first order in South Korea, for twenty thousand colour TV kits. Along with the order came a manager to train the workers. Local contribution involved the assembly of the TVs, the wooden cabinet, testing, selling and service. The final product was advertised as manufactured under foreign supervision, long after the initial foreigner had left.\n\nFour times a year Hemant travelled. The glamour of international references entered the house, as he flew to South Korea and Japan looking for the best deals. He always went alone, always made sure his trips included at least two weekends, which he claimed he needed in order to establish personal contacts. He invariably came back in great good humour, with generous presents for everyone: perfume, chocolate, sweaters, jeans, toys, Japanese dolls, games for the children, underwear for Astha, toiletries, soaps, creams, shampoo, kitchen and electronic equipment. Gradually their house acquired the gloss of a house with money.\n\n*\n\nAstha was now virtually a single mother. Beleaguered by job, small children and house, she sometimes toyed with the idea of resigning from school, but between her marriage and the birth of her children, she too had changed from being a woman who only wanted love, to a woman who valued independence. Besides there was the pleasure of interacting with minds instead of needs.\n\nAt school she had grown to be her principal's right-hand woman, appreciated and valued for one tenth the work she did at home, and paid for it too. Her salary meant she didn't have to ask Hemant for every little rupee she spent. With two children, family obligations, entertainment and holiday costs, the travelling involved in a new business, the uncertainty of business itself, rising prices, she knew Hemant would prefer her to bear her small expenses herself. As it was he spent enough on her clothes and jewellery that she always looked well turned out.\n\nAnd so the once looked-down-upon job had become dear. She couldn't leave it. Nor could she go on relying too much on her mother-in-law for help with the children, it led to remarks from mother-in-law to Hemant to Astha which left her seething with anger and resentment.\n\n*\n\nThus began the search for a maid. A succession of women filed through their flat, but they either came with large families, or they had insufficient references, or they stole, or they were lazy.\n\nHemant felt Astha was guilty of mismanagement, it could surely not be that no ayah was right? After all he managed a factory with four hundred workers.\n\n'Why can't you train these servants properly?' he demanded.\n\n'I do try'' she said, not liking to acknowledge how inadequate she felt with all of them. 'I was all right with Bahadur, (their cook) and the two part-timers.' (To wash clothes, clean the dishes, swab the floor, and dust the rooms.)\n\n'Then don't hire one.'\n\n'I need someone to help me'' said Astha bitterly, wondering how much her husband really knew of her life.\n\n'Have all the help you want'' went on Hemant carelessly, 'only learn how to manage it.'\n\nThe search continued till Bahadur, their cook, went home to Nepal on annual leave, and brought back a widow.\n\n'My sister'' he said, introducing her laconically. 'See if you like her.'\n\nAstha looked at the woman. She had a broad flat face, slitted eyes and a wooden expression.\n\n'Have you done domestic work before?' asked Astha, beginning with the standard questions, while wondering whether this woman was Bahadur's blood sister, cousin sister or village sister, and whether they were sleeping together.\n\n'Mala knows everything'' said Bahadur interrupting. 'Try her.' There was something about the woman's straight gaze that appealed, and she was employed. Mala's appeal grew when Astha discovered how quick and capable she was. She was fast, she was clean, she needed to be told nothing twice. When Astha and Hemant went out she made sure the children had their meals on time, and that they were in bed by nine. She even made sure Anuradha finished her homework, and this while being illiterate.\n\nMala had some bad qualities. She stole food and clothes, she answered back, she took her time coming from her quarter upstairs, she became deaf when it suited her, and on Bahadur's days off she tended to develop illnesses from which she did not fully recover till he came back.\n\nUnfortunately for Astha this usually happened on weekends, when Hemant was around.\n\n'I am going to fire that bloody woman'' ranted Hemant the last time Mala had fever.\n\n'She can't help it'' defended Astha.\n\n'She is shamming.'\n\n'How can we prove that?'\n\n'She is like this because you encourage her.'\n\n'How do I encourage her?'\n\n'I saw her going out with Bahadur.'\n\n'He said he was taking her to the doctor. Do you want me to take her to the doctor instead?'\n\n'She thinks she can get away with anything.'\n\n'I'm sure shell be all right soon.'\n\n'Where's Mala?' whined Himanshu, who was listening.\n\n'See? You make the children too dependent on her.'\n\n'She helps look after them, it's natural they should like her.'\n\n'You treat her as though she was one of the family. You have to know how to handle servants.'\n\n'I can't behave in any other way.'\n\n'She's shamming'' Himanshu piped up insistently, wanting to be heard.\n\n'She's sick darling, don't you get sick sometimes?' said Astha.\n\nIt was in this two children, husband, servants, job scenario that Astha started to have headaches. Years after she would remember the first time it happened, thinking that as a herald of what was to come, it might have announced its arrival in her life a little more gently, allowing her time to get used to this pain in her forehead, this throbbing at her temples, this stretching of the skin around her eyes.\n\nShe had laid the table for dinner, and they had all sat down to eat when she discovered she had forgotten the water. She rose from her chair, and in that moment, between getting up and standing, in the moment that hung between a bent body and a straightened back, it appeared. Just above her nose, at the inner corner of her eyebrow. She pressed the spot, and the pain promptly shot off in neat lines across her eye socket. It will disappear as suddenly as it came, she thought, carefully pouring the water into everybody's glasses.\n\nThe heaviness in her head increased as she ate. If she didn't lie down soon, she might fall headlong into her plate, banging herself against the table, startling the family.\n\n'I'm going to lie down'' she managed.\n\n'Are you ill?' asked the husband, looking at his wife's wrinkled eyebrows and drawn face.\n\n'I'll be all right. Just a little headache.'\n\nHow the children were put to bed, when Hemant came to the room, Astha did not know. Through the night the pain grew worse. Nausea came upon her, she could no longer stay lying. She got up and sat outside, maybe the cooler air would help. It didn't.\n\nAs she bent to retch in the toilet, she hoped that now she would feel better. But though the queasy feeling gradually subsided, the throbbing was still there. Her limbs were shaking, she had to lie down again. Sometimes it seemed, if she lay on the hurting side, that felt better, sometimes she felt that no, the other side was better, and she kept gingerly turning her head trying to pin the point of meagre comfort.\n\nGradually towards morning, when the sky lightened, and the pain began to recede, she fell asleep.\n\nThe next day, the whole world seemed new. She was still in one piece, that terrible thing had gone. Her head felt delicate, it had gone through some bad times and needed to be treated gently.\n\n'Are you all right now?' asked Hemant, looking concerned.\n\n'Yes, I'm better'' she replied.\n\n'What happened to you?'\n\n'I don't know.'\n\nShe took leave from school and sat around the whole day, not using her eyes to read, not using her mind to think. She dusted and tidied, mindless labour that soothed and kept her busy. She hoped that what had happened to her the night before was a one time thing.\n\n*\n\nSoon it became clear that her headaches had arrived to stay. Stress made them worse, going out in the sun made them worse, sleeping too little, too late made them worse, eating the wrong kinds of food made them worse. Slowly her life changed to accommodate her headaches. She learned to dread each small twinge, was it going to be bad or medium? Maybe she was tired, should she lie down and rest? Or maybe it was anxiety, should she meditate, shut her eyes, ignore the throbbing, clear her mind of images, and focus on a spot of light between her eyebrows? The last was the most difficult, but her GP had said there was nothing physiologically wrong with her, it was all in her mind. He prescribed some painkillers, but they only gave momentary relief, making her dull and drowsy, with greater chances of having a headache the next day.\n\nHer mother took her to a homeopath in her neighbourhood in Jangpura. 'My daughter is not well, doctor, she suffers from tension. Little things upset her, and she gets a headache.'\n\nThe homeopath, a well-known one in that area, looked concerned. 'Tension'' he stated, 'the disease of modern life. The secret of health is a balanced mind.'\n\n'I try and be calm'' said Astha earnestly, 'but still I have headaches, and the pain lasts quite long.'\n\n'Right side or left?'\n\n'Usually right.'\n\n'Front or back of the head?'\n\n'Eyebrows. One or the other, never both.'\n\n'Morning or evening?'\n\n'Any time. Occasionally I wake up with a headache, at other times it comes in the afternoon or evening.'\n\n'Which season?'\n\n'All.'\n\n'Hot or cold suits you?'\n\n'Cold.'\n\n'Sun or shade?'\n\n'Shade.'\n\nEtc. etc. etc.\n\nAstha left the homeopath clutching Sanguinaria and Belladonna, 30X. Four times a day, alternately. Come after two weeks.\n\nShe dutifully took the Sanguinaria and Belladonna four times a day alternately. She kept a diary of her headaches. Once to twice a week. Hemant felt homeopathy was mumbo-jumbo, and took her to an ENT specialist. The specialist looked up Astha's nose and informed her husband that with such a deviated septum, it was a wonder that Astha could breathe properly, in fact if you notice, her mouth is open.\n\nAstha shut her mouth quickly.\n\nAnd of course she is going to have headaches. Time will not improve her condition.\n\nAt the thought of everything going from bad to worse, all power of decision left Astha.\n\nThe family took a second opinion, and surgery was decided upon.\n\n*\n\nAstha was in hospital four days. Her nose was heavily bandaged and hugely swollen. She could hardly breathe. It was not a good beginning to a life of easy breathing, and a head that didn't pain.\n\nHemant spent a part of every evening with her, while Papaji supervised in the factory.\n\n'Poor little baby'' he murmured as he stroked her hand, 'does it hurt?'\n\nAstha nodded, and tears rolled down either side of her bandaged nose. She tried to talk, but then her nose moved, it hurt more, and the tears came faster.\n\n'Baby, don't talk'' said Hemant tenderly. Astha wished to capture his expression in her heart for ever. She looked more beseeching, more piteous, and Hemant pressed a soft kiss under the swollen lump, lingering long on the salty lips.\n\n'How are the children doing?' croaked Astha.\n\n'Do not worry'' said Hemant, head of the household, the type of person his wife could depend on, poor little thing. 'Mala is very reliable when you are not there. She knows she can't try her funny business with me. Besides they love being with Mummy, she thinks they are not dressed well enough, and has bought both of them new sets of clothes.'\n\nAfter he left, 'How good Sa'ab is'' said the day nurse with a sigh. 'Coming to see you every day. Not every husband is so nice.'\n\n'Yes, he is'' said Astha.\n\n'Love marriage?' asked the night nurse.\n\n'No.'\n\n'Arranged is best'' said the night nurse with an even larger sigh, and then proceeded to tell the story of how her husband had first seduced and then married her sister. She could hardly bear to speak to him when he came home at night, that is why she had taken up this job, otherwise she came from a respectable family where the women didn't work, but now what else could she do, it was very bad madam, her sister looked after all the children and ran the house.\n\n*\n\nAfter her operation, Astha came home, waited for her headaches to go and life to become pain free. But the headaches continued, and Hemant was naturally not as attentive as he had been in the hospital.\n\nIf that nurse could see her now, her envy would be greatly diluted, thought Astha as she fretted over absent husband, and often absent children as well.\n\nWhere were they? Upstairs. Five days had been enough to establish this pattern. When she called them down, this was seen as objecting to their being with the grandparents. She tried talking to Hemant about this.\n\n'It upsets the children's routine if they are up for so long'' she protested. 'And if they eat so much junk, their appetite is ruined for dinner.'\n\n'You fuss too much. Besides their Dada Dadi are lonely. They complain they do not see enough of the children.'\n\n'I send them up whenever I can, Hem, you know that.'\n\n'Yes, but you know how it is with old people, they think they have little time left, all rubbish of course, but if it cheers them to have the children, why not?'\n\n'What about me? As it is when I am in school Himanshu is upstairs. When I come home I want the children. I hardly have you, I should have them.'\n\nTears came to her eyes. More tears for Astha, poor thing. She was climbing a mountain, and when she reached the top her face sweating, her heart going at its fastest, all she could see was another mountain. As she gazed at the jagged edges, her head began to ache, and the blood that was pounding in her heart obliged by moving to her head and pounding there.\n\nHemant rolled his eyes, and drew out a handkerchief to dry her face. 'What rubbish'' he repeated. 'It is all your imagination. When don't you have me? You are the one who keeps wanting to stay at home with the children, or your school work, or your books when we are invited to parties, or when I want to go to the club.'\n\n'How can you say that? I always come with you.'\n\n'And hate it, don't deny it. Half these invitations I refuse because of you. I am the one who is lonely, and without company.'\n\nBy what sleight of hand had their problems become identical?\n\nShe continued with her sketching, but found herself scribbling poetry, her father's encouragement more firmly in her mind now than when it was first given. She wrote about gardens and flowers, the silent dark faces of gardeners tending plants and never getting credit. She wrote about love, rejection, desire and longing. The language was oblique, but it was her own experience endlessly replayed.\n\nWriting alleviated the heaviness within her, a heaviness she found hard to deal with. Discussing her feelings with Hemant usually led to argument, distance, and greater misery. In the struggle to express herself she found temporary relief.\n\nAfter Astha had written about two hundred poems, she felt she needed to go somewhere with them. Publication would make her work seem less futile, but how to get there? She started revising them, typing them out on the small portable typewriter Hemant had brought back from the States.\n\nAfter she did twenty she showed them to Hemant. As a man of the world, she trusted his sense of how to do things.\n\n'Poems?' he remarked, looking pleased. 'I didn't know you were still writing.'\n\nAstha smiled and said, yes, she was still writing.\n\nThe last he had seen her poems had been on their honeymoon, he reminisced, while Astha smiled some more. 'That was about a lake'' he went on.\n\n'I don't write about things like that now.'\n\n'You don't?'\n\n'I've lost interest in Nature. I'm older, I think differently'' said Astha.\n\n'But you look as young'' responded Hemant automatically. He put his arm around her for a moment before turning his attention to her writing.\n\nAstha waited nervously. It was the first time anyone was seeing her poems. Hemant frowned, shuffling through the twenty typewritten sheets. To his wife's horror he started reading one out in a puzzled voice:\n\nChanges\n\nThe eventual release from pain\n\nIn the tearing relentless separation\n\nFrom those in habit loved\n\nCan come so slowly\n\nIt seems there will never be a day\n\nOf final peace and tranquillity\n\nWho promised me, that if I\n\nDid gaze upon reality\n\nAccept it, embrace it, befriend it\n\nI would never suffer again\n\nBut no matter how many times\n\nI heave the doorways of my soul\n\nTo let the chill light in\n\nThe darkness grows silently\n\nTo hide me in the break of day.\n\nHemant stared at her. Astha cringed. 'Actually, forget it'' she said. 'They probably need more working over.'\n\n'But I am here to help you'' said Hemant genially. 'I personally thought the one you wrote in Srinagar was very good. I said so at the time, didn't I?'\n\nYes, you did, you did, you did. But now it's all changed, and I want to bang my head against the wall because you never understand anything. 'I thought you might help me in deciding what to do with them'' she said tense and calm.\n\nHemant continued riffling through the papers, sparing her the embarrassment of more loud reading.\n\n'You don't like them?'\n\n'I don't know what to make of them. Look, I am no reader, but they sound rather bleak, don't you think?'\n\n'Do they?'\n\n'Good heavens, Az, they are all about cages and birds, and mice, and suffering in situations that are not even clear. There is not one happy poem here.'\n\n'Poems are about emotions'' defended Astha. Maybe now he would ask her why she felt sad and they could really talk.\n\n'What kind of emotions? This person sounds positively neurotic.'\n\n'I don't think so.'\n\n'If others read these poems, they might actually think you weren't happy.'\n\n'No, no, they are not about me'' said Astha quickly.\n\n'I know that. But people are so quick to put two and two together and come up with five, quick to gossip, you know Az.'\n\n'Perhaps I should test that by sending them somewhere'' said Astha looking down, not wanting to see his face.\n\nHemant looked doubtful. 'Well, I don't know, it's up to you.' He held out the poems and she took them forlornly.\n\nThat night she thought long and hard of 'Changes'. How self-indulgent it had sounded when Hemant had read it out. And this was one she had considered her best, evocative and moving. Maybe he was right, they were all too alike, she would be exposing herself to the world.\n\nShe gave up writing and continued rather sadly to draw, sketching with the soft pencils and coloured charcoal that Hemant got her from Japan. Nobody could put two and two together about painting, say it was negative rather than positive, say she should paint lakes in Kashmir instead of mice, birds and cages. Maybe one day she could do something with her art, but for now her school and herself were audience enough.\n\nThat summer Astha's mother announced, 'I am going to Rishikesh for a month.'\n\n'Why?' asked Astha.\n\n'Swamiji is giving a course.'\n\n'So? You listen to him here, don't you?'\n\n'His ashram is by the banks of a river. It will be a different experience.'\n\n'I think you should stay here,' said Astha uneasily.\n\n'In my stage of life one is free from places. Soon I will be retiring. I have to think of what to do \u2013 where to go.'\n\n'You can stay with us'' said Astha, who had not learned the futility of making this statement.\n\n'Why don't you come too?' asked her mother with equal futility. 'It will help your headaches.'\n\n'I'm all right'' said Astha. She looked at her mother, who was smiling benignly. Astha became suspicious, it was not like her mother to smile, and that too at nothing in particular.\n\n*\n\n'Ma is going to Rishikesh'' said Astha to Hemant that evening.\n\n'Why?'\n\n'She says she is free of places.'\n\n'Very foolish of your mother.'\n\n'Talk to her.'\n\n'I will, as soon as I find the time'' said Hemant.\n\nWhich turned out not to be before she left.\n\n*\n\nFrom the banks of the Ganga in Rishikesh Astha's mother sent her a parcel containing a letter, a commentary on the _Gita,_ and a small booklet entitled _The_ _Purpose_ _of_ _Life._\n\nDearest daughter,\n\nHow are you?\n\nThe air here is pure, and the scenery is beautiful. Hemant, you and the children should come. I will book a room. Everything is on me. It will do you good to meet Swamiji. He is so wise, just seeing him is satisfaction. He is also asking you to come. Everything is on me.\n\nI am sending you two books that Swamiji has written. Read them every day. In ourselves alone is peace. Even when we know how difficult it is to change ourselves, still we expect others to change, and are unhappy when our expectations are not met. Remember that. It will help with your headaches also.\n\nIf you were to hear Swamiji you would realise that to keep a relationship going I should ignore the dark side, i.e. weaknesses of a person. Accept without condition if you want to live in peace. Any relationship can be beautiful if you nurture it. In time of difficulty don't lose heart. Freedom from all complexes is essential. Don't assert your ego \u2013 don't argue. Employ wisdom to solve the problem. You are committed to ME says Lord Krishna.\n\nAccommodation and acceptance keep families together. What you cannot change accept gracefully, cheerfully as prasad for the Lord. Create a home where you are. Such a person is free from sorrow. Every understanding requires composed mind. Worst thing in life is anger. Read the _Gita,_ especially chapter xiv.\n\nWith a thousand blessings for a long and happy life,\n\nMa\n\nAstha stared at this communication. Where did these thoughts come from, what was happening to her mother, a helpless widow, with her child too caught up in the web of daily life to go and free her parent from another web. If only Hemant had talked to her mother, but then why should she rely on Hemant every time.\n\n*\n\nWhen the mother came back after her month in Rishikesh, she made it a point to have her stay over often. The mother prowled around, pointing out the wasteful habits of the servants, the dirt in various corners of the house, the children's thinness and bad eating habits, and Astha's neglect of her in-laws.\n\nReduced to a nervous wreck, Astha took her anger out on the children. 'Don't scold them'' her mother's soft voice filtered unctuously through her shouts, making the children behave worse than ever. 'They are only children.'\n\nHow come, thought Astha resentfully, this thought never occurred when she was young?\n\n'Swamiji has taught me a great deal'' continued Astha's mother, reading into her daughter's silence. 'In the old days I was ignorant. Now I know better. If I made mistakes with you, I do not want you to make them with your children. All too soon this time will go. Let them enjoy their childhood.'\n\nAstha felt hunted. Nothing she could do was right.\n\nHer mother introduced her to Mrs Reddy, short, plump, grey hair pulled back, widowed colleague and original introducer to Swamiji. 'Tell her'' she ordered, 'how much going to the classes will help her.'\n\n'Behen, it is all right. When the time is right for her, she will come herself'' said Mrs Reddy.\n\n'Tell her'' insisted the mother, concerned about her daughter's happiness.\n\n'The Hindu religion'' opined Mrs Reddy, 'is wide, is deep, capable of endless interpretation. Anybody can get anything they want from it, ritual, stories, thoughts that sustain. But first you have to realise your need.'\n\n'She is always so tense and angry'' complained the mother.\n\n'I don't need religion, whatever I am'' said Astha firmly, while the two older women looked sorrowful.\n\n*\n\nThe time came for Astha's mother to retire.\n\n'I must leave this flat, beti'' said the mother. 'It is too expensive for me.'\n\n'Of course you will come and live with us, Ma'' said Astha.\n\nTradition reared its obdurate head. 'What'll his mother think?'\n\n'What'll she think? Nothing. She lives upstairs. It is not as though you are taking away her space. Besides Hemant is doing well enough for one mother-in-law not to be considered a drain on his resources.'\n\n'It doesn't look nice.'\n\n'To whom? To whom doesn't it look nice?'\n\n'To me.'\n\n'I wish you wouldn't be so stick in the mud, Ma. Why didn't you have a son to look after you when you were old, if you cannot take anything from a daughter? Why did you stop with me?'\n\n'I have talked to Swamiji'' responded the mother. 'He also thinks one must be independent.'\n\n'What does Swamiji know? Parents belong to their children.' By now Astha was grinding her teeth with impatience. When had this swami become so important that all Astha was saying meant nothing.\n\n'I am thinking of moving to Rishikesh.'\n\n'Rishikesh? You are going to live there all your life?' Astha was appalled.\n\n'Arre, who knows how long one is going to live? The atmosphere should be pure, one should lead a life of virtue and truth, where and how does not matter.'\n\n'What if you fall ill? Who will look after you? Swamiji?'\n\n'One cannot live in fear'' said the mother severely.\n\n'Nor in isolation. You will be lonely.'\n\nThe mother was silent. So was Astha, what could one say about loneliness?\n\n'Swamiji is insisting that I take my time and think about it'' said the mother finally, 'he is not agreeing for right now.'\n\n'And a good thing too'' said Astha baffled. She felt that in some way she had been tested and found wanting. She envied Hemant his relatively straightforward relationship with his parents. They demanded from him material care \u2013 which he gave, grandchildren \u2013 which he gave, emotional concern and physical presence \u2013 which he gave. Duties, responsibilities, obligations, all seemed clear.\n\n*\n\nA few weeks later Astha's mother gave up the lease of her flat, and got rid of most of her belongings. 'Material possessions are a burden'' she informed her daughter.\n\nHer daughter did not feel the same way. She loved the pretty things that decorated her home, her books, her lamps, her carpets, her cutlery, tableware, linen, furniture, everything that Hemant and she had bought together. Now she wanted to add the twelve boxes of books that had formed the beds and the divan at her parents' place.\n\n'Are you mad? We don't have the room'' declared Hemant.\n\n'We do, we can build shelves.'\n\n'Come on, Az, donate them to a library. We can't clutter up our house with a lot of old books. And you know you don't read them.'\n\n'That's not the point.'\n\n'What is the point? Books are meant to be read, and in a library they will be of use. Better looked after too.'\n\n'Please, Hemu, my father's books.'\n\n'Don't be so sentimental, Az. I will talk to Ma, you will see she will agree.'\n\nAstha's mother agreed to such an extent that the books were donated to a library before Astha even knew about it.\n\n*\n\nAstha was devastated. 'Why did you do that?' she screamed at her mother. 'They were mine as well. I loved them.'\n\n'But you never showed any great interest in them when you were growing up'' protested the mother.\n\n'That was then. This is now. Don't you care about Papa's memory? How could you do this to him, to me?'\n\n'People do not live in their things, beti. Besides'' added the mother, 'it is Hemant's house, and he said there was no room.'\n\n'Then who am I? The tenant? We could have found room, we could have built bookshelves, done something, we could at least have discussed it.'\n\n'You know how much work they were. Every year take them out, dust them, and then they get infested by silverfish, accumulate dust and space. In a library at least they will be read.'\n\n'You sound like him. At the very least I would have kept a few, or do you think I too should not be weighed down by material possessions?'\n\nThe mother sniffed, looked martyred and misunderstood. What was the use of saying anything, thought the daughter, the books had gone, and all the screaming in the world was not going to bring them back. But together her husband and her mother had deprived her of the dearest part of her father, and continued before her eyes to be oblivious of their crimes.\n\n*\n\nAstha's mother was now free to leave for Rishikesh.\n\n'When will you be back?' asked Astha anxiously, as she dropped her mother at the station.\n\n'I don't know beti, let me see how it goes.'\n\n'I wish you hadn't turned to religion, Ma'' said Astha feeling as though her mother had cheated on her, manifesting a strange turn of mind that her daughter could neither follow nor understand.\n\n'We are all looking for peace of mind'' said her mother. 'Swamiji will guide me.'\n\nThe train came and she left. Astha stood on the platform and watched her mother leave the city where she had spent all her working and married life. Now with just a bedroll and a trunk she was embarking on a pilgrimage, searching for a community she could call her own, with no possessions to weigh her down.\n\n*\n\nThe months passed. Astha's mother showed no signs of returning. Her letters, about love, peace, renunciation and knowledge, revealed nothing.\n\nDear Beta,\n\nPerform action with the full understanding that you have no control over the result. Success and failure have to be faced by everyone. By being thoughtful, reflective and prayerful we can overcome the spirit of 'I'ness that dominates all our actions. This approach keeps families intact and we don't become insecure. We have a set up to relax in, this paves the way to security, and to self understanding.\n\nThe meaning of life is struggle. There are challenges in all walks of life, how to tackle them is the question, not to run away from home, work, society and obligations. Perform your duties with detachment. Learn to give and not take. When you develop the spirit of giving intelligently, there is peace in the mind. Most of our problems are due to discontent with what we have.\n\nGive my love and blessings to dear Hemant, Anu, little Himanshu, and to your mother and father-in-law. With many more blessings to my dearest daughter,\n\nMa\n\nOnce or twice Astha asked Hemant, 'Won't you go and see her, convince her that her place is with us?' but Hemant was clearly not concerned enough for action. Astha's suspicions hardened, maybe her mother was right, it would not be so good for her to live with her daughter. She wished she had a house that was more clearly hers.\n\n'I need to go and see my mother'' she finally said to her husband. 'She might end up staying in Rishikesh. She probably feels neglected.'\n\n'That's absurd'' said Hemant, 'why should she feel neglected? Old people turn to religion. It is natural.'\n\n'It is not'' said Astha indignantly, 'only when they have no other choices.'\n\nHemant looked at her. 'Religion is a choice as much as any other thing'' he said. 'If she decides to stay in Rishikesh, it must be because she is happy there. Besides I have told you I will talk to her.'\n\n'Like when?'\n\n'When she comes.'\n\nAstha paused. She felt her mother's condition was partly Hemant's fault. Had he shown more concern... Tersely she pointed out, 'I know you had no time, but this cannot be left any longer, I need to see if she is all right.'\n\nHemant took umbrage. 'If that was the way you felt, you should have gone before'' he said. 'I have enough things on my plate.'\n\n'And so I will. As soon as the children's exams are over.'\n\n*\n\nIt was five o'clock on a Saturday morning of the following week, when Hemant took his wife and two children to the New Delhi Railway Station. He bought his wife a _Femina_ and _Stardust,_ for his children chips and chocolate, and sat with them in the compartment till the train left. 'Bye-bye Papa, bye Papa'' said his children. 'Why aren't you coming with us, Papa? See you soon, Papa.' Papa wrapped his arms around them, gave Astha a brief pat and jumped onto the platform.\n\nThe children passed the five hours to Haridwar having their breakfast, playing games, fighting, eating rubbish, dozing, and going to the bathroom, while Astha was divided between looking after them, and looking out of the window. The fields on either side had wheat growing in them. Her mother must have looked at this scene and felt alone. If she were not weighed down by children, husband, job, she too might become nothing, no different from the dots of people they were passing, lost on the flat plains of northern India.\n\n*\n\nAt Haridwar they got down, and walked across the road to the depot, from where they were to catch a bus to Rishikesh.\n\n'Bus to Rishikesh?' said Information. 'Half an hour. I will announce.'\n\nAstha and her children settled on one of the benches watching the others sitting, squatting, or sleeping on the floor. The hall was large and spacious. Already the air felt cooler than in Delhi, the breeze less polluted.\n\nThey sat and sat and watched bus after bus leave. Finally after twenty-five minutes Astha asked Information, 'When will you announce the bus that you said was leaving for Rishikesh in half an hour?'\n\n'It is already leaving'' he said languidly pointing to a bus lumbering away.\n\nThere was no time to get angry. 'Quick'' shouted Astha, grabbing the one suitcase, and shoving the smaller bags at her children.\n\nThey ran towards the slowly moving bus, their feet slamming the dust while the passengers stared at them curiously. Astha banged on its side, 'Stop, stop,' and the passengers hands echoed theirs in the banging, and the bus did in fact stop as it turned towards the exit.\n\nFeeling stupid and incompetent \u2013 you can't even catch a bus \u2013 Astha pushed her children up the steps and clambered on. Inside she distributed Anuradha and Himanshu where she found space on the hard shiny rexine seats, each packed with three to four people. The suitcase she manoeuvred in the crowded aisle, the packages she held in her lap, and with her attention wandering between her children, the green trees, the butterflies, the narrow road slowly rising, the mountains beyond, the tightly oiled plaits and shiny magenta nylon ribbons of the little girl in front, she passed the hour to Rishikesh.\n\n*\n\nThey finally stopped in a small and dusty square, which appeared to be the depot. Lugging their baggage to a group of waiting scooters, Astha gave the address of the ashram, and they bumped their way through narrow roads, lined with refuse and running sewers, the scooter wallah blaring away at every pig, cow, mongrel, rickshaw, two-wheeler and pedestrian in his way.\n\n'What's that?' asked Himanshu pointing to some enormous black creatures, rooting in the profuse garbage, ugly as sin.\n\n'Pigs, darling.'\n\n'But they are not pink'' he objected.\n\n'That is just in books, stupid'' said Anuradha. She herself was seeing a black pig for the first time, but her grasp of the difference between reality and theoretical knowledge was infinitely quicker than her brother's.\n\nIt was late afternoon by the time they reached the ashram doors, set in the middle of high walls. As they stepped into the compound, it seemed another world, clean, green, spacious, its long low buildings hidden by trees and shrubs on either side of a central open space. At the far end they could see benches, more trees and a paved terrace overlooking mountains across the river.\n\nAstha's mother was waiting with her arms open to receive her children, to show them her place, peaceful, serene, and at its centre a swami who contained the clues to life.\n\n*\n\nIn the ashram Astha could see how her mother had changed. Her movements were confident, her smile less tentative. She had made friends, she spent a lot of time walking around the terrace, and many hours reading the notes she had taken during Swamiji's lectures. 'Look'' she said, showing her notebook to Astha:\n\n*\n\nSleep, the state of being most pure. In sleep there is no thought, no emotion, no subject, no object. Sleep is the state where there is no 'I'. The state in itself no different from death, or previous lives in which we are in identical states \u2013 we need sleep not only to survive (you can't be awake if there is no sleep) but in order to understand reality.\n\n*\n\n'What on earth does all this mean?'\n\nAstha's mother looked conspiratorial. 'Ask Swamiji, he will tell you. He's a very learned man, he studied fifteen years before his own swami sent him into the world.'\n\n'But when he lectures he does so with a mike'' criticised Astha. 'That is not very unworldy.'\n\n'If you live in this world, you make it serve your aims. It is hard for him to speak continuously and loudly to such large audiences'' pointed out the mother protectively. 'So we insisted he have the mike.'\n\n'A present from one of his disciples?' inquired Astha, thinking in an idle way, that as a teacher she too could do with a mike, and she never talked as much as this man.\n\n'A present from me'' said her mother smiling that little smile again.\n\n'He asked?'\n\n'He never asks.'\n\n*\n\nThat evening Astha spent a lot of time staring at the swami upon the dais, who, after his lecture was immediately surrounded by his devotees, many women, some men, some resident and some from town.\n\nThey were sitting in the pleasant lecture hall, next to the river. All sixteen fans were whirring. Groups of people, while waiting their turn with the swami, were comparing notes on what he had said: today he explained very well, today he used a lot of Sanskrit words, difficult to understand, but then really you need ten years to understand. What was it Swamiji said, when that man asked a question about the mind \u2013 to answer will take me six years, to comprehend will take you twelve. Swamiji was in form all right, how he makes you laugh sometimes, and how my life has changed since I started coming to the lectures, yes, you get mental peace, no doubt about that.\n\nOne man in so many lives. Certainly in her mother's. She turned to look at her. 'Don't you want to ask him anything?' she asked.\n\nHer mother shook her head shyly. 'These people know so much more than me. Let them ask.'\n\n*\n\nAs they came down the steps a breeze was blowing, and a pink tinge on the water reflected the sunset.\n\n'Let us go to the temple'' said the mother.\n\nAstha stared at her. 'Since when have you started going to the temple?' she asked. Her father had not believed in going to temples, and as a consequence nobody ever went.\n\n'There is arti in the evenings, and one of the women here is a very good singer'' said the mother as Astha's question slid by her. 'Come'' she said, calling to the children, 'Anu, Himu, come, we are going to the temple.'\n\nThe temple was in another compound, small, white, with pink decorated columns and roof, facing the river front, lit with tube lights, and floored with marble. It was exquisitely clean with devotees waiting quietly for the evening prayers to begin. The mother sat at the back, Astha sat next to her, the children fidgeted and looked around.\n\nThe service lasted for forty-five minutes. Bhajan singing, praying, arti, offering bhog, receiving prasad, drinking holy water, and smoothing wet __ hand over head and eyes.\n\nIn the queue to receive prasad Anuradha asked, 'How come we don't do this at home?'\n\nAstha didn't know what to say. We don't believe was not strictly true, I don't have the time trivialised religion in a way that might be bad for her children, saying only old people prayed like this suggested that religion was only useful when you were feeble and decrepit. Instead she said, 'God is in our hearts, beti, and some of us do not believe in ritual. Maybe when Nani comes to Delhi, you can pray with her.'\n\n'We will all do it together'' said Nani firmly, her eyes gleaming with the prospect of inducting her grandchildren into puja, ritual, Vedanta, and the sound beginnings of a Hindu life.\n\n*\n\nIt was towards the end of Astha's visit that her mother said, 'I'm thinking of selling my land, and building a set of rooms in the ashram. Swamiji has agreed.'\n\n'Live with us, Ma'' Astha said hopelessly, 'it is the best solution.'\n\n'It doesn't look nice. Mother-in-law comes and never leaves.' Here the mother sighed, and looked at the waters of the river with a melancholy eye. 'It is so beautiful here, so peaceful'' she went on.\n\n'You must be lonely, Ma'' said Astha. They were sitting on one of the benches overlooking the river. The children were running up and down the steps. The heat of the day had gone, the light was gentle, the water below them was turning dark.\n\n'It is a lonely life'' said the mother, filling Astha with a dreadful sense of guilt.\n\n'It is my house too. If people mind it is just too bad. I don't believe in all this shit about parents being the responsibility only of the sons.'\n\n'After all Hemant's parents are staying with him, aren't they, not with their daughters.'\n\n'His parents can't stay with the daughters, one is in America, and one\u2014' Astha was going to say and one is staying with her in-laws, but changed it to, 'and one has a bad marriage, with a small house.'\n\n*\n\nNext morning at the lecture Astha again looked at her mother's teacher carefully. His beard was grey, there were little white spikes sticking up from his shaven head. He wore glasses, and the eyes behind them were gleaming, sharp, intelligent, she supposed compassionate \u2013 he was a swami after all \u2013 and how does one describe a swami's eyes? His legs were crossed, his foot waggled, his clothes all saffron. His voice was deliberate and quiet:\n\n'There is pain and suffering in every life. When the burden becomes intolerable, we seek distractions, which in turn trap us. We develop a craving for pleasure and sensation, till finally we are at the complete mercy of our desires, which out of ignorance we have encouraged to grow into monsters.\n\n'With desire comes dissatisfaction, and a dissatisfied man is full of misery, even if he has at hand the pleasures that the world can give him.\n\n'We mistake gratifying our senses for living in the world. We act in order to be happy, and then we are surprised that the happiness does not last, and we look for other things, and the same pattern is repeated. Discontent is the cause of restlessness.\n\n'All our pleasures are connected with our deeds. They have a beginning and an end. The fruits of our actions similarly have a beginning and an end. They are transient and can therefore never quench our longing.\n\n'We breathe to live, but every breath draws us one step nearer to our end. In our body is our decay. We cannot alter this decay, the richest man in the world shares the fate of the poorest.\n\n'Against the world we are weak. Hunger, thirst, cold, heat, flood, famine, storms, all these things create fear. We run seeking protection here and there, but the strongest protection against the world comes from knowledge that comes from within.\n\n'It is only in a state of self-realisation that we can draw from the reservoir within to gain happiness. If we find contentment within ourselves, we will find good in all things. As the sun shines so shall the contentment within us light our lives and the lives around us.\n\n'We protect our feet with shoes, we protect our body with clothes. We cannot be harmed by the stones in our paths, nor by the sun or the rain that falls on us. Similarly, those who have achieved self-realisation are contented in all circumstances. The troubles they encounter on their journey through life cannot hurt them.'\n\n*\n\nAstha listened, caught up in his words, like everybody else in the room. The swami looked beyond time, because he was bowed down by nothing. If examples were what one had to go by, he was a good example of what he preached. His face shone with non-attachment, though his disciples hung on to every word he uttered with fierce attachment.\n\nAll these people here were looking, looking for shoes to protect them from the rough paths they had to tread. She wanted shoes as well. She sat in front of the swami trying them on. For a wild moment she wanted to go up to him and beg, tell me what to do.\n\nAnd he would tell her, what? She already knew. Misery springs from desire, desire springs from attachment, and that if she gave up all these things, she would be happy.\n\nThe weight in her chest increased. She had come to rescue her mother, and yet seeing her mother in that place, the person who seemed to need rescuing was herself.\n\nShe tried no more to prevent her mother from living in Rishikesh, or from selling her plot of land. Clearly her mother needed quite a bit of money if she were to live respectably in the ashram. It seemed crazy to sell a piece of property, whose value, now that the bridge was built, doubled practically every year, but when one gave up material possessions, one also gave up speculation in the future.\n\nAnother three days and Astha left.\n\nIn October that year, the sale of the plot went through.\n\n'Dear child'' said Astha's mother, who was in Delhi for the signing of the papers, 'I have given Hemant part of the proceeds of the house.'\n\n'Why? The money is for you, Ma.'\n\n'I don't need so much. You can consider this your father's legacy.'\n\n'They why give it to Hemant?' asked Astha bridling.\n\n'Why not? He is a man, he knows about money. He will invest it for you and the children. I have discussed the whole thing with him.'\n\nHow had this happened? Hemant had found a buyer and checked the legalities of the sale, but even if he was the man of business, she wanted to participate in any decision concerning the money her mother chose to give her.\n\n'Really, Ma, don't you think women can be responsible for their own investments?'\n\n'Of course, but this was a lot. Are you suggesting I hand the whole thing over to you?'\n\nNo, Astha wasn't. The sad thing was that she herself would have felt nervous handling a large sum. Suppose she did something foolish, and it did not multiply fast enough, it would be through her arrogance that the money had not functioned in the optimum manner.\n\n'Hemant is very clever, look at the way he does business, with no background'' went on the mother. 'You yourself have said he manages everything financial. It was the same with your father, I only did the household accounts.'\n\n'You were earning too, Ma.'\n\n'Yes, yes. But he looked after my tax saving, my provident fund, decided how much we should spend, how much to save, all that. After him, Hemant took over.'\n\n'Yes, Ma.'\n\n'He has promised to double the amount in a few years.'\n\nCould Astha ever have made such a promise? Never, not even if the gates of hell opened and the stock market collapsed in her lap. She had better stick to her job, and what it earned her. Nobody thought it was anything. Nobody discussed it, speculated with it, promised to increase it at fantastic rates. She could do with it what she liked, take it to bed, chew it, shit rupees in the morning and nobody would bat an eyelid.\n\nHer mother had delivered her into Hemant's hands. If her mother was at fault, so was her father, for managing the money, and teaching his wife that this was normal behaviour, so was her mother-in-law for bringing up Hemant to never regard women as beings to be consulted in their own lives, so was the Swamiji for teaching that only in detachment lies happiness, which lesson can be read in as many different ways as there are people and attachments.\n\n*\n\nAfter Astha's mother left, the money was discussed briefly and bitterly.\n\n'Darling?'\n\n'Dearest.'\n\n'You know Ma's money?'\n\n'I have several plans for it. It will be well invested, don't you worry. Long term for the children, shorter term for you.'\n\n'Thank you darling. But I was wondering, you know, whether I could also have a say in what you do with it.'\n\nHemant began to frown. 'Don't you trust me?'\n\n'Of course, of course, I trust you. It's not a question of trust, surely. You are my husband.'\n\n'Exactly. So what's going on?'\n\n'I wish to feel\u2014' Here Astha paused and treaded carefully among the thickly laid minefields of income, expenditure, rights, responsibilities, knowledge, power, and dependency. 'I mean if I wasn't so ignorant about things concerning money, I wouldn't feel so stupid.'\n\nHemant relaxed. 'When I have finished I will explain everything to you. In fact I am glad you have brought this up. I have been thinking you should know what is going on. That way if anything happens to me, you will not be left in the dark.'\n\n'But Hemu'' said Astha, 'I don't wish to be enlightened only because you might die, which I hope will not be for a long long time, and certainly not before me.'\n\nHemant smiled, 'We will die together in old age, huh?'\n\n'Yes'' she replied, 'yes'' she repeated, 'yes'' she faltered. 'We will die together, I hope, but meanwhile, I feel so clueless about our financial situation. I know that in business things can be uncertain, so I thought that now that I have some money, it would be useful if I looked after it. That way I will gain experience.'\n\n'Your mother gave me this money to manage, I didn't ask for it'' said Hemant coldly. 'She trusts me even if you don't.'\n\n'That's not what I mean. I know she trusts you, certainly much more than she trusts me, but is it such a bad thing if I know how much is in my name and how I can have access to it?' Astha was pleading now, begging Hemant to understand. She meant nothing personal. She didn't want to feel dependent, that was all. Surely equals could relate better than master and slave?\n\n'What has gotten into you I don't understand. I will tell your mother to give the whole thing to you, you will handle it yourself. She should have consulted you first, before she handed anything to me. In fact why didn't she ask you to look for a buyer and get a lawyer to check the sale deeds? You have been missing out on so many things that life is not worth living, isn't that so?'\n\nAstha sat stunned. What kind of fool had she been to expect Hemant to understand? She had a good life, but it was good because nothing was questioned. This boat could not be rocked. She should paint that on a canvas and put it up on the wall, and stare at it day and night, so that its message burnt its way through her brain into her heart. This boat cannot be rocked.\n\nBesides if the boat could not be rocked, what need did she have of money, or knowledge of investments? Hands that had grasped money, and felt it pass through their fingers were the ones capable of rocking boats. Hers were not.\n\n*\n\nThe next morning, quickly she got her children ready and sent them off to school, quickly she had her tea, packed her breakfast to eat later, jamming an omelette between two slices of fridge cold bread and dripping violent red chilli sauce over the insides. Quickly, quickly she did all this, smiling, smiling all the while, so that no distress was palpable.\n\nIt was only in the staff room in school that Astha could be alone with her thoughts.\n\n'Why so silent?' they asked? 'Are you ill?'\n\nAstha shook her head. She looked at her colleagues, women she met every day, women whom she liked, whose lives ran smoothly, women who had no shadows between their husbands and themselves, whose husbands were 'him' and 'he'' and whose in-laws were 'they'. Whom among them could she tell that she had not been able to sleep? What reason could she give that they would not think self-indulgent?\n\n'I'm fine'' she repeated, and opened the usual stack of brown-paper covered notebooks that laced each day's work. \nChapter IV\n\nIt was early in the year 1987, that the principal of Astha's school invited The Street Theatre Group to hold a workshop on their premises. The workshop would be held in the break between the final exams in March and the opening of the new school year in April.\n\nThe staff were not pleased.\n\nAs usual the Principal wants to attract attention to herself.\n\nJust because she is interested in theatre, we are forced to be interested too.\n\nThey'll want staff volunteers, wait and see.\n\nWe have to correct exam scripts, prepare report cards, see to the merit lists, file an account of each child's progress in the school records \u2013 where is the time to do all this extra-curricular activity?\n\nA gloomy silence descended. The Principal was not known to respect the convenience of her teachers.\n\n*\n\nAstha wondered whether she would be asked, she did not relish working in the holidays while her children were at home. She was very fond of Mrs Dubey but sometimes she felt that their special relationship caused her to be exploited. She had done enough for the school, the Principal should look elsewhere, she decided, readying herself for a tussle.\n\nIn which she lost.\n\n'You need someone with more experience if an outsider is coming'' she tried objecting.\n\n'With Aijaz you don't need experience'' said the Principal. 'For him any place is a stage, any person an actor. He has performed at factory gates, outside offices, at bus stops, in front of shops. He has dramatised issues like unemployment, atrocities against women and urban poverty. Indeed he is the voice of the underprivileged. That is his genius.'\n\nHe can take his genius elsewhere, thought Astha, why is he bringing it here.\n\n'He is my brother's friend and is coming here on my personal request'' went on the Principal, somewhat coyly. 'This is a great opportunity.'\n\n'It's my children's holidays.' The woman-to-woman approach.\n\n'Bring them, they will benefit. Aijaz is a wizard. He is actually a history lecturer, but his knowledge of drama is immense. Besides writing his own plays and songs, he has adapted Brecht, Shakespeare, and Greek tragedy into Hindi. People grumble about the lack of activity in the school, but when it comes to giving our students exposure they come up with all kinds of objections. Where is the school spirit?'\n\nAstha had no option but to agree.\n\n*\n\nHemant was not pleased. He timed his trips to be free for his children's holidays.\n\n'Why can't you stay at home? And why drag the children into this?'\n\n'I had no choice'' said Astha. 'Anyway it will be good for the children to see schools not as elite as theirs. Anu was actually asking when were we taking her to Disneyland. All her school friends have been, she says. I don't believe her. Disneyland! Imagine!'\n\n'Nothing wrong with wanting to go to Disneyland'' said Hemant.\n\n'At this age! Why, I haven't been abroad yet.'\n\n'We are not talking about you. If parents can afford to show their children the world, why not?' said Hemant. 'This is the eighties. We are not deprived Indians any longer.'\n\nAstha felt there was something morally wrong with getting things without struggling for them, but she knew this view irritated Hemant. He was making more money at his age than their combined fathers at their retirement, and he didn't seem to have any intention of letting his children struggle. She turned the conversation to the topic at hand.\n\n'Apparently Aijaz Akhtar Khan, the founder of The Street Theatre Group is very well known. He teaches history, and during the holidays he performs in slums, factories, streets, villages and small towns.'\n\n'What's the point of that?'\n\n'Create empathy, generate social awareness by having workshops that involve workers and students, bridge the class divide'' said Astha glibly, replicating that morning's exchange with her Principal.\n\n'Culture-vultures'' snorted Hemant, 'why don't they do something real about the class divide, like creating jobs?'\n\n'Not everybody can be a factory owner.'\n\n*\n\nHimanshu was delighted. His face broke into a slow and gleaming smile that went straight to his mother's heart. He was always wanting to come to his mother's school instead of his own.\n\nAnuradha registered her brother's pleasure and loudly protested against the injustice being done to her.\n\n'Why should I spend my holidays going to your school?' she demanded. 'Don't I go enough to my own?'\n\n'I can't leave you here alone the whole morning. It's not classes, it's a drama workshop. You'll be doing fun things.'\n\n'I don't want to do fun things. Besides Papa said he was going to spend fewer hours at the factory and take us out.'\n\n'Well let him actually make the programme and then we will see'' said Astha with some irritation.\n\n'I won't'' said Anuradha her eyes flashing, getting ready for a confrontation that would continue till collapse or victory. 'You can't make me. I'll spend my holidays with Dadi upstairs.'\n\nHimanshu looked on piously, while Anuradha waited for the next round. 'Please beta'' said Astha, 'your Dadi then complains to me that she gets tired. You have so much energy she doesn't really know how to keep up with you. Come for a few days, if you don't like it you needn't continue. Promise.'\n\nOnce it was established that Anuradha was doing her mother a favour, it was easier to take her.\n\nAt first Astha did not pay much attention to Aijaz. He seemed quite capable of managing thirty-two children without her. He sat them in a circle on the stage. Do you know why people sit in a circle \u2013 so that there is no hierarchy \u2013 all of us have something to offer from backstage to front \u2013 what is the theatre about \u2013 communication \u2013 what kind? \u2013 drama \u2013 older than the written word \u2013 what did they think was the subject of drama \u2013 where did they find it...\n\nHow pedantic, thought Astha, is he giving the history of drama, are they going to do an exam, or is he going to get on with the workshop, which is why we are all here in the first place, I'm sure all the children are bored. And her mind wandered, till it came back ten minutes later to Aijaz explaining that the way man lived in society was politics and this affected everybody, literate, illiterate, powerful, powerless, poor, rich. He read out sections of the newspaper and asked how they would translate what was happening into drama for people who couldn't read? For example what would they do with the Babri Masjid-Ram Janambhoomi controversy?\n\nThe spot where Ram was born thousands of years ago some say is the exact spot where a masjid stands today. Is this fact or faith? If it is faith, is it sacrosanct? Are there ways in which faith can be motivated and played upon by political forces...\n\nHis voice faded, and Astha's mind turned to the religion consumed at home on one of her husband's TVs. Ever since the Ramayan was serialised, viewing it had become a ritual, insisted upon by the grandparents, strongly supported by Hemant.\n\nAnd so, every Sunday morning, the family gathered upstairs before a ClearVision TV, twenty-inch screen, manufactured by the son of the house, and watched the story of the Ramayan. Week after week they agreed, this was the golden age of India, this is our noble heritage, now thoroughly debased, when justice flourished, when Hindus had pride, when a king showed responsibility towards his people, when duty, honour, devotion, truth and loyalty had a place in Ram Rajya. And today the birthplace of this king, our Lord, is occupied by a mosque, the shame of it, dismissing as nonsense the protest that it was not possible to really place the exact spot of a man's birthplace so many thousands of years ago.\n\nSuddenly Astha saw the long arm of history twisted and refracted, till it popped out of a TV box, took them to Ayodhya and planted them on Ram Kot in front of the Babri Masjid.\n\nShe was sitting at the back of the stage, her arms around her knees, thinking all this, when she looked up and saw Aijaz looking at her. Uncertainly she smiled. 'What do you think, Astha?' he asked.\n\nHow had he found out her name? And from being indifferent to Aijaz, the single use of her name, created a pleasure in what she, unused to the ways of men outside marriage, saw as interest rather than a communication strategy.\n\n'Do you think you can write the script?' he went on.\n\n'Um'' Astha hesitated, 'I don't know anything about the Babri Masjid.'\n\nAijaz leaned towards her and said, 'Just a working script. Your daughter has volunteered your name. She says you write.'\n\n'I am not really a writer, just a few poems'' said Astha surprised, her eyes on her daughter's back, with the hair curling down the white shirt.\n\nAijaz was used to persuading people. 'Just a simple working script which we can improvise on, Astha.'\n\nHe was focusing on her. She blushed.\n\nHimanshu frowned. Was his mother being forced to do something unpleasant, but no, she was agreeing, she was participating in extra-curricular activities, doing the bit that wasn't necessary, volunteering despite her uncertainty about her capacities, because everything was worth trying.\n\nAijaz smiled, showing his even pearly teeth. Why does he smile like that, he knows he is charming, thought the newly appointed writer of scripts.\n\n*\n\nGoing back in the scooter, Astha thought of the India International Centre, where her parents-in-law were members, and the library that only she was interested enough to use. There was bound to be something on the Babri Masjid there. As if reading her thoughts, Himanshu piped up, 'I'll help you, Mama.'\n\nAnuradha snorted. 'You? You are so stupid. What can you do? Do you even know what the Babri Masjid is? Do you know where it is?'\n\nHimanshu turned around and hit Anuradha in the stomach. Anuradha hit him back twice as hard, then once on the back for good measure. Astha slapped Anuradha's hand. Anuradha glared at her mother. Himanshu began to cry. Just then the scooter took a wrong turn inside the colony, and in the middle of shouting at her children, Astha had to break off and redirect the scooter wallah through the maze of Vasant Vihar. He insisted on charging ten rupees more, and that was the end of their first morning at the theatre workshop.\n\n*\n\nLater Astha had a talk with Anuradha. 'We are going to be together for fifteen days,' she said. 'And in that time I forbid you to call your brother stupid.'\n\nAnuradha looked cunning. 'And after that?'\n\n'Even after that. You can't go on calling someone stupid. It hurts their feelings.'\n\n'But he is.'\n\n'Even if he is.'\n\nAnuradha looked victorious. 'See, you also think so.'\n\nAstha stared at her daughter, 'Anu, what's the matter with you? Four years younger, what comparison can there be?'\n\n'You are always taking his side.'\n\nWhy was it, thought Astha wearily, that love always had to be balanced by its opposite? She had a secret tenderness for Himanshu that her daughter targeted unerringly, battering her mother, shouting out her dislike, making even the love and hate in the world. She looked at Anuradha's contorted face, and angry eyes, and cajoled, 'I need help in writing a script. Himanshu can't help me.'\n\nAnuradha looked wary. 'Don't try and flatter me'' she said.\n\n'You mean what I'm saying is not true?'\n\nFor a moment Anuradha was out-manoeuvred.\n\n'So, it'll have to be you'' continued Astha.\n\n'When do we start?'\n\n'This evening. We'll go to the library and get some facts first.'\n\n'And leave Himanshu behind.'\n\n'Absolutely. I'll send him upstairs.'\n\n*\n\nThat evening Astha and Anuradha made for the library. As Anuradha looked at magazines, Astha quickly browsed through the books in the history section. There seemed to be no end of fuss around this mosque. Had there been a temple on this site, claimed to be the birthplace of Lord Ram? Had Babur ordered this temple destroyed? Had he compounded the arrogance of conquest by building a mosque bearing his name using materials from the temple? Zealous historians, pursuing evidence and rationality had gone into its structure, pillars, stones, inscriptions, had investigated Babur's diary, his religious and building habits, had cited examples of British divisive policies, but nothing had been able to quiet the controversy.\n\nAstha stared at the picture of the Babri Masjid. What was it about this monument that had created so much bloodshed and fighting over two centuries? It was not even remarkable, squat and three domed, surrounded by trees. How could she effectively present its history, long and tortured, in a manner that was simple without distorting?\n\nOver the weekend as she read through books and photo-copies she had made in the library, she thought that controversies need places, disputes need sites, not the other way around, and the Babri Masjid was one of them. Babri Masjid-Ram Janambhoomi. The amount of blood, hate, and passion for ownership these words evoked bathed each stone with a corrosive mixture, slashing through the surface so that it was no longer an old mosque. It was a temple, a birthplace, a monument to past glory, anything but a disused nesting place for bats. Despite all this it had endured for over four hundred years.\n\nIt was too much to handle as a play. She felt like giving up, but the thought of not having anything to show Aijaz drove her on. She gripped her pen, took a deep breath, and plunged.\n\nShe was still plunging when Hemant returned from the Sunday tea spent upstairs with his parents, bonding over business and politics.\n\n'Back already?' she asked.\n\n'It's been two hours'' he replied.\n\n'Oh, I hadn't realised. This whole thing is very complicated'' said Astha.\n\n'People make it so'' replied the husband. 'Otherwise what is there in an abandoned mosque? The government is too bloody soft on these Muslims, that is the problem.'\n\n'Surely that is not the issue. Power seekers \u2013 on both sides \u2013 use religion quite blatantly. How can beliefs about god be compatible with violence?'\n\n'You don't know _their_ religion.'\n\n'As though ours is so much better. Ram would have hated what was going on in his name \u2013 a man who sacrificed everything to keep his father's honour, who left his home, his palace, his kingdom in order to make sure his brother inherited, he would be the last to appreciate the fuss over his birthplace.'\n\n'Times have changed. We are preserving his honour as it needs to be done now.'\n\nAstha stared at her husband. Was he agreeing that people should be killed in the name of God? She didn't want to know what he thought.\n\n'Wasn't Aijaz going to write this play'' continued Hemant. 'Didn't you tell me he was a history teacher? Surely this is his area of expertise, not yours. How have you got so involved?'\n\n'He wants everybody to participate'' said Astha thinking quickly. 'Besides you forget I am the teacher volunteer.'\n\n'Volunteer, not donkey.'\n\n'Translating history into theatre is hardly work a donkey can do.'\n\n'Nor can you. What is your experience?'\n\n'I don't need experience.' She felt she was being denied something, not understood, throttled, and choked. And yet it was just a play. He was right, she had no experience. Though Aijaz was in a better position to write about masjids and controversies, still she would hold her own, paltry though that own might be. 'Aijaz doesn't think experience is necessary'' she went on in defence.\n\n'Oh pardon me'' he said, and his wife hated the mockery in his tone, 'he clearly knows how to get work out of you.'\n\n*\n\n'Can I speak to you a moment?' Astha asked Aijaz on Monday during the fifteen-minute break he allowed the kids.\n\n'Trouble with the masjid?' he smiled.\n\nAstha nodded briefly.\n\n'Shall we go to the canteen?'\n\n*\n\nIn the canteen she opened her bag and took out flurries of photostats. 'I don't know where to begin'' she started. 'It's such a tangled history, and leaving one piece out makes it lopsided. Besides it is used for many different political purposes in the present as well.' This Astha had only realised yesterday. So far the Babri Masjid had been something mentioned in the news with the irritating air of a problem that wouldn't go away. 'I do wish you would write it, or conceive it. I am sure you are far more knowledgeable.'\n\nAijaz looked at her clutching her photostats. 'Do you think it is only the so-called experts that should be allowed to deliver opinions? You are looking at it from the outside. Your perspective is fresh, it is invaluable.'\n\n'But I am very ignorant and I cannot possibly do it justice'' she said, quick as a flash putting herself down.\n\n'It doesn't matter, Astha'' he said. His voice was coming at her, his eyes were looking at her, any second and his teeth would glow at her. She was married, she should not be registering these things. She shifted uneasily on the hard canteen bench, clutching her bag in her lap. 'The thing is'' he went on, 'we have to create awareness. There may be differences of interpretation, it doesn't matter. If our players and our audience think for one moment about this issue, we have done our job.'\n\n'You have already created awareness in one'' she mumbled daringly.\n\n'And you will create it in many.'\n\n'I don't know'' she replied, 'I have no experience.'\n\nThe smile, the teeth, the hand that lightly touched the phototstats. 'What is all this?'\n\n'Material I gathered. I sketched out a few ideas, though I am not sure\u2014'\n\n'Let's see'' he interrupted, leaning forward. She could smell him, a faint sharp smell. She shifted uneasily again, clutching her bag still more firmly to her stomach, riffling through her papers.\n\n'I thought of starting in 1528, you know when Mir Baqi decrees that a mosque be built at the highest point in Ayodhya in the name of his most noble ruler Emperor Babur, a brief two-line scene. We could have a boy with a placard announcing dates and locations. Perhaps the same boy could double as the mosque, a mosque that just wants to be left alone thinking each fight will be the last.'\n\n'Himanshu would be good for the part. He is the youngest.'\n\nHer own thoughts exactly. She looked up and smiled, he smiled back, she quickly looked down, he must think she found her paunch fascinating, she looked at it so much. 'Do go on'' he said after a moment's silence.\n\n'Then a short scene set in 1855. The Muslims think the Ayodhya ruler is showing favours to the Hindus. They claim that the temple at Hanuman Garhi is built on a mosque, they march towards it, the Hindus retaliate by saying the Babri Masjid is built on a temple and they march upon it\u2014' she paused. 'Actually there was more but I have pared it down to the essentials, everybody thinking they have been done in, and asserting their power through temples and mosques.' She looked at Aijaz anxiously. 'I hope I have got it right?'\n\n'Absolutely. Then?'\n\n'A lot of people were killed during this time, Hindus as well as Muslims, and the whole thing became openly political. There was an enquiry committee consisting of Hindus and Muslims, presided over by the British Resident. But after 1857 power equations changed, and two years later, the British declared that access to the Babri Masjid would be bifurcated. The Hindus were to enter from the east, and the Muslims from the north.'\n\n'Then?'\n\n'This state continues till the British leave. Then in 1949, some idols appear. The Hindus claim this is a miracle, while the constable on duty states that about fifty to sixty people broke into the masjid on the night of December 22. The next day the District Magistrate declared the area disturbed and locks are put on the masjid. At this point I stopped.'\n\n'You haven't written more?' Aijaz sounded disappointed.\n\n'Well last February the district court ordered the locks open. Rajiv Gandhi is probably involved, but I don't know how far to go in showing the masjid as a tool in modern political equations'' said Astha, pleased at his tone.\n\n'We'll work something out.'\n\nAijaz took out his wallet, while Astha groped around for change. 'If you don't let me pay for one sweet and overcooked cup of tea I'll be very upset'' said Aijaz as they rose to go.\n\n*\n\nThe appreciation that Aijaz had shown moved Astha so much she couldn't help talking about it at dinner.\n\n'Aijaz liked the script'' she started.\n\n'He told us it was a wonderful script'' put in Himanshu 'but we could change it any way we wanted because we are to bring our own \u2013 own \u2013 what, Mama?'\n\n'Interpretations.'\n\n'Yes that \u2013 to our parts. And I'm to be the mosque and carry placards. I have to keep crying and getting hit. Everybody wants me.'\n\nHis parents looked at him indulgently. 'Really beta?' said Hemant. 'I must come and see you.'\n\n'Yes, Papa. We are going to do it the last day of the holidays. All our families and friends should come, Aijaz said.'\n\n'Aijaz Uncle'' corrected the father. 'He is older than you.'\n\n'No Papa, Aijaz does not believe in hi \u2013 hi\u2014' He looked at his mother again.\n\n'Hierarchy.'\n\n'The girls in Mama's school don't call him anything'' said Anuradha. 'They are so shy, can you imagine? It is a very good thing that Himanshu and I go, otherwise poor Aijaz would have a hard time with them.'\n\n'Anu'' reproved Astha, 'they are not that bad.'\n\n'Oh, Mama, you don't know.'\n\nOnly later did Astha realise that Hemant had not actually said anything about her script. Well, it didn't matter, he would see the play performed, and recognise his wife's hidden talents. At night, lying in bed, she drifted off to sleep with thoughts of Aijaz and the days ahead.\n\n*\n\nAstha loved looking at Aijaz on stage, allowing herself frequent covert glances. He was of mediun height, his body compact. His face was the clear delicate luminous brown of freshly rained-on earth. His lips were a darker brown than his skin, and his eyes were black and narrow. While working he rolled up the sleeves of his shirt, allowing Astha to view at her leisure his round strong arms, hairless, smooth and muscular. He had prematurely grey hair, which, thick and springy, fell about his face and neck in ways that suggested a good barber. He must be vain of his hair, thought Astha, knowing how attractive the grey made a young face look.\n\n*\n\nThrough those fifteen days, Astha saw the little thing she had penned transformed, and her admiration for him grew. Song, dance, mime, action, improvisation, actor involvement, he fused all these elements effortlessly into a fast-moving, absorbing piece.\n\nShe and her children talked of nothing but the play, the rehearsals, the way everybody was acting, who was good, who was bad, who came late, who not, who had team spirit, who not, and what Aijaz had said. Every day Astha was called upon to add a bit of information, to corroborate some piece of evidence, suddenly she was the Babri Masjid expert, and this she felt was Aijaz's doing \u2013 he who was the history teacher, allowing her to parade her knowledge when surely his own was greater.\n\nHe looked at her, he wanted her opinion even when it wasn't necessary, he smiled when there was no occasion. Perhaps she shouldn't think of him so much, but soon it would be over, where was the harm, it made her happy, and that in itself was worth something?\n\n*\n\nSometimes as Astha sat on the stage she absently sketched the scenes before her, wanting to capture her son as the mosque, her daughter as a rabble rouser, and Aijaz as their teacher. By now she knew by heart his perfect teeth, his full lips, the smoothness of his cheeks, the deep dimple near his mouth, the curl in his hair, the glint in his eyes. She tried to translate these things on paper, but only registered pale copies. Her activities attracted his attention.\n\n'What are you doing?' he asked during one break.\n\n'Nothing much.'\n\n'No let's see'' and he gently tugged at the papers she had turned face down on her lap. For a moment his curled fist rested on her knee.\n\nHastily she shoved the drawings at him, repeating the mandatory, 'It's nothing much.'\n\nAijaz turned the papers over. Astha drew fast and there were ten sketches in all. 'For how long have you been drawing?' he asked.\n\n'On and off since I was young. Mostly off.'\n\n'You should continue. You capture whatever is going on well.'\n\nShe found his immediate presence too disturbing for conversation.\n\n'Why don't you come to my place sometime, you can have a look at what I do.'\n\nParalysed silence on her part. After a second he dropped the papers back into her lap and shouted 'Time's up', clapping his hands to get the children's attention.\n\nWhat did it mean, did he like her, did he want to have an affair with her, why had she been so startled by his hand on her knee, why hadn't she responded, but she was a married woman, with two children and those right before her eyes.\n\n*\n\n'What was Aijaz saying to you, Mama?' asked Anuradha, the sharp eyed one, in the scooter back home.\n\n'Nothing much, beta. He was looking at my drawings, that's all.'\n\n'Did he like them?'\n\n'There is nothing much to like'' said Astha, teaching her daughter how to devalue her work, and passing on the tradition from woman to woman.\n\nAnuradha lost interest. Himanshu having just grasped their topic of conversation demanded, 'What? What? What did Aijaz say to you, Mama?'\n\n'Nothing much beta.'\n\n'Then what was Didi saying?'\n\nAnuradha cast him her usual you're so stupid look.\n\n'She wanted to know what he said about my drawings, that's all.'\n\n'What did he say?'\n\n'Nothing.'\n\n'What did he _say_?'\n\n'He said they weren't bad, that I should continue, continue to draw'' repeated Astha quickly before Himanshu could say what again.\n\n*\n\nThat night, lying awake in bed, Astha went over everything Aijaz had said, she relived that touch on her knee, his head bent over her drawings.\n\nIn a few more days the workshop would end. Would he repeat his invitation? Had it been a spur of the moment thing, or was he attracted to her? Why was she so shy? Maybe she should phone him, call him over, but how, with everybody watching, it was so difficult, after this would she ever see him? How could they meet again?\n\nShe tossed and turned, trying not to disturb Hemant. If an accidental brush against her knee was so dislocating what would anything else be? And then she felt stupid, had Aijaz asked her to elope with him? No, he had merely asked her over to look at his drawings. What connection did that have with her marriage? She was a fool, a fool, a fool.\n\nOne thing was clear though, he liked her drawings, he thought she had something. He was also an artist, he must know what he was talking about. Suddenly she glimpsed possibilities, suddenly her life seemed less constricted.\n\nShe sighed, and closed her eyes, willing sleep to come, pressing herself firmly against her husband, hoping for the comfort of habit.\n\n*\n\nThe auditorium was dark. The parents in the hall fidgeted, making allowances for the twenty-minute delay in the rise of the curtain on _Babri_ _Masjid:_ _Fact,_ _Fiction_ _and_ _You._\n\n'It sounds like a bloody political tract'' said Hemant.\n\n'Don't you like it?' asked Astha, sitting next to him in the front row. 'The title was mine. Aijaz thought it was a good one.'\n\n'Darling, you would hardly go and see a play called _Babri_ _Masjid:_ _Fact,_ _Fiction_ _and_ _You_ if your children were not in it.'\n\n'No, I wouldn't, but maybe I should. There are too many people like me in this country who are not paying attention to what is happening.'\n\nHemant raised his eyebrows. 'What is happening?'\n\n'The locks on the masjid were opened to appease Hindu sentiments. Then the Muslim Women's Bill was introduced twenty-five days later in Parliament to appease Muslim sentiments. Basically both communities were pandered to as an election ploy.'\n\nHer husband stared at her. 'Are you all right?'\n\nAstha looked self-conscious. 'Of course I'm all right, why shouldn't I be?'\n\n'You sound like a parrot.'\n\n'To have an opinion is to sound like a parrot?'\n\n'Please. Keep to what you know best, the home, children, teaching. All this doesn't suit you.'\n\n*\n\nThe play was over. Himanshu came rushing over to them. 'Did you see me?' he cried. 'I was under the sheet.'\n\n'Beta, you were the best mosque anyone has ever seen'' said Hemant swinging him up in his arms. 'No wonder everybody was fighting over you.'\n\n'And me?' cried Anuradha tugging at his sleeve, 'Did you see me?'\n\n'Of course I did. You were soooo good, sweetheart.'\n\n'Come and meet Aijaz, Papa'' went on Anuradha dragging him to where the director stood, surrounded by parents congratulating him, telling him how their children had enjoyed the workshop, how they could talk of nothing but their play, and when was he going to do another?\n\nAstha watched as Hemant met Aijaz, watched as they shook\n\nhands, exchanged a few words, as Aijaz ruffled Himanshu's hair in a parting gesture, watched as he turned towards other parents. A few months later she heard he was going around with a woman working in an NGO.\n\nHer name was Pipeelika Trivedi. She lived alone in Delhi, sufficiently isolated from conventional society to believe her choice of partner concerned only herself. Her mother was horrified when she learnt of her engagement.\n\n'You can't do this'' she told her daughter.\n\n'Why not? You're the one who is always going on about me getting married.'\n\n'But not to a Muslim.'\n\n'He's sweet. So what if he's a Muslim?'\n\nHer mother clicked her tongue. 'They marry four times.'\n\n'How do you know?'\n\n'It's part of their religion.'\n\n'Do you, you personally, know any Muslim who has married four times?'\n\n'How is that relevant?'\n\n'It shows you are speaking out of prejudice. Meet him and then decide.'\n\n'It has nothing to do with meeting him. You like him, he must be nice. But everybody knows that all they have to do is say Talak, talak, talak, and the girl is out on the streets.'\n\n'She is not.'\n\n'How do you know?'\n\n'The Qu'ran says.'\n\n'How do you know?'\n\n'Aijaz says.'\n\n'It's not true. He is lying.'\n\n'Does he know more about the Qu'ran or do you?'\n\n'I know more about the world'' said the mother, tight and tense.\n\n'Well I know more about Aijaz.'\n\nThe mother looked at her stubborn daughter. 'You kept saying no to any boy I suggested for this? For this I struggled after your father died, so that you throw yourself away?'\n\n'He is not a heap of dung, you know. Besides I am almost twenty-nine, you've always said you want to see me married, now is your chance. I'm not going to find anyone else. He's intelligent, sensitive, socially committed, a history lecturer, a theatre activist, but all you can see is a Muslim who is going to both divorce me and marry four times.'\n\nThe mother stayed silent, hating to be so opposed to her daughter. The daughter wondered at the unreasonableness of her mother. They had always been so close.\n\n*\n\nMrs Trivedi, the mother, had been a widow for much of her adult life. Her parents' apprehensions about their daughter's marriage to her Delhi University teacher, twenty years her senior, were fully justified when he dropped dead one morning in front of the blackboard in his classroom. The widow was left with two small children.\n\nShe tried for a while to manage in Delhi, but it was difficult. She was too gentle, too pretty, too meek, too young and her circumstances too straightened. Her parents called her to live with them in Bangalore.\n\nMrs Trivedi went. Gentle and meek though she was, she had also been a wife, and she found it galling to fit into the daughter mode again. Added to this was her parents' obvious sense of her doom. The years with Jyotin had been the best in her life, it was an insult to his memory to be treated like a cornucopia of tragedy. She made enquiries about a teaching job in a boarding school where she could live with her children.\n\n'I am moving to Shiksha Kendra'' she announced. 'Board, lodging, and the children's education all will be taken care of.'\n\nThe parents were opposed to further movement. 'You have not suffered enough?' they asked. 'These poor children have not been knocked around enough?'\n\n'We will manage. Their home is with me, their mother, not in a place'' said the widowed daughter, showing all the signs of marriage to an intellectual husband.\n\n'On what money?'\n\n'At Shiksha Kendra I won't need to spend on the essentials. My salary will pay for the extras.'\n\nShiksha Kendra, set in a forest, miles away from nowhere, the brainchild of the philosopher S Swaminathan, a school which emphasised harmony with nature, respect for every form of life, and the all-round development of body and mind. All this the brochure said, and all this Mrs Trivedi felt when she visited the school. A home for herself and her children was what she was looking for, and at Shiksha Kendra she found it.\n\n'This school will not equip Ajay and Pipeelika for the competitive world'' warned the grandparents. 'They need to get ahead. They have no father, they are starting out with a disadvantage.'\n\n'Swaminathan got ahead'' said their daughter, somewhat elliptically, 'if fame and reputation are anything to go by.'\n\n*\n\nAjay, the son, showed his determination to succeed from a very early age. No need to tell him the disadvantages of his situation, he felt them all himself. A boy with competition in his blood, he stood first all his life, in school, in IIT, making straight to the US as soon as he possibly could with a wonderful scholarship to MIT. His success evoked tears of joy in all concerned. The widowed mother's sacrifices had borne fruit. He departed amid great jubilation, and never came back.\n\n*\n\nPipeelika, the daughter, was left to fulfil the hopes of her mother on native soil. After school, she moved up North, to Miranda House, college and hostel, to do an Honours degree in Sociology. After that an MA from the Delhi School of Economics.\n\nHer brother thought she should come to the States and do a PhD, increase her market value, he would sponsor her, but I do not wish to join the diaspora, and what about Ma, said Pipee, morally the superior. Instead, after a brief teaching stint, she joined an NGO run by three women, dealing with alternative education for slum children.\n\n*\n\nPipee had been working in Ujjala for four years when she met Aijaz Akhtar Khan, at a conference. She was reading a paper on the effects of communalism on the education of Muslim children in the basti. They were discriminated against, made to feel stupid and backward, were told their loyalties were to Pakistan, and looked upon with suspicion. Aijaz was reading on the use of street theatre in the dissemination of social and political awareness in educational institutions. Clearly their interests were similar.\n\nAfter the sessions were over, he sought an introduction, inviting her to a nearby dhaba for a glass of tea if she were free. She looked at him, the clear warm reddish light brown colour, the long thin nose, the gleaming even teeth, the thick grey hair, and then she smiled, her mouth turning in, making dents at either edge. Yes, she was free.\n\nThey talked for hours, it became dark and Aijaz insisted on escorting her back to her flat. She lay awake at night thinking of him. He seemed so gay and lighthearted, with many interests besides teaching. Not only did he manage and encourage drama activities in his college, but he was the prime mover and shaker of The Street Theatre Group. Drama was an effective way of addressing communal issues and dealing with social evils, if she liked he could bring the group to her basti. Pipee liked, and she was sure Neeraj, her friend and colleague would also approve. Ujjala was already involved in introducing drama to their children through the helpers, it would be a wonderful opportunity.\n\n*\n\nThe courtship took six months. Now Pipee wanted to marry him. Mrs Trivedi tried an old tack.\n\n'Your father would not have liked it'' she said.\n\n'He would have. My father was a secularist'' said the daughter firmly. 'Any father who names his daughter after an ant proves that.'\n\n(Pipeelika? That's not a proper name, that's a word.\n\nWhat does it mean? How do you spell it, pronounce it?\n\nIs this a real name? Never heard of it.\n\nIsn't that the Sanskrit for ant? How can you be named after an ant?\n\nAnd so on through the years.)\n\nPipee saw her mother momentarily at a loss and pursued her advantage. 'You can't deny it'' she said, 'my father didn't want his daughter's life cluttered up with references to goddesses, and now I am going to live up to his legacy. He married whom he liked, so did you, now it's my turn.'\n\n'You don't know what you are letting yourself in for. It is only later that you will realise.'\n\n'We'll wait for that day. Right now we are getting married.'\n\n'What does his family say?'\n\nPipee hesitated.\n\n'See. They are not happy either'' her mother quickly pointed out.\n\n'You are all the same. We don't care.'\n\n'Oh, Pip, everybody else will care'' sighed her mother.\n\n'Neeraj doesn't. She likes him, he is so charismatic it is hard not to like him, you would too if you gave yourself the chance.'\n\n'I keep telling you it is not him.'\n\n'So much the worse for you, Ma. Besides _Neeraj_ thinks, even if _you_ don't, that I will be happy with him, she encouraged me.'\n\nMrs Trivedi was silent for a moment. Neeraj was somebody she respected, whose interest in her daughter she had been hitherto grateful for. Now she said bitterly, 'You all work in your own NGO, and think you don't have to answer to anybody. My child you are swimming in a very small pond.'\n\n'Small pond! When I've been working with women for five years, going to all kinds of slums, seeing all manner of injustices done to people I have actually met. If we help them too overtly we alienate the community, and lose whatever influence we have. It's so frustrating. Ma, you haven't even seen a slum.'\n\n'I hope my daughter will not judge her partner by the men in slums'' said Mrs Trivedi crossly. 'And don't tell me what Neeraj thinks. I had no idea she would encourage you to go against your family and religion.'\n\nThis was not the time for Pipee to point out that she didn't give a shit about religion. 'Come on, Ma'' she said instead, wrapping her arms around her mother's neck, 'the world has changed, you don't realise it living in this tiny place. When you meet Aijaz you will love him, you'll see.'\n\n'At least make sure my grandchildren are Hindus. Once you marry God knows what he'll make you do.'\n\n'Ma! They will be his children too. He's not that sort of person, and do you think I would love him if he were? He never mentions religion, except politically, never suggested conversion, nothing. In fact you are the one obsessed with the whole thing.'\n\n*\n\nAijaz to Pipee in Delhi, 'Was it bad?'\n\n'It could have been worse.'\n\n'Poor thing, it must be very hard for her'' said Aijaz, shifting Pipee's head more comfortably on his shoulder. They had finished making love, and were talking about their marriage.\n\n'When she sees you, she will come around.'\n\n'The whole world may not have your faith in me.'\n\n'But my world does, and she is a big part of it.'\n\n'You are very close to her, aren't you?' asked Aijaz wistfully.\n\n'Of course. She is all I have. My father's family don't like my mother, we are not in touch with them, my grandparents disapprove of my lifestyle, and Ajay shows no signs of coming back. What's the use of having a son and brother if all he does is write patronising letters from the States?'\n\n'Well, you have me now, and so does your mother.' Aijaz pulled Pipee on top of him, and pushed his hands through her hair, pulling her head back so that he could look at her milky skin, and pink mouth with its indented corners that smiled in a peculiar way. She smiled now, loving the feel of his hands in her hair, the way he massaged her scalp.\n\nPipee had a lot of hair, it sprung up all around her head in waves and curls and frizzes. Aijaz loved it, loved it almost as much as he loved her breasts, large and full of give. He shifted his hands to them, and Pipee squirmed a little. She was still not used to how much sex Jazu seemed to need, but men were like that, and all the time before and after was the stuff of happiness, when they were talking, wrapped in each other's arms.\n\n*\n\nAfterwards Aijaz cast a nostalgic look around, 'I will miss this room'' he said.\n\n'I won't'' said Pipee. 'The landlord is an extortionist. This must be the smallest room in all of Delhi.' In fact it was one of six tiny ones, built around a spiral staircase in the back spaces of one of the larger houses of Greater Kailash II.\n\n'Our love grew here'' pronounced Aijaz.\n\nPipee laughed, 'Well it can flourish somewhere larger.'\n\n'Yes'' said Aijaz thoughtfully. 'Sometimes I wish I had my own flat \u2013 but out of the house all day, teaching, travelling, theatre \u2013 being a paying guest was the most convenient thing.'\n\n'Soon both of us will have a proper home. I am sick of living in hostels and rented rooms. It's been almost ten years, but now all of that is over.'\n\nPipee flung an arm out, the future glinting in her eyes. Aijaz smiled and kissed the waving arm.\n\n'You have to promise to spend more time with me'' went on Pipee. 'I refuse to be a nagging wife. You have to promise and keep your promise, and never break it, without my saying a single word.'\n\n'Of course. Why do you think I am getting married?'\n\n'Sex every minute, seems like.'\n\n'You think one needs to get married for that?' laughed Aijaz.\n\nPipee remained silent.\n\n'What's the matter?'\n\n'Nothing.' But the number of women Aijaz had had bothered her sometimes.\n\n'I want to settle down, I want a home, I want you'' said Aijaz turning to Pipee impatiently again.\n\nPipee pushed him away, 'Really Jazu, sometimes I think you just have one thing on your mind.'\n\nAijaz looked proud and manly. 'Wait till we are living together \u2013 then you will really see.'\n\n'Yes, let's see if marriage will cool your ardour.'\n\n'My ardour, as you put it, will never be cooled. And we must really start looking, it's very difficult without a company lease, or months of rent in advance.'\n\n'I've a surprise for you.'\n\n'You've found a place!'\n\n'In a way.'\n\nAijaz looked wary. 'What way?'\n\n'Now listen \u2013 listen properly\u2014'\n\n'I'm listening, I'm listening.'\n\n'You know what Premlata said to her father, when he was going to marry her off? She said thirteen was under-age and against the law, and if necessary, she would call the police! Wasn't that brave of her?'\n\n'Very. But what's the connection?'\n\n'Our efforts are bearing fruit, that's what. After three years of going to our centre at Salempuri, more children have reached literacy level, more girls are going to school, and you wouldn't believe how some of them have changed! They always worked hard, these girls, they cook, wash clothes, look after the cows, buffaloes, younger brothers and sisters, send them to school, help in the family business, they embroider, make envelopes, necklaces, sew sequins on, but are often made to feel worthless. But at the centre they develop self-confidence, look at Premlata! We want to open more centres.'\n\n'I still don't see what that has to do with us'' repeated Aijaz patiently. He had heard Pipee about her work many times.\n\n'Since we are expanding, we are going to apply for permission from the home ministry for foreign funding. Then Ujjala will hire me a flat in lieu of a raise in pay. They know I'm getting married\u2014'\n\n'I take it they don't disapprove of me, no don't tell me, our marriage is a strike for communal harmony.'\n\n'What's wrong with their approval?'\n\n'Your mother hates me because I am Muslim. Your friends love me because I am Muslim, I don't know which is worse.'\n\n'How does it matter? Look what they are doing for us, isn't that nice of them?'\n\n'Very'' said Aijaz with reserve. 'And I am sure they will extract their pound of flesh. Make you work ten times harder, demand your presence so much you will hardly be in your precious flat.' There were times when he resented the women in Pipee's life, especially Neeraj.\n\n'You don't know how women operate. Just think, we will have enough space to have my mother visit us in Delhi during her holidays, and of course your family too.'\n\nAijaz yawned and turned away. Pipee tried to suppress her annoyance. Why was the man so unwilling to discuss his family?\n\n'Have you told them yet?' she demanded.\n\n'I'll tell them, I'll tell them, what's the hurry?'\n\n'They're your family, I want to meet them, know them.'\n\n'You are so idealistic'' remarked Aijaz.\n\n'It must be nice to have so many people belonging to you.'\n\n'It's a total pain in the ass. You can deal with one person's expectations, but here there is the whole community.'\n\n'So?'\n\n'So they all take my father's side. He has never accepted my theatre activities. If his eldest son wanted to be a lecturer, the least he could do was help with the mango orchards in Shahjehanpur during the summer, instead of getting involved in some silly drama-shama. It doesn't even pay, which makes it that much harder to understand.'\n\n'We can both help with the orchards from time to time'' said Pipee enthusiastically and ignorantly.\n\n'That's not all. They wanted me to marry my cousin Azra. My mother was especially keen since she had brought Azra up after my aunt died. I suppose they were trying to make sure I eventually returned. They don't understand my life, they don't realise I have no time for all this fuss.'\n\n'I am sure they will hate me'' said Pipee in a small voice.\n\n'We'll take it as it comes. Why worry now?'\n\n*\n\nIt took six months for the grant to come through. The accommodation they ended up with belonged to Neeraj's sister's husband settled in the States. He had bought a flat in Vasant Kunj as an investment, and was now looking for a reliable tenant (i.e. one who would leave when asked). Neeraj convinced him that Ujjala and Pipee were what he was looking for.\n\nIt had two bedrooms, two bathrooms, a kitchen with built-in closed shelves, a dining area at right angles from the sitting area. Outside the sitting-dining there was one big balcony, outside the bedrooms there were two smaller ones.\n\n'I'm going to have a potted garden here'' said Pipee, stretching out her arms to the hot white sky above the verandah. 'I'm going to have everything. I can't wait to show it to my mother.'\n\n'Look at all the space! How clever you are, darling'' exclaimed Aijaz, putting his hands under her shirt, and unhooking her bra as she walked about.\n\n'Well it was really Neeraj. She always manages to find solutions to problems.'\n\n'I prefer to think it was you.'\n\n'Are you jealous?' laughed Pipee. 'You shouldn't be, she loves you.'\n\n'She hardly knows me.'\n\n'I talk to her sometimes.'\n\n'About us?' Aijaz looked appalled.\n\n'One can't be talking of work all the time'' temporised Pipee, and then to change the subject, 'Oh, I never told you, we will also have a phone, think how nice that will be.'\n\n'We could have got that on our own.'\n\n'We would have had to wait years.'\n\n'I have connections too, you know, I could have got us a phone. Now stop moving.'\n\n'Jazu, do you ever think of anything else?' murmured Pipee, as she so frequently had to.\n\n'No'' said Aijaz pinning her against the wall, seriously this time.\n\nIt was in September 1988 that the marriage between Aijaz Akhtar Khan and Pipeelika Trivedi was solemnised in Tees Hazari. The bride and groom paid for their own wedding, the whole thing came to five hundred rupees. No relatives were present from either side, a colleague of Aijaz's and Neeraj acted as witnesses, while the theatre crowd, a few of Aijaz's colleagues, and the staff of Ujjala, later gathered at Karim's to complete the celebratory aspects.\n\nPipee had arranged her work so that she would be free the two weeks of Aijaz's autumn break. They were going to Shiksha Kendra, and as Mrs Trivedi's winter holidays started in mid October, they would all come back together.\n\n'I think we will avoid my grandparents, they needn't really know we are coming. Besides they are very orthodox, and will fuss like mad over the Mozzie issue.'\n\n'I can pretend to be a Hindu if you wish.'\n\n'I wouldn't dream of it, why should you? You are not a pariah, after all.'\n\n'It's not a question of pariah, what difference does it make? Old people need to be treated carefully.'\n\nPipee needed only a second to realise the possible personal application of this remark. 'Will your family look upon me as a pariah? Shouldn't we visit them so that they get to know me?'\n\n'No, let's give them time to get used to it first'' said Aijaz. 'Besides your mother is coming back with us, and we can't complicate matters.'\n\n'If your mother came too, they could be company for each other'' said Pipee, showing how little she knew of the science of in-laws.\n\n'Another time.'\n\n*\n\n'You have to travel quite a bit to this school of yours, Pip'' said Aijaz on the second day of their train ride to Bangalore.\n\n'It's the best school in India'' said Pipee proudly.\n\n'And like all shrines, difficult to reach'' replied Aijaz looking deadpan.\n\nPipee smiled in the way Aijaz loved to see, the corners of her mouth turned in, the deepening dimples. She pinched his side several times in the crowded second class compartment. 'You'll see'' she said loftily, 'I will say no more.'\n\n'Promise?'\n\nThis time Pipee pinched him so hard, he cried out, and everybody looked at them with curiosity and disapproval. Young, alone and enjoying themselves.\n\nIn Bangalore they took a bus to Madanapalle. 'From there we will take a taxi'' said Pipee. 'It is sixteen kilometres.'\n\n*\n\nAs they drove away in the taxi towards Shiksha Kendra, Pipee grew thoughtful, the dimples and the smile went.\n\n'Why so quiet, dearest?' asked Aijaz, 'I'm not used to it.'\n\n'Nothing much.'\n\n'Come on, tell me.'\n\n'It was in school that I first fell in love, and now I am coming here on my honeymoon. I feel strange when I think about it, that's all.'\n\n'Your first love! You never told me.'\n\n'There was nothing to tell.'\n\nAijaz ignored this. 'Who was he?' he went on.\n\n'She.'\n\n'She?'\n\n'Her name was Samira.'\n\n'You were in love with a woman?'\n\n'Woman? Hardly that. Schoolgirl really. She was only seventeen.'\n\n'That's not so young. In my village girls marry at sixteen, how old were you?'\n\n'Well Shiksha Kendra is not Shahjehanpur'' said Pipee a little coldly. 'And what does it matter how old I was? It was so long ago I do not remember.'\n\n'Did your mother know?'\n\n'What was there to know? We were schoolgirls'' said Pipee withdrawing from the conversation.\n\n'Where is she now?'\n\n'She married'' said Pipee shortly. 'We lost touch after college.'\n\nAijaz fell silent. Pipee was so unlike her usual self that he didn't know what to think. It must have been like those crushes that girls had on filmstars or their teachers. She was young and inexperienced and imagined her feelings to be love.\n\nHe looked sideways at her, she was still looking remote. Did she think he was narrow minded enough to disapprove of a schoolgirl crush, he who knew of the strong ties that existed between women in the zenana? He reached for her hand. 'Don't feel sad, Pip. I am here. After all this is our honeymoon.'\n\nPipee smiled at him and thought there were some things that could not be shared, no matter how understanding the other person. All said and done she was lucky to have found him. So many of her acquaintances were still struggling, looking for love and companionship, rejecting arranged marriages, only to experience a series of heartbreaks on their own.\n\n*\n\nAt the gates of Shiksha Kendra, Pipee stopped and paid the taxi. 'I want to walk'' she said to Aijaz. 'I want you to see it slowly, take it all in.'\n\nThe path leading inside was wide with thin trees lining the way, shady, green. 'The school planted these'' said Pipee gesturing around her. 'This is a drought area, even now the leaves are drooping.'\n\nAijaz looked carefully and could see that indeed the leaves were drooping. 'They have their own dairy, bakery, their own gardens, fruit trees, imli trees, mango trees, which they lease out'' continued Pipee.\n\n'U-huh'' said Aijaz.\n\n'We were not allowed to touch any of the fruit because it wasn't the school's.'\n\n'But of course you did.'\n\n'Of course.'\n\n'Where are we going?'\n\n'They will be eating now'' said Pipee looking at her watch, 'It's one.' And she started to walk faster, though laden with bags, afraid to miss her mother in the dining hall, anticipating the surprise and pleasure on her face.\n\nThe din in the dining hall was deafening, though Pipee didn't seem to notice. Aijaz hung back as she scanned the rows of tables and benches.\n\n'There she is'' she said, making unerringly towards a particular back.\n\nHer husband remained in the doorway watching the reunion.\n\n*\n\nLater in Mrs Trivedi's two rooms in Peacock House. 'Mama, don't you like Aijaz? Isn't he all I promised?'\n\n'Very much beta'' said the mother. 'He is your husband after all.'\n\n'Let's show him where I grew up'' Pipee went on.\n\n'What is there to show? These two rooms.'\n\n'And where Ajay didn't grow up.'\n\nMrs Trivedi shot a glance at her son-in-law, who was careful to look as bland and harmless as possible. 'My son stayed at the hostel'' she said. 'It was better, he could participate more in the activities, and of course he came every day.'\n\n'For fifteen minutes'' said Pipee.\n\n'And every weekend.'\n\n'Just to eat.'\n\n'Pipee thinks he should have stayed with me like she did'' went on Mrs Trivedi. 'She doesn't realise boys need to be with their own kind. He had a housemaster who was like a father to him. And I had Pipee.'\n\nPipee pressed her cheek against her mother's, 'And now you have Jazu.'\n\nJazu looked charming.\n\n'Indeed'' said Mrs Trivedi.\n\n*\n\nThey spent ten days at Shiksha Kendra. Pipee took Aijaz over the entire campus, the banyan tree, the rocks they had to climb, the place where they watched the sun rise every morning, the art, music and dance rooms, the playgrounds, the senior school, the library, the lab where she had sat for her exams.\n\n'It's a whole world in itself, isn't it?' wondered Aijaz.\n\n'Some parents are not happy about how cut off it is, they think their children will not be able to survive outside. But look at me. I've survived perfectly well.'\n\n'Indeed you have'' said Aijaz kissing her. 'A perfectly untouched specimen.'\n\n*\n\nPipee was right, once Mrs Trivedi came to know Aijaz, she loved him. They had been in Delhi two weeks when Pipee said triumphantly, 'Well Ma, what do you have to say about Muslims now?'\n\n'He is a very good boy, beta'' responded Mrs Trivedi.\n\n'Then?' said Pipee sharply for she knew what was coming.\n\n'I am sure his family like you equally'' said Mrs Trivedi smoothly.\n\n'They will when we go to meet them in the holidays. They are a very large family, and his mother is old and cannot travel easily'' said Pipee, with a guile to match her mother's.\n\nLater to Aijaz, 'I've told her that we are going to meet your family in the holidays.'\n\n'What was the need to do that?'\n\n'She happens to think the lack of your family presence in our marriage very strange. Don't mind, she is just an old worry pot.'\n\nAijaz said nothing. Pipee felt a pang of guilt. What did it matter about his family anyway? Let them think whatever they wanted, she should not make it more of an issue than he did. Besides Aijaz had been so sweet to her mother, coaxing her from her prejudice, never seeming to mind her oblique references to Muslims, four wives, large families, instant divorce, inter-community marriages, the religion of babies from such unions. The more she relaxed with him the more she wanted to know.\n\n'There is a limit to your questions'' Pipee shouted one day. 'Is he your son-in-law or the whole Muslim community dating from Babur's time to now?'\n\n'It's all right'' said Aijaz soothingly. 'Let her ask. After all I am the first Muslim she has had anything much to do with.'\n\n'Still.'\n\n'I'm used to this. She is not alone.'\n\nSuch gentleness deserved to be rewarded by total belief. Pipee vowed that she would never mention Aijaz's family unless he himself brought them up.\n\nAfter Mrs Trivedi left, the young couple settled into the joys of living on their own.\n\nPipee's hours were flexible, and she tried to be home by the time Aijaz arrived. This was usually not difficult. After a morning of teaching Aijaz was often at college rehearsals, or working out programmes with The Street Theatre Group. And, thought Pipee indignantly, everybody imagines academics to have nothing but free time.\n\n*\n\nUjjala was by now in the process of establishing a community centre at another basti. This second place had a greater number of facilities. It had sewing machines for women to acquire a skill to increase their earning potential, it had a library, toys, and art and craft supplies for the fifteen to twenty children who came every afternoon from three to five.\n\nSoon they extended their activities by organising trips outside Delhi. The results were encouraging. Girls, helpers and administrators bonded, and the girls' sense of themselves strengthened. Each of them wrote a piece on how she had experienced the trip, what she had felt being away from home with others from the basti, for the first time not part of a family structure. Pipee put these together in a series of booklets called _Yatra_ _aur_ _Vichar_ that she spent many hours over. She was filled with a sense of achievement, all day with Ujjala, every other moment with Aijaz, she thought life could have no more to offer.\n\n*\n\nIt was almost a year after their marriage that Aijaz made a casual announcement. 'We have to go to Shahjehanpur. They want to see you.'\n\nPipee, lounging on the cushions of the cane double-seater they had recently bought, looked up, astonished. In all this time Aijaz's family had shown no signs of her existence.\n\n'That's nice'' she said carefully.\n\nAijaz stared moodily at the balcony. Pipee gazed at him, and for the thousandth time thought how she loved the way he looked. His wavy grey hair, his clean brown colour, his sharp nose, his warm eyes. 'You know my mother also had her reservations'' went on Pipee encouragingly.\n\n'Mine would have too, had she known'' muttered Aijaz.\n\n'What? What did you say?'\n\n'You heard me.'\n\n'Do you mean your mother \u2013 family \u2013 didn't know we were getting married.'\n\n'Something like that.'\n\n'You didn't tell them?'\n\n'How could I tell them?' demanded Aijaz. 'You knew the problems.'\n\n' Still, you could have _told_ them. They must be feeling awful now, much worse than if you had _told_ them.'\n\n'For God sakes, Pip, stop going on.'\n\n'You hide things from them, from me, and you accuse me of going on'' shouted Pipee. 'How do you think I feel?'\n\n'You have to take me as I am'' shouted Aijaz back. 'Me, alone. If I didn't tell them it was to spare them pain, and you trouble.'\n\nPipee tried to tell herself that Aijaz was an exemplary human being, socially committed, personally tender, but this palliative irritated her further. He had no moral right to behave in a way that didn't add up.\n\nAll the things her mother used to say about Hindu-Muslim marriages came unpleasantly to her mind. For a moment she stared at him with revulsion. What was the use of him looking like a dream if he could behave like a nightmare?\n\n'What the hell, Aijaz'' she said, 'you have a poor idea of trouble. You have not been fair to your family or to me.'\n\n'I'm sorry, Pip, I really am, don't be angry. My family is not like yours. There are so many, and they all want to be part of things, they would never have tolerated a Tees Hazari wedding, we would have had to go there and get married amid five thousand people at least, God it's enough to put anyone off. And then there might have been fuss about the conversion thing \u2013 I didn't wish to put you through all that.'\n\n'Or yourself'' said Pipee dryly.\n\n'Whatever'' said Aijaz, looking charming.\n\n'Well, why now?'\n\n'They heard rumours. Made enquiries.'\n\n'So I am going to meet them with the guarantee that they will hate me.'\n\n'They'll adore you.'\n\n'With this background?'\n\n'You don't know my family. Once they know they can't change things, they just accept them.'\n\nThis time Pipee kept her reservations to herself. 'When are we going?' she finally asked.\n\n'We are on the waiting list. As soon as I can get confirmed bookings.'\n\n'What about my work? We have a big meeting with the community helpers from both centres next week. Neeraj, the others, won't like it.'\n\n'Tell them you are going to visit your Muslim in-laws. They will love it.'\n\n*\n\nTheir reactions were reserved, which was just as I expected, thought Pipee, and shows how little Aijaz knows of families in general.\n\n'I hope they like me soon'' she said to him on their first night in Shahjehanpur.\n\nIt was summer, and their beds were spread on the terrace, in deference to their married status, a little separately to one side of a storeroom. As soon as it was late enough they squeezed together in one under a mosquito net.\n\nAijaz yawned. Pipee poked him. 'Do you think they will?' she asked.\n\n'Give it time, Pip, now let me sleep.'\n\n'I told you they wouldn't like me.'\n\n'They are so glad I'm married, they would have liked anyone.'\n\n'But they would have preferred a Muslim?'\n\n'Come on, Pip, be reasonable. After all your mother would have preferred a Hindu. Anyway who has the time to worry about such things?'\n\n'I suppose'' said Pipee forlornly, thinking of his mother and the jewellery box she had pushed in front of her.\n\n'For my eldest son's wife'' she had said.\n\n'No'' said Pipee, embarrassed, yet dying to look at what was inside.\n\n'Take'' said Ammi, with a trace of reproach, her hands busy with the lid. Pipee gazed at the plump, rounded fingers, studded with gold rings, the short nails which gleamed with clear nail polish, the kurta sleeves long and fitting, and the many bangles that tinkled at her wrists. She looked very sure of herself, unlike her own poor mother, who lived in two rooms at Shiksha Kendra, with no one to boss over except some very small children.\n\nEventually, since she would not take, she was given a heavy gold necklace, thick gold bangles embossed with flowers, and a set of jhumkas set with pearls and rubies.\n\nShe held them, admiring their beauty, marvelling at their heaviness before returning them, I have no locker, I will have to worry about their safety, keep them for me please.\n\n*\n\nIn the days that followed, Pipee realised for the first time she had married a Muslim. Everything was strange, the large haveli, the dishes they ate from, the food they ate, their paan making, the way they dressed, the way they greeted each other, As salamalaikum \u2013 Wa Alaikum Assalam, their manner of speaking, the kh's that made her Hindi tongue seem crude and unsophisticated.\n\nAnd then there were so many relatives. How many people lived in that house, till the end of her visit she did not know. They were a world complete unto themselves, so different from anything she had known while growing up. Occasionally when eating in the long dining hall, she would gather as many as she could within a single glance and feel a great longing for the day when she would be completely accepted as one of their own.\n\nIt was the year 1989, and bricks were being collected for the Ram Mandir \u2013 collected, worshipped, and escorted out of towns, wrapped in silk and saffron, on their way to Ayodhya. If communal disturbances occurred in the wake of these processions, that was not the fault of the bricks, but the fault of the narrow-mindedness of minority communities, who couldn't bear to feel that their domination in this country was over.\n\nIt was in this atmosphere that Aijaz and The Street Theatre Group travelled to Rajpur fifty kilometres outside Delhi to put up a play.\n\n'I wish you wouldn't go'' said Pipee, 'Rajpur is a sensitive area. It is not safe.'\n\n'If I only went to places where it was safe, I would never go anywhere'' said Aijaz. 'Theatre is a limited medium, but what else do people like us have?'\n\n'Then don't go'' said Pipee, 'don't go if it is no use.'\n\nAijaz looked depressed. 'One has to do what one has to do'' he said. 'Of course it is so much easier working with people from schools and colleges, they even write the scripts, and do the research.'\n\n'Well stay here, and go to schools and colleges, instead of dashing out on weekends to some town or mohalla, or factory, god knows where all. Now you are married you have a responsibility to me, to us'' said Pipee, and then felt guilty. Here she was sounding like a nagging wife. Would she like it if Aijaz stopped her from going to the bastis? Or decided that her work with Ujjala led her into dangerous situations?\n\n'What is the use of confining oneself to the middle classes where it is safe \u2013 safe and cowardly'' went on Aijaz reflectively.\n\n'At least wait till the whole fuss about the bricks is over'' amended Pipee. 'I will come with you next time. I've never travelled with you. Besides you will be leaving me alone on New Year's Eve.'\n\nAijaz looked at her in astonishment, 'I never knew this day meant anything to you, Pip. It's just a capitalist device for making money.'\n\n'If it can keep you home, then I am a committed capitalist.'\n\n'You are being totally neurotic. When I go somewhere nice, then you come. The mohallas of this township are dirty and crowded, there is nothing much to see or do. I'll be worrying about you, instead of concentrating on the play and the group.'\n\n'I can take care of myself'' said Pipee with great dignity.\n\n'So can I'' said Aijaz ruffling her hair.\n\n'Didn't you know this man?' asked Hemant looking through the papers three days later.\n\n'Which man?' asked Astha indifferently, her life an arid desert so far as men were concerned.\n\nHemant flapped the papers in front of her. There, in the middle of page three were the headlines, THEATRE GROUP BURNED ALIVE IN VAN, and below the story:\n\nA horrendous incident took place here last night, in the township of Rajpur. Aijaz Akhtar Khan, noted theatre activist, and his troupe were dragged from the site of their performance, and taken away in a Matador. Later the charred remains of the Matador along with the bodies were found near the river. The culprits are still absconding.\n\nIt is surmised that rising tensions between two communities led to this action. Aijaz Akhtar Khan, leader of the well-known Street Theatre Group was in town to perform in the mohallas. The issues dealt with were of a sensitive nature, and it is surprising that in this time of communal unrest he got permission to stage a piece involving the Babri Masjid-Ram Janambhoomi controversy. The District Magistrate says he was deliberately misled about the contents.\n\nAccording to our sources, a procession containing bricks for the proposed Ram temple in Ayodhya was routed through a gully adjacent to a minority community mohalla earlier in the afternoon. Despite the presence of the police, slogans were shouted. Untoward incidents were then avoided, but that evening violence, possibly premeditated, broke out during a performance by The Street Theatre Group. Unruly elements in the crowd started heckling the actors. Other elements responded. In the confusion the members of the group were driven away in a van, ostensibly for safety. This seems to have been a ploy.\n\nAijaz Akhtar Khan has left behind a wife.\n\nThere followed a list of the other members of the theatre group, along with their survivors, but Astha could not read further for the tears in her eyes. What a way to die, what a horrible, horrible way to die \u2013 and for what? Because the man was trying to reach people and do some good. She hadn't even known he was married. She turned away her head to cry some more.\n\nHemant, watching her, immediately lost his temper. 'Why are you crying?' he demanded. 'What was he to you?'\n\n'Some murderers trap and burn a whole theatre group in a van and you ask me why I am crying?'\n\n'This kind of thing happens all the time, I don't see you wasting your tears.'\n\n'I can't weep for the whole world, only when it means something to me. Maybe I am deficient, but I knew him, he was always working for everybody's good, even the children loved him. And he has been burnt to death. Isn't that reason enough?' she sobbed rocking to and fro with rage and grief.\n\n'Don't get me wrong, this should not have happened. But if you meddle in things that do not concern you, you have to take the consequences. He was a Muslim, he should have kept to the issues within his own religion.'\n\nAstha stared at her husband in revulsion. Ten men had died in the most ghastly way possible, and this was all he could say. Did he have no feelings?\n\n*\n\nAfter Hemant left for work she started phoning. Identification of the bodies was being done at Willingdon Hospital, they would probably be released the next day. A condolence meeting was being held that afternoon at the Constitution Club. The next day there would be a funeral procession that would start at the Club and go all the way to the electric crematorium.\n\nNumbly Astha put on a white sari, she would go straight from school to the meeting, at least she would be with people who felt as she did. She would meet his wife, what would it be like to be her at this moment, and to have your husband dead like this. Could you ever get over it, should she arrange for the driver to bring her children there after school, they had known Aijaz, they would grieve with her, they should be exposed to the political realities of this country, but then to be exposed to such violence, such mindless hate, how could she explain it, she could barely deal with it herself. Political realities could wait, Mala would look after them, if she was late they could go upstairs.\n\n*\n\nAt the Constitution Club mourners were gathered on dusty lawns, standing on sidewalks, dressed in white, with black armbands, sombre faced. There were many speeches:\n\nWe are witnessing crimes deliberately stoked by the forces of communalism. Neutral voices are seen as threatening, the voice of secularism is not tolerated. Can ten men be burned alive, taken from the mohalla in full view of everybody without connivance from the authorities? What has the State done so far, what have the police done so far to apprehend the criminals? Is this the message for the citizens of this country, live in fear, do not raise your voices for they will be stifled by fire, murder and violence.\n\n_This_ is what the state provides, _this_ lawlessness, _this_ disregard for life, _this_ brute force. _This_ is its protection for its citizens.\n\nTo speak and be heard is the freedom that is at the heart of a secular nation, this is the right for which these brave young men gave their lives. Now we must carry on as though they were in our midst, forcing us to resist repressive fascist forces. This is the struggle that lies before us.\n\n*\n\nAstha saw Mrs Dubey, her eyes damp and swollen, she went to her and touched her on the shoulder, they stared wordlessly at each other, and then Astha's own tears, soaking her hanky, her nose running.\n\nIt grew dark. Candles were lit and passed around. They started singing. Songs of protest, songs that Aijaz had penned, songs that many had sung in different circumstances. They ended with _We_ _Shall_ _Overcome_ in Hindi. Word went around about the funeral arrangements. Tomorrow they would start at noon from the Club and walk all the way to the Crematorium with the ten bodies. Let the city see the atrocity that had been committed, let the traffic come to a standstill, let the line of death be visible in slow motion.\n\n*\n\nNext day there was a crowd of thousands waiting for the bodies to be released from the hospital. Many had not known the ten men, but it was not necessary to have known them. They came to protest an outrage, to arouse similar protests from an anaesthetised public. Artists and innocent men have been murdered without any provocation during a performance in broad daylight. Today them, tomorrow us. How can this happen? What can we do?\n\nFinally the procession started. On and on they walked, blocking traffic, creating havoc, silent, disciplined and determined. The police tried to stop them, they did not have permission, they would have to turn back. The news spread \u2013 they are trying to stop us, we are going to defy them, nothing can turn us back, we will fight if necessary and then the police had to give in, escorting them instead, as they walked down the streets of Daryaganj, past the Jama Masjid, turning right towards Ring Road, then on to the electric crematorium, where thousands more were waiting to receive them.\n\nIt took six hours to reach their destination. The vast room quickly filled while the rest of the crowd waited outside. The families of the men laid the bodies out, and two by two their charred remains, indistinguishable from one another, were slid into the massive fires and the doors clanged shut. They had been together in life, and they were together now. Silence occupied the hall. Astha watching from a squeezed-in place near the door remembered the Aijaz she had known, and that once she had thought he smiled too much.\n\n*\n\nFour days later a massive protest rally was organised from the Red Fort to the Prime Minister's house.\n\n'I shall be late coming home from school today'' said Astha to her husband that morning. Her tone was cold; she had still not forgiven him.\n\n'Why?' he asked busy with his own preparations for the factory. 'Where are you going?'\n\n'To a rally to protest the circumstances of ten men's deaths.'\n\nHemant looked at Astha. Astha returned the look.\n\n'Whenever did rallies do any good? Goondas hire people from neighbouring villages at ten rupees a day to come and make trouble, block traffic and show their muscle.'\n\n'It's not the political, made-up kind of rally. We want to draw attention to what has happened. How does one speak so that one is heard? You tell me a better way.'\n\n'Rallies!' snorted Hemant ignoring the question. 'No matter how big \u2013 who cares \u2013 who remembers what they are about?'\n\n'Besides, we don't want their memories to die.'\n\n'I'm sure you don't.'\n\nAstha left the house without a further word.\n\n*\n\nBy the time school had finished and Astha reached Red Fort, the air was thick with banners. Some of the marchers were carrying posters with Aijaz's photograph hugely blown up. Some were carrying banners with Leftist slogans. Black armbands were being passed around.\n\nThe rally set off. Down the road, shouting slogans, they marched, blocking traffic in a way that Astha found most satisfying. Cars were standing still, motorists were fuming, and people were getting late because of her. She shouted with the others:\n\n_Sampradayakta_\n\n_Down_ _Down_\n\n_Down_ _Down_\n\n_Communalism_\n\n_Will_ _not_ _succeed_\n\n_Will_ _not_ _succeed_\n\n_The_ _Street_ _Theatre_ _Group_\n\n_Martyrs_ _All_\n\n_Aijaz_ _Akhtar_ _Khan_\n\n_Remembered_ _Forever_\n\nWhy did they have to die like this, thought Astha, trapped in a van, what were his last thoughts, he who had lived for others. How was there any fairness in the world when such a man could be murdered so brutally? Tears came to her eyes, but tears were not an adequate tribute to Aijaz, they were too ephemeral.\n\nHe had seen talent in her, what was it like to live with a man who saw you as having something to offer? If only there was some cause to which she could devote herself, maybe she would not feel so lost and dissatisfied, but what, and how? Knowing what to do was so difficult, and brooding over her life she continued to shout and raise her fist with the others. Down Red Fort Road, past the Asian Circus, past the Centre for Tibetan Refugees, past the Kashmiri outlet for woollen shawls, past the police chowki, past water sellers, lemonade sellers, past bhelpuri wallahs down Connaught Place and Janpath marched the procession. Compressed into half the road, cars were inching along, staring at them, curious, sympathetic, frustrated, annoyed.\n\nThey reached the boat club. Astha sank under one of the trees, extremely hot and tired. She had not realised her clothes were unsuitable for marching in the sun, she was wearing a thick black polo neck sweater, with Hemant's vest on underneath. This meant that though damp and hot, she couldn't possibly take it off and be exposed in her underwear.\n\nThe speakers on the stage were beginning to talk about state atrocities, an endless list. After that were impassioned recitals of Brecht's poetry in Hindi. Fists were clenched, defiance was hurled towards parliament looming above the tree tops behind the boat club.\n\nAn hour later the procession set off towards the prime minister's residence. Three roads away they met a police block. 'No further'' said the policemen. 'Question of security.'\n\nThey handed over their memorandum, and were forced to disperse.\n\n*\n\nAs Astha was leaving, her principal stopped her. 'Astha meet Reshana, she used to be a singer for The Street Theatre Group. She was especially close to Aijaz.'\n\nAstha stared at the direct eyes, the face still with sorrow. Especially close \u2013 how close was that? What about his marriage \u2013 was she close before or after?\n\n'I am trying to meet all those who worked with him'' Reshana was saying through swollen lips. 'We have to make sure his memory does not die, are you interested?'\n\n'There is nothing I wouldn't do for him'' breathed Astha.\n\n'Good'' said Reshana. 'I will inform you of our first meeting.'\n\nAs Reshana left, Astha turned to Mrs Dubey, 'Who is she?'\n\n'Reshana Singh. She is a classical singer from an old and established family. She has many connections, it is good she is taking such an interest in this cause.'\n\n'Is Aijaz's wife not here?'\n\n'Poor thing, I only saw her at the funeral. I don't think she is able to cope with the shock of it all.'\n\n'I would have liked to meet her.'\n\n'When she recovers, we can arrange something.'\n\n*\n\nIn the evening Hemant asked somewhat testily how it had gone. Astha was too full of the day to continue angry with him. If he was limited, that was his misfortune, she could be generous. Where should she begin, the crowd shouting slogans, the palpable determination to do something, singing _We_ _Shall_ _Overcome,_ the sense of togetherness, her excitement at Reshana asking her to be part of the new society.\n\n'The traffic arrangements were terrible as usual'' said Hemant, not realising she had an answer. 'I had a meeting with the distributor in Connaught Place, and getting there was totally impossible. Why do they allow rallies in the middle of the day, in central Delhi, I'll never know. Arre, you want to protest, protest, who is stopping you? Let the ordinary tax-payer lead his life, that's all I ask, but no.'\n\nAstha's generosity was not required, her sharing could keep. She could not enter into his frustrations, he could not share her enthusiasm.\n\nFor the rest of the evening, they talked of the children, Hemant's concern about his mother's arthritis, his father's blood pressure, his forthcoming trip to South Korea, and maybe they could all go abroad next year for a holiday, and finally something that was beginning to bother him more and more, the increasing competition in colour TVs.\n\nIn Noida alone where Hemant had his factory, eight others had come up. He was making 1,500 black and white, and 1,200 colour TVs a month, but the market had become so cut-throat that he was forced to reduce his profit margin to maintain his position.\n\nNever mind, Astha tried to console, dragging her mind to business concerns, now that the government has allowed religion on TV, there will be no end to the shows that will have the same kind of popularity as the _Ramayana_ and the _Mahabharat._ There was a captive audience of millions, with a big enough market for all.\n\nHemant grunted. That very day he had heard that a rival factory was trying to instigate a strike amongst his workers. He had managed to bribe the men in question, but the general atmosphere of suspicion made it difficult to go on bribing. He would discuss the problem with his father in the morning. \nChapter V\n\nA few days later at the meeting of the Sampradayakta Mukti Manch, a forum set up in memory of The Street Theatre Group, it was decided that painters should donate a painting for an exhibition devoted to worker unity and secularism. Astha was busy staring out of the dirty windows, where was his wife, was she still getting over her grief, how come she wasn't there, would she never get to see her, when she heard Reshana address her.\n\n'Astha, what about you?'\n\nAstha panicked. Why was Reshana asking, had Mrs Dubey made claims about her talent, but she was no good, she was a beginner, a drawer, a sketcher, nothing serious, her support was absolute, but she could do nothing on her own.\n\n'I'm not sure'' she managed.\n\nReshana smiled warmly, 'Just try'' she suggested.\n\nAstha felt the disapproval of the gathering at her delayed consent. There were too many people looking. She nodded, and sank back into her chair.\n\n*\n\nHer anxiety over her task was so great, she had to start immediately. After school the next day she sketched crowd scenes, patterning them on Rajasthani miniatures, trying to choose between a funeral and a procession. Finally she decided there would be more colour and interest in a procession.\n\nShe painted a broad road, on one side lines of figures, dots of black hair, holding banners, on the other side, rows of cars, scooters, taxis, cycles, and bordering this the white shops of Connaught Place, the trees in the central section, the massed pedestrians, the large blue sky.\n\nShe could think of nothing but her painting. When she was teaching, her mind was on her figures, the spaces, the colours of her canvas. At home, after lunch, she painted, and as a result there was no time to take the naps she had been used to. Her headaches became worse and often in the evening, after the children's homework, she lay on the sofa, balm smothered, dopey with pain killers. When the pain was very bad she threw up. She tried to keep this from her husband, to participate enthusiastically in his social life. But he did notice, and he did mind.\n\n'Why are you doing this to yourself?' asked Hemant one evening, when it was obvious she was in pain, the smell of balm all pervading, eyes drooping, brow furrowed, face contorted. 'You can't paint and teach, every time I come home you are lying on the sofa. You are suffering, we are all suffering.'\n\nCertainly suffering was involved, it was true. Astha remained silent, her shoulders hunched. Assent and helplessness.\n\n'Your body cannot stand the strain. Mummy said you are neglecting the children, you do not sleep in the afternoons, you are exhausted in the evenings, you are spreading mess in the house, everything smells of turpentine. And all for what? Some dead man.' He never mentioned the nine others.\n\n'It's not for some dead _men,'_ flashed Astha, 'it's for a cause. And I'm sorry your mother found it more convenient to complain to you instead of me.'\n\n'What is it to her? She has your interests at heart.'\n\n'I have a better idea of my interests.'\n\n'It seems not. You can't do everything. Leave your job if you insist on painting. It never brought in enough money to justify your going out of the house.'\n\n'You were the one who thought I should work.'\n\n'But now you need not, dearest, I am making enough money.'\n\n'I want something of my own'' murmured Astha.'\n\n'What?'\n\n'My own money'' though she knew it was contrary to the spirit of good marriages for a wife to hang on to things and say they were her own.\n\n'You make me sound like a stingy husband, Az'' said Hemant in some hurt.\n\n'No, no, that's not what I meant'' she said weakly.\n\n'Then? Quit for heaven's sake.'\n\nBut she was not yet enough of a painter to risk giving up a job she had had for ten years. It represented security, not perhaps of money, but of her own life, of a place where she could be herself.\n\n*\n\nReshana phoned frequently, inquiring about the progress of the painting, once dropping by, flattering Astha by her interest and her praise.\n\nAt this visit Astha asked, 'Reshana, how come I never see nor hear anything of Aijaz's wife? She must still be really devastated, poor thing'' she added in case her remark was construed as criticism.\n\nReshana made a face, 'Just between you and me Astha, some people have a problem working with others. I do not wish to say more.'\n\n'Oh.'\n\n*\n\nSix months after the massacre, the exhibition was ready to be held.\n\n'Ten thousand rupees'' said Reshana.\n\n'Ten thousand!'\n\n'We need the money.'\n\n'But will people buy? I'm not known.'\n\n'It's a large canvas. It is good. If you were known I would have priced it at fifty thousand.'\n\nTen thousand rupees \u2013 the outrage among spectators, each one saying, I wouldn't take this canvas if you _paid_ me ten thousand. She would be tested, tried and rejected. But the money was for the Manch, she couldn't protest too much.\n\n*\n\nReshana was right. The painting sold on the second day. The crowd was large, many people wanted to help the anti-communal cause, especially if they could get something in exchange.\n\n'I have told all my friends that my mother has sold a painting'' said Anuradha, looking important.\n\n'Only one, darling'' said Astha.\n\n'So what? They were very impressed. None of their mothers is painting.'\n\n'Well don't say too much about it, it was not because of me. People bought it to help those harmed by state violence.'\n\nAnuradha's face darkened as she stared at her mother, and Astha knew she had ruined her satisfaction. She wanted to say yes, I have done it, I have sold my first painting, I have achieved something, let us celebrate, but the number of 'I's' involved ensured that the words refused to leave her mind.\n\n*\n\nThe Manch was anxious not to lose the impetus it had gained and efforts were made to chalk out a long-term plan of action. Unfortunately the Manch also had its fair share of members who could not agree, and valuable time was spent in arguing. Some talked excitedly of the international recognition their cause could get with a film that would document communal atrocities in the villages of North India. It could end with the murder of The Street Theatre Group so that the middle class could also relate to the theme.\n\nSome wanted to start at a more grass roots level, doing the kinds of things Aijaz had done, street plays, slogans, posters, meetings, pamphlets, consciousness raising.\n\nSome wanted to bring anti-communal activists and academics together to exploit the forum of the written word, maybe start a journal. Others thought this was too elitist, too far from the spirit of the theatre group.\n\nSome wanted to concentrate on bringing out a collection of the writings of various members of the group, while objectors felt that since Aijaz was the main person who wrote, it would be like bringing out his writings, and such individualism was inimical to the spirit of the Manch.\n\nSome felt that all their energies should go towards bringing the killers to book. Not a single arrest had been made so far, and this just mirrored the complicity of the police in communal riots and murders.\n\nMost felt this would only end in frustration, and with the rampant corruption of the government they might as well bang their heads against a brick wall for the rest of their lives. The need of the hour was for positive action.\n\nAt last a sub-committee was formed. They would present a report, everybody would meet again.\n\n*\n\nAstha sat silently at the back, her head bent steadily on the moving hands of her watch, and as the hour advanced so did her alarm. It was getting late, the children were upstairs, their homework had to be attended to, Hemant would be coming home.\n\nAs she got up to go, Reshana, near the door, put up her fingers. 'Five minutes'' she mouthed. Astha sank back in her seat. She felt the familiar pain marching across her temples to the tune of what were five minutes.\n\nIt took twenty-five. Astha was in agony. Reshana turned to her once in the middle, winked and smiled, enclosing her in a conspiratorial glance from which Astha was powerless to escape. She thought of the dinner, they could order some chicken from the neighbourhood restaurant. Rice, a salad, potatoes fried in cumin and coriander, it would only take a minute, there was the dal from the afternoon.\n\nThe meeting over, Astha made her way to Reshana. 'I have to go'' she whispered, 'it's getting very late.'\n\n'Stay a moment, I want to introduce you to someone who really liked what you did.'\n\n'His wife?'\n\n'No. This man is a film maker.'\n\nPyjama-kurta, grey beard, grey hair, 'I loved the emotion portrayed in your painting. I wish I could have afforded it, but Reshana here had priced it too high.'\n\n'Very funny, Arjun'' said Reshana distractedly, 'If we start buying our own work we might as well kiss the Manch goodbye. And here, Astha, meet Kabir, he was a good friend of Aijaz's, they used to perform together. He is on the sub-committee.'\n\nKabir blew smoke through his cigarette, and smiled at Astha, 'Tell me about your other work.'\n\n'I have not exhibited anything else.'\n\n'You must do more.'\n\n'Thank you'' said Astha in some confusion. She could barely keep her voice from trembling.\n\nReshana looked at her. 'Are you all right?' she asked.\n\n'I have a headache'' said Astha, clenching her teeth, and carefully enunciating her words.\n\n'Oh you poor thing. Why didn't you say? Come let me walk you to a scooter stand.' On the road Reshana gave Astha a brief hug. 'Take care.'\n\nAstha replied, feeling foolish, 'You too, see you, bye'' and turned to a scooter man who was cleaning his teeth with a neem twig.\n\n'How much to Vasant Vihar?'\n\n'Thirty.'\n\n'Too much.'\n\n'That's what it costs.'\n\n'I'll pay by the metre.'\n\n'Metre not working.' The scooter wallah spat on the road to emphasise his point.\n\n'Twenty-five'' argued Astha, 'I pay twenty-five every time.'\n\n'Thirty. At this time I won't get a fare back'' he added to make the defeat easier for her.\n\nAstha sat inside. The scooter wallah, galvanised into action, threw away his neem twig, and jumped vigorously up and down on the pedal. It didn't start. He flung open the seat, took out his tools along with a rag, fiddled with something underneath, carefully wiping his hands every five seconds.\n\n'I can take another auto'' Astha pleaded. She didn't dare look at her watch.\n\nThe scooter wallah glared at her. 'I'm fixing'' he stated. 'Nothing wrong. Just fixing.'\n\nFinally the vehicle coughed and shuddered. Astha's head throbbed along with everything else.\n\nAt the next red light more stalling. With cars furiously honking behind it, the scooter was reluctantly pushed to the side of the road, and tinker, tinker, on and on before it sputtered into life, only to collapse on the bridge over the tracks of the rail museum. Astha could stand it no longer. She jumped out, opened her purse defiantly, and thrust fifteen rupees towards the man's hand.\n\n'What's this?'\n\n'Your fare. Or don't you want it? This is the worst scooter I have seen in my life. You have made me late, very late.'\n\n'What can I do? The scooter is about to start. Just fixing.'\n\n'No. Here.'\n\n'Twenty.'\n\n'You make me late, and now you are arguing about the fare.' Astha was almost beside herself.\n\nThe scooter wallah was not impressed. 'I'm a poor man'' he insisted, scratching his balls. 'What can I do? The fare is twenty till here.'\n\nAstha lacked the courage to throw fifteen rupees at the poor man and walk away. She thrust another five towards him, and walked down the bridge towards the next stop light, where there was a cab stand. She was coated with dust. The sound of traffic roared in her ears, there would be the problem of dinner waiting for her and the children's homework which would not have been touched.\n\n*\n\nIt was clear from the moment she stepped inside that she was in trouble. Hemant received her frostily, no question as to how was the meeting, you are looking tired, are you all right, I will look after things, you go and lie down.\n\nInstead there was silence through the hastily put together meal, silence as she went through the children's notebooks after dinner.\n\nHimanshu wrinkled his nose at the balm on her forehead. 'You smell, Mama'' he complained.\n\n'Sorry darling, my head is hurting'' murmured Astha.\n\n'Shall I press it?' he asked. Himanshu liked pressing his mother's head, and she liked having him do it, the touch of his small inept hands soothing to her.\n\n'All right'' she allowed. He scrambled into her lap, and put his face next to hers, managing to jab her eye with his finger. The discomfort was slight, but the tears still came. The day had been too much.\n\n*\n\nThe homework was finished at last, the school bags packed and the children asleep. Before sinking her head onto her own pillow and blanking out the whole horrible day, Astha had to try and make amends with Hemant. He had come home, she had not been there, he must have been surprised, wondering, maybe even worried.\n\nFrom the passageway she could see him in his reclining chair, with his newspaper, feeling lonely. She was his wife. Still she looked, feeling exposed in her thin nightie, breasts hanging loose and obvious, eyes watering with fresh balm. She lifted her feet to go towards him, but found herself walking to her bed. She was tired, her feet were telling her, and tired women cannot make good wives.\n\nThat night as the pain receded and she fell asleep, she dreamt. She and another person were riding close together in a scooter \u2013 rickshaw. The person turned, it was Aijaz with long silky hair, which brushed across her face. Astha leaned closer, the corners of their mouths met and pressed, alone against the commotion of the street. Slowly Astha opened her mouth, and bit on the hair. She didn't let go, even when the scooter stopped, and they got out, her mouth firmly clamped on the rich, long, black, thick, sweetly smelling, dusty hair. This made her dumb, she could not argue with the scooter wallah, who was charging too much, but Aijaz took care of him. Aijaz took care of everything. Together, they walked into a room full of doors and windows, with a huge double bed in the centre. Blue and white curtains waved in the breeze, sunlight came flowing through, the bed was covered with soft, printed Rajasthani quilts. Doors opened, people walked in and out, but they were invisible.\n\nSlowly they fell on the bed, kissing all the while, when Aijaz, entwined around her, turned into Reshana.\n\nReshana?\n\nShe woke. It was early morning, the sky was lightening, she could hear the birds beginning outside. Deeply unsettled, she turned to Hemant, opened his pyjamas, gave him an erection and climbed on top.\n\nHe forgave her sins of the evening before by responding.\n\n*\n\nThe disturbance lingered with Astha all next day, the vividness and strong emotions of her dream demanding some kind of recognition. Hesitantly she started making sketches. Two women faced each other in a scooter, their noses covered because of the pollution, only the eyes visible. The scooter wallah was a dark Sardarji with a striking red turban. Perched next to him was a young man, taking a ride. Around the edges of the canvas, traffic, buildings, road, but in the centre the scooter with its passengers bent towards each other, the devouring eyes, the Sardarji and the young man.\n\n'How's it going?' Reshana phoned to ask.\n\n'Fine'' said Astha briefly, not wanting to engage with Reshana when her head was full of other things. She would think about the Manch canvas when she had finished this one.\n\n*\n\nNow that Astha was devoting practically all her afternoons to painting she found it difficult to work inside the house. There were too many interruptions, the servant, the children, the phone, the kitchen, her own restless mind. Besides which she was continually observed by whoever happened to be around, watched intently as she made preliminary sketches, prepared canvas, squeezed paint, mixed and applied colour, cleaned brushes. She could not say go away, that was rude as well as selfishly withholding of herself.\n\nThe canvases also meant that when they entertained guests, certain conversational sequences were invariably set in motion \u2013 who paints, my wife, oh really, very good hobby for a woman, my daughter also paints very nicely, or my sister, or wife's sister \u2013 you name it, there was always somebody who knew somebody who painted. Each time this happened Astha was forced to make her work the subject of idle gossip, a thing she hated doing.\n\n*\n\nShe mentioned this to Hemant one weekend. They were in the bedroom lying off a heavy lunch eaten upstairs.\n\n'I need more space.'\n\nHemant drew her close. 'The whole house is yours, Az.'\n\n'I was thinking of something more specific. You know, a place to work in peace, spread my stuff about.'\n\nShe knew it sounded presumptuous and unfamily-like to want space that was hers and hers alone. Hemant clearly thought so too, as he said, 'You don't need more, you have all you can use here.'\n\n'Not quite. I get in everybody's way.'\n\n'Many women would die to have the space you do. We could never afford anything like this now. If only your father had done the same\u2014'\n\n'Maybe I could have the other room on the barsati?' Astha interrupted in a rush, a room so uncomfortable, distant, remote, and undesirable that she could ask with equanimity, and hopefully be given without hesitation.\n\n'What?'\n\n'Nobody is using it.'\n\n'But it belongs to Sangeeta, she may feel insecure. You know how touchy she already is.'\n\nA wave of anger hit Astha, Sangeeta sitting in Meerut was to be given greater consideration than herself.\n\n'I will vacate it whenever necessary, besides the servants are already there, and presumably she tolerates that.'\n\n'But darling, it has no electrical connections, how can you use it?'\n\n'I'll get it wired, all we have to do is extend the connection from the servants' room.'\n\n'It'll kill you, with your headaches, that's for sure.'\n\n'Please, please, please.'\n\nHemant looked distinctly annoyed. His wife on the roof, next to the servants' quarters, painting.\n\n'What is wrong with working down here? I let you work \u2013 I don't stop you \u2013 I say nothing about the smell, about the canvases all over.'\n\n'The smell, the canvases, the inconvenience are exactly why. Please, darling.'\n\nHemant talked to his parents. They did not agree. Sangeeta would be very sensitive to a family member encroaching on her territory, servants were different.\n\nAstha vowed bitterly to earn enough money to rent her own studio one day. In the meantime if there was no area available to her, she would try and make do with the wide ranges inside her head. Constantly reminded of the space nobody thought enough of her to give, she became very bad tempered during interruptions. Finally she steeled herself, she shut the door, and if disturbed too often locked it. In this way a certain uneasy privacy was granted her.\n\n*\n\nAfter _Women_ _Travelling,_ Astha's imagination increasingly worked in pictures. For the Manch painting she decided to experiment with an issue she felt strongly about. She would deal with the Rath Yatra, with the journey a Leader was making across the Hindu heartland in the name of unifying the nation. Like the religious leaders of old, he drove a chariot, identical to Arjun's in the serialised _Mahabharat,_ familiar to millions of viewers. That the chariot was really a DCM Toyota was a necessary concession to the 10,000 kilometres to be done in thirty-six days. His journey was to start from Somnath, one of the first places to be destroyed by Muslim mauraders (Mahmud of Ghazni) in 1025, and end in Ayodhya, where Lord Ram was born, the hallowed spot that needed to be reappropriated to assuage the feelings of 700 million Hindus. It was also a journey to political prominence.\n\nTo portray this Astha chose a large canvas, four by six, and again drew inspiration from Rajasthani miniatures. On one end was a temple, on the other was the Babri Masjid, on its little hill. Between the two the leader travelled, in a rath, flanked by holy men, wearing saffron, carrying trishuls, some old, some young, their beards flowing over their chests. Besides the rath on motorbikes were younger men, with goggles and helmets, whose clothes she painted saffron as well, to suggest militant religion. She sketched scenes of violence, arson and stabbing that occurred in towns on the way, people fighting, people dying; she showed young men slashing their bodies, and offering a tilak of blood to the Leader; she showed young men offering even more blood in a vessel; she showed the arrest of the Leader as he approached Ayodhya.\n\n*\n\nThe day Astha finished her Manch canvas, called simply _Yatra,_ she took a deep breath and stared at it for a long time. This was good, she felt it was. The Manch had promised her half the money for the painting, she wondered how much that would be.\n\nThis time Reshana priced Astha's painting at 20,000 rupees. 'It's very strong. A bit bloody, but the scale is so small it is not offensive. And it certainly adds to the colour.'\n\n'Thanks'' said Astha, feeling warm and glowy.\n\n'I had no idea you were doing the yatra. A controversial issue will be noticed in the reviews.'\n\nAstha saw respect on her face, which pleased her, but unfortunately it also made her remember her dream. Desire for Hemant darted through her, the safe, solid, stable, secure thing in her life.\n\n'Come back tomorrow and see where we have put it'' continued Reshana, and Astha returned from the exhibition hall with an empty feeling in her chest. The canvas she had worked on and thought about all these months was gone.\n\n*\n\nAgain Reshana proved right. Astha's painting was mentioned in the reviews, one paper even printed a photograph of it, and it was sold before the end of the exhibition.\n\nHemant said, 'Congratulations, you must be really pleased, I am happy for you'' as though they had met at a party, instead of sharing the same bed for years.\n\nAstha said, equally politely, 'Thank you, Hemant.' She put out of her mind an idle romance, that he would be the one to buy it, give it pride of place in house or office, and tell everyone that this was an example of his wife's work. She knew this was impossible, and that people who expect the impossible are setting themselves up for misery, and Astha would rather die than be such a pathetic woman.\n\nInstead she hugged the vision of herself as a woman who had sold two paintings in one year, sum total thirty thousand rupees, of which ten thousand was hers. She felt rich and powerful, so what if this feeling only lasted a moment.\n\nOne day she would get so famous that Hemant would feel obliged to display something she had done, and somebody, friend? banker? associate? would see it and, impressed, would ask to meet her. Unlike Hemant, he would find her fascinating. Would he want to have an affair with her? What would he be like in bed? Here Astha firmly drew a line across the remaining part of her fantasy, it exceeded anything remotely credible.\n\nSummer holidays. Everything that was touched or breathed was dust laden. The heat was its usual, intense and unbearable.\n\nThere was no question of Astha painting, her children were all over the place, she was busy with things to occupy them, summer workshops, the transportation involved, and the impending visit of Sangeeta with her children.\n\n'Will you show Sangeeta Bua your paintings, Ma?' asked Anuradha.\n\n'Right now I have nothing to show.'\n\n'You have the picture of it from the newspaper, and the mention in the review.'\n\n'Let it be, babu, she might think I am showing off.'\n\n' So? Shefali is always boasting about all the things she has.'\n\n'Poor thing. Sweetie, there is a lot of trouble in Shefali's house, her parents fight, and maybe she talks like that because she is insecure. Let's not say anything about my paintings it might make Sangeeta Bua feel bad.'\n\n'You mean jealous.'\n\nThat was what Astha meant, but this was the child's aunt they were talking about. 'Painting is not everybody's cup of tea'' she temporised.\n\n*\n\nThrough the summer, and the trials with Sangeeta, Sangeeta's children, Shefali and Samir, and her own children, her painting remained with her, at the back of her mind. She yearned for the moments when her hand, her eye, her brain fused into one, and her daily life was blocked out. She had experienced this increasingly with the second and third canvas, and she was impatient to experience it again.\n\nMeanwhile the six of them shopped, went to the zoo, went to films, went to restaurants, went to Appu Ghar, went to the science museum, went to the crafts museum, went swimming. For a week the nine of them went to Nainital, where Hemant rented a cottage. Here they boated, roamed around the lake, took long walks, had pony rides, and Astha was wife, mother, sister-in-law, daughter-in-law.\n\nHemant was happy with her. He found this easier when his relatives were there, and Astha spending so much time with them. When their anniversary came, he bought her a ring, an emerald surrounded by tiny diamonds. The quality was excellent, and the ring looked well on her hand.\n\n'Brings out your colour'' said Hemant turning the hand around in his own, smiling at her.\n\n'Such a husband'' murmured her mother-in-law in the background. Sangeeta looked on registering each gesture.\n\nThere was no need for Astha to say anything.\n\n*\n\nThe summer over, Sangeeta and her children departed, school about to start. Astha stood on the verandah overlooking her tiny garden, thinking her forced exile from paint, turpentine and linseed oil was at last over. She looked at the scene in front of her, wondering how she could catch even a fraction of it on canvas. The sky was heavy with dark clouds, the air had a grey yellow quality to it that made the grass and trees more luminous, the red flowers of the gulmohar tree more vivid, the waxy white flowers of the champa tree more arresting against their large dark green leaves. There was so much moisture in the air, that as the breeze blew, it brushed her face with dampness.\n\nMughal miniatures were full of monsoon scenes, lovers on the roof, the man's hand fondling the woman's breast, while the woman leans heavily against him, a grey sky above with white birds flying in a V-formation against the clouds. How about a monsoon urban scene, children splashing in puddles, kites flying, jamun, bhutta and phalsa sellers squatting in front of their baskets on pavements, and on the roof, a solitary woman looking towards the heavy darkness above. Melancholy filled her. After the deadness of summer, the monsoon was a time of awakening and desire, but what was one to do with one's longing?\n\nShe wished she could share her feelings with someone, but with only Hemant to fall back upon it was certain that her loneliness was secured. Still he was all she had, and she made an attempt when he came home and settled down to his drink.\n\n'It was really pretty today.'\n\n'I suppose. I didn't have time to notice.'\n\n'That's why I am telling you. I want to share it.' But already the tone was edgy, and Hemant responded promptly.\n\n'Yes, it's nice when you have time to admire nature.'\n\nThe offensive implications were clear. Astha forced a sketchy smile to her lips, then turned to study the label on the whisky bottle. More than this she could not lie.\n\n'I have a surprise for you'' he said.\n\nShe was grateful, 'Oh, really? What?'\n\n'We are going to Goa.'\n\n'Goa! Why Goa? The monsoon has begun there.'\n\n'Arre! You were the one who wanted to go.'\n\n'That was in winter. In season.'\n\n'Exactly. And do you know how expensive it is in season?'\n\n'Not if we had stayed in a cheap place. There are plenty of those.'\n\n'Why go if we have to slum it. Now I've got an excellent package deal.' His eyes softened and he squeezed Astha's arm. 'It's been fifteen years since we married. It's an anniversary present for you.'\n\n'Our anniversary is over.'\n\n'O-ho, May-July same thing. Either it's hot or it's raining. And the rates are off-season. I've got reservations for the Taj. When one goes to a five-star, the hotel becomes the destination then you really get your money's worth.' Hemant looked pleased with himself. 'Off season rates'' he repeated as they settled down to dinner.\n\n'But Hem'' said Astha, managing to get excited at the idea of staying in a five-star hotel, if it was raining outside, so what, five-star was five-star. 'It will take two days to go there, two days to get back, almost as long as the stay itself, is it worth it?'\n\n'We are flying'' and pride swelled his chest, and filled the room.\n\n'What? Have you won a lottery?'\n\n'I have to go to Bombay to see a dealer, the children's tickets will cost half, yours is the only ticket we have to pay for. We will spend the money you earned for your painting.'\n\n'But darling, you could have asked me if I wanted to spend the money on a plane ticket, and that too when it is off season.'\n\n'You have a bee in your bonnet about seasons. I am telling you it will be very nice, you don't trust me.'\n\n'I do, really I do.'\n\n'Then show it.'\n\n*\n\nIt was fair, she told herself later, that her money should go towards paying for a family holiday, after all why should Hemant have to pay for everything. There was no question of any choice in the matter. Everything was already decided. They reached Goa in the rain, they drove to the hotel in the rain, the children ran towards the beach in the rain.\n\n'Why don't we go too?' asked Astha. 'It might never stop raining.'\n\n'No, you go.'\n\n'I don't want to go without you,' said Astha. There was a possibility he would remind her they were on holiday, and why did they holiday if not to be together? She glanced wistfully outside. In the distance was the sound of the sea, and she could make out a thick grey and white undulating line.\n\n'You are such a child'' said Hemant indulgently moving to her. 'Remember this trip was to celebrate our anniversary?' He started tugging at her sari.\n\n'What are you doing? The children may come in any minute.'\n\n'Just a quickie. They won't come for another fifteen minutes at least.'\n\nQuickies. It seemed that was all they ever were. They completed the act within the specified time, the sound of the rain and the more distant noise of the sea mingling with Hemant's breathing in Astha's ears.\n\n*\n\nThe next day it was clear in the morning.\n\n'I have been talking to reception and they say that we should sight-see now as it will probably rain in the afternoon. I have hired a taxi.'\n\n'Where are we going?' demanded Anuradha.\n\n'Mapusa, and then some beaches.'\n\nThey set off. Husband, wife, two handsome children, riding in a taxi, sightseeing in Goa.\n\nThe town of Mapusa was small and barring a few traces of Portuguese influence, not very interesting. The Mediterranean colonial style of architecture could be seen here and there in old houses surrounded by lush green gardens, colourful bougainvillaea and hibiscus spilling over boundary walls, or flinging themselves with abandon on the houses.\n\nAfter driving them around a bit, the taxi stopped in front of a hideous shopping arcade with concrete circles plastered all over for decoration. The traffic was chaotic and noisy, taxis, cars, cycles and motorbikes driven by scantily clad foreigners whizzing around.\n\n'Cashew nuts, Goan wines'' said the driver firmly as the family hesitated inside the car. 'Antiques, silver, jewellery'' he continued gesturing at the dark spaces behind open doors.\n\n'Might as well see what this town has to offer'' said Hemant.\n\n*\n\nPerhaps that was a mistake. Because one of the things the town offered was an antique silver box, priced at five thousand rupees. It was so beautiful Astha fell in love with it immediately \u2013 old, blackened, intricately carved, and totally useless.\n\n'Please, can I have that box?' she asked Hemant.\n\n'You must be out of your mind'' said Hemant.\n\nThe tone, the refusal both hurt her. She was an earning woman, why couldn't she have a say in how some of their money was spent? She never said anything when he chose to squander money on airline tickets, why couldn't she buy a box she liked? Maybe it was too expensive, but she was sure if they bargained, it would become cheaper. Besides silver was silver.\n\n'It's so pretty. It would also be a memento of Goa.'\n\n'It's too expensive, these people are all cheats.'\n\nThe shopkeeper sensing indecision, urged the box upon them, very nice, old box, old price, now it will be twice as expensive if you go to buy.\n\n'See?' said Astha. 'Old prices.'\n\n'How can you believe him? They all lie.'\n\n'I also earn. Can't I buy a box if I want, even if it is a little overpriced?'\n\n'You earn!' snorted Hemant. 'What you earn, now that is really something, yes, that will pay for this holiday.'\n\nI have earned for my ticket she thought, but this was not the place to bring it up. The children pottering about in the shop had fallen silent. Anuradha went and stood at the doorway staring at the traffic. Himanshu was fiddling with the cashew nuts they had bought to take back to Delhi.\n\nAstha let out her breath in jerks so that nothing was audible. 'Let's go'' she said almost to herself.\n\n*\n\nThey went to see the other beaches, and on the way back from Vagator, Hemant put his arm around her for a conciliatory moment in the taxi. She could feel the solidness of his body next to hers. She felt limp, attacked and baffled. She didn't want his touch, his nearness to compete with the pureness of her despair.\n\nShe got through the rest of the day somehow, sick and wretched. The beaches were lovely, and she felt resentful of their beauty, resentful at being forced to register anything besides the pain within.\n\nBack in the hotel, the children beat against her mind, forcing attention from her through their shells. 'Look, look at this one \u2013 you're not looking \u2013 see, mine, put it to your ear, can you hear the sound? \u2013 not like that, you have to put it like this, can you hear the sea now? \u2013 I want to take all these shells to Delhi, they will look so pretty \u2013 I'm not putting sand everywhere, they are perfectly clean \u2013 that's my shell \u2013 she took my shell \u2013 it's mine \u2013 I saw it first \u2013 no he did not \u2013 she's always taking my things \u2013 you are always taking his side...'\n\nAnother hour and Astha's head was splitting. By the time the children had eaten their dinner and changed she was ready for the waves of pain that submerged her consciousness.\n\nThe night passed. Twice, thrice she staggered to the bathroom, clutching the walls for support to retch into the pot. Each time she hoped the pain would lessen, but it didn't, and her nausea continued until the birds started chirping, and the dark sky turned silver with the day. Finally with nothing left in her stomach, nothing left of her, she managed to close her eyes and sink into a calm exhaustion.\n\nOnce or twice she was aware of Hemant asking from his side of the bed, expressing concern in a strained voice, 'Are you all right?'\n\nShe acknowledged its tokenness by replying in a voice hoarse from vomiting, 'I'm fine.'\n\n'Are you sure?'\n\n'Yes. The pain will soon go.'\n\n*\n\nIt was late next morning. Hemant had given the children breakfast, and he was now sitting with Astha on the verandah, in front of a tray of tea and papaya. Astha looked over the undulating grassy patches to the sea line. She could hear the thundering of the waves. Above, the sky was rolling with heavy grey clouds. She couldn't remember seeing a miniature of the sea. Maybe miniature painters traditionally lived inland.\n\n'Feeling better?' Hemant asked. She nodded. He held out his hand, and she put her own in it. The feel of it was dry and warm. There was a certain comfort she associated with this hand.\n\n'I had hoped that with the sea air your headaches might become better, not worse'' he continued, a careful, mock blame in his voice.\n\n'That would have been nice'' she managed just as carefully.\n\nInside, the children's voices could be heard trading shells, with a few brief snatches of argument.\n\n'I think the children really like it here'' remarked Hemant, letting go of her hand to pour her a cup of tea.\n\nIt was tepid, but Astha sipped it gratefully, aware of the residual heaviness in her head with every motion.\n\n*\n\nThe bill for five days and four nights was nine thousand five hundred rupees. Hemant was triumphant.\n\n'That was money well spent'' he said as he came back to the room after settling all the accounts at the front desk.\n\nNine thousand five hundred rupees spent on one of the worst weeks of my life, thought Astha, as she stepped into the hotel bus for the airport. She thought hopelessly of all the things she could have done with that money, of the beautiful silver box she could have possessed and admired for ever. But their money spending was decided by him, not by her.\n\n'Oh, God, you look terrible. Have you been ill?'\n\nThus was Astha greeted by her colleagues on the first day of school.\n\n'The holidays were tiring'' she replied. 'My maid was away.'\n\nEverybody understood what this meant.\n\n'I tell you, after one's servant takes a holiday, it should be understood that we get a holiday too. Look at Astha, poor thing, it is obvious she needs a break.'\n\nTake a break, how they all said it like a mantra, as if taking a break would make any difference when you always came back to the same thing.\n\nColleague two was talking of her sister-in-law, settled in America, who had discovered that her husband was cheating on her, and who now wanted a divorce. This brought about the usual virtuous reactions centring around Us and Them, East versus West.\n\n'There they go on divorcing \u2013 marrying till the age of 60\u201370.'\n\n'They do not understand the concept of family. They only think of themselves.'\n\n'The divorce rate is three out of four.'\n\n'They don't know what it is to be a woman, what it is to sacrifice.'\n\nWell, Astha was a woman, and she was sick of sacrifice. She didn't want to be pushed around in the name of family. She was fed up with the ideal of Indian womanhood, used to trap and jail. Excuse me, stop the juggernaut and let me off. I have had enough.\n\n'It may not be a bad thing'' she said tentatively. 'If a marriage is terrible, it is good to be able to leave.'\n\nEverybody stared at her. Astha fiddled with her notebooks. They would be wondering whether her marriage was all right. 'Take my sister-in-law, for example'' she added quickly. 'Her only time off is with us in the summer. She is not allowed to work, rather her in-laws make her slave inside the house, she is nothing but an unpaid servant. If she complains, her husband sides with his parents. If she were in the West she could contemplate divorce without the social and economic death that would follow here.'\n\nThe bell rang. Astha got up carrying the forty notebooks of her students and headed for class. She had a job, there was no doubt as to that, but she doubted whether that made her any less trapped than poor Sangeeta. She should have kept her mouth shut about divorce. Its sole result would be speculation about her.\n\n*\n\nMeanwhile Anuradha turned thirteen and started menstruating. She did not take kindly to this, and Astha grew to dread her periods, interspersed as they were with bouts of rage, pain and depression. She could not remember ever attracting so much attention to herself during these times, even when it had hurt unbearably.\n\n'It is a woman's lot'' she explained.\n\n'Why, why is it a woman's lot, it's not fair'' moaned Anuradha, as she clutched her hot water bottle, tears flowing from her eyes, wetting the corners of her face, disappearing into her hair.\n\nWhere does fairness come into it, thought Astha. It hurts, you bear it. That was the end of the matter. 'It happens so you can have children'' she tried again.\n\n'I'm never going to have children, I'm going to adopt.'\n\n'We'll see when the time comes. You might want your own children.'\n\nAnuradha glared at her mother and did not deign to reply.\n\n'What's the matter with Didi?' piped up Himanshu, who was watching his sister wail and scream with great interest.\n\n'She's got a stomach ache. Go and see what is on TV, beta.'\n\n'Nothing is on TV. Why can't we get a dish and watch the Gulf War?'\n\nAstha turned to stare at her son. Anuradha forgot her pain long enough to point out how spoilt he was.\n\n'What's wrong with _Chitrahaar?_ You have always liked it.'\n\n'I want to watch the Gulf War. In school everybody watches the CNN and the BBC.'\n\n'I doubt everybody in school has a dish. You can talk to your father when he comes.'\n\n*\n\n'How was the day?' asked Hemant when he came home that night.\n\n'Terrible. Anu has her period.'\n\n'Oh? Poor little thing. Was it very bad?'\n\n'Yes. I had to give her two Brufen. I hope she doesn't become dependent on them. How will she bear pain in later life?'\n\n'She's still a little thing. Why should she have to suffer so much?'\n\n'She's not so little, and it's part of nature.'\n\n'Where is she now?'\n\n'In bed with a hot water bottle, reading Nancy Drew. And she has a test tomorrow.'\n\n'Poor baby, let her be'' repeated Hemant quickly, pouring himself a drink and making for Anu's bedside.\n\n'Oh Papa, I want some chocolate'' murmured Anuradha in a babyish voice, snuggling next to her father.\n\n'Tomorrow, all right?'\n\n'All right.'\n\nHe never sounds or looks like that when I have a headache, thought Astha, and then struck that thought from her consciousness. The father-daughter bond could not be compared to the rocky terrain of a marital relationship.\n\n*\n\nA few weeks later a dish appeared on the terrace. Astha was informed of this casually the night before.\n\n'A dish? But it is so expensive.'\n\n'It is good for the children. They will see the BBC, the CNN, they will know what is happening in the world. Who can watch Doordarshan? Two channels, I ask you. Now at least they will have competition.'\n\n'But such a lot of money, to have a dish in our own house. We are not a hotel, or something.'\n\n'Arre, I am in the TV business, I have to keep up with these things.'\n\nAstha's mind travelled to the little silver box in Mapusa, only five thousand, while the dish was at least eight times that. But it was useless to say or feel anything, the children and the business ensured the non-comparable nature of any\n\nargument. If she knew how much money they had, she might be on surer ground, but she never did.\n\n31 December. Constitution Club. 6.30 p.m. A slight mist was beginning to add to the general chill. Astha had not realised it would be this cold, and she stood shivering in her sweater and shawl. Nearby was a peanut seller, roasting his peanuts over a small fire but she didn't dare advance towards him, in case it looked as though she thought more of her appetite than the cause.\n\nIt was the anniversary of the massacre of The Street Theatre Group. It was also a protest against the Hindu Samaj Andolan decision to construct a temple at the site of the Babri Masjid.\n\n'Come and help me, Astha'' said Reshana, approaching with two big plastic bags. Squatting on the pavement, they poked candles through tiny foil-coated plates to prevent wax from dripping onto the hands that carried them. Absorbed, Astha could forget the scene she had had with Hemant before she left.\n\n*\n\nTen days ago, Hemant had asked, 'What shall we do this New Year's Eve?'\n\nAstha looked wary. Last year they had spent over two thousand to go to a five-star hotel with friends, and Astha had disgraced herself by getting a headache and throwing up at one o'clock in the morning with the discomfort of everyone's concern directed towards her. 'Leave me at home'' she had pleaded when Hemant had taken it up with her. 'I can't help myself.' But that was not socially acceptable either.\n\n'I don't know'' she now said. 'What did you have in mind?'\n\n'I'm not sure'' said Hemant leafing through brightly coloured invitations sent by various hotels and clubs about Xmas Nite, New Year's Eve Nite, dinner and dance. 'The Delhi distributor has invited us, they have booked a hall at the Sayonara club.\n\nBut it's not very personal, they call all their clients, and it is one big tamasha'' said Hemant looking disgusted.\n\n'Can't we stay at home'' asked Astha tentatively. 'That's really personal.'\n\n'Stay at home on New Year's Eve? No thank you.'\n\n'Tell me then, where are we going?'\n\n'We've got several invitations, let's see how many we can take in'' said Hemant, his pride at being socially sought after showing in his voice.\n\n'All right'' said Astha, not bothering to ask who the invitations were from. Some friends, some place. Eating, drinking, laughing, talking. It made no difference to her. Her mind was always not quite there.\n\nShe didn't tell him about the demonstration, also planned for New Year's Eve. She felt this information would not be well received.\n\n*\n\nNow she was about to be proved right. Hemant saw her getting ready to leave and demanded, 'Where are you going? I am free, you know that.'\n\nAstha thought of all the evenings she had been free and waiting, and wondered if there would ever be a day when she could feel the same right to complain that Hemant did. Now she tried to be conciliatory, she didn't want tension on a night of heavy duty partying. 'I am not going to be away long, just an hour.'\n\n'Where are you going?' Hemant repeated.\n\n'To a demonstration outside Rashtrapati Bhavan. It is the anniversary of the massacre.'\n\n'You seem to forget that your place as a decent family woman is in the home, and not on the streets. You also forget that this is New Year's Eve, and we are going out.'\n\n'No, I do not forget. I will come back in time, what does it matter what I do one or two hours before?'\n\nHemant's face assumed its shut-in aspect. Astha knew she was equivocating. It mattered because going out with her husband must be the highlight of the day, not something she was squeezing into the rest of her activities, unregarded, unimportant, done for the sake of doing. She left the house, hoping the anticipation of parties would do its bit in removing Hemant's ill humour.\n\n*\n\nBack at the Constitution Club. By 7 p.m. about three hundred people had gathered. 'Good turnout'' said Reshana to Astha as they finished with the last of the candles, and gathered themselves up from the pavement. 'And that too on New Year's Eve. We did contact everybody but you can never be sure.'\n\n'Many might think this is the best way to spend it'' said Astha with feeling. 'To do something you believe in makes other things a little easier.'\n\nReshana drew back. Astha flushed. There she was trying to give Reshana her heart and soul, behaving inappropriately. She must remember that everybody was here for the cause, and if the cause also had a personal impetus, discretion demanded this be shrouded in silence.\n\n*\n\nDown Rajpath they marched, candles glowing. They carried placards that declared they were for a united India, that secularism was part of our Constitution and traditions, that communalism was the scourge of the nation.\n\nThey chanted as they went:\n\n_Raise_ _your_ _voices_ _\u2013_ __ _We_ _are_ _one_\n\n_We_ _will_ _fight_ _injustice_ \u2013 _We_ _will_ _fight_ _together_\n\n_Communalism_ _will_ _\u2013_ __ _Never_ _succeed,_ _never_ _succeed_\n\n_These_ _are_ _false_ _weapons_ \u2013 _Of_ _the_ _true_ _god_ __ _Ram_\n\n_False_ _Ram-lovers_ __ \u2013 _False_ _weapons_\n\n_Temple,_ _church_ _\u2013_ __ _Mosque,_ _gurudwara,_\n\n_All_ _the_ _same_ \u2013 __ _The_ _same_ _for_ _all_\n\nUp towards Raisina Hill, candles dripping wax on the paper plates, holding a memorandum, on which they had been gathering signatures for the past month, protesting the attacks on the Muslims, protesting the bid to demolish the Babri Masjid.\n\nAround them swirled cars and pedestrians, irritated at having to stop in front of aggressive placards and glowing candles, while the procession marched across as many roads as it could, hindering traffic, drawing attention to its message.\n\nTowards Rashtrapati Bhavan, home of the President, home of previous Viceroys. Huge, mammoth, it towered in the distance, far from the high and massive wrought-iron gates that barred unauthorised entry. There the processionists halted, lit by TV cameras that dimmed the candles they were focused on. From Raisina Hill Astha could see the lights of cars swishing up and down Rajpath. How few we are, how many indifferent on this one road. She looked towards the former Viceroy's Lodge. Designed as a regal sandstone testimony to British glory, it had served its purpose for only seventeen years, before becoming a testimony to illusion.\n\nThe protest songs and slogans continued. Finally an official arrived, and a side gate opened a suspicious crack. The memorandum along with two spokespersons squeezed in; two people and a thousand signatures of mainly school and college teachers, artists, painters, and film people. A lot of the marchers had brought their children, who looked as convinced as their parents as to the justice of their cause and the usefulness of their protest, never mind how few they were.\n\nWhile they were waiting a letter was read out. Worded in English \u2013 objection \u2013 it should be in Hindi. The writer, an English academic, quickly explained in Hindi that the letter would be in both languages, and sent to all the leading newspapers. For now he begged their indulgence, he would read the letter in English, pending its immediate translation.\n\nThe letter proclaimed that the Sampradayakta Mukti Manch, and the teacher and artist community were united in condemning both the BJP and the Congress in encouraging Fascist forces in the country, and in failing to take quick action against the threats to the Babri Masjid. Were these threats actualised, secularism would be at grave risk, and communal hatred unleashed on a scale that would be difficult to control. To take no action was tantamount to encouraging social divisions along religious lines. Weaker sections would suffer. This was not to be tolerated. We appeal to the government to do something before it is too late.\n\nSigned\u2014\n\n*\n\nA resolution was then formed to establish a core group that would see to further action, the first being to circulate more petitions.\n\nThis done, the songs resumed:\n\n_For_ _how_ _long_ _will_ _they_ _loot_ _my_ _village?_\n\n_Taking_ _a_ _torch_ _I_ _will_ _go_\n\n_Through_ _the_ _world_ _I_ _will_ _wander_\n\n_To_ _make_ _my_ _village_ _safe_ _for_ _me._\n\nHalf an hour passed without any sign of the spokespersons. The last night of the year was wearing on. Astha kept surreptitiously looking at her watch.\n\n'Arre, will we ring in the New Year here?'\n\n'Let's go, they can come later.'\n\n'No point hanging around.'\n\nThe TV crew began to pack up. The candles had burnt down. As the procession started back towards the Manch office, Astha lagged behind, keeping a sharp look out for an empty scooter. She had to be home by eight-thirty, or things would be worse with Hemant. At one point where the procession had stopped the traffic she found one, and by quickly agreeing to pay his price, bumped her way home.\n\nIt turned out they were going to at least three of the parties they had been invited to. Senior bank manager, dear friend, and American NRI come to visit his parents.\n\nAstha hurried towards a thick dressy silk sari, peeling off her woollens. The sari was green with a broad red and gold border with woven flowers, hearts and peacocks. A matching deep green blouse was dotted with tiny gold paisleys. Her ordinary jewellery would have to do, she hadn't had time to go to the bank locker to take out some of her heavier stuff. Hopefully Hemant wouldn't notice. She threw her mother-in-law's maroon pashmina shawl casually over her shoulder, and thus guaranteed to freeze in the manner of women partying in Delhi winters, she was ready.\n\nReady to feel cold, ready to drink, dance, smile, laugh, talk, ready for anything the last day of the year might bring. She had made a gesture of some significance before Rashtrapati Bhavan, it made her more amenable to the evening now. Having something of your own makes you strong, she thought.\n\n'All set?' asked Hemant picking up his wallet and car keys.\n\n'Yes'' replied Astha. She was pleased that he had put the unpleasantness over her involvement in the demonstration behind him. They were going to have a nice time.\n\n'You look very nice'' he commented admiringly.\n\n'Thank you'' said Astha, feeling a small flush of pleasure.\n\n'I thought we would go to the manager's house first. It is bound to be the dullest, we can leave quickly.'\n\n*\n\nThe senior bank manager lived in a large house in the older part of South Delhi. Clearly he believed in doing things big. The little front lawn, shamiana draped, the verandah, the drawing, dining room, all were devoted to the party.\n\n'Gosh'' said Astha, as they were bumped constantly by people, avoiding glasses, and lighted cigarettes, 'there must be five thousand people here.'\n\n'More like two hundred'' said Hemant smiling indulgently.\n\nAstha could see his mood had further lightened. It had taken two hundred people but still she was glad.\n\nShe put out her hands to warm over an angeethi and tried not to breathe the thin spirals of acrid smoke coming from it, the coals were obviously wet. Other women holding soft drinks and glasses of juice were also standing around the angeethi. Astha smiled at them uncertainly, noticing the jamawars and pashminas flung over their shoulders, their smooth white waxed arms, glittering jewellery, and beauty parlour done hair. They looked perfect, perfect in a way she could never hope to look, lacquered and finished. She wished she could say she despised that look, and she did despise it, in theory, while crumbling wordlessly before it in practice, never able to hold her own.\n\nShe took a deep breath, turned to the woman next to her, and remarked, 'Cold isn't it?'\n\nThe woman smiled her agreement, 'What does your husband do?' she asked in her turn.\n\n'He manufactures television sets. And yours?'\n\nAnd so the phrases flowed on, till Hemant, one double whisky down, gestured to her that it was time to leave.\n\n*\n\nThe next party was at the house of the parents of the NRI. The place was full of men slapping each other on the back, counting the years they had been acquainted, walking down memory lanes, those lanes always so evident at this time of year with the foreign returned, the come back for two-three weeks when the weather is good and the kids can stand it, returned.\n\nThe food was mostly chaat and snacks. There was all kinds: papri, gol guppa, and for those who couldn't eat cold things in cold weather, hot tikkis, with green sour chutney and red sweet chutney, fat and swollen bhatura served with spicy channa, laced with halved green chillies and onion rings, dosas, idlis and vadas, and finally jalebis floating in hot oil, crisp, sweet, inviting to be crunched up ring by ring. There was even tea in earthen mugs which all those who weren't drinking sipped gratefully.\n\n'Oh, can't we go home, now?' moaned Astha, who thought she would burst if she had to eat another thing.\n\n'One more party, darling'' said Hemant over his whisky glass. 'Chin up.'\n\nI hope he is not too drunk to drive, thought Astha, the glow the food had given her fading, as she thought of the drive to Greater Kailash II where Hemant's closest school friend resided.\n\n*\n\nAnkur's party was on the terrace of his two-roomed barsati. Ankur had divorced after ten years of marriage, and was now discovering the joys of an affluent single life in an emphatically male environment. He fancied himself a cook, and with flushed face offered earthen mugs of mulled wine, mulled wine going ethnic, he said genially. On one side of the terrace a barbecue was set up. Seekh kababs, paneer, mutton and fish tikkas were being served with thin romali rotis, folded into triangles, on flimsy silver paper plates. There were four dead-looking salads, all smothered in shiny glutinous mayonnaise: pink (thousand island?) green (herb?) two whites (garlic? yoghurt?). As Astha jabbed at bits of paneer, it was easier to seem to eat than to argue with her host, she wondered it was only five hours ago that she had stared at wax dripping onto an identical foil plate.\n\nInside the music was loudly drowning everything out.\n\n'Come, darling, let's dance.' Hemant's alcohol-aided spirits were high.\n\nAstha obediently swayed, her sari palla slipping, looking covertly at the others doing their stuff to popular numbers pounding through the dimly lit smoky room.\n\n'Are you OK?' shouted Hemant above the din.\n\n'Yes'' she shouted back, automatic response-cum-smile.\n\n'Good'' he said, his voice slightly slurred, and in the dark he came towards her and pecked her cheek. His breath smelt of whisky, and she let her head tilt towards him, imitating reciprocity, before a couple bumped into them and forced them apart.\n\n'Now, now'' they shrieked, 'no kissing between husbands and wives.'\n\nHow stupid they all are, thought Astha. No kissing between husbands and wives. As though we were something besides conservative, strait-laced, middle-aged Indians. Should an unmarried couple kiss, I would like to see the reaction, I would just like to.\n\n*\n\nEarly next morning Astha rose, made herself a cup of tea and went out. It was another year, and she wanted to mark it in some way special to her. New Years should be private affairs, she thought, thinking of all the partying she had done last night. They had screeched Happy New Year, hugged, kissed, danced, eaten relentlessly, drunk this and that, and finally at 2 a.m. made their way home, Hemant slow and careful because he was trying to appear in control, and Astha silent, because she knew Hemant had drunk more than he should have, and there was no tone sufficiently neutral in which she could convey this.\n\nIt was cold in the tiny garden. Astha grabbed the mali's jhaaroo, and began to sweep the dead leaves into a pile. She wanted to make a fire. A fire was a good New Year's thing. Burning all the old year debris away.\n\nAs the flames smoked through the wet leaves, Astha cupped her hands around the mug of tea. It was Flowery Orange Pekoe, a delicate and flavourful smell. She smiled, thinking of the year ahead. She had found what she really wanted to do, something she was good at, she was lucky. She now felt established enough as a painter to give her art the time and energy that was its due. She was ready to leave her job. She had been teaching almost fifteen years, staying because it had been a good occupation for a woman.\n\nShe finished her tea, and went into the spare room. It was early, but she wanted to begin the first day of the New Year with work important to her. She took out her file and started visualising scenes for a March for Justice. The idea had grown last night among the candles. The canvas would be dark, with a group of people huddled before the gates of Rashtrapati Bhavan, which loomed remote and massive in the background. The bright spots were going to be the candles the marchers held, the yellow of the halogen street lamps and the red and white lights of the cars on Rajpath. The rest would be in shadow. Astha hummed as she worked. There was no one to tell her how tuneless her singing was. \nChapter VI\n\nPipee stumbled into the New Year alone in her flat, staring at the two-rod heater, nursing a small rum and coke. It had been a year since Aijaz's death, and as every day in the past year, she had been fierce in her desire to be alone, turning down well-meaning invitations that friends, colleagues, relatives and acquaintances showered her with.\n\nHer mother-in-law had phoned from Shahjehanpur asking her to visit. But she couldn't. Not yet. The one time Pipee went, she had hardly been able to stand the memories that swept her every inch of the way. In every face she saw traces of Aijaz, and their sweetness to her had made it even harder.\n\nShe and her mother-in-law had cried and cried together, but conversation had been difficult, everything they had in common was in the past. She only stayed a few days, and as she was leaving, the mother-in-law gave her a cheque for one lakh. 'I didn't spend on your marriage, now you take this.'\n\n'I don't want it'' Pipee's voice trembled, there seemed no limit to the number of times they could cry together.\n\n'Please, for him'' replied the mother. 'He would not like to think we did not look after his wife. I want you to know you will always have a home with us.'\n\nDully, and with Neeraj's help, Pipee bought a flat in Vasant Kunj. She had the money from Shahjehanpur, life insurance, and dollars sent to her by her brother.\n\nIt usually takes a lifetime to possess a place of one's own.\n\n*\n\n'You can't go on like this'' Neeraj had remonstrated on New Year's Eve at the office. 'It is not healthy. You are still young.'\n\n'I don't feel young. I don't feel anything.'\n\n'Make an effort, you are not even trying.'\n\nPipee turned away. What did Neeraj think, that she liked feeling the way she did? In fact she would give anything to be free from the thoughts that haunted her. Only since Aijaz's death did she realise that how you die is as important as the loss itself, and can make all the difference to the ones left behind.\n\nThere was no relief from the pain of his final moments. She couldn't get rid of the thought of him trapped in the Matador, suffocating with the heat, burning bit by bit, screaming for help perhaps, trying to break the windows, wrench open the doors, and then the terrible moment when he realised he was going to die, him along with nine others, those nine there because of him. What had it felt like? Had he been able to think of her, their love, their lost future?\n\nTill now not a single culprit had been brought to book. Perhaps if the assassins had been identified and punished, a bit of the horror might be stilled; she didn't know, she only knew it wasn't likely to happen. As for that Sampradayakta Mukti Manch, she hated it more than anything. What had they done to ensure justice? Had they worked on bringing pressure on any government organisation? No, they had a platform in his name which they called freedom from communalism, and all they did was hold exhibitions, raise money, and indulge in cultural nonsense. She hated them, each and every one of them individually, but above all she hated Reshana Singh, who had surfaced out of the woodwork, from god knows where, after Aijaz's death and taken over his memory. She managed to imply that theirs had been a deep connection, she was practically masquerading as his wife. How well had she known Aijaz, she was so much older than him, that any attraction on her husband's part must have been a purely passing phase.\n\nWhat should she do, should she leave Delhi? Her mother had tried to get her to relocate in the south, you can find an NGO in Bangalore or Madras, there are slums here as well. You need to put the past behind you, start a new life, you will be near me, come, come, she persuaded in letter after letter. Pipee now looked at the last one:\n\nMy darling daughter,\n\nEvery day I miss you, think of you, pray to God for your well being, and the courage that will see you through this crisis. Aijaz was a wonderful man, a loving husband, and you were lucky in your marriage. I say this despite the terrible tragedy, because what you two had can never leave you. You have been a wife, you have been loved, and this will stay with you for the rest of your life.\n\nI know what you are going through and darling I would have given my right hand for the same thing not to have happened to you that happened to me. But it seems we cannot escape our destiny, whether our husbands are young or old.\n\nMaybe it is a blessing in disguise that you have no children. When your father left me, I had my Pipeelika, and my Ajay, I needed no one else, but you with your youth, your intelligence, your personality, you need other outlets. Aijaz would not want you to be unhappy or alone. I know that. Life has its own laws that will be heard and felt.\n\nYou are always, always in my heart,\n\nYour loving Ma.\n\nMaybe, she thought, staring at her mother's letter, she really should make more of an effort to go out. Although it had been a year she didn't feel any better, perhaps she never would. But to go on refusing to meet people, always to be alone, that was not the answer either. Her life stretched before her, long and dreary. What her mother was advising was to form a new relationship. But how? Aijaz had been hard enough to find. And there had been Samira when she was young. She had never loved anybody else.\n\nPerhaps she should go to the States, leave Aijaz to the Reshanas of this world. The whole of last year Ajay had been calling her insistently for a Ph.D. programme, you will be surprised what a difference a complete change of place will make.\n\nYes, she would be surprised. Ajay had no imagination, but still she, who had lost everything and had nothing more to lose, could give it a try. In the meantime she might travel with Ujjala.\n\nWith these thoughts, in front of the heater, eating her dinner of scrambled egg on toast, Pipee passed into the new year.\n\nWaving saffron flags, Hinduism marched across the country in the following months, marched in time to film songs converted into bhajans, to Leaders trying to convince the masses that the glory of an ancient land could be resurrected by their united hands. Young men, show your manhood, rescue mother India from the influence of the Muslim invaders, whose long shadow falls over us even now. The wrongs of the past have to be righted.\n\nThese hoards, gathered mostly from the Hindi heartland, become the face of militant Hinduism, armed with tridents, swords, and a determination to die if necessary for the cause entrusted to them. This behemoth turns it head towards Ram's Janambhoomi. A temple needs to be constructed on the sacred soil of Ram's birthplace, burdened for so many years by a mosque. A date is fixed for the event.\n\nAs they converge upon Ayodhya, a cordon is drawn around the city, roads are blocked, trains and buses denied entry, any leader suspected of creating trouble is carefully watched.\n\nBut there are always the fields and villages, always people to give food, water, rest, and show the way.\n\nAnd likewise there are leaders to hide in the lanes of Ayodhya to mastermind the breaking of the cordon around the city, there are officials in the state police who feel it their duty to personally assist all those similarly inclined.\n\nThe kar sevaks declare that neither guns nor bullets can stop them. They prove this when in defiance of all barriers they climb the mosque, plant a saffron flag on the highest dome and claim it for their own. They are fired upon by the police, hundreds of them are injured, many are killed. Videos are made of this, and are later shown around the country as an example of the threat to Hinduism.\n\nThe government falls, and for the time being further crisis is averted, but only for the time being, promise the forces for Hindu Restoration in India.\n\nGive us three places in India, that is all we want, Ayodhya, Varanasi, and Mathura where the Muslim invader built mosques on our sacred sites. If necessary we will bathe these mosques in blood. Why should Hindus give up their position of dominance in the only Hindu country in the world? If it is mosques the Muslims want, let them go to the many countries where Islam is the official religion, we are not stopping them.\n\n*\n\nThe Sampradayakta Mukti Manch were doing what they could in the face of resurgent communalism. They prepared pamphlets, organised marches with other Left groups, and decided to go to the banks of the Saryu to talk directly to the people of Ayodhya to counter the growing rhetoric of religious fanaticism. As they planned their trip, Reshana suggested that Astha also come. 'Between you and me I wonder if academics sometimes have the impact we desire. How I wish Aijaz was with us today. He could capture a crowd like no one else.' She sighed and continued, 'If you could give a small five-ten minute speech? I think it might make a difference to the women. If they realise they have some kind of voice, it will be a useful counter thrust to violence and aggression. After all they stand to lose the most. It's worth a try.'\n\nAstha agreed. Now that she was no longer teaching she welcomed brief respites from the house. And yes, any contribution to the cause was worth a try. In her association with the Manch she had been exposed to detail after detail of atrocities perpetuated in the name of religion. She had made paintings for this cause, she had been part of debates that worried about the far-reaching implications of fundamentalism, she had seen the spread of the worst kind of jingoistic rhetoric and it gave her both a platform and a focus around which she built her work. When she looked back it seemed amazing that she had come such a long way in two years. The detour she had taken between home and school had now become the road she travelled.\n\n'I hope it won't be a problem, leaving your children'' Reshana ended.\n\nIt was a rhetorical statement, but Astha responded with a dry laugh, 'Since when has the personal been allowed to interfere with the need of the hour?'\n\n*\n\nSo far her mother-in-law had not commented about her activities. But Astha's going to Ayodhya was a different matter.\n\n'You know I never interfere in whatever you decide to do. Today young people feel they must live their own lives. But there are times when it is necessary to listen to the advice of elders. What is the need to leave your family, and roam about like a homeless woman on the streets of some strange city?'\n\n'To protest.'\n\nMummy looked nonplussed. 'But why go to Ayodhya?' she returned after a pause. 'You want to say something you write a letter to the newspapers. That is much better. People get to hear. You used to write.'\n\n'Long ago.'\n\n'This is all politics, you should not get involved. Besides have you thought about what you are going to protest? Lord Ram's Janamsthan is in Ayodhya, is there any country in the world where the birthplace of their god is not honoured? Hindu tolerance does not mean you accept everything and anything. Is this the pride we have in ourselves?'\n\n'But Mummy, if the temple is constructed, thousands of people will die agitating over it. Why they could feed hundreds  of poor children on the money they are collecting for the bricks.'\n\nHer mother-in-law looked at her. 'It is not a woman's place to think of these things'' she said firmly.\n\nAstha remained silent, her mind full of her husband. She had mentioned her trip as a possibility in a casual conversation with Hemant. Was this his way of letting her know he did not want her to go? He did not even have time for a discussion with her.\n\nMeanwhile Mummy was repeating, 'You know I never try and stop you from doing anything. Even when you neglect the children, and are busy in your paintings and meetings, I do not say anything. I am not the type to interfere. I am glad my daughter-in-law does not feel she has to sit at home. Till I have the use of my hands and feet I will help you, but it is my duty to point out that you are going too far.'\n\n'You won't have to help with the children this time, I will take them'' said Astha wildly thinking of Anuradha's sulky face, and Himanshu's bewildered expression. 'It is good if they are exposed to such things early.'\n\n'Exposing them to what? Filth and crowds? Don't you care about your children or husband? But he is too good, he will say nothing. If you were living in the conditions Sangeeta is, you would better value what you have. I hope you never regret this.'\n\nAstha was struck dumb. Her mother-in-law had never spoken so openly before. And where did Hemant have the time to notice what she was doing, let alone mind? But he had noticed, he had minded, and so had others. Mumbling something non-committal she retreated downstairs shaking with rage and hopelessness. With a mother like that, what chance that Hemant would ever support her? She dreaded trying to convince him and the possible scene. And because she dreaded these things, she became all the more determined to go.\n\n*\n\nThe argument started that night when they were getting ready to go to bed.\n\n'I have decided to go to Ayodhya'' she said.\n\n'As my wife, you think it proper to run around, abandoning home, leaving the children to the servants?'\n\nAstha went into familiar distress. As his wife? Was that all she was?\n\nShe tried to interest him in the issue, pulling out a pamphlet from her bedside drawer, 'Look at the stuff they are publishing. It's so inflammatory but people fall for it.'\n\n'You should see the stuff they publish against India and Hindus in Pakistan. Why don't you protest against that?'\n\n'I do protest. I happen to think that any religion that incites violence is bad, ours, theirs, everybody's. Listen to this:\n\nThis is not a 'new' political struggle. It is the 77th attempt in the history to restore the Ramjanambhoomi, our heritage. Thus far over 300,000 kar sewaks have laid down their lives in the 400 years.\n\nPseudo-secularists want the mosque declared a national monument forgetting that Ram was an Indian and Babur an invader. It is a national dishonour if a symbol of invasion is so declared:\n\n'Now Ask Yourself!'\n\nCan even the most tolerant, most reasonable and peace-loving Indian run away from his pride \u2013 the reason for his being? The time has come to fight for our threatened faith.\n\n'Hindus unite! Act as one.\n\nNot against anyone!\n\nBut in defence of our motherhood.'\n\nShe watched as Hemant reached out and turned the Ramjanambhoomi Nyas pamphlet over in his hands. She liked his hands. They were so square, so competent, they smelled nice, they felt nice on her. His palms were soft and pink, his nails always short and clean. Why was it like this between them? She sidled next to him, and put her hands under his kurta, rubbing his soft stomach. 'I do so wish they hadn't planned it around New Year's. I hate to leave you alone, but darling what to do?' Plaintive, appealing, emphatic.\n\nHemant grunted. 'Say no, what else is there to do?'\n\n'But I have committed, it'll look bad.'\n\n'They don't own you.'\n\n'Just for two days. I'll be back so soon, you won't realise I have gone'' said Astha trying to be playful.\n\n'I won't be here.'\n\n'Why?'\n\n'I have to go away.'\n\n'But you never said.'\n\n'I only knew of this today.'\n\nShe did not believe him. How would she leave the children? She would have to move them upstairs, and that too after her dignified statement of taking them with her. He was doing this to punish her.\n\n'Where are you going?'\n\n'Bombay. To see a dealer. It's important.'\n\nAstha did not ask how and why, and nor did Hemant elaborate. 'What about the children?' she asked a little forlornly. They had never been without both parents before, without her really, Hemant was frequently away.\n\n'That's your responsibility'' he replied. 'I have work to do, a factory to run, I can't be both mother and father.'\n\nShe would have to be conciliatory with Mummy, she would have to sit down and explain why she was going instead of getting angry, she would have to tell the children she was leaving them with their grandmother, and hope the grandmother would not bad mouth her while she was away. She might as well have spared herself the worry of what Hemant was going to do, he was going to manage just fine.\n\n*\n\nThat night she couldn't sleep. Her mind refused to rest, roaming restlessly among the things that made up her life, her home, children, husband, painting, the Sampradayakta Mukti Manch. Was it too much for a woman to handle; was her mother-in-law right? But why? Her children were well taken care of, she had trustworthy servants, she had someone who cooked better than she, she had left her teaching. And yet she was chained.\n\nHer thoughts grew darker and darker. Restlessly she tossed to and fro, looking for a position that would force her mind to imitate her closed eyes, and free her into sleep. Hemant snored next to her, and his impenetrability irritated her further.\n\nNext morning, tired and bewildered, she got up, looked at her husband, who appeared fresh and lovely. He glanced at her, and she smiled, her lips stretched across her face, cracking her skull, but still her lips would stretch, and her eyes would look up at him.\n\n*\n\nHemant left for Bombay, departing one day before she did, destroying the fantasy she had had that he might drop her at the station, and they could part tenderly with many expressions of I will miss you, hurry back, phone me when you reach.\n\n'You are also leaving?' Himanshu asked, round eyes.\n\n'Yes darling, only for two days.'\n\n'But why?'\n\n'I have some work.'\n\nBut this explanation did not resonate the way the father's did, and both mother and son felt a little unconvinced.\n\n*\n\nNext night, the train to Ayodhya from Old Delhi at 9.30 p.m. Both children insisted on accompanying her to the station. Mala was taken to escort them back. They left the house at 8.00 and at 8.30 were caught in a religious procession starting from a gurudwara.\n\nMother, son and daughter watched the green dial of the dashboard clock tick the minutes away as they waited and waited. For the first time Astha felt the impatience Hemant did in traffic, but there was nothing she could do, blaming the government did not come so easily to her, nobody to blame in fact, but God above who had made them Indians in an overcrowded land.\n\nPeople darted in and out of the traffic, bumping against rickshaws, cars, buses, weaving in and out all over the road. From time to time cars, scooters and scooter-rickshaws inched forwards squeezing themselves wherever they could, but they could not squeeze themselves as small as people did.\n\n'Will Mama miss the train?' asked Himanshu interestedly.\n\n'Don't be stupid'' said Anuradha.\n\nAstha clenched her fists. 'I think I can see the traffic lights now'' she said after the car had crawled along for twenty minutes.\n\nIt was five to nine. There were the traffic lights visible at last, the end of the intersection was almost in sight, it was the last major light before the station.\n\nFinally they reached. Station. Parking lot. Platform. They might as well have saved themselves all that anxious clockwatching. The train was one hour late. They hung around the platform, surrounded by standing, sitting, squatting, lying, waiting people. There was hardly any room to move. By the year 2010 standing room only in India. Make way, make way, squeeze in more, that year is lurking around the corner.\n\n*\n\n'How long do we have to wait for that stupid train?' complained Anuradha, while Himanshu clung to her. Astha felt his body through her sari, felt his arms around her waist, his hand resting on the bit of bare back between her sari and her blouse.\n\n'Do you want an orange?' she asked.\n\nHe nodded. Astha reached into her sling bag and started peeling one.\n\n'I also want'' said Anuradha indignantly. Astha handed her half.\n\nAnnouncement. The train was delayed another hour. The people on the platform stirred, rippled, and then settled down to waiting again.\n\n'Go home'' said Astha, 'Mala take them home. It is getting very late.'\n\n'No, I'm not going, I'm waiting with you'' wailed Himanshu.\n\n'We will wait, it's all right'' said Anuradha gruffly. She demanded some money to buy _Stardust,_ and settling herself on her mother's suitcase, began to read. Himanshu picked his nose, and looked vacantly at the train tracks beyond his feet.\n\nAt last the whistle, the clang, the arrival. The platform woke, and a huge beast sprang into motion. It pushed, it shoved, it jostled. Sharp cornered boxes and heavy suitcases were lugged onto the heads of coolies while the parcels and bags slung from its arms jut, poke, obstruct, protrude, and threaten with injury. Astha clutched Himanshu with one hand and dragged Anuradha along with the other, trying to keep up with the coolie looking for her compartment.\n\nThere was her name and berth number, pasted outside a second class AC coach. More squeeze and push till they reached the berth.\n\nFinally. The coolie was paid, and Mala and the children sat around in a listless sort of way, listening to more announcements of delayed trains, before they all agreed that the family had seen Astha off and now they could go home.\n\n'Bye darlings, bye dearest ones'' she said, 'I'll be back before you know it, and I will phone, all right. Be good, don't give Dadi any trouble.'\n\nThe children jumped off and led by Mala fought their way out of the crowd.\n\nEventually, three hours after it was supposed to, the departure whistle blew, and the train gave a little jerk.\n\nAstha sank back into solitude. She laid out the pillow and sheets that an attendant had thrown at her, and settled down for the night, rocking with the quickening rhythm of the train, not yet wanting to close her eyes and go to sleep.\n\nNext morning, and the U.P. landscape through the purple film plastered over the train windows. The land on either side was flat and dry, with patches of green fields. Uttar Pradesh, home to eighty million people, many of them leading poor, illiterate, and harsh lives, but ready to leave their fields, villages, and towns to converge upon the Babri Masjid, to protect their faith and motherland, something that would not have occurred to them before.\n\n*\n\nFaizabad, Ayodhya's twin city, 11 a.m. The Sampradayakta Mukti Manch had made arrangements for the women to stay at a guest house they frequently used. Astha got into a rickshaw and gave the address.\n\n'Have you come to do Ram darshan at the masjid?' asked the rickshaw wallah, as Astha put her feet on her bag to prevent it from falling on the road.\n\n'Yes'' she answered cautiously.\n\nThe rickshaw wallah nodded, it was the expected answer, Astha could see.\n\n*\n\nThe guest house was a large white washed bungalow set away from the road, in what used to be the Faizabad Civil Lines. A middle-aged lady came out to greet her.\n\n'Astha Vadera? The others from your organisation are out. They will be back soon.'\n\nShe was taken to a high-ceilinged, dark drawing room and served tea. The lady launched smoothly into a brief history of her life, she owned the house, she didn't really need the money, running a guest house was a time pass, one must be active, her son and daughter were in America, she didn't want to burden their lives. See, here they are, gesturing at pictures in ornate silver and wooden frames on the massive Burma teak sideboard.\n\nA house, thought Astha, if my mother had a house, she too could have done something like this, instead of going to Rishikesh and losing herself in an ashram.\n\nThe widow got up, adjusting her sari palla around her head. 'Your room is upstairs. Come.'\n\nThe uncovered staircase was next to the outer wall, and led up to five small rooms in a row. There was a verandah running the length, a nice view of the garden, in one corner were the bathrooms, in another, a bit of terrace to sit on.\n\n'Food to be ordered two hours in advance'' said the widow, unlocking the door of a small room, one little window, red floor, one bed, chair, table and cupboard.\n\nAs Astha sat there, eleven forty-five in the morning, the sense of adventure she had experienced in the train fell away. The room was neat, clean, without character and totally remote from everything that made up her days. She felt strange and dislocated. What would her children be doing? She missed them, she hoped Anuradha wasn't fighting too much with Himanshu, she hoped that their grandmother wasn't feeding them too much rubbish, but it didn't matter, it was just two days, she hoped they weren't watching too much TV, but then that didn't matter either, it was just two days.\n\nTonight will be better, she thought, trying to argue away her depression, tonight at the function she would be where the action was, she would make her speech, feel the purpose of her visit more.\n\n*\n\nA little later, when Astha washed and went down she discovered that in the widow's estimation there were seven thousand temples in Ayodhya.\n\n'Seven thousand? Are you absolutely sure?' she demanded incredulously.\n\nThe widow looked at her sternly. 'It's Ram's birthplace. There is a need for so many. When there is a festival like Ram Navmi, lakhs of pilgrims visit. Many temples double as dharmsalas. They charge one rupee to ten, and the pilgrims sleep wherever they can.'\n\n'So many temples. And they want one more.' The figures startled her into being na\u00efve, she knew the agitation had nothing to do with numbers.\n\n'Of course'' said the widow. 'Ram was born right on the exact spot where the Babri Masjid is. You can even see from the pillars inside that there was a temple there. Eight pillars with Hindu carvings, mango leaves, goddesses, apsaras, kalash in black stone. Where did they come from? They built the mosque around them to mock us.'\n\n'Mock us?'\n\nThe widow glanced at her pityingly, and spelt it out. To remind us that they have the power to destroy our temples.'\n\n'I don't think it's quite like that'' began Astha when the widow interrupted. 'Even now, Muslims living here really have their allegiance somewhere else. You will see during cricket matches they want Pakistan to win, this is not their soil.'\n\nAstha knew it was useless to protest. Opinions like this, based on preconceptions, did not change. What did it take, a lifetime? A whole new history? What?\n\nThe widow seemed nice, even educated, she would not condone violence, no. Hers would be the gentle voice declaring 'they' were all the same, and these were words that would have a longer reach than any missile thrown.\n\nWas it like this in Pakistan, Astha wondered. Did Muslims look upon Hindus with suspicion? Ah, but where were the Hindus in Pakistan? All dead or gone, leaving scars that rankled even now.\n\n*\n\nReshana came, 'Have you been waiting long?' and didn't wait for an answer. 'They are escorting a fresh lot of bricks into Ayodhya, bricks wrapped in saffron, silk, cotton, with tikka on them, stamped with the name of Ram, as though they were an object of worship, bricks to build the temple, high hysteria around the whole thing. We have been trying to make sure our function tonight is well attended, but\u2014' her voice trailed off, a little hopeless.\n\nAstha understood. As an artist the visual and symbolic appeal of saffron clad bricks would be far stronger than any appeals to reason and history. Still one had to do what one had to do.\n\nAfternoon shaded into evening, as not far from the banks of the Saryu, on a platform in front of a mike a thin academic gave the history of the Babri Masjid.\n\nThe audience spread before him had been gathered through posters and advertisements, with the promise of entertainment and songs. Despite Reshana's fears the turnout was large, though it was debatable whether they had come for the spectacle or from a willingness to be converted to a historical point of view.\n\nThere is no evidence, thundered the academic, punctuating the air with an excited fist, no evidence that Babur, busy fighting the Afghans, ever came to Ayodhya, let alone destroyed a temple.\n\nDo you think Babur, founder of an empire in India, would have come here to build this little mosque? Yes, there is an inscription inside saying he ordered it, but the close set writing is of a much later style, carved to strengthen rumours of imperial destruction. The wooden beam below the arch is not a remnant of a temple, but put there by local masons, using local materials, unskilled in building arches. There are others like it in Jaunpur.\n\nBrothers and sisters, I have not come from Delhi to bore you with historical details, only to show you that for every bit of evidence used to prove there was a temple to Lord Ram here, there is a counter-argument to prove there wasn't.\n\nHistory can be used to build or to destroy. We choose the lessons we wish to learn from it. For years Muslims and Hindus have lived peacefully together. It is the British who suggested that an ancient temple was destroyed so that Hindu would turn against Muslim. Brothers and sisters, we have seen what the British succeeded in doing. They believed in Divide and Rule. They ploughed rivers of blood through our country. The same dark forces threaten us now. It is politicians who are creating religious insecurities to get votes. Do not let them succeed.\n\n*\n\nAstha was sitting in front, nervously waiting her turn, clutching in her cold palm the piece of paper on which her rehearsed points were written. She looked around to see the reaction of the audience. He may have been passionate, but he was still an academic. 'Do you think they understand what he is saying?' she whispered to Reshana.\n\n'It's all we can do, though I doubt we are any match for organisations that have been working the fundamentalist rhetoric at the grass roots level for years.'\n\n'I think this speaker should appeal to their emotions, instead of talking about beams, arches and inscriptions'' said Astha.\n\n'He's a very respected historian'' replied Reshana stiffly. 'And he is showing the relevance of beams and inscriptions.'\n\nHer tone annoyed Astha, Reshana was so easily offended. How come love for the people did not translate itself into tolerance for individuals? She looked around for a more congenial sight.\n\nThey were in front of a canal, next to a bridge. Across the modern park on the other side of the water lay the old town, its interspersed domes and spires clearly conveying its mixed heritage. It looked old and fragile in the yellowish rose of the falling light.\n\nFinally the academic finished. 'It's your turn now'' whispered Reshana.\n\nAstha got up. Her irritation had given her energy. When she spoke her voice was firm and clear.\n\n'Brothers and sisters'' she started, 'In essence women all over the world are the same, we belong to families, we are affected by what affects our husbands, fathers, brothers and children. In history many things are not clear, the same thing that is right for one person is wrong for another, and it is difficult to decide our path of action. We judge not by what people tell us, but by what we experience in our homes. And that experience tells us that where there is violence, there is suffering, unnecessary and continuous suffering. When we look to righting wrongs committed hundreds of years ago, we look to the past. But that past cannot feed us, clothe us, or give us security. History cannot be righted easily, but lives are lost easily, pain and trauma to women and children come easily. Tomorrow your sacrifice will have been forgotten because the duty of life is towards the living.'\n\nShe saw some people nodding, and she ended by repeating that nothing except misery and suffering were to be gained by violence.\n\nA song, followed by a street play, and the evening concluded with an invocation to Gandhiji:\n\nGandhiji was a devout Hindu \u2013 none more devout than he. But he knew the true meaning of religion. All men are brothers. Hatred between communities led to his death, and in listening to the voice of hate we kill him all over again.\n\nThe speaker was speaking in terms everybody could understand \u2013 Gandhiji, father of the Nation \u2013 love \u2013 hate \u2013 oneness. But were these strong enough to drown out \u2013 exploited for centuries \u2013 awake \u2013 defend \u2013 protect \u2013 Motherland \u2013 Ram \u2013 God, faith, and Love of Him?\n\nIn the middle of all this Astha looked up and saw someone staring at her. The woman caught in the act, went on staring instead of looking away. Then she smiled slowly, squirrel front teeth advancing slightly from the rest.\n\nI wonder who she is, thought the one being stared at.\n\n*\n\nLater. 'I really liked your speech.'\n\n'Oh, thank you. It was nothing much.'\n\n'It made sense. The basti women who are with me related to it.'\n\n'It was my first. I am not used to making speeches.'\n\nA pause.\n\n'What is your name?'\n\n'Pipeelika.'\n\n'Pipeelika? Ant?'\n\n'Yes. My father's legacy. He liked the sound and he had a sense of humour. It does mean I have spent my life explaining it. Maybe it has affected me, I don't know. What's your name?'\n\n'Astha.'\n\n'Faith?'\n\n'Yes. I don't know if I have been touched by it. The faith in my family centres around my mother and her swami.'\n\n'Don't you have faith in anything?'\n\n'I don't know. Perhaps my brush.'\n\n'You paint?'\n\n'Yes.'\n\n'Can I see?'\n\n'Any time.'\n\nThey arranged to meet the next morning. 'Do you know the place?' hinted Astha at parting. 'I thought I'd see something of Ayodhya before my train leaves tomorrow night.'\n\n'Of course I know the place. I'll show you around.'\n\n*\n\nShe thought about her later. Her hair was like a halo round her face, springing away from it, black, brown, red, orange, and copper, her skin was a pale milky coffee colour. She liked the way she smiled, but she looked sad at the same time, why was that? Had she herself sounded interesting, why hadn't she brought something nicer to wear, suppose she didn't come to the park at ten like planned, why hadn't she asked her where she was staying?\n\nShe tried shaking herself, if she didn't come, she would see Ayodhya on her own.\n\n*\n\nThe next day as she hurried in a rickshaw to the meeting place, she saw her waiting under a tree. Immediately she felt stupid. A stranger she had hardly spoken to, to bother about her clothes, what was wrong with her? They would meet, they would part, she would catch the evening train home.\n\n'Hi.'\n\n'Hi.'\n\nThey said nothing much as they walked through the small town. In every lane were shops crammed with representations of gods, pictures and figures, small, medium, large kalashes, bells for doing arti, prayer beads, green, yellow, black, blue with pearls, and the mandatory rudraksha in every possible size and colour from pale blonde to dark brown. This was a town of serious religious buyers judging from the number of shops.\n\n'We are near Hanuman Garhi, it's on the way to the masjid, do you want to see it?'\n\n'If you think it's worth seeing,' said Astha, humble in her being guided mode.\n\n'It's one of the biggest and richest temples here. Hindus and Muslims fought over it too, though that is not so well publicised.'\n\nThey climbed steps lined with beggars, mostly old people dressed in white or saffron, begging bowls in front, in which people were dropping money, coconuts, sweets, prashad. Overtaking them were eager pilgrims bounding up, shouting, 'Jai Shri Ram, Jai Siya Ram'.\n\n'This is supposed to be the temple with the most steps,' said Pipee, as they passed an old lady bent over her cane, her eyes on her bony, bare brown feet, with their spread-out toes. They could hear her murmuring Ram, Ram, with every step she took.\n\nInside Pipee hung back as Astha advanced towards the shrine flanked by huge donation boxes. A long line of devotees queued before the priest, clutching their offerings, boxes of sweets, coconuts, flower garlands, small thalis with tikka and incense. The priest, swift and practised, set aside their garlands and coconuts, deftly opened each box, dumped half the sweets in the bucket next to him, and returned the rest. The devotee then took a parikrama of the shrine, lingered in the courtyard and rang the bell while leaving.\n\nIf only I could feel like that, thought Astha, looking at the expression on some of their faces, coming to this temple would mean so much. Her eyes fell on the daan box. She opened her purse and took out five rupees, I wish I had something more in my life, I wish an end to this hollow feeling. She shoved the money in the box and rejoined Pipee.\n\n'Do you not believe?' asked Astha as they passed through the inner courtyard, and down the steps.\n\n'No,' she spat out. 'I believe in nothing. I hate religion. You wanted to see, and I am showing you,'\n\n'I'm sorry,' said Astha, a little alarmed, 'that you are doing something you don't want to.'\n\nThe woman drew a breath, and touched her arm briefly, 'No, I'm sorry I was like that, it's nothing to do with you. Come, let's go to Kanak Bhavan.'\n\nOn the way, Astha hesitantly, 'If you hate religion, doesn't it upset you to come to places like these, where there is nothing but?'\n\n'Oh, who cares how upset I get?' she said flippantly. 'I have to come. We are based in a slum, and this kind of field trip works very well to sensitise women to communal issues, which in moments of crisis get totally out of hand. Besides I don't like staying in Delhi much.'\n\n'Why is that?'\n\n'No particular reason. I live alone, I like to travel.'\n\nAstha looked sideways at Pipee and encountered nothing but her hair.\n\n*\n\nIn Kanak Bhavan a small guide greeted them.\n\n'Five rupees, I show you.'\n\nAnd for five rupees they saw the room where Ram slept, where Sita played her sitar, where they played chess, where they bathed, where they dressed, the cupboard where those clothes were kept, where Sita got ready to receive Ram in the evenings, where Kekayi did Sita's muh dekhayii when she came a bride into this house.\n\n'Wasn't all this some thousands of years, BC?' whispered Astha to Pipee, amazed that such anachronisms could be taken seriously.\n\n'Nothing here is archaeologically or historically accurate,' whispered Pipee back.\n\nThe boy gauged what they were saying, though in English.\n\n'Who knows what is real or not?' he said, smooth beyond his years. 'What matters is the feeling of devotion,'\n\nAstha felt ashamed of herself, and tipped him ten rupees as they climbed down the narrow stairs, into the main courtyard below, repeating her earlier wish for something, she knew not what.\n\n'I take it you are religious?' asked Pipee, observing the size of the tip.\n\n'I gave because I want something.'\n\n'He's not a wishing well.'\n\n'He will do for one.'\n\n'What did you wish for?'\n\n'There are many hollows in my life, and I wanted them filled.'\n\nPipee fell silent, and Astha wondered about her empty spaces, with eyes like that, there could be many. 'Are we going to the masjid now?' she ventured as they left Kanak Bhavan.\n\nPipee sighed, 'We should have gone there first, but I always find it so depressing.'\n\n'Why?'\n\n'You'll see.'\n\n*\n\nThey walked up Ramkot, the slight incline that led to the mosque. The way was lined with temples. Temples, houses, houses that doubled as temples, temples that doubled as dharamsalas, all needed for the lakhs of pilgrims that descended on holy occasions. Ramnavmi, Diwali, Navratra.\n\n'It makes me sick the way Ram is being associated with Hindu-Indian-nationalism. It was terrible when the locks of the masjid were opened some years ago. The Muslims were not even given a hearing, considering this is waqf land. Millions of pilgrims poured in to see statues they believed were placed there divinely because God wanted his birthplace back. People will believe anything.'\n\n'We did a play about it,' put in Astha.\n\n'Really?'\n\n'With Aijaz.'\n\n'You knew him?'\n\n'He came to our school. He put together a brilliant piece about the Babri Masjid. Then I never saw him again.'\n\n'His life was short.'\n\n'Yes. That's why I am part of this group.'\n\n'The S double M?'\n\n'Is that what you call it?'\n\n'When I call it anything. I prefer never to think about it.'\n\n'Why? Don't you think they do good work?'\n\n'It's so elitist, and Aijaz was nothing if not one of the people. Now they sell art in his name.'\n\n'I did something they sold.'\n\n'You are part of their core group?'\n\nAstha laughed dryly. 'Hardly that. I can barely make it to a few meetings. But the Manch was happy to have my canvas, I was happy to sell it, and the cause benefited, surely.'\n\n'I wonder. Preaching to the converted. Working through songs, art, literature.'\n\n'But that was what he himself did, I saw him, and he was very effective.'\n\n'There is now such strong feeling about Hindu manhood, pride, valour, protection of the motherland, redressing the wrongs of history, that I wonder whether any street play, song or poster can make a point beyond general entertainment.'\n\n'There were lots of people last night.'\n\n'Of course they were there. They know how to promote themselves all right.'\n\n'If you think like this, why are you here? Why did you bring your women?' challenged Astha.\n\n'To attend the picnic,' said Pipee facetiously before lapsing into silence, leaving Astha to wonder how much to tread in these murky waters.\n\nShe turned her gaze to the bare feet of the women in front. It was not necessary to walk barefoot so far from the shrine, but for these women the very hill was sacrosanct, and their bare feet honoured a faith Astha could never have.\n\nIf she did, she would not be throwing money around, wishing for elusive fulfilment. Faith would do it. She too would walk barefoot up Ramkot not minding the stones, the heat, the germs, the piss of dogs, the shit of monkeys, the spit of people. Wearing no skin of dead animals to pollute the purity of the place, no leather, no shoes, no belt, no bag, no wallet.\n\nBy this time they had reached the top. The nicest thing about the mosque was its location. On the highest spot in Ayodhya, it overlooked the town, with its collection of spires, domes and houses crowded together. Beneath them swayed the trees, a mild calm breeze blew about, a breeze that seemed to suggest that there were many ways to worship.\n\nIn a mosque built in 1528 there was now a Hindu image. Was this not enough to make it a temple? Courts had declared that Hindus had the right to worship here. But now the worship had extended beyond the deity, so that the shape of the enclosing structure had become an obstacle to faith, and every barefoot pilgrim a warrior.\n\nAt the entrance they stopped to take off their shoes. Outside bhajans were blaring on loudspeakers, declaring that the name of god is more effective if all can hear. Inside, under the central dome, hardly visible under a mound of flowers, were the images flanked on either side by men in khaki armed with guns. In front of them a line of devotees streamed past, stuffing money into large donation boxes. Pipee refused to join the line while Astha, made uneasy by the guns, hurried past the little figure that had suddenly appeared on the night of December 22nd, 1949.\n\nThe two women walked down from Ramkot to a song shrieking from a cassette on full volume, _'We_ _will_ _go_ _to_ _Ayodhya_ _\/_ __ _We_ _will_ _build_ _Ram's_ _temple'._ They neared the main road and a waiting rickshaw wallah started cycling towards them. As he did so a policeman detached himself from a patrol group. His khaki clad belly hung over his belt, his truncheon swung from a thick hand. Leisurely he walked over to the approaching rickshaw wallah, grabbed him by his kurta, pulled so hard that the women could hear a tearing sound, forced him from his seat and kicked him. Once, twice. The grizzled rickshaw wallah looked around, smiled and slowly, quietly, began to pull his rickshaw away. Not a word was exchanged. The policeman walked back to his group.\n\n'God! Did you see? How could he behave like that?' demanded Astha, her tone shrill and naive.\n\n'Maybe he thought he was a Muslim,' shrugged Pipee.\n\n'So?'\n\n'So? I don't know. Perhaps the cops think Muslims shouldn't tread on this sacred soil. At any rate they generally don't come here. That man must have been desperate for customers.'\n\n'And why shouldn't Muslims... it's a mosque as well. He should have hit him back.'\n\n'And be beaten into pulp?' inquired Pipee. 'No, I don't think so.'\n\nAstha stared at the ground moodily, and pulled her sari palla further over her head to protect herself against the sun.\n\n'But why are you so upset?' demanded Pipee in her turn. 'These things happen all the time. Surely you know that.'\n\n'He looked like Himanshu,' said Astha suffering into her sari.\n\n'Himanshu?'\n\n'My son.'\n\n'Your son is so old?'\n\n'Of course not. But that look \u2013 when he gets out in cricket for example \u2013 he is not very good, and when he gets out \u2013 that is when he smiles. Just like that.'\n\n'Mother-son,' said Pipee somewhat gloomily. 'An obsessive over-protective phenomenon.'\n\nAstha felt defensive. 'Hardly. I am careful not to smother him. He is the one who clings to me.'\n\n'Despite yourself, you must be liking that.'\n\nSomehow Astha didn't mind this comment from Pipee, it was so non-judgemental. 'I do rather,' she confessed, and then, 'he is the only one in the whole world who smiles whenever he sees me, no matter what.'\n\n'Women. So pathetic in their hunger for love,' remarked Pipee sapiently, guiding Astha into a tea stall.\n\n'Isn't there someone you love?' asked Astha seizing this opportunity, hoping Pipee wouldn't draw back, that it was not too early to be exchanging of themselves.\n\n'I married for it,' said Pipee, and again that reserve. Yesterday it was 'I live alone'. Was she divorced, had her husband been unfaithful, or cruel, was it a problem of in-laws? How soon before she could ask?\n\n*\n\nThey left the tea stall, and had stopped for a moment under the shade of a tree to exchange phone numbers, when a monkey jumped on Astha's back. The sudden weight, the shock of her sari being pulled from her shoulder, her own scream, left her collapsed with fright.\n\nPipee grabbed Astha and examined her arms, her back, her neck, pushing the hair to one side, looking minutely for any scratch the monkey might have left.\n\nAt last she drew her away from a crowd that was beginning to gather, curiosity gleaming in their eyes. 'It's all right,' she declared, 'no scratch, you won't need rabies injections.'\n\nRabies injections. This thought had not occurred to Astha, she had registered nothing besides panic and the fact that Pipee's arm around her had tightened for an unnecessary second.\n\n*\n\nIt was getting on for five in the evening when they took separate rickshaws to go to their separate guest houses. Looking at the face before her, Astha said, 'Please keep in touch.'\n\nThe eyes crinkled. 'Of course. I'll be visiting my in-laws for a few days, they are not far from here. So sometime next week?'\n\nI'll look forward. I mean it,' Astha went on babbling, hating herself. Why did she always have to sound so stupid? And how come she was visiting in-laws when she was no longer married? Perhaps it wasn't divorce but death, and she so young and attractive, with her smile, her hair, her skin, her eyes.\n\nThe mouth folded inwards. 'Well, see you then.'\n\nThey smiled at each other and parted.\n\n*\n\nBack in the guest house. 'I see you have met Aijaz's wife,' said Reshana.\n\n'No, I haven't,' replied Astha apprehensively.\n\nPipeelika \u2013 wife. The one you were talking to last night.'\n\nOh no. What must she have thought, why didn't she say, why didn't she guess, what kind of impression did she make \u2013 oh no, oh no, oh no. Aijaz. Aijaz's wife. What must it be like to be Aijaz's wife? Widow \u2013 widow, not wife. So that was why she looked like that, and spoke like that.\n\n'It's obvious she didn't tell you,' Reshana continued. 'Just like her. Very strange woman. After his death she became totally neurotic. Wanted to own his memory. How can anyone do that? Aijaz was one of the people, if he was anything, but she resented everything we did to keep his memory alive, accusing us of all sorts of things. It was so perverse.'\n\n'To have your husband die like that must be very difficult,' murmured Astha.\n\n'It was hard for everybody, not only her,' shot out Reshana.\n\n'But why didn't you tell me?'\n\n'Me tell you? Why should I?' demanded Reshana annoyed. 'When I saw her with you, I imagined she must have introduced herself like any normal person. Besides she avoids us''\n\n'She did, she did introduce herself,' Astha was already defending Pipee.\n\nReshana looked at her, 'Then how come you didn't know she was Aijaz's wife?'\n\n'She only said her first name, and that was so unusual I started commenting on it \u2013 oh, I don't know.'\n\n'Huh. She never forgets she was his wife when it comes to the Manch. Always bad-mouthing us. I'm surprised she didn't try it with you.'\n\n'She said nothing about the Manch.'\n\n'Surprising. She usually does. Probably trying to impress you with her tolerance. Or maybe she is a little better now, I hoped so when I saw her in the audience,' said Reshana briskly packing her last item, her bathroom slippers, and then sitting on the suitcase to shut it. She was leaving by bus for Lucknow to do some field work.\n\n*\n\nAlone Astha sat in a daze. She had met Aijaz's wife. She couldn't believe she had spent so many hours with her without knowing. She didn't think Pipee would phone her in Delhi, it now seemed too improbable. But she felt wretched, and in this mood passed the remaining few hours before leaving for the station.\n\n'What is your train, beti?' asked the widow when she went down.\n\n'The Sarva Yamuna.'\n\nThe widow sighed.\n\n'It is not a good train?' Astha asked uneasily.\n\n'It will be late,' said the widow.\n\n'How late?'\n\n'Two \u2013 three hours. Maybe four.'\n\n'How can you be so sure?'\n\n'It comes through Bihar. No law and order in Bihar. So the train is always late.'\n\n'Always?'\n\n'Without fail,' said the widow looking ghoulish and satisfied. 'These Biharis keep pulling the chain when and where they feel like it, getting off, getting on, so the train gets later and later. Simple.'\n\nThere was nothing Astha could do, but continue as she had planned. She was afraid of waiting at the station so many hours alone, she thought of Hemant who had not wanted her to come, and who might consider himself justified if she was dragged into a corner of a deserted platform and raped.\n\nShe gritted her teeth against her unreason, while the widow sent her gardener, Hanif, to call a rickshaw.\n\n'Here is your packed dinner,' she said handing her the box Astha had forgotten she had paid for.\n\n'Thank you,' she said and at nine-thirty departed into the still small-town night. The stars above were more brilliant than they ever could be in the polluted skies of Delhi.\n\n*\n\nThe usual damp, stale, wet coal, urine station smells met her as she stepped onto the platform. She sat desolately on her suitcase and waited. Soon, it would be ten. Maybe the widow was wrong.\n\nAt 11 p.m. the train was announced to be three hours late, and the widow was proved right. The numbness that had been seeping into Astha during the past hour intensified. Mosquitoes big as flies buzzed around her. She slapped them off, and started walking up and down the platform. There were mostly people sleeping, covered from head to foot in sheets, shawls, durries or sacking to keep the mosquitoes away; shapeless bundles, unidentifiable as man and woman, the length alone indicating child or adult. Water pipes strung along the side of the track dripped through the hoses that were attached to each one.\n\nShe drank a kulhar full of sweet tea and went on walking up and down the long platform in a kind of daze. There were a few other passengers waiting, but most were the bundles scattered abundantly about. At the far end in the darkest corner of the station, the only sign of life, a bundle with an elbow raised, jerking frenziedly and rhythmically beneath the cloth, in an action Astha immediately recognised. She hurried back to the tube lights of the central platform, where amid the mosquitoes and the tinkling of water dripping onto rail tracks, she had another kulhar of sweet earth-smelling tea, and then waited, waited in one spot for the train to come.\n\n*\n\nAstha passed the night in the train restlessly. Not for her the easy sleep on the way out. Was it only two days ago that she had left? She thought of the meeting, the speech she had given, all the temples she had seen, the security around the masjid, Pipee protecting her from the monkey, thinking of rabies injections. She wished she had known her connection to Aijaz, she wouldn't feel so foolish now, but Pipee clearly hadn't wanted to tell, that much was obvious. If she had had the foresight to take her number she could at least phone and apologise.\n\nHer thoughts turned to home. When would Hemant come back? When the work finishes was all he had said, maybe he would have called his mother and informed her. How would it be between them? Would he still be annoyed that she had gone away? Had he missed her?\n\n*\n\nNext morning, Delhi. Quickly she hired a coolie and jumped into a three-wheeler with her suitcase. She hadn't seen her children for two days and three nights, and now every thought was fastened onto them.\n\nVasant Vihar at last. She rang the doorbell, and there they were, her precious children.\n\nHimanshu rushed to hug her, clutching a drawing in his hand that said _Welcome_ _Home_ _Mama._ Anuradha complained about the book she had to read in the holidays.\n\nI am home, thought Astha. Emotion grabbed her firmly by the throat.\n\n'Do you like my drawing?' shouted Himanshu, tugging her hand, feeling not enough attention had been paid to it.\n\n'I love your drawing,' responded Astha automatically. She lifted it close to her face to illustrate her total concentration. 'Such peacocks, so colourful, and my, what a sun. And so many cars, you have done well with the cars. Are these blue streaks rain?'\n\nHimanshu nodded.\n\n'So interesting. My goodness, Himu, you've got everything in this drawing.'\n\n'Didi said you wouldn't like it,' said Himanshu.\n\n'Well, Anu doesn't know everything, does she?' replied Astha.\n\nHimanshu beamed, and lost no time in informing his sister, 'Mama likes it. So there. You were wrong.'\n\nAnuradha did not even have to think. 'She's saying that to be nice to you, stupid.'\n\n'That was an entirely unnecessary thing to say,' Astha informed her daughter.\n\n'You mean you are not being nice to him?'\n\n'It's not that. I like the painting irrespective of whether I am his mother or not.'\n\nAnuradha did not bother to reply, while Himanshu said nothing more about his art work.\n\n'Have you heard from Papa, Anu? When is he coming home?'\n\n'Don't you know?' asked Anuradha, puzzled.\n\n'Of course,' said Astha quickly. 'I just wondered whether he was sticking to his original plans.'\n\n'Well I don't know. Dadi might. Ask her.'\n\n'I will. Now I'll just go up and see her, all right?'\n\nAs she climbed the steps to make sure no cause for offence would be found in postponed expressions of gratitude, or delayed news giving, Astha thought yes, indeed, she was home.\n\nHemant arrived next evening, smiling, genial and pleasant. Astha was relieved. Clearly their misunderstanding was a thing of the past. 'So,' he asked, after her questions about Bombay, 'the mosque still standing?'\n\n'Yes indeed.'\n\n'And Ayodhya? How was that? Did you see anything?'\n\n'It was very nice,' said Astha, pleased at his interest. 'I met a girl. She showed me around.'\n\n'What kind of girl?' smiled Hemant. 'You are no more than a girl yourself.'\n\n'Oh Hemant, don't be silly, I am old. She is much younger than me.'\n\n'Well, what kind of girl?' he repeated indulgently.\n\nAstha changed the topic. 'The mosque is quite unremarkable, you know. It is old, but that's it. Half its beauty comes from the little hill where it is, overlooking the town. But it's full of policemen with guns, it's not possible to worship there, though people do \u2013 lots.'\n\n'I told you there was no need for you to go.'\n\n'I had to make a speech, don't you remember?'\n\n'Why do you have to travel to Ayodhya to make a speech? It's not as though you were a religious leader, or a politician or a public figure.'\n\n'But it is important for everyone to do what they can, to make things better, you have to try, whether ultimately it makes a difference is not in your hands,' said Astha earnestly.\n\n'Well, I hope you are not going to indulge in more rabblerousing.'\n\nHis fingers were twisting the ring he had given her so that her hand hurt. Hopelessness settled in its familiar place in her chest. He belittled her, yet if she pointed this out, he would deny it. It was better to stay silent.\n\nLater they made love.\n\nThe ritual enacted before partings, after homecomings, this establishment of the marital tie, this coming together of flesh that had been sundered.\n\n*\n\nOr so Astha thought, until next morning, while unpacking his suitcase, she came across a condom.\n\nShe stared at it for a long time, its implications running through her head. What should she do? Leave it in the suitcase, throw it, or confront him? Who had he slept with, he who was never in any place for very long, it could not be that he was in love \u2013 or had a relationship \u2013 or maybe he did. Some woman might travel with him, how would she ever know? Maybe the distributor had supplied him with someone, she had read somewhere that women were often a part of business deals.\n\nBut why now? Was this his message to her? Was this why he seemed in good spirits? Why he had asked her about Ayodhya, and expressed an interest in what she had seen?\n\nFinally she left the suitcase on the bed, the lid closed and buckled, the children should not see and ask what is this, the servants should not see and jump to unnecessary conclusions.\n\nShe waited till he came home. It was 9 p.m., he was late as usual. As usual he first poured himself a drink before settling down in the drawing room. The children were interacted with, while Astha moved between the drawing room, kitchen, and dining area, unable to sit anywhere, the condom firmly in her heart.\n\n'How are things at work?' she asked after a while. Not that he ever discussed business with her. For that he had his father.\n\nTo her surprise he didn't brush aside the question. 'There is trouble in some factories in Noida, all the TV ones.'\n\nAstha had to drag her mind to this. 'Are you worried about ours?'\n\nHemant got irritated. 'Obviously I am worried. Different unions compete for power over the workers, and we get caught in the middle. Everybody suffers but who sees that?'\n\n'You pay a fair wage, the workers will realise that making trouble will benefit no one.'\n\n'Even if they don't come to their senses, I can't pay more than I do. Five thousand for the men with overtime, and four thousand for the women with benefits. Paying four hundred and fifty salaries is no joke,' brooded Hemant, 'and these are the rates.'\n\n'Then what is the problem?'\n\n'The Communist Party Union tells them they can ensure they get more benefits and a higher wage. Well, let's see. So far they have not been able to make inroads.'\n\n'Maybe nothing will happen.'\n\nHemant grunted, slowly sipped his drink, threw back his head on the sofa and closed his eyes. It seemed a bit difficult to bring up the condom in these circumstances, yet it had to be done. There were problems in her life as well as his.\n\nThere's something in your suitcase,' she said. 'Perhaps you would like to take it out yourself. I left it on the bed.'\n\n'Yes?' he said indifferently. 'What is it?'\n\n'A condom.'\n\nAt this he opened his eyes. 'Ah, yes.'\n\n'I take it when you travel you have sex, and that is why there are condoms in your suitcase,' Astha could barely keep her voice from breaking.\n\n'As usual, your imagination runs away with you.'\n\n'That is not an answer.'\n\n'For your information, I don't.'\n\nShe didn't believe him, and yet the hurt eased a little. 'You carry condoms just like that?'\n\n'Of course not. The dealer wanted to give me a girl, was very insistent, forced this condom on me, but I'm not that kind of guy, I left for the bar before the girl came. As you can see, it is not used.'\n\nAs a story it was thin, but yes, the condom was not used. Hemant got up and stroked her cheek. 'Even if you behave badly I love you, he said.'\n\nAstha forced herself to be content with this. It was too dangerous to venture further. \nChapter VII\n\nPipee called a week later.\n\n'I'm sorry, I'm sorry, I didn't know. You must have thought me horrible. I'm sorry,' Astha rushed to say.\n\n'How were you to know? I didn't tell you. You could be the one angry with me.'\n\n'Of course not, how could I be angry with you? You spent so much time with me, you showed me places you hate, you protected me from the monkey.'\n\n'Hardly,' she laughed, 'I can't stop monkeys from jumping onto people, much as I would like to.'\n\n'I have been waiting and waiting to tell you I'm sorry if I upset you in any way.'\n\n'Well now you've told me. And you didn't upset me in any way \u2013 I'm over that kind of stuff. Don't worry about it.'\n\nA pause. Then, 'You were going to show me your paintings.'\n\n'Please, come over. I would love that.'\n\n'Tomorrow?'\n\n'Please.'\n\nShe had called, she had called, and for a moment despite the condom, and all the wretchedness of the past week, Astha felt a little lighter.\n\n*\n\nShe looked at the work she was doing for the Manch, trying to prepare a readable memorandum that would combine historical accuracy with emotional appeal. It was proving uphill work. How could she make the nation care about the fact that no destruction of a temple had been chronicled in Babur, or in any other contemporary source, be it Abdul Qadir Badauni of the 16th centrury, or Goswami Tulsidas, in Ramcharitmanas.\n\nAstha stared rebelliously at her writing: _The_ _un-Islamic_ _black_ _stone_ _pillars_ _within_ _the_ _mosque_ _are_ _not_ _proof_ _that_ _a_ _temple_ _was_ _destroyed_ _on_ _the_ _Babri_ _Masjid_ _site._ _As_ _they_ _are_ _not_ _load_ _bearing,_ _they_ _were_ _probably_ _taken_ _from_ _a_ _Hindu_ _or_ _Jain_ _temple,_ _ravaged_ _by_ _Shah_ _Juran_ _Ghori_ _and_ _brought_ _for_ _decoration._ _Seeing_ _their_ _location_ _as_ _a_ _sign_ _of_ _contempt_ _for_ _Hindu_ _feelings_ _is_ _a_ _political_ _interpretation._\n\nIt sounded so uninteresting. Yet she had to go on sifting, sieving, fact from fact, fiction from fiction, and in the end not be sure of anything. It was lonely working on these pamphlets, it was not like painting where she required no mind to bounce her thoughts off. If only she had some of Aijaz's magic.\n\nAs she looked at what she was writing, her old hostility to words rose in her. She couldn't do it, she was a painter, not a writer.\n\n*\n\n'But it's not bad,' said Pipee the next day, when Astha showed it to her.\n\n'Pedantic, dry and boring,' said Astha.\n\nPipee pulled in the corners of her mouth while Astha stared in fascination at the dents it made in her cheeks. 'Now don't bother so much, just finish it. No one will read it anyway. The Manch excels in preaching to the converted.'\n\n'But it's for the nation.'\n\n'Please. Give me a break.'\n\nThey went to Astha's work room. Pipee's eyes flickered over the canvases. 'I know nothing about painting,' she said. 'You must teach me.'\n\n'There is nothing to learn. I've always responded to colours. It's words I find so slippery,' said Astha, the burden of _The_ _Testimony_ _of_ _the_ _Black_ _Pillars_ lying heavy upon her.\n\n'How do you manage to fit so many people in?'\n\n'It's something I learned from the miniatures. They are both very full and very detailed, I love that.'\n\n'It must take for ever.'\n\n'It does, rather. The one the Manch sold took almost six months. Now I am getting faster, but still \u2013 I can't work on them as much as I wish.'\n\n'You've got a pretty fancy set-up, it couldn't be that difficult. Doesn't your husband help you?'\n\n'My husband spends a lot of time at the factory and he travels too, so he can't really help with the children. And the setup...' her voice trailed off miserably. It was hard to explain her life, especially when she herself barely understood it.\n\nThe usual female trap, it's all right, you are not alone, we all experience it in one way or another,' said Pipee putting her hand on Astha's and pressing it gently. 'So if you want to do anything of your own I guess you have to work your ass off. You are like an ant too. I shall call you Ant, I'm not sure I like this faith business.'\n\nAstha blushed with pleasure, 'So we can be ants together.'\n\n'Exactly.'\n\n*\n\nAnuradha and Himanshu stared at Pipee over lunch.\n\n'Is that your scooter outside?' asked Himanshu.\n\n'Yes.'\n\n'How come you ride a scooter?'\n\n'To get around. Do you think only men should drive scooters?' asked Pipee.\n\nAstha felt embarrassed at her son's ideas, maybe she hadn't been sensitizing him to gender issues. She blushed into her roti, while Anuradha asked accusingly, 'How come you are called ant?'\n\n'My father thought I should work like an ant for the good of the community.'\n\n'And do you?'\n\nPipee smiled at the assembled mother and children. 'Sometimes,' she said, 'when I feel like it. I'm not a very good ant I am afraid.'\n\n*\n\nPipee left after tea when Astha began to worry about her children's homework. Driving home on her scooter, she thought I want to know her better, at least she doesn't remind me of Aijaz. Her house is quite near mine, that is convenient, I wonder if she realises she is attractive. Her marriage sounds horrible. I'm sure her husband is a jerk.\n\nShe thought of Astha's painting. She clearly had a political sensibility, which made her acquiescence in a traditional domestic set-up even stranger. Maybe she was just unawak-ened. And she loved her hair, it was so thick and curled around her face even when tied back, and her skin was so pretty, clear pink and white.\n\nAs for Astha who had shown such eagerness to know Pipee, how was she to realise that given certain circumstances, there was no aphrodisiac more powerful than talking, no seduction more effective than curiosity.\n\nThey began to meet more often. Astha was circumspect in revealing the amount of time she spent with Pipee. She knew it would be frowned upon as excessive. When the boundaries of what might be considered normal interaction passed, she started to lie. Thus an element of secrecy entered the relationship and gave it an illicit character.\n\nThey met on weekdays; evenings and weekends were out. Still Hemant caught a whiff of this new interest in his wife's life and was free with his disapproval. Since Pipee was a woman this disapproval was tinged with contempt, and the assurance of no real threat, indeed had Pipee been a man, Astha would have found it impossible to stray so far down the road of intimacy, or be so comfortable on it.\n\n'Women,' said the husband emphatically after a somewhat long phone conversation the wife had had with her friend, 'always mind-fucking.'\n\nAstha cringed. Mind-fucking. Not the excitement of the real thing. The organ penetrated, the ears, the weapon of penetration, words. Words, that left no mark but in the mind,where they mingled with others that had been used to describe someone else's past, till those experiences became your own, and you saw with other eyes, because you were no longer one person, but two. Listening upon listening, fucking upon fucking. In full view.\n\nThen she grew angry. How dare Hemant be so derogatory. Would he prefer her to be like him, with condoms in her suitcase, which a friend had put there by accident? She refused to engage with him on any issue, he was capable of nothing but the very crudest understanding. Instead she related the whole to Pipee who said that men were so pathetic, so fucked up themselves, they only understood the physical, and in this way she felt soothed.\n\n*\n\n'Have you ever been in a relationship with a woman?' asked Pipee one day.\n\nThey were lingering at the caf\u00e9 at the Tagore Arts Centre, after a lunch of kebabs and roti. It was late February, there were people sitting on the steps of the lawn next to them, on the walls white rose creepers were blooming. It was almost four, and the sunny spot they had originally chosen had long gone cold. A little boy was swabbing at the tables with a dirty cloth, a waiter was tilting the chairs against the tables, to enable the sweeper to clean properly. Pipee's voice had dropped to a murmur, Astha leaned forward to catch her words.\n\nAstha felt uneasy and didn't answer.\n\n'Well?'\n\nShe tried to laugh. 'I'm married,' she said.\n\n'So? Are you telling me you are happy, fulfilled, and what have you?'\n\nUnexpected tears came to Astha's eyes. Pipee was instantly contrite. 'I'm sorry. I didn't mean to make you cry.'\n\n'It's all right,' said Astha wiping her nose on the edge of her sari palla.\n\n'No, it's not,' said Pipee. 'If you are unhappy, it's not all right.'\n\nAstha went on sniffing. 'I don't usually think about it,' she offered.\n\n'Who would think about anything if they could help it?' said Pipee gloomily, 'God knows I have tried...'\n\nThere was a silence while Pipee drew squiggles in the rings of water left by their glasses on the table, and Astha watched her fingers. 'Have you?' she finally asked.\n\n'Once. Met her in school, continued in college, on and off for three years. Eventually she got married. Much later I did too.'\n\n'Oh.'\n\n'What was her name?'\n\n'Samira.'\n\n'Was she nice?'\n\n'Not often. She seduced me, and then when I fell in love, triumphed in that power. It was not so different from being with a man, though I am sure it can be.'\n\n'Oh?'\n\n'It is more a question of choice than people make out. That is what I believe at any rate. Besides sex is sex, don't you think? It is other things that become important.'\n\n'Yes, yes \u2013 of course. Did your husband know?'\n\n'I told him. But you know what men are like\n\n'No, I don't think I do,' said Astha forlornly. 'I have actually only known my husband, and now I am not even sure of that.'\n\nShe thought of the condom again \u2013 would it go on coming up in her mind at every point of sadness in her life, she wondered. She could tell Pipee about it, but Pipee might think she was inadequate in her responses, or weak in her understanding, or a fool. For now she preferred to keep this wound to herself.\n\n'Does your husband have affairs?'\n\n'I don't know.' Then quickly, 'Did yours?'\n\n'Well there were several women before we got married, I knew that.' Astha thought of the little gesture Aijaz had offered her, and now realised that it was in fact an invitation. 'I think he must have had an affair with Reshana Singh, the way she goes on. I know she thinks I am jealous, and maybe,' went on Pipee reflectively, 'I am.' She shook her head. There is no escape from jealousy, is there? We are all embryonic Othellos.'\n\n'I know what you mean,' said Astha gloomily.\n\n'Yes, well. I don't know why I am delving into the past today,' said Pipee, hauling her heavy bag onto her shoulder and getting up to go as the sweeper reached their table.\n\n'Maybe so I can get to know you.'\n\n*\n\n'You're so pretty, Ant.'\n\n'Do you really think so?'\n\nThey were sitting in Pipee's flat drinking beer before an early lunch. Pipee had made arrangements to go to work late, and now she pulled Astha by the hand and led her to the bathroom mirror.\n\n'Are we going to do mirror, mirror on the wall \/ Who is the fairest one of all?' laughed Astha nervously. She often felt an underlying tension when talking to Pipee, as they swooped and dived among their lives, offering bits to the other to share.\n\n'A modern version of it,' said Pipee putting on the light and pushing Astha's head gently forwards. 'Look.'\n\nAstha tried to turn away, 'I don't like looking at my face, especially so close.'\n\nThen she felt Pipee's hands in her hair, her clip undone, her hands framing the oval of her face. Lightly from behind she traced her eyebrows with her fingers, her nose, cheeks and mouth.\n\nThe two women said nothing looking at their reflections in the small water-stained mirror. 'See?' whispered Pipee.\n\nAstha saw nothing, and abruptly left the bathroom. Later taking a scooter-rickshaw home, she felt lost and confused, the image of the two of them in the mirror often returning when she thought of Pipee.\n\n*\n\nOne day, in Astha's house, a rare occasion. Pipee preferred to meet Astha anywhere else than in her house.\n\n'So this is the marital bed,' said Pipee, surveying Astha's room, full of double bed. 'The marital bed in the marital room.'\n\n'Like in most people's houses,' replied Astha, not particularly liking Pipee's tone.\n\n'I know. It's how I used to live. Are you happy here? Do you have good sex?'\n\n'Good enough, I suppose.'\n\n'Don't you know?'\n\n'Well he was my first, and only.'\n\n'You're joking.'\n\n'Not really.'\n\n'What about the other two?'\n\n'They were crushes. One I kissed a lot, with the other there were only letters.'\n\n'Have you ever wanted more lovers?'\n\nWhat could Astha say? She was living, the way people like her lived, where was the question of more lovers, or love for that matter?\n\nPipee stretched out her palm for Astha's hand. Gently she held it, fingering her thumb nail. Round and round the stubby nail Pipee's finger went, lightly tracing the pink part, the white part, the skin part. Astha looked at their two hands together, and inched a little closer to the woman on her bed.\n\nPipee took a firmer grip of the hand in hers, and turned it over, stroking the back of it, gently sliding her rings off, and putting them on her own fingers, manoeuvring her bangles off and slipping them on to her own more narrow wrist.\n\n'I look so bare without them,' murmured Astha.\n\n'All the better,' murmured Pipee even more softly. Her breath quickened, and she pressed the tips of Astha's fingers into her mouth, sucking each one gently before letting them go. Astha hardly dared breathe.\n\n'What would your precious spouse say, if he saw us together now?' asked Pipee.\n\nAstha swallowed and did not reply.\n\n'Did you say he was a faithful husband?'\n\n'I didn't say anything.'\n\n'Is he good in bed?'\n\n'I suppose.'\n\n'If you have to suppose, he is not,' said Pipee severely.\n\nAstha decided she knew nothing about love making, that she was inexperienced and stupid. 'What about you?' she responded in a low tone, 'You yourself have only had two lovers.'\n\n'Yes, that's true,' sighed Pipee. 'But I'm looking for a third.'\n\n*\n\n'Why so silent?' asked Hemant that evening.\n\n'Silent? Am I?'\n\n'You need me to tell you that?'\n\n'Sorry. I hadn't realised.'\n\nMore silence. What should she talk about? What had she talked about before silence came upon her? Their days, his day certainly. Now she made enquiries.\n\n'I have managed to bribe our union leader this time, but bribing is difficult, the workers are watchful and suspicious, I won't be able to do it again.'\n\nAstha hated it when Hemant talked about bribing, and yet the way he described it, it seemed necessary.\n\n'Pipee came over today,' she said, changing the subject.\n\n'That woman,' said Hemant.\n\nAstha's heart sank. Things would be difficult if Hemant became violent about his dislike. She tried to change the topic again, but Hemant was having none of it. 'What did you say she did?' he continued.\n\n'She works with basti children,' said Astha proudly. 'She helps them get through school, she gives them a sense of self-confidence, and strength.'\n\n'Who finances this?'\n\n'She's part of an NGO called Ujjala.'\n\nHemant grunted, 'One of those types.'\n\n'What does that mean?'\n\n'Take money from here and there, and pretend they are working.'\n\nIt seemed there was nothing Astha could say, and yet he wanted her to talk. She started on the children. That was always safe. It was what they were united upon, and it served its purpose now.\n\n*\n\nThat night, Hemant started his sex routine.\n\n'No,' said Astha, 'I don't feel like it.'\n\nHemant paused. This was the first time his wife had not felt like it. 'What's up?' he demanded.\n\n'Nothing.'\n\n'Then?'\n\n'Then what? Do I have to give it just because you are my husband? Unless I feel close to you I can't \u2013 I'm not a sex object, you have others for that.'\n\nHemant relaxed. That old thing. He took her face in his hands. 'Sweetheart, why do you upset yourself over nothing? You are my wife, I love you, there has never been another woman for me, never. On business trips people don't understand commitments to wife and family, they assume their clients want a good time. If I had had sex, would the condom not have been used? You only tell me,' he whispered, his hands falling to her breasts and circling them in the way that was so familiar, kneading them, pressing them, as he continued, 'you only tell me,' then pulling up her nightie, and fondling her, 'does what you imagine have any logic?'\n\nWithout her willing it her body responded. Hemant became even more ardent. 'Baby, you are the only one for me, what's the matter, are you jealous?'\n\n'No,' she said, trying to push him away, but it was of no use.\n\nAfter the marital function had been performed, Astha got up to wash herself. Looking up from her wet and soapy hands, she caught sight of a sad and haggard face. How old she looked, and yet she wasn't old. She was thirty-six, but all the life seemed gone. She leaned over the sink, and examined her face more carefully, certain to increase her wretchedness. Around her eyes tiny wrinkles were beginning to form. She stretched her mouth in imitation laughter, and they became more pronounced. She stared at her nose and saw the blackheads there. Her skin looked yellow and sallow, when she put her head up to look at the folds in her neck more clearly, she could see the white line at the base of her scalp, where the new hair had come, and the dyed parts grown out.\n\nWhy should anyone love her, she thought hopelessly. She was so ugly. She thought of Pipee sucking her fingers. She looked at them, and put them experimentally in her mouth. They didn't taste very nice \u2013 of soap and sex. What had Pipee thought of them? And what would Pipee's own fingers taste like? What had Pipee seen when she had pushed her face towards the mirror? Certainly not what she saw now. Slowly she went back to bed.\n\n*\n\nHer meetings with Pipee increased. When she was alone in the home in the mornings Pipee dropped by on her way to work, she phoned her at least five times a day, short brief conversations, but which drew each of them firmly into the nitty gritty of the other's life. And the days when she didn't see or talk to her were days with something missing, and not even extra hours at the canvas could fill the vacuum Astha felt. She started to fantasise about touching her, imagined her hair between her fingers, her skin beneath her own, her hands on the back of her neck.\n\n*\n\nAstha frantically trying to reach an appointed meeting place.\n\n'I'm sorry, I'm sorry, I'm sorry.'\n\n'I was about to go.'\n\n'I'm sorry. I couldn't help it.'\n\n'What happened?'\n\n'I forgot it was a bandh. Not a single scooter wallah agreed to come. Not one. They said they would be beaten up. I even offered fifty bucks.'\n\n'Then?'\n\n'In the end he took eighty.'\n\n'Eighty! Three times! You shouldn't have paid it, Ant.'\n\n'I had no choice. I would have given him anything.'\n\n'He was taking advantage of the situation,' said Pipee sternly.\n\n'What could I do? You were waiting, I kept thinking of that, but I was on the road, and there was no way to tell you.'\n\n'Oh sweetie, it's all right. Now, forget about it. I thought you must be having a problem. I can't imagine where I'd be without my scooter.'\n\n'I don't see why you haven't bought a car,' said Pipee later, as she was stirring her cold coffee. The meal had ended, and Astha was worrying about how she was to get home. 'One needs to be mobile. I learned how to drive a scooter, it was all I could afford, but with you it's different.'\n\n'We have a car.'\n\n'Hemant's.'\n\n'Which I use.'\n\n'Only when he doesn't.'\n\n'He sends it back from the factory with the driver whenever I need it.'\n\n'I'm sure he does, but you can be more independent with your own.'\n\n'A car costs over a lakh.'\n\n'You could hold an exhibition, and earn.'\n\n'Not lakhs.'\n\n'Ant, why are you being like this? Didn't you tell me your mother left you some money?'\n\n'With Hemant.' And the old hurt comes to choke her.\n\n'Hemant is not a monster. Have you tried asking him? Since ask you must.'\n\n'He'll say whenever I want the car, I have it. Also I can ask my in-laws for theirs.'\n\n'If Hemant can keep a car for his parents, he can keep one for you.'\n\n'Well, I, the children that is, use it as well. For tuitions, classes, and stuff.'\n\n'The point is'' went on Pipee patiently, 'if you had a car you would not have to do all this asking business.'\n\n'I can't ask for a car for myself.'\n\n'Why not?'\n\n'Hemant says there is going to be trouble in the factory.'\n\n'How long has this factory been running?'\n\n'Ten years.'\n\n'I imagine he has made enough money to buy you a car.'\n\n'He is very generous to me.'\n\n'Good. Now come let me drop you home otherwise you'll be cheated all over again.'\n\nAs they roared through the streets of Delhi, Astha leaned against Pipee, with her arms around her waist. Once or twice Pipee turned to ask, are you all right, Astha merely nodded, too happy to speak, even had the sound of the vehicle allowed it.\n\n*\n\n'But why? The car is there for you whenever you want it.'\n\n'Please, Hemant. I am thirty-six. I need to be independent. I am always adjusting to everybody else's needs.'\n\n'And the money?'\n\n'We could use what my mother gave.'\n\n'You know I have invested that for the children, and in five years the amount has grown nicely.' Hemant looked satisfied. Astha had heard all this before, heard when the bonus shares came, heard when the dividends came, when the debentures were bought, heard as it doubled, trebled, quadrupled. There was no question of touching it, she knew that. Only somewhere surely there was money she could touch? She said as much.\n\nHemant looked at her. 'Who is putting these ideas into your head?'\n\n'Nobody'' said Astha offended. 'Does somebody have to tell me to want a car?'\n\n*\n\n'Mama, Papa is getting you a car?' Anuradha. Hemant had told her first.\n\n'It is also my money'' said Astha suddenly angry. The children turned towards her, slightly shocked. Only prices were discussed in their house, never money, and certainly not whose money was whose. It was all common money because they were a family.\n\n'Papa's money too'' said Anuradha quickly.\n\n'Of course Papa's money too. But if necessary I will hold an exhibition to help pay for this car.'\n\nHimanshu put his hand into hers. 'When I earn I will buy you a car'' he proclaimed. Astha tightened her hold on his thin interwoven fingers and stared at the overgrown nails, at the fine hair glinting blondly, at the sun exposed brown skin.\n\n'And I will buy a car for Papa'' said Anuradha.\n\n'But Papa and Mama are not separate'' said Astha, quickly. 'Whatever you buy will be for both of us. Don't I use the car we already have? It is not Papa's or mine. And now we will have a second car, besides the one upstairs, neither Papa's nor mine, but for everybody. We are a family with growing needs.'\n\nIt was the end of the term, before the summer holidays, and PTA day. Astha was in her children's school, trying not to stare at the fathers and mothers around her, united and content.\n\nThere was a whole list of teachers she had to see.\n\nHimanshu had done badly in his mid-terms. He hadn't finished his papers, but really he knew everything. Astha was waiting to tell his class teacher this, something she had been saying since nursery. The teacher in turn would tell her that even so, he had to increase his speed, other children managed. If he didn't, he was going to find it very difficult at the higher levels. Astha could predict the conversation verbatim, but these motions had to be gone through.\n\nAnuradha was doing badly in science and maths. She didn't understand the method of explanation, and Astha had to find out why in a way that didn't compromise her daughter's intelligence, attentiveness, or abilities. Both her children were dead against her discussing any of their problems in school. Before she had left Anu had screamed, don't say anything, don't Mama, then in class the teacher says, so you are having difficulties, why don't you ask me during the lesson instead of complaining later, and the kids stare at me, as though I am a moron. Besides, I keep telling you, there is no _point_ asking for explanations, they all repeat the same thing in the same way, only slower.\n\nHimanshu had looked equally worried, you are not going to say anything to my teacher, are you, Mama? In class, she'll say something to me, she will get angry.\n\n*\n\nWith the futility of it all firmly established, she waited in line after line to see various instructors, behind parents busy pumping them for the secret of success in that particular subject. She was going to be late for her meeting with Pipee, why had she, against all experience, allotted two hours to school? Now precious minutes were being wasted in the corridors of this huge and unfeeling institution.\n\nThe maths teacher, Mr Sharma, before whom there was a line half a mile long. Anu obviously not the only one having problems. After an hour of waiting, her turn.\n\nAnuradha? Yes, a very bright child, but she should work harder, if she has a problem in following, she should ask me, I tell them, ask me, there is no excuse for not understanding. She should do one hour timed exercise every day, maths is only practice.\n\nAstha pulled out Anu's mid-term exam from her bag, 59 out of 100. Mr Sharma's temper rose the moment he saw it, look at this, correct method, but a mistake in adding and the answer is wrong. And here, she has copied the sum incorrectly. How will she get marks? Crooked margins, untidy rough work. Careless, careless.\n\nAstha stared at the paper, she understood nothing of it. Were Anu's crimes so bad? A crooked margin, a sum added wrong, another carelessly copied, did that result in 59 and feeling a failure?\n\nSurely it doesn't matter if the rough work is not neat? she queried.\n\nIt is the attitude that matters, the attitude, thundered Mr Sharma.\n\nIn the face of this, further comment seemed redundant.\n\n*\n\nFinally, all the teachers met, the Vadera children ticked off in various registers, her participation in the learning process marked, her children's faults pointed out and noted, and she was free. Frantically she ran out of the school gates, she was over an hour late already. Pipee would have cooked for her, she would be wondering.\n\nAs she rang the bell to Pipee's apartment, she could hear footsteps coming towards the door. Her heart beat faster, explanations trembled on her lips. The door opened, and before her, the face, always in her mind, always indistinct, the long narrow eyes, the hair which sprang back wild and unruly, the voice she could drown in, the mouth that pulled inwards as she smiled, the little mole hanging under her nose like a dew drop.\n\nThere she was, and there Astha stood, and nothing else mattered. Silently Pipee motioned her in, took her bag, and closed the door.\n\n'What took you so long? I was getting worried.'\n\n'Sorry, Pip, I'm sorry, I couldn't help it, there were millions of parents, and the teachers took ages. I kept thinking of you waiting, I felt terrible the whole time.'\n\nThey were standing. Slowly Pipee put her arms around her. She could feel her hands on the narrowness of her back, on the beginning spread of her hips. Gently she undid her blouse hooks, and her bra, looking at her face as she did so and slowly she continued, feeling her back with her palm, coming round up towards her breasts, feeling their softness, especially where the nipples were, feeling them again and again, in no hurry to reach any conclusion. They were enclosed in a circle of silence, the only sound, the sound of their breaths, close together and mingled.\n\nIn the small bedroom, Astha tense with nervousness. She was afraid, yet there was no going back. Sensing how she felt, Pipee took her time, touching every crevice of her body with her mouth. The sweaty patches of her armpits with small stiff hair beginning to poke out, the soft fold of flesh where the arm joined the torso, the hard bony part behind her ears, the deep crease between her buttocks, the hairiness between her thighs.\n\nIn between they talked, the talk of discovery and attraction, of the history of a three month relationship, the teasing and pleasure of an intimacy that was complete and absolute, expressed through minds as much as bodies.\n\nAfterwards Astha felt strange, making love to a woman took getting used to.\n\nAnd it also felt strange, making love to a friend instead of an adversary.\n\n*\n\nShe returned home in a daze. As she neared her house, she succumbed to panic, she was a mother, nothing should disturb that. For a brief and guilty moment she wished she was like Pipee, alone and free, but she checked herself. A large part of her belonged to her children, that was how she lived her life. She couldn't imagine any other way.\n\nShe was a wife too, but not much of her was required there. A willing body at night, a willing pair of hands and feet in the day and an obedient mouth were the necessary prerequisites of Hemant's wife.\n\n*\n\nA few days later. 'Hemant should be pleased'' said Astha to her lover, 'he says women are always mind-fucking.'\n\nThey both laughed at the wife's revenge.\n\nAstha was in love. All day she thought of her, visualising the turn of her neck, long, sloping, unornamented, the collar bones on either side of the small hollow at the base of her throat, the screws of her hair latticed, as she had once seen them against the dark, heavy, green of the trees of the Tagore Arts Centre. And her fingers, long and so narrow the bones showed, with stubby nails, and a snake ring, three silver bands, two small turquoise eyes, two black painted dots for a nose. And her eyelids, that fragile tender area where she wanted to press her lips, compressing them to the size of peas to enclose that space. And the mouth with its inward-turning corners, she could gaze at those dents for ever.\n\nFrom time to time she brooded about her own sexual nature, but her desire for Pipee was so linked to the particular person, that she failed to draw any general conclusions. So far as her marriage was concerned, they were both women, nothing was seriously threatened. Meanwhile her best time at home was when she was fantasising about the one she loved without interruptions, lost in her thoughts, wallowing in her feelings.\n\nAll this made it difficult for her to focus on what was going on around her. She was able to forget she had another life only when she was absorbed in her painting or her children's homework, an echo of an earlier simplicity that now appeared to have some advantages.\n\n*\n\nAstha was surprised when Hemant noticed.\n\n'You seem distracted'' he pointed out. 'What is it?'\n\nShe felt a flash of fear, but then an affair with a woman was not an easy thing for a husband to suspect. Caution drew her lips into a smile, and put a hood across her eyes. 'Nothing'' she said. 'What do you imagine?'\n\nA look of dissatisfaction that it seemed must always be on one of their two faces, crossed his brow, giving her a momentary sense of control. She was not there. How right he was. But when had he acquired the sensitivity?\n\nWhat about the times he had not been there, and the reasons had always been such that her own claims seemed selfish. Now sexually involved with another, she realised how many facets in the relationship between her husband and herself reflected power rather than love. Hemant had managed to ignore her because ultimately he filled his own landscape. That her discontent had been expressed in nuances that were minor, only helped him in his disregard.\n\nIn the days that followed, Hemant began to watch Astha. Let him watch, thought Astha, he who had not looked since the early days of marriage, was now looking and found that what he saw did not add up.\n\nHer lies grew skilful. Her desperation and her need ensured that they tripped off her tongue, as though she had rehearsed them for hours.\n\n*\n\nFed by right-minded parents, Astha had believed that never, ever must one lie. There was a Pinocchio lurking in her moral self, waiting and watching. Her nose would grow, her eyes cross themselves in vain attempts to hide the gruesome deed, her skin would turn yellow and pimples sprout all over her. Her inner ugliness would be reflected for all to see.\n\nShe had lied about the boys she had known, and each time she had been punished. They had left her, she had not deserved better.\n\nWhen she married she had wanted to tell her husband about those boys, but she had been afraid he would not accept her, and that tiny seed, usually forgotten, was still inside, telling her she was unworthy. She had compromised by being excessively truthful; she knew her husband trusted her implicitly.\n\nNow, she lied all day. The strongest thing in her was the most secret. Pipee encouraged her. Not for her the moral values of George Washington, the boy on the burning deck, Eklavya, Ram, Sita, Lakshman, those for whom words translated into codes of honour never to be broken no matter what.\n\n'Of course you have to lie. They don't own you.'\n\n'I know, but... I wish I didn't feel the way I did.'\n\n'What way?' asked her lover very naturally, and when she didn't reply, very insistently, 'what way?'\n\n'Oh you know'' Astha became vague. 'So much of the real stuff is with you, and since I can't talk about it with my family, it makes me feel pretty schizo.'\n\n'Why do you have to talk about it with them? You talk about it with me. Are they your guardians or something?'\n\nIt was hard to explain. Pipee lived on a grander, more open scale then she did.\n\n*\n\nMeanwhile this grand and open creature was growing jealous of other claims. She had even wondered, to Astha's horror, when she was going to inform Hemant about them.\n\n'He is not your owner, you know, he'll have to face up to his inadequacies.'\n\n'No, no \u2013 I can't do that.'\n\n'Why not?'\n\nIt seemed so unthinkable, how could she explain. 'Maybe I'm a coward'' she remarked thoughtfully.\n\n'Oh dearest'' sighed Pipee, 'Don't be so hard on yourself. You've lived a certain way, you are used to certain ideas, you can't suddenly be different. If I am impatient with your situation, it's because I want you to be happy.'\n\nShe turned the other's face towards her, took out her pins and stroked her open hair, reaching into the scalp, in a way that reminded Astha of her mother oiling her hair every Sunday when she was young. She closed her eyes and sank against her, feeling as though she were in a warm bath. With Pipee there was no battering against something hard and ununderstanding, she was all warmth and intuition. She thanked God again for this love in her life, when she had thought all chance of love was over.\n\n*\n\nIf God had given her love, there was no time supplement with this gift, so Astha often found herself wishing despairingly she could live each day twice, once with Pipee, and once on the ordinary plane.\n\nShe dreaded the occasions when her lives clashed, and was at no time more at the mercy of her circumstances than weekends.\n\nOne Wednesday Pipee said firmly, 'There is a gay and lesbian film festival this Saturday. I know, I know, Saturdays are difficult for you, but I want to go, and I think you should too.'\n\n'But Pip...'\n\n'Don't say anything. I want to see the films with you. Don't you think they have a special relevance?'\n\nThis was undeniable.\n\n'Do you mind if I ask Hemant? Don't worry, he is bound not to come.'\n\nPipee made a face. 'Then why ask him?'\n\n'He's going to be home this weekend.' went on Astha hurriedly. 'He will find it strange if I make a programme without him.'\n\n'Let him.'\n\n'He is beginning to complain.'\n\n'Of what?'\n\n'Oh, just'' said Astha, avoiding specifics. 'You know. He feels something is not quite right.'\n\n'Well, it's not. It's time he woke up.'\n\n'I wouldn't go so far'' said Astha quickly.\n\n'You wouldn't, huh?'\n\n'No. Everything is all right the way it is.'\n\n'Don't ignore the obvious, Ant'' was all Pipee said.\n\n*\n\n'Go with you and Pipeelika Khan to a gay film show? Are you out of your mind, Az?'\n\n'Well, I am going with her. You can for once come to something I am interested in.'\n\nHemant stared. 'I'm not interested in homosexuals'' he said. 'And I thought neither were you. But I'm learning something every day.' He held out his hand, and Astha slowly put her own in his. 'Stay home. We can rent a video. We haven't done that in a long time.'\n\nIt was not fair. It needed his wife's having an affair for Hemant to promise to see a video with her, something he knew she loved. Such an evening might have made her happy a year ago, now it seemed like blackmail.\n\n'I've promised Pipee...' she said.\n\n'So? Unpromise her.'\n\n'Maybe next Saturday, but not this.'\n\nA sullen look settled on Hemant's face. Astha could see resentment, and she felt sorry. But not half as sorry as she would feel if she didn't go, didn't sit next to Pipee in a dark hall, with their arms, hands, knees touching.\n\n'Why can't we do this some other time?' she went on, 'often you have not been here for me. I think you should be understanding about one day.'\n\n'I was busy, Az, I was establishing myself.'\n\n'For ten years?'\n\n'It takes that long, you knew that, you supported me.'\n\nTheir disagreements had the history of their marriage hanging onto them, and Astha had no time for this now. 'We can continue our discussion when I come back'' she said.\n\nHemant refused to respond, and Astha could think of nothing further to say. She didn't want to leave him like this, but he was giving her no choice. Quickly she put on the first sari that came to hand, grabbed a shawl and left.\n\n*\n\nIt was only when she walked down the road to the scooter stand that she realised it was in fact a chilly day, and she was going to feel cold. Should she go back for a sweater? No, she didn't want to encounter Hemant again, better to freeze. She wrapped her shawl and palla firmly around her, and hoped it wouldn't rain.\n\nThe cold bit through her in the three-wheeler, and her nose began to run. She looked through her purse for a hanky \u2013 no, she had forgotten to bring a hanky as well. She bent down and wiped her nose on her petticoat.\n\nAt the community centre, Astha made her hesitant way into the hall. She was late, and it took her a few minutes to adjust to the darkness. She felt her sari being tugged, and there was Pipee leaning against the wall by the entrance, waiting. She sank down next to her, feeling exhausted after the battle with Hemant, looking with cursory interest at the screen, registering indifferently the men and women, speaking broad American about the discrimination they faced as gays. In between some social scientist gave his opinion, in between that there were clips of marches and demonstrations. Astha looked at the faces on the screen. All of them open, none of them living a life of lies.\n\nPipee's voice breathed through the craven recesses of her mind, 'We have to struggle for acceptance and the right to love as we feel. Don't you think so, Ant?' But try as Astha could, she could not connect to what she was seeing. Her own situation was different, though if Pipee didn't think so she would keep that information to herself.\n\nIn the intermission, Astha broke the news to Pipee as they were drinking coffee, 'I have to go.'\n\n'Why?' she could hear the concern in Pipee's voice, feel her hand as it lay in the crook of her elbow. 'Are you feeling all right? Don't you find it interesting?'\n\n'Anuradha has a test. It was difficult for me to get away today.' She didn't want to go into the whole Hemant thing.\n\nAt the sounds of domesticity, Pipee's face twisted slightly, but she merely said, 'Why can't the father sit with her for a change? Why do you have to do it all the time?'\n\n'That's the way it is.'\n\n'Go then.' She gave her a little push.\n\n'Dearest, don't be offended, I'll make it up to you, I swear.'\n\n'How will you go home?'\n\n'Scooter'' said Astha dully. How did it matter how she got home? Maybe she should crawl to that shrine of marital and maternal bliss on her belly, or drag herself on her behind with the stumps of her legs sticking out straight out in front of her, pulling herself along with her hands the way lepers did, begging for alms around the crowded intersections of Delhi.\n\n'Pipeee'' Astha could hear somebody shouting. 'Hurry up, the next film has started.'\n\n'I'll call you when I get home.' She squeezed Astha's arm again, kissed her cheek, and vanished into the hall warm with its comradeship for her, cold with its indifference towards Astha.\n\n*\n\n'How come you are back so early?' asked Hemant in a jocular tone, as he saw her descend from the scooter, her ears red and her nose now running like a river. Without looking at him, Astha held out a hand and said, 'Hanky.' Hemant took out his own and gave it to her. She blew her nose, at long last, through the tortures of the day, she was able to blow her nose.\n\nThen she looked at him. He was smiling. He thought he had won, and now was trying to be nice to her.\n\n'Just like that'' she said.\n\n'Gay and lesbian films not your cup of tea, huh?'\n\n'Not at all. They were very good.'\n\n'Then?'\n\n'Then what? The hall was crowded, and I didn't have enough to wear, and I was feeling cold, and I came home because I didn't want to sit on the floor too long.'\n\nHemant looked disbelieving.\n\n'Have the children had their lunch?' asked Astha.\n\n'Yes, for all you care.'\n\nIt didn't bother Astha, his tone, nothing bothered her. She went inside, she was hungry, she had had no breakfast, and she now ate some leftover lunch. Then she made herself a cup of tea, she felt a sore throat coming, and for now would think of nothing but her physical wellbeing.\n\n*\n\nThat evening Hemant solicitously offered her a brandy for her cold. He talked of her painting, he talked about the children, he talked about her mother, his parents, he said maybe next year they would go on a holiday to America, by then the factory problems would be sorted, he worked very hard, and he needed to take it easy. And the car, what about that?\n\nShe, which car?\n\nHe looked hurt. The car she had demanded, had said she needed in order to be independent, he had arranged to get one in the company name.\n\nIn the company name. It was that easy after all.\n\nWhat did she think of a white Maruti?\n\nAstha's feelers went up. She could not remember when her opinion had been sought about a major purchase. Why was he being so considerate? Was he trying to buy her? True, she would be able to rush to Pipee's whenever she liked, in the car her husband had bought for her, but how was that going to make her feel? She didn't want a car, she realised, it would end up making her feel more guilty, and were she to express all this to Pipee she would say, but it is your car, why do you feel you have to pay for it with mind, body and soul, she could hear her voice even now.\n\n'I don't know how to drive'' she temporised.\n\n'I'll teach you, we can learn every Sunday'' he said, caressing her. 'We will be together.'\n\n'Yes, yes, I suppose.'\n\n'Try and sound a little enthusiastic, will you?'\n\n'I am enthusiastic, why do you look for meanings in everything I say? They say husbands should not teach wives, the relationship deteriorates'' said Astha irritably.\n\n'We'll see. I don't want to waste money on lessons.'\n\nThe car came. Hemant in fact had no time. In the end it was Ram Singh, the driver, who taught Astha in the colony lanes.\n\n*\n\nThat summer Rajiv Gandhi was assassinated by a suicide bomber in Tamil Nadu during an election campaign. Political uncertainty meant that Pipee's work in the bastis grew more demanding and she could not see Astha much in the day. Astha felt her absence every minute, and when Pipee called her in the evenings, she went, but her home situation was such that the meetings had to be hurried, not more than an hour, not much for lovers, certainly not much for Pipee.\n\n'Ajay has written. He is keen to sponsor me. He has been suggesting it ever since I finished my MA.'\n\n'What?'\n\n'A Ph.D.'\n\n'You never told me.'\n\n'It was before I met you'' said Pipee.\n\n'How come you never mentioned it? And why are you telling me now?'\n\n'Because his letter has come'' said Pipee in limpid accents.\n\nAstha slid Pipee's hand from under her shoulder and looked at it. There was the ring, there were the bones, there were the long thin fingers. She placed their palms together, and thought it was an illusion, you could never be one with another, no matter how hard you tried. It was better to realise and accept that, life became easier once you did.\n\n'Yes'' she said, 'you better explore doing a Ph.D. It's a good thing for the future.'\n\n'Well let me see. It will mean leaving you.'\n\nAnd Astha's poor heart rejoiced to hear she was important.\n\n*\n\nDriving home, Astha brooded over Ajay's letter. She was not stupid, she knew why Pipee had brought up the letter. She wants a full life, after six months she wants commitment, if I can't give it to her, why shouldn't she look elsewhere, but she didn't want Pipee to look elsewhere, she wanted her to stay with her for ever, as she was, as they were.\n\nResentfully she thought of Pipee's Ph.D. Suddenly it was a burning desire. Well, she knew why. She was saying if Astha had her children, she had her Ph.D., as though you could equate the two.\n\n*\n\nAstha: I have a fantasy, listen my love, and do not laugh. It is not much, I think it is not much.\n\nI have a room, small but private, where my family pass before my eyes. It is very light, before me is a wall which divides the house, but I can see my children, that satisfies me, though to them I am invisible, that satisfies me too.\n\nThis room will be our room, you with me, living in harmony. Our lives are separate, different things call to us, different demands are made on us, but always that solid base beneath us, like two flies caught in a sticky pool they cannot leave.\n\n*\n\n'Sticky flies? You must be mad.'\n\n'All right the image is bad. Still, you know what I mean.'\n\n'You are a hopeless romantic. You want me and you want not to leave your old life. It's a nice fantasy, I wish it were possible. I also wish'' added Pipee after a little thought, 'that it had taken place in my house.'\n\n'Does that show something?' asked Astha, wishing that she with whom she shared everything, was not quite so into analysis.\n\n'Well, what do you think?'\n\n'I think nothing. It was a dream, an idle dream, for God's sake, something I know can never happen.'\n\n'But it can, don't you see, even in a dream you are in your precious Vasant Vihar. There are other places in the world, Ant, if you would only consider them. Instead you allow yourself to be shut up by that man, who neither knows nor appreciates you, and for what? I do not understand.'\n\n'There are my children.'\n\n'Your children don't have to be stuck in that house any more than you do.'\n\nAstha's mind boggled. What about their school, their routine, their friends in the colony, their grandparents, their father, who whatever his faults, did love his kids? Maybe she was deeply conventional but for her the business of raising children had a set of dynamics that were the standard ones. That those dynamics did not include companionship and understanding was regrettable, but she had grown used to it. She saw herself as a bird pecking at a few leftover crumbs from the feast of life. She said as much. Pipee stared at her.\n\n'I never thought of myself as a crumb'' she said dryly.\n\nThis drove Astha on to further explanations.\n\n'I love you, you know how much you mean to me, I try and prove it every moment we have together, but I can't abandon my family, I can't. Maybe I should not have looked for happiness, but I couldn't help myself. I suppose you think I should not be in a relationship, but I had not foreseen... Oh Pipee, I'm sorry I am not like you.'\n\n'What do you mean? Don't you want an honest aboveboard life?'\n\n'You are being unfair.'\n\n'When do you ever think of me? Always their needs, your needs, before mine.'\n\n'That's not true.'\n\n'It is. You can't see me in the evenings\u2014'\n\n'How can you say that? Just the other day I spent the whole evening with you, I went home at twelve, I told endless lies\u2014'\n\n'Who asked you to tell lies? I didn't. Don't you see, Ant, I want an end to all this deception.'\n\n'My whole life is a fabric of lies'' said Astha sadly, 'you are the one true thing I have.'\n\n'And you don't want to change it. That's the trouble with married people'' said Pipee gloomily, 'there are always others involved. Why did I think with a woman it would be different?'\n\nPanic rose in Astha. Tears came to her eyes, and she felt a headache coming on. All she wanted to do was drive back, shut herself in her room, and sleep till the end of time. She got up.\n\n'What are you doing?'\n\n'Going home, since you ask.'\n\nPipee reached out and pulled her dupatta. 'Don't you get it? That I love you, I want you, I miss you?'\n\n'What about your other friends and your work?' asked Astha in a small voice.\n\n'What about it? Work never kept one warm at night, and yes, I have friends, but they are not people I choose to be intimate with. Either I spend my time here moping, or I go out with them, talk, laugh, then come home to a flat which holds the moments I have had with you. It reminds me\u2014' Here she paused, Astha looked tortured, and Pipee continued quickly, 'whatever it is, I don't wish to experience that kind of emptiness again. Sometimes I go crazy with longing, and I can't even pick up the phone.'\n\n'You can.'\n\n'I can't. I don't want to hear your husband's voice, I don't want to put the phone down if he picks it up, I don't want to share your life of lies.'\n\nAstha thought that if husband and wife are one person, then Pipee and she were even more so. She had shared parts of herself she had never shared before. She felt complete with her. But this was not the time to say these things.\n\n'I'm sorry, I don't mean to be harsh'' said Pipee contritely. 'Leaving a marriage, even like yours, could not be easy. I do feel that away from that house and those people you will be able to lead a fuller life. You have so much in you, so much to give, but take your time. Whatever you do it'll be all right.'\n\nAstha turned towards her gratefully. That her lover should understand how she was feeling was enough. She sank down to the sofa into her arms.\n\n*\n\nIt was dark before Astha got up to leave. With the thought of Pipee in her mind, the scent of Pipee on her body she moved trance-like towards her car, driving slowly and automatically down the roads of Delhi.\n\nAs Astha parked the car outside the gates of her house Anuradha came rushing out. 'I'm failing'' she gasped. 'You have to see my teacher. Where were you? I have been waiting hours and hours.'\n\n'In what subject?'\n\n'Maths.'\n\nOf course. It had to be maths.\n\nAnuradha was biting her nails. 'I'll fail the year, I'll fail the year, I'll fail the year, I'll fail the year'' she chanted in a frenzy.\n\n'Can we wait for the results before you decide that?' demanded Astha.\n\n'You don't care! Why can't I have tuition, all my friends have tuition, but you want me to do it by myself, because you did.'\n\n'Anu, don't be unreasonable. When did I say...?'\n\n'You did, and now you have forgotten. You want me to be like you. I am not, but you don't care.'\n\nAnuradha stared at her mother, tears streaming down her young cheeks. Dear God, thought Astha, when did I say she had to be like me? When did I say she couldn't have tuition because I hadn't? When?\n\n'Sweetie, can we talk about this later? I have just come home, I am tired, I don't remember what I said and when and why. If it is necessary of course you will have a tutor, but you also have to learn to do things on your own.'\n\nAnuradha ran inside, scowling. 'I had to phone Papa, you weren't here, he said to wait till you came home, you would know what to do, you know the teachers, but what's the use? I have been saying I need tuition. No one listens to me.'\n\nAstha walked in wearily. In the face of Ami's maths her own feelings seemed an indulgence rather than a necessity. They would have to wait, wait in the wings, wait on a more permanent basis.\n\n*\n\nAfter dinner Astha watched her daughter sit in her favourite chair and read unconcernedly. The lamp shone on her hair, highlighting its copper shades. Her socks lay untidily on the carpet, she was wiggling her toes in front of the heater, the light catching the lurid colours of her nail polish. Her troubles were over, her friend's tutor had been phoned, he was going to start from next week.\n\nAstha sighed and took another sip of her tea. On the sofa next to her, Himanshu laboured over his homework, his notebook getting more and more smudged as he continually rubbed out what he had written, shredding bits of the page.\n\nHer thoughts wandered, to the series she had imagined on mosques and temples in Ayodhya, Kashi and Mathura. Pipee had thought it was a brilliant idea, but there was no space in her head to execute any idea, be it ever so brilliant. \nChapter VIII\n\nNovember 26th\n\nPip called. I want you to come with me for three weeks. Will you?\n\nWhere?\n\nThe Ekta Yatra, from December 10 to January 26. It starts from Kanyakumari and ends in Kashmir.\n\nI thought of the Rath Yatra canvas I had painted for the Manch. Here was another journey, taken by another Leader.\n\nWhat's this Yatra all about I ask her.\n\nThe usual. This Leader says he wants to unify the country.\n\nThey all say that.\n\nWell, he is going to spread his message from north to south, east to west. I know it's a political stunt, yatras like this create nothing but trouble, but it's an excuse to be together. Will you come?\n\nOf course.\n\n(My heart is beating, my hands begin to sweat, of course I will come as though it is the easiest thing in the world, of course, as though I can get up and go anywhere I like, anytime I like, of course, because I love you, and at times love makes life simple because it demands you worship at its altar, dragged though you may be kicking and screaming.)\n\nGood. Talk to you later. I have to work on this, her voice breaks in on my thoughts.\n\nI put the phone down. Three weeks with her. My mind is whirring, how will I manage it, what will I say, but I have to go, I have to. She has never asked me to do anything so directly. Is this a test, I wonder?\n\n*\n\nNext day\n\nWith P. Why a Yatra? I ask. It's not like Ayodhya. You are not taking Ujjala helpers to sensitise them to communal issues.\n\nI thought of this as a way to be together. Then she hesitated. I am learning the language of her voice, she was about to say something she thought might hurt me.\n\nAnd?\n\nShe was silent.\n\nCome on, Pip, tell me, though actually I don't want to hear. I want to stay with the pleasure of proof that she wants to be with me, but I force myself to ask because shadows between us are ten times worse.\n\nShe started talking of communal issues as an area of research, something that now interests her. (Due to Aijaz's death, I imagine.) With an interpreter she could get some field work done.\n\nSo she thought I might be upset because she is still thinking of her wretched higher studies.\n\nNice, I said enthusiastically, what a good idea.\n\nShe looked pleased, and then told me how she was arranging it. I waited for Neeraj's name to appear, which it did. I am sure Neeraj is pushing her towards this Ph.D. I hate Neeraj, though we met only once. She is fake arty, has a deep voice, smokes endless cigarettes, and of course has a marriage where her husband is deeply involved in all she does. He is a lawyer, and helps her when necessary, and in Ujjala it is always necessary. She looks upon Pipee as her prot\u00e9g\u00e9. Pip says she gets on her nerves sometimes, but she has been so kind to her, she can never say anything. I personally think she is insensitive and power hungry. I wonder if Pip has told her about us.\n\nNeeraj's cousin is a journalist, she is arranging that we go in the accompanying bus. Pip has to produce articles for her newspaper (I am sure Neeraj thinks it will help her CV). The cousin herself is going to make an initial brief appearance. After a week or so we go to Bangalore, then to Shiksha Kendra.\n\nShiksha Kendra, her school, her past, her mother. I want to absorb everything to do with her, because it is her.\n\n*\n\nNext day\n\nDreading talk with H. Yet why should I be nervous, hasn't he travelled, it is my turn, but even as I think this, I know it is the wrong argument to use. I shouldn't seem to want justice, it will create endless arguments, I must seem to want his compassion, his magnanimity. He is doing me a favour, but I must also be firm, he is not going to be compassionate and magnanimous if he has a choice.\n\nYesterday. Pip, why can't the two of us go on a holiday for the weekend, why the Ekta Yatra?\n\nI want more time with you. After this we will go on a holiday if you wish.\n\n*\n\nNight\n\nHemant could not believe his ears. What, go where? Do what?\n\nGo on the Ekta Yatra, cover it for the Manch. (Since he has never taken much interest in the Manch, he doesn't know this is not the kind of thing they would do.)\n\nThen he started. And went on and on. I was running off on a wild goose chase, neglecting my family and burdening his poor mother with my responsibilities. I had no sense of what was fitting for a woman, I hadn't bothered to ask him whether it was appropriate or convenient. Ever since Aijaz had died, and I had started being exploited by the Manch, and gone to Ayodhya, and met Pipeelika Khan, I had no sense of home, duty, wifehood or motherhood.\n\nI said nothing. What should I reply to? The text or the subtext? How calm my relationship with Pipee has made me! There was a time when had he said half so much I would have started crying. Now all I said was I am leaving on December 8th, and will call my mother to help with the children, I didn't want to bother his mother, or even him.\n\nThis made him even angrier. 'Why stop at this yatra?' he practically shouted. 'The Dalits have called a Nyaya Yatra, they want justice, some mill workers have called a Roti Yatra, they want employment, the Indian Save the Cow Federation has called a Cow Yatra, to prevent cow slaughter, every Tom, Dick and Harry is going to march up and down India demanding something. Join them all.'\n\nAgain I said nothing.\n\n'Who will protect you? Suppose you get raped?'\n\nHe doesn't care how low he hits. 'Why would I get raped?' I asked after a moment.\n\n'Anything can happen. All these yatras have goondas attached to them. You think everybody who is going is so moved by the desire to unite our country? Our country is better united by you staying at home, so that there is one less incident to cope with.'\n\nEvery day the papers are full of crimes against women. Yet I have to learn to not be so afraid. There are other women in this world. They live.\n\n*\n\nNovember 27th\n\nWhen I told P. about the rape she got quite angry. Tell that sod to stuff his fantasies of rape up his ass. What does he mean by scaring you like this? It is his way of keeping you at home.\n\nShe takes what I feel, clothes it into words, and there it is, for us to look at, and for me to feel better.\n\nHave to phone my mother tomorrow. Another session of blackmail and guilt.\n\n*\n\nNovember 29th\n\nFinally phoned. She was all against my going, of course. Little does Hemant realise how much her thinking matches his.\n\nShe started out by blessings \u2013 from both God and the swami, swiftly moving to blame. 'I have been expecting your call. It has been a long time since I heard from you, but then I know how busy you always are.'\n\n'I'm sorry, Ma.'\n\n'Give my love to Himanshu and Anuradha, give my regards to Hemant'' she went on, messages too important to be left to the end.\n\n'I'm going\u2014' I started.\n\n'With the family?' Sharp as a knife, when it comes to protecting their interests.\n\n'Is that the only reason you can think of for going somewhere?'\n\n'Then why are you going?'\n\n'For an assignment.'\n\n'With who?'\n\n'By myself. Please come and stay here.'\n\n'How long?'\n\n'Three weeks.'\n\n'Three weeks! Why are you leaving your family for three weeks?'\n\n'It's an assignment, I told you. Assignments don't adjust themselves for my convenience.'\n\n'Then don't take such assignments.'\n\n'Ma, will you come or not?'\n\n'I'll see. What does Hemant think?'\n\n'Why don't you ask him?'\n\nLet them both see together.\n\n*\n\nNovember 30th\n\nTaking large supply of headache medicines, couldn't bear to have a headache even one day. Just imagine in two weeks, I will be away from pollution, stress, tension, strain, I will be rolling along in a bus, staring out of the window, sitting next to her, our bodies touching.\n\n*\n\nDecember 1st\n\nPip talks about nothing else but the Yatra, I thought education of slum children was her speciality, but it seems she is diversifying. She is full of this as a political ploy, the Hindu vote bank under the pseudo secular banner of national unity, the Rath Yatra last year, the increase in communal tension, the rise in violent incidents, the number of towns under curfew. And incidentally, one Leader trying to replace another by doing his own journey.\n\nMaybe I can do another canvas on this Yatra \u2013 it will be fun seeing first-hand what it is all about.\n\n*\n\nP. has visited Neeraj's cousin twice. You also come.\n\nNo, you go. I shall see her on the trip.\n\nWhen I am with her and others I feel marginal and excluded. It's stupid, I know, but what we have is so intense I can't bear for it to be diluted, I can't bear for her not to give me her full attention. This is not _good,_ I know. Maybe if we were together all the time, it would be different.\n\nBut we are not because of me, not her, then I am the one who complains.\n\nHow do people have affairs? They seem very complicated businesses.\n\n*\n\nNext day\n\nTold H. my ticket has come, hoping to involve him in my going. He demanded to see it. 'Why do you want to see it?' I asked suspiciously. 'I don't have it.'\n\n'Why don't you have it?' he asked suspiciously in his turn. (The perfect marriage.)\n\n'The Manch has it.'\n\n'I want to check if it is all right'' he said.\n\n'Don't worry, it is all right. But thank you for your concern. I know you want me to develop myself and stand on my own two feet.'\n\n'Since my wife understands her duties so well, why should I worry?'\n\nHa, ha. Why don't we get divorced.\n\nI hope my children aren't tainted by his idea of my duty. I don't want them to think I am abandoning them. What if they are taught that while I am away?\n\n'In this day and age no child can think anything if their mother travels once in a blue moon'' said Pip.\n\nI wonder.\n\n*\n\nDecember 3rd\n\nAs the time comes to go I am tense and anxious. I have never left the children for so long. I told them this evening I was going for three weeks, and I'll phone you every day \u2013 I promise.\n\n'I don't care'' said Anu flicking her hair around. 'You can go. It doesn't matter.'\n\nI wanted to slap her. It is so difficult to reach her in her adult mode.\n\nHimu said, 'Go, Mama, we should learn to be without you.' (!) Sometimes he sounds so grown up.\n\nI wish things didn't seem so muddled and confused. Nothing is sure except that I might be raped. I walked to the church nearby where it is usually peaceful.\n\n'Teach me how to live, God'' I prayed, casting an uneasy eye at the Christ hanging bloodily before me. His own life had been short and violent, but presumably successful. 'I am not asking for happiness, but I would welcome some stability, so I need not run all over the place looking for love and confirmation. Give me substance, God, give me a life that has not been lived for nothing. And protect my children'' I added as I got up to leave.\n\nI thought of Pip on the way home. She has her future plans, her study, Ujjala, Neeraj. Is there really a place for me in her life? Though even as I write this, I can hear her saying you have your painting, your children, your home. If there is neediness in love, is it more or less genuine? If you need, you want, you search, you cling. You reward the person you have found with all your feelings.\n\n*\n\nDecember 5th\n\nMy mother has come radiating disapproval. She considers the whole trip unnecessary. She who has turned to God, while her daughter is running after human love, how can I reassure her?\n\n*\n\nDecember 6th\n\nYesterday Pip asked, 'Does Hemant think you are having an affair? Why else would he be so suspicious about your ticket?'\n\n'I don't know. I don't care.'\n\n'Maybe he indulges himself when he travels.'\n\nI felt a tightness in my chest, and then annoyance. Why does she keep bringing this up? 'I don't know what he does. It could be, I suppose.'\n\n'Most men do.'\n\n'Do they?'\n\n'Don't be such a child.'\n\nI thought of all the late nights at the factory, the trips out of town, the extended trips to South East Asia, the condom, the many opportunities there must have been, but I said nothing.\n\n'Sometimes one doesn't want to know. It's painful or inconvenient. But now you are not so dependent on him, now it is all right to see.'\n\n'I suppose.'\n\n'Does he suspect you are having an affair?'\n\n'It's not the same thing.'\n\n'Why not?'\n\n'You're a woman.'\n\n'And that makes you a faithful wife?'\n\n'No. But it is different, surely.'\n\n'What you mean is you don't feel guilty.'\n\nWhat could I say to this? This love of mine would have not been possible had she been a man, and yes, I don't feel guilty.\n\n'Would you mind if Hemant was having an affair?' she went on, probing.\n\n'Of course not. He can do what he likes.'\n\n'After all you do, don't you?'\n\n'Yes. Yes' I do.' __\n\n*\n\nTook the children and my mother to a restaurant in the evening. A last outing before I left. They wanted dosas so we went to Sagar. Hemant was working late as usual.\n\nIt was not a nice meal. I was not giving the children my full attention; they felt it and began to fight. My mother fingered everything unhappily. No doubt she was calculating the owner's profits, seeing how the place was jammed with customers. But because she is so spiritually oriented she was forced to remain silent. I couldn't wait to get home.\n\n*\n\nDecember 7th\n\nMa keeps saying in a puzzled way, why doesn't the Manch send a man, it's not safe for a woman, what kind of place is this, should I talk to them and explain the situation, you have a family, maybe they don't have families, why isn't Reshana going, why is she so keen to send you? Finally I lost my temper, and had to shout are men the only ones who can do things, nothing is going to happen to me, will you stop talking like this, you are making everything worse.\n\nShe continues in a different register, don't talk to strange men, don't wear any jewellery on your trip, not even your watch, be careful of what you eat and drink, keep on phoning.\n\nI have to remind myself of my three weeks with P.\n\n*\n\nLater.\n\nAm leaving shortly. More lies as to why they can't drop me at the station, Reshana, Manch, gathering early, train leaving late. I want to leave before Hemant comes home. Remember journey to Ayodhya, when the children came to leave me at the station, and waited for hours with me, the pre-Pipee time, the non-lying, looking for the key to happiness time.\n\n*\n\nStill later\n\nAt P.'s place we prepare for departure. We shut the windows, shut the fridge, leave two lights burning. I have had puris made for us and aaloo ki sabzi, along with pickles. We take all the fruit she has, plus water, glasses, steel plates, napkins, a knife.\n\nWe don't say much, but already I feel she and I are enclosed in our own special world. Is this feeling on call to those who are happily married?\n\nThis is what she is offering me if I leave Hemant, this togetherness. Dearest, is this why you were so insistent that I come? You have already proved your point, we don't have to get on that train at all, don't have to go to Kanyakumari via Madras, with my supposed Reshana at all.\n\nI think we are ready she said. The taxi will come soon.\n\nWe lock up and go down.\n\n*\n\nDecember 9th, night\n\nAt last, Kanyakumari. The train to Madras took for ever, and from there a bus. Felt complete and peaceful the whole way; I think she felt the same. No wonder marriages start with going away, cutting off from the old, entering the new with a journey, just the two of you \u2013 even in an ocean of people \u2013 just the two of you. It seemed so wonderful, we kept looking at each other and smiling.\n\nI am waiting for her to finish her bath, then we walk down to the beach. The Yatra starts tomorrow.\n\n*\n\nLater\n\nThe beach half a kilometre from hotel. We could see the gulls, smell the sea air. As we leave the hotel after tea, I babble, the tip of the continent, the tip of the continent. P. laughs at me, grabs my hand, and we run, our feet sliding in the softness of the sand. We run to the shore line, where we can see the waters of the Arabian Ocean, Bay of Bengal, and Indian Ocean merge, grey, blue and green. The sands are three distinct\n\ncolours too, red, black and pale yellow flowing into one another. There is something about the sea, its smells, its sounds, you feel small but liberated. There it is before you, vast and eternal. My troubles felt trivial.\n\nWe were together, we were happy. We walked along the water, me with my polyester sari tucked high into my petticoat, handbag with our money heavy but safe under arm, chappals in hand, P. with her salwar rolled up, taking turns with the bag.\n\nLittle boys ran up and down hawking packets of the separate coloured sands. I bought some for Anu and Himu.\n\nP. pointed out the Vivekananda Rock in the water. Apparently that is where Vivekananda stood one December a century ago and moved to great emotion by the sight of India across him, pledged to work for the upliftment of the masses and the unity of the country. The Leader of this Yatra is also big on unity and saving the country, still not unified; the masses, still not uplifted.\n\nI stare at the sunset as though I had never seen one before. I felt every second of its sinking in my bones. I am scared. No one can be so happy and have it last. When am I going to pay?\n\n*\n\nEarly morning, December 10th\n\nWe stayed awake the whole night. I kept telling Pipee she had to go to sleep, for me it was a holiday but she was here on work. She looked at me and said when will you learn anything, the whole thing was a way to be with you. She closed her hands over me, and I could scarcely breathe with the pleasure. I often find it hard to accept that she could desire someone like me, but when I am with her the doubts fade, and I feel strong and loved.\n\nMuslim and Sikh relatives of martyrs who have died for the country are gathered here to hand the Leader the flag that will be hoisted in Srinagar's Lal Chowk, 47 days, 14 states and 15,000 kilometres later.\n\n*\n\nDecember 10th, night\n\nThe mood of last night completely gone. Five hours in the hot sun. P. was a wreck. If she arranged this trip to be with me, if I need this kind of plan to leave home, then we pay for our sins in sweat and irritation.\n\nBut we are together \u2013 no denying \u2013 would I have had the imagination to think of something like this? Why am I so passive, why can't I bristle with initiative, maybe this is what she hates about me.\n\nHer Ph.D. rears its ugly head whenever I see her talk to someone or take out her notebook. She has already made contact with several journalists while I watch her.\n\nThe Leader was late, the auspicious moment came and went, and still we waited, sweat pouring down, 10,000 of us boiling away. Then finally the Leader spoke for one hour, then all the martyr's relatives spoke, then every Tom, Dick and Harry took his turn.\n\nAt 1.47 we started. The coconut was broken, lemons put under the wheels of the two vehicles made to look like a temple and a houseboat. South and north. Inside there are two rooms, storage, water tanks, etc. The Leader refused air-conditioning, he was taking this journey not for his comfort, but for the unity of India. We could have done with some air-conditioning, but then we are not leaders.\n\n*\n\nDecember 15th, night\n\nWe cross at least five villages or towns a day. Whenever I can I phone home from an STD booth. At appointed stops, the Leader emerges to the front of the houseboat he is riding in and addresses the people over loudspeakers. He indicates the flag in the Bharat Mata Temple perched on the bonnet of each vehicle. He tells them about the pride every Indian must have in his nation, the pride that has been trampled upon in the past. He announces that India is one, and that is the meaning of his journey. He declares that India will not tolerate terrorism in Punjab or Kashmir. He reiterates that no Indian can accept the separate status given to Kashmir, that Article 370 of the Constitution is now irrelevant. He describes the water he is carrying with him, the water of all of India's sacred rivers; the soil he is carrying belonging to the birthplaces of India's noble sons. He allows them to have darshan of the vessels in which the water and the soil is kept. Amazingly they want to. They rush to touch them, to put tikka on them, to garland them. They also want to touch the Leader's feet, but this the security men do not allow.\n\nAt night we eat what has been arranged for us at the circuit house or dak bungalow, and fall into bed, weary as hell. Perhaps it is just as well we are so tired for we do not have a room to ourselves. All intimacy is confined to the bathroom. In the bus our hands enjoy a limited freedom, no one can see what we do, but still, was there an easier way to be together?\n\n600 kilometres in 4 days.\n\n*\n\nDecember 18th\n\nWho would have thought one state was so large? We are still in Tamil Nadu. We are visiting, glimpsing rather, all the temple towns in a cavalcade, flanked by two security jeeps, rifle butts poking out through the windows. The Leader has to be protected. The heat of the air is sharp, this is their winter, so strange to never be cold. From the bus window, the landscape flashes by, the greens and the browns brighter than the ones I am used to, with an occasional rock or hill. I think of the flat plains of the north, and I think Ah, the diversity of India. Soon I will talk like the Leader, of Unity in Diversity, of The Oneness underlying The Difference.\n\nI fantasise about food constantly. The food provided for us is too hot, and I am forced to eat dry rice. Whenever I can I buy fruit for both of us. P. doesn't care what she eats, but if I go on with this stuff, I shall be sick.\n\nToday she gave a banana I had kept for her to a journalist. I wanted to kill that woman. In the bus P. said, I didn't want it, and her stomach is upset. What could I say? I kept my jealousy to myself.\n\n*\n\nDecember 20th\n\nWe are now in Karnataka. Phoned children from an STD booth near the tea stall where we had halted, while P. finishes her cold drink. We then walk down the road bordered by red earth. The cacti on the edge come up to my shoulders. There are fields and fields of tomatoes, light green against the leaves, supported by trellises, or simply sticks. I can see women picking them. Green tomatoes wait in piles next to the road, for buyers. They are obviously reddened somewhere else. I remember my father used to like green tomato chutney, a recipe he taught my mother. My own children will never be able to think of my cooking, only Bahadur's. I don't care, I am too happy to worry about anything.\n\nThe Deccan Plateau. Hills popping out of the landscape. The bus weaves to and fro and I feel sick. I take Avomine, and drowse against P.'s shoulder. I love her smell.\n\nDays merge one into another, the landscape changes, I too have fallen into the rhythm of the journey. My mind is stilled. At night we roll into beds that are provided for us at the circuit house or dak bungalow. How many more days before we can share a bed???\n\n*\n\nDecember 22nd\n\nThere are two buses following the Leader. One is security, aides and party workers. The other is publicity and journalists.\n\nThe woman who had a stomach upset continually hounds us. She is a correspondent for a paper based in Madras. Periodically, when the convoy stops, Pip and she disappear for their interviews. At these times I take out my pad and sketch. It will be a record of our journey when I return, and maybe a base for a canvas. I want to feel productive, that I did something besides stare besottedly at one woman all day. It's not easy being in love every single minute. Resentment creeps in, especially when the other person is talking to someone else.\n\nMeanwhile we pass through Mother India, who impassively stares at this cavalcade of temple, houseboat, and gun-toting security men. Nothing is new for India. Doesn't the Leader say that again and again, India is our mother. Her qualities are patience, tolerance, love and resignation. Her rewards are that she is forced to suffer over Kashmir the recalcitrant child, Punjab the rebellious one. The father \u2013 i.e. the Leader \u2013 will not stand for this any longer. Time to take a firm hand.\n\nHow can we listen to this rubbish day after day? I complained to P. when she was looking at my drawing pad that night.\n\nOnly three more days before we take the bus for Bangalore. Then it will just be you and me, she replied, carefully examining each sketch. I am continually flattered by her attention and comments:\n\nAre these scenes for your Ekta Yatra canvas, I like the houseboat and temple and the way you have captured these crowds, but isn't this a lot for one painting, and so on.\n\nLiving with someone interested in the details of your work is companionship at the deepest level. I long to create the canvas I have in my head so she can see it too.\n\n*\n\nDecember 24th\n\nBangalore at last! In the guest house of the Y. Our room, our bed, on which we spend hours. Maybe this is what good marriages are like. To be able to express what comes into your head, and know it will be understood as you meant it. To be more yourself because all of you is able to love in a way the other responds to.\n\nShe goes to sleep, and I pass my hand over her breasts. At first it had seemed odd, after years of being made love to by a man, to have one's breasts met by a similar pair, though larger. No wonder men like them so much. You can do much with a pair of breasts. These loose, hanging, swinging items, breasts, penis \u2013 objects of passion and anxiety. Stuff you can hold in your hands, squeeze, maul, make yours, like playing with clay \u2013 taking you back to your childhood.\n\nThe rubber trees are enormous and green outside, the bougainvillaea is blooming, it is warm, fragrant, pleasant, far from the cold of Delhi. Why can't I live here for ever with her, forget I have a life outside this room, this bed, these arms, this mind that sees me the way I am and loves me still.\n\nShe looks at my face, puts her arms around me, don't look so sad, we have each other, we are the lucky ones.\n\n*\n\nDecember 25th\n\nChristmas.\n\nShe pointed out her grandparents' house as we passed by in a scooter.\n\n'Can't we visit?'\n\n'No.'\n\n'Why? I want to see them. I want to see where you spent your childhood.'\n\n'Well, you can't. They'll pester me to stay, and ask a lot of questions.'\n\nI could make out a small house, a little garden and a huge tree, studded with white champa blossoms. Pretty, but I couldn't imagine Pipee in it, the antithesis of suburban.\n\n'What was it like, growing up here?'\n\n'All right'' she said non-committally. Getting into her past is sometimes a problem. Especially the death of her husband. She never talks about that.\n\nSpent the day roaming Bangalore. Talking, talking to fill the time our lives were separate \u2013 oh this happened when I was here, didn't I tell you, and she said and he said, tell me, tell me how it was?\n\nWe laugh because we are together, doesn't matter where, or how, cemented by our nights and words together.\n\n*\n\nJanuary 2nd, 1992\n\nBack from a week at Pip's school. Idyllic place, with all the usual about idylls. Trees, millions of butterflies, thousands of birds, lap of nature, the works. The most miraculous thing about the place, I had no headaches. Pip, no headache, I said every evening, and she smiled, the corners deepening, dimple appearing, eyes warming. My painlessness I offered as a gift, she accepted it as her due.\n\nShe showed me her butterfly tree, the walks her mother and she used to take, her classrooms, her library, she was even nostalgic about the din in the dining hall.\n\nI had no idea P. was so involved in her school. The usual pangs with every teacher she threw herself on, with every old friend she talked about. This kind of jealousy, however slight, makes no sense. I think I need my head examined.\n\nWe stayed with P.'s mother. I slept on the divan in the big room, she with her mother in the bedroom. Her mother didn't say much to me, she is a house parent, besides being a middle-school class teacher and quite busy. Does she know we are lovers??? I ask P.\n\n'I think so.'\n\n'You told her?'\n\n'She has eyes.'\n\nSo does my mother but even if I told her, I bet a thousand to one she would not believe it. I said as much.\n\n'She has always known how I am feeling, that is the important thing.'\n\nIt seems Pip has the ideal mother-daughter relationship, just as she had the ideal marriage. I wonder how these things operate?\n\nPip organised a street play around interpretations of history. Among other things she used my pamphlet, _The_ _Testimony_ _of_ _the_ _Black_ _Pillars._\n\nLast night we went to P.'s favourite restaurant in Bangalore, a great relief after school food, though to listen to P. it was manna from heaven.\n\n(n.b. If I am jealous of every thing about P. that doesn't include me, perhaps I should not mind so much her attitude to my family.)\n\n*\n\nJanuary 4th\n\nWe are leaving on the Karnataka Express. The Yatra has reached Gujarat, then it is going to Rajasthan, Madhya Pradesh, Uttar Pradesh.\n\nIn the evening P. said, I don't want you to go home.\n\nWhat is she saying, it is almost a month, is this another test? Did I not pass the first one?\n\nStay with me a few days in Delhi, please. You can always go back to them a bit later.\n\nI agreed, but for the first time, the thought crossed that perhaps P. was not always wholly reasonable. Maybe I should assert myself. (How?)\n\n*\n\nJanuary 7th\n\nWe came yesterday. Took a scooter to her flat, stopping at the market on the way to buy provisions: milk, bread, eggs, fruit and vegetables. It seems strange to come to an empty place, no one waiting, nothing done. You have to do everything yourself the minute you come, clean, organise, buy, cook. If it is so tiring in winter, what will it be like in summer? But this is Pip's life, and she doesn't complain.\n\nPip has gone to Ujjala, I make the bed, dust, clean the cobwebs, cook lunch, and then haul out this diary to write.\n\nI wonder how Anu and Himu are managing. I can't tell on the phone. Their school is opening today. Did they finish their holiday homework? Does my mother manage to get them up and off in time? Are they all right? They say yes to everything. P. says I worry to feel needed.\n\nI feel disturbed here. Why isn't Pip coming? She promised to come quickly, she might have gotten caught up with meeting her colleagues, Neeraj probably, while I am here waiting. It was much better in Bangalore.\n\n*\n\nJanuary 8th\n\nAwful, awful. Couldn't sleep. Last night we fought. She left this morning without telling me where she was going. What did I do, it was nothing.\n\n'In a few days you will be gone'' she started over the dinner we had cooked together.\n\nOh no. 'Yes.'\n\n'And then?'\n\n'Then?'\n\n'Back to the way it was?'\n\nShe was spoiling for a fight. I was determined to say nothing, but she went on, 'You don't really want to be here.'\n\n'I do'' I said quickly.\n\nShe started withdrawing. Leaving a trail that I followed. 'Would I be staying if I didn't?'\n\nShe glared at me, pointedly left the table and began clearing away the dishes. Doesn't she realise what I go through because I want to be with her? I am in the same city as my children and I cannot meet them. Still she broods. Is this how she wants to spend our time in Delhi? To fight, sulk and turn away from me?\n\nWhy is she like this? I wish Aijaz were still alive, but then she would never have been interested in me. They had the perfect marriage, she hankers after that wholeness. What can I do? I live my life in fragments, she is the one fragment that makes the rest bearable. But a fragment, however potent, is still a fragment.\n\nThis morning I got up, made her breakfast, but she would not relent, continued cold. If she wants to punish me she certainly doesn't have to try very hard. I am in such misery, I don't care what I do. To be with her, yet distant, anything is better than that. She has left me alone here, God knows for how long. I might as well go home.\n\nI wish I had the energy to hate her, but I don't. I feel sick.\n\n*\n\nJanuary 9th\n\nHome. They all exclaimed how thin I was.\n\nI left without saying goodbye, or leaving a note. What will she think when she comes back and finds me not there?\n\n*\n\nJanuary 13th\n\nJaundice. I vomit all the time.\n\nP. is all right, then how come me? We drank the same things, but some germ from some water drop has lain inside me, waiting for me to be safe at home before moving in for the kill.\n\n*\n\nFebruary 15th\n\nWhat was the point? I can still barely eat. I look yellow and horrible. I smell.\n\nI have travelled from P.'s house to my own via the tip of the continent, a long detour.\n\nThis is what happens when you leave your home. The in-laws, the mother, the husband, the servants all unite on this.\n\nI feel exhausted.\n\nMy mother is still here because I am ill.\n\nH. grates on my nerves. It's all my fault, does he never get tired of finding different ways to say this. He likes me to be ill and dependent.\n\nP. comes to visit in spite of their hostile attitude.\n\nI am sorry, she said, I'm sorry I left you like that.\n\nI am sorry, I replied, that I didn't wait for you.\n\nWe talk of other things.\n\nShe told me there was a bomb blast attacking the Yatra in the Punjab, two people were killed.\n\nSuppose it had been us?\n\nHave I been struck by this dreadful illness because I left my home to be with the one I love? I feel so weak I can't get out of bed. When Hemant comes home and puts his heavy arm around me, I want to tell him everything just to see the look on his face. But then I'll have to cope with the rest of it.\n\nMy children draw pictures with huge Get well soon Mamas on them. I keep them by my bed and look at them often. Pip calls, concern in her voice.\n\nI can't deal with my life. I want a safe place, a warm place, a loved place.\n\n(n.b. Who doesn't?)\nChapter IX\n\nGradually Astha's bilirubin count came back to normal, as did her diet. Her mother departed for Rishikesh, yet she remained tired. When Pipee dropped by on her way to work, she did her best to be amusing and interesting. Pipee should not feel she was in love with an invalid, but it was so much effort, she almost wished she wouldn't come. Yet the days she didn't, she felt unloved and anxious.\n\nEvery morning she gazed piercingly and objectively in the mirror. She looked haggard, yellow, ugly and undesirable, she would perfectly understand if Pipee never wanted to see her again. When her lover left, she again checked the mirror, despite her better judgement. Maybe in the interim she had grown more beautiful, maybe Pipee had spotted something attractive that had missed her eye in the morning.\n\n'I wish I didn't feel so exhausted'' she permitted herself to moan occasionally.\n\n'It's only natural.'\n\n'Yes, but it's so boring for you.'\n\n'Let me decide that.'\n\nA pause.\n\n'How's your work going? How is Neeraj?' asked Astha to cover up the anxiety of the silence.\n\nEverything was fine, Pipee assured her, as she got up to leave.\n\nAfter these visits, Astha felt depressed and gloomy \u2013 why can't we be like we were during the trip \u2013 what's the point \u2013 I wish I were dead \u2013 while her family put her listlessness down to her fragile state of health.\n\n*\n\nMeanwhile tension in the house gathered. The workers of the factory went on strike, despite ClearVision offering fifty thousand rupees to the strike leaders, couched as temporary relief measures. It was clear that the rival union meant business, and soon another six TV factories in the area saw labour unrest.\n\nThese factory owners were not united. Meetings ended acrimoniously. They could not decide on an incentive package, though all of them felt that the demands of the union were unreasonable.\n\nEvery day that passed meant greater losses for the company, as well as an erosion of their market share. It was more than the owner could bear.\n\n'Half pay'' Hemant fumed, 'we still have to pay them half their wages. Where do they think I am going to get this money if there is no production? The company will be ruined. Bloody fuckers.'\n\nHe spent his days running around looking for a solution, meeting lawyers, representing his case before the Labour Commissioner of Noida, trying to get the strike declared illegal. Meanwhile they were losing their share of the market at a time when there were over four hundred TV manufacturers in India.\n\n*\n\nTwo months later the Labour Commissioner declared the strike to be a lock out. No work, no pay.\n\nTriumph reigned in the Vadera household, it was seen as the silver lining in the dark cloud that had lain across their home.\n\nThe next day the manager's car was damaged, and every window of the factory broken. The number of guards were increased, but a few days later a fire broke out on the premises. It was detected before great damage could be done, but Hemant could not risk further vandalism and was forced to hire a private security agency, with instructions for twenty-four-hour surveillance. More money spent without any sales to cover the costs.\n\nDespite being declared illegal, the strike continued. Too many workers, owners, factories were affected for there to be any immediate resolution.\n\n*\n\nHemant developed chest pain. The doctors diagnosed hypertension, told him change your food habits, quit smoking, cut down drinking, exercise every day, and avoid anxiety. The early forties was a vulnerable time for men with stress.\n\nHemant was seeing the work of the past eleven years go down the drain, and he wasn't able to respond to this advice.\n\nHis parents went into damage control.\n\nIt was decided that as soon as school shut for summer, he, his wife and children would go on a holiday, and spend a relaxed time with Hemant's sister Seema in the US. When Hemant came back, they would work on the lifestyle-food habits-exercise thing. Meanwhile Papaji would manage things in the factory.\n\n*\n\nAstha told Pipee of these plans while they were having lunch at a restaurant in Connaught Place.\n\n'How long will you be away?'\n\n'I don't know yet.'\n\n'I suppose you have to go?' asked Pipee a little hesitantly.\n\nAstha remained silent. If only she didn't have to put her husband's health over the companionship of her lover. But not going was like getting divorced, a public statement of difference and separation.\n\n'Look, it's not working out'' said Pipee suddenly.\n\n'What is not working out?' asked Astha desperately.\n\n'One should never have affairs with married people, they are the worst.'\n\nAstha looked at the face she had kissed lovingly and in such detail at least a thousand times, and said resentfully, 'Why did you, then? You want to spoil what we have.'\n\n'I had thought that with a woman it would be different\u2014'\n\n'So did I. With a woman\u2014'\n\nA silence fell, in which the air-conditioners fought audibly against the April heat. The glass on the windows let in blue-tinted light. At certain places the glaze had peeled and spots of glare came through. Astha dabbed at the breadcrumbs left on the table from their soup rolls. Pipee looked moody. 'You can tell me all about your nice little domestic holiday when you come back'' she remarked coldly.\n\nAstha stared at Pipee anxiously, 'You know how it is. The workers are on strike, he has got high blood pressure'' then she stopped, hearing the words of a devoted wife in her ears.\n\nPipee concentrated on her empty glass. 'No. I don't know how it is.'\n\n'You are independent'' said Astha bitterly, 'so you can talk like this.'\n\n'And somebody is holding your hands, preventing you from being the same?'\n\n'You need money'' flashed Astha, 'or do you think I should be independent on his money? Stand in the streets with a begging bowl? Live in an ashram like my mother? What about my children?'\n\n'Your children, your children, don't hide behind them. Live with me. Bring them.'\n\nThat old thing.\n\n'But no \u2013 you don't even try \u2013 Ant why don't you even try?' Pipee swallowed once or twice. 'Have an exhibition, do something on your own, or are you waiting for Hemant to give you permission?'\n\n'You are not being fair.'\n\n'Yes. Well.'\n\n*\n\nThe anticipated vacation split Astha more decisively than anything else since she had got to know Pipee. There was her lover and her lover's feelings. But there was also the visas for the USA and the UK, the foreign exchange, the getting ready, choosing suitable clothes and shoes, the packing and shopping for presents.\n\nWith their holiday abroad Hemant and Astha joined the have-gone-abroad club, whose denizens created envy and ill-concealed curiosity about how much money they were going to spend, where had they got it from, even with the factory in trouble they can afford to go, they must have stashed it away all these years.\n\nMany people took their proposed trip badly. The most immediate was Sangeeta who was there as usual for the summer holidays. She insisted on being part of the discussion and planning that revolved around itineraries, addresses of friends of friends, cheap fares, cheap central hotels, foreign exchange. Astha had to brace herself against the flow of her resentment and curiosity.\n\n'One day I too will go abroad. Seema is always inviting me'' she said.\n\nIt has nothing to do with me, thought Astha, if she is angling for a trip let her angle directly. Sangeeta sighed, announced Poison was her favourite perfume and disappeared upstairs for the day.\n\n*\n\nAnuradha said now her friends would not be able to act so superior, she too could tell stories of abroad, and Himanshu said now he could have the latest in Nintendo and Sega, and could they please go to Hamleys.\n\n'Hamleys? What is Hamleys?' asked Astha.\n\n'A shop in London'' said Himanshu. 'Everybody goes there.'\n\n'He is so retarded'' said Anuradha.\n\nAstha hoped the trip wasn't feeding into her children's materialist desires.\n\nAstha's mother was delighted. She wrote from her ashram: God bless you my little one and your family. Poor Hemant needs a break from all his troubles. You do not give him enough attention. Remember men have to bear the burdens of the outside world, home is their refuge.\n\n*\n\nPipee retreated further into herself, getting ready for her summer, Shahjehanpur, Shiksha Kendra and Ayodhya, we'll compare notes when I get back, bye, no need to drop me to the station, have a nice time, call me on your return.\n\nAstha felt Pipee's abandonment, but maybe she thinks I have left her, she brooded in the middle of the night, when the electricity went, and the couple lay sweating.\n\n'I will be glad to leave this fucking country'' muttered Hemant.\n\n'So will I'' muttered his wife.\n\nDelhi, the trap in summer, with power cuts, water shortages, heat waves, dusty winds, and pollution emanating from all its pores. Not the garden city of their youths, but fourth, third, creeping up to second, now coughing and wheezing its way to first, yes, almost the first most polluted city in the world.\n\nA trip abroad would be nice, no matter whom one loved and whom one left behind.\n\n*\n\nFinally the family took off on their cheap flight to Miami, Florida, with a stopover at London on the way back.\n\nHour after hour into the dark night they flew. Four abreast, in the central section of the plane: father, mother, daughter, son going to holiday on Western shores.\n\n'Are you all right?' Hemant would ask from time to time. Astha nodded, her eyes closed. She wondered at the great silence concerning the discomfort of planes, the torture one had to undergo to get to the lands of milk and honey. Her knees were hurting in the small cramped space, her shoulders and back were aching, a headache was coming on, would she make it to the bathroom to throw up if she had to. Excuse me, I am sick, I have to throw up, madam use the bag in the pocket in front of your seat, ah, there it is, sorry, not at all.\n\nThe rest were enjoying themselves. Himanshu was absorbed in the child kit the airline had given him, Anuradha had her headset glued to her ears, and fiddled with the dials constantly. Hemant was nursing his drink, chewing with relish on the peanuts that came with it, tinkling the ice and the alcohol in his glass, twitching his toes in the airline socks, his shoes neatly stowed away under the seat in front of him.\n\nHe shouldn't be drinking, thought his wife, but she was in too much pain to comment or persuade.\n\n*\n\nThey stayed for three weeks in Florida. Hemant talked incessantly of his life as a student, and how he had slummed it, how he had worked to earn a little extra money, how he had slept two hours a night, how the great American tradition encouraged self-reliance from babyhood, how you had to sink or swim, how the whole society was geared towards meritocracy, not towards blackmailing people by going on strike. Loafers wanting something for nothing were not tolerated here.\n\nSeema and Suresh sympathised completely, never mind, you have family, family still means something, and they talked of here and there, there and here, till Astha felt her ears would fall off.\n\nThree weeks crammed in their guest room, three weeks of Anuradha feeling jealous of everything that Sushma (the daughter) had to show her.\n\n'School in the USA is like no school at all'' she announced to her mother. 'They get hardly any homework, they choose what they want to study. Her maths, I can do it with my eyes shut.'\n\n'I am sorry, darling'' said Astha looking at her daughter's angry face.\n\n'Why should you be sorry?' said Anuradha turning upon her mother, the easiest person in the world for her to turn upon.\n\n'The system here is not so demanding, that's all I meant.'\n\n'She thinks she is so clever, but she is not, Mama, I know much more than she does. Her handwriting and spelling are so bad, you wouldn't believe, but she doesn't care, and neither do her teachers. She says in the computer everything comes out OK, so what is the point? Imagine!'\n\n'You are better off beti, you can write, you can spell, you can do maths, when you come here for higher studies you will be at an advantage.'\n\nAnuradha looked mollified. 'I'll show her'' she muttered.\n\n'Quite'' said Astha, 'and while you are about it, do remember that we are guests in their house, and that she is your cousin.'\n\n'She has an American accent.'\n\n'That is not something she can help, she only knows this country, poor thing.'\n\nMother and daughter smiled slightly at one another. Nothing is so much a bond as criticising relatives.\n\n*\n\nThe marriage of Seema and Suresh was a source of great amazement to the brother and sister-in-law. Seema and Suresh constantly deferred to each other. Suresh cleared up after meals, ran the dishwasher, did the grocery shopping, mowed the lawn on weekends, and went to the park with his son to kick a few balls in the evening, almost as a duty.\n\n'What has happened to Suresh'' wondered Hemant. 'He was never like this at home.'\n\n'This is not home'' replied Astha.\n\n'Poor chap'' went on Hemant. 'You should have seen him when he was just married. Boozing and smoking with the rest of us. Now he doesn't even touch a cigarette.' Hemant fumbled for his own packet and lit one, to further express his disgust.\n\n'Perhaps it would be better if you took a leaf out of his book'' said Astha. 'Suresh looks just fine to me, at least he is not a source of worry to his family.'\n\n'He is ashamed to look me in the eye'' declared Hemant, surrounding those very eyes with smoke.\n\n*\n\nThe high point of their US holiday was a trip to Disney World.\n\n'It's built on 27,000 acres. Acres of fun'' said Suresh, while Seema sketched the delights of the fairy tale park, water park, animal park, future park, past park, sports park. She spoke with all the pride of ownership.\n\nThey planned to drive to Orlando and spend three days there. The hotels were expensive, but to absorb such wonders money was necessary.\n\nHemant offered to participate in the driving, but Suresh did some more back slapping, this was America, not your India, where a visitor could drive without an International Driving Licence or indeed without any kind of licence at all, just a bribe.\n\n*\n\nDisney World, Orlando, Florida, USA.\n\nIs such a thing possible in your India? There was no end to this question, as Hemant was forced time and again, to say no, such a thing was not possible in their India.\n\nSo organised, such crowds, such a money-making machine, such technological marvels, such fantasy, such going through tunnels, haunted houses and castles, such an onslaught of souvenirs, such marvelling, such eating of hamburgers, hot dogs, Kentucky Fried Chicken, tacos, and thick milkshakes. Around they wandered with those milkshakes which never seemed to end, sipping the cold sweet stuff through giant straws. Was there anything in this country that wasn't big?\n\nAnuradha and Himanshu loved it, Hemant loved it, Suresh, Seema plus two kids, their millionth visit with Indian tourist and wonder seeker in tow, they loved it all over again. Even Astha managed to be caught up in what she saw and experienced. They were all children together, all Mickey Mousers in a Disney World.\n\nBesides families everywhere there were couples embracing, couples walking with their hands in each others pockets, kissing, eating, conversing, laughing.\n\nSuresh and Seema became even more of a couple here. They walked holding hands. For our benefit, or because they are on vacation, or because they have lived in America so long, or because they love each other so much? It was the last possibility that Astha could bear the least. Anything but that Hemant's sister should live in bliss while she lived in misery.\n\n'I thought Disney World was for children'' she remarked to Seema.\n\nSeema and Suresh both grinned at her.\n\n'Arre, people come to enjoy'' said Suresh.\n\n'Relax, have fun, spend quality time together'' clarified Seema for Astha's greater understanding.\n\n*\n\n'Well, wife'' said Hemant, the second night in the hotel, at his most affectionate, swept by emotion at having seen Disney World, and recorded it on a thousand pictures taken for the benefit of back home, 'it's been quite an experience, no?'\n\n'Yes, it has.'\n\nIt was late, the children had fallen asleep, exhausted by so much pleasure and walking around. Hemant sat next to Astha, and put his arm around her.\n\n'How's your head?' he enquired tenderly.\n\n'OK.'\n\nThey sat on in silence. After a while Astha dislodged herself. 'I have to pee'' she said.\n\n'OK'' said Hemant, getting up as well.\n\n'What are you doing?'\n\n'Coming with you.'\n\n'Don't be silly.'\n\n'What's so silly about it?'\n\nIt was easier to let him come, and Astha sat on the toilet seat, feeling a bit strange. It had been a long time since they had shared any intimacy.\n\n'Go away'' she said at last, 'I can't pee.'\n\nHe ran the tap.\n\n'Now?'\n\nA small trickle. Hemant tore a piece of toilet paper and advanced his hand towards her legs. The trickle stopped. Her legs tightened. 'Please leave the bathroom'' she stammered.\n\n'Why? I'm your husband.'\n\n'So what?'\n\n'So everything.'\n\n'You think marriage is just sex.'\n\n'Of course I don't. What do you want that I don't give you?'\n\n'Interest. Togetherness. Respect.'\n\n'Baby, I respect you'' said Hemant soothingly, 'you are my wife. As for togetherness, that's just what I want.'\n\n'Why all of a sudden?'\n\n'We are on holiday. This is what people do on holiday.'\n\n'I don't want to. I am out of practice.'\n\n'Well, let's get into practice'' said Hemant stretching out his hand again towards her legs.\n\n'I am not able to switch on and off like you'' said Astha.\n\n'It is not as though you were the most willing creature. Each time I try and come near you, you say you have a headache. A man is tired, he can't be doing the chasing all the time.'\n\n'Is that what you call it, chasing? Not having sex on demand? There has to be something more between us. I have to feel it is me you want.'\n\nHemant looked baffled. 'Of course it's you I want. You are my wife'' he repeated.\n\n'That's the problem. Anybody could be your wife.'\n\n'What rubbish. I picked you, didn't I?'\n\n'Picking is not the same as knowing.'\n\n'Why do you always make things so complicated? You are my wife, that is enough for me, I would have thought it is enough for you. Or is it someone else?'\n\n'Are you referring to my life or yours?' asked Astha flushing slightly.\n\n'Come on darling'' replied Hemant, ignoring her barb, 'we are on holiday. I want this to bring us closer, as a family, as a couple.'\n\nHe had felt her distance, he wanted her back. There seemed to be no way out, unless she decided to leave the marriage there and then. Slowly she moved towards him. With sleeping children in the room they would of course have sex in the bathroom. He spread a towel on the mat and waited for her to undress.\n\nOn and on marched the holiday, relentless, inexorable, eating up money, energy, rolls of film, pushing them to cheap eating places, and suitcases that grew heavier by the day.\n\n'Shopping on the way back, shopping on the way back'' Hemant kept saying but it didn't quite work like that. There were so many souvenirs, the Disney World ones alone filled half a suitcase. Besides there were the presents Seema and Suresh were sending back for the rest of the family, and clothes for everybody, so much cheaper in the States than anywhere else.\n\n*\n\nLondon. They were met at the airport by Hemant's cousin.\n\nAstha had always liked this cousin. He had gone abroad to do well, since he couldn't do well in India, and ended up owning a shop in the suburbs of London. Just what this meant was only now becoming clear as they drove, drove and drove, and finally stopped in front of a house, which was a double storied, very narrow building, identical to the entire row on the street. Naked houses on a treeless street.\n\n'Welcome to my humble abode'' said Jagdish, edging the car near the curb, and jumping out to take their suitcases. 'I'll see where Liz is'' he panted, lugging them inside.\n\nLiz, the unenthusiastic wife. 'Hello, would you like a cup of tea?' she asked, and they could feel the indifference, and they could understand why Jagdish was being so effusive.\n\n'Their house is so small, Mama'' whispered Anuradha, awed by such discomfort in the West.\n\n'They don't have much money'' whispered Astha back.\n\nThe bags, the guests, and the host struggled up the narrow stairs, what a nice house you have Jagdish, well, it's all right, and they went down to have the tea that Liz had prepared.\n\n*\n\nOne week in London, of learning how to take the Tube, of don't worry, Jagdish, we will take care of ourselves, no, no, please do not bother, Liz, we will manage, and Jagdish's reply, well, if that's all right then.\n\nEvery morning Astha got up and made sandwiches so they could save money on eating. They bought the ingredients and the drinks at the corner store, because Liz clearly did not understand the imperatives of Indian hospitality, and they didn't want to burden Jagdish's marriage further. They gritted their teeth and managed to not all bathe every morning, the house only had one complete bathroom.\n\nThere was some disagreement as to how they would spend this precious week. Astha wanted to see all the art treasures London had to offer, she was willing to go on her own while her family did whatever they wanted. But Hemant would not hear of this \u2013 we are here to be together \u2013 and as a compromise they spent a morning at the Tate, a morning at the British Museum, and then covered the famous sights of London in a couple of day tours. Many photographs were taken as proof of the good time they were having.\n\nAll this over, they devoted themselves to shopping. There had to be much looking, exclaiming, comparing, soul searching, and converting of currencies before they could buy.\n\n'I must say London is a very expensive place'' said Hemant, as they emerged from Marks and Spencer, arms laden, a light rain falling, a cold wind blowing.\n\n'I wish we didn't feel the need to buy everything we see'' moaned Astha, exhaustion reducing her to the desire to lie in front of the department store door, and be trampled to death by all the Indians rushing in and out, buying, buying.\n\nAnuradha and Himanshu looked at her reproachfully. They could hardly contain themselves in this material paradise. Floors and floors of merchandise with Hemant the indulgent father. The trouble, thought Astha, was that she too could hardly contain herself when she saw the kitchenware, gadgets, art supplies, bed linen, children's toys, clothes, underwear, stationery. Was there anything that did not move her with the urge to possess? No, such shopping was not morally good, she felt her sense of perspective and focus vanish amidst its successful assault on her greed. It was just as well these trips were rare.\n\n*\n\nOn the evening of their fifth night. 'There seems to be trouble in India'' said Jagdish, a held back pleasure edging the notes of concern. He was entitled to a revenge so small, that he was in the safe place, the sane country, something in return for his unsatisfactory house, job, career, marriage and neighbourhood.\n\nAstha and Hemant looked at each other. At home trouble was part of the atmosphere, outside it assumed more sinister proportions.\n\n'What's happened?' asked Hemant\n\n'On the BBC. They are going to build the temple'' continued Jagdish.\n\nHemant relaxed. Oh, the temple. 'These politicians keep stirring things up'' he replied, uninterested.\n\nWhile the family ate, Astha hung around the TV waiting for the news.\n\nThere it was. A brief visual of the Babri Masjid at night, floodlights beaming, sounds of bhajans in the background, thousands of kar sevaks surrounded by security forces, clearing the ground, laying the foundation for the temple, working, working, round the clock.\n\nThings are tense in this ancient temple town, said the commentator, where a mosque stands on the site that Hindus claim to be the birthplace of the Lord Ram. While six thousand pilgrims work day and night, an estimated fifty thousand more have assembled here. The kar sevaks swear that this time they will rather die than stop. There have been protest marches by groups concerned with saving the Babri Masjid but so far the laying of the temple's foundation continues at a lower spot on the hill. The Prime Minister has called Hindu holy leaders to Delhi to discuss the issue.\n\nHow awful, thought Astha, what was going to happen? She wanted to go home. Her political self, her intelligent self functioned best there, here she felt isolated, saturated with things rather than thoughts.\n\nWhat was Pipee doing? Each day she had been aware of her absence, yet she had enjoyed being with her family, enjoyed the comparative ease between Hemant and herself.\n\nShe dreaded what Pipee would say when she sensed this. As she tried to defend herself, I am married, she felt the betrayal Pip would feel, but by now betrayal was a second skin.\n\nAstha had often imagined the breaking of her relationship with Pipee. What she hadn't realised was how slow the process would be, and in what infinitesimal stages.\n\nThere were differences, she thought miserably, but they hadn't seemed so important. This was no longer the case. After she came back they were clearly not in harmony.\n\n*\n\n'You won't like abroad'' remarked Astha to Pipee. 'It is awful.'\n\n'Who would have thought it?' said Pipee dryly.\n\n'You know what I mean'' said Astha impatiently.\n\n'No, I don't. How could I? And anything is better than the things I saw.'\n\n'What did you see?'\n\n'For ten days total frenzy, policemen jeered at, control rooms smashed, loudspeakers blaring out prayers and bhajans \u2013 in such an atmosphere \u2013 pandemonium at the building site, and kar sewaks all over.'\n\n'You mean you went to Ayodhya?'\n\n'Yes.'\n\n'But why? You didn't tell me.'\n\n'Where were you to tell?'\n\n'It might have been dangerous, Pip.'\n\n'Oh Ant, one can't always be safe. It was no more dangerous for me than for all those other poor women there. Besides I wanted to go. I am thinking of a conference on how families are affected in communal riots.'\n\nMore PhD stuff, thought Astha. 'Well, how was it?' she asked.\n\n'They are going to build the temple in the masjid area. That kind of energy, so deliberately stoked doesn't go away. It's only a matter of time.'\n\nThere was a silence. Pipee leaned back in her chair, and stared at the clouds that were running against the sky of her Vasant Kunj flat. Astha looked at her, she seemed so distant. She had felt closer thousands of miles away, thinking of her, writing to her.\n\nThen Pipee said, 'Enough about Ayodhya. How was Disney?'\n\n'Fine.'\n\n'And you and Hemant?' she asked. 'How was that?'\n\nAstha kept her face still. 'Also fine'' she said.\n\nPipee looked at her sharply, 'You have had sex with him'' she stated flatly.\n\nHemant's face rose before Astha's eyes, the moments in the bathroom, the appeal he would never verbalise, her own realisation that somewhere he still had the power to affect her. She felt her face going red.\n\n'You've never really liked it any other way, have you?' persisted Pipee, her voice dry and hard.\n\n'That's not true'' pleaded Astha.\n\n'Yes it is. What you really want is your husband's cock.'\n\nAstha winced and tried to retaliate. 'It's not that. You resent that I am not leaving him. You want a full-time partner. I understand that.'\n\n'You would. It is what you have, after all.'\n\nAstha was silent for a moment. If her husband represented more than just a cock, so much the worse, but how was she to explain to Pipee? It was better not to advance into these murky waters. She went on, 'It has nothing to do with us.'\n\n'You went away with your family, that was bad enough, and I didn't say anything, because it's no use, and then you do this, why have me?'\n\nEverything Pipee said was a distortion. Words were raising their ugly heads, and Astha could do nothing. No matter how hard she tried, she was not going to succeed.\n\nPipee kept that transgression in her heart and used it as a foundation for the separation she saw ahead. A good memory is always useful when something needs to be destroyed.\n\n*\n\nPipee and Astha continued to see each other, but there was now a carefulness between them. For Astha everything became dull, the grass looked ordinary, the sky looked bleak, the paint on her canvas colourless.\n\nA thousand times she said to herself, confront her, tell her you want it like it was, or not at all, but she was too afraid. Pipee might say not at all, then what would happen to her, worse than this, much worse.\n\nThings would become all right on their own. Love would triumph, even in circumstances like these. Love had to, that was its nature.\n\nBut Pipee behaved as though love had had its day. Even moments of affection contained references to endings. Pipee to Astha, tucking her hair behind her ears.\n\n'I'm so grateful to you, Ant, never forget that, no matter what happens. From you I got the energy to go on.'\n\n'To do what? Leave me?'\n\n'You really want to go into who left whom?'\n\nAstha couldn't say she did. 'You see?' said Pipee. 'We both gave each other something. Let us leave it at that.'\n\nAshta couldn't say anything. Words made what was between them so small.\n\n'And of course whatever happens, we will always be friends'' went on Pipee.\n\n'Yes, always'' replied Astha gratefully, too inexperienced to know that that is what breaking up people say to each other to make it more bearable.\n\nMeanwhile the strike was resolved after six months. There was a tremendous backlog to be made up, and a market share to be recaptured. Hemant made an effort to resume his previous pace of work, when the chest pains started again. The doctors were very severe, he was not paying enough attention to his health. Collapse was imminent if he continued smoking, drinking, not exercising, eating red meat and heavy food. Furthermore he had to try and control his levels of stress, a very modest working day was all his body could tolerate. Angina had to be taken as a warning, a serious warning.\n\nThe whole family was alarmed. His father insisted another manager be hired, money wasn't everything. As for Astha, a brief survey of the literature on heart disease established that permanent changes were required in their living habits. Diet \u2013 exercise, diet \u2013 exercise, there was no getting away from these pillars of health and longevity. It was up to her, Hemant was not going to change on his own.\n\nEvery morning she made sure they went for a walk. All those years ago, exercising and resentful with her parents, she was now doing the same with her husband, with feelings so much more complicated with the years that had passed. Was this where her life had led her, this the space she had travelled between those walks and these? Striding briskly to still the thoughts in her head, speaking to mask the feelings in her heart. She looked at Hemant, swinging his arms, concentrating on getting his heart rate up. Perhaps he was disappointed too, perhaps he had looked for something different in marriage. They didn't talk about such things, she would never know.\n\nShe changed her family's way of eating. She bought books on low cholesterol diets, she studied recipes demanding no fat and little salt. When the children complained they were compensated privately during lunch and tea.\n\nHemant was bad tempered about having to give up his favourite foods, but he had no choice. And if he had to eat porridge for breakfast instead of his usual green chilli-onion-tomato omelette, he could not complain, his wife was eating the hated porridge too.\n\nAstha spent a lot of time thinking about herself. Was she a traditional wife as Pipee had alleged? She flinched at the idea, but she was certainly doing what devoted wives did, putting a great deal of effort into protecting their husband's insides. When she saw him tired, afraid, depressed at having to change, unprepared mentally for the betrayal of his body, she felt sorry for him, and wanted to help him live. She told herself it was for the children, but sometimes she wondered bleakly at the nature of the bond between them.\n\nHemant was touched by her efforts. Occasionally he would enquire, 'Well wife, how are you?' in a proprietary kind of way.\n\n'Fine, fine'' Astha replied in a monotone. Hemant was not adept at noticing discrepancies between the apparent and the stated, and this quality was conspicuous now.\n\n*\n\nThe monsoon came and went. The muggy days marched into October to become more human.\n\n'I'm going to get an air-conditioner'' said Pipee. 'That is if my scores are so lousy I get no aid, and am forced to remain in this dump.'\n\nAstha turned absolutely still. 'The PhD?' she asked.\n\n'I give my GRE next month.'\n\nOh Pip, you didn't tell me and not telling used to be felt a deception between us, but I see no more, mourned Astha silently, as Pipee continued, 'I need a change.'\n\nAstha made a heroic effort. 'Yes. I'm sure you do. In an academic environment you are bound to flourish. They'll love the work you do.'\n\nPipee looked at her and smiled, 'I'll miss you, Ant.'\n\nAstha didn't believe her. Pipee went on talking, and Astha heard all the things she wasn't saying, her loneliness, her desire for steady companionship, the need for commitment.\n\nThey smiled at each other. Astha said she understood. They drank tea, they exchanged a goodbye kiss, they did all this before Astha ran to her car, buried her face in the steering wheel, and took a good, long look at the void she had desperately tried to plug through loving Pipee.\n\nWhat would it be like to be painfully separate having known togetherness?\n\nHow would she live? But she had to, she had that rock of stability women had, her husband and her children.\n\nDrearily she turned the ignition and let out the clutch. The car rattled and jerked. She had to start it three times in the two metres she backed it. The clutch seemed to be slipping, the car's servicing was long overdue. This knowledge had hovered on the edges of her mind for a long time, but the imperatives of her life had not allowed her to pay attention. No longer. Her life was made up of these things.\n\n*\n\nAt home she threw herself into a frenzy of house cleaning. Every nook and cranny, every book, every mote of dust, layer of dirt, every inch of carpet, every remote cupboard high and low she attacked.\n\n'Mama's gone mad'' Anuradha informed her father conversationally, 'all she does is clean. And she makes me polish and clean too.'\n\n'You will thank me for it later, when you have your own home'' snapped Astha. Everybody looked surprised.\n\n'Do you have a headache?' asked Hemant.\n\n'No.'\n\n'What is it then?'\n\n'Nothing. Nothing.' And without her wanting or willing, the tears started pouring down her face.\n\nAnuradha's face contorted. 'Why are you crying, Mama?'\n\n'No reason, sweetie'' gulped Astha.\n\n'Shall I get you your medicine?'\n\nAstha continued to sob, while Hemant said, 'Do, Anu, there's a good girl'' and as the daughter ran off, he turned to his wife and said, 'Don't cry. You are upsetting the children. They will think something is really wrong with you.'\n\n'Let them'' wailed Astha. 'Let them know mothers also can feel.'\n\n'Az? Are you all right? Stop cleaning, if it upsets you.'\n\n'You always say how dirty the house is.'\n\nShe sounded unreasonable to her own ears. Hemant sat quietly by. Anuradha ran back with the pills and a glass of water, dragging a worried Himanshu with her.\n\n'She says you are crying'' he accused.\n\n'It's all right, baby'' said the mother, swallowing the pills, though she had no headache. 'It's nothing really.' Himanshu stared at her, his face opaque. 'It's nothing, baby'' repeated Astha. The boy turned and walked away while Anuradha looked at him with contempt.\n\n'He's so thick'' she said. 'He never understands anything.'\n\n*\n\nInto Astha's mind came a memory, dredged from her subconscious. Her mother coming from the bedroom, the bedroom that had been locked, unusually, for a whole hour.\n\n'Why are you crying, Mama?'\n\n'It's nothing.'\n\n'How can you be crying for nothing?' persisted Astha.\n\n'I am not crying. What gave you that idea, beta?'\n\nOh well, if she wasn't crying, then those couldn't be tears, nor could those be signs of grief around her eyes and mouth. They could go on being happy, everything in its place.\n\n*\n\nAstha was amazed at how much work Pipee devoted to applying for a Ph.D. She spent hours at the US Educational Foundation studying profiles of universities, their faculties, their requirements, the ones most likely to give aid, the cost of living in small towns, cities, East Coast, West Coast, balanced against the cost of living with Ajay if she got into his university, the cost of clothing in cold places, hot places.\n\n'I can afford to apply to only five'' she said, looking at her finances, calculating how much for GRE, how much for application money, how much for postage. 'Though that is taking a risk. Well let us see. Maybe I won't get aid, or maybe I won't get a visa.'\n\nShe would talk of something else, and Astha would seize these flimsy possibilities, clutching them in her grip, where they lay mangled and inert, of no real comfort.\n\nWho has seen tomorrow, she often thought, and this with tomorrow staring her in the face.\n\nWhen she was with Hemant she felt like a woman of straw, her inner life dead, with a man who noticed nothing, with whom for that very reason it was soothing to be with. Her body was his, when they made love it was Pipee's face Astha saw, her hands she felt. She accepted the misery of this dislocation as her due for being a faithless wife.\n\n*\n\nPipee's GRE scores were announced. She had got 23.4 on 24, and could now walk into any university she chose, or so her congratulators said. Astha hugged her, and felt she hadn't known misery till then.\n\n'Oh, Ant'' was all Pipee said, but Astha could see how happy she was, how vindicated she felt with her score, how much further down the road towards the USA she was treading.\n\nThen Pipee began to talk. 'You can join me for a bit if you feel like it.'\n\n'Of course I'll feel like it.'\n\n'No, I mean it, Ant. I know you'll be sad without me. You come for a holiday.'\n\n'Sure.'\n\nBut they did not explore this topic further. Too much water had flowed under this particular bridge for it to be a comfortable one to tread on now.\n\n*\n\nIt was December. The initial response from American universities was positive. 'I'm so glad'' said Astha looking at the face before her, and the shy glow on it. She re-read the letter Pipee had shown her. The University of Illinois had received Pipeelika Khan's application, and were acknowledging this. She would hear from them in the very near future. 'They have realised how bright you are'' went on Astha, thinking true love had no element of self in it, and she had better measure up.\n\n'Not everyone thinks of me like you do, Ant.'\n\n'Your application, your recommendations, your NGO work, your academic record, that paper you wrote and published, your GRE scores, why else would they reply so fast?'\n\n'You're biased.'\n\n'Not without reason, surely?'\n\nPipee laughed, and took the letter back, carefully folding it along the original creases, before sliding it into the browny-yellow envelope. Astha watched her.\n\n'When will you finally hear?' she asked.\n\n'Hopefully by January.'\n\nMaybe by January a bomb would fall on Urbana Champaign and blow the university off the face of the earth, thought Astha, but what would be the point, there would be other places. The real act of leaving was in the decision, not in the departure.\n\n'Maybe they won't give me a visa'' Pipee broke in on the pause. 'After all, I'm single.'\n\n'Yes'' said Astha slowly, 'you're single.' How soon before she would find someone?\n\n'You've not asked me what I'm going to work on.'\n\n'Education of slum children, I imagine?'\n\n'No. The politics of communalism and how it is represented. I am more interested in that now, maybe because of what is happening around us. It might also help me come to terms with things in my life. If you realise you are not alone...' She did not complete her sentence, and Astha felt more than ever removed from her life, from a pain so horrifying she bowed before it and shut her mouth.\n\n'The only trouble is there are so many aspects, all of such relevance that it is a bit hard to choose a specific area'' went on Pipee.\n\nFor a moment Astha felt an intense stab of envy, not just for Pipee, but for anyone who had the possibility of a new life. She had to remind herself sternly that if she wanted, she too had choices. \nChapter X\n\nBy the end of the year, there was plenty of material being generated for Pipee's thesis. Kar seva had been stopped in summer on the condition that the Prime Minister solve the Babri Masjid problem in four months. Those four months were up with no solution in sight.\n\nThousands of kar sevaks were again being mobilised for what was termed symbolic kar seva, starting 6 December. The central government sent 135 companies of its security forces to Ayodhya and Faizabad despite the protests of the U.P. government, who claimed the law and order of their state was their responsibility.\n\n'This time they are not going to give up easily, they have been stopped twice before'' said Pipee worriedly.\n\n'Are you going?' asked Astha.\n\n'Too much to do. I hope nothing happens.'\n\n'The Babri Masjid has survived almost five hundred years. Why should something happen now?'\n\n*\n\nMeanwhile Ayodhya is witnessing the unprecedented influx of thousands of kar sevaks from all over the country. Religious leaders issue press statements declaring that religion is above politics, above nation, above courts and any restraining orders passed.\n\nBy 5 December the city has swelled by 200,000 kar sevaks, and there are not enough places to put them. Schools and colleges are declared shut, while the kar sevaks storm various institutions for accommodation. The area around the masjid is littered with garbage and human excreta. Food prices go up, the U.P. government declares they have in stock 8,ooo tonnes of rice, 100 tonnes of sugar, 45,000 litres of kerosene, as well as an ample supply of life-saving drugs.\n\nThe BJP declares that no harm will come to the masjid, the kar seva will only be symbolic.\n\nThe Union Minister sends extra paramilitary forces to Ayodhya.\n\n*\n\n7 December, Astha's house. Headlines: A NATION'S SHAME there on the folded newspaper lying on the verandah, waiting to be read, digested, somehow understood.\n\nAstha picked it up and stared at the front page. It was not possible, this could not have happened, but there it was:\n\n*\n\nA NATION'S SHAME: BABRI MASJID DEMOLISHED\n\nCentre sacks Kalyan Singh's Government. 500,000 kar sevaks armed with pickaxes, crowbars, pipes and uprooted barbed wire barricades, attacked the disputed site yesterday. All domes collapsed under the onslaught. Between 11.50 to 4.50 the central dome collapsed. Between 2.00 to 4.00 p.m. the two side ones were destroyed. 50 people injured. Hundreds of kar sevaks carted away bricks, pillars, and large stones. BJP leaders urged restraint through megaphones.\n\nAngry kar sevaks singled out photographers and foreign correspondents beating some brutally with sticks and leaving them bleeding on the road.\n\nCurfew in many U.P. towns. Muslim MPs seek the Prime Minister's resignation. Muslim houses set ablaze. Kar sevaks not allowing fire engines in many places.\n\nArmy alerted in six states.\n\n*\n\n'They've broken the mosque'' she found herself telling Hemant. 'They have done it at last.'\n\n'I know.'\n\n'You know?' Astha stared at him.\n\n'It was on the BBC last night.'\n\n'Why didn't you tell me?'\n\n'What was there to say? I didn't know you were interested.'\n\nHe was lying. She had gone to Ayodhya twice, painted the masjid at least five times, scripted a play about it, and he didn't know she was interested? This was his revenge for being concerned in things other than him.\n\nShe turned away, sickened by everything.\n\n'I never knew you were such a Muslim lover'' said Hemant watching her. 'Do you know what happens to our shrines in Pakistan, Bangladesh, not only to our shrines but to Hindus? Why doesn't your precious Manch ever protest about that? Or any of your activist friends?'\n\n'The fact that shrines are desecrated there, doesn't make it acceptable here. It's not a Muslim thing, it's a secular thing, a human thing.'\n\n'It's a cowardly thing, a fool thing'' he said mockingly.\n\n*\n\nThere was no point talking to him. Her one thought was to call Pipee, Pipee who felt like she did, with whom there would be no arguing at this moment. Quickly she dialled her number.\n\n'Have you heard?'\n\n'Yes. They've done it.'\n\n'You were right.'\n\n'What are we going to do?'\n\n'What can we do?'\n\nBoth women fell silent, their own lives dwarfed by what was happening around them.\n\n'Neeraj phoned. There's a demonstration on at the BJP office, we might as well go. It must have been planned, such a thing cannot happen without careful planning.'\n\n'You said it was only a matter of time.'\n\n'It was just a thing to say. I had no idea I would be proved right so quickly.'\n\n'Don't cry, sweetheart. Come to the church crossing. I'll pick you up in half an hour.'\n\n*\n\nThey talked little as they drove. As they came nearer Central Delhi they could see that the streets were lined with throngs of weeping men and women, dressed in black, faces covered. Near the BJP office, they were forced to walk, the whole area was cordoned off, lined with policemen, ready to lathi charge at any provocation.\n\nJournalists were there, TV crew, academics and activists, all shocked and numb. They shared their information in broken sentences: paramilitary forces hand in glove with the kar sevaks \u2013 the police were helping \u2013 the leaders shouting on megaphones don't destroy the mosque \u2013 but pre-arranged that such messages to be ignored \u2013 many killed in the falling rubble \u2013 absolute pandemonium with 500,000 kar sevaks \u2013 the situation going to worsen \u2013 the government in U.P. had no political will to protect the mosque \u2013 only a matter of time before something like this happened\u2014\n\nThey waited for one hour, two hours. Nobody came out of the BJP office to address them. Was there anybody there? They courted arrest, were put into waiting buses, taken to the local police station, kept for half an hour and sent away.\n\n*\n\nThree days later the United Left Front organised a march to protest the demolition of the masjid.\n\nOn the morning of 10 December Astha said to her husband, 'There is a march today.'\n\nThe husband said nothing.\n\nAstha persisted with her information. 'It's going to be a tremendously big march. Traffic will be blocked around Red Fort and Connaught Place for hours. Do try and avoid those places if you wish to save yourself trouble.'\n\nNothing.\n\n'OK?'\n\nThe husband saw a female bull charging from the distance and his body tautened. He lifted a wary face and looked at his wife. Astha carefully patted the tea tray cloth.\n\n'I always admired your sense of proportion'' he said at last.\n\nAstha raised her eyebrows and looked inquiring.\n\n'Out in the streets, jostling with goondas, neglecting your family, all for some fool masjid you didn't even know existed before your great friend Aijaz chose to educate you.'\n\n'It has nothing to do with Aijaz'' said Astha, choking on the rage she had kept inside her the last three days.\n\n'Then his widow.'\n\n'I suppose I have no mind of my own.'\n\n'I didn't say that.'\n\n'You meant it.'\n\n'I refuse to talk to hysterical women'' said Hemant, 'especially when I have got a busy day ahead. Some people, work, you know.' He got up and went into the bathroom, firmly closing the door.\n\nWhen Astha reached the Red Fort her eyes were red with the hour-long cry she had had after Hemant left. Pipee saw her and linked her arm through hers, lacing her fingers through Astha's own clammy hand. 'Dearest, don't be this upset, it's terrible, but you can't afford to take it so personally.'\n\nAstha nodded dumbly. Everything in the world was terrible.\n\n*\n\nA mild winter sun shone on the gathered marchers as they stood around waiting, while truck after truck of United Front activists and associates drove in.\n\nThe line started. It was so long that by the time it was Astha's turn, forty minutes had passed.\n\nThey marched out of the Red Fort into the middle of the road, blocking all traffic. As she walked Astha could see various people, Pipee included, handing out leaflets to onlookers, scooter wallahs, passengers in rickshaws, men scratching their balls, women holding children on their hips, women with plastic shopping baskets in their hands. Down the line the familiar slogans were shouted: _Down_ _with_ _communalism,_ _down_ _down;_ _BJP_ _down,_ _down;_ _False_ _followers_ _of_ _Ram,_ _you_ _will_ _never_ _succeed;_ _Mandir_ __ \u2013 __ _masjid,_ _all_ _one._\n\nAstha was overcome with futility. Maybe Hemant was\n\nright. What was the use of forcing motorists, passengers and pedestrians to listen to the voice of tolerance and peace? It had not prevented anything. Maybe the true victory of fundamentalism was the total despair of the secularist.\n\nThe line reached Delhi Gate, and turned right towards the Ram Lila maidan. Down another empty street with bad-tempered traffic gathered on the other side of the divide, and then the line poured into the Ram Lila grounds, to mill around a platform erected for the speakers.\n\nOne after the other they spoke, leaders from the Congress, from the Left parties, activists who had seen what had happened in Ayodhya. They expressed anguish, regret, sorrow, they issued warnings, predicted consequences:\n\n*\n\nWhat had happened was a betrayal of trust. Millions of Muslims would now feel insecure in their homeland. The assurances of the U.P. government had meant nothing, the assurances of the central government had meant nothing.\n\nThe law had been blatantly, openly flouted, what was going to prevent it from being flouted again? What was going to prevent the two disputed sites in Kashi and Mathura from going the way of the Babri Masjid? Was this a government or a passive instrument in the hands of thugs? Without delay the government should acquire all the land around the Babri Masjid.\n\nIt behoved every citizen in the land to be vigilant so that anti-communal forces did not gain ascendancy. How was it possible to demolish a masjid in broad daylight in little over four hours? And that too with home-made tools, pickaxes, crowbars, the implements of farmers and peasants. No, there was organisation and planning, there was the connivance of the authorities.\n\nVarious Leaders had been arrested, but was it all for show, like the security forces that were sent to protect the Babri Masjid, and helped in its destruction?\n\nThe nation and its people demanded answers.\n\n*\n\nIt was late afternoon by the time Astha left. When Hemant came home he did not ask about the rally. And Astha was only able to sleep towards morning.\n\nAfter the demolition:\n\nNationwide, 1,801 people were murdered in communal clashes in the next two months. 226 places in 17 states and 1190.18 lakh people were affected by curfew.\n\nIn Pakistan 240 temples were targeted by mobs.\n\nIn Bangladesh attempts were made to destroy 305 temples, 1,300 houses, and 270 shops belonging to Hindus.\n\nIn the United Kingdom 18 temples and cultural centres were damaged.\n\nIn Afghanistan 4 temples were attacked.\n\n*\n\nOver the next two months major riots broke out in Bombay. 41 areas were affected, 31 per cent of the deaths were caused by the police. Pipee decided she needed to gather first-hand material.\n\n'Why are you going?' asked Astha.\n\n'I have to go. Awful things are happening there.'\n\n'I know. I also read the papers'' said Astha irritably, 'and that is why I wish you wouldn't go. It's not safe.'\n\nPipee looked at her for a moment, then gave a strange laugh. 'One has to do what one has to do.'\n\nAstha looked bewildered.\n\nPipee ruffled her hair with a slow unsteady hand, 'Don't worry, I can take care of myself.'\n\n*\n\nShe came back unharmed but terribly shaken. 'It's worse than you realise, the police are actually firing on the innocent, making false arrests, and refusing to register complaints. How can Muslims have any security or protection when the forces of law are among those who beat and kill? Go back to Pakistan they keep taunting, when were they ever in Pakistan, that they should go _back?_ And nothing is done, nothing. What kind of country can these people feel part of? To be a Muslim here is a curse.'\n\n*\n\n'They started it'' said Hemant. 'After the Babri Masjid fell they were the ones who first took to stoning temples in Bombay. What did they expect, that this is the time of the Muslim rulers, where Hindus will sit down and not retaliate? Who set fire to the temple in Govandi? Who started throwing stones at buses, and police stations, and the BMC offices?'\n\nAstha listened. If it was quite clear that there were many ways to regard what was happening, it was equally clear that she and Hemant held opposite views. Whose voice would be stronger remained to be seen.\n\nThe most effective way she had of making a statement was with paint, and she focused on that. It took her mind off her personal predicament, with such violence around her, her problems seemed small. She turned to brush and canvas to make her contribution to her country, she hoped it would be noticed. It was only a drop in a large, large ocean, but drops added up.\n\n*\n\nPipee ended up making several trips to distressed areas. Work, she said briefly, and maybe, thought Astha, she keeps coming back because she misses me, though she knew Pipee would never say anything to this purpose. Once she had decided that they were going to break up, that was it.\n\nShe called Pipee over one day, and had the pleasure of her approval when she saw her canvases.\n\n'They are strong and make a very effective statement. I can see how you have evolved, Ant.' So what if Pipee was leaving, at that moment Astha felt they could never be parted.\n\n'They really are good'' continued the friend and activist, 'perhaps you can hold an exhibition on your own. It's time you emerged from the shadow of the Manch.'\n\nThen she returned to her travels, so much is happening.\n\nAstha noticed Pipee didn't ask her to join her even for a weekend. She would have gone anywhere if Pipee had only asked her.\n\nIt was in January that Pipee got the letter confirming her admission to the University of Illinois, Urbana Champaign.\n\n'So soon, you have got to hear so soon, how wonderful'' said Astha over the phone, glad that Pipee could not see her face. 'They must really want you.'\n\n'I don't know about that. The amount they offer will show.'\n\n'When will you know?'\n\n'Soon, I hope.'\n\nAfterwards Astha scolded herself, this is the dress rehearsal for the real thing, why should I care, what is she to me, someone I loved, but we both have our own lives. She has chosen larger horizons, it's her life, this is mine. She told herself this firmly and repeatedly and was surprised that the information did not make her feel better.\n\n*\n\nNext morning she woke up with a headache. So, what's new, she asked herself, quickly swallowing a decongestant and a painkiller.\n\nAs the pills took effect and the pain receded, she drearily made for the spare room, to try that other cure, work. She pressed her turpentine rag against her face and breathed its sharp smells. She imagined it around her eyes, running under her eyebrows, through her temples, pushing all the throbbing in front of it, sweeping it away, throwing it out, so that it lost the power to affect, now and ever after.\n\nIt was a month before the axe finally fell. 'Oh, Pip, I'm so glad. How much are they giving you?'\n\n'A full fee waiver, and twelve hundred dollars a month.'\n\n'You deserve every bit of it. I hope you will be very happy.'\n\n'Hey, I'm not going yet. And I want to see your exhibition before I go.'\n\n'I'm doing my best.'\n\n*\n\n'I will get Ravi's wife to review it'' said Hemant. 'She is an art critic for _The_ _Indian_ _Express._ She probably knows others as well.'\n\n'Thank you.'\n\n'I'm sure they will be impressed'' said Hemant smiling at Astha.\n\nAstha knew Hemant was being helpful because the Manch was not involved and he welcomed the breach between her and any activism, but she had been too long married to linger over the source of his appreciation.\n\n*\n\nEvery morning, the children in school, the servants supervised, Hemant safe with his diet lunch in the factory, and Astha would shut herself inside her painting room. She needed to feel closed in and protected, if by nothing else than walls. There she was with shrouded canvasses, bottles of turpentine and linseed oil, tubes of colour lying in baskets around the easel, and grey rags stiff with dried paint. These were the tools of her trade, these were the things that established her separate life, touching them was comfort.\n\nAs her brush moved carefully over the canvas, her hand grew sure, her back straightened, she sat firmer on her stool, her gaze became more concentrated, her mind more focused. A calmness settled over her, tenuous, fragile, but calmness nevertheless. She thought of her name. Faith. Faith in herself. It was all she had.\n\n*\n\nPipee too was working. She was looking for a tenant, getting her papers ready, packing away her books, winding up her affairs. She required no interaction with Astha in arranging these things.\n\nThere was a time when Astha's day revolved around being\n\navailable to Pipee. The children, Hemant and all her obligations were frantically juggled so that she could see and talk to Pipee whenever possible.\n\nNow they talked, but it was on the surface, both of them reluctant to work at letting go of a connection that would naturally cease when Pipee left.\n\n'I can't wait to see your work.'\n\n'Come.'\n\n'I'll come, I'll come. I'm going to Shahjehanpur. They have called me. I think I ought to go before I leave.'\n\n'How come you didn't tell me?' quavered Astha, the weaker of the two, remembering the time when she knew what Pipee was going to do the second she thought of doing it.\n\n'You are so busy these days.'\n\nAt this prevarication, for the first time Astha felt relief that in a few months she would not have to talk to Pipee anymore.\n\n*\n\nIt was May. The amaltas trees were blooming. Every morning when Astha stepped onto the road, she was forced to step on fallen, perfect, clear yellow flowers, pale green buds, and scattered curving yellow stamens. She stepped on them because there was no way to avoid them, and the flowers forgave her by looking just the same.\n\nThis morning Astha was going to Vasant Kunj to pick up Pipee, despite her protests, to take her to the station for the train to Shahjehanpur.\n\nAs she drove her hands felt heavy on the wheel. How many times had she travelled down this road in hope and longing, and then rushed back dreading the demands and questions of her children, husband, in-laws. Where have you been, we were waiting for you, this that and the other happened, and you weren't here to fulfil your place in this house. Soon nobody would have cause for complaint, if there had been neglect, she would make up for it now.\n\nIn Pipee's flat, third floor, no fans or cooler because the electricity had gone. 'How hot it is'' remarked Astha for something to say.\n\n'Thank God, I'll soon be leaving'' said Pipee carrying a small suitcase and locking, then double locking the doors.\n\n'Indeed.'\n\nPipee looked at her. 'I don't mean I want to leave you. You know that.'\n\nAt these words hope sprang up in Astha's breast. The eternal stuff, hope. She looked at it in disgust. Hope looked back coy and stubborn. It did not have to say anything. Its presence spoke for itself.\n\nThe suitcase thrown in the car, the two got in and started the long drive to the New Delhi Railway Station.\n\nThat summer was the hottest Delhi had ever known. The temperature hit the 40s and stayed there day after day. Astha only felt fresh enough to work in the early mornings. She now woke at 4.45, made herself a cup of tea, and was at her easel by 5.00. As a consequence it became necessary to sleep by 10 p.m. Every day she and Hemant fought about this.\n\n'This is crazy, you are crazy, your life revolves around those canvases.'\n\nHow could she make him understand? Work was the only place she could forget everything, where she could become her mind, her hand, and the vision inside her head. At any rate she was sleeping badly, only by working hours every morning before the demands of the house took over did she know some peace. All this was not explainable.\n\n'Only for a little while more'' she tried coaxing.\n\n'You've been saying that for ages.'\n\n'You were the one who encouraged me to hold an exhibition, you showed interest, you said you would speak to Ravi's wife.'\n\n'But what's the big hurry? You can have your exhibition later, anyway winter is a better time.'\n\n'I want it now.'\n\nAstha was feeling too sick at heart to give being sweet and coaxing more than the briefest try.\n\n*\n\nThe hall at the Tagore Arts Centre was rented for five days. There were twenty canvases in all. It was two years work, and from December on, she had worked almost every day. Six canvases were devoted to the Babri Masjid and different forms of protest, another six to various aspects of Pipee and herself, though she hoped they were so disguised no one would be able to identify the women. There were four of her children, and two of men she had modelled on Hemant, one of Mala and Bahadur. Basically my life, thought Astha as she, Hemant, and the children worked in the gallery, putting them up, placing them to the best advantage.\n\n'Why is this so small?' asked Hemant picking up one the size of a sheet of paper.\n\n'It's for Pipee to take with her'' said Astha. 'I made it small on purpose.' Then as Hemant said nothing, she continued, 'Do you think she will like it?'\n\nThe painting was an interior, two women sitting on a charpai. The patches of colour came from a red cushion, an open window, the white of a pillow on the bed, the bangles of one, the bag and chappals of the other thrown on the floor. The figures themselves were indistinct and shadowy, one had a drooping head, the other had her face turned away. The small canvas added to the sense of claustrophobia.\n\n'It's kind of sad'' said Hemant.\n\nAstha was always surprised when Hemant said anything she could remotely relate to. 'I suppose it is rather'' she said. 'Maybe she should only be surrounded by happy things in Illinois?'\n\n'I'm sure she'll like it. Are these women you two?'\n\n'Of course not'' said Astha quickly. 'They are imaginary. You can't see their faces. Could be anyone.'\n\n'Hummm.'\n\nThe last paintings to be hung were the Babri Masjid series, six in all, ending with a bare hillock, a trishul and a saffron flag planted on empty earth amid scattered stones, a peepul tree hanging forlornly on one side.\n\n'I hope it is not too obvious'' fussed Astha, knots gathering inside her from the stress of exhibiting, displaying, exposing.\n\n'Not at all'' said Hemant. 'You need obvious symbols to say obvious things.'\n\nWhy is he being so nice to me, thought Astha. He even seems to have changed his political opinions. Is it because all the work is over, no more early nights, early mornings, is it because she's going in another two weeks?\n\n'Mama, I'm going to give out the price lists'' said Anuradha firmly.\n\n'Of course darling, I shall depend on you.'\n\n*\n\nPipee had returned from Shahjehanpur, but she still had to get her tenant, and decide what to do about her possessions. She couldn't help Astha with the setting up of her exhibition, Astha couldn't help her with the disposing of her things.\n\n'It is rather badly timed'' said Astha on the phone, 'but I am holding this exhibition for you, Pip. So that, before you go...' She stopped. She didn't need to finish her sentences with Pipee.\n\n'I know, I know, Ant. I wish I could be more there for you.'\n\n'And I for you.'\n\n*\n\nPrivately Astha thought perhaps it was just as well, she couldn't bear to witness the disbanding of Pip's house, where they had been skin on skin, mind on mind with nothing in between.\n\nAnd Pipee thought, it is just as well Ant is not here when I am packing to leave. I don't think she could take it, and I couldn't take her not taking it. I wish she hadn't come with so much baggage, but she did, and well, there it is.\n\nThe opening of the exhibition, 1 August.\n\nPipee said, 'I had no idea you had been painting so much. It's wonderful, just wonderful.'\n\nAll Astha could say was, 'Did you like no. 12? It's for you.'\n\nPipee looked at her, squeezed her hand, and after half an hour of hanging about, affectionately said, 'I have to go dearest, I will see you later.'\n\n'So soon?'\n\n'I have to. A tenant is coming to see the flat, and could come at no other time. I am getting frantic, I hope this one works out.'\n\nAstha thought of the very long distance between Pipee's flat and the Tagore Arts Centre, of the rush hour traffic, of the hour it would take her to get back home, and decided she could only be grateful that Pipee had come at all.\n\n'Bye, see you.' She blew her a kiss and was gone, leaving Astha to listen to what Hemant was saying and who he was introducing her to.\n\nMore than half the paintings sold. Astha made almost two lakhs.\n\n'It is a good beginning'' said Hemant, quite the manager of his wife's career. 'Ravi said his wife is going to give it a positive review, and talk to some other art critics so they mention it too. Exposure is what counts at this stage.'\n\n*\n\n'You must mail me the reviews'' said Pipee. 'I'm sure they will be very good. I look forward to reading them.'\n\nOh, Pipee, don't talk like a stranger to me, I can't stand it. I only want to talk about how sad I am feeling.\n\nBut the wall between them was by now quite high, and from time to time they both threw another brick on it. They were doing this now.\n\n'How is the tenant?'\n\n'Just what I wanted.'\n\nAnd so on.\n\n*\n\n'I will take you to the airport'' said Astha on the phone.\n\n'Are you sure?'\n\n'Please, Pip, don't be insulting.'\n\n'I only meant what about Hemant, won't he mind?'\n\n'Hemant will understand.'\n\nWith Pipee about to go, it was guaranteed that Hemant would understand anything.\n\n*\n\nThe night of 6 August. The last time Astha would drive to Vasant Kunj. The weather was hot and still, it hadn't rained since the night of her opening. She parked, climbed the three flights to Pipee's flat, rang the bell, and contemplated the bars, bolts and locks on the wooden and screen doors. Last time, for the last time, rang irritatingly in her mind. Was there anything about this night that was not going to be drenched in significance? She wished it was over, that she did not have to go through it step by painful step, Pipee's departure from her life.\n\nShe thought of how they had both been ants together. And now Pipee was journeying eight hours to London, ten hours to Chicago, two hours by bus to Urbana, to be an ant somewhere else.\n\n'Hi!'\n\n'Hi.'\n\nShe entered. The flat was bare, with just the things Pipee had sold the tenant. The bed, the cane chairs, the small wooden dining table.\n\nAstha sat down and looked around. 'What'll you do when you come back?' she asked. 'About bedding and stuff?'\n\n'Oh'' Pipee sounded vague. 'Buy new, I guess.' Astha could tell it was far from her mind, her return, why was she hurting herself by looking for clues?\n\n'Are you ready?' she asked.\n\n'The chowkidar is coming. I have to give him the keys and he'll take down the stuff.' Pipee wasn't quite looking at her, and Astha realised she was making Pipee uneasy, the way she sounded, sad, heavy, teary. She said nothing more as she watched Pipee doing last-minute things.\n\n'It's a good thing that weight is not important when you fly to the US'' said Pipee as the chowkidar staggered out first with one heavy suitcase, then another. 'They go by the number of bags.'\n\n'Yes, I suppose.'\n\n'Come, let's go.'\n\n*\n\nThe long drive, their car one of a stream going to the International Airport late at night, all saying good-bye to people they loved. As Astha drove, she imagined the misery in the cars around her. Join the queue, Astha, join the queue.\n\nThe crowd at the Indira Gandhi International Airport was as usual overwhelming. Astha drove up the ramp to Departure, nosing through the cars, coaches, taxis, and thousands of people.\n\n'I'll get a trolley'' said Pipee, jumping out.\n\n'I'll get the luggage'' said Astha moving towards the dickey, and fumbling with the key. There was Pipee with the trolley, the luggage unloaded, there was a policeman waving her car away \u2013 no standing allowed, and Pipee saying go, sweetie \u2013 where will you park \u2013 it's so crowded, and Astha, wailing but I want to see you inside \u2013 they won't allow you \u2013 and the policeman \u2013 not moving towards any of the other parked vehicles on the ramp but threatening her with traffic violation \u2013 Pipee propelling her into her car \u2013 a last kiss, goodbye, goodbye, take care, and she was lost to the eye even before she had wheeled her trolley through the entrance door.\n\n*\n\n'So, she's gone'' said Hemant when Astha returned. Awake at that late hour and witness to his wife's face and eyes.\n\n'Yes.'\n\n'Was the plane on time?'\n\n'I don't know. I didn't stay.'\n\n'How was she?'\n\n'Who?'\n\n'Your friend, who else?'\n\nAstha could not reply. 'I'm tired'' she said, 'I want to sleep.'\n\nMechanically she changed, brushed her teeth, put cream on, got into her side of the bed, pulled the sheet up, and turning to the very edge lay absolutely still. Motion of any kind was painful to her. Her mind, heart and body felt numb.\n\nIt continued like this for days. She felt stretched thin, thin across the globe. \n\n# Acknowledgements\n\nMany people have been particularly kind in sharing information that I found invaluable. In this regard I would particularly like to mention Anshu Balbir, Mita Bose, Ranjan Dhawan, Jamal Kidwai, Vinay Minucha, and Jay a Srivastava. This book could not have been written without their help.\n\nMasooma Ali, Sunanda Ali, Bharati Bhargava, Nidhi Dalmia, Nilanjana Dalmia, Christopher Fruean of Walt Disney World, Vijay Kapur, Vimla Kapur, Fauzia Khan, Angela Koreth, M. K. Raina, Maseeh Rehman, Saswati Sen Gupta and Jaya Sharma were badgered about numerous details, and responded with patience and generosity.\n\nJaya Srivastava, Sharmila Purkayastha and V. Karthika were generous with pamphlets and books I would not have been able to get otherwise.\n\nPenelope Anderson, Janet Chawla, Katyayani Dalmia, Anuradha Marwah, Ira Singh, Ramya Sreenivasan, and Addison Ullrich contributed encouragement, interest, enthusiasm and criticism.\n\nAnuradha Marwah helped in crucial ways during the final stages.\n\nJulian Loose, my editor at Faber, bestowed a clarity and vision upon these pages that much benefited them. Heather Schroder showed faith in me by becoming my agent on trust.\n\nRoma Bhagat Baraya helped with legal aspects of the text. Sanjeev Saith, my publisher at Indialnk was a model of patience and tact. His meticulous attention to detail was greatly appreciated.\n\nIra Singh's repeated readings and comments helped shape the characters. Her other contributions defy exact description.\n\nGun Nidhi Dalmia, my husband, was astute and reassuring in his reading of my manuscript. My hours at the computer would not have been possible without his support.\n\nMy children, Maya, Amba, Katyayani and Agastya were with me throughout in body and spirit.\n\nDuring the research for my novel I consulted the following books for their spiritual commentary: _The_ _Secret_ _of_ _the_ _Kath_ _Upanishad,_ Swami Krishnananda, The Divine Life Society: Tehri-Garhwal, UP, India, 1974; Maharishi Mahesh Yogi on the _Bhagavad-Gita,_ _A_ _New_ _Translation_ _and_ _Commentary,_ Chapters 1-6, Penguin, 1967.\n\nFor the political events that form the background of the novel I consulted: _Ayodhya_ _Imbroglio:_ T. P. Jindal, Ashish Publishing House, New Delhi, 1995; _The_ _Demolition:_ _India_ _at_ _the_ _Crossroads,_ Nilanjan Mukhopadhyay, Harper Collins, 1994; _The_ _People's_ _Verdict:_ An inquiry into the December '92 & January '93 riots in Bombay by the Indian People's Human Rights Tribunal Conducted by Justice S. M. Daud & Justice H. Suresh published by The Indian People's Human Rights Commission, August 1993; Pradeep Nayak, _The_ _Politics_ _of_ _the_ _Ayodhya_ _Dispute,_ Commonwealth Publishers, New Delhi, 1993; and _Cry_ _the_ _Beloved_ _Country,_ a PUDR pamphlet.\n\nIn section VI, the advertisement by Ramjanambhoomi Nyas (a non-political body affiliated to the VHP) in The Pioneer, May 11, 1991, has been taken from Pradeep Nayak, _The_ _Politics_ _of_ _the_ _Ayodhya_ _Dispute,_ (details mentioned above) p. 167.\n\nThe Hindustan Times was consulted on microfilm, accessed from the Teen Murti Memorial Library. I am grateful to the library for allowing me this facility.\n\n*\n\nThe actual events leading to the destruction of the Babri Masjid have either been fictionalised or used in imaginative reconstructions.\n\n*\n\nThis book went through several of its many drafts during a three month stint at the Universities of Kent and Stirling in the UK. I am indebted to the Charles Wallace Trust, India for granting me a fellowship to these places. \n\n# About the Author\n\nManju Kapur is the author of four novels. Her first, _Difficult Daughters_ , received tremendous international acclaim, won the Commonwealth Prize for First Novels (Eurasia Section), and was a number one bestseller in India. Her second novel _A Married Woman_ was called 'fluent and witty' in the _Independent_ , while her third, _Home_ , was described as 'engaging, glistening with detail and emotional acuity' in the _Sunday Times_. Her most recent novel, _The Immigrant_ , was called 'intensely readable' in the _Daily Mail_ and 'admirable and enjoyable' by the _Guardian_. She lives in New Delhi. \n\n# Copyright\n\nThis ebook edition published in 2012  \nby Faber and Faber Ltd  \nBloomsbury House  \n74\u201377 Great Russell Street  \nLondon WC1B 3DA\n\nAll rights reserved  \n\u00a9 Manju Kapur, 2003\n\nThe right of Manju Kapur to be identified as author of this work has been asserted in accordance with Section 77 of the Copyright, Designs and Patents Act 1988\n\nThis ebook is copyright material and must not be copied, reproduced, transferred, distributed, leased, licensed or publicly performed or used in any way except as specifically permitted in writing by the publishers, as allowed under the terms and conditions under which it was purchased or as strictly permitted by applicable copyright law. Any unauthorised distribution or use of this text may be a direct infringement of the author's and publisher's rights, and those responsible may be liable in law accordingly\n\nISBN 978\u20130\u2013571\u201326780\u20134\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nTHE SHAPE OF SPECTATORSHIP\n\nFILM AND CULTURE\n\nJohn Belton, Editor\nFILM AND CULTURE\n\n_A series of Columbia University Press_\n\nEdited by John Belton\n\nFor the list of titles in this series, see Series List.\nTHE\n\nSHAPE\n\nOF\n\nSPECTATORSHIP\n\nART, SCIENCE, AND EARLY CINEMA IN GERMANY\n\nSCOTT CURTIS\n\nCOLUMBIA UNIVERSITY PRESS\n\nNEW YORK\n\nColumbia University Press\n\n_Publishers Since 1893_\n\nNew York Chichester, West Sussex\n\ncup.columbia.edu\n\nCopyright \u00a9 2015 Columbia University Press\n\nAll rights reserved\n\nE-ISBN 978-0-231-50863-6\n\nLibrary of Congress Cataloging-in-Publication Data\n\nCurtis, Scott.\n\nThe shape of spectatorship : art, science, and early cinema in Germany \/ Scott Curtis.\n\npages cm. \u2014 (Film and culture)\n\nIncludes bibliographical references and index.\n\nISBN 978-0-231-13402-6 (cloth : alk. paper) \u2014 ISBN 978-0-231-13403-3 (pbk. : alk. paper) \u2014 ISBN 978-0-231-50863-6 (ebook)\n\n1. Motion pictures\u2014Germany\u2014History\u201420th century. 2. Motion picture audiences\u2014Germany\u2014History\u201420th century. 3. Motion pictures\u2014Aesthetics. 4. Motion pictures in science\u2014Germany. 5. Documentary films\u2014Germany\u2014History\u201420th century. I. Title.\n\nPN1993.5.G3C88 2015\n\n791.430943\u2014dc23\n\n2015010546\n\nA Columbia University Press E-book.\n\nCUP would be pleased to hear about your reading experience with this e-book at cup-ebook@columbia.edu.\n\nCover Design: Jordan Wannemacher\n\nCover Image: From Wilhelm Braune and Otto Fischer, \"Versuche am unbelasteten und belasteten Menschen,\" _Abhandlungen der Mathematisch-Physischen Klasse der K\u00f6niglich S\u00e4chsischen Gesellschaft der Wissenschaften_ 21, no. 4 (1895): 151\u2013322\n\nReferences to websites (URLs) were accurate at the time of writing. Neither the author nor Columbia University Press is responsible for URLs that may have expired or changed since the manuscript was prepared.\nTo my parents\n\nCONTENTS\n\nList of Illustrations\n\nAcknowledgments\n\nINTRODUCTION\n\n1. SCIENCE'S CINEMATIC METHOD: MOTION PICTURES AND SCIENTIFIC RESEARCH\n\nEarly Scientific Filmmaking: An Overview\n\nBergson, Cinema, and Science\n\nThe Science of Work and the Work of Science\n\nBrownian Motion and \"the Space Between\"\n\nNerve Fibers, Tissue Cultures, and Motion Pictures\n\n2. BETWEEN OBSERVATION AND SPECTATORSHIP: MEDICINE, MOVIES, AND MASS CULTURE\n\nThe Multiple Functions of the Medical Film\n\nMotion Pictures and Medical Observation\n\nTime, Spectatorship, and the Will\n\n3. THE TASTE OF A NATION: EDUCATING THE SENSES AND SENSIBILITIES OF FILM SPECTATORS\n\nCinema and the Spirit of Reform\n\nChildren, Crowds, and the Education of Vision and Taste\n\n\"Cinematic Lesson Plans\" in Elementary and Adult Education\n\n4. THE PROBLEM WITH PASSIVITY: AESTHETIC CONTEMPLATION AND FILM SPECTATORSHIP\n\nAgency and Temporality in the Aesthetic Experience of Cinema\n\n_Einf\u00fchlung_ , Identity, and Embodied Vision\n\nThe Politics of Contemplation\n\nCONCLUSION: TOWARD A TACTILE HISTORIOGRAPHY\n\nNotes\n\nBibliography\n\nIndex\nLIST OF ILLUSTRATIONS\n\nFigure 1.1. | Braune and Fischer's military recruit in the experimental suit  \n---|---  \nFigure 1.2. | The subject at rest with the grid superimposed  \nFigure 1.3. | Braune and Fischer's camera placement  \nFigure 1.4. | The resulting chronophotograph  \nFigure 1.5. | Determination of the coordinates of a point _P_ from the projections of _P_ on two planes as seen from the two cameras  \nFigure 1.6. | Side and top views of the instrument used to measure coordinates  \nFigure 1.7. | Measurement of a coordinate  \nFigure 1.8. | A table of the coordinates derived from experiment 1  \nFigure 1.9. | The graph of the coordinates (view from the right side)  \nFigure 1.10. | The graph of the coordinates (view from above of different body parts)  \nFigure 1.11. | Left and back views of the tridimensional model representing the attitudes of the human body during walking  \nFigure 1.12. | Seddig's cinematic apparatus for measuring Brownian motion  \nFigure 1.13. | Seddig's photographic rendering of Brownian motion  \nFigure 1.14. | The irregular paths of Brownian motion  \nFigure 1.15. | Hermann Braus  \nFigure 1.16. | Ross Granville Harrison  \nFigure 1.17. | Harrison's sketches of the elongation of frog nerve fibers grown in culture  \nFigure 2.1. | Frames from Ludwig Braun's film of a dog's beating heart  \nFigure 2.2. | Groedel's serial cassette X-ray apparatus  \nFigure 2.3. | Max Nordau  \nFigure 2.4. | A hypnotist practicing his craft, France, circa 1900  \nFigure 3.1. | A typical storefront movie theater ( _Ladenkino_ ) from the pre\u2013World War I era  \nFigure 3.2. | _The Wanderings of Odysseus_ , touted to be a \"Reformfilm\"  \nFigure 3.3. | \"The Cultural Work of the Film Theater: Thoughts from the Year 1784 by Friedrich von Schiller\"  \nFigure 3.4. | The geometry of taste  \nFigure 3.5. | Children at Luisen-Kino in Berlin, circa 1910  \nFigure 3.6. | A page from Lemke's _Die kinematographische Unterrichtsstunde_ (The Cinematic Lesson Plan, 1911)  \nFigure 3.7. | Hermann H\u00e4fker  \nFigure 4.1. | Aesthetic experience as a series of interlocking dichotomies  \nFigure 4.2. | A prewar audience enjoying a night at the Union-Theater in Berlin, 1913  \nFigure 4.3. | Kammer-Lichtspiele Theater in Berlin, 1912\nACKNOWLEDGMENTS\n\nThis has been, to use an inappropriate medical metaphor, a long and difficult birth, but certainly not for lack of attendants or painkillers. After so many years of conceptions, reconceptions, and labor, however, it is hard to know who to thank or how far back to go, so I will simply begin to recite and offer my deepest apologies to anyone I inadvertently leave out. Various versions took shape in various locations, where I owe debts to the institutions and people who supported me and this project. The initial research was made possible by a stipend from the German Academic Exchange Service. Several people made my stay in Frankfurt more productive than it might have been. Heide Schl\u00fcpmann, Helmut Diederichs, and Martin Loiperdinger offered good advice and kind, if somewhat bewildered encouragement as I struggled to define my topic. Diederichs also provided the rare image of Hermann H\u00e4fker for chapter 3. Special thanks go to the late Eberhard Spiess for allowing me access to the periodicals and holdings of the Deutsches Institut f\u00fcr Filmkunde, and to Brigitte Capitain for her patience as I took advantage of this generosity. In Iowa City, I appreciate the attention David Depew, Kathleen Farrell, and Hanno Hardt gave to the first version of this project, while John Durham Peters and Dudley Andrew deserve special thanks for their guidance through the years. In Los Angeles, I am grateful to Linda Mehr and the late Robert Cushman of the Academy of Motion Picture Arts and Sciences' Margaret Herrick Library; I learned much from their example. Joe Adamson, Val Almendarez, Anne Coco, Steve Garland, Harry Garvin, Barbara Hall, Doug Johnson, Janet Lorenz, David Marsh, Howard Prouty, Lucia Schultz, Matt Severson, Warren Sherk, and all of my fellow librarians at the Herrick deserve thanks as well.\n\nIn Evanston, I owe thanks to the Department of Radio\/Television\/Film\u2014especially my colleagues Bill Bleich, Michelle Citron, Laura Kipnis, Chuck Kleinhans, Larry Lichty, Hamid Naficy, Eric Patrick, Jeff Sconce, Jacob Smith, Jacqueline Stewart, Deb Tolchinsky, and the rest of the faculty\u2014for having faith in me. Chairs Annette Barbier, Mimi White, Lynn Spigel, and David Tolchinsky were steadfast in their support. I thank Lynn Spigel, especially, for all she has done on my behalf (as far back as Los Angeles) as a mentor, model, and friend. Deans David Zarefsky and Barbara O'Keefe at the School of Communication offered resources in various forms, and I am grateful for their time, money, and patience. In Berlin, stays at the Max Planck Institute, thanks to Lorraine Daston, significantly sharpened my thinking about physics and observation, especially. In Weimar, Karl Sierek, Friedrich Balke, Daniel Eschk\u00f6tter, and the graduate students at Bauhaus-Universit\u00e4t welcomed me and gave me space and time to work. In Innsbruck, Mario Klarer, Gudrun Grabher, Christian Quendler, Erwin Feyersinger, Robert Tinkler, Cornelia Klecker, Johannes Mahlknecht, Monika Datterl, and Maria Meth likewise gave time, space, and warm companionship freely, as well as trips to mountain cabins. In Doha, the entire staff and faculty of Northwestern University in Qatar made me feel welcome, especially program directors Mary Dedinsky and Sandra Richards, and colleagues Greg Bergida, David Carr, Susan Dun, Elizabeth Hoffman and Bob Vance, Joe Khalil, Muqeem Khan, John Laprise, Jocelyn Mitchell, Sue Pak, Christina Paschyn and Alex Demianczuk, Barry Sexton, Bianca Simon, Anne and Adam Sobel, Allwyn Tellis, Tim Wilkerson, and Ann Woodworth. Deans Jim Schwoch, Jeremy Cohen, and Everette Dennis displayed an inordinate amount of trust and confidence in my abilities; Dean Dennis, especially, offered whatever it took\u2014and it took a lot\u2014to get it done, and I am deeply indebted to him.\n\nAlong the way, a number of people deserve commendation for having read or commented on various parts of this project in various forms. Chapter 1 benefited greatly from the insights of Nancy Anderson, Charlotte Bigg, Thomas Haakenson, Andreas Mayer, and others at the Max Planck Institute; Hannah Landecker kindly shared her research and insights on cell biology; Martin Carrier and Alfred Nordmann sharpened my thinking about atomistic physics; Richard Kremer and Ken Alder offered face-saving corrections from the historian of science's point of view; and Dan Morgan, Oliver Gaycken, and Frank Kessler were kind enough to read the chapter at various points and help me clarify the argument. Chapter 2 profited from the attention of Ken Alder and the Science in Human Culture Group at Northwestern, who prompted me to rethink it, while Nancy Anderson and Mike Dietrich provided time and space at Dartmouth College to rewrite it. Lisa Cartwright, Oliver Gaycken, Andreas Killen, Kirsten Ostherr, and Henning Schmidgen were at various points inspirational and instrumental in shaping this chapter; Lisa Cartwright was especially helpful at a key point in the process. Chapter 3 owes its life to Steve Wurtzler, Jennifer Barker, and John Belton, and I owe Frank Kessler and Sabine Lenk for keeping it alive. A much earlier, different version of chapter 4 was lucky to have the scrutiny of Ben Singer, David Bordwell, David Levin, and Marc Silberman. For its current form, I must thank Robin Curtis, Gertrud Koch, Dan Morgan, Inga Pollmann, and especially Kaveh Askari and Tony Kaes for all of their insight and encouragement. There are still others who provided valuable assistance along the way, including research assistants David Gurney, Dan Bashara, and Rebecca Barthel. John Carnwath kindly and expertly corrected my translations, rescuing me from many infelicities. Stefanie Harris, J\u00f6rg Schweinitz, and various anonymous readers offered important insights that prompted revisions and changes in argument. For their stalwart professional support and friendship over the years, I must offer my heartfelt thanks to Richard Abel, Rick Altman, Matthew Bernstein, Jane Gaines, Dilip Gaonkar, the late Miriam Hansen, Tom Levin, Charlie Musser, Jan Olssen, Patrice Petro, Lauren Rabinovitz, Eric Rentschler, Mark Sandberg, Vivian Sobchack, and Virginia Wexman.\n\nAll of the people named so far I count as my friends, but some friends deserve special mention for their unselfish and nonjudgmental acceptance of me and my book. Tom Gunning has been a friend and mentor for a very long time; more than anyone, he has shaped the contours of this ongoing investigation, usually without even knowing it. Tony Kaes has been a loyal fan and inspiration since I was a student. Both Tony and Tom have offered insightful, transformative commentary on several versions. Oliver Gaycken, Vinzenz Hediger, and Kirsten Ostherr are my fellow travelers on this interdisciplinary journey; I don't often take a step without consulting them. Oliver read every word I have given him and always came back for more. Greg Waller and Brenda Weber have always offered valuable moral support and close friendship at just the right times. Ken Alder, Joe Carli, Lisa Cuklanz, Tracy Davis, Doug Johnson, Charlie Keil, and Will Schmenner have all been steady, life-long friends on whom I have leaned especially heavily at times. All the graduate students who have attended my seminars deserve note for their role in shaping my thinking over the years, but I thank especially Dan Bashara, Catherine Clepper, Beth Corzo-Duchardt, Alla Gadassik, Leslie Ann Lewis, Jason Roberts, Jocelyn Szczepaniak-Gillece, Kati Sweaney, and Meredith Ward. For their continuing friendship, I thank Richard Abel, Charles Acland and Haidee Wasson, Dana Benelli, Joanne Bernardi, Bill Bleich, Jeremy Cohen and Catherine Jordan, Kelley Conway and Matthew Sweet, Mark Garrett Cooper and Heidi Rae Cooley, Don Crafton and Susan Ohmer, Nick Davis, Leslie Midkiff DeBauche, Nico de Klerk, Carol Donelan and Shannon Spahr, Nata\u0161a \u010eurovi\u010dov\u00e1, Dirk and Myrna Eitzen, Jen Fay, Andr\u00e9 Gaudreault, Philippe Gauthier, Frank Gray, Alison Griffiths and William Boddy, Barbara Hall and Val Almendarez, Sara Hall and Monty George, Stefanie Harris, Micaela Hester, Chris Horak, Laura Horak and Gunnar Iverson, Rembert Hueser, Jenn Horne and Jonathan Kahana, Christopher Hurless and Rachel Henriquez, Zara Kadkani and Axel Schmitt, Jim Lastra, Tom Levin, Melody Marcus, Caitlin McGrath, Christie Milliken, Priska Morrissey, Tania Munz, Bill Palik, Anna Parkinson, Jennifer Peterson, Sarah Projansky, Christian and Grace Quendler, Isabelle Raynauld, Mark Sandberg, Ben Singer, Blane Skowhede, Jake Smith and Freda Love Smith, Stefan Soldovieri, Matthew Solomon, Shelley Stamp, Bing Stickney, Claudia Swan, Steve Tremble, Alison Trope, Mark Williams and Mary Desjardins, Tami Williams, Michael and Julia Wilson, Pam Robertson Wojcik, Robb Wood and Hanaa Issa, Steve Wurtzler, Harvey Young, and Josh Yumibe.\n\nI should stress that this book would not have been published except for the efforts of John Belton and Jennifer Crewe, whose stubborn determination to wrest the manuscript from me finally overmatched my stubborn refusal to give it up. I also thank my editorial team at Columbia University Press: Ben Kolstad, Roy Thomas, Jennifer Jerome, Anne McCoy, and Kathryn Schell.\n\nOf course, my family also deserves a large share of credit, not the least for their good-natured and bemused acceptance of a project that seemed never to end. Cindy and Randy Lee have been unconditional in their love and acceptance; Brandon and Noelle Lee, Trevor, Josh, and Michael Lee all are great relatives to have. Grant and Donna Boyles deserve mention for their love, care, and hospitality. I know my grandmother, Louise, and my aunt, Dian, would have been proud. I thank the Pike and Kelly families for welcoming me into their close-knit web of love and kindness: Ken and Elnora Kelly, Clayton and Carol Pike, Kerry and Kelli Graf (and Kaitlyn and Kayla), and Kory Pike; Kevin and Meredith Pike, especially, are not just relatives, but friends, which is a rare thing to say. But I owe most to Kirsten Pike, who has been my anchor, sail, and compass since I met her; this book and I would not be the same without her years of love and encouragement. If I had another book in me, I would dedicate it to her. But I must dedicate this work to my parents, Ray and Pat Curtis, whose inexhaustible patience, support, and love made everything possible.\n\nThere have been publications of parts of this project along the way, but what lies in the pages ahead is usually significantly different from what came before. Even so, we should note that parts of chapter 1 originally appeared as \"Die kinematographische Methode. Das 'Bewegte Bild' und die Brownsche Bewegung,\" _montage\/AV: Zeitschrift f\u00fcr Theorie & Geschichte audiovisueller Kommunikation_ 14, no. 2 (2005): 23\u201343; and \"Science Lessons,\" _Film History_ 25, nos. 1\u20132 (2013): 45\u201354. Parts of chapter 2 originally appeared as \"Between Observation and Spectatorship: Medicine, Movies, and Mass Culture in Imperial Germany,\" in _Film 1900: Technology, Perception, Culture_ , edited by Annemone Ligensa and Klaus Kreimeier (New Barnet, U.K.: Libbey, 2009), 87\u201398; and \"Dissecting the Medical Training Film,\" in _Beyond the Screen: Institutions, Networks and Publics of Early Cinema_ , edited by Marta Braun, Charlie Keil, Rob King, Paul Moore, and Louis Pelletier (New Barnet, U.K.: Libbey, 2012), 161\u2013167. Parts of chapter 3 originally appeared as \"The Taste of a Nation: Training the Senses and Sensibility of Cinema Audiences in Imperial Germany,\" _Film History_ 6, no. 4 (Winter 1994): 445\u2013469. Parts of chapter 4 originally appeared as \"Einf\u00fchlung und die fr\u00fche deutsche Filmtheorie,\" in _Einf\u00fchlung. Zur Geschichte und Gegenwart eines \u00e4sthetischen Konzepts_ , edited by Robin Curtis and Gertrud Koch (Paderborn: Fink, 2009), 61\u201384.\nINTRODUCTION\n\n_Probably no contemporary invention has generated quite as much discussion in the daily press and in daily conversation as the cinema. Everywhere new theaters shoot up overnight like mushrooms. Our cities at night can no longer be imagined without the beaming portals of the movie houses. But it's not just the simple folk pushing themselves through these \"narrow gates of grace.\" The educated class, as well as science and the schools, the state, the city, and rural communities have all grasped the cultural significance of cinema and have taken a step closer to the establishment and utilization of their own film theaters. Who would look upon this burning question with indifference?_\n\nADOLF SELLMANN (1912)\n\nWhatever cinema is, it has always been many things to many people. Even in 1912, it was clear to a reformer such as Adolf Sellmann that a variety of interest groups and interested parties, from scientists to educators to town councils, were using motion picture technology. Each group recognized cinema's \"cultural significance\" and power or acknowledged its inevitability, but not every group agreed on how motion pictures should be used, either in the public sphere or, especially, within the boundaries of the group or discipline. Everybody had their reasons for using motion pictures, and those reasons often diverged. This book attempts to understand how various disciplines or communities used motion pictures. What did cinema mean to these groups? What were the criteria for the acceptance of motion pictures as a tool within a given discipline? What problems presented themselves such that motion pictures were considered a solution? This book explores these questions to discover the criteria for the legitimacy of a new media technology within the disciplines of science (specifically, human motion studies, physics, and biology), medicine, education, and aesthetics in Germany before World War I.\n\nThese disciplines correspond roughly to the familiar historical trajectory of early cinema, from its roots in scientific research to its early bids for acceptance as an art form. Taken together, they also represent the heterogeneity of early cinema, not only in terms of the many types of films available during this period but also with respect to the varied venues, audiences, and uses of the medium. Additionally, they typify relatively well-defined communities with strong, native traditions, where, outside of the entertainment industry, the liveliest discussion of motion pictures took place during cinema's early period. This listing obviously leaves out the entertainment industry, but questions of appropriation and legitimacy are less interesting in this area (at least to me), where the criterion for acceptance of film within that industry was clear, even tautological, in that the medium only had to prove its commercial viability and little else. So with its focus on the way groups used film for purposes other than entertainment, this study is aligned primarily with recent work in nontheatrical uses of film.\n\nHowever, even within the framework of \"useful cinema\" and the good work that has been done to define that area of film history, questions of appropriation and legitimacy are not often explicitly asked. While we know much about the use of motion pictures in the classroom in the 1920s, for example, we still know comparatively little about the state of pedagogical theory and practice at that time and why some groups within the discipline saw motion pictures as a partial solution to a variety of problems, and why others did not. We know little of their disciplinary agenda. Perhaps inevitably, we approach these questions as film historians, not as historians of education. Yet to understand fully any given appropriation, we must fully understand the agenda that shapes it. There is an intimate and complex relationship between any technology and the agenda that makes use of it. The technology is not merely applied to a problem; the problem presents itself in part because of the technology. What any scientist investigates, for example, is partly due to what the available technology makes available for investigation. Historians of science are very good at demonstrating the dialectical relationship between tools, theories, and representations, which shows us that we cannot take \"use\" for granted; the criteria for use of any given tool within a given discipline are not obvious. As different groups used media technologies for their different purposes, the nature of those appropriations changed, and in a significant way, so did the medium. \"What cinema is\" for one group was not necessarily the same as for another. Indeed, the nature of the appropriation often depended on what the agents _thought_ film was. So there was more to \"use\" than simply taking a camera, recording an event, and projecting it; the representational problems faced by any given discipline shaped its appropriation of (media) technology. Understanding those representational problems demands an intimate knowledge of the historical contexts and camps of that discipline.\n\nSo this project is not just about the encounter between other disciplines and film but the encounter between other disciplines and _film studies_. Specifically, each chapter stages a meeting between methods or approaches common to film studies and those of the history of science, the philosophy of medicine, the history of education, or the history of aesthetics. What does the result of such an assignation look like? What can we take away from such an encounter? Or, to put it another way, what can we reasonably expect from interdisciplinary research? Max Weber has his hand up: \"With every piece of work that strays into neighboring territory... we must resign ourselves to the realization that the best we can hope for is to provide the expert with useful _questions_ of the sort that he may not easily discover for himself from his own vantage point inside his discipline.\" \"Useful questions,\" however, are rarely presented as such; they are instead approaches or agendas that seem foreign at first, yet bear on our own. To formulate them as questions, we need to know the discipline well enough to recognize the pattern common to the approaches. So interdisciplinary research should be more than cherry-picking a few juicy quotations from an exciting discovery in another field; it must entail some significant level of immersion. What useful questions does the history of science, for example, provide film and media studies, and vice versa? As hinted above, one broad question could be: What is the relationship between technology and a disciplinary agenda? This ambitious question might be answered only after an accumulation of case studies, but it is useful for film and media studies, because it leads us to speculate about the tangible relationship between a representational technology and a community's conception of the object of study. Another question might be something like: What is the relationship between a technology and other elements of the experimental system? This question forces us away from our habitual focus on film and toward an understanding of film and media technologies as part of a larger experimental arrangement or as part of a technological group along the lines of what Germans call a _Medienverbund_ , or \"media ensemble.\" With the help of the history of science, we can see film in these contexts as an important but nevertheless interdependent part of an experimental system or a larger institutional project. Each disciplinary encounter in the following chapters results, I hope, in a different set of similarly useful questions.\n\nWhat can film historians bring to the table? We can bring different kinds of answers. Our training in close analysis heightens our sensitivity to formal relationships that rely on analogies and homologies as well as causal or empirical connections. This would be useful, because the acceptance or legitimacy of any given technology for a discipline hinges not simply on the tool's function for a particular task but also on how that technology fits into a larger disciplinary system. A wrench is useful because it fits the bolt; certain parts of the bolt and wrench have similar shapes. In fact, the formal features of each determine their use, or vice versa; there is no reason to use the wrench on the bolt, except for this formal relationship. Likewise, film as a tool must have fit the object, task, and system of which it was a part. This fit was variable, because any given technology as complex as motion pictures is not a single thing but multiply adaptable to various agendas, so the relationship between them might be successful or not. The relationship was shaped discursively, too: the convergence between a technology and a discipline depended on the successful appropriation of the technology but also on the successful preparation of the discipline or agenda for that technology. If wrench and bolt are designed together from the start, the mutual shaping of film and discipline happened over time and was just as discursive as practical. That is, as mentioned earlier, the fit between a research task and film often depends on what the agents _think film is_ , as well as the analogical relationship between a technology and a system (such as how the film frame isolates an object in a way similar to how experiments attempt to isolate objects and variables). Film and media studies is especially good at teasing out film-theoretical assumptions and finding formal connections. So a useful question could be: To what extent can the _formal_ features of a technology and its representational products explain the relationship between that technology and the research agenda? This is a useful question for historians of science, because the significance of formal features of their objects of study rarely takes precedence.\n\nMore than a book about Germany, science, aesthetics, or even cinema, then, this is a book about historiography. How should we approach the relationship between a (media) technology and the group or discipline that uses it? This book insists that the formal features of the technology matter. It argues that the disciplinary legitimacy of a technology or the successful appropriation of a technology by an expert group depends on _a correspondence between the logic of the discipline and the formal features of the technology_. In this case, filmic form refers to two manifestations of the image: the projected image and the frames on the celluloid filmstrip. The logic of the discipline refers to its problem-solving protocols, its investigatory methods, and its ideological (in terms of its discipline) assumptions. Medical logic, for example, refers to a method of arriving at a diagnosis, etiology, prognosis, and treatment of a disease. It is a way of thinking that provides for a (more or less) consistent and balanced understanding of what is peculiar to the individual case and what is generalizable from it. As we shall see in chapter 2, this logic relied on series of cases for training and context. We will also see that the ambidexterity of film, its ability to be useful in both its still and moving forms, corresponded in unique ways to this medical logic and observational practice. The goal of that chapter, then, is in part to sketch the correlation between a \"medical way of thinking,\" as philosopher of science Ludwik Fleck put it, and the specific formal features of film. Likewise, in some scientific disciplines, the ability of film to frame and isolate the object under study and film's temporal malleability (its ability to expand or compress time) were analogous to features of scientific experiments, such as the ability to isolate variables and extend observational duration. In others, film's temporal discontinuity, indicated by the gap between film frames, matched theoretical ideals about the physical world. For education and aesthetics, the logic of the discipline was less about solving problems than about describing goals or mental operations to attain those goals. In education, the consonance between the detail and duration of the moving image and the richness of the natural world created a homology between the (perceived) realism of the image and the goal of visual instruction, which was to teach students to recognize objects and generalize from them. Aesthetics describes the terms of artistic production and, especially pertinent to this project, the conditions of aesthetic experience; it describes the set of presumed cognitive and emotional operations that accompany aesthetic experience, along with their ideological significance. With regard to film, then, the formal features of the object and the accompanying experience were compared with the prevailing understanding of aesthetic experience. In this case, the pace of the moving image worked against ideals of free will and agency embedded in common conceptions of the aesthetic experience, while the ability of the moving image to encourage emotional projection corresponded to different theories of aesthetic experience popular at the time.\n\nFrame, gap, detail, duration, and pace: each discipline or group, then, saw something useful (or counterproductive, as the case may be) about filmic form but emphasized different features for different goals. In the early period, as motion picture technology emerged as a real possibility for various applications, individual researchers, teachers, reformers, or cultural pundits took it upon themselves to justify motion pictures as a tool or good object to their colleagues. They endeavored to demonstrate that motion pictures could indeed conform to the logic and practices of the discipline. This is why the early period of film's cultural dissemination is so productive for this kind of project: the reasons for appropriation\u2014what the champions thought film was\u2014are often clearly stated, before the use of motion pictures becomes naturalized and obvious, requiring no justification at all (which is the case today when scientists, for example, use moving images as a matter of course without discussion). This is also why this project is less concerned about individual films than the contours of the discussion about the use of film in general; close analysis of filmic style can tell us very little about the justifications mounted for film or the correlation between disciplinary logic, practice, and film form. Such rationalizations can tell us much, however, about the state of the discipline at the time. Whether the use of film addressed common representational problems within a discipline (as in the case of cell biology at the turn of the century) or the discipline faced larger philosophical problems that film seemed to exemplify (as in the case of fin de si\u00e8cle aesthetics), the engagement with motion pictures tells us quite a lot about the priorities and changes that faced any given discipline. Again, this is especially the case during film's early period, when groups latched onto this new technology because of a pressing need or because of its prevalence. In other words, the timing of these appropriations is not coincidental: each group needed motion pictures in some way\u2014even as a scapegoat, which was often the case. Within any given discipline, these maneuvers were far from unanimously approved; dissent was more common in the cultural than scientific spheres, but the debates sharpen our understanding of what was at stake for each group.\n\nWhat was at stake exactly? As a primarily visual technology, motion pictures presented an aid or a challenge to _expert modes of viewing_ , which were the most common manifestation of disciplinary logic and practice. Around what practices, ideals, and \"ways of thinking\" do disciplines coalesce? There are many answers, ranging from laboratory procedures to nationalist rhetoric. But a very common way to train new members of a discipline is to teach them what to look for and how to look for it. Ludwik Fleck insisted that \"one has first to learn to look in order to be able to see that which forms the basis of the given discipline,\" but I would further insist that \"learning to look\" _is_ what often forms the basis of a discipline. Whether it is medical observation or aesthetic contemplation, disciplines have ways of looking that are assumed; if you have to ask what it is or how to do it, you obviously are not part of that group. It is a badge of honor or a barrier to entry. Scientific and medical disciplines, for example, invested much time and energy into training recruits to their modes of viewing, to what Fleck called \"the directed readiness to certain perceptions.\" British surgeon Sir James Paget emphasized in 1887 that \"becoming scientific in our profession [requires] the training of the mind in the power and habit of accurately observing facts.... The main thing for progress and for self-improvement is accurate observation.\" Likewise, Swiss pedagogue Johann Heinrich Pestalozzi organized his entire, influential educational program around close examination; he believed that direct perception of the world was \" _the foundation of all knowledge_.\" And perhaps it goes without saying that aesthetic experience, as expounded by German philosophers since Immanuel Kant, relied heavily on a mode of viewing that was leisurely, probing, and attentive\u2014in a word, contemplative. Above all, these disciplinary modes of looking are more than individual, as Fleck notes: \"We look with our own eyes, but we see with the eyes of the collective body, we see the forms whose sense and range of permissible transpositions is created by the collective body.\" So expert modes of vision and disciplines orbit each other closely and inextricably, held together by ideological bonds.\n\nThis is not to say there is only one mode of viewing for each discipline. Indeed, there are two obvious examples of different modes of expert viewing that obtain regardless of discipline: the holistic, all-at-once, instant appraisal; and the roaming, penetrating, leisurely, detail-oriented contemplation of the object. These two modes can be found in a range of disciplines from scientific observation to medical observation to aesthetics. In art history, for example, Robert Vischer distinguished between _Sehen_ and _Schauen_ , two modes of viewing artworks, which can be translated as \"seeing\" (by which he meant the synthetic, intuitive, instant appraisal) and \"scanning\" (by which he meant analytic, detail-oriented contemplation). These correspond roughly to our \"glance\" and \"gaze,\" as long as they are not confused with their current alignment in media studies with distraction and contemplation. For Vischer and experts across disciplines, glance and gaze were complementary modes of viewing, both requiring expertise. In medicine, physicians in turn-of-the-century Germany who were troubled by the increasingly scientific approach to healing\u2014represented in their minds by the analytic, focused gaze\u2014advocated a more holistic view that took in the entire patient at glance rather than a penetrating gaze that examined only the localized disease in detail. Even if in this case these modes of viewing were taken up as flags symbolizing different professional stances, in practice medical training encompassed both modes. An expert both synthesizes and analyzes.\n\nIn fact, as Michel Foucault has noted about medical observation, these modes of expert viewing were more than merely visual\u2014they were also a set of logical operations. Specifically, I find that expert viewing is largely a process of _correlation_. What happens during expert observation? The expert sees the parts and the whole and finds patterns between them and correlates those patterns to expert knowledge of the disciplinary context. Scientific observation, for example, connotes an analytic gaze, but the most important aspect of scientific observation is the context the scientist brings to it; the researcher assimilates observed data into an existing framework of knowledge. What the scientific observer already knows frames what he or she observes, and he or she incorporates or juxtaposes new data with old and thereby generates new insights. Observation therefore implies _the production of knowledge through correlative insight_. This is true even with art, in which the lynchpin of post-Kantian conceptions of aesthetic experience is the \"free play of associations.\" As the expert gazes upon the artwork, the parts work together in pleasing ways to prompt in the viewer a correlation of textual and contextual patterns. As these associations come together, interlock, separate, and recombine with others, the viewer supposedly experiences a pleasant, even moving state of being between artwork, world, and his or her own subjectivity\u2014all of which has been known as aesthetic experience, which depends fundamentally on a particular kind of viewing that is leisurely, active, attentive, and _correlative_.\n\nEspecially with regard to aesthetic contemplation, expert viewing was not only a disciplinary practice or a logical operation, it was also an exhortation. That is, expert viewing involved a measure of training and hence status, so it also often implied a way of looking at the world or an ideological stance. Immanuel Kant's and Friedrich Schiller's philosophies, for example, gave aesthetic experience a prominent ethical and moral position, in that it was both a mode of engaging with art and a way of bridging an impasse between competing forms of knowledge (Kant) or between conflicting human impulses (Schiller). Schiller's system, especially, suggested that aesthetic experience could lead the way to a better social organization, if only we applied this mode of engaging with art to our way of engaging with the world. Arthur Schopenhauer was even more insistent: aesthetic experience, for him, was one of the few ways by which we could escape the dreary consequences of our everyday impulses, and therefore in his system it became the fundamental model for interacting with the world in general. In the twentieth century, the discussion about the relative merits of contemplation and distraction was, as we know, highly politically charged precisely because of their implications for how one should conduct oneself; recall, for example, Georg Luk\u00e1cs's dismissal of the contemplative life as unconscionable in the face of changes that called for political action and initiative. Even scientific observation became a global ideal as \"objectivity\" came to mean more than refraining from theoretical speculation. I am not suggesting that these systems work or that they are universal. I am arguing that the operations and stances required by these expert modes of viewing also functioned widely as rhetorical instruments, as presumptions about how one should engage with the world.\n\nHow did motion pictures fit into these traditions of expert viewing? That is exactly the question the present project explores. If my larger argument claims that the legitimacy or success of a media technology depended on a correspondence between its formal features and the logic of the discipline, then the more specific argument holds that a discipline's expert viewing ideals and practices exemplified its logic and that _a discipline_ ' _s successful appropriation of film hinged on its ability to accommodate the new technology to its mode of expert viewing_. To put it another way, the acceptance of film as an appropriate tool or good object within a discipline depended on an alignment between the formal features of the filmic image and the practices and ideals of that discipline's modes of expert viewing. The acceptance of film as a tool rested in part on the advocate's ability to adapt\u2014physically and\/or rhetorically\u2014motion pictures to established modes of viewing. If proponents could prove, either through example or argument, that the filmic image could accommodate or even amplify established methods of observation, then the community accepted the justification and elaborated on it in subsequent literature.\n\nFor example, when teachers needed to justify the educational use of motion pictures, they aligned its formal features to already established conceptions of expert viewing and visual instruction; in Germany, these conceptions were encapsulated by the term _Anschauungsunterricht_ , which translates roughly as \"visual means of instruction.\" Their justifications were persuasive to the extent that certain features of film could match or accommodate the methods of this means of instruction. When cultural pundits argued against the idea that film could be art, their arguments were less often about its mechanical reproduction of nature and more about the relationship between the projected moving image and the possibility of aesthetic contemplation. In each case, the question was, can film be used in a manner familiar to our way of viewing and our way of viewing the world? Motion pictures also famously offered a view _different_ from what individual viewers or disciplines were accustomed to. My argument about the correspondence between film and disciplinary logic accommodates this difference as well, because historically the tug-of-war between the familiar and the novel stretched across the literature in a given field; advocates of film technology came to it because it was new and different, yet understood it in terms of what had already come before. Indeed, any given experimental system or discipline moves forward in the same way: methods, technologies, and inscriptions generate new insights that are folded into the existing system, which is thereby subtly changed. In other words, the patterns of justification and dissemination of this new media technology, like that of knowledge itself, were incremental and correlative.\n\nThis implies that expert vision is not the only criterion by which disciplines judged motion picture technology. A survey of the literature in each of these fields reveals that the advantages of film for their projects were often expressed in two ways: in terms of inscription (what the camera can record) and reception (how the image can be viewed). Chapter 1, unlike the rest of the chapters, will focus on inscription. It will examine how researchers adapted motion pictures as an inscription technology to their objects of study, their theories, and their disciplinary needs. The following chapters will explore how experts adapted motion pictures to their different modes of viewing: medical observation, visual methods of instruction, and aesthetic contemplation.\n\nNeedless to say, expert viewing requires training that derives from and reinforces a community. Whether that community is a scientific discipline or an educated class of citizens or both, training to partake of the privileges of that group often came in the form of _visual_ training as well as education in the relevant literature or canon. Sometimes that training was explicit, as in _Anschauungsunterricht_ , which was specifically designed to teach children how to observe; this was not considered disciplinary training so much as one criterion for good citizenship and cultivation. Sometimes it was implicit, as in aesthetic contemplation (although the German tradition of _Bildbetrachtung_ , or \"image viewing,\" explicitly instilled principles of looking, art appreciation, and aesthetic contemplation). In any case, experts and their apprentices came together to form a community (disciplinary, class based, national) through shared values and practices, especially practices of viewing. These bonds were shaped not only by the training and values themselves but also by comparison with their opposite: the untrained, \"valueless\" spectator. Indeed, the values of a group were often not explicitly stated except through reference to what was unacceptable. At stake was the proper method of processing visual information to achieve the desired end, yet the idea of _proper_ viewing was articulated equally often, if not more often, via denunciations of _improper_ viewing. The most obvious example, described in chapter 2, would be the medical experts' description of film audiences as passive, agog, and addicted, whereas their own practices of film viewing emphasized activity, detachment, and control. Yet the latter model was rarely explicitly stated. Instead, we must surmise that \"proper\" viewing model from descriptions of their practice as well as their explicit denunciations of \"improper\" viewing.\n\nIt is hardly surprising that a community of privileged white men in imperial Germany would condescend to audiences of a perceived lower-class amusement such as the movies. Yet perhaps we miss something if we leave it at that. Our understanding of the discursive construction of film spectatorship too often ignores the obvious _duality_ of these constructions. If \"proper\" modes of viewing relied on \"improper\" modes as a useful foil, then the opposite is also true: our understanding of spectatorship is incomplete without an understanding of expert observation. Observation and spectatorship depended on each other (perhaps still) in a figure\/ground relationship wherein the dominance of any given term was often difficult to determine. Spectatorship and observation functioned as each other's \"negative space,\" the outline of one shaping the other by default. This book contends that the \"shape of spectatorship\" can be truly determined only by simultaneously examining the \"shape of observation.\" Observation and spectatorship function akin to stillness and movement or analysis and synthesis: they are oscillating, dynamic dichotomies that are not opposites so much as necessary halves of a hermeneutic process. In this case the hermeneutic is one of self-identity; as experts worked to define their position with regard to disciplinary viewing, \"spectatorship\" was something of a by-product of the work process. More precisely, the work of (explicitly or implicitly) defining spectatorship or observation resulted in a coproduction of the opposite term in a strikingly consistent logic of the supplement. So yes, we know how privileged white men characterized spectators, and we have known that since audiences were aligned with crowds in discussions of early theater. But if we want to know _why_ any given characterization of spectators took its particular form, then we should be aware of the modes of viewing preferred _by the experts writing the description_.\n\nIf theories of film spectatorship in the 1970s and 1980s were preoccupied with the ideological dominance of Hollywood, then the historiographic innovations of film studies during the 1980s and 1990s demonstrated that the Hollywood model was, of course, not always dominant in production and reception; the paradigm of early cinema was not simply Hollywood in nuce but a positive model in its own right: these viewers saw film differently, and early film production and exhibition was heterogeneous. This heterogeneity also implied multiple types of textual address and hence multiple kinds of spectator in terms of gender, race, ethnicity, and class. So if the transitional era codified and standardized filmic address, then it also in a way standardized the spectator; the individual spectator of 1970s film theory was, then, an abstraction generated by filmmaking practice of the Hollywood era, just as the spectator of 1990s film history is a product of textual analysis of early film, or the latest theory of film spectatorship might be a derivation from journalistic accounts of audience experience or of the theorist's own experience. Theories of film spectatorship derive from either filmic textual address or written textual address, and these derivations are separate from historical, empirical viewers and audiences. While contemporary written discussions of viewers and audiences may seem closer to the historical viewer, the written account also produces an abstract notion of spectatorship. My point is that we must take into account the role and interest of the expert in the abstraction. We need to recognize our own expertise and role in producing spectators, but we also need to acknowledge the _disciplinary_ context of historical experts as they wrote about observation and spectatorship. If histories of early film helped to establish the multiplicity of film spectatorship through the categories of gender, race, ethnicity, and class, I would like to add one more category, expert\/lay, which seems a powerful determinate of the character of written discussions of audiences and of disciplinary modes of viewing. Expert\/lay appears closely aligned with high\/low class differences, but I would argue that it is more specific, because educational or disciplinary training provides a traceable, historiographically clear path to the fount of class-based distinctions.\n\nYet as Walter Benjamin argued, the line between expert observer and lay spectator was often fuzzy, especially as film viewers adopted \"an attitude of expert appraisal.\" Benjamin described a mode of reception in which the reactions of the individual were simultaneously amplified and regulated by the reactions of the mass, thereby giving the spectator the pleasure of viewing but also the detachment of the expert. Similarly, chapter 2 describes an expert mode of reception that was also simultaneously distanced from and thrilled by the moving image. Hence the dichotomy between expert and lay is heuristic; most often it was very fluid in practice, but the descriptions of spectatorship often functioned to chart a perceived difference that was helpful for self-identity or class identity. The significance of such descriptions has implications for the \"modernity thesis\" debate, which pivots on the validity of generalizing Benjamin's claim\u2014that our \"mode of perception\" changes over historical periods\u2014to questions of film history and style. That is, can we claim that any film style of a particular era expressed the \"mode of perception\" of the time? This debate has often been preoccupied with details of film style during the early period, or with periodization, or with the question of whether cognitive structures really change, but too often the debate has overlooked the obvious: experts of the time described the cinematic experience as precisely an expression of the state of urban life and as a reflection of the modern mode of perception. Writing about film was a primary means by which modern problems (of representation, observation, education, aesthetic standards) were articulated. In other words, Benjamin's claim did not come out of a blue sky: writers in the 1910s and 1920s described film in exactly these terms, so trying to _explain_ those historical descriptions by constructing a homology between filmic style and modern urban experience\u2014without overgeneralizing to include all films at all times during this period\u2014is not only valid, but logical.\n\nSo descriptions of the cinematic experience in Germany before World War I often functioned as a means to articulate modern problems of representation, observation, education, or aesthetic standards. To describe the experience of attending a film screening was to work through questions associated with modern urban upheaval and transformation. It was so in many places during the early period, when cinema was not really \"cinema\" but a heterogeneous mix of screen practices and ideals\u2014one of many \"cultural series,\" in Andr\u00e9 Gaudreault's term. In Germany, however, the lines between expert and lay were perhaps more starkly drawn (hence more visible) than elsewhere. At the same time, the deeply ambivalent character of discourse during this time of transition, as Germany struggled to balance rapid modernization with traditional values, makes it an especially interesting case study for the mutual construction of expert observation and lay spectatorship. Historians of Wilhelmine Germany have long recognized the dichotomous yet strikingly fluid relationship between left and right political positions, progressive and reactionary attitudes toward the role of the state, male and female roles in public service, or middle- and working-class positions on the social hierarchy. The same ambivalence is evident in the reactions to new media forms such as motion pictures, which makes it an intriguing and challenging case from which we can perhaps infer broader principles about the mechanism of cultural legitimacy. Of course, arguments for and against the use of motion pictures in Germany at this time are similar, perhaps identical, to arguments for and against in other countries. Yet every agent marshals evidence and rhetorical weight from his or her native tradition in support of his or her argument; the Germans were no different. So this book straddles the line between a specialist study on early cinema or German culture and a broader investigation of the nature of expert appropriations of new technologies. The mixture of these two goals differentiates it sufficiently, I hope, from other works on the discourse on early cinema in Germany or elsewhere.\n\nThis project therefore takes up the dual challenge of describing _how_ experts cleared a cultural space for motion picture technology while explaining _why_ that work (rhetorical or practical) took the form it did. Both questions require not just primary research into specific appropriations but an understanding of the disciplinary expectations and needs faced by the experts. So each chapter will explore the _history_ of that discipline (such as biology) by also using the _method_ of a corresponding discipline (such as the history of science). Each chapter, then, is intended to be a hybrid between the methods of film studies and those of another discipline. Explaining why the work of appropriation took the form that it did also requires an understanding of the historical and cultural context of the experts, in this case imperial Germany. So each chapter also attempts to provide a social context, but because arguments for and against carried the weight of past traditions and values, a detailed philosophical context is sometimes required as well. For example, to understand fully the visceral reaction to film's perceived accelerated pace, we will need to examine that reaction's relationship to Schiller's temporal understanding of aesthetic experience, which was accepted by experts and the middle class implicitly. Over time, philosophical ideas sometimes spread and harden into ideological dogma.\n\nI should emphasize that this project focuses on the question of cultural legitimacy or the acceptance of a new media technology by a given discipline at a historical moment. It therefore concentrates on the _correspondence between_ film form and discipline, rather than the _impact_ of form _on_ that discipline. Each chapter charts the relationship between the material or form of cinema (which may include the celluloid image, the projected image, the apparatus, or the venue) and the logic of a discipline by demonstrating how experts conformed that material to their disciplinary agenda. Charting this relationship synchronically is, I believe, preliminary to surveying the diachronic impact of film on any given discipline's understanding of its object of study. That is, any determination of the influence of moving images on a discipline's agenda or conception of its object would require, in my opinion, an examination of the changes in the discipline over time. An accumulation of case studies on a particular topic in which motion pictures were vital would be necessary to determine the change in research questions and the extent to which moving images shaped those questions. The conclusion hints at what such a diachronic or longitudinal study might look like, but the chapters ahead focus instead on the criteria for acceptance (or refusal) of film within each expert group, rather than on any transformation in that group's thinking as a result of using film. I believe we need first to clarify the pertinent elements of film form and disciplinary logic and their potential points of contact before we can argue for the impact of film form on that logic, if any.\n\nChapter 1 examines the use of chronophotography and motion picture technology for scientific research in the fields of human motion studies, physics, and biology. After a survey of early scientific filmmaking, the chapter leverages Henri Bergson's critique of modern science's \"cinematographical thinking\" to underscore the philosophical affinity between film and scientific method. Given that \"science\" is just as varied as \"film,\" however, the chapter finds that different disciplines used motion picture technology for different reasons and in ways that emphasized film's different formal features. In these case studies, the filmic material consisted of the chronophotographic image, the projected image, and the technological apparatus, all of which were adjusted to conform to various elements and processes of scientific inquiry. Accordingly, the chapter examines how film became evidence in relation to the discipline's object of study, its theories, and its representational options. This chapter does not focus on observation but emphasizes experiment instead.\n\nChapter 2 continues this investigation of film for research purposes but extends the argument to the field of medicine. The chapter provides an overview of early medical filmmaking that emphasizes not the different uses by different specialties but instead the common functions governing the use of film technology, such as the use of film to explore, document, or educate. Any given film functioned multiply, depending on who was using it for what purpose at the time; this rubric allows us to glimpse research films in tandem with their other uses. The chapter then explores the relationship between the formal features of film technology and expert medical observation. The use of motion pictures as both still and moving images\u2014the way researchers described their viewing of and extraction of data from these images\u2014subtly expressed the researchers' mode of observation. In this case, the chapter explores the similarity between a \"medical way of thinking\"\u2014its disciplinary logic\u2014and how investigators took advantage of the unique features of film technology, especially the celluloid image and the projected image. It also underlines the ideological investment in that mode of viewing, which was a sign of training and disciplinary allegiance. That investment explains diagnoses of film spectators that accompanied the general medicalization of society during the Wilhelmine period. That physicians dismissed movies in their mass cultural incarnation but lauded their use in the laboratory is best explained through the dual, mutually dependent character of observation and spectatorship.\n\nChapter 2 functions as a pivot between the specialized, rarified use of motion pictures in the laboratory (which was more common than we might think) and the emergence of film in the public sphere. If physicians diagnosed the cultural ills of a nation, reformers of all sorts took to the streets to carry out the treatment. Chapter 3 surveys the film reform movement of the early period and its relationship to larger reform movements, especially those that emphasized a renewal of cultural priorities through education. In this case, then, the filmic material was the cinema venue itself as reformers and educators attempted to mold exhibitor and spectator habits to certain educational standards or aspirations. The art education movement was especially pertinent for efforts to \"ennoble\" cinema and its audiences through an aesthetic education. The history of German pedagogy, moreover, reveals a deep investment increasing over the nineteenth century in a mode of visual instruction known as _Anschauungsunterricht_ , which spread quickly across Europe and the United States, where it was known in English as \"the object lesson.\" Film reformers such as Hermann Lemke worked hard to fashion film screenings for educational use that would conform to this method of visual instruction and observational training. Reformers such as Hermann H\u00e4fker, on the other hand, created film presentations that attempted to conform to models of aesthetic contemplation, which were not incompatible with these educational and ideological goals.\n\nH\u00e4fker's efforts to create a film screening that would make any middle-class aesthete proud speak to the perceived importance of aesthetic values for unifying German culture and ameliorating its social ills. Chapter 4 argues that the explosion of writings about cinema around 1912, which Anton Kaes called the _Kino-Debatte_ , was not only about the danger cinema presented to literary values and territory, as has long been argued, but also about the relationship between the cinematic experience and aesthetic reception in general. How could film be considered an aesthetic object? What aspects of film viewing fit or not with established conceptions of aesthetic contemplation? It is something of a critical commonplace that the advent of mass reception, of which the filmic experience was emblematic, prompted a change in the standards of \"traditional aesthetics.\" This chapter examines the validity of that claim by demonstrating what was precisely at stake in aesthetic contemplation as it was usually understood, and what exactly about the cinematic experience presented a challenge. So this chapter divides the ideological components of aesthetic contemplation into four main areas\u2014issues of agency, identity, time, and space\u2014to show where the objections to and applause for film borrowed from \"traditional aesthetics\" and where they diverged. Furthermore, the chapter shows that professional aesthetics of the day\u2014as encapsulated in discussions of _Einf\u00fchlung_ , or emotional projection during the aesthetic experience\u2014was wrestling with some of the same issues that challenged traditional aesthetics, such as emotional projection, synesthesia, and movement, even while descriptions of the cinematic experience also touched upon these issues. The _Kino-Debatte_ , then, anticipated the theoretical proclamations of the 1920s and 1930s that declared contemplation inadequate for modern living. Chapter 4 explicates the precise nature of the change in preference from contemplation to distraction and suggests that the _Kino-Debatte_ was an important turning point for this transition. Chapter 4 therefore focuses on how film's projected image could be discursively managed to fit (or not) into the logic of aesthetic contemplation. Finally, the conclusion emphasizes the importance of disciplinary context and expert\/lay distinctions for histories of nontheatrical film and suggests a historiographic approach that highlights how experts \"handled\" motion picture images, technologies, and audiences\u2014and the impressions those efforts left on their own institutions and on the history of cinema.\n\nEach chapter functions more or less on its own, but together I hope they will add to the conversation about nontheatrical film, spectatorship, and interdisciplinary work in film and media studies. Indeed, each chapter demonstrates that the appropriation of motion pictures was a series of constant adjustments between writing and reading, or between inscription and reception. Looking at film in this way, we can begin to explore how cinema fit into a cultural space, the work that went into making it fit, and what that work tells us about the people and institutions who did the work. In other words, how experts managed to call film their own while using it to manage the scientific and social problems they faced is the subject of this book.\n\nSCIENCE'S CINEMATIC METHOD\n\nMOTION PICTURES AND SCIENTIFIC RESEARCH\n\nTime would flee, I subdue it.\n\n\u2014CHARLES CROS (1877)\n\nIn the early 1890s, after \u00c9tienne-Jules Marey's successes with chronophotography ensured the continued viability of his \"graphic method\" of recording motion, two German physiologists, Wilhelm Braune and Otto Fischer, set out to improve upon Marey's techniques for the study of human locomotion. No mere dilettantes, Braune and Fischer were already well known in their field for studies of the human center of gravity and other investigations into the fundamentals of human motion. From 1889 to 1904 they published a series of studies of human locomotion, culminating in their landmark work, _Der Gang des Menschen_ (The Human Gait). In their Leipzig laboratory, they dressed their experimental subject, a military recruit, in a black jersey and attached to him an elaborate mechanical scaffolding of incandescent tubes designed to illuminate his stride with short, intensely bright bursts of light (fig. 1.1). They then photographed the subject as he walked, as comfortably as could be expected, across the darkened room. As he walked, the tubes fired, producing a strobe effect that was recorded through the camera's open shutter. The resulting image\u2014a series of white lines across a black background\u2014became the basis for hundreds of calculations concerning the specific mechanics of human motion (fig. 1.4). Interested in the economy of the laboring body\u2014its most efficient conservation and expenditure of energy\u2014they hoped their scientific analysis of human movement would lead eventually to a more efficient and productive society (or, at least, a more efficient soldier). Likewise, their improvements over Marey's system of photographic measurement led to a much more analyzable and therefore \"productive\" chronophotographic image. After Braune and Fischer, photographs became even more efficient and productive tools of scientific research.\n\nIn Marburg in 1907, a young physicist named Max Seddig presented his dissertation on \"Measurement of the Temperature Dependency of Brownian Motion.\" In an attempt to clarify the arguments for and against the atomic\u2013kinetic model of heat, while also trying to provide experimental confirmation of Albert Einstein's theories of Brownian motion, Seddig fashioned a device that could record chronophotographic images of microscopic particles affected by molecular activity. Seddig's microscope\u2013cinematograph combination supplied an objective record of Brownian motion from which he could calculate the velocity of these particles. Yet it was not the objective record that Seddig emphasized and praised but the machine's ability to measure time intervals precisely. Alternatively, Heidelberg biologist Hermann Braus presented in 1911 the results of his application of a similar microscope\u2013cinematograph combination to record and explore the growth of a tissue culture of a frog's heart. Using time-lapse cinematography, Braus sought to demonstrate that the culture actually grew rather than merely survived outside the organism. Unlike Seddig, he was not so concerned with measurement but with the temporal record of the event. With his motion picture record Braus was able to challenge competing claims about the growth of nerve cells.\n\nThese three cases\u2014from human motion studies, physics, and biology, respectively\u2014represent a fair sampling of the scientific use of film for research purposes in Germany before 1914. From these case studies we can draw some preliminary conclusions about the practical and philosophical connections between film and science. We might also be tempted, of course, to make general theoretical claims about the nature of cinema's relationship to modern science and temporality. But if we are to understand film's role in modern scientific inquiry, we must temper expansive claims with more historically localized analyses of how film technology was actually _used_. What \"film\" meant to any given scientist depended very little on a theoretical conception of cinema in general; instead it depended primarily on how some specific incarnations of motion picture technology could be applied to the specific and historically contingent problems the scientist faced in his or her discipline. Hence any study of scientific research films must incorporate methods common to both the history of science and film history by investigating the manner in which research questions and media technology mutually influenced each other. What questions does a given discipline privilege at a given time? How do those questions shape\u2014indeed, how are they shaped by\u2014experimental design and available instruments? The appropriation of motion picture technology as a scientific research tool, its specific use within the laboratory, reveals the researcher's assumptions about that which the camera is designed to capture. And these assumptions vary as widely as the different uses of film. But rather than making broad claims about what \"science\" or \"cinema\" is, thereby concealing this variety under top-down theoretical categories, we should let individual experiments reveal their assumptions and make our generalizations from those, if necessary. Design and deployment are themselves implicit theoretical statements.\n\nIf the cinematograph were an especially flexible tool that could be adapted to a number of different needs, these needs existed in the first place because of changes in the sciences themselves at the turn of the century. Biologists interested in cell development, for example, searched for new modes of visualization that would allow researchers a clear view of movement in time to resolve some heated disputes about the nature of cell growth; the techniques of tissue culture, on one hand, and those of motion pictures, on the other, offered two kinds of solutions, as we shall see. Physicists, too, looked to cinematography as a tool for investigating previously invisible phenomena, such as the effects of molecular movement, a topic of debates about the behavior of atomic phenomena. Seddig's use of chronophotography and motion pictures reflects a common application of this technology in scientific research; while scientists admired cinematography's ability to capture fleeting phenomena, they prized most highly film's ability to decompose the event into discrete, regular units, which permitted measurement of its temporal and spatial components. Indeed, this particular use of motion picture technology, especially in the physical sciences, betrays an assumption about the event or phenomenon under study as itself discrete, divisible, determined by classical laws, regular, and\u2014just like the filmstrip\u2014reversible. Seddig's use of motion pictures therefore demonstrates at once the value of film for scientific research and his (and Einstein's) commitment to a particular understanding of the relationships between movement, time, and space\u2014an understanding that French philosopher Henri Bergson was criticizing at precisely this moment in _Creative Evolution_ , his 1907 landmark study of the philosophy of biology. In short, as scientific disciplines reconceptualized the nature of matter, time, space, and the organism, they seized upon tools that could visualize these phenomena in accordance with these new concepts.\n\nBut film has never been just a convenient device, patiently waiting on the shelf as the scientist thinks up a new use for it as a solution to a new problem. Its availability and its existence also generated questions. Investigators' interest in temporal phenomena was in part spurred by the arrival of a machine that could make the phenomena amenable for analysis. Furthermore, those scientific research programs that featured film technology were not only shaped by that technology, but their scientific method itself had certain features in common with the filmic apparatus. Scientific experiment shares with motion pictures an impulse to record immediately and directly, a willingness to manipulate time, and an inclination to isolate and quantify phenomena. There are good practical reasons for using motion picture technology, of course. But in the late nineteenth century there developed also a _philosophical_ affinity between science and film that went beyond mere convenience. Cinema and science have come to share a certain way of thinking, so the history of the application of motion pictures in science can offer us a valuable opportunity to explore the relationship between science and modernity. Bergson was the first to suggest that science, through its parsing of continuous movement into discontinuous moments, proceeds in a way analogous to cinematography. If we are to understand fully the implications of the appropriation of motion picture technology by the scientific community, we must maintain a balance between the theoretical and the practical by considering this philosophical affinity alongside the way investigators put film to work in the laboratory. Bergson's conception of science as inherently \"cinematic\" offers us a logical point of departure for such considerations. True enough, Bergson was not as popular in Germany as he was in France and the United States. But because I am interested in his thoughts on cinema and science in general\u2014and not in his thoughts (if any) on particular applications in Germany or elsewhere\u2014his historical impact on German culture is actually not relevant to my approach. Bergson's theoretical framework can help us answer the question \"What did cinema and science see in each other at this particular moment?\" The individual case studies can reveal _how_ film was used; reading Bergson alongside these cases can help to reveal _why_ film was used. This chapter, then, will survey the use of motion pictures in the three case studies mentioned above.\n\nLet me first reiterate the role of this chapter in the book's overall goals. As I mentioned in the introduction, the larger argument (the \"general theory,\" if I may) concerns the correspondence and mutual accommodation between the logic of a discipline\u2014its problem-solving patterns, its investigatory methods, its ideological assumptions\u2014and film's formal characteristics. This implies not just taking advantage of a medium-specific formal feature, such as temporal malleability (for example, time-lapse cinematography), to solve a representational problem, it also implies a functional or productive homology between this formal feature (for example, the linear regularity of time-lapse recording as a statistical sample of a larger unit of time) and a way of solving problems (the primacy of mathematics, for example) or of viewing the world (as naturally divisible into equal, regular units, for example). This match\u2014between the formal features of the representational technology and the investigatory presumptions\u2014matters, because it provides the researcher, community, or discipline with the reassuring sense that the tool will fit the task to which it is assigned. However, we must note that the match is ideally never perfect; otherwise there would be no new information. Experimental systems are designed to generate new questions, so there should be a dislocation or displacement between the more or less ideological assurance of this tool's \"worldview,\" so to speak, and the strangeness of the view it provides. Time-lapse cinematography can, for example, confirm an understanding of nature as regular and divisible, but it also offers a surprising, even thrilling new image that prompts new questions. The larger argument of this book is that the acceptance of any new (media) technology depends in part on this correspondence between some set of its formal features and the logic of the discipline. The specific argument (the \"special theory\") is that expert vision expresses this disciplinary logic especially well and that film's legitimacy within disciplines depended on its accommodation to expert modes of viewing.\n\nBut expert viewing is not the only way that disciplinary logic is expressed; the experimental system itself is also a manifestation of the discipline's problem-solving patterns and theoretical presumptions. As Gaston Bachelard and others have argued, instruments are \"theories materialized\": the design and deployment of experimental technology carries a preconception or preunderstanding of the phenomenon it is designed to isolate. If we extend this system to include chronophotography or motion picture technology, we can see how the work of accommodating their formal features already made a statement about the relationship between the system and the object of study. Indeed, the work of creating a legible image\u2014that is, understandable and acceptable to the discipline\u2014reveals this relationship between system and object quite clearly. Likewise, the theory guiding the experimental observations is an expression of disciplinary logic. If instruments are theories materialized, then, as Hans-J\u00f6rg Rheinberger has noted, theories are also \"machines idealized.\" Einstein's theory of Brownian motion, for example, made several \"shortcuts\"\u2014including the presumption of molecular two-dimensionality and velocity without direction, as we shall see\u2014to manage the object mathematically and accommodate the phenomenon to both experimental confirmation and a particular disciplinary understanding of nature. Einstein's mathematics, in other words, simultaneously became an \"instrument\" to guide observation _and_ a theoretical rendering of the experimental situation. To offer one more example, disciplinary logic can be expressed through the experimental system's representational options. Film is only one part of an experimental system, of course, and only one in a range of representational tools that includes writings, sketches, graphs, and photographs. Selecting film as part of this media ensemble already implied a certain set of questions or needs. Hermann Braus, for example, found that using film was an especially powerful means of engendering belief among colleagues, while at the same time expressing, through its formal features (such as the duration of the projected image and its temporally forward motion) theoretical presumptions about cell growth. In this case, film and the new technology of tissue culture\u2014also a representational tool\u2014combined to express a new direction in the discipline's visual needs and strategies.\n\nSo the correspondence and accommodation between experimental system or discipline and technology can happen in several ways, depending on the researcher's or discipline's goals and needs, which are locally and historically determined. This chapter will explore the \"general theory\" rather than the \"special theory\" of media technology's disciplinary legitimacy. Specifically, this chapter will focus on three ways in which experimental systems incorporated chronophotographic or motion picture technology, especially on the correspondence and mutual accommodation between a given set of formal characteristics (of photography and\/or film) and (1) an _object of study_ , (2) a _theory_ , and (3) the _representational options_ of an experimental system and discipline. Or, to put it another way, each of these three adaptations required a certain kind and amount of work to make the chronophotographic or filmic image into evidence. A close analysis of Braune and Fischer's method will show how they created evidence in relation to their object of study (in this case, the human body). The discussion of Seddig's experiment stresses the creation of evidence in relation to a theory (Einstein's theory of Brownian motion). And the section on Braus emphasizes the creation of evidence in relation to a set of representational options within a discipline (here, cell biology). Each example deals to some extent with all three adaptations, of course, because they are intertwined, but the emphasis changes. In general, I will demonstrate that while the scientific community readily accepted chronophotography and film as valid instruments, investigators still had to perform considerable labor to adapt these devices to their tasks and to transform the resulting images into acceptable scientific evidence. Motion pictures may have allowed scientists to manage time and movement, but researchers first needed to manage film's temporal rush and excessive detail. The nature of this work was shaped by its historical context. The subject of this chapter is therefore the way that investigators were continuously obliged to adapt as they juggled chronophotographic or motion picture technology, the needs of the individual experiment, the theories shaping the experiment, and the discipline's priorities shaping the representational choices.\n\nWhile observation is not an explicit focus of this chapter, it is inescapable. Braune and Fischer decomposed movement to train expert vision to phenomena that it might not see or have been able to see. Einstein's theory of Brownian motion focused experimenters' attention, showing them what to look for. For Braus, film was an observational tool that forced researchers to abandon previous theories and modes of analysis (which emphasized discontinuity) for an approach that emphasized synthesis and continuity. These case studies demonstrate that, as Ian Hacking has noted, experiment and observation are only heuristically separable. Nevertheless, this chapter emphasizes other ways, besides observation, that disciplines accommodated the formal features of film and chronophotography. Expert vision is the explicit focus of the following chapters.\n\nThe chapter has five sections. It starts with a general overview of scientific research films (as opposed to popular science films) before World War I, in which I outline various prominent applications as well as some hypotheses about how motion pictures fit with the rhetorical goals of scientific enterprise. (A majority of scientific applications of film during the early period were devoted to various medical fields, which I will explore separately in chapter 2.) The next section continues to explore why motion picture technology was intriguing to researchers; it focuses on Bergson's discussion in _Creative Evolution_ of the relationship between cinema and science. In the third section, I place Braune and Fischer's work on human locomotion in the context of both social modernity and the science of work. This section contains a detailed explication of Braune and Fischer's method to show exactly how the merging of apparatus, image, and object actually occurs. The fourth section relates Seddig's work to the general problems of atomistic physics at the turn of the century. In the fifth section, I sketch Braus's cinematic contribution within the context of fin de si\u00e8cle changes in the discipline of cell biology. The concluding paragraphs of this chapter compare the different cases directly to emphasize the mutual dependence between motion picture technology and emerging scientific agendas.\n\nEARLY SCIENTIFIC FILMMAKING: AN OVERVIEW\n\nScientists from a wide variety of disciplines\u2014including but not limited to botany, military engineering, meteorology, neurology, psychology, and medicine, as well as the three already mentioned\u2014were among the earliest users of motion picture technology. Most histories of research films start with Jules Janssen, Eadweard Muybridge, and \u00c9tienne-Jules Marey, who were important transitional figures between scientific photography and motion pictures. The initial adoption of moving images was relatively smooth because so much work had already gone into creating scientifically valid photographic images during the middle decades of the nineteenth century. The slow process by which still photography had been standardized\u2014setting norms for emulsions, exposure times, preparation techniques, image interpretation, and so on\u2014meant that the enthusiasm for photography often collided with rapidly evolving disciplinary requirements for scientific documentation. As Jennifer Tucker and others have shown, photography was not immediately and unconditionally accepted as a completely objective and scientific record. Photography's evidentiary status depended on its ability to meet several criteria of production and reception. Photographs had to withstand cross-examination by experts from any discipline to which they wished to testify; they were as subject to expert judgment as drawings or other illustrations. In Germany, for example, microphotographs were not generally accepted as proper evidence by the scientific community until respected bacteriologist Robert Koch's innovations and rhetorical efforts made \"reading\" photographs of bacteria a truly collective activity among an international group of scientists and physicians. Yet by the 1890s, the protocols for generating acceptable scientific photographs had been established in most disciplines, so the innovation of motion pictures was greeted enthusiastically, their way smoothed by Marey's renowned chronophotographs and graphic inscription methods.\n\nMarey is indeed the most important figure in any history of early scientific film, because his efforts and resources shaped the application of motion pictures to scientific experiment. Marey was intrigued by Muybridge's serial photography when he first encountered it in 1881 but ultimately disappointed in its scientific value: Muybridge's 24-camera, trip-wire method of recording was prone to mechanical inaccuracy and incapable of managing the exact time intervals required for careful research. So at the Coll\u00e8ge de France in the 1880s and 1890s Marey explored photographic and chronophotographic methods for visualizations of movement that accounted for distances and times more precisely. The Institut Marey was founded in 1901 to carry on his work; researchers such as Lucien Bull, Pierre Nogu\u00e8s, and Joachim-L\u00e9on Carvallo continued to explore the relationship between experiment and visualization, especially in the areas of slow and high-speed cinematography and X-ray cinematography. Marey's assistant at the Coll\u00e8ge de France, Charles \u00c9mile Fran\u00e7ois-Franck, continued his work on microcinematography in particular; collaborating with Lucienne Chevroton, Fred Vl\u00e8s, and others, Fran\u00e7ois-Franck published widely on the chronophotographic and cinematographic recording of microscopic and macroscopic movement. These two sites were also magnets for individual researchers searching for novel ways to make visible their objects of study; French biologist Antoine Pizon and Swiss biologist Julius Ries both worked with the team at the Institut Marey to capture visually the process of cell division, for example, while Fran\u00e7ois-Franck helped French physicist Victor Henri with his research into Brownian motion (I will have more to say about all of these examples later in the chapter).\n\nWhether scientific cinema grew out of team efforts focusing on new visualization techniques or out of the work of individual researchers focused on experimental problems that motion pictures might solve, both models had one thing in common: the need for resources. Needless to say, the early motion picture apparatus was expensive, often cumbersome, and usually difficult to adapt. Its use in scientific circles, therefore, was generally restricted to established researchers who had the necessary financial and technical resources at their disposal. Indeed, the distribution of resources largely dictated the spread of motion picture technology in the laboratory. Thanks to Marey's considerable political and scientific skill, France could boast at least two sites with the necessary budget and expertise to pursue such a program. Germany also had a university research infrastructure in place that allowed individual researchers to explore the use of motion picture technology, but no single site dominated. Aside from the many medical applications, which we will examine closely in the next chapter, we can count botanist Wilhelm Pfeffer's time-lapse chronophotography of plant growth at Leipzig University and Carl Cranz's high-speed studies of ballistics at the military academy in Berlin-Charlottenburg as notable intersections of science and film at the university level.\n\nResearch film also received a boost from manufacturers who recognized that lending their equipment and funds provided not only a measure of legitimacy and good press but entertainment for movie-going audiences as well. Path\u00e9 Fr\u00e9r\u00e8s, for example, funded the filmmaking of microcinematographer Jean Comandon and then distributed his films to theaters around the world. Occasionally a German manufacturer would lend a hand to researchers; film pioneer Oskar Messter, for example, had pretensions in this arena, and companies such as Ernemann were sometimes acknowledged as technical patrons. But even researchers who purchased the basic apparatus from a manufacturer such as Lumi\u00e8re or Ernemann were still obliged to make modifications to the equipment in their own laboratory setting. Carl Zeiss's optical laboratory in Jena, for example, seems to have been especially suited to this kind of work. Generally speaking, despite the involvement of some film manufacturers, these films were usually made by specialists to be shown to other specialists at scientific conferences or in the lecture hall.\n\nThis is not to say, however, that scientific films were limited exclusively to an elite audience. Scientific titles were often part of the program at the local movie theater. Companies such as Path\u00e9, Gaumont, and \u00c9clair in France or Urban in England, especially, produced their own series of scientific films for general audiences. German audiences would have been familiar with this genre from the titles imported by these companies. (Foreign manufacturers generally dominated the German exhibition market until the 1910s.) In addition, the German periodical _Film und Lichtbild_ (a German _Popular Science_ for film enthusiasts) acted as something of a clearinghouse and ad hoc distribution company by publishing lists of scientific films and offering discounts to its readers. There were also dedicated screenings occasionally, as well as theaters devoted to the genre, such as the Fata Morgana theater in Dresden, which opened in August 1912 and showed only scientific, industrial, and nonfiction films.\n\nBut what of the research films of a Seddig or a Braus? Did they ever find their way to the public? It is very difficult to say without a complete survey and correlation of all films made in the laboratory with those screened in public or semipublic venues\u2014a task that is likely impossible to complete. There are good reasons for both possible answers, yes and no. On one hand, these films were the result of considerable expense and effort, so investigators might have been reluctant to give them up for duplication. (Comandon and others like him were exceptions, because they made legitimate research films for hire.) Also, some scientists might have hesitated to cross the line between serious research and its popularization. Motion pictures already had acquired the yellow tinge of mass culture, so some investigators were probably reluctant to have their work completely jaundiced by a matinee showing at the local nickelodeon. Not that the scientific application of motion pictures encountered serious objection; while film histories often repeat legends of academic hostility to researchers using motion pictures in their experiments (usually limited to French medical films and the Doyen controversy, described in chapter 2), evidence of such protests is actually rare in comparison with the general enthusiasm displayed for the new technology. On the other hand, research films were the topic of a considerable number of screenings and discussions, and teachers, reformers, and even scientists encouraged these ventures into the popular realm as important efforts at public outreach. Furthermore, there were already a number of screenings of these films in the semipublic realm of the university lecture, so it would not have been too much of a leap to take the next step. And as manufacturers such as Messter and Ernemann lent their equipment and expertise to researchers, they may have asked for copies of the films, which might have been made available for rental in the manufacturer's catalog. It seems that the public screening of any given laboratory film depended on the resources and predilections of the individual researcher or the manufacturer; I have not yet found a consistent conduit in Germany between the scientific laboratory and the movie theater.\n\nBy and large, then, these films served primarily as a form of scientific evidence. Despite its mass culture connotations, its high cost, the high level of technical expertise required, and the often futile results, motion picture technology offered several tempting benefits to the researcher. Like still photography, the motion picture camera provided a mechanical, automatic, hence \"objective\" record, thereby adding substantial evidentiary weight to scientific claims. The photographic image, like other graphic inscription devices (such as the electrocardiograph), seemed to provide researchers with an unmediated and permanent record of any given phenomenon, one that could be stored and disseminated with ease. And because it could be projected and reproduced, the photographic image proved useful for demonstrations as well as experiments; indeed, a motion picture projector was just as likely to be found in a lecture hall as in a laboratory.\n\nUnlike still photography, however, the motion picture had the capacity to record events as they occurred over time. This singular feature offered several advantages. The camera itself could act as a mechanical, indefatigable prosthesis of the scientist's eye, a tireless observer of events that was able to catch the slightest change without the interruption of a blink. Furthermore, motion pictures offered the scientist the option of manipulating time by recording (or projecting) the film at different speeds. Slow-moving events could be sped up with time-lapse techniques; fast-moving phenomena could be slowed down through high-speed cinematography. As a result, temporal events invisible to our ordinary perception became \"visible\"; film became a kind of temporal microscope or telescope, bringing nature's aloof wonders closer to our level. Finally, as implied above, the motion picture camera could also act as a precise measuring tool. By controlling the rate at which the film passed through the camera as the phenomenon traveled a set distance, the scientist could then calculate the speed of the recorded movement. This ability was by far the most intriguing aspect of cinema's scientific potential, and researchers spent considerable energy perfecting it.\n\nMotion picture technology, then, was an especially flexible tool that could be adapted to a number of different tasks. In this respect, however, it is no different than any number of technical innovations adapted for scientific use, from perspective drawing to the computer. Successful adaptation depends on a variety of circumstances, but as sociologist of science Bruno Latour has argued, the most salient predictor of which technologies the scientific community will adopt is the extent to which the adoption will aid the community in rhetorical struggles. Latour maintains that technologies that serve as inscription devices, or \"writing and imaging procedures,\" function rhetorically in debates between authors and groups as tools that help \"in the mustering, the presentation, the increase, the effective alignment, or ensuring the fidelity of new allies.\" Struggles between scientists are the same as those between generals and politicians, Latour argues; those with the most allies win. Therefore, the essential characteristics of any inscription device\u2014the qualities that will ensure its success in the scientific arena\u2014have less to do with its ability to provide accurate inscription (visualization, writing) per se than whether those properties can be put to use in rhetorical struggles. Latour explains:\n\nIn other words, it is not _perception_ which is at stake in this problem of visualization and cognition. New inscriptions, and new ways of perceiving them, are the results of something deeper. If you wish to go out of _your_ way and come back heavily equipped so as to force others to go out of _their_ ways, the main problem to solve is that of _mobilization_. You have to go and to come back _with_ the \"things\" if your moves are not to be wasted. But the \"things\" have to be able to withstand the return trip without withering away. Further requirements: the \"things\" you gathered and displaced have to be presentable all at once to those you want to convince and who did not go there. In sum, you have to invent objects which have the properties of being _mobile_ but also _immutable_ , _presentable_ , _readable_ , and _combinable_ with one another.\n\nMotion picture technology had all of the qualities of this sort of immutable mobile, a good indicator of its success as a scientific instrument. The instrument itself was mobile, but more importantly, so were the films. \"Immutability,\" in Latour's sense, refers to the permanency of both the inscription process and the object or condition represented. Simply, the inscription must be relatively permanent, as films were, while providing a translation without any seeming corruption of the thing represented. The photographic image's necessary physical connection (via light and chemical processes) to the object represented served to guarantee that the object was relatively uncorrupted by the recording process. The films were meant to be projected, so they were of course presentable, but because they were also photographs, they could be presented in a wide variety of ways, as illustrations in journal articles or as lecture slides, for example. In this way, the films could also be combined with other technologies, such as print technology, but the apparatus itself could be combined with others as well, such as the microscope. The \"readability\" or legibility of the technology is the most contentious aspect of any innovation, because the interpretation of new forms of inscription always requires negotiation within a discipline. Experts and innovators haggle over the meaning of signs until standards of production and protocols of interpretation emerge. Generally speaking, however, motion pictures, like photography before them, were considered very legible for scientific purposes.\n\nBERGSON, CINEMA, AND SCIENCE\n\nWithout a doubt, motion pictures presented scientists with new analytic techniques that could manipulate time and space. However, at the same time, science's very mode of analysis was newly subject to debate. At least since the romantics, some German thinkers had come to associate science's empirical, experimental, and materialist method with a cold, mechanistic, and \"disenchanted\" view of the world. As Anne Harrington has argued, the rebellion against mechanistic, reductionist science often consisted of calls for \"wholeness\" in various forms, from Hans Driesch's biological vitalism to Gestalt psychology to Richard Wagner's _Gesamtkunstwerk_. This reaction was not limited to Germany, of course; perhaps the most prominent figurehead of this rebellion was Henri Bergson, whose philosophy\u2014or its popularization\u2014spread like wildfire through the parlors and lecture halls of Europe and the United States in the years before World War I. If film was still a relatively marginal research technology at this time, Bergson's _Creative Evolution_ brought it to the center of the debates about scientific method.\n\n_Creative Evolution_ , which appeared in 1907 and quickly became Bergson's most famous work, was concerned with evolutionary biology in the same manner that his earlier _Matter and Memory_ (1896) was concerned with psychology and his later _Duration and Simultaneity_ (1922) dealt with physics. In _Creative Evolution_ , he took the evolution of life as a fact, but expressed dissatisfaction with scientific explanations of it. According to Bergson, the mechanistic approach to biology perpetuates a common analytic mistake: it divides up organisms and living processes in order to understand their parts, but does not, because of this division, fully understand their total, living reality. Consequently, the analytic, mechanistic, reductionist approach cannot satisfactorily explain change or the creation of new forms or new solutions. Bergson was intent to bind the organism's \"living reality\" to the flow of time. That is, Bergson made the flow of time, which we all experience as incessant and forward moving, the model of life itself. The central concept here is his notion of _dur\u00e9e_ , or \"duration.\" Bergson insisted that we make a category mistake when we divide the continuity of lived experience into the discontinuity of a series of separate points. Bergson reversed the traditional or commonsensical view that change is a succession of states, a view that makes those states logically anterior to change. Bergson's view, instead, was that change is primary and that to analyze change by breaking it into a succession of states misrepresents the true reality of the world, which exists in constant flux, \"becoming,\" or _dur\u00e9e_. Organisms exist in time, in _dur\u00e9e_ , and cannot be separated from it without losing an understanding of their lived reality.\n\nThis same logic applies to the examination of motion. Bergson saw the universe in a constant state of flux; change and movement are the only constants, the only true reality. Matter, form, or solidity are only stable views of this essential instability. Unfortunately, our common, ordinary, analytic perception cannot grasp this flux; it can only extract determinate moments, which we then mistake for an accurate picture of reality. Even our body is not solid but \"is changing form at every moment; or rather, there is no form, since form is immobile and the reality is movement. What is real is the continual _change of_ form: _form is only a snapshot view of a transition_.... [Therefore,] our perception manages to solidify into discontinuous images the fluid continuity of the real\" (328, emphasis in original). It is not so much that this mode of perception is wrong, it is just that it is incomplete and should not be mistaken for a true understanding of the world. Bergson acknowledged that human beings cannot help but think this way, dividing the flow of the world and time into discrete intervals, but he insisted that this habit of thought should be overcome: \"to invert the habitual direction of the work of thought.\" That is, given that analysis or this way of viewing the world is more or less habitual and necessary, Bergson declared that to recognize _dur\u00e9e_ , \"The mind must do violence to itself, has to reverse the direction of the operation by which it habitually thinks, has perpetually to revise, or rather to recast, all its categories.\" Bergson's philosophy, then, was an attempt to save us from our logical errors of thought so that we could see the world in a different way. Specifically, he argued against two category mistakes: mistaking discontinuity for continuity and space for time.\n\nBergson used the example of motion pictures, which consist of a series of still images projected to imitate real movement, to illustrate the relationship between the continuity of _dur\u00e9e_ and the discontinuity of our ordinary, analytic perception. The mechanism of cinema was analogous to, even expressed, our ordinary perceptual process. This process, Bergson elaborated,\n\nconsists in extracting from all the movements peculiar to all the figures an impersonal movement abstract and simple, _movement in general_ , so to speak: we put this into the apparatus, and we reconstitute the individuality of each particular movement by combining this nameless movement with the personal attitudes. Such is the contrivance of the cinematograph. And such is also that of our knowledge.... Whether we would think becoming, or express it, or even perceive it, we hardly do anything else than set a kind of cinematograph inside us. We may therefore sum up what we have been saying in the conclusion that the _mechanism of our ordinary knowledge is of a cinematographical kind_. (332, emphasis in original)\n\nWhat does it mean to \"set a kind of cinematograph inside us\"? It means that when we try to think movement or change, we cannot help but to conceive it first as a series of individual states of being or \"snapshots\" of form. The accumulation of these \"snapshots\" provides us with a conception of movement, a conception as illusory as the movement of the cinematic image at twenty-four frames per second. Thinking of movement or change as a series of states\u2014rather than as a thing in itself, as Bergson urged us to do\u2014forces us to extrapolate movement from the series. With that extrapolation, movement becomes universal, ideal, or \"movement in general.\" We attempt to reconstitute the particularity of the movement by an effort of synthesis, which is what the accumulation of snapshots amounts to. Out of habit, our ordinary way of thinking is cinematic.\n\nIn the same way, modern science \"proceeds according to the cinematographical method,\" in that \"it is the essence of science to handle _signs_ , which it substitutes for the objects themselves\" (357). These signs \"denote a fixed aspect of the reality under an arrested form. To think movement, a constantly renewed effort of the mind is necessary. Signs are made to dispense us with this effort by substituting, for the moving continuity of things, an artificial reconstruction which is its equivalent in practice and has the advantage of being easily handled\" (ibid.). According to Bergson, science is not _really_ able to understand movement per se, movement as continual flux. Yes, it can understand it as the difference between the changes of two stable states, but it cannot grasp the essential dynamism of change. Instead, the scientific approach creates abstractions\u2014the geometric understanding of movement as a line or the mathematical understanding of movement as an equation\u2014to help us grasp that which is in constant motion. For Bergson, the normal scientific approach to movement proceeds by leaps from moment to moment, from arrangement to rearrangement; science may increase the number of moments it isolates \"but it always isolates moments.... It does not bear on the interval, but only on extremities\" (357\u2013358). To the extent that science followed this tendency to divide movement into stable units and to treat the interval as an enabling ellipsis and not as precisely the problem, modern science for Bergson was essentially cinematic.\n\nBergson characterized the history of science in these terms as well. Aristotelian science was also cinematic, but it differed from modern science in the way it broke up time. For the ancients, the time of the movement of a falling body, for example, had certain determinate periods, whose natural articulations, like puberty, presented moments when there occurred the natural release of a new form. For modern science, according to Bergson, time \"has no natural articulations. We can, we ought to, divide it as we please. All moments count. None of them has the right to set itself up as a moment that represents or dominates the others. And, consequently, we know a change only when we are able to determine what it is about at any one of its moments\" (360). Time, then, has become democratic in the modern age; there are no more privileged moments. This democratization marks the difference between the _qualitative_ description of the ancients and the _quantitative_ measurement of the moderns (361). Modern science works only with a view to measure, and it selects as its objects only those phenomena that can be measured. \"It retains only the events or systems of events that can be thus isolated without being made to undergo too profound a deformation, because only these lend themselves to the application of its method. Our physics dates from the day when it was known how to isolate such systems\" (371\u2013372). In the same way, motion picture technology divides temporal events evenly, into a neat twenty-four frames per second, for example. The precise and democratic division of time common to the cinematic apparatus mimics modern science's insistence that no moment be privileged over another.\n\nIn this way, Bergson outlined the deep affinity between cinema and science, one that at least partially explains their immediate mutual attraction. But if Bergson condemned our \"ordinary perception\" and therefore cinema (and scientific method) as incomplete\u2014thereby joining the rebellion against mechanistic science\u2014he nevertheless recognized that cinema had potential beyond its analytic character. In an interview from 1914, Bergson regarded motion pictures somewhat more sympathetically:\n\nSeveral years ago, I went to the cinema. I saw it at its origins. Obviously, this invention, a complement to instant photography, can suggest new ideas to a philosopher. It could be an aid to the synthesis of memory, or even of thought. If the circumference [of a circle] is composed of a series of points, memory is, like cinema, a series of images. Immobile, it is in a neutral state; in movement, it is life itself.\n\nBergson recognized that cinema had the capacity to provide both analysis and synthesis, that it had an inherently ambiguous character precisely because it is both still and moving. Indeed, Bergson's invocation of the cinematographic apparatus was never an outright condemnation of the cinema but instead a way of describing a tendency in our habitual, everyday way of thinking that science had codified into experimental method. If the point of his philosophical project was to \"reverse\" this way of thinking toward a truer, less alienating view of the world\u2014toward an embrace of \"intuition\"\u2014then there was also the possibility that cinema could somehow participate in that reversal. In other words, if modernity\u2014\"the alienating, blinding experience of the age of large-scale industrialism\" to which, Walter Benjamin claims, Bergson's work responded\u2014exacerbated this alienating perception, cinema's analytic character aligned it with that alienation, while its synthetic nature associated it with authentic experience. Cinema was both a social irritant and amelioration. In other words, the Bergsonian tension between continuity and discontinuity, between unity and fragmentation, was an expression of \"the alienating, blinding experience of the age of large-scale industrialism,\" but it was also a tension expressed in the very form of film itself. The aspect of cinema that made it a significant tool for scientific analysis existed alongside the aspect of cinema that made it useful for philosophical thinking.\n\nWhile Bergson's philosophy struck a chord throughout Europe and the United States, its resonance was muted in Germany, where the reception of his work was not quite as overwhelming. Nevertheless, many German thinkers struck notes in much the same key. Wilhelm Dilthey, for example, argued that positivist philosophy and science alone could not grasp the whole of life's rich variety: \"The basic conception of my philosophy is that up to now no one has put whole, full, and unmutilated experience at the basis of philosophizing, that is to say, the whole and full reality.\" According to Dilthey, this \"whole and full reality\" could not be fully understood using the \"mutilating\" knives of reductionist analysis and causal explanation. So he proposed his famous division between the physical sciences, which rely on causal explanation, and the human sciences, which derive their insights from empathetic or hermeneutic understanding: \"The basis of the human studies is not conceptualization but total awareness of a mental state and its reconstruction based on empathy.\" This recalls Bergson's conclusion about the incompleteness of the scientific enterprise:\n\nIt seems then that, parallel to this physics, a second kind of knowledge ought to have grown up, which could have retained what physics allowed to escape. On the flux itself of duration, science neither would nor could lay hold, bound as it was to the cinematographical method. This second kind of knowledge would have set the cinematographical method aside. It would have called upon the mind to renounce its most cherished habits. It is within becoming that it would have transported us by an effort of sympathy. (372)\n\nIn other words, both philosophers were interested in opposing the machine of scientific method with the wholeness offered by a certain intuitive, empathetic understanding. Cinema, via Bergson, stood squarely in the middle of these debates about scientific method in fin de si\u00e8cle Germany. Its inherently ambiguous character was reflected in the different ways scientists used it in their experiments and in the different ends to which it was employed. It is true that Bergson in _Creative Evolution_ emphasized only the analytic potential of motion picture frames\u2014that is, the use of celluloid as a \"mutilating\" dissection knife trained on living movement. Yet scientists in their actual use of film also effectively employed the _projected_ film as an end in itself, revealing what Bergson would later recognize (and Gilles Deleuze would still later describe) as cinema's and science's capacities for synthesis within a persuasive image of movement itself. In the rest of this chapter, I will outline some of these different means and ends by examining a series of case studies of the appropriation of cinematic technology in the science of work, in physics, and in cell biology.\n\nTHE SCIENCE OF WORK AND THE WORK OF SCIENCE\n\nTo illustrate his point that \"with immobility set beside immobility, even endlessly, we could never make movement,\" Bergson suggested this thought experiment:\n\nSuppose we wish to portray on a screen a living picture, such as the marching past of a regiment. There is one way in which it might first occur to us to do it. That would be to cut out jointed figures representing the soldiers, to give each of them the movement of marching, a movement varying from individual to individual although common to the human species, and to throw the whole on the screen. We should need to spend on this little game an enormous amount of work, and even then we should obtain but a very poor result: how could it, at its best, reproduce the suppleness and variety of life? (331)\n\nThis \"little game\" bears a remarkable similarity to the serious experiments of Braune and Fischer, who used the marches of a military recruit to extract principles of motion \"common to the human species\" from his \"individual movement.\" Their results, after \"an enormous amount of work,\" look very much like \"jointed figures,\" although Braune and Fischer would protest that their representations were not meant to convey \"the suppleness and variety of life\" but rather the universal laws that subtend it. Interested primarily in isolating moments in motion and reducing them to universal principles, they were unapologetic contributors to the mechanistic, positivist, analytic trend in the sciences. In this section, I will use their work as an example of the \"cinematographical method\" in science that Bergson interrogates. But I also want to demonstrate how their agenda fits into larger social concerns about modernity. So I will first survey the science of work and its promise to offer a way to manage social problems and class relations. The military heritage of this discipline leads to a consideration of the \"docile body\" in the science of work and scientific photography or cinematography. Then I will turn to the actual process by which scientific images are rendered, by reading their technique as exemplary of the transformation of the chronophotographic image into acceptable evidence.\n\nBefore the image could be accepted as evidence, the subject itself had to be made manageable. At each stage of Braune and Fischer's experiment, they had to make adjustments to the subject, the apparatus, and the means of analysis. The recruit's body, for example, had to be continually manipulated and adjusted to create a legible image. Indeed, just as adapting the subject's body to the apparatus, and vice versa, played a large role in creating and legitimating a scientific image, so adapting labor to the machine age was the primary goal of the science of work. Likewise, Braune and Fischer's intricate efforts to prepare the military recruit for his mission correspond nicely with the mathematical contortions required to translate the image into acceptable scientific data. Understanding the mechanics of the human body required its submission and disassembly on both sides of the equation, before and after the chronophotographic inscription. As we shall see, disassembly, or analysis, was the first step to rebuilding the body and society. Braune and Fischer used the apparatus (both the camera and its accompanying scholarly translations) to break down human movement into its constituent parts. But the painstaking work of analyzing each movement would be rewarded only if they could find a way to eliminate needless effort and bring all the elements back together again so that they worked more efficiently, all in the name of conserving energy and forestalling fatigue.\n\nAs Anson Rabinbach has persuasively demonstrated, the twin concepts of \"energy\" and \"fatigue\" were enormously powerful tropes for scientists and reformers of fin de si\u00e8cle Europe. New scientific models of energy consumption and conservation, along with the ubiquitous technologies and techniques of mass production that accompanied the second industrial revolution, led many scientists to think of the human body as a motor, a machine governed by the same laws of physics and chemistry as its man-made counterparts. The centuries-old battle between vitalism and mechanism heated up once again as this new mechanistic theory of the human motor encountered the critical vitalism, or _Lebensphilosophie_ , of Bergson, Driesch, and others. As the demands of adapting human labor to new industrial techniques grew, a new science\u2014the science of work\u2014adopted these mechanistic principles. The science of work studied the \"human motor\" in order to define its laws and track down the origin of fatigue.\n\nYet the trope of fatigue was more than a scientific mania of the age; it expressed a profound concern over decline and social disintegration. As the structural changes wrought by the Industrial Revolution rippled through society, there arose a tendency \"to locate the body as the site where social deformations and dislocations can be most easily observed\" (21). Metaphors of health and sickness were used to express national anxiety. Fatigue became more than a physical ailment, it became a _moral_ problem, a sign of weakness and absence of will. According to Rabinbach, \"In fatigue the physical horizon of the body's forces was identified with the moral horizon of the species; the moral infirmity of the population was directly proportional to the debilitating effects of fatigue.... [Furthermore,] fatigue represented the membrane between morally sacrosanct labor and the violent, irrational impulses that constantly threatened to disrupt social order\" (43). The tensions between human labor, capital, and the machine age demanded some sort of solution, and European scientists studying the nature of work believed that greater productivity was the key to social harmony. Investigations into the nature of fatigue were hopeful steps toward resolving class conflict scientifically, that is to say, \"neutrally.\"\n\nThe science of work, which emerged in the nineteenth century on the periphery of scientific study, compensated for this marginal status by becoming a self-consciously international phenomenon. The relatively small group of researchers interested in this topic around the 1880s had very little support, institutional or otherwise. Ernest Solvay, a Belgian chemist, who conceived of society as \"an enormous industrial enterprise dedicated to increasing overall productivity while encouraging social justice,\" endowed the Institut de Sociologie in Brussels as part of his plan to study the nature of energy and fatigue. Angelo Mosso of Turin, an Italian physiologist and educational reformer, invented the first efficient and accurate measure of fatigue, the ergograph, in 1884. But it was Hermann von Helmholtz of Germany and \u00c9tienne-Jules Marey of France, in particular, who gave direction and means to the fledgling science.\n\nHelmholtz's contributions to science are legion, but he is perhaps best known for his version of the first law of thermodynamics, or the law of conservation of energy, which holds that energy is neither created nor destroyed but simply transferred. His mathematical formulation of this fundamental law of physics provided science with its most substantive version yet, and it soon became more than a truism. This law's companion axiom, the second law of thermodynamics, which was formulated by other scientists, including Rudolf Clausius and William Thomson (Lord Kelvin), also gained credence beyond the world of physics. The second law explained the process of _entropy_ , which holds that although energy cannot be destroyed, it tends over time to be degraded from useful forms to uselessness. The best example of entropy is the phenomenon of heat transference: when a hot body is placed next to a cold one, the potentially useful energy of the former will transfer to the latter until both have equal, and less useful, temperatures. The relevance of these laws of conservation and entropy was soon extended from the study of molecules to that of the human body and even to society as a whole. Fatigue became the corporal analog of the second law of thermodynamics, and degeneration became its social equivalent (45\u201348). Scientists, reformers, and opinion makers of all sorts soon began to depict the nature and problems of society in terms of energy, attention, will, and utility, on one hand, and fatigue, degeneration, entropy, and uselessness, on the other. The laws of thermodynamics therefore provided the metaphors that motivated social plans for managing the conflicts and excesses of industrial society.\n\nOn the other hand, Marey provided the means by which scientists could study the laws of energy and entropy as they were played out through the human body. Marey, a physiologist who made lasting contributions to the fields of cardiology, physiological instrumentation, and aviation, as well as to the craft of photography and the science of work, provided not only the means for a minute analysis of the movements of the human body but the basis upon which these studies could be counted as legible and legitimate areas of inquiry. His transformation of ephemeral phenomena, such as movement, into scientifically acceptable (that is, legible by disciplinary standards) visual evidence through his \"graphic method\" prompted a flood of motion studies that overran journals in the 1890s. In this respect, Marey is an exemplary figure in late nineteenth-century European positivism. His graphic method signaled the ascendancy of the process-oriented approach in physiology that has dominated the field for the past 150 years. His focus on the disassembly and reformation (in the broadest possible sense) of the human body is representative of the general goal of positivistic, inductive, physical sciences: analyzing individual instances of natural phenomena and provisionally concluding from them an ideal model or set of laws. As Rabinbach demonstrates, Marey represents the forging of a crucial link between cultural and social modernity, between late nineteenth-century disruptions in the perception of time and space and the efforts to manage the contemporary social crises (84\u201388). His kymograph and chronophotographs codified, even embodied, the historical confluence of these forces, giving the science of work direction, means, and legitimacy.\n\nDesigned to calculate \"the mechanical work expended in different movements,\" Marey's chronophotographs were also a link between cultural and social modernity, providing the basis for both a new leisure activity and a science of labor. Marey intended his \"ergonomics\"\u2014or science of efficient movement\u2014to lead to greater productivity. The concept of \"training\" was very important for Marey's ergonomics and the science of work in general. Marey maintained that all animal locomotion is characterized by the transformation of abrupt and disjointed movements into consistent motion. (This is also an accurate description of the technological principle of cinema.) The central feature of all work\u2014whether of humans or machines\u2014is the transformation of irregular, inconsistent, and jarring shocks into regular and uniform activity. The body's own elasticity permits the suppression of shock into regular effort. Muscles, for example, act to turn abrupt movements into dynamic work. Marey believed that animal and human motors are naturally efficient yet capable of improvement. His chronophotography attempted to demonstrate the potential for greater economy to be attained from \"training\"\u2014essentially a program of scientific and bodily discipline. Through careful study of the movement of the body in various stages of a work process\u2014whether forging iron, for example, or pole-vaulting\u2014scientists could spot and correct inefficient movements, thereby showing the worker how he or she might expend the least amount of force and consequently accomplish the task with the least amount of fatigue.\n\nBraune and Fischer's studies of human motion extended this tradition. To \"investigate the influence of a relatively heavy load on gait,\" they asked their experimental subject to carry \"an army regulation knapsack, three full cartridge pouches and an 88 rifle in the 'shoulder-arms' position\" in a number of different experimental settings. These studies, ostensibly dealing \"only with the experimental determination of the process of movement, without considering the cause,\" were designed to recreate the process of labored movement in the hopes that these activities could be improved ergonomically. In this they shared the approach and goal of other practitioners of the physiological branch of the German science of work, or _Arbeitswissenschaft_ , such as Nathan Zuntz and Wilhelm Schumberg, whose _Studien zu einer Physiologie des Marsches_ \"surveyed all aspects of military drill\" in the hopes of pinpointing the causes and consequences of fatigue and march-related illnesses. The military orientation was a common motif in the science of work: Braune and Fischer's experiments were supported by the German High Ministry; Marey stressed the potential benefits of motion study for military training in his numerous appeals for support from the French government; Wilhelm Weichardt tested his infamous \"fatigue vaccine\" on the Austro-Hungarian army; and Mosso tested his ideas about the relationship of mental and physical fatigue on Italian soldiers.\n\nIt is not too surprising that European theories of efficiency, elasticity, and fatigue often took an explicitly military orientation, considering that a military motto might be \"efficiency through training.\" As a matter of fact, the science of work was the industrial application of forms of discipline first deployed in the military. The temporal decomposition and reconstitution of the human body through chronophotographic or cinematic means also fit well into such agendas. _Moving Picture World_ once noted, \"The United States Army has had [motion] pictures taken of a soldier going through the manual of arms. Thumb books with these pictures are made up and furnished to the recruit, who by looking carefully through them can easily trace every minute movement that goes to make up the completed action.\" In this example the recruit was expected to incorporate the lessons of the cinematic image in much the same way that he was expected to embody military ideology. The point is not so much that cinema and the science of work cooperated with the military but that the military application of motion pictures and of these scientific principles point to a coincidence of techniques that provide further insight into the relationship between science and cinema. Michel Foucault cast some light onto the larger intellectual history connecting the science of work and the state:\n\nThe great book of Man-the-Machine was written simultaneously on two registers: the anatomico-metaphysical register, of which Descartes wrote the first pages and which the physicians and philosophers continued, and the technico-political register, which was constituted by a whole set of regulations and by empirical and calculated methods relating to the army, the school and the hospital, for controlling and correcting the operations of the body. These two registers are quite distinct, since it was a question, on one hand, of submission and use, and on the other, of functioning and explanation: there was a useful body and an intelligible body.\n\nBut these registers, Foucault continues, overlap in the notion of \"'docility,' which joins the analyzable body to the manipulable body. A body is docile that may be subjected, used, transformed and improved.\" The scientist's efforts to survey the human body and the colonel's attempts to modify it both required that the body submit to a regimen of exercises. This may have included a series of measurements, tests, recordings, or it might have meant calisthenics in the morning and full-load drills in the afternoon. Either way, whether under the rubric of science or the state, docility, according to Foucault, \"implies an uninterrupted, constant coercion, supervising the processes of the activity rather than its result and it is exercised according to a codification that partitions as closely as possible time, space, movement.\"\n\nThis last sentence is certainly an apt description of Braune and Fischer's experimental procedure, especially in that their chronophotographs \"partition as closely as possible time, space, movement.\" Foucault called these methods \"disciplines,\" and we would include science among them, because its work is essentially that of domestication. We may, after Ian Hacking, divide the work of the scientist into two types: representing and intervening. Hacking equated this division, generally speaking, with the split between theory and experiment, even while acknowledging that the two are inseparable. Indeed, experiment is tightly bound with the process of representing, as sociologists of science have shown. But whether through representation or experimentation, any phenomenon to be studied must be \"tamed\" before it can become scientific data. In the attempt to analyze (or, more accurately, render analyzable) natural phenomena, the work of the scientist involves any number of phases, such as selecting, partitioning, measuring, or representing. It is impossible to present a phenomenon in its \"natural\" state; it must be rendered into material images, such as graphs, photographs, tables, charts, and diagrams\u2014representations that function as Bergson's \"snapshot\" of duration. Sociologist of science Michael Lynch, following Foucault, calls the product of these scientific procedures a \"docile object\":\n\nIt is an object that \"behaves\" in accordance with a programme of normalization. This does not mean that it fails to resist, or that its recalcitrance does not serve to adumbrate its objective news for science. It is to say that, when an object becomes observable, measurable and quantifiable, it has already become _civilized_ ; the disciplinary organization of civilization extends its subjection to the object in the very way it makes it knowable. The docile object provides the material template that variously supports or frustrates the operations performed on it. Its properties become observable-reportable in reference to the practices for revealing them. If the object was not compliant to such a programme, its attributed properties would be incompletely or \"unscientifically\" observable.\n\nThis same civilizing process applies equally to the human body in motion studies, the paramecium under the biologist's microscope, or the astronomer's optical pulsars. I would argue that it also applies to the apparatus created or transformed to view or inscribe these phenomena. Motion picture technology also underwent a certain domestication as its image was transformed into acceptable scientific evidence. For the cinematic image to function legitimately as scientific evidence, it must undergo a transformation, a _rendering_ that is common to all scientific practice and \"docile objects.\" The army's flip books, for example, were used to train the soldier's body, to corral its forces into a productive and useful activity. The books were part of the system of discipline. Yet the very creation of the flip books was also one way of making the cinematic image analyzable by controlling its size, speed, and impact. In the same way, Braune and Fischer took specific, formal steps to domesticate both the human body and their chronophotographic apparatus. The next section will describe these steps in some detail and discuss how they fit into the broader process of representing natural phenomena in science.\n\nBRAUNE AND FISCHER'S _THE HUMAN GAIT_\n\n_The Human Gait_ appeared as a series of papers published between 1895 and 1904 in the _Proceedings of the Royal Saxon Society for Sciences_ (Abhandlungen der k\u00f6niglich s\u00e4chsischen Gesellschaft der Wissenschaften). Although Braune died immediately after the experiments described in their first chapter (perhaps testifying to the exhausting nature of the work), Fischer honored his teacher by making him the first author, and both names have been associated with the studies ever since. Chapter 1 appeared in 1895 and chapters 2 through 6 appeared in 1899, 1900, 1901, 1903, and 1904. Sponsored by the German High Ministry of War, the essays were the first quantitative analysis of human locomotion, and their precision set a standard that is still frequently cited today. That an English translation appeared in 1987\u2014and is still considered an important text in the modern field of human motion studies\u2014testifies to their continued relevance in the science of human movement. In addition to formulating important axioms for this field, Braune and Fischer developed new techniques and instruments for analyzing images. In fact, they are recognized as the originators of analytic, close-range photogrammetry, the science of measurement from photographs.\n\nPrevious investigators of human motion, such as the Weber brothers, Marey, Muybridge, Carlet, and Hermann Vierordt, were primarily concerned with qualitative, two-dimensional motion studies. That is, their largely descriptive studies focused on motion through only two axes\u2014the horizontal and vertical lift of the human leg, for example. Braune and Fischer, on the other hand, realized that animal locomotion takes place along a _z_ -axis as well, that is, _sideways_. They strove to create, through experimentation and exact measurement, an ideal, three-dimensional model of human movement, something that had not been done up to that point. Several researchers had tried to represent the intricacies of human movement graphically, but their methods lacked the scientific rigor to which Braune and Fischer aspired. Braune and Fischer noted, for example, that the achievements of Muybridge, Ansch\u00fctz, and Londe \"are very important for artists, particularly those who depict people and animals in motion,\" but \"the use of photography as a scientific research tool and the improvement of cameras to this end are due, above all, to Marey.\"\n\nThe primary difference between studies \"important for artists\" and those that could be counted as \"scientific\" was measurement. Muybridge and Londe had in common the instinct for \"automatic writing,\" that is, some method of creating automatic, mechanically inscribed signs of movement. But such a sign alone would be useless if it could not be held up to ever stricter scientific protocols. Marey's single-camera setup represented considerable progress over Muybridge's series of cameras, simply because, for Muybridge's series method to be successful, \"the distances between axes of the cameras, standing side by side, had to correspond to the phases of movement and the different cameras had to be optically similar\" (6). This distance depended upon the velocity of the moving body, which could not be known beforehand and is different for each body and each type of movement among the various body parts. Therefore, precise comparison between the pictures was nearly impossible. Yet Marey's single-camera system had problems as well. \"If all the points of a human or animal body moved in one plane during walking or running, these series of pictures would represent not only a one-sided projection but also a true picture of the whole process of movement. However, movement in space has to be taken into account, whereby the centers of all joints describe double curves. Thus, the projection achieved by Marey's method is insufficient to describe completely the movement in space\" (8). In other words, while Marey's pictures provided information about the horizontal and vertical motions of the entire body during walking, they provided no insight into its sideways oscillations. Braune and Fischer's contribution, then, was a four-camera system that produced \"two-sided\" chronophotography, because \"two simultaneous photographic exposures of the same movement are sufficient to determine the movement in any direction\" (10).\n\nTwo-sided chronophotography had its own peculiar difficulties, however. It \"requires the cameras to be opened and shut at short intervals at precisely the same time. This requirement can only be achieved using a highly complicated mechanism. Therefore, to interrupt the exposure, we relied not on shutting the camera but on _altering the photographic object itself_ , so that it was possible to dispense with a particular mechanism for shutting the camera\" (10, emphasis added). From the beginning, then, Braune and Fischer confronted the dilemma of simultaneously regulating subject and apparatus. This was not an either\/or situation; the subject was not completely subdued in favor of an implacable apparatus and method. Rather, both were altered until their properties _merged_ in the representation. That is, features of the body that could be graphically enhanced (such as a limb's straight line) were coordinated with characteristics of the camera (such as the two-dimensional film plane) to create a usable image. This stage of their experiment, then, was crucial for adapting the body to the chronophotographic image and vice versa.\n\nBraune and Fischer's alterations included, first of all, highlighting the recruit's body with a series of strategically placed Geissler tubes\u2014long, thin, straight tubes filled with rarefied nitrogen that, when exposed to an electric current, become incandescent. The recruit wore a black jersey, similar to the one used by Marey, which \"provided a dark background for the tubes and permitted better attachment of the tubes to the body\" (12) (fig. 1.1). When fired by a regulated electrical circuit, the flashes of light provided an ideally manageable strobe effect. The black jersey offered not only a dark background for the tubes, it effectively erased all extraneous details\u2014meaning most of the body itself\u2014from the picture and presented only the most graphic qualities of the process under examination. Significantly, Braune and Fischer also took into account the points that were not illuminated (fig. 1.2).\n\nFIGURE 1.1. Braune and Fischer's military recruit in the experimental suit\n\nFIGURE 1.2. The subject at rest with the grid superimposed\n\nIn some places the Geissler tubes were surrounded by narrow rings of black Japan varnish; these places thereby appeared as short interruptions in the line of light in the pictures. They were located near the ends of the tubes and at the same level as the center of the joint. They therefore marked the corresponding positions of the joints as isolated points of light in the photographs. Similar black rings were also located at the places corresponding to the center of gravity of each segment of the body; the different centers of gravity appeared, then, as black dots on the white lines in the photograph (14).\n\nGenerally speaking, Braune and Fischer were \"marking\" the object in preparation for its representation, a process that is common in scientific rendering practices. In recounting how a \"natural\" space is rendered into a geometricized workplace, Lynch identified a number of themes or processes. Exploring how representations\u2014graphs, tables, diagrams, and photographs\u2014come to embody the \"natural object,\" he asks \"how science initially determines what is natural on the basis of what its graphic qualities disclose.\" \"Marking\" is a first step toward identifying and cultivating those graphic qualities. Lynch finds that marking occurs in two phases: \"labeling\" and \"upgrading visibility.\" In the first phase, the visibility of the object is initially consolidated by first-order techniques, for example, dyeing a cell so that its constituent parts show up under microscopic study. In the second, scientists mark instances that stand out as clear examples, which is a process of selective perception while the project is underway. Braune and Fischer's Geissler tubes served this dual purpose, then, by both enabling the visibility of the object and making visible only what already had been identified as significant.\n\nThe second step common to this rendering process, so closely related to the first as to be only theoretically distinct, is \"the constitution of graphic space.\" Here a \"'mathematical' space comes to dwell within the 'natural' terrain.\" Scientists mark the space of the experiment in such a way that it can be measured or formally structured. In the happy coincidence of anatomy and classification, or the way a specimen is appropriated so that its visible properties are brought together with the graphic qualities of the representational medium, natural objects are prepared for \"mathematicization.\" In other words, the surfaces of the natural object are brought in line with ideal, geometric properties that can be mathematically useful. The placement of the Geissler tubes illustrates nicely how the protomathematical properties of the human body merge with the graphic qualities of the chronophotographic image to become an intelligible, analyzable, measurable scientific representation.\n\nBraune and Fischer also constituted the graphic space by careful camera placement and superimposition of a coordinate grid. They placed two cameras across from each other and perpendicular to the _x_ -axis (the recruit's path) and two other cameras at 30-degree angles to the _x_ -axis (fig. 1.3). The two simultaneous exposures (of each side of the body) could therefore determine the position of any given point in three-dimensional space. To ensure that the points on the different exposures were registered exactly in relation to one another, Braune and Fischer created a grid that could be superimposed upon the image, so that the points of light could be matched to an established set of coordinates (see fig. 1.2).\n\nFIGURE 1.3. Braune and Fischer's camera placement\n\nTo enable us to draw the trajectories in a system of tridimensional co-ordinates, after photographing the phases of movement we photographed on the same plate a network of 1-cm squares printed on a glass plate covered with Japan varnish. In order to improve accuracy we built a large wooden frame on which a 1-m square table of co-ordinates could rotate about a vertical axis. This frame rested on four small screws for which four metal recesses were prepared in the floor. The frame and the table of co-ordinates could thus be brought into exactly the same position at any time. (16)\n\nSo the wooden frame held a one-meter-square glass grid that could rotate parallel to the plane of each of the four cameras. After the initial exposures of the recruit were finished, Braune and Fischer would cover the cameras, screw the frame into its predetermined spot on the _x_ -axis, and superimpose the grid onto each of the four photographic plates, thereby creating a \"graphic space.\"\n\nIt was also necessary that the different phases of movement be regulated _temporally_. To coordinate the firing of the strobe effect, Braune and Fischer connected the primary circuit of the induction coil to a large tuning fork (15). They determined that the fork vibrated at a frequency of twenty-six vibrations per second, which meant that between any two phases of movement approximately 0.04 seconds elapsed (16). The frequency of the fork did not matter so much as its regularity.\n\nConnecting the subject to an electrical source also regulated him in a more indirect way. Braune and Fischer sewed long strips of gutta-percha (a substance resembling rubber but containing more resin) into the jersey where the tubes were to be placed to protect the recruit from electric shock. \"Perhaps we were overconcerned with regard to the insulation and could have saved ourselves a great deal of work since it usually took us between 6 and 8 hours to dress the experimental subject. However, we thought that the subject would walk naturally if he knew that the electric current, of which so many people are afraid, would not come into contact with his body\" (14). Indeed, it would be surprising if the recruit actually did walk naturally. The eleven Geissler tubes were connected in series and powered by a large induction coil in the laboratory. Wires from the coil hung from the ceiling, were draped over a light wooden rod fixed across the shoulders of the recruit (see fig. 1.1), and connected to the circuit on the body. The recruit could therefore walk freely for about ten meters, \"the length of the room necessary for the experiment\" (14). Camera placement played a crucial role in the success of the experiment, so it was essential that the subject not waver from a specified path, which the constraints of the wires and the threat of electrical shock certainly ensured.\n\nFIGURE 1.4. The resulting chronophotograph\n\nClearly, then, by the time the exposures were made, the phenomenon of human movement had already been transformed from an unreadable, \"natural\" object into a regulated, mathematicized process. The photographic image's unruly detail had also been restrained in a number of ways: the strobe effect reduced the image to only its most graphic, mathematical components; the camera placement and the coordinate grid regulated the space of the experiment; the tuning fork and Geissler tubes marked the temporality of the images. The resulting chronophotographic image, however, was far from self-explanatory (fig. 1.4). No image is meaningful without another set of interpretive routines.\n\nIf they were to create a three-dimensional model from these images, Braune and Fischer needed to measure the distances between the points of light and darkness. Figure 1.5 can only hint at some of the spectacular trigonometric operations involved in determining three-dimensional coordinates from two photographs, operations that I will not gloss in any detail. Generally, however, they used triangulation, the basic principle of photogrammetry, to find the position of any point in space from the bearings of two fixed points a known distance apart. This required, of course, that they take measurements directly from the photographic plates. To accomplish this task, they created an instrument designed especially for this purpose (fig. 1.6).\n\nFIGURE 1.5. Determination of the coordinates of a point _P_ from the projections of _P_ on two planes as seen from the two cameras\n\nThe photographic plate was fixed upon a mobile ring and viewed through a microscope, which could slide along a track and bring any point on the plate into view. The microscope gave a view of both the image on the photographic plate and a ruler placed alongside the plate, thereby allowing easy measurement of the points on the plate (fig. 1.7).\n\nFIGURE 1.6. Side ( _A_ ) and top ( _B_ ) views of the instrument used to measure coordinates\n\nFIGURE 1.7. Measurement of a coordinate: The instrument trained on the photograph ( _A_ ) and the view through the instrument ( _B_ )\n\nThese measurements for all four views were then collected and, \"to avoid an accumulation of data\" (!) (43), tabulated for only nine points on the human body: the shoulder, elbow, wrist, hip, knee, ankle, center of gravity of foot, tip of foot, and the point on the head (fig. 1.8). This reduction of the recorded points to a workable number was the first step in transforming the wealth of data into an ideal, that is, a theoretical model. It should also be noted that the gaps in the tables represent coordinates that for some reason could not be determined (a forearm blocking a point on the hip, for example); likewise, question marks indicate indeterminate data. These statistics, although varying by individual experiments, were consistent over time. \"Thus, the results reported in the tables of co-ordinates are valid not only for an individual. They also represent general laws of the movements of the limbs in human gait\" (80). At once more than and less than a human body, the numbers provided the principal tool by which Braune and Fischer reconstructed human movement.\n\nFIGURE 1.8. A table of the coordinates derived from experiment 1\n\nThe second step in \"polishing\" the rough data involved plotting the coordinate numbers onto a graph, essentially recreating the recruit's movement on another grid (fig. 1.9). Idiosyncrasies were reduced by the use of straight lines and points. Comparing one of the photographs (see fig. 1.2) with the graph, we can see clearly how theoretical presuppositions informed the creation of the image. Yet Braune and Fischer maintained, \"It must be stressed that the lines drawn in the diagram do not represent the Geissler tubes used for the experiment but the long axes situated inside the limbs\" (81). Their coordinates also provided for a series of horizontal, or overhead, views of human movement. Figure 1.10 provides successive overhead views of the movements of the nine sections of the body. The views were separated, of course, to make them easier to read.\n\nFIGURE 1.9. The graph of the coordinates (view from the right side)\n\nEdmund Husserl, writing on \"The Origin of Geometry,\" outlined how material objects in the real world are identified with idealized geometric forms:\n\nFirst to be singled out from the thing-shapes are surfaces\u2014more or less \"smooth,\" more or less perfect surfaces; edges, more or less rough or fairly \"even\"; in other words, more or less pure lines, angles, more or less perfect points; then, again, among the lines, for example, straight lines are especially preferred, and among the surfaces the even surfaces; for example, for practical purposes boards limited by even surfaces, straight lines, and points are preferred, whereas totally or partially curved surfaces are undesirable for many kinds of practical interests. Thus the production of even surfaces and their perfection (polishing) always plays its role in praxis.\n\nFIGURE 1.10. The graph of the coordinates (view from above of different body parts)\n\nLynch suggests that we read Husserl's account \"as a description, not of a once-and-for-all historical movement from proto-science to science, but as an account of what scientists do every time they prepare a specimen for analysis in actual laboratory work.\" We can see this mathematization, this \"polishing,\" take place as Braune and Fischer smoothed out the rough edges of their data from table to graph. The body\/image became progressively less recalcitrant as the process continued. Like a student who erases the rough pencil sketches and calculations only after finishing the final graph in pen, so Braune and Fischer's polishing erased the contingent, individual body for a generalized, ideal one. It became an _eidetic_ image. _Eidos_ is the Greek word for \"idea\" or \"species,\" but Bergson elaborated on this definition:\n\nWe might, and perhaps we ought to, translate _eidos_ by \"view\" or rather by \"moment.\" For _eidos_ is the stable view taken of the instability of things: the _quality_ , which is a moment of becoming; the _form_ , which is a moment of evolution; the _essence_ , which is the mean form above and below which the other forms are arranged as alterations of the mean; finally, the intention or _mental design_ which presides over the action being accomplished, and which is nothing else, we said, than the _material design_ , traced out and contemplated beforehand, of the action accomplished. To reduce things to Ideas is therefore to resolve becoming into its principle moments, each of these being, moreover, by the hypothesis, screened from the laws of time and, as it were, plucked out of eternity. That is to say that we end in the philosophy of Ideas when we apply the cinematographical mechanism of the intellect to the analysis of the real.\n\nThe culmination of Braune and Fischer's experiment, the final eidetic image, the reduction of real movement to an Idea\u2014arrived at through an application of the cinematographical mechanism\u2014was their three-dimensional model of human movement (fig. 1.11). With this model, they attempted to reconstruct movement by means of a series of discontinuous images. Their final operation, then, was essentially cinematic.\n\nIn this case, science's successful appropriation of cinema or chronophotography seems to have depended upon the ability to analyze the image and derive translatable information from it. The indeterminate detail of the image needed to be regulated, interpreted, and translated into data that could be mathematicized, tabulated, graphed, and modeled. Quantitative frame analysis, then, has been the traditional means of extracting this information or of disciplining the image. Anthony Michaelis, in his 1955 survey of the history of the scientific applications of motion pictures, declared that \"only the quantitative use of cinematography, combined with frame analysis, has produced the maximum amount of research data of which the motion picture film is inherently capable.\" In other words, if the image were to be a fully productive member of the scientific community, it needed to be, like the worker, broken down and reconstituted.\n\nFIGURE 1.11. Left ( _A_ ) and back ( _B_ ) views of the tridimensional model representing the attitudes of the human body during walking\n\nBraune and Fischer's efforts confirm Bergson's idea of the cinematic nature of science. In their attempts to chart the movement of the human body over time, Braune and Fischer were not concerned with duration so much as with isolation of particular moments. (They were also very interested in training the expert eye; their three-dimensional model in this respect functions as a guide to what to look for when studying human movement.) Film's success as a scientific instrument depended above all on its malleability and on the ability of scientists to manage the excessive detail within the image by reconfiguring the apparatus and image through various methods of frame analysis. Braune and Fischer's example is instructive because it provides a relatively clear view of the work involved in making a malleable and legitimate image. They also provide a good example because their simultaneous work on the image, the apparatus, and the human body replayed certain themes significant to cinema's relation to science and especially the science of work: the domestication of the object and apparatus, the isolation of physiological functions, and the disassembly and theoretical re-formation of movement and process. Braune and Fischer's example suggests that chronophotography's legitimacy as a scientific instrument rested not so much on its ability to record movement through time as its ability to stop that movement\u2014that is, not so much its continuity as its _dis_ continuity. Above all, the ability to divide processes into ever smaller units appealed to the late nineteenth-century researcher.\n\nBROWNIAN MOTION AND \"THE SPACE BETWEEN\"\n\nBraune and Fischer's method depended upon finding the geometry inherent in their subject\u2014in this case, the human body\u2014and translating it into mathematical terms. Chronophotography was essential to this operation, because it could isolate the \"determinate moments\" (in Bergson's words) of the flux of movement so that the inherently geometric aspects of the body could be highlighted and studied. As Bergson argues,\n\nThere is an order approximately mathematical immanent in matter, an objective order, which our science approaches in proportion to its progress. For if matter is a relaxation of the inextensive into the extensive and, thereby, of liberty into necessity, it does not indeed wholly coincide with pure homogenous space, yet is constituted by the movement which leads to space, and is therefore on the way to geometry. It is true that laws of mathematical form will never apply to it completely. For that, it would have to be pure space and step out of duration.\n\nHere Bergson is building on his argument that matter, or more precisely form, is a \"snapshot view\" of reality in transition. Yes, everything is in continual flux, but this constant movement (hence contingency, which, for Bergson, implies liberty) sometimes, to our perception, \"relaxes\" into a more or less stable shape. So from the perspective of this moment\u2014the \"moment\" during which the chair I am sitting on appears to be and acts solid and stable, which is a longer or shorter moment, depending on one's viewpoint\u2014we can map and measure this form and make use of this momentary \"interruption\" in duration. We can therefore make use of the geometry inherent in form, or what Bergson calls the \"mathematical immanent in matter.\"\n\nBoth Ren\u00e9 Descartes and Isaac Newton recognized this \"mathematical immanent in matter.\" Descartes worked toward a mathematical physics when he argued that certain qualities (such as extension, shape, and motion) were more knowable, objective, or certain than other qualities (such as color, sound, heat, and cold). He posited that we could know these \"primary\" qualities through measurement and ground our knowledge in the more certain world of mathematics. He thereby mapped geometry onto the physical world more thoroughly than his predecessors and opened the way for Newton's mechanics as mathematical formulations of motion. Classical physics was therefore founded upon matching the material world to mathematics. But, as Bergson noted, the pure form and space of geometry never completely match the matter of the object under study, simply because the object exists in time; in math, the elements of the equation do not age and change. Braune and Fischer's method indicates the lengths to which scientists must go to reconcile the _duration_ of the object (however momentary) with the _space_ of geometry and thereby harness the \"mathematical immanent in matter.\" Hence Bergson's conclusion that \"there is something artificial in the mathematical form of a physical law\" (238).\n\nYet science succeeds. Bergson offered this explanation: \"One hypothesis only, therefore, remains plausible, namely, that the mathematical order is nothing positive, that it is the form toward which a certain _interruption_ tends of itself, and that materiality consists precisely in an interruption of this kind.\" In other words, Bergson recognized a homology between science's method of arresting time and the momentary interruption of duration that is materiality. Or, to put it another way, science's tendency to _spatialize_ time is analogous to the \"relaxation\" of duration into form and matter. In any case, according to Bergson, science's success in finding mathematical correspondences depends on a temporal interruption.\n\nMax Seddig's attempt to confirm Einstein's theory of Brownian motion is an interesting case study, because his cinematic and chronophotographic method corresponded so closely to important features of the theory. Einstein's theory, as I will explain later, emphasized a gap or interruption: an elision of the path of particles in favor of two independent but related points along that path. In other words, Einstein stressed the _displacement_ of particles rather than their actual path. Likewise, Seddig's experimental method focused on the interval _between_ filmic exposures, so that two separate exposures represented two independent but related points. Seddig's results captured more fully than most the specifics of Einstein's theory precisely because his method was able to accommodate this gap or displacement that previous researchers missed or ignored. Seddig harnessed the inherent discontinuity of the chronophotographic\/cinematographic method to create an experimental system that corresponded to Einstein's important displacement equation. He experimentally created a \"temporal interruption\" akin to what Einstein had created theoretically, thereby instantiating the spatialization of time and form that Bergson found to be central to science's success. More than opening up new realms of the visible or even confirming important theories, this use of motion pictures in physics also illuminates certain features and limits of the scientific method as it had been traditionally understood. Indeed, examining Seddig's experiments alongside Bergson's critique allows us to see the deep compatibility between this application of motion pictures and the state of the physical sciences at the turn of the century. This section will proceed in the following manner: after a general discussion of the stakes involved in the discussion of atomistic physics, I will explicate Einstein's theory of Brownian motion as a watershed moment that collected and focused scientific observation. A survey of the experimental attempts to capture or confirm this phenomenon will lead us to Seddig's case and a close analysis of the connection between his experimental system and Einstein's theory. Finally, we will return to Bergson in order to ruminate on the homology between film form and the physics of Einstein and Seddig.\n\nEinstein's theorization of the phenomenon of Brownian motion, for which he assumed the atomic\u2013kinetic theory of matter, is generally considered to be \"one of the fundamental pillars (or even the main one) supporting atomism in its victorious struggle against phenomenological physics in the early years of this century.\" Throughout the nineteenth century, two approaches to scientific explanation had been in competition in classical physics: the atomic\u2013kinetic theory of heat, on one hand, and on the other, a phenomenological thermodynamics, which assumed that all natural phenomena could be encompassed by the application of Newton's laws of mechanics, but which did not presume to have access to the ultimate constituents of matter. (This approach to physics is called \"phenomenological\" because it deals with the description and classification of phenomena while refusing to indulge in any claims about causation.) This approach allowed physicists to describe certain phenomena (such as heat transference) and derive laws (such as the first and second laws of thermodynamics) from observation without explaining _why_ matter and energy acted in this way. These laws of thermodynamics therefore functioned much like other laws of classical physics, such as Newton's law of gravity, in that they presumed that phenomena would obey these laws no matter what their size or situation. These laws, in other words, were generally regarded as both scalable and absolute.\n\nThe rival approach to explaining thermal phenomena was the atomic\u2013kinetic theory of matter, which started with specific assumptions about the constituents of matter, that is, \"that it was discrete, molecular, ultimately atomic, and that heat was a 'concealed' form of motion associated with the molecules of a substance.\" This theory had the advantage of actually trying to _explain_ the nature of heat, but it had the disadvantage of being unobservable. While atomic theories of matter date back to the Greeks, many eminent nineteenth-century scientists, such as Ernst Mach and Wilhelm Ostwald, resisted building theories of physics on the supposed behavior of matter consisting of particles so small that they were invisible. If advocates of the kinetic theory were to succeed, they would need to deduce the existence of these particles from their effects. Einstein and others theorized that these effects could indeed be deduced mathematically from the random fluctuations in an observable system, such as a container full of a certain kind of gas. These random fluctuations would not occur, by definition, if the absolute laws of thermodynamics held. In other words, the proof of the real existence of molecules was tied to proving that classical thermodynamics was true only in a statistical\u2014not absolute\u2014sense. And at the turn of the century, the atomic theory of matter and its statistical correlation were far from universally accepted. The theory and confirmation of Brownian motion was a crucial victory in this struggle.\n\nBrownian motion, the irregular motion of microscopic particles suspended in fluid, had been known long before Scottish botanist Robert Brown described it in 1828, when he turned his microscope to cytoplasmic granules extracted from pollen. His achievement was \"to show that the motion could not be attributed to any supposed vitality of the particles themselves, since all kinds of inorganic as well as organic substances behaved similarly.\" This meant that the cause of the movement had to be external to the particles themselves. There were a variety of attempts to explain this phenomenon, but it was not until the close of the century that French physicist Louis-Georges Gouy suggested that Brownian motion constituted a clear demonstration of the existence of molecules in continuous, albeit random movement. However, Gouy did not work out any mathematical theory that could lead to experimental confirmation. This was Einstein's contribution in 1905. Not that previous scientists hadn't tried. But the attempts to clarify the phenomenon experimentally were hindered by the lack of agreement among the observations. Researchers simply could not agree on the principal features of Brownian motion. Not only were the movements too quick and irregular to submit themselves to steady observation, but researchers could not concur on even such basic presumptions as whether the movements were dependent on the temperature of the fluid. This tangle of confusion indicates, to Roberto Maiocchi at least, \"how difficult it is to make a meaningful and conclusive scientific 'observation' and, as a result, how any inductivist conception, which claims to start from an empirical base in order to then construct theories of some importance, is unsustainable.\"\n\nPhotography and cinematography offered at least the possibility of creating an \"objective\" record that could be compared with previous and current descriptions of Brownian motion. But again, without sufficient technical means and a theory to guide them, this hope was fool's gold. Take, for example, the case of Austrian scientist Felix Exner, who attempted \"to measure the size of the particles and their speed\" via direct photographic exposure. Unfortunately, the photographic plates at the time were not sensitive enough to register the small amount of light coming through the microscope lens. Undaunted, he observed the movements and traced them _manually_ onto a blackened photographic emulsion, projected these traces onto a screen, measured the distances between the points, and divided by the observation time. \"Of course, the values are not very exact,\" he admitted. Indeed. Exner was interested in establishing the numerical relationship between the temperature of the fluid and the velocity of the suspended particles. His article was the first attempt to quantify features of Brownian motion that previously had been described only qualitatively. Even though his results were the most accurate studies to date, given his method, it is not entirely surprising that they were not decisive. But he did spotlight the need for systematic observation and measurements and the role photography could play in this process.\n\nBut previous observation had very little impact on Einstein's formulation of his theory. The jumble of conflicting observations was of limited use. Einstein was not interested primarily in generating a theoretical explanation of a puzzling phenomenon. Instead, he saw Brownian motion as an observable system that could conceivably resolve the debate between phenomenological thermodynamics and the atomic\u2013kinetic model. Specifically, he derived equations governing motion based on the assumption of random molecular movement and then argued that their confirmation would demonstrate the existence of molecules and also show that the laws of thermodynamics did not apply _absolutely_ to particles of molecular dimensions. In his 1905 paper on the topic, Einstein indicated both the inadequacy of previous observations and the stakes involved for future observations:\n\nIn this paper it will be shown that according to the molecular-kinetic theory of heat, bodies of microscopically visible size suspended in a liquid will perform movements of such magnitude that they can be easily observed in a microscope, on account of the molecular motions of heat. It is possible that the movements to be discussed here are identical with the so-called \"Brownian molecular motion\"; however, the information available to me regarding the latter is so lacking in precision, that I can form no judgment in the matter.\n\nIf the movement discussed here can actually be observed (together with the laws relating to it that one would expect to find), then classical thermodynamics can no longer be looked upon as applicable with precision to bodies even of dimensions distinguishable in a microscope: an exact determination of actual atomic dimensions is then possible. On the other hand, had the prediction of this movement proved to be incorrect, a weighty argument would be provided against the molecular-kinetic conception of heat.\n\nEinstein tackled the big problems; his equations would either prove or disprove the atomic theory of matter.\n\nBecause he did not invoke previous studies or conduct experiments of his own, Einstein's vision of Brownian motion was not empirically based; instead, \"Brownian motion\" was for him a system that he had mathematically pared down, even created, which could then be observed and measured. Einstein did not so much _describe_ Brownian motion as _manage_ it mathematically, thereby providing researchers with the theoretical guidance they needed, giving them mathematical pointers so that they would know what to look for. As Maiocchi notes, \"Only when Einstein had constructed, _independently_ of the experimental accounts, a sufficiently articulated theory, did the experimenters know _what had to be observed_ and only after this theoretical clarification did the observations turn out to be conclusive.\" Einstein's theory, more than any technology, corralled subsequent observations into a stable of usable data.\n\nOf course, before they could be so guided, the researchers first had to _understand_ Einstein's work, and this was far from a foregone conclusion. As Mary Jo Nye remarks, Einstein's 1905 and 1906 papers included mathematical derivations that \"were certainly beyond the ken of even the more precocious experimentalist.\" The novelty of Einstein's equations lies in his emphasis on the _displacement_ of the particles rather than their actual path or velocity. As noted earlier, Einstein was interested in proving the real existence of molecules and also in demonstrating that thermodynamics was true only statistically and not absolutely. These two goals were intimately related, in that any kinetic theory of molecules had to incorporate a thorough understanding of statistical mechanics. The observed properties of a gas, for example, depend on the _average behavior_ of its molecules. But in any system that is sufficiently random\u2014and the model of gas as a system of molecules assumes this randomness\u2014there will be fluctuations from this average behavior. If such fluctuations were large enough, then they would put into doubt the stability of the measured properties and thereby raise questions about classical thermodynamics. In Brownian motion, Einstein had found a system of observable fluctuations that could prove the existence of molecules.\n\nThese fluctuations were expressed in his equations as the mean square displacement (\u03bb _x_ )2 of a particle in any given direction in any given time interval. Einstein realized that particle velocities were so great that they would never be observed directly and therefore could not be measured accurately. So his solution was to factor the displacement of the particles. Maiocchi explains:\n\nWhile previously the attempt had always been made to estimate the length of the trajectory actually traversed by a particle, Einstein's theory deals with the _displacement_ effected in a given time, i.e., the intervening distance between the points of departure and arrival, _independently of the path followed_. This is a change of radical importance because it changes completely _the object of the observation_ : it is no longer a matter of trying to measure the velocity of the Brownian movements (obtained by dividing the length actually traversed during the observation time by the time itself), but of a different quantity.\n\nThis solved the problem that had bedeviled researchers in their previous attempts to describe the phenomenon, as Stephen Brush argues:\n\nEinstein showed that there is no possibility of observing this velocity.... Hence any attempt to measure the \"instantaneous\" velocity of particles in Brownian movement will give erratic and meaningless results. It is for just this reason that all the efforts of the experimentalists... had failed to lead to any definite conclusion about the average speeds of suspended particles. They were simply measuring the wrong thing until Einstein pointed out that only the ratio of mean square displacement to time could be expected to have any theoretical significance.\n\nPrevious experiments, such as Exner's in 1900, had focused incorrectly on the _path_ of the particles. Since the invention of the ultramicroscope in 1903, researchers had turned to Brownian motion with renewed interest, further spurred by Einstein's paper in 1905. Some followed Exner's lead in measuring the actual paths, not catching Einstein's insight. French physicist Victor Henri's results, for example, published in 1908, are notable in that he too used cinematography to record particle movement and to trace particles' actual paths. His results did not agree with Einstein's theories and momentarily cast doubt on Einstein's equations. But it soon became clear that experimental error negated Henri's results. The winner in the Brownian motion sweepstakes was Henri's countryman Jean Perrin, who ran a series of experiments that were also published in 1908. Initially, Perrin paid little attention to Einstein's papers, but when he examined them more closely after 1908, he realized that his experiments confirmed Einstein's predictions. Further work by Perrin produced the most persuasive experimental confirmation of Einstein's theory; after that, virtually no one in the scientific community doubted the existence of atoms, and in 1926 Perrin won the Nobel Prize for his efforts.\n\nWhich brings us to Seddig. Even though Seddig achieved results comparable with those of Perrin, they were, according to Nye, \"neither as accurate nor as convincing.\" While Seddig is therefore little more than a footnote in the traditional accounts of this moment in the history of science, he did discover an important homology between an element of his experimental method (cinematography and chronophotography) and an aspect of the new theory of physics (displacement). Like Exner, Seddig was interested in verifying the dependence of particle velocity on the temperature of the fluid. And like Exner, Henri, and others, Seddig first attempted to record the actual paths of the particles via photographic exposure through an ultramicroscope. He hoped that with long exposures of the particles, the traces left on the photographic plates would correspond to the paths:\n\nIt seemed obvious to attempt to photograph the moving particles, which show up in the ultramicroscope as luminous points, on a stationary plate for a certain exposure time (around one second). The luminous, moving points should then sketch black lines on the plate, which should correspond to the horizontal path covered during this time. The lengths of the resulting curves obtained at _different_ temperatures but during _identical_ intervals should then stand in the reciprocal relation predicted by the theory.\n\nBut again, as with Exner, this method failed due to the weak light from the ultramicroscope and the relatively insensitive photographic emulsions available at the time.\n\nHowever, after this failure, Seddig employed a regular microscope, which provided stronger light and therefore better exposures, and an attached cinematograph. Again, like Henri, he tried to record the movements of the particles cinematographically. As he notes in his first presentation of his work, \"A cinematic method with a precision camera gave some results that were also sufficiently precise.\" While his modified cinematic apparatus seemed to achieve a greater degree of exactitude, he was eventually dissatisfied with the instability of the cinematic image in his efforts to trace the paths. So his final results were determined with the use of an ultramicroscope and a series of multiply exposed photographic plates (fig. 1.12). But while experimenting with cinematography, Seddig came upon the solution to a vexing problem. He had noticed that the strong light required to illuminate the particles also heated the suspending fluid, generating a temperature increase that could not be predicted and that therefore jeopardized his data. He minimized this problem by substituting an intermittent light source for a continuous one. The lamp fired two successive flashes one-tenth of a second apart, while the camera was rigged with a system to measure exactly the time interval between the flashes (fig. 1.13). The light source moreover was projected through a series of cooling and polarizing filters and reflected off a mirror into the microscope, which was attached to the camera. The camera was attached to an electrical circuit so that with each exposure the circuit would open and close, creating an electrical spark that was graphically recorded on a rotating drum covered with blackened paper. The graphic record served as a control for the frame rate so that the interval between exposures could be measured precisely.\n\nFIGURE 1.12. Seddig's photographic rendering of Brownian motion\n\nFIGURE 1.13. Seddig's cinematic apparatus for measuring Brownian motion\n\nFIGURE 1.14. The irregular paths of Brownian motion\n\nWith this setup, Seddig could record the movement of any given particle, but not completely: the intermittent flash recorded only two points along that highly irregular path (for an example, see fig. 1.14)\u2014say, point _A_ 1 and point _A_ 2. The path of the particle between these two points, which occurred _between_ the frames, left no trace and was therefore experimentally elided. All that could be measured was the straight line between the two points\u2014 _which was exactly the empirical translation of Einstein_ ' _s displacement equation_. In other words, Seddig had fashioned an experimental method that corresponded to (and therefore partially confirmed) Einstein's theory. Einstein praised Seddig's efforts, even if he was a bit befuddled by Seddig's method: \"I have read Seddig's paper. He has done it very well. I cannot quite make head or tail of his descriptions of the results.\"\n\nWhy was \"displacement\" so crucial to Einstein's theory? Because it worked around one of the major objections to the kinetic theory: the disparity between theoretical velocity and observational velocity at the molecular level. As Polish physicist Maryan Smoluchowski pointed out, experimental observation of particle velocity would never correspond to the theory, because theoretical velocity was simply not measurable by observational procedures: \"What we see is only the mean position of the particle, driven 10\u201320 times a second, each time in a different direction, by that velocity. Its center will describe an unpredictable zig-zag path made up of straight lines _much shorter in length than the size of the particle_. Its displacement becomes visible only when the geometric sum of these lines is raised to an appreciable value.\" Jean Perrin put it in another way:\n\nThe apparent mean speed of a grain during a given time varies _in the wildest way_ in magnitude and direction, and does not tend to a limit as the time taken for an observation decreases, as may easily be shown by noting, in the camera lucida, the positions occupied by a grain from minute to minute, and then every five seconds, or, better still, by photographing them every twentieth of a second, as has been done by Victor Henri, Comandon, and de Broglie when kinematographing the movement.\n\nThat is, the particle, buffeted randomly by molecules, constantly moves ever so slightly in a variety of directions\u2014these zigzags, smaller than the particle itself, cannot be observed and therefore measured or empirically confirmed. With the reduction of the equation to a question of displacement _without direction_ , velocity (speed + direction) was no longer an issue.\n\nThus the gap. Einstein's theory elided the actual path of the particles over time, erasing it from the equation, thereby creating a theoretical _interruption_ (\u00e0 la Bergson) that was matched by Seddig's cinematic interruption. By focusing on the dark space _between_ the frames\u2014or, chronophotographically, between exposures\u2014Seddig's method mimicked Einstein's theoretical attempt to erase the space between two points along a particle's unpredictable path. The gap between exposures was long enough to allow Seddig to measure the displacement of the particle from point _A_ 1 to point _A_ 2. If it had been too small a gap, Seddig would have traced (erroneously) the path of the particle, as Henri did. Seddig therefore found in his apparatus a specific use that serendipitously suited his needs.\n\nSeddig's application of photography, chronophotography, and motion pictures to the problem of Brownian motion represents a particularly resourceful partnership between film and physics. We could leave it at that and still note with satisfaction the especially neat fit of technology and theory that this application evinces. Seddig's techniques matched certain elements of Einstein's equations and while lacking the precision and expertise of Perrin's experiments, still counted as partial confirmation. But this correspondence of technology and theory also testifies to Bergson's insight into the deeper connection between motion pictures and science. If science's success in finding mathematical correspondences takes advantage of what might be considered a \"temporal interruption\" in the constant flux of becoming\u2014a \"temporal interruption\" we call \"form\"\u2014by finding the mathematical, atemporal line of geometry in that form, then the scientific use of motion picture technology often matched this process by creating its own temporal interruption\u2014the space between frames, for example\u2014and finding mathematical correspondences to the resulting form. The temporal character of motion was therefore reduced to a series of static, discrete, two-dimensional images; time and motion were then \"understood\" by the measurement of differences between them. Motion pictures, in this application, reinforced the tendency to mistake the category of space for the category of time.\n\nFurthermore, the spatialization of time allows us to conceive of time as reversible. Our felt experience of time is that it is irreversible, but our experience of space is that we can pursue it in any direction. So when we represent the flow of time as displacement along a homogenous axis, \"Nothing prevents us in this abstract representation,\" as French physicist Louis de Broglie remarked, \"from supposing that we may reverse the course of time, contrary to the most certain property of real duration.\" Likewise, a mathematical equation also implies this equivalence and reversibility. As Suzanne Guerlac explains, \"In geometrical terms, if we depict a movement from left to right, we can reformulate it as passing from right to left. In algebraic terms, equations are commutable.\" But when we conceive physical processes as reversible, we might fail to consider certain essential properties of real time, namely its directionality. Bergson, of course, argued for a dynamic ontology of irreversible time so that important properties of duration do not escape our understanding.\n\nBy erasing the direction of the particle\u2014whether the movement is from point _A_ 1 to point _A_ 2 or from point _A_ 2 to point _A_ 1 is irrelevant for the purposes of the theory\u2014Einstein's equations implied a temporal reversibility that was also mimicked by Seddig's use of photographic and motion picture technology. In more than one way, the scientific use of film (especially in the physical sciences) was mathematical; in Seddig's hands, film functioned mathematically. How was this accomplished? Three aspects of this kind of application\u2014the use of a flash, the use of the frame, and the form of the celluloid strip\u2014combined to create a formal, almost geometric support for the mathematical rendering of the phenomenon. Seddig used his flash as an intermittent light source that would not raise fluid temperature as much as constant illumination. But the flash also immobilized the particle by reducing exposure time to an instant and the spatial fullness of the phenomenon to a two-dimensional position. The flash focused both time and space to the sharpness of a point. Furthermore, the regularity of the flash created a series of equivalent points\u2014 _A_ 1, _A_ 2, _A_ 3...\u2014each of which corresponded to a single frame. The frame, in other words, determined the boundaries of the event, both spatially, in that the event took place within the field of the frame, and temporally, in that the frame represented a unit of time. So the flash and the frame reduced, limited, and focused the fullness of the event to a series of separable, measurable points\u2014which is to say that the technology accommodated a mathematical understanding of the phenomenon.\n\nWhile the photographic series is not strictly commutable\u2014one cannot rearrange the order of the units and still obtain the same result\u2014it is reversible, in that [ _A_ 1, _A_ 2, _A_ 3...] is mathematically the same as [... _A_ 3, _A_ 2, _A_ 1]. In this respect, the celluloid strip itself functioned as a line, as a trajectory of completed motion. Bergson distrusted the representation of movement as a line, because such a rendering confuses movement as we experience it\u2014that is, as something indivisible\u2014with our imaginative recreation of that movement as a series of stages or points. More precisely, it confuses movement with the line traversed; it conflates what is happening right now with what just happened. The path is past, while movement exists in the moment. We may divide the path into points, but these points have no reality outside of the line drawn. Bergson argued that the difference is simple: \"At a stage we _halt_ , whereas at these points the moving body _passes_.\" We therefore mistake immobility for mobility, discontinuity for continuity, and past for present. Representing motion as a line, especially one that predicts future motion, is an illusion, which rises from the conception that \"motion, _once completed_ , has deposited along its course an immobile trajectory on which one may count as many immobilities as one wishes. From this, one concludes that motion deposits at each instant a position with which it coincides.\" In truth, however, no such deposit occurs, because movement itself does not halt; otherwise it would not be movement. But motion pictures reinforce this illusion, not so much through the projected image, which, as Gilles Deleuze pointed out, _is_ movement, but through the celluloid strip of individual frames. What is an exposed film frame except a \"deposit\" of a position coinciding with movement? And just like a line, the series of immobile instants on the celluloid strip is completely reversible. Bergson might therefore have argued that cinematic exposure itself is a category mistake.\n\nNERVE FIBERS, TISSUE CULTURES, AND MOTION PICTURES\n\nToward the end of his dissertation, Seddig remarked on the close similarity of Brownian motion and cell movement, calling for further investigation into the relation between temperature fluctuation and \"biological motion\" that might lead to an understanding of the deeper \"causes of movement itself.\" Even if this particular research program never really caught on, in a way it points to the interesting history of cooperation between biologists and physicists during the early days of microcinematography. The precise date of the first moving pictures taken through a microscope has not been recorded, but Marey devotes a chapter to the technique in his 1894 book, _Le mouvement_. Marey's experiments were continued by various disciples, most of them working at the Institut Marey in France. Lucien Bull, for example, is known best for his high-speed cinematography of fast-moving objects, which was an important technique in physics for research on surface tension phenomena. In 1903 he helped biology Professor Antoine Pizon microcinematically record the multiplication of a colony of _Botryllus_ , or sea squirts. Another disciple of Marey and a pioneer in the field, Charles Fran\u00e7ois-Franck, adapted Marey's apparatus and made chronophotographic plates of a variety of biological phenomena from 1902 to 1908. Lucienne Chevroton, who worked at the Coll\u00e8ge de France alongside Fran\u00e7ois-Franck, similarly added a motion picture camera to a microscope apparatus in 1909 to record microcinematographic investigations of a sea urchin egg. Fran\u00e7ois-Franck and Chevroton offered their apparatus to their colleague Victor Henri, a physicist at the Coll\u00e8ge de France, who used it to record Brownian motion, as we saw in the previous section. He, in turn, may have introduced it to Jean Comandon, who during the early years became the most successful microcinematographer of cell movement. Lucien Bull also helped outfit Swiss biologist Julius Ries with a Lumi\u00e8re microscope\u2013cinematograph combination for his research on the fertilization of sea urchin eggs. As early as 1900, with the help of Charles Gaumont, French physicist Henri B\u00e9nard used cinematography to study dynamic, convective systems\u2014specifically, the spontaneous formation of inorganic matter into hexagonal shapes in a heated liquid\u2014and went so far to comment on the analogy between this type of motion and biological phenomena. In Germany, physicist Henry Siedentopf, who ran Carl Zeiss's optical laboratory in Jena and collaborated with Richard Zsigmondy on the invention of the ultramicroscope in 1903, actively used motion picture techniques to investigate crystallography. Siedentopf also helped Heidelberg biologist Hermann Braus with his cinematic investigations of tissue culture, thereby continuing this trend of cinematographic collaboration between biology and physics.\n\nThis cooperation between physicists and biologists interested in adapting motion picture technology is not surprising, given that investigators in both fields had set out to study motion as a way of answering difficult questions about causality. Brownian motion, as we have seen, functioned as an observable system that could provide clues to the constituents of matter itself. Biologists similarly searched for observable, temporally continuous systems in which the analysis of motion could resolve previously intractable disputes about the processes governing the organism. Braus, a morphologist and experimenter, used motion pictures for precisely this purpose. His case is interesting because it illustrates the close formal relation between cinema and biology at the turn of the century; motion picture technology not only helped solve evidentiary problems in biology, but the introduction of cinematic means into experimental technique matched broader changes in the discipline itself. Furthermore, Braus's use of motion pictures demonstrates especially dramatically cinema's _rhetorical_ power in science; Braus altered his viewpoint in a dispute because of what he witnessed in a filmic record. This section will place Braus's cinematic contribution in the context of certain broader debates and experimental trends in the discipline of biology.\n\nThe dispute in question\u2014between Braus (fig. 1.15) and American experimenter Ross Granville Harrison (fig. 1.16)\u2014concerned the development of nerve fibers. Braus and Harrison were not the only figures in the debate, which had been brewing since the mid-nineteenth century. The problem was this: while it was clear that nerve fibers grew, it was not clear _how_ they grew. Did a fiber grow from one cell or from a chain of cells? How was the connection between fibers and tissues established? Did the nerve grow from the center to the periphery, or was the connection already there in the embryo? The disputants offered at least three theories. Many believed that the formation of the nerve required the participation of many cells, not just one. This was known as the \"multicellular\" theory of nerve development. Others, such as Braus, maintained that nerve fibers had already been present in embryonic form and that as the organism developed and cells grew and multiplied, the nerves stretched across them, making bridges of nerve fibers that differentiated as they became functional. This was the \"protoplasmic bridge\" theory. Still others, such as Harrison, argued that nerve fibers grew outward from a single nerve cell into the interstices between other cells. This became known as the \"outgrowth\" theory, and most biologists would come to agree, but not until around 1930, that this theory was the most accurate and persuasive depiction of nerve fiber development (fig. 1.17).\n\nFIGURE 1.15. Hermann Braus\n\nCourtesy of the National Library of Medicine\n\nFIGURE 1.16. Ross Granville Harrison\n\nCourtesy of the National Library of Medicine\n\nFIGURE 1.17. Harrison's sketches of the elongation of frog nerve fibers grown in culture\n\nFrom Ross G. Harrison, \"The Outgrowth of the Nerve Fiber as a Mode of Protoplasmic Movement,\" _Journal of Experimental Zoology_ 9, no. 4 (December 1910): 787\u2013846\n\nThere were a number of reasons why investigators in the early twentieth century could not agree on this topic, including prevailing concepts of the structure of the nervous system. But a large part of the problem derived from contemporary methods of observation. As Harrison complained in 1906, \"Prior to the year 1904 all attempts to solve these problems were based on observations made upon successive stages of normal embryos. When one compares the careful analyses of their observations, as given by various authors, one cannot but be convinced of the futility of trying by this method to satisfy everyone that any particular view is correct.\" That is, researchers attempted to observe the growth of nerve fibers by slicing, staining, and mounting successive stages of an embryo, which were then viewed through a microscope. This technique had long been dominant in biology and developmental embryology. Crucial questions about the origin, direction, and character of growth had to be inferred from what was effectively a series of still images.\n\nBecause investigators basically had to speculate about any movement taking place between images, disputes over interpretation were common. Among the most famously frustrating was the debate between Santiago Ram\u00f3n y Cajal, who subscribed to the outgrowth theory, and Hans Held, who held onto the protoplasmic bridge theory. Susan Billings explains, \"Cajal claimed that _Plasmodesmen_ were artifacts of Held's preparative method, while Held, in turn, thought perhaps pale staining or alcohol fixation prevented Cajal from seeing them. The impression one might get from reading their papers is that Cajal and Held were discussing vastly different material.\" Yet when Cajal visited Held's laboratory in Germany, he said that he \"had the pleasure of examining Held's excellent preparations. Just as we expected, they are very successful, and to our great surprise, they show very much the same picture as ours.\" As with Brownian motion, researchers were unable to agree on what they saw in what were nearly identical preparations. (Indeed, this is another example of the influence of theory on observation.)\n\nThe histological technique of slicing, solidifying, staining according to type, and then mounting the preparation on a microscope slide had been the foundation of cell research up through the nineteenth century. It served biology well; as Hannah Landecker notes, \"the technique of staining, by suspending the cell in time, freed the experimenter from the temporal exigencies of a living subject.\" Precisely because the subject did not move, the researcher could examine the preparation at his or her leisure. \"Cellular movement, or lack thereof,\" Landecker continues, \"was a matter of inference when using static representations, the hardened moments preserved in histological specimens. It was about inferring what was happening in the spaces between the sequential slices of preserved moments.\" Indeed, perhaps because this method was so dominant, movement itself was not important to biology's research questions\u2014one could even argue that biologists' questions were often tailored to their technique. However, once questions of movement and growth could not be answered, new techniques were required; alternatively, we could argue that once the new techniques were available, new questions came to the foreground. Physicists had solved this problem of observation and interpretation in the case of Brownian motion by emphasizing the gap or discontinuity, thereby excising from the theory that which was not observable. Biologists could not do this, however, because their ostensible object of study was continuity itself. So they had to develop techniques that would allow them to study cell movement directly.\n\nHarrison believed that \"the only hope of settling these problems definitely lies, therefore, in experimentation.\" In 1905, Braus approached these questions experimentally by transplanting limbs of very young tadpoles to various areas of other tadpoles' bodies. He then examined the nerves in the transplanted limbs, finding evidence in favor of the protoplasmic bridge theory. Harrison repeated these experiments and came to completely different conclusions. In fact, Harrison argued, \"The same facts [in Braus] may be interpreted quite readily, if not more so, in accordance with outgrowth theory. The experiments do not approach the problem directly enough to determine questions of histogenesis [the origin of the nerve fiber], and there are too many loopholes left to permit a rigid proof.\" What was needed, apparently, was a means of isolating the nerve cell itself so that the growth of the nerve fiber could be observed directly. In fact, this is exactly what Harrison did in a now-famous experiment that he first described in a June 1907 announcement:\n\nThe immediate object of the following experiments was to obtain a method by which the end of a growing nerve could be brought under direct observation while alive.... The method employed was to isolate pieces of embryonic tissue, known to give rise to nerve fibers.... The pieces were taken from frog embryos about 3 mm. long.... After carefully dissecting it out, the piece of tissue is removed by a fine pipette to a cover slip upon which is a drop of lymph freshly drawn from one of the lymph sacs of an adult frog. The lymph clots very quickly, holding the tissue in a fixed position. The cover slip is then inverted over a hollow slide and the rim sealed with paraffin. When reasonable aseptic precautions are taken, tissues will live under these conditions for a week and in some cases specimens have been kept alive for nearly four weeks. Such specimens may be readily observed from day to day under highly magnifying powers.\n\nBy successfully maintaining a tissue specimen outside of the body (known as \"in vitro,\" whereas the state of tissue inside the body is known as \"in vivo\"), Harrison initiated the technique of \"tissue culture,\" a biological method whereby fragments of tissue from an organism are transferred to an artificial environment in which they can continue to survive and function in some form. With the cell population thereby isolated, the researcher could better examine and manipulate cell behavior. The histologist could observe the growth of the cell directly under the microscope and even slow down or accelerate that growth by manipulating the environment. Investigators from all quarters immediately took notice. Within the next few years, the literature on tissue culture exploded; its most famous experiments were those conducted by Alexis Carrel and Montrose Burrows at the Rockefeller Institute, where they managed to keep the culture from a chicken embryo alive for decades.\n\nThis method also presented a major break with traditional methods of visualization in biology. Not only did the in vitro experimental technique offer a different way of isolating the cells, but this form of representation also replaced the discontinuous form of representation then common to biology (for example, sequential slides, illustrations of stages of development, still photographs) with temporal continuity (movement, growth). Tissue culture allowed the researcher to observe movement in time; there still remained the challenge of representing this movement, of course, given the traditional means (slides at conferences or illustrations in texts and journals). Harrison's experiment also showed fairly conclusively that the nerve fiber grew from a single cell. It demonstrated that the fiber could and did grow without the help of other cells and that nerve growth did not require the presence of preformed protoplasmic bridges. Even so, biologists such as Hans Held challenged Harrison's results for years afterward, maintaining that the nerve fiber grew on its own in vitro but grew with the aid of bridges in vivo.\n\nSo in September 1911, Braus's demonstration before the Society of German Natural Scientists and Physicians spoke to both the interest in the new technique and the enduring controversy about nerve fiber development. His demonstration also addressed new challenges in representation in biology, in that he used motion pictures to depict these two issues. At the start of the talk, Braus discussed the still-new technique of tissue culture as Harrison and Carrel had developed it to that point. He then noted, \"I thought that many of you would be interested in seeing the phenomena of growth and movement in these cultures, if not in a live culture, at least via cinematographic images. I have films that were made in order to study the more detailed movements in the cultures of my many preparations, in which the transformations were too fast or too slow to be observed in the objects themselves.\" This offering to his colleagues at the conference deserves further examination, because it indicates two ways in which film had a crucial role in the experiment.\n\nFirst, Braus used film to _communicate_ the results of his work. In this respect, the medium of motion pictures functioned as one of any number of forms of communication that describe or depict experimental procedures so that the (expert) reader can judge the validity and value of the project. Some have called this expert communication and evaluation \"virtual witnessing\"; in the modern era, when there is no way to inspect or replicate every experiment personally, the significance of such reports for the dissemination and acceptance of new knowledge cannot be underestimated. In this early example, film promoted \"virtual witnessing\" in an important new way, not only because film made the cultures present to the expert witnesses in a more than virtual manner but because it was portable and repeatable in a way that live cultures were not.\n\nSecond, Braus used film as a _substitute_ for his object of study (the tissue from a frog's heart). He made the films to study movements that were \"too fast or too slow to be observed in the objects themselves\"; his films were therefore the object of his analysis. In fact, the capabilities of motion pictures so closely matched the capabilities of tissue culture that they became an experiential substitute for the technique. That is, the projected image moves and can thereby present movement or growth, but film was also, like tissue culture, fragmentary and easily malleable, both temporally and spatially. If the histologist could isolate, dissect, transplant, graft, and retard or accelerate the growth of cells, then the cinematographer could match these actions perfectly by framing, cutting, splicing, editing together, and slowing down or speeding up the film itself. (Of course, motion pictures were quantitatively different than tissue cultures because of the ease with which images could be multiplied or accelerated.) In this respect, Braus's use of film became what we might call a \"virtual experiment\"\u2014manipulation of the film could isolate and analyze aspects of the tissue that could also be manipulated, isolated, and analyzed in the technique of tissue culture itself.\n\nBraus then described the star attraction of his films: a beating heart extracted from a frog embryo and maintained in culture for days. That achievement was in itself noteworthy, but Braus was interested in proving something more: \"As far as I can tell, observations about the _growth_ of organs in vitro are not addressed in the literature.\" That is, he wanted to demonstrate that specimens actually grew in culture, they did not merely survive. So he used staining techniques and time-lapse cinematography: \"To prove that the culture, whose pulsation I've shown, is actually growing, I have filmed isolated, stained cells from the culture at greater magnifications. The exposures occurred, on average, every 10 minutes over the course of 10 hours. Here in the projection they have been compressed into just a few minutes, so that we can see the movements of the stained cells unfold rapidly, although in reality they are very slow.\" The stains allowed Braus to distinguish the movement of the cells and their changing form as they grew and multiplied. His films thereby functioned not just as a substitute for the culture, but as a new form of evidence.\n\nThis is a particularly potent example of the power of cinema to present evidence forcefully, because the cinematic record of his experiments with tissue culture caused Braus to reverse his position on the origin of nerve fibers.\n\nTo supplement this, I am showing images of growing nerves, in which\u2014just as with the mesoderm cells\u2014many of these growth phenomena can be observed under the microscope _directly_. The nerves are of particular interest because of the _in vitro_ heart movements. At the time of extraction and the end of the experiment, the heart structure had no ganglia cells.... But from our experiences with nerves grown _in vitro_ , we can rule out the claim that actual nerves originate in any other way than from the central neuroblasts. I want to discuss here objections that could be raised against this conclusion, reached by Harrison based on his preparations, which prompted me to verify his findings. This investigation has persuaded me that Harrison is basically correct.\n\nHaving witnessed these processes for himself through his filmic record, he was able to declare that \"only the isolation of neuroblasts [the nerve cells] gives total certainty; the nerve fiber grows out of the neuroblast like a mold grows out of an isolated spore.\" Braus went on to defend Harrison against the objections of other colleagues in the field, with whom he previously agreed.\n\nSo Braus also used film as a _confirmation_ of Harrison's theory. Braus's motion pictures were more than a demonstration; they _enacted_ the technique and Harrison's theory of growth. Harrison posited the nerve cell fiber as a plant-like \"outgrowth\" rather than a differentiated element of a multiplying mass. He argued that nerve fiber growth moved forward incrementally, outwardly, and linearly. Likewise, Braus's time-lapse technique in this instance\u2014with its statistical sampling of time through regular exposures and its amplification of the forward push of incremental growth\u2014enacted and insisted on an incremental, regular, mathematical, and resolutely _linear_ theory of growth. In other words, this set of techniques (the frame's isolation of the tissue, the temporal manipulation of growth, and the magnification and projection of the image) enacted, amplified, and confirmed Harrison's theory of growth and technique of tissue culture with incomparable rhetorical power. While cell growth, when illustrated by sequential images (such as slides), could have been seen as reversible and thus disputable when direction was at issue, the temporal continuity of the cinematic record _pushed_ the observer (including Braus) to a particular conclusion\u2014that the growth of the nerve fiber happens in _this_ way. If other scientific uses of cinema try to stop time, this application stressed duration and the teleology of growth. As a means of visualization and as an experimental tool, this application of motion pictures precisely matched broader changes in experimental technique and representation in biology at the turn of the century.\n\nAs we can see, motion picture technology presented itself as a potential research instrument to biologists involved in tissue culture, but film became a powerful tool for researchers because the research program in biology had changed to accommodate it. That is, Braus came to film not only because it solved a particular problem but because the problem itself was shaped by changes in conceptions of growth and time that motion picture technology could articulate (indeed, that the moving image might have prompted). Of course, the practical benefits outlined at the beginning of this chapter provided compelling reasons to use cinematography. In Braus's case, it offered the opportunity to record his experiments and then to present some of his findings to a wide audience. The photographic or filmic image also projected the impression of objectivity and thereby provided a potent rhetorical weapon in his arguments with the conclusions of other biologists. And film's ability to manipulate time allowed Braus to present quickly evidence that would have taken hours to observe directly. But beyond these obvious advantages, we could argue that the growth Braus documented was researchable only _because of_ cinema. The motion picture camera\u2014because it could alter the scale of cellular time through slow-motion or time-lapse cinematography\u2014could capture events that were not otherwise visible. These events were not merely difficult to observe without the camera, in a real sense they did not exist without it. They became, as Walter Benjamin noted, our \"optical unconscious,\" revealed to us by cinematography in the same way that our psychic unconscious is revealed by\u2014 _and comes into being by virtue of_ \u2014psychoanalysis.\n\nBut there is more to the success of cinema in science and its relatively rapid appropriation than even these good (practical) reasons. Not only were filmic techniques increasingly available, they were also adaptable to various scientific enterprises. If Seddig focused on the space between the frames, on cinema's _analytic_ ability, Braus similarly used the time between exposures to his advantage. But for Braus the image of the object was paramount. The images made little sense analytically; instead, Braus concentrated on their _synthesis_. For Braus the gap was crucial for the construction of the film, but the ellipsis was elided in favor of a temporal continuity or biological teleology. Growth is not reversible. This is not to say that film's analytic abilities were confined to (or more suitable for) the physical sciences, while its synthetic nature was best suited to the life sciences. The variety of applications in a range of disciplines makes any such generalization rash. In fact, as Latour might argue, cinema's adaptability was no greater or less than that of any technology\u2014from the microscope to the computer. Nor was Braus's \"synthetic\" use of cinema any less scientific than Seddig's \"analytic\" emphasis. Braus was doing more than \"simply\" recording a phenomenon and sharing it with others, even though he did not explicitly extract quantitative data from the images. Braus's use of time-lapse cinematography recorded only certain, regularly paced moments over the continuity of an event, thereby creating, I would argue, _a statistical sampling_ of the temporal dimension of the phenomenon. We can therefore see Braus's particular appropriation of motion picture technology as a statistical application of cinematography very much in keeping with the contemporary adoption of statistical thinking for the study of systems in both physics and biology. Indeed, part of the persuasiveness of these records could be found in their \"objective,\" mathematical construction of a representative sample.\n\nBergson, however, articulated the limits of analysis with respect to dynamic processes, and an extended comparison of the use of motion pictures in physics and biology would expose the paradox of analysis and synthesis as a problem of comprehending continuity through discontinuity. Thinking about Bergson and motion pictures together, in other words, highlights fundamental assumptions in certain scientific approaches about the nature of time and movement. Seddig's use of motion picture technology to study Brownian motion certainly reveals a commitment to interruption, reversibility, and immobility in the study of dynamic systems\u2014a commitment that was obviously present in Einstein's equations as well. The ambidexterity of motion picture technology\u2014its dual nature as an instrument of both still and moving images\u2014not only explains its success in science, but its varied and subtle permutations in scientific application can themselves be understood as a form of commentary on scientific theory and method. How researchers used motion picture technology in their experiments tells us much about their disciplinary and philosophical investments.\n\nAbove all, I want to stress the mutually reinforcing relationship between science and cinema. The problems and solutions that the technology and the research program presented to each other were entwined and dependent. For Braus and others, motion picture technology fit quite well with other elements of this experimental ensemble, in part because certain formal features\u2014its temporal malleability; its ability to frame and isolate; the forward, teleological motion of its projected image; and so on\u2014seemed to match similar features of the technique of tissue culture. There are other, broader homologies as well. The technique of tissue culture extracted life, reproduced it technologically, and prolonged it, allowing researchers to artificially accelerate or slow life down. And what is cinema except a technology that extracts bodies from their natural time and space, reproducing them mechanically, reanimating them repeatedly (faster or slower as needed) long after the host has expired? The idea that tissues, organs, and life itself are separable from the body\u2014and, by implication, that death is foreign to the organism\u2014is, like cinema, an especially modern notion. So we could say that cinema's singular facility with time, space, and life made it an ideal instrument to capture and confront these concepts in modern science. But we need not go so far. In fact, such expansive analogies tempt us to think of cinema and science as single entities, when the point of these case studies has been to demonstrate the opposite: that film is not monolithic and that science, like other broad umbrellas, contains many different disciplinary agendas, each of which has adopted some aspects of film technology while ignoring others. That so many different researchers found something to use in motion pictures indeed indicates a special relationship between scientists and film. Braune and Fischer's deconstruction and reconstruction of the human body, adapting it to modern labor systems; Einstein's and Seddig's flexible notions of time expressed in the invisible, random, and reversible world of molecular physics; Harrison's and Braus's conceptions of life as separable, temporally malleable, and technologically reproducible\u2014in each case, these scientists turned to motion pictures not merely to observe and record phenomena, but because in cinema they found a kindred spirit, an amiable partner that shared their vision, that could look at the world as they did. The cinematograph, then, was not just a handy tool. In more ways than Bergson imagined, the form and application of motion pictures articulated science's modern agendas.\n\nIn this chapter, we have examined the merger or accommodation of film form and objects, theories, and disciplinary agendas. In each case, film proved its worth because of this correspondence\u2014not exclusively, of course, because the amount of work necessary to adapt film to these aspects of scientific endeavor is prodigious and varied. But we can credit the researchers for recognizing the homology between film and experiment in the first place. The next chapter will examine medical research films and the correspondence between film form and _observational_ practices or ideals.\n\nBETWEEN OBSERVATION AND SPECTATORSHIP\n\nMEDICINE, MOVIES, AND MASS CULTURE\n\nStillness is the unattainable value.\n\n\u2014PAUL VAL\u00c9RY (1932)\n\nFrom around 1904 to 1914, motion pictures endured a difficult and very public transition between their good standing as a scientific tool and their growing notoriety as an instrument of mass culture. During the early period in Germany especially, there was a strong contrast between the enthusiasm for motion pictures as a scientific or pedagogical tool and the simultaneous condemnation of its public, commercial incarnation. Physicians in particular wrote many editorials and studies depicting the threat that motion pictures posed to the psychic, physical, and moral health of the nation. To a certain extent, this is unsurprising: the German complaints about cinema follow more or less the same pattern that we can find in the United Kingdom, France, Russia, and the United States at this time. There seems to be no reason to think that the discussions in the scientific community and those in the public sphere were related or exceptional. But if we look closely, we find that the scientific and public debates about cinema were in fact closely related: the reasons scientists and physicians accepted film as a scientific instrument were rooted in the same logic that prompted them and others to reject cinema's public manifestation. This logic concerned, to put it too simply, the perceived difference between _observation_ and _spectatorship_. That is, participants in both discussions seemed to agree on the advantages and dangers of the moving image, but many of these advantages and dangers appear to stem from _different ways of viewing the image_.\n\nIf the previous chapter focused primarily on the negotiation between object and apparatus\u2014such as Braune and Fischer's adjustments to their recruit, to their working space, and to their camera to create a scientifically productive image\u2014this chapter emphasizes the relationship between researcher and image. How the researcher _studies_ the image may be just as significant as how he or she produces it. Indeed, \"observation\" is a woefully underused key to understanding medical practice and identity. Many physicians felt that their professional reputation and livelihood depended on carefully cultivated observational skills. No less a personage than the eminent British surgeon Sir James Paget emphasized in 1887 that \"becoming scientific in our profession [requires] the training of the mind in the power and habit of accurately observing facts.... The main thing for progress and for self-improvement is accurate observation.\" As Paget indicates, part of this investment in observation stemmed from the contemporary adoption of scientific principles in medical practice, which some contested, but which ultimately bolstered the field's legitimacy. As physicians identified with scientists, \"observation\" became an important common link. But it is not simply a borrowed scientific practice, it is also key to understanding modern medical logic. As Michel Foucault and others have shown, modern medicine's understanding and approach to disease is inextricably tethered to specific habits of perception. The process of observation was not simply preliminary to diagnosis but actually rehearsed the logic by which medicine grasped its object.\n\nWe can see this process especially clearly when physicians encountered the cinematic image. How they used motion pictures and what they said about them can reveal the subtleties and complexities of their observational practice. As I have argued elsewhere, modern medicine's reliance on the correlation between life and death, or between movement and stillness, if you will, makes motion pictures a privileged representational technology for exploring medical hermeneutics. In their attempt to understand and locate pathology, physicians find in the corpse lesions that might be blamed for death and then compare these to signs in the living body that might be harbingers of disease. In the stilled corpse, researchers are able to grasp some elements of disease that are elusive in life. Similarly, physicians take advantage of the duality of motion pictures by going back and forth between the stilled and the moving image to grasp elusive properties of the recorded object or event. Exploring this process in detail reveals not only the doctor's cultural investment in observational practice but also the foundations of modern medical logic. Examining the medical community's reaction to movies, on the other hand, also reveals this investment in expert observation and their characterization of its opposite\u2014spectatorship\u2014as a pathological condition.\n\nNearly from the beginning, motion pictures were enlisted as an aid to medical research and education. German doctors, in particular, were captivated by the potential of the cinematic image. Certainly, the application of motion picture technology to medicine was not qualitatively different in Germany than in any of the other Western, industrialized countries. Germany, however, did enjoy a quantitative distinction: between 1900 and 1920, Germans wrote far more and far more frequently about medical cinematography than their counterparts in the United Kingdom, France, the United States, and the rest of the world. One could say that Germans wrote more about everything, but Germany's leadership in certain specialties that made extensive use of cinematography, notably radiology and neurology, helps validate this claim. It could also be argued that in the late nineteenth and early twentieth centuries, Germany's institutional research infrastructure was among the best in the world, which allowed its scientists to incorporate new instruments and agendas more quickly than, for example, researchers in the United States at this time. Of course, the number of journal articles praising cinema's potential as a new medical technology should not imply that the general practitioner put it to use on a daily basis. On the contrary, given the expense, skill, and patience required to operate a motion picture camera\u2014much less one adapted for medical use\u2014the average doctor never came in contact with one or even considered the possibility. During this period, the medical application of cinema remained confined primarily to university research laboratories and some medical school lecture halls, with a few privileged physicians leading the way. Still, German researchers enthusiastically applied motion pictures to a range of specialties from ophthalmology to gynecology.\n\nWhat did cinematography promise the intrepid researcher? This physician waxes poetic about microcinematography, but his claims are representative of all medical and scientific uses of motion pictures:\n\nMicrocinematography captures [ _fixiert_ ] what a scholar has witnessed and researched in his quiet laboratory, so that he can then demonstrate his collected research to a larger group, be it at a conference, before a medical society, or in an auditorium of students. But microcinematography not only documents movement processes, it is also a helpful tool for research itself. It captures every phase of movement and thereby allows the researcher to examine each image at length and to study the temporal and spatial relationships of the movements carefully and calmly. Microcinematography does not require hasty progress, as living movement often does; it allows the scholar the time he needs to grasp all worthwhile detail and lets him determine when to move to the next image.\n\nCompressed in this paragraph are most of the advantages typically cited in the early literature on the scientific application of motion picture technology. To recap briefly: first, the cinematic image, like the magic lantern slide, was projectable, and therefore could be shared among colleagues and students. This made it ideal for educational purposes, which required an efficient dissemination of information, but also for rhetorical purposes; the scientific film, like all scientific illustration, existed to present evidence meant to persuade others. The size and spectacle of the cinematic image had a powerful persuasive effect in itself, but part of that power came from the photographic character of the image. This speaks to the second advantage, its ability to \"capture\" or to document moving phenomena. Because its photographic base allowed it to record the variety and randomness of the natural world, the motion picture had an evidentiary status that was very useful for presentations and publications, because it could therefore act as a substitute for the living patient. This also meant that it could function as an object of study itself; it \"provides the researcher the opportunity to examine each image at length.\" This is the third advantage: the spatial and temporal malleability of the motion picture extended the senses and allowed the researcher to explore new domains previously unavailable. In addition, the individual frames of the film effectively managed the data, especially the informational complexity of any kind of movement. Indeed, the most noteworthy aspect of this excerpt is its enthusiasm for the researcher's ability to control the rate of movement of the motion picture. The author lauds the temporal control that came with the moving image and emphasizes the slow pace of scientific study, the almost contemplative stance before the image: \"in his quiet laboratory... to examine each image at length... carefully and calmly... the time he needs.\" The rest of this chapter will explore these advantages in detail.\n\nBut for now, this testimonial will stand in contrast to other descriptions of film common in the medical community\u2014those that decried cinema as a threat to public health. As medicine's prestige grew, physicians expanded their expertise to broader social questions and offered their prescriptions for a healthy nation. So alongside the enthusiasm for the scientific potential of motion pictures there was a separate yet related discourse that was not so enthusiastic. Between 1904 and 1914, reformers and physicians in Germany consistently viewed the new medium of motion pictures with suspicion. They not only protested the dank, unventilated storefronts in which many motion pictures were then being screened, but also felt that the motion pictures themselves presented a serious danger to public safety. The sensational subject matter of many of the movies threatened to corrupt the taste of the nation. But a particularly insidious form delivered this content, as this doctor attests:\n\nThe effect of this sensational subject matter is heightened by the _temporal concentration of events_. The cinema concentrates the sensations of a detective story or thick trashy novel into 10 or 15 minutes. The resulting psychological effect is thus completely different. When reading, we can pause at will, critically reflect on the text, relieve ourselves of the internal pressure through contemplation, digest the scary parts. In the cinema, such excitation of the passions is intensified and multiplied by the rapid succession of vivid images that passes before our eyes. There is no time for reflection or release, no time to find an inner balance. These spine-chilling and grotesque \"dramas\" distress the nervous systems of young and sensitive persons to the point of suffering, but without providing the spectator any of the means with which he usually defends himself against attacks on his nervous system: quiet _contemplation_ , _intellectual assimilation_ , and sober criticism are not possible.\n\nThe contrast with the previous quotation is instructive. If cinema provided the _researcher_ the opportunity \"to examine each image at length and to study the temporal and spatial relationships of the movements carefully and leisurely,\" it did not afford _the average moviegoer_ that same luxury\u2014\"quiet _contemplation_ , _intellectual assimilation_ , and sober criticism are not possible.\" The difference between the (boring) research film and the (exciting) film drama perhaps played a role in this characterization, but even more important for the purposes of this chapter was cinema's temporal rush, its insistent and impatient movement forward, which this doctor felt could have a lulling, even hypnotic effect. Those without sufficient training or education were susceptible to cinema's lurid charms. But the educated, professional class apparently had the tools to manage this onslaught. What were these tools? Most obviously, researchers, unlike the average moviegoing audience, were in control of the image and its projection\u2014they could literally manipulate their films however they pleased. But this ability to control the rate of projection went hand in hand with a mode of viewing that was very much part of a nineteenth-century researcher's training and identity, which distinguished itself from another mode of viewing attributed to the lay public, especially the cinema audience.\n\nExpert training depends on developing a set of skills, but it also distinguishes itself from those who lack the skills. In the medical profession, this dynamic determined the nature of the doctor\u2013patient relationship, or the nature of internal debates about midwives and quacks, for example; the medical profession relied on the constant maintenance of that hazy line between \"normal\" and \"abnormal,\" or between \"healthy\" and \"diseased\" in the broadest sense. Medical observation similarly depended on its opposite\u2014in this case, spectatorship. The debates about motion pictures are an excellent example of this dynamic. Comparing these two discourses allows us to see quite clearly the criteria for cinema's legitimacy. By understanding what German doctors considered to be the _proper_ mode of observation, we come to understand more fully what they thought was an _improper_ way of viewing images, and vice versa. We also see that this proper mode was not simply the result of disciplinary training. The \"objectivity\" of the scientific eye arose not merely out of professionalization but also in contrast to the \"subjectivity\" of the untrained other. That is, disciplinary modes of viewing relied on _class_ distinctions, as well as professional training.\n\nSo to appreciate the medical community's diagnosis of film spectatorship as passive, weak-willed self-abandonment to the flow of images\u2014a characterization that was not uniquely made by doctors, but which is best understood with reference to them\u2014we must take into account its investment in observational practice, especially as it relates to experiment and temporality. Physicians and their educated brethren distinguished themselves from the layperson via their viewing practices: by incorporating film into experimental method, for example, they observed and contemplated the image at a leisurely, disinterested pace, while ordinary spectators were swept up and seduced by it. These observational practices were therefore more than cultural capital, more than a sign of expert learning and status\u2014they were a badge of honor that also served as a shield against the encroachment of mass culture and the rush of modern life. How one approached cinema's temporality, it seems, determined whether one was an observer or a spectator.\n\nThe rest of this chapter consists of three sections. Because the early history of medical filmmaking is not yet common knowledge in film studies, the first section provides an introductory survey of the early use of motion picture technology in medical research and training, focusing on such work in Germany. This section will examine the multiple functions of film in the medical context by focusing on the needs that motion pictures served in the international medical community. (This discussion will focus on medical research and training films\u2014films by physicians for physicians\u2014rather than medical films intended to educate the general public.) The second section explores how the use of motion pictures corresponded to established principles of observation in medicine. It will also demonstrate the professional investment in this particular mode of viewing. While the first section emphasizes film and the history of medicine, this section focuses on film and the _philosophy_ of medicine. It therefore investigates the use of film in relation to medical logic. Finally, the last section takes a closer look at the medical editorials about the dangers of cinema, especially its hypnotic power, in order to illuminate the moral, ethical, and class dimensions of medical observation and cinematic spectatorship.\n\nTHE MULTIPLE FUNCTIONS OF THE MEDICAL FILM\n\nMost histories of research films recount individual cases within various disciplines or fields (just as I did in chapter 1), but it would be worthwhile at this point to step back and sketch broader patterns of use across specialties. Describing these patterns in terms of \"function\" has the advantage of accounting for the full life of a film beyond the immediate task. As we have seen, motion pictures functioned in various ways for researchers; they extended the senses, of course, but they also documented and displayed data. Even though a researcher made a film for a specific purpose\u2014to examine the mechanics of an organ, for example\u2014that same film also functioned as a record of the case and as an educational aid; it was a rare medical film indeed that had only one use. I would argue that there are three main functions of the research film: _exploratory_ , _documentary_ , and _educational_. These broad functions could be applied to much nonfiction filmmaking as well, but as a triad they are especially applicable to research films of all sorts, medical or scientific. Generally speaking, research films are made to explore unknown domains or relationships; to document and display results; and to train students, teachers, or, sometimes, the general public. These functions are by no means mutually exclusive; any film can have multiple, even simultaneous, functions. In fact, understanding how a film functions necessarily entails pinpointing who is using the film, when, and for what purpose.\n\nWe can group various cases in the history of the medical film around function, a retrospective critical category, but also around specific, historical needs that motion pictures satisfied for medical research. For example, as we have seen in physiology and biology, researchers at the turn of the century were increasingly interested in process and function over structure and morphology; the discovery of X-rays only heightened the prevailing interest in the mechanics of organ function, for instance. Motion pictures also proved to be a valuable technology for visualizing, documenting, and measuring the functions of the human body. Likewise, just as photography was an important method of documenting cases, motion pictures also proved indispensable for recording, for example, pathological movement. Such documents served as entries in an archive of images that medical professionals recognized to be vital to the health of their discipline. Physicians established image archives in hospitals and other research centers in Europe and the United States, and calls for similar motion picture archives of medical films were common at this time. Physicians recognized the importance of these films for medical education as well. While the profession was very self-conscious of the modernization of medical education that had taken place since the Enlightenment\u2014replacing the medieval, scholastic model of book learning with an emphasis on hands-on training and observation\u2014the practice of live demonstration of patient symptoms in medical lectures was logistically and ethically troublesome. Motion pictures offered a way out of this dilemma and presented a novel solution to issues in medical training. The rhetoric surrounding the use of film as an educational tool mirrored the broader discourse on pedagogical applications of cinema, but the medical training film illustrates especially clearly the contemporary emphasis on efficiency and modernization. So this section presents a brief survey of early medical filmmaking using the three functions as a preliminary heuristic, but it will also chart film's meanings in the context of specific issues (organ function, image archives, live demonstration, and efficiency) important to the medical profession at this historical moment.\n\nTHE EXPLORATORY FUNCTION\n\nOf course, the motion picture camera can be used to explore facets of the natural world below or beyond the threshold of human perception, such as the extremely small (via microcinematography) or the extremely slow (via time-lapse cinematography). The various means by which motion picture cameras can manipulate time and space have been very important to research films. For many medical professionals exploring the rhythms and movements of the human body, this ability to manipulate time also contributed to its use as an _experimental_ instrument, more than simply as an observational aid or recording device. Since the mid-nineteenth century, branches of the medical profession in the Western world had incorporated a scientific ethos into their research and practice. Even though this trend was often contested, by the beginning of the twentieth century, the role of experiment in medical research was firm, if not universal. The experimental use of motion pictures certainly had precedent in the chronophotographic research in physiology of Marey or of Braune and Fischer, but also in the increasingly frequent use of photography in medicine. How can the motion picture camera function as an experimental tool? For that matter, what does an experiment do? To answer these questions, let us consider a fairly well-known example in the history of medical filmmaking: Austrian cardiologist Ludwig Braun's use of a motion picture camera in 1897 to record the beating heart of a dog. A specialist in cardiac dynamics and mechanics, Braun employed frame-by-frame analysis to gauge changes in the size and position of the heart as it beat by measuring the shape and displacement of shadows and other markings on his filmed images. From these short strips he was able to extract conclusive information about, for example, the nature of the apex beat (the lowermost and outermost prominent cardiac pulsation) (fig. 2.1).\n\nFIGURE 2.1. Frames from Ludwig Braun's film of a dog's beating heart\n\nBraun used the camera in a way that instruments are often used in scientific experiments, that is, to test a theory, to measure constants, or to explore new domains. A scientific experiment is usually designed first to isolate and stabilize variables (such as temperature, air pressure, etc.) surrounding an event or phenomenon. An experiment is also designed to be consistently repeatable, not only so that other researchers can check the results for themselves, but, more importantly, so that the event itself can be scrutinized closely. And an experiment rarely stands alone\u2014usually it is one of a series of experiments, so repeatability is crucial for the success of the series as well. Time itself cannot be manipulated, of course, but an experiment allows the duration of an event to be significantly extended through repeated viewings. These three features of an experiment\u2014isolation, stabilization, and repeatability\u2014also allow the researcher to register quantifiable changes in the environment under study. A thermometer is an example of an experimental instrument that isolates and quantifies one particular variable, temperature. Other instruments are used to control or to create an (often artificial) environment for the experiment; the air pump, which creates a useful vacuum, is a good example. Film does not control or alter an environment to a great degree, but it can create something of a _new_ environment, which acts as a record of or substitute for the event. Through its framing and focus, for example, film isolated and stabilized the phenomenon for closer study; a beating heart is difficult to examine, so the filmed record of the heart became a \"working object\" that could be examined repeatedly. The motion picture was the best kind of repeatable experiment: if the record could function as a substitute, it could be endlessly repeated without the work involved in setting up the actual experiment again and again. In this way, as noted in the previous chapter, the motion picture functioned as a \"virtual experiment.\" Researchers also employed motion pictures to register and measure changes by using the constant frame size and frame rate of the film to calculate changes in time and space. Braun used the displacement between frames to measure the size and position of the heart. Finally, the success or legitimacy of an experiment is enhanced if its results can be somehow inscribed (via a published report or visual demonstration) and easily presented or exchanged. In the absence of the actual experiment, the researcher's report of the experiment allows readers a kind of \"virtual witnessing,\" as Steven Shapin calls it. Film, of course, facilitates this \"witnessing\" in an extremely persuasive way. All these features contributed to the motion picture camera's success as an experimental instrument.\n\nBraun's work also exemplified long-standing medical attention to the mechanics of organ function, which was only intensified with the discovery of X-rays and the development of motion pictures. As we know, the discovery of X-rays generated incredible enthusiasm\u2014in 1896 alone there were more than 1,000 articles published on the phenomenon. Much of this excitement came from the medical community, which immediately recognized benefits; engineers and experimenters sought to combine X-rays and cinematography. In 1897, Scottish physician John Macintyre used a cinematograph to take several X-ray exposures of a frog's leg on motion picture film to create the first moving X-ray image. Others continued to work on the combination to visualize organ function. For example, as early as 1903, Dutch physician P. H. Eykman used X-ray cinematography to study the swallowing mechanism in humans. Similarly, between 1903 and 1906, Joachim-L\u00e9on Carvallo at the Institut Marey made a series of X-ray films of swallowing and digestion in small animals. Starting around 1909, the German radiological team of K\u00e4stle, Rieder, and Rosenthal made X-ray films of joint, lung, and peristaltic movements. And American Lewis Gregory Cole likewise made films of gastric phenomena. But the combination was always plagued by difficulties: the low X-ray-tube power and slow emulsions of the early days resulted in weak exposures or overlong exposure times, on one hand, or the mechanical problems involved in capturing the image, either directly with large sheets of X-ray film (known as the \"direct method\") or indirectly through fluoroscopic screens (the \"indirect method\"), led to unstable images, on the other. Franz Groedel was the acknowledged leader in X-ray cinematography; between 1909 and 1915 he worked tirelessly to develop a viable direct method, including a mechanism consisting of a series of falling cassettes (fig. 2.2). His commitment to the idea of moving X-ray images never wavered: \"In spite of [the] present inadequate apparatus,\" he said in 1915, \"important questions have already been solved, or the solution thereof brought nearer.\"\n\nFIGURE 2.2. Groedel's serial cassette X-ray apparatus\n\nMany physicians also hoped that cinematography, and especially X-ray cinematography, could be used diagnostically, that is, as a way of searching for clues that aid the identification of a disease or injury, usually via the correlation of visible symptoms with known disease entities or classifications. But even as Groedel and others lauded its potential, the actual diagnostic value of X-rays and X-ray cinematography remained in doubt for many years. In 1912, one prominent leader in radiology said of X-ray cinematography, \"Of that, I don't expect too much. It will be didactically valuable, explain some processes, but it will not be able to serve diagnostic purposes in every instance.\" Nevertheless, researchers and engineers put considerable energy into crafting X-ray images that moved. French radiologist Andr\u00e9 Lomon collaborated in 1911 with scientific filmmaker Jean Comandon, known best for his masterful microcinematography, to produce moderately successful results with indirect X-ray cinematography. They worked on this approach up to the outbreak of World War I, which \"relegated all roentgen cinematographs to storerooms and museums.\" Groedel, Lomon, and others continued to work on the technical problems after the war, but it was not until the 1930s that the direct method proved regularly feasible, and it took until the 1950s for the indirect method to have daily clinical potential. Despite the inherent difficulties, X-ray cinematography promised at the very least the possibility of viewing hidden processes and movements that would help in understanding organ function and perhaps in diagnosing illness and injury. Indeed, what distinguished the exploratory function, whether manifested in experiment or diagnosis, was the possibility of _discovery_ \u2014the image presented a view that revealed something hidden from human perception. Yet to see something hidden, researchers often limited some variables and expanded others, a process made more convenient through motion picture technology, which could isolate or enlarge spatially (through lens or framing choices) or temporally (through recording or projection speed choices), or, as in the case of X-ray cinematography, film could be combined with other technologies that offered new ways of seeing. So film's exploratory function emphasized discovery, but it also underlined and depended upon film technology's formal analog to the functions of experimental restraint and modularity.\n\nTHE DOCUMENTARY FUNCTION\n\nIf the exploratory function charts new domains and reveals hidden spaces in order to help to understand processes or to identify disease, at some point the domain has been identified and the diagnosis agreed upon. At that point, film serves as an illustration of that which is already known; it becomes equivalent to the statement \"Here is an example of X.\" This is the documentary function of medical cinema. For anything to function as a document, it must have some evidentiary value; the evidential authority of the motion picture derives primarily from its photographic base, which has a well-known rhetorical power. Such is the nature of this image that it served meaningfully as a substitute for the object filmed, especially if the phenomenon was rare or difficult to reproduce in demonstration. This would explain the popularity of motion pictures among neurologists at the turn of the century, who used film to document examples of pathological movement associated with neurological disorders. For example, in 1897, neurologist Paul Schuster utilized the resources of Berlin clinics to create a series of short films of patients with a variety of neurological diseases, including Parkinson's, multiple sclerosis, myoclonus, hemichorea, and ataxia. These single-shot films, between three and ten seconds each, emphasized pathological movement and were designed primarily to illustrate Schuster's lectures without having to rely on live demonstrations of the patients. Schuster also hoped that these films and others like it would someday form an archive of material for medical educators. There were many other physicians and neurologists in Europe and the United States who saw the same potential: Gheorghe Marinescu in Bucharest, Paul Richer at La Salpetri\u00e8re in Paris, Walter Chase in Boston, Arthur Van Gehuchten in Louvain, Emil Kraepelin in Munich, Hans Hennes in Bonn, Osvaldo Polimanti in Naples and Perugia, and Theodore Weisenburg in Philadelphia\u2014all used motion pictures to document nervous disorders.\n\nThese films were especially helpful in medical demonstrations, for which the use of live patients was always troublesome. While the live demonstration was a major advance in medical education\u2014a huge step from the centuries-long, scholastic tradition of learning medicine only from ancient texts\u2014it had its own set of challenges. According to an American student taking classes at the Allgemeines Krankenhaus in Vienna in 1865, each professor was provided with a lecture room near his ward: \"At the time of lecture this room is filled in with 'specimens' in the shape of men and women who are transferred from the other wards for the occasion. These patients are looked upon and spoken of as 'material' for the medical instruction and as such are submitted to examination by the students without much reference to any feelings which they as men and women may have on the subject.\" Patients did not submit gladly, apparently. In another letter, the student draws a sketch of his routine at the hospital, which includes \"scolding and pitching into the patients for coming late (wh. they always do in Vienna).\" Motion pictures promised an alternative, as an obviously frustrated Hans Hennes noted in 1910:\n\nHow often does it happen to the professor that a patient fails during a lecture, that a manic suddenly changes his mood, a catatonic suddenly fails to perform his stereotyped movements. Although he executed his pathological movements without disturbance on the ward, the changed environment of the lecture hall has the effect of not letting him produce his peculiarities\u2014so that he does not display precisely what the professor wanted him to demonstrate. Other patients show their interesting oddities \"maliciously,\" only when there are no lectures, continuing education courses, and so on. Such occurrences, which are frequently disturbing to the clinical lecturer, are almost completely corrected by the cinematograph. The person doing the filming is in a position to wait calmly for the _best possible moment_ to make the recording. Once the filming is done, the pictures are available for reproduction at any moment. Film is always \"in the mood.\" There are no failures.\n\nHennes seemed not to be bothered by any ethical dilemmas; his concerns were purely practical. Motion pictures might have accidentally solved some ethical predicaments, but as Lisa Cartwright has demonstrated, filming patients often led to others. Film technology, in this respect, followed a pattern in medical history in which new instruments create a psychic and physical distance between doctor and patient and thereby further tip the power dynamic in the physician's favor. Physicians were already pleased to substitute photographs and slides for patients in lecture, if not to alleviate the obvious ethical concerns, then at least to present all the students with a larger, projected view. Schuster intended his films to substitute for live demonstrations as well: \"We wanted to use recordings of typical movement complexes to allow students to see the theoretically oriented lecture illustrated with motion pictures, regardless of the clinic's available material [i.e., patients].\"\n\nNeurologists such as Schuster also found motion pictures especially exciting, because the records could be used for further analysis of pathological movement. (This dual use is an example of the function of the film changing according to who is looking at it, students or researchers.) It was also not uncommon to use photography and motion pictures to document intervention outcomes. Orthopedic surgeons, for example, used motion pictures to film \"before and after\" records of corrective surgeries. These films were presented at congresses to persuade other physicians of a given diagnosis and therapy, or they were presented to students as examples of a particular surgical strategy. Even Comandon's motion pictures of microorganisms could be classified under this general rubric of \"demonstration.\"\n\nMedical films, like medical photographs, also appealed to researchers and educators because they could be part of a disciplinary archive of images. Hospitals such as the Saint-Louis in Paris, Bellevue in New York, and the Charit\u00e9 in Berlin established photographic departments for just this purpose. A report from Bellevue in 1869 indicates that a photographic archive and department could be a magnet for the discipline: \"Members of the medical profession begin to visit the Department periodically, for the purpose of obtaining such photographs as pertain to each one's more especial class of investigation. Many interesting cases of skin disease, fractures, and results of important surgical operations have been fully illustrated by series of photographs, which give opportunity for comparison and study not offered by any other means.\" Motion pictures promised the same opportunity for comparison \"fully illustrated by a series of photographs,\" so to speak. Calls for an archive of film records were common. In his important essay on the use of motion pictures in neurology, Hennes saw the immediate value of a permanent storehouse of medical films: \"If one were to collect the most interesting cases in this way with universal participation and support, a cinematic archive, analogous to the phonographic one, could be created, the lasting value of which could certainly not be denied.\" Franz Goerke went beyond medical cinema in his call for an archive:\n\nIs it not therefore obvious that we should take up the idea of creating a central collection place, a state archive for the deposit of motion pictures and so save them from destruction? Just as museums are collection places for works in the fine arts, just as the developments in the sciences have their place in city and state collections, just as libraries are protecting the intellectual achievements of mankind, and just as documents and government papers are kept in state archives, so equally important is a state archive for motion pictures, which will allow researchers in future with the help of other sources and sharply drawn conclusions to reconstruct a visual image of former times.\n\nFor researchers of all sorts, not just physicians, the idea of motion pictures as documents led inevitably to the impulse to collect and store them. As we shall see, this impulse was not exclusive but was especially pertinent to medical thinking. Indeed, of all the possible hopes the medical community had for photography and film, the dream of a universal and portable archive of cases was the most persistent.\n\nTHE EDUCATIONAL FUNCTION\n\nYet showing films was not as easy as it might seem. Despite the obvious suitability of motion pictures for demonstration or reference, film was simply not used as often as magic lantern slides in the lecture hall during the early period. Most universities lacked projection equipment, finding it expensive, cumbersome, and difficult to master. The films themselves, despite dreams of a ready archive, lacked reliable distribution outlets; other than the lists of films available from the major producers (Path\u00e9, Gaumont, Eclipse, Urban), there was no systematic information about available films on medical topics. The large firms generally distributed only the films they produced, so researchers such as Hennes had few options for distributing their films; procuring copies of someone else's usually involved a personal connection of some sort. Or, often, the films just did not exist anymore. Goerke complains, \"For a learned institute it is nearly impossible to obtain a film for an important learned lecture: we know of the film's existence, but in the meantime it has become a victim of destruction or oblivion.\" This is why the calls for an archive were so common: educators just wanted a reliable source of scientific and medical cinema. So if the use of film as an experimental tool or as a document attracted the interest of just a few dedicated (and historiographically visible) pioneers\u2014usually attached to well-funded research institutions in centers such as Berlin and Paris\u2014we cannot say the same for the clinical or educational applications of medical film, which were scarce, geographically scattered, and sparsely documented.\n\nStill, most medical filmmakers at this time cite as their motivation a desire to improve teaching. In fin de si\u00e8cle Paris, Eug\u00e8ne Louis Doyen, a maverick surgeon known for his innovative techniques and disdain for the academy, wrote, \"It has been with the object of completing our means of teaching the art of surgery that I have been led to study and employ the cinematograph.\" Doyen hired two cameramen to film his surgeries. Three of these films\u2014depicting a hysterectomy and a craniectomy\u2014were first presented at the July 1898 meeting of the British Medical Association in Edinburgh, where they were enthusiastically received. Despite this successful screening in the United Kingdom, the French Academy of Medicine refused to show Doyen's films to the membership; he had already run afoul of traditional medicine with his unusual (some would say risky) techniques. His colleagues were not about to let him have a forum. Doyen eventually made dozens of films, including one depicting his surgical separation of conjoined twins Radica and Doodica in 1902. During this time, one of his cameramen, Ambroise-Fran\u00e7ois Parnaland, tried to sell his prints of Doyen's films to Path\u00e9 and other companies, but it is not clear whether they were actually widely screened to the public at this early date. In any case, Doyen's controversial techniques and the graphic quality of the films further deteriorated his relationship with academic medicine. Doyen insisted that his films were didactically valuable; they were meant primarily to illustrate and publicize Doyen's tools and techniques, but they were also intended to serve as training films for surgeons.\n\nIn Germany, the watershed moment came at a February 1910 demonstration of \"Film in the Service of Medicine,\" in which the \"service\" was primarily educational. Representatives from the Berlin medical establishment were so tightly packed into the Kaiserin-Friedrich-Haus that the organizers had to turn people away. Among the films shown were two surgical films made by Doyen, the first featuring famed German surgeon Ernst von Bergmann and the second focused on Doyen himself. The principle organizer, Robert Kutner from Berlin, also showed his film demonstrating CPR techniques, which drew praise for their potentially wide appeal. Other films included James Fr\u00e4nkel's results of corrective orthopedic surgery, Braun's heart film, X-ray films by Carvallo and others, and Karl Reicher's microcinematographic records of single-celled organisms. While it was not the first time that films such as these had been shown in Germany, the event received much attention from the national and international medical community and helped to focus awareness on the educational potential of medical film. In a later speech, Kutner sang cinema's praises:\n\nAnd how convenient, how effortless!... The evidentiary power of [cinema] is more persuasive than that of any other document, exceeding even the most vivid description.... The motion picture projector demonstrates its most spectacular educational applications in auditorium demonstrations of microscopic or macroscopic images of movement. In a normal lecture-room demonstration of movement, especially that of small objects (think, for example, of a frog's beating heart), only a small part of the audience really sees anything, while everyone present can observe a film presentation equally well. Without the assistance of the motion picture projector, almost all X-ray motion pictures and certainly all motion pictures taken from a microscope could only be shown to a small group or even just to one person at a time.\n\nKutner described the pedagogical advantages of motion pictures in a language common to advocates of the educational film at the time, a rhetoric found in Germany but also elsewhere in Europe and in the United States. First, the rhetoric emphasized the \"vividness\" of the motion picture, usually in opposition to \"description\" or, more pointedly, book learning. What exactly is this \"vividness\"? The clarity, texture, and abundant detail of the photographic image combine with projected movement to give the image a _presence_ unlike any previous representational form. Its level of detail allows the photographic image to reproduce patterns of texture and variation, hence to represent the structure and randomness of the natural world, while the movement of the image presents this world in real time in a particularly striking way. The object \"lives\" onscreen. This is perhaps obvious, but it is all to say that \"vividness\" referred to the sense of presence that the moving image evokes. For early advocates of educational film, it was as if the thing itself were there in the room, available to direct perception. Film thereby functioned as an object lesson, an acceptable substitute for the thing itself (in chapter 3 I will discuss the educational implications of motion pictures as an \"object lesson\" more thoroughly).\n\nThis permitted Doyen to extol the virtues of the motion picture over books and corpses. Complaining about the inadequacy of \"surgery of the dead\"\u2014the practice of rehearsing operative techniques on cadavers\u2014to confer a sense of the surgical experience, Doyen asked, \"Do our books fill the gap thus left? Certainly not. The most detailed descriptions, the best diagrams or photographs of the various steps of an operation are inadequate.... It is not sufficient to follow the operation, as it were, secondhand; rather, the author of the technique, the master himself, must be seen at work. The surgeon is judged by his work, and no text-books, however well-illustrated, can sufficiently express his personality.\" In motion pictures, on the other hand, Doyen found a perfect medium to express vividly the personality of \"the master himself.\" Movies were not \"secondhand\"; they allowed Doyen to be \"present\" to the students. Especially noteworthy is Doyen's concept of \"personality.\" Doyen was not publicity shy, by any means, but he was not concerned here with conveying via a medical film his charisma and good looks, or not only those things. Primarily, his films were meant to present his technique and his tools. More specifically, they demonstrated how Doyen held himself and how he moved to accomplish his task. Film provided, better than any other previous medium, a demonstration of the actual movements required in surgery. Doyen's \"personality\" is his \"posture\" or \"attitude\"\u2014his _embodied_ technique. To convey that \"personality\" was to presume that the student would copy it, that while the student watched the film, a kind of kinesthetic empathy took place whereby the movements seen were somehow felt or incorporated into the student's own body. This is the mimetic presumption of most training films, it seems; many training films expect that we will copy their movements and that the student will take on the \"personality\" or \"attitude\" of the master.\n\nSecond, Kutner emphasized the _efficiency_ of the moving image for the pedagogical task. \"Efficiency\" was a mantra repeated by all sorts of social engineers in Europe and the United States around 1900; the medical community also chanted its name, especially but not exclusively in the United States. Efficiency was a key concept in transforming the turn-of-the-century hospital from \"a well of sorrow and charity\" into a \"workplace for the production of health.\" In the United States, for example, from around 1900 to 1920, health officials were increasingly dissatisfied with the duplication of services, the lack of coordination of units, and the general low level of effectiveness in patient care among clinics, dispensaries, and hospitals nationwide. \"Efficiency\" became an institutional logic to promote standardization of facilities, services, and administration. In fact, in the United States at least, efficiency was the rubric through which the modern hospital adopted business practices to establish itself as a desirable place for treatment and to attract paying patients. _Modern Hospital_ , the organ of the American Hospital Association, devoted itself to promoting economy and efficiency in hospital management, while the American College of Surgeons was established initially to focus on the standardization of tools and techniques within surgical practice.\n\nKutner's praise of film also relied on this concept of efficiency. Here he refers, of course, to economies of scale: the simple claim that more people could see a large projected image than could see a small image. As medical school enrollments grew steadily toward the turn of the century, this claim gained traction\u2014lecturers used projected images more and more from the 1870s onward. But Kutner had in mind another form of efficiency; when he wrote \"how convenient, how effortless!,\" he was probably not referring to the motion picture apparatus, which was definitely not convenient and effortless. Instead, he was referring to the efficiency of the image itself. It had a \"persuasive evidentiary power beyond that of any other document, beyond even the most vivid description.\" For Kutner and others, that power came naturally to the image, especially to the photographic image. It worked quickly and effortlessly on the spectator. When Doyen insisted that \"with the cinematograph we can make hundreds of people follow in one minute what a whole lecture could not make clear to a limited number of students,\" he made a similar claim about cinema's efficiency: not simply about numbers of students but about the immediacy of the image versus the indirectness of the spoken word. If the image was direct, instantaneous, vivid, and penetrating, then the written or spoken description was aloof, dull, and circuitous. In a way, Kutner's and Doyen's preferences echo a bias common in modern medical education. The nineteenth century continued a long transformation in medical education (and education in general) that emphasized direct perception of the objects of study over their mediated presentation in books. The discussion of film as an educational tool made this bias even more explicit. The direct perception of objects was seen as a much more effective and efficient mode of learning. The image was perceived as efficient because it could affect the viewer immediately, like a drug, or a blow to the head, whereas the word must be read or spoken and then processed cognitively, all of which takes time. The image was understood as physical and immediate, while the word was intellectual.\n\nPrecisely these perceived qualities of the filmic image\u2014its presence, its directness, its immediacy, and its ability to prompt mimesis\u2014made it an especially efficient training tool in the eyes of physicians and others. Protected by the twin firewalls of expertise and closed community, medical professionals could extol the virtues of film for educational goals. Once films such as these escaped these protections, however, they had the potential to become scandalous, as Doyen's example demonstrates. Furthermore, these cinematic properties _themselves_ were condemned as part of a system of commercial amusements. As we shall see, the objection to cinematic spectatorship revolved around precisely the same constellation of properties; the difference was not only the type of films (fiction versus nonfiction) but also the protections in place for regulating cinematic projection. Under the watchful eye of the educator, film's immediacy could be mediated, paradoxically, by the written or spoken word. The educator provided an expert framework, such as a lecture and projection control, that corralled and attenuated cinema's more vivid effects. In the movie theater, however, the content and its immediacy were left unchecked, which often led, these experts complained, to mimesis of a criminal sort. The paradox of the rhetoric of the educational film was that even though the moving image exemplified the very image of efficiency, that efficiency cut both ways, for good and for ill. The difference between these two visions of film was also the difference between observation and spectatorship, which is the subject of the following sections.\n\nMOTION PICTURES AND MEDICAL OBSERVATION\n\nThe exploratory, documentary, and educational functions of the medical film give some indication of how motion pictures were and are used by the medical community. But they do not, by themselves, present a clear picture of how motion pictures fit established, disciplinary observational models, nor do they explain the shrill and regular objections to movies in the theaters. The next section will examine those protests in detail, but the argument of this section is that to understand those complaints, we must first understand the researcher's investment in a particular mode of viewing related to medical and scientific observational practices. The condemnation of \"movies\" in the public sphere and the praise of \"motion pictures\" in the scientific context have in common a stance about the proper way to process visual information. This section outlines that position by exploring how researchers processed data in the motion pictures they used for medical purposes. On one hand, as we have already seen, medical researchers found the ability to analyze movement frame by frame to be an enormous advantage of motion pictures. On the other hand, the uses for motion pictures were not exclusively analytic\u2014the moving image itself was also very important for medical purposes; the efforts to perfect cineradiography are a good example of this application. In either case, however, the patterns of use reveal a need to regulate the insistent temporal push of the moving image. The patterns also correspond to certain common features of medical observation, especially the importance of the series to medical logic and the claim that observational accuracy depends on the observer's willingness to be methodical and leisurely.\n\nIn other words, I argue for a homology between observation and film; more precisely, there are certain features of film\u2014namely, its repeatability, its photographic detail, and its ambidexterity between movement and stillness\u2014that correspond to features of observational practice emphasized in the medical literature. Moreover, certain aspects of observation, such as correlation and description, could be easily adapted to film technology. Selected observational practices appear to have found purchase in the cinematic image. This section will chart these correspondences.\n\nMy approach to the argument will be unorthodox, however. Rather than survey the history of medical observation and compare it with early uses of motion pictures in medicine, I will take one statement as representative of all. In an 1897 presentation to the Society of German Natural Scientists and Physicians (Gesellschaft deutscher Naturforscher und \u00c4rzte), Ludwig Braun listed six characteristics of cinematography most important for his work. Braun provided, at the very beginning of the medical community's interest in the technology, an articulate and representative example of the discourse on medical film. Taken collectively, all six theses recognized and applauded the exploratory, documentary, and educational functions of motion pictures in the medical arena. The film provided a \"good, detailed\" document that could be used as a working object or substitute for the heart itself. The nature of the medium allowed frame-by-frame exploration of the movement of the heart. And the film could be presented in an educational setting, where everyone could see the movement clearly. Throughout the history of medical film, these have been the primary advantages that advocates have listed in favor of using this technology. Braun's list can therefore function synecdochially for the larger discourse on scientific and medical film. But it also demonstrates quite clearly the elements of medical observation most germane to his use of film. So in what follows, I examine each of his theses as a pretext to an explanation of medical logic, the relationship between analysis and synthesis in medical observation, and the temporality of the medical gaze.\n\n1. _The cinematograph delivers a long series of chronophotographic images. Each individual image has the virtues of a good, detailed photograph. The film can record the movements of a normal-colored, physiologically moving heart just as well as those of a heart that has been purposefully manipulated in various ways that serve the specific experiment._\n\nAt first, the last sentence appears out of place; its discussion of film's ability to record a manipulated organ equally as well as one that has not been altered seems both obvious and odd, given the emphasis on the photographic character of the medium in the previous sentences. But it is precisely the history of the use of photography in medicine to this point that prompts the statement. Despite the popular conception of photography as a reliable document, in 1897 there was a wide gap between what one saw in an organ and what one got in a photograph. The extreme variability in film emulsions, for instance, along with equal variability in photographic expertise meant that \"standardization was not one of photography's strong points\" in the nineteenth and early twentieth centuries. For example, until the introduction of orthochromatic and panchromatic photographic emulsions in 1884 and 1905, respectively, the reddish-yellow color of anatomical specimens came out too dark in photographs unless special precautions were taken. So if a researcher were interested in the color or texture of an organ, some sort of manipulation was required to create a good photograph. Braun emphasized here, however, that when using motion pictures, movement is the object of study, so one does not _need_ to make these adjustments, although that option is still available. For the purposes of recording a \"physiologically _moving_ heart,\" film's problems with color and texture were less relevant, thereby giving motion pictures a hidden advantage over the \"good, detailed photograph.\"\n\nNevertheless, we cannot ignore the importance of the \"good, detailed photograph\" in Braun's scheme. The photographic character of the image was the foundation upon which Braun built his argument for motion pictures. But the most important aspect of the image was its _repeatability_ : that \"each individual image\" could be examined as a photograph and that there were a series of them available for that purpose. This statement is one of the clearest indications that, as Lisa Cartwright has noted, medical researchers at this time used film primarily as series photography. But why was series photography interesting for physicians? How did it fit their disciplinary needs? Series photography presented, in a representational form, what we might call the \"statistical\" aspect of medical logic. Ludwik Fleck, in a 1927 article on \"Some Specific Features of the Medical Way of Thinking,\" argued that the fundamental cognitive task of medicine is to find a law for irregular phenomena. Medicine attends to the human body, which is a wildly variable and stubbornly individual object. Finding some sort of consistency in this variability, especially when focusing on atypical, morbid phenomena associated with disease, requires the examination of many, many cases. The clinic was the prototypical site where this observation occurred; physicians used \"statistical juxtaposition and comparison of many such phenomena\" to arrive at an \"ideal\" picture of the disease. Hence, \"the role of statistics in medicine is immense. It is only numerous, very numerous observations that eliminate the individual character of the morbid element\" and allow a regular \"law\" to emerge, to the extent that this is possible. Medical logic relies on series of cases found in clinics to come to an understanding of the \"facts\" or symptoms of a particular disease. Through the repetition of these facts and their variation, the researcher begins to see the pattern that becomes the ideal picture of the disease.\n\nCertainly, observing a patient in a clinic is much different than looking at a photograph or a film. The physical presence of the patient calls for not just a visual examination but palpation, percussion, auscultation, and other physical diagnostic techniques, not to mention the active questioning of the patient, none of which can be done to a photograph. But as Bernike Pasveer explained, part of the process of making images diagnostically viable involved encoding the images with information gathered by physical examination. That is, a successful X-ray image of a patient with pulmonary tuberculosis was able to show information (such as position of the lesions) that one could render or confirm from auscultation, for example. Learning to read the image required correlating sight and sound. The image thereby became a visual analog of the sounds and sensations obtained by physical examination. At that point, the photograph became a legitimate working object and could be compared with _other_ photographs. Photographs could be compared like cases, but the photo could even become the basis of comparison of these cases, in that the image eventually standardized a vocabulary and a way of looking at the disease (and the patient).\n\nSo series photography acted as a formal analog to the series of cases in a clinic, without having either the clinic or the actual cases present. Photography functioned \"statistically\" by presenting a series of images of different cases to be compared, or different views of the same case, either spatially (for example, different angles of view) or temporally (for example, tracking the stages of a disease as it develops). However, it is very important to emphasize that at this point in the development of medical cinema, the diagnostic value of photography, series photography, or cinematography was very limited. (Even the diagnostic value of X-ray images at this time was restricted.) Usually the image functioned as a document of an already established diagnosis or as the presentation of a question for others to answer. But the image itself did very little work in the detection required to establish an etiology of disease. Yes, the archive of images did help in the recognition of _patterns_ of disease\u2014a series of photographs might declare, \"Here is the course smallpox takes over nine days,\" for example. But the cause of the disease remained hidden to the camera. In this respect, the value of series photography and motion pictures for Braun and others was more experimental than diagnostic; the images functioned to control variables, to manipulate time, and to measure phenomena more than they detected causes or invisible relationships. It was only with the development of radiography and cineradiography that this possibility was broached, but the true diagnostic potential of the moving medical image was still decades away.\n\nSeries photography was much different than still photography in that the series was capable of depicting the duration and transformation of disease. But series photography and Braun's use of cinematography were similar to still photography in one clinic-like respect: they all functioned as an aid to _correlation_ , by which I mean establishing a mutual or reciprocal relationship between objects or events, such as correlating lesions in a cadaver to disease elements in a living patient. The clinic, as a collection of similar cases in a single location, functioned as an archive of the common symptoms and development of a disease. Series photography worked in that way as well. Photography's ability to isolate, frame, and repeat similar cases was a powerful aid in the standardization and multiplication of observational views. In addition, the arrangement of photographs in a series allowed not only their sequential organization but also their _simultaneous_ presentation. Georges Didi-Huberman has argued that Charcot's arrangement of his patients into living tableaux functioned like tables of data by organizing their signs into simultaneous events. Photography allowed this same organization, and much more easily. In series photography, the sequence was important, because it suggested a causal order or chronology, but the simultaneous display of images was arguably equally essential to the process of comparison and correlation. Series photography, as a research tool that allowed both sequential analysis and simultaneous display, thereby succinctly articulated the ideals of medical observation and logic. Braun's use of motion pictures as series photography also matched this logic, especially in the way he compared the images, as the next theses demonstrate.\n\n2. _The study of the resulting\u2014especially the enlarged\u2014images allows the analysis of movement, the recognition of every intermediate state and every phase of the process, and with that, a more exact assessment of the resulting transitional steps than was possible before._\n\n3. _The individual photograms are very uniform. If one lays two successive images on top of each other, those parts that remain motionless match perfectly, while the moving parts show positions that correspond to the differences in their movement. From this, one can perceive the spatial displacements, judge them better than before, and to a certain extent measure them and calculate the speed of the displacement in space from the time of exposure and the number of exposed frames._\n\nClearly, medical observation is not simply passive looking. Observation calls on more than vision alone\u2014it requires all of the senses combined with an extensive linguistic and logical apparatus. Foucault described \"the clinical gaze\" as a perceptual act sustained by logical operations, and correlation is one of its primary logical maneuvers. Medical photography and film functioned in many ways, but they operated primarily, I argue, as an aid to correlation. In fact, with his detailed description in thesis three of how he compares individual images, Braun offered an example of how correlation works. He laid two images on top of each other, aligning their similarities; from this alignment, the salient differences emerged. Likewise in the clinic, the repetition of cases over time created a cumulative, collective \"image\" of the disease, in which the more or less constant aspects of the entity were stabilized and the variations were set aside for further study, then reintegrated into the \"picture\" of the malady. Medical photography abstracted and accelerated this process; the number of photos of a particular affliction accumulated and created something of a \"virtual\" clinic, an archive of cases to be aligned and compared. Series photography worked the same way on a smaller scale by providing images to align and compare so that variations in the single case could be correlated. Cinematography worked on an even smaller scale: it created images not just of a single case but of a single moment. But the point here is that one of the key features of motion pictures in this mode was not necessarily the depiction of movement but _the generation of images for comparison_. That is, for Braun's minute research on the mechanics of the heart, ever more images were needed to make meaningful correlations. Cinema was useful in this case not because it depicted movement but because it functioned as an \"instant archive\" of images for comparison.\n\nBraun was mostly interested in measurement; he was working toward a precise tabulation of the dynamics of the heart. So his second thesis emphasized cinema's ability to aid the \"analysis of movement.\" The comparison of images that he outlined was designed, like Braune and Fischer's apparatus discussed in chapter 1, to extract data for the measurement of movements. The paradox, of course, is that his analysis of movement entailed, as we learned from Bergson in chapter 1, the comparison and measurement of different states of rest. When Braun emphasized \"the analysis of movement,\" he focused on various stages of motion or \"transitional steps.\" Those intermediate stages, however, were actually states of rest, or at least they functioned that way as still images. The analysis of movement was therefore a carefully designed comparison of various states of stillness. Just as paradoxical is the equation of \"intermediate states\" of movement with the film frame. That is, if analysis implied the \"recognition of every intermediate state,\" then it is not going too far to suggest that those states, those \"resulting transitional steps,\" to some extent resulted from the process of analysis itself. As the heart beat, the film frames divided and presented its natural articulations. The smallest possible \"step\" was between two frames; another step might have been between several. But in every case, the frame was the measure of the transition; the \"event\" was equated with the temporal boundaries implied by the spatial boundaries of the frame. Chronophotography, for example, was essentially a record of successive phases of motion: in deciding the time between exposures, the researcher already conceded that X amount of time would equal a single \"phase.\" This is as clear an indication as any that the technology the researcher used helped to shape his or her conception of the object. Braun's understanding of movement depended at least partly on his understanding of motion pictures. Indeed, it also appears at first glance that Braun's understanding of movement and continuity depended entirely on a method based on stillness and discontinuity, \u00e0 la Bergson. This, however, was not always the case, as the next three theses indicate.\n\n4. _The cinematographic shot can be used to present to the observer any movement of the heart synthetically, at will, and even decelerated to a rather great extent without impairing the clarity of the images._\n\nWhen Braun wrote that cinematography can present \"any movement of the heart synthetically,\" he meant it can show actual movement not the decomposed movement of analysis. If analysis breaks down movement, synthesis is the reconstitution of movement, the projection of a moving image. We could argue that the relationship between film's still images and their movement in projection corresponds to the relationship between analysis and synthesis in scientific method. In that tradition, synthesis functioned as an important check on analysis; analysis decomposed a phenomenon into its constituent parts, and synthesis reconstituted those parts to verify that the analysis was correct. Synthesis appears therefore to be only a supplement to the more valuable labor of analysis. This was how Marey used cinematography: as a way of training expert vision by decomposing movement and building it slowly back up through synthesis, which taught the viewer what to look for when the movement was encountered again. If he was dismissive of the moving image, having no interest in it other than as a verification of his analytic results, historians of the scientific film have often taken that dismissal to be definitive. But that ignores the real scientific value\u2014not to mention the pedagogical value\u2014of the moving image. Many other researchers, including Braun, have noted the importance of the moving image for understanding their objects of study. Thesis four gives an indication of that value, which in this case was primarily correlative. As we saw in the second and third theses, motion picture technology generated an \"instant archive\" of images for comparison; in frame-by-frame analysis, this comparison was used most often for measurement: the images were used to track spatial displacement. Braun used moving images not for measurement but for correlation of a different sort. The temporal malleability of film\u2014it could be \"decelerated to a rather great extent\"\u2014created a new set of views for comparison. That is, just as the proliferation of images generated many _spatial_ displacements to be compared, so the variability of projection speed generated many _temporal_ \"views\" to be evaluated. Different projection speeds revealed different aspects of the movement, which were collected and compared for a better understanding of the whole. This is a rather common procedure in the scientific appraisal of research films, which demonstrates that while frame-by-frame analysis may be the best way to derive quantitative data from film, it is clearly not the only kind of information that researchers get from motion pictures. Synthesis was not merely a supplement to analysis, but an equal partner.\n\n5. _The reproduction of movement in slow motion grants the observer more time to recognize significant features, to evaluate precisely the elements of movement, to determine whether or not all the parts of the heart start moving at the same time, and to compare the contraction process of the various parts of the heart, especially of both ventricles._\n\nIn thesis 5, Braun goes on to describe exactly how slow-motion projection of the images helped his project. It allowed him to recognize, compare, evaluate, and decide on various features of the heart's actions. How did it do this? The ability to manipulate the speed of the presentation gave him \"more time\" ( _l\u00e4ngere Zeitr\u00e4ume_ ). This is arguably the key feature of scientific and medical filmmaking, the reason any researcher chooses motion pictures over other representational technologies. To comprehend\u2014from the Latin \"to grasp\"\u2014any complex action requires that we slow it down and bring it within reach of our senses and limited processing abilities. Motion picture technology allowed that to happen. It gave the researcher \"more time\" to contemplate the details of the actions under study, especially when he or she exercised his or her exclusive privilege to control the pace of projection. In scientific filmmaking, as in experiment, the inexorable forward push of time and of the motion picture can be manipulated, slowed, stopped, accelerated. This was also true for narrative or any other kind of filmmaking, but the difference rested not in the technique but in the ability of the scientist to control the image, an ability not available to the spectator in the audience. I will have more to say later in the chapter about how this scientific approach to time becomes normative when applied to movies in the theater, but for now we can simply note the importance Braun and others give to cinema's ability to make \"more time.\"\n\nThis ability to control or to extend the time for examination of the event was a crucial advantage for researchers. Before exploring its manifestations in the medical research film, I want to point out how this ability related to established practices of medical observation. I am especially interested in the temporality of observation, which is addressed in historical discussions about proper method in medical observation, in Foucault's discussion of \"the clinical gaze,\" and in Michael Hau's concept of \"the holistic gaze.\" First, method: throughout the nineteenth century, physicians insisted on the difference between careful and careless observation, which depended on the ability to diligently apply a prescribed method. Textbooks at the time sought to outline this method for students and junior practitioners. British physiologist Thomas Laycock wrote one such text, in which he made it clear: \"The foundation of medical experience is observation of disease, and the requisites to successful observation are minuteness and accuracy. The clinical student must therefore make up his mind to be sedulously minute and carefully accurate in investigating the cases under his notice.\" German pathologist Rudolph Virchow described in detail his method of performing autopsies. Autopsies conducted in a haphazard way promoted interpretive error, he argued, so he \"drew particular attention to the necessity of insisting\u2014in autopsies for medico-legal purposes, as in everything else now\u2014upon completeness of examination and exactness of method, both in the investigation and in note-taking, so that it might be decided subsequently, though not in anticipation, whether there was any significance or importance in what was observed, or whether it was accidental and unessential.\" Virchow was particularly careful to describe exactly what should be observed in an autopsy and the order in which observations should be made. Method therefore precluded haste. An observation could be made quickly, to be sure, but over and over physicians warned their students against the dangers of hasty observation. Instead, observation was to be practiced carefully, leisurely, with an eye to detail (at least until one was trained).\n\nIndeed, detail itself prohibited hasty observation; if one properly attended to and assimilated the details, it took time to do so. Foucault argued that \"the clinical gaze\" emerged from precisely this intersection of vision, detail, and language. With the nineteenth century, according to Foucault, \"Rational discourse is based less on the geometry of light than on the insistent, impenetrable density of the object.\" That is, if Western thought since Descartes equated truth with light and light with an abstract ideal, then during the nineteenth century \"the solidity, obscurity, the density of things closed in upon themselves, have powers of truth that they owe not to light, but to _the slowness of the gaze_ that passes over them, around them, and gradually into them, bringing them nothing more than its own light.\" The clinical gaze cast its own light, searching across the dense landscape of information in the clinic and separating the essential from the inessential. So for Foucault, this careful gaze was inseparable from the process of analysis. Foucault characterized the analytic aspect of medical perception as a process of simultaneously recognizing, separating, naming, and acting upon some disease element. The clinical gaze was a process of seeing, thinking, and naming, which therefore became a deeply contemplative maneuver that took its time. German surgeon and professor Theodor Billroth emphasized the imaginative element in this process: \"Solitary, meditative observation is the first step in the poetry of research, in the formation of scientific phantasies, the reality of which we then test with the tools of logic, mathematics, physics and chemistry. Our tests will be the more successful the better we have learned to handle these tools. The diseased organism, the patient, must be observed in just this way, thoughtfully, and in a state of mental solitude and meditation.\" The abundance of detail, the richness of the information\u2014what Foucault calls \"the plentitude of concrete things\"\u2014was at least partly the reason for this emphasis on a contemplative, solitary, slow, and steady gaze.\n\nBut physicians characterized medical observation not only in these terms. Michael Hau has described medical observation as an active, gestalt-like process of holistic apprehension; he argues that many German physicians at the turn of the century objected to the analytic, overly scientific approach to observation that had become fashionable in the medical community. Threatened by the new technologies and methods common to laboratory science, many physicians took refuge in an explicitly artistic approach to observation, emphasizing their \"ability to 'see' and to synthesize seemingly fragmented characteristics of a human body into an aesthetic, coherent whole.\" By making proclamations about the relationship between health and beauty, these physicians stressed an aesthetic approach to the whole body to differentiate themselves from the more specialized and decidedly more fragmented approach of, say, bacteriologists. If the \"holistic\" gaze was more immediate and intuitive, that might distinguish it from the contemplative and correlative gaze described earlier, but it could also describe the brand of medical perception Foucault calls \"the glance.\" Foucault explains the difference between the clinical, analytic gaze and the more synthetic glance: \"The gaze implies an open field, and its essential activity is of the successive order of reading.... The glance, on the other hand, does not scan a field: it strikes at one point, which is central or decisive; the gaze is endlessly modulated, the glance goes straight to its object. The glance chooses a line that instantly distinguishes the essential.\" This form of the expert eye is similar to what Lorraine Daston describes as \"all-at-onceness\" in scientific observation: an intuitive, immediate understanding of the relationships presented by an image or by phenomenon. This distinction between types of medical perception is useful for our purposes, because it highlights the duration or temporality of observation: the difference between the searching, contemplative gaze of the explorer and the immediate, intuitive glance of the connoisseur. With regard to motion pictures, Braun found the first mode of observation especially valuable, because it corresponded with film's ability to make \"more time\" to recognize, compare, evaluate, and decide. But neither Marey nor Braun dismissed the glance or the synthesis of images entirely.\n\n6. _Finally, the cinematograph can project single images to a large auditorium, and can also present animated images as \"living photography\" by showing the entire chain of images synthetically._\n\nWhile they perhaps do not immediately correspond to the experience of motion pictures, Foucault's \"gaze\" and \"glance\" remind us that medical (and scientific) observation was not only analytic but also encompassed a more synthetic, holistic, or intuitive element. Likewise, Braun's use of motion pictures both as still photographs and as moving images should stress the equal value of both for medical cinema. The individual frame was very helpful for measurement or correlation, while the projected image had important rhetorical, educational, and even documentary uses. It makes little sense to separate them; we should, in fact, consider the relationship between the still image and the moving image as a vital dialectic for medical understanding. Physicians went back and forth between the two forms of the medical film in a productive, yet prescribed way that preserved and proved their expertise. Indeed, Braun's sixth thesis is all about expertise, however obliquely. Motion pictures could be used in the auditorium, where they could instruct and dazzle the audience with the latest in medical knowledge and technology. Imagine Braun (or others) giving a medical lecture on heart function, in which he has control of the film image. Braun could present his material one frame at a time, as a slide show, carefully delineating for his audience the precise difference between one phase and another. He could take as much time as he needed, with the image still projected, to describe the salient features of the organ's mechanism. But Braun could also capitalize on the rhetorical power of motion pictures by turning, with a flourish, the single image into \"living photography.\" This provided the medical students with another kind of information, while at the same time powerfully persuading them of Braun's argument. But even the spectacular aspect of the moving image was attenuated and contained by two barriers mentioned earlier in this chapter: expertise (\"we know what we are looking for in the image\") and closed community (\"this knowledge is exclusive to us\"). That is, the seriousness of purpose implied by the specialized educational setting contained the full power of the moving image. Even so, presenting \"the animated image as 'living photography'\" implied an almost Doyen-like commitment to showmanship, mastery, and self-promotion. Through the alternation of still and moving images, of analytic and synthetic approaches, surgeon became magician and back again, experts both.\n\nIt is also worthwhile to note the cultural capital invested in the gaze and the glance. In textbooks and treatises on medical observation, authors stressed that the expert eye did not come easily: \"Aspiring physicians will never be able to develop an eye for diagnostics from books. That can only be learned through careful observation and constant contact with his patients.\" Here the author emphasized a truism of medical education in the nineteenth century: direct observation was preferable to, even precluded, the scholastic, bookish approach that ruled medicine for so long. But he also confirmed several aspects of the clinical gaze: it was contemplative, it relied on the accumulation of cases, it was learned only through vigorous training. On the other hand, another author described the glance: \"It is that piercing gaze, which unconsciously separates the essential from the unessential and penetrates to the bottom of things with ease and certainty, without requiring difficult research, which is easily misled by irrelevancies that are easily apparent, while important things that elude perception are overlooked.\" The glance was decisive, confident, second nature. Like the gaze, the glance was acquired only after long training and experience. While Foucault characterized the distinction between the gaze and the glance as a historical progression from one mode of observation to another, and Hau described it as a cultural or disciplinary differentiation, I would suggest that the duo, like stillness and movement, constituted a set of alternatives or a dialectic, each always part of the larger category called observation. But the character of the two does not ultimately concern us here, nor even their relationship to each other, as much as their function as badges of professional expertise. The cultural investment in these modes of observation is palpable, as indicated by the numerous texts on observational method or by the number of debates about the specificity of clinical observation, especially when threatened by new technology (the sphygmomanometer, for example). Observation, in whatever form, was central to the German physician's professional and class identity. The explorer and the connoisseur may have had different skills, but they invariably came from the same place and were quite comfortable in each other's company (or even in the same skin).\n\nThis exercise in textual analysis has been designed to highlight the three-way correspondence between certain formal features of motion pictures, particular techniques used to examine them, and specific principles of observation. The way Braun used individual film frames, for example, illustrates and corresponds to the principle of correlation in observation. These correspondences demonstrate that the acceptance and legitimacy of motion pictures in science and medicine depended not merely on the availability of the technology but also on its ability to conform to established disciplinary imperatives. At this historical moment in the development of Western medicine, motion picture technology partly addressed a number of representational and observational issues, including the increased demand for rigorous observation in medicine and the rising interest in the documentation and analysis of complex movements. (So it is perhaps not surprising that the number of research projects on pathological movement, for example, increased as film was accepted as a tool for studying it.) Beyond the question of legitimacy, these correspondences also demonstrate, I hope, that observation is a complex operation and that we can learn much about it by studying how film is used. Specifically, it is clear that for nineteenth- and twentieth-century scientists, medical researchers, and scholars, \"observation\" was considered to be primarily _a self-disciplinary method of ordering thought_. As a means of protecting the community against the chaos of irrational and biased subjectivity, observation was closely aligned with experiment as a universally adopted method of enforcing professional standards. Observation was a prescribed method of organizing information through such means as stabilizing and isolating variables; measuring, describing, and classifying objects and events; and arranging and correlating these items or events to arrive at verifiable conclusions. This process is what it meant for physicians to be \"scientific\" as Paget so forcefully insisted: \"In our calling careful practice and scientific study should be inseparable\" (which also emphasizes the futility of separating experiment and observation).\n\nFilm worked as a research instrument to the extent that it could conform to or inform this method. In particular, film was an aid to correlation. The motion picture camera was a tool for generating images or views for comparison, either comparison of the individual frames or comparison of different temporal \"views\" via slow-motion or time-lapse cinematography. Alongside this feature, film was praised as something that gives the user \"more time,\" either by storing material for convenient retrieval, by reproducing and repeating events, or by offering variable filming and projection speeds. One expert applauded film's utility in the medical classroom:\n\nCinematography could help with that. The images can be shot with all the time in the world, using specimens that have been prepared with the utmost care, and are then permanently available. Instead of the uncertain outcomes of fleeting experiments, the serial images can be shown repeatedly, which not only makes a greater impression on students, but by presenting the frames individually, can also provide an analysis of the phenomenon.\n\nMotion pictures gave more time to the researcher and the student alike\u2014more time to prepare, more time to analyze and comprehend. Especially important was the flexibility of film with regard to the dialectic between analysis and synthesis; if researchers have always had analysis and synthesis, only with motion pictures did they have that choice in the same representational system. Alternating back and forth between the still image and the moving image, researchers took advantage of cinema's ambidexterity to present an \"impressive\" yet scientific picture of the phenomenon. Yet this ambidexterity was not merely an advantage of film\u2014it signaled a hermeneutic strategy that in part became standard procedure because of it. The cognitive gains that resulted from going back and forth between the still and moving image\u2014which have been known ever since a mental image could be sketched\u2014were codified into experimental and observational method with the advent of motion pictures. But more to the point, the scientific legitimacy of this alternation\u2014that stillness and movement could be _equated with_ analysis and synthesis\u2014was to some extent due to the availability of motion pictures as a research tool that could offer both options. So film was not simply a useful tool for understanding complex events: the use of film actually made manifest a mode of understanding. The ambidexterity of film operationalized a way of thinking and made it visible to the community in both images and practice. The motion picture camera was not just an instrument, but a theoretical approach.\n\nBut I want to return to the researcher's obvious glee, which the quotation above by K. W. Wolf-Czapek vividly demonstrates, in being able to manage the flow of time by controlling motion pictures. Wolf-Czapek's cheerleading betrays a certain emotional investment in this potential, as if it were crucial to the lecturer's pedagogical success or the physician's identity as a researcher. Controlling the passage of time almost became the very definition of \"scientific\" because of all it implied about careful preparation, leisurely observation, accurate description, and deliberate correlation. But as we saw with Braun earlier, the giddiness that the researcher felt toward the moving image was not only about the control of time, but the submission to it as well. Or more precisely, the moving image was thrilling in itself: its invitation to mimesis, its temporal rush, its presence. Controlling the moving image completely would also make it less spectacular, less effective as a persuasive representation. So the researcher had to negotiate this duality by deciding how and when to use the motion picture in its two forms. Various techniques in the laboratory and the lecture hall\u2014such as frame-by-frame analysis, slow-motion and time-lapse cinematography, varying projection speeds, looping a single shot so that it repeats, or tracing an image to abstract and manage its detail\u2014help to manage the thrill and temporality of the moving image (even if these techniques come with their own sensory pleasure). But even when the image was projected in all its glory as a thrilling example of \"living photography,\" the conventions of the lecture attenuate that potential. And it goes without saying that a film about pathological movement did not have the same immersive potential as a feature narrative. Physicians recognized the full impact of the motion picture, to be sure, but they did their best to manage that impact in the laboratory, the classroom, and the professional meeting. Was it simply a generic demand? That is, was it simply the case that any representational form would have the same restrictions? Other media did have similar boundaries of decorum and professionalization. But cinema was a special case unrelated to its content\u2014it _moved_ , and this meant that there was a little bit of carnival even in the most staid presentation or most boring research film. With the possibility of this kind of presentation, what signified \"scientific\" or \"professional\" depended especially _on how one managed the passage of time_. If scientific experiment always implied some control of the temporal variable, with motion pictures that control became explicit and essential to scientific identity, especially in the face of its mass culture incarnation. Hence the medical community's swift and vicious reaction to \"the movies.\"\n\nTIME, SPECTATORSHIP, AND THE WILL\n\nMovies were only one of many ailments of modernity, of course, and in the scheme of things, they took up no more editorial space\u2014probably considerably less\u2014than complaints about trashy literature, modern art, lurid advertising, trolleys, trade unions, or other by-products of industrialized life in the metropolis. But with their forceful entry into public consciousness, starting around 1907, motion pictures touched a very particular nerve among the educated elite. The previous section exposed one tendril of that nerve: the professional and emotional investment in methodical, leisurely, observational practice attached to scientific method. The pace of motion pictures threatened that practice; more precisely, for most commentators the (perceived) quickness of movies endangered a cultural tradition of learning, expertise, and status that invested heavily in studious attention, methodical description, and logical reasoning. Modernity jeopardized that investment in a variety of ways, but motion pictures best represented the problem of social acceleration: movies, like change itself, moved too quickly for comfort. They therefore put attention and reason at risk and, in that way, exemplified a problem of national importance. In their editorials about movies, physicians discussed the problem of cinema's pace in the context of this investment in attention and reasoning. They also addressed the objectionable subject matter of many films and the unhygienic conditions of many movie theaters. Physicians joined reformers in their protest against the number of \"trashy films\" ( _Schundfilme_ ) that exposed themselves to a dazed public in seedy storefront cinemas ( _Ladenkinos_ ). This final section will examine some of these complaints about the physical and psychic or moral threat of motion pictures. I want to provide a survey of the variety of complaints, but I also want to emphasize _pace_ as the primary source of nagging discomfort with movies felt by doctors and other members of the educated middle-class ( _Bildungsb\u00fcrgertum_ ). This section, then, serves as a segue between the focus on film as a research instrument in this and the previous chapter and the exploration of film as a social tool and phenomenon in the following chapters. Let us start this examination, however, with a brief check of credentials: How did German physicians position themselves as experts on movies, of all things?\n\nAs late as the 1850s, academically trained physicians in Germany were not considered experts of much. Their therapies were, by and large, no more effective than those available to the lay public; barber-surgeons, midwives, and folk medicine presented plenty of competition; and a physician's social standing was usually lower than the upper classes he served, which meant that medical advice was routinely questioned and rejected. Claudia Huerkamp has demonstrated that the turnaround was a result of the Prussian state bureaucracy's policy to take control of all guild organizations, to standardize medical education, to license medical practice, to thereby suppress folk alternatives, and to prioritize hygienic practices among the population. These policy decisions effectively eliminated the competition and, especially through the establishment of health insurance in the 1880s, expanded the clientele for academic physicians, particularly into the working classes. As doctors took on more patients from the working classes, their social standing was above the majority of their patients, thereby reversing the previous patient\u2013doctor power dynamic. In addition, the state directives promoted a general \"medicalization\" of society, in which academic concepts of health and disease were adopted by society at large; social problems that were not previously seen as medical problems, such as alcoholism, were gradually treated as diseases. Furthermore, the academic medical community stabilized and enforced professional standards, especially by adopting scientific methods and techniques, which led to new approaches to diagnostics, anesthetics, and surgery, for example. As their expertise and their association with the craft of science grew, so did their authority. So state policies, the expansion of clientele, the growing professionalization of medical ranks, and the association with scientific principles, all combined not only to create a medicalization of society but to place academically trained physicians in position as its chief diagnosticians.\n\nBy the 1890s this authority began to consolidate, and by 1914 it seemed perfectly natural to view many social problems within a medical framework and equally natural that physicians were qualified to comment on them. Doctors examined and prescribed for the body politic as well as for the individual. They saw themselves not merely as professionals or experts, but as _Kulturtr\u00e4ger_ (bearers of culture) who had a responsibility to support, carry on, and protect the nation's heritage. This intervention into the social sphere manifested itself in two ways: more doctors became involved in social medicine or public health, and more doctors wrote cultural critiques from a medical standpoint. For Munich psychiatrist Emil Kraepelin, it was simply a matter of applying one's skills in diagnosis, prophylactics, and therapy to the growing need for public medicine. Such public health issues as alcoholism, tuberculosis, and sexually transmitted diseases increasingly occupied the expertise of academically trained physicians in Germany at the turn of the century. While Germany lacked a national ministry of health until after World War I, individual doctors were very much involved in national campaigns. Kraepelin, for example, was committed to the temperance movement of the day, a typical example of the growing medicalization of social issues in industrialized nations. Likewise, motion pictures were swept up in this trend as certain aspects of the cinematic experience came to be regarded as a public health threat.\n\nKraepelin's prot\u00e9g\u00e9, Robert Gaupp, also directed his skills toward public health issues, such as child psychology and welfare, but he combined his specialized training with his cultural capital and status as a University of T\u00fcbingen professor to bring less obviously medical issues, such as motion pictures and pulp fiction ( _Schundliteratur_ ), under the rubric of medicine. Perhaps the most famous example of the medical diagnosis of culture, however, is Max Nordau's _Degeneration_ (1892). With chapters titled \"The Symptoms,\" \"Diagnosis,\" \"Etiology,\" and \"Prognosis,\" and various case studies of modernist endeavor, from the symbolists to Leo Tolstoy, Henrik Ibsen, Richard Wagner, and Friedrich Nietzsche, _Degeneration_ argued that cultural products of the day displayed classic symptoms of degeneration, decadence, and hysteria, which were \"the consequences of the excessive organic wear and tear suffered by the nations through the immense demands on their activity, and through the rank growth of large towns.\" Drawing in part on the contemporary discourse of \"nervousness,\" Nordau (fig. 2.3) found that the excesses of modernity had physical consequences, the symptoms of which could be read in cultural products. Just as Italian psychiatrist Cesare Lombroso theorized that social deviance was based on biological retrogression\u2014criminality, he surmised, must be due to some primitive remainder or reversal in the normal progression of the species\u2014so Nordau argued that the cultural avant-garde was not progressive and forward-thinking but instead atavistic and regressive, displaying traits common to deviants and hysterics. Using medical norms to evaluate literature and the arts, Nordau combined physiological theories of social deviance, Darwinian evolutionary models, and liberal ideals of ordered progress and rationality into an old-fashioned, curmudgeonly disapproval of modernism. Nordau's work was extremely visible\u2014it was one of Europe's ten best-selling books of the 1890s\u2014but certainly not unique. Darwinian theories of heredity and degeneration; medical or psychological investigations into hysteria, nervousness, and hypnotism; social ferment due to the rise of social democracy, industrialization, and modernization; all combined to create an uneasy mood of cultural pessimism among the educated middle classes of Wilhelmine Germany, which the commentaries of Gaupp and Nordau exemplify.\n\nSo the metaphorical extension of \"health\" and \"disease\" to all aspects of social life became more than a metaphor for doctors and their readers in imperial Germany. Likewise, the moral and physical threats that motion pictures and other cultural forms presented were also very real for these commentators. One physician's complaint about the dangers of representing the erotic and sexual in film presents a common stance:\n\nThe arrival of the cinematograph brought much damage in its wake.... While the danger, which undermines serious moral principles and hence the moral foundations of a man, must be fought, it is, of course, only a threat to the masses who don't think for themselves, for the minors, the young and easily impressionable, who are not used to having their own opinion about things or probing things with critical method. And because these masses make up the majority of our nation, this public, easily accessible sexual fare, which is apt to damage the sexual apparatus [ _sexuellen Tractus_ ], represents a serious _sexual danger_.\n\nFIGURE 2.3. Max Nordau\n\nOf course, the idea that sexually arousing images or narratives might damage one's moral fiber was not new. But this complaint is especially representative and remarkable for its insistence that \"critical method\" ( _kritischer Methode_ ) was the difference between the protected and the unprotected. Without it, apparently the masses were at modernity's mercy. But it is not simply that the masses lacked the tools to defend themselves; again and again, medical and lay pundits claimed that movies corroded \"critical method\" itself.\n\nWe find this idea even in editorials that outlined the mere physical threat that motion pictures presented to the average viewer. A Dr. Paul Schenk, for instance, discussed the flicker effect of early cinema, warning teachers who hoped to use motion pictures for educational purposes:\n\nFrom the standpoint of visual hygiene, the usefulness of cinema as an educational tool remains dubious. Modern man systematically ruins his eyes. We suffer from an excess of visual stimuli.... The much-maligned \"flicker\" of the cinematic image is a malaise that presently, and probably forever, deprives the cinema of the claim to be a \"hygienic\" means of instruction.... This impression is further strengthened by the unnaturally fast changes of scenery. Our eyes cling intently to the screen, where, in addition to the flicker of the images, a change of scenery takes place almost every minute within a time period that lasts 8 to 12 or up to 15 minutes.\n\nCinema's educational utility, according to Schenk, was hampered by the threat its \"flicker\" presented to student health. On one hand, this flicker supposedly damaged the eyes; there was apparently a physical connection between the flicker and nerve damage of some sort. On the other hand, the _pace_ of the film intensified this damage. In fact, this physical damage acted as a metaphor or outward symptom of a deeper, psychic damage caused by the temporal push of cinema. Reformers or _Kinogegner_ (enemies of the cinema) also took up this diagnosis in a general medicalization of cinema in the discourse of the time. Albert Hellwig, not a doctor but one of Germany's most prolific _Kinogegner_ , relied on medical terminology and cited an Italian doctor's discussion of cinema's assault on the sensitive mind: \"Of a group of neurasthenics, d'Abundo observed that frequenting cinematographic presentations brought about all sorts of ailments, especially insomnia. It was not so much the influence of the contents of the presentation, but rather an effect of the rapid, constantly moving action and attendant flickering.\" Another nonphysician, O. G\u00f6tze, put a finer point on it: \"The hasty tempo of the images, and especially the Leipziger-medley-like program [i.e., one resembling a mixed vegetable dish], seduce the child into superficial _viewing and observation_ habits. Even the images that could offer intellectual enrichment go by so quickly that it is impossible to form clear impressions.\" With this observation, G\u00f6tze articulated the position of the medical community and reformers with regard to the movies. Medically speaking, the cramped, oppressive atmosphere of the theaters, together with the flicker effect, presented a threat to physical health, while the immoral content of the films and their delivery by means of a relentless temporality presented psychological dangers. Cinema's quick pace threatened reason itself.\n\nThis complaint about tempo corresponded to a broader contemporary discussion about modernity and \"social acceleration.\" We could trace this feeling that \"the world is moving faster\" back to the eighteenth century, of course, but objections piled up toward the mid-nineteenth, especially at the fin de si\u00e8cle. Nordau complained about a lot, but close examination of _Degeneration_ reveals that the root cause of the state of things was, for him, the heightened pace of modern life:\n\nIts own new discoveries and progress have taken civilized humanity by surprise. It has had no time to adapt itself to its changed conditions of life. We know that our organs acquire by exercise an ever greater functional capacity, that they develop by their own activity, and can respond to nearly every demand made upon them; but only under one condition\u2014that this occurs gradually, that time be allowed them. If they are obliged to fulfill, without transition, a multiple of their usual task, they soon give out entirely.... To speak without metaphor, statistics indicate in what measure the sum of work of civilized humanity has increased during the half-century. It had not quite grown to this increased effort. It grew fatigued and exhausted, and this fatigue and exhaustion showed themselves in the first generation, under the form of acquired hysteria; in the second, as hereditary hysteria.\n\nSo, according to Nordau, the human body was not able to keep up with the increased demands and pace of modern life and grew fatigued, which led to a corrosion of the nerves and, subsequently, hysteria and degeneration, which showed itself in daily life and in modern art. \"All the symptoms enumerated are the consequences of states of fatigue and exhaustion, and these, again, are the effect of contemporary civilization, of the vertigo and whirl of our frenzied life, the vastly increased number of sense impressions and organic reactions, and therefore of perceptions, judgments, and motor impulses, which at present are forced into a given unity of time,\" writes Nordau. Too much information compressed into too little time was Nordau's recipe for decline.\n\nNordau's antidote for sensory overload was not withdrawal but \"attention.\" Attention was the savior, the critical faculty that brought order to chaos:\n\nThus we see it is only through attention that the faculty of association becomes a property advantageous to the organism, and attention is nothing but the faculty of the will to determine the emergence, degree of clearness, duration and extinction of presentations in consciousness.... Culture and command over the powers of nature are solely the result of attention; all errors, all superstition, the consequence of defective attention. False ideas of the connection between phenomena arise through defective observation of them, and will be rectified by a more exact observation. Now, to observe means nothing else than to convey deliberately determined sense-impressions to the brain, and thereby raise a group of presentations to such clearness and intensity that it can acquire preponderance in consciousness, arouse through association its allied memory-images, and suppress such as are incompatible with itself. Observation, which lies at the root of all progress, is thus the adaptation through attention of the sense-organs and their centers of perception to a presentation or group of presentations predominating in consciousness.\n\nAs Jonathan Crary has demonstrated, Nordau's views here represented a vast literature in experimental psychology and social commentary focused on attention. Crary has covered this ground well, but Nordau's remarks nevertheless bring up two issues especially salient to this chapter. First, we should note the absolute centrality of attention for Nordau's Enlightenment project: \"Culture and command over the powers of nature are solely the result of attention; all errors, all superstition, the consequence of defective attention.\" Attention was not simply the ability to focus, it was the motor of human progress. The stakes were high, precisely because Nordau, as others before and after, tied attention to human volition. \"Attention is nothing but the faculty of the will,\" which meant that human action relied almost entirely on attention in Nordau's scheme. Attention, as choice, expressed free will; to focus is to choose. But more than that, attention, as Crary points out, is also inhibition; it is the suppression of stimuli and the ordering of information. It is not simply choice that makes us human, according to Nordau, but the kind of choices we make for the good of progress. Attention, then, had a powerful ethical charge. Without the proper exercise of attention, according to Nordau and this tradition, we would be automatons and decadents, abandoning our future to the caprice of chance, fate, or worse, the flow of modernity.\n\nWhat did the proper exercise of attention look like? For Nordau, it looked exactly like \"observation,\" which brings us to the second issue: observation as the practical manifestation of attention and volition. According to Nordau, attention determined \"the emergence, degree of clearness, duration and extinction\" of stimuli, thus controlling the flow of information to consciousness. Attention was an important gatekeeper against the rush of modern life; significantly, it could control the _duration_ of the presentation. Attention, or its equivalent in observational practice, controlled the rate at which consciousness is exposed to stimuli, just as a scientific experiment isolates and controls variables and the rate at which the researcher receives information. The difference between overstimulation and an unacceptably quick pace is sometimes difficult to distinguish, but if we think of observation and experiment as a set of practices that manage the flow of data, then it really does not matter. In fact, attention itself, as Crary emphasizes, was a notoriously slippery concept; not only did it bleed easily into distraction, but psychologists more or less failed to locate or define it, except as an \"imprecise way of designating the relative capacity of a subject to selectively isolate certain contents of a sensory field at the expense of others in the interests of maintaining an orderly and productive world.\" Attention still received significant play, but precisely because of its ambiguity it makes good sense now to focus instead on observation. In the context of the hailstorm of modern life, attention functioned for Nordau and others as a \"stimulus shield,\" as well as a metaphor for bourgeois values of order, rationality, and discipline. But most of all, it functioned as an abbreviation for a set of observational practices derived, at least in part, from scientific method. These practices, which I outlined in the previous section, could be learned; the medical student or the young scientist was constantly required to adopt and perfect this mode of viewing. Once incorporated, this training became second nature, of course, but more importantly, as we see in Nordau and others, it carried an intense moral responsibility: \"Observation... lies at the root of all progress.\" For Nordau and the _Bildungsb\u00fcrgertum_ , that which threatened these practices, or risked the possibility of passing these practices to another generation, attacked the very foundation of human endeavor.\n\nSo it is easy to see why physicians and other members of the educated elite greeted cinema's temporal rush, emblematic of the incessant pace of modern life, with something less than enthusiasm. Stuck in a theater seat, with no way to slow or stop its continuously unfurling movement, they could only helplessly watch film corrode their (and their children's) \"critical faculties.\" We have already seen how psychiatrist Robert Gaupp, along with his colleague at T\u00fcbingen, art historian Konrad Lange, condemned \"the temporal concentration\" of cinema: \"In the cinema... excitation of the passions is intensified and multiplied by the rapid succession of vivid images that passes before our eyes. There is no time for reflection or release, no time to find an inner balance... quiet _contemplation_ , _intellectual assimilation_ , and sober criticism are not possible.\" Others echoed his medical opinion that the rush of images was too much for the average viewer: \"All eyes greedily fix on the living picture as it flits by. And the mind must hurriedly link together the connections between images that were only briefly glimpsed.... In the theater I must _mentally exert myself_ more throughout [than in the cinema]. I've got to pay attention to the words and thoughts, which often appear in a very terse, sharply focused form. Such a tight line of thought is not necessary in the movies.\" For some commentators, cinema's rush of images threatened reason itself:\n\nThe mere habituation to the darting, convulsive, twitching images of the flickering screen slowly and surely corrodes man's mental and, ultimately, moral strength. First, one gets used to switching quickly and abruptly from one impression to the next; one loses the slow continuity of the succession of ideas, the ability to grasp, which is a precondition for all sound judgment. Second, one becomes accustomed to yielding to and unthinkingly following its random string of images and ideas; one no longer misses the logical structure of an overarching thought, which is indeed what binds individual impressions together into what is generally referred to as \"thought.\"... Third, as a result of the rapid passing of images, one gets used to absorbing only approximations of an impression; one does not grasp the image clearly and consciously, in all its details.\n\nFor these writers, cinema's threat was ultimately a question of mental hygiene: methodical observation as taught by the scientific method was one good practice, akin to flossing, that would help insure the health of the psychic mechanism. It was only one of many good observational practices\u2014aesthetic contemplation was another\u2014that reflected a well-ordered mind. Motion pictures, on the other hand, encouraged sloppy habits of thought. Instead of the active self-control that these hygienic practices implied, motion pictures suggested disrepair and self-abandonment.\n\nWe can see these implications especially clearly in the characterization of cinema as a hypnotic agent. The ground between cinema and hypnotism has been well tilled, but I return to it because hypnotism provides not only a succinct and well-known model of spectatorship, but because, surprisingly enough, it also offers a model of observation. That is, if we consider the diagnostic value of hypnotism, or its function as an experimental tool, the double-edged value of this trope will become especially clear. So let us first briefly explore the discussion of hypnosis as a metaphor for film spectatorship. As Stefan Andriopolous has shown, the comparison of cinema with hypnosis tapped into contemporary debates about hypnotic crime that flourished during the initial heyday of hypnosis, between 1885 and 1900. In the clinical literature, researchers such as Hippolyte Bernheim in France and Auguste Forel in Switzerland wondered whether the power the hypnotist had over the subject could extend to forcing the hypnotized person to commit crimes against his or her will. To investigate this phenomenon, they even staged fake crimes, in which the hypnotized subject was asked to shoot or stab another person with blank bullets or wooden daggers. If the results were inconclusive, the \"belief in the possibility of perfectly camouflaged suggestions produced the paranoia that there might be an unlimited number of unknown hypnotic crimes.\" The idea that an agent could plant a suggestion in a hypnotized subject to be carried out later excited the already overwrought imaginations of many.\n\nFor many physicians and reformers, cinema acted as a hypnotist, sending impressionable subjects to the streets with powerful, posthypnotic suggestions to commit crimes of all varieties. Or it functioned as a \"trigger\" for latent psychopathies. One physician cited a 1911 case in Frankfurt, in which a twelve-year-old boy stole a purse. When interrogated, the boy confessed, \"I admit to having carried out the purse-snatching: I by chance ended up in a crowd where Frau K. stood with her purse. I had once seen the presentation of a purse-snatching in a movie theater. I was thereby compelled to try something like that, too.\" This criminal sounds rather literate for a twelve-year-old, but nevertheless the cinematic depiction of a crime apparently presented a suggestion too powerful to resist. The doctor commented, \"The effect of the dramatic, moving images on the child's psyche must have been especially strong because the purse-snatching took place on a crowded street in the middle of Frankfurt.\" The medical literature on cinema during this time was littered with such cases in which a youngster who has committed a crime admits to visiting the movie theater at least once a week or even to learning the technique at the movies. Apparently, like Drs. Caligari and Mabuse, cinema invisibly enlisted an army of impressionable youth to do its bidding.\n\nHow was this accomplished? Gaupp explained the connection between cinema and hypnotism:\n\nAdd to this the well-known psychological fact that, when hearing or reading about exciting events, few people have sufficient imagination to visualize the events graphically before the mind's eye. Cinema, however, brings everything right before our eyes in embodied form, and does so under psychological conditions that are conducive to deep and often lasting suggestive effect: the darkened room, the monotonous sound, the forcefulness of exciting scenes following each other beat by beat lull every critical faculty to sleep in impressionable souls, and thus, not infrequently the content of the drama becomes a fateful suggestion for the complaisant youthful mind. We know that _all suggestions adhere more strongly when the critical faculties sleep_.\n\nGaupp was not the first or the last to note the structural similarity between cinema and hypnosis; as Rae Beth Gordon notes, critics from Ycham to Raymond Bellour have been fascinated by the parallels: the darkened room and the luminous screen of the theater mimic or encourage the intense focus and narrowed consciousness of the hypnotized subject; the monotonous noise of projection and the \"scenes following each other beat by beat\" are like the lulling rhythms of the hypnotist. Everyone from Hugo M\u00fcnsterberg to Jean Epstein has remarked on film's suggestive power. Noteworthy, however, is the image that emerges from this varied commentary of the spectator as suggestible, childlike, and feminized\u2014a characterization that was fairly commonplace, especially in social psychology. The idea of \"suggestibility,\" especially, was a lynchpin that connected the trope of \"the child\" to other groups, namely women and crowds. That is, reformers and physicians made a conceptual leap from children to crowds (and to the cinema audience) via the idea of suggestibility, which had become an important concept in crowd psychology. For example, Gustave Le Bon, the most well-known popularizer of nineteenth-century crowd psychology, characterized the masses as \"an enraged child.\" Furthermore, according to Le Bon,\n\nIt will be remarked that among the special characteristics of crowds there are several\u2014such as impulsiveness, irritability, incapacity to reason, the absence of judgment and of the critical spirit, the exaggeration of the sentiments, and others besides\u2014which are almost always observed in beings belonging to inferior forms of evolution\u2014in women, savages, and children, for instance.\n\nDarwin's evolutionary scheme provided a quasi-scientific basis for comparing crowds to children, but \"suggestibility\" was even more significant for this comparison. Le Bon devoted a chapter to \"the suggestibility and credulity of crowds,\" arguing that the crowd is \"perpetually hovering on the borderland of unconsciousness, readily yielding to all suggestions,\" a mental state most commonly found in women, children, and the hypnotized subject.\n\nVia Charcot, hypnotism had already come to be associated with female hysteria. Under his supervision, female patients (and even some male patients) fell into a deep trance and, through the power of suggestion, disassociated themselves from their bodies, allowing Charcot to practically sculpt them into various positions. Likewise, in Le Bon's crowd psychology, this pliability and impressionability was a distinctly feminine trait. Those susceptible to suggestion\u2014or more precisely, those unable to withstand suggestion\u2014were feminized, namely women, children, savages, crowds, and weak-willed men (fig. 2.4). This is also why hypnosis was such a powerful trope for characterizing the cinema audience: for Gaupp and others, the viewer was in a state of self-abandonment and pliability, swept away by the moment like a hysteric or hypnotized subject. Film viewers\u2014whether actual children or merely uneducated and therefore childlike\u2014lacked the will to distance themselves from the psychological and physiological effects of suggestion. Precisely this lack of will was at issue; _willenlos_ is the word Gaupp uses to describe impressionable youth. His diagnosis described both pathology and morality; the lack of will was a moral weakness that manifested itself in mental impressionability and physical mimesis. Those who _could_ resist were, of course, male experts. Against the feminized subject, physicians such as Charcot, Le Bon, and Gaupp pitted the moral strength of the masculine professional.\n\nFIGURE 2.4. A hypnotist practicing his craft, France, circa 1900\n\nThis image of the film spectator is a familiar picture. But what of the hypnotist himself? That is, if hypnotism were a valuable trope for describing the spectator, could it be equally valuable to describe the observer? What properties of hypnotism expressed the hypnotist's stake in the process, especially with regard to his mode of viewing? What was the medical purpose of hypnosis? Psychoanalysis and the medical community abandoned serious work in hypnosis early in the twentieth century for a variety of reasons; not only was it becoming too much of a sensational sideshow, but the power dynamic it implied was indefensible within humanist inquiry. Nevertheless, before this crisis and in its modern form, hypnosis held therapeutic potential, because it allowed the investigator to probe deeply into the patient's psyche. It had the effect of anesthesia\u2014it allowed the physician to explore the psyche without the interference of a \"live\" consciousness. It was considered a form of \"psychic dissection.\" In this respect, it _suspended time_. More precisely, it functioned _experimentally_ : like an experiment, it controlled variables, stabilized the environment, and provided enough time to examine. Berlin psychiatrist Albert Moll declared, \"Hypnotism is a mine for the psychological investigator, for hypnosis is nothing but a mental state. When we think that psychologists have always used dreams so much in their investigations of mental life, and that experiments can be better made in hypnosis than in ordinary sleep, _because it can be regulated at pleasure_ , we cannot deny the value of hypnosis to psychology.\" One prominent French psychiatrist put it this way:\n\nThrough numerous examples, I will try to show that with hypnotic processes we may practice, if I may express it in this way, an actual _moral vivisection_ (if the reader is not too frightened by the word) and witness with our own eyes and _make_ function the intellectual mechanism just as the physiologist sees and _makes_ function the organic machine.\n\nWhen did a physiologist \"make\" a body function like a machine? In the course of an experiment, of course. Similarly, hypnotism was a process that stabilized the subject; isolated important variables; and allowed the time to inventory, describe, and classify those variables in a scientific way. Like a filmstrip, the subject's consciousness under hypnotism could be paused, rewound, slowed down, or inspected bit by bit. It gave the researcher time to explore, to accumulate data, to contemplate the details, and to correlate their patterns. In this way, the comparison between cinema and hypnosis goes further than what we find in the discourse itself: it is not just that cinema was a hypnotist or even that hypnotism was a constant theme in the films themselves, but that hypnotism and cinema were functionally equivalent in the eyes of the researcher. Both were experimental apparatuses that had a flexible, controllable regulatory mechanism in the hands of the scientist, and both had, in relation to the layperson, a potentially dangerous pace precisely because it could not be controlled.\n\nHence, proper observational method and experimental control functioned as mental prophylactics against the sweep of time. \"Will\" was the name they often gave to the power to resist, but if \"will\" ever existed beyond the realm of bourgeois fantasy, it existed as ingrained, scientific, observational method. For example, hypnosis was often characterized as an extreme \"narrowing\" of consciousness with an attenuation of peripheral awareness; the paradox, of course, being that those most capable of intense focus and attention were most easily hypnotized. Scientific observation, on the other hand, was emphatically _not_ considered a \"narrowing\" of consciousness; this was not what Nordau or others meant when they lauded \"attention.\" Indeed, given the emphasis on logical operations such as correlation, observation was nothing if not an \"expansion\" of awareness; the researcher constantly mentally compared the object under study to other objects in past and present experience. So the prevailing view of observation held that logical operations attached to a viewing protocol\u2014the finest aspect of \"attention\" as the educated elite understood it\u2014were the best defense against slipping into a dangerous immersion.\n\nThe masculine, medical professional was distinguished not only by his will, but by his training in scientific observation. Indeed, this training was his primary defense against the onslaught and seductive power of (moving) images. The scientific appropriation of motion pictures depended on the ability of the researcher to halt this onslaught, to forestall its assault and to read _slowly_. It must be so: if the researcher were to correlate the data, he had to be able to control its flow. Ultimately, then, the negative comparison between cinema and hypnosis was about mastery: mastery not only of the hypnotized subject but of the ability to hypnotize. Hypnotism and motion pictures were perfectly acceptable scientific tools, as long as they were in the qualified hands of (male) professionals. Hypnotism, for example, was condoned (and still is) as a legitimate procedure for psychic exploration. Likewise, physicians in Germany used motion pictures to help diagnose or visualize disease. Like motion pictures, hypnosis was used to isolate, stabilize, and present\u2014in this case, psychic or somatic trauma. Hypnosis was a tool for acquiring the distance and time necessary to observe and to diagnose. Physicians used motion pictures to record the human body, but then they slowed the images down or stopped them to examine the phenomenon frame by frame, also giving them the time and distance they needed to master the event. In the same way, hypnosis allows access to different temporal registers. In both cases, the scientific use of hypnosis or motion pictures implied a mastery of time and of the human body. Film's popular incarnation, however, implied a lack of mastery of both.\n\nIn his essay \"What Is Enlightenment?\" (1784), Immanuel Kant made a compelling argument for self-mastery: \"Enlightenment is man's release from his self-incurred tutelage  _selbstverschuldete Unm\u00fcndigkeit_ ]. Tutelage is man's inability to make use of his understanding without direction from another. It is self-incurred when its cause lies not in lack of understanding but in lack of resolution and courage to use it without direction from another.\" For Kant, true freedom meant being able to think (and speak) for oneself. Those who did not or could not were condemned to political and social infancy. \"Tutelage\" is an important term, in this respect; literally translated, _Unm\u00fcndigkeit_ , means \"minority\" in the sense of not yet being \"of age.\" ( _M\u00fcndigkeit_ means \"majority,\" as in \"the age at which full civil rights are accorded.\") Kant basically argued that those who refused to assume this responsibility to think for themselves were politically no more than children. So in this essay he gave license to society's \"enlightened\" to lead the others. Hence the importance of education, especially _Bildung_ (self-cultivation), for attaining this state of political and social responsibility. The emphasis we have seen so far on observation and attention reveals that _educating the eye_ was absolutely central to discharging this ethical duty. Self-cultivation was not limited to training in literature, music, or history; training _vision_ was also very much a part of this process. Scientific or medical observation was only one element of a range of viewing practices that could be considered _Bildung_ -worthy. Pedagogical practices of visual education and practices of aesthetic contemplation, which the next two chapters respectively explore, were two others. Their opposite number\u2014what we call \"spectatorship\"\u2014can be fully appreciated only by examining the infantilization of the \"unenlightened,\" which is the task of [chapter 3.\n\nTHE TASTE OF A NATION\n\nEDUCATING THE SENSES AND SENSIBILITIES OF FILM SPECTATORS\n\nI step in; intermission has just begun. An oppressive, damp draft hits me, even though the doors are open. The spacious room (500 capacity) is filled to the last seat with children. There is an incredible ruckus: running, yelling, shrieking, laughing, chatting. Boys scuffle. Orange peels and empty bon-bon boxes fly through the air. The floor is studded with candy wrappers. Along the windowsills and radiators young toughs roughhouse. Girls and boys sit together, densely packed. Fourteen-year-old boys and girls with hot, excited faces tease each other in unchildlike ways.... Children of all ages, even two- and three-year-olds, sit there with glistening cheeks. Women walk among them selling sweets. Many children sneak candy and drink soda [ _Brause_ ]; young boys smoke furtively.\n\nThen the movie begins.\n\n\u2014A SCHOOLTEACHER FROM BREMEN (1913)\n\nFor the cinema reformers of imperial Germany, this was a scene from hell. This is what German _Kultur_ had come to, what modernity had wrought: children melting and spoiled like day-old candy on the floor of a movie theater. Like Professor Rath of _Der Blaue Engel_ (The Blue Angel, 1930), who follows his students into a seedy nightclub and is initially shocked by the sexuality and degeneracy within, this schoolteacher from Bremen walked into a matinee and was horrified by what she saw. Her emphasis on the corporeality of the scene, on the _body_ of the audience, so to speak\u2014fighting, eating, sweating, and awakening sexuality\u2014attests to the perception that cinema presented a grave danger to the children's emotional and moral fitness and especially their physical health. Given the traditional bourgeois association of sensuality with the lowly masses, this scene also represented a threat to the well-being of the nation, of the body politic. As this chapter will demonstrate, \"children\" and \"the masses\" were often interchangeable concepts. Indeed, while cinema was often portrayed as a gaping Moloch devouring innocent children in some pagan ritual, the children depicted in this passage are far from sacrificial lambs. They present something of a veiled threat to the narrator, as if she had entered a strange, chaotic culture. In the contemporary literature on children and cinema, scenes like this one provoked twin paternal urges: to protect and to control.\n\n\"Women walk among them selling sweets.\" Along with the concern for sensuality (and its tacit partner, capitalism), the numerous references to sweets stand out in this description. Implicit was the assumption that cinema was spoiling the \"taste\" of the children for financial gain. Reformers complained constantly about the \"tastelessness\" of both the theater atmosphere and the films themselves. Figuratively speaking, the concession candy that ate at the children's teeth was also rotting their aesthetic sensibilities. Konrad Lange, a noted art historian at the University of T\u00fcbingen and a ferocious enemy of film, put it more bluntly: \"If one were to judge the artistic understanding of our good, middle-class citizens, one would have to say that their taste is rotten to the core. They have a morbid taste for the slick, the effeminate, the sentimental, and the sugary. They display a demoralizing aversion to the healthy dark bread of true art.\" Lange made the relationship between taste, class, and the body as explicit as possible. Taste, as the simultaneous expression of individual judgment and social distinction, serves as a connection between the private and the public spheres. It is, as Pierre Bourdieu notes, \"a class culture turned into nature, that is, _embodied_.\" It therefore provides a link between individual consumption and a national agenda. Protective of the masculine ideals at the heart of that agenda, Lange condemned the feminization of culture accompanying the onslaught of modernity; he was not alone: most middle-class males of the Western world seemed to share his concerns.\n\nLange's solution, like that of many teachers and educators involved in cinema reform, stressed the education of children and adults. The abiding faith in the ability of education to overcome social ills and promote social progress was a fundamental plank in the platform of many fin de si\u00e8cle movements, from the socialists to the progressives. But the tradition of _aesthetic_ education, with its promise to harmonize the senses and sharpen judgment, offered a quintessentially German solution. By pointing the way to a \"true\" and \"pure\" aesthetic experience, aesthetic education also pledged to counteract the corrupting influence of the cinema. While many reformers, such as Lange, steadfastly refused to be seduced by cinema's charms, some flirted with this particular Lola, courting her in hopes of making her an honest woman by giving her an aesthetic education (or an education in aesthetics). Even while the film industry used the reformers for its own ends, it revealed the contradictions of their ideology. Just as Professor Rath's affair with Lola reveals the indefensibility of his position\u2014a teacher who, ultimately, does not have the best interests of his students at heart\u2014so, too, the reformers' involvement with film shows that there were larger issues at stake than the health of the children.\n\nEven if Professor Rath portrays the hypocrisy of his class, I see him as a distinctly modern character caught between\u2014and profoundly ambivalent about\u2014the solemn textbooks of his classroom and the cunning spectacle of the nightclub, or more broadly, between _Kultur_ and _Zivilisation_. As Norbert Elias and others have noted, these terms played an important function in German self-identity; by the beginning of the twentieth century, they formed a contrasting pair. If _Kultur_ implied \"inwardness, depth of feeling, immersion in books, development of the individual personality,\" or the general cultivation of one's inner life, then _Zivilisation_ implied \"superficiality, ceremony, formal conversation.\" Or, as Raymond Geuss puts it, _Zivilisation_ \"has a mildly pejorative connotation and was used to refer to the external trappings, artifacts, and amenities of an industrially highly advanced society and also to the overly formalistic and calculating habits and attitudes that were thought to be characteristic of such societies.\" By the 1910s and 1920s, _Zivilisation_ came to be associated with the primary burdens of industrialized society: capitalism and technology, or the utilitarian, material, even decadent world of commercial interests. By contrast, _Kultur_ was associated with not only inner, spiritual values but a specifically German configuration of ethical and moral ideals. Professor Rath is caught between his allegiance to learning and teaching the subtleties of the German cultural heritage, to cultivating his soul and those of his students to become good Germans, on one hand, and his attraction to the superficial, voyeuristic pleasures of a crass milieu that combines the worst of sensuality and capitalism, on the other. His downward spiral, of course, begins with his choice between them.\n\nGerman educators interested in reforming film and its theaters felt they faced the same dilemma. But unlike Rath, they did not approach it as an either\/or choice. Instead, the appropriation of motion pictures for educational purposes was a negotiation between aesthetic and moral values, on one hand, and modern technology or commercial interests, on the other. Such negotiations necessarily implied that commercial cinema was not viewed negatively across the board. Just like the reform movements in general, film reform was nothing if not multiple, consisting of many different voices expressing the full range between progressive optimism and cultural pessimism. Yet a picture emerges of a well-meaning but deeply ambivalent group committed to preserving traditional ideals while modernizing the social curriculum. Motion pictures, in this case, were for many not merely a problem to be solved but perhaps also a solution in themselves; if they could be reformed, or \"ennobled\" as reformers often put it, then they might serve as a means to reconcile _Zivilisation_ and _Kultur_ , a goal shared by most of the reformers of imperial Germany, no matter what their chosen object of improvement. The educational use of motion picture technology reveals this attempt at reconciliation, especially in the way that films were made to fit established pedagogical practices. As reformers pressed film into an educational mold, their work also highlighted the ideological and practical contours of that framework.\n\nSpecifically, the use of motion pictures in education followed the contours of a pedagogical trend or approach known as _Anschauungsunterricht_ , or \"visually based means of instruction.\" Popularized in the nineteenth century, this method exemplified the reformist impulse in education, in that it attempted to counter the perceived overrationalization and rote memorization of the traditional pedagogical approach with a self-consciously modern strategy that emphasized the immediate visual perception of things themselves, as opposed to their description in books. Known as \"the object lesson\" in English-speaking countries, the approach asked teachers to present an object, such as a seashell, and ask the students a series of simple questions about it. The questions were designed to lead the students from concrete description to a higher, abstract understanding of the object in relation to its environment and to other things. Many claimed that motion pictures could function as outstanding representations of objects that could not be brought into the classroom, if and only if film could be made to conform to this and other methods of visual training. This chapter, then, continues some of the concerns traced in the previous chapter, especially the investment of the expert class in specific methods of processing visual information. At stake here is not merely the type of viewing, such as medical or scientific observation, but the culture's interest in _what counts as learning. Anschauungsunterricht_ , in this respect, was not just one more pedagogical method but a way of acquiring _Kultur_ ; it was a strong component of _Bildung_ , or the process of self-cultivation. What counted as learning, in this case, was a way of forging _vision_ and _thought_ into _taste_ ; if \"taste\" were the sign of cultivation, then \"vision\" was the means by which taste was trained.\n\nThis chapter argues that the goal of reform was not merely to align film and its theaters with standards of taste and morality but to conform motion pictures to specific modes of viewing; _Anschauungsunterricht_ serves as an obvious and convenient method of training vision to which reformers and educators adapted motion pictures and their projection. While _Anschauungsunterricht_ was explicitly a method of training observation in school-age children, the ideal pertained to adults as well, as we see in all sorts of adult education programs at the time, especially those concerned with aesthetic education. Hence, the image of spectatorship or of ordinary movie audiences at this time must be understood in contrast to this attempt to assimilate motion pictures into a larger ideological arena, signaled most strongly by the desire to train audiences to a particular mode of viewing associated with _Anschauungsunterricht_ and aesthetic education.\n\nSo this chapter charts the work involved in fitting motion pictures to educational and reformist agendas; it also surveys the cultural and ideological landscape on which these efforts took place. Three sections alternate between practical efforts and historical and theoretical context. The first section discusses the general contours of reform in Wilhelmine Germany before outlining some of the concrete steps reformers took to protect child audiences from the hazards of cinema. Censorship, taxes, and child protection laws were accompanied by attempts to create an alternative film system by controlling means of production, distribution, and exhibition. I will also describe reformers' efforts to persuade production companies and theater owners to support reform films and exhibition values, which led to the creation of reform theaters and community cinemas. The second section examines some of the guiding principles or assumptions behind these efforts, including _Anschauungsunterricht_ , aesthetic education, and the discourse on \"the child\" and \"the masses.\" The urge to protect children from the \"degeneracy\" of mass entertainments went hand in hand with the desire to educate the general public. Both concerns drew life from child and crowd psychology popularized at the turn of the century. Reformers saw aesthetic education and _Anschauungsunterricht_ , entwined in theory and practice, as potential solutions to the problematic picture of spectators the experts painted. The third section looks closely at the work of two reformers who set out to create alternative, educational, and edifying exhibition spaces in response to the perceptions and problems outlined in the previous section. Educator and reformer Hermann Lemke articulated principles for the educational use of film for school-age children, especially at commercial theaters that arranged special screenings for elementary schools. Another reformer, Hermann H\u00e4fker, hoped to offer educational or edifying film presentations for adults. H\u00e4fker's attempts to create \"model presentations\" ( _Mustervorstellungen_ ) exemplify the use of film as an instrument of ideological solidarity. Increasingly worried about the \"bad taste\" of mass entertainments, H\u00e4fker enlisted film in a program of aesthetic education designed to raise the sensibility of the people to a unified, national level. Taking his cue from a long tradition of aesthetic education dating back to Schiller, as well as the art education movement then taking place, H\u00e4fker wanted to use motion pictures to train the tastes of the nation. Finally, this section takes a longer look at Schiller's ideas about aesthetic education to lay bare Germany's long-standing ideological investment in the relationships between vision, taste, pedagogy, and nation.\n\nCINEMA AND THE SPIRIT OF REFORM\n\nIn their desire to make a change for the better, the men and women involved in film reform were part of a much larger set of movements sweeping the industrialized world around the turn of the century. As increased industrialization and urbanization brought one social upheaval after another, \"reform\" expressed the mood of the times in a variety of ways throughout Europe and the United States. In the United Kingdom, constitutional reforms swept through Parliament as groups demanded suffrage throughout the last third of the century. In the United States, the agrarian Populist movement of the 1890s and the Progressive movement of the 1900s reflected a broad impulse toward criticism and change. Progressivism, in particular, captured the spirit of reform through its outrage over the excesses of capitalism, its faith in progress, and its interventionist policies. During the 1880s, the pressures of industrialization and democracy prompted the French parliament to create the only state-run, compulsory, secular primary school system in the world. The growing confrontations between the forces of labor and capital also prodded republican politicians to campaign for social legislation, such as regulation of working conditions, to ensure social peace.\n\nGermany, in particular, was deluged by swelling transformations in the public sphere provoked by rapidly changing demographics. The Industrial Revolution and national unification came relatively late to Germany and accelerated very quickly. The resulting discord between the classes and between rural and urban lifestyles seemed especially acute. During the high tide of these changes, which occurred from around 1890 to 1920, the concept of \"reform\" as an expression of the sense of transition and as a plan for managing it took on special significance for self-understanding. Germany's groundbreaking legislation providing for compulsory insurance for workers' sickness, workplace accidents, and retirement pensions became an influential model for the United Kingdom, the United States, and France. These measures, dealing in some form with physical conditions and consequences of the workplace, illustrate the strong connection between class and somatic issues in reform movements during the late nineteenth century. Reform manifested itself in everything from _Jugendstil_ decor, to a new, more \"natural\" style in women's clothing ( _Reformkleidung_ ), to nutrition reform ( _Ern\u00e4hrungsreform_ ). \"Reform\" implied a battle against tradition, against perceived cultural and social stagnation; as such, it provided a plan for the formation of new, more \"authentic\" concepts for living. In fact, the connections between the reform movements and the more general tradition of _Kulturkritik_ are very strong; from Jean-Jacques Rousseau and Johann Heinrich Pestalozzi in the eighteenth century to Friedrich Nietzsche and Paul de Lagarde in the nineteenth, the critique of society paved the way for a general re-evaluation of values in the twentieth.\n\nVery often, the critique of society focused on the educational system. Education reform was among the first movements to sweep across Germany. The kaiser himself had set the agenda on a December morning in 1890 while addressing a congress of educators in Berlin; Wilhelm II claimed he grasped \"the spirit of an expiring century\" with his calls for school reform. In answering the question of how the German schools of the nineteenth century could be reshaped to meet the needs of the twentieth, the kaiser echoed sentiments that had been expressed throughout Europe during the often rocky transition from the Victorian age to the modern. Specifically, he voiced his fears that the _Gymnasium_ (high school) failed to train its students adequately for the requirements of Germany's rapid industrialization. Second, he complained about the \"excess of mental work\" in the schools, arguing that such \"overburdening\" was threatening the physical health of Germany's youth. Finally, he insisted that German schools devote more time and energy to fostering specifically national values, thus turning away from the traditional emphasis on the classics: \"We must make German the basis of the _Gymnasium_ ; we should raise young Germans, not young Greeks and Romans.\"\n\nThe kaiser's concerns about \"the modern,\" \"the healthy,\" and \"the national\" reflected and reinforced similar fixations of the European elite. Like them, he found the educational system to be both the problem and the solution to perceived crises. By inadequately preparing the nation's children for the demands of the future, the system risked irrelevance, according to reformers of the time. Swift reform promised both a brighter future and a greater measure of control over the rapid changes taking place. Among the different examples of education reform were the country boarding schools ( _Landerziehungsheime_ ), which were experimental schools located in the countryside as an explicit rejection of urban culture. Their emphasis on physical education echoed the hopes of the youth movement for a spiritually renewing combination of countryside, fresh air, and _Volk_. Likewise, work schools ( _Arbeitsschule_ ) hoped to renew the creative (and ethical, hence political) spirit through manual labor, such as gardening, and handicrafts, such as wood sculpting or leatherwork. The art education movement ( _Kunsterziehungsbewegung_ ) similarly stressed the creative capacities of children, advocating renewal through art and education of the aesthetic sensibility.\n\nFilm reformers shared the kaiser's interest in \"the modern,\" \"the healthy,\" and \"the national.\" At the center of their concerns lay motion pictures, which they also found to be both scourge and cure. An emblem of modernity, cinema represented a plague, especially toxic to children, and proper education of the public was the only hope to halt the epidemic. At the same time, cinema was emerging as the most powerful instrument of mass education and therefore potentially provided the surest treatment for whatever ills modernity had spread. Before treatment could begin, however, commercial interests had to be persuaded to participate in this remedy. Moreover, film needed a stamp of legitimacy to have any authority in this rescue mission. Film reform was the process by which these goals were attempted, if not completely achieved. It shared roots, objectives, and ideology with other reform movements of the day, especially, and not surprisingly, educational reform.\n\nIt does seem rather unusual that the head of the German empire, almost by definition the representative of a conservative status quo, would come out so strongly in favor of reform. A mixture of progressive reforms and reactionary politics indicated the ambivalent attitude of the bourgeoisie toward the troubling issues of the day. All reform movements revealed, in one way or another, the fundamental irony of the kaiser's position. The calls for clothing reform in Germany exemplified this contradiction. In his 1901 book, _The Culture of the Female Body as a Foundation for Women_ ' _s Clothing_ , Paul Schultze-Naumberg made an extensive study of the debilitating physical effects resulting from methods of forcing the female form into an aesthetic ideal. In a graphic and impassioned plea to eliminate the corset, in particular, Schultze-Naumberg demonstrated how its use eventually caused deformation of the muscles, bones, and internal organs. He called for a more functional clothing in keeping with \"a new concept of corporeality.\" While consistent with similar efforts by women's movements to liberate themselves from the pressures of social constraints, Schultze-Naumberg's \"new concept\" of more \"natural\" bodies included only those from healthy German stock. Carl Heinrich Stratz, another strong advocate of clothing reform, took a similar approach in his book _Women_ ' _s Clothing and Its Natural Development_ , grounding his arguments for the elimination of the corset on the conclusion that it threatened the racial superiority of European women. Schultze-Naumberg and Stratz, whose worries about the integrity of the Fatherland were cloaked in concerns for the health of women, are excellent examples of the fusion of progressive goals of more liberal movements with reactionary, nationalist politics.\n\nClearly, Schultze-Naumberg and Stratz shared with their fellow guardians of culture a rather paternalistic attitude toward the role of women in the changing public sphere of modernizing Germany. Women in general, and female sexuality in particular, were often targets of public disapproval about modern life, as in Stratz's work above, or in the complaints about Asta Nielsen's \"unladylike\" behavior in her films, or in the concerns about the number of women in cinema audiences. For some, these complaints are signs of a deeper anxiety about the increasing role of women in public life in Germany; the women's movement or the youth movement in their various manifestations are often cited as prime motivators in the perceived decline of male cultural authority. According to this version, the women's movement exemplified a menace that appeared to surround German intellectuals; actions such as the youth movements seemed to threaten paternal credibility even in the home. Faced with such massive structural changes, the cultural elite embraced reform as a way of coping with, and controlling, modernization. While it is true that most voices that survive in print are male, we must not forget that women were not simply the objects but the agents of reform. In fact, late nineteenth-century imperial Germany saw a proliferation of volunteer organizations\u2014instigating reform, helping the poor, and exchanging information\u2014that provided structure to a society in transition. Women entered the public sphere not only as consumers but also through volunteer organizations that functioned in tandem with, even as a substitute for, the state and provided steady and urgent pressure for changes in social policy. So social policy in imperial Germany was not simply a top-down affair instigated by male experts but grew out of women's culture. Likewise, while Germany's \"crisis in culture\" might have been acutely felt by male intellectuals as a decline of male authority, this perception must be tempered by the knowledge that the social mobility that women and workers gained from modernization also benefited males and the middle class not only through the greater economic power that came with a consumer society but the political power that came with the alignment of state policy with middle-class goals\u2014an alignment that occurred partly as a result of the intervention of women's and volunteer organizations. So while the hyperbole of public debate sometimes provides an easy example of male, middle-class \"anxiety,\" we should not take it entirely at face value.\n\nThe _Schund_ campaigns of late nineteenth-century Germany are a great example of the battle between modern consumer culture and the goals of middle-class volunteer organizations. _Schund_ is usually translated as \"trash\" or \"rubbish,\" but reformers of the late nineteenth century applied it to serialized novels and pamphlets purveyed by colporteurs or kiosks, especially literature perceived to lack any redeeming value yet not obscene. While the serialized novel emerged in the mid-nineteenth century, the market grew exponentially thereafter; by 1890 it was so popular that _Schundliteratur_ accounted for fully two-thirds of all German literary sales. By the time of the Weimar Republic, _Schund_ referred in general to any thin, mass-market paperback novel sold by the millions at kiosks and stationery shops.\n\nEducators and school administrators were the first to launch campaigns against this form of entertainment; according to Corey Ross, \"the roots of the campaign against _Schund_ can indeed be traced back to the efforts of elementary school teachers in the 1870s and 1880s to influence the reading habits of their pupils by drawing up lists of recommended titles.\" Teachers continued to be the primary force behind these campaigns, even as they comprised a diverse group of interested parties, including temperance groups, religious associations, women's commissions, leagues against public vice, and local police forces. Yet, as Kara Ritzheimer has argued, the rhetoric of reform unified these groups, despite their diverse goals and methods. Ritzheimer paraphrases, for example, a professor who warned that\n\nchildren who read excessively were likely to develop a lust for reading ( _Lesewut_ ) that might \"effeminize the body,\" \"cause the senses to lose their acuteness,\" \"weaken the memory,\" \"over-excite the imagination,\" and \"destroy a will to pay attention to serious matters.\" Furthermore, this \"reading lust\" was capable of breeding indolence ( _Schlaffheit_ ), indifference ( _Gleichg\u00fcltigkeit_ ), absentmindedness ( _Zerfahrenheit_ ), mental laziness ( _Denkfaulheit_ ), extravagance ( _\u00dcberspanntheit_ ), and slack behavior when it came to work and play ( _Unlust zu Arbeit und Spiel sich einstellen_ ).\n\nThis rhetoric of sensuality, addiction, feminization, and weakness was a portrait of media effects that transformed apparently normal children into the worst caricature of the lower class. Beyond the vocabulary of media effects, these groups also promoted _children_ as the primary focus of their efforts, thereby unifying their protectionist rhetoric around a class-neutral issue, rather than waging a campaign in the name of _adults_ , which could barely evade paternalistic connotations, if not all-out class warfare. In any case, the rhetoric was incredibly pliable, in that it was applied equally vociferously to books and motion pictures.\n\nCinema reform not only patterned itself after the educational reform movements, the instigators were often hardened veterans of the anti- _Schund_ campaigns. Karl Brunner, for example, was a _Gymnasium_ professor and a leading anti- _Schund_ campaigner who eventually became the head film censor in Berlin. The Hamburg commission discussed later started as a response to _Schund_. Most of the reformers were teachers and educators, so their close ties to the education reform groups of the period were also formative. Hermann Lemke, a _Gymnasium_ professor from Storkow and one of the founders of _Kinoreform_ , was well connected to the Society for the Dissemination of Popular Education (Gesellschaft zur Verbreitung von Volksbildung), the leading educational organization in Germany. Hermann H\u00e4fker, the most articulate representative of film reform and arguably Germany's first film theorist, was a journalist and writer who was also close to the leaders of the art education movement. Konrad Lange, one of the leading voices of the art education movement, taught art history at the University of T\u00fcbingen.\n\nDespite their similar backgrounds, these reformers were not all of one mind. The disparate views and priorities of all involved, as well as the absence of a central organization or platform, make it difficult even to characterize _Kinoreform_ as a movement. Scattered around mostly northern and small-town Germany, the representatives worked primarily at the local level, trying to coordinate national efforts through friendly trade journals, such as _Der Kinematograph_ (1907\u20131935) out of D\u00fcsseldorf and especially _Bild und Film_ (1912\u20131914) out of M\u00f6nchen-Gladbach. The birth of trade magazines devoted exclusively to film coincides with the birth of the reform movement in 1907; during its earliest years, _Der Kinematograph_ acted as a willing partner in _Kinoreform_. The range of viewpoints in its pages, and in the other magazines that followed shortly thereafter, testifies to the difficulty the reformers had in choosing the most effective course of action.\n\nIf they did not agree on methods, their approaches at least reflected the experience and infrastructure already gained in the work against _Schund_. Basically, the efforts of the film reform movement overall can be divided into \"negative\" and \"positive\" solutions, in the parlance of the day: those that emphasized regulation, taxes, police enforcement, and censorship, and those that offered alternatives to the objectionable fare; this chapter will focus on the latter, \"positive\" approach. We can also divide these approaches further based on the object of reform. _Filmreform_ , for example, expressed an emphasis on the _content_ of films, with an accompanying strategy that focused on production. _Kinoreform_ , on the other hand, expressed an emphasis on the _space_ of film exhibition, and with it a strategy to clean up and \"uplift\" film theaters (fig. 3.1). _Filmreformers_ such as Brunner and Lange spent their energy devising negative methods to control film production and reception, while Lemke and H\u00e4fker developed positive strategies for both _Filmreform_ and _Kinoreform_.\n\nFIGURE 3.1. A typical storefront movie theater ( _Ladenkino_ ) from the pre\u2013World War I era\n\nCourtesy Deutsches Institut f\u00fcr Filmkunde, Wiesbaden\n\nYet these reformers had a set of common objectives framed in ways similar to those in the _Schund_ campaigns. First and foremost, they felt compelled to protect children from what they perceived to be the dangerous effects of cinema. This was first explicitly stated in 1907, when a teacher's group in Hamburg, the Society of Friends of the Schools and Instruction for the Fatherland (Gesellschaft der Freunde des vaterl\u00e4ndischen Schul- und Erziehungswesens), formed a commission to study the effects of cinema on schoolchildren. Its conclusions were predictable and familiar: both the films themselves and the theaters produced physical and moral side effects in school-age children. The combination of the \"flicker effect\" and the lack of adequate ventilation in theaters caused \"eyestrain, nausea, and vomiting,\" according to the commission. Emphasizing the connection between the body and ethical judgment, the symptoms were a sign of a deeper moral sickness, manifested in school by \"apathy for learning, carelessness, and a tendency to daydream.\" Jurist Albert Hellwig, certainly the most prolific reformer who advocated the negative approach, echoed these concerns in 1914, when he argued that \"a promotion of a certain superficiality and inattentiveness, as well as a retardation of concentration and aesthetic cultivation\" could be counted among the psychological dangers to young moviegoers\u2014a diagnosis familiar from the discourse examined in the previous chapter.\n\nSecond, the reformers made it clear from the very beginning that they hoped to use cinema for educational purposes. In this and many other ways, the German reformers were very similar to their American counterparts, who also took it upon themselves to \"uplift\" both the theaters and the films for the good of the masses. The Hamburg commission concluded its study with the recommendation that\n\nTechnically and thematically impeccable cinematic presentations can be an outstanding instrument for education and entertainment. Pedagogically and artistically minded groups must advocate for better, nobler uses of the cinematograph by encouraging the big corporations in this industry to present good, child-oriented productions in special screenings for children.\n\nHermann Lemke answered this call to arms independently in the summer of 1907, when he persuaded a Friedenau cinema theater owner to host Germany's first \"reform theater,\" which was likely merely a \"film reform\" night or series at a commercial theater, given that it did not last very long in this form. Lemke gave the opening address, making the goals of cinema reform clear to the mostly middle-class audience of teachers, press, and community leaders. He charged that the current state of cinema had caused the aesthetic sensibilities of the people to regress. Calling on the combined power of educators and the press, he maintained that \"when the taste of the people is so backward, it's the duty of the intellectual [ _geistigen_ ] leaders to influence them and put their aesthetic taste back on the right track.\" Cleaning up the cinema theaters was the first order of business in this project, making it one of the earliest examples of _Kinoreform_. Lemke applauded the improvements the theater owner had already made:\n\nThis reformation is already apparent in the way this auditorium has been given a worthy interior decoration. Gone is the small, narrow room where everyone is crammed and squeezed together; in its place we find a larger, airier hall... so that the patron's sense that he is in a second-rate establishment vanishes all on its own. Good ventilation has been provided in order to reduce health risks.\n\nLemke's concerns demonstrate how closely \"taste\" and \"respectability\" were tied to \"the body,\" and especially the body of \"the masses.\" He was preaching to the converted, however. _Der Kinematograph_ later reported, \"it seems that the middle class is more interested than the working class in the direction the reform theater is taking. While the seats in the third section show hardly any patrons, the first section (50 cents admission) is mostly sold out.\" Still, Lemke was sufficiently encouraged to organize a Cinema Reform Association the following autumn. Represented by teachers, members of the press, theater owners, and production companies, the association was one of many throughout Germany that hoped to coordinate efforts from these quarters toward their educational goals. Indeed, composed of businessmen, teachers, council members, and theater owners in the community, local _Kino-Kommissions_ such as this were the primary means through which reformers organized their efforts, disseminated their results, and created larger networks. Lemke's society received contributions from such firms as the German branches of Eclipse and Gaumont. While cleaning up the _Kinos_ , the reformers turned their attention to the films themselves.\n\nEnjoying an easy fraternity with producers during the early years, film reformers hoped to capitalize upon their good relations with the motion picture companies to increase the number and availability of reform-type productions. In a particularly idealistic move indicative of the \"positive\" approach, Lemke in 1908 suggested that his reform association act as a \"Film-Idea-Central,\" a clearinghouse of sorts for reform-minded scripts. Members of the association could submit ideas for scenarios, and the society would negotiate with the studios on the writers' behalf. Lemke explained, \"Because we're in constant contact with the manufacturers, such an exchange will be relatively easy to arrange, particularly because we know what is being demanded. We would provide distribution free of charge and only require that those who use it become members. In this way, we may be able to bring the film companies to the cutting edge of this moment and also have a productive influence internationally.\" Unfortunately, while the members of the movement might have held some early enthusiasm for this plan, the film companies themselves apparently did not take to it; the idea never went beyond the initial stages, and no further mention is made of the Film-Idea-Central in the trade press or reform publications.\n\nThe failure of the Film-Idea-Central and the film reform theater in Friedenau established something of a frustrating pattern for the reformers. Film companies and exhibitors expressed early interest in reform projects, even going so far as to sponsor events, but eventually refused more meaningful and lasting support. The end of 1908 saw the opening of Germany's first film trade show\/exhibition at Berlin's Zoological Gardens. Jointly sponsored by Lemke's reform party and the leading film companies at the time, and with the rather obvious motto of Refining the Cinema ( _Veredelung des Kinos_ ), it was nonetheless heavily criticized even by friendly periodicals for its lack of organization. Exhibitors, manufacturers, and production companies declined the reformers' help for the next exhibit in 1912. Likewise, when Lemke and H\u00e4fker attempted to muster support for their special exhibitions, the film companies were initially supportive but lost interest fairly quickly. Realizing that domestic companies could not or would not produce sufficient numbers and variety of educational films, H\u00e4fker went so far as visiting foreign film companies, such as the Charles Urban Trading Company in London and Eclipse in Paris, to find suitable nature films for his exhibitions. Lemke even went to England and wrote film treatments to jump-start some sort of interest in his program. Very early on, it was quite clear that the production companies were cautious about backing the reformers and their schemes.\n\nThis did not mean, however, that the film companies wanted nothing to do with the reform movement. They were certainly willing to use the reform movement for their own ends; despite their difficulties, the reformers were still a legitimating presence\u2014they were, after all, educators, clergy, journalists, and otherwise pillars of their respective communities. Film companies were eager to cash in on this allegiance. Advertising trumpeted this relationship, even if the reformers had nothing to do with the making of the film. A 1912 Italian film distributed by the German company PAGU, _Die Irrfahrten des Odysseus_ (The Wanderings of Odysseus), was rather disingenuously labeled a \"Reformfilm\" and carried this blurb: \"From a special press screening, which was attended by the most respected Berlin literary figures and art critics, came the unanimous decision: 'This film signals the long-awaited reform of cinema'\" (fig. 3.2). Aware of the potential directions cinema could take, the film companies initially went along with the reformers, especially if they could be used as a selling point. But as soon as it became apparent that the vast majority of the viewing public was more interested in narrative entertainment free from \"ennobling\" connotations, the companies brushed off the reform societies' efforts to influence the product directly.\n\nThe extreme positions of some reformers did little to help the overall cause with the production companies. Lange and Brunner, for example, were steadfast in refusing film any legitimacy whatsoever as a medium of entertainment. Their regular denunciations of \"film drama\" ( _Kinodrama_ ) merely antagonized an industry leaning heavily toward narrative films. This prejudice against narrative films often disguised stronger rhetoric against international domination of the German film market. \"In the international film drama, the wildest passions of all nations come together for a gruesome rendezvous,\" clergyman Paul Samuleit charged. Likewise, their complaints about capitalist interests tainting cinema's potential were thinly veiled laments about the presence of _foreign_ capital. Some reformers, such as H\u00e4fker and Lange, dismissed film drama because of aesthetic concerns. It did not offend their sensibilities because of sloppy production qualities, although these did attract attention. Rather, the filmed drama betrayed what they saw to be cinema's primary mission: to record movement and \"real life.\" The argument for filmic realism, of course, coincided with their desire to use cinema for educational purposes, as we will see in the next section. As Sabine Hake notes, they did not dismiss the possibility of story elements in their educational films, but the excesses of the \"trashy film\" so contradicted their stated ideals that many rallied against film drama altogether, for both political and aesthetic reasons.\n\nFIGURE 3.2. _The Wanderings of Odysseus_ (L'Odissea, Italy, Milo, 1911), touted to be a \"Reformfilm\"\n\nLemke hoped to reform the cinema through example, stressing cooperation with and from the industry, and to rally schools together to create a distribution system. Others were not so willing to rely on this teamwork. One faction of the reform movement, led by Albert Hellwig and Karl Brunner, saw censorship and regulation (the \"negative\" approach) as the best way to combat the onslaught of _Schundfilme_. Both Hellwig and Brunner advocated a series of legal restrictions on the cinema, including censorship, entertainment taxes ( _Lustbarkeitsteuer_ ), poster censorship, safety regulations, and child protection laws ( _Kinderschutz_ ). Authorities tried to maintain some control over child audiences (and, consequently, theaters) by restricting their visits to specific hours of the day, regulating the length of the matinees, and requiring that they be accompanied by an adult, that police should have unlimited access to the theater during the matinees, that the day's program must be given prior approval, or that a \"suitable pause\" separate the films.\n\nReformers recognized early on the importance of establishing a distribution network for their educational films. For this task, Lemke and his circle enlisted the help of the Society for the Dissemination of Adult Education (Gesellschaft zur Verbreitung von Volksbildung, hereafter referred to as the GVV). An umbrella organization for more than 8,000 local education groups, clubs, associations, and societies, it was a formidable partner in _Kinoreform. Bildungs-Verein_ , the house publication, had a circulation of 13,000\u2014many times that of any film trade magazine. Yet the GVV leadership was hesitant about cinema's importance as an educational tool. Even though Johannes Tews, the director of the GVV and editor of _Bildungs-Verein_ , had attended the opening of Lemke's Friedenau reform theater, he still considered cinema to be of minor significance. The GVV resisted involvement with cinema until 1912, when it established a film distribution center of 180 films in 16 categories, from history of the Fatherland to educational films on biology.\n\nThe reformers found a more willing and beneficial partner in the Lichtbilderei, established in 1909 as a foundation of the Association for Catholic Germany (Volksverein f\u00fcr das katholische Deutschland). The Lichtbilderei was Germany's largest educational film institute before World War I, with an extensive catalog of titles. It began as a rental source for magic lantern slides, which could be used for public lectures, but started collecting films as well after 1911. By the end of 1912, it reportedly had around 900 titles and was collecting more at about 30 films per week, and by 1913 offered 400 slide series and 1,400 film titles. The Lichtbilderei was not limited to providing films for schools, churches, and clubs; it also provided programming for many commercial theaters. Approximately 40 weekly theaters and 50 to 60 Sunday _Kinos_ showed Lichtbilderei films regularly. The Lichtbilderei was also involved in the distribution of more commercial dramas, actually acquiring \"monopoly\" rights over such established hits as _Quo Vadis?_ (Italy, 1913), _Giovanna d_ ' _Arco_ (The Maid from Orleans, 1913), and _Tirol in Waffen_ (Tirol in Arms, 1914). From 1912 to 1915, the Lichtbilderei was something of an organizational center for the cinema reform movement. Its stock of films gave life to the community cinemas and private _Reformkinos_ , and its publications\u2014the periodical _Bild und Film_ (Image and Film) and the series of books from the association's Volksvereins publishing company\u2014were the principal forum for the discussion of _Kinoreform_ issues after 1912.\n\nIn 1912, the GVV, in association with the Lichtbilderei, established the funds for two educationally oriented _Wanderkino_. These traveling cinemas toured from town to town, playing for four to six weeks in each place, in an effort to offset the influence of commercial cinemas and unify aesthetic and educational standards across the nation. Showing between nine and eleven films an evening, accompanied by lectures concerning such topics as \"A Modern Factory,\" the enterprise was basically modeled after the GVV's successful _Wandertheater_ and public lecture series. Between the fall of 1912 and the outbreak of war, the _Wanderkinos_ offered a total of 1,279 such evenings.\n\nReformers had most success with their exhibition experiments. In addition to the reform theaters and _Wanderkinos_ already mentioned, a number of communities established their own public cinemas. The first was founded in the town of Eickel at a cost to the community of 14,000 German marks. Others opened soon afterward, in such towns as Altona, Wiesbaden, Osterfeld, Frankfurt (Oder), Gleiwitz, and Stettin. These cinemas became the center of local reform activity and provided the precedent for the state-run cinemas of modern Germany, which continue to illustrate the relation between taste and nation. The proclamations of the early _kommunale Kinos_ articulated this relationship and the goals of the reform movement in general:\n\nTo oppose, for aesthetic, cultural, and patriotic reasons, the trash that is generally offered in the private theaters; to replace it with films of scientific, entertaining, and educational value; to exert, in association with established institutions with similar principles, a gradual influence on the film market, which is currently almost entirely dependent on foreign countries, and thereby keep here the millions that are flowing out of the country. Finally, to enlist cinema in the service of youth organizations and colleges by providing suitable presentations.\n\nTo the modern observer, the cinema reformers of imperial Germany might seem a bit quixotic. Tilting their lances to such impassive windmills as capitalism and narrative, they only reluctantly and belatedly conceded that they were charging against the wind of public opinion. As the movies became more popular and an evening's entertainment began to look less and less like a lecture series, instead relying more heavily on _Kinodrama_ , the reformers began to look more and more irrelevant. Their own Dulcinea\u2014the children of the nation\u2014seemed oblivious to their activities. Even those sympathetic to their cause, like this reviewer of a book on cinema and theater reform, found their efforts somewhat naive.\n\n[The author] is certainly entitled to his opinion in this terribly serious matter. However, he will surely also understand the skepticism of those who do not believe at all in the \"ennobling\" of films towards a literary bent [ _literarischen Seite_ ], because they see completely heterogeneous things being forced into an unhappy marriage. The idea that benevolent corporations will free the theaters of commercial interests altogether is too pretty to be given much credence.\n\nOthers were not so kind. Speaking on behalf of the industry in 1911, a trade journal editor pointedly replied to the reformers, \"We ourselves know what we need, and we don't need tutelage.\" One theater owner from the 1920s remembered them as \"sanctimonious folks and hypocrites, morality sleuths in male and female guise.\" Film histories, until recently, have been equally dismissive. Siegfried Kracauer charged simply that, with their zealous efforts to defend the literary canon of the nation, \"they yielded to the truly German desire to serve the established powers.\" Even if a bit condescending, Kracauer was not far off the mark. While the proclamations of the various _Kinoreformers_ embraced a wide range of opinion, they never strayed far from the status quo. As Sabine Hake noted, \"In sharp contrast to the intellectuals, the reformers aligned themselves openly with the existing power structure.\" We must not, however, underestimate the reformers' contribution to German culture. In trying to sway what Kracauer called \"the salutary indifference of the masses,\" the reformers succeeded in dominating the discourse on cinema in the years before World War I. In addition to the permanent impression they left on German film culture, German mass communication research owes them an especially heavy debt: their focus on media effects had a lasting influence on the vocabulary and goals of modern mass media studies in Germany.\n\nUltimately, of course, cinema reform was not completely successful. The reformers failed to meet their stated goals and, considering the extreme position of many reformers, this is perhaps all for the best. World War I abruptly changed the nation's priorities, and even though the calls for reform were heard again through the Weimar years, the urgency of the moment had passed. In 1913, lances heavy with disappointment, the movement clearly appeared to be running out of breath. Sighed Lemke, \"I was always hoping that someone would take over the chairmanship from me, assist me, and further expand the [Cinema Reform] Association, but no one was willing to do so and the result was that the Association remained in its infancy [ _in seinen Kinderschuhen stecken blieb_ , or literally, \"stuck in its children's shoes\"].\"\n\nCHILDREN, CROWDS, AND THE EDUCATION OF VISION AND TASTE\n\nLemke's metaphor was apt, because it reveals the extent to which the reformers thought about the cinema (and themselves) through the metaphor of \"the child.\" Because they were educators and teachers, this is perhaps to be expected. It is noteworthy, however, that they applied this trope to adult audiences. References to their audiences as \"children,\" especially in connection with mention of \"the masses,\" are scattered throughout the discourse. One reformer, looking for the underlying causes of cinematic drama's continued popularity, maintained that \"just as much blame belongs to the audience, the people [ _das Volk_ ], this 'big child,' whose alarmingly spoiled taste craves for cinema's dramatic trash and silly comedies, practically forcing the theater owners to present them with aesthetic and moral duds week after week.\" Even Georg Luk\u00e1cs thought about cinema spectatorship in terms of children: \"In the 'cinema' we should forget these heights [of great drama] and become irresponsible. The _child_ in every individual is set free and becomes lord of the psyche of the spectator.\" Whether Luk\u00e1cs's \"inner child\" was inherently good or evil depends upon one's viewpoint, of course, and there were many to choose from at this time.\n\nThis section will survey some of the prevailing assumptions about child psychology and pedagogy to clarify the underlying ideological presumptions about child and adult moviegoers. \"Suggestibility\" was the common denominator linking children and crowds; studies of children were even the source of theories about social psychology. So this section will demonstrate how the portrait of (film) spectatorship usually derived from expert analyses of children and crowds. At the same time, I will show that symptoms of this spectatorship were problems to be solved by training in observational methods, specifically aesthetic education exemplified by the kind of programs promoted by Hamburg museum director Alfred Lichtwark, who was the driving force behind the art education movement of the time. His approach was popular and well known to film reformers\u2014especially because its nationalist flavor appealed to the taste of many experts of the day\u2014but it was also familiar because it exemplified the observational approach to general education known as _Anschauungsunterricht_. Looking closely at this approach or trend in pedagogy reveals it was a self-conscious response to the perceived disorder and quickened pace of modern life; observational training was a way of ordering thought that countered pace and disorder by emphasizing \"dwelling\" and correlation. This section thereby connects psychology, reform movements, and pedagogy to explain the ideological and practical emphasis on expert modes of viewing as a solution to the multiple problems spectatorship posed to film reform.\n\nChild psychology of the period provides a partial explanation for the equation of children and the masses. Swedish author Ellen Key's _Century of the Child_ , an enormously popular children's rights manifesto published originally in 1900, advocated a reassessment of the prevailing view that children were inherently evil. Summing up a trend in child psychology that emphasized the creative nature of the child, it called for new teaching methods to correspond to the new century, leaving behind the authoritarian methods of the old school and reassessing pedagogy \"from the child outward\" ( _vom Kinde aus_ ). If adult society, utilitarianism, and the demands of the \"real world\" had determined the standards of pedagogy before, now attention turned to the child's needs and inner nature. Whereas the old pedagogy might have emphasized uniformity, now the child could expect to be treated as \"the measure of itself.\" As Key insisted, \"instruction should only cultivate the child's own individual nature,\" which Key and others assumed to be creative, good, and even wise.\n\nFreud, however, was less optimistic about the life of the child. His essay on \"Infantile Sexuality,\" published in his 1905 _Three Essays on Sexuality_ , painted a darker picture of childhood as a \"hothouse of nascent psychopathology.\" His explanation of the importance of the child's body\u2014describing the oral, anal, and phallic stages\u2014on mental development was groundbreaking. Its lasting contribution is manifold, but most immediately it underlined the influence of childhood development on adult mental life. While there is little indication that Freud's theories were wholeheartedly accepted by garden-variety reformers, the new child psychology of both Key and Freud provides a clue to the urgency reformers felt when they argued for aesthetic cultivation and against the influence of sexually charged _Kinodramen_.\n\nDespite Sigmund Freud's seminal contributions, Darwin's evolutionary theories of child development still had a strong grip on the public imagination during this period. In particular, Darwin argued that child development recapitulated the mental evolution of the species. Accordingly, the maturing child was expected to exhibit mental characteristics of subhuman species. In _The Descent of Man_ , Darwin observed, \"We daily see these faculties developing in every infant; and we may trace a perfect gradation from the mind of an utter idiot, lower than that of an animal low in the scale, to the mind of a Newton.\" Discussions of crowd psychology latched onto this teleological comparison between children and primitive mentalities. Gustave Le Bon, the most well known popularizer of nineteenth-century crowd psychology, characterized the masses as \"an enraged child.\" Furthermore, according to Le Bon,\n\nIt will be remarked that among the special characteristics of crowds there are several\u2014such as impulsiveness, irritability, incapacity to reason, the absence of judgment and of the critical spirit, the exaggeration of the sentiments, and others besides\u2014which are almost always observed in beings belonging to inferior forms of evolution\u2014in women, savages, and children, for instance.\n\nDarwin's evolutionary scheme provided a quasi-scientific basis for comparing crowds with children, but even more significant for this comparison was the concept of \"suggestibility.\" Le Bon devoted a chapter to \"the suggestibility and credulity of crowds,\" arguing that the crowd is \"perpetually hovering on the borderland of unconsciousness, readily yielding to all suggestions\" (21), a mental state most commonly found in women and children (29). Most serious psychologists of his time dismissed Le Bon's rather crude arguments, but the metaphorical connection between children and the masses was still quite powerful for researchers. In fact, one could argue that social psychology has its roots in child study. Alfred Binet, a disciple of La Salp\u00eatri\u00e8re's Charcot and one of the founders of experimental social psychology, used the observational opportunities provided by public school classrooms to test his evolving theories of suggestibility. His conclusions about children and suggestibility worked their way into his formative studies of social behavior, which had a profound impact on the direction of modern social psychology.\n\nReformers borrowed the concept of \"suggestibility\" as they described the cinema audiences and their scopophilia or _Schaulust_. The Hamburg commission noted this condition in their report, complaining that\n\nmany cinema presentations endanger children morally as well. Let's assume, for example, that a young boy with a tendency towards thievery were to see crimes presented with elegance and brilliant success. Wouldn't that arouse his imitative instinct? A young girl could easily learn how to get easy money and enjoy an apparently carefree and, in her eyes, wonderful life by selling her honor. When she needs to earn a living later in life, she might ask herself: \"why work at a sewing machine for 10 pfennigs an hour, why work at a factory for 10 marks a week?\"\n\nWhy, indeed? These remarks prefigure persistent themes in the discourse on cinema during this period, especially the preoccupations with suggestibility, crime, and female sexuality. Emilie Altenloh, author of the first sociological study of cinema, even found parts of this equation in the nature of female spectatorship:\n\nThe female sex, of which it is generally said that it always purely and emotionally absorbs an impression in its entirety, must be particularly receptive to filmic presentation. By contrast, it seems very difficult for people who are highly developed intellectually to project themselves into the sequences of events, which are often strung together haphazardly. On various occasions people who were used to grasping things on a purely intellectual level said that it was extremely hard for them to comprehend the logic of a movie plot.\n\nAltenloh equated holistic or synthetic approaches to the image with female spectatorship, and analytic approaches with expert or educated observation. She suggested that, on one hand, this open or holistic approach to the image enabled an empathetic projection that is unavailable to those who approach the film analytically. On the other hand, this empathetic mode of viewing left the spectator vulnerable to suggestion, and in this step she equated feminine and childlike modes of spectatorship. She further maintained that, in the absence of a strong family life or education, cinema held a mesmerizing influence on its young patrons, especially young male workers: \"It is undeniable that the cinema's lack of all higher interests has a certain influence on the entire way of thinking and living for these unstable people [ _ungefestigen Menschen_ ],\" she concluded. \"From the lives of outlaws, the morals of Apaches, and the fearlessness of heroes in cowboy films they take a philosophy of life that forces them into trajectories similar to that of their celebrated idols.\"\n\nAlbert Hellwig also wrote often on the suggestive power of cinema and its dangers for the criminally inclined or morally weak. In one article, he described a neurasthenic woman's response to a night at the movies. In the film, a postal clerk dreams that he is attacked by robbers: \"there appear a series of threatening faces and ghostlike hands, which reach out to others in their sleep.\" This made such an impression on the young lady that she began to see hallucinations of these hands day and night. \"The rather intelligent lady was initially fully aware that it was merely a hallucination, a product of her imagination. She was nonetheless quite upset because she saw this group of gigantic hands appear suddenly and in a variety of circumstances.\" Hellwig implied that the cause of the woman's hallucinations was a combination of cinema's suggestive power and the woman's pathological condition, neurasthenia, a vague, yet debilitating nervous condition in vogue during this time. It left its victims incapable of work and inflicted upon them a dazzling array of symptoms, including headaches, the fear of responsibility, graying hair, and insomnia. According to Anson Rabinbach, \"neurasthenics were identifiable by their impoverished energy and by the excessive intrusion of modern urban society on their physical and mental organization.\" It was a form of mental fatigue that left its victims unable to resist the stimuli of the modern world; it was characterized, in short, as a weakness of the will, as moral exhaustion.\n\nThe combination of pathology and morality is significant, because the concept of \"moral weakness\" metaphorically connected judgment and physical strength. The reformers' focus on both the unhealthy atmosphere of the nickelodeons and suggestive power of film reveals an underlying concern for both the bodies of the audiences and their moral judgment. This concern manifested itself as a problem of \"taste\"\u2014taste lies between the realms of sensuality and reason. As with the question of the nature of the child, reformers were divided over the nature of the masses, especially their judgment. Against those who argued that the masses were not ready for reform, that they were not interested in what interested the educated classes, Hermann Lemke argued, \"I for one cannot imagine that the general population has such bad taste; and even if the people were not yet mature enough for it [cinema's reform], they would have to be educated. But one must never indulge their lowest instincts\u2014that is harmful to the community and must be prevented.\" Hellwig was less willing to entertain the idea that the masses were inherently good: \"It is the bad taste of the audience that ultimately makes the trashy film.\" The solution to this problem of taste and, by extension, the crisis of moral judgment, was aesthetic education.\n\nSince Schiller, aesthetic education has offered a solution to the twin problems of sensuality and suggestibility. That is, Schiller suggested the category of the aesthetic as a medium between alienated Nature and Reason. In an alienated world, the aesthetic provided Schiller with hope for reintegration and, hence, social harmony. The aesthetic category acted as a corridor between raw nature and a higher morality. \"In a word,\" Schiller wrote, \"there is no other way of making sensuous man rational except by first making him aesthetic.\" The reformers were very interested in making \"sensuous man rational.\" Schiller's importance for the reformist agenda is illustrated by an editorial in the trade periodical, _Lichtbild-B\u00fchne_ (fig. 3.3). The headline reads, \"The Cultural Work of the Cinema Theater: Thoughts from the Year 1784, by Friedrich von Schiller.\" The essay invoked Schiller's \"The Stage Considered as a Moral Institution\" to argue that cinema could function in the same manner. The aesthetic, however, was a precondition to the moral, and cinema must first go through that transformation. An illustration from a 1918 reform pamphlet illustrates the axiomatic nature of this relationship between the aesthetic and the moral (fig. 3.4). The upper-left sphere represents \"entertainment with immoral effect\" and \"morally unobjectionable entertainment,\" while the upper-right sphere signifies \"art\" and \"non-art.\" A transubstantiation occurs when the rather plain problems of morality (left) and aesthetics (right) are superimposed to reveal the nature and proportion of \"art,\" \"trash,\" and _Kitsch_. We could say that this new sphere represents the issue of \"taste.\"\n\nFIGURE 3.3. \"The Cultural Work of the Film Theater: Thoughts from the Year 1784 by Friedrich von Schiller\"\n\nFIGURE 3.4. The geometry of taste: the superimposition of morality (\"entertainment with immoral effect\" [ _Unmoralisch wirkende Unterhaltung_ ] and \"morally unobjectionable entertainment\" [ _Moralisch einwandfrei Unterhaltung_ ]) and aesthetics (\"art\" [ _Kunst_ ] and \"non-art\" [ _Unkunst_ ]) reveals the nature of taste (\"art\" [ _Kunst_ ], \"trash\" [ _Schund_ ], and _Kitsch_.)\n\nSchiller represents the beginning of a long tradition of aesthetic education in Germany, one that eventually became grafted onto questions of nationalism. The most famous, or infamous, example of this development was August Julius Langbehn's _Rembrandt as Educator_ , first published anonymously and with enormous success in 1890. Like Schiller and de Lagarde before him, Langbehn reacted against the perceived excessive rationalization of the Enlightenment. The preoccupation with systemization, objectivity, and book learning had, in his opinion, brought about \"the decline of the spiritual life of the German people\" (7). Specialization, he complained, precluded exercise of creative power: \"one thirsts for synthesis\" in overeducated Germany, he wrote, and so \"one turns to art!\" (8). The German people could be rescued from this \"systematic, scholarly, cultured barbarism\" by \"going back to their original source of power, their individualism\" (9). Furthermore, if individualism was the root of all art, and he claimed it was, and if education should correspond to the nature of its students, then art education would be the most effective and natural form of instruction. Rembrandt, even though he was Dutch, was for Langbehn \"the most individual of all German artists\" (11). \"The scholar is characteristically international, the artist national,\" he wrote, underlining the difference between science and art, word and image (11). The goal of art education, as Langbehn saw it, was to effect a spiritual regeneration of the German people by reacquainting them with their own inner nature as it was exemplified by the masterpieces of national art. But if this program sounds reasonable, most of Langbehn's essay was noxiously antimodern, antiliberal, and even anti-Semitic, which unfortunately struck a chord among the reading public and resounded throughout German culture of the time. He advocated approaches to aesthetic education rooted in national (as opposed to international\/foreign) sources, but he understood the local and national in biological, racial, and ethnic terms. Progressive agendas had no place in his conception of aesthetic education.\n\nAlfred Lichtwark, generally recognized to be the driving force behind the art education movement, also valued local artistic traditions in his address to the 1901 art education conference in Dresden. \"Our education still lacks a _firm national foundation_ ,\" he declared. The basis for a national culture, as with Langbehn, could be found in German art. \"Up to now,\" Lichtwark said, \"the schools have not considered it their task to acquaint youth not only with the names, but the works of the great artists who express the German character\" (104). And he blamed this lack of attention to \"national art\" for the lack of \"formative power\" in German culture. But Lichtwark's idea of _Heimat_ was not rooted in blood and soil, like Langbehn's, but rather in the historical and cultural environment in which talent and cultural forms develop; Lichtwark's more historically contingent conception of _Heimat_ therefore led to a more liberal and modernist understanding of the relationship between local and national culture. Yet like Langbehn, Lichtwark held that \"the challenge of art education\" was inseparable from \"a moral renewal of our life\" (99). This hope was certainly not limited to Lichtwark; most representatives of the art education movement held it as their ultimate goal.\n\nBut Konrad Lange was cautious of such sanguine hopes, asking at that same conference whether \"with ' _Kunsterziehung_ ' [art education] we've actually found the magic word to solve all social questions.\" If he seemed less concerned about the spiritual state of the people, he was, like Lichtwark, still very anxious about the state of German art. He acknowledged that \"we actually have masters of the first order in all the areas of the fine arts, men who are living proof that the creative German spirit is not yet dead,\" but claimed that this was not enough. For this relative good health to survive, it needed good soil in which to grow. \"And this soil can only be the people's understanding of art.\" Worried that the elements of modern urban life could undermine children's sense of culture, Lange advocated leading them to art to maintain a sense of artistic tradition, to bring \"the artistic education of our youth... in closer connection to the living, creative art of the present.\"\n\nThe education of taste was also very important to Lichtwark. To his contemporaries, he was even better known as the director of the Hamburg _Kunsthalle_ , which came into international prominence during his tenure. There he was instrumental in organizing groundbreaking exhibits of amateur and artistic photography, promoting local artists, and rediscovering forgotten local talents, such as the romantic painter Philipp Otto Runge. \"We do not want a museum that simply stands and waits,\" proclaimed Lichtwark upon assuming the directorship of the Hamburg _Kunsthalle_ in 1886. \"Rather, we want an institution that actually works for the aesthetic education of our population.\" Aesthetic education, for Lichtwark, was about individual self-cultivation, of course, but it also had value for the community. Summarizing Lichtwark, Eckard Schaar writes, \"Artistic cultivation is not an innate ability... but rather a participation in a national collective property that, carried by the spirit of the people, influences the soul of the individual.\" Lichtwark envisioned his museum as an educational center for the artistic life of the region that would focus and direct this process of individual and communal aesthetic development. It would be a clearinghouse of taste, wherein exhibits of art from around the world would help raise the sensibilities of the general public and teach new techniques to local artists. For instance, in his introduction to the first exhibit of amateur photography in 1893, Lichtwark stated that the show's goal was to \"raise the artistic taste of the public and stir interest\" in the new art. The development of a national art depended upon the aesthetic education of both the public and the artists\u2014his museum would take up that task.\n\nLichtwark's influence on actual educational practices came through one of his most popular books, _Exercises in the Contemplation of Art Works_. The drills consisted of Aristotelian question-and-answer sessions between teacher and student, demonstrating by example how the child's inherent aesthetic taste could be cultivated and guided to acceptable standards. The student would gaze upon a painting and answer the teacher's questions about its form and content until the work's meaning would reveal itself to the child. For this process to be successful, Lichtwark stressed the importance of extended contemplation of single artworks in a quiet, conducive environment. Consistent with the _vom Kinde aus_ philosophy mentioned earlier, this method of aesthetic training soon gained wide favor among German art educators. Lichtwark's system also confronted the important issue of national taste.\n\nThe typical modern German has a weakness in the area of aesthetic education. He lacks an external refinement and solidity of form as well as an inner connection to the visual arts. He has no desire for artistic pleasures that require an education of the eye and heart. His eyes see poorly and his soul not at all. For the preservation of our nation as well as our national economy, these inadequacies must be forcefully redressed.\n\nLichtwark designed his _Exercises_ to provide a training program for children and others who were \"aesthetically weak.\" By teaching youngsters how to look, gaze, and, ultimately, _see_ , Lichtwark was following a set of presumptions common to aesthetic education: train the eye and the heart follows. For the art education reformers of imperial Germany, then, educating public taste was a project in nation building. Simply, education _through_ art was a way of building a distinctly national art, while education _to_ art was designed to build consensus and therefore national unity, as well as maintain traditional standards and methods of evaluation. These two directions were and are common for all art education programs from Friedrich Schiller to John Dewey.\n\nLichtwark's approach caught on. A number of books in the following years staged an encounter between an imaginary viewer, a painting, and a helpful questioner. _Bildbetrachtung_ (\"image viewing\" or \"image contemplation\") became a common method of teaching art appreciation, which emphasized the contemplative, especially the spiritual side of aesthetic education and experience. Schoolteacher Heinrich Wolgast echoed the views of many in Hamburg and elsewhere who stressed the connection between vision and spiritual development through disciplined viewing exercises: \"the child who conquers the world with these more sensitive organs will reap greater rewards than one with untrained eyes,\" he wrote, and \"intellectual understanding of the world based on a deeper grasp of its appearances will guarantee an improved preparedness for life.\" Reformers such as Wolgast hoped that training the observational acuity of children and adults would result not just in spiritual but ultimately national renewal; the art education movement is therefore a good example of the explicit, parallel investment in visual education and national goals that we see consistently in discussions of positive film reform. Lichtwark's program enjoyed national visibility and increasing popularity, especially among educators interested in current ideas about visual education. The parallels were not lost on film reformers such as Lemke, Sellmann, and H\u00e4fker, who had similar educational aspirations for film.\n\nYet these pedagogical impulses sprang from a broader trend in visual education that suffused nineteenth-century pedagogy, for which Wolgast's formulation could have easily served as the motto. This trend, known generally as _Anschauungsunterricht_ (\"observational instruction\" or \"visual means of instruction\" or simply \"visual education\"), started with the educational principles of Swiss pedagogue Johann Heinrich Pestalozzi (1746\u20131827), who believed that _Anschauung_ , or \"sense impression\" was \" _the foundation of all knowledge_.\" _Anschauung_ is a difficult word even for Germans: in Kant's _Critiques_ it is usually translated as \"intuition,\" but it also means \"sense perception,\" which are two very different things. In German dictionaries the two meanings of the word, one cognitive, one perceptual, sit side by side, matching the epistemological and phenomenological (or the knowable and visible) sides of the experiential coin. Yet _Anschauugsunterricht_ strove to develop the child's innate sense of form (in the Kantian sense) through observational exercises focused on the visible world, so it thereby embraced _Anschauung_ 's seemingly opposed connotations. It is perhaps best understood as a process through which the pupil attains an appreciation of both the detail of the individual object and its place in a larger system, whether philosophical, taxonomic, or social. Pestalozzi's program was notable for its then-radical approach to education. Rather than a top-down, authoritarian, deductive, and speculative method that demanded the student listen to the lecturer and recite back what was said or read in books, Pestalozzi advocated a bottom-up, democratic, inductive, and empirical approach to learning that asked the teacher to interact with the student\u2014similar to the process in Lichtwark's _Exercises_ \u2014and trust that the student could come to a higher understanding of the material through careful observation of the object in front of him or her. It countered mere book learning and rote memorization with a program that emphasized the pedagogical power of natural objects themselves.\n\nBut it was not merely about bringing the child to an object; the method was primarily about teaching the child how to _observe_. Yet as we saw in the previous chapter, observation is above all a complex logical operation, as Clive Ashwin notes about Pestalozzi's _ABC der Anschauung_ (1803): \"Its content was designed to activate and exercise the child's faculties, first by distinguishing separate entities and isolating them within the perceptual manifold; secondly by enabling the child to observe and note their peculiarities of form; and thirdly by associating each form with its correct name. This integration of number, form, and word led Pestalozzi to put the _ABC der Anschauung_ at the centre of his educational scheme.\" In practice, this involved a specific, step-by-step process. The teacher and the student would look at an object, such as a seashell, and the child would be asked to describe what he or she saw and be encouraged to point out individual characteristics of the object (\"distinguishing separate entities and isolating them\"). A series of questions asked by the teacher would bring the student to understand the basic form of the object, sometimes by drawing the object or its geometrical shape (\"enabling the child to observe and note their peculiarities of form\"). Naming each element and its form would be an important part of the exercise (\"associating each form with its correct name\"). Equally as important, the student would come to an understanding of the salient elements of the object (such as the seashell's function as both protective covering and housing) and its relationship to other such objects or abstract principles. Through this \"object lesson,\" the child would learn from direct perception but would also learn the method for correct observation \"in which essential qualities of the object are distinguished from the accidental ones.\"\n\nPestalozzi perceived this process as a solution to the problem of rote memorization and excessive book learning, but also to the problem of society's perceived acceleration. He sought to advance learning through direct, visual encounters as opposed to verbal or written encounters, but he also implicitly designed this approach to _decelerate_ the learning experience. The entry on Pestalozzi in the _Encyclopedia of Philosophy_ summarizes the goal: \"According to Pestalozzi, it is the curse of modern civilization that its hasty and primarily verbal education does not give man enough time for the process of _Anschauung_ , a term perhaps best translated as 'internalized apperception,' or as dwelling on the meaning and challenge of an impression.\" Written and spoken information were too abstract for Pestalozzi, but the spoken lecture also expressed the \"hasty\" pace of modern life. The object lesson allowed the child to dwell on the salient features and relationships, or on _details_ and their _correlation_ to principle and form. Indeed, Pestalozzi's method embodied the modern educator's contested relationship to detail, which on one hand exemplified the confusion, ornamentation, and overstimulation that many saw as the problem with modernity's visual field. Melanie Keene, in her dissertation on the object lesson in England, discusses the example of palaeontologist Gideon Algernon Mantell, whose distaste for the crowds and hyperstimuli of the 1851 Great Exhibition in London fortified his belief in the value of attention to single, small objects as doorways to general principles. Mantell's motivation for adopting Pestalozzi's pedagogical principles was common; in the profusion of detail, the object lesson commanded _time to dwell_ on the form underlying the confusion.\n\nYet on the other hand, this very detail functioned as the object's ground of authenticity. Not just any object could serve as the focus of a lesson. Preferably, the object came from \"nature,\" a term which for Pestalozzi was synonomous with all that is genuine, authentic, and free from artificiality. A useful contrast in this scheme is the drawing, which was often an important _part_ of the lesson\u2014students learned about form by drawing shapes\u2014but which was explicitly eschewed as the _basis_ of a lesson. That is, students could draw an object and thereby learn about its form, but they could not use a drawing as the lesson's object itself. Pestalozzi was unhappy with rote memorization of words, but he was \"equally dissatisfied with the use of pictures as a substitute for the direct experience of objects.\" The difference, of course, is in the details, which in this method signified the randomness, contingency, and authenticity of nature; \"authentic\" meant \"free from human influence,\" or \"that which resists us\" (one German word for \"object\" is _Gegenstand_ , or \"that which stands against\"). Pestalozzi and his disciples complained about the overabundance of stimuli and detail in the modern world\u2014almost invariably associated with the man-made world and its objects\u2014but they clung to the detail of nature as the ground for their method.\n\nThe _Encyclopedia of Philosophy_ 's entry on Pestalozzi translates _Anschauung_ as \"internalized apperception,\" which Christopher Ritter helpfully unpacks as \"self-directed understanding through which newly observed qualities of an object are related to past experience.\" The editor and translator of Johann Herbart's 1804 elaboration of Pestalozzi's method agreed that apperception was key: \"He [Herbart] teaches that the chief object of instruction is to secure the reaction of the mind upon what is offered to sense-perception. We must understand what we see. We must explain it by what we know already.\" Looking at _Anschauungsunterricht_ in this way, it is clear that Pestalozzi's method signaled a distinctly modern moment when self-identity was aligned with observation, or more precisely, with the logical operation most closely associated with the practice of observation: _correlation_ , which we explored in the previous chapter. To relate newly observed qualities to past experience and then to order those qualities and experience into a hierarchy and network of relationships\u2014this was what modern educators and experts defined as _understanding_. And this is what they demanded from their students as they trained their vision to be expert. Observation and correlation ruled as all fields committed to visualization and new forms of seeing developed across the disciplines. Herbert Spencer, who popularized Pestalozzi's method in England, exclaimed, \"Of new practices that have grown up during the decline of these old ones, the most important is the systematic culture of observation.\" So educators versed in _Anschauungsunterricht_ understood observation as a twofold operation: to _dwell_ on authentic, natural detail and to _correlate_ that newly observed detail to past experience with form and function. The object lesson was therefore a reaction to the perceived _quickness_ and _disorder_ of modern life.\n\nPestalozzi and his followers soon and quickly disseminated his method throughout Europe, where it found fertile ground. The commitment to visualization in the sciences encouraged visual instruction in education, which was the logical extension of Pestalozzi's approach. And obviously, given its emphasis on nature, _Anschauungsunterricht_ was perceived to be an especially effective method for teaching natural history and science. Science educators, like nearly all those interested in visual education whatever their discipline or national origin, expressed the belief that images worked _quickly_ and _efficiently_ on the viewer's mind. Chapter 2 explored this assumption about the presence and power of images; briefly, the trope presumed a homology between image, idea, and the structure of the human mind. Because of this presumed structural similarity, images were thought to leave a stronger impression than verbal or written descriptions. German physiologist Carl Jacobj put it best:\n\nSymbolically descriptive word images [ _Wortbilder_ ] of concepts can be replaced by the simultaneously created visual image [ _Anschauungsbild_ ] that represents the factual object of observation immediately and in all its details, so that it [the visual image] is imprinted in a faster, stronger, and more sustainable way on the conscious mind and, as a consequence, on memory.\n\nEven if the image worked as a fine substitute for a wordy description, Jacobj indicated that no educator ever left images to work on their own. Indeed, without proper guidance, images were almost always considered anathema to proper understanding. So the second major trope or guideline in visual education, beyond the vividness and efficiency of images, was that the real power remained with the instructor's words. But this relationship to words was fraught, as we have seen: a discomfort with words as the basis of a lesson but a recognition that words needed to frame a lesson. As Ashwin notes, \"In its most fundamental sense, then, _Anschauung_ meant something like 'direct and correct observation': observation which was closely _associated with_ language to the extent that the child was given the correct verbal equivalent at the point of making the observation, but which was not _mediated by_ language.\" These dicta remained unchanged as motion picture technology became an option for visual education, as we will see in the next section.\n\n\"CINEMATIC LESSON PLANS\" IN ELEMENTARY AND ADULT EDUCATION\n\nYet this tidy portrait of Pestalozzi and _Anschauungsunterricht_ should not deceive us into envisioning nineteenth-century pedagogy as an orderly sequence of inheritances. Over the course of the century, these principles were not only diffused so broadly that some did not even know their origin (if they could be said to have one), but there were so many interpreters, objections, and tweaks to theories of educational praxis that average reformers could be forgiven if they were sometimes confused about which practice belonged to which theory. For example, the art education movement positioned itself against what it perceived to be the excessive emphasis on science and mathematics in the approach of Herbart and his disciples, even as it adopted his method for ordering the lesson plan and even though Herbart himself argued against excessive rationalism in education, too. Nor should we be confused about the degree of impact these theories actually had in the authoritarian, state-run classrooms of imperial Germany\u2014which is to say, hardly at all. We should, however, recognize that all of these reform efforts hoped to provide _balance_ and _order_ to the educational experience, even if the understanding of proper balance and order was constantly in dispute. So when faced with the educational potential of film, which hardly anyone disputed, and the chaotic application of such potential, which everyone noted, advocates such as Lemke and H\u00e4fker drew upon the trends in pedagogical theory at the time in order to place film into an orderly and recognizable system of practice. This section will discuss precisely how this was accomplished with regard to elementary education (Lemke) and adult education (H\u00e4fker).\n\nFilms in the elementary school classroom were very exceptional, if not unheard of, during the early period in Germany. The apparatus was cumbersome and difficult for the untrained, but the expense was prohibitive as well. Because hardly any elementary schools could afford motion picture technology, the only solution, if film were to be a modern addition to their visual instruction agenda, was to take students to the local theaters. These excursions would generally take place during special screening times set aside during classroom hours, when the children and teachers could attend a matinee showing of a program of films deemed suitable by teachers, reformers, and school administrators. Most teachers and reformers, however, balked at the idea, having already decided that the movie theater was inherently corrupting, no matter what the content. Even so, commercial theaters were undaunted, especially after decrees in 1910 and 1912 in Breslau and Prussia, which restricted entry for children to film theaters except for special children's matinees. Most provinces eventually adopted these laws, leaving theaters without an important part of the market and prompting many exhibitors to cooperate more readily with schools and educators. So a proprietor of a film theater in Berlin, in a pamphlet promoting the educational use of (his) commercial theaters, assessed the situation, stressing the importance of film for visual instruction:\n\nIt is hardly necessary to mention the value of cinematic presentations for visual instruction [ _Anschauungsunterricht_ ]. For some time there have been efforts everywhere to use this important element in teaching. Institutions of higher education, which have the necessary resources, have their own screening rooms equipped with cinematographic equipment. Similar plans have repeatedly been made for elementary schools, but they have not yet been carried out, mainly because of high costs.\n\nThe word _Anschauungsunterricht_ was commonly used in connection to film's potential place in the curriculum. It was especially popular with regard to teaching natural history, for which motion pictures seemed to enjoy a preternatural inclination. Yet almost as often as educators suggested that motion pictures could function well as an object lesson, they questioned the \"motion\" part of \"motion pictures,\" as in this declaration by a teacher named R\u00fcswald: \"Based on my experience, I would only support the use of film in teaching in the following circumstance: namely, that slides and film are used simultaneously. This demand is based on the facts that the impressions of the cinematographic image are too quick and thus too superficial, and that now more than ever a calm, quiet, measured dwelling on a single subject matter of education is bitterly needed.\" This equation of movement, haste, and superficiality was common, and it was certainly consistent with the principles outlined in the previous section (we will see more of it in chapter 4). But educator Adolf Sellmann would have none of it:\n\nThis objection is unjustified. Does the observation of movements in nature make one superficial? Not at all. On the contrary, this kind of observation can and must lead in many cases to an especially acute attention and thoroughness. If I want to grasp the action precisely, I must look closely, observe keenly, and turn focused attention to the moving process. Motion processes observed in nature can imprint themselves deeply on the soul, so that the observed motion becomes an inner experience. This can, of course, also be the case with motion that is observed on film. If I have focused on it with all my attention and therefore with all my soul, the impression lasts longer. _The living picture can often have a longer-lasting impact than the still picture._ Why should observed movement only fleetingly be remembered? It surely depends entirely on the mind [ _Seele_ ] that looks at it.\n\nSellmann and other advocates argued that movement actually _sharpened_ the attention and that film's ability to replicate that movement\u2014and, crucially, to reveal hidden aspects of it\u2014gave it a privileged role among the _Medienverbund_ (media ensemble) of early visual instruction. Educators argued back and forth about the role of filmed movement in the classroom, but their discussion could be distilled to a question that was rarely articulated: What is the role of observed movement in understanding the natural world? Or better: What does it mean to understand movement? Fundamentally, the split in camps corresponded to a choice between discontinuity and continuity, with teachers such as R\u00fcswald advocating the use of slides (and film, in his case, although some argued for slides only), while Lemke, Sellmann, and others supported the use of _motion_ pictures. Lobbyists for film saw themselves on the side of modern pedagogy, while critics of film defended against unnecessary and potentially dangerous technology.\n\nYet the primary justification for the use of film as an object lesson resided not necessarily in its movement but in the ability of the photographic image to replicate in its detail the randomness, variety, and disorder of the natural world. Educators signaled this by consistently calling film's image \"faithful to nature\" ( _Naturgetreu_ ), which was a term often used by scientists and researchers when justifying their faith in the photographic image. Of course, not every aspect of the photograph is perfectly faithful to nature, but in these discussions, _Naturgetreu_ referred most often to the level of detail that allowed the photographic image to reproduce patterns of texture and variation. Physician Richard Kretz wrote, \"Photography is perfectly faithful to nature [ _Naturgetreu_ ], that is, the images reproduce... all forms and proportions, the distribution of light and shadow in a completely correct manner.\" Later, a geography instructor declared,\n\nThe cinematograph offers an excellent substitute for student hikes and field trips. It leads the student not only through the wider area of his home province, but also through the most distant latitudes. He gets to know countries and peoples through his own observation [ _aus eigener Anschauung_ ]. The most accurate description of a landscape, the most in-depth, vivid [ _anschaulichste_ ] description of life and the activity in it, the most detailed painting cannot replace what film, with true fidelity to nature [ _Naturgetreu_ ], parades before the eyes of the students.\n\nThis \"fidelity\" referred not to color or depth or emphasis or any of the aspects of nature and observation that many complained photography could not represent well. Instead, it referred to the same qualities that brought scientific curiosity to bear on nature in the first place: the abundant variations on patterns of similarity and difference found in the forms and random textures of natural phenomena. Because photography could replicate these forms and textures with such detail, it could act as a substitute or a representation of the object of study. Like \"vividness\" ( _Anschaulichkeit_ ), _Naturgetreu_ referred to the advantage of images over words. But _Naturgetreu_ was finally a stronger justification for film's role in the object lesson. Even if some disagreed on the role of movement in films seen in the classroom, _Naturgetreu_ was a description of film that nearly all could agree upon, especially given the importance of nature's details to the principles of _Anschauungsunterricht_ , as we have seen.\n\nNevertheless, if the image itself was more or less pedagogically controversial, educators objected, often rightly, to the lack of films specifically made for their curricula or the lack of rigor in most special school presentations at commercial theaters. Many complained that the _wissenschaftliche_ (\"scientific\" or \"academic\") presentations programmed by commercial exhibitors were often nothing more than inoffensive nonfiction titles randomly strung together. _Reformkinos_ , or theaters especially procured in order to offer more edifying screenings, were sometimes a solution to this problem, but they required financial and institutional support not often forthcoming. In his 1909 pamphlet, _Kinematographie und Schule_ , Georg Victor Mendel argued for creating standing theaters devoted to educational films for schoolchildren, a goal later achieved in the Urania theaters and the _kommunales Kinos_ discussed earlier. Ernemann, the Dresden-based equipment manufacturer, dedicated in 1909 its standing exhibition space to educational or scientific screenings for children three afternoons per week. Also in Dresden in 1910, civil engineer August Kade funded twice-yearly educational and scientific screenings\u2014with live musical and song accompaniment\u2014in a city-owned exhibition space, which was dubbed the Kosmographia \"scientific theater.\" Overall, at least eight reform-oriented theaters or screening spaces opened in Germany between 1909 and 1914, all of which were touted as alternatives to educational programs at commercial theaters. Even with these efforts, commercial cinemas remained the most widely available option for elementary school educational film screenings (fig. 3.5).\n\nFIGURE 3.5. Children at Luisen-Kino in Berlin, circa 1910\n\nYet the problem of curricular integration remained. Educators complained that the film programs at these theaters were well intentioned, but offered far too much far too quickly:\n\nIt is precisely the broad scope of the programs, which are all condensed into one to two hours, that must stir the most concern from a pedagogical point of view, as students do not arrive at calm observation or reflection, and none of the images can leave a lasting impression on them.... Not until projectors and film are cheaper and each school has its own cinematograph, or a special device can be attached to the projection apparatus that occasionally allows cinematographic images to be shown, which the teacher can present himself and explain as the lesson requires\u2014only then can film have a profitable application in the school.\n\nFrom the typical teacher's point of view, school screenings at commercial theaters fundamentally lacked the _control_ they required: motion pictures moved at a pace that could not be controlled, but also the program itself was out of their hands. According to this teacher, film had a future in the classroom only if that control\u2014in a literally hands-on manner\u2014could be assured. Lemke, whose efforts represented the most serious attempt to accommodate films and schools, took up the difficult challenge to integrate motion pictures into the grade-school learning experience. He not only had to take into account these complaints but also to assuage implicit anxieties about the role of the teacher in a technologically mediated classroom; admittedly, his and other utopian proclamations about the coming ascendancy of visual media and the subsequent decline of lectures did not help matters in that regard.\n\nEven if teachers such as Lemke assumed that film's fidelity to nature (its _Naturgetreu_ qualities) allowed it to adequately substitute for the object, that was only half of _Anschauungsunterricht_ , or the \"object lesson\"; film needed to be integrated into the lesson plan as well. Lemke attacked this problem by following the principles for guided apperception that developed in the late nineteenth century, especially after a new generation of pedagogues in the 1880s further refined Herbart's interpretation of Pestalozzi. As a concession to those against commercial screenings for children, Lemke argued for a distinction between special screenings for schoolchildren ( _Sch\u00fclervorstellungen_ ) and screenings that incorporated methods unique to the classroom, which he dubbed _Schulvorstellungen_ ; Lemke thereby put the film program in the teacher's control. He advocated that teachers understand the available films, actively work with the exhibitor to curate the programs, and then incorporate the screenings into lesson plans according to accepted Herbartian principles. Lemke also emphasized discussion sessions before and after film screenings in the commercial theaters. This plan required much more preparation: preselection of the films, discussion among the faculty about modifying or accommodating the program to the existing curriculum, and teacher training in using the films and leading discussion. For a brief time, Lemke even held intensive teacher-training seminars on educational film issues and techniques.\n\nIn fact, Lemke's suggestions for organizing film into a lesson plan followed Herbartian principles step by step. One of Herbart's enduring legacies in pedagogy is a five-step program for guiding apperception within a lesson:\n\n1. preparation ( _Vorbereitung_ )\n\n2. presentation ( _Darbietung_ )\n\n3. association ( _Verkn\u00fcpfung_ )\n\n4. generalization ( _Zusammenfassung_ )\n\n5. application ( _Anwendung_ )\n\nThrough these reform-minded and theoretical steps, teachers would introduce new knowledge of an object to a student by first reminding the student of already known material (preparation); then presenting the new material, repeating as necessary (presentation); encouraging associations between what is known already and what is new speculating on abstract principles linking the two objects (generalization); and finally thinking about how to apply this knowledge to new objects. In his own plan for the educational use of film, which he outlined in detail in _Die kinematographische Unterrichtsstunde_ (The Cinematic Lesson Plan, 1911), Lemke followed these steps closely by explaining how film could be deployed through each (see fig. 3.6).\n\nFIGURE 3.6. A page from Lemke's _Die kinematographische Unterrichtsstunde_ (The Cinematic Lesson Plan, 1911), in which he provides a Herbartian lesson plan for a specific film\n\nHe also recommended specific films and groupings of films that followed the idea of apperception: introducing new objects and concepts through abstract connections to the familiar. In this way, Lemke and others accommodated film to principles of _Anschauungsunterricht_ that were already widely accepted by reform-minded teachers: first, by connecting the specific features of the cinematic image to the visual emphasis in that tradition ( _Naturgetreu_ and the _object_ lesson), then by demonstrating that film could be incorporated into the curriculum in a familiar way (apperception, film, and the object _lesson_ ). With these efforts, and his own journal devoted to educational film and slide material ( _Die Lichtbildkunst in Schule, Wissenschaft und Volksleben_ , Storkow 1912\u20131914), Lemke was on the leading edge of the educational use of film in Europe and the United States. For an excellent example of the use of film for adult edification, however, we must turn to Hermann H\u00e4fker (fig. 3.7).\n\nAfter the failure or, at best, limited success of the attempts to create an alternative production and distribution system, reformers realized that focusing on exhibition held the most promise for fulfillment of their program. So like Lemke, H\u00e4fker looked to existing commercial theaters to establish an alternative exhibition venue to create a suitable educational or edifying environment. Miriam Hansen has argued that the peculiarities of early cinema exhibition presented the structural possibility of an alternative public sphere. The variety format; the sense of theatrical space; the combination of lectures, live music, sound effects, and so on; and the uneven development of modes of production, distribution, and exhibition\u2014all contributed to \"overlapping types of public sphere, of 'nonsynchronous' layers of cultural organization.\" Between the \"fissures of institutional development,\" alternative modes of reception and experience could emerge. The reformers, of course, hoped to \"synchronize\" these layers, not only by coordinating the modes of production, distribution, and exhibition, but also by integrating the various cultural spheres that commercial cinema was already grafting upon itself: literature, science, the tradition of the lecture series, and art.\n\nFIGURE 3.7. Hermann H\u00e4fker\n\nHermann H\u00e4fker's \"model presentations\" ( _Mustervorstellungen_ ) are the best example of the reformist exhibition program aimed at adults. Some have called him Germany's first film theorist\u2014he was certainly one of the very first to write regularly about the cinema. He began the century working as a writer, journalist, and translator for a number of periodicals, covering a range of topics, from Shakespeare's sonnets to his own bicycle tour of Finland. He was one of the first writers for _Der Kinematograph_ and a spirited contributor to and editor of _Bild und Film_ , eventually writing three books on film for the Volksvereins publishing company. His _Bild und Wort_ (Image and Word) society film exhibitions were prototypes for many \"model presentations\" that reformers tried to implement on a regular basis around Germany. His 1913 book, _Kino und Kunst_ (Cinema and Art), was an elaborate justification of the artistic potential of cinema and an extension of his earlier work in the reform journals. In this monograph, he describes his attempts to create aesthetically pleasing and educationally effective cinema programs. As we have seen, H\u00e4fker was not alone in these attempts, but he was unique in providing theoretical justifications of his programs.\n\nLike Lange and the other reformers, H\u00e4fker was concerned with the aesthetic sensibility of the masses and the influence of bad taste. His comments about taste were directed particularly to the contemporary state of film exhibition. Of the nickelodeons of the 1910s, H\u00e4fker noted, \"the educated circles have been repulsed by the tastelessness of the programs.\" Further, \"it's not the What of the program, but the How of the presentation that makes the impression.\" Of course, he certainly did not withhold complaints about the \"sensational\" films the producers presented to the audience. But unlike many of his contemporaries, such as Willi Warstat, who felt that censorship was the proper solution, H\u00e4fker continued to express his concern for the \"tasteless\" exhibition. This tastelessness referred, most generally, to the intrusion of modern life's hectic pace into the auditorium, where spectators were assaulted with a \"breathless chase of one number after another, accompanied by intertitles, the uninterrupted noise of the projector, the lights, etc.\" In this regard, H\u00e4fker's goals were consonant with those of _Anschauungsunterricht_. H\u00e4fker demanded an exhibition that avoided the exciting and the extraordinary and instead tried to establish \"a quiet and natural mood.\" He advised exhibitors to program their films in accordance with classical aesthetic principles, building tension and then release by alternating comedies with dramas and \"scenes from the life of nature and simple people.\" The exhibitor should also refrain from putting all the films on one reel, allowing instead a short pause between them so that \"the spectator's eyes would receive their necessary recovery time and the nerves a moment to relax.\"\n\nThis last bit of advice points to a range of literature dealing with visual fatigue and the motion pictures. In this discussion, the equation of cinema with modernity was most explicit. H\u00e4fker expressed the concerns of the day quite well when he complained, \"image and form, word and sound, color and line... rain like a hailstorm on the nerves of modern man, especially in, but not limited to, the city.\" Cinema came to epitomize this hailstorm. Some of the first articles written on cinema in Germany were medical papers on the harmful results of the \"flicker\" effect. Other medical investigations dealt with the threat of eyestrain in the film theaters. Nearly all reformers or opponents of cinema criticized its threat to public health and vision, as we saw in chapter 2. And as we saw in chapter 1, this outcry represented the larger preoccupation with fatigue that characterized debates coming out of the late nineteenth century. As Anson Rabinbach has shown, the trope of fatigue was more than a scientific mania of the age; it expressed a profound anxiety of decline and social disintegration. In the medical, scientific, and even literary study of fatigue, there was \"a tendency to equate the psychological with the physical and to locate the body as the site where social deformations and dislocations can be most easily observed.\" In other words, metaphors of health and sickness were used to express national anxiety. Fatigue, together with neurasthenia, was more than a physical ailment\u2014it was also perceived as a _moral_ disorder, a sign of weakness and the absence of will. Neurasthenia, mentioned before in connection to cinema's suggestive power, was the most typical metaphor for the delicate condition of the national psyche.\n\nH\u00e4fker viewed modernity as a series of \"shocks\"; he sought a haven to which he could escape the hailstorm of modernity. He just wanted to rest for a while, to give his nerves time to recuperate, and he wanted to make cinema such a haven. But cinema would never be this sanctuary, he said in 1908, \"so long as the corresponding sense of illusion is missing and the correct mood is lacking.\" There is so much in the modern world to disturb this mood, but treating film as an art form, especially exhibiting films \"tastefully,\" could slow this flood of \"the much-too-much\" ( _eine Eind\u00e4mmung des Vielzuvielen_ ). He planned to do just this with his \"model presentations.\" In 1910 he presented to the Image and Word association in Dresden a model program that was to be the prototype for other cities. The selection consisted mostly of nature films, but plans for further exhibitions included travelogues, scientific films, and _actualit\u00e9s_. Originally, he intended to continue the exhibitions in coordination with local schools, but the project fizzled due to lack of readily available films for continuous programming.\n\nThe 1910 presentation, entitled \"Spectacles of the Earth,\" highlighted H\u00e4fker's preferred form, the nature film: \"The first part showed high mountains and deserts; the second part concerned ethnological subjects (Laplanders, Chinese, Arabs, Indians, cannibals, etc.). The third part dealt with 'The Thousand Games of Water' (Victoria Falls, Niagara Falls, storms on the coast, surfs, rapids, geysers, underwater volcanoes from New Zealand).\" The films were accompanied by lectures, slides, music, and nature sound effects, all of which H\u00e4fker tried to orchestrate into a _Gesamtkunstwerk_ of Wagnerian proportions. The presentation began with a lecture of what to expect, what to look for, and \"in which sense to take it.\" It would then alternate films with slides and lectures, carefully presenting each. H\u00e4fker provides a detailed\u2014and obviously quite proud\u2014description of the final section of the program:\n\nThen it became dark once again. You could hear the sound of water, and as the curtain parted, you could see an actual waterfall, etc. At the end of this section there was a beautiful image\u2014one of the few that were also artistically impeccable. [\"Trip on the Avon River in New Zealand\"]. The spectators didn't know at first exactly where they were, when, as if by magic an invisible, delicate music sounded, completely in rhythm and harmony, as if made for the image (and, of course, purposefully arranged), accompanying the scene to its conclusion. As the lights came up in the auditorium in front of the closed curtain, the loud applause was not only for all that had been seen up to that point, but for the last scene and the genuine musical enjoyment that accompanied it. The proscenium seemed a magical realm, a mysterious land of light, life, and music.\n\nH\u00e4fker's further descriptions show the pains he took to assure a proper environment and mood. He reports that three men worked the slides to guarantee precise timing; curtains hung all around the auditorium to dampen the sound; colored stage lights splashed the proscenium as the audience seated themselves (59\u201360).\n\nThese preparations certainly have many precedents in traditions of theatrical and orchestral performance, and the format is adapted from the long tradition of lecturing in performance halls. Like other reformers, H\u00e4fker insisted that focusing on the viewing environment was the first step toward cinema's eventual aesthetic respectability. But H\u00e4fker set himself apart from his contemporaries with his claim that the entire cinematic apparatus\u2014image, light, music, sound effects, lectures\u2014could be used in combination for the artistic presentation of film, calling this Wagnerian use of cinema _Kinetographie_. H\u00e4fker's efforts to guarantee the proper conditions show his concern was primarily with the spectator's relation to the film. The conditions of reception were vitally important to his program and his conception of the function of art. The full effect of the \"total presentation\" ( _Gesamt-vorf\u00fchrung_ )\u2014here illustrated by the audience's reported confusion\/illusion that they were in New Zealand\u2014required the spectators' complete and undistracted attention. It required, in short, their _contemplating_ the film as they would an artwork. He hoped that he could educate audiences to this way of viewing films.\n\nH\u00e4fker took his cue from Lichtwark's _Exercises in the Contemplation of Art Works_ , which provided the foundation for the training of taste and vision, a way of viewing art that H\u00e4fker transferred to film. This way of viewing was certainly not unique, having immediate precedent in the German tradition of _Bildbetrachtung_ , as exemplified by Lichtwark's exercises. His presentations did not simply provide an environment conducive to the passive reception of art; they set out to actually train the audience's vision. Through the lectures, H\u00e4fker guided the audience to what was important and \"in which sense to take it\"\u2014that sense being, primarily, vision. But he did not want to stop there: \"In order to draw attention to especially interesting images, perhaps one should _occasionally_ employ little _signal lights_. They could be colored incandescent lamps placed around the screen that light up shortly before surprising scenes or scenes that are difficult to see\" (57\u201358). These visual cues would reinforce his verbal guidance, perhaps eventually creating some sort of physiological response. (Apparently, H\u00e4fker did not consider that the lights might have been a distraction.) There was also a moral dimension to this way of looking. In H\u00e4fker's discussion of approaches to art, contemplation was exemplary of a certain economy of energy, in that focused attention on the artwork is a way of exercising the will against the excessive stimuli of modernity. If neurasthenia was a type of mental fatigue caused by the difficulties of dealing every day with modern life, art provided not only a haven of unity and harmony in a distracted and disorganized world, it also offered an opportunity to train the attention and exercise the taste. Art and the artistic presentation of film were workouts for the mind; museums and film theaters could be mental health clubs.\n\nLemke's and H\u00e4fker's education of vision and taste exemplifies the strikingly ambivalent tone of contemporary reform movements, in that they were both nationalistic and progressive, both protective of traditional values and open to modern innovations. Yet that combination of tradition and modernity was and is common for all early adopters of new (media) technology who try to incorporate their new tool into an established disciplinary method, or especially an established mode of expert viewing. In their case, their technophilia or excitement over the new medium prompted them to find within it some potential for compromise or appeal to their less excited colleagues. It is hardly coincidental that the problems most cited about film and its exhibition\u2014its quick tempo, excessive detail, and jumbled programs\u2014corresponded to some of the most likely complaints about modernity in general, namely, quick pace and disorder. Lemke's and H\u00e4fker's plans, anxiously aware of these charges, attempted to contain them by enveloping film within a protective casing of order, dwelling, and observational method, whether called _Anschauungsunterricht_ or contemplation. Whether motion pictures, per se, could be incorporated into \"traditional aesthetics\" is the question of the next chapter. But before we move to that topic, let us briefly review the ideological connection between vision, taste, morality, and education.\n\nSchiller, unlike most philosophers, was not suspicious of the senses, least of all of the sense of sight. According to Schiller, knowledge of the physical world passes through the senses and is therefore contingent on them, but vision provides the opportunity to transcend the physical world and enter the aesthetic on the way to the moral realm. The key to this journey is \"contemplation.\" Schiller declared, \"As long as man, in that first physical state, is merely a passive recipient of the world of sense... he is still completely One with that world.... Only when, at the aesthetic stage, he puts it outside himself, or _contemplates_ it, does his personality differentiate itself from it.\" Upon entering the aesthetic, the subject renounces his or her passions and creates the possibility of becoming a _moral_ being. Contemplation is the exercise through which this process begins. The very act of perception, the very apparatus of vision is both inextricably implicated in the sensual world and ironically outside of it. \"From the moment a man _sees_ an object, he is no longer in a merely physical state,\" Schiller noted. That is, while exercise of the other senses testifies to one's _proximity_ to the natural world, vision offers the opportunity for _distance_. The aesthetic of contemplation, exemplified by what Benjamin called the \"aura\" of an artwork, is based on distance. The aesthetic of distraction, again illustrated by Benjamin's discussion of cinema and architecture's tactile qualities, is based on proximity. Schiller again: \"If desire seizes directly upon its object, contemplation removes its object to a distance, and makes it into a true and inalienable possession by putting it beyond the reach of passion.\"\n\nHence the whole concept of subjectivity\u2014becoming a knowing subject by objectifying and therefore \"possessing\" Nature\u2014is dependent upon the refusal of passion and sensuality. Once \"outside\" this sphere, the moral becomes possible. For Schiller, the renunciation of Nature is not a goal in itself as much as a necessary step toward the fulfillment of humanity's moral potential. Like the act of vision, always in the physical world while simultaneously having the potential to transcend it, humanity balances on the fine line between the sensual and the moral. Schiller called this line \"the aesthetic.\"\n\nTaste, like vision, is both embedded in Nature and somehow removed from it. Even more than vision, taste implies participation in the social world. An artwork affects the individual, but the exercise of aesthetic judgment implies universality. When we find ourselves agreeing that something is beautiful or sublime, we are exercising a unique and precious form of intersubjectivity based on our recognition of shared capacities for aesthetic experience. This is what Kant meant when he, following Vico, called taste a _sensus communis_ \u2014a communal sense.\n\nThe concept of taste provides the ideal illustration of the relation between aesthetics and ideology. While society could impose moral behavior on its subjects by appealing only to Reason, it is more efficient to employ the emotions in this task. As the medium between Nature and Reason, the aesthetic allows this operation. Schiller explained:\n\nThe ethical State can merely make it (morally) necessary, by subjecting the individual will to the general; the aesthetic State alone can make it real, because it consummates the will of the whole through the nature of the individual. Though it may be his needs which drive man into society, and reason which implants within him the principles of social behavior, beauty alone can confer upon him a _social character_. Taste alone brings harmony into society, because it fosters harmony in the individual.\n\nFaced with a society that they felt was becoming more alienated and fractured, reformers latched onto the promise of harmony and unity offered by the aesthetic realm. Lichtwark and H\u00e4fker focused on vision to effect a renewal in taste. Their exercises in the contemplation of artworks were, like Lemke's Herbartian ordering of the \"cinematic lesson plan,\" attempts to ward off the distractions of modernity, prophylactics against the \"much-too-much.\" If spectatorship had been characterized as an addiction that lulled audiences into an impressionable somnambulism, H\u00e4fker, Lemke, and other reformers hoped to counteract this state by inscribing cinema into an aesthetic of contemplation and reflection. The audience's vision required _training_ so that mental and physical fatigue would not set in; it was a way of \"pumping up\" moral weaklings. While H\u00e4fker's _Gesamtkunstwerkeffekt_ would provide the illusion necessary for the aesthetic experience, it was not intended to lull the audience into distractedness. Rather, it provided access to the _sensus communis_ through a _disinterested_ , distanced aesthetic experience. Nature films were both safely asexual and reminders of potential harmony. Yet the use of nature films is ironic; the reformers' emphasis on vision and distance and disavowal of _Kinodrama_ and \"sensational\" films amounted to a refusal of sensuality and corporeality\u2014in short, a refusal of Nature. Training audiences to conform to certain rules of observation\u2014an ascetic education of their vision\u2014was part of an ideology that combined educational practices and Kantian aesthetics to establish some sense of social order.\n\nThis legitimation strategy\u2014anesthetizing\/aestheticizing cinema and its audiences\u2014was a response to modernity's perceived assault on the body and the body politic, often exemplified by cinema's \"flicker.\" H\u00e4fker and others felt that training the aesthetic sensibility could fend off the \"shocks\" of modern life. The combined concepts of \"the child\" and \"taste\" served as a fulcrum for the reformers, allowing them to \"uplift\" the motion pictures and incorporate cinema into their ideology. As Germany's _Kinoreformers_ attempted to redeem and legitimate cinema as Art, they recognized within it the potential for recovering a lost utopia of unity and, ultimately, a means for social control. Yet even by 1912, presenting contemplation as a solution to modernity's ills seemed slightly old-fashioned, as we shall see.\n\nTHE PROBLEM WITH PASSIVITY\n\nAESTHETIC CONTEMPLATION AND FILM SPECTATORSHIP\n\nBut the difficulties which photography caused for traditional aesthetics were child's play compared to those presented by film.\n\n\u2014WALTER BENJAMIN (1936)\n\nAs cinema's bandwagon\u2014already heavy with reformers and trade journal reporters\u2014rolled toward World War I, literary intellectuals, pundits, and other belletrists climbed aboard (sometimes climbing down again after their thousand words) just to see what the ride was like. Judging from the sharp spike in the number of film essays written between 1912 and 1914, it was apparently de rigueur to offer a learned editorial on the way modern life and culture found expression through this new phenomenon. Just as a range of opinions comprised the reformist discourse explored in chapter 3, so the tone of this collection of articles, which Anton Kaes felicitously dubbed the _Kino-Debatte_ , extended from peevish outrage and haughty condescension to diplomatic concession or even roguish delight in the new medium. This expansion of the discourse in Germany corresponded to cinema's more visible public profile at this time, due to a stronger domestic film industry; the production of longer and more emotionally involving story films; the rise of film stars, such as Asta Nielsen and Henny Porten; the emergence of picture palaces and the successful \"embourgeoisement\" of the cinematic experience; and, most tellingly, the development of the _Autorenfilm_ , a short-lived strategy that attached literary and theatrical luminaries to industry projects. Perhaps because those weighing in\u2014including Ernst Bloch, Max Brod, Alfred D\u00f6blin, Georg Luk\u00e1cs, Kurt Pinthus, Walter Serner, and others\u2014were or were to become such prominent names in the German literary tradition, this part of the conversation about film has received the most attention in secondary surveys of the period. We should be quick to note, however, that reformers and trade journal writers did not disappear during this time\u2014on the contrary, they exerted a clear influence on the direction of the discussion, even if negatively\u2014but the sheer number of new voices in the mix has tended to shift our attention from the pedagogical and commercial sections of the debate.\n\nThis chapter will be no different in that respect. It will, however, back away from the scholarly emphasis on the relation between literature and film. While depictions of the _Kino-Debatte_ have been as varied as the debate itself, scholars usually focus on the battle between image and word in the contemporary discussion of cinema's relationship to German culture. It is indeed hard to ignore the incessant complaints about the supposed decline in literacy attributed to the consumption of sensational and superficial images rather than great literature or the many declarations that cinema could never be considered genuine art as long as it lacked words to express the depths of the human soul. Anton Kaes was absolutely correct when he noted that the tension between old and new \"found expression in a vigorous discourse about the relationship between literature and cinema,\" or, simply, that the debate about cinema was \"a debate about the literature of the time.\" While theories of film started to disengage themselves from literary or theatrical models by the 1920s, before World War I, cinema and its champions felt the need to justify themselves in terms of literature. For Sabine Hake, literature was \"the primary reference point\" for writers coming to terms with modernity through their essays on cinema. Peter Jelavich emphasized the attempts to conform film to \"traditional bourgeois aesthetics, which demanded clarity of authorial voice and rootedness in the written word.\" Helmut Diederichs similarly charted the ways in which this group divided the ground between literature and film. Stefanie Harris has demonstrated, on the other hand, how cinema's unique form shaped the literary work of such writers as Kurt Pinthus. As Heinz-B. Heller usefully pointed out, these were _literary_ intellectuals after all, so it comes as no surprise that their response to film would be from a position firmly grounded in their chosen medium.\n\nFocusing so closely on early cinema's relationship to literary form and turf, however, unintentionally narrows our understanding of cinematic experience to that particular relationship, leaving relatively unexplored the question of film and aesthetic reception in general. The above quotation from Benjamin\u2014his point that the challenge that photography presented to traditional aesthetics was \"child's play\" compared with film\u2014summarizes a fairly common conception: that film was emblematic of a change in aesthetic standards at the fin de si\u00e8cle as mass reception and distraction replaced individual contemplation as the dominant or most appropriate mode of aesthetic reception. Indeed, in film and media studies this account is more or less taken on faith in Benjamin's word alone. I have no argument with the general outlines of this story, but even his well-known \"Work of Art\" essay compresses the events considerably. So we have a dual historiographical problem, as I see it: the emphasis on the literary misses a large swath of the cinematic experience, specifically the relationship between viewer and image, while the leap from contemplation to distraction in film history and theory is too often taken for granted without spelling out the character of \"traditional aesthetics\" and the transition to whatever replaced it. This chapter argues that a close, renewed examination of the _Kino-Debatte_ is essential to solving both historiographical problems.\n\nPrevious chapters were concerned with the criteria for film's legitimacy within any given discipline and the adaptability of motion picture technology to an expert mode of viewing as a major factor in establishing that legitimacy; this chapter will explore film's legitimacy within the realm of aesthetics and its degree of adaptability to the expert mode of viewing known as aesthetic contemplation. It presumes that, following a trend in German aesthetics since Kant, the question of aesthetic value hinged on _reception_ more than form. That is, the major statements on aesthetics in the long nineteenth century\u2014especially those of Kant, Schiller, Schopenhauer, and others, but excluding those within the Hegelian tradition, which was concerned with how meaning inhered in form rather than how we experienced it\u2014were concerned primarily with the role of aesthetics as a way of being in the world, more than whether any particular form was more artistic than another. They were concerned with the function of art within a moral, ethical, social, and philosophical system. The value and legitimacy of art in this system depended primarily on the singularity of the experience it aroused. The exact nature of that experience has been a topic of constant exploration since these statements, but the significance of that experience for the system in general\u2014whether philosophical, moral, or political\u2014cannot be underestimated. Form prompts experience, to be sure, but these and other statements emphasized the universal character of the aesthetic experience rather than the variety of experiences created by various forms. More often than not, evaluations of any given form, such as music, rested on the ability of that form to catapult the reader\/listener\/viewer into a particular kind of aesthetic experience. Better experience usually equaled more hallowed form.\n\nSo to understand fully the relationship between early cinema and aesthetics, we must focus on the relationship between the cinematic experience and aesthetic experience as it was understood, or between watching movies and the expert mode of viewing called aesthetic contemplation. This chapter asks the following questions: to what extent did the cinematic experience, as these authors described it, conform to their understanding of aesthetic experience? Specifically, how did the experience of film align with their understanding of aesthetic contemplation as an implicitly expert mode of viewing? To what extent did the descriptions of cinematic experience participate in changes to expert conceptions of this mode of viewing?\n\nTo answer these questions requires first understanding what the authors presumed about aesthetic experience. The trouble with aesthetic experience, of course, as countless analytic philosophers have complained, is its notoriously slippery surface. It is very difficult to define logically, or even on an individual basis. Fortunately, for the purposes of this project we need not come to a philosophically rigorous conclusion; instead, we need only outline what the authors _thought_ aesthetic experience was. Even that is elusive, because any given writer borrowed ideas or presumptions, often haphazardly, from a long and varied aesthetic tradition. Indeed, we might find it more historiographically productive to think of descriptions of aesthetic experience (rather than aesthetic experience per se) as statements often expressing competing values. If one writer latched onto Kant's idea of disinterest and detachment as the point of aesthetic experience, another might champion, after Schopenhauer, the idea of losing oneself in the artwork. If one author saw repose as the fundamental criterion of any aesthetic encounter, another might have emphasized the value of the free play of associations while engaged with a work of art. My point is that these key ideas or terms\u2014detachment, loss of self, repose, free play, and others\u2014describing any aesthetic experience (not just cinematic) _functioned within a system of implicit dichotomies_. Let me explain.\n\nIf we gather common ideas about the nature of aesthetic experience in the major statements of Kant, Schiller, Schopenhauer, and others within the tradition of nineteenth-century German aesthetics, it appears that aesthetic contemplation entailed (1) _detachment or disinterest_ , meaning that aesthetic judgment is without desire, passion, or self-interest; (2) _repose_ , in that the object is lingered over without haste, giving enough time for the free play of the imagination; (3) _activity_ , meaning that the mind is alive with associations and correlations; and (4) _loss of self_ , in that the contemplation of the object can lead to either a transcendental immersion into the object (Schopenhauer) or a moment of awareness of one's participation in a larger community (Kant). Under this category we can also subsume discussions about the role of the body in aesthetic experience; for some, aesthetic contemplation was a purely mental operation; others looked to the body as a model for understanding aesthetic pleasure, while avoiding the purely sensual. Some terms are mutually reinforcing (disinterest and participation in a community, for example), while others are contradictory (detachment and immersion seem to be opposite ideas).\n\nBut we can arrange these terms\u2014detachment, repose, activity, and loss of self\u2014into spheres or larger categories that also contain their opposites. These categories are: _space_ (distance\/proximity or detachment\/immersion), _identity_ (loss of self\/self-awareness), _time_ (repose\/haste), and _agency_ (activity\/passivity or free will\/determination) (see fig. 4.1). This arrangement shows especially clearly the _ideological_ implications of these terms and of the descriptions of aesthetic experience they evoked. That is, to give aesthetic experience its moral or philosophical weight within a larger system\u2014the pivotal role of Kant's _Critique of Judgment_ with regard to his other _Critiques_ comes to mind\u2014the terms writers and philosophers used to describe aesthetic experience were rhetorically linked to fundamental ideological values (or ideologically inflected categories of experience) such as agency and identity. Understanding the rhetorical role these terms played allows us to glimpse the stakes of any historical debate regarding aesthetic experience. Pairing the terms with their opposites also underlines their mutual dependence in those historical debates.\n\nFIGURE 4.1. Aesthetic experience as a series of interlocking dichotomies\n\nThis scheme also allows us to escape the analytic philosophers' futile attempts to logically reconcile descriptions, an evasion much more in keeping with the sharpest understandings of aesthetic experience as indeterminate. Schiller argued, for example, that the aesthetic state is a _mediating_ moment \"midway between matter and form, passivity and activity,\" wherein indeterminacy is the preferred state of being. Accepting the indeterminacy of the aesthetic state should turn us away from pat definitions and toward an understanding of aesthetic contemplation as dichotomous, fluid, oscillating, or unfixed. (Again, I want to stress that questions about what the aesthetic state actually is and even whether it exists are irrelevant when considering historical descriptions of it.) For example, descriptions of contemplation almost always preferred repose over haste\u2014thereby presuming that both the object and the subject were more or less in a state of stillness\u2014but the quick, gestalt-like expert glance over the artwork as a whole was not ruled out as part of the process. However, detachment was almost a universal feature of these descriptions, so its opposites, self-interest or passion, were hardly ever included as viable elements of the experience. On the other hand, descriptions of the aesthetic experience often oscillated between sides of a dichotomy: claims that emotional detachment or psychic distance were crucial to aesthetic experience (Kant, Schiller) must be balanced or reconciled with descriptions that emphasized emotional projection or immersion into the artwork (Schopenhauer, Adolf Hildebrand). Likewise, the relationship between active and passive stances in aesthetic experience is often unclear from the descriptions; mastery alternates with surrender (as in many descriptions of the experience of the sublime), while focused attention alternates with the free play of associations or loss of self in the experience. This approach, then, matches a theoretical understanding of aesthetic experience as indeterminate to a historiographical scheme that does not try to resolve contradictions but allows them to stand to highlight their ideological function.\n\nObviously these categories overlap; questions of agency and identity are indeed hard to separate. But that is precisely the point: the overlapping conceptual categories strengthen the ideological weight of any given description. So this scheme might be useful for an investigation of _any_ historical description of aesthetic experience; it is definitely useful to understanding the early debates about motion pictures. To summarize, viewing the historical claims for aesthetic experience in terms of a series of dichotomies, rather than as a series of firm characteristics, has three advantages. First, we avoid the futile philosophical debate about what the aesthetic experience actually is and replace it with the recognition that historical descriptions of aesthetic contemplation\u2014what writers and aestheticians _thought_ it was\u2014have been varied and contradictory, but not infinitely so. Second, by subsuming the dichotomies under the general categories of identity and agency, we emphasize that most philosophers (in the neo-Kantian tradition) recognized aesthetics as a moral category that mediated between opposite poles of their choice. For example, the Kantian\/Schillerian tradition of putting Art between Reason and Sensuality extends in spirit to at least Theodor Adorno, if not also Jacques Ranci\u00e8re and other contemporary philosophers who hope to find for Art a place apart and some emancipatory potential. Seeing the aesthetic experience as a series of unstable dichotomies recognizes not only Art's socially mediating function but also that historical descriptions of aesthetic experience emphasized its productive indeterminacy.\n\nFinally, with regard to this specific project, these dichotomies allow us to see more clearly how the discussion of motion pictures was ultimately an _aesthetic_ debate, in the sense that these essays on cinema may be about the status of literature, the content of the films, or whatever, but the core issue was the larger ideological problem of _the moral significance of the aesthetic experience_. These writers obviously cared about the decline in reading skills, about the loss of ability to concentrate, the superficiality of the mass audience, the novelty of the moving image, and many other things that may or may not have indicated the decline of civilization. But, at bottom, all these complaints can be traced to the _character_ of the aesthetic experience, which apparently mattered most to these writers. \"Film presentations are generally rejected from an artistic perspective,\" wrote Emilie Altenloh about the taste of the upper classes. While Altenloh undoubtedly refers to judgments of artistic _form_ (\"von _k\u00fcnsterlischen_ Gesichtspunkten\"), this chapter agues that complaints or judgments of form were ultimately in the service of a better aesthetic experience. Modernity was therefore an affront to their aesthetic sensibilities and values, which were rooted in a philosophical tradition that privileged a particular, expert mode of viewing a stationary image or object. The extent to which cinema challenged or could be accommodated to these values is the question of this chapter.\n\nTo burrow further, we could argue that these dichotomies grew out a more fundamental, historical division between active and passive viewers. As we have seen, film spectators were often characterized as hypnotized by the cinematic image, even addicted to it, and captivated by the ease with which it entered the stream of consciousness, while cinema's fragmented temporality, quick pace, and motley form matched the habits and psyche of the modern city dweller, who was at once impatient and distracted, unable to concentrate on a single, main idea. Hence movies were perceived as effortless, whereas real art required work; movies were a jumble, whereas real art had a distinct form; movie spectators needed their next film like a drug, while real art encouraged a detached observer; movies moved by far too quickly, disallowing the continuous, leisurely attention required of real art. If these sets seem familiar, it is partly because they have been repeated over the centuries whenever audiences are described; theater audiences were probably the first victims of such characterizations. Ranci\u00e8re's explanation of the paradox of spectatorship sheds some light on this issue:\n\nThere is no theatre without a spectator.... But according to the accusers, being a spectator is a bad thing for two reasons. First, viewing is the opposite of knowing: the spectator is held before an appearance in a state of ignorance about the process of production of this appearance and about the reality it conceals. Second, it is the opposite of acting: the spectator remains in her seat, passive. To be a spectator is to be separated from both the capacity to know and the power to act.\n\nThe idea of aesthetic contemplation from Kant and Schiller to early twentieth-century professional aestheticians looks like a deliberate attempt to counter the problem of passivity in spectatorship by endowing the viewer with both a protocol and a philosophical justification that connect aesthetic enjoyment to reasoning and knowledge production (\"the capacity to know\"), on one hand, and political maturity or citizenship (\"the power to act\") on the other. Endowing aesthetic contemplation with agency and linking it to individual or civic identity gave this mode of viewing the moral weight it needed to counter a mode of viewing perceived to be common to the passive, irresponsible crowd. The contemporary discussion of the cinematic experience could not avoid confronting this weighty connection between agency and contemplation\u2014often to film's disadvantage.\n\nPrecisely because contemplation had long been endowed with high moral import\u2014with values such as reason and citizenship\u2014any changes to the conditions of contemplation put these values at risk, especially as mass reception became more common with new urban demographics and forms. If cinema were not at the heart of these transformations, for many essayists it was a worst-case scenario. At the same time, however, the discussion about cinema was not merely a reactionary defense of Enlightenment-era ideals; it was a collective attempt to find some common ground, to ask (often implicitly) whether contemplation was the proper way to engage with these new forms, and if not, what was? In fact, Benjamin's own position on the relative value of contemplation and distraction was not limited to his words in the \"Work of Art\" essay. As Carolin Duttlinger has shown, Benjamin's conception of attention ( _Aufmerksamkeit_ ) played a central and complex role from his earliest essays, and if he advocated distraction ( _Zerstreuung_ ) as the perceptual stance most appropriate toward modernity in the \"Work of Art\" essay, that was only one part of a long conversation he had with himself about the relationship between a mode of viewing and one's position toward and within modern life. Ultimately, this conversation was about the relationship between \"a way of being with images,\" in Dudley Andrew's ingenious phrase, and a way of being in the world. For Benjamin as well as the writers of the _Kino-Debatte_ decades earlier, the two stances were one and the same.\n\nSo this chapter will explore the status of aesthetic reception in the context of discussions about cinematic spectatorship. The first section will trace the broad themes in the _Kino-Debatte_ about movies, modernity, and spectatorship in relation to the rhetorical or ideological categories of time and agency. Through an examination of Schiller and Schopenhauer, it will stress the philosophical and subsequently ideological connection between temporality, indeterminacy, and aesthetic experience, and how this constellation persisted in the descriptions of early cinema. The second section will inspect discussions of aesthetic experience in professional aesthetics of the time, especially around the concept of _Einf\u00fchlung_ , or \"feeling into\" the artwork. These discussions about the relationship of emotions and the body to aesthetic experience were attempting to come to grips with, on one hand, unresolved ideas about reception in nineteenth-century aesthetics, and on the other, modern aesthetic reception characterized by turn of the century democratic access to art. This section will demonstrate that discussions of the cinematic experience also pressed on some of the same troubling issues\u2014such as emotional projection, immersion and detachment, and visual pleasure\u2014regarding the nature of aesthetic contemplation in the modern world that professional aestheticians discussed. This section explores the categories of space and identity. The third section focuses on Georg Luk\u00e1cs's famous early essay on film, \"Thoughts Toward an Aesthetic of the Cinema\" (1913), to demonstrate the nascent political critique of contemplation that was to be such a central feature of his and other theorists' stance after World War I. This final section will also return to Benjamin, Siegfried Kracauer, and the discussion of contemplation and distraction to realign our history of these theories in light of the _Kino-Debatte_ 's discursive construction of spectatorship.\n\nAGENCY AND TEMPORALITY IN THE AESTHETIC EXPERIENCE OF CINEMA\n\nThe scheme outlined above could apply to any historical description of aesthetic experience after Kant, but this section will recast some familiar themes in the _Kino-Debatte_ in terms of the roles time and agency played in contemporary understandings of aesthetic experience. Let us begin with a relatively obvious case: motion pictures as an emblem of social acceleration. This was a bright, visible thread in early debates about cinema all over the world. Basically, commentators complained that the incessant tempo of motion pictures\u2014its relentless push forward and its quick change from one view to the next\u2014exemplified the problems of modern social acceleration. Educator Georg Kleib\u00f6mer discussed cinema's pace at length in his 1909 essay, \"Cinema and Schoolchildren.\" He opened his attack by describing the Great Lisbon Earthquake's profound effect on young Johann Wolfgang von Goethe, and then wondered whether the eruption of Mount Pel\u00e9e in 1902 had the same effect:\n\nAnd let us search for just one child for whom this natural event had an indelible significance. How is it that we all well know that this search will be in vain? Because in our time all the world's grand strokes of fate are immediately known to us in every detail; because one harrowing incident displaces the other before it can be \"worked over\" by our mind. Now even the most tragic events only touch the surface of the soul.\n\nSuperficiality abounds, according to Kleib\u00f6mer, because we do not have enough time to absorb the emotional significance of events before we are compelled to move to the next one. \"The moral education of a child,\" he instructed, \"should include a _few_ events that give him an understanding of certain emotional values,\" but \"above all, such emotions must not confront other feelings in close temporal proximity.\" And in the cinema? \"In rapid succession jest and seriousness alternate on the screen, since variety must reign.... So first _horror_ , _fear_ in the highest degree, then _compassion_ increased to the utmost.\" He therefore criticized \"the danger that this rapid traversal of the entire scale of feelings poses for the truth and depth of children's emotions.\" Kleib\u00f6mer saw filmic form as a synecdoche for the rapidity with which modern information was conveyed; he assumed that because of that rapidity, children could not absorb the significance of events, and this was the root cause of their perceived superficiality (as opposed to, perhaps, the possibility that they were six-year-olds not named Goethe).\n\nKleib\u00f6mer was not alone in his opinion; writers of all sorts, not just cranky reformers, connected movies and modernity in this way. Austrian author Karl Hans Strobl similarly argued, \"The cinema is one of the most perfect expressions of our time. Its hasty, erratic tempo corresponds to the nervousness of our lives; the anxious flickering, the whisking away of its scenes is the extreme opposite of a measured, regular stride, of confident persistence.\" In the pages of _Der Kinematograph_ a commentator noted matter of factly, \"And so the cabaret, with its indiscriminate, democratic disposition, appeals best to the nervous haste and variety of our time, which of course is not say that the theater is doomed. But now the cinema, the projection house, successfully competes with the cabaret for the affection of the people just as the latter surpassed the legitimate stage.\" Right-wing editor of _Der Kunstwart_ and proto-Nazi Wilhelm Stapel was even more explicit:\n\nBecause of the rush of images, you get used to absorbing only an approximation of the impression; you do not get a clear and conscious understanding of the image in its details. Therefore only the coarse, surprising, sensational impressions stick. The sense for the intimate, the precise, the delicate is lost. The patrons of the cinema \"think\" only in garish, vague ideas. Any image that lights up their mind's eye takes up all of their attention; they no longer mull and reconsider it, they no longer indulge in the particularities and the reasons.\n\nDevotees of the connection between early cinema and modernity will find this theme familiar. Noteworthy, however, is not simply this well-known correspondence but the strong implication that cinema's tempo interfered with the usual aesthetic experience of viewing images. Stapel was very clear in this regard: the spectator does not \"get a clear and conscious understanding of the image in its details.... they no longer mull and reconsider it.\" If one is to understand the image, one must attend to the details slowly. Now this could just be a commonsensical caution that one cannot simultaneously do anything hastily and well, but it was not usually put in those terms. Instead, the complaint was specifically about the superficiality that results from processing (or being forced to process) visual imagery too rapidly. What was the connection between viewing tempo and shallowness?\n\nTo appreciate the moral implications of this equation, we must return to Schiller and Schopenhauer, who based their philosophies on a common understanding of time. The importance of temporality for Schiller's aesthetics is not only rarely discussed, it is key to understanding later ideological objections to film. So the following will explicate their aesthetic schemes while emphasizing temporality and indeterminacy, which will clarify certain aspects of the debates about cinema. Even if these early nineteenth-century philosophers were in many ways considered obsolete by the early twentieth century, their ideas about the moral and cultural importance of aesthetic experience\u2014and the relationship between that significance and the passing of time\u2014were widely and implicitly assumed. Schiller, especially, presumed that everyday, sensuous existence is necessarily tied to the inexorable succession of moments, while abstract thought and reason\u2014or the Platonic idea\u2014exist outside this temporal course in a timeless, static, changeless existence. Schiller called our animal nature the \"sense drive\" and the spiritual or intellectual side of our existence the \"form drive.\" He described the sense drive:\n\nSince everything that exists in time exists as a succession, the very fact of something existing at all means that everything else is excluded.... when man is sensible of the present, the whole infinitude of his possible determinations is confined to this single mode of being. Wherever, therefore, this drive functions exclusively, we inevitably find the highest degree of limitation. Man in this state is nothing but a unit of quantity, an occupied moment of time\u2014or rather, he is not at all, for his Personality is suspended as long as he is ruled by sensation, and swept along by the flux of time.\n\nHere and elsewhere, Schiller defined his understanding of the different facets of human existence in terms of their relationship to time. In the sense drive, one is nothing but \"an occupied moment of time\"; by being occupied by (in terms of both concerned with and controlled by) sensuous existence\u2014ruled only by needs and desires of the moment\u2014one is \"swept along by the flux of time.\" This conception of temporality, for Schiller and others, implied a limit on human potential: \"Wherever, therefore, this drive functions exclusively, we inevitably find the highest degree of limitation.\" The form drive, however, \"embraces the whole sequence of time, which is as much as to say: it annuls time and annuls change. It wants the real to be necessary and eternal, and the eternal to be necessary and real\" (81). Ideas aspire to timelessness, being outside of the succession of events and therefore outside of perception. Reason is timeless, in Schiller's scheme, but also ethereal. So Schiller invented another motive force to harmonize, balance, or counter the other two:\n\nFor as long as he only feels, his Person, or his absolute existence, remains a mystery to him; and as long as he only thinks, his existence in time, or his Condition, does likewise. Should there, however, be cases in which he were to have this twofold experience _simultaneously_ , in which he were to be at once conscious of his freedom and sensible of his existence, were, at one and the same time, to feel himself matter and to know himself as mind, then he would in such cases, and in such cases only, have a complete intuition of his human nature. (95, emphasis in original)\n\nFor Schiller, the desiring, physical side and the spiritual, intellectual side of our nature have their limitations, but when we are able to experience both at the same time we are without limitations. That is, if our lopsided, specialized existence is determined by our physical desires (that is, as matter without form), or by our intellectual aspirations (that is, form without matter), then the third way escapes determination entirely; it is indeterminate. Schiller called this third mode of existence \"the play drive,\" which is exemplified by aesthetic experience. It is a moment of indeterminacy, of oscillation between Sense and Reason, or between physical necessity and law. That moment of indeterminacy holds the greatest human potential, when free will is exercised most productively in the service of \"a complete intuition of [our] human nature.\" Schiller concluded:\n\nOur psyche passes, then, from sensation to thought via a middle disposition in which sense and reason are both active _at the same time_. Precisely for this reason, however, they cancel each other out as determining forces, and bring about a negation by means of an opposition. This middle disposition, in which the psyche is subject neither to physical nor to moral constraint, and yet is active in both these ways, pre-eminently deserves to be called a free disposition... we must call this condition of real and active determinability the _aesthetic_. (141, emphasis in original)\n\nThe exercise of free will, then, is the goal of aesthetic experience. But not entirely: the goal is to be \"sensible of existence\" without being determined by it; to be aware of all aspects of our nature while they are held in abeyance. Aesthetic experience is a balancing act between physical necessity and moral obligation, actively taking part in both sides of the human condition without letting either determine us. Schiller thereby gave the aesthetic experience a strong ethical charge; it serves as the moment when we are most fully human, overly determined by neither of our natures: \"For, to mince matters no longer, man only plays when he is in the fullest sense of the word a human being, and _he is only fully a human being when he plays_ \" (107, emphasis in original). For Schiller, aesthetic experience was the grandest kind of play, which brings our fullest human potential into relief.\n\nEspecially interesting and rarely noted, however, is the emphasis Schiller placed on the temporal dimension of this experience. The sense drive is \"swept along by the flux of time,\" the formal drive \"annuls time,\" but the play drive is repeatedly described as an experience of simultaneity: \"a middle disposition in which sense and reason are both active _at the same time._ \" The importance of simultaneity in his scheme marks Schiller's move as a very modern gesture; both simultaneous and indeterminate (\"this condition of real and active determinability [ _Bestimmbarkeit_ ]\"), his category feels very modern indeed. It is not too great a leap to argue that simultaneity creates this indeterminacy; we are both in and outside of time, which puts us in a position of being in neither one drive nor the other, but both states at the same time. This clarifies his formulation of the play drive: \"the play-drive, therefore, would be directed towards annulling time _within time_ , reconciling with absolute being and change with identity\" (97, emphasis in original). The annulment of time experienced in the formal drive is reconciled with the experience of time within the sense drive, thereby resulting in the \"annulling time _within time_.\" Whatever one makes of the validity of his interpretation, there can be no doubt that Schiller's conception of the aesthetic experience was ultimately temporal.\n\nSchopenhauer also took pains to underline the temporality of the experience, even if it was different from Schiller's: \"It is the state where, simultaneously and inseparably, the perceived individual thing is raised to the idea of its species, and the knowing individual to the pure subject of will-less knowing, and now the two, as such, no longer stand in the stream of time and of all other relations.\" Both subject and object are raised to a transcendent state in Schopenhauer's conception of contemplation, during which the object becomes an idea and the subject loses itself and its desire in the process, thereby also finding temporary relief from the misery of longing and striving. The emancipatory potential of aesthetic experience derives from its ability to provide a momentary escape from time; this is different from Schiller's more moral and community-oriented aesthetics, but it affirms rather than cancels Schiller's conception of the aesthetic as a primarily temporal experience. For both philosophers, and in the most common conceptions of aesthetic experience in the long nineteenth century, human limitation was tied to \"the flux of time,\" and its emancipatory potential was linked to the escape from that flux or being able to maintain a balance between competing temporalities.\n\nThis review of Schiller's and Schopenhauer's philosophies is necessary, because it reminds us of the _ethical or moral mission_ they assigned to aesthetic experience, thereby giving it an ideological weight that it carried through the nineteenth and twentieth centuries. It also establishes _the importance of time_ within traditional aesthetics, especially the notion of repose in the aesthetic stance. Finally, it emphasizes that Schiller and others calibrated their conception of contemplation to highlight its _indeterminacy_ , or more precisely, the importance of the indeterminate quality of the aesthetic experience as an emblem of human liberty. All of these features of aesthetic experience\u2014especially the focus on experience, rather than artistic form\u2014spread and took root in everyday understandings of aesthetic issues that we see in the _Kino-Debatte_ and other, early discussions about viewing motion pictures. Again, the goal is to demonstrate that these debates were largely discussions about aesthetic (as opposed to narrowly literary) experience, especially this model of contemplation, to which modern spectatorship was detrimentally compared. Indeed, the unfavorable comparisons functioned primarily as a defense of contemplation's ideological value. We could dismiss the comparisons out of hand as ideologically compromised, but we would lose the deeply ambivalent and complex character of aesthetic reception. Contemplation was not yet dead; whether the values associated with it actually ever died is still open to question.\n\nThose values were primarily ethical in imperial Germany. Modernity, it appears, was too pushy; it shoved its spectators along, giving them no pause for reflection, which was an offense to values of repose. So when writers such as Gaupp, Sellmann, Stapel, Strobl, or others noted the potential effect of the fast pace of movies and modernity, they were not simply complaining about the vapid superficiality of \"kids these days\"; they were defending a very long tradition that insisted on leisurely approaches to the image. To allow oneself to succumb to the \"flux of time\" while attending to an image\u2014or to let the image impose such limitations\u2014was an abdication of the duty to self-cultivation, of the obligation to become fully human. Contemplation was depicted as a moment of escape, but also a moment of self-awareness; the aesthetic experience had a special place in state and cultural self-conceptions in Germany, which the righteous defended vigorously. But because of its pivotal role between emancipation, civic duty, and self-awareness, contemplation represented a complex, ambivalent stance toward the world that was not so easily dismissed, as Benjamin recognized (and which I will discuss at greater length later in the chapter).\n\nAnother good example within the discussion concerns the fragmentary, disjointed quality authors attributed to modern life and to cinema. If movies and modernity were too pushy and quick, they were also disjunctive: the cinema \"is short, rapid, even encrypted, and it stops for nothing. There is something compact, precise, even military about it. This fits very well with our age, an age of extracts.\" These comments on form were often framed as an analogy between movies, modernity, and the mind of its spectators; modern life was disordered, as were movies and the thought processes of its audience. This homology\u2014a discursive construction of film's spectator\u2014was usually an implicit defense of aesthetic ideals of detachment and effort. Theater critic Hermann Kienzl's brief commentary is exemplary:\n\nThe psychology behind the triumph of cinema is urban psychology. Not only because the metropolis is the natural focal point from which all social life radiates, but especially because the metropolitan soul\u2014this inquisitive and inscrutable soul perpetually on the run, stumbling from one fleeting impression to the next\u2014is really the soul of the cinema! Indeed, some city dwellers even lack the stamina and concentration for mental and emotional absorption\u2014and probably also the time, especially in Berlin, a metropolis agitated by work-fever. The same trivial drive for relaxation that leads city dwellers to the operetta or farce after work to replenish their exhausted energies leads them likewise to seek the effortless pleasures offered by movie theaters.\n\nHere we have the usual stereotypes about the metropolis and its inhabitants: they are driven by passion, desire, necessity; their actions are hurried and jumbled. Needless to say, these were also transgressions against the aesthetic values of detachment and repose. Kienzl pretended to offer merely a description of the situation, but it was an implicit defense of the principles outlined earlier.\n\nThere is also the implication, well before Hugo M\u00fcnsterberg, that movies were popular and effective because they somehow mimicked the structure of the modern mind, or at least the mind of its audience. For many commentators, the link between movies, modernity, and mind made film \"the most psychological representation in our time.\" The \"effortless pleasures\" of the cinema replenished the modern soul in a way\u2014a trivial way, they would say\u2014that other entertainments could not. Most writers disparaged movies and the mind of the masses in a dual dismissal, but others saw it differently:\n\nOnly film technology permits the rapid sequence of images that roughly corresponds to our own imaginative faculty and in some measure imitates its jerky unpredictability [ _Sprunghaftigkeit_ ]. Part of the fatigue to which we finally fall prey while watching theatrical works of art results not from the noble effort of aesthetic enjoyment, but rather from the exertion in adapting to the plodding, affected movement of life on stage. Spared this effort in the cinema, one is free to devote a considerably more uninhibited commitment to the illusion.\n\nWhereas Kienzl and others assumed that the mind of an intellectual is ordered and disciplined, Lou Andreas-Salom\u00e9 concluded, after spending a year with Sigmund Freud and Viktor Tausk, that the mind is no such thing. When Kienzl remarked that \"city dwellers lack the stamina and concentration for mental and emotional absorption,\" he compared the stalwart literary mind with the flighty, disordered, lazy mind of the moviegoer. But Andreas-Salom\u00e9 acknowledged that movies moved quickly and disjunctively, which matched her understanding of the \"imaginative faculty.\" This put both in a favorable, or at least neutral, light. Indeed, her sympathetic view of cinema's aesthetic potential depended on this structural similarity; the speed at which we think\u2014slowed down considerably by theater\u2014was matched only by cinema. Likewise, she saw in early cinema's unpredictability\u2014we can imagine that to early, novice audiences the continual surprise at the next view might have given them the impression that film narrative or editing was capricious (which it sometimes was)\u2014a disconnectedness that fit her understanding of human thought. Actually, the primary difference between her idea of mind and Kienzl's was that she refused to create two models divided by class, one for elite intellectuals and one for the mass audience. Her model assumed that, when it comes to thinking, we are all a hot mess, trained or not. M\u00fcnsterberg also developed a single model, but presumed that all our imaginative faculties are well ordered. In any case, many authors assumed a structural similarity between movies and mind, but their aesthetic judgment depended on whose mind movies resembled.\n\nBut Kienzl and Andreas-Salom\u00e9 shared one assumption: true art required effort and movies were effortless. The \"noble effort of aesthetic enjoyment\" was a persistent trope in these and other discussions about the difference between high and low cultural forms, in which high art was usually differentiated by its complexity or difficulty and low art by its simplicity. For Andreas-Salom\u00e9, cinema's effortlessness was a result of its structural similarity with the mind, especially with regard to its motion and pace. It had also something to do with the age-old comparison of images and ideas, a comparison common to the _Kino-Debatte_ , too, as when Hermann Duenschmann, apparently a big fan of Gustave Le Bon, compared the film audience with a crowd: \"The crowd thinks only in images and can only be influenced through images that act suggestively on their imagination.\" This implied that the learned think in words, but the visual status of ideas has always been ambiguous. At least since Descartes, many philosophers have defined \"ideas\" as images, not words. Descartes consistently made the analogy that ideas are \"like portraits drawn from Nature.\" Or Locke, in his _Essay Concerning Human Understanding_ , wrote that ideas are \"Pictures drawn in our Minds,\" or that the \"Idea is just like that Picture, which the Painter makes of the visible Appearances joyned together.\" Now an important caveat here is that Descartes and Locke were thinking not of concepts, but of sensory ideas\u2014the images that our brain creates from the sensory information gathered by normal perception. But the leap to concepts is not too hard to make and has been made since Plato, perhaps. The point is that one support for this notion that films were \"effortless\" was the isomorphism between images and the way the mind is thought to work. (This was also a support for the assumption within discussion about educational media that visual instruction is more efficient than instruction with other kinds of materials.)\n\nYet images per se cannot be easier to comprehend; otherwise the aesthetic contemplation of paintings would have no pedestal upon which to place itself. So there must have been something other than the similarity between images and mind that supported the claim that movies were effortless. Ulrich Rauscher gives a clue: \"I fear the cinema has one disadvantage for the audience: because it tells its story so comfortably, because it takes over the operations of sense-making itself, the cinema, which could foster the creativity of our literati, will bring about a general laziness of the public's imagination.\" Film is effortless, because it \"takes over the operations of sense-making itself\" ( _weil er die Versinnlichung der Vorg\u00e4nge selber \u00fcbernimmt_ ) for the audience, leaving nothing for the imagination to work on. The \"noble effort\" of aesthetic enjoyment was perceived as the free play of associations that accompanied a productively ambiguous artwork; the imagination filled in interpretive gaps with its own associations, thereby bringing subject and object together as the hermeneutic circle between part, whole, and subject tightened. Just as photography was not considered legitimate aesthetic material because of its mechanical nature and its excessive detail\u2014which left nothing to the imagination, presumably\u2014so, too, motion pictures were dismissed because their detail and their narrative patterns left the viewer nothing to do but watch. There was more to it than that, however, because these same movies were disdained for their chaotic, confusing stories, which doubtless would leave much to the imagination. The problem, then, was that the succession of images was not merely obvious, but _preordained_. Whether the images were confusing or crystal clear, they were always _someone else_ ' _s_ images, and if we combine this concept with the \"images = ideas\" formula, we have a series of views _imposed_ upon the viewer in a preordained temporal sequence, which does not happen with paintings or literary images. Hence the loss of free play, or more importantly, _free will_. The cinematic experience, in this view, was not _indeterminate_ , but rather _determined_. (Benjamin quotes Georges Duhamel: \"I can no longer think what I want to think. My thoughts have been replaced by moving images.\") By showing us where to look, editing and cinematic narrative forced these writers to look there; this was for many patrons a step too far.\n\nThose patrons would presumably never allow themselves to enjoy cinema's flow of images. But the inadequacy of this rigid decree\u2014that the flux of time jeopardizes agency and thereby imposes limits on human potential\u2014for modern life becomes apparent in the _Kino-Debatte_ (and elsewhere) as well, especially in discussions that emphasized _alertness_ and _somnambulism_. Essays often invoked these tropes when describing the tolls of modern life. Recalling Georg Simmel's vivid portrait of \"The Metropolis and Mental Life\" (1905) or Freud's idea of the \"stimulus shield\" (1920), these essays also assumed that life in the big city dulled the senses and made one blas\u00e9, requiring ever greater thrills to be satisfied. For many writers, cinema was just the ticket: \"But every day and every hour cinema restores to our pampered senses, which in the heyday of technology have forgotten how to be astonished, the feeling of Pygmalion's enchantment with Galatea.\" Anticipating film theory of the 1920s, this author and others claimed that cinema renewed vision by offering a point of view we otherwise could not have. For writers such as Rauscher, cinema's preordained succession of images was oppressive and imprisoning, whereas for this writer, it was liberating to see with another's eyes. Max Brod, too, was rejuvenated by a visit to the cinema:\n\nThe vividness with which so much happens [in the film] has finally shaken me out of my semi-somnolent state. Now on the way home I become an inventor, imagining new images for the Biograph: a pursuit in which, instead of automobiles, locomotives or trolleys, two ships, a cruiser and a pirate ship, race against each other over the wide surface of the sea, the gap between them narrowing amid the most furious shooting.\n\nOr this early editorial: \"I believe that through the cinema we have only now learned to see. We have been awakened to the pleasure of watching [ _Schauen_ ].... Reality appears much more clearly before us, and the interest is twice as great. We can almost let our mind fall asleep and reap with our eyes what the soul desires.\" If these writers were drifting through their modern lives, semi-awake to its challenges and blas\u00e9 to its pleasures, cinema roused them from this aesthetic slumber to give them a new outlook. Here, then, a trip to the cinema did exactly what any aesthetic experience should do: renew our perception. This idea, however, was relatively new in aesthetics and already expressed the waning dominance of the Schillerian model.\n\nIndeed, that aesthetic contemplation was undergoing a transformation is evident in essays that evoked alertness and somnambulism in terms of modern life. Strobl both admired and was uneasy with social acceleration and its effect on our perception:\n\nBut wherever the streets of world commerce run, wherever the billions cross paths, wherever goods are converted into money and money into power\u2014at these hubs of human and commercial relations a sustained readiness is necessary. Constant presence of mind has replaced the old contemplativeness that let one's mind wander, because we no longer need to have it at hand.\n\nAccording to Strobl, a leisurely approach to the traffic of images or to the city was simply not viable. One required instead a \"presence of mind\" that was very much embedded in the present but not necessarily \"swept away by the flux of time.\" Alertness, as we will see in our discussion of Benjamin, was considered a more modern response to time and succession than repose, yet was no less complex in its relationship to time.\n\nAesthetic experience, as discussed by Kant, Schiller, and Schopenhauer in the early nineteenth century, was a complex and contradictory structure that was nonetheless required to bear a particularly heavy ethical load. If it collapses logically with the barest nudge, it was still home to a family of values related to agency and identity. As we have seen, early discussions about film were often couched in terms of agency, especially with regard to the spectator's relation to the passing of time. Complaints about cinema's incessant temporal push and its preordained succession of imagery continued at least one line of thinking about aesthetic experience, which insisted that the viewer's stance before the image was also a stance before the world, one that recognized our limits but also strove to glimpse human potential in an ephemeral moment of transcendence. Aesthetic contemplation was understood as an exercise of free will, so to surrender oneself completely to the image or to the flow of time was an abdication of duty; indeterminacy meant liberty, but the rush of preordained images meant constraint. The temporal complexity of Schiller's and Schopenhauer's visions of aesthetic experience was therefore at odds with conceptions of cinematic temporality as constraining or controlling. But with statements attesting to the importance of alertness in modern life or positing that seeing the world from another viewpoint could reawaken the senses, the discussion began to question the utility of a leisurely aesthetic stance in favor of an approach to images\u2014and the world\u2014that was not merely cognitively active and unhurried, but _reactive_ and _quick_. So in the _Kino-Debatte_ we can begin to see the \"difficulties\" new forms might have posed for what Benjamin called \"traditional aesthetics.\"\n\n_EINF\u00dcHLUNG_ , IDENTITY, AND EMBODIED VISION\n\nBut an understanding of aesthetic contemplation as inherently tied to leisure and free will\u2014or, basically, the cultured gentleman's prerogative\u2014is only partially correct. We must also consider the relationship between aesthetic experience and identity or subjectivity. One might think that the gentleman's prerogative also presumed a stable, unified self, which would be confirmed in aesthetic experience. Kant's theory of the beautiful, for example, held that pleasure arose from the harmony between the artwork and our cognitive faculties, which assumed a resonance between them that was stable and stabilizing. His theory of the sublime, however, maintained that its pleasure derived from the oscillation between the physical threat of nature's awesome power or infinitude and our cognitive framing of that boundlessness. Faced with this power, the threat of physical annihilation presented a literal loss of self, while the mastery implied by cognitively framing such power was self-affirming. So the existence of that particular kind of aesthetic experience, especially given the importance of the sublime in aesthetic theory, meant that aesthetic experience per se was never inherently stabilizing.\n\nThis split between body and mind, or between loss of self (whether through physical danger or an experience of transcendence) and self-awareness (whether cognitive or corporeal), was exacerbated by the structure of Schopenhauer's system. On one hand, Schopenhauer insisted that a loss of self was absolutely necessary to any experience calling itself aesthetic, if it were to have any mollifying effect at all. As Schopenhauer described it, the viewer attends to the object and gradually loses self-awareness as the associations and Ideal become prominent. Likewise, his description of aesthetic contemplation emphasized the purely visual and cognitive\u2014as opposed to corporeal\u2014experience of viewing. For him, the aesthetic gaze divorced itself from the world, from the flux of time, and even from the artwork itself, the specifics of which did not figure largely in his understanding of aesthetic experience. Indeed, Kant, Schiller, Schelling, Schopenhauer, and Hegel all ignored the specific features of any given artwork in favor of using the object as a prompt for abstract thought and for mounting ever more complex philosophical systems. Johann Friedrich Herbart continued this approach in his formalist aesthetics, which \"proposed a simplified theory of form, one that defined aesthetics essentially as the science of elementary relations to lines, tones, planes, colors, ideas, and so on,\" without reference to the specter of \"content\" or \"meaning.\" Similarly, \"the perfect aesthetic frame of mind for Herbart was a state of absolute indifference, that 'quiet seriousness' that lies 'between depression and excitation.'\" In this tradition, their understanding of the gaze matched their understanding of the object: neither emphasized detail or materiality; both could be characterized as disembodied or ephemeral.\n\nOn the other hand, Schopenhauer also simultaneously pursued another approach, which he did not see as contradictory, but complementary. Part of the second volume of _The World as Will and Representation_ speculated on the _physiology_ of the perceptual act, an approach that aesthetically minded scientists such as Hermann von Helmholtz, Gustav Fechner, and Wilhelm Wundt pursued further in their experiments on the nature of (aesthetic) perception. Their findings, however, indicated that the human body was a transient, variable, and unstable field upon which any subjectivity rested at its own risk. That is, if a stable and unified subjectivity previously depended on a happy conception of the body as similarly stable and unified, experimental physiology and psychology of the mid-nineteenth century invalidated that guarantee. As Jonathan Crary has argued, the fleeting nature of the human body left barely any solid ground for subjectivity.\n\nMany aestheticians during the late nineteenth century looked for an approach to questions about aesthetic perception that would navigate between the Scylla of abstract, disembodied philosophizing about art and the Charybdis of fragmenting, \"scientistic\" approaches to the aesthetic experience. They were less interested in philosophical or physiological questions about how we perceive form and space than in the psychological problem of how we take delight in the specific characteristics of form and space. Robert Vischer's seminal 1873 dissertation, _On the Optical Sense of Form_ , coined the term _Einf\u00fchlung_ , or \"feeling into,\" as an answer to this question. Vischer meant it to explain \"the symbolism of form,\" or how spatial form has meaning for us: the subject \"unconsciously projects its own bodily form\u2014and with this also the soul\u2014into the form of the object\" (92). But _Einf\u00fchlung_ eventually came to mean emotional projection in general, and then, with Edward B. Titchener's translation of the term as \"empathy,\" it transformed into a broad psychological concept. In aesthetics, however, _Einf\u00fchlung_ was discussed in terms of several problems, including a renewed interest in analyzing the specific formal and material features of artworks and the experience they prompted; describing the emotional content of the aesthetic experience in terms of projection of self in relation to distinct forms; and understanding the role of the body as a measure of both art and aesthetic experience, especially in terms of aesthetic pleasure. That is, writers taking up the idea of _Einf\u00fchlung_ explained aesthetic pleasure as a resonance between the structure of the body and the structure of the artwork, thereby explicitly acknowledging the embodied nature of perception.\n\nThis brief history is necessary, because it clarifies \"traditional aesthetics\" and its \"challenges,\" which were twofold: (1) new artistic _forms_ , such as photography, fit uncomfortably in the canon of fine or even applied art; and (2) new aesthetic _experiences_ , especially mass reception. The question of form is outside this chapter's purview, but we can address the nature of mass reception, which differs from individual contemplation in at least two pertinent ways: (1) the emotions of the audience amplify those of the individual, thereby making emotional projection into the work much more prominent; and (2) because of this amplification, and the awareness of being in an audience, the sense of embodiment is more pronounced as well. We could therefore view the discussion of _Einf\u00fchlung_ in the late nineteenth and early twentieth centuries as an exploration of conflicting dichotomies nascent in \"traditional aesthetics\" (such as the relations between mind and body, or between free will and destabilizing aesthetic experience), while also attempting to reconcile the contradictions inherent in these grand aesthetic systems with changes in class demographics as more and more people gained access to and were interested in art. Specifically, I would argue that _Einf\u00fchlung_ aesthetics could be seen as an attempt to reconcile issues especially pertinent to mass reception (embodiment, emotional projection) with individual contemplation and its ideological implications. While mass reception was not explicitly acknowledged in discussions of _Einf\u00fchlung_ , the theories pave the way for a deeper understanding of the role of emotion and corporeality in aesthetic experience. Indeed, many contemporary theorists have looked back to _Einf\u00fchlung_ aesthetics as a possible framework for new theories of embodied spectatorship.\n\nThe dichotomies circling around the category of identity were less fraught ideologically than those concerning agency. Whereas no one argued seriously for \"determination\" against \"free will\" (although \"surrender\" might have been an option), philosophers were able to describe aesthetic experience in terms of \"loss of self\" just as persuasively as \"self-awareness.\" The issue of embodied perception, however, brushed against the almost universal requirement of disinterest, in that any hint of sensuality immediately lowered the status of aesthetic experience to mere pleasure. Like paper and scissors, physicality always lost against reason in this ideological game. But because the ideological stakes were generally not as high in this category, this section will not focus on those stakes; instead it will emphasize descriptions of aesthetic experience. Specifically, in an attempt to understand precisely the nature of film's \"challenge\" to \"traditional aesthetics,\" this section will draw upon theories of _Einf\u00fchlung_ , which were particularly adept at describing emotional projection and embodied perception in aesthetic experience, two issues also present in the debates about cinema. In fact, if the legitimacy of cinema as an aesthetic form in part hinged on the answers to these questions, it was also true that these two issues\u2014which I want to stress were present in \"traditional aesthetics\" from the beginning, if we look closely\u2014mounted the strongest theoretical and practical challenge to individual contemplation as it was generally understood. Emotional projection and embodied perception threatened to change the questions about aesthetic legitimacy itself. If _Einf\u00fchlung_ aesthetics tried to reconcile individual contemplation with features of aesthetic experience that were also prominent in mass reception, then the presence of a form such as cinema pressed harder on those questions.\n\nThis is not to say, however, that early discussions of cinema participated directly in debates within professional aesthetics or that professional aestheticians participated in the _Kino-Debatte_ ; to my knowledge no writings on film during this period in Germany ever mentioned the term _Einf\u00fchlung_. But the early discussions of film brought up the same issues. To be sure, salons crowded with paintings and people had already prompted much complaint as mass reception elbowed its way into the art world and demanded a seat at the table. But cinema was not merely one of many emblems of modernity; the nature of the experience itself\u2014the feeling of movement, synesthesia, emotional projection, and spatial depth\u2014coincided precisely with issues in aesthetic reception that professional aesthetics hoped to pin down. We can therefore read the _Kino-Debatte_ as part of an extended conversation in German culture about the nature of aesthetic experience. The debates were not just a series of complaints about film encroaching on literary form and territory; they were also a very sophisticated debate about the nature of aesthetic reception, about what it meant, in a broad sense, to contemplate an image.\n\nAs an opening example, consider this quotation from a 1912 article describing an _actualit\u00e9_ that featured Wilhelm II on one of his hunting trips:\n\nThe Kaiser sits motionlessly\u2014only the image twitches slightly here and there, flickering and spotting up as if something were boiling under the projected surface. Suddenly, he raises his rifle and my ear hears the gunshot without its being fired. The audience silently rejoices over their Kaiser's fine shot; a wave-like movement goes through the room. The mountain goat rolls.\n\nThis otherwise incidental passage is exemplary of the early debates about film in its focus on audience reaction. This writer described, for example, a process of emotional projection: the audience understood the representation by projecting itself into the scene of the hunt. They rejoiced _with_ the kaiser as well as _over_ his excellent aim. The writer also emphasized the _physical_ response of the audience, demonstrating the importance of the body for this process of projection. The \"wave-like movement\" that ripples through the room was an effect of this projection. But here we should note what is different about the cinematic experience over the solitary enjoyment of a painting. This writer described the mutual amplification of audience emotion (which Benjamin later theorized in his \"Work of Art\" essay) that is inherent to the cinematic experience and that manifested itself here as a physical reaction. There is a reciprocity between spectator and screen, just as _Einf\u00fchlung_ aesthetics described the mutual relationship between viewer and artwork, but here that feeling of resonance was amplified by the number of people in the room all feeling it at once (fig. 4.2).\n\nFIGURE 4.2. A prewar audience enjoying a night at the Union-Theater in Berlin, 1913\n\nNote, too, that if _Einf\u00fchlung_ aestheticians thought about that resonance between painting or sculpture and viewer in terms of \"movement,\" here that term is no longer metaphorical but strikingly literal: the movement of the image, of the kaiser, and of the audience were all working in concert to provide the basis for emotional projection. If before\u2014as I will explain later\u2014the idea of \"movement\" in aesthetics was limited to what we might term \"inner movement,\" or emotional resonance, now that same reciprocity depended on a _physical_ sensation of movement. Furthermore, just as writers on _Einf\u00fchlung_ argued that our senses were not strictly delimited and that stimuli for one sense could also stimulate another, so we see here a similar recognition of the confusion of the senses. In noting that he \"hears the gunshot,\" the writer emphasized a _synesthetic_ response; he recognized that the film experience is not merely visual, but that vision itself is embodied and connected to the other senses. (This \"gunshot\" could have been a sound effect, of course, but we should also consider that it could have been psychosomatic, an effect of deep immersion.)\n\nThis speaks to another theme in _Einf\u00fchlung_ aesthetics: the idea of contemplation as the ground for aesthetic experience. Our usual assumption that the spectator is held in continuous rapt attention is troubled by this passage, for it also emphasizes the _materiality_ of the experience: \"the image twitches slightly here and there, flickering and spotting up as if something were boiling under the projected surface.\" The film experience seemed to require the audience, like the motionless kaiser and the rolling mountain goat, to hold opposite attitudes\u2014stasis and movement, contemplation and distraction\u2014at the same time. At once immersed in the illusion while blithely noting the twitching image, the audience alternated between attention and detachment. Already, the cinematic experience highlighted contradictions at the heart of contemplation. The rest of this section, then, will explore the issues of emotional projection, synesthesia, and movement as they relate to identity and embodied perception in both _Einf\u00fchlung_ aesthetics and early discussions of cinema in the _Kino-Debatte_.\n\nEmotional projection was a key issue in aesthetics during this period. An essay from the debates on film will help explicate this topic. Alfred Polgar felt that attending the cinema was a sensual experience. In his 1911 essay, \"Drama in the Film Theaters,\" he described seeing pretty girls on the screen; they smiled at him and he felt that he should smile back, even wait for them outside the theater once the show was over. He knows it is silly to think of such things, but there is something strangely compelling in the way the film takes on a life of its own, the way it \"smiles back at me.\" This could be the sigh of another lonely film guy, or the first glimmer of a phenomenology of film, in which the film itself has a subjectivity that engages with the spectator, such as that suggested by Vivian Sobchack. But it also hints at the connection between emotional projection, _Einf\u00fchlung_ , and identity.\n\nPolgar's emotional investment in the girls on the screen recalls one of the central concepts of Vischer's dissertation and of this trend in German aesthetics: the imputation of one's inner life to an inanimate object, which Vischer termed _Einf\u00fchlung_. Vischer wrote:\n\nWe have seen how the perception of a pleasing form evokes a pleasurable sensation and how such an image symbolically relates to the idea of our own bodies\u2014or conversely, how the imagination seeks to experience itself through the image. We thus have a wonderful ability to project and incorporate our own physical form into an objective form.... What can that form be other than the form of a content identical with it? It is therefore our own personality that we project into it. (104)\n\nAccording to Vischer, we project our emotions and sense of our body (such as our orientation in space) onto an inanimate object, just as we project these aspects of ourselves onto other people to understand their expressions. To us, the work of art itself is permeated by human sensations. In fact, subjectivity is present not only in our attitude toward the work but also within the work itself. \"There is also a will _within_ the image\" (114), Vischer wrote, meaning that we project our experience on to the relation _between_ the parts of the work as well. Vischer's concept of _Einf\u00fchlung_ described how the projection of human feelings onto inanimate objects plays a role in the creation, shaping, and reception of artworks.\n\nLikewise, Adolf Hildebrand's _The Problem of Form in the Fine Arts_ (1893)\u2014which had already gone through nine editions by 1914\u2014was an incredibly popular exploration of the problems posed by sensual perception. He was especially interested in legibility\u2014how we can understand the relation between changing appearances in people, nature, and art. Hilde-brand explained:\n\nNature in its movements and transformations produces changes in its appearance, which we interpret as the visible signs of those processes. We perceive the signs and imagine the process; we participate in it, so to speak, perform along with it, and accept the internal activity as the cause of the external appearance. Just as the child learns to understand laughter and tears by joining the process and is able to feel, through muscular activity that he himself calls forth, the inner cause of the pleasure or pain, so does all gestural expression and all movement on the part of others become for us a comprehensible expression of internal processes, a comprehensible language.\n\nHildebrand extended this analogy to inanimate objects and representations as well. We understand images by imputing them with a \"story\" of their processes, similar to our own; our own bodily sensations and experiences enliven the images and make them comprehensible to us. We understand art through a process of empathy. Taken together, the theories of Vischer and Hildebrand helped to establish _Einf\u00fchlung_ as a popular explanation of the aesthetic experience.\n\nPolgar also made a comparison between film and music to further his thesis about the power of the cinematic image. Music, he wrote, has the power to \"move\" us; it moves our \"inner humanity\" in a way quite mysterious. It could also provoke a synesthetic response: \"a powerful suite of images, color, feeling, and idea.\" Film, Polgar argued, functions in the same way: it is \"a similar fertilization of all the other senses through the stimulation of an optical sense.\" This idea of synesthesia was important to _Einf\u00fchlung_ theorists, because it helped to describe what actually happens during aesthetic reception, but also because it indicated the centrality of the body as a measure of aesthetic response. _Einf\u00fchlung_ theory postulated that the harmonious correlation of form to the physical or sensory structure of the viewer was the key to aesthetic pleasure. That is, according to the theories, if the object has a structure similar to our body, it is pleasing to us; if not, it strikes us as unpleasant. By extension, the symmetry of the body accounts for the tendency toward symmetry in art. A sympathetic response to this harmony could take place on a number of levels. There is, for example, in terms of symmetry, the physical correspondence of the structure of sense organs and various objects (a horizontal line matches the horizontality of our eyes, for instance). In terms of regularity, the formal arrangement in series matches the rhythmic function of our organs. In other words, \"the body, in effect, imposes an 'organic norm' in viewing the world, according to which regularity, symmetry, and proportion induce pleasing sensations by emulating the normal human body.\"\n\nBut this \"viewing\" is not merely visual. Vischer noted that sometimes \"a visual stimulus is experienced not so much with our eyes as with a different sense in another part of our body\" (98). Sunglasses could have the effect of \"cooling\" the skin, while \"loud\" colors might offend our auditory nerves. \"For in the body there is, strictly speaking, no such process as localization.\" The embodiment of vision meant that \"the whole body is involved; the entire physical being is moved\" (99). On one hand, this embodiment implied the possibility\u2014even necessity\u2014of synesthesia, at least at some level. The senses resonate with one another; the boundaries between them are not simply blurred, but porous, causing a stimulus to one sense to spark a response in the others. On the other hand, Vischer also argued that this resonance between the senses is the model for the relation between the arts: \"These reflexes or reciprocal vibrations of the senses are the physical cause of the unity of the arts\" (99). While this parallel between the structure of the body and the relation between the arts was just a footnote to his argument, it points to an ambiguity in Vischer's book and in subsequent discussions of _Einf\u00fchlung_ , as we will see: the question of embodied spectatorship did not always imply physical participation in the act of viewing; instead, it often implied shared forms and orientations between the artwork and the human body per se. That is, the body was more often a _model for_ art than a _participant in_ art. Vischer's evocation of synesthesia functioned ambiguously by pointing both to embodied spectatorship and to the body as an idealized model.\n\nHowever, Ernst Bloch's 1914 essay, \"Melody in the Cinema, or Immanent and Transcendental Music,\" pressed harder on this issue via the example of silent film accompanied by music. A musician by training, Bloch placed greater emphasis on the redemptive and sensual nature of sound. But he argued that, while it was exclusively visual, the film image gained another dimension when combined with the proper music. Indeed, his article was a plea for music that was appropriate for the film and not simply random notes played to cover the sound of the projector. He found synesthetic possibilities in film music, which could present a complete sensual experience initiated by the ear, not the eye.\n\nAs visitors to the cinema we must initially rely exclusively upon our eyes. Now, the sense of touch conveys the impression of reality most immediately; in front of the film screen, however, we must renounce everything\u2014pressure, warmth, scent, sound, and the feeling of being sensually encompassed [ _sinnliches Mittendarinsein_ ]\u2014that gives the appearance of things its fully \"real\" character. The skin, the nose, ears, and all other senses are switched off, while the eyes are overwhelmed. Only an optical impression of black and white is excerpted from the real world, and since this impression is presented in the most confusing momentary motion and without any stylization, it produces the uncanny impression of a solar eclipse, a silent and sensuously deprived reality that is heightened only in its speed and its concentration, but without departing from our world aesthetically and ideally. But now the ear takes on an unusual function: it fills in as the replacement for all the other senses. Because the rustling, rubbing, and noisy collision of objects, because above all, human voices (which are themselves ringing with emotion) can blend seamlessly into [musical] tones\u2014indeed, precisely because there is nowhere a natural or manufactured sound in the world that competes with music\u2014this art form [film music] is able to reflect the colorfulness of lived reality and achieve at one stroke a sensuous totality that never makes us aware of our individual senses.\n\nLike many writers during this period, Bloch presented film as a lack: in front of the screen we must \"renounce everything\" that gives the world its \"fully 'real' character,\" such as \"pressure, warmth, scent, sound.\" We may disagree with his assessment of the cinematic experience\u2014we might rightly wonder where he found such a cool, odorless, silent, and spacious film theater\u2014but we can also see this as a condemnation of the poverty of any form that relies on vision alone. In this case, Bloch argued that film's silent, black-and-white world presented only an attenuated version of the world, a meager representation that could not satisfy the need for sensuality until it was matched with music. Then, as if music were color itself, this world came alive, and the filmic representation was able to regain the \"colorfulness of lived reality.\" It seems at first glance that music was the sole hero here, rescuing bored spectators everywhere from the paucity of filmed reality. But it was precisely the _combination_ of film and music that created this embodied experience. Bloch called film \"a silent and sensuously deprived reality that is heightened only in its speed and its concentration,\" but it was exactly that speed and concentration that allowed a \"sensuous totality,\" in that film and music are both temporal forms. It was their combination (as in opera, for example) that swept the spectator into a powerful representation of \"the colorfulness of lived reality.\"\n\nBut film is not like opera; \"its speed and its concentration\" are like no other forms. What did he mean by \"concentration\" ( _Konzentration_ )? On one hand, he referred to the temporal density of filmic representation: its ability to create temporal ellipses through editing and to pack more events into a limited amount of time\u2014film as a precipitate of time. On the other hand, it recalled the concentration of the film viewer, the spectator's absorption and immersion in the film, the kind of contemplation that almost allowed the viewer to forget \"pressure, warmth, scent, sound,\" and so on. Both were required for this new form of embodied spectatorship. Almost despite himself, Bloch argued for a synesthetic experience unique to cinema. In so doing, he extended further the issue that Vischer and others explored: the role of the body in the aesthetic experience. Bloch's description of film and music's \"sensuous totality\" ( _Gesamtsinnlichkeit_ ) certainly pushed synesthetic principles of artistic formation and reception much further than what we find in most theories of _Einf\u00fchlung_.\n\nBloch's description of film and music and Polgar's case of the pretty girls bring up an interesting question in relation to this theory of aesthetic response. Generally speaking, Vischer, Hildebrand, and other theorists of _Einf\u00fchlung_ limited their discussion to _static_ forms, such as painting, sculpture, and architecture. We cannot blame them for not including cinema in their investigations, but even theater and dance or other temporal forms were rarely, if ever, evoked. Music featured prominently, but only because it is abstract; these writers were working on the problem of legibility\u2014how is it that we understand something that is not like us?\u2014so music's abstractness, not its temporality, was intriguing to them. Polgar's pretty girls were just a representation, certainly\u2014black and white, silent, two-dimensional\u2014but they _moved_ , and the moving image's precise role in _Einf\u00fchlung_ theory was still indeterminate. To what extent could _Einf\u00fchlung_ aesthetics accommodate a temporal form such as moving pictures? Vischer, Hildebrand, and others frequently discussed \"movement,\" but we should first understand what they meant by this term.\n\nMovement is implied by the very term _Einf\u00fchlung_ , which translates literally as \"feeling into.\" Yet it also goes without saying that the combination of \"feeling\" and \"into\"\u2014that is, the comparison of emotion and space\u2014must be metaphorical. Nevertheless, the emphasis in _Einf\u00fchlung_ aesthetics on physiology and on spectator response left open the possibility that \"movement\" during the aesthetic experience might be more than merely metaphorical. This was indeed the case for both Vischer and Hildebrand, who discussed physical movement of the spectator in terms of the movement of the eye over the object before it. But this activity, as Vischer explained, is largely confined to the eye: \"We achieve this muscular activity, by moving the eye while looking at the object: that is, by scanning [ _Schauen_ ]\" (94). Hildebrand agreed; both theorists distinguished seeing ( _Sehen_ ) from scanning ( _Schauen_ ) and likened the latter to touch as a more sensual, tactile appropriation of the image. Vischer allowed that there might be a more visceral or holistic physical response to art when he ventured that \"mental stimuli can bring about motor stimuli in the lower organs, and vice versa. The whole body is involved; the entire physical being is moved\" (99). Ultimately, however, this response was not what he had in mind when discussing the aesthetic experience. Instead, both he and Hildebrand stressed the importance of the imagination as a mediator between raw stimuli and a purely aesthetic response. \"I might imagine myself,\" Vischer wrote, \"moving along the line of a range of hills guided by kinesthetic imagination (be it direct or mediated by the reflex stimuli of sensitized nerves). In the same way, fleeting clouds might carry me far away. This is no longer seeing [ _Sehen_ ] but a _watching_ [ _Zusehen_ ]: the forms appear to move, but only _we_ move in the imagination\" (101, emphasis in original). In other words, our projection into the work\u2014and our physical response to it\u2014depends on an act of imagination, ensuring the aesthetic experience remains within the domain of the \"higher\" functions. The movement of the eye initiates the process, but in our response any \"movement\" is to be construed as virtual, not real.\n\nIf _Einf\u00fchlung_ was a combination of emotional projection and (a limited form of) embodied spectatorship, the work of architectural historian August Schmarsow is especially appropriate. His inaugural lecture from 1893, \"The Essence of Architectural Creation,\" outlined a theory of architecture as \"the creatress of space\": \"Our sense of space [ _Raumgef\u00fchl_ ] and spatial imagination [ _Raumphantasie_ ] press toward spatial creation [ _Raumgestaltung_ ]; they seek their satisfaction in art. We call this art architecture; in plain words, it is the _creatress of space_ [ _Raumgestalterin_ ].\" Schmarsow argued that we understand architectural form by projecting ourselves into it\u2014by correlating our bodily axis, for example, to the vertical lines of a building. For Schmarsow, this was a process of spatial creation ( _Raumgestaltung_ ), closely linked to proprioception, our sense of spatial orientation, or what he called \"the intuited form of space\": \"The intuited form [ _Anschauungsform_ ] of space... consists of the residues of sensory experience to which the muscular sensations of our body, the sensitivity of our skin, and the structure of our body all contribute\" (286). Our bodily activity creates or contributes to a physical \"memory\" of sensation, which serves as the basis for a physiological understanding of architectural form. Like Vischer and Hildebrand, Schmarsow found the imagination central to this process: \"It is an act of free aesthetic contemplation when, with the aid of our imagination, we transport ourselves from the exterior that we see before us into the center of the interior space; when, by inquiring into its axial system, we strive to open up a remote organism to the analogous feeling within ourselves\" (293). This \"striving\" to open up the interior of a building by means of our already existing sense of space is to Schmarsow a creative process, and \"movement\" was vital to it.\n\nAs in the work of Vischer and Hildebrand, this \"movement\" seemed virtual or imagined, detached from real muscular tension, but in Schmarsow's essays something more seems to be at play: \"We cannot express its relation to ourselves in any way other than by imagining that we are in motion, measuring the length, width, and depth, or by attributing to the static lines, surfaces, and volumes the movement that our eyes and our kinesthetic sensations suggest to us, even though we survey the dimensions while standing still\" (291). Here Schmarsow hinted that our \"kinesthetic sensations\" play a part, that in the appropriation of architectural form there is a productive tension between stillness and movement in the body at the moment of apperception. Indeed, architecture may be the \"creatress of space,\" but Schmarsow emphasized the creative aspect of spectatorship as well:\n\nJust as, in response to external events, shared emotional feelings intensify in their rise and fall into moods or press in their growth toward blissful delight or convulsive pain in order to move farther out and fill the immediate surroundings with the vibrations of the inner life and to influence those surroundings, if only through the fleeting sound of the human voice\u2014so, too, the purely imagined impressions and their integration or combination into three-dimensional visual forms involuntarily project themselves into the world outside and develop further into a sensorially perceptible reality. (292)\n\nSchmarsow argued that _emotion_ was key to understanding space and that this emotion projected outward to create a larger reality. Furthermore, when he wrote that \"the spatial construct is, so to speak, an emanation of the human being present, a projection from within the subject\" (289), Schmarsow suggested not simply that architectural space was a mutual construction of architect and viewer (\u00e0 la Kant), but that the apperception of architecture involved a simultaneous construction of lived space, that the architectural representation of space was not merely imagined but felt emotionally and physically\u2014and that this emotional and physical \"movement,\" in a real way, _creates_ the space represented. (In this respect, Schmarsow's ideas might be useful for thinking about our bodily and emotional engagement with cinematic space.)\n\nOn the other end of the spectrum, however, we find Munich psychologist Theodor Lipps, regarded as the chief proponent and theorist of _Einf\u00fchlung_ aesthetics. Between 1890 and 1914, he was a prolific\u2014not to say incontinent\u2014explicator of _Einf\u00fchlung_ , both in aesthetic terms (emotional projection of ourselves into art) and psychological terms (as the fundamental way we understand others). On the question of movement, Lipps was contradictory. In a typically confusing passage, Lipps defined his terms: \"Finally, one could figuratively say that 'activity' is the inner breath or heartbeat, or more generally, it is the inner motion. But 'motion' here is not meant as a simple event within me, but rather the fact that I move. Of course, this 'motion' has nothing to do with space.\" In postulating that \"I move\" but that \"this 'motion' has nothing to do with space,\" Lipps hoped to have it both ways, succinctly expressing the tantalizing tension in the definition of movement in _Einf\u00fchlung_ aesthetics. Yet, in the end, Lipps was fairly clear that this movement was limited to \"some inner, striving motion\" ( _der inneren strebenden Bewegung_ ). Indeed, we might assume (or hope) that kinesthetic affect is an important aspect of the aesthetic experience, but Lipps was adamant that if we _feel_ any bodily response, if we are aware of our bodies in any way, then we are no longer in the realm of the aesthetic proper. Real bodily movement or physical response may be mimicry or erotic pleasure, according to Lipps, but at the point that it is physical, it ceases to be aesthetic. He stated it flatly:\n\nBut the pleasure which I feel while I am paying attention to my physical states or the processes in my physical organs cannot be identical, neither wholly nor in part, with the joy which I feel when I do _not_ pay attention to the processes in my physical organs, but devote my whole attention to the aesthetic object. In short, _A_ cannot equal non- _A_.... _Einf\u00fchlung_ means, not a sensation in one's _body_ , but feeling something, namely, _oneself_ , into the aesthetic object.... Actually, the sensations of my own bodily state are only present in aesthetic contemplation in order to be entirely absent to me.\n\nThis, of course, meant that any sexual feeling or arousal could not be part of the aesthetic experience:\n\nThe sexual has nothing, absolutely nothing, to do with the aesthetic. Those who employ it to explain aesthetic feeling know as little of the meaning of beauty and aesthetic contemplation as those who warn against \"nudity\" in art because they fear that even chaste nudity will threaten morality: first their own morality, then that of others to whom they ascribe their own crudity.\n\nIn sum, Lipps insisted that the movement implied by _Einf\u00fchlung_ was metaphorical and must remain so if it were to be aesthetic; he firmly stood by a strict, conservative delimitation of art and body that required the terms of aesthetic pleasure to be divorced from bodily response.\n\nThe cinematic experience, however, brings us back to the question. As we become immersed in the cinematic image, we might, as Lipps demanded, forget about our bodies, but this very forgetfulness might result in involuntary movement, such as gripping the arm of the chair (or the person next to you) during a car chase or moving slightly side to side during a well-choreographed fight sequence. It might even result in erotic pleasure. During the time that Lipps wrote about _Einf\u00fchlung_ , these pleasures in the cinema would not have even been included in the category of \"the aesthetic,\" but the character of the moving image\u2014among other things\u2014may have forced a reconsideration of the definition of aesthetic pleasure. We can see this renegotiation start to take place with Walter Serner's 1913 essay, \"Kino und Schaulust,\" which is one of the first to suggest the sexual dimension of visual pleasure. Serner's discussion of the link between cinematic movement and bodily reaction implicitly challenged the division between visual stimuli and visceral response that we have seen in traditional and _Einf\u00fchlung_ aesthetics. I am not suggesting that Serner was actively intervening in this discussion; indeed, his attitude toward cinema was ambivalent at best. But his essay indicates that what remains virtual and latent in _Einf\u00fchlung_ aesthetics\u2014movement in the image and in the spectator\u2014was even at the time visceral and literal in the experience of cinema.\n\nSerner was explicit that the _Schaulust_ he described was not the polite, distanced gaze of the aesthete: \"Not the harmless kind, for which everything is only movement or only color or both, but a frightening desire that is no less powerful than the very deepest kind. It is a desire that boils the blood and makes it rage, until an unfathomably powerful excitement common to all desire races through the flesh.\" This \"unfathomably powerful excitement\" was, of course, sexual desire. Serner was at pains to point out that this brand of sexual desire was not merely erotic but was tinged as well with the thirst for violence. Cinema appealed to our basest instincts, our darkest needs. But it was not simply the content of the films that made this appeal; it was cinematic movement itself:\n\nIt was neither the quick victory over something that was dying nor an astonishing profitability that with one stroke put this [cinema] into place from the outset. It was the exciting adventurousness of a tiger hunt, of a daring mountain run, of a death-defying automobile ride; it was the breathtaking pursuit of a wounded, bleeding thug over the dizzyingly high rooftops of New York; it was the eerie district filled with misery, sickness, and crime, and the entire horrific detective genre with murder and violence, Browning and Navajo; and it was all the bloody, blazing images of fire and death, of horror and terror, on which all eyes sate themselves after long privation. A gaze, it was, that had rhythm and life and was desire.\n\nThe tiger hunt, the mountain ride, the chase\u2014these were the components of cinema as much as the bloody pictures of fire and death. Serner named the _thrill_ of cinema as an explicitly visceral reaction in which the spectator was swept away by the movement of the image. The eye was no longer detached from the body but connected \"through the sheer endless rush, which is all the more welcome, since it knows that movement is the greatest component of the desire to look and its innermost essence.\" _Einf\u00fchlung_ aesthetics saw the body as a model for art, as a category for understanding, but mostly not as an active participant in the aesthetic experience, except for the initial roaming gaze. And there is no indication that Serner or anyone else at this moment viewed what he described as aesthetic. But as the filmic image and cinematic experience _became_ associated with Art through other means (for example, the immersive form of film narrative, movie palaces, and so on), this residual experience of movement and visceral response could not be logically left behind, even if it were attenuated in practice or, as Tom Gunning argues, took its place in lower forms of cinema. In any case, after cinema, aesthetics could no longer logically justify the well-mannered separation between contemplation and physical response.\n\nTHE POLITICS OF CONTEMPLATION\n\nWhatever their differences regarding the role of the entire body in aesthetic response, all writers on _Einf\u00fchlung_ (indeed, most writers on aesthetics in general) assumed that contemplation was absolutely central to the aesthetic experience. Recall that Schmarsow declared, \"It is an act of _free aesthetic contemplation_ when, with the aid of our imagination, we transport ourselves from the exterior that we see before us into the center of the interior space\" (293, emphasis added). Likewise, Lipps consistently invoked contemplation as the prerequisite to, even the ground for, _Einf\u00fchlung_ : \"The contemplation of the observed movement awakens the tendency to a corresponding inner response [ _Selbstbet\u00e4tigung_ ]; and by corresponding we mean that which would be connected with the execution of such a movement if I were to perform it. And this tendency is at the same time _actualized_ in the act of contemplation.\" Lipps argued that whatever movement we might feel when encountering a work (be it a painting or dance or another form), we do not actually move, nor do we need to move. Instead, the impulse toward imitation of that movement in the work is resolved or dissipated by the act of contemplation itself. In Lipps's version, the contemplative act transmutes _motion_ into _emotion_. Contemplation thereby allows emotional projection to take place, undisturbed by the intrusion of physical sensation. It is a surrender to the object and to the process: \"the aesthetic experience is the way that I feel moved when I engage in aesthetic contemplation, when I surrender myself completely to the representation,\" wrote Lipps. In this respect, _Einf\u00fchlung_ aesthetics grafted common notions of aesthetic contemplation onto the process of _Einf\u00fchlung_. We could even say that the theory of _Einf\u00fchlung_ was constructed to explain what happens during the contemplation of artworks, but without disturbing the moral and ideological significance of the operation. That is, the meaning of \"contemplation\" in _Einf\u00fchlung_ aesthetics was not markedly different from what we might find in aesthetics generally, even if elements of _Einf\u00fchlung_ theory tread close to an expansion of that meaning.\n\nCould this model of contemplation accommodate the cinematic experience? An initial answer might be no, it could not, especially given the standard history, which dictates that contemplation faded and distraction took its place. The evidence in this chapter elaborates that history. The first section demonstrated that objections to cinema's rush of images were grounded not simply in contemplation's leisurely approach but also in the historically strong tension between the flux of time and free will. That is, writers were not just _unaccustomed to_ the rush of imagery but felt _determined by_ the choice and flow of imagery that seemed to push them along ever more impatiently through the present, giving them a strong, lived sense of time's passage. Meanwhile, other writers reveled in the sense of presence and the present that motion pictures gave them, which sharpened their senses and put them on alert, as if caught on foot in midday traffic. Likewise, the second section sketched the trouble with contemplation from another angle as aesthetics tried to accommodate aspects of the aesthetic experience that were nascent in early theories but never fully accounted for, especially emotional projection and embodied perception. As these features became more prominent in the theories, it was more difficult to graft a common conception of contemplation (as largely inattentive to the body) onto this model. The cinematic experience, however, was varied. As we saw in chapter 3, reformers such as H\u00e4fker worked hard to fit film presentations into the ideologically dominant model of contemplative experience. We could also argue, after Charles Musser, that some film genres, such as landscape films, were designed from the start to provide for a contemplative gaze.\n\nWe could further contend, after Miriam Hansen and Heide Schl\u00fcpmann, that cinema's public sphere had already created an alternative to detached aesthetic contemplation. They argue persuasively that the usual interpretation of these early discussions of film audiences in terms of class misreads to a certain extent the masculine anxieties of the moment, which should more properly be classified in terms of gender and sexuality. The presence of women in the public sphere in general and in the movie theater specifically, according to Hansen and Schl\u00fcpmann, indicates not only a change in gender dynamics in pre\u2013World War I Germany, but also a change in aesthetic modes of reception\u2014specifically, a shift from detached aesthetic contemplation to a more immersive, emotionally involved mode of viewing that has often been characterized as female. Their point is that, if we are going to look to the debates as evidence of a shift in aesthetic reception, we would do well to consider that most interlocutors were male and that many in the audience were female; contemplation was an ideological tool in service of not just class-based but also gender-based goals.\n\nHansen and Schl\u00fcpmann provide an important corrective and an avenue of research that has been enormously productive. I would only indicate that, as chapter 3 demonstrated, children also presented an important model of spectatorship. In fact, this book argues that gender, class, and age were only some of the categories through which spectatorship was discursively constructed, and that our understanding of this process is best grasped via the larger division between _expert_ and _lay_ viewers. An expert could very well have an immersive, emotionally involved reaction to a film or a painting, as in Diderot's famous description of one of his encounters with a painting: \"I was motionless, my eyes wandered without fixing themselves on any object, my arms fell to my sides, my mouth opened.... I shall not tell you how long my enchantment lasted. The immobility of being, the solitude of a place, its profound silence, all suspend time; time no longer exists, nothing measures it, man becomes as if eternal.\" Apart from the final venture into metaphysics (\"man becomes as if eternal\"), this description recalls numerous reports of ordinary moviegoers, female and otherwise. But because it was written by an expert, it is free of the damning taint of passivity, which characterized all spectators\u2014young and old, female and male. Diderot was absorbed by the image, but he wandered through it of his own accord. Passivity, not necessarily immersion or emotion, was the feminized trait that characterized lay spectators of all sorts.\n\nYet class does matter; if contemplation held most aesthetic theories in a tight ideological grip, it did so in service of implicit class privilege and agency. After World War I, however, when class barriers were subject to attack, the reaction to contemplation took on a more overtly political tone. While nineteenth-century philosophers might have viewed contemplation (and themselves) as active, revolutionaries saw contemplation (and the philosophers) as bourgeois and passive in favor of the status quo. This critique of contemplation was voiced most vociferously by the Dadaists, who vowed to no longer assume the role of mere \"passive spectator [ _passiver Beschauer_ ] of this comical world, of an aesthete like Oscar Wilde,\" for, as Richard Huelsenbeck warned at the time, \"to sit in a chair for a single moment is to risk one's life.\" For the Dadaists, contemplation was irredeemably tied to passivity. They condemned the class dimensions of contemplation as a bourgeois appropriation of art for art's sake that ignored the revolutionary potential of the aesthetic experience. Early discussions of cinema reversed this complaint. Various _Kinogegner_ rejected motion pictures for their perceived association with the working classes and further argued that the nature of the filmic medium precluded the possibility of contemplation, which they defended as if protecting class boundaries. Either way, it seemed, motion pictures were not usually connected to contemplation. If they were, as in H\u00e4fker's case, it was in defense of film, contemplation, and middle-class values. In fact, the extent to which film was considered suitable for either contemplation or a distracted mode of viewing depended precisely on the class alignments of any given writer.\n\nAfter World War I, then, the passive and active connotations of contemplation and distraction traded places. The rehabilitation of distraction began, at least in film theory, with Siegfried Kracauer's 1926 essay, \"The Cult of Distraction,\" in which he described film presentations in Berlin's lush movie palaces (fig 4.3). For Kracauer, these inherently fragmented, varied, and artistically incoherent presentations (which included live revues, advertisements, films, lighting schemes, music, and other spectacular elements) structurally matched \"the disorder of society\" and, hence, had the potential to present the working classes with a true picture of \"the uncontrolled anarchy of our world\" (327). That picture could also function as an image of the mass audience or masses ( _die Masse_ ) of and for itself; that is, for Kracauer, if the mass audience could recognize itself in an image of itself, it would be a step closer to political consciousness: \"Here, in pure externality, the audience encounters itself; its own reality is revealed in the fragmented sequence of splendid sense impressions. Were this reality to remain hidden from the viewers, they could neither attack nor change it; its disclosure in distraction is therefore of _moral_ significance\" (326, emphasis in original). But Kracauer complained that picture palaces in 1926 papered over this fragmentation with \"the glue of sentimentality\" or an \"artistic\" unity, thereby hiding the true picture of alienation behind a surface sheen. They functioned in this case, for Kracauer, as distraction _from_ , whereas he advocated distraction _for_ political awakening. Although never named in this essay, contemplation was implicitly no more than a phantom appendage of an \"idealist culture that haunts us today only as a specter\" (327).\n\nFIGURE 4.3. Kammer-Lichtspiele Theater in Berlin, 1912\n\nBenjamin also gave distraction an explicitly political valence. In his 1929 \"Surrealism\" essay, for example, he discussed the \"transformation of a highly contemplative attitude into revolutionary opposition.\" But Benjamin's understanding of the nature of that political charge differed from Kracauer's. His \"Work of Art\" essay (1935\u20131939) remains the definitive sign of the rising political stock of distraction and the devalued currency of contemplation between the wars. This essay has received more than its share of commentary, so I only want to underline certain keys differences between Benjamin and Kracauer. For Kracauer, distraction functioned as an image of the masses for the masses; the homology between leisure and labor, between the disjunctive form of entertainment and the alienated character of work and modern, urban life was something that the audience _could see_ or picture through their experience in the picture palace. Indeed, Kracauer assumed that, once they grasped this picture, the mass audience would _reflect_ on it and come to an understanding of its political situation. So the image that distraction presented would be structurally no different than any image the audience would contemplate and reflect on.\n\nBenjamin, on the other hand, argued that modernity and technological reproducibility had destroyed the cult value of images, so he searched for an alternative mode of thinking that would not rely on images per se, or even vision. Distraction, for him, was this revolutionary mode that struck at the heart of knowledge as usual. For Benjamin, contemplation comes at knowledge through the encounter with\u2014or reliance on\u2014a transcendental Ideal (recall Schopenhauer's understanding of aesthetic experience as loss of oneself and a nirvana-like communion with the Ideal). It is a top-down approach that already presumes form, like a Kantian a priori. That \"already-given\" rigs the game of knowledge production, stacking the deck against experience. Contemplation, for Benjamin, succumbed to the tyranny of forms of knowledge already given. Distraction, however, represented another way of thinking that worked from the bottom up; it takes the world and experience for what they are, not what they are presumed to be: \"A person who concentrates before a work of art is absorbed by it.... By contrast, the distracted masses absorb the work of art into themselves.\" The tables have turned: contemplation is now passive, in that the viewer is ruled by the rules and form of the work of art. Distraction, on the other hand, is now active, in that the masses control the work and care not for its form or the usual ways of appropriating it. Benjamin evoked touch and habit\u2014senses other than vision\u2014to describe this mode; distraction was a famously tactile form of knowledge: \"The values of distraction should be defined with regard to film, just as the values of catharsis are defined with regard to tragedy... Distraction, like catharsis, should be conceived as a _physiological_ phenomenon.\" Like Bergson, he hoped to turn us away from our common, ordinary perception that actually keeps us chained to the status quo.\n\nGeorg Luk\u00e1cs also offered a thorough and trenchant analysis of the class dimensions of contemplation. From his early _Theory of the Novel_ (1916) to _History and Class Consciousness_ (1923), Luk\u00e1cs presented a strong critique of the _vita contemplativa_. For Luk\u00e1cs, contemplation was not merely a mode of aesthetic viewing, but a general reaction to the alienating character of modern life that must be surmounted for any transformation of the world to take place. It was a \"pattern of consciousness in which man contemplates from a position of formal freedom his own integration in a system of alien compulsions and confuses this formal 'freedom' of his contemplation with an authentic freedom.\" Contemplation, for Luk\u00e1cs, was simultaneously tied to both transcendence and complacency. In the preface to the 1967 edition of _History and Class Consciousness_ , Luk\u00e1cs wrote, \"Above all, I was absolutely convinced of one thing: that the purely contemplative nature of bourgeois thought had to be radically overcome.\" Even in _Theory of the Novel_ , Luk\u00e1cs organized his argument around two modes: contemplation and action. Specifically, he found two broad trends in the history of the novel, \"abstract idealism, which concentrates on pure action, and Romanticism, which interiorizes action and reduces it to contemplation.\" Novels such as _Don Quixote_ (1605) emphasized action without reflection, while novels such as Novalis's _Heinrich von Ofterdingen_ (1802) expressed the inward turn of romanticism, away from the disappointment and failures of modern alienation. Luk\u00e1cs explained:\n\nIn Romanticism, the literary nature of the _a priori_ status of the soul vis-\u00e0-vis reality becomes conscious: the self, cut off from transcendence, recognizes itself as the source of the ideal reality, and, as a necessary consequence, as the only material worthy of self-realization. Life becomes a work of literature; but, as a result, man becomes the author of his own life and at the same time the observer of that life as a created work of art.\n\nLuk\u00e1cs set up agency\/identity dichotomies similar to Benjamin's: the paths to identity or self-realization were twofold, either through transcendence or self-awareness. If one denied the possibility of transcendence, then only a materialist, humanist approach would be possible, which was what Luk\u00e1cs advocated. Likewise, once this path was chosen, agency returned and one could be an active \"author\" of one's life rather than a passive \"observer,\" which Luk\u00e1cs condemned. In _Theory of the Novel_ , he advocated humanism (exemplified by Goethe's _Wilhelm Meister_ ) as a middle ground between action and contemplation, but in his later, more explicitly Marxist works, Luk\u00e1cs focused primarily on a radical critique of contemplation.\n\nBut the first rumblings of this dissatisfaction could be felt before World War I, especially in Luk\u00e1cs's 1913 essay, \"Thoughts Toward an Aesthetic of the Cinema\" (\"Gedanken zu einer Aesthetik des 'Kino'\"). This brief but provocative article for the _Frankfurter Zeitung_ \u2014it was chosen for publication over Bloch's piece on film and music\u2014has been given insightful consideration by a number of scholars over the years, but it is worth another look, because it both addresses the most insistent themes of the _Kino-Debatte_ (the distinction between theater and cinema, word and image) and looks past them to anticipate themes in film theory of the 1920s. While it does not speak to most issues common in _Einf\u00fchlung_ aesthetics (for example, emotional projection, synesthesia, or embodied spectatorship), it offers in nascent form a critique of contemplation that Luk\u00e1cs developed in his later work; in this way, it articulates cinema's implicit challenge to \"traditional aesthetics\" that will come out more forcefully in Luk\u00e1cs's later work and the writings of Benjamin, Kracauer, and others.\n\nLuk\u00e1cs did this through what at first appears to be a standard comparison of theater and cinema, a strategy common to much early writing on film. The difference between theater and film, according to Luk\u00e1cs, is the difference between presence and absence, specifically, the presence of the actor on stage. But it is not the gestures or words of the actor that give theater its effect, nor the action of the drama. Instead, it is \"the power with which a person, the living will of a person, without mediation and without restraining direction, streams forth onto an equally living mass or multitude\" (13). The very _presence_ of the actor\u2014the force of his or her personal will, his or her drive toward the fulfillment of his or her being, his or her destiny\u2014is the essence of theater. This living, breathing essence of another person is inherently transitory\u2014tied to the now of the moment, to the _present_ \u2014but its very ephemerality is \"not a deplorable weakness, but rather a productive limit, the necessary correlate and most observable expression of destiny in drama\" (13\u201314). That is, the presence of the actor in the transitory moment shared by everyone in the theater is the most visible expression of the enactment of becoming that is drama itself. This sense of becoming, of the lived moment, is available to us all merely through the act of living intensely, but \"this so-called 'life' never attains an intensity that could raise everything up to the sphere of destiny\" (14). Drama, in other words, frames and concentrates this sense of destiny and fate for us: \"Throughout the presentation of the drama this metaphysical feeling grows in immediacy and perceptibility: out of the deepest truth of people and their position in the universe grows a self-evident reality\" (14). The stage, by virtue of its emphasis on _presence_ and _the moment_ , becomes a vehicle for the profound truths of human existence. Actors are an instantiation of these metaphysical truths in that they focus the force of living, or the direction of fate, into an ephemeral here-and-now shared by the actor and the audience; in fact, the mere presence of a great actor is \"already and without any great drama destiny, divine service, tragedy, mystery\" because of his or her unique ability to be \"absolutely present\" (14).\n\nCinema, on the other hand, does not share this emphasis on \"presence\" or \"the present.\" Motion pictures lack this \"presence\" not because of any inherent defect, but simply because in films \"there are only movements and actions of people\u2014but _no people_ \" (14, emphasis in original). The people on the screen are not present\u2014nor do they exist in the present\u2014so they have no presence. Even so, the representations in cinema are \"no less organic and alive than those images of the stage\" (14). They are simply different: \"they maintain a life of a completely different kind. In a word, they become _fantastic_ \" (14, emphasis in original). They present \"a new aspect\" of life: \"one without the present, a life without fate, without reasons, without motives, a life without measure or order, without essence or value, a life without soul, of pure surface, a life with which the innermost of our soul does not want to coincide.... The world of the 'cinema' is thus a world without background or perspective, without any difference in weight or quality, as only the present gives things fate and weight, light and lightness\" (14). At first glance, this judgment sounds very much like it might have come from a _Kinoreformer_ , like a condemnation of the cinema as \"soulless\" and suitable for only the basest amusements that require no thought or reflection. But Luk\u00e1cs was after something else here. He saw in cinema the same utopian potential that one could find in fantasy: \"' _Everything is possible_ ': this is the worldview of the 'cinema'\" (15, emphasis in original). Cinema was not just the poor cousin of the stage; it offered a completely different set of possibilities.\n\nThe basis for this difference lies in their formal properties as well as their dissimilar relationship to time and the word. The stage is populated with living, breathing people; drama focuses on exactly that moment of their shared presence with the audience. Motion pictures, however, do not picture people, but representations of people. Nevertheless, \"not only in their technique, but also in their effect, cinematic images\" are \"equal in their essence to nature\" (14). The \"technique\" Luk\u00e1cs speaks of here is the photographic quality of the cinematic image, while its \"effect\" is the illusion of movement that the apparatus creates. This movement, this temporal flow, also distinguishes cinema from theater. The stage has a paradoxical relation to time: \"it is the flow of grand moments, something internally deep at rest, almost 'arrested,' something become eternal, as a direct result of the painfully strong 'present'\" (14). Theater's emphasis on presence and the present gives it, oddly, a sense of stillness. The essence of cinema, on the other hand, \"is movement in itself, an eternal variability, the never-resting change of things\" (15). These different concepts of time, according to Luk\u00e1cs, correspond to the fundamental difference between the stage and cinema: \"The one is purely metaphysical, distancing itself from all that is empirically alive; the other is so strongly unmetaphysical, so exclusively empirically alive, that through this sheer extremity of its nature another entirely different metaphysics arises\" (15). One is fate, thought, stillness, presence, metaphysics. The other is fantasy, surface, movement, absence, action. But, as Tom Levin has noted in his reading of this essay, the intriguing aspect of cinema is that it is both \"empirical\" and \"fantastic.\" That is, its photographic image grounds it in reality, in the everyday, while its temporality\u2014its rush of images, as well as their arbitrary relation to laws of physics and causality\u2014frees it of necessity and allows it flights of fantasy. As Luk\u00e1cs declares, \"The fundamental law of connection for the stage and the theater is inexorable necessity; for the 'cinema' it is unlimited possibility\" (15). The uniqueness of cinema lies not simply in its ability to portray fantastic worlds but, as Levin argued, in its dialectical relation between the real and the possible. \"Everything is true and real, is equally true and equally real\" in the cinema, according to Luk\u00e1cs. The combination of reality and possibility and the _equality_ of the two even gives cinema a utopian dimension.\n\nBut I would also suggest that, in Luk\u00e1cs's essay, cinema provided a critical dimension alongside its utopian potential. In his comparison of stage and film in terms of \"metaphysics\" and \"soullessness,\" there was, I think, a critique of contemplation that emerged more forcefully in his later work, especially _History and Class Consciousness_ (1923). Consistently throughout the essay, Luk\u00e1cs distinguished the \"soul\" of the theater (its stillness, its metaphysical depth, its emphasis on destiny and the human condition, its presence) with the \"soullessness\" of the cinema (its constant movement, its emphasis on surface, its metaphysical \"lightness,\" its fantastic quality, its absence). In this distinction, Luk\u00e1cs was not alone. Many other writers of this period also called cinema \"soulless.\" Franz Pfemfert, editor of _Die Aktion_ , bemoaned in 1911 cinema's role in the \"soullessness\" of modern society. For Pfemfert, the naked reality presented by the cinematic image\u2014which he blamed for the _lack_ of fantasy in the filmic world\u2014could best be utilized for educational purposes. But Kurt Pinthus, introducing his 1913 _Kinobuch_ , a collection of scenarios written specifically for cinema, warned his readers, \"Always remember: this is not high art, not 'soulful' art, not theatrical art. We write only pieces for cinema, not for theater.\"\n\nHere we find what Luk\u00e1cs meant by \"soul.\" Pinthus wrote, \"not 'soulful' art, not theatrical art,\" as if they are one and the same. Pinthus and Luk\u00e1cs referred, of course, to German inwardness ( _Innerlichkeit_ ), that national tendency toward contemplation and reflection (and the demand for it in art). In his consistent opposition of \"action\" (cinema) to \"soul\" (theater), Luk\u00e1cs indicated that cinema offered an alternative to this cul-de-sac of reflection. His distinction between cinema and theater corresponds to the dichotomy between abstract idealism and romantic inwardness found in his _Theory of the Novel_ , which in turn was the foundation for his critique of contemplation in _History and Class Consciousness_. The latter book, especially, condemned as bankrupt the alienated, contemplative character, in favor of political or radical action. His discussion of cinema's \"soullessness,\" then, was not a condemnation, not even a slight in favor of the theater, but the articulation of a new possibility, a new form that more closely corresponded to the practical needs of the people. Cinema's emphasis on surface and lightness was an opportunity to escape from the suffocating and futile tradition of _Innerlichkeit_ , a formulation remarkably similar to those of Kracauer and Benjamin. Luk\u00e1cs thereby subtly pointed to the emergence of a new aesthetic, which Kracauer and Benjamin later articulated as \"distraction.\" So when Luk\u00e1cs declared that, with cinema, \"Man has lost his _soul_ ; in return, however, he gains his _body_ \" (16), he celebrated the potential for a new aesthetic standard that incorporated physiological, even tactile, responses to cinema, thereby pointing the way for film theorists of the 1920s and beyond.\n\n\"What does all this mean? It means that real presentation banishes contemplation,\" Benjamin announced, only slightly ironically, in 1930. It certainly seems that way, with Luk\u00e1cs proclaiming that our soul has been replaced by our body and Benjamin theorizing a physiological and tactile model of knowledge. Yet as Carolin Duttlinger has shown, Benjamin's own conception of contemplation and distraction was much more complex and dialectical than the version even the \"Work of Art\" essay provides. In his essays from the early 1920s, Benjamin saw in religious contemplation a presence of mind that allowed for both absorption and alertness, which was different from what he saw in the secular, solipsistic immersion of the modern age. Indeed, he and Bertolt Brecht agreed that modern contemplation and distraction were two sides of the same coin: both precluded critical engagement and response and therefore impeded political action. If his discussion of distraction in the \"Work of Art\" essay reversed this view, it is only because he dialectically leavened distraction with both habit ( _Gewohnheit_ ) and alertness or presence of mind ( _Geistesgegenwart_ ). Yet later, in some of his historical works, Benjamin's conception of modern contemplation again adds _Geistesgegenwart_ as the essential ingredient. So Duttlinger concludes that Benjamin's writings consistently reject \"solipsistic contemplation in favor of a more flexible, perpetually alert presence of mind.\"\n\nThe balance between immersion and detachment has been historically difficult to find, not only because it is challenged with every new art form, but because it is, as Benjamin's work so clearly shows, a _moral and political_ stance as well as an aesthetic stance, so finding the balance also implies finding a proper stance toward the historical moment. This is why the task continues to fascinate, and why contemplation, no matter what opprobrium it receives, will never die. For example, as Jocelyn Szczepaniak-Gillece has shown, contemplation and immersion made a comeback in film theory in the 1930s through the 1950s, as modernist architects attempted to design movie theaters that would create an ideal, immersive viewing experience and hence an ideal spectator, who was contemplative yet alert and responsive. Indeed, the change in ideas about contemplation seems to respond to Benjamin and Brecht's complaints that the inherently individualist nature of contemplation separates the viewer from the collective, thereby inhibiting communication, solidarity, and political action. These architect-theorists, such as Benjamin Schlanger and Frederick Kiesler, hoped to induce an almost Schopenhauerian disembodied viewing experience, while also making that experience resolutely communal. Like Benjamin, they accepted and worked with cinema's collectivity. This is the primary difference between the writers of the _Kino-Debatte_ and later theorists: for the most part, the earlier writers were unwilling to relinquish the historically strong connection between contemplation and individual viewing. Mass reception, for them, held little aesthetic potential (and the fear of social democracy often shut down any political potential). Yet we have seen glimmers of this possibility in various individual essays across the debates; if they often argued individually against film's aesthetic potential, as a group the essays helped clear the way. If _single_ writers clung to models of individual, expert viewing, the _collection_ of essays put mass reception on the film-theoretical agenda. Later shifts in film theory were therefore anticipated collectively by the film essays that appeared in the years before World War I.\nCONCLUSION\n\nTOWARD A TACTILE HISTORIOGRAPHY\n\nOf course, after World War I everything changed. On 1 July 1918, the newly established state film production company, UFA, created a _Kulturabteilung_ , or cultural division, to make and distribute a regular series of educational films. The leader of this division, Ernst Krieger, had been a major in the imperial army and had served alongside Alexander Grau, the wartime minister of culture. The war had offered a number of lessons, but in this case it had persuaded Krieger, Grau, and others beyond a doubt of the value of motion pictures as a means of mass education and indoctrination\u2014which after the war could be directed toward peacetime social issues. Krieger hired a number of doctors, scientists, and educators fresh from the front to make educational films encompassing titles from _The Alps_ (1918) to _Marital Hygiene_ (1922). The division also funded scientific research films, thereby giving the state's seal of approval and support to scientific and medical cinematography. The number of universities and institutes using motion pictures to document and to explore increased dramatically after the war, as film found a secure place in laboratories and lecture halls. Hermann H\u00e4fker's dream of a state-run, educational cinema had come true, even if\u2014in UFA's case, at least\u2014it had to make room for fictional films as well. Indeed, cinema's aesthetic pretensions received a boost after 26 February 1920, when _The Cabinet of Dr. Caligari_ opened in Berlin to much hype and wide acclaim. Its remarkable mise-en-sc\u00e8ne, acting, and narrative framework struck a chord with all audiences, even those beyond Germany's borders; its self-consciously artistic presentation drew from a distinctly German style, marking it immediately as \"Art.\" _Caligari_ became a lightning rod of debate about the nature of film, even while people lined up at the box office. Ushering in German cinema's golden era, _Caligari_ gave cinematic form some measure of aesthetic legitimacy. Meanwhile, the credibility of film in terms of aesthetic reception received more attention and validation with new theories (Rudolf Arnheim, Bal\u00e0zs, Benjamin, Kracauer) that acknowledged its significance for the transformation of aesthetic standards.\n\nSo in terms of disciplinary legitimacy, cinema had much less to worry about after the war. Debates continued about its proper function in society or how its artistic potential could be best realized, but they were not nearly as heated as in the prewar years. German society seemed to have adjusted itself to cinema, and cinema to it. In addition to prompting UFA's role as state-sanctioned protector of educational (and propaganda) film, the war had also cut off the supply of imported films, thereby allowing domestic production companies to flourish; the complaint about foreign films could no longer fuel debates, and the supply of \"quality\" German films also rose. The international success of _Caligari_ awakened the establishment to the artistic and economic potential of film, giving it a stamp of legitimacy that before had been only grudgingly offered at home. Firmly entrenched in German culture by the 1920s, cinema spent considerably less time defending its right to exist.\n\nOn the other hand, little has changed. We are still as preoccupied as ever\u2014perhaps even more\u2014with social acceleration and the dangers of distraction. If motion pictures exemplified the quick, relentless pace of modernity and the social consequences of inattentiveness, their secure place in mass culture did nothing to calm nervousness about their perceived effects. In fact, those concerns have doubled or tripled in our new century, predictably, as the emergence of new media forms seems to underline the lack of control we have over the pace of change and the technologies that come with it. The perceived effects have not changed, only the media to which they are attached. The rhetorical patterns we have seen over the course of this investigation have remained remarkably steady over the century; with each new media form we have a new round of experts decrying the infantilization of its audience while appropriating it for their own disciplinary projects. The difference between expert observers and lay spectators, as we have seen, resides in the amount of control one can wield over the technology and its product. The audience is infantilized precisely because it is perceived to surrender to the technology without maintaining a proper distance or detachment from it; the expert is able to use the same technology because he or she is able to master the medium\u2014which testifies to his or her own self-mastery as well. This dichotomy between resistance and surrender seems at first indistinguishable from middle-class ideals of self-mastery or restraint. These class ideals definitely reinforce the disciplinary distinctions, but because experts have the technology _in their hands_ , often literally exercising their control in this way, their disciplinary alignment is a more historiographically proximate explanation of their ideological stance. Hence the importance of examining these rhetorical patterns, which still persist in our media landscape today, in terms of _disciplines_ as well as class.\n\nExpert\/lay distinctions are perhaps just as intractable as class distinctions, and they can be just as blunt. Yet by implicating education, training, and disciplinary communities, they offer an obvious and clear point of entry for historical investigation. Indeed, this project has argued that a full description of the rhetorical dichotomies common to debates about media\u2014which would allow us to see their porousness and mutual dependence\u2014requires an acknowledgment of the heterogeneity of film technologies, products, and experiences. Taken as a whole, the expressions of film's potential or dangers are dynamic and conflicted\u2014a result of the many-sided quality of the products or technologies the historical agents encounter on the ground. Any given agent might be intransigent in this or that essay or speech, but the collection of voices within a discipline or community are, like the object itself, heterogeneous. The heterogeneity of object and audience is, in my opinion, best (but not exclusively) expressed through the specific institutional appropriations of film in the nontheatrical context, and if we want to understand these applications, it seems apparent that we need to understand the disciplinary agendas at play and their relationship to the application. I am convinced that the historical interplay between these heterogeneous agendas and objects holds more insight into \"what cinema is\" than our own top-down theoretical proclamations.\n\nSo the way that experts _handled_ the technology, both literally and figuratively, fills out the history of nontheatrical uses of media such as film. Production, distribution, and exhibition histories are traditionally important paths of inquiry for film and media studies, but we can also add descriptions of how experts have studied, manipulated, and adapted the image to their own ends, which reveals presumptions and expectations about their audience, their own expertise, and the role of film in their discipline. Experts have emphasized different aspects of film form in their use or study of the image: the still image is good for some tasks, the moving image is appropriate for other approaches; the detail of an image is an intriguing landscape open to discovery for some, while others (or even the same) gravitated toward the usefully abstract animated image. We are certainly aware of all these various facets of film, but we could focus more on patterns of use and their relationship to historically persistent rhetorical patterns. Investigating those varied uses also allows us to see the precise relationship between film form and disciplinary agendas. Moreover, the variety of approaches or entry points for expert analysis of the image also testifies to the ambidexterity of film, or its usefulness as both a still and moving image, among its other, varied manifestations. By emphasizing film's variety of forms and ways of \"handling\" it, film's malleability prompts what we might call a _tactile historiography_.\n\nIndeed, it was precisely film's adaptability that appealed to experts, that brought them to the idea that they could appropriate it, transform it, and make it their own. These experts sculpted cinema's _dispositif_ \u2014the technology, the image, the audience, and the relationship between all three\u2014in markedly different ways. The scientific disciplines had the most rigid framework within which motion pictures could fit; the technology was used primarily to control duration and to aid correlation. From the aesthetic debates, we can see the negotiation between the cinematic experience\u2014as varied as it was\u2014and ideals of individual contemplation. In any case, the extent to which film could be shaped according to preexisting standards and practices determined its acceptability. What was this shaping, exactly? For researchers, it involved a very literal hands-on tinkering with the technology to adapt it to standards of evidence and imaging. It also involved, as we have seen, tinkering with the object of study to adapt it to the representational technology. For someone like Lemke or H\u00e4fker, on the other hand, it sometimes meant tinkering with the technology or the screening venue, but more often corralling funders, exhibitors, town councils, and other members of the community and driving them toward a common goal. Persuading these groups required a similar negotiation between the film experience or _dispositif_ and established, often disciplinary agendas or ideals (\"modernizing education,\" for example). For essayists and aestheticians, it meant thinking through how some aspect of film form could fit established or emerging ideals. To press film into service, then, meant shaping it to an existing framework of institutional resources, policies, practices, and ideals.\n\nSo tactile historiography is sensitive to the historical impressions left on cinema (the filmic \"material\" that includes the technology, the image, and the audience). These impressions are of at least two types. On one side, we have the institutional or disciplinary framework to which film is made to fit; this is a historically specific but more or less stable configuration of disciplinary ideals, established practices, rules, policies, norms, and conventions. In education, for example, limited funding meant that elementary schools would not include film projection equipment in the classroom until the 1920s or so. So in the early 1910s educational screenings were held in commercial film theaters, which of course shaped the educational experience, sometimes objectionably. Yet teachers undoubtedly managed these educational screenings in such a way that the norms of the classroom were imported to the theater, so the use of film in this context expressed a combination of two institutional frameworks or two sets of institutional or disciplinary norms: commercial and educational.\n\nOn the other side, historical agents such as Lemke, who worked to make film fit into educational norms, left on these experiences the impression of their specific sculpting, tinkering, or negotiating. In this case, Lemke's specific adjustments in part consisted of his arrangement of the films and discussion into a program designed around Herbartian principles. A discursive capitulation to disciplinary norms in education, this move shaped the experience in a local, perhaps fleeting way, in that the audience might have been only vaguely aware of this justification, so its impression was likely not durable. Still, the effort itself led to the acceptance of film as a tool that _could_ be managed, as opposed to the earlier conception of film as totally unsuitable. This shift did indeed leave a lasting impression on the practices of educational film in Germany. Norms change, as do specific, local adjustments to them, which then incrementally extend or change the norms (or not). This historiographic approach sees film as an adaptable material, the form of which at any given historical moment expresses these norms and adjustments.\n\nYet if form matters, _how_ does it matter? How can an understanding of film form contribute to an understanding of broader historical trends, or vice versa? Usually film historians illustrate this relationship through close analyses of individual films, the best of which present analogies between textual and historical structures, as in Tom Gunning's essay on D. W. Griffith's _The Lonedale Operator_ (1909), which demonstrates the homology between the mise-en-sc\u00e8ne, the editing patterns of the film, and the emergence of new gender roles and modern forms of transportation and communication. In this approach, the historian illuminates film style and historical moment simultaneously\u2014each expresses the other. I advocate, in addition to this method, an approach that similarly demonstrates the mutually expressive relationship between disciplinary practice, ideals, and agendas, on one hand, and film form broadly speaking, on the other, which could include the specific quality of the image, the structure of the technology at the time, or the experience itself. The particular style of, say, a research film could be pertinent, but the way the technology is used in a research setting might reveal more substantial and specific patterns that help to explain the medium's use in that particular laboratory at that particular time. The style of the research film could also express these patterns of use, so that is an area worthy of further investigation, but this project has stressed the experts' own understanding of film form expressed through their patterns of use.\n\nWe must acknowledge, however, that all of these aspects of film form\u2014style, technology, image, experience, and so on\u2014are unstable and subjective; they change shape with the historical moment, of course, but what counts most as \"form\" depends on what the experts take it to be. This historiographic approach acknowledges that disciplines may see film differently than we do. Ludwig Braun, for example, saw motion pictures as an extension of serial photography; he therefore understood the technology as an image generator that could be best used to examine minute differences in the rhythmic function of rapidly moving objects such as the heart. In this case, Braun took film form to be a series of slightly different exposures, and the specific photographic quality of the image, for example, counted very little for his task. (If the technology could have generated successive line drawings of the heart just as easily, he would have been thrilled.) He took what he needed and left the rest. This unspoken selection or emphasis usually follows disciplinary logic, so this understanding of form\u2014of what film is\u2014also expresses that logic; form and discipline illuminate each other. Braun understood film to be \"incremental exposures,\" which expressed the serial logic in scientific and especially medical thinking at the time. The analogy between \"incremental exposure\" and \"series\" has explanatory power, because it demonstrates\u2014more powerfully than, for example, just listing advantages and disadvantages\u2014 _why a tool would be useful_. If the form of a wrench is crucial to understanding why it works on a bolt, the relationship between film and any given discipline is not so straightforward; analogies and correspondences can help us see how film fit an agenda.\n\nSo this historiographic strategy differs from the usual, first, in its emphasis on patterns of use rather than style. Style is important and can be helpful, but patterns of use apply to adjustments to the technology, the circulation of images, the multiple function of any given film, or the role of moving images in a selection of representational options\u2014a partial list at best. Patterns of use provide many more points of contact with the organization's goals or the discipline's logic than style alone. An analysis of style helps us understand, for example, the conventions within a given genre, which is good, but it limits us to the genre, when patterns of use take us into the heart of the discipline itself. Second, this approach emphasizes the notion of film form that arises from patterns of use. These patterns of use make sense only against a background of disciplinary problems and solutions. So the third way that this approach differs from the norm is in its emphasis on the disciplinary (as opposed to economic, technological, biographical, or aesthetic) context. This requires, as I have noted, some significant immersion into those contexts, but we would be rewarded with a deeper understanding of why this or that media technology mattered to the discipline. Having this insight into a discipline also helps us, as film historians, better argue the significance of film for that discipline. If we are determined to venture into the realm of the nontheatrical film, and we are convinced of its importance for this or that task, then we cannot back away from the possibility that film had some impact on the way an organization, community, or even discipline understood its agenda, its object, and itself. Having expertise in the discipline helps make the case for the significance of moving images on collective \"thought styles,\" in Ludwik Fleck's phrase, of which disciplinary logic is one expression.\n\nThis project has emphasized the correspondence between material and discipline. If we are interested in the mechanism of cultural legitimacy\u2014the process by which a new technology or form comes to be accepted by a group or discipline\u2014then this approach makes sense: any new form is not only understood in terms of what came before, but it must play by the rules set out by the discipline. The rules of evidence or legibility in the physical sciences, for example, are to a certain extent malleable or accommodating to new technologies, but if the technology does not offer representational solutions at least somewhat familiar with more or less agreed-upon conventions, it will not be considered useful. But if we are to explain not just legitimacy and acceptance but also the extent to which the technology _transforms_ the discipline, then we must also consider the _strangeness_ or difference the new form brings to the endeavor. This strangeness evokes wonder, as we have seen in the revelationist film theory of Bal\u00e1zs, Epstein, or Benjamin, or in the descriptions of researchers as they gaze into the microscopic sublime. This wonder may propel agents to replicate the experiments or experiences; it may reveal new vistas that shift the discipline's horizon of experience. In other words, the strangeness of the experience of film might have acted as an engine of change in a given discipline. Cell biology provides a good example; descriptions of cell \"behavior\"\u2014and the conclusions about cell life that rested upon such observations\u2014could not have existed without the temporal manipulations afforded by time-lapse cinematography. That is, the descriptions and therefore understanding of cell behavior depended on the technology's ability to match one timescale\u2014that of the cells\u2014to another, our human perception. Cell biologists really saw life through film's eye, and it changed what they thought life was; it changed their \"thought style.\" The strangeness of film transformed biological conceptions of life, even if only slightly.\n\nYet these and other transformations are incremental, not revolutionary. Hans-J\u00f6rg Rheinberger's idea of \"differential reproduction\" in experimental method is pertinent here; experiments are designed to be reproducible, to be sure, but that is not so important as the slight difference that each experiment affords. The difference between what is known and what is unexpected is usually only slight, but the gap between them is the real goal and product of experimental practice, because that is the space where new knowledge is produced. Likewise, the use of a new media technology as a tool is almost always experimental, and the difference between current understanding and the new view it provides is sometimes a surprisingly productive opening. The use of this tool is an attempt to solve a problem, and we can see over the course of the disciplinary appropriations of this tool a series of linked solutions to a common problem\u2014like a series of experiments. Looking at the history of the disciplinary uses of film helps us to understand its role in a larger series of linked solutions and to see film's incremental effect on the problem itself. It is as if Braun's method of comparing minute differences in his incremental exposures and then projecting them to get a sense of the whole in its duration could be applied to historiography: we note the slight differences in applications of film technology in a discipline, but we are able to see the cumulative effect only by running the series through the flip book of history, so to speak. Or, to apply a biological metaphor, film technology could be the mutant gene in the disciplinary organism, the true effects of which we notice only after a generation or two.\n\nSo this project has envisioned film's _dispositif_ \u2014its technology, its image, its audience\u2014as material in a grand, cross-disciplinary series of experiments. Experts shaped this \"material,\" adjusted or trimmed it to fit a set of needs, but the crucial difference that film provided to these disciplines came from its resistance to these efforts. The material was not infinitely malleable. The technology could be made to do only so much, the image was only so informative, the audience only so agreeable. We see this resistance in the strangeness of the view to the experts, but also in its inappropriate fit; some disciplines, such as geology, found little use for it. We see it also in the excess or residue, such as pleasure or thrill, which spills sloppily over the disciplinary framework; pleasure continues to be difficult to fit even into aesthetics. Expert vision wrestles with this remainder, even as it depends on it. Aesthetic contemplation, for example, is at once surrender and mastery; the oscillation between them relies on the irreducibly material, sensual, and singular character of the artwork, which is the basis for immersion and pleasure, but also the ground of its resistance to rationalization and abstraction\u2014both of which are also part of the aesthetic experience. Similarly, the filmic _dispositif_ as \"material\" partly resists expert efforts to sculpt it; this is especially true when it comes to the audience, which has been subject to much rationalization and abstraction. The \"shape of spectatorship\" in this regard was not just the negative outline or boundary with expert vision but the product of expert modeling. The form spectatorship has taken depended on the discursive and practical molding of experts but also on the resistance of the audience and cinema itself to this kneading and on the spectators' willingness to take the experiment to places beyond which they were prodded.\nNOTES\n\nINTRODUCTION\n\n 1. Adolf Sellmann, \"Das Geheimnis des Kinos,\" _Bild und Film_ 1, nos. 3\u20134 (1912): 65\u201367, here 65. All translations are my own unless otherwise noted.\n\n 2. The scholarship on \"useful film\" and educational film is growing. See, e.g., the special issues on \"Gebrauchsfilm\" in _montage\/AV: Zeitschrift f\u00fcr Theorie & Geschichte audiovisueller Kommunikation_ 14, no. 2 (2005), and no. 3 (2006); Vinzenz Hediger and Patrick Vonderau, eds., _Films That Work: Industrial Film and the Productivity of Media_ (Amsterdam: Amsterdam University Press, 2009); Charles R. Acland and Haidee Wasson, eds., _Useful Cinema_ (Durham, N.C.: Duke University Press, 2011); and Devin Orgeron, Marsha Orgeron, and Dan Streible, eds., _Learning with the Lights Out_ (New York: Oxford University Press, 2012).\n\n 3. Max Weber, \"Science as a Vocation,\" in _The Vocation Lectures_ , ed. and with an introduction by David Owen and Tracy B. Strong, trans. Rodney Livingstone (Indianapolis: Hackett, 2004), 1\u201331, here 7 (emphasis in original). This and the following paragraph have been adapted from my essay, \"Science Lessons,\" _Film History_ 25, nos. 1\u20132 (2013): 45\u201354.\n\n 4. See Norbert Bolz, _Am Ende der Gutenberg-Galaxis. Die neuen Kommunikationsverh\u00e4ltnisse_ (Munich: Fink, 1993), 115, or chap. 3; or Thomas Elsaesser, \"Die Stadt von Morgen: Filme zum Bauen und Wohnen in der Weimarer Republik,\" in _Geschichte des dokumentarischen Film in Deutschland, Band 2: Weimarer Republik (1918\u20131933)_ , ed. Klaus Kreimeier, Antje Ehmann, and Jeanpaul Goergen (Stuttgart: Reclam-Verlag, 2005), 381\u2013410; as well as Vinzenz Hediger and Patrick Vonderau, \"Record, Rhetoric, Rationalization: Industrial Organization and Film,\" in Hediger and Vonderau, _Films That Work_ , 35\u201349; and Petr Szczepanik, \"Modernism, Industry, Film: A Network of Media in the Ba\u0165a Corporation and the Town of Zl\u00edn in the 1930s,\" in Hediger and Vonderau, _Films That Work_ , 349\u2013376.\n\n 5. On experimental systems, see Hans-J\u00f6rg Rheinberger, _Toward a History of Epistemic Things: Synthesizing Proteins in the Test Tube_ (Stanford, Calif.: Stanford University Press, 1997).\n\n 6. Ludwik Fleck, \"Some Specific Features of the Medical Way of Thinking [1927],\" in _Cognition and Fact: Materials on Ludwik Fleck_ , ed. Robert S. Cohen and Thomas Schnelle (Dordrecht, Netherlands, and Boston: Reidel, 1986), 39\u201346.\n\n 7. Ludwik Fleck, \"Scientific Observation and Perception in General [1935],\" in Cohen and Schnelle, _Cognition and Fact_ , 59\u201378, here 60.\n\n 8. Fleck, \"Scientific Observation and Perception in General [1935],\" 61.\n\n 9. Sir James Paget, \"An Address on the Utility of Scientific Work in Practice,\" _British Medical Journal_ (15 October 1887): 811\u2013814, here 811.\n\n 10. Johann Heinrich Pestalozzi, _How Gertrude Teaches Her Children_ [1801], ed. Ebenezer Cooke, trans. Lucy E. Holland and Frances C. Turner, 2d ed. (Syracuse, N.Y.: Bardeen, 1898), tenth letter, 220 (emphasis in original).\n\n 11. Ludwik Fleck, \"To Look, To See, To Know [1947],\" in Cohen and Schnelle, _Cognition and Fact_ , 129\u2013151, here 137.\n\n 12. Robert Vischer, _On the Optical Sense of Form_ [1873], in _Empathy, Form, and Space: Problems in German Aesthetics, 1873\u20131893_ , ed. and trans. Harry Francis Mallgrave and Eleftherios Ikonomou (Santa Monica, Calif.: Getty Center for the History of Art and the Humanities, 1994), 89\u2013124, esp. 93\u201395. In art history, see Norman Bryson's discussion of the terms in _Vision and Painting: The Logic of the Gaze_ (New Haven: Yale University Press, 1983). For the use of \"gaze\" and \"glance\" in media studies, see John Ellis, _Visible Fictions: Cinema, Television, Video_ (London: Routledge, 1982); and Timothy Corrigan, _A Cinema Without Walls: Movies and Culture After Vietnam_ (New Brunswick, N.J.: Rutgers University Press, 1991).\n\n 13. Michael Hau, \"The Holistic Gaze in German Medicine, 1890\u20131930,\" _Bulletin of the History of Medicine_ 74, no. 3 (Fall 2000): 495\u2013524.\n\n 14. Michel Foucault, _The Birth of the Clinic: An Archaeology of Medical Perception_ , trans. A. M. Sheridan Smith (New York: Vintage, 1973), 109.\n\n 15. See William Egginton, \"Intimacy and Anonymity, or How the Audience Became a Crowd,\" in _Crowds_ , ed. Jeffrey T. Schnapp and Matthew Tiews (Stanford, Calif.: Stanford University Press, 2006), 97\u2013110.\n\n 16. \"The Work of Art in the Age of Its Technological Reproducibility (Second Version),\" in _Walter Benjamin: Selected Writings_ , vol. 3, _1935\u20131938_ , ed. Howard Eiland and Michael W. Jennings, trans. Edmund Jephcott, Howard Eiland, and others (Cambridge, Mass.: Belknap, 2002), 101\u2013136, here 116.\n\n 17. Benjamin, \"The Work of Art in the Age of Its Technological Reproducibility (Second Version),\" 104.\n\n 18. Representative interventions in the debate about the role of modernity in a history of film style include the collection _Cinema and the Invention of Modern Life_ , ed. Leo Charney and Vanessa Schwartz (Berkeley: University of California Press, 1995); David Bordwell, _On the History of Film Style_ (Cambridge, Mass.: Harvard University Press, 1997), 141\u201346; Noel Carroll, \"Modernity and the Plasticity of Perception,\" _Journal of Aesthetics and Art Criticism_ 59, no. 1 (Winter 2001): 11\u201318; Ben Singer, _Melodrama and Modernity: Early Sensational Cinema and Its Contexts_ (New York: Columbia University Press, 2001); Charlie Keil, \"'To Here from Modernity': Style, Historiography, and Transitional Cinema,\" in _American Cinema_ ' _s Transitional Era: Audiences, Institutions, Practices_ , ed. Charlie Keil and Shelley Stamp (Berkeley: University of California Press, 2004), 51\u201365; Tom Gunning, \"Modernity and Cinema: A Culture of Shocks and Flows,\" in _Cinema and Modernity_ , ed. Murray Pomerance (New Brunswick, N.J.: Rutgers University Press, 2006), 297\u2013315; and Frank Kessler, \"Viewing Change, Changing Views: The 'History of Vision' Debate,\" in _Film 1900: Technology, Perception, Culture_ , ed. Annemone Ligensa and Klaus Kreimeier (New Barnet, U.K.: Libbey, 2009), 23\u201335.\n\n 19. Among many other surveys discussed in chap. 4, see esp. Sabine Hake, _The Cinema_ ' _s Third Machine: Writing on Film in Germany, 1907\u20131933_ (Lincoln: University of Nebraska Press, 1993).\n\n 20. Andr\u00e9 Gaudreault, _Film and Attraction: From Kinematography to Cinema_ , trans. Timothy Barnard (Urbana: University of Illinois Press, 2011).\n\n 21. See, e.g., Kevin Repp, _Reformers, Critics, and the Paths of German Modernity: Anti-Politics and the Search for Alternatives, 1890\u20131914_ (Cambridge, Mass.: Harvard University Press, 2000); Andrew Lees, _Cities, Sin, and Social Reform in Imperial Germany_ (Ann Arbor: University of Michigan Press, 2002); and Dennis Sweeney, \"Reconsidering the Modernity Paradigm: Reform Movements, the Social and the State in Wilhelmine Germany,\" _Social History_ 31, no. 4 (November 2006): 405\u2013434.\n\n 22. Ben Singer names this ambivalence \"ambimodernity\" and brings it to the attention of the film studies community in \"The Ambimodernity of Early Cinema: Problems and Paradoxes in the Film-and-Modernity Discourse,\" in Ligensa and Kreimeier, _Film 1900_ , 37\u201351.\n\n1. SCIENCE'S CINEMATIC METHOD\n\n 1. Charles Cros, \"Inscription,\" in _Oeuvres completes_ , ed. Louis Forestier and Pascal Pia (Paris: Pauvert, 1964), 135\u2013136.\n\n 2. On the graphic method, see Merriley Borell, \"Extending the Senses: The Graphic Method,\" _Medical Heritage_ 2, no. 2 (March\/April 1986): 114\u2013121; Robert G. Frank Jr., \"The Telltale Heart: Physiological Instruments, Graphic Methods, and Clinical Hopes, 1854\u20131914,\" in _The Investigative Enterprise: Experimental Physiology in Nineteenth-Century Medicine_ , ed. William Coleman and Frederic L. Holmes (Berkeley: University of California Press, 1988), 211\u2013290; Soraya de Chadarevian, \"Graphical Method and Discipline: Self-Recording in Nineteenth-Century Physiology,\" _Studies in History and Philosophy of Science_ 24, no. 2 (June 1993): 267\u2013291; and Robert M. Brain, \"Representation on the Line: Graphic Recording Instruments and Scientific Modernism,\" in _From Energy to Information: Representation in Science and Technology, Art, and Literature_ , ed. Bruce Clarke and Linda Dalrymple Henderson (Stanford, Calif.: Stanford University Press, 2002), 155\u2013177.\n\n 3. Wilhelm Braune and Otto Fischer, _The Human Gait_ , trans. Paul Maquet and Ronald Furlong (Berlin and New York: Springer, 1987). In addition to _The Human Gait_ , Braune and Fischer's human motion studies included _Das Gesetz der Bewegungen an der Basis der mittleren Finger und im Handgelenk des Menschen_ (Leipzig, 1887); _\u00dcber den Schwerpunkt des menschlichen K\u00f6rpers mit R\u00fccksicht auf die Ausr\u00fcstung des deutschen Infanteristen_ (Leipzig, 1889), translated as _On the Centre of Gravity of the Human Body as Related to the Equipment of the German Infantry Soldier_ by Paul Maquet and Ronald Furlong (Berlin and New York: Springer, 1985); _Bestimmung der Tr\u00e4gheitsmomente des menschlichen Koerpers und seiner Glieder_ (Leipzig, 1892), translated as _Determination of the Moments of Inertia of the Human Body and Its Limbs_ by Paul Maquet and Ronald Furlong (Berlin and New York: Springer, 1988); and Fischer's _Theoretische Grundlagen f\u00fcr eine Mechanik der lebenden K\u00f6rper_ (Leipzig, 1906).\n\n 4. Max Seddig, \"Ueber Abh\u00e4ngigkeit der Brown'schen Molekularbewegung von der Temperatur,\" _Sitzungsberichte der Gesellschaft zur Bef\u00f6rderung der gesammten Naturwissenschaften zu Marburg_ 18 (1907): 182\u2013188; Seddig, \"\u00dcber die Messung der Temperaturabh\u00e4ngigkeit der Brownschen Molekularbewegung,\" _Physikalische Zeitschrift_ 9, no. 14 (15 July 1908): 465\u2013468; Seddig, \"Messung der Temperatur-Abh\u00e4ngigkeit der Brown'schen Molekularbewegung,\" Habilitationsschrift, Akademie in Frankfurt a. M., 1909; Seddig, \"Exacte Messung des Zeitintervalles bei kinematographischen Aufnahmen,\" _Jahrbuch f\u00fcr Photographie und Reproduktionstechnik_ 26 (1912): 654\u2013657; and Seddig, \"Messung der Temperatur-Abh\u00e4ngigkeit der Brown-Zsigmondyschen Bewegung,\" _Zeitschrift f\u00fcr Anorganische Chemie_ 73\u201374 (1912): 360\u2013384.\n\n 5. Hermann Braus, \"Mikro-Kino-Projektionen von in vitro gez\u00fcchteten Organanlagen,\" _Verhandlungen der Gesellschaft deutscher Naturforscher und \u00c4rzte_ 83, part 2 (1911): 472\u2013475.\n\n 6. For a fascinating account of cinema's relationship to time and science, see Mary Ann Doane, _The Emergence of Cinematic Time: Modernity, Contingency, the Archive_ (Cambridge, Mass.: Harvard University Press, 2002).\n\n 7. Examinations of the relationship between illustrative materials and the agendas of scientists are increasingly popular in the history of science; a good, representative example is Martin Kemp, \"Temples of the Body and Temples of the Cosmos: Vision and Visualization in the Vesalian and Copernican Revolutions,\" in _Picturing Knowledge: Historical and Philosophical Problems Concerning the Use of Art in Science_ , ed. Brian S. Baigrie (Toronto and Buffalo, N.Y.: University of Toronto Press, 1996), 40\u201384. Two especially influential collections are Michael Lynch and Steve Woolgar, eds., _Representation in Scientific Practice_ (Cambridge, Mass.: MIT Press, 1990); and Caroline A. Jones and Peter Galison, eds., _Picturing Science, Producing Art_ (New York: Routledge, 1998). Catelijne Coopmans, Janet Vertesi, Michael E. Lynch, and Steve Woolgar, eds., _Representation in Scientific Practice Revisited_ (Cambridge, Mass.: MIT Press, 2014) is an excellent recent collection.\n\n 8. For more on the relationship between motion pictures and scientific experiment, see the section on \"The Multiple Functions of the Medical Film\" in chap. 2 in this volume.\n\n 9. On experimental systems and \"dislocation\" or \"differential reproduction,\" see Hans-J\u00f6rg Rheinberger, _Toward a History of Epistemic Things: Synthesizing Proteins in the Test Tube_ (Stanford, Calif.: Stanford University Press, 1997), esp. chap. 5. Generally speaking, experimental systems are designed to produce incremental differences, which ultimately produce new inquiries and systems. But the balance between old and new is not symmetrical; my theory of disciplinary appropriation accommodates both the _correspondence_ between disciplinary ideals and film and the _difference_ between them, but the history I tell in this book favors the former. For more on the latter, see the conclusion in this volume.\n\n 10. Gaston Bachelard, _The New Scientific Spirit_ , trans. Arthur Goldhammer (Boston: Beacon, 1984), 13. See also Davis Baird, _Thing Knowledge: A Philosophy of Scientific Instruments_ (Berkeley: University of California Press, 2004).\n\n 11. Rheinberger, _Toward a History of Epistemic Things_ , 21.\n\n 12. Ian Hacking, _Representing and Intervening: Introductory Topics in the Philosophy of Natural Science_ (Cambridge: Cambridge University Press, 1983).\n\n 13. For surveys of scientific uses of photography and film, see Karl Wilhelm Wolf-Czapek, ed., _Angewandte Photographie in Wissenschaft und Technik_ (Berlin: Union Deutsche Verlagsgesellschaft, 1911); Martin Weiser, _Medizinische Kinematographie_ (Dresden and Leipzig: Steinkopff, 1919); F. Paul Liesegang, _Wissenschaftliche Kinematographie_ (D\u00fcsseldorf: Liesegang, 1920); Anthony R. Michaelis, _Research Films in Biology, Anthropology, Psychology, and Medicine_ (New York: Academic, 1955); Virgilio Tosi, _Cinema Before Cinema: The Origins of Scientific Cinematography_ , trans. Sergio Angelini (London: British Universities Film & Video Council, 2005); Timothy Boon, _Films of Fact: A History of Science in Documentary Films and Television_ (New York: Wallflower, 2008); and Kelly Wilder, _Photography and Science_ (London: Reaktion, 2009).\n\n 14. The literature on this transition between chronophotography and film is vast, but one place to start is Deac Rossell, _Living Pictures: The Origins of the Movies_ (Albany: State University of New York Press, 1998). On Janssen, see esp. the work of Jimena Canales, who describes the history of Janssen's photographic revolver in the context of emerging cinematographic forms of representation in the following: \"Photogenic Venus: The 'Cinematographic Turn' in Science and Its Alternatives,\" _Isis_ 93 (2002): 585\u2013613; \"Sensational Differences: The Case of the Transit of Venus,\" _Cahiers Fran\u00e7ois Vi\u00e8te_ 1, nos. 11\/12 (September 2007): 15\u201340; _A Tenth of a Second: A History_ (Chicago: University of Chicago Press, 2009), 87\u2013115; and \"Desired Machines: Cinema and the World in Its Own Image,\" _Science in Context_ 24, no. 3 (September 2011): 329\u2013359.\n\n 15. Jennifer Tucker, _Nature Exposed: Photography as Eyewitness in Victorian Science_ (Baltimore, Md.: Johns Hopkins University Press, 2005); or Tucker, \"Photography as Witness, Detective, and Imposter: Visual Representation in Victorian Science,\" in _Victorian Science in Context_ , ed. Bernard Lightman (Chicago: University of Chicago Press, 1997), 378\u2013408; see also Joel Snyder, \" _Res Ipsa Loquitur_ ,\" in _Things That Talk: Object Lessons from Art and Science_ , ed. Lorraine Daston (New York: Zone, 2004), 195\u2013221; as well as Wilder, _Photography and Science_.\n\n 16. An excellent discussion of one scientist's dissatisfaction with photography can be found in Sarah de Rijcke, \"Drawing Into Abstraction. Practices of Observation and Visualisation in the Work of Santiago Ram\u00f3n y Cajal.\" _Interdisciplinary Science Reviews_ 33, no. 4 (2008): 287\u2013311. Ram\u00f3n y Cajal found that photography could not capture the three-dimensionality of nerve cells as well as drawings.\n\n 17. Thomas Schlich, \"'Wichtiger als der Gegenstand selbst': Die Bedeutung des fotografischen Bildes in der Begr\u00fcndung der bakteriologischen Krankheitsauffassung durch Robert Koch,\" in _Neue Wege in der Seuchengeschichte_ , ed. Martin Dinges and Thomas Schlich (Stuttgart: Steiner, 1995), 143\u2013174. See also Olaf Breidbach, \"Representation of the Microcosm: The Claim for Objectivity in 19th Century Scientific Microphotography,\" _Journal of the History of Biology_ 35 (2002): 221\u2013250.\n\n 18. On the similarities between the graphic method and cinema, see Lisa Cartwright, \"'Experiments of Destruction': Cinematic Inscriptions of Physiology,\" _Representations_ 40 (Fall 1992): 129\u2013152; and Cartwright, _Screening the Body: Tracing Medicine_ ' _s Visual Culture_ (Minneapolis: University of Minnesota Press, 1995).\n\n 19. On Germany's research infrastructure, see, e.g., Margit Sz\u00f6ll\u00f6si-Janze, \"Science and Social Space: Transformations in the Institutions of _Wissenschaft_ from the Wilhelmine Empire to the Weimar Republic,\" _Minerva_ 43 (2005): 339\u2013360.\n\n 20. Carl Cranz, \"\u00dcber einen ballistischen Kinematographen,\" _Deutscher Mechaniker-Zeitung_ 18 (15 September 1909): 173\u2013177. See also P. W. W. Fuller, \"Carl Cranz, His Contemporaries, and High-Speed Photography,\" _Proceedings of SPIE_ , no. 5580, 26th International Congress on High-Speed Photography and Photonics (25 March 2005): 250\u2013260; and Wilhelm Pfeffer, \"Die Anwendung des Projectionsapparates zur Demonstration von Lebensvorg\u00e4ngen,\" _Jahrb\u00fccher wissenschaftliche Botanik_ 35 (1900): 711\u2013745. On Pfeffer, see esp. Oliver Gaycken, \"'The Swarming of Life': Moving Images, Education, and Views Through the Microscope,\" _Science in Context_ 24, no. 3 (September 2011): 361\u2013380; and Gaycken, \"The Secret Life of Plants: Visualizing Vegetative Movement, 1880\u20131903,\" _Early Popular Visual Culture_ 10, no. 1 (2012): 51\u201369.\n\n 21. The best overview of Comandon's life and work is B\u00e9atrice de Pastre and Thierry Lefebvre, eds., _Filmer la science, comprendre la vie: Le cinema de Jean Comandon_ (Paris: Centre national de la cin\u00e9matographie, 2012).\n\n 22. On Messter, see Christian Ilgner and Dietmar Linke, \"Filmtechnik: Vom Malteserkreuz zum Panzerkino,\" in _Oskar Messter: Filmpioneer der Kaiserzeit_ , ed. Martin Loiperdinger (Basel and Frankfurt: Stroemfeld\/Roter Stern, 1994), 93\u2013134, esp. 128\u2013134; as well as Frank Kessler, Sabine Lenk, and Martin Loiperdinger, eds., _Oskar Messter_ \u2014 _Erfinder und Gesch\u00e4ftsmann_ , KINtop Schriften 3 (Basel and Frankfurt: Stoemfeld\/Roter Stern, 1994). On Ernemann, see, e.g., the nod to the manufacturer in Hans Hennes,\"Die Kinematographie der Bewegungsst\u00f6rungen,\" _Die Umschau_ 15, no. 29 (1911): 605\u2013606; as well as Hanns G\u00fcnther, \"Mikrokinematographische Aufnahmeapparate,\" _Film and Lichtbild_ 1, no. 1 (1912): 4\u20136; 1, no. 2 (1912): 13\u201314.\n\n 23. On popular scientific cinema, see Thierry Lefebvre, \"The Scientia Production (1911\u20131914): Scientific Popularization Through Pictures,\" _Griffithiana_ no. 47 (May 1993): 137\u2013153; Oliver Gaycken, \"'A Drama Unites Them in a Fight to the Death': Some Remarks on the Flourishing of a Cinema of Scientific Vernacularization in France, 1909\u20131914,\" _Historical Journal of Film, Radio and Television_ 22, no. 3 (2002): 353\u2013374; and Gaycken, _Devices of Curiosity: Early Cinema and Popular Science_ (Oxford and New York: Oxford University Press, 2015). For an excellent account of popular science films in the United Kingdom, see Boon, _Films of Fact_. On the entwinement of scientific experiment, projection, popular science, and Victorian physics, see Simon Schaffer, \"Transport Phenomena: Space and Visibility in Victorian Physics,\" _Early Popular Visual Culture_ 10, no. 1 (February 2012): 71\u201391. On popularization in science in general, see Roger Cooter and Stephen Pumfrey, \"Separate Spheres and Public Places: Reflections on the History of Science Popularization and Science in Popular Culture,\" _History of Science_ 32, no. 97 (1994): 237\u2013267.\n\n 24. Readers were asked to write the editor for details; see \"Verzeichnis wissenschaftlich und technisch wertvoller Films,\" _Film und Lichtbild_ 1, no. 2 (1912): 16; or \"An unsere Abonnenten!\" _Film und Lichtbild_ 2, no. 5 (1913): 84.\n\n 25. For reports of screenings, see \"Wissenschaft und Lichtspiele,\" _Bild und Film_ 1, no. 2 (1912): 49\u201350; \"Kino und Wissenschaft,\" _Bild und Film_ 1, no. 2 (1912): 55; W. Thielemann, \"Kinematographie und biologische Forschung,\" _Bild und Film_ 3, no. 7 (1913\/1914): 171\u2013172; and \"Wissenschaftliche Abende,\" _Film und Lichtbild_ 1, no. 3 (1912): 30\u201331.\n\n 26. \"Ein neues wissenschaftliches Kino,\" _Film und Lichtbild_ 1, no. 5 (1912): 62. It is possible that the Fata Morgana was the only one of its kind.\n\n 27. See Thierry Lefebvre, _La chair et le celluloid: Le cin\u00e9ma chirurgical du Docteur Doyen_ (Brionne: Jean Doyen \u00e9diteur, 2004), for a discussion of the controversy surrounding the theft and possible unauthorized exhibition of Parisian doctor Eug\u00e8ne-Louis Doyen's surgical films in the early 1900s. See also chap. 2 in this volume.\n\n 28. For example, Wilhelm Richter, a Berlin teacher and school reformer, often wrote on scientific cinema in the popular press, cheering all efforts to bring new views to public perception. See \"Der Kinematograph als naturwissenschaftliches Anschauungsmittel,\" _Naturwissenschaftliche Wochenschrift_ 12, no. 52 (28 December 1913): 817\u2013820.\n\n 29. On the use of film in university teaching, see Franz Bergmann, \"Der Kinematograph im Hochschulunterricht,\" _Bild und Film_ 2, no. 2 (1912\/1913): 48; Wilhelm Richter, \"Hochschulkinematographie,\" _Bild und Film_ 2, nos. 11\/12 (1912\/1913): 253\u2013257; and L. Segmiller, \"Das Skizzieren nach Lichtbildern bei Tageslicht und k\u00fcnstlicher Beleuchtung,\" _Film und Lichtbild_ 1, no. 4 (1912): 35\u201339.\n\n 30. One survey of the contemporary use of microcinematography notes, \"Films of the latter [Ernst Sommerfeldt] are available commercially from Ernemann and depict crystallographic phenomenon,\" so apparently whether a research film made it into theaters depended on both the researcher's willingness and the manufacturer's evaluation of its popular appeal. On this example, see Ernst Sommerfeldt, \"\u00dcber fl\u00fcssige und scheinbar lebende Kristalle; mit kinematographischen Projektionen,\" _Verhandlungen der Gesellschaft deutscher Naturforscher und \u00c4rzte_ 79, part 2 (1907): 202. The survey of microcinematography is Engelhard Wychgram, \"Aus optischen und mechanischen Werkst\u00e4tten IV,\" _Zeitschrift f\u00fcr wissenschaftliche Mikroskopie und f\u00fcr mikroskopishe Technik_ 28 (1911): 337\u2013361, esp. 351\u2013361.\n\n 31. On \"mechanical objectivity,\" see Lorraine Daston and Peter Galison, \"The Image of Objectivity,\" _Representations_ 40 (Fall 1992): 81\u2013128; and Daston and Galison, _Objectivity_ (New York: Zone, 2007).\n\n 32. On the difference between instruments for experimentation and for demonstration, see Thomas L. Hankins and Robert J. Silverman, \"The Magic Lantern and the Art of Demonstration,\" in _Instruments and the Imagination_ (Princeton, N.J.: Princeton University Press, 1995), 37\u201371.\n\n 33. Bruno Latour, \"Visualization and Cognition: Thinking with Eyes and Hands,\" _Knowledge and Society: Studies in the Sociology of Culture Past and Present_ 6 (1986): 1\u201340, here 5. Brian Winston has also considered Latour's work in relation to film: \"The Documentary Film as Scientific Inscription,\" in _Theorizing Documentary_ , ed. Michael Renov (London and New York: Routledge, 1993), 37\u201357.\n\n 34. Latour, \"Visualization and Cognition,\" 7 (emphasis in original).\n\n 35. See esp. Nicolas Rasmussen, _Picture Control: The Electron Microscope and the Transformation of Biology in America, 1940\u20131960_ (Stanford, Calif.: Stanford University Press, 1997); Bernike Pasveer, \"Knowledge of Shadows: The Introduction of X-Ray Images in Medicine,\" _Sociology of Health and Illness_ 11, no. 4 (December 1989): 361\u2013381; and Edward Yoxen, \"Seeing with Sound: A Study of the Development of Medical Images,\" in _The Social Construction of Technological Systems: New Directions in the Sociology and History of Technology_ , ed. Wiebe E. Bijker, Thomas P. Hughes, and Trevor J. Pinch (Cambridge, Mass.: MIT Press, 1987), 281\u2013303.\n\n 36. Anne Harrington, _Reenchanted Science: Holism in German Culture from Wilhelm II to Hitler_ (Princeton, N.J.: Princeton University Press, 1996). For an account of earlier debates about mechanistic approaches in science, see Peter Hanns Reill, _Vitalizing Nature in the Enlightenment_ (Berkeley: University of California Press, 2005).\n\n 37. Henri Bergson, _Creative Evolution_ , trans. Arthur Mitchell (New York: Random House, 1944). Hereafter cited parenthetically. Bergson indicates in the opening footnote of chap. 4 that he began thinking about science and film during his 1902\u20131903 course on the \"History of the Idea of Time\" at the Coll\u00e8ge de France.\n\n 38. The mechanistic approach to biology was represented in Germany by such \"biophysicists\" as Hermann von Helmholtz and Emil Du Bois-Reymond, who hoped to demonstrate that life operated under the same physical and chemical laws as other phenomena. See Helmholtz's _\u00dcber die Erhaltung der Kraft_ (Berlin: Reimer, 1847) for a comparison of muscles and mechanics, or Du Bois-Reymond's _Untersuchungen \u00fcber thierische Elektricit\u00e4t_ (Berlin: Reimer, 1848\u20131884) for an exploration of life's basic physical forces.\n\n 39. Henri Bergson, _An Introduction to Metaphysics_ , trans. T. E. Hulme (New York: Macmillan, 1955), 52.\n\n 40. Bergson, _Metaphysics_ , 51.\n\n 41. At this point we should note the influence of Bergson's philosophy on film theory from the work of Jean Epstein to Gilles Deleuze. For Epstein, see representative selections in _French Film Theory and Criticism_ , vols. 1 and 2, ed. Richard Abel (Princeton, N.J.: Princeton University Press, 1988); for Deleuze, see _Cinema 1: The Movement-Image_ and _Cinema 2: The Time-Image_ , trans. Hugh Tomlinson and Barbara Habberjam (Minneapolis: University of Minnesota, 1986 and 1989).\n\n 42. Michel Georges-Michel, \"Henri Bergson nous parle au cin\u00e9ma,\" _Le Journal_ (20 February 1914): 7; translated by Louis-Georges Schwartz as \"Henri Bergson Talks to Us About Cinema,\" _Cinema Journal_ 50, no. 3 (Spring 2011): 79\u201382, here 81. See also Frank Kessler's German translation and discussion of this article: \"Henri Bergson und die Kinematographie,\" _KINtop_ 12 (2003): 12\u201316.\n\n 43. I want to thank Paula Amad, whose doctoral dissertation \"Archiving the Everyday: A Topos in French Film History, 1895\u20131931\" (University of Chicago, 2002) led me in this direction. See also Amad's _Counter-Archive: Film, The Everyday, and Albert Kahn's_ Archives de la Plan\u00e8te (New York: Columbia University Press, 2010). For more on Bergson's understanding of movement, see Jimena Canales, \"Movement Before Cinematography: The High-Speed Qualities of Sentiment,\" _Journal of Visual Culture_ 5, no. 3 (December 2006): 275\u2013294.\n\n 44. Walter Benjamin, \"On Some Motifs in Baudelaire,\" in _Walter Benjamin: Selected Writings_ , vol. 4, _1938\u20131940_ , ed. Howard Eiland and Michael W. Jennings, trans. Edmund Jephcott and others (Cambridge, Mass.: Belknap, 2003), 313\u2013355, here 314.\n\n 45. This is not to say that Bergson had no effect in Germany. Georg Simmel, e.g., was impressed and influenced by Bergson. See Gregor Fitzi, _Soziale Erfahrung und Lebensphilosophie. Georg Simmels Beziehung zu Henri Bergson_ (Konstanz, Germany: UVK Verlagsgesellschaft, 2002). For Bergson's reception by such thinkers as Max Scheler, Roman Ingarden, Hans Driesch, Max Horkeimer, and others, see Rudolf W. Meyer, \"Bergson in Deutschland. Unter besonderer Ber\u00fccksichtigung seiner Zeitauffassung,\" in _Studien zum Zeitproblem in der Philosophie des 20. Jahrhunderts_ , ed. Rudolf W. Meyer et al. (Munich: Alber, 1982), 10\u201364; and G\u00fcnther Pflug, \"Die Bergson-Rezeption in Deutschland,\" _Zeitschrift f\u00fcr philosophische Forschung_ 45, no. 2 (April\u2013June 1991): 257\u2013266.\n\n 46. Cited in Michael Ermarth, _Wilhelm Dilthey: The Critique of Historical Reason_ (Chicago: University of Chicago Press, 1978), 24.\n\n 47. Harrington, _Reenchanted Science_ , 27.\n\n 48. Wilhelm Dilthey, \"The Construction of the Historical World in the Human Studies,\" in _Selected Writings_ , ed., trans., and with an introduction by H. P. Rickman (Cambridge: Cambridge University Press, 1976), 168\u2013245, here 181.\n\n 49. My discussion of the science of work is indebted to Anson Rabinbach, _The Human Motor: Energy, Fatigue, and the Origins of Modernity_ (Berkeley and Los Angeles: University of California Press, 1990). Hereafter cited parenthetically. Closely related to this idea of fatigue was the modern notion of \"nervousness\": see Andreas Killen, _Berlin Electropolis: Shock, Nerves, and German Modernity_ (Berkeley: University of California Press, 2006).\n\n 50. In addition to Harrington's _Reenchanted Science_ , a good overview of the debate, from the vitalist's point of view, is Frederick Burwick and Paul Douglass, eds., _The Crisis in Modernism: Bergson and the Vitalist Controversy_ (Cambridge: Cambridge University Press, 1992). With regard to cinema and vitalism, see also Inga Pollmann, \"Cinematic Vitalism: Theories of Life and the Moving Image\" (PhD diss., University of Chicago, 2011).\n\n 51. Rabinbach, _Human Motor_ , 181; see also Ernest Solvay, _Notes sur le productivisme et le comptabilisme_ (Brussels: Misch and Thron, 1900) 2: 323. On the institute, see Daniel Warnotte, _Ernest Solvay et l_ ' _Institut de Sociologie: Contribution \u00e0 l_ ' _histoire de l_ ' _\u00e9nerg\u00e9tique sociale_ (Brussels: Bruylant, 1946).\n\n 52. Rabinbach, _Human Motor_ , 134; see also Angelo Mosso, _Fatigue_ , trans. Margaret Drummond and W. B. Drummond (New York: Putnam, 1904); and Karen J. Fleckenstein, \"The Mosso Plethysmograph in 19th-Century Physiology,\" _Medical Instrumentation_ 18, no. 6 (November\u2013December 1984): 330\u2013331.\n\n 53. See Leo Koenigsberger, _Hermann von Helmholtz_ (Braunschweig, Germany: Viewig, 1911); as well as David Cahan, ed., _Hermann von Helmholtz and the Foundations of Nineteenth-Century Science_ (Berkeley and Los Angeles: University of California Press, 1993).\n\n 54. See especially Greg Myers, \"Nineteenth-Century Popularizations of Thermodynamics and the Rhetoric of Social Prophecy,\" in _Energy & Entropy: Science and Culture in Victorian Britain_, ed. Patrick Brantlinger (Bloomington: Indiana University Press, 1988), 307\u2013338; Ed Block Jr., \"T. H. Huxley's Rhetoric and the Popularization of Victorian Scientific Ideas, 1854\u20131874,\" in Brantlinger, _Energy & Entropy_, 205\u2013228; and _Degeneration: The Dark Side of Progress_ , ed. J. Edward Chamberlin and Sander L. Gilman (New York: Columbia University Press, 1985). The concept of degeneration will receive much more attention in chap. 2 of this volume.\n\n 55. Recent studies of Marey include Marta Braun, _Picturing Time: The Work of \u00c9tienne-Jules Marey_ (Chicago: University of Chicago Press, 1992); and Francois Dagognet, _\u00c9tienne-Jules Marey: A Passion for the Trace_ , trans. Robert Galeta with Jeanine Herman (New York: Zone, 1992). See also Marey's _La methode graphique dans les sciences \u00e9xperimentales et principlement en physiologie et en m\u00e9decine_ (Paris: Masson, 1885); _Movement_ , trans. Eric Pritchard (London: Heinemann, 1895); and _La chronophotographie_ (Paris: Gauthier-Villars, 1899).\n\n 56. \u00c9tienne-Jules Marey, \"\u00c9tudes sur la marche de l'homme,\" _Revue Militaire de M\u00e9decine et de Chirurgie_ 1 (1880): 244\u201346, cited in Rabinbach, _Human Motor_ , 116.\n\n 57. For the culmination of this approach, see Frederick Winslow Taylor, _The Principles of Scientific Management_ (1911; New York: Norton, 1947); and Frank B. Gilbreth, _Motion Study: A Method for Increasing the Efficiency of the Workman_ (New York: Van Nostrand, 1911). On the use of motion pictures in this program, see Richard Lindstrom, \"'They All Believe They Are Undiscovered Mary Pickfords': Workers, Photography, and Scientific Management,\" _Technology and Culture_ 41, no. 4 (2000): 725\u2013751; Ram\u00f3n Reichert, \"Der Arbeitstudienfilm: Eine verborgene Geschichte des Stummfilms,\" _medien & zeit: Kommunikation in Vergangenheit und Gegenwart_ 5 (2002): 46\u201357; Philipp Sarasin, \"Die Rationalisierung des K\u00f6rpers: \u00dcber 'Scientific Management' und 'biologische Rationalisierung'\" in _Geschichtswissenschaft und Diskursanalyse_ (Frankfurt: Suhrkamp, 2003), 61\u201399; Elspeth H. Brown, _The Corporate Eye: Photography and the Rationalization of American Commercial Culture, 1884\u20131929_ (Baltimore: Johns Hopkins University Press, 2005); Florian Hoof, \"'The One Best Way': Bildgebende Verfahren der \u00d6konomie und die Innovation der Managementtheorie Nach 1860,\" _montage\/AV: Zeitschrift f\u00fcr Theorie und Geschichte audiovisueller Kommunikation_ 15, no. 1 (2006): 123\u2013138; and Scott Curtis, \"Images of Efficiency: The Films of Frank B. Gilbreth,\" in _Films That Work: Industrial Film and the Productivity of Media_ , ed. Vinzenz Hediger and Patrick Vonderau (Amsterdam: Amsterdam University Press, 2009), 85\u201399.\n\n 58. This tradition goes back as far as 1836, with the publication of the work of Wilhelm Weber, who with his brothers Ernst and Eduard investigated human motion using a variety of innovative visual and graphic technologies. See _Mechanik der menschlichen Gehwerkzeuge: Eine anatomisch-physiologische Untersuchung_ , in _Wilhelm Weber_ ' _s Werke_ , vol. 6, ed. Der k\u00f6niglichen Gesellschaft der Wissenschaften zu G\u00f6ttingen (Berlin: Springer, 1894), 1\u2013305. (Wilhelm Braune and Otto Fischer were closely involved in the selection and editing of Weber's works.) For a comprehensive history of human motion studies, see Andreas Mayer, _Wissenschaft vom Gehen. Die Erforschung der Bewegung im 19. Jahrhundert_ (Frankfurt: Fischer, 2013).\n\n 59. Braune and Fischer, _The Human Gait_ , 18.\n\n 60. Braune and Fischer, _The Human Gait_ , 4.\n\n 61. Rabinbach, _Human Motor_ , 189; see also Nathan Zuntz and Wilhelm Schumberg, _Studien zu einer Physiologie des Marsches_ (Berlin: Hirschwald, 1901).\n\n 62. Rabinbach, _Human Motor_ , 104; see also \u00c9tienne-Jules Marey, _Animal Mechanism: A Treatise on Terrestrial and Aerial Locomotion_ (New York: Appleton, 1874).\n\n 63. Rabinbach, _Human Motor_ , 144; see also Wilhelm Weichardt, \"Erm\u00fcdungsbek\u00e4mpfung durch Antikenotoxin,\" _Deutsche milit\u00e4r\u00e4rztliche Zeitschrift_ 42, no. 1 (5 January 1913): 12\u201313.\n\n 64. Rabinbach, _Human Motor_ , 135; see also Mosso, _Fatigue_ , 121.\n\n 65. \"Novel Uses for Moving Pictures,\" _Moving Picture World_ 1, no. 3 (23 March 1907): 39\u201340.\n\n 66. Michel Foucault, _Discipline and Punish_ , trans. Alan Sheridan (New York: Vintage, 1979), 136.\n\n 67. Foucault, _Discipline and Punish_ , 137.\n\n 68. Ian Hacking, _Representing and Intervening: Introductory Topics in the Philosophy of Natural Science_ (Cambridge: Cambridge University Press, 1983).\n\n 69. Michael Lynch, \"Discipline and the Material Form of Images: An Analysis of Scientific Visuality,\" _Social Studies of Science_ 15, no. 1 (February 1985): 43\u201344 (emphasis in original). Daston and Galison, in their work on objectivity (cited in note 31), call it a \"working object,\" which has become the more common term in the history and sociology of science.\n\n 70. Some of the most notable early sociological studies of the scientific process include Harold Garfinkel, Michael Lynch, and Eric Livingston, \"The Work of a Discovering Science Construed with Materials from the Optically Discovered Pulsar,\" _Philosophy of the Social Sciences_ 11, no. 2 (June 1981): 131\u2013158; Karin Knorr-Cetina, _The Manufacture of Knowledge: An Essay on the Constructivist and Contextual Nature of Science_ (Oxford: Pergamon, 1981); Bruno Latour and Steve Woolgar, _Laboratory Life: The Social Construction of Scientific Facts_ (London and Beverly Hills, Calif.: Sage, 1979); and Latour, _Science in Action_ (Cambridge, Mass.: Harvard University Press, 1987).\n\n 71. Chapter 1 appeared as \"Versuche am unbelasteten und belasteten Menschen,\" _Abhandlungen der Mathematisch-Physischen Klasse der K\u00f6niglich S\u00e4chsischen Gesellschaft der Wissenschaften_ 21, no. 4 (1895): 151\u2013322; chap. 2 as \"Die Bewegung des Gesamtschwerpunktes und die \u00e4u\u00dferen Kr\u00e4fte\" in 25, no. 1 (1899): 1\u2013130; chap. 3 as \"Betrachtungen \u00fcber die weiteren Ziele der Untersuchung und \u00dcberblick \u00fcber die Bewegungen der unteren Extremit\u00e4ten\" in 26, no. 3 (1900): 85\u2013170; chap. 4 as \"\u00dcber die Bewegung des Fu\u00dfes und die auf denselben einwirkenden Kr\u00e4fte\" in 26, no. 7 (1901): 467\u2013556; chap. 5 as \"Die Kinematik des Beinschwingens\" in 28, no. 5 (1903): 319\u2013418; and chap. 6 as \"\u00dcber den Einflu\u00df der Schwere und der Muskeln auf die Schwingungsbewegung des Beins\" in 28, no. 7 (1904): 531\u2013617.\n\n 72. Herman J. Wohlring, review of _The Human Gait_ , _Human Movement Science_ 8, no. 1 (February 1989): 79\u201383. Aerial photography, the counterpart to close-range photogrammetry, also developed from military applications. See Teodor J. Blachut and Rudolf Burkhardt, _Historical Development of Photogrammetric Methods and Instruments_ (Falls Church, Va.: American Society for Photogrammetry and Remote Sensing, 1989); and Paul Virilio, _War and Cinema: The Logistics of Perception_ , trans. Patrick Camiller (London: Verso, 1989).\n\n 73. Eadweard Muybridge, _Animal Locomotion: An Electro-Photographic Investigation of Consecutive Phases of Animal Movement_ (Philadelphia: University of Pennsylvania, 1887); or _Muybridge_ ' _s Complete Human and Animal Locomotion_ (New York: Dover, 1979); and Albert Londe, _Photographie m\u00e9dicale_ (Paris: Gauthiers-Villars, 1893). There also seems to have been surprisingly little crossover between Braune and Fischer's work and that of other German chronophotographers, such as Ottomar Ansch\u00fctz and Ernst Kohlrausch, but that is primarily a disciplinary issue; as the quotation above indicates, Braune and Fischer saw Ansch\u00fctz's decision to pursue the popular potential of his devices, rather than any research applications, as a markedly different path. A potentially interesting overlap might be the German Ministry of War, with which Ansch\u00fctz worked in 1884, but more research needs to be done here. On Ansch\u00fctz, see Deac Rossell, _Faszination der Bewegung: Ottomar Ansch\u00fctz zwischen Photographie und Kino_ , KINtop Schriften 6 (Frankfurt: Stroemfeld\/Roter Stern, 2001).\n\n 74. Braune and Fischer, _The Human Gait_ , 6. Hereafter cited parenthetically.\n\n 75. Lynch, \"Discipline and the Material Form of Images,\" 43.\n\n 76. Lynch, \"Discipline and the Material Form of Images,\" 42.\n\n 77. Edmund Husserl, _The Crisis of European Sciences and Transcendental Phenomenology_ , trans. David Carr (Evanston, Ill.: Northwestern University Press, 1970), 376.\n\n 78. Michael Lynch, \"The Externalized Retina: Selection and Mathematization in the Visual Documentation of Objects in the Life Sciences,\" in _Representation in Scientific Practice_ , ed. Michael Lynch and Steve Woolgar (Cambridge, Mass.: MIT Press, 1990), 153\u2013186, here 170.\n\n 79. Bergson, _Creative Evolution_ , 342 (emphasis in original).\n\n 80. Michaelis, _Research Films in Biology, Anthropology, Psychology, and Medicine_ , 371.\n\n 81. Bergson, _Creative Evolution_ , 238.\n\n 82. Jill Vance Buroker, \"Descartes on Sensible Qualities,\" _Journal of the History of Philosophy_ 29, no. 4 (October 1991): 585\u2013611.\n\n 83. Bergson, _Creative Evolution_ , 240 (emphasis in original).\n\n 84. An earlier, shorter version of this section appeared as \"Die kinematographische Methode. Das 'Bewegte Bild' und die Brownsche Bewegung,\" _montage\/AV: Zeitschrift f\u00fcr Theorie & Geschichte audiovisueller Kommunikation_ 14, no. 2 (2005): 23\u201343.\n\n 85. Roberto Maiocchi, \"The Case of Brownian Motion,\" _British Journal for the History of Science_ 23 (September 1990): 257\u2013283, here 257. See also Stephen G. Brush, \"A History of Random Processes: I. Brownian Movement from Brown to Perrin,\" _Archive for History of Exact Sciences_ 5, no. 1 (1968): 1\u201336; Mary Jo Nye, _Molecular Reality: A Perspective on the Scientific Work of Jean Perrin_ (New York: American Elsevier, 1972); Milton Kerker, \"Brownian Movement and Molecular Reality Prior to 1900,\" _Journal of Chemical Education_ 51, no. 12 (December 1974): 764\u2013768; Peter Clark, \"Atomism Versus Thermodynamics,\" in _Method and Appraisal in the Physical Sciences_ , ed. Colin Howson (Cambridge: Cambridge University Press, 1976), 41\u2013106; Brush, _Statistical Physics and the Atomic Theory of Matter from Boyle and Newton to Landau and Onsager_ (Princeton, N.J.: Princeton University Press, 1983), 79\u2013104; and John Stachel, \"Einstein on Brownian Motion,\" in _The Collected Papers of Albert Einstein_ , vol. 2, _The Swiss Years: Writings, 1900\u20131909_ , ed. John Stachel (Princeton, N.J.: Princeton University Press, 1989), 206\u2013222. On the modern implications and theoretical offspring of Brownian motion, see Erwin Frey and Klaus Kroy, \"Brownian Motion: A Paradigm of Soft Matter and Biological Physics,\" _Annalen der Physik_ 14, nos. 1\u20133 (February 2005): 20\u201350.\n\n 86. Clark, \"Atomism Versus Thermodynamics,\" 42.\n\n 87. For Mach's objections to the kinetic theory, see Ernst Mach, _Die Principien der W\u00e4rmlehre: Historisch-kritisch Entwickelt_ (Leipzig: Barth, 1896), 428\u2013429; for a discussion of Mach and the historical context of the debate, see Stephen G. Brush, _The Kind of Motion We Call Heat: A History of the Kinetic Theory of Gases in the 19th Century_ (Amsterdam and New York: North-Holland and American Elsevier, 1976), 274\u2013299; for more on Mach, see John T. Blackmore, _Ernst Mach: His Work, Life, and Influence_ (Berkeley: University of California Press, 1972). For Ostwald and other anti-atomists, see Mary Jo Nye, _Molecular Reality_ ; and Nye, \"The Nineteenth-Century Atomic Debates and the Dilemma of an 'Indifferent Hypothesis,'\" _Studies in History and Philosophy of Science_ 7, no. 3 (1976): 245\u2013268.\n\n 88. Brush, _Statistical Physics_ , 97.\n\n 89. Louis-Georges Gouy, \"Le mouvement brownien et les mouvements mol\u00e9culaires,\" _Revue g\u00e9n\u00e9rale des sciences pures et appliqu\u00e9es_ 6, no. 1 (15 January 1895): 1\u20137.\n\n 90. Maiocchi, \"The Case of Brownian Motion,\" 260.\n\n 91. Felix M. Exner, \"Notiz zu Brown's Molecularbewegung,\" _Annalen der Physik_ 2, no. 8 (1900): 843\u2013847.\n\n 92. Exner, \"Notiz,\" 844\u2013845.\n\n 93. Albert Einstein, \"On the Movement of Small Particles Suspended in a Stationary Liquid Demanded by the Molecular-Kinetic Theory of Heat,\" in _Investigations on the Theory of the Brownian Movement_ , ed. R. F\u00fcrth, trans. A. D. Cowper (New York: Dover, 1956), 1\u201318, here 1\u20132.\n\n 94. The best explanation of Einstein's use of Brownian motion as a statistical system is Martin J. Klein, \"Fluctuations and Statistical Physics in Einstein's Early Work,\" in _Albert Einstein: Historical and Cultural Perspectives_ , ed. Gerald Holton and Yehuda Elkana (Princeton, N.J.: Princeton University Press, 1982), 39\u201358. Klein remarks, \"Einstein had _invented_ the Brownian motion. To say anything less, to describe this paper in the usual way, that is, as his _explanation_ of the Brownian motion, is to undervalue it\" (47, emphasis in original). See also J\u00fcrgen Renn, \"Einstein's Invention of Brownian Motion,\" _Annalen der Physik_ 14, supplement (2005): 23\u201337.\n\n 95. Maiocchi, \"The Case of Brownian Motion,\" 260 (emphasis in original).\n\n 96. Nye, _Molecular Reality_ , 111.\n\n 97. Significantly, Brownian motion described an observable system in liquid, whereas up to this point discussions of random systems and fluctuations had focused only on gases. That is, the kinetic theory to this point applied only to gases, not liquids or solids. By removing that particular restriction in his focus on Brownian motion, Einstein also raised the stakes of the debate over the existence of atoms.\n\n 98. Einstein further simplified the system by limiting his mathematical derivations to two dimensions, thereby only taking into account the horizontal displacement of particles. In this way, Einstein's theory corresponds to experimental practice in that the field of observation corresponds to the flat, two-dimensional field of the microscope.\n\n 99. Maiocchi, \"The Case of Brownian Motion,\" 263\u2013264 (emphasis in original).\n\n. Brush, \"A History of Random Processes,\" 22\u201323.\n\n. On the invention and use of the ultramicroscope, see David Cahan, \"The Zeiss Werke and the Ultramicroscope: The Creation of a Scientific Instrument in Context,\" in _Scientific Credibility and Technical Standards in 19th and Early 20th Century German and Britain_ , ed. Jed Z. Buchwald (Dordrecht, Netherlands: Kluwer Academic, 1996), 67\u2013115.\n\n. Victor Henri, \"\u00c9tudes cin\u00e9matographique des mouvements browniens,\" _Comptes rendus hebdomadaires des s\u00e9ances de l_ ' _Academie des Sciences_ 146 (18 May 1908): 1024\u20131026; and Henri, \"Influence du milieu sur les mouvements browniens,\" _Comptes rendus hebdomadaires des s\u00e9ances de l_ ' _Academie des Sciences_ 147 (6 July 1908): 62\u201365. For a later use of motion pictures to record the distribution of particles in a gaseous system, which builds upon Seddig's and Henri's work, see Richard Lorenz and W. Eitel, \"\u00dcber die \u00f6rtliche Verteilung von Rauchteilchen,\" _Zeitschrift f\u00fcr anorganische Chemie_ 87, no. 1 (12 May 1914): 357\u2013374.\n\n. For a critique of Henri's results, see Aim\u00e9 Cotton, \"Recherches r\u00e9centes sur les mouvements browniens,\" _La Revue du Mois_ 5 (10 June 1908): 737\u2013741.\n\n. The most influential paper of many is Jean Perrin, \"Mouvement brownien et mol\u00e9cules,\" _Annales de chimie et de physique_ 18 (September 1909): 1\u2013114, translated by F. Soddy as \"Brownian Movement and Molecular Reality,\" in _The Question of the Atom_ , ed. Mary Jo Nye (Los Angeles, Calif.: Tomash, 1984), 507\u2013601. The best commentary on Perrin is Mary Jo Nye, _Molecular Reality_. For Perrin's own discussion of Henri, Seddig, and others, see _Atoms_ , trans. D. L. Hammick, 2d English ed. rev. (London: Constable, 1923), 109\u2013133. On Perrin's visualization techniques, see Charlotte Bigg, \"Evident Atoms: Visuality in Jean Perrin's Brownian Motion Research,\" _Studies in History and Philosophy of Science_ 39 (2008): 312\u2013322. See also Bigg, \"A Visual History of Jean Perrin's Brownian Motion Curves,\" in _Histories of Scientific Observation_ , ed. Lorraine Daston and Elizabeth Lunbeck (Chicago: University of Chicago Press, 2011), 156\u2013180.\n\n. Nye, _Molecular Reality_ , 97.\n\n. Seddig, \"Messung der Temperatur-Abh\u00e4ngigkeit der Brown'schen Molekularbewegung,\" 12 (emphasis in original).\n\n. Seddig, \"Ueber Abh\u00e4ngigkeit,\" 185.\n\n. Marey came upon this solution as early as 1891. See Braun, _Picturing Time_ , 166.\n\n. Einstein to Jakob Laub, 30 July 1908, quoted in Stachel, \"Einstein on Brownian Motion,\" 220. We should also note that in 1907 there was another attempt to verify Einstein's theories: Theodor Svedberg, _Studien zur Lehre von der Kolloiden L\u00f6sungen_ (Uppsala: Akademische Buchdruckerei Edv. Berling, 1907). Einstein was not so kind: \"The errors in Svedberg's method of observation and also in his theoretical treatment became clear to me at once. I wrote a minor correction at the time, which only addressed the worst, as I couldn't bring myself to detract from Mr. S's great pleasure in his work.\" (Einstein to Jean Perrin, 11 November 1909, quoted in Stachel, \"Einstein on Brownian Motion,\" 220).\n\n. Maryan Smoluchowski, \"Essai d'une th\u00e8orie cin\u00e8tique du movement brownien et des milieux troubles,\" _Krakau Anzeiger_ 7 (1906): 585\u2013586, quoted in Maiocchi, \"The Case of Brownian Motion,\" 264.\n\n. Jean Perrin, _Les atoms_ , 4th ed. rev. (Paris: Librarie F\u00e9lix Alcan, 1914), 157; _Atoms_ , 2d English ed. rev., 109\u2013110 (emphasis in original). On Jean Comandon's Brownian motion films, see Jean Comandon with Albert Dastre, \"Cin\u00e9matographie, \u00e0 l'ultra-microscope, de microbes vivants et des particules mobiles,\" _Comptes rendus hebdomadaires des s\u00e9ances de l'Acad\u00e9mie des Sciences_ 149 (22 November 1909): 938\u2013941. Perrin showed the films of Henri and Comandon to the Soci\u00e9t\u00e9 des Amis de l'Universit\u00e9 de Paris in 1911: Jean Perrin, \"La realit\u00e9 des molecules,\" _Revue Scientifique_ 49, no. 2 (1911): 774\u2013784, quoted in Nye, _Molecular Reality_ , 153. See also Hannah Landecker, \"Cellular Features: Microcinematography and Film Theory,\" _Critical Inquiry_ 31, no. 4 (2005): 903\u2013937.\n\n. On the debate in the 1920s between Bergson and Einstein regarding this category mistake, see Jimena Canales, \"Einstein, Bergson, and the Experiment That Failed: Intellectual Cooperation at the League of Nations,\" _Modern Language Notes_ 120, no. 5 (2006): 1168\u20131191.\n\n. Louis de Broglie, \"The Concepts of Contemporary Physics and Bergson's Ideas on Time and Motion,\" in _Bergson and the Evolution of Physics_ , ed. and trans. P. A. Y. Gunter (Knoxville: University of Tennessee Press, 1969), 45\u201362, here 49.\n\n. Suzanne Guerlac, _Thinking in Time: An Introduction to Henri Bergson_ (Ithaca, N.Y.: Cornell University Press, 2006), 19.\n\n. Bergson was not alone in this stance; even Nobel Prize\u2013winning Belgian physicist and chemist Ilya Prigogine argued that physics should account for the irreversibility of time. See, e.g., _From Being to Becoming: Time and Complexity in the Physical Sciences_ (San Francisco: Freeman, 1980); and with Isabelle Stengers, _Order Out of Chaos: Man_ ' _s New Dialogue with Nature_ (Boulder, Colo.: New Science Library, 1984); and _The End of Certainty: Time, Chaos, and the New Laws of Nature_ (New York: Free Press, 1997). For an introduction to this contradiction between time in physics and time as it is experienced, see David Z. Albert, _Time and Chance_ (Cambridge, Mass.: Harvard University Press, 2000).\n\n. For different interpretations of the instant and the point in photography and cinema, see Thierry de Duve, \"Time Exposure and Snapshot: The Photograph as Paradox,\" _October_ 5 (Summer 1978): 113\u2013125; and Doane, _The Emergence of Cinematic Time_ , 208\u2013230.\n\n. Henri Bergson, _Matter and Memory_ , trans. Nancy Margaret Paul and W. Scott Palmer (Cambridge, Mass.: Zone, 1988), 247 (emphasis in original).\n\n. Bergson, _Matter and Memory_ , 250 (emphasis in original).\n\n. Deleuze, _Cinema 1_.\n\n. Seddig, \"Messung der Temperatur-Abh\u00e4ngigkeit der Brown'schen Molekularbewegung,\" 36\u201337.\n\n. Marey, _Movement_.\n\n. _Lucien Bull_ (Brussels: Hayez, 1967), 11.\n\n. For a representative sampling, see Charles Fran\u00e7ois-Franck, \"La chronophotographie simultan\u00e9e du coeur et des courbes cardiographiques chez les mammif\u00e8res,\" _Comptes rendus hebdomadaires des s\u00e9ances et m\u00e9moires de la Soci\u00e9t\u00e9 de Biologie_ 54 (8 November 1902): 1193\u20131197; \"Note sur quelques points de technique relatifs \u00e0 la photographie et \u00e0 la chronophotographie avec le magn\u00e9sium \u00e0 deflagration lente,\" _Comptes rendus hebdomadaires des s\u00e9ances et m\u00e9moires de la Soci\u00e9t\u00e9 de Biologie_ 55 (5 December 1903): 1538\u20131540; \"\u00c9tudes graphiques et photographiques de m\u00e9canique respiratoire compar\u00e9e,\" _Comptes rendus hebdomadaires des s\u00e9ances et m\u00e9moires de la Soci\u00e9t\u00e9 de Biologie_ 61 (28 July 1906): 174\u2013176; and \"D\u00e9monstrations de microphotographie instantan\u00e9e et de chronomicrophotographie,\" _Comptes rendus hebdomadaires des s\u00e9ances et m\u00e9moires de la Soci\u00e9t\u00e9 de Biologie_ 62 (25 May 1907): 964\u2013967. Although many of his citations are incorrect, Thierry Lefebvre, \"Contribution \u00e0 l'histoire de la microcin\u00e9matographie: De Fran\u00e7ois-Franck \u00e0 Comandon,\" _1895_ 14 (June 1993): 35\u201343, is an essential introduction.\n\n. L. Chevroton and F. Vl\u00e8s, \"La cin\u00e9matique de la segmentation de l'oeuf et la chronophotographie du d\u00e9veloppement de l'Oursin,\" _Comptes rendus hebdomadaires des s\u00e9ances de l_ ' _Academie des Sciences_ 149 (8 November 1909): 806\u2013809.\n\n. Henri, \"\u00c9tudes cin\u00e9matographique des mouvements browniens.\"\n\n. Isabelle do O'Gomes, \"L'oeuvre de Jean Comandon,\" in _Le cin\u00e9ma et la science_ , ed. Alexis Martinet (Paris: CNRS \u00c9ditions, 1994), 78\u201385.\n\n. Julius Ries, \"Kinematographie der Befruchtung und Zellteilung,\" _Archiv f\u00fcr Mikroskopische Anatomie und Entwicklungsgeschichte_ 74 (1909): 1\u201331.\n\n. Henri B\u00e9nard, \"Les tourbillons cellulaires dans une nappe liquide. II. Proc\u00e9d\u00e9s m\u00e9caniques et optiques d'examens, lois num\u00e9riques des ph\u00e9nom\u00e8nes,\" _Revue g\u00e9n\u00e9rale des sciences pures et appliqu\u00e9es_ 11 (1900): 1309\u20131328; and B\u00e9nard, \"Formation de centres de giration \u00e0 l'arri\u00e8re d'un obstacle en movement,\" _Comptes rendus de l_ ' _Acad\u00e9mie des Sciences_ 147 (1908): 839\u2013842. See also Jos\u00e9 Eduardo Wesfreid, \"Scientific Biography of Henri B\u00e9nard (1874\u20131939),\" in _Dynamics of Spatio-Temporal Cellular Structures: Henri B\u00e9nard Centenary Review_ , ed. Innocent Mutabazi, Jos\u00e9 Eduardo Wesfreid, and \u00c9tienne Guyon (New York: Springer, 2006), 9\u201340; and David Aubin, \"'The Memory of Life Itself': B\u00e9nard's Cells and the Cinematography of Self-Organization,\" _Studies in History and Philosophy of Science_ 39, no. 3 (2008): 359\u2013369.\n\n. H. Siedentopf and E. Sommerfeldt, \"\u00dcber die Anfertigung kinematographischer Mikrophotographien der Kristallisationserscheinungen,\" _Zeitschrift f\u00fcr Elektrochemie_ 13, no. 24 (14 June 1907): 325\u2013326; and Siedentopf, \"\u00dcber ultramikroskopische Abbildung,\" _Zeitschrift f\u00fcr wissenschaftliche Mikroskopie und mikroskopische Technik_ 26 (1909): 391\u2013410.\n\n. A contemporary overview of Braus's career can be found in Walther Vogt's eulogy, \"Hermann Braus,\" _M\u00fcnchener medizinische Wochenschrift_ 72, no. 8 (20 February 1925): 304\u2013305. An assessment of his legacy in morphology is in Lynn K. Nyhart, \"Learning from History: Morphology's Challenges in Germany ca. 1900,\" _Journal of Morphology_ 252 (April 2002): 2\u201314.\n\n. A shorter version of this section appeared as \"Science Lessons,\" _Film History_ 25, nos. 1\u20132 (2013): 45\u201354.\n\n. My presentation of these theories and the history of this debate is indebted to Susan M. Billings's excellent survey, \"Concepts of Nerve Fiber Development, 1839\u20131930,\" _Journal of the History of Biology_ 4, no. 2 (Fall 1971): 275\u2013305.\n\n. Ross Granville Harrison, \"Further Experiments on the Development of Peripheral Nerves,\" _American Journal of Anatomy_ 5, no. 2 (31 May 1906): 121\u2013131. A brief overview of Harrison's career can be found in J. S. Nicholas, \"Ross Granville Harrison, 1870\u20131959,\" _Yale Journal of Biology and Medicine_ 32, no. 6 (June 1960): 407\u2013412.\n\n. Peter J. Taylor and Ann S. Blum, \"Pictorial Representation in Biology,\" _Biology and Philosophy_ 6, no. 2 (April 1991): 125\u2013134; and Nick Hopwood, \"Producing Development: The Anatomy of Human Embryos and the Norms of Wilhelm His,\" _Bulletin of the History of Medicine_ 74 (2000): 29\u201379.\n\n. Billings, \"Concepts,\" 293\u2013294.\n\n. Santiago Ram\u00f3n y Cajal, \"New Observations on the Development of Neuroblasts, with Comments on the Neurogenetic Hypothesis of Hensen-Held (1908),\" in _Studies on Vertebrate Neurogenesis_ , trans. Lloyd Guth (Springfield, Ill.: Thomas, 1960), 71\u201376, quoted in Billings, \"Concepts,\" 294. See also De Rijcke, \"Drawing Into Abstraction.\"\n\n. Hannah Landecker, \"New Times for Biology: Nerve Cultures and the Advent of Cellular Life in Vitro,\" _Studies in the History and Philosophy of Biological and Biomedical Sciences_ 33, no. 4 (2002): 667\u2013694, here 672.\n\n. Landecker, \"New Times,\" 673.\n\n. Landecker examines this formulation with regard to the rise of microcinematography in \"Creeping, Drinking, Dying: The Cinematic Portal and the Microscopic World of the Twentieth-Century Cell,\" _Science in Context_ 24, no. 3 (September 2011): 381\u2013416; and \"The Life of Movement: From Microcinematography to Live-Cell Imaging,\" _Journal of Visual Culture_ 11, no. 3 (December 2012): 378\u2013399.\n\n. Harrison, \"Further Experiments,\" 121.\n\n. Hermann Braus, \"Experimentelle Beitr\u00e4ge zur Frage nach der Entwickelung peripherer Nerven,\" _Anatomischer Anzeiger_ 26, nos. 17\/18 (1 April 1905): 433\u2013479.\n\n. Ross Granville Harrison, \"Experiments in Transplanting Limbs and Their Bearing Upon the Problems of the Development of Nerves,\" _Journal of Experimental Zoology_ 4, no. 2 (June 1907): 239\u2013281, here 241.\n\n. Ross Granville Harrison, \"Observations on the Living Developing Nerve Fiber,\" _Anatomical Record_ 1, no. 5 (1 June 1907): 116\u2013118, here 116. Harrison describes the method and its results more fully in \"The Outgrowth of the Nerve Fiber as a Mode of Protoplasmic Movement,\" _Journal of Experimental Zoology_ 9, no. 4 (December 1910): 787\u2013846.\n\n. Leo Loeb is known to have accomplished in vivo tissue culture as early as 1897. See Lewis Philip Rubin, \"Leo Loeb's Role in the Development of Tissue Culture,\" _Clio Medica_ 12, no. 1 (1977): 33\u201356. See also H. M. Carleton, \"Tissue Culture: A Critical Summary,\" _British Journal of Experimental Biology_ 1, no. 1 (October 1923): 131\u2013151.\n\n. On this experiment, see the announcements by Alexis Carrel and Montrose T. Burrows, including \"Cultivation of Adult Tissues and Organs Outside of the Body,\" _Journal of the American Medical Association_ 55, no. 16 (15 October 1910): 1379\u20131381; \"Cultivation of Sarcoma Outside of the Body: A Second Note,\" _Journal of the American Medical Association_ 55, no. 18 (29 October 1910): 1554; \"Human Sarcoma Cultivated Outside of the Body: A Third Note,\" _Journal of the American Medical Association_ 55, no. 20 (12 November 1910): 1732; and \"Cultivation of Tissues in Vitro and Its Technique,\" _Journal of Experimental Medicine_ 13, no. 3 (1 March 1911): 387\u2013396.\n\n. The definitive argument for this break in biological representation, to which I am indebted, is Hannah Landecker, _Culturing Life: How Cells Became Technologies_ (Cambridge, Mass.: Harvard University Press, 2007); see also her \"Technologies of Living Substance: Tissue Culture and Cellular Life in Twentieth Century Biomedicine\" (PhD diss., Massachusetts Institute of Technology, 1999).\n\n. This shift in the discipline's attention was not just a result of representational issues but also to a certain extent due to philosophical discomfort as a result of Bergson's critiques. For a good example of a biologist wrestling with these issues, albeit much later, see P. Lecomte du No\u00fcy, _Biological Time_ , with a foreword by Alexis Carrel (New York: Macmillan, 1937).\n\n. Billings, \"Concepts,\" 301\u2013302.\n\n. Braus, \"Mikro-Kino-Projektionen\"; this was reprinted in slightly revised form in _Wiener medizinische Wochenschrift_ 61, no. 44 (1911): 2809\u20132812.\n\n. Braus, \"Mikro-Kino-Projektionen,\" 472\u2013473.\n\n. Steven Shapin, \"Pump and Circumstance: Robert Boyle's Literary Technology,\" _Social Studies of Science_ 14, no. 4 (November 1984): 481\u2013520. See also Steven Shapin and Simon Schaffer, _Leviathan and the Air-Pump: Hobbes, Boyle, and the Experimental Life_ (Princeton, N.J.: Princeton University Press, 1985).\n\n. Braus, \"Mikro-Kino-Projektionen,\" 473.\n\n. Braus, \"Mikro-Kino-Projektionen,\" 474 (emphasis in original).\n\n. Braus, \"Mikro-Kino-Projektionen,\" 474.\n\n. Walter Benjamin, \"Little History of Photography,\" in _Walter Benjamin: Selected Writings_ , vol. 2, _1927\u20131934_ , ed. Michael W. Jennings, Howard Eiland, and Gary Smith, trans. Rodney Livingstone and others (Cambridge, Mass.: Belknap, 1999), 507\u2013530, here 512.\n\n. For another example of the statistical use of film frames, see Scott Curtis, \"'Tangible as Tissue': Arnold Gesell, Infant Behavior, and Film Analysis,\" _Science in Context_ 24, no. 3 (September 2011): 417\u2013442.\n\n2. BETWEEN OBSERVATION AND SPECTATORSHIP\n\n 1. Paul Val\u00e9ry, _Id\u00e9e Fixe_ , trans. David Paul (New York: Pantheon, 1965), 22.\n\n 2. For similar debates, see Michael Chanan, _The Dream That Kicks: The Prehistory and Early Years of Cinema in Britain_ (London and Boston: Routledge & Kegan Paul, 1980); Richard Abel, _The Cin\u00e9 Goes to Town: French Cinema, 1896\u20131914_ (Berkeley: University of California Press, 1994); Yuri Tsivian, _Early Cinema in Russia and Its Cultural Reception_ , ed. Richard Taylor, trans. Alan Bodger (London and New York: Routledge, 1994); Lee Grieveson, _Policing Cinema: Movies and Censorship in Early-Twentieth-Century America_ (Berkeley: University of California Press, 2004).\n\n 3. Sir James Paget, \"An Address on the Utility of Scientific Work in Practice,\" _British Medical Journal_ (15 October 1887): 811\u2013814, here 811.\n\n 4. On physicians' adoption of scientific method, see W. F. Bynum, _Science and the Practice of Medicine in the Nineteenth Century_ (Cambridge: Cambridge University Press, 1994); on the resistance of doctors to science, see John Harley Warner, \"Ideals of Science and Their Discontents in Late Nineteenth-Century American Medicine,\" _Isis_ 82, no. 3 (September 1991): 454\u2013478.\n\n 5. Michel Foucault, _The Birth of the Clinic: An Archaeology of Medical Perception_ , trans. A. M. Sheridan Smith (New York: Vintage, 1973); Michael Hau, \"The Holistic Gaze in German Medicine, 1890\u20131930,\" _Bulletin of the History of Medicine_ 74, no. 3 (Fall 2000): 495\u2013524.\n\n 6. Scott Curtis, \"Still\/Moving: Digital Imaging and Medical Hermeneutics,\" in _Memory Bytes: History, Technology, and Digital Culture_ , ed. Lauren Rabinovitz and Abraham Geil (Durham, N.C.: Duke University Press, 2004), 218\u2013254.\n\n 7. Like chapter 1, this chapter will focus on research films, but for more on the early medical education film in Germany, see Waldemar Schweisheimer, _Die Bedeutung des Films f\u00fcr soziale Hygiene und Medizin_ (Munich: M\u00fcller, 1920). On early American medical education films, see Martin Pernick, _The Black Stork: Eugenics and the Death of_ \" _Defective_ \" _Babies in American Medicine and Motion Pictures Since 1915_ (New York: Oxford University Press, 1999); and esp. Kirsten Ostherr's valuable studies, _Cinematic Prophylaxis: Globalization and Contagion in the Discourse of World Health_ (Durham, N.C.: Duke University Press, 2005); and _Medical Visions: Producing the Patient Through Film, Television, and Imaging Technologies_ (New York: Oxford University Press, 2013).\n\n 8. The most comprehensive survey of early literature on medical research films remains Anthony R. Michaelis, _Research Films in Biology, Anthropology, Psychology, and Medicine_ (New York: Academic, 1955). Specific statistics about Germany's output compared with other countries can be found on p. 326. See also Adolf Nichtenhauser, \"History of Motion Pictures in Medicine\" (unpublished manuscript, MS C 380, History of Medicine Division, National Library of Medicine, Bethesda, Md.). The definitive contemporary treatment of the subject is Lisa Cartwright, _Screening the Body: Tracing Medicine_ ' _s Visual Culture_ (Minneapolis: University of Minnesota Press, 1995).\n\n 9. A particularly vivid indication of the rise of German radiology is the change in the rosters of editorial board advisors of the major radiological journals. _The Archives of the Roentgen Ray_ (London), e.g., had no one from Germany on its board in 1907, but it had seven members from Germany and Austria by 1913.\n\n 10. On Germany's research infrastructure, see, e.g., Margit Sz\u00f6ll\u00f6si-Janze, \"Science and Social Space: Transformations in the Institutions of _Wissenschaft_ from the Wilhelmine Empire to the Weimar Republic,\" _Minerva_ 43 (2005): 339\u2013360.\n\n 11. The best survey of the early literature in Germany is Martin Weiser, _Medizinische Kinematographie_ (Dresden and Leipzig: Theodor Steinkopff, 1919). See also Karl Wilhelm Wolf-Czapek, _Die Kinematographie: Wesen, Entstehung und Ziele des lebenden Bildes_ (Berlin: Union Deutsche Verlagsgesellschaft, 1908; 2d enl. ed., 1911); and relevant sections in Wolf-Czapek, ed., _Angewandte Photographie in Wissenschaft und Technik_ (Berlin: Union Deutsche Verlagsgesellschaft, 1911); Hans Lehmann, _Die Kinematographie: Ihren Grundlagen und ihre Anwendungen_ (Leipzig: Teubner, 1911); Oswald Polimanti, \"Der Kinematograph in der biologischen und medizinischen Wissenschaft,\" _Naturwissenschaftliche Wochenschrift_ 10, no. 49 (3 December 1911): 769\u2013774; and Polimanti, \"Die Anwendung der Kinematographie in den Naturwissenschaften, der Medizin und im Unterricht,\" in _Wissenschaftliche Kinematographie_ , by Franz Paul Liesegang, with Karl Kieser and Oswald Polimanti (D\u00fcsseldorf: Liesegang, 1920), 257\u2013310.\n\n 12. Weiser, _Medizinische Kinematographie_ , 62.\n\n 13. Robert A. Nye, _Crime, Madness, and Politics in Modern France: The Medical Concept of National Decline_ (Princeton, N.J.: Princeton University Press, 1984). See esp. his final chapter on the United Kingdom and Germany.\n\n 14. Robert Gaupp, \"Der Kinematograph vom medizinischen und psychologischen Standpunkt,\" in _Der Kinematograph als Volkunterhaltungsmittel_ , by Robert Gaupp and Konrad Lange (Munich: D\u00fcrer-Bund-Flugschrift zur Ausdruckskultur 100, 1912), 9 (emphasis in original).\n\n 15. The definitive statement on the role of experiment in medicine is Claude Bernard, _Introduction \u00e0 l_ ' _\u00e9tude de la m\u00e9decine exp\u00e9rimentale_ (Paris: Bailli\u00e8re, 1865), translated as _An Introduction to the Study of Experimental Medicine_ by Henry Copley Greene (New York: Macmillan, 1927). See also Bynum, _Science and the Practice of Medicine in the Nineteenth Century_ ; and Warner, \"Ideals of Science.\"\n\n 16. For overviews of the origins of medical photography, see Alison Gernsheim, \"Medical Photography in the Nineteenth Century,\" _Medical and Biological Illustration_ (London) 11, no. 2 (April 1961): 85\u201392; Renata Taurek, _Die Bedeutung der Photographie f\u00fcr die medizinische Abbildung im 19. Jahrhundert_ (Cologne: Hansen, 1980); Daniel M. Fox and Christopher Lawrence, _Photographing Medicine: Images and Power in Britain and America Since 1840_ (New York: Greenwood, 1988); Andreas-Holger Maehle, \"The Search for Objective Communication: Medical Photography in the Nineteenth Century,\" in _Non-verbal Communication in Science Prior to 1900_ , ed. Renato G. Mazzolini (Florence: Olschki, 1993), 563\u2013586; Monique Sicard, Robert Pujade, and Daniel Wallach, _\u00c0 corps et \u00e0 raison: Photographies m\u00e9dicales, 1840\u20131920_ (Paris: Marval, 1995); and Gunnar Schmidt, _Anamorphotische K\u00f6rper: Medizinische Bilder vom Menschen im 19. Jahrhundert_ (Cologne: B\u00f6hlau, 2001). On the impact of photography on medical practice, see Stanley Joel Reiser, _Medicine and the Reign of Technology_ (Cambridge: Cambridge University Press, 1978).\n\n 17. Ludwig Braun, _\u00dcber Herzbewegung und Herzstoss_ (Jena: Fischer, 1898). On Braun, see Nichtenhauser, \"History of Motion Pictures in Medicine,\" 35\u201338; Michaelis, _Research Films_ , 133; Cartwright, _Screening the Body_ , 20\u201324; and Peter Geimer, \"Living and Non-living Pictures,\" in _Undead: Relations Between the Living and the Lifeless_ , ed. Peter Geimer (Berlin: Max-Planck-Institut f\u00fcr Wissenschaftsgeschichte, 2003), 39\u201351. For other early attempts to film the action of the heart, see \u00c9tienne-Jules Marey, _Movement_ , trans. Eric Pritchard (London: Heinemann, 1895), chap. 16; and Charles Fran\u00e7ois-Franck, \"La chronophotographie simultan\u00e9e du coeur et des courbes cardiographiques chez les mammif\u00e8res,\" _Comptes rendus hebdomadaires des s\u00e9ances et m\u00e9moires de la Soci\u00e9t\u00e9 de Biologie_ 54 (8 November 1902): 1193\u20131197. For other uses of film in experimental physiology, see, e.g., Martin Philippson, _L'autonomie et la centralisation dans le syst\u00e8me nerveux des animaux: \u00e9tude de physiologie exp\u00e9rimentale et compare_ (Brussels: Falk, 1905); and esp. the work of Oswald Polimanti, \"\u00dcber Ataxie cerebralen und cerebellaren Ursprungs,\" _Archiv f\u00fcr Physiologie_ (1909): 123\u2013134; and \"Zur Physiologie der Stirnlappen,\" _Archiv f\u00fcr Physiologie_ (1912): 337\u2013342.\n\n 18. On the nature of scientific experiment, see Hans-J\u00f6rg Rheinberger, _Toward a History of Epistemic Things: Synthesizing Proteins in the Test Tube_ (Stanford, Calif.: Stanford University Press, 1997); Ian Hacking, _Representing and Intervening: Introductory Topics in the Philosphy of Natural Science_ (Cambridge: Cambridge University Press, 1983); and Hans Radder, esp. _The Material Realization of Science_ (Assen and Maastricht: Van Gorcum, 1988), 59\u201369; _In and About the World: Philosophical Studies of Science and Technology_ (Albany: State University of New York, 1996), 11\u201320; and his edited volume, _The Philosophy of Scientific Experimentation_ (Pittsburgh, Pa.: University of Pittsburgh Press, 2003). See also Theodore Arabatzis, \"Experiment,\" in _The Routledge Companion to Philosophy of Science_ , ed. Stathis Psillos and Martin Curd (London and New York: Routledge, 2008), 159\u2013170.\n\n 19. On the notion of \"working objects,\" see Lorraine Daston and Peter Galison, \"The Image of Objectivity,\" _Representations_ 40 (Fall 1992): 81\u2013128; and _Objectivity_ (New York: Zone, 2007).\n\n 20. Steven Shapin, \"Pump and Circumstance: Robert Boyle's Literary Technology,\" _Social Studies of Science_ 14, no. 4 (November 1984): 481\u2013520. See also Steven Shapin and Simon Schaffer, _Leviathan and the Air-Pump: Hobbes, Boyle, and the Experimental Life_ (Princeton, N.J.: Princeton University Press, 1985).\n\n 21. See the bibliography in Otto Glasser, _Wilhelm Conrad Roentgen and the Early History of the Roentgen Rays_ (Springfield, Ill.: Thomas, 1934).\n\n 22. Excellent reviews of the subject include Hans A. Jarre, \"Roentgen Cinematography,\" in _The Science of Radiology_ , ed. Otto Glasser (Springfield, Ill.: Thomas, 1933), 198\u2013209; and L. J. Ramsey, \"Early Cineradiography and Cinefluorography,\" _History of Photography_ 7, no. 4 (October\u2013December 1983): 311\u2013322. See also Monika Dommann, _Durchsicht, Einsicht, Vorsicht: Eine Geschichte der R\u00f6ntgenstrahlen, 1896_ \u2013 _1963_ (Z\u00fcrich: Chronos, 2003). For early cinema and the X-ray, see Cartwright, _Screening the Body_ ; and Solveig J\u00fclich, \"Seeing in the Dark: Early X-Ray Imaging and Cinema,\" in _Moving Images: From Edison to the Webcam_ , ed. John Fullerton and Astrid S\u00f6derberg Widding (Sydney: Libbey, 2000), 47\u201358.\n\n 23. John Macintyre, \"X-Ray Records for the Cinematograph,\" _Archives of Skiagraphy_ 1, no. 2 (April 1897): 37; and the report of his screening for \"Ladies Night\" of the Royal College of Surgeons of England in \"The Royal Society Conversazione,\" _Lancet_ 149 (19 June 1897): 1706.\n\n 24. P. H. Eykman, \"Der Schlingact, dargestellt nach Bewegungsphotographien mittelst R\u00f6ntgen-Strahlen,\" _Pfl\u00fcger_ ' _s Archiv f\u00fcr die gesammte Physiologie des Menschen und der Thiere_ 99 (1903): 513\u2013571.\n\n 25. Virgilio Tosi, _Cinema Before Cinema: The Origins of Scientific Cinematography_ , trans. Sergio Angelini (London: British Universities Film & Video Council, 2005), 170.\n\n 26. C. K\u00e4stle, H. Rieder, and J. Rosenthal, \"Ueber kinematographisch aufgenommene R\u00f6ntgenogramme (Bio-R\u00f6ntgenographie) der inneren Organe des Menschen,\" _M\u00fcnchener medizinsiche Wochenschrift_ 56, no. 6 (9 February 1909): 280\u2013283; \"The Bioroentgenography of the Internal Organs,\" _Archives of the Roentgen Ray_ 15, no. 1 (June 1910): 3\u201312.\n\n 27. Lewis Gregory Cole, \"The Gastric Motor Phenomena Demonstrated with the Projecting Kinetoscope,\" _American Quarterly of Roentgenology_ 3, no. 4 (1912): 1\u201311.\n\n 28. Franz M. Groedel, \"The Present State of Roentgen Cinematography and Its Results as to the Study of the Movements of the Inner Organs of the Human Body,\" _Interstate Medical Journal_ 22 (March 1915): 281\u2013290, here 290. Of his numerous texts on the topic, see esp. \"Roentgen Cinematography and Its Importance in Medicine,\" _British Medical Journal_ (24 April 1909): 1003; and his three-part series \"Die Technik der R\u00f6ntgenkinematographie,\" _Deutsche medizinsiche Wochenschrift_ 35 (11 March 1909): 434\u2013435; 39 (6 February 1913): 270\u2013271; and 39 (24 April 1913): 798\u2013799. See also W. Bruce Fye, \"Franz M. Groedel,\" _Clinical Cardiology_ 23, no. 2 (February 2000): 133\u2013134.\n\n 29. On diagnosis, see Carlo Ginzburg, \"Clues: Roots of an Evidential Paradigm,\" in _Clues, Myths, and the Historical Method_ , trans. John and Ann C. Tedeschi (Baltimore: Johns Hopkins University Press, 1989), 96\u2013125; or Caroline Whitbeck, \"What Is Diagnosis? Some Critical Reflections,\" _Metamedicine_ 2, no. 3 (October 1981): 319\u2013329.\n\n 30. On doubts about the clinical application of X-rays, see Andrew Warwick, \"X-Rays as Evidence in German Orthopedic Surgery, 1895\u20131900,\" _Isis_ 96, no. 1 (March 2005): 1\u201324.\n\n 31. Friedrich Dessauer, _Die neuesten Fortschritte in der R\u00f6ntgenphotographie_ (Leipzig: Nemnich Verlag, 1912), 15. For a more optimistic view, see Carl Bruegel, \"Bewegungsvorg\u00e4nge am pathologischen Magen auf Grund r\u00f6ngenkinematographischer Untersuchungen,\" _M\u00fcnchener medizinische Wochenschfrift_ 60, no. 4 (28 January 1913): 179\u2013181.\n\n 32. Andr\u00e9 Lomon and Jean Comandon, \"Radiocin\u00e9matographie par la photographie des \u00e9crans intensificateurs,\" _La Presse M\u00e9dicale_ 35 (3 May 1911): 359.\n\n 33. Jarre, \"Roentgen Cinematography,\" 202.\n\n 34. For a more complete survey, see K. Podoll and J. L\u00fcning, \"Geschichte des wissenschaftlichen Films in der Nervenheilkunde in Deutschland 1895\u20131929,\" _Fortschritte der Neurologie, Psychiatrie_ 66 (1998): 122\u2013132; and Genevi\u00e8ve Aubert \"From Photography to Cinematography: Recording Movement and Gait in a Neurological Context,\" _Journal of the History of the Neurosciences_ 11, no. 3 (2002): 255\u2013264. See also Juliet Clare Wagner, \"Twisted Bodies, Broken Minds: Film and Neuropsychiatry in the First World War\" (PhD diss., Harvard University, 2009).\n\n 35. Paul Schuster, \"Vorf\u00fchrung pathologischer Bewegungscomplexe mittelst des Kinematographen und Erl\u00e4uterung derselben,\" _Verhandlungen der Gesellschaft deutscher Naturforscher und \u00c4rzte_ 69, part 1 (1898): 196\u2013199. For more on Schuster and his context, see Bernd Holdorff, \"Die privaten Polikliniken f\u00fcr Nervenkranke vor und nach 1900\" and \"Zwischen Hirnforschung, Neuropsychiatrie und Emanzipation zur klinischen Neurologie bis 1933,\" in _Geschichte der Neurologie in Berlin_ , ed. Bernd Holdorff and Rolf Winau (Berlin and New York: de Gruyter, 2001), 127\u2013137, and 157\u2013174.\n\n 36. Gheorghe Marinescu, \"Les troubles de la marche dans l'h\u00e9mipl\u00e9gie organique \u00e9tudi\u00e9s \u00e0 l'aide du cin\u00e9matographe,\" _La semaine M\u00e9dicale_ (1899): 225\u2013228; see also Alexandru C. Barboi, Christopher G. Goetz, and Radu Musetoiu, \"The Origins of Scientific Cinematography and Early Medical Applications,\" _Neurology_ 62 (June 2004): 2082\u20132086.\n\n 37. See Albert Londe's report of his work with Richer in Albert Londe, _Notice sur les titres et travaux scientifiques_ (Paris: Masson, 1911).\n\n 38. Walter Greenough Chase, \"The Use of the Biograph in Medicine,\" _Boston Medical and Surgical Journal_ 153, no. 21 (23 November 1905): 571\u2013572. See also Cartwright, _Screening the Body_ , chap. 3.\n\n 39. Arthur Van Gehuchten, \"Coup de couteau dans la moelle lombaire. Essai de physiologie pathologique,\" _Le N\u00e9vraxe_ 9 (1907): 208\u2013232. See also Genevi\u00e8ve Aubert, \"Arthur Van Gehuchten Takes Neurology to the Movies,\" _Neurology_ 59 (November 2002): 1612\u20131618.\n\n 40. Emil Kraepelin, \"Demonstration von Kinematogrammen,\" _Centralblatt f\u00fcr Nervenheilkunde und Psychiatrie_ 32 (1909): 689.\n\n 41. Hans Hennes, \"Die Kinematographie im Dienste der Neurologie und Psychiatrie, nebst Beschreibung einiger selteneren Bewegungsst\u00f6rungen,\" _Medizinische Klinik_ 6, no. 51 (18 December 1910): 2010\u20132014.\n\n 42. See Polimanti titles cited in note 11.\n\n 43. T. H. Weisenburg, \"Moving Picture Illustrations in Medicine, with Special Reference to Nervous and Mental Diseases,\" _Journal of the American Medical Association_ 59, no. 26 (28 December 1912): 2310\u20132312.\n\n 44. This section on live demonstration draws from my essay, \"Photography and Medical Observation,\" in _The Educated Eye: Visual Pedagogy in the Life Sciences_ , ed. Nancy Anderson and Michael R. Dietrich (Hanover, N.H.: Dartmouth College Press, 2012), 68\u201393.\n\n 45. Clare [Blake] to Dear Pater, Vienna, 9 November [1865], Clarence John Blake Papers, Francis A. Countway Library of Medicine, Harvard University, quoted in John Harley Warner, _Against the Spirit of System: The French Impulse in Nineteenth-Century American Medicine_ (Princeton, N.J.: Princeton University Press, 1998), 304.\n\n 46. Clare to Sister Agnes, Vienna, 29 March 1869, Clarence John Blake Papers, Francis A. Countway Library of Medicine, Harvard University, quoted in Warner, _Against the Spirit of System_ , 311.\n\n 47. Hennes, \"Die Kinematographie im Dienste der Neurologie,\" 2012, quoted in Friedrich A. Kittler, _Gramophone, Film, Typewriter_ , trans. and with an introduction by Geoffrey Winthrop-Young and Michael Wutz (Stanford, Calif.: Stanford University Press, 1999), 145 (emphasis in original).\n\n 48. See Cartwright, _Screening the Body_ , chap. 3: \"An Etiology of the Neurological Gaze.\"\n\n 49. See Reiser, _Medicine and the Reign of Technology_. See also Joel D. Howell, _Technology in the Hospital: Transforming Patient Care in the Early Twentieth Century_ (Baltimore: Johns Hopkins University Press, 1995).\n\n 50. For discussion of the projection of images in medical education, see A. Wassermann, \"Die medizinische Fakult\u00e4t,\" in _Die Universit\u00e4ten im Deutschen Reich_ , ed. W. Lexis (Berlin: Asher, 1904), 146\u2013147; or Erwin Christeller, \"Die Bedeutung der Photographie f\u00fcr den pathologisch-anatomischen Unterricht und die pathologisch-anatomische Forschung,\" _Berliner klinische Wochenschrift_ 55, no. 17 (29 April 1918): 399\u2013401; for broader overviews, see Sigmund Theodor Stein, _Das Licht im Dienste wissenschaftlicher Forschung: Handbuch der Anwendung des Lichtes und der Photographie in der Natur- und Heilkunde_ (Leipzig: Spamer, 1877); Sigmund Theodor Stein, _Die optische Projektionskunst im dienste der exakten Wissenschaften: ein Lehr- und Hilfsbuch zur unterst\u00fctzung des naturwissenschaftlichen Unterrichts_ (Halle an der Saale, Germany: Knapp, 1887); see also Henning Schmidgen, \"Pictures, Preparations, and Living Processes: The Production of Immediate Visual Perception ( _Anschauung_ ) in Late-19th-Century Physiology,\" _Journal of the History of Biology_ 37, no. 3 (October 2004): 477\u2013513.\n\n 51. Schuster, \"Vorf\u00fchrung pathologischer Bewegungscomplexe,\" 196\u2013197.\n\n 52. See, e.g., Albert E. Stein, \"Ueber medizinisch-photographische und -kinematographische Aufnahmen,\" _Deutsche medizinische Wochenschrift_ 38 (20 June 1912): 1184\u20131186. For early reviews of the use of photography and cinematography for the study of pathological movement, with implications for therapy, see Ernst Jendrassik, \"Klinische Beitr\u00e4ge zum Studium der normalen und pathologischen Gangarten,\" _Deutsche Archiv f\u00fcr klinische Medizin_ 70 (1901): 81\u2013132; and James Fr\u00e4nkel, \"Kinematographische Untersuchung des normalen Ganges und einiger Gangst\u00f6rungen,\" _Zeitschrift f\u00fcr orthop\u00e4dische Chirurgie_ 20 (1908): 617\u2013646.\n\n 53. An example of a German report on Comandon's work that stresses the importance of his films as both exploratory and documentary is \"Die Kinematographie des Unsichtbaren,\" _Prometheus_ 21, no. 1054 (5 January 1910): 218\u2013220.\n\n 54. J. Frey, \"Report of the Photographic Department of Bellevue Hospital for the Year 1869,\" in _Tenth Annual Report of the Commissioners of Public Charities and Correction of the City of New York for the Year 1869_ (Albany: van Benthuysen, 1870), 85. www.artandmedicine.com\/ogm\/1869.html.\n\n 55. Hennes, \"Die Kinematographie im Dienste der Neurologie und Psychiatrie,\" 2014. For other calls for a central agency to handle medical or scientific films, see Robert Kutner, \"Die Bedeutung der Kinematographie f\u00fcr medizinische Forschung und Unterricht sowie f\u00fcr die volkshygienische Belehrung,\" _Zeitschrift f\u00fcr \u00c4rztliche Fortbildung_ 8, no. 8 (15 April 1911): 249\u2013251; or Weiser, _Medizinische Kinematographie_ , 4. This call was eventually answered after World War I with the formation of the Kulturfilm section of UFA. See Universum-Film A. G., _Das medizinische Filmarchiv bei der Kulturabteilung der Universum-Film A.G._ (Berlin: Gahl, 1919).\n\n 56. Franz Goerke, \"Proposal for Establishing an Archive for Moving Pictures (1912),\" trans. Cecilie L. French and Daniel J. Leab, _Historical Journal of Film, Radio and Television_ 16, no. 1 (March 1996): 9\u201312, here 10. Originally published as \"Vorschlag zur Einrichtung eines Archives f\u00fcr Kino-films\" in _Der Deutsche Kaiser im Film. Zum 25 j\u00e4hrigen Regierungs-Jubil\u00e4um Seiner Majest\u00e4t des Deutschen Kaisers K\u00f6nigs von Preu\u00dfen Wilhelm II_ , ed. Paul Klebinder (Berlin: Klebinder, 1912), 63\u201368.\n\n 57. Goerke, \"Proposal for Establishing an Archive,\" 9\u201310.\n\n 58. Eug\u00e8ne Doyen, \"Le cin\u00e9matograph et l'enseignement de la chirurgie,\" _Revue critique de m\u00e9decine et de chirurgie_ 1, no. 1 (15 August 1899): 1\u20136; partially translated as \"The Cinematograph and the Teaching of Surgery,\" _British Gyn\u00e6cological Journal_ 15 (1899): 579\u2013586, here 581. On Doyen, see Robert Didier, _Le Docteur Doyen: Chirurgien de la Belle \u00c9poque_ (Paris: Librairie Maloine, 1962); and esp. the work of Thierry Lefebvre, including \"Le cas \u00e9trange du Dr Doyen, 1859\u20131916,\" _Archives_ 29 (February 1990): 1\u201312; \"Le Dr Doyen, un pr\u00e9curseur,\" in _Le cin\u00e9ma et la science_ , ed. Alexis Martinet (Paris: CNRS \u00c9ditions, 1994), 70\u201377; \"Die Trennung der Siamesischen Zwillinge Doodica und Radica durch Dr. Doyen,\" _KINtop 6_ (1997): 97\u2013101; and _La chair et le celluloid: Le cin\u00e9ma chirurgical du Docteur Doyen_ (Brionne: Jean Doyen \u00e9diteur, 2004).\n\n 59. Although the films were not mentioned in the _British Medical Journal_ 's proceedings of the July 1898 meeting, there is a letter to the editor that remarks on the strong impression they made: G. P. Coldstream, \"The Cinematoscope as an Aid in Teaching,\" _British Medical Journal_ (3 September 1898): 658. On Doyen's films, see Thierry Lefebvre, \"La collection des films du Dr Doyen,\" _1895_ 17 (December 1994): 100\u2013114; and Tiago Baptista, \"' _Il faut voir le ma\u00eetre_ ': A Recent Restoration of Surgical Films by E.-L.Doyen (1859\u20131916),\" _Journal of Film Preservation_ 70 (November 2005): 42\u201350. After 1907, French film manufacturer Eclipse distributed Doyen's films throughout Europe (see Lefebvre, _La chair et le celluloid_ , 67\u201376).\n\n 60. Doyen, \"Le cin\u00e9matograph et l'enseignement de la chirurgie,\" 3.\n\n 61. For a catalog of Doyen's films to that point, see Eug\u00e8ne Louis Doyen, _L'enseignement de la technique op\u00e9ratoire par les projections anim\u00e9es_ (Paris: Soci\u00e9t\u00e9 g\u00e9n\u00e9rale des cin\u00e9matographes Eclipse, ca. 1911).\n\n 62. For Doyen's detailed discussion of the medical case, the surgical technique, and its ethical aftermath, see Eug\u00e8ne Louis Doyen, _La cas des xiphopages hindoues Radica et Doodica_ (Paris: Bourse de Commerce, 1904). This pamphlet also reprints a heated exchange of letters between Doyen and a Dr. Legrain, who accused Doyen of besmirching the profession with the film. See also Lefebvre, \"Die Trennung der Siamesischen Zwillinge.\" On the Parnaland incident, see Lefebvre, _La chair et le celluloid_ , 39\u201359.\n\n 63. The concern about spectacle in medicine was more pronounced in France\u2014probably due to the Doyen controversy\u2014but there were grumblings from England and Germany as well, especially with regard to Doyen's films. See \"A Surgical Showman,\" _British Medical Journal_ (19 January 1907): 163, which reports from Germany and elsewhere about screenings of Doyen films that left doctors disgusted; at one of his demonstrations, \"it is said that he was hissed at Brussels.\"\n\n 64. See the report in \"Verwandte Gebiete,\" _Zentralblatt f\u00fcr R\u00f6ntgenstrahlen, Radium und verwandte Gebiete_ 1, no. 2 (1910): 78\u201380.\n\n 65. Kutner was a proponent of medical photography in general, having taken some of the earliest images from a cytoscope with famed urologist Max Nitze. See Max Nitze, _Kystophotographischer Atlas_ (Wiesbaden: Bergmann, 1894). See also Harry W. Herr, \"Max Nitze, the Cystoscope and Urology,\" _Journal of Urology_ 176, no. 4 (October 2006): 1313\u20131316. Kutner was also a strong advocate of continuing education for physicians, having founded the journal _Zeitschrift f\u00fcr \u00c4rztliche Fortbildung_.\n\n 66. See also James Fr\u00e4nkel, \"Kinematographische Demonstration,\" _Verhandlungen der freien Vereinigung der Chirurgen Berlins_ 20, part 1 (1907): 12\u201313.\n\n 67. Karl Reicher, \"Kinematographie in der Neurologie,\" _Verhandlungen der Gesellschaft deutscher Naturforscher und \u00c4rzte_ 79, part 2 (1907): 235\u2013236.\n\n 68. See \"Special Correspondence: Berlin,\" _British Medical Journal_ (5 March 1910): 598, for another report of the evening. In England, Dr. William Stirling was Kutner's counterpart as a champion of the educational value of medical films. See reports of Stirling's presentations of a variety of films in medicine and biology in \"Medical News,\" _Lancet_ 177 (27 May 1911): 1470; and _Lancet_ 182 (11 October 1913): 1083\u20131084.\n\n 69. Kutner, \"Die Bedeutung der Kinematographie,\" 250.\n\n 70. Alongside this discourse of \"seeing as\" (film as an extension of direct perception and the moving image as a substitute for the thing itself) there was another discourse of \"seeing differently\" (film as a technology for representing things in ways the naked eye could not perceive). Educators regarded both features of cinema to be pedagogically useful, while film theorists such as Jean Epstein and B\u00e9la Bal\u00e0zs prioritized the latter. This difference also corresponds to the \"documentary\" and \"exploratory\" functions outlined so far.\n\n 71. Doyen, \"The Cinematograph and the Teaching of Surgery,\" 580\u2013581 (translation modified).\n\n 72. A good overview is Jennifer Karns Alexander, _The Mantra of Efficiency: From Waterwheel to Social Control_ (Baltimore: Johns Hopkins University Press, 2008). See also Evelyn Cobley, _Modernism and the Culture of Efficiency: Ideology and Fiction_ (Toronto and Buffalo, N.Y.: University of Toronto Press, 2009).\n\n 73. Paul Starr, _The Social Transformation of American Medicine_ (New York: Basic, 1982), 146.\n\n 74. George Rosen, \"The Efficiency Criterion in Medical Care, 1900\u20131920,\" _Bulletin of the History of Medicine_ 50, no. 1 (Spring 1976): 28\u201344; Margarete Arndt and Barbara Bigelow, \"Toward the Creation of an Institutional Logic for the Management of Hospitals: Efficiency in the Early Nineteen Hundreds,\" _Medical Care Research and Review_ 63, no. 3 (June 2006): 369\u2013394.\n\n 75. See the issue devoted to the \"Conference on Hospital Standardization,\" _Bulletin of the American College of Surgeons_ 3, no. 1 (1917); or Frank B. Gilbreth, \"Scientific Management in the Hospital,\" _Modern Hospital_ 3 (1914): 321\u2013324.\n\n 76. My essay on \"Photography and Medical Observation\" (see note 44) likewise examines correspondences between formal features of still photography and practices\/ideals of medical observation.\n\n 77. Ludwig Braun, \"Ueber den Werth des Kinematographen f\u00fcr die Erkenntniss der Herzmechanik,\" _Verhandlungen der Gesellschaft deutscher Naturforscher und \u00c4rzte_ 69, part 1 (1898): 185\u2013186. This list is included and elaborated in his _\u00dcber Herzbewegung und Herzstoss_ (Jena: Fischer, 1898).\n\n 78. Kelly Wilder, _Photography and Science_ (London: Reaktion, 2009), 23.\n\n 79. Gernsheim, \"Medical Photography in the Nineteenth Century,\" 87.\n\n 80. Cartwright, _Screening the Body_ , 38, 48.\n\n 81. Ludwik Fleck, \"Some Specific Features of the Medical Way of Thinking [1927],\" in _Cognition and Fact: Materials on Ludwik Fleck_ , ed. Robert S. Cohen and Thomas Schnelle (Dordrecht, Netherlands, and Boston: Reidel, 1986), 39\u201346, here 39\u201340. Foucault also discusses the importance of the series in medical thinking in _The Birth of the Clinic_ , 97. On the importance of comparing pathological states to find some sense of consistency, see Georges Canguilhem, _The Normal and the Pathological_ , trans. Carolyn R. Fawcett in collaboration with Robert S. Cohen (New York: Zone, 1991), 51. On medical logic, see Friedrich Oesterlen, _Medical Logic_ , trans. G. Whitley (London: Sydenham Society, 1855); Frederick P. Gay, \"Medical Logic,\" _Bulletin of the History of Medicine_ 7 (1939): 6\u201327; Lester S. King, \"Medical Logic,\" _Journal of the History of Medicine and Allied Sciences_ 33, no. 3 (July 1978): 377\u2013385; and King, _Medical Thinking: A Historical Preface_ (Princeton, N.J.: Princeton University Press, 1982).\n\n 82. Bernike Pasveer, \"Representing or Mediating: A History and Philosophy of X-Ray Images in Medicine,\" in _Visual Cultures of Science: Rethinking Representational Practices in Knowledge Building and Science Communication_ , ed. Luc Pauwels (Lebanon, N.H.: Dartmouth College Press\/University Press of New England, 2006), 41\u201362.\n\n 83. For an example of the use of series photography to track the development of smallpox in a patient over several days, see Samuel A. Powers, _Variola: A Series of Twenty-One Heliotype Plates Illustrating the Progressive Stages of the Eruption_ (Boston: Samuel A. Powers, 1882). My thanks to Mark Rowland for pointing me to this text. For examples of different angles of a patient during the same session, see Albert Londe's photographs of female patients documented in the journal _Nouvelle iconographie de la Salp\u00eatri\u00e8re_ (Paris: Masson, 1888\u20131918).\n\n 84. For an interesting discussion of X-ray cinematography's diagnostic value, see the exchange between Drs. Fr\u00e4nkel, von Bergmann, Levy-Dorn, Albu, and Kutner in Albert Fr\u00e4nkel, \"R\u00f6ntgendiagnosen und R\u00f6ntgenfehldiagnosen beim Magenkarzinom; diagnostischer Fortschritt durch R\u00f6ntgenkinographie.\" _Zentralblatt f\u00fcr R\u00f6ntgenstrahlen, Radium und verwandte Gebiete_ 3, no. 4 (1912): 149\u2013150.\n\n 85. Warwick, \"X-Rays as Evidence in German Orthopedic Surgery, 1895\u20131900.\"\n\n 86. Photographs were often used as illustrations exchanged between attendees of a lecture. An example, picked more or less at random, is Adolf Magnus-Levy, \"Ueber Organ-Therapie beim endemischen Kretinismus,\" _Verhandlungen der Berliner medicinischen Gesellschaft_ 34, part 2 (1903): 350\u2013357. See esp. the discussion of this presentation on 22 July 1903 in part 1, pp. 246\u2013249. No photos are published with the paper, but they discuss the photographs that were passed around among the audience. Many such uses of photographs can be found in the _Verhandlungen_ and similar proceedings.\n\n 87. Cartwright, _Screening the Body_ , 36.\n\n 88. Georges Didi-Huberman, _Invention of Hysteria: Charcot and the Photographic Iconography of the Salp\u00eatri\u00e8re_ , trans. Alisa Hartz (Cambridge, Mass.: MIT Press, 2003), 24\u201325.\n\n 89. For more on this topic, see Scott Curtis, \"Photography and Medical Observation.\"\n\n 90. Foucault, _Birth of the Clinic_ , 109.\n\n 91. Foucault, _Birth of the Clinic_ , esp. 107\u2013123.\n\n 92. Neurologists such as Schuster needed movement to recognize the disorder\u2014which would otherwise be lost in the individual frames\u2014so they consistently slowed (but did not stop) the image in projection. See Schuster, \"Vorf\u00fchrung pathologischer Bewegungskomplexe.\"\n\n 93. Many researchers emphasized this feature of motion picture technology, but for an extended discussion of the applications of film's temporal malleability, with interesting responses from the expert audience, see the report of Herr v. Gr\u00fctzner's presentation at the T\u00fcbingen Society of Medical and Natural Scientists: \"Medizinisch-Naturwissenschaftlicher Verein T\u00fcbingen,\" _M\u00fcnchener medizinische Wochenschrift_ 56, no. 3 (19 January 1909): 154\u2013155.\n\n 94. Thomas Laycock, _Lectures on the Principles and Methods of Medical Observation and Research_ (Philadelphia: Blanchard and Lea, 1857), 64.\n\n 95. Rudolph Virchow, _Post-mortem Examinations_ , trans. T. P. Smith, 3d ed. (Philadelphia: Blakiston, 1895), 12.\n\n 96. Foucault, _Birth of the Clinic_ , xiv.\n\n 97. Foucault, _Birth of the Clinic_ , xiii (emphasis added).\n\n 98. Foucault, _Birth of the Clinic_ , esp. 107\u2013122.\n\n 99. Theodor Billroth, _The Medical Sciences in the German Universities: A Study in the History of Civilization_ , trans. William H. Welch (New York: Macmillan, 1924), 52\u201353. Originally published as _\u00dcber das Lehren und Lernen de medicinischen Wissenschaften an den Universit\u00e4ten der deutschen Nation nebst allgemeinen Bemerkungen \u00fcber Universit\u00e4ten; eine culturhistorische Studie_ (Vienna: Gerold, 1876).\n\n. Foucault, _Birth of the Clinic_ , xiii.\n\n. Hau, \"The Holistic Gaze in German Medicine, 1890\u20131930,\" 503. For more on the resistance of clinicians to laboratory methods in medicine, see Russell C. Maulitz, \"Physician Versus Bacteriologist: The Ideology of Science in Clinical Medicine,\" in _The Therapeutic Revolution: Essays in the Social History of American Medicine_ , ed. Morris J. Vogel and Charles E. Rosenberg (Philadelphia: University of Pennsylvania Press, 1979), 91\u2013107.\n\n. Foucault, _Birth of the Clinic_ , 121.\n\n. Lorraine Daston, \"On Scientific Observation,\" _Isis_ 99, no. 1 (2008): 97\u2013110. See also Lorraine Daston and Elizabeth Lunbeck, eds., _Histories of Scientific Observation_ (Chicago: University of Chicago Press, 2011).\n\n. There are perhaps some connections here to be made between the glance of the connoisseur and theories of cinephilia. See Christian Keathley, _Cinephilia and History; or, The Wind in the Trees_ (Bloomington: Indiana University Press, 2006).\n\n. Again, see Curtis, \"Still\/Moving: Digital Imaging and Medical Hermeneutics,\" for more on the relationship between film and medical hermeneutics.\n\n. Which should remind us of similar patterns of showmanship in early entertainment film described by Tom Gunning in \"An Aesthetic of Astonishment: Early Film and the (In)Credulous Spectator,\" _Art and Text_ 34 (Spring 1989): 31\u201345.\n\n. My thanks to Christian Quendler for stating this so succinctly for me.\n\n. Erwin Risak, _Der klinische Blick_ , 7th and 8th eds. (Vienna: Springer, 1943), 4.\n\n. Carl F. Flemming, _Pathologie und Therapie der Psychosen_ (Berlin: Hirschwald, 1859), 281\u2013282.\n\n. For a good overview of the relationship between clinical observation and scientific methods, see Kenneth D. Keele, _The Evolution of Clinical Methods in Medicine_ (London: Pitman, 1963). On the debates about clinical observation and the sphygmomanometer specifically, see Jeremy Booth, \"A Short History of Blood Pressure Measurement,\" _Proceedings of the Royal Society of Medicine_ 70, no. 11 (November 1977): 793\u2013799; Reiser, _Medicine and the Reign of Technology_ , 101\u2013106; and Hughes Evans, \"Losing Touch: The Controversy Over the Introduction of Blood Pressure Instruments Into Medicine,\" _Technology and Culture_ 34, no. 4 (October 1993): 784\u2013807.\n\n. Paget, \"An Address on the Utility of Scientific Work in Practice,\" 811.\n\n. Karl Wilhelm Wolf-Czapek, \"Die Kinematographie im medizinische Unterricht,\" _Jahrbuch f\u00fcr Photographie und Reproduktionstechnik_ 22 (1908): 58\u201359, here 58.\n\n. A close review of the primary literature on scientific film shows that analysis and synthesis were always considered two sides of the same coin. See \u00c9tienne-Jules Marey, _Movement_ , trans. Eric Pritchard (London: Heinemann, 1895), esp. chap. 18: \"Synthetic Reconstruction of the Elements of an Analyzed Movement.\" See also Hannah Landecker, \"Microcinematography and the History of Science and Film,\" _Isis_ 97, no. 1 (2006): 121\u2013132; and Oliver Gaycken, \"'The Swarming of Life': Moving Images, Education, and Views Through the Microscope,\" _Science in Context_ 24, no. 3 (September 2011): 361\u2013380.\n\n. On the theoretical implications of experimental apparatuses, see Davis Baird, _Thing Knowledge: A Philosophy of Scientific Instruments_ (Berkeley: University of California Press, 2004).\n\n. More could be said about the relationship between control of and submission to or pleasure in the scientific image. A good place to start would be Anne Secord, \"Botany on a Plate: Pleasure and the Power of Pictures in Promoting Early Nineteenth-Century Scientific Knowledge,\" _Isis_ 93, no. 1 (March 2002): 28\u201357.\n\n. Claudia Huerkamp, _Der Aufstieg der Arzte im 19. Jahrhundert: Vom gelehrten Stand zum professionellen Experten: Das Beispiel Preu\u00dfens_ (G\u00f6ttingen: Vandenhoeck & Ruprecht, 1985). See also Ute Frevert, _Krankheit als politisches Problem 1770\u20131880. Soziale Unterschichten in Preu\u00dfen zwischen medizinischer Polizei und staatlicher Sozialversicherung_ (G\u00f6ttingen: Vandenhoeck & Ruprecht, 1984).\n\n. Huerkamp stresses this medicalization of culture in her essay, \"The Making of the Modern Medical Profession, 1800\u20131914: Prussian Doctors in the Nineteenth Century,\" in _German Professions, 1800\u20131950_ , ed. Geoffrey Cocks and Konrad H. Jarausch (New York and Oxford: Oxford University Press, 1990), 66\u201384.\n\n. On the social prestige and authority of physicians, see Alfons Labisch, _Homo Hygienicus: Gesundheit und Medizin in der Neuzeit_ (Frankfurt and New York: Campus, 1992); Michael H. Kater, \"Professionalization and Socialization of Physicians in Wilhelmine and Weimar Germany,\" _Journal of Contemporary History_ 20 (1985): 677\u2013701; Paul Weindling, \"Bourgeois Values, Doctors and the State: The Professionalization of Medicine in Germany 1848\u20131933,\" in _The German Bourgeoisie_ , ed. David Blackbourn and Richard J. Evans (London and New York: Routledge, 1991), 198\u2013223; Charles E. McClelland, \"Modern German Doctors: A Failure of Professionalization?\" in _Medicine and Modernity: Public Health and Medical Care in Nineteenth- and Twentieth-Century Germany_ , ed. Manfred Berg and Geoffrey Cocks (Cambridge and New York: Cambridge University Press, 1997), 81\u201397. For a discussion of this phenomenon from the viewpoint of medical ethics, see Robert M. Veatch, \"Generalization of Expertise,\" _Hastings Center Studies_ 1, no. 2 (1973): 29\u201340.\n\n. On scientists, especially, as _Kulturtr\u00e4ger_ , see Fritz K. Ringer, _The Decline of the German Mandarins: The German Academic Community, 1890\u20131933_ (Cambridge, Mass.: Harvard University Press, 1969), 6; and Russell McCormmach, \"On Academic Scientists in Wilhelmian Germany,\" in _Science and Its Public: The Changing Relationship_ , ed. Gerald Horton and William A. Blanpied (Dordrecht, Netherlands, and Boston: Reidel, 1976), 157\u2013171.\n\n. Paul Weindling, \"Public Health in Germany,\" in _The History of Public Health and the Modern State_ , ed. Dorothy Porter (Amsterdam and Atlanta, Ga.: Rodopi, 1994), 119\u2013131. See also Weindling, _Health, Race, and German Politics Between National Unification and Nazism, 1870\u20131945_ (Cambridge and New York: Cambridge University Press, 1989).\n\n. Eric J. Engstrom, \"Emil Kraepelin: Psychiatry and Public Affairs in Wilhelmine Germany,\" _History of Psychiatry_ 2, no. 6 (June 1991): 111\u2013132. See also Engstrom's _Clinical Psychiatry in Imperial Germany: A History of Psychiatric Practice_ (Ithaca, N.Y.: Cornell University Press, 2003); and Emil Kraepelin, _Memoirs_ , ed. H. Hippius, G. Peters, D. Ploog in collaboration with P. Hoff and A. Kreuter, trans. Cheryl Wooding-Deane (Berlin and New York: Springer-Verlag, 1987). Kraepelin was also swept up by the degeneration craze described below: see Emil Kraepelin, \"Zur Entartungsfrage,\" _Zentralblatt f\u00fcr Nervenheilkunde und Psychiatrie_ 31 (1908): 745\u2013751, translated as \"On the Question of Degeneration,\" _History of Psychiatry_ 18, no. 3 (2007): 399\u2013404.\n\n. According to Andreas Killen, as early as 1912 the Reich Health Office started to collect materials documenting the educational benefits of scientific films and the health risks of commercial cinema. Andreas Killen, \"Psychiatry, Cinema, and Urban Youth in Early-Twentieth-Century Germany,\" _Harvard Review of Psychiatry_ 14, no. 1 (2006): 38\u201343.\n\n. Robert Gaupp, _Psychologie des Kindes_ (Leipzig: Teubner, 1910), and \"Das Pathologische in Kunst und Literatur,\" _Deutsche Revue_ 36, no. 2 (April 1911): 11\u201323. For an especially explicit statement of the physician's duty to society, see Gaupp, \"Der Arzt als Erzieher seines Volkes,\" _Medicinisches Correspondenz-Blatt_ 89, no. 32 (9 August 1919): 295\u2013296. On Gaupp, see William Mayer, \"Robert Gaupp,\" _American Journal of Psychiatry_ 108, no. 10 (April 1952): 724\u2013725.\n\n. Max Nordau, _Degeneration_ (London: Appleton, 1895), 43. Originally published as _Entartung_ , 2 vols. (Berlin: Duncker, 1892\u20131893). Nordau, an Austro-Hungarian physician living in Paris, was a prolific writer of fiction and cultural criticism as well as a foreign correspondent for German-language newspapers in Berlin, Vienna, and Budapest.\n\n. For contemporary discussions of nervousness, see, e.g., Wilhelm Heinrich Erb, _Ueber die wachsende Nervosit\u00e4t unserer Zeit_ (Heidelberg: Universit\u00e4ts Buchdruckerei von J. H\u00f6rning, 1893); Auguste Forel, _Hygiene der Nerven und des Geistes im gesunden und kranken Zustande_ (Stuttgart: Moritz, 1903); and Robert Gaupp, \"Die Nervosit\u00e4t unserer Zeit im Lichte der Wissenschaft,\" _Medicinisches Correspondenz-Blatt_ 77, no. 31 (3 August 1907): 633\u2013639. On nervousness and modernity in Germany, see Joachim Radkau, _Das Zeitalter der Nervosit\u00e4t: Deutschland zwischen Bismarck und Hitler_ (Munich: Hanser, 1998); Andreas Killen, _Berlin Electropolis: Shock, Nerves, and German Modernity_ (Berkeley: University of California Press, 2006); Michael Cowan, _Cult of the Will: Nervousness and German Modernity_ (University Park: Pennsylvania State University Press, 2008).\n\n. On Lombroso and Nordau, see Charles Bernheimer, \"Decadent Diagnostics,\" in _Decadent Subjects: The Idea of Decadence in Art, Literature, Philosophy, and Culture of the_ Fin de Si\u00e8cle _in Europe_ , ed. T. Jefferson Kline and Naomi Schor (Baltimore: Johns Hopkins University Press, 2002), 139\u2013162. For more on Nordau, see George L. Mosse, \"Max Nordau and His Degeneration,\" in Max Nordau, _Degeneration_ (New York: Fertig, 1968), xiii\u2013xxxvi; Thomas Anz, \"Gesundheit, Krankheit und literarische Norm: Max Nordaus 'Entartung' als Paradigma pathologisierender Kunstkritik,\" in _Gesund oder Krank?: Medizin, Moral und \u00c4sthetik in der deutschen Gegenwartsliteratur_ (Stuttgart: Metzler, 1989), 33\u201352; Hans-Peter S\u00f6der, \"Disease and Health as Contexts of Modernity: Max Nordau as a Critic of Fin-de-Si\u00e8cle Modernism,\" _German Studies Review_ 14, no. 3 (October 1991): 473\u2013487; Christoph Schulte, _Psychopathologie des Fin de Si\u00e8cle: Der Kulturkritiker, Arzt und Zionist Max Nordau_ (Frankfurt: Fischer Taschenbuch, 1997); C\u00e9line Kaiser, _Rhetorik der Entartung: Max Nordau und die Sprache der Verletzung_ (Bielefeld, Germany: Transcript, 2007).\n\n. To be fair, even those friendly to modern art and culture often saw it in similar, especially primitivist terms. See Doris Kaufmann, \"'Pushing the Limits of Understanding': The Discourse on Primitivism in German _Kulturwissenschaften_ , 1880\u20131930,\" _Studies in History and Philosophy of Science_ 39 (2008): 434\u2013443.\n\n. S\u00f6der, \"Disease and Health as Contexts of Modernity,\" 474.\n\n. Surveys of cultural pessimism in Germany include Fritz Stern, _The Politics of Cultural Despair: A Study in the Rise of the Germanic Ideology_ (Berkeley: University of California Press, 1961); and Ringer, _The Decline of the German Mandarins_.\n\n. Ike Spier, \"Die sexuelle Gefahr des Kinos,\" _Die neue Generation_ 8 (1912): 192\u2013198, here 192 (emphasis in original). I will discuss the battle against _Schundfilms_ (\"trash films\") in chap. 3, but for representative works, see Albert Hellwig, _Schundfilms: Ihr Wesen, ihre Gefahren und ihre Bek\u00e4mpfung_ (Halle an der Saale, Germany: Waisenhaus, 1911); Max Grempe, \"Gegen die Frauenverbl\u00f6dung im Kino,\" _Gleichheit_ 23, no. 5 (1912): 70\u201372; Malwine Rennert, \"Die Zaung\u00e4ste des Lebens im Kino,\" _Bild und Film_ 4, no. 11 (1914\/1915): 217\u2013218; and, for a reasonable rebuttal, Joseph Landau, \"Mechanisierte Unsterblichkeit,\" in _Der Deutsche Kaiser im Film. Zum 25j\u00e4hrigen Regierungs-Jubil\u00e4um Seiner Majest\u00e4t des Deutschen Kaisers K\u00f6nigs von Preu\u00dfen Wilhelm II_ , ed. Paul Klebinder (Berlin: Klebinder, 1912), 18\u201322.\n\n. Paul Schenk, \"Der Kinematograph und die Schule,\" _Aerztliche Sachverst\u00e4ndigen-Zeitung_ 14, no. 15 (1 August 1908): 312\u2013313. For an entertaining \"experiment\" in which a writer submits three men to hours of continuous, flickering projection with predictable results, see Naldo Felke, \"Die Gesundheitssch\u00e4dlichkeit des Kinos,\" _Die Umschau_ 17, no. 1 (1 January 1913): 254\u2013255.\n\n. A survey of the (mostly French) discussion of \"flicker\" in early cinema can be found in Thierry Lefebvre, \"Flimmerndes Licht: Zur Geschichte der Filmwahrnehmung im fr\u00fchen Kino,\" _KINtop_ 5 (1996): 71\u201380.\n\n. For excellent surveys of this trend, see Killen, \"Psychiatry, Cinema, and Urban Youth in Early-Twentieth-Century Germany\"; and Killen, \"The Scene of the Crime: Psychiatric Discourses on the Film Audience in Early Twentieth Century Germany,\" in _Film 1900: Technology, Perception, Culture_ , ed. Annemone Ligensa and Klaus Kreimeier (New Barnet, U.K.: Libbey, 2009) 99\u2013111.\n\n. Albert Hellwig, \"\u00dcber die sch\u00e4dliche Suggestivkraft kinematographischer Vorf\u00fchrung,\" _Aerztliche Sachverst\u00e4ndigen-Zeitung_ 20, no. 6 (15 March 1914): 122; I will discuss Hellwig's work in more depth in chap. 3, but for a taste of his reliance on medical terminology and audiences, see Hellwig, \"Zur Psychologie kinematographischer Vorf\u00fchrungen,\" _Zeitschrift f\u00fcr Psychotherapie und medizinische Psychologie_ 6 (1916): 88\u2013120; and \"Hypnotismus und Kinematograph,\" _Zeitschrift f\u00fcr Psychotherapie und medizinische Psychologie_ 6 (1916): 310\u2013315.\n\n. O. G\u00f6tze, \"Jugendpsyche und Kinematograph,\" _Zeitschrift f\u00fcr Kinderforschung_ 16 (1911): 418 (emphasis in original).\n\n. Thierry Lefebvre quotes a similar, French objection from 1913 to film's temporal malleability: \"The cinema, with its rapid unfolding, its somewhat brutal speed of images which follow one another, distort the slow and progressive work of nature. Here is a film showing a seed which suddenly sprouts, becomes stem, flower, fruit all in just a couple of seconds. Nature does not do this; nature 'does not jump,' as told to us by the old philosophy.\" \"The Scientia Production (1911\u20131914): Scientific Popularization Through Pictures,\" _Griffithiana_ no. 47 (May 1993): 137\u2013153, here 145.\n\n. A good survey is Hartmut Rosa and William E. Scheuerman, eds., _High-Speed Society: Social Acceleration, Power, and Modernity_ (University Park: Pennsylvania State University Press, 2009).\n\n. Nordau, _Degeneration_ , 40. For more on Nordau's critique of the speed of modernity, see G\u00fcnther A. H\u00f6fler, \"La naissance de la 'nervosit\u00e9' issue de l'esprit de la modernit\u00e9 technologique. D\u00e9g\u00e9n\u00e9rescence et nomadisation chez Max Nordau et Adolph Wahrmund,\" in _Max Nordau (1849\u20131923): Critique de la D\u00e9g\u00e9n\u00e9rescence, M\u00e9diateur Franco-Allemand_ , _P\u00e8re Fondateur du Sionisme_ , ed. Delphine Bechtel, Dominique Bourel, and Jacques Le Rider (Paris: Cerf, 1996), 149\u2013160.\n\n. Nordau, _Degeneration_ , 42.\n\n. Nordau, _Degeneration_ , 55.\n\n. Jonathan Crary, _Suspensions of Perception: Attention, Spectacle, and Modern Culture_ (Cambridge, Mass.: MIT Press, 1999).\n\n. Crary, _Suspensions of Perception_ , 25. See also Friedrich Nietzsche's discussion of attention and the will to mastery in _Beyond Good and Evil_ , sect. 19, in _Basic Writings of Nietzsche_ , trans. Walter Kaufmann (New York: Modern Library, 1966), 215\u2013217.\n\n. Crary, _Suspensions of Perception_ , 17. If \"attention\" were a difficult concept to define, at least it remained ideologically productive through the twentieth century; the same cannot be said for \"will\" or \"volition,\" which lost currency after around 1900. See G. E. Berrios and M. Gili, \"Will and Its Disorders: A Conceptual History,\" _History of Psychiatry_ 6 (1995): 87\u2013104.\n\n. Gaupp, \"Der Kinematograph vom medizinischen und psychologischen Standpunkt,\" 9 (emphasis in original).\n\n. Adolf Sellmann, \"Das Geheimnis des Kinos,\" _Bild und Film_ 1, nos. 3\u20134 (1912): 65\u201367, here 66 (emphasis in original).\n\n. Wilhelm Stapel, \"Der homo cinematicus,\" _Deutsches Volkstum_ 21 (October 1919): 319\u2013320, here 319.\n\n. See the articles by Andreas Killen (notes 122 and 133 above), as well as Stefan Andriopoulos, _Possessed: Hypnotic Crimes, Corporate Fiction, and the Invention of Cinema_ (Chicago: University of Chicago Press, 2008); and Rae Beth Gordon, _Why the French Love Jerry Lewis: From Cabaret to Early Cinema_ (Stanford, Calif.: Stanford University Press, 2001).\n\n. Stefan Andriopoulos, \"Spellbound in Darkness: Hypnosis as an Allegory of Early Cinema,\" _Germanic Review_ 77, no. 2 (Spring 2002): 102\u2013116, here 103.\n\n. Leopold Laquer, \"\u00dcber die Sch\u00e4dlichkeit kinematographischer Veranstaltungen f\u00fcr die Psyche des Kindesalters,\" _Aerztliche Sachverst\u00e4ndigen-Zeitung_ 27, no. 11 (1 June 1911): 221.\n\n. Laquer, \"\u00dcber die Sch\u00e4dlichkeit kinematographischer Veranstaltungen,\" 222.\n\n. The most famous of these was the case of the Borbacher Knabenmord, in which a young man accused of killing a little boy recounted the films he saw leading up to the crime. This case was a touchstone for medical and reformist literature on cinema through the 1920s. See Killen, \"The Scene of the Crime,\" 104; and Hellwig, \"\u00dcber die sch\u00e4dliche Suggestivkraft kinematographischer Vorf\u00fchrung.\"\n\n. Gaupp, \"Der Kinematograph vom medizinischen und psychologischen Standpunkt,\" 9 (emphasis in original).\n\n. Gordon, _Why the French Love Jerry Lewis_ , 128.\n\n. Gustave Le Bon, _The Crowd: A Study of the Popular Mind_ (Atlanta, Ga.: Cherokee, 1982), 16. Originally published as _Psychologie des foules_ (Paris: Alcan, 1895). Translated and published in German as _Psychologie der Massen_ (Leipzig: Klinkhardt, 1908).\n\n. Le Bon, _The Crowd_ , 21. For a version of this argument applied to film audiences, see Hermann Duenschmann, \"Kinematograph und Psychologie der Volksmenge. Eine sozialpolitische Studie,\" _Konservative Monatsschrift_ 69, no. 9 (June 1912): 920\u2013930.\n\n. Le Bon, _The Crowd_ , 29.\n\n. Didi-Huberman, _Invention of Hysteria_.\n\n. L\u00e9on Chertok and Isabelle Stengers, _A Critique of Psychoanalytic Reason: Hypnosis as a Scientific Problem from Lavoisier to Lacan_ , trans. Martha Noel Evans (Stanford, Calif.: Stanford University Press, 1992). For a detailed study of the historical relationship between hypnosis and psychoanalysis, see Andreas Mayer, _Sites of the Unconscious: Hypnosis and the Emergence of the Psychoanalytical Setting_ (Chicago: University of Chicago Press, 2013).\n\n. Albert Moll, _Hypnotism_ (London: Scott, 1890), 333 (emphasis added). Originally published as _Der Hypnotismus_ (Berlin: Fischer's Medicinische, 1889). To be fair, it should be noted that this particular application of hypnosis is relatively uncommon during the nineteenth and early twentieth centuries, when the therapeutic technique is used overwhelmingly for somatic ailments. For an example, see W. P. Carr, \"Suggestion as Used and Misused in Curing Disease,\" in _Hypnotism and Hypnotic Suggestion_ , ed. E. Virgil Neal and Charles S. Clark (Rochester: New York State Publishing, 1900), 5\u201317. For more on the experimental and therapeutic uses of hypnosis, see Mayer, _Sites of the Unconscious_.\n\n. [Henri-\u00c9tienne] Beaunis, \"L'exp\u00e9rimentation en psychologie par le somnambulisme provoqu\u00e9,\" _Revue philosophique_ 10, no. 7 (1885): 2 (emphasis in original).\n\n. Immanuel Kant, \"What Is Enlightenment?,\" in _German Aesthetic and Literary Criticism_ , ed. David Simpson (Cambridge: Cambridge University Press, 1984), 29\u201334, here 30.\n\n3. THE TASTE OF A NATION\n\n 1. \"Die Bremer Lehrerinnen und die Kinogefahr,\" _Die Lehrerin_ 30 (1913): 156, quoted in Albert Hellwig, _Kind und Kino_ (Langensalza: Beyer, 1914), 71.\n\n 2. Stephen Kern discusses the bourgeois attitude toward sexuality in _Anatomy and Destiny: A Cultural History of the Human Body_ (Indianapolis: Bobbs-Merrill, 1975). Cf. Peter Gay, _The Education of the Senses_ , vol. 1 (New York: Oxford University Press, 1984); and Michel Foucault, _The History of Sexuality_ , trans. Robert Hurley (New York: Vintage, 1990).\n\n 3. Konrad Lange, _Die k\u00fcnstlerische Erziehung der deutschen Jugend_ (Darmstadt: Bergstrae\u00dfer, 1893), 12.\n\n 4. Pierre Bourdieu, _Distinction_ (Cambridge, Mass.: Harvard University Press, 1984), 190 (emphasis in original).\n\n 5. A more complete treatment of this theme can be found in Patrice Petro, _Joyless Streets: Women and Melodramatic Representation in Weimar Germany_ (Princeton, N.J.: Princeton University Press, 1989). Ann Douglas discusses a similar concern in the United States in _The Feminization of American Culture_ (New York: Knopf, 1977).\n\n 6. See Josef Chytry, _The Aesthetic State: A Quest in Modern German Thought_ (Berkeley and Los Angeles: University of California Press, 1989).\n\n 7. Norbert Elias, _The Civilizing Process_ , vol. 1, _The History of Manners_ , trans. Edmund Jephcott (New York: Pantheon, 1978), 19. For other discussions of _Kultur_ and _Zivilisation_ in the German context, see Fritz Ringer, _The Decline of the German Mandarins: The German Academic Community, 1890\u20131933_ (Cambridge, Mass.: Harvard University Press, 1969); and Jeffrey Herf, _Reactionary Modernism: Technology, Culture, and Politics in Weimar and the Third Reich_ (Cambridge and New York: Cambridge University Press, 1984). See also J\u00f6rg Fisch, \"Zivilisation, Kultur,\" in _Geschichtliche Grundbegriffe. Historisches Lexikon zur politisch-sozialen Sprache in Deutschland_ , ed. Otto Brunner, Werner Conze, and Reinhardt Koselleck (Stuttgart: Klett, 1992), 7: 679\u2013774.\n\n 8. Raymond Geuss, \"Kultur, Bildung, Geist,\" _History and Theory_ 35, no. 2 (May 1996): 151\u2013164, here 153.\n\n 9. Another dichotomy, Ferdinand T\u00f6nnies's _Gemeinschaft_ (community) and _Gesellschaft_ (society), struck a similarly antimodernist tone. Contrasting the unity of the small, rural community, which he felt was disappearing in the modern industrial transformation, with the alienation and fragmentation of the metropolis, T\u00f6nnies was perhaps more elegiac than staunchly antimodernist. Still, the tendency to describe or criticize modern bourgeois society through reference to a precapitalist past was described by Georg Luk\u00e1cs as \"romantic anti-capitalism,\" capturing the deeply ambivalent, contradictory character of nineteenth-century reactions to industrialization. See T\u00f6nnies, _Gemeinschaft und Gesellschaft_ (Leipzig: Fues, 1887), translated by Charles Loomis as _Community and Society_ (East Lansing: Michigan State University Press, 1957); Georg Luk\u00e1cs, \"\u00dcber den Dostojewski-Nachlass,\" _Moskauer Rundschau_ 17 (22 March 1931): 4; Robert Sayre and Michael L\u00f6wy, \"Figures of Romantic Anti-capitalism,\" _New German Critique_ 32 (Spring\/Summer 1984): 42\u201392. For a good, general overview of Germany's ambivalent reaction to modernity, see Kenneth D. Barkin, \"The Crisis of Modernity, 1887\u20131902,\" in _Imagining Modern German Culture, 1889\u20131910_ , ed. Fran\u00e7oise Forster-Hahn (Washington, D.C.: National Gallery of Art, 1996), 19\u201335.\n\n 10. Dennis Sweeney, \"Reconsidering the Modernity Paradigm: Reform Movements, the Social and the State in Wilhelmine Germany,\" _Social History_ 31, no. 4 (November 2006): 405\u2013434, here 406. See also Kevin Repp, _Reformers, Critics, and the Paths of German Modernity: Anti-Politics and the Search for Alternatives, 1890\u20131914_ (Cambridge, Mass.: Harvard University Press, 2000); and Andrew Lees, _Cities, Sin, and Social Reform in Imperial Germany_ (Ann Arbor: University of Michigan Press, 2002).\n\n 11. Repp, _Reformers, Critics, and the Paths of German Modernity_ , 278.\n\n 12. This chapter began life as \"The Taste of a Nation: Training the Senses and Sensibility of Cinema Audiences in Imperial Germany,\" _Film History_ 6, no. 4 (Winter 1994): 445\u2013469.\n\n 13. On reform movements in general, see Norman Rich, _The Age of Nationalism and Reform, 1850\u20131890_ (New York: Norton, 1970), 103\u2013122; Maureen A. Flanagan, _America Reformed: Progressives and Progressivisms, 1890s_ \u2013 _1920s_ (New York: Oxford University Press, 2007); or Judith F. Stone, _The Search for Social Peace: Reform Legislation in France, 1890\u20131914_ (Albany: State University of New York Press, 1985).\n\n 14. See, e.g., such general surveys as Hans-Ulrich Wehler, _The German Empire, 1871\u20131918_ (Leamington Spa, U.K.: Berg, 1985); or David Blackbourn, _History of Germany, 1780\u20131918: The Long Nineteenth Century_ , 2d ed. (Malden, Mass.: Blackwell, 2003). See also studies of German urbanization, such as J\u00fcrgen Reulecke, _Geschichte der Urbanisierung in Deutschland_ (Frankfurt: Suhrkamp, 1985); or surveys of European modernity, such as Andrew Lees and Lynn Lees, eds., _The Urbanization of European Society in the Nineteenth Century_ (Lexington, Mass.: Heath, 1976); and Hans J\u00fcrgen Teuteberg, ed., _Urbanisierung im 19. und 20. Jahrhundert: historische und geographische Aspekte_ (Cologne: B\u00f6hlau, 1983).\n\n 15. On reform in Germany, in addition to Repp ( _Reformers, Critics, and the Paths of German Modernity_ ) and Lees and Lees ( _The Urbanization of European Society_ ), see R\u00fcdiger vom Bruch, _Wissenschaft, Politik und \u00f6ffentliche Meinung: Gelehrtenpolitik im Wilhelminischen Deutschland, 1890\u20131914_ (Husum, Germany: Matthiesen, 1980); J\u00fcrgen Reulecke, _Sozialer Frieden durch soziale Reform: Der Centralverein f\u00fcr das Wohl der Arbeitenden Klassen in der Fr\u00fchindustrialisierung_ (Wuppertal: Hammer, 1983); and vom Bruch, ed., _Weder Kommunismus noch Kapitalismus: B\u00fcrgerliche Sozialreform in Deutschland vom Vorm\u00e4rz bis zur \u00c4ra Adenauer_ (Munich: Beck, 1985). On educational reform in particular, see Christa Berg, ed., _Handbuch der deutschen Bildungsgeschichte_ (Munich: Beck, 1991); and Wolfgang Scheibe, _Die Reformp\u00e4dagogische Bewegung, 1900\u20131932: Eine einf\u00fchrende Darstellung_ , 9th ed. (Weinheim and Basel: Beltz, 1984). Some have rightly argued that, despite the implications of the concept of \"reform,\" we should be careful not to view the educational or social theories and practices that came out of this period as complete breaks with tradition. See J\u00fcrgen Oelkers, _Reformp\u00e4dagogik. Eine kritische Dogmengeschichte_ (Weinheim and Munich: Juventa, 1989).\n\n 16. A good survey of late nineteenth-century _Kulturkritik_ is David L. Gross, \" _Kultur_ and Its Discontents: The Origins of a 'Critique of Everyday Life' in Germany, 1880\u20131925,\" in _Essays on Culture and Society in Modern Germany_ , ed. Gary D. Stark and Bede Karl Lackner (College Station: Texas A&M University Press, 1982), 70\u201397.\n\n 17. _Verhandlung \u00fcber Fragen des h\u00f6heren Unterrichts, Berlin 4. bis 17. Dezember 1890_ (Berlin, 1891), 770, quoted in James C. Albisetti, _Secondary School Reform in Imperial Germany_ (Princeton, N.J.: Princeton University Press, 1983), 4. For an especially compelling discussion of the debates about the value of Greek ideals in imperial Germany, see Suzanne L. Marchand, _Down from Olympus: Archaeology and Philhellenism in Germany, 1750\u20131970_ (Princeton, N.J.: Princeton University Press, 1996).\n\n 18. See, e.g., Wilhelm Frei, _Landerziehungsheime: Darstellung und Kritik einer modernen Reformschule_ (Leipzig: Klinkhardt, 1902); or Herbert Bauer, _Zur Theorie und Praxis der ersten deutschen Landerziehungsheime: Erfahrungen zur Internats- und Ganztagserziehung aus den Hermann-Lietz-Schulen_ (Berlin: Volk und Wissen, 1961).\n\n 19. See Georg Kerschensteiner, \"Begriff der Arbeitsschule,\" in _Die deutsche Reformp\u00e4dagogik_ , ed. Wilhelm Flitner and Gerhard Kudritzki (D\u00fcsseldorf and Munich: K\u00fcpper, 1961), 222\u2013238.\n\n 20. Essays expressing the themes of \"the modern,\" \"the healthy,\" and \"the national\" might be, respectively: Hermann Kienzl, \"Theater und Kinematograph,\" _Der Strom_ 1, no. 7 (October 1911): 219\u2013221; Robert Gaupp, \"Die Gefahren des Kino,\" _S\u00fcddeutsche Monatshefte_ 9, no. 9 (1911\/1912): 363\u2013366; and Albert Hellwig, \"Kinematograph und Zeitgeschichte,\" _Die Grenzboten_ 72, no. 39 (1913): 612\u2013620. These and other representative essays can be found in J\u00f6rg Schweinitz, ed., _Prolog vor dem Film_ : _Nachdenken \u00fcber ein neues Medium, 1909\u20131914_ (Leipzig: Reclam, 1992). Essays from this period are also collected in Anton Kaes, ed., _Kino-Debatte: Texte zum Verh\u00e4ltnis von Literatur und Film 1909\u20131929_ (T\u00fcbingen: Niemeyer, 1978); and Fritz G\u00fcttinger, ed., _Kein Tag ohne Kino: Schriftsteller \u00fcber den Stummfilm_ (Frankfurt: Deutsches Filmmuseum, 1984). Among the numerous commentaries, see especially Kaes's introduction, revised and translated as \"Literary Intellectuals and the Cinema: Charting a Controversy (1909\u20131929),\" _New German Critique_ 40 (Winter 1987): 7\u201334; Heide Schl\u00fcpmann, _Unheimlichkeit des Blicks: Das Drama des fr\u00fchen deutschen Kinos_ (Frankfurt: Stroemfeld\/Roter Stern, 1990), 189\u2013243; and Sabine Hake, _The Cinema_ ' _s Third Machine: Writing on Film in Germany, 1907\u20131933_ (Lincoln: University of Nebraska Press, 1993), 27\u201342.\n\n 21. Repp ( _Reformers, Critics, and the Paths of German Modernity_ ) and Sweeney (\"Reconsidering the Modernity Paradigm\") emphasize imperial Germany's diversity of approaches to the problems of modernity, as well as the deep ambivalence toward the modern that most elites felt. On the other hand, for histories of ideas that stress the retrograde elements of German society and the reactionary responses that, for some, foreshadow National Socialism, see Fritz Stern, _The Politics of Cultural Despair: A Study in the Rise of the Germanic Ideology_ (Berkeley and Los Angeles: University of California Press, 1961); or George L. Mosse, _The Crisis of German Ideology: Intellectual Origins of the Third Reich_ (New York: Grosset & Dunlap, 1964).\n\n 22. Paul Schultze-Naumberg, _Die Kultur des weiblichen K\u00f6rpers als Grundlage der Frauenkleidung_ , quoted in Kern, _Anatomy_ , 15. Schultze-Naumberg shifted easily from advocating \"natural clothing\" to supporting art fashioned after natural bodies; during the Third Reich he was an architect of the campaign against \"degenerate\" art. See Kern, _Anatomy_ , 223\u2013226.\n\n 23. Carl Heinrich Stratz, _Die Frauenkleidung und ihre nat\u00fcrliche Entwicklung_ (Stuttgart: Enke, 1900).\n\n 24. See Schl\u00fcpmann, _Unheimlichkeit_ , 8\u201325; Beth Irwin Lewis, \" _Lustmord_ : Inside the Windows of the Metropolis,\" in _Berlin: Culture and Metropolis_ , ed. Charles W. Haxthausen and Heidrun Suhr (Minneapolis: University of Minnesota Press, 1990), 111\u2013140; and the essays included in J. Edward Chamberlin and Sander L. Gilman, eds., _Degeneration: The Dark Side of Progress_ (New York: Columbia University Press, 1985).\n\n 25. Along with Kaes (\"Literary Intellectuals and the Cinema\"), Schl\u00fcpmann ( _Unheimlichkeit_ ), and Hake ( _Third Machine_ ), see Miriam Hansen, \"Early Silent Cinema: Whose Public Sphere?,\" _New German Critique_ 29 (Spring\/Summer 1983): 147\u2013184. On women and reform in Germany, see Christoph Sach\u00dfe, _M\u00fctterlichkeit als Beruf: Sozialarbeit, Sozialreform und Frauenbewegung, 1871\u20131929_ (Frankfurt: Suhrkamp, 1986); and Ann Taylor Allen, _Feminism and Motherhood in Germany, 1800\u20131914_ (New Brunswick, N.J.: Rutgers University Press, 1991).\n\n 26. Susanne Asche, \"F\u00fcrsorge, Partizipation und Gleichberechtigung\u2014die Leistungen der Karlsruherinnen f\u00fcr die Entwicklung zur Gro\u00dfstadt (1859\u20131914),\" in _Karlsruher Frauen, 1715\u20131945: Eine Stadtgeschichte_ (Karlsruhe: Badenia, 1992), 171\u2013256.\n\n 27. Eventually exemplified by Oswald Spengler's _The Decline of the West_ , trans. Charles Francis Atkinson (New York: Knopf, 1926\u201328). There are numerous commentaries, but see, e.g., Ringer, _The Decline of the German Mandarins_ ; Klaus Vondung, \"Zur Lage der Gebildeten in der wilhelminischen Zeit,\" in _Das wilhelminische Bildungsb\u00fcrgertum Zur Sozialgeschichte seiner Ideen_ , ed. Klaus Vondung (G\u00f6ttingen: Vandenhoeck & Ruprecht, 1976), 20\u201333; or Charles E. McClelland, \"The Wise Man's Burden: The Role of Academicians in Imperial German Culture,\" in Stark and Lackner, _Essays on Culture and Society in Modern Germany_ , 45\u201369. An excellent expression of the perceived loss of cultural authority of experts in an age of mass culture and mediocre scientists is Ernst Meumann, \"Wilhelm Wundt. Zu seinem achtzigsten Geburtstag,\" _Deutsche Rundschau_ 38, no. 11 (August 1912): 193\u2013224.\n\n 28. On the vital role of feminist activists in shaping the direction of reform movements and the public sphere in general, see Allen, _Feminism and Motherhood in Germany, 1800\u20131914_ ; and Kathleen Canning, _Languages of Labor and Gender: Female Factory Work in Germany, 1850\u20131914_ (Ithaca, N.Y.: Cornell University Press, 1996).\n\n 29. Mirjam Storim, \"'Einer, der besser ist, als sein Ruf': Kolportageroman und Kolportagebuchhandel um 1900 und die Haltung der Buchbranche,\" in _Schund und Sch\u00f6nheit: Popul\u00e4re Kultur um 1900_ , ed. Kaspar Maase and Wolfgang Kaschuba (Cologne: B\u00f6hlau, 2001), 252\u2013282, here 255\u2013256. See also Rudolf Schenda, _Die Lesestoffe der kleinen Leute: Studien zur popul\u00e4ren Literatur im 19. und 20. Jahrhundert_ (Munich: Beck, 1976).\n\n 30. Luke Springman, \"Poisoned Hearts, Diseased Minds, and American Pimps: The Language of Censorship in the _Schund und Schmutz_ Debates,\" _German Quarterly_ 68, no. 4 (Autumn 1995): 408\u2013429, here 413.\n\n 31. Corey Ross, _Media and the Making of Modern Germany: Mass Communications, Society, and Politics from the Empire to the Third Reich_ (New York: Oxford University Press, 2008), 66.\n\n 32. Kaspar Maase, \"Krisenbewu\u00dftsein und Reformorientierung: Zum Deutungshorizont der Gegener der modernen Popul\u00e4rk\u00fcnste 1880\u20131918,\" in Maase and Kaschuba, _Schund und Sch\u00f6nheit: Popul\u00e4re Kultur um 1900_ , 290\u2013342. See also his \"Struggling About 'Filth and Trash': Educationalists and Children's Culture in Germany Before the First World War,\" _Paedagogica Historica_ 34, no. 1 (1998): 8\u201328.\n\n 33. Professor Dr. Fri\u00df Johannesson, \"Das Lesen der Jugend au\u00dferhalb der Schule,\" _Die Hochwacht_ no. 2 (November 1911), quoted in Kara L. Ritzheimer, \"Protecting Youth from 'Trash': Anti- _Schund_ Campaigns in Baden, 1900\u20131933,\" PhD diss. (State University of New York\u2013Binghamton, 2007), 25.\n\n 34. Class warfare, however, was not unknown in these campaigns, especially given the historical coincidence of the rise of mass entertainment and the rise of the Social Democratic Party of Germany (SPD), which many (such as Karl Brunner) saw as uncoincidental and fought both with equal vigor. For more, see Ross, _Media and the Making of Modern Germany_.\n\n 35. A fine articulation of Brunner's position with regard to film (and the SPD) is _Der Kinematograph von heute_ \u2014 _eine Volksgefahr_ (Berlin: Vaterl\u00e4ndischen Schriftenverbandes, 1913).\n\n 36. Other trade periodicals included _Der deutsche Lichtspiel-Theater-Besitzer_ (Berlin, 1909\u20131914), _Erste Internationale Film-Zeitung_ (Berlin, 1908\u20131920), _Film und Lichtbild_ (Stuttgart, 1912\u20131914), and _Die Lichtbild-B\u00fchne_ (Berlin, 1908\u20131940). Helmut H. Diederichs provides a more complete survey of the trade press in his _Anf\u00e4nge deutscher Filmkritik_ (Stuttgart: Fischer, 1986). On _Der Kinematograph_ in particular, see Thomas Schorr, \"Die Film- und Kinoreformbewegung und die Deutsche Filmwirtschaft. Eine Analyse des Fachblatts _Der Kinematograph_ (1907\u20131935) unter p\u00e4dagogischen und publizistischen Aspekten,\" PhD diss. (Universit\u00e4t der Bundeswehr, Munich, 1990). Hake also discusses the trade press in _Third Machine_ , 3\u201326.\n\n 37. C. H. Dannmeyer, _Bericht der Kommission f\u00fcr_ \" _Lebende Photographien_ \" (Hamburg: Kampen, 1907), 27\u201328.\n\n 38. Hellwig, _Kind und Kino_ , 22. Hellwig wrote much on _Schundfilms_ and censorship, including \"Die Beziehungen zwischen Schundliteratur, Schundfilms und Verbrechen,\" _Archiv f\u00fcr Kriminal-Anthropologie und Kriminalistik_ 51, no. 1 (24 January 1913): 1\u201332; \"Die ma\u00dfgebenden Grunds\u00e4tze f\u00fcr Verbote von Schundfilms nach geltendem und k\u00fcnstigem Rechte,\" _Verwaltungsarchiv_ 21 (1913): 405\u2013455; and _Die Filmzensur: Eine rechtsdogmatische und rechtpolitische Er\u00f6rterung_ (Berlin: Frankenstein, 1914).\n\n 39. See Eileen Bowser, _The Transformation of Cinema, 1907\u20131915_ (New York: Scribner, 1990), 37\u201352. See also J. A. Lindstrom, \"'Getting a Hold Deeper in the Life of the City': Chicago Nickelodeons, 1905\u20131914.\" PhD diss. (Northwestern University, 1998); Lee Grieveson, _Policing Cinema: Movies and Censorship in Early-Twentieth-Century America_ (Berkeley: University of California Press, 2004); and Jennifer Lynn Peterson, _Education in the School of Dreams: Travelogues and Early Nonfiction Film_ (Durham, N.C.: Duke University Press, 2013), esp. chap. 3.\n\n 40. Dannmeyer, _Bericht der Kommission_ , 39.\n\n 41. \"Die Er\u00f6ffnung des Reform-Kinematographentheater,\" _Der Kinematograph_ no. 32 (7 August 1907). _Der Kinematograph_ was not paginated. For more on the sometimes stuffy discussion of ventilation, see a translation of the American Society of Heating and Ventilation Engineers' \"Report of Committee on Standards for Ventilation Legislation for Motion Picture Show Places,\" in _Gesundheits-Ingenieur_ 36, no. 22 (31 May 1913): 409\u2013410; and a German response, Konrad Meier, \"Vorschriften \u00fcber L\u00fcftung von Kinotheatern,\" _Gesundheits-Ingenieur_ 36, no. 26 (28 June 1913): 483\u2013484.\n\n 42. \"Ein kurzer R\u00fcckblick auf die erste Woche des Reform-Kinematographen-Theaters,\" _Der Kinematograph_ no. 33 (14 August 1907).\n\n 43. \"Kinematographische Reformvereinigung,\" _Der Kinematograph_ no. 43 (23 October 1907).\n\n 44. For more on _Kino-Kommissions_ , see Sabine Lenk and Frank Kessler, \"The Institutionalization of Educational Cinema: The Case of the _Kinoreformbewegung_ in Germany,\" in _The Institutionalization of Educational Cinema: Educational Cinemas in North America and Europe in the 1910s and 1920s_ , ed. Marina Dahlquist and Joel Frykholm (Bloomington: Indiana University Press, forthcoming). See also Rudolf W. Kipp, _Bilddokumente zur Geschichte des Unterrichtsfilms_ (Gr\u00fcnwald, Germany: Institut f\u00fcr Film und Bild in Wissenschaft und Unterricht, 1975), 13\u201315.\n\n 45. \"Kinematographische Reformvereinigung.\" Oskar Kalbus, one of the driving forces behind UFA's _Kulturfilm_ division, later intimated that these donations were not uncontroversial: \"Although this association was soon sharply criticized because of the close relationship between Lemke and the French film industry, it can nevertheless take credit for having given the first important impetus for the introduction of film in schools.\" Kalbus, \"Abri\u00df einer Geschichte der deutschen Lehrfilmbewegung,\" in _Das Kulturfilmbuch_ , ed. Edgar Beyfu\u00df and Alexander Kossowsky (Berlin: Chryselius'scher, 1924), 1\u201313, here 3.\n\n 46. Hermann Lemke, \"Die Verwertung und Nutzbarmachung neuer Film-Ideen\u2014K\u00fcnstlerische Films,\" _Der Kinematograph_ no. 57 (29 January 1908).\n\n 47. Ludwig Brauner, \"Die Kino-Ausstellung in Berlin,\" _Der Kinematograph_ no. 104 (25 December 1908).\n\n 48. Indeed, by this time the relations between the exhibitors and the reformers and trade journals were downright hostile. See \"Die Kino-Austellung und 'Wir,'\" _Erste Internationale Film-Zeitung_ 6, no. 50 (14 December 1912): 52.\n\n 49. Hermann H\u00e4fker, \"Eine Reise an die Quellen der Kinematographie,\" _Der Kinematograph_ no. 163 (9 February 1910); and 172 (13 April 1910).\n\n 50. Hermann Lemke, \"Volkst\u00fcmliche Reisebeschreibungen,\" _Der Kinematograph_ no. 34 (21 August 1907).\n\n 51. _Der Kinematograph_ no. 258 (6 December 1911).\n\n 52. Paul Samuleit and Emil Borm, _Der Kinematograph als Volks- und Jugendbildungsmittel_ (Berlin: Gesellschaft f\u00fcr Verbreitung von Volksbildung, 1912), 23\u201324, quoted in Hake, _Third Machine_ , 36 (translation modified).\n\n 53. Hake, _Third Machine_ , 36\u201338.\n\n 54. Unwilling to rely just on production companies, by 1909 Lemke hoped to create a cost-sharing distribution cooperative among interested schools. See Hermann Lemke, _Praktische Forderungen f\u00fcr die Verwertung der Kinematographie im Unterricht_ (Friedenau: Schule und Technik, 1909). Georg Victor Mendel agreed and followed up with a plan to open a \"purely scientific [ _wissenschaftlichen_ ] theater\": Georg Victor Mendel, _Kinematographie und Schule: Plan zur Gr\u00fcndung eines rein wissenschaftlichen Theaters f\u00fcr Kinematographie und Projektion_ (Berlin: privately printed, 1909).\n\n 55. The legal discourse on cinema in Germany is far too vast to even attempt a survey here. Albert Hellwig's reviews are the best place to start, however: _Rechtsquellen des \u00f6ffentlichen Kinematographenrechts_ (M. Gladbach [M\u00f6nchengladbach]: Volksvereins, 1913); and _\u00d6ffentliches Lichtspielrecht_ (M. Gladbach: Volksvereins, 1921). Other contemporary surveys include Bruno May, _Das Recht des Kinematographen_ (Berlin: Falk, 1912); and Hans M\u00fcller-Sanders, \"Die Kinematographenzensur in Preu\u00dfen,\" PhD diss. (Badischen Ruprecht-Karls-Universit\u00e4t, Heidelberg, 1912). See also Gary D. Stark, \"Cinema, Society, and the State,\" in Stark and Lackner, _Essays on Culture and Society in Modern Germany_ , 122\u2013166; and Kaspar Maase, \"Massenkunst und Volkserziehung: Die Regulierung von Film und Kino im deutschen Kaiserreich,\" _Archiv f\u00fcr Sozialgeschichte_ 41 (2001): 39\u201377.\n\n 56. Hellwig, _\u00d6ffentliches Lichtspielrecht_ , 32\u201333. Not all regulations applied to the same theaters at the same time, of course. For an excellent case study of the variety of local tactics, see Amelie Duckwitz, Martin Loiperdinger, and Susanne Theisen, \"'Kampf dem Schundfilm!': Kinoreform and Jugendschutz in Trier,\" _KINtop_ 9 (2000): 53\u201363.\n\n 57. My presentation of the GVV is indebted to Schorr, \"Die Film- und Kinoreformbewegung,\" 81\u201394; and Horst Dr\u00e4ger, _Die Gesellschaft f\u00fcr Verbreitung von Volksbildung: Eine historisch-problemgeschichtliche Darstellung von 1871\u20131914_ (Stuttgart: Klett, 1975), 226\u2013237.\n\n 58. Dr\u00e4ger, _Gesellschaft f\u00fcr Verbreitung_ , 236.\n\n 59. Willi Warstat and Franz Bergmann, _Kino und Gemeinde_ (M. Gladbach: Volksvereins, 1913), 114\u2013116.\n\n 60. Heiner Schmitt, _Kirche und Film: Kirchliche Filmarbeit in Deutschland von ihren Anf\u00e4ngen bis 1945_ (Boppard: Boldt, 1979), 41. For more on the role of Catholicism in this sphere, see Margaret Stieg Dalton, _Catholicism, Popular Culture, and the Arts in Germany, 1880\u20131933_ (Notre Dame, Ind.: University of Notre Dame Press, 2005).\n\n 61. The \"monopoly\" system, established in Germany between 1910 and 1911, allowed distributors to acquire sole rights to a film and pass this exclusivity to cinema managers in the form of local exhibition rights. The theater owner's local monopoly enabled him to charge more and make, for the first time in Germany, a considerable profit. See Corinna M\u00fcller, _Fr\u00fche deutsche Kinematographie: formale, wirtschaftliche und kulturelle Entwicklungen 1907\u20131912_ (Stuttgart and Weimar: Metzler, 1994), 126\u2013158; as well as her essay, \"The Emergence of the Feature Film in Germany Between 1910 and 1911,\" in _Before Caligari: German Cinema, 1895\u20131920_ , ed. Paolo Cherchi Usai and Lorenzo Codelli (Madison: University of Wisconsin Press, 1990), 94\u2013113.\n\n 62. The best survey of the role of the _Lichtbilderei_ in the reform movement is Diederichs, _Anf\u00e4nge deutscher Filmkritik_ , 84\u201388.\n\n 63. Dr\u00e4ger, _Gesellschaft f\u00fcr Verbreitung_ , 234\u2013235.\n\n 64. Volker Schulze, \"Fr\u00fche kommunale Kinos und die Kinoreformbewegung in Deutschland bis zum Ende des ersten Weltkriegs,\" _Publizistik_ 22, no. 1 (January\u2013March 1977): 61\u201371.\n\n 65. Minutes from the meeting of the community representatives of Eickel, 14 May 1912 (archive of the City of Wanne-Eickel), quoted in Schulze, \"Fr\u00fche kommunale Kinos,\" 64.\n\n 66. Rudolf Pechel in _Literarischen Echo_ 16 (1913\/1914): 582, quoted in Ludwig Greve, Margot Pehle, and Heidi Westhoff, eds., _H\u00e4tte ich das Kino! Die Schriftsteller und der Stummfilm_ (Munich: K\u00f6sel, 1976), 68. Pechel reviewed Willy Rath's _Kino und B\u00fchne_ (M. Gladbach: Volksvereins, 1913).\n\n 67. Arthur Mellini, \"Die ganze Richtung passt uns nicht!\" _Lichtbild-B\u00fchne_ 5 (4 February 1911): 3\u20134, quoted in Karen J. Kenkel, \"The Nationalisation of the Mass Spectator in Early German Film,\" in _Celebrating 1895: The Centenary of Cinema_ , ed. John Fullerton (Sydney: Libbey, 1998), 155\u2013162, here 158.\n\n 68. Max Kullmann, \"Die Entwicklung des deutschen Lichtspieltheater,\" PhD diss. (University of Nuremberg, 1935), quoted in Hake, _Third Machine_ , 27. Kullmann quoted a film theater owner.\n\n 69. Siegfried Kracauer, _From Caligari to Hitler_ (Princeton, N.J.: Princeton University Press, 1947), 19.\n\n 70. Hake, _Third Machine_ , 28.\n\n 71. Helmut Kommer, _Fr\u00fcher Film und sp\u00e4te Folgen: Zur Geschichte der Film- und Fernseherziehung_ (Berlin: Basis, 1979).\n\n 72. Hermann Lemke, _Die Kinematographie der Gegenwart, Vergangenheit und Zukunft_ (Leipzig: Demme, 1911), 24.\n\n 73. Many scholars have stressed the connection between \"the masses\" and \"the feminine\" as an indication of the anxieties and spirit of the age. This line of reasoning is indeed extremely significant, but the connection between \"the masses\" and \"children\" (as a similarly charged rhetorical construction) deserves a closer look. On the masses as feminine, see esp. Susanna Barrows, _Distorting Mirrors: Visions of the Crowd in Late Nineteenth-Century France_ (New Haven: Yale University Press, 1981). For another, parallel examination of German reformers on cinema, children, and the masses, see Karen J. Kenkel, \"The Adult Children of Early Cinema,\" _Women in German Yearbook_ (2000): 137\u2013160.\n\n 74. Lorenz Pieper, \"Kino und Drama,\" _Bild und Film_ 1, no. 1 (1912): 5.\n\n 75. Georg Luk\u00e1cs, \"Thoughts Toward an Aesthetic of the Cinema,\" trans. Janelle Blankenship, _Polygraph_ 13 (2001): 13\u201318, here 16 (emphasis in original). Originally published as \"Gedanken zu einer Aesthetik des 'Kino,'\" _Frankfurter Zeitung_ 251 (10 September 1913): 1\u20132.\n\n 76. This phrase and _vom Kinde aus_ are attributed to Hamburg pedagogue Johannes Gl\u00e4ser, one of many who popularized and realized Key's suggestions. See Scheibe, _Die Reformp\u00e4dagogische Bewegung, 1900\u20131932_ , 65.\n\n 77. Ellen Key, \"Erziehung,\" in _Das Jahrhundert des Kindes_ (Berlin, 1905), in Flitner and Kudritzki, _Die deutsche Reformp\u00e4dagogik_ , 52\u201354, here 52.\n\n 78. Stephen Kern, \"Freud and the Emergence of Child Psychology, 1880\u20131910,\" PhD diss. (Columbia University, 1970), 264.\n\n 79. Charles Darwin, _The Descent of Man_ (London, 1871), quoted in Kern, \"Freud,\" 212.\n\n 80. The importance of \"primitivism\"\u2014of which Darwin was a prime but not uncommon example\u2014for this connection between children (or the feminine) and the masses cannot be underestimated. For its role in shaping turn of the century cultural agendas in Germany, see Doris Kaufmann, \"'Pushing the Limits of Understanding': The Discourse on Primitivism in German _Kulturwissenschaften_ , 1880\u20131930,\" _Studies in History and Philosophy of Science_ 39 (2008): 434\u2013443. For examinations in relation to cinema, see Assenka Oksiloff, _Picturing the Primitive: Visual Culture, Ethnography, and Early German Cinema_ (New York: Palgrave, 2001); and Beth Corzo-Duchardt, \"Primal Screen: Primitivism and American Silent Film Spectatorship,\" PhD diss. (Northwestern University, 2013).\n\n 81. Gustave Le Bon, _The Crowd: A Study of the Popular Mind_ (Atlanta, Ga.: Cherokee, 1982), 16. Hereafter cited parenthetically. Originally published as _Psychologie des foules_ (Paris: Alcan, 1895). Translated and published in German as _Psychologie der Massen_ (Leipzig: Klinkhardt, 1908). See also Robert A. Nye, _The Origins of Crowd Psychology: Gustave Le Bon and the Crisis of Mass Democracy in the Third Republic_ (London and Beverly Hills, Calif.: Sage, 1975)..\n\n 82. Erika Apfelbaum and Gregory R. McGuire, \"Models of Suggestive Influence and the Disqualification of the Social Crowd,\" in _Changing Conceptions of Crowd Mind and Behavior_ , ed. C. F. Graumann and S. Moscovici (New York and Berlin: Springer, 1986), 27\u201350.\n\n 83. See Walter Serner, \"Kino und Schaulust,\" in Schweinitz, _Prolog vor dem Film_ , 208\u2013214. Originally published in _Die Schaub\u00fchne_ 9, nos. 34\/35 (1913): 807\u2013811. Chapter 4 will discuss _Schaulust_ and this essay in more detail.\n\n 84. Dannmeyer, _Bericht der Kommission_ , 27\u201328.\n\n 85. Emilie Altenloh, _Zur Soziologie des Kino: Die Kino-Unternehmung und die sozialen Schichten ihrer Besucher_ (Jena: Diederichs, 1914), 91.\n\n 86. Altenloh, _Zur Soziologie des Kino_ , 65.\n\n 87. Albert Hellwig, \"\u00dcber die sch\u00e4dliche Suggestivkraft kinematographischer Vorf\u00fchrung,\" _Aerztliche Sachverst\u00e4ndigen-Zeitung_ 20, no. 6 (15 March 1914): 122. Hellwig was reviewing and citing from an article by Italian psychiatrist Giuseppe d'Abundo, \"Sopra alcuni particolari effetti delle projezioni cinematografiche nei nevrotici,\" _Rivista Italiana di Neuropatologia, Psichiatria ed Elettroterapia_ 4, no. 10 (October 1911): 433\u2013442; for another German review, see \"Kinematograph als Krankheitsstifter,\" in _Fortschritte der Medizin_ 30 (1912): 302. For more on Italian uses of cinematography in the human sciences, see Silvio Alovisio, _L'occhio sensibile. Cinema e scienze della mente nell'Italia del primo Novecento. Con una antologia di testi d'epoca_ (Turin: Edizioni Kaplan, 2013).\n\n 88. Anson Rabinbach, _The Human Motor: Energy, Fatigue, and the Origins of Modernity_ (Berkeley and Los Angeles: University of California Press, 1990), 157.\n\n 89. Hermann Lemke, \"Die kinematographische Reformpartei, ihre Aufgaben und Ziele,\" _Der Kinematograph_ no. 42 (16 October 1907).\n\n 90. Albert Hellwig, _Schundfilms: Ihr Wesen, ihre Gefahren und ihre Bek\u00e4mpfung_ (Halle an der Saale, Germany: Waisenhaus, 1911), 33, quoted in Hake, _Third Machine_ , 39.\n\n 91. Friedrich Schiller, _On the Aesthetic Education of Man_ , trans. Elizabeth M. Wilkinson and L. A. Willoughby (Oxford: Oxford University Press, 1967), 161.\n\n 92. \"Die Kultur-Arbeit des Kinematographen-Theaters,\" _Die Lichtbild-B\u00fchne_ 2, no. 41 (4 February 1909).\n\n 93. Erwin Ackerknecht, _Das Lichtspiel im Dienste der Bildungspflege: Handbuch f\u00fcr Lichtspielreformer_ (Berlin: Weidmannsche, 1918), 66.\n\n 94. August Julius Langbehn, \"Rembrandt als Erzieher,\" in _Die Kunsterziehungsbe-wegung_ , ed. Hermann Lorenzen (Bad Heilbrunn, Germany: Klinkhardt, 1966), 7\u201317. Hereafter cited parenthetically. Stern's _The Politics of Cultural Despair_ provides the standard account of Langbehn's place in history.\n\n 95. For more on the reverberations of Langbehn's essay through Wilhelmine Germany, see Corona Hepp, _Avantgarde: Moderne Kunst, Kulturkritik und Reformbewegungen nach der Jahrhundertwende_ (Munich: Deutscher Taschenbuch, 1987).\n\n 96. Alfred Lichtwark, \"Der Deutsche der Zukunft,\" in Flitner and Kudritzki, _Die deutsche Reformp\u00e4dagogik_ , 99\u2013110, here 104 (emphasis in original). Hereafter cited parenthetically. Lichtwark and Langbehn were acquaintances; Lichtwark introduced Langbehn to the work of Rembrandt in 1887. See Gisela Wilkending, _Volksbildung und P\u00e4dagogik_ \" _vom Kinde aus_ \" _: Eine Untersuchung zur Geschichte der Literaturp\u00e4dagogik in den Anf\u00e4ngen der Kunsterziehungsbewegung_ (Weinheim, Germany: Beltz, 1980), 79\u201385. For more on Lichtwark, see Julius Gebhard, _Alfred Lichtwark und die Kunsterziehungsbewegung in Hamburg_ (Hamburg: Hoffmann und Campe, 1947); Hans Pr\u00e4ffcke, _Der Kunstbegriff Alfred Lichtwarks_ (Hildesheim, Z\u00fcrich, and New York: Olms, 1986); Carolyn Kay, _Art and the German Bourgeoisie: Alfred Lichtwark and Modern Painting in Hamburg, 1886\u20131914_ (Toronto and Buffalo, N.Y.: University of Toronto Press, 2002); but esp. Jennifer Jenkins, _Provincial Modernity: Local Culture and Liberal Politics in Fin-de-Si\u00e8cle Hamburg_ (Ithaca, N.Y.: Cornell University Press, 2003).\n\n 97. Jenkins, _Provincial Modernity_ , 76.\n\n 98. Konrad Lange, \"Das Wesen der k\u00fcnstlerischen Erziehung,\" in Lorenzen, _Kunsterziehungsbewegung_ , 21\u201326, here 22. For more on the art education movement, see Peter Joerissen, _Kunsterziehung und Kunstwissenschaft im wilhelminischen Deutschland, 1871\u20131918_ (Cologne and Vienna: B\u00f6hlau, 1979).\n\n 99. Lange, _Die k\u00fcnstlerische Erziehung der deutschen Jugend_ , 10.\n\n. Lange, \"Das Wesen der k\u00fcnstlerischen Erziehung,\" 26.\n\n. Alfred Lichtwark, \"Die Aufgaben der Kunsthalle: Antrittsrede den 9. December 1886,\" In _Drei Programme_ , 2d ed. (Berlin: Cassirer, 1902), 11\u201331, here 29.\n\n. Eckard Schaar, \"Zust\u00e4nde,\" in Alfred Lichtwark, _Erziehung des Auges: Ausgew\u00e4hlte Schriften_ , ed. Eckard Schaar (Frankfurt: Fischer, 1991), 8, quoted in Jenkins, _Provincial Modernity_ , 64.\n\n. Alfred Lichtwark, _Die Bedeutung der Amateur-Photographie_ (Halle an der Saale, Germany: Knapp, 1894), 1.\n\n. Alfred Lichtwark, \"Museen als Bildungsst\u00e4tten,\" _Der Deutsche der Zukunft_ (Berlin: Cassirer, 1905), 89\u2013107.\n\n. Alfred Lichtwark, _\u00dcbungen in der Betrachtung von Kunstwerken_ (Dresden: K\u00fchtmann, 1900), 17.\n\n. See, for instance, John Dewey, _Art as Experience_ (New York: Putnam, 1984).\n\n. See, e.g., Walter Geisel, _Wie ich mit meinen Jungens Kunstwerke betrachte_ (Gl\u00fcckstadt: Geisel, 1904); Paul Quensel, _Meisterbilder und Schule: Anregungen zu praktischen Versuchen_ (Munich: Kunstwart-Verl, 1905); Leipziger Lehrerverein, ed., _Bildbetrachtungen: Arbeiten aus der Abteilung f\u00fcr Kunstpflege des Leipziger Lehrervereins_ (Leipzig: Teubner, 1906); and Ulrich Diem, _Bildbetrachtung: Eine Wegleitung f\u00fcr Kunstfreunde_ (St. Gallen: Fehr'sche, 1919). _Bildbetrachtung_ was even more popular as a teaching method after World War II.\n\n. Heinrich Wolgast, \"Die Bedeutung der Kunst f\u00fcr die Erziehung,\" in Lorenzen, _Die Kunsterziehungsbewegung_ , 17\u201320, here 19.\n\n. For a European history of the movement, along with a clear explication of the principles of _Bild_ \\- _und Kunstbetrachtung_ , see Ludwig Praehauser, _Erfassen und Gestalten: Die Kunsterziehung als Pflege formender Kr\u00e4fte_ (Salzburg: M\u00fcller, 1950).\n\n. Pestalozzi, _How Gertrude Teaches Her Children_ [1801], ed. Ebenezer Cooke, trans. Lucy E. Holland and Frances C. Turner, 2d ed. (Syracuse, N.Y.: Bardeen, 1898), tenth letter, 220 (emphasis in original).\n\n. Clive Ashwin, \"Pestalozzi and the Origins of Pedagogical Drawing,\" _British Journal of Educational Studies_ 29, no. 2 (June 1981): 138\u2013151, here 146.\n\n. Keiichi Takaya, \"The Method of _Anschauung_ : From Johann H. Pestalozzi to Herbert Spencer,\" _Journal of Educational Thought_ 37, no. 1 (2003): 77\u201399, here 84.\n\n. Robert Ulich, \"Pestalozzi, Johann Heinrich,\" in _The Encyclopedia of Philosophy_ , vol. 6, ed. Paul Edwards (New York: Macmillan, 1967), 121\u2013122.\n\n. Melanie Judith Keene, \"Object Lessons: Sensory Science Education, 1830\u20131870,\" PhD diss. (University of Cambridge, 2008), 51\u201354.\n\n. Ulich, \"Pestalozzi, Johann Heinrich,\" 122.\n\n. Ashwin, \"Pestalozzi and the Origins of Pedagogical Drawing,\" 146.\n\n. Christopher Owen Ritter, \"Re-presenting Science: Visual and Didactic Practice in Nineteenth-Century Chemistry,\" PhD diss. (University of California, Berkeley), 2001, 128.\n\n. W. T. Harris, editor's preface to _Herbart_ ' _s ABC of Sense Perception and Minor Pedagogical Works_ , by Johann Friedrich Herbart, ed. and trans. William J. Eckoff (New York: Appleton, 1896), vii.\n\n. Herbert Spencer, _Education: Intellectual, Moral, and Physical_ [1861] (New York: Appleton, 1896), quoted in Takaya, \"The Method of _Anschauung_ ,\" 81.\n\n. The best survey of its dissemination in Germany is Gottlieb Gustav Deussing, \"Der Anschauungsunterricht in der deutschen Schule von Comenius bis zur Gegenwart,\" PhD diss. (Universit\u00e4t Jena, 1884).\n\n. On _Anschauungsunterricht_ in the natural sciences, see Massimiano Bucchi, \"Images of Science in the Classroom: Wallcharts and Science Education, 1850\u20131920,\" _British Journal for the History of Science_ 31, no. 2 (1998): 161\u2013184; Lynn K. Nyhart, \"Science, Art, and Authenticity in Natural History Displays,\" in _Models: The Third Dimension of Science_ , ed. Soraya de Chadarevian and Nick Hopwood (Stanford, Calif.: Stanford University Press, 2004), 307\u2013335; Nyhart, _Modern Nature: The Rise of the Biological Perspective in Germany_ (Chicago: University of Chicago Press, 2009), esp. chap. 5, \"The 'Living Community' in the Classroom\"; Henning Schmidgen, \"Pictures, Preparations, and Living Processes: The Production of Immediate Visual Perception ( _Anschauung_ ) in Late-19th-Century Physiology,\" _Journal of the History of Biology_ 37, no. 3 (October 2004): 477\u2013513; and Schmidgen, \"1900\u2014The Spectatorium: On Biology's Audiovisual Archive,\" _Grey Room_ 43 (2011): 42\u201365.\n\n. I have examined the trope of efficiency in visual education in \"The Efficiency of Images: Educational Effectiveness and the Modernity of Motion Pictures,\" in _The Visual Culture of Modernism_ , SPELL: Swiss Papers in English Language and Literature 26, ed. Deborah L. Madsen and Mario Klarer (T\u00fcbingen: Narr, 2011), 41\u201359; and \"Dissecting the Medical Training Film,\" in _Beyond the Screen: Institutions, Networks and Publics of Early Cinema_ , ed. Marta Braun et al. New Barnet, U.K.: Libbey, 2012), 161\u2013167.\n\n. Carl Jacobj, \"Anschauungsunterricht und Projektion,\" _Zeitschrift f\u00fcr wissenschaftliche Mikroskopie und mikroskopische Technik_ 36, no. 4 (1919): 273\u2013314, here 275, quoted in Schmidgen, \"1900\u2014The Spectatorium,\" 51.\n\n. Ashwin, \"Pestalozzi and the Origins of Pedagogical Drawing,\" 146 (emphasis in original).\n\n. For a contemporary assessment of the gap between reform ideals and actual practice, see I. L. Kandel, \"Germany,\" in _Comparative Education: Studies of the Educational Systems of Six Modern Nations_ , ed. Peter Sandiford (London and Toronto: Dent, 1918), 121\u2013130; or, to a lesser extent, the apologetic William S. Learned, _An American Teacher_ ' _s Year in a Prussian Gymnasium_ (New York: Educational Review, 1911).\n\n. \"Besuch kinematographischer Vorf\u00fchrung durch Sch\u00fcler h\u00f6herer Lehranstalten (Breslau 1910)\" and \"Besuch der Kinematographentheater durch Sch\u00fcler und Sch\u00fclerinnen sowie durch die Z\u00f6glinge der Seminare und Pr\u00e4parandenanstalten (Berlin 1912)\" in _Dokumente zur Geschichte der Schulfilmbewegung in Deutschland_ , ed. Fritz Terveen (Emsdetten: Lechte, 1959), 16\u201317.\n\n. Otfrid von Hanstein, _Kinematographie und Schule. Ein Vorschlag zur Reform des Anschauungs-Unterrichts_ (Berlin: Lichtspiele Mozartsaal, 1911), 3. Other major statements about the educational use of film before World War I include Samuleit and Borm, _Der Kinematograph als Volks- und Jugendbildungsmittel_ ; Adolf Sellmann, _Der Kinematograph als Volkserzieher?_ (Langensalza, Germany: Beyer, 1912); Friedrich Murawski, _Die Kinematographie und ihre Beziehungen zu Schule und Unterricht_ (Dresden: Bieyl and Kaemmerer, 1914); Sellmann, _Kino und Volksbildung_ (M. Gladbach: Volksvereins Verlag, 1914); and esp. Sellmann, _Kino und Schule_ (M. Gladbach: Volksvereins Verlag, 1914).\n\n. For a friendly assessment, see Wilhelm Richter, \"Der Kinematograph als naturwissenschaftliches Anschauungsmittel,\" _Naturwissenschaftliche Wochenschrift_ 12, no. 52 (28 December 1913): 817\u2013820.\n\n. K. R\u00fcswald, \"Der Film im Erdkundlichen und Naturwissenschaftlichen Unterricht,\" in Terveen, _Dokumente zur Geschichte der Schulfilmbewegung in Deutschland_ , 43\u201344. A fine summary of the arguments forwarded against educational uses of film can be found in H. Graupner, \"Unterrichtshygiene,\" in _Handbuch der deutschen Schulhygiene_ , ed. Hugo Selter (Dresden and Leipzig: Steinkopff, 1914), 174\u2013321, esp. his section on \"Kinematograph und Unterrichtshygiene,\" 302\u2013307.\n\n. Sellmann, _Kino und Schule_ , 15 (emphasis in original).\n\n. Richard Kretz, \"Die Anwendung der Photographie in der Medicin,\" _Wiener klinische Wochenschrift_ 7, no. 44 (1 November 1894): 832.\n\n. Paul Knospe, _Der Kinematograph im Dienste der Schule. Unter besonderer Ber\u00fccksichtigung des erdkundlichen Unterrichts_ (Halle an der Saale, Germany: Waisenhaus, 1913), 9. There are many more examples, but see also Bastian Schmid, \"Kinematographie und Schule,\" _Die Naturwissenschaften_ 1, no. 6 (7 February 1913): 145\u2013146.\n\n. Oskar Kalbus summarizes and quotes from such complaints in _Der Deutsche Lehrfilm in der Wissenschaft und im Unterricht_ (Berlin: Heymanns, 1922), 6.\n\n. Mendel, _Kinematographie und Schule_.\n\n. \"Zur Er\u00f6ffnung des Ernemann-Kino in Dresden,\" _Der Kinematograph_ no. 134 (21 July 1909). It was not unusual for film equipment manufacturers in Germany to have exhibition storefronts that were open to the public in a quasi-museum-like setting. See Deac Rossell, \"Beyond Messter: Aspects of Early Cinema in Berlin,\" _Film History_ 10 (1998): 52\u201369.\n\n. \"Die Dresdner 'Kosmographia,'\" _Bild und Film_ 1, no. 1 (1912): 19. See also Uli Jung, \"Film f\u00fcr Lehre und Bildung,\" in _Geschichte des dokumentarischen Films in Deutschland_ , vol. 1, ed. Uli Jung and Martin Loiperdinger (Stuttgart: Reclam, 2005), 333\u2013340.\n\n. Those venues were the Ernemann-Kino (1909) and the Kosmographia (1910) in Dresden; the Reform-Kino in Braunschweig (1910); the Reformtheater in Bremen (1911); the Gemeindekino in Eickel (1912); the Germania Saal in Hagen (1912); the Musterlichtbildb\u00fchne in Altona (1912); and the Urania in Stettin (1914).\n\n. Hermann Bredtmann, \"Kinematographie und Schule,\" _P\u00e4dagogisches Archiv_ 56, no. 3 (1914): 154\u2013163, here 161 and 163.\n\n. On Lemke, see Schorr, \"Die Film- und Kinoreformbewegung,\" 56ff; M\u00fcller, _Fr\u00fche deutsche Kinematographie_ , 71\u201376; and M\u00fcller, \"Der fr\u00fche Film, das fr\u00fche Kino und seine Gegner und Bef\u00fcrworter,\" in _Schund und Sch\u00f6nheit: Popul\u00e4re Kultur um 1900_ , ed. Kaspar Maase und Wolfgang Kaschuba (Cologne, Weimar, and Vienna: B\u00f6hlau, 2001), 62\u201391.\n\n. See the technophilia of Hermann Lemke's _Durch die Technik zur Schulreform. Zwei modern-technische Lehrmethoden und Veranschaulichungsmittel in der Schule der Zukunft_ (Leipzig: Demme, 1911), in which he predicts that the combination of films and phonographs will make teachers obsolete. See also Sellmann's vision of a \"complete revolution\" in teaching once projectors are universally installed; _Kino und Schule_ , 39.\n\n. Hermann Lemke, _Die kinematographische Unterrichtsstunde_ (Leipzig: Demme, 1911), 5.\n\n. M\u00fcller, \"Der fr\u00fche Film, das fr\u00fche Kino und seine Gegner und Bef\u00fcrworter,\" 75. Sellmann held similar workshops in Eickel, but they also did not last in the long run. See \"Bericht \u00fcber eine Besprechung der Kinokommssion des Westf\u00e4lischen Landgemeindetages anl\u00e4\u00dflich der Er\u00f6ffnungsfeier des Gemeindelichtspielhauses in Eickel,\" _Bild und Film_ 2, no. 3 (1912): 70\u201371. See also Lenk and Kessler, \"The Institutionalization of Educational Cinema.\"\n\n. Actually, Herbart was not this clear or consistent, so these steps are the result of refinements by later Herbartians such as Wilhelm Rein, esp. _P\u00e4dagogik im Grundri\u00df_ (1890), 4th ed. (Leipzig: G\u00f6schen, 1907), 109. For a helpful discussion of Herbart and Herbartianism, see Harold B. Dunkel, _Herbart and Education_ (New York: Random House, 1969).\n\n. Miriam Hansen, _Babel and Babylon: Spectatorship in American Silent Film_ (Cambridge, Mass.: Harvard University Press, 1991), 93.\n\n. Helmut H. Diederichs, \"Naturfilm als Gesamtkunstwerk: Hermann H\u00e4fker und sein 'Kinetographie'-Konzept,\" _Augenblick_ 8 (1990): 37\u201360. If H\u00e4fker is known at all to English-speaking readers, it is through Kracauer's characterization of him in _From Caligari to Hitler_ as the man who \"praised war as the salvation from the evils of peace\" (28). H\u00e4fker saw World War I mainly as an opportunity for the state to take control of cinema and put his plans into action. While there is no doubt that H\u00e4fker was conservative, nationalistic, and blind to the horrors of war, it would be unfair to depict him as a warmonger with the prefascist tendencies implied by Kracauer. H\u00e4fker earned a \"heart attack\" in a concentration camp for his resistance to the Nazi government. All biographical information comes from Diederichs's article and his entry on H\u00e4fker in _Cinegraph_ , ed. Hans-Michael Bock (Munich: edition text + kritik, 1984). My presentation of H\u00e4fker is indebted to these essays and my conversations with Diederichs.\n\n. Hermann H\u00e4fker, \"Zur Dramaturgie der Bilderspiele,\" _Der Kinematograph_ no. 32 (7 August 1907).\n\n. Hermann H\u00e4fker, \"Meisterspiele,\" _Der Kinematograph_ no. 56 (22 January 1908).\n\n. H\u00e4fker, \"Zur Dramaturgie der Bilderspiele.\"\n\n. Hermann H\u00e4fker, _Kino und Kunst_ (M. Gladbach: Volksvereins, 1913), 5.\n\n. See, for instance, Robert Gaupp and Konrad Lange, _Der Kinematograph als Volksunterhaltungsmittel_ (Munich: Callwey, 1912); R. Stigler, \"\u00dcber das Flimmern der Kinematographen,\" _Archiv f\u00fcr die gesamte Physiologie des Menschen und der Tiere_ (Bonn) 123 (1908): 224\u2013232; or Naldo Felke, \"Die Gesundheitssch\u00e4dlichkeit des Kinos,\" _Die Umschau_ 17, no. 1 (1 January 1913): 254\u2013255.\n\n. Rabinbach, _Human Motor_ , 21.\n\n. H\u00e4fker, \"Meisterspiele.\"\n\n. H\u00e4fker, _Kino und Kunst_ , 9. Hereafter cited parenthetically.\n\n. Ernst Schultze, _Der Kinematograph als Bildungsmittel_ (Halle an der Saale, Germany: Waisenhaus, 1911), 118.\n\n. Max Brethfeld, \"Neue Versuche, die Kinematographie f\u00fcr die Volksbildung und Jugenderziehung zu verwerten,\" _Neue Bahnen_ (1910): 422, quoted in Diederichs, \"Naturfilm,\" 41.\n\n. H\u00e4fker, _Kino und Kunst_ , 61. Hereafter cited parenthetically.\n\n. On the Urania lecture hall influence, see Gerhard Ebel and Otto L\u00fchrs, \"Urania\u2014eine Idee, eine Bewegung, eine Institution wird 100 Jahre alt,\" in _100 Jahre Urania: Wissenschaft heute f\u00fcr morgen_ (Berlin: Urania Berlin, 1988), 15\u201374. There are also structural similarities between H\u00e4fker's presentations and the presentation of early cinema's passion plays. See Charles Musser, _The Emergence of Cinema: The American Screen to 1907_ (New York: Scribner, 1990), 208\u2013218.\n\n. Jonathan Crary offers a concise overview of the attention psychologists paid to attention in his \"Unbinding Vision,\" _October_ 68 (Spring 1994): 21\u201344, and offers a more lengthy treatment in _Suspensions of Perception: Attention, Spectacle, and Modern Culture_ (Cambridge, Mass.: MIT Press, 1999). With regard to the perceived increase in sensual diversions, see, e.g., George M. Beard, _American Nervousness: Its Causes and Consequences_ (New York: Putnam, 1881); and Tom Lutz, _American Nervousness, 1903: An Anecdotal History_ (Ithaca, N.Y.: Cornell University Press, 1991).\n\n. See Martin Jay, _Downcast Eyes: The Denigration of Vision in Twentieth-Century French Thought_ (Berkeley and Los Angeles: University of California Press, 1993).\n\n. Schiller, _On the Aesthetic Education of Man_ , 183 (emphasis in original)\n\n. Walter Benjamin, \"The Work of Art in the Age of Its Technological Reproducibility (Third Version),\" in _Walter Benjamin, Selected Writings_ , vol. 4, _1938\u20131940_ , ed. Howard Eiland and Michael W. Jennings, trans. Edmund Jephcott and others (Cambridge, Mass.: Belknap, 2003), 251\u2013283.\n\n. Schiller, _On the Aesthetic Education of Man_ , 183.\n\n. Immanuel Kant, _Critique of Judgement_ , trans. James Creed Meredith (Oxford: Oxford University Press, 1952), 150\u2013154, \u00a740.\n\n. Schiller, _On the Aesthetic Education of Man_ , 215 (emphasis in original). For my discussion of aesthetics and ideology, I am indebted to Terry Eagleton, _The Ideology of the Aesthetic_ (Cambridge, Mass.: Blackwell, 1990).\n\n4. THE PROBLEM WITH PASSIVITY\n\n 1. Walter Benjamin, \"The Work of Art in the Age of Its Technological Reproducibility (Second Version),\" in _Walter Benjamin: Selected Writings_ , vol. 3, _1935\u20131938_ , ed. Howard Eiland and Michael W. Jennings, trans. Edmund Jephcott, Howard Eiland, and others (Cambridge, Mass.: Belknap, 2002), 101\u2013133, here 109.\n\n 2. For representative essays, see the following anthologies: Anton Kaes, ed., _Kino-Debatte: Texte zum Verh\u00e4ltnis von Literatur und Film, 1909\u20131929_ (T\u00fcbingen: Niemeyer, 1978); Ludwig Greve, Margot Pehle, and Heidi Westhoff, eds., _H\u00e4tte ich das Kino! Die Schriftsteller und der Stummfilm_ (Munich: K\u00f6sel, 1976); Fritz G\u00fcttinger, ed., _Kein Tag ohne Kino: Schriftsteller \u00fcber den Stummfilm_ (Frankfurt: Deutsches Filmmuseum, 1984); J\u00f6rg Schweinitz, ed., _Prolog vor dem Film: Nachdenken \u00fcber ein neues Medium, 1909\u20131914_ (Leipzig: Reclam, 1992); Helmut H. Diederichs, ed., _Geschichte der Filmtheorie: Kunsttheoretische Texte von M\u00e9li\u00e8s bis Arnheim_ (Frankfurt: Suhrkamp, 2004); and Richard W. McCormick and Alison Guenther-Pal, eds., _German Essays on Film_ (New York: Continuum, 2004).\n\n 3. The best overview of the early German film industry remains Corinna M\u00fcller, _Fr\u00fche deutsche Kinematographie: formale wirtschaftliche und kulturelle Entwicklungen, 1907\u20131912_ (Stuttgart and Weimar: Metzler, 1994). On early German cinema in general, see the following anthologies: Paolo Cherchi Usai and Lorenzo Codelli, eds., _Before Caligari: German Cinema, 1895\u20131920_ (Madison: University of Wisconsin Press, 1990); Thomas Elsaesser with Michael Wedel, eds., _A Second Life: German Cinema_ ' _s First Decades_ , (Amsterdam: Amsterdam University Press, 1996); and Thomas Elsaesser and Michael Wedel, eds., _Kino der Kaiserzeit: Zwischen Tradition und Moderne_ (Munich: edition text + kritik, 2002). On the _Autorenfilm_ , see esp. Helmut H. Diederichs, _Anf\u00e4nge deutscher Filmkritik_ (Stuttgart: Fischer, 1986); Diederichs, \"The Origins of the _Autorenfilm_ ,\" in Cherchi Usai and Codelli, _Before Caligari_ , 380\u2013401; and Leonardo Quaresima, \" _Dichter, Heraus!_ The _Autorenfilm_ and German Cinema of the 1910s,\" _Griffithiana_ 38\/39 (October 1990): 101\u2013120.\n\n 4. Anton Kaes, \"Literary Intellectuals and the Cinema: Charting a Controversy (1909\u20131929),\" _New German Critique_ 40 (Winter 1987): 7\u201334, here 7\u20138.\n\n 5. Sabine Hake, _The Cinema_ ' _s Third Machine: Writing on Film in Germany, 1907\u20131933_ (Lincoln: University of Nebraska Press, 1993), 63.\n\n 6. Peter Jelavich, \"'Am I Allowed to Amuse Myself Here?': The German Bourgeoisie Confronts Early Film,\" in _Germany at the Fin de Si\u00e8cle: Culture, Politics, and Ideas_ , ed. Suzanne Marchand and David Lindenfeld (Baton Rouge: Louisiana State University Press, 2004), 227\u2013249, here 247.\n\n 7. Helmut H. Diederichs, \"Kino und die Wortk\u00fcnste: Zur Diskussionen der deutschen literarischen Intelligenz 1910 bis 1915,\" _KINtop_ 13 (2004): 9\u201323. See also Diederichs, \"Fr\u00fchgeschicht deutscher Filmtheorie. Ihre Entstehung und Entwicklung bis zum Ersten Weltkrieg,\" unpublished Habilitationsschrift (J. W. Goethe-Universit\u00e4t Frankfurt am Main, 1996).\n\n 8. Stefanie Harris, _Mediating Modernity: German Literature and the_ \" _New_ \" _Media, 1895\u20131930_ (University Park: Pennsylvania State University Press, 2009).\n\n 9. Heinz-B. Heller, _Literarische Intelligenz und Film: Zu Ver\u00e4nderungen der \u00e4sthetischen Theorie und Praxis unter dem Eindruck des Films 1910\u20131930 in Deutschland_ (T\u00fcbingen: Niemeyer, 1985). See also Thomas Koebner, \"Der Film als neue Kunst: Reaktionen der literarischen Intelligenz: Zur Theorie des Stummfilms (1911\u20131924),\" in _Literaturwissenschaft-Medienwissenschaft_ , ed. Volker Canaris and Helmut Kreuzer (Heidelberg: Quelle und Meyer, 1977), 1\u201331. While they do not focus as much on the relationship between literature and film, Miriam Hansen and Heide Schl\u00fcpmann set the agenda for the study of this era in other ways, which will be discussed later in the chapter: Miriam Hansen, \"Early Silent Cinema: Whose Public Sphere?,\" _New German Critique_ 29 (Spring\/Summer 1983): 147\u2013184; Heide Schl\u00fcpmann, _Unheimlichkeit des Blicks: Das Drama des fr\u00fchen deutschen Kinos_ (Frankfurt: Stroemfeld\/Roter Stern, 1990).\n\n 10. This applies, of course, to descriptions of the _beautiful_ , rather than to those of the _sublime_ , which often emphasized an immediate overwhelming of the imagination. My thanks to Dan Morgan for reminding me of this distinction.\n\n 11. Friedrich Schiller, _On the Aesthetic Education of Man_ , trans. Elizabeth M. Wilkinson and L. A. Willoughby (Oxford: Oxford University Press, 1967), 123.\n\n 12. See, for example, Robert Vischer's description of the difference between _Sehen_ and _Schauen_ (\"seeing\" and \"scanning\") in _On the Optical Sense of Form_ (1873), in _Empathy, Form, and Space: Problems in German Aesthetics, 1873\u20131893_ , ed. and trans. Harry Francis Mallgrave and Eleftherios Ikonomou (Santa Monica, Calif.: Getty Center for the History of Art and the Humanities, 1994), 89\u2013124, esp. 93\u201395. See also Adolf Hildebrand's elaboration of this idea in _The Problem of Form in the Fine Arts_ (1893), in Mallgrave and Ikonomou, _Empathy, Form, and Space_ , 227\u2013279, esp. 229\u2013232.\n\n 13. For a complete survey, see Steve Odin, _Artistic Detachment in Japan and the West: Psychic Distance in Comparative Aesthetics_ (Honolulu: University of Hawaii Press, 2001).\n\n 14. Emilie Altenloh, _Zur Soziologie des Kino: Die Kino-Unternehmung und die sozialen Schichten ihrer Besucher_ (Jena: Diederichs, 1914), 91.\n\n 15. William Egginton, \"Intimacy and Anonymity, or How the Audience Became a Crowd,\" in _Crowds_ , ed. Jeffrey T. Schnapp and Matthew Tiews (Stanford, Calif.: Stanford University Press, 2006), 97\u2013110.\n\n 16. Jacques Ranci\u00e8re, _The Emancipated Spectator_ , trans. Gregory Elliott (London and New York: Verso, 2009), 2.\n\n 17. Carolin Duttlinger, \"Between Contemplation and Distraction: Configurations of Attention in Walter Benjamin,\" _German Studies Review_ 30, no. 1 (February 2007): 33\u201354.\n\n 18. Dudley Andrew, \"Film and Society: Public Rituals and Private Space,\" in _Exhibition: The Film Reader_ , ed. Ina Rae Hark (New York: Routledge, 2001), 161\u2013172, here 165. Originally published in _East-West Film Journal_ 1, no. 1 (1986): 7\u201322.\n\n 19. A good introduction to the broader social problem, minus cinema, is Hartmut Rosa and William E. Scheuerman, eds., _High-Speed Society: Social Acceleration, Power, and Modernity_ (University Park: Pennsylvania State University Press, 2009).\n\n 20. Georg Kleib\u00f6mer, \"Kinematograph und Schuljugend,\" _Der Kinematograph,_ no. 124 (12 May 1909) (emphasis in original in this and subsequent quotations).\n\n 21. Karl Hans Strobl, \"Der Kinematograph,\" in _Kein Tag ohne Kino: Schriftsteller \u00fcber den Stummfilm_ , ed. Fritz G\u00fcttinger (Frankfurt: Deutsches Filmmuseum, 1984), 50\u201354, here 52. Originally published in _Die Hilfe_ 17, no. 9 (2 March 1911).\n\n 22. Ph. Sommer, \"Zur Psychologie des Kinematographen,\" _Der Kinematograph_ no. 227 (3 May 1911).\n\n 23. Wilhelm Stapel, \"Der homo cinematicus,\" _Deutsches Volkstum_ 21 (October 1919): 319\u2013320, here 319.\n\n 24. Schiller, _On the Aesthetic Education of Man_ , 79. Hereafter cited parenthetically.\n\n 25. Arthur Schopenhauer, _The World as Will and Representation_ , trans. E. F. J. Payne (New York: Dover, 1966), vol. 1, \u00a738, p. 197.\n\n 26. Josef Chytry, _The Aesthetic State: A Quest in Modern German Thought_ (Berkeley and Los Angeles: University of California Press, 1989). See also David Aram Kaiser, _Romanticism, Aesthetics, and Nationalism_ (Cambridge: Cambridge University Press, 1999), esp. his chapter on \"Schiller's Aesthetic State.\"\n\n 27. Egon Friedell, \"Prolog vor dem Film,\" in Kaes, _Kino-Debatte_ , 42\u201347, here 43. Originally published in _Bl\u00e4tter des Deutschen Theaters_ 2 (1912): 509\u2013511.\n\n 28. Hermann Kienzl, \"Theater und Kinematograph,\" in Schweinitz, _Prolog vor dem Film_ , 230\u2013234, here 231. Originally published in _Der Strom_ 1, no. 7 (October 1911): 219\u2013221.\n\n 29. Walter Hasenclever, \"Der Kintopp als Erzieher: Eine Apologie,\" in Kaes, _Kino-Debatte_ , 47\u201349, here 48. Originally published in _Revolution_ 1, no. 4 (1 December 1913): n.p.\n\n 30. Lou Andreas-Salom\u00e9, _In der Schule bei Freud: Tagesbuch eines Jahres 1912\/1913_ , ed. Ernst Pfeiffer (Z\u00fcrich: Niehans, 1958), 102\u2013103.\n\n 31. Hermann Duenschmann, \"Kinematograph und Psychologie der Volksmenge. Eine sozialpolitische Studie,\" _Konservative Monatsschrift_ 69, no. 9 (June 1912): 920\u2013930, here 924.\n\n 32. Ren\u00e9 Descartes, \"The Search for Truth,\" _The Philosophical Writings of Descartes_ , trans. John Cottingham, Robert Stoothoff, and Dugald Murdoch, vol. 2 (Cambridge and New York: Cambridge University Press, 1984\u20131991), 400\u2013420, here 406.\n\n 33. John Locke, _An Essay Concerning Human Understanding_ , ed. and with an introduction by Peter H. Nidditch (Oxford: Oxford University Press, 1975), 152, book 2, chap. 10, sect. 5.\n\n 34. Locke, _Essay_ , 607, book 4, chap. 7, sect. 16.\n\n 35. Lex Newman, \"Ideas, Pictures, and the Directness of Perception in Descartes and Locke,\" _Philosophy Compass_ 4, no. 1 (2009): 134\u2013154.\n\n 36. Ulrich Rauscher, \"Die Kino-Ballade,\" in G\u00fcttinger, _Kein Tag ohne Kino_ , 143\u2013149, here 148. Originally published in _Der Kunstwart_ 26, no. 13 (1 April 1913): 1\u20136.\n\n 37. Georges Duhamel, _Sc\u00e8nes de la vie future_ (Paris: Mercure de France, 1930), 52, quoted in Walter Benjamin, \"The Work of Art in the Age of Its Technological Reproducibility (Third Version),\" in _Walter Benjamin: Selected Writings_ , vol. 4, _1938\u20131940_ , trans. Edmund Jephcott and others, ed. Howard Eiland and Michael W. Jennings (Cambridge, Mass.: Belknap, 2003), 267.\n\n 38. Georg Simmel, \"The Metropolis and Mental Life,\" in _The Sociology of Georg Simmel_ , ed. and trans. Kurt H. Wolff (New York: Free Press, 1950), 409\u2013424; Sigmund Freud, _Beyond the Pleasure Principle_ , trans. James Strachey (London: Hogarth, 1950).\n\n 39. Joseph Landau, \"Mechanisierte Unsterblichkeit,\" in _Der Deutsche Kaiser im Film_ , ed. Paul Klebinder (Berlin: Klebinder, 1912), 18\u201322, here 20.\n\n 40. Max Brod, \"Kinematographentheater,\" in Kaes, _Kino-Debatte_ , 39\u201341, here 41. Originally published in _Die neue Rundschau_ 20, no. 2 (February 1909): 319\u2013320.\n\n 41. \"Neuland f\u00fcr Kinematographentheater,\" in Kaes, _Kino-Debatte_ , 41. Originally published in _Lichtbild-B\u00fchne_ 3 (September 1910): 3.\n\n 42. Strobl, \"Der Kinematograph,\" in _Kein Tag ohne Kino_ , 51.\n\n 43. Juliet Koss describes the gentleman art historian in terms of a unified, stable subjectivity in \"On the Limits of Empathy,\" _Art Bulletin_ 88, no. 1 (March 2006): 139\u2013157; see also Koss, _Modernism After Wagner_ (Minneapolis: University of Minnesota Press, 2010).\n\n 44. Harry Francis Mallgrave and Eleftherios Ikonomou, \"Introduction,\" _Empathy, Form, and Space_ , 10\u201311. Herbart's aesthetics were widely influential in the nineteenth century (but not as influential as his pedagogical ideas, as we saw in the previous chapter). See his 1813 _Lehrbuch zur Einleitung in die Philosophie_ , third edition (K\u00f6nigsberg: Unzer, 1834) or his 1831 _Kurze Encyklop\u00e4die der Philosophie_ (Hamburg: Voss, 1884). Robert Zimmermann later expanded Herbartian aesthetics into an equally influential comprehensive system devoted to the \"the science of form.\" See his _Allgemeine Aesthetik als Formwissenschaft_ (Vienna: Braum\u00fcller, 1865).\n\n 45. Many of Schopenhauer's arguments in \"Supplements to the Third Book\" take into account questions of perception and physiology, esp. chap. 30, \"On the Pure Subject of Knowing,\" and chap. 39, \"On the Metaphysics of Music.\" Schopenhauer, _The World as Will and Representation_ , vol. 2, pp. 367\u2013375 and 447\u2013457, respectively.\n\n 46. Looking for \"laws\" of beauty as a scientist might look for laws of nature, Fechner measured hundreds of paintings to find a statistical, scientific basis for the perfect format. See Gustav Fechner, _Vorschule der Aesthetik_ (Leipzig: Breitkopf & H\u00e4rtel, 1897 [1876]). Likewise, Wundt, the father of experimental psychology, tried to link pleasure in the perception of forms to the physiological structure of the eye by noting the ease with which the eye traced the contours of various forms. See Wilhelm Wundt, _Grundz\u00fcge der physiologischen Psychologie_ , 5th ed. (Leipzig: Engelmann, 1902\u20131903 [1874]), 1: 486ff. For a discussion of the implications of the work of these and other researchers for modern subjectivity, see Jonathan Crary, _Techniques of the Observer: On Vision and Modernity in the Nineteenth Century_ (Cambridge, Mass.: MIT Press, 1990). For more on the relationship between the sciences, especially physiology, and aesthetics, see Robert Michael Brain, \"The Pulse of Modernism: Experimental Physiology and Aesthetic Avant-Gardes Circa 1900,\" _Studies in History and Philosophy of Science_ 39, no. 3 (2008): 393\u2013417.\n\n 47. Mallgrave and Ikonomou, \"Introduction,\" 2.\n\n 48. Vischer, _On the Optical Sense of Form_ , 89\u2013124. Hereafter cited parenthetically.\n\n 49. Gustav Jahoda, \"Theodor Lipps and the Shift from 'Sympathy' to 'Empathy,'\" _Journal of the History of the Behavioral Sciences_ 4, no. 2 (2005): 151\u2013163; and Susan Lanzoni, \"Empathy in Translation: Movement and Image in the Psychological Laboratory,\" _Science in Context_ 25, no. 3 (September 2012): 301\u2013327.\n\n 50. The early \"formal-analytic\" approach to art, so important for the formation of the discipline of art history, is exemplified by Conrad Fiedler, _\u00dcber die Berurtheilung von Werken der bildenden Kunst_ (Leipzig: Hirzel, 1876). Translated by Henry Schaefer-Simmern and Fulmer Mood as _On Judging Works of Visual Art_ (Berkeley: University of California Press, 1949).\n\n 51. The _Zeitschrift f\u00fcr \u00c4sthetik und allgemeine Kunstwissenschaft_ (Journal for Aesthetics and Art History), which began in 1906 under the editorship of Max Dessoir, was the leading forum in Germany for discussions of _Einf\u00fchlung_ and its implications for aesthetics and reception.\n\n 52. These questions were not limited to Germany, as many ideas and approaches spread to the United Kingdom, France, and the United States during the nineteenth century. Representative English-language examples of psychological aesthetics include Herbert Spencer, _The Principles of Psychology_ , 2 vols. (London: Williams and Norgate, 1855); Grant Allen, _Physiological Aesthetics_ (London: King, 1877); James Sully, \"Pleasure of Visual Form,\" _Mind: A Quarterly Review of Psychology and Philosophy_ 5, no. 18 (April 1880): 181\u2013201; and Vernon Lee and C. Anstruther-Thomson, _Beauty and Ugliness and Other Studies in Psychological Aesthetics_ (London and New York: Lane, 1912). On Lee, see esp. Hilary Fraser, \"Women and the Ends of Art History: Vision and Corporeality in Nineteenth-Century Critical Discourse,\" _Victorian Studies_ 42, no. 1 (Autumn 1998): 77\u2013100. Applications of these ideas to film include Hugo M\u00fcnsterberg, _The Photoplay: A Psychological Study_ (London and New York: Appleton, 1916); Victor Oscar Freeburg, _Pictorial Beauty on the Screen_ (New York: Macmillan, 1923); and Frances Taylor Patterson, _Scenario and Screen_ (New York: Harcourt, Brace, 1928). Contemporary analyses of this trend in English-language (film) aesthetics include Laura Marcus, _The Tenth Muse: Writing About Cinema in the Modernist Period_ (Oxford and New York: Oxford University Press, 2007); Lynda Nead, _The Haunted Gallery: Painting, Photography, Film c. 1900_ (New Haven and London: Yale University Press, 2007); and Kaveh Askari, _Making Movies into Art: Picture Craft from the Magic Lantern to Early Hollywood_ (London: British Film Institute, 2014), esp. chap. 3.\n\n 53. Joseph Imorde, \"Einf\u00fchlung in der Kunstgeschichte,\" in _Einf\u00fchlung. Zur Geschichte und Gegenwart eines \u00e4sthetischen Konzepts_ , ed. Robin Curtis and Gertrud Koch (Paderborn, Germany: Fink, 2009), 127\u2013142.\n\n 54. In addition to my own essay on \"Einf\u00fchlung und fr\u00fche deutsche Filmtheorie,\" in Curtis and Koch, _Einf\u00fchlung. Zur Geschichte und Gegenwart eines \u00e4sthetischen Konzepts_ , 61\u201384, see Robin Curtis, \"Einf\u00fchlung and Abstraction in the Moving Image: Historical and Contemporary Reflections,\" _Science in Context_ 25, no. 3 (September 2012): 425\u2013446; and Robert Michael Brain, \"Self-Projection: Hugo M\u00fcnsterberg on Empathy and Oscillation in Cinema Spectatorship,\" in the same issue, pp. 329\u2013353.\n\n 55. Even when one of the preeminent names in aesthetic and _Einf\u00fchlung_ theory, Max Dessoir, was given the opportunity to discuss the phenomenon of film from a theoretical standpoint, he chose, rather uninterestingly, to extol the virtues of words for literature and denounce the lack of words in silent film. See \"Kino und Buchhandel,\" in Schweinitz, _Prolog vor dem Film_ , 284\u2013285.\n\n 56. Walter von Molo, \"Im Kino,\" in Schweinitz, _Prolog vor dem Film_ , 28\u201339, here 31. Originally published in _Velhagen & Klasings Monatshefte_ 26, no. 8 (April 1912): 618\u2013627.\n\n 57. Alfred Polgar, \"Das Drama im Kinematographen,\" in _Kein Tag ohne Kino_ , 56\u201361, here 60. Originally published in _Der Strom_ 1, no. 2 (May 1911).\n\n 58. Vivian Sobchack, _The Address of the Eye: A Phenomenology of Film Experience_ (Princeton, N.J.: Princeton University Press, 1992).\n\n 59. Hildebrand, _The Problem of Form in the Fine Arts_ , 261.\n\n 60. Polgar, \"Das Drama im Kinematographen,\" 59.\n\n 61. Mallgrave and Ikonomou, \"Introduction,\" 23.\n\n 62. Ernst Bloch, \"Die Melodie im Kino oder immanente und transzendentale Musik,\" in Schweinitz, _Prolog vor dem Film_ , 326\u2013334, here 328\u2013329. Originally published in _Die Argonauten_ 1, no. 2 (1914): 84\u201385. This translation, while largely my own, borrows some phrases from Ernst Bloch, \"On Music in the Cinema,\" in _Literary Essays_ , trans. Andrew Joron and others (Stanford, Calif.: Stanford University Press, 1998), 157\u2013158.\n\n 63. See also Hildebrand, _The Problem of Form in the Fine Arts_ , 229. Hildebrand further equates _Schauen_ with _Abtasten_ (\"probing\" or \"touching\").\n\n 64. See Hildebrand, _The Problem of Form in the Fine Arts_ , 261, for a concurring opinion: \"What we simply call the life of nature is actually the animation of nature through the imagination.\"\n\n 65. August Schmarsow, \"The Essence of Architectural Creation\" (1893), in Mallgrave and Ikonomou, _Empathy, Form, and Space_ , 281\u2013297, here 287 (emphasis in original). Hereafter cited parenthetically.\n\n 66. For a survey of his output, see Niels W. Bokhove and Karl Schuhmann, \"Bibliographie der Schriften von Theodor Lipps,\" _Zeitschrift f\u00fcr philosophische Forschung_ 45, no. 1 (January\u2013March 1991): 112\u2013130.\n\n 67. Theodor Lipps, \"Einf\u00fchlung und \u00e4sthetischer Genu\u00df,\" _Die Zukunft_ 54, no. 14 (20 January 1906): 100\u2013114, here 101.\n\n 68. Lipps, \"Einf\u00fchlung und \u00e4sthetischer Genu\u00df,\" 103.\n\n 69. Theodor Lipps, \"Einf\u00fchlung, innere Nachahmung, und Organempfindungen,\" _Archiv f\u00fcr die gesamte Psychologie_ 1, nos. 2\/3 (1903): 185\u2013204, here 201, 202, 203. Translated as \"Empathy, Inner Imitation, and Sense-Feeling,\" in _A Modern Book of Aesthetics_ , ed. Melvin M. Rader (New York: Holt, 1935), 291\u2013304, here 302\u2013303 (translation modified).\n\n 70. Theodor Lipps, _\u00c4sthetik. Psychologie des Sch\u00f6nen und der Kunst. Erster Teil: Grundlegung der \u00c4sthetik_ (Hamburg and Leipzig: Voss, 1903), 148.\n\n 71. Other writers on _Einf\u00fchlung_ explored the relation between empathy and bodily response, however tentatively. See Karl Groos, \"Das \u00e4sthetische Miterleben und die Empfindungen aus dem K\u00f6rperinnern,\" _Zeitschrift f\u00fcr \u00c4sthetik und allgemeine Kunstwissenschaft_ 4 (1909): 161\u2013182; Vernon Lee, \"Weiteres \u00fcber Einf\u00fchlung und \u00e4sthetisches Miterleben,\" _Zeitschrift f\u00fcr \u00c4sthetik und allgemeine Kunstwissenschaft_ 5 (1910): 145\u2013190; and later, Johannes Volkelt, _System der \u00c4sthetik, Erster Band: Grundlegung der \u00c4sthetik_ , 2d ed. (Munich: Beck, 1927), esp. 186\u2013201, in which he discusses motor sensations as way of facilitating _Einf\u00fchlung_. See also Christian G. Allesch, _Geschichte der psychologischen \u00c4sthetik_ (G\u00f6ttingen: Verlag f\u00fcr Psychologie, 1987) for a complete discussion of this topic.\n\n 72. _Schaulust_ has been translated as \"scopophilia\" by Freud's translators, but this rendering gives it a clinical connotation that neither Freud nor Serner intended. It literally means \"the desire to look\" or \"the sexual pleasure in looking,\" but \"voyeurism\" tends to narrow its meaning as well. Not having an adequate English word at hand, I will simply refer to the concept in German. (On the inadequacies of the standard translation of Freud, see Bruno Bettelheim, _Freud and Man_ ' _s Soul_ [New York: Knopf, 1983]).\n\n 73. Walter Serner, \"Kino und Schaulust,\" in Schweinitz, _Prolog vor dem Film_ , 208\u2013214, here 208. Originally published in _Die Schaub\u00fchne_ 9, nos. 34\/35 (1913): 807\u2013811.\n\n 74. Serner, \"Kino und Schaulust,\" 210.\n\n 75. Serner, \"Kino und Schaulust,\" 211.\n\n 76. Tom Gunning, \"The Cinema of Attractions: Early Film, Its Spectator and the Avant-Garde,\" in _Early Cinema: Space, Frame, Narrative_ , ed. Thomas Elsaesser with Adam Barker (London: British Film Institute, 1990), 56\u201362.\n\n 77. Lipps, \"Einf\u00fchlung, innere Nachahmung, und Organempfindungen,\" 195 (emphasis in original). Translated as \"Empathy, Inner Imitation, and Sense-Feeling,\" in Rader, _A Modern Book of Aesthetics_ , 300\u2013301 (translation modified).\n\n 78. Lipps, \"Einf\u00fchlung und \u00e4sthetischer Genu\u00df,\" 113.\n\n 79. Charles Musser, \"A Cinema of Contemplation, A Cinema of Discernment: Spectatorship, Intertextuality and Attractions in the 1890s,\" in _The Cinema of Attractions Reloaded_ , ed. Wanda Strauven (Amsterdam: Amsterdam University Press, 2006), 159\u2013179.\n\n 80. Miriam Hansen, \"Early Silent Cinema: Whose Public Sphere?,\" _New German Critique_ 29 (Spring\/Summer 1983): 147\u2013184; Heide Schl\u00fcpmann, _Unheimlichkeit des Blicks: Das Drama des fr\u00fchen deutschen Kinos_ (Frankfurt: Stroemfeld\/Roter Stern, 1990).\n\n 81. Diderot, _Salons_ , III, ed. Jean Seznec and Jean Adh\u00e9mar (Oxford, 1963), 134\u2013135, quoted in Michael Fried, _Absorption and Theatricality: Painting and Beholder in the Age of Diderot_ (Chicago: University of Chicago Press, 1980), 125.\n\n 82. Richard Huelsenbeck, _Wozu Dada. Texte 1916\u20131936_ (Giessen: Anabas, 1994), 35, quoted in David C. Durst, _Weimar Modernism: Philosophy, Politics, and Culture in Germany 1918\u20131933_ (Lanham, Md.: Lexington, 2004), 48. Durst's book was instrumental in crafting my argument about the politics of contemplation.\n\n 83. Siegfried Kracauer, \"Cult of Distraction: On Berlin's Picture Palaces,\" in _The Mass Ornament: Weimar Essays_ , trans. and ed. Thomas Y. Levin (Cambridge, Mass.: Harvard University Press, 1995), 323\u2013328. Hereafter cited parenthetically.\n\n 84. Walter Benjamin, \"Surrealism: The Last Snapshot of the European Intelligensia,\" in _Walter Benjamin: Selected Writings_ , vol. 2, _1927\u20131934_ , ed. Michael W. Jennings, Howard Eiland, and Gary Smith, trans. Rodney Livingstone and others (Cambridge, Mass.: Belknap, 1999), 207\u2013221, here 213.\n\n 85. Benjamin, \"The Work of Art in the Age of Its Technological Reproducibility (Third Version)\"; and Benjamin, \"The Work of Art in the Age of Its Technological Reproducibility (Second Version),\" 101\u2013136.\n\n 86. Benjamin, \"The Work of Art in the Age of Its Technological Reproducibility (Second Version),\" 119.\n\n 87. Benjamin, \"Theory of Distraction,\" in _The Work of Art in the Age of Its Technological Reproducibility, and Other Writings on Media_ , ed. Michael W. Jennings, Brigid Doherty, and Thomas Y. Levin, trans. Edmund Jephcott, Rodney Livingstone, Howard Eiland, and others (Cambridge, Mass.: Belknap, 2008), 56\u201357 (emphasis added).\n\n 88. Of the many commentaries on Benjamin and Kracauer on distraction, see esp. Miriam Bratu Hansen, _Cinema and Experience: Siegfried Kracauer, Walter Benjamin, and Theodor W. Adorno_ (Berkeley: University of California Press, 2012); and Paul North, _The Problem of Distraction_ (Stanford, Calif.: Stanford University Press, 2012).\n\n 89. The standard postwar discussion of this topic is Hannah Arendt, _The Human Condition_ (Chicago: University of Chicago Press, 1958).\n\n 90. Georg Luk\u00e1cs, _History and Class Consciousness_ , trans. Rodney Livingstone (Cambridge, Mass.: MIT Press, 1971), 319.\n\n 91. Luk\u00e1cs, \"Preface to the New Edition [1967],\" _History and Class Consciousness_ , xviii.\n\n 92. Georg Luk\u00e1cs, _The Theory of the Novel_ , trans. Anna Bostock (Cambridge, Mass.: MIT Press, 1971), 135.\n\n 93. Luk\u00e1cs, _The Theory of the Novel_ , 118.\n\n 94. Georg Luk\u00e1cs, \"Gedanken zu einer Aesthetik des 'Kino,'\" _Frankfurter Zeitung_ 251 (10 September 1913): 1\u20132. There is also an earlier version, \"Gedanken zu einer Aesthetik des 'Kino,'\" which appeared in the German-Hungarian journal _Pester Lloyd_ (16 April 1911): 44\u201346. I will be using Janelle Blankenship's excellent translation, \"Thoughts Toward an Aesthetic of the Cinema,\" _Polygraph_ 13 (2001): 13\u201318. Hereafter cited parenthetically.\n\n 95. See especially Tom Levin, \"From Dialectical to Normative Specificity: Reading Luk\u00e1cs on Film,\" _New German Critique_ 40 (Winter 1987): 35\u201361; and Janelle Blankenship, \"Futurist Fantasies: Luk\u00e1cs's Early Essay 'Thoughts Toward an Aesthetic of the Cinema,'\" _Polygraph_ 13 (2001): 21\u201336.\n\n 96. Janelle Blankenship notes that this formulation recalls Bergson's concept of _dur\u00e9e_ , which might have decisively shaped Luk\u00e1cs's later work. Blankenship \"Futurist Fantasies,\" 22.\n\n 97. Levin, \"From Dialectical to Normative Specificity,\" 35\u201361.\n\n 98. We must here note that this designation of cinema\u2014or any form for that matter\u2014as utopian is very provisional in Luk\u00e1cs's work. Luk\u00e1cs's early work sometimes endorsed the possibility of a glimpse in art of unalienated, authentic life and, at other times, foreclosed that possibility, depending on his mode of analysis, metaphysical or historical. See Gy\u00f6rgy M\u00e1rkus, \"Life and the Soul: The Young Luk\u00e1cs and the Problem of Culture,\" in _Luk\u00e1cs Revalued_ , ed. Agnes Heller (London: Blackwell, 1983), 1\u201326.\n\n 99. Franz Pfemfert, \"Kino als Erzieher,\" in Schweinitz, _Prolog_ , 165\u2013169. Originally published in _Die Aktion_ 1, no. 18 (19 June 1911): 560\u2013563.\n\n. Kurt Pinthus, _Das Kinobuch_ (Frankfurt: Fischer Taschenbuch, 1983), 27.\n\n. Benjamin, \"Garlanded Entrance: On the 'Sound Nerves' Exhibition at the Gesundheitshaus Kreuzberg,\" in Jennings, Doherty, and Levin, eds., _The Work of Art in the Age of Its Technological Reproducibility, and Other Writings on Media_ , 60\u201366, here 62.\n\n. Duttlinger, \"Between Contemplation and Distraction,\" 51.\n\n. Jocelyn Szczepaniak-Gillece, \"Machines for Seeing: Cinema, Architecture, and Mid-century American Spectatorship,\" PhD diss. (Northwestern University, 2013).\n\nCONCLUSION\n\n 1. For a history of the _Kulturfilm_ , see Oskar Kalbus, _Pionere des Kulturfilms: Ein Beitrag zur Geschichte des Kulturfilmschaffens in Deutschland_ (Karlsruhe: Neue Verlags-Gesellschaft, 1956).\n\n 2. For a clear explanation of _dispositif_ as a historiographical concept, see Frank Kessler, \"La cin\u00e9matographie comme dispositif (du) spectaculaire,\" _CiN\u00e9MAS_ 14, no. 1 (2003): 21\u201334; and \"The Cinema of Attractions as _Dispositif_ ,\" in _The Cinema of Attractions Reloaded_ , ed. Wanda Strauven (Amsterdam: Amsterdam University Press, 2006), 57\u201369.\n\n 3. Tom Gunning, \"Systematizing the Electric Message: Narrative Form, Gender, and Modernity in _The Lonedale Operator_ ,\" in _American Cinema_ ' _s Transitional Era: Audiences, Institutions, Practices_ , ed. Charlie Keil and Shelley Stamp (Berkeley: University of California Press, 2004), 15\u201350. See also Tom Gunning, \"Film History and Film Analysis: The Individual Film in the Course of Time,\" _Wide Angle_ 12, no. 3 (July 1990): 4\u201319.\n\n 4. Ludwik Fleck, _Genesis and Development of a Scientific Fact_ , ed. Thaddeus J. Trenn and Robert K. Merton, trans. Fred Bradley and Thaddeus J. Trenn (Chicago: University of Chicago Press, 1979); and _Cognition and Fact: Materials on Ludwik Fleck_ , ed. Robert S. Cohen and Thomas Schnelle (Dordrecht, Netherlands, and Boston: Reidel, 1986).\n\n 5. Nicolas Rasmussen follows the development of a representational technology and its disciplinary conventions in _Picture Control: The Electron Microscope and the Transformation of Biology in America, 1940\u20131960_ (Stanford, Calif.: Stanford University Press, 1997).\n\n 6. See my \"Vergr\u00f6sserung und das mikroskopische Erhabene,\" _Zeitschrift f\u00fcr Medienwissenschaft_ 5 (2011): 96\u2013110. See also Hannah Landecker, \"Cellular Features: Microcinematography and Film Theory,\" _Critical Inquiry_ 31, no. 4 (2005): 903\u2013937.\n\n 7. Hannah Landecker, \"Creeping, Drinking, Dying: The Cinematic Portal and the Microscopic World of the Twentieth-Century Cell,\" _Science in Context_ 24, no. 3 (September 2011): 381\u2013416.\n\n 8. Hans-J\u00f6rg Rheinberger, _Toward a History of Epistemic Things: Synthesizing Proteins in the Test Tube_ (Stanford, Calif.: Stanford University Press, 1997).\n\n 9. These ideas about linked solutions and mutant genes I owe to George Kubler, _The Shape of Time: Remarks on the History of Things_ (New Haven: Yale University Press, 1962).\n\n 10. A particularly compelling statement of this dynamic is in Andreas Gailus, \"Of Beautiful and Dismembered Bodies: Art as Social Discipline in Schiller's _On the Aesthetic Education of Man_ ,\" in _Impure Reason: Dialectic of Enlightenment in Germany_ , ed. W. Daniel Wilson and Robert C. 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Anton Kaes, 39\u201341. T\u00fcbingen: Niemeyer, 1978. Originally published in _Die neue Rundschau_ 20, no. 2 (February 1909): 319\u2013320.\n\nBruegel, Carl. \"Bewegungsvorg\u00e4nge am pathologischen Magen auf Grund r\u00f6ngenkinematographischer Untersuchungen.\" _M\u00fcnchener medizinische Wochenschfrift_ 60, no. 4 (28 January 1913): 179\u2013181.\n\nBrunner, Karl. _Der Kinematograph von heute_ \u2014 _eine Volksgefahr_. Berlin: Vaterl\u00e4ndischen Schriftenverbandes, 1913.\n\nCarr, W. P. \"Suggestion as Used and Misused in Curing Disease.\" In _Hypnotism and Hypnotic Suggestion_ , ed. E. Virgil Neal and Charles S. Clark, 5\u201317. Rochester: New York State Publishing, 1900.\n\nCarrel, Alexis, and Montrose T. 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H. \"Der Schlingact, dargestellt nach Bewegungsphotographien mittelst R\u00f6ntgen-Strahlen.\" _Pfl\u00fcger_ ' _s Archiv f\u00fcr die gesammte Physiologie des Menschen und der Thiere_ 99 (1903): 513\u2013571.\n\nFechner, Gustav. _Vorschule der Aesthetik_. Leipzig: Breitkopf & H\u00e4rtel, 1897.\n\nFelke, Naldo. \"Die Gesundheitssch\u00e4dlichkeit des Kinos.\" _Die Umschau_ 17, no. 1 (1 January 1913): 254\u2013255.\n\nFiedler, Conrad. _\u00dcber die Berurtheilung von Werken der bildenden Kunst_. Leipzig: Hirzel, 1876. Translated by Henry Schaefer-Simmern and Fulmer Mood as _On Judging Works of Visual Art_ (Berkeley: University of California Press, 1949).\n\nFischer, Otto. _Theoretische Grundlagen f\u00fcr eine Mechanik der lebenden K\u00f6rper_. Leipzig, 1906.\n\nFlemming, Carl F. _Pathologie und Therapie der Psychosen._ Berlin: Hirschwald, 1859.\n\nForel, Auguste. _Hygiene der Nerven und des Geistes im gesunden und kranken Zustande_. Stuttgart: Moritz, 1903.\n\nFran\u00e7ois-Franck, Charles. \"D\u00e9monstrations de microphotographie instantan\u00e9e et de chronomicrophotographie.\" _Comptes rendus hebdomadaires des s\u00e9ances et m\u00e9moires de la Soci\u00e9t\u00e9 de Biologie_ 62 (25 May 1907): 964\u2013967.\n\n\u2014\u2014. \"\u00c9tudes graphiques et photographiques de m\u00e9canique respiratoire compar\u00e9e.\" _Comptes rendus hebdomadaires des s\u00e9ances et m\u00e9moires de la Soci\u00e9t\u00e9 de Biologie_ 61 (28 July 1906): 174\u2013176.\n\n\u2014\u2014. \"La chronophotographie simultan\u00e9e du coeur et des courbes cardiographiques chez les mammif\u00e8res.\" _Comptes rendus hebdomadaires des s\u00e9ances et m\u00e9moires de la Soci\u00e9t\u00e9 de Biologie_ 54 (8 November 1902): 1193\u20131197.\n\n\u2014\u2014. \"Note sur quelques points de technique relatifs \u00e0 la photographie et \u00e0 la chronophotographie avec le magn\u00e9sium \u00e0 deflagration lente.\" _Comptes rendus hebdomadaires des s\u00e9ances et m\u00e9moires de la Soci\u00e9t\u00e9 de Biologie_ 55 (5 December 1903): 1538\u20131540.\n\nFr\u00e4nkel, Albert. \"R\u00f6ntgendiagnosen und R\u00f6ntgenfehldiagnosen beim Magenkarzinom; diagnostischer Fortschritt durch R\u00f6ntgenkinographie.\" _Zentralblatt f\u00fcr R\u00f6ntgenstrahlen, Radium und verwandte Gebiete_ 3, no. 4 (1912): 149\u201350.\n\nFr\u00e4nkel, James. \"Kinematographische Demonstration.\" _Verhandlungen der freien Vereinigung der Chirurgen Berlins_ 20, part 1 (1907): 12\u201313.\n\n\u2014\u2014. \"Kinematographische Untersuchung des normalen Ganges und einiger Gangst\u00f6rungen.\" _Zeitschrift f\u00fcr orthop\u00e4dische Chirurgie_ 20 (1908): 617\u2013646.\n\nFreeburg, Victor Oscar. _Pictorial Beauty on the Screen_. New York: Macmillan, 1923.\n\nFrei, Wilhelm. _Landerziehungsheime: Darstellung und Kritik einer modernen Reformschule_. Leipzig: Klinkhardt, 1902.\n\nFreud, Sigmund. _Beyond the Pleasure Principle_. Translated by James Strachey. London: Hogarth, 1950.\n\nFrey, J. \"Report of the Photographic Department of Bellevue Hospital for the Year 1869.\" In _Tenth Annual Report of the Commissioners of Public Charities and Correction of the City of New York for the Year 1869_ , 85. Albany: van Benthuysen, 1870. www.artandmedicine.com\/ogm\/1869.html.\n\nFriedell, Egon. \"Prolog vor dem Film.\" In _Kino-Debatte: Texte zum Verh\u00e4ltnis von Literatur und Film 1909\u20131929_ , ed. Anton Kaes, 42\u201347. T\u00fcbingen: Niemeyer, 1978. Originally published in _Bl\u00e4tter des Deutschen Theaters_ 2 (1912): 509\u2013511.\n\nGaupp, Robert. \"Das Pathologische in Kunst und Literatur.\" _Deutsche Revue_ 36, no. 2 (April 1911): 11\u201323.\n\n\u2014\u2014. \"Der Arzt als Erzieher seines Volkes.\" _Medicinisches Correspondenz-Blatt_ 89, no. 32 (9 August 1919): 295\u2013296.\n\n\u2014\u2014. \"Der Kinematograph vom medizinischen und psychologischen Standpunkt.\" In _Der Kinematograph als Volkunterhaltungsmittel_ , by Robert Gaupp and Konrad Lange, 1\u201312. Munich: D\u00fcrer-Bund-Flugschrift zur Ausdruckskultur 100, 1912.\n\n\u2014\u2014. \"Die Gefahren des Kino.\" In _Prolog vor dem Film: Nachdenken \u00fcber ein neues Medium, 1909\u20131914_ , ed. J\u00f6rg Schweinitz, 64\u201369. Leipzig: Reclam, 1992. Originally published in _S\u00fcddeutsche Monatshefte_ 9, no. 9 (1911\/1912): 363\u2013366.\n\n\u2014\u2014. \"Die Nervosit\u00e4t unserer Zeit im Lichte der Wissenschaft.\" _Medicinisches Correspondenz-Blatt_ 77, no. 31 (3 August 1907): 633\u2013639.\n\n\u2014\u2014. _Psychologie des Kindes_. Leipzig: Teubner, 1910.\n\nGaupp, Robert, and Konrad Lange, _Der Kinematograph als Volksunterhaltungsmitte._ Munich: Callwey, 1912.\n\nGeisel, Walter. _Wie ich mit meinen Jungens Kunstwerke betrachte_. Gl\u00fcckstadt: Geisel, 1904.\n\nGeorges-Michel, Michel. \"Henri Bergson nous parle au cin\u00e9ma.\" _Le Journal_ (20 February 1914): 7. Translated by Louis-Georges Schwartz as \"Henri Bergson Talks to Us About Cinema.\" _Cinema Journal_ 50, no. 3 (Spring 2011): 79\u201382.\n\nGilbreth, Frank B. _Motion Study: A Method for Increasing the Efficiency of the Workman_. New York: Van Nostrand, 1911.\n\n\u2014\u2014. \"Scientific Management in the Hospital.\" _Modern Hospital_ 3 (1914): 321\u2013324.\n\nGoerke, Franz. \"Proposal for Establishing an Archive for Moving Pictures (1912).\" Translated by Cecilie L. French and Daniel J. Leab. _Historical Journal of Film, Radio and Television_ 16, no. 1 (March 1996): 9\u201312. Originally published as \"Vorschlag zur Einrichtung eines Archives f\u00fcr Kino-films,\" in _Der Deutsche Kaiser im Film: zum 25j\u00e4hrigen Regierungs-Jubil\u00e4um Seiner Majest\u00e4t des Deutschen Kaisers K\u00f6nigs von Preu\u00dfen Wilhelm II_ , ed. Paul Klebinder, 63\u201368 (Berlin: Klebinder, 1912).\n\nG\u00f6tze, O. \"Jugendpsyche und Kinematograph.\" _Zeitschrift f\u00fcr Kinderforschung_ 16 (1911): 418.\n\nGouy, Louis-Georges. \"Le mouvement brownien et les mouvements mol\u00e9culaires.\" _Revue g\u00e9n\u00e9rale des sciences pures et appliqu\u00e9es_ 6, no. 1 (15 January 1895): 1\u20137.\n\nGraupner, H. \"Unterrichtshygiene.\" In _Handbuch der deutschen Schulhygiene_ , ed. Hugo Selter, 174\u2013321. Dresden and Leipzig: Theodor Steinkopff, 1914.\n\nGrempe, Max. \"Gegen die Frauenverbl\u00f6dung im Kino.\" _Gleichheit_ 23, no. 5 (1912): 70\u201372.\n\nGroedel, Franz M. \"Die Technik der R\u00f6ntgenkinematographie.\" _Deutsche medizinsiche Wochenschrift_ 35 (11 March 1909): 434\u2013435.\n\n\u2014\u2014. \"Die Technik der R\u00f6ntgenkinematographie.\" _Deutsche medizinsiche Wochenschrift_ 39 (6 February 1913): 270\u2013271.\n\n\u2014\u2014. \"Die Technik der R\u00f6ntgenkinematographie.\" _Deutsche medizinsiche Wochenschrift_ 39 (24 April 1913): 798\u2013799.\n\n\u2014\u2014. \"The Present State of Roentgen Cinematography and Its Results as to the Study of the Movements of the Inner Organs of the Human Body.\" _Interstate Medical Journal_ 22 (March 1915): 281\u2013290.\n\n\u2014\u2014. \"Roentgen Cinematography and Its Importance in Medicine.\" _British Medical Journal_ (24 April 1909): 1003.\n\nGroos, Karl. \"Das \u00e4sthetische Miterleben und die Empfindungen aus dem K\u00f6rperinnern.\" _Zeitschrift f\u00fcr \u00c4sthetik und allgemeine Kunstwissenschaft_ 4 (1909): 161\u2013182.\n\nG\u00fcnther, Hanns. \"Mikrokinematographische Aufnahmeapparate.\" _Film and Lichtbild_ 1, no. 1 (1912): 4\u20136; 1, no. 2 (1912): 13\u201314.\n\nH\u00e4fker, Hermann. \"Eine Reise an die Quellen der Kinematographie.\" _Der Kinematograph_ no. 163 (9 February 1910); 172 (13 April 1910).\n\n\u2014\u2014. _Kino und Kunst_. M. Gladbach [M\u00f6nchengladbach]: Volksvereins, 1913.\n\n\u2014\u2014. \"Meisterspiele.\" _Der Kinematograph_ no. 56 (22 January 1908).\n\n\u2014\u2014. \"Zur Dramaturgie der Bilderspiele.\" _Der Kinematograph_ no. 32 (7 August 1907).\n\nHarris, W. T. Editor's preface to _Herbart_ ' _s ABC of Sense Perception and Minor Pedagogical Works_ , by Johann Friedrich Herbart. Translated and edited by William J. Eckoff, vii\u2013xi. New York: Appleton, 1896.\n\nHarrison, Ross Granville. \"Experiments in Transplanting Limbs and Their Bearing Upon the Problems of the Development of Nerves.\" _Journal of Experimental Zoology_ 4, no. 2 (June 1907): 239\u2013281.\n\n\u2014\u2014. \"Further Experiments on the Development of Peripheral Nerves.\" _American Journal of Anatomy_ 5, no. 2 (31 May 1906): 121\u2013131.\n\n\u2014\u2014. \"Observations on the Living Developing Nerve Fiber.\" _Anatomical Record_ 1, no. 5 (1 June 1907): 116\u2013118.\n\n\u2014\u2014. \"The Outgrowth of the Nerve Fiber as a Mode of Protoplasmic Movement.\" _Journal of Experimental Zoology_ 9, no. 4 (December 1910): 787\u2013846.\n\nHasenclever, Walter. \"Der Kintopp als Erzieher: Eine Apologie.\" In _Kino-Debatte: Texte zum Verh\u00e4ltnis von Literatur und Film 1909\u20131929_ , ed. Anton Kaes, 47\u201349. T\u00fcbingen: Niemeyer, 1978. Originally published in _Revolution_ 1, no. 4 (1 December 1913): n.p.\n\nHellwig, Albert. \"Die Beziehungen zwischen Schundliteratur, Schundfilms und Verbrechen.\" _Archiv f\u00fcr Kriminal-Anthropologie und Kriminalistik_ 51, no. 1 (24 January 1913): 1\u201332.\n\n\u2014\u2014. \"Die ma\u00dfgebenden Grunds\u00e4tze f\u00fcr Verbote von Schundfilms nach geltendem und k\u00fcnstigem Rechte.\" _Verwaltungsarchiv_ 21 (1913): 405\u2013455.\n\n\u2014\u2014. _Die Filmzensur: Eine rechtsdogmatische und rechtpolitische Er\u00f6rterung_. Berlin: Frankenstein, 1914.\n\n\u2014\u2014. \"Hypnotismus und Kinematograph.\" _Zeitschrift f\u00fcr Psychotherapie und medizinische Psychologie_ 6 (1916): 310\u2013315.\n\n\u2014\u2014. _Kind und Kino_. Langensalza: Beyer, 1914.\n\n\u2014\u2014. \"Kinematograph und Zeitgeschichte.\" In _Prolog vor dem Film: Nachdenken \u00fcber ein neues Medium, 1909\u20131914_ , ed. J\u00f6rg Schweinitz, 97\u2013109. Leipzig: Reclam, 1992. Originally published in _Die Grenzboten_ 72, no. 39 (1913): 612\u2013620.\n\n\u2014\u2014. _\u00d6ffentliches Lichtspielrecht_. M. Gladbach [M\u00f6nchengladbach]: Volksvereins, 1921.\n\n\u2014\u2014. _Rechtsquellen des \u00f6ffentlichen Kinematographenrechts_. M. Gladbach [M\u00f6nchengladbach]: Volksvereins, 1913.\n\n\u2014\u2014. _Schundfilms: Ihr Wesen, ihre Gefahren und ihre Bek\u00e4mpfung_. Halle an der Saale, Germany: Waisenhaus, 1911.\n\n\u2014\u2014. \"\u00dcber die sch\u00e4dliche Suggestivkraft kinematographischer Vorf\u00fchrung.\" _Aerztliche Sachverst\u00e4ndigen-Zeitung_ 20, no. 6 (15 March 1914): 122.\n\n\u2014\u2014. \"Zur Psychologie kinematographischer Vorf\u00fchrungen.\" _Zeitschrift f\u00fcr Psychotherapie und medizinische Psychologie_ 6 (1916): 88\u2013120.\n\nHelmholtz, Hermann von. _\u00dcber die Erhaltung der Kraft_. Berlin: Reimer, 1847.\n\nHennes, Hans. \"Die Kinematographie der Bewegungsst\u00f6rungen.\" _Die Umschau_ 15, no. 29 (1911): 605\u2013606.\n\n\u2014\u2014. \"Die Kinematographie im Dienste der Neurologie und Psychiatrie, nebst Beschreibung einiger selteneren Bewegungsst\u00f6rungen.\" _Medizinische Klinik_ 6, no. 51 (18 December 1910): 2010\u20132014.\n\nHenri, Victor. \"\u00c9tudes cin\u00e9matographique des mouvements browniens.\" _Comptes rendus hebdomadaires des s\u00e9ances de l_ ' _Academie des Sciences_ 146 (18 May 1908): 1024\u20131026.\n\n\u2014\u2014. \"Influence du milieu sur les mouvements browniens.\" _Comptes rendus hebdomadaires des s\u00e9ances de l_ ' _Academie des Sciences_ 147 (6 July 1908): 62\u201365.\n\nHerbart, Johann Friedrich. _Kurze Encyklop\u00e4die der Philosophie_. Hamburg: Voss, 1884.\n\n\u2014\u2014. _Lehrbuch zur Einleitung in die Philosophie_ , third edition (K\u00f6nigsberg: Unzer, 1834).\n\nHildebrand, Adolf. _The Problem of Form in the Fine Arts_ (1893). In _Empathy, Form, and Space: Problems in German Aesthetics, 1873\u20131893_ , ed. and trans. Harry Francis Mallgrave and Eleftherios Ikonomou, 227\u2013279. Santa Monica, Calif.: Getty Center for the History of Art and the Humanities, 1994.\n\nJacobj, Carl. \"Anschauungsunterricht und Projektion.\" _Zeitschrift f\u00fcr wissenschaftliche Mikroskopie und mikroskopische Technik_ 36, no. 4 (1919): 273\u2013314.\n\nJendrassik, Ernst. \"Klinische Beitr\u00e4ge zum Studium der normalen und pathologischen Gangarten.\" _Deutsche Archiv f\u00fcr klinische Medizin_ 70 (1901): 81\u2013132.\n\nKalbus, Oskar. \"Abri\u00df einer Geschichte der deutschen Lehrfilmbewegung.\" In _Das Kulturfilmbuch_ , ed. Edgar Beyfu\u00df and Alexander Kossowsky, 1\u201313. Berlin: Chryselius'scher, 1924.\n\n\u2014\u2014. _Der Deutsche Lehrfilm in der Wissenschaft und im Unterricht_. Berlin: Heymanns, 1922.\n\nKandel, I. L. \"Germany.\" In _Comparative Education: Studies of the Educational Systems of Six Modern Nations_ , ed. Peter Sandiford, 121\u2013130. London and Toronto: Dent, 1918.\n\nKant, Immanuel. _Critique of Judgement_. Translated by James Creed Meredith. Oxford: Oxford University Press, 1952.\n\n\u2014\u2014. \"What Is Enlightenment?\" In _German Aesthetic and Literary Criticism_ , ed. David Simpson, 29\u201334. Cambridge: Cambridge University Press, 1984.\n\nK\u00e4stle, C., H. Rieder, and J. Rosenthal. \"The Bioroentgenography of the Internal Organs.\" _Archives of the Roentgen Ray_ 15, no. 1 (June 1910): 3\u201312.\n\n\u2014\u2014. \"Ueber kinematographisch aufgenommene R\u00f6ntgenogramme (Bio-R\u00f6ntgenographie) der inneren Organe des Menschen.\" _M\u00fcnchener medizinsiche Wochenschrift_ 56, no. 6 (9 February 1909): 280\u2013283.\n\nKey, Ellen. \"Erziehung,\" _Das Jahrhundert des Kindes_ (Berlin, 1905). In _Die deutsche Reformp\u00e4dagogik_ , ed. Wilhelm Flitner and Gerhard Kudritzki, 52\u201354. D\u00fcsseldorf and Munich: K\u00fcpper, 1961.\n\nKienzl, Hermann. \"Theater und Kinematograph.\" In _Prolog vor dem Film: Nachdenken \u00fcber ein neues Medium, 1909\u20131914_ , ed. J\u00f6rg Schweinitz, 230\u2013234. Leipzig: Reclam, 1992. Originally published in _Der Strom_ 1, no. 7 (October 1911): 219\u2013221.\n\n\"Kinematograph als Krankheitsstifter.\" _Fortschritte der Medizin_ 30 (1912): 302.\n\n\"Kinematographische Reformvereinigung.\" _Der Kinematograph_ no. 43 (23 October 1907).\n\n\"Kino und Buchhandel.\" In _Prolog vor dem Film: Nachdenken \u00fcber ein neues Medium, 1909\u20131914_ , ed. J\u00f6rg Schweinitz, 272\u2013289. Leipzig: Reclam, 1992.\n\n\"Kino und Wissenschaft.\" _Bild und Film_ 1, no. 2 (1912): 55.\n\nKleib\u00f6mer, Georg. \"Kinematograph und Schuljugend.\" _Der Kinematograph_ no. 124 (12 May 1909).\n\nKnospe, Paul. _Der Kinematograph im Dienste der Schule. Unter besonderer Ber\u00fccksichtigung des erdkundlichen Unterrichts_. Halle an der Saale, Germany: Waisenhaus, 1913.\n\nKracauer, Siegfried. \"Cult of Distraction: On Berlin's Picture Palaces.\" In _The Mass Ornament: Weimar Essays_ , trans. and ed. Thomas Y. Levin, 323\u2013328. Cambridge, Mass.: Harvard University Press, 1995.\n\nKraepelin, Emil. \"Demonstration von Kinematogrammen.\" _Centralblatt f\u00fcr Nervenheilkunde und Psychiatrie_ 32 (1909): 689.\n\n\u2014\u2014. _Memoirs_. Edited by H. Hippius, G. Peters, and D. Ploog in collaboration with P. Hoff and A. Kreuter. Translated by Cheryl Wooding-Deane. Berlin and New York: Springer-Verlag, 1987.\n\n\u2014\u2014. \"Zur Entartungsfrage.\" _Zentralblatt f\u00fcr Nervenheilkunde und Psychiatrie_ 31 (1908): 745\u2013751. Translated as \"On the Question of Degeneration,\" _History of Psychiatry_ 18, no. 3 (2007): 399\u2013404.\n\nKretz, Richard. \"Die Anwendung der Photographie in der Medicin.\" _Wiener klinische Wochenschrift_ 7, no. 44 (1 November 1894): 832.\n\nKutner, Robert. \"Die Bedeutung der Kinematographie f\u00fcr medizinische Forschung und Unterricht sowie f\u00fcr die volkshygienische Belehrung.\" _Zeitschrift f\u00fcr \u00c4rztliche Fortbildung_ 8, no. 8 (15 April 1911): 249\u2013251.\n\nLandau, Joseph. \"Mechanisierte Unsterblichkeit.\" In _Der Deutsche Kaiser im Film. Zum 25j\u00e4hrigen Regierungs-Jubil\u00e4um Seiner Majest\u00e4t des Deutschen Kaisers K\u00f6nigs von Preu\u00dfen Wilhelm II_ , ed. Paul Klebinder, 18\u201322. Berlin: Klebinder, 1912.\n\nLangbehn, August Julius. \"Rembrandt als Erzieher.\" In _Die Kunsterziehungsbewegung_ , ed. Hermann Lorenzen, 7\u201317. Bad Heilbrunn, Germany: Klinkhardt, 1966.\n\nLange, Konrad. \"Das Wesen der k\u00fcnstlerischen Erziehung.\" In _Die Kunsterziehungsbewegung_ , ed. Hermann Lorenzen, 21\u201326. Bad Heilbrunn, Germany: Klinkhardt, 1966.\n\n\u2014\u2014. _Die k\u00fcnstlerische Erziehung der deutschen Jugend_. Darmstadt: Bergstrae\u00dfer, 1893.\n\nLaquer, Leopold. \"\u00dcber die Sch\u00e4dlichkeit kinematographischer Veranstaltungen f\u00fcr die Psyche des Kindesalters.\" _Aerztliche Sachverst\u00e4ndigen-Zeitung_ 27, no. 11 (1 June 1911): 221\u2013222.\n\nLaycock, Thomas. _Lectures on the Principles and Methods of Medical Observation and Research_. Philadelphia: Blanchard and Lea, 1857.\n\nLe Bon, Gustave. _The Crowd: A Study of the Popular Mind_. Atlanta, Ga.: Cherokee, 1982. Originally published as _Psychologie des foules_ (Paris: Alcan, 1895). Translated and published in German as _Psychologie der Massen_ (Leipzig: Klinkhardt, 1908).\n\nLearned, William S. _An American Teacher_ ' _s Year in a Prussian Gymnasium_. New York: Educational Review, 1911.\n\nLecomte du No\u00fcy, P[ierre]. _Biological Time_. With a foreword by Alexis Carrel. New York: Macmillan, 1937.\n\nLee, Vernon. \"Weiteres \u00fcber Einf\u00fchlung und \u00e4sthetisches Miterleben.\" _Zeitschrift f\u00fcr \u00c4sthetik und allgemeine Kunstwissenschaft_ 5 (1910): 145\u2013190.\n\n\u2014\u2014, and C. Anstruther-Thomson. _Beauty and Ugliness and Other Studies in Psychological Aesthetics_. London and New York: Lane, 1912.\n\nLehmann, Hans. _Die Kinematographie: ihren Grundlagen und ihre Anwendungen_. Leipzig: Teubner, 1911.\n\nLeipziger Lehrerverein, ed. _Bildbetrachtungen: Arbeiten aus der Abteilung f\u00fcr Kunst-pflege des Leipziger Lehrervereins_. Leipzig: Teubner, 1906.\n\nLemke, Hermann. _Die Kinematographie der Gegenwart, Vergangenheit und Zukunft_. Leipzig: Demme, 1911.\n\n\u2014\u2014. \"Die kinematographische Reformpartei, ihre Aufgaben und Ziele.\" _Der Kinematograph_ no. 42 (16 October 1907).\n\n\u2014\u2014. _Die kinematographische Unterrichtsstunde_. Leipzig: Demme, 1911.\n\n\u2014\u2014. \"Die Verwertung und Nutzbarmachung neuer Film-Ideen\u2014K\u00fcnstlerische Films.\" _Der Kinematograph_ no. 57 (29 January 1908).\n\n\u2014\u2014. _Durch die Technik zur Schulreform. Zwei modern-technische Lehrmethoden und Veranschaulichungsmittel in der Schule der Zukunft_. Leipzig: Demme, 1911.\n\n\u2014\u2014. _Praktische Forderungen f\u00fcr die Verwertung der Kinematographie im Unterricht._ Friedenau: Schule und Technik, 1909.\n\n\u2014\u2014. \"Volkst\u00fcmliche Reisebeschreibungen.\" _Der Kinematograph_ no. 34 (21 August 1907).\n\nLichtwark, Alfred. \"Der Deutsche der Zukunft.\" In _Die deutsche Reformp\u00e4dagogik_ , ed. Wilhelm Flitner and Gerhard Kudritzki, 99\u2013110. D\u00fcsseldorf and Munich: K\u00fcpper, 1961.\n\n\u2014\u2014. \"Die Aufgaben der Kunsthalle: Antrittsrede den 9. December 1886.\" In _Drei Programme_ , 2d ed., 11\u201331. Berlin: Cassirer, 1902.\n\n\u2014\u2014. _Die Bedeutung der Amateur-Photographie_. Halle an der Saale, Germany: Knapp, 1894.\n\n\u2014\u2014. \"Museen als Bildungsst\u00e4tten.\" _Der Deutsche der Zukunft_ , 89\u2013107. Berlin: Cassirer, 1905.\n\n\u2014\u2014. _\u00dcbungen in der Betrachtung von Kunstwerken_. Dresden: K\u00fchtmann, 1900.\n\nLiesegang, F. Paul. _Wissenschaftliche Kinematographie_. D\u00fcsseldorf: Liesegang, 1920.\n\nLipps, Theodor. _\u00c4sthetik. Psychologie des Sch\u00f6nen und der Kunst. Erster Teil: Grundlegung der \u00c4sthetik_. Hamburg and Leipzig: Voss, 1903.\n\n\u2014\u2014. \"Einf\u00fchlung, innere Nachahmung, und Organempfindungen.\" _Archiv f\u00fcr die gesamte Psychologie_ 1, nos. 2\/3 (1903): 185\u2013204. Translated as \"Empathy, Inner Imitation, and Sense-Feeling,\" in _A Modern Book of Aesthetics_ , ed. Melvin M. Rader, 291\u2013304 (New York: Holt, 1935).\n\n\u2014\u2014. \"Einf\u00fchlung und \u00e4sthetischer Genu\u00df.\" _Die Zukunft_ 54, no. 14 (20 January 1906): 100\u2013114.\n\nLocke, John. _An Essay Concerning Human Understanding_. Edited and with an introduction by Peter H. Nidditch. Oxford: Oxford University Press, 1975.\n\nLomon, Andr\u00e9, and Jean Comandon. \"Radiocin\u00e9matographie par la photographie des \u00e9crans intensificateurs.\" _La Presse M\u00e9dicale_ 35 (3 May 1911): 359.\n\nLonde, Albert. _Notice sur les titres et travaux scientifique_. Paris: Masson, 1911.\n\n\u2014\u2014. _Nouvelle iconographie de la Salp\u00eatri\u00e8re_. Paris: Masson, 1888\u20131918.\n\n\u2014\u2014. _Photographie m\u00e9dicale_. Paris: Gauthiers-Villars, 1893.\n\nLorenz, Richard, and W. Eitel. \"\u00dcber die \u00f6rtliche Verteilung von Rauchteilchen.\" _Zeitschrift f\u00fcr anorganische Chemie_ 87, no. 1 (12 May 1914): 357\u2013374.\n\nLuk\u00e1cs, Georg. _History and Class Consciousness_. Translated by Rodney Livingstone. Cambridge, Mass.: MIT Press, 1971.\n\n\u2014\u2014. _The Theory of the Novel_. Translated by Anna Bostock. Cambridge, Mass.: MIT Press, 1971.\n\n\u2014\u2014. \"Thoughts Toward an Aesthetic of the Cinema.\" Translated by Janelle Blankenship. _Polygraph_ 13 (2001): 13\u201318. Originally published as \"Gedanken zu einer Aesthetik des 'Kino,'\" _Frankfurter Zeitung_ 251 (10 September 1913): 1\u20132.\n\n\u2014\u2014. \"\u00dcber den Dostojewski-Nachlass.\" _Moskauer Rundschau_ 17 (22 March 1931): 4.\n\nMach, Ernst. _Die Principien der W\u00e4rmlehre: Historisch-kritisch Entwickelt_. Leipzig: Barth, 1896.\n\nMacintyre, John. \"X-Ray Records for the Cinematograph.\" _Archives of Skiagraphy_ 1, no. 2 (April 1897): 37.\n\nMagnus-Levy, Adolf. \"Ueber Organ-Therapie beim endemischen Kretinismus.\" _Ver-handlungen der Berliner medicinischen Gesellschaft_ 34, part 2 (1903): 350\u2013357; 34, part 1 (1903): 246\u2013249.\n\nMarey, \u00c9tienne-Jules. _Animal Mechanism: A Treatise on Terrestrial and Aerial Locomotion_. New York: Appleton, 1874.\n\n\u2014\u2014. \"\u00c9tudes sur la marche de l'homme.\" _Revue Militaire de M\u00e9decine et de Chirurgie_ 1 (1880): 244\u2013246.\n\n\u2014\u2014. _La chronophotographie_. Paris: Gauthier-Villars, 1899.\n\n\u2014\u2014. _La methode graphique dans les sciences \u00e9xperimentales et principlement en physiologie et en m\u00e9dicine_. Paris: Masson, 1885.\n\n\u2014\u2014. _Movement_. Translated by Eric Pritchard. London: Heinemann, 1895. Originally published as _Le movement_. Paris: Masson, 1894.\n\nMarinescu, Gheorghe. \"Les troubles de la marche dans l'h\u00e9mipl\u00e9gie organique \u00e9tudi\u00e9s \u00e0 l'aide du cin\u00e9matographe.\" _La semaine M\u00e9dicale_ (1899): 225\u2013228.\n\nMay, Bruno. _Das Recht des Kinematographen_. Berlin: Falk, 1912.\n\n\"Medical News.\" _Lancet_ 177 (27 May 1911): 1470.\n\n\"Medical News.\" _Lancet_ 182 (11 October 1913): 1083\u20131084.\n\n\"Medizinisch-Naturwissenschaftlicher Verein T\u00fcbingen.\" _M\u00fcnchener medizinische Wochenschrift_ 56, no. 3 (19 January 1909): 154\u2013155.\n\nMeier, Konrad. \"Vorschriften \u00fcber L\u00fcftung von Kinotheatern.\" _Gesundheits-Ingenieur_ 36, no. 26 (28 June 1913): 483\u2013484.\n\nMellini, Arthur. \"Die ganze Richtung passt uns nicht!\" _Lichtbild-B\u00fchne_ 5 (4 February 1911): 3\u20134.\n\nMendel, Georg Victor. _Kinematographie und Schule: Plan zur Gr\u00fcndung eines rein wissenschaftlichen Theaters f\u00fcr Kinematographie und Projektion_. Berlin: privately printed, 1909.\n\nMeumann, Ernst. \"Wilhelm Wundt. Zu seinem achtzigsten Geburtstag.\" _Deutsche Rundschau_ 38, no. 11 (August 1912): 193\u2013224.\n\nMoll, Albert. _Hypnotism_. London: Scott, 1890. Originally published as _Der Hypnotismus_ (Berlin: Fischer's Medicinische, 1889).\n\nMosso, Angelo. _Fatigue_. Translated by Margaret Drummond and W. B. Drummond. New York: Putnam, 1904.\n\nM\u00fcller-Sanders, Hans. \"Die Kinematographenzensur in Preu\u00dfen.\" PhD diss., Badischen Ruprecht-Karls-Universit\u00e4t, Heidelberg, 1912.\n\nM\u00fcnsterberg, Hugo. _The Photoplay: A Psychological Study_. London and New York: Appleton, 1916.\n\nMurawski, Friedrich. _Die Kinematographie und ihre Beziehungen zu Schule und Unterricht_. Dresden: Bieyl and Kaemmerer, 1914.\n\nMuybridge, Eadweard. _Animal Locomotion: An Electro-Photographic Investigation of Consecutive Phases of Animal Movement_. 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Princeton, N.J.: Princeton University Press, 1998.\n\n\u2014\u2014. \"Ideals of Science and Their Discontents in Late Nineteenth-Century American Medicine.\" _Isis_ 82, no. 3 (September 1991): 454\u2013478.\n\nWarnotte, Daniel. _Ernest Solvay et l_ ' _Institut de Sociologie: Contribution \u00e0 l_ ' _histoire de l_ ' _\u00e9nerg\u00e9tique sociale_. Brussels: Bruylant, 1946.\n\nWarwick, Andrew. \"X-Rays as Evidence in German Orthopedic Surgery, 1895\u20131900.\" _Isis_ 96, no. 1 (March 2005): 1\u201324.\n\nWeber, Max. \"Science as a Vocation.\" In _The Vocation Lectures_ , ed. and with an introduction by David Owen and Tracy B. Strong, trans. Rodney Livingstone, 1\u201331. Indianapolis: Hackett, 2004.\n\nWehler, Hans-Ulrich. _The German Empire, 1871\u20131918_. Leamington Spa, U.K.: Berg, 1985.\n\nWeindling, Paul. \"Bourgeois Values, Doctors and the State: The Professionalization of Medicine in Germany 1848\u20131933.\" In _The German Bourgeoisie_ , ed. David Blackbourn and Richard J. Evans, 198\u2013223. New York: Routledge, 1991.\n\n\u2014\u2014. _Health, Race, and German Politics Between National Unification and Nazism, 1870\u20131945_. Cambridge and New York: Cambridge University Press, 1989.\n\n\u2014\u2014. \"Public Health in Germany.\" In _The History of Public Health and the Modern State_ , ed. Dorothy Porter, 119\u2013131. Atlanta, Ga.: Rodopi, 1994.\n\nWesfreid, Jos\u00e9 Eduardo. \"Scientific Biography of Henri B\u00e9nard (1874\u20131939).\" In _Dynamics of Spatio-Temporal Cellular Structures: Henri B\u00e9nard Centenary Review_ , ed. Innocent Mutabazi, Jos\u00e9 Eduardo Wesfreid, and \u00c9tienne Guyon, 9\u201340. New York: Springer, 2006.\n\nWhitbeck, Caroline. \"What Is Diagnosis? Some Critical Reflections.\" _Metamedicine_ 2, no. 3 (October 1981): 319\u2013329.\n\nWilder, Kelly. _Photography and Science_. London: Reaktion, 2009.\n\nWilkending, Gisela. _Volksbildung und P\u00e4dagogik_ \" _vom Kinde aus_ \" _: Eine Untersuchung zur Geschichte der Literaturp\u00e4dagogik in den Anf\u00e4ngen der Kunsterziehungsbewegung_. Weinheim, Germany: Beltz, 1980.\n\nWinston, Brian. \"The Documentary Film as Scientific Inscription.\" In _Theorizing Documentary_ , ed. Michael Renov, 37\u201357. London and New York: Routledge, 1993.\n\nWohlring, Herman J. Review of _The Human Gait. Human Movement Science_ 8, no. 1 (February 1989): 79\u201383.\n\nYoxen, Edward. \"Seeing with Sound: A Study of the Development of Medical Images.\" In _The Social Construction of Technological Systems: New Directions in the Sociology and History of Technology_ , ed. Wiebe E. Bijker, Thomas P. Hughes, and Trevor J. Pinch, 281\u2013303. Cambridge, Mass.: MIT Press, 1987.\nINDEX\n\n**Page numbers refer to the print edition but are hyperlinked to the appropriate location in the e-book**.\n\n_ABC der Anschauung_ (Pestalozzi),\n\nabstraction: of movement, , 37\u201338, 49\u201363, 74\u201375; of spectators, 11\u201312\n\nactive and passive viewers, , . _See also_ expert\/lay distinctions\n\nAdorno, Theodor,\n\nadult education and motion pictures, 184\u201390\n\naesthetic contemplation, , , , , , 197\u2013202; agency or free will and, , , ; alertness ( _Geistesgegenwart_ ) and, ; Benjamin and, , 240\u201341; distraction and, , 233\u201335; as expert vision or observation, , , ; gender and, 231\u201332; as ground for aesthetic experience, , ; indeterminacy and, ; Kracauer and, ; Luk\u00e1cs and, 235\u201340; mass reception and, , , ; moral significance of, ; motion pictures and, , , 195\u201396, 230\u201331; movement and, , 230\u201331; as observational training, 170\u201372; passivity and, ; political implications of, ; Schiller and, ; Schopenhauer and, , . _See also_ aesthetic experience\n\naesthetic education, 143\u201347; film reform and, ; as German tradition, ; moral renewal and, , 169\u201370; museums and, ; nationalism and, 168\u201371; observational training and, , 168\u201371; Schiller and, 167\u201368, ; suggestibility and,\n\naesthetic experience, , , , __ ; aesthetic education and, ; agency within, , 202\u201314; descriptions of, 196\u2013201; detachment and, 196\u201398, , , , ; embodied vision and, 214\u201330; as emotional projection, , 220\u201321; identity and, 214\u201316; moral significance of, 197\u2013201, ; movement and, 224\u201330; as renewal of perception, ; repose and, ; Schiller and, 205\u201307; Schopenhauer and, , ; synesthesia and, 221\u201322; as system of dichotomies, 196\u2013201; taste and, ; temporality and, 204\u20137. _See also_ aesthetic contemplation\n\naesthetics, , 4\u20135; _Einf\u00fchlung_ as solution to problems in, 215\u201316; formalist and scientific approaches to, ; reception and form in, ; \"traditional,\" , , , , , 216\u201317,\n\nagency, , , 197\u201399, , 202\u201314. _See also_ free will; volition; will\n\n_Die Aktion_ ,\n\nalertness or presence of mind ( _Geistesgegenwart_ ), 212\u201314,\n\nall-at-onceness, . _See also_ observation: gaze and glance\n\nalignment of image, object, and technology, , 43\u201362\n\nAllgemeines Krankenhaus,\n\n_The Alps_ (film),\n\nAltenloh, Emilie, ,\n\nalternative public sphere, early film exhibition as,\n\nAmerican College of Surgeons,\n\nAmerican Hospital Association,\n\nanalysis and synthesis, , , 36\u201337, , , 117\u201318, ; medical observation and, 120\u201325; motion pictures and, , ; scientific cinematography's relationship to, , 117\u201318; spectatorship and,\n\nanalysis (close reading), 4\u20135, , 247\u201349\n\nanalysis of motion, , , , , , 115\u201316; Bergson's critique of, 32\u201333, ; frame-by-frame, , , ,\n\nAndreas-Salom\u00e9, Lou, 209\u201310\n\nAndrew, Dudley,\n\nAndriopolous, Stefan,\n\nanesthesia,\n\nanimal locomotion, ,\n\n_Anschaulichkeit_ (vividness), 107\u20138,\n\n_Anschauung_ (sense impression), 172\u201376\n\n_Anschauungsunterricht_ (visual means of instruction), , , , 145\u201346, 172\u201384; as apperception, 174\u201375; correlation and, 174\u201375; detail and, 173\u201374; elementary and adult education and, 176\u201377; Herbartian principles of, 182\u201383; Lichtwark and, ; motion pictures and, , 182\u201383; natural sciences and, ; as object lesson, 173\u201375; as observational training, , , , 173\u201375; Pestalozzi and, , 174\u201375; relationship to images and words, 175\u201376; self-cultivation ( _Bildung_ ) and, ; social acceleration and, 173\u201374. _See also_ object lesson\n\nAnsch\u00fctz, Ottomar, , 265n73\n\napperception, 173\u201374, 181\u201382, 226\u201327\n\n_Arbeitsschule_ (works schools),\n\n_Arbeitswissenschaft_ (the science of work),\n\narchitecture, , 224\u201327,\n\narchives, of images, , 103\u20135, 114\u201316\n\nAristotelian science,\n\nArnheim, Rudolf,\n\nart education movement ( _Kunsterziehungsbewegung_ ), , , 169\u201371,\n\nAshwin, Clive, ,\n\natomic-kinetic model of heat, , ,\n\natomic-kinetic theory of matter, 64\u201365\n\nattention ( _Aufmerksamkeit_ ), , 132\u201333, 139\u201340, , , , 198\u201399, ,\n\naura,\n\nautomatic writing,\n\nautopsies,\n\n_Autorenfilm_ (author's film),\n\nBachelard, Gaston,\n\nBal\u00e0zs, B\u00e9la, , , 279n70\n\nbearers of culture ( _Kulturtr\u00e4ger_ ), . _See also_ physicians\n\nbeautiful, theory of the, , 304n10\n\nBellevue Hospital, 104\u20135\n\nBellour, Raymond,\n\nB\u00e9nard, Henri,\n\nBenjamin, Walter, , , , 236\u201337, , ; alertness and, ; aura and, ; Bergson and, ; contemplation and, , , , 240\u201341; distraction and, , , ; Duhamel and, ; expert\/lay distinctions and, ; mode of perception and, ; optical unconscious and, ; traditional aesthetics and, ,\n\nBergson, Henri, , , , , , , , , , ; analysis and, 87\u201388; Benjamin and, ; cinematographical thinking and, , , 32\u201337; critique of science and, 32\u201337, 62\u201364; _eidos_ and, ; German counterparts to, 36\u201337; movement and, 33\u201334; synthesis and, ; time and, 74\u201376; United States and, , ,\n\nBernheim, Hippolyte,\n\n_Bildbetrachtung_ (image viewing), , ,\n\n_Bild und Film_ (trade journal), , ,\n\n_Bild und Wort_ (Image and Word society) (film exhibitions),\n\n_Bildung_ (self-cultivation), ; visual training and,\n\n_Bildungsb\u00fcrgertum_ (educated middle-class), ,\n\nBillings, Susan,\n\nBillroth, Theodor,\n\nBinet, Alfred,\n\nbiology, , , , ; cell, , , , , 76\u201388,\n\n_Der Blaue Engel_ (The Blue Angel) (film)\n\nBloch, Ernst, , 222\u201324\n\n_The Blue Angel_ (film),\n\nBorbacher Knabenmord (child murder of Borbach), 287n151\n\n_Botryllus_ (sea squirts),\n\nBourdieu, Pierre,\n\nBraun, Ludwig, 98\u2013100, 111\u201321, , ,\n\nBraune, Wilhelm, 19\u201320, 24\u201326, , 41\u201344, 62\u201363, , , , . _See alsoThe Human Gait_\n\nBraus, Hermann, , , , , __ ; nerve fibers and, , , 82\u201386,\n\nBrecht, Bertolt, ,\n\nBritish Medical Association,\n\nBrod, Max, ,\n\nBrown, Robert,\n\nBrownian motion, , 24\u201325, , 62\u201376, __ , __ , , , , 266n97; Einstein's theory of, 64\u201369, 266n94, 266n98; importance of displacement to Einstein's theory, 68\u201369\n\nBrunner, Karl, 152\u201353, 157\u201358, 292n34\n\nBrush, Stephen,\n\nBull, Lucien, ,\n\nBurrows, Montrose,\n\n_The Cabinet of Dr. Caligari_ (film),\n\ncapitalism, , , ,\n\ncardiac dynamics and mechanics, , __ ,\n\nCarlet, H. M.,\n\nCarrel, Alexis,\n\nCartwright, Lisa, ,\n\nCarvallo, Joachim-L\u00e9on, , ,\n\ncell biology, , , , , 76\u201388,\n\ncensorship, , ,\n\n_Century of the Child_ (Key),\n\nCharcot, Jean-Martin, , ,\n\nCharit\u00e9 Hospital,\n\nCharles Urban Trading Company,\n\nChase, Walter,\n\nChevroton, Lucienne, ,\n\nchild psychology, , , 162\u201364. _See also_ children and crowds: analogies between; crowd psychology\n\nchildren and crowds: analogies between, , ; aesthetic sensibilities of, ; as models for descriptions of film spectatorship, , _See also_ crowd psychology\n\nchronophotography, , , , , , 60\u201361, , ; Brownian motion and, , , ; ergonomics and, ; for _The Human Gait_ , 44\u201362, __ ; human locomotion and, 19\u201320; two-sided,\n\ncinema: theater compared to, 236\u201339\n\n_Cinema and Art_ ( _Kino und Kunst_ ) (H\u00e4fker),\n\n\"Cinema and Schoolchildren\" (Kleib\u00f6mer),\n\nCinema Reform Association, ,\n\n_The Cinematic Lesson Plan_ ( _Die kinematographische Unterrichtsstunde_ ) (Lemke), , __\n\ncinematographical thinking, , , , 32\u201337. _See also_ Bergson, Henri\n\ncinematography, 21\u201322; adaptation of for biology and physics, 76\u201377; as aid to correlation, , ; analysis and synthesis and, ; Bergson's analogy of, 32\u201337; Braun and, 111\u201314; Braus and, , 84\u201387; Brownian motion and, , 69\u201370; as confirmation of biological theory, ; high-speed, , , ; medical, 92\u201393,  , ; microcinematography, , , 92\u201393, ; science and, 32\u201337; slow-motion, , , ; as substitute for object of study, ; time-lapse, , , 84\u201385, , , , ; as optical unconscious, ; as virtual experiment, , ; as virtual witness, , ; X-rays and, , 100\u2013102, _101. See also_ chronophotography; _The Human Gait_ ; medical filmmaking; research film\n\nclass: educated middle-class, , ; ideals, ; identity, , ; warfare, 292n34; working,\n\nclass-based distinctions, , ,\n\nclassical physics, , ,\n\nClausius, Rudolf,\n\nclinical gaze, , 118\u201319, . _See also_ observation: gaze and glance\n\nclothing reform, 149\u201350\n\nCole, Lewis Gregory,\n\nColl\u00e8ge de France, ,\n\nComandon, Jean, , , , ,\n\ncommunal sense ( _sensus communis_ ),\n\ncontinuity and discontinuity, , , 32\u201334, , , , , , , 86\u201387, ,\n\ncorrelation, , , , , , , ; definition of, ; medical observation and, 115\u201317; film as aid to, , ,\n\ncountry boarding schools ( _Landerziehungsheime_ ),\n\nCranz, Carl,\n\nCrary, Jonathan, 132\u201333,\n\n_Creative Evolution_ (Bergson), , , ,\n\ncrime, hypnotism and, 135\u201336\n\ncritical method ( _kritischer Methode_ ), . _See also_ observation\n\n_Critique of Judgment_ (Kant),\n\n_Critiques_ (Kant), ,\n\nCros, Charles,\n\ncrowd psychology, , ; theories of, 164\u201365; suggestibility and, 165\u201366; will or volition and, . _See also_ children and crowds: analogies between\n\n\"The Cult of Distraction\" (Kracauer),\n\ncultural capital,\n\ncultural pessimism,\n\ncultural series,\n\n_The Culture of the Female Body as a Foundation for Women's Clothing_ (Schultze-Naumberg), 149\u201350\n\ncurricular integration,\n\nDadaists,\n\nDarwin, Charles, , , , 296n80\n\nDaston, Lorraine,\n\nde Broglie, Louis,\n\n_Degeneration_ (Nordau),\n\ndegeneration (concept),  , 131\u201332\n\nde Lagarde, Paul, ,\n\nDeleuze, Gilles, , 75\u201376\n\nDescartes, Ren\u00e9, , , 210\u201311\n\n_The Descent of Man_ (Darwin),\n\nDessoir, Max, 307n51,308n55\n\ndetachment: aesthetic,191, 196\u201398, , , , ; expert, , , . _See also_ observation: aesthetic experience\n\ndetail: as fidelity to nature, , , 107\u201308, 111\u201313, ; as ground for authenticity, 173\u201375, ; as ground for observational practice, 119\u201320, , , ; scientific management of, , , , , ; as symptom of modernity, , ,\n\nDewey, John,\n\nDiderot, Denis,\n\nDidi-Huberman, Georges,\n\nDiederichs, Helmut,\n\ndifferential reproduction, , 257n9\n\nDilthey, Wilhelm,\n\ndirect perception (immediate or sensual), , , , , , , , , , 279n70\n\ndisciplinary agendas, 2\u20133, , , , 245\u201346, ; technology and, 3\u20134, 14\u201315,\n\ndisciplinary logic, 4\u20136, 9\u201310, 14\u201315, 23\u201324, , 248\u201349; film form and, 5\u20136, 23\u201324, 111\u201325. _See also_ medical logic; patterns of use: film form and\n\ndiscontinuity. _See_ continuity and discontinuity\n\ndisease, , , , , 127\u201328; cinematography and, ; neurological, ; series photography and, 113\u201315\n\ndisinterest. _See_ detachment; aesthetic\n\ndisplacement, of particles, , 68\u201370, 72\u201373. _See also_ Brownian motion\n\n_dispositif_ , ; as experimental material, 250\u201351\n\ndissemination of media technologies, ,\n\ndistraction, , , , , ; attention and, ; Benjamin and, , , ; Kracauer and, ; proximity and,\n\nD\u00f6blin, Alfred,\n\ndocile bodies ( _aka_ working objects), 42\u201344\n\ndoctor\u2013patient relationship, ,\n\ndocumentary function, of medical filmmaking, 102\u20135\n\ndomestication of the image, 42\u201344, 51\u201362; of the body, 42\u201344. _See also_ docile bodies\n\n_Don Quixote_ (novel),\n\nDoodica (conjoined twin),\n\nDoyen, Eug\u00e8ne Louis, 106\u201310, 279n63\n\n\"Drama in the Film Theaters\" (Polgar),\n\nDriesch, Hans, ,\n\nDuenschmann, Hermann,\n\nDuhamel, Georges,\n\nduration, , , , ; cinematography and, , , . _See alsodur\u00e9e_\n\n_Duration and Simultaneity_ (Bergson),\n\n_dur\u00e9e_ (duration), 32\u201333, , ; form as interruption of, 62\u201364\n\nDuttlinger, Carolin, ,\n\nEclipse (film company), 155\u201356\n\neducated middle-class ( _Bildungsb\u00fcrgertum_ ), ,\n\neducation, , , ; art education movement, , , 169\u201371, . _See also_ adult education and motion pictures; aesthetic education, _Anschauungsunterricht_ , elementary education and motion pictures\n\neducational function, of medical filmmaking, 105\u201310\n\neffects of media technologies, 244\u201345\n\nefficiency, ; as social goal, 38\u201342; and the human body, 38\u201342; in medicine, ; of the filmic image, 109\u201310, 175\u201376, 210\u201311\n\neidetic images,\n\n_eidos_ ,\n\n_Einf\u00fchlung_ (emotional projection, feeling into), , , 215\u201322, 224\u201329, , 307n51; body as model for, ; contemplation and, , ; _Kino-Debatte_ and, , ; movement and, 224\u201328; physical response and, 228\u201330; space and, 225\u201328\n\nEinstein, Albert, 20\u201321, 24\u201325, , 66\u201369, 72\u201374, 87\u201388\n\nelementary education and motion pictures, 176\u201384\n\nElias, Norbert,\n\nembodied perception or vision, 214\u201330, . See also aesthetic experience; _Einf\u00fchlung_\n\nembourgeoisement,\n\nemotional projection. _SeeEinf\u00fchlung_\n\nempathy, . See also _Einf\u00fchlung_\n\nenemies of the cinema ( _Kinogegner_ ),\n\nenergy, conservation of, 39\u201340\n\nEnlightenment, , , ,\n\nentropy,\n\nEpstein, Jean, , , 279n70\n\nergograph,\n\nergonomics,\n\nErnemann (film company), 28\u201329,\n\n_Essay Concerning Human Understanding_ (Locke),\n\n\"The Essence of Architectural Creation\" (Schmarsow),\n\nEurope, , , , ,\n\nevidence, creation of, 24\u201325\n\nevil, children as,\n\nevolutionary theory,\n\n_Exercises in the Contemplation of Art Works_ (Lichtwark), 170\u201372,\n\nExner, Felix, , 69\u201370\n\nexperiment: in biology, 82\u201383; characteristics of, 98\u201399; differential reproduction in, 250\u201351; disciplinary logic and, 23\u201324; historiographic implications of, 250\u201351; infrastructure for, 27\u201328; in medicine, 98\u2013102; motion picture equipment manufacturers and, ; motion pictures and, 22\u201324, , ; popular audiences for, 28\u201329; time and, 139\u201340\n\nexperimental systems, 3\u20134, , 23\u201324, , 257n9\n\nexpert modes of viewing, 6\u201310, 24\u201325. _See also_ observation\n\nexpert observation. _See_ observation\n\nexpert training, , ; advantages of cinematography for, ; and professional identity, ; and research films, ,\n\nexpert viewers. _See_ observation\n\nexpert vision. _See_ observation\n\nexpert\/lay distinctions, 11\u201313, 94\u201395, ; historiographic implications of,\n\nexploratory function, of medical filmmaking, 98\u2013102\n\nEykman, P. H.,\n\nfaithful to nature ( _Naturgetreu_ ), , ,\n\nFata Morgana (film theater),\n\nfatigue, 39\u201342, ,\n\nFechner, Gustav,\n\nfeeling into. _SeeEinf\u00fchlung_\n\nfemale hysteria,\n\nfemale spectatorship, , 231\u201332\n\nfilm drama ( _Kino-drama_ ), , , ,\n\nfilm exhibition, legal restrictions on,\n\nfilm form, 4\u20135, , , , ; correspondences between form and logic, 23\u201324; disciplinary logic and, ; historiography and,\n\nfilm frame, as spatial and temporal boundary,\n\nfilmic realism, , . _See also_ _Naturgetreu_ (faithful to nature)\n\nFilm-Idea-Central,\n\n\"Film in the Service of Medicine\" (demonstration),\n\nfilm reform, 152\u201362; distribution and, 159\u201360; exhibition and, , 184\u201390; foreign films and, ; goals of, ; Kino-Kommissions and, ; Kinoreform and, ; legacy of, 160\u201362; narrative and, ; objections to motion pictures, ; positive and negative approaches to, ; production and, 156\u201359; trade journals and 152\u201353; uplift and,\n\nfilm strip as line, 74\u201376\n\nfilm studies,\n\n_Film und Lichtbild_ (periodical),\n\nfirst law of thermodynamics, ,\n\nFischer, Otto, 19\u201320, 24\u201326, , 41\u201344, . _See alsoThe Human Gait_\n\nFleck, Ludwik, 5\u20137, ,\n\nflicker effect, 130\u201331, , ,\n\nflip books,\n\nfolk medicine,\n\nForel, Auguste,\n\nform: geometry inherent in, 62\u201363, ; as temporal interruption in duration, . _See also_ abstraction: of movement\n\nform drive, ,\n\nFoucault, Michel, , 42\u201343, , , 118\u201322\n\nframe-by-frame analysis, , , ,\n\nFrance, 27\u201328, , , ,\n\nFran\u00e7ois-Franck, Charles \u00c9mile, ,\n\nFr\u00e4nkel, James,\n\n_Frankfurter Zeitung_ ,\n\nfree play of associations,\n\nfree will, , ; aesthetic contemplation and, , , ; aesthetic experience and, , 202\u201314; crowd psychology and, ; identity and, . _See also_ agency; volition; will\n\nFrench Academy of Medicine,\n\nFreud, Sigmund, 163\u201364, , , 309n72\n\n_From Caligari to Hitler_ (Kracauer), 301n145\n\n_Der Gang des Menschen_ (Braune and Fischer), . _See alsoThe Human Gait_\n\nGaudreault, Andr\u00e9, ,\n\nGaumont (film company),\n\nGaumont, Charles,\n\nGaupp, Robert, 127\u201328, 136\u201337,\n\ngaze: clinical or medical, , , 118\u201319; glance and, , 120\u201322; holistic, , , 120\u201321, . _See also_ aesthetic contemplation, all-at-onceness, observation, _Sehen_ , _Schauen_\n\n\"Gedanken zu einer Aesthetik des 'Kino' (Thoughts Toward an Aesthetic of the Cinema)\" (Luk\u00e1cs), ,\n\nGeissler tubes, , 48\u201349, 51\u201352,\n\n_Geistesgegenwart_ (alertness or presence of mind), 212\u201314,\n\ngender. _See_ women\n\ngender dynamics,\n\ngeometry, ; inherent in form, 62\u201363, . _See also_ abstraction: of movement\n\nGerman High Ministry of War,\n\nGermany, reaction to modernity, 13\u201314\n\n_Gesamtkunstwerk_ (Wagner),\n\n_Gesamtsinnlichkeit_ (sensuous totality),\n\n_Gesamtvorf\u00fchrung_ (total presentation),\n\nGesellschaft der Freunde des vaterl\u00e4ndischen Schul- und Erziehungswesens (Society of Friends of the Schools and Instruction for the Fatherland),\n\nGesellschaft deutscher Naturforscher und \u00c4rzte (Society of German Natural Scientists and Physicians), ,\n\nGesellschaft zur Verbreitung von Volksbildung (Society for the Dissemination of Popular Education) (GVV), ,\n\nGeuss, Raymond,\n\n_Gewohnheit_ (habit), ,\n\n_Giovanna d'Arco_ ( _The Maid from Orleans_ ) (film),\n\nglance, and gaze, , 120\u201322\n\nGoerke, Franz, ,\n\nGoethe, Johann Wolfgang von, ,\n\nGordon, Rae Beth,\n\nG\u00f6tze, O.,\n\nGouy, Louis-Georges,\n\nGrau, Alexander,\n\nGreat Exhibition (1851),\n\nGreat Lisbon Earthquake,\n\nGriffith, D. W.,\n\nGroedel, Franz, 101\u20132\n\nGuerlac, Suzanne,\n\nGunning, Tom, ,\n\ngutta-percha,\n\nGVV. _See_ Gesellschaft zur Verbreitung von Volksbildung\n\n_Gymnasium_ (high school), ,\n\nhabit ( _Gewohnheit_ ), ,\n\nHacking, Ian, ,\n\nH\u00e4fker, Hermann, , 152\u201353, , , 176\u201377, ; contemplation and, , ; _Kinetographie_ and, ; Lichtwark and, ; model presentations and, , ; photograph of, __ ; taste and, ; World War I and, 301n145\n\nHake, Sabine, , ,\n\nHamburg commission, , ,\n\nHansen, Miriam, , 231\u201332\n\nHarrington, Anne,\n\nHarris, Stefanie,\n\nHarrison, Ross Granville, , __ , , 82\u201383, ,\n\nHau, Michael, , ,\n\nhealth campaigns,\n\nhealth insurance,\n\nheat transference, ,\n\nHegel, Georg Wilhelm Friedrich,\n\n_Heimat_ ,\n\n_Heinrich von Ofterdingen_ (Novalis),\n\nHeld, Hans, ,\n\nHeller, Heinz-B.,\n\nHellwig, Albert, , , ,\n\nHennes, Hans, 103\u20136\n\nHenri, Victor, , 69\u201370,\n\nHerbart, Johann, , , 181\u201382, ,\n\nheterogeneity, ; of early cinema, , ,\n\nhigh school ( _Gymnasium_ ), ,\n\nhigh-speed cinematography, , ,\n\nHildebrand, Adolf, , , 224\u201325\n\nhistoriography, , ; analogy and homology in, ; disciplinary or expert use of film and, , , 246\u201347; _Kino-Debatte_ and, ; \"tactile,\" 246\u201350\n\n_History and Class Consciousness_ (Luk\u00e1cs), 235\u201336, 239\u201340\n\nholistic gaze, , , 120\u201321,\n\nHollywood, mode of production and reception,11\u201312\n\nHuelsenbeck, Richard,\n\nHuerkamp, Claudia,\n\n_The Human Gait_ (Braune and Fischer), , 44\u201362; camera placement for, __ ; coordinates graph for, __ , __ ; coordinates table for, __ ; determination of coordinates for, __ ; instrument for coordinate measurements, __ ; measurement of coordinate, __ ; military recruit for, __ ; resulting chronophotograph for, __ ; subject at rest, __ ; tridimensional model for, __ , __\n\nhuman locomotion, 19\u201320, , 38\u201339, 41\u201344. _See alsoThe Human Gait_\n\nhuman sciences,\n\nHusserl, Edmund, 58\u201359\n\nhypnotism, __ ; cinema and, 136\u201337; as model for observation, 138\u201340; as model for spectatorship, 135\u201337\n\nhysteria, ,\n\nIbsen, Henrik,\n\nidentity: aesthetic experience and, , 214\u201330; class, , ; free will and, ; professional, , , , , ; self-identity, , , , ,\n\n_Image and Word_ ( _Bild und Wort_ society) (film exhibitions),\n\nimage archives, , 103\u20135, 114\u201316\n\nimage viewing ( _Bildbetrachtung_ ), , ,\n\nimagination, , , ; architecture and, ; repose and,\n\nimaginative faculty, 209\u201310\n\nimmutability,\n\nimproper viewing, , 94\u201395. _See also_ spectatorship\n\nincremental exposures, . _See also_ series photography\n\nIndustrial Revolution, , 147\u201348\n\n\"Infantile Sexuality\" (Freud),\n\ninfantilization, of audience, . _See also_ children and crowds: analogies between\n\ninner child,\n\n_Innerlichkeit_ (inwardness), 239\u201340\n\ninner movement, . See also movement and: aesthetic contemplation; movement and: aesthetic experience\n\ninscription devices, , 30\u201331\n\nInstitut de Sociologie,\n\nInstitut Marey, ,\n\ninterdisciplinary research,\n\ninwardness ( _Innerlichkeit_ ), 239\u201340\n\n_Die Irrfahrten des Odysseus_ ( _The Wanderings of Odysseus_ ) (film), , __\n\nJacobj, Carl,\n\nJanssen, Jules,\n\nJelavich, Peter,\n\njournals, trade 152\u201353\n\nKade, August,\n\nKaes, Anton, , ,\n\nKaiserin-Friedrich-Haus,\n\nKammer-Lichtspiele Theater, __\n\nKant, Immanuel, , , 214\u201315; aesthetic contemplation and, ; aesthetic experience and, , 195\u201399; citizenship and, 140\u201341; taste and,\n\nK\u00e4stle, C.,\n\nKeene, Melanie,\n\nKelvin, Lord. _See_ Thomson, William\n\nKey, Ellen,\n\nKienzl, Hermann, 209\u201310\n\nKiesler, Frederick,\n\nKillen, Andreas, 284n122\n\n_Der Kinematograph_ (trade journal), 152\u201353, , ,\n\n_Kinematographie und Schule_ (Mendel),\n\n_Die kinematographische Unterrichtsstunde_ ( _The Cinematic Lesson Plan_ ) (Lemke), , __\n\n_Kinetographie_ ,\n\n_Kinobuch_ (Pinthus),\n\n_Kino-Debatte_ , 16\u201317, 193\u201395, 201\u20132, , ; aesthetic experience and, , ; alertness and, ; _Einf\u00fchlung_ and, , ; historiography and, ; literary emphasis and, ; somnambulism and, ; traditional aesthetics and,\n\n_Kino-drama_ (film drama), , , ,\n\n_Kinogegner_ (enemies of the cinema),\n\n_Kino-Kommissions_ ,\n\n_Kinoreform_ , 153\u201355, , , . _See also_ film reform\n\n_Kino und Kunst_ ( _Cinema and Art_ ) (H\u00e4fker),\n\n\"Kino und Schaulust\" (Serner),\n\n_Kitsch_ ,\n\nKleib\u00f6mer, Georg, 202\u20133\n\nKoch, Robert,\n\nKosmographia,\n\nKracauer, Siegfried, , , 233\u201334, , , 301n145\n\nKraepelin, Emil, ,\n\nKretz, Richard,\n\nKrieger, Ernst,\n\n_kritischer Methode_ (critical method), . _See also_ observation\n\n_Kultur_ and _Zivilisation_ , 144\u201345\n\n_Kulturtr\u00e4ger_ (bearers of culture), . _See also_ physicians\n\n_Kunsterziehungsbewegung_ (art education movement), , , 169\u201371,\n\n_Der Kunstwart_ ,\n\nKutner, Robert, ,\n\nkymograph,\n\nlaboratories, , , ; motion pictures in, , , , 27\u201330, ,\n\n_Ladenkinos_ (storefront cinemas), , __\n\nLandecker, Hannah,\n\n_Landerziehungsheime_ (country boarding schools),\n\nLangbehn, August Julius, 168\u201369\n\nLange, Konrad, , , , 169\u201370, ; _Kino-drama_ and, ; taste and,\n\nLatour, Bruno, 30\u201331,\n\nLaycock, Thomas,\n\nlay viewers, , . _See also_ spectatorship\n\n_Lebensphilosophie_ ,\n\nLe Bon, Gustave, ,\n\nlegibility, , , ,\n\nlegitimacy, , , , , ; of motion pictures within a discipline, 1\u20132, 4\u20135, , , , , 122\u201323, ,\n\nLemke, Hermann, , , , , , , , ; Cinema Reform Association and, , ; Film-Idea-Central and, ; Herbart and, 181\u201382; Society for the Dissemination of Adult Education and, ; Society for the Dissemination of Popular Education and,\n\nLevin, Tom,\n\n_Lichtbild-B\u00fchne_ (trade periodical), , __\n\nLichtbilderei (film institute), 159\u201360\n\nLichtwark, Alfred, , 169\u201372,\n\nLipps, Theodor, 227\u201328,\n\nlocal exhibition rights, 295n61\n\nLocke, John, 210\u201311\n\nlocomotion: animal, , ; human, 19\u201320, , 38\u201339, 41\u201344. _See alsoThe Human Gait_\n\nlogic: disciplinary, 4\u20136, 9\u201310, 14\u201315, 23\u201324, , 248\u201349; film form and, 5\u20136, 23\u201324, 111\u201325; medical, , , 90\u201391, 111\u201325. _See also_ patterns of use: film form and\n\nLombroso, Cesare,\n\nLomon, Andr\u00e9,\n\nLonde, Albert,\n\n_The Lonedale Operator_ (Griffith),\n\nLuisen-Kino, __\n\nLuk\u00e1cs, Georg, , , , , 288n9, 311n98; child psychology and, ; contemplation and, 235\u201337, ; Romanticism and, ; time and,\n\nLumi\u00e8re (film company),\n\nLynch, Michael, , ,\n\nMach, Ernst,\n\nMacintyre, John,\n\nmagazines, film 152\u201353\n\n_The Maid from Orleans_ ( _Giovanna d'Arco_ ) (film),\n\nMaiocchi, Roberto, ,\n\nMantell, Gideon Algernon,\n\nMarey, \u00c9tienne-Jules, , 40\u201342, , , ; graphic method of, , , ; movement and, ; single-camera system of,\n\nMarinescu, Gheorghe,\n\n_Marital Hygiene_ (film),\n\nMarxism,\n\nmasculinity, , 150\u201351,\n\nmathematical immanent in matter. _See_ geometry\n\n_Matter and Memory_ (Bergson),\n\nmean square displacement, . _See also_ Brownian motion\n\n\"Measurement of the Temperature Dependency of Brownian Motion\" (Seddig),\n\nmedia ensemble ( _Medienverbund_ ), ,\n\nmedia technology, 1\u20134, , ,\n\nmedical demonstrations, 103\u20134\n\nmedical filmmaking, 92\u201393, 90\u201396, , , ; documentary function of, 102\u20135; educational function of, 105\u201310; exploratory function of, 98\u2013102; multiple functions of, 96\u2013110; observation and, 110\u201325. See also cinematography, research film\n\nmedical gaze, , , 118\u201319. _See also_ gaze; medical perception; observation\n\nmedical hermeneutics,\n\nmedicalization, of society,\n\nmedical logic, , , 90\u201391, 111\u201325. _See also_ disciplinary logic, patterns of use: film form and\n\nmedical perception, 119\u201320. _See also_ gaze; medical gaze; observation\n\nmedical way of thinking, . _See also_ medical logic; disciplinary logic; patterns of use: film form and\n\nmedicine, , , 15\u201316, , , , ; Doyen and, 106\u201307; folk, ; photography and, , ; social, ; spectacle and, 279n63\n\n_Medienverbund_ (media ensemble), ,\n\n\"Melody in the Cinema, or Immanent and Transcendental Music\" (Bloch),\n\nMendel, Georg Victor,\n\nmental development,\n\nMesster, Oskar, 28\u201329\n\nmetaphysics,\n\n\"The Metropolis and Mental Life\" (Simmel),\n\nMichaelis, Anthony,\n\nmicrocinematography, , , 92\u201393; medical filmmaking and,\n\nmicrophotographs,\n\nmilitary,\n\nmind, comparisons with motion pictures and modernity, 208\u201312\n\nmodel presentations ( _Mustervorstellungen_ ), , ,\n\nmode of perception (Benjamin), 12\u201313\n\n_Modern Hospital_ ,\n\nmodernity, , , , , , , ; cultural and social, ; excesses of, , , , , , , ; German reaction to, 13\u201314, 149\u201351; motion pictures as emblem of, 125\u201326, , , , , 208\u20139, ; motion pictures as potential haven from, ; pace of change within, 131\u201334, 203\u20134, ; as series of shocks, , . _See also_ social acceleration\n\nmodernity thesis,\n\nMoll, Albert,\n\nmonopoly system, 295n61\n\nmoral weakness,\n\nMosso, Angelo, ,\n\n_Le mouvement_ (Marey),\n\nmovement, , ; aesthetic education and, 178\u201379; analysis of, , , , , , 115\u201316; animal locomotion, , ; architecture and, ; Bergson and, 33\u201334, , ; contemplation and, , 230\u201331; education and, ; _Einf\u00fchlung_ and, 224\u201328; human locomotion, 19\u201320, , 38\u201339, 41\u201344; immobility and, 37\u201338; inner, ; Marey and, ; pathological, . _See also_ Brownian motion; _The Human Gait_\n\n_Moving Picture World_ ,\n\nmulticellular theory of nerve development,\n\nM\u00fcnsterberg, Hugo, ,\n\nmusic, 222\u201324\n\nMusser, Charles,\n\n_Mustervorstellungen_ (model presentations), , ,\n\nMuybridge, Eadweard, 26\u201327,\n\nNature, 190\u201391\n\nnature films, ,\n\n_Naturgetreu_ (faithful to nature), , ,\n\nnegative space, spectatorship as, ,\n\nneo-Kantian tradition,\n\nnerve fibers, 76\u201388, __\n\nnervousness,\n\nneurasthenia,\n\nneurological diseases,\n\nneurology, ; cinematography and, 103\u20135\n\nNewton, Isaac, ,\n\nNielsen, Asta, ,\n\nNietzsche, Friedrich, ,\n\nNogu\u00e8s, Pierre,\n\nnontheatrical uses of film, ; disciplinary agendas and,\n\nNordau, Max, , __ , 131\u201334,\n\nNovalis,\n\nnovels, ; serialized,\n\nNye, Mary Jo, 68\u201369\n\nobjectivity, , ,\n\nobject lesson, , , , 173\u201375, 177\u201379, , 183\u201384. _See alsoAnschauungsunterricht_\n\nobservation: the accommodation of motion pictures to, 5\u20136, 9\u201311, 111\u201312, ; aesthetic contemplation and, , , 170\u201372, ; aesthetic education and, , 168\u201371; _Anschauungsunterricht_ and, , , , 172\u201376; analysis and synthesis and, , ; attention and, ; in biology, 80\u201382; as correlation, , 111\u201317, ; detail and, 119\u201320, , , ; disciplinary logic and, , 9\u201310, 23\u201324, , , 111\u201325; as engine of progress, ; experiment and, , ; gaze and glance as, , 119\u201322; gender and, , , 231\u201332; hypnotism and, , 138\u201340; as ideology, 8\u20139, ; in medicine, , , , 110\u201325; as method of ordering thought, ; modernity and, 132\u201334, , , 191\u201392; photography and, ; as practice, , , 118\u201320; spectatorship and, , , , , , , , , 139\u201340, , , 191\u201392, , ; temporality of, 118\u201326; theory and, , 66\u201367, , 80\u201381; training and, 6\u20137, 10\u201311, , , , , 145\u201346. _See also_ all-at-onceness, analysis and synthesis; \"critical method\"; expert modes of viewing\n\n_On the Optical Sense of Form_ (Vischer),\n\nopera,\n\noptical unconscious,\n\n\"ordinary perception\" (Bergson), , 33\u201336, , , ,\n\norgan function,\n\n\"The Origin of Geometry\" (Husserl),\n\nOstwald, Wilhelm,\n\noutgrowth theory of nerve development, , ,\n\npace: human liberty or potential and, ; of modern life. , , , , , , , , ; of motion pictures, , , , , , , , , , , , ; of observation, , , ; superficiality and, , 203\u20134; tastelessness and, . _See also_ modernity; observation; social acceleration\n\nPaget, Sir James, , ,\n\nPAGU (film company),\n\nParnaland, Ambroise-Fran\u00e7ois,\n\nparticle displacement, , 68\u201370, 72\u201373\n\npassive and active viewers, , . _See also_ expert\/lay distinctions\n\npassivity, ,\n\nPasveer, Bernike,\n\nPath\u00e9 Fr\u00e9r\u00e8s, ,\n\npathological movement,\n\npathology,\n\npatterns of use: film form and, 20\u201322, 248\u201351; historiography and, 20\u201322, , 248\u201349; medicine and, , 121\u201323; professional identity and,\n\nperception: apperception, 173\u201374, 181\u201382, 226\u201327; embodied, 215\u201317, , ; direct (immediate or sensual), , , , , , , , ; mode of (Benjamin), 12\u201313; \"ordinary\" (Bergson), , 33\u201336, , , , . _See also_ aesthetic experience; medical gaze; medical perception; observation\n\nPerrin, Jean, ,\n\nPestalozzi, Johann Heinrich, , , 172\u201376,\n\nPfeffer, Wilhelm,\n\nPfemfert, Franz,\n\nphotogrammetry,\n\nphotography, , 29\u201330, , , , ; Brownian motion and, 66\u201367, 69\u201370, __ , 73\u201375; medicine and, , , 104\u201305, 112\u201314, ; _Naturgetreu_ and, ; series, 112\u201315, , . _See also_ chronophotography\n\nphysical sciences,\n\nphysicians: as bearers of culture ( _Kulturtr\u00e4ger_ ), ; doctor\u2013patient relationship, , ; as experts, 126\u201327\n\nphysics, , , 76\u201377; classical, , , ; metaphysics,\n\nphysiology, , , ,\n\nPinthus, Kurt, ,\n\nPizon, Antoine, ,\n\nPlato,\n\nplay drive, 206\u20137\n\npleasure, visual, 228\u201329\n\nPolgar, Alfred, , ,\n\nPolimanti, Osvaldo,\n\npolitical consciousness,\n\npolitics, of contemplation, 230\u201342\n\nPopulist movement,\n\nPorten, Henny,\n\npositivism,\n\npresence, , , , 237\u201338\n\nprimitivism, 296n80\n\n_The Problem of Form in the Fine Arts_ (Hildebrand),\n\n_Proceedings of the Royal Saxon Society for Sciences_ ,\n\nprofessional identity, , , , ,\n\nProgressive movement,\n\nprojection, , , , , ; emotional. _See_ Einf\u00fchlung; slow-motion, ; speeds of, , ,\n\nproper viewing, , 94\u201395. _See also_ observation\n\nprotoplasmic bridge theory of nerve development, , ,\n\npsychoanalysis, ,\n\npsychology: of children, , , 162\u201364; of crowds, , , 164\u201366; social, ,\n\npublic health, ,\n\npublic sphere, , ; women in,\n\npulp fiction ( _Schundliteratur_ ), ,\n\nquantitative frame analysis, , , ,\n\n_Quo Vadis?_ (film),\n\nRabinbach, Anson, , , ,\n\nRadica (conjoined twin),\n\nradiology, ,\n\nRam\u00f3n y Cajal, Santiago,\n\nRanci\u00e8re, Jacques, 199\u2013200\n\n_Raumgestaltung_ (spatial creation),\n\nRauscher, Ulrich, 211\u201312\n\nreadability. _See_ legibility\n\nrealism. _See_ filmic realism\n\nRefining the Cinema ( _Veredelung des Kinos_ ),\n\nreform: of clothing, 149\u201350; of education, 148\u201349, ; of film, , , 152\u201362, , ; in Germany, 147\u201352; _Kinoreform_ , , 154\u201355, , , ; _Kulturkritik_ and, ; modernity and, ; spirit of, 147\u201362\n\n_Reformkinos_ ,\n\nregulation, of cinemas, 158\u201359\n\nReicher, Karl,\n\nReich Health Office, 284n122\n\n_Rembrandt as Educator_ (Langbehn), 168\u201369\n\nrepeatability, ,\n\nrepose, , ,\n\nresearch film: , 19\u201326, , , , ; Bergson and, 32\u201337; Brownian motion and, 62\u201376, __ , __ ; disciplinary contexts of, ; distribution of, ; documentary function of, 102\u20135; educational function of, 105\u201310; exploratory function of, 98\u2013102; medical, 92\u201393, 96\u2013125; nerve fibers and, 76\u201388, __ ; overview of, 26\u201331; public screening of, ; science of work and, 37\u201344; tissue cultures and, 76\u201388. _See also_ chronophotography; cinematography; _The Human Gait_ ; medical filmmaking\n\nRheinberger, Hans-J\u00f6rg, ,\n\nRicher, Paul,\n\nRieder, H.,\n\nRies, Julius, ,\n\nRitter, Christopher,\n\nRitzheimer, Kara,\n\nRockefeller Institute,\n\nRomanticism,\n\nRosenthal, J.,\n\nRoss, Corey,\n\nRousseau, Jean-Jacques,\n\nRunge, Philipp Otto,\n\nRussia,\n\nR\u00fcswald, K. (teacher),\n\nSaint-Louis Hospital,\n\nSamuleit, Paul,\n\nscanning ( _Schauen_ ), ,\n\nSchaar, Eckard,\n\n_Schauen_ (scanning), ,\n\n_Schaulust_ (scopophilia), , 228\u201330, 309n72\n\nSchelling, Friedrich Wilhelm Joseph,\n\nSchenk, Paul,\n\nSchiller, Friedrich, , , , ; aesthetic contemplation and, , ; aesthetic education and, 167\u201368, ; aesthetic experience and, , , 195\u201398, 204\u20137; form drive and, , ; play drive and, 206\u20137; sense drive and, , ; temporality and, 204\u20137; vision and,\n\nSchlanger, Benjamin,\n\nSchl\u00fcpmann, Heide, 231\u201332\n\nSchmarsow, August, 225\u201327,\n\nSchopenhauer, Arthur, , , , , ; aesthetic experience and, , , , ; self and, ; temporality and,\n\nSchultze-Naumberg, Paul,\n\n_Sch\u00fclervorstellungen_ (commercial screenings for children),\n\n_Schulvorstellungen_ (screenings for classes),\n\nSchumberg, Wilhelm,\n\n_Schund_ campaigns, 151\u201352,\n\n_Schundfilme_ (trashy films), , ,\n\n_Schundliteratur_ (pulp fiction), ,\n\nSchuster, Paul, ,\n\nscientific filmmaking. _See_ research film. _See also_ chronophotography; cinematography; _The Human Gait_ ; medical filmmaking\n\nscientific method, , , , , , , , , , , . _See also_ experiment; observation\n\nscientific observation. _See_ observation\n\nscientific theater,\n\nscopophilia ( _Schaulust_ ), , 228\u201330, 309n72\n\nsea squirts ( _Botryllus_ ),\n\nsecond law of thermodynamics, ,\n\nSeddig, Max, 20\u201321, 25\u201326, , , 69\u201375, 87\u201388\n\n_Sehen_ (seeing), ,\n\nself, loss of, ,\n\nself-awareness, ,\n\nself-cultivation ( _Bildung_ ), ,\n\nself-identity, , ,\n\nself-mastery,\n\nSellmann, Adolf, , , ,\n\nsense drive, ,\n\nsense impression ( _Anschauung_ ), 172\u201373,\n\nsensuality, 142\u201343\n\nsensuous totality ( _Gesamtsinnlichkeit_ ),\n\n_sensus communis_ (communal sense),\n\nsequential images,\n\nserialized novels,\n\nseries photography, 112\u201315, , . _See also_ chronophotography; medicine; photography\n\nSerner, Walter, , 228\u201329\n\nsexual arousal, 228\u201329\n\nsexual images, ,\n\nsexuality,\n\nShapin, Steven,\n\nshocks, modernity as series of, ,\n\nSiedentopf, Henry,\n\nSimmel, Georg,\n\nslow-motion cinematography, , ,\n\nslow-motion projection,\n\nSmoluchowski, Maryan,\n\nSobchack, Vivian,\n\nsocial acceleration, , , , , , ; _Anschauungsunterricht_ and, 173\u201374. _See also_ modernity; pace: human liberty or potential and\n\nSocial Democratic Party of Germany (SPD), 292n34\n\nsocial psychology, ,\n\nSociety for the Dissemination of Popular Education (Gesellschaft zur Verbreitung von Volksbildung) (GVV), ,\n\nSociety of Friends of the Schools and Instruction for the Fatherland (Gesellschaft der Freunde des vaterl\u00e4ndischen Schul- und Erziehungswesens),\n\nSociety of German Natural Scientists and Physicians (Gesellschaft deutscher Naturforscher und \u00c4rzte), ,\n\nSolvay, Ernest,\n\n\"Some Specific Features of the Medical Way of Thinking\" (Fleck),\n\nsomnambulism, 212\u201313\n\nsoullessness, 239\u201340\n\nspatial creation ( _Raumgestaltung_ ),\n\nSPD. _See_ Social Democratic Party of Germany\n\nspectacle, in medicine, 279n63\n\nspectatorship, , , 200\u2013202, ; analysis, synthesis and, ; children, crowds and, 162\u201365; embodied, , 220\u201325; gender and, , 231\u201332; hypnotism and, 135\u201337; observation and, , , , , , , , , 139\u201340, , , 191\u201392, , ; theories of film, 11\u201312. _See also_ improper viewing; lay viewers; negative space\n\n\"Spectacles of the Earth\" (H\u00e4fker),\n\nSpencer, Herbert,\n\n\"The Stage Considered as a Moral Institution\" (Schiller),\n\nstaining techniques, 84\u201385\n\nStapel, Wilhelm, ,\n\nstatistical aspect, of medical logic,\n\nstillness and movement, , , ,\n\nstimulus shield,\n\nstorefront cinemas ( _Ladenkinos_ ), , __\n\nStratz, Carl Heinrich,\n\nstrobe effect, 51\u201352\n\nStrobl, Karl Hans, , ,\n\n_Studien zu einer Physiologie des Marsches_ (Zuntz and Schumberg),\n\nsubjectivity, , , , , ; objectivity and, ,\n\nsublime, theory of the,\n\nsuggestibility, , , 164\u201365,\n\nsurgery of the dead,\n\n\"Surrealism\" (Benjamin),\n\nsynesthesia, 221\u201322,\n\nsynthesis. _See_ analysis and synthesis\n\nSzczepaniak-Gillece, Jocelyn,\n\ntaste, , , , , , , , ; aesthetics; morality and, 167\u201368; geometry of, __ ; H\u00e4fker and, 186\u201389; ideology and, ; nation and, 170\u201371; as _sensus communis_ , ; and vision, 145\u201346,\n\nTausk, Viktor,\n\ntechnological reproducibility,\n\ntechnology, ; formal relationships in, ; media, 1\u20134,\n\ntemperance movement,\n\ntemporal discontinuity. _See_ continuity and discontinuity\n\ntemporal interruption, ,\n\ntemporality: of aesthetic experience, 202\u201314; of observation, 118\u201326\n\ntemporal phenomena,\n\nTews, Johannes,\n\ntextual analysis. _See_ analysis (close reading)\n\ntheater: cinema compared to, 236\u201339; scientific,\n\ntheory: disciplinary logic and, ; observation and, , 66\u201367, , 80\u201381\n\n_Theory of the Novel_ (Luk\u00e1cs), 235\u201336,\n\nthermodynamics, , ,\n\nThomson, William (Lord Kelvin),\n\n\"Thoughts Toward an Aesthetic of the Cinema (Gedanken zu einer Aesthetik des 'Kino')\" (Luk\u00e1cs), ,\n\nthought styles, , . _See also_ disciplinary logic\n\n_Three Essays on Sexuality_ (Freud),\n\ntime, , ; Luk\u00e1cs and, ; reversibility of, 74\u201375; spatialization of, 62\u201364, ; spectatorship and will and, 125\u201341. _See also_ duration; _dur\u00e9e_ ; temporality\n\ntime-lapse cinematography, , , 84\u201385, , , , , ; Braus and, ; medical filmmaking and,\n\n_Tirol in Waffen_ ( _Tirol in Arms_ ) (film),\n\ntissue cultures, 76\u201388\n\nTitchener, Edward B.,\n\nTolstoy, Leo,\n\nT\u00f6nnies, Ferdinand, 288n9\n\ntotal presentation ( _Gesamtvorf\u00fchrung_ ),\n\ntrade journals and magazines, 152\u201353\n\ntraditional aesthetics. _See_ aesthetics\n\ntraining film,\n\ntrashy films ( _Schundfilme_ ), , ,\n\ntriangulation,\n\nTucker, Jennifer,\n\ntutelage,\n\ntwo-sided chronophotography, . _See also_ chronophotography; _The Human Gait_\n\nUFA, , ; cultural division of ( _Kulturabteilung_ ),\n\nultramicroscope, ,\n\nUnion-Theater, __\n\nUnited Kingdom, , , , 147\u201348\n\nUnited States, , , , , , 147\u201348; Bergson and, , , ; object lesson and,\n\nUniversity of T\u00fcbingen, , , ,\n\nuplift and educational film,\n\nuseful cinema,\n\nVal\u00e9ry, Paul,\n\nVan Gehuchten, Arthur,\n\n_Veredelung des Kinos_ (Refining the Cinema),\n\nVierordt, Hermann,\n\nVirchow, Rudolph,\n\nvirtual experiments, ,\n\nvirtual witnessing, ,\n\nVischer, Robert, , , , , 224\u201325\n\nvisual means of instruction. _SeeAnschauungsunterrich_\n\nvisual pleasure, 228\u201329\n\nvitalism and mechanism, 39\u201340\n\nvividness ( _Anschaulichkeit_ ), 107\u20138,\n\nVl\u00e8s, Fred,\n\nvolition, , 286n143. _See also_ agency; free will\n\nvon Bergmann, Ernst,\n\nvon Helmholtz, Hermann, ,\n\nWagner, Richard, ,\n\n_The Wanderings of Odysseus_ ( _Die Irrfahrten des Odysseus_ ) (film), , __\n\n_Wanderkino_ ,\n\nWarstat, Willi,\n\nWeber, Max,\n\nWeber, Wilhelm, 263n58\n\nWeber brothers,\n\nWeichardt, Wilhelm,\n\nWeimar Republic,\n\nWeisenburg, Theodore,\n\n\"What Is Enlightenment?\" (Kant),\n\nwholeness and fragmentation, 36\u201337\n\nWilhelm II (kaiser), ,\n\n_Wilhelm Meister_ (Goethe),\n\nwill 139. _See also_ agency; free will\n\nWolf-Czapek, K. W.,\n\nWolgast, Heinrich,\n\nwomen, ; clothing reform and, 149\u201350; elite and, ; hysteria and, ; in public sphere, 150\u201351, ; spectatorship by, , 231\u201332\n\n_Women's Clothing and Its Natural Development_ (Stratz),\n\nworking class,\n\n\"Work of Art\" (Benjamin), , , , ,\n\nworking objects. _See_ docile bodies\n\n_The World as Will and Representation_ (Schopenhauer),\n\nWorld War I, , , 161\u201362, , , , ; class and, ; gender dynamics and, ; H\u00e4fker and, 301n145; health campaigns and, ; _Kulturabteilung_ and, ; X-ray cinematography and,\n\nWundt, Wilhelm,\n\nX-rays, , 113\u201314; cinematography, , 100\u2013102, __\n\nZeiss, Carl, ,\n\n_Zivilisation_ and _Kultur_ , 144\u201345\n\nZoological Gardens,\n\nZsigmondy, Richard,\n\nZuntz, Nathan, \nFILM AND CULTURE\n\nA series of Columbia University Press\n\nEdited by John Belton\n\n_What Made Pistachio Nuts? Early Sound Comedy and the Vaudeville Aesthetic_ , Henry Jenkins\n\n_Showstoppers: Busby Berkeley and the Tradition of Spectacle_ , Martin Rubin\n\n_Projections of War: Hollywood, American Culture, and World War II_ , Thomas Doherty\n\n_Laughing Screaming: Modern Hollywood Horror and Comedy_ , William Paul\n\n_Laughing Hysterically: American Screen Comedy of the 1950s_ , Ed Sikov\n\n_Primitive Passions: Visuality, Sexuality, Ethnography, and Contemporary Chinese Cinema_ , Rey Chow\n\n_The Cinema of Max Ophuls: Magisterial Vision and the Figure of Woman_ , Susan M. White\n\n_Black Women as Cultural Readers_ , Jacqueline Bobo\n\n_Picturing Japaneseness: Monumental Style, National Identity, Japanese Film_ , Darrell William Davis\n\n_Attack of the Leading Ladies: Gender, Sexuality, and Spectatorship in Classic Horror Cinema_ , Rhona J. Berenstein\n\n_This Mad Masquerade: Stardom and Masculinity in the Jazz Age_ , Gaylyn Studlar\n\n_Sexual Politics and Narrative Film: Hollywood and Beyond_ , Robin Wood\n\n_The Sounds of Commerce: Marketing Popular Film Music_ , Jeff Smith\n\n_Orson Welles, Shakespeare, and Popular Culture_ , Michael Anderegg\n\n_Pre-Code Hollywood: Sex, Immorality, and Insurrection in American Cinema, 1930\u20131934_ , Thomas Doherty\n\n_Sound Technology and the American Cinema: Perception, Representation, Modernity_ , James Lastra\n\n_Melodrama and Modernity: Early Sensational Cinema and Its Contexts_ , Ben Singer\n\n_Wondrous Difference: Cinema, Anthropology, and Turn-of-the-Century Visual Culture_ , Alison Griffiths\n\n_Hearst Over Hollywood: Power, Passion, and Propaganda in the Movies_ , Louis Pizzitola\n\n_Masculine Interests: Homoerotics in Hollywood Film_ , Robert Lang\n\n_Special Effects: Still in Search of Wonder_ , Michele Pierson\n\n_Designing Women: Cinema, Art Deco, and the Female Form_ , Lucy Fischer\n\n_Cold War, Cool Medium: Television, McCarthyism, and American Culture_ , Thomas Doherty\n\n_Katharine Hepburn: Star as Feminist_ , Andrew Britton\n\n_Silent Film Sound_ , Rick Altman\n\n_Home in Hollywood: The Imaginary Geography of Hollywood_ , Elisabeth Bronfen\n\n_Hollywood and the Culture Elite: How the Movies Became American_ , Peter Decherney\n\n_Taiwan Film Directors: A Treasure Island_ , Emilie Yueh-yu Yeh and Darrell William Davis\n\n_Shocking Representation: Historical Trauma, National Cinema, and the Modern Horror Film_ , Adam Lowenstein\n\n_China on Screen: Cinema and Nation_ , Chris Berry and Mary Farquhar\n\n_The New European Cinema: Redrawing the Map_ , Rosalind Galt\n\n_George Gallup in Hollywood_ , Susan Ohmer\n\n_Electric Sounds: Technological Change and the Rise of Corporate Mass Media_ , Steve J. Wurtzler\n\n_The Impossible David Lynch_ , Todd McGowan\n\n_Sentimental Fabulations, Contemporary Chinese Films: Attachment in the Age of Global Visibility_ , Rey Chow\n\n_Hitchcock's Romantic Irony_ , Richard Allen\n\n_Intelligence Work: The Politics of American Documentary_ , Jonathan Kahana\n\n_Eye of the Century: Film, Experience, Modernity_ , Francesco Casetti\n\n_Shivers Down Your Spine: Cinema, Museums, and the Immersive View_ , Alison Griffiths\n\n_Weimar Cinema: An Essential Guide to Classic Films of the Era_ , Edited by Noah Isenberg\n\n_African Film and Literature: Adapting Violence to the Screen_ , Lindiwe Dovey\n\n_Film, A Sound Art_ , Michel Chion\n\n_Film Studies: An Introduction_ , Ed Sikov\n\n_Hollywood Lighting from the Silent Era to Film Noir_ , Patrick Keating\n\n_Levinas and the Cinema of Redemption: Time, Ethics, and the Feminine_ , Sam B. Girgus\n\n_Counter-Archive: Film, the Everyday, and Albert Kahn's Archives de la Plan\u00e8te_ , Paula Amad\n\n_Indie: An American Film Culture_ , Michael Z. Newman\n\n_Pretty: Film and the Decorative Image_ , Rosalind Galt\n\n_Film and Stereotype: A Challenge for Cinema and Theory_ , J\u00f6rg Schweinitz\n\n_Chinese Women's Cinema: Transnational Contexts_ , Edited by Lingzhen Wang\n\n_Hideous Progeny: Disability, Eugenics, and Classic Horror Cinema_ , Angela M. Smith\n\n_Hollywood's Copyright Wars: From Edison to the Internet_ , Peter Decherney\n\n_Electric Dreamland: Amusement Parks, Movies, and American Modernity_ , Lauren Rabinovitz\n\n_Where Film Meets Philosophy: Godard, Resnais, and Experiments in Cinematic Thinking_ , Hunter Vaughan\n\n_The Utopia of Film: Cinema and Its Futures in Godard, Kluge, and Tahimik_ , Christopher Pavsek\n\n_Hollywood and Hitler, 1933\u20131939_ , Thomas Doherty\n\n_Cinematic Appeals: The Experience of New Movie Technologies_ , Ariel Rogers\n\n_Continental Strangers: German Exile Cinema, 1933\u20131951_ , Gerd Gem\u00fcnden\n\n_Deathwatch: American Film, Technology, and the End of Life_ , C. Scott Combs\n\n_After the Silents: Hollywood Film Music in the Early Sound Era, 1926\u20131934_ , Michael Slowik\n\n_\"It's the Pictures That Got Small\": Charles Brackett on Billy Wilder and Hollywood's Golden Age_ , Edited by Anthony Slide\n\n_Plastic Reality: Special Effects, Technology, and the Emergence of 1970s Blockbuster Aesthetics_ , Julie A. Turnock\n\n_Maya Deren: Incomplete Control_ , Sarah Keller\n\n_Dreaming of Cinema: Spectatorship, Surrealism, and the Age of Digital Media_ , Adam Lowenstein\n\n_Motion(less) Pictures: The Cinema of Stasis_ , Justin Remes\n\n_The Lumi\u00e8re Galaxy: Seven Key Words for the Cinema to Come_ , Francesco Casetti\n\n_The End of Cinema? A Medium in Crisis in the Digital Age_ , Andr\u00e9 Gaudreault and Philippe Marion\n\n_Studios Before the System: Architecture, Technology, and the Emergence of Cinematic Space_ , Brian R. Jacobson\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nPraise for\n\nA UNIVERSE FROM NOTHING\n\n\"Krauss possesses a rare talent for making the hardest ideas in astrophysics accessible to the layman, due in part to his sly humor . . . one has to hope that this book won't appeal only to the partisans of the culture wars\u2014it's just too good and interesting for that. Krauss is genuinely in awe of the 'wondrously strange' nature of our physical world, and his enthusiasm is infectious.\"\n\n\u2014Associated Press\n\n\"An eloquent guide to our expanding universe . . . There have been a number of fine cosmology books published recently but few have gone so far, and none so eloquently, in exploring why it is unnecessary to invoke God to light the blue touchpaper and set the universe in motion.\"\n\n\u2014Financial Times\n\n\"How physicists came up with the current model of the cosmos is quite a story, and to tell it in his elegant A Universe from Nothing, physicist Lawrence Krauss walks a carefully laid path . . . It would be easy for this remarkable story to revel in self-congratulation, but Krauss steers it soberly and with grace . . . His asides on how he views each piece of science and its chances of being right are refreshingly honest . . . unstable nothingness, as described by Krauss . . . is also invigorating for the rest of us, because in this nothingness there are many wonderful things to see and understand.\"\n\n\u2014Nature\n\n\"[An] excellent guide to cutting-edge physics . . . As Krauss elegantly argues in A Universe from Nothing, the accelerating expansion, indeed the whole existence of the cosmos, is most likely powered by 'nothing.' Krauss is an exemplary interpreter of tough science, and the central part of the book, where he discusses what we know about the history of the universe\u2014and how we know it\u2014is perfectly judged. It is detailed but lucid, thorough but not stodgy . . . Space and time can indeed come from nothing; nothing, as Krauss explains beautifully, being an extremely unstable state from which the production of \"'something\"' is pretty much inevitable . . . A Universe from Nothing is a great book: readable, informative and topical.\"\n\n\u2014New Scientist\n\n\"With its mind-bending mechanics, Krauss argues, our universe may indeed have appeared from nowhere, rather than at the hands of a divine creator. There's some intellectual heavy lifting here\u2014Einstein is the main character, after all\u2014but the concepts are articulated clearly, and the thrill of discovery is contagious. 'We are like the early terrestrial mapmakers,' Krauss writes, puzzling out what was once solely the province of our imaginations.\"\n\n\u2014Mother Jones\n\n\"His arguments for the birth of the universe out of nothingness from a physical, rather than theological, beginning not only are logical but celebrate the wonder of our natural universe. Recommended.\"\n\n\u2014Library Journal\n\n\"Lively and humorous as well as informative . . . Readers will find the result of Krauss's '[celebration of our] absolutely surprising and fascinating universe' as compelling as it is intriguing.\"\n\n\u2014Publishers Weekly\n\n\"The author delivers plenty of jolts in this enthusiastic and lucid but demanding overview of the universe, which includes plenty of mysteries\u2014but its origin isn't among them. A thoughtful, challenging book\u2014but not for the faint of heart or those not willing to read carefully.\"\n\n\u2014Kirkus Reviews\n\n\"Krauss is a lucid . . . writer, as well as a sparkling speaker and wit, an all-purpose science communicator . . . [I]t is an account of how to untie a paradox, scientifically. And it's also a scientist's hymn\u2014a song of secular appreciation\u2014to the unseen.\"\n\n\u2014cbcnews.ca\n\n\"In A Universe from Nothing, Lawrence Krauss has written a thrilling introduction to the current state of cosmology\u2014the branch of science that tells us about the deep past and deeper future of everything. As it turns out, everything has a lot to do with nothing\u2014and nothing to do with God. This is a brilliant and disarming book.\"\n\n\u2014Sam Harris, author of The Moral Landscape\n\n\"People always say you can't get something from nothing. Thankfully, Lawrence Krauss didn't listen. In fact, something big happens to you during this book about cosmic nothing, and before you can help it, your mind will be expanding as rapidly as the early universe.\"\n\n\u2014Sam Kean, author of The Disappearing Spoon\n\n\"Nothing is not nothing. Nothing is something. That's how a cosmos can be spawned from the void\u2014a profound idea conveyed in A Universe From Nothing that unsettles some yet enlightens others. Meanwhile, it's just another day on the job for physicist Lawrence Krauss.\"\n\n\u2014Neil deGrasse Tyson, astrophysicist, American Museum of Natural History\n\n\"With characteristic wit, eloquence, and clarity Lawrence Krauss gives a wonderfully illuminating account of how science deals with one of the biggest questions of all: how the universe's existence could arise from nothing. It is a question that philosophy and theology get themselves into muddle over, but that science can offer real answers to, as Krauss's lucid explanation shows. Here is the triumph of physics over metaphysics, reason and enquiry over obfuscation and myth, made plain for all to see: Krauss gives us a treat as well as an education in fascinating style.\"\n\n\u2014A. C. Grayling, author of The Good Book\n\n\"Astronomers at the beginning of the twentieth century were wondering whether there was anything beyond our Milky Way Galaxy. As Lawrence Krauss lucidly explains, astronomers living two trillion years from now, will perhaps be pondering precisely the same question! Beautifully navigating through deep intellectual waters, Krauss presents the most recent ideas on the nature of our cosmos, and of our place within it. A fascinating read.\"\n\n\u2014Mario Livio, author of Is God A Mathematician? and The Golden Ratio\n\n\"In this clear and crisply written book, Lawrence Krauss outlines the compelling evidence that our complex cosmos has evolved from a hot, dense state and how this progress has emboldened theorists to develop fascinating speculations about how things really began.\"\n\n\u2014Sir Martin Rees, author of Our Final Hour\n\n\"A series of brilliant insights and astonishing discoveries have rocked the Universe in recent years, and Lawrence Krauss has been in the thick of it. With his characteristic verve, and using many clever devices, he's made that remarkable story remarkably accessible. The climax is a bold scientific answer to the great question of existence: Why is there something rather than nothing?\"\n\n\u2014Frank Wilczek, Nobel Laureate and Herman Feshbach professor of Physics at MIT, and author of The Lightness of Being\nThank you for purchasing this Free Press eBook.\n\n* * *\n\nSign up for our newsletter and receive special offers, access to bonus content, and info on the latest new releases and other great eBooks from Free Press and Simon & Schuster.\n\nCLICK HERE TO SIGN UP\n\nor visit us online to sign up at  \neBookNews.SimonandSchuster.com\n\n## CONTENTS\n\nPreface to the Paperback Edition\n\nPreface\n\nChapter 1: A Cosmic Mystery Story: Beginnings\n\nChapter 2: A Cosmic Mystery Story: Weighing the Universe\n\nChapter 3: Light from the Beginning of Time\n\nChapter 4: Much Ado About Nothing\n\nChapter 5: The Runaway Universe\n\nChapter 6: The Free Lunch at the End of the Universe\n\nChapter 7: Our Miserable Future\n\nChapter 8: A Grand Accident?\n\nChapter 9: Nothing Is Something\n\nChapter 10: Nothing Is Unstable\n\nChapter 11: Brave New Worlds\n\nEpilogue\n\nAfterword by Richard Dawkins\n\nAbout the Author\n\nQ & A with the Author\n\nIndex\n_To Thomas, Patty, Nancy, and Robin, for helping inspire me to create something from nothing . . ._\n_On this site in 1897,_  \n _Nothing happened._  \n\u2014Plaque on wall of Woody Creek Tavern,  \nWoody Creek, Colorado\n\n## PREFACE TO THE PAPERBACK EDITION\n\nSince the hardcover version of this book first appeared, a visceral negative reaction among some commentators to the very idea of a universe arising from nothing has been balanced by a major scientific discovery that supports this possibility. The confirmation of the Higgs boson refines our understanding of the relationship between seemingly empty space and our existence. I want to elaborate on both the Higgs boson and the negative reactions to A Universe from Nothing in this new preface.\n\nWhen I chose to subtitle this book Why There Is Something Rather Than Nothing, I wanted to connect the remarkable discoveries of modern science to a question that has fascinated theologians, philosophers, natural philosophers, and the general public for more than two millennia. But I wasn't fully aware of how my choice of words might lead to the same kind of confusion that occurs whenever one says in public that Evolution is a theory.\n\nIn popular parlance, theory means something very different from its scientific sense. So too nothing is a hot-button issue for some people, a line in the sand that some people are not willing to cross, so that even using the word, just as using the word God, can be so polarizing that it obfuscates more important issues. A similar remark can be made about the question \"Why?\": using why and nothing together can be as explosive as mixing diesel fuel and fertilizer.\n\nIn chapter 9 of this book I mention a fact that I now want to introduce first here. Whenever one asks \"Why?\" in science, one actually means \"How?\". \"Why?\" is not really a sensible question in science because it usually implies purpose and, as anyone who has been the parent of a small child knows, one can keep on asking \"Why?\" forever, no matter what the answer to the previous question. Ultimately, the only way to end the conversation seems to be to say \"Because!\"\n\nScience changes the meaning of questions, especially why-like questions, as it progresses. Here is an early example of this fact, which illustrates a number of features in common with the more recent revelations I treat in this book.\n\nThe renowned astronomer Johannes Kepler claimed in 1595 to have had an epiphany when he suddenly thought he had answered a profoundly important why question: \"Why are there six planets?\" The answer, he believed, lay in the view of the five Platonic solids, those sacred objects from geometry whose faces can be composed of regular polygons\u2014triangles, squares, etc.\u2014and that could be circumscribed by spheres whose size would increase as the number of faces of the solid increased. If these spheres then separated the orbits of the six known planets, he conjectured, perhaps their relative distances from the sun and the fact that there were just six of them could be understood as revealing, in a profound and deep sense, the mind of God, the mathematician. (The idea that geometry was sacred goes back as far as Pythagoras.) \"Why are there six planets?\"\u2014then, in 1595\u2014was considered a meaningful question, one that revealed purpose to the universe.\n\nNow, however, we understand the question is meaningless. In the first place, we know there are not six planets, there are nine planets. (Pluto will always be a planet for me. Not only do I like to annoy my friend Neil deGrasse Tyson by so insisting, but my daughter did her fourth-grade science project on Pluto, and I don't want that to have been in vain!) More important, however, we know our solar system is not unique, which Kepler and his era did not know. More than two thousand planets orbiting other stars have been discovered (by a satellite named Kepler, coincidentally!).\n\nThe important question then becomes not \"Why?\" but \"How does our solar system have nine planets?\" (or, eight planets, depending upon your count). Since clearly lots of different solar systems exist, with very different features, what we really want to know is how typical we are, what specific conditions might have existed allowing our solar system to have four rocky planets closest to the sun, surrounded by a number of far larger gas giants. The answer to this question might shed light on the likelihood of finding life elsewhere in the universe, for example.\n\nMost important, however, we realize that there is nothing profound about six (or eight or nine), nothing that points to purpose or design . . . no evidence of \"purpose\" in the distribution of planets in the universe. Not only has \"why\" become \"how\" but \"why\" no longer has any verifiable meaning.\n\nSo too, when we ask \"Why is there something rather than nothing?\" we really mean \"How is there something rather than nothing?\" This brings me to the second confusion engendered by my choice of words. There are many seeming \"miracles\" of nature that appear so daunting that many have given up trying to find an explanation of how we came to be and, instead, blame it all on God. But the question I really care about, and the one that science can actually address, is the question of how all the \"stuff\" in the universe could have come from no \"stuff,\" and how, if you wish, formlessness led to form. That is what seems so astounding and nonintuitive. It seems to violate everything we know about the world\u2014in particular the fact that energy in its various forms, including mass, is conserved. Common sense suggests that \"nothing,\" in this sense the absence of \"something,\" should have zero total energy. Therefore, where did the 400 billion or so galaxies that make up the observable universe come from?\n\nThe fact that we need to refine what we mean by \"common sense\" in order to accommodate our understanding of nature is, to me, one of the most remarkable and liberating aspects of science. Reality liberates us from the biases and misconceptions that have arisen because our intellects evolved through our animal ancestors, whose survival was based on whether predators might lurk behind trees or in caves and not on understanding the wave function of electrons in atoms.\n\nOur modern conception of the universe is so foreign to what even scientists generally believed a mere century ago that it is a tribute to the power of the scientific method and the creativity and persistence of humans who want to understand it. That is worth celebrating. As I describe in this book, the question and the possible answers to how something might come from nothing are even more interesting than merely the possibility of galaxies manifesting from empty space. Science provides a possible road map for the creation of space (and time) itself\u2014and perhaps also an understanding of how the laws of physics that govern the dynamics of space and time can arise haphazardly.\n\nFor many people, however, the fascinating possible resolutions of these age-old mysteries are not sufficient. The deeper question of nonexistence overwhelms them. Can we understand how absolute nothingness, without even the potential for anything at all to exist, does not still reign supreme? Can one ever say anything other than the fact that the nothing that became our something was a part of \"something\" else, in which the potential for our existence, or any existence, was always implicit?\n\nIn the book I take a rather flippant attitude toward this concern, because I don't think it adds anything to the productive discussion, which is \"What questions are actually answerable by probing the universe?\" I have discounted this philosophical issue, but not because I think those people who occupy themselves with certain aspects of it are not trying hard to define logical questions. Rather, I discount this aspect of philosophy here because I think it bypasses the really interesting and answerable physical questions associated with the origin and evolution of our universe. No doubt some will view this as my own limitation, and maybe it is. But it is within that context that people should read this book. I don't make any claims to answer any questions that science cannot answer, and I have tried very carefully within the text to define what I mean by \"nothing\" and \"something.\" If those definitions differ from those you would like to adopt, so be it. Write your own book. But don't discount the remarkable human adventure that is modern science because it doesn't console you.\n\nNow, the good news! This past summer, physicists around the world, including me, were glued to computers at very odd hours to watch live as scientists at the Large Hadron Collider, outside Geneva, announced that they had found one of the most important missing pieces of the jigsaw puzzle that is nature\u2014the Higgs particle (or Higgs boson).\n\nProposed almost fifty years ago to allow for consistency between theoretical predictions and experimental observations in elementary particle physics, the Higgs particle's discovery caps one of the most remarkable intellectual adventures in human history\u2014one that anyone interested in the progress of knowledge should at least be aware of\u2014and makes even more remarkable the precarious accident that allowed our existence to form from nothing, the subject of this book. The discovery is further proof that the universe of our senses is just the tip of a vast, largely hidden cosmic iceberg and that seemingly empty space can provide the seeds for our existence.\n\nThe prediction of the Higgs particle accompanied a remarkable revolution that completely changed our understanding of particle physics in the latter part of the twentieth century. Just fifty years ago, in spite of the great advances of physics in the previous half century, we understood only one of the four fundamental forces of nature\u2014electromagnetism\u2014as a fully consistent quantum theory. In just one subsequent decade, however, not only had three of the four known forces surrendered to our investigations, but a new elegant unity of nature had been uncovered. It was found that all of the known forces could be described using a single mathematical framework\u2014and that two of the forces, electromagnetism and the weak force (which governs the nuclear reactions that power the sun), were actually different manifestations of a single underlying force.\n\nHow could two such different forces be related? After all, the photon, the particle that conveys electromagnetism, has no mass, while the particles that convey the weak force are very massive\u2014almost one hundred times as heavy as the particles that make up atomic nuclei, a fact that explains why the weak force is weak.\n\nBritish physicist Peter Higgs and several others showed that, if there exists an otherwise invisible background field (Higgs field) permeating all of space, then the particles that convey some force like electromagnetism can interact with this field and effectively encounter resistance to their motion and slow down, like a swimmer moving through molasses. As a result, these particles can behave as if they are heavy, as if they have a mass. The physicist Steven Weinberg (and somewhat later Abdus Salam) applied this idea to a model of the weak and electromagnetic forces previously proposed by Sheldon L. Glashow, and everything fit together.\n\nThis idea can be extended to the rest of particles in nature, including particles like those that make up the protons and neutrons, as well as fundamental particles like electrons, all of which combine to make up the atoms in our bodies. If some particle interacts more strongly with this background field, it ends up acting heavier. If it interacts more weakly, it acts lighter. If it doesn't interact at all, like the photon, it remains massless.\n\nIf anything sounds too good to be true, this is it. The miracle of mass\u2014indeed, of our very existence (because if not for the Higgs, there would be no stars, no planets, and no people)\u2014is apparently possible because of some otherwise hidden background field whose only effect seems to be to allow the world to look the way it does.\n\nBut relying on invisible miracles is the stuff of religion, not science. To ascertain whether this remarkable accident was real, physicists relied on another facet of the quantum world. Associated with every background field is a particle, and if you pick a point in space and hit it hard enough, you may whack out real particles. The trick is hitting it hard enough over a small enough volume. And that's the rub. After fifty years of trying, including a failed attempt in the United States to build an accelerator to test these ideas, no sign of the Higgs had appeared. In fact, I was betting against it, since a career in theoretical physics has taught me that nature usually has a far richer imagination than we do.\n\nUntil July.\n\nThe apparent discovery of the Higgs boson may not result in a better toaster or a faster car. But it provides a remarkable celebration of the human mind's capacity to uncover nature's secrets, and of the technology we have built to control them. Hidden in what seems like empty space\u2014indeed, like nothing\u2014appear to be the very elements that allow for our existence.\n\nThe discovery of a Higgs field further validates many of the ideas I discuss in this book. The idea that the very early universe went through a period of superluminal expansion, called inflation, that basically produced almost all the space and matter in the observable universe from almost nothing relies heavily on the possibility that another field, much like the Higgs field we seem to have discovered this past year, momentarily held sway in early times.\n\nThe existence of a Higgs field permeating all of space today also begs several important questions, most notably \"What conditions in the early universe led to such a cosmic accident?\" \"Why does the field have the value it is measured to have?\" \"Could it have been different?\" \"Could the laws of physics, had initial conditions been slightly different, have resulted in a universe without matter as we observe it today?\" These are precisely the kind of questions I discuss near the end of this book.\n\nWhatever the ultimate resolution of these puzzles, and others that I shall discuss in this book, our discoveries in fundamental physics and astronomy over the past forty years have changed our understanding of our place in the universe in profound ways, by changing not only the questions we ask, but the very meaning of the questions we have asked. That, as I want to stress once again, is perhaps the greatest legacy of modern science, a legacy it shares with great music, great literature, and great art, and one that needs to be shared more widely.\n\n## PREFACE\n\n_Dream or nightmare, we have to live our experience as it is, and we have to live it awake. We live in a world which is penetrated through and through by science and which is both whole and real. We cannot turn it into a game simply by taking sides._\n\n\u2014JACOB BRONOWSKI\n\nIn the interests of full disclosure right at the outset I must admit that I am not sympathetic to the conviction that creation requires a creator, which is at the basis of all of the world's religions. Every day beautiful and miraculous objects suddenly appear, from snowflakes on a cold winter morning to vibrant rainbows after a late-afternoon summer shower. Yet no one but the most ardent fundamentalists would suggest that each and every such object is lovingly and painstakingly and, most important, purposefully created by a divine intelligence. In fact, many laypeople as well as scientists revel in our ability to explain how snowflakes and rainbows can spontaneously appear, based on simple, elegant laws of physics.\n\nOf course, one can ask, and many do, \"Where do the laws of physics come from?\" as well as more suggestively, \"Who created these laws?\" Even if one can answer this first query, the petitioner will then often ask, \"But where did that come from?\" or \"Who created that?\" and so on.\n\nUltimately, many thoughtful people are driven to the apparent need for First Cause, as Plato, Aquinas, or the modern Roman Catholic Church might put it, and thereby to suppose some divine being: a creator of all that there is, and all that there ever will be, someone or something eternal and everywhere.\n\nNevertheless, the declaration of a First Cause still leaves open the question, \"Who created the creator?\" After all, what is the difference between arguing in favor of an eternally existing creator versus an eternally existing universe without one?\n\nThese arguments always remind me of the famous story of an expert giving a lecture on the origins of the universe (sometimes identified as Bertrand Russell and sometimes William James), who is challenged by a woman who believes that the world is held up by a gigantic turtle, who is then held up by another turtle, and then another . . . with further turtles \"all the way down!\" An infinite regress of some creative force that begets itself, even some imagined force that is greater than turtles, doesn't get us any closer to what it is that gives rise to the universe. Nonetheless, this metaphor of an infinite regression may actually be closer to the real process by which the universe came to be than a single creator would explain.\n\nDefining away the question by arguing that the buck stops with God may seem to obviate the issue of infinite regression, but here I invoke my mantra: The universe is the way it is, whether we like it or not. The existence or nonexistence of a creator is independent of our desires. A world without God or purpose may seem harsh or pointless, but that alone doesn't require God to actually exist.\n\nSimilarly, our minds may not be able to easily comprehend infinities (although mathematics, a product of our minds, deals with them rather nicely), but that doesn't tell us that infinities don't exist. Our universe could be infinite in spatial or temporal extent. Or, as Richard Feynman once put it, the laws of physics could be like an infinitely layered onion, with new laws becoming operational as we probe new scales. _We simply don't know!_\n\nFor more than two thousand years, the question, \"Why is there something rather than nothing?\" has been presented as a challenge to the proposition that our universe\u2014which contains the vast complex of stars, galaxies, humans, and who knows what else\u2014might have arisen without design, intent, or purpose. While this is usually framed as a philosophical or religious question, it is first and foremost a question about the natural world, and so the appropriate place to try and resolve it, first and foremost, is with science.\n\nThe purpose of this book is simple. I want to show how modern science, in various guises, can address and _is_ addressing the question of why there is something rather than nothing: The answers that have been obtained\u2014from staggeringly beautiful experimental observations, as well as from the theories that underlie much of modern physics\u2014all suggest that getting something from nothing is not a problem. Indeed, something from nothing may have been _required_ for the universe to come into being. Moreover, all signs suggest that this is how our universe _could_ have arisen.\n\nI stress the word _could_ here, because we may never have enough empirical information to resolve this question unambiguously. But the fact that a universe from nothing is even plausible is certainly significant, at least to me.\n\nBefore going further, I want to devote a few words to the notion of \"nothing\"\u2014a topic that I will return to at some length later. For I have learned that, when discussing this question in public forums, nothing upsets the philosophers and theologians who disagree with me more than the notion that I, as a scientist, do not truly understand \"nothing.\" (I am tempted to retort here that theologians are experts at nothing.)\n\n\"Nothing,\" they insist, is not any of the things I discuss. Nothing is \"nonbeing,\" in some vague and ill-defined sense. This reminds me of my own efforts to define \"intelligent design\" when I first began debating with creationists, of which, it became clear, there is no clear definition, except to say what it isn't. \"Intelligent design\" is simply a unifying umbrella for opposing evolution. Similarly, some philosophers and many theologians define and redefine \"nothing\" as not being any of the versions of nothing that scientists currently describe.\n\nBut therein, in my opinion, lies the intellectual bankruptcy of much of theology and some of modern philosophy. For surely \"nothing\" is every bit as physical as \"something,\" especially if it is to be defined as the \"absence of something.\" It then behooves us to understand precisely the physical nature of both these quantities. And without science, any definition is just words.\n\nA century ago, had one described \"nothing\" as referring to purely empty space, possessing no real material entity, this might have received little argument. But the results of the past century have taught us that empty space is in fact far from the inviolate nothingness that we presupposed before we learned more about how nature works. Now, I am told by religious critics that I cannot refer to empty space as \"nothing,\" but rather as a \"quantum vacuum,\" to distinguish it from the philosopher's or theologian's idealized \"nothing.\"\n\nSo be it. But what if we are then willing to describe \"nothing\" as the absence of space and time itself? Is this sufficient? Again, I suspect it would have been . . . at one time. But, as I shall describe, we have learned that space and time can themselves spontaneously appear, so now we are told that even this \"nothing\" is not really the nothing that matters. And we're told that the escape from the \"real\" nothing requires divinity, with \"nothing\" thus defined by fiat to be \"that from which only God can create something.\"\n\nIt has also been suggested by various individuals with whom I have debated the issue that, if there is the \"potential\" to create something, then that is not a state of true nothingness. And surely having laws of nature that give such potential takes us away from the true realm of nonbeing. But then, if I argue that perhaps the laws themselves also arose spontaneously, as I shall describe might be the case, then that too is not good enough, because whatever system in which the laws may have arisen is not true nothingness.\n\nTurtles all the way down? I don't believe so. But the turtles are appealing because science is changing the playing field in ways that make people uncomfortable. Of course, that is one of the purposes of science (one might have said \"natural philosophy\" in Socratic times). Lack of comfort means we are on the threshold of new insights. Surely, invoking \"God\" to avoid difficult questions of \"how\" is merely intellectually lazy. After all, if there were no potential for creation, then God couldn't have created anything. It would be semantic hocus-pocus to assert that the potentially infinite regression is avoided because God exists outside nature and, therefore, the \"potential\" for existence itself is not a part of the nothingness from which existence arose.\n\nMy real purpose here is to demonstrate that in fact science _has_ changed the playing field, so that these abstract and useless debates about the nature of nothingness have been replaced by useful, operational efforts to describe how our universe might actually have originated. I will also explain the possible implications of this for our present and future.\n\nThis reflects a very important fact. When it comes to understanding how our universe evolves, religion and theology have been at best irrelevant. They often muddy the waters, for example, by focusing on questions of nothingness without providing any definition of the term based on empirical evidence. While we do not yet fully understand the origin of our universe, there is no reason to expect things to change in this regard. Moreover, I expect that ultimately the same will be true for our understanding of areas that religion now considers its own territory, such as human morality.\n\nScience has been effective at furthering our understanding of nature because the scientific ethos is based on three key principles: (1) follow the evidence wherever it leads; (2) if one has a theory, one needs to be willing to try to prove it wrong as much as one tries to prove that it is right; (3) the ultimate arbiter of truth is experiment, not the comfort one derives from one's a priori beliefs, nor the beauty or elegance one ascribes to one's theoretical models.\n\nThe results of experiments that I will describe here are not only timely, they are also unexpected. The tapestry that science weaves in describing the evolution of our universe is far richer and far more fascinating than any revelatory images or imaginative stories that humans have concocted. Nature comes up with surprises that far exceed those that the human imagination can generate.\n\nOver the past two decades, an exciting series of developments in cosmology, particle theory, and gravitation have completely changed the way we view the universe, with startling and profound implications for our understanding of its origins as well as its future. Nothing could therefore not be more interesting to write about, if you can forgive the pun.\n\nThe true inspiration for this book comes not so much from a desire to dispel myths or attack beliefs, as from my desire to celebrate knowledge and, along with it, the absolutely surprising and fascinating universe that ours has turned out to be.\n\nOur search will take us on a whirlwind tour to the farthest reaches of our expanding universe, from the earliest moments of the Big Bang to the far future, and will include perhaps the most surprising discovery in physics in the past century.\n\nIndeed, the immediate motivation for writing this book now is a profound discovery about the universe that has driven my own scientific research for most of the past three decades and that has resulted in the startling conclusion that most of the energy in the universe resides in some mysterious, now inexplicable form permeating all of empty space. It is not an understatement to say that this discovery has changed the playing field of modern cosmology.\n\nFor one thing, this discovery has produced remarkable new support for the idea that our universe arose from precisely nothing. It has also provoked us to rethink both a host of assumptions about the processes that might govern its evolution and, ultimately, the question of whether the very laws of nature are truly fundamental. Each of these, in its own turn, now tends to make the question of why there is something rather than nothing appear less imposing, if not completely facile, as I hope to describe.\n\nThe direct genesis of this book hearkens back to October of 2009, when I delivered a lecture in Los Angeles with the same title. Much to my surprise, the YouTube video of the lecture, made available by the Richard Dawkins Foundation, has since become something of a sensation, with nearly a million viewings as of this writing, and numerous copies of parts of it being used by both the atheist and theist communities in their debates.\n\nBecause of the clear interest in this subject, and also as a result of some of the confusing commentary on the web and in various media following my lecture, I thought it worth producing a more complete rendition of the ideas that I had expressed there in this book. Here I can also take the opportunity to add to the arguments I presented at the time, which focused almost completely on the recent revolutions in cosmology that have changed our picture of the universe, associated with the discovery of the energy and geometry of space, and which I discuss in the first two-thirds of this book.\n\nIn the intervening period, I have thought a lot more about the many antecedents and ideas constituting my argument; I've discussed it with others who reacted with a kind of enthusiasm that was infectious; and I've explored in more depth the impact of developments in particle physics, in particular, on the issue of the origin and nature of our universe. And finally, I have exposed some of my arguments to those who vehemently oppose them, and in so doing have gained some insights that have helped me develop my arguments further.\n\nWhile fleshing out the ideas I have ultimately tried to describe here, I benefitted tremendously from discussions with some of my most thoughtful physics colleagues. In particular I wanted to thank Alan Guth and Frank Wilczek for taking the time to have extended discussions and correspondence with me, resolving some confusions in my own mind and in certain cases helping reinforce my own interpretations.\n\nEmboldened by the interest of Leslie Meredith and Dominick Anfuso at Free Press, Simon & Schuster, in the possibility of a book on this subject, I then contacted my friend Christopher Hitchens, who, besides being one of the most literate and brilliant individuals I know, had himself been able to use some of the arguments from my lecture in his remarkable series of debates on science and religion. Christopher, in spite of his ill health, kindly, generously, and bravely agreed to write a foreword. For that act of friendship and trust, I will be eternally grateful. Unfortunately, Christopher's illness eventually overwhelmed him to the extent that completing the foreword became impossible, in spite of his best efforts. Nevertheless, in an embarrassment of riches, my eloquent, brilliant friend, the renowned scientist and writer Richard Dawkins, had earlier agreed to write an afterword. After my first draft was completed, he then proceeded to produce something in short order whose beauty and clarity was astounding, and at the same time humbling. I remain in awe. To Christopher, Richard, then, and all of those above, I issue my thanks for their support and encouragement, and for motivating me to once again return to my computer and write.\n\n## CHAPTER 1\n\nA COSMIC MYSTERY STORY: BEGINNINGS\n\n_The Initial Mystery that attends any journey is: how did the traveler reach his starting point in the first place?_\n\n\u2014LOUISE BOGAN, _Journey Around My Room_\n\n_It was a dark and stormy night._\n\nEarly in 1916, Albert Einstein had just completed his greatest life's work, a decade-long, intense intellectual struggle to derive a new theory of gravity, which he called the general theory of relativity. This was not just a new theory of gravity, however; it was a new theory of space and time as well. And it was the first scientific theory that could explain not merely how objects move through the universe, but also how the universe itself might evolve.\n\nThere was just one hitch, however. When Einstein began to apply his theory to describing the universe as a whole, it became clear that the theory didn't describe the universe in which we apparently lived.\n\nNow, almost one hundred years later, it is difficult to fully appreciate how much our picture of the universe has changed in the span of a single human lifetime. As far as the scientific community in 1917 was concerned, the universe was static and eternal, and consisted of a single galaxy, our Milky Way, surrounded by a vast, infinite, dark, and empty space. This is, after all, what you would guess by looking up at the night sky with your eyes, or with a small telescope, and at the time there was little reason to suspect otherwise.\n\nIn Einstein's theory, as in Newton's theory of gravity before it, gravity is a purely attractive force between all objects. This means that it is impossible to have a set of masses located in space at rest forever. Their mutual gravitational attraction will ultimately cause them to collapse inward, in manifest disagreement with an apparently static universe.\n\nThe fact that Einstein's general relativity didn't appear consistent with the then picture of the universe was a bigger blow to him than you might imagine, for reasons that allow me to dispense with a myth about Einstein and general relativity that has always bothered me. It is commonly assumed that Einstein worked in isolation in a closed room for years, using pure thought and reason, and came up with his beautiful theory, independent of reality (perhaps like some string theorists nowadays!). However, nothing could be further from the truth.\n\nEinstein was always guided deeply by experiments and observations. While he performed many \"thought experiments\" in his mind and did toil for over a decade, he learned new mathematics and followed many false theoretical leads in the process before he ultimately produced a theory that was indeed mathematically beautiful. The single most important moment in establishing his love affair with general relativity, however, had to do with observation. During the final hectic weeks that he was completing his theory, competing with the German mathematician David Hilbert, he used his equations to calculate the prediction for what otherwise might seem an obscure astrophysical result: a slight precession in the \"perihelion\" (the point of closest approach) of Mercury's orbit around the Sun.\n\nAstronomers had long noted that the orbit of Mercury departed slightly from that predicted by Newton. Instead of being a perfect ellipse that returned to itself, the orbit of Mercury precessed (which means that the planet does not return precisely to the same point after one orbit, but the orientation of the ellipse shifts slightly each orbit, ultimately tracing out a kind of spiral-like pattern) by an incredibly small amount: 43 arc seconds (about 1\/ **100** of a degree) per century.\n\nWhen Einstein performed his calculation of the orbit using his theory of general relativity, the number came out just right. As described by an Einstein biographer, Abraham Pais: \"This discovery was, I believe, by far the strongest emotional experience in Einstein's scientific life, perhaps in all his life.\" He claimed to have heart palpitations, as if \"something had snapped\" inside. A month later, when he described his theory to a friend as one of \"incomparable beauty,\" his pleasure over the mathematical form was indeed manifest, but no palpitations were reported.\n\nThe apparent disagreement between general relativity and observation regarding the possibility of a static universe did not last long, however. (Even though it did cause Einstein to introduce a modification to his theory that he later called his biggest blunder. But more about that later.) Everyone (with the exception of certain school boards in the United States) now knows that the universe is not static but is expanding and that the expansion began in an incredibly hot, dense Big Bang approximately 13.72 billion years ago. Equally important, we know that our galaxy is merely one of perhaps 400 billion galaxies in the observable universe. We are like the early terrestrial mapmakers, just beginning to fully map the universe on its largest scales. Little wonder that recent decades have witnessed revolutionary changes in our picture of the universe.\n\nThe discovery that the universe is not static, but rather expanding, has profound philosophical and religious significance, because it suggested that our universe had a beginning. A beginning implies creation, and creation stirs emotions. While it took several decades following the discovery in 1929 of our expanding universe for the notion of a Big Bang to achieve independent empirical confirmation, Pope Pius XII heralded it in 1951 as evidence for Genesis. As he put it:\n\nIt would seem that present-day science, with one sweep back across the centuries, has succeeded in bearing witness to the august instant of the primordial Fiat Lux [Let there be Light], when along with matter, there burst forth from nothing a sea of light and radiation, and the elements split and churned and formed into millions of galaxies. Thus, with that concreteness which is characteristic of physical proofs, [science] has confirmed the contingency of the universe and also the well-founded deduction as to the epoch when the world came forth from the hands of the Creator. Hence, creation took place. We say: \"Therefore, there is a Creator. Therefore, God exists!\"\n\nThe full story is actually a little more interesting. In fact, the first person to propose a Big Bang was a Belgian priest and physicist named Georges Lema\u00eetre. Lema\u00eetre was a remarkable combination of proficiencies. He started his studies as an engineer, was a decorated artilleryman in World War I, and then switched to mathematics while studying for the priesthood in the early 1920s. He then moved on to cosmology, studying first with the famous British astrophysicist Sir Arthur Stanley Eddington before moving on to Harvard and eventually receiving a second doctorate, in physics from MIT.\n\nIn 1927, before receiving his second doctorate, Lema\u00eetre had actually solved Einstein's equations for general relativity and demonstrated that the theory predicts a nonstatic universe and in fact suggests that the universe we live in is expanding. The notion seemed so outrageous that Einstein himself colorfully objected with the statement \"Your math is correct, but your physics is abominable.\"\n\nNevertheless, Lema\u00eetre powered onward, and in 1930 he further proposed that our expanding universe actually began as an infinitesimal point, which he called the \"Primeval Atom\" and that this beginning represented, in an allusion to Genesis perhaps, a \"Day with No Yesterday.\"\n\nThus, the Big Bang, which Pope Pius so heralded, had first been proposed by a priest. One might have thought that Lema\u00eetre would have been thrilled with this papal validation, but he had already dispensed in his own mind with the notion that this scientific theory had theological consequences and had ultimately removed a paragraph in the draft of his 1931 paper on the Big Bang remarking on this issue.\n\nLema\u00eetre in fact later voiced his objection to the pope's 1951 claimed proof of Genesis via the Big Bang (not least because he realized that if his theory was later proved incorrect, then the Roman Catholic claims for Genesis might be contested). By this time, he had been elected to the Vatican's Pontifical Academy, later becoming its president. As he put it, \"As far as I can see, such a theory remains entirely outside of any metaphysical or religious question.\" The pope never again brought up the topic in public.\n\nThere is a valuable lesson here. As Lema\u00eetre recognized, whether or not the Big Bang really happened is a scientific question, not a theological one. Moreover, even if the Big Bang had happened (which all evidence now overwhelmingly supports), one could choose to interpret it in different ways depending upon one's religious or metaphysical predilections. You can choose to view the Big Bang as suggestive of a creator if you feel the need or instead argue that the mathematics of general relativity explain the evolution of the universe right back to its beginning without the intervention of any deity. But such a metaphysical speculation is independent of the physical validity of the Big Bang itself and is irrelevant to our understanding of it. Of course, as we go beyond the mere existence of an expanding universe to understand the physical principles that may address its origin, science can shed further light on this speculation and, as I shall argue, it does.\n\nIn any case, neither Lema\u00eetre nor Pope Pius convinced the scientific world that the universe was expanding. Rather, as in all good science, the evidence came from careful observations, in this case done by Edwin Hubble, who continues to give me great faith in humanity, because he started out as a lawyer and then became an astronomer.\n\nHubble had earlier made a significant breakthrough in 1925 with the new Mount Wilson 100-inch Hooker telescope, then the world's largest. (For comparison, we are now building telescopes more than ten times bigger than this in diameter and one hundred times bigger in area!) Up until that time, with the telescopes then available, astronomers were able to discern fuzzy images of objects that were not simple stars in our galaxy. They called these nebulae, which is basically Latin for \"fuzzy thing\" (actually \"cloud\"). They also debated whether these objects were in our galaxy or outside of it.\n\nSince the prevailing view of the universe at the time was that our galaxy was all that there was, most astronomers fell in the \"in our galaxy\" camp, led by the famous astronomer Harlow Shapley at Harvard. Shapley had dropped out of school in fifth grade and studied on his own, eventually going to Princeton. He decided to study astronomy by picking the first subject he found in the syllabus to study. In seminal work he demonstrated that the Milky Way was much larger than previously thought and that the Sun was not at its center but simply in a remote, uninteresting corner. He was a formidable force in astronomy and therefore his views on the nature of nebulae held considerable sway.\n\nOn New Year's Day 1925, Hubble published the results of his two-year study of so-called spiral nebulae, where he was able to identify a certain type of variable star, called a Cepheid variable star, in these nebulae, including the nebula now known as Andromeda.\n\nFirst observed in 1784, Cepheid variable stars are stars whose brightness varies over some regular period. In 1908, an unheralded and at the time unappreciated would-be astronomer, Henrietta Swan Leavitt, was employed as a \"computer\" at the Harvard College Observatory. (\"Computers\" were women brought in to catalogue the brightness of stars recorded on the observatory's photographic plates; women were not allowed to use the observatory telescopes at the time.) Daughter of a Congregational minister and a descendant of the Pilgrims, Leavitt made an astounding discovery, which she further illuminated in 1912: she noticed that there was a regular relationship between the brightness of Cepheid stars and the period of their variation. Therefore, if one could determine the distance to a single Cepheid of a known period (subsequently determined in 1913), then measuring the brightness of other Cepheids of the same period would allow one to determine the distance to these other stars!\n\nSince the observed brightness of stars goes down inversely with the square of the distance to the star (the light spreads out uniformly over a sphere whose area increases as the square of the distance, and thus since the light is spread out over a bigger sphere, the intensity of the light observed at any point decreases inversely with the area of the sphere), determining the distance to faraway stars has always been the major challenge in astronomy. Leavitt's discovery revolutionized the field. (Hubble himself, who was snubbed for the Nobel Prize, often said Leavitt's work deserved the prize, although he was sufficiently self-serving that he might have suggested it only because he would have been a natural contender to share the prize with her for his later work.) Paperwork had actually begun in the Royal Swedish Academy to nominate Leavitt for the Nobel in 1924 when it was learned that she had died of cancer three years earlier. By dint of his force of personality, knack for self-promotion, and skill as an observer, Hubble would become a household name, while Leavitt, alas, is known only to aficionados of the field.\n\nHubble was able to use his measurement of Cepheids and Leavitt's period-luminosity relation to prove definitively that the Cepheids in Andromeda and several other nebulae were much too distant to be inside the Milky Way. Andromeda was discovered to be another island universe, another spiral galaxy almost identical to our own, and one of the more than 100 billion other galaxies that, we now know, exist in our observable universe. Hubble's result was sufficiently unambiguous that the astronomical community\u2014including Shapley, who, incidentally, by this time had become director of the Harvard College Observatory, where Leavitt had done her groundbreaking work\u2014quickly accepted the fact that the Milky Way is not all there is around us. Suddenly the size of the known universe had expanded in a single leap by a greater amount than it had in centuries! Its character had changed, too, as had almost everything else.\n\nAfter this dramatic discovery, Hubble could have rested on his laurels, but he was after bigger fish or, in this case, bigger galaxies. By measuring ever fainter Cepheids in ever more distant galaxies, he was able to map the universe out to ever-larger scales. When he did, however, he discovered something else that was even more remarkable: the universe is expanding!\n\nHubble achieved his result by comparing the distances for the galaxies he measured with a different set of measurements from another American astronomer, Vesto Slipher, who had measured the spectra of light coming from these galaxies. Understanding the existence and nature of such spectra requires me to take you back to the very beginning of modern astronomy.\n\nOne of the most important discoveries in astronomy was that star stuff and Earth stuff are largely the same. It all began, as did many things in modern science, with Isaac Newton. In 1665, Newton, then a young scientist, allowed a thin beam of sunlight, obtained by darkening his room except for a small hole he made in his window shutter, through a prism and saw the sunlight disperse into the familiar colors of the rainbow. He reasoned that white light from the sun contained all of these colors, and he was correct.\n\nA hundred fifty years later, another scientist examined the dispersed light more carefully, discovered dark bands amidst the colors, and reasoned that these were due to the existence of materials in the outer atmosphere of the sun that were absorbing light of certain specific colors or wavelengths. These \"absorption lines,\" as they became known, could be identified with wavelengths of light that were measured to be absorbed by known materials on Earth, including hydrogen, oxygen, iron, sodium, and calcium.\n\nIn 1868, another scientist observed two new absorption lines in the yellow part of the solar spectrum that didn't correspond to any known element on Earth. He decided this must be due to some new element, which he called helium. A generation later, helium was discovered on Earth.\n\nLooking at the spectrum of radiation coming from other stars is an important scientific tool for understanding their composition, temperature, and evolution. Starting in 1912, Slipher observed the spectra of light coming from various spiral nebulae and found that the spectra were similar to those of nearby stars\u2014except that all of the absorption lines were shifted by the same amount in wavelength.\n\nThis phenomenon was by then understood as being due to the familiar \"Doppler effect,\" named after the Austrian physicist Christian Doppler, who explained in 1842 that waves coming at you from a moving source will be stretched if the source is moving away from you and compressed if it is moving toward you. This is a manifestation of a phenomenon we are all familiar with, and by which I am usually reminded of a Sidney Harris cartoon where two cowboys sitting on their horses out in the plains are looking at a distant train, and one says to the other, \"I love hearing that lonesome wail of the train whistle as the magnitude of the frequency changes due to the Doppler effect!\" Indeed, a train whistle or an ambulance siren sounds higher if the train or ambulance is moving toward you and lower if it is moving away from you.\n\nIt turns out that the same phenomenon occurs for light waves as sound waves, although for somewhat different reasons. Light waves from a source moving away from you, either due to its local motion in space or due to the intervening expansion of space, will be stretched, and therefore appear redder than they would otherwise be, since red is the long-wavelength end of the visible spectrum, while waves from a source moving toward you will be compressed and appear bluer.\n\nSlipher observed in 1912 that the absorption lines from the light coming from all the spiral nebulae were almost all shifted systematically toward longer wavelengths (although some, like Andromeda, were shifted toward shorter wavelengths). He correctly inferred that most of these objects therefore were moving away from us with considerable velocities.\n\nHubble was able to compare his observations of the distance of these spiral galaxies (as they were by now known to be) with Slipher's measurements of the velocities by which they were moving away. In 1929, with the help of a Mount Wilson staff member, Milton Humason (whose technical talent was such that he had secured a job at Mount Wilson without even having a high school diploma), he announced the discovery of a remarkable empirical relationship, now called Hubble's law: There is a linear relationship between recessional velocity and galaxy distance. Namely, galaxies that are ever more distant are moving away from us with faster velocities!\n\nWhen first presented with this remarkable fact\u2014that almost all galaxies are moving away from us, and those that are twice as far away are moving twice as fast, those that are three times away three times as fast, etc.\u2014it seems obvious what this implies: _We are the center of the universe!_\n\nAs some friends suggest, I need to be reminded on a daily basis that _this is not the case_. Rather, it was consistent with precisely the relationship that Lema\u00eetre had predicted. Our universe is indeed expanding.\n\nI have tried various ways to explain this, and I frankly don't think there is a good way to do it unless you think outside the box\u2014in this case, outside the universal box. To see what Hubble's law implies, you need to remove yourself from the myopic vantage point of our galaxy and look at our universe from the outside. While it is hard to stand outside a three-dimensional universe, it is easy to stand outside a two-dimensional one. On the next page I have drawn one such expanding universe at two different times. As you can see, the galaxies are farther apart at the second time.\n\nNow imagine that you are living in one of the galaxies at the second time, t2 which I shall mark in white, at time t2.\n\nTo see what the evolution of the universe would look like from this galaxy's vantage point, I simply superimpose the right image on the left, placing the galaxy in white on top of itself.\n\nVoil\u00e0! From this galaxy's vantage point every other galaxy is moving away, and those that are twice as far have moved twice the distance in the same time, those that are three times as far away have moved three times the distance, etc. As long as there is no edge, those on the galaxy feel as if they are at the center of the expansion.\n\nIt doesn't matter what galaxy one chooses. Pick another galaxy, and repeat:\n\nDepending upon your perspective, then, either _every place_ is the center of the universe, or _no place_ is. It doesn't matter; Hubble's law is consistent with a universe that is expanding.\n\nNow, when Hubble and Humason first reported their analysis in 1929, they not only reported a linear relationship between distance and recession velocity, but also gave a quantitative estimate of the expansion rate itself. Here are the actual data presented at the time:\n\nAs you can see, Hubble's guess of fitting a straight line to this data set seems a relatively lucky one. (There is clearly some relationship, but whether a straight line is the best fit is far from clear on the basis of this data alone.) The number for the expansion rate they obtained, derived for the plot, suggested that a galaxy a million parsecs away (3 million light-years)\u2014the average separation between galaxies\u2014is moving away from us with a speed of 500 kilometers\/second. This estimate was not so lucky, however.\n\nThe reason for this is relatively simple to see. If everything is moving apart today, then at earlier times they were closer together. Now, if gravity is an attractive force, then it should be slowing the expansion of the universe. This means the galaxy we see moving away from us at 500 kilometers\/second today would have been moving faster earlier.\n\nIf for the moment, though, we just assume that the galaxy had always been carried away with that velocity, we can work backward and figure out how long ago it would have been at the same position as our galaxy. Since galaxies twice as far away are moving twice as fast, if we work backward we find out that they were superimposed on our position at exactly the same time. Indeed, the entire observable universe would have been superimposed at a single point, the Big Bang, at a time that we can estimate in this way.\n\nSuch an estimate is clearly an upper limit on the age of the universe, because, if the galaxies were once moving faster, they would have gotten where they are today in less time than this estimate would suggest.\n\nFrom this estimate based on Hubble's analysis, the Big Bang happened approximately 1.5 billion years ago. Even in 1929, however, the evidence was already clear (except to some scriptural literalists in Tennessee, Ohio, and a few other states) that the Earth was older than 3 billion years old.\n\nNow, it is embarrassing for scientists to find that the Earth is older than the universe. More important, it suggests something is wrong with the analysis.\n\nThe source of this confusion was simply the fact that Hubble's distance estimates, derived using the Cepheid relations in our galaxy, were systematically incorrect. The distance ladder based on using nearby Cepheids to estimate the distance of farther away Cepheids, and then to estimate the distance to galaxies in which yet more distant Cepheids were observed, was flawed.\n\nThe history of how these systematic effects have been overcome is too long and convoluted to describe here and, in any case, no longer matters because we now have a much better distance estimator.\n\nOne of my favorite Hubble Space Telescope photographs is shown below:\n\nIt shows a beautiful spiral galaxy far far away, long long ago (long long ago because the light from the galaxy takes some time\u2014more than 50 million years\u2014to reach us). A spiral galaxy such as this, which resembles our own, has about 100 billion stars within it. The bright core at its center contains perhaps 10 billion stars. Notice the star on the lower left corner that is shining with a brightness almost equal to these 10 billion stars. On first sighting it, you might reasonably assume that this is a much closer star in our own galaxy that got in the way of the picture. But in fact, it is a star in that same distant galaxy, more than 50 million light-years away.\n\nClearly, this is no ordinary star. It is a star that has just exploded, a supernova, one of the brightest fireworks displays in the universe. When a star explodes, it briefly (over the course of about a month or so) shines in visible light with a brightness of 10 billion stars.\n\nHappily for us, stars don't explode that often, about once per hundred years per galaxy. But we are lucky that they do, because if they didn't, we wouldn't be here. One of the most poetic facts I know about the universe is that essentially every atom in your body was once inside a star that exploded. Moreover, the atoms in your left hand probably came from a different star than did those in your right. We are all, literally, star children, and our bodies made of stardust.\n\nHow do we know this? Well, we can extrapolate our picture of the Big Bang back to a time when the universe was about 1 second old, and we calculate that all observed matter was compressed in a dense plasma whose temperature should have been about 10 billion degrees (Kelvin scale). At this temperature nuclear reactions can readily take place between protons and neutrons as they bind together and then break apart from further collisions. Following this process as the universe cools, we can predict how frequently these primeval nuclear constituents will bind to form the nuclei of atoms heavier than hydrogen (i.e., helium, lithium, and so on).\n\nWhen we do so, we find that essentially no nuclei\u2014beyond lithium, the third lightest nucleus in nature\u2014formed during the primeval fireball that was the Big Bang. We are confident that our calculations are correct because our predictions for the cosmic abundances of the lightest elements agree bang-on with these observations. The abundances of these lightest elements\u2014hydrogen, deuterium (the nucleus of heavy hydrogen), helium, and lithium\u2014vary by 10 orders of magnitude (roughly 25 percent of the protons and neutrons, by mass, end up in helium, while 1 in every 10 billion neutrons and protons ends up within a lithium nucleus). Over this incredible range, observations and theoretical predictions agree.\n\nThis is one of the most famous, significant, and successful predictions telling us the Big Bang really happened. _Only a hot Big Bang can produce the observed abundance of light elements and maintain consistency with the current observed expansion of the universe._ I carry a wallet card in my back pocket showing the comparison of the predictions of the abundance of light elements and the observed abundance so that, each time I meet someone who doesn't believe that the Big Bang happened, I can show it to them. I usually never get that far in my discussion, of course, because data rarely impress people who have decided in advance that something is wrong with the picture. I carry the card anyway and reproduce it for you later in the book.\n\nWhile lithium is important for some people, far more important to the rest of us are all the heavier nuclei like carbon, nitrogen, oxygen, iron, and so on. These were _not_ made in the Big Bang. The only place they can be made is in the fiery cores of stars. And the only way they could get into your body today is if these stars were kind enough to have exploded, spewing their products into the cosmos so they could one day coalesce in and around a small blue planet located near the star we call the Sun. Over the course of the history of our galaxy, about 200 million stars have exploded. These myriad stars sacrificed themselves, if you wish, so that one day you could be born. I suppose that qualifies them as much as anything else for the role of saviors.\n\nIt turns out a certain type of exploding star, called a Type Ia supernova, has been shown by careful studies performed over the 1990s to have a remarkable property: with high accuracy, those Type Ia supernovae that are intrinsically brighter also shine longer. The correlation, while not fully understood theoretically, is empirically very tight. This means that these supernovae are very good \"standard candles.\" By this we mean that these supernovae can be used to calibrate distances because their intrinsic brightness can be directly ascertained by a measurement that is independent of their distance. If we observe a supernova in a distant galaxy\u2014and we can because they are very bright\u2014then by observing how long it shines, we can infer its intrinsic brightness. Then, by measuring its apparent brightness with our telescopes, we can accurately infer just how far away the supernova and its host galaxy are. Then, by measuring the \"redshift\" of the light from the stars in the galaxy, we can determine its velocity, and thus can compare velocity with distance and infer the expansion rate of the universe.\n\nSo far so good, but if supernovae explode only once every hundred years or so per galaxy, how likely are we ever to be able to see one? After all, the last supernova in our own galaxy witnessed on Earth was seen by Johannes Kepler in 1604! Indeed, it is said that supernovae in our galaxy are observed only during the lifetimes of the greatest astronomers, and Kepler certainly fits the bill.\n\nStarting out as a humble mathematics teacher in Austria, Kepler became assistant to the astronomer Tycho Brahe (who himself had observed an earlier supernova in our galaxy and was given an entire island by the king of Denmark in return), and using Brahe's data on planetary positions in the sky taken over more than a decade, Kepler derived his famous three laws of planetary motion early in the seventeenth century:\n\n1. Planets move around the Sun in ellipses.\n\n2. A _line_ connecting a planet and the Sun sweeps out equal _areas_ during equal intervals of time.\n\n3. The _square_ of the _orbital period_ of a planet is directly _proportional_ to the _cube_ (3rd power) of the _semi-major axis_ of its orbit (or, in other words, of the \"semi-major axis\" of the ellipse, half of the distance across the widest part of the ellipse).\n\nThese laws in turn lay the basis for Newton's derivation of the universal law of gravity almost a century later. Besides this remarkable contribution, Kepler successfully defended his mother in a witchcraft trial and wrote what was perhaps the first science fiction story, about a journey to the moon.\n\nNowadays, one way to see a supernova is simply to assign a different graduate student to each galaxy in the sky. After all, one hundred years is not too different, in a cosmic sense at least, from the average time to do a PhD, and graduate students are cheap and abundant. Happily, however, we don't have to resort to such extreme measures, for a very simple reason: the universe is big and old and, as a result, rare events happen all the time.\n\nGo out some night into the woods or desert where you can see stars and hold up your hand to the sky, making a tiny circle between your thumb and forefinger about the size of a dime. Hold it up to a dark patch of the sky where there are no visible stars. In that dark patch, with a large enough telescope of the type we now have in service today, you could discern perhaps 100,000 galaxies, each containing billions of stars. Since supernovae explode once per hundred years per, with 100,000 galaxies in view, you should expect to see, on average, about three stars explode on a given night.\n\nAstronomers do just this. They apply for telescope time, and some nights they might see one star explode, some nights two, and some nights it might be cloudy and they might not see any. In this way several groups have been able to determine Hubble's constant with an uncertainty of less than 10 percent. The new number, about 70 kilometers per second for galaxies on average of 3 million light-years apart, is almost a factor of 10 smaller than that derived by Hubble and Humason. As a result, we infer an age of the universe of closer to 13 billion years, rather than 1.5 billion years.\n\nAs I shall describe later, this too is in complete agreement with independent estimates of the age of the oldest stars in our galaxy. From Brahe to Kepler, from Lema\u00eetre to Einstein and Hubble, and from the spectra of stars to the abundance of light elements, four hundred years of modern science have produced a remarkable and consistent picture of the expanding universe. Everything holds together. The Big Bang picture is in good shape.\n\n## CHAPTER 2\n\nA COSMIC MYSTERY STORY: WEIGHING THE UNIVERSE\n\n_There are known knowns. These are things we know that we know. There are known unknowns. That is to say, there are things that we know we don't know. But there are also unknown unknowns. There are things we don't know we don't know._\n\n\u2014DONALD RUMSFELD\n\nHaving established that the universe had a beginning, and that that beginning was a finite and measurable time in the past, a natural next question to ask is, \"How will it end?\"\n\nIn fact, this was the very question that led me to move from my home territory, particle physics, into cosmology. During the 1970s and 1980s, it became increasingly clear from detailed measurements of the motion of stars and gas in our galaxy, as well as from the motion of galaxies in large groups of galaxies called clusters, that there was much more to the universe than meets either the eye or the telescope.\n\nGravity is the chief force operating on the enormous scale of galaxies, so measuring the motion of objects on these scales allows us to probe the gravitational attraction that drives this motion. Such measurements took off with the pioneering work of the American astronomer Vera Rubin and her colleagues in the early 1970s. Rubin had graduated with her doctorate from Georgetown after taking night classes while her husband waited in the car because she didn't know how to drive. She had applied to Princeton, but that university didn't accept women into their graduate astronomy program until 1975. Rubin rose to become only the second woman ever to be awarded the Gold Medal of the Royal Astronomical Society. That prize and her many other well-deserved honors stemmed from her groundbreaking measurements of the rotation rate of our galaxy. By observing stars and hot gas that were ever-farther from the center of our galaxy, Rubin determined that these regions were moving much faster than they should have been if the gravitational force driving their movement was due to the mass of all the observed objects within the galaxy. Due to her work, it eventually became clear to cosmologists that the only way to explain this motion was to posit the existence of significantly more mass in our galaxy than one could account for by adding up the mass of _all_ of this hot gas and stars.\n\nThere was a problem, however, with this view. The very same calculations that so beautifully explain the observed abundance of the light elements (hydrogen, helium, and lithium) in the universe also tell us more or less how many protons and neutrons, the stuff of normal matter, must exist in the universe. This is because, like any cooking recipe\u2014in this case nuclear cooking\u2014the amount of your final product depends upon how much of each ingredient you start out with. If you double the recipe\u2014four eggs instead of two, for example\u2014you get more of the end product, in this case an omelet. Yet the initial density of protons and neutrons in the universe arising out of the Big Bang, as determined by fitting to the observed abundance of hydrogen, helium, and lithium, accounts for about twice the amount of material we can see in stars and hot gas. Where are those particles?\n\nIt is easy to imagine ways to hide protons and neutrons (snowballs, planets, cosmologists . . . none of them shines), so many physicists predicted that as many protons and neutrons lie in dark objects as visible objects. However, when we add up how much \"dark matter\" has to exist to explain the motion of material in our galaxy, we find that the ratio of total matter to visible matter is not 2 to 1, but closer to 10 to 1. If this is not a mistake, then the dark matter cannot be made of protons and neutrons. There are just not enough of them.\n\nAs a young elementary particle physicist in the early 1980s, learning of this possibility of the existence of exotic dark matter was extremely exciting to me. It implied, literally, that the dominant particles in the universe were not good old-fashioned garden-variety neutrons and protons, but possibly some new kind of elementary particle, something that didn't exist on Earth today, but something mysterious that flowed between and amidst the stars and silently ran the whole gravitational show we call a galaxy.\n\nEven more exciting, at least for me, this implied three new lines of research that could fundamentally reilluminate the nature of reality.\n\n1. If these particles were created in the Big Bang, like the light elements I have described, then we should be able to use ideas about the forces that govern the interactions of elementary particles (instead of the interactions of nuclei relevant to determine elemental abundance) to estimate the abundance of possible exotic new particles in the universe today.\n\n2. It might be possible to derive the total abundance of dark matter in the universe on the basis of theoretical ideas in particle physics, or it might be possible to propose new experiments to detect dark matter\u2014either of which could tell us how much total matter there is and hence what the geometry of our universe is. The job of physics is not to invent things we cannot see to explain things we can see, but to figure out how to see what we cannot see\u2014to see what was previously invisible, the known unknowns. Each new elementary particle candidate for dark matter suggests new possibilities for experiments to detect directly the dark matter particles parading throughout the galaxy by building devices on Earth to detect them as the Earth intercepts their motion through space. Instead of using telescopes to search for faraway objects, if the dark matter particles are in diffuse bunches permeating the entire galaxy, they are here with us now, and terrestrial detectors might reveal their presence.\n\n3. If we could determine the nature of the dark matter, and its abundance, we might be able to determine how the universe will end.\n\nThis last possibility seemed the most exciting of all, so I will begin with it. Indeed, I got involved in cosmology because I wanted to be the first person to know how the universe would end.\n\nIt seemed like a good idea at the time.\n\nWhen Einstein developed his theory of general relativity, at its heart was the possibility that space could curve in the presence of matter or energy. This theoretical idea became more than mere speculation in 1919 when two expeditions observed starlight curving around the Sun during a solar eclipse in precisely the degree to which Einstein had predicted should happen if the presence of the Sun curved the space around it. Einstein almost instantly became famous and a household name. (Most people today think it was the equation E = mc2, which came fifteen years earlier, that did it, but it wasn't.)\n\nNow, if space is potentially curved, then the geometry of our whole universe suddenly becomes a lot more interesting. Depending upon the total amount of matter in our universe, it could exist in one of three different types of geometries, so-called _open, closed,_ or _flat_.\n\nIt is hard to envisage what a curved three-dimensional space might actually look like. Since we are three-dimensional beings, we can no more easily intuitively picture a curved three-dimensional space than the two-dimensional beings in the famous book _Flatland_ could imagine what their world would look like to a three-dimensional observer if it were curved like the surface of a sphere. Moreover, if the curvature is very small, then it is hard to imagine how one might actually detect it in everyday life, just as, during the Middle Ages at least, many people felt the Earth must be flat because from their perspective it looked flat.\n\nCurved three-dimensional universes are difficult to picture\u2014a closed universe is like a three-dimensional sphere, which sounds pretty intimidating\u2014but some aspects are easy to describe. If you looked far enough in one direction in a closed universe, you would see the back of your head.\n\nWhile these exotic geometries may seem amusing or impressive to talk about, operationally there is a much more important consequence of their existence. General relativity tells us unambiguously that a closed universe whose energy density is dominated by matter like stars and galaxies, and even more exotic dark matter, _must_ one day recollapse in a process like the reverse of a Big Bang\u2014a Big _Crunch,_ if you will. An open universe will continue to expand forever at a finite rate, and a flat universe is just at the boundary, slowing down, but never quite stopping.\n\nDetermining the amount of dark matter, and thus the total density of mass in the universe, therefore promised to reveal the answer to the age-old question (at least as old as T. S. Eliot anyway): Will the universe end with a bang or a whimper? The saga of determining the total abundance of dark matter goes back at least a half century, and one could write a whole book about it, which in fact I have already done, in my book _Quintessence_. However, in this case, as I shall now demonstrate (with both words _and_ then a picture), it is true that a single picture is worth at least a thousand (or perhaps a hundred thousand) words.\n\nThe largest gravitationally bound objects in the universe are called _superclusters_ of galaxies. Such objects can contain thousands of individual galaxies or more and can stretch across tens of millions of light-years. Most galaxies exist in such superclusters, and indeed our own galaxy is located within the Virgo supercluster of galaxies, whose center is almost 60 million light-years away from us.\n\nSince superclusters are so large and so massive, basically anything that falls into anything will fall into clusters. So if we could weigh superclusters of galaxies and then estimate the total density of such superclusters in the universe, we could then \"weigh the universe,\" including all the dark matter. Then, using the equations of general relativity, we could determine whether there is enough matter to close the universe or not.\n\nSo far so good, but how can we weigh objects that are tens of millions of light-years across? Simple. Use gravity.\n\nIn 1936, Albert Einstein, following the urgings of an amateur astronomer, Rudi Mandl, published a short paper in the magazine _Science_ titled \"Lens-Like Action of a Star by the Deviation of Light in the Gravitational Field.\" In this brief note Einstein demonstrated the remarkable fact that space itself could act like a lens, bending light and magnifying it, just like the lenses in my own reading glasses.\n\nIt was a kindlier, gentler time in 1936, and it is interesting to read the informal beginning of Einstein's paper, which after all was published in a distinguished scientific journal: \"Some time ago, R. W. Mandl paid me a visit and asked me to publish the results of a little calculation, which I had made at his request. This note complies with his wish.\" Perhaps this informality was accorded to him because he was Einstein, but I prefer to suppose that it was a product of the era, when scientific results were not yet always couched in language removed from common parlance.\n\nIn any case, the fact that light followed curved trajectories if space itself curved in the presence of matter was the first significant new prediction of general relativity and the discovery that led to Einstein's international fame, as I have mentioned. So it is perhaps not that surprising (as was recently discovered) that in 1912, well before Einstein had in fact even completed his general relativity theory, he had performed calculations\u2014as he tried to find some observable phenomenon that would convince astronomers to test his ideas\u2014that were essentially identical to those he published in 1936 at the request of Mr. Mandl. Perhaps because he reached the same conclusion in 1912 that he stated in his 1936 paper, namely \"there is no great chance of observing this phenomenon,\" he never bothered to publish his earlier work. In fact, after examining his notebooks for both periods, we can't say for sure that he later even remembered having done the original calculations twenty-four years before.\n\nWhat Einstein did recognize on both occasions is that the bending of light in a gravitational field could mean that, if a bright object was located well behind an intervening distribution of mass, light rays going out in various directions could bend around the intervening distribution and converge again, just as they do when they traverse a normal lens, producing either a magnification of the original object or the production of numerous image copies of the original object, some of which might be distorted (see figure below).\n\nWhen he calculated the predicted effects for lensing of a distant star by an intervening star in the foreground, the effect was so small that it appeared absolutely unmeasurable, which led him to make the remark mentioned above\u2014that it was unlikely that such a phenomenon could ever be observed. As a result, Einstein figured that his paper had little practical value. As he put it in his covering letter to the editor of _Science_ at the time: \"Let me also thank you for your cooperation with the little publication, which Mister Mandl squeezed out of me. It is of little value, but it makes the poor guy happy.\"\n\nEinstein was not an astronomer, however, and it would take one to realize that the effect Einstein had predicted might be not only measurable, but also useful. Its usefulness came from applying it to the lensing of distant objects by much larger systems such as galaxies or even clusters of galaxies, not to the lensing of stars by stars. Within months of Einstein's publication, the brilliant Caltech astronomer Fritz Zwicky submitted a paper to the _Physical Review_ in which he demonstrated the practicality of precisely this possibility (and also indirectly put down Einstein for his ignorance regarding the possible effect of lensing by galaxies rather than stars).\n\nZwicky was an irascible character and way ahead of his time. As early as 1933 he had analyzed the relative motion of galaxies in the Coma cluster and determined, using Newton's laws of motion, that the galaxies were moving so fast that they should have flown apart, destroying the cluster, unless there was far more mass in the cluster, by a factor more than 100, than could be accounted for by the stars alone. He thus should properly be considered as having discovered dark matter, though at the time his inference was so remarkable that most astronomers probably felt there might be some other less exotic explanation for the result he got.\n\nZwicky's one-page paper in 1937 was equally remarkable. He proposed three different uses for gravitational lensing: (1) testing general relativity, (2) using intervening galaxies as a kind of telescope to magnify more distant objects that would otherwise be invisible to telescopes on earth, and, most important, (3) resolving the mystery of why clusters appear to weigh more than can be accounted for by visible matter: _\"Observations on the deflection of light around nebulae may provide the most direct determination of nebular masses and clear up the above-mentioned discrepancy.\"_\n\nZwicky's paper is now seventy-four years old but reads instead like a modern proposal for using gravitational lensing to probe the universe. Indeed, each and every suggestion he made has come to pass, and the final one is the most significant of all. Gravitational lensing of distant quasars by intervening galaxies was first observed in 1987, and in 1998, sixty-one years after Zwicky proposed weighing nebulae using gravitational lensing, the mass of a large cluster was determined by using gravitational lensing.\n\nIn that year, physicist Tony Tyson and colleagues at the now defunct Bell Laboratories (which had such a noble and Nobel tradition of great science, from the invention of the transistor to the discovery of the cosmic microwave background radiation) observed a distant large cluster, colorfully labeled CL 0024 + 1654, located about 5 billion light-years away. In this beautiful image from the Hubble Space Telescope, a spectacular example of the multiple image of a distant galaxy located another 5 billion light-years behind the cluster can be seen as highly distorted and elongated images amidst the otherwise generally rounder galaxies.\n\nLooking at this image provides fuel for the imagination. First, every spot in the photo is a galaxy, not a star. Each galaxy contains perhaps 100 billion stars, along with them probably hundreds of billions of planets, and perhaps long-lost civilizations. I say long-lost because the image is 5 billion years old. The light was emitted 500 million years before our own Sun and Earth formed. Many of the stars in the photo no longer exist, having exhausted their nuclear fuel billions of years ago. Beyond that, the distorted images show precisely what Zwicky argued would be possible. The large distorted images to the left of the center of the image are highly magnified (and elongated) versions of this distant galaxy, which otherwise would probably not be visible at all.\n\nWorking backward from this image to determine the underlying mass distribution in the cluster is a complicated and complex mathematical challenge. To do so, Tyson had to build a computer model of the cluster and trace the rays from the source through the cluster in all possible different ways, using the laws of general relativity to determine the appropriate paths, until the fit they produced best matched the researchers' observations. When the dust settled, Tyson and collaborators obtained a graphical image that displayed precisely where the mass was located in this system pictured in the original photograph:\n\nSomething strange is going on in this image. The spikes in the graph represent the location of the visible galaxies in the original image, but most of the mass of the system is located _between_ the galaxies, in a smooth, dark distribution. In fact, more than 40 times as much mass is between the galaxies as is contained in the visible matter in the system (300 times as much mass as contained in the stars alone with the rest of visible matter in hot gas around them). Dark matter is clearly not confined to galaxies, but also dominates the density of clusters of galaxies.\n\nParticle physicists like myself were not surprised to find that dark matter also dominates clusters. Even though we didn't have a shred of direct evidence, we all hoped that the amount of dark matter was sufficient to result in a flat universe, which meant that there had to be more than 100 times as much dark matter as visible matter in the universe.\n\nThe reason was simple: a flat universe is the only mathematically beautiful universe. Why? Stay tuned.\n\nWhether or not the total amount of dark matter was sufficient to produce a flat universe, observations such as these obtained by gravitational lensing (I remind you that gravitational lensing results from the local curvature of space around massive objects; the flatness of the universe relates to the global average curvature of space, ignoring the local ripples around massive objects) and more recent observations from other areas of astronomy have confirmed that the total amount of dark matter in galaxies and clusters is far in excess of that allowed by the calculations of Big Bang nucleosynthesis. We are now virtually certain that the dark matter\u2014which, I reiterate, has been independently corroborated in a host of different astrophysical contexts, from galaxies to clusters of galaxies\u2014must be made of something entirely new, something that doesn't exist normally on Earth. This kind of stuff, which isn't star stuff, isn't Earth stuff either. But it _is_ something!\n\nThese earliest inferences of dark matter in our galaxy have spawned a whole new field of experimental physics, and I am happy to say that I have played a role in its development. As I have mentioned above, dark matter particles are all around us\u2014in the room in which I am typing, as well as \"out there\" in space. Hence we can perform experiments to look for dark matter and for the new type of elementary particle or particles of which it is comprised.\n\nThe experiments are being performed in mines and tunnels deep underground. Why underground? Because on the surface of the Earth we are regularly bombarded by all manner of cosmic rays, from the Sun and from objects much farther away. Since dark matter, by its very nature, doesn't interact electromagnetically to produce light, we assume that its interactions with normal material are extremely weak, so it will be extremely difficult to detect. Even if we are bombarded every day by millions of dark matter particles, most will go through us and the Earth, without even \"knowing\" we are here\u2014and without our noticing. Thus, if you want to detect the effects of the very rare exceptions to this rule, dark matter particles that actually bounce off atoms of matter, you had better be prepared to detect very rare and infrequent events. Only underground are you sufficiently shielded from cosmic rays for this to be possible even in principle.\n\nAs I write this, however, an equally exciting possibility is arising. The Large Hadron Collider, outside of Geneva, Switzerland, the world's largest and most powerful particle accelerator, has just begun running. But we have many reasons to believe that, at the very high energies at which protons are smashed together in the device, conditions similar to those in the very early universe will be re-created, albeit over only microscopically small regions. In such regions the same interactions that may have first produced what are now dark matter particles during the very early universe may now produce similar particles in the laboratory! There is thus a great race going on. Who will detect dark matter particles first: the experimenters deep underground or the experimentalists at the Large Hadron Collider? The good news is that, if either group wins the race, no one loses. We all win, by learning what the ultimate stuff of matter really is.\n\nEven though the astrophysical measurements I described don't reveal the identity of dark matter, they do tell us how much of it exists. A final, direct determination of the total amount of matter in the universe came from the beautiful inferences of gravitational lensing measurements like the one I have described combined with other observations of X-ray emissions from clusters. Independent estimates of the clusters' total mass is possible because the temperature of the gas in clusters that are producing the X-rays is related to the total mass of the system in which they are emitted. The results were surprising, and as I have alluded, disappointing to many of us scientists. For when the dust had settled, literally and metaphorically, the total mass in and around galaxies and clusters was determined to be only about 30 percent of the total amount of mass needed to result in a flat universe today. (Note that this is more than 40 times as much mass as can be accounted for by visible matter, which therefore makes up less than 1 percent of the mass needed to make up a flat universe.)\n\nEinstein would have been amazed that his \"little publication\" ultimately was far from useless. Supplemented by remarkable new experimental and observational tools that opened new windows on the cosmos, new theoretical developments that would have amazed and delighted him, and the discovery of dark matter that probably would have raised his blood pressure, Einstein's small step into the world of curved space had ultimately turned into to a giant leap. By the early 1990s, the holy grail of cosmology had apparently been achieved. Observations had determined that we live in an open universe, one that would therefore expand forever. Or had they?\n\n## CHAPTER 3\n\nLIGHT FROM THE BEGINNING OF TIME\n\n_As it was in the beginning, is now, and shall ever be._\n\n\u2014GLORIA PATRI\n\nIf you think about it, trying to determine the net curvature of the universe by measuring the total mass contained within it and then using the equations of general relativity to work backward has huge potential problems. Inevitably, you have to wonder whether matter is hidden in ways that we cannot uncover. For instance, we can only probe for the existence of matter within these systems using the gravitational dynamics of visible systems like galaxies and clusters. If significant mass somehow resided elsewhere, we'd miss it. It would be much better to measure the geometry of the whole visible universe directly.\n\nBut how can you measure the three-dimensional geometry of the whole visible universe? It's easier to start with a simpler question: How would you determine if a two-dimensional object like the Earth's surface was curved if you couldn't go around the Earth or couldn't go above it in a satellite and look down?\n\nFirst, you could ask a high school student, What is the sum of the angles in a triangle? (Choose the high school carefully, however . . . a European one is a good bet.) You would be told 180 degrees, because the student no doubt learned Euclidean geometry\u2014the geometry associated with flat pieces of paper. On a curved two-dimensional surface like a globe, you can draw a triangle, the sum of whose angles is far greater than 180 degrees. For example, consider drawing a line along the equator, then making a right angle, going up to the North Pole, then another right angle back down to the equator, as shown below. Three times 90 is 270, far greater than 180 degrees. Voil\u00e0!\n\nIt turns out that this simple, two-dimensional thinking extends directly and identically to three dimensions, because the mathematicians who first proposed non-flat, or so-called non-Euclidean, geometries realized that the same possibilities could exist in three dimensions. In fact, the most famous mathematician of the nineteenth century, Carl Friedrich Gauss, was so fascinated by the possibility that our own universe might be curved that he took data in the 1820s and '30s from geodetic survey maps to measure large triangles between the German mountain peaks of Hoher Hagen, Inselberg, and Brocken to determine if he could detect any curvature of space itself. Of course, the fact that the mountains are on the curved surface of the Earth means that the two-dimensional curvature of the surface of the Earth would have interfered with any measurement he was performing to probe for curvature in the background three-dimensional space in which the Earth is situated, which he must have known. I assume he was planning to subtract any such contribution from his final results to see if any possible leftover curvature might be attributable to a curvature of the background space.\n\nThe first person to try to measure the curvature of space definitively was an obscure mathematician, Nikolai Ivanovich Lobachevsky, who lived in remote Kazan in Russia _._ Unlike Gauss, Lobachevsky was actually one of two mathematicians who had the temerity to propose in print the possibility of so-called hyberbolic curved geometries, where parallel lines could diverge. Remarkably, Lobachevsky published his work on hyperbolic geometry (which we now call \"negatively curved\" or \"open\" universes) in 1830.\n\nShortly thereafter, when considering whether our own three-dimensional universe might be hyperbolic, Lobachevsky suggested that it might be possible to \"investigate a stellar triangle for an experimental resolution of the question.\" He suggested that observations of the bright star Sirius could be taken when the Earth was on either side of its orbit around the Sun, six months apart. From observations, he concluded that any curvature of our universe must be _at least_ 166,000 times the radius of the Earth's orbit.\n\nThis is a big number, but it is trivially small on cosmic scales. Unfortunately, while Lobachevsky had the right idea, he was limited by the technology of his day. One hundred and fifty years later, however, things have improved, thanks to the most important set of observations in all of cosmology: measurements of the cosmic microwave background radiation, or CMBR.\n\nThe CMBR is nothing less than the afterglow of the Big Bang. It provides another piece of direct evidence, in case any is needed, that the Big Bang really happened, because it allows us to look back directly and detect the nature of the very young, hot universe from which all the structures we see today later emerged.\n\nOne of the many remarkable things about the cosmic microwave background radiation is that it was discovered in New Jersey, of all places, by two scientists who really didn't have the slightest idea what they were looking for. The other thing is that it existed virtually under all our noses for decades, potentially observable, but was missed entirely. In fact, you may be old enough have seen its effects without realizing it, if you remember the days before cable television, when channels used to end their broadcast days in the wee morning hours and not run infomercials all night. When they went off the air, after showing a test pattern, the screen would revert to static. About 1 percent of that static you saw on the television screen was radiation left over from the Big Bang.\n\nThe origin of the cosmic microwave background radiation is relatively straightforward. Since the universe has a finite age (recall it is 13.72 billion years old), and as we look out at ever more distant objects, we are looking further back in time (since the light takes longer to get to us from these objects), you might imagine that if we looked out far enough, we would see the Big Bang itself. In principle this is not impossible, but in practice, between us and that early time lies a wall. Not a physical wall like the walls of the room in which I am writing this, but one that, to a great extent, has the same effect.\n\nI cannot see past the walls in my room because they are opaque. They absorb light. Now, as I look out in the sky back further and further in time, I am looking at the universe as it was younger and younger, and also hotter and hotter, because it has been cooling ever since the Big Bang. If I look back far enough, to a time when the universe was about 300,000 years old, the temperature of the universe was about 3,000 degrees (Kelvin scale) above absolute zero. At this temperature the ambient radiation was so energetic that it was able to break apart the dominant atoms in the universe, hydrogen atoms, into their separate constituents, protons and electrons. Before this time, neutral matter did not exist. Normal matter in the universe, made of atomic nuclei and electrons, consisted of a dense \"plasma\" of charged particles interacting with radiation.\n\nA plasma, however, can be opaque to radiation. The charged particles within the plasma absorb photons and reemit them so that radiation cannot easily pass through such a material uninterrupted. As a result, if I try to look back in time, I cannot see past the time when matter in the universe was last largely comprised of such a plasma.\n\nOnce again, it is like the walls in my room. I can see them only because electrons in atoms on the surface of the wall absorb light from the light in my study and then reemit it, and the air between me and the walls is transparent, so I can see all the way to the surface of the wall that emitted the light. So too with the universe. When I look out, I can see all the way back to that \"last scattering surface,\" which is the point at which the universe became neutral, where protons combined with electrons to form neutral hydrogen atoms. After that point, the universe became largely transparent to radiation, and I can now see the radiation that was absorbed and reemitted by the electrons as matter in the universe became neutral.\n\nIt is therefore a _prediction_ of the Big Bang picture of the universe that there should be radiation coming at me from all directions from that \"last scattering surface.\" Since the universe has expanded by a factor of about 1,000 since that time, the radiation has cooled on its way to us and is now approximately 3 degrees above absolute zero. And that is precisely the signal that the two hapless scientists found in New Jersey in 1965, and for whose discovery they were later awarded the Nobel Prize.\n\nActually a second Nobel Prize was given more recently for observations of the cosmic microwave background radiation, and for good reason. If we could take a photo of the surface of the last scattering surface, we would get a picture of the neonatal universe a mere 300,000 years into its existence. We could see all the structures that would one day collapse to form galaxies, stars, planets, aliens, and all the rest. Most important, these structures would have been unaffected by all the subsequent dynamical evolution that can obscure the underlying nature and origin of the first tiny primordial perturbations in matter and energy which were presumably created by exotic processes in the earliest moments of the Big Bang.\n\nMost important for our purpose, however, on this surface there would be a characteristic scale, which is imprinted by nothing other than time itself. One can understand this as follows: If one considers a distance spanning about 1 degree on the last scattering surface as seen by an observer on Earth, this would correspond to a distance on that surface of about 300,000 light-years. Now, since the last scattering surface reflects a time when the universe itself was about 300,000 years old, and since Einstein tells us that no information can travel through space faster than the speed of light, this means that no signal from one location could travel across this surface at that time by more than about 300,000 light-years.\n\nNow consider a lump of matter smaller than 300,000 light-years across. Such a lump will have begun to collapse due to its own gravity. But a lump larger than 300,000 light-years across won't even begin to collapse, because it doesn't yet even \"know\" it is a lump. Gravity, which itself propagates at the speed of light, cannot have traveled across the full length of the lump. So just as Wile E. Coyote runs straight off a cliff and hangs suspended in midair in the Road Runner cartoons, the lump will just sit there, waiting to collapse when the universe becomes old enough for it to know what it is supposed to do!\n\nThis singles out a special triangle, with one side 300,000 light-years across, a known distance away from us, determined by the distance between us and the last scattering surface, as shown below:\n\nThe largest lumps of matter, which will have already begun to collapse and in so doing will produce irregularities on the image of the microwave background surface, will span this angular scale. If we are able to obtain an image of this surface as it looked at that time, we would expect such hot spots to be, on average, the largest significant lumps we see in the image.\n\nHowever, whether the angle spanned by this distance is precisely 1 degree will in fact be determined by the geometry of the universe. In a flat universe, light rays travel in straight lines. In an open universe, however, light rays bend outward as one follows them back in time. In a closed universe, light rays converge as one follows them backward. Thus, the actual angle spanned on our eyes by a ruler that is 300,000 light-years across, located at a distance associated with the last scattering surface, depends upon the geometry of the universe, as shown below:\n\nThis provides a direct, clean test of the geometry of the universe. Since the size of the largest hot spots or cold spots in the microwave background radiation image depends just upon causality\u2014the fact that gravity can propagate only at the speed of light, and thus the largest region that can have collapsed at that time is simply determined by the farthest distance a light ray can have propagated at that time\u2014and because the angle that we see spanned by a fixed ruler at a fixed distance from us is just determined by the curvature of the universe, a simple picture of the last scattering surface can reveal to us the large-scale geometry of space-time.\n\nThe first experiment to attempt such an observation was a ground-launched balloon experiment in Antarctica in 1997 called BOOMERANG. While the acronym stands for _B_ alloon _O_ bservations _o_ f _M_ illimetric _E_ xtragalactic _R_ adiation _an_ d _G_ eophysics, the real reason it was called this name is simpler. A microwave radiometer was attached to a high-altitude balloon as shown below:\n\nThe balloon then went around the world, which is easy to do in Antarctica. Actually, at the South Pole it is really easy to do, since you can just turn around in a circle. However, from McMurdo Station the round trip around the continent, aided by the polar winds, took two weeks, after which the device returned to its starting point, hence the name BOOMERANG.\n\nBoomerang path around Antarctica.\n\nThe purpose of the balloon trip was simple. To get a view of the microwave background radiation, reflecting a temperature of 3 degrees above absolute zero (Kelvin scale), which is not contaminated by the far hotter material on Earth (even in Antarctica temperatures are more than two hundred degrees hotter than the temperature of the cosmic microwave background radiation), we want to go as far as possible above the ground, and even above most of the atmosphere of the Earth. Ideally we use satellites for this purpose, but high-altitude balloons can do much of the job for far less money.\n\nIn any case, after two weeks, BOOMERANG returned an image of a small part of the microwave sky displaying hot and cold spots in the radiation pattern coming from the last scattering surface. Shown below is one image of the region the BOOMERANG experiment observed (with \"hot spots\" and \"cold spots\" being shaded dark and light respectively), superimposed upon the original photo of the experiment:\n\nThis image serves two purposes as far as I am concerned. First, it displays the actual physical scale of the hot and cold spots as seen in the sky by BOOMERANG, with the foreground images for comparison. But it also illustrates another important aspect of what can only be called our cosmic myopia. When we look up on a sunny day, we see a blue sky, as shown in the previous image of the balloon. But this is because we have evolved to see visible light. We have done so, no doubt, both because the light from the surface of our Sun peaks in the visible region, and also because many other wavelengths of light get absorbed in our atmosphere, so they cannot reach us on the Earth's surface. (This is fortunate for us, since much of this radiation could be harmful.) In any case, if we had instead evolved to \"see\" microwave radiation, the image of the sky we would see, day or night, as long as we weren't looking directly at the Sun, would take us directly back to an image of the last scattering surface, more than 13 billion light-years away. This is the \"image\" returned by the BOOMERANG detector.\n\nThe first flight of BOOMERANG, which produced this image, was remarkably fortunate. Antarctica is a hostile, unpredictable environment. On a later flight, in 2003, the entire experiment was nearly lost due to a balloon malfunction and subsequent storm. A last-minute decision to cut free from the balloon before it was blown to some inaccessible location saved the day and a search-and-rescue mission located the payload on the Antarctic plain and recovered the pressurized vessel containing the scientific data.\n\nBefore interpreting the BOOMERANG image, I want to emphasize one more time that the actual physical size of the hot spots and cold spots recorded on the BOOMERANG image are fixed by simple physics associated with the last scattering surface, while the _measured_ sizes of the hot spots and cold spots in the image derive from the geometry of the universe. A simple two-dimensional analogy may help further explain the result: In two dimensions, a **closed geometry** resembles the surface of a sphere, while an **open geometry** resembles the surface of a saddle. If we draw a triangle on these surfaces, we observe the effect I described, as straight lines converge on a sphere, and diverge on a saddle, and, of course, remain straight on a flat plane:\n\nSo the million-dollar question now is, How big _are_ the hot spots and cold spots in the BOOMERANG image? To answer this, the BOOMERANG collaboration prepared several simulated images on their computer of hot spots and cold spots as would be seen in closed, flat, and open universes, and compared this with (another false color) image of the actual microwave sky.\n\nIf you examine the image on the lower left, from a simulated closed universe, you will see that the average spots are larger than in the actual universe. On the right, the average spot size is smaller. But, just like Baby Bear's bed in Goldilocks, the image in the middle, corresponding to a simulated flat universe, is \"just right.\" The mathematically beautiful universe hoped for by theorists seemed to be vindicated by this observation, even though it appears to conflict strongly with the estimate made by weighing clusters of galaxies.\n\nIn fact, the agreement between the predictions for a flat universe and the image obtained by BOOMERANG is almost embarrassing. Examining the spots and searching for the largest ones that had time to collapse significantly inward at the time reflected in the last scattering surface, the BOOMERANG team produced the following graph:\n\nThe data are the points. The solid line gives the prediction for a flat universe, with the largest bump occurring close to 1 degree!\n\nSince the BOOMERANG experiment published its results, a far more sensitive satellite probe of the microwave background radiation was launched by NASA, the Wilkinson Microwave Anisotropy Probe (WMAP). Named after the late Princeton physicist David Wilkinson, who was one of the original Princeton physicists who should have discovered the CMBR had they not been scooped by the Bell Labs scientists, WMAP was launched in June 2001. It was sent out to a distance of one million miles from the Earth, where, on the far side of the Earth from the Sun, it could view the microwave sky without contamination from sunlight. Over a period of seven years it imaged the whole microwave sky (not just a portion of the sky as BOOMERANG did, since BOOMERANG had to contend with the presence of the Earth below it) with unprecedented accuracy.\n\nHere the entire sky is projected on a plane, just as the surface of a globe can be projected on a flat map. The plane of our galaxy would lie along the equator, and 90 degrees above the plane of our galaxy is the North Pole on this map and 90 degrees below the plane of our galaxy is the South Pole. The image of the galaxy, however, has been removed from the map in order to reflect purely the radiation coming from the last scattering surface.\n\nWith this kind of exquisite data a much more precise estimate can be made of the geometry of the universe. A WMAP plot that is analogous to the one shown for the BOOMERANG image confirms to an accuracy of 1 percent that we live in a flat universe! The expectations of theorists were correct. Yet once again, we cannot ignore the apparent obvious inconsistency of this result with the result I described in the last chapter. Weighing the universe by measuring the mass of galaxies and clusters yields a value a factor of 3 smaller than the amount needed to result in a flat universe. Something has to give.\n\nWhile theorists may have been patting themselves on the back for guessing that the universe is flat, almost no one was prepared for the surprise that nature had in store to resolve the contradictory estimates of the geometry of the universe coming from measuring mass versus measuring curvature directly. The missing energy needed to result in a flat universe turned out to be hiding right under our noses, literally.\n\n## CHAPTER 4\n\nMUCH ADO ABOUT NOTHING\n\n_Less is more._\n\n\u2014LUDWIG MIES VAN DER ROHE,  \nAFTER ROBERT BROWNING\n\nOne step forward, two steps back, or so it seemed in our search for understanding our universe and accurately giving it a face. Even though observations had finally definitively determined the curvature of our universe\u2014and in the process validated long-held theoretical suspicions\u2014suddenly, even though it was known that ten times as much matter exists in the universe as could be accounted for by protons and neutrons, even that massive amount of dark matter, comprising 30 percent of what was required to produce a flat universe, was nowhere near sufficient to account for all the energy in the universe. The direct measurement of the geometry of the universe and the consequent discovery that the universe is indeed flat meant that 70 percent of the energy of the universe was still missing, neither in nor around galaxies or even clusters of galaxies!\n\nThings were not quite as shocking as I have made them out to be. Even before these measurements of the curvature of the universe, and the determination of the total clustered mass within it (as described in chapter 2), there were signs that the by-then conventional theoretical picture of our universe\u2014with sufficient dark matter (three times as much as we now know exists, in fact) to be spatially flat\u2014was just not consistent with observations. Indeed, as early as 1995, I wrote a heretical paper with a colleague of mine, Michael Turner, from the University of Chicago, suggesting that this conventional picture couldn't be correct, and in fact the only possibility that appeared consistent with both a flat universe (our theoretical preference at the time) and observations of the clustering of galaxies and their internal dynamics was a universe that was far more bizarre and that hearkened back to a crazy theoretical idea Albert Einstein had in 1917 to solve the apparent contradiction between the predictions of his theory and the static universe he thought we lived in and which he later abandoned.\n\nAs I recall, our motivation at the time was more to show that something was wrong with the prevailing wisdom than it was to suggest a definitive solution to the problem. The proposal seemed too crazy to really believe, so I don't think anyone was more surprised than we were when it turned out, three years later, that our heretical suggestion was precisely on the money after all!\n\nLet's return to 1917. Recall that Einstein had developed general relativity and had heart palpitations of joy over discovering that he could explain the precession of the perihelion of Mercury, even as he had to confront that fact that his theory couldn't explain the static universe in which he thought he was living.\n\nHad he had greater courage of his convictions, he might have _predicted_ that the universe couldn't be static. But he didn't. Instead, he realized that he could make a small change in his theory, one that was completely consistent with the mathematical arguments that had led him to develop general relativity in the first place, and one that looked like it might allow a static universe.\n\nWhile the details are complex, the general structure of Einstein's equations in general relativity is relatively straightforward. The left-hand side of the equations describes the curvature of the universe, and with it, the strength of the gravitational forces acting on matter and radiation. These are determined by the quantity on the right-hand side of the equation, which reflects the total density of all kinds of energy and matter within the universe.\n\nEinstein realized that adding a small extra constant term to the left-hand side of the equation would represent a small extra constant _repulsive_ force throughout all of space in addition to the standard gravitational attraction between distant objects that falls off as the distance between them increases. If it were small enough, this extra force could be undetectable on human scales or even on the scale of our solar system, where Newton's law of gravity is observed to hold so beautifully. But he reasoned that, because it was constant throughout all of space, it could build up over the scale of our galaxy and be large enough to counteract the attractive forces between very distant objects. He thus reasoned that this could result in a static universe on the largest scales.\n\nEinstein called this extra term the _cosmological term._ Because it is simply a constant addition to the equations, it is now, however, conventional to call this term the _cosmological constant._\n\nOnce he recognized that the universe is actually expanding, Einstein dispensed with this term and is said to have called the decision to add it to his equations his biggest blunder.\n\nBut getting rid of it is not so easy. It is like trying to put the toothpaste back in the tube after you have squeezed it out. This is because we now have a completely different picture of the cosmological constant today, so that, if Einstein had not added the term, someone else would have in the intervening years.\n\nMoving Einstein's term from the left-hand side of his equations to the right-hand side is a small step for a mathematician but a giant leap for a physicist. While it is trivial mathematically to do so, once this term is on the right-hand side, where all the terms contributing to the energy of the universe reside, it represents something completely different from a physical perspective\u2014namely a new contribution to the total energy. But what kind of stuff could contribute such a term?\n\nThe answer is, _nothing._\n\nBy _nothing,_ I do not mean nothing, but rather _nothing_ \u2014in this case, the nothingness we normally call empty space. That is to say, if I take a region of space and get rid of everything within it\u2014dust, gas, people, and even the radiation passing through, namely absolutely _everything_ within that region\u2014if the remaining empty space _weighs something,_ then that would correspond to the existence of a cosmological term such as Einstein invented.\n\nNow, this makes Einstein's cosmological constant seem even crazier! For any fourth grader will tell you how much energy is contained in nothing, even if they don't know what energy is. The answer must be nothing.\n\nAlas, most fourth graders have not taken quantum mechanics, nor have they studied relativity. For when one incorporates the results of Einstein's special theory of relativity into the quantum universe, empty space becomes much stranger than it was before. So strange in fact that even the physicists who first discovered and analyzed this new behavior were hard-pressed to believe that it actually existed in the real world.\n\nThe first person to successfully incorporate relativity into quantum mechanics was the brilliant, laconic British theoretical physicist Paul Dirac, who himself had already played a leading role in developing quantum mechanics as a theory.\n\nQuantum mechanics was developed from 1912 to 1927, primarily through the work of the brilliant and iconic Danish physicist Niels Bohr and the brilliant young hot-shots Austrian physicist Erwin Schr\u00f6dinger and German physicist Werner Heisenberg. The quantum world first proposed by Bohr, and refined mathematically by Schr\u00f6dinger and Heisenberg, defies all commonsense notions based on our experience with objects on a human scale. Bohr first proposed that electrons in atoms orbit around the central nucleus, as planets do around the Sun, but demonstrated that the observed rules of atomic spectra (the frequencies of light emitted by different elements) could only be understood if somehow the electrons were restricted to have stable orbits in a fixed set of \"quantum levels\" and could not spiral freely toward the nucleus. They could move between levels by absorbing or emitting only discrete frequencies, or quanta, of light\u2014the very quanta that Max Planck had first proposed in 1905 as a way of understanding the forms of radiation emitted by hot objects.\n\nBohr's \"quantization rules\" were rather ad hoc, however. In the 1920s, Schr\u00f6dinger and Heisenberg independently demonstrated that it was possible to derive these rules from first principles if electrons obeyed rules of dynamics that were different from those applied to macroscopic objects like baseballs. Electrons could behave like waves as well as particles, appearing to spread out over space (hence, Schr\u00f6dinger's \"wave function\" for electrons), and the results of measurements of the properties of electrons were shown to yield only probabilistic determinations, with various combinations of different properties not being exactly measurable at the same time (hence, Heisenberg's \"Uncertainty Principle\").\n\nDirac had shown that the mathematics proposed by Heisenberg to describe quantum systems (for which Heisenberg won the 1932 Nobel Prize) could be derived by careful analogy with the well-known laws governing the dynamics of classical macroscopic objects. In addition, he was also later able to show that the mathematical \"wave mechanics\" of Schr\u00f6dinger could also be so derived and was formally equivalent to Heisenberg's formulation. But Dirac also knew that the quantum mechanics of Bohr, Heisenberg, and Schr\u00f6dinger, as remarkable as it was, applied only to systems where Newton's laws, and not Einstein's relativity, would have been the appropriate laws governing the classical systems that the quantum systems were built with by analogy.\n\nDirac liked to think in terms of mathematics rather than pictures, and as he turned his attention to trying to make quantum mechanics consistent with Einstein's laws of relativity, he started playing with many different sorts of equations. These included complicated multicomponent mathematical systems that were necessary to incorporate the fact that electrons have \"spin\"\u2014that is to say they spin around like small tops and have angular momentum, and they also can spin both clockwise and anticlockwise around any axis.\n\nIn 1929, he hit pay dirt. The Schr\u00f6dinger equation had beautifully and accurately described the behavior of electrons moving at speeds much slower than light. Dirac found that if he modified the Schr\u00f6dinger equation into a more complex equation using objects called matrices\u2014which actually meant that his equation really described a set of four different coupled equations\u2014he could consistently unify quantum mechanics with relativity, and thus in principle describe the behavior of systems where the electrons were moving at much faster speeds.\n\nThere was a problem, however. Dirac had written down an equation meant to describe the behavior of electrons as they interacted with electric and magnetic fields. But his equation appeared also to require the existence of new particles just like electrons but with opposite electric charge.\n\nAt the time, there was only one elementary particle in nature known with a charge opposite that of the electron\u2014the proton. But protons are not at all like electrons. To begin with, they are 2,000 times heavier!\n\nDirac was flummoxed. In an act of desperation he argued that the new particles were in fact protons, but that somehow when moving through space the interactions of protons would cause them to act as if they were heavier. It didn't take long for others, including Heisenberg, to show that this suggestion made no sense.\n\nNature quickly came to the rescue. Within two years of the time Dirac proposed his equation, and a year after he had capitulated and accepted that, if his work was correct, then a new particle must exist, experimenters looking at cosmic rays bombarding the Earth discovered evidence for new particles identical to electrons but with an opposite electric charge, which were dubbed positrons.\n\nDirac was vindicated, but he also recognized his earlier lack of confidence in his own theory by later saying that his equation was smarter than he was!\n\nWe now call the positron the \"antiparticle\" of the electron, because it turns out that Dirac's discovery was ubiquitous. The same physics that required an antiparticle for the electron to exist requires one such particle to exist for almost every elementary particle in nature. Protons have antiprotons, for example. Even some neutral particles, like neutrons, have antiparticles. When particles and antiparticles meet, they annihilate into pure radiation.\n\nWhile all this may sound like science fiction (and indeed antimatter plays an important role in _Star Trek_ ), we create antiparticles all the time at our large particle accelerators around the world. Because antiparticles otherwise have the same properties as particles, a world made of antimatter would behave the same way as a world of matter, with antilovers sitting in anticars making love under an anti-Moon. It merely is an accident of our circumstances, due, we think, to rather more profound factors we will get to later, that we live in a universe that is made up of matter and not antimatter or one with equal amounts of both. I like to say that while antimatter may seem strange, it is strange in the sense that Belgians are strange. They are not really strange; it is just that one rarely meets them.\n\nThe existence of antiparticles makes the observable world a much more interesting place, but it also turns out to make empty space much more complicated.\n\nLegendary physicist Richard Feynman was the first person to provide an intuitive understanding of why relativity requires the existence of antiparticles, which also yielded a graphic demonstration that empty space is not quite so empty.\n\nFeynman recognized that relativity tells us that observers moving at different speeds will make different measurements of quantities such as distance and time. For example, time will appear to slow down for objects moving very fast. If somehow objects could travel faster than light, they would appear to go backward in time, which is one of the reasons that the speed of light is normally considered a cosmic speed limit.\n\nA key tenet of quantum mechanics, however, is the Heisenberg Uncertainty Principle, which, as I have mentioned, states that it is impossible to determine, for certain pairs of quantities, such as position and velocity, exact values for a given system at the same time. Alternatively, if you measure a given system for only a fixed, finite time interval, you cannot determine its total energy exactly.\n\nWhat all this implies is that, for very short times, so short that you cannot measure their speed with high precision, quantum mechanics allows for the possibility that these particles act as if they are moving faster than light! But, if they are moving faster than light, Einstein tells us they must be behaving as if they are moving backward in time!\n\nFeynman was brave enough to take this apparently crazy possibility seriously and explore its implications. He drew the following diagram for an electron moving about, periodically speeding up in the middle of its voyage to faster-than-light speed.\n\nHe recognized that relativity would tell us that another observer might alternatively measure something that would appear as shown below, with an electron moving forward in time, then backward in time, and then forward again.\n\nHowever, a negative charge moving backward in time is mathematically equivalent to a positive charge moving forward in time! Thus, relativity would require the existence of positively charged particles with the same mass and other properties as electrons.\n\nIn this case one can reinterpret Feynman's second drawing as follows: a single electron is moving along, and then at another point in space a positron-electron pair is created out of nothing, and then the positron meets the first electron and the two annihilate. Afterward, one is left with a single electron moving along.\n\nIf this doesn't bother you, then consider the following: for a little while, even if you start out with just a single particle, and end with a single particle, for a short time there are three particles moving about:\n\nIn the brief middle period, for at least a little while, something has spawned out of nothing! Feynman beautifully describes this apparent paradox in his 1949 paper, \"A Theory of Positrons,\" with a delightful wartime analogy:\n\nIt is as though a bombardier watching a single road through the bomb-sight of a low-flying plane suddenly sees three roads and it is only when two of them come together and disappear again that he realizes that he has simply passed over a long switchback in a single road.\n\nAs long as this time period during this \"switchback\" is so short that we cannot measure all the particles directly, quantum mechanics and relativity imply that not only is this weird situation allowed, it is required. The particles that appear and disappear in timescales too short to measure are called _virtual_ particles.\n\nNow inventing a whole new set of particles in empty space that you cannot measure sounds a lot like proposing a large number of angels sitting on the head of a pin. And it would be about as impotent an idea if these particles had no other measurable effects. However, while they are not directly observable, it turns out their _indirect_ effects produce most of the characteristics of the universe we experience today. Not only this, but one can calculate the impact of these particles more precisely than any other calculation in science.\n\nConsider, for example, a hydrogen atom\u2014the system Bohr tried to explain by developing his quantum theory and Schr\u00f6dinger later tried to describe by deriving his famous equation. The beauty of quantum mechanics was that it could explain the specific colors of light emitted by hydrogen when it was heated up by arguing that electrons orbiting around the proton could exist only in discrete energy levels, and when they jumped between levels they absorbed or emitted only a fixed set of frequencies of light. The Schr\u00f6dinger equation allows one to calculate the predicted frequencies, and it gets the answer almost exactly right.\n\nBut not exactly.\n\nWhen the spectrum of hydrogen was observed more carefully, it was seen to be more complicated than had previously been estimated, with some additional small splittings between levels observed, called the \"fine structure\" of the spectrum. While these splittings had been known since Bohr's time, and it was suspected that perhaps relativistic effects had something do to with them, until a fully relativistic theory was available, no one could confirm this suspicion. Happily, Dirac's equation managed to improve the predictions compared to Schr\u00f6dinger's equation and reproduced the general structure of the observations, including fine structure.\n\nSo far so good, but in April of 1947, United States experimentalist Willis Lamb and his student Robert C. Retherford performed an experiment that might otherwise seem incredibly ill motivated. They realized that they had the technological ability to measure the energy-level structure of the level of hydrogen atoms with an accuracy of 1 part in 100 million.\n\nWhy would they bother? Well, whenever experimentalists find a new method to measure something with vastly greater precision than was possible before, that is often sufficient motivation for them to go ahead. Whole new worlds are often revealed in the process, as when the Dutch scientist Antonie Philips van Leeuwenhoek first stared at a drop of seemingly empty water with a microscope in 1676 and discovered it was teeming with life. In this case, however, the experimenters had more immediate motivation. Up until the time of Lamb's experiment, the available experimental precision could not test Dirac's prediction in detail.\n\nThe Dirac equation did predict the general structure of the new observations, but the key question that Lamb wanted to answer was whether it predicted it in detail. This was the only way to actually test the theory. And when Lamb tested the theory, it seemed to give the wrong answer, at a level of about 100 parts per billion, well above the sensitivity of his apparatus.\n\nSuch a small disagreement with experiment may not seem like a lot, but the predictions of the simplest interpretation of the Dirac theory were unambiguous, as was the experiment, and they differed.\n\nOver the next few years, the best theoretical minds in physics jumped into the fray and tried to resolve the discrepancy. The answer came after a great deal of work, and when the dust had settled, it was realized that the Dirac equation actually gives precisely the correct answer, but only if you include the effect of virtual particles. Pictorially, this can be understood as follows. Hydrogen atoms are usually pictured in chemistry books something like this, with a proton at the center and an electron orbiting around it, jumping between different levels:\n\nHowever, once we allow for the possibility that electron-positron pairs can spontaneously appear from nothing for a bit before annihilating each other again, over any short time the hydrogen atom really looks like this:\n\nAt the right of the figure I have drawn such a pair, which then annihilate at the top. The virtual electron, being negatively charged, likes to hang around closer to the proton, while the positron likes to stay farther away. In any case, what is clear from this picture is that the actual charge distribution in a hydrogen atom is _not,_ at any instant, described by simply a single electron and proton.\n\nRemarkably, we physicists have learned (after all the hard work by Feynman and others) that we can use Dirac's equation to calculate to an arbitrarily high precision, the impact on the spectrum of hydrogen of all the possible virtual particles that may exist intermittently in its vicinity. And when we do, we come up with _the best, most accurate prediction in all of science_. All other scientific predictions pale in comparison. In astronomy, the most recent observations of the cosmic microwave background radiation allow us to compare with theoretical predictions at the level of perhaps 1 part in 100,000, which is remarkable. However, using Dirac's equation, and the predicted existence of virtual particles, we can calculate the value of atomic parameters and compare them with observations and have remarkable agreement at the level of about 1 part in a billion or better!\n\nVirtual particles therefore exist.\n\nWhile the spectacular precision available in atomic physics is hard to match, there is nevertheless another place where virtual particles play a key role that may actually be more relevant to the central issue of this book. It turns out that they are responsible for most of your mass, and that of everything that is visible in the universe.\n\nOne of the great successes in the 1970s in our fundamental understanding of matter came with the discovery of a theory that accurately describes the interactions of quarks, the particles that make up the protons and neutrons that form the bulk of material from which you and everything you can see are made. The mathematics associated with the theory is complex, and it took several decades before techniques were developed that could handle it, particularly in the regime where the strong interaction between the quarks became appreciable. A herculean effort was launched, including building some of the most complicated parallel processing computers, which simultaneously utilize tens of thousands of individual processors, in order to try to calculate the fundamental properties of protons and neutrons, the particles we actually measure.\n\nAfter all of this work, we now have a good picture of what the inside of a proton actually looks like. There may be three quarks contained therein, but there is also a lot of other stuff. In particular, virtual particles reflecting the particles and fields that convey the strong force between quarks are popping in and out of existence all the time. Here is a snapshot of how things actually look. It is not a real photograph of course, but rather an artistic rendering of the mathematics governing the dynamics of quarks and the fields that bind them. The odd shapes and different shadings reflect the strength of the fields interacting with one another and with the quarks inside the proton as virtual particles spontaneously pop in and out of existence.\n\nThe proton is intermittently full of these virtual particles and, in fact, when we try to estimate how much they might contribute to the mass of the proton, we find that the quarks themselves provide very little of the total mass and that the fields created by these particles contribute most of the energy that goes into the proton's rest energy and, hence, its rest mass. The same is true for the neutron, and since you are made of protons and neutrons, the same is true for you!\n\nNow, if we can calculate the effects of virtual particles on the otherwise empty space in and around atoms, and we can calculate the effects of virtual particles on the otherwise empty space inside of protons, then shouldn't we be able to calculate the effects of virtual particles on truly empty space?\n\nWell, this calculation is actually harder to do. This is because, when we calculate the effect of virtual particles on atoms or on the proton mass, we are actually calculating the total energy of the atom or proton _including_ virtual particles; then, we calculate the total energy that the virtual particles would contribute without the atom or proton present (i.e., in empty space); and _then_ we subtract the two numbers in order to find the net impact upon the atom or proton. We do this because it turns out that each of these two energies is formally infinite when we attempt to solve the appropriate equations, but when we subtract the two quantities, we end up with a finite difference, and moreover one that agrees precisely with the measured value!\n\nHowever, if we want to calculate the effect of virtual particles on empty space alone, we have nothing to subtract, and the answer we get is therefore infinite.\n\nInfinity is not a pleasant quantity, however, at least as far as physicists are concerned, and we try to avoid it whenever possible. Clearly, the energy of empty space (or anything else, for that matter) cannot be physically infinite, so we have to figure out a way to do the calculation and get a finite answer.\n\nThe source of the infinity is easy to describe. When we consider all possible virtual particles that can appear, the Heisenberg Uncertainty Principle (which I remind you says that the uncertainty in the measured energy of a system is inversely proportional to the length of time over which you observe it) implies that particles carrying ever more energy can appear spontaneously out of nothing as long as they then disappear in ever-shorter times. In principle, particles can therefore carry almost infinite energy as long as they disappear in almost infinitesimally short times.\n\nHowever, the laws of physics as we understand them apply only for distances and times larger than a certain value, corresponding to the scale where the effects of quantum mechanics must be considered when trying to understand gravity (and its associated effects on space-time). Until we have a theory of \"quantum gravity,\" as it is called, we can't trust extrapolations that go beyond these limits.\n\nThus, we might hope that the new physics associated with quantum gravity will somehow cut off the effects of virtual particles that live for less time than the \"Planck-time,\" as it is called. If we then consider the cumulative effects of only virtual particles of energies equal to or lower than that allowed by this temporal cutoff, we arrive at a finite estimate for the energy that virtual particles contribute to nothing.\n\nBut there is a problem. This estimate turns out to be about 1,000,\u00ad000,\u00ad000,\u00ad000,\u00ad000,\u00ad000,\u00ad000,\u00ad000,\u00ad000,\u00ad000,\u00ad000,\u00ad000,\u00ad000,\u00ad000,\u00ad000,\u00ad000,\u00ad000,\u00ad000,\u00ad000,\u00ad000,\u00ad000,\u00ad000,\u00ad000,\u00ad000,\u00ad000,\u00ad000,\u00ad000,\u00ad000,\u00ad000,\u00ad000,\u00ad000,\u00ad000,\u00ad000,\u00ad000,\u00ad000,\u00ad000,\u00ad000,\u00ad000,\u00ad000,\u00ad000 times larger than the energy associated with all the known matter in the universe, including dark matter!\n\nIf the calculation of the atomic energy level spacings including virtual particles is the best computation in all of physics, this estimate of the energy space\u2014120 orders of magnitude larger than the energy of everything else in the universe\u2014is undoubtedly the worst! If the energy of empty space were anywhere near this large, the repulsive force induced (remember the energy of empty space corresponds to a cosmological constant) would be large enough to blow up the Earth today, but more important, it would have been so great at early times that everything we now see in our universe would have pushed apart so quickly in the first fraction of a second of the Big Bang that no structure, no stars, no planets, and no people would ever have formed.\n\nThis problem, appropriately called the Cosmological Constant Problem, has been around since well before I was a graduate student, first made explicit by the Russian cosmologist Yakov Zel'dovich around 1967. It remains unsolved and is perhaps the most profound unsolved fundamental problem in physics today.\n\nIn spite of the fact that we have had no idea how to solve the problem for more than forty years, we theoretical physicists knew what the answer had to be. Like the fourth grader who I suggested would have guessed that the energy of empty space had to be zero, we too felt that when an ultimate theory was derived, it would explain how the effects of virtual particles would cancel, leaving empty space with precisely zero energy. Or nothing. Or rather, Nothing.\n\nOur reasoning was better than the fourth grader's, or so we thought. We needed to reduce the magnitude of the energy of empty space from the truly gargantuan value that the na\u00efve estimate suggested to a value consistent with the upper limits allowed by observation. This would require some way to subtract from a very large positive number another very large positive number so the two would cancel to 120 decimal places, leaving something non-zero in the 121st decimal place! But there is no precedent in science for canceling two large numbers to such accuracy, with only something minuscule left over.\n\nHowever, zero is a number that is easy to produce. Symmetries of nature often allow us to demonstrate that there are precisely equal and opposite contributions coming from different parts of a calculation, canceling out exactly, with precisely nothing left over. Or, again, Nothing.\n\nThus, we theorists were able to rest easy and sleep at night. We didn't know how to get there, but we were sure what the final answer had to be.\n\nNature, however, had something different in mind.\n\n## CHAPTER 5\n\nTHE RUNAWAY UNIVERSE\n\n_It is mere rubbish, thinking at present of the origin of life; one might as well think of the origin of matter._\n\n\u2014CHARLES DARWIN\n\nWhat Michael Turner and I argued in 1995 was heretical in the extreme. Based on little more than theoretical prejudice, we presumed the universe was flat. (I should stress once again here that a \"flat\" three-dimensional universe is not flat like a two-dimensional pancake is flat, but is rather the three-dimensional space that all of us intuitively picture, in which light rays travel in straight lines. This is to be contrasted with the much harder to picture curved three-dimensional spaces in which light rays, which trace the underlying curvature of space, do not travel in straight lines.) Then we inferred that all available cosmological data at the time were consistent with a flat universe only if about 30 percent of the total energy resided in some form of \"dark matter,\" as observations seemed to indicate existed around galaxies and clusters, but much more strangely than even this, that the remaining 70 percent of the total energy in the universe resided not in any form of matter, but rather in empty space itself.\n\nOur idea was crazy by any standards. In order to result in a value for the cosmological constant consistent with our claim, the estimated value for this quantity described in the last chapter would have to be reduced somehow by 120 orders of magnitude and still not be precisely zero. This would involve the most severe fine-tuning of any physical quantity known in nature, without the slightest idea how to adjust it.\n\nThis was one of the reasons that, as I lectured at various universities about the quandary of a flat universe, I evoked mostly smiles and no more. I don't think many people took our proposal seriously, and I am not even sure Turner and I did. Our point in raising eyebrows with our paper was to illustrate graphically a fact that was beginning to dawn not just on us, but also on several of our theorist colleagues around the world: something looked wrong with the by-then \"standard\" picture of our universe, in which almost all the energy required by general relativity to produce a flat universe today was assumed to reside in exotic dark matter (with a pinch of baryons\u2014i.e., us Earthlings, stars, visible galaxies\u2014to salt the mix).\n\nA colleague recently reminded me that for the two years following our modest proposal, it was referenced only a handful of times in subsequent papers, and apparently all but one or two of these were in papers written by Turner or me! As perplexing as our universe is, the bulk of the scientific community believed it couldn't be as crazy as Turner and I suggested it was.\n\nThe simplest alternative way out of the contradictions was the possibility that the universe wasn't flat but open (one in which parallel light rays today would curve apart if we traced their trajectory backward. This was of course before the cosmic microwave background measurements made it clear that this option was not viable.) However, even this possibility had its own problems, though the situation there remained far from clear as well.\n\nAny high school physics student will happily tell you that gravity sucks\u2014that is, it is universally attractive. Of course, like so many things in science, we now recognize that we have to expand our horizons because nature is more imaginative than we are. If for the moment we assume that the attractive nature of gravity implies that the expansion of the universe has been slowing down, recall that we get an upper limit on the age of the universe by assuming that the velocity of a galaxy located at a certain distance from us has been constant since the Big Bang. This is because, if the universe has been decelerating, then the galaxy was once moving away faster from us than it is now, and therefore it would have taken less time to get to its current position than if it had always been moving at its current speed. In an open universe dominated by matter, the deceleration of the universe would be slower than in a flat universe, and therefore the inferred age of the universe would be greater than it would be for a flat universe dominated by matter, for the same current measured expansion rate. It would in fact be much closer to the value we estimate by assuming a constant rate of expansion over cosmic time.\n\nRemember that a non-zero energy of empty space would produce a cosmological constant\u2014like gravitational repulsion\u2014implying that the expansion of the universe would instead speed up over cosmic time, and therefore galaxies would previously have been moving apart more slowly than they are today. This would imply that it would have taken even longer to get to their present distance than it would for a constant expansion. Indeed, for a given measurement of the Hubble constant today, the longest possible lifetime of our universe (about 20 billion years) is obtained by including the possibility of a cosmological constant along with the measured amount of visible and dark matter, if we are free to adjust its value along with the density of matter in the universe today.\n\nIn 1996, I worked with Brian Chaboyer and our collaborators Pierre Demarque at Yale and postdoc Peter Kernan at Case Western Reserve to put a lower limit on the age of these stars to be about 12 billion years. We did this by modeling the evolution of millions of different stars on high-speed computers and comparing their colors and brightness with actual stars observed in globular clusters in our galaxy, which were long thought to be among the oldest objects in the galaxy. Assuming about a billion years for our galaxy to form, this lower limit effectively ruled out a flat universe dominated by matter and favored one with a cosmological constant (one of the factors that had weighed on the conclusions in my earlier paper with Turner), while an open universe teetered on the hairy edge of viability.\n\nHowever, the ages of the oldest stars involve inferences based on observations at the edge of the then current sensitivity and, in 1997, new observational data forced us to revise our estimates downward by about 2 billion years, leading to a somewhat younger universe. So the situation became much murkier, and all three cosmologies once again appeared viable, sending many of us back to the drawing board.\n\nAll of this changed in 1998, coincidentally the same year that the BOOMERANG experiment demonstrated that the universe is flat.\n\nIn the intervening seventy years since Edwin Hubble measured the expansion rate of the universe, astronomers had worked harder and harder to pin down its value. Recall that in the 1990s they had finally found a \"standard candle\"\u2014that is, an object whose intrinsic luminosity observers felt they could independently ascertain, so that, when they measured its apparent luminosity, they could then infer its distance. The standard candle seemed to be reliable and was one that could be observed across the depths of space and time.\n\nA certain type of exploding star called Type Ia supernova had recently been demonstrated to exhibit a relationship between brightness and longevity. Measuring how long a given Type Ia supernova remained bright required, for the first time, taking into account time dilation effects due to the expansion of the universe, which imply that the measured lifetime of such a supernova is actually longer than its actual lifetime in its rest frame. Nonetheless, we could infer the absolute brightness and measure its apparent brightness with telescopes and ultimately determine the distance to the host galaxy in which the supernova had exploded. Measuring the redshift of the galaxy at the same time allowed us to determine velocity. Combining the two allows us to measure, with increasing accuracy, the expansion rate of the universe.\n\nBecause supernovae are so bright, they provide not only a great tool to measure the Hubble constant, they also allow observers to look back to distances that are a significant fraction of the total age of the universe.\n\nThis offered a new and exciting possibility, which observers viewed as a much more exciting quarry: measuring how Hubble's constant changes over cosmic time.\n\nMeasuring how a constant is changing sounds like an oxymoron, and it would be except for the fact that we humans live such brief lives, at least on a cosmic scale. On a human timescale the expansion rate of the universe is indeed constant. However, as I have just described, the expansion rate of the universe will change over cosmic time due to the effects of gravity.\n\nThe astronomers reasoned that if they could measure the velocity and distance of supernovae located far away\u2014across the far reaches of the visible universe\u2014then they could measure the rate at which the expansion of the universe was slowing down (since everyone assumed the universe was acting sensibly, and the dominant gravitational force in the universe was attractive). This in turn they hoped would reveal whether the universe was open, closed, or flat, because the rate of slowing as a function of time is different for each geometry.\n\nIn 1996, I was spending six weeks visiting Lawrence Berkeley Laboratory, lecturing on cosmology and discussing various science projects with my colleagues there. I gave a talk about our claim that empty space might have energy, and afterward, Saul Perlmutter, a young physicist who was working on detecting distant supernovae, came up to me and said, \"We will prove you wrong!\"\n\nSaul was referring to the following aspect of our suggestion of a flat universe, 70 percent of the energy of which should be contained in empty space. Recall that such energy would produce a cosmological constant, leading to a repulsive force that would then exist throughout all of space and that would dominate the expansion of the universe, causing its expansion to _speed up,_ not slow down.\n\nAs I have described, if the expansion of the universe was speeding up over cosmic time, then the universe would be older today than we would otherwise infer had the expansion been slowing down. This would then imply that the look-back in time to galaxies with a given redshift would be longer than it would otherwise be. In turn, if they have been receding from us for a longer time, this would imply that the light from them originated from farther away. The supernovae in galaxies at some given measured redshift would then appear fainter to us than if the light originated closer. Schematically, if one was measuring velocity versus distance, the slope of the curve for relatively nearby galaxies would allow us to determine the expansion rate today, and then whether the curve bent upward or downward for distant supernovae would tell us whether the universe was speeding up or slowing down over cosmic time.\n\nTwo years after our meeting, Saul and his collaborators, part of an international team called the Supernova Cosmology Project, published a paper based on early preliminary data that indeed suggested we were wrong. (Actually, they did not argue that Turner and I were wrong, since they, along with most of the other observers, really didn't give much credence to our proposal.) Their data suggested that the distance-versus-redshift plot curved downward, and thus that an upper limit on the energy of empty space had to be well below what would have been required to make a significant contribution to the total energy today.\n\nHowever, as often happens, the first data that come in might not be representative of all the data\u2014either you are simply statistically unlucky, or unexpected systematic errors might affect the data, which are not manifest until you have a much bigger sample. This was the case with data that the Supernova Cosmology Project published, and so the conclusions were incorrect.\n\nAnother international supernova search project, called the High-Z Supernova Search Team, led by Brian Schmidt at Mount Stromlo Observatory in Australia, was carrying out a program with the same goal, and they began to obtain different results. Brian recently told me that when their first significant High-Z Supernova determination came in, suggesting an accelerating universe with significant vacuum energy, they were turned down for telescope time and informed by a journal that they must be wrong because the Supernova Cosmology Project had already determined that the universe was indeed flat, and dominated by matter.\n\nThe detailed history of the competition between these two groups will undoubtedly be replayed many times, especially after they share the Nobel Prize, which they undoubtedly will.* This is not the place to worry about priority. Suffice it to say that by early 1998, Schmidt's group published a paper demonstrating that the universe appeared to be accelerating. About six months later, Perlmutter's group announced similar results and published a paper confirming the High-Z Supernova result, in effect acknowledging their earlier error\u2014and lending more credence to a universe dominated by the energy of empty space or, as it is now more commonly called, dark energy.\n\nThe speed with which these results were adopted by the scientific community\u2014even though they required a wholesale revision of the entire accepted picture of the universe\u2014provides an interesting study in scientific sociology. Almost overnight, there appeared to be universal acceptance of the results, even though, as Carl Sagan has emphasized, \"Extraordinary claims require extraordinary evidence.\" This was certainly an extraordinary claim if ever one was.\n\nI was shocked when, in December 1998, _Science_ magazine called the discovery of an accelerating universe the Scientific Breakthrough of the Year, producing a remarkable cover with a drawing of a shocked Einstein.\n\nI wasn't shocked because I thought that the result wasn't worthy of a cover. Quite the contrary. If true, it was one of the most important astronomical discoveries of our time, but the data at the time were merely strongly suggestive. They required such a change in our picture of the universe that I felt that we should all be more certain that other possible causes of the effects observed by the teams could be ruled out definitively before everyone jumped on the cosmological constant bandwagon. As I told at least one journalist at the time, \"The first time I didn't believe in a cosmological constant was when observers claimed to discover it.\"\n\nMy somewhat facetious reaction may seem strange, given that I had been promoting the possibility in one form or another for perhaps a decade. As a theorist, I feel that speculation is fine, especially if it promotes new avenues for experiment. But I believe in being as conservative as possible when examining real data, perhaps because I reached scientific maturity during a period when so many new and exciting but tentative claims in my own field of particle physics turned out to be spurious. Discoveries ranging from a claimed new fifth force in nature to the discovery of new elementary particles to the supposed observation that our universe is rotating as a whole have come and gone with much hoopla.\n\nThe biggest concern at the time regarding the claimed discovery of an accelerating universe was that distant supernovae may appear dimmer than they would otherwise be expected to be, not because of an accelerated expansion, but merely because either (a) they _are_ dimmer, or (b) perhaps some intergalactic or galactic dust present at early times partially obscures them.\n\nIn the intervening decade, it has nevertheless turned out that the evidence for acceleration has become overwhelming, almost unimpeachable. First, many more supernovae at high redshift have been measured. From these, a combined analysis of the supernovae from the two groups done within a year of the original publication yielded the following plot:\n\nAs a guide to the eye, to help you see whether the distance-versus-redshift curve bends upward or downward, the observers have drawn a dotted straight line in the upper half of the plot from the bottom left to the top right corner that goes through the data that represent nearby supernovae. The slope of this line tells us the expansion rate today. Then, in the lower half of the figure they have made that same straight line horizontal, to guide the eye. If the universe were decelerating, as had been expected in 1998, the distant supernovae at a redshift (z) close to 1 would fall below the straight line. But as you can see, most of them fall above the straight line. This is due to either one of two reasons:\n\n1. the data are wrong, or\n\n2. the expansion of the universe _is_ accelerating.\n\nIf we take, for the moment, the second alternative and ask, \"How much energy would we have to put in empty space in order to produce the observed acceleration?\" the answer we come up with is remarkable. The solid curve, which fits the data best, corresponds to a flat universe, with 30 percent of the energy in matter and 70 percent in empty space. This is, remarkably, precisely what is needed in order to make a flat universe consistent with the fact that only 30 percent of the required mass exists in and around galaxies and clusters. An apparent concordance has been achieved.\n\nNevertheless, because the claim that 99 percent of the universe is invisible (1 percent visible matter embedded in a sea of dark matter surrounded by energy in empty space) fits into the category of an extraordinary claim, we should seriously consider the first of the two possibilities I mention above: namely, that the data are wrong. In the intervening decade, all the rest of the data from cosmology has continued to solidify the general concordance picture of a cockamamie, flat universe in which the dominant energy resides in empty space and in which everything we can see accounts for less than 1 percent of the total energy, with the matter we can't see being composed mostly of some yet unknown, new type of elementary particles.\n\nFirst, new data on stellar evolution have improved as new satellites have provided us with information on the elemental abundances in old stars. Using these, my colleague Chaboyer and I were able, in 2005, to demonstrate definitively that the uncertainties in the estimates of the age of the universe using these data were now small enough to rule out lifetimes younger than about 11 billion years. This was inconsistent with any universe in which empty space itself contained a significant amount of energy. Again, since we are not certain that this energy is due to a cosmological constant, it now goes by the simpler name \"dark energy,\" in analogue to the moniker of \"dark matter\" that dominates galaxies.\n\nThis estimate for the age of our universe was vastly improved in about 2006 when new precision measurements of the cosmic microwave background using the WMAP satellite allowed observers to precisely measure the time since the Big Bang. We now know the age of the universe to four significant figures. It is 13.72 billion years old!\n\nI would never have figured that, in my lifetime, we would obtain such accuracy. But now that we have it, we can confirm that there is no way that a universe with the measured expansion rate today could be this old without dark energy, and in particular, dark energy that behaves essentially like the energy represented by a cosmological constant would behave. In other words, it is energy that appears to remain constant over time.\n\nIn the next scientific breakthrough, observers were able to measure accurately how matter, in the form of galaxies, has clustered together over cosmic time. The result depends upon the expansion rate of the universe, as the attractive force pulling galaxies together has to compete with the cosmic expansion driving matter apart. The larger the value of the energy of empty space, the sooner it will come to dominate the energy of the universe, and the sooner the increasing expansion rate will eventually stop the gravitational collapse of matter on ever larger scales.\n\nBy measuring gravitational clustering, therefore, observers have been able to confirm, once again, that the only flat universe that is consistent with observed large-scale structure in the universe is one with approximately 70 percent dark energy and, once again, that dark energy behaves more or less like the energy represented by a cosmological constant.\n\nIndependent of these indirect probes of the expansion history of the universe, the supernova observers have done extensive tests of possibilities that could induce systematic errors in their analysis, including the possibility of increased dust at large distances that make supernovae look dimmer, and ruled them out one by one.\n\nOne of their most important tests involved searching back in time.\n\nEarlier in the history of the universe, when what is now our currently observable region was much smaller in size, the density of matter was much greater. However, the energy density of empty space remains the same over time if it reflects a cosmological constant\u2014or something like it. Thus, when the universe was less than about half its present size, the energy density of matter would have exceeded the energy density of empty space. For all times before this time matter, and not empty space, would have produced the dominant gravitational force acting on the expansion. As a result, the universe would have been decelerating.\n\nIn classical mechanics there is a name for the point at which a system changes its acceleration and, in particular, goes from decelerating to accelerating. It is called a \"jerk.\" In 2003, I organized a conference at my university to examine the future of cosmology and invited one of the High-Z Supernova survey members, Adam Riess, who had told me he would have something exciting to report at the meeting. He did. The next day, the _New York Times,_ which was reporting on the meeting, ran a photo of Adam accompanied by the headline \"Cosmic Jerk Discovered.\" I have kept that photo and turn to it for amusement from time to time.\n\nThe detailed mapping of the expansion history of the universe, demonstrating that it shifted from a period of deceleration to acceleration, added substantial weight to the claim that the original observations, which implied the existence of dark energy, were in fact correct. With all of the other evidence now available, it is very difficult to imagine that, by adhering to this picture, somehow we are being led on a cosmic wild-goose chase. Like it or not, dark energy seems here to stay, or at least to stay until it changes in some way.\n\nThe origin and nature of dark energy is without a doubt the biggest mystery in fundamental physics today. We have no deep understanding of how it originates and why it takes the value it has. We therefore have no idea of why it has begun to dominate the expansion of the universe and only relatively recently, in the past 5 billion years or so, or whether that is a complete accident. It is natural to suspect that its nature is tied in some basic way to the origin of the universe. And all signs suggest that it will determine the future of the universe as well.\n\n* * *\n\n* Indeed, as this book goes to print I just learned that Saul and Brian, along with Adam Reiss, who was part of the High-Z Supernova project, were awarded the Nobel Prize in Physics for 2011 for their discovery.\n\n## CHAPTER 6\n\nTHE FREE LUNCH AT THE END OF THE UNIVERSE\n\n_Space is big. Really big. You just won't believe how vastly, hugely, mind-bogglingly big it is. I mean, you may think it's a long way down the road to the chemist's, but that's just peanuts to space._\n\n\u2014DOUGLAS ADAMS, _The Hitchhiker's Guide to the Galaxy_\n\nOne out of two isn't bad, I suppose. We cosmologists had guessed, correctly it turned out, that the universe is flat, so we weren't that embarrassed by the shocking revelation that empty space indeed has energy\u2014and enough energy in fact to dominate the expansion of the universe. The existence of this energy was implausible, but even more implausible is that the energy is not enough to make the universe uninhabitable. For if the energy of empty space were as large as the a priori estimates I described earlier suggested it should be, the expansion rate would have been so great that everything that we now see in the universe would quickly have been driven beyond the horizon. The universe would become cold, dark, and empty well before stars, our Sun, and our Earth could have formed.\n\nOf all the reasons to suppose that the universe was flat, perhaps the simplest to understand arose from the fact that the universe had been well-known to be almost flat. Even in the early days, before dark matter was discovered, the known amount of visible material in and around galaxies accounted for perhaps 1 percent of the total amount of matter needed to result in a flat universe.\n\nNow, 1 percent may not seem like much, but our universe is very old, billions of years old. Assuming that the gravitational effects of matter or radiation dominate the evolving expansion, which is what we physicists always thought was the case, then if the universe is not precisely flat, as it expands, it moves further and further away from being flat.\n\nIf it is open, the expansion rate continues at a faster rate than it would for a flat universe, driving matter farther and farther apart compared to what it would be otherwise, reducing its net density and very quickly yielding an infinitesimally small fraction of the density required to result in a flat universe.\n\nIf it is closed, then it slows the expansion down faster and eventually causes it to recollapse. All the while, the density first decreases at a slower rate than for a flat universe, and then as the universe recollapses, it starts to increase. Once again, the departure from the density expected for a flat universe increases with time.\n\nThe universe has increased in size by a factor of almost a trillion since it was 1 second old. If, at that earlier moment, the density of the universe was not almost exactly that expected of a flat universe but was, say, only 10 percent of that appropriate for a flat universe at the time, then today the density of our universe would differ from that of a flat universe by at least a factor of a trillion. This is far greater than the mere factor of 100 that was known to separate the density of the visible matter in the universe from the density of what would produce a flat universe today.\n\nThis problem was well-known, even in the 1970s, and it became known as the Flatness Problem. Considering the geometry of the universe is like imagining a pencil balancing vertically on its point on a table. The slightest imbalance one way or the other and it will quickly topple. So it is for a flat universe. The slightest departure from flatness quickly grows. Thus, how could the universe be so close to being flat today if it were not _exactly_ flat?\n\nThe answer is simple: it must be essentially flat today!\n\nThat answer's actually not so simple, because it begs the question, How did initial conditions conspire to produce a flat universe?\n\nThere are two answers to this second, more difficult question. The first goes back to 1981, when a young theoretical physicist and postdoctoral researcher at Stanford University, Alan Guth, was thinking about the Flatness Problem and two other related problems with the standard Big Bang picture of the universe: the so-called Horizon Problem and the Monopole Problem. Only the first need concern us here, since the Monopole Problem merely exacerbates both the Flatness and Horizon problems.\n\nThe Horizon Problem relates to the fact that the cosmic microwave background radiation is extremely uniform. The small temperature deviations, which I described earlier, represented density variations in matter and radiation back when the universe was a few hundred thousand years old, of less than 1 part in 10,000 compared to the otherwise uniform background density and temperature. So while I was focused on the small deviations, a deeper, more urgent question was, How did the universe get to be so uniform in the first place?\n\nAfter all, if instead of the earlier image of the CMBR (where temperature variations of a few parts in 100,000 are reflected in different colors), I showed a temperature map of the microwave sky on a linear scale (with variations in shades representing variations in temperature, of say, \u00b10.03 degree [Kelvin] about the mean background temperature of about 2.72 degrees above absolute zero, or a variation of 1 part in 100 about the mean), the map would look like this:\n\nCompare this image, which contains nothing discernible in the way of structure, to a similar projection of the surface of the Earth, with only slightly greater sensitivity, with color variations representing variations about the mean radius by about 1 part in 500 or so:\n\nThe universe is, therefore, on large scales, _incredibly uniform_!\n\nHow could this be? Well, one might simply assume that, at early times, the early universe was hot, dense, and in thermal equilibrium. This means that any hot spots would have cooled, and cold spots would have heated up until the primordial soup reached the same temperature throughout.\n\nHowever, as I pointed out earlier, when the universe was a few hundred thousand years old, light could have traveled only a few hundred thousand light-years, representing a small percentage of what is now the total observable universe (this former distance would represent merely an angle of about 1 degree on a map of the complete cosmic microwave background last scattering surface as it is observed today). Since Einstein tells us that no information can propagate faster than light, in the standard Big Bang picture, there is simply no way that one part of what is now the observable universe at that time would have been affected by the existence and temperature of other parts on angular scales of greater than about 1 degree. Thus, there is no way that the gas on these scales could have thermalized in time to produce such a uniform temperature throughout!\n\nGuth, a particle physicist, was thinking about processes that could have occurred in the early universe that might have been relevant for understanding this problem when he came up with an absolutely brilliant realization. If, as the universe cooled, it underwent some kind of phase transition\u2014as occurs, for example, when water freezes to ice or a bar of iron becomes magnetized as it cools\u2014then not only could the Horizon Problem be solved, but also the Flatness Problem (and, for that matter, the Monopole Problem).\n\nIf you like to drink really cold beer, you may have had the following experience: you take a cold beer bottle out of the refrigerator, and when you open it and release the pressure inside the container, suddenly the beer freezes completely, during which it might even crack part of the bottle. This happens because, at high pressure, the preferred lowest energy state of the beer is in liquid form, whereas once the pressure has been released, the preferred lowest energy state of the beer is the solid state. During the phase transition, energy can be released because the lowest energy state in one phase can have lower energy than the lowest energy state in the other phase. When such energy is released, it is referred to as \"latent heat.\"\n\nGuth realized that, as the universe itself cooled with the Big Bang expansion, the configuration of matter and radiation in the expanding universe might have gotten \"stuck\" in some meta-stable state for a while until ultimately, as the universe cooled further, this configuration then suddenly underwent a phase transition to the energetically preferred ground state of matter and radiation. The energy stored in the \"false vacuum\" configuration of the universe before the phase transition completed\u2014the \"latent heat\" of the universe, if you will\u2014could dramatically affect the expansion of the universe during the period before the transition.\n\nThe false vacuum energy would behave just like that represented by a cosmological constant because it would act like an energy permeating empty space. This would cause the expansion of the universe at the time to speed up ever faster and faster. Eventually, what would become our observable universe would start to grow faster than the speed of light. This is allowed in general relativity, even though it seems to violate Einstein's special relativity, which says nothing can travel faster than the speed of light. But one has to be like a lawyer and parse this a little more carefully. Special relativity says nothing can travel _through space_ faster than the speed of light. But _space itself_ can do whatever the heck it wants, at least in general relativity. And as space expands, it can carry distant objects, which are at rest in the space where they are sitting, apart from one another at superluminal speeds.\n\nIt turns out that the universe could have expanded during this inflationary period by a factor of more than 1028. While this is an incredible amount, it amazingly could have happened in a fraction of a second in the very early universe. In this case, everything within our entire observable universe was once, before inflation happened, contained in a region much smaller than we would have traced it back to if inflation had not happened, and most important, so small that there would have then been enough time for the entire region to thermalize and reach exactly the same temperature.\n\nInflation made another relatively generic prediction possible. When a balloon gets blown up larger and larger, the curvature at its surface gets smaller and smaller. Something similar happens for a universe whose size is expanding exponentially, as can occur during inflation\u2014driven by a constant and large false vacuum energy. Indeed, by the time inflation ends (solving the Horizon Problem), the curvature of the universe (if it is non-zero to begin with) gets driven to an absurdly small value so that, even today, the universe appears essentially flat when measured accurately.\n\nInflation is the only currently viable explanation of both the homogeneity and flatness of the universe, based on what could be fundamental and calculable microscopic theories of particles and their interactions. But more than this, inflation makes another, perhaps even more remarkable prediction possible. As I have described already, the laws of quantum mechanics imply that, on very small scales, for very short times, empty space can appear to be a boiling, bubbling brew of virtual particles and fields wildly fluctuating in magnitude. These \"quantum fluctuations\" may be important for determining the character of protons and atoms, but generally they are invisible on larger scales, which is one of the reasons why they appear so unnatural to us.\n\nHowever, during inflation, these quantum fluctuations can determine when what would otherwise be different small regions of space end their period of exponential expansion. As different regions stop inflating at slightly (microscopically) different times, the density of matter and radiation that results when the false vacuum energy gets released as heat energy in these different regions is slightly different in each one.\n\nThe pattern of density fluctuations that result after inflation\u2014arising, I should stress, from the quantum fluctuations in otherwise empty space\u2014turns out to be precisely in agreement with the observed pattern of cold spots and hot spots on large scales in the cosmic microwave background radiation. While consistency is not proof, of course, there is an increasing view among cosmologists that, once again, if it walks like a duck and looks like a duck and quacks like a duck, it is probably a duck. And if inflation indeed is responsible for all the small fluctuations in the density of matter and radiation that would later result in the gravitational collapse of matter into galaxies and stars and planets and people, then it can be truly said that we all are here today because of quantum fluctuations in what is essentially _nothing_.\n\nThis is so remarkable I want to stress it again. Quantum fluctuations, which otherwise would have been completely invisible, get frozen by inflation and emerge afterward as density fluctuations that produce everything we can see! If we are all stardust, as I have written, it is also true, if inflation happened, that we all, literally, emerged from quantum nothingness.\n\nThis is so strikingly nonintuitive that it can seem almost magical. But there is at least one aspect of all of this inflationary prestidigitation that might seem particularly worrisome. Where does all the energy come from in the first place? How can a microscopically small region end up as a universe-sized region today with enough matter and radiation within it to account for everything we can see?\n\nMore generally, we might ask the question, How is it that the density of energy can remain constant in an expanding universe with a cosmological constant, or false vacuum energy? After all, in such a universe, space expands exponentially, so that if the density of energy remains the same, the total energy within any region will grow as the volume of the region grows. What happened to the conservation of energy?\n\nThis is an example of something that Guth coined as the ultimate \"free lunch.\" Including the effects of gravity in thinking about the universe allows objects to have\u2014amazingly\u2014\"negative\" as well as \"positive\" energy. This facet of gravity allows for the possibility that positive energy stuff, like matter and radiation, can be complemented by negative energy configurations that just balance the energy of the created positive energy stuff. In so doing, gravity can start out with an empty universe\u2014and end up with a filled one.\n\nThis may also sound kind of fishy, but in fact it is a central part of the real fascination that many of us have with a flat universe. It is also something that you might be familiar with from high school physics.\n\nConsider throwing a ball up in the air. Generally, it will come back down. Now throw it harder (assuming you are not indoors). It will travel higher and stay aloft longer before returning. Finally, if you throw it hard enough, it will not come down at all. It will escape the Earth's gravitational field and keep heading out into the cosmos.\n\nHow do we know when the ball will escape? We use a simple matter of energy accounting. A moving object in the gravitational field of the Earth has two kinds of energy. One, the energy of motion, is called _kinetic energy,_ from the Greek word for motion. This energy, which depends upon the speed of the object, is always positive. The other component of the energy, called _potential energy_ (related to the potential to do work), is generally negative.\n\nThis is the case because we define the total gravitational energy of an object located at rest infinitely far away from any other object as being zero, which seems reasonable. The kinetic energy is clearly zero, and we define the potential energy as zero at this point, so the total gravitational energy is zero.\n\nNow, if the object is not infinitely far away from all other objects but is close to an object, like the Earth, it will begin to fall toward it because of the gravitational attraction. As it falls, it speeds up, and if it smacks into something on the way (say, your head), it can do work by, say, splitting it open. The closer it is to the Earth's surface when it is let go, the less work it can do by the time it hits the Earth. Thus, potential energy _decreases_ as you get closer to the Earth. But if the potential energy is zero when it is infinitely far away from the Earth, it must get more and more negative the closer it gets to the Earth because its potential to do work decreases the closer it gets.\n\nIn classical mechanics, as I defined it here, the definition of potential energy is arbitrary. I could have set the potential energy of an object as zero at the Earth's surface, and then it would be some large number when the object is infinitely far away. Setting the total energy to zero at infinity does make physical sense, but it is, at least at this point in our discussion, merely a convention.\n\nRegardless of where one sets the zero point of potential energy, the wonderful thing about objects that are subject to only the force of gravity is that the _sum_ of their potential and kinetic energies remains a constant. As objects fall, potential energy is converted to the kinetic energy of motion, and as they bounce back up off the ground, kinetic energy is converted back to potential, and so on.\n\nThis allows us a marvelous bookkeeping tool to determine how fast one needs to throw something up in the air in order to escape the Earth, since if it eventually is to reach infinitely far away from the Earth, its total energy must be greater than or equal to zero. I then simply have to ensure that its total gravitational energy at the time it leaves my hand is greater than or equal to zero. Since I can control only one aspect of its total energy\u2014namely the speed with which it leaves my hand\u2014all I have to do is find the magic speed where the positive kinetic energy of the ball equals the negative potential energy it has due to the attraction at the Earth's surface. Both the kinetic energy and the potential energy of the ball depend precisely the same way on the mass of the ball, which therefore cancels out when these two quantities are equated, and one finds a single \"escape velocity\" for all objects from the Earth's surface, namely about 7 miles per second, when the total gravitational energy of the object is precisely zero.\n\nWhat has all this got to do with the universe in general, and inflation in particular, you may ask? Well, the exact same calculation I just described for a ball that I throw up from my hand at the Earth's surface applies to every object in our expanding universe.\n\nConsider a spherical region of our universe centered on our location (in the Milky Way galaxy) and large enough to encompass a lot of galaxies but small enough so that it is well within the largest distances we can observe today:\n\nIf the region is large enough but not too large, then the galaxies at the edge of the region will be receding from us uniformly due to the Hubble expansion, but their speeds will be far less than the speed of light. In this case, the laws of Newton apply, and we can ignore the effects of special and general relativity. In other words, every object is governed by physics that is identical to that which describes the balls that I have just imagined trying to eject from the Earth.\n\nConsider the galaxy shown above, moving away from the center of the distribution as shown. Now, just as for the ball from the Earth, we can ask whether the galaxy will be able to escape from the gravitational pull of all the other galaxies within the sphere. And the calculation we would perform to determine the answer is precisely the same as the calculation we performed for the ball. We simply calculate the total gravitational energy of the galaxy, based on its motion outward (giving it positive energy), and the gravitational pull of its neighbors (providing a negative energy piece). If its total energy is greater than zero, it will escape to infinity, and if less than zero, it will stop and fall inward.\n\nNow, remarkably, it is possible to show that we can rewrite the simple Newtonian equation for the total gravitational energy of this galaxy in a way that reproduces _exactly_ Einstein's equation from general relativity for an expanding universe. And the term that corresponds to the total gravitational energy of the galaxy becomes, in general relativity, the term that describes the curvature of the universe.\n\nSo what do we then find? In a flat universe, and _only_ in a flat universe, the total average Newtonian gravitational energy of each object moving with the expansion is _precisely zero_!\n\nThis is what makes a flat universe so special. In such a universe the positive energy of motion is exactly canceled by the negative energy of gravitational attraction.\n\nWhen we begin to complicate things by allowing for empty space to have energy, the simple Newtonian analogy to a ball being thrown up in the air becomes incorrect, but the conclusion remains essentially the same. In a flat universe, even one with a small cosmological constant, as long as the scale is small enough that velocities are much less than the speed of light, the Newtonian gravitational energy associated with every object in the universe is zero.\n\nIn fact, with a vacuum energy, Guth's \"free lunch\" becomes even more dramatic. As each region of the universe expands to ever larger size, it becomes closer and closer to being flat, so that the total Newtonian gravitational energy of everything that results after the vacuum energy during inflation gets converted to matter and radiation becomes precisely zero.\n\nBut you can still ask, Where does all the energy come from to keep the density of energy constant during inflation, when the universe is expanding exponentially? Here, another remarkable aspect of general relativity does the trick. Not only can the gravitational energy of objects be negative, but their relativistic \"pressure\" can be negative.\n\nNegative pressure is even harder to picture than negative energy. Gas, say in a balloon, exerts pressure on the walls of the balloon. In so doing, if it expands the walls of the balloon, it does work on the balloon. The work it does causes the gas to lose energy and cool. However, it turns out that the energy of empty space is gravitationally repulsive precisely because it causes empty space to have a \"negative\" pressure. As a result of this negative pressure, the universe actually does work _on_ empty space as it expands. This work goes into maintaining the constant energy density of space even as the universe expands.\n\nThus, if the quantum properties of matter and radiation end up endowing even an infinitesimally small region of empty space with energy at very early times, this region can grow to be arbitrarily large and arbitrarily flat. When the inflation is over, one can end up with a universe full of stuff (matter and radiation), and the total Newtonian gravitational energy of that stuff will be as close as one can ever imagine to zero.\n\nSo when all the dust is settled, and after a century of trying, we have measured the curvature of the universe and found it to be zero. You can understand why so many theorists like me have found this not only very satisfying, but also highly suggestive.\n\nA universe from Nothing . . . indeed.\n\n## CHAPTER 7\n\nOUR MISERABLE FUTURE\n\n_The future ain't what it used to be._\n\n\u2014YOGI BERRA\n\nIn one sense it is both remarkable and exciting to find ourselves in a universe dominated by nothing. The structures we can see, like stars and galaxies, were all created by quantum fluctuations from nothing. And the average total Newtonian gravitational energy of each object in our universe is equal to nothing. Enjoy the thought while you can, if you are so inclined, because, if all this is true, we live in perhaps the worst of all universes one can live in, at least as far as the future of life is concerned.\n\nRemember that barely a century ago, Einstein was first developing his general theory of relativity. Conventional wisdom then held that our universe was static and eternal. In fact, Einstein not only ridiculed Lema\u00eetre for suggesting a Big Bang, but also invented the cosmological constant for the purpose of allowing a static universe.\n\nNow, a century later we scientists can feel smug for having discovered the underlying expansion of the universe, the cosmic microwave background, dark matter, and dark energy.\n\nBut what will the future bring?\n\nPoetry . . . of a sort.\n\nRecall that the domination of the expansion of our universe by the energy of seemingly empty space was inferred from the fact that this expansion is speeding up. And, just as with inflation, as described in the last chapter, our observable universe is at the threshold of expanding faster than the speed of light. And with time, because of the accelerated expansion, things will only get worse.\n\nThis means that, the longer we wait, the less we will be able to see. Galaxies that we can now see will one day in the future be receding away from us at faster-than-light speed, which means that they will become invisible to us. The light they emit will not be able to make progress against the expansion of space, and it will never again reach us. These galaxies will have disappeared from our horizon.\n\nThe way this works is a little different than you might imagine. The galaxies do not suddenly disappear or twinkle out of existence in the night sky. Rather, as their recession speed approaches the speed of light, the light from these objects gets ever more redshifted. Eventually, all their visible light moves to infrared, microwave, radio wave, and so on, until the wavelength of light they emit ends up becoming larger than the size of the visible universe, at which point they become officially invisible.\n\nWe can calculate about how long this will take. Since the galaxies in our local cluster of galaxies are all bound together by their mutual gravitational attraction, they will not recede with the background expansion of the universe discovered by Hubble. Galaxies just outside our group are about 1\/5000th the distance out to the point where the recession velocity of objects approaches the speed of light. It will take them about 150 billion years, about 10 times the current age of the universe, to get there, at which point all the light from the stars within the galaxies will have redshifted by a factor of about 5,000. By about 2 trillion years, their light will have redshifted by an amount that will make their wavelength equal to the size of the visible universe, and the rest of the universe will literally have disappeared.\n\nTwo trillion years may seem like a long time, and it is. In a cosmic sense, however, it is nowhere near an eternity. The longest-lived \"main sequence\" stars (which have the same evolutionary history as our Sun) have lifetimes far longer than our Sun and will still be shining in 2 trillion years (even as our own Sun dies out in about only 5 billion years). And so in the far future there may be civilizations on planets around those stars, powered by solar power, with water and organic materials. And there may be astronomers with telescopes on those planets. But when they look out at the cosmos, essentially everything we can now see, all 400 billion galaxies currently inhabiting our visible universe, will have disappeared!\n\nI have tried to use this argument with Congress to urge the funding of cosmology now, while we still have time to observe all that we can! For a congressperson, however, two years is a long time. Two trillion is unthinkable.\n\nIn any case, those astronomers in the far future would be in for a big surprise, if they had any idea what they were missing, which they won't. Because not only will the rest of the universe have disappeared, as my colleague Robert Scherrer of Vanderbilt and I recognized a few years ago, but essentially all of the evidence that now tells us we live in an expanding universe that began in a Big Bang will also have disappeared, along with all evidence of the existence of the dark energy in empty space that will be responsible for this disappearance.\n\nWhile less than a century ago conventional wisdom still held that the universe was static and eternal, with stars and planets coming and going, but on its largest scales the universe itself perduring, in the far future, long after any remnants of our planet and civilization have likely receded into the dustbin of history, the illusion that sustained our civilization until 1930 will be an illusion that will once again return, with a vengeance.\n\nThere are three main observational pillars that have led to the empirical validation of the Big Bang, so that, even if Einstein and Lema\u00eetre had never lived, the recognition that the universe began in a hot, dense state would have been forced upon us: the observed Hubble expansion; the observation of the cosmic microwave background; and the observed agreement between the abundance of light elements\u2014hydrogen, helium, and lithium\u2014we have measured in the universe with the amounts predicted to have been produced during the first few minutes in the history of the universe.\n\nLet's begin with the Hubble expansion. How do we know the universe is expanding? We measure the recession velocity of distant objects as a function of their distance. However, once all visible objects outside of our local cluster (in which we are gravitationally bound) have disappeared from our horizon, there will no longer be any tracers of the expansion\u2014no stars, galaxies, quasars, or even large gas clouds\u2014that observers could track. The expansion will be so efficient that it will have removed all objects from our sight that are actually receding from us.\n\nMoreover, on a timescale of less than a trillion years or so, all the galaxies in our local group will have coalesced into some large meta-galaxy. Observers in the far future will see more or less precisely what observers in 1915 thought they saw: a single galaxy housing their star and their planet, surrounded by an otherwise vast, empty, static space.\n\nRecall also that all evidence that empty space has energy comes from observing the rate of speed-up of our expanding universe. But, once again, without tracers of the expansion, the acceleration of our expanding universe will be unobservable. Indeed, in a strange coincidence, we are living in the only era in the history of the universe when the presence of the dark energy permeating empty space is likely to be detectable. It is true that this era is several hundred billion years long, but in an eternally expanding universe it represents the mere blink of a cosmic eye.\n\nIf we assume that the energy of empty space is roughly constant, as would be the case for a cosmological constant, then in much earlier times the energy density of matter and radiation would have far exceeded that in empty space. This is simply because, as the universe expands, the density of matter and radiation decreases along with the expansion because the distance between particles grows, so there are fewer objects in each volume. At earlier times, say earlier than about 5 billion to 10 billion years ago, the density of matter and radiation would have been far greater than it is today. The universe at this time and earlier was therefore dominated by matter and radiation, with their consequent gravitational attraction. In this case, the expansion of the universe would have been slowing down at these early times, and the gravitational impact of the energy of empty space would have been unobservable.\n\nBy the same token, far in the future, when the universe is several hundred billion years old, the density of matter and radiation will have decreased even further, and one can calculate that dark energy will have a mean energy density far in excess of a thousand billion times greater than the density of all remaining matter and radiation in the universe. It will, by then, completely govern the gravitational dynamics of the universe on large scales. However, at that late age, the accelerating expansion will have become essentially unobservable. In this sense, the energy of empty space ensures, by its very nature, that there is a finite time during which it is observable, and, remarkably, we live during this cosmological instant.\n\nWhat about the other major pillar of the Big Bang, the cosmic microwave background radiation, which provides a direct baby picture of the universe? First, as the universe expands ever faster in the future, the temperature of the CMBR will fall. When the presently observable universe is about 100 times larger than it is now, the temperature of the CMBR will have fallen by a factor of 100, and its intensity, or the energy density stored within it, will have fallen by a factor of 100 million, making it about 100 million times harder to detect than it currently is.\n\nBut, after all, we have been able to detect the cosmic microwave background amidst all the other electronic noise on Earth, and we can imagine that observers in the far future will be 100 million times smarter than those we are blessed with today, so that all hope is not lost. Alas, it turns out that even the brightest observer one could imagine, with the most sensitive instrument one could build, will still be essentially out of luck in the distant future. This is because in our galaxy (or the meta-galaxy that will form when our galaxy merges with its neighbors, beginning with Andromeda in about 5 billion years) there is hot gas between stars, and this gas is ionized, so that it contains free electrons, and thus behaves like a plasma. As I described earlier, such a plasma is opaque to many types of radiation.\n\nThere is something called a \"plasma frequency,\" below which radiation cannot permeate a plasma without absorption. Based on the currently observed density of free electrons in our galaxy, we can estimate the plasma frequency in our galaxy, and if we do this, we find that the bulk of the CMB radiation from the Big Bang will be stretched, by the time the universe gets to be about 50 times its present age, to long enough wavelengths, and hence low enough frequencies, that it will be below our future (meta-) galaxy's plasma frequency at that time. After that, the radiation will essentially not be able to make it into our (meta-)galaxy to be observed, no matter how tenacious the observer. The CMBR, too, will have disappeared.\n\nSo no observed expansion, no leftover afterglow of the Big Bang. But what about the abundance of the light elements\u2014hydrogen, helium, and lithium\u2014which also provides a direct signature of the Big Bang?\n\nIndeed, as I described in chapter 1, whenever I meet someone who doesn't believe in the Big Bang, I like to show them the following figure that I keep as a card in my wallet. I then say: \"See! There was a Big Bang!\"\n\nThis figure looks very complicated, I know, but it actually shows the relative predicted abundance of helium, deuterium, helium-3, and lithium, compared to hydrogen, based on our current understanding of the Big Bang. The green curve, going up and to the right, displays the predicted abundance of helium, the second most abundant element in the universe, by weight, compared with hydrogen (the most abundant element). The red and dark blue curves, going down and to the right represent the predicted abundances of deuterium and helium-3, respectively, not by weight but by number of atoms compared to hydrogen. Finally, the purple lower curve represents the predicted abundance of the next lightest element, lithium, again by number.\n\nThe predicted abundances are plotted as functions of the assumed total density of normal matter (made of atoms) in the universe today. If varying this quantity produced no combination of all the predicted elemental abundances that fit with our observations, it would be strong evidence against their production in a hot Big Bang. Note that the predicted abundances of these elements vary by almost 10 orders of magnitude.\n\nThe unshaded boxes associated with each curve represent the allowed range of the actual estimated primordial abundance of these elements based on observations of old stars and hot gas in and outside of our galaxy\n\nThe vertical blue band then represents that region where all the predictions and observations _do_ agree. It is hard to imagine more concrete support than this agreement between predictions and observations, again for elements whose predicted abundances vary by 10 orders of magnitude, for an early, hot Big Bang where all the light elements were first produced.\n\nIt is worth repeating the implications of this remarkable agreement more forcefully: Only in the first seconds of a hot Big Bang, with an initial abundance of protons and neutrons that would result in something very close to the observed density of matter in visible galaxies today, and a density of radiation that would leave a remnant that would correspond precisely to the observed intensity of the cosmic microwave background radiation today, would nuclear reactions occur that could produce precisely the abundance of light elements, hydrogen and deuterium, helium and lithium, that we infer to have comprised the basic building blocks of the stars that now fill the night sky.\n\nAs Einstein might have put it, only a very malicious (and, therefore, in his mind unimaginable) God would have conspired to have created a universe that so unambiguously points to a Big Bang origin without its having occurred.\n\nIndeed, when the rough agreement between the inferred helium abundance in the universe with the predicted helium abundance arising from a Big Bang was first demonstrated in the 1960s, this was one of the key bits of data that helped the Big Bang picture win out over the then very popular steady-state model of the universe championed by Fred Hoyle and his colleagues.\n\nIn the far future, however, things will be quite different. Stars burn hydrogen, producing helium, for example. At the present time only about 15 percent or so of all the observed helium in the universe could have been produced by stars in the time since the Big Bang\u2014once again, a compelling bit of evidence that a Big Bang was required to produce what we see. But in the far future this will not be the case, because many more generations of stars will have lived and died.\n\nWhen the universe is a trillion years old, for example, far more helium will have been produced in stars than will have been produced in the Big Bang itself. This situation is displayed in the following chart:\n\nWhen 60 percent of the visible matter in the universe is comprised of helium, there will be no necessity for production of primordial helium in a hot Big Bang in order to produce agreement with observations.\n\nObservers and theorists in some civilization in the far future will, however, be able to use this data to infer that the universe must have had a finite age. Because stars burn hydrogen to helium, there will be an upper limit on how long stars could have existed in order not to further deplete the ratio between hydrogen and helium. Thus, future scientists will estimate that the universe in which they live is less than about a trillion years old. But any direct signature that the beginning involved a Big Bang, rather than some other kind of spontaneous creation of our future single (meta-)galaxy, will be lacking.\n\nRemember that Lema\u00eetre derived his claim of a Big Bang purely on the basis of thinking about Einstein's general relativity. We can assume that any advanced civilization in the far future will discover the laws of physics, electromagnetism, quantum mechanics, and general relativity. Will some Lema\u00eetre of the far future therefore be able to derive a similar claim?\n\nLema\u00eetre's conclusion that our universe had to begin in a Big Bang was unavoidable, but it was based on an assumption that will not be true for the observable universe of the far future. A universe with matter stretching out uniformly in all directions, one that is isotropic and homogenous, cannot be static, for the reasons Lema\u00eetre and eventually Einstein recognized. However, there is a perfectly good solution of Einstein's equations for a single massive system surrounded by an otherwise empty static space. After all, if such a solution did not exist, then general relativity would not be able to describe isolated objects like neutron stars or, ultimately, black holes.\n\nLarge mass distributions like our galaxy are unstable, so eventually our (meta-)galaxy will itself collapse to form a massive black hole. This is described by a static solution of Einstein's equation called the Schwarzschild solution. But the time frame for our galaxy to collapse to form a massive black hole is much longer than the time frame for the rest of the universe to disappear. Thus, it will seem natural for scientists of the future to imagine that our galaxy could have existed for a trillion years in empty space without significant collapse and without requiring an expanding universe surrounding it.\n\nOf course, speculations about the future are notoriously difficult. I am writing this, in fact, while at the World Economic Forum in Davos, Switzerland, which is full of economists who invariably predict the behavior of future markets and revise their predictions when they turn out to be horribly wrong. More generally, I find any predictions of the far future, and even the not-so-far future, of science and technology to be even sketchier than those of \"the dismal science.\" Indeed, whenever I'm asked about the near future of science or what the next big breakthrough will be, I always respond that if I knew, I would be working on it right now!\n\nThus, I like to think of the picture I have presented in this chapter as something like the picture of the future presented by the third ghost in Dickens's _A Christmas Carol_. This is the future as it _might be_. After all, since we have no idea what the dark energy permeating empty space is, we also therefore cannot be certain that it will behave like Einstein's cosmological constant and remain constant. If it doesn't, the future of the universe could be far different. The expansion may not continue to accelerate, but instead may once again slow down over time so that distant galaxies will not disappear. Alternatively, perhaps there will be some new observable quantities we cannot yet detect that may provide astronomers in the future with evidence that there was once a Big Bang.\n\nNevertheless, based on everything we know about the universe today, the future I have sketched out is the most plausible one, and it is fascinating to consider whether logic, reason, and empirical data might still somehow induce future scientists to infer the correct underlying nature of our universe, or whether it will forever remain obscured behind the horizon. Some brilliant future scientist exploring the fundamental nature of forces and particles might derive a theoretical picture that will suggest that inflation must have happened, or that there must be an energy in empty space, which would further explain why there are no galaxies within the visible horizon. But I am not so sanguine about this.\n\nPhysics is, after all, an empirical science, driven by experiment and observation. Had we not observationally inferred the existence of dark energy, I doubt any theorist would have been bold enough to suggest its existence today. And while it is also possible to imagine tentative signatures that might suggest something is wrong with the picture of a single galaxy in a static universe without a Big Bang\u2014perhaps some observation of elemental abundances that appears anomalous\u2014I suspect that Occam's razor will suggest that the simplest picture is the correct one, and that the anomalous observations might be explained by some local effects.\n\nEver since Bob Scherrer and I laid out the challenge that future scientists will use falsifiable data and models\u2014the very paragon of good science\u2014but in the process that they will come up with a false picture of the universe, many of our colleagues have tried to suggest ways to probe that the universe is actually expanding in the far future. I too can imagine possible experiments. But I cannot see that they would be well motivated.\n\nFor example, you would need to eject bright stars from our galaxy and send them off into space, wait a billion years or so for them to explode, and try to observe their recession velocities as a function of the distance they reach before they explode in order to probe to see if they are getting any extra kick from a possible expansion of space. A tall order, but even if you could imagine somehow pulling this off, I cannot see the National Science Foundation of the future actually funding the experiment without at least some other motivation for arguing on behalf of an expanding universe. And if somehow stars from our galaxy are naturally ejected and detectable as they move out toward the horizon, it is not clear to me that observing an anomalous acceleration of some of these objects would be interpreted in terms of such a bold and strange proposal as an expanding universe dominated by dark energy.\n\nWe can consider ourselves lucky that we live at the present time. Or as Bob and I put it in one of the articles we wrote: \"We live at a very special time . . . the only time when we can observationally verify that we live at a very special time!\"\n\nWe were being somewhat facetious, but it is sobering to suggest that one can use the best observational tools and theoretical tools at one's disposal and nevertheless come up with a completely false picture of the large-scale universe.\n\nI should point out, nevertheless, that even though incomplete data _can_ lead to a false picture, this is far different from the (false) picture obtained by those who choose to ignore empirical data to invent a picture of creation that would otherwise contradict the evidence of reality (young earthers, for example), or those who instead require the existence of something for which there is no observable evidence whatsoever (like divine intelligence) to reconcile their view of creation with their a priori prejudices, or worse still, those who cling to fairy tales about nature that presume the answers before questions can even be asked. At least the scientists of the future will be basing their estimates on the best evidence available to them, recognizing as we all do, or at least as scientists do, that new evidence may cause us to change our underlying picture of reality.\n\nIn this regard, it is worth adding that perhaps we are missing something even today that might have been observable had only we lived 10 billion years ago or perhaps could see if we lived 100 billion years into the future. Nevertheless, I should stress that the Big Bang picture is too firmly grounded in data from every area to be proved invalid in its general features. But some new, nuanced understanding of the fine details of the distant past or distant future, or of the origin of the Big Bang and its possible uniqueness in space, might easily emerge with new data. In fact, I hope it will. One lesson that we can draw from the possible future end of life and intelligence in the universe is that we need to have some cosmic humility in our claims, even if such a thing is difficult for cosmologists.\n\nEither way, the scenario I have just described has a certain poetic symmetry, even if it is equally tragic. Long into the future, scientists will derive a picture of the universe that will hearken back to the very picture we had at the beginning of the last century, which itself ultimately served as the catalyst for investigations that led to the modern revolutions in cosmology. Cosmology will have come full circle. I for one find that remarkable, even if it underscores what some may view as the ultimate futility of our brief moment in the sun.\n\nRegardless, the fundamental problem illustrated by the possible future end of cosmology is that we have only one universe to test\u2014the one we live in. While test it we must if we want to have any hope of understanding how what we now observe arose, we nevertheless are limited in both what we can measure and in our interpretations of the data.\n\nIf many universes exist, and if we could somehow probe more than one, we might have a better chance of knowing which observations are truly significant and fundamental and which arise only as an accident of our circumstances.\n\nAs we shall see next, while the latter possibility is unlikely, the former is not, and scientists are pressing forward with new tests and new proposals to further our understanding of the unexpected and strange features of our universe.\n\nBefore proceeding, however, it is perhaps worth ending with another, more literary picture of the likely future I have presented here and one that is particularly relevant to the subject of this book. It comes from Christopher Hitchens's response to the scenario I have just described. As he put it, \"For those who find it remarkable that we live in a universe of Something, just wait. Nothingness is heading on a collision course right toward us!\"\n\n## CHAPTER 8\n\nA GRAND ACCIDENT?\n\n_Once you assume a creator and a plan, it makes humans objects in a cruel experiment whereby we are created to be sick and commanded to be well._\n\n\u2014CHRISTOPHER HITCHENS\n\nWe are hardwired to think that everything that happens to us is significant and meaningful. We have a dream that a friend is going to break her arm, and the next day we find out that she sprained her ankle. Wow! Cosmic! Clairvoyant?\n\nThe physicist Richard Feynman used to like to go up to people and say: \"You won't believe what happened to me today! You just won't believe it!\" And when they would inquire what happened, he would say, \"Absolutely nothing!\" By this he was suggesting that when something like the dream I described above happens, people ascribe significance to it. But they forget the myriad nonsense dreams they had that predicted absolutely nothing. By forgetting that most of the time nothing of note occurs during the day, we then misread the nature of probability when something unusual does occur: among any sufficiently large number of events, something unusual is bound to happen just by accident.\n\nHow does this apply to our universe?\n\nUntil the discovery that, inexplicably, the energy of empty space is not only not zero, but takes a value that is 120 orders of magnitude smaller than the estimate I described based on ideas from particle physics suggests, the conventional wisdom among physicists was that every fundamental parameter we measured in nature _is_ significant. By this I mean that, somehow, on the basis of fundamental principles, we would eventually be able to understand things such as why gravity is so much weaker than the other forces of nature, why the proton is 2,000 times heavier than the electron, and why there are three families of elementary particles. Put another way, once we understood the fundamental laws that govern the forces of nature at its smallest scales, all of these current mysteries would be revealed as natural consequences of these laws.\n\n(A purely religious argument, on the other hand, could take significance to an extreme by suggesting that each fundamental constant is significant because God presumably chose each one to have the value it does as part of a divine plan for our universe. In this case, nothing is an accident, but by the same token, nothing is predicted or actually explained. It is an argument by fiat that goes nowhere and yields nothing useful about the physical laws governing the universe, other than perhaps providing consolation for the believer.)\n\nBut the discovery that empty space has energy started a revision in thinking among many physicists about what is required in nature and what may be accidental.\n\nThe catalyst for this new gestalt originates from the argument I gave in the last chapter: dark energy is measurable today because \"now\" is the only time in the history of the universe when the energy in empty space is comparable to the energy density in matter.\n\nWhy should we be living at such a \"special\" time in the history of the universe? Indeed, this flies in the face of everything that has characterized science since Copernicus. We have learned that the Earth is not the center of the solar system and that the Sun is a star on the lonely outer edges of a galaxy that is merely one out of 400 billion galaxies in the observable universe. We have come to accept the \"Copernican principle\" that there is nothing special about our place and time in the universe.\n\nBut with the energy of empty space being what it is, we _do_ appear to live at a special time. This is shown best by the following illustration of a \"brief history of time.\"\n\nThe two curves represent the energy density of all matter in the universe, and the energy density of empty space (presuming it is a cosmological constant) as a function of time. As you can see, the density of matter falls, as the universe expands (as the distance between galaxies becomes ever greater and matter therefore gets \"diluted\"), just as you would expect. However, the energy density in empty space remains constant, because, one might argue, with empty space there is nothing to dilute! (Or, as I have somewhat less facetiously described, the universe does work on empty space as it expands.) The two curves cross relatively close to the present time, which is the source of the strange coincidence I have described.\n\nNow consider what would happen if the energy in empty space were, say, 50 times greater than the value we estimate today. Then the two curves would cross at a different, earlier time, as shown in the figure below.\n\nThe time that the two curves cross for the upper, enlarged value of the energy of empty space is the time when galaxies first formed, about a billion years after the Big Bang. But remember that the energy of empty space is gravitationally repulsive. If it had come to dominate the energy of the universe before the time of galaxy formation, the repulsive force due to this energy would have outweighed (literally) the normal attractive gravitational force that caused matter to clump together. And galaxies would never have formed!\n\nBut if galaxies hadn't formed, then stars wouldn't have formed. And if stars hadn't formed, planets wouldn't have formed. And if planets hadn't formed, then astronomers wouldn't have formed!\n\nSo, in a universe with an energy of empty space merely 50 times bigger than that we observe, apparently no one would have been around today to try to measure the energy.\n\nCould this be telling us something? Shortly after the discovery of our accelerating universe, physicist Steven Weinberg proposed, based on an argument he had developed more than a decade earlier\u2014before the discovery of dark energy\u2014that the \"Coincidence Problem\" could therefore be solved if perhaps the value of the cosmological constant that we measure today were somehow \"anthropically\" selected. That is, if somehow there were many universes, and in each universe the value of the energy of empty space took a randomly chosen value based on some probability distribution among all possible energies, then only in those universes in which the value is not that different from what we measure would life as we know it be able to evolve. So maybe we find ourselves in a universe with a tiny energy in empty space because we couldn't find ourselves in one with a much larger value. Put another way, it is not too surprising to find that we live in a universe in which we can live!\n\nThis argument, however, makes mathematical sense only if there is a possibility that many different universes have arisen. Talking about many different universes can sound like an oxymoron. After all, traditionally the notion of universe has become synonymous with \"everything that exists.\"\n\nMore recently, however, _universe_ has come to have a simpler, arguably more sensible meaning. It is now traditional to think of \"our\" universe as comprising simply the totality of all that we can now see and all that we could ever see. Physically, therefore, our universe comprises everything that either once could have had an impact upon us or that ever will.\n\nThe minute one chooses this definition for a universe, the possibility of other \"universes\"\u2014regions that have always been and always will be causally disconnected from ours, like islands separated from any communication with one another by an ocean of space\u2014becomes possible, at least in principle.\n\nOur universe is so vast that, as I have emphasized, something that is not impossible is virtually guaranteed to occur somewhere within it. Rare events happen all the time. You might wonder whether the same principle applies to the possibility of many universes, or a _multiverse,_ as the idea is now known. It turns out that the theoretical situation is actually stronger than simply a possibility. A number of central ideas that drive much of the current activity in particle theory today appear to require a multiverse.\n\nI want to stress this because, in discussions with those who feel the need for a creator, the existence of a multiverse is viewed as a cop-out conceived by physicists who have run out of answers\u2014or perhaps questions. This may eventually be the case, but it is not so now. Almost every logical possibility we can imagine regarding extending laws of physics as we know them, on small scales, into a more complete theory, suggests that, on large scales, our universe is not unique.\n\nThe phenomenon of inflation provides perhaps the first, and perhaps best, rationale. In the inflationary picture, during the phase when a huge energy temporarily dominates some region of the universe, this region begins to expand exponentially. At some point, a small region within this \"false vacuum\" may exit inflation as a phase transition occurs within the region and the field within it relaxes to its true, lower energy value; the expansion within this region will then cease to be exponential. But the space _between_ such regions will continue to expand exponentially. At any one time, unless the phase transition completes through all of space, then almost all of space lies within an inflating region. And the inflating region will separate those regions that first exit inflation by almost unfathomable distances. It is like lava pouring out of a volcano. Some of the rock will cool and solidify, but those rocks will be carried far apart from one another as they float on a sea of liquid magma.\n\nThe situation can be even more dramatic. In 1986, Andrei Linde, who along with Alan Guth has been one of the chief architects of modern inflationary theory, promoted and explored a possibly even more general scenario. This was also anticipated in some sense by another inventive Russian cosmologist in the United States, Alex Vilenkin. Both Linde and Vilenkin have the inner confidence that one finds in great Russian physicists, but their history is quite different. Linde thrived in the old Soviet physics establishment before immigrating to the United States after the fall of the Soviet Union. Brash, brilliant, and funny, he has continued to dominate much of theoretical particle cosmology in the interim. Vilenkin emigrated far earlier, before he was a physicist, and worked in the United States in various jobs, including as a night watchman, while he studied. And while he was always interested in cosmology, he accidentally applied to the wrong school for graduate work and ended up doing a thesis in condensed matter physics\u2014the physics of materials. He then got a job as a postdoctoral researcher at Case Western Reserve University, where I later became Chair. During this period, he asked his supervisor, Philip Taylor, if he could spend several days a week working on cosmology in addition to his assigned projects. Philip later told me that, even with this part-time labor, Alex was the most productive postdoc he had ever had.\n\nIn any case, what Linde recognized is that, while quantum fluctuations during inflation may often push the field that drives inflation toward its lowest energy state, and thus provide a graceful exit, there is always the possibility that, in some regions, quantum fluctuations will drive the field toward yet higher energies, and hence away from values where inflation will end, so that inflation will continue unabated. Because such regions will expand for longer periods of time, there will be far more space that is inflating than that which is not. Within these regions, quantum fluctuations again will drive some subregions to exit inflation and thus stop expanding exponentially, but again there will be regions where quantum fluctuations will cause inflation to persist even longer. And so on.\n\nThis picture, which Linde dubbed \"chaotic inflation,\" indeed resembles more familiar chaotic systems on Earth. Take boiling oatmeal, for example. At any point a bubble of gas may burst from the surface, reflecting regions where liquid at high temperature completes a phase transition to form a vapor. But between the bubbles the oatmeal is roiling and flowing. On large scales there is regularity\u2014there are always bubbles popping somewhere. But locally, things are quite different depending upon where one looks. So it would be in a chaotically inflating universe. If one happened to be located in a \"bubble\" of true ground state that had stopped inflating, one's universe would appear very different from the vast bulk of space around it, which would still be inflating.\n\nIn this picture, inflation is eternal. Some regions, indeed most of space, will go on inflating forever. Those regions that exit inflation will become separate, causally disconnected universes. I want to stress that a multiverse is _inevitable_ if inflation is eternal, and eternal inflation is by far the most likely possibility in most, if not all, inflationary scenarios. As Linde put it in his 1986 paper:\n\nThe old question why our universe is the only possible one is now replaced by the question in which theories [of] the existence of mini-universes of our type [are] possible. This question is still very difficult, but it is much easier than the previous one. In our opinion, the modification of the point of view on the global structure of the universe and on our place in the world is one of the most important consequences of the development of the inflationary universe scenario.\n\nAs Linde emphasized, and has since become clear, this picture also provides another new possibility for physics. It could easily be that there are many possible low-energy quantum states of the universe present in nature that an inflating universe might ultimately decay into. Because the configuration of the quantum states of these fields will be different in each such region, the character of the fundamental laws of physics in each region\/universe can then appear different.\n\nHere arose the first \"landscape\" in which the anthropic argument, provided earlier, could play itself out. If there are many different states in which our universe could end up in after inflation, perhaps the one we live in, one in which there is non-zero vacuum energy that is small enough so galaxies could form, is just one of a potentially infinite family and the one that is selected for inquisitive scientists because it supports galaxies, stars, planets, and life.\n\nThe term \"landscape\" did not, however, first arise in this context. It was promoted by a much more effective marketing machine associated with the juggernaut that has been driving particle theory for much of the past quarter century\u2014string theory. String theory posits that elementary particles are made up of more fundamental constituents, not particles, but objects that behave like vibrating strings. Just as string vibrations on a violin can create different notes, so too in this theory different sorts of vibrations produce objects that might, in principle, behave like all the different elementary particles we find in nature. The catch, however, is that the theory is not mathematically consistent when defined in merely four dimensions, but appears to require many more to make sense. What happens to the other dimensions is not immediately obvious, nor is the issue of what other objects besides strings may be important to define the theory\u2014just some of the many unsolved challenges that have presented themselves and dulled some of the early enthusiasm for this idea.\n\nHere is not the place to thoroughly review string theory, and in fact a thorough review is probably not possible, because if one thing has become clear in the past twenty-five years, it is that what was formerly called string theory is clearly something much more elaborate and complicated, and something whose fundamental nature and makeup is still a mystery.\n\nWe still have no idea if this remarkable theoretical edifice actually has anything to do with the real world. Nevertheless, perhaps no theoretical picture has ever so successfully permeated the consciousness of the physics community without having yet demonstrated its ability to successfully resolve a single experimental mystery about nature.\n\nMany people will take the last sentence as a criticism of string theory, but although I have been branded in the past as a detractor, that is not really my intent here, nor has it been my intent in the numerous lectures and well-intentioned public debates I have had with my friend Brian Greene, one of string theory's main proponents, on the subject. Rather, I think it is simply important to cut through the popular hype for a reality check. String theory involves fascinating ideas and mathematics that might shed light on one of the most fundamental inconsistencies in theoretical physics\u2014our inability to cast Einstein's general relativity in a form that can be combined with the laws of quantum mechanics to result in sensible predictions about how the universe behaves on its very smallest scales.\n\nI have written a whole book about how string theory has attempted to circumvent this problem, but for our purposes here, only a very brief summary is necessary. The central proposal is simple to state, if difficult to implement. On very small scales, appropriate to the scale where the problems between gravity and quantum mechanics might first be encountered, elementary strings may curl up into closed loops. Amidst the set of excitations of such closed loops there always exists one such excitation that has the properties of the particle that, in quantum theory, conveys the force of gravity\u2014the graviton. Thus, the quantum theory of such strings provides, in principle, the playing field on which a true quantum theory of gravity might be built.\n\nSure enough, it was discovered that such a theory might avoid the embarrassing infinite predictions of the standard quantum approaches to gravity. There was one hitch, however. In the simplest version of the theory, such infinite predictions can be obviated only if the strings that make up elementary particles are vibrating, not merely in the three dimensions of space and one of time that we are all familiar with, but rather in twenty-six dimensions!\n\nYou might expect that such a leap of complexity (and, perhaps, faith) would be enough to turn off most physicists about the theory, but in the mid-1980s some beautiful mathematical work by a host of individuals, most notably Edward Witten at the Institute for Advanced Study, demonstrated that the theory could in principle do far more than just provide a quantum theory of gravity. By introducing new mathematical symmetries, most notably a remarkably powerful mathematical framework called \"supersymmetry,\" it became possible to reduce the number of dimensions required for consistency of the theory from twenty-six to merely ten.\n\nMore important, however, it looked like it might be possible, within the context of string theory, to unify gravity with the other forces in nature in a single theory, and moreover possible to explain the existence of every single elementary particle known in nature! Finally, it appeared as if there might be a single unique theory in ten dimensions that would reproduce everything we see in our four-dimensional world.\n\nClaims of a \"Theory of Everything\" began to propagate, not just in the scientific literature, but in popular literature as well. As a result, perhaps more people are familiar with \"superstrings\" than are familiar with \"superconductivity\"\u2014the latter being the remarkable fact that when some materials are cooled to extremely low temperatures, they can conduct electricity without any resistance whatsoever. This is not only one of the most remarkable properties of matter ever observed, but it has already transformed our understanding of the quantum makeup of materials.\n\nAlas, the intervening twenty-five years or so have not been kind to string theory. Even as the best theoretical minds in the world began to focus their attention on it, producing volumes of new results and a great deal of new mathematics in the process (Witten went on to win the highest prize in mathematics, for example), it became clear that the \"strings\" in string theory are probably not the fundamental objects at all. Other, more complicated structures, called \"branes,\" named after membranes in cells, which exist in higher dimensions, probably control the behavior of the theory.\n\nWhat is worse, the uniqueness of the theory began to disappear. After all, the world of our experience is not ten-dimensional, but rather four-dimensional. Something has to happen to the remaining six spatial dimensions, and the canonical explanation of their invisibility is that they are somehow \"compactified\"\u2014that is, they are curled up on such small scales that we cannot resolve them on our scales or even on the tiny scales that are probed by our highest energy particle accelerators today.\n\nThere is a difference between these proposed hidden domains and the domains of spirituality and religion, even though they may not appear so different on the surface. In the first place, they are accessible in principle if one could build a sufficiently energetic accelerator\u2014beyond the bounds of practicality perhaps, but not beyond the bounds of possibility. Second, one might hope, as one does for virtual particles, to find some indirect evidence of their existence via the objects we can measure in our four-dimensional universe. In short, because these dimensions were proposed as part of a theory developed to actually attempt to explain the universe, rather than justify it, they might ultimately be accessible to empirical testing, even if the likelihood is small.\n\nBut beyond this, the possible existence of these extra dimensions provides a huge challenge to the hope that our universe is unique. Even if one starts with a unique theory in ten dimensions (which, I repeat, we do not yet know exist), then every different way of compactifying the invisible six dimensions can result in a different type of four-dimensional universe, with different laws of physics, different forces, different particles, and governed by differing symmetries. Some theorists have estimated that there are perhaps 10500 different possible consistent four-dimensional universes that could result from a single ten-dimensional string theory. A \"Theory of Everything\" had suddenly become a \"Theory of Anything\"!\n\nThis situation was exemplified sarcastically in a cartoon from one of my favorite scientific comic strips, called _xkcd_. In this strip one person says to another: \"I just had an awesome idea. What if all matter and energy is made of tiny vibrating strings.\" The second person then says, \"Okay. What would that imply?\" To which the first person responds: \"I dunno.\"\n\nOn a slightly less facetious note, the Nobel Prize\u2013winning physicist Frank Wilczek has suggested that string theorists have invented a new way of doing physics, reminiscent of a novel way of playing darts. First, one throws the dart against a blank wall, and then one goes to the wall and draws a bull's-eye around where the dart landed.\n\nWhile Frank's comment is an accurate reflection of much of the hype that has been generated, it should be stressed that at the same time those working on the theory are honestly trying to uncover principles that might govern the world in which we live. Nevertheless, the plethora of possible four-dimensional universes, which used to be such an embarrassment for string theorists, has now become a virtue of the theory. One can imagine that, in a ten-dimensional \"multiverse\" one can embed a host of different four-dimensional universes (or five-dimensional ones, or six-dimensional ones, or so on . . .), and each one can have different laws of physics, and moreover, in each one the energy of empty space can be different.\n\nWhile it sounds like a convenient fabrication, it appears to be an automatic consequence of the theory, and it does create a true multiverse \"landscape\" that might provide a natural framework for developing an anthropic understanding of the energy of empty space. In this case, we do not need an infinite number of possible universes separated in three-dimensional space. Rather, we can imagine an infinite number of universes stacked up above a single point in our space, invisible to us, but each of which could exhibit remarkably different properties.\n\nI want to emphasize that this theory is not as trivial as the theological musing of Saint Thomas Aquinas about whether several angels could occupy the same place, an idea that was derided by later theologians as fruitless speculations on how many angels could fit on the point of a needle\u2014or most popularly, on the head of a pin. Aquinas actually answered this question himself by saying that more than one angel could not occupy the same space\u2014of course, without any theoretical or experimental justification! (And if they were bosonic quantum angels, he would have been wrong in any case.)\n\nPresented with such a picture, and adequate mathematics, one might hope, in principle, to actually make physical predictions. For example, one might derive a probability distribution describing the likelihood of finding different types of four-dimensional universes embedded in a larger dimensional multiverse. One might find, for example, that the bulk of such universes that have small vacuum energy also have three families of elementary particles and four different forces. Or one might find that only in universes with small vacuum energy could there exist a long-range force of electromagnetism. Any such result might provide reasonably compelling evidence that a probabilistic anthropic explanation of the energy of empty space\u2014in other words, finding that a universe that looks like ours with small vacuum energy is not improbable\u2014makes solid physical sense.\n\nYet the mathematics has not yet brought us this far, and it may never do so. But in spite of our current theoretical impotence, this does not mean that this possibility is not actually realized by nature.\n\nNevertheless, in the meantime, particle physics has taken anthropic reasoning a step further.\n\nParticle physicists are way ahead of cosmologists. Cosmology has produced one totally mysterious quantity: the energy of empty space, about which we understand virtually nothing. However, particle physics has not understood many more quantities for far longer!\n\nFor example: Why are there three generations of elementary particles\u2014the electron, and its heavier cousins the muon and tauon, for example, or the three different sets of quarks, of which the lowest energy set makes up the bulk of matter we find on Earth? Why is gravity so much weaker than the other forces in nature, such as electromagnetism? Why is the proton 2,000 times heavier than the electron?\n\nSome particle physicists have now jumped on the anthropic bandwagon in the extreme, perhaps because their efforts to explain these mysteries according to physical causes have not yet been successful. After all, if one fundamental quantity in nature is actually an environmental accident, why aren't most or all of the other fundamental parameters? Maybe all of the mysteries of particle theory can be solved by invoking the same mantra: if the universe were any other way, we could not live in it.\n\nOne might wonder if such a solution of the mysteries of nature is any solution at all or, more important, whether it describes science as we understand it. After all, the goal of science, and in particular physics, over the past 450 years has been to explain why the universe must be the way we measure it to be, rather than why in general the laws of nature would produce universes that are quite different.\n\nI have tried to explain why this is not quite the case, namely why many respectable scientists have turned to the anthropic principle and why a number have worked quite hard to see if we might learn something new about our universe based on it.\n\nLet me now go further and try to explain how the existence of forever undetectable universes\u2014either removed from us by virtually infinite distances in space or, right beyond the tip of our noses, removed from us by microscopic distances in possible extra dimensions\u2014might nevertheless be subject to some kind of empirical testing.\n\nImagine, for example, that we devised a theory based on unifying at least three of the four forces of nature in some Grand Unified Theory, a subject of continued intense interest in particle physics (among those who have not given up looking for fundamental theories in four dimensions). Such a theory would make predictions about the forces of nature that we measure and about the spectrum of elementary particles that we probe at our accelerators. Should such a theory make a host of predictions that are subsequently verified in our experiments, we would have very good reason to suspect that it contains a germ of truth.\n\nNow, suppose this theory also predicts a period of inflation in the early universe, and in fact predicts that our inflationary epoch is merely one of a host of such episodes in an eternally inflating multiverse. Even if we could not explore the existences of such regions beyond our horizon directly, then, as I have said earlier, if it walks like a duck and quacks like a duck . . . Well, you know.\n\nFinding possible empirical support for the ideas surrounding extra dimensions is more far-fetched but not impossible. Many bright young theorists are devoting their professional careers to the hope of developing the theory to the point where there might be some evidence, even indirect, that it is correct. Their hopes might be misplaced, but they have voted with their feet. Perhaps some evidence from the new Large Hadron Collider near Geneva will reveal some otherwise hidden window into this new physics.\n\nSo, after a century of remarkable, truly unprecedented progress in our understanding of nature, we have found ourselves able to probe the universe on scales that were previously unimaginable. We have understood the nature of the Big Bang expansion back to its earliest microseconds and have discovered the existence of hundreds of billions of new galaxies, with hundreds of billions of new stars. We have discovered that 99 percent of the universe is actually invisible to us, comprising dark matter that is most likely some new form of elementary particle, and even more dark energy, whose origin remains a complete mystery at the present time.\n\nAnd after all of this, it may be that physics will become an \"environmental science.\" The fundamental constants of nature, so long assumed to take on special importance, may just be environmental accidents. If we scientists tend to take ourselves and our science too seriously, maybe we also have taken our universe too seriously. Maybe literally, as well as metaphorically, we are making much ado about nothing. At least we may be making too much of the nothing that dominates our universe! Maybe our universe is rather like a tear buried in a vast multiversal ocean of possibilities. Maybe we will never find a theory that describes why the universe has to be the way it is.\n\nOr maybe we will.\n\nThat, finally, is the most accurate picture I can paint of reality as we now understand it. It is based on the work of tens of thousands of dedicated minds over the past century, building some of the most complex machines ever devised and developing some of the most beautiful and also the most complex ideas with which humanity has ever had to grapple. It is a picture whose creation emphasizes the best about what it is to be human\u2014our ability to imagine the vast possibilities of existence and the adventurousness to bravely explore them\u2014without passing the buck to a vague creative force or to a creator who is, by definition, forever unfathomable. We owe it to ourselves to draw wisdom from this experience. To do otherwise would do a disservice to all the brilliant and brave individuals who helped us reach our current state of knowledge.\n\nIf we wish to draw philosophical conclusions about our own existence, our significance, and the significance of the universe itself, our conclusions should be based on empirical knowledge. A truly open mind means forcing our imaginations to conform to the evidence of reality, and not vice versa, whether or not we like the implications.\n\n## CHAPTER 9\n\nNOTHING IS SOMETHING\n\n_I don't mind not knowing. It doesn't scare me._\n\n\u2014RICHARD FEYNMAN\n\nIsaac Newton, perhaps the greatest physicist of all time, profoundly changed the way we think about the universe in many ways. But perhaps the most important contribution he made was to demonstrate the possibility that the entire universe is explicable. With his universal law of gravity, he demonstrated for the first time that even the heavens might bend to the power of natural laws. A strange, hostile, menacing, and seemingly capricious universe might be nothing of the sort.\n\nIf immutable laws governed the universe, the mythical gods of ancient Greece and Rome would have been impotent. There would have been no freedom to arbitrarily bend the world to create thorny problems for mankind. What held for Zeus would also apply to the God of Israel. How could the Sun stand still at midday if the Sun did not orbit the Earth but its motion in the sky was actually caused by the revolution of the Earth, which, if suddenly stopped, would produce forces on its surface that would destroy all human structures and humans along with them?\n\nOf course, supernatural acts are what miracles are all about. They are, after all, precisely those things that circumvent the laws of nature. A god who can create the laws of nature can presumably also circumvent them at will. Although why they would have been circumvented so liberally thousands of years ago, before the invention of modern communication instruments that could have recorded them, and not today, is still something to wonder about.\n\nIn any case, even in a universe with no miracles, when you are faced with a profoundly simple underlying order, you can draw two different conclusions. One, drawn by Newton himself, and earlier espoused by Galileo and a host of other scientists over the years, was that such order was created by a divine intelligence responsible not only for the universe, but also for our own existence, and that we human beings were created in her image (and apparently other complex and beautiful beings were not!). The other conclusion is that the laws themselves are all that exist. These laws themselves require our universe to come into existence, to develop and evolve, and we are an irrevocable by-product of these laws. The laws may be eternal, or they too may have come into existence, again by some yet unknown but possibly purely physical process.\n\nPhilosophers, theologians, and sometimes scientists continue to debate these possibilities. We do not know for certain which of them actually describes our universe, and perhaps we shall never know. But the point is, as I emphasized at the very beginning of this book, the final arbiter of this question will not come from hope, desire, revelation, or pure thought. It will come, if it ever does, from an exploration of nature. Dream or nightmare, as Jacob Bronowski said in the opening quote in the book\u2014and one person's dream in this case can easily be another's nightmare\u2014we need to live our experience as it is and with our eyes open. The universe is the way it is, whether we like it or not.\n\nAnd here, I think it is _extremely significant_ that a universe from nothing\u2014in a sense I will take pains to describe\u2014that arises naturally, and even inevitably, is increasingly consistent with everything we have learned about the world. This learning has _not_ come from philosophical or theological musings about morality or other speculations about the human condition. It is instead based on the remarkable and exciting developments in empirical cosmology and particle physics that I have described.\n\nI want thus to return to the question I described at the beginning of this book: Why is there something rather than nothing? We are now presumably in a better position to address this, having reviewed the modern scientific picture of the universe, its history, and its possible future, as well as operational descriptions of what \"nothing\" might actually comprise. As I also alluded to at the beginning of this book, this question too has been informed by science, like essentially all such philosophical questions. Far from providing a framework that forces upon us the requirement of a creator, the very meaning of the words involved have so changed that the sentence has lost much of its original meaning\u2014something that again is not uncommon, as empirical knowledge shines a new light on otherwise dark corners of our imagination.\n\nAt the same time, in science we have to be particularly cautious about \"why\" questions. When we ask, \"Why?\" we usually mean \"How?\" If we can answer the latter, that generally suffices for our purposes. For example, we might ask: \"Why is the Earth 93 million miles from the Sun?\" but what we really probably mean is, \"How is the Earth 93 million miles from the Sun?\" That is, we are interested in what physical processes led to the Earth ending up in its present position. \"Why\" implicitly suggests purpose, and when we try to understand the solar system in scientific terms, we do not generally ascribe purpose to it.\n\nSo I am going to assume what this question really means to ask is, \"How is there something rather than nothing?\" \"How\" questions are really the only ones we can provide definitive answers to by studying nature, but because this sentence sounds much stranger to the ear, I hope you will forgive me if I sometimes fall into the trap of appearing to discuss the more standard formulation when I am really trying to respond to the more specific \"how\" question.\n\nEven here, from the perspective of actual _understanding,_ this particular \"how\" question has been supplanted by a host of operationally more fruitful questions, such as, \"What might have produced the properties of the universe that most strikingly characterize it at the present time?\" or, perhaps more important, \"How can we find out?\"\n\nHere I want to once again beat what I wish were a dead horse. Framing questions in this way allows the production of new knowledge and understanding. This is what differentiates them from purely theological questions, which generally presume the answers up front. Indeed, I have challenged several theologians to provide evidence contradicting the premise that theology has made no contribution to knowledge in the past five hundred years at least, since the dawn of science. So far no one has provided a counterexample. The most I have ever gotten back was the query, \"What do you mean by knowledge?\" From an epistemological perspective this may be a thorny issue, but I maintain that, if there were a better alternative, someone would have presented it. Had I presented the same challenge to biologists, or psychologists, or historians, or astronomers, none of them would have been so flummoxed.\n\nThe answers to these sorts of fruitful questions involve theoretical predictions that can be tested via experiments to drive our operational knowledge of the universe forward more directly. Partly for this reason, I have focused on such fruitful questions up to this point in this book. Nevertheless, the \"something from nothing\" question continues to have great currency, and therefore probably needs to be confronted.\n\nNewton's work dramatically reduced the possible domain of God's actions, whether or not you attribute any inherent rationality to the universe. Not only did Newton's laws severely constrain the freedom of action of a deity, they dispensed with various requirements for supernatural intervention. Newton discovered that the motion of planets around the Sun does not require them to be continually pushed along their paths, but rather, and highly nonintuitively, requires them to be pulled by a force acting toward the Sun, thus dispensing of the need for the angels who were often previously invoked as guiding the planets on their way. While dispensing with this particular use of angels has had little impact on people's willingness to believe in them (polls suggest far more people believe in angels in the United States than believe in evolution), it is fair to say that progress in science since Newton has even more severely constrained the available opportunities for the hand of God to be manifest in his implied handiwork.\n\nWe can describe the evolution of the universe back to the earliest moments of the Big Bang without specific need for anything beyond known physical laws, and we have also described the universe's likely future history. There are certainly still puzzles about the universe that we don't understand, but I am going to assume that readers of this book are not wedded to a \"God of the Gaps\" picture, whereby God is invoked whenever there is something specific about our observations that seems puzzling or not fully understood. Even theologians recognize that such recourse not only diminishes the grandeur of their supreme being, but it also opens that being up to being removed or further marginalized whenever new work explains or removes the puzzle.\n\nIn this sense, the \"something from nothing\" argument really tries to focus on the original act of creation and asks whether a scientific explanation can ever be logically complete and fully satisfying in addressing this specific issue.\n\nIt turns out that, given our current understanding of nature, there are three different, separate meanings for the \"something from nothing\" question. The short answer to each is \"quite plausibly yes,\" and I shall discuss each in turn in the rest of this book as I attempt to explain why or, as I have argued just now, better yet how.\n\nOccam's razor suggests that, if some event is physically plausible, we don't need recourse to more extraordinary claims for its being. Surely the requirement of an all-powerful deity who somehow exists outside of our universe, or multiverse, while at the same time governing what goes on inside it, is one such claim. It should thus be a claim of last, rather than first, resort.\n\nI have already argued in the preface to this book that merely defining \"nothingness\" as \"nonbeing\" is not sufficient to suggest that physics, and more generally science, is not adequate to address the question. Let me give an additional, more specific argument here. Consider an electron-positron pair that spontaneously pops out of empty space near the nucleus of an atom and affects the property of that atom for the short time the pair exists. In what sense did the electron or positron exist before? Surely by any sensible definition they didn't. There was potential for their existence, certainly, but that doesn't define _being_ any more than a potential human being exists because I carry sperm in my testicles near a woman who is ovulating, and she and I might mate. Indeed, the best answer I have ever heard to the question of what it would be like to be dead (i.e., be nonbeing) is to imagine how it felt to be before you were conceived. In any case, if potential to exist were the same as existence, then I am certain that by now masturbation would be as hot button a legal issue as abortion now is.\n\nThe Origins Project at Arizona State University, which I direct, recently ran a workshop on the Origin of Life, and I cannot help but view the present cosmological debate in this context. We do not yet fully understand how life originated on Earth. However, we have not only plausible chemical mechanisms by which this might be conceivable, but we are also homing in closer and closer every day to specific pathways that might have allowed biomolecules, including RNA, to arise naturally. Moreover, Darwinian evolution, based on natural selection, provides a compellingly accurate picture of how complex life emerged on this planet following whatever specific chemistry produced the first faithfully self-replicating cells with a metabolism that captured energy from their environment. (As good a definition of life as I can come up with for the moment.)\n\nJust as Darwin, albeit reluctantly, removed the need for divine intervention in the evolution of the modern world, teeming with diverse life throughout the planet (though he left the door open to the possibility that God helped breathe life into the first forms), our current understanding of the universe, its past, and its future make it more plausible that \"something\" can arise out of nothing without the need for any divine guidance. Because of the observational and related theoretical difficulties associated with working out the details, I expect we may never achieve more than plausibility in this regard. But plausibility itself, in my view, is a tremendous step forward as we continue to marshal the courage to live meaningful lives in a universe that likely came into existence, and may fade out of existence, without purpose, and certainly without us at its center.\n\nLet's now return to one of the most remarkable features of our universe: it is as close to being flat as we can measure. I remind you of the unique facet of a flat universe, at least on scales where it is dominated by matter in the form of galaxies, and where a Newtonian approximation remains valid: in a flat universe, and only in a flat universe, the average Newtonian gravitational energy of every object participating in the expansion is precisely zero.\n\nI emphasize that this was a falsifiable postulate. It didn't have to be this way. Nothing required this except theoretical speculations based on considerations of a universe that could have arisen naturally from nothing, or at the very least, from _almost nothing_.\n\nI cannot overstress the importance of the fact that, once gravity is included in our considerations of nature, one is no longer free to define the total energy of a system arbitrarily, nor the fact that there are both positive and negative contributions to this energy. Determining the total gravitational energy of objects being carried along by the expansion of the universe is _not_ subject to arbitrary definition any more than the geometric curvature of the universe is a matter of definition. It is a property of space itself, according to general relativity, and this property of space is determined by the energy contained within it.\n\nI say this because it has been argued that the statement that the average total Newtonian gravitational energy of every galaxy in a flat, expanding universe is zero is arbitrary, and that any other value would be just as good, but that scientists \"define\" the zero point to argue against God. So claimed Dinesh D'Souza, anyway, in his debates with Christopher Hitchens on the existence of God.\n\nNothing could be further from the truth. The effort to determine the curvature of the universe was an undertaking carried out over half a century by scientists who devoted their lives to determining the actual nature of the universe, not to imposing their own desires upon it. Even well after the theoretical arguments about why the universe should be flat were first proposed, my observational colleagues, during the 1980s and even early 1990s, remained bent on proving otherwise. For, after all, in science one achieves the greatest impact (and often the greatest headlines) not by going along with the herd, but by bucking against it.\n\nNevertheless, the data have had the last word, and the last word is in. Our observable universe is as close to being flat as we can measure. The Newtonian gravitational energy of galaxies moving along with the Hubble expansion _is_ zero\u2014like it or not.\n\nI would now like to describe how, if our universe arose from nothing, a flat universe, one with zero total Newtonian gravitational energy of every object, is precisely what we should expect. The argument is a little subtle\u2014subtler than I have been able to describe in my popular lectures on the subject\u2014so I am happy to have the space here to carefully try to lay it out.\n\nFirst, I want to be clear about what kind of \"nothing\" I am discussing at the moment. This is the simplest version of nothing, namely empty space. For the moment, I will assume space exists, with nothing at all in it, and that the laws of physics also exist. Once again, I realize that in the revised versions of nothingness that those who wish to continually redefine the word so that no scientific definition is practical, this version of nothing doesn't cut the mustard. However, I suspect that, at the times of Plato and Aquinas, when they pondered why there was something rather than nothing, empty space with nothing in it was probably a good approximation of what they were thinking about.\n\nAs we saw in chapter 6, Alan Guth has explained precisely how we can get something from this kind of nothing\u2014the ultimate free lunch. Empty space can have a non-zero energy associated with it, even in the absence of any matter or radiation. General relativity tells us that space will expand exponentially, so that even the tiniest region at early times could quickly encompass a size more than large enough to contain our whole visible universe today.\n\nAs I also described in that chapter, during such a rapid expansion, the region that will eventually encompass our universe will get flatter and flatter even as the energy contained within empty space grows as the universe grows. This phenomenon happens without the need for any hocus pocus or miraculous intervention. This is possible because the gravitational \"pressure\" associated with such energy in empty space is actually negative. This \"negative pressure\" implies that, as the universe expands, the expansion dumps energy _into_ space rather than vice versa.\n\nAccording to this picture, when inflation ends, the energy stored in empty space gets turned into an energy of real particles and radiation, creating effectively the traceable beginning of our present Big Bang expansion. I say the traceable beginning because inflation effectively erases any memory of the state of the universe before it began. All complexities and irregularities on initially large scales (if the initial preexisting universe or metaverse were large, even infinitely large) get smoothed out and\/or driven so far outside our horizon today that we will always observe an almost uniform universe after enough inflationary expansion has taken place.\n\nI say almost uniform because I also described in chapter 6 how quantum mechanics will always leave some residual, small-density fluctuations that get frozen during inflation. This results in the second amazing implication of inflation, that small-density fluctuations in empty space due to the rules of quantum mechanics will later be responsible for all the structure we observe in the universe today. So we, and everything we see, result out of quantum fluctuations in what is essentially nothingness near the beginning of time, namely during the inflationary expansion.\n\nAfter all the dust is settled, the generic configuration of the matter and radiation will be that of an essentially flat universe, one in which the average Newtonian gravitational energy of all objects will appear to be zero. This will almost always be the case, unless one could very carefully fine-tune the amount of inflation.\n\nTherefore, our observable universe can start out as a microscopically small region of space, which can be essentially empty, and still grow to enormous scales containing eventually lots of matter and radiation, all without costing a drop of energy, with enough matter and radiation to account for everything we see today!\n\nThe important point worth stressing in this brief summary of the inflationary dynamics discussed in chapter 6 is that something can arise from empty space _precisely_ because the energetics of empty space, in the presence of gravity, are _not_ what common sense would have guided us to suspect before we discovered the underlying laws of nature.\n\nBut no one ever said that the universe is guided by what we, in our petty myopic corners of space and time, might have originally thought was sensible. It certainly seems sensible to imagine that a priori, matter cannot spontaneously arise from empty space, so that _something,_ in this sense, cannot arise from _nothing_. But when we allow for the dynamics of gravity and quantum mechanics, we find that this commonsense notion is no longer true. This is the _beauty_ of science, and it should not be threatening. Science simply forces us to revise what is sensible to accommodate the universe, rather than vice versa.\n\nTo summarize then: the observation that the universe is flat and that the local Newtonian gravitational energy is essentially zero today is strongly suggestive that our universe arose though a process like that of inflation, a process whereby the energy of empty space (nothing) gets converted into the energy of something, during a time when the universe is driven closer and closer to being essentially exactly flat on all observable scales.\n\nWhile inflation demonstrates how empty space endowed with energy can effectively create everything we see, along with an unbelievably large and flat universe, it would be disingenuous to suggest that empty space endowed with energy, which drives inflation, is really _nothing_. In this picture one must assume that space exists and can store energy, and one uses the laws of physics like general relativity to calculate the consequences. So if we stopped here, one might be justified in claiming that modern science is a long way from really addressing how to get something from nothing. This is just the first step, however. As we expand our understanding, we will next see that inflation can represent simply the tip of a cosmic iceberg of nothingness.\n\n## CHAPTER 10\n\nNOTHING IS UNSTABLE\n\nFiat justitia\u2014ruat caelum. _(Do justice, and let the skies fall.)_\n\n\u2014ANCIENT ROMAN PROVERB\n\nThe existence of energy in empty space\u2014the discovery that rocked our cosmological universe and the idea that forms the bedrock of inflation\u2014only reinforces something about the quantum world that was already well established in the context of the kinds of laboratory experiments I have already described. Empty space is complicated. It is a boiling brew of virtual particles that pop in and out of existence in a time so short we cannot see them directly.\n\nVirtual particles are manifestations of a basic property of quantum systems. At the heart of quantum mechanics is a rule that sometimes governs politicians or CEOs\u2014as long as no one is watching, anything goes. Systems continue to move, if just momentarily, between all possible states, including states that would not be allowed if the system were actually being measured. These \"quantum fluctuations\" imply something essential about the quantum world: nothing always produces something, if only for an instant.\n\nBut here's the rub. The conservation of energy tells us that quantum systems can misbehave for only so long. Like embezzling stockbrokers, if the state that a system fluctuates into requires sneaking some energy from empty space, then the system has to return that energy in a time short enough so that no one measuring the system can detect it.\n\nAs a result, you might presume to safely argue that this \"something\" that is produced by quantum fluctuations is ephemeral\u2014not measurable, unlike, say, you or I or the Earth on which we live. But this ephemeral creation, too, is subject to the circumstances associated with our measurements. For example, consider the electric field emanating from a charged object. It is definitely real. You can feel the static electric force on your hair or watch a balloon stick to a wall. However, the quantum theory of electromagnetism suggests that the static field is due to the emission, by the charged particles involved in producing the field, of virtual photons that have essentially zero total energy. These virtual particles, because they have zero energy, can propagate across the universe without disappearing, and the field due to the superposition of many of them is so real it can be felt.\n\nSometimes conditions are such that real, massive particles can actually pop out of empty space with impunity. In one example, two charged plates are brought close together and, once the electric field gets strong enough between them, it becomes energetically favorable for a real particle-antiparticle pair to \"pop\" out of the vacuum, with the negative charge heading toward the positive plate and the positive charge toward the negative one. In so doing, it is possible that the reduction in energy arising from reducing the net charge on each of the plates and hence the electric field between them can be greater than the energy associated with the rest mass energy required to produce two real particles. Of course, the strength of the field has to be huge for such a condition to be possible.\n\nThere is actually a place where strong fields of a different kind might allow a phenomenon similar to that described above to occur\u2014but in this case due to gravity. This realization actually made Stephen Hawking famous among physicists in 1974, when he showed that it might be possible for black holes\u2014out of which, in the absence of quantum mechanical considerations at least, nothing can ever escape\u2014to radiate physical particles.\n\nThere are many different ways to try to understand this phenomenon, but one of these is strikingly familiar to the situation I described above with electric fields. Outside of the core of black holes is a radius called the \"event horizon.\" Inside an event horizon, no object can classically escape because the escape velocity exceeds the speed of light. Thus, even light emitted inside this region will not make it outside the event horizon.\n\nNow imagine a particle-antiparticle pair nucleates out of empty space just outside of the event horizon due to quantum fluctuations in that region. It is possible, if one of the particles actually falls within the event horizon, for it to lose enough gravitational energy by falling into the black hole that this energy exceeds twice the rest mass of either particle. This means that the partner particle can fly off to infinity and be observable without any violation of energy conservation. The total positive energy associated with the radiated particle is more than compensated by the loss of energy experienced by its partner particle falling into the black hole. The black hole can therefore radiate particles.\n\nThe situation is even more interesting, however, precisely because the energy lost by the infalling particle is greater than the positive energy associated with its rest mass. As a result, when it falls into the black hole, the net system of the black hole plus the particle actually has less energy than it did before the particle fell in! The black hole therefore actually gets _lighter_ after the particle falls in by an amount that is equivalent to the energy carried away by the radiated particle that escapes. Eventually the black hole may radiate away entirely. At this point we do not know because the final stages of black hole evaporation involve physics on such small distance scales that general relativity alone cannot tell us the final answer. On these scales, gravity must be treated as a fully quantum mechanical theory, and our current understanding of general relativity is not sufficient to allow us to determine precisely what will happen.\n\nNevertheless, all of these phenomena imply that, under the right conditions, not only can nothing become something, it is required to.\n\nAn early example in cosmology of the fact that \"nothing\" can be unstable and form something comes from efforts to understand why we live in a universe of matter.\n\nYou probably don't wake up each morning wondering about this, but the fact that our universe contains matter is remarkable. What is particularly remarkable about this is that, as far as we can tell, our universe does not contain substantial amounts of antimatter, which you will recall is required to exist by quantum mechanics and relativity, so that for every particle that we know of in nature, there can exist an equivalent antiparticle with opposite charge and the same mass. Any sensible universe at its inception, one might think, would contain equal amounts of both. After all, the antiparticles of normal particles have the same mass and similar other properties, so if particles were created at early times, it would have been equally easy to create antiparticles.\n\nAlternatively, we could even imagine an antimatter universe in which all of the particles that make up the stars and galaxies were replaced with their antiparticles. Such a universe would appear to be almost identical to the one we live in. Observers in such a universe (themselves made of antimatter) would no doubt call what we call antimatter as matter. The name is arbitrary.\n\nHowever, if our universe began sensibly, with equal amounts of matter and antimatter, and stayed that way, we wouldn't be around to ask \"Why?\" or \"How?\" This is because all particles of matter would have annihilated with all particles of antimatter in the early universe, leaving nothing but pure radiation. No matter or antimatter would be left over to make up stars, or galaxies, or to make up lovers or antilovers who might otherwise one day gaze out and be aroused by the spectacle of the night sky in each other's arms. No drama. History would consist of emptiness, a radiation bath that would slowly cool, leading ultimately to a cold, dark, bleak universe. Nothingness would reign supreme.\n\nScientists began to understand in the 1970s, however, that it is possible to begin with equal amounts of matter and antimatter in an early hot, dense Big Bang, and for plausible quantum processes to \"create something from nothing\" by establishing a small asymmetry, with a slight excess of matter over antimatter in the early universe. Then, instead of complete annihilation of matter and antimatter, leading to nothing but pure radiation today, all of the available antimatter in the early universe could have annihilated with matter, but the small excess of matter would have had no comparable amount of antimatter to annihilate with, and would then be left over. This would then lead to all the matter making up stars and galaxies we see in the universe today.\n\nAs a result, what might otherwise seem a small accomplishment (establishing a small asymmetry at early times) might instead be considered almost as the moment of creation. Because once an asymmetry between matter and antimatter was created, nothing could later put it asunder. The future history of a universe full of stars and galaxies was essentially written. Antimatter particles would annihilate with the matter particles in the early universe, and the remaining excess of matter particles would survive through the present day, establishing the character of the visible universe we know and love and inhabit.\n\nEven if the asymmetry were 1 part in a billion there would be enough matter left over to account for everything we see in the universe today. In fact, an asymmetry of 1 part in a billion or so is precisely what was called for, because today there are roughly 1 billion photons in the cosmic microwave background for every proton in the universe. The CMBR photons are the remnants, in this picture, of the early matter-antimatter annihilations near the beginning of time.\n\nA definitive description of how this process could have happened in the early universe is currently lacking because we have not yet fully and empirically established the detailed nature of the microphysical world at the scales where this asymmetry was likely to have been generated. Nevertheless, a host of different plausible scenarios has been explored based on the current best ideas we have about physics at these scales. While they differ in the details, they all have the same general characteristics. Quantum processes associated with elementary particles in the primordial heat bath can inexorably drive an empty universe (or equivalently an initially matter-antimatter symmetric universe) almost imperceptibly toward a universe that will be dominated by matter or antimatter.\n\nIf it could have gone either way, was it then just a circumstantial accident that our universe became dominated by matter? Imagine standing on top of a tall mountain and tripping. The direction you fall was not preordained, but rather is an accident, depending upon which direction you were looking in or at what point in your stride you tripped. Perhaps similarly our universe is like that, and even if the laws of physics are fixed, the ultimate direction of the asymmetry between matter and antimatter was driven by some random initial condition (just as in the case of tripping down the mountain, the law of gravity is fixed and determines that you will fall, but your direction may be an accident). Once again, our very existence in that case would be an environmental accident.\n\nIndependent of this uncertainty, however, is the remarkable fact that a feature of the underlying laws of physics can allow quantum processes to drive the universe away from a featureless state. Physicist Frank Wilczek, who was one of the first theorists to explore these possibilities, has reminded me that he utilized precisely the same language I have used previously in this chapter, in the 1980 _Scientific American_ article he wrote on the matter-antimatter asymmetry of the universe. After describing how a matter-antimatter asymmetry might plausibly be generated in the early universe based on our new understanding of particle physics, he added a note that this provided one way of thinking about the answer to the question of why there is something rather than nothing: _nothing_ is unstable.\n\nThe point Frank was emphasizing is that the measured excess of matter over antimatter in the universe appears on first glance to be an obstacle to imagining a universe that could arise from an instability in empty space, with nothingness producing a Big Bang. But if that asymmetry could arise dynamically after the Big Bang, that barrier is removed. As he put it:\n\nOne can speculate that the universe began in the most symmetrical state possible and that in such a state no matter existed; the universe was a vacuum. A second state existed, and in it matter existed. The second state had slightly less symmetry, but was also lower in energy. Eventually a patch of less symmetrical phase appeared and grew rapidly. The energy released by the transition found form in the creation of particles. This event might be identified with the big bang . . . The answer to the ancient question \"Why is there something rather than nothing?\" would be that \"nothing\" is unstable.\n\nBefore I proceed, however, I am again reminded of the similarities between the discussion I have just given of a matter-antimatter asymmetry and the discussions we had at our recent Origins workshop to explore our current understanding of the nature of life in the universe and its origin. My words were different, but the fundamental issues are remarkably similar: What specific physical process in the early moments of the Earth's history could have led to the creation of the first replicating biomolecules and metabolism? As in the 1970s in physics, the recent decade has seen incredible progress in molecular biology. We learned of natural organic pathways, for example, that could produce, under plausible conditions, ribonucleic acids, long thought to be the precursors to our modern DNA-based world. Until recently it was felt that no such direct pathway was possible and that some other intermediate forms must have played a key role.\n\nNow few biochemists and molecular biologists doubt that life can arise naturally from nonlife, even though the specifics are yet to be discovered. But, as we discussed all of this, a common subtext permeated our proceedings: Did the life that first formed on Earth _have_ to have the chemistry that it did, or are there many different, equally viable possibilities?\n\nEinstein once asked a question that, he said, was the one thing he really wanted to know about nature. I admit it is the most profound and fundamental question that many of us would like answered. He put it as follows: \"What I want to know is whether _God_ [ _sic_ ] had any choice in the creation of the universe.\"\n\nI have annotated this because Einstein's God was not the God of the Bible. For Einstein, the existence of order in the universe provided a sense of such profound wonder that he felt a spiritual attachment to it, which he labeled, motivated by Spinoza, with the moniker \"God.\" In any case, what Einstein really meant in this question was the issue I have just described in the context of several different examples: Are the laws of nature unique? And is the universe we inhabit, which has resulted from these laws, unique? If you change one facet, one constant, one force, however slight, would the whole edifice crumble? In a biological sense, is the biology of life unique? Are we unique in the universe? We will return to discuss this most important question later in this book.\n\nWhile such a discussion will cause us to further refine and generalize notions of \"nothing\" and \"something,\" I want to return to taking an intermediate step in making the case for the inevitable creation of something.\n\nAs I have defined it thus far, the relevant \"nothing\" from which our observed \"something\" arises is \"empty space.\" However, once we allow for the merging of quantum mechanics and general relativity, we can extend this argument to the case where space itself is forced into existence.\n\nGeneral relativity as a theory of gravity is, at its heart, a theory of space and time. As I described in the very beginning of this book, this means that it was the first theory that could address the dynamics not merely of objects moving through space, but also how space itself evolves.\n\nHaving a quantum theory of gravity would therefore mean that the rules of quantum mechanics would apply to the properties of space and not just to the properties of objects existing in space, as in conventional quantum mechanics.\n\nExtending quantum mechanics to include such a possibility is tricky, but the formalism Richard Feynman developed, which led to a modern understanding of the origin of antiparticles, is well suited to the task. Feynman's methods focus on the key fact to which I alluded at the beginning of this chapter: quantum mechanical systems explore all possible trajectories, even those that are classically forbidden, as they evolve in time.\n\nIn order to explore this, Feynman developed a \"sum over paths formalism\" to make predictions. In this method, we consider all possible trajectories between two points that a particle might take. We then assign a probability weighting for each trajectory, based on well-defined principles of quantum mechanics, and then perform a sum over all paths in order to determine final (probabilistic) predictions for the motion of particles.\n\nStephen Hawking was one of the first scientists to fully exploit this idea to the possible quantum mechanics of space-time (the union of our three-dimensional space along with one dimension of time to form a four-dimensional unified space-time system, as required by Einstein's special theory of relativity). The virtue of Feynman's methods was that focusing on all possible paths ends up meaning that the results can be shown to be independent of the specific space and time labels one applies to each point on each path. Because relativity tells us that different observers in relative motion will measure distance and time differently and therefore assign different values to each point in space and time, having a formalism that is independent of the different labels that different observers might assign to each point in space and time is particularly useful.\n\nAnd it is most useful perhaps in considerations of general relativity, where the specific labeling of space and time points becomes completely arbitrary, so that different observers at different points in a gravitational field measure distances and times differently, and all that ultimately determines the behavior of systems are geometric quantities like curvature, which turn out to be independent of all such labeling schemes.\n\nAs I have alluded to several times, general relativity is not fully consistent with quantum mechanics, at least as far as we can tell, and therefore there is no completely unambiguous method to define Feynman's sum-over-paths technique in general relativity. So we have to make some guesses in advance based on plausibility and check to see if the results make sense.\n\nIf we are to consider the quantum dynamics of space and time then, one must imagine that in the Feynman \"sums,\" one must consider every different possible configuration that can describe the different geometries that space can adopt during the intermediate stages of any process, when quantum indeterminacy reigns supreme. This means we must consider spaces that are arbitrarily highly curved over short distances and small times (so short and so small that we cannot measure them so that quantum weirdness can reign free). These weird configurations would then not be observed by large classical observers such as us when we attempt to measure the properties of space over large distances and times.\n\nBut let's consider even stranger possibilities. Remember that, in the quantum theory of electromagnetism, particles can pop out of empty space at will as long as they disappear again on a time frame determined by the Uncertainty Principle. By analogy, then, in the Feynman quantum sum over possible space-time configurations, should one consider the possibility of small, possibly compact spaces that themselves pop in and out of existence? More generally, what about spaces that may have \"holes\" in them, or \"handles\" like donuts dunking into space-time?\n\nThese are open questions. However, unless one can come up with a good reason for excluding such configurations from the quantum mechanical sum that determines the properties of the evolving universe, and to date no such good reason exists that I know of, then under the general principle that holds everywhere else I know of in nature\u2014namely that anything that is not proscribed by the laws of physics must actually happen\u2014it seems most reasonable to consider these possibilities.\n\nAs Stephen Hawking has emphasized, a quantum theory of gravity allows for the creation, albeit perhaps momentarily, of space itself where none existed before. While in his scientific work he was not attempting to address the \"something from nothing\" conundrum, effectively this is what quantum gravity may ultimately address.\n\n\"Virtual\" universes\u2014namely the possible small compact spaces that may pop into and out of existence on a timescale so short we cannot measure them directly\u2014are fascinating theoretical constructs, but they don't seem to explain how something can arise from nothing over the long term any more than do the virtual particles that populate otherwise empty space.\n\nHowever, recall that a nonzero real electric field, observable at large distances away from a charged particle, can result from the coherent emission of many virtual zero energy photons by the charge. This is because virtual photons that carry zero energy do not violate energy conservation when they are emitted. The Heisenberg Uncertainty Principle, therefore, does not constrain them to exist for only very brief times before they must be reabsorbed and disappear back into nothingness. (Again recall that the Heisenberg Uncertainty Principle states that the uncertainty with which we measure the energy of a particle, and hence the possibility that its energy may change slightly by the emission and absorption of virtual particles, is inversely proportional to the length of time over which we observe it. Hence, virtual particles that carry away zero energy can do so essentially with impunity\u2014namely they can exist for arbitrarily long times and travel arbitrarily far away before being absorbed . . . leading to the possible existence of long-range interactions between charged particles. If the photon was not massless, so that photons always carried away non-zero energy due to a rest mass, the Heisenberg Uncertainty Principle would imply that the electric field would be short range because photons could propagate only for short times without being reabsorbed again.)\n\nA similar argument suggests that one can imagine one specific type of universe that might spontaneously appear and need not disappear almost immediately thereafter because of the constraints of the Uncertainty Principle and energy conservation. Namely, a compact universe with zero total energy.\n\nNow, I would like nothing better than to suggest that this is precisely the universe we live in. This would be the easy way out, but I am more interested here in being true to our current understanding of the universe than in making an apparently easy and convincing case for creating it from nothing.\n\nI have argued, I hope compellingly, that the average Newtonian gravitational energy of every object in our flat universe is zero. And it is. But that is not the whole story. Gravitational energy is not the total energy of any object. To this energy we must add its rest energy, associated with its rest mass. Put another way, as I have described earlier, the gravitational energy of an object at rest isolated from all other objects by an infinite distance is zero, because if it is at rest, it has no kinetic energy of motion, and if it is infinitely far away from all other particles, the gravitational force on it due to other particles, which could provide potential energy to do work, is also essentially zero. However, as Einstein told us, its total energy is not merely due to gravity, but also includes the energy associated with its mass, so that, as is famously known, E = mc2.\n\nIn order to take this rest energy into account, we have to move from Newtonian gravity to general relativity, which, by definition, incorporates the effects of special relativity (and E = mc2) into a theory of gravity. And here things get both subtler and more confusing. On small scales compared to the possible curvature of a universe, and as long as all objects within these scales are moving slowly compared to the speed of light, the general relativistic version of energy reverts to the definition we are familiar with from Newton. However, once these conditions no longer hold, all bets are off, almost.\n\nPart of the problem is that it turns out that energy as we normally think of it elsewhere in physics is not a particularly well-defined concept on large scales in a curved universe. Different ways of defining coordinate systems to describe the different labels that different observers may assign to points in space and time (called different \"frames of reference\") can lead, on large scales, to different determinations of the total energy of the system. In order to accommodate this effect, we have to generalize the concept of energy, and, moreover, if we are to define the total energy contained in any universe, we must consider how to add up the energy in universes that may be infinite in spatial extent.\n\nThere is a lot of debate over precisely how to do this. The scientific literature is replete with claims and counterclaims in this regard.\n\nOne thing is certain, however: There is one universe in which the total energy is definitely and precisely zero. It is not, however, a flat universe, which is in principle infinite in spatial extent, and therefore the calculation of total energy becomes problematic. It is a closed universe, one in which the density of matter and energy is sufficient to cause space to close back upon itself. As I have described, in a closed universe, if you look far enough in one direction, you will eventually see the back of your head!\n\nThe reason the energy of a closed universe is zero is really relatively simple. It is easiest to consider the result by analogy with the fact that in a closed universe the total electric charge must also be zero.\n\nSince the time of Michael Faraday we think of electric charge as being the source of an electric field (due in modern quantum parlance to the emission of the virtual photons I described above). Pictorially, we imagine \"field lines\" emanating out radially from the charge, with the number of field lines being proportional to the charge, and the direction of field lines being outward for positive charges and inward for negative charges, as shown below.\n\nWe imagine these field lines going out to infinity, and as they spread out, getting farther apart. This implies that the strength of the electric field gets weaker and weaker. However, in a closed universe, the field lines associated with a positive charge, for example, may start out spreading apart but eventually, just as the lines of longitude on a map of the Earth come together at the North and South Poles, the field lines from the positive charge will come together again on the far side of the universe. When they converge, the field will get stronger and stronger again until there is enough energy to create a negative charge that can \"eat\" the field lines at this antipodal point of the universe.\n\nIt turns out a very similar argument, in this case associated not with the \"flux\" of field lines but with the \"flux\" of energy in a closed universe, tells us that the total positive energy, including that associated with the rest masses of particles, must be exactly compensated for by a negative gravitational energy, so that the total energy is precisely zero.\n\nSo if the total energy of a closed universe is zero, and if the sum-over-paths formalism of quantum gravity is appropriate, then quantum mechanically such universes could appear spontaneously with impunity, carrying no net energy. I want to emphasize that these universes would be completely self-contained space-times, disconnected from our own.\n\nThere is a hitch, however. A closed expanding universe filled with matter will in general expand to a maximum size and then recollapse just as quickly, ending up in a space-time singularity where the no-man's land of quantum gravity at present cannot tell us what its ultimate fate will be. The characteristic lifetime of tiny closed universes will therefore be microscopic, perhaps on the order of the \"Planck time,\" the characteristic scale over which quantum gravitational processes should operate, about 10\u221244 seconds or so.\n\nThere is a way out of this dilemma, however. If, before such a universe can collapse, the configuration of fields within it produces a period of inflation, then even an initially tiny closed universe can rapidly, exponentially expand, becoming closer and closer to an infinitely large flat universe during this period. After one hundred or so doubling times of such inflation, the universe will be so close to flat that it could easily last much longer than our universe has been around without collapsing.\n\nAnother possibility actually exists, one that always gives me a slight twinge of nostalgia (and envy), because it represented an important learning experience for me. When I was first a postdoc at Harvard, I was playing with the possible quantum mechanics of gravitational fields, and I learned of a result by a good friend from graduate school, Ian Affleck. A Canadian who had been a graduate student at Harvard when I was at MIT, Affleck joined the Society of Fellows a few years before I did and had used the mathematical theory of Feynman that we now use for dealing with elementary particles and fields, called quantum field theory, to calculate how particles and antiparticles could be produced in a strong magnetic field.\n\nI realized that the form of the solution that Ian had described, something called an \"instanton,\" resembled very much an inflating universe, if one took over his formalism to the case of gravity. But it looked like an inflating universe that began from nothing! Before writing up this result, I wanted to address my own confusion about how to interpret what physics such a mathematical solution might correspond to. I soon learned, however, that while I was cogitating, just down the road the very creative cosmologist I mentioned earlier, Alex Vilenkin, who has since become a friend, had actually just written a paper that described in exactly this fashion how quantum gravity indeed might create an inflating universe directly from nothing. I was scooped, but I couldn't be that upset because (a) I frankly didn't understand in detail at that point what I was doing, and (b) Alex had the boldness to propose something that at the time I didn't. I have since learned that one doesn't have to understand all the implications of one's work in order to publish. Indeed, there are several of my own most important papers that I only fully understood well after the fact.\n\nIn any case, while Stephen Hawking and his collaborator Jim Hartle have proposed a very different scheme for trying to determine the \"boundary conditions\" on universes that may begin from nothing at all, the important facts are these:\n\n1. In quantum gravity, universes can, and indeed always will, spontaneously appear from nothing. Such universes need not be empty, but can have matter and radiation in them, as long as the total energy, including the negative energy associated with gravity, is zero.\n\n2. In order for the closed universes that might be created through such mechanisms to last for longer than infinitesimal times, something like inflation is necessary. As a result, the only long-lived universe one might expect to live in as a result of such a scenario is one that today appears flat, just as the universe in which we live appears.\n\nThe lesson is clear: quantum gravity not only appears to allow universes to be created from nothing\u2014meaning, in this case, I emphasize, the absence of space and time\u2014it may require them. \"Nothing\"\u2014in this case no space, no time, no anything!\u2014 _is_ unstable.\n\nMoreover, the general characteristics of such a universe, if it lasts a long time, would be expected to be those we observe in our universe today.\n\nDoes this prove that our universe arose from nothing? Of course not. But it does take us one rather large step closer to the plausibility of such a scenario. And it removes one more of the objections that might have been leveled against the argument of creation from nothing as described in the previous chapter.\n\nThere, \"nothing\" meant empty but preexisting space combined with fixed and well-known laws of physics. Now the requirement of space has been removed.\n\nBut, remarkably, as we shall next discuss, even the laws of physics may not be necessary or required.\n\n## CHAPTER 11\n\nBRAVE NEW WORLDS\n\n_It was the best of times. It was the worst of times._\n\n\u2014CHARLES DICKENS\n\nThe central problem with the notion of creation is that it appears to require some externality, something outside of the system itself, to _preexist,_ in order to create the conditions necessary for the system to come into being. This is usually where the notion of God\u2014some external agency existing separate from space, time, and indeed from physical reality itself\u2014comes in, because the buck seems to be required to stop somewhere. But in this sense _God_ seems to me to be a rather facile semantic solution to the deep question of creation. I think this is best explained within the context of a slightly different example: the origin of morality, which I first learned from my friend Steven Pinker.\n\nIs morality external and absolute, or is it derived solely within the context of our biology and our environment, and thus can it be determined by science? During a debate on this subject organized at Arizona State University, Pinker pointed out the following conundrum.\n\nIf one argues, as many deeply religious individuals do, that without God there can be no ultimate right and wrong\u2014namely that God determines for us what is right and wrong\u2014one can then ask the questions: What if God decreed that rape and murder were morally acceptable? Would that make them so?\n\nWhile some might answer yes, I think most believers would say no, God would not make such a decree. But why not? Presumably because God would have some reason for not making such a decree. Again, presumably this is because reason suggests that rape and murder are not morally acceptable. But if God would have to appeal to reason, then why not eliminate the middleman entirely?\n\nWe may wish to apply similar reasoning to the creation of our universe. All of the examples I have provided thus far indeed involve creation of something from what one should be tempted to consider as nothing, but the _rules_ for that creation, i.e., the laws of physics, were preordained. Where do the rules come from?\n\nThere are two possibilities. Either God, or some divine being who is not bound by the rules, who lives outside of them, determines them\u2014either by whim or with malice aforethought\u2014or they arise by some less supernatural mechanism.\n\nThe problem with God determining the rules is that you can at least ask what, or who, determined God's rules. Traditionally the response to this is to say that God is, among the Creator's many other spectacular attributes, the _cause of all causes,_ in the language of the Roman Catholic Church, or the _First Cause_ (as per Aquinas), or in the language of Aristotle, moving the _prime mover_.\n\nInterestingly, Aristotle recognized the problem of a first cause, and decided that for this reason the universe must be eternal. Moreover, God himself, whom he identified as pure self-absorbed thought, the love of which motivated the prime mover to move, had to be eternal, not causing motion by creating it, but rather by establishing the end purpose of motion, which itself Aristotle deemed had to be eternal.\n\nAristotle felt that equating First Cause with God was less than satisfying, in fact that the Platonic notion of First Cause was flawed, specifically because Aristotle felt every cause must have a precursor\u2014hence, the requirement that the universe be eternal. Alternatively, if one takes the view of God as the cause of all causes, and therefore eternal even if our universe is not, the _reductio ad absurdum_ sequence of \"why\" questions does indeed terminate, but as I have stressed, only at the expense of introducing a remarkable all-powerful entity for which there is simply no other evidence.\n\nIn this regard, there is another important point to stress here. The apparent logical necessity of First Cause is a real issue for any universe that has a beginning. Therefore, on the basis of logic alone one cannot rule out such a deistic view of nature. But even in this case it is vital to realize that this deity bears no logical connection to the personal deities of the world's great religions, in spite of the fact that it is often used to justify them. A deist who is compelled to search for some overarching intelligence to establish order in nature will not, in general, be driven to the personal God of the scriptures by the same logic.\n\nThese issues have been debated and discussed for millennia, by brilliant and not-so-brilliant minds, many of the latter making their current living by debating them. We can return to these issues now because we are simply better informed by our knowledge of the nature of physical reality. Neither Aristotle nor Aquinas knew about the existence of our galaxy, much less the Big Bang or quantum mechanics. Hence the issues they and later medieval philosophers grappled with must be interpreted and understood in the light of new knowledge.\n\nConsider, in the light of our modern picture of cosmology, for example, Aristotle's suggestion that there are no First Causes, or rather that causes indeed go backward (and forward) infinitely far in all directions. There is no beginning, no creation, no end.\n\nWhen I have thus far described how something almost always can come from \"nothing,\" I have focused on either the creation of something from preexisting empty space or the creation of empty space from no space at all. Both initial conditions work for me when I think of the \"absence of being\" and therefore are possible candidates for nothingness. I have not addressed directly, however, the issues of what might have existed, if anything, before such creation, what laws governed the creation, or, put more generally, I have not discussed what some may view as the question of First Cause. A simple answer is of course that either empty space or the more fundamental nothingness from which empty space may have arisen, preexisted, and is eternal. However, to be fair, this does beg the possible question, which might of course not be answerable, of what, if anything, fixed the rules that governed such creation.\n\nOne thing is certain, however. The metaphysical \"rule,\" which is held as an ironclad conviction by those with whom I have debated the issue of creation, namely that \" _out of nothing nothing comes,_ \" has no foundation in science. Arguing that it is self-evident, unwavering, and unassailable is like arguing, as Darwin falsely did, when he made the suggestion that the origin of life was beyond the domain of science by building an analogy with the incorrect claim that matter cannot be created or destroyed. All it represents is an unwillingness to recognize the simple fact that nature may be cleverer than philosophers or theologians.\n\nMoreover, those who argue that out of nothing nothing comes seem perfectly content with the quixotic notion that somehow God can get around this. But once again, if one requires that the notion of true nothingness requires not even the _potential_ for existence, then surely God cannot work his wonders, because if he does cause existence from nonexistence, there must have been the potential for existence. To simply argue that God can do what nature cannot is to argue that _supernatural_ potential for existence is somehow different from regular natural potential for existence. But this seems an arbitrary semantic distinction designed by those who have decided in advance (as theologians are wont to do) that the supernatural (i.e., God) must exist so they define their philosophical ideas (once again completely divorced from any empirical basis) to exclude anything but the possibility of a god.\n\nIn any case, to posit a god who could resolve this conundrum, as I have emphasized numerous times thus far, often is claimed to require that God exists outside the universe and is either timeless or eternal.\n\nOur modern understanding of the universe provides another plausible and, I would argue, far more physical solution to this problem, however, which has some of the same features of an external creator\u2014and moreover is logically more consistent.\n\nI refer here to the multiverse. The possibility that our universe is one of a large, even possibly infinite set of distinct and causally separated universes, in each of which any number of fundamental aspects of physical reality may be different, opens up a vast new possibility for understanding our existence.\n\nAs I have mentioned, one of the more distasteful but potentially true implications of these pictures is that physics, at some fundamental level, is merely an environmental science. (I find this distasteful because I was brought up on the idea that the goal of science was to explain why the universe had to be the way it is and how that came to be. If instead the laws of physics as we know them are merely accidents correlated to our existence, then that fundamental goal was misplaced. However, I will get over my prejudice if the idea turns out to be true.) In this case, the fundamental forces and constants of nature in this picture are no more fundamental than the Earth-Sun distance. We find ourselves living on Earth rather than Mars not because there is something profound and fundamental about the Earth-Sun distance, but rather simply if Earth were located at a different distance, then life as we know it could not have evolved on our planet.\n\nThese anthropic arguments are notoriously slippery, and it is almost impossible to make specific predictions based on them without knowing explicitly both the probability distribution among all possible universes of the various fundamental constants and forces\u2014namely, which may vary and which don't, and what possible values and forms they may take\u2014and also exactly how \"typical\" we are in our universe. If we are not \"typical\" life forms, then anthropic selection, if it occurs at all, may be based on different factors from those we would otherwise attribute it to.\n\nNevertheless, a multiverse, either in the form of a landscape of universes existing in a host of extra dimensions, or in the form of a possibly infinitely replicating set of universes in a three-dimensional space as in the case of eternal inflation, changes the playing field when we think about the creation of our own universe and the conditions that may be required for that to happen.\n\nIn the first place, the question of what determined the laws of nature that allowed our universe to form and evolve now becomes less significant. If the laws of nature are themselves stochastic and random, then there is no prescribed \"cause\" for our universe. Under the general principle that anything that is not forbidden is allowed, then we would be guaranteed, in such a picture, that some universe would arise with the laws that we have discovered. No mechanism and no entity is required to fix the laws of nature to be what they are. They could be almost anything. Since we don't currently have a fundamental theory that explains the detailed character of the landscape of a multiverse, we cannot say. (Although to be fair, to make any scientific progress in calculating possibilities, we generally assume that certain properties, like quantum mechanics, permeate all possibilities. I have no idea if this notion can be usefully dispensed with, or at least I don't know of any productive work in this regard.)\n\nIn fact, there may be no fundamental theory at all. Although I became a physicist because I hoped that there was such a theory, and because I hoped that I might one day help contribute to discovering it, this hope may be misplaced, as I have already lamented. I take solace in the statement by Richard Feynman, which I summarized briefly before, but want to present in its entirety here:\n\nPeople say to me, \"Are you looking for the ultimate laws of physics?\" No, I'm not. I'm just looking to find out more about the world, and if it turns out there is a simple ultimate law that explains everything, so be it. That would be very nice to discover. If it turns out it's like an onion with millions of layers, and we're sick and tired of looking at layers, then that's the way it is . . . My interest in science is to simply find out more about the world, and the more I find out, the better it is. I like to find out.\n\nOne can carry the argument further and in a different direction, which also has implications for the arguments at the core of this book. In a multiverse of any of the types that have been discussed, there could be an infinite number of regions, potentially infinitely big or infinitesimally small, in which there is simply \"nothing,\" and there could be regions where there is \"something.\" In this case, the response to why there is something rather than nothing becomes almost trite: there is something simply because if there were nothing, we wouldn't find ourselves living there!\n\nI recognize the frustration inherent in such a trivial response to what has seemed such a profound question throughout the ages. But science has told us that anything profound or trivial can be dramatically different from what we might suppose at first glance.\n\nThe universe is far stranger and far richer\u2014more wondrously strange\u2014than our meager human imaginations can anticipate. Modern cosmology has driven us to consider ideas that could not even have been formulated a century ago. The great discoveries of the twentieth and twenty-first centuries have not only changed the world in which we operate, they have revolutionized our understanding of the world\u2014or worlds\u2014that exist, or may exist, just under our noses: the reality that lies hidden until we are brave enough to search for it.\n\nThis is why philosophy and theology are ultimately incapable of addressing by themselves the truly fundamental questions that perplex us about our existence. Until we open our eyes and let nature call the shots, we are bound to wallow in myopia.\n\nWhy is there something rather than nothing? Ultimately, this question may be no more significant or profound than asking why some flowers are red and some are blue. \"Something\" may always come from nothing. It may be required, independent of the underlying nature of reality. Or perhaps \"something\" may not be very special or even very common in the multiverse. Either way, what is really useful is not pondering this question, but rather participating in the exciting voyage of discovery that may reveal specifically how the universe in which we live evolved and is evolving and the processes that ultimately operationally govern our existence. That is why we have science. We may supplement this understanding with reflection and call that philosophy. But only via continuing to probe every nook and cranny of the universe that is accessible to us will we truly build a useful appreciation of our own place in the cosmos.\n\nBefore concluding, I want to raise one more aspect of this question that I haven't touched upon, but which strikes me as worth ending with. Implicit in the question of why there is something rather than nothing is the solipsistic expectation that \"something\" will persist\u2014that somehow the universe has \"progressed\" to the point of our existence, as if we were the pinnacle of creation. Far more likely, based on everything we know about the universe, is the possibility that the future, perhaps the infinite future, is one in which nothingness will once again reign.\n\nIf we live in a universe whose energy is dominated by the energy of nothing, as I have described, the future is indeed bleak. The heavens will become cold and dark and empty. But the situation is actually worse. A universe dominated by the energy of empty space is the worst of all universes for the future of life. Any civilization is guaranteed to ultimately disappear in such a universe, starved of energy to survive. After an unfathomably long time, some quantum fluctuation or some thermal agitation may produce a local region where once again life can evolve and thrive. But that too will be ephemeral. The future will be dominated by a universe with nothing in it to appreciate its vast mystery.\n\nAlternatively, if the matter that makes us up was created at the beginning of time by some quantum processes, as I have described, we are virtually guaranteed that it, too, will disappear once again. Physics is a two-way street, and beginnings and endings are linked. Far, far into the future, protons and neutrons will decay, matter will disappear, and the universe will approach a state of maximum simplicity and symmetry.\n\nMathematically beautiful perhaps, but devoid of substance. As Heraclitus of Ephesus wrote in a slightly different context, \"Homer was wrong in saying: 'Would that strife might perish from among gods and men!' He did not see that he was praying for the destruction of the universe; for if his prayers were heard, all things would pass away.\" Or, as Christopher Hitchens has restated it, \"Nirvana _is_ nothingness.\"\n\nA more extreme version of this eventual retreat into nothingness may be inevitable. Some string theorists have argued, on the basis of complex mathematics, that a universe like ours, with a positive energy in empty space, _cannot_ be stable. Eventually, it must decay to a state in which the energy associated with space will be negative. Our universe will then recollapse inward to a point, returning to the quantum haze from which our own existence may have begun. If these arguments are correct, our universe will then disappear as abruptly as it probably began.\n\nIn this case, the answer to the question, \"Why is there something rather than nothing?\" will then simply be: \"There won't be for long.\"\n\n## EPILOGUE\n\n_The sanction of experienced fact as a face of truth is a profound subject, and the mainspring which has moved our civilization since the Renaissance._\n\n\u2014JACOB BRONOWSKI\n\nI began this book with another quote from Jacob Bronowski:\n\nDream or nightmare, we have to live our experience as it is, and we have to live it awake. We live in a world which is penetrated through and through by science and which is both whole and real. We cannot turn it into a game simply by taking sides.\n\nAs I have also argued, one person's dream is another person's nightmare. A universe without purpose or guidance may seem, for some, to make life itself meaningless. For others, including me, such a universe is invigorating. It makes the fact of our existence even more amazing, and it motivates us to draw meaning from our own actions and to make the most of our brief existence in the sun, simply because we are here, blessed with consciousness and with the opportunity to do so. Bronowski's point, however, is that it doesn't really matter either way, and what we would like for the universe is irrelevant. Whatever happened, happened, and it happened on a cosmic scale. And whatever is about to happen on that scale will happen independent of our likes and dislikes. We cannot affect the former, and we are unlikely to affect the latter.\n\nWhat we can do, however, is try to understand the circumstances of our existence. I have described in this book one of the most remarkable journeys of exploration humanity has ever taken in its evolutionary history. It is an epic quest to explore and understand the cosmos on scales that simply were unknown a century ago. The journey has pushed the limits of the human spirit, combining the willingness to follow evidence wherever it might lead with the courage to devote a lifetime to exploring the unknown with the full knowledge that the effort might go nowhere, and finally requiring a mixture of creativity and persistence to address the often tedious tasks of sorting through endless equations or endless experimental challenges.\n\nI have always been attracted to the myth of Sisyphus and have likened the scientific effort at times to his eternal task of pushing a boulder up a mountain, only to have it fall back each time before he reaches the top. As Camus imagined, Sisyphus was smiling, and so should we. Our journey, whatever the outcome, provides its own reward.\n\nThe phenomenal progress we have made in the past century has brought us to the cusp, as scientists, of operationally addressing the deepest questions that have existed since we humans took our first tentative steps to understand who we are and where we came from.\n\nAs I have described here, in the process the very meaning of these questions has evolved along with our understanding of the universe. \"Why is there something rather than nothing?\" must be understood in the context of a cosmos where the meaning of these words is not what it once was, and the very distinction between something and nothing has begun to disappear, where transitions between the two in different contexts are not only common, but required.\n\nAs such, the question itself has been sidelined as we strive in our quest for knowledge. Instead, we are driven to understand the processes that govern nature in a way that allows us to make predictions and, whenever possible, to affect our own future. In so doing, we have discovered that we live in a universe in which empty space\u2014what formerly could have passed for nothing\u2014has a new dynamic that dominates the current evolution of the cosmos. We have discovered that all signs suggest a universe that could and plausibly did arise from a deeper nothing\u2014involving the absence of space itself\u2014and which may one day return to nothing via processes that may not only be comprehensible but also processes that do not require any external control or direction. In this sense, science, as physicist Steven Weinberg has emphasized, does not make it impossible to believe in God, but rather makes it possible to not believe in God. Without science, everything is a miracle. With science, there remains the possibility that nothing is. Religious belief in this case becomes less and less necessary, and also less and less relevant.\n\nThe choice to turn to the notion of divine creation falls to each of us, of course, and I don't expect the ongoing debate to die down anytime soon. But as I have stressed, I believe that if we are to be intellectually honest, we must make an informed choice, informed by fact, not by revelation.\n\nThat has been the purpose of this book, to provide an informed picture of the universe as we understand it and to describe the theoretical speculations that currently are driving physics forward as we scientists attempt to separate the wheat from the chaff in our observations and theories.\n\nI have made clear my own predilection: the case that _our_ universe arose from nothing seems by far the most compelling intellectual alternative to me at the present time. You will draw your own conclusion.\n\nI want to end my discussion by returning to a question that I personally find even more intellectually fascinating than the question of something from nothing. It is the question Einstein asked about whether God had any choice in the creation of the universe. This question provides the basic motivation for almost all research into the fundamental structure of matter, space, and time\u2014the research that has occupied me for much of my professional life.\n\nI used to think there was a stark choice in the answer to this question, but in the process of writing this book, my views have altered. Clearly, if there is a single theory involving a unique set of laws that describes and, indeed, prescribes how our universe came into being and the rules that have governed its evolution ever since\u2014the goal of physics since Newton or Galileo\u2014then the answer would appear to be, \"No, things had to be the way they were, and are.\"\n\nBut if our universe is not unique, and it is a part of a vast and possibly infinite multiverse of universes, would the answer to Einstein's question be a resounding \"Yes, there is a host of choices for existence\"?\n\nI am not so sure. It could be that there is an infinite set of different combinations of laws and varieties of particles and substances and forces and even distinct universes that may arise in such a multiverse. It may be that only a certain very restricted combination, one that results in the universe of the type in which we live or one very much like it, can support the evolution of beings who can ask such a question. Then the answer to Einstein will still remain negative. A God or a Nature that could encompass a multiverse would be as constrained in the creation of a universe in which Einstein could ask the question as either would be if there is only one choice of a consistent physical reality.\n\nI find oddly satisfying the possibility that, in either scenario, even a seemingly omnipotent God would have no freedom in the creation of our universe. No doubt because it further suggests that God is unnecessary\u2014or at best redundant.\n\n## AFTERWORD\n\nby Richard Dawkins\n\nNothing expands the mind like the expanding universe. The music of the spheres is a nursery rhyme, a jingle to set against the majestic chords of the Symphonie Galactica. Changing the metaphor and the dimension, the dusts of centuries, the mists of what we presume to call \"ancient\" history, are soon blown off by the steady, eroding winds of geological ages. Even the age of the universe, accurate\u2014so Lawrence Krauss assures us\u2014to the fourth significant figure at 13.72 billion years, is dwarfed by the trillennia that are to come.\n\nBut Krauss's vision of the cosmology of the remote future is paradoxical and frightening. Scientific progress is likely to go into reverse. We naturally think that, if there are cosmologists in the year 2 trillion AD, their vision of the universe will be expanded over ours. Not so\u2014and this is one of the many shattering conclusions I take away on closing this book. Give or take a few billion years, ours is a very propitious time to be a cosmologist. Two trillion years hence, the universe will have expanded so far that all galaxies but the cosmologist's own (whichever one it happens to be) will have receded behind an Einsteinian horizon so absolute, so inviolable, that they are not only invisible but beyond all possibility of leaving a trace, however indirect. They might as well never have existed. Every trace of the Big Bang will most likely have gone, forever and beyond recovery. The cosmologists of the future will be cut off from their past, and from their situation, in a way that we are not.\n\nWe know we are situated in the midst of 100 billion galaxies, and we know about the Big Bang because the evidence is all around us: the redshifted radiation from distant galaxies tells us of the Hubble expansion and we extrapolate it backward. We are privileged to see the evidence because we look out on an infant universe, basking in that dawn age when light can still travel from galaxy to galaxy. As Krauss and a colleague wittily put it, \"We live at a very special time . . . the only time when we can observationally verify that we live at a very special time!\" The cosmologists of the third trillennium will be forced back to the stunted vision of our early twentieth century, locked as we were in a single galaxy which, for all that we knew or could imagine, was synonymous with the universe.\n\nFinally, and inevitably, the flat universe will further flatten into a nothingness that mirrors its beginning. Not only will there be no cosmologists to look out on the universe, there will be nothing for them to see even if they could. Nothing at all. Not even atoms. Nothing.\n\nIf you think that's bleak and cheerless, too bad. Reality doesn't owe us comfort. When Margaret Fuller remarked, with what I imagine to have been a sigh of satisfaction, \"I accept the universe,\" Thomas Carlyle's reply was withering: \"Gad, she'd better!\" Personally, I think the eternal quietus of an infinitely flat nothingness has a grandeur that is, to say the least, worth facing off with courage.\n\nBut if something can flatten into nothing, can nothing spring into action and give birth to something? Or why, to quote a theological chestnut, is there something rather than nothing? Here we come to perhaps the most remarkable lesson that we are left with on closing Lawrence Krauss's book. Not only does physics tell us how something could have come from nothing, it goes further, by Krauss's account, and shows us that nothingness is unstable: something was almost bound to spring into existence from it. If I understand Krauss aright, it happens all the time: The principle sounds like a sort of physicist's version of two wrongs making a right. Particles and antiparticles wink in and out of existence like subatomic fireflies, annihilating each other, and then re-creating themselves by the reverse process, out of nothingness.\n\nThe spontaneous genesis of something out of nothing happened in a big way at the beginning of space and time, in the singularity known as the Big Bang followed by the inflationary period, when the universe, and everything in it, took a fraction of a second to grow through twenty-eight orders of magnitude (that's a 1 with twenty-eight zeroes after it\u2014think about it).\n\nWhat a bizarre, ridiculous notion! Really, these scientists! They're as bad as medieval Schoolmen counting angels on pinheads or debating the \"mystery\" of the transubstantiation.\n\nNo, not so, not so with a vengeance and in spades. There is much that science still doesn't know (and it is working on it with rolled-up sleeves). But some of what we do know, we know not just approximately (the universe is not mere thousands but billions of years old): we know it with confidence and with stupefying accuracy. I've already mentioned that the age of the universe is measured to four significant figures. That's impressive enough, but it is nothing compared to the accuracy of some of the predictions with which Lawrence Krauss and his colleagues can amaze us. Krauss's hero Richard Feynman pointed out that some of the predictions of quantum theory\u2014again based on assumptions that seem more bizarre than anything dreamed up by even the most obscurantist of theologians\u2014have been verified with such accuracy that they are equivalent to predicting the distance between New York and Los Angeles to within one hairsbreadth.\n\nTheologians may speculate about angels on pinheads or whatever is the current equivalent. Physicists might seem to have their own angels and their own pinheads: quanta and quarks, \"charm,\" \"strangeness,\" and \"spin.\" But physicists can count their angels and can get it right to the nearest angel in a total of 10 billion: not an angel more, not an angel less. Science may be weird and incomprehensible\u2014more weird and less comprehensible than any theology\u2014but science works. It gets results. It can fly you to Saturn, slingshotting you around Venus and Jupiter on the way. We may not understand quantum theory (heaven knows, I don't), but a theory that predicts the real world to ten decimal places cannot in any straightforward sense be wrong. Theology not only lacks decimal places: it lacks even the smallest hint of a connection with the real world. As Thomas Jefferson said, when founding his University of Virginia, \"A professorship of Theology should have no place in our institution.\"\n\nIf you ask religious believers why they believe, you may find a few \"sophisticated\" theologians who will talk about God as the \"Ground of all Isness,\" or as \"a metaphor for interpersonal fellowship\" or some such evasion. But the majority of believers leap, more honestly and vulnerably, to a version of the argument from design or the argument from first cause. Philosophers of the caliber of David Hume didn't need to rise from their armchairs to demonstrate the fatal weakness of all such arguments: they beg the question of the Creator's origin. But it took Charles Darwin, out in the real world on HMS _Beagle,_ to discover the brilliantly simple\u2014and non-question-begging\u2014alternative to design. In the field of biology, that is. Biology was always the favorite hunting ground for natural theologians until Darwin\u2014not deliberately, for he was the kindest and gentlest of men\u2014chased them off. They fled to the rarefied pastures of physics and the origins of the universe, only to find Lawrence Krauss and his predecessors waiting for them.\n\nDo the laws and constants of physics look like a finely tuned put-up job, designed to bring us into existence? Do you think some agent must have caused everything to start? Read Victor Stenger if you can't see what's wrong with arguments like that. Read Steven Weinberg, Peter Atkins, Martin Rees, Stephen Hawking. And now we can read Lawrence Krauss for what looks to me like the knockout blow. Even the last remaining trump card of the theologian, \"Why is there something rather than nothing?\" shrivels up before your eyes as you read these pages. If _On the Origin of Species_ was biology's deadliest blow to supernaturalism, we may come to see _A Universe from Nothing_ as the equivalent from cosmology. The title means exactly what it says. And what it says is devastating.\n\n## ABOUT THE AUTHOR\n\nLawrence M. Krauss is Foundation Professor in the School of Earth and Space Exploration and the Physics Department at Arizona State University, as well as Co-Director of the Cosmology Initiative and Inaugural Director of the Origins Project. The Origins program involves new and wide-ranging interdisciplinary research, teaching, and outreach focusing on all aspects of origins: from the origins of the cosmos to human origins, to the origins of consciousness and culture. Krauss is an internationally known theoretical physicist with broad research interests, including the interface between elementary particle physics and cosmology. He received his PhD in physics from the Massachusetts Institute of Technology in 1982, and joined the Harvard Society of Fellows. In 1985, he joined the faculty of physics at Yale University, and then moved to Case Western Reserve University as Ambrose Swasey Professor in 1993. From 1993 to 2005, he served as chairman of the physics department at Case. He is the recipient of numerous international awards for his research and writing, and is the only physicist to receive awards from all three major US physics societies, the American Physical Society, the American Association of Physics Teachers, and the American Institute of Physics.\n\nKrauss is also one of the few prominent scientists today to have actively crossed the chasm between science and popular culture, causing him to be heralded as a unique \"public intellectual\" by _Scientific American_ magazine. For example, besides his books and radio and television work, and his newspaper and magazine commentaries, Krauss has performed solo with the Cleveland Orchestra, narrating Gustav Holst's _The Planets_ at the Blossom Music Center in the most highly attended concert at that venue, and he was nominated for a Grammy Award for his liner notes for a Telarc CD of music from _Star Trek_. In 2005, he also served as a jury member at the Sundance Film Festival.\nAlso by Lawrence M. Krauss\n\n_The Fifth Essence_\n\n _Fear of Physics_\n\n _The Physics of Star Trek_\n\n_Beyond Star Trek:_  \n _From Alien Invasions to the End of Time_\n\n_Quintessence:_  \n _The Mystery of the Missing Mass_\n\n_Atom:_  \n _A Single Oxygen Atom's Journey from the Big Bang to Life on Earth . . . and Beyond_\n\n_Hiding in the Mirror:_  \n _The Quest for Alternate Realities, from Plato to String Theory_\n\n_Quantum Man:_  \n _Richard Feynman's Life in Science_\n\n## Q & A WITH THE AUTHOR\n\n1. What do you really mean by \"nothing\"?\n\nAs I describe in the book, I believe it is most useful to base our definitions on empirically discovered realities rather than abstract philosophical precepts. For me, independent of the question of \"non-existence,\" which takes one off on lots of potentially deep philosophical issues but rather impotent physics ideas, the really seemingly miraculous aspect of our universe, which I also believe has inspired much of the debate about this topic over the centuries, is how all of the stuff we see could have arisen from a universe in which that stuff did not already exist. It seems at the very least to violate the conservation of energy, and more significantly, common sense. But one of the great things about science that I want to convey is that common sense is not necessarily a good guide to understanding nature at the forefront. Our common sense should derive from the universe, rather than vice versa. And the remarkable non-miraculous miracle is that combining quantum mechanics with gravity allows stuff to arise from no-stuff.\n\nNow, that state of no-stuff may not be \"nothing\" in a classical sense, but it is a remarkable transformation nevertheless. So, the first form of \"nothing\" is just empty space. But one is perfectly reasonable in questioning whether this is really \"nothing\" because space is there, as is time. I then describe how it is possible that space and time themselves could have arisen from no space and time, which is certainly closer to absolute nothing. Needless to say, one can nevertheless question whether that is nothing, because the transition is mediated by some physical laws. Where did they come from? That is a good question, and one of the more modern answers is that even the laws themselves may be random, coming into existence along with universes that may arise. This may still beg the question of what allows any of this to be possible, but at some level it is, as I describe at the beginning of the book, \"turtles all the way down.\" There are questions we can address effectively via empirical methods and questions we can ask that don't lead to physical insights and predictions. The trick is to tell the difference between the two.\n\n2. Why \"How?\" and not \"Why?\"\n\n\"Why\" questions are laced with intellectual baggage that is usually unintended. We can ask \"Why are there nine planets around our sun?\" (since for me Pluto will always be a planet!) but by that we don't ascribe significance or purpose to the number nine, as if the universe was designed so there would be nine planets around the sun. If our sun was the only star, then one might ascribe some significance to that particular number (as Kepler did when he tried to explain six planets in terms of Platonic solids). But we are much more interested in the different ways solar systems can arise and how. Asking \"Why?\" presumes some purpose, which need not exist. Ultimately one can keep asking why forever, and the ultimate answer may simply be \"Because,\" but that doesn't illuminate much.\n\n3. If you get rid of God, then does life lose all purpose?\n\nFor me it certainly doesn't\u2014quite the opposite. I would find little purpose living in a world ruled by some divine Saddam Hussein\u2014like character, as my late friend Christopher Hitchens put it, who not only makes all the rules, but punishes those who disobey them with eternal damnation. I find living in a universe without purpose to be amazing, because it makes the accident of our existence and our consciousness even more precious\u2014something to be valued during our brief moment in the sun.\n\n4. What do you mean by \"flat\"? Is the universe flat as a pancake?\n\nI wish I had described this a little more carefully in the hardcover edition, and I have expanded the discussion in this edition. A flat three-dimensional space is just the kind of space you already thought you lived in, where light rays travel in straight lines, and perpendicular axes (x,y, and z) remain perpendicular. In curved three-dimensional spaces neither statement is true. Since mass and energy can curve space locally (i.e., around the sun and earth for example), the big question is what about the global structure of space on the largest scales: Is it curved or not? And it turns out on the largest observable scales, it isn't. And that fact is very telling, as I describe in the book, because it is what one would expect from a universe that arose from nothing.\n\n5. Isn't science just another kind of faith?\n\nAbsolutely not. Scientists change their minds, admit they are wrong, and are happy and eager to throw out ideas that turn out not to work. We don't presume to know for certain the answers to questions before we ask them. So yes, we have faith that the universe is comprehensible, but the greatest thing about science is that our faith is shakable. At any moment we can give up believing in anything we once believed in, if nature suggests otherwise.\n\n6. Does the search for the Higgs boson at the Large Hadron Collider have cosmological significance? What if we discover it? What if we don't?\n\nI discuss this issue in the new preface to this paperback edition. The search for the Higgs boson reflects the capstone of a remarkable intellectual journey that began over fifty years ago, and if it is discovered at the LHC as initial results reported in 2011 suggest, it would validate a theoretical edifice that otherwise would be on shaky ground. In that sense it would be remarkable if our ideas about the Higgs are correct, because usually nature surprises us. Most theories are in fact wrong. If that weren't the case, anyone could do physics. But in any case if the Higgs exists, it means that another aspect of our existence is a cosmic accident. Particles would get their masses by interactions with a background, otherwise invisible, field, like trying to swim through molasses. That means if such a field had not become established in the early universe, we wouldn't be here . . . yet more something from nothing! At the same time, a Higgs discovery at the LHC will likely raise more questions than answers: Why does it have the mass it does? How can we understand its existence in the context of all four known forces in nature? And so on.\n\n7. I have read it claimed that the fundamental laws of nature have nothing to say on the subject of where observed forces came from, or of why the world should have consisted of the particular kinds of particles and fields it does, or why there should have been a world in the first place. Can you comment?\n\nIn fact, it is one of the great developments in particle physics in the past forty years to realize that the properties of the universe we see, which forces are manifest, which particular kinds of fields can exist on observable scales, and which particles have mass and which don't, can arise spontaneously as an accident of our circumstances. This phenomenon is called \"spontaneous symmetry breaking,\" and it basically says that as the universe evolves and cools, some background field can develop throughout space, just like an ice crystal spontaneously forms on your window sill and just as the Higgs field is predicted to have done. (The nature of the specific patterns on your window sill on a frosty day is not predetermined at the beginning of time but arises dynamically.)\n\nWhen this background field develops, it causes some particles to become massive (and therefore become unstable to decay to other particles and disappear) and others to remain massless. It also determines which forces operate at long distances, like electromagnetism, and which don't, like the weak interaction. As for why a world can exist in the first place, once again, spontaneous symmetry breaking\u2014in this case including the possibility of gravity acting\u2014can cause some universes to expand indefinitely and be long lived, while others will disappear in an instant. Thus it can also explain why some worlds exist long enough to ask the question: \"Why is there something rather than nothing?\"\n\n8. Isn't it presumptuous to claim that we know the universe came from nothing, and that science has answered all the outstanding questions of cosmology?\n\nIt is amusing to read this criticism, usually launched by people who haven't read the book. One of the central points of my book is that we don't know all the answers, but what we have learned is remarkably tantalizing, while at the same time butting us up against some profound fundamental questions that may never be truly amenable to empirical falsification.\n\n9. Isn't science compatible with religion? After all, they both explore the same questions, don't they?\n\nScience is compatible with some basic form of deism\u2014namely, we cannot say that a universe, even one that comes from nothing by natural physical processes, was not created with some underlying purpose that may not be evident. (The fact that there is no evidence of purpose makes it of course harder to argue for one, but never mind.) But having said that, science is not compatible with all the strict doctrines of all the world's major religions, and that includes Christianity, Judaism, Islam, as well as some of the minor ones, like Mormonism and Buddhism. And there is good reason for this: The doctrines were written down by people who didn't know how the world worked. Except for Mormonism, which is recent, they were written down when we didn't know that the Earth orbited the Sun!\n\n10. Are you an atheist?\n\nNot in the sense that I can claim definitively that there is no God or purpose to the universe. I cannot claim definitively that there isn't a teapot orbiting Jupiter, as Bertrand Russell once said. It is highly unlikely, of course. But what I can claim definitively is that I wouldn't want to live in a universe with a God\u2014that makes me an anti-theist, as my friend Christopher Hitchens was.\nWe hope you enjoyed reading this Free Press eBook.\n\n* * *\n\nSign up for our newsletter and receive special offers, access to bonus content, and info on the latest new releases and other great eBooks from Free Press and Simon & Schuster.\n\nCLICK HERE TO SIGN UP\n\nor visit us online to sign up at  \neBookNews.SimonandSchuster.com\n\n## INDEX\n\nabsorption lines, , \u201311\n\nAdams, Douglas,\n\nAffleck, Ian, \u201369\n\nAndromeda, , , ,\n\nAntarctic, , ,\n\nanthropic arguments, \u201326, , , \u201337\n\non fundamental forces and constants, \u201376\n\non something-rather-than-nothing question, \u201378\n\nantiparticles:\n\nas appearing to move backward in time, \u201365\n\nasymmetry between particles and, \u201362, \u201360\n\ncreation in electric field of, ,\n\nin Dirac's electron equation, \u201361\n\nas required by quantum mechanics, \u201365, , \u201362\n\nas required by relativity theory, \u201365,\n\n_see also_ positrons; virtual particles\n\nAristotle, \u201373\n\nAtkins, Peter,\n\nbaryons,\n\nBig Bang, xvii, , , , ,\n\nCMBR left from, _see_ cosmic microwave background radiation\n\ndating of, , \u201316, ,\n\ndensity of protons and neutrons in, \u201325\n\nelements created in, , \u201325, , \u201314\n\nevidence for, , , , , \u201315, ,\n\nfuture disappearance of evidence for, \u201319,\n\nGod and, , \u20136\n\nknown physical laws as explanation of,\n\nnucleosynthesis in,\n\nas predicted by relativity theory, , , \u201315\n\nwall between us and, \u201344\n\nBig Crunch,\n\nbiology, , \u201391\n\nblack holes, , \u201356\n\nBogan, Louise,\n\nBohr, Niels, \u201359,\n\nBOOMERANG (Balloon Observations Of Millimetric Extragalactic Radiation and Geophysics), \u201354,\n\nbosons,\n\nBrahe, Tycho, ,\n\nbranes,\n\nBronowski, Jacob, xi, ,\n\nBrowning, Robert,\n\nCamus, Albert,\n\nCepheid variable stars, \u20139,\n\nChaboyer, Brian, , \u201387\n\nchaotic inflation,\n\n_Christmas Carol, A_ (Dickens),\n\nCL 0024 + 1654,\n\nclosed universe, \u201328\n\nas appearing flat, ,\n\nhot and cold spots in, \u201352\n\ninflation in, , \u201370\n\nlight rays as converging in, ,\n\nrate of expansion in, ,\n\ntotal energy in, \u201368\n\nclusters, , , , , , ,\n\ngravitational lensing by, , \u201333\n\nmeasurement of, \u201388\n\nComa cluster,\n\nCopernican principle,\n\ncosmic microwave background radiation (CMBR):\n\nBOOMERANG's attempt to photograph, \u201354\n\ndiscovery of, , , , , , ,\n\nas evidence for Big Bang, , , ,\n\nflat universe implied by, \u201354,\n\nhot spots and cold spots in, , \u201349, \u201352, , ,\n\nmeasurements of, \u201343, \u201354,\n\nobservation vs. prediction of, \u201369\n\nphotons in,\n\nuniformity of, \u201395\n\nas unobservable in far future, \u201311\n\ncosmic myopia, \u201350\n\ncosmic rays, ,\n\ncosmological constant,\n\nas \"anthropically selected,\" \u201326\n\ndark energy as represented by, , \u201373, \u201376, , , , \u201384, , , , , \u201324,\n\nEinstein's introduction and regret of, , \u201358,\n\nCosmological Constant Problem, \u201373\n\ncosmology, xii, xvi, xviii, , ,\n\npossible future end of, \u201319, \u201388\n\ncreator, creationism, xi, xii, xiv, , ,\n\n_see also_ God; theology\n\ndark energy, xii,\n\nage of universe and,\n\namount of, ,\n\nin anthropic argument,\n\nbleak future implied by, \u201319, \u201380\n\ncosmic jerk and, \u201389\n\ncosmological constant as representative of, , \u201373, \u201376, , , , \u201384, , , , , \u201324,\n\ndensity of, , \u201324\n\ndiscovery of, xviii, , \u201317, , ,\n\nevidence for, \u201389, \u20139\n\nexpansion of universe as dominated by, , \u201389, , , , \u20139\n\nfalse vacuum energy as,\n\nin flat universe, , , \u201388,\n\nKrauss and Turner's proposal of, , \u201376,\n\nmeasurement of, \u201373,\n\norigin of, , ,\n\nreal particles and radiation created from, ,\n\nin string theory,\n\nzero gravitational energy allowed by,\n\ndark matter, ,\n\namount of, \u201326, \u201328, , , \u201356\n\nin closed universe,\n\ncomposition of, , \u201335\n\ndensity of galaxy clusters as dominated by,\n\ndetection of, , , , \u201337, , ,\n\nexperiments on, \u201336\n\nflat universe as caused by, , , , , ,\n\nratio of visible matter to, , , , ,\n\nZwicky's early prediction of,\n\nDarwin, Charles, , , , \u201391\n\nDawkins, Richard, xix, \u201391\n\nDemarque, Pierre,\n\ndeuterium, abundance of, ,\n\nDickens, Charles, ,\n\ndimensions, extra:\n\npossible evidence for, \u201338\n\nin string theory, , , , , \u201335\n\nundetectable universes in, ,\n\nDirac, Paul, , \u201360, \u201369,\n\nDoppler effect,\n\nD'Souza, Dinesh,\n\nE = mc2, ,\n\nEddington, Arthur Stanley, \u20135\n\nEinstein, Albert, , ,\n\nexperiment as important to, \u20133\n\ngravitational lensing paper of, \u201331, \u201337\n\nidea of expanding universe rejected by, , \u201357,\n\nand question of God's choice, \u201361, \u201385\n\nuniversal speed limit discovered by,\n\n_see also_ relativity, general theory of; relativity, special theory of\n\nelectric fields, , ,\n\nelectromagnetism, , ,\n\nquantum theory of, ,\n\nelectrons, , , ,\n\nantimatter form of, \u201360\n\nin early universe,\n\nmass of, , ,\n\nas moving faster than light, \u201365\n\npotential for existence of,\n\nin quantum levels,\n\nspin of,\n\nwave function of, , , \u201367\n\nelementary particles, ,\n\n_see also specific particles_\n\nempty space, xvi\n\nenergy of, _see_ dark energy\n\nreal particles created in,\n\nvirtual particles and fields in, \u201365, \u201371, \u201398, ,\n\n_see also_ nothing\n\nenergy:\n\nconservation of, \u2013104, , , ,\n\ndark, _see_ dark energy\n\ndefinition of,\n\nkinetic, \u2013101\n\nnegative, \u2013104,\n\npotential, \u2013101\n\nrest,\n\nenergy density, , , \u20134\n\nevent horizon,\n\nevolution, biological, xiii, ,\n\nfalse vacuum energy, , , , ,\n\nFaraday, Michael,\n\nFeynman, Richard, ,\n\non accuracy of quantum theory, \u201390\n\non fundamental laws, xiii,\n\nnecessity of antiparticles demonstrated by, \u201365, \u201362\n\nsum over paths formalism developed by, \u201363, \u201369\n\nfield lines, \u201367\n\n_Flatland_ (Abbott),\n\nFlatness Problem, ,\n\nflat universe, , ,\n\nclosed universe as giving appearance of, ,\n\ndark energy in, , , \u201388,\n\ndark matter as cause of, , , , , ,\n\nevidence against, \u201354, \u201356,\n\nevidence for, , , \u201356, \u201376, , \u201393, ,\n\nexpansion rate of, , \u201388,\n\nas expected of universe arising from nothing, \u201352\n\nas explained by inflation, , , \u201351, ,\n\nhot and cold spots in,\n\nKrauss and Turner's prediction of, , \u201376,\n\nlight rays as straight in, ,\n\nslowed expansion of,\n\ntotal mass needed for, , ,\n\nzero gravitational energy in, ,\n\nzero total energy in,\n\nfundamental constants of nature, , , \u201376\n\nfundamental forces, , , \u201376\n\n_see also specific forces_\n\ngalaxies:\n\ngravitational lensing by,\n\ninstability of,\n\nredshift of, \u201311, , \u20137\n\nrotation of, \u201324\n\nspiral, _see_ spiral galaxies (spiral nebulae)\n\nsuperclusters of,\n\n_see also_ clusters; Hubble's Law\n\nGalileo Galilei, ,\n\nGauss, Carl Friedrich, \u201341\n\nGenesis, ,\n\ngeodetic survey maps, \u201341\n\ngeometry, \u201341\n\nof universe, _see_ closed universe; flat universe; open universe\n\nGod, ,\n\nBig Bang and, , \u20136\n\nDarwin's removal of need for, , \u201391\n\nEinstein's \"choice\" question to, \u201361, \u201385\n\nfundamental constants and,\n\ninfinite regress and, xii, xv\n\nin Newton's work, ,\n\nOccam's Razor and,\n\nand origin of morality,\n\nPius XII's attempt to prove existence of, ,\n\nredundancy of,\n\nsomething from nothing as created by, xv, \u201375\n\nand total gravitational energy,\n\n_see also_ theology\n\nGod of the Gaps, \u201346\n\nGrand Unified Theory,\n\ngravitational lensing, \u201335\n\ndark matter detected by, ,\n\nEinstein's paper on, \u201331, \u201337\n\nZwicky's proposed use for, \u201332,\n\ngraviton,\n\ngravity, xvi\n\nas attractive, , ,\n\nof dark matter,\n\nand expansion of universe, , \u201380,\n\nmotion of galaxies affected by, \u201324\n\nnegative energy allowed by, \u2013104,\n\nnegative pressure of,\n\nNewton's theory of, , , , , , \u201366\n\nspeed of, ,\n\nas unified with other forces,\n\nweakness of, ,\n\nweighing superclusters of galaxies with,\n\n_see also_ relativity, general theory of\n\nGreene, Brian,\n\nGuth, Alan, xviii, , , , , ,\n\nHarris, Sydney,\n\nHartle, Jim,\n\nHawking, Stephen, , \u201364, ,\n\nHeisenberg, Werner, \u201359\n\n_see also_ Uncertainty Principle\n\nhelium, ,\n\nabundance of, , \u201325, , \u201312,\n\nHeraclitus of Ephesus,\n\nHigh-Z Supernova Search Team, ,\n\nHilbert, David, \u20133\n\nHitchens, Christopher, xviii\u2013xix, , , ,\n\n_Hitchhiker's Guide to the Galaxy, The_ (Adams),\n\nHomer,\n\nHorizon Problem, , ,\n\nHoyle, Fred,\n\nHubble, Edwin, ,\n\nage of universe estimated by, \u201316\n\nexpansion of universe discovered by, , , , , , , ,\n\nnew galaxies discovered by, \u20139\n\nHubble's constant, \u201316, , \u201380\n\nHubble's law, \u201316, ,\n\nHubble Space Telescope, ,\n\nHumason, Milton, , ,\n\nHume, David,\n\nhydrogen, , , \u201368\n\nabundance of, , \u201325, , \u201312,\n\nhyperbolic geometry,\n\ninfinite regress, xii, xv\n\ninfinities, xii\u2013xiii,\n\ninflation, , , ,\n\nas allowed in general relativity, \u201397\n\nas beginning from nothing,\n\nin closed universe, , \u201370\n\nconservation of energy and, \u2013104\n\nenergy density and, , \u20134\n\nas eternal, \u201329, ,\n\nflat universe explained by, , , \u201351, ,\n\nhomogeneous universe explained by, , \u201351\n\nHorizon Problem solved by,\n\nmultiverses created by, \u201329\n\nquantum fluctuations and, \u2013104,\n\nsmall-density fluctuations in, \u201351\n\ntotal gravitational energy and, \u20133\n\nzero gravitational energy and, , ,\n\ninstanton,\n\niron, ,\n\nJames, William, xii\n\nJefferson, Thomas,\n\njerk, \u201389\n\n_Journey Around My Room_ (Bogan),\n\nKepler, Johannes, \u201320,\n\nthree laws of planetary motion discovered by,\n\nKernan, Peter,\n\nkinetic energy, \u2013101\n\nKrauss, Lawrence, xvi\u2013xviii, , \u201376, , , , \u201387, \u201318\n\nLamb, Willis, \u201367\n\nlandscape, \u201330, \u201335, \u201377\n\nLarge Hadron Collider, \u201336,\n\nlast scattering surface, \u201347, , \u201354,\n\nBOOMERANG's attempt to picture, \u201354\n\ncharacteristic scale of, \u201345\n\nas earliest visible point of universe, \u201344\n\ngeometry of universe revealed by, \u201347,\n\nLeavitt, Henrietta Swan, \u20138\n\nLeeuwenhook, Antonie Philips van,\n\nLema\u00eetre, Georges, \u20136, , , , , \u201315\n\n\"Lens-Like Action of a Star by the Deviation of Light in the Gravitational Field\" (Einstein), \u201331, \u201337\n\nlife, origin of, , , ,\n\nlight:\n\nbending of, \u201327, \u201330, , ,\n\ngeometry of universe and,\n\nhuman ability to see, \u201350\n\nquanta of,\n\nwavelengths of, , ,\n\nlight speed, as cosmic speed limit, , , , , ,\n\nLinde, Andrei, \u201328,\n\nlithium:\n\nabundance of, , \u201325, , \u201312,\n\nLobachevsky, Nikolai Ivanovich, \u201342\n\nMandl, Rudi, , ,\n\nmass:\n\ndensity in universe of,\n\nof electrons, , ,\n\nof photons,\n\nof protons, \u201361, ,\n\nrest, ,\n\ntotal amount of, , , , , ,\n\nmatter:\n\nconstant creation of,\n\nas created by quantum fluctuation, \u201351,\n\nas created in electric field, ,\n\ndensity of, , , , , ,\n\ndisappearance of,\n\nenergy density of, \u201324\n\norigin of, ,\n\nratio of dark matter to, , , , ,\n\nMercury, orbit of, ,\n\nmetabolism, ,\n\nmeta-galaxy, , \u201311, ,\n\nMies van der Rohe, Ludwig,\n\nMilky Way, , , , , ,\n\nmiracles, \u201342\n\nMonopole Problem, ,\n\nmorality, xvi, \u201372\n\nmotion, Newton's laws of,\n\nMount Stromlo Observatory,\n\nMount Wilson Observatory, ,\n\nmultiverse, , \u201329, \u201335, \u201377,\n\nGod's choice and, \u201385\n\nmuons,\n\nnebulae, \u20137, \u201332\n\n_see also_ spiral galaxies (spiral nebulae)\n\nnegative pressure, ,\n\nneutrons, ,\n\namount of, ,\n\nantiparticle of,\n\ndecay of,\n\ninitial density of,\n\nnuclear reactions of,\n\nneutron stars,\n\nNewton, Isaac:\n\nGod as viewed by, ,\n\ngravitational theory of, , , , , , \u201366\n\nlaws of motion formulated by,\n\nprism experiment of,\n\nsingle theory sought by,\n\nNobel Prize, , ,\n\nnothing:\n\ndefinitions of, xiii\u2013xv,\n\nas effected by virtual particles, \u201371,\n\nenergy in,\n\nas possible end of universe, \u201380,\n\nuniverse as originating from, xvii, \u201343, , \u201370\n\nas unstable, \u201370, , ,\n\nweight of,\n\n_see also_ empty space; something-rather-than-nothing question\n\n_On the Origin of Species_ (Darwin),\n\nopen universe,\n\nin computer simulation,\n\ndeceleration of,\n\neternal expansion of, ,\n\nevidence for, \u201337,\n\nexpansion rate of, ,\n\nhot and cold spots in,\n\nlight rays as bent in, , ,\n\nLobachevsky's investigation of, \u201342\n\nOrigin of Life workshop, ,\n\nOrigins Project, ,\n\noxygen, ,\n\nPais, Abraham,\n\nparticle accelerators, \u201336,\n\nparticle physicists, , ,\n\nparticle theory, xvi\n\nperihelion, ,\n\nperiod-luminosity relationship, \u20138\n\nPerlmutter, Saul, , ,\n\nphase transitions, \u201396, ,\n\nphilosophy, xiii\u2013xiv,\n\nphotons, , , \u201367\n\nphysical laws, xiii, , ,\n\nas arising from nothing, xv, , \u201380\n\nand asymmetry of matter and antimatter, \u201359\n\nas complete and necessary for describing universe's evolution,\n\nmiracles vs., \u201342\n\norigins of, xi\u2013xii\n\npossible randomness of, xvii, \u201376\n\n_Physical Review,_\n\nPinker, Steven, \u201372\n\nPius XII, Pope, ,\n\nPlanck, Max,\n\nPlanck-time, ,\n\nplanets, motion of, ,\n\nplasma, , , \u201311\n\nPlato, xii, ,\n\npositrons, \u201360, \u201365, \u201368,\n\npotential energy, \u2013101\n\npressure, negative, ,\n\nprotons,\n\namount of, ,\n\nantimatter of,\n\ndecay of,\n\nin early universe, ,\n\nmass of, \u201361, ,\n\nnuclear reactions of,\n\npositrons mistaken for, \u201361\n\nsmashing of, \u201336\n\nstructure of, \u201370\n\nquantum fluctuations, \u201398\n\nconservation of energy as limitation to, \u201354\n\nas explanation of everything, , , \u201351, , , ,\n\nas important in inflation, \u2013104,\n\n_see also_ inflation\n\nquantum gravity, ,\n\nblack hole radiation and,\n\nand fate of closed expanding universes,\n\ninflation created by,\n\nspace created by, \u201364\n\nvirtual particles and, \u201372\n\nquantum levels,\n\nquantum mechanics, ,\n\naccuracy of predictions of, \u201369, \u201390\n\ndevelopment of, \u201359\n\nrelativity theory and, \u201359, ,\n\nof space-time, ,\n\n_see also_ Uncertainty Principle\n\nquantum vacuum, xiv\n\nquarks, \u201370,\n\nquasars, \u201332\n\n_Quintessence_ (Krauss),\n\nradiation:\n\nas caused by meeting of particle and antiparticle,\n\nas created by quantum fluctuation, \u201351\n\ndensity of, ,\n\nin early universe, \u201344\n\nquanta of,\n\nredshift, \u201311, , \u201382, \u201386,\n\ndistance-versus-, , \u201382, \u201386,\n\nin far future, \u20137\n\nRees, Martin,\n\nrelativity, general theory of:\n\napparent problem with, , ,\n\nBig Bang predicted by, , , \u201315\n\ncollapse of closed universe predicted in,\n\ncurvature of universe in, , ,\n\ndark matter required for flat universe in,\n\ndensity of universe in,\n\nequations of, \u201358\n\nevolution of universe explained by, , ,\n\nexperimental evidence for, \u20133, \u201327\n\nfaster-than-light growth of space allowed by, \u201397\n\nlabeling of space and time as arbitrary in,\n\nlight bent in, \u201327, \u201334\n\nnegative energy in, , ,\n\nnegative pressure in,\n\nnonstatic universe predicted by, \u20132, , , \u201357\n\nquantum mechanics and, \u201360, ,\n\nrest energy in,\n\nspace as curved in, \u201327, \u201334, , \u201342,\n\ntotal energy in, \u201366\n\nweighing the universe with,\n\n_see also_ cosmological constant\n\nrelativity, special theory of, ,\n\nantiparticles required by, \u201365,\n\nand observer-dependent measurements, ,\n\nreligion, _see_ God; theology\n\nRetherford, Robert C.,\n\nRichard Dawkins Foundation, xvii\n\nRiess, Adam,\n\nRNA, ,\n\nRoman Catholic Church, xii, , ,\n\nRubin, Vera,\n\nRumsfeld, Donald,\n\nRussell, Bertrand, xii\n\nSagan, Carl, \u201383\n\nScherrer, Robert, , \u201318\n\nSchmidt, Brian,\n\nSchr\u00f6dinger, Erwin, \u201359\n\n_see also_ electrons, wave function of\n\nSchwarzchild solution,\n\nscience, xiii\u2013xiv\n\nGod and,\n\npurposes of, xv\n\nsomething-rather-than-nothing question probed by, xiii\n\nthree key principles of, xvi\n\n_Science,_ , ,\n\n_Scientific American,_\n\nself-replicating cells,\n\nShapley, Harlow, \u20137,\n\nsingularity,\n\nSlipher, Vesto, ,\n\nsnowflakes, xi\n\nsolar eclipse, \u201327\n\nsomething-rather-than-nothing question, \u201345, \u201389,\n\nanthropic answer to, \u201378\n\nboundary conditions for, \u201370\n\nas \"how\" question,\n\nand instability of nothing,\n\nmetaphysical answer to, \u201375\n\norigin of universe and, xiii\n\nphysical laws as complete for, \u201352\n\nphysical laws unnecessary for, , \u201380\n\nas possibly insignificant, \u201379\n\nquantum fluctuations as answer to, \u201351, , ,\n\nquantum gravity's answer to,\n\nas scientific, xiii\n\nshifting meaning of, \u201383\n\nspace unnecessary for, \u201370\n\nspace:\n\nas arising from nothing, xiv\u2013xv, \u201370\n\nas created by quantum gravity, \u201364\n\ncurvature of, \u201327, \u201335, , \u201342, ; _see also_ closed universe; flat universe; open universe\n\nfaster-than-light growth of, \u201397\n\nin general theory of relativity, , \u201327, \u201329, ,\n\ninflation of, _see_ inflation\n\nas possibly infinite, xiii\n\nquantum theory as applying to,\n\nspace-time, quantum mechanics, ,\n\nspectrum, \u201310, ,\n\nSpinoza, Baruch,\n\nspiral galaxies (spiral nebulae), ,\n\nabsorption lines of, \u201311\n\n\"standard candle,\" , \u201379\n\nstars:\n\nbrightness of, \u20138,\n\ncomposition of, ,\n\nelements made in, \u201319, \u201314\n\nmain sequence,\n\nvariable,\n\n_Star Trek,_\n\nstatic electricity,\n\nStenger, Victor,\n\nstring theory, , \u201335,\n\nsum over paths formalism, \u201363, \u201369\n\nSun, spectrum of,\n\nsuperclusters,\n\nsuperconductivity,\n\nsupernaturalism, \u201342\n\nsupernova:\n\nBrahe's observation of,\n\nelements created in, \u201318\n\nnumber of, , \u201321\n\nand rate of expansion of universe, \u201382, \u201386,\n\nType a, , , \u201382, \u201386,\n\nSupernova Cosmology Project, \u201382\n\nsuperstrings,\n\nsupersymmetry,\n\nsymmetry, , ,\n\nSysiphus,\n\ntauons,\n\ntheology, xi, xiii\u2013xiv, xvi, , \u201346, ,\n\n_see also_ God\n\nTheory of Everything, ,\n\n\"Theory of Positrons, A\" (Feynman), \u201365\n\nThomas Aquinas, xii, , , ,\n\nthought experiments,\n\ntime:\n\nantiparticles as appearing to move backward in, \u201365\n\nas arising from nothing, xiv\u2013xv\n\ncharacteristic scale imprinted on last scattering surface by, \u201345\n\nin general theory of relativity, ,\n\nas possibly infinite, xiii\n\n_see also_ space-time\n\ntotal energy:\n\nin closed universe, \u201368\n\nin flat universe,\n\nproblems in measurement of,\n\nas total gravitational energy plus energy associated with mass,\n\ntotal gravitational energy, \u20133,\n\ndefinition of,\n\nin inflation, \u20133\n\nNewtonian equation for,\n\nas nonaribtrary, \u201349\n\nin total energy,\n\nin universe arising from nothing, \u201352,\n\ntriangles, sum of angles in, \u201340,\n\nTurner, Michael, , \u201376, ,\n\nType Ia supernova, , , \u201382, \u201386,\n\nTyson, Tony, , \u201334\n\nUncertainty Principle, , \u201365, , , ,\n\nuniverse:\n\nage of, , \u201316, , , , \u201387, , ,\n\nalleged rotation of,\n\naverage density of,\n\nboundary conditions of creation from nothing of, \u201370\n\ncooling of, ,\n\nend of, , , ,\n\nas eternal, \u201374\n\nhomogeneity of, ,\n\nmeasurement of curvature of, \u201354, , ,\n\nmultiverse and, , \u201329\n\norigins of, xv, xvi, xvii, xviii, , , , ; _see also_ Big Bang\n\nphase transition of, \u201396,\n\nas possibly infinite, xiii\n\nrecollapse and disappearance of,\n\nsize of, \u20139\n\nstatic view of, , , , \u201357, , , ,\n\ntotal mass in, , , , , ,\n\nweight of, \u201337,\n\n_see also_ closed universe; flat universe; inflation; open universe\n\nuniverse, expansion of, xii, xvii, \u20134, , , , , , , ,\n\ndark energy and, , \u201389, , , , \u20139,\n\ngravity and, , \u201380,\n\nrate of, \u201315, , \u201389, , , ,\n\nas unobservable in far future, \u201319\n\n_see also_ Hubble's law\n\nVilenkin, Alex, ,\n\nVirgo supercluster,\n\nvirtual particles:\n\nDirac equation and, \u201369\n\nempty space effected by, \u201369, \u201371, \u201398, , , ,\n\nas explanation of static electricity,\n\nFeynman's proof of, \u201365\n\nindirect evidence for, \u201367, \u201371,\n\nin quantum gravity,\n\nshort lifetime of, \u201368,\n\nzero total energy of,\n\nvirtual photons, , \u201367\n\nvirtual universes,\n\nWeinberg, Steven, ,\n\nWilczek, Frank, xviii, \u201335,\n\nWilkinson, David,\n\nWilkinson Microwave Anisotropy Probe (WMAP), \u201354,\n\nWitten, Edward, \u201333\n\nX-ray emissions,\n\nZel'dovich, Yakov, \u201373\n\nZwicky, Fritz, \u201332,\n\nFREE PRESS  \nA Division of Simon & Schuster, Inc.  \n1230 Avenue of the Americas  \nNew York, NY 10020  \nwww.SimonandSchuster.com\n\nCopyright \u00a9 2012 by Lawrence M. Krauss\n\nAll rights reserved, including the right to reproduce this book or portions thereof in any form whatsoever. For information address Free Press Subsidiary Rights Department, 1230 Avenue of the Americas, New York, NY 10020.\n\nFirst Free Press hardcover edition January 2012\n\nFREE PRESS and colophon are trademarks of Simon & Schuster, Inc.\n\nJacket design by James Perales  \nCover photograp (stars) \u00a9 Getty\/Luke Peterson Photography\/Flickr  \nAuthor photo \u00a9 Nancy Dahl-Tacconi\n\nThe Simon & Schuster Speakers Bureau can bring authors to your live event. For more information or to book an event contact the Simon & Schuster Speakers Bureau at 1-866-248-3049 or visit our website at www.simonspeakers.com.\n\nLibrary of Congress Cataloging-in-Publication Data  \nKrauss, Lawrence Maxwell.  \nA universe from nothing : why there is something rather than nothing\/ Lawrence M. Krauss ; with an afterword by Richard Dawkins.  \np. cm.  \nIncludes index.  \n1. Cosmology. 2. Beginning. 3. End of the universe. I. Title.  \nQB981.K773 2012 2011032519  \n523.1'8\u2014dc23\n\nISBN 978-1-4516-2445-8  \nISBN 978-1-4516-2447-2 (ebook)\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":"\n\n_This book is, in essence, a love story. It is a falling in love with self, with another, and with Earth story. The journey of being in love\u2014of letting love in to shape you, grow you, change you\u2014takes you through many landscapes, territories unknown and unfamiliar to you, to the windy, rocky precipice of deep change. The journey is punctuated with surprises, brilliant insights, deep joy, and a discovery of rich, dynamic, internal strengths, and genius. Travel well._\n\n _Dedicated to  \nHailey Elizabeth Kennington  \nand  \nAraina Marie Kennington.  \nAnd for Darian Sahara Marie Ronan\u2014to the  \nday we meet on this extraordinary journey.  \nAlso, to the young girl inside each of us._\n\n _May your spirits stay wild, unbroken, and joined to Earth.  \nMay your bodies be strong and your hearts beat passionately.  \nMay your minds always be hungry.  \nMay you know deep love and ecstatic joy.  \nMay wonder and awe be your constant companions.  \nAnd may your souls be whole and thrum from the adventure._\n\nThanks are extended to the following authors, translators, and publishers for granting permission to reprint:\n\n\"Who Makes These Changes?\" by Rumi, which appears on page 67, is from _The Essential Rumi_ , translated by Coleman Barks, and is used by permission of the translator. _The Essential Rumi_ was originally published by HarperSan Franciso.\n\n\"A Third Body\" and \"One Source of Bad Information,\" which appear on pages 198\u201399 and 213\u201314, are from _Loving a Woman in Two Worlds_ and _Morning Poems_ by Robert Bly, and are used by permission of the author. _Loving a Woman in Two Worlds_ was originally published by Dial\/Doubleday. _Morning Poems_ was originally published by HarperCollins.\n\n\"Flowers\" and \"The Movement of Great Things,\" which appear on pages 98\u201399 and 169\u201370, are from _The Taste of Wild Water: Poems and Stories Found While Walking in Woods_ by Stephen Harrod Buhner, and are used by permission of the author and the publisher, Raven Press, Silver City, New Mexico.\n\n\"Fearing for My Life,\" \"Do You Recognize Me Now,\" and \"A Third Body,\" which appear on pages 33, 187\u201389, and 202\u20133, are from _Dancing with the Beloved_ by Paul Ferrini, and are used by permission of the author and the publisher, Heartways Press, Inc.\n\nThe text from _The Sexual Teachings of the White Tigress: Secrets of the Female Taoist Masters_ by Hsi Lai, which appears on page 118, is used by permission of the author. _The Sexual Teachings of the White Tigress_ was published by Destiny Books, Rochester, Vermont.\n\nThe poem by Rumi beginning \"If you want to know god,\" which appears on page 166, is from _The Spiritual Practices of Rumi_ by Will Johnson, and is used by permission of the author. This version of the poem is Johnson's rewording of the Coleman Barks translation published in _The Essential Rumi_. _The Spiritual Practices of Rumi_ was published by Inner Traditions, Rochester, Vermont.\n\nSEX AND THE INTELLIGENCE  \nOF THE HEART\n\n\"A beautifully written and honest book about exploring the depths of sensuality and consciousness and how this awareness affects our lives, relationships, and the planet.\"\n\nBRIGITTE MARS, AUTHOR OF _T HE SEXUAL HERBAL_\n\n\"With unflinching honesty, Julie McIntyre beautifully delivers a critically important message for all of us on the path to self-realization: a rarified spirituality divorced from the human energies and impulses of the erotic simply doesn't work. Ecstasy is our birthright, and the shaming of our bodies needs to stop now.\"\n\nWILL JOHNSON, AUTHOR OF _B REATHING THROUGH THE WHOLE BODY_ AND _R UMI'S FOUR ESSENTIAL PRACTICES_\n\n\"Sex, Earth, and Spirit are one. Fearless sex, healed earth, healed soul: all one. The afflicted modern mind has forgotten this truth; contemporary sages merely glimpse it. McIntyre has done the profound work to reclaim and thoroughly live it. With this book, she has created a glorious bible with innumerable inroads to joy.\"\n\nAPHRODITE PHOENIX, AUTHOR OF _A RE THEY BAD GIRLS OR BRILLIANT?_\n_Acknowledgments_\n\nFor all those who've awakened my heart, which beats stronger because of you.\n\nThis being my first book, there are many people\u2014friends, companions, associates, partners-in-crime, pioneers, and believers\u2014for whom I am deeply grateful and who I would like to acknowledge. To all of you, who in no small way encouraged me to lift the bushel basket and reveal what has been tenderly hidden.\n\nTo Stephen Buhner, more than you know, I am eternally, deeply grateful to you, for believing in me, for your love, companionship, deep friendship, and for claiming my soul from the lost and found. Thanks to Trishuwa for your love, for transformation, and for friendship; and to Margaret Rhode for your friendship, love, and all the treats you left at my door. I thank M. John Fayhee for encouraging me to hold nothing back; Jaxon Burgess for your enduring and caring friendship; and Mark Heffernan for your original songs, stories, and friendship. Thanks to Julie and Tanya for giving me a column in _Tapestry_ magazine month after month in which to practice. And thanks to Phil for being a great pen pal.\n\nI thank the men in my life: my lovers, husbands, and partners; James P. Ronan for marriage and friendship and Jon (Hawk) Stravers Sr.; for the Driftless adventures; and Jon Stravers Jr., whose spirit kept showing up in moments of deep grief\u2014may you be whole in spirit on the other side, you are missed dearly. And I thank my son, Garrett Ronan who helped me along the way.\n\nI thank the gang at Isaac's Bar, the Buffalo Bar, and Diane's Parlor in Silver City, New Mexico; the Buckhorn Saloon in Pinos Altos, New Mexico; and the Main Entrance in Prairie du Chien, Wisconsin. I thank you for your companionship and laughter, support and friendship. I thank you all for being part of my life and encouraging me to be who I am: Allan, Shawn, Fred, Farhad, Luan, Jean, Gay, Erika, Hans, Aleisha, Colette, Benjamin, Challa, Calixte, Kevin, Sahuara, Stephanie, Lisa, Jay, Rebecca, Merla, my Irish family\u2014Lucy, Joseph, Michael, and especially Nikki, for your friendship and support and for reading the manuscript.\n\nI thank Flick, who with aging arthritic joints and loss of hearing traveled countless miles up and down the stairs to my loft and napped on the floor near me while I wrote, fretted, and, at times, excavated this book.\n\nI gratefully acknowledge John Seed, Arne Naes, Johanna Macy, Dolores La Chapelle, and the early Deep Ecologists whose fearless devotion has been a well of inspiration and influence. I thank Rosemary Gladstar for giving me a place to start teaching this work, and the women at the Women's Herbal Conferences, my apprentices and students, who sat in my classes and said to me afterward: \"You have to write this book.\" I am grateful to the life-drawing classes at Western New Mexico University, where I was paid to pose nude while I spent time in interior work.\n\nI thank Jon Graham whose initial letter of acceptance of the manuscript I referred to countless times to keep me going; and Laura and Kate, editors from heaven, who corrected my grammatical inadequacies, made suggestions, and who work to make authors' dreams come true.\n\nAnd thanks to my birth family\u2014for provoking my destiny.\n\nTo the invisibles: I am deeply indebted to the spirits of the Driftless Region of the Upper Mississippi River Valley, the Land of Enchantment, those of Southwest New Mexico who have heard and seen all of me and my internal world and who have walked through the writing of this book side by side with me; Venus, the goddess of love, beauty, and sexuality who rules my Libra sun and Taurus moon; Eros, the god of sexual love (and the chaos that ensues) and beauty, and from whom the energy and word _erotic_ originates; Psyche, the soul, for drawing forth character; the Muses whose names inspire art, science, and wonder; Dionysus, god of the vine, half-mortal son of Zeus, born of fire and nursed by rain; Artemis, lover of woods and the wild chase over the mountain; and Pan, whose home is all the wild places\u2014thickets and forests and mountains.\n\nI pray I have not disappointed any of you.\n**_Contents_**\n\nCover Image\n\nTitle Page\n\nDedication\n\nEpigraph\n\nAcknowledgments\n\nIntroduction: It Stops with Me\n\n **PART ONE**\n\n ** _The Fall from Earth_**\n\n **Chapter 1:** Intimacy: Food That Feeds the Soul of Love\n\nTHE YEARNING FOR CONNECTION\n\nIN THE LUMINOUS WORLD OF INTIMACY\n\nFOOD FOR THE SOUL OF LOVE\n\nLEARNING TO HIDE\n\nBIRTHING THE AUTHENTIC SELF\n\nNO SUCH THING AS SAFE\n\n **Chapter 2:** Autonomous Personhood\n\nWHAT IS THE SOUL?\n\nWHO IS \"I\"?\n\nFILLING THE LONG BAG\n\nTHROWING OUT THE OLD SCRIPTS\n\nRETRIEVING YOUR SOUL\n\n **Chapter 3:** Getting to Know You\n\nTHE PARENT, ADULT, AND CHILD IN YOU\n\nTHE PREDATOR INSIDE\n\nACKNOWLEDGING ALL OF YOU\n\nBEGINNINGS ARE SUCH DELICATE TIMES\n\nTHE COUNCIL WITHIN\n\nFINDING THE NURTURING PARENT\n\nRECOGNIZING AND INTEGRATING ALL THE PARTS OF YOU\n\nDEVELOPING CHILD-TO-CHILD INTIMACY\n\nCROSS TRANSACTIONS\n\n **Chapter 4:** The Numinous\n\nSHAKEN BY THE NUMINOUS\n\nIN THE CHURCH OF EARTH\n\nWITNESSED BY SPIRIT\n\n **PART TWO**\n\n ** _Earthly Sexual Body_**\n\n **Chapter 5:** Human Beings\n\nCOUNSEL WITH THE STAR PEOPLE\n\nORIGINS OF THE SEPARATION\n\nIT'S ALL FOR THE CHILDREN\n\nQUELLING THE SONG OF THE WATERS\n\nBECOMING WHOLE\n\nA WORKING DEFINITION OF WILD\n\nDEATH AND DISEASE THROUGH SEPARATION\n\n **Chapter 6:** Finding the Wild\n\nA SACRED VOICE IS CALLING YOU\n\nGOING WILD\n\nCOMING TO LIFE\n\nA DICHOTOMY THAT DAMAGES\n\nTHE SENSING BODY\n\n **Chapter 7:** Choosing Another Way\n\nHEALING OURSELVES, HEALING EARTH\n\nRECLAIMING THE BODY'S SENSING: THE HEART OF THE MATTER\n\nEXPERIENCING THE INTERWORLD\n\nANCIENT HISTORY\n\n **Chapter 8:** The Language of Love\n\nLOVING WITHOUT RESERVATION\n\nTHE SMELL OF LOVE\n\nLEARNING TO LET GO\n\nTHE LANGUAGE OF SENSUALITY\n\n **Chapter 9:** Sacred Sex\n\nLOVING THROUGH SPACE AND TIME\n\nNURTURING THE THIRD BODY\n\nA JOINING OF SOULS\n\nEGO STATES AND SEXUAL POSITIONS\n\n **PART THREE**\n\n ** _Challenges You May Meet on the Road_**\n\n **Chapter 10:** The Dance of Trust\n\nBREAKING THE TRUST\n\nMENDING THE TRUST\n\nINJUNCTIONS\n\nSAY IT OUT LOUD\n\nSEXUAL SECRETS\n\nTRUST, INTIMACY, AND SACRED SEX\n\n **Chapter 11:** Healing Shame\n\nWORKING THROUGH SHAME\n\nTHE SHAME OF SEX\n\nTOUCHING WITH LOVE AND ATTENTION\n\nSEEING BELOW THE SURFACE\n\n **Chapter 12:** Freeing Ourselves from Sexual Tyranny\n\nSACRED SEX AND THE WORLD'S OLDEST PROFESSION\n\nTHE BEGINNINGS OF SEXUAL LIBERATION\n\nTHE POLITICS OF SEXUALITY: AN ANCIENT ARENA FOR CONFLICT\n\nPORNOGRAPHY: HARMFUL OR CRUCIAL?\n\nSEXUAL PLEASURE IS OUR BIRTHRIGHT\n\nREADING THE INVISIBLE INTO VISIBILITY\n\n **Chapter 13:** Healing the Human Soul\n\nSEEING WHO YOU ARE\n\nSELF-NURTURING\n\nECOSEXUALITY\n\nFinal Words\n\nANOTHER NEW SEXUAL REVOLUTION\n\n **APPENDIX:** Talking to Children and Teens about Sex\n\nADOLESCENTS, SEX, AND INTIMACY\n\nFootnotes\n\nEndnotes\n\nBibliography and Recommended Reading\n\nIndex\n\nAbout the Author\n\nAbout Inner Traditions \u2022 Bear & Company\n\nBooks of Related Interest\n\nCopyright & Permissions\nINTRODUCTION\n\n _It Stops with Me_\n\n_The basic problem is that everybody has sexual thoughts._\n\nJUNE JORDAN, \"A COUPLE OF WORDS ON BEHALF OF SEX (ITSELF),\" IN SOME OF US DID NOT DIE\n\n _Everything that people do is connected with \"sex\": politics, religion, art, the theater, music, is all \"sex.\"_\n\nG. I. GURDJIEFF, _P ARABOLA_\n\nI placed my feet on the trailhead, half hidden under leaf litter. Ideas, thoughts, and images of where this trail might lead, written on disjointed notes, were scattered about the forest floor. Some were stuck on branches set there by winds, still others continued dancing about, flitting here then there. Once the start of this journey called \"writing my first book\" began, there was no turning back, no u-turns allowed, no way of knowing where it would lead, who would appear, or what news they would bear from distant lands. And though I could feel the presence of invisibles\u2014those of myth, of Earth, early travelers of old, writers of the territory and my own destiny\u2014falling in stride behind and alongside me, not knowing where this trail was going was as important as the trail itself.\n\nWriting this book was an exercise in personal psychological restructuring. The experience was the best, and most difficult, therapy I've had, ever. At times I was torn up inside, and at other times, the experience brought me unbounded joy and confidence. There were days when the only reason I wrote was because I had given my word. I have always wanted to be a writer; to not write would be an abomination to my soul. To say it was a love-hate relationship is trite; however, true it is. This book, and the process of it, became my therapist, my lover, my companion. We loved and hated each other. I thought about it all the time; I talked to it, dreamed of it, woke with predawn thoughts of it, with whole chapters lingering in the interworld between sleep and wakefulness.\n\nEros and Psyche spiraled through this journey with me. Eros, the god of love and passion, irresistible and clever, seduces the psyche with temptations of pleasure and occasions for suffering. Love always leads to the interior world of reflection and contemplation.\n\nEros always leads to Psyche. Touching fingers to keyboard, each stroke took me deeper into the meaning of this book, the spirit of which came to sit in my room day by day, taking me around this bend and that, exposing unusual connections and insights, showing things I hadn't, wouldn't have, thought of on my own. Often, it would stand over my shoulder and whisper sweet inspirations and musings in my ear. We walked, trotted, strolled, ran down back alleys and side streets and down the stairwell into the cavern (or abyss as it felt on some days) of my psyche to rattle her awake, to shake the foundation of my beliefs, hang-ups, fears, dreams, history, to emerge fresh and ragged, awake, and drunk from the long, strange trip.\n\n _All serious daring starts from within._\n\nEUDORA WELTY\n\nUntil I gave myself over to the writing and the magic and the struggle, I did not know, could not know, where this book would lead and what it would ultimately become. When you first discover that you are pregnant, you nurture the fetus, the baby, as it grows, and you pray that the baby is healthy, has all its functions and parts, but until the baby is born, you have no idea what she'll look like, how healthy she'll be, what sort of person she'll mature into, and, especially, what'll she do in the world. Such has been my relationship with this book. What it will do in the world I do not know. And like children, it does not belong to me; it never did. I have been merely the messenger.\n\nSaying out loud my personal and family secrets sent me into a three-year spin before I got clear enough that I could be devoted to writing this book. As I wrote the chapter on sacred sex, a wee voice inside me insisted \"You can't say that,\" \"No, don't say that either,\" \"What are you doing describing that intimate thing?\" That part of me and I were able to reach an agreement so I could continue. Conversations with myself continued throughout as I also found myself writing things that confront social, cultural, religious, and political values around sex, sexuality, and our relationship with Earth.\n\nI'm making two assumptions at the beginning of this book. The first one is that if you're reading this, you feel something is missing from your relationships, your love life, and maybe even your life. We have all felt such a sense of something missing\u2014of emptiness\u2014at least once in our lives. Many feel it most poignantly in those quiet moments just after sex, or sometimes after a conversation with a friend: a bit of emptiness like something is absent. There is a knowing in our deep selves that there is the possibility of something more, and perhaps, from time to time, you strive to name it and sometimes even strive to find it. This book is about that missing thing. It explores the nature of deep intimacy, the juice of life, the food that feeds the soul of love and relationships. It also (and necessarily) challenges deeply held beliefs about sex, intimacy, and relationships that are often carried from a young age. It challenges the way we see, think, and feel about sex personally and as a culture, for all these things are involved in the generation of that emptiness that we sometimes feel. And it explores the ecological function of sexuality and our relationship with Earth and the world of the invisibles.\n\nSo, I think it's safe to say we all carry baggage, some lack of clarity, and (to greater or lesser extents) shame about sex and intimacy and about our sexuality, our bodies, our relationships, and our feelings. It is nearly impossible to grow up without some of that detritus, particularly in the United States. We also carry an ecological dysfunction, a disharmony within our personal ecology and with our relationship with the deep ecology of Earth. Thus, the second assumption I am making is that you are looking for some resolution, some healing around all of this.\n\nMost of our parents had issues around sex. They didn't talk to us about sex or bodies or relationships. Many of us grew up thinking, if we ever even heard the word, that intimacy only meant the act of sex or making love. Intimacy often involves sex, but it does not mean sex. Nor is sex the only arena for intimacy. Real intimacy takes place between real people, or rather, people being real\u2014unguarded, undefended, unconcealed, and vulnerable. To be real, without a fa\u00e7ade, requires a willingness to be na\u00efve, to see what's in front of you as if for the first time.\n\nEven if we did get some sex education in school or from our parents, it was almost never from a place of deep caring and genuine sharing. Often, it was guarded, haphazard, or superficial. It almost never included talk about intimacy, emotional openness, masturbation, pleasure, or consciousness. A lot was missing from those teachings, and in the West, it still is.\n\nAll of us grew up being lied to, blatantly or by omission. Shame grew around our sexual longings, sexual interest, and our bodies. Shame grew because it was all a secret and because those who told us about it carried shame in themselves. The baggage and misinformation around sex has not changed in any meaningful way. And since the sexual revolution of the '60s, it has not gotten better. It's gotten worse.\n\nThe use of sex toys\u2014dildos, vibrators\u2014was all hidden. We found out about them from friends and magazines and through our own curiosity driving our need to know. We covertly sought them out where information and access to them hid\u2014in shadows. In seven states (Alabama, Georgia, Texas, Indiana, Mississippi, Louisiana, and Virginia), it is illegal to sell sex toys. Information is systematically withheld from our children. Government, schools, and church programs deliberately lie to children and withhold health services and health products in order to promote abstinence-only propaganda.\n\nThe most powerful thing you can do for yourself, the most healing thing you can do for your children, nieces and nephews, grandchildren, and students is to become a subversive activist. Someone, who by means of thinking for yourself, will cause the system of lies and regulations to disintegrate from within bit by bit. But thinking for yourself is only one part of the dynamic; you must think behaviorally and act differently from new choices. Sexual patterns _are_ social constructions; as Carol Hanisch is credited with saying, \"the personal is political.\"\n\nThe way to begin is to inventory and take an honest, penetrating look at your beliefs and values around sex, sexuality, intimacy, eroticism, shame, masturbation, and your body and your relationship with Earth and all her inhabitants. Begin to clear out all the old impediments and deformities around being a sexual, sensual, erotic, fully alive human being. Where did your beliefs and ideas come from? Why do you still hold them, or why do they still hold you? What do they serve? Is there another way of seeing things? What is true for you and why is it true for you? What beliefs and behaviors support you being a fully alive, fully sexual, empowered human being? What are your fears and how do they dictate your beliefs and behaviors?\n\nWhen we stop contaminating sex, love, and intimacy with lies and misinformation, we start shaping our own ecstatic sexual experiences and we begin to decolonize the ideas we carry around sex and intimacy. Until we have thoughts and ideas that are our own, that we have come to through feeling them, through analysis, experimentation, and contemplation, only then can we be free to travel in the realm of the sacred, a realm that our sexuality opens up to us, the realm of the gods. The veil between their realm and this world is thin; sacred sex is one of the ways we can pass through the veil. And traveling in that realm is our birthright. Let no one tell you different.\n\n _To tell the truth is to become beautiful, to begin to love yourself, value yourself. And that's political, in its most profound way._\n\nJUNE JORDAN\n\nSometimes you just have to say: Fuck it, it stops with me. Owning our sexual nature is a subversive act. Giving our children permission and encouragement to think for themselves, to own their sexuality, to be curious is an act of political, emotional, spiritual, and cultural restoration. It is an act of relationship fostering and trust building. And it is a reclamation of the sacred. Giving our children honest information is empowering to them and to our relationship with them. They have a right to know. We have an obligation and a responsibility to tell them and to be present for them when they have questions. It is a loving thing to do. We cannot begin to tell the truth, to step up to the plate of honest information, if the beliefs and information we carry are contaminated. Thus the need for unearthing and sifting through what is true for us and discarding what is not, of separating the wheat from the chaff.\n\nIt means healing the splits between our sexuality, our sexual behaviors, the sexualized Earth, and how sex is pretended to be in our culture. At the time of this writing, Christine O'Donnell, Republican candidate for the Delaware senate (thankfully, she lost), has been on an antisex campaign for ten years. She is antisex, antimasturbation: \"It is not enough to be abstinent with other people, you also have to abstain alone. The Bible says that lust in your heart is committing adultery, so you can't masturbate without lust.\" Tiger Woods, the world's best golfer, finalized his divorce and announced that he was taking a break from golf to work on his \"infidelities.\" As a result, he has entered sex addiction therapy. AT&T, one of his largest corporate sponsors, withdrew sponsorship without stating why. The \"public\" is outraged at his \"immoral\" behavior; yet, we are secretly fascinated with the juicy details. This is part of the hidden world of sex\u2014our secret fascinations.\n\nMagazines use sex, eroticism, and sexiness to sell their products; yet, we refuse to admit we are sexual beings, that we have sexual longings and fantasies, except in those magazines and pornographic Internet sites openly devoted to doing so.\n\nImagine this (fantastic) scenario: People take responsibility for their needs and talk with each other honestly about them. What if we (those of us who feel this way) said to each other that one person is not enough to satisfy all our needs, that we need multiple sex partners, and that those who wish to be monogamous should do so without putting fears and moral codes onto other people? What if we gave up that disintegrating toxin, shame, and lived a life of empowerment rather than one of victimhood?\n\nHow can we ever expect to change anything if we are unable to shape our own sexual and core experiences? If we continue to feel oppressed, repressed, and victimized by something that happened chapters ago in our life, by the rules of behavior fed to us by the religious right, neoconservatives, our parents, our culture, and the media, how can we imagine ever changing or having an influence on anything? When will we get our priorities in order and start saying no to political and religious meddling that is mucking up the sanctity of relationships, pleasure, and our spiritual destinies?\n\n _You've got to have something to eat and a little love in your life before you can hold still for any damn body's sermon on how to behave._\n\nBILLIE HOLIDAY\n\nAny energy or inspiration we may have to influence our own lives, communities, schools, and government is diminished while we insist on feeling shame, guilt, and unworthiness about ourselves and while we keep ourselves repressed in the bedroom. As long as we persist in letting others think for us and tell us what appropriate behavior is and acquiesce to scare tactics that threaten to withhold love or money if we don't behave, we will remain utterly powerless in the face of any real or imagined power outside ourselves. If we don't own and take charge of our sexuality, someone else will. (Oh, wait, they already have.) It's time to bring it all back to its rightful owners, to each of us as autonomous individuals. For you must understand, Nature, Gaia, is sexual, sensual, and highly erotic. Nature is having sex all the time; that's one of the reasons why it feels so alive, and it's one of the reasons why, when we are immersed in Nature, we feel alive. Birds and bees are pollinating flowers every day. And flowers? They are the reproductive organs of plants.\n\nTrees, heavy and dripping pollen, rub their branches against each other in sexual friction. Plants have gone through countless metamorphoses in their sexual organs since before the beginning of time, developing ingenious and innovative ways to spread their pollen and propagate their species. Every flower we put on our dining table is the sex organ of a plant. Each time we eat corn on the cob, wrapped in pubic corn silk, we are ingesting corn ovules, which hold the ovary that becomes a seed when fertilized by corn pollen. Flowers exude a seductive odor when ready for mating, causing birds, bees, and butterflies to join in ritual dances of reproduction. Some male plants exude an odor that remarkably resembles the seminal emissions of men and animals. The ailanthus species (tree of heaven) will produce flower clusters that are either female, male, or both. Only the male and male\/female flowers produce the odor that fills the air with the unmistakable scent of a man's ejaculate.\n\nHuman sexuality and our reproductive organs have evolved from plants. Plant reproductive systems were the template or prototype that Gaia innovated on as animals and humans evolved. The semen of animals and men performs the same function in almost precisely the same manner as does the pollen of plants. Pollen enters the sticky folds of the flower's stigma, much like a human vulva, and traverses the whole length of the style, which is analgous to a vagina, until it enters the ovary and comes in contact with the ovule.\n\nSlugs, hermaphroditic (possess both sex organs) and slow moving, make love for hours. Each slug inserts a penis into the other and are then simultaneously impregnated. Bonobos, or pygmy chimpanzees, are one of the most peaceful groups of mammals on the planet. They have evolved a unique system of peacekeeping and bartering: exchanging sex for food. Bonobos engage in tongue kissing, mutual masturbation, face-to-face sex, homosexuality, anal sex, and oral sex. And instead of fighting, they have sex, lots of sex.\n\nWe are inspired and transfixed by mating songs and the tandem flights of birds during courtship. How can we deny our curiosity upon seeing animals being tender and affectionate with each other and even having sex? None of us is unmoved upon seeing spectacular sunrises or the way a full moon enlightens and casts shadows in a dark forest. Nor are we unmoved while sitting on the porch and smelling the rain as a storm thunders toward us across the horizon. Sometimes we are so moved we take the feelings that have been stirred up into the bedroom or to a bed of leaves in the forest.\n\nIf you have ever let your hands caress the slow, smooth curves of water-worn boulders, you have been touched by the erotic, sensual, elemental power of Gaia. However much we try to deny or control Nature, we cannot separate ourselves from her; we are part of Nature. Nature is our nature.\n\nGaia is in a constant state of heat: expanding, reproducing, and expressing herself. Volcanoes and earthquakes\u2014Earth orgasms\u2014shudder and quiver in ecstatic rhythms. The moon pulls on the waters of Earth, causing the ebb and flow of tides. Women's wombs respond to the moon's influence on their bodies and psyches as they circle through menstrual cycles. Semen of the gods rides the ocean waves as foam. Sex is a basic drive in all living organisms. Without sacred sex, how can we imagine the sacredness in land and ecosystems? Indigenous peoples around the world have an ongoing concept of sacred sex that is incorporated into their seasonal and yearly ceremonies to increase abundance. West African tribes have elaborate sex ceremonies that may last for weeks at a time.\n\n _Oh, what a catastrophe, what a maiming of love when it was made a personal, merely personal feeling, taken away from the rising and the setting of the sun, and cut off from the magic connection of the solstice and equinox! This is what is the matter with us, we are bleeding at the roots, because we are cut off from the earth and sun and stars, and love is a grinning mockery, because, poor blossom, we plucked it from its stem on the tree of Life, and expected it to keep blooming in our civilized vase on the table._\n\nD. H. LAWRENCE,  \n _L ADY CHATTERLEY'S LOVER_\n\nPerhaps our deepest fear is not that we would be uncivilized and out of control but that we would be free to rediscover the sacredness of being alive. The enchantment of Gaia and the split between wild Earth and our wild natures would be healed. Do we fear the power that is there in the deep Earth, in our deep sexual selves, would be too great? It would seem that we are more afraid of sex, of our sexuality, than we are of drugs, prescription medicines, corporations, or governments.\n\nThere is a tremendous amount of rhetoric coming from the pulpits and benches of religious and political edifices admonishing us on \"right\" or \"moral\" behavior and what is customarily acceptable sexual behavior in and out of the bedroom. Jeffrey Escoffier writes that \"[t]housands of case histories have been researched and published to illustrate the dangers, penalties and pitfalls of any deviation from the prevailing sexual morality.\"\n\nThere are equally similar admonitions regarding our behaviors and relationships to the environment, though those who would have us behave accordingly are incapable, at least not demonstrably capable, of following their own precepts. We need not look any further than current politics, policies, and legislation regarding logging, mining, oil drilling, water-use rights, and so on to find a fairly comprehensive picture of our relationship to the natural world and finite resources. There are just too many rules, rules that never have been and never will be the solution to the problem. If they were, we would all be sexually and ecologically healthier and happier and more secure in the future survivability of the human species. No, rules are not the solution, and moreover, the rules are creating more problems. If they serve anything, it is in forcing us to examine what works, what needs re-working and what needs abolishing.\n\nI assert in the pages that follow that there is a direct relationship between our beliefs, values, and behaviors about sex, sexuality, and intimacy and our beliefs, values, and behaviors toward and about the environment and Earth. I also assert that healing one relationship will heal the other, that the two are intimately, inherently intertwined; how we treat our bodies and ourselves is how we treat Earth. If our sex isn't sacred, we won't treat Earth as sacred.\n\nSex has been around since the first single-celled bacteria duplicated through mitosis in the primordial ooze 3.85 billion years ago. Gaia took that single-celled organism, split it into two, and inserted a drive to reconnect, to be one again. It is that drive to reconnect that urges us toward a more intimate sexual experience. It is that same drive that pulls us out of our four walls and into the wild. In _Dazzle Gradually,_ Lynn Margulis and Dorion Sagan explore the origins of sex\u2014that drive to reconnect.\n\nSex has many origins: evolutionary, sociolinguistic, and perhaps even unconconsious or metaphysical origins that are not really origins at all, since they stand, at least psychologically, outside of time. Metaphysically, the conjunction of two individuals in an act of mating recalls the original split of each individual in his or her essential solitude from the universe of which he or she is a part. Thus, biology aside, the union of opposites resembles a sort of awkward \"healing\" of the primordial condition in which each of us finds ourselves: separate and alone.\n\nHonoring that innate drive is to acknowledge the gift of life, and it calls us to do deeper healing of the damage caused from secularizing and marginalizing sex.\n\nWhat we would see around us in the natural world, if we choose to take in the view with wonder and awe as if for the first time, is what is inside us, what has always been inside us: our true nature. We would see, as Thoreau said, \"The earth which is like a map spread out around me is but my inmost soul revealed.\"\n\n _This leads us to the concept of most primitives, that \"the sacred\" is power. The \"power\" flows through animals, plants, waterfalls, mountains, humans, etc. in endless abundance._\n\nGREGORY BATESON,  \nQUOTED BY DOLORES LACHAPELLE  \nIN _S ACRED LAND, SACRED SEX_\n\nMany are the attempts to heal that separation, to restore the sacredness and spirit of sex, often through a resurgence of Eastern tantric practices and pagan Earth fertility rites. However, even in these practices there is a tendency to reduce them to techniques and mechanical uses. Western practices of tantric sex are more a hybridization of new age techniques, including yoga, aromatherapy, massage techniques\u2014\"a single invented tradition,\" says David Gordon White in _Kiss of the Yogini._ \"What passes for Tantric sexuality in the West has almost no connection with its original inspiration in medieval India.\" And in _Shamans, Mystics and Doctors,_ Dr. Sudhir Kakar says, \"New Age Tantra is to medieval Tantra what finger painting is to fine art . . . a remarkably unimaginative series of yogic exercises applied to the sexual act . . . a _coitus reservatus par excellence_ . . . a sad attempt to mechanize the mysteries of sexual love.\"\n\nOur separation from our sexuality is equivalent to damaged ecosystems, so eloquently described by Aldo Leopold, John Muir, and so many Deep Ecologists. And like damaged ecosystems, our damaged human ecosystem, separated from sex, is in need of repair. As Deep Ecology principles are applied to heal ecosystems, healing the damaged human ecosystem requires that we be sensitive to our spiritual, physical, and psychological needs and evolution. It is important to seriously consider and explore what role a sense of place, a sense of our place, plays within the circle of all life.\n\nThe healing begins when we shift our perspective from seeing sex as some _thing_ (the character of being either male or female) or a _thing_ (technique) that happens between two beings and move sex into the realm of relationship. Seeing and experiencing sex from a place of communication and relationship takes us into the realm of the sacred and a state of sacredness. Sex as technique will always maintain a level of one-dimensional superficiality, the flat and parched landscape of human secularized experience. Dropping below the surface level of sex brings us to the source of all healing, in the deep waters of relationship and intimacy. Holding sex, our bodies, our relationships, and our place in the great scheme of biological diversity and evolution will return us to the ancient and archaeological roots of our species origins, to the primordial ooze where the first single-celled bacteria split. It returns us to the realm of the invisibles, the gods and goddesses of myth and legend who are more than storybook characters.\n\nSacred sex heals the split that has occurred between spirit and matter, between male and female, between physical and the numinous. It heals the dichotomy between humans and the sexuality of Earth. Sacred sex transmutes the sex act into the rejoining of the original split into male and female and the secularized split of body and soul.\n\nMost of us today are hungering for intimacy in our lives, to feel a deep connection that was lost when extended families broke up and when we fell away from the wild sexuality of Earth that expresses itself through every life-form. First, we must feel connected to and be intimate with ourselves and get to know all the voices inside. Georg Feuerstein writes, \"Intimacy is conditional on our acceptance of embodiment. We must be intimate with ourselves before we can be intimate with another person.\" Once we are intimate with ourselves, we can be present with another human being, and then we can sense the sacred, the numinous, the spirit that enlivens all of life.\n\nSex is a union of the energy of two bodies, two lives, two spirits, two souls. Sex is explosive. It's the process for creating life and for rewriting a life. Sacred sex transforms a physical act into a prayer, a devotion. And in sacred sex, a life, a relationship, a story is transformed. Old wounds are healed. Stories become narratives and not predetermined fatalism for the way our life unfolds. Each becomes something new, something different, something other than what he or she was before. Each being becomes whole, and as a result of that wholeness, Earth moves closer to a state of wholeness.\n\nIn the pages that follow, we'll go beyond and below superficiality, traveling into the territory of the sacred as we repossess and restore the lost connections of intimate relationships with another human being, with Nature, with Gaia, with wildness. In the territory of the sacred, we will once again return to our lives and our world the sacredness of sex and the healing that comes from knowing our true nature and Nature outside us, as well as how the two natures are inextricably interwoven. We will learn to navigate the world with a sense of wonder and awe as we once did as children. Do not let time and familiarity destroy the mystery.\n\nAs I wrote this book, I was primarily telling all this to four people: my three granddaughters Araina and Hailey Kennington and Darian, and the fourth person is the young girl I was that so desperately needed to hear these words so long ago.\n\nThis book is meant for anyone wanting to have a life ensouled with sexual vitality, intimate human and nonhuman relationships, and for those wanting to heal the separation between humans and Earth. It is written from one woman to women in heterosexual relationships, though the work is meant to be used for anyone in any form of relationship: It is transferrable. In the interest of full disclosure, I want to say that this book does not affirm the limited reality you may currently live in. But, it does affirm a reality that you could live in if you give up limitations.\n>  \n> \n> **PART 1**\n> \n>  **The Fall from Earth**\n1\n\n _ **Intimacy**_\n\n **Food That Feeds the Soul of Love**\n\n_Love, I know, is essential if death is to be put in its place, and it has a place, but love is essential even if I do not know the words that give it flesh and scent. That is why we find it so difficult to write about sex. Not because we are so inhibited and prudish but because when we write about sex, we get acts and organs, a breast, a vagina, a cock, juices and tongues and thrusts\u2014and wind up with recipes but no food. Orgasm is just a word. We have a hunger and love fills it, however briefly, and our accounts of having sex do not catch what drives us into the night seeking light._\n\nCHARLES BOWDEN,  \n _B LUES FOR CANNIBALS_\n\nWhy isn't love enough? For over fifteen years I have been asking the Universe this question. I had the naive notion that if you love someone enough, have enough integrity and keep your agreements, everything else will take care of itself. I have loved much, deeply, and passionately. But something has always been missing. I hungered for more depth in my relationships, for talks deep into the night about the meaning of life, talks of the personal struggles we all go through. I hungered for soul-to-soul bonding and to be seen at the deepest parts of myself. As much as I ached and fought for more, the missing feeling was always present, pushing on my psyche like a bone spur in the bottom of my foot.\n\nWhat was this hunger, this consuming passion and desire? I wanted to name it, to know its shape and texture. I wanted to know its taste and smell. But it was elusive, some invisible thing I chased in the dream corners of my heart. The reality that was present in my family and relationships never satisfied my heart and soul. I wanted, needed to know the truth, what was real.\n\nThis yearning, this appetite scared me at times for it seemed insatiable. I felt its powerful force pushing on me from the pit of my stomach; a soul hunger always wanting the next meal. Nevertheless, I was driven by an inborn, deep-seated belief that the depth I sensed was possible; it really could be.\n\nWhen many people think of intimacy, it's often oriented around thoughts of making love or having sex, but this was not enough for me. It was emotional and spiritual intimacy that I longed for. Ideally, I wanted to marry it with physical sharing. I wanted a depth of intimacy, a bonding of spirit with another person. Through this bond, in the act of sexual union, my partner and I would travel together in the world of Spirit. That is what I wanted. And eventually, that is what I found.\n\nThe journey of learning the steps to the dance of intimacy was long and well worth the work of transforming myself, of unlearning and dropping defenses that kept intimacy out of my reach.\n\nTHE YEARNING FOR CONNECTION\n\nI was born with a belief that love was supposed to be this way, not the way I saw it in the faces and distant behaviors of my parents, lovers, husbands, and friends, people who said they loved me. I felt love from them. Still, I was unsatisfied. Something was missing, and there was too much space, too much drama, too many games, too much unspoken, between people who said they loved each other.\n\nWhen I was six years old, each night at bedtime my younger sister and I kissed my parents good night. It was an organic movement on my part, an impulse that arose naturally from within, for I had not witnessed that behavior in any of my older siblings. At six years old, I didn't understand why my father wouldn't show his love. Why, when I kissed him on the cheek, didn't he return the kiss, or look me in the eye, or hold me? I needed him to return the affection. I needed to feel his love for me. But he couldn't bring himself to do it. He never allowed himself to drop the fa\u00e7ade of strength that he had spent a lifetime building. I felt that he didn't love me. But to my child's mind, it didn't make any sense. How could a father not love his children? So I believed that something was wrong with me, that I was somehow damaged and unlovable.\n\nAs a child, I spent a great deal of time and energy trying to show my father how much I loved him, trying to get below the surface, hoping my love would be returned. As a young adult and well into adulthood, I would bake his favorite desserts and take them to him on Sundays. My parents came to depend on me visiting them with fresh-from-the-oven home-baked cakes and cookies in hand. My mother was often gracious. On a few occasions, my father thanked me. I knew he enjoyed the treats but what I needed was to hear him say something genuine and sincere. At that time I didn't have the skills or permission to ask directly.\n\nWhen I was thirty-eight years old, I moved two hours away from my parents. After years of being a devoted daughter, going through two marriages and two divorces, and raising a son, who was then seventeen, I was finally starting my own life (I'm a late bloomer, but bloom I did). I stopped baking for them, and my visits became irregular. My mother made it clear to me how disappointed my father was. I felt like the bad daughter, which is how I was supposed to feel. Families do that; they have an ingenious way of trying to get you to keep up the old, familiar behavior and family script, to maintain the status quo.\n\nFor forty-seven years, my father and I walked in each other's life, but we never got to know each other. We didn't know how. For seventeen years we were strangers to each other, living under the same roof, eating the same food, working side by side in pained silence. I ask myself if I knew him. Did I know what made him happy, what fed his soul? What drove him to rise each morning? Did intimacy terrify him so much that he'd do about anything to avoid it? Or was it simply that he didn't know how to go there? How did he deal with the pain of his own family, I mean, deeply, inside himself? Outwardly, he didn't hide the rage and betrayal he felt. How did he feel watching his wife of fifty-five years suffering a slow death, helpless to make her well? How could I have made better our time together here on Earth?\n\nIN THE LUMINOUS WORLD OF INTIMACY\n\nWithout intimacy to feed love, relationships are lonely and desperate. In the deepest sense, even among friends, there should be an invisible, yet palpable, flow of trust and love between people. Though rare, I did sometimes experience it, like the time a friend shared a part of himself he had long kept secret.\n\nThe two of us had made a conscious, out-loud agreement to have a deep and intimate relationship where we would actively work to reveal hidden parts of ourselves. In the beginning of our friendship, he found the courage to ask me to sit with him so he could tell me this thing he had carried inside him, hidden. We sat under the grape arbor partly shaded from the afternoon midsummer sun. Sitting across from him, looking into his eyes, I could see how afraid he was, how difficult it was for him to tell me his innermost secrets. His body tensed up as it responded to the inner voices telling him that saying secrets out loud was too risky, that something bad would happen if he did. He sat, uncomfortably at first, on the edge of his chair, facing me with the sun at his back.\n\nSeeing him in distress, I felt the invisible fingers of my heart go out to touch him. My entire being swelled with love and compassion for this man. Gently, I wrapped my heart field around him and held him in it as he began to speak. His body slowly began to relax. It seemed as if with each word he uttered, some resistance flowed out of his body with it. The stress left his face, and then his shoulders relaxed. We held each other's gaze. As the sounds and meanings of each word came up to meet me, I greeted every one, breathed it in, and let it find a place to rest inside me.\n\nAs he talked, I silently acknowledged him with frequent nods of my head. I leaned forward, resting my elbows on my knees or my chin in my hand. Only a few moments had passed when I realized that he and I had moved from the mundane world into some extraordinary, other, and sacred territory. The sounds of the world around us began to fade from our awareness. Water flowing in the nearby stream, an occasional car passing on the road, even the slight breeze in the trees, all these sounds retreated from the world we had entered. My senses were heightened, the warmth of the sun moved from the surface of my skin into the marrow of my bones. Colors took on an iridescent glow. My gaze softened, and the hard edges of the physical world became soft and out of focus, like the edges of a ball of cotton.\n\nWithout diverging from the telling of his own story, my friend and I shared an aware, unspoken understanding that we had slid through an opening between worlds. As we held taut the unseen realm we found ourselves engaged in, a small wild bee flew over to us and then stopped. She hovered at the edge of us, seeming to touch, to taste the bond that was being birthed between the two of us. She hovered, and then moved along the edge, the arc, of an invisible force field from me, to my friend, and back again. As if drinking in nectar from a beautiful, invisible flower, the bee would stop and taste this wonderful food. She never altered her course as she traveled in a curved sweep back and forth from my friend to me. She would stop for a moment, near each of us, in midair, the way a kite pauses momentarily in the air held by the tension of the string and air currents before some imperceptible force gently alters its position.\n\nIn that moment, the three of us were immersed in an ecstatic, luminous world. We had found the key to a door that, until this moment, had been locked. We had entered the world I had hungered for. And in that moment I realized that that world exists side by side with ours. Intimate, deep sharing and a communion of souls collapses the separation of the two worlds. I sensed that all of creation responds to such intimacy as we did. And I wanted that closeness, that experience of the luminous world as a way of life, every day of my life. It took me a long time to get to that place where the two worlds overlap.\n\nFOOD FOR THE SOUL OF LOVE\n\nAfter enough relationships had gone awry, with me going away feeling empty and disappointed, I realized I had taken on the smell of resentment. The need I carried stirred some aspect of my soul, or some need in my soul stirred up the question. It's difficult to know which happened first and doesn't, in this case, really matter. What matters is that I followed it.\n\nI revisited the question.\n\nSitting in a coffeehouse with pen to journal, I held the question in my heart and awareness. \"What is this depth I hunger for?\" \"Why isn't love enough?\" As I wrote, authentic knowledge from the Universe entered my conscious mind, traveled down through my fingertips, out my pen, and appeared on the page:\n\n\"Intimacy is food that feeds the soul of love.\"\n\nI looked at the words I had written. I kept writing to fill out my understanding and to anchor it inside me. Love has a soul that needs tending, feeding, and nurturing. Like a newborn infant that needs to suckle milk from her mother's breast, the food of intimacy is a vital food, a life essence that is chosen daily and given freely. When the infant leaves the womb, she enters a new world of unknown terrain. If we choose the path of intimacy, we must leave the womb of familiar old habits that prevent us from having the love we deserve. Intimacy is our birthright.\n\nIntimacy requires taking risks, going beyond your comfort level. To have a life of intimacy, you must do that which you are most afraid of. The intimate, sacred life asks you to be vulnerable and exposed. It requires sensitizing yourself to the full range of feelings and emotions; from gross to subtle in increasing elegance. Being present with what you are feeling is the ground from which intimacy grows.\n\nYou must be willing to be seen in the deepest part of you and to give up hiding and lying. It requires rigorous self-examination (or as Data from _Star Trek_ says, self-diagnostics) to see yourself clearly so that all of who you are can be present. And it asks that you see and marvel at the world and all that is part of it through the eyes of wonder.\n\nHarriet Goldhor Lerner in _The Dance of Intimacy_ says: \"An intimate relationship is one in which neither party silences, sacrifices, or betrays the self and each party expresses strength and vulnerability, weakness and competence in a balanced way.\"\n\nI had learned not to let love in too far or too deep. I adopted this from early childhood based on my belief (at that time) that there was not enough love to go around anyhow, so someone was going to get shortchanged and be brokenhearted. And I was afraid that if love got in all the way, or if I allowed myself to open to it, someone would see that something truly was wrong with me, that I wasn't loveable. The risk of someone finding that out was, for too long, far too great. When love started to come into the hidden places inside, like water finding its way through sand, rock, and crevices, I created dams that prevented it from going too far or too deep.\n\nTo have the level of intimacy I longed for, I had to break deeply held beliefs and patterns of behavior. I began to give up ideas of who I thought I was. I gave up the romance of being alone, of marginalizing and privatizing my feelings. I gave up the fa\u00e7ade of strength that I had crafted over the years, the strong, tough part of me that people first met. I crafted that skill from watching my father. It seemed to work for him, and the world is, after all, a dangerous place. I created that strong part to hide how terrified I was underneath. But my strength shielded the secret, tender heart inside me. I kept my natural childlike self in a closet. And as I gave these things up, I found that not many people wanted to look past the illusions or the pain to see the truth of who lay beneath. Eventually, I had to give up wanting to keep others comfortable to have the life I knew I wanted and was meant to live.\n\nI gave up these things because, like razor wire, they hindered the path to what I wanted most: intimacy and the freedom to be intimate. Primarily, I wanted to be free, truly free, the kind of freedom that comes only from being awake, aware, and in charge of my own life; the freedom that is birthed when I am present inside my own body and thoughts and feelings; and the freedom that comes from strength of character, flexibility of options, and not feeling like a victim of anything that is happening around me so that I can choose how I feel, what I think, and how I am in each moment. I gave up hiding, gave up creating fights to make distance because being close was too scary. I chose over and over to do the things that frightened me the most. I could measure the level of intimacy each thing hid by the degree of fear I had around it. If there was tremendous energy running inside me, if I was shaking or felt pressure building, it was a clue that saying something out loud was important, for in my birth family, I had learned at a young age to not make noise and to become invisible. How many of us do that? Learn to remain silent as our deep needs and right to be alive are sacrificed to maintain the status quo?\n\n _I don't want the cheese, I just want to get out of the trap._\n\nSPANISH PROVERB\n\nLEARNING TO HIDE\n\nWhen my father's farming partner, an old man who had been a state champion wrestling coach, grabbed my newly blossomed breasts while we were on the hay wagon together, I said nothing.\n\nWhen I was molested and sodomized at age eleven by the hired hand, an all-star eighteen-year-old high-school wrestler, I said nothing.\n\nWhen my first husband beat me and created a bone spur in my hip, I lied and said I had slipped down the stairs. I believed I had to make up something to tell the physician and my family to explain why I couldn't walk or move without pain for days.\n\nI didn't know what to say, how to say it, or whom to say it to. As a survival mechanism, I learned to freeze up and shut down emotionally. Anything else was far too frightening. Though my body was still present, I was somewhere else deep inside my mind, unreachable.\n\nNever did my mother or my older sisters ever talk about a woman's body and the changes it goes through, about sexuality or being sexual. We didn't talk about moon-time cycles. When my first menstrual cycle began at age twelve, I didn't know what was happening. I was visiting a friend when it happened, and she knew. Her stepmother took care of me. She talked to me about what was happening and how to take care of myself. It was the shorthand version about how long the bleeding might last, how I might feel, and what to do about it. What she left out was all the mystery and potential power inherent in the bleeding time. I didn't know it was a rite-of-passage time and that in indigenous cultures a young girl's first moon time is met with great celebration and the young girl is honored and welcomed into the circle of womanhood. What I knew was that I felt alone, confused, and ashamed. I wanted to hide. I spent time wishing I was a boy and doing as many boy things as I could. Not until I was much older did I come to love and work with my female body and the sacredness of her cycles.\n\nI knew my friend's mother had called to tell my mom that I had started to bleed. But when I returned home, not a word was said. After a few hours, I finally asked, \"Did Sharon call you?\" My mother said yes. That was the end of the talk about menstrual cycles. When I was thirteen and wanted to use tampons, my mother accused me of being sexually active. She didn't know that you didn't need to have had sex to be able to use them. Thankfully, my sister, ten years older than me, informed her that it was possible to use tampons without first having had sex. Where my sister got that information I'll never know, but thankfully, she had it and offered it in my defense.\n\nA year into my first marriage my mother took me into her bedroom, fished around in her underwear drawer, and pulled out of one of her size 36 C bras. She handed it to me and suggested I wear it for something that sounded like a clich\u00e9: to show off what God had given me. It worked. If you like the torpedo look. I got the point.\n\nI have to thank my mother here, for despite our problems and dysfunction, she had a sexiness that she carried and a classic, movie-star attractiveness to her. She was a strong woman with a sense of justice. She loved wearing barely enough clothes to keep herself covered, and in the protestant Midwest, she stood out. Only now, after years of working on myself, can I see the impact her behavior had on me, which outweighed the difficult and ill-equipped verbal communication between us. On an unconscious level, she was giving me permission to have my own sense of inner strength and sexuality in the world.\n\nWhen I reached puberty and came into my sexuality, my father pulled away from me completely. I was emotionally cast away from him. An unconscious part of me was learning the power of sex, and without elders or mentors to guide me through it, I was in free fall. For some reason, unknown and unbelievable to me for a long time, older men were attracted to me. I learned quickly, and painfully, that sexuality is a form of power. For me, it became a sideways attempt to get love from men that I couldn't get from my father. But because I thought I was unlovable, I put myself in more than a few dangerous situations proving to myself that I was what I believed I was: damaged goods. The damaged part of me acted out fiercely and desperately.\n\nI watched my family. No one seemed happy. None of us would show genuine love and caring daily as a way of life. We were too afraid, too ignorant, too much asleep and shut down. I lived in an atmosphere of emotional and spiritual poverty, starved of love and affection. We would have fun now and then; but even in the croquet, volleyball, and softball games we played, there was an air of seriousness to them. And we all held back some part of us, as if it were illegal to abandon ourselves to spontaneous, childlike fun and let unrestrained joy be a part of it all. It's taken years for me to be able to play cards again. There was always a deck of cards on the dining table alongside the condiments, and if we weren't working or watching television, which was on all the time, there was a card game going on. It was not fun to me. It was serious, cutthroat, with fists of cards slamming the table. It was passionate, as if the card game were the only place for passion to have expression. Playing cards was how we dissociated from ourselves and each other. We could pretend we were close. My mother dealt with the pain of it by disappearing into the world of crossword puzzles. She would be bent over them at the kitchen table early in the morning, at the end of each meal, and into the late hours of night.\n\nSomething was deeply wrong. I could taste the acridness that hung in the air with each inhalation of breath. I actively began to find something different, unconsciously and clumsily at first. Without a guide or elder, I created my own tattered map from scraps of existential poems, glue sniffing, self-tattooing, my mother's stash of prescription diet pills, Jack Daniels, and advice from the older friends I gravitated toward. They at least had a few more years of time in the territory than I had and they taught me how to swear.\n\nIn my sixteenth year, I began to fight for my own life. I began to make noise and demand to be noticed and I caused outward disturbances. It felt good. No, it felt great. Sixteen years of repressed rage began to leak out the edges as I rebelled against the silence, family lies, and games. I rebelled against being put to sleep, dulled by unspoken codes of behavior and family scripts. The waking up was slow for some years as I fumbled my way along. Incrementally, I began to have some sense of myself, to feel my own boundaries, and to take charge of my life, though from the outside I'm sure it resembled nothing like taking control of my life. From my parents' and counselor's perspectives, I was definitely not in control. I started the habit of running away from home, which dominoed into a ridiculous cycle that no one seemed to know how to break. No one seemed to care enough to find out short of grounding me.\n\nI quit going home after school on Fridays. As soon I was absent without leave for twenty-four hours, my parents reported me missing. Eventually, the police caught up with me, and I was deposited at the police station to wait for my parents to pick me up. The next weekend, it happened all over again. They assigned a probation officer to my case for a year. It was a valuable experience, I must say. I learned things during my brief stint in \"the system\" that showed me the template for how our society operates. I learned how to be crafty, how to play their superficial games and still get what I wanted: older boys in butch cars, sex in backseats, rock and roll, and drugs.\n\nAt the end of that year, I was beginning to feel desperate for a way out of the bottomless, directionless, meaningless existence I had carved out. It was exceedingly uncomfortable and becoming increasingly dangerous. A few of my friends had begun to seek drug and alcohol counseling and attend meetings. Well, I thought, I could surely use someone to talk to, so I went weekly to a counseling session. I also went to that AA meeting place where you say, \"Hi, my name is Julie and I'm an alcoholic.\" During that time, I was not so politely asked to leave school and not come back. Twice. At seventeen I moved out of my parents' house, rented a room, and took a shit job at a capon factory. Was fired twice. I managed to get fired from a capon factory. Twice. That is fucked up.\n\nI thought about one of those twelve steps where you say: \"I am powerless (over alcohol).\" It left a bad taste in my mouth. I didn't like feeling powerless (and I still don't), so I set out on a journey to feel powerful (and I still am).\n\nI admitted myself to alcohol and drug rehab. More than once. I was in and out of detox, the psyche ward, and half-way houses for two years. During one of my tours in the detox ward, a remark my counselor made had a lasting impression on me. \"You have to learn how to talk.\" Maybe yes, maybe no, I thought. There is power in silence. But I respected her, so I ruminated on that and concluded that it would be nice to know how to interact, to be socially adept and expressive. It became a personal project and a life goal. I watched people all the time. I watched how people changed when given the opportunity to talk about themselves, to say what they felt out loud. I learned how to engage them and keep the conversation going in directions we were mutually interested in. I began to understand the importance of talking from a simple place about how I felt and what I needed. The daily practice of refining my expression and language choices and calibrating the impact was tiring at first, and frustrating. It's a choice I made again and again even when it made others uncomfortable.\n\nI had to learn discretion and how to do \"readings\" on people and circumstances. I would go into a tailspin, terrified afterward that I had said too much, made someone uncomfortable to the point of him not returning, had revealed too much of myself, and as a result, as sure as hell never freezes, something really bad was going to happen to me.\n\nBIRTHING THE AUTHENTIC SELF\n\nLooking in the rearview mirror at my life, I can see now just how brave, how determined I was. I didn't know I was those things at the time, but that's why a rearview mirror is helpful\u2014it gives you a perspective of the distance and the terrain that you've covered. It may be that you have friends who are able to see the calluses on your feet from walking the path and are willing to remind you.\n\nTo make changes takes courage and fortitude: those qualities of soul that enable you to go head on into the darkness, into uncertainty and difficulty. You must have _spiritus_ and _coeur,_ inspiration and heart _._ Being intimate takes you into uncharted and uncultivated territory. There is no place there for survival mechanisms of defensiveness, paranoia, or making up what other people are thinking or saying. To be in the presence of true intimacy, one must lay down the sword that maintains distance, separation, and game dynamics. In tilling the soil of intimacy, you prepare the ground by doing your own internal work, faithfully, devotedly, as if your life depends on it.\n\nIn the process, you meet your _intimus;_ your inmost nature. Over time you become your own best friend advocating for your needs, wants, and happiness. You learn to craft a life of joy so that all you do, how you make your home and life, brings you a sense of wholeness. The child parts of you begin to relax and trust you to take care of them. You speak on their behalf so their needs and wants get tended to in the open. The old and familiar habits of behavior begin to break down as you choose to do something different.\n\nThis new way is frightening. And liberating. Simple and most difficult. There is a very young part of each one of us who knows that life can be more full, enriching, and deeply rewarding. Begin by discerning wants from deep needs. Needs can be felt in the body somewhat like a slight pressure. The more fear you feel about speaking out loud on your behalf, the more energy there is around the need. And the greater the freedom there is in voicing it. For example: \"I need a hug.\" Adding the phrase _will you_ empowers you and activates the other person. \"Will you give me a hug? I really need one.\" There is natural childlike energy and na\u00efvet\u00e9 in those two words. Say them out loud. Or practice saying them inside yourself to get the feel of them. You can feel how the energy of the words comes from a younger part inside you. Notice how you hold your body when you say it. How is your breathing? What expression does your face take on? With practice you can call up this part of you at will. Allow that part of you to look through your eyes, to speak through your voice. The self of you, who is at the center of all you have been and will ever be, becomes a witness to this small person you once were being born into the world once again.\n\nIt's not unusual to feel a bit shaky inside after speaking so vulnerably. The shakiness is normal for a while until you flex those muscles often enough that the new behavior patterns find their own groove. You literally shake up your internal structure to allow something new to replace the old. Like bushwhacking a new trail in a deep forest, the first swath is difficult. Subsequent trips get easier as the path becomes clearer, well marked. You are essentially creating new neurological pathways. Think of it as adding a new software program to the mainframe of your consciousness. You are breaking the rules about intimacy, about sharing secrets and trusting another human being with your heart and the truth of you. It's important that you understand that you are breaking covert agreements about being a certain way with those close to you, for there may be repercussions. Those close to you may not be happy with the new you. Keep going. Others have gone this way before you and have left maps of the territory. But they are just maps, not the territory itself.\n\nYou are birthing your authentic self. The birthing of self is frightening, uncomfortable, and awkward. It really is like going through the birth canal when you were first born into the world. The journey was fraught with distress, pressure, and uncertainty; an unknown world lay before you. Even so, you made it through (even caesarian babies find a way through and out). And that is exhilarating, to express yourself through the birth canal and continue to be authentic in each moment thereafter.\n\nIt's important to say what you're feeling out loud. \"I feel shaky and kind of scared.\" Ask from the little and undefended parts of you all the questions you need to ask so there is nothing unsaid. Simple and direct questions as a child would ask: \"Do you still love me?\" \"Will you hold me?\" \"Do you still want to be with me?\" \"Do you like being with me?\" \"Do you think I'm pretty?\" \"Will you tell me what you like about me?\" \"I feel funny. Will you sit and talk with me?\" This is a scary experience for many of us, finding a new way to be often is. But, there is life on the other side.\n\nVery few of us were raised in families where talking honestly about how we feel was encouraged. Most of us grew up feeling scared of saying how we feel, and we learned to lie or hide to keep ourselves safe, to not disrupt the unspoken agreements we'd made with those we lived with.\n\nNO SUCH THING AS SAFE\n\n _Security is mostly superstition. It does not exist in nature, nor do children of men as a whole experience it. Avoiding danger is no safer in the long run than outright exposure. Life is either a daring adventure, or nothing._\n\nHELEN KELLER\n\nI want to say something about the word _safe_. We like to think we can create an environment in which we feel safe, where our life and well-being are not threatened. Safe does not exist. At most, we can take a calculated risk, which means we consider all the possible variables and potential outcomes and, based on an analysis of the circumstances and risk factors, we decide on a particular action. Take learning to skydive for example. We may decide to skydive after we've taken into account each factor and its potential for danger. When we consider an instructor, we look at his years of training and experience, number of accidents, attitude, awareness, and attention to detail. We inquire about the integrity of the parachute equipment, the condition of the airplane and the pilot's experience, and the wind and terrain. We balance our need for this particular adventure against our fears of it and disappointment if we decide not to do it. When we finally decide, it's a result of weighing all the possible outcomes and agreeing to the risk involved. We take responsibility for the choice.\n\nIn intimate situations, we may want to ask, \"Is it safe to say something?\" First of all, it's a trick question. It's a setup: No matter how the person responds\u2014yes, no, or maybe\u2014the outcome will always be their fault. Responsibility for your well-being is unfairly put in the other person's hands, letting you off the hook of any personal responsibility. Whether or not you feel a situation is safe for you to proceed is ultimately your decision and responsibility. Deciding to take personal responsibility is the first step toward ultimate freedom.\n\n _There's no safety in numbers . . . or anything else._\n\nJAMES THURBER\n\nWhen we were young and chose to hide or not say something, we felt it was the safest thing to do to keep from being beaten or to help us fit in. Making those decisions as children was a smart way to survive; it got us through painful and frightening situations. And it was survival. As grown-ups, we get to do things differently, and the price we pay for not saying or doing something is often our self-respect and dignity and a postponement of freedom as we continue to keep parts of us shut down, hidden, and compromised.\n\nEric Berne said, \"Parents, deliberately or unaware, teach their children from birth how to behave, think, feel and perceive. Liberation from these influences is no easy matter, since they are deeply ingrained and are necessary during the first two or three decades of life for biological and social survival. Indeed, such liberation is only possible at all because the individual starts off in an autonomous state, that is, capable of awareness, spontaneity and intimacy, and he has some discretion as to which parts of his parents' teachings he will accept. At certain specific moments early in life he decides how he is going to adapt to them. It is because his adaptation is in the nature of a series of decisions that it can be undone, since decisions are reversible under favorable circumstances.\"\n\n _The significance of being intimate is that in being close, you are thrown back to a time when you decided that being close was too scary, so you folded in on yourself. When you go back to that time, you give yourself the opportunity to be a child again, but this time with the power of an adult. You learn that you no longer have to hide your feelings to survive. You learn that it's ok to have feelings, needs, wants and desires and to say them out loud to another person who will receive all of you without judgment or fear. You learn that you no longer have to hide your feelings to survive but that you can feel alive. You learn to take your place in the relationship rather than adapting to what is there. In so doing, you reclaim the precious parts of yourself-your trust, your faith, your honesty, your integrity, your child like joy and enthusiasm for life. All the things that you locked away in a place where they would not be touched by the devastation in your family._\n\nGENEEN ROTH, _W HEN FOOD IS LOVE_\n\nWhen you choose a life of intimacy and all that it requires and affords, a door opens to some new adventure, to awareness and intuitive development. Light is able to penetrate the shadowed places in your interior house. There comes a point on each journey when you know, without doubt, without hesitation, that you cannot go back to your old life; you cannot be who you once were and also have a new life with new riches. If it were possible, I would have found how to do it. I tried to keep one foot in each life, the old and the new. It's a recipe for crazy making and relationship messes. You have to choose one or the other, and the old life is like the line from _The Matrix:_ \"You've been down that road before, Neo.\"\n\nSexuality is not a thing, an act, or a behavior, but rather a state of being who you are, what your nature is. And it is bound tightly to creativity. Sexuality is used to create, not just life but art, poetry, food, a home. It is given from Eros\u2014the God of sexual passion, that longing for the divine, and it is the instinctive drive to connect to the larger world\u2014and when Eros is made part of all that we do, all that we do becomes alive, enhanced, animated. Food tastes juicier, colors are brighter, and life takes on a luminous sexiness.\n\nHans Hofmann, in _Sex Incorporated,_ writes about the broader meaning of sex.\n\nSex is the action through which we accomplish what sexuality prompts us to do. The term _sex_ should be rescued from its promiscuous meanings in common usage. Restored to its precise significance, sex connotes the interaction by which persons express their most intimate union. _Expression_ is here the most crucial term. The intimacy and mutuality of two people's relationship with each other is not limited to or by sex. But sex expresses most intensely the character of such a union, for better or for worse. That is why sexual intercourse represents the quintessence of sex. From this center, sex radiates in a descending line of significance into all other forms of human interaction and intercourse.\n\nIf you choose a life of awareness and attention to detail, a life dripping with sensuality and intimacy will belong to your true self as you emerge into the light of day. That life asks you to pay attention to the details, to awaken your senses, to relinquish limiting behaviors and beliefs, to be present with what is right in front of you, with what's real from moment to moment.\n**2**\n\n _ **Autonomous Personhood**_\n\n**Fearing for My Life**\n\n _Meeting you,_\n\n _I shuddered,_\n\n _fearing for my life._\n\n _Now I understand why._\n\n _Death has come_\n\n _in the guise of the beloved._\n\n _The one I used to be_\n\n _is gone forever!_\n\nPAUL FERRINI,  \n _D ANCING WITH THE BELOVED_\n\n _All my life I've wanted nothing but to bring sex and friendship together\u2014and I seem to be farther away from it than ever._\n\nERICA JONG,  \n _A NY WOMAN'S BLUES_\n\n _The Self is not a known territory, but a wilderness. Too often we forget that. Too often we reach the boundaries of what we know about ourselves and turn back._\n\nPAUL FERRINI,  \n _T HE WISDOM OF THE SELF_\n\n _The privilege of a lifetime is being who you are._\n\nJOSEPH CAMPBELL,  \n _R EFLECTIONS ON THE ART OF LIVING_\n\n _The gods cannot give you anything that you cannot imagine for yourself._\n\nJULIE MCINTYRE\n\nSo you want intimacy and meaning in your life. The want, the desire to have something, is the beginning, the acorn from which all else will grow. In the wanting, inside that acorn, is an intimation of what it might look like when the want is made manifest, comes to fruition, like the great oak that is inside the acorn. Some part of you has suggestions about how to have that thing in your life, what it will look like, and having the desire sets the creating of it in motion. There becomes a movement in the direction of that thing, in this instance, a relationship. Not just any relationship but one that has a particular feeling and quality to it, and when you imagine it, you get a warm and fuzzy feeling inside and parts of you relax just imagining it, and you say, \"Yes, that's it, that's what I want.\" Anything that is created begins in the imagination with images.\n\nCreating the life you want begins with desire and imagination. Imagining the love and intimacy you desire and the life you want, a rich life, ecstatic life, creative life, interesting life, romantic life, passionate life, ensouled life. To be in a life other than the life you have been living, imagination and desire to be a person other than the person you have been must be the initiator. Imagination and desire can be the key that turns the ignition of change. You know the definition of insanity\u2014doing the same thing over and over and expecting different results. To get different results, you must do something different. You have an image of what you desire, but you haven't been able to have that in your life. Something has to change at a fundamental level. We keep looking for Mister Right to come along; but even if he did, we're still the same person, doing the same things, with the same unhappiness, the same feelings of unworthiness, the same psychoses, and we're still reading from the old scripts our family gave us.\n\nIntimacy comes from the Latin _intimus,_ meaning \"innermost.\" To be intimate with another, we first must become intimate with ourselves, our primary relationship, which is the foundation of intimately relating to others, to Earth, to anyone and anything. There is a problem with being intimate with ourselves when we are more concerned with what others are feeling than what we are feeling. Lacking intimacy with ourselves creates or at least maintains a fragmentation within our self, an incompleteness, an unfinished business of the soul to make itself whole. The soul has its own desires and purposes. It is here to make itself, to remake itself, to be educated in the journey of becoming a human being.\n\nIntimate moments assert themselves into our lives when we least expect it. Consider how uncomfortable it is to see a friend or acquaintance in the grocery store. It's an intimate thing to have someone see what we are buying and taking into our homes. Some of the things in the cart are . . . intimate. Remember the experience of visiting a friend's home for the first time? There is almost always the requisite tour of the house. As you move more deeply into the heart of the home, a little unease begins to push on you for you see intimately how people live and what is important to them. Those feelings of unease build as you approach the bedroom, since we know what goes on in there, in that bed, under those blankets, and suddenly we turn to leave the room, quickly. These are two examples of how uncomfortable we can get simply with superficial intimacy.\n\n _To be what we are, and to become what we are capable of becoming, is the only end in life._\n\nROBERT LOUIS STEVENSON\n\nA primary focus of this book is to go beyond and below casual superficiality and assumptions to the place where real and lasting transformation happens. My goal is to get to the source of those uncomfortable feelings and eventually, ultimately, not let them unhinge you.\n\nWHAT IS THE SOUL?\n\nBefore I go further, I feel it's necessary to come to a working understanding and definition of the word _soul_ for the duration of this book.\n\nWe know what love is, and we know it has different meanings with respect to the relationship. \"I love my granddaughter\" has a particular kind of meaning, and you know it when you read it or hear the words. \"I love steamed artichokes\" has another kind of meaning, different from \"I love my granddaughter.\" \"I love my horse\" has still another meaning in it, particular to my relationship with my horse.\n\nBut soul is something that, after more than 3 million years of human habitation on Earth, we cannot seem to agree on. There are other invisibles that we can agree not only exist, we can agree on the meaning of them as well. Wind, for example, and heat and cold are invisible, but we know them through feeling and how our bodies experience them. We know love through feeling it and experiencing it, just as we have the experience of heat and cold.\n\nSoul, like love, is an invisible thing; you can't see it, but you can feel it. The soul of a song that stirs you to the core, the soul of a great poem or story or painting. When we are touched by a great work of art, it is a soul-to-soul touching. A person who is a good soul walks into the room, and you feel the soul of that person walking in. But feeling, knowing our own soul, is even more elusive to us than the soul of a song that moves us deeply. Most of us can agree that the thing that is present in a living body that is not present in a dead body is soul. Soul is not a flat, two-dimensional phenomenon.\n\nThe concept of soul is ubiquitous around the globe and has been since antiquity. Socrates made the appeal that it is the soul that animates the body of a living thing. If a person is clinically brain dead from an accident is the soul still present? Does the soul need a fully functioning body to be present? I have met paraplegics and people with missing limbs and have sat with my dying father-in-law while he was unconscious. I have experienced their souls to be intact, present, and available. We've heard from scientists and from people who have come out of a coma that comatose people are not only able to hear conversations around them but remember them and the feelings in the room. Who is remembering? Contrarily, I have met people who are healthy in body and sick in soul. Soul is the bearer of moral qualities and moves toward ideals of justice, courage, and truth.\n\nThe idea that the soul animates the body doesn't fit when we consider stones, mountains, or trees or even in the way we think of humans and animals as being animated and moving about. Have you met a stone outcropping or stood gazing out from a mountaintop that moved you to stillness and awe? What was it that struck you, caused you to stop in your tracks and stand still, to inhale deeply in those moments? The energy coming off the stone face or rising up from the valley below, from the breathtaking view\u2014is that the soul of the stone or the mountain? Is it the spirit of that place?\n\nSomething has reached out and touched us despite our hurry to get somewhere. And what in us is responding to such spectacle, to such a mystifying feeling? When we stand in an old-growth forest, amid giant redwoods and thousand-year-old trees, what is happening in the silence of the forest that can humble us to tears or raise the hairs on our arms? What happens inside us when a tree we planted as a sapling and watered, nurtured, and loved is suddenly uprooted by a storm or taken out by fire or a neighbor's chainsaw? The part of us that bonded with the tree is deeply affected, changed by its absence; we've lost a dear friend to death. Something invisible is presenting itself to our senses, to our feeling body. Something bigger and outside us, some self-organized whole that cannot be found in its parts has stopped us in our tracks.\n\nMichael Perlman describes the impact trees have on our psyche, our soul, as \"[t]he reforesting of the soul.\" When we sit with a favorite tree or walk in the forest, the impact on our soul is a sense of renewal and of deep connection with someone more than human. I've seen students sit with trees and witnessed the transformative powers trees have on their psyche. The erect straight trunks of pine trees infuse our souls, causing us to stand more erect and treelike; we develop a more profound sense of ourselves, of our boundaries, and feel stronger. Alligator junipers that grow in the high deserts of the Southwest lift our spirits to greater heights, bringing a sense of hope and a feeling of falling in love with all of life. Trees are erotic and sexy; it's not an accident that while walking in a forest with your beloved you are often moved to make love on the forest floor, immersed in the fecund, musky smell of leaf litter and pine needles, beneath a great towering canopy. Pine trees produce a great deal of pine pollen, one of the most abundant and ready sources of testosterone. It increases testosterone in the body and balances the androgen\/estrogen ratio.\n\nThe animistic worldview asserts that everything is ensouled, alive with energy. _Anima_ is Latin for \"soul,\" thus _anima mundi_ is the world soul or soul-infused world, a mysterious life-force energy ensouling all that is created. From archaeological reports, we can surmise that so-called primitive peoples not only believed in an ensouled world but also saw the energies, souls, and spirits of things. They left a visible record of their encounters on cave walls and rock faces and erected monuments to the unseen, yet ensouled forces of the Universe.\n\n _The soul should always stand ajar, ready to welcome the ecstatic experience._\n\nEMILY DICKINSON\n\n _Good for the body is the work of the body and good for the soul is the work of the soul, and good for either is the work of the other._\n\nHENRY DAVID THOREAU\n\nI have a sense of what soul is in me, in others, in animals, in plants, and in ecosystems. I cannot definitively define and point to it, nor has science been able to dissect it. I do believe that some things are meant to remain indefinable and mysterious. We understand very little if we can only understand what can be explained. Perhaps soul is one thing that we are not meant to fully grasp, though I think my theory and understanding of it is as workable as anyone's. We are each in our own private, trembling lives, fumbling toward understanding mystery and the mystery of our lives as much as possible.\n\nI believe soul to be the accumulation of all the parts that make up the \"I\" of us.\n\n _[T]he absence of the soul is far more terrible in the living man than in a dead one._\n\nCHARLES DICKENS,  \nBARNABY RUDGE\n\nThe Oxford English Dictionary (OED) defines soul as \"the principle life in man and animals; animate existence; the principle of thought and action in man commonly thought of as an entity distinct from the body; the spiritual part of man in contrast to the purely physical. Also, occasionally the corresponding or analogous principle in animals; the personification of some quality; the inspirer or leader of some business cause, movement, etc. The chief agent, prime mover or leading spirit.\"\n\nSoul has gone through some interesting semantic evolutions. From the Greek, Psyche is soul. It is through trials and tribulations that we are forced into the arena of psyche to craft responses and build character. In love relationships, Eros is a prime mover creating chaos in our interior world, forcing us downward and into the ground of our psyche. The Celts have defined spiritual wholeness by three conditions. According to the Encyclopedia of Celtic Wisdom, _cra' bhadh_ is the trust of the soul, or devout observation; _creideamh_ is the heart's consent, or belief; and _iris_ is the mind's pledge, or faith: \"When these three are as one, then there is true strength and power within the _coich anama,_ or the soul-shrine, as the body is termed. The body is like the cover of a triptych which unfolds its panels to reveal a landscape full of wonders.\" _Anam,_ Gaelic for soul, insinuates life's vigor and strength of character and is related to _anal,_ the breath of life. The soul weaves in and out of consciousness and wanders in and out of life. It is seen as dynamic, fluid, and flexible and can travel in and out of the body, and it is \"intimately moved by the mind and the heart, and cannot often be seen as separate from either.\" The Celtic soul is not seen as superior to the body or subservient to a divine master as in Christian definitions of soul. The soul is viewed as a personal responsibility; we are responsible for maintaining its wholeness and integrity.\n\nSoul knowledge is the basis of all self-knowledge. It is intelligent, aware, mysterious, and fascinating, and it influences and shapes who we become. Soul has its own destiny. Soul seeks to know the truth of itself and strives to have meaning, has will toward meaning. When we are unable to find the meaning of our lives, when there is meaninglessness, we do meaningless things, and it is in the midst of meaninglessness that people resort to violence. If we feel our life has no meaning, then what we do\u2014the choices we make and what we think\u2014does not matter.\n\nViktor Frankl articulated three drives, or wills, to meaning in our lives, and two of them occur when the basic will to meaning is frustrated. The will to power and the will to pleasure are \"substitutes of frustrated will to meaning. The search for meaning is our basic concern. Only a man who has been frustrated in his basic concern resorts either to will to power or will to pleasure.\" Living out a search for power ends in violence. The will to seek pleasure or the pursuit of happiness is self-defeating in that it is the very pursuit of happiness that derails us. We think that we can find happiness if we have enough money, enough things, enough love, enough security, and then we won't be afraid or sad any longer. Happiness is a by-product not an end in itself. Happiness is the effect of finding meaning in our lives, though it is related to happenstance; it comes and then it's gone. Joy, however, endures. It is the payoff, the effect of being in love, of loving another human being, of understanding the meaning in our work, our relationship with Earth, and in the experience of being alive. Plotinus asserted that happiness could not be found in the physical world, but that even daily, physical acts were determined by the \"higher phase of the soul.\"\n\nOur lack of having a sense, and an understanding of what soul is in ourselves; what its function is, its role in who we are and what we are drawn to, and what the soul is in all living things means we live in a world with no intimacy. Everything then becomes cold, dead, lifeless material to be used or consumed. Living an ensouled life is to have a relationship with not merely the physicality of Earth, with wilderness, and with sex, but to have joyful interrelations with the livingness of each, with the differentiating soul of each. It is a lack of understanding and intimacy that causes us to fear things we don't understand that leads us to clear-cut forests, to dominate and have power over what we do not know intimately.\n\nSex is not merely some thing or an event in the midst of a circumstance. It is a primal, life-force energy that happens in a soul-filled sea of details\u2014of wants, desires, and hormones. Sex is soul driven to connect with the ancient powers of life, death, and procreation. Owning our wild sexual natures involves self-understanding: bringing our faintly appreciated feelings to full consciousness and becoming more aware of how we individually and as a species are related to other species.\n\n _There is no easy or quick plan to happiness, there is no single spot where you can start. Where you are right now is the best place to begin. Be careless in your dress if you must, but keep a tidy soul._\n\nMARK TWAIN\n\n _Through soul you build your own world._\n\nJOHANN WOLFGANG VON GOETHE\n\nOnly what you know to be true from your own experiments, feelings, and perceptions has value to your current life. The soul is always seeking to be expressed\u2014that is its goal. It responds in each moment to what is presented to it. As you respond newly in the midst of old, familiar situations, new patterns are laid down, patterns that verify, ascertain, and reveal the soul's true essence. With each pattern, like setting mosaic tiles, the soul becomes ever more sophisticated and true in itself.\n\nThe soul is shaped through life experiences, difficult and challenging family and work relationships\u2014seemingly dead-end jobs or demanding and fulfilling jobs and the trials and joys of marriage and raising children. It is educated in the fabric of life experiences, whether on solitary treks to the wilderness or on crowded subway trains. Through conflicting demands, illness, contrasting experiences, and accidents that when rightly seen may not be accidents at all, the soul discovers what its own values and belief systems are. It is through these life experiences that the soul knows what feels good, right, and true and what is not supportive to its life force and destiny. It is being in the world and through the world that our soul takes on its own unique shape and character. It is helped along if we are present in its making, home in our bodies using our internal sensory guidance system.\n\nThe more our soul, the I of us, is taken into account and accounted for, the more we are able to hold ourselves responsible. The little or large indignities that we at one time accepted become insufferable, and we are moved to respond to restore and maintain our essential dignity and character, to keep our soul intact, so that we are no longer the enemy of ourselves. With the OED definitions we can extrapolate that the I of our soul is the \"chief agent or prime mover.\" The soul is the \"leading spirit\" of our lives, always moving in the direction to complete its wholeness. Ultimately, when we reach the end of our Earthly days, will we be able to face ourselves and answer \"yes\" to the questions: \"Did I become myself?\" \"Did I live my own life?\"\n\nWHO IS \"I\"?\n\nTake a few, slow moments to ponder the following questions. Do you ever eat in secret? Are you afraid to ask for what you want? Does your partner know you masturbate? Do you know what you want? What inspires you? Are you afraid to ask for help when you need it? Do you feel guilty when you ask for help or get what you want? Are you afraid to let your lover see you naked? Are you uncomfortable with your sexuality? Does sex scare you? Have you been waiting for someone else to make you happy? Are your other relationships the primary source of comfort and security? Do you treat yourself often with massage, hot baths, a day off in bed? Do you give yourself what the Italians call the sweetness of doing nothing? Do you enjoy your own company or do you avoid being alone? Do you hide yourself or practice invisibility to feel safe? Have you ever looked at photos of naked women? Naked men? Do you enjoy it? Does anyone else know? What's your favorite type of nude photo? How do you feel about masturbation? Are you comfortable walking around your house naked? What excites you sexually, passionately, gets your juices flowing? Are you pleased with your answers?\n\nWho inside you answered these questions? Did you answer with your head or your heart? Were you honest with yourself? What will you change so you'll feel good about your answers and yourself? What will it take for you to be autonomous, self-ruling, and free?\n\nAsking yourself these questions and others initiates the process of getting to know yourself. We take ourselves for granted and operate unconsciously on autopilot a good bit of the time. We have a surface relationship with ourselves, one that gets us from home to our jobs and back home again. The labor involved in getting to know ourselves is not unlike peeling an onion, one layer at a time. The onion is not really the best metaphor because our psyche is not organized in neat, peel-away-able layers. Though there are tears. The way I work with the parts of me is more like a model of the Native American Medicine Wheel with the I that is I in the center. All the other parts of me, my ego states, are around me like spokes in a wheel. That was my basic working model to begin with, but over time as I worked to integrate and harmonize my fragmented selves, the relationship between my different selves has become a more free-form, organic, and spontaneous working\/playing relationship. Krishnamurti talks about it as the center and says:\n\nThe centre is a bundle of memories, a bundle of traditions, and the centre has been brought about by tension, through pressure, through influence. The centre is the result of time, within the field of culture\u2014and so on. So that is the centre. Now that centre, because it is a centre, has space outside it, obviously. And because of the movement, it has space in itself. If it had no movement it would have no space. So there is space, outside the centre and in it. And the centre is always seeking wider space, to move more widely. To put it differently, the centre is consciousness. That is, the centre has the borders which it recognizes as \"the me.\"\n\nTo get to the center, to the I, we begin to look at how the scripts we were given and the ones we made up for ourselves as children are part of our programming; they've become beliefs about how to act and think and what sort of person we are. They are not subconscious in the clinical sense of the term, but they do run our show without our awareness much of the time. Our subconscious has been running on autopilot for thousands of miles over many years. The conscious is the creative, thinking part of us, as well as the part that can notice what the subconscious is doing and saying. When consciousness is active, we say we are aware or awake. We are actively noticing. When we go unconscious, our default mechanism is to act out the old programming, often with detrimental, unpleasant effects. When the conscious, aware part of you quits paying attention, you stop thinking and essentially go to sleep, then the unconscious and unintegrated parts that have been storing up feelings of mad, sad, scared, and enraged use your mouth to form words that fall out like glass marbles tumbling to the floor. Noisily at that. If we are unconscious, it's nearly impossible to stop it from happening. Then you have a relationship mess on your hands to clean up. If you're very lucky nothing gets broken like pottery or glass or trust.\n\nConsciousness is the interaction of physical and cognitive processes, the faculty of perceiving. Ayn Rand wrote, \"A consciousness conscious of nothing but itself is a contradiction in terms: before it could identify itself as consciousness, it had to be conscious of something.\" Consciousness is a degree of awareness as opposed to alertness. For example, we've all had the experience of driving along in our car and then realize fifteen miles of road have passed under our wheels without our conscious awareness. Suddenly we \"come to\" and realize we have no memory of driving the car, but we were driving the car. Often we don't remember what we were thinking or where our minds were during the past fifteen miles. Something, or some part, drove the car; some part of us that knew how to drive the car got us safely along the miles that are now behind us while we were doing something else. Were we conscious while we were driving the car as our \"minds\" were someplace else? Were we simply alert enough to prevent an accident? The witnessing self, the I, went someplace else, took a nap, perhaps, or traveled to some distant planet to gather inspiration or work out a problem we've been having. That part came back just in time, and it was the reentry into our bodies that jolted us to awareness of our circumstance and, as it were, good fortune. Maybe we were daydreaming. The point is that during those miles, we can say that we \"went unconscious.\" Our awareness took a detour while we mundanely operated a motor vehicle.\n\nA solid foundation of intimacy with another person begins here, inside yourself, the I that is I. The work is difficult and challenging, and through it you'll confront hidden value systems and beliefs about yourself, about sex, and about how your world and the larger world works as well as your beliefs, values, and relationship with Nature. You'll meet yourself face-to-face, one-on-one. As one of my apprentices has often said, \"This work is changing me.\" Indeed, it does change you; it is a transformation of character. And it is work. It requires making time in your life for you and for any sequestered parts of you to become unsequestered and in the open so they can have a life with you and through you. It is the work of what Jung called making the darkness conscious; bringing into conscious awareness those aspects of ourselves that have been hidden. It is bringing all that has been unconscious into our full awareness.\n\nFILLING THE LONG BAG\n\nEach human being is a complex, interesting, multifaceted system with a multiple personality. As infants, we come into the world as a vortex of energy, unabashed and unrepressed. All our basic needs for survival get met, and if we are lucky, our innate need for love and bonding are met as well. As we grow, we learn to get most of our wants met. As children, we know very well what we want and that we want it now. And we say no to what repels us.\n\nBut it doesn't take long for an infant to discover how to fit into her family or social environments. Between birth and six years old, we are most impressionable, and a great deal of the programming of our subconscious happens between these years. During these early years, we start accepting things that aren't true. This is how the child of us becomes corrupted. We're programmed to devalue and disempower ourselves. By the time we are two or three, we have a growing list of what we are to do to fit in\u2014brush your teeth, eat all your food, pick up your toys\u2014and an even longer list of what not to do\u2014don't eat the dirt, don't hit the dog, don't cry, don't talk in church, don't ask questions, don't play with yourself, don't color outside the lines, don't hit your brother, don't be noisy.\n\nSlowly, the vortex of energy we once were begins to lose vitality as parts of us are put in a bag. The young child learns quickly that parts of her personality are not acceptable to those who love her, so she puts the unlovable or inappropriate parts of herself in a secret room inside her, or they get put in what Robert Bly calls the long bag. In _A Little Book on the Human Shadow,_ he says, \"We spend our life until we're twenty deciding what parts of ourselves to put into the bag, and we spend the rest of our lives trying to get them out again.\"\n\nWe learned to sequester parts of ourselves because they made others feel uncomfortable and were unacceptable in our family, in school, or in social settings. As we grow, we are socialized (made friendly or cooperative) to fit into the narrow definitions of what it means to be a human being, what it means to be happy, and what it means to be a contributing member of society. Often, part of that socialization means that we begin to believe what others say about us either explicitly (\"You cry too easily\" or \"You're lazy\") or indirectly through their behavior toward us.\n\n _Intimate relating begins with the self. It is a toxic fantasy to believe that we can be intimate with others when we have not learned (or are afraid) to be intimate with ourselves. Self-intimacy develops naturally when we have not been excessively poisoned by the toxic attitudes of others toward us or by toxic patterns that we inflict on ourselves._\n\nJERRY GREENWALD, _C REATIVE INTIMACY_\n\nUnfortunately, women and girls are too often told and trained to be nice and to look pretty. Unfortunate because inside those admonitions are messages that intuition is not to be trusted, that who they are and what they are feeling are not trustworthy and that anything besides \"nice\" is not acceptable and certainly wildishness is out of the quesion. They are trained to be powerless. Clarissa Pinkola Est\u00e9s notes: \"Women's curiosity is given a negative connotation, whereas men were called investigative. Women were called nosy, whereas men were called inquiring. In reality, the trivialization of women's curiosity so that it seems like nothing more than irksome snooping denies women's insight, hunches, intuitions. It denies all her senses. It attempts to attack her fundamental power.\"\n\nFor men, as well, intuition is power, is genderless and a function of a healthy soul. Little boys are also given some version of these messages: be nice, tuck in your shirt, comb your hair, don't cry. With these messages planted, boys grow to mistrust their intuitive natures and are unable to discern what's real or what they are feeling and unable to say \"the emperor has no clothes.\" Attempts to kill the willful, wild spirit in children are many and come disguised as teachers, parents, and religious leaders.\n\nWe'll be spending some time in the next chapter getting to know the parts of ourselves we've put in the secret room or the bag we drag behind us. Either metaphor can work, but Bly's metaphor of the long bag expresses how putting parts of ourselves away acts as a weight on who we are; it restricts our movements because often the creative parts, the predator, or the outrageous parts are in that bag. It's time for them to come out, to have a life and to be part of your life. It's time for you once again to be a vortex of energy, a spontaneous, 360-degree personality.\n\nTHROWING OUT THE OLD SCRIPTS\n\nYou can get a sense of how important it is to develop an intimate relationship with yourself if you begin to notice the internal self-critical chatter as you're tending to work or chores, as you interact with other people, as you walk by the mirror and see a reflection of yourself. Take note of how many times you hear a voice inside you that says \"You're getting fat,\" or \"You shouldn't have said that out loud,\" or \"You're not smart enough to do that.\" These lines are scripts; that is, early programming that was given to us and internalized as truth. For some time you've believed the script, and it's influenced who you are and the choices you've made. Now, those admonishments are old, battered, overused; the expiration date has long expired, and they've begun to taste sour in your mouth. They no longer work, nor should they.\n\nEric Berne, founder of transactional analysis, defines scripts as lines and codes of behavior given to us by others, usually family members, but scripts can come from anyone whose authority we take more seriously than our own. The term _scripts_ expresses Berne's idea that each of us follows personal life scripts from early childhood decisions and parental programming. Maintaining the scripts into adulthood restricts our movement and flexibility of responses.\n\nWhen I hear the word _script,_ I have this image of standing on a stage and being handed pages filled with dialogue that is meant for me. I take a cursory look at what is written there and throw it back at the stagehand saying something to the effect of \"Over my dead body I'll say these lines; give me some new ones.\" But there isn't time for anyone to write new lines for me, so I ad-lib (you must be able to improvise). I take a deep, inspiring breath that expands my belly. I inhale inspiration from the air spirit as the in-breath spirals in and down into my womb. A valve opens up, screwed open by the spiraling breath. Some old, ancient matriarchal energy begins to rise up from the primordial womb that connects all sentient, sacred, sexual beings and moves through my womb, up through my belly, dusting off the wild, instinctual, intuitive, truth-telling part of me. I laugh uproariously, a full-bellied, breast-shaking, table-pounding, tears-flowing laugh. I can't stop. Something has been set free, my voice has found its sound in the world, and I hear it echo off the theater walls. I fall in love with the sound of it. I have a vision of other women and children throwing their scripts off the stage or burning them. My soul begins to sing through the instrument of my body.\n\nThrowing out old scripts is going against the unspoken rules of behavior set down by our family and culture. Breaking the rules, speaking out, doing something different, making noise is disconcerting at first. Notice if you feel scared after you've had an outburst of laughter, witticism, or anger, as if you have done something wrong. Est\u00e9s writes about the meaning behind a sudden wild laugh.\n\nIn the sacred, the obscene, the sexual, there is always a wild laugh waiting, a short passage of silent laughter, or crone-nasty laughter, or the wheeze that is a laugh, or the laugh that is wild and animal, or the trill that is like a run on the musical scale. Laughter is a hidden side of women's sexuality; it is physical, elemental, passionate, vitalizing, and therefore arousing. It is a kind of sexuality that does not have a goal, unlike genital arousal. It is a sexuality of joy, just the moment, a true sensual love that flies free and lives and dies and lives again on its own energy. It is sacred because it is so healing. It is sensual for it awakens the body and the emotions. It is sexual because it is exciting and causes waves of pleasure. It is not one dimensional, for laughter is something one shares with oneself as well as with many others. It is a woman's wildest sexuality.\n\n _I think what a joy it is to be alive and I wonder if I'll ever leap inward to the root of this flesh and know myself as once I was._\n\nFRANK HERBERT,  \n _D UNE_\n\n _Man's task in life is to give birth to himself, to become what he potentially is, the most important product of his effort is his own personality._\n\nERICH FROMM,  \n _M AN FOR HIMSELF_\n\nRETRIEVING YOUR SOUL\n\nThis work, essentially, is doing your own soul retrieval. It is immediate, empowering, and rich and can ultimately bring renewed creativity and joy. The child parts of you that were put away, shut out, and marginalized are given a place to live inside you, an opportunity to be part of your daily life, and given permission to live loud and free and imaginatively. The door through which we wish to walk into freedom and into wholeness is not outside ourselves; we are the door. And once the door is found, one must then get up and walk through it. You may travel to faraway lands and seek a guru, but you will not find what you are ultimately seeking. What you seek is in a place that is too obvious and ordinary to consider: the soul that is called _I am_.\n\nI don't need a minister to mediate between me and my gods or Creator or the wind or Mother Earth or my deceased grandmother or anyone for that matter. I go directly. I sought out intermediaries early on because they were the ones I found to talk to. Also, I was looking for someone else to make decisions for me or think for me. It felt important then in a distorted sense. In hindsight, I see how disempowered I kept myself. Until I found a spiritual path, I didn't know I could do it myself or how. On rare occasions now, I'll ask for external validation and an outside perspective, someone other than my partner or a close friend. Primarily, I go directly to the source, feel for the truth to sound its bell inside me. I no longer want anyone making decisions for me or telling me what to do. I believe in personal responsibility as a catalyst for deep and lasting changes that will echo into the larger society, one concentric ring of change at a time.\n\n _I believe that we are solely responsible for our choices, and we have to accept the consequences of every deed, word, and thought throughout our lifetime._\n\nELISABETH K\u00dcBLER-ROSS\n\nHaving said all that, there are times we need someone else, another human being, to help us find our way. There are times when we are so deep in our emotional swamps of despair, grief, and fear that we are unable to see, think, or feel. In those times, by all means seek out someone to hear you and help you find your way. Be certain, however, that you are clear about why you are going to that person, what you are asking for, and what you need and that you are always feeling for the truth of it. Having clarity going into a situation can prevent drama and minimize the possibility of a disappointment or disaster.\n\nBeing vulnerable is the hardest thing of all. Sometimes the mere mention of a word brings up thoughts, feelings, and emotions that stop us in our tracks. We make a 180-degree turn and run as fast and far as we can in the other direction. Being vulnerable and self-revealing are tantamount to living an ensouled, intimate life.\n\n _As human beings our greatness lies not so much in being able to remake the world\u2014that is the myth of the atomic age\u2014as in being able to remake ourselves._\n\nMOHANDAS K. GANDHI\n\n _We all walk in mysteries. We do not know what is stirring in this atmosphere that surrounds us, nor how it is connected with our own spirit. So much is certain\u2014that at times we can put out feelers of our soul beyond its bodilylimits; and a presentiment, an actual insight is accorded to it._\n\nJOHANN WOLFGANG VON GOETHE\n\nBeing aware is one thing. Knowing what your options are in any given moment and circumstance is another thing. And choosing which of several options to act on from a place of wholeness is something else. The necessity for rigorous self-examination is not an academic exercise. You have to actually do it. No one can do it for you. It is the struggle for consciousness. It is a struggle because the parts of you that are invested in your being unconscious, in keeping a safe distance and maintaining old beliefs, will compete to maintain their reality; they will escalate, act out, and have temper tantrums. Those parts will offer you all sorts of justifications for maintaining the old beliefs and behaviors. When this happens, you must sit down with them and hear them out. Then you tell them all the reasons the old ways no longer work, how you want more from life and a deeper understanding of who you are. Tell them you want to be free. You must find inside yourself motivations for the change and hold on to that. Ultimately, you need their agreement, need to have them in alignment with your decisions for lasting change to take root.\n\nSignificant motivations for me are that I want to be free from reactionary, outdated beliefs. I want to create my own value system, not carry ones that belonged to other people who were afraid and didn't care to find another way. I want to know who I am and I want my soul to have its life in its most authentic, gratifying, passionate manner. I want to be honorable in word and deed. I want to live large, be outrageous, outspoken, defiant, incorrigible, irrepressible, unpredictable, creative, a force to be reckoned with. I want to know my interior world and all its inhabitants, all their gifts and virtues and idiosyncrasies, what moves them to tears and makes them shudder with joy and pleasure. These are things that motivate me, keep me going forward through the brambles, rubble and scree, and worn-out collections of cultural and familial scripts and values. These are the forces moving me onward toward clarity of understanding and freedom from habituated manners and fear of movement. I want to be called beloved by myself and to call Earth beloved. I want to feel alive. I want to be free.\n\nBeing intimate with yourself is the beginning of being intimate with another being. Not the new age rhetoric of simply loving yourself more or for the first time, though that is a small and beginning part of it. Intimacy is more than love, for love is never enough. Intimacy is self-knowledge and self-understanding; it is befriending yourself, companioning yourself, and caring enough to notice who you are, who you have been, and who you've always wanted to be. It is caring enough to get your hands dirty digging up the dreams you had as a child, those adventures you've always wanted to have, the projects you wanted to create, the books you wanted to write that have never left you. It is finding a way, if it's the last thing you do, to have them come alive in your life, in this lifetime, so the soul and spirit of you are awake and engaged with living.\n\nIt involves descending the treacherous steps downward into the basement of your psyche to find what makes your soul sing, to discover where your soul would have you go if you were to, at last, ask and listen for the answer that comes in the silent inner knowing deep in the intuitive heart of your child.\n\nIntimacy begins in your inner world, between your legs, in those sensitive nipples, in your bedroom, on Earth\u2014experimenting, exploring what is sexually exciting and pleasingly satisfying to you. It's discovering what makes your soul thrum, what inspires you, what holds you back. It is understanding the meanings in the stride of your walk, the carriage of your shoulders, the tilt of your chin, the look behind your eyes, and, more importantly, the who of you that is looking out of your eyes. This is the work of becoming conscious.\n\n _Making your unknown known is the important thing._\n\nGEORGIA O'KEEFE\n\nConsciousness\u2014that state of being aware of your internal world\u2014begins the journey of becoming an integrated, whole human being from the inside out. It is being present with and aware of how you are feeling and what you are thinking in the midst of doing. Integration and coming to full consciousness begins with where you are. So, where are you right now? In your head? In your heart? Are you aware of your feet, your breathing, the aches and pains in your body? Are you happy, sad, mad, scared, pissed off, frustrated? Take inventory right now of how you feel. Go deeper inside. What voices are you hearing? What are they saying? Are they shouting, joyful, raging, whispering, whimpering? Not least of all: What are you seeing? And who is doing the seeing?\n**3**\n\n _ **Getting to Know You**_\n\n_Without deviation from the norm, progress is not possible._\n\nFRANK ZAPPA\n\nI have been trained to use Eric Berne's Transactional Analysis model for identifying and gaining facility with interior ego states. Transactional Analysis is a modality that Berne, a Canadian-born psychiatrist, called interpersonal interactions. Berne described an ego state as \"a coherent system of feelings, and operationally as a set of coherent behavior patterns; or pragmatically, as a system of feelings which motivates a related set of behavior patterns.\" He noted that \"e]go states are normal psychological phenomena.[*1 The human brain is the organ or organizer of psychic life, and its products are organized and stored in the form of ego states.\"\n\nHuman beings are a kaleidoscope of multiple ego states. All of these ego states possess energy. Sometimes a great deal of energy. Have you ever seen a two-year-old having a temper tantrum? Comments Jungian analyst James Hollis: \"It is not that we have a single child within, perhaps hurt, frightened, codependent or withdrawn in compensation, but a whole host of children, a veritable kindergarten, including the class clown, the artist, the rebel, the spontaneous child at one with the world. . . . Virtually all have been neglected or suppressed.\"\n\nEach ego state has a function, and each one will perform its function whether you want it to or not\u2014or are conscious of it or not. It is crucial, over time, to be able to identify the ego states; free any that are repressed or locked away; understand their functions; and make friends with them so that you exist as a coherent, integrated whole. Robert Bly says in _A Little Book on the Human Shadow: \"_ But why would we give away, or put into the bag, so much of ourselves? Why would we do it so young? And if we have put away so many of our angers, spontaneities, hungers, enthusiasms, our rowdy and unattractive parts, then how can we live? What holds us together?\" And many of us are not held together very well, but we find ways to compensate and keep going. There is another way.\n\nThe goal is to have facility with interior ego states so that we walk as whole, integrated human beings in the world.\n\nTHE PARENT, ADULT, AND CHILD IN YOU\n\nBerne thought of every individual as being three different people: a Parent, who is the moral consultant and who may be critical, sentimental, or nurturing; a rational, factual Adult who sets up contracts and commitments with other people and says yes or no; and a compliant, rebellious, or spontaneous Child\u2014the instinctual, intuitive aspect of us, the one who \"takes the trip\" in sex and in life. In the sexual arena, her vocabulary consists of \"Wow, that was great!\" (The Child is a natural aphrodisiac.) And in some cases when the Parent or Adult has used bad judgment, \"ugh, ouch, or yuck.\" When I use the term _Child,_ I'm referring to any aspect of ourselves from infancy to about age twelve but truly, the Child never stops growing. It becomes more sophisticated in the world into the teens, twenties, fifties, and old age. I use the terms _ego state, aspect,_ and _part_ interchangeably.\n\nThe Parent ego state is the voice in our head, which also sometimes uses our mouth to say things out loud. It uses words and phrases such as \"Well done,\" \"I'm proud of you,\" \"I'm sorry you got hurt,\" \"You're the greatest.\" The Parent takes two forms, one direct, one indirect or one nurturing (natural) and one critical (adapted). The nurturing Parent is an actual ego state that is _cathected,_ which means you put energy into that ego state so that a person becomes a nurturing parent. The critical Parent is a Child adaptation, not a natural Parent ego state but an indirect influence of the Parent. This one responds the way the real mother or father actually responded\u2014\"Do as I say, not as I do.\"\n\nThe Adult part of us is matter-of-fact and operates very much like a computer, calculating variables of speed, time, space, and outcomes. It tells you when to cross the street and how much speed is needed to cross safely, when to hold 'em or fold 'em, how to analyze outside data for optimum decision making, and how to operate an iPod and MP3 player. The Adult uses words such as _now, wait, need more information, in ten minutes_ \u2014factual statements. It has no interest in maturity but it is interested in accuracy. The Adult mediates objectively between the Parent and the Child.\n\nThere are also two forms of Child: the natural Child and the adapted Child. The Child personality of us is the little boy or little girl we once were. Each of us has a predominant child ego state; in some it may be the four-year-old, in others it may be the two-year-old or nine-month-old. Berne doubts that it is ever older than six years old. When the natural Child is cathected, we are spontaneous and childlike. The Child is the best part of us, the most creative, joy-filled, intimate, loving, just as real children are. The natural Child is best described as the little girl who has an energy coming off her that is magical and magnetic. When she walks into a room and your heart just goes out to her; you're naturally drawn to her. In men, it's the Child that makes them fun, witty, charming, charismatic (I love charisma), and playful. In women, the Child looks very similar: charming, witty, fun, and playful\u2014these are sexy characteristics. It is the Child in us that loves Nature, loves the elements, loves socializing, loves to have fun, loves to feel, to be nurtured, soothed, held, and stroked.\n\nWhen an adapted Child walks into a room, you immediately know there is something off-putting about her. Adapted Children react to their circumstances and their environment by withdrawing, whining, or pouting; they are disagreeable to be around. They are complicit, behaving as mother or father wants them to behave so they tend to take on the voice of their real parents. This behavior masks painful or scared feelings and their inability to deal with the world around them. An adapted Child might say critical things such as, \"You shouldn't have done that,\" \"Stop being so immature,\" \"That was childish,\" \"Don't be ridiculous.\" Berne says, \"The Parental influence is the cause, and the adapted Child the effect.\n\nTHE PREDATOR INSIDE\n\nAnother part of us, not specifically identified by Berne in his model, is the Predator. Human beings are a predator species with a capacity for unkindness, and cruelty. As Elisabeth K\u00fcbler-Ross was told by a young Jewish woman, a Holocaust survivor who had lost all her brothers and sisters, parents and grandparents, \"The Nazis taught me this, there is a Hitler inside each of us, and if we do not come to terms with the Hitler inside us, the violence will never cease.\"*2 Each of us has a Predator living inside, a Mr. Hyde who is the worst in us, and until we face the predator inside, it will always lurk in the murky backwaters of our psyche planning a way out. There is a part of each of us that has wanted, may still want, to cause harm to something or someone. It has tremendous energy, often driven by unresolved anger about a perceived injustice, a discount to the Child, a recent lie told to you by a friend that you have stuffed inside yourself. Or this part may be driven by archaic rage\u2014a deep-seated anger that has been fermenting and bubbling inside you while you've been holding onto something from the distant past. Rage is the kind of anger that the Predator can really sink its hooks into. When motivated by archaic rage, the predator may lob bombs at the person who originally caused you harm or at someone who in some way resembles the original perpetrator. Left unattended, either unresolved anger or archaic rage will result in bitterness, resentment, and soul sickness.\n\nThe human Predator takes two forms\u2014the natural Predator and the adapted Predator\u2014and comes in all sizes. The natural Predator has an ecological function within our structure. Its job is to be alert to danger that may threaten its survival. When life is going along swimmingly with the current, everything seems to be in order, and the Universe is gracing you with relatively calm seas, the Predator relaxes its watch some. Then suddenly the winds pick up and the current becomes rough; the Predator is alerted to danger and engages its 360-degree radar for incoming trouble. Its function is to fight for your life either by hunting for food, a job, or a mate or by defending your life and property, your rights, and your dignity. It fights for what is rightfully yours. When we fail to integrate and work with our natural Predator we become easy prey, falling victim to crime, abuse, theft, financial scams.\n\nHuman beings are carnivores by nature, vegetarian by choice. We have pointed eyeteeth that we expose when we open our mouths to eat meat or vegetables or to smile. We are hunters of the animal order. We have an innate Predator and when that part is integrated its actions feel completely congruent and fluid, in the natural order of things.\n\nThe adapted Predator is another unfavorable adaptation of the Child. The threatened Child can take on and express adapted aspects of the Predator in the same way that a repressed Child expresses the critical Parent. The adapted Predator is the part of us that tears legs off spiders, clear-cuts forests, wages war, kicks dogs, or squashes bugs when there is another option. When given an outlet through hunting animals, whether for sport or food, the natural Predator (the hunter) is given release. When the Predator is afraid it lashes out in an attempt to protect itself. The adapted Predator is learned in childhood when our parents yell at us or strike us. The watching part of us wakes up and begins to look out for danger in the world around us. The Child learns how to attack when it's scared or mad. When our Child feels scared in intimate relationships it's apt to attack the other person in the way that it was attacked when young, which effectively shuts down intimacy. Often what happens is the Predator in the recipient attacks back. Then there is a real relationship mess.\n\nAdapted Predators have not learned how to get their needs met without resorting to games or attacking others; they believe that the world is full of dangers and that people have ill intent. There are power issues involved; adapted Predators want power and control over others and over their environment, and they'll use manipulation to get it. Those who prey on the elderly or the poor for financial gain are expressing the adapted Predator. They'll engage in insurance scams and Ponzi schemes in a misguided effort to come out on top. This type of Predator has no sense of empathy, no connection between themselves and their prey. They see people only as an external means to an end. The adapted Predator is one who has not learned how to get its needs met without causing damage and feels there is no other recourse available when the Parent and Adult are not supplying critical information or tending to fears and terrors about survival.\n\nIntimacy with self requires that we be willing to see each part of ourselves for what it is. Not just the bright, shiny, generous, loving, funny parts, but also the parts that have been hidden, denied, imagined not to exist, overshadowed, and, yes, especially, the parts of you that you have hated and come to resent and be fearful of. Being willing to see the goodness in me was almost as difficult as it was to see that I am certainly capable of being an asshole. We all have a part of us that has lashed out at a loved one or a store clerk or has had temper tantrums. When the pressure of stress builds to the boiling point, it's often the Predator that takes over. There are things that feed the Predator, that arm it with a sword or loaded gun or the verbal equivalent of each. Resentment, jealousy, unresolved anger, deep grief, and shame are some of them. Sometimes, it takes the form of the sexual predator. It prowls nightclubs and dance halls, hunting its prey. Sex isn't always its goal; the game of seduction is. Sex as the goal depends on how serious the player is.\n\nWe're all capable of it. It took some time, but I was finally willing to truly see that part in myself. Then to own it, make a relationship with it, and set limits on its behavior. I have been unkind, and I have hurt people I love and care about. A teacher told me once that there are two kinds of people: assholes and those who know they are assholes. I believed that was true. I also believed that I was a third kind\u2014I wasn't an asshole. Eventually, it came time to own up.\n\nI liked to think I was unkind unintentionally, but the truth is when the Predator is kept in the bag of shadow, it intends to cause hurt. Until I could come to terms with that truth, with the Predator in me, with this capacity to be unkind, I couldn't trust myself. Until I owned this part of me that was unclear, terrified, and strictly interested in survival, I couldn't stop watching, knowing that part could be let loose in the world.\n\nWe've all done something that inflicted some hurt on someone. My sister has a part that, when she was young, used to bite kittens' ears. Thankfully she stopped biting the ears of kittens, but whether she has acknowledged and integrated the Predator part of her is unknown to me. When my brother's girlfriend came to visit for the first time when they were sixteen (they've been married ever since), I, who was seven years younger, watched in horror as she delightedly pulled the legs off a daddy longlegs. This wasn't simply benign curiosity, I could tell by her skill and focus that she had done it before.\n\nWhen my granddaughters at ages nine and six came to visit me one time, it took no time at all to discover their new catch-all phrase: \"It was an accident.\" The first and second time they used it, I let it ride. When I noticed it was becoming a habit and realized they were using the phrase to conveniently excuse some behavior, I had to talk about it. Going unconscious is not an acceptable behavior for me. Nor, dear one, is hitting your little sister an accident. Words don't fall out of our mouths, nor do our hands of their own volition strike at someone accidentally or without intention. The words come out, the hand moves through the air because we've gone unconscious and some part of us that intended to say those words noticed that we went unconscious and took advantage of the moment.\n\nRudeness has become an acceptable violent behavior. We in the Western world seem to be much too willing to shrug off a comment, a behavior, ours or others, as accidents or \"I didn't mean to\" or \"It wasn't my intention.\" This is a significant part of the problem in our world, in our relationships to others and to ourselves. Discounting a person's dignity and lying have become acceptable codes of conduct. Violence begins in our thoughts and is then is expressed in our speech. It escalates to behaviors. If you choose to take on this work and let it change you, you will be in a minority.\n\n _[W]hat is imposed on you from outside is of no value whatever. It doesn't count._\n\nBERTRAND RUSSELL\n\nJust because there is an invitation to incite a riot is no reason to incite a riot. Unkindness does not need to be met with unkindness. What happens \"out there\" is none of our business. What we do with external events that impact us on the inside is of consequence. How we respond, what choices we make, what behaviors we choose are, most certainly, our business. In spite of the circumstances we find ourselves in, we are free to act decisively one way or another. Ultimate freedom is reserved to each human being. How we respond to unchanging conditions and environments is the test of human character. When our soul is fragmented, when we have not found the meaning of our lives, we are powerless and unable to access what Viktor Frankl called \"the defiant power of the human spirit.\"\n\nGandhi had the Predator in him; the difference is he was aware of its existence and worked to keep it from acting out in a way that caused damage. He used its energy to effect change. He may have even given it a new job description, one that would support his work of _ahimsa_. Reading his autobiography, you hear the rage against injustices that he transmuted into a voice of authority to be reckoned with. And he was quite clear that nonviolence is not synonymous with passivity. To be nonviolent requires thought, decisions, and action. Just as there is a Hitler inside us, there is also a Gandhi; we have the capacity for great compassion, forgiveness, truth, and jusice.\n\n _Personal change and the ability to bring about social change are linked. There is no use striving to implement principles such as nonviolence or justice in public affairs as long as one neglects them in one's personal life._\n\nGANDHI,  \n _G ANDHI: AN AUTOBIOGRAPHY_\n\nACKNOWLEDGING ALL OF YOU\n\nHaving a Predator inside does not make you a bad person. It's as much a part of your nature as the Child, the Parent, and the Adult. The degree to which you are unwilling to become conscious of the Predator and give it a new job description will determine how frequently and to what degree you let the Predator control your behavior. You cannot alter its behavior if you are unaware of its existence. It's really a matter of self-awareness and setting limits on acting-out behaviors. The Predator is activated when the grown-up or Adult part of you is not paying attention in situations that threaten the Predator's survival. The Predator is interested only in surviving.\n\nThese are not roles we play but ecological and psychological realities. The voices we hear in our head are actual people. It is in your best interest to cultivate relationships with these parts of you; acknowledge them or not, integrate them or not, they are with you for the rest of your life. Ignoring these parts or pretending they don't exist will cause you added grief, and the tremendous sources of energy, creativity, wise council, and personal power these states provide will be unavailable to you.\n\nEveryone has a sense of these many states, but rarely do we take it much further and really look at what it means. We have all heard ourselves or a friend say: \"One part of me wants to do this and another part wants to do this other thing.\" The meditations and exercises in this book will help you get to know who those parts are, and then how to work with competing wants and needs.\n\nThe crucial things in interior work are:\n\n  1. Identify the ego states. Who is talking, poking at me inside, trying to get my attention?\n  2. Free any that are repressed or locked away. We live in a culture that does not support being a fully functioning, self-possessed, confident human being. We live in a Prozac nation, a culture that bombards us from infancy with messages that tell us feeling is bad and talking about feeling is worse.\n  3. Understand their function. Each ego state may or may not know what its function is. Or its function may need to be redirected to another area, given a new job, so it has a place to put its energy into what supports all of you.\n  4. Make friends with them. This takes time and a commitment on your part to work with them daily. There will be parts of yourself that you do not want to see, do not want to even acknowledge exist, much less make friends with. All the more important to accept and befriend them.\n\n _[T]he child is the forerunner of humanity\u2014forerunner in the sense that the child is the possessor of all those traits which, when healthily developed, lead to a healthy and fulfilled human being and thus to a healthy and fulfilled humanity._\n\nASHLEY MONTAGU,  \n _G ROWING YOUNG_\n\n  **MEDITATION**    \n**I've Been Waiting for You**\n\nSet aside about thirty minutes to do this meditation. Find a quiet place where you will not be disturbed; ask family members not to interrupt you during this time. Turn off your cell phone. Eliminate as many distractions as you can. Create a place with soft lighting and a comfortable place to sit: it's best if you're not lying down. Settle into an overstuffed chair or make a nest of pillows and blankets on the floor to support you so that you feel held and nurtured. Have a journal and pen close by, but not on your lap. (Note: You may want to first record the following instructions, allowing pauses between questions, and listen to them the first time or two you do this exercise.)\n\nNow, close your eyes, letting them rest into the back of your head. Take some deep breaths: inhaling deeply, filling your lungs, your diaphragm, and hold it, hold it, hold it. Exhale completely. Again. And one more time.\n\nSee standing in front of you the Child that you once were. Just see her there. Notice everything about her. What is she wearing? What's her posture? Is she looking at you? Does she seem happy? Is she sad?\n\nHow do you feel seeing her? Will she make eye contact with you? Ask her if she has anything she wants to say to you. Is there anything you want to say to her? Ask her if you can give her a hug. If yes, then for real, reach out your arms and hold her to you. Bring her so close to you that your arms are wrapped completely around your shoulders. Hold your child and yourself close for a few moments. Notice how you feel.\n\nWhen you are ready, thank your Child for coming to be with you. Bring yourself back to where you are sitting, pick up your journal, and write down everything you saw and felt and what she said. Do you like her? Are you comfortable with her? Uncomfortable? There is much information embedded in your response to seeing her. It will give you insights into how and why you have felt the way you have about yourself for many years.\n\nBEGINNINGS ARE SUCH DELICATE TIMES\n\nAnything can happen in this first meeting. You may see your Child clearly or just have a sense of her. For some, the first meeting will be easy. There may be a sense of homecoming, spontaneous joy, and love when you first meet. For those whose denied Child has been put away so deeply and for so long, the work will be difficult. She may be upset that it's been so long since you've spent time with her. She may refuse to look at you. She may have pain to share with you and stories to tell you, happy ones and sad ones. There may be a lot of energy in what she has to say as she's been waiting a long time for you to ask to see her. As well, the reunion may be genuinely joyful.\n\nThere is the story of Carl Jung as he was going through his middle years. He was sitting on the shores of Lake Zurich building sand castles and playing with toy figures, shaping stones, giving neglected regions of his psyche room to be. He knew that \"when we are stuck we are saved by what is within.\"\n\nBeginnings are delicate times. Your Child will be watching to see if you will show up again and want to spend time with her. She may not trust grown-ups, and you are a grown-up. It's important to develop the relationship at her pace, and I suggest you do this work with her daily so trust can begin to be fostered. If she asks you to do something, take it seriously; do not make agreements you are unwilling or unable to keep. That will do more damage than saying, \"I'm sorry, I'm unable to do that. Is there something else that would work for you?\" Offer an alternative that will tend to her need or want. Learning the art of negotiation will be very helpful, for children will ask for things that in the moment you may not be able to deliver. Negotiate time frames or treats. If you ask her to do something she doesn't want to do, take into account that she may want a reward afterward. Ask her what she needs in return. Spending time with your Child and letting her have her wants and needs will bring you more joy and confidence than anything else you could do.\n\nIntimacy depends on freeing up the Child, letting her have her voice and giving her a place to be alive and out loud in the world. To have true and deep intimacy with another human being, you must have true and deep intimacy with yourself. Get to know yourself, reclaim parts of you that have been shut away, repressed, alienated, and marginalized. This is the beginning of the journey to wholeness, of being an integrated, whole human being. This is more important than I can say. It's awkward, and in the beginning, it may seem silly or stupid, even. This work has the power to change your life, to heal wounds, to empower you to speak truth to power. This work\u2014and it is hard work that takes a lot of practice\u2014gets to the core of you, the truth of you, so you can live a life filled with meaning.\n\n _I wanted to change the world. But I have found that the only thing one can be sure of changing is oneself._\n\nALDOUS HUXLEY\n\nFrom this position of empowerment and strength, no one can tell you what your truth is, or tell you to do or be anything but what you know is in you to do or be.\n\nThe child is the major source of energy. I have clients do this work and have seen physiological changes occur before my eyes. It's not unusual to feel different, feel happier, have more hope, and feel less alone even after the first meeting with your Child.\n\nThis relationship is real. The Child of you, the little girl of you is real, and she has been waiting for you to show up for her. It's important to talk over your goals and options with your Child. Include her in the process. If you make a relationship with your Child and then leave her out of your planning, she can and will likely make your life difficult and disruptive. If your Child is not aligned with your goals, you are going to have trouble achieving them. Children are clever at getting you to take notice. If the Child is not worked with consciously, she will break through; once she has tasted freedom, her only goal, if she is ignored, is to get out. She has no investment then in keeping your life orderly if you betray her trust.\n\n _If you bring forth what is within you, what you bring forth will save you. If you do not bring forth what is within you, what you do not bring forth will destroy you._\n\nTHE GOSPEL ACCORDING TO THOMAS\n\n ** MEDITATION   \nStarting at the Beginning**\n\nLet's take this deeper still. Return to your sitting place as before and begin again with deep breathing and relaxation. This time, see lying in front of you on the floor, the Infant of you. How does she look? Is she breathing? What color is her skin? Is she moving? How do you feel seeing your Infant? Pick up your Infant, swaddle her in your arms at your breast, and let her nurse. Notice whether you held her in your left or right arm. Allow yourself to experience how the two of you feel being together. Really see her. How does it feel to feed the Infant of you? Let anything you feel rise to the surface. Wrap your love and caring around her as you nurse. Gaze into her eyes and notice the mothering instinct awaken inside you. Notice the degree of hunger she has. Tell her you are glad that she was born.\n\nWhen you are ready, put her down and pick up your journal. Write down everything that happened and everything you felt. Many people feel a sensation in their breasts while nursing their Infant. It's extraordinary, really.\n\nBoth men and women are able to do this meditation. It is as important for men as well to nurse their Infant. Men have mammary glands just the same as women, and there are several instances, historical and recent, where men have either nursed their own children or become wet nurses for surrogate or adopted children.*3 The greatest, most immediate benefit to nursing, for both men and women, is that it begins the emotional bonding of nurturing Parent to Infant, of you to yourself. Nursing your Infant is awkward at first, but it is imperative to bonding with yourself. Occasionally an Infant will look withdrawn, pale, even lifeless or blue. It can be alarming. Trust that you can bring more life to her. Breathe the life and love gently into her. Hold her near you as you fall asleep and again when you wake up in the morning.*4\n\n **Who Makes These Changes?**\n\n _Who makes these changes?_\n\n _I shoot an arrow right._\n\n _It lands left._\n\n _I ride after a deer and find myself_\n\n _Chased by a hog._\n\n _I plot to get what I want_\n\n _And end up in prison._\n\n _I dig pits to trap others_\n\n _And fall in._\n\n _I should be suspicious_\n\n _Of what I want._\n\nRUMI,  \nTRANSLATED BY COLEMAN BARKS\n\nTHE COUNCIL WITHIN\n\n _Ninety-nine percent of who you are is invisible and untouchable._\n\nBUCKMINSTER FULLER\n\nWorking with our Child and Infant is the beginning of knowing the multifaceted personality that we are. There are many ego states and personalities inside of us. As human beings, we have been given an elaborate and fascinating inner guidance system.\n\n ** MEDITATION   \nMeeting the Council Members**\n\nBegin as you did with the first meditation, getting comfortable in a quiet place with your journal and pen nearby. Go through a few cycles of deep breathing and relaxation exercises.\n\nSee yourself standing in a forest glade. It's a warm summer day, and there is a slight breeze rustling the leaves. Sunlight is filtering through the canopy, and you watch the dance of light and shadow on the forest floor. You notice a path beneath your feet, and you begin to walk along it. Eventually, the path leads out of the forest into a small meadow of wildflowers and grasses. Following the path down a meadow hill, you come to a stream. There are a few stones placed just so, and you are able to easily step across the stream to the other bank. You walk a few paces on the path along the stream and see just up ahead a one-room structure. Walk to the front door and step inside. Notice everything about this room. Take note of the round table in the middle of the room. How does the room feel? Is there a source of light? Where is it coming from? Take your time noticing; no need to hurry. It's taken awhile for you to arrive at this place. There are chairs, some around the table, some along the wall. Beings are there in that room. Some of them are standing; some are sitting in chairs.\n\nLet your gaze travel the room noticing all who are there. How do you feel seeing them? Are they looking at you? How many of them are there? Do you recognize any of them? Ask them if they have anything they want to say to you. Pay attention to what they are saying to you and how it feels to hear their words. Is there anything you want to say to them, to the members of your inner council, in response?\n\nIt's nearly time to leave. Thank them for being here and talking with you. Do not make any promises or agreements unless you are certain you will keep them. They take these things seriously. Breaking agreements with any members of your inner council exacts a cost.\n\nLeave through the door and step on the trail upstream to the crossing. Pick up the trail and head up the hill through the meadow and back into the forest to where you began. When you are ready, bring yourself back to the place you are sitting. Pick up your journal and write down everything that happened and all that was said.\n\nEvery age you have been and will be is inside you\u2014the child ego state, the teenager, the young adult, the middle-aged adult. You also have inside you your own internal mother and father, your self-nurturing parents. The elder you will become is there too, as well as your future self. You don't have to wait to be that age to give them an active part and voice in your life. There are many times you may need to call on an elder from your inner council for the kind of grounded, mature advice only an elder can offer. We have nonhuman allies too. Some of us may have plant and animal allies. Anyone and anything can show up in our inner council. There are no rules here. Work with any of them individually as you need to.\n\nSome people I've worked with had characters from _The Wonderful Wizard of Oz_ and another from _Star Trek_. Some members are lifetime guides, and others may show up for a short time to help us through some growth period. I recently had the main character from an Elizabeth Moon novel spontaneously show up while I was in council. She's the captain of a space vessel. \"Kylara Vatta, what are you doing here?\" I asked, in surprise. \"I'm your new captain,\" she announced, smiling. Indeed, I have called on her several times when in need of her particular strengths of quick analysis and decisiveness\u2014strengths that only a captain possesses. If you need someone or a specific teacher to be on your inner council, ask that being from within your council if he or she would be willing to be a member, or call them in. They are similar in function to a board of directors in a corporate body whose mission ideally is the integrity and wholeness of the corporation. I've also enlisted single aspects, certain characteristics that I needed or wanted to have incorporated into my personality. I wanted Spenser from the Robert Parker novels to be on my inner council and Jay from Michelle West's _Hunter's Death_ and _The Hidden City._ I have called on them at times when I felt their particular virtues were needed.\n\nFINDING THE NURTURING PARENT\n\nIf you don't have a good model for a nurturing Parent, find someone you consider fills that role. We all need good models, mentoring in how to be a human being. As examples, Andy Griffith from the forty-year-old television series, the _Andy Griffith Show,_ Jean-Luc Picard from _Star Trek_ 's _Next Generation,_ and the most famous nurturing, wise, compassionate father of all (in my humble opinion), Harper Lee's Atticus Finch. He's strong, yet tender, and willing to stand up for what he believes. Richard Farnsworth in _Anne of Green Gables_ stole my heart with the warm knowing twinkle in his eyes. My favorite has been Fred MacMurray who played Steve Douglas, the father in _My Three Sons._ I couldn't wait to watch that television show when I was growing up. For thirty minutes, several days a week, I would completely lose myself in that world. Mr. Douglas was my father\u2014or the father I wanted. He was kind, funny, compassionate, strong in his values and beliefs, tender, and nurturing. Uncle Charley was curmudgeonly and scared me a little. He masked his nurturing side under the facade of tough love. Tough love is not nurturing; it's tough love.\n\nAs for female role models, it's been a challenge to find comparable traits in female characters, to find actresses who exemplify the combination of compassion, nurturing, and strength. I am tempted to call on Harper Lee herself and Ethel Barrymore. Aunt Bee in the _Andy Griffith Show,_ was matronly, loving, and doting, though I prefer a little more softness and less \"parent.\" And there was some attractive nurturing energy about Gloria Stuart when she played the elderly Rose in _Titanic_ that made me want to be in her presence.\n\nYou can get a sense of what a nurturing Parent looks and feels like. Many of us grow up lacking a consistent nurturing parent. Most of the parents I know are critical; they mask their strength and compassion by being critical and stern. Nurturing Parent role models and the ones you create or call on inside yourself possess genuine caring. You must know yourself well enough so that anything hidden or repressed is cleared up, healed, and not left to contaminate the nurturing part of you.\n\nRECOGNIZING AND INTEGRATING ALL THE PARTS OF YOU\n\nRigorous self-examination describes the process of interior work. You must be able to notice, account for, and be accountable to each part of you that lives within. The more you work with these parts, hear their voices, and understand each one's gifts and virtues and what each can bring to your life, the more whole and integrated you will become, the more alive you will feel, and the more mastery you will have over your life. When all of your inner council can make a decision together, in agreement with one another, the more ease and less drama you will have in your life.\n\nYour heart must be in this\u2014a deep desire and dedication to knowing yourself, to awakening your latent powers of intuition, motive force and creativity. You have a right to do this work, to be whole and happy in yourself and your relationships. And if there is to be any salvation for the human species, each of us must do this work. Now, without delay. Gaining facility with interior ego states takes time, devotion, and practice. But it won't cost you hundreds of dollars and time in therapy sessions, and the effects are immediate, the rewards innumerable.\n\n _All that we have ever been and all that we ever need to be is known in the eternal place inside ourselves where all is quiet._\n\nRAM DASS\n\n _From this moment I am prepared to control whatever personality awakes in me each day. . . . Today I control my destiny . . . I will become master of myself. I will become great._\n\nOG MANDINO SCROLL VI\n\nThe goal of this work is to exist as a coherent, integrated whole. Understand that any ego state can emerge at any time it is needed. In the interior world, all these ego states are interacting whether you are aware of them or not. Sometimes they interact well; often they are in conflict. In the exterior human world, people's many ego states are interacting with each other, sometimes well, often not so well. When they interact well, your life in those moments flows smoothly.\n\nCulturally, pressure exists to locate consciousness in the brain and the adult ego state. In actuality, consciousness is limited to no particular part of the body and no particular ego state. Consciousness can become habituated to body location or ego state by restricting it to particular locations like the adult ego state in the brain. The most fundamental ego state is the Child.\n\n _Being a human being\u2014in the sense of being born to the human species\u2014must be defined also in terms of becoming a human being . . . a baby is only potentially a human being, and must grow into humanness in the society and the culture, the family._\n\nABRAHAM MASLOW, _M OTIVATION AND PERSONALITY_\n\n _What is important to me is not the truth outside myself, but the truth within me._\n\nKONSTANTIN STANISLAVSKY\n\nDEVELOPING CHILD-TO-CHILD INTIMACY\n\nAll of this work with interior ego states is important in the context of sacred sex because of the nature of relationship and intimacy. The point of this work is to get to a place of actively noticing your interior world and to then be able to notice your partner's. The more elegant you become with your own interior world, the more intuitive you will become and the more present you can be with your lover. The Child in you understands the nature of sex; she has been pleasuring herself since she was developed enough in the womb to move her hands to her genitals.\n\nBerne describes intimacy as \"a candid Child to Child relationship with no games and no mutual exploitation.\" To get to the place of Child-to-Child relating, a sequence of things happens inside each participant. The Adult assesses the situation, reads the contracts, and commitments to each, and once this is all understood by the Child, the Adult retreats to the background yet continues to monitor the agreements and is in charge of keeping the Parent from interfering and spoiling the fun. Once the Parent is out of the way, the Child becomes freer to engage in spontaneous and fun-filled intimacy. Ideally, the Parent gives its blessing and encouragement, freeing the child of any fear of intimacy and reducing any possible feelings of guilt.\n\nYou can check this out yourself when you embark on an intimate relationship. Listen carefully to the voices inside you, and you will hear the Child exclaiming excitement at the possibility and the Adult reading over the commitments and any reservations or cautions the Parent may be inserting into the conversation. It can happen fairly quickly, so be attentive, but even after the moment has passed you can replay it and slow it down to see what took place.\n\nAs you become sensitized to subtle shifts in your interior world, you will become more attuned to shifts in your partner. You'll recognize the Child in him and be able to communicate directly with that Child. Just as you did when you began to work with the Child in you, there will be a time of trust building and finding your way together. It's quite natural for the Child in him to see the Child in you being happy and present in the relationship, even if on a conscious level he's unaware of it happening. You'll see the changes he's going through, the ego states he moves through, so you can match yours with his.\n\nRemember when you were fifteen years old and you met a boy. The attraction, the awkwardness, the spontaneity, the giggles, the explorations, and the energy of that fifteen-year-old is still there and can be part of your current relationship. Use that energy to bring freshness to the relationship as you become good friends, exploring the world together with wonder and curiosity. Sixteen-year-old ego states are much more daring and exploratory with sex and each other's bodies. It's exciting and clumsy and potent.\n\n _It is vital to see that everything is attended to, lest something goes unattended._\n\nThe difference between sex and lovemaking is that in the latter, we tend to be the most unshielded, the most vulnerable and childlike. It is the Child who knows no distance, no space between desire and action. Sacred sex can be the catalyst for healing the wounds of our birth and our family of origin. It can heal sexual trauma, shame, and disgust. Because sacred sex happens in the arms of love, compassion, and intimacy, it becomes a portal, a threshold between what was and what can be. They don't have to be mutually exclusive. Sometimes I'll ask my lover to fuck me and make love to me at the same time.\n\n _Stepping through to the other side can be frightening._\n\nIt is the Child part of us who decides whether or not someone can be trusted. Respect comes from the Adult, after the Child gives her permission. The Child is always monitoring relationships and situations. She is the one who decides whether or not she likes someone, likes their smell and the sound of their voice. She is highly sensitive to the slightest perturbations, affronts, discounts, and all manner of meanings coming to her from the world; other people and relationships are part of the world.\n\nWhen there is a strong foundation of trust between you and your Child and your Child and your partner, very young ego states may take the opportunity to emerge at will. For the health of that ego state, for healing to happen, it's critical to notice in yourself and in your lover when this is happening. Memories, pictures, visions, smells, and words\u2014communications in various forms\u2014will arise spontaneously to the conscious mind. Follow whatever is happening without censoring. Let them speak, cry, wail, laugh, ask questions. Inside yourself, have the older, nurturing part of you witness and be present with her without interfering. That little one needs to know you are tending to her, companioning her, holding her, hearing her. She needs to know that the grown-up of you is with her and she is not alone inside.\n\nAs wounds begin to open up and the child shows herself\u2014her fears, what she didn't understand long ago (\"Why are grown-ups so mean?\")\u2014you do well to have answers for her. It's all right to not know all the answers; it's not all right to lie about them. Examples of responses are: \"I don't know why grown-ups are mean. Sometimes people get scared, and they don't know what else to do and being mean makes them feel safe.\" \"You know sweetie, your father was not a good person. And I'm sorry he was unkind, but he's not here now, and he can't hurt you anymore.\" \"I wish I could shield you from all the ugliness of the world, but that can never be.\" If you know the truth, say the truth. Primarily, the Child needs to know that you are hearing her and that she can ask for and say what she needs to. Children are children, inside you or born from your loins. Answer their questions without evasion. Children can spot evasion faster than adults so don't confuse them.\n\nAs the Child emerges, there will often be subtle changes in voice, eye movement, word choices, or cadence of speech. It's not unusual for a two-year-old ego state, for example, to emerge. Remember having a conversation with a two-year-old you've known. They don't use big words. Often speech is rapid if there is excitement, slower if there is fear or shyness, and the voice may be nearly a whisper. The more you work with each one, the more you'll come to recognize who is presenting herself. The deeper you are willing to go, the more focused you can be in this work, the more you will come to know yourself, understand yourself, be intimate with yourself, and, thus, be intimate with anyone or anything you wish.\n\nBefore I was deep into this work on myself, I cherished early mornings together with a former lover. It was in those quiet, soft moments, as the sweetness of sleep is drifting away, before full alertness takes hold, when he wasn't fully awake and hadn't yet had time to put up defenses, that he had that little boy, sleepy look on his face. It was then that he was his most vulnerable, sweet, available, spontaneous, and natural childlike self. And so was I. The beauty of this work is that it allows you to enter those states at will at any time and in any place. One day, you'll be moving about your business and suddenly notice that it is the Child of you who is out in the world as the primary ego state.\n\n _And, ah, what joyful ease overcomes you._\n\nYou don't have to be a psychotherapist or a trained mental health professional to do this work. All that is required is that you _do_ the work. We all walk around with unexamined beliefs, prejudices, and assumptions that impact our ability to be intimate, to be real, to be compassionate, to be free. Those unexamined psychological structures contaminate our reading of the map and ultimately the territory of the sacred.\n\nIt helps to read and have resources available. Children (I'm referring to the ones inside you in this instance) love having information; they are hungry for it. They need it to navigate the world, to help make sense of things. If I read a book that is helpful, I go immediately to the bibliography and order anything that catches my eye. I love books; the Child in me loves books. They are friends and teachers, bedtime companions, and trips to other galaxies, other worlds, and the world of fantasy.\n\nI want to clarify a fine point in all this, and that is, if you were to walk around saying, \"my child needs this, my child wants that,\" you may not get the outcome you're expecting. That's what dissociation is. The distinction is _you_ are the one who is responsible for hearing what your Child needs and wants. It's up to you, to use your power to get needs met, to speak on her behalf; you do the work of integration. To use the above phrases as a habit diminishes your power, puts responsibility where it doesn't belong, and will likely irritate the people around you. It's fine to say \"What my Child needs is . . .\" in intimate relationships, especially if there has been a betrayal of some sort.\n\nCROSS TRANSACTIONS\n\nTo understand how ego states interact requires an understanding of transactions. Spending a little time on this will help you understand how communications can go sour and how to avoid it in the future. Eric Berne describes this process in detail in _Games People Play_ and _Sex in Human Loving._ They are worth looking at, if for no other reason than the clarity of the diagrams. As stated earlier, there are three main ego states in each human being: the Child, the Adult, and the Parent. Healthy or straight-across transactions can be described as Child to Child, Adult to Adult, and Parent to Parent. Cross transactions occur, for example, when the Child in one person initiates a conversation or movement and the other person responds from the Adult or Parent ego state.\n\nInitiator (Child): \"I want to play the box drum. It looks like so much fun.\"\n\nRespondent (Parent): \"I'd be glad to buy you one if you agree to play an hour a day with me.\"\n\nChild: \"Um, OK. I can do that, that would be fun.\"\n\nParent: \"I'm serious. I don't want it to sit in your closet like your guitar.\"\n\nUh-huh. Feel that? The Child feels chastised, shamed, and heartbroken. At once, all the fun, excitement, and anticipation of having a drum and playing music with her friend are dashed. She no longer has any interest in playing music with him in any way, shape, or form.\n\nThe Child's initial response of \"Um, OK\" indicated a thoughtful processing of the condition that was put on the gift of the drum: \"If you'll play an hour a day with me.\" She decided that the one hour a day was agreeable and within her parameters of fun. Then came the critical Parent's voice out of her friend's mouth, sucking all the energy out of the room. The Child felt hurt, ashamed, and deeply disappointed: Her friend was mad at her for not playing the guitar and told her it was unacceptable behavior to let the guitar sit in the closet. But it wasn't her friend's voice she heard, it was her mother's voice, or it could be her father's voice\u2014the voice of the Parent\u2014telling her that her behavior wasn't acceptable and to be acceptable you have to do it this other way. To prevent an old pattern of responding with \"Fuck you,\" or holding onto feeling ashamed and being mad at a betrayal of friendship, she went to her friend and still from the Child ego state said, \"Are you mad at me for not playing the guitar regularly?\" The friend is taken aback as he realizes what he had said and responds by saying, \"No, sweetie, no, I'm not mad at you. I'm so sorry I sounded like I was criticizing you, and I understand why you'd feel that way, but no, I'm not mad at you. I'm so sorry, I apologize. It's your guitar; you can play it or not. I really want to play music with you, and I know how busy you are, and I would be heartbroken if we couldn't.\" He went on: \"Are we OK? Do you need anything from me?\"\n\nThe apology was genuine; the Child heard it and felt its sincerity. Moreover, the Child felt heard and cared for. Because the Child trusted her friend and wanted to feel close again, she went to him undefended. Disaster was averted, shame disrupted, and trust and intimacy restored.\n\nWhen we are whole in ourselves, undefended, and willing to maintain trust and intimacy, then we are able to make decisions from a place of wholeness.\n**4**\n\n _ **The Numinous**_\n\n_Numinous: of, pertaining to, or of the nature of a numen. Evoking awe or reverence, as the presence of something holy or divine. Irrational; mysterious, inscrutable. The numinous is the part of spiritual and sacred experience that is characterized by feelings of fascination and awe._\n\n _Sacred: set apart or dedicated to religious use; hallowed; pertaining or related to deity, religion, or hallowed places or things. Consecrated or dedicated to a person or purpose; entitled to reverence or respect; not to be profaned; inviolable._\n\n _T HE READER'S DIGEST  \nGREAT ENCYCLOPEDIC DICTIONARY_\n\nIn all Neolithic, Paleolithic, primitive, and indigenous cultures around the world, life was infused with the sacred, the numinous, a sense of other. Reality for those people, and for many modern people today, was a blending of the holy in all they did. What Western culture considers profane, daily acts\u2014such as eating, having sex, preparing food, and building houses\u2014were, for ancient, indigenous, and primitive peoples, enacted with a sense of the pervasive invisible world around them. To them, the invisible world was sacred; spirit was revered, accounted for, and brought into daily acts.\n\nThe sacred in daily life was not a project, not a means to an end, but a pathway given freely from divine grace and a pathway, as well, leading to divine grace. It took into conscious consideration the intricate weaving together of human and nonhuman, physical and nonphysical, seen and unseen, soul and spirit. The life of the senses breathed the sacred into mundane acts, birthing a life inseparable from anything inside or outside us. The concept of outside was inconceivable as the sacred wove the physical and nonphysical worlds together into a cohesive whole.\n\nSHAKEN BY THE NUMINOUS\n\nIn 1957 Mircea Eliade wrote _The Sacred and the Profane,_ in which he described the world we now live in as having two separate realities. Religious people believe in, have experiences with, and orient their lives with a sense of the _hierophanies,_ that is, the physical manifestation of the holy. \"But for the primitive, such an act is never simply physiological; it is, or can become a sacrament, that is, a communion with the sacred.\" Nonreligious people reduce daily acts as mundane and simply physiological. And the cosmos becomes a place to conquer, reducing life-forms to mathematical and scientific studies as well as a place for dissecting the parts in order to grasp the reality of the whole. But is it possible to live a life that is wholly and completely desacralized?\n\nA lot has changed and transpired since Eliade wrote in 1957, and it would appear that, in fact, it is possible to live without a relationship to the sacred or a sense of anything manifesting holy virtues. Our world order, governments, and secular and religious institutions are seeped in deceit, secrecy, and abuse of power. But these things are nothing new; they have been happening since man created the state as an organizational structure. Eliade gives examples of \"crypto-religious\" behavior on the part of the nonreligious man.\n\nThere are for example, privileged places, qualitatively different from all others\u2014a man's birthplace, or the scenes of his first love, or certain places in the first foreign city he visited in youth. Even for the most frankly nonreligious man, all these places still retain an exceptional, a unique quality; they are the \"holy places\" of his private universe, as if it were in such spots that he had received the revelation of a reality _other_ than that in which he participates through ordinary daily life.\n\nI suspect, also, there are moments of profound, extraordinary sexual experiences that touch and move nonreligious people. The sacred and numinous know no boundaries conjured up by humanity's belief or nonbelief in their existence. The Earth quakes in laughter at our absurd religious doctrines and our tendency to draw physical and invisible lines. Lines that cut us off from sources of life-affirming mystery, from each other, from our birth-given, soul-charged experiences of intimacy. We humans have become our own sacred clowns, but we refuse to see our own joke. Listen closely in the still of the night; you'll hear coyote laughing.\n\nOur world, the world of man and woman, of Homo sapiens, is a dendritic spiraling pattern weaving us in, around, and through the (mostly) invisible matrix that holds our very existence tenuously intact. It is arrogant to believe we can render the strands of connection nonexistent. It is possible for some to deny and be unaware of their presence. It is possible to live with no acknowledgement or place for them in one's life.\n\nEven for the nonreligious man, an awakening, an epiphany occurs in desperate situations or during a crisis, altering the sense of reality at once. Times of feeling utterly helpless when our children are seriously ill, a parent is dying, or a life-threatening disease or accident befalls us. It is in these times that a door inside, a portal to the Great Mystery, is revealed, and we, by chance of circumstance, see the way through, see a new chance, a glimmer of hope, as we feel the influence of some invisible force at work in our crisis. A desperate clinging to life, ours or another's, aborts previously held beliefs, and we grasp for the nearest raft that may prolong life.\n\nAs if waking from a deep sleep or comatose state, we see reality with new eyes. What was previously hidden from view by our own making is present and palpable. At least for a while, if not permanently, the person we were has morphed into someone who has become aware of being woven into mystery. Whatever the reason, our prayers were heard, the medicine worked, the rescue workers arrived at the eleventh hour, and something has given us a chance for more time with life. The impact we feel of a life extended, of death averted, remains imprinted on our psyche. Even if, after the crisis has passed, we choose to return to an ordinary and mundane life, are tempted to discount intervention by the numinous, and with each passing day travel farther from the mystery, there has been inserted into the psyche the unexplainable, the nonrational experience that shook us aware for a few, brief moments. Where there was once irrefutable belief in a nonspiritual, nonsacred world, there is now a sliver of doubt; a questioning of attitudes and ideas has entered the mind and soul of the nonreligious. What remains is the impression of a map, delineating a new territory where the sacred has become part of our interior landscape. Whether we choose to pick up the map and rechart our life course or put the whole mysterious experience in a box and slide it under the bed in our interior house is our decision alone.\n\nIN THE CHURCH OF EARTH\n\nThe moments of Spirit intervention are often too tritely called a religious experience, suggesting a philosophical conversion. The words _religious experience_ are too simplistic and overused to describe what is often indescribable. It puts the experience into a category where it doesn't really belong.\n\nPart of this is my own prejudice. The word _religion_ leaves a bad taste in my mouth, left over from my experience with religions, particularly Christianity, and their public aspect of doctrines, hierarchies, and rules of conduct. Religion comes from the Latin _religare,_ meaning \"to bind or restrain,\" as in to be obligated. Religion is man-made, built on promoting the idea that God is \"out there,\" outside us; furthermore, he makes an appearance in specific buildings at prescribed times and days, as set forth in the precepts of the religion's manifesto. The followers of the religion need a middleman to intercede on their behalf, to dictate how they should behave, to absolve them when they wander from the prescribed behavior, to bless, to give permission for marriage, to acknowledge life and death, and to be self-appointed determiners of life and when it begins.\n\nI prefer _spiritual,_ which is a private set of beliefs coming out of personal experiences with the sacred, with Nature, with the numinous. Those times of Spirit intervention, whether during a crisis or while walking through a forest, are spiritual experiences. My use of the word _religion_ is personal and in the context of Cicero's etymology of the word, who traced it to _relegere,_ which means \"to read over again\" or a \"linking back.\" My religion is a spirituality that is by definition a relationship with the powers of the Universe, with incorporeal, immaterial, supernatural essences. When I work with Spirit, I am working with the vital powers and subtle energies that animate the material world.\n\nI'm not religious in the sense of dogma or doing things according to an external authority. I link back and read over and over again the sacred texts or the original word inscribed on the land and in trees, in tracks left by animals passing, sleeping, and eating. The hierographology of pictographs, standing stones, cairns, pot shards or the way a sweat lodge fire is tended\u2014all these speak of ancient relationships to the sacred, to the land, to Nature. I ascribe to no hierarchy in my personal cosmology; there is no overarching sacred leader. I am on equal footing with all that inhabit this earthly life; we each just have different job descriptions. That said, I am in service to something outside myself, greater than myself, and that is to be in service to the _anima mundi,_ the soul of the world, to the Creator and Gaia. To do that, I must simultaneously remake myself into a whole human being.\n\nMy spirituality is not confined to a particular building, creed, or obligations to an organization. My church is the wilderness of southwest New Mexico, the Mississippi River, the forests, mountains, wild water\u2014wherever I find myself. In this, I am of the Dionysian school of the madwomen, the maenads, who had no temples. Edith Hamilton writes about the Maenads and their worship of Dionysus in the wild outdoors.\n\nThey went to the wilderness to worship, to the wildest mountains, the deepest forests, as if they kept to the customs of an ancient time before men had thought of building houses for their gods. They went out of the dusty, crowded city, back to the clean purity of the untrodden hills and woodlands. There Dionysus gave them food and drink: herbs and berries and the milk of the wild goat. Their beds were on the soft meadow grass; under the thick-leaved trees; where the pine needles fall year after year. They woke to a sense of peace and heavenly freshness; they bathed in a clear brook. There was much that was lovely, good, and freeing in this worship under the open sky and the ecstasy of joy it brought in the wild beauty of the world.\n\nI take my communion with the plants, the green nations, the holy waters of hot springs, the sweat lodge. I engage in a dialogue with the spirit keepers of this land, or whatever land I sit on in a sacred manner to share the prayers and smoke of the sacred pipe. Sitting with my pipe, feeling the presence of the spirit keepers of the four winds, the ancestors and the old ones of this land, and my inner council that lives inside me, I am reminded of the mystery of all things. The Great Mystery, the numinous, the incomprehensible largeness of this immense, expansive, extraordinary, beautiful Universe. I am humbled at once by my smallness and my significance at the same time. Feeling them all come in to sit in a circle with me\u2014invisible, palpable to my senses\u2014to hear my prayers, to \"break bread\" together; this is my sacrament.\n\nI am an ordained minister of the Church of Gaia, a nonprofit organization exploring and participating with the nonlinear intelligence of Nature. In the opinion of the organization's founders, \"survivability of the human species depends on human beings once more reconnecting to the Earth. Without this rekindling of our ability to care for the nonhuman world from which we emerged, our behavior will continue to be careless of Earth and its preservation.\" This means that each one of us must learn to \"reinhabit our interbeing with the world,\" as the rain-forest activist John Seed puts it. \"This reinhabiting is essential, not an academic or rhetorical pursuit, and always a personal one.\"\n\nAs a practitioner of Earth-centered spirituality, I continually expose and immerse myself in the numinous territory of sacred lovemaking and with the wild land of southwest New Mexico. Inexhaustibly, I look for novel and deeper ways to participate with life, to understand my place and my responsibilities, and to somehow speak on behalf of those who have no human voice. But to speak on others' behalf, I must first hear what they are saying not only with my ears but with my whole body\u2014my skin, my hair, my feet, my heart.\n\nThey speak in vibrations, images, and feelings. Those forms of communication and expression are the many winds that breathe over the land: tree-bending gale-force winds; sand-blasting dust-storm winds; suck-the-moisture-from-your-skin winds; dry, hot, southwest June winds that take the water out of your lungs. The winds that come in July bring moisture and foretell of the monsoons near arrival, winds that carry a fecund, erotic smell of rain in the forest and release essential oils from the trees. The smell of pi\u00f1on in the early morning air tells of humidity on the rise. The many songs and howls of coyotes tell stories of their lovemaking, their kill, their grieving, and their celebrations. Javelina tracks in red dirt and bites taken out of prickly pear cactus suggest migration patterns and movements. The voice is not solitary and cannot speak without some other: the trees speak through the wind; the wind speaks through the grasses; a coyote's voice speaks through a primordial echo in our soul; fire speaks through wood and air; rivers and streams speak through stones as water journeys off mountains and through valleys. They all speak directly to our hearts and imaginations through the invisible sea of energy we all share and swim in together. To think we can create and sustain a dichotomy, extract ourselves from this sea, creates a kind of insanity, a craziness, in our psyche.\n\nOur own voices speak through vibration, air, breath, earth of our tongues, teeth, vocal chords, mouth, and fire of passion. I must listen attentively, objectively, as if for the first time. I must listen with my whole body, not a part of me, but the whole of me, hearing, feeling, smelling, tasting, touching, seeing. Only then can I begin to understand the meanings in the communications and how I can be in service to all those voices. Only then can I know if there is anything to translate or if it's time to simply listen, sit on Earth in a sacred manner, and take communion with them.\n\n _Unless we see or hear phenomena or things_\n\n _From within the things themselves,_\n\n _we shall never succeed in recording them in our hearts._\n\nBASHO\n\nWITNESSED BY SPIRIT\n\nWhen I was searching for a spiritual path that felt right to me, I tried on the traditions that were available where I lived. For two years, I was a card-carrying, dues-paying, active member of a Wiccan church in the upper Midwest. After some truly mind-blowing psychic experiences with those women and discovering latent gifts of my own, I came to the conclusion that, that particular feminist path was not for me. I didn't like male bashing or even leaving out the male aspect of the Universe. I said thanks but no thanks, dropped out, quit paying my dues, and tore up my card.\n\nAfter that, I talked to a friend of mine about his Buddhist practices. He invited me to join him at a Tuesday evening _sesshin_ or _zazen_ sitting meditation. We drove along the Mississippi River near New Albin, Iowa. As we drove the long, winding, and rutted gravel road through the sandstone bluffs and deciduous forests to the _zendo,_ I felt a sense of something bigger than and outside of me getting involved in my life.\n\nThe first night of sitting was uncomfortable. I was fidgety, nervous, uncertain, and afraid of doing something wrong. The Buddhist monk gave a brief instruction on how and why to sit on the cushion so that our knees were lower than our hips, taking the stress off our lower backs and straightening the spine. He described how to hold our hands: left hand resting in the right palm with thumb tips lightly touching. He talked about watching our breath and letting thoughts that appear rise and float off like clouds dispersing. It was a very long forty minutes. I was shocked to notice just how much chatter was going on in my mind at any given moment.\n\nZazen agreed with me, and I loved it in return. I enjoyed the experience of sitting weekly with a small group of people, and I added it as a daily practice between the weekly sesshins. During a New Year's Eve sesshin, my experience radically changed. We sat in the usual manner for the prescribed forty minutes, after which there was a ten-minute _kinhin_ or silent walking meditation followed by another forty minutes of sitting. This went on for three hours.\n\nSometime into the second hour, I found myself completely relaxing into the experience. All tension in my body had dissolved. I sat effortlessly as some invisible support held me up. I felt good in myself, almost happy, and quite calm. My eyes were softly focused on a point on the wall in front of me. It wasn't work this time to find the still, silence inside. Each breath took me deeper into being relaxed until I noticed something internally begin to shift. My senses became acutely fine tuned as my peripheral vision expanded. I noticed my skin begin to tingle. The point on the wall and the wall itself began to merge and expand; boundaries blurred. I noticed some new thing was happening in the area of my heart, and I felt something move out from and encircle me. I was acutely aware of being larger than myself. And, I was not myself. I sat still, breathing, soft focused, feeling, watching. The experience grew in strength. A lightness of heart and spirit overtook my senses, and I realized I felt happy, joyful. The physical elements of the room fell away, and I found myself in a strange and wonderful landscape. My body no longer had boundaries; no beginning, no end, nothing between me and the world I had just entered. Or had it entered me? The landscape was barren of any physical markers though there was a sourceless light that filled all space. I experienced a pervading sense of peace, calm, serenity, and an unexplainable, overriding sense of growing joy. Pure, radiant, blissful joy. And love. I was the only human being, the only being in that place, and I was overcome with tremendous, unconditional love. I felt loved. And I felt that I deserved this love. Nowhere inside me was there hesitation, doubts, or feelings of unworthiness.\n\nIt was my first experience of being seen, being witnessed by Spirit. And I was being loved. I held that ecstatic, mystical state, unwavering, for the remainder of the forty minutes. I didn't want to stop, not ever. My visible, physical world was being nourished by the invisible. From that moment forward, I tried to re-create that feeling in my life, and I found I could while I was alone in the woods or sitting along the river. Someplace inside me I knew I could have, was supposed to have, this feeling, this experience with another human being. Simultaneously, I knew this experience was available when connecting with Earth as much as it was here, in the sacred space of meditation, that this was a dynamic experience and as such, it was transferable to other places.\n\nOne afternoon, a few hours before I would be sleeping with my lover, I asked the ancestors if they would show me how to make love like a prayer. What happened was astonishing. As I lay down with him, I reached out my awareness to feel them with me. There was an invisible, palpable presence in the room, between us, with us. The room took on a soft, luminescence. It felt as if they became part of my body, they were so close. I was fully present, watching, participating, and aware of these magnificent beings in the room with us, wrapped around us, between us almost. I gave myself over to this ecstatic experience. It felt as though we were water, fluidly unencumbered by boundaries of any sort. As our two bodies were enraptured, our souls were ravishing each other, spiraling in and out, twining around, sharing fluids, memories, and DNA. All the while, being held in a numinous orb nearly floating off the bed.\n\nRudolf Otto in writing _Das Heilige_ (The Sacred) in 1917 set afire schools of religious and philosophical thought. He set forth profound and, at the time, befuddling concepts of awe-inspiring mystery and human fascination with the mysterious. Otto frequently used the expression _the numinous feeling_. Though the phrase implies the subjective feeling, Otto was quite clear that numinous feelings were both subjective and objective realities.\n\nThe word \"numinous\" has been widely received as a happy contribution to the theological vocabulary, as standing for that aspect of deity which transcends or eludes comprehension in rational or ethical terms. But it is Otto's purpose to emphasize that this is an objective reality, not merely a subjective feeling in the mind; and he uses the word feeling in this connexion not as equivalent to emotion but as a form of awareness that is neither that of ordinary perceiving nor of ordinary conceiving . . . The ambiguity attaching both to the English feeling and the German Gefuhl should not therefore mislead us. We do after all speak of feeling the beauty of a landscape or feeling the presence of a friend, and our \"feeling\" in these cases is not merely an emotion engendered or stimulated in the mind but also a recognition of something in the objective situation awaiting discovery and acknowledgement. It is analogously to such uses that Otto speaks of the \"feeling of the numinous\" or (less aptly) the \"numinous feeling.\"\n>  \n> \n> **PART 2**\n> \n>  **Earthly Sexual Body**\n**5**\n\n _ **Human Beings**_\n\n **The Ground Where the Gods Reside**\n\n_There are some who can live without wild things, and some who cannot._\n\nALDO LEOPOLD,  \n _A S AND COUNTY ALMANAC_\n\n _The universe is dead for us, and how is it to come alive again? \"Knowledge\" has killed the sun, making it a ball of gas, with spots; \" knowledge\" has killed the moon, it is a dead little earth fretted with extinct craters as with smallpox; the machine has killed the earth for us, making it a surface, more or less bumpy, that you travel over. How, out of all this, are we to get back the grand orbs of the soul's heavens, that fill us with unspeakable joy? How are we to get back Apollo, and Attis, Demeter, Persephone, and the halls of Dis? How even see the star Hesperus, or Betelgeus? We've got to get them back, for they are the world our soul, our greater consciousness, lives in. The world of reason and science, the moon, a dead lump of earth, the sun, so much gas with spots; this the dry and sterile little worldthe abstracted mind inhabits. The world of our little consciousness, which we know in our pettifoggin apartness. This is how we know the world when we know it apart from ourselves, in the mean separateness of everything._\n\nD. H. LAWRENCE, \"A PROPOS OF _L ADY CHATTERLEY'S LOVER_\"\n\n _We are so little at peace with ourselves and our neighbors because we are so little at peace with our place in the world, our land. American history has been to a considerable extent the history of our warfare against the natural life of the continent. Until we end our violence against the earth . . . how can we hope to end our violence against each other? The earth, which we all have in common, is our deepest bond, and our behavior toward it cannot help but be an earnest of our consideration for each other and for our descendants.  \nAs long as man relates only to other men, he can be a specialist with impunity; . . . Once he is joined to the earth with permanence of expectation and interest, his concerns ramify in proportion to his understanding of his dependence on the earth and his consequent responsibility toward it. He realizes, because the demands of his place make it specific and inescapable, that his responsibility is not merely that of an underling, a worker at his job, but also moral, historical, political, aesthetic, ecological, domestic, educational, and so on._\n\nWENDELL BERRY,  \n _T HE LONG-LEGGED HOUSE_\n\n _Until I can know what other men know when they say, \"this is where I live,\" I will know nothing of worth\u2014andwhen I can say that & feel it deeply, I'll know most of what I'll ever be able to know. \"This is where I live.\" What a thing to know!_\n\nLEW WELCH,  \nQUOTED BY DOLORES LACHAPELLE  \nIN _S ACRED LAND, SACRED SEX_\n\n _If there can be such a thing as instinctual memory, the consciousness of land and water must lie deeper in the core of us than any knowledge of our fellow beings. We were bred of the earth before we were born of our mothers. Once born, we can live without our mothers or our fathers or any other kin or friend, or even human love. We cannot live without the earth or apart from it, and something is shriveled in man's heart when he turns away from it and concerns himself only with the affairs of men._\n\nMARJORIE KINNAN RAWLINGS,  \nQUOTED BY DOLORES LACHAPELLE  \nIN _S ACRED LAND, SACRED SEX_  \nAND BY GABRIEL MILLER  \nIN _T HE FILMS OF MARTIN RITT_\n\nIndigenous and ancient cultures on Earth had, and still have, elaborate systems of ceremonies and rituals that revered the natural world as sacred. The realm of spirit, the invisibles, and the gods came to play and lived among them. These cultures engaged themselves in the changing seasons, made medicine with the phases of the moon, and created rituals to celebrate and mark transitions from one stage of life to another. During ceremonies of harvest and abundance, fermented plant brews flowed freely while the autumn air filled with songs and people danced. They made prayers and offerings to the spirit that moves through all things and to the souls of plants and animals for food, meat, and clothing so these things would continue to be abundant. They took care to maintain balance and harmony in all they did.\n\nTime was set aside for thanking the gods for the growing season, for the stores of food for the winter. And it was a time of thanking Earth and plants and the spirits of growing. Prayers of beseechment for next year's planting season as well as help getting them through the winter months wafted up on spirals of smoke. A keen understanding and awareness of all the beings that were helpful and partly responsible for the tribe's wellbeing and failure, abundance and loss, health and illness were acknowledged and fed to keep them happy and nearby. It was understood that nothing could be done well without assistance and blessings from the unseen world of spirits and ancestors.\n\nCentral to indigenous cosmology is, naturally, the desire to maintain balance in all things to ensure a sustainable future for the generations to come. Ralph Metzner in _Green Psychology,_ says, \"Once we recognize that the human exploitation and destruction of the biosphere is related to a dissociative split within human consciousness between the spiritual and the natural, then the question becomes\u2014how did the separation come about?\"\n\nCOUNSEL WITH THE STAR PEOPLE\n\nThe Inuit live on the treeless, windswept tundra of northern Canada, where the night sky is illuminated only by the Milky Way, far-off galaxies, and star formations whose ancient mythologies are expressed in their names. The Inuit believe in animism. All things, living and nonliving, have a spirit: people, animals, plants, material objects, forces of Nature, and the elements\u2014fire, wind, earth, and air. As David Suzuki and David Knudtson write, \"The heavens high above, they say, are the sacred abode of a mighty spirit. The _anatkut_ (wise ones) say that it is a woman. To this place in the skies and to the potent feminine spirit, the souls of all who die are conveyed.\"\n\n _To these Arctic peoples, the soul embodies the very essence of each form of life. They envision the soul as a tiny being, a minute version of the creature that it animates and transforms. Appropriately, they believe that the soul islocated in a bubble of air in the groin, the same general anatomical area to which the modern biologist assigns the gene-laden germ cells, egg and sperm, bristling with DNA-encoded instruction for assembling a new life. Thus, the soul of a human being is a tiny human being, the soul of caribou a tiny caribou, and that of seal is a tiny seal._\n\nDAVID SUZUKI AND PETER KNUDTSON,  \n _W ISDOM OF THE ELDERS_\n\nThe sky ceiling in that part of the world is unimpeded by artificial light. There are places in the Gila Wilderness, where I live in southwestern New Mexico, that are completely uncontaminated by artificial light. Nights when the moon is dark, the sky people can be seen in their full glory. To plant feet on the ground, head tilted back, and gaze up into a night sky dazzling with trillions of stars and planets is to be humbled. It becomes glaringly apparent why ancient people called them Star People and their ancestors. Staring up at them, there is a sense of their sentience and a keen awareness of being watched and listened to. There is a wisdom there, an aliveness palpable to the senses, heart, and mind, and suddenly, feelings of being alone in the world dissipate into the evening air.\n\nHuman beings have a long history of living in harmony and intimately in relationship with the wild world. Francis Weller notes that \"[i]t is the part of our psychic life that we hold in communion with the life that moves around us.\" When we are in the wild, some primordial, elemental part of us awakens to this truth. Holding that truth, we are able to move from disingenuousness to authentic awareness of the life teeming around us. We become awakened to the innate need in us to recover our lost connections to wild things and places.\n\n _For I was born a thousand years ago, born in the culture of bows and arrows . . . born in an age when people loved the things of nature and spoke to it as though it has a soul._\n\nCHIEF DAN GEORGE,  \nOPEN LETTER, 1975\n\nWe are biologically wired to communicate with, be in relationship with, and live harmoniously as equal members of Earth's species. Edward Goldsmith in _The Way_ states that \"it is part of that intuitive heritage that enables man to be cognitively adjusted to the world in which he lives. However, with the development of the world view of modernism, and in particular of the paradigm of science, the world became 'disenchanted,' secularized and mechanomorphized.\"\n\nFor a short time after my son and I moved to Vernon County in southwest Wisconsin so that he could attend a Waldorf high school, we lived in a one-hundred-year-old brownstone house. We heated the house and water for cooking and bathing with wood. An artesian spring flowed year-round through a pipe under its own underground pressure about ten paces from our front door. The nearest town of two hundred people was six miles from our house. Neighbors were scattered over hills and behind small sections of old tree stands and forest remnants. I had been living in a college town for fifteen years before, and the move to the brownstone in Wisconsin, reawakened something ancient and primordial. Each night, I would unceremoniously bundle myself up in sweaters, gloves, and hat and go stand under the sky. Each night, I would gaze up at those Star People as if I had never seen such a sight before that night while simultaneously feeling an ancient kinship with them. I was awestruck, humility struck, and I prayed and prayed to those ancestors, my ancestors, the ancient ones who live there. Every night for the three months that I lived there, I would go out and talk with them, admire them, pay homage to them.\n\nOne night in the dream time, after weeks of talking with them, I was abruptly transported through a white cloud tube to the place where the Star People live. They were in council. They wanted to meet \"in person\" this being who was talking to them each night, and they wanted me to see them straight on. We held each other in high regard as I stood among them, motionless in the presence of such wise, magnificent, and luminous beings. I wasn't there long before I was ushered, or more like dropped, down a spiraling cloud tube again, and the moment I slammed back into my body, I was met with a raging, spontaneous, full-body orgasm jolting me awake. Spirituality and sex are intimately related.\n\nThe ancestors are real. And they respond.\n\nORIGINS OF THE SEPARATION\n\n _The Christian religion lost, in Protestantism finally, the togetherness with the universe, the togetherness of the body, the sex, the emotions, the passions with the earth and sun and stars._\n\nD. H. LAWRENCE,  \n\"A PROPOS OF _L ADY CHATTERLEY'S LOVER_\"\n\nSexuality is the sensation of Nature in one's own organism. In primitive and indigenous religions, religion and sexuality were one. When natural sexual expressions were repressed in the human animal during the development of religion and agriculture, this produced an unbridgeable contradiction between sexuality as a sin and religion as a liberation from sin.\n\nHow did we get so far from our relationship with and dependence on the natural world? David Suzuki writes about the origins of the mind-body divide.\n\nThe movement away from the natural world was made possible by a quite remarkable train of thought\u2014ideas that shaped our civilization. Today we take those ideas so much for granted that we see them not as ideas (which can be rethought, revisited, discarded) but as reality. Many thinkers trace the origins of our particular and violent fall from grace, our exile from the garden, back to Plato and Aristotle, who began a powerful process of separating the world-as-abstract-principle from the world-as-experience\u2014dividing mind, that is, from body, and human beings from the world they inhabit. In the process they laid the groundwork for experimental science.\n\nThe move from hunting and gathering to agriculture happened around 11,000 BCE, long before Plato and Aristotle. It began the long repression of sexuality in general and in women's sexuality specifically, which continues to this day, and directly parallels 10,000 years of abusive and oppressive environmental policies toward Earth, toward the environment, and toward wildness. The shift from hunter-gatherer to farmer\u2014from wild mobility to agrarian domestication\u2014changed the interior emotional and psychic landscapes as well as the physical landscape.\n\nOnce the reality of owning land was inculcated in the minds of overlords, boundaries were set up and lines were drawn, and the struggle to accumulate property and material wealth was on. It didn't stop with controlling and domesticating the land; it seeped into the consciousness of our forefathers and spilled over to owning women and slaves.\n\nWith the advent of farming, women were quickly domesticated along with farm animals. Archaeologist Timothy Taylor argues that \"[a] major event in the development of sexual inequality occurred when farming was invented, a system by which people could produce food when they wanted it rather than relying, like every other species, on natural availability.\" Although women were central in the early development of farming, it quickly led to their oppression. With the increased availability of animal milk, along with breast milk, children were birthed more frequently. As sources of milk became more available, the ties that bound women to hearth and home also increased. And the economic value of women decreased as they were taken out of the labor force, unable to generate an income or sense of independence of their own.\n\nMy mother was a farm wife who much preferred sitting on \"her\" tractor and bailer or hayrake and feeding livestock to cooking, cleaning, and changing diapers. My younger sister (born eleven months after me) and I were primarily raised by our two sisters, eight and ten years older. By the time my mother was severely ill with emphysema, she was unable to get disability payments since my father never paid her as a wage earner, and therefore no social security payments were made on her behalf. It was a source of endless arguments and resentment between them.\n\n _How womankind, who are confined to the house still more than men, stand it, I do not know; but I have ground to suspect that most of them do not stand it at all._\n\nHENRY DAVID THOREAU,  \n\"WALKING\"\n\nTaylor, in _The Prehistory of Sex,_ continues, \"While hunter-gatherer sex had been modeled on the idea of sharing and complementarity, early agriculturalist sex was voyeuristic, repressive, homophobic, and focused on reproduction. Afraid of the wild, farmers set out to destroy it.\" Paternity certainty has no value or importance in preagricultural cultures where concern over sexual fidelity was also relatively unimportant.\n\nTaylor argues, \"How a society treats the natural environment and how it views food are both closely connected to its attitudes toward sex and to the particular quality of the relations between the sexes. The rapidly expanding agricultural populations of Neolithic Europe swamped the hunting and foraging peoples who had lived there before by sheer force of numbers. Farming set in motion a cycle of ecological devastation\u2014immediately connected with human sexual and reproductive aims\u2014that seems set to continue until the world's last surviving forests vanish under the plow.\"\n\nAn interesting concept worth pondering is that _fork,_ an Indo-European word, was originally at one with _fuck._ Often the phrase \"spreading my seed\" is used to describe impregnating a woman with the male seed, as in planting seeds in farm furrows. Taylor notes that _\"_ [t]he idea of the female sex as a field into which grain is sown is common among farming cultures and can be found in Talmudic, Egyptian, and Vedic writings. The idea of the female earth mound being entered by the male force is startlingly embodied at Newgrange.\"\n\n _Semen is Latin_\n\n _For a dormant, fertilized,_\n\n _Plant ovum\u2014_\n\n _A seed._\n\n _Men's ejaculate_\n\n _Is chemically more akin_\n\n _To plant pollen._\n\n _See,_\n\n _It is really_\n\n _More accurate_\n\n _To call it_\n\n _Mammal pollen._\n\n _To call it_\n\n _Semen_\n\n _Is to thrust_\n\n _An insanity_\n\n _Deep inside our culture:_\n\n _That men plow women_\n\n _And plant their seed_\n\n _When, in fact,_\n\n _What they are doing_\n\n _Is pollinating_\n\n _Flowers._\n\n _Now_\n\n _Doesn't that change everything between us?_\n\nSTEPHEN HARROD BUHNER,\n\n _T HE TASTE OF WILD WATER_\n\nNewgrange at County Meath, Ireland, is a megalithic tomb built around 3200 BCE, about three hundred years before the pyramids. Construction of the Passage Tomb is estimated to have taken three hundred laborers at least twenty years to construct. The mound covers an acre of land. The inner passage is 19 meters long leading to an inner cruciform chamber. The mound is oriented such that at the winter solstice the sun penetrates the passageway illuminating the inner chamber. The sun was considered to have male penetrating and fertilizing properties.\n\nIn 1890, the massacre at Wounded Knee stripped native peoples of their myths and stories, turning them into property and their land into real estate to be parceled up. The lands conquered by the U.S. government were divided, bought, and sold as private property. Detached from the spirit of the land, new rectangular boundary lines were drawn with no regard for natural landmarks, the sacred or holy. What was once commonly shared wildlands were plowed under, deforested, and desecrated. It happened in other places that European settlers came in contact with: Canada, Australia, Hawaii. In modern times, rain forests around the globe are being stripped of their power. The spiritual order, the ceremonies, the invisibles, whole native cosmologies have been burned, bulldozed, and paved over. We've cut down native trees only to name streets after them in some hollow attempt to immortalize what has been lost. It is a feeble attempt to assuage guilt and loss of something greater than the sum of its parts, as if a street sign serves as a monument to what has been lost. Taking it further, the roots of future generations of trees are sealed off in concrete graves.\n\nRupert Sheldrake writes, \"The scientific and technological conquest of nature expresses a mentality of dominion that had been widespread in the ancient world but was vastly increased in power by technology and amplified by the mechanistic theory of nature has taken the place of Christian missionaries in justifying the dispossession of native peoples and the disregard of their sacred places. Since nature is inanimate, their animistic relationship to the living world around them must be superstitious, their attitudes backwards. They cannot be allowed to stand in the way of progress. And now, like the Buffalo hunters, we can hardly believe what we have done.\"\n\nThere is widespread alienation from ourselves, each other, Earth, and the Spirit that breathes through and animates all things. We have severed the spirit from place, soul from body, sex from sexuality, and truth from power. In places where the wild is cut, mowed, and paved over, the soul of that place goes underground, waiting, while those left behind deal with the aftermath, inhabiting a soul-less landscape and suffering from diseases that arise from living on land that has no soul, that is empty of vitality and life-force energy, that is spiritually dead, where the livingness of the land is unavailable to infuse human souls and food that is grown there is devoid of the sustenance that can only come from ensouled landscapes. And underground is where the soul of place remains until the day when its rightful place is returned to it, until the day some person or peoples are willing to breathe and dance the soul to life once again, not unlike the Native American Ghost Dance of 1889.\n\nWhen the soul of a culture is usurped, stolen, and marginalized, its soul undergoes severe distortions, showing up as diminished vitality, poverty, homelessness, mental disorientation, and dispiritedness. Loss of soul manifests as diseases of the spirit: depression, suicide, overeating, and addictions. In indigenous cultures, shamans, healers, and medicine people would be called in to restore the broken connections. The soul of a culture goes underground until the day of restoration and breathing it into life comes about, if it comes at all.\n\nAfter hominids shifted from hunting and gathering to agriculture, from nomadic to settled, the wildness of life began to vanish. Christopher Ryan and Cacilda Jeth\u00e1 write about the effect settled life had on concepts of property and paternity.\n\nOnce people were farming the same land season after season, private property quickly replaced communal ownership as the modus operandi in most societies. For nomadic foragers, personal property\u2014anything needing to be carried\u2014is kept to a minimum, for obvious reasons. There is little thought given to who owns the land, or the fish in the river, or the clouds in the sky. Men (and often, women) confront danger together. An individual male's _parental investment,_ in other words\u2014the core element of the standard narrative\u2014tends to be diffuse in societies like those in which we evolved, not directed toward one particular woman and her children, as the conventional model insists.\n\nNot only did Earth need domestication in their view, but also women, wild and earthy, needed limitations set on their provocative sexual power, their sexual nature.\n\nAt the height of the witch inquisitions between 1550 and 1650, a paradigm shift took place. The move from an intuitive, magical, mystical, and visionary worldview to an objective, mechanistic, medical, and science-based view formed the basis for the hysteria and fear of Nature that followed. The shift coincided with the worst outbreak of syphilis in European history.\n\nThe epidemic was shocking. Scholars began to perceive Nature as threatening, no longer seeing it as the divine feminine soul of all creation, as Isis, Artemis, or Sophia. Nature was now seen as deceitful and dangerous. Sexual repression became epidemic as the new disease poisoned the innate trust between men and women.\n\n _When once the woman has tempted us, and we have tasted the forbidden fruit, there is no such thing as checking our appetites, whatever the consequences may be._\n\nGEORGE WASHINGTON,  \nLETTER TO MRS. RICHARD STOCKTON, 1783\n\nIT'S ALL FOR THE CHILDREN\n\nWilhelm Reich attained his medical degree in Germany and studied with Freud, becoming one of Freud's favorite students until Reich expanded on Freud's theories. His research on sexual repression was substantial and he wrote _The Sexual Revolution._ Reich was a pioneer in body psychotherapy, founded somatic psychology, and influenced, among others, Fritz Perls's Gestalt therapy. Reich traced civilization's suppression of biological (sexual) functioning and saw how it became perverted into social institutions: war, torture, racial hatred, and slavery. He believed that if \"people were using their morality to repress their sexuality, but sexuality made life worth living\u2014was in fact, life itself\u2014then they needed to change their morality and have more satisfying sex; and psychoanalysis had to use whatever means were necessary to get the patient not merely to see this, but to live it.\" Reich was a proponent of contraceptives, divorce, abortion, and the importance of economic independence for women. He believed that the split between mind and body causes us to destroy each other and the planet and allows us to go to war. He believed that we protect ourselves by \"armoring\" from expressing things society says we must not express. He went on to say that if men and women are unable to have orgasms it would lead to neurosis, anti-Semitism, hate, greed, racism and fascism.\n\nSexual energy, like any energy, doesn't stop; it can't be stopped. It has to come out somewhere. It's like trying to keep steam in a boiling kettle. No matter how tightly you hold the lid, as the pressure builds, roiling steam always finds a way out. In the West, suppressed sexual energy is escaping to the tune of $200 billion per year spent on pornography, prostitutes. and Direct TV pay-per-view sex films.\n\nWhen we look closely at the habit of Western culture to clear-cut old-growth forests, turn wilderness into open grazing and residential areas and deserts into gambling meccas, and build locks and damns to control rivers, you'll begin to see that it's not a far stretch at all to then institute laws and regulations that dictate and control our behavior around sexuality and the expression of it. It's not a far leap, hardly a skip, to book banning or the National Endowment of the Arts imposing a \"decency standard.\"*5 It's not far from cancelling the homecoming dance at Vermont's Montpelier High School because of concerns about dirty dancing and student drug use. In September 2010, CraigsList blocked access to the \"adult services\" section and replaced it with a black bar stamped \"censored.\" Craigslist was criticized for allegedly facilitating prostitution and \"sex trafficking\" in the United States. In the summer of 2010, the organization Pornography Harms congratulated itself for being responsible for Facebook's removal of the \"Our Porn Ourselves\" Facebook page, claiming that the material was \"inappropriate\" and too easily available to children. Who's in charge here?\n\nOf course, it's all for the children. We need to protect our children. Who and what are we protecting them from? I know the arguments: We're protecting them from pedophiles, axe murderers, mother rapers, father rapers, pornography, immorality, premarital sex, teenage pregnancy, sex trafficking, cults, drug use, and drug dealers\u2014the arguments go on ad nauseam. As Marty Klein points out numerous times in his Internet newsletter, _Sexual Intelligence,_ \"And then we have completely bogus numbers (bound to be reprinted endlessly)\u2014like the Rebecca Project's 'An estimated 100,000\u2013300,000 American children are at risk for becoming victims of commercial sexual exploitation.' 'At risk!' Not in any way harmed, just vulnerable! The technical word for this is 'nonsense.'\"\n\nTeens are being prosecuted and charged as adults for \"sexting,\" sending sexual photos of _themselves_ to their friends via cell phones. The laws designed to prevent exploitation of minors by adults are being used to destroy the lives of teenagers\u2014the ones the laws are supposed to be protecting. A Michigan man is facing twenty years in prison for stupidly redoing a video of himself singing a children's song to a group of first graders at their school, under the watchful eyes of their teacher. At his home, he took the video of the performance, spliced in shots of himself singing sexually explicit lyrics to the children. Then, he couldn't resist himself; he posted it on YouTube, disclaiming that \"no actual children have been exposed\" to the song. He's being charged with manufacturing child porn, even though he didn't actually manufacture child porn. He made it all up. No crime was actually committed (unless being tasteless and having a juvenile sense of humor is a crime), and no harm was inflicted.\n\nDavid Sobel, talking about environmental education for second and third graders, writes:\n\nThey hear the story of the murder of activist Chico Mendez and watch videos about the plight of indigenous forest people displaced by logging and exploration for oil. They learn that between the end of morning recess and the beginning of lunch, more than 10,000 acres of rainforest will be cut down, making way for fast food 'hamburgerable' cattle . . . In response to physical and sexual abuse, children learn distancing techniques, ways to cut themselves off from the pain. My fear is that our environmentally correct curriculum will end up distancing children from, rather than connecting them with, the natural world. The natural world is being abused and they just don't want to deal with it.\n\nThe problems are not as simple or as superficial as we'd like to make them out to be. We are teaching our children to be disconnected from their feelings, from the ground of their own experience, to be afraid of Nature and the wild. We are giving them the right and wrong information at the wrong time or not at all. They are losing touch with what's real inside them and with the world in which they live.\n\nThe best way to protect your children is to empower them; give them good, true information and be their friend. Be honest with them. Children love having information, and only with good information can they can begin to make good choices. Be clear about not putting your insecurities and fears on them. Make certain that you do better than your best to keep their wild, intuitive, inquisitive, imaginative, and, yes, sexual selves intact. They will be exposed plenty to institutions, religious leaders, friends, schools, and governments that believe it is their duty to break a child's spirit and strong will and good heartedness. It's your job as a parent, guardian, grandmother, aunt, or uncle to nurture and feed their whole, wild spirit from the moment they take their first breath.\n\n _Children are educated by what the grown-up is and not by his talk._\n\nCARL JUNG\n\nQUELLING THE SONG OF THE WATERS\n\nI grew up in the upper Mississippi River Valley of Northeast Iowa. My bones and blood were formed from farm soil and the waters of the Upper Iowa and Mississippi Rivers. My partner for five years before I moved to New Mexico was a raptor researcher along the Mississippi River, so over that time I logged a few hours in his boat with him as we trolled the lakes, backwaters, and islands of that part of the river in pools 9 and 10 (the river between locks are called pools).\n\nOn a quiet, early morning trip in late summer when the river was, what Mark Twain called, \"a lazy river\" and turning leaves of autumn were beginning to scatter across the surface and shoreline, I was becoming entranced by the gentle rocking of the boat against the slow, deliberate, current of the river. Just ahead in my soft vision was a lock, and in my entranced state, I could see how locks and dams act as river birth control, much like an IUD or a diaphragm. The river cannot flow freely, naturally, of her own volition. And there is a reason why so many of us have difficulty crying; our tears are dammed. Too often we feel unable to cry with abandon and without apology. Too often we become afraid of the power of our emotions to express them unrestrained. So, we dam them up; lock them away.\n\n _He gives water to the dead._\n\nFRANK HERBERT, DUNE\n\n _This song of the waters is audible to every ear, but there is other music in these hills, by no means audible to all. To hear even a few notes of it you must first live here for a long time, and you must know the speech of hills and rivers. Then on a still night, when the campfire is low and the Pleiades have climbed over rimrocks, sit quietly and listen for a wolf to howl, and think hard of everything you have seen and tried to understand. Then you may hear it\u2014a vast pulsing harmony\u2014its score inscribed on a thousand hills, its notes the lives and deaths of plants and animals, its rhythms spanning the seconds and the centuries._\n\nALDO LEOPOLD,  \nFROM \"SONG OF THE GAVILAN\"\n\nBuilt below the surface of the Mississippi River are structures called wing dams, which prevent the river from cutting a new channel. The Army Corps of Engineers controls the river\u2014the river's habitats, ecosystems, channel depth, and island formations\u2014via wing dams and locks, damming, and dredging\u2014the process of digging out sand and silt that gets disturbed as tows move barges from city to city.\n\nThe increase in barge traffic on large rivers has nearly killed the Illinois River. One hundred years ago, it was as ecologically diverse as the Mississippi but is now considered \"bleak\" by the corps. The murky brown plumes seen trailing behind a barge in aerial photographs are the result of sand and sediment churned up from the river bottom. The sand, being heavier, settles quickly, but it takes two hours for the sediment to settle, and with anywhere from eleven tows a day near La Crosse to thirty-two a day at Alton, Illinois, the water never clears. Invertebrates are buried by the debris, and fish gills cannot cope with constant high concentrations of silt.\n\nIn 1781, Thomas Jefferson wrote: \"The Mississippi will be one of the principal channels of future commerce for the country westward of the Alleghaney . . . This river yields turtle of a peculiar kind, perch, trout, gar, pike, mullets, herrings, carp, spatula fish of 50 lb. weight, cat-fish of 100 lb. weight, buffalo fish and sturgeon.\"\n\nCarp, a hardy fish introduced to the United States in the 1880s, has dominated the commercial fishery of the Mississippi since the early 1900s. All others have declined, including buffalo, catfish, freshwater drum, lake sturgeon, and bullhead. Walleye, sauger, yellow perch, and white bass have disappeared entirely from the commercial catch.\n\nSadly, the Army Corps of Engineers is under heavy pressure from corporations that use the river to transport coal and grain, making the corps blind to other solutions that would work with the river. Rather than shaping the river to suit barge traffic, imagine building barges and boats that fit the river. So much has been lost in the unwillingness to see the sacredness in the largest river on the North American continent and the third largest river in the world. As long as we continue to see Nature as something \"out there,\" we will continue looking for solutions from the outside and from the top down\u2014the reductionist's seat in the spectator's box. \"Solutions\" from this seat have no enduring qualities.\n\nJerry Mander in _The Absence of the Sacred_ addresses this when he says:\n\nThe assumptions have been gaining in strength for thousands of years, fed both by Judeo-Christian religious doctrines that have desanctified the earth and placed humans in domination over it; and by technologies that, by their apparent power, have led us to believe we are some kind of royalty over nature, exercising Divine will. We have lost the understanding that existed in all civilizations prior to ours, and that continues to exist on Earth today in societies that live side by side with our own; we have lost a sense of the sacredness of the natural world. The new technologies don't accept this notion; they live in a world that is removed from it; they themselves have lost touch with the source of that knowledge. They find it silly.\n\nThere has been continual and growing exertion from governments, institutions, religious organizations, and lobbyists to suppress and control the wild in ecosystems and in human behavior. Diminishing biological diversity is analogous to diminishing diversity in human expression as our first amendment rights are undermined with each redefinition.\n\nBECOMING WHOLE\n\n _I would like to recommend that in judging the rightness of our actions toward the natural world, we be guided by a fundamental respect for the dignity of wild Nature. Dignity is the intrinsic quality in all beings that we are morally obligated to uphold. If our behavior does not infringe on the dignity of animals, plants, rocks, rivers, and the relationships among them, our actions are proper and sustainable, both ethically and ecologically._\n\nMOLLIE MATTESON,  \n\"THE DIGNITY OF WILD THINGS,\" WILD EARTH\n\nWe all belong to Earth. We are of Earth, and Earth is inside each of us. Our existence depends on Earth and the elements. The separations we feel are man-made; we took ourselves out of the relationship. We invent labels and hierarchies, we draw lines of separation, and we even manage to separate ourselves from the ground we walk on, the trees we depend on for building our homes and cleaning the air and the plants we harvest for food and medicine. Separation is easy. But it isn't in the natural scheme of things.\n\nThe human predicament is that we can't seem to find the connection between ourselves and Earth. The predicament is that without that connection we are doomed as a species. If we were to know this connection in the deepest part of us, let awareness of that connection permeate and infuse every cell of our three bodies, we would necessarily have to make other choices; it would alter our entire orientation. The predicament continues\u2014we are so deep in the quagmire we can't seem to imagine something else.\n\nUnless you know that the ecosystem of you is intimately connected to the ecosystem of Earth, you won't be whole. Unless you are able to reconcile and integrate all the parts that make up your ecosystem, you will not be able to be integrated into the sensuous ecosystems of Earth. An ecosystem is a biological environment. It consists of all the organisms (ego states) living in a particular area (you or me in our body), as well as the physical components of the environment with which the organisms (ego states) interact, such as the air we breathe and the blood, sweat, piss, tears, light, and shadow in our bodies. _Biocoenosis_ or _biocenosis_ was coined by Karl Mobius in 1877. It refers to all the interacting organisms living together in a specific habitat. An ecosystem is not only the physical and biological components of an environment but also the mental isolates. Our histories, illnesses, and personal and family stories are also part of our inner ecosystem.\n\nIt is the collection of all the parts of us, physical, mental, emotional, and spiritual. It is the sum total of all our parts.\n\nIt doesn't matter if we are red, black, white, tall, short, Muslim, Catholic, pagan, Buddhist, communist, capitalist, or Rosicrucian, each of us belongs to Earth; we all breathe in the same air, the breath of the Great Mystery, and we _get_ to live on Earth together. These words have no meaning without the experience of living on Earth, without \"Gaiaphilia\"\u2014the feeling of love for Earth. Without the experience of belonging to the livingness of Earth, our lives become rhetorical and theoretical like environmentalists who never have dirt under their fingernails. We think that because we've said something, the problem is solved. The problem isn't solved; there are still problems, and we're still fighting, warring, and killing, mutilating each other and destroying other species. We still don't believe or act as if Earth is alive and sentient and aware. Nor do we act from the truth that we have emerged as a species from the womb\u2014the soil and microbes, the water and air\u2014of Gaia. Neither do we want to know that we are sexual, have sex; even as you read these words, someone is having sex, right now. Maybe it's your neighbor or your parents or your sister across the country. It could be your mail carrier, the clerk at the store, your children. It could be you.\n\nA WORKING DEFINITION OF WILD\n\nI've been using the words _wild, wilderness,_ and _Nature_ rather loosely up to this point. Before going further, I want to explore the meanings of each word. Our language lets us be lazy in our descriptions of things, people, and events. For example, to say \"the wind blew wild\" conjures up images in your mind. Other words that could be used and may be more accurate and for certain more descriptive might be: _tumultuous, ferocious, tempestuous, violent, furious_. _The Reader's Digest Great Encyclopedic Dictionary_ gives the following definitions of the word _wild:_\n\n1. Inhabiting the forest or open field; not domesticated or tamed; living in a state of nature. 2. Growing or produced without care or culture; not cultivated. 3. Being without civilized inhabitants or cultivation; desert; waste. 4. Living in a primitive or savage way. 5. Boisterous; unruly; unrestrained. 6. Immoral; dissolute; orgiastic.\n\nIt goes on to describe _wild_ as, \"originating violent disturbances; stormy; turbulent; rashly imprudent; showing reckless want of judgment.\"\n\nLooking closer at the origin of the word _wild_ in the Oxford English Dictionary, we find _wild_ with intriguing, old English spellings that conjure up images and feelings about the word's origins.\n\n _The Reader's Digest Great Encyclopedic Dictionary_ defines _wilderness_ as follows:\n\nWilderness: 1. an uncultivated, uninhabited, or barren region. 2. A waste, as of an ocean. 3. A multitudinous and confusing collection; a wilderness of curiosities. 4. Wildness. Nature: 1. the fundamental qualities or characteristics that together define the identity of something; essential character. 3. The entire material universe and its phenomena.\n\nWildness is an idea that has moved immensely through time, notes author Robert MacFarlane. \"And in that time, two great and conflicting stories have been told about it. According to the first of these, wildness is a quality to be vanquished, according to the second, it is a quality to be cherished.\"\n\nThe etymology of the word _wild_ is vexed and subtle, but the most persuasive past proposed for it involves the Old High German _wildi,_ and the Old Norse _vilr,_ as well as the pre-Teutonic _ghweltijos_. All three of these terms carry implications of disorder and irregularity, and as Roderick Nash has written, they bequeathed to the English root word _will_ \"a descriptive meaning of . . . willful, or uncontrollable. Wildness, according to this etymology, is an expression of independence from human direction, and wild land can be said to be self-willed land, land that proceeds according to its own laws and principles, land whose habits, the growth of its trees, the movements of its creatures, the free descent of its streams through its rocks are of its own devising and own execution. Land that, as the contemporary definition of wild continues, 'acts or moves freely without restraint; is unconfined, unrestricted.'\"\n\nThe basic definition of wildness has remained constant since those first appearances, but the values ascribed to this quality have diverged dramatically.\n\nOn the one hand, wildness has been perceived as a dangerous force that confounds the order-bringing pursuits of human culture and agriculture. Wildness, according to this story, is cognate with wastefulness. Wild places resist conversion to human use, and they must therefore be destroyed or overcome.\n\nThe soul has a need for the wild. David Abrams in _Becoming Animal_ says it this way:\n\nA calloused coldness, or meanness, results when our animal senses are cut off for too long from the animate earth, when our ears\u2014inundated by the whooping blare of car alarms and the muted thunder of subways\u2014no longer encounter the resonant silence, as our eyes forget the irregular wildness of things green and growing behind the rectilinear daze.\n\nThe definition of that need is personal, but it is there, inside us. Jay Griffiths notes that \"[t]he human spirit has a primal allegiance to wildness, to really live, to snatch the fruit and suck it, to spill the juice. We may think we are domesticated, but we are not. Feral in pheromone and intuition, feral in our sweat and fear, feral in tongue and language, feral in cunt and cock. This is the first command: \"to live fealty to the feral angel.\"\n\nWe get our need for the wild met however we can. Some of us go camping, hike hundreds of miles, go fishing, watch porn, or go to strip clubs. Eros is intimately involved in the ways we fill the need for the wild for Eros is the connector. We don't seek fulfillment from something unless there is a measure of attraction, need, or affection for the thing, whether it's camping, hiking, or sex.\n\nRupert Sheldrake explores the meanings of nature, from inborn characteristics to the wider natural world.\n\nOne of the primary meanings of nature is an inborn character or disposition, as in the phrase _human nature._ This in turn is linked to the idea of nature as an innate impulse or power. On a wider scale, nature is the creative and regulative power operating in the physical world, the immediate cause of all its phenomena. And hence nature comes to mean the natural or physical world as a whole. When nature in this sense is personified, she is Mother Nature, an aspect of the Great Mother, the source and sustainer of all life, and the womb to which all life returns.\n\nOur concept of Nature is entwined with our concepts of and relationships between men and women, nature and humanity, humanity and animals. When we reject the idea of Nature as organic and motherlike and perceive it as cold and inanimate, our relationship to Nature becomes lifeless. But can we really claim Nature is inanimate and lifeless given all the research and writings on the teeming lives of bacteria and fungi, the studies on plant communication, the reproductive forces happening around us, and the mystical experiences most of us have had?\n\nEvery ancient culture on Earth had relationships with the animating forces of the Universe. Pythagoreans took into account the five great elements from which they believed all things were fashioned. The Chinese I Ching and feng shui, the system of ordering homes, gardens, buildings, are set on the foundations of the five elements. Nothing is built or undertaken without consideration of the elements and how they influence wealth, health, happiness, prosperity. Altars are erected to honor the ancestors and deities and to prevent misfortune from coming about as a result of dishonoring the dead, the spirits, and the elements. In Japan, mountains themselves are considered shrines connecting Heaven and Earth.\n\nDEATH AND DISEASE THROUGH SEPARATION\n\nApproximately one hundred years (depending on how you calculate it) of environmentalism and conservation have gone by, and things are getting worse. One hundred percent of the air in the Lower Forty-eight is now contaminated with eight cancer-causing industrial chemicals at levels that exceed safety standards. We are living with an epidemic of cancer, antibiotic-resistant bacteria, Lyme disease, and other debilitating diseases Rachel Carson warned of in her groundbreaking work _Silent Spring._ What does it mean when Monsanto sends thiram-treated tomato seeds (the EPA ruled thiram too toxic for home garden use and application requires the use of gloves) and Maxim XL\u2013treated corn and vegetable seeds to earthquake-devastated Haiti? This \"gift\" of 60,000 hybrid seeds was distributed by a $127 million project funded by the U.S. Agency for International Development (USAID). The program, called Winner, was designed to promote \"agricultural intensification.\" Monsanto, the same neighborhood corporation that brought us Agent Orange, recorded more than $11.7 billion in sales in 2009. They hold over 650 biotechnology patents, most of them for corn, cotton, and soy. In 2004, in Brazil, Monsanto sold a farm to a U.S Senator for one-third its price in exchange for his work to legalize glyphosate, the world's most widely used herbicide.\n\nExporting hybrid seeds that devastate and toxify soil, water, and the bodies (human and nonhuman) that ingest genetically modified seeds creates a culture of dependence and sickness. \"The genetically modified seeds, such as those donated and later immolated, cannot be saved from year to year. Some so-called terminator seeds\u2014the DNA of which is altered so as to not drop seed after harvest\u2014require the farmer to buy new seeds from Monsanto the following year in a legally binding contract, instead of collecting the seeds that would have naturally developed on the plant before its DNA was modified.\" This increases poverty and indebtedness; legalized slavery.\n\nWhere there is no free choice, when our natural connections to the land that feeds our bodies and souls are severed, the soil is tilled for feudalism, imperialism, and fascism to emerge. On June 4, 2010, 10,000 Haitian farmers walked 7 kilometers to Hinche to receive the gift of seeds from their benefactor. Upon arrival, it was World Environment Day; the farmers took the 400 tons of vegetable seeds and burned them all. Sometimes you have to say _no_.\n\nIf you are unable to integrate and be in relationship with wild ecosystems, with ego states that naturally occur in Nature just as they do in human beings, with the aliveness and sexuality of Earth, you will never be able to integrate your own sexuality, your inner ecosystems.\n\nReferring to Aphrodite, Michael Perlman comments:\n\nIt is strange how rarely the Goddess of luxuriant sensuality and love is linked with the love of nature, with ecological concerns, with the power of beauty. Her world of sensual and erotic display, of nightclub and bedroom, seems so distant from the backpack trail; we're not in the habit of juxtaposing thoughts of wild nights and the wildness Henry David Thoreau celebrates in his famous essay on walking. We don't generally think of ecological concern as having erotic and sensual power (even though the image of the woods as a sexy place is hardly confined to Greek myth), and recent ecological thought has begun to appreciate the loss in that omission. If, for instance, we are to talk about Greek mythological figures in relation to contemporary ecological concerns, shouldn't we be talking instead about Artemis, goddess of pristine wilderness and inviolate, virgin forest?\n\nAs long as we maintain separation from all that is Earth and sensual, we will never know the kind of majesty and peace that comes from being whole and integrated as a sacred, sexual being, a playing member of the circle of life. And there will be continued assaults on Earth and continued sexual repression.\n**6**\n\n _ **Finding the Wild**_\n\n_Now is the time for you to mark your entry through the spirited gate. You need to find your crossroads, the gateway that can take you anywhere in this world or in any other worlds._\n\nBRADFORD KEENEY,  \n _T HE BUSHMAN WAY OF TRACKING GOD_\n\n _Authentic tidings of invisible things!_\n\nWILLIAM JAMES, _O N SOME OF LIFE'S IDEALS_\n\n _The great sun burning with light, the strong earth\u2014dear earth\u2014the warm sky, the pure air, the thought of ocean, the inexpressible beauty of all filled me with a rapture, an ecstasy, an inflatus._\n\nRICHARD JEFFRIES,  \n _T HE STORY OF MY HEART_\n\n _Our judgments concerning the worth of things, big or little, depends on the feelings the things arouse in us._\n\nWILLIAM JAMES,  \n _O N A CERTAIN BLINDNESS IN HUMAN BEINGS_\n\nA SACRED VOICE IS CALLING YOU\n\nAbraham Maslow termed mystical experiences _peak experiences_. He said they frequently occur in well-integrated, mature people. Often a mystical experience results from a deep need to feel the numinous, to have an experience that transcends the understanding of the intellect. When we allow ourselves to fall totally, madly in love with the other, to give ourselves over completely to the feelings we have for him or her, the opportunity for the ecstatic to enter our lovemaking is increased. It's as if an invitation has been sent out for the mystical to enter the relationship. As we go deeper and deeper into feeling the love we have for the other, physical boundaries begin to fall away. Love in that deeply intimate place is the catalyst for the mysterious, ecstatic event to unfold between the two, and in our psyches.\n\nJames Hillman describes eloquently the _daimon_ or _fate, genius, calling, soul, destiny_ (he uses these terms interchangeably depending on the context), some invisible guiding force that is with us during our lifetime. Before we take on a human body there is a time when we sit in counsel with the Universe, with God, Creator, the cosmos, with Gaia and whoever else you might recall being there. In this counsel, before we make the journey to Earth, we make agreements about how we can best spend our time here during this chapter of the soul's making. At that time, we are each given a soul companion\u2014a daimon\u2014that guides us here to Earth. The daimon is necessary because on the way to becoming born, we lose our memory of the agreements we made. The daimon is a life companion, a soul guide who helps us navigate this life and to slowly remember why we've come here. Remembering those agreements; to become who we were meant to be, to make the soul's journey, to let the Genius out of the bottle, to enliven our destiny, to participate in making the world a better place, whatever our agreements are, our daimon guides us, pulls us here, pushes us there, brings that experience to us, helping us to be who we came here to be.\n\nThe daimon is alert to movements of our soul's purpose, and those early movements, intimations, can often be seen in early childhood. Edward O. Wilson talks of it this way, \"You start by loving a subject. Birds, probability theory, explosives, stars, differential equations, storm fronts, sign language, swallowtail butterflies\u2014the odds are that your obsession will have begun in childhood. The subject will be your lodestar and give sanctuary in the shifting mental universe. . . . A man's work is nothing but this slow trek to rediscover, through the detours of art, those two or three great and simple images in whose presence his heart first opened.\"\n\nThere has been a fairly obvious theme in my life of relationship experiences, sexual experiences, sexual abuse, sexual violence, sexual ecstasy, wilderness ecstasy, bonding with Earth, Earth sexuality that has undoubtedly helped shape who I've become, who I was born to be, and which has heavily influenced the writing of this book. I chose at each juncture to keep going, to say \"yea\" or \"nay.\" And I am keenly aware of the touch of an invisible guide having a hand in my life and in helping me stay alive in dangerous situations.\n\n _Reason flows from the blending of rational thought and feeling. If the two functions are torn apart, thinking deteriorates into schizoid intellectual activity and feeling deteriorates into neurotic life-damaging passions._\n\nERICH FROMM,  \n _T HE REVOLUTION OF HOPE_\n\nThe ancient Greeks had a word (they had a word for everything) for making the invisible visible, _opathe_. We have a passionate desire to make what lies just below the surface of a thing visible to our senses, to our seeing, to our feeling. It is the work of the scientist to see the intimate workings of cells, mitochondria, bacteria, and organs and how all things are in relationship to each other. We need and want to feel the presence of the mysterious in our lives, inside our waking and dreaming. To know, without doubt, that which we came from is here around us and inside us, that we are not alone. We need a sense of mystery, of the unknown, yet we strive diligently to unravel, make sense, to prove or disprove the existence of spiritual forces, numinosity, invisible force fields. We want to know if our prayers and dreams are being heard and answered. We need to know this to have our lives makes some sense, have meaning beyond the mundane. If not, what has been the purpose for the inextinguishable existence of stories, legends, myths from every culture on Earth of gods and goddesses, spirit messages, trees that talk and walk about? We need the myths of old as well as to create new myths; those stories that are made up that have truth in them, help us find our own truths. We endeavor to unravel and unearth the magic and mystery of not only the Universe but, especially, our own humble inner and outer workings. As Richard Earnheart says, \"It is the goal of every serious artist, scientist, and writer: unbridling the unseen, unearthing the undiscovered, unleashing a great tempest of the heretofore unexpected.\"\n\nIt is through sacred sex and relationship with the wild that we strive to satisfy our deeply felt need for the numinous. These deep fully-present connections provide the moments when we glimpse the movement of some great thing inside us, or the movement of a barely visible form skittering past the corner of our eyes, seen but not quite seen. The moments when we feel so much love for a grandchild or lover that we are taken outside ourselves, moved in the core of ourselves. Feeling is at the core of becoming whole, is the center spiral from which sacred sex spins. It is the balm that heals the split between humans and Gaia, and it is this feeling that will repair damage done from the mechanistic, scientific world-view and return us to a state of wholeness. Sacred, ecstatic lovemaking is a gift from the gods; it is a portal through which we experience a spiritual reality, experience our lover as a sacred, holy being. It is in these sacred moments that our ordinary awareness becomes heightened, illuminated, and we experience the all-encompassing presence of spirit and our interrelationship with all beings.\n\n _If you cannot face directly into your sexuality,  \nYou will never discover your true spirituality.  \nYour earthly spirit leads to discovering your heavenly spirit.  \nLook at what created you to discover what will immortalize you._\n\nHSI LAI,  \n _T HE SEXUAL TEACHINGS OF THE WHITE TIGRESS_\n\nIt is not just the moments after sex or, sadly, during sex that we often feel something is missing. Human beings have been feeling that something is missing for a very long time. Rather than try to name what it is, rather than turning to face the feeling and follow it back to its source, we turn to Prozac, television, and therapy and fill our homes with stuff. We think if only we have more stuff, the right stuff, then some psychological shift will magically happen, and we'll fill the loss with the meanings and connections we've been looking for. Stuff dulls our senses so we don't have to feel how much our life doesn't work while we're looking for meaning. What gives our life meaning is something invisible. But meaning doesn't happen in a vacuum, it comes out of being in relationship to the work we are meant to do\u2014with our lovers, with ourselves, with the first plant that communicated with us.\n\nOur relationship with the wildness of Earth is one doorway into the mystical experience. I had been walking the eighty acres of a friend's land for nearly three hours trying to find the right spot to put myself for my third vision quest, which I planned to begin a week later. I crossed the stream, hiked up the hill on the north, then meandered back down and back across the stream, up the southern hill, and around the forest edge. After a few hours, I made my way back to my van. Instinctively, I touched my left earlobe. Gasping in shock, I discovered one of my earrings was gone. It was a special pair, made by a friend who was a Mayan sculptor in the Yucatan. I was horrified and spent a few moments kicking myself in the ass for wearing them on a walk through the forest. I quickly gave it up as I decided that since I was asking a lot of the land and the spirit of the land\u2014to hold me and guide me during my coming four-day fast\u2014I was glad to leave the earring as an offering of gratitude. When leaving a gift for the spirits, it's good to give something that has value and meaning so it's not an idle gesture. I was happy to think of it being out there, somewhere, as one of my giveaways.\n\nA week later, I returned to the land with my backpack, sleeping bag, camp pad, prayer pipe, and water jugs to fast for four days and nights. I began to walk toward the spot I had chosen earlier when I sensed it was now wrong; it didn't feel right today. I continued to walk, but as I prepared to cross the stream, I was stopped suddenly in my tracks, held still, prevented from moving by some invisible force. Instinctively I looked down as if to see what was holding me still. It took a moment for the reality of what I was seeing to register in my awareness. Next to my right foot was the lost earring. The impact of what was happening continued to move deeper into my interior world. Out of eighty acres, the earring could have fallen anywhere. I could have crossed the stream at any point, but here I was and here it was.\n\nAs I stood there feeling everything I was feeling, my mind whirling at the sight of it, an energy force seemed to rise up from the ground and stream, slowly wrapping around me a phenomenal sense of love. Oh, what a rapture to be in the midst of this experience\u2014the numinous, the real, extraordinary. To feel unconditionally loved by the spirit of this land and the spirit of the vision quest. Certain my heart would burst, water flowed freely from my eyes. Something told me to pick it up, to take it back, and place it on my quest altar as a reminder during my time here, a reminder that I was not alone. I had invisible companions who loved me and this was a demonstration of their love and support for what I was doing. My four-day quest became a rich tapestry of meaning.\n\nThere was no doubt, to my way of seeing, that my daimon had a hand in that event. I have no doubt that Gaia, Earth, the spirits of place and of vision quest were instrumental in the confluence of that sequence of events.\n\nThe meaning of that moment when I was stopped in my tracks still percolates and affects my life. I needed that personal experience with those invisibles on that land at that time for my soul work and destiny. It may take years after such an event for the full meaning to come to fruition.\n\nPeople are hungry for the invisibles. We are desperate to know they share our lives and to feel their presence. Note the popularity of the Harry Potter books and movies, _The Lord of the Rings_ trilogy, _Stardust, Avatar, The Chronicles of Narnia, American Gods,_ and so on. We are like Alice, of _Alice in Wonderland._ When a rabbit walked by as she sat under a tree reading, she knew by the waistcoat he was wearing that he was no ordinary rabbit; she could tell that he was intelligent and aware. She followed him down the rabbit hole into another world because the invisible had spoken to her.\n\nThe use of psychotropics such as marijuana, ayahuasca, peyote, or mushrooms is illegal but the illegality has nothing to do with addiction. These substances are illegal because they open the doors of perception, allowing us to see the invisibles underneath the form of the visible world. Publishers such as Inner Traditions publish New Age and esoteric books on subjects that mainstream publishers won't touch because these subjects validate the existence of the invisibles. The normal response is to deny their existence. As accounts of the invisible realm rise up from the underworld and from the dusty shelves of archaic libraries, they are being met by an equal force ready to discount, discredit, and censor them, as evidenced by the continuing war on drugs and the medical marijuana debate. In recent years there has been an increasing rise in the use of traditional herbal medicine, shamanism, energy work, and channeling and a resurgence of the peyote church and ayahuasca ceremonies as more and more people seek to meet needs that are not being met by allopathic medicine or pharmaceutical drugs. The degree to which people are willing to pay out-of-pocket expenses for non-Western medicine and healers, and risk legal recriminations to participate in \"illegal\" ceremonies speaks volumes of the need for invisibles. What if we actually lived as if the gods of Earth and myth lived among us? What would our life be like if we worked directly with our daimon every day and called on the invisible ancestors\u2014spirits of place, of mountain, of ocean\u2014to help guide us? What would our lives look like? What would it do to our level of hope? How would it change our orientation to Nature, to wilderness, to each other? How would we begin?\n\nGOING WILD\n\nWe need regular infusions of the wild, of the numinous and the touch of the invisibles. We need to seek refuge or renewal in physical wilderness. We may need to smell the sweet vanilla scent of a pine forest or hear the waves rise and fall on the ocean. Wild is known through our senses. Going into wild ecosystems forces us to become aware; that's part of their function, to cause us to shift our focus from the prosaic and ordinary to the heightened, otherworldly, and extraordinary. Each of our senses becomes sharp, fine-tuned; we listen for the deep sounds, look for the unseen as our peripheral vision expands farther and farther out. Our nervous systems are heightened and each hair on our body acts as a kinesthetic radar receptor. We become acutely aware of our own sounds, our breathing, the heaviness of foot on the forest floor, the scratching sound of Gore-Tex, and the tension we are holding in our body. The primordial part of us knows that we are both predator and prey. There are bears, mountain lions, rattlesnakes, scorpions, and loose rocks to be mindful of. Before we even set foot in areas designated as wilderness or even go to a campground, some internal shift happens\u2014we enroll the Predator part of us, the part that's interested in surviving. If we were to slow down a bit, we would notice that our concerns turn toward providing for what we were taught are the basic necessities for survival: food, shelter, water, and clothing. We begin by taking inventory of equipment and gear on hand. Is it in good working order? Do I need to replace or add anything? How will I protect and defend myself if a bear or mountain lion comes upon my scent, then my camp, then me?\n\nWe ask ourselves questions about the weather, about food and shelter. Do we need a tent or will a tarp be sufficient? Do we want to sleep on the ground sheltered only by the night sky? What if it rains or the temperature drops? Do we have rain gear, enough warm clothing? Is there potable water where we are going? Do we need to bring a purification system? What about food and cooking? The process of preparing for our trip increases our awareness; our senses become more acute. And once we've arrived, we need to know how to read the geography of the land to keep us from pitching camp in the middle of an animal path or in the floodplain of a creek.\n\nAs critically important as the physical needs are, we also need to know how to read the invisible geography of our chosen campsite. How does it feel? Is it benevolent, supportive, friendly, ominous? Some part of us is reading this landscape as that part is always looking for what feels good and avoiding places that feel funny, weird, or scary. We do it each time we go into a coffee shop or restaurant but rarely do we pay attention to what we are actually doing as we search the place to find just the right spot to sit, the spot that feels good. We actually imagine sitting there before our body arrives in the chair. Once in the wilderness, have we acknowledged that the wild has been called home by many others before we hominids showed up? Have we asked permission to camp in the middle of someone else's living room and has it been granted? Have we told the other inhabitants; the plants, the animals, the stones, the trees, the elementals, what we are doing there, how long we intend to stay? Have we asked them if they would grant us a drama-free stay? Have we regarded the spirit of the place and made offerings, or in some way reciprocated, given back in exchange for staying, as we would take a bottle of fine wine to dinner at a friend's house?\n\nDespite our brilliance as a species, we still think the disharmony we feel within ourselves, with each other, and with Earth can be assuaged, ignored, or somehow altered by iPods, the Internet, instant messaging, and bigger, flatter, sharper television screens. Though I enjoy having those things in my life to some degree, none of them has been able to replace intimacy. None of them replaces my experience with the wild or the soul food I find there. The World Wide Web connects me to far-off lands, loved ones living abroad, hard-to-find books, new and old music and research papers. YouTube brings Viktor Frankl, Eric Berne, Virginia Satir and Occupy Wall Street into my office, but these things do not bring the livingness, the soul, of wild ecosystems into my home. They do not bring the scent of pine forest, the spontaneous call of a red-tailed hawk overhead, a lizard sunning itself on the rock in my path. They do not render me awestruck in the same way the soul of the mountains does or the experience of being seen by an ancient, invisible teacher come to walk by my side for a spell. And they certainly do not render my senses alert to the livingness of being immersed in changing landscapes and ecosystems.\n\n _But the Genius which, according to the old belief, stands at the door by which we enter, and gives us the lethe to drink, that we may tell no tales, mixed the cup too strongly, and we cannot shake off the lethargy now at noonday. Sleep lingers all our lifetime about our eyes, as night hovers all day in the boughs of the fir tree. All things swim and glitter. Our life is not so much threatened as our perception. Ghost-like we glide through nature, and should not know our place again._\n\nRALPH WALDO EMERSON,  \n\"EXPERIENCE,\" _S ELF-RELIANCE AND OTHER ESSAYS_\n\nOur electronics keep our eyes engaged in looking while we lose our ability to see. We have become so geometrically single focused that when we look at the M\u00fcller-Lyer arrows pictured below, we can't tell whether one line is longer than the other.\n\n _The M\u00fcller-Lyer arrows_\n\nThe modern Western brain, has been trained by exposure to photographs and representational art and also by exposure to street corners, square rooms, and other \"carpentered\" features of the environment to see cues of depth and distance on two-dimensional pictorial surfaces. This training confuses the Western perception of the M\u00fcller-Lyer diagram. But studies such as one published by Philip L Kilbride and H. W. Leibowitz,\"The Ponzo Illusion among the Baganda of Uganda,\" show that those living an indigenous lifestyle in the non-\"carpentered\" natural world are able to judge the length of the lines accurately. That is, they are able to see the lines as they are: equal in length.\n\nFor the Western brain, the slope of the arrows in the M\u00fcller-Lyer diagram creates the illusion of depth. If you look at a corner inside your house you'll notice that the juncture where two walls meet the ceiling or the floor visually reproduces the M\u00fcller-Lyer line with the inverted arrows, or \"V\" shapes pointing in toward the line. When you position yourself across the street from the corner of a building you'll be able to see that the M\u00fcller-Lyer line with the arrows pointing out away from it is repeated where the vertical line of the corner meets the roof and again at the bottom corner of the building.\n\nMany books have been written by people who travel to exotic lands, go deep into the jungle or to remote islands, to live and study with indigenous peoples. Overwhelmingly, they recount how they were transformed and their senses awakened, how they were finally able to see as indigenous people see. They saw minute details in butterfly larvae and in the patterns of leaves and tree bark. They were able to sense from a distance, when someone was approaching. Then they return to their ordinary life with the stories and myths that reveal some essential truth and even warnings, to write a book telling of their adventures benefiting all who read the accounts. I'm glad they are willing to be messengers.\n\nIt is helpful to travel to other countries and experience other cultures, to be offered a very different perspective. But not all of us can afford to take the time, commit the resources necessary, and then be lucky enough to be accepted into an ancient society that still has the old ways intact, unblemished by missionaries and Western culture.\n\nWhat are we to do, those of us who are homebound, unable to travel? How are we to replicate and access the experience of restoring the indigenous way of seeing and feeling? We can do what indigenous people have always done: What they do is available to us, right here and now. There is not a single thing I can think of (short of being in a coma or a politician) that is stopping us from becoming indigenous to this place, the place where we are living in any given moment.\n\nThe knowledge indigenous people pass on is a wisdom that comes not from schooling, not from higher education, but from being intrinsically part of their habitat and using their heart field as an organ of perception. Indigenous people are as much a part of their landscape as the mosquito larvae, the mountain, the grasses, the u\u00f1a de gato vine. It is wisdom, a body of knowledge that is dynamic and adaptable and transferrable and available. It is wisdom that is transmitted, if you will, directly from the jungle, directly from the mountain, from forests or desert mesas, from the trees and plants in the backyard, to anyone willing and choosing to be the receiver. The more time we spend in the forest, walking through streams and wild landscapes with a hungry heart, a desire to learn, and enough humility to allow ourselves to be educated, the closer we are to becoming indigenous. Humility must be part of the asking; the invisibles must feel, must know\u2014for feeling is knowing\u2014your desire to be touched, to be taught, to be shown.\n\n _In every walk with nature one receives far more than he seeks._\n\nJOHN MUIR\n\nThe more time we spend in uncivilized landscapes, the more we know, the more intelligent we become, the more tuned to the life of the forest, to the intricate, delicate, minute breathing of the trees, the imperceptible movement of stone. It is unavoidable, then, to become aware of our own internal movements. And one day, when we're not looking, we realize we know things we didn't know we knew and have no memory of learning. It happens spontaneously as you move through your day. Interesting thoughts and insights are there, inside us, bubbling to the surface. We begin to see connections between and among things. We begin to sense and perceive things; we notice the smell of things and places and people. Authentic knowledge is percolating through gray matter. There is a noticed sense of calm that begins to relax our musculature. Our breathing deepens, our heartbeat regulates, and our peripheral vision expands more consistently. We may feel a sense of inner strength and a kinship with other life-forms. With each visit to uncivilized landscapes, we become more present in our bodies and more present with the experience of being there.\n\n _I am alarmed when it happens that I have walked a mile into the woods bodily, without getting there in spirit. In my afternoon walk I would fain forget all my morning occupations and my obligations to society. But it sometimes happens that I cannot easily shake off the village. The thought of some work will run in my head and I am not where my body is\u2014I am out of my senses. In my walks I would fain return to my senses. What business have I in the woods, if I am thinking of something out of the woods?_\n\nHENRY DAVID THOREAU,  \n\"WALKING\"\n\nSome part of us knows that to give life to that primordial, ancient faculty inside us, the indigenous heart of us, the part that knows how to walk silently through a forest, knows that we have to break a few rules of the culture we live in. \"It's not polite to stare, dear.\" We're wearing a cultural \"chastity belt,\" and I am quite done with the irritation and chaffing that it causes to my sensibilities and free will.\n\n _Dare to be seduced by the senses_.\n\nCultures still living closely connected to nature, who depend on it for survival and livelihood, don't live in carpentered houses with square corners. Because they have not been schooled out of their native senses, their feelings not dulled or shut down, they are able to see when one line is longer than the other. Their seeing hasn't been adulterated nor their vision become pinpoint, forced into straight lines and corners.\n\n _The story of Romulus and Remus being suckled by a wolf is not a meaningless fable. The founders of every state which has risen to eminence have drawn their nourishment and vigor from a similar wild source. It was because the children of the Empire were not suckled by the wolf that they were conquered and displaced by the children of the northern forests who were._\n\nHENRY DAVID THOREAU,  \n\"WALKING\"\n\nTo reclaim the body's sensuality and sexuality requires reclaiming the body's sensory perceiving and feeling. Our senses evolved in concert with the wildness of the world. Sensory acuity is the ground where the wild is fostered. We are not encouraged to feel in our culture, to sense, to be sensual and full of sensations, or to find meaning in sexual experiences. Marjorie Hope Nicolson eloquently states it this way: \"Like men of every age, we see in Nature what we have been taught to look for, we feel what we have been prepared to feel.\" The experience of wilderness depends on, is fundamental to, and is accessed through our senses. The singularly most dissident, nonconformist, rebellious thing you can do and the most immediate entrance to the wild state of mind is\n\nto\n\ncome\n\nto\n\nyour\n\nsenses,\n\nnow.\n\nCOMING TO LIFE\n\nWilheim Reich developed somatics, a system of body therapy used in conjunction with psychotherapy, to release held emotions and trauma. Reich wanted a full-body emotional response to life. He believed that if you covered yourself up, armored yourself, you could deaden pain, but you would rob yourself of a life full of joy. Reich asserted there was a direct correlation between aggression and body armoring. He held that the longer a person armors herself, the tenser the musculature becomes, the deeper the psychosis goes. Armoring leads to cancer, arthritis, and rheumatism and dulls awareness, intuition, and creativity. When emotions get backed up inside a person and the musculature becomes more tense and atrophied, the more likely it is for aggression to build and war to occur.\n\nThe armored body is not unlike an animal that has been taken out of its natural habitat and put into a cage. The animal paces back and forth, back and forth, getting more and more tense, feeling the bars of the cage getting closer and the space getting tighter. The animal's territorial rights have been usurped; it needs space to hunt and run. It needs to move, stretch, run free and uninhibited. But it is being confined, its movements restricted artificially against its nature, and it becomes, as a result, violent and rebellious. Krishnamurti describes it this way: \"There is the desire to expand. And when society presses me in, drives me into a certain corner, I explode\u2014which is again a revolt in order to expand. And when one lives in a small flat in a very crowded street and there is no open country to breathe in and no opportunity to go there, I become violent. The animals do this.\"\n\nEverybody wants to be more alive, to feel more alive, to thrive rather then merely live or survive. Psychobiologist Stanley Keleman writes, \"Being more alive means being more sexual, more sensuous. To be more sexual is to broaden one's range of feeling and expressive action.\" Eros, sexual energy, is life-force energy permeating all of creation and influencing our own creativity and ecstatic experiences.\n\n _Wild is sexual; we must become wild once again_.\n\nWilderness is sexual. Jay Griffiths in _Wild_ says, \"[I]f you had to choose part of the human anatomy as an analogy for wilderness, you'd have to go for the loins\u2014and we humans lose an acute and vital part of our sensuality when we ignore the wild world; the grinding of shoots thrusting up into the light; the hungry torsion as snake squeezes snake, birds flightily dipping as they twang an orgasm between wing beats, the delicate incipience of young sexuality in bud and blossom, lizards eyeing each other up for a darting lick of quick sex or basking with satisfied lust.\"\n\nWhen we exercise our senses, feeling into the world as we move through it, smelling, tasting, touching, hearing, we awaken the wild inside us. In _A Blue Fire,_ James Hillman makes reference to this when he says, \"When we move with senses acute, listening, watching, breathing in tune with the world about us, recognizing its priority and ourselves as guests, witnessing its 'God-givenness,' then we have made a wilderness area or moment. The restoration of the pristine stars in a fresh attitude toward what is, whatever and wherever it is.\"\n\nWe take back the original meaning of the word _wyld;_ that feeling that is out of the ordinary; that is fresh, unwearied and is present in old-growth forests, magnificent stone outcroppings, the sound of ocean waves cresting and crashing, and our sexual longings. We can be wild in any place or event, and when we do, we subvert the dominant culture's attempts to dwarf our senses and pigeonhole our feelings and colonize our minds. We come alive once again; we become the free, spiraling balls of energy that we were as young children, moving spontaneously and without hesitation through time and space.\n\nBecause the wild is dangerous to so many people's way of thinking (yes, eating the wild will change you), we are taught it must be repressed, suppressed, manicured, trimmed, tamed, cut, sprayed, fenced out, and controlled. The thing about fences is: when you fence something out, something else is being fenced in.\n\nThe soulful wild is a place where our soul can be reclaimed. Once you truly let yourself feel again after a few meals of the wild and sensuous Earth, a shift in consciousness penetrates the foundations of your beliefs and value systems\u2014sometimes slowly and gently over time, other times an unexpected shift in principles and perceptions is thrust into your mind and all that you were before suddenly becomes pregnant with uncivilized thoughts unlike anything you've thought previously. That sudden shift is like that \"aha!\" moment that comes abruptly after you've been mulling over a difficult concept. One moment you're vexed by it, and the next you have clarity that rings pure like temple bells. When wyld inserts itself in our consciousness, a part of us that has been asleep is suddenly awakened. We begin to shift out of unconscious programming into conscious awareness.\n\n _You must get rid of what is commonly called knowledge of them [things seen in Nature]. Not a single scientific term or distinction is the least to the purpose for you would fain perceive something and you must approach the object totally unprejudiced. You must be aware that no thing is what you have taken it to be._\n\nHENRY DAVID THOREAU,  \n _T RUE HARVEST_\n\nOur intuitions and perceptions become clear, and suddenly we are able to see why, for instance, we've had a particular, ongoing problem and why that person in our life has been such an irritation. You begin to see below the water's surface, beyond whatever sparkling reflections have been catching your gaze and distracting you. Moreover, the surface is no longer a satisfying place to be spending your brief life; a hunger for more begins to drive you now out of the familiar into deeper, darker waters. A willingness and need to explore and push the edges of conformity excites and animates your life.\n\nAnd sometimes the shift comes with a painful, life-changing price, as it did with Aldo Leopold the day he killed a wolf.\n\nIn those days we had never heard of passing up a chance to kill a wolf. In a second we were pumping lead into the pack, but with more excitement than accuracy: how to aim a steep downhill shot is always confusing. When our rifles were empty, the old wolf was down, and a pup was dragging a leg into the impassable slide-rocks.\n\nWe reached the old wolf in time to watch a fierce green fire dying in her eyes. I realized then, and have known ever since, that there was something new to me in those eyes\u2014something known only to her and to the mountain. I was young then and full of trigger-itch; I thought that because fewer wolves meant more deer, that no wolves would mean hunters' paradise. But after seeing the green fire die, I sensed that neither the wolf nor the mountain agreed with such a view.\n\nMany of us turn to writers and poets whose works were stimulated and inspirited while spending time in the forests, mountains, and deserts. They have taken into themselves some nourishment that kneads human souls and thus gives rise to the great works of the likes of Thoreau, Goethe, Leopold, Muir, Schauberger, and LaChapelle. They went into the wild to service an internal hunger and became servants to something outside and greater than themselves.\n\nThere are those of us who make a pilgrimage to a mountain, who need to feel the process of finding wilderness intensely in our bodies\u2014legs pushing our weight up serpentine trails, lungs burning with the cold fire of clear, high air, skin flushing with the heat of exertion, as we ascend, laying this mile beneath our feet and then the next. Or we may go to the seashore for the spray of water as wave after wave is spent on stones. We may need the gentle, low, rhythmic tides on a sandy South Pacific shore. We choose places that fill a need, that match our psychological states or shake us out of them. We go to be stirred up, calmed down, soothed, inspired, renewed, healed. To make love, to be made love to, to be called beloved by Earth. Conversely, we may avoid places like deserts for their irrepressible ability to show us the desert that has for too long existed inside us. Taking the wild inside you, tasting the scent on your tongue, comingling your soul essence with the soul of Gaia, stirs primordial faculties, from groin to reptilian brain to conscious awareness, seducing the senses to shake off the stupor induced by the drugs called culture and materialism.\n\nJohn Seed, the rainforest activist said, \". . . as the implications of evolution and ecology are internalized . . . we begin to identify with all life. . . . alienation subsides . . . 'I am protecting the rain forest' develops into 'I am part of the rain forest protecting myself. I am that part of the rain forest recently emerged into thinking.'\"\n\nAs your perceptions become acute and your seeing more clear, your sense of self grows more whole, sturdy, and potent. Your soul takes on larger dimensions, filling out, dropping baby fat, maturing and becoming sophisticated. You may even feel bigger, taller, taking up more space. You may find yourself changing the way you wear your clothes\u2014tight-fitting jeans are replaced by garments that allow freedom of movement. Unwilling to be constrained, we choose clothes that are loose on our bodies, full of color, and outrageous. You may take on more affect in speech and behavior. Suddenly you notice your arms are moving spontaneously, your hands gesticulating. Your hips begin to loosen and sway, and there is an energy now, coming off your body, and your core scent has changed, becoming musky. You have more energy and improved memory. You shed the garments of civilization, and soon your favorite pair of shoes or your cool leather harness boots are squeezing and constricting your feet. You can wear those favorite shoes only occasionally now as your feet take on a hissing sort of dialect from walking naked between, up, over, and around the valleys and mountains of Gaia's bosom; feet skin touching Earth skin. And over time, we are able to bring life-giving water to our internal desert, transforming it into an oasis.\n\nAnd speaking of shoes . . . The Mamas, or spiritual elders of the Tayrona living in the Sierra Nevada de Marta in Columbia, say shoes break the contact between people and Earth. Walking barefoot in the forest or even around your home has psychological and physical benefits. It connects you immediately to the electrical energy of Earth, grounding you in your body, causing you to be more present. Life-force energy from Earth is directly transmitted through the soles of the feet, increasing awareness and energy. Walking barefoot reduces inflammation in the lower extremities, regulates thyroid function, and reduces stress, and since the bottoms of our feet are full of acupuncture points, the whole body is brought into alignment. Walking shoeless reduces the chances of sprains and deformed feet and toes, as well as strengthening the muscles in the feet and legs and increasing circulation. We are healthier physically, emotionally, and psychologically the more we move about sans shoes. In fact, there is no record of feet disorders before the Renaissance. The introduction of the elevated heel was a brilliant invention to depict wealth (Louis XIV had four-inch heels made for him) and to protect feet from the filth that plagued ancient Egypt and Rome. Butchers were known to wear heels to avoid the carcass debris that littered the floor.\n\nAside from their fashion and functional uses (keeps your feet in the stirrups), heels keep us off balance and insulate our naked feet from the sensual, erotic, wet, gritty, slippery muddy, warm, sandy, and rich soil of Earth.\n\nA DICHOTOMY THAT DAMAGES\n\n _When the great crusade against sex and the body started in full blast with Plato, it was a crusade for \" ideals,\" and for this \"spiritual\" knowledge in apartness. Sex is the great unifier. In its big, slower vibration it is the warmth of heart which makes people happy and together, in togetherness._\n\nD. H. LAWRENCE,  \n\"A PROPOS OF _L ADY CHATTERLEY'S LOVER_\"\n\nMany spiritual traditions and religious practices have wrongly sought to separate the body and soul; the body is regarded as a place of denigration, an obstacle to enlightenment, while more value is placed on advancing and purifying the soul. Though we all seem to disagree on where exactly the soul resides in the body, or exactly what the soul is, it needs purifying whether or not it can be located.\n\nPlato's philosophy was based on his theory of a soul divided into three components, reason, will, and appetite. He contended that one can identify the parts of the soul because they sometimes clash with one another. He regarded the body and soul as separate entities. As a dualist, he also posited an \"unreal\" world of the senses and physical processes, and a \"real\" world of ideal forms. This way of viewing natural phenomena has had dire consequences.\n\nGeorg Feuerstein writes: \"[Y]et how many spiritual seekers have struggled to realize truth, God or a higher consciousness by escaping what they called 'the prison of the body.' In treating the body as an enemy, that antagonist of the Spirit, they doomed themselves to experiences of an amputated God. They failed to see the body as part of the Great Mystery. We can learn from their mistake.\" The body is the temple of the gods and the vehicle through which soul has life.\n\nBradford Keeney observes that **\"** [c]elibacy makes no sense to a Bushman. 'You've got to be kidding! Are there really people who think that not having sex makes them closer to the gods? What god would want to hang around them? Maybe it's better if you don't go home. Your trickster gods sound boring because they don't know how to have a good time. Hang out with us and you'll learn that God loves sex!'\" The original cultures never had sexual hang-ups or phobias about touching. Keeney goes on to note that this mind-body schism is \"a new development in the scheme of things, and it seems to have started as a word game. Mind got separated from body. Good was separated from evil. Mind married good, while body married evil. Thereafter, whenever mind experiences itself as inseparable from body, a sin of inappropriate union is declared. Being touched by God shifted to being heard by God. But God isn't listening to your words; God wants to be touched. The same is true for you, so go find someone to hug.\"\n\nI was between my fourteenth and sixteenth years. I was in a two-year love-hate relationship with Christianity and the Lutheran church I sporadically attended a mile from our farm. There was something there that I was looking for, some need I had I thought could be met there. So many of the tenets and teachings were a source of rage for me, and still I returned hoping to find something to feed my unnamed hunger, which compelled me to attend. I didn't understand the reason for being born, for being dropped in this family in the middle of no-fucking-where.\n\nInteresting to me was that the single most often misquoted passage from the book of Genesis is that human beings were \"given dominion over all things.\" There was no room in the interpretation of the doctrines for Earth to be alive, aware, and sentient, yet that's how I experienced Earth. Everything, everyone, except human beings, was unfeeling and ignorant. It fell way short of answering how other life-forms were able to last and evolve for billions of years before the \"superior\" human species was created out of the genius of an oversixty, white, lonely, loving, upright, and vengeful god. Something was wrong with what I was being indoctrinated in.\n\nThrough the teachings of Christianity, only humans were created in the image of God, only humans have a soul. But prophetically it is written also in Genesis 9:2: \"[T]he fear of you and the dread of you shall be upon every beast of the earth, and upon every fowl of the air, and upon all that moveth on the earth, and upon all the fishes of the sea; into your hands they are delivered.\" This is the passage from Genesis that is often underemphasized, the part that puts responsibility for caretaking the Earth in the hands of humans. It warns us that however we treat Earth and all that moves on Earth, we shall be treating ourselves for we are part of \"all that moveth upon the Earth.\"\n\nIn whose image are the fishes, the seas, grains of sand, microbes, fungi, and snail made in? Why should human beings presume to think humans are the only species made in the image of God Earth-maker?\n\n _God Yahweh formed man out of the soil of the earth and blew into his nostrils the breath of life, and man became a living soul. And God Yahweh planted a garden in Eden in the east and placed the man therein . . . God Yahweh took the man and put him in the Garden of Eden to serve and preserve it._\n\nDANIEL HILLEL,  \n _O UT OF THE EARTH_\n\nThe word _animal_ comes from the Latin _animalis,_ meaning \"having breath.\" Creator, Earth-maker breathed life into all beings. Can we extrapolate, then, that all animals are living souls, having breath? And further, _spirit_ comes from the Latin _spiritus,_ meaning \"breath.\" All living things have a soul, and it is the spirit, the breath of life, that is given from the Great Spirit that animates and connects all living things. We all breathe the same breath, our inhale is that of the cosmic exhale. Each of us sit, lie, move, creep, crawl, live, die in the same matrix of spirit breath. When we feel _inspired,_ we are filled, imbued with the spirit that moves through all things, and we are aroused to act, to do something with the inspiration. We know it is inspiration for it's as if we have been taken over by some benevolent force that insists we _do_ something without a moment's hesitation.\n\nIt was in the middle of this time of my life that I would often ride my horse into the fields and open spaces behind our house. I loved riding my horse. It was one way I escaped the insanity of my family. I could breathe and smell the fresh air and think clearly without the contamination of our family psychosis.\n\nI was indomitable and unconquerable with my legs wrapped around this magnificent, powerful animal, as mighty muscles, thighs, and shoulders moved and ass swayed beneath me with each stride. I became intoxicated with the primal scent of horse hair wet with sweat. And on one such day, as my horse was walking along with me on her back, the sun warm and low on the horizon on that late summer day, I was lingering deep in thought, as I often did when I was alone, when a peculiar feeling began to take over my senses. It touched the entirety of my body at once as if being bathed in it, a light touch at first, just light enough to get my attention. My skin began to flush and prickle, and I could feel the hairs on my arms and neck stand up. We walked slowly for a few more moments as the feeling got stronger and slid beneath my skin to my muscles, my bones, then each organ was infused with it; my blood took on its pattern until it settled into my bone marrow. I was utterly taken over by it, possessed by it. I stopped my horse. I was taken over by the potency of the feeling so much so it disabled me from movement. A sweet, unfamiliar exotic aroma filled my nostrils briefly. I became acutely aware of a presence near me. Slowly, and with effort, I turned in each direction fully expecting to see someone standing near me. I was surprised to see no one. I sat on my horse, still, and feeling this curious, unsettling phenomena.\n\nFrom somewhere between my head and the space around me, not entirely inside me and not entirely outside me, I heard a voice. It wasn't my voice I heard; it was not any voice I recognized. It was strong and soothing, and in it I felt a deep sense of reverence and deep, abiding love. Like the sun on my skin, a unique warmth soothed my every cell. The words \"I have given you all that you need\" floated like ripe pollen finding ground upon my fertile, virgin consciousness. There was no doubt at the time that I had just heard the voice of God. Even now, decades later, I know it was the voice of God I heard that day, but it was not Yahweh, the standing, upright God of Christianity. My understanding of God is not a self-limiting concept, nor does it exempt me from taking personal responsibility as I'm waiting for the rapture. It is the God of nature, of Spirit; the One who weaves the matrix of life.\n\nFor thirty-five years, those words germinated, grew, composted, grew anew as I worked out various and possible meanings of it. I began initially to look outside myself, attempting to define at first what it was I needed. I had food, clothing, shelter, water, a dog, a horse, a farm, a family, schooling. I had a spiritual hunger; I needed food for that. I used drugs and alcohol in excess, attempting to understand the meanings of it. Later, I sought out teachers and ecstatic experiences until a realization that rang a brilliant tone of truth began slowly to emerge in my understanding. My body is the vehicle that enables me to have this human, incarnate, magnificent experience of being alive on Earth. Without a body, there are no senses, no feelings to get you to take notice, no feelings to follow, no anchor for the invisibles to touch you in their voiceless manner. Keeney writes \"Your feelings must trigger the ropes to pull you in the right direction.\" Feelings are our genius, our unique sensing apparatus that we use to find our way and follow what feels good and joyful. Feelings helps us know when we are in the presence of the Divine, the numinous, and the ecstatic. When we have a \"funny\" feeling, it informs us that something someplace inside is out of harmony. Feelings help us navigate life and the world around us and inside us. Without feelings we would be, what would we be . . . robots . . . a ship at sea with no compass or rudder.\n\nThe body is the vehicle for spirit and soul to have experiences, for them to evolve through a collective, intimate dance. Keleman says, \"A person is not in or out of _touch_ with his body. He _is_ his body. We need to get rid of this crazy idea, 'I have a body.' It's the other way around. This is a fact we may not want to swallow, but the head is not the chief cook and bottle washer; the whole body is.\"\n\nOnce you feel, and you must be in your body to feel, you are no body but yourself. No one can tell you what is truth and what is not; you feel it in your body. You feel truth in your body. Your body does not lie. It cannot lie. It doesn't know how to lie. And once you feel, behaviors are then invented out of imagination as a result of the feeling rather than an imitation or some archaic learned performance.\n\n _In my case Pilgrim's Progress consisted in my having to climb down a thousand ladders until I could reach out my hand to the little clod of earth that I am._\n\nCARL JUNG\n\nTHE SENSING BODY\n\n _Sex is one of the nine reasons for reincarnation . . . the other eight don't count._\n\nHENRY MILLER,  \n _B IG SUR AND ORANGES OF HIERONYMUS BOSCH_\n\nWhen I feel pain or stress, I feel it; I don't think pain or stress. Because I want to know and understand things, I trace the body feeling back to the first intimation of it, when the pain or stress were just hanging out on the periphery of my energy field before they became so engrained in my physical state, constricting muscles and causing one of my too-frequent tension headaches. I'm able to trace it back to a teeny, tiny, whispering voice and the beginnings of tension, which I recognize as irritation or a nagging feeling. Irritation begins like a breeze ruffling the surface of a still pond; it's just enough in the early stages to get me to pay attention, to notice that something is amiss and I need to tend to it, discover what it is and whether to do something about it or not.\n\nOne thing I hate most in my life is when my body is uncomfortable or restricted in some way, so I pay more attention to the early warning signs, those psychic pinches that, if unattended to, can throw me off my game. Paying attention to the subtlest messages sensitizes my nervous system to what and from where something is incoming, or if something is coming from my unconscious to the surface.\n\nWhen the source of the feeling is coming from outside me, I feel it first in my body as the energy of someone's anger, or their fear touches on the electromagnetic field of my heart and causes an almost undetectable shudder as it continues on and penetrates my physical body. As fast as the speed of light, a psychological part of me feels it. If it's fear that is coming in, a deep part of me gets scared; I hold my breath and take care not to make sudden or large movements until whatever it is has passed or an older part of me takes care to reduce or redirect its impact one way or another.\n\nWe know when we are under stress because we feel it; our muscles tense up and get rigid. We know we are sick because we don't feel well. We buy just the right avocados and pineapples and melons by feeling their ripeness and sensing whether they are happy and alive. We choose activities and experiences that make us feel happy and good. We avoid as much as possible experiences that don't feel good. And it's the feeling of \"doesn't feel good\" that alerts us to look for something different to do.\n\nWe enjoy a cocktail or a glass of wine or a cold beer at the end of the day to unwind and relax after a stressful day at work. Sex feels good; it releases tension and stress. We feel close to another person, are held and loved, and often the afterglow can keep us going into the next day. We are hardwired to seek out what feels good, to repeatedly put ourselves in the company of good feelings.\n\nConditioning out of feeling begins early in life. Each time we tell a child that what she wants is bad for her, will rot her teeth and her mind, we are inserting messages that tell the child she doesn't know what's good for her, that her wants and needs are not trustworthy. When we dictate mealtimes rather than feeding on demand, we are sending messages that the feeling body is untrustworthy and not sophisticated enough to know what is best. Scheduled meals send messages that say natural hunger instincts are not important and they disrupt the orderly functioning of the family. We feel one thing in our body and hear contrary words coming to us from outside. Distrust of innate feelings, needs, and hungers are programmed into us by family, teachers, ministers, and friends who tell us \"Don't be sad,\" \"Don't be angry,\" \"It's not time to eat yet,\" \"You'll spoil your appetite,\" \"Eat all your food,\" \"Don't climb that tree,\" \"Get over it,\" \"Snap out of it,\" \"Masturbation is bad,\" \"Don't eat dirt.\"\n\n _Our education from the start has taught us a certain range of emotions, what to feel and what not to feel, and how to feel the feelings we allow ourselves to feel. All the rest is just non-existent._\n\nD. H. LAWRENCE,  \n\"A PROPOS OF _L ADY CHATTERLEY'S LOVER_\"\n\nFrom the first day of school, children are taught to sit still. The message is don't move, don't be noticed, don't feel (for if I feel, I'll have to move). We assume that children sitting still cause less disturbance in the classroom. But, there is more going on. In _Your Body Speaks Its Mind,_ Keleman says, \". . . the stilling of the children's bodies induces individual patterns of alertness which enhance and sustain one another. The energy of each child's alertness can then be directed into forming the role of the ideal student.\"\n\nWe have a spiritual imperative and responsibility to encourage ourselves and our children to feel, to trust in the innate wisdom of the body. Giving children time outdoors, in backyards, and in the forests around their homes, encourages self-confidence and trust in their growing intuition and natural instincts\u2014things that cannot be learned in books or classrooms. We owe it to them to foster their innate instinct to bond with Earth in the most natural way inherent in them\u2014play. They love finding secret spots or building forts where they are free to explore, imagine, and dream. They need contact with the real world around them and in their neighborhoods.\n\nChildren are naturally empathic with animals until some adult tells them that animals are dirty or that they have fleas and mites. We pass on our own fears of the outdoors, of ticks, Lyme disease, snakes, and falling from trees. By the time children reach six years of age and are in public school, they are afraid of their own shadows and of a little dirt. Trust in their instinctual movements toward wilderness, trust in their hungers and intuition are put in the bag of shadow.\n\nWe spend on average six hours a day watching television or in front of a computer screen, on our iPods, with MP3 players, or with cell phones clamped to our heads like skull implants. We have instant messaging, Skype messaging, text messaging, and social networks such as Facebook that give us the illusion of having deep, meaningful relationships. We no longer understand the distinction between friend and acquaintance or have real communication, real dialogue, as we become ever more superficial, our lives speeding up, hurling us toward death. Internet social networks give the illusion of intimacy while we reduce ourselves to characters and caricatures. We become a data stream, links of superficial sound and data bites, like medical intake forms that tell a physician very little about the human being filling out the form.\n\nAfter having thousands of images flit across our computer screens every day at an unimaginable speed, we then turn off the screens at some hour of the night for sleep and find ourselves in the midst of relative silence and a field of vision with no moving images. Everything suddenly is s l o w e d w a y d o w n. Then what do we do with ourselves? How do we sleep surrounded by silence? How do we sleep when our brain is processing all those images and information, our immune systems compromised by the nonstop bombardment of radiation from computer screens?\n\nOur values and moral compasses were once formed from living in direct contact with the natural world and the world of invisibles and in tight-knit communities of cooperation and preservation. We understood the intrinsic, delicate balance of living in right relationship with all our relatives, not just upright two-leggeds but every created thing we could see and feel and sense and know.\n\nWe've created a world where our understanding of and relationship to the natural world is becoming squeezed into cubes and bites of information we can access from our homes, Blackberries, or iPads as we run from one mechanical place to the other. Nearly all we do is on the run; we eat on the run, talk on the run, try to keep relationships going on the run, have sex on the run. Slowing down is frightening, and we're not even certain we know how to do it any longer. Jerry Mander says, \"We hear people say that nature is boring, and it is clear why they say this. We don't know how to be with it. We are not slow enough.\"\n\nWe seem unable to accept that we are part of Nature. If we could accept that fact, it would change our fundamental belief about ourselves; we'd realize that we are not special. We're simply an expression of Gaia, one species among approximately thirty million.\n\nBeing of the animal kingdom, human beings are not made to live on a diet of megabytes and gigabytes and artificial frequencies. We are at our best, calmest, and most creative, influential, self-confident, balanced, and healthy when we have regular time walking, sitting, hiking, and being in nature, especially wild ecosystems. Our brains and bodies evolved in the dynamic field of the natural world; it's there that we function at our optimum best physically, psychologically, intuitively, and spiritually. Our senses come alive when we are in natural and wild ecosystems; we are immersed in unusual sounds, interesting smells, and our bodies move differently across unpaved and uneven landscapes.\n\nDenying our feelings, turning away from them because they are too painful or discomforting or we don't know what to do with them, desensitizes us to subtleties in feeling and the meanings they contain. We've been so conditioned to not feel. Lacking awareness of our feelings and letting them atrophy grows a kind of callus on our sensing body. If you've ever learned to play guitar, you know the experience of watching your fingertips get thick with layers of skin. Or have you ever had the experience of having Super Glue dry on your fingertips? Being taught at a young age that our feelings are not to be trusted or acted upon desensitizes our psyche in the same way that getting a little Super Glue on our fingertips dulls our sense of touch. Until the glue wears off, there is a numbness, a desensitization wherever it has dried. Feeling objects with those glue-muffled fingertips produces an odd sensation; feeling is there, but it is dull and distant, almost like a memory. A similar phenomenon occurs with our feelings over time. You sense something is there, but you can't quite put your finger on it. Emotions, which are not the same as feelings, get dumbed down as well, so much so that it takes focus of will to say what you feel. \"I think . . . I feel . . . kinda . . . um . . . don't know . . . sort of . . . like, ah . . . oh, sad . . . I think I feel kind of sad.\"\n\nWhen I work with clients and students, I often begin by getting them into their bodies. I had a student who was very intelligent and articulate, and when she talked, she was very animated. Her arms flew here and there, her eyes were often cast upward, and her chin tilted slightly upward as well. Her voice was high in her throat near her sinuses, and the authority of anything she had to say went drifting up and out of the room like a puff of smoke. She had interesting insights and valuable experiences to share, but when she talked from her head, it was difficult to take her seriously. She wasn't home to own her authenticity, her body of experiences.\n\nI had her feel into her feet, without looking at them and then asked her to talk from her diaphragm. Her energy immediately and visibly shifted; her shoulders relaxed, her chin dropped, her cheeks flushed. She looked more present because she was more present; she was grounded in her body. Then I had her breathe through her heart and bring her breath all the way into her diaphragm. Her eyes grew a little larger and softer, and suddenly there was a beautiful, wise, mature ego state looking out of her eyes. She let go a long, deep sigh. When she spoke from that place, her voice came from deep in her belly (in the belly of Earth), as if the Earth herself were speaking. Through body presence, she brought an authority and aliveness to her life force and into the room. And when she spoke from that place, I noticed my own body shift in response to the power that was now there in the room. I felt her authority and the truth she was speaking. Once a person speaks from inside her own body, from her diaphragm, from her loins, there is sexual power attached to it that we call charisma. That is sexy.\n\nOur sexual health and expression are woven into our hearts and our souls and are anchored through gravity and attraction in our body; the interplay of tension between Eros and Psyche. They touch and are touched by Earth and all beings who share this planet with us. To know we are touching and being touched, we must feel and be at home in our bodies.\n\nOur bodies, our brains, every part of us was designed to be impressed upon by the natural world. We need immersion in the world of nature, the world of invisibles where _spiritus mundi_ is easily and readily accessible. To come back to our sensing bodies, we need to return to the body the source of natural impressions, to have our souls imprinted with the visions of seeing butterflies alight on flowers in bloom, hear birdsong, drink from the deep well of direct knowledge that is found only in wild and natural ecosystems. But not just imprinted, our souls are taught, reshaped out of linear programming and technological programming into something more.\n\nSimply living does not equate with being alive. In Hawaii there is a word for the energy or life force that activates all living beings and the elements and is true vitality\u2014 _mana._ Describing this force, Hawaiian musician Israel Kamakawiwo'ole says, \"Mana is like an energy that you get. We believe we get ours from the elements first, the Earth, your sky, your ocean, your God, and all that is inside us. And when we open our mouth to speak, to sing, to play, that's what we let out.\"\n\n _Freud and his psychoanalytic descendants are no doubt correct in their assessment that the search for ideal love\u2014for that one perfect soul mate\u2014is the futile wish of not-fully-developed selves. But it also seems true that the longing for a profound, all-consuming erotic connection (and the heightened state of awareness that goes with it) is in our very wiring. The yearning for fulfillment through love seems to be to our psychic structure what food and water are to our cells._\n\nBARBARA GRAHAM,  \n\"THE FUTURE OF LOVE,\" _U TNE READER_\n\nThe absence of experiential processes reinforces feelings of separation. It is not just another human being that we long to feel the touch of intimacy with. There is longing to feel a deep, innate, natural connection to the living Earth and all that we share our life with. There is a primordial need built into the human species for fenceless open country, for the musky, erotic smells of the forest floor, for the sounds of birds and insects, and for the sight of expansive vistas. In ancient and indigenous cultures, there was no separation, no divide between human beings and Earth. All was sacred, alive, intelligent, aware, and sensual. Plants, trees, stones, and the elements were part of daily relationships for food, medicine, and wisdom. Foragers of the Kalahari and aboriginal tribes in Australia are intimately connected to and have a deep sense of and relationship with the places they inhabit and hunt. They use song lines to connect and bind them to place and to nonhuman beings. As David Abram reports in the _Spell of the Sensuous,_ \"Language is inseparable from song and story, and the songs and stories, in turn, are inseparable from the shapes and features of the land. The chanting of any part of a song cycle links the human singer to one of the animals or plants or powers within the landscape, to Crocodile Man or Pandanus Tree Woman or Thunderstorm Man\u2014to whatever more-than-human being first chanted those verses as he or she wandered across the dreaming Earth. But it also binds the human singer to the land itself, to the specific hills, rocks, and streambeds that are the visible correlate of those sung stanzas.\"\n\nFar from our Western orientation of anthropocentrism, indigenous and aboriginal peoples tend more toward _Gaiacentrism,_ if we must put a \"centrism\" label on the human\/nonhuman relationship of those cultures. It is our anthropocentrism, the political philosophical mind-set that human beings are the center of all things, that is at the center of the separation. Human exceptionalism has various meanings in different contexts. To the ancient Athenians, the ability to reason distinguished humans from the natural world. For Christians, it has been spirit\u2014the human spirit, the Holy Spirit, and the spiritual (Christian) life.\n\nIn modern times, human beings have created a Grand Canyon\u2013size physical, emotional, and spiritual chasm that is growing deeper and taking us farther away from our bodies and Earth as holy places. Reclaiming the soul, our inherent, intuitive, instinctual nature, opens the door to remembering and reclaiming the body of Earth and our bodies as sacred, alive, aware, intelligent, sexual, and caring. The gravity of humility pulls us below superficiality, shedding our robes of grandiosity and anthropocentrism. Viewing ourselves as a superior species, for which we see the results of as we look around, has led us to the edge of the shaky precipice of becoming an endangered species. Dorion Sagan and Lynn Margulis postulate in _Dazzle Gradually_ that the human mind is not as unique or special as we like to believe.\n\nPerhaps the greatest psychological stumbling block in the way of widespread scholarly acceptance of Gaia is the implicit shadow of doubt it throws over the concept of the uniqueness of humanity in nature. Gaia denies the sanctity of human attributes. If intricate planning, for instance can be mimicked by cunning arrays of sub-visible entities, what is so special about Homo sapiens and our most prized congenital possession, the human intellect? The Gaian answer to this is probably that nothing is so very special about the human species or mind. Indeed, recent research points suggestively to the possibility that the physical attributes and capacities of the brain may be a special case of symbiosis among modified bacteria.\n\nFocusing on the deep ecology of Earth and our own deep ecology begins to uproot archaic beliefs and values that prevent us from living an inhabited, authentic life. Working directly with the intelligence of plants, with all the intelligence and graceful beauty of our bodies and through ceremonies, we begin to heal and reclaim the sacredness of Earth, our earthly sexuality, and the bond that has been severed.\n\nHow do we go about reclaiming an intimate connection with the wild Earth, with a stone, a tree, or the animals that live with us? We begin by understanding that connection is a feeling thing, not a thinking thing. Secondly, the mind cannot feel; the mind analyzes and categorizes data, and it interprets feelings. Our feeling sense is the instrument of true knowing. Whether we are creating intimacy with our lover, our children, our friends, or other animals or beings, intimacy begins by teaching yourself to see, to perceive, and to feel as we each did naturally as infants and young children.\n**7**\n\n ** _Choosing Another Way_**\n\n_The earth does not belong to man; man belongs to the earth. All things are connected like the blood which unites one family. Man does not weave the web of life; he is merely a strand in it. Whatever he does to the web, he does to himself._\n\nCHIEF SEATTLE,  \nLETTER TO ALL, 1854\n\n _The language of the body is the key that can unlock the soul._\n\nKONSTANTIN STANISLAVSKY\n\n _If you understand what is being said and live it then you will be in a totally different world. But if you don't live it, daily, then you will just be living as you are. That's all._\n\nJ. KRISHNAMURTI\n\nNone of us ever truly forgets the nurturing, warm comfort of the womb. We spend our lives trying to re-create that feeling of being held and protected from real or imagined demons and threats. We yearn to be fed all the food that sustains our three bodies. Nor do we forget the pain of individuation, of leaving Mother and home, peers, and social circles. Individuation causes a pain of separation, and we seek throughout our lives to have the separation and the resulting loneliness filled with meaningful relationships. We seek to fill our need to have purpose and significance in our work and meaning in our lives. If we succeed in finding those things, they hold little meaning without someone to share them with.\n\nThis desire for intimacy with lovers and friends, to feel that we are not alone in the Universe, is innate in each of us. I do still feel lonely at times, but never alone. If I feel alone, it's simply because I've shut myself down and have not called on the powers of the Universe. They never abandon us; it's we who take ourselves away from them.\n\nThere is a cost, however, in choosing to do and be something different. A willingness to give up whatever the gods ask of you is integral to having a new life. Sometimes their asking is a demand.\n\nFriends may drop away, go away emotionally, or start a fight to create distance when the fingers of intimacy reach beyond their comfort level. Maybe you feel a need to move, one that painfully takes you away from children and grandchildren, the geography that shaped you during your formative years, and all that is familiar.\n\nOne of my lovers told me he was afraid of me. Another said that he didn't know what to do with me. Those statements feel funny, don't they? They meant that I was different; I wanted things they did not. They sensed a thing in me that set me apart from them, and it was that thing that I followed even as I left the relationships and moved 1,100 miles away from my granddaughters and the Upper Mississippi River Valley. Regrettably, at the time I was still young in my learning and becoming aware. I did not know what to do with those statements. I felt alienated and alone in my need. I didn't have the skills then to talk about it, to open up a dialogue to begin to bring the other into my world. What I was certain of then, and still am, is that they did not want the discomfort that depth of intimacy brings. One of my lovers had a habit of losing his cell phone when intimacy began to stretch beyond his comfort level. I had my own habit of distancing: I'd start a fight. My family fought; it's what I know how to do. It's what I did when I was afraid. Fights release pressure that's been building from too much or too little intimacy, not getting our needs met. The aftermath creates space and time. Berne named this game uproar. It's a dance of getting close, fighting, creating distance, getting closer, fighting. Nothing was ever resolved. We didn't have the skill or willingness to analyze what had happened, why it happened, or what to do differently to get to the next level, to deepen our relationship. We painfully maintained the status quo. I painfully left the relationships.\n\nThese relationships could have grown, become more intimate, if I had been able to say: \"When you lose your cell phone I feel abandoned and hurt. I get afraid that you will go away forever, so I do what I know how to do and that is argue.\" He could have said: \"I've always run away, that's why I chose a career that kept me moving for long periods of time.\" Or, \"You're really strong in yourself and I get afraid that I am going to be a disappointment to you so I go away.\" Willingness to self-examine to root out the source of the behavior and willingness to apologize, and mean it, are vital to a healthy relationship.\n\nIt is important to know what you need in a relationship. Is the person able to give you those things, meet your needs? Is he able and willing to go to the places you need him to go to? If not, you're headed for a heartbreak or drama or both. Are you willing to compromise, take what you can get, and still maintain integrity with yourself?\n\nFilling a role for another person that is not the truth of you betrays who you are, your authentic self, and creates a wound in the child of you. The wound that is created takes significant work to mend. Your child self feels you have abandoned her needs, put her back in a bag, and sent the message that it is not acceptable for her to be out in the world. The trust the two of you had will be ruined, and that is difficult to repair. It can be mended over time, when she sees that you will fight for her right and need to be alive and have a life with you. This is one of the basic, most necessary aspects to healing old wounds. Apologize, negotiate, and hear her voice, listen to her needs and wants. Trust can be mended.\n\nWhen it comes to the wholeness of the relationship or the wholeness of self, fight for wholeness of self first. You are your primary relationship, and wholeness in yourself is your primary responsibility. You cannot have a healthy relationship with other people or any aspect of Gaia if you have no internal integrity. The need for rigorous self-examination is of utmost importance. Being honorable with yourself leads to becoming honorable in other relationships. Intimacy begins with being intimate with yourself, all parts of you, and, most especially, your deep self. Self-examination\u2014being attentive to the voices inside, articulating your needs, finding ways to get them met, speaking on behalf of your child\u2014is ecological reclamation of the soul. This work is, in reality, doing your own soul retrieval. It's simply accessing and integrating parts of you that have been marginalized and put, as Robert Bly writes, \"in the long bag of shadow.\" The work is to reclaim the child without destroying the child. Thomas Patrick Malone writes about the importance of the unconscious in establishing intimacy.\n\n\"Unconscious\" is not really a useful or descriptive word. It simply means \"that which is not conscious,\" which really tells us nothing about what it _is._ The negative term belies the power of the concept. Perhaps \"unconscious\" should be renamed the \"intimator,\" the natural connector, the natural spirit, the nexus, or perhaps, in Paul Tillich's phrase, the \"ground substance.\" Whatever it is, it is a universal natural constant, not an absence of something else. Movement into the unconscious while you are awake is intimacy. Without the unconscious, we cannot have intimacy, only knowledge. Only in the intimate, unconscious experience can we know our true ecology, this is the intimate self.\n\nHEALING OURSELVES, HEALING EARTH\n\nAfter I left therapy and counseling, I became captivated with indigenous teachings at about the same time I began working more fully with plant medicines. Something powerful tugged at my heart and mind, something familiar and resonant. I went from one workshop and intensive to another, gathering pieces that worked for me, inspired me, and fueled me until the next one. But always I had a feeling that something important and integral to where I wanted to go was missing. Finally, I heard a teacher speak about the spirit of plants and how to work with them beyond the physical. I wrote to him, and the following year I flew out to spend a weeklong intensive. It was my introduction to the work that has shaped my life and, eventually, this book.\n\nDuring that week, the six of us were introduced to the exercises presented in the following pages. I hated the exercises at first and didn't really believe in them. The last thing I wanted to do was fucking inner child work. I had gotten so sick of hearing about \"my inner child\" from my sister I wanted to projectile vomit on her if I heard it one more time. I went into the work kicking and screaming. But like other therapies and workshops, my bottom line was, \"Well, hell, I've traveled all this distance and paid all this money, I'm going to get my money's worth.\" I would take all the teachings inside me and do what I could with them later. If they stuck and felt right, I'd keep working with them to more or less degrees. If not, they got shit canned.\n\nHere I was, in a small room with just five other people, strangers at that. Intimate? Hell, yes. It was uncomfortable, vulnerable, and emotional. I did the exercises, spent the week giving my all to it, and flew home. I continued to do the work half-heartedly for a few years until I began to see and experience the value of it and the changes I was making\u2014deep, lasting changes. My intuition and senses became sharper, my thinking quicker. My ability to analyze interactions became more elegant. I felt more confident, self-assured, and aware. After some years, it went from being second nature to first; in a healthy family environment, it could have been first to begin with. I can't imagine doing anything different again. It would be impossible to do anything different.\n\nI teach it to my students and apprentices and some clients. It's astonishing to see the changes it makes in people, instantaneously. The goal is to integrate it, let the work change you, take you on, shape the truth of who you are, and bring you into sexual loving. Ultimately, you become the author of your life through this practice. And it does take practice.\n\nMuch of our current dysfunction, lassitude, and discomposure is the result of and can be attributed to the loss of our connection with Nature, with old-growth forests, wild streams, and mountains. And it is the result of the loss of our connection with our own nature, our sexual nature, and our bodies\u2014that sensate, sensing organ of perception. Will Johnson talks about this when he says, \"By losing touch with the energies and sensations inside our bodies, we've severed our connection to the greater world of nature of which we're so intimately a part. The domain of union that spiritual seekers strive to contact is not some kind of exotic or esoteric condition. It is our simplest and most natural state; but we need to be here, fully in our bodies, and we need to heal our alienation from nature, both within and without, in order to experience it.\n\nThe more we take ourselves out of the ecosystem, distance ourselves from Nature, and from our own nature, the more unstable we and our culture become. If some part is cut off, or alienated from the whole, it is no longer whole. When our sexuality is cut off from the body of Earth sexuality, we are unable to function as whole beings; one part does not contain all the parts of the whole. Earth is unable to function and produce as a healthy organism because our relationship is dangerously out of balance. If we are going to live on Earth, then shall we be alive and _live_ on Earth? As long as we live on Earth, we need to be whole; a complete, fully functioning whole.\n\nMany people and environmental groups insist Earth needs our help in healing. What Earth needs from us upright two-leggeds, if anything, is for us to be whole, to own every part of what it means to be part of the biota. That means the feeling aspect of the human species. And sex is a feeling thing. Sexuality, sensuousness, erotica are feeling things; when we allow ourselves to feel, we feel more alive, ensouled. It is a cruel disconnect to deny our innate feeling function. We are sensate beings. We cannot live healthfully denying our feelings.\n\nIf human beings didn't exist on Earth, all species, all life on Earth would do remarkably well without us. How would we do without other species? Where would we get our inspiration, our medicines? Where would we get our air? Our food? Our houses? We could not live at all.\n\nFeelings are not socially constructed; that is, we were feeling the world before we began to think. While we were in our mother's womb, we could feel if we were loved and wanted, or not. Thinking our way through life without feeling cuts off part of our innate intelligence. Our imaginations and feelings have been stifled and subdued. Let's bring them out into the world, shall we? Set them free in the world, let their fingers reach out and touch the other, out there, and watch what adventures rise up to meet you.\n\n _If I spent enough time with the tiniest creature\u2014even a caterpillar\u2014  \nI would never have to prepare a sermon. So full of God is every creature._\n\nMEISTER ECKHART\n\nRECLAIMING THE BODY'S SENSING: THE HEART OF THE MATTER\n\nMost of us are more or less aware of invisible energy fields. We feel it most when someone walks up to meet us. There comes a certain point at which, as he gets closer in physical proximity, you begin to get a little uncomfortable, when you feel he is now \"in your space.\" You feel a little invaded; he is too close, and it's a discomforting feeling. You may even pull back a bit, stepping out of and away from that space. Often when someone, especially someone we don't know, gets too close, we stop breathing a bit. Once a \"safe\" distance has been reestablished, we can breathe normally again.\n\nThe ancients knew that the organ of perception is the heart. It is through the heart that we are connected to things by means of the senses. Apprehending images is the role of the heart. James Hillman says, \"But the heart's way of perceiving is both a sensing and an imagining: to sense penetratingly we must imagine, and to imagine accurately we must sense.\"\n\nWith each beating of our heart, electrical and magnetic energy is created, and it radiates from our body as the electromagnetic field of the heart. There has been much scientific evidence of this over the past fifteen years. Stephen Buhner writes that the \"electromagnetic field that the heart produces is some five thousand times more powerful than that created by the brain.\" The field of energy that is emitted is measurable with the use of magnetic field meters and is strongest in most people at the surface of the body to about eighteen inches from the body. Even the most sensitive electromagnetic measuring instruments can still detect the field up to ten feet from the body. Being an electromagnetic field, there is no way to determine just how far the heart's electromagnetic field actually extends or whether there are limits to it.\n\nEverything emits a field of energy. What that means is that we are literally swimming in fields of electromagnetic energy. These fields of energy are transmitting meanings to us via our own heart field. The heart perceives the transmission and immediately images appear on the screen of our vision. The soul of the world, the soul of our lover, is not perceived if we are unconscious to our heart as a sensing organ. Keeping the heart in the reductionistic model as a pumping mechanism dooms us to a life of unconsciousness. It's important to begin to work consciously with the energy fields and, more importantly, with the heart as an organ of perception and imagining.\n\nWhen we are in the depths of sacred sex and sexual loving, we are in an ocean of senses, images, and meanings. These images and meanings are communications that flow between two lovers, between human beings and Earth. Because everything in Nature gives off electromagnetic frequencies and Nature is having sex all the time, the sexual energy of Earth is part of the sea of communications, the linguistic medium and meanings we live in every day.\n\nLet's begin to play and work with the heart field that each of us possesses.\n\n ** EXERCISE   \nExpanding the Heart Field**\n\nGet comfortable in a cozy chair, feet on the floor, and take some deep, relaxing breaths. Now, bring your awareness into the area of your heart. Let your breathing move into your heart, as if you are breathing through your heart. In the beginning, it may feel a little tingly in that area, which is a good indication that you're there. If you don't feel it the first time, don't panic, just keep breathing through your heart. You're using a new sensing awareness, one that has been dulled through a lifetime of atrophy. When you feel that you can hold your attention in your heart, stand up and notice how you feel. Do you feel more present, grounded in yourself? Notice that your peripheral vision is expanded; that is, as you look straight ahead, you can see things to your far left and far right. With practice, you'll be able to feel your awareness move into different parts of your body.\n\nTake a walk outside and find a tree that captures your attention. Sit down beside it and take a few deep breaths, relaxing more fully with each one. Let your eyes become soft focused on the tree you've chosen. Notice everything about it: its colors, shapes, textures. Now, ask yourself, \"How does it feel?\" What's the first thing that comes to you? Where do you feel it in your body? Are there any sensations, images, words that accompany the feeling? Sit with the feeling for a moment.\n\nStand up and walk away from the tree about twenty feet. It's helpful to pull your shoulders back and hold your arms straight out at your sides. When you do this, you may feel vulnerable and exposed. You are, that's why many of us become habituated to rounding our shoulders and slouching so we feel that our chests, our hearts, are protected. It's also a way to feel less noticeable. Pulling your shoulders back and up really opens up the heart and lungs and can release emotions we've held there. As you hold your shoulders back, with your heart open, slowly begin to walk toward the tree with your heart field extended. Pay particular attention to the moment your heart field and the \"heart field\" of the tree touch. This is nonphysical kinesthetic touching; touching with the invisible, nonphysical field of the heart. Pause there for a moment and notice how you feel to be touched this way. Continue on toward the tree, noting if the field grows in potency as you move closer to it.\n\nNow, ask your Child to come and sit with you. As you keep your eyes soft focused, ask your Child to tell you everything about this tree. She is the one who has direct access to everything she comes in contact with and to the world trees live in. She knows how to talk with them. Notice, as well, any feelings or emotions in your body. How do you feel doing this?\n\nNext, ask your Infant to be with you and hold her in your arms or on your lap. Infants have no language, but they perceive the world directly. Notice everything your Infant does. If you need to, ask your Child if she would translate for you. Children often translate for preverbal infants; they understand that language. Whatever movements, faces, or changes in color the Infant makes are all communications to you about the tree and its medicine or teachings or power.\n\nIt's helpful to keep a journal of your experiences, your \"readings.\" Just when you think you're not becoming more sensitive, you can compare earlier entries and see how much more keen your perceptions are getting.\n\nYou can do this with anything, but trees are so generous, have potent energy fields, and love to play this way. You may find if you do this with trees of various ages, saplings to very old trees, and of different species, that each has a very different feeling to it. Everything in Nature has it own intelligence, sexuality, and personality, and each moves through ego states just as human beings do. Some trees are sexier than others, just like humans.\n\nThere will come a time when you suddenly realize that this way of feeling and seeing has become a part of how you move in the world. You no longer have to think about it; it just is, you are. Occasionally you'll notice that you are moving too fast, getting wrapped up in and consumed by some mundane event or deadline, and a soft voice inside will whisper to you or somehow make noise or a movement to remind you that you know how to feel better, different, more present, and you'll just make the shift then, in that moment, to breathing through your heart. Once you do that, you'll be in your body again, aware of your feet and surroundings, hearing birdsong and wind.\n\nIn the beginning, you will want to stop soon after you start as it can be unsettling. It just takes practice. You are using sight and sensory feeling that has been malnourished. It's never been completely gone you know. We use this every time we walk into a room or walk in the forest or talk with a friend on the phone. There are immediate psychic \"hits\" we get, but we have learned to ignore them, to let them stay below conscious awareness. With practice and patience with yourself, you will be able to understand just what those hits mean. You'll begin to decipher the meanings in them.\n\n _I am the wind that breathes upon the sea,_\n\n _I am the wave on the ocean,_\n\n _I am the murmur of leaves rustling,_\n\n _I am the rays of the sun,_\n\n _I am the beam of the moon and the stars,_\n\n _I am the power of the trees growing,_\n\n _I am the bud breaking into blossom,_\n\n _I am the movement of the salmon swimming,_\n\n _I am the courage of the wild boar fighting,_\n\n _I am the speed of the stag running,_\n\n _I am the speed of the stag running,_\n\n _I am the strength of the ox pulling the plough,_\n\n _I am the size of the mighty oak,_\n\n _And I am the thoughts of all people_\n\n _Who praise my beauty and grace._\n\nTHE BLACK BOOK OF CARMARTHEN\n\nEXPERIENCING THE INTERWORLD\n\nYou will be astonished to find what happens when you comingle your attention, seeing, and feeling with another living thing in nature. Repeatedly, I feel the thing I'm gazing at in this way responding. They are as equally responsive to this touch, to this silent communication, as we are hungry for it. It is their adventure too, you know. They need us to be part of their life experience. When I sit with a plant and ask it for its body to make medicine and then administer the medicine to myself or to a client, the plants are able to do their work in the world. We have an obligation to do this work, to reclaim our place in the scheme of the natural, wild Earth and the place we call home.\n\nOne fine day, without expectation, something from nature touches you back. You feel a \"hi\" as you walk by the plant. The blue boulders shaped and softened by water spiraling through the canyon touch you like a lover and ask you to lay naked on their cool bodies. Walking through my house, I feel the plants communicating their thirst and hunger, that it's time for watering and feeding.\n\nYou'll be walking along, and suddenly a tree will reach out and touch you. Maybe it's not even a tree in your field of vision, but you know that what is touching you is a tree. As you are touched by the invisible energy of that tree, a rapid series of things happens without you being aware of it. Some invisible part of the tree touches an invisible part of you, and a memory, recognition, and perhaps even a picture is formed in your mind's eye, and you see the tree you've sat with and talked to and wrapped your arms around. All of the feelings and experiences of being with that tree flood your entire being at lightning speed.\n\nOne day, you decide to take an alternate route that bypasses the tree by a hill and dale. You feel the touch of something, and you know in your most earnest self that it is that tree, not just any tree in the park. And something inside switches on automatically, a subconscious decision is made, as if some large hand reached out from the Universe to pivot your body, and you turn, change your direction and plans, and make tracks to that tree. The touch grows in strength and the pull on you stronger, more insistent, and you pick up your pace, and ah, there she is, a magnificent, large, old red oak. As you approach the tree, you feel her move inside you. Her strength touches a primordial and infantile part of you, and tears of reunion and more fall out the corners of your eyes for that love is touching places inside you that need this medicine, medicine you didn't know you needed until it finds a resting place inside, and it's there inside you now because you've been sitting with this tree, getting to know it. Unbeknownst to you, it was getting to know you, see you, all of you, what you have, who you are, and what you need to be whole or to do your work in the world. And for the love of sharing this way, of going to the tree with childlike curiosity, a bond of intimacy was formed, and the tree reciprocated by touching you back. As you feel this awe and wonder overtake you, you say thank you, and you raise your face to the sky and say, \"Yes, I want more of this in my life.\" You affirm the experience and anchor it in this moment by saying yes and thank you out loud.\n\nThis is what an intimate meeting at the interworld between the human world and the world of trees feels like. I was touched by something invisible, yet palpable, and of the ground I stood on beneath that great, sheltering canopy. The mythic power of trees shows up in cultures the world over; the Bodhi Tree, the Tree of Life, the tree that connects the underworld of legend to the upper world of the gods. And we humans are in the middle of those worlds, as they pass through us, calling us to take our place in holding the balance between worlds, to hold the balance of human and nonhuman, between caretaking and dominion. We are not smart enough to walk this world without the intermediary help of those who abide in the other worlds. We pass by, casually taking in the scenery, but do we stop to acknowledge the ancient voices whispering to us as leaves rustle on a wind-less day?\n\nAs Michael Perlman says, \"The trees are psychological presences that move the soul\u2014sometimes to right them, sometimes to cut them down, always to somehow imagine them. So it is misleading to speak of 'animism,' or the projection of human qualities onto trees. To speak of 'powerful trees' means addressing the ways in which trees animate the human.\" Nature is ensouled, filled with an aliveness, intelligence, and consciousness. That knowledge has never left us; it's just been buried deep in our subconscious under layers of false information and schooling, religion, the work of putting food on the table, and unsupportive family scripts. We live in a culture that puts very little value on living an inhabited life. And the practice of seeing and feeling in this way is the root of living an inhabited life, a life of purpose directed by soul. It is the beginning of interbeingness, of mutually weaving your being with another. Doing this with parts of Nature starts the process of growing intercommunities. You are engaging in an ecological reclamation that our ancestors never forgot or buried. It is a simple wisdom addressing the heart and of recognizing our kinship with all of creation. And it is paramount to our survival as a species and to the survival of Earth as we know her.\n\nAccording to William Blake, the move to abandon the polytheistic animism of antiquity, what was at the time seen and experienced through the lens of spiritual perception, came about when a system was formed that \"enslaved the vulgar by attempting to realize or abstract the mental deities from their objects: thus began priesthood.\" In other words, Ralph Metzner extrapolates, \"the loss of direct perceptual communion with the spirits of nature was brought about by a political move\u2014the institution of priesthoods as intermediaries between the human and the divine.\"\n\nRight now, where you're sitting, look around you; what is the part of Earth that is nearest to you? Some people will get a confused look on their face and uncomfortably look around the room. Perhaps it's the clay pot on the bookshelf? The potted plant sitting on the windowsill? The wood stacked neatly near the fireplace? Maybe it's the beam in the ceiling? These are parts of Earth. But, did you touch your own skin, your own sweet face? That is the part of Mother Earth that is nearest to you. It doesn't get any closer or any more intimate. Touching our own skin and realizing that we are able to do so because we are Earth; we are the pulsing, moving, sweating, shitting, crying, laughing red-blooded, breath of air, fire of digestion and passion, and bones of long dead human, plant, and animal ancestors. Each of us is a microcosm of the macrocosm. Everything that makes up Earth makes up us; we are made of the same substances.\n\nWe are all made up of the soil, rocks, minerals, plants, air, water, and animals that share Earth with us. It's an intimate thing to take into our bodies the bones of our ancestors whose bodies have become part of Earth in the plants and animals we eat and the water we drink. Because of this, we have an evolutionary connection to everything that has been created.\n\nNow, touch your hand to your cheek and hold it there for a moment knowing that you are the result of billions of years of evolution. How do you feel?\n\nGaia, Earth, breathes, reproduces, composts all manner of waste\u2014physical, psychological, and emotional. We breathe oxygen created by plants' respiration. The water we drink sustains the water and electrical systems in our bodies. Going barefoot grounds us into our bodies. When we wear rubber soles on our feet, we are separated, insulated from the electromagnetic field of Earth. As soon as we kick off our shoes and walk barefoot on Earth, we are instantly grounded in ourselves and to Earth. It resets our circadian rhythms, mineralizes our bodies, and reduces inflammation in the lower extremities. The negative charge from Earth through our feet dissolves calcium buildup from positive charges in tumors, heart, kidneys, and gallbladder.\n\n _When we look at the world around us, we find that we are not thrown into chaos and randomness but are part of the great order, a grand symphony of life. Every molecule in our body was once a part of previous bodies\u2014living or nonliving\u2014and will be part of future bodies. In this sense, our body will not die but will live on, again and again, because life lives on. Moreover, we share not only life's molecules, but also its basic principles of organization with the rest of the living world. And since our mind, too, is embodied, our concepts and metaphors are embedded in the web of life together with our bodies and brains. Indeed, we belong to the universe, we are at home in it, and this experience of belonging can make our lives profoundly meaningful._\n\nFRITJOF CAPRA,  \n\"IS THERE ROOM FOR SPIRIT IN SCIENCE?\"\n\nWith each turning of the seasons, we innately feel it in our bodies. As spring begins its stirrings from deep in the heart of Earth, we feel a renewed sense of hope, our energies rise with each young plant that slowly emerges from the warming ground. As warm spring winds cleanse the air of winter, we tend to do our own spring-cleaning of our homes and bodies. We change our diets to eat lighter foods, such as fresh spring greens, and we may be reinspired to start walking or running again. When we feel the first hint of fall in the air, we gather wood, store food, seal windows, and weatherproof our homes. The hunter-gatherer still lives in our DNA; it knows how to live with each changing season. We are influenced deeply by the cycles of Earth, the moon, the sun, thunderstorms, winds, and wild places. Inextricably woven into our DNA is the DNA of Earth and all life-forms that have lived and died.\n\nThink how often we use images of nature to describe our sexual experiences, our orgasms? Try to describe these without metaphors from nature. It can be done, I suppose, but our natural inclination is to reach for the elements and raw elemental power to relate our experiences. Just think of Carole King's \"I feel the earth move under my feet,\" or Barry White's \"It may be winter outside (but in my heart it's spring).\"\n\n _People like you and I, though mortal of course like everyone else, do not grow old no matter how long we live . . . for we never cease to stand like curious children before the great mystery into which we were born._\n\nALBERT EINSTEIN\n\nConversations between our bodies, the invisible field of energy we emanate and are immersed in, and the things and people we surround ourselves with are happening all the time. It doesn't stop; there are just gaps between words, sentences. Even the gaps, the rests, the pauses are rich in meanings and feelings.\n\n _Men of earlier times do not as yet separate their own soul experience from the life of nature. They do not feel that they stand as a special entity beside nature. They experiencethemselves in nature as they experience lightning and thunder in it, the drifting of clouds, the course of the stars or the growth of plants. What moves man's hand on his own body, what places his foot on the ground and makes him walk, for the prehistoric man, belongs to the same sphere of world forces that also causes lightning, cloud formations and all other external events._\n\nRUDOLF STEINER,  \n _T HE RIDDLES OF PHILOSOPHY_\n\n _I have lost, as I have said, some sense of myself. I no longer require as much. And though I am hopeful of recovery, an adjustment as smooth as the way the river lies against the earth at this point, this is no longer the issue with me. I am more interested in this: from above, to a hawk, the bend must appear only natural and I for the moment inseparably a part, like salmon or a flower. I cannot say well enough how this single perception has dismantled my loneliness._\n\nBARRY HOLSTUN LOPEZ,  \n _R IVER NOTES_\n\n ** EXERCISE   \nHeart Field Practice**\n\nBring your breathing awareness to the area of your heart as you did earlier. Notice your peripheral vision. This immediately drops your sensory feeling into your heart, the organ of perception and communication. Physiologically, it is the fastest way to change your heart rate, breathing, and blood pressure, and it changes the cascade of stress hormones to healing hormones.\n\nChoose some object to look at. As much as possible, stop thinking and simply feel, simply see. If thoughts try to lure you to an unresolved dilemma, return to looking and feeling. Notice everything about the thing you are looking at. How does it feel? Is it mad, sad, happy, scared, lonely? Do you like it? Do you like one part of it more than another? Can you articulate why? Can you see the hands that made it? How did they feel about making it? When we do this with anything, the touch of our heart field and caring begins to slowly awaken the sleeping spirit of the thing. Some things take longer than others to awaken.\n\nKeep your heart wrapped around it. If you have a fondness or appreciation for this thing, you can send that to it. As soon as you do this, the electromagnetic field of the thing you are gazing at and the electromagnetic field of your heart begin to merge. Stay with this, keep feeling.\n\nBefore you know it, the thing you are seeing will begin to alter ever so slightly revealing itself to you in response to your heart. Its history, its medicine, its maker, its needs, if any, will appear on the screen of your vision, and you will feel it in your body. Notice everything that is happening; how you feel, where you feel it in your body, how your breathing is, how your heart feels. Pay attention to any attempt to censor or discount the information that comes to you in any part of you. This is how the thing is communicating to you, through your body. The thing you are seeing and you become intertwined, and the separation, the space between you and the object, has become filled with directed meanings. This is the ecstatic path toward revisioning and living an ensouled life. This is our birthright, and it belongs to everyone.\n\nWhen my partner and I were building our house, each tile, stone, artwork, board, and beam that was brought in was worked with in this way until we could feel the spirit of it begin to awaken and come into the room. We prayed with, talked to, and brought Eros into each piece until everything in our home became alive, our home ensouled. In anything you undertake, you can direct the flow of heart into it.\n\n _This form of wisdom will enable the physician to discern the Unity of Nature and to recognize man as a faithful copy of the great Universe, governed by the same laws and expressing them in his own being. As this is a meta-physical truth, every physician must be also a philosopher. And astrue wisdom comes from within, the physician must posses the faculty of intuition, the handmaiden of self-reliance. Therefore the true physician is one who does his own thinking and is not satisfied merely to repeat the thoughts of others. As intuition and self-reliance are developed in the physician, the secret doors of Nature will open to him._\n\nPARACELSUS\n\n ** EXERCISE   \nExpanding the Heart Field with Your Lover**\n\nAsk your lover to sit with you so that you are facing each other. Breathe through your heart for a few moments. Release any tension you feel in your body. Feel your love for him. Let yourself be filled up with all the love you have inside for your partner. Keep feeling the love you have for him. Hold it there, allow it to grow and expand; reach out with it from your body to touch his. Allow your love to move inside him, gently, as far as he will allow it to. Notice where it stops. Notice how his body shifts, his breathing alters. Does it speed up or slow down? Does he sigh? When love touches deep inside, there is a sigh of acknowledgment, a deep sigh similar to when an infant nurses and has been taking in milk and love from the mother. The love you are giving and he is receiving is a specific kind of food, a kind that we all need. With each deep sigh, love goes in deeper feeding and nourishing his body and soul. There is a point when the deep part of him becomes satiated, and there is that deep, satisfied sigh and often a little quiver will accompany it as he relaxes into receiving your love.\n\nNow, let your love wrap around him and hold it there gently. Nothing to force here, you're just allowing. Notice whatever comes to you without censoring. Notice whatever feelings you have. You may see parts of him you hadn't seen before, discover new things you love about him. You may find yourself falling in love again and deeper still, becoming entranced and dreamy. And you may see and know things about yourself you hadn't before now. You may have stood before your lover with no clothes on, but has he seen you naked? There are many ways to be naked before your loved one in the moments of revelation. The energies of our souls transmit meanings in the invisible force field that is created when heart meets heart.\n\nNow move to more specific areas of his body. Look at his hands and touch one of them with your heart field. How does it feel? How does it look? Is it happy, sad, mad? Does it have significant male or female energy in it? Now do the same thing with his other hand. How does it feel? You may find that assumptions you had are not true and that possibly you have never really looked at his hands this way, with the gaze of your heart, before. Do the same thing with each of his feet. Hold each foot in your hands as you do this, feeling each one, loving each one. Yes, now, you hold his penis in your hand, send your heart field, your love, your caring there. How does it feel? How do you feel seeing him this way? How does his member respond to this kind of intimate touching?\n\nAs you do this over time, you will notice the energy in each part of his body changes in response to your love. Love infused with intimacy and heart changes everything it touches. Each part of him will become more alive, ensouled; the intelligence that is there will wake up and become part of your lovemaking and soon, a marvelous conversation begins between your bodies.\n\nMy thighs were the ugliest part of my body. I didn't like them and tried to change their shape over the years, and I did. But changing the shape of them outwardly never changed the feeling of them. They didn't feel like my thighs; I didn't recognize them. I was dissociated from them, so of course they didn't feel like mine. It was one of the first of many secrets I told my lover and was astonished to hear how incredibly erotic he thought they were. As he spent time loving them, adoring them, talking to them and being friends with them, they began to change, and my relationship with my own thighs changed. I began to appreciate them, to see them in a new light, to see everything wonderful about them I hadn't been willing to see. I noticed the shape of them, the strength of them, the texture and color of them.\n\nWhen you give love to parts of his body, to his hands, for instance, love doesn't linger on the skin, it moves below the surface and goes in as deep as he will let it. Each time you spend a few moments loving them, holding them, love goes in a little deeper, below the skin into the intelligence of muscles and tendons altering holding patterns. It goes into the blood, to the heart, to all the organs, and the ego states, as well, eventually get fed with this love, and they begin to let go of holding patterns.\n\n _If you want to know God,_\n\n _Then turn your face to your friend,_\n\n _and don't look away._\n\nRUMI\n\nTo feel the touch of Spirit on your body, the touch of another human heart touching your heart, you must go to your lover with all defenses put aside. Vulnerability allows access to feelings. Letting down your defenses is the hardest thing of all at times, for the part of us that enjoys our defenses and the space they create has tremendous tenacity. You may need to negotiate with this part of you that is afraid, for it is the fearful part that holds onto armoring. We learn to wear many masks and to develop personalities as we go through life to give us a sense of protection. Those masks and personalities are developed to protect the small Child inside us. But it is just that Child in us that needs to feel this touch, needs to be touched, be seen by another human soul and by the Spirit that moves through all things. It is the Child in us that remembers what it felt like to be immersed in, surrounded by, bathed in that energy field, the Great Mystery, the heartbeat of Earth, the original mother, before coming into the world in human form, and it is that feeling we search for all the days of our lives. It is that feeling, that touch, that we need, in part, to be whole. We need that feeling from our partner, our parents, our friends, for it is love without judgments, it is seeing without condemnation, and then we can begin to feel our presence in life is welcomed and needed.\n\nANCIENT HISTORY\n\nThe concept of the relationship as a vehicle for the sacred is thousands of years old. A bronze sculpture of Shiva's consort, circa 950\u2013960 CE, is known for its blending of sacred sensuality, sexuality, and spirituality. Elaborate statues such as these were commissioned by kings to depict and honor the gods and goddesses of sexuality and fertility. The most famous Indian statue is of Lord Shiva reaching out to touch the breast of his consort, Uma-Parvarti. Hindu sculpture can often be explicitly and unembarrassedly erotic. Physical grace and sexual prowess in kings and queens were regarded as vital and admirable attributes in a ruler. In Hindu tradition, outlined in the Rig Veda, the erotic _rasa,_ or flavor was considered one of the nine _rasas_ comprising the Hindu aesthetic system. The Rig Veda, an ancient Indian text of sacred hymns, begins with the creation of _kama,_ sexual desire: In the beginning was desire, and desire was with God. In the Hindu scheme of things, the gratification of _kama_ remains one of the three fundamental goals of human existence, along with _dharma\u2014_ duty or religion\u2014and _artha,_ the creation of wealth. The sacred poetry of the Vedas was primarily written by people overwhelmed by the beauty and power of nature, which they personified and deified as a pantheon of gods and goddesses. Nature was alive in a pantheistic and animistic worldview.\n\nThe explicitly erotic sculptures along with the long Indian literary tradition of erotic devotional poetry may at one level be read as metaphors for the longing of the soul for the divine and of the devotee for God. Yet such poems and sculptures are also clearly a frank expression of pleasure in life and love and sex.\n\nIn the ecstatic poetry of Kabir, Mirabai, and Hafez, God appears frequently as a lover. It is seen in the writings of Rumi, the Persian poet, whose love affair with Shams demonstrates the sacred fabric of love and devotion, mysticism and eroticism. The Kama Sutra, India's sacred love text, describes the finer points of sexuality and love, the art of living and of sensual pleasures including food, perfume, lotions, incense, flowers, and fine attire, not simply acrobatic sexual positions.\n\nWell known for his interest and explorations in sex and erotic literature, Sir Richard Burton (1821\u20131890) brought the Kama Sutra to the English-speaking world, eighty years before the sexual revolution of the 1960s. In response to the \"Obscene Publications Act of 1857,\" he founded the Kama Shastra Society, devoted to publishing and circulating erotic and sexual literature that would be illegal to publish publicly. In _Pleasure Bound: Victorian Sex Rebels and the New Eroticism,_ Deborah Lutz examines two groups of sex rebels devoted to challenging the morality of Victorian England. The Cannibal Club and the Aesthetes included prominent artists, poets, socialists, and clergy. Their activities were often met with ostracism, arrest, censorship, and jail time. Oscar Wilde spent two years of hard labor for being gay. The members were freethinkers and rebels challenging the rigid morality of the day. They championed unpopular ideas and stereotypes of sexuality and women's suffrage. Cole Riley wrote that \"[t]hey saw how difficult it was for women to break out of the stifling laws and stereotypes of the time. They knew that a society could never be truly progressive if women do not equal legal and voting rights, and full rights over their own bodies, including birth control and abortion.\"\n\n _Yes, I am a free lover. I have an inalienable, constitutional and natural right to love whom I may, to love as long or as short a period as I can; to change that love every day if I please and with that right neighbor you nor any law you can frame have any right to interfere. And I have the further right to demand a free and unrestricted exercise of that right, and it is your duty not only to accord it, but, as a community, to see that I am protected in it. I trust that I am fully understood, for I mean just that, and nothing less._\n\nVICTORIA WOODHULL,  \n\"AND THE TRUTH SHALL MAKE YOU FREE\"\n**8**\n\n _ **The Language of Love**_\n\n**The Movement of Great Things**\n\n _There is memory of ocean,_\n\n _the swelling of waves,_\n\n _the movement of great things,_\n\n _just beneath the surface._\n\n _My conscious mind staggers,_\n\n _a part sleeping begins to waken._\n\n _What is this great thing?_\n\n _That has caught us up?_\n\n _Beloved . . ._\n\n _Shall we find out together?_\n\n _Shall we travel to a land_\n\n _where two-dimensionality does not rule?_\n\n _Where all that we encounter gazes back at us?_\n\n _Where directions for the journey_\n\n _are written in the shape and textures of the land?_\n\n _Where we see, as far as the eye can touch,_\n\n _the soul of us opening outward?_\n\n _Shall we take that step together?_\n\n _Leave the comfort of the porch,_\n\n _and strike cross country,_\n\n _to find the place where the Teacher lives,_\n\n _the place where the big and the little become one,_\n\n _the place from which we came long ago,_\n\n _the place we have heard calling since before we were born?_\n\n _Shall we go out Beloved and take the path before us?_\n\n _Shall we let the perfume of our love_\n\n _fill all our three bodies?_\n\n _Come, take my hand,_\n\n _it has awaited the deep you to fill it,_\n\n _a length of time too long for remembering._\n\n _Come Beloved, let us take this journey together._\n\n _My feet are hungry for the first step._\n\nSTEPHEN HARROD BUHNER,  \n _T HE TASTE OF WILD WATER_\n\nThere is a language to love, and it involves more than words. It is a communication in which directed meanings are passed from one person into another. It flows through the heart field in which love is wrapped and then into the other; an exquisite dialogue that transcends our common language. When two people are in love, this is the linguistic medium they share.\n\nEach part of us has a different story to tell. As we touch with the fingers of our hearts, we read the text of each other's lives. There are stories waiting to be received that live in hands, shoulders, eyes, feet, knees, breasts, and intonations of voice. Our stories are breathed in by the other, and we are then taken to places we have never known, places we have imagined and longed for. Many of our stories have been held secret, never shared before. When another person receives the stories, takes them deeply in, in a sense he lives them experientially. Over time, the threads of one person's life are woven into the fabric of another.\n\nEach part of a person's body has its own distinct feeling and meaning. When you explore your lover's body, you explore the terrain of his life; you enter the geography of his deepest self. We spend time with and hold in wonder the meanings of the other.\n\nIt is the saddest thing in the world to have love to give, to feel the urge to love until you feel your heart nearly burst from it, and have no place to set it, no ground tilled to receive it. Or to have it only partially received or censored so that only a tiny portion of your giving is received. The most joyful thing is to have your love welcomed, recognized, and given a home, a resting place. A place it has longed to be for all eternity.\n\nYou can't have, or give, too much love.\n\n _A journey makes us vulnerable, takes us from our more secure environments and commits us to the unknown. Perhaps this is why the journey has so often been our basic metaphor for life itself. Our life journey is a precarious pilgrimage, a passage through landscapes of promise and peril, a crossing from the darkness of the womb to the shadows of death. We travel in the hope that the light will not fail to guide us, that the star will not be lost, that homecoming will be granted and love not withheld._\n\nTHOMAS MERTON\n\nOur body, the lines in our face, and the look in our eyes hold the stories of who we are, how we became who we are as well as all the wounds and the wonder of us. Stories are the narratives of our soul's journey. They tell of the events, people, and places that touched us and how we were impacted by them. When the stories held in our body are received by another, there is a softening in the soul, an opening to feel the touch of another human heart. There is a release, then, of the burden of carrying stories alone. There is an ebb and flow in all this, a back and forth, a rising and settling, as each part of us opens to receive the other; a knowing of the other's interior world opens and love deepens. Someplace in the middle the two meet, and together they discover a new world. After awhile, who I am begins to move inside the other, and the other inside me; his story becomes mine and mine, his. And the lives we had before are changed.\n\nThe language of love has sound, shape, texture, movement. There are sighs, tears, a tensing and relaxing of the shoulders and breath. The person hearing the thing receives it in his heart field, and there it has a place to live.\n\n _When one finds her voice, her life takes on grace._\n\n _L ADY IN THE WATER_\n\nLOVING WITHOUT RESERVATION\n\nIt's all too easy to be casual about a thing, a life, a concept. It's all too easy to fail to follow the golden thread all the way to the finish, to see it through to the end, as far as it will go, heeding the ebb and flow, rise and fall, obvious and subtle oracles. Out of fear is born the habit to hold something back. We never know when we might, once again, find ourselves with a long swim, alas, back to shore.\n\nIt takes a long time to be willing to love without reserve.\n\nAnd thus, as intimacy feeds the soul of love, a new verse, a new story is written. We become something new, something joined, influenced by the other, changing from moment to moment. As we walk in the world with this new thing inside us, those who encounter us can experience this new song that is being written.\n\n _Love is the willingness and ability to be affected by another human being and to allow that effect to make a difference in what you do, say, become._\n\nGENEEN ROTH, _W HEN FOOD IS LOVE_\n\nJames Hillman writes: \"A heart's image lies within each person. It is what we truly reveal when we fall helplessly in love, for then we are opened to display who we most truly are, giving a glimpse of our soul's genius. People say: 'He looks so different\u2014he must be in love.' 'She's fallen in love; she's truly changed.' When love moves the heart, something else is perceived in the idolized object, which poetic language tries to capture.\"\n\nFalling in love causes the imagination to unfurl. As the imagination expands, we fall in love deeper, wilder, more passionately, hungrily. Love is of the spirit, drawing the soul toward images painted on the heart.\n\nSuch love is not unlike reading a good novel; you find yourself in the story, engaged with the characters. You become invested in their lives and their adventures. You find yourself thinking about them during your day, wondering what they are doing, how they are doing, what's happening next in their lives. You look forward to sitting down with them again to look into their eyes and receive the story of their lives and travels. You witness the transformation of the characters.\n\nLove is like this. It is to become obsessed, wild, madly, chaotically, deeply in love. Each waking moment is filled with thoughts of the beloved. I smell him, feel the touch of him wrapped around me as I move through my day. As my beloved opens to show me more of himself, his stories find a place inside me to live. My caring and love for him grows. I want to call him every hour to hear his voice and to say how much I love him. I leave love notes on his pillow, taped to the mirror, or tucked in his travel bag. I think of him during the day and dream of him at night. When I'm sitting with a cup of my favorite morning drink, I recall our time together, the way he looks at me and sees me, the sound of his breathing, and the feel of him inside me.\n\nI want to touch him, caress his face, rub his tired shoulders, and hold him as he lays his head on my shoulder and falls carelessly to sleep. I allow myself each moment to fall in love with him over and over again. I tell him all the ways I love him, all the details of noticing him and his innuendoes. The various laughs he has: the nasal snorts of fifteen-year-old humor, the snicker, the chuckle, the chortle, and my favorite, the deep, full-out belly laugh. Each laugh has a different quality to it, a different ego state that lets it loose in the room. Each laugh has a meaning in it, the meaning of which part of him is in the room and engaging with me or whoever else is present.\n\nI want to know everything about him, and so I watch as he stands at his closet, doors open. He ponders each thing that catches his eye, and he imagines trying on each piece of clothing to see if it matches how he's feeling that particular day. It's a joy to watch him stand in front of the full-length mirror, looking at himself, turning this way and then the other. He settles into the clothes as each part of him responds to what he's put on. Then, if they all agree on the garments, he's set in his wardrobe for the day. If not, back to the closet they go.\n\nWhen he walks into a room with his personal\/sexual power entering with him, a force to be reckoned with, he has the power to take down any walls of China within me. And I give him that power gladly. I fall in love over and over again with abandon. I tell him all these things, describing in as much detail as I can my noticing of how he carries himself, how he feels in his body, how much I love watching him put on his favorite clothes that are like good friends wrapped around him. I tell him not only the details of my noticing him, but the meanings of them and how they impact me. Watching him prepare food with Eros and exquisite detail, then watching how much he enjoys eating the food is to read chapters, verses of his passion, his gusto, his patience, his reverence for life and love of food and sensory experiences.\n\nTo learn the language of love requires that we, out of love, out of desire, pay attention. The subtleties of this language are easily missed, passed over, ignored. As with the details in anything we are doing, there must be interested consideration. The ecstatic journey of sacred sex begins long before the physical act. Attraction and desire awaken us. A part of my soul reaches out to touch his, and in mutual desire, his touches mine. The dance of getting to know a lover is chaotic, frightening, and exciting. Old pains and hurts rise to the surface, and we may doubt our choice to enter into a relationship again. Eros causes chaos in our previously ordered life. In the midst of chaos, we are forced to travel to our interior. Psyche, the goddess of soul, is excited at the possibility of new experiences, but we struggle between our fears and a new opportunity for romance, love, sex, and intimacy and the changes that may result.\n\nI had been dating a man sporadically for three months vacillating between, \"I want this,\" \"I don't want this.\" Talking to each part of me on both sides of that fence, we all finally came to an agreement that yes, we want to explore this with him. Why not, after all? I was recently divorced after a sixteen-year relationship. He was tall, ruggedly handsome, strong, intelligent, interesting, very sexy, charismatic, fun. He had great lips. _And_ he had his own boat on the Mississippi River. It would be an adventure if nothing else. We had a date planned, so I drove the hour and a half down river to his house. Excitement was building. I had packed an overnight bag and decided that this would be our first night of sleeping together and having sex. We hadn't talked about it until we were sitting on a park bench overlooking the river. He asked me if I would spend the night, if we could just sleep together.\n\nSurprisingly, my fears came up again, but I went back to the agreement I had made with myself to explore this. \"Yes, I think I'd like that very much.\" Of course, I thought \"just sleep together\" was code for \"let's sleep together and have sex.\" Much to my intrigue, surprise, and a little disappointment (I needed to get laid), we actually . . . slept together. We hugged, kissed, caressed, told stories about our lives, and fell asleep in each other's arms. We had a sweet, intimate sharing and the movement toward a deeper relationship. I woke up feeling shy. He'd seen me naked and heard some of my life stories. Would he still want to have sex with me? He was leaving the next day for a month-long river trip. Desire built and hunger grew. I wanted this man.\n\nWhen we were together again on the houseboat, passions were high, hunger nearly overcame us. Still, we didn't have sex. Damn. Until the next night. And it was sweet as honey dripping from the comb.\n\nI didn't tell this story to advocate waiting. It's a personal choice how the dance of having sex for the first time goes and when. The sexless nights we spent together were windows into understanding who this man was. He was romantic, sensual, tender, and strong. It was important to him to build desire, to share something of ourselves, to set a foundation for whatever might follow. We slipped into a comfortability with each other. Our bodies fit together, our lips and tongues somehow already knew each other, knew exactly how to move and talk to each other the way lips and tongues do. We both enjoyed touching, kissing, snuggling, and cuddling immensely. I wanted him. I loved the way he looked at me and the smile that lit up his face when he saw me. I loved the feel of his hands on my body. I loved the smell of him. We began the relationship making love without the in-ing and outing and sweating and moaning. However, once we started in-ing and outing, we didn't stop. There were wonderfully few sexless nights after that.\n\nI surely loved that man but I think we grew mutually frustrated with each other over time; he with how I always wanted more from him, the fights I started, the growing chasm between our value systems, and I with his unwillingness to go to deeper levels of intimacy. We were both strong and independent. He had a well-established career and I was just beginning to understand who I was and what I wanted my life and work to look like. Our lives overlapped infrequently and he understood me less as I began to make deep changes. The lack of deep intimacy in the relationship, my rudimentary relationship skills, and the changes I was making left me feeling unhappy and unsatisfied. I hate endings but I knew that I would never get what I wanted there and I began to feel a pull in another direction, away from that relationship.\n\nI left the relationship less than gracefully. I was in the middle of my fourth decade and had begun to create a body of work that was mine. I could feel my time in the Midwest coming to an end, something new beginning. When I decided to let the relationship go, I resigned myself to not knowing if there would be another partner or other lovers, but I knew I would be devoted to building my work and teaching.\n\nTHE SMELL OF LOVE\n\nEach one of us has our own unique smell, the smell that is our soul signature alone. Sexual attraction is much about smell. Gustav Jager, a German physician and hygienist was the first to formalize the concept of pheromones, which he named anthropines. He correctly identified them as lipophilic compounds that are associated with skin and follicles that determine the individual signature of human odors.\n\nSome deep part of us can smell the other even if it's unconscious; it's part of our unconscious attraction but often comes to conscious awareness during sexual intimacy. You can tell a great deal about a person if you learn to smell below the surface, to the core of a person. And you'll know if his smell is a good match for you, if it's seductive, intoxicating, erotic. It's important to the health of a relationship if you like your partner's smell and if you like him in general. The smell of a person changes as his internal world becomes more whole. A musty smell can indicate that there hasn't been much love in the deep part of him. Lack of intimacy can smell old and rancid. When love is there, feeding a person's soul and whole being, the smell is sweet and fresh as clover.\n\nAlmost as soon as I told the Universe that, lover or not, I would not stop doing my work, my new lover appeared in my life. As he and I grew over the months and years together, I watched him change, grow, and become more himself with my love, with part of my spirit living inside him. I could see him relaxing in the trust that had grown between us, a dependability of one upon the other. There is trust in our love, in our friendship, a knowing that the other will be here: we will be at each other's back and come to each other's aid; we will listen to each other's struggles, joys, dreams, and insights. The desire and hunger is still there, passion is still there, but without the urgency, without clinging in desperation for fear that it will end. Desire and passion need tending to; they need spices and seasonings as well as attention to subtle moods and needs, yours and his.\n\nI often go to the watching place inside me to see myself from new perspectives, a perspective that changes as I change. I hardly recognize who I was even a few years ago. My posture is more erect. More often than not, I walk with confidence through the world. Parts of me have awakened\u2014parts that were asleep and in danger of becoming bitter in my old life because they did not have a place to be alive in the world. They are living in the world now, learning to walk and talk, being educated, becoming part of a fully integrated human being.\n\nI've learned how to love myself, how to fall in love with myself. Such self-love happens most easily when the love of another human being is let inside, is granted permission to pass all the defenses, the objections, and arguments against this act. When I allowed love to pass through all the gatekeepers and journey as rivulets into the deepest parts of myself, parts of me that had never been touched by love, then the journey of loving myself began.\n\nI became my own advocate, knowing what my needs and wants are, saying them out loud as needed, and working to get them met either by myself or by someone else. I learned to do this straightforwardly without the drama of games or cons or sideways attempts. I learned to say no first by giving myself permission to say no. It takes patience and focus and in the beginning is challenging. The closer I came to resolution, to truly giving up game dynamics, the more the scared parts of me escalated, the more focus it took to let them know I was serious. I seriously wanted to be free of old scripts and dramas.\n\nThis is an uncomfortable time when love flows inside to the deepest parts of the self that have never been touched by love; the places inside that are bent and crooked. Love doesn't travel in a straight line. Like the tip of the ocean wave after it's crested and fallen and now meets the shore and the two create a whole new thing, never straight, and the water changes the beach, the rocks, the sand, and the muddy shore, and the shore changes how the water moves, its speed, its impact, and how it moves back away from the shore into the ocean again.\n\nDoing this repeatedly, I let his love come into me, touch me, fill me, grow me, expand me, change me. His love fills all the spaces inside me. He lives inside me, in my heart, and is in my soul. All of him, every part of him, every story of him, each heartbreak and the original wound of his birth family are now part of me. Nothing of him is hidden or secret from me, and nothing of me is hidden or secret from him.\n\nAs the original wound in each of us is healed, made whole by the other's love, our psychological structure is altered. The defenses and games I played have been given up, seen for what they are, and rendered obsolete. I'm not willing to sacrifice myself to keep peace and, rarely, to make someone else happy, though there are times when it's important to put my needs aside temporarily to strengthen our relationship or to tend to something that is more urgent.\n\nI reach for words to describe the depth and breadth of my love for him. I do the best I can with the words at hand. I search the lexicon grasping for others. A thousand words I say each night, and none come near to the depth and breadth of my love for him. There are no words big enough, strong enough, or whole enough to hold the meaning of my feelings for him, though I try. I say \"I love you,\" words that I have and are the truth of my feelings, and yet, they are pale ashes compared to the kaleidoscope of colors that are in the feelings.\n\n _Attempting to fill three words, three syllables_\n\n _With the depth of my feelings for you_\n\n _Resembles mining the secrets of the ocean_\n\n _Holding still the moon_\n\n _Or painting rainbows in the sky_\n\n _With crayons_\n\nFROM THE AUTHOR'S JOURNAL\n\nI want the words that hold the meanings of how I feel. I want to make my feelings known to him, and when I least expect it, the language of love is made known to me. Here inside me, inside him, flowing between us like a sweet, gentle stream it has been all along. The bee was telling me where it was, what it was, and how it tasted. It is that invisible field that pulses from my heart to his and back again. How ignorant of me.\n\nThe language of love is the language of falling in love each day. To learn this language, you must allow yourself to feel, to be aware. Each time I look into his eyes, hear one of his many laughs, or see him walk into a room, I'm aware of many subtle and visible responses in me. My posture changes, I feel the petals of my heart unfold, I feel my soul smile inside. That is love talking.\n\nLove is spoken when I do things that make him happy, those moments when I'm in service to his happiness and joy. His happiness is inextricably woven into mine. Love speaks through the little things I do, just by him taking joy in me. The part of me that came alive in our friendship is free to voice the running commentary she has on life's absurdities. She knows she can be herself, spontaneously without censorship.\n\nBeauty is love. When beauty and love are more than skin deep, a luminous glow radiates from a deep well, sourced from the love they have for themselves; it comes from deep inside them. They're magnetic and attractive and upon seeing them, your eyes are transfixed, following them across the street. And this language is more than skin deep; it's preverbal, beyond the surface, and past fantasy. The language is real, rising up from the deep ocean of self; it travels across the surface of the body and is taken up in the breath of the spirit that moves through and enlivens all things.\n\nLetting the love inside speak, giving myself over to it, to its textures, sights, sounds, and poetry.\n\n **This Simple Thing**\n\n _I awaken slowly, reluctantly,_\n\n _from a deep, and glorious sleep._\n\n _What is this thing that awakens me?_\n\n _What is this invisible touch on my spirit?_\n\n _Ah, it's you my love._\n\n _No need even to open my eyes_\n\n _I feel what you are doing_\n\n _Touching without fingers_\n\n _I smile knowing you are here._\n\n _You've been watching me while I sleep_\n\n _love_\n\n _ecstatic I_\n\n _in this simple thing._\n\nFROM THE AUTHOR'S JOURNAL\n\nLEARNING TO LET GO\n\nWhen I met the man who wanted to take this intimate, sacred, ecstatic journey with me, I was forty-four chronologically and around three years old psychologically. I wasn't very grown-up, educated, or sophisticated in my interactions. I played a lot of games and had few life skills. I was emotionally shut down for the most part, and my fallback survival plan when things got too challenging was to emotionally run away by withdrawing. I fought the urge to withdraw, to shut down. I struggled with my demons and kept going until I reached the other side of it.\n\nI knew a lot about sex and, to a degree, love. But I knew precious little, experientially, about going all the way with love and intimacy, about trusting another or of giving myself into his keeping, and less about how to nurture a healthy relationship. I only imagined how letting that much love in all the way could change me, for I had only seen a few hints of those changes in a few people who had begun to go there. I was about to learn the true meaning of vulnerability, that searing openness and exposure that is sourced from utter defenselessness. I was about to learn the true meaning of surrender. I was about to discover that my understanding of self-love was immature. Many times on the journey, I wanted to run away, to go back to my old life or create a new one that didn't involve having to face the best, and worst, parts of me. Feeling that vulnerable and tender was terrifying and painfully uncomfortable. I'd often rather chew tacks. It distorted my sense of self. Only in the midst of the journey could I see how my soul was distorted, fragmented, and a little crooked in places.\n\nThe prospect of letting go of myself completely, of trusting another so completely, someone who could truly see, who wanted to see every part of me, petrified me with fear and shook me at my core as if a great meteor smacked down in the middle of my idea of myself. Just because you want something doesn't exclude being afraid of it. It filled me with so much terror that I stopped having deep, full-body orgasms. I became emotionally and sexually frustrated and was forced to own how deeply I wanted to have this experience with this man who could love me all the way without holding anything back, who was patient and nonjudgmental, who encouraged me to have that sort of full-body shivering pleasure. To do this, I had to change at the core level of me and go through this deep fear to see what was on the other side, what was so terrifying.\n\nWilhelm Reich used the term _body armoring_ to refer to self-protection, or at least a sense of protection and safety. But body armoring is not localized; it affects our musculature and our nervous system, our ability to think clearly as it constricts our entire physiology. When impulses or emotions are inhibited, the body responds by tensing muscles. \"Inhibited libido,\" he said, \"is tense muscles and relaxed libido is sexual charm.\" Body armoring affects our ability to have open, meaningful, relaxed, and passionate sex. The fear paralyzed my groins and created a tension that was impossible to unlock until I worked with the fear and the parts of me, the Child of me, that was terrified. I had created a split between my mind and body; my body embodied the fear and rage I had around vulnerability, changing behavior patterns, being sexual in a sacred and intimate way rather than just having sex or fucking. Vulnerability unleashes a great outpouring of human potential.\n\nI wanted intimate relating, but I didn't know there was a price to pay. And the price I paid was everything I was so I could become the person I was always meant to be. I changed inside and out. My favorite clothes no longer fit the person I was becoming. My favorite earrings didn't look right. My favorite hangouts changed, and friends dropped away as my voice, my values, and my beliefs were restructured. There is a cost to freedom. I left my birth family and refused to be part of their psychosis or pretend that I fit in.\n\nThe language of love is felt in the kiss on the forehead or top of the head. The endearing moments when your lover is asleep, and the part of him that is bonded with you wakes inside him just enough to reach over to hold you, to feel you close and touch his foot to yours, to know you are still there. Love is there in the darkest night when in the midst of a nightmare, he's there, holding you, telling you, \"Everything's all right.\" Love silently speaks in the still, quiet, dreamy hours of the night.\n\nWhen it comes to sex, making love, and intimacy, our culture is still in kindergarten compared to other cultures such as the Japanese and Chinese, who have entire societies devoted to teaching young men and women not only about sex but about the body and how to make love to another with reverence as an expression of the holy.\n\nThe language of those cultures to describe erotica is marvelous, romantic, and delicate, using words such as _jade dragon, white tigress, green dragon, jade stalk, palace gates, yoni,_ and _lingam_. Compare this to Western language that describes sex and reinforces lovemaking as a heroic performance with the hard-cock approach ignoring delicate desires, imaginations, subtleties, and those all too rare and precious moments of ecstatic intimacy and mystery. Instead, we use the words _private parts, pussy, dick, ball sack,_ and _cunt,_ as if to somehow insulate us from intimacy and tenderness. Those words are suitable to bar talk, but not always suitable in intimate moments. Intimate talk, no matter the words, has a feeling tone to it. It feels like intimacy, or it feels like distance. It draws you in or it repels you; it honors you or it discounts you.\n\nWithout the erotic, sensual Earth to inspire our choices and our architecture, our language, our life, where would we reach for words to describe our experiences? If we weren't sexual beings, what would the Washington monument and space shuttles look like? We know a phallic symbol when we see one. We recognize sperm as a prototype for ships and weapons to invade foreign soil. Our skin cells are modeled after the skin cells of leaves and bark and animal hides; they die and shed and are replaced by new ones.\n\n _Virtually no one has enough accurate, judgment-free information about sex. We don't watch other people do it, we can't find many accurate representations of it in the media, and few of us talk honestly about it with each other. It's almost impossible to know the full range of sexual thoughts and behavior of the people around us\u2014and, therefore, impossible to know how much we have in common._\n\nMARTY KLEIN,  \n _Y OUR SEXUAL SECRETS_\n\nTHE LANGUAGE OF SENSUALITY\n\nThe word _sensuous_ comes from the Latin _sentire,_ \"to feel,\" and the related word _sensus,_ \"to feel and perceive.\" Sensuous refers not only to the physical senses but to any means of feeling, as intellectual or aesthetic sensitivity and intuition, as in the sensuous pleasure of walking in the rain, the sight of soft snow falling in the moonlight, and the way the air smells fresh and clean after a spring rain or monsoon. Sensuous powerfully appeals to the senses in a sexual or quasi-sexual way. John Milton invented the word _sensuous_ to have a synonym of _sensual,_ minus the association with sex. He used the word in an often-quoted formulation of what poetry should be: simple, sensuous, and passionate.\n\nSensuality is a language of love and sex. It is a language of our body and of our nature. When something is sensual, we respond bodily and experience it through our senses. Skin is sensual when it is soft, smooth, and supple. Well-formed muscles are the sensual, erotic landscape of the human form. As skin and our bodies age, we begin to take on the look of stone and tree and landscape that has been shaped and worn by the elements. Georgia O'Keefe is well known for capturing the sensuality and eroticism of stone cathedrals and landscapes in her paintings. Best known of her collection is her flower series capturing the resemblance between the petals of flowers and the female vulva.\n\n _I had to create an equivalent for what I felt about what I was looking at\u2014not copy it._\n\nGEORGIA O'KEEFE\n\nA warm summer breeze on our exposed skin and the way warm bath-waters wrap around our bodies are sensual and erotic experiences. The sway of a woman's hips, the breath escaping softly in a sigh, and the way food is prepared slowly with care and attention to details can all be sensual experiences. Anything that is sensual is erotic and satisfying to our physical body and senses. As we grew in the womb, we were infused in the warm, sensual fluid of prebirth waters. We have an innate understanding of the meaning of and a need for the sensual.\n\nWe all know what sexy is when we see it; we feel it when it walks into the room. We feel the impact of it, our heads turn in that direction; we feel pleasure, are intrigued, interested. Having been trained out of looking and seeing, we watch out the corner of our eyes, trying not to be noticed noticing. There is an energy that comes off a body when sexual energy is either part of who the person is or is part of who the person is trying to be, or what they are wanting to communicate. It's sexual energy, attractive, magnetic, irresistible energy. We can't help but turn our heads and be drawn into its orbit.\n\nSexuality has a language of its own; it speaks in the subtle movements of the eyes shifting and darting, the movement of a thigh, carriage of the body, long, supple movements involving posture and shape; the way a low-cut blouse exposes the mysterious valley between bosomy mountains, or a skirt slides up to show the inside of a thigh. Whether our look is sly or withdrawn or direct and substantial, if our hair is flowing or pulled tightly back, and the ways we hug and touch; spontaneous, passionate, full-body embraces or one arm barely touching a shoulder are speaking the languages of intimacy, sexuality, and invitations. Standing with one hip slightly forward, the set of shoulders, the tilt of the head, use of makeup around the eyes and mouth are sensual ways we present ourselves in our bodies and speak a voiceless communication about how we feel about our sexuality.\n\n _Sexual and creative: these are important words and concepts. You are a sexual being\u2014all humans are. It is up to you to determine how free or how stifled you will be as a sexual being, but you are a sexual being. This sexuality is the core of your creative process, for it is the core of YOU. Sexuality is the deepest part of self-identity, for it is how you relate to SELF and how you relate to others._\n\nALAN SEALE,  \n _I NTUITIVE LIVING AS A SACRED PATH_\n\nThe language of love is a quality, an essence. It is not a form or technique. It's not learning to speak Spanish or Russian. It's a language that is multisensory, multilinguistic, not confined to the voice box and a few hand gestures\u2014though the erotically placed and rhythmic hand gesture is linguistic.\n\nEach of the languages love speaks has meaning. Snuggling and cuddling are filled with meaning. How our bodies and psyches respond to touch is meaning rich. Rigidity in his body or yours and where it's held tells you much about holding patterns and armoring. How he kisses, what position he likes most, and whether he likes to look at your vulva or not, gives you glimpses at his inner world and how he feels about sex and himself. One lover preferred entering me from behind while standing up. The intimacy of face to face was often more than his comfort level would tolerate.\n\nBeing aware of the language of love and sexuality increases our sensitivity to the pervasiveness of these communications happening around and in us. If we understand that we are sexual beings transmitting sexual messages we can shape them directly, become conversant in the dialects and raise them to sophisticated degrees of finesse, nuance, and subtleties. Sacred lovemaking is the divine alchemy of the numinous, physical, spiritual, sexual and soul. The language of love is that of beauty, sex, art of nudity, and earthy sensuality.\n\nThe language of love has many forms and expressions. Whether it is expressed through the spoken word, poetry, song, paintings, or the sacred gaze, the language of love is an essence, a meaning-filled communication. It always comes from and through the heart, so the form matters only in as much as it is your medium of expressing the essence in any moment. In sacred sex, the language of love is expressed through the heart, the joyful attention to details, heightened perceptions, and the physical embrace.\n**9**\n\n _ **Sacred Sex**_\n\n**Do You Recognize Me Now?**\n\n _Can you be completely one with me_\n\n _And still come back to yourself?_\n\n _Can you live in our surrender_\n\n _and carry it back and forth_\n\n _between the world of time_\n\n _and the world where time does not exist?_\n\n _Can you live on the undefended edge,_\n\n _where the breezes of love blow_\n\n _not just for a day or a month or a year, but for an eternity?_\n\n _Can you be defined by that wave_\n\n _into which our two waves merge?_\n\n _Can you live in its trajectory_\n\n _and die in its embrace?_\n\n _Many men have asked you to dance,_\n\n _but has anyone asked you to dance like this?_\n\n _With or without bodies, it does not matter!_\n\n _Will you be all that you are in its fullness_\n\n _and let yourself spill over into me?_\n\n _Will you harvest your tears with mine_\n\n _and water the ground with them?_\n\n _Will you bring me into the secret caves_\n\n _of your doubts and your fears,_\n\n _allowing the moonlight to guide us_\n\n _through the snowy woods?_\n\n _You have come to me from deep waters,_\n\n _ascending from the center of the circle._\n\n _Now as I dance, moving around the circle_\n\n _from one partner to the next,_\n\n _I once again await the depth of your eyes_\n\n _and the electricity of your hand._\n\n _But I am lucky that the dance ends_\n\n _before we come face to face._\n\n _I know when that moment comes_\n\n _I will be lost forever, and so will you!_\n\n _You see, I am not just the one_\n\n _who comes to you in the dance. Rumi says:_\n\n _\"Lovers don't finally meet somewhere._\n\n _They're in each other all along.\"_\n\n _Do you recognize me now?_\n\n _Rumi says: \"Gone inner and outer,_\n\n _no moon, no ground or sky.\"_\n\n _Has anyone asked you to dance like this?_\n\nPAUL FERRINI,  \n _D ANCING WITH THE BELOVED_\n\nWhen I met him, I didn't know much. I knew just enough to feel a spiritual imperative to keep him in my life if even from a distance. We had in us the drive to reconnect, and like shy teenagers, our first meeting was awkward, an undercurrent of unknown passion made balance difficult. There was that powerful attraction, the way metal filings respond to a magnet. His smell, his movements, his cells were astoundingly familiar, and though I didn't understand at the moment, or for a long time, I recognized a kinship in him. There was a great urgency in me to get his attention, engage him in a conversation; some part of me needed to keep him within arm's reach, the manner or form did not matter.\n\nThe urge to reunite, an invisible compass pointing each in the direction of the other, came at just the right time to save each other's life. Ten years came and went before we became lovers. It felt as if we had waited our entire lives to do this dance of ecstatic, sacred love with each other. The first night we made love, I saw the place before time, before the two of us were born, the agreement we made to find one another. I understood the tension, the awkwardness, the discomfort, the attraction, the fear I'd felt all those years. I was suddenly catapulted out of the half-sleeping state I'd been in.\n\nRabbi Zalman Schachter-Shalomi, founder of the Spiritual Eldering Institute, had these reflections about sex.\n\nI remember thinking of sexuality as that lousy trick that God played on us. How could God do such a terrible thing as to implant in us an urge that is so difficult to resist? This very same urge has been reinforced time and time again since we stopped being amoebas and turned into humans. Just think about how this was reinforced. I had two parents, four grandparents, eight great-grandparents. They all did it. Now see how it spreads out to countless beings who all did it. Now think of the children who will have children who will have children who will have children who will all do it. And all this fantastic amount of genetic information is concentrated in one flesh, as it were, of transmission. How else could it be but ecstatic? How else could it be when so much of the past has to transfer itself to so much of the future through such narrow orifices? It can't be but ecstatic.\n\nLOVING THROUGH SPACE AND TIME\n\nWhen we became lovers, we were living in different states, he on the East Coast and me in the Midwest. We talked on the phone several times a week. He often began the conversation telling me something funny that had happened or that he'd read. Laughing together closed the distance, and we slipped into the growing comfort of our companionship. Hearing his laughter, the ache and longing of missing him squeezed around my heart. I so wanted to hold him in my arms.\n\nA thing happens when you miss someone so desperately; the intensity of feeling and need becomes a solvent to all armoring; the shielding we've put up for protection is ineffective. With the armoring and shielding taken down, it becomes possible to surrender to what is there. Somatically, the body responds by letting go of tensions and relaxing the musculature. There is yielding to feeling, and you let yourself follow it, feel it all, everything that is there between the two, inside you, inside him, growing and expanding, filling the room you're sitting in, expanding to fill each room in the entire house, and it continues expanding to fill the neighborhood and the town and beyond. There is no end to its expansiveness.\n\nOur phone conversations took us deeper and deeper into another reality, a place very much like my time sitting in zazen, though there were 1,100 miles, 23 hours of driving, or 8 hours of air travel between us. As we talked and gave ourselves over to feeling love for the other, we slipped between the cracks of reality and illusion into the numinous. Boundaries unraveled. Distance dissipated. Time and space transcended, and we were in each other's presence. Reality began to shift, and soon we could see the other strongly, clearly. Simultaneously, we each felt the other's presence near us, felt the rhythm of our breathing and passion growing between us. We reached out with our love and touched the other's cheek. The sweet, fecund, anciently familiar scent of him filled my senses. Pictures began to appear before my eyes, and inspirations rose up in the gaps between words. \"This is what we do when our soul leaves our bodies, when we are gone from the Earth plane, this is how it feels,\" I whispered. I felt his hands on my skin, his breath on my face, felt him penetrate me, slowly, deeply, as we breathed together, whispering declarations of desire, love, and longing. Years later, we still engage in sacred phone sex, e-mail sex, shower sex, keeping the livingness of our bond whole.\n\nI believe that this is what we are meant to do, are born to do; dance ecstatically with the beloved in the throes of sacred sex, of giving oneself over to sacred traveling in other worlds, suspending time and space. This is the state we are yearning for, aching for, longing for. This is the feeling we are looking for in our relationships, that awe-fullness to experience something wholly other, completely outside ourselves and our normal experience, transported, transmuted, enraptured, and caught up in the sublime.\n\nSometimes the numinous catches hold of us spontaneously. Paying attention to it or not, the feeling gets us to pay attention, to sit up and take notice. Reality begins to shift like the movement of Earth's tectonic plates. Perceptions are altered during this shift; the room becomes luminous, filled with light. Colors shimmer; the edges of things that were a moment ago solid become blurred. You are in nonordinary reality, taken through yourself, outside yourself, into the realm of the gods where the gate is opened by the simple joy of the expansive heart, the feeling of the bond between the two of you: all that you are, all the other is, is in this moment where nothing else exists but the exquisite warmth from the fire of love, birthed from the depth of intimacy. In this moment joy rises up, your heart spilling over with so much love you can no longer contain it, and it has nowhere else to go but to spill out the corners of your eyes.\n\nWhen you go to your lover, go with a sense of wonder and awe as if for the first time. Let your hunger for him, your love for him crack open the shell to reveal the soft, tender center of the heart. With the armoring fallen away, you gain access to the more subtle feelings. You are able to see and feel with your heart. From this place awakens the natural wonder of the child. The same wonder you had as a child long ago, exploring the world around you, splashing in mud puddles and intently watching insects, following their minute movements. Allow that natural curiosity and interest to overtake you as you look into your lover's eyes.\n\nRemember your first times together if you must, to bring that freshness and newness, all that attracted you to him. Bring that to this place until you can marry the familiar and newness spontaneously. Let your eyes soften as your gaze falls on him. See who he is, this being sitting before you: see his goodness, his fears, his passions, his soul, and the child he once was. Become naive once again. The words _naive_ and _na\u00efvet\u00e9_ come from French _na\u00eff,_ \"natural, inborn,\" which comes from Latin _nasci,_ \"to be born,\" and _natura,_ \"birth.\" They describe inborn characteristics. The same Latin word gives us our words _native_ and _nation_. To be na\u00efve is to be childlike, natural, and native, aboriginal in your own skin.\n\nSeeing with your soul is what Rumi referred to as \"gazing raptly.\" In most social situations and even during lovemaking, we rarely make eye contact, rarely see who we are in conversation with. We learn to not look, to cast our eyes away, shut down our hearts and are barely present with the other. Sacred sex means being fully present in body, feeling with skin and heart, attending with mind, and seeing with eyes. Will Johnson in describing this experience of depth seeing says, \"Just stay connected through the gaze, keep surrendering to the current, and accept whatever's around the next bend. Where does this river run to? If anyone knows, there would be no way to tell you. The only way you can find out is to jump in yourself, surrender to its current, and trust its wisdom. Rumi and Shams must have surrendered completely, and the journey they took together is still rightly revered.\"\n\n ** EXERCISE   \nDepth Seeing**\n\nFor this practice, eliminate as many distractions as possible, relax as deeply as you can. Holding the gaze and heart field are hindered if there is tension held anywhere in the body. Tension in the body will become a distraction, and your mind will want to focus there. Breathe into it and free your body of it with each exhalation.\n\nSit face-to-face across from your lover. Look directly into each other's eyes. When you begin, you may find you are naturally drawn to the left or right eye or both simultaneously. There are no rules here. What's important is that you relax into this, keep your eyes soft-focused. It's helpful here as well to have a sense of your peripheral vision. It's impossible to have pinpoint vision while your peripheral vision is enhanced. Keep in mind that your heart field is not unidirectional; it radiates 360 degrees around your body. No matter what sexual position you choose in any moment, you will both be wrapped in the field of the heart.\n\nAllow your heart field to expand while you are gazing. Let it reach out and touch your lover, noticing when the two fields meet and how that feels. Hold this gaze as the fields of your heart touch and then pass into each other. You may feel a little \"buzz\" in the center of your chest as his heart field touches you. Keep breathing easily. With your gaze and the field between you, you'll begin to feel a magnetic pull bringing the two of you closer as your souls begin to merge. Shifts will begin to occur in your body and your awareness. Colors will become brighter, luminous, filling the room. You'll feel strange sensations in your limbs, your head, and every cell of your being as the life force begins to pulsate and radiate throughout. As if a great cloud has been penetrated, your vision will be clearer, your perceptions sharper.\n\nNotice also how your lover's body looks and feels to you as well as everything that is happening inside and around your own skin. Let your seeing become your thinking. The moment your mind becomes active, you will lose the gaze. It's nothing to fret about; gently, not willfully, return to your gaze and heart-field awareness.\n\nWhen our eyes meet in that soulful gaze, it is the feeling in the gaze, the feelings that result in the gaze, that distinguish it from merely looking. It may seem as if you've opened a great book of memories with all the attendant feelings and emotions. You're beginning to ride the wave of ecstatic union, where physical boundaries begin to fall away easily from you and from him. As the physical boundaries fall away, the two of you feel as if you have touched the source of all that is and you've entered the territory of the gods.\n\nIt is the place where day and night entwine as lovers, where wave after wave falls in upon itself, where the real of day slips past and the numinous presents itself. The longer you hold this state and the deeper into the bliss of ecstatic, soul-to-soul communion you go, the sooner you will no longer be riding the wave but will be the wave. The movement will sometimes be fluid, smooth, and calm; at other times a turbulence will rise up inside you. Calm or turbulent, it does not matter, you are going somewhere, to a land where colors have taste and smells have sound. In the soft, sweet gaze of your lover's outward eyes, you begin to feel drunk with love, drinking deep, deeper into his inward eyes, the place of his soul looking back at you.\n\nThere is a feeling in the room now, between and around the two of you; it's palpable. Feel it. Drink it in. Anchor it into the body memory. This is what is real, this feeling, this experience. It is beyond ordinary, day-to-day living. This feeling can be brought into all that you do. It is what an ensouled life feels like. An ensouled life is a real life.\n\nTo get comfortable with this kind of intimate seeing, practice it often for as long as you can. This can be done in any position and during any activity. You'll begin to know the territory and maintain your balance in the presence of whatever arises inside you the more you practice. You'll fall in love with this feeling of being in the gaze\/heart field, wanting to do it with all those you love. With practice, you'll soon be moving in the world gazing at everyone you meet, if you wish.\n\n ** EXERCISE   \nSacred Sex**\n\nTake your lover by his hand and bed down with him. Keep your heart field wrapped around him as you lay naked body to naked body, eye to eye, breath to breath. Bring your lips to softly touch his as you gaze into his eyes, feel the ebb and flow, the in and out of his breath. Match your breathing, inhaling his exhales, exhaling as he inhales your sweet breathing. Breathe together like this as you hold each other's gaze and until you feel you have the rhythm of breathing together, sharing the atmosphere between the two of you. It's okay if you miss a breath, just return to it. Return to his eyes, his face, his heart, his breathing until you feel your two souls embrace. Be aware of that feeling of the numinous, of your two souls, as you slowly begin to be enchanted in the sacred space you've entered into. Hold it as long as you can, breathing naturally. In that rapturous gaze, you are enthralled, enchanted, absorbed. This is a feeling state and it is invisible. The experience within the gaze is the entwining of two souls. Holding the gaze as naturally as you breathe affords a shift in consciousness. Let it shift without restriction; follow where it leads. Looking with only the physical eyes is one dimensional. To gaze is to look with the eyes of the soul.\n\nHold that feeling of rapture in your heart, your energy field\u2014that sophisticated organ of imaginal seeing we call the heart and your awareness as you begin sensuously to travel the landscape of his body with your hands and eyes, your lips and tongue. Take your time. Let your heart feel and see. Feel the textures and curves of each part of him. Let your hands travel around his body as if for the first time, getting to know each part. Explore with other parts of your body, the back of your hand, your forearm, and your feet. Notice that each part of his body has its own personality and that each is communicating with you, with your body and each part of your body. Notice what they are saying, communicating. Notice where he holds tension, which parts are holding back or are afraid, take extra time breathing love into those places, holding them in your heart and its extended field.\n\nKeep in mind you are feeling and sensing your way. Don't fret if you can't hold his gaze at the same time. There is no dogma about what is right and wrong. When working and playing in the realm of the sacred, there are no rules; there is simply following the feeling that presents itself. Follow that, wherever you are drawn to look and gaze there. Follow whatever feelings arise, notice all that is happening. Notice that when you are in this state, your peripheral vision is greatly expanded.\n\nWhere do you feel you want to spend more time, lingering, savoring? Continue dropping deeper and deeper into the soul union, giving yourself over to this journey, this exploration, this ecstatic dance of sacred loving. Gaze deeply at each part of him. Engage yourself with the sweet fragrance of his body, of his sex. Become intoxicated with love and longing to have his soul, his smell, his love, him, inside you. Take his fingers in your mouth, his lips, his nipple, his penis. Each movement you make is slow and deliberately organic, following the energy of each moment, filled with the rapture of love and the depth of your union. Make love as if it's the last time you will ever make love. As your desire and excitation builds, slowly slide him inside you as you continue breathing and holding his eyes in your gaze. Hold him in your heart, sending your deep love for him through the invisible field of energy and your deep seeing. Feel everything, hear everything, smell everything. Become inebriated with feeling, swept up in the ocean of sensing. Feel your sex organs talking to each other, send your love for him from your heart, through your belly, down into your vagina, wrapping around his penis. Pay keen attention to his movements, his breathing, sounds, and smells; his responses are communications from his deep self through his body. Follow the waves of ecstatic love and passion, of sexual energy, as they travel throughout your body. Your vagina will begin to feel more ensouled, more sensitive, alive, and aware. It has its own intelligence as any part of your body does.\n\nPermit whatever images, feelings, emotions to arise and fall away. Allow any sounds that rise out of your deep self, your body, your passion to be part of your lovemaking. As he moves inside you, talk to him with your hands, send your love through them, explore his body, light touch here, more pressure there. Let your fingers and the intelligence in them slide between your two bodies to touch the base of his penis, then his testicles, and that sweet soft crease between his leg and groin. Let them slip along the length of his penis inside you. With each movement notice what that part of his body is communicating to you. Try out different ways of moving with him as he moves inside you. Raise and lower your hips, move side to side, slow down, speed up as he thrusts, penetrating all your bodies. Allow your erotic, sensual, sexual self be completely, freely involved in sacred sex.\n\nYou may discover emotional wounds or physical scars you hadn't felt until now. Send your caring into those places, fall in love with each one. You'll touch parts of his body he doesn't really like. Take your time here. They especially need to be noticed, caressed, kissed, and loved. Keep breathing your love into them with your breath and heart. Really notice him in ways you have not in the past. If you find your mind beginning to chatter on, gently return your attention to breathing through your heart and to gazing. In essence, you give yourself over completely to loving him, to being in service to the sacred communion of sex as holy devotion.\n\nAfter lovemaking, continue holding this space you're in, hold him in your arms, gently kissing his forehead, pronouncing your love for him and how wonderful it is to be with him, how good he feels, all the while keeping your heart field wrapped around the two of you.\n\nPractice often until having your heart field involved intimately in your life becomes as natural to you as breathing. You can have sacred sex and wildly passionate sex at the same time. Sacred sex isn't about holding anything back. That's what fucking is; it's holding back parts of ourselves, holding back love, and, especially, holding back feeling and intimacy. Sacred sex means holding nothing back in order to transform mechanical sex to sex as a divine, fluid, ecstatic experience of two bodies becoming one numinous, boundary-less body.\n\nSacred sex creates a bond between two people, as does sharing heart fields with anyone you are intimate with. To bond with someone is to bring a particular kind of power into the world. A bond is an exchange of soul food. As you bond with someone, a link is created. It must be worked with consciously, with intention and focus.\n\nWhether or not you practice the gaze\/heart field in a sexual relationship or with an intimate friend, you can have this experience, this soul-to-soul meeting with the energy that is birthed between the two. Let the energy take you deeper inside your own feeling body and into the psychic field of the other. You are moving away from \"knowledge of\" to what we call \"knowledge with.\" The gods speak to us through sexual potency and the bond of intimacy, which in turn makes possible an awareness of the world of the gods.\n\nNURTURING THE THIRD BODY\n\nA numen is a local divinity or presiding spirit. When people commit to each other, there is a third entity that is created, that is birthed into existence from the bond of you. The third being created from the bond of the two is an indwelling force or quality that animates or guides the relationship. It is a spiritually alive being. Something that comes into being that is more than the sum of its parts, more than the two of you. Sex is more than the sum of its parts. Just as Earth is more than the sum of its parts. Sacred sex is a ceremony, if you will. It is a communication with the invisibles inside us and the spirit of the relationship.\n\nThe third that is created is birthed through and because of the relationship. It is brought into being, into essence, as an invisible thing, the _anima_ of the relationship. You can't see it, but you can feel it. And your feeling of it will allow you to work with it directly, to shape it, to tend it, nurture it, and feed it with the love between you and your awareness of its presence. Once you become aware of this third being, bring it into sexual touching, into sacred sex. It's a palpable presence, one that gets stronger the more you nourish it through the relationship.\n\nThe two people in the relationship hold this third being in the fore of their hearts and minds. All decisions must be deferred to the whole so the choices that are made ideally enhance the whole or at a minimum maintain the energy and never take energy away from it. Be certain that the soul of the relationship is taken care of and accounted for moment to moment. The third being lives not just in the bedroom; it is alive and becomes part of your day-to-day living. Even if the two of you are in separate households, or separated by miles, the third being that is born out of your bond is still a viable and integral part of the relationship.\n\n **A Third Body**\n\n _A man and a woman sit near each other, and they do not long_\n\n _at this moment to be older, or younger, nor born_\n\n _in any other nation, or time, or place._\n\n _They are content to be where they are, talking or not-talking._\n\n _Their breaths together feed someone whom we do not know._\n\n _The man sees the way his fingers move;_\n\n _he sees her hands close around a book she hands to him._\n\n _They obey a third body that they share in common._\n\n _They have made a promise to love that body._\n\n _Age may come, parting may come, death will come._\n\n _A man and a woman sit near each other;_\n\n _as they breathe they feed someone we do not know,_\n\n _someone we know of, whom we have never seen._\n\nROBERT BLY,  \n _L OVING A WOMAN IN TWO WORLDS_\n\nA JOINING OF SOULS\n\nSacred sex calls us to participate directly in the immense and infinite boundary-less spiritual landscape. It calls us to participate in, within and to transform physical boundaries as we deconstruct, re-create, and recover our authentic being. Sacred sex is not a technique; it's a living communication or conversation, not a dead language of antiquity. It's a communion, a holy sharing of the sacred texts of our mind, body, and soul. In that communion, we are filled with awe, our spirit lifted and moved to stillness.\n\nAs trust in each other grows, as a deep friendship and bond sets anchor, an interesting thing begins to happen. A conversation between each part of me and each part of him begins. During lovemaking, I'm aware of a conversation going on between my vagina and his penis, as they have their own conscious personalities. I feel his love channeling through his organ to places deep inside me. My hands talk to his skin, his skin talks back, our feet chat to each other as they caress and explore the other. As various ego states present themselves through parts of our body and varying sexual positions, there are opportunities to have conversations with each of them.\n\nBeing reunited after an absence, the parts of me that have especially missed him are eager to be reacquainted. The hunger I feel for him is all consuming. I can't wait to touch him, to have my hands on him, to smell him and taste him. Each part of me reunites with each part of him.\n\nAt times I feel like a flip book with a picture that changes as you quickly flip the pages; one ego state after another rises up inside me to meet him and to be seen by him, rekindling and deepening our friendship.\n\nSacred sex is the closest we come to joining our souls with another human being. It is a primordial and biologically encoded urge to marry our spirits during lovemaking, to feel the boundaries of our bodies disappear; to not know where one body ends and another begins, to be inextricably bound spirit to spirit, soul to soul, body to body, as two strands of DNA spiraling in and around the other. There is an innate need, a hunger inside us to know God, to know the infinite and boundless Universe through our sensing body, through the field of our hearts, through the waters of sex and semen and sweat.\n\nBradford Keeney writes about how the Bushmen in Kenya experience their kinship with Nature and God.\n\nWhen Bushmen say they own something, it means not only that they own the feeling for it but also that the feeling has transmitted its essence, its complex nexus of relationships, into their very being. We become the other\u2014whether a friend or a butterfly, redwood forest, giraffe, or seahorse\u2014through our intensely felt union with it. . . . You have to make love with God. The rest of the world seems to have had a marriage ceremony with God, but their marriage hasn't been consecrated. . . . Feeling God is akin to having sex or making love with God. It is a transmission and a reception of the highest and most powerful love. . . . If God is love and we get close to this big love, then how could it not be as amazing as the most intimate experiences of sexuality?\n\nSacred sex happens when we are home in our bodies, and we allow any feelings to arise without censoring and allow the dreamer and the imaginal facility of the heart to be present in our lovemaking. Those parts of us are uninhibited, untethered, and move easily between the worlds of the ordinary and the nonordinary. They are the parts that travel in the world of the written word, when a good novel is able to take us into that world of fiction, making us completely unaware of the physical world where we are sitting with a book in our hands.\n\nThe dreamer in us is able to transcend physical boundaries, go beyond physical touch to the invisible, numinous, touch of spirit on our soul: the sublime. When we experience the numinous, we experience something external to ourselves that is greater than ourselves. The sublime and the transcendent are counterparts to the numinous. The sublime and numinous cannot be analyzed or dissected; they cannot be explicated. These are not matters solely for the mind to ruminate on. These belong to the realm of feeling. Rudolf Otto proposes that \"the sublime may stimulate the capacity to perceive the numinous.\" There is a tendency for the sublime and numinous to pass over into each other.\n\nThe major transformative experience of sex is to reach the state of the _numinous,_ to fall away from the bondage of thought and religious dogma into the open palms of God, of the ancestors waiting for your arrival into the _mysterium tremendum,_ the wild ecstatic state of bliss where your thinking mind is quieted, given over to the power of the moment. The body and soul surrender to the rapture and to the gods that have come to play, flirt, kiss, suck, nibble, love, and tease your mind out of linear, dogmatic thinking, indoctrinization, and walled schooling. The mind stops thinking and makes room for witnessing. \"Some say Human Beings are the ground where the gods reside.\" Joan Halifax in _The Fruitful Darkness,_ says, \"But I am sure that it is not in us but in the interworld between us and sacred space that the gods finally arise.\"\n\n _Consider re-entry into the wild. Become a wild shaman, a wild pagan, a wild Christian, a wild Buddhist, a wild Jew, a wild agnostic, a wild artist, a wild performer, a wild whatever you want to call it because the name is less important than the experience of being wild in this natural though always uncommon way of giving priority to mystery over mastery._\n\nBRADFORD KEENEY,  \n _S HAKING MEDICINE_\n\n _Eros: love-desire connective energy through erotic passion we overcome our habitual egoic insularity and reach out into the core of other beings. Blazing Eros recognizes no barriers; it is the organic impulse toward wholeness.That wholeness is the holy, the sacred. The word holy is etymologically related to \"whole,\" both of which refer to a condition of completeness or fullness._\n\nGEORG FEUERSTEIN,  \n _S ACRED SEXUALITY_\n\n _Holiness is present at every dawn and at every sunset . . . Holiness is in the rain, the snow, every season for the person who embraces his or her own seasons . . . holiness is in the smile we wear and the tears we shed. Holiness is the smile we appreciate on others and their tears that we care for._\n\nBRUCE DAVIS\n\nTo begin with what is sacred, what is holy, I bring it down, Earth into the soil of my body. It is not only something outside me, or apart from me. It is not found only in temples, synagogues, or churches, and it does not start and stop, no ending or beginning. The kingdom of sacred divinity is inside me and inside you. It resides in the space between us where our divine spirits comingle, where our souls intertwine and the boundaries of our bodies diffuse.\n\nUnexpected things happen in the rapture of sacred sex. He's inside you, and the two of you are breathing, looking into each other's eyes. You feel his love moving inside you, going deeper as he moves inside. You feel overwhelming love and trust, and the room takes on a luminous glow. Spirit arrives; the third being has come to be with the two of you.\n\n **A Third Body**\n\n _Lovers give birth to a third body._\n\n _Its lungs have the capacity_\n\n _to breathe for two,_\n\n _although they breathe_\n\n _something other than air._\n\n _Its heart has the strength_\n\n _to keep two alive,_\n\n _although it pumps something_\n\n _other than blood._\n\n _This body is not made of flesh,_\n\n _but of thought and feeling._\n\n _It is the labor of two hearts_\n\n _and two minds_\n\n _that have learned_\n\n _to dance together._\n\n _Although it is created by two_\n\n _who live in separate bodies,_\n\n _those two_\n\n _inhabit this body_\n\n _they have made together_\n\n _when those separate bodies die._\n\nPAUL FERRINI,  \n _D ANCING WITH THE BELOVED_\n\nThe place inside you where you have been hiding grief and sorrow and fears begins to open up with the touch of his love. The place inside you may be a room. It has a flexible door that responds to the touch of his love. It opens, and all that you have been holding back, keeping safe and secret, begins to flow out in a river of tears. From the watching place inside you, notice as the room is emptied of its contents, and you see each part you tucked away behind the door. All the while you are making love with your man inside you. And he is making love with you, breathing with you, holding you. He holds you and loves you while you do the sacred work of letting the sorrow and fear leave your body, your psyche, all the holding in you've been doing is released as the touch of his love, his penis inside you touches the ground of you, and a great torrent of release sweeps through the two of you. What has been emptied can now be filled with something new, something luminous, a wholeness is created in the wake of exiting pain.\n\nEGO STATES AND SEXUAL POSITIONS\n\nAs you travel deeper into the sacred territory of your lovemaking, you may notice him taking on a younger ego state. Listen to his choice of words; notice his playfulness and innocence, the feelings and emotions rising up from deep inside. If he is willing to make love from this place, a younger ego state, he needs to feel you are there to receive him, feel your love and caring of him.\n\nChanging sexual positions facilitates a change in ego states. And these different positions can alter, or alternate, the male\/female energies inside us. For example, the male on top, face-to-face position puts the woman in the traditional place of being receptive, open, inviting, and receiving the male. Being on the bottom is a position of vulnerability, submissiveness, if you will, and it naturally allows for younger ego states to emerge. The man on top from behind can bring to the surface unresolved issues of powerlessness and the rape element. It can also satisfy a need to be dominated or overpowered, to feel the strength of him pushing on you while in a position that compromises your strength.\n\nThe position most likely to access and facilitate _his_ feminine nature is, as you might have surmised, the female-on-top position. He is then not only symbolically and emotionally, but also for real, in the position of receptivity. If he is willing, he can move from symbolically receptive to authentically receptive. That is to say that he makes a conscious choice to be receptive, to open up to receiving your love and letting it fill him.\n\nThere were a few times I became keenly aware of, and quite surprised by, the male in me inside my body, the strength of that part and, at times, its forcefulness. I felt the male in me using my arms and hands to hold my lover strong and tight. It took some time for my partner to get used to so much male energy between the sheets, but he spent time talking to the man of me, so much so that it feels at home being with him and is not shy about influencing our lovemaking. When I feel more male energy in me, I still love having female sexual organs. When I'm on top making love, I tend to have more male energy, but it is not the rule. Often I am the older, mature woman of me. Rarely, in my experience, does that position facilitate a younger ego state.\n\nWhat is really fascinating is that as the relationship develops in strength, trust, and friendship, as your bodies and sexual organs become friends, it's fascinating to notice that his penis feels like it's yours. There often comes the moment that you can't tell who has the vagina and who has the penis. They are, and the two of you are, deep in the ecstatic state of lovemaking. The more your bodies speak to each other, the more your movements are exquisite and flawless as a finely choreographed dance, the more ambiguous ownership of sexual organs.\n\nWhich ego state is present during fellatio? It may be quite a different one during the giving than the receiving. Ingestion of semen has a mystical and potent history. In many cultures, it is considered empowering to the woman and feeds her subtle bodies. In every culture, it has the potential to increase intimacy in the relationship. It also has the potential to extend lovemaking.\n\nBut, don't take my word for it. Find out for yourself; that's where the fun is. As you play and experiment with changing positions, notice how you feel when you are on top. Do you feel more or less masculine, more or less vulnerable or powerful? More deliberate in your moves? How does he feel to you when you are on top? Does he feel softer to you, more yielding, more feminine, open, and receptive even? When you are on the bottom, beneath him, how do you feel? Do you feel more or less vulnerable? Which one do you favor and is the most fun? Which one is more frightening, uncomfortable, and brings up the most \"stuff \"? Does he look younger, older, harder, softer, or more mature in one position than another?\n\nThere may be physical limitations that prescribe which positions are the most comfortable if, for example, you have a physical disability of some kind. Level of fatigue may dictate the choice of positions as well. If your energy levels are low, try facing each other on your side, front to front or front to back. The important thing is to keep your heart field present in each position.\n\nGive each position your attentive noticing. Which ego states come out in which position? Follow that energy; let whoever wants to be present, be present. You're not in charge; you become, if you will, your own spiritual midwife, witnessing, holding, allowing younger parts of you to emerge, say what they want to say, ask questions they need answers to. Let your Child's wonder, curiosity, and enjoyment of sex rise to the surface. All the while, your heart fields are wrapped around each other, your spirits entwined as much or more than your physical bodies are, and engaged in soul-to-soul sharing. Return often to a state of vulnerability and na\u00efvet\u00e9 (it does get less scary). Notice if any defenses begin to arise in you, any tensions in your physical body as well as emotional and psychological armoring that attempts to lock into place. Over time, you may notice that ego states shift position; they are not locked into one position. They are dynamic, evolving, and can move around as they become more integrated in who you are.\n\nAct as if you're making love for the first time, and you're being present with every nuance, each movement, noticing everything and every detail in yourself. It's crucial to undefend moment by moment, opening more of yourself, more of your body and your deepest self, letting him enter every part of you. As his penis enters physically farther, his spirit, his soul essence, his love, his caring, his gaze are also penetrating the deepest oceans of you.\n\nNotice each ego state that makes herself known. The farther he gets inside you, physically, emotionally, and spiritually, the greater the chance for deeply held ego states to emerge from being touched. See your lover through these eyes, the eyes of a young child who loves to look, to feel, and to see. The one who can see deep into your lover's interior world is intuitive and sensitive to what is happening.\n\nAs a fine woodworker knows the defining characteristics of each unique piece of wood he works with\u2014the grain, color, smell, and texture\u2014so too can you know the intimate and defining details of your lover inside and outside. The woodworker takes care to listen to the medium of his expression, waiting for the moment the thing reveals itself. It is no different with a human being; it is relationship and communication.\n\nDon't be afraid to talk, to say how you feel, how much you enjoy the feel of him inside you. The more you talk, the less you'll stay hidden, and the deeper the sharing can go. Continue to be aware of the energy of your heart wrapping around him. Say out loud all the things you enjoy; your favorite positions, how his lips feel, how the two of you kiss. Explore each other's skin, levels of touch, the contours of your bodies. You're beginning the journey of knowing your lover, the geography of his body and the secret territory inside. Tell him all the things you enjoy about him\u2014the way he touches your skin or the way he enters a room. Say out loud what you love about his body and his skin. Go into detail; find details in him that you love, that you adore, that you look forward to meeting each day. Murmur admirations as your fingers whisper slowly along the luscious, wonderful, erotic, curves of his body. Let your fingertips fluidly trace the movement of curvature where neck and shoulder meet and caress each other; continue along his arm to the curve of his hips as he lies on his side; linger in the valley, that sweet sway in the small of his back, drawing spirals with the tip of your finger. Follow the serpentine curves as your two bodies' entwine. How is he responding? Does he like his ears rubbed? Does he let loose a sigh when you touch certain parts of him? As you're tracing the lines of his body, you're also noticing how different areas of his skin have various textures. There are the soft parts of his body, his stomach, his ass, under his arms. Thicker, rougher skin is on his hands, his forearms, maybe his thighs. It's your caring, your love for him, your desire to know him intimately, to know the details of his inner and outer life, that slows you down to notice, to see. Adore each part of him. Fall in love with each one over and over, with abandon.\n\nHe will show you what shadows he casts that illuminate parts of his interior world. The behaviors he acts out, his habits, the choices he makes whisper at what is in the depths of his character. He may show you his thirst for intimacy and fear of it. Is he afraid of speaking in public? Is he uncomfortable being alone? Does he prefer to start his morning quietly and slowly, or does he wake ready to meet the day? What is his relationship with food? Does he love romance? Does he cry at movies? Is he comfortable with you crying at movies? What angers him consistently or spontaneously? What is his relationship with his family? Do you know what soothes him, inspires him, and excites him?\n\nWhen you begin to see below the surface to places that were once hidden from view, when he begins to let your love travel inside to touch his deep child-self, an invisible thing is present there, between the two of you. That invisible thing is trust, unseen yet casting a shadow, a hint of its presence in the look of an eye, the touch of a hand, a secret shared. Sometimes fragile, sometimes strong as steel, trust is a foundational structure upon which the building of the relationship grows.\n>  \n> \n> **PART 3**\n> \n>  **Challenges You May Meet on the Road**\n**10**\n\n _The Dance of Trust_\n\n_Trust yourself. Create the kind of self that you will be happy to live with all your life. Make the most of yourself by fanning the tiny, inner sparks of possibility into flames of achievement._\n\nGOLDA MEIR\n\nAs you may have discovered by now, the process of getting clear, becoming aware, and knowing the deep parts of you as you practice sacred sex and intimacy can occasionally get a little rocky. You may feel off balance, disoriented, and you may even feel that you are shaken to your core. The thing is, when the boat rocks, you never know which way it is going to tip. Rigorous self-examination and trust in the process, yourself, and your partner are crucial. Then, it matters not which way the boat tips if trust is in the framework. Trust cannot exist without the possibility of betrayal, the two coexist on a continuum. Only those we trust at a primal level\u2014ourselves, lovers, husbands, wives, sisters\u2014have the power to betray us. To think it is possible to have any sort of relationship with absolute trust, containment, and security is to place yourself in a world that does not exist.\n\nThe degree of trust between people directly relates to the degree of intimacy they share. Trust and intimacy are dance partners, spiraling, embracing, and weaving the couple into a tighter matrix of relationship. Inside the spiraling dance is the embryo of freedom; freedom to voice that which has been silenced, freedom to bend, to choose, to ask, to feel, to express. There is freedom from games, codependency and drama. And when you feel free to be spontaneous, your life becomes less a life filled with regrets, resentments, and searching and becomes more a life of expressing who you were born to be. Your life is authentic, and you are authentically being. All that you do is a reflection of you, and _you_ are in all that you do.\n\nThe most immediate and deep healing I've experienced was with someone I completely trusted\u2014someone who I knew could betray me because of that very trust, but who I chose to trust nonetheless\u2014someone who saw all the broken and lost parts of me, as well as the intelligent, witty, intuitive, and charming parts. It was having all of me seen, acknowledged, and welcomed into the world, having my secrets received, that allowed parts of me to wake up, that enabled me to come home and begin healing and integrating.\n\nBREAKING THE TRUST\n\nTrust isn't a word or concept that normally comes up in everyday conversations. When it does, it has weight to it. Often, it's mentioned in the context of feelings of betrayal, broken trust, and broken promises or agreements. Trust is often taken for granted, meaning that it is not actively maintained. As we move into any new relationship, there is often a brief period of suspicion, doubt, and hesitation while the Child of you is checking out the other person. At some point, she'll make a decision that she is willing to extend trust to this person and explore the relationship.\n\nTrust is always a decision. If you were to trace back incrementally how you got to the point of trusting someone, you would find a series of moments when you scrutinized the other person out of the corner of your eye, ran all the feelings you were having through your inner council, and got your Adult's information and the Parent's blessing. You would find the moment at which you made the decision based on intuition and feeling and whether or not the person's language was congruent with his behavior. Your Child is calibrating all these things whether you are aware of it or not.\n\nThere are few gray areas in matters of trust; either it is actively being cultivated or it is eroding; either there is trust or there is betrayal. You may find that you trust someone with your deepest self and be watching them at the same time.\n\nMost often we don't consciously think about trust unless we feel someone has done something that causes us to distrust them. They have abused our trust or gone unconscious and let a mean or mad part of themselves say hurtful things out loud. It takes time to rebuild trust after it's been damaged. This is true whether the damage to trust is to the Child in us or the Child in the other person. We fracture or break the trust of the Child in ourself by not keeping our agreements with her, by abandoning her, or by undermining her trust in any number of small ways. We seldom consider the importance of building and maintaining trust in our own self to take care of our own security. James Hillman says, \"What we long for is a situation where one is _protected from one's OWN treachery and ambivalence._ \" Trust should be treated as a real thing that needs tending for it to grow and strengthen the relationship. Getting to the place in a relationship where the two of you are able to relax into your bond, with trust as the adhesive, liberates both of you to be who you are with each other. Trust in yourself frees you to be spontaneous with a diverse palate of responses in your repertoire. Being in a relationship with someone whom you don't trust, or where trust has been damaged and not repaired, diverts energy away from intimacy.\n\nWhen two people have a falling out, it usually starts with one person getting irritated about something. Say you are upset and decide to say something to your partner. But without the skills and experience of interior work and knowing yourself, a part of you, an unconscious part, takes over and says something unkind or damaging, even. Typically, your partner will respond with something equally hurtful, and then the two of you are off to the races. It's common in this scenario for old stuff to come out in the argument. You may bring out things you don't like about your partner, or all the other times they said something or did something hurtful. The reason this happens is that the part of you that spoke has been keeping a file on all the little irritations or slips of the tongue, broken agreements, or promises not fulfilled. In the Transactional Analysis model, it's called saving stamps.\n\nThat part of you\u2014a paranoid part, most often the Predator or the unhappy Child\u2014is keeping score, building an arsenal, and sharpening her blade. If a counterpart in him has been doing the same thing, the argument escalates, one of you leaves slamming the door, and all is quiet for a few hours. This is a version of what Eric Berne called uproar. You may be able to maintain distance while a cool breeze wafts between the two of you for a day or two, until it gets too uncomfortable, and one of you has to say _something._ Often it's just enough to break the ice that's been growing thicker: \"I'm sorry we had a fight.\" \"I'm sorry I yelled at you. I've been really stressed lately.\" Then you hug and go about your lives. Or, you have the most amazing, intense sex of your relationship. Often one of those or a combination is enough to return you to the status quo. A lot of us live our lives just that way.\n\nUnfortunately, in this scenario, even though the two of you are able to cohabitate and return to \"normal\" life, there is a tremendous amount of accumulated debris between you. The part of you that was stressed\u2014if in fact that is the truth, for we often lie or gloss things over to avoid the real problem\u2014and lashed out has not gotten what she needs and has unresolved anger, rage, grief, or fear. The pressure has been released for the time being, but the underlying issue is, once again, being put under a lid. Russell Brand, the English comedian and disc jockey, said, \"You know, these relationships we 'ave, everything sort of bubbles under the surface. No one ever says what they actually mean, do they? It's all a bit pappy and rubbish.\"\n\n **One Source of Bad Information**\n\n _There's a boy in you about three_\n\n _Years old who hasn't learned a thing for thirty_\n\n _Thousand years. Sometimes it's a girl._\n\n _This child had to make up its mind_\n\n _How to save you from death. He said things like:_\n\n _\"Stay home. Avoid elevators. Eat only elk.\"_\n\n _You live with this child, but you don't know it._\n\n _You're in the office, yes, but live with this boy_\n\n _At night. He's uninformed, but he does want_\n\n _To save your life. And he has. Because of this boy_\n\n _You survived a lot. He's got six big ideas._\n\n _Five don't work. Right now, he's repeating them to you._\n\nROBERT BLY,  \n _M ORNING POEMS_\n\n ** EXERCISE   \nFollowing a Feeling to Its Source**\n\nWhenever you feel something pushing at you from inside, something trying ever so persistently to get your attention, don't ignore the feeling. Follow the feeling to its source inside you. Start by asking yourself, \"What _is_ that? What is going on?\" Turn to face the feeling. If you had to give the feeling a name, what would it be? Mad, sad, glad, or scared? Depressed, grieving, terrified, or lonely? Ask your Child to come and sit with you; ask her to tell you what's going on. She will know. There will be a need she has and wants your help with it. If she's scared, she needs to feel not scared or at least she needs to know that the grown-up part of you will do everything it can to help her feel less scared. She may be sad, in which case you ask her, \"What do you need to feel happy?\"\n\nIf you've been working long hours, she may need physical strokes. If so, ask your partner if he would lie down with you and give you some strokes: stroke your forehead, rub your feet, rub your back, massage your head. Notice how you begin to relax; your breathing lets go in deep exhalations. If you've been holding a lot of emotion back to get through a difficult time, there may be an upwelling of uncontrolled emotion when the love coming through his touch reaches the Child inside you. You're being vulnerable and being held in the love of your partner, and your Child is being held by you, taken care of by you. It may feel so good to her, she may just want to weep a little. Let her; let the tears flow, or the laughter, or the joy rise up and the stress leave.\n\nIt's absolutely not necessary to wait until you know exactly what's going on for you. In fact, it's important in the beginning, until you gain facility with interior ego states, feelings, and emotions, and become intimate with yourself, to start saying out loud that you don't feel like yourself and it's distracting you from being present with your partner.\n\nYou may even say to your partner, for he has likely noticed that you're not feeling yourself, \"I'm a little distracted right now. I feel funny, just not myself. Will you sit and talk with me so I can figure this out?\" Or, \"Will you sit and hold me?\" It may come to light as you're describing how you're feeling that you've been afraid of the upcoming business trip he's about to take. Or you're disappointed about something. Or you're afraid about your finances. It could be anything, but you won't know if you don't look and feel. Trying to work it out inside yourself is possible, may be necessary, and is a valuable skill to have. However, if you are in an intimate relationship, enlisting your partner's help and support\u2014a shoulder to lean on and a loving ear willing to listen\u2014strengthens the relationship and takes the pressure off you. Unshielding yourself to create intimate moments can be a wonderful aphrodisiac.\n\nMENDING THE TRUST\n\nOnce you have betrayed the trust of another, you must do interior work to analyze the dynamics of your behavior; what went unattended, what part was upset, what needs were not met? Then you must own your behavior and then do the necessary work to clean up the mess you've made in the relationship and make amends. It's important, if at all possible, assuming your partner is still talking to you, that he is involved in the repair; that he gets what he needs to feel better with you. It may take time to decide to trust someone again, to reestablish intimacy, but it can happen.\n\nTrust can be repaired if the other person, and the Child inside him, sees that you are taking responsibility for your actions. Saying \"I'm sorry, I didn't mean to say that,\" might work for some people. It no longer works for me to say it or accept it. When I look at my transaction honestly, with a desire to truly know myself and be whole, I know some part of me meant to say, or do, that thing. It might be that I have been letting pressure build with no strategy in place to relieve it. Parts of me were getting really tired of not having a break; I went unconscious and quit thinking and paying attention. Replaying the event puts me in relation with the movements and communications. Only when I'm in relation to it will I have options from which to do something different.\n\nThis is where maintaining eye contact is essential. It's awkward and uncomfortable in the beginning. Scary, really. Maintaining eye contact while you make amends communicates your willingness to repair trust and intimacy. There will be a strong tendency to want to hide and feel ashamed for screwing up. It's important to not give in to the lure of indulging in feelings of shame but to proactively work to make the relationship whole again.\n\nIf your partner has hurt you, tell him what you need from him to feel better and begin to be friends again. Say everything out loud: \"What you said or did really hurt me. I need to know why you did that so you will not do it, or any version of it, again.\" Sometimes a gift is needed to make up such as: \"I want you to buy me a new futon,\" or \"I want you to take me out for a really nice dinner.\" What you ask for should be equivalent to the damage done. It's important to have your Child involved in the request. What does she need? What would it take for her to feel counted again? Breaking trust, intentionally causing hurt in someone, is a discount to the Child and to the relationship. So reparations need to take both those things into account.\n\nWe all make mistakes and we have all been unkind at times. To heal the damage done from going unconscious, it helps to work at not feeling bad about it, to not wallow in the hurtful thing you did, not indulge yourself in carrying the bad feelings around. Take some time to talk to your Child so she doesn't walk around feeling bad. It doesn't change anything, and it expands the bad feelings between everyone involved for longer periods of time. Resolve it inside yourself as quickly as possible so movement toward intimacy can happen sooner. If necessary, put the bad feelings aside until later, work to heal the damage and the relationship first. That is primary in this scenario.\n\nI had a habit of beating myself up for days after an incident of going unconscious and being unkind. I was a student of self-loathing; it was a familiar behavior. It took time to come to terms with what I had done, but mostly it took time to give up feeling like a piece of shit and reinforcing old scripts about my self-worth. It took a lot of work to come to terms with the predator in me, to integrate that part so she didn't act out. It took time for me to really get that I am capable of hurting someone emotionally. There is a need for rigorous self-examination, of listening to the voices inside, of watching your own movements and taking action to prevent any part of you that is upset from using your mouth. And then you must learn to self-correct in the moment. It's work, particularly in the beginning, and it gets easier. The more you do this work, the fewer mistakes you'll make and the less hurt you'll cause, and the more secure and confident you'll become in yourself.\n\n _You can be in a huge crowd, but if you don't feel like you can trust anybody  \nor talk to anybody, you feel like you're really alone._\n\nFIONA APPLE\n\n _Today I trust my instinct, I trust myself. Finally._\n\nISABELLE ADJANI\n\n _It is impossible to go through life without trust: that is to be imprisoned in the worst cell of all, oneself._\n\nGRAHAM GREENE,  \n _T HE MINISTRY OF FEAR_\n\n _Someone to tell it to is one of the fundamental needs of human beings._\n\nMILES FRANKLIN\n\nINJUNCTIONS\n\nAn injunction is a writ or an agreement some part of you made to refrain from or forbid an act or speech. Injunctions around behaviors or stories are often made by a very young part of ourselves who made a declaration as the consequence of some painful experience. We have all made some vow, some agreement with ourselves. Injunctions range from never being vulnerable, never depending on anyone, never getting married, never trusting someone ever again, never sharing any intimate part of yourself or your story. Mine were to never trust or depend on anyone and to keep everything to myself. I had injunctions about how to behave in a relationship. I believed that the less that was said, the less opportunity there was for drama, misunderstandings, and miscommunications. The more I kept to myself, secret and private, the more likely the fragile little boat of the relationship and my life would withstand the storm.\n\nI had injunctions about trusting men specifically and people in general. I had decided that whatever I was going to accomplish in my life, I would do it alone. I was the only one I could count on, who would show up and do all the work. A friend called it the Lone Ranger syndrome. It worked pretty well for a while, until I began to burn out. I was on the verge of complete exhaustion, with kidney pain, adrenal fatigue, and massive headaches. My hair was falling out by the handful, and my feet hurt all the time. I had to reexamine how I had structured my life before I became seriously ill. I began to really look at my birth family, our dynamics and unspoken rules about behavior, and consciously chose to do something different, to break the rules, including the ones I had made. As you can see, I've renegotiated those old family agreements. Saying forbidden things, breaking the rules, has terrified me but also liberated me.\n\nBreaking injunctions, conscious and unconscious ones, changes the dynamic with yourself. Breaking an injunction about saying secrets out loud without talking it over with the parts of you that originally made the injunction may have serious consequences. Those parts of you can and will make a mess of things for you to get even and to get your attention. They don't like being left out of such life-changing decisions.\n\nYou may hear things like: \"You shouldn't have said that. You're going to be sorry.\" \"That'll come back to you.\" \"What were you thinking?\" \"You moron!\" The part of us that is scared gets activated and escalates the fear inside. Panic ensues. And you start bargaining with parts of yourself to calm the fear.\n\nHans Hofmann said, \"The real challenge consists in being so true to oneself that sharing oneself nakedly with another person will be unselfconscious and honest, not marred by exaggerated expectations or apprehensions based on past disappointments or unfulfilled fantasies.\"\n\nSAY IT OUT LOUD\n\nEveryone has secrets. Secrets are kept for essentially one reason: fear. Fear of abandonment, fear of being hurt, fear of being seen, fear of being too weird, fear of someone you care for being hurt are all powerful motivations to keep secrets. Secrets themselves can be a source of fear, fear of being found out. This does create a dilemma for anyone who wants to have intimacy and yet hasn't found a way to work with these fears. The human animal is a tribal species. We need other human beings for physical and emotional touch. We are not meant to do this dance of life alone. We may choose to retreat to the wilderness on a solitary quest to heal ourselves and, while out there alone, experience the spirits pushing on internal wounds and the crooked places inside, helping us to grow. But we cannot remain in the wildland alone. Without the touch and love of another human being, we would become physically and psychologically ill.\n\nIn a _A Blue Fire,_ James Hillman expresses it this way:\n\nHow can we know ourselves by ourselves? We can be known to ourselves through another, but we cannot go it alone. This is the hero's way, perhaps appropriate during a heroic phase. The opus of the soul needs intimate connections, not only to individuate but simply to live. For this we need relationships of the profoundest kind through which we can realize ourselves, where self-revelation is possible, where interest in and love for soul is paramount, and where eros may move freely\u2014whether it be in analysis, in marriage and family, or between lovers and friends.\n\nWe all have a thing or two we want to keep close to our heart, something we may even take to the grave with us. If you have such a secret, it's important to find a way to have true resolution and forgiveness in yourself. Then make the conscious choice to keep the secret. The important thing is to choose from free will and not out of guilt or shame. Be attentive to the nature of the secret. Secrets have the potential to do damage to your deep self and to your relationship. Talk it over carefully with all the members of your inner council. The movie _Get Low,_ with Robert Duvall, is a magnificent story of one man's redemption from a secret he carried for forty years, one that he wanted peace with before he died. It's well worth watching.\n\nSecrets have a power of their own. Family secrets that have been kept for generations have a way of insinuating themselves into the psyche of current or future family members. Secrets such as the fact that your mother had an abortion before you were born, your aunt was hidden away in a psychiatric hospital, your great, great grandfather kept slaves, all have a way of slowly eating away at the fabric of family life as well as at the individual family members.\n\nThere is a distinct difference between keeping secrets, which is intentionally hiding information, and maintaining privacy. Privacy is \"the right to maintain a nonrelational sphere of existence.\" For example, when you keep sexual acts between you and your lover, husband, or partner away from the eyes of your neighbors, you are exercising your right to privacy.\n\nSecrets can be used to create distance; for example, if you keep secret that you were beaten in a previous relationship, you might fear being physically vulnerable. If there was emotional abuse, you may be calculating how far you're willing to extend yourself emotionally. In these scenarios, you are holding back parts of yourself. If you secretly have judgments about your body, your sexual organs, an ugly part of your body, the inhibition you feel as a result will directly impact your sexual relations and your enjoyment of them as well as the energy you have available for enjoying life.\n\nIn relationships, keeping secrets about how you're feeling, what's working, what isn't working, what you need and want in the relationship and for yourself leads to loss of self-dignity, erodes intimacy, and creates loneliness. To have the depth of intimacy and sacred sex we've been exploring requires that we say things out loud. Speaking things out loud can be scary if you feel you are breaking personal beliefs, family, and cultural scripts about what is acceptable to say out loud and what is not.\n\nSecrets _can_ make you feel like a phony\u2014inauthentic, isolated, angry\u2014and increase anxiety. Being shut down from our feelings makes us feel disconnected and faraway. If you are secretly dissatisfied with how you and your lover have sex but feel too afraid to talk about it, resentment will start creeping into the relationship. Talking with your partner in a loving way about what each of you likes, doesn't care for, or would really love to have increases intimacy and sexual satisfaction. Sharing things that trouble you, scare you, or bring you joy and excitement increases trust and strengthens the bond.\n\nSharing secrets can build intimacy, but care still needs to be taken about what you tell your partner and how you tell him. Asking yourself the following questions can help you get clear about what to share and the motivations behind your decision to share it:\n\nIs sharing this going to bring us closer or will it cause a rift between us?\n\nWill saying this add to the life and wholeness of the relationship?\n\nHow can I say this thing without threatening our relationship?\n\nDo I need to tell this for my own health and wholeness?\n\nWill this secret push on his sensitivities?\n\nWill this cause harm to my partner?\n\nIf you decide it's going to bring you closer, practice inside yourself saying it to your beloved or whoever it is you're going to tell it to. Imagine the impact you want it to have, the potential outcome, and how it might be received. Own the secret as yours without blaming or projecting.\n\nSEXUAL SECRETS\n\nThe scripts we were given as children are often involved in the dynamics of secrets. They can be used to maintain the script\u2014the beliefs we have about our bodies, ourselves, our sexuality and sex. For example, not telling your lover you have herpes or that you need added lubrication are shame secrets we have about our bodies. Marty Klein says, \"Finally, using sexual secrets to fulfill childhood scripts is very costly . . . scripts emerge as the mind grapples with a single goal: preventing childhood pain. So scripts specifically ignore the needs of contemporary, adult situations.\" When we hide our condition or our needs, we feel isolated, and it reinforces old beliefs that something is wrong with us and that something bad will happen if we reveal our secret. We keep secrets, we unconsciously think, to protect the child in us from harm. It actually creates the situation we are trying to avoid\u2014feeling lonely, isolated, and unworthy. We can't protect the child from pain that happened in the past. Attempting to use secrets, sexual secrets, to protect the child gives us a false sense of power over our situation and relationships. It goes nowhere good. It degrades trust in the relationship and mistrust from the child to you since she's not getting her real needs met. Trust produces intimacy. Giving up old scripts produces freedom to choose.\n\nTelling our sexual secrets and fantasies dictates that we give up ideas about what is proper and appropriate. That can be challenging since our society has many cultural rules about what is proper and appropriate behavior.\n\nAbout every six months (usually after two glasses of wine), much to my partner's fascination, I loosen up my restrictions about telling parts of my past sexual experiences to him. Just when I thought I'd said everything, another few months would pass and we would be having fun\u2014enjoying each other's company, bantering witticisms, telling stories, and sexually flirting\u2014and I would spontaneously tell him about more of my experiences. When I first began doing this, it was fun in the moment, but the next day, afraid I had said too much, I would shyly go to him and ask if he still liked me and did he still want to be with me. I had to ask so the uncertainty would not become a caustic eating at my conscience.\n\nMany of us have sexual fantasies that trigger shame and guilt inside us. This often has roots in religion, when children are led to believe that God magically hears _everything,_ even our most secret and private thoughts. Having sexual fantasies is normal though there are many messages in the media and from religion that tell us otherwise. It's all right to have fantasies about actors since they are unavailable and can be objectified. But having a sexual thought about your neighbor, that's a little too close to home. Fantasies and thoughts are not bad, having them does not make you a bad person; however, acting out fantasies can be damaging to our relationships, our families, our lives.\n\nTalking about sexual fantasies can actually bring people closer; they are intimate thoughts that can enhance a relationship and deepen trust. Noticing that someone looks attractive, sexy, or seductive doesn't mean you're going to run off with him. It means you're not dead. Making comments about it all to your lover or partner is a sign of trust and comfort-ability. You can make it playful and fun. Go back and forth: \"Well, what about that woman. Do you find her sexy? What do you find attractive?\" I know this has potential for escalating into something you don't want. So, be clear in yourself before engaging your mouth. Playful is the best approach; the more playful, the more fun it will be. If you are unable to be playful, it's best to wait until you are filled with love and feel self-possessed; that is, confident and not threatened by outside forces. If you and your lover are able to do this, it can turn into a kind of flirtation between the two of you.\n\nTRUST, INTIMACY, AND SACRED SEX\n\nIt's actually quite a lot of fun when you are able to have a relationship with all the parts of you, to have facility with ego states and their needs, wants, and desires, and to know how each one expresses herself. Even more fun when you can cathect\u2014put energy into or activate\u2014any ego state or state of mind at will. Saying things out loud is a pathway to building trust and engages the other person. Share your thoughts, hopes, dreams, and insights with him. Getting to know each other this way, showing yourself to another person, communicates that you want to have a deeper relationship, that you trust enough to share your deepest self with him. Talking out loud is a way of thinking out loud. When you talk with your awareness in your heart, it opens up the imaginal realm to seeing things you hadn't previously.\n\nDuring sacred sex, as you are arousing each other\u2014talking about what feels good, sharing your love and feeling your hunger and desire building\u2014maintain eye contact, especially when the temptation to close your eyes is greatest. Keeping your eyes open opens wounds, brings them to the surface to be seen and received by the other person. There can be tremendous emotional releasing. It's important, in fact, that you allow everything you are feeling to be felt.\n\nMaintaining eye contact increases trust, healing, and bonding. You can see the other person, see his love of you, see that he is trustworthy. With your eyes open, gazing into your lover's eyes, it's easier to stay present with your partner and with your feelings and body. Notice everything that comes up and stay with it; let it move through you. Let shame, discomfort, and fears have their place, their time for healing. Allow your child to be fully present here.\n\nClosing your eyes takes you inside yourself, and it can be used to shut down the feelings that are arising when things get too uncomfortable. Honor everything that is happening, attend to every detail. Once you have facility with the territory of the sacred in lovemaking, it gets easier to stay present with what's going on between the two of you, what's happening inside you, what's happening inside him, where you travel inside him, the communications between your bodies.\n\nAs you continue to work with the different parts of you, you'll become more experienced, sophisticated, and adept in your responses to your child's need and wants. You'll also be more flexible in responding to old wounds, memories, and issues of trust and shame that will inevitably come to the surface at some, or several, points along the way. As well, you'll become more elegant in your communication skills, analysis, and responses with your partner as you become more whole in yourself, and over time, you will have a greater repertoire of verbal and behavioral options.\n\nYou may doubt what your Child is saying, your perceptions and sensing. It's always that way in the beginning. It's tenuous as the relationship begins and you're sorting out who's who inside. Trust yourself; you'll discover you know more than you think you do. You will find your way. Keep going. Ask questions and feel the responses in your body. Your ability to calibrate and attune to the nuances and subtleties will evolve into a system of sophisticated conversations and insights the more you work with ego states, feeling, and perceiving.\n\n _Once you trust yourself, then you will know how to live._\n\nJOHANN WOLFGANG VON GOETHE, FAUST\n**11**\n\n ** _Healing Shame_**\n\n_I never learned hate at home, or shame. I had to go to school for that._\n\nDICK GREGORY\n\nRecognizing shame is easy; you feel the impact of it immediately. Your flushed cheeks may demonstrate outwardly your feelings, and the last thing you want at this uncomfortable moment is to be betrayed by your own physiology. Shame nags and gnaws at you, erodes your dignity, and inserts questions and doubts about your character. Feelings of being a bad person or an inadequate human being or having done something wrong permeate your thoughts and drain energy away from you. You'll spend hours, days, years trying to understand what happened and, mostly, how to unhook yourself from that miserable feeling inside you. The more complete your intimacy is with yourself and the more whole you are in yourself, the quicker you will find resolution to the matter and be able to derail shame before it becomes a runaway train carrying your dignity with it.\n\nAs you begin work to heal shame, an interesting thing happens as it does with any issue you are bringing to conscious awareness to resolve. Inevitably, because the Universe has a sick sense of humor and impeccable timing, people and events will magically plop themselves in front of you for the sole purpose of bringing up the issues you've decided you're ready to be done with. These people and events have an uncanny ability to shoot a stinging arrow smack in the middle of your shame button. It is one of the most irksome phenomena of healing psychological wounds.\n\nWORKING THROUGH SHAME\n\nMy primary shame was around my sexuality, sex, and my body and feelings of inadequacy, of taking up too much space, and of not wanting to be a burden. Of course, there came into my life a woman who knew exactly how to shame me just as I was beginning to taste freedom from it. Everything I had shame about was pushed on in a single transaction. The particulars of what happened aren't important. What is important is the meaning in her transaction with me, which informed me that I didn't have a right to have needs, to be sexual, to ask for what I wanted, or to be happy. What it felt like to my Child was that I didn't have a right to be alive and to be who I am.\n\nIt was impossible to keep my balance, so after the shock of it wore off, I fell apart. I wanted to hide, run away, escape, and shrink into oblivion. I withdrew and nearly went catatonic, unable to move or breathe. I believed that if I breathed or moved I'd be seen, and once seen, something terrible would happen again. I shut down and was unable to think or analyze. My feelings became a tangled mess like those balls of fishing line left caught in a snag of branches.\n\nIt's difficult work to get out of that space. But it is possible. I untangled the transaction in reverse, one inch of line at a time, until I saw where and how it all began. Only then could I see that it wasn't my fault; I had done nothing to provoke it. \"It's my fault\" is the first place I would normally go. The woman who had hurt me had probably been scared or upset. But that didn't matter to me; what mattered was that she had done and said those deeply hurtful things. I told my Child that people are just mean sometimes and that we hadn't done anything wrong. I told her that I and others were immensely glad that she is alive.\n\nI learned to go through this laborious process every time I experienced shame. I would feel it, go into shock and shut down almost entirely, then come back to myself, analyze the transaction, unhook it, and get mad.\n\n _When will my shame fall away?_\n\n _When will I accept being mocked_\n\n _and let my robe of dignity burn up?_\n\n _When the wandering pony inside_\n\n _comes calm to my hand._\n\nLALLA\n\nShame has a physical response in the body and is a crisis of the soul. It's similar to embarrassment, but shame insinuates there is something abhorrently wrong with your moral compass. Simple embarrassment passes; you've committed a social faux pas, a breach of etiquette. Interestingly, it's usage in medical context is to indicate physiological distress as in respiratory embarrassment. And in Portuguese it comes from _embaracar_ meaning \"a noose\" or \"rope.\" Indeed, embarrassment and shame do feel like a noose, choking the life out of us, stealing our dignity and our energy. We feel humiliated, mortified, disgusted with ourselves, unworthy, disempowered, degraded. These are ugly, debilitating feelings to have in any moment, worse to carry them as a way of life. We lose our sense of self, and access to our strength is severely limited when those feelings are present.\n\nShame sticks like black tar. It causes significant physiological responses in the body. Your face gets red and feels warm as the blood vessels dilate, the sensation sliding up your face to the top of your head. Breathing gets shallow and moves up into your chest, with accompanied heaviness and tensing in the shoulders and limbs as if a tremendous weight has descended on you. It seems impossible to look straight ahead; your head drops and you stare at the floor. Moving becomes nearly impossible. Your only wish is to vaporize or die. Nausea may accompany the symptoms as your stomach tenses up. Your head feels like it's being squeezed in a vice grip. You become keenly aware that the only sound you hear is that of your heartbeat thrumming in your ears. Shame isolates you in the depth of deep loneliness. The urge to cover yourself, to hide, run, or disappear, is paramount. Rodin's sculpture _Eve after the Fall,_ where Eve is depicted as lowering her head and covering herself in shame, is thought to be the origin of the word _shame,_ meaning \"to cover.\"\n\n _Girls blush, sometimes, because they are alive, half wishing they were dead to save the shame. The sudden blush devours them, neck and brow; they have drawn too near the fire of life, like gnats, and flare up bodily, wings and all. What then? Who's sorry for gnat or girl?_\n\nELIZABETH BARRETT BROWNING\n\nShame is that horrible feeling that I have no control over my life; that someone else has control over how I feel and behave, and I am a mere puppet flailing in response. To save myself from shame, I unlocked each belief about myself that I encountered in the midst of shame. With time, self-loving diligence, and a little rage to keep me focused, I could see the roots of shame and where they were anchored inside. \"Oh, that feels like the time my uncle grabbed my breasts and wanted to have sex with me.\" Most of my shame around sex and sexuality I internalized from my family's discomfort with it and their belief that sex should be a secret. And because I had a strong sexuality, they seemed to feel that there was something demented about me, something inherently and morally wrong with me. Family secrets are a common source of shame. Not talking about your aunt being in a mental institution brings up the shame you feel about being different, or the internalized belief that being in an institution is shameful. Just because something is not talked about does not mean it doesn't exist. It is as present as the elephant in the room that everyone ignores.\n\nLooking back to year eleven, if I had felt that I could tell my mom about the hired hand, I imagine I would have made different life choices. I would have felt more empowered in my life to do so. The event wouldn't have had any power in my life.\n\nAt sixteen, I was raped. Telling my parents about it was the most hideous thing I remember having to do as a young girl. Their response was outrage at the man. Fortunately, they didn't blame me. The way they chose to support me was to notify the sheriff. What I really needed was to be held and nurtured and to feel their love. I wanted to be asked what I needed. My family was skill-less in the nurturing department. I hated knowing they knew that I had sex, forced sex, yes, but sex it was. I felt ashamed, weak, stupid, incompetent, and utterly fucking alone. After the sheriff left, the whole subject seemed to be erased. It was never talked about again. It was gone, deleted, vaporized, as if it never happened. I wondered if I was overreacting and began to question if it even really happened. The lack of humanity and intimacy from my parents when I needed it most warped some part of my psyche, and I always felt a little crazier after that, a little more feral.\n\nTheir response seared scars into the depths of me. As a survival mechanism, the sixteen-year-old I was at the time became my primary ego state. She was the tough cowboy (not cowgirl) and biker (not biker chick) and full of rage. She is still a major part of my personality, an active part of me, though she is more balanced, more whole in herself and integrated. She is the one who can smell bullshit before it walks in the door. She protected me for a long time and is still a significant part of me, but now she is integrated with all the other parts of me, and they are all friends.\n\nShe has her own wardrobe, you see: harness or cowboy boots, several black leather jackets including a vintage biker jacket, two pairs of Levis and two pairs of Gap blue jeans (each with a different attitude), and dark sunglasses. And she hand rolls her own cigarettes. She's most comfortable in bars and saloons where the wood is darkened from years gone by when you could still smoke cigarettes in pubs. The lights are soft, low, and kind. The men and women there have a kind of wild hunger in their eyes, nostrils flaring; they've come looking for something. The gods who love this kind of scene are there always. Eros is sitting on one beam in the ceiling and Dionysus on another, and Psyche is hanging out on a bar stool, waiting for someone to sit down and start chatting.\n\nI don't have regrets about any of the things from my childhood. Nor do I carry hate or rage toward any of my family; sadness has taken its place. I'm certain I wouldn't be who I am without them or those childhood experiences. I have grieved for the young girl of me, for the rape, the sodomy, and for the fact that I was so utterly alone.\n\nShame has its functions. People who have experienced shame have a sense of humility. They have an understanding of suffering, humiliation, and their own humanity. It can also teach us about ourselves, our limitations, our weaknesses, the limits to our competence and autonomy. It can show us where we've let ourselves down and can strongly motivate us to act. Also, it teaches us to have a sense of humor about ourselves. It's a fast way to break the shame cycle.\n\nI hadn't thought about shame for some time when something interesting happened. It was a nonevent, really. A person just walked through the room, didn't even make contact with me. It happened to be someone for whom I have a tremendous amount of respect. I was having a beer with the guys a few days before I was to take part in a ceremony. Part of preparing for that particular ceremony is to eliminate alcohol among other things. I had participated in this ceremony several times over the last year, was familiar with it and how to prepare for it. It was still five days out, but here I was, having a beer. When I saw this person (the facilitator of the ceremony), I felt ashamed, like I had done something wrong. Later that night as I was lying in bed trying to sleep, I kept looking at this feeling and the images it brought up; then analyzing it. Why would I feel that? He didn't even see me, as far as I knew. And the preparations for the ceremony are \"suggestions\" (yes, that's justifying my behavior and it's true). I called up my inner council to ask why I was feeling so ashamed. The answer was there immediately: \"You feel this way because you strive for and have a perfection complex. You don't want to fuck up or lose your integrity. Primarily, the root of it is that you want to be the good girl. And, you didn't do anything wrong.\" Ah, that was it indeed, for it rang a clear, resonate tone. I talked to the young part of me that strived diligently in my family to be the golden child, the peacekeeper, and the perfect daughter. It was the little girl of me that felt shame at breaking or stretching the \"rules.\" I let her know that nothing bad would happen, that I understood what she was feeling, that I was sorry I hadn't talked it over with her and for putting her in that situation. I asked if she would like to give up that old script. There was an immediate and vigorous nod and a clear, \"Yes, I want to.\" She didn't have to \"try to be good\" any longer. She could relax and be herself and let her natural goodness flow unimpeded, unforced. She could give up being good for others for the sake of being good.\n\nShame numbs us to our sexual feelings. We feel we are not good enough or have no right to enjoy sex. On the other hand, we may feel we have to \"put out\" in order to be loved, or we overcompensate sexually in order to not feel. We may have internalized the message that \"good girls\" don't have sex, or if they do have sex, they don't enjoy it, and when we do, we feel a deep conflict and shame about it.\n\nTHE SHAME OF SEX\n\nShame is tightly wound into our sexuality, our feelings about sex, and our sexual organs. Shame seems to have a built-in navigation system that heads directly to our sexual organs\u2014the vagina, uterus, penis, testicles. These are organs that, when engaged, make us feel alive, so it's not unusual for shame to bring up \"right to be alive\" issues side by side with \"right to be or feel sexual\" issues. We become ashamed of having breasts or a dick, ashamed of being alive, and we want to apologize for being a man or a woman. We apologize for our life. It's not an accident that shame issues are intimately attached to illness and diseases of the reproductive systems in men and women.\n\nWhen I got the diagnosis of cervical dysplasia, every atom in the room vaporized. The only thing in my line of vision was each sexual relationship, encounter, rape, indiscretion, seduction, and heartbreak (the ones I received and the ones I caused). I felt dirty, betrayed, ashamed. I didn't need to be a rocket scientist to see that my sexuality, my feelings about it, and my sexual history were all, directly and plainly, laid out on the delta of my diseased cervix.\n\nBren\u00e9 Brown researched shame through hundreds of interviews and came up with this definition of shame: \"the intensely painful feeling or experience of believing we are flawed and therefore unworthy of acceptance and belonging.\" When we feel shame, we feel a deep disconnect, a separation from the experience of being alive, wanted, needed, acknowledged. We feel we don't belong, that something is wrong with us, and that \"wrongness\" casts us out.\n\nShame breaks the contact between people and is sometimes employed to disrupt intimacy when it gets too uncomfortable. Or we project it onto other people, it's hard not to when all we see is seen through the lens of shame. People sometimes use it to mask their own sense of unworthiness and unhappiness. It's a cruel game to play. As Nietzsche said, \"What do you regard as most humane? To spare someone shame.\"\n\nWhat I have done that has worked for me is everything that I describe in this book. You can't prevent people from being unkind. You can heal old shame. There will be occasions when you feel shamed or are ashamed, either as a response to something you've done or said in an unconscious moment or from someone else. Recognizing shame can stop you from sliding down into the abyss of self-loathing. These are times when self-love, self-compassion, and self-awareness are crucial. To recover your authentic self from the wounds of shame, begin with what's real. The real is what we are always seeking in our lives. The real is what is true in any moment. It is also that which holds up over time.\n\nThis is not an academic or intellectual exercise. We're going for what's real, and to get the pieces of, to get a sense of the real, before anything else we must feel and think at the same time. Feeling is our body's innate intelligence; it is our internal GPS, our Grounding Positional System. Without it, we are left to our judgments, fears, and old scripts. We are easily influenced and slide (or are jolted in some cases) from our center by outside forces. Truth begins with and inside the body.\n\n ** EXERCISE   \nIdentifying Shame**\n\nThe following statements help reveal internal beliefs. As you say each one, notice how you feel. Does it feel true for you?\n\nI deserve what the person did\/said.\n\nI'm not worthy.\n\nI'm a piece of shit.\n\nI don't belong anywhere.\n\nI deserve to be abandoned.\n\nI am unwanted.\n\nI am unlovable.\n\nI am ugly.\n\nI am dirty.\n\nI am not important.\n\nI am a burden to others.\n\nI don't deserve to live.\n\nI don't have a right to be alive.\n\nBeing born was a mistake.\n\nI am not smart enough, not thin enough, not strong enough, not cute enough, not funny enough, not fun enough, not [fill in the blank] enough.\n\nNotice how you feel with each statement. What comes up inside you? Do you feel angry, justified, arrogant? Do you shut down and deny the statements?\n\nThese are defenses against shame, so we don't have to feel shame's real effect on us. We create defenses to survive the impact. We may resolve to be perfect, so it doesn't ever happen again; we get enraged to divert attention from our flaws; we withdraw to separate ourselves from others and intimate interactions.\n\nThese are all archaic belief systems that have poisoned how we are, who we are, and how we behave. The wounds they cause are deep. We've believed them for so long we have contempt for ourselves. The choices we've made with those beliefs inside have been contaminated. Shame scripts go to the core identity of being a human being. Like computer software programs, they can be rewritten. And it does take time to learn to be a software programmer. Have patience with yourself, be kind and nurturing. Do things that feel good. Shame is part of the cult of secrecy. Talking about it out loud to another person breaks open the secret, diffusing the power of shame. Ask for support and regularly get physical and emotional strokes. When you hear these scripts inside you, it's important to _hear_ them and what the part of you that is saying them needs from you to feel better. Create new, supportive, and life-affirming scripts. Accept your humanity. Being born is not shameful.\n\nWhen we begin to heal shame, whether it came from our family of origin, from relationships, or ourselves, grief will be part of the healing. Grief is a natural response to the losses we experience in life. There is grieving the loss of childhood that never was. We grieve the love and affection from parents that weren't there. We grieve for the pain we caused others. We grieve for how we abandoned ourselves and created a false self. We grieve that we've lived so long without truly knowing ourselves. We may grieve that we haven't grieved. This is especially true for men who are told some form of \"it's not manly to cry or to show grief or sadness.\"\n\nHere is where working with your Child, which is essentially reparenting yourself, is deeply healing. Ask her if she would come and talk with you. Hold her in your lap, close to you. Say to her all the things you needed to hear at her age: \"I love you.\" \"I want you in my life.\" \"I need you in my life.\" Tell her that you're sorry she has carried so much shame and sadness. Tell her that the two of you can talk about things, take walks and explore the world together. Tell her that you are proud of her, how cool, smart, and fun she is. Ask her if she has anything she wants to say, if there is anything she really wants to do. As you have this conversation, really be present, in your heart, so the feeling in the words goes to the deep parts of you, which is where _she_ is.\n\nTo reparent ourselves is to override the damaging scripts and messages we took in when we were children. In this process we can give ourselves new words, new messages, new ways of thinking, and feeling. We give ourselves permission to be alive. Being alive is having feelings, needs, and wants. In this process, we write ourselves anew.\n\nI had decided to take a job as a nude model for the life-drawing class at the university. This decision stirred up every belief I had about myself. The four months before the job began were spent in deep conversations with every part of me, including my body. I was terrified. Every unkind and critical view I had of myself and my body came up. Each day, I would stand naked in front of the mirror and look at what the students would be seeing. It was painful and frightening. I talked endlessly with my Child who was afraid something bad would happen, who was terrified of being naked in front of others. It was one of those life-altering decisions, the kind that you don't fully understand as you make it. Yet you do know that someday its meaning will become clear. I was crying with fear as I left my house the first night.\n\nThe professor hadn't arrived yet, so I sat on the beat-up old couch across from his office. Holding my Child and talking to her, a peculiar thing occurred; I felt a shift begin to happen. I felt bigger, stronger, and more capable, and a sense of deep peace seeped into me. I was proud of myself. Still, I was a little shaky as I walked into the studio awhile later. As I stood in front of the class and let my robe fall away; I was keenly aware that fear and shame were falling to the ground with it.\n\nAlong with reparenting, we are also giving ourselves new behaviors. The words alone won't change much. The old beliefs are anchored in how we hold our body, how fluidly or awkwardly we move. They are held in our breathing patterns, the way we hold our hands, curve our shoulders, and move our hips when we walk. The sum of all past moments is held in _totalidad_ in our current physical body. Though the moments are past, the body remembers the patterns and follows their dictates. Keleman says, \"It's been my experience, as others', that those who are not held enough have a fear of falling and hold themselves stiffly away from earth. Those who feel shame for their sexuality and dislike for their bodily responses never really hold their ground with others. They are always proving themselves or shrinking. They are weak-kneed.\" Until I had intimacy and deep love, I had a paralyzing fear of heights. I couldn't even drive the mountain roads without nearly seizing up. Interestingly, when I felt less than powerful and wobbly, my left knee slipped out of joint easily and painfully. Both of these experiences have been transmuted through the practice of sacred sex.\n\nAs you do the internal work with all the parts of you, I strongly encourage you to get deep tissue bodywork to release the holding patterns. Weekly massages, Rolfing, tai chi, and Feldenkrais work can all facilitate repatterning and letting go of how your body has held beliefs. Watch someone whose walk and movements you particularly enjoy. Follow them without being noticed and begin to take on their walk and body movements, copying them as closely as you can. Notice how you feel with this new movement. When you are out for a walk or stroll, try altering your stride, the movement of your hips, the swing of your arms, the length of your stride until you find something you want to incorporate. Changes will occur in your body as you work with your ego states, as well as with the following exercises.\n\nThe following exercise will take you deeper inside your own body, deepening your relationship with your own piece of Earth, giving you a sense of place inside your own feet. We explored earlier how each part of the body has its own intelligence. Different parts of our body can hold tremendous amounts of emotional, psychological, and spiritual, cuts, scrapes, and bruises, as well as physical stories, injuries, wounds, and pains. Parts of our body can also be sources of tremendous power, wisdom, and strength.\n\nEach of us has a part of our body we would rather not have touched or noticed. Every one of us has a part of our body that we wish were different or completely gone. Most of us have a part of our body that we hate. We walk around hoping no one will notice. We wish we wouldn't notice or have to look at those parts, or that some magic cream will make it all pretty. A friend used to call me _mujer de mas cremas,_ woman of many creams. Creams are superficial. Beauty comes from within, when, at last, love fills all the unloved places.\n\n  **MEDITATION   \nLoving the Body**\n\nSit in a comfortable, quiet place where you won't be disturbed for a few moments. Take some deep, relaxing breaths. Now, see before you the ugliest part of your body. How does that part of your body feel? Is it mad, sad, scared, glad? Ask that part of you if it has anything it needs to say to you. Do you feel shame in or toward that part of your body? Does that part need anything from you? What would it take for that part to be happy? Is there anything you want to say to it? How do you feel seeing a part of your body that has repulsed you and has been discounted for so long?\n\nWhen you're ready, thank that part for coming to be with you and talk with you.\n\nSpend time each day working with this part of your body until it begins to feel happy, and you can feel it integrating into your body. Treat this part just as you have your ego states\u2014as intelligent, alive, aware. It has needed something from you. For you to be healthy, this part needs to be tended, loved, held, and appreciated. Our bodies work hard for us; they are the vehicles through which we enjoy and experience sacred sex, the touch of a loved one, the embrace of a child, earth, sand, grass under our feet, and the sun on our skin. Journaling about this relationship can take you deeper into the meanings, insights, and understandings of that part of your body. Your body is the temple of your soul.\n\nTOUCHING WITH LOVE AND ATTENTION\n\nWhen you touch your lover in that loving way, you may have noticed how he, and different parts of his body, respond to being touched. If you touch on a place he's not comfortable with, he may tense up, pull away, or hold his breath. You may do something similar if he touches, or comments on, a part of you that you don't want to be seen.\n\nHe may volunteer a comment on it, or not. The important thing is to not force the subject but keep the communication open. You could say what you're skittish about, what you love, and what you're shy about. What parts of your body are really sensitive to being touched, sensitive in the sense of \"it makes me really uncomfortable to have my tummy rubbed.\" Actually, it frightened me in the beginning. Abdomens are so soft and vulnerable. Since the first tentative touching, I've come to place his hand when I need to be touched there or when I feel like I want to work more with my fear around it. I place his hand on my lower abdomen (that's so not a sexy word); I breathe deeply, breathing in his touch, all the time talking to the part of me that is afraid. In the beginning, I didn't go too far past the moment of serious discomfort, just enough. Now I can let his hands go there uninterrupted. I'm aware of the love with which he touches me there, how much he loves that part of my body as he communicates all this through his hand and heart, talking to my belly. It's a conversation that flows between his hand and my belly and between each part of our bodies.\n\nThis next exercise will experientially give you an idea of how people hold stories, beliefs, and values in their bodies.\n\n ** EXERCISE   \nDepth Reading on a Stranger**\n\nFind a full-body clothed photograph of someone. You can use one from a newspaper or magazine. Choose one that shows the whole person, head to toe. I discourage you from using a photograph of a relative or close friend. In the beginning, it's too challenging to separate out your own feelings for this person and what you are perceiving. So, at least for the first few times, choose a photograph of someone unknown to you.\n\nHold the photograph in front of you. When you look at the photograph, you won't see the whole person all at once. Your eyes are going to be drawn to one place, then another, then another. When we look at something, we see it in a series of snapshots. Where were your eyes drawn to first? Stay there for a moment. How does that part of the person feel to you? Write down everything that comes to you. Ask your Child to tell you things about that part. Feel that part with your heart field, how does it feel? What's the second part you were drawn to? How does that part feel? Do this with each part you are drawn to, asking your Child to tell about it, then your heart field. Were you drawn to the top or bottom of the body or to particular body parts, like the hands? Are there any parts of the body that seem \"invisible\"?\n\nNext, take a sheet of paper and hold it vertically over the photograph so that part of the paper is covering half the person. Notice everything about this half. What is the hand doing? How does the eye look and feel? What kind of \"energy\" is coming off that side of the person? How does the foot feel? Write down everything. Move the paper so it's vertically covering the other half of the photograph. How does this half feel? What are you drawn to? Does this eye look the same or different from the other one? The foot? Is the energy on this side the same or different from the first side? How does this person feel about their sexuality? Are you uncomfortable asking this question? How do you feel looking at this person as a sexual being?\n\nRepeat the exercise as you move the paper so it's covering the bottom half of the person, horizontally. Go through noticing and asking questions. Then cover the top half and again ask yourself how various parts feel.\n\nRemove the paper and look at the person in the picture again. Do you see anything different or new that you didn't the first time? How do you feel seeing these things?\n\n** EXERCISE   \nDepth Reading on Yourself**\n\nGet a full-body, clothed, head-to-toe photograph of yourself. Keep the picture facedown. Take some deep breaths. Now, turn the picture over and hold it in front of you. What is the first thing you notice? What stands out, gets your attention? How does this part feel? What are you drawn to next? How does that part feel? Is it mad, sad, scared, happy? Really see yourself in your body. How do you feel seeing yourself? Look at your eyes. How do they feel? Can you tell who, of you, is looking out of them?\n\nOften people have splits in their bodies, discernible demarcations where parts are unintegrated and disconnected from the whole person. Can you see where yours are? It can be revealing to cover half your body photograph vertically with a piece of paper, noticing how one side looks. Notice everything: the look in your eye, how you're holding your hand, the direction of your leg and foot, if there is any energy coming off that side of your body, How does it feel? Then move the paper to cover the other side. How does that side look? Are there differences? Is one side more alive than the other? It's not unusual for one side to have more masculine energy than the other, leaning forward ready to move with fist clenched or pulled back trying to be unseen and diminutive. You may have to move the paper back and forth a couple of times to really notice your eyes. Are they the same on each side? Are there different feelings coming from each one? Before integration it's not unusual for a different ego state to be looking out of each eye.\n\nNow move the paper horizontally covering the top half of your body in the photograph. What do you notice? How does that part feel? Are you stable in your feet, grounded, or about to be airborne? Cover the bottom half and repeat the process, really noticing and seeing what is or isn't there. From your photograph, can you tell how you feel about your own sexuality? How do you feel seeing yourself this way?\n\nSEEING BELOW THE SURFACE\n\nThis will give more insight into your body dynamics, physiology, and holding patterns. It can be both instructive and disconcerting but doing the exercise with a desire to know yourself, and for the love of yourself, will help ease the tension around it. Give yourself permission to really see. With practice, you can begin to see below the surface to the truth of what lies there waiting to be revealed.\n\n _The knowledge of nature as it is\u2014not as we imagine it to be\u2014constitutes true philosophy. He who merely sees the external appearance of things is not a philosopher. Thetrue philosopher sees the reality, not merely the outward appearance._\n\nPARACELSUS\n\nTaking a few moments every day to work with different parts of your body will, over time, deepen your relationship with yourself. You'll feel more integrated, more alive, as parts of you, inside and out, begin to be part of your life, part of you.\n\nThe more you are integrated in body and soul, the more resilient and flexible you will be in dealing with shame\u2014whether old or new, self-shaming or from the outside. As you work with trust and shame and healing from the inside, you'll discover an openness of spirit, a healing of the soul, and have new ways of responding to the challenges of becoming a human being in diverse cultural, social, and interpersonal relationships.\n**12**\n\n ** _Freeing Ourselves from Sexual Tyranny_**\n\n_If a woman hasn't got a tiny streak of harlot in her, she's a dry stick as a rule._\n\nD. H. LAWRENCE,  \n _T HE POSTHUMOUS PAPERS OF D. H. LAWRENCE_\n\nI present this chapter for calm, reasoned study of pornography and prostituion\u2014topics that are sticky, discomforting, and hairy ones to navigate. The mere mention of the word _pornography_ stimulates an array of responses from various quarters. Finding ways to address these topics without shutting down the discourse forced me to get clear inside myself about it. I'm not flat about it. I have strong opinions about pornography and prostitution, which directly relate to free speech, expression, and the right to choose. These are not the same opinions I held twenty years ago, which is one reason I've included these topics.\n\nFor five decades now I have labored as an apprentice human being. As the apprenticeship enters its the fifth decade, I can still see the remains of a few contaminated beliefs, which, in the continuation of my self-training, I am working to detoxify.\n\nThere are costs to this dance, and regrettably, I fear my son and my lovers paid a good share of the price of me becoming who I am, but primarily of who I wasn't. The issue of pornography, _my_ issue with pornography, came slowly to the surface when my son reached adolescence. My husband found copies of _Playboy_ and _Penthouse_ magazines in his room and reacted by blowing a gasket. I timidly concurred, agreeing that these magazines demeaned women and were somehow going to turn him into a pervert.\n\nEven then, something about my husband's reaction and my compliance with it felt very wrong. Something about our insistence that he not have them in the house was terribly misplaced. For a young boy reaching adolescence, having _Penthouse_ and _Playboy_ in his room is a rite of passage of sorts, and there is nothing wrong with him having them. How else would he find that his interests and curiosity are normal? Moreover, every boy and girl deserves to have their own room, their personal space that is just theirs, away from parental tyranny. The way we handled it was poor at best. The worst is that it may have caused shame around his natural curiosity and emerging hormones and sexuality. That is especially regrettable given that my husband had nude photos of me in his dresser drawer at the time. It could have been an opportunity to talk about sex, sexuality, intimacy, and relationships. I never took the time to ask him what he thought of looking at nude women or photos of other people having sex. How did he feel about it? Did he enjoy it? We could have talked about why finding the magazines was so upsetting to my husband. About how he was trying to recover from Catholicism and his own contaminated baggage around sexuality. I didn't take the time to point out to my son that these were actors and though, yes, some people enjoy bondage, being submissive, being overpowered, not everyone does, and certainly not everyone looks the way porn actors do nude. They get paid very well to keep those hard bodies in shape for being photographed.\n\nPornography and prostitution are social quagmires where passions erupt from both sides of the slippery spectrum of controversy. While passions rise over the topics, understanding of them gains little foothold. The debates over pornography are embedded in moral and political ideologies, and as long as the debate stays under those blankets, misunderstandings of the nature, meanings, and functions of pornography flourish. As long as the debate remains in the hands of the church, that bastion of morality, and politicians who wrongly assume that morality can be regulated, it will be difficult to see the possibility that pornography and prostitution may actually have redeeming social functions and value. But as we lose sight of what our individual and social values are, the topic of pornography becomes ever more confusing. How do we begin to sort through these issues when our social value system is corrupt\u2014with banks, Wall Street, corporations, and the government engaging in deceit as policy\u2014and any recourse through the Constitution is undermined?\n\nSex has been difficult to talk about, not in the least because our culture is not clear about it. We have conflicting messages coming to us from a variety of sources. Our own feelings about sex and our sexuality are often convoluted and contaminated with toxic messages. Sex is challenging to talk about because it is moist and leaves a wet spot. And as Eric Berne says: \"In fact, it is more than wet, it is slippery. Anyone who ignores that is going to feel a little sticky talking about it.\"\n\nI have not always been a proponent of pornography or prostitution, though prostitution held a special allure for me when, at sixteen, I considered it as a profession that might be able to get me away from my home. It took an entire twenty-seven seconds for me to dismiss it, as I was too damned scared of life to get myself past the Iowa border. Nonetheless, it required a certain level of devotion to clarity and freedom to sort through my beliefs, thoughts, fears, and assumptions, as well as the debates and opinions around those subjects, and create my own beliefs and values.\n\nSACRED SEX AND THE WORLD'S OLDEST PROFESSION\n\nMy efforts to find writings on sexuality, ecopsychology, nature writings, **D** eep Ecology writings, sacred sex, tantric sex, and sexual history that made connections between Earth, Nature, and human sexuality were nearly futile. Those connections are seldom made though there were a few noteworthy exceptions. Interestingly, it was a woman who chose prostitution as a way of life and made a very good living at it who made the clearest connection. In her book _A Woman Whose Calling Is Men,_ the pseudonymous Aphrodite Phoenix writes about the divine side of prostitution.\n\nBehind every situation that feels earthy and sensually connected, good to oneself and others, and naturally health-bestowing, atonement with Goddess can be fathomed as the underlying spiritual cause. I believe that all prostitutes who experience an increase in feminine power are women who, consciously or not, partake of psychic nourishment from what Sue Monk Kidd calls \"Feminine Divine.\" It's the reason we insist that the work doesn't harm us, and that there's more to the work than just raking in money from meaningless, anonymous sex. In our work we're experiencing a spiritual feminism.\n\nProstitution has been present in nearly every culture and age of the human race. In some cultures, prostitution was viewed as a valuable and healthy part of the culture and economy, such as in ancient Greece where it was a cult of beauty and profit. In some countries it was a royal enterprise, a principal source of revenue for the state, and women and boy prostitutes were trained in the finer points of their profession. Prostitution was common among the Yuma Indians of California and the Colorado tribes. In China, Japan, and Thailand, prostitution is a common form of barter. Even married women participate, sharing the proceeds with their husbands.\n\nScenes depicting sex on undulating beds and mattresses as well as seated positions were found etched on wine-drinking buckets, or _situlae,_ from the Iron Age in Europe. Archaeological evidence suggests that prostitution existed in ancient Rome, as seen in brothel tokens, called _spintriae_. Each token, slightly smaller than a U.S. quarter, depicts a sex scene on one side and a roman numeral, I through XVI, on the other. It's been suggested that the number on the coin related to the cost of the sex scene on the obverse side. Coin specialist Aleksander Bursche of the University of Warsaw undertook his own study and surveyed modern-day prostitutes about the coins, inquiring which acts they charge more for. They told him that positions that afforded deeper penetration, such as sex from behind, caused more vaginal soreness and thus they cost more. Geoffrey Fishburn at the University of New South Wales argued against this theory, noting that the same sex scene appears on coins of differing numerical values and identical scenes show up in Pompeian murals. Another argument suggests that the coins were merely poker chips. Regardless, it does raise titillating questions and suggests the openness of sex, as well as a sexual repertoire, in ancient Rome. Or perhaps, coins, goblets, and murals depicting sex scenes were an homage to the power and eroticism of sex, the beauty and art of naked human bodies entwined in sacred sexual acts. They may even be the earliest versions of pornography.\n\n _Obscenity only comes in when the mind despises and fears the body, and the body hates and resists the mind._\n\nD. H. LAWRENCE,  \n\"A PROPOS OF _L ADY CHATTERLEY'S LOVER_\"\n\nThe word _pornography_ has a long, convoluted, and interesting history. It comes from the Greek _porne,_ meaning \"to sell,\" and _graphos, \"_ to write.\" The sex workers in ancient Greece were referred to as the _porne_ , which included both women of importance and boys. They were required to wear elaborate clothing and pay taxes. And in the ancient Middle East, prostitutes were the first to have contact with men returning from battle in order to help them assimilate back into society and recover from the shock of war. In seventeenth-century Japan, the _oiran_ were courtesans and considered \"women of pleasure.\" The highest ranks of courtesan were available only to the wealthy. They were often practiced in fine art, dance, music, and calligraphy, as well as the art of sexual fulfillment, and were well educated.\n\nThe Egyptians had sacred prostitution as did the Persians, who learned the trade from Libya. In Israel both male and females practiced prostitution. The cult of Venus or Mylitta (the Assyrian name for Venus and counterpart to Ishtar, in Greece her name was Aphrodite) in Babylonia passed into Cyprus and Phoenicia. In Cyprus, temples sacred to Astarte, the goddess of fertility, love, and reproduction, dotted the landscape. She had a strong presence until the cult of Yahweh emerged among the Hebrews. The Hebrews knew Astarte as the goddess of the Sidonians and continued to worship her at temples in Mizpah built alongside those of Yahweh, which angered him greatly. Worship of the ancient goddesses formed the basis of many cultures and civilizations who oriented life around them with ritual sex, seasonal goddess celebrations, and elaborate rituals honoring the deities. \"The advent of Christianity and its relatively puritanical views on sex all but killed off the sex cult, but during the world's polytheistic zenith it was one of the firmest pillars of society.\" In Armenia, Venus, was worshipped under the name Ana\u00eftas. Young women lived in Ana\u00eftas's temple and offered amorous services to foreigners. The women left all their earnings on the altar as an offering to the love goddess in hopes of securing a husband.\n\nInnana, the Sumerian goddess of sexual love, fertility, and warfare, can be dated to the city of Uruk, known as the \"city of courtesan,\" circa 4000 to 3100 BCE. Innana was celebrated and honored at sacred sites and temples along the Euphrates and Tigris Rivers. Eanna, meaning \"house of heaven,\" is the name of the temple in Uruk dedicated to Inanna, where sacred prostitution was a common practice. Innana, considered an aspect of the Great Mother, was a fierce warrior; she was often depicted on the back of a lion. And she is the center of pleasure, \"the one who makes women and men turn to one another in the night.\" Innana gave the gift of blood to women, the healing blood that flows at the new or dark moon that cleanses the body and prepares the womb for the next cycle of the moon's influence on a woman's cycle.\n\nAfter the fall of organized prostitution during the Roman Empire, many prostitutes became slaves until the movement to abolish slavery swept the land, and prostitution was reinstated as a legitimate business during the Middle Ages. Though the medieval Roman Catholic Church viewed sexual relations outside marriage as sinful, prostitution was seen as a tolerable alternative to the greater evils of masturbation, rape, and sodomy; however, the church often urged prostitutes to reform their wicked ways.\n\nCourtiers played an essential part in Renaissance Europe. Royal marriages were made primarily to preserve the bloodline; love and passion was not part of the equation. So it was common for members of the royalty to seek sexual companionship from other people at the court, hence the origin of the terms _to court_ and _courtesan_.\n\nOne of the oldest documents having to do with prostitution was written in 1266 in Venice. The Maggior Consiglio, or Greater Council, decreed the night watch to expel every \"woman of evil life\" from the houses of citizens. After the women were forced out of public houses, they took their business to private houses, or houses of prostitution, which initially were off limits to the night watch's watchful eyes But then a second council decree gave the night watch the power to expel evil women and men from private domiciles and fine them: They were now being forced out of business from their own homes. After a time, wisdom prevailed; the authorities conceded the futility of outlawing prostitution, and the Maggior Consiglio decreed (they liked to decree in the day) that a place in Venice should be found to house the prostitutes.\n\nWhereas, by reason of the multitude of people constantly coming and going, it behooves our State to see to providing some place in Venice proper for the habitation of sinful women:\n\nIt is hereby commanded to the Captains of the City Wards to examine diligently all places on the Rialto which might be suited for such a purpose, and after due examination, to make report to us in writing, namely, concerning the most suitable place and the conditions under which the women in question may be kept there, with which report let them come before the Council of Forty and make their findings known.\n\nProstitution will always have a place in human society. St. Augustine, demonstrating his understanding and knowledge of mankind observed: \"Do away with the prostitute in the human scheme and you will upset everything through an incursion of lust; put them in the matron's place, and you will bring injury and dishonor upon the latter.\"\n\nTHE BEGINNINGS OF SEXUAL LIBERATION\n\nWilheim Reich coined the phrase _sexual revolution,_ though its roots may be found in the eighteenth century during the Enlightenment, and later the term _free love_ came into being during the same century. An early proponent at that time was Mary Wollstonecraft. Wollstonecraft argued that women should not give up freedom and sexuality. She thus made a personal choice to not marry her partner. Later free love proponents included Mary Gove Nichols and Hannah R. Brown, as well as male advocates of the movement such as William Francis Barry, in the mid-to-late 1800s and Gloria Steinem in the twentieth century.\n\nThe early feminists spoke out and wrote against marriage and were radical in their thinking of the time and had a profound impact on sexual relations of their eras. In 1798, the radical Swedenborgians August Nordenski\u00f6ld and C. B. Wadstr\u00f6m published the _Plan for a Free Community_ in which they proposed the establishment of a society of sexual liberty, where slavery was abolished and the European and Negro lived together in harmony. In the treatise, marriage is criticized as a form of political repression. The challenge to traditional morality and religion brought about by the Age of Enlightenment and the emancipatory politics of the French Revolution created an environment where such ideas could flourish.\n\nCharles Fourier, in France in the early nineteenth century, coined the word _feminism_ and said that the suppression of passions is not only destructive to the individual but to society as a whole. He argued that all sexual expressions should be enjoyed as long as people are not abused and that \"affirming one's difference\" can actually enhance social integration.\n\nThough considered scandalous at the time, out-of-wedlock children and sexual liaisons seemed acceptable behavior for admired artists, who were following the dictates of their own wills rather than those of social convention. In this way, these artists were in step with their era's liberal philosophers of the cult of passion, such as Fourier, and their actual or eventual openness can be understood to be a prelude to the freer ways of the twentieth century.\n\nJosiah Warren and the experimental societies viewed sexual freedom as a clear, direct expression of an individual's self-ownership. Free love particularly stressed women's rights since marriage laws and measures that discouraged birth control discriminated against women. Discrimination against women continues to this day. Sarah Seltzer, writing for _AlterNet,_ reported on the House's passage of H.R.358, the notorious antiabortion and \"Let Women Die\" bill. According to Seltzer, a major proponent of the bill was the Council of Catholic Bishops, who heavy-handedly and successfully lobbied for some of the worst measures in the bill, taking rights away from women\u2014while the bishops were in the midst of fighting charges of child sex abuse. Meanwhile, in June 2010, in Phoenix, Arizona, a woman, eleven weeks pregnant with her fifth child, was admitted to St. Joseph's Hospital and Medical Center. She had a right-sided heart failure and was told by her physicians that if she continued the pregnancy she would die. The woman agreed to an abortion. Unfortunately, she was in a Catholic hospital, which forbids abortions. The doctrine of the Catholic Church would have let both mother and child die. Physicians turned to Sr. Margaret McBride, the administrator at the hospital, who gave her approval for an abortion that would save the life in front of them, the mother. By making that decision, she was automatically excommunicated by Bishop Thomas J. Olmsted of the Phoenix Diocese.\n\nIn speaking to the sixth International Pre- and Perinatal Psychology Association of North America, Jeannine Parvati Baker, midwife and founder of Hygieia College, spoke movingly about how the dominant culture insulates people from their own vitality.\n\nFrom the current \"war on drugs\" to the obstetrical theater, to the church and temple, people are seeking safety from the raw power of life. Yet birth is as safe as life gets. The ways we scare ourselves from being wild woman, mother, midwife and healer are rooted in fears by the dominant culture. To revision God from being only Father, or Father and Son, or even Divine Parents can help us become free to be fully inspired lovers, connected to our power to be our own healers.\n\nRecognizing that sexuality is spirituality and that ecstasy is a divine right will begin to move us back into a harmonious relationship with the other gender, with our sexuality as a life force, and with the body and ground of our own being. This is the first healing we are called to do.\n\nIf there is anything to be ashamed of, and I use the term cautiously, it is in denying our sexuality and feelings and that Earth is alive and intelligent. It is in not doing something to alter the rate and velocity of our march to self-destruction. It is in not holding our governments, schools, religions, and ourselves accountable for the choices made and the behaviors that come of them.\n\nReverence for our lovers, partners, and children can translate to reverence for Earth, for the nonhuman, numinous dimension of wilderness, of Nature. If the separation that is extant among human relations can be made whole and intimate, then our relationship with the community of Earth can be. Regaining our sense of awe and wonder and our ability to fall in love over and over again from sunrise to sunset will begin to bring the sacred back to our daily lives. Experiences that draw us up and out of our egocentric lives into the larger reality, the community of all beings, must become integrated into our daily lives. Integrating sex into our lives, unapologetically, will inevitably integrate our personal power, our creativity, our voices and bodies for deep and lasting changes in our lives and in our society.\n\n _When I speak of the erotic, I speak of it as an assertion of the life force of women; of that creative energy empowered, the knowledge and use of which we are now reclaiming in our language, our history, our dancing, our loving, our work, our lives._\n\nAUDRE LORDE\n\nWe have been taught, programmed if you will, to distrust our eroticism, our sexual power, our allure. Some of us have accepted part and parcel the archaic myth of the fall of man by woman. By accepting the programming and the stories, we have committed a crime against our basic human nature, our bodies and psyches as sexual. The old arguments are beginning to crumble as women speak out and begin to take back and own their rightful place in the arena of sexual freedom. There is hope and a rumbling in the subconscious of women slowly awakening.\n\n _When sleeping women wake, mountains move._\n\nCHINESE PROVERB\n\nTHE POLITICS OF SEXUALITY: AN ANCIENT ARENA FOR CONFLICT\n\n _To discuss the nature and meaning of obscenity is almost as difficult as to talk about God. Until I began delving into the literature which has grown up about the subject I never realized what a morass I was wading into._\n\nHENRY MILLER,  \n _H ENRY MILLER ON WRITING_\n\nThe politics of sexuality is the most ancient arena for human conflict, on par with the domain of religion as a source of conflict, large and small, but then, the two have been bedfellows for thousands of years. It also in large part causes intrapersonal conflict, that is, conflict inside the self. Many, many people watch porn, look at porn magazines, even as they lie to themselves, their partners, coworkers, or priests about it or in some way hide the fact. Many people talk about the subject of pornography, but even talking _about_ pornography is an intimate, self-revealing act and is wrought with fear, anxiety, and suspicion.\n\n _And no, I do not believe it is blasphemous to compare oppression of sexuality to oppressions of race and ethnicity: Freedom is indivisible or it is nothing at all besides sloganeering and temporary, short-sighted, and short-lived advancement for a few. Freedom is indivisible, and either we are working for freedom or you are working for the sake of your self-interests and I am working for mine._\n\nJUNE JORDON,  \n\"A NEW POLITICS OF SEXUALITY\"\n\nThe politics of sexuality, more than any other oppression, is the exploitation of sexuality for power. In fact, religion has been the vehicle driving the oppression and politics of sexuality. It could be a religion in its own right, the religion of the sexual oppressors or the religion of the sexually oppressed. I do not believe that any one man or woman has the right to tell me what I can or cannot do or what I shall like or dislike or attempt to dictate my behavior regarding my sexuality and how I express it. I do not believe I am blasphemous in saying that pornography has a valid and legitimate place in the scheme of human sexuality. I will go further and say that pornography plays a vital role in relieving the pressure of religious and politically sanctioned morality.\n\nIn _Ensouling Language_ , Stephen Harrod Buhner writes, \"Sex must be _integrated_ into Western cultures as an active and accepted part of what it means to be human for its repression is inherently connected to the ecological and cultural problems we face.\" Integration\u2014to merge, to infuse, to bring together\u2014means no part can be left out. When sexual expression in the form of pornography is left out of something as basic and essential as our sexuality and as inherent to our human beingness, there is a hole left in its place. A hole is there in the place where sexual expression should be, and where holes are, degradation of the wholeness of human beings takes place.\n\n _I kind of like occasional acts of public lewdness. A little bit of real obscenity and indecency actually makes me feel more secure. I get nervous always being on the extreme by myself. Public acts of sex or penis fondling all add to the \"wow, I'm not in Kansas anymore\" feeling._\n\nDALE PENDELL,  \n _I NSPIRED MADNESS_\n\nPart of the problem and legalities around pornography is that it is difficult to define what is objectionable, what goes too far. D. H. Lawrence was right when he said, \"nobody knows what the word obscene means.\" In 1964, Supreme Court Justice Potter Stewart admitted that he couldn't define what material is obscene: \"But I know it when I see it.\" In America, all pornography is legal except that which is deemed \"obscene.\" The legal definition of obscenity in the United States is that it \"must be shown that the average person, applying contemporary community standards and viewing the material as a whole, would find (1) that the work appeals to predominately 'prurient' interest; (2) that it depicts or describes sexual conduct in a patently offensive way; and (3) that it lacks serious literary, artistic, political or scientific value.\"\n\nAs you can see, there are problems with that definition. Contemporary community standards are changing. Obscenity is subjective, that is, its meaning lies in what we bring to it\u2014how we view it and what we think of it. I'm more offended by rudeness, unconscious behavior, and sex without intimacy. What I find truly obscene is an endless war based on lies and misinformation. Our foreign policies, U.S. banking system, public schooling, Catholic priests sexually abusing children, people losing their homes and jobs, and children starving\u2014these things are obscene, vulgar, indecent, cruel, and morally reprehensible.\n\nI don't find pornography offensive or obscene; I do find fuck films distasteful perhaps, extreme maybe, raunchy sometimes. I do find pornography and nude photos titillating, interesting, exciting, yes. Always it gives me permission to live outside the box. \"But in legal parlance, obscenity is a category of speech\u2014speech about sex\u2014that falls outside the protection of the First Amendment. Or so, at least, the Supreme Court has said; many legal scholars find no basis in history or logic for this 'obscenity exception' to the First Amendment.\"\n\n _No argument for the suppression of obscene literature has ever been offered which by unavoidable implications will not justify, and which has not already justified, every other limitation that has ever been put upon mental freedom._\n\nTHEODORE SHROEDER,  \n _C HALLENGE TO SEX CENSORS_\n\n _As we move toward sexual maturity, hyped-up sexual excitation becomes secondary, and the quality of our feelings determine sexual identity. That's the difference between adolescent and adult sexuality. What we see around us is a great deal of fantasy stimulation to make us feel excited. We have manipulated ourselves into accepting stimulation rather than feeling. We've done this by sacrificing our bodies as the sources of our feelings of aliveness, and becoming addicted to the brain and nerves as the pleasure center.  \nOr, conversely, if your heart and your honest body can be controlled by the state, or controlled by community taboo, are you not then, and in that case, no more than a slave ruled by outside force?_\n\nJUNE JORDON,  \n\"A NEW POLITICS OF SEXUALITY\"\n\nSex writer Violet Blue has organized a new movement, Our Porn, Ourselves, whose goal is to bring sex back to its rightful place, to ourselves. To that end, the organizers wrote a manifesto setting themselves apart from right-wing or old-guard feminists who maintain the dominant culture's rules of behavior regarding sexual repression, freedom of speech, self-expression, and the pursuit of sexual health and liberation. Blue writes, \"We women are tired of people trying to control our sexuality by telling us what we should or shouldn't like sexually (porn) based on what someone else thinks is best for us. It's like keeping women in a perpetual state of being children about sex. And women who say they are feminists make it worse by discounting all the women who find porn to be an empowering sex toy. Or if not, to at least give us the benefit of the doubt that we can make that decision for ourselves, thank you very much.\"\n\n **Our Porn, Ourselves Manifesto:  \nPro-Porn Principles**\n\nWE who declare that organizations such as Feminists Against Pornography do not speak for us.\n\nWE who want the world to know that organizations such as Feminists Against Pornography do not represent feminists as a group.\n\nWE who believe that every woman has the right and power to enjoy her sexuality as she decides.\n\nWE who believe that to tell a woman how she may or may not enjoy her sexuality in any way is to deny that woman of her rights over her sexuality.\n\nWE who state that any woman who attempts to control the way another woman enjoys, explores or expresses her sexuality is in fact creating a world that is harmful for all women.\n\nWE who state that we are women, and we like pornography.\n\nWE who state that as women, we are not harmed or threatened by the creation or viewing of pornography, and we wholly support the rights of gender to view, create, and enjoy pornography without judgment.\n\nWE who want a world in which pornography is simply a sex toy enjoyed by all genders and sexual orientations, where women and men view porn within their own self-defined healthy sexuality, without being considered sick, twisted, wrong or morally ill, and that men who enjoy pornography are no more likely to beat their wives, rape women or become pedophiles than anyone else in society.\n\nWE hereby declare ourselves as adult women capable of making our own choices about our bodies and enjoyment of explicit visual stimulation for our sexual health and well-being.\n\nWE hereby demand that our voices be heard.\n\nOn July 22, 2010, Facebook removed the Our Porn, Ourselves Facebook campaign page, according to Our Porn, Ourselves official web-site. The organization Pornography Harms claimed victory and thanked Facebook for removing \"a very inappropriate pro-porn page with links to pornography that our children had easy access to.\"\n\nPornography plays a vital function in the sexual health of human beings. It has been one of the most important places of discourse on sex and sexuality. Pornography shamelessly exposes that which has been kept secret and hidden. It gives our unnamed desires an arena. Pornography exposes us to sex that turns us on, turns us off, intrigues, satisfies, helps expand our ideas of what we like, what we don't like, what we might like if we had an opportunity to try it, and it increases the likelihood and frequency of orgasms, with ourselves or others\u2014a vital function of vitally functioning people. And, who would think of killing or fighting while looking at naked breasts or cocks?\n\nThrough Facebook and other Internet tools, we have the ability to have community discussions around any topic of our choosing. It may be that Internet porn sites will be the arena where community standards regarding porn are set, not by legislators or bishops.\n\nAbout every forty to fifty years, or two generations, there has been a sexual revolution. July 19 and 20, 1848, Lucretia Mott and Elizabeth Cady Stanton organized the first public gathering in the United States to address the rights of women. They were joined by Susan B. Anthony and Frederick Douglass, a former slave. It was the Seneca Falls Convention located in Seneca Falls, N.Y. There they drafted the Seneca Falls Declaration of Sentiments calling for women's right to vote, to own property, and to have equal access to education and employment. The late nineteenth century birthed a trifecta of women's organizations: the Suffragists who worked for the right of women to vote; The Social Feminists who birthed the Women's Trade Union League, the General Federation of Women's Clubs, the National Council of Jewish Women, and the National Council of Colored Women; and the Radical Feminists who argued that for social and economic equality between men and women. This latter group produced Alice Paul, who introduced the first Equal Rights Amendment in 1910.\n\nElizabeth Cady Stanton developed the \"Bloomer Costume,\" a proactive alternative to the restrictive Victorian dress of the day. The \"freelovers\" of the nineteen century advocated that sex had functions outside of procreation and encouraged the relaxation of external controls in order to experience more personally responsible sexual expression and experimentation.\n\nThe 1920s ushered in the radio, talkies, women's suffrage, and those wild women, the flappers who came to represent the independent woman of the twenties. The Nineteenth Amendment was ratified granting women the right to vote. As attitudes toward women changed and birth control became available, a sexual revolution followed.\n\nThe years following World War II found people less likely to defer to external authority on behavior and morality than they did in the previous fifteen years of depression and war. The decades of the 1940s and 1950s saw the rate of single mothers more than double, Alfred Kinsey researching and reporting on _Sexuality and the Human Male_ in 1948, Elvis Presley gyrating on the television screen, the appearance of Hugh Hefner's inaugural issue of _Playboy_ magazine in 1953, and Lucy and Ricky Ricardo sleeping in separate beds on _I Love Lucy_. In the sixties, the first films depicting nudity and sexual intercourse were _Midnight Cowboy_ and _Romeo and Juliette_. Woodstock, love-ins and social activism rose alongside rock and roll, the Vietnam War protests, and the Civil Rights movement.\n\nThe twenty-first century is showing another sexual revolution as the new feminists are writing erotica and producing pornographic films. Sexting, home erotica, and porno films are available to anyone with a camera and Internet access as YouTube has made self-broadcasting possible. The World Wide Web makes available nearly any form of erotica, pornography, nude photos right in the comfort of your own home. Meanwhile, Occupy Wall Street is the largest, most organized, peaceful protest in our country's history as social revolutions are fueled by the energy of sexual revolutions: Eros and Psyche.\n\nPORNOGRAPHY: HARMFUL OR CRUCIAL?\n\nIn the United States alone, a pornographic video is produced every thirty-nine minutes; 11,000 adult movies are released per year\u2014more than twenty times mainstream movie releases. In 2006, the sum of international revenues from pornographic videos, sexual novelties, magazines, \"dance\" clubs, pay-per-view television, and the Internet was approximately $97 billion. That figure is more than the combined annual revenues of the NFL, NBA, and Major League Baseball. In the United States, revenues for pornography are larger than the revenues of Microsoft, Google, Amazon, eBay, Yahoo!, Apple, Netflix, and EarthLink combined. It's rather mind-boggling to consider that $3,075,64 is being spent on pornography internationally every second. The United States produces more Internet porn than anybody else on the planet.\n\nRecent studies point to porn-induced sexual dysfunction.*6 _Psychology Today_ reported on a growing problem with porn-induced sexual dysfunction. Sexual desire and erections respond to dopamine signals. Dopamine is the reward chemical in the hypothalamus and the central nervous system. Hundreds of young men are reporting erectile dysfunction due to overstimulation from viewing Internet porn. There are enough studies now to show that, indeed, continual stimulation of dopamine from masturbating to Internet porn _can_ cause erectile dysfunction due to desensitization and rewiring of the brain. It is reversible when stimulation from porn is removed.\n\nThese studies suggest porn addiction is the cause. It may be involved, but the more I read on the subject, the more red flags are raised. For one thing, if you do anything repeatedly, including working at a repetitive office or factory job, desensitization can occur. If I eat the same meal every day, my taste buds get desensitized and the food no longer looks appetizing or tastes good to me; I get no enjoyment out of it. Porn addiction, or any addiction, is a _symptom,_ not the cause. If we focus on overuse of Internet pornography, we do a severe disservice to young men. There may be women having similar problems with overstimulation from or habituation to Internet porn (are women having porn-induced clitoral erectile dysfunction?), but men are the most studied since, without an erection, the possibility of great sex with an intimate partner is severely limited, which causes a cascade of other problems, including anxiety, shame, self-esteem issues, relationship difficulties, and functionality.\n\nAre the arguments that porn increases pedophiliac tendencies and makes men want to rape and therefore increases the risk of child abuse and rape legitimate? Aside from any studies done on these issues, think about it reasonably for yourself. Would a person with a healthy moral compass about right and wrong behavior be even remotely interested in viewing films that depict children in sexual acts? Assuming you could even get them to sit still for it. No, people who engage in pedophilia are pedophiles to begin with.\n\nAs Theodore Shroeder states in his work, _Challenge to Sex Censors,_ \"obscenity exists only in the minds that discover it and charge others with it.\" If we were to reflect objectively on the subject matter we would find the law of nature; that is everyone performs the very acts they attribute to others.\n\nPorn films are films and as such are incapable of forcing anyone to do anything he or she doesn't want to do or of being a leading cause of divorce. The leading cause of divorce is the _decision_ to divorce. Pornography is a pressure valve, releasing social and cultural oppression. Studies have repeatedly shown that pornography actually decreases the incidence of rape and aggressive sexual acts. There are more incidences of rape and violence against women in countries where there is no porn than there are in countries where pornography is available. And there are statistics that reveal, without mention of pornography being an instigator, that: \"one out of eight women will be raped while in college and 84% of women who were raped knew the assailant.\" A story in the _Huffington Post_ reported, \"Rape within the US military has become so widespread that it is estimated that a female soldier in Iraq is more likely to be attacked by a fellow soldier than killed by enemy fire.\" Rape in the military is not gender specific, it crosses lines. \"According to the Veterans Affairs Office 37% of the sexual trauma cases reported last year (2010) were men.\"\n\nThe questions not being asked are: Why so much \"porn addiction\" in the first place? And why so much emphasis on it? Why such high statistics of rape in the military? What is missing in our lives? Where is the meaning to life? What do we need to have fulfilling lives? If our life is empty, we may turn to anything to feel better momentarily. Porn is not the problem. The problems we face and apply Band-Aid treatments to are much, much deeper. Perhaps we need to look at the deeper issues of a lack of intimacy, the economic debacle, endless war permeating all of our psyches, rampant lack of hope and opportunities, and the mess we are creating for the next generations. Perhaps porn use is the morphine of the masses so we don't have to feel how bad our lives are. Pornography has always been around in some version. It will always be here.\n\nSEXUAL PLEASURE IS OUR BIRTHRIGHT\n\nIsn't it curious that the clitoris is designed solely for pleasure? It has nothing to do with reproducing the species. It's a sign from God, a direct communication that we are meant to have and enjoy sex outside of procreation. A woman's vagina has its own cycle of sexual response, lubricating about every fifteen minutes during the sleep cycle. Another signpost saying, \"Have sex, a lot of sex, often.\" Studies have shown that as a man ages, or becomes depressed or bored in his relationship, his testosterone levels drop. Studies also show that just talking to a woman he finds attractive and sexy will increase a man's testosterone levels by 12 percent. The most obvious and familiar male sexual response is the erection. There are other clues males generate that indicate sexual interest and arousal such as the steady but not penetrating look in his eyes, the way he holds his body, and the movement of some invisible thing that leaves him and travels to the woman.\n\nPornography is not shameful, nor is watching porn, though there are many people and institutions that would prefer we thought it so. They work very hard to make it shameful and to make those of us who enjoy it feel shameful. Shaming does work temporarily as a behavioral modification tool, but the cost is detrimental to the shamed.\n\nPornography educates us about bodies, showing us there are many kinds of bodies and that sex organs are not all the same. Like fingerprints, every human body is different and unique. There are numerous shapes and sizes and colors of labia majora and minora, penises, clitorises, breasts, areolas, and asses. In some, the labia majora are predominant; in others, they are nearly nonexistent and the labia minora are the most distinguishable features of the vulva. The subtle shading and colorations that occur between the clitoris and the sacrum are fascinating and beautiful. Pubic hair comes curly, straight, kinky, long, short, red, black, blonde, brown, groomed, trimmed, and shaved.\n\nAdults often use pornography with their partners for sexual stimulation\u2014to break up monotony or to increase intimacy. For many people, watching other people have sex is, well, sexy and a turn on. People who view porn tend to choose porn that is within their comfort zone, or they may go beyond what is familiar to them and use it to explore, to answer questions they might have about what other people do, and, then, perhaps, to experiment. They want to watch someone else do what they have only thought about trying out before doing it in the privacy of their own bedrooms.\n\nIsabella Rossellini took the porn world to new heights with her short _Green Porno_ films, which she made for the Sundance Channel. She has combined the elements of short stories, animals, and sex. \"Not everyone is interested in animals, but everyone is interested in sex.\" The _Green Porno_ short (one-minute long) films depict animals, sea creatures, and insects reproducing, yes, having sex. \"They are quite scandalous,\" says Rossellini. \"They make love in funny ways. Some of them are hermaphrodites, some change sex during their lifetime\u2014things that if we would do it as human beings, we would be arrested. But they do it naturally.\" These fantastic clips can introduce children to sex and reproduction in the animal world. They're short, frank, funny, and refreshingly candid. The costumes and props are all made of paper, so it's like watching a live puppet show.\n\nThe problem for some people is the disconnect they experience between what their body is telling them and what their internal programming is telling them\u2014that it's wrong and perverse to view porn. We have the same feelings about pornography that we have about anything that is too revealing, embarrassing, obscene, private, and explicit as well as exciting, interesting, and life affirming. Porn allows us to go to our imaginative extremes. Author Dale Pendell says, \"There _should_ be extremes that are in bad taste. Extremes by definition are in bad taste. You have to allow that. That's what tolerance is. And tolerance is the basis of any kind of free society. And the only hope for people who want to live without Big Brother.\"\n\nListen to your body; it knows what feels good. There is nothing inherently wrong with pornography or watching it. Well, except when you let it interfere with your job, and you miss busting a $550 million Ponzi scheme. Investigations revealed that during one week in 2008, a supervisor at the Securities and Exchange Commission attempted 196 times to view porn on his office computer.\n\nREADING THE INVISIBLE INTO VISIBILITY\n\nPersonally, I don't really like a lot of the porn that's available. It's poorly filmed and acted. Fortunately, that is changing as more women get involved in the industry. More art porn is available, more erotica, which is what I prefer over hard-core porn, though I do occasionally wander outside my porn comfort zone.\n\nLooking at artful, nude photographs of men and women is something I enjoy from time to time. I love looking at the human form and find that looking at nude photos of women, or two women entwined, or a man and a woman entwined, or two men wrapped around each other is erotic, stimulating, and pleasurable. Porn has helped me be more accepting and understanding of myself and my differences and similarities to other women. I feel sexier looking at photos of nudes. The more intimacy I have with myself and others, the more permission and freedom I give myself to express my sexual desires and needs, as well as my sexuality, the more fulfilled I am. Not only am I more fulfilled, I feel more powerful, stronger, and focused and am able to effect change from that place. Personal power and sexuality are inextricably connected.\n\nMy lover and I enjoy occasionally looking at photos of nude women posing in provocative positions. Sometimes it's just men, sometimes men and women. We do \"readings\" on the models. We do readings on nearly everything, so this is no different. If there is a page of thumbnail photos, we let our gaze scan the page and pick the one that gets our attention, that stands out the most. Frequently, the same one gets both our attentions. We then click on that thumbnail for a closer look. In the photo some part of her body will get our attention, and we'll say out loud everything we see. Our eyes become soft focused, allowing us to see how she feels about that part of her body. Typically, the part of her body she really loves, the part she gives the most attention to, puts more energy into, will be more noticeable to the onlooker's gaze.\n\nMost people are adept at hiding, making invisible, essentially, the parts they don't like. You can notice this as you sit waiting in airports or doctors' offices; one person will get your attention over the others, then, as you continue looking, some part of his body will get your attention over other parts. Looking deeper still, you'll begin to see how he feels about that part of his body, and then how that part of his body feels. If you put your gaze on the part that is hidden, you'll see just how that person feels about this part of his body.\n\nLooking at nudes is a good practice for doing readings (developing intuition). You'll be able to see which people really enjoy their bodies, which parts of their bodies are more alive than others. My lover and I talk about what we like, what's appealing, what isn't, and how we can bring it into our relationship. We talk about my posing for him, or dressing up or down, which can lead to playful foreplay and wonderfully passionate, intimate lovemaking as our desire rises.\n\nConsider looking at nude photos of men and women online\u2014there's so much available in the privacy of our own homes now. You'll get a sense of how magnificently, marvelously, variously the human body is crafted in men and women. And, if you allow your seeing, your gaze, to look at the models, you can easily see if they enjoy their bodies and what parts of their bodies they really like and inhabit.\n\nYou'll discern more for yourself about what is attractive to you beyond superficial looks, what inside of them is attractive that is coming to the surface. Consider it research if you must, or just enjoy the show. If you are able to get comfortable looking at naked bodies, you'll become a healthier sexual being. Talking to children, or anyone, about sex will be easier as a natural extension of it.\n\nIf we had a more pluralistic sexuality in our culture, deeper intimacy and connections with lovers, partners, family, friends, and Earth, and more freedoms, would this not translate to reducing the denial of sexuality, sex, and the freedom to express it? A pluralistic sexual environment would lead to fewer power struggles as equality became more the norm.\n\nHaving pornography in the relationship is problematical when one of you feels a need to keep it secret from the other. Secrets and shame are thugs lurking, hand in hand, in the shadows, where they like it best. Feeling as if your lover isn't present while making love right after he watched a porn film isn't pornography's fault. There is a flaw in the relationship, which may be an unwillingness to be intimate or vulnerable. Also, a lot of people really don't know how or what it means to be truly present while making love. There may be an inability to consciously ask for what one needs in a relationship.\n\nThe problem is not pornography. Or pornographers. Or addictions. Or drugs. The source of the problem lies in our inner beings. We are terrified of ourselves, of intimacy, of our power. We are terrified of seeing the truth or speaking our truth. We are afraid of saying no and even of saying yes. Afraid of unveiling ourselves. The crimes against essential human dignity stop with me. The sin of lying to our children, of deceiving them and ourselves must become something else. Placing the blame on something outside ourselves will not change our behavior or circumstances or how we feel. Each of us has our own private ways of dealing with or avoiding the pain of our lives, the pain we see in the world, the pain of growing old. If watching porn helps, don't deny us that as we find ways to be alive in the midst of the heartbreaks and pain.\n**13**\n\n ** _Healing the Human Soul_**\n\n_I became a visitor to some wounded part of myself._\n\nCHARLES BOWDEN,  \n _B LUES FOR CANNIBALS_\n\n _The longest and most exciting journey is the journey inwards._\n\nKONSTANTINE STANISLAVSKY\n\nHaving intimacy in our life necessitates that we be selfish. The word and concept of _selfish_ has gotten a lot of bad press; anyone who is selfish, acts selfish, or takes care of themselves is seen as greedy, negative, or bad. When I talk about being selfish, I'm talking about putting your needs and wants first. When you do that, you are being intimate with yourself, which enables you to be intimate with others. It brings you into alignment with, and accountable to, yourself. This is one of the hardest things to do if we have been raised to put the needs of others before our own. It is also often used to pretend to be intimate. Taking care of others, putting their needs before your own, is a way to withdraw, to take yourself out, of true intimate relating. Taking care of others, worrying about their happiness, tending to their needs is a fa\u00e7ade of responsibility preventing you from _being_ with others.\n\nPretending we're something we're not precludes us from being real. Taking care of someone can sustain a relationship, but it does not feed it. Intimacy nourishes, deepens, and changes relationships and each participant.\n\nThere are times when we want to and need to put our needs on hold. If a friend has become ill or is having a difficult time, you can be of service in the true sense. When it becomes a way of life that interferes with your being present, real, and intimate, it is a problem.\n\nIt took me a long time to understand this and the importance of it, longer still to exorcise the belief and habit of putting myself in second or third place. Once I did, the sense of freedom I experienced propelled me to experimenting with this until it became a way of life, an expression of my intimate self. From that place, I could see how being second or third was unkind to me. It reinforced childhood injunctions about not being seen or taking up too much space.\n\nGandhi was once asked by a friend if his reason for living in a village and serving the people there was purely humanitarian. Gandhi replied, \"I am here to serve no one else but myself; to find my own self-realization through the service of these people.\" Gandhi's liberation was India's liberation. His insistence that \"personal change and the ability to bring about social change are linked,\" is one of his legacies. To know, to feel, to experience myself in the presence and contrast of others is empowering, sometimes joyful, sometimes extremely challenging.\n\nThere is genius in each one of us. Every person is born with a soul, and that soul has an urge, a drive to do something, to be something, to create, to labor, to teach, to become itself, to express itself, to awaken our innate gifts and talents from dormancy. It's not so much the form of what you do that matters; what matters is that you do it in response to the movement of the soul. And that in that movement toward soul fulfillment, you do it well, with finesse, with _el mundo_ of you. Genius is simply to become an expert at being yourself, mastering the subject of your soul. It is not the same thing as being intelligent or having a high IQ; it is a specific _kind_ of intelligence. Genius is natural talent, creative power, heightened intuition, excellence, and imagination. The genius in each of us embodies all those things, and it demands that we let the genie out of the bottle; let it impose itself onto the world, let its power into the room and through your life. When we are able to do this, we bring Eros and Psyche, sexual energy, love, our hearts, our imaginations, our understanding of spirituality and the numinous into all that we do; to become masters of the soul's journey, to become geniuses in sexual and intimate arenas, to become geniuses of our interior worlds. To bring the genius of your heart's perceptions into the world is a legacy to leave those who will come after. Living by the genius of the heart allows your soul to grow. When we are able to do this, we stop looking for meanings and begin to give meaning to our life in relation to the forms or vehicles the soul needs. That's a really nice legacy to leave, especially where soul and genius are concerned. Our legacy is the imprint of our soul on the world.\n\nThe form through which these things are expressed changes according to the needs of the soul; the form itself is part of the expression. It is not the actual expression itself, but the essence of the person that matters. For example, artists use various mediums to express their soul's urge to create and inside the medium\u2014the form\u2014is expression\u2014the act. Out of that action comes the essence or meaning. It's important to have flexibility with the form to facilitate higher development of the soul. Soul needs the form to be dynamic and fluid in response to its own growth.\n\nWhen I teach, consult with clients, work with plants, spend time in the wilderness, hold my granddaughters, or make sacred love I experience some of the happiest and most ecstatic and joy-filled moments in my life. These are areas where my genius lies, where it is given expression. I'm in service to what is fulfilling to me, challenges me, and moves me toward being all I imagine I can be. Being in service to my soul means to take in soul food, to nourish and sustain my soul so that, in turn, my soul can sustain and nourish this body to which it gives form. As well, I tend to the caretaking of my body, that temple to my soul. When I teach or talk to the ancestors or make pilgrimage to the sacred waters of the hot springs, or when I am able to help a client my soul is fed. These things fill a need in me that would be unmet if I let my fear make decisions for me\u2014fears of teaching, of writing, of fucking up. Especially if I am afraid going into a teaching weekend, it gives me pause to self-examine, to work in interior time becoming the master of my destiny, and to remember why I'm doing it\u2014why I _must_ do it. I calibrate the level of fear and level of joy I have as I approach something to the level of importance it holds for my soul.\n\nI have a well-developed, irritating and exasperating habit of putting myself in situations that look uncannily like a box from which there is only one way out\u2014one I have not discovered until I am in the box\u2014and where the old ways out no longer work. Moreover, the old ways are not supposed to work. I have to go through the distress of finding new solutions to new paradoxes, reinventing myself continuously. It forces me to make choices from a grown-up place and a place of vulnerability, maturity, and freedom from reactionary behavior. It is in those difficult moments that character is defined; it is the difference between seeing a thing through to the end and abandoning oneself. This is what Michael Meade calls \"the right kind of trouble.\" The right kind of trouble is deeply troubling to the soul: it can manifest as a period of heightened irritability, restlessness, and perturbation or can even bring on full-blown depression or a psychotic break. These are signals that the soul is disturbed with the status quo, that there is a great need to go to the depths of yourself to find your way through the difficulty.\n\nWe all have our unique ways to innovate and grow. The best choices are those that are made from free choice, free will. Choices made from fear, from old habits, or from \"should\" are doomed to failure and regret. Freedom comes from the prices paid and the time put in to rewrite yourself over and over again, moment to micromoment. It's hard work; the rewards are immeasurable. It is the work that defines and distinguishes between Homo sapiens, the species to which we belong, and more fully developed human beings who have a complete range of emotional, spiritual, feeling responses\u2014who have become the \"I\" that is I from living and being truly alive.\n\nIf one more person chooses to become undefended, to become more whole, the circle of life becomes more healed. Hope has new life, extending out like concentric rings reaching distant shores.\n\nIt is only since healing the original wounds of my family that I can look back and say I know that I have been blessed. I have a fierce determination and a voracious hunger that drives me on to seek out, find the root of, and harvest my own truth. I want to be free.\n\nI knew freedom was possible, is possible still. Though I had excuses and apologies for my behavior, for my searching, for my dissatisfaction, eventually I gave them up. They served me not. Holding on to them depressed my motive force, weighing heavy on my soul. They were coping skills that kept me stuck in old patterns. Giving them up enabled me to take responsibility and ownership of my choices and behaviors. Letting them go meant giving up being a victim and taking up being empowered.\n\nIt took time to be confident in myself. Even after going through torturous decision-making processes, I would second-guess myself. I believed that if I made the wrong choice something horrible would happen. But more than that, I had no certainty, no self-possession, no sense of where I ended and someone else began; other people's needs were convoluted and tangled up in my own. From years of doing this work, I've retrained myself, watched how my body changed shape in response. I dropped \"baby fat\" when I was in my early forties. I've fallen in love with myself, become self-assured, possessed of self. My boundaries are more clear as well as my \"seeing.\" I've given myself permission to feel everything, to live a life of the senses and of sensuousness, to live outside the conventions of the city and to join Dionysus in the wild woodlands and hillsides.\n\nSEEING WHO YOU ARE\n\nIt's valuable to be able to see the impact you have on people in your life and in the world. Try stepping just outside yourself, as if hovering just above your head witnessing your movements through the world. Take note of how you've influenced people. Really see what their life would be like without you in it; see the void your absence would create. Come to know that you matter. _You matter_. The substance of you, the physical, spiritual, psychological aspects of you, is significant. If you feel less than significant, what would it take for you to feel consequential, substantial, influential, noteworthy? Be of consequence. Be outstanding. Be a force to be reckoned with, or will you live an unlived life?\n\n _Well-behaved women seldom make history._\n\nLAUREL THATCHER ULRICH,  \n _W ELL-BEHAVED WOMEN SELDOM MAKE HISTORY_\n\nYou are the authority of your life, the author who gets to pen a new story, a personal myth. Becoming the authority of your life necessitates you taking personal responsibility. You must show up for yourself and be accountable to yourself. Writing a life requires drafts, edits, rewrites. It begins with imagining what you want, remembering who you've always wanted to be, and making choices that support and move you in that direction, toward that person you know in your deepest self you are meant to be. That person has been inside you all these years, getting you to notice at times, waiting for you to say, \"Okay, it's time. Let's do this dance. I'm ready, already.\"\n\nThere is a difference between being a writer and an author. Writing is writing, and being a writer is being a writer. As author, you are the original creator of a piece of work; you have given it existence and authority. Authorship determines, points to, and carries responsibility for what is created.\n\nFeel the difference. \"I'm writing my life.\" How does it feel? Where do you feel it? Now, try this: \"I'm authoring my life.\" How does that feel? It has more weight, more gravity, more authority. \"I'm writing my life\" feels more heady, almost as if it's coming out of the eyes. The second one is felt closer to the heart and womb, where creativity is birthed from the fires of passion and imagination.\n\nReaching a state of grace is maturity of soul, courage of heart, and strength of character. Having a sense of humor and self-forgiveness and self-compassion, and being attentive to your intimate self will get you through difficult circumstances. Courage is needed for the obvious, to see inside yourself things you've been afraid to see. Loving yourself takes fierce devotion and courage, for many of us were not raised to love ourselves, to understand what that means and be curious about how self-love feels inside us. The willingness to be changed by the work, to follow it through to the end of your life\u2014that is grace.\n\nIt is not tensile strength that is called for, but strength that is already there inside you waiting to be tapped and used. When I pray about strength, I don't ask for more strength. I ask to use the strength that is already there, inside me, and I ask for help in shaping the strength to fit the shape of the thing I'm working with. It calls for reliance on your self, self-analysis, self-correction, and an indomitable will to make yourself your primary relationship.\n\nThere must be a commitment to give up lying in all its forms; lying to yourself through the games you play and the justifications and excuses that support their continuation. We are ingenious at manufacturing excuses and justifications, but they are forms of self-imprisonment. It is beneficial if you possess a desire to get to the truth of things, to aim a searing hot arrow at pretense, burning away the masks, the self-delusions that cover the truth. The truth that comes from seeing and perceiving directly.\n\nMy definition of lying was expanded, much to my horror, by a plant who blatantly called me a liar. I was shocked to hear it and asked in reply, \"Just what do you mean by that? I'm one of the most honorable people I know.\" She was not humored and responded with, \"In many ways you are; however, you lie by omission. You lie by keeping your needs a secret. You lie by not speaking out. You lie by adapting to circumstances. And in so doing, you betray yourself. Therefore, you are a liar.\"\n\nOuch. That was painful to hear, but she was right. Oh, the ways we delude ourselves. I saw myself in each one of those statements, and I felt what a cruelty it was to myself, to me. Lying sucks the energy out of relationships and undermines agreements to be intimate. It's a hustle to present yourself as something you're not. People just don't look favorably upon being hustled. I thought about each lie, felt into each one, and set about to correct them. This is a good place to talk about devotion since it is easily derailed when we engage in self-deception. _Devotion_ is a word that often conjures up images of religious fervor since it is often used in religious contexts.\n\nThere is a common misunderstanding about devotion and commitment. We think once we have committed to something or someone, everything is taken care of. Devotion and commitment begin first with deciding you want that person to be part of your life; you decide to be devoted and committed to him. You then confirm your commitment each day. Commitment is not something to be taken for granted; you must put energy into it to keep it spiritually alive. Deciding every day to be devoted to self-awareness and self-reflection engages all of you and keeps what you are doing in front of you. Devotedly doing the work is holding context for yourself. To hold context is to put energy into something; you think about, ponder, contemplate, pray if you pray, dream, and imagine the thing you are holding context for. You hold a macrocontext of the work overall and a microcontext for specific problems, issues, behaviors, and motivations you want to understand and resolve.\n\nI knew I was beginning to heal when I found myself being spontaneous. Spontaneously, I was suddenly making fun of my own behaviors and joking about my family: \"My sister's breasts are so huge they are registered as weapons of mass destruction.\" Being able to laugh at the absurdities of life and circumstances allows you to enjoy your own company. Explore yourself and your family from the perspective of a healed, whole person. Ask members of your inner council to join you in this. They see things you haven't. If there is an issue or person in your family that is particularly irritating, bring them to your inner council. You can bring anything to your council, by the way.\n\n _You grow up the day you have your first real laugh\u2014at yourself._\n\nETHEL BARRYMORE\n\nUnderstanding that you are human, and understanding what it means to be human, lays open the threshold of forgiving yourself for your human frailties, which is to have compassion for all your mistakes, the hurtful words that you uttered, the ways you failed your children. Allow yourself to be educated by life and by your mistakes. What happened before you became conscious is in the past. The past as it was has no place and no power now or in the future except in how it shaped you, mentored you, and allowed you to become who you are now. It doesn't have to order your life or condemn you to be who you were when you started out. You get to be different, and you can't get there without forgiving yourself. In forgiveness, we are no longer affected by our past.\n\nThis work heals the past **.** In time, the stories of our lives become memories, echoes, an imprint on the retina of the heart. They no longer have power over us. The scripts our families gave us can no longer influence us. The new life has no tolerance for old behaviors or indulging ourselves in what happened so long ago. The stories and the wounds that we carried are no matter. It's not the wound that matters; it's what happens afterward. And now is afterward.\n\nBeing the initiator of your life is a feeling thing. What feels good? What makes you happy? What will it take for \"I will\" to become \"I am\"? Imagine yourself on your deathbed and looking back on your life. What does it look like? What does it feel like? Do you hear a voice inside yourself saying \"I could have\"? What is the thing you could have done and didn't? Where did you stop yourself? What stopped you? Was it fear? Does it feel good to see these things? Are you willing to let fear and hesitation dictate your life? Are you ready to be the leading lady of your own life?\n\nWhat will it take for you to feel good about who you became, the work you did, the legacy you left? As you look back on your footprints, do you like the imprint you left in your path? What choices must you make now to feel good when the end comes\u2014and come it does to each one of us.\n\n _Is it not time to throw off the shroud? Is it not time to speak out, to cry out, to fly, to test wings, to fall, and to laugh with joy over the divine bruises?_\n\nGERRY SPENCE,  \n _G IVE ME LIBERTY_\n\nYou can walk around unconscious (read, coma), or you can choose to be conscious, aware, awake, alive. You are getting closer to death, to biodegrading, to becoming compost and worm food, even as you read the words on this page. So why not go for it? Create your own myth; become the heroine of your own story. Choosing a life of awareness means giving up the option of going unconscious.\n\nAt some point you'll know you've crossed an invisible line that makes taking up archaic behaviors impossible. You'll know this has happened if you imagine yourself, who you are now, in the old life. When the new you is imagined in the old life, there is a feeling to it, an uneasiness, a disconnect, a misfit. It's sort of a prickly, queasy feeling in the pit of your stomach; your breathing gets shallow or you may even stop breathing. These feelings are communications from the new you.\n\nThen there is grieving. You grieve for the childhood you never had, for the life you've left, and for those whom you've left, or who have left you, who have helped you along the way. There is grief for the unkindnesses toward yourself, your body, your lover, your children, your parents. You grieve for the things you needed to hear as a young girl and the love you needed to feel but didn't. Grieve until you are wrung out and empty of grief, and you will find yourself on the other side of it. You'll know when you're there; it has a feeling to it. Say a silent thank you, I love you. And, good-bye.\n\nBecome the epicenter of your universe. You are the initiator. Initiate. Find your passion, your hunger, and follow it from moment to moment. Ask someone who loves you and who you trust to tell you how he sees you, what he sees and knows to be your strengths and gifts. Simultaneously feeling and thinking your way through this will take you far in crafting a life that brings you joy, fulfillment, and satisfaction.\n\nTry on various scenarios in your imagination. Fill in as many details as you possibly can. Ask for help in whatever spiritual tradition that suits you, or create your own spiritual tradition and ceremony. Go to the mountain or the ocean and drink in the power that is there by the bucketful. Watch the birds; listen to them sing up the sun each morning. That is ceremony. It can be simple or elaborate.\n\nAsk the ancestors of your land, of your heritage for help. In Africa, they say they talk to the ancestors because they are closest to them, closest to Earth. God is very busy, they say, but the ancestors, they are here to help. They remember us. They want to help, so sit down at their feet, or do a walk about, or a medicine walk, and have a conversation with them. Give them details. As Sun Bear, the Ojibway holy man said, \"If you ask for crumbs, you'll get a crumby life.\" This is your _life_ we're talking about, your happiness, your joy, your soul's journey.\n\nThe ancestors want to help you. It's their adventure too, you know. They need you to live your life so they can experience what living is like through you. Until I began in devotion, to follow the Earth-centered path, I had a foolish idea that God and the spirits were above Earth, hovering or hanging out in some ethereal, shimmering, invisible-to-my-eye, holier-than-thou, untouchable-by-human hands heavenly realm, a cloud perhaps. I asked myself, Why would they hang out there when there are trees and waterfalls and the smell of cottonwood buds and lavender and a lover's perfume and sex all here on Earth? If I was God or a spirit, this is where I would be spending all my time, invisible or not. This is the place where things are happening. There are many things happening in the realm of the invisibles that I am interested in, but as far as sensate, kinesthetic, erotic, and sensual experiences go, Earth is the place. I fall in love with being here every day as I fall in love with my lover, deeper and fuller.\n\nIt brings our ancestors joy to see us having our dreams fulfilled. They need us to move spontaneously and freely through time and space. They love the sound of laughter and the lightning that flashes out your eyes when you are expressing the truth of who you are in the world.\n\nIn many cultures there are ceremonies that last for days, sometimes weeks, ceremonies with a never-ending fire and long dances with drumming and singing. Often a person dancing is taken over by a spirit; the spirits come to have the ecstatic, sexual experience once again, of blending soul and spirit and body. They come to the human world to help us, and they come to the human world to be helped, to play, to feel. And they come to share the journey and the wisdom of the ages.\n\nWhy is creating your own life important in intimate relationships? Intimacy begins with you. Intimate relationships are best fostered in an environment where both people are self-assured and self-possessed. Each person in the relationship must have a sense of their work in the world, something they take joy in, something that is their creation and a manifestation of their dreams. As a partnership, there must be some joint adventure, a shared dream and shared goals that you hold in your hearts and give energy to, things that strengthen the bond you share.\n\n _Self-awareness is the architect of authenticity_.\n\nMaking unilateral decisions in a partnership, about the partnership, is a recipe for disaster. It is discounting the other and the spirit of the relationship. If you are in a partnership, engage your partner in the process. You must understand how important this is. Say what you have been thinking about, and ask for feedback. Hear, with all of you, what is being said. Ask for clarification if you don't understand something, don't let it lie there and ferment into assumptions and misunderstandings. Saying things out loud is a way of thinking and feeling out loud. And it's a way to fight for the life of the relationship. Not saying things out loud leads to secrets and these kinds of secrets are deadly to your soul and to the relationship. As you talk, you are often able to see more details or get excited about your possibilities, and it sparks the imaginative fires of the other. Speak up when something isn't working for you. Needs cannot get met if you don't know, or won't say, what they are.\n\nDepending on your partner to make you happy, to be the source of your self-esteem, is most certainly a formula for disaster. His presence in your life, his happiness, and the joy and adventures you share together are all sources of and additions to your happiness. His joys and successes infect you, as yours do him. They cannot help but influence you; the two of you are bonded in body, soul, and spirit by the invisible connections between and around you. But he is an addition to you, not your source; you are the most important thing in your life.\n\n _If another person is the most important thing in your life, then you're in trouble and they're in trouble because they become responsible for your suffering and your successes. But if consciousness is the most important thing in our lives and relationship is means toward that end . . . ah! Then we are approaching paradise. We are approaching the possibility of actually becoming a human being before we die._\n\nSTEPHEN LEVINE,  \n _E MBRACING THE BELOVED_\n\nMaturity is taking personal responsibility in crafting a new life. Growing up, being a grown-up, has immeasurable rewards. There are few experiences that compare to having a sense of yourself moving in the world, capable, clear, and certain. There will be times when you feel wimpy, scared, and uncertain. Being a grown-up is being aware of all these feelings as each one comes up, and then tending to them. It means listening to the small part of you that is feeling these things. Take care to listen to what she's feeling; what she needs you to do. You won't know what to do for her if you are unwilling to listen. To listen is to be changed by what you hear.\n\nYou may need to ask your lover to rub your feet or your neck or to just hold you when you're not feeling like yourself. Maybe you need to say, \"I feel wimpy, will you tell me all the things you like about me?\" Yes, it's hard; it feels weird, uncomfortable, unnatural, and completely foreign. We have not been raised this way; we have not been given permission or encouragement to have feelings, to say them out loud in the world. And in our culture, all those uncomfortable things are in place to discourage exploration of that territory. The last thing our culture and our governments want is for us to be walking around feeling good, following our feelings, and noticing that life in these times does not feel good.\n\nLet the Child of you do the asking. She is the one who needs to hear these things. Pay attention to the responses and let them trickle down deep inside you to touch her. With practice, you will be able to feel love moving inside you. You'll feel the changes it makes in your physiology. These changes come in response to love filling you up, lifting your spirit and enlivening you. This looks good on paper and in theory, but when it becomes as natural as breathing, it is glorious. After some time, after the deep parts of you have been fed love and know now that nothing bad will happen by being alive, by taking up space, by getting their needs met, your need to ask for support will diminish and become circumstantial. Emotional incontinence is temporary; you will build new skills, new muscles, and become familiar with the territory.\n\nIn the beginning, it is tiresome; you'll want to stop many times. You're breaking from old behaviors; secret agreements you made with yourself when you were very young are being changed and renegotiated. Part of you will be frightened when you break from family scripts and long-held values. I encourage you to hold that part of yourself, reassure her that you are watching, that you will let nothing bad happen.\n\nSELF-NURTURING\n\nConsult with yourself daily, throughout the day. If you feel overwhelmed, take time out or away. This depth work takes a lot of focus and uses energy, and at times you will feel depleted, wrung out, and you'll get irritated and pissed off at everything as you become exhausted. Stress on yourself and the relationship, upsets, and misunderstandings can be prevented by attending to how you are feeling and what you need and finding ways to renew yourself, to fill up your dwindling reserves of energy.\n\nConsider giving yourself a no-value day\u2014a day when you do nothing that counts toward anything but taking care of yourself. One day might look like this: Don't dress in street clothes; wear your robe and lounge pants all day long. Linger in bed for as long as you can or want to. Read for a bit, or sip tea in bed. Though, honestly, as romantic as that sounds and looks in the movies, in reality it can be awkward and clumsy. Nonetheless, you have to try it at least twice. Linger in bed extra long, bask in the afterglow of great, predawn lovemaking with your beloved or yourself. Read your favorite sci-fi or mystery novel for a few hours. Lie in bed until you choose the one thing that will next make you happy. It could be cooking your favorite breakfast or brunch or lunch. What would make your palate and stomach and the Child in you really happy? Maybe it's taking yourself out for a meal in a cozy, nurturing caf\u00e9. After the meal, maybe back to bed to read more of your sci-fi or mystery novel.\n\nBe sensitive to situational renewal time. If I've been working with people a lot after a teaching weekend, I need time alone or to be held in silence and feel my beloved wrap his love and arms around me. Other times, I need to go alone to the hot springs or on a hike and talk to the ancestors, sit on stone and be filled up with wild landscapes. Maybe I need to gaze at Tree or Stone or River, and feel their gaze on me. If I've been writing all day and straining my eyes and brain, what I need is a glass of fine wine or a shot of tequila (with lime, of course, and no salt, please) and something between my ears besides what I've been writing about, so a good movie is in order. If I feel nostalgic and melancholy, then definitely a romance is in order. If it's been a difficult day, I may want an action, adventure, or karate flick where the bad guys get their asses kicked real good. What I need depends on what I'm feeling.\n\nAsk your lover to take a day off with you. Let the Child in each of you choose something that would be really fun to do together. Plan several days over the next month. And stick to them.\n\nAs you are getting used to the idea, maybe even excited about authoring your life, I give you an exercise that will help you see things about yourself and take you into the territory of your values, beliefs, and deep feelings. Before beginning this exercise, consider making an agreement with yourself to not censor anything, to not hold anything back. You may take a few days or weeks to work on this, for as you move throughout your life in the midst of this assignment, you will find more things to add to each part of it. This exercise is designed to help you get to know yourself better, to see things on paper that maybe you hadn't thought of before. Writing also exercises your faculties of analysis and insight, and helps you to go deeper.\n\n ** EXERCISE   \nExploring Who You Are**\n\nLove\n\nWhat do you love about sex?\n\nWhat do you love about yourself?\n\nWhat do you love about your body?\n\nWhat do you love about your lover?\n\nWhat do you love about your life?\n\nHate\n\nWhat do you hate about sex?\n\nWhat do you hate about yourself?\n\nWhat do you hate about your body?\n\nIs there anything you hate about your lover?\n\nWhat do you hate about your life?\n\nAuthoring\n\nIf you could have any life you wanted, what would it look like?\n\nWhat will it take for you to have that?\n\nWhat steps do you need to take to get there?\n\nIntimacy\n\nHow do you define intimacy?\n\nHow does intimacy feel to you?\n\nDoes creating intimacy frighten you?\n\nDo you know why?\n\nAre you intimate with yourself?\n\nDo you want more intimacy in your life?\n\nWith your partner?\n\nHow will you create that?\n\nI encourage you to use the word _hate_ rather than substituting it with _I don't like_. They are not the same thing, you see. Hate has a particular feel to it; it provides energy to work with, and in lists such as these it lets you know which areas of your life stand out as needing to be changed, altered, redirected, and made whole. \"I don't like\" doesn't offer nearly the same energy, impact, or fuel for change. If you want to make real changes, deep changes, lasting changes in your life, \"I don't like\" isn't going to get you there. You need the passion that hate possesses. If you find yourself avoiding the word _hate_ , you might add it to your list: Why I hate using the word _hate_.*7\n\n _I feel ready to follow even the most trivial hunch._\n\nWILLIAM STAFFORD\n\nECOSEXUALITY\n\nIt is impossible to be whole human beings, fully, consciously, functional human beings, as long as we are disconnected from our sexuality and as long as we deny that our sexuality is implicitly bound up with Earth's sexuality. As long as we continue refusing to see Earth and all her inhabitants and residents as sexual, the disconnect and impoverishment of spirit will continue.\n\nAll of life is sexual; human sex organs have evolved from the prototypes of plants and animals as evolution occurred. Peter Tompkins and Christopher Bird note that \"plants have female organs in the form of vulva, vagina, uterus and ovaries, serving precisely the same functions as they do in woman, as well as distinct male organs in the form of penis, glans, and testes, designed to sprinkle the air with billions of spermatozoa, were facts quickly covered by the eighteenth-century establishment with the almost impenetrable veil of Latin nomenclature, which stigmatized the labiate vulva, and mis-styled the vagina; the former being called 'stigma,' the latter 'style.' Penis and glans were equally disfigured into 'filament' and 'anther.'\"\n\nThen there are the bees who instinctively know where the pollen is and its quality and quantity. They take that information back to the hive, and with the shaking of their rear ends, they communicate all that information. Bees are prodigious and libidinous; they shake their sexual energy all over the place, and when we eat honey or bee pollen or royal jelly, we're eating sexual energy. They communicate by dancing. The male bees, called drones, carry \"the wisdom of the hive\" and keep it in harmony by swarming and \"chanting.\" They also wait for females to fly by.\n\nThey fight until there is one who is able to mate with the female. They have sex in flight, then the penis breaks off and the male bee dies. There are advantages to being human.\n\nDucks have an interesting method of copulation. Forced sex is common, but the female duck has an elaborate system of vaginas that tricks the penis into one of several false sheaths. Only when the drake she chooses enters her vagina with his corkscrew penis will she allow her eggs to be fertilized. The human cervix has a complementary function. It is able to discern which sperm would be a good match; Hippocrates believed that a woman could regulate the acceptance or rejection of sperm. The cervix is able to \"upsuck\" or expel semen. Midwives have been studying female sexuality and the reproductive patterns of women for 100,000 years. Midwives today carry gathered knowledge of ancient cultures of women who have intimate knowledge of their bodies, know when they are ovulating, and can say when conception occurred. Few women in the West are so in touch with their body's rhythms and cycles to be able to say when much of anything is happening.\n\nBreast feeding prevents ovulation and suppresses a woman's menstrual cycle. Suckling infants stimulate the breast, increasing hormonal opiates that suppress the production of hormones involved in ovulation. I wonder if men who choose to nurse infants and children have a corresponding hormonal response; that is, does male lactation from breastfeeding suppress sperm-producing hormones?\n\nWe've become a culture separated from our own body of Earth, turning to cultural solutions for everything from birth control to stimulation, relaxation, sleep, and wakefulness. Plant-based birth control and knowledge of one's body has been found in the archaeological records dating as far back as 630 BCE. In the Greek city of Cyrene on Africa's northern tip, women used the plant silphium or laserwort (wild fennel). It became a prized plant for the freedom it provided to enjoy sex without the worry of pregnancy. The plant's importance is evident on a Cyrenian coin, that depicts a regal-looking woman sitting in a chair. One hand is touching the plant and the other is pointing to her genitals. From the fifth century BCE in Greece, lemon halves were applied to the cervix as a spermicide. Plants as a source of bio-available hormones whose uses have stopped and started menstruation and alleviated premenstrual and menopausal symptoms were first taken seriously by the scientific community in 1933 when Boleslaw Skarynski found trihydroxyoestrin in willow, a substance that resembles estrogen. Hormone-containing plants have been used as abortifacients, aphrodisiacs, and contraceptives.\n\nThe use of plants as contraceptives is new information to humans. Animals have been using plants for all manner of healing and controlling population since they appeared on the scene (plants were here first). Plants as contraceptives have been used by animals since they evolved side-by-side on Earth's grand stage. A certain species of red clover, _Trifolium subterraneum,_ is rich in an isoflavone that disrupts reproduction. Over three hundred plant species have been found to contain levels of phytoestrogens (plant compounds that mimic the female reproductive hormone, estrogen). Phytoestrogens are capable of turning reproduction on or off, depending on the timing of their ingestion. In birds and animals, they tend to decrease fertility. All animal reproduction is dependent on plant chemistry.\n\nTompkins and Bird note that, in Teutonic mythology, \"Baldur, god of light, had secretly gazed upon the naked form of the flower princess Nanna as she bathed in a stream. When her natural loveliness was enhanced by the energy over which Baldur ruled, his heart, said the legend, was pierced, and the marriage of Light and Flowers became a foregone conclusion.\"\n\nGustav Theodor Fechner postulated the concept that believing whether plants have a soul or not changes one's whole insight into nature. If humanity admitted to an omnipresent, all-knowing god who bestowed animation on all things, then nothing in the world could be excluded from this munificence, neither plant nor stone nor crystal nor wave. Why would universal spirit, he asked, sit less firmly in nature than in human beings and not be as much in command of nature's power as it is of human bodies? \"That it is a dark and cold world we sit in if we will not open the inward eyes of the spirit to the inward flame of nature.\"\n\nThen there is George Washington Carver, of slave descent, who became an agricultural chemist, later known as the Black Leonardo. Through his unorthodox methods, he was able to turn the peanut, useful only as hog food, into peanut butter and he used peanut oil to heal the atrophied muscles of polio patients. \"Nature is the greatest teacher and I learn from her best when others are asleep. In the still dark hours before sunrise God tells me of the plans I am to fulfill. The secrets are in the plants. To elicit them you have to love them enough,\" he said.\n\nNot long before Carver's death, a visitor to his laboratory saw him reach out his long sensitive fingers to a little flower on his workbench. \"When I touch that flower,\" he said rapturously, \"I am touching infinity. It existed long before there were human beings on this earth and will continue to exist for millions of years to come. Through the flower, I talk to the Infinite, which is only a silent force. This is not a physical contact. It is not in the earthquake, wind or fire. It is in the invisible world. It is that still small voice that calls up the fairies.\"\n\nThe December 11, 1995, online issue of _High Country News_ reported the deaths of 342 migrating snow geese. Their last stop was a lake that filled in after the Berkeley open-pit copper mine in Butte, Montana, quit operations and closed the mine. The edge of the lake was lined with rocks heavy in pyrite, and when the water hit the rocks, it turned into sulfuric acid. In turn, the sulfuric acid leached metals from the ore, causing the lake water to be filled with arsenic, cadmium, copper, gold, zinc, and silver. When the geese drank the water, they were poisoned to death.\n\nIt has since been discovered that a curious, dark, goopy substance identified as live yeast appeared near the edge of the lake. Testing of the yeast found that it actually absorbed the metals in the lake, which is high in sulfuric acid. And not a mere 5 or 10 percent of heavy metals but as much as 85 to 90 percent. Further testing discovered that this particular yeast is found only in the rectal swabs of . . . snow geese.\n\nAs I write this, there has been a frightening and as yet mysterious and unknown phenomenon: massive deaths of birds and fish have been reported around the world. Thousands of birds have dropped spontaneously en masse from the sky. Explanations that range from Orwellian conspiracy theories, end-of-world times, prophesies come true, solar flares, and government testing in the ionosphere are rampant. I don't have an answer. What I know is that we have no time to put off making changes, real changes, from deep inside each of us. Real changes cannot begin any other place\u2014the kind of changes that move personal mountains and tap the reservoir of human potential buried beneath centuries of oppression, repression, line drawing, and all manner of violence to the human spirit and soul and to the soul of Earth.\n\n _Why not go out on a limb? That's where the fruit is._\n\nWILL ROGERS\n\nIt is impossible to live in the world and not be in relationship to everything in and around you, to everyone and everything you come in contact with. It may be in balance or out of balance, but a relationship you do have. You can get a sense of this if you are able to drop labels for a few moments. For example, if you don't say, \"I have an eating disorder,\" then what do you have? You have a relationship with eating; you have a relationship with food. You have a relationship with your body and with hunger. The next question is: What is the relationship with my body? Without using the words _good_ and _bad,_ try to describe the relationship and how it is. That takes you into the waters of your psyche to plumb the meaning of the relationship and how it affects your movement and your happiness. Try this on: \"I have a relationship with food and my body that has been out of balance for some time. I use food so I don't have to feel or think. I've mistreated my body by overeating or undereating. I've acted like an enemy of my soul and body. I want to change those relationships.\"\n\nIf we suspend the use of labels such as _Muslim_ and _homosexual,_ then what are we left with? What is your relationship to men who love men? Would you know what a Muslim is if we didn't have a word for Muslim? What would your relationship be to someone from Pakistan or Afghanistan? To your brother who sleeps with another man? Relationships are not static; they are living and dynamic, mutable within circumstances and as values and beliefs change. You have a relationship with your children, and the relationship changes as they grow and become independent. You have a relationship with yourself, but do you know yourself, what that relationship is?\n\n _When your priority becomes consciousness, even more than relationship, then conscious relationship is possible._\n\nSTEPHEN LEVINE\n**_Final Words_**\n\n_Nosce te ipsum. Know thyself._\n\nTEMPLE OF APOLLO, DELPHI\n\n _Deconditioning also involves risk and suffering. But it is transformative, freeing the self from helplessness and fear. It unleashes the fifth freedom, the right to an autonomous consciousness. That makes deconditioning about as individual and personal act as is possible. Maybe the only genuine individual act._\n\nJOE BAGEANT,  \n\"AMERICA: Y UR PEEPS B SO DUM?\"\n\nUntil we do the deep work of healing our individual psychoses, owning our sexuality, and calling on Eros and Dionysus, Aphrodite and Artemis, the ancestors, and our own inner guidance systems to create a change, all the words and lip service in the world are not going to change one iota of it. If it could, it would have happened far sooner than the time we find ourselves in. As Dale Pendell says in the _Los Angeles Times:_\n\nIt's not that if you make a place for Dionysian energy, that kind of wild and unpredictable God, that everything will go ok. It won't. That's not true at all. But the cost of trying to suppress it is even worse. Then you end up sacrificing your own children. In the United States today we have more people in prison than any other country on a per capita basis. The majority of these are drug crimes. It's a war against ourselves. It's a war against our children. It was a problem for the Greeks but at least they came to realize you had to admit a certain amount of chaos. You can't try to live risk free. If you try to live completely risk free you're going to destroy what you had. What's a really secure environment? San Quentin is pretty secure.\n\nIt is characteristic of human nature to reach for, search for something outside ourselves, to transcend our experience and make some sense of it. Be cautious of becoming overindulgent and insistent upon giving up personal power and deferring to the great Oz (Oz was, as we know, a timid little man\u2014it could have been a woman\u2014behind the machine, not unlike a lot of us), the Christian God, or governments. Frank Herbert in his classic novel _Dune_ showed us the tragedy of the \"messiah delusion.\" There is no messiah, no savior waiting in the wings for just the right moment of despair to crack open and slip in to rescue us. No one will come down from the heavens and rescue us from the path of illusion and irresponsibility we've insisted on treading. There is no rapture coming. The Hopi have been telling us for thousands of years, \"We are the ones we've been waiting for.\"\n\nCharles Bowden poignantly asks us to examine what the real dilemma is.\n\nImagine the problem is not physical, imagine that the problem has never been physical. It is not biodiversity, it is not the ozone layer, it is not the greenhouse effect, the whales, the old growth forests, the loss of jobs, the crack in the ghetto, the abortions, the tongue in the mouth, the diseases stalking everywhere as love goes on, unconcerned. Imagine the problem is not some syndrome of our society. Not something that can be solved by some commission or laws or redistribution of what we call wealth. Imagine that it goes deeper; right to the core of what we call our civilization. And that nothing outside of ourselves can effect real change; that our civilizations, our governments, are sick. And that we are mentally ill and spiritually dead. And that all our issues and crises are symptoms of this deeper sickness. Imagine the problem is not physical, and no amount of driving, no amount of road will help deal with the problem. Imagine that the problem is not that we are powerless, or that we are victims, but that we have lost the fire and belief, and courage to act. We hear whispers of the future but we slap our hands against our ears. We catch glimpses but we turn our faces swiftly aside. The whistle is always blowing. There is no denying what is before my eyes. We all know the future; we only must say it and face it. There will be no first hundred days for this future; there will be no five year plans, there will be no program. Imagine the problem is that we cannot imagine a future where we possess less but are more. Imagine the problem is a future that terrifies us because we lose our machines, but gain our feet and pounding hearts.\n\nImagine a world, our private world, where we refuse to lie and a world where rudeness is not common behavior. Imagine a world where each of us takes responsibility for our own lives and well-being. Buckminster Fuller articulated it best when he said, \"All of humanity is in peril of extinction if each one of us does not dare, now and henceforth, always to tell only the truth, and all the truth, and to do so promptly\u2014right now.\"\n\nHuman beings are flesh-and-blood biological animals, and though we have access to the invisibles to work with, to pray to, to ask for help, they cannot do the work for us; it's not their responsibility to fix the mess we've gotten ourselves into. They come to aid us, to help us find our way, to regain our balance, to open doors and opportunities. It's up to us, then, to do the work.\n\nThere are pockets of people beginning to create change, willing to be innovative and creative and to say, \"Fuck it, it stops here. I take personal responsibility.\"\n\n _You've got to jump off cliffs all the time and build your wings on the way down._\n\nRAY BRADBURY\n\n _Take your life in your own hands and what happens? A terrible thing: no one to blame._\n\nERICA JONG\n\nWe need a growing body of people who are willing to make conscious choices and to stand up and speak truth to power, who are willing to own their personal power and sexuality and heal the wound between human beings and Earth. We need people who are willing to get their hands dirty doing the work, willing to stand outside mainstream thought and dogma, and especially, willing to go into the murky waters of their own psyches and heal from inside their own skins. We know so much but do so little. As Buckminster Fuller said, \"We are called to be architects of the future, not its victims.\" So we begin by deconstructing, dismantling what has not worked in our lives, in our interior world.\n\nJ. Krishnamurti said, \"When you are really learning you are learning throughout your life and there is no one special teacher to learn from. Then everything teaches you\u2014a dead leaf, a bird in flight, a smell, a tear, the rich and the poor, those who are crying, the smile of a woman, the haughtiness of a man. You learn from everything, therefore there is no guide, no philosopher, no guru. Life itself is your teacher, and you are in a state of constant learning.\"\n\nIn every new movement that begins, its conception is traceable back to an idea. Ideas are at the heart of any new movement. The idea is not always clearly defined, but it is there, an ember, a spark; first in the imaginal then spoken and written. It's always waiting to be given form, made manifest, through the actions and in the lives of men and women.\n\nIncorporating sacred sex more fully into our lives and living as the sexual beings that we are provides us with the challenge and opportunity to be a wholly integrated human being. Despite edicts to focus on the family, divorce and remarriage are common occurrences. Though not officially sanctioned, people are experimenting with alternatives to the nuclear family and traditional marriages\u2014having open marriages, living with extended families. There are movements to incorporate pornography and Earth sexuality into our relationships. The question is no longer whether or not to explore something different; it is how honest, creative, innovative, and clear-sighted are we willing to be. We can no longer afford to ignore the sexual crisis and the need to be openhearted and open-minded in our understanding of sexual dynamics.\n\nIt would appear that each of us is evolving into a more responsible human being who can make choices from a place of deep wholeness. A human being who can make plans according to her own needs and desires and not fit into someone else's plan for her.\n\nIn the end, it may be our distrusting, fearful, inhumane relationships with one another and with Earth, with Nature, that will be the fall of the human species. To keep civilization going, we must invent new values about human beings and sexuality, bringing the wildness of Earth, inviting Eros and Dionysus, Aphrodite and Artemis into all that we do.\n\nI revisited my family story rewriting it while in the arms of my beloved. That story is old and it no longer fits who I've become. I traveled the lands of gods and goddesses of antiquity (they're all still around, you know). I met some of the new ones coming up from the still-fertile, if latent, soil of our sexuality\u2014that ruled by Eros and Psyche. I traced the roots of our repressed sexuality through religious and political oppression, through sexual revolutions, to the brilliance and courage of Lenny Bruce, James Hillman, Eric Berne, Marty Klein, George Carlin, Mary Roach, Aphrodite Phoenix, Laura Agustin, Erica Jong, Stanley Keleman, Henry Miller, D. H. Lawrence, and Ana\u00efs Nin, to name a few.\n\nThe extremism, fanaticism, repressive attitudes, constipated beliefs, and hypocritical absurdities we insist we hold onto with tooth and claw continue to be bizarre and frustrating. I will be relieved when I see people leaving the church and linking back to Gaia's temples\u2014those formed of green and flesh.\n\n _The people should not be afraid of their government; government should be afraid of their people._\n\n _V FOR VENDETTA_\n\nANOTHER NEW SEXUAL REVOLUTION\n\nJames Baldwin said Shakespeare's \"bawdiness\" mattered to him once he realized that bawdiness signified \"respect for the body.\" Bawdiness will be part of the new sexual, political, and ecological revolution. This is the new sexual revolution; a revolution of authentic voices. A revolution of earthy lustiness, boldness, out-of-the-closet complete ownership of sexual, sensual energy infusing everything we touch and breathe upon. The kind of revolution that pulls us out of the secret holding in and takes us out to Earth to self-baptism in the wild waters, mud between our toes and red clay under our fingernails. This will be a revolution that redefines _pretty_ and _sexy_ and one where beauty comes out of the body from self-love and adoration; the beauty that is created when we abandon ourselves to falling in love\u2014with men, with women, with children, with nonhuman beings\u2014and when that love permeates all that we do.\n\nThis revolution will link us back to our feeling, sensing bodies, our ground of being, and free us to say what is true and real. And it will fill the dry, calcified cracks separating humans and Earth with the slippery healing balm of sexual, sensual energies. It will call on the gods and goddesses of the old ways, those of the new ways, and the spirit beings of the places we call home to be part of healing the wild bond between humans and Earth.\n\nThe new revolution brings mythic elements out of the dreamworld and back to the interworld where humans and gods comingle. It will heal the biodegradation of _anima mundi_ and the soul of each of us that has been happening since we humans took ourselves out from the wild glades. It will of necessity begin inside each of us. When you are able to say, \"Enough is quite enough,\" and \"What I've been doing hasn't worked out so well,\" then you have options available to you. Seek not happiness; it is surely a derailment. Rather, seek what makes your heart thrum and vibrate, what makes you come alive. Happiness comes when we engage in our soul's work. This is the arena in which healing of the Earth begins and takes hold.\n\n _Until one is committed, there is hesitancy, the chance to draw back, always ineffectiveness. Concerning all acts of initiation (and creation), there is one elementary truth . . . that the moment one definitely commits, then Providence moves too. All sorts of things occur to help one that would never otherwise have occurred. . . . Whatever you can door dream you can do, begin it. Boldness has genius, power and magic in it. Begin now._\n\nJOHANN WOLFGANG VON GOETHE\n\nLive inside your own feet and partake of the adventure of your own soul. Follow the essence of things, the feelings inside, and the form will take shape organically. Have sacred sex, lots of sex, wild sex, passionate sex. Feel deeply and with abandon. Owning your personal power, your sexual power, and doing the work of personal healing through intimacy helps everyone. It does not happen until you free yourself and are then able to make choices as a free woman. As Erica Jong says in her novel _Sappho's Leap,_ \"Choice is the luxury of the free.\"\n\nNothing we do happens in a vacuum. All that we do radiates out to the larger society. You are not finished falling down. Growth is the act of returning to your feet and dusting yourself off, gently. It is taking the next step, the unknown step, the one that opens the heart to the remarkable mystery of life, of being. Sooner or later, your life will be over. Two millennia ago Rabbi Hillel said; \"If I don't do it\u2014who will do it? And if I don't do it right now\u2014when will I?\"\n\nIn closing\u2014finally\u2014I quote Eric Berne who continues to help clarify and inspire:\n\nThe somber picture . . . in which human life is mainly a process of filling in time until the arrival of death, or Santa Claus, with very little choice, if any, of what kind of business one is going to transact during the long wait, is a commonplace but not the final answer. For certain fortunate people there is something which transcends all classifications of behavior, and that is awareness; something which rises above the programming of the past, and that is spontaneity; and something that is more rewarding than games, and that is intimacy. But all three of these may be frightening and even perilous to the unprepared. Perhaps they are better off as they are, seeking their solutions in popular techniques of social action, such as \"togetherness.\" This may mean that there is no hope for the human race, but there is hope for individual members of it.\n**APPENDIX**\n\n ** _Talking to Children and Teens about Sex_**\n\n_It is easier to build strong children than to repair broken men._\n\nFREDERICK DOUGLASS\n\n _In a time of universal deceit, telling the truth is a revolutionary act._\n\nGEORGE ORWELL\n\n _Saguaros are almost never found in the open. Underhill . . . quoted some Papago words that indicate they may have had the nurse plant concept before it was suggested by Anglo botanists. The Papago belief was \"saguaros have trouble getting started in life. They are so big. Like a fat child, they get sick. But paloverde grows fast like a mother. It bends down the leaves and keeps the wind away.\"_\n\nRUTH UNDERHILL,  \nQUOTED IN _D ESERT PLANTS_ 2 (AUTUMN 1980)\n\nH _omo sapiens_ are not the only species to grow through various ego states and stages of life. We are not the only species to be young, have adolescence, mature, and grow old and sometimes wise. Every species, whether it has roots and flowers, stands on two legs or four, needs elders to teach the young how to be that species. Elder trees teach young trees how to be tree, the archetype. Jacqueline Memory Paterson writes about European and Native American beliefs about the role of elder trees.\n\nThe unique personality of the elder was anciently believed to come from the spirit of the \"Elder Mother\" who dwelt within the tree. The Elder Mother, called Elle or Hyldemoer in Scandinavian and Danish myth, worked strong earth magic and according to legend avenged all who harmed her host trees. No forester of old would touch elder, let alone cut it, before asking the Elder Mother's permission three times over and even then he was still in dread of her possible wrath. Likewise, in many country districts of Europe and Britain, wise people still show respect by touching their hats when passing elder trees, in continuance of ancient custom. Certain North American tribes also believe that elder is the Mother of the human race.\n\nElder bears teach cubs how to be bear. In his book _The Presence of the Past,_ Rupert Sheldrake writes: \"When harvesting plants, I never take the elder plants, insuring that their wisdom and genetic strength can be passed on to the younger generations. According to the morphogenesis view, living organisms, such as badgers, willow trees, or earthworms, inherit not only genes but also habits of development and behaviour from past members of their own species and also from the long series of ancestral species from which their species has arisen.\" We need sages, wise elders to take their place among us, to teach the children. We need elders who have owned their own sexuality unashamedly, unabashedly, humbly, unapologetically. And we need them to pass it on to the children through the teachings and by modeling healthy sexuality.\n\nWe tend often to take ourselves too seriously. This is especially true when talking to children about a topic we consider serious. Or scary. Or difficult. Or bad. Or naughty. Or seriously uncomfortable. One thing we all need to be aware of is that children are inherently intuitive and psychic. They feel vibrations, they sense our tentativeness, how we truly feel about something, and they know when we are lying. Their bullshit meters are very sensitive. Lying is damaging to them and to your relationship with them. When they sense you are lying, it brings your trustworthiness into question and it degrades their self-respect: Why would someone who loves me lie to me?\n\nHow do we talk to children about sex and sexuality? Start where we are, where they are. It's helpful if you can remember when you were their age, what was happening for you and your sexual curiosity. Be honest. Understand that children, at a very young age, are fascinated with their bodies, with our bodies, with sex. To raise sexually healthy children, capable of sexually intimate and healthy relationships, we need to be sexually healthy in ourselves and with our partners. Accepting that your children are sexual beings from the beginning is a very good start.\n\nWith that said, be cautious talking about sex with children until you have tried and enjoyed all kinds of sex\u2014various positions, wet sex, oral sex, licking, touching, caressing, standing up sex, backward sex, sex under the sun and moon, sex on and in a river, battery-operated sex\u2014at least a bazillion times. You ought to know what you are talking about. In other words, to the best of your ability, know what you are talking about.\n\nWhen talking to children about sex, remember, it's about them. I recommend offering personal information only when asked for, or if you have an anecdote that seems to be supportive of the conversation and what was asked. Otherwise, stick to what's real. If you realize some of your fear about talking straight is that you don't have all the facts, now is the time to find them. Don't know where your cervix is and why you have one? Sadly, I had only a vague idea until mine got sick. On one visit with my gynecologist, she inserted a tiny tube with a camera on the end. I watched the monitor as I got an up-close-and-personal look at my cervix as it was healing. It was amazing and revealing to meet that part of me, to say hello \"face-to-face,\" to know what it looked like, what I had been seeing as I prayed and talked with it over the six weeks of healing.\n\nI don't know what the odds are that your child is going to ask you about masturbation, but in the event it does come up, are you comfortable talking about it? Most people aren't comfortable talking about masturbation though most everyone does it. Jocelyn Elders, surgeon general from 1994 to 1995, lost her job after fifteen months for suggesting that teaching masturbation wouldn't be out of place in our school system. Elders said: \"I think that masturbation never got anybody pregnant, does not make anybody go crazy, and what we're about is preventing HIV in our bright young people. Nobody has to teach anybody how to masturbate, God taught us how. So I think that now, even in our society, they're saying that maybe this is something that we should stress more for couples and we know that they do already.\"\n\nSince they were old enough to understand language, I've been telling my granddaughters three primary things: how much I love them, that they can ask me anything, and that I will never lie to them. It's my job to encourage whatever thread of interest they want to follow for as long as they want to follow it. I support their natural curiosity and expose them to wild ecosystems. This includes collecting, and petting, snails, slugs, and worms. In those moments, I feel the trust and bond between the three of us grow deep roots. I take care not to impose my fears onto them. I give them room to explore with parameters, and I give them information on how to fight against what they don't want and for what they do want, to say what they need and want. I talk to them about what to do if they turn around in a crowd and can't see me or their mom: stay where you are. More people get lost by wandering off in a panic. Panic, from Pan, is the imaginal, and it stops rational thought in its tracks. James Hillman says, \"The imaginal is never more vivid than when we are connected to it instinctually.\" When in a panic, the imagination runs without rational thought or adult information.\n\nI began having anatomy talks with my granddaughters when they were four and seven. They referred to both boy's and girl's genitals as \"wieners.\" They giggled when I gave them the words labia, clitoris, and penis. The next day, the seven-year-old, pointing between her legs, asked me: \"What was the el word for this part that girls have and boys don't?\"\n\nDon't fool yourself into thinking that you can wait until adolescence or puberty to have \"the talk.\" Curiosity about the body and what feels good begins in the first year of life. We first learn that the breast and I are separate, that I can move my body this way and get from here to there, and that dirt is interesting to taste.\n\nAs soon as children discover they have a body with physical boundaries, they become fascinated with what it does and can do. Children begin exploring their body parts, including the genitals. It starts very young, younger than you may be comfortable knowing about. I learned this when my son was two years old. I heard him giggling to himself in his crib. I walked in to see him sitting up and playing with his penis. He was enjoying himself very much. He didn't see me, so I quietly backed out of the room and left him to enjoy himself. Toddlers, both boys and girls, find ways of pleasuring themselves before they are a year old. They engage in genital play, fingering, or simple handling of their genitals.\n\nSensual touching begins in the moments immediately following birth when baby is placed on mom's breast. If she chooses to breast feed, the bond is reinforced. Fathers need to bond with infants immediately as well. Holding the infant in his arms next to his chest imprints his heartbeat and smell in the baby.\n\nBabies need ample cuddling, skin-to-skin contact, and nurturing. This is their first experience of intimacy. Dressing them, diaper changing, and bathing them is their first experience of the sensuality of the body. Feeding time is nurturing and cuddling time. If your baby is a boy, you may notice an automatic response of erection during feeding, whether nursing or bottle feeding. It's a normal response. Take care to not discourage or interfere with it. Baby girls have clitoral erections, which, as we know, are less noticeable. It's not unusual for it to be accompanied by vaginal discharges so don't be alarmed if you notice extra fluid in the diaper.\n\nAt around four and five years old, children like to play doctor. They want to see and touch each other's bodies, though the idea of intercourse doesn't occur to them. It's normal behavior for that age. What's not normal or healthy is if older siblings or adults touch their bodies. Parents can and do until children learn how to bathe themselves. This is the age when kids begin learning how to navigate life, and how to resolve conflicts and put their things away. Marty Klein says, \"That's when their sexuality starts developing, too.\" If you want to raise healthy, empowered children, let them explore their sexuality in a safe, comfortable environment; in their own home. If children are healthy, they \"will become more sexual rather than less, and probably on a faster timetable than you're comfortable with.\"\n\nWhen I visit my granddaughters, we snuggle up together at bedtime with a storybook in my hands, I hug them, kiss them all over their faces, tell them how much I love them, how wonderful they are, and we giggle a lot. The conversation flows unhindered in the direction they want it to. They ask about breasts, pubic hair, and kissing boys. \"Yuck,\" says the four-year-old.\n\nWe have fun with the teaching and learning. They are interested and curious, and I encourage those things in them. My job as their grandmother, our job as parents, is to show up and to tell the truth. Our job also is to make certain we are not talking with shame or shaming them for their natural curiosity. Hold the space for them as intelligent beings, capable of thinking and making decisions. The more honest information we give them, the better able they will be to make intelligent, informed decisions.\n\nOne time my granddaughters and I were having dinner out on our first night together. We looked at the menu, and I read the descriptions of meal choices and asked if anything sounded good to them. They decided what they would like to eat. When the waitress came to our table to take our order, I indicated for them to tell her. They both got shy and said, \"You tell her. Mom always orders for us.\" Very quickly I assessed the situation, decided what response would best support them, support our relationship, and help them get the food they wanted without any drama. Thoughtfully, I took a deep breath, leaned forward, looked at them with grandmother eyes, and said: \"Okay girls, here's the deal. When you're at home with your mom and your other grandma, you have certain ways of doing things that works for all of you. And when we're together we have ways of being together that work for us. Since we don't see each other very often, we need to figure it out more quickly. What I'm asking you to do is this: If you want something, anything, it's up to you to ask for it. So you get to tell the waitress what you want to order. It's really fun to do it. Will you both do that?\" They looked at each other hesitantly, so I said, \"It's all right, nothing bad will happen. I'm right here.\" As the sense of freedom and independence percolated, sly smiles crossed their faces. Having a restaurant meal just became even more interesting.\n\nTalking with children about things that matter is to empower them with life skills, tools, and information. It's to build a foundation for them to be free, independent, autonomous, authentic, and shame free. Encourage them to feel and think at the same time. Children are like little detectives, inquisitively asking questions they want answers to. Without your help and an environment where they know they can ask you anything, they will absolutely find out for themselves from other sources, and there are many other sources.\n\nCensoring gives the message that they are not capable of making their own decisions. Children need to be able to make mistakes, to discover where their own comfort zones are, what their limitations are. The stronger the foundation of trust, love, and intimacy you have with them, the healthier that process will be for all of you.\n\nIf you watch television, take note of how many obvious and subtle references there are to sex and sexuality in one hour. Sex is used as a sales pitch so much so that we equate a new car, certain beers, and clothing with being sexual or having sex. As adults, we screen out the meanings in them, but children are exposed to thousands of images and messages about sex long before their senses are dulled, and they are trained out of seeing and analyzing. Watch television with your children and be sensitive to opportunities to talk about what you see, not only in advertisements but in movies, documentaries, and television series. Point out the difference between actors and real people.\n\nThey are like sponges taking in the values and beliefs from the media and of the adults in their lives. Seven-year-olds are well enough aware of adult standards and expectations to be sensitive to their own failures, fears, and mistakes. They are quite skilled at keeping secret their own interest in sex, though they may ask sideways with questions about pregnancy and birth. Seven- to nine-year-olds continue to engage in same-gender sex games, girls primarily, while boys channel it into secret clubs and build clubhouses.\n\nIt's important to keep in mind that the onset of puberty is declining in age. Though studies are primarily aimed at girls, boys are not immune to the effects of pseudoestrogens and hormones in cows' milk, beef, and plastics, which have been found to initiate early puberty. Typically, girls reach puberty between the ages of 9 and 12 and boys between 10 and 14. In the last decade, there has been an increase in the number of 7-yearold girls developing breasts, acne, and pubic hair. Exposure to estrogen in food, plastics, and chemicals is common, and \"[e]strogens do stimulate breast development,\" according to Stanley Korenman, an endocrinologist at the University of California, Los Angeles. In the 1700s, the average onset of menses was between 17 and 18 years of age. It's been speculated that malnutrition in that time had something to do with the later onset. With the availability of more food and the increase of added hormones to meat and dairy products and chemicals in our water and lotions, the onset of puberty is happening at younger ages.\n\nChildren who are obese are prone to go through puberty at an earlier age. Mt. Sinai School of Medicine studied six- to eight-year-old girls in Cincinnati, San Francisco, and New York City. Urine samples were collected, and the presence of hormone-altering chemicals used in shampoo and lotions were found. Exposure to phthalates and fragrances were associated with early breast and pubic hair development.\n\nIf you notice your child showing signs of early puberty understand that it may have psychological and emotional impacts for them. Imagine being the one girl in your second-grade class with breasts developing.\n\nChildren are the most creative about finding answers on their own and learn early to navigate the Internet. In an era of texting, Facebook, and Twitter, parents are outmaneuvered quickly if they aren't prepared and haven't set a solid foundation of open communication early on. If you don't know the answers, tell your child so and that you will find it for her or the two of you will search it out together. Having a healthy relationship with your own sexuality and intimacy\u2014that is, owning your sexuality, what you know, what you don't know, and being honest with a sense of humor about it\u2014goes a very long way to raising sexually healthy children. Hardly anything compares to it in strengthening the bond between you and your children. Having your own strong, clear foundations around sexuality and intimacy will be invaluable in maintaining close, intimate relationships with your children and as a family.\n\nParents do get nervous as they watch their children grow into sexual beings (as if they hadn't been sexual beings all along). We carry so much fear about touching and hugging and snuggling that, as our children get older, too often we pull back.\n\nYou'll have to find a balance that works to maintain a close relationship with your children, your need for privacy, and their need for sex education. What you decide to share should be based on what feels right to you, is supportive to their healthy development, and is not based on shame, guilt, or fear about sexuality.\n\nSaying no demonstrates that in intimate relationships you can set limits and have a sense of modesty. You don't have to be angry when you say no; saying no can come from the love of the relationship, the child, or yourself and doesn't need to be hurtful or defensive or cause damage to the child or the relationship. Saying no models healthy limit setting, instills trust that you will answer when you feel good about it while encouraging the child to ask whatever he wants.\n\nIn all that my granddaughters and I do together, I encourage them to think and feel for themselves, and I repeatedly give them permission to say no. We cannot start early enough encouraging _them_ to say no. They certainly understand the word's meaning. It's a valuable word two-year-olds learn and use very well and often.\n\nEveryone needs to have permission to say no. If you haven't given it to yourself or the children in your lives, try it right now. Make it a meditation. My inability to say no was convoluted and confused with wanting to be the good girl. Not a very attractive or sexy habit at that. Reflexively, I gave up my needs and wants and happiness to please others. I habitually disempowered myself. Conversely, it is as important to be able to say yes consciously from free will and have some skill in thinking ahead to the consequences of a yes, or no.\n\nBefore we became puritanical about sex and sexuality, extended families in indigenous cultures shared the same sleeping space and sex just happened as a fact of life. There was no separation; everything was a part of life's natural rhythm. Everyone in the household knew how often sex happened, the sounds that were made, and children grew up seeing their parents' genitals. Because this way of life is foreign to us, just the thought of it can make us uncomfortable. Each of us has to examine our own feelings and find what works for us. Some families sleep together in a family bed when their children are small. Others have separate bedrooms. Some people are more comfortable with nudity than others. Whatever your personal comfort level, it helps to be calm and matter-of-fact about asserting your need for privacy. It is important to answer children's questions about sexuality as openly as possible with as little anxiety and fear as possible, for children pick that up very easily. There are ways to answer questions you find difficult with honesty and without the contamination of shame.\n\nHowever you choose to respond to your children's inquiries and curiosities around sex and your sexual privacy, you might find it useful to first consider a few things:\n\n  * What will be the impact of your decisions on your child's sexual health and maturity?\n  * What will be the impact on you and your child's friendship and bond of trust?\n  * What is motivating your decision?\n  * Are you able to step outside cultural fears to make your own considered decisions?\n  * Your children will find answers to their questions, with or without your blessing and support; which would you prefer?\n\nUnderstand the difference between privacy and secrecy; they are not even remotely the same thing though the two do get mixed up. Privacy is a conscious choice to be selective about what information you disclose about yourself. Secrecy, and I'm talking specifically about sexual secrets, keeps the past alive and toxic and it holds us as victims. Secrets isolate us, keep parts of us hidden from our partners and friends, prevent healing, cause doubt and low self-esteem, and can increase paranoid tendencies and fear of being found out. Our fears of conflict and of being rejected all get fed by secrets.\n\nWhen we understand that increasing children's experience within the natural world increases the chances of them becoming healthy, sexual, sensual adults with healthier imaginations, more whole in themselves and with greater capacity for problem solving, only then can we truly see our role in facilitating that growth. As we grow more distant from nature, the physical and emotional distance between each other becomes greater. The natural environment is where children are sensitized; it is the world of sensations and beauty\u2014where air and sun kiss skin, where bare feet touch the sensual Earth, where textures, tastes, smells, and colors are rich and vibrant and ever changing. It is the natural world where a child learns to inhabit her interbeing with the world into which she was born. In natural ecosystems, environments teeming with life, microbes, and the sexual reproduction of plants and animals, children learn the interplay of organisms, and the experience of their life among other life-forms is generated. They need to feel bonded to and inseparable from the experience of living, which uncensored play in Nature fosters. Spending too much time in artificial environments, looking at life on the Internet or under microscopes, causes a sense of schizophrenia where a child's sense of place in the biotic community is never encouraged to grow. Without regular time in nature, a child's senses, sensibilities, and sensory acuity retreats.\n\nChildren learn lessons from the natural world that cannot be taught in classrooms or through lecturing or Internet searches: They are able to experience directly the changing seasons and how other life-forms respond to changes in the environment; how they prepare for winter, make nests, gather food, tend to their young\u2014lessons that translate to human life. They learn that all animals have sex and can see it as a natural function of life and of being alive. More importantly, the education of their senses takes place formatively in the wild or Nature. It is through sensing, developing a keen awareness of subtle, sensory inputs, that they become discerning and learn to trust their bodies and their intuition. They learn what their fears are, how to assess danger, how each fear feels in their bodies and how to work it out in themselves. Nature is a great teacher of values; she is judgment free and unattached to outcomes.\n\nWatch children at play in a natural environment. They are immersed in the now of the experience, their senses keenly tuned to the rhythmic movements and music of nature, their inner world and the outer world flow and intertwine unobstructed, one with the other. There is a silent, elegant communication that is taking place. They are \"sitting\" at the feet of Master and Mother of all that is in rapt attention; it is an attentiveness not found while sitting at desks in school buildings with fluorescent lights humming overhead. They sit on the bosom of the Great Mother, as she feeds and nourishes their bodies with myriad scents and colors of flowers, sounds of birdsong, the sacred dance of air, heat and cold, sun and moon. The elemental world of gnomes, sylphs, undines, and salamanders feeds their imaginations and fosters the knowing of other worlds and beings. This is the village it takes to raise a child. This is where their parents, our parents, and other parents and family are. We come through our human parents, but we are born to our family in the wild.\n\nThe ecological crisis we find ourselves in will not be remedied by more rules, more parental admonitions. The ecological crisis, as well as psychological dysfunctions, is mended by the balm of time in the natural world. When we are born, we need to be placed not only on the breast of our human parent, but we also need to be bonded with Earth.\n\nADOLESCENTS, SEX, AND INTIMACY\n\nAdolescence comes from the Latin _adoescere,_ meaning \"to grow up.\" The term has been in common usage only since 1904 when Stanley Hall \"discovered\" this stage of growth, which he attributed to social changes in the early twentieth century. The National Child Labor Committee was formed in 1904 to abolish all child labor. Child labor laws keep children under sixteen out of the workforce, and universal education laws keep them in secondary schools longer, prolonging the period of dependence.\n\nAdolescence is the time of life between puberty and being an adult. Because it is a time of becoming an adult, a grown-up, adolescents need adults who model healthy sexuality and maturity. Overcoming some of the culture's shames and fears are necessary to being an elder.\n\nThey need to see and experience the adults in their lives feeling and thinking, which then supports and encourages young adults to feel and think for themselves. Because being ill informed about sex can have serious consequences, particular attention needs to be paid to it.\n\nSex hormones are fully activated as children reach adolescence. During teenage years, adolescents begin to assert their independence, and rightfully so. They insist on being autonomous and try in earnest to separate their identity from their parents and siblings. People in this age group tend not to want to hear anything about sex from their parents unless they've had an intimate relationship before the terrible teen years. The teen years are the time when adolescents are learning to grow their own internal mother and father, and it's when they craft and adjust their own moral compass. They need permission to explore their bodies and experiment with dress, hair color, piercings, and tattoos if they wish. They need to find out for themselves what their limits are. Ideally, its best if this can all happen in an environment of love and caring and open communication. Teens are interested in sex; it's a biologically encoded reality that they become more focused on as hormones begin the cascade of bodily changes, altering the geography of their bodies.\n\nThere is an interesting phenomenon that happens between fathers and daughters and mothers and sons during adolescence. When our preteens reach adolescence, the changes that take place inside and externally can be quite dramatic. For girls, it's not just the visible physical changes\u2014blossoming breasts and widening hips\u2014but the hormonal changes that begin the onset of the menstrual cycle and make them fertile and ready to bear children. Pheromones\u2014those chemicals released by the secreting individual that impact the behavior of the receiving individual\u2014are released into the air bringing a whole new element into the family dynamics. There is young and newly awakened sexual energy in the house. The young woman experiments with her cleavage: How much can she show without being sent to her room to change into something else. Literally, she is being asked to change into something more acceptable so that everyone feels more comfortable while the sexual part of her is repressed and denied a life.\n\nWhen this begins to happen and fathers and mothers have little or no information, relationship skills, or comfort with sexuality, they do the one thing they know, the one thing that damages the father\/daughter or mother\/son relationship\u2014and sadly causes damage in young adults coming into sexual power\u2014they shut down emotionally and create distance. The response that would be the most whole for everyone involved is for adults to grow up; work through their discomfort and lack of clarity about being sexual; understand and accept that their children are sexual beings; and start talking\u2014break the silence and stigma around sexuality and sex.\n\nChildren and teens are human beings, read: sexual beings. They have fantasies, desires, needs, and questions. They need to feel good in their bodies, accepted and supported in the changes they are going through. We all know how challenging a time it can be; let's not make it any more painful and confusing.\n\nHans Hofmann notes that \"[s]exuality is for the young a symbol of emancipation from family control and a foretaste of important, independent actions to come. Adolescent expectations about the creativity of sexuality are generally high and their criticisms of the failures of their elders are disarmingly perceptive.\" What else must be emphasized is the intimate relationship between sexuality and self-understanding. Young adults need to unify what they know of themselves as human beings, how they feel and think about themselves and the reality of being a sexual being. An understanding of the body and its role in expressing sexuality is so much more satisfying if there is self-acceptance, admiration of physical attributes and shame free. As Hofmann writes, \"Sex can never be successfully abstracted from earthly enjoyment of the fleshly nature of man.\"\n\nAdolescents want and need to know about safe sex, how to use a condom, the pros and cons of various forms of birth control, where to get birth control, how to love and care for themselves and their evolving bodies, and how to nurture the relationships they will be exploring. One of the most loving things you can do is to give them information on what to do in the event of rape or sexual assault, such as where to get tested for sexually transmitted diseases.\n\nThey need to know and it's our job to give them accurate, up-to-date, honest, and truthful information. We cannot fairly expect them to take care of themselves, to act in their own best interests, to make informed choices, to advocate for themselves if we are withholding valuable information, information they must have. Research on teenage males published by the Urban Institute in 2000 suggests that although sex education has become almost universal, students are not receiving even general information early enough to fully protect themselves against unintended pregnancy and STDs.\n\nIf you are a parent depending on schools and teachers to provide your child with good and comprehensive sex education, you're fooling yourself and committing a grave disservice to your sons and daughters. Abstinence-only programs were originally instituted by the Clinton administration in 1996 as part of welfare reform; the Bush administration boosted these programs, which expired in June 2009. In March 2010, at the eleventh hour of the Health Reform Bill under the Obama administration, funding of abstinence-only programs was resurrected to the tune of $50 million dollars a year for five years. What's frightening is the definition of abstinence-only education, which states that it must teach that \"a mutually faithful monogamous relationship in the context of marriage is the expected standard of human sexual activity\" and that \"sexual activity outside of the context of marriage is likely to have harmful effects.\" This should more accurately be called, abstinence-only propaganda. \"Expected standard\"\u2014by whom?\n\nCurrent and extensive research pointing out that abstinence-only programs don't work. In fact, they fail miserably. For the first time in more than a decade, teen pregnancy rates rose 1 percent between 2005 and 2006 according to the Guttmacher Institute. It's nonpartisan and nonprofit, by the way. The rates rose again from 2006 to 2007. A congressionally mandated study conducted over nine years at a cost of almost $8 million concluded that these programs are not effective in stopping or even delaying teen sex and have no beneficial impact on young people's sexual behavior.\n\n _Boys and girls in America have such a sad time together; sophistication demands that they submit to sex immediately without proper preliminary talk. Not courting talk\u2014real straight talk about souls, for life is holy and every moment is precious._\n\nJACK KEROUAC, _O N THE ROAD_\n\nThe earlier the foundation of self-esteem, self-worth, and a relationship of friendship that is created between you and your children, the easier the adolescent transition and all its inherent quandaries and milieu will be. They are human beings trying to find their way just as we did and still are. Regard them as thinking, intelligent beings, and they are likely to act as that. Regard them in ways that maintains their essential human dignity, whole and intact.\n\nLearning the art of negotiation and making and keeping agreements are essential to healthy relating. You'll have a very difficult time trying to get your teenagers to keep agreements if they see you breaking yours all over the place. Only make agreements you're able and willing to keep. If you find that you are unable to keep an agreement you've made, renegotiate as soon as possible. When negotiating, it's important to acknowledge the agreement, the other person's investment in the agreement, and any inconvenience renegotiating may have caused and then offer alternatives.\n\nIf you and your teen can talk about whatever either of you needs to talk about, transition through these years will be much easier on all of you.\n\nIt's never too late to begin an intimate relationship with your teen if you don't have one already. It does get more difficult with surly teenagers who have a dozen years of repressed rage accumulated. In some instances, it will take tremendous self-effacement on your part to repair any damage that's been done to the relationship. Rage, which is primarily directed at you (it always goes for the closest target), is a natural and healthy response in teens who find themselves in a family and a culture that governs by lies, deceit, power struggles, and lack of accountability while they are being admonished to be something different.\n\nI believe that parents and guardians need to be actively involved in what schools are teaching our children. Schools are businesses, and they don't always have the best interests of our children in mind. While many of us believe our children have a right to full disclosure when it comes to sex education, few of us have the emotional maturity to present the facts objectively. As long as parents, \"intellectuals,\" academia, religious leaders, and politicians are confused themselves about the facts and nature of sex, confusion will be passed on. The prevalence of sexual activity and interest in sex among children and young adults will always be part of life. Meanwhile, as they are kept in the middle of groups competing for power over morality, nothing gets done and people continue to be repressed by widespread ignorance and fear around sexuality.\n\n _By the time many people are fourteen or fifteen, they have been divested of their loves, their ancient and intuitive tastes, one by one, until when they reach maturity there is no fun left, no zest, no gusto, no flavor. Others have criticized and they have criticized themselves, into embarrassment. When the circus pulls in at five of a dark, cold summer morn, and the calliope sounds, they do not rise and run, they turn in their sleep, and life passes by._\n\nRAY BRADBURY,  \n _Z EN IN THE ART OF WRITING_\n\nA 2004 study by National Public Radio, the Kaiser Family Foundation (a private, nonpartisan, nonprofit organization focusing on major health-care issues), and Harvard's Kennedy School of Government found that only 7 percent of Americans say sex education should not be taught in schools. Among the remaining 93 percent, most parents seem to be generally content with whatever kind of sex education is taught by their local schools. However, there is disagreement about what kind of sex education should be taught. Fifteen percent of Americans believe abstinence from sexual intercourse should be the only thing taught with no information about how to obtain and use condoms or other forms of contraception. Forty-six percent believe that teaching \"abstinence-plus\" is the most appropriate approach. Abstinence-plus is an approach that considers that while some teens do not abstain from sexual intercourse, sex education should include information on condoms and contraceptives. Thirty-six percent believe that teaching abstinence is not the most important thing but that sex education should focus on teaching teens about responsible decision making regarding sex.\n\nUltimately, the decision school administrators make regarding what sort of sex education program will be taught in their school district is far too often motivated by the greenback. Federal funding is the vehicle that drives the sex education our children are getting in school.\n\nSex education in schools was historically initiated by health advocates to teach children how to avoid pregnancy and sexually transmitted diseases, whereas abstinence-only education was entered into and pushed for by evangelical or born-again Christians (NPR\/Kaiser\/Kennedy School National Survey on Sex Education 2004) who believe it is morally wrong to engage in sexual relations before marriage. Seventy-eight percent of born-again or evangelical Christians believe that having sexual activity outside marriage is likely to have a detrimental affect on physical and psychological health.\n\nSex education in Sweden's education system is compulsory and has been since 1956. Sweden believes in open communication as a primary foundation for healthy sexual and marital relations. And they don't pretend that people aren't having sex, teenagers included. (Is the United States even operating in the same century?) \"Swedes generally view sexual intercourse as a natural and expected occurrence during teen years.\"\n\nThe curriculum starts out at age 6 with information on sperm, eggs, and anatomy. From age 12 on the curriculum focus leans toward disease prevention, contraception, sex positions, and same-sex relations. The curriculum is intended to reach all students before age 15, the age of consent. There is a moral dimension involved as well; sex within loving relationships, gender and sexual equality are encouraged.\n\nJuxtaposed to the increase in teen pregnancy in the United States in the apron strings of abstinence-only programs, Sweden's teen pregnancy and sexually transmitted disease rates are among the lowest in the world. Sweden's teenage birthrate is 7 in 1,000 compared to 49 per 1,000 in the United States. Among fifteen- to nineteen-year-olds, the cases of gonorrhea in the United States are nearly six hundred times as great on a per capita basis. Pierre-Andre Michaud, chief of the Multidisciplinary Unit for Adolescent Health at the University of Lausanne Hospital in Switzerland and a leading researcher in European teen sexuality, dismisses the idea\u2014widely held in the United States\u2014that sex constitutes risky behavior for teens. In an editorial in May 2006, _Journal of Adolescent Health,_ he wrote: \"In many European countries\u2014Switzerland in particular\u2014sexual intercourse, at least from the age of 15 or 16 years, is considered acceptable and even part of normative adolescent behavior.\" Switzerland has one of the world's lowest rates of abortion and teen pregnancy. Teens in Switzerland, Sweden and the Netherlands, have easy access to contraceptives, confidential health care and comprehensive sex education. In Sweden, teens have access to free medical care, free condoms and prescriptions for inexpensive oral contraceptives, and general advice at youth clinics\u2014all without parental consent.\n\n\"Abstinence,\" says Michaud, \"is not something the Swiss press on teens. We think it's unfair. It's useless. It's inefficient. We have been advocating the use of condoms . . . and I think that we tend to be successful.\" And Joan-Carles Suris, head of the research group on adolescent medicine at the University of Lausanne, says it this way: \"The main difference is that in the United States sexual activity is considered a risk. Here we consider it a pleasure.\"\n\nThe message of responsible sex education is not restricted to the schoolroom. Throughout Sweden's cities there are youth clinics that provide continued education on and availability to low-cost contraception. Teens are given tools and information, even encouragement and permission to decide for themselves when they are ready for sex and then (horror of horrors) left to behave responsibly.\n\nMuslims who have immigrated to Sweden in the years since the Iraqi war are prohibiting their children from attending the sex education classes. Between 2003 and 2007, Sweden granted full refugee status to 24,799 Iraqis compared to 260 in Britain. With the increase in Muslim parents invoking a decades-old provision designed to give parents the option of taking their children out of Christian instruction, the Swedish government is set to abolish that provision. Sweden has a reputation for being sexually liberal, and in 2008, the state-run pharmaceutical stores launched a line of sex toys aimed at women. The initiative was funded by tax dollars, and within a few days the products became the chain's bestsellers.\n\nFindings in a 2010 study done by the National Foundation for Educational Research in the UK (NFER) found a variety of approaches to sex education around the globe. Astonishingly, the United States isn't listed and if it were, it would be the most conservative and abusive approach on the planet. Norway begins sex education at age six and a few other countries like Finland and Japan delay it a few years, with most countries averaging sex and relationship education between ten and twelve years old. As students get older the curriculum becomes more sophisticated. In Finland, fifteen-year-olds receive an introductory sexual package including a condom.\n\nInterestingly, in the majority of countries parents do not have the right to withdraw their children from sex and relationship classes, though it is permitted in British Columbia and Singapore. Switzerland believes that sex education relies on the parents as well as institutions to \"combat myths.\" In France, sex education is \"one of the core social and civil competencies to be acquired in the course of mandatory education,\" in Victoria, sexuality education is seen as a \"whole-school learning approach,\" while in Hungary schools have \"an unavoidable duty to address the questions of sexual culture and behaviour.\"\n\nWhile schools extol critical thinking skills as part of the curriculum, it doesn't extend to sex education. Sex education that does exist in the United States primarily teaches abstinence only and sex values, it does not explore abstinence as an option, sexual activity as an option, values, relationships, intimacy, communication skills, the fluidity of human sexual responses, or sexual diversity and identity. Sex education is a top-down formula not community building, empowering, or pedagogical.\n\nChange will come when we begin to genuinely care about giving genuine, heart-felt information for the benefit and welfare of our children. It will come when we care about empowering them through the dissemination of factual and objective information that encourages the transformation of information into knowledge and then wisdom. It will come about when we give up our positions that have roots in fear, ignorance, and power struggles.\n_**Footnotes**_\n\n*1 Berne viewed human beings as psychological phenomena, but ego states are really ecological phenomena because human beings have evolved within an ecological context. Every species has built into it a system of multiple ego states. Every species on Earth has an interior world that is a multiple personality where specific ego states emerge at specific times. Take, for example, cats. Everyone knows that a kitten has distinctively different energy than an old cat. The terms _kitten_ and _cat_ have inherently different meanings. Tree saplings have a distinctly different energy than a 100-year-old Juniper or a 4,000-year-old bristlecone pine. Further, there is a decidedly different _feeling_ to the various ages of different species; how we relate to a kitten is very different from how we relate to an old cat, how we interact with a three-year-old girl is very different from how we relate to a ninety-year-old woman. When we see the parents of different species tending to their young, we instinctively understand the part of them (and the part of us) that nurtures.\n\n*2 As told to me by Stephen Harrod Buhner, who studied with K\u00fcbler-Ross. Her story of how she came to recognize the Hitler within is in _Quest: The Life of Elisabeth K\u00fcbler-Ross_ by Derek Gil.\n\n*3 All male mammals, including humans, have rudimentary mammae, and the male nipple is nearly perfect in function. Prolactin, the hormone responsible for milk production, is present in both men and women. Evolutionarily, it makes sense that both sexes be capable of providing nourishment to the young to ensure survival of the species. Darwin speculated that among early mammalian ancestors both sexes may have nursed the young, but that over time the mammae in male mammals evolved to be inactive. Today, male lactation is found in at least one species: In an article in the 1996 issue of _Compleat Mother,_ Patty Stuart Macadam of the University of Toronto states that male lactation \"is somewhat common in Dayak fruit bats, a rare species found in Malaysia.\" In a _Scientific American_ article (September 6, 2007), _\"_ Strange but True: Males Can Lactate,\" author Nikhil Swaminathan writes about several instances of human male lactation. For men to be able to produce milk, it takes strong desire coupled with nipple stimulation, with successful lactation happening usually within one to two weeks.\n\n*4 When I work with women with disordered eating there is always a starving Infant inside, one starved for soul food, love, intimacy, being wanted and needed.\n\n*5 In 1989, as a result of the NEA being embroiled in the controversy between \"freedom of expression\" and the right of taxpayers to determine the use of public funds, Congress mandated the Decency Standard (public law 101-512), which has devalued freedom of expression in society.\n\n*6 Note that there are also contrary studies that refute these findings.\n\n*7 This exercise is a modified version of the love\/hate\/admire list found on page 65 of _Ensouling Language_ by Stephen Harrod Buhner.\n_**Endnotes**_\n\nINTRODUCTION. IT STOPS WITH ME\n\n1. Hillman, _A Blue Fire,_ 266.\n\n2. Klein, _America's War on Sex,_ 23 and 89.\n\n3. Hanisch, \"The Personal Is Political.\"\n\n4. _Sex in the 90s,_ MTV news special, 1996.\n\n5. LaChapelle, _Sacred Land, Sacred Sex,_ 254.\n\n6. Escoffier, _Sexual Revolution,_ 176.\n\n7. Margulis and Sagan, _Dazzle Gradually,_ 121.\n\n8. Henry David Thoreau, _The Journal of Henry David Thoreau,_ vol. 6, 740. The full sentence from Thoreau's journal reads, \"This earth which is like a map spread out around me is but my inmost soul exposed.\"\n\n9. White, _Kiss of the Yogini,_ xiii.\n\n10. Kakar, _Shamans, Mystics and Doctors,_ 151.\n\n11. Feuerstein, _Sacred Sexuality,_ 210.\n\nCHAPTER 1. INTIMACY: FOOD THAT FEEDS THE SOUL OF LOVE\n\n1. Lerner, _The Dance of Intimacy,_ 3 _._\n\n2. Berne, _Games People Play,_ 182.\n\n3. Hofmann, _Sex Incorporated,_ 7.\n\nCHAPTER 2. AUTONOMOUS PERSONHOOD\n\n1. Perlman, _The Power of Trees,_ 2.\n\n2. Buhner, _The Natural Testosterone Plan,_ 22.\n\n3. Matthews and Matthews, _The Encyclopaedia of Celtic Wisdom,_ 299.\n\n4. Ibid.\n\n5. Frankl, _Man's Search for Meaning,_ 121.\n\n6. Ibid., 122.\n\n7. Krishnamurti, _The Awakening of Intelligence,_ 267.\n\n8. Rand, _For the New Intellectual,_ 124.\n\n9. Bly, _A Little Book on the Human Shadow,_ 18.\n\n10. Est\u00e9s, _Women Who Run with the Wolves,_ 51.\n\n11. Berne, _Sex in Human Loving,_ 139.\n\n12. Est\u00e9s, _Women Who Run with the Wolves,_ 342.\n\nCHAPTER 3. GETTING TO KNOW YOU\n\n1. Berne, _Intuition and Ego States,_ 123.\n\n2. Berne, _Games People Play,_ 27.\n\n3. Hollis, _The Middle Passage,_ 103.\n\n4. Bly, _A Little Book on the Human Shadow,_ 24.\n\n5. Frankl, _Man's Search for Meaning,_ 147.\n\n6. Hollis, _The Middle Passage,_ 104.\n\nCHAPTER 4. THE NUMINOUS\n\n1. Eliade, _The Sacred and the Profane,_ 14 _._\n\n2. Ibid., 24.\n\n3. Hamilton, _Mythology,_ 57.\n\n4. From www.gaianstudies.org.\n\n5. Otto, _The Idea of the Holy,_ xvi.\n\nCHAPTER 5. HUMAN BEINGS: THE GROUND WHERE THE GODS RESIDE\n\n1. Metzner, _Green Psychology,_ 98.\n\n2. Suzuki and Knudtson, _Wisdom of the Elders,_ 49.\n\n3. Weller, \"Reclaiming Our Indigenous Soul.\"\n\n4. Goldsmith, _The Way,_ 124.\n\n5. Suzuki, _The Sacred Balance,_ 276.\n\n6. Taylor, _The Prehistory of Sex,_ 9.\n\n7. Ibid., 142.\n\n8. Ibid., 11.\n\n9. Ibid., 187.\n\n10. Sheldrake, _The Rebirth of Nature,_ 59.\n\n11. Ibid., 60.\n\n12. Ryan and Jeth\u00e1, _Sex at Dawn,_ 13.\n\n13. Phillips, \"Am I a Spaceman?\"\n\n14. Wilhem Reich, lecture on somatic psychology at www.sonoma.edu\/us\u00aders\/d\/daniels\/reich.\n\n15. Moats, \"MHS Homecoming Dance Canceled.\"\n\n16. Klein, \"CraigsList, Sex Trafficking, & the Next Moral Panic.\"\n\n17. Sobel, \"Beyond Ecophobia.\"\n\n18. Davis, _The Father of Waters,_ 87.\n\n19. Ibid.\n\n20. Ibid.\n\n21. Mander, _In the Absence of the Sacred,_ 187.\n\n22. MacFarlane, _The Wild Places,_ 30.\n\n23. Ibid.\n\n24. Ibid.\n\n25. Abrams, _Becoming Animal,_ 134.\n\n26. Griffiths, _Wild,_ 2.\n\n27. Sheldrake, _The Rebirth of Nature,_ 10.\n\n28. Bell, \"Haitians Challenge Monsanto's Influence.\"\n\n29. Stock, \"Manifest Haiti: Monsanto's Destiny.\"\n\n30. Perlman, _The Power of Trees,_ 12.\n\nCHAPTER 6. FINDING THE WILD\n\n1. Hillman, _The Soul's Code,_ 13.\n\n2. Wilson, _Biophilia,_ 65\u201366.\n\n3. Earnheart, Richard, \"About the Artist,\" at www.richardearnhea\u00adrt.com\/aboutheartist\/htm.\n\n4. For more information on the M\u00fcller-Lyer arrows see \"94 Visual Phenomena & Optical Illusions by Michael Bach\" at www.michaelbach.de\/ot\/. Click on the link for the M\u00fcller-Lyer Illusion.\n\n5. Philip L Kilbride and H. W. Leibowitz, \"The Ponzo Illusion among the Baganda of Uganda.\"\n\n6. Nicolson, _Mountain Gloom and Mountain Glory,_ 1.\n\n7. Krishnamurti, _The Awakening of Intelligence,_ 268.\n\n8. Keleman, _The Human Ground,_ 31.\n\n9. Griffiths, _Wild,_ 48.\n\n10. Hillman, _A Blue Fire,_ 9.\n\n11. Leopold, _A Sand County Almanac,_ 138.\n\n12. Seed, \"Think Like a Mountain.\"\n\n13. Ferry, \"Keepers of the World.\"\n\n14. For more on Plato's worldview see the Indiana University webpage on Human Intelligence atwww.indian\u00ada.edu\/~intell\/plato.shtml.\n\n15. Feuerstein, _Sacred Sexuality,_ 207.\n\n16. Keeney, _The Bushman Way of Tracking God,_ 215.\n\n17. Ibid., 215.\n\n18. Ibid., 249.\n\n19. Keleman, _The Human Ground,_ 35.\n\n20. Keleman, _Your Body Speaks its Mind,_ 58.\n\n21. Mander, _In the Absence of the Sacred,_ 85.\n\n22. Ibid., 86.\n\n23. Montagne, \"Israel Kamakawiwo'ole: The Voice of Hawaii.\"\n\n24. Abram, _Spell of the Sensuous,_ 172.\n\n25. Margulis and Sagan, _Dazzle Gradually,_ 181.\n\nCHAPTER 7. CHOOSING ANOTHER WAY\n\n1. Malone and Malone, _The Art of Intimacy,_ 260.\n\n2. Johnson, _The Spiritual Practices of Rumi,_ 140.\n\n3. Hillman, _City and Soul,_ 37.\n\n4. Buhner, _The Secret Teachings of Plants,_ 86.\n\n5. Perlman, _The Power of Trees,_ 24.\n\n6. Blake, \"The Marriage of Heaven and Earth,\" 38.\n\n7. Metzner, _Green Psychology,_ 108.\n\n8. Dehiia, _Chola: Sacred Bronzes of Southern India._\n\n9. Dalrymple, \"India: The Place of Sex,\" 33.\n\n10. Riley, \"Victorian Sex Rebels and Atheists.\"\n\nCHAPTER 8. THE LANGUAGE OF LOVE\n\n1. Hillman, _The Soul's Code,_ 146.\n\n2. From www.anthropin\u00ade.eu\/Anthropine\/Med\u00adia\/praesentation\/an\u00adthropine_english.\n\n3. Wilhelm Reich, lecture on somatic psychology at www.sonoma.edu\/u\u00adsers\/d\/daniels\/r\u00adeich.\n\nCHAPTER 9. SACRED SEX\n\n1. Schachter-Shalomi and Eve Ilsen, \"Sacred Sex.\"\n\n2. Johnson, _Rumi's Four Essential Practices,_ 117.\n\n3. Roach, _Bonk_.\n\n4. Keeney, _The Bushman Way of Tracking God,_ 80.\n\n5. Otto, _The Idea of the Holy,_ 42.\n\n6. Halifax, _The Fruitful Darkness,_ 18.\n\nCHAPTER 10. THE DANCE OF TRUST\n\n1. Hillman, _A Blue Fire,_ 277.\n\n2. Hofmann, _Sex Incorporated,_ 13.\n\n3. Hillman, _The Myth of Analysis_ , 92.\n\n4. Jack Morin, _The Erotic Mind,_ 293.\n\n5. Klein, _Your Sexual Secrets,_ 48.\n\nCHAPTER 11. HEALING SHAME\n\n1. Brown, _I Thought It Was Just Me (but it isn't),_ 30.\n\n2. Nietzsche, from www.brainyquot\u00ade.com\/quotes\/author\u00ads\/f\/friedrich_n\u00adietzsche_10.html.\n\n3. Keleman, _The Human Ground_ , 53.\n\nCHAPTER 12. FREEING OURSELVES FROM SEXUAL TYRANNY\n\n1. Berne, _Sex in Human Loving_ , 1.\n\n2. Phoenix, _A Woman Whose Calling Is Men,_ book 2, 27\u201328.\n\n3. \"Coins of Ancient Rome,\" www.dengedeng\u00ade.com\/2010\/02\/coins-of-a\u00adncient-rome.\n\n4. Adams, \"Pay for Play.\"\n\n5. \"The Sex Cult of Venus,\" http:\/\/heritag\u00ade-key.com\/rome\/sex-c\u00adult-venus.\n\n6. Diamant, _The Red Tent,_ 158.\n\n7. Mantegazza, _The Sexual Relations of Mankind,_ 272\u201373.\n\n8. Ibid., 270.\n\n9. Seltzer, \"Bishops Are Behind the 'Let Women Die' Act.\"\n\n10. Baker, \"The Deep Ecology of the Family.\"\n\n11. Buhner, _Ensouling Language,_ 271.\n\n12. Miller, _Henry Miller on Writing,_ 175.\n\n13. \"Obscene, Obscenity,\" The 'Lectric Law Library, www.lectla\u00adw.com\/def2\/o\u00ad002.htm.\n\n14. Heins, _Sex, Sin, and Blasphemy,_ 17.\n\n15. From http:\/\/ourporno\u00adurselves.org\/about.\n\n16. Ibid.\n\n17. From www.pornharms.com.\n\n18. From http:\/\/news.ufl.edu\/2\u00ad004\/11\/29\/sexual\/r\u00adevolution.\n\n19. Cheesman, \"6 Ways That Porn Runs the World.\"\n\n20. Robinson and Wilson, \"Porn-Induced Sexual Dysfunction Is a Growing Problem.\"\n\n21. Shroeder, _Challenge to Sex Censors,_ 31. Quoting an anonymous clergyman a century ago.\n\n22. From http:\/\/abacus.ba\u00adtes.edu\/admin\/off\u00adices\/scs\/salt7.html.\n\n23. From www.huffingto\u00adnpost.com\/rep-jane-ha\u00adrman\/finally-some-progres\u00ads-in_b_125504.html.\n\n24. From www.va.gov.\n\n25. From www.orlandosen\u00adtinel.com\/news\/local\/or\u00adl-blogs,0,2664973.htmlpage.\n\n26. Pendell, _Inspired Madness,_ 69.\n\n27. Cohn, \"Did SEC Staffer Surf Porn?\"\n\nCHAPTER 13. HEALING THE HUMAN SOUL\n\n1. Gandhi, _An Autobiography_.\n\n2. Ibid, xvii.\n\n3. From www.mosaicvoices.org.\n\n4. Tompkins and Bird, _The Secret Life of Plants,_ 107.\n\n5. Taylor, _The Prehistory of Sex,_ 60\u201361.\n\n6. Bellows, \"The Birth Control of Yesteryear.\"\n\n7. Engel, _Wild Health,_ 178.\n\n8. Tompkins and Bird, _The Secret Life of Plants,_ 121.\n\n9. Ibid., 134.\n\n10. Ibid., 137.\n\n11. Ibid., 142.\n\n12. Dobb, \"New Life in a Death Trap.\"\n\nEPILOGUE\n\n1. Dale Pendell, interview with Emily Green, _Los Angeles Times,_ 19 October 2003. From http:\/\/quantumtant\u00adra.com\/pendell\/html.\n\n2. Bowden, _Blood Orchid,_ 138, 140.\n\n3. Fuller, _Critical Path,_ xi\n\n4. Fuller, from www.muzz.com\/Buck\u00adminster_Fuller.aspx.\n\n5. Krishnamurti, _Think on These Things,_ 14.\n\n6. Jong, _Sappho's Leap,_ 88.\n\n7. Reeves, _The Character of Leadership,_ 119.\n\n8. Berne, _Games People Play,_ 184.\n\nAPPENDIX. TALKING TO CHILDREN AND TEENS ABOUT SEX\n\n1. Paterson, _Tree Wisdom,_ 279.\n\n2. Sheldrake, _The Presence of the Past,_ 71.\n\n3. From http:\/\/blogs.alte\u00adrnet.org\/speakeasy\/2010\/10\/20\/d\u00adr-jocelyn-elders-ma\u00adrijuana-masturbation-a\u00adnd-medicine.\n\n4. Hillman, _A Blue Fire,_ 98.\n\n5. Calderone and Ramey, _Talking with Your Child about Sex,_ 5.\n\n6. Klein, \"'Catching' Your Kid Playing Doctor.\"\n\n7. Carroll, \"Growing Up Too Soon? Puberty Strikes 7-year-old Girls.\"\n\n8. From www.niehs.hi\u00adh.gov\/research\/supporte\u00add\/sep\/2010\/prenatal.\n\n9. Hofmann, _Sex Incorporated,_ 6.\n\n10. Ibid., 11.\n\n11. See www.advocatesfo\u00adryouth.org\/publicati\u00adons\/429?task=view.\n\n12. For information on sexual and reproductive health, policy analysis, and public education, see www.guttmacher.org\n\n13. From www.kff.org\/news\u00admedia\/upload\/Sex-Educatio\u00adn-in-America-Summary.pdf. See also www.npr.org\/template\u00ads\/story\/story.php?storyI\u00add=1622610.\n\n14. Ibid.\n\n15. Reiss, _The End to Shame,_ 62.\n\n16. Grose, \"Straight Facts about the Birds and Bees.\"\n\n17. Agnvall, \"Is Teen Sex Bad?\"\n\n18. Ibid.\n\n19. Ibid.\n\n20. Grose, \"Straight Facts about the Birds and Bees.\"\n\n21. Vaughan, \"Swedish Muslims and Strawberry Condoms.\"\n\n22. Stockholm Newsroom, \"Swedes Convince Their State Shops to Sell Sex Toys.\"\n\n23. Oscarsson, \"Coming Home from School with Strawberry Condoms.\"\n\n24. From www.nfer.ac.uk.\n**_Bibliography and Recommended Reading_**\n\nAbram, David. _Becoming Animal: An Earthly Cosmology_. New York: Pantheon Books, 2010.\n\n______. _The Spell of the Sensuous: Perception and Language in a More-Than-Human World._ New York: Random House, 1996.\n\nAdams, Cecil. \"Pay for Play: Did the Romans Issue Sexually Depictive Tokens for Use in Foreign Brothels?\" _The Straight Dope,_ 18 January 2008. www.straightdop\u00ade.com\/columns\/read\/23\u00ad55\/pay-for-play.\n\nAgnvall, Elizabeth. \"Is Teen Sex Bad? Americans and Western Europeans Don't Agree on What's Normal and Acceptable But Many Health Experts Do.\" Special to _The Washington Post,_ Tuesday, 16 May 2006.\n\nAgustin, Laura Maria. _Sex at the Margins: Migration, Labour, Markets and the Rescue Industry_. London: Zed Books, 2007.\n\nAnand, Margo. _The Art of Sexual Ecstasy: The Path of Sacred Sexuality for Western Lovers_. Los Angeles: Jeremy P. 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New York: Avenel, 1981.\n\nTompkins, Peter, and Christopher Bird. _The Secret Life of Plants._ New York: Harper & Row, 1973.\n\nVaughan, Hal. \"Swedish Muslims and Strawberry Condoms.\" The Hal Blog, 29 June 2009. http:\/\/halmasonber\u00adg.wordpress.com\/tag\/iraqis.\n\nWeller, Francis. \"Reclaiming Our Indigenous Soul.\" _Sacred Fire Magazine: The Heart of the Living World_ 13 (2011): 32.\n\nWhite, David Gordon. _Kiss of the Yogini: \"Tantric Sex\" in its South Asian Contexts._ Chicago: University of Chicago Press, 2006.\n\nWilson, Edward O. _Biophilia: The Human Bond with Other Species._ Cambridge, Mass.: Harvard University Press, 1984.\n\n **WEBSITES**\n\nCenter for Sexual Pleasure and Health. <http:\/\/thecsph.org>\n\nDamn Interesting. www.damninteresting.com\n\nExaminer.com: National Sex & Relationships. www.examine\u00adr.com\/sex-and-relati\u00adonships-in-national?=1\n\nThe Foundation for Gaian Studies: www.gaianstudies.org\n\nGay History & Literature: Essays by Rictor Norton. <http:\/\/rictornorton.co.uk>\n\nGuttmacher Institute. www.guttmacher.org\n\nHeritage Key, a website about ancient history. <http:\/\/heritage-key.com>\n\nIndiana University, series on human intelligence. www.indiana.edu\/~i\u00adntell\/plato.shtml\n\n _Journal of Ultrasound in Medicine_. www.jultrasoundmed.org\n\nKaiser Family Foundation. www.kff.org\n\nThe 'Lectric Law Library. www.lectlaw.com\/d\u00adef2\/o002.htm\n\nLive Strong. www.livestrong.com\n\nMen Stuff: The National Men's Resource. www.menstuff.org\n\nMiss Maggie Mayhem. <http:\/\/missmaggiemayhem.com>\n\nNational Foundation for Educational Research. www.nfer.ac.uk\n\nNational Institute of Environmental Health Sciences. _www.niehs.nih.gov_\n\n94 Visual Phenomena & Optical Illusions by Michael Bach. www.michaelbach.de\/ot\/\n\nOur Porn, Ourselves. www.ourpornourselves.org\n\nRichard Earnheart, artist's website. www.richardea\u00adrnheart.com\/abouth\u00adeartist\/htm\n\n _Scientific American._ www.scientifica\u00admerican.com\/articl\u00ade.cfm?id=strange-but-tr\u00adue-males-can-lactate&pr\n\n _Sexual Intelligence._ www.sexualint\u00adelligence.org\n\nSexuality & Modernity. www.isis.aus\u00adt.com\/stephan\/writin\u00adgs\/sexuality\/vict.htm\n\n _Susie Bright's Journal._ www.susiebright.com\n\nThe Straight Dope. www.straightdope.com\n\nTruthout. www.truth-out.org\n\nThe University at Texas, Digital Writing & Research Lab. www.cwrl.utexas.edu\n\nWilhem Reich's somatic psychology lecture on Victor Daniels's website in the psychology department at Sonoma State University. www.sonoma.edu\/u\u00adsers\/d\/daniels\/reich\n\n _Yes!_ magazine. www.yesmagazine.org\n_**Index**_\n\nAborgines, 144\u201345, 200  \nAbram, David, 111, 144\u201345  \n _Absence of the Sacred, The_ , 107  \nabstinence, 6, 304\u20135, 307  \nabuse, 23  \naccidents, 41  \nactivism, need for, 4  \nadapted Child, 56\u201357  \nadapted Predator, 57\u201359  \naddiction, 257\u201359, 263  \nAdjani, Isabelle, 217  \nadolescents, 302\u20139  \nAdult, 55\u201357, 61\u201362  \nAesthetes, 167\u201368  \naggression, 128  \nalienation, 100\u2013101  \naloneness, romance of, 22  \nanam, 39  \nancestors, 273\u201374  \nanger, 57  \nanimals, 279\u201380, 281, 282\u201383  \nanimism, 38, 84\u201385, 93\u201394, 158\u201359  \nAnthony, Susan B., 255\u201356  \nanthropocentrism, 145\u201346  \nAphrodite, 114  \nApple, Fiona, 217  \naromatherapy, 11  \nassholes, 59  \nAugustine, Saint, 247  \nauthentic self, 27\u201329  \nauthorship, 269, 277\u201378  \nautonomous personhood, 34\u201353, 274  \nfilling long bag and, 45\u201347  \nknowing self and, 42\u201345  \nmeaning of soul and, 36\u201342  \nold scripts and, 47\u201349  \nsoul retrieval and, 49\u201353  \nawareness, 51, 52\u201353  \nawe, 191, 249\u201350\n\nBageant, Joe, 284  \nBaker, Jeannine Parvati, 249  \nBaldur, 281  \nBaldwin, James, 288\u201389  \nbarefeet, 132\u201333, 160  \nBarry, William Francis, 247\u201348  \nBarrymore, Ethel, 271  \nBasho, 85  \nBateson, Gregory, 11  \nbawdiness, 288\u201389  \nbeauty, 179, 236, 289  \n _Becoming Animal_ , 111  \nbees, 279\u201380  \nbeginnings, 64\u201365  \nBerne, Eric, 30, 47, 54, 73, 213, 290  \nBerry, Wendell, 91  \nbetrayal, 210\u201314  \nbiocoenosis, 109  \nBird, Christopher, 279, 281  \nbirds, 282\u201383  \nbirth control, 280\u201381  \nBlake, William, 159  \nbloomers, 256  \n _Blue Fire_ , A, 129, 219  \nBly, Robert, 45\u201346, 55, 150, 198\u201399, 213\u201314  \nbody  \nlies and, 137\u201338  \nmeditation for, 236  \nsensing of, 138\u201346, 153\u201357  \nbody armoring, 128, 181, 190  \nbodywork, 235  \nbonobos, 8  \nBowden, Charles, 16, 264, 285\u201386  \nBradbury, Ray, 286, 306  \nBrand, Russell, 213  \nbreastfeeding, 66\u201367, 280, 295  \nBrown, Bren\u00e9, 231  \nBrown, Hannah R., 247\u201348  \nBrowning, Elizabeth Barrett, 228  \nBuddhism, 86  \nBuhner, Stephen Harrod, 98\u201399, 153, 169\u201370, 251\u201352  \nBursche, Aleksander, 244  \nBurton, Richard, 167  \nBushmen, 134, 200\n\nCampbell, Joseph, 34  \ncamping, 122\u201323  \nCannibal Club, 167\u201368  \nCapra, Fritjof, 160  \nCarson, Rachel, 113  \nCarver, George Washington, 281\u201382  \ncathected, 56  \ncelibacy, 134  \nCelts, 39  \nceremonies, 92\u201393  \ncervix, 280  \nChallenge to Sex Censors, 258  \nchange, 34\u201335  \nChild, 55\u201357, 63\u201365, 73\u201376  \nrelationships and, 149\u201350  \nshame and, 233\u201334  \ntrust and, 212\u201313, 224  \nchildren, 5\u20136, 102\u20135, 140  \nfilling long bag and, 45\u201347  \ntalking about sex with, 292\u2013309  \nChristianity, 134\u201335, 145  \ncoich anama, 39  \ncommitment, 270\u201371  \nconsciousness, 43\u201344, 52\u201353, 72  \nCouncil, 68\u201370  \ncourage, 269  \ncra'bhadh, 39  \nCraigslist, 103  \ncreideamh, 39  \ncriticism, 47\u201349  \ncross transactions, 76\u201378  \ncuriosity, 46\n\ndaily life, 80  \ndaimons, 116\u201317  \n _Dance of Intimacy, The_ , 21  \ndarkness, 45  \nDavis, Bruce, 202  \n _Dazzle Gradually_ , 10\u201311, 145\u201346  \ndeath, 112\u201314  \nDeep Ecology, 11\u201312, 146  \ndefensiveness, 27  \nDepth Readings, 237\u201340  \nDepth Seeing, 192\u201394  \ndevotion, 270  \nDickens, Charles, 39  \nDickinson, Emily, 38  \nDionysus, 83\u201384, 284\u201385  \ndisease, 112\u201314, 128  \ndistance, 190\u201392  \ndivorce, 258, 287  \ndopamine, 257  \nDouglass, Frederick, 255\u201356, 291  \n\"Do You Recognize Me Now?\" 187\u201389  \ndrugs, 120\u201321  \nducks, 280  \n _Dune_ , 285\n\nEarnheart, Richard, 118  \nEarth, 11\u201312, 108\u20139, 145\u201346, 150\u201352, 160. _See also_ Nature  \nEckhart, Meister, 152  \necosexuality, 279\u201383  \nego states, 54\u201355, 62, 204\u20137  \nEgypt, 245  \nEinstein, Albert, 161  \nElders, Jocelyn, 293\u201394  \nelectronics, 123\u201324, 140\u201342  \nEliade, Mircea, 80\u201381  \nEmerson, Ralph Waldo, 123  \nemptiness, 119  \nenergy fields, 153\u201357  \n _Ensouling Language_ , 251\u201352  \nerectile dysfunction, 257\u201358  \nEros, 31\u201332, 39, 129  \nEscoffier, Jeffrey, 9  \nEst\u00e9s, Clarissa Pinkola, 46, 48  \nexercises  \nDepth Reading on a Stranger, 237\u201338  \nDepth Seeing, 192\u201394  \nExpanding the Heart Field, 154\u201357  \nExpanding the Heart Field with Your Lover, 164\u201365  \nExploring Who You Are, 278\u201379  \nFollowing a Feeling to Its Source, 214\u201315  \nHeart Field Practice, 162\u201364  \nIdentifying Shame, 232\u201333  \nSacred Sex, 194\u201397  \n_See also_ meditations  \nExpanding the Heart Field, 154\u201357  \nExploring Who You Are, 278\u201379  \nexpression, 32  \nextremes, 260\u201361  \neye contact, 216, 223\u201324\n\nFacebook, 103, 141, 255  \nfantasies, 223  \nfarming, 97, 101  \nfears, 5, 9  \nFechner, Gustav Theodor, 281  \nfeelings, body and, 137\u201346  \nfeminism, 247\u201348, 254\u201355  \nFeminists Against Pornography, 254\u201355  \nFerrini, Paul, 33, 34, 187\u201389, 202\u20133  \nFeuerstein, Georg, 12, 134, 201\u20132  \nfights, 148\u201349, 212  \nfish, 106\u20137, 282\u201383  \nFishburn, Geoffrey, 244  \nFollowing a Feeling to Its Source, 214\u201315  \nFourier, Charles, 248  \nFrankl, Viktor, 40  \nFranklin, Miles, 217  \nfreedom, 30, 211, 267\u201368  \nfree love, 247  \nFromm, Erich, 49, 117  \nFruitful Darkness, The, 201  \nfuck and fucking, 74, 98  \nFuller, Buckminster, 68, 286\n\nGaia, 8\u20139, 108\u20139, 145\u201346, 150\u201352, 160. _See also_ Nature  \nGaiacentrism, 145  \n _Games People Play_ , 76  \nGandhi, Mohandas K., 50, 61, 265  \nGenesis, Book of, 135  \ngenitals  \nof plants, 279  \nwords for, 182, 294  \ngenius, 265\u201366  \nGeorge, Dan, 94  \ngifts, 216  \nGila Wilderness, 94  \nGod, 135, 137, 167, 200  \nGoethe, Johann Wolfgang von, 41, 50\u201351, 224, 289\u201390  \nGoldsmith, Edward, 95  \ngrace, 269  \nGraham, Barbara, 144  \nGreene, Graham, 217  \n _Green Porno_ , 260  \n _Green Psychology_ , 93  \nGreenwald, Jerry, 46  \nGregory, Dick, 225  \ngrief, 59, 272\u201373  \nGriffiths, Jay, 111, 129  \nguilt, 7\n\nHaitian farmers, 113  \nHalifax, Joan, 201  \nHamilton, Edith, 83\u201384  \nHanisch, Carol, 4  \nhappiness, 40, 289\u201390  \nhealing, 10\u201312, 150\u201352  \nheart, 27, 153\u201357  \nHeart Field Practice, 162\u201366  \nHerbert, Frank, 49, 105, 285  \n _High Country News_ , 282  \nHillel, Daniel, 135, 290  \nHillman, James, 116, 129, 153, 172\u201373, 212, 219, 294  \nHippocrates, 280  \nHofmann, Hans, 32, 219, 303\u20134  \nHoliday, Billie, 7  \nHollis, James, 55  \nHopi, 285  \n _Huffington Post_ , 258  \nhuman beings, 90\u2013101  \nbecoming whole, 108\u20139  \nchildren, 102\u20135  \ncounsel with Star People, 93\u201395  \ndisconnection from Nature, 104  \nseparation and, 96\u2013101  \n_See also_ soul  \nhumility, 125, 229  \nhunter-gatherer cultures, 96\u201397, 98, 161  \nHuxley, Aldous, 65  \nhybrid seeds, 113\n\n\"I,\" 42\u201345  \nIdentifying Shame, 232\u201333  \nIllinois River, 106  \nimagination, 34\u201335  \nInfant, 66\u201367  \ninjunctions, 218\u201319  \nInnana, 246  \ninner child, 27\u201328, 151  \ninspiration, 27  \nintegrity, 149\u201350  \nintelligence, 265\u201366  \ninternal work, 27, 61\u201364  \ninternet, 256  \ninterpersonal interactions, 54  \ninterworld, 157\u201362  \nintimacy, 5, 16\u201332  \nauthentic self and, 27\u201329, 274  \nChild-to-Child and, 73\u201376  \ndefinitions of, 21, 35  \nhunger for, 12  \nlearning to hide, 23\u201327  \nlove and, 20\u201322  \nluminous world of, 19\u201320  \nsecrets and, 219\u201321  \nwith self, 35, 52\u201353  \nyearning for connection, 16\u201317  \n_See also_ relationships  \nintimus, 27\u201328, 35  \nintuition, 46\u201347, 130, 132  \nInuit, 93\u201395  \ninvisibility, 22, 23\u201327  \nirritation, 138\n\nJager, Gustav, 176  \nJames, William, 115  \nJapan, 245  \njealousy, 59  \nJefferson, Thomas, 106  \nJeffries, Richard, 115  \nJeth\u00e1, Cacilda, 101  \nJohnson, Will, 192  \nJong, Erica, 33, 287, 290  \nJordon, June, 5, 251, 253  \njoy, 40, 48  \nJung, Carl, 45, 64, 105, 138\n\nKakar, Sudhir, 11  \nKamakawiwo'ole, Israel, 144  \nKama Sutra, 167  \nKeeney, Bradford, 115, 134, 137, 200, 201  \nKeleman, Stanley, 128\u201329, 137, 140, 235  \nKerouac, Jack, 305  \nKilbride, Philip L., 123  \nKinsey, Alfred, 256  \n _Kiss of the Yogini,_ 11  \nKlein, Marty, 103, 183, 222, 295  \nKnudtson, David, 93\u201394  \nKorenman, Stanley, 297\u201398  \nKrishnamurti, J., 43, 128, 147, 287  \nK\u00fcbler-Ross, Elisabeth, 50, 57\n\nLai, Hsi, 118  \nlanguage, 144\u201345  \nlaughter, 48, 271  \nLawrence, D. H., 9, 90\u201391, 96, 133, 140, 252  \nLeivowitz, H. W., 124  \nLeopold, Aldo, 11\u201312, 90, 106, 130\u201331  \nLerner, Harriet Goldhor, 21  \nletting go, 180\u201383  \nLevine, Stephen, 275, 283  \nliberation, sexual, 247\u201350, 288\u201390  \nlies and lying, 137\u201338, 270, 293  \nlife  \ncoming alive, 128\u201333  \ncreating, 34\u201335  \nmeaning and, 39\u201340  \n_See also_ human beings  \n _Little Book on the Human Shadow, A_ , 45\u201346, 55  \nlong bag, 45\u201347, 150  \nLopez, Barry Holstun, 162  \nLorde, Audre, 250  \nlove, 169\u201386  \nintimacy, 16\u201317, 20\u201322  \nletting go, 180\u201383  \nsensuality, 183\u201386  \nsmell of, 176\u201380  \nwithout reservations, 172\u201376  \nLoving the Body, 236  \nLutz, Deborah, 167\u201368\n\nMacFarlane, Robert, 110  \nmaking love, 74  \nMalone, Thomas Patrick, 150  \nMamas, 132\u201333  \nmana, 143\u201344  \nMander, Jerry, 107, 141  \nMargulis, Lynn, 10\u201311, 145\u201346  \nmarriage, 248, 287  \nMaslow, Abraham, 72, 116  \nmassage, 11, 235  \nmasturbation, 5, 293\u201395  \nMatteson, Mollie, 108  \nmaturity, 275  \nMcBride, Margaret, 249  \nMcIntyre, Julie  \nbirthing authentic self, 27\u201329  \nchildhood of, 23\u201327  \nearring story of, 119\u201320  \nfather and, 17\u201319, 23  \ngrowth of, 266\u201368  \nissue with pornography, 242  \nlearning to hide, 23\u201327  \nletting go and, 180\u201382  \nneed for intimacy, 16\u201317, 20\u201322  \nnude modeling experience of, 234  \npoetry of, 178\u201379, 180  \nreligion and, 134\u201338  \nshame and, 226\u201330, 231  \nspirituality of, 82\u201388  \nStar People and, 95  \nMeade, Michael, 267  \nmeaning, 39\u201340  \nmeditations  \nI've Been Waiting for You, 63\u201364  \nLoving the Body, 236  \nMeeting the Council Members, 68\u201370  \nStarting at the Beginning, 66\u201367  \n_See also_ exercises  \nMeir, Golda, 210  \nMendez, Chico, 104  \nmenstruation, 23\u201324, 281  \nMerton, Thomas, 171  \nMetzner, Ralph, 93, 159  \nMichaud, Pierre-Andre, 308  \nmidwives, 280  \nMiller, Henry, 138, 250  \nMilton, John, 183  \nmind-body divide, 96\u2013101, 102, 133\u201338  \nMississippi River, 105\u20137  \nMobius, Karl, 109  \nMonsanto, 113  \nMontagu, Ashley, 63  \nmorality, 9\u201310, 102  \nMott, Lucretia, 255\u201356  \n\"Movement of Great Things,\" 169\u201370  \nMuir, John, 11\u201312, 126  \nM\u00fcller-Lyer arrows, 123\n\nnaivete, 192  \nnarratives, 13  \nNash, Roderick, 110\u201311  \nnatural Child, 56\u201357  \nnatural Predator, 57\u201358  \nNature  \nchildren and, 300\u2013302  \nloss of the sacred and, 107  \nas mirror of sexuality, 104  \nreligion and, 107  \nseparation from, 96\u2013101  \nsexuality of, 7\u20139  \nwalks in, 126\u201328  \nneeds, 28  \nNew Age movement, 11  \nNewgrange, 99  \nNichols, Mary Gove, 247\u201348  \nNicolson, Marjorie Hope, 127  \nno, importance of saying, 298\u201399  \nNordenski\u00f6ld, August, 248  \nnuminous, 79\u201388  \ncalling of, 116\u201321  \nEliade on, 80\u201381  \nexperiencing interworld, 157\u201362  \ngoing wild, 121\u201328  \nOtto on, 88  \nsacred sex and, 201\u20132  \nspirituality and, 82\u201388\n\nobesity, 298  \nobscenity, 252\u201353  \nOccupy Wall Street, 256  \nO'Donnell, Christine, 6  \nO'Keefe, Georgia, 52, 183\u201384  \nOlmsted, Thomas J., 249  \n\"One Source of Bad Information,\" 213\u201314  \nopathe, 117  \nopen marriages, 287\u201388  \noptions, 51  \nOrwell, George, 291  \nOtto, Rudolf, 88, 201  \nOur Porn, Ourselves, 254\u201355\n\nParacelsus, 163\u201364, 239\u201340  \nparanoia, 27  \nParent, 55\u201357, 70\u201371  \npassion, 273  \npaternity, 98, 101  \nPaterson, Jacqueline Memory, 292  \nPaul, Alice, 256  \npeak experiences, 116  \npedophilia, 258  \nPendell, Dale, 252, 260\u201361, 284\u201385  \nPenthouse, 242  \nperception, 130, 132  \nPerlman, Michael, 37\u201338, 114, 158\u201359  \npersonal responsibility, 30, 41\u201342, 50, 269, 286\u201387  \npheromones, 176, 303  \nPhoenix, Aphrodite, 243\u201344  \n _Plan for a Free Community_ , 248  \nplants, 7\u20138, 279  \nPlato, 133  \n _Playboy_ , 242  \npleasure, 40, 259\u201361  \n _Pleasure Bound_ , 167\u201368  \nPlotinus, 40  \npolyamory, 6, 287\u201388  \npornography, 6, 245  \ndebate over value of, 257\u201359  \npolitics of, 250\u201357  \nreading people and, 261\u201363  \nsexual pleasure and, 259\u201361  \nPornography Harms, 103, 255  \npositions, sexual, 204\u20137  \npower, 23, 40  \nPredator, 57\u201362, 122, 213  \n _Prehistory of Sex, The_ , 98  \n _Presence of the Past, The_ , 292  \nprivacy, 220, 300  \nprogramming, 43\u201344, 47\u201349  \nprostitution, sacred sex and, 243\u201347  \nPsyche, 39  \nPsychology Today, 257  \npsychotropic drugs, 120\u201321  \npuberty, 297\u201398  \npygmy chimpanzees, 8\n\nrage, 57, 305\u20136  \nRam Dass, 71  \nRand, Ayn, 44  \nrape, 258\u201359  \nRawlings, Majorie Kinnan, 92  \nRebecca's Project, 103  \nred clover, 281  \nReich, Wilhelm, 102, 128, 181, 247  \nrelationships  \ncreating, 34\u201335  \nexercises for, 162\u201365  \nhistory of, 166\u201368  \nold scripts and, 147\u201350  \nself-examination and, 147\u201350, 274\u201375  \n_See also_ intimacy  \nreligion, 82\u201383, 86, 107, 134  \nreparenting, 234  \nrepercussions, 28  \nrepression, 102\u20135  \nresentment, 59  \nresponsibility, 41\u201342, 50, 269, 286\u201387  \nreverence, 249\u201350  \nRig Veda, 167  \nRiley, Cole, 168  \nrituals, 92\u201393  \nRogers, Will, 283  \nRoman Empire, 244\u201345, 246  \nRossellini, Isabella, 260  \nRoth, Geneen, 31, 172  \nrudeness, 60  \nRumi, 67, 166, 167, 192  \nRussell, Bertrand, 60  \nRyan, Christopher, 101\n\n _Sacred and the Profane, The_ , 80  \nsacred sex, 187\u2013207  \nexercises for, 192\u201397  \nhealing and, 12\u201313  \njoining of souls and, 199\u2013203  \nloving through space and time, 190\u201392  \nprostitution and, 243\u201347  \nthird body and, 197\u201399  \ntreatment of Earth and, 10\u201311  \ntrust and, 223\u201324  \nsafety, 29\u201332  \nSagan, Dorion, 10\u201311, 145\u201346  \n _Sappho's Leap_ , 290  \nSchachter-Shalomi, Zalman, 189\u201390  \nschools, 306  \nscripts, 43\u201344, 47\u201349  \nSeale, Alan, 185  \nSeattle, Chief, 147  \nsecrets, 19\u201320, 28, 219\u201323, 263  \nSeed, John, 132  \nself, birthing of, 27\u201329  \nself-examination, 21, 39\u201340, 51, 54\u201378, 274  \nacknowledging all of self, 61\u201364  \nintegrating parts and, 71\u201372  \nmeditations for, 63\u201364, 66\u201367, 68\u201370  \nPredator and, 57\u201361  \nrelationships and, 147\u201350  \nTransactional Analysis and, 54\u201357, 70\u201378  \nselfishness, 264\u201365  \nself-love, 177, 269  \nself-nurturing, 276\u201379  \nSeltzer, Sarah, 248  \nsemen, 7\u20138  \nsenses, 127\u201328, 153\u201357  \nsensing body, 138\u201346  \nsensuality, 183\u201386  \nseparation, 10\u201313, 96\u2013101  \ndamages of, 133\u201338  \ndeath and disease through, 112\u201314  \nsex  \nhealing and, 12\u201313  \norigins of, 10\u201311  \npolitics of, 250\u201357  \npositions and ego states, 204\u20137  \n_See also_ sacred sex  \nsex education, 5\u20136, 292\u2013309  \n _Sex Incorporated_ , 32  \n _Sex in Human Loving,_ 76  \nsexting, 103\u20134, 256  \nsex toys, 4  \nsex trafficking, 103  \nsexual abuse, 23  \nsexual education, 103\u20135  \n _Sexual Intelligence_ , 103  \n _Sexuality and the Human Male_ , 256  \nsexual repression, 102\u20135  \nsexual revolution, 247\u201350, 288\u201390  \nsexual secrets, 221\u201323  \nshadow, 45\u201346  \n _Shamans, Mystics and Doctors_ , 11  \nshame, 4, 5, 7, 59, 225\u201340  \nexercises for, 232\u201333  \nsex and, 231\u201336  \nworking through, 226\u201330  \nSheldrake, Rupert, 100, 112, 292  \nShiva, 167  \nshoes, 132\u201333  \nShroeder, Theodore, 253, 258  \n _Silent Spring_ , 113  \nsilphium, 280\u201381  \nSkarynski, Boleslaw, 281  \nslugs, 8  \nsmell, 176\u201380  \nsnow geese, 282  \nSobel, David, 104  \nsocialization, 46  \nSocrates, 36  \nsomatics, 128  \nsoul  \nhealing of, 268\u201376  \njoining of souls, 199\u2013203  \nmeanings of, 36\u201342  \nseeing with, 192  \nselfishness and, 264\u201368  \nseparation and, 133\u201338  \nwork of, 289\u201390  \nsoul retrieval, 49\u201353, 150  \n _Spell of the Sensuous_ , 144\u201345  \nSpence, Gerry, 272  \nSpirit, witnessing by, 86\u201388  \nspirituality, 82\u201385  \nStafford, William, 279  \nStanislavsky, Konstantin, 72, 147, 264  \nStanton, Elizabeth Cady, 255\u201356  \nStar People, 93\u201395  \nSteinem, Gloria, 247\u201348  \nSteiner, Rudolf, 161\u201362  \nStevenson, Robert Louis, 35  \nstress, 138\u201339  \nSun Bear, 273  \nSuris, Joan-Carles, 308  \nsurrender, 181  \nsurvival mechanisms, 27, 30  \nSuzuki, David, 93\u201394  \nSweden, 307\u20139\n\ntantric sex, 11  \nTaylor, Timothy, 97, 98  \ntears, 105  \nteenagers, 103\u20134, 302\u20139  \ntemper tantrums, 51  \ntension, 138\u201339  \nthird body, 197\u201399  \n\"This Simple Thing,\" 180  \nThoreau, Henry David, 11, 38, 97, 126, 127, 130  \nthree drives, 40  \nThurber, James, 30  \ntolerance, 260\u201361  \nTompkins, Peter, 279, 281  \ntouching, 236\u201339  \nTransactional Analysis, 47, 54  \ntransformation, 35  \ntravel, 124\u201325  \ntrees, 37\u201338, 157\u201359  \ntrust, 177, 207, 210\u201324  \nbreaking of, 211\u201314  \nexercises for, 214\u201315  \ninjunctions and, 218\u201319  \nmending of, 215\u201317  \nsacred sex and, 223\u201324  \nsecrets and, 219\u201323  \nTwain, Mark, 41\n\nUlrich, Laurel Thatcher, 268  \nunconsciousness, 43\u201344, 51, 60, 150  \nUnderhill, Ruth, 291  \nunkindness, 60\u201361  \nuproar, 213\n\nvalidation, 49  \nVenus, 246  \nviolence, 60, 258  \nViolet Blue, 254\u201355  \nvitality, 143\u201344  \nvulnerability, 28, 50, 180\u201381\n\nWadstr\u00f6m, C. B., 248  \nwalking, 126\u201328  \nwants, 28  \nWarren, Josiah, 248  \nWashington, George, 102  \n _Way, The_ , 95  \nWelch, Lew, 91\u201392  \nWeller, Francis, 94  \nWhite, David Gordon, 11  \nwholeness, 149\u201350  \n\"Who Makes These Changes?\" 67  \nWicca, 86  \n _Wild_ , 129  \nWilde, Oscar, 168  \nwilderness, 110, 129  \nwild\/wildness  \ncalling of, 116\u201321  \ncoming to life, 128\u201333  \ngoing wild, 121\u201328  \nmeanings of, 109\u201312  \nwillow, 281  \nWilson, Edward O., 116\u201317  \nwitch hunts, 101  \nWollstonecraft, Mary, 247\u201348  \n _Woman Whose Calling Is Men, A_ , 243\u201344  \nwomen, domestication of, 97  \nwonder, 191, 249\u201350  \nWoodhill, Victoria, 168  \nWoods, Tiger, 6  \nWounded Knee, 99\u2013100\n\nyoga, 11  \n _Your Body Speaks Its Mind_ , 140\n\nZappa, Frank, 54\n_About the Author_\n\nJulie McIntyre is an Earth Ceremonialist and spiritual teacher who leads Earth Medicine apprenticeships, wilderness retreats, and Deep Ecology intensives throughout the United States, Canada, and Ireland. She lives in Silver City, New Mexico.\n_About Inner Traditions \u2022 Bear & Company_\n\nFounded in 1975, Inner Traditions is a leading publisher of books on indigenous cultures, perennial philosophy, visionary art, spiritual traditions of the East and West, sexuality, holistic health and healing, self-development, as well as recordings of ethnic music and accompaniments for meditation.\n\nIn July 2000, Bear & Company joined with Inner Traditions and moved from Santa Fe, New Mexico, where it was founded in 1980, to Rochester, Vermont. 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Powell_\n\nINNER TRADITIONS \u2022 BEAR & COMPANY  \nP.O. Box 388  \nRochester, VT 05767  \n1-800-246-8648 **  \nwww.InnerTraditions.com**\n\nOr contact your local bookseller\nDestiny Books\n\nOne Park Street\n\nRochester, Vermont 05767\n\nwww.DestinyBooks.com\n\nDestiny Books is a division of Inner Traditions International\n\nCopyright \u00a9 2012 by Julie McIntyre\n\nAll rights reserved. No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the publisher.\n\n **Library of Congress Cataloging-in-Publication Data**\n\nMcIntyre, Julie.\n\nSex and the intelligence of the heart : nature, intimacy, and sexual energy \/ Julie McIntyre.\n\np. cm.\n\nIncludes bibliographical references and index.\n\nprint ISBN 978-1-59477-397-6\n\nebook ISBN 978-1-59477-698-4\n\n1. Sex. 2. Nature. 3. Mind and body. I. Title.\n\nHQ21.M4634 2012\n\n306.7\u2014dc23\n\n2012002238\n\nFor permissions information please see page vi.\n\nTo send correspondence to the author of this book, mail a first-class letter to the author c\/o Inner Traditions r Bear & Company, One Park Street, Rochester, VT 05767, and we will forward the communication, or visit the author's websites at **www.sexandtheintell\u00adigenceoftheheart.com** or **www.gaianstudies.org**.\nElectronic edition produced by\n\nwww.dmiepub.com\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nE-text prepared by the Online Distributed Proofreading Team\n(http:\/\/www.pgdp.net) from scanned images of public domain material\ngenerously made available by the Google Books Library Project\n(http:\/\/books.google.com\/)\n\n\n\nNote: Project Gutenberg also has an HTML version of this\n      file which includes the original illustrations.\n      See 38128-h.htm or 38128-h.zip:\n      (http:\/\/www.gutenberg.org\/files\/38128\/38128-h\/38128-h.htm)\n      or\n      (http:\/\/www.gutenberg.org\/files\/38128\/38128-h.zip)\n\n\n      Images of the original pages are available through\n      the the Google Books Library Project. See\n      http:\/\/books.google.com\/books?vid=MaYBAAAAQAAJ&id\n\n\nTranscriber's note:\n\n      Obvious typographical errors have been corrected, but\n      otherwise the original spelling has generally been retained,\n      even where several different spellings have been used to\n      refer to the same person.\n\n      The printed book contained footnotes and endnotes. The\n      endnotes have been treated as footnotes, and marked with\n      anchors prefixed by E, as in [E01]. When one endnote is\n      referenced twice, the second occurence is marked by adding\n      a b, as in [E12b], and the text of the endnote is repeated\n      in the appropriate place.\n\n      The printed book contained a few features, such as Greek\n      text and illustrations, that could not be reproduced in this\n      format. These have been marked in the text using {curly\n      braces}.\n\n      A list of corrections is at the end of this e-book.\n\n\n\n\n\nMEMOIRS OF  LEONORA CHRISTINA\n\nDaughter of Christian IV. of Denmark\n\nWritten During Her Imprisonment in the Blue Tower at Copenhagen\n1663-1685\n\nTranslated by F. E. Bunnett\n\n\n\n\n\n\n\nLondon\nHenry S. King & Co., 65 Cornhill\n1872\n\nLondon: Printed by\nSpottiswoode and Co., New-Street Square\nand Parliament Street\n\nAll rights reserved\n\n\n\n\nPREFACE.\n\n\nIn placing the present translation of LEONORA CHRISTINA ULFELDT'S\nMemoirs before the English reading public, a few words are due from\nthe Publishers, in order to explain the relation between this edition\nand those which have been brought out in Denmark and in Germany.\n\nThe original autograph manuscript of Leonora Christina's record of\nher sufferings in her prison, written between the years 1674 and\n1685, belongs to her descendant the Austrian Count Joh. Waldstein,\nand it was discovered only a few years ago. It was then, at the\ndesire of Count Waldstein, brought to Copenhagen by the Danish\nMinister at Vienna, M. Falbe, in order that its authenticity might be\nthoroughly verified by comparison with documents preserved in the\nDanish archives and libraries, and known to be in the hand-writing of\nthe illustrious authoress. When the existence of this interesting\nhistoric and literary relic had become known in Denmark, a desire to\nsee it published was naturally expressed on all sides, and to this\nthe noble owner most readily acceded.\n\nThus the first Danish edition came to light in 1869, promoted in\nevery way by Count Waldstein. The editor was Mr. Sophus Birket-Smith,\nassistant librarian of the University Library at Copenhagen, who\nenriched the edition with a historical introduction and copious\nnotes. A second Danish edition appeared a few months later; and in\n1871 a German translation of the Memoir was edited by M. Ziegler,\nwith a new introduction and notes, founded partly on the first Danish\nedition, partly on other printed sources, to which were added\nextracts from some papers found in the family archives of Count\nWaldstein, and which were supposed to possess the interest of\nnovelty.\n\nThe applause with which this edition was received in Germany\nsuggested the idea of an English version, and it was at first\nintended merely to translate M. Ziegler's book into English. During\nthe progress of the work, however, it was found preferable to adopt\nthe second Danish edition as the basis of the English edition. The\ntranslation which had been made from M. Ziegler's German, has been\ncarefully compared with the Danish original, so as to remove any\ndefects arising from the use of the German translation, and give it\nthe same value as a translation made direct from the Danish; a new\nintroduction and notes have been added, for which the Danish editor,\nMr. Birket-Smith has supplied the materials; and instead of the\nfragments of Ulfeldt's Apology and of an extract from Leonora\nChristina's Autobiography found in the German edition, a complete\ntranslation of the Autobiography to the point where Leonora's Memoir\nof her sufferings in prison takes up the thread of the narrative, has\nbeen inserted, made from the original French text, recently published\nby Mr. S. Birket-Smith. As a matter of course the preface of Count\nWaldstein, which appears in this edition, is the one prefixed to the\nDanish edition. The manuscript itself of the record of Leonora\nChristina's sufferings in prison was commenced in 1674, and was at\nfirst intended to commemorate only what had happened during the\npreceding ten years of her captivity; it was afterwards extended to\nembrace the whole period down to 1685, and subjected to a revision\nwhich resulted in numerous additions and alterations. As, however,\nthese do not seem to have been properly worked in by the authoress\nherself, the Memoir is here rendered, as in the Danish edition, in\nits original, more perfect shape, and the subsequent alterations made\nthe subject of foot notes.\n\n\n\n\n PREFACE\n TO\n THE DANISH EDITION.\n\n\nWhen, in the summer of 1858, I visited the graves of my Danish\nancestors of the family of Ulfeldt, in the little village church at\nQuaerndrup, near the Castle of Egeskov, on the island of Fyn, I\nresolved to honour the memory of my pious ancestress Leonora\nChristina, and thus fulfil the duty of a descendant by publishing\nthis autograph manuscript which had come to me amongst the heirlooms\nleft by my father.\n\nIt is well known that the last male representative of the family of\nUlfeldt, the Chancellor of the Court and Realm of Her Majesty the\nEmpress Maria Theresia, had only two daughters. One of them,\nElizabeth, married Georg Christian, Count Waldstein, while the\nyounger married Count Thun.\n\nOut of special affection for her younger son Emanuel (my late\nfather), my grandmother bequeathed all that referred to the Ulfeldts\nto him, and the manuscript which I now--in consequence of requests\nfrom various quarters, also from high places--give to publicity by\nthe learned assistance of Mr. Sophus Birket-Smith, thus came to me\nthrough direct descent from her father:\n\n'Corfitz, Count of Ulfeldt of the holy Roman Empire, Lord of the\nlordships Koeltz-Jenikau, Hof-Kazof, Broedlich, Odaslowitz, and the\nfief Zinltsch, Knight of the Golden Vliess, First Treasurer of the\nhereditary lands in Bohemia, Ambassador at the Ottoman Porte,\nafterwards Chancellor of the Court and the Empire, sworn Privy\nCouncillor and first Lord Steward of his Imperial and Royal Majesty\nCarolus VI., as well as of His Imperial Roman and Royal Majesty of\nHungary, Bohemia,' &c.\n\nWe add: the highly honoured paternal guide of Her Majesty the Queen\nEmpress Maria Theresia, of glorious memory, during the first year of\nher government, until the time when the gifted Prince Kaunitz, whose\ngenius sometimes even was too much for this, morally noble lady,\nbecame her successor.\n\nI possess more than eleven imposing, closely written folio volumes,\nwhich contain the manuscripts of the Chancellor of the Empire, his\nnegociations with the Sublime Porte, afterwards with the\nStates-General of the Netherlands, as well as the ministerial\nprotocols from the whole time that he held the office of Imperial\nChancellor; all of which prove his great industry and love of order,\nwhile the original letters and annotations of his exalted mistress,\nwhich are inserted in these same volumes, testify to the sincere,\nalmost childlike confidence with which she honoured him.\n\nBut this steady and circumspect statesman was the direct grandson of\nthe restless and proud\n\nCORFITZ, first Count of Ulfeldt of the Roman Empire, High Steward of\nthe Realm in Denmark, &c., and of his devoted and gifted wife LEONORA\nCHRISTINA, through their son\n\nLEO, Imperial Count Ulfeldt, Privy Councillor, Field-marshal, and\nViceroy in Catalonia of the Emperor Carl VI., and his wife, a born\nCountess of Zinzendorf.\n\nI preserved, therefore with great care this manuscript, as well as\nall other relics and little objects which had belonged to my Danish\nancestress, whose exalted character and sufferings are so highly\ncalculated to inspire sympathy, interest, and reverence. Amongst\nthese objects are several writings, such as fragments of poems,\nprayers, needlework executed in prison (some embroidered with hair of\na fair colour); a christening robe with cap worked in gold, probably\nused at the christening of her children; a very fine Amulet of\nChristian IV. in blue enamel, and many portraits; amongst others the\noriginal picture in oil of which a copy precedes the title page, &c.\n&c.\n\nConsidering that the manuscript has been handed down directly from my\nancestors from generation to generation in direct line, I could not\npersonally have any doubt as to its genuineness. Nevertheless I\nyielded to the suggestions of others, in order to have the\nauthenticity of the manuscript thoroughly tested. In what way this\nwas done will be seen from the Introduction of the Editor.\n\nThough the final verdict of history may not yet have been given on\nCorfitz Ulfeldt, yet--tempus omnia sanat--yon ominous pillar, which\nwas to perpetuate the memory of his crime into eternity, has been put\naside as rubbish and left to oblivion. Noble in forgetting and\npardoning, the great nation of the North has given a bright example\nto those who still refuse to grant to Albert, Duke of Friedland--the\ngreat general who saved the Empire from the danger that threatened it\nfrom the North--the place which this hero ought to occupy in the\nWalhalla at Vienna.\n\nBut as to the fiery temper of Corfitz and the mysterious springs\nwhich govern the deeds and thoughts of mankind, it may be permitted\nto me, his descendant, to cherish the belief, which is almost\nstrengthened into a conviction, that a woman so highly gifted, of so\nnoble sentiments, as Leonora appears to us, would never have been\nable to cling with a love so true, and so enduring through all the\nchanges of life, to a man who was unworthy of it.\n\n                                    JOH. COUNT WALDSTEIN.\n\n  Cairo: December 8, 1868.\n\n\n\n\nCONTENTS.\n\n\n                                                         PAGE\n\n INTRODUCTION                                               1\n\n AUTOBIOGRAPHY                                             31\n\n A RECORD OF THE SUFFERINGS OF THE IMPRISONED COUNTESS:--\n\n   PREFACE (TO MY CHILDREN)                                87\n\n   A REMINISCENCE OF ALL THAT OCCURRED TO ME, LEONORA\n     CHRISTINA, IN THE BLUE TOWER, FROM AUGUST 8 OF THE\n     YEAR 1663, TO JUNE 11 OF THE YEAR 1674               102\n\n\n\n\n MEMOIRS\n OF\n LEONORA CHRISTINA.\n\n\n\n\nINTRODUCTION.\n\n\nAmongst the women celebrated in history, LEONORA CHRISTINA, the\nheroine as well as the authoress of the Memoirs which form the\nsubject of this volume, occupies a conspicuous place, as one of the\nnoblest examples of every womanly virtue and accomplishment,\ndisplayed under the most trying vicissitudes of fortune. Born the\ndaughter of a King, married to one of the ablest statesmen of his\ntime, destined, as it seemed, to shine in the undisturbed lustre of\nposition and great qualities, she had to spend nearly twenty-two\nyears in a prison, in the forced company--more cruel to her than\nsolitary confinement--of male and female gaolers of the lowest order,\nand for a long time deprived of every means of rendering herself\nindependent of these surroundings by intellectual occupation. She had\nto suffer alone, and innocently, for her husband's crimes; whatever\nthese were, she had no part in them, and she endured persecution\nbecause she would not forsake him in his misfortune. Leonora\nChristina was the victim of despotism guided by personal animosity,\nand she submitted with a Christian meekness and forbearance which\nwould be admirable in any, but which her exalted station and her\ngreat mental qualities bring out in doubly strong relief.\n\nIt is to these circumstances, which render the fate of Leonora so\ntruly tragic, as well as to the fact that we have her own authentic\nand trustworthy account before us, that the principal charm of this\nrecord is due. Besides this, it affords many incidental glimpses of\nthe customs and habits of the time, nor is it without its purely\nhistorical interest. Leonora and her husband, Corfits Ulfeldt, were\nintimately connected with the principal political events in the North\nof Europe at their time; even the more minute circumstances of their\nlife have, therefore, a certain interest.\n\nNo wonder that the history of this illustrious couple has formed, and\nstill forms, the theme both of laborious scientific researches and of\npoetical compositions. Amongst the latter we may here mention in\npassing a well-known novel by Rousseau de la Valette,[01] because it\nhas had the undeserved honour of being treated by a modern writer as\nan historical source, to the great detriment of his composition.\nDocuments which have originated from these two personages are of\ncourse of great value. Besides letters and public documents, there\nexist several accounts written by both Corfits Ulfeldt and Leonora\nreferring to their own life and actions. Ulfeldt published in 1652 a\ndefence of his political conduct, and composed, shortly before his\ndeath, another, commonly called the 'Apology of Ulfeldt,' which has\nnot yet been printed entirely, but of which an extract was published\nin 1695 in the supplement of the English edition of Rousseau de la\nValette's book. Some extracts from an incomplete copy discovered by\nCount Waldstein in 1870, in the family archives at the Castle of\nPalota, were published with the German edition of Leonora's Memoir;\ncomplete copies exist in Copenhagen and elsewhere. Leonora Christina,\nwho was an accomplished writer, has composed at least four partial\naccounts of her own life. One of them, referring to a journey in\n1656, to be mentioned hereafter, has been printed long ago; of\nanother, which treated of her and Ulfeldt's imprisonment at Bornholm,\nno copy has yet been discovered. The third is her Autobiography,\ncarried down to 1673, of which an English version follows this\nIntroduction; it was written in the Blue Tower, in the form of a\nletter to the Danish antiquarian, Otto Sperling, jun., who wished to\nmake use of it for his work, 'De feminis doctis.'[02]\n\n [01] _Le Comte d'Ulfeld, Grand Maistre de Danemarc._ _Nouvelle\n historique_, i.-ii. Paris, 1678. 8vo. An English translation, with\n a supplement, appeared 1695: _The Life of Count Ulfeldt, Great\n Master of Denmark, and of the Countess Eleonora his Wife._ Done out\n of French. With a supplement. London. 1695. 8vo.\n\n Another novel by the same author, called _Casimir King of Poland_,\n is perhaps better known in this country, through a translation by\n F. Spence in vol. ii. of _Modern Novels_, 1692.\n\n [02] It is by a slip of memory that Mr. Birket Smith, in his first\n Danish edition of Leonora Christina's memoir of her life in prison,\n describes this work under the name of _De feminis eruditis_.\n\nAbout a century ago a so-called Autobiography of Leonora was\npublished in Copenhagen, but it was easily proved to be a forgery; in\nfact, the original of her own work existed in the Danish archives,\nand had been described by the historian Andreas Hoeier. It has now\nbeen lost, it is supposed, in the fire which destroyed the Castle of\nChristiansborg in 1794, but a complete copy exists in Copenhagen, as\nwell as several extracts in Latin; another short extract in French\nbelongs to Count Waldstein. Finally, Leonora Christina wrote the\nmemoir of her sufferings in the prison of the Blue Tower from\n1663-1685, of which the existence was unknown until discovered by\nCount Waldstein, and given to the public in the manner indicated in\nthe Preface.\n\nIn introducing these memoirs to the English public, a short sketch of\nthe historical events and the persons to whom they refer may not be\nunwelcome, particularly as Leonora herself touches only very lightly\non them, and principally describes her own personal life.\n\n_Leonora Christina_ was a daughter of _King Christian IV._ of Denmark\nand _Kirstine Munk_. His Queen, Anna Catherine, born a princess of\nBrandenburg, died in 1612, leaving three princes (four other children\ndied early), and in 1615 the King contracted a morganatic marriage\nwith Kirstine Munk, a lady of an ancient and illustrious noble\nfamily. Leonora was born July 18 (new style), 1621, at the Castle of\nFredriksborg, so well known to all who have visited Denmark, which\nthe King had built twenty miles north of Copenhagen, in a beautiful\npart of the country, surrounded by smiling lakes and extensive\nforests. But little is known of her childhood beyond what she tells\nherself in her Autobiography. Already in her eighth year she was\npromised to her future husband, Corfits Ulfeldt, and in 1636 the\nwedding was celebrated with great splendour, Leonora being then\nfifteen years old. The family of Ulfeldt has been known since the\nclose of the fourteenth century. Corfits' father had been Chancellor\nof the Realm, and somewhat increased the family possessions, though\nhe sold the ancient seat of the family, Ulfeldtsholm, in Fyen, to\nLady Ellen Marsvin, Kirstine Munk's mother. He had seventeen\nchildren, of whom Corfits was the seventh; and so far Leonora made\nonly a poor marriage. But her husband's great talents and greater\nambition made up for this defect. Of his youth nothing is known with\nany certainty, except that he travelled abroad, as other young\nnoblemen of his time, studied at Padua, and acquired considerable\nproficiency in foreign languages.[03] He became a favourite of\nChristian IV., at whose Court he had every opportunity for displaying\nhis social talents. At the marriage of the elected successor to the\nthrone, the King's eldest son, Christian, with the Princess Magdalene\nSibylle of Saxony, in 1634, Corfits Ulfeldt acted as marechal to the\nspecial Ambassador Count d'Avaux, whom Louis XIII. had sent to\nCopenhagen on that occasion, in which situation Ulfeldt won golden\nopinions,[04] and he was one of the twelve noblemen whom the King on\nthe wedding-day made Knights of the Elephant. After a visit to Paris\nin 1635, in order to be cured of a wound in the leg which the Danish\nphysicians could not heal, he obtained the sanction of the King for\nhis own marriage with Leonora, which was solemnised at the Castle of\nCopenhagen, on October 9, 1636, with as much splendour as those of\nthe princes and princesses. Leonora was the favourite daughter of\nChristian IV., and as far as royal favour could ensure happiness, it\nmight be said to be in store for the newly-married pair.\n\n [03] La Valette's account of his participation in the Thirty Years'\n War is entirely fictitious, as almost all that he tells of\n Ulfeldt's travels, &c.\n\n [04] See _Caroli Ogerii Ephemerides sive, Iter Danicum, Svecicum,\n Polonicum, &c._ Paris, 1656. 8vo. p. 36, 37, 40, by D'Avaux's\n secretary, Ogier.\n\nAs we have stated, Ulfeldt was a poor nobleman; and it is\ncharacteristic of them both that one of her first acts was to ask him\nabout his debts, which he could not but have incurred living as he\nhad done, and to pay them by selling her jewels and ornaments, to the\namount of 36,000 dollars, or more than 7,000_l._ in English\nmoney--then a very large sum. But the King's favour soon procured him\nwhat he wanted; he was made a member of the Great Council, Governor\nof Copenhagen, and Chancellor of the Exchequer.\n\nHe executed several diplomatic missions satisfactorily; and when, in\n1641, he was sent to Vienna as special Ambassador, the Emperor of\nGermany, Ferdinand III., made him a Count of the German Empire.\nFinally, in 1643, he was made Lord High Steward of Denmark, the\nhighest dignity and most responsible office in the kingdom. He was\nnow at the summit of power and influence, and if he had used his\ntalents and opportunities in the interests of his country, he might\nhave earned the everlasting gratitude of his King and his people.\n\nBut he was not a great man, though he was a clever and ambitious man.\nHe accumulated enormous wealth, bought extensive landed estates,\nspent considerable sums in purchasing jewels and costly furniture,\nand lived in a splendid style; but it was all at the cost of the\ncountry. In order to enrich himself, he struck base coin (which\nafterwards was officially reduced to its proper value, 8 per cent.\nbelow the nominal value), and used probably other unlawful means for\nthis purpose, while the Crown was in the greatest need of money. At\nthe same time he neglected the defences of the country in a shameful\nmanner, and when the Swedish Government, in December 1643, suddenly\nordered its army, which then stood in Germany, engaged in the Thirty\nYears' War, to attack Denmark without any warning, there were no\nmeans of stopping its victorious progress. In vain the veteran King\ncollected a few vessels and compelled the far more numerous Swedish\nfleet to fly, after a furious battle near Femern, where he himself\nreceived twenty-three wounds, and where two of Ulfeldt's brothers\nfell fighting at his side; there was no army in the land, because\nCorfits, at the head of the nobility, had refused the King the\nnecessary supplies. And, although the peace which Ulfeldt concluded\nwith Sweden and Holland at Broemsebro, in 1645, might have been still\nmore disastrous than it was, if the negotiation had been entrusted to\nless skilful hands, yet there was but too much truth in the\nreproachful words of the King, when, after ratifying the treaties, he\ntossed them to Corfits saying, 'There you have them, such as you have\nmade them!'\n\nFrom this time the King began to lose his confidence in Ulfeldt,\nthough the latter still retained his important offices. In the\nfollowing year he went to Holland and to France on a diplomatic\nmission, on which occasion he was accompanied by Leonora. Everywhere\ntheir personal qualities, their relationship to the sovereign, and\nthe splendour of their appearance, procured them the greatest\nattention and the most flattering reception. While at the Hague\nLeonora gave birth to a son, whom the States-General offered to grant\na pension for life of a thousand florins, which, however, Ulfeldt\nwisely refused. In Paris they were loaded with presents; and in the\nMemoirs of Madame Langloise de Motteville on the history of Anna of\nAustria (ed. of Amsterdam, 1783, ii. 19-22) there is a striking\n_recit_ of the appearance and reception of Ulfeldt and Leonora at\nthe French Court. On their way home Leonora took an opportunity of\nmaking a short trip to London, which capital she wished to see, while\nher husband waited for her in the Netherlands.\n\nIf, however, this journey brought Ulfeldt and his wife honours and\npresents on the part of foreigners, it did not give satisfaction at\nhome. The diplomatic results of the mission were not what the King\nhad hoped, and he even refused to receive Ulfeldt on his return. Soon\nthe turning-point in his career arrived. In 1648 King Christian IV.\ndied, under circumstances which for a short time concentrated\nextraordinary power in Ulfeldt's hands, but of which he did not make\na wise use.\n\nDenmark was then still an elective monarchy, and the nobles had\navailed themselves of this and other circumstances to free themselves\nfrom all burdens, and at the same time to deprive both the Crown and\nthe other Estates of their constitutional rights to a very great\nextent. All political power was virtually vested in the Council of\nthe Realm, which consisted exclusively of nobles, and there remained\nfor the king next to nothing, except a general supervision of the\nadministration, and the nomination of the ministers. Every successive\nking had been obliged to purchase his election by fresh concessions\nto the nobles, and the sovereign was little more than the president\nof an aristocratic republic. Christian IV. had caused his eldest son\nChristian to be elected successor in his own lifetime; but this\nprince died in 1647, and when the King himself died in 1648, the\nthrone was vacant.\n\nAs Lord High Steward, Ulfeldt became president of the regency, and\ncould exercise great influence on the election. He did not exert\nhimself to bring this about very quickly, but there is no ground for\nbelieving that he meditated the election either of himself or of his\nbrother-in-law, Count Valdemar, as some have suggested. The children\nof Kirstine Munk being the offspring of a morganatic marriage, had\nnot of course equal rank with princes and princesses; but in\nChristian IV.'s lifetime they received the same honours, and Ulfeldt\nmade use of the interregnum to obtain the passage of a decree by the\nCouncil, according them rank and honours equal with the princes of\nthe royal house.\n\nBut as the nobles were in nowise bound to choose a prince of the same\nfamily, or even a prince at all, this decree cannot be interpreted as\nevidence of a design to promote the election of Count Valdemar. The\novertures of the Duke of Gottorp, who attempted to bribe Ulfeldt to\nsupport his candidature, were refused by him, at least according to\nhis own statement. But Ulfeldt did make use of his position to extort\na more complete surrender of the royal power into the hands of the\nnobility than any king had yet submitted to, and the new King,\nFredrik III., was compelled to promise, amongst other things, to fill\nup any vacancy amongst the ministers with one out of three candidates\nproposed by the Council of the Realm. The new King, Fredrik III.,\nChristian IV.'s second son, had never been friendly to Ulfeldt. This\nlast action of the High Steward did not improve the feelings with\nwhich he regarded him, and when the coronation had taken place (for\nwhich Ulfeldt advanced the money), he expressed his thoughts at the\nbanquet in these words: 'Corfitz, you have to-day bound my hands; who\nknows, who can bind yours in return?' The new Queen, a Saxon\nprincess, hated Ulfeldt and the children of Kirstine Munk on account\nof their pretensions, but particularly Leonora Christina, whose\nbeauty and talents she heartily envied.\n\nNevertheless Ulfeldt retained his high offices for some time, and in\n1649 he went again to Holland on a diplomatic mission, accompanied by\nhis wife. It is remarkable that the question which formed the\nprincipal subject of the negotiation on that occasion was one which\nhas found its proper solution only in our days--namely, that of a\nredemption of the Sound dues. This impost, levied by the Danish Crown\non all vessels passing the Sound, weighed heavily on the shipping\ninterest, and frequently caused disagreement between Denmark and the\ngovernments mostly interested in the Baltic trade, particularly\nSweden and the Dutch republic.\n\nIt was with especial regard to the Sound dues that the Dutch\nGovernment was constantly interfering in the politics of the North,\nwith a view of preventing Denmark becoming too powerful; for which\npurpose it always fomented discord between Denmark and Sweden, siding\nnow with the one, now with the other, but rather favouring the design\nof Sweden to conquer the ancient Danish provinces, Skaane, &c., which\nwere east of the Sound, and which now actually belong to Sweden.\nCorfits Ulfeldt calculated that, if the Dutch could be satisfied on\nthe point of the Sound dues, their unfavourable interference might be\ngot rid of; and for this purpose he proposed to substitute an annual\npayment by the Dutch Government for the payment of the dues by the\nindividual ships. Christian IV. had never assented to this idea, and\nof course the better course would have been the one adopted in\n1857--namely, the redemption of the dues by all States at once for a\nproportionate consideration paid once for all. Still the leading\nthought was true, and worthy of a great statesman.\n\nUlfeldt concluded a treaty with Holland according to his views, but\nit met with no favour at Copenhagen, and on his return he found that\nin his absence measures had been taken to restrict his great power;\nhis conduct of affairs was freely criticised, and his enemies had\neven caused the nomination of a committee to investigate his past\nadministration, more particularly his financial measures.\n\nAt the same time the new Court refused Leonora Christina and the\nother children of Kirstine Munk the princely honours which they had\nhitherto enjoyed. Amongst other marks of distinction, Christian IV.\nhad granted his wife and her children the title of Counts and\nCountesses of Slesvig and Holstein, but Fredrik III. declined to\nacknowledge it, although it could have no political importance, being\nnothing but an empty title, as neither Kirstine Munk nor her children\nhad anything whatever to do with either of these principalities.\nUlfeldt would not suffer himself to be as it were driven from his\nhigh position by these indications of disfavour on the part of the\nKing and the Queen (the latter was really the moving spring in all\nthis), but he resolved to show his annoyance by not going to Court,\nwhere his wife did not now receive the usual honours.\n\nThis conduct only served to embolden those who desired to oust him\nfrom his lucrative offices, not because they were better patriots,\nbut because they hoped to succeed him. For this purpose a false\naccusation was brought against Ulfeldt and Leonora Christina, to the\neffect that they had the intention of poisoning the King and the\nQueen. Information on this plot was given to the Queen personally, by\na certain Dina Vinhowers, a widow of questionable reputation, who\ndeclared that she had an illicit connection with Ulfeldt, and that\nshe had heard a conversation on the subject between Corfits Ulfeldt\nand Leonora, when on a clandestine visit in the High Steward's house.\nShe was prompted by a certain Walter, originally a son of a\nwheelwright, who by bravery in the war had risen from the ranks to\nthe position of a colonel, and who in his turn was evidently a tool\nin the hands of other parties. The information was graciously\nreceived at Court; but Dina, who, as it seems, was a person of weak\nor unsound mind, secretly, without the knowledge of her employers,\nwarned Ulfeldt and Leonora Christina of some impending danger, thus\ncreating a seemingly inextricable confusion.\n\nAt length Ulfeldt demanded a judicial investigation, which was at\nonce set on foot, but in which, of course, he occupied the position\nof a defendant on account of Dina's information. In the end Dina was\ncondemned to death and Walter was exiled. But the statements of the\ndifferent persons implicated, and particularly of Dina herself at\ndifferent times, were so conflicting, that the matter was really\nnever entirely cleared up, and though Ulfeldt was absolved of all\nguilt, his enemies did their best in order that some suspicion might\nremain. If Ulfeldt had been wise, he might probably have turned this\nwhole affair to his own advantage; but he missed the opportunity.\nUtterly absurd as the accusation was, he seems to have felt very\nkeenly the change of his position, and on the advice of Leonora, who\ndid not doubt that some other expedient would be tried by his\nenemies, perhaps with more success, he resolved to leave Denmark\naltogether.\n\nAfter having sent away the most valuable part of his furniture and\nmovable property, and placed abroad his amassed capital, he left\nCopenhagen secretly and at night, on July 14, 1651, three days after\nthe execution of Dina. The gates of the fortress were closed at a\ncertain hour every evening, but he had a key made for the eastern\ngate, and ere sunrise he and Leonora, who was disguised as a valet,\nwere on board a vessel on their way to Holland. The consequences of\nthis impolitic flight were most disastrous. He had not laid down his\nhigh offices, much less rendered an account of his administration;\nnothing was more natural than to suppose that he wished to avoid an\ninvestigation. A few weeks later a royal summons was issued, calling\nupon him to appear at the next meeting of the Diet, and answer for\nhis conduct; his offices, and the fiefs with which he had been\nbeneficed, were given to others, and an embargo was laid on his\nlanded estates.\n\nLeonora Christina describes in her Autobiography how Ulfeldt\nmeanwhile first went to Holland, and thence to Sweden, where Queen\nChristina, who certainly was not favourably disposed to Denmark,\nreceived Ulfeldt with marked distinction, and promised him her\nprotection. But she does not tell how Ulfeldt here used every\nopportunity for stirring up enmity against Denmark, both in Sweden\nitself and in other countries, whose ambassadors he tried to bring\nover to his ideas. On this painful subject there can be no doubt\nafter the publication of so many authentic State Papers of that time,\namongst which we may mention the reports of Whitelock, the envoy of\nCromwell, to whom Ulfeldt represented that Denmark was too weak to\nresist an attack, and that the British Government might easily obtain\nthe abolition of the Sound dues by war.\n\nIt seems, however, as if Ulfeldt did all this merely to terrify the\nDanish King into a reconciliation with him on terms honourable and\nadvantageous to the voluntarily exiled magnate. Representations were\nseveral times made with such a view by the Swedish Government, and in\n1656 Leonora Christina herself undertook a journey to Copenhagen, in\norder to arrange the matter. But the Danish Government was\ninaccessible to all such attempts.\n\nThis attitude was intelligible enough, for not only had Ulfeldt left\nDenmark in the most unceremonious manner, but in 1652 he published in\nStralsund a defence against the accusations of which he had been the\nsubject, full of gross insults against the King; and in the following\nyear he had issued an insolent protest against the royal summons to\nappear and defend himself before the Diet, declaring himself a\nSwedish subject. But, above all, the influence of the Queen was too\ngreat to allow of any arrangement with Ulfeldt. The King was entirely\nled by her; she, from her German home, was filled with the most\nextravagant ideas of absolute despotism, and hated the free speech\nand the independent spirit prevailing among the Danish nobility, of\nwhich Ulfeldt in that respect was a true type. Leonora Christina was\ncompelled to return in 1656, without even seeing the King, and as a\nfugitive. It is of this journey that she has given a Danish account,\nbesides the description in the Autobiography.\n\nIt may be questioned whether it would not have been wise, if\npossible, to conciliate this dangerous man; but at any rate it was\nnot done, and Ulfeldt was, no doubt, still more exasperated. Queen\nChristina had then resigned, and her successor, Carl Gustav, shortly\nafter engaged in a war in Poland. The Danish Government, foolishly\noverrating its strength, took the opportunity for declaring war\nagainst Sweden, in the hope of regaining some of the territory lost\nin 1645. But Carl Gustav, well knowing that the Poles could not carry\nthe war into Sweden, immediately turned his whole force against\nDenmark, where he met with next to no resistance. Ulfeldt was then\nliving at Barth, in Pommerania, an estate which he held in mortgage\nfor large sums of money advanced to the Swedish Government. Carl\nGustav summoned Ulfeldt to follow him, and Ulfeldt obeyed the summons\nagainst the advice of Leonora Christina, who certainly did not desire\nher native country to be punished for the wrongs, if such they were,\ninflicted upon her by the Court.\n\nThe war had been declared on June 1, 1657; in August Ulfeldt issued a\nproclamation to the nobility in Jutland, calling on them to transfer\ntheir allegiance to the Swedish King. In the subsequent winter a most\nunusually severe frost enabled the Swedish army to cross the Sounds\nand Belts on the ice, Ulfeldt assisting its progress by persuading\nthe commander of the fortress of Nakskov to surrender without\nresistance; and in February the Danish Government had to accept such\nconditions of peace as could be obtained from the Swedish King, who\nhad halted a couple of days' march from Copenhagen. By this peace\nDenmark surrendered all her provinces to the east of the Sound\n(Skaane, &c.), which constituted one-third of the ancient Danish\nterritory, and which have ever since belonged to Sweden, besides her\nfleet, &c.\n\nBut the greatest humiliation was that the negotiation on the Swedish\nside was entrusted to Ulfeldt, who did not fail to extort from the\nDanish Crown the utmost that the neutral powers would allow. For\nhimself he obtained restitution of his estates, freedom to live in\nDenmark unmolested, and a large indemnity for loss of income of his\nestates since his flight in 1651. The King of Sweden also rewarded\nhim with the title of a Count of Solvitsborg and with considerable\nestates in the provinces recently wrested from Denmark. Ulfeldt\nhimself went to reside at Malmo, the principal town in Skaane,\nsituated on the Sound, just opposite Copenhagen, and here he was\njoined by Leonora Christina.\n\nIn her Autobiography Leonora does not touch on the incidents of the\nwar, but she describes how her anxiety for her husband's safety did\nnot allow her to remain quietly at Barth, and how she was afterwards\ncalled to her mother's sick-bed, which she had to leave in order to\nnurse her husband, who fell ill at Malmo. We may here state that\nKirstine Munk had fallen into disgrace, when Leonora was still a\nchild, on account of her flagrant infidelity to the King, her\nparamour being a German Count of Solms. Kirstine Munk left the Court\nvoluntarily in 1629,[05] shortly after the birth of a child, whom the\nKing would not acknowledge as his own; and after having stayed with\nher mother for a short time, she took up her residence at the old\nmanor of Boller, in North Jutland, where she remained until her death\nin 1658.\n\n [05] La Valette's account of a lawsuit instituted by the King\n against Kirstine Munk, in which she was defended by Ulfeldt--of\n Ulfeldt's duel with Hannibal Sehested, afterwards his\n brother-in-law, &c.--is entirely fictitious. No such things took\n place.\n\nVarious attempts were made to reconcile Christian IV. to her, but he\nsteadily refused, and with very good reason: he was doubtless well\naware that Kirstine Munk, as recently published diplomatic documents\nprove, had betrayed his political secrets to Gustav Adolf, the King\nof Sweden, and he considered her presence at Court very dangerous.\nHer son-in-law was now openly in the service of another Swedish king,\nbut the friendship between them was not of long duration. Ulfeldt\nfirst incurred the displeasure of Carl Gustav by heading the\nopposition of the nobility in the newly acquired provinces against\ncertain imposts laid on them by the Swedish King, to which they had\nnot been liable under Danish rule. Then other causes of disagreement\narose. Carl Gustav, regretting that he had concluded a peace, when in\nall probability he might have conquered the whole of Denmark,\nrecommenced the war, and laid siege to Copenhagen. But the Danish\npeople now rose as one man; foreign assistance was obtained; the\nSwedes were everywhere beaten; and if the Dutch, who were bound by\ntreaty to assist Denmark, had not refused their co-operation in\ntransferring the Danish troops across the Sound, all the lost\nprovinces might easily have been regained.\n\nThe inhabitants in some of these provinces also rose against their\nnew rulers. Amongst others, the citizens of Malmo, where Ulfeldt at\nthe time resided, entered into a conspiracy to throw off the Swedish\ndominion; but it was betrayed, and Ulfeldt was indicated as one of\nthe principal instigators, although he himself had accepted their\nforced homage to the Swedish King, as his deputy. Very probably he\nhad thought that, if he took a part in the rising, he might, if this\nwere successful, return to Denmark, having as it were thus wiped out\nhis former crimes, but having also shown his countrymen what a\nterrible foe he could be. As it was, Denmark was prevented by her own\nallies from regaining her losses, and Ulfeldt was placed in custody\nin Malmo, by order of Carl Gustav, in order that his conduct might be\nsubjected to a rigorous examination.\n\nUlfeldt was then apparently seized with a remarkable malady, a kind\nof apoplexy, depriving him of speech, and Leonora Christina conducted\nhis defence. She wrote three lengthy, vigorous, and skilful replies\nto the charges, which still exist in the originals. He was acquitted,\nor rather escaped by a verdict of Not Proven; but as conscience makes\ncowards, he contrived to escape before the verdict was given. Leonora\nChristina describes all this in her Autobiography, according to which\nUlfeldt was to go to Lubeck, while she would go to Copenhagen, and\ntry to put matters straight there. Ulfeldt, however, changed his plan\nwithout her knowledge, and also repaired to Copenhagen, where they\nwere both arrested and sent to the Castle of Hammershuus, on the\nisland of Bornholm in the Baltic, an ancient fortress, now a most\npicturesque ruin, perched at the edge of perpendicular rocks,\noverhanging the sea, and almost surrounded by it.\n\nThe Autobiography relates circumstantially, and no doubt truthfully,\nthe cruel treatment to which they were here subjected by the\ngovernor, a Major-General Fuchs. After a desperate attempt at escape,\nthey were still more rigorously guarded, and at length they had to\npurchase their liberty by surrendering the whole of their property,\nexcepting one estate in Fyen. Ulfeldt had to make the most humble\napologies, and to promise not to leave the island of Fyen, where this\nestate was situated, without special permission. He was also\ncompelled to renounce on the part of his wife the title of a Countess\nof Slesvig-Holstein, which Fredrik III. had never acknowledged. She\nnever made use of that title afterwards, nor is she generally known\nby it in history. Corfits Ulfeldt being a Count of the German Empire,\nof course Leonora and her children were, and remained, Counts and\nCountesses of Ulfeldt. This compromise was effected in 1661.\n\nHaving been conveyed to Copenhagen, Ulfeldt could not obtain an\naudience of the King, and he was obliged, kneeling, to tender renewed\noath of allegiance before the King's deputies, Count Rantzow, General\nHans Schack, the Chancellor Redtz, and the Chancellor of the\nExchequer, Christofer Gabel, all of whom are mentioned in Leonora's\naccount of her subsequent prison life.\n\nA few days after, Corfits Ulfeldt and Leonora Christina left\nCopenhagen, which he was never to see again, she only as a prisoner.\nThey retired to the estate of Ellensborg, in Fyen, which they had\nstill retained. This was the ancient seat of the Ulfeldts, which\nCorfits' father had sold to Ellen Marsvin, Leonora Christina's\ngrandmother, and which had come to Leonora through her mother. In the\nmeanwhile it had been renamed and rebuilt such as it stands to this\nday, a picturesque pile of buildings in the Elizabethan style. Here\nUlfeldt might have ended his stormy life in quiet, but his thirst for\nrevenge left him no peace. Besides this, a great change had taken\nplace in Denmark. The national revival which followed the renewal of\nthe war by Carl Gustav in 1658 led to a total change in the form of\ngovernment.\n\nIt was indisputable that the selfishness of the nobles, who refused\nto undertake any burden for the defence of the country, was the main\ncause of the great disasters that had befallen Denmark. The abolition\nof their power was loudly called for, and the Queen so cleverly\nturned this feeling to account, that the remedy adopted was not the\nrestoration of the other classes of the population to their\nlegitimate constitutional influence, but the entire abolition of the\nconstitution itself, and the introduction of hereditary, unlimited\ndespotism. The title 'hereditary king,' which so often occurs in\nDanish documents and writings from that time, also in Leonora's\nMemoir, has reference to this change. Undoubtedly this was very\nlittle to Ulfeldt's taste. Already, in the next year after his\nrelease, 1662, he obtained leave to go abroad for his health. But,\ninstead of going to Spaa, as he had pretended, he went to Amsterdam,\nBruges, and Paris, where he sought interviews with Louis XIV. and the\nFrench ministers; he also placed himself in communication with the\nElector of Brandenburg, with a view of raising up enemies against his\nnative country. The Elector gave information to the Danish\nGovernment, whilst apparently lending an ear to Ulfeldt's\npropositions.\n\nWhen a sufficient body of evidence had been collected, it was laid\nbefore the High Court of Appeal in Copenhagen, and judgment given in\nhis absence, whereby he was condemned to an ignominious death as a\ntraitor, his property confiscated, his descendants for ever exiled\nfrom Denmark, and a large reward offered for his apprehension. The\nsentence is dated July 24, 1663. Meanwhile Ulfeldt had been staying\nwith his family at Bruges. One day one of his sons, Christian, saw\nGeneral Fuchs, who had treated his parents so badly at Hammershuus,\ndriving through the city in a carriage; immediately he leaped on to\nthe carriage and killed Fuchs on the spot. Christian Ulfeldt had to\nfly, but the parents remained in Bruges, where they had many friends.\n\nIt was in the following spring, on May 24, 1663, that Leonora\nChristina, much against her own inclination, left her husband--as it\nproved, not to see him again alive. Ulfeldt had on many occasions\nused his wealth in order to gain friends, by lending them\nmoney--probably the very worst method of all. It is proved that at\nhis death he still held bonds for more than 500,000 dollars, or\n100,000_l._, which he had lent to various princes and noblemen, and\nwhich were never paid. Amongst others he had lent the Pretender,\nafterwards Charles II., a large sum, about 20,000 patacoons, which at\nthe time he had raised with some difficulty. He doubted not that the\nKing of England, now that he was able to do it, would recognise the\ndebt and repay it; and he desired Leonora, who, through her father,\nwas cousin of Charles II., once removed, to go to England and claim\nit. She describes this journey in her Autobiography.\n\nThe Danish Government, hearing of her presence in England, thought\nthat Ulfeldt was there too, or hoped at any rate to obtain possession\nof important documents by arresting her, and demanded her\nextradition. The British Government ostensibly refused, but underhand\nit gave the Danish minister, Petcum, every assistance. Leonora was\narrested in Dover, where she had arrived on her way back,\ndisappointed in the object of her journey. She had obtained enough\nand to spare of fair promises, but no money; and by secretly giving\nher up to the Danish Government, Charles II. in an easy way quitted\nhimself of the debt, at the same time that he pleased the King of\nDenmark, without publicly violating political propriety. Leonora's\naccount of the whole affair is confirmed in every way by the light\nwhich other documents throw upon the matter, particularly by the\nextracts contained in the Calendar of State Papers, Domestic Series,\nof the reign of Charles II., 1663-64.\n\nLeonora was now conducted to Copenhagen, where she was confined in\nthe Blue Tower--a square tower surmounted by a blue spire, which\nstood in the court of the royal castle, and was used as a prison for\ngrave offenders (see the engraving). At this point the Memoir of her\nsufferings in the prison takes up the thread of her history, and we\nneed not here dwell upon its contents.\n\nAs soon as Ulfeldt heard that the Brandenburg Government had betrayed\nhim, and that sentence had been passed on him in Copenhagen, he left\nBruges. No doubt the arrest of Leonora in England was a still greater\nblow to him. The Spanish Government would probably have surrendered\nhim to the Danish authorities, and he had to flee from place to\nplace, pursued by Danish agents demanding his extradition, and men\nanxious to earn the reward offered for his apprehension, dead or\nalive. His last abode was Basle, where he passed under a feigned\nname, until a quarrel between one of his sons and a stranger caused\nthe discovery of their secret. Not feeling himself safe, Ulfeldt left\nBasle, alone, at night, in a boat descending the Rhine; but he never\nreached his destination. He was labouring under a violent attack on\nthe chest, and the night air killed him. He breathed his last in the\nboat, on February 20, 1664. The boatmen, concluding from the gold and\njewels which they found on him that he was a person of consequence,\nbrought the body on shore, and made the matter known in Basle, from\nwhence his sons came and buried him under a tree in a field--no one\nknows the spot.\n\nMeanwhile the punishment of beheading and quartering had been\nexecuted on a wooden effigy in Copenhagen. His palace was demolished,\nand the site laid out in a public square, on which a pillar of\nsandstone was erected as an everlasting monument of his crimes. This\npillar was taken away in 1842, and the name was changed from Ulfeldt\nSquare to Greyfriars Square, as an indication of the forgetting and\nforgiving spirit of the time, or perhaps rather because the treason\nof Ulfeldt was closely connected with the ancient jealousy between\nDanes and Swedes, of which the present generation is so anxious to\nefface the traces.\n\nHis children had to seek new homes elsewhere. Christian, who killed\nFuchs, became a Roman Catholic and died as an abbe; and none of them\ncontinued the name, except the youngest son Leo, who went into the\nservice of the German Emperor, and rose to the highest dignities. His\nson Corfits likewise filled important offices under Charles VI. and\nMaria Theresa, but left no sons. His two daughters married\nrespectively a Count Waldstein and a Count Thun, whose descendants\ntherefore now represent the family of Ulfeldt.\n\nLeonora Christina remained in prison for twenty-two years--that is,\nuntil the death of Sophia Amalia, the Queen of Fredrik III. This\nKing, as well as his son Christian V., would willingly have set her\nat liberty; but the influence of the Queen over her husband and son\nwas so strong that only her death, which occurred in 1685, released\nLeonora.\n\nThe Memoir of her life in prison terminates with this event, and her\nafter-life does not offer any very remarkable incidents.\nNevertheless, a few details, chiefly drawn from a MS. in the Royal\nLibrary at Copenhagen, recently published by Mr. Birket Smith, may\nserve to complete the historical image of this illustrious lady. The\nMS. in question is from the hand of a Miss Urne, of an ancient Danish\nfamily, who managed the household of Leonora from 1685 to her death\nin 1698. A royal manor, formerly a convent, at Maribo, on the island\nof Laaland, was granted to Leonora shortly after her release from the\nBlue Tower, together with a sufficient pension for a moderate\nestablishment.\n\n'The first occupation of the Countess,' says Miss Urne, 'was\ndevotion; for which purpose her household was assembled in a room\noutside her bed-chamber. In her daily morning prayer there was this\npassage: \"May the Lord help all prisoners, console the guilty, and\nsave the innocent!\" After that she remained the whole forenoon in her\nbedchamber, occupied in reading and writing. She composed a book\nentitled the \"Ornament of Heroines,\" which Countess A. C. Ulfeldt and\nCount Leon took away with them, together with many other rare\nwritings. Her handiwork is almost indescribable, and without an\nequal; such as embroidering in silk, gold embroidery, and turning in\namber and ivory.'\n\nIt will be seen from Leonora's own Memoir that needlework was one of\nher principal occupations in her prison. Count Waldstein still\npossesses some of her work; in the Church of Maribo an altar-cloth\nembroidered by her existed still some time ago; and at the Castle of\nRosenborg, in Copenhagen, there is a portrait of Christian V. worked\nby Leonora in silk, in return for which present the King increased\nher annual pension. Miss Urne says that she sent all her work to\nElizabeth Bek, a granddaughter of Leonora, who lived with her for\nsome years. But she refused to send her Leonora's Postille, or manual\nof daily devotion, which had been given Leonora on New Year's Day, in\nthe last year of her captivity, by the castellan, Torslev, who is\nmentioned in Leonora's Memoir, and who had taught her to turn ivory,\n&c. This book has disappeared; but amongst the relics of Leonora\nChristina, the Royal Library at Copenhagen preserves some leaves\nwhich had been bound up with it, and contain verses, &c., by Leonora,\nand other interesting matter.\n\nHer MS. works were taken to Vienna after her death. It is not known\nwhat has become of some of them. A copy of the first part of the book\non heroines exists in Copenhagen. Miss Urne says that she possessed\nfragments of a play composed by her and acted at Maribo Kloster; also\nthe younger Sperling speaks of such a composition in Danish verse;\nbut the MS. seems to be lost now.\n\nSeveral of Leonora's relations stayed with her from time to time at\nMaribo; amongst them the above-mentioned Elizabeth Bek, whose mother,\nLeonora Sophie, famous for her beauty, had married Lave Bek, the head\nof an ancient Danish family in Skaane. After Ulfeldt's death Lave Bek\ndemanded of the Swedish Government the estates which Carl Gustav had\ngiven to Ulfeldt in 1658, but which the Swedish Government had\nafterwards confiscated, without any legal ground. Leonora Christina\nherself memorialised the Swedish King on the subject, and at least\none of her memorials on the subject, dated May 23, 1693, still\nexists; but it was not till 1735 that these estates were given up to\nLave Bek's sons. Leonora's eldest daughter, Anne Catherina, lived\nwith her mother at Maribo for several years, and was present at her\ndeath. She had married Casetta, a Spanish nobleman, mentioned by\nLeonora Christina in her Memoir, who was with her in England when she\nwas arrested. After the death of Casetta and their children, Anne\nCatherina Ulfeldt came to live with her mother. She followed her\nbrother to Vienna, where she died. It was she who transmitted the MS.\nof Leonora's Memoir of her life in the Blue Tower to the brother,\nwith the following letter, which is still preserved with the MS.:--\n\n  'This book treats of what has happened to our late lady mother in\n  her prison. I have not been able to persuade myself to burn it,\n  although the reading of it has given me little pleasure, inasmuch\n  as all those events concern her miserable state. After all, it is\n  not without its use to know how she has been treated; but it is\n  not needful that it should come into the hands of strangers, for\n  it might happen to give pleasure to those of our enemies who\n  still remain.'\n\nThe letter is addressed 'A Monsieur, Monsieur le Comte d'Ulfeldt,'\n&c., but without date or signature. The handwriting is, however, that\nof Anne Catherina Ulfeldt, and she had probably sent it off to Vienna\nfor safety immediately after her mother's death, before she knew that\nher brother would come to Maribo himself. Miss Urne says, in the MS.\nreferred to, that the King had ordered that he was to be informed\nimmediately of Leonora's demise, in order that she might be buried\naccording to her rank and descent; but she had beforehand requested\nthat her funeral might be quite plain. Her coffin, as well as those\nof three children who had died young, and whose coffins had been\nprovisionally placed in a church at Copenhagen, was immured in a\nvault in the church of Maribo; but when this was opened some forty\nyears ago, no trace of Leonora's mortal remains could be found,\nthough those of the children were there: from which it is concluded\nthat a popular report, to the effect that the body had been secretly\ncarried abroad, contains more truth than was formerly supposed. Count\nWaldstein states that in the family vault at Leitsmischl, there is\none metal coffin without any inscription, and which may be hers. If\nso, Leonora has, as it were, after her death followed her husband\ninto exile. At any rate, the final resting-place of neither of them\nis known with certainty.\n\n\n\n\n AUTOBIOGRAPHY\n OF\n LEONORA CHRISTINA\n\n 1673.\n\n\n\n\nAUTOBIOGRAPHY.\n\n\nSir,[06]--To satisfy your curiosity, I will give you a short account\nof the life of her about whom you desire to be informed. She was born\nat Fredericksborg, in the year 1621, on June 11.[07] When she was six\nweeks old her grandmother took her with her to Dalum, where she\nremained until the age of four years; her first master there being\nMr. Envolt, afterward a priest at Roeskild. About six months after\nher return to the Court, her father sent her to Holland to his\ncousin, a Duchess of Brunswick, who had married Count Ernest of\nNassau, and lived at Lewarden.\n\n [06] This autobiographical sketch is written in the form of a\n letter to Dr. Otto Sperling the younger, the son of Corfits\n Ulfeldt's old friend, who was for some years Leonora's\n fellow-prisoner in the Blue Tower.\n\n [07] It is curious that Leonora seems for a long time to have been\n under a mistake as to the date of her birthday. The right date is\n July 18, new style.\n\nHer sister Sophia, who was two years and a half older than herself,\nand her brother, who was a year younger, had gone to the aforesaid\nDuchess nearly a year before. I must not forget to mention the first\nmischances that befell her at her setting out. She went by sea in one\nof the royal ships of war; having been two days and a night at sea,\nat midnight such a furious tempest arose that they all had given up\nany hope of escaping. Her tutor, Wichmann Hassebart (afterwards\nBishop of Fyn), who attended her, woke her and took her in his arms,\nsaying, with tears, that they should both die together, for he loved\nher tenderly. He told her of the danger, that God was angry, and that\nthey would all be drowned. She caressed him, treating him like a\nfather (after her usual wont), and begged him not to grieve; she was\nassured that God was not angry, that He would see they would not be\ndrowned, beseeching him again and again to believe her. Wichmann shed\ntears at her simplicity, and prayed to God to save the rest for her\nsake, and for the sake of the hope that she, an innocent girl,\nreposed in Him. God heard him, and after having lost the two\nmainmasts, they entered at dawn of day the harbour of Fleckeroe,[08]\nwhere they remained for six weeks.\n\n [08] On the South Coast of Norway.\n\nHaving received orders to proceed by sea, they pursued their route\nand arrived safely. Her sister being informed of her arrival, and\nbeing told that she had come with a different retinue to\nherself--with a suite of gentlemen, lady preceptor, servants and\nattendants, &c.--she burst into tears, and said that she was not\nsurprised that this sister always insinuated herself and made herself\na favourite, and that she would be treated there too as such. M.\nSophia was not mistaken in this; for her sister was in greater favour\nwith the Duchess, with her governess, and with many others, than she\nwas herself. Count Ernest alone took the side of M. Sophia, and this\nrather for the sake of provoking his wife, who liked dispute; for M.\nSophia exhibited her obstinacy even towards himself. She did all the\nmischief she could to her sister, and persuaded her brother to do the\nsame.\n\nTo amuse you I will tell you of her first innocent predilections.\nCount Ernest had a son of about eleven or twelve years of age; he\nconceived an affection for her, and having persuaded her that he\nloved her, and that she would one day be his wife, but that this must\nbe kept secret, she fancied herself already secretly his wife. He\nknew a little drawing, and by stealth he instructed her; he even\ntaught her some Latin words. They never missed an opportunity of\nretiring from company and conversing with each other.\n\nThis enjoyment was of short duration for her; for a little more than\na year afterwards she fell ill of small-pox, and as his elder\nbrother, William, who had always ridiculed these affections, urged\nhim to see his well-beloved in the condition in which she was, in\norder to disgust him with the sight, he came one day to the door to\nsee her, and was so startled that he immediately became ill, and died\non the ninth day following. His death was kept concealed from her.\nWhen she was better she asked after him, and she was made to believe\nthat he was gone away with his mother (who was at this time at\nBrunswick), attending the funeral of her mother. His body had been\nembalmed, and had been placed in a glass case. One day her preceptor\nmade her go into the hall where his body lay, to see if she\nrecognised it; he raised her in his arms to enable her to see it\nbetter. She knew her dear Moritz at once, and was seized with such a\nshock that she fell fainting to the ground. Wichmann in consequence\ncarried her hastily out of the hall to recover her, and as the dead\nboy wore a garland of rosemary, she never saw these flowers without\ncrying, and had an aversion to their smell, which she still retains.\n\nAs the wars between Germany and the King of Denmark had been the\ncause of the removal of his aforesaid children, they were recalled\nto Denmark when peace was concluded. At the age of seven years and\ntwo months she was affianced to a gentleman of the King's Chamber.\nShe began very early to suffer for his sake. Her governess was at\nthis time Mistress Anne Lycke, Qvitzow's mother. Her daughter, who\nwas maid of honour, had imagined that this gentleman made his\nfrequent visits for love of her. Seeing herself deceived, she did not\nknow in what manner to produce estrangement between the lovers; she\nspoke, and made M. Sophia speak, of the gentleman's poverty, and\namused herself with ridiculing the number of children in the family.\nShe regarded all this with indifference, only declaring once that she\nloved him, poor as he was, better than she loved her rich\ngallant.[09]\n\n [09] Count Christian Pentz, to whom Sophia was married in 1634.\n\nAt last they grew weary of this, and found another opportunity for\ntroubling her--namely, the illness of her betrothed, resulting from a\ncomplaint in his leg; they presented her with plaisters, ointments,\nand such like things, and talked together of the pleasure of being\nmarried to a man who had his feet diseased, &c. She did not answer a\nword either for good or bad, so they grew weary of this also. A year\nand a half after they had another governess, Catharina Sehestedt,\nsister of Hannibal.[10] M. Sophia thus lost her second, and her\nsister had a little repose in this quarter.\n\n [10] Hannibal Sehestedt afterwards married Leonora's younger sister\n Christiana; he became a powerful antagonist of Ulfeldt, and is\n mentioned often in the following Memoir.\n\nWhen our lady was about twelve years old, Francis Albert, Duke of\nSaxony,[11] came to Kolding to demand her in marriage. The King\nreplied that she was no longer free, that she was already betrothed;\nbut the Duke was not satisfied with this, and spoke to herself, and\nsaid a hundred fine things to her: that a Duke was far different to a\ngentleman. She told him she always obeyed the King, and since it had\npleased the King to promise her to a gentleman, she was well\nsatisfied. The Duke employed the governess to persuade her, and the\ngoverness introduced him to her brother Hannibal, then at the Court,\nand Hannibal went with post-horses to Moen, where her betrothed was,\nwho did not linger long on the road in coming to her. This was the\nbeginning of the friendship between Monsieur and Hannibal, which\nafterwards caused so much injury to Monsieur. But he had not needed\nto trouble himself, for the Duke never could draw from her the\ndeclaration that she would be ready to give up her betrothed if the\nKing ordered her to do so. She told him she hoped the King would not\nretract from his first promise. The Duke departed ill satisfied, on\nthe very day the evening of which the betrothed arrived. (Four years\nafterwards they quarrelled on this subject in the presence of the\nKing, who appeased them with his authority.)\n\n [11] Frantz Albrecht, Duke of Saxe-Lauenburg, the same who in the\n Thirty Years' War alternately served the Protestants and the\n Imperialists. In the battle of Luetzen he was near Gustav Adolf when\n he fell, and he was regarded by many as the one who treacherously\n fired the fatal shot.\n\nIt happened the following winter at Skanderborg that the governess\nhad a quarrel with the language-master, Alexandre de Cuqvelson, who\ntaught our lady and her sisters the French language, writing,\narithmetic, and dancing. M. Sophia was not studious; moreover, she\nhad very little memory; for her heart was too much devoted to her\ndolls, and as she perceived that the governess did not punish her\nwhen Alexandre complained of her, she neglected everything, and took\nno trouble about her studies. Our lady imagined she knew enough when\nshe knew as much as her sister. As this had lasted some time, the\ngoverness thought she could entrap Alexandre; she accused him to the\nKing, said that he treated the children badly, rapped their fingers,\nstruck them on the hand, called them bad names, &c., and with all\nthis they could not even read, much less speak, the French language.\nBesides this, she wrote the same accusations to the betrothed of our\nlady. The betrothed sent his servant Wolff to Skanderborg, with\nmenaces to Alexandre. At the same time Alexandre was warned that the\nKing had sent for the prince,[12] to examine his children, since the\nfather-confessor was not acquainted with the language.\n\n [12] That is, the King's eldest son Christian, who was elected his\n successor, but died before him.\n\nThe tutor was in some dismay; he flattered our lady, implored her to\nsave him, which she could easily do, since she had a good memory, so\nthat he could prove by her that it was not his fault that M. Sophia\nwas not more advanced. Our lady did not yield readily, but called to\nhis remembrance how one day, about half a year ago, she had begged\nhim not to accuse her to the governess, but that he had paid no\nattention to her tears, though he knew that the governess treated\nthem shamefully. He begged her for the love of Jesus, wept like a\nchild, said that he should be ruined for ever, that it was an act of\nmercy, that he would never accuse her, and that from henceforth she\nshould do nothing but what she wished. At length she consented, said\nshe would be diligent, and since she had yet three weeks before her,\nshe learnt a good deal by heart.[13] Alexandre told her one day,\ntowards the time of the examination, that there was still a great\nfavour she could render him: if she would not repeat the little\nthings which had passed at school-time; for he could not always pay\nattention to every word that he said when M. Sophia irritated him,\nand if he had once taken the rod to hit her fingers when she had not\nstruck her sister strongly enough, he begged her for the love of God\nto pardon it. (It should be mentioned that he wished the one to\nstrike the other when they committed faults, and the one who\ncorrected the other had to beat her, and if she did not do so\nstrongly enough, he took the office upon himself; thus he had often\nbeaten our lady.)\n\n [13] In the margin the following addition is inserted: 'She had at\n that time an unusual memory. She could at one and the same time\n recite one psalm by heart, write another, and attend to the\n conversation. She had tried this more than once, but I think that\n she has thereby spoilt her memory, which is not now so good.'\n\nShe made excuses, said that she did not dare to tell a lie if they\nasked her, but that she would not accuse him of herself. This promise\ndid not wholly satisfy him; he continued his entreaties, and assured\nher that a falsehood employed to extricate a friend from danger was\nnot a sin, but was agreeable to God; moreover, it was not necessary\nfor her to say anything, only not to confess what she had seen and\nheard. She said that the governess would treat her ill; so he replied\nthat she should have no occasion to do so, for that he would never\ncomplain to her. Our lady replied that the governess would find\npretext enough, since she was inclined to ill-treat the children; and\nanyhow, the other master who taught them German was a rude man, and\nan old man who taught them the spinette was a torment, therefore she\nhad sufficient reason for fear. He did not give way, but so persisted\nin his persuasion that she promised everything.\n\nWhen the prince arrived the governess did not forget to besiege him\nwith her complaints, and to beg him to use his influence that the\ntutor might be dismissed. At length the day of the examination having\ncome, the governess told her young ladies an hour before that they\nwere to say how villanously he had treated them, beaten them, &c. The\nprince came into the apartments of the ladies accompanied by the\nKing's father-confessor (at that time Dr. Ch(r)estien Sar); the\ngoverness was present the whole time.\n\nThey were first examined in German. M. Sophia acquitted herself very\nindifferently, not being able to read fluently. The master\nChristoffre excused her, saying that she was timid. When it came to\nAlexandre's turn to show what his pupils could do, M. Sophia could\nread little or nothing. When she stammered in reading, the governess\nlooked at the prince and laughed aloud. There was no difference in\nthe gospel, psalms, proverbs, or suchlike things. The governess was\nvery glad, and would have liked that the other should not have been\nexamined. But when it came to her turn to read in the Bible, and she\ndid not hesitate, the governess could no longer restrain herself, and\nsaid, 'Perhaps it is a passage she knows by heart that you have made\nher read.' Alexandre begged the governess herself to give the lady\nanother passage to read. The governess was angry at this also, and\nsaid, 'He is ridiculing me because I do not know French.' The prince\nthen opened the Bible and made her read other passages, which she\ndid as fluently as before. In things by heart she showed such\nproficiency that the prince was too impatient to listen to all.\n\nIt was then Alexandre's turn to speak, and to say that he hoped His\nHighness would graciously consider that it was not his fault that M.\nSophia was not more advanced. The governess interrupted him saying,\n'You are truly the cause of it, for you treat her ill!' and she began\na torrent of accusations, asking M. Sophia if they were not true. She\nanswered in the affirmative, and that she could not conscientiously\ndeny them. Then she asked our lady if they were not true. She replied\nthat she had never heard nor seen anything of the kind. The\ngoverness, in a rage, said to the Prince, 'Your highness must make\nher speak the truth; she dares not do so, for Alexandre's sake.'\n\nThe Prince asked her if Alexandre had never called her bad names--if\nhe had never beaten her. She replied, 'Never.' He asked again if she\nhad not seen nor heard that he had ill-treated her sister. She\nreplied, 'No, she had never either heard or seen it.' At this the\ngoverness became furious; she spoke to the prince in a low voice; the\nprince replied aloud, 'What do you wish me to do? I have no order\nfrom the King to constrain her to anything.' Well, Alexandre gained\nhis cause; the governess could not dislodge him, and our lady gained\nmore than she had imagined in possessing the affection of the King,\nthe goodwill of the Prince, of the priest, and of all those who knew\nher. But the governess from that moment took every opportunity of\nrevenging herself on our lady.\n\nAt length she found one, which was rather absurd. The old Jean\nMeinicken, who taught our lady the spinette, one day, in a passion,\nseized the fingers of our lady and struck them against the\ninstrument; without remembering the presence of her governess, she\ntook his hand and retaliated so strongly that the strings broke. The\ngoverness heard with delight the complaints of the old man. She\nprepared two rods; she used them both, and, not satisfied with that,\nshe turned the thick end of one, and struck our lady on the thigh,\nthe mark of which she bears to the present day. More than two months\nelapsed before she recovered from the blow; she could not dance, nor\ncould she walk comfortably for weeks after. This governess did her so\nmuch injury that at last our lady was obliged to complain to her\nbetrothed, who had a quarrel with the governess at the wedding of M.\nSophia, and went straight to the King to accuse her; she was at once\ndismissed, and the four children, the eldest of which was our lady,\nwent with the princess[14] to Nikoping, to pass the winter there,\nuntil the king could get another governess. The King, who had a good\nopinion of the conduct of our lady, who at this time was thirteen\nyears and four months old, wrote to her and ordered her to take care\nof her sisters. Our lady considered herself half a governess, so she\ntook care not to set them a bad example. As to study, she gave no\nthought to it at this time; she occupied herself in drawing and\narithmetic, of which she was very fond, and the princess, who was\nseventeen years of age, delighted in her company. Thus this winter\npassed very agreeably for her.\n\n [14] Namely, Magdalena Sybilla of Saxony, then newly married\n (October 5, 1634) to Prince Christian, the eldest son and elected\n successor of Christian IV. M. Sophia's wedding to Chr. Pentz was\n celebrated on the 10th of the same month.\n\nAt the approach of the Diet, which sat eight days after Pentecost,\nthe children came to Copenhagen, with the prince and princess, and\nhad as governess a lady of Mecklenburg of the Blixen family, the\nmother of Philip Barstorp who is still alive. After the Diet, the\nking made a journey to Glueckstad in two days and a half, and our lady\naccompanied him; it pleased the King that she was not weary, and that\nshe could bear up against inconveniences and fatigues. She afterwards\nmade several little journeys with the King, and she had the good\nfortune occasionally to obtain the pardon of some poor criminals, and\nto be in favour with the king.\n\nOur lady having attained the age of fifteen years and about four\nmonths, her betrothed obtained permission for their marriage, which\nwas celebrated (with more pomp than the subsequent weddings of her\nsisters), on October 9, 1636. The winter after her marriage she was\nwith her husband at Moen, and as she knew that her husband's father\nhad not left him any wealth, she asked him concerning his debts, and\nconjured him to conceal nothing from her. He said to her, 'If I tell\nyou the truth it will perhaps frighten you.' She declared it would\nnot, and that she would supply what was needful from her ornaments,\nprovided he would assure her that he had told her everything. He did\nso, and found that she was not afraid to deprive herself of her gold,\nsilver, and jewels, in order to pay a sum of thirty-six thousand\nrix-dollars. On April 21, 1637, she went with her husband to\nCopenhagen in obedience to the order of the king, who gave him the\npost of V.R.[15] He was again obliged to incur debt in purchasing a\nhouse and in setting up a larger establishment.\n\n [15] V.R. probably stands for Viceroy, by which term Leonora no\n doubt indicates the post of Governor of Copenhagen.\n\nThere would be no end were I to tell you all the mischances that\nbefell her during the happy period of her marriage, and of all the\nsmall contrarieties which she endured; but since I am assured that\nthis history will not be seen by anyone, and that you will not keep\nit after having read it, I will tell you a few points which are\nworthy of attention. Those who were envious of the good fortune of\nour lady could not bear that she should lead a tranquil life, nor\nthat she should be held in esteem by her father and King; I may call\nhim thus, for the King conferred on her more honours than were due to\nher from him. Her husband loved and honoured her, enacting the lover\nmore than the husband.\n\nShe spent her time in shooting, riding, tennis, in learning drawing\nin good earnest from Charles v. Mandern, in playing the viol, the\nflute, the guitar, and she enjoyed a happy life. She knew well that\njealousy is a plague, and that it injures the mind which harbours it.\nHer relations tried to infuse into her head that her husband loved\nelsewhere, especially M. Elizabet, and subsequently Anna, sister of\nher husband, who was then in her house. M. Elizabet began by\nmentioning it as a secret, premising that no one could tell her and\nwarn her, except her who was her sister.\n\nAs our lady at first said nothing and only smiled, M. Elis... said:\n'The world says that you know it well, but that you will not appear\nto do so.' She replied with a question: 'Why did she tell her a\nthing as a secret, which she herself did not believe to be a secret\nto her? but she would tell her a secret that perhaps she did not\nknow, which was, that she had given her husband permission to spend\nhis time with others, and when she was satisfied the remainder would\nbe for others; that she believed there were no such jealous women as\nthose who were insatiable, but that a wisdom was imputed to her,\nwhich she did not possess; she begged her, however, to be wise enough\nnot to interfere with matters which did not concern her, and if she\nheard others mentioning it (as our lady had reason to believe that\nthis was her own invention) that she would give them a reprimand. M.\nElis... was indignant and went away angry, but Anna, Monsieur's\nsister, who was in the house, adopted another course. She drew round\nher the handsomest women in the town, and then played the procuress,\nspoke to her brother of one particularly, who was a flirt, and who\nwas the handsomest, and offered him opportunities, &c. As she saw\nthat he was proof against it, she told him (to excite him) that his\nwife was jealous, that she had had him watched where he went when he\nhad been drinking with the King, to know whether he visited this\nwoman; she said that his wife was angry, because the other woman was\nso beautiful, said that she painted, &c.\n\nThe love borne to our lady by her husband made him tell her all, and,\nmoreover, he went but rarely afterwards to his sister's apartments,\nfrom which she could easily understand that the conversation had not\nbeen agreeable to him; but our lady betrayed nothing of the matter,\nvisited her more than before, caressed this lady more than any other,\nand even made her considerable presents. (Anna remained in her house\nas long as she lived.)\n\nAll this is of small consideration compared with the conduct of her\nown brother. It is well known to you that the Biel... were very\nintimate in our lady's house. It happened that her brother made a\njourney to Muscovy, and that the youngest of the Biel... was in his\nsuite. As this was a very lawless youth, and, to say the truth, badly\nbrought up, he not only at times failed in respect to our lady's\nbrother, but freely expressed his sentiments to him upon matters\nwhich did not concern him; among other things, he spoke ill of the\nHolstein noblemen, naming especially one, who was then in waiting on\nthe King, who he said had deceived our lady's brother. The matter\nrested there for more than a year after their return from this\njourney. The brother of our lady and Biel... played cards together,\nand disputed over them; upon this the brother of our lady told the\nHolstein nobleman what Biel... had said of him more than a year\nbefore, which B. did not remember, and swore that he had never said.\nThe Holstein nobleman said insulting things against Biel....\n\nOur lady conversed with her brother upon the affair, and begged him\nto quiet the storm he had raised, and to consider how it would cause\nan ill-feeling with regard to him among the nobility, and that it\nwould seem that he could not keep to himself what had been told him\nin secret; it would be very easy for him to mend the matter. Her\nbrother replied that he could never retract what he had said, and\nthat he should consider the Holstein nobleman as a villain if he did\nnot treat B. as a rogue.\n\nAt length the Holstein nobleman behaved in such a manner as to\nconstrain B. to send him a challenge. B. was killed by his adversary\nwith the sword of our lady's brother, which she did not know till\nafterwards. At noon of the day on which B. had been killed in the\nmorning, our lady went to the castle to visit her little twin\nsisters; her brother was there, and came forward, laughing loudly and\nsaying, 'Do you know that Ran... has killed B...?' She replied, 'No,\nthat I did not know, but I knew that you had killed him. Ran... could\ndo nothing less than defend himself, but you placed the sword in his\nhand.' Her brother, without answering a word, mounted his horse and\nwent to seek his brother-in-law, who was speaking with our old\nfriend,[16] told him he was the cause of B.'s death, and that he had\ndone so because he had understood that his sister loved him, and that\nhe did not believe that his brother-in-law was so blind as not to\nhave perceived it. The husband of our lady did not receive this\nspeech in the way the other had imagined, and said, 'If you were not\nher brother, I would stab you with this poniard,' showing it to him.\n'What reason have you for speaking thus?' The good-for-nothing fellow\nwas rather taken aback at this, and knew not what to say, except that\nB... was too free and had no respect in his demeanour; and that this\nwas a true sign of love. At length, after some discussion on both\nsides, the brother of our lady requested that not a word might be\nsaid to his sister.\n\n [16] The old friend is Dr. Otto Sperling, sen., a physician in\n extensive practice at Copenhagen, and intimate friend of Ulfeldt.\n Mr. Biel... signifies most probably a certain Christian Bielke,\n whose portrait still exists at Rosenborg Castle, in Copenhagen,\n with an inscription that he was killed in a duel by Bartram Rantzau\n on Easter eve 1642. If this date is true, Bielke cannot have\n accompanied Leonora's brother Count Valdemar on his journey to\n Russia, as this journey only took place in 1643. Count Valdemar was\n to marry a Russian princess, but it was broken off on his refusing\n to join the Greek church.\n\nAs soon as she returned home, her husband told her everything in the\npresence of our old friend, but ordered her to feign ignorance. This\nwas all the more easy for her, as her husband gave no credence to it,\nbut trusted in her innocence. She let nothing appear, but lived with\nher brother as before. But some years after, her brother ill-treated\nhis own mother, and her side being taken by our lady, they were in\nconsequence not good friends.\n\nIn speaking to you of the occupations of our lady, after having\nreached the age of twenty-one or thereabouts, I must tell you she had\na great desire to learn Latin. She had a very excellent master,[17]\nwhom you know, and who taught her for friendship as well as with good\nwill. But she had so many irons in the fire, and sometimes it was\nnecessary to take a journey, and a yearly accouchement (to the number\nof ten) prevented her making much progress; she understood a little\neasy Latin, but attempted nothing difficult; she then learnt a little\nItalian, which she continued studying whenever an opportunity\npresented itself.\n\n [17] Dr. Otto Sperling, senior.\n\nI will not speak of her short journeys to Holstein, Jutland, &c.; but\nin the year 1646 she made a voyage with her husband by sea, in the\nfirst place to Holland, where she gave birth to a son six weeks after\nher arrival at the Hague. From thence she went with her husband to\nFrance, first to Paris and afterwards to Amiens; there they took\nleave of the King and of the Queen Mother, Regent, and as they were\nreturning by Dunkirk she had the curiosity to see England, and\nbegged her husband to permit her to cross over with a small suite, to\nwhich he consented, since one of the royal vessels lay in the roads.\nShe took a nobleman with her who knew the language, our old friend, a\nservant, and the valet of the aforesaid nobleman, and this was the\nwhole of her retinue. She embarked, and her husband planned to pass\nthrough Flanders and Brabant, and to await her at Rotterdam. As she\nwas on the vessel a day and night, and the wind did not favour them,\nshe resolved to land and to follow her husband, fancying she could\nreach him in time to see Flanders and Brabant; she had not visited\nthese countries before, having passed from Holland by sea to Calais.\n\nShe found her husband at Ostend, and travelled with him to Rotterdam;\nfrom thence she pursued her former plan, embarked at Helvoot-Sluys,\nand arrived at Duns, went to London, and returned by Dover, making\nthe whole voyage in ten days, and she was again enceinte. She was an\nobject of suspicion in London. The Prince Palatine, then Elector of\nHeidelberg,[18] belonged to the party opposed to the beheaded King,\nwho was then a prisoner; and they watched her and surrounded her with\nspies, so she did not make a long sojourn in London. Nothing else was\nimagined, when it was known she had been there, but that she had\nletters from the King of Dan... for the King of Engl.... She returned\nwith her husband to Dan....\n\n [18] Prince Ruprecht, Duke of Cumberland, nephew of Charles I.\n\nIn the year 1648 fortune abandoned our lady, for on February 28 the\nKing was taken from her by death. She had the happiness, however, of\nattending upon him until his last breath. Good God, when I think of\nwhat this good King said to her the first day, when she found him\nill in bed at Rosenborg, and wept abundantly, my heart is touched. He\nbegged her not to weep, caressed her, and said: 'I have placed you so\nsecurely that no one can move you.' Only too much has she felt the\ncontrary of the promise of the King who succeeded him, for when he\nwas Duke and visited her at her house, a few days after the death of\nthe King, finding her in tears, he embraced her, saying: 'I will be a\nfather to you, do not weep.' She kissed his hand without being able\nto speak. I find that some fathers have been unnatural towards their\nchildren.\n\nIn the year 1649 she made another voyage with her husband to Holland,\nand at the Hague gave birth to a daughter. When her husband returned\nfrom this journey, he for the first time perceived the designs of\nHannibal, of Gerstorp, and Wibe, but too late. He absented himself\nfrom business, and would not listen to what his wife told him. Our\nold friend shared the opinion of our lady, adducing very strong\nreason for it, but all in vain; he said, that he would not be a\nperpetual slave for the convenience of his friends. His wife spoke as\na prophet to him, told him that he would be treated as a slave when\nhe had ceased to have authority, that they would suspect him, and\nenvy his wealth; all of which took place, though I shall make no\nrecital of it, since these events are sufficiently known to you.\n\nWe will now speak a little of the events which occurred afterwards.\nWhen they had gained their cause,[19] our lady feared that the strong\nparty which they had then overcome would not rest without ruining\nthem utterly at any cost; so she advised her husband to leave the\ncountry, since he had the King's permission to do so,[20] and to save\nhis life, otherwise his enemies would contrive some other invention\nwhich would succeed better. He consented to this at length, and they\ntook their two eldest children with them, and went by sea to\nAmsterdam. At Utrecht they left the children with the servants and a\nfemale attendant, and our lady disguised herself in male attire and\nfollowed her husband, who took the route to Lubeck, and from thence\nby sea to Sweden, to ask the protection of Queen Christina, which he\nreceived; and as the Queen knew that his wife was with him in\ndisguise, she requested to see her, which she did.\n\n [19] Namely, the process against Dina. _See_ Introduction.\n\n [20] Ulfeldt had not really the permission of the King to leave the\n country in the way he did. These words must therefore be understood\n to mean that the favourable termination of the trial concerning\n Dina's accusations had liberated Ulfeldt from the special\n obligation to remain in Copenhagen, which his position in reference\n to that case imposed upon him.\n\nThe husband of our lady purposed to remain some time in Pomerania,\nand the Queen lent him a vessel to convey him thither. Having been\nthree days at sea, the wind carried them towards Dantzig, and not\nbeing able to enter the town, for it was too late, they remained\noutside the gates at a low inn. An adventure fit for a novel here\nhappened to our lady. A girl of sixteen, or a little more, believing\nthat our lady was a young man, threw herself on her neck with\ncaresses, to which our lady responded, and played with the girl, but,\nas our lady perceived what the girl meant, and that she could not\nsatisfy her, she turned her over to Charles, a man of their suite,\nthinking he would answer her purpose; he offered the girl his\nattentions, but she repelled him rudely, saying, she was not for him,\nand went again to our lady, accosting her in the same way. Our lady\ngot rid of her, but with difficulty however, for she was somewhat\nimpudent, and our lady did not dare to leave her apartment. For the\nsake of amusing you, I must tell you, what now occurs to me, that in\nthe fort before Stade, the name of which has escaped me, our lady\nplayed with two soldiers for drink, and her husband, who passed for\nher uncle, paid the expenses; the soldiers, willing to lose for the\nsake of gaining the beer, and astonished that she never lost, were,\nhowever, civil enough to present her with drink.\n\nWe must return to Dantzig. The husband of our lady, finding himself\nnear Thoren, desired to make an excursion there, but his design was\ninterrupted by two men, one who had formerly served in Norway as\nLieutenant-Colonel, and a charlatan who called himself Dr. Saar, and\nwho had been expelled from Copenhagen. They asked the Mayor of the\ntown to arrest these two persons, believing that our lady was Ebbe\nWl....[21] They were warned by their host that these persons said\nthey were so-and-so, and that these gentlemen were at the door to\nprevent their going out. Towards evening they grew tired of keeping\nguard, and went away. Before dawn the husband of our lady went out of\nthe house first, and waited at the gate, and our lady with the two\nservants went in a coach to wait at the other gate until it was\nopened; thus they escaped this time.\n\n [21] That is, Ebbe Ulfeldt,--a relative of Corfitz who left Denmark\n in 1651 and afterwards lived in Sweden.\n\nThey went by land to Stralsund, where our lady resumed her own\nattire, after having been in disguise twelve weeks and four days, and\nhaving endured many inconveniences, not having gone to bed all the\ntime, except at Stockholm, Dantzig, and Stettin. She even washed the\nclothes, which inconvenienced her much. The winter that they passed\nat Stralsund, her husband taught her, or rather began to teach her,\nSpanish. In the spring they again made a voyage to Stockholm, at the\ndesire of Queen Chr.... This good Queen, who liked intrigue, tried to\nexcite jealousy and to make people jealous, but she did not succeed.\nThey were in Sweden until after the abdication of the Queen, and the\nwedding and coronation of King Charles and Queen Hedevig, which was\nin the year 1654. They returned to Pomerania for a visit to Barth,\nwhich they possessed as a mortgage. There, our lady passed her time\nin study, sometimes occupied with a Latin book, sometimes with a\nSpanish one. She translated a small Spanish work, entitled _Matthias\nde los Reyos_; but this book since fell into the hands of others, as\nwell as the first part of _Cleopatre_, which she had translated from\nthe French, with matters of greater value.\n\nIn the year 1657,[22] her husband persuaded her to make a voyage to\nDannem... to try and gain an audience with the King, and see if she\ncould not obtain some payment from persons who owed them money. Our\nlady found various pleas for not undertaking this voyage, seeing a\nhundred difficulties against its successful issue; but her husband\nbesought her to attempt it, and our old friend shared her husband's\nopinion that nothing could be done to her, that she was under the\nprotection of the King of Sweden, and not banished from Dan... with\nsimilar arguments. At length she yielded, and made the journey in the\nwinter, travelling in a coach with six horses, a secretary, a man on\nhorseback, a female attendant, a page and a lacquey--that was all.\nShe went first to see her mother in Jutland, and remained there three\ndays; this was immediately known at the Court.\n\n [22] This date is erroneous; the journey took place in November and\n December 1656.\n\nWhen she had passed the Belt, and was within cannon-shot of Corsor,\nshe was met by Uldrich Chr. Guldenl...,[23] who was on the point of\ngoing to Jutland to fetch her. He returned with his galley and\nlanded; she remained in her vessel, waiting for her carriage to be\nput on shore. Guld... impatient, could not wait so long, and sent the\nburgomaster Brant to tell her to come ashore, as he had something to\nsay to her. She replied that if he had anything to say to her, he\nought to show her the attention of coming to her. Brant went with\nthis answer; awaiting its issue, our lady looked at her attendants\nand perceived a change in them all. Her female attendant was seized\nwith an attack from which she suffers still, a trembling of the head,\nwhile her eyes remained fixed. The secretary trembled so that his\nteeth chattered. Charles was quite pale, as were all the others. Our\nlady spoke to them, and asked them why they were afraid; for her they\nhad nothing to fear, and less for themselves. The secretary answered,\n'They will soon let us know that.' Brant returned with the same\nmessage, with the addition that Gul... was bearer of the King's\norder, and that our lady ought to come to him at the Castle to hear\nthe King's order. She replied that she respected the King's order\nthere as well as at the castle; that she wished that Gul... would\nplease to let her know there the order of His Majesty; and when\nBrant tried to persuade her, saying continually, 'Oh! do give in, do\ngive in!' she used the same expression, and said also, 'Beg Gul... to\ngive in,' &c. At length she said, 'Give me sufficient time to have\ntwo horses harnessed, for I cannot imagine he would wish me to go on\nfoot.'\n\n [23] U.C. Gyldenlove, illegitimate son of Christian IV. and\n half-brother of Leonora.\n\nWhen she reached the castle she had the coach pulled up. Brant came\nforward to beg her to enter the castle; she refused, and said she\nwould not enter; that if he wished to speak to her he must come to\nher, that she had come more than half-way. Brant went, and returned\nonce again, but she said the same, adding that he might do all that\nseemed good to him, she should not stir from the spot. At length the\ngood-for-nothing fellow came down, and when he was ready to speak to\nher, she opened the coach and got out. He said a few polite words to\nher, and then presented her with an order from the King, written in\nthe chancery, the contents of which were, that she must hasten to\ndepart from the King's territory, or she would have to thank herself\nfor any ill that might befall her. Having read the order she bowed,\nand returned him the order, which was intended to warn her, saying,\n'That she hoped to have been permitted to kiss the King's hand, but\nas her enemies had hindered this happiness by such an order, there\nwas nothing left for her but to obey in all humility, and thanking\nHis Majesty most humbly for the warning, she would hasten as quickly\nas possible to obey His Majesty's commands. She asked if she were\npermitted to take a little refreshment, for that they had had\ncontrary winds and had been at sea all day. Gul... answered in the\nnegative, that he did not dare to give her the permission; and since\nshe had obeyed with such great submission, he would not show her the\nother order that he had, asking her at the same moment if she wished\nto see this other order? She said, no; that she would abide by the\norder that she had seen, and that she would immediately embark on\nboard her ferry-boat to return. Gul... gave her his hand, and begged\nher to make use of his galley.\n\nShe did so. They went half the way without speaking; at length Gul...\nbroke the silence, and they entered into conversation. He told her\nthat the King had been made to believe that she had assembled a\nnumber of noblemen at her mother's house, and that he had orders to\ndisperse this cabal. They had a long conversation together, and spoke\nof Dina's affair; he said the King did not yet know the real truth of\nit. She complained that the King had not tried to know it. At length\nthey arrived by night at Nyborg. Gul... accompanied her to her\nhostelry, and went to his own, and an hour afterwards sent\nScherning[24] to tell her that at dawn of day she must be ready, in\norder that they might arrive at Assens the next evening, which it was\nimpossible to do with her own horses, as they did not arrive till\nmorning. She assented, saying she would act in obedience to his\norders, began talking with Scherning, and conversed with him about\nother matters. I do not know how, but she gained his good graces, and\nhe prevailed so far with Gul... that Gul... did not hasten her\nunduly. Towards nine o'clock the next morning he came to tell her\nthat he did not think it necessary to accompany her further, but he\nhoped she would follow the King's order, and begged her to speak with\nKay v. Ahlefeld at Haderslef, when she was passing through; he had\nreceived orders as to what he had to do. She promised this, and\nGul... returned to Copenhagen, placing a man with our lady to watch\nher.\n\n [24] Probably Povl Tscherning, a well-known man of the time, who\n held the office of Auditor-General.\n\nOur lady did not think it necessary to speak to Kay v. Ahlefeld, for\nshe had nothing to say to him, and she did not want to see more\norders; she passed by Haderslef, and went to Apenrade, and awaited\nthere for ten days[25] a letter from Gul... which he had promised to\nwrite to her; when she saw that he was not going to keep his word she\nstarted on her way to Slesvig, halting half way with the intention of\ndining. Holst, the clerk of the bailiwick of Flensborg, here arrived\nin a coach with two arquebuses larger and longer than halberds. He\ngave orders to close the bar of Boy..., sent to the village, which is\nquite close, that the peasants should hold themselves ready with\ntheir spears and arms, and made four persons who were in the tavern\ntake the same arms, that is, large poles. Afterwards he entered and\nmade a long speech, with no end of compliments to our lady, to while\naway the time. The matter was, that the governor[26] desired her to\ngo to Flensborg, as he had something to say to her, and he hoped she\nwould do him the pleasure to rest a night at Flensborg.\n\n [25] In order to understand how she could wait for ten days at\n Apenrade, it must be borne in mind that the duchy of Slesvig was at\n that time divided into several parts, of which some belonged to the\n King, others to the Duke of Gottorp. Haderslev and Flensborg\n belonged to the King, but Apenrade to the Duke; in this town,\n therefore, she was safe from the pursuit of the Danish authorities.\n\n [26] The governor of Flensborg at that time was Detlef v. Ahlefeld,\n the same who in 1663 was sent to Koenigsberg to receive information\n from the court of Brandenburg on the last intrigues of Ulfeldt.\n\nOur lady replied that she had not the pleasure of his acquaintance,\nand therefore she thought he took her for someone else; if she could\noblige him in anything she would remain at Slesvig the following day,\nin order to know in what she could serve him. No, it was not that; he\nrepeated his request. She ordered Charles to have the horses put to.\nHolst understood this, which was said in French, and begged her for\nthe love of God not to set out; he had orders not to let her depart.\n'You,' said she, in a somewhat haughty tone, 'who are you? With what\nauthority do you speak thus?' He said he had no written order, but by\nword of mouth, and that his governor would soon arrive; he begged her\nfor the love of God to pardon him. He was a servant, he was willing\nto be trodden under her feet. She said: 'It is not for you to pay me\ncompliments, still less to detain me, since you cannot show me the\nKing's order, but it is for me to think what I ought to do.'\n\nShe went out and ordered her lacquey, who was the only determined one\nof her suite, to make himself master of Holst's chariot and\narquebuses. Holst followed her, begging her a hundred times, saying,\n'I do not dare to let you pass, I do not dare to open the bar.' She\nsaid, 'I do not ask you to open;' she got into the coach. Holst put\nhis hand upon the coach-door and sang the old song. Our lady, who had\nalways pistols in her carriage when she travelled, drew out one and\npresented it to him saying, 'Draw back, or I will give you the\ncontents of this.' He was not slow in letting go his hold; then she\nthrew a patacoon to those who were to restrain her, saying, 'Here is\nsomething for drink; help in letting the carriage pass the fosse!'\nwhich they immediately did.\n\nNot a quarter of an hour after she had gone, the governor arrived\nwith another chariot. There were two men and four guns in each\nchariot. Our lady was warned of the pursuit; she begged her two\ncoachmen, whom she had for herself and her baggage, to dispute them\nthe road as much as they could; she ordered Charles always to remain\nat the side of her carriage, in order that she might throw herself\nupon the horse if she saw that they gained ground. She took off her\nfurred robe. They disputed the road up to the bridge, which separated\nthe territory of the King from that of the Duke.\n\nWhen she had passed the bridge she stopped, put on her robe, and\nalighted. The others paused on the other side of the bridge to look\nat her, and thus she escaped again for this time.[27] But it was\namusing to see how the secretary perspired, what fright he was in; he\ndid not afterwards pretend to bravery, but freely confessed that he\nwas half dead with fear. She returned to Barth, and found her husband\nvery very ill. Our old friend had almost given up all hope of his\nrecovery, but her presence acted as a miracle; he was sufficiently\nstrong in the morning to be taken out of bed, to the great surprise\nof our old friend.\n\n [27] The clerk Holst was shortly after, when the Swedes occupied\n Flensborg, put to a heavy ransom by Ulfeldt, in punishment of his\n conduct to Leonora. Documents which still exist show that he\n applied to the Danish Government for compensation, but apparently\n in vain.\n\nJust as our lady was thinking of passing some days in tranquillity,\noccupied in light study, in trifling work, distillations,\nconfectionery, and such like things, her husband mixed himself in the\nwars. The King of Sweden sent after him to Stettin; he told his wife\nthat he would have nothing to do with them. He did not keep his word,\nhowever; he did not return to Barth, but went straight off with the\nKing. She knew he was not provided with anything; she saw the danger\nto which he was exposed, she wished to share it; she equipped herself\nin haste, and, without his sending for her, went to join him at\nOttensen. He wished to persuade her to return to Hamburgh, and spoke\nto her of the great danger; she said the danger was the reason why\nshe wished to bear him company, and to share it with him; so she went\nwith him, and passed few days without uneasiness, especially when\nFriderichsodde was taken; she feared for both husband and son. There\nshe had the happiness of reconciling the C. Wrangel and the C.\nJaques,[28] which her husband had believed impossible, not having\nbeen able to succeed. She had also the good fortune to cure her\neldest son and eight of her servants of a malignant fever named\nSprinckeln; there was no doctor at that time with the army, our old\nfriend having left.\n\n [28] Count Jakob Casimir de la Gardie, a Swedish nobleman. Count\n Wrangel was the Swedish General.\n\nWhen her husband passed with the King to Seeland, she remained at\nFyen. The day that she had resolved to set out on the following to\nreturn to Schone, a post arrived with news that her mother was at the\npoint of death and wished to speak to her; she posted to Jutland,\nfound Madame very ill and with no hope of life. She had only been\nthere one night, when her husband sent a messenger to say that if she\nwished to see him alive she must lose no time. Our lady was herself\nill; she had to leave her mother, who was already half dead; she had\nto take her last farewell in great sorrow, and to go with all speed\nto seek her husband, who was very ill at Malmoe. Two days afterwards\nshe received the tidings of her mother's death, and as soon as the\nhealth of her husband permitted it, she went to Jutland to give the\nnecessary orders for her mother's funeral. She returned once more to\nSchone before the burial; after the funeral[29] she went to\nCopenhagen and revisited Malmoe one day before the King of Sweden\nbegan the war for the second time and appeared before Kopenh....\n\n [29] The funeral took place with great pomp in the church of St.\n Knud, at Odense, on June 23, 1658, together with that of Sophia\n Elizabeth, Leonora's sister, who is mentioned in the beginning of\n the Autobiography.\n\nIn the year 1659 the King of Sweden ordered her husband to be\narrested at Malmoe. She went immediately to Helsingor to speak to the\nKing, but had not the happiness of speaking to him; on the contrary,\nthe King sent two of his counsellors to tell her that she was free to\nchoose whether she would return to her estates and superintend them,\nor go back to Malmoe and be arrested with her husband. She thanked\nHis Majesty very humbly for the favour of the choice; she chose to\nsuffer with her husband, and was glad to have the happiness of\nserving him in his affliction, and bearing the burden with him which\nwould lighten it to him.\n\nShe returned to Malmoe with these news; her husband exhibited too\nmuch grief that she was not permitted to solicit on his behalf, and\nshe consoled him as well as she was able. A few days after, an\nofficer came to their house and irritated her husband so much by his\nimpertinent manner that he had a fit of apoplexy. Our lady was\noverwhelmed with sorrow; she sent for the priest the next morning,\nmade her husband receive the holy communion, and received it herself.\nShe knew not at what hour she might be a widow; no one came to see\nher, no one in consequence consoled her, and she had to console\nherself. She had a husband who was neither living nor dead; he ate\nand drank; he spoke, but no one could understand him.\n\nAbout eight months after, the King began to take proceedings against\nher husband, and in order to make her answer for her husband they\nmixed her up in certain points as having asked for news: whence the\nyoung lady was taken whom her husband brought to Copenhagen? who was\nTrolle? and that she had kept the property of a Danish nobleman in\nher house.[30] Since her husband was ill, the King graciously\npermitted her to answer for him; thus they proceeded with her for\nnine weeks in succession; she had no other assistance in copying her\ndefence than her eldest daughter, then very young. She was permitted\nto make use of Wolff, for receiving the accusations and taking back\nthe replies, but he wrote nothing for her. If you are interested in\nknowing the proceedings, Kield[31] can give you information\nrespecting them.\n\n [30] The young lady was Birgitte Rantzau, who was engaged to\n Korfits Trolle, a Danish nobleman, who had been very active in\n preparing the intended rising of the citizens of Malmoe against the\n Swedes. Ulfeldt was accused of having favoured and assisted this\n design (_see_ the Introduction), and he had brought Trolle's bride\n over to Copenhagen, or accompanied them thither.\n\n [31] Wolf and Kield were servants of Ulfeldt.\n\nWhen the proceedings had lasted so many weeks, and she had answered\nwith regard to the conversations which it was said her husband had\nhad with one and another, they fancied that her husband feigned\nillness. Four doctors were sent with the commandant to visit the sick\nman, and they found that he was really ill; not content with this,\nthey established the Court in his house, for they were ashamed to\nmake her come to them. They caused the city magistrate to come,\nplacing him on one side of the hall, and on the other the Danish\nnoblemen who were under arrest, all as witnesses; eight Commissioners\nsat at a round table, the lawyer in front of the table and two clerks\nat another table; having made these arrangements, our lady was\ndesired to enter.\n\nWe must mention, in the first place, that two of the delinquents who\nwere executed afterwards, and another, together with one of the\nservants of her husband, were brought there. The principal\ndelinquents were summoned first, and afterwards the others, to take\nan oath that they would speak the truth. We must mention that these\ngentlemen were already condemned, and were executed a few days\nafterwards. When the lawyer had said that they had now taken their\noaths according to the law, our lady said, 'Post festum! After having\nproceeded against my husband so many weeks, having based everything\non the tattle of these delinquents, you come, after they are\ncondemned to suffer for their trespasses, and make them take an oath.\nI do not know if this is conformable to law!'\n\nThe lawyer made no reply to this, and, thinking to confuse our lady,\nsaid that he found things contrary the one to the other, cited\npassages, leaves, lines, and asked her if she could make these things\nagree. She, having at that time a good memory, remembered well what\nher own judgment had dictated to her, and said that they would not\nfind her replies what the lawyer said, but so-and-so, and asked that\nthey should be read openly, which was done. The lawyer made three\nattempts of the same kind; when they saw there was nothing to be\ngained by this, the Commissioners attacked her three at a time, one\nputting one question and another, another. She said to them quietly,\n'Messieurs, with your permission, let one speak at a time, for I am\nbut one, and I cannot answer three at once!' At which they were all a\nlittle ashamed.\n\nThe principal point to which they adhered was, that her husband was a\nvassal by oath, and a servant of the King, with which assertion they\nparried every objection. She proved that it was not so, that her\nhusband was neither vassal nor a servant; he had his lands under the\nKing just as many Swedes had elsewhere, without on that account being\nvassals; that he had never taken an oath of fidelity to the King of\nSweden, but that he had shown him much fidelity; that he owed him no\nobligation--this she showed by a letter from the King, in which he\nthanked him for his services, and hoped so to act that he would\nrender him still more. She shut the mouth of the delinquent,[32] and\nbegged the Commissioners to reflect on what she had said.\n\n [32] The person alluded to is a Bartholomaeus Mikkelsen, who was\n executed as ringleader of the conspiracy.\n\nWhen all was over, after the space of three hours, she requested that\nthe protocol might be read before her. The President said that she\nneed have no doubt the protocol was correct, that she should have a\ncopy of it, that they now understood the matter, and would make a\nfaithful report of it to the King. No sentence was passed, and they\nremained under arrest. The King of Sweden died, and peace was\nconcluded, but they remained under arrest. A friend came to inform\nthem, one day, that there was a vessel of war in the roads, which was\nto take them to Finland. When she saw her husband a little recovered,\nthat he could use his judgment, she advised him to escape and go to\nLubeck. She would go to Copenhagen and try to arrange the matter. He\nconsented to it, and she contrived to let him out in spite of all\nthe guards round the house (thirty-six in number).\n\nWhen she received the news that he had passed and could reckon that\nhe was on his way to Lubeck, she escaped also, and went straight to\nCopenh.... Having arrived there, she found her husband arrived before\nher; she was much surprised and vexed, fearing what happened\nafterwards, but he had flattered himself so with the comfortable hope\nthat he would enter into the good graces of the King. The next day\nthey were both arrested and brought to Borringh...[33]; her husband\nwas ill; on arriving at Borr... they placed him on a litter and\nbrought him from the town to the castle, a distance of about two\nleagues.\n\n [33] Bornholm. (_See_ the Introduction.)\n\nIt would weary you to tell you of all that passed at Borr... If you\ntake pleasure in knowing it, there is a man in Hamburgh who can tell\nit you.[34] I will tell you, however, a part and the chief of what I\nremember concerning it. At Ronne, the town where they disembarked at\nBorringh----, our lady wrote to the King and to the Queen in the name\nof her husband, who was ill, as I have already said, and gave the\nmemorials to Colonel Rantzou, who promised to deliver them, and who\ngave hopes of success.[35] There Fos arrived and conveyed them to the\nCastle of Hammershuus. The governor Fos saw that our lady had a small\nbox with her, and was seized with the desire to know what was in it\nand to possess himself of it. He sent one Dina, the wife of the\nwarder to our lady, to offer to procure a boat for their escape.\nThere is no doubt she accepted the offer, and promised in return\nfive hundred crowns. This was enough for Fos; he went one night with\nthe Major to their apartment, thundered like a madman, said that they\nwished to betray him, &c.; the end of the farce was, that he took the\nbox, but, for the sake of a little ceremony, he sealed it with her\nhusband's seal, promising to keep it for its safety.\n\n [34] She refers no doubt to a servant who accompanied them of the\n name of Pfluegge.\n\n [35] The original of this letter to the King exists still.\n\nAbout three weeks after, he took the two prisoners to walk a little\nin the fields; the husband would not go, but the wife went out to\ntake the air. The traitor gave her a long history of his past\nadventures, how many times he had been in prison, some instances of\nhow great lords had been saved by the assistance of those they had\ngained over, and made their fortune. He thought they would do the\nsame. She said she had not much to dispose of, but besides that, they\nwould find other means for rewarding such a service. He said he would\nthink of it, that he had nothing to lose in Dan....\n\nAfter various discussions from day to day, her husband wished her to\noffer him 20,000 rix-dollars; this sum seemed to him too little, and\nhe asked 50,000 dollars. She said that she could easily promise it,\nbut could not keep her word, but provided it was twenty she would pay\nit. He asked for a security; her husband had a note which would give\nsecurity, but our lady did not think it good that he should see this\nnote, and told Fos that in her box there was a letter that could\nsecure it; she did not know that he had already opened the box. Some\ndays after, she asked him if he had made up his mind? He said, 'I\nwill not do it for less than 50,000, and there is no letter in your\nbox which would secure it to me. I have opened it; to-morrow I will\nsend it to Copenh....' She asked him quietly if he had done right in\nbreaking her husband's seal; he answered rudely that he would take\nthe responsibility.\n\nTowards autumn, Hannibal and the other heirs of our lady's mother\nsent to her husband to notify to him that they could not longer delay\ndividing the inheritance, and since they knew that he had in his\npossession papers of importance, they requested to be informed of\nthem. Her husband stated in his reply that Fos had taken his letters,\nand that in a rude manner. This answer having been read in the\npresence of Fos, he flew in a thundering rage, used abusive language\nfirst to the husband and then to the wife, her husband having firmly\npromised our lady not to dispute with this villain, for she feared\nsome evil might result, but to leave her to answer, for Fos would be\nanswered.\n\nShe was not angry; she ridiculed him and his invectives. At length he\ntold her that she had offered him 20,000 dollars to induce him to\nbecome a traitor; she replied with calmness, 'If it had been 50,000,\nwhat then?' Fos leapt into the air like an enraged animal, and said\nthat she lied like a ----, &c. She was not moved, but said 'You speak\nlike an ass!' Upon this he loaded her with abuse, and then retracted\nall that he had just said. She said quite quietly, 'I am not going to\nappeal to these gentlemen who are present (there were four) to be\nwitnesses, for this is an affair that will never be judicially\nsettled, and nothing can efface this insult but blood.' 'Oh!' said\nhe, seizing his sword, and drawing it a little out of the scabbard,\n'this is what I wear for you, madam.' She, smiling, drew the bodkin\nfrom her hair, saying, 'Here are all the arms at present which I\nhave for you.' He manifested a little shame, and said that it was not\nfor her but her sons, if she still had four.[36] She, moreover,\nridiculed him, and said that it was no use his acting the brave\nthere. In short, books could be filled with all the quarrels between\nthese two persons from time to time. He shouted at times with all his\nmight, he spoke like a torrent, and foamed at the mouth, and the next\nmoment he would speak low like another man. When he shouted so\nloudly, our lady said, 'The fever is attacking him again!' He was\nenraged at this.\n\n [36] It will be remembered from the Introduction that Fuchs was\n killed two years after by one of Leonora's sons at Bruges.\n\nSome weeks afterwards he came to visit them, and assumed a humble\nmanner. Our lady took no notice of it, and spoke with him on\nindifferent subjects; but her husband would not speak to him, and\nnever afterwards was he able to draw from him more than a few words.\nTowards Christmas, Fos treated the prisoners very ill, more so than\nformerly, so that Monsieur sent the servant to beg him to treat him\nas a gentleman and not as a peasant. Fos went to them immediately,\nafter having abused Monsieur's servant; and as he entered, Monsieur\nleft the apartment and went into another, and refused to give him his\nhand. Fos was enraged at this, and would not remain, nor would he\nspeak a word to our lady, who begged him to hear her. A moment after,\nhe caused the door to be bolted, so that they could not go out to\ntake the air, for they before had free access to a loft. At every\nFestival he devised means of annoying them; he closed all the\nwindows, putting to some bars of iron, and to others wooden framework\nand boxes; and as to their food, it was worse than ever. They had to\nendure that winter in patience; but as they perceived that Fos's\ndesign was that they should die of hunger, they resolved to hazard an\nescape, and made preparation through the winter, in order to escape\nas soon as the thaw would set in.\n\nOur lady, who had three pairs of sheets that her children had sent\nher, undid some articles of clothing and made cordage and a sail; she\nsewed them with silk, for she had no thread. Her husband and the\nservant worked at the oars. When the moon was favourable to them in\nthe month of April, they wished to carry out the plan they had been\nprojecting for so long a time. Our lady was the first to make the\ndescent: the height was seventy-two feet; she went on to the ravelin\nto await the others. Some time elapsed before her husband came, so\nshe returned, and at last she heard a great noise among the ropes,\nher husband having lost a shoe in his descent. They had still to wait\nfor the valet; he had forgotten the cord, and said that he could not\ncarry it with him.\n\nIt was necessary to descend the rampart into the moats, which were\ndry; the height is about forty feet. Our lady was the first to\ndescend; she helped her husband, for his strength was already\nfailing. When they were all three in the fosse, the moon was obscured\nand a little rain fell. This was unfortunate, as they could not see\nwhich road to take. Her husband said it would be better to remain\nwhere they were till daylight, for they might break their necks in\ndescending the rocks. The servant said he knew the way, as he had\nobserved it when the window was free; that he would go in front. He\nwent in advance, gliding in a sitting position, after him our lady,\nand then her husband; they could not see an inch before them; the man\nfell from an incredible height, and did not speak; our lady stopped,\nshouted to him, and asked him to answer if he was alive.\n\nHe was some time before he answered, so she and her husband\nconsidered him dead; at length he answered, and said he should never\nget out of this ravine; our lady asked him if he judged the depth to\nbe greater than one of the cords could reach? She would tie two\ntogether, and throw the end to him to draw him up. He said that one\ncord would be sufficient, but that she could not draw him up, that\nshe would not be strong enough; she said she could, she would hold\nfirm, and he should help himself with his knees. He took courage, and\nshe drew him up; the greatest marvel was, that on each side of her\nthere was a precipice deeper than that over which he fell, and that\nshe had nothing by which to support herself, except a small\nprojection, which they believed to be of earth, against which she\nplaced her left foot, finding no resting-place for the right one.\n\nWe can truly say that God had granted her his protection, for to\nescape from such a danger, and draw another out of it, could not have\nbeen done by unaided man. Our fool Fos explained it otherwise, and\nused it for his own purposes, saying that without the assistance of\nthe devil it would have been impossible to stand firm in such a\nplace, still less to assist another; he impressed this so well on the\nQueen, that she is still of the opinion that our lady exercises\nsorcery. Fos would take the glory from God to give it to the devil,\nand this calumny has to be endured with many others. But let us\nreturn to our miserable fugitives, whom we left in the fosse. Our\nlady, who had shouted to her husband not to advance, as soon as she\nheard the valet fall, called to him to keep back, turn quietly, and\nto climb upwards, for that there was no passage there; this was done,\nand they remounted the fosse and kept themselves quiet. Her husband\nwished that they should remain there, since they did not know which\nroad to take.\n\nWhile they were deliberating, the moon shone forth a little, and our\nlady saw where she was, and she remembered a good passage which she\nhad seen on the day when she walked out with the governor; she\npersuaded her husband to follow her; he complained of his want of\nstrength; she told him that God would assist him, and that he did not\nrequire great strength to let himself glide down, that the passage\nwas not difficult, and that in ascending on the opposite side, which\nwas not high, the valet and herself could assist him. He resolved,\nbut he found it difficult enough; at length, however, they succeeded;\nthey had then to go half a quarter of a league to reach the place\nwhere the boats were.\n\nHer husband, wearied out, could not walk, and begged her, for the\nlove of God, to leave him where he was; he was ready to die; she\nconsoled him, and gave him restoratives, and told him that he had but\na little step to make; he begged her to leave him there, and to save\nherself with the servant: she would find means afterwards to rescue\nhim from prison. She said no, she would not abandon him; that he knew\nwell the opportunities she had had to escape before, if she had\nwished to forsake him; that she would never quit him nor leave him in\nthe hands of this tyrant; that if Fos ventured to touch him, she was\nresolved on avenging herself upon him.\n\nAfter having taken a little breath, he began again to proceed. Our\nlady, who was loaded with so many ropes and clothes, could scarcely\nwalk, but necessity gave her strength. She begged her husband to lean\non her and on the valet, so he supported himself between them, and in\nthis way arrived where the boats were; but too late, for it was\nalready day. As our lady saw the patrol coming in the distance, she\nbegged her husband to stop there with the valet, saying that she\nwould go forward in advance, which she did. She was scarcely a\nmusket-shot distant from a little town where the major lodged, when\nshe spoke with the guard, and asked them after the major. One of them\nwent for the major, whose name was Kratz.\n\nThe major saw our lady with great consternation; he asked after her\nhusband. She told him where he was, and in a few words she requested\nthat he would go to the castle and tell Major-General Fos that his\nill-treatment had been the cause of the desperate resolution they had\ntaken, and to beg him not to ill-treat them; they were at present\nsick at heart; they could not endure anything; she begged him to\nconsider that those who had resolved to face more than one form of\ndeath, would not fear it in any shape. Kratz conducted the prisoners\nto his house, mounted his horse, and went in search of the governor,\nwho was still in bed, and told him the affair.\n\nThe governor got out of bed like a furious creature, swore, menaced;\nafter having recovered a little, the major told him what our lady had\nbegged him to say. Then he was for some time thoughtful, and said, 'I\nconfess it; they had reason to seek their liberty, for otherwise\nthey would never have had it.' He did not immediately come for the\nprisoners, for he had another apartment prepared for them. As he\nentered, he assumed a pleasant manner, and asked if they ought to be\nthere; he did not say an unkind word, but, on the contrary, said he\nshould have done the same. They were conducted to the Royal Hall to\nwarm themselves, for they were all wet with the rain; our lady had\nthen an opportunity of speaking to the valet, and of taking from him\nthe papers that he had, which contained all that had passed during\nthe time of their imprisonment,[37] and she counselled the valet to\nlay aside the arms that he had upon him, and that if he had anything\nwhich he wished to secure that he would deliver it up to her keeping.\nThe valet gave her what she asked, followed her orders, threw away\nhis arms, but as regarded his own papers he would not give them up,\nfor he did not share her fears; but he knew afterwards, for Fos\ncaused him to be entirely stripped, and took away everything from\nhim, and made him pay well for having noted down the dishes that they\nhad on the first day of the Festivals, and on the rest.\n\n [37] This account of what happened during their imprisonment at\n Hammershuus, written by Leonora herself, is also mentioned in her\n Record of her prison-life in the Blue Tower. But no copy of it has\n yet come to light. Uhlfeldt's so-called apology contains much\n information on this subject.\n\nAt length towards evening our lady and her husband were conveyed into\nanother apartment, and the valet into the body-guard loaded with\nirons. They were there together thirteen weeks, until Fos received\norders from the Court to separate them; meanwhile, he encased the\nprisons in iron. I may well use such a term, for he caused plates of\niron to be placed on the walls, double bars and irons round the\nwindows.[38] When he had permission to separate them, he entered one\nday to begin a quarrel, and spoke of the past; our lady begged him\nnot to say more, but he would go on; he was determined to quarrel. He\nsaid to her, 'Madame, you are so haughty, I will humble you; I will\nmake you so--so small,' and he made a measurement with his hand from\nthe floor. 'You have been lifted up and I will bring you down.' She\nlaughed, and said, 'You may do with me whatever you will, but you can\nnever humble me so that I shall cease to remember that you were a\nservant of a servant of the King my father;' at last, he so forgot\nhimself as to hold his fist in her face. She said to him, keeping her\nhand on her knife which she had in her pocket, 'Make use of your foul\nmouth and accursed tongue, but keep your hands quiet.' He drew back,\nand made a profound bow in ridicule, calling her 'your grace,' asked\nher pardon, and what he had to fear. She said, 'You have nothing to\nfear; if you take liberties, you will meet with resistance--feeble\nenough, but such as I have strength to give you.'\n\n [38] Fuchs' own report on this subject still exists, and in it he\n estimates the iron employed at three tons.\n\nAfter some further invectives, he said farewell, and begged they\nmight be good friends; he came once more and conducted himself in the\nsame manner, but less violently. He said to a captain who was\npresent, of the name of Bolt, that he did it expressly in order to\nhave a quarrel with her husband, that he might revenge himself for\nher conduct upon him, but that her husband would not speak to him. At\nlength the unhappy day of their separation came, and Fos entered to\ntell them that they must be prepared to bid each other a final\nfarewell, for that he had orders to separate them, and in this life\nthey would never see each other again; he gave them an hour to\nconverse together for the last time. You can easily imagine what\npassed in this hour; but as they had been prepared for this\nseparation weeks before, having been warned of it by their guard with\nwhom they could talk, it did not surprise them. Our lady had gained\nover four of the guards, who were ready to let them escape easily\nenough, but her husband would not undertake it, always saying that he\nhad no strength, but that she might do it. Well, they had to abide by\nit; after this sad day[39] they were separated, he in one prison\nbelow and she in another above, one above another, bars before the\nwindows, he without a servant, and she without a waiting woman.\n\n [39] The precise date was June 15, 1661, but the order for their\n separation is dated already on the 4th of April.\n\nAbout three weeks after, our lady fell ill; she requested a woman or\ngirl to wait upon her, and a priest. Fos sent answer, with regard to\na woman or girl to wait upon her, he did not know anyone who would do\nit, but that there was a wench who had killed her child, and who\nwould soon be beheaded, and if she wished for her, she could have\nher. As to a priest, he had no orders, and she would have no priest\neven if death were on her lips. Our lady said nothing but 'Patience;\nI commend it to God.' Our lady had the happiness of being able to\ngive her husband signs daily, and to receive such, and when the wind\nwas not too strong they could speak to one another. They spoke\nItalian together, and took their opportunity before the reveille.\nTowards the close of the governorship of this villain, he was\ninformed of this. He then had a kind of machine made which is used to\nfrighten the cattle from the corn in the summer, and which makes a\ngreat noise, and he desired the sentinel to move this machine in\norder to hinder them hearing each other.\n\nFifteen days before Count Rantzow came to Borringholm to treat with\nthem, Fos had news of it from Copenhagen from his intimate friend\nJaques P...; he visited our lady, told her on entering that her\nchildren had been expelled from Skaane by the Swedes; our lady said,\n'Well, the world is wide, they will find a place elsewhere.' He then\ntold her that Bolt had come from Copenhagen with the tidings that\nthey would never be let at liberty; she replied, 'Never is a long\ntime; this imprisonment will not last a hundred years, much less an\neternity--in the twinkling of an eye much may change; the hand of\nGod, in whom are the hearts of kings, can change everything.' He\nsaid, 'You have plenty of hope; you think perhaps if the King died,\nyou would be free?' She replied, 'God preserve the King. I believe\nthat he will give me liberty, and no one else.' He chatted about a\ngreat many things, and played the flatterer.\n\nAt length Count Rantzow came and made a stay at Borringh... of eleven\nweeks. He visited the prisoners, and did them the favour of having\nthe husband to dine with him, and in the evening our lady supped with\nhim, and he conferred with them separately. Our lady asked him of\nwhat she was accused; he replied, 'Will you ask that? that is not the\nway to get out of Borringholm; do you know that you have said the\nKing is your brother? and kings do not recognise either sisters or\nbrothers.' She replied, 'To whom had I need to say that the King is\nmy brother? who is so ignorant in Denmark as not to know that? I have\nalways known, and know still, the respect that is due to the King; I\nhave never given him any other title than my King and Lord; I have\nnever called him my brother, in speaking of him; kings are gracious\nenough to recognise their sisters and brothers as such; for example,\nthe King of England gives the title of sister to his brother's wife,\nalthough she is of very mediocre extraction.[40] Rantzow replied,\n'Our King does not wish it, and he does not know yet the truth about\nDina's affair.' She said, 'I think the King does not wish to know.'\nHe replied, 'Indeed, by God he desires with all his heart to be\ninformed of it.' She answered, 'If the King will desire Walter to\ntell him, and this with some earnestness, he will be informed of it.'\nRantzow made no reply.\n\n [40] Leonora alludes to the wife of the then Duke of York,\n afterwards James II., who was the daughter of Lord Edward\n Clarendon.\n\nWhen he had concluded everything with her husband, whom he had\nobliged to yield up all his possessions, Rantzow acquainted our lady\nwith the fact; she said that her husband had power to give up what\nwas his, but that the half belonged to her, and that this she would\nnot give up, not being able to answer for it before God nor before\nher children; she had committed no crime; liberty should be given to\nher husband for the half of their lands, and that if the King thought\nhe could retain her with a good conscience she would endure it.\nRantzow with a serious air replied, 'Do not think that your husband\nwill ever be set at liberty, if you do not sign with him.' She said\nthat the conditions were too severe; that they should do better for\ntheir children to die as prisoners, God and all the world knowing\ntheir innocence, than to leave so many children beggars. Rantzow\nsaid, 'If you die in prison, all your lands and property are\nforfeited, and your children will have nothing; but at this moment\nyou can have your liberty, live with your husband; who knows, the\nKing may still leave you an estate, and may always show you favour,\nwhen he sees that you yield to his will.' Our lady said that since\nthere was no other prospect for her husband's liberty, she would\nconsent. Rantzow ordered her husband and herself separately to place\nin writing the complaints they had to bring forward against Fos, and\nall that had happened with regard to their attempt at escape; which\nwas done. Our lady was gracious in her demeanour to Fos, but her\nhusband could not make up his mind even to speak to him. Rantzow\nreturned to Copenh... and eighteen days afterwards the galley of\nGabel came with orders to the new governor (Lieutenant-Colonel\nLytkens, a very well-bred man and brave soldier, his wife a noble\nlady of the Manteuffel family, very polite and pretty), that he\nshould make the prisoners sign the papers sent, and when the\nsignature was done, should send them on together.\n\nThe governor sent first to the husband, as was befitting, who made\ndifficulties about signing because they had added points here and\nthere, and among other things principally this, that they were never\nto plead against Fos. The husband said he would rather die. The good\ngovernor went in search of the wife and told her everything, begging\nher to speak to her husband from the window; when he knew that she\nhad spoken to him, he would return. She thanked the governor, and\nwhen he had gone out she spoke to her husband, and persuaded him to\nsign. Then the governor made her sign also; and after that, towards\nnine o'clock in the evening, her husband came to her, having been\nseparated just twenty-six weeks.[41] They were separated on a\nSaturday, and they met again on a Saturday. Fos was still at the\ncastle; it is easy to believe that he was in great rage. Time does\nnot permit to dwell on it. Two days afterwards they embarked and came\nto Copenhagen, and were received on the Custom-house pier by C.\nRantzow and Gabel. The Queen knew nothing of it. When she was told of\nit she was so angry that she would not go to table. In a few words\nthe King held his ground, and as she would not accept the thanks of\nMonsieur and his wife, the King ordered her to receive them in\nwriting. They spent the Christmas of 1660 in the house of C. Rantzow.\nAfterwards they went to Fyen, to the estate of Ellensborg, which was\ngraciously left to them.[42]\n\n [41] The apology of Uhlfeldt contains an account of this whole\n transaction. He states that when he asked his wife through the\n window whether they ought to sign and live rather than die in\n prison, which would otherwise be their lot, Leonora answered with\n the following Latin verse:\n\n     Rebus in adversis facile est contemnere mortem,\n     Fortius ille facit, qui miser esse potest.\n     Accidit in puncto, quod non speratur in anno.\n\n [42] Ellensborg was the ancient seat of the Ulfeldt family, which\n had been sold to Ellen Marsvin, Leonora's grandmother, and Leonora\n inherited it from her mother. It is now called Holckenhavn, and the\n seat of Count Holck.\n\nHer husband having permission to go to France to take the waters for\neighteen months, left Ell... with his family in the month of June\n1662, and landed at Amsterdam. Our lady went from thence to Bruges to\nhire a house, and returned to Amsterdam. Her daughter Helena fell ill\nof the small-pox; she remained with her, and her husband and the\nother children went to Bruges. When her daughter had recovered, she\nwent to rejoin her husband and children. She accompanied her husband,\nwho went to France. Having arrived at Paris, the doctors did not\nfind it advisable that he should take the waters, and he returned to\nBruges. Her husband begged our lady to make a journey to England, and\nto take her eldest son with her. She raised obstacles, and showed him\nplainly that she should obtain nothing; that she should only be at\ngreat expense. She had examples before her which showed her that the\nKing of England would never pay her husband. He would not have been\nturned from his purpose at this time but for their son's rencontre\nwith Fos, which prevented the journey that winter, and postponed the\nmisfortunes of our lady, though it did not ultimately prevent them.\n\nBut towards the spring the same design was again brought forward; our\nlady was assisted by the nobleman who followed her afterwards[43] in\ndissuading her husband; but no reasoning could avail; he believed the\nKing could not forget the benefits received, and refuse to pay his\ncousin. Our lady prepared for her departure, since her husband wished\nit. The day that she bade him her last farewell--a fatal day,\nindeed--her husband's heart did not tell him that these would be the\nlast embraces he would give her, for he was so satisfied and so full\nof joy that she and all were astonished. She, on the contrary, was\nsad. The last day of their intercourse was May 24, 1663. She had many\ncontretemps at first, and some time elapsed before she had the honour\nof speaking to the King.\n\n [43] Namely Casetta, a Spanish nobleman, who afterwards married\n their daughter Anna Katherine, but both he and their children died\n soon. (_See_ the Introduction.)\n\nThe King greeted her after the fashion of the country, treated her as\nhis cousin,[44] and promised her all sorts of satisfaction; that he\nwould send his secretary[45] to her to see her papers, which he did.\nThe secretary made her fine promises, but the time was always\npostponed. The minister resident, Petkum, minister of the King of\nDanem..., came to visit her (he had placed some obstacles in the way\nof her demands, from what was told her). She showed him her papers,\ninformed him of the affair, told him that the King of Denmark had had\nall the papers in his hands, and had graciously returned them. The\ntraitor made a semblance of understanding the affair, and promised\nthat he would himself help in securing the payment of her demands.\nBut this Judas always intended to betray her, asking her if she did\nnot like to make excursions, speaking to her of beautiful houses,\ngardens and parks, and offering her his coach. But our lady was not\ninclined to make excursions.\n\n [44] Charles the Second's Grandmother, Anna, the Queen of James I.\n was sister of Leonora Christina's father, Christian IV.\n\n [45] Sir Henry Bennet, afterwards Lord Arlington.\n\nWhen he saw that he could not catch her in this way, he obtained an\norder to arrest her. Our poor lady knew nothing of all this; she had\nletter upon letter from her husband requesting her return. She took\nleave of the King by letter, gave her papers to a lawyer[46] upon a\nreceipt, and set out from London. Having arrived at Dover, and\nintending to embark the same evening for Flanders, a lieutenant of\nthe name of Braten[47] appeared, who came to show her an order from\nthe King of Anglet... which she read herself, the purport of which\nwas that the governor was to arrest such a lady, and to place her in\nthe castle till further orders. She asked the reason why. He said\nthat she had left without permission from the King. She told him\nthat she had taken leave of the King by letter, and had spoken the\nday before her departure with the Prime Minister and Vice-Admiral\nAschew,[48] who had bade her farewell.[49]\n\n [46] A certain Mr. Mowbray.\n\n [47] Elsewhere she writes the name Broughton.\n\n [48] Sir George Askew.\n\n [49] Compare with this account the following extracts in the\n _Calendar of State Papers_, domestic series, 1663, 1664, pp. 196,\n 197, 200:--\n\n 1663--_July 8._--Warrant to Captain Strode, governor of Dover\n Castle, to detain Elionora Christiana, Countess of Uhlfeldt, with\n her husband, if he be found with her, and their servants; to keep\n her close prisoner, and secure all her papers, according to\n instructions to be given by Thos. Parnell.\n\n _July 8._--Warrant to Thos. Parnell to observe the movements of the\n said Countess of Uhlfeldt; to seize her should she attempt to embark\n at Gravesend with her papers, and to detain her close prisoner.\n\n (_July_).--Instructions (by Sec. Bennet) to Thos. Parnell, to go to\n Dover Castle to deliver instructions, and assist in their execution,\n relative to a certain lady (the Countess of Uhlfeldt), who is not to\n be permitted to depart, whether she have a pass or not; but to be\n invited, or if needful compelled, to lodge at the castle, where the\n best accommodation is to be provided for her. It is suspected that\n her husband lies concealed in the kingdom, and will also try to pass\n with his lady, but he also is to be detained, and her servants also.\n\n _July 11._--Thos. Parnell to Williamson. 'Found the Countess (of\n Uhlfeldt) at Dover, and by the aid of the Lieut.-Governor sent the\n searcher to her inn, to demand her pass. She said she had none, not\n knowing it would be wanted. She submitted patiently to be taken to\n the castle, and lodged there till a message was sent to town. The\n Regent's gentleman, the bearer will give an account of all things.'\n\nWhen she came to the castle, the emissary of Petkum presented\nhimself, by name Peter Dreyer. Then the Lieutenant said, 'It is the\nKing of Danemarc who has ordered you to be arrested.' She asked the\ncause. He replied, 'You undoubtedly set out incognito from\nDanemarck.' She replied to this that the King of Danem... had given\nher husband leave of absence for a term of eighteen months, which had\nnot yet expired. They ordered her boxes and those of the nobleman who\naccompanied her to be opened, and they took all the papers.\nAfterwards Dreyer spoke to her, and she asked him why she was\ntreated thus? He said he did not know the real cause, but that he\nbelieved it was for the death of Fos, and that she was believed to\nhave been the cause of his death. They always mentioned this to her,\nand no other cause.\n\nThis double traitor Braten enacted the gallant, entertained her, made\nher speak English (as she was bolder in speaking this language than\nany other), for she had just begun to learn it well, having had a\nlanguage-master in London. One day he told that they intended\nconducting her to Danemarck. She told him there was no need to send\nher to Danem...; she could go there very well by herself. He said,\n'You know yourself what suits you; if you will not go there\nwillingly, I will manage so that you may go to Flanders.' She did not\nsee that this was feasible, even if he was willing; she spoke with\nhim as to the means, saw that he did not satisfy her, and did not\ntrust his conversation; as he was cunning, he made her believe that\nthe King wished her to go secretly, and that he would take it all\nupon himself; that the King had his reasons why he did not wish to\ndeliver her into the hands of the King of Danem....\n\nThis deception had such good colouring, for she had written several\ntimes to the King during her arrest, and had begged him not to reward\nher husband's services by a long arrest, only speaking of what she\nhad done at the Hague for him: she had taken her jewels and rings and\ngiven them to him, when his host would not any longer supply him with\nfood.[50] Her claim was not small; it exceeded 20,000 patacoons.[51]\n\n [50] Several letters written by Leonora during her imprisonment at\n Dover to Charles II., Sir Henry Bennet, &c., are printed in a\n Danish periodical, _Danske Samlinger_, vol. vi.\n\n [51] Reckoning the patacoon to 4s. 8d., this claim would be nearly\n 5,000_l._\n\nOur lady allowed herself to be persuaded that the King of England\nwished her to leave secretly. The traitor Braten told her that he\nthought it best that she should disguise herself as a man. She said\nthat there was no necessity she should disguise herself; that no one\nwould pursue her; and even if it were so, that she would not go in\ndisguise with any man who was not her husband. After having been\ndetained seventeen days at Dover, she allowed herself to be conducted\nby Braten, at night, towards the ramparts, descended by a high ladder\nwhich broke during her descent, passed the fosse, which was not\ndifficult; on the other side there was a horse waiting for her, but\nthe nobleman, her attendant, and the nobleman's valet, went on foot;\nthey would not allow her valet to go with them; Braten made an excuse\nof not being able to find him, and that time pressed; it was because\nthey were afraid that there would be an effort at defence.\n\nWhen she arrived where the traitors were, her guide gave a signal by\nknocking two stones one against another. At this, four armed men\nadvanced; Petkum and Dreyer were a little way off; one held a pistol\nto her breast, the other a sword, and said, 'I take you prisoner.'\nThe other two traitors said, 'We will conduct you to Ostend.' She had\nalways suspected treachery, and had spoken with her companion, in\ncase it happened, what it would be best to do, to give herself up or\nto defend herself? She decided on allowing herself to be betrayed\nwithout a struggle, since she had no reason to fear that her life\nwould be attempted because her son had avenged the wrong done to his\nparents. Thus she made no resistance, begged them not to take so much\ntrouble, that she would go of herself; for two men held her with so\nmuch force that they hurt her arm. They came with a bottle of dry\nwine to quench her thirst, but she would not drink; she had a good\nway to go on foot, for she would not again mount the horse.\n\nShe showed some anger towards her guide, begged him in English to\ngive her respects to the governor,[52] but to convey to the traitor\nBraten all the abuse that she could hurriedly call to mind in this\nlanguage, which was not quite familiar to her. She advanced towards\nthe boat; the vessel which was to convey her was in the roads, near\nthe Downs. She bade farewell to the nobleman. She had two bracelets\nwith diamonds which she wished to give him to convey to her children;\nbut as he feared they would be taken from him, she replaced them\nwithout troubling him with them. She gave a pistol to her servant,\nand a mariner then carried her to the boat; she was placed in an\nEnglish frigate that Petkum had hired, and Dreyer went with her.[53]\nShe was thirteen days on the road, and arrived near the Custom-house\npier on August 8, 1663, at nine o'clock in the morning.\n\n [52] Leonora did not know that the governor of the castle was in\n the plot.\n\n [53] Additional light is thrown on the arrest of Leonora Christina\n at Dover by the following extracts in the _Calendar of State\n Papers_, p. 224, 225:--\n\n _August 1_, _Whitehall_.--(Sec. Bennet) to Capt. Strode. The King is\n satisfied with his account of the lady's escape and his own\n behaviour; continue the same mask, of publishing His Majesty's\n displeasure against all who contributed to it, especially his\n lieutenant, and this more particularly in presence of M. Cassett,\n lest he may suspect connivance. Cassett is to continue prisoner some\n time. The Danish Resident is satisfied with the discretion used, but\n says his point would not have been secured had the lady gone to sea\n without interruption.\n\n _August 1_?--Account (proposed to be sent to the Gazette?) relative\n to Count Uhlfeldt--recording his submission in 1661, the present\n sentence against him, his further relapse into crime after a solemn\n recantation, also signed by his wife who was his accomplice, though\n her blood saved her from sharing his sentence, but who has now\n betrayed herself into the hands of the King of Denmark. She was in\n England when the conspiracy against the King of Denmark's life was\n detected. The King of England had her movements watched, when she\n suddenly went off without a pass, for want of which she was stayed\n by the Governor of Dover Castle, who accommodated her in the castle.\n The Resident of Denmark posted to Dover, and secured the master of a\n ship then in the road, with whom he expected her to tamper, which\n she did, escaped through the castle window, and entering a shallop\n to go on board, was seized and conveyed to Denmark. With note (by\n Lord Chancellor Clarendon) that he is not satisfied with this\n account, but will prepare a better for another week.\n\n[The remaining part of the Autobiography treats of the commencement\nof her imprisonment in the Blue Tower, which forms the subject of the\nfollowing Memoir.]\n\n\n\n\n A RECORD\n OF\n THE SUFFERINGS OF THE IMPRISONED COUNTESS\n LEONORA CHRISTINA.\n\n\n\n\nPREFACE.\n\n_TO MY CHILDREN._\n\n\nBeloved children, I may indeed say with Job, 'Oh, that my grief were\nthoroughly weighed, and my calamity laid in the balances together!\nFor now it would be heavier than the sand of the sea.' My sufferings\nare indeed great and many; they are heavy and innumerable. My mind\nhas long been uncertain with regard to this history of my sufferings,\nas I could not decide whether I ought not rather to endeavour to\nforget them than to bear them in memory. At length, however, certain\nreasons have induced me, not only to preserve my sorrow in my own\nmemory, but to compose a record of it, and to direct it to you, my\ndear children.[54]\n\n [54] In the margin is added: 'As I now hope that what I write may\n come into your hands, my captivity during the last three years also\n having been much lightened.'\n\nThe first of these reasons is the remembrance of the omnipotence of\nGod; for I cannot recall to mind my sorrow and grief, my fears and\ndistresses, without at the same time remembering the almighty power\nof God, who in all my sufferings, my misery, my affliction, and\nanxiety, has been my strength and help, my consolation and\nassistance; for never has God laid a burden upon me, without at the\nsame time giving me strength in proportion, so that the burden,\nthough it has weighed me down and heavily oppressed me, has not\noverwhelmed me and crushed me; for which I praise and extol through\neternity the almighty power of the incomprehensible God.\n\nI wish, therefore, not alone to record my troubles and to thank God\nfor His gracious support in all the misfortunes that have befallen\nme, but also to declare to you, my dear children, God's goodness to\nme, that you may not only admire with me the inconceivable help of\nthe Almighty, but that you may be able to join with me in rendering\nHim thanks. For you may say with reason that God has dealt\nwonderfully with me; that He was mighty in my weakness and has shown\nHis power in me, the frailest of His instruments. For how would it\nhave been possible for me to resist such great, sudden, and\nunexpected misfortunes, had not His spirit imparted to me strength?\nIt was God who Himself entered with me into the Tower-gate; it was He\nwho extended to me His hand, and wrestled for me in that prison cell\nfor malefactors, which is called 'the Dark Church.'\n\nSince then, now for almost eleven years, He has always been within\nthe gate of my prison as well as of my heart; He has strengthened me,\ncomforted me, refreshed me, and often even cheered me. God has done\nwonderful things in me, for it is more than inconceivable that I\nshould have been able to survive the great misfortunes that have\nbefallen me, and at the same time should have retained my reason,\nsense, and understanding. It is a matter of the greatest wonder that\nmy limbs are not distorted and contracted from lying and sitting,\nthat my eyes are not dim and even wholly blind from weeping, and from\nsmoke and soot; that I am not short-breathed from candle smoke and\nexhalation, from stench and close air. To God alone be the honour!\n\nThe other cause that impels me is the consolation it will be to you,\nmy dear children, to be assured through this account of my sufferings\nthat I suffer innocently; that nothing whatever has been imputed to\nme, nor have I been accused of anything for which you, my dear\nchildren, should blush or cast down your eyes in shame. I suffer for\nhaving loved a virtuous lord and husband, and for not having\nabandoned him in misfortune. I was suspected of being privy to an act\nof treason for which he has never been prosecuted according to law,\nmuch less convicted of it, and the cause of the accusation was never\nexplained to me, humbly and sorrowfully as I desired that it should\nbe. Let it be your consolation, my dear children, that I have a\ngracious God, a good conscience, and can boldly maintain that I have\nnever committed a dishonourable act. 'This is thankworthy,' says the\napostle St. Peter, 'if a man for conscience toward God endure grief,\nsuffering wrongfully.' I suffer, thank God, not for my misdeeds, for\nthat were no glory to me; yet I can boast that from my youth up I\nhave been a bearer of the cross of Christ, and had incredibly secret\nsufferings, which were very heavy to endure at such an early age.\n\nAlthough this record of my sufferings contains and reveals nothing\nmore than what has occurred to me in this prison, where I have now\nbeen for eleven years, I must not neglect in this preface briefly to\nrecall to your minds, my dear children, my earlier misfortunes,\nthanking God at the same time that I have overcome them.\n\nNot only you, my dear children, know, but it is known throughout the\nwhole country, what great sorrow and misfortune Dina and Walter, with\ntheir powerful adherents, inflicted on our house in the year 1651.\n\nAlthough I will not mention the many fatiguing and difficult\njourneys, the perils by sea, and various dangers which I have endured\nin foreign countries, I will only remind you of that journey which my\nlord requested me to undertake to Denmark, contrary to my wish, in\nthe year 1657.[E01] It was winter time, and therefore difficult and\ndangerous. I endured scorn and persecution; and had not God given me\ncourage and taken it from him who was to have arrested me, I should\nnot at that time have escaped the misery of captivity.\n\n [E01] This journey really took place in November and December,\n 1656.\n\nYou will remember, my dear children, what I suffered and endured\nduring fourteen months in custody at Malmoe; how the greatest favour\nwhich His Majesty, King Charles X. of Sweden, at that time showed me,\nwas that he left it to my free will, either to remain at liberty,\ntaking care of our property, or to be in prison with my lord. I\nacknowledged the favour, and chose the latter as my duty, esteeming\nit a happiness to be allowed to console and to serve my anxious\nhusband, afflicted as he subsequently was by illness. I accepted it\nalso as a favour that I was allowed (when my lord could not do it\nhimself on account of illness) to appear before the tribunal in his\nstead. What anxiety and sorrow I had for my sick lord, what trouble,\nannoyance and distress, the trial caused me (it was carried on daily\nfor more than nine weeks), is known to the most high God, who was my\nconsolation, assistance, and strength, and who inspired me with\nheart and courage to defend the honour of my lord in the presence of\nhis judges.\n\nYou will probably not have forgotten how quickly one misfortune\nfollowed another, how one sorrow was scarcely past when a greater one\nfollowed in its track; we fared, according to the words of the poet:\n\n    Incidit in Scyllam, qui vult vitare Charibdin.\n\nWe escaped custody and then fell into strict captivity, without doubt\nby the dispensation of God, who inspired my lord with the idea of\nrepairing, contrary to our agreement, to Copenhagen instead of\nLuebeck. No pen can describe how sorrowful I was when, contrary to all\nexpectation, I met my lord in Copenhagen, when I had imagined him\nescaped from the power and violence of all his enemies. I expected\njust that which my lord did not believe would happen, but which\nfollowed immediately--namely, our arrest. The second day after my\narrival (which they had waited for) we were apprehended and conveyed\nto Bornholm, where we were in close imprisonment for seventeen\nmonths. I have given a full description of what I suffered, and this\nI imagine is in your keeping, my dear children; and from it you see\nwhat I and my sick lord endured; how often I warded off greater\nmisery, because my lord could not always brook patiently the bad\ntreatment of the governor, Adolf Foss, who called himself Fux.\n\nIt was hard and bitter indeed to be scorned and scoffed at by a\npeasant's son; to have to suffer hunger at his will, and to be\nthreatened and harassed by him; but still harder and more bitter was\nit to be sick beneath his power, and to hear from him the words that\neven if death were on my lips no minister of God's word should come\nto me. Oh monstrous tyranny! His malice was so thoroughly beyond all\nbounds, that he could not endure that we should lighten each other's\ncross; and for this reason he contrived, after the lapse of eleven\nmonths, to have us separated from each other, and to place us each in\nthe hardest confinement.\n\nMy husband (at that time already advancing in years) without a\nservant, and I without an attendant, was only allowed a light so long\nas the evening meal lasted. I cannot forbear bitterly recalling to\nmind the six months of long and hard separation, and the sad farewell\nwhich we took of each other; for to all human sight there was no\nother prospect than that which the governor announced to us--namely,\nthat we were seeing and speaking with each other for the last time in\nthis world. God knows best how hard our sufferings were, for it was\nHe who consoled us, who gave us hope contrary to all expectation, and\nwho inspired me with courage when the governor visited me and\nendeavoured to fill me with despair.\n\nGod confirmed my hope. Money and property loosened the bonds of our\ncaptivity, and we were allowed to see and speak with each other once\nmore. Sad as my lord had been when we were separated at Borringholm,\nhe was joyous when two years afterwards he persuaded me to undertake\nthe English journey, not imagining that this was to part us for ever.\nMy lord, who entertained too good an opinion of the King of England,\nthought that now that he had come to the throne he would remember not\nonly his great written and spoken promises, but that he would also\nbear in mind how, at the time of his need and exile, I had drawn the\nrings from my fingers and had pawned them for meals for him and his\nservants. But how unwillingly I undertook this journey is well known\nto some of you, my dear children, as I was well aware that from an\nungrateful person there is nothing else to be expected but\ningratitude. I had the example of others by whom to take warning; but\nit was thus destined to be.\n\nBitter bread was in store for me, and bitter gall was to fill my cup\nin the Blue Tower of Copenhagen Castle; thither was I to go to eat it\nand drink it out. It is not unknown to you how falsely the King of\nEngland acted towards me; how well he received me on my arrival; how\nhe welcomed me with a Judas kiss and addressed me as his cousin; and\nhow both he himself and all his high ministers assured me of the\nroyal favour, and promised me payment of the money advanced. You know\nhow cunningly (at the desire of His Majesty the King of Denmark) he\nhad me arrested at Dover, and subsequently sent me word through the\ntraitor Lieutenant Braten that he would let me escape secretly, at\nthe same time delivering me into the hand of the Danish Minister\nSimon Petcon, who had me arrested by eight armed men; keeping aloof,\nhowever, himself, and never venturing to come near me. They held\nsword and pistol to my breast, and two of them took me between them\nand placed me in a boat, which conveyed me to a vessel held in\nreadiness by the said Minister; a man of the name of Peter Dreyer\nhaving received orders to conduct me to Copenhagen.\n\nFrom this period this record of my suffering begins. It contains all\nthat happened to me within the gates of the Blue Tower. Reflect, my\ndear children, on these hard sufferings; but remember also God's\ngreat goodness towards me. Verily, He has freed me from six\ncalamities; rest assured that He will not leave me to perish in the\nseventh. No! for the honour of His name, He will mightily deliver me.\n\nThe narrative of my sufferings is sad to hear, and must move the\nhardest heart to pity; yet in reading it, do not be more saddened\nthan can be counterbalanced by joy. Consider my innocence, courage,\nand patience; rejoice over these.\n\nI have passed over various petty vexations and many daily annoyances\nfor the sake of brevity, although the smallest of them rankled sore\nin the wounds of my bitter sorrow.\n\nI acknowledge my weaknesses, and do not shrink from confessing them\nto you. I am a human being, and am full of human imperfections. Our\nfirst emotions are not under our own power; we are often overhasty\nbefore we are able to reflect. God knows that I have often made\nmyself deaf and blind, in order not to be carried away by passion. I\nam ashamed to mention and to enumerate the unchaste language, bad\nwords and coarse invectives, of the prison governor Johan Jaeger, of\nKresten Maansen, the tower warder, of Karen the daughter of Ole, and\nof Catharina Wolff; they would offend courtly ears. Yet I can assure\nyou they surpass everything that can be imagined as indecent, ugly,\nchurlish and unbecoming; for coarse words and foul language were the\ntokens of their friendliness and clemency, and disgusting oaths were\nthe ornament and embellishment of their untruthfulness; so that their\nintercourse was most disagreeable to me. I was never more glad than\nwhen the gates were closed between me and those who were to guard me.\nThen I had only the woman alone, whom I brought to silence,\nsometimes amicably, and at others angrily and with threats.\n\nI have also had, and have still, pleasant intercourse with persons\nwhose services and courtesies I shall remember as long as I live.\nYou, my dear children, will also repay them to every one as far as\nyou are able.\n\nYou will find also in this record of my sufferings two of the chief\nfoes of our house, namely Jorgen Walter and Jorgen Skroder,[E02] with\nregard to whom God has revenged me, and decreed that they should have\nneed of me, and that I should comfort them. Walter gives me cause to\nstate more respecting him than was my intention.\n\n [E02] This man was a German by birth, but settled in Denmark, where\n he was nobilitated under the name of Loevenklau. His bad conduct\n obliged him to leave the country, and he went to Sweden, where he\n had lived before he came to Denmark, and where Ulfeldt, then in\n Sweden, procured him an appointment as a colonel in the army. This\n kindness he repaid by informing the Danish Government against\n Ulfeldt in 1654, in consequence of which he was not only allowed to\n return to Denmark, but even obtained a lucrative office in Norway.\n Here he quarrelled with the viceroy, Niels Trolle, and tried to\n serve him as he had served Ulfeldt; but he failed to establish his\n accusations against Trolle, and was condemned into the forfeiture\n of his office and of his patent of nobility. He then left Denmark\n at least for a season, and how he came to apply to Leonora\n Christina for assistance is not known, as she has omitted to\n mention it in the Memoir itself, though she evidently intended to\n do so.\n\nOf the psalms and hymns which I have composed and translated, I only\ninsert a few, in order that you, my dear children, may see and know\nhow I have ever clung steadfastly to God, who has been and still is\nmy wall of defence against every attack, and my refuge in every kind\nof misfortune and adversity. Do not regard the rhymes; they are not\naccording to the rules which poets make; but regard the matter, the\nsense, and the purport. Nor have I left my other small pastime\nunmentioned, for you may perceive the repose of my mind from the fact\nthat I have had no unemployed hours; even a rat, a creature so\nabominable to others, affording me amusement.\n\nI have recorded two observations, which though they treat of small\nand contemptible animals, yet are remarkable, and I doubt whether any\nnaturalist hitherto has observed them. For I do not think it has been\nrecorded hitherto that there exists a kind of caterpillar which\nbrings forth small living grubs like itself, nor either that a flea\ngives birth to a fully-formed flea, and not that a nit comes from a\nnit.[55]\n\n [55] A pen has afterwards been drawn through this paragraph, but\n the observations occur in the manuscript.\n\nIn conclusion, I beg you, my dear children, not to let it astonish\nyou that I would not avail myself of the opportunity by which I might\nhave gained my freedom. If you rightly consider it, it would not have\nbeen expedient either for you or me. I confess that if my deceased\nlord had been alive, I should not only have accepted the proposal,\nbut I should have done my utmost to have escaped from my captivity,\nin order to go in quest of him, and to wait on him and serve him till\nhis last breath; my duty would have required this. But since he was\nat that time in rest and peace with God, and needed no longer any\nhuman service, I have with reason felt that self-obtained liberty\nwould have been in every respect more prejudicial than useful to us,\nand that this would not be the way to gain the possessions taken from\nus, for which reason I refused it and endeavoured instead to seek\nrepose of mind and to bear patiently the cross laid upon me. If God\nso ordains it, and it is His divine will that through royal mercy I\nshould obtain my freedom, I will joyfully exert myself for you, my\nbeloved children, to the utmost of my ability, and prove in deed that\nI have never deviated from my duty, and that I am no less a good and\nright-minded mother than I have been a faithful wife. Meanwhile let\nGod's will be your will. He will turn and govern all things so that\nthey may benefit you and me in soul and body, to whose safe keeping I\nconfidently recommend you all, praying that He will be your father\nand mother, your counsellor and guide. Pray in return for me, that\nGod may direct me by His good spirit, and grant me patience in the\nfuture as heretofore. This is all that is requested from you by,\n\nMy dearly beloved children, your affectionate mother,\n\n                                    LEONORA CHRISTINA, V.E.G.\n\nWritten in the Blue Tower, anno 1674, the 18th of July, the eleventh\nyear of imprisonment, my birthday, and fifty-third year of my\nage.[56]\n\n [56] The conclusion of the Preface, from the words 'Meanwhile let\n the will of God,' etc. has afterwards been erased, when the\n manuscript was continued beyond the date assigned in the Preface;\n and the following paragraphs, 'I bear also in mind,' etc. were\n intended to form a new conclusion, but do not seem to have been\n properly worked in.\n\n       *       *       *       *       *\n\nI bear also in mind, with the greatest humility and gratitude, our\ngracious hereditary King's favour towards me, immediately after His\nMajesty came to the throne. I remember also the sympathy of our most\ngracious Queen Regent, and of Her Highness the Electoral Princess of\nSaxony in my unfortunate fate; also the special favour of Her Majesty\nthe Queen.\n\nI have also not forgotten to bear duly in mind the favour shown\ntowards me by Her Majesty the Queen Mother, the virtuous Landgravine\nof Hesse.\n\nI have also recorded various things which occurred in my imprisonment\nduring the period from the year 1663 to the year 1674, intending with\nthese to conclude the record of my sufferings; as I experienced a\npleasure, and often consoled myself, in feeling that it is better to\nremain innocently in captivity than to be free and to have deserved\nimprisonment. I remember having read that captivity has served many\nas a protection from greater dangers, and has guarded them from\nfalling into the hands of their enemies. There have been some who\nhave escaped from their prison and immediately after have been\nmurdered. There have also been some who have had a competence in\nprison and afterwards have suffered want in freedom. Innocent\nimprisonment does not diminish honour, but rather increases it. Many\na one has acquired great learning in captivity, and has gained a\nknowledge of things which he could not master before. Yes,\nimprisonment leads to heaven. I have often said to myself: 'Comfort\nthyself, thou captive one, thou art happy.'\n\nSince the year 1674 constituted only half the period of my captivity,\nI have added in this record of my sufferings some facts that occurred\nsince that time within my prison-gates. I am on the eve of my\nliberty, May 19, 1685. To God alone be the honour, who has moved His\nRoyal Majesty to justice! I will here mention those of whose death I\nhave been informed during my captivity.\n\n1. The Prime Minister of His Majesty, Count Christian of\nRantzow[E03], died in the month of September, 1663. He did not live\nto drink the health of our Princess and of the Electoral Prince of\nSaxony at the feast of their betrothal. Still less did he live long\nenough to see a wooden effigy quartered in mockery of my lord,\naccording to his suggestion. Death was very bitter to him.\n\n [E03] This Count Rantzow was the same who had negotiated the\n compromise with Ulfeldt and Leonora at Bornholm in 1661, and in\n fact brought it about. It was currently reported in Copenhagen at\n the time that he had received a large sum of money from Ulfeldt on\n that occasion, and he afterwards showed his friendly disposition\n towards him by promising him to intercede with the King for\n Christian Ulfeldt when the latter had killed Fuchs. Leonora,\n however, speaks of him as an enemy probably because he presided in\n the High Court of Appeal which condemned Ulfeldt as a traitor. But\n the facts of the case left him scarcely any other alternative than\n that of judging as he did, nor would it have been surprising if\n Ulfeldt's last conduct had altered Rantzow's feelings towards him.\n Rantzow also presided in the commission which examined Leonora in\n the Blue Tower.\n\n2. The Mistress of the Robes of the Queen Dowager, who was so severe\non me in my greatest sorrow, had a long and painful illness; she said\nwith impatience that the pain of hell was not greater than her pain.\nHer screams could often be heard in the tower. She was carried on a\nbed into the town, and died there.\n\n3. The death of Able Catherine was very painful. As she had formerly\nsought for letters on the private parts of my person, so she was\nafterwards herself handled by the surgeons, as she had boils all over\nher. She was cut and burnt. She endured all this pain, hoping to\nlive, but neither the art of the surgeons nor the visits of the Queen\ncould save her from death.[E04]\n\n [E04] Abel Catharina is mentioned in the Memoir itself as the\n person who searched Leonora when she first entered her prison, and\n did so in a very unbecoming manner; she acted, however, under the\n orders of the Mistress of the Robes, M. v. Haxthausen. Abel\n Catharina is otherwise chiefly known as the founder of a charity\n for old women in Copenhagen, which still bears her name.\n\n4. Secretary Erich Krag, who had displayed the malice of his heart in\nmy imprisonment in the 'Dark Church,' was snatched away by death in a\nplace of impurity. He was lively and well, had invited guests to\ndinner, sat and wrote at his table, went out to obey the necessities\nof nature, and was found dead by his attendants when they had waited\nsome time for him.\n\n5. Major-General Fridrich von Anfeldte,[E05] who had more than once\nmanifested his delight at my misfortunes, died as he had lived. He\nwas a godless man and a blasphemer. He fell a victim to jealousy, and\nwent mad, because another obtained an honorary title which he had\ncoveted; this was indeed little enough to deprive him of sense and\nreason. He would hear nothing of God, nor would he be reconciled with\nGod. Both Queens, the Queen Dowager and the Queen Regent, persuaded\nhim at length to be so. When he had received the sacrament, he said,\n'Now your Majesties have had your desire; but what is the good of\nit?' He continued to curse and to swear, and so died.\n\n [E05] This name is mis-spelt for Ahlefeldt. This officer received\n Leonora on her arrival at Copenhagen, as she relates herself. He\n had distinguished himself in the siege of Copenhagen in 1659, and\n died as a Lieutenant-General.\n\n6. General Schak died after a long illness.\n\n7. Chancellor Peter Retz likewise.\n\n8. His Royal Majesty King Friedrich III.'s death accelerated the\ndeath of the Stadtholder Cristoffer Gabel. He felt that the hate of\nthe Queen Dowager could injure him greatly, and he desired death. God\nheard him.[E06]\n\n [E06] Christoffer Gabel is mentioned several times in the\n Autobiography. He was an influential man at the time, in great\n favour at court, and he had a great part in effecting the release\n of Ulfeldt from the prison at Bornholm, for which he, according to\n Leonora's statement, received 5,000 dollars from Ulfeldt. Both he\n and Reedtz were members of the court which condemned Ulfeldt.\n\n9. It has pleased God that I should be myself a witness of Walter's\nmiserable death; indeed, that I should compassionate him. When I\nheard him scream, former times came to my mind, and I often thought\nhow a man can allow himself to be led to do evil to those from whom\nhe had only received kindness and honour.\n\n10. Magister Buch, my father-confessor, who acted so ill to me,\nsuffered much pain on his bed of languishing. He was three days\nspeechless before he died.\n\n11. When the rogue and blasphemer, Christian, who caused me so much\nannoyance in my captivity, had regained his liberty and returned to\nhis landlord, Maans Armfeld in Jutland, he came into dispute with the\nparish priest, who wanted him to do public penance for having seduced\na woman. The rogue set fire to the parsonage; the minister's wife was\nburnt to death in trying to save some of her property, and all the\nminister's possessions were left in ashes. The minister would not\nbring the rogue to justice. He commended him to the true Judge, and\nleft vengeance to Him. The incendiary's conscience began to be\nawakened; for a long time he lived in dread, and was frightened if he\nsaw anyone coming at all quickly, and he would call out and say\ntremblingly, 'Now they are going to take me!' and would run hither\nand thither, not knowing where to go. At length he was found dead on\nthe field, having shot himself; for a long rifle was found lying\nbetween his legs, the barrel towards his breast, and a long ramrod in\nhis hand, with which he had touched the trigger. He did not,\ntherefore, die in as Christian a manner as if he had perished under\nthe hand of the executioner, of which he had so lightly said that he\nshould not care for it at all, so long as he could bring someone else\ninto trouble.\n\n\n\n\nA RECORD OF SUFFERING;\n\n_OR, A REMINISCENCE OF ALL THAT OCCURRED TO ME, LEONORA CHRISTINA, IN\nTHE BLUE TOWER, FROM AUGUST 8 OF THE YEAR 1663, TO JUNE 11[57] OF THE\nYEAR 1674._\n\n [57] Afterwards altered to anno 1685, the 19th of May.\n\n\nThe past is rarely remembered without sorrow, for it has been either\nbetter or worse than the present. If it was more joyous, more happy,\nand full of honour, its remembrance justly saddens us, and in\nproportion as the present is full of care, unhappiness, and\ndishonour. If past times were sadder, more miserable, and more\ndeplorable than the present, the remembrance of them is equally\nsorrowful, for we recover and feel once more all the past misfortunes\nand adversities which have been endured in the course of time. But\nall things have, as it were, two handles by which they may be raised,\nas Epictetus says. The one handle, he says, is bearable; the other is\nnot bearable; and it rests with our will which handle we grasp, the\nbearable or the unbearable one. If we grasp the bearable one, we can\nrecall all that is transitory, however sad and painful it may have\nbeen, rather with joy than with sorrow.[E07] So I will seize the\nbearable handle, and in the name of Jesus I will pass rapidly through\nmy memory, and recount all the wretchedness and misery, all the\ngrief, scorn and suffering, contempt and adversity, which have\nbefallen me in this place, and which I have overcome with God's help.\nI will, moreover, in no wise grieve over it; but, on the contrary, I\nwill remind myself at every step of the goodness of God, and will\nthank the Most High who has been constantly near me with His mighty\nhelp and consolation; who has ruled my heart, that it should not\ndepart from God; who has preserved my mind and my reason, that it has\nnot become obscured; who has maintained my limbs in their power and\nnatural strength, and even has given, and still gives me, repose of\nmind and joyfulness. To Thee, incomprehensible God, be honour and\npraise for ever!\n\n [E07] The passage alluded to occurs in Epictet's Encheiridion,\n chap. 43 (in some editions chap. 65), where he says: 'Every matter\n has two handles, one by which it may be carried (or endured), the\n other by which it cannot be carried (or endured). If thy brother\n has done thee injury, do not lay hold of this matter from the fact\n that he has done thee an injury, for this is the handle by which it\n cannot be carried (or endured); but rather from this side: that he\n is thy brother, educated with thee; and thou wilt lay hold of the\n matter from that side from which it may be managed.' It is easily\n seen how Leonora makes use of the double meaning of the Greek word\n {phoretos}, which is equally well used of an object which can be\n carried in the literal physical sense, and of a matter which can be\n endured or borne with.\n\n {Illustration:\n  DAS ALTE SCHLOSS IN COPENHAGEN MIT DEM BLAUEN THURM.\n  THE OLD CASTLE OF COPENHAGEN.\n  SHOWING THE BLUE TOWER IN THE MIDDLE OF THE BACK-GROUND.}\n\nAnd now to proceed with my design. I consider it necessary to begin\nthe record of my sufferings with the commencement of the day which\nconcluded with the fatal evening of my captivity, and to mention\nsomewhat of that which befell me on the vessel. After the captain had\ncast anchor a little outside the pier of St. Anna, on August 8, 1663,\nat nine o'clock in the forenoon, he was sent on shore with letters by\nPeter Dreyer, who was commissioned by Petcon, at that time the\nminister resident in England, of his Majesty the King of Denmark, to\ntake charge of me. I dressed myself and sat down in one of the cabins\nof the sailors on the deck, with a firm resolution to meet\ncourageously all that lay before me;[58] yet I in no wise expected\nwhat happened; for although I had a good conscience, and had nothing\nevil with which to reproach myself, I had at various times asked the\nbefore-mentioned Peter Dreyer the reason why I had been thus brought\naway. To this question he always gave me the reply which the traitor\nBraten had given me at Dover (when I asked of him the cause of my\narrest); namely, that I was, perhaps, charged with the death of\nMajor-General Fux, and, that it was thought I had persuaded my son\nto slay him; saying, that he knew of no other cause. At twelve\no'clock Nils Rosenkrantz, at that time Lieutenant-Colonel, and Major\nSteen Anderson Bilde, came on board with some musketeers.\nLieutenant-Colonel Rosenkrantz did not salute me. The Major walked up\nand down and presently passed near me. I asked him, en passant, what\nwas the matter? He gave me no other answer than, 'Bonne mine, mauvais\njeu;' which left me just as wise as before. About one o'clock Captain\nBendix Alfeldt came on board with several more musketeers, and after\nhe had talked some time with Peter Dreyer, Dreyer came to me and\nsaid, 'It is ordered that you should go into the cabin.' I said,\n'Willingly;' and immediately went. Soon after, Captain Alfeldt came\nin to me, and said he had orders to take from me my letters, my gold,\nsilver, money, and my knife. I replied, 'Willingly.' I took off my\nbracelets and rings, gathered in a heap all my gold, silver, and\nmoney, and gave it to him. I had nothing written with me, except\ncopies of the letters which I had addressed to the King of England,\nnotes respecting one thing or another relating to my journey, and\nsome English vocabularies; these I also gave up to him. All these\nAlfeldt placed in a silver utensil which I had with me, sealed it in\nmy presence, and left the vessel with it. An hour, or somewhat more,\nafterwards, Major-General Friderich von Anfeldt,[59] Commandant in\nCopenhagen, arrived, and desired that I should come to him outside\nthe cabin. I obeyed immediately. He greeted me, gave me his hand, and\npaid me many compliments, always speaking French. He was pleased to\nsee me in health, he feared the sea might have inconvenienced me; I\nmust not allow the time to seem long to me; I should soon be\naccommodated otherwise. I caught at the last word and said, smiling,\n'Monsieur says otherwise, but not better.' 'Yes, indeed,' he replied,\n'you shall be well accommodated; the noblest in the kingdom will\nvisit you.' I understood well what he meant by this, but I answered:\n'I am accustomed to the society of great people, therefore that will\nnot appear strange to me.' Upon this, he called a servant and asked\nfor the before-mentioned silver utensil (which Captain Alfeldt had\ntaken away with him). The paper which Captain Alfeldt had sealed over\nit was torn off. The Major-General turned to me, and said: 'Here you\nhave your jewels, your gold, silver, and money back; Captain Alfeldt\nmade a mistake--they were only letters which he had orders to demand,\nand these only have been taken out, and have been left at the Castle;\nyou may dispose of the rest as you wish yourself.' 'In God's name,' I\nanswered, 'am I, therefore, at liberty to put on again my bracelets\nand rings?' 'O Jesus,' he said, 'they are yours; you may dispose of\nthem as you choose.' I put on the bracelets and rings, and gave the\nrest to my attendant. The Major-General's delight not only appeared\nin his countenance, but he was full of laughter, and was overflowing\nwith merriment. Among other things he said that he had had the\nhonour of making the acquaintance of two of my sons; that he had been\nin their society in Holland; and he praised them warmly. I\ncomplimented him in return, as was proper, and I behaved as if I\nbelieved that he was speaking in good faith. He indulged in various\njokes, especially with my attendant; said that she was pretty, and\nthat he wondered I could venture to keep such a pretty maiden; when\nHolstein ladies kept pretty maids it was only to put their husbands\nin good humour; he held a long discourse on how they managed, with\nother unmannerly jests which he carried on with my attendant. I\nanswered nothing else than that he probably spoke from experience. He\nsaid all kinds of foolish jokes to my servant, but she did not answer\na word. Afterwards the prison governor told me that he (von Anfeldt)\nhad made the King believe, at first, that my attendant was my\ndaughter, and that the King had been long of that opinion. At length,\nafter a long conversation, the Major-General took his leave, saying\nthat I must not allow the time to seem long to me; that he should\nsoon come again; and he asked what he should say to his Majesty the\nKing. I begged him to recommend me in the best manner to their\nMajesties' favour, adding that I knew not well what to say or for\nwhat to make request, as I was ignorant of what intentions they had\nwith regard to me. Towards three o'clock Major-General von Anfeldt\nreturned; he was full of laughter and merriment, and begged me to\nexcuse him for being so long away. He hoped the time had not appeared\nlong to me; I should soon get to rest; he knew well that the people\n(with this he pointed to the musketeers, who stood all along both\nsides of the vessel) were noisy, and inconvenienced me, and that\nrest would be best for me. I answered that the people did not\ninconvenience me at all; still I should be glad of rest, since I had\nbeen at sea for thirteen days, with rather bad weather. He went on\nwith his compliments, and said that when I came into the town his\nwife would do herself the honour of waiting on me, and, 'as it seems\nto me,' he continued, 'that you have not much luggage with you, and\nperhaps, not the clothes necessary, she will procure for you whatever\nyou require.' I thanked him, and said that the honour was on my side\nif his wife visited me, but that my luggage was as much as I required\nat the time; that if I needed anything in the future, I hoped she\nmight be spared this trouble; that I had not the honour of knowing\nher, but I begged him, nevertheless, to offer her my respects. He\nfound various subjects of discourse upon Birgitte Speckhans[E08] and\nother trifles, to pass away the time; but it is not worth the trouble\nto recall them to mind, and still less to write them down. At last a\nmessage came that he was to conduct me from the vessel, when he said\nto me with politeness: 'Will it please you, madame, to get into this\nboat, which is lying off the side of the ship?' I answered, 'I am\npleased to do anything that I must do, and that is commanded by His\nMajesty the King.' The Major-General went first into the boat, and\nheld out his hand to me; the Lieutenant-Colonel Rosenkrantz, Captain\nAlfeldt, Peter Dreyer, and my attendant, went with me in the boat.\nAnd as a great crowd of people had assembled to look at the\nspectacle, and many had even gone in boats in order to see me as they\nwished, he never took his eyes off me; and when he saw that I turned\nsometimes to one side and sometimes to another, in order to give them\nthis pleasure, he said, 'The people are delighted.' I saw no one\ntruly who gave any signs of joy, except himself, so I answered, 'He\nwho rejoices to-day, cannot know that he may not weep to-morrow; yet\nI see, that, whether for joy or sorrow, the people are assembling in\ncrowds, and many are gazing with amazement at one human being.' When\nwe were advanced a little further, I saw the well-known wicked\nBirgitte Ulfeldt,[E09] who exhibited great delight. She was seated in\nan open carriage; behind her was a young man, looking like a student.\nShe was driving along the shore. When I turned to that side, she was\nin the carriage and laughed with all her might, so that it sounded\nloudly. I looked at her for some time, and felt ashamed of her\nimpudence, and at the disgrace which she was bringing on herself; but\nfor the rest, this conduct did not trouble me more than the barking\nof the dogs, for I esteemed both equally.[60] The Major-General went\non talking incessantly, and never turned his eyes from me; for he\nfeared (as he afterwards said) that I should throw myself into the\nwater. (He judged me by himself; he could not endure the change of\nfortune, as his end testified, for it was only on account of an\nhonorary title which another received in his stead that he lost his\nmind. He did not know that I was governed by another spirit than he,\nwhich gave me strength and courage, whilst the spirit he served led\nhim into despair.[61]) When the boat arrived at the small pier near\nthe office of the Exchequer, Captain Alfeldt landed and gave me his\nhand, and conducted me up towards the castle bridge. Regiments of\nhorse and foot were drawn up in the open place outside the castle;\nmusketeers were standing on both sides as I walked forwards. On the\ncastle bridge stood Jockum Walburger, the prison governor, who went\nbefore me; and as the people had placed themselves in a row on either\nside up to the King's Stairs, the prison governor made as if he were\ngoing thither; but he turned round abruptly, and said to Alfeldt,\n'This way,' and went to the gate of the Blue Tower; stood there for\nsome time and fumbled with the key; acted as if he could not unlock\nit, in order that I might remain as long as possible a spectacle to\nthe people. And as my heart was turned to God, and I had placed all\nmy confidence in the Most High, I raised my eyes to heaven, sought\nstrength, power, and safety from thence, and it was graciously\nvouchsafed me. (One circumstance I will not leave unnoticed--namely,\nthat as I raised my eyes to heaven, a screaming raven flew over the\nTower, followed by a flock of doves, which were flying in the same\ndirection.) At length, after a long delay, the prison governor opened\nthe Tower gate, and I was conducted into the Tower by the\nbefore-mentioned Captain Alfeldt. My attendant, who was preparing to\nfollow me, was called back by Major-General von Anfeldt, and told to\nremain behind. The prison governor went up the stairs, and showed\nAlfeldt the way to a prison for malefactors, to which the name of the\n'Dark Church' has been given. There Alfeldt quitted me with a sigh\nand a slight reverence. I can truly say of him that his face\nexpressed pity, and that he obeyed the order unwillingly. The clock\nwas striking half-past five when Jockum closed the door of my prison.\nI found before me a small low table, on which stood a brass\ncandlestick with a lighted candle, a high chair, two small chairs, a\nfir-wood bedstead without hangings and with old and hard bedding, a\nnight-stool and chamber utensil. At every side to which I turned I\nwas met with stench; and no wonder, for three peasants who had been\nimprisoned here, and had been removed on that very day, and placed\nelsewhere, had used the walls for their requirements. Soon after the\ndoor had been closed, it was opened again, and there entered Count\nChristian Rantzow, Prime Minister, Peter Zetz, Chancellor,\nChristoffer von Gabel, at that time Chancellor of the Exchequer, and\nErich Krag, at that time Secretary, all of whom gave me their hands\nwith civility. The Chancellor spoke and said: 'His Royal Majesty, my\ngracious master and hereditary king, sends you word, madame, that His\nMajesty has great cause for what he is doing against you, as you will\nlearn.' I replied: 'It is much to be regretted by me, if cause should\nbe found against me; I will, however, hope that it may not be of such\na kind that His Majesty's displeasure may be lasting. When I know the\ncause I can defend myself.' Count Rantzow answered: 'You will obtain\npermission to defend yourself.' He whispered something to the\nChancellor, upon which the Chancellor put a few questions: first,\nWhether on my last journey I had been in France with my husband? To\nwhich I answered in the affirmative. Then, What my husband was doing\nthere? To which I replied, that he was consulting physicians about\nhis health, whether it would be serviceable to him to use the warm\nbaths in the country, which no one would advise him to do; he had\neven been dissuaded from trying them by a doctor in Holland of the\nname of Borro,[E10] when he had asked his opinion. Thirdly, What I\nhad purposed doing in England? To this I replied that my intention\nhad been to demand payment of a sum of money which the King of\nEngland owed us, and which we had lent him in the time of his\nmisfortune. Fourthly, Who had been in England with me? I mentioned\nthose who were with me in England--namely, a nobleman named Cassetta,\nmy attendant who had come hither with me, a lacquey named Frantz, who\nhad remained in England, and the nobleman's servant. Fifthly, Who\nvisited my husband in Bruges? I could not exactly answer this, as my\nlord received his visits in a private chamber, where I was not\nadmitted. Count Rantzow said, 'You know, I suppose, who came to him\noftenest?' I answered, that the most frequent visitors among those I\nknew were two brothers named Aranda,[E11] the before-mentioned\nCassetta, and a nobleman named Ognati. Sixthly the Chancellor asked,\nWith whom I had corresponded here in the country? To which I\nanswered, that I had written to H. Hendrick Bielcke, to Olluff\nBrockenhuuss, Lady Elsse Passberg, and Lady Marie Ulfeldt;[E12] I did\nnot remember any more. Count Rantzow enquired if I had more letters\nthan those which I had given up? To which I answered in the negative,\nthat I had no more. He asked further, Whether I had more jewels with\nme than those he had seen? I answered that I had two strings of\nsmall round pearls on my hat, and a ring with a diamond, which I had\ngiven a lieutenant named Braten in Dover (it was he who afterwards\nbetrayed me). Count Rantzow asked, How much the pearls might have\nbeen worth? This I could not exactly say. He said, that he supposed I\nknew their approximate value. I said they might be worth 200\nrix-dollars, or somewhat more. Upon this they were all silent for a\nlittle. I complained of the severity of my imprisonment, and that I\nwas so badly treated. Count Rantzow answered, 'Yes Madame, His Royal\nMajesty has good cause for it; if you will confess the truth, and\nthat quickly, you may perhaps look for mercy. Had Marechal de\nBirron[E13] confessed the matter respecting which he was interrogated\nby order of the King, when the royal mercy was offered to him if he\nwould speak the truth, it would not have fared with him as it did. I\nhave heard as a truth that the King of France would have pardoned him\nhis crime, had he confessed at once; therefore, bethink yourself,\nmadame!' I answered, 'Whatever I am asked by order of His Majesty,\nand whatever I am cognizant of, I will gladly say in all submission.'\nUpon this Count Rantzow offered me his hand, and I reminded him in a\nfew words of the severity of my imprisonment. Count Rantzow promised\nto mention this to the King. Then the others shook hands with me and\nwent away. My prison was closed for a little. I therefore profited by\nthe opportunity, and concealed here and there in holes, and among the\nrubbish, a gold watch, a silver pen which gave forth ink and was\nfilled with ink, and a scissor-sheath worked with silver and\ntortoiseshell. This was scarcely done when the door was again\nopened, and there entered the Queen's Mistress of the Robes, her\nwoman of the bed-chamber, and the wife of the commissariat clerk,\nAbel Catharina. I knew the last. She and the Queen's woman of the\nbed-chamber carried clothes over their arm; these consisted of a long\ndressing-gown stitched with silk, made of flesh-coloured taffeta and\nlined with white silk, a linen under-petticoat, printed over with a\nblack lace pattern, a pair of silk stockings, a pair of slippers, a\nshift, an apron, a night-dress, and two combs. They made me no\ngreeting. Abel Cath. spoke for them, and said: 'It is the command of\nHer Majesty the Queen that we should take away your clothes, and that\nyou should have these in their place.' I answered, 'In God's name!'\nThen they removed the pad from my head, in which I had sown up rings\nand many loose diamonds. Abel Cath. felt all over my head to see if\nanything was concealed in my hair; then she said to the others,\n'There is nothing there; we do not require the combs.' Abel Cath.\ndemanded the bracelets and rings, which were a second time taken from\nme. I took them off and gave them to them, except one small ring\nwhich I wore on the last joint of my little finger, and which could\nnot be worth more than a rix-dollar, this I begged to be allowed to\nkeep. 'No,' said the Mistress of the Robes, 'You are to retain\nnothing.' Abel Cath. said, 'We are strictly forbidden to leave you\nthe smallest thing; I have been obliged to swear upon my soul to the\nQueen that I would search you thoroughly, and not leave you the\nsmallest thing; but you shall not lose it; they will all be sealed up\nand kept for you, for this I swear the Queen has said.' 'Good, good,\nin God's name!' I answered. She drew off all my clothes. In my\nunder-petticoat I had concealed some ducats under the broad gold\nlace; there was a small diamond ornament in my silk camisole, in the\nfoot of my stockings there were some Jacobuses', and there were\nsapphires in my shoes. When she attempted to remove my chemise, I\nbegged to be allowed to retain it. No; she swore upon her soul that\nshe dared not. She stripped me entirely, and the Mistress of the\nRobes gave Abel Cath. a nod, which she did not at once understand; so\nthe Mistress of the Robes said: 'Do you not remember your orders?'\nUpon this, Abel Cath. searched my person still more closely, and said\nto the lady in waiting: 'No, by God! there is nothing there.' I said:\n'You act towards me in an unchristian and unbecoming manner.' Abel\nCath. answered: 'We are only servants; we must do as we are ordered;\nwe are to search for letters and for nothing else; all the rest will\nbe given back to you; it will be well taken care of.' After they had\nthus despoiled me, and had put on me the clothes they had brought,\nthe servant of the Mistress of the Robes came in and searched\neverywhere with Abel Cath., and found every thing that I had\nconcealed. God blinded their eyes so that they did not observe my\ndiamond earrings, nor some ducats which had been sown into leather\nround one of my knees; I also saved a diamond worth 200 rix-dollars;\nwhile on board the ship I had bitten it out of the gold, and thrown\nthe gold in the sea; the stone I had then in my mouth.[62]\n\n [58] In the margin is added: 'I had a ring on with a table-diamond\n worth 200 rix-dollars. I bit this out, threw the gold in the sea,\n and kept the stone in my mouth. It could not be observed by my\n speech that there was anything in my mouth.'\n\n [59] That is the Aulefeldt mentioned in the Preface under the name\n of Anfeldt.\n\n [E08] Birgitte Speckhans was the wife of Frants v. Speckhans,\n master of ceremonies, afterwards Privy Councillor, &c. She had\n formerly been in the service of Leonora Christina, who was then at\n the height of her position, and ever afterwards proved herself a\n friend of her and Ulfeldt. It was in her house that they stayed\n after escaping from Malmoe, and she kept some of their movable\n goods for them during their imprisonment at Hammershuus.\n\n [E09] Birgitte Ulfeldt was a younger sister of Corfitz, who, in a\n letter to Sperling, declares her to be his and Leonora's bitterest\n enemy. What is known of her life is certainly not to her advantage.\n\n [60] In the margin is added: 'The sorrow manifested by many would\n far rather have depressed me; for several people, both men and\n women, shed tears, even those whom I did not know.'\n\n [61] This paragraph was afterwards struck out, the contents being\n transferred to the Preface.\n\n [E10] This is the famous Jos. Borro or Burrhus, physician and\n alchymist. He is often mentioned in books of the seventeenth\n century, on account of his wonderful cures and alleged knowledge of\n the art of making gold. In 1667 he came to Denmark, where King\n Fredrik III. spent considerable sums on the establishment of large\n laboratories for him, in a building which is still known as 'The\n Gold-house.'\n\n [E11] D'Aranda was one of the most influential families in Bruges.\n One of them, by name Bernard, was some time in the Danish army,\n afterwards secretary to Corfitz Ulfeldt, and employed by him in\n diplomatic missions. He died in 1658, but when Ulfeldt came to\n Bruges in 1662 he lived for some time with one of Bernard's\n brothers.\n\n [E12] H. Bielke was Admiral of the realm; his wife was an Ulfeldt,\n and it was he who procured Corfitz Ulfeldt his leave of absence in\n 1662, of which he made such regretable use. He, too, was one of the\n judges that convicted him. Oluf Brokkenhuus was Corfitz Ulfeldt's\n brother-in-law; Elizabeth Parsbjerg was the widow of his elder\n brother Lauridts Ulfeldt. Marie Ulfeldt was sister of Corfitz.\n\n [E13] Charles de Goutant, Duc de Biron, a celebrated French\n General, some time favourite of Henry IV. King of France, was found\n guilty of conspiring against his master with the courts of Spain\n and Savoy. Henry IV. forgave him, but he recommenced his intrigues.\n It is supposed that the King would have forgiven him a second time\n if he had confessed his crime; but he refused to do so, and was\n beheaded in 1602.\n\n [62] This passage was afterwards altered thus: 'God blinded their\n eyes so that they did not perceive my earrings, in each of which\n there is a large rose diamond, and from which I have now removed\n the stones. The gold, which is in form of a serpent, is still in my\n ears. They also did not perceive that something was fastened round\n my knee.'\n\nThe Mistress of the Robes was very severe; they could not search\nthoroughly enough for her. She laughed at me several times, and\ncould not endure that I sat down, asking whether I could not stand,\nand whether anything was the matter with me. I answered, 'There is\nonly too much the matter with me, yet I can stand when it is\nnecessary.' (It was no wonder that the Mistress of the Robes could so\nwell execute the order to plunder, for she had frequently accompanied\nher deceased husband. Colonel Schaffshaussen[E14], in war.) When she\nhad searched every part thoroughly, they took all my clothes, except\na taffeta cap for the head, and went away. Then the prison governor\ncame in with his hat on, and said, 'Leonora, why have you concealed\nyour things?' I answered him not a word; for I had made the\nresolution not to answer him, whatever he might say; his qualities\nwere known to me; I was aware that he was skilful in improving a\nreport, and could twist words in the manner he thought would be\nacceptable, to the damage of those who were in trouble. He asked\nagain with the same words, adding 'Do you not hear?' I looked at him\nover my shoulder, and would not allow his disrespect to excite me.\nThe table was then spread, and four dishes were brought in, but I had\nno appetite, although I had eaten little or nothing the whole day.\n\n [E14] This lady is known under the name of Haxthausen; and\n Schaffshausen is probably a mistake on Leonora's part, although of\n course she may have been married to an officer of this name before\n she married N. v. Haxthausen. She was a German by birth.\n\nAn hour afterwards, when the dishes had been carried away, a girl\ncame in named Maren Blocks, and said that she had orders from the\nQueen to remain the night with me. The prison governor joked a good\ndeal with the before-mentioned Maren, and was very merry, indulging\nin a good deal of loose talk. At last, when it was nearly ten\no'clock, he said good night and closed the two doors of my prison,\none of which is cased with copper. When Maren found herself alone\nwith me, she pitied my condition, and informed me that many, whom she\nmentioned by name (some of whom were known to me) had witnessed my\ncourage with grief and tears, especially the wife of H. Hendrick\nBielcke[E12b], who had fainted with weeping. I said, 'The good people\nhave seen me in prosperity; it is no wonder that they deplore the\ninstability of fortune;' and I wished that God might preserve every\none of those from misfortune, who had taken my misfortune to heart. I\nconsoled myself with God and a good conscience; I was conscious of\nnothing wrong, and I asked who she was, and whom she served? She said\nshe was in the Queen's private kitchen, and had the silver in her\nkeeping (from which I concluded that she had probably to clean the\nsilver, which was the case). She said that the Queen could get no one\nwho would be alone with me, for that I was considered evil; it was\nsaid also that I was very wise, and knew future events. I answered,\n'If I possessed this wisdom, I scarcely think that I should have come\nin here, for I should then have been able to guard myself against\nit.' Maren said we might know things and still not be able to guard\nagainst them.\n\n [E12b] H. Bielke was Admiral of the realm; his wife was an Ulfeldt,\n and it was he who procured Corfitz Ulfeldt his leave of absence in\n 1662, of which he made such regretable use. He, too, was one of the\n judges that convicted him. Oluf Brokkenhuus was Corfitz Ulfeldt's\n brother-in-law; Elizabeth Parsbjerg was the widow of his elder\n brother Lauridts Ulfeldt. Marie Ulfeldt was sister of Corfitz.\n\nShe told me also that the Queen had herself spoken with her, and had\nsaid to her, 'You are to be this night with Leonora; you need not be\nafraid, she can now do no evil. With all her witchcraft she is now in\nprison and has nothing with her; and if she strikes you, I give you\nleave to strike her back again till the blood comes.' Maren said\nalso, 'The Queen knows well that my mind has been affected by acute\nillness, and therefore she wished that I should be with you.' So\nsaying she threw her arms round my neck as I was sitting, and\ncaressed me in her manner, saying, 'Strike me, dear heart, strike\nme!' 'I will not,' she swore, 'strike again.' I was rather alarmed,\nfearing that the frenzy might come on. She said further that when she\nsaw me coming over the bridge, she felt as if her heart would burst.\nShe informed me with many words how much she loved me, and how the\nmaid of honour, Carisius, who was standing with her in the window,\nhad praised me, and wished to be able to do something for my\ndeliverance, with many such words and speeches. I accepted the\nunusual caress, as under the circumstances I could not help it, and\nsaid that it would be contrary to all justice to offer blows to one\nwho manifested such great affection as she had done, especially to\none of her sex; adding, that I could not think how the Queen had\nimagined that I struck people, as I had never even given a box on the\nears to a waiting-woman. I thanked her for her good opinion of me,\nand told her that I hoped all would go well, dark as things looked;\nthat I would hold fast to God, who knew my innocence, and that I had\ndone nothing unjustifiable; that I would commend my cause to Him, and\nI did not doubt that He would rescue me: if not immediately He would\ndo so some day, I was well assured.\n\nMaren began to speak of different things; among others of my sister\nElizabeth Augusta[E15], how she had sat in her porch as I had been\nconveyed past as a prisoner, and had said that if I were guilty there\nwas nothing to say against it, but that if I were innocent they were\ngoing too far. I said nothing to this, nor did I answer anything to\nmuch other tittle-tattle. She began to speak of her own persecution,\nwhich she did with great diffuseness, interspersing it with other\nstories, so that the conversation (in the present circumstances) was\nvery wearisome to me; I was besides very tired, and worn out with\ncare, so I said I would try to sleep and bid her good-night. My\nthoughts prevented me from sleeping. I reflected on my present\ncondition, and could in no wise reconcile myself to it, or discover\nthe cause of such a great misfortune. It was easy to perceive that\nsomewhat besides Fux's death was imputed to me, since I was treated\nwith such disrespect.\n\n [E15] Elizabeth Augusta, a younger sister of Leonora, married Hans\n Lindenow, a Danish nobleman, who died in the siege of Copenhagen,\n 1659.\n\nWhen I had long lain with my face to the wall, I turned round and\nperceived that Maren was silently weeping, so I asked her the reason\nof her tears. She denied at first that she was crying, but afterwards\nconfessed that she had fallen into thinking over this whole affair.\nIt had occurred to her that she had heard so much of Lady Leonora and\nher splendour, &c., of how the King loved her, and how every one\npraised her, &c., and now she was immured in this execrable thieves'\nprison, into which neither sun nor moon shone, and where there was a\nstench enough to poison a person only coming in and out, far more one\nwho had to remain in it. I thought the cause of her weeping was that\nshe should be shut up with me in the terrible prison; so I consoled\nher, and said that she would only remain with me until another had\nbeen fixed upon, since she was in other service; but that I for my\npart did not now think of past times, as the present gave me\nsufficient to attend to; if I were to call to mind the past, I would\nremember also the misfortunes of great men, emperors, kings,\nprinces, and other high personages, whose magnificence and prosperity\nhad far exceeded mine, and whose misfortunes had been far greater\nthan mine; for they had fallen into the hands of tyrants, who had\ntreated them inhumanly, but this king was a Christian king, and a\nconscientious man, and better thoughts would occur to him when he had\ntime to reflect, for my adversaries now left him no leisure to do so.\nWhen I said this, she wept even more than before, but said nothing,\nthinking in herself (as she declared to me some days afterwards) that\nI did not know what an infamous sentence had been pronounced upon my\nlate lord,[E16] and weeping all the more because I trusted the King\nso firmly. Thus we went on talking through the night.\n\n [E16] That Leonora here speaks of her husband as her 'late lord,'\n is due only to the fact that the Memoir was not written till after\n his death; at the time of these events he was still alive.\n\nOn the morning of August 9, at six o'clock, the prison governor came\nin, bade me good morning, and enquired whether we would have some\nbrandy. I answered nothing. He asked Maren whether I was asleep; she\nreplied that she did not know, came up to my bed, and put the same\nquestion to me. I thanked her, adding that it was a kind of drink\nwhich I had never tasted. The prison governor chattered with Maren,\nwas very merry considering the early hour, told her his dreams, which\nhe undoubtedly invented merely for the sake of talking. He told her,\nsecretly, that she was to come to the Queen, and ordered her to say\naloud that she wished to go out a little. He said that he would\nremain with me in the meanwhile, until she returned, which he did,\nspeaking occasionally to me, and asking me whether I wished for\nanything? whether I had slept? whether Maren had watched well? But\nhe got no answer, so that the time seemed very long to him. He went\nout towards the stairs and came back again, sang a morning psalm,\nscreamed out sometimes to one, and sometimes to another, though he\nknew they were not there.\n\nThere was a man named Jon who helped to bring up the meals with\nRasmus the tower warder, and to him he called more than forty times\nand that in a singing tone, changing his key from high to low, and\nscreaming occasionally as loud as he could, and answering himself\n'Father, he is not here! by God, he is not here!' then laughing at\nhimself; and then he began calling again either for Jon or for\nRasmus, so that it seemed to me that he had been tasting the brandy.\nAbout eight o'clock Maren came back, and said that at noon two women\nwould come to relieve her. After some conversation between the prison\ngovernor and Maren, he went out and shut the doors. Maren told me how\nthe Queen had sent for her, and asked her what I was doing, and that\nshe answered that I was lying down quietly, and not saying anything.\nThe Queen had asked whether I wept much. Maren replied, 'Yes indeed,\nshe weeps silently.' 'For,' continued Maren, 'if I had said that you\ndid not weep, the Queen would have thought that you had not yet\nenough to weep for.' Maren warned me that one of the two women who\nwere to watch me was the wife of the King's shoemaker, a German, who\nwas very much liked by the Queen. Her Majesty had employed her to\nattend Uldrich Christian Gyldenlowe in the severe and raving illness\nof which he died, and this woman had much influence with the Queen.\nWith regard to the other woman, Maren had no idea who she might be,\nbut the last-mentioned had spoken with the Queen in Maren's presence,\nand had said that she did not trust herself to be alone with me. The\nwomen did not come before four o'clock in the afternoon. The prison\ngovernor accompanied them, and unlocked the door for them. The first\nwas the wife of the shoemaker, a woman named Anna, who generally\nwould not suffer anybody else to speak. The other was the wife of the\nKing's groom, a woman named Catharina, also a German. After greeting\nme, Anna said that her Majesty the Queen had ordered them to pass a\nday or two with me and wait upon me. 'In God's name,' I answered.\n\nAnna, who was very officious, asked me, 'Does my lady wish for\nanything? She will please only say so, and I will solicit it from the\nQueen.' I thanked her, and said that I should like to have some of my\nclothes, such as two night-jackets, one lined with silk and another\nbraided with white, my stomacher, something for my head, and above\nall my bone box of perfume, which I much needed. She said she would\nat once arrange this, which she did, for she went immediately and\nproffered my request. The things were all delivered to me by the\nprison governor at six o'clock, except my box of perfume, which had\nbeen lost, and in its place they sent me a tin box with a very bad\nkind of perfume. When the time arrived for the evening meal,\nCatharina spread a stool by the side of my bed, but I had no desire\nto eat. I asked for a lemon with sugar, and they gave it me. The\nprison governor sat down at the table with the two women, and did the\npart of jester, so much so that no one could have said that they were\nin a house of mourning, but rather in one of festivity. I inwardly\nprayed to God for strength and patience, that I might not forget\nmyself. God heard my prayer, praised be His name. When the prison\ngovernor was tired of the idle talking and laughing, he bade good\nnight after ten o'clock, and told the women to knock if they wanted\nanything, as the tower warder was just underneath. After he had\nlocked both the doors, I got up, and Catharina made my bed. Anna had\nbrought a prayer-book with her, from which I read the evening prayer,\nand other prayers for them; then I laid down and bid them good night.\nThey laid on a settle-bed which had been brought in for them. I\nslumbered from time to time, but only for short intervals.\n\nAbout six o'clock on the morning of August 10 the prison governor\nopened the door, to the great delight of the women, who were\nsincerely longing for him, especially Catharina, who was very stout;\nshe could not endure the oppressive atmosphere, and was ill almost\nthe whole night. When the prison governor, after greeting them, had\ninquired how it fared with them, and whether they were still alive,\nhe offered them brandy, which they readily accepted. When it was\nseven o'clock, they requested to go home, which they did, but they\nfirst reported to the Queen all that had happened during the half-day\nand the night. The prison governor remained with me.\n\nWhen it was near nine o'clock, he brought in a chair without saying\nanything. I perceived from this that visitors were coming, and I was\nnot wrong; for immediately afterwards there entered Count Rantzow,\nprime minister, chancellor H. Peter Retz, Christoffer Gabel, the\nchancellor of the exchequer, and secretary Erick Krag, who all shook\nhands with me and seated themselves by my bed. Krag, who had paper,\npen and ink with him, seated himself at the table. Count Rantzow\nwhispered something to the chancellor. The chancellor upon this began\nto address me as on the previous occasion, saying that his Majesty\nthe King had great cause for his treatment of me. 'His Majesty,' he\nwent on to say, 'entertains suspicion with regard to you, and that\nnot without reason.' I inquired in what the suspicion consisted. The\nchancellor said, 'Your husband has offered the kingdom of Denmark to\na foreign lord.' I inquired if the kingdom of Denmark belonged to my\nhusband, that he could thus offer it, and as no one answered, I\ncontinued and said, 'Good gentlemen, you all know my lord; you know\nthat he has been esteemed as a man of understanding, and I can assure\nyou that when I took leave of him he was in perfect possession of his\nsenses. Now it is easy to perceive that no sensible man would offer\nthat which was not in his own power, and which he had no right to\ndispose of. He is holding no post, he has neither power nor\nauthority; how should he, therefore, be so foolish as to make such an\noffer, and what lord would accept it?'\n\nCount Rantzow said: 'Nevertheless it is so, madame; he has offered\nDenmark to a foreign potentate; you know it well.' I answered, 'God\nis my witness that I know of no such thing.' 'Yes,' said Count\nRantzow, 'your husband concealed nothing from you, and therefore you\nmust know it.' I replied, 'My husband certainly never concealed from\nme anything that concerned us both. I never troubled myself in former\ndays with that which related to his office; but that which affected\nus both he never concealed from me, so that I am sure, had he\nentertained any such design, he would not have held it a secret from\nme. And I can say, with truth, that I am not the least aware of it.'\nCount Rantzow said: 'Madame, confess it while the King still asks you\nto do so.'\n\nI answered, 'If I knew it I would gladly say so; but as truly as God\nlives I do not know it, and as truly am I unable to believe that my\nhusband would have acted so foolishly, for he is a sick man. He urged\nme to go to England in order to demand the money that had been lent;\nI undertook the journey, unwillingly, chiefly because he was so very\nweak. He could not go up a few steps of the stairs without resting to\nget his breath; how should he, then, undertake a work of such labour?\nI can say with truth that he is not eight days without an attack,\nsometimes of one kind sometimes of another.' Count Rantzow again\nwhispered with the chancellor, and the chancellor continued: 'Madame,\nsay without compulsion how the matter stands, and who is privy to it;\nsay it now, while you are asked freely to do so. His Majesty is an\nabsolute Sovereign; he is not fettered by law; he can do as he will;\nsay it.' I answered: 'I know well that his Majesty is an absolute\nSovereign, and I know also, that he is a Christian and a\nconscientious man; therefore, his Majesty will do nothing but what he\ncan justify before God in heaven. See, here I am! You can do with me\nwhat you will; that which I do not know I cannot say.'\n\nCount Rantzow began again to bring forward the Marechal de Birron,\nand made a long speech about it. To this I at length replied, that\nthe Marechal de Birron in nowise concerned me; that I had no answer\nto make on the matter, and that it seemed to me that it was not a\ncase in point. Count Rantzow asked me why, when I was demanded with\nwhom I had corresponded in the kingdom, I had not said that I had\nwritten to him and to the treasurer Gabel. To this I replied that I\nthought those who asked me knew it well, so that it was not necessary\nfor me to mention it; I had only said that of which they probably did\nnot know. Count Rantzow again whispered to the chancellor, and the\nchancellor said: 'In a letter to Lady Elsse Passberg you have written\nrespecting another state of things in Denmark,' (as he said this, he\nlooked at Count Rantzow and asked if it was not so, or how it was);\n'what did you mean by that, madame?' I replied that I could not\nrecollect what cause her letter had given me to answer it in this\nway; what came before or what followed, would, without a doubt,\nexplain my meaning; if I might see the letter, it would prove at once\nthat I had written nothing which I could not justify.\n\nNothing more was said with regard to it. Count Rantzow asked me what\nforeign ministers had been with my lord in Bruges. 'None,' I\nanswered, 'that I am aware of.' He asked further whether any Holstein\nnoblemen had been with him. I answered, 'I do not know.' Then he\nenumerated every Prince in Germany, from the Emperor to the Prince of\nHolstein, and enquired respecting each separately whether any of\ntheir Ministers had been with my husband. I gave the same answer as\nbefore to each question, that I was not aware that any one of them\nhad been with him. Then he said, 'Now, madame, confess! I beg you;\nremember Marechal de Birron! you will not be asked again.' I was\nsomewhat tired of hearing Birron mentioned so often, and I answered\nrather hastily: 'I do not care about the Marechal de Birron; I\ncannot tell what I do not know anything about.'\n\nSecretary Krag had written somewhat hurriedly it seemed, for when at\nmy desire he read aloud what he had written, the answers did not\naccord with the questions; this probably partly arose from hurry, and\npartly from malice, for he was not amicably inclined towards my late\nlord. I protested against this when he read the minutes. The\nchancellor agreed with me in every item, so that Krag was obliged to\nre-write it. After this they got up and took their leave. I requested\nto beg His Majesty the King to be gracious to me, and not to believe\nwhat he had been informed with regard to my husband. I could not\nimagine they would find that he had ever deviated from his duty.\n'Yes,' answered Count Rantzow, 'if you will confess, madame, and tell\nus who is concerned in this business and the details of it, you might\nperhaps find him a gracious lord and king.' I protested by the living\nGod that I knew nothing of it; I knew of nothing of the kind, much\nless of accomplices. With this they went away, after having spent\nnearly three hours with me, and then the prison governor and the\nwomen entered. They spread the table and brought up the meal, but I\ntook nothing but a draught of beer. The prison governor sat down to\ntable with the women. If he had been merry before, he was still more\nso now, and he told one indecent story after another.\n\nWhen they had had enough of feasting and talking he went away and\nlocked the door; he came as usual again about four o'clock in the\nafternoon, and let the women go out, staying with me until they\nreturned, which generally was not for two hours. When the women were\nalone with me, Anna told Catharina of her grief for her first\nhusband, and nothing else was talked of. I behaved as if I were\nasleep, and I did the same when the prison governor was alone with\nme, and he then passed the time in singing and humming. The evening\nmeal was also very merry for the women, for the prison governor\namused them by telling them of his second marriage; how he had wooed\nwithout knowing whom, and that he did not know it until the\nbetrothal. The story was as ludicrous as it was diffuse. I noticed\nthat it lasted an hour and a quarter.\n\nWhen he had said good night, Anna sat down on my bed and began to\ntalk to Catharina, and said, 'Was it not a horrible story of that\ntreacherous design to murder the King and Queen and the whole royal\nfamily?' Catharina answered, 'Thank God the King and Queen and the\nwhole family are still alive!' 'Yes,' said Anna, 'it was no merit of\nthe traitors, though, that they are so; it was too quickly\ndiscovered; the King knew it three months before he would reveal it\nto the Queen. He went about sorrowfully, pondering over it, unable\nquite to believe it; afterwards, when he was quite certain of it, he\ntold the Queen; then the body-guard were doubled, as you know.'\nCatherina enquired how they had learnt it. Anna answered, 'That God\nknows; it is kept so secret that no one is allowed as much as to ask\nfrom whom it came.' I could not help putting in a word; it seemed to\nme a pity that they could not find out the informer, and it was\nremarkable that no one ventured to confess having given the\ninformation. Catherina said, 'I wonder whether it is really true?'\n'What do you mean?' answered Anna; 'would the King do as he is doing\nwithout knowing for certain that it is true? How can you talk so?' I\nregarded this conversation as designed to draw some words from me,\nso I answered but little, only saying that until now I had seen\nnothing which gave credibility to the report, and that therefore I\nfelt myself at liberty not to believe it until I saw certain proof of\nit. Anna adhered to her statement, wondered that there could be such\nevil people as could wish to murder the good King, and was very\ndiffuse on the matter.[E17] She could be at no loss for material, for\nshe always began again from the beginning; but at last she had to\nstop, since she spoke alone and was not interrupted either by\nCatharina or by me.\n\n [E17] When the sentence on Ulfeldt had become publicly known, the\n most absurd rumours circulated in Copenhagen, and found their way\n to foreign newspapers. For instance _the kingdom's_ Intelligencer,\n No. 33, Aug. 10-17, 1663, says, in a correspondence from Hamburg:\n 'They say the traitors intended to set Copenhagen on fire in divers\n places, and also the fleet, to destroy the King and family, to blow\n up the King's palace, and deliver the crown over to another.' The\n Government itself, on hearing of Ulfeldt's plots, made great\n military preparations.\n\nI got up and requested to have my bed made, which Catharina always\ndid. Anna attended to the light during the night, for she was more\nwatchful than Catharina. I read aloud to them from Anna's book,\ncommended myself to God, and laid down to sleep. But my sleep was\nlight, the promenades of the rats woke me, and there were great\nnumbers of them. Hunger made them bold; they ate the candle as it\nstood burning. Catharina, moreover, was very uncomfortable all night,\nso that this also prevented my sleeping. Early on the morning of\nAugust 11 the prison governor came as usual with his brandy\nattentions, although they had a whole bottle with them. Catharina\ncomplained a good deal, and said she could not endure the oppressive\nair; that when she came in at the door it seemed as if it would\nstifle her; if she were to remain there a week she was certain that\nshe would be carried out dead. The prison governor laughed at this.\n\nThe women went away, and he remained with me. He presented me\nMajor-General von Anfeldt's compliments, and a message from him,\nthat I 'should be of good courage; all would now soon be well.' I\nmade no reply. He enquired how I was, and whether I had slept a\nlittle; and answered himself, 'I fancy not much.' He asked whether I\nwould have anything, again answering himself, 'No, I do not think you\nwish for anything.' Upon this he walked up and down, humming to\nhimself; then he came to my bedside and said: 'Oh, the dear King! he\nis indeed a kind master! Be at peace; he is a gracious sovereign, and\nhas always held you in esteem. You are a woman, a weak instrument.\nPoor women are soon led away. No one likes to harm them, when they\nconfess the truth. The dear Queen, she is indeed a dear Queen! She is\nnot angry with you. I am sure if she knew the truth from you, she\nwould herself pray for you. Listen! if you will write to the Queen\nand tell her all about the matter, and keep nothing back, I will\nbring you pen, ink, and paper. I have no wish, on my soul! to read\nit. No, God take me if I will look at it; and that you may be sure of\nthis, I will give you wax that you may seal it. But I imagine you\nhave probably no seal?' As I answered him not a word, he seized my\nhand and shook it rather strongly, saying, 'Do you not hear? Are you\nasleep?' I raised my head threateningly; I should like to have given\nhim a box on the ears, and I turned round to the wall.\n\nHe was angry that his design had failed, and he went on grumbling to\nhimself for more than an hour. I could not understand a word beyond,\n'Yes, yes! you will not speak.' Then he muttered somewhat between his\nteeth: 'You will not answer; well, well, they will teach you. Yes, by\nGod! hum, hum, hum.' He continued thus until the tower warder,\nRasmus, came and whispered something to him; then he went out. It\nseemed to me that there was someone speaking with him, and so far as\nI could perceive it must have been someone who asked him if the ink\nand paper should be brought up, for he answered, 'No, it is not\nnecessary; she will not.' The other said, 'Softly, softly!' The\nprison governor, however, could not well speak softly, and I heard\nhim say, 'She cannot hear that; she is in bed.' When he came in again\nhe went on muttering to himself, and stamped because I would not\nanswer; he meant it kindly; the Queen was not so angry as I imagined.\nHe went on speaking half aloud; he wished the women would come; he\ndid nothing else but beg Rasmus to look for them.\n\nSoon after Rasmus came and said that they were now going up the\nKing's Stairs. Still almost an hour passed before they came in and\nreleased him. When they had their dinner (my own meal consisted of\nsome slices of lemon with sugar) the prison governor was not nearly\nso merry as he was wont to be, though he chattered of various things\nthat had occurred in former times, while he was a quarter-master. He\nalso retired sooner than was his custom. The women, who remained,\ntalked of indifferent matters. I also now and then put in a word, and\nasked them after their husbands and children. Anna read some prayers\nand hymns from her book, and thus the day passed till four o'clock,\nwhen the prison governor let them out. He had brought a book with\nhim, which he read in a tolerably low tone, while he kept watch by\nme. I was well pleased at this, as it gave me rest.\n\nAt the evening meal the prison governor began amongst other\nconversation to tell the women that a prisoner had been brought here\nwho was a Frenchman; he could not remember his name; he sat\ncogitating upon the name just as if he could not rightly hit upon it.\nCarl or Char, he did not know what he was called, but he had been\nformerly several years in Denmark. Anna enquired what sort of a man\nhe was. He replied that he was a man who was to be made to sing,[63]\nbut he did not know for a certainty whether he was here or not.\n(There was nothing in all this.) He only said this in order to get an\nopportunity of asking me, or to perceive whether it troubled me.\n\n [63] That is, give information.\n\nHe had undoubtedly been ordered to do this; for when he was gone Anna\nbegan a conversation with Catharina upon this same Carl, and at last\nasked me whether we had had a Frenchman in our employ. I replied that\nwe had had more than one. She enquired further whether there was one\namong them named Carl, who had long been in our service. 'We had a\nservant,' I answered, 'a Frenchman named Charle; he had been with us\na long time.' 'Yes, yes,' she said, 'it is he. But I do not think he\nhas arrived here yet; they are looking for him.' I said, 'Then he is\neasy to find, he was at Bruges when I left that town.' Anna said she\nfancied he had been in England with me, and she added, 'That fellow\nknows a good deal if they get him.' I answered, 'Then it were to be\nwished that they had him for the sake of his information.' When she\nperceived that I troubled myself no further about him she let the\nconversation drop, and spoke of my sister Elizabeth Augusta, saying\nthat she passed her every day. She was standing in her gateway or\nsitting in the porch, and that she greeted her, but never uttered a\nword of enquiry after her sister, though she knew well that she was\nwaiting on me in the Tower. I said I thought my sister did not know\nwhat would be the best for her to do. 'I cannot see,' said Anna,\n'that she is depressed.' I expressed my opinion that the less we\ngrieved over things the better. Other trifles were afterwards talked\nof, and I concluded the day with reading, commended myself to the\ncare of Jesus, and slept tolerably well through the night.\n\nAugust 12 passed without anything in particular occurring, only that\nAnna tried to trouble me by saying that a chamber next to us was\nbeing put in order, for whom she did not know; they were of course\nexpecting someone in it. I could myself hear the masons at work. On\nthe same day Catharina said that she had known me in prosperity, and\nblessed me a thousand times for the kindness I had shown her. I did\nnot remember having ever seen her. She said she had been employed in\nthe storeroom in the service of the Princess Magdalena Sybille, and\nthat when I had visited the Princess, and had slept in the Castle, I\nhad sent a good round present for those in the storeroom, and that\nshe had had a share in it, and that this she now remembered with\ngratitude. Anna was not pleased with the conversation, and she\ninterrupted it three times; Catharina, however, did not answer her,\nbut adhered to the subject till she had finished. The prison governor\nwas not in good humour on this day also, so that neither at dinner\nnor at supper were any indecent stories related.\n\nOn August 13, after the women had been into the town and had\nreturned, the prison governor opened the door at about nine o'clock,\nand whispered something to them. He then brought in another small\nseat; from this I perceived that I was to be visited by one more\nthan on the previous occasion. At about ten o'clock Count Rantzow,\nGeneral Skack, Chancellor Retz, Treasurer Gabel, and Secretary Krag\nentered. They all saluted me with politeness; the four first seated\nthemselves on low seats by my bedside, and Krag placed himself with\nhis writing materials at the table. The Chancellor was spokesman, and\nsaid, 'His royal Majesty, my gracious Sovereign and hereditary King,\nsends you word, madame, that his Majesty has great cause for all that\nhe is doing, and that he entertains suspicions with regard to you\nthat you are an accomplice in the treason designed by your husband;\nand his royal Majesty had hoped that you would confess without\ncompulsion who have participated in it, and the real truth about it.'\n\nWhen the Chancellor ceased speaking, I replied that I was not aware\nthat I had done anything which could render me suspected; and I\ncalled God to witness that I knew of no treason, and therefore I\ncould mention no names. Count Rantzow said, 'Your husband has not\nconcealed it from you, hence you know it well.' I replied, 'Had my\nhusband entertained so evil a design, I believe surely he would have\ntold me; but I can swear with a good conscience, before God in\nHeaven, that I never heard him speak of anything of the kind. Yes, I\ncan truly say he never wished evil to the King in my hearing, and\ntherefore I fully believe that this has been falsely invented by his\nenemies.' Count Rantzow and the Chancellor bent their heads together\nacross to the General, and whispered with each other for some time.\nAt length the Chancellor asked me whether, if my husband were found\nguilty, I would take part in his condemnation. This was a remarkable\nquestion, so I reflected a little, and said, 'If I may know on what\ngrounds he is accused, I will answer to it so far as I know, and so\nmuch as I can.' The Chancellor said, 'Consider well whether you\nwill.' I replied as before, that I would answer for him as to all\nthat I knew, if I were informed of what he was accused. Count Rantzow\nwhispered with Krag, and Krag went out, but returned immediately.\n\nSoon afterward some one (whom I do not know) came from the\nChancellor's office, bringing with him some large papers. Count\nRantzow and the Chancellor whispered again. Then the Chancellor said,\n'There is nothing further to do now than to let you know what sort of\na husband you have, and to let you hear his sentence.' Count Rantzow\nordered the man who had brought in the papers to read them aloud. The\nfirst paper read was to the effect that Corfitz, formerly Count of\nUlfeldt, had offered the kingdom of Denmark to a foreign sovereign,\nand had told the same sovereign that he had ecclesiastical and lay\nmagnates on his side, so that it was easy for him to procure the\ncrown of Denmark for the before-mentioned sovereign.\n\nA paper was then read which was the defence of the clergy, in which\nthey protested that Corfitz, Count of Ulfeldt, had never had any\ncommunication with any of them; that he had at no time shown himself\na friend of the clergy, and had far less offered them participation\nin his evil design. They assured his royal Majesty of their fidelity\nand subjection, &c. Next, a paper was read, written by the\nBurgomaster and council in Copenhagen, nearly similar in purport,\nthat they had had no correspondence with Count Corfitz Ulfeldt, and\nequally assuring his royal Majesty of their humble fidelity. Next\nfollowed the reading of the unprecedented and illegal sentence which,\nwithout a hearing, had been passed on my lord. This was as unexpected\nand grievous as it was disgraceful, and unjustifiable before God and\nall right-loving men. No documents were brought forward upon which\nthe sentence had been given. There was nothing said about prosecution\nor defence; there was no other foundation but mere words; that he had\nbeen found guilty of having offered the crown of Denmark to a foreign\nsovereign, and had told him that he had on his side ecclesiastical\nand lay magnates, who had shown by their signed protestations that\nthis was not the case, for which reason he had been condemned as a\ncriminal.\n\nWhen the sentence with all the names subjoined to it had been read,\nthe reader brought it to me, and placed it before me on the bed.\nEveryone can easily imagine how I felt; but few or none can conceive\nhow it was that I was not stifled by the unexpected misery, and did\nnot lose my sense and reason. I could not utter a word for weeping.\nThen a prayer was read aloud which had been pronounced from the\npulpit, in which Corfitz was anathematised, and God was prayed not to\nallow his gray hair to go to the grave in peace. But God, who is\njust, did not listen to the impious prayer of the unrighteous,\npraised be His name for ever.\n\nWhen all had been read, I bemoaned with sighs and sorrowful tears\nthat I had ever lived to see this sad day, and I begged them, for\nJesus' sake, that they would allow me to see on what the hard\njudgment was based. Count Rantzow answered, 'You can well imagine,\nmadame, that there are documents upon which we have acted: some of\nyour friends are in the council.' 'May God better it!' I said. 'I beg\nyou, for God's sake, to let me see the documents. Les apparences sont\nbien souvent trompeuses. What had not my husband to suffer from that\nSwede in Skaane, during that long imprisonment, because he was\nsuspected of having corresponded with his Majesty, the King of\nDenmark, and with his Majesty's ministers? Now, no one knows better\nthan his Majesty, and you my good lords, how innocently he suffered\nat that time, and so this also may be apparently credible, and yet\nmay not be so in truth. Might I not see the documents?' To this no\nanswer was given. I continued and said, 'How is it possible that a\nman who must himself perceive that death is at hand should undertake\nsuch a work, and be so led away from the path of duty, when he did\nnot do so at a time when he acknowledged no master, and when such\ngreat promises were made him by the Prince of Holstein, as the\nPrince's letters show, which are now in his Majesty's hands.' Count\nRantzow interrupted me and said, 'We did not find those letters.'\n'God knows,' I replied, 'they were there; of that I am certain.' I\nsaid also, 'At that time he might have done something to gratify a\nforeign sovereign; at that time he had power and physical vigour, and\nalmost the entire government was in his hands; but he never looked to\nhis own advantage, but pawned his own property to hasten the King's\ncoronation, so that no impediment might come between.[64] This is his\nreward! Good gentlemen, take an example of me, you who have seen me\nin prosperity, and have compassion on me. Pray his royal Majesty to\nbe mild, and not to proceed to such severity.'\n\n [64] In the margin the following explanatory note is added: 'When\n his Majesty (Christian IV.) was dead, there was no prince elected,\n so that the States were free to choose the king whom they desired,\n wherefore the Duke of Holstein, Duke Frederick, promised my\n deceased lord that if he would contrive that he should be elected\n king, the land of Fyen should belong to him and a double alliance\n between his children and ours should be concluded. But my lord\n rejected this proposal and would not assist in dispossessing the\n son of Christian IV. of the kingdom. The prince had obtained\n several votes, but my lord contested them.'\n\nThe Chancellor and Treasurer were moved by this, so that the tears\ncame into their eyes. Count Rantzow said to the General and the\nChancellor, 'I think it is a fortnight ago since the sentence was\npublished?' The Chancellor answered, 'It is seventeen days ago.'[E18]\nI said, 'At that time I was still in England, and now I am asked for\ninformation on the matter! Oh, consider this, for God's sake! and\nthat there was no one present to speak on my husband's behalf.' Count\nRantzow enquired whether I wished to appeal against it? I replied,\n'How am I to appeal against a judicial decree? I only beg for Jesus'\nsake that what I say may be considered, and that I may have the\nsatisfaction of seeing the documents upon which the sentence is\nbased.'\n\n [E18] The sentence on Ulfeldt was given on July 24, but probably\n not published till a few days later.\n\nCount Rantzow answered as before, that there were documents, and that\nsome of my friends had sat in the council, and added that all had\nbeen agreed, and that not one had had anything to say against it. I\ndared not say what I thought. I knew well how matters are done in\nsuch absolute governments: there is no such thing as opposition, they\nmerely say, 'Sign, the King wishes it; and ask not wherefore, or the\nsame condemnation awaits thee.'[65] I was silent, and bewailed my\nunhappiness, which was irremediable. When Krag read aloud the minutes\nhe had written, namely, that when I was asked whether I would\nparticipate in my husband's sentence, I had answered that I would\nconsider of it. I asked, 'How was that?' The Chancellor immediately\nreplied, 'No, she did not say so, but she requested to know the\naccusation brought against her husband.' I repeated my words\nagain,[66] I know not whether Krag wrote them or not; for a great\npart of that which I said was not written. Krag yielded too much to\nhis feelings in the matter, and would gladly have made bad worse. He\nis now gone where no false writings avail; God took him away suddenly\nin an unclean place, and called him to judgment without warning. And\nCount Rantzow, who was the principal mover and inventor of that\nillegal sentence, the like of which was never known in Denmark, did\nnot live to see his desire fulfilled in the execution of a wooden\nimage.[E19] When this was done, they rose and shook hands with me.\nThis painful visit lasted more than four hours.\n\n [65] It had happened as I thought. There were some in the council\n who refused to sign, some because they had not been present at the\n time of the procedure, and others because they had not seen on what\n the sentence was founded; but they were nevertheless compelled to\n sign with the others, on the peril of the king's displeasure.\n [Marginal note.]\n\n [66] In the margin is added, 'and asked whether I was permitted to\n appeal against this sentence. All were silent.'\n\n [E19] A line has been drawn in the MS. through the two last\n paragraphs, and their contents transferred to the continuation of\n the Preface.\n\nThey went away, leaving me full of anxiety, sighing and weeping--a\nsad and miserable captive woman, forsaken by all; without help,\nexposed to power and violence, fearing every moment that her husband\nmight fall into their hands, and that they might vent their malice on\nhim. God performed on that day a great miracle, by manifesting His\npower in my weakness, preserving my brain from bewilderment, and my\ntongue from overflowing with impatience. Praised be God a thousand\ntimes! I will sing Thy praise, so long as my tongue can move, for\nThou wast at this time and at all times my defence, my rock, and my\nshield!\n\nWhen the gentlemen were gone away, the prison governor came and the\nwomen, and a stool was spread by the side of my bed. The prison\ngovernor said to me, 'Eat, Leonora; will you not eat?' As he said\nthis, he threw a knife to me on the bed. I took up the knife with\nangry mind, and threw it on the ground. He picked up the knife,\nsaying, 'You are probably not hungry? No, no! you have had a\nbreakfast to-day which has satisfied you, have you not? Is it not\nso?' Well, well, come dear little women (addressing the two women),\nlet us eat something! You must be hungry, judging from my own\nstomach.' When they had sat down to table, he began immediately to\ncram himself, letting it fall as if inadvertently from his mouth, and\nmaking so many jokes that it was sad to see how the old man could not\nconceal his joy at my unhappiness.\n\nWhen the meal was finished, and the prison governor had gone away,\nAnna sat down by my bed and began to speak of the sorrow and\naffliction which we endure in this world, and of the joy and delights\nof heaven; how the pain that we suffer here is but small compared\nwith eternal blessedness and joy, wherefore we should not regard\nsuffering, but should rather think of dying with a good conscience,\nkeeping it unsullied by confessing everything that troubles us, for\nthere is no other way. 'God grant,' she added, 'that no one may\ntorment himself for another's sake.' After having repeated this\nremark several times, she said to me, 'Is it not true, my lady?'\n'Yes, certainly it is true,' I replied; 'you speak in a Christian\nmanner, and according to the scriptures.' 'Why will you, then,' she\nwent on to say, 'let yourself be tormented for others, and not say\nwhat you know of them?' I asked whom she meant. She answered, 'I do\nnot know them.' I replied, 'Nor do I.' She continued in the same\nstrain, however, saying that she would not suffer and be tormented\nfor the sake of others, whoever they might be; if they were guilty\nthey must suffer; she would not suffer for them; a woman was easily\nled away, but happiness was more than all kindred and friends.\n\nAs she seemed unable to cease chattering, I wished to divert her a\nlittle, so I asked whether she were a clergyman's daughter; and since\nshe had before told me of her parentage, she resented this question\nall the more, and was thoroughly angry; saying, 'If I am not a\nclergyman's daughter, I am the daughter of a good honest citizen, and\nnot one of the least. In my time, when I was still unmarried, I never\nthought that I should marry a shoemaker.' I said, 'But your first\nhusband, too, was also a shoemaker.' 'That is true,' she replied,\n'but this marriage came about in a very foolish manner,' and she\nbegan to narrate a whole history of the matter, so that I was left in\npeace. Catharina paced up and down, and when Anna was silent for a\nlittle, she said, with folded hands, 'O God, Thou who art almighty,\nand canst do everything, preserve this man for whom they are seeking,\nand never let him fall into the hands of his enemies. Oh God, hear\nme!' Anna said angrily to her, 'Catharina, do you know what you are\nsaying? How can you speak so?' Catharina answered, 'Yes, I know well\nwhat I am saying. God preserve him, and let him never fall into the\nhands of his enemies. Jesus, be Thou his guide!' She uttered these\nwords with abundant tears. Anna said, 'I think that woman is not in\nher senses.' Catharina's kind wish increased my tears, and I said,\n'Catharina shows that she is a true Christian, and sympathises with\nme; God reward her, and hear her and me!' Upon this Anna was silent,\nand has not been so talkative ever since. O God, Thou who art a\nrecompenser of all that is good, remember this in favour of\nCatharina, and as Thou heardest her at that time, hear her prayer in\nfuture, whatever may be her request! And you, my dear children, know\nthat if ever fortune so ordains it that you can be of any service\neither to her or her only son, you are bound to render it for my\nsake; for she was a comfort to me in my greatest need, and often took\nan opportunity to say a word which she thought would alleviate my\nsorrow.\n\nThe prison governor came as usual, about four o'clock, and let the\nwomen out, seating himself on the bench and placing the high stool\nwith the candle in front of him. He had brought a book with him, and\nread aloud prayers for a happy end, prayers for the hour of death,\nand prayers for one suffering temporal punishment for his misdeeds.\nHe did not forget a prayer for one who is to be burnt; in reading\nthis he sighed, so religious had he grown in the short time. When he\nhad read all the prayers, he got up and walked up and down, singing\nfuneral hymns; when he knew no more, he began again with the first,\ntill the women released him. Catharina complained that her son had\nbeen ill, and was greatly grieved about it. I entered into her\nsorrow, and said that she ought to mention her son's illness to the\nQueen, and then another would probably be appointed in her place; and\nI begged her to compose herself, as the child would probably be\nbetter again. During the evening meal the prison governor was very\nmerry, and related all sorts of coarse stories. When he was gone,\nAnna read the evening prayer. I felt very ill during this night, and\noften turned about in bed; there was a needle in the bed, with which\nI scratched myself; I got it out, and still have it.[67]\n\n [67] In the margin: 'The feather-bed had an old cover, and was\n fresh filled when I was lying in the roads; the needle, in the\n hurry, had therefore been left in.'\n\nOn August 14, when the prison governor opened the door early, the\nwomen told him that I had been very ill in the night. 'Well, well,'\nhe answered, 'it will soon be better.' And when the women were ready\nto go to the Queen (which they were always obliged to do), Anna said\nto Catharina, outside the door, 'What shall we say to the Queen?'\nCatharina answered: 'What shall we say, but that she is silent and\nwill say nothing!' 'You know very well that the Queen is displeased\nat it.' 'Nevertheless, we cannot tell a lie;' answered Catharina;\n'she says nothing at all, so it would be a sin.'[68] Catharina came\nback to the mid-day meal, and said that the Queen had promised to\nappoint another in her stead; in the afternoon, she managed secretly\nto say a word to me about the next chamber, which she imagined was\nbeing put in readiness for me and for no one else; she bid me good\nnight, and promised to remember me constantly in her prayers. I\nthanked her for her good services, and for her kind feeling towards\nme.\n\n [68] In the margin: 'I myself heard this conversation.'\n\nAbout four o'clock the prison governor let her and Anna out. He sang\none hymn after another, went to the stairs, and the time appeared\nlong to him, till six o'clock, when Anna returned with Maren Blocks.\nAt the evening meal the prison governor again told stories of his\nmarriage, undoubtedly for the sake of amusing Maren. Anna left me\nalone, and I lay quiet in silence. Maren could not find an\nopportunity of speaking with me the whole evening, on account of\nAnna. Nothing particular happened on August 15 and 16.\n\nWhen the prison governor let out Anna in the morning and afternoon,\nMaren Blocks remained with me, and the prison governor went his own\nway and locked the door, so that Maren had opportunity of talking\nwith me alone. She told me different things; among others, that the\nQueen had given my clothes to the three women who had undressed me,\nthat they might distribute them amongst themselves. She asked me\nwhether I wished to send a message to my sister Elizabeth. I thanked\nher, but said that I had nothing good to tell her. I asked Maren for\nneedles and thread, in order to test her. She replied she would\ngladly procure them for me if she dared, but that it would risk her\nwhole well-being if the Queen should know it; for she had so strictly\nforbidden that anyone should give me either pins or needles. I\ninquired 'For what reason?' 'For this reason,' she replied, 'that you\nmay not kill yourself.' I assured her that God had enlightened me\nbetter than that I should be my own murderer. I felt that my cross\ncame from the hand of the Lord, that He was chastising me as His\nchild; He would also help me to bear it; I trusted in Him to do so.\n'Then I hope, dear heart,' said Maren, 'that you will not kill\nyourself; then you shall have needles and thread; but what will you\nsew?' I alleged that I wished to sew some buttons on my white\nnight-dress, and I tore off a pair, in order to show her afterwards\nthat I had sewn them on.\n\nNow it happened that I had sewn up some ducats in a piece of linen\nround my knee; these I had kept, as I pulled off the stockings myself\nwhen they undressed me, and Anna had at my desire given me a rag, as\nI pretended that I had hurt my leg. I sewed this rag over the\nleather. They all imagined that I had some secret malady, for I lay\nin the linen petticoat they had given me, and went to bed in my\nstockings. Maren imagined that I had an issue on one leg, and she\nconfided to me that a girl at the court, whom she mentioned by name,\nand who was her very good friend, had an issue of which no one knew\nbut herself, not even the woman who made her bed. I thought to\nmyself, you keep your friend's secret well; I did not, however, make\nher any wiser, but let her believe in this case whatever she would. I\nwas very weak on those two days, and as I took nothing more than\nlemon and beer, my stomach became thoroughly debilitated and refused\nto retain food. When Maren told the prison governor of this, he\nanswered, 'All right, her heart is thus getting rid of its evil.'\nAnna was no longer so officious, but the prison governor was as merry\nas ever.\n\nOn August 17 the prison governor did not open the door before eight\no'clock, and Anna asked him how it was that he had slept so long. He\njoked a little; presently he drew her to the door and whispered with\nher. He went out and in, and Anna said so loudly to Maren, that I\ncould hear it (although she spoke as if she were whispering), 'I am\nso frightened that my whole body trembles, although it does not\nconcern me. Jesus keep me! I wish I were down below!' Maren looked\nsad, but she neither answered nor spoke a word. Maren came softly up\nto my bed and said, 'I am sure some one is coming to you.' I\nanswered, 'Let him come, in God's name.' Presently I heard a running\nup and down stairs, and also overhead, for the Commissioners came\nalways through the apartments, in order not to cross the square. My\ndoors were closed again. Each time that some one ran by on the\nstairs, Anna shuddered and said, 'I quite tremble.'\n\nThis traffic lasted till about eleven. When the prison governor\nopened the door, he said to me, 'Leonora, you are to get up and go to\nthe gentlemen.' God knows that I could hardly walk, and Anna\nfrightened me by saying to Maren, 'Oh! the poor creature!' Maren's\nhands trembled when she put on my slippers. I could not imagine\nanything else than that I was to be tortured, and I consoled myself\nwith thinking that my pain could not last long, for my body was so\nweary that it seemed as if God might at any moment take me away. When\nMaren fastened the apron over my long dress, I said: 'They are indeed\nsinning heavily against me; may God give me strength.' The prison\ngovernor hurried me, and when I was ready, he took me by the arm and\nled me. I would gladly have been free of his help, but I could not\nwalk alone. He conducted me up to the next story, and there sat Count\nRantzow, Skack, Retz, Gabel, and Krag, round the table.\n\nThey all rose when I entered, and I made them a reverence as well as\nI was able. A small low seat had been placed for me in the middle, in\nfront of the table. The Chancellor asked me whether I had not had\nmore letters than those taken from me in England. I answered that I\nhad not had more; that all my letters had been then taken from me.\nHe asked further, whether I had at that time destroyed any letters.\n'Yes,' I answered, 'one I tore in two, and threw it in a closet.'\n'Why did you do so?' enquired Count Rantzow. 'Because' I replied,\n'there were cyphers in it; and although they were of no importance, I\nfeared, notwithstanding, that they might excite suspicion.' Count\nRantzow said: 'Supposing the pieces were still forthcoming?' 'That\nwere to be wished,' I replied, 'for then it could be seen that there\nwas nothing suspicious in it, and it vexed me afterwards that I had\ntorn it in two.' Upon this the Chancellor drew forth a sheet of paper\nupon which, here and there, pieces of this very letter were pasted,\nand handed it to Krag, who gave it to me. Count Rantzow asked me if\nit were not my husband's handwriting. I answered that it was. He\nsaid: 'A part of the pieces which you tore in two have been found,\nand a part are lost. All that has been found has been collected and\ncopied.' He then asked the Chancellor for the copy, who gave it to\nCount Rantzow, and he handed it to me, saying, 'See there what is\nwanting, and tell us what it is that is missing.' I took it, and\nlooked over it and said: 'In some places, where there are not too\nmany words missing, I think I can guess what is lost, but where a\nwhole sentence is wanting, I cannot know.'\n\nMost of the letter had been collected without loss of intervening\npieces, and it all consisted of mirth and jest. He was telling me\nthat he had heard from Denmark that the Electoral Prince of Saxony\nwas to be betrothed with the Princess of Denmark;[E20] and he joked,\nsaying that they would grease their throats and puff out their cheeks\nin order that with good grace and voice they might duly trumpet\nforth each their own titles, and more of the same kind, all in high\ncolouring. He described the way in which Count Rantzow contrived to\nlet people know his titles; when he had a dinner-party, there was a\nman employed to read aloud his titles to the guests, asking first\neach separately, whether he knew his titles; if there was anyone who\ndid not know them, the secretary must forthwith come and read them\naloud.\n\n [E20] Leonora refers to the betrothal of Prince Johan George of\n Saxony and Anna Sophia, the eldest daughter of Fredrik III., of\n which an account occurs in the sequel.\n\nIt seemed that Count Rantzow referred all this to himself, for he\nasked me what my husband meant by it. I replied that I did not know\nthat he meant anything but what he had written; he meant undoubtedly\nthose who did such things. The Chancellor averted his face from Count\nRantzow, and his lips smiled a little; Gabel also did the same. Among\nother things there were some remarks about the Electoral Prince, that\nhe probably cherished the hope of inheriting the Crown of Denmark;\n'mais j'espere ... cela ne se fera point.' Count Rantzow enquired as\nto the words which were wanting. I said, if I remembered rightly, the\nwords had been, 'qu'en 300 ans.' He enquired further as to the\nexpressions lacking here and there, some of which I could not\nremember exactly, though they were of no importance. I expressed my\nopinion that they could easily gather what was wanting from the\npreceding and following words; it was sufficiently evident that all\nwas jest, and this was apparent also to Gabel, who said, 'Ce n'est\nque raillerie.' But Count Rantzow and the General would not allow it\nto pass as jest.\n\nSkack said: 'One often means something else under the cloak of jest,\nand names are used when others are intended.' For in the letter there\nwas something said about drinking out; there was also an allusion\nmade to the manners of the Swiss at table, and all the titles of the\ncanton nobles were enumerated, from which Skack thought that the\nnames of the cities might have another signification. I did not\nanswer Skack; but as Count Rantzow continued to urge me to say what\nmy husband had meant by it, I replied that I could not know whether\nhe had had another meaning than that which was written. Skack shook\nhis head and thought he had, so I said: 'I know no country where the\nsame customs are in vogue at meals as in Switzerland; if there are\nother places where the same customs prevail, he may perhaps have\nmeant these also, for he is only speaking of drinking.'\n\nGabel said again, 'It is only jest.' The cyphers, for the sake of\nwhich I had torn the letter in two, were fortunately complete, and\nnothing was missing. Count Rantzow gave me a sheet of paper, to which\npieces of my lord's letter were pasted, and asked me what the cyphers\nmeant. I replied, 'I have not the key, and cannot solve them out of\nmy head.' He expressed his opinion that I could do it. I said I could\nnot. 'Well, they have been read,' he said, 'and we know what they\nsignify.' 'All the better,' I answered. Upon this, he gave me the\ninterpretation to read, and the purport of it was that our son had\nwritten from Rome, asking for money, which was growing short, for the\nyoung nobleman was not at home. I gave the paper back to Count\nRantzow without saying anything. Count Rantzow requested the\nTreasurer that he should read the letter, and Rantzow began again\nwith his questions wherever anything was wanting, requesting that I\nshould say what it was. I gave him the same answer as before; but\nwhen in one passage, where some words were missing, he pressed me\nhard to say them, and it was evident from the context that they were\nironical (since an ironical word was left written), I said: 'You can\nadd as much of the same kind as pleases you, if one is not enough; I\ndo not know them.' Gabel again said, 'Ce n'est que raillerie.'[E21]\n\n [E21] A copy of the fragments which had been recovered of this\n letter is still in existence.\n\nNo further questions were then made respecting the letters; but Count\nRantzow enquired as to my jewels, and asked where the large diamond\nwas which my husband had received in France.[E22] I replied that it\nhad long been sold. He further asked where my large drop pearls\nwhere, which I had worn as a feather on my hat, and where my large\npearl head-ornament was. 'All these,' I replied, 'have long been\nsold.' He asked further whether I had then no more jewels. I\nanswered, 'I have none now.' 'I mean,' he said, 'elsewhere.' I\nreplied, 'I left some behind.' 'Where, then?' he asked. 'At Bruges,'\nI replied. Then he said: 'I have now somewhat to ask you, madame,\nthat concerns myself. Did you visit my sister in Paris the last time\nyou were there?' I replied, 'Yes.' He asked whether I had been with\nher in the convent, and what was the name of the convent. I informed\nhim that I had been in the convent, and that it was the Convent des\nFilles Bleues. At this he nodded, as if to confirm it. He also wished\nto know whether I had seen her. I said that no one in the convent\nmight be seen by anyone but parents; even brothers and sisters were\nnot allowed to see them.[E23] 'That is true,' he said, and then rose\nand gave me his hand. I begged him to induce his gracious Majesty to\nhave pity on me, but he made no answer. When the Treasurer Gabel\ngave me his hand, I begged the same favour of him. He replied, 'Yes,\nif you will confess,' and went out without waiting for a reply.\n\n [E22] Ulfeldt received this present probably in 1647, when in\n France as ambassador, on which occasion Queen Anna is known to have\n presented to Leonora a gold watch set with diamonds of great value.\n\n [E23] The lady alluded to is Helvig Margaretha Elizabeth Rantzow,\n widow of the famous General Josias Rantzow, who died as a marechal\n of France. She had become a Romanist, and took the veil after her\n husband's death. Subsequently she founded the new order of the\n Annunciata. In 1666 the first convent of this order, of which she\n was abbess, removed to Hildesheim, where she died in 1706.\n\nFor more than three hours they had kept up the interrogation. Then\nthe prison governor came in and said to me: 'Now you are to remain in\nhere; it is a beautiful chamber, and has been freshly whitewashed;\nyou may now be contented.' Anna and Maren also came in. God knows, I\nwas full of care, tired and weary, and had insufferable headache;\nyet, before I could go to rest, I had to sit waiting until the\nbedstead had been taken out of the 'Dark Church' and brought hither.\nAnna occupied herself meanwhile in the Dark Church, in scraping out\nevery hole; she imagined she might find something there, but in vain.\nThe woman who was to remain with me alone then came in. Her pay was\ntwo rix-dollars a week; her name is Karen, the daughter of Ole. After\nthe prison governor had supped with the woman and Maren, Anna and\nMaren Blocks bade me good night; the latter exhibited great\naffection. The prison governor bolted two doors before my innermost\nprison. In the innermost door there is a square hole, which is\nsecured with iron cross-bars. The prison governor was going to attach\na lock to this hole, but he forebore at Karen's request, for she said\nshe could not breathe if this hole were closed. He then affixed locks\nto the door of the outer chamber, and to the door leading to the\nstairs; he had, therefore, four locks and doors twice a day to lock\nand unlock.\n\nI will here describe my prison. It is a chamber, seven of my paces\nlong and six wide; there are in it two beds, a table, and two stools.\nIt was freshly whitewashed, which caused a terrible smell; the floor,\nmoreover was so thick with dirt, that I imagined it was of loam,\nthough it was really laid with bricks. It is eighteen feet high, with\na vaulted ceiling, and very high up is a window which is two feet\nsquare. In front of it are double thick iron bars, besides a\nwire-work, which is so close that one could not put one's little\nfinger into the holes. This wire-work had been thus ordered with\ngreat care by Count Rantzow (so the prison governor afterwards told\nme), so that no pigeons might bring in a letter--a fact which he had\nprobably read in a novel as having happened. I was weak and deeply\ngrieved in my heart; I looked for a merciful deliverance, and an end\nto my sorrow, and I sat silent and uncomplaining, answering little\nwhen the woman spoke to me. Sometimes in my reverie I scratched at\nthe wall, which made the woman imagine that I was confused in my\nhead; she told this to the prison governor, who reported it to the\nQueen, and during every meal-time, when the door was open, she never\nfailed to send messengers to enquire how it fared with me, what I\nsaid, and what I was doing.\n\nThe woman had, however, not much to tell in obedience to the oath\nshe, according to her own statement, had taken in the presence of the\nprison governor. But afterwards she found some means to ingratiate\nherself. And as my strength daily decreased, I rejoiced at the\nprospect of my end, and on August 21 I sent for the prison governor,\nand requested him to apply for a clergyman who could give me the\nsacrament. This was immediately granted, and His Majesty's Court\npreacher, Magister Mathias Foss, received orders to perform for me\nthe duties of his office, and exhorted me, both on behalf of his\noffice and in consequence of the command he had received, not to\nburden my conscience; I might rest assured, he said, that in this\nworld I should never see my husband again, and he begged me to say\nwhat I knew of the treason. I could scarcely utter a word for\nweeping; but I said that I could attest before God in heaven, from\nwhom nothing is hidden, that I knew nothing of this treason. I knew\nwell I should never see my husband again in this life; I commended\nhim to the Almighty, who knew my innocence; I prayed God only for a\nblessed end and departure from this evil world; I desired nothing\nfrom the clergyman but that he should remember me in his prayers,\nthat God might by death put an end to my affliction. The clergyman\npromised faithfully to grant my request. It has not pleased God to\nhear me in this: He has willed to prove my faith still further, by\nsending to me since this time much care, affliction, and adversity.\nHe has helped me also to bear the cross, and has Himself supported\nits heaviest end; His name be praised for ever. When I had received\nthe Lord's Supper, M. Foss comforted me and bid me farewell.\n\nI lay silently for three days after this, taking little or nothing.\nThe prison governor often enquired whether I wished for anything to\neat or drink, or whether he should say anything to the King. I\nthanked him, but said I required nothing.\n\nOn August 25 the prison governor importuned me at once with his\nconversation, expressing his belief that I entertained an evil\nopinion of the Queen. He inferred it from this: the day before he had\nsaid to me that His Majesty had ordered that whatever I desired from\nthe kitchen and cellar should be at once brought to me, to which I\nhad answered, 'God preserve His Majesty; he is a good sovereign; may\nhe show clemency to evil men!' He had then said, 'The Queen is also\ngood,' to which I had made no answer. He had then tried to turn the\nconversation to the Queen, and to hear if he could not draw out a\nword from me; he had said: 'The Queen is sorry for you that you have\nbeen so led away. It grieves her that you have willed your own\nunhappiness; she is not angry; she pities you.' And when I made no\nanswer, he repeated it again, saying from time to time, 'Yes, yes, my\ndear lady, it is as I say.' I was annoyed at the talk, and said,\n'Dieu vous punisse!' 'Ho, ho!' he said, misinterpreting my words, and\ncalling Karen, he went out and closed the doors. Thus unexpectedly I\ngot rid of him. It was ridiculous that the woman now wanted to oblige\nme to attend to what the prison governor had said. I begged her to\nremember that she was now not attending on a child (she had before\nbeen nurse to children). She could not so easily depart from her\nhabit, and for a long time treated me as a child, until at length I\nmade her comprehend that this was not required.\n\nWhen I perceived that my stomach desired food and could retain it, I\nbecame impatient that I could not die, but must go on living in such\nmisery. I began to dispute with God, and wanted to justify myself\nwith Him. It seemed to me that I had not deserved such misfortune. I\nimagined myself far purer than David was from great sins, and yet he\ncould say, 'Verily I have cleansed my heart in vain, and washed my\nhands in innocency. For all the day long have I been plagued, and\nchastened every morning.' I thought I had not deserved so exceedingly\ngreat a chastisement as that which I was receiving. I said with Job,\n'Show me wherefore thou contendest with me. Is it good unto thee that\nthou shouldest oppress, that thou shouldest despise the work of thine\nhands?' I repeated all Job's expressions when he tried to justify\nhimself, and it seemed to me that I could justly apply them to\nmyself. I cursed with him and Jeremiah the day of my birth, and was\nvery impatient; keeping it, however, to myself, and not expressing it\naloud. If at times a word escaped me, it was in German (since I had\ngenerally read the Bible in German), and therefore the woman did not\nunderstand what I was saying. I was very restless from coughing, and\nturned from side to side on the bed. The woman often asked me how I\nwas. I begged her to leave me quiet and not to speak to me. I was\nnever more comfortable than in the night when I observed that she was\nsleeping; then, unhindered, I could let my tears flow and give free\nvent to my thoughts. Then I called God to account. I enumerated\neverything that I had innocently suffered and endured during my life,\nand I enquired of God whether I had deviated from my duty? Whether I\nought to have done less for my husband than I had done? Whether the\npresent was my recompense for not having left him in his adversity?\nWhether I was to be now tortured, tormented, and scorned for this?\nWhether all the indescribable misfortunes which I had endured with\nhim were not enough, that I had been reserved for this irremediable\nand great trouble? I do not wish to conceal my unreasonableness. I\nwill confess my sins. I asked if still worse misfortunes were in\nstore for me for which I was to live? Whether there was any\naffliction on earth to be compared to mine? I prayed God to put an\nend to my sufferings, for it redounded in no wise to his honour to\nlet me live and be so tormented. I was after all not made of steel\nand iron, but of flesh and blood. I prayed that He would suggest to\nme, or inform me in a dream, what I was to do to shorten my misery.\n\nWhen I had long thus disputed and racked my brains, and had also wept\nso bitterly that it seemed as if no more tears remained, I fell\nasleep, but awoke with terror, for I had horrible fancies in my\ndreams, so that I feared to sleep, and began again to bewail my\nmisery. At length God looked down upon me with his eye of mercy, so\nthat on August 31 I had a night of quiet sleep, and just as day was\ndawning I awoke with the following words on my lips: 'My son, faint\nnot when thou art rebuked of the Lord; for whom the Lord loveth he\nchasteneth, and scourgeth every son whom he receiveth.' I uttered the\nlast words aloud, thinking that the woman was sleeping; possibly she\nawoke at the moment, and she asked me whether I wished for anything.\nI answered 'No.' 'You were speaking,' she said, 'and you mentioned\nyour stockings; I could not understand the rest.' I replied, 'It must\nhave been then in my sleep. I wish for nothing.'\n\nI then lay quietly thinking. I perceived and confessed my folly, that\nI, who am only dust and ashes, and decay, and am only fit for the\ndunghill, should call God to account, should dispute with my Creator\nand his decrees, and should wish to censure and question them. I\nbegan to weep violently, and I prayed fervently and from my heart for\nmercy and forgiveness. While I had before boasted with David, and\nbeen proud of my innocence, now I confessed with him that before God\nthere is none that doeth good; no, not one. While before I had spoken\nfoolishly with Job, I now said with him that I had 'uttered that I\nunderstood not; things too wonderful for me which I knew not.' I\nbesought God to have mercy on me, relying on his great compassion. I\ncited Moses, Joshua, David, Jeremiah, Job, Jonah, and others, all\nhighly endowed men, and yet so weak that in the time of calamity\nthey grumbled and murmured against God. I prayed that He would in his\nmercy forgive me, the frailest of earthen vessels, as I could not\nafter all be otherwise than as He had created me. All things were in\nhis power; it was easy to Him to give me patience, as He had before\nimparted to me power and courage to endure hard blows and shocks. And\nI prayed God (after asking forgiveness of my sins) for nothing else\nthan good patience to await the period of my deliverance. God\ngraciously heard me. He pardoned not only my foolish sins, but He\ngave me that also for which I had not prayed, for day by day my\npatience increased. While I had often said with David, 'Will the Lord\ncast off for ever? and will he be favourable no more? Is his mercy\nclean gone for ever? doth his promise fail for evermore? Hath God\nforgotten to be gracious? Hath He in anger shut up his tender\nmercies?' I now continued with him, 'This is my infirmity, but I will\nremember the years of the right hand of the Most High.' I said also\nwith Psalm cxix.: 'It is good for me that I have been afflicted; that\nI might learn thy statutes.'\n\nThe power of God was working within me. Many consolatory sentences\nfrom the Holy Scriptures came into my mind; especially these:--'If so\nbe that we suffer with Christ, that we may be also glorified\ntogether.' Also: 'We know that all things work together for good to\nthem that love God.' Also: 'My grace is sufficient for thee, for my\nstrength is made perfect in weakness.' I thought especially often of\nChrist's words in St. Luke, 'Shall not God avenge his own elect,\nwhich cry day and night unto him, though he bear long with them? I\ntell you that he will avenge them speedily.' I felt in my trouble how\nuseful it is to have learned psalms and passages from the Bible in\nyouth. Believe me, my children, that it has been a great consolation\nto me in my misery. Therefore, cultivate now in your youth what your\nparents taught you in childhood; now, while trouble visits you less\nseverely, so that when it comes, you may be ready to receive it and\nto comfort yourselves with the Word of God.\n\nI began by degrees to feel more at peace, and to speak with the\nwoman, and to answer the prison governor when he addressed me. The\nwoman told me sundry things, and said that the prison governor had\nordered her to tell him everything that I spoke or did, but that she\nwas too wise to do such a thing; that she understood now better than\nshe had done at first how to behave. He went out, but she remained\nshut up with me, and she would be true to me. And as it appeared that\nI did not at once believe what she said, she swore it solemnly, and\nprayed God to punish her if ever she acted falsely towards me. She\nstroked and patted my hand, and laid it against her cheek, and begged\nthat I would believe her, using the words, 'My dearest lady, you can\nbelieve me; as truly as I am a child of God, I will never deceive\nyou! Now, is not that enough?' I answered, 'I will believe you;'\nthinking at the same time that I would do and say nothing but what\nshe might divulge. She was very glad that she had induced me to\nspeak, and said, 'When you lay so long silent, and I had no one with\nwhom I could speak, I was sad, and determined that I would not long\nlead this life, even if they gave me double as much, for I should\nhave become crazed. I was afraid for you, but still more for myself,\nthat my head would give way.'\n\nShe went on talking in this way, introducing also various merry\nstories. When she was young she had been in the service of a\nclergyman, who encouraged his domestics in the fear of God, and there\nshe had learned prayers and sentences from the Bible by heart; she\nknew also the Children's Primer, with the explanatory remarks, and\nsang tolerably well. She knew in some measure how she should walk\nbefore God and behave towards her neighbour; but she acted contrary\nto her knowledge--for she had a malicious temper. She was an elderly\nwoman, but she liked to reckon herself as middle-aged. It appeared\nthat in her youth she had been pretty and rather dissolute, since\neven now she could not lay aside her levity, but joked with the\ntower-warder, and the prison governor's coachman, a man of the name\nof Peder, and with a prisoner named Christian (more will presently be\nsaid with regard to this prisoner; he was free to go about the\ntower).[69]\n\n [69] When I took my meals, the woman had opportunity of talking\n with the three men. The coachman helped the tower-warder Rasmus to\n bring up the food. [Marginal note.]\n\nMaren Blocks often sent me a message through this coachman, besides\nvarious kinds of candied sugar and citron, letting me know from time\nto time whether anything new was occurring. All this had to be done\nthrough the woman. One day she came in when the doors were closed,\nand brought me a message from Maren Blocks, saying, 'My lady, if you\nwill now write to your children in Skaane, there is a safe\nopportunity for you to do so.' I answered, 'My children are not in\nSkaane, yet if I can send a message to Skaane, I have a friend there\nwho will probably let me know how it fares with my children.' She\ngave me a piece of crumpled paper and a pencil. I wrote a few words\nto F. Margrete Rantzow,[E24] saying that she probably knew of my\nmiserable condition, but supposing that her friendship was not\nlessened by it, and begging her to let me know how my children were,\nand from what cause they had come to Skaane, as I had been informed\nwas the case, though I did not believe it. This was what I wrote and\ngave to the woman. I heard nothing further of it, and I imagine that\nshe had been ordered to find out to whom I wrote, &c. (They have been\nbusy with the idea that some of you, my dear children, might come to\nSkaane.) I sewed up the letter or slip of paper in such a manner that\nit could not be opened without making it apparent. I asked the woman\nseveral times if she knew whether the letter had been sent away. She\nalways answered that she did not know, and that with a morose\nexpression, and at last she said (when I once more asked her to\nenquire of Peder), 'I suppose that the person who ought to have it\nhas got it.' This answer made me reflect, and since then I asked no\nfurther.\n\n [E24] Margrete Rantzow was the sister of that Birgitte Rantzow to\n whom there is an allusion in the Autobiography of Leonora, where\n she relates the examination to which she was subjected at Malmoe.\n Margrete's husband was Ove Thott, a nobleman in Skaane, who had\n taken an important part in the preparations for a rising against\n the Swedes, in which Corfitz Ulfeldt was implicated.\n\nI remained all this time in bed, partly because I had nothing with\nwhich to beguile the time, and partly because of the cold, for no\nstove was placed in my prison till after the New Year. Occasionally I\nrequested the woman to manage, through Peder, that I should have a\nlittle silk or thread, that I might beguile the time by embroidering\na piece of cloth that I had; but the answer I received was that he\ndared not. A long time afterwards it came to my knowledge that she\nhad never asked Peder for it. There was trouble enough, however, to\noccupy my thoughts without my needing to employ the time in\nhandiwork.\n\nIt was on September 2 that I heard some one moving early overhead, so\nI asked the woman if she knew whether there was a chamber there (for\nthe woman went up every Saturday with the night-stool). She answered\nthat there was a prison there like this, and outside was the rack\n(which is also the case). She observed that I showed signs of fear,\nand she said, 'God help! Whoever it is that is up there is most\nassuredly to be tortured.' I said, 'Ask Peder, when the doors are\nunlocked, whether there is a prisoner there.' She said she would do\nso, and meanwhile she kept asking herself and me who it might be. I\ncould not guess; still less did I venture to confess my fear to her,\nwhich she nevertheless perceived, and therefore increased; for after\nshe had spoken with Peder, about noon,[70] and the doors were locked,\nshe said, 'God knows who it is that is imprisoned there! Peder would\ntell me nothing.' She said the same at the evening meal, but added\nthat she had asked him, and that he would give no answer. I calmed\nmyself, as I heard no more footsteps above, and I said, 'There is no\nprisoner up there.'[71] 'How do you know that?' she asked. 'I gather\nit from the fact,' I said, 'that since this morning I have heard no\none above; I think if there were anyone there, they would probably\ngive him something to eat.' She was not pleased that my mind was\nquieted, and therefore she and Peder together endeavoured to trouble\nme.\n\n [70] I could not see when she spoke with any one, for she did so on\n the stairs. [Marginal note.]\n\n [71] In the margin is added: 'There was none.'\n\nOn the following day, when the doors were being locked after the\nmid-day dinner (which was generally Peder's task), and he was pulling\nto my innermost door, which opens inside, he put in his head and\nsaid, 'Casset!' She was standing beside the door, and appeared as if\nshe had not rightly understood him, saying, 'Peder spoke of some one\nwho is in prison, but I could not understand who it is.' I understood\nhim at once, but also behaved as if I had not. No one knows but God\nwhat a day and night I had. I turned it over in my mind. It often\nseemed to me that it might be that they had seized him, although\nCassetta was a subject of the King of Spain; for if treason is\nsuspected, there is no thought given as to whose subject the man\nsuspected may be. I lay in the night secretly weeping and lamenting\nthat the brave man should have come into trouble for my sake, because\nhe had executed my lord's will, and had followed me to England, where\nwe parted, I should say, when Petcon and his company separated us and\ncarried me away.\n\nI lay without sleep till towards day, then I fell into a dream which\nfrightened me. I suppose my thoughts caused it. It came before me\nthat Cassetta was being tortured in the manner he had once described\nto me that a Spaniard had been tortured: four cords were fastened\nround his hands and feet, and each cord was made secure in a corner\nof the room, and a man sometimes pulled one cord and sometimes\nanother; and since it seemed to me that Cassetta never screamed, I\nsupposed that he was dead, and I shrieked aloud and awoke. The woman,\nwho had long been awake, said: 'O God! dear lady, what ails you? Are\nyou ill? You have been groaning a long time, and now you screamed\nloudly.' I replied, 'It was in my dream; nothing ails me.' She said\nfurther, 'Then you have had a bad dream?' 'That may well be,' I\nanswered. 'Oh, tell me what you have dreamt; I can interpret\ndreams.' I replied, 'When I screamed I forgot my dream, otherwise no\none can interpret dreams better than I.' I thank God I do not regard\ndreams; and this dream had no other cause than what I have said. When\nthe door was locked after the mid-day meal, the woman said of herself\n(for I asked no further respecting the prisoners), 'There is no one\nimprisoned there; shame on Peder for his nonsense!' I asked him who\nwas imprisoned there, and he laughed at me heartily. 'There is no one\nthere, so let your mind be at peace.' I said, 'If my misfortunes were\nto involve others, it would be very painful to me.'\n\nThus matters went on till the middle of September, and then two of\nour servants were brought as prisoners and placed in arrest; one Nils\nKaiberg, who had acted as butler, and the other Frans, who had been\nin our service as a lacquey. After having been kept in prison for a\nfew weeks and examined they were set at liberty. At the same time two\nFrenchmen were brought as prisoners: an old man named La Rosche, and\na young man whose name I do not know. La Rosche was brought to the\ntower and was placed in the witch-cell; a feather-bed had been thrown\ndown, and on this he lay; for some months he was never out of his\nclothes. His food consisted of bread and wine; he refused everything\nelse. He was accused of having corresponded with Corfitz, and of\nhaving promised the King of France that he would deliver Crooneborg\ninto his hands.[72] This information had been given by Hannibal\nSehested, who was at that time in France, and he had it from a\ncourtesan who was then intimate with Hannibal, but had formerly been\nin connection with La Rosche, and probably afterwards had quarrelled\nwith him. There was no other proof in favour of the accusation.\nProbably suspicion had been raised by the fact that this La Rosche,\nwith the other young man, had desired to see me when I was in arrest\nin Dover, which had been permitted, and they had paid me their\nrespects. It is possible that he had wished to speak with me and to\ntell me what he had heard in London, and which, it seemed to him,\nexcited no fears in me. But as I was playing at cards with some\nladies who had come to look at me, he could not speak with me; so he\nasked me whether I had the book of plays which the Countess of\nPembroke had published.[E25] I replied, 'No'. He promised to send it\nme, and as I did not receive it, I think he had written in it some\nwarning to me, which Braten afterwards turned to his advantage.\n\n [72] Did not this accord well with the statement that my lord had\n offered the kingdom of Denmark to two potentates? [Marginal note.]\n\n [E25] The book in question is probably Philip Sidney's work, 'The\n Countess of Pembroke's Arcadia,' a famous book of its time, which\n Leonora, who does not seem to have known it, has understood to be a\n book by the Countess of Pembroke. It is true, however, that\n Philip's sister, Mary Sidney, Countess of Pembroke, had translated\n a French play, Antonius (1592, and again 1595).\n\nHowever all this may be, La Rosche suffered innocently, and could\nprove upon oath that he had never spoken with my lord in his life,\nand still less had corresponded with him.[73] In short, after some\nmonths of innocent suffering, he was set at liberty and sent back to\nFrance. The other young man was confined in an apartment near the\nservants' hall. He had only been apprehended as a companion to the\nother, but no further accusation was brought against him.[E26] At\nfirst, when these men were imprisoned, there was a whispering and\ntalking between the prison governor and the woman, and also between\nPeder and her; the prison governor moreover himself locked my door. I\nplainly perceived that there was something in the wind, but I made no\nenquiries. Peder at length informed the woman that they were two\nFrenchmen, and he said something about the affair, but not as it\nreally was. Shortly before they were set at liberty the prison\ngovernor said, 'I have two parle mi franco in prison; what they have\ndone I know not.' I made no further enquiries, but he jested and\nsaid, 'Now I can learn French.' 'That will take time,' said I.\n\n [73] In the margin is noted: 'I had never seen La Rosche nor his\n companion till I did so at Dover.'\n\n [E26] La Roche Tudesquin had some time been in the Danish army, but\n had returned to France when Hannibal Sehested, while in Paris as\n Ambassador from the King of Denmark, received information from a\n certain Demoiselle Langlois that La Roche was implicated in a\n conspiracy for surrendering the principal Danish fortresses to a\n foreign prince. He and a friend of his, Jaques Beranger, were\n arrested in Brussels in September 1663, but not, as Leonora says,\n immediately brought to Copenhagen. The Spanish Government did not\n consent to their extradition till the following year, and they were\n not placed in the Blue Tower till June 1664. La Roche seems to have\n been guilty of peculation while in the Danish service, but the\n accusation of treason seems to have been unfounded.\n\nIn the same month of September died Count Rantzow. He did not live to\nsee the execution of an effigy, which he so confidently had hoped\nfor, being himself the one who first had introduced this kind of\nmockery in these countries.[E27]\n\n [E27] In the MS. a pen is drawn through this paragraph, of which\n the contents were to form part of the Preface. The date of Count\n Rantzow is moreover not correctly given; he died on November 8,\n five days before the execution of Ulfeldt's effigy.\n\nOn October 9 our Princess Anna Sophia was betrothed to the Electoral\nPrince of Saxony. On the morning of the day on which the festivities\nwere to take place I said to the woman, 'To-day we shall fast till\nevening.' For I thought they would not think of me, and that I should\nnot receive any of the remains until the others had been treated, at\nany rate, to dinner. She wished to know the reason why we were to\nfast. I answered, 'You shall know it this evening.' I lay and thought\nof the change of fortune: that I, who twenty-eight years ago had\nenjoyed as great state as the Princess, should now be lying a\ncaptive, close by the very wall where my bridal chamber had been;\nthank God, that it afflicted me but little. Towards noonday, when the\ntrumpets and kettledrums were sounding, I said, 'Now they are\nconducting the bride across the square to the great hall.' 'How do\nyou know that?' said the woman. 'I know it,' I said; 'my spirit tells\nme so.' 'What sort of spirit is that?' she asked. 'That I cannot\ntell you,' I replied. And as the trumpets blew every time that a new\ncourse of dishes and sweets were produced, I mentioned it; and before\nthey were served the kettledrums were sounded. And as they were\nserved on the square in front of the kitchen, I said each time, 'We\nshall have no dinner yet.' When it was nearly three o'clock, the\nwoman said, 'My stomach is quite shrunk up; when shall we have\ndinner?' I answered, 'Not for a long time yet; the second course is\nonly now on the table; we shall have something at about seven\no'clock, and not before.' It was as I said. About half-past seven the\nprison governor came and excused himself, saying that he had asked\nfor the dinner, but that all hands in the kitchen were occupied. The\nwoman, who had always entertained the idea that I was a witch, was\nnow confirmed in her opinion.[74]\n\n [74] In the margin is added: 'The prison governor told the woman\n about the magnificence of the festivity and Peder also told her of\n it, so that it seemed to her that I could know somewhat from\n customs of former times.'\n\nOn the following day knights were dubbed, and each time when the\ntrumpets blew I did not only say, 'Now they have made a knight' (for\nI could hear the herald calling from the window, though I could not\nunderstand what he said), but even who had been made a knight; for\nthis I guessed, knowing who were in the Council who were not knights\nbefore; and because it was as I said, the woman believed for certain\nthat I was an enchantress. I perceived this, as she put questions to\nme concerning things which I could not know, and to which I often\ngave equivocal answers. I thought perhaps that the fear she had that\nI could know what would happen might hinder her from entangling me\nwith lies. Since then she whispered much less with the prison\ngovernor. She told of a person whom she regarded as a witch, whose\npower, however, consisted in nothing else than in the science of\ncuring French pox, and causing the miscarriage of bad women, and\nother improprieties. She had had much intercourse with this woman.\n\nSome time after the departure of the Electoral Prince it was\ndetermined that a wooden effigy should be subjected to capital\npunishment, and on the forenoon my chamber was opened, swept,\ncleaned, and strewed with sand.[75] When it was opened, towards noon,\nand the woman had been on the stairs, talking with the coachman, she\ncame in, and walking up to my bed, stood as if startled, and said\nhurriedly, 'Oh, Jesus! Lady, they are bringing your husband!' The\nnews terrified me, which she observed; for as she uttered it, I\nraised myself in the bed and stretched out my right arm, and was not\nable to draw it back again at once. Perhaps this vexed her, for I\nremained sitting in this way and not speaking a word; so she said,\n'My dearest lady, it is your husband's effigy.' To this I said, 'May\nGod punish you!' She then gave full vent to her evil tongue, and\nexpressed her opinion that I deserved punishment, and not she, and\nused many unprofitable words. I was quite silent, for I was very\nweak, and scarcely knew where I was. In the afternoon I heard a great\nmurmuring of people in the inner palace square, and I saw the effigy\nbrought across the street by the executioner on a wheelbarrow, and\nplaced in the tower below my prison.\n\n [75] The Queen wished that this wooden statue should be brought\n into my outer chamber, and so placed in front of the door that it\n would tumble into me when my inner door was opened; but the King\n would not permit it. [Addition in the margin.]\n\nThe next morning, at about nine o'clock, the effigy was wofully\ntreated by the executioner, but no sound came from it. At the mid-day\nmeal the prison governor told the woman how the executioner had cut\noff its head, and had divided the body into four quarters, which were\nthen placed on four wheels, and attached to the gallows, while the\nhead was exhibited on the town hall. The prison governor stood in the\nouter chamber, but he narrated all this in a loud tone, so that I\nmight hear it, and repeated it three times.[E28] I lay and thought\nwhat I should do; I could not show that I made but little of it, for\nthen something else perhaps would be devised to trouble me, and in\nthe hurry I could think of nothing else than saying to the woman with\nsadness, 'Oh, what a shame! speak to the prison governor and tell him\nto beg the King to allow the effigy to be taken down and not to\nremain as it is!' The woman went out, and spoke softly with the\nprison governor; but he answered aloud and said, 'Yes, indeed, taken\ndown! There will be more put up; yes, more up;' and kept on repeating\nthese words a good while.\n\n [E28] The execution took place on November 13. The King's order\n concerning it to the prison governor, Jochum Waltpurger, exists\n still. It is to this effect: 'V. G. T., Know that you have to\n command the executioner in our name, that to-day, November 13, he\n is to take the effigy of Corfitz, formerly called Count of Ulfeldt,\n from the Blue Tower where it is now, and bring it on a car to the\n ordinary place in the square in front of the castle; and when he\n has come to the place of justice, strike off the right hand and the\n head, whereafter he is to divide the body into four parts on the\n spot, and carry them away with him, whilst the head is to be placed\n on a spike on the Blue Tower for remembrance and execration.' The\n order was afterwards altered in this particular, that the head was\n to be placed on the town hall, and the four parts of the body one\n at each of the gates of the city. The executioner was subsequently\n ordered to efface the arms of Corfitz and his wife wherever they\n occurred in the town; for instance, on their pews in the churches.\n Leonora states in her Autobiography that the prison governor some\n time after told her that the Queen had desired that the effigy\n should be placed in the antechamber of Leonora's prison, and that\n she should be ordered to see it there; but that the king refused\n his consent.\n\nI lay silently thinking; I said nothing, but indulged in my own\nreflections. Sometimes I consoled myself, and hoped that this\ntreatment of the effigy was a token that they could not get the man;\nthen again fear asserted its sway. I did not care for the dishonour,\nfor there are too many instances of great men in France whose\neffigies have been burnt by the executioner, and who subsequently\narrived again at great honour.\n\nWhen the door was unlocked again for the evening meal, there was a\nwhispering between the prison governor and the woman. A lacquey was\nalso sent, who stood outside the outer door and called the prison\ngovernor to him (my bed stands just opposite the doors, and thus when\nall three doors are opened I can see the staircase door, which is the\nfourth). I do not know what the woman can have told the prison\ngovernor, for I had not spoken all day, except to ask her to give me\nwhat I required; I said, moreover, nothing more than this for several\ndays, so that the prison governor grew weary of enquiring longer of\nthe woman; for she had nothing to communicate to him respecting me,\nand she tormented him always with her desire to get away; she could\nnot longer spend her life in this way.\n\nBut as she received no other consolation from him than that he swore\nto her that she would never get away as long as she lived, for some\ndays she did nothing else than weep; and since I would not ask her\nwhy she wept, she came one day up to my bedside crying, and said, 'I\nam a miserable being!' I asked her why? what ailed her? 'I ail\nenough,' she answered; 'I have been so stupid, and have allowed\nmyself to be shut up here for the sake of money, and now you are\ncross with me and will not speak with me.' I said, 'What am I to say?\nyou wish perhaps to have something to communicate to the prison\ngovernor?' Upon this she began to call down curses on herself if she\nhad ever repeated to the prison governor a word that I had said or\ndone; she wished I could believe her and speak with her; why should\nshe be untrue to me? we must at any rate remain together as long as\nwe lived. She added many implorations as to my not being angry; I had\nindeed cause to be so; she would in future give me no cause for\nanger, for she would be true to me. I thought, 'You shall know no\nmore than is necessary.'\n\nI let her go on talking and relating the whole history of her\nlife--such events as occur among peasants. She had twice married\ncottagers, and after her last widowhood she had been employed as\nnurse to the wife of Holger Wind, so that she had no lack of stories.\nBy her first husband she had had a child, who had never reached\nmaturity, and her own words led me to have a suspicion that she had\nherself helped to shorten the child's days; for once when she was\nspeaking of widows marrying again, she said among other things,\n'Those who wish to marry a second time ought not to have children,\nfor in that case the husband is never one with the wife.' I had much\nto say against this, and I asked her what a woman was to do who had a\nchild by her first husband. She answered quickly, 'Put a pillow on\nits head.' This I could only regard as a great sin, and I explained\nit to her. 'What sin could there be,' she said, 'when the child was\nalways sickly, and the husband angry in consequence?' I answered as I\nought, and she seemed ill at ease. Such conversation as this gave me\nno good reason to believe in the fidelity which she had promised me.\n\nThe woman then took a different tack, and brought me word from the\ncoachman of all that was occurring. Maren Blocks sent me a\nprayer-book through her, and that secretly, for I was allowed no book\nof any kind, nor any needles and pins; respecting these the woman had\nby the Queen's order taken an oath to the prison governor. Thus the\nyear passed away. On New Year's day, 1664, the woman wished me a\nhappy year. I thanked her, and said, 'That is in God's hands.' 'Yes,'\nshe said, 'if He wills it.' 'And if He does not will it,' I answered,\n'it will not be, and then He will give me patience to bear my heavy\ncross.' 'It is heavy,' she said, 'even to me; what must it not be to\nyou? May it only remain as it is, and not be worse with you!' It\nseemed to me as if it could not be worse, but better; for death, in\nwhatever form, would put an end to my misery. 'Yes,' she said, 'is it\nnot all one how one dies?' 'That is true,' I answered; 'one dies in\ndespair, another with free courage.' The prison governor did not say\na word to me that day. The woman had a long talk with the coachman;\nshe no doubt related to him our conversation.\n\nIn the month of March the prison governor came in and assumed a\nparticularly gentle manner, and said, among other things, 'Now you\nare a widow; now you can tell the state of all affairs.' I answered\nhim with a question, 'Can widows tell the state of all affairs?' He\nlaughed and said, 'I do not mean that; I mean this treason!' I\nanswered, 'You can ask others about it who know of it; I know of no\ntreason.' And as it seemed to him that I did not believe that my\nhusband was dead,[E29] he took out a newspaper and let me read it,\nperhaps chiefly because my husband was badly treated in it. I did not\nsay much about it--nothing more than, 'Writers of newspapers do not\nalways speak the truth.' This he might take as he liked.\n\n [E29] The date of Ulfeldt's death is variously given as the 20th or\n the 27th of February, 1664. The latter date is given in a letter\n from his son Christian to Sperling, and elsewhere, (for instance,\n in a short Latin Biography of Ulfeldt called 'Machinationes\n Cornificii Ulefeldii,' published soon after); but the better\n evidence points to the earlier date. Christian Ulfeldt was not, it\n seems, at Basle at the time, and may have made a mistake as to the\n date, though he indicates the right day of the week (a Saturday),\n or he may have had reason for purposely making a misleading\n statement. In Copenhagen the report of his death was long suspected\n to be a mere trick.\n\nI lay there silently hoping that it might be so, that my husband had\nby death escaped his enemies; and I thought with the greatest\nastonishment that I should have lived to see the day when I should\nwish my lord dead; then sorrowful thoughts took possession of me, and\nI did not care to talk. The woman imagined that I was sad because my\nlord was dead, and she comforted me, and that in a reasonable\nmanner; but the remembrance of past times was only strengthened by\nher consolatory remarks, and for a long time my mind could not again\nregain repose. Your condition, my dearest children, troubled me. You\nhad lost your father, and with him property and counsel. I am captive\nand miserable, and cannot help you, either with counsel or deed; you\nare fugitives and in a foreign land. For my three eldest sons I am\nless anxious than for my daughters and my youngest son.[E30] I sat up\nwhole nights in my bed, for I could not sleep, and when I have\nheadache I cannot lay my head on the pillow. From my heart I prayed\nto God for a gracious deliverance. It has not pleased God to grant\nthis, but He gave me patience to bear my heavy cross.\n\n [E30] Ulfeldt and Leonora had twelve children in all, of which\n seven were alive when Corfitz died; and it so happened as,\n explained before, that the youngest, Leo, was the only one who\n continued the name. It is from him that Count Waldstein, the owner\n of the MS., is descended.\n\nMy cross was so much heavier to me at first, as it was strictly\nforbidden to give me either knife, scissors, thread, or anything that\nmight have beguiled the time to me. Afterwards, when my mind became a\nlittle calmer, I began to think of something wherewith to occupy\nmyself; and as I had a needle, as I have before mentioned, I took off\nthe ribands of my night-dress, which were broad flesh-coloured\ntaffeta. With the silk I embroidered the piece of cloth that I had\nwith different flowers worked in small stitches. When this was\nfinished, I drew threads out of my sheet, twisted them, and sewed\nwith them. When this was nearly done, the woman said one day, 'What\nwill you do now when this is finished?' I answered, 'Oh, I shall get\nsomething to do; if it is brought to me by the ravens, I shall have\nit.' Then she asked me if I could do anything with a broken wooden\nspoon. I answered, 'Perhaps you know of one?' After having laughed a\nwhile, she drew one forth, the bowl of which was half broken off. 'I\ncould indeed make something with that,' I said, 'if I had only a tool\nfor the purpose. Could you persuade the prison governor or Peder the\ncoachman to lend me a knife?' 'I will beg for one,' she answered,\n'but I know well that they will not.' That she said something about\nit to the prison governor I could perceive from his answer, for he\nreplied aloud, 'She wants no knife; I will cut her food for her. She\nmight easily injure herself with one.'[76]\n\n [76] In the margin is this note: 'Once when I asked the prison\n governor for some scissors to cut my nails, he answered, and that\n loudly, \"What! what! her nails shall grow like eagles' claws, and\n her hair like eagles' feathers!\" I know well what I thought--if I\n had only claws and wings!'\n\nWhat she said to the coachman I know not (this I know, that she did\nnot desire me to obtain a knife, for she was afraid of me, as I\nafterwards discovered). The woman brought the answer from the\ncoachman that he dared not for his life. I said, 'If I can but have a\npiece of glass, I will see what I can make that is useful with the\npiece of spoon.' I begged her to look in a corner in the outermost\nroom, where all rubbish was thrown; this she did, and found not only\nglass, but even a piece of a pewter cover which had belonged to a\njug. By means of the glass I formed the spoon handle into a pin with\ntwo prongs, on which I made riband, which I still have in use (the\nsilk for this riband I took from the border of my night-dress). I\nbent the piece of pewter in such a manner that it afterwards served\nme as an inkstand. It also is still in my keeping. As a mark of\nfidelity, the woman brought me at the same time a large pin, which\nwas a good tool for beginning the division between the prongs, which\nI afterwards scraped with glass.\n\nShe asked me whether I could think of anything to play with, as the\ntime was so long to her. I said, 'Coax Peder, and he will bring you a\nlittle flax for money and a distaff.' 'What!' she answered, 'shall I\nspin? The devil may spin! For whom should I spin?' I said, 'To\nbeguile the time, I would spin, if I only had what is necessary for\nit.' 'That you may not have, dear lady,' said she; 'I have done the\nvery utmost for you in giving you what I have done.' 'If you wish\nsomething to play with,' said I, 'get some nuts, and we will play\nwith them.' She did so, and we played with them like little children.\nI took three of the nuts, and made them into dice, placing two kinds\nof numbers on each, and we played with these also. And that we might\nknow the {circled dot} which I made with the large pin,[77] I begged\nher to procure for me a piece of chalk, which she did, and I rubbed\nchalk into it. These dice were lost, I know not how; my opinion is\nthat the coachman got possession of them, perhaps at the time that he\ncheated the woman out of the candles and sugar left. For he came to\nher one day at noon quite out of breath, and said she was to give him\nthe candles and the sugar which he had brought her from Maren Blocks,\nand whatever there was that was not to be seen, as our quarters were\nto be searched. She ran out with the things under her apron, and\nnever said anything to me about it until the door was locked. I\nconcealed on myself, as well as I was able, my pin, my silk, and the\npieces of sewing with the needle and pin. Nothing came of the search,\nand it was only a _ruse_ of the coachman, in order to get the\ncandles that were left, for which she often afterwards abused him,\nand also for the sugar.\n\n [77] I removed my nails with the needle, scratching them till they\n came away. I let the nail of the little finger of my right hand\n grow, in order to see how long it would become; but I knocked it\n off unawares, and I still have it. [Marginal note.]\n\nI was always at work, so long as I had silk from my night-dress and\nstockings, and I netted on the large pin, so that it might last a\nlong time. I have still some of the work in my possession, as well as\nthe bobbins, which I made out of wooden pegs. By means of bags filled\nwith sand I made cords which I formed into a bandage (which is worn\nout), for I was not allowed a corset, often as I begged for one; the\nreason why is unknown to me. I often beguiled the time with the piece\nof chalk, painting with it on a piece of board and on the table,\nwiping it away again, and making rhymes and composing hymns. The\nfirst of these, however, I composed before I had the chalk. I never\nsang it, but repeated it to myself.\n\nA morning hymn, to the tune, 'Ieg wil din Priiss ud Synge'[E31]:--\n\n [E31] This hymn-tune is still in use in the Danish Church.\n\n        I\n\n    God's praise I will be singing\n      In every waking hour.\n    My grateful tribute bringing\n      To magnify his power;\n    And his almighty love,\n      His angel watchers sending,\n      My couch with mercy tending,\n    And watching from above.\n\n        II\n\n    In salt drops streaming ever\n      The tears flowed from my eyes;\n    I often thought I never\n      Should see the morning rise.\n    Yet has the Lord instilled\n      Sleep in his own good pleasure;\n      And sleep in gracious measure\n    Has his command fulfilled.\n\n        III\n\n    Oh Christ! Lord of the living,\n      Thine armour place on me,\n    Which manly vigour giving,\n      Right valiant shall I be,\n    'Gainst Satan, death, and sin.\n      And every carnal feeling,\n      That nought may come concealing\n    Thy sway my heart within.\n\n        IV\n\n    Help me! Thy arms extending;\n      My cross is hard and sore:\n    Support its heaviest ending,\n      Or I can bear no more.\n    Too much am I oppressed!\n      My trust is almost waning\n      With pain and vain complaining!\n    Thine arrows pierce my breast.\n\n        V\n\n    In mercy soothe the sorrow\n      That weighs the fatherless;\n    Vouchsafe a happier morrow,\n      And all my children bless!\n    Strength to their father yield,\n      In their hard fate respect them,\n      From enemies protect them;\n    My strength, be Thou their shield.\n\n        VI\n\n    I am but dust and ashes,\n      Yet one request I crave:\n    Let me not go at unawares\n      Into the silent grave.\n    With a clear mind and breast\n      My course in this world closing,\n      Let me, on Thee reposing,\n    Pass to Thy land of rest.\n\nI composed the following hymn in German and often sang it, as they\ndid not understand German; a hymn, somewhat to the air of 'Was ist\ndoch auff dieser Welt, das nicht fehlt?' &c.:--\n\n        I\n\n    Reason speaketh to my soul:\n          Fret not Soul,\n    Thou hast a better goal!\n      It is not for thee restricted\n        That with thee\n        Past should be\n      All the wrongs inflicted.\n\n        II\n\n    Why then shouldst thou thus fret thee,\n          Anxiously,\n    Ever sighing, mournfully?\n      Thou canst not another sorrow\n        Change with this,\n        For that is\n      Which shall be on the morrow.\n\n        III\n\n    Loss of every earthly gain\n          Bringeth pain;\n    Fresh courage seek to obtain!\n      Much was still superfluous ceded,\n        Nature's call\n        After all\n      Makes but little needed.\n\n        IV\n\n    Is the body captive here?\n          Do not fear:\n    Thou must not hold all too dear;\n      Thou art free--a captive solely;\n        Can no tower\n        Have the power\n      Thee to fetter wholly?\n\n        V\n\n    All the same is it at last\n          When thou hast\n    The long path of striving past,\n      And thou must thy life surrender;\n        Death comes round,\n        Whether found\n      On couch hard or tender.\n\n        VI\n\n    Courage then, my soul, arise!\n          Heave no sighs\n    That nought yet thy rest supplies!\n      God will not leave thee in sorrow:\n        Well He knows\n        When He chose\n      Help for thee to borrow.\n\nThus I peacefully beguiled the time, until Doctor Otto Sperling[E32]\nwas brought to the tower; his prison is below the 'dark church.' His\nfate is pitiable. When he was brought to the tower his feet and hands\nwere chained in irons. The prison governor, who had formerly not been\nfriendly with him, rejoiced heartily at the doctor's misfortune, and\nthat he had fallen into his hands, so that the whole evening he did\nnothing but sing and hum. He said to the woman, 'My Karen, will you\ndance? I will sing.' He left the doctor to pass the night in his\nirons. We could hear that a prisoner had been brought in from the\nmurmuring, and the concourse of people, as well as from the locking\nof the prison, which was below mine (where iron bolts were placed\nagainst the door).[78] The joy exhibited by the prison governor\nexcited my fear, also that he not only himself opened and shut my\ndoor, but that he prevented the woman from going out on the stairs,\nby leaning against the outermost door of my prison. The coachman\nstood behind the prison governor making signs; but as the prison\ngovernor turned from side to side, I could not rightly see him.\n\n [E32] Dr. Otto Sperling, the elder, is often alluded to in the\n Autobiography of Leonora as 'notre vieillard;' he was a faithful\n friend of Ulfeldt, and in 1654 he settled in Hamburg, where he\n educated Corfitz's youngest son Leo. He was implicated in Ulfeldt's\n intrigues, and a compromising correspondence between them fell into\n the hands of the Spanish Government, which placed it at the\n disposal of Hannibal Sehested when he passed through the\n Netherlands on his way home from his mission to France in 1663. In\n order to obtain possession of Sperling's person, the Danish\n authorities used the ruse of sending a Danish officer to his house\n in Hamburg, and request him to visit professionally a sick person\n just across the Danish frontier, paying in advance a considerable\n fee. Sperling, who did not suspect the transaction, was arrested\n immediately on crossing the boundary, and brought to Copenhagen. He\n was condemned to death July 28, 1664; but the sentence was\n commuted, and he died in the Blue Tower December 25, 1681. Otto\n Sperling, jun., to whom Leonora sent the MS. of her Autobiography,\n and who often visited her at Maribo, was his son.\n\n [78] The prison cell is outside that in which the doctor is\n immured. It is quite dark where he is. [Note in the margin.]\n\nOn the following day, at about eight o'clock, I heard the iron bolts\ndrawn and the door below opened; I could also hear that the inner\nprison was opened (the doctor was then taken out for examination).\nThe woman said, 'There is certainly a prisoner there; who can it be?'\nI said: 'It seems indeed that a prisoner has been brought in, for the\nprison governor is so merry. You will find it out from Peder; if not\nto-day, another time. I pity the poor man, whoever he may be.' (God\nknows my heart was not as courageous as I appeared.) When my door was\nopened at noon (which was after twelve o'clock, for they did not open\nmy door till the doctor had been conveyed to his cell again), the\nprison governor was still merrier than usual, and danced about and\nsang, 'Cheer up! courage! It will come to pass!'\n\nWhen he had cut up the dinner, he leaned against the outer door of my\nprison and prevented the woman from going out, saying to me, 'I am to\nsalute you from the Major-General von Alfeldt; he says all will now\nsoon be well, and you may console yourself. Yes, yes, all will now\nsoon be well!' I behaved as if I received his words in their apparent\nmeaning, and I begged him to thank the Major-General for his\nconsolation; and then he repeated the same words, and added, 'Yes,\nindeed! he said so.' I replied with a question: 'What may it arise\nfrom that the Major-General endeavours to cheer me? May God cheer him\nin return! I never knew him before.' To this the prison governor made\nno answer at all. While the prison governor was talking with me, the\ncoachman was standing behind him, and showed by gestures how the\nprisoner had been bound hand and foot, that he had a beard and a\ncalotte on his head, and a handkerchief round his neck. This could\nnot make me wiser than I was, but it could indeed grieve me still\nmore. At the evening meal the woman was again prevented speaking with\nthe coachman, and the coachman again made the same signs, for the\nprison governor was standing in his usual place; but he said nothing,\nnor did I.[79] On the following morning the Doctor was again brought\nup for examination, and the prison governor behaved as before. As he\nstood there ruminating, I asked him who the prisoner below was. He\nanswered that there was no one below. I let the matter rest for the\ntime, and as we proceeded to speak of other things, the woman slipped\nout to Peder, who told her quickly who it was. Some days went by in\nthe same manner. When sentence had been pronounced on the Doctor, and\nhis execution was being postponed,[80] and I said nothing to the\nprison governor but when he accosted me, he came in and said: 'I see\nthat you can judge that there is a prisoner below. It is true, but I\nam forbidden to tell you who it is!' I answered: 'Then I do not\ndesire to know.' He began to feel some compassion, and said: 'Don't\nfret, my dear lady; it is not your husband, nor your son, nor\ndaughter, nor brother-in-law, nor any relative; it is a bird which\nought to sing,[81] and will not, but he must, he must!' I said: 'I\nought to be able to guess from your words who it is. If the bird can\nsing what can ring in their ears, he will probably do so; but he\ncannot sing a melody which he does not know!' Upon this he was\nsilent, and turned away and went out.\n\n [79] In the margin is added: 'When the prison governor was singing\n to himself on those first days, he said, \"You must sing, my bird;\n where is your velvet robe?\" laughing at the same time most\n heartily. I inferred from that song who it was.'\n\n [80] In the margin is added: 'In order to grieve the Doctor and to\n frighten him, the prison governor unlocked his cell early on the\n morning after sentence had been passed, and behaved as if the\n priest were coming to him.'\n\n [81] That is, give information.\n\nBy degrees all became quiet with regard to the Doctor, and no more\nwas said about the matter, and the prison governor came in from time\nto time when the door was opened, and often made himself merry with\nthe woman, desiring her to make a curtsey to him, and showing her how\nshe should place her feet and carry her body, after the fashion of a\ndancing-master. He related also different things that had occurred in\nformer times, some of them evidently intended to sadden me with the\nrecollection of my former prosperity: all that had happened at my\nwedding, how the deceased King had loved me. He gave long accounts of\nthis, not forgetting how I was dressed, and all this he said for the\nbenefit of no one else but myself, for the woman meanwhile stood on\nthe stairs talking with the tower warder, the coachman, and the\nprisoner Christian.\n\nMaren Blocks, who constantly from time to time sent me messages and\nkept me informed of what was going on, also intimated to me that she\nwas of opinion that I could practise magic, for she wrote me a slip\nof paper[82] with the request that I should sow dissension between\nthe Lady Carisse and an Alfelt, explaining at length that Alfelt was\nnot worthy of her, but that Skinckel was a brave fellow (Carisse\nafterwards married Skinckel). As the letter was open, the coachman\nknew its contents, and the woman also. I was angry at it, but I said\nnothing. The woman could easily perceive that I was displeased at it,\nand she said, 'Lady, I know well what Maren wishes.' I replied, 'Can\nyou help her in it?' 'No,' she declared, and laughed heartily. I\nasked what there was to laugh at. 'I am laughing,' said she, 'because\nI am thinking of the clever Cathrine, of whom I have spoken before,\nwho once gave advice to some one desiring to sow discord between good\nfriends.' I enquired what advice she had given. She said that they\nmust collect some hairs in a place where two cats had been fighting,\nand throw these between the two men whom it was desired to set at\nvariance. I enquired whether the trick succeeded. She replied, 'It\nwas not properly tried.' 'Perhaps,' I said, 'the cats were not both\nblack?' 'Ho, ho!' said she, 'I see that you know how it should be\ndone.' 'I have heard more than that,' I replied; 'show her the trick,\nand you will get some more sugar-candy, but do not let yourself be\nagain cheated of it by Peder as you were lately. Seriously, however,\nPeder must beg Maren Blocks to spare me such requests!' That she as\nwell as Maren believed that I could practise magic was evident in\nmany ways. My own remarks often gave cause for this. I remembered how\nmy deceased lord used to say (when in his younger days he wished to\nmake anyone imagine that he understood the black art), that people\nfeared those of whom they had this opinion, and never ventured to do\nthem harm. It happened one day at the mid-day meal, when the prison\ngovernor was sitting talking with me, that the woman carried on a\nlong conversation on the stairs with the others respecting the\nwitches who had been seized in Jutland, and that the supreme judge in\nJutland at that time sided with the witches and said they were not\nwitches.[E33] When the door was locked we had much talk about\nwitches, and she said, 'This judge is of your opinion, that it is a\nscience and not magic.' I said, as I had before said, that some had\nmore knowledge than others, and that some used their knowledge to do\nevil; although it might happen naturally and not with the devil's\nart, still it was not permitted in God's Word to use nature for evil\npurposes; it was also not fair to give the devil the honour which did\nnot belong to him. We talked on till she grew angry, laid down and\nslept a little, and thus the anger passed away.\n\n [82] In the margin is added: 'Peder had some time before thrown\n into me eight ducats in a paper, saying, as he closed the door,\n \"Your maid!\" And as the woman knew it, I gave her one of them and\n Peder one. I know not whether my maid had given him more; she had\n many more concealed on her person.'\n\n [E33] The name of this judge was Villum Lange, and it is a curious\n coincidence that a letter from him of a somewhat later time (1670),\n has been found in one of the archives, in which he speaks of this\n very affair, and in which he expresses himself very much in the\n sense here indicated.\n\nSome days after she said: 'Your maid is sitting below in the prison\ngovernor's room, and asks with much solicitude after you and what you\nare doing. I have told Peder of what you have sewed, and of the\nribbons you have made, but he has promised solemnly not to mention it\nto anyone except to Maren, Lars' daughter; she would like so much to\nbe here with you.' I replied: 'It would be no good for her to sit\nwith me in prison; it would only destroy her own happiness; for who\nknows how long I may live?' I related of this same waiting-maid that\nshe had been in my employ since she was eight years old, all that I\nhad had her taught, and how virtuous she was. To this she replied,\n'The girl will like to see what you have sewed; you shall have it\nagain directly.' I handed it to her, and the first time the doors\nwere unlocked she gave it to the prison governor, who carried it to\nthe Queen. (Two years afterwards the prison governor told me this\nhimself, and that when the King had said, 'She might have something\ngiven her to do,' the Queen had answered, 'That is not necessary. It\nis good enough for her! She has not wished for anything better.') I\noften enquired for the piece of sewing, but was answered that Peder\nwas not able to get it back from the girl.\n\nLate in the autumn the prison governor began to sicken: he was ill\nand could not do much, so he let the coachman frequently come alone\nto lock and unlock both the doctor's door below and mine. The iron\nbars were no longer placed before the outermost prison below, but\nfour doors were locked upon me. One day, when Peder was locking up,\nhe threw me a skein of silk,[83] saying, 'Make me some braces for my\nbreeches out of it.' I appeared not to have heard, and asked the\nwoman what it was that he had said. She repeated the same words. I\nbehaved as if I did not believe it, and laughed, saying, 'If I make\nthe braces for him, he will next wish that you should fasten them to\nhis breeches.' A good deal of absurd chatter followed. As meal-time\nwas approaching, I said to the woman, 'Give Peder back his silk, and\nsay that I have never before made a pair of braces; I do not know how\nthey are made.' (Such things I had to endure with smiles.)\n\n [83] In the margin is added: 'As my linen was washed in the\n servants' hall, it once happened that a maid there must unawares\n have forgotten a whole skein of thread in a clean chemise, at which\n I said to the woman: \"You see how the ravens bring me thread!\" She\n was angry and abused me; I laughed, and answered her jestingly.'\n\nAt the time that our former palace here in the city (which we had\nceded by a deed when we were imprisoned at Borringholm) was pulled\ndown, and a pillar (or whatever it is) was raised to my lord's shame,\nthe prison governor came in when he unlocked at noon, and seated\nhimself on my bed (I was somewhat indisposed at the time), and began\nto talk of former times (I knew already that they were pulling down\nthe palace), enumerating everything the loss of which he thought\nmight sadden me, even to my coach and the horses. 'But,' he said,\n'all this is nothing compared with the beautiful palace!' (and he\npraised it to the utmost); 'it is now down, and not one stone is left\non another. Is not that a pity, my dear lady?' I replied: 'The King\ncan do what he will with his own; the palace has not been ours for\nsome time.' He continued bewailing the beautiful house and the garden\nbuildings which belonged to it. I asked him what had become of\nSolomon's temple? Not a stone of that beautiful building was now to\nbe found; not even could the place be pointed out where the temple\nand costly royal palace had once stood. He made no answer, hung his\nhead, and pondered a little, and went out. I do not doubt he has\nreported what I said. Since that day he began to behave himself more\nand more courteously, saying even that His Majesty had ordered him to\nask me whether I wished for anything from the kitchen, the cellar, or\nthe confectioner, as it should be given me; that he had also been\nordered to bring me twice a week confectionery and powdered sugar,\nwhich was done.[84] I begged the prison governor to thank the King's\nMajesty for the favour shown me, and praised, as was proper, the\nKing's goodness most humbly. The prison governor would have liked to\npraise the Queen had he only been able to find cause for so doing; he\nsaid, 'The Queen is also a dear Queen!' I made no answer to this. He\ncame also some time afterwards with an order from the King that I\nshould ask for any clothes and linen I required: this was written\ndown, and I received it later, except a corset, and that the Queen\nwould not allow me. I never could learn the cause of this. The Queen\nalso was not well pleased that I obtained a bottle-case with six\nsmall bottles, in which was sprinkling-water, headwater, and a\ncordial. All this, she said, I could well do without; but when she\nsaw that in the lid there was an engraving representing the daughter\nof Herod with the head of St. John on a charger, she laughed and\nsaid, 'That will be a cordial to her!' This engraving set me thinking\nthat Herodias had still sisters on earth.\n\n [84] In the margin is added: 'I wrote different things from the\n Bible on the paper in which the sugar was given me. My ink-bottle\n was made of the piece of pewter lid which the woman had found, the\n ink was made from the smoke of the candle collected on a spoon, and\n the pen from a fowl's feather cut by the piece of glass. I have\n this still in my possession.'\n\nThe prison governor continued his politeness, and lent me at my\ndesire a German Bible, saying at the same time, 'This I do out of\nkindness, I have no order to do so; the Queen does not know it.' 'I\nbelieve that,' I replied, and thanked him; but I am of opinion that\nthe King knew it well. Some days afterwards Maren Blocks sent for her\nprayer-book back again. I had taught the woman a morning and evening\nprayer by heart, and all the morning and evening hymns, which she\nrepeated to me night and morning. I offered to teach her to read if\nshe would procure an A B C. She laughed at this jeeringly, and said,\n'People would think me crazed if I were to learn to read now.' I\ntried to persuade her by argument, in order that I might thus get\nsomething to beguile the time with; but far from it; she knew as much\nas she needed. I sought everywhere for something to divert my\nthoughts, and as I perceived that the potter, when he had placed the\nstove, had left a piece of clay lying outside in the other room, I\nbegged the woman to give it to me.\n\nThe prison governor saw that she had taken it, but did not ask the\nreason. I mixed the clay with beer, and made various things, which I\nfrequently altered again into something else; among other things I\nmade the portraits of the prison governor and the woman, and small\njugs and vases. And as it occurred to me to try whether I were able\nto make anything on which I could place a few words to the King, so\nthat the prison governor should not observe it (for I knew well that\nthe woman did not always keep silence; she would probably some time\nsay what I did), I moulded a goblet over the half of the glass in\nwhich wine was brought to me, made it round underneath, placed it on\nthree knobs, and wrote the King's name on the side--underneath the\nbottom these words ... il y a un ... un Auguste.[E34]\n\n [E34] The words 'under the bottom ... to ... Auguste,' inclusive,\n have been struck out in the MS., and it has been impossible to read\n more than what here is rendered. In the Autobiography, where the\n same occurrence is related, Leonora says that she put on it the\n names both of the King and of the Queen; that on the bottom she\n wrote to the Queen, and that it was the Queen who discovered the\n inscription; from which it would appear that the Queen at all\n events was included in her ingeniously contrived supplique.\n\nI kept it for a long time, not knowing in what way I could manage to\nget it reported what I was doing, since the woman had solemnly sworn\nto me not to mention it: so I said one day: 'Does the prison governor\nask you what I am doing?' 'Yes, indeed he does,' she replied, 'but I\nsay that you are doing nothing but reading the Bible.' I said: 'You\nmay ingratiate yourself in his favour and say that I am making\nportraits in clay; there is no reason that he should not know that.'\nShe did so, and three days after he came to me, and was quite gentle,\nand asked how I passed my time. I answered, 'In reading the Bible.'\nHe expressed his opinion that I must weary of this. I said I liked at\nintervals to have something else to do, but that this was not allowed\nme. He enquired what I had wanted the clay for, which the woman had\nbrought in to me; he had seen it when she had brought it in. I said,\n'I have made some small trifles.' He requested to see them. So I\nshowed him first the woman's portrait; that pleased him much, as it\nresembled her; then a small jug, and last of all the goblet. He said\nat once: 'I will take all this with me and let the King see it; you\nwill perhaps thus obtain permission to have somewhat provided you for\npastime,'[85] I was well satisfied. This took place at the mid-day\nmeal. At supper he did not come in. The next day he said to me:\n'Well, my dear lady, you have nearly brought me into trouble!' 'How\nso?' I asked. 'I took the King a petition from you! the Queen did not\ncatch sight of it, but the King saw it directly and said, \"So you are\nnow bringing me petitions from Leonora?\" I shrank back with terror,\nand said, \"Gracious King! I have brought nothing in writing!\" \"See\nhere!\" exclaimed the King, and he pointed out to me some French\nwriting at the bottom of the goblet. The Queen asked why I had\nbrought anything written that I did not understand. I asserted that I\nhad paid no attention to it, and begged for pardon. The good King\ndefended me, and the _invention_ did not please him ill. Yes, yes, my\ndear lady! be assured that the King is a gracious sovereign to you,\nand if he were certain that your husband were dead, you would not\nremain here!' I was of opinion that my enemies well knew that my\nhusband was dead. I felt that I must therefore peacefully resign\nmyself to the will of God and the King.\n\n [85] In the margin is added: 'The prison governor told me\n afterwards that the clay things were placed in the King's\n art-cabinet, besides a rib of mutton, which I used as a knife,\n which he also gave to the King; hoping (he said) in this way to\n obtain a knife for me.'\n\nI received nothing which might have beguiled the time to me, except\nthat which I procured secretly, and the prison governor has since\nthen never enquired what I was doing, though he came in every\nevening and sat for some time talking with me; he was weak, and it\nwas a labour to him to mount so many steps. Thus we got through the\nyear together.\n\nThe prison governor gradually began to feel pity for me, and gave me\na book which is very pretty, entitled 'Wunderwerck.'[E35] It is a\nfolio, rather old, and here and there torn; but I was well pleased\nwith the gift. And as he sat long of an evening with me, frequently\ntill nine o'clock, talking with me, the malicious woman was\nirritated.[86] She said to Peder, 'If I were in the prison governor's\nplace, I would not trust her in the way he does. He is weak; what if\nshe were now to run out and take the knife which is lying on the\ntable outside, and were to stab him? She could easily take my life,\nso I sit in there with my life hanging on a thread.'\n\n [E35] This book was doubtless the German translation of Conr.\n Lycosthenes' work, 'Prodigiorum ac Ostentorum Chronicon.' It is an\n amusing illustrated volume, much read in its time. The translation\n in question appeared in Basle, 1557.\n\n [86] In the margin is added: 'The day that the prison governor had\n taken away the clay things the woman was very angry with me,\n because I gave him a small jug which I had made; she said it was\n made in ridicule of her, the old slut with the jug! I ought to have\n given him the cat which I had also made. I said, \"I can still do\n so.\"'\n\nAbsurd as the idea was, the knife was not only in consequence hidden\nunder the table, but the prison governor for a long time did not\nventure to come to me, but sat outside by my outermost door and\ntalked there just as long as before, so that I was no gainer.[87] (I\ndid not know what the woman had said till three years afterwards,\nwhen it was mentioned by the prisoner Christian, who had heard the\nwoman's chatter.)\n\n [87] In the margin is added: 'At first when the prison governor's\n fear was so great, he did not venture to be alone in the outer\n room. Peder and the tower warder were not allowed both to leave him\n at the same time. I did not know the reason for this.'\n\nOne day when the prison governor intended to go to the holy\ncommunion, he stood outside my outermost door and took off his hat,\nand begged for my forgiveness; he knew, he said, that he had done\nmuch to annoy me, but that he was a servant. I answered, 'I forgive\nyou gladly!' Then he went away, and Peder closed the door. The woman\nsaid something to Peder about the prison governor, but I could not\nunderstand what. Probably she was blaming the prison governor, for\nshe was so angry that she puffed; she could not restrain her anger,\nbut said: 'Fye upon the old fool! The devil take him! I ought to beg\npardon too? No' (she added with an oath), 'I would not do it for\nGod's bitter death! No! no!' and she spat on the ground. I said\nafterwards: 'What does it matter to you that the prison governor asks\nme for my friendship? Do you lose anything by it? If you will not\nlive like a Christian and according to the ordinances of the Church,\ndo not at any rate be angry with one who does. Believe assuredly that\nGod will punish you, if you do not repent of what evil you have done\nand will not be reconciled with your adversaries before you seek to\nbe reconciled with God!'\n\nShe thought that he had done nothing else than what he was ordered to\ndo. I said, 'You good people know best yourselves what has been\nordered you.' She asked, 'Do I do anything to you?' I answered, 'I\nknow not what you do. You can tell any amount of untruths about me\nwithout my knowing it.' Upon this she began a long story, swearing by\nand asserting her fidelity; she had never lied to anyone nor done\nanyone a wrong. I said: 'I hear; you are justifying yourself with the\nPharisee.' She started furiously from her seat and said, 'What! do\nyou abuse me as a Pharisee?' 'Softly, softly!' I said; 'while only\none of us is angry, it is of no consequence; but if I get angry also,\nsomething may come of it!' She sat down with an insolent air, and\nsaid, 'I should well imagine that you are not good when you are\nangry! It is said of you that in former days you could bear but\nlittle, and that you struck at once. But now'----(with this she was\nsilent). 'What more?' I said. 'Do you think I could not do anything\nto anyone if I chose, just as well as then, if anyone behaved to me\nin a manner that I could not endure? Now much more than then! You\nneed not refuse me a knife because I may perhaps kill you; I could do\nso with my bare hands. I can strangle the strongest fellow with my\nbare hands, if I can seize him unawares, and what more could happen\nto me than is happening? Therefore only keep quiet!'\n\nShe was silent, and assumed no more airs; she was cast down, and did\nnot venture to complain to the prison governor. What she said to the\nothers on the stairs I know not, but when she came in, when the room\nwas locked at night, she had been weeping.[88]\n\n [88] In the margin is added: 'Some time after this dispute I had a\n quarrel with her about some beer, which she was in the habit of\n emptying on the floor, saying, \"This shall go to the subterraneous\n folk.\" I had forbidden her to do so, but she did it again, so I\n took her by the head and pushed it back with my hand. She was\n frightened, for this feels just as if one's head was falling off. I\n said, \"That is a foretaste.\"'\n\nOn Sunday at noon I congratulated[E36] the prison governor and said:\n'You are happy! You can reconcile yourself with God, and partake of\nHis body and blood; this is denied to me (I had twice during two\nyears requested spiritual consolation, but had received in answer\nthat I could not sin as I was now in prison; that I did not require\nreligious services). And as I talked upon this somewhat fully with\nthe prison governor, I said that those who withheld from me the\nLord's Supper must take my sins upon themselves; that one sinned as\nmuch in thought as in word and deed; so the prison governor promised\nthat he would never desist from desiring that a clergyman should come\nto me; and asked whom I wished for. I said: 'The King's Court\npreacher, whom I had in the beginning of my troubles.' He said: 'That\ncould scarcely be.' I was satisfied whoever it was.\n\n [E36] This custom of congratulating persons who intend to\n communicate, or just have done so, is still retained by many of the\n older generation in Denmark.\n\nA month afterwards I received the holy communion from the German\nclergyman, M. Hieronimus Buk, who behaved very properly the first\ntime, but spoke more about the law than the gospel. The prison\ngovernor congratulated me, and I thanked him, for he had brought it\nabout.\n\n1665. In this year, on Whitsun-eve, the prison governor ordered\nMay-trees to be placed in my inner prison, and also in the anteroom.\nI broke small twigs from the branches, rubbed off the bark with\nglass, softened them in water, laid them to press under a board,\nwhich was used for carrying away the dirt from the floor, and thus\nmade them flat, then fastened them together and formed them into a\nweaver's reed. Peder the coachman was then persuaded to give me a\nlittle coarse thread, which I used for a warp. I took the silk from\nthe new silk stockings which they had given me, and made some broad\nribbons of it (The implements and a part of the ribbons are still in\nmy possession.) One of the trees (which was made of the thick end of\na branch which Peder had cut off) was tied to the stove, and the\nother I fastened to my own person. The woman held the warp: she was\nsatisfied, and I have no reason to think that she spoke about it, for\nthe prison governor often lamented that I had nothing with which to\nbeguile the time, and he knew well that this had been my delight in\nformer times, &c.[89]\n\n [89] In the margin is added: 'I made the snuffers serve as\n scissors. When Balcke came to me and brought me at my desire\n material for drawers, and requested to know the size, I said I\n could make them myself. He laughed, and said, \"Who will cut them\n out?\" I replied I could do it myself with the snuffers. He begged\n to see me do it, and looked on with no little astonishment.'\n\nHe remained now again a long time with me after meals, for his fear\nhad passed away, or he had, perhaps, forgotten, as his memory began\nto fail him. He said then many things which he ought not. He declined\nperceptibly, and was very weak; he would remain afterwards sitting\noutside, reading aloud, and praying God to spare his life. 'Yes,' he\nwould say, 'only a few years!' When he had some alleviation, he\ntalked unceasingly. Creeping along the wall to the door, he said, 'I\nshould like to know two things: one is, who will be prison governor\nafter me? The other is, who is to to have my Tyrelyre?' (That was\nTyre, his wife.) I replied: 'That is a knowledge which you cannot\nobtain now, especially who will woo your wife. You might, perhaps,\nhave already seen both, but at your age you may yet have long to\nlive.' 'Oh!' said he, 'God grant it!' and looked up to the window.\n'Do you think so, my dear lady?' 'Yes, I do,' I replied. A few days\nafterwards, he begged me again to forgive him, if he had done me any\nwrong since the last time, for he wished to make reconciliation with\nGod before he became weaker, and he wept and protested, saying, 'It\nindeed grieves me still that I should have often annoyed you, and you\ncomfort me.' On Sunday at noon I congratulated him on his spiritual\nfeast.\n\nThus he dragged on with great difficulty for about fourteen days,\nand as I heard that two men were obliged almost to carry him up the\nstairs, I sent him word that he might remain below on the ground\nfloor of the tower, and that he might rest assured I would go\nnowhere. He thanked me, crawled up for the last time to my door, and\nsaid, 'If I did that and the Queen heard of it, my head would answer\nfor it.' I said: 'Then confess your weakness and remain in bed. It\nmay be better again; another could meanwhile attend for you.' He took\noff his cap in recognition of my advice, and bade me farewell. I have\nnever seen him again since then. One day afterwards he crawled up in\nthe tower-chamber, but came no farther.\n\nA man of the name of Hans Balcke was appointed in his place\nto keep watch over the prisoners. He was very courteous. He\nwas a cabinet-maker by trade; his father, who had also been a\ncabinet-maker, had worked a good deal for me in the days of my\nprosperity. This man had travelled for his trade both in Italy and\nGermany, and knew a little Italian. I found intercourse with him\nagreeable, and as he dined in the anteroom outside, in the tower, I\nbegged him to dine with me, which he did for fourteen days. One day,\nwhen he carved the joint outside, I sent him word requesting him to\ncome in. He excused himself, which appeared strange to me.\n\nAfter he had dined, he said that Peder the coachman had jeered at\nhim, and that he had been forbidden to dine with me. When he\nafterwards remained rather long with me talking, I begged him myself\nto go, so that this also might not be forbidden. He had on one\noccasion a large pin stuck in his sleeve, and I begged him for it. He\nsaid, 'I may not give it you, but if you take it yourself, I can't\nhelp it.' So I took it, and it has often been of use to me. He gave\nme several books to read, and was in every way courteous and polite.\nHis courtesy was probably the reason why the prisons were not long\nentrusted to him, for he was also very good to Doctor Sperling,\ngiving him slices of the meat which came up to me, and other good\nfood. In his childhood he had been a playfellow of the doctor's\nchildren. He talked also occasionally a long time with the doctor,\nboth on unlocking and locking his door, which did not please the\nservants.[90] The prison governor lay constantly in bed; he\nendeavoured as often as he could to come up again, but there was\nlittle prospect of it. So long as the keys were not taken from him,\nhe was satisfied.\n\n [90] In the margin is added: 'While Balcke filled the place of\n prison governor, he drank my wine at every meal, which had formerly\n fallen to the tower warder, the coachman, or the prisoner\n Christian, when the old prison governor had not wished for it, so\n that this also contributed to Balcke's dismissal.'\n\nMy maid Maren, Lars' daughter, had risen so high in favour at court,\nthat she often sat in the women's apartment, and did various things.\nOne day the woman said to me, 'That is a very faithful maiden whom\nyou have! She speaks before them up there in a manner you would never\nbelieve.' I replied: 'I have permitted her to say all she knows. I\nhave no fear of her calumniating me.' 'Have you not?' she said\nironically. 'Why does she throw herself, then, on her bare knees, and\ncurse herself if she should think of returning to you?' I said: 'She\nwished to remain with me (according to your own statement), but she\nwas not allowed; so she need not curse herself.' 'Why then do you\nthink,' said she, 'that she is so much in favour at court?' 'Do you\nmean,' I replied, 'that if anyone is in favour at court, it is\nbecause their lips are full of lies? I am assured my maid has\ncalumniated no one, least of all me; I am not afraid.'\n\nThe woman was angry, and pouted in consequence for some time. Some\nweeks afterwards Maren, Lars' daughter, was set at liberty, and\nbecame waiting-maid to the Countess Friis: and Balcke brought me some\nlinen which she still had belonging to me. The woman was not a little\nangry at this, especially as I said: 'So faithful I perceive is my\nmaid to me, that she will not keep the linen, which she might easily\nhave done, for I could not know whether it had not been taken from\nher with the rest.'\n\nAll my guards were very ill satisfied with Balcke, especially the\nwoman, who was angry for several reasons. He slighted her, she said,\nfor he had supplied a basin for the night-stool which was heavier\nthan the former one (which leaked); but she was chiefly angry because\nhe told her that she lived like a heathen, since she never went to\nthe sacrament. For when I once received the holy communion, while\nBalcke was attending to me, he asked her if she would not wish to\ncommunicate also, to which she answered, 'I do not know German.'\nBalcke said, 'I will arrange that the clergyman shall come to you\nwhose office it is to administer the Lord's Supper to the prisoners.'\nShe replied that in this place she could not go with the proper\ndevotion: if she came out, she would go gladly. Balcke admonished her\nseverely, as a clergyman might have done. When the door was closed,\nshe gave vent to puffing and blowing, and she always unfastened her\njacket when she was angry.\n\nI said nothing, but I thought the evil humour must have vent, or she\nwill be choked; and this was the case, for she abused Balcke with the\nstrongest language that occurred to her. She used unheard-of curses,\nwhich were terrible to listen to: among others, 'God damn him for\never, and then I need not curse him every day.' Also, 'May God make\nhim evaporate like the dew before the sun!' I could not endure this\ncursing, and I said, 'Are you cursing this man because he held before\nyou the word of God, and desires that you should be reconciled with\nGod and repent your sins?' 'I do not curse him for that,' she said,\n'but on account of the heavy basin which the accursed fellow has\ngiven me, and which I have to carry up the steep stairs;[91] the\ndevil must have moved him to choose it! Does he want to make a priest\nof himself? Well, he is probably faultless, the saucy fellow!' and\nshe began again with her curses.\n\n [91] In the margin: 'It is indeed a bad flight of stairs to the\n place where the basin was emptied.'\n\nI reproved her and said: 'If he now knew that you were cursing him in\nthis way, do you not think he would bring it about that you must do\npenitence? It is now almost two years since you were at the Lord's\ntable, and you can have the clergyman and you will not.' This\nsoftened her a little, and she said, 'How should he know it, unless\nyou tell him?' I said, 'What passes here and is said here concerns no\none but us two; it is not necessary that others should know.' With\nthis all was well; she lay down to sleep, and her anger passed away;\nbut the hate remained.\n\nThe prison governor continued to lie in great pain, and could neither\nlive nor die. One day at noon, when Balcke unlocked (it was just\ntwenty weeks since he had come to me), a man came in with him, very\nbadly dressed, in a grey, torn, greasy coat, with few buttons that\ncould be fastened, with an old hat to which was attached a drooping\nfeather that had once been white but was now not recognisable from\ndirt. He wore linen stockings and a pair of worn-out shoes fastened\nwith packthread.[92] Balcke went to the table outside and carved the\njoint; he then went to the door of the outer apartment, stood with\nhis hat in his hand, made a low reverence, and said, 'Herewith I take\nmy departure; this man is to be prison governor.' I enquired whether\nhe would not come again to me. He replied, 'No, not after this time.'\nUpon this I thanked him for his courteous attendance, and wished him\nprosperity.[93]\n\n [92] In the margin is added: 'Gabel had said (I was afterwards\n informed) that I was frightened at the appearance of the man, and\n thought it was the executioner. I did not regard him as such, but\n as a poor cavalier, and I imagined he was to undertake the duties\n which Peder the coachman performed.'\n\n [93] In the margin: 'Balcke has waited upon me for twenty weeks,\n and he was accused of having told me what happened outside. In\n proof of this it was alleged that he had told me that Gabel had\n been made Statholder, to whom I afterwards gave this title in M.\n Buck's hearing. Balcke one day could not restrain himself from\n laughing, for while he was standing and talking with me, the woman\n and the man were standing on the stairs outside, chuckling and\n laughing; and he said, \"Outside there is the chatter market. Why\n does not Peder so arrange it that it is forbidden? You can get to\n know all that goes on in the world without me.\"'\n\nPeder the coachman locked the door, and the new prison governor,\nwhose name was Johan Jaeger,[E37] never appeared before me the whole\nday, nor during the evening. I said to the woman in the morning, 'Ask\nPeder who the man is;' which she did, and returned to me with the\nanswer that it was the man who had taken the Doctor prisoner; and\nthat now he was to be prison governor, but that he had not yet\nreceived the keys. Not many days passed before he came with the Lord\nSteward to the old prison governor, and the keys were taken from the\nold man and given to him. The old man lived only to the day after\nthis occurred. In both respects his curiosity was satisfied; he saw\nthe man who was to be prison governor after him (to his grief), and\nthe doctor who attended him obtained his Tyrelyre before the year was\nended.\n\n [E37] It was a Colonel Hagedorn that entrapped and arrested Dr.\n Sperling, and Jaeger played only a subordinate part in that\n transaction. He is stated to have been a cousin of Gabel, and to\n have been formerly a commander in the navy. He was appointed prison\n governor on June 12, 1665, and Balcke therefore doubtless only held\n the appointment provisionally.\n\nThe new prison governor Jaeger[E37b] did not salute me for several\nweeks, and never spoke to me. He rarely locked my doors, but he\ngenerally opened them himself. At length one day, when he had got new\nshoes on, he took his hat off when he had opened the door, and said\n'Good morning.' I answered him, 'Many thanks.' The woman was very\npleased while this lasted. She had her free talk with Peder the\ncoachman (who still for a couple of months came to the tower as\nbefore) and with the prisoner Christian, who had great freedom, and\nobtained more and more freedom in this prison governor's time,\nespecially as Rasmus the tower-warder was made gatekeeper, and a man\nof the name of Chresten was appointed in his place. Among other idle\ntalk which she repeated to me, she said that this prison governor was\nforbidden to speak with me. I said, 'I am very glad, as he then can\ntell no lies about me.' I am of opinion that he did not venture to\nspeak with me so long as Peder brought up the food to the tower, and\nwas in waiting there; for when he had procured Peder's dismissal on\naccount of stealing, he came in afterwards from time to time. The\nvery first time he was intoxicated. He knew what Peder had said of\nBalcke, and he informed me of it.[94]\n\n [E37b] It was a Colonel Hagedorn that entrapped and arrested Dr.\n Sperling, and Jaeger played only a subordinate part in that\n transaction. He is stated to have been a cousin of Gabel, and to\n have been formerly a commander in the navy. He was appointed prison\n governor on June 12, 1665, and Balcke therefore doubtless only held\n the appointment provisionally.\n\n [94] In the margin is added: 'While Balcke waited on me, a folding\n table was brought in for the bread and glasses, and also for the\n woman's food, which she did not take till the doors had been\n locked. There was nothing there before but the night-stool to place\n the dishes on: that was the woman's table.'\n\nBefore I mention anything of the prisoner Christian's designs\nagainst me, I will in a few words state the crime for which he was in\nprison. He had been a lacquey in the employ of Maans Armfelt. With\nsome other lacqueys he had got into a quarrel with a man who had been\na father to Christian, and who had brought him up from his youth and\nhad taken the utmost care of him. The man was fatally wounded, and\ncalled out in the agonies of death: 'God punish thee, Christian! What\na son you have been! It was your hand that struck me!' The other\nlacqueys ran away, but Christian was seized. His dagger was found\nbloody. He denied, and said it was not he who had stabbed the man. He\nwas sentenced to death; but as the dead man's widow would not pay for\nthe execution, Christian remained for the time in prison, and his\nmaster paid for his maintenance. He had been there three years\nalready when I came to the prison, and three times he was removed;\nfirst from the Witch Cell to the Dark Church; and then here where I\nam imprisoned.[95] When I was brought here, he was placed where the\nDoctor is, and when the Doctor was brought in, Christian was allowed\nto go freely about the tower. He wound the clock for the\ntower-warder, locked and unlocked the cells below, and had often even\nthe keys of the tower.\n\n [95] In the margin is added: 'At that time there was a large double\n window with iron grating, which was walled up when I was brought\n here; and Christian told me afterwards how the maids in the\n store-room had supplied him with many a can of beer, which he had\n drawn up by a cord.'\n\nI remember once, when Rasmus the tower-warder was sitting at dinner\nwith the prison governor in my outermost cell, and the prison\ngovernor wished to send Peder on a message, he said to Rasmus: 'Go\nand open! I want Peder to order something. 'Father,' said Rasmus,\n'Christian has the key.' 'Indeed!' said the prison governor; 'that is\npretty work!' And there it rested, for Rasmus said, 'I am perfectly\nsure that Christian will not go away.' Thus by degrees Christian's\nfreedom and power increased after Peder the coachman left, and he\nwaited on the prison governor at meals in my outermost room.\n\nOne day, when the woman had come down from above, where she had been\nemptying the utensils in my room, and the doors were locked, she said\nto me: 'This Christian who is here has been just speaking with me\nupstairs. He says he cannot describe the Doctor's miserable\ncondition, how severe is his imprisonment, and what bad food he gets,\nsince Balcke left. He has no longer any candle except during\nmeal-time, and no light reaches him but through the hole in his door\nleading into the outer room. He begged me to tell you of it; his eyes\nwere full of tears, such great pity had he for him.' I said: 'That is\nall that one can do, and it is the duty of a Christian to sympathise\nwith the misfortune of one's neighbour. The poor man must have\npatience as well as I, and we must console ourselves with a good\nconscience. The harder he suffers the sooner comes the end; he is an\nold man.'\n\nTwo days afterwards she came again with some talk from Christian. The\nDoctor sent me his compliments, and he asked constantly if I was\nwell; she said also, that Christian would give him anything I liked\nto send him. I regarded this as a snare, but I said that Christian\ncould take a piece of roast meat when the prison governor was with\nme, and that he should look about for something into which wine could\nbe poured, and then she could secretly give some from my glass, and\nbeg Christian to give my compliments to the Doctor. This was\naccepted, and I had rest for a few days. Christian conformed entirely\nto the woman, caused a dispute between her and the tower-warder, and\nmade it immediately right again; so that there was no lack of\nchatter. At last she said one day: 'That is an honest fellow, this\nChristian! He has told me how innocently he got into prison and was\nsentenced. He is afraid that you may think he eats and drinks all\nthat you send to the Doctor. He swore with a solemn oath that he\nwould be true to you, if you would write a word to the Doctor.[96] I\nhope you do not doubt my fidelity!' and she began to swear and to\ncurse herself if she would deceive me. She said, he had taken a no\nless solemn oath, before she believed him. I said: 'I have nothing to\nwrite to him. I do not know what I have to write.' 'Oh!' said she,\n'write only two words, so that the old man may see that he can trust\nhim! If you wish for ink, Christian can give you some.' I replied: 'I\nhave something to write with, if I choose to do so, and I can write\nwithout ink and paper.'\n\n [96] In the margin is this note: 'Christian had at that time given\n me some pieces of flint which are so sharp that I can cut fine\n linen with them by the thread. The pieces are still in my\n possession, and with this implement I executed various things.'\n\nThis she could not understand; so I took some pieces of sugared\nalmonds, and made some letters on them with the large pin, placing on\nfour almonds the words: _non ti fidar_! I divided the word _fidar_,\nand placed half on each almond. I had in this way rest for a day, and\nsomewhat to beguile the time. Whether the Doctor could not see what\nwas written on the almonds, or whether he wished to test Christian's\nfidelity, I know not, but Christian brought the woman a slip of paper\nfrom the Doctor to me, full of lamentations at our condition, and\nstating that my daughter Anna Cathrina, or else Cassetta, were the\ncause of his misfortune.\n\nI wished to know more of this, so I wrote to him desiring information\n(we wrote to each other in Italian). He replied that one or the other\nhad left his letter lying somewhere on the table, where it was found\nand despatched; for that a letter of his was the cause of his\nmisfortune. I wrote back to him that it was not credible, but that he\nwas suspected of having corresponded with my lord, and hence his\nletters had been seized. The more I tried to impress this upon him\nthe more opinionated he became,[97] and he wrote afterwards saying\nthat it was a scheme of Cassetta's to get him into the net, in order\nto bring me out of it. When he began to write in this way, I acquired\na strange opinion of him, and fancied he was trying to draw something\nout of me which he could bring forward; and I reflected for some days\nwhether I should answer. At last I answered him in this strain, that\nno one knew better than he that I was not aware of any treason; that\nthe knowledge as to how his correspondence with my lord had become\nknown was of no use to him; that I had no idea why he was sentenced,\nand that no sentence had been passed on me. Some weeks elapsed before\nthe Doctor wrote. At last he communicated to me in a few words the\nsentence passed upon him, and we corresponded from time to time with\neach other.\n\n [97] In the margin is added: 'Such is his character.'\n\nThe prison governor became gradually more accessible, came in at\nevery meal-time, and related all sorts of jokes and buffooneries,\nwhich he had carried on in his youth: how he had been a drummer, and\nhad made a Merry Andrew of himself for my brother-in-law Count Pentz,\nand how he had enacted a dog for the sake of favour and money, and\nhad crawled under the table, frightening the guests and biting a dog\nfor a ducat's reward. When he had been drinking (which was often the\ncase) he juggled and played Punch, sometimes a fortune-teller, and\nthe like.\n\nWhen Chresten the tower-warder, and Christian the prisoner, heard the\nprison governor carrying on his jokes, they did the same, and made\nsuch a noise with the woman in the antechamber that we could not hear\nourselves speak. She sat on Christian's lap, and behaved herself in a\nwanton manner. One day she was not very well, and made herself some\nwarm beer and bread, placing it outside on the stove. The prison\ngovernor was sitting with me and talking, Chresten and Christian were\njoking with her outside, and Christian was to stir the warm beer and\nbread, and taste if it was hot enough. Chresten said to Christian,\n'Drink it up if you are thirsty.' The words were no sooner said than\nthe deed was done, and almost at the same moment the prison governor\ngot up and went away. When the door was locked, the woman seemed to\nbe almost fainting. I thought she was ill, and I was fearful that she\nmight die suddenly, and that the guilt of her death might be laid on\nme, and I asked quickly, 'Are you ill?' She answered, 'I am bad\nenough,' confirming it with a terrible oath and beginning to unbutton\nher jacket. Then I saw that she was angry, and I knew well that she\nwould give vent to a burst of execrations, which was the case.\n\nShe cursed and scolded those who had so treated her; a poor sick\nthing as she was, and she had not had anything to eat or drink all\nday. I said, 'Be quiet, and you shall have some warm beer.' She swore\nwith a solemn oath, asking how it was to be got here? it was summer\nand there was no fire in the stove, and it was no use calling, as no\none could hear. I said, 'If you will be silent, I will cause the pot\nto boil.' 'Yes,' and she swore with another fearful oath, 'I can\nindeed be silent, and will never speak of it.' So I made her take\nthree pieces of brick, which were lying behind the night-stool, and\nplace on these her pot of beer and bread (everything that she was to\ndo was to be done in silence; she might not answer me with words but\nonly with signs, when I asked her anything). She sat down besides the\npot, stirring it with a spoon. I sat always on my bed during the day,\nand then the table was placed before me. I had a piece of chalk, and\nI wrote various things on the table, asking from time to time whether\nthe pot boiled. She kept peeping in and shaking her head. When I had\nasked three times and she turned to me and saw that I was laughing,\nshe behaved herself like a mad woman, throwing the spoon from her\nhand, turning over the stool, tearing open her jacket, and\nexclaiming, 'The devil may be jeered at like this!' I said, 'You are\nnot worthy of anything better, as you believe that I can practise\nmagic.' 'Oh (and she repeated a solemn oath) had I not believed that\nyou could practise magic, I should never have consented to be locked\nup with you; do you know that?' I reflected for a moment what answer\nto give, but I said nothing, smiled, and let her rave on.\n\nAfterwards she wept and bemoaned her condition. 'Now, now,' I said,\n'be quiet! I will make the pot boil without witchcraft.' And as we\nhad a tinder-box, I ordered her to strike a light, and to kindle\nthree ends of candles, which she was to place under the pot. This\nmade the pot boil, and she kissed her hand to me and was very merry.\nOnce or twice afterwards I gave her leave to warm beer in this way:\nit could not always be done, for if the wind blew against the window\n(which was opened with a long pike) the smoke could not pass away. I\nsaid, 'Remember your oath and do not talk of what takes place here,\nor the lights will be taken from us; at any rate we shall lose some\nof them.' She asserted that she would not. I heard nothing of it at\nthe time, but some years afterwards I found that she had said that I\nhad taken up two half-loose stones from the floor (this was\nafterwards related in another manner by a clergyman, as will be\nmentioned afterwards). She had also said that I had climbed up and\nlooked at the rope-dancers in the castle square, which was true. For\nas Chresten one day told the woman that rope-dancers would be\nexhibiting in the inner castle yard, and she informed me of it and\nenquired what they were, and I explained to her, she lamented that\nshe could not get a sight of them. I said it could easily be done, if\nshe would not talk about it afterwards. She swore, as usual, with an\noath that she would not. So I took the bedclothes from the bed and\nplaced the boards on the floor and set the bed upright in front of\nthe window, and the night-stool on the top of it. In order to get\nupon the bedstead, the table was placed at the side, and a stool by\nthe table in order to get upon the table, and a stool upon the table,\nin order to get upon the night-stool, and a stool on the night-stool,\nso that we could stand and look comfortably, though not both at once.\nI let her climb up first, and I stood and took care that the bed did\nnot begin to give way; she was to keep watch when I was on the top. I\nknew, moreover, well that the dancers did not put forth their utmost\nskill at first.[98]\n\n [98] In the margin is added: 'These rope-dancers did things that I\n had never seen before. One had a basket attached to each leg, and\n in each basket was a boy of five years of age, and a woman fell\n upon the rope and jumped up again. But during the time of the other\n woman, I saw a man suspended by his chin and springing back upon\n the rope.'\n\nI could see the faces of the King and Queen: they were standing in\nthe long hall, and I wondered afterwards that they never turned their\neyes to the place where I stood. I did not let the woman perceive\nthat I saw them. During this woman's time I once had a desire to see\nthe people go to the castle-church and return from it. The bed was\nagain placed upright, and I sat for a long time on the top, until\neveryone had come out of church. The woman did not venture to climb\nup; she said that she had been afraid enough the last time, and was\nglad when she had come down.\n\nThe first time I received the holy communion during this prison\ngovernor's time, two brass candlesticks which did not match were\nbrought in, with tallow candles. This displeased the woman, though\nshe said nothing to me. But when at length she was compelled to take\nthe sacrament, after more than three years had elapsed since she had\nbeen at the Lord's table, she begged Chresten, the tower-warder, to\ngo to her daughter (who was in the service of a carpenter in the\ntown), and to get the loan of a pair of beautiful brass candlesticks\nand a couple of wax candles. If she could also procure for her a fine\nlinen cloth, she was to do her best; she would pay for it.\n\nWhether the woman had before thought of the candlesticks and candles\nwhich had been placed for me, or whether Chresten himself thought\nthat it would not be proper to provide better for her, I know not,\nbut shortly before the priest came, Chresten unlocked the outer door\nof my prison and said, 'Karen, hand me out the candlestick you have,\nand two candles.' Her behaviour is not to be described: she asked if\nhe had not spoken with her daughter, and much of the same kind (I did\nnot at the time know what she had desired of Chresten). He made no\nreply to her question, but asked for the candlestick and candles. For\na long time she would not give them, but cursed and scolded. I was\nstill lying down, and I asked her if I should be her maid, and should\ndo it for her? whether she could withhold from him what he requested?\nSo she handed them to him through the hole of the inner door, with so\nmany execrations against him that it was terrible to listen to. He\nlaughed aloud, and went away. This made her still more angry. I did\nmy best to appease her, telling her that such conduct was a most\nimproper preparation, and holding before her the sinfulness of her\nbehaviour. She said she thought that the sin belonged to him who had\ngiven cause for it. I asked her, at last, in what the Lord's Supper\nconsisted? whether it consisted in candlesticks and candles? I\nrebuked her for looking to externals and not to the essential; and I\nbegged her to fall on her knees and pray heartily to God for\nforgiveness of her sins, that He might not impute her folly to her.\nShe answered that she would do so, but she did not do it at once.\n\nI imagine that the clergyman[99] was well informed by Chresten of all\nthat concerned her, as he put to her so many questions: where she was\nborn? whom she had served? and more of the same kind, and finally,\nwhether she had her certificate of confession, and how long it was\nsince she had received the Lord's Supper? After this he confessed her\nin a strange manner; at first as one who had deserved to do public\npenance for great sins, then as a criminal under sentence of death\nwho was preparing for her end; at last consoling her, and performing\nhis office. When all was over and she came in to me, I wished her\njoy. 'Joy, indeed' (she answered); 'there is not much good in it!\nThis does me more harm than good! If I could only get out, I would\nindeed go straight to the sacrament; I reckon this as nothing!' I\ninterrupted her quickly, and said: 'Reflect upon what you are saying!\nblaspheme not God--I will not hear that! You know well what God's\nWord says of those who receive Christ's body and blood unworthily and\nhave trodden under foot his body?' 'Under foot?' said she. 'Yes,\nunder foot!' I said, and I made a whole sermon upon it. She listened\ndecently; but when I was silent, she said: 'He looked upon me as a\nmalefactor, and as one under sentence of death. I have never murdered\nanyone (I thought, we know not what);[100] why should I die? God\nAlmighty grant'----and with this she was silent. I preached to her\nagain, and said that she had deserved eternal death on account of her\nsins, and especially because she had so long kept aloof from the\nLord's table. 'This confession,' she said, 'I have to thank Chresten\nfor; Balcke was also probably concerned in it.' And she began to\ncurse them both. I threatened her with a second confession, if she\ndid not restrain such words. I told her I could not justify myself\nbefore God to keep silence to it, and I said, 'If you speak in this\nway to Chresten, you may be sure he will inform against you.' This\nkept her somewhat in check, and she did not go out upon the stairs\nthat noon.[101]\n\n [99] In the margin is added: 'This was the priest who attended to\n the prisoners, and as he confessed her in the anteroom, I heard\n every word said by him, but not her replies.'\n\n [100] In the margin is added: 'Her child.'\n\n [101] In the margin is added: 'She was in every respect a malicious\n woman, and grudged a little meat to any prisoner. A poor sacristan\n was my neighbour in the Dark Church, and I gave her a piece of meat\n for him. She would not take it to him, which she could easily have\n done without anyone seeing. When I saw the meat afterwards, I found\n fault with her. Then she said, \"Why should I give it to him? He has\n never given me anything. I get nothing for it.\" I said, \"You give\n nothing of your own away.\" This sacristan was imprisoned because he\n had taken back his own horse, the man to whom he had sold it not\n having paid him. He sang all day long, and on Sunday he went\n through the service like a clergyman, with the responses, &c.'\n\nAfter that time she was not so merry by far with the man. She often\ncomplained to me that she was weak, and had strained herself lifting\nthe new basin which Balcke had given her; she could not long hold\nout, she said, and she had asked the prison governor to let her go\naway, but that he had answered that she was to die in the tower. I\nsaid, 'The prison governor cannot yet rightly understand you; ask\nChresten to speak for you.' This she did, but came back with the same\nanswer. One day she said: 'I see well, dear lady, that you would be\nas gladly free of me as I should be to go. What have I for all my\nmoney? I cannot enjoy it, and I cannot be of service to you.' I said:\n'Money can do much. Give some money to the prison-governor, and then\nhe will speak for you. Request one of the charwomen to carry the\nbasin instead of you, and this you could pay with very little.' She\ndid the latter for some weeks; at length one day she said to me, 'I\nhave had a silver cup made for the prison governor. (Her daughter\ncame to her on the stairs as often as she desired, and she had\npermission to remain downstairs the whole afternoon, under pretext of\nspeaking with her daughter. Whether she gave him presents for this, I\nknow not, but I was well contented to be alone. She was, however,\nonce afraid that I should tell the priest of it.) The fact was, the\nprison-governor did not dare to speak for her with the King. She\nasked my advice on the matter. I said, 'Remain in bed when the\ndinner is going on, and I will go out and speak with the\nprison-governor.' This was done. At first he raised some\ndifficulties, and said, 'The Queen will say that there is some trick\nat the bottom of it.' I said they could visit and examine the woman\nwhen she came out; that we had not been such intimate friends; that I\nknew the woman had been sent to wait on me; when she could do so no\nlonger, but lay in bed, I had no attendance from her, and still less\nwas I inclined to wait on her; she did her work for money, and there\nwere women enough who would accept the employment.\n\nThree days afterwards, when the King came from Fridrichsborg, the\nprison-governor came in and said that the woman could go down in the\nevening; that he had another whom Chresten had recommended, and who\nwas said to be a well-behaved woman (which she is).\n\nKaren the daughter of Ole therefore went down, and Karen the daughter\nof Nels came up in her place. And I can truly say that it was one of\nthe happiest days during my severe imprisonment; for I was freed of a\nfaithless, godless, lying[102] and ill-behaved woman, and I received\nin her stead a Christian, true, and thoroughly good (perhaps too\ngood) woman. When the first took her departure, she said, 'Farewell,\nlady! we are now both pleased.' I answered, 'That is perhaps one of\nthe truest words you have ever spoken in your life.' She made no\nreply, but ran as fast as she could, so that no weakness nor illness\nwere perceptible in her. She lived scarcely a year afterwards,\nsuffering severe pain for six weeks in her bed, before she died; the\nnature of her malady I know not.\n\n [102] In the margin is added: 'She had begged Chresten, for more\n than half a year before she left, to tell the prison-governor that\n her life hung on a thread; that I had a ball of clay in my\n handkerchief, and that I had threatened to break her head to pieces\n with it (I had said one day that a person with a ball of that kind\n could kill another). She invented several similar lies, as I\n subsequently heard.'\n\nOn the day after this Karen's arrival, she sat thoroughly depressed\nall the afternoon. I asked her what was the matter. She said, 'Oh! I\nhave nothing to do, and I might not bring work with me! I weary to\ndeath.' I enquired what work she could do. 'Spinning,' she answered,\n'is my work principally; I can also do plain needlework and can knit\na little.' I had nothing to help her in this way; but I drew out some\nends of silk, which I had kept from what I cut off, and which are too\nshort to work with, and other tufts of silk from night-jackets and\nstockings; I had made a flax-comb of small pins,[103] fastened to a\npiece of wood; with this I combed the silk and made it available for\ndarning caps; and I said to her, 'There is something for you to do;\ncomb that for me!' She was so heartily pleased that it was quite a\ndelight to me. I found from her account of this and that which had\noccurred in her life, that she had a good heart, and that she had\noften been deceived owing to her credulity. She had also known me in\nmy prosperity; she had been in the service of a counsellor's lady who\nhad been present at my wedding, and she could well remember the\ndisplay of fireworks and other festivities; she wept as she spoke of\nit, and showed great sympathy with me. She was a peasant's daughter\nfrom Jutland, but had married the quarter-master of a regiment. By\ndegrees I felt an affection for her, and begged her to speak to\nChristian and to enquire how the Doctor was; I told her that\nChristian could occasionally perform small services for us, and could\nbuy one thing or another for us; for he had a lad, in fact sometimes\ntwo, who executed commissions for him, but that I had never trusted\nthe other woman, so that he had never bought anything for me;\nbesides, the other woman had not cared to spin; but that Christian\nshould now procure us what we wanted in return for our candles. And\nas she did not care to drink wine (for at each meal the woman\nreceived at that time half-a-pint of French wine), I said: 'Give\nChresten your wine as I give wine to Christian, then Chresten can let\nit stay with the cellar-clerk and can take it weekly, which will give\nhim a profit on it, and then he will see nothing even if he remarks\nanything.'\n\n [103] In the margin is added: 'The pins I had obtained some time\n ago from the first woman. She had procured them with some needles,\n and, thinking to hide them from me, she carried them in her bosom\n in a paper and forgot them. In the evening when she dropped her\n petticoat to go to bed, the paper fell on the floor. I knew from\n the sound what it was. One Saturday, when she went upstairs with\n the night-stool, I took the pins out of her box, and she never\n ventured to ask for them; she saw me using them afterwards, and\n said nothing about them.'\n\nThis was done, and Christian got us two hand-distaffs. Mine was but\nsmall, but hers was a proper size. I spun a little and twisted it\ninto thread, which is still in my possession. Christian procured her\nas much flax as she desired, and brought her up a whole wreath in his\ntrousers. She spun a good deal on the hand-distaff, and I arranged my\nloom on a stool, which I placed on the table, fastening one beam with\nribbon and cord which I had made myself, so that when the key was put\ninto the staircase-door, I could in one pull loosen my loom and\nunfasten the other beam which was fastened to myself, and put all\naway before the inner door was opened. I made myself also a wooden\nskewer (I had before used a warp), so that I could weave alone; I had\nalso obtained a real weaver's comb; so we were very industrious,\neach at her own work.\n\nThe prison governor was full of foolish jokes, and played tricks such\nas boys enjoy; he tried to jest with the woman, but she would not\njoin him. Almost every day he was drunk at dinner-time when he came\nup. Afterwards he came rarely of an evening, but sent a servant\ninstead, who would lie and sleep on the wall in the window. He wanted\nto jest with me also, and opened his mouth, telling me to throw\nsomething in and see if I could hit his mouth. I laughed and said,\n'How foolish you are!' and begged him to come nearer, and I would see\nif I could hit him. 'No, no,' said he; 'I am not such a fool; I\ndaresay you would box my ears.' One day he came up with a peculiar\nkind of squirt, round in form like a ball, and he placed a small tube\nin it, so small as scarcely to be seen; it was quite pretty. When\npressed in any part, the water squirted out quite high and to a\ndistance. He was saucy, and squirted me. When he saw that I was\nangry, he came to me with the squirt, ran away and sat down with his\nmouth as wide open as possible and begged me to squirt into it if I\ncould. I would not begin playing with him, for I knew his coarseness\nwell from his stories, and I gave him back the squirt. When Karen was\nbringing in the meat, the prison governor had the squirt between his\nlegs, and was seated on a low stool, from which he could squirt into\nthe woman's face; he was some distance from her, and the ball was not\nlarger than a large plum. She knew nothing of the squirt (she is\nsomewhat hasty in her words), and she exclaimed, 'May God send you a\nmisfortune, Mr. governor! Are you insulting me?' The prison governor\nlaughed like an insane man, so pleased was he at this.\n\nBy degrees he became less wild; he rarely came up sober, and he would\nlie on the woman's bed and sleep while I dined, so that Chresten and\nthe woman had to help him off the bed when they had woke him. The\nkeys of the prisons lay by his side, and the principal key close by\n(did he not take good care of his prisoners?).[104] He was not afraid\nthat I should murder him. One evening he was intoxicated, and behaved\nas such; and began, after his fashion, to try and caress me,\nendeavouring to feel my knee and seized the edge of my petticoat. I\nthrust him away with my foot, and said nothing more than: 'When you\nare intoxicated, remain away from me, and do not come in, I tell\nyou.' He said nothing, got up and went away; but he did not come in\nafterwards when he was tipsy, but remained outside in the anteroom,\nlying down in the window, where there was a broad stone bench against\nthe wall; there he lay and slept for some time after my doors were\nlocked, then the coachman and Chresten came and dragged him down.\nOccasionally he came in when he was not drunk, and he gave me at my\nrequest some old cards, which I sewed together and made into a box.\nChristian covered it with thin sticks of fir, which I afterwards\nstitched over, and I even secretly contrived to paint it. I have it\nin my possession. The prison governor saw it afterwards, but he never\nasked where the covering had come from.[105] In this box (if I may\ncall it so) I keep all my work and implements, and it stands by day\non my bed.\n\n [104] In the margin is noted: 'I said one day to the woman, \"Were\n it not for the Queen, who would make the King angry with me, I\n would retaliate upon the prison governor for having decoyed Doctor\n Sperling. I would take the keys when he was sleeping, and wait for\n Chresten to come with the cups, and then I would go up the King's\n stairs and take the keys to the King, just as the lacquey did with\n the old prison-governor. But I should gain nothing from this King,\n and perhaps should be still more strictly confined.\"'\n\n [105] In the margin is noted: 'At first, when this Karen did not\n know the prison governor, she did not venture so boldly to the\n prisoners in the Dark Church to give them anything, for she said,\n \"The prison governor stares at me so.\" I said, \"It is with him as\n with little children; they look staring at a thing, and do not know\n what it is.\" It is the case with him, he does not trouble himself\n about anything.'\n\nChristian's power increased. He waited not only outside at dinner,\nbut he even locked my door in the face of the tower-warder. He came\nwith the perfuming-pan into my room when the woman took away the\nnight-stool; in fact, he subsequently became so audacious that he did\neverything he chose, and had full command over the prisoners below.\nChresten availed himself also of the slack surveillance of the prison\ngovernor, and stayed sometimes the whole night out in the town, often\ncoming in tipsy to supper. One evening Chresten was intoxicated, and\nhad broken some panes of glass below with his hand, so that his\nfingers were bloody; he dashed my wine-cup on the ground, so that it\ncracked and was bent; and as the cup was quite bloody outside when he\ncame in to me, and some blood seemed to have got into the wine, I\nspoke somewhat seriously with the prison governor about it. He said\nnothing but 'The man is mad,' took the cup and went himself down into\nthe cellar, and had the cup washed and other wine put in it. How they\nafterwards made it up I know not. The indentations on the cup have\nbeen beaten out, but the crack on the edge is still there; this suits\nthe cellar-clerk well, for now scarcely half a pint goes into the\ncup. Christian held his own manfully against the prison governor,\nwhen he had a quarrel with some of the prisoners below; and Chresten\ncomplained of this to the prison governor, who came in and wanted to\nplace Christian in the Witch Cell; but he thrust the prison governor\naway, and said that he had nothing to do with him, and that he had\nnot put him into the prison; and then harangued him in such a style\nthat the Governor thanked God when he went away. Christian then\ncalled after him from the window, and said, 'I know secret tricks of\nyours, but you know none of mine.' (One I knew of, of which he was\naware, and that not a small one. There was a corporal who had stabbed\na soldier, and was sought for with the beating of drums: the prison\ngovernor concealed him for several weeks in the tower.) On the\nfollowing morning Christian repented, and he feared that he might be\nlocked up, and came to my door before it had been opened[106] (it\noften happened that the anteroom was unlocked before the food was\nbrought up, and always in the winter mornings, when a fire was made\nin the stove outside), and he begged me to speak for him with the\nprison governor, which I did; so that things remained as they were,\nand Christian was as bold as before.\n\n [106] In the margin is added: 'The hinges of my outer door are so\n far from the wall that they are open more than a hand's breadth, so\n that I have got in large things between them; and above they are\n still more open, and when I put my arm through the peep-hole of the\n inner door and stretch it out, I can reach to the top of the outer\n one, though the woman cannot.'\n\nThe woman and I lived in good harmony together. Occasionally there\nwere small disputes between Christian and her, but at that time they\nwere of no importance. I quieted his anger with wine and candles.\nThis woman had a son, who died just after she had come to me, and a\ndaughter who is still alive; at that time she was in the service of a\ntailor, but she is now married to a merchant. The daughter received\npermission occasionally to come and speak with her mother on the\nstairs. This annoyed Christian, as he thought that through her all\nsorts of things were obtained; and he threatened often that he would\nsay what he thought, though he did not know it, and this frequently\ntroubled the woman (she easily weeps and easily laughs). I could soon\ncomfort her. We spent our time very well. I taught her to read,\nbeginning with A B C, for she did not know a single letter. I kept to\nfixed hours for teaching her. She was at the time sixty years of age.\nAnd when she could spell a little,[107] she turned the book one day\nover and over, and began to rub her eyes and exclaimed, 'Oh God, how\nstrange it is! I do not know (and she swore by God) a single letter.'\nI was standing behind her, and could scarcely keep from laughing. She\nrubbed her eyes again, and (as she is rather hasty with her words)\nshe pointed quickly to an O, and said, 'Is not that an O?' 'Yes,' I\nsaid, and I laughed when she turned to me. She then for the first\ntime perceived that she was holding the book upside down; she threw\nherself on the bed and laughed till I thought she would burst.\n\n [107] In the margin: 'She has a curious manner of spelling. She\n cannot spell a word of three syllables; for when she has to add the\n two syllables to the third, she has forgotten the first. If I urge\n her, however, she can read the word correctly when she has spelt\n the first syllable. She spells words of two syllables and reads\n those of four.'\n\nOne day when she was to read, and did not like to lay aside her\ndistaff, it did not go smoothly, and she gave it up, and said, 'Am I\nnot foolish to wish to learn to read in my old age? What good does it\ndo me? I have spent much money on my son to have him taught to read,\nand see, is he not dead?' I knew how much she was able to do, and I\nlet her go on speaking. She threw the book on her bed, sat down to\nher work, and said, 'What do I need to learn to read in a book? I\ncan, thank God, read my morning and evening prayer.' (I thought to\nmyself, 'badly enough.' She knew very little of her catechism.) I\nsaid (gently): 'That is true, Karen. It is not necessary for you to\nlearn to read a book, as you can read very nicely by heart.' I had\nscarcely said this than she jumped up, took her book again, and began\nto spell. I neither advised her nor dissuaded her, but treated her\nlike a good simple child.[108]\n\n [108] In the margin: 'Once she asked me whether she could not get a\n book in which there was neither _q_ nor _x_, for she could not\n remember these letters. I answered, \"Yes, if you will yourself have\n such a book printed.\"'\n\nI fell ill during this year,[109] and as the prison governor no\nlonger came in to me and sent the servant up of an evening, I begged\nthe woman to tell him that I was ill, and that I wished a doctor to\ncome to me. The woman told him this (for by this time he understood\nDanish, and the woman understood a little German), and when she said,\n'I am afraid she will die,' he answered, 'Why the d---- let her die!'\nI had daily fevers, heat, but no shivering; and as an obstruction was\nthe chief cause of my illness, I desired a remedy. The prison\ngovernor ridiculed the idea. When I heard this, I requested he would\ncome to me, which he did. I spoke to him rather seriously; told him\nthat it was not the King's will that he should take no more care of\nme than he did, that he had more care for his dog than for me (which\nwas the case). Upon this his manner improved, and he enquired what I\nwished for, and I said what I desired, and obtained it. I had become\nrather excited at the conversation, so that I felt weak. The woman\ncried and said: 'I am afraid you will die, dear lady! and then the\nbad maids from the wash-house will wash your feet and hands.' (One of\nthe maids below had sent very uncivil messages to me.) I replied that\nI should not say a word against that. 'What?' said she angrily, 'will\nyou suffer that? No,' she added with an asseveration, 'I would not! I\nwould not suffer it if I were in your place.' So I said, like that\nphilosopher, 'Place the stick with the candlestick at my side, and\nwith that I can keep them away from me when I am dead.'[110] This\nbrought her to reason again, and she talked of the grave and of\nburial. I assured her that this did not trouble me at all; that when\nI was dead, it was all one to me; even if they threw my body in the\nsea, it would, together with my soul, appear before the throne of God\nat the last day, and might come off better perhaps than many who were\nlying in coffins mounted with silver and in splendid vaults. But that\nI would not say, as the prison governor did in his levity, that I\nshould like to be buried on the hill of Valdby, in order to be able\nto look around me. I desired nothing else than a happy end. We spoke\nof the prison governor's coarseness; of various things which he did,\non account of which it would go badly with him if the Queen knew it;\nof his godlessness, how that when he had been to the Lord's Supper,\nhe said he had passed muster; and other things. There was no fear of\nGod in him.\n\n [109] In the margin of the MS. is added: 'When this Karen came to\n me she left me no peace till I allowed her to clean the floor; for\n I feared that which happened, namely that the smell would cause\n sickness. In one place there was an accumulation of dirt a couple\n of feet thick. When she had loosened it, it had to remain till the\n door was opened. I went to bed, threw the bed-clothes over my head,\n and held my nose.'[E38]\n\n [E38] 'Anno 1666, soon after Karen, Nil's daughter, came to me, we\n first discovered that there was a stone floor to my prison chamber,\n as she broke loose a piece of rubbish cemented together, and the\n stones were apparent. I had before thought it a loam floor. The\n former Karen, Ole's daughter, was one of those who spread the dirt\n but do not take it away. This Karen tormented me unceasingly,\n almost daily, that we must remove it everywhere, and that at\n once--it would soon be done. I was of opinion that it would make us\n ill if it was done all at once, as we required water to soften it,\n and the stench in this oppressive hole would cause sickness, but\n that it would be easier and less uncomfortable to remove one piece\n after another. She adhered to her opinion and to her desire, and\n thought that she could persuade the prison governor and the\n tower-warder to let the door remain open till all had been made\n clean. But when the tower-warder had brought in a tub of water, he\n locked the door. I went to bed and covered my face closely, while\n she scraped and swept up the dirt. The quantity of filth was\n incredible. It had been collecting for years, for this had been a\n malefactors' prison, and the floor had never been cleaned. She laid\n all the dirt in a heap in the corner, and there was as much as a\n cartload. It was left there until evening at supper-time, when the\n doors were opened. It was as I feared: we were both ill. The woman\n recovered first, for she could get out into the air, but I remained\n in the oppressive hole, where there was scarcely light. We gained\n this from it, that we were tormented day and night with numbers of\n fleas, and they came to her more than to me, so much so that she\n was often on the point of weeping. I laughed and made fun of it,\n saying that she would now have always something to do, and would\n have enough to beguile the time. We could not, however, work. The\n fleas were thick on our stockings, so that the colour of the\n stockings was not to be perceived, and we wiped them off into the\n water-basin. I then discovered that one flea produces another. For\n when I examined them, and how they could swim, I perceived that\n some small feet appeared behind the flea, and I thought it was a\n peculiar kind. At last I saw what it was, and I took the flea from\n which the small one was emerging on my finger, and it left behind\n evidences of birth: it hopped immediately, but the mother remained\n a little, until she recovered herself, and the first time she could\n not hop so far. This amusement I had more than once, till the fleas\n came to an end. Whether all fleas are born in this manner I cannot\n tell, but that they are produced from dirt and loam I have seen in\n my prison, and I have observed how they become gradually perfect\n and of the peculiar colour of the material from which they have\n been generated. I have seen them pair.'\n\n It is scarcely necessary to say that, as far as natural history is\n concerned, Leonora has committed a mistake.\n\n [110] In the margin is added: 'On the stick there was a tin\n candlestick, which was occasionally placed at the side of my bed. I\n used it for fixing my knitting.'[E39]\n\n [E39] Leonora alludes to an anecdote told by 'Cicero in Tuscul.\n Quaest. lib. i. c. 43.' He recounts that the cynic Diogenes had\n ordered that his body should not be buried after his death but left\n uninterred. His friends asked, 'As a prey to birds and wild\n beasts?' 'Not at all,' answered Diogenes; place a stick by me,\n wherewith I may drive them away.' 'But how can you?' rejoined\n these; 'you won't know!' 'But what then,' was his reply, 'concern\n the attacks of the wild beasts me, when I don't feel them?'\n\nI requested to have the sacrament, and asked M. Buck to come to me at\nseven o'clock in the morning, for at about half-past eight o'clock\nthe fever began. The priest did not come till half-past nine, when\nthe fever heat had set in (for it began now somewhat later). When I\nhad made my confession, he began to preach about murder and homicide;\nabout David, who was guilty of Uriah's death, although he had not\nkilled him with his own hand. He spoke of sin as behoved him, and of\nthe punishment it brings with it. 'You,' he said, 'have killed\nGeneral Fux, for you have bribed a servant to kill him.' I replied,\n'That is not true! I have not done so!' 'Yes, truly,' he said; 'the\nservant is in Hamburg, and he says it himself.' I replied: 'If he has\nso said, he has lied, for my son gave Fux his death-blow with a\nstiletto. I did not know that Fux was in Bruges until I heard of his\ndeath. How could the servant, then, say that I had done it? It was\nnot done by my order, but that I should not have rejoiced that God\nshould have punished the villain I am free to confess.' To this he\nanswered, 'I should have done so myself.' I said: 'God knows how Fux\ntreated us in our imprisonment at Borringholm. That is now past, and\nI think of it no more.' 'There you are right,' he said, as he\nproceeded in his office. When all was over, he spoke with the prison\ngovernor outside the door of my anteroom, just in front of the door\nof the Dark Church, and said that I made myself ill; that I was not\nill; that my face was red from pure anger; that he had spoken the\ntruth to me, and that I had been angry in consequence. Christian was\nstanding inside the door of the Dark Church, for at this time there\nwere no prisoners there, and he heard the conversation, and related\nit to me when I began to get up again and spoke with him at the\ndoor.\n\nSome time afterwards Christian said to me, quite secretly, 'If you\nlike, I will convey a message from you to your children in Skaane.' I\nenquired how this could be done. He said: 'Through my girl; she is\nthoroughly true; she shall go on purpose.' He knew that I had some\nducats left, for Peder the coachman had confided it to him, as he\nhimself told me. I accepted his offer and wrote to my children, and\ngave him a ducat for the girl's journey.[111] She executed the\ncommission well, and came back with a letter from them and from my\nsister.[E40] The woman knew nothing of all this.\n\n [111] In the margin: 'The girl was a prostitute to whom he had\n promised marriage, and the tower-warder--both the former one and\n Chresten--let her in to Christian, went out himself, and left them\n alone.'\n\n [E40] This sister was Hedvig, who married Ebbe Ulfeldt, a relative\n of Corfitz Ulfeldt. He was obliged to leave Denmark in 1651, on\n account of irregularities in the conduct of his office, and went to\n Sweden, where he became a major-general in the army. He is the\n person alluded to in the Autobiography. Several of Leonora's\n children lived in Sweden with their relatives after the death of\n Corfitz Ulfeldt; but in 1668 the Danish Government obtained that\n they were forbidden the country.\n\nBy degrees Christian began to be insolent in various ways. When he\ncame with his boy's pouch, in which the woman was to give him food,\nhe would throw it at her, and he was angry if meat was not kept for\nhimself for the evening; and when he could not at once get the pouch\nback again, he would curse the day when he had come to my door and\nhad spoken with me or had communicated anything to me. She was sad,\nbut she said nothing to me. This lasted only for a day, and then he\nknocked again at the door and spoke as usual of what news he had\nheard. The woman was sitting on the bed, crossing herself fifteen\ntimes (he could not see her, nor could he see me). When he was gone,\nshe related how fearfully he had been swearing, &c. I said: 'You must\nnot regard this; in the time of the other Karen he has done as much.'\nHis courage daily increased. The dishes were often brought up\nhalf-an-hour before the prison-governor came. In the meanwhile\nChristian cut the meat, and took himself the piece he preferred\n(formerly at every meal I had sent him out a piece of fish, or\nanything else he desired). The stupid prison governor allowed it to\ngo on; he was glad, I imagine, that he was spared the trouble, and\npaid no attention to the fact that there was anything missing in the\ndish. I let it go on for a time, for it did not happen regularly\nevery day. But when he wanted food for his boy, he would say nothing\nbut 'Some food in my boy's pouch!' We often laughed over this\nafterwards, when he was away, but not at the time, for it grew worse\nfrom day to day. He could not endure that we should laugh and be\nmerry; if he heard anything of the kind outside, he was angry. But if\none spoke despondingly, he would procure what was in his power.[112]\nOne day he listened, and heard that we were laughing; for the woman\nwas just relating an amusing story of the mother of a schoolboy in\nFrederichsborg (she had lived there); how the mother of the boy did\nnot know how to address the schoolmaster, and called him Herr\nWillas.[E41] He said, 'I am no Herr.' 'Then Master,' said the woman.\n'I am no Master either,' he said; 'I am plain Willas.' Then the woman\nsaid: 'My good plain Willas! My son always licks the cream from my\nmilk-pans when he comes home. Will you lick him in return, and that\nwith a switch on his back?' While we were laughing at this, he came\nto the door and heard the words I was saying: 'I don't suppose that\nit really so happened; one must always add something to make a good\nstory of it.' He imagined we were speaking of him, and that we were\nlaughing at him. At meal-time he said to the woman, 'You were very\nmerry to-day.' She said, 'Did you not know why? It is because I\nbelong to the \"Laetter\"'[E42] (that was her family name). 'It would be\na good thing,' he said, 'to put a stop to your laughter altogether;\nyou have been laughing at me.' She protested that we had not, that\nhis name had not been mentioned (which was the case); but he would\nnot regard it. They fell into an altercation. She told me of the\nconversation, and for some days he did not come to the door, and I\nsent him nothing; for just at that time a poor old man was my\nneighbour, and I sent him a drink of wine. Christian came again to\nthe door and knocked. He complained very softly of the woman; begged\nthat I would reprove her for what she had said to him, as he had\nheard his name mentioned. I protested to him that at the time we were\nnot even thinking of him, and that I could not scold her for the\nwords we had spoken together. I wished to have repose within our\nclosed door. 'Yes,' he answered; 'household peace is good, as the old\nwoman said.' With this he went away.\n\n [112] In the margin: 'In the time of his good humour he had\n procured me, for money and candles, all that I desired, so that I\n had both knife and scissors, besides silk, thread, and various\n things to beguile the time. This vexed him afterwards.'\n\n [E41] The title 'Herr' was then only given to noblemen and clergy.\n Master means 'magister,' and was an academical title.\n\n [E42] The original has here an untranslatable play upon words.\n _Leth_ is a family name; and the woman says 'I am one of the Letter\n (the Leths),' but laughter is in Danish 'Latter.'\n\nAfterwards he caused us all sorts of annoyance, and was again\npacified. Then he wished again that I should write to Skaane.[113] I\nsaid I was satisfied to know that some of my children were with my\nsister; where my sons were, and how it fared with them, I did not\nknow: I left them in God's care. This did not satisfy him, and he\nspoke as if he thought I had no more money; but he did not at that\ntime exactly say so. But one day, when he had one of his mad fits, he\ncame to the door and had a can with wine (which I gave him at almost\nevery meal) in his hand, and he said: 'Can you see me?' (for there\nwas a cleft in the outermost door, but at such a distance one could\nnot clearly see through). 'Here I am with my cup of wine, and I am\ngoing to drink your health for the last time.' I asked: 'Why for the\nlast time?' 'Yes,' he swore, coming nearer to the door and saying: 'I\nwill do no more service for you; so I know well that I shall get no\nmore wine.' I said, 'I thank you for the services you have rendered\nme; I desire no more from you, but nevertheless you may still get\nyour wine.' 'No!' he said; 'no more service! there is nothing more to\nbe fetched.' 'That is true,' I answered. 'You do not know me,' said\nhe; 'I am not what you think; it is easy to start with me, but it is\nnot easy to get rid of me.' I laughed a little, and said: 'You are\nfar better than you make yourself out to be. To-morrow you will be of\nanother mind.'\n\n [113] In the margin: 'Immediately after the girl had been in\n Skaane, he gave her a box full of pieces of wax, on which were the\n impressions of all the tower keys; and amongst them was written,\n \"My girl will have these made in Skaane.\" I had this from the\n woman, who was just then carrying up the night-stool, and on the\n following Saturday I gave the box back with many thanks, saying I\n did not care to escape from the tower in this way. This did not\n please him, as I well saw.'\n\nHe continued to describe himself as very wicked (it was, however, far\nfrom as bad as he really is). I could do nothing else but laugh at\nhim. He drank from the can, and sat himself down on the stool\noutside. I called him and begged him to come to the door, as I wanted\nto speak with him. There he sat like a fool, saying to himself:\n'Should I go to the door? No,' and he swore with a terrible oath,\n'that I will not do! Oh yes, to the door! No, Christian, no!'\nlaughing from time to time immoderately, and shouting out that the\ndevil might take him and tear him in pieces the day on which he\nshould go to my door or render me a service. I went away from the\ndoor and sat down horrified at the man's madness and audacity. Some\ndays passed in silence, and he would accept no wine. No food was\noffered to him, for he continued, in the same way as before, to cut\nthe meat before the prison governor came up. As the prison governor\nat this time occasionally again came in to me and talked with me, I\nrequested him that Christian, as a prisoner, should not have the\nliberty of messing my food. This was, therefore, forbidden him in\nfuture.\n\nSome days afterwards he threw the pouch to the woman on the stairs,\nand said: 'Give me some food for to-night in my lad's pouch.'[114]\nThis was complied with with the utmost obedience, and a piece of meat\nwas placed in the pouch. This somewhat appeased him, so that at noon\nhe spoke with the woman, and even asked for a drink of wine; but he\nthreatened the woman that he would put an end to the laughing. I did\nnot fear the evil he could do to me, but this vexatious life was\nwearisome. I allowed no wine to be offered to him, unless he asked\nfor some. He was in the habit every week of procuring me the\nnewspapers[E43] for candles, and as he did not bring me the\nnewspapers for the candles of the first week, I sent him no more. He\ncontinued to come every Saturday with the perfuming-pan, and to lock\nmy door. When he came in with the fumigating stuff, he fixed his eyes\nupon the wall, and would not look at me. I spoke to him once and\nasked after the doctor, and he made no reply.\n\n [114] In the margin is added: 'At this time there was a peasant\n imprisoned in the Dark Church for having answered the bailiff of\n the manor with bad language. I sent him food. He was a great rogue.\n I know not whether he were incited by others, but he told Karen\n that if I would write to my children, he would take care of the\n letter. I sent him word that I thanked him; I had nothing to say to\n them and nothing to write with. The rogue answered, \"Ah so! Ah\n so!\"'\n\n [E43] The newspapers in question were probably German papers which\n were published in Copenhagen at that time weekly, or even twice a\n week; the Danish _Mercurius_ (a common title for newspapers) was a\n monthly publication.\n\nThus it went on for some weeks; then he became appeased, and brought\nthe woman the papers from the time that he had withheld them, all\nrolled up together and fastened with a thread. When the prison\ngovernor came in during the evening and sat and talked (he was\nslightly intoxicated), and Chresten had gone to the cellar, the woman\ngave him back the papers, thanking him in my name, and saying that\nthe papers were of no interest to me; I had done without them for so\nmany weeks, and could continue to do so. He was so angry that he tore\nthe papers in two with his teeth, tore open his coat so that the\nbuttons fell on the floor, threw some of the papers into the fire,\nhowled, screamed, and gnashed with his teeth. I tried to find\nsomething over which I could laugh with the prison governor, and I\nspoke as loud as I could, in order to drown Christian's voice.[115]\nThe woman came in as pale as a corpse, and looked at me. I signed to\nher that she should go out again. Then Christian came close to my\ndoor and howled, throwing his slippers up into the air, and then\nagainst my door, repeating this frequently. When he heard Chresten\ncoming up with the cups, he threw himself on the seat on which the\nprison governor was accustomed to lie, and again struck his slippers\nagainst the wall. Chresten gazed at him with astonishment, as he\nstood with the cups in his hand. He saw well that there was something\namiss between the woman and Christian, and that the woman was afraid;\nhe could not, however, guess the cause, nor could he find it out; he\nthought, moreover, that it had nothing to do with me, since I was\nlaughing and talking with the prison governor. When the doors were\nclosed, the lamentations found free vent. The woman said that he had\nthreatened her; he would forbid her daughter coming on the stairs and\ncarrying on her talk, and doing other things that she ought not. I\nbegged her to be calm; told her he was now in one of his mad fits,\nbut that it would pass away; that he would hesitate before he said\nanything of it, for that he would be afraid that what he had brought\nup to her would also come to light, and then he would himself get\ninto misfortune for his trouble; that the prison governor had given\nher daughter leave to come to her, and to whom therefore should he\ncomplain? (I thought indeed in my own mind that if he adhered to his\nthreat, he would probably find some one else to whom he could\ncomplain, as he had so much liberty; he could bring in and out what\nhe chose, and could speak with whom he desired in the watchman's\ngallery.) She wept, was very much affected, and talked with but\nlittle sense, and said: 'If I have no peace for him, I will--yes, I\nwill--.' She got no further, and could not get out what she would do.\nI smiled, and said at last: 'Christian is mad. I will put a stop to\nit to-morrow: let me deal with him! Sleep now quietly!'\n\n [115] In the margin: 'It was wonderful that the governor did not\n hear the noise which Christian made. He was telling me, I remember,\n at the time, how he had frightened one of the court servants with a\n mouse in a box.'\n\nShe fell asleep afterwards, but I did not do so very quickly,\nthinking what might follow such wild fits. Next day towards noon I\ntold her what she was to say to Christian; she was to behave as if\nshe were dissatisfied, and begin to upbraid him and to say, 'The\ndevil take you for all you have taught her! She has pulled off her\nslippers just as you do, and strikes me on the head with them. She is\nangry and no joke, and she took all the pretty stuff she had finished\nand threw it into the night-stool. \"There,\" said she, \"no one shall\nhave any advantage of that.\"' At this he laughed like a fool, for it\npleased him. 'Is she thoroughly angry?' he asked. 'Yes,' she replied;\n'she is indeed.' At this he laughed aloud on the stairs, so that I\nheard it. For a fortnight he behaved tolerably well, now and then\ndemanding wine and food; and he came moreover to the door and\nrelated, among other things, how he had heard that the prince (now\nour king) was going to be married. I had also heard it, though I did\nnot say so, for the prison governor had told me of it, and besides I\nreceived the papers without him. And as I asked him no questions, he\nwent away immediately, saying afterwards to the woman, 'She is angry\nand so am I. We will see who first will want the other.' He\nthreatened the woman very much. She wished that I would give him fair\nwords. I told her that he was not of that character that one could\nget on with him by always showing the friendly side.[116] As he by\ndegrees became more insolent than could be tolerated, I said one day\nto the prison governor that I was surprised that he could allow a\nprisoner to unlock and lock my doors, and to do that which was really\nthe office of the tower-warder; and I asked him whether it did not\noccur to him that under such circumstances I might manage to get out,\nif I chose to do so without the King's will? Christian was a\nprisoner, under sentence of death; he had already offered to get me\nout of the tower. The prison governor sat and stared like one who\ndoes not rightly understand, and he made no reply but 'Yes, yes!' but\nhe acted in conformity with my warning, so that either he himself\nlocked and unlocked, or Chresten did so. (I have seen Christian\nsnatching the keys out of Chresten's hand and locking my door, and\nthis at the time when he began to make himself so angry.)\n\n [116] In the margin is added: 'He enticed the prison governor to\n throw a kitten that I had down from the top of the tower, and he\n laughed at me ironically as he told the woman of his manly act, and\n said, \"The cat was mangy! the cat was mangy!\" I would not let him\n see that it annoyed me.'\n\nIf Christian had not been furious before, he became so now,\nespecially at the time that Chresten came in with the perfuming-pan\nwhen the woman was above. He would then stand straight before me in\nthe anteroom, looking at me like a ghost and gnashing his teeth; and\nwhen he saw that I took the rest of the fumigating stuff from\nChresten's hand (which he had always himself given me in paper), he\nburst into a defiant laugh. When the doors were unlocked in the\nevening, and Christian began talking with the woman, he said: 'Karen,\ntell her ladyship that I will make out a devilish story with you\nboth. I have with my own eyes seen Chresten giving her a letter. Ay,\nthat was why she did not let me go in with the perfuming-pan, because\nI would not undertake her message to Skaane. Ay, does she get the\nnewspapers also from him? Yes, tell her, great as are the services I\nhave rendered her, I will now prepare a great misfortune for her.'\nGod knows what a night I had! Not because I feared his threat, for I\ndid not in the least regard his words; he himself would have suffered\nthe most by far. But the woman was so sad that she did nothing but\nlament and moan, chiefly about her daughter, on account of the\ndisgrace it would be to her if they put her mother into the Dark\nChurch, nay even took her life. Then she remembered that her daughter\nhad spoken with her on the stairs, and she cried out again: 'Oh my\ndaughter! my daughter! She will get into the house of correction!'\nFor some time I said nothing more than 'Calm yourself; it will not be\nas bad as you think,' as I perceived that she was not capable of\nlistening to reason, for she at once exclaimed 'Ach! ach!' as often\nas I tried to speak, sitting up in bed and holding her head between\nher two hands and crying till she was almost deluged. I thought,\n'When there are no more tears to come, she will probably stop.'\n\nI said at length, when she was a little appeased: 'The misfortune\nwith which the man threatens us cannot be averted by tears. Calm\nyourself and lie down to sleep. I will do the same, and I will pray\nGod to impart to me His wise counsel for the morrow.' This quieted\nher a little; but when I thought she was sleeping, she burst forth\nagain with all the things that she feared; she had brought in to me\nslips of paper, knife and scissors, and other things furnished by him\ncontrary to order. I answered only from time to time: 'Go to sleep,\ngo to sleep! I will talk with you to-morrow!' It was of no avail. The\nclock struck two, when she was still wanting to talk, and saying, 'It\nwill go badly with the poor old man down below!'[117] I made as if I\nwere asleep, but the whole night, till five o'clock and longer, no\nsleep came to my eyes.\n\n [117] In the margin is added: '1666. While Karen, Nil's daughter,\n waited on me, a Nuremberger was my neighbour in the Dark Church; he\n was accused of having coined base money. She carried food to him\n every day. He sang and read day and night, and sang very well. He\n sang the psalm 'Incline thine ear unto me, O Lord,' slowly at my\n desire. I copied it, and afterwards translated it into Danish. And\n as he often prayed aloud at night and confessed his sins, praying\n God for forgiveness and exclaiming again and again, 'Thou must help\n me, God! Yes, God, thou must help me, or thou art no God. Thou must\n be gracious;' thus hindering me from sleep, I sent him word through\n Karen to pray more softly, which he did. He was taken to the Holm\n for some weeks, and was then set at liberty.\n\nWhen the door was unlocked at noon, I had already intimated to her\nwhat she was to say to Christian, and had given her to understand\nthat he thought to receive money from her and candles from me by his\nthreats, and that he wanted to force us to obey his pleasure; but\nthat he had others to deal with than he imagined. She was only to\nbehave as if she did not care for his talk, and was to say nothing\nbut 'Good day,' unless he spoke to her; and if he enquired what I had\nsaid, she was to act as if she did not remember that she was to tell\nme anything. If he repeated his message, she was to say: 'I am not\ngoing to say anything to her about that. Are you still as foolish as\nyou were last night? Do what you choose!' and then go away. This\nconversation took place, and he threatened her worse than before. The\nwoman remained steadfast, but she was thoroughly cast down when our\ndoors were locked; still, as she has a light heart, she often laughed\nwith the tears in her eyes. I knew well that Christian would try to\nrecover favour again by communicating me all kind of news in writing,\nbut I had forbidden the woman to take his slips of paper, so that he\ngot very angry. I begged her to tell him that he had better restrain\nhimself if he could; that if he indulged his anger, it would be worse\nfor him. At this he laughed ironically, and said, 'Tell her, it will\nbe worse for her. Whatever I have done for her, she has enticed me to\nby giving me wine: tell her so. I will myself confess everything; and\nif I come to the rack and wheel, Chresten shall get into trouble. He\nbrought her letters from her children.' (The rogue well knew that I\nhad not allowed the woman to be cognisant neither of the fact that he\nhad conveyed for me a message to Skaane to my children, nor of the\nwax in which the tower keys were impressed; this was why he spoke so\nfreely to her.) When our doors were locked, this formed the subject\nof our conversation. I laughed at it, and asked the woman what\ndisgrace could be so great as to be put on the wheel; I regarded it\nas thoughtless talk, for such it was, and I begged her to tell him\nthat he need not trouble himself to give himself up, as I would\nrelieve him of the trouble, and (if he chose) tell the prison\ngovernor everything on the following day that he had done for me; he\nhad perhaps forgotten something, but that I could well remember it\nall.\n\nWhen the woman told him this, he made no answer, but ran down, kept\nquiet for some days, and scarcely spoke to the woman. One Saturday,\nwhen the woman had gone upstairs with the night-stool, he went up to\nher and tried to persuade her to accept a slip of paper for me, but\nshe protested that she dare not. 'Then tell her,' he said, 'that she\nis to give me back the scissors and the knife which I have given her.\nI will have them, and she shall see what I can do. You shall both\ntogether get into trouble!' She came down as white as a corpse, so\nthat I thought she had strained herself. She related the conversation\nand his request, and begged me much to give him back the things, and\nthat then he would be quiet. I said: 'What is the matter with you?\nare you in your senses? Does he not say that we shall get into\ntrouble if he gets the scissors and knife back again? Now is not the\ntime to give them to him. Do you not understand that he is afraid I\nshall let the things be seen? My work, he thinks, is gone, and the\npapers are no longer here, so that there is nothing with which he can\nbe threatened except these things. You must not speak with him this\nevening. If he says anything, do not answer him.' In the evening he\ncrept in, and said in the anteroom to her, 'Bring me the scissors and\nthe knife!' She made no answer. On the following morning, towards\nnoon, I begged her to tell him that I had nothing of his; that I had\npaid for both the scissors and knife, and that more than double their\nvalue. He was angry at the message, and gnashed with his teeth. She\nwent away from him, and avoided as much as possible speaking with him\nalone. When he saw that the woman would not take a slip of paper from\nhim, he availed himself of a moment when the prison governor was not\nthere, and threw in a slip of paper to me on the floor. A strange\ncircumstance was near occurring this time: for just as he was\nthrowing in the paper, the prison governor's large shaggy dog passed\nin, and the paper fell on the dog's back, but it fell off again in\nthe corner, where the dog was snuffling.\n\nUpon the paper stood the words: 'Give me the knife and scissors back,\nor I will bring upon you as much misfortune as I have before rendered\nyou good service, and I will pay for the knife and scissors if I have\nto sell my trousers for it. Give them to me at once!' For some days\nhe went about like a lunatic, since I did not answer him, nor did I\nsend him a message through the woman; so that Chresten asked the\nwoman what she had done to Christian, as he went about below gnashing\nhis teeth and howling like a madman. She replied that those below\nmust best know what was the matter with him; that he must see he was\nspoken with in a very friendly manner here. At noon on Good Friday,\n1667,[118] he was very angry, swore and cursed himself if he did not\ngive himself up, repeating all that he had said before, and adding\nthat I had enticed him with wine and meat, and had deceived him with\ncandles and good words. That he cared but little what happened to\nhim; he would gladly die by the hand of the executioner; but that I,\nand she, and Chresten, should not escape without hurt.\n\n [118] In the MS. this date '1667' is in the _margin_, not in the\n text.\n\nThe afternoon was not very cheerful to us. The woman was depressed. I\nbegged her to be calm, told her there was no danger in such madness,\nthough it was very annoying, and harder to bear than my captivity;\nbut that still I would be a match for the rogue. She took her book\nand read, and I sat down and wrote a hymn upon Christ's sufferings,\nto the tune 'As the hart panteth after the water-springs.'[119]\n\n [119] In the margin is added: 'This very hymn was afterwards the\n cause of Christian's being again well-behaved, as he subsequently\n himself told me, for he heard me one day singing it, and he said\n that his heart was touched, and that tears filled his eyes. I had\n at that time no other writing-materials than I have before\n mentioned.'\n\nChristian had before been in the habit of bringing me coloured eggs\non Easter-Eve; at this time he was not so disposed. When the door was\nlocked, I said to Chresten, 'Do not forget the soft-boiled eggs\nto-morrow.' When the dinner was brought up on Easter-Day, and the\neggs did not come at once (they were a side dish), Christian looked\nat me, and made a long nose at me three or four times. (I was\naccustomed to go up and down in front of the door of my room when it\nwas unlocked.) I remained standing, and looked at him, and shrugged\nmy shoulders a little. Soon after these grimaces, Chresten came with\na dish full of soft-boiled eggs. Christian cast down his eyes at\nfirst, then he raised them to me, expecting, perhaps, that I should\nmake a long nose at him in return; but I intended nothing less. When\nthe woman went to the stairs, he said, 'There were no coloured eggs\nthere.' She repeated this to me at once, so that I begged her to say\nthat I ate the soft-boiled eggs and kept the coloured ones, as he\nmight see (and I sent him one of the last year's, on which I had\ndrawn some flowers; he had given it to me himself for some candles).\nHe accepted it, but wrote me a note in return, which was very\nextraordinary. It was intended to be a highflown composition about\nthe egg and the hen. He tried to be witty, but it had no point. I\ncannot now quite remember it, except that he wrote that I had sent\nhim a rotten egg; that his egg would be fresh, while mine would be\nrotten.[120] He threw the slip of paper into my room. I made no\nanswer to it. Some days passed again, and he said nothing angry; then\nhe recommenced. I think he was vexed to see Chresten often receive my\nwine back again in the cup. At times I presented it to the prison\ngovernor. Moreover, he received no food, either for himself or his\nboy. One day he said to the woman, 'What do you think the prison\ngovernor would say if he knew that you give the prisoners some of his\nfood to eat?' (The food which came from my table was taken down to\nthe prison governor.) 'Tell her that!' The woman asked whether she\nwas to say so to me, as a message from him. 'As whose message\notherwise?' he answered. I sent him word that I could take as much as\nI pleased of the food brought me: that it was not measured out and\nweighed for me, and that those who had a right to it could do what\nthey liked with what I did not require, as it belonged to no one. On\nthis point he could not excite our fear. Then he came back again one\nday to the old subject, that he would have the scissors and the\nknife, and threatening to give himself up; and as it was almost\napproaching the time when I received the Lord's Supper, I said to the\nwoman: 'Tell him once for all, if he cannot restrain himself I will\ninform against him as soon as the priest comes, and the first Karen\nshall be made to give evidence; she shall, indeed, be brought\nforward, for she had no rest on his account until I entered into his\nproposals. Whether voluntarily or under compulsion, she shall say the\ntruth, and then we shall see who gets into trouble.' He might do, I\nsent word, whatever he liked, but I would be let alone; he might\nspare me his notes, or I would produce them. When the woman told him\nthis, he thought a little, and then asked, 'Does she say so?' 'Yes,'\nsaid the woman, 'she did. She said still further: \"What does he\nimagine? Does he think that I, as a prisoner who can go nowhere, will\nsuffer for having accepted the services of a prisoner who enjoys a\nliberty which does not belong to him?\"' He stood and let his head\nhang down, and made no answer at all. This settled the fellow, and\nfrom that time I have not heard one unsuitable word from him. He\nspoke kindly and pleasantly with the woman on the stairs, related\nwhat news he had heard, and was very officious; and when she once\nasked him for his cup to give him some wine, he said sadly, 'I have\nnot deserved any wine.' The woman said he could nevertheless have\nsome wine, and that I desired no more service from him. So he\nreceived wine from time to time, but nothing to eat.[121] On the day\nthat I received the Lord's Supper, he came to the door and knocked\nsoftly. I went to the door. He saluted me and wished me joy in a very\nnice manner, and said that he knew I had forgiven those who had done\naught against me. I answered in the affirmative, and gave no further\nmatter for questions; nor did he, but spoke of other trivialities,\nand then went away. Afterwards he came daily to the door, and told me\nwhat news he had heard; he also received wine and meat again. He told\nme, among other things, that many were of opinion that all the\nprisoners would be set at liberty at the wedding of the prince (our\npresent king) which was then talked of; that the bride was to arrive\nwithin a month (it was the end of April when this conversation took\nplace), and that the wedding was to be at the palace.\n\n [120] What he meant by it I know not; perhaps he meant that I\n should die in misery, and that he should live in freedom. That\n anticipation has been just reversed, for his godless life in his\n liberty threw him subsequently into despair, so that he shot\n himself. Whether God will give me freedom in this world is known to\n Him alone.\n\n [121] In the margin is added: 'He could not prevent his boy Paaske\n from having a piece of meat placed for him in front of the door.'\n\nThe arrival of the bride was delayed till the beginning of June, and\nthen the wedding was celebrated in the palace at Nykjobing in\nFalster. Many were of opinion that it took place there in order that\nthe bride might not intercede for me and the doctor.[122] When the\nbride was to be brought to Copenhagen, I said to Christian: 'Now is\nthe time for you to gain your liberty. Let your girl wait and fall on\none knee before the carriage of the bride and hold out a\nsupplication, and then I am sure you will gain your liberty.' He\nasked how the girl should come to be supplicating for him. I said,\n'As your bride--' 'No (and he swore with a terrible oath), she is not\nthat! She imagines it, perhaps, but (he swore again) I will not have\nher.' 'Then leave her in the idea,' I said, 'and let her make her\nsupplication as for her bridegroom.' 'Yes,' he said, in a crestfallen\ntone, 'she may do that.' It was done, as I had advised, and Christian\nwas set at liberty on June 11, 1667. He did not bid me good-bye, and\ndid not even send me a message through the tower-warder or the boy.\nHis gratitude to the girl was that he smashed her window that very\nevening, and made such a drunken noise in the street, that he was\nlocked up in the Town-hall cellar.[123] He came out, however, on the\nfollowing day. His lad Paaske took leave of his master. When he asked\nhim whether he should say anything from him to us, he answered, 'Tell\nthem that I send them to the devil.' Paaske, who brought this\nmessage, said he had answered Christian, 'Half of that is intended\nfor me' (for Christian had already suspected that Paaske had rendered\nservices to the woman). We had a hearty laugh over this message; for\nI said that if Paaske was to have half of it, I should get nothing.\nWe were not a little glad that we were quit of this godless man.\n\n [122] In the margin is added: 'The bride had supplicated for me at\n Nykjobing, but had not gained her object. This was thought to be\n dangerous both for the land and people.'\n\n [123] In the margin is added: 'It was a Sunday; this was the honour\n he showed to God. He went into the wine-house instead of into God's\n house. He came out about twelve o'clock.'\n\nWe lived on in repose throughout the year 1668. I wrote and was\nfurnished with various handiwork, so that Chresten bought nothing for\nme but a couple of books, and these I paid doubly and more than\ndoubly with candles. Karen remained with me the first time more than\nthree years; and as her daughter was then going to be married, and\nshe wished to be at the wedding, she spoke to me as to how it could\nbe arranged, for she would gladly have a promise of returning to me\nwhen the woman whom I was to have in her stead went away. I did not\nknow whether this could be arranged; but I felt confident that I\ncould effect her exit without her feigning herself ill. The prison\ngovernor had already then as clerk Peder Jensen Totzloff,[E44] who\nnow and then performed his duties. To this man I made the proposal,\nmentioning at the same time with compassion the ill health of the\nwoman. I talked afterwards with the prison governor himself about it,\nand he was quite satisfied; for he not only liked this Karen very\nmuch, but he had moreover a woman in the house whom he wished to\nplace with me instead.\n\n [E44] His name was Torslev; see the Introduction and the\n Autobiography.\n\nKaren, Nils' daughter, left me one evening in 1669, and a German\nnamed Cathrina ----[E45] came in her place. Karen took her departure\nwith many tears. She had wept almost the whole day, and I promised to\ndo my utmost that she should come to me when the other went away.\nCathrina had been among soldiers from her youth up; she had married a\nlieutenant at the time the prison governor was a drummer, and had\nstood godmother to one of his sons. She had fallen into poverty after\nher husband's death, and had sat and spun with the wife of the prison\ngovernor for her food. She was greatly given to drinking, and her\nhands trembled so that she could not hold the cup, but was obliged to\nsupport it against her person, and the soup-plate also. The prison\ngovernor told me before she came up that her hands occasionally\ntrembled a little, but not always--that she had been ill a short time\nbefore, and that it would probably pass off. When I asked herself how\nit came on, she said she had had it for many years. I said, 'You are\nnot a woman fit to wait upon me; for if I should be ill, as I was a\nyear or somewhat less ago, you could not properly attend to me.' She\nfell at once down on her knees, wept bitterly, and prayed for God's\nsake that she might remain; that she was a poor widow, and that she\nhad promised the prison governor half the money she was to earn; she\nwould pray heartily to God that I might not be ill, and that she\nwould be true to me, aye, even die for me.\n\n [E45] The name is in blanco; she was probably the Catharina Wolf\n which is mentioned in the Preface.\n\nIt seemed to me that this last was too much of an exaggeration for me\nto believe it (she kept her word, however, and did what I ordered\nher, and I was not ill during her time). She did not care to work.\nShe generally laid down when she had eaten, and drew the coverlid\nover her eyes, saying 'Now I can see nothing.' When she perceived\nthat I liked her to talk, she related whole comedies in her way,\noften acting them, and representing various personages. If she began\nto tell a story, and I said in the middle of her narrative, 'This\nwill have a sorrowful ending,' she would say, 'No, it ends\npleasantly,' and she would give her story a good ending. She would do\nthe reverse, if I said the contrary. She would dance also before me,\nand that for four persons, speaking as she did so for each whom she\nwas representing, and pinching together her mouth and fingers. She\ncalled comedians 'Medicoants.' Various things occurred during her\ntime, which prevented me from looking at her and listening to her as\nmuch as she liked.[124]\n\n [124] In the margin is added: 'A few months after she had come to\n me, she had an attack of ague. She wept, and was afraid. I was well\n satisfied with her, and thought I would see what faith could do, so\n I wrote something on a slip of paper and hung it round her neck.\n The fever left her, and she protested that all her bodily pains\n passed all at once into her legs when I hung the paper round her\n neck. Her legs immediately became much swollen.'\n\nIt happened that Walter,[E46] who in consequence of Dina's affair had\nbeen exiled from Denmark, came over from Sweden and remained\nincognito at Copenhagen. He was arrested and placed in the tower\nhere, below on the ground floor. He was suspected of being engaged\nin some plot. At the same time a French cook and a Swedish baker were\nimprisoned with him, who were accused of having intended to poison\nthe King and Queen. The Swede was placed in the Witch Cell,\nimmediately after Walter's arrest. Some days elapsed before I was\nallowed to know of Walter's arrival, but I knew of it nevertheless.\nOne day at noon, when Walter and the Frenchman were talking aloud\n(for they were always disputing with each other), I asked the prison\ngovernor who were his guests down below, who were talking French. He\nanswered that he had some of various nations, and related who they\nwere, but why they were imprisoned he knew not, especially in\nWalter's case.\n\n [E46] Walter's participation in the plot of Dina is mentioned in\n the Introduction. He was then ordered to leave the country, but\n afterwards obtained a pardon and permission to return. He does not\n seem to have availed himself of this till the year 1668; but his\n conduct was very suspicious, and he was at once arrested and placed\n in the Blue Tower, where he died towards the end of April 1670.\n\nThe two before-mentioned quarrelled together, so that Walter was\nplaced in the Witch Cell with the Swede, and the Frenchman was\nconveyed to the Dark Church, where he was ill, and never even came to\nthe peep-hole in the door, but lay just within. I dared not send him\nanything, on account of the accusation against him. Walter was\nimprisoned for a long time, and the Frenchman was liberated. When M.\nBock came to me, to give me Christ's body and blood, I told him\nbefore receiving the Lord's Supper of Walter's affair, which had been\nproved, but I mentioned to him that at the time I had been requested\nto leave Denmark through Uldrich Christian Gyldenlove. Gyldenlove had\nsworn to me that the king was at the time not thoroughly convinced of\nthe matter, and I had complained that his Majesty had not taken pains\nto convince himself; and I requested the priest to ask the\nStadtholder to manage that Walter should now be examined in Dina's\naffair, and that he and I should be confronted together in the\npresence of some ministers; that this could be done without any\ngreat noise, for the gentlemen could come through the secret passage\ninto the tower. The priest promised to arrange this;[125] he did so,\nand on the third day after Walter was placed in the Dark Church, so\nthat I expected for a long time every day that we should be examined,\nbut it was prevented by the person whose interest it was to prevent\nit.[E47]\n\n [125] In the margin is added: 'When the priest left me, he spoke\n with Walter in front of the grated hole, told him of my desire, and\n its probable result. Walter laughed ironically, and said, \"My hair\n will not stand on end for fear of that matter being mooted again.\n The Queen knows that full well. Say that too!\" While Walter was in\n the Witch Cell hole, he had written to the Queen, but the King\n received the paper.'\n\n [E47] Leonora alludes, no doubt, to the Queen Sophia Amalia.\n\nWalter remained imprisoned,[126] and quarrelled almost daily with\nChresten, calling him a thief and a robber. (Chresten had found some\nducats which Walter had concealed under a stool; the foolish Walter\nallowed the Swede to see that he hid ducats and an ink-bottle between\nthe girths under the stool, and he afterwards struck the Swede, who\nbetrayed him.) Chresten slyly allowed Walter to take a little\nexercise in the hall of the tower, and in the meanwhile he searched\nthe stool. It may well be imagined that at the everlasting scolding\nChresten was annoyed, and he did not procure Walter particularly good\nfood from the kitchen; so that sometimes he could not eat either of\nthe two dishes ordered for him; and when Walter said one day, 'If you\nwould give me only one dish of which I could eat, it would be quite\nenough,' Chresten arranged it so that Walter only received one dish,\nand often could not eat of that. (This was to Chresten's own damage,\nfor he was entitled to the food that was left; but he was ready to\nforego this, so long as he could annoy the others.)\n\n [126] In the margin is noted: 'I looked through a hole in my\n outermost door at the time that Walter was brought up in the Dark\n Church. He wept aloud. I afterwards saw him once in front of the\n hole of the door of his cell. He was very dirty, and had a large\n beard full of dirt, very clotted.'\n\nOnce Chresten came to him with a dish of rice-porridge, and began at\nonce to quarrel with him, so that the other became angry (just as\nchildren do), and would eat nothing. Chresten carried the porridge\naway again directly, and laughed heartily. I said to Chresten, in the\nprison governor's presence, 'Though God has long delayed to punish\nWalter, his punishment is all the heavier now, for he could scarcely\nhave fallen into more unmerciful hands than yours.' He laughed\nheartily at this, and the prison governor did the same. And as there\nis a hole passing from the Dark Church into the outer room, those who\nare inside there can call upstairs, so that one can plainly hear what\nis said. So Walter one day called to the prison governor, and begged\nhim to give him a piece of roast meat; the prison governor called to\nhim, 'Yes, we will roast a rat for you!' I sent him a piece of roast\nmeat through Chresten; when he took it, and heard that I had sent it\nto him, he wept.\n\nThus the time passed, I had always work to do, and I wrote also a\ngood deal.[127] The priest was tired of administering the Lord's\nSupper to me, and he let me wait thirteen and fourteen days; when he\ndid come, he performed his office _par maniere d'acquit_. I said\nnothing about it, but the woman, who is a German, also received the\nLord's Supper from him; she made much of it, especially once (the\nlast time he confessed her); for then I waited four days for him\nbefore it suited him to come, and at last he came. It was Wednesday,\nabout nine o'clock. He never greeted us, nor did he wish me joy to\nthe act I intended to perform. This time he said, as he shook hands,\n'I have not much time to wait, I have a child to baptise.' I knew\nwell that this could not be true, but I answered 'In God's name!'\nWhen he was to receive the woman's confession, he would not sit down,\nbut said 'Now go on, I have no time,' and scarcely gave her time to\nconfess, absolved her quickly, and read the consecrating service at\nposthaste speed. When he was gone, the woman was very impatient, and\nsaid that she had received the holy communion in the field from a\nmilitary chaplain, with the whole company (since they were ready to\nattack the enemy on the following day), but that the priest had not\nraced through God's word as this one had done; she had gained nothing\nfrom it.\n\n [127] In the margin is added: 'From books which had been secretly\n lent me, and I did so with the pen and ink I have before mentioned,\n on any pieces of paper which I happened to procure.'\n\nI comforted her as well as I could, read and sang to her, told her\nshe should repent and be sorry for her sins, and labour to amend her\nways, and not be distracted by the want of devotion in the priest;\nshe could appropriate to herself Christ's sufferings and merits for\nthe forgiveness of her sins, for the priest had given her his body\nand blood in the bread and wine. 'Yes,' she answered, 'I shall, with\nGod's will, be a better Christian.' I said 'Will you keep what you\nhave promised me?' Her vow was, not to drink herself tipsy, as she\nhad once done. I will not omit to mention this. She received, as I\nhave before said, half a pint of French wine at each meal, and I half\na measure of Rhine wine. She could drink both portions without being\nquite intoxicated, for at her meal she drank the French wine and lay\ndown; and when she got up in the afternoon she drank my wine.[128] In\nthe evening she kept my wine for breakfast, but once she had in her\ncup both my wine and her own, so that at noon she had two half-pints\nof wine; she sat there and drank it so quietly, and I paid no\nattention to her, being at the moment engaged in a speculation about\na pattern which I wanted to knit; at length I looked at her because\nit was so long before she laid down; then she turned over all the\nvessels, one after another, and there was nothing in them. I accosted\nher and said, 'How is it? have you drank all the wine?' She could\nscarcely answer. She tried to stand up, and could not. 'To bed, you\ndrunken sow,' said I. She tried to move, but could not; she was sick,\nand crept along by the wall to fetch a broom. When she had the broom,\nshe could do nothing with it. I told her to crawl into bed and lie\ndown; she crawled along and fell with her face on the bed, while her\nfeet were on the ground. There she was sick again, and remained so\nlying, and slept. It is easy to imagine how I felt.\n\n [128] In the margin is noted: 'Chresten was not well satisfied with\n the woman, for in her time he never received a draught of wine, so\n that he once stole the wine from her can and substituted something\n impure in its place; at this she made a great noise, begged me for\n God's sake to give her leave to strike Chresten with the can. She\n did not gain permission to do so; she told Chresten afterwards that\n she had not dared to do it, for my sake. She had a great scar on\n one cheek, which a soldier had once given her for a similar act.'\n\nShe slept in this way for a couple of hours, but still did not quite\nsleep off her intoxication; for when she wanted afterwards to clean\nherself and the room, she remained for a long time sitting on a low\nstool, the broom between her knees and her hair about her ears. She\ntook off her bodice to wash it, and so she sat with her bosom\nuncovered, an ugly sight; she kept bemoaning herself, praying to God\nto help her, as she was nigh unto death. I was angry, but I could\nscarcely help laughing at this sad picture. When the moaning and\nlamenting were over, I said angrily, 'Yes, may God help you, you\ndrunkard; to the guards' station you ought to go; I will not have\nsuch a drunkard about me; go and sleep it out, and don't let me hear\nyou talk of God when you are not sober, for then God is far from you\nand the d----l is near!' (I laughed afterwards at myself.) She laid\ndown again, and about four o'clock she was quite sober, made herself\nperfectly clean, and sat quietly weeping. Then she threw herself with\ngreat excitement at my feet, clung to them, howled and clamoured, and\nbegged for God's sake that I would forgive her this once, and that it\nshould never happen again; said how she had kept the wine &c.; that\nif I would only keep her half a year, she would have enough to\npurchase her admission into the hospital at Luebeck.\n\nI thought I would take good care that she did not get so much again\nat once, and also that perhaps if I had another in her place she\nmight be worse in other things. Karen could not have come at this\ntime, for her daughter was expecting her confinement, and I knew that\nshe would then not be quiet. So I promised her to keep her for the\ntime she mentioned. She kept her word moreover, and I so arranged it\nsix weeks later that she received no more wine, and from this time\nthe woman received no wine; my wine alone could not hurt her. She was\nquite intimate with Walter. She had known him formerly, and Chresten\nwas of opinion that he had given her all his money before he was ill;\nfor he said that Walter had no money any longer. What there was in it\nI know not. Honest she was not, for she stole from me first a brass\nknitting-pin, which I used at that time; it was formed like a bodkin,\nand the woman never imagined but that it was gold. As my room is not\nlarge, it could soon be searched, but I looked for three days and\ncould not find the pin. I was well aware that she had it, for it is\nnot so small as not to be seen, so I said afterwards, 'This brass pin\nis of no great importance; I can get another for two pence.' The next\nday she showed me the pin, in a large crevice on the floor between\nthe stones. But when she afterwards, shortly before she left, found\none of my gold earrings which I had lost, and which undoubtedly had\nbeen left on the pillow, for it was a snake ring, this was never\nreturned, say what I would about it. She made a show of looking for\nit in the dirt outside; she knew I dared not say that I had missed\nit.\n\nThe prison governor at this time came up but rarely; Peder Jensen\nwaited on me.[129] His Majesty was ill for a short time, and died\nsuddenly on February 9, 1670. And as on the same day at twelve\no'clock the palace bell tolled, I was well aware what this indicated,\nthough the woman was not. We conversed on the subject, who it might\nbe. She could perceive that I was sad, and she said: 'That might be\nfor the King, for the last time I saw him on the stairs, getting out\nof the carriage, he could only move with difficulty, and I said to\nmyself that it would soon be over with him. If he is dead, you will\nhave your liberty, that is certain.' I was silent, and thought\notherwise, which was the case. About half-past four o'clock the fire\nwas generally lighted in the outside stove, and this was done by a\nlad whom Chresten at that time employed. I called him to the door and\nasked him why the bell had tolled for a whole hour at noon. He\nanswered, 'I may not say; I am forbidden.' I said that I would not\nbetray him. He then told me that the King had died in the morning. I\ngave free vent to my tears, which I had restrained, at which the\nwoman was astonished, and talked for a long time.\n\n [129] In the margin is added: 'At this time I had six prisoners for\n my neighbours. Three were peasants from Femeren, who were accused\n of having exported some sheep; the other three were Danish. They\n were divided in two parties, and as the Danes were next the door, I\n gave them some food; they had moreover been imprisoned some time\n before the others. When the Danes, according to their custom, sang\n the morning and evening psalms, the Germans growled forth with all\n their might another song in order to drown their voices; they\n generally sang the song of Dorothea.' [E48]\n\n [E48] The song of St. Dorothea exists in many German and Danish\n versions.\n\nI received all that she said in silence, for I never trusted her. I\nbegged her to ask Chresten, when he unlocked the door, what the\ntolling intimated. She did so, but Chresten answered that he did not\nknow. The prison governor came up the same evening, but he did not\nspeak with me. He came up also the next day at noon. I requested to\nspeak with him, and enquired why the bell had sounded. He answered\nironically, 'What is that to you? Does it not ring every day?' I\nreplied somewhat angrily: 'What it is to me God knows! This I know,\nthat the castle bell is not tolled for your equals!' He took off his\nhat and made me a bow, and said, 'Your ladyship desires nothing\nelse?' I answered, 'St. Martin comes for you too.'[E49] 'St. Martin?'\nhe said, and laughed, and went away and went out to Walter, standing\nfor a long time whispering with him in front of the hole; I could see\nhim, as he well knew.[130] He was undoubtedly telling him of the\nKing's death, and giving him hope that he would be liberated from\nprison. God designed it otherwise. Walter was ill, and lay for a long\ntime in great misery. He behaved very badly to Chresten; took the\ndirt from the floor and threw it into the food; spat into the beer,\nand allowed Chresten to see him do so when he carried the can away.\nEvery day Chresten received the titles of thief and rogue, so that it\nmay easily be imagined how Chresten tormented him. When I sent him\nsome meat, either stewed or roasted, Chresten came back with it and\nsaid he would not have it. I begged Chresten to leave it with him,\nand he would probably eat it later. This he did once, and then\nChresten showed me how full it was of dirt and filth.[131]\n\n [E49] The feast of St. Martin is supposed the proper time for\n killing pigs in Denmark. It is reported that when Corfitz Uldfeldt,\n in 1652, had published a defence of his conduct previously to his\n leaving Denmark the year before, he sent a copy to Peder Vibe, one\n of his principal adversaries, with this inscription:--\n\n     Chaque pourceau a son St. Martin;\n     Tu n'echapperas pas, mais auras le tien.\n\n [130] In the margin is added: 'As I was to receive clothes, I asked\n for mourning clothes. Then the prison governor asked me for whom I\n wished to mourn, and this in a most ironical manner. I answered:\n \"It is not for your aunt; it is not for me to mourn for her,\n although your aunt has been dead long. I think you have as good\n reason for wearing mourning as I.\" He said he would report it. I\n did not receive them at once.'\n\n [131] In the margin is added: 'Chresten showed me once some bread,\n from which Walter had taken the crumb, and had filled it full of\n straw and dirt, in fact, of the very worst kind.'\n\nWhen Chresten had to turn Walter in bed, the latter screamed so\npitifully that I felt sympathy with him, and begged Chresten not to\nbe so unmerciful to him. He laughed and said, 'He is a rogue.' I\nsaid, 'Then he is in his master's hands.' This pleased Chresten well.\nWalter suffered much pain; at length God released him. His body was\nleft in the prison until his brother came, who ordered it to be\nburied in the German Church. When I heard that Karen could come to me\nagain, and the time was over which I had promised the other to keep\nher, Cathrina went down and Karen returned to me. This was easily\neffected, for the prison governor was not well pleased with Cathrina;\nshe gave him none of her money, as she had promised, but only empty\nwords in its place, such as that he was not in earnest, and that he\nsurely did not wish to have anything from her, &c.[132] The prison\ngovernor began immediately to pay me less respect, when he perceived\nthat my liberation was not expected.\n\n [132] In the margin is added; 'The prison governor also severely\n reprimanded the woman because she had told me that the King was\n dead; that it would not go as well with me as I thought. She gave\n him word for word.'\n\nWhen the time came at which I was accustomed to receive the holy\ncommunion, I begged the prison governor that he should manage that I\nshould have the court preacher, D. Hans Laet, as the former court\npreacher, D. Mathias Foss, had come to me on the first occasion in my\nprison. The prison governor stated my desire, and his Majesty\nassented. D. Hans Laet was already in the tower, down below, but he\nwas called back because the Queen Dowager (who was still in the\npalace) would not allow it; and the prison governor sent me word,\nthrough Peder Jensen, that the King had said I was to be content with\nthe clergyman to whom I was accustomed, so that the necessary\npreparation for the Lord's Supper was postponed till the following\nday, when Mag. Buck came to me and greeted me in an unusual manner,\ncongratulating me in a long oration on my intention, saluting me\n'your Grace.' When he was seated, he said, 'I should have been glad\nif D. Hans Laet had come in my place.' I replied, 'I had wished it\nalso.' 'Yes,' he said, 'I know well why you wished it so. You wish to\nknow things, and that is forbidden me. You have already caused one\nman to lose his employ.' I asked him whether I had ever desired to\nknow anything from him? 'No,' he replied, 'you know well that you\nwould learn nothing from me; for that reason you have asked me\nnothing.' 'Does the Herr Mag, then,' I said, 'mean that I desired D.\nHans Laet in order to hear news of him?' He hesitated a little, and\nthen said, 'You wanted to have D. Hans Laet in order that he might\nspeak for you with the King.' I said, 'There may perhaps be something\nin that.' Upon this he began to swear all kinds of oaths (such as I\nhave never heard before),[133] that he had spoken for me. (I thought:\n'I have no doubt you have spoken of me, but not in my favour.') He\nhad given me a book which I still have; it is 'St. Augustini\nManuali;' the Statholder Gabel had bought it, as he said more than\nonce, protesting by God that it had cost the Herr Statholder a\nrix-dollar. (I thought of the 5,000 rix-dollars which Gabel received,\nthat we might be liberated from our confinement at Borringholm, but I\nsaid nothing; perhaps for this reason he repeated the statement so\noften.) I asked him whom I had caused to lose his employ. He\nanswered, 'Hans Balcke.[134] He told you that Treasurer Gabel was\nStatholder, and he ought not to have done so.' I said, 'I do not\nbelieve that Balcke knew that he ought not to say it, for he did not\ntell it to me as a secret. One might say just as well that H.\nMagister had caused Balcke to lose his place.' He was very angry at\nthis, and various disputes arose on the subject. He began again just\nas before, that I wanted to have D. Laet, he knew why. I said, 'I did\nnot insist specially on having D. Laet; but if not him, the chaplain\nof the castle, or another.' He asked, 'Why another?' I replied,\n'Because it is not always convenient to the Herr Magister. I have\nbeen obliged to wait for him ten, twelve, and even fourteen days, and\nthe last time he administered his office in great haste, so that it\nis not convenient for him to come when I require him.' He sat turning\nover my words, not knowing what to answer, and at last he said; 'You\nthink it will go better with you now because King Frederick is dead.\nNo, you deceive yourself! It will go worse with you, it will go worse\nwith you!' And as he was growing angry, I became more composed and I\nasked gently why so, and from what could he infer it? He answered, 'I\ninfer it from the fact that you have not been able to get your will\nin desiring another clergyman and confessor; so I assure you things\nwill not be better with you. If King Frederick is dead, King\nChristian is alive.' I said: 'That is a bad foundation; your words of\nthreatening have no basis. If I have not this time been able to\nobtain another confessor, it does not follow that I shall not have\nanother at another time. And what have I done, that things should go\nworse with me?' He was more and more angry, and exclaimed aloud\nseveral times, 'Worse, yes, it will be worse!' Then I also answered\nangrily, 'Well, then let it come.'\n\n [133] In the margin is added: 'Among his terrible curses was one\n that his tongue might be paralysed if he had not spoken for me. The\n following year God struck him with paralysis of the tongue; he had\n a stroke from anger, and lived eight days afterwards; he was in his\n senses, but he was not able to speak, and he died; but he lived to\n see the day when another clergyman administered the holy communion\n to me.'\n\n [134] In the margin is added: 'I saw now that this was the cause of\n Balcke's dismissal.'\n\nUpon this he was quite silent, and I said: 'You have given me a good\npreparation; now, in God's name!' Then I made my confession, and he\nadministered his office and went away without any other farewell than\ngiving me his hand. I learned afterwards that before M. Buck came to\nme he went to the prison governor, who was in bed, and begged him to\ntell Knud, who was at that time page of the chamber,[E50] what a\nsacramental woman I was; how I had dug a hole in the floor in order\nto speak with the doctor (which was an impossibility), and how I had\npractised climbing up and looking out on the square. He begged him\nseveral times to tell this to the page of the chamber: 'That is a\nsacramental woman!'[135]\n\n [E50] This Knud was the favourite of King Christian V., Adam Levin\n Knuth, one of the many Germans who then exercised a most\n unfavourable influence on the affairs of Denmark.\n\n [135] In the margin is added: 'Chresten, who was ill satisfied both\n with Karen and with me, gave us a different title one day, when he\n was saying something to one of the house-servants, upon which the\n latter asked him who had said it? Chresten answered, 'She who is\n kept up there for her.' When I was told of this, I laughed and\n said, 'That is quite right, we are two \"shes.\"'\n\nIn the end of April in the same year my door was opened one\nafternoon, and the prison governor came in with some ladies, who kept\nsomewhat aside until he had said, 'Here are some of the maids of\nhonour, who are permitted to speak to you.' There came in first a\nyoung lady whom I did not know. Next appeared the Lady Augusta of\nGluecksburg, whom I recognised at once, as she was but little altered.\nNext followed the Electoral Princess of Saxony, whom I at once\nrecognised from her likeness to her royal father, and last of all our\ngracious Queen, whom I chiefly looked at, and found the lineaments of\nher countenance just as Peder Jensen had described them. I saw also a\nlarge diamond on her bracelet, and one on her finger, where her glove\nwas cut. Her Majesty supported herself against the folding table as\nsoon as she had greeted me. Lady Augusta ran up and down into every\ncorner, and the Electoral Princess remained at the door. Lady Augusta\nsaid: 'Fye, what a disgusting room this is! I could not live a day in\nit. I wonder that you have been able to endure it so long.' I\nanswered, 'The room is such as pleases God and his Majesty, and so\nlong as God will I shall be able to endure it.' She began a\nconversation with the prison governor, who was half tipsy, and spoke\nwith him about Balcke's marriage, whose wedding with his third wife\nwas taking place on that very day; she spoke against marrying so\noften, and the prison governor replied with various silly speeches.\nShe asked me if I was plagued with fleas. I replied that I could\nfurnish her with a regiment of fleas, if she would have them. She\nreplied hastily with an oath, and swore that she did not want them.\n\nHer question made me somewhat ironical, and I was annoyed at the\ndelight she exhibited at my miserable condition; so when she asked me\nwhether I had body or wall lice, I answered her with a question, and\nenquired whether my brother-in-law Hanibal Sehested was still alive?\nThis question made her somewhat draw in, for she perceived that I\nknew her. She made no answer. The Electoral Princess, who probably\nhad heard of my brother-in-law's intrigues with Lady Augusta,[E51]\nwent quickly up to the table (the book lay on it, in which Karen used\nto read, and which she had brought in with her), took the book,\nopened it and asked whether it was mine. I replied that it belonged\nto the woman whom I had taught to read, and as I gave the Electoral\nPrincess her fitting title of Serene Highness, Lady Augusta said:\n'You err! You are mistaken; she is not the person whom you think.' I\nanswered, 'I am not mistaken.' After this she said no more, but gave\nme her hand without a word. The gracious Queen looked sadly on, but\nsaid nothing. When her Majesty gave me her hand, I kissed it and held\nit fast, and begged her Majesty to intercede for me, at any rate for\nsome alleviation of my captivity. Her Majesty replied not with words,\nbut with a flood of tears. The virtuous Electoral Princess cried\nalso; she wept very sorrowfully. And when they had reached the\nanteroom and my door was closed, both the Queen and the Electoral\nPrincess said, 'It is a sin to treat her thus!' They shuddered; and\neach said, 'Would to God that it rested with me! she should not stay\nthere.' Lady Augusta urged them to go away, and mentioned it\nafterwards to the Queen Dowager, who said that I had myself to thank\nfor it; I had deserved to be worse treated than this.\n\n [E51] Hannibal Sehested was dead already in 1666, as Leonora was no\n doubt well aware. The whole passage seems to indicate that he is\n supposed to have had some love-intrigue with the duchess. Nothing\n has transpired on this subject from other sources, but it is\n certain that her husband, Duke Ernst Gynther, for some time at\n least, was very unfriendly disposed to Hannibal Sehested.\n\nWhen the King's funeral was over, and the Queen Dowager had left the\ncastle, I requested the prison governor that he should execute my\nmessage and solicit another clergyman for me, either the chaplain of\nthe castle or the arsenal chaplain, or the one who usually attended\nto the prisoners; for if I could get no other than M. Buck, they must\ntake the sin on their own heads, for that I would not again confess\nto him. A short time elapsed, but at length the chaplain of the\ncastle, at that time M. Rodolff Moth, was assigned me. God, who has\never stood by me in all my adversity, and who in my sorrow and\ndistress has sent me unexpected consolation, gave me peculiar comfort\nin this man. He consoled me with the Word of God; he was a learned\nand conversable man, and he interceded for me with his Majesty. The\nfirst favour which he obtained for me was, that I was granted another\napartment on July 16, 1671, and Bishop D. Jesper's postil.\n\nHe afterwards by degrees obtained still greater favours for me. I\nreceived 200 rix-dollars as a gift, to purchase such clothes for\nmyself as I desired, and anything I might wish for to beguile the\ntime.[136]\n\n [136] In the margin is noted: 'Some of my money I expended on\n books, and it is remarkable that I obtained from M. Buck's books\n (which were sold by auction) among others the great Martilegium, in\n folio, which he would not lend me. I excerpted and translated\n various matters from Spanish, Italian, French, and German authors.\n I especially wrote out and translated into Danish the female\n personages of different rank and origin, who were mentioned with\n praise by the authors as valiant, true, chaste and sensible,\n patient, steadfast and scholarly.' [E52]\n\n [E52] The Martilegium was probably a German history of Martyrs,\n entitled 'Martilogium (for martyrologium) der Heiligen' (Strasburg\n 1484, fol.). The extracts to which she refers were no doubt her\n earliest collections for her work on Heroines.\n\nIn this year her Majesty the Queen became pregnant, and her Majesty's\nmother, the Landgravine of Hesse, came to be with her in her\nconfinement. On September 6 her Serene Highness visited me in my\nprison, at first wishing to remain incognito. She had with her a\nPrincess of Curland, who was betrothed to the son of the Landgravine;\nher lady in waiting, a Wallenstein by birth; and the wife of her\nmaster of the household. The Landgravine greeted me with a kiss, and\nthe others followed her example. I did not at that time recognise the\nwife of the master of the household, but she had known me formerly in\nmy prosperity at the Hague, when she had been in the service of the\nCountess Leuenstein, and the tears stood in her eyes.\n\nThe Landgravine lamented my hard fate and my unhappy circumstances. I\nthanked her Serene Highness for the gracious sympathy she felt with\nme, and said that she might help much in alleviating my fetters, if\nnot in liberating me from them entirely. The Landgravine smiled and\nsaid, 'I see well you take me for another than I am.' I said, 'Your\nSerene Highness's deportment and appearance will not allow you to\nconceal your rank, were you even in peasant's attire.' This pleased\nher; she laughed and jested, and said she had not thought of that.\nThe lady in waiting agreed with me, and said that I had spoken very\njustly in saying that I had recognised her by her royal appearance.\nUpon this the Landgravine said, 'You do not know her?' pointing to\nthe Princess of Curland. She then said who she was, and afterwards\nwho her lady in waiting was, and also the wife of the master of the\nhousehold, who was as I have before mentioned. She spoke of the pity\nwhich this lady felt for me, and added 'Et moy pas moins.' I thanked\nher 'Altesse tres-humblement et la prioit en cette occasion de faire\nvoir sa genereuse conduite.' Her Serene Highness looked at the prison\ngovernor as though she would say that we might speak French too long;\nshe took off her glove and gave me her hand, pressing mine and\nsaying, 'Croyez-moy, je fairez mon possible.' I kissed her Serene\nHighness's hand, and she then took leave of me with a kiss.\n\nThe virtuous Landgravine kept her word, but could effect nothing.\nWhen her Majesty the Queen was in the perils of childbirth, she went\nto the King and obtained from him a solemn promise that if the Queen\ngave birth to a son I should receive my liberty. On October 11, in\nthe night between one and two o'clock, God delivered her Majesty in\nsafety of our Crown Prince. When all present were duly rejoicing at\nthe Prince's birth, the Landgravine said, 'Oh! will not the captive\nrejoice!' The Queen Dowager enquired 'Why?' The Landgravine related\nthe King's promise. The Queen Dowager was so angry that she was ill.\nShe loosened her jacket, and said she would return home; that she\nwould not wait till the child was baptised. Her coach appeared in the\npalace square. The King at length persuaded her to remain till the\nbaptism was over, but he was obliged to promise with an oath that I\nshould not be liberated. This vexed the virtuous Landgravine not a\nlittle, that the Queen should have induced her son to break his\npromise; and she persisted in saying that a king ought to keep his\nvow. The Queen Dowager answered, 'My son has before made a vow, and\nthis he has broken by his promise to your Serene Highness.' The\nLandgravine said at last: 'If I cannot bring about the freedom of the\nprisoner, at least let her, at my request, be removed to a better\nplace, with somewhat more liberty. It is not to the King's\nreputation that she is imprisoned there. She is, after all, a king's\ndaughter, and I know that much injustice is done to her.' The Queen\nDowager was annoyed at these words, and said, 'Now, she shall not\ncome out; she shall remain where she is!' The Landgravine answered,\n'If God will, she will assuredly come out, even though your Majesty\nmay will it not;' so saying, she rose and went out.\n\nOn October 18 the lady in waiting, Wallenstein, sent for Peder Jensen\nTotzloff, and delivered to him by command a book entitled, D.\nHeinrich Mueller's 'Geistliche Erquickstunden,'[E53] which he gave me\nwith a gracious message from the Landgravine. On the same day I sent\nher Serene Highness, through Totzloff, my dutiful thanks, and\nTotzloff took the book back to the lady in waiting, with the request\nthat she would endeavour to prevail on her Highness to show me the\ngreat favour of placing her name and motto in the book, in\nremembrance of her Highness's generosity and kindness. I lamented my\ncondition in this also, that from such a place I could not spread\nabroad her Serene Highness's praise and estimable benefits, and make\nthe world acquainted with them; but that I would do what I could, and\nI would include her Serene Highness and all her family in my prayers\nfor their welfare both of soul and body. (This I have done, and will\ndo, so long as God spares my life.)\n\n [E53] 'Hours of Spiritual Refreshment.' This very popular book of\n devotion was first published in 1664, and had an extraordinary run\n both in Germany and, through translations, in Denmark. The last\n Danish extract of it was published in 1846, and reached the third\n edition in 1856.\n\nOn October 23 I received the book back through Totzloff, and I found\nwithin it the following lines, written by the Landgravine's own\nhand:\n\n          1671.\n\n      Ce qui n'est pas en ta puissance\n      Ne doit point troubler ton repos;\n      Tu balances mal a propos\n      Entre la crainte et l'esperance.\n      Laisse faire ton Dieu et ton roy,\n      Et suporte avec passience ce qu'il resoud pour toy.\n\n  Je prie Dieu de vous faire cette grace, et que je vous puisse\n  tesmoigner combien je suis,\n\n          Madame, vostre tres-affectionee a vous servir,\n                                            {Monogram}\n\nThe book is still in my possession, and I sent word through Totzloff\nto the lady in waiting to request her to convey my most humble thanks\nto her Highness; and afterwards, when the Landgravine was about to\nstart on her journey, to commend me to her Serene Highness's favour.\n\nIn the same year, 1671, Karen, Nils' daughter, left me on account of\nill health. For one night a woman was with me named Margrete, who was\na serf from Holstein. She had run away from her master. She was a\nvery awkward peasant woman, so towards evening on the following day\nshe was sent away, and in her place there came a woman named Inger, a\nperson of loose character. This woman gave herself out as the widow\nof a non-commissioned officer, and that she had long been in service\nat Hamburg, and nursed lying-in women. It happened with her, as is\noften the case, that one seeks to obtain a thing, and that to one's\nown vexation. Chresten had spoken for this woman with the prison\ngovernor, and had praised her before me, but the prison governor took\nupon another recommendation the before-mentioned Margrete. So long as\nthere was hope that the Landgravine might obtain my freedom, this\nwoman was very amenable, but afterwards she began by degrees to show\nwhat was in her, and that it was not for nothing that she resembled\nDina.\n\nShe caused me annoyance of various kinds, which I received with\npatience, thinking within myself that it was another trial imposed by\nGod upon me, and Dina's intrigues often came into my mind, and I\nthought, 'Suppose she should devise some Dina plot?' (She is capable\nof it, if she had only an instigator, as Dina had.) Among other\nannoyances, which may not be reckoned among the least, was this: I\nwas one day not very well, having slept but little or not at all\nduring the night, and I had lain down to sleep on the bed in the day;\nand she would give me no rest, but came softly past me in her socks,\nand in order to wake me teased a dog which I had,[137] so that he\ngrowled. I asked her why she grudged my sleeping? She answered, 'I\ndid not know that you were asleep.' 'Why, then,' I said, 'did you go\nby in your stockings?' She replied, 'If you saw that, then you were\nnot asleep,' and she laughed heartily by herself. (She sat always in\nfront of my table with her back turned to me; whether it was because\nshe had lost one eye that she sat in that position to the light, I\nknow not.)\n\n [137] In the margin is added: 'This dog was of an Icelandic breed,\n not pretty, but very faithful and sagacious. He slept every\n afternoon on the stool, and when she had fallen asleep, she let her\n hands hang down. Then the dog would get up and run softly and bite\n her finger till the blood came. If she threw down her slippers, he\n would take one and sit upon it. She never got it back again without\n a bloody finger.'\n\nI did not care for any conversation with her, so I lay still; and\nwhen she thought I was asleep, she got up again and teased the dog. I\nsaid, 'You tax my patience sorely; but if once my passion rises, you\nwill certainly get something which will astonish you, you base\naccursed thing!' 'Base accursed thing,' she repeated to herself with\na slight laugh. I prayed to God that he would restrain me, so that I\nmight not lay violent hands on this base creature. And as I had the\nother apartment (as I have before mentioned),[138] I went out and\nwalked up and down between four and five o'clock. She washed and\nsplashed outside, and spilled the water exactly where I was walking.\nI told her several times to leave her splashing, as she spilled the\nwater in all directions on the floor, so that I made my clothes\ndirty, and often there was not a drop of water for my dog to drink,\nand the tower-warder had to fetch her water from the kitchen spring.\nThis was of no avail. One day it occurred to her, just as the bell\nhad sounded four, to go out and pour all the water on the floor, and\nthen come back again. When I went to the door, I perceived what she\nhad done. Without saying a word, I struck her first on one cheek and\nthen on the other, so that the blood ran from her nose and mouth, and\nshe fell against her bench, and knocked the skin from her shin-bone.\nShe began to be abusive, and said she had never in her life had such\na box on her ears. I said immediately, 'Hold your tongue, or you will\nhave another like it! I am now only a little angry, but if you make\nme really angry I shall strike you harder.' She was silent for the\ntime, but she caused me all the small annoyance she could.\n\n [138] In the margin is this note: 'In the year 1672, on the 4th\n May, one of the house-servants was arrested for stealing. Adam\n Knudt, at that time gentleman of the chamber, himself saw him take\n several ducats early one morning from the King's trousers, which\n were hanging against the walls. He was at first for some hours my\n neighbour in the Dark Church. He was then placed in the Witch Cell,\n and as he was to be tortured, he received secret warning of it\n (which was forbidden), so that when the executioner came he was\n found to have hung himself. That is to say, he was said to have\n hung himself, though to all appearance this was not possible; he\n was found with a cloth round his neck, which was a swaddling-cloth\n belonging to one of Chresten, the tower-warder's, children.\n Chresten became my neighbour, and was ostensibly brought to\n justice, but he was acquitted and reinstated in his office.\n\nI received it all with gentleness, fearing that I might lay violent\nhands on her. She scarcely knew what to devise to cause me vexation;\nshe had a silver thimble on which a strange name was engraved; she\nhad found it, she said, in a dust-heap in the street. I once asked\nher where she had found some handkerchiefs which she had of fine\nDutch linen, with lace on them, which likewise were marked with\nanother name; they were embroidered with blue silk, and there was a\ndifferent name on each. She had bought them, she said, at an auction\nat Hamburg.[139] I thought that the damage she had received on one of\nher eyes might very likely have arisen from her having 'found'\nsomething of that kind,[E54] and as I soon after asked her by what\naccident she had injured her eye, she undoubtedly understood my\nquestion well, for she was angry and rather quiet, and said, 'What\ninjury? There is nothing the matter with my eye; I can, thank God,\nsee with both.' I let the matter rest there. Soon after this\nconversation she came down one day from upstairs, feeling in her\npocket, though she said nothing until the afternoon, when the doors\nwere locked, and then she looked through all her rubbish, saying 'If\nI only knew where it could be?' I asked what she was looking for. 'My\nthimble,' she said. 'You will find it,' I said; 'only look\nthoroughly!' And as she had begun to look for it in her pockets\nbefore she had required it, I thought she might have drawn it out of\nher pocket with some paper which she used, and which she had bought.\nI said this, but it could not be so.\n\n [139] In the margin is added: 'She was so proud of her knowledge of\n German that when she sang a morning hymn (which, however rarely\n happened) she interspersed it with German words. I once asked her\n if she knew what her mother's cat was called in Danish, and I said\n something at which she was angry.\n\n [E54] It was a common superstition that persons who understood the\n art of showing by magic the whereabouts of stolen goods, had the\n power, by use of their formulas alone, to deprive the thief of an\n eye.\n\nOn the following day, towards noon, she again behaved as if she were\nlooking for it upstairs; and when the door was closed she began to\ngive loose to her tongue, and to make a long story about the thimble,\nwhere it could possibly be. 'There was no one here, and no one came\nin except us two;' and she gave me to understand that I had taken it;\nshe took her large box which she had, and rummaged out everything\nthat was in it, and said, 'Now you can see that I have not got it.' I\nsaid that I did not care about it, whether she had it or no, but that\nI saw that she accused me of stealing. She adhered to it, and said,\n'Who else could have taken it? There is no one else here, and I have\nlet you see all that is mine, and it is not there.' Then for the\nfirst time I saw that she wished that I should let her see in the\nsame manner what I had in my cardbox, for she had never seen anything\nof the work which I had done before her time. I said, 'I do not care\nat all what you do with your thimble, and I respect myself too much\nto quarrel with you or to mind your coarse and shameless accusation.\nI have, thank God, enough in my imprisonment to buy what I require,\n&c. But as you perhaps have stolen it, you now imagine that it has\nbeen stolen again from you, if it be true that you have lost it.' To\nthis she made no answer, so that I believe she had it herself, and\nonly wanted by this invention to gain a sight of my things. As it was\nthe Christmas month and very cold, and Chresten was lighting a fire\nin the stove before the evening meal, I said to him in her presence,\n'Chresten, you are fortunate if you are not, like me, accused of\nstealing, for you might have found her thimble upstairs without\nhaving had it proclaimed from the pulpit; it was before found by\nInger, and not announced publicly.'\n\nThis was like a spark to tinder, and she went to work like a frantic\nbeing, using her shameless language. She had not stolen it, but it\nhad been stolen from her; and she cursed and swore. Chresten ordered\nher to be silent. He desired her to remember who I was, and that she\nwas in my service. She answered, 'I will not be silent, not if I were\nstanding before the King's bailiff!' The more gently I spoke, the\nmore angry was she; at length I said, 'Will you agree with me in one\nwish?--that the person who last had the thimble in her possession may\nsee no better with her left eye than she sees with her right.' She\nanswered with an oath that she could see with both eyes. I said,\n'Well, then, pray God with me that she may be blind in both eyes who\nlast had it.' She growled a little to herself and ran into the inner\nroom, and said no more of her thimble, nor did I. God knows that I\nwas heartily weary of this intercourse.\n\nI prayed God for patience, and thought 'This is only a trial of\npatience. God spares me from other sorrow which I might have in its\nstead.' I could not avail myself of the occasion of her accusing me\nof theft to get rid of her, but I saw another opportunity not far\noff. The prison governor came one day to me with some thread which\nwas offered for sale, rather coarse, but fit for making stockings and\nnight-waistcoats. I bought two pounds of it, and he retained a pound,\nsaying, 'I suppose the woman can make me a pair of stockings with\nit?' I answered in the affirmative (for she could do nothing else but\nknit). When he was gone, she said, 'There will be a pair of stockings\nfor me here also, for I shall get no other pay.' I said, 'That is\nsurely enough.' The stockings for the prison governor were finished.\nShe sat one day half asleep, and made a false row round the stocking\nbelow the foot. I wanted her to undo it. 'No,' said she, 'it can\nremain as it is; he won't know but that it is the fashion in\nHamburg.'[140]\n\n [140] In the margin is added: 'There was no similar row on the\n other stocking. The prison governor never mentioned it.'\n\nWhen his stockings were finished, she began a pair for herself of the\nsame thread, and sat and exulted that it was the prison governor's\nthread. This, it seemed to me, furnished me with an opportunity of\ngetting rid of her. And as the prison governor rarely came up, and\nshe sent him down the stockings by Totzloff, I begged Totzloff to\ncontrive that the prison governor should come up to me, and that he\nshould seat himself on the woman's bed and arrange her pillow as if\nhe wanted to lean against it (underneath it lay her wool). This was\ndone. The prison governor came up, took the knitting in his hand, and\nsaid to Inger, 'Is this another pair of stockings for me?' 'No, Mr.\nPrison governor,' she answered, 'they are for me. You have got yours.\nI have already sent you them.' 'But,' said he, 'this is of my thread;\nit looks like my thread.' She protested that it was not his thread.\nAs he went down to fetch his stockings and the scales, she said to\nme, 'That is not his thread; it is mine now,' and laughed heartily. I\nthought, 'Something more may come of this.'\n\nThe prison governor came with the scales and his stockings, compared\none thread with the other, and the stockings weighed scarcely half a\npound. He asked her whether she had acted rightly? She continued to\nassert that it was her thread; that she had bought it in Hamburg, and\nhad brought it here. The prison governor grew angry, and said that\nshe lied, and called her a bitch. She swore on the other hand that\nit was not his thread; that she would swear it by the Sacrament. The\nprison governor went away; such an oath horrified him. I was\nperfectly silent during this quarrel. When the prison governor had\ngone, I said to the woman, 'God forbid! how could you say such words?\nDo you venture to swear a falsehood by the Sacrament, and to say it\nin my presence, when I know that it is the prison governor's thread?\nWhat a godless creature you are!' She answered, with a half\nridiculous expression of face, 'I said I would take the Sacrament\nupon it, but I am not going to do so.' 'Oh Dina!' I thought, 'you are\nnot like her for nothing; God guard me from you!' And I said, 'Do you\nthink that such light words are not a sin, and that God will not\npunish you for them?' She assumed an air of authority, and said, 'Is\nthe thread of any consequence? I can pay for it; I have not stolen it\nfrom him; he gave it to me himself. I have only done what the tailors\ndo; they do not steal; it is given to them. He did not weigh out the\nthread for me.' I answered her no more than 'You have taken it from\nhim; I shall trouble myself no more about it;' but I begged Totzloff\nto do all he could that I should be rid of her, and have another in\nher place of a good character.\n\nTotzloff heard that Karen had a desire to return to me; he told me\nso. The prison governor was satisfied with the arrangement. It was\nkept concealed from Inger till all was so settled that Karen could\ncome up one evening at supper-time. When the prison governor had\nunlocked the door, and had established himself in the inner room, and\nthe woman had come out, he said: 'Now, Inger, pack your bundle! You\nare to go.' 'Yes, Mr. Prison governor,' she answered, and laughed,\nand brought the food to me, and told me what the prison governor had\nsaid, saying at the same time, 'That is his joke.' 'I heard well,' I\nanswered, 'what he said; it is not his joke, it is his real\nearnestness.' She did not believe it; at any rate she acted as if she\ndid not, and smiled, saying, 'He cannot be in earnest;' and she went\nout and asked the prison governor whether he was in earnest. He said,\n'Go! go! there is no time for gossip!' She came into me again, and\nasked if I wished to be rid of her. I answered, 'Yes.' 'Why so?' she\nasked. I answered: 'It would take me too long to explain; the other\nwoman who is to remain here is below.' 'At any rate,' said she, 'let\nme stay here over the night.' ('Ah, Dina!' I thought.) 'Not a quarter\nof an hour!' I answered; 'go and pack your things! That is soon\ndone!' She did so, said no word of farewell, and went out of the\ndoor.\n\nThus Karen came to me for the third time, but she did not remain an\nentire year, on account of illness.[141]\n\n [141] In the margin is noted: 'I must remember one thing about\n Karen, Nil's daughter. When anything gave her satisfaction, she\n would take up her book directly and read. I asked her whether she\n understood what she read. \"Yes, of course,\" she answered, \"as truly\n as God will bless you! When a word comes that I don't understand, I\n pass it over.\" I smiled a little in my own mind, but said nothing.'\n\nIn the year 1673 M. Moth became vice-bishop in Fyn. I lost much in\nhim, and in his place came H. Emmeke Norbye, who became court\npreacher, and who had formerly been a comrade of Griffenfeldt; but\nGriffenfeldt did not acknowledge him subsequently, so that he could\nachieve nothing for me with Griffenfeldt.[E55] He one day brought me\nas answer (when I sent him word among other things that his Majesty\nwould be gracious if only some one would speak for me), 'It would be\nas if a pistol had been placed at the King's heart, and he were to\nforgive it.'\n\n [E55] Griffenfeldt, who was then at the height of his power, was\n the son of a wine-merchant, by name Schumacher, but had risen by\n his talents alone to the highest dignities. He was ennobled under\n the name of Griffenfeldt, and was undoubtedly the ablest statesman\n Denmark ever possessed. Eventually he was thrust from his high\n position by an intrigue set on foot by German courtiers and backed\n by foreign influence. He was accused of treason and kept in prison\n from 1676 to 1698, the year before he died, to the great, perhaps\n irreparable damage, of his native country. The principal witness\n against him was a German doctor, Mauritius, a professional spy, who\n had served the Danish Government in this capacity. The year after\n the fall of Griffenfeld, he was himself arrested on a charge of\n perjury, forgery, and high treason, and placed in the Blue Tower;\n he was convicted and conducted to Bornholm, where he died. But\n Griffenfeldt, who had been convicted on his false testimony, was\n not liberated. Griffenfeldt's ability and patriotism cannot be\n doubted, but his personal character was not without blemish; and it\n is a fact that in his prosperity he disclaimed all connection with\n his earlier friends, and even his near relations.\n\nIn the same year my sister Elisabeth Augusta sent me a message\nthrough Totzloff and enquired whether I had a fancy for any fruit, as\nshe would send me some. I was surprised at the message, which came to\nme from my sister in the tenth year of my captivity, and I said,\n'Better late than never!' I sent her no answer.\n\nOne funny thing I will yet mention, which occurred in the time of\nKaren, Nil's daughter. Chresten, who had to make a fire in the stove\nan hour before supper (since it had no flue), so that the smoke could\npass out at the staircase door before I supped, did not come one\nevening before six o'clock, and was then quite tipsy. And as I was\nsitting at the time near the stove in the outer apartment on a log of\nwood, which had been hewed as a seat, I said it was late to make the\nfire, as he must now go into the kitchen. He paid no attention to my\ngentle remark, until I threatened him with hard words, and ordered\nhim to take the wood out. He was angry, and would not use the tongs\nto take the wood out, nor would he permit Karen to take them out with\nthe tongs; but he tore them out with his hands, and said, 'Nothing\ncan burn me.' And as some little time elapsed before the wood was\nextinguished, he began to fear that it would give little satisfaction\nif he so long delayed fetching the meal. He seated himself flat on\nthe ground and was rather dejected; presently he burst out and said,\n'Oh God, you who have had house and lands, where are you now\nsitting?' I said, 'On a log of wood!' He answered, 'I do not mean\nyour ladyship!' I asked, 'Whom does your worship mean, then?' He\nreplied, 'I mean Karen.' I laughed, and said no more.\n\nTo enumerate all the contemptuous conduct I endured would be too\nlengthy, and not worth the trouble. One thing I will yet mention of\nthe tower-warder Chresten, who caused me great annoyance at the end\nof this tenth year of my imprisonment. Among other annoyances he once\nstruck my dog, so that it cried. I did not see it, but I heard it,\nand the woman told me it was he who had struck the dog. I was greatly\ndispleased at it. He laughed at this, and said, 'It is only a dog.' I\ngave him to understand that he struck the dog because he did not\nventure to strike me. He laughed heartily at the idea, and I said, 'I\ndo not care for your anger so long as the prison governor is my\nfriend' (this conversation took place while I was at a meal, and the\nprison governor was sitting with me, and Chresten was standing at the\ndoor of my apartment, stretching out his arms.) I said, 'The prison\ngovernor and you will both get into heavy trouble, if I choose. Do\nyou hear that, good people?' (I knew of too many things, which they\nwished to hide, in more than one respect.) The prison governor sat\nlike one deaf and dumb, and remained seated, but Chresten turned away\nsomewhat ashamed, without saying another word. He had afterwards some\nfear of me, when he was not too intoxicated; for at such times he\ncared not what he said, as regards high or low. He was afterwards\ninsolent to the woman, and said he would strike the dog, and that I\nshould see him do so. This, however, he did not do.\n\nChresten's fool-hardiness increased, so that Peder Totzloff informed\nthe prison governor of his bad behaviour, and of my complaints of the\nwild doings of the prisoners, who made such a noise by night that I\ncould not sleep for it, for Chresten spent the night at his home, and\nallowed the prisoners to do as they chose. Upon this information, the\nprison governor placed a padlock upon the tower door at night, so\nthat Chresten could not get out until the door was unlocked in the\nmorning. This annoyed him, and he demanded his discharge, which he\nreceived on April 24, 1674; and in his place there came a man named\nGert, who had been in the service of the prison governor as a\ncoachman.\n\nIn this year, the ---- May, I wrote a spiritual 'Song in Remembrance\nof God's Goodness,' after the melody 'Nun ruhen alle Waelder.'\n\n        I.\n\n    My heart! True courage find!\n    God's goodness bear in mind,\n    And how He, ever nigh,\n    Helps me my load to bear,\n    Nor utterly despair\n    Tho' in such heavy bonds I lie.\n\n        II.\n\n    Ne'er from my thoughts shall stray\n    How once I lingering lay\n    In the dark dungeon cell;\n    My cares and bitter fears,\n    And ridicule and tears,\n    And God the Lord upheld me well.\n\n        III.\n\n    Think on my misery\n    And sad captivity\n    Thro' many a dreary year!\n    Yet nought my heart distresses;\n    The Lord He proves and blesses,\n    And He protects me even here!\n\n        IV.\n\n    Come heart and soul elate!\n    And let me now relate\n    The wonders of God's skill!\n    He was my preservation\n    In danger and temptation,\n    And kept me from impending ill.\n\n        V.\n\n    The end seemed drawing near,\n    I wrung my hands with fear,\n    Yet has He helped me e'er;\n    My refuge and my guide,\n    On Him I have relied,\n    And He has ever known my care.\n\n        VI.\n\n    Thanks to Thee, fount of good!\n    Thou canst no evil brood,\n    Thy blows are fatherly;\n    When cruel power oppressed me,\n    Thy hand has ever blessed me,\n    And Thou has sheltered me!\n\n        VII.\n\n    Before Thee, Lord, I lie;\n    Give me my liberty\n    Before my course is run;\n    Thy Gracious Hands extend\n    And let my suffering end!\n    Yet not my will, but Thine, be done.\n\nIn this year, on July 25, his royal Majesty was gracious enough to\nhave a large window made again in my inner apartment; it had been\nwalled up when I had been brought into this chamber. A stove was also\nplaced there, the flue of which passed out into the square. The\nprison governor was not well satisfied at this, especially as he was\nobliged to be present during the work; this did not suit his\nlaziness. My doors were open during the time; it was twelve days\nbefore the work was finished. He grumbled, and did not wish that the\nwindow should be made as low as it had been before I was imprisoned\nhere; I persuaded the mason's journeyman to cut down the wall as low\nas it had before been, which the prison governor perceived from the\npalace square, and he came running up and scolded, and was thoroughly\nangry. But it was not to be changed, for the window-frame was already\nmade. I asked him what it mattered to him if the window was a stone\nlower; it did not go lower than the iron grating, and it had formerly\nbeen so. He would have his will, so that the mason walled it up a\nstone higher while the prison governor was there, and removed it\nagain afterwards, for the window-frame, which was ready, would not\notherwise have fitted.\n\nIn the same year Karen, Nil's daughter, left me for the third and\nlast time, and in her stead came a woman named Barbra, the widow of a\nbookbinder. She is a woman of a melancholy turn. Her conscience is\naroused sometimes, so that she often enumerates her own misdeeds (but\nnot so great as they have been, and as I have found out by enquiry).\nShe had two children, and it seems from her own account that she was\nto some extent guilty of their death, for she says: 'Who can have any\ncare for a child when one does not love its father?' She left her\nhusband two years before he died, and repaired to Hamburg, supporting\nherself by spinning; she had before been in the service of a princess\nas a spinning-maid. Her father is alive, and was bookbinder to the\nKing's Majesty; he has just now had a stroke of paralysis, and is\nlying very ill. She has no sympathy with her father, and wishes him\ndead (which would perhaps be the best thing for him); but it vexes me\nthat she behaves so badly to her sister, who is the wife of a tailor,\nand I often tell her that in this she is committing a double sin; for\nthe needy sister comes from time to time for something to eat. If she\ndoes not come exactly on the evening which she has agreed upon, she\ngets nothing, and the food is thrown away upstairs. When at some\nlength I place her sin before her, she says, 'That meat is bad.' I\nask her why she let it get bad, and did not give it in time to her\nsister. To this she answers that her sister is not worthy of it. I\npredict evil things which will happen to her in future, as they have\ndone to others whom I enumerate to her. At this she throws back her\nhead and is silent.\n\nAt this time her Majesty the Queen sent me some silkworms to beguile\nthe time. When they had finished spinning, I sent them back to her\nMajesty in a box which I had covered with carnation-coloured satin,\nupon which I had embroidered a pattern with gold thread. Inside, the\nbox was lined with white taffeta. In the lid I embroidered with black\nsilk a humble request that her Majesty would loose my bonds, and\nwould fetter me anew with the hand of favour. Her Majesty the\nvirtuous Queen would have granted my request had it rested with her.\n\nThe prison governor became gradually more sensible and accommodating,\ndrank less wine, and made no jokes. I had peace within my doors. The\nwoman sat during the day outside in the other apartment, and lay\nthere also in the night, so that I began not to fret so much over my\nhard fate. I passed the year with reading, writing, and composing.\n\nFor some time past, immediately after I had received the yearly\npension, I had bought for myself not only historical works in various\nlanguages, but I had gathered and translated from them all the famous\nfemale personages, who were celebrated as true, chaste, sensible,\nvalorous, virtuous, God-fearing, learned, and steadfast; and in anno\n1675, on January 9, I amused myself with making some rhymes to M.\nThomas Kingo, under the title, 'To the much-famed Poet M. Thomas\nKingo, a Request from a Danish Woman in the name of all Danish\nWomen.' The request was this, that he would exhibit in befitting\nhonour the virtuous and praiseworthy Danish women. There are, indeed,\nvirtuous women belonging to other nations, but I requested only his\npraise of the Danish. This never reached Kingo; but if my good friend\nto whom I entrust these papers still lives, it will fall probably\ninto your hands, my beloved children.\n\nIn the same year, on May 11, I wrote in rhyme a controversial\nconversation between Sense and Reason; entitled, 'Controversial\nThoughts by the Captive Widow, or the Dispute between Sense and\nReason.'\n\nNothing else occurred this year within the doors of my prison which\nis worth recording, except one event--namely, when the outermost door\nof the anteroom was unlocked in the morning for the sake of sweeping\naway the dirt and bringing in fresh water, and the tower-warder\noccasionally let it stand open till meal-time and then closed it\nagain, it happened that a fire broke out in the town and the bells\nwere tolled. I and the woman ran up to the top of the tower to see\nwhere it was burning.\n\nWhen I was on the stairs which led up to the clock-work, the prison\ngovernor came, and with him was a servant from the silver-chamber. He\nfirst perceived my dog, then he saw somewhat of the woman, and\nthought probably that I was there also; he was so wise as not to come\nup the stairs, but remained below at the lowest holes, from whence\none can look out over the town, and left me time enough to get down\nagain and shut my door. Gert was sorry, and came afterwards to the\ndoor and told me of his distress. I consoled him, and said there was\nnothing to fear. Before the prison governor opened the door at noon,\nhe struck Gert with his stick, so that he cried, and the prison\ngovernor said with an oath, 'Thou shalt leave.' When the prison\ngovernor came in, I was the first to speak, and I said: 'It is not\nright in you to beat the poor devil; he could not help it. The\nexecutioner came up as he was going to lock my door, and that made\nhim forget to do so.' He threatened Gert severely, and said, 'I\nshould not have minded it so much had not that other servant been\nwith me.'\n\nThe words at once occurred to me which he had said to me a long time\nbefore, namely that no woman could be silent, but that all men could\nbe silent (when he had asserted this, I had thought, if this be so,\nthen my adversaries might believe that I, had I known of anything\nwhich they had in view, should not have been able to keep silence).\nSo I now answered him thus: 'Well, and what does that signify? It was\na man; they can all keep silence; there is no harm done.' He could\nnot help laughing, and said, 'Well, you are good enough.' I then\ntalked to him, and assured him that I had no desire to leave the\ntower without the King's will, even though day and night all the\ntower doors were left open, and I also said that I could have got out\nlong ago, if that had been my design. Gert continued in his service,\nand the prison governor never told Gert to shut me in in the\nmorning.[142]\n\n [142] In the margin is noted: 'At my desire the prison governor\n gave me a rat whose tail he had cut off; this I placed in a\n parrot's cage, and gave it food, so that it grew very tame. The\n woman grudged me this amusement; and as the cage hung in the outer\n apartment, and had a wire grating underneath, so that the dirt\n might fall out, she burned the rat with a candle from below. It was\n easy to perceive it, but she denied it.'\n\nAt this time I had bought myself a clavicordium, and as Barbra could\nsing well, I played psalms and she sang, so that the time was not\nlong to us. She taught me to bind books, so far as I needed.[E56]\n\n [E56] The MS. itself is bound in a very primitive manner, which\n renders it probable that Leonora has done it herself.\n\nMy father confessor, H. Emmeke, became a preacher at Kioge anno 1676.\nIn the same year my pension was increased, and I received yearly 250\nrix-dollars. It stands in the order that the 200 rix-dollars were to\nbe used for the purchase of clothes and the remaining fifty to buy\nanything which might beguile the time.[E57] God bless and keep his\ngracious Majesty, and grant that he may live to enjoy many happy\nyears.\n\n [E57] It appears from the State accounts that ever since the year\n 1672 a sum of 250 dollars a year had been placed at her disposal.\n It would seem, therefore, that somehow or other a part of them had\n been unlawfully abstracted by someone during the first years.\n\nBrant was at this time treasurer.\n\nOn December 17 in this same year Barbra left me, and married a\nbookbinder's apprentice; but she repented it afterwards. And as her\nhusband died a year and a half after her marriage, and that suddenly,\nsuspicion fell upon Barbra. She afterwards went to her brother's\nhouse and fell ill. Her conscience was awakened, and she sent for\nTotzloff and told almost in plain terms that she had poisoned her\nhusband, and begged him to tell me so. I was not much astonished at\nit, for according to her own account she had before killed her own\nchildren; but I told Peder Totzloff that he was not to speak of it;\nif God willed that it should be made known, it would be so\nnotwithstanding; the brother and the maid in the house knew it; he\nwas not to go there again, even if she sent a message to him. She\nbecame quite insane, and lay in a miserable condition. The brother\nsubsequently had her removed to the plague-house.\n\nIn Barbra's place there came to me a woman named Sitzel, daughter of\na certain Klemming; Maren Blocks had brought about her employment, as\nSitzel owed her money. She is a dissolute woman, and Maren gave her\nout as a spinster; she had a white cap on her head when she came up.\nSitzel's debt to Maren had arisen in this way: that Maren--since\nSitzel could make buttons, and the button-makers had quarrelled with\nher--obtained for her a royal licence in order to free her from the\nopposition of the button-makers, under the pretext that she was\nsickly. When the door was locked in the evening, I requested to see\nthe royal licence which Maren had obtained for her. And when I saw\nthat she was styled in it the sickly woman, I asked her what her\ninfirmity was. She replied that she had no infirmity. 'Why, then,' I\nasked, 'have you given yourself out as sickly?' She answered, 'That\nwas Maren Block's doing, in order to get for me the royal licence.'\n'In the licence,' I said, 'you are spoken of as a married woman, and\nnot as a spinster; have you, then, been seduced?' She hung her head\nand said softly, 'Yes.'\n\nI was not satisfied. I said, 'Maren Block has obtained the royal\nlicence for you by lies, and has brought you to me by lies; what,\nthen, can I expect from your service?' She begged my pardon, promised\nto serve me well, and never to act contrary to my wishes. She is a\ndangerous person; there is nothing good in her; bold and shameless,\nshe is not even afraid of fighting a man. She struck two\nbutton-makers one day, who wanted to take away her work, till they\nwere obliged to run away. With me she had no opportunity of thus\ndisplaying her evil passions, but still they were perceptible in\nvarious ways. One day I warded off a scuffle between her and Maren\nBlocks; for when Maren Blocks had got back the money which she had\nexpended on the royal licence for Sitzel, she wanted to remove her\nfrom me, and to bring another into her place; but I sent word to\nMaren Blocks that she must not imagine she could send me another whom\nI must take. It was enough that she had done this time.[143]\n\n [143] In the margin stood originally the following note, which has\n afterwards been struck out: 'In this year, 1676, the prison\n governor married for the third time; he married a woman who herself\n had had two husbands. Anno 1677, Aug. 9, died my sister Elisabeth\n Augusta.'\n\nIn the place of H. Emmeke Norbye, H. Johan Adolf Borneman became\npalace-preacher; a very learned and sensible man, who now became my\nfather confessor, and performed the duties of his office for the\nfirst time on April 10, 1677.\n\nOn October 9, in the same year, my father confessor was Magister\nHendrich Borneman, dean of the church of Our Lady (a learned and\nexcellent man), his brother H. Johan Adolf Borneman having\naccompanied the King's Majesty on a journey.\n\nI have, thank God, spent this year in repose: reading, writing, and\ncomposing various things.\n\nAnno 1678 it was brought about for me that my father-confessor, H.\nJohan Adolf Borneman, should come to me every six weeks and preach a\nshort sermon.\n\nIn this year, on Easter-Day, Agneta Sophia Budde was brought to the\ntower. Her prison was above my innermost apartment. She was accused\nof having designed to poison the Countess Skeel; and as she was a\nyoung person, and had a waiting-woman in her attendance who was also\nyoung, they clamoured to such an extent all day that I had no peace\nfor them. I said nothing, however, about it, thinking she would\nprobably be quiet when she knew that her life was at stake. But no!\nshe was merry to the day on which she was executed![144]\n\n [144] On a piece of paper which is fastened to the MS. by a pin is\n the following note referring to the same matter: 'On March 4, in\n the same year 1678, a woman named Lucia, who had been in the\n service of Lady Rigitze Grubbe, became my neighbour. She was\n accused by Agneta Sophia Budde, as the person who at the\n instigation of her mistress had persuaded her to poison Countess F.\n Birrete Skeel, and that Lucia had brought her the poison. There was\n evidence as to the person from whom Lucia had bought the poison.\n This woman was a steady faithful servant. She received everything\n that was imposed upon her with the greatest patience, and held out\n courageously in the Dark Cell. She had two men as companions, both\n of whom cried, moaned and wept. From the Countess Skeel (who had to\n supply her with food) meat was sent her which was full of maggots\n and mouldy bread. I took pity on her (not for the sake of her\n mistress, for she had rendered me little good service, and had\n rewarded me evil for the benefits of former times, but out of\n sympathy). And I sent her meat and drink and money that she might\n soften Gert, who was too hard to her. She was tortured, but would\n not confess any thing of what she was accused, and always defended\n her mistress. She remained a long time in prison.[E58]\n\n [E58] The acts of this famous trial are still in existence.\n Originally the quarrel arose out of the fact that the Countess\n Parsberg (born Skeel) had obtained a higher rank than Lady Grubbe,\n and was further envenomed by some dispute about a window in the\n house of the latter which looked down on the courtyard of the\n Countess's house. Regitze Grubbe (widow of Hans Ulrik Gyldenlove,\n natural son of Christian IV. and half-brother of Ulrik Christian\n Gyldenlove, as well as of Leonora Christina), persuaded another\n noble lady, Agnete Budde, through a servant, to poison Countess\n Parsberg. Miss Budde was beheaded, the girl Lucie was exiled, and\n Lady Grubbe relegated for life to the island of Bornholm.\n\nIn the same year, on the morning of July 9, the tower-warder Gert was\nkilled by a thief who was under sentence of death, and to whom he had\nallowed too great liberty. I will mention this incident somewhat more\nin detail, as I had advised Gert not to give this prisoner so much\nliberty; but to his own misfortune he paid no attention to my advice.\nThis thief had broken by night into the house of a clergyman, and had\nstolen a boiling-copper, which he had carried on his head to\nCopenhagen; he was seized with it at the gate in the morning, and was\nplaced here in the tower. He was condemned to be hanged (he had\ncommitted various other thefts). The priest allowed the execution to\nbe delayed; he did not wish to have him hanged. Then it was said he\nwas to go to the Holm; but he remained long in prison. At first, and\nuntil the time that his going to the Holm was talked of, he was my\nneighbour in the Dark Church; he behaved quite as a God-fearing man,\nread (apparently) with devotion, and prayed to God for forgiveness of\nhis sins with most profound sighs. The rogue knew that I could hear\nhim, and I sent him occasionally something to eat. Gert took pity on\nhim, and allowed him to go by day about the basement story of the\ntower, and shut him up at night again.\n\nAfterwards he allowed him also at night to remain below. And as I had\nseen the thief once or twice when my door stood open, and he went\npast, it seemed to me that he had a murderous countenance; and for\nthis reason, when I heard that the thief was not placed of an evening\nin the Dark Church, I said to Gert that he ventured too far, in\nletting him remain below at night; that there was roguery lurking in\nhim; that he would certainly some day escape, and then, on his\naccount, Gert would get into trouble. Gert was not of opinion that\nthe thief wished to run away; he had no longer any fear of being\nhanged; he had been so delighted that he was to go to the Holm, there\nwas no danger in it. I thought 'That is a delight which does not\nreach further than the lips,' and I begged him that he would lock him\nup at night. No; Gert feared nothing; he even went farther, and\nallowed the thief to go up the tower instead of himself, and attend\nto the clock-work.\n\nThree days before the murder took place, I spoke with Gert, when he\nunlocked my door in the morning, of the danger to which he exposed\nhimself by the liberty he allowed the thief, but Gert did not fear\nit. Meanwhile my dog placed himself exactly in front of Gert, and\nhowled in his face. When we were at dinner, the dog ran down and\nhowled three times at the tower-warder's door. Never before had I\nheard the dog howl.\n\nOn July 19 (as I have said), when Gert's unfortunate morning had\narrived, the thief came down from the clock-work, and said that he\ncould not manage it alone, as the cords were entangled. The rogue had\nan iron rod ready above, in order to effect his project. Gert went\nupstairs, but was carried down. The thief ran down after Gert was\ndead, opened his box, took out the money, and went out of the tower.\n\nIt was a Friday, and the bells were to be rung for service. Those\nwhose duty it was to ring them knocked at the tower door, but no one\nopened. Totzloff came with the principal key and opened, and spoke to\nme and wondered that Gert was not there at that time of the day. I\nsaid: 'All is not right; this morning between four and five I was\nrather unwell, and I heard three people going upstairs and after a\ntime two coming down again.' Totzloff locked my door and went down.\nJust then one of the ringers came down, and informed them that Gert\nwas lying upstairs dead. When the dead man was examined, he had more\nthan one wound, but all at the back of the head. He was a very bold\nman, courageous, and strong; one man could not be supposed to have\ndone this to him.\n\nThe thief was seized the same evening, and confessed how it had\nhappened: that, namely, a prisoner who was confined in the Witch\nCell, a licentiate of the name of Moritius, had persuaded him to it.\nThis same Moritius had great enmity against Gert. It is true that\nGert took too much from him weekly for his food. But it is also true\nthat this Moritius was a very godless fellow; the priest who\nconfesses him gives him no good character. I believe, indeed, that\nMoritius was an accessory, but I believe also that another prisoner,\nwho was confined in the basement of the tower, had a hand in the\ngame. For who should have locked the tower-door again after the\nimprisoned thief, had not one of these done so? For when the key was\nlooked for, it was found hidden above in the tower; this could not\nhave been done by the thief after he was out of the tower. The thief,\nmoreover, could not have unlocked Gert's box and taken his money\nwithout the knowledge of Moritius. The other prisoner must also have\nbeen aware of it. It seems to me that it was hushed up, in order that\nno more should die for this murder; for the matter was not only not\ninvestigated as was befitting, but the thief was confined down below\nin the tower. He was bound with iron fetters, but Moritius could\nspeak with him everyday: and for this reason the thief departed from\nhis earlier statement, and said that he alone had committed the\nmurder. He was executed on August 8, and Moritius was taken to\nBorringholm, and kept as a prisoner there.[E55b]\n\n [E55b] Griffenfeldt, who was then at the height of his power, was\n the son of a wine-merchant, by name Schumacher, but had risen by\n his talents alone to the highest dignities. He was ennobled under\n the name of Griffenfeldt, and was undoubtedly the ablest statesman\n Denmark ever possessed. Eventually he was thrust from his high\n position by an intrigue set on foot by German courtiers and backed\n by foreign influence. He was accused of treason and kept in prison\n from 1676 to 1698, the year before he died, to the great, perhaps\n irreparable damage, of his native country. The principal witness\n against him was a German doctor, Mauritius, a professional spy, who\n had served the Danish Government in this capacity. The year after\n the fall of Griffenfeld, he was himself arrested on a charge of\n perjury, forgery, and high treason, and placed in the Blue Tower;\n he was convicted and conducted to Bornholm, where he died. But\n Griffenfeldt, who had been convicted on his false testimony, was\n not liberated. Griffenfeldt's ability and patriotism cannot be\n doubted, but his personal character was not without blemish; and it\n is a fact that in his prosperity he disclaimed all connection with\n his earlier friends, and even his near relations.\n\nIn Gert's place a tower-warder of the name of Johan, a Norwegian, was\nappointed--a very simple man. The servants about court often made a\nfool of him. The imprisoned young woman and her attendant did so the\nfirst time after his arrival that the attendant had to perform some\nmenial offices upstairs. The place to which she had to go was not far\nfrom the door of their prison. The tower-warder went down in the\nmeanwhile, and left the door open. They ran about and played. When\nthey heard him coming up the stairs, they hid themselves. He found\nthe prison empty, and was grieved and lamented. The young woman\ngiggled like a child, and thus he found her behind a door. Johan was\nglad, and told me the story afterwards. I asked why he had not\nremained with them. 'What,' he answered, 'was I to remain at their\ndirty work?' There was nothing to say in reply to such foolish talk.\n\nI had repose within my doors, and amused myself with reading, writing\nand various handiwork, and began to make and embroider my shroud, for\nwhich I had bought calico, white taffeta, and thread.\n\nOn April 7 a young lad escaped from the tower, who had been confined\non the lower story with iron fetters round his legs. This prisoner\nfound opportunity to loosen his fetters, and knew, moreover, that the\nbooby Johan was wont to keep the tower key under his pillow. He kept\nan iron pin in readiness to unlock the door of the room when the\ntower-warder was asleep; he opened it gently, took the key, locked in\nthe booby again, and quitted the tower. The simple man was placed in\nconfinement, but after the expiration of six weeks he was set at\nliberty.\n\nIn his place there came a man named Olle Mathison, who was from\nSkaane; he had his wife with him in the tower. Towards the end of\nthis year, on December 25, I became ill of a fever, and D. Mynchen\nreceived orders to visit me and to take me under his care--an order\nwhich he executed with great attention. He is a very sensible man,\nmild and judicious in his treatment. Ten days after I recovered my\nusual health.\n\nIn the beginning of the year 1680 Sitzel, Klemming's daughter, was\npersuaded by Maren Blocks to betroth herself to one of the King's\nbody-guard. She left me on November 26. In her place I had a woman\nnamed Margrete. When I first saw her, she appeared to me somewhat\nsuspicious, and it seemed to me that she was with child; however, I\nmade no remark till the last day of the month of January. Then I put\na question to her from which she could perceive my opinion. She\nanswered me with lies, but I interrupted her at once; and she made\nuse of a special trick, which it is not fit to mention here, in order\nto prove her false assertion; but her trick could not stand with me,\nand she was subsequently obliged to confess it. I asked her as to the\nfather of the child (I imagined that it was the King's groom of the\nchamber, who had been placed in arrest in the prison governor's room,\nbut I did not say so). She did not answer my question at the time,\nbut said she was not so far advanced; that her size was owing rather\nto stoutness than to the child, as it was at a very early stage.\n\nThis woman, before she came to me, had been in the service of the\nprison governor's wife, and the prison governor had told me she was\nmarried. So it happened that I one day asked her of her life and\ndoings; upon which she told me of her past history, where she had\nserved, and that she had had two bastards, each by a different\nfather; and pointing to herself, she added: 'A father shall also\nacknowledge this one, and that a brave father! You know him well!' I\nsaid, 'I have seen the King's groom of the chamber in the square, but\nI do not know him.' She laughed and answered (in her mother-tongue),\n'No, by God, that is not he; it is the good prison governor.' I truly\ndid not believe it. She protested it, and related some minute details\nto me.\n\nI thought I had better get rid of her betimes, and I requested to\nspeak with the prison governor's wife, who at once came to me. I\ntold her my suspicion with regard to the woman, and on what I based\nmy suspicion; but I made no remark as to what the woman had confessed\nand said to me. I begged the prison governor's wife to remove the\nwoman from me as civilly as she could. She was surprised at my words,\nand doubted if there was truth in them. I said, 'Whether it be so or\nnot, remove her; the sooner the better.' She promised that it should\nbe done, but it was not. Margrete seemed not to care that it was\nknown that she was with child; she told the tower-warder of it, and\nasked him one day, 'Ole, how was it with your wife when she had\ntwins?' Ole answered: 'I know nothing about it. Ask Anne!' Margrete\nsaid that from certain symptoms she fancied she might have twins.\n\nOne day, when she was going to sew a cloth on the arms of my\narm-chair, she said, 'That angel of God is now moving!' And as the\nwife of the prison governor did not adhere to her word, and\nMargrete's sister often came to the tower, I feared that the sister\nmight secretly convey her something to remove the child (which was no\ndoubt subsequently the case), so I said one day to Margrete: 'You say\nthat the prison governor is your child's father, but you do not\nventure to say so to himself.' 'Yes!' she said with an oath, 'as if I\nwould not venture! Do you imagine that I will not have something from\nhim for the support of my child?' 'Then I will send for him,' I said,\n'on purpose to hear what he will say.' (It was at that time a rare\noccurrence for the prison governor to come to me.) She begged me to\ndo so; he could not deny, she said, that he was the father of her\nchild. The prison governor came at my request. I began my speech in\nthe woman's presence, and said that Margrete, according to her own\nstatement, was with child; who the father was, he could enquire if he\nchose. He asked her whether she was with child? She answered, 'Yes,\nand you are the father of it.' 'O!' he said, and laughed, 'what\nnonsense!' She adhered to what she had said, protested that no other\nwas the child's father, and related the circumstances of how it had\noccurred. The prison governor said, 'The woman is mad!' She gave free\nvent to her tongue, so that I ordered her to go out; then I spoke\nwith the prison governor alone, and begged him speedily to look about\nfor another woman for me, before it came to extremities with her. I\nsupposed he would find means to stop her tongue. I told him the truth\nin a few words--that he had brought his paramour to wait on me. He\nanswered, 'She lies, the malicious woman! I have ordered Totzloff\nalready to look about for another. My wife has told me what you said\nto her the other day.' After this conversation the prison governor\nwent away. Peder Totzloff told me that an English woman had desired\nto be with me, but could not come before Easter.\n\nFour days afterwards Margrete began to complain that she felt ill,\nand said to me in the forenoon, 'I think it will probably go badly\nwith me; I feel so ill.' I thought at once of what I had feared,\nnamely of what the constant visits of her sister indicated, and I\nsent immediately to Peder Totzloff, and when he came to me I told him\nof my suspicion respecting Margrete, and begged him to do his utmost\nto procure me the English woman that very day. Meanwhile Margrete\nwent up stairs, and remained there about an hour and a quarter, and\ncame down looking like a corpse, and said, 'Now it will be all right\nwith me.' What I thought I would not say (for I knew that if I had\nenquired the cause of her bad appearance she would have at once\nacknowledged it all, and I did not want to know it), so I said, 'If\nyou keep yourself quiet, all will be well. Another woman is coming\nthis evening.' This did not please her; she thought she could now\nwell remain. I paid no regard to this nor to anything else she said,\nbut adhered to it--that another woman was coming. This was arranged,\nand in the evening of March 15 Margrete left, and in her place came\nan English woman, named Jonatha, who had been married to a Dane named\nJens Pedersen Holme.\n\nWhen Margrete was gone, I was blamed by the wife of the prison\ngovernor, who said that I had persuaded Margrete to affirm that her\nhusband was the father of Margrete's child.\n\nAlthough it did not concern me, I will nevertheless mention the\ndeceitful manner in which the good people subsequently brought about\nthis Margrete's marriage. They informed a bookbinder's apprentice\nthat she had been married, and they showed both him and the priest,\nwho was to give them the nuptial benediction, her sister's marriage\ncertificate.[145]\n\n [145] In the margin is added: 'Ole the tower-warder was cudgelled\n on his back by the prison governor when Margrete was gone, and he\n was charged with having said what Margrete had informed him\n respecting her size.'\n\nIn the same year, on the morning of Christmas Day, God loosened D.\nOtto Sperling's heavy bonds, after he had been imprisoned in the Blue\nTower seventeen years, eight months, twenty-four days, at the age of\neighty years minus six days. He had long been ill, but never confined\nto his bed. Doctor Muenchen twice visited him with his medicaments. He\nwould not allow the tower-warder at any time to make his bed, and was\nquite angry if Ole offered to do so, and implied that the doctor was\nweak. He allowed no one either to be present when he laid down. How\nhe came on the floor on Christmas night is not known; he lay there,\nknocking on the ground. The tower-warder could not hear his knocking,\nfor he slept far from the doctor's room; but a prisoner who slept on\nthe ground floor heard it, and knocked at the tower-warder's door and\ntold him that the doctor had been knocking for some time. When Ole\ncame in, he found the doctor lying on the floor, half dressed, with a\nclean shirt on. He was still alive, groaned a good deal, but did not\nspeak. Ole called a prisoner to help him, and they lifted him on the\nbed and locked the door again. In the morning he was found dead, as I\nhave said.\n\nA.D. 1682, in the month of April, I was sick and confined to my bed\nfrom a peculiar malady which had long troubled me--a stony matter had\ncoagulated and had settled low down in my intestines. Doctor Muenchen\nused all available means to counteract this weakness; but he could\nnot believe that it was of the nature I thought and informed him; for\nI was perfectly aware it was a stone which had settled in the duct of\nthe intestines. He was of opinion, if it were so, that the\nmedicaments which he used would remove it.[146] At this time the\ndoctor was obliged to travel with his Majesty to Holstein. I used the\nremedies according to Doctor Muenchen's directions, but things\nremained just as before. It was not till the following morning that\nthe remedies produced their effect; and then, besides other matter, a\nlarge stone was evacuated, and I struck a piece out of it with a\nhammer in order to see what it was inside; I found it to be composed\nof a substance like rays, having the appearance of being gilded in\nsome places and in others silvered. It is almost half a finger in\nlength and full three fingers thick, and it is still in my\npossession. When Doctor Muenchen returned, I sent him word how it was\nwith me. He was at the time with the governess of the royal children,\nF. Sitzele Grubbe. Doctor Muenchen desired Totzloff to request me to\nlet him see the stone. I sent him word that if he would come to me,\nhe should see it. I would not send it to him, for I well knew that I\nshould never get it again.\n\n [146] In the margin is added: 'Other natural matter was evacuated,\n but the stone stuck fast in the duct, and seemed to be round, for I\n could not gain hold of it with an instrument I had procured for the\n purpose.'\n\nA.D. 1682, June 11, I wrote the following spiritual song.\n\nIt can be sung to the melody, 'Siunge wii af Hiaertens-Grund.'[E59]\n\n [E59] This tune is still in use in Denmark; it is known in the\n Latin church as 'in natali Domini.'\n\n        I.\n\n    What is this our mortal life\n    Otherwise than daily strife?\n    What is all our labour here,\n    The servitude and yoke we bear?\n    Are they aught but vanity?\n    Art and learning what are ye?\n    Like a vapour all we see.\n\n        II.\n\n    Why, then, is thy anxious breast\n    Filled with trouble? Be at rest!\n    Why, then, dost thou boldly fight\n    The phantoms vain that mock thy sight?\n    Is there any, small or grand,\n    Who can payment duly hand\n    At the creditor's demand?\n\n        III.\n\n    Naked to the world I came,\n    And I leave it just the same;\n    The Lord has given and He takes;\n    It is well whate'er He makes.\n    To the Lord all praises be;\n    I will trust Him heartily!\n    And my near deliverance see.\n\n        IV.\n\n    One thing would I ask of Thee.\n    That Thy House I once may see,\n    And once more with song and praise\n    May my pious offering raise,\n    And magnify Thy grace received,\n    And all that Jesus has achieved\n    For us who have in Him believed.\n\n        V.\n\n    If Thou sayest unto me,\n    'I have no desire in thee,\n    There is no place for thee above;'\n    Oh Jesus! look Thou down in love!\n    Can I not justly to Thee say\n    'Let me but see Thy wounds, I pray:'\n    God's mercy cannot pass away.\n\nOn June 27, the Queen sent me some silk and silver, with the request\nthat I would embroider her a flower, which was traced on parchment;\nshe sent also another flower which was embroidered, that I might see\nhow the work should be done, which is called the golden work. I had\nnever before embroidered such work, for it affects the eyes quickly;\nbut I undertook it, and said I would do it as well as I could. On\nJuly 9, I sent the flower which I had embroidered to the governess of\nthe royal children, F. Sitzele Grubbe, with the request that she\nwould present it most humbly to her Majesty the Queen. The Queen was\nmuch pleased with the flower, and told her that it excelled the\nothers which certain countesses had embroidered for her.\n\nI afterwards embroidered nine flowers in silver and silk in this\ngolden work, and sent them to the Queen's mistress of the robes, with\nthe request that she would present them most humbly to her Majesty\nthe Queen. The mistress of the robes assured me of the Queen's\nfavour, and told me that her Majesty was going to give me two silver\nflagons, but I have not heard of them yet. In the same year I\nembroidered a table-cover with floss silk, in a new design devised by\nmyself, and I trimmed it with taffeta and silver fringe; this also I\nbegged Lady Grubbe, the governess of the King's children, to present\nmost humbly to her Majesty, and it was graciously received. On\nNovember 29, I completed the work which I had made for my death-gear.\nIt was embroidered with thread. On one end of the pillow I worked the\nfollowing lines:\n\n    Full of anxiety and care, in many a silent night,\n    This shroud have I been weaving with sorrowful delight!\n\nOn the other end I embroidered the following: (N.B. The pillow was\nstuffed with my hair).\n\n    When some day on this hair my weary head will lie,\n    My body will be free and my soul to God will fly.\n\nOn the cloth for the head I embroidered:\n\n    I know full well, my Jesus, Thou dost live,\n    And my frail body from the dust wilt give,\n    And it with marvellous beauty will array\n    To stand before Thy throne on the great day.\n    Fulfilled with heavenly joy I then shall be,\n    And Thee, great God, in all Thy splendour see.\n    Nor unknown wilt Thou to mine eyes appear!\n    Help Jesus, bridegroom, be Thou ever near!\n\nHer Majesty the Queen was always gracious to me, and sent me again a\nnumber of silkworms that I might amuse myself with feeding them for\nher, and I was to return what they spun. The virtuous Queen also sent\nme sometimes oranges, lemons, and some of the large almanacs, and\nthis she did through a dwarf, who is a thoroughly quick lad. His\nmother and father had been in the service of my deceased sister\nSophia Elizabeth and my brother-in-law Count Pentz.\n\nThe governess of the royal children, F. Sitzel Grubbe, was very\ncourteous and good to me, and sent me several times lemons, oranges,\nmulberries, and other fruits, according to the season of the year.\n\nA young lady, by birth a Donep, also twice sent me fruit.\n\nThe maids of honour once sent me some entangled silk from silkworms,\nwhich they wanted to spin, and did not rightly know how to manage it;\nthey requested me to arrange it for them. I had other occupation on\nhand which I was unwilling to lay aside (for I was busy collecting my\nheroines), but nevertheless I acceded to their wish.[E60] My\ncaptivity of nearly twenty years could not touch the heart of the\nQueen Dowager (though with a good conscience I can testify before God\nthat I never gave her cause for such inclemency). My most gracious\nhereditary King was gracious enough several times in former years to\nintercede for me with his royal mother, through the high ministers of\nthe State. Her answer at that time was very hard; she would entitle\nthem 'traitors,' and, 'as good as I was,' and would point them to the\ndoor. All the favours which the King's majesty showed me--the outer\napartment, the large window, the money to dispose of for\nmyself--annoyed the Queen Dowager extremely; and she made the King's\nmajesty feel her displeasure in the most painful manner. And as she\nhad also learned (she had plenty of informers) that I possessed a\nclavicordium, this annoyed her especially, and she spoke very angrily\nwith the King about it; on which account the prison governor came to\nme one day and said that the King had asked him how he had happened\nto procure me a clavicordium. 'I stood abashed,' said the prison\ngovernor, 'and knew not what to say.' I thought to myself, 'You know\nbut little of what is happening in the tower.' I did not see him more\nthan three times a year. I asked who had told the King of the\nclavicordium. He answered: 'The old Queen; she has her spies\neverywhere, and she has spoken so hardly to the King that it is a\nshame because he gives you so much liberty;' so saying, he seized the\nclavicordium just as if he were going to take it away, and said, 'You\nmust not have it!' I said, 'Let it alone! I have permission from his\nMajesty, my gracious Sovereign, to buy what I desire for my pastime\nwith the money he graciously assigns me. The clavicordium is in no\none's way, and cannot harm the Queen Dowager.' He pulled at it\nnevertheless, and wanted to take it down; it stood on a closet which\nI had bought. I said, with rather a loud voice, 'You must let it\nremain until you return me the money I gave you for it; then you may\ndo with it what you like.' He said, 'I will tell the King that.' I\nbegged him to do so. There was nothing afterwards said about it,[147]\nand I still have the clavicordium, though I play on it rarely. I\nwrite, and hasten to finish my heroines, so that I may have them\nready, and that no sickness nor death may prevent my completing them,\nnor the friend to whom I confide them may leave me, and so they would\nnever fall into your hands, my dearest children.\n\n [E60] 'I have in my imprisonment also gained some experience with\n regard to caterpillars. It amused me at one time to watch their\n changes. The worms were apparently all of one sort, striped alike,\n and of similar colour. But butterflies did not come from all. It\n was quite pretty to see how a part when they were about to change,\n pressed against something, whatever it might be, and made\n themselves steady with a thread (like silkworm's silk) on each\n side, passing it over the back about fifty times, always at the\n same place, and often bending the back to see if the threads were\n strong enough; if not, they passed still more threads round them.\n When this was done, they rapidly changed their form and became\n stout, with a snout in front pointed at the end, not unlike the\n fish called knorr by the Dutch; they have also similar fins on the\n back, and a similar head. In this form they remain for sixteen\n days, and then a white butterfly comes out. But of some\n caterpillars small worms like maggots come out on both sides,\n whitish, broad at one end and pointed at the other. These surround\n themselves with a web with great rapidity, each by itself. Then the\n worm spins over them tolerably thickly, turning them round till\n they are almost like a round ball. In this it lies till it is quite\n dried up; it eats nothing, and becomes as tiny as a fly before it\n dies. Twelve days afterwards small flies come out of the ball, and\n then the ball looks like a small bee-hive. I have seen a small\n living worm come out of the neck of the caterpillar (this I\n consider the rarest), but it did not live long, and ate nothing.\n The mother died immediately after the little one had come out.'\n\n It is perhaps not unnecessary to add that this observation, which is\n correct as to facts, refers to the habits of certain larvae of wasps\n which live as parasites in caterpillars.\n\n [147] In the margin is added: 'The prison governor told me\n afterwards that the King laughed when he had told his Majesty my\n answer about the clavicordium, and had said, \"Yes, yes.\"'\n\nOn September 24, M. Johan Adolf, my father confessor, was promoted;\nhe became dean of the church of Our Lady. He bade me a very touching\nfarewell, having administered the duties of his office to me for\nnearly six years, and been my consolation. God knows how unwillingly\nI parted with him.\n\nAt the beginning of this year H. Peder Collerus was my father\nconfessor; he was at the time palace-preacher. He also visited me\nwith his consolatory discourse every six weeks. He is a learned man,\nbut not like Hornemann.\n\nOn April 3, an old sickly dog was sent to me in the Queen's name. I\nfancy the ladies of the court sent it, to be quit of the trouble. A\nmarten had bit its jaw in two, so that the tongue hung out on one\nside. All the teeth were gone, and a thin film covered one eye. It\nheard but little, and limped on one side. The worst, however, was,\nthat one could easily see that it tried to exhibit its affection\nbeyond its power. They told me that her Majesty the Queen had been\nvery fond of the dog. It was a small 'King Charles;' its name was\n'Cavaillier.' The Queen expressed her opinion that it would not long\ntrouble me. I hoped so also.[E66b]\n\n [E66b] This poem still exists, and is printed in the second volume\n of Hofman's work on Danish noblemen. It is intended to convey an\n account of her own and her husband's fate.\n\nOn August 12 of this year I finished the work I had undertaken, and\nsince my prefatory remarks treated of celebrated women of every kind,\nboth of valiant rulers and sensible sovereigns, of true, chaste,\nGod-fearing, virtuous, unhappy, learned, and steadfast women, it\nseemed to me that all of these could not be reckoned as heroines; so\nI took some of them out and divided them into three parts, under the\ntitle, 'The Heroines' Praise.' The first part is to the honour of\nvaliant heroines. The second part speaks of true and chaste heroines.\nThe third part of steadfast heroines. Each part has its appendix. I\nhope to God that this my prison work may come into your hands, my\ndearest children. Hereafter I intend, so God will, to collect the\nothers: namely, the sensible, learned, god-fearing, and virtuous\nwomen; exhibiting each to view in the circumstances of her life.[E61]\n\n [E61] It has been stated already that a copy of the first part of\n this work is still preserved. Amongst the heroines here treated of\n are modern historical personages, as Queen Margaret of Denmark,\n Thyre Danobod who built the Dannevirke, Elizabeth of England, and\n Isabella of Castilia, besides mythical and classic characters, as\n Penthesilea, Queen of the Amazons, Marpesia, Tomyris, Zenobia,\n Artemisia, Victorina, etc. There existed not a few works of this\n kind--we need only mention Boccacio's 'Donne Illustri,' in which\n many of these last personages also occur.\n\nI will mention from her own statement somewhat of Jonatha, who now\nattended on me. I will pass over the long story of how she left her\nmother; the fact is, that against her mother's will she married a\nDanish merchant, named Jens Pedersen Holme. But her life and doings\n(according to her own statement) are so strange, that it may be worth\nwhile to record somewhat of them. After they were married, she says,\nit vexed her, and was always in her mind that she had made her mother\nangry, and had done very wrong. Her mother had sent her also a hard\nletter, which distressed her much; and she behaved refractorily\ntowards her husband, and in many ways like a spoilt unreasonable\nchild, sometimes even like one who had lost her reason and was\ndesperate.\n\nIt seems also that her husband treated her as if her mind was\naffected, for he had her looked after like a child, and treated her\nas such. She told him once that she was intending to drown herself in\nthe Peblingeso,[E62] and at another time that she would strike him\ndead. The husband feared neither of these threats; still he had her\nwatched when she went out, to see which way she took. Once she had\nfirmly resolved to drown herself in the Peblingeso, for this place\npleased her; she was even on her way there, but was brought back. She\nstruck her husband, too, once after her fashion. He had come home one\nday half intoxicated, and had laid down on a bed, so that his legs\nrested on the floor. She says she intended at the time to strike him\ndead; she took a stick and tried to see if he were asleep, talking\nloudly to herself and scolding, and touching him softly on the\nshinbone with the stick. He behaved as if he were asleep. Then she\nstruck him a little harder. Upon this he seized the stick and took it\naway from her, and asked what she had in her mind. She answered, 'To\nkill you.' 'He was grieved at my madness,' she said, 'and threw\nhimself on his knees, praying God to govern me with His good spirit\nand give me reason.' The worst is that it once came into her mind not\nto sleep with her husband, and she laid down on a bench in the room.\nFor a long time he gave her fair words, but these availed nothing. At\nlast he said, 'Undress yourself and come and lie down, or I shall\ncome to you.' She paid no attention to this; so he got up, undressed\nher completely, slapped her with his hand, and threw her into bed.\nShe protested that for some days she was too bruised to sit; this\nproved availing, and she behaved in future more reasonably.\n\n [E62] The Peblingeso is one of three lakes which surround\n Copenhagen on the land-side, in a semicircle.\n\nLittle at peace as she was with her husband when she had him with\nher, she was greatly grieved when he left her to go to the West\nIndies. He sent by return vessels all sorts of goods to sell, and\nshe thus maintained herself comfortably.\n\nIt happened at last that the man died in the West Indies, and a\nperson who brought her the news stated that he had been poisoned by\nthe governor of the place named ----, at an entertainment, and this\nbecause he was on the point of returning home, and the governor was\nafraid that Holme might mention his evil conduct. These tidings\nunsettled her mind so, that she ran at night, in her mere\nnight-dress, along the street, and squabbled with the watchmen. She\nwent to the admiral at the Holm, and demanded justice upon the absent\nculprit, and accused him, though she could prove nothing.\n\nThus matters went on for a time, until at last she gained repose, and\nGod ordained it that she came to me. My intercourse with her is as\nwith a frail glass vessel, for she is weak in many respects. She\noften doubts of her salvation, and enumerates all her sins. She\nlaments especially having so deeply offended her mother, and thus\nhaving drawn down a curse upon her. When this fear comes upon her, I\nconsole her with God's word, and enter fully into the matter, showing\nher, from Holy Scripture, on what a repentant sinner must rely for\nthe mercy of God. Occasionally she is troubled as to the\ninterpretation of Holy Scripture, as all passages do not seem to her\nto agree, but to contradict each other. In this I help her so far as\nmy understanding goes, so that sometimes she heartily thanks God that\nshe is come to me, where she finds rest and consolation.\n\nAfter she had been with me for a year or two, she learned that the\ngovernor, whom she suspected, had come to Copenhagen. She said to me,\n'I hear the rogue is come here; I request my dismissal.' I asked her\nwhy. 'Because,' she replied, 'I will kill him.' I could scarcely keep\nfrom laughing; but I said, 'Jesus forbid! If you have any such\ndesign, I shall not let you go.' And as she is a person whose like I\nhave never known before--for she could chide with hard words, and yet\nat the same time she was modest and well-behaved--I tried to make her\ntell me and show me how she designed to take the governor's life.\n(She is a small woman, delicately formed.) Then she acted as if her\nenemy were seated on a stool, and she had a large knife under her\napron. When he said to her, 'Woman, what do you want?' she would\nplunge the knife into him, and exclaim, 'Rogue, thou hast deserved\nthis.' She would not move from the place, she would gladly die, if\nshe could only take his life. I said, 'Still it is such a disgrace to\ndie by the hand of the executioner.' 'Oh, no!' she replied, 'it is\nnot a disgrace to die for an honourable deed;' and she had an idea\nthat any one thus dying by the hand of the executioner passed away in\na more Christian manner than such as died on a bed of sickness; and\nthat it was no sin to kill a man who, like a rogue, had murdered\nanother. I asked her if she did not think that he sinned who killed\nanother. 'No,' she replied, 'not when he has brought it upon\nhimself.' I said, 'No one may be his own judge, either by the law of\nGod or man; and what does the fifth commandment teach us?'[E63] She\nanswered as before, that she would gladly die if she could only take\nthe rogue's life. (I must add that she said she could not do it on my\naccount, for I would not let her out.) She made a sin of that which\nis no sin, and that which is sin she will not regard as such. She\nsays it is a sin to kill a dog, a cat, or a bird; the innocent\nanimals do no harm; in fact, it is a still greater sin to let the\npoor beasts hunger. I asked her once whether it was a sin to eat\nmeat. 'No,' she answered; 'it is only a sin to him who has killed the\nanimal.' She protested that if she were obliged to marry, and had to\nchoose between a butcher and an executioner, she would prefer the\nlatter. She told me of various quarrels she had had with those who\nhad either killed animals or allowed them to hunger.\n\n [E63] The Lutheran Church has retained the division of the\n Commandments used in the Roman Church; and the Commandment against\n murder is therefore here described as the fifth, whilst in the\n English catechism it is the sixth.\n\nOne story I will not leave unmentioned, as it is very pretty. She\nsold, she said, one day some pigs to a butcher. When the butcher's\nboy was about to bind the pigs' feet and carry them off hanging from\na pole, she was sorry for the poor pigs, and said, 'What, will you\ntake their life? No, I will not suffer that!' and she threw him back\nhis money. I asked her if she did not know that pigs were killed, and\nfor what reason she thought the butcher had bought them. 'Yes,' she\nreplied, 'I knew that well. Had he let them go on their own legs, I\nshould have cared nothing about it; but to bind the poor beasts in\nthis way, and to hear them cry, I could not endure that.' It would\ntake too long to enumerate all the extravagant whims which she\nrelated of herself. But with all this she is not foolish, and I well\nbelieve she is true to any one she loves. She served me very well,\nand with great care.\n\nThe above-mentioned governor[E64] was killed by some prisoners on\nboard the vessel, when he was returning to the West Indies. By a\nstrange chance the vessel with the murderers came to Copenhagen.\n(They were sentenced to death for their crime.) Jonatha declared\nthat the governor had had only too good a death, and that it was a\nsin that any one should lose his life on account of it. I practise\nspeaking the English language with Jonatha. She has forgotten\nsomewhat of her mother tongue, since she has not spoken it for many\nyears; and as she always reads the English Bible, and does not at\nonce understand all the words, I help her; for I not only can\nperceive the sense from the preceding and following words, but also\nbecause some words resemble the French, though with another accent.\nAnd we often talk together about the interpretation of Holy\nScripture. She calls herself a Calvinist, but she does not hold the\nopinions of Calvinists. I never dispute with her over her opinions.\nShe goes to the Lord's Supper in the Queen's church[E65]. Once, when\nshe came back to me from there, she said she had had a conversation\nupon religion with a woman, who had told her to her face that she was\nno Calvinist. I asked her of what religion the woman imagined that\nshe was. She replied: 'God knows that. I begged her to mind her own\nbusiness, and said, that I was a Christian; I thought of your grace's\nwords (but I did not say them), that all those who believe on Christ\nand live a Christian life, are Christians, whatever name they may\ngive to their faith.'\n\n [E64] The name of this governor, which is not mentioned by Leonora,\n was Jorgen Iversen, the first Danish governor of St. Thomas. In\n 1682 he returned to the colony from Copenhagen on board a vessel\n which was to bring some prisoners over to St. Thomas. Very soon\n after their departure, some of the prisoners and of the crew raised\n a mutiny, killed the captain and some of the passengers, amongst\n them the ex-governor Iversen. But one of the prisoners who had not\n been in the plot afterwards got the mastery of the vessel, and\n returned to Copenhagen. The vessel struck on a rock, near the\n Swedish coast, but the crew were saved and sent home to Copenhagen\n by the Swedish Government, and the murderers were then executed.\n\n [E65] The Queen's church was a room in the castle where service was\n held according to the Calvinist rite.\n\nIn this year 1684 I saw the Queen Dowager fall from the chair in\nwhich she was drawn up to the royal apartment. The chair ran down the\npulleys too quickly, so that she fell on her face and knocked her\nknee. During this year her weakness daily increased, but she thought\nherself stronger than she was. She appeared at table always much\ndressed, and between the meals she remained in her apartments.\n\nI kept myself patient, and wrote the following:--\n\n_Contemplation on Memory and Courage, recorded to the honour of God\nby the suffering Christian woman in the sixty-third year of her life,\nand the almost completed twenty-first year of her captivity._\n\n    The vanished hours can ne'er come back again,\n    Still may the old their youthful joys retain;\n    The past may yet within our memory live,\n    And courage vigour to the old may give.\n    Yet why should I thus sport with Memory's truth,\n    And harrow up the fairer soil of youth?\n    No fruit it brings, fallow and bare it lies,\n    And the dry furrow only pain supplies!\n    In my first youth, in honourable days\n    Upon such things small question did I raise.\n    Then years advanced with trouble in their train,\n    And spite of show my life was fraught with pain.\n    The holy marriage bond--my rank and fame,\n    Increased my foes and made my ill their aim.\n    Go! honour, riches, vanish from my mind!\n    Ye all forsook me and left nought behind.\n    'Twas ye have brought me here thro' years to lie;\n    Thus can man's envy human joy deny!\n    My God alone, He ne'er forsook me here,\n    My cross He lightened, and was ever near;\n    And when my heart was yielding to despair,\n    He spoke of peace and whispered He was there.\n    He gave me power and ever near me stood,\n    And all could see how truly God was good.\n\n    What Courage can achieve I next will heed;\n    He who is blessed with it, is blest indeed.\n    To the tired frame fresh power can Courage give,\n    Raising the weary mind anew to live;\n    I mean that Courage Reason may instil\n    Not the foolhardiness that leads to ill.\n    Far oftener is it that the youth will lie\n    Helpless, when Fortune's favours from him fly,\n    Than that the old man should inactive stay,\n    Who knows full well how Fortune loves to play.\n    Fresh Courage seizes him; from such a shield\n    Rebound the arms malicious foes may wield.\n    Courage imparts repose, and trifles here,\n    Beneath its influence, as nought appear;\n    But a vain loan, which we can only hold\n    Until the lender comes, and life is told.\n    Courage pervades the frame and vigour gives,\n    And a fresh energy each part receives;\n    With appetite and health and cheerful mind,\n    And calm repose in hours of sleep we find,\n    So that no visions in ill dreams appear,\n    And spectre forms filling the heart with fear.\n    Courage gives honied sweetness to our food\n    And prison fare, and makes e'en death seem good.\n    'Tis well! my mind is fresh, my limbs are sound,\n    And no misfortune weighs me to the ground.\n    Reason and judgment come from God alone,\n    And the five senses unimpaired I own.\n    The mighty God in me His power displays,\n    Therefore join with me in a voice of praise\n    And laud His name: For Thou it is, oh God,\n    Who in my fear and anguish nigh me stood.\n    Almighty One, my thanks be ever thine!\n    Let me ne'er waver nor my trust resign.\n    Take not the courage which my hope supplies,\n    Till my soul enters into Paradise.\n\nWritten on February 28, 1684, that is the thirty-sixth anniversary\nsince the illustrious King Christian the Fourth bade good-night to\nthis world, and I to the prosperity of my life.\n\nI have now reached the sixty-third year of my age, and the twentieth\nyear, sixth month, and fifteenth day of my imprisonment. I have\ntherefore spent the third part of my life in captivity. God be\npraised that so much time is past. I hope the remaining days may not\nbe many.\n\nAnno 1685, January 14, I amused myself with making some verses in\nwhich truth was veiled under the cloak of jest, entitled: 'A Dog,\nnamed Cavaillier, relates his Fate.'\n\nThe rhymes, I suppose, will come into your hands, my dearest\nchildren.[E66]\n\n [E66] This poem still exists, and is printed in the second volume\n of Hofman's work on Danish noblemen. It is intended to convey an\n account of her own and her husband's fate.\n\nOn February 20, the Queen Dowager Sophia Amalia died. She did not\nthink that death would overtake her so quickly; but when the doctor\nwarned her that her death would not be long delayed, she requested to\nspeak with her son. But death would not wait for the arrival of his\nMajesty, so that the Queen Dowager might say a word to him. She was\nstill alive; she was sitting on a chair, but she was speechless, and\nsoon afterwards, in the same position, she gave up her spirit.\n\nAfter the death of this Queen I was much on the lips of the people.\nSome thought that I should obtain my liberty; others believed that I\nshould probably be brought from the tower to some other place, but\nshould not be set free.\n\nJonatha, who had learned from Ole the tower-warder, some days before\nthe death of the Queen, that prayers were being offered up in the\nchurch for the Queen (it had, however, been going on for six weeks,\nthat this prayer had been read from the pulpit), was, equally with\nOle the tower-warder, quite depressed. Ole, who had consoled himself\nand her hitherto with the tidings from the Queen's lacqueys, that the\nQueen went to table and was otherwise well, though she occasionally\nsuffered from a cough, now thought that there was danger, that death\nmight result, and that I, if the Queen died, might perhaps leave the\nprison. They did their best to conceal their sorrow, but without\nsuccess. They occasionally shed secretly a few tears. I behaved as if\nI did not remark it, and as no one said anything to me about it, I\ngave no opportunity for speaking on the subject. A long time\npreviously I had said to Jonatha (as I had done before to the other\nwomen) that I did not think I should die in the tower. She remembered\nthis and mentioned it. I said: 'All is in God's hand. He knows best\nwhat is needful for me, both as regards soul and body; to Him I\ncommend myself.' Thus Jonatha and Ole lived on between hope and fear.\n\nOn March 15, the reigning Queen kept her Easter. Jonatha came quite\ndelighted from her Majesty's church, saying that a noble personage\nhad told her that I need not think of getting out of the prison,\nalthough the Queen was dead; she knew better and she insisted upon\nit. However often I asked as to who the personage was, she would not\ntell me her name. I laughed at her, and said, 'Whoever the personage\nmay be, she knows just as much about it as you and I do.' Jonatha\nadhered to her opinion that the person knew it well. 'What do you\nmean?' I said; 'the King himself does not know. How should others\nknow?' 'Not the King! not the King!' she said quite softly. 'No, not\nthe King!' I answered. 'He does not know till God puts it into his\nheart, and as good as says to him, \"Now thou shalt let the prisoner\nfree!\"' She came somewhat more to herself, but said nothing. And as\nshe and Ole heard no more rumours concerning me, they were quite\ncomforted.\n\nOn March 26, the funeral of the Queen Dowager took place, and her\nbody was conveyed to Roskild.\n\nOn April 21, I supplicated the King's Majesty in the following\nmanner. I possessed a portrait engraving of the illustrious King\nChristian the Fourth, rather small and oval in form. This I\nilluminated with colours, and had a carved frame made for it, which\nI gilded myself. On the piece at the back I wrote the following\nwords:--\n\n    My grandson, and great namesake,\n    Equal to me in power and state;\n    Vouchsafe my child a hearing,\n    And be like me in mercy great!\n\nBesides this, I wrote to his Excellency Gyldenlove, requesting him\nhumbly to present the Supplique to the King's majesty, and to\ninterest himself on my behalf, and assist me to gain my liberty. His\nExcellency was somewhat inconvenienced at the time by his old\nweakness, so that he could not himself speak for me; but he begged a\ngood friend to present the engraving with all due respect, and this\nwas done on April 24.[E67]\n\n [E67] This picture is still preserved at the Castle of Rosenbourg,\n in Copenhagen.\n\nOf all this Jonatha knew nothing. Peder Jensen Totzloff was my\nmessenger. He has been a comfort to me in my imprisonment, and has\nrendered me various services, so that I am greatly bound to him. And\nI beg you, my dearest children, to requite him in all possible ways\nfor the services he has rendered me.\n\nOn May 2, it became generally talked of that I should assuredly be\nset at liberty, and some asked the tower-warder whether I had come\nout the evening before, and at what time; so that Ole began to fear,\nand could not bear himself as bravely as he tried to do. He said to\nme in a sad tone: 'My good lady! You will certainly be set at\nliberty. There are some who think you are already free.' I said, 'God\nwill bring it to pass.' 'Yes,' said he, 'but how will it fare with me\nthen?' I answered, 'You will remain tower-warder, as you now are.'\n'Yes,' said he, 'but with what pleasure?' and he turned, unable to\nrestrain his tears, and went away. Jonatha concluded that my\ndeliverance was drawing near, and endeavoured to conceal her sorrow.\nShe said, 'Ole is greatly cast down, but I am not.' (And the tears\nwere standing in her eyes.) 'It is said for certain that the King is\ngoing away the day after to-morrow. If you are set at liberty, it\nwill be this very day.' I said, 'God knows.' Jonatha expressed her\nopinion that I was nevertheless full of hope. I said I had been\nhopeful ever since the first day of my imprisonment; that God would\nat last have mercy on me, and regard my innocence. I had prayed to\nGod always for patience to await the time of His succour; and God had\ngraciously bestowed it on me. If the moment of succour had now\narrived, I should pray to God for grace to acknowledge rightly His\ngreat benefits. Jonatha asked if I were not sure to be set free\nbefore the King started for Norway; that it was said for certain that\nthe King would set out early on the following morning. I said: 'There\nis no certainty as to future things. Circumstances may occur to\nimpede the King's journey, and it may also happen that my liberty may\nbe prevented, even though at this hour it may perhaps be resolved\nupon. Still I know that my hope will not be confounded. But you do\nnot conceal your regret, and I cannot blame you for it. You have\ncause for regret, for with my freedom you lose your yearly income and\nyour maintenance.[148] Remember how often I have told you not to\nthrow away your money so carelessly on your son. You cannot know what\nmay happen to you in your old age. If I die, you will be plunged\ninto poverty; for as soon as you receive your money, you expend it\non the apprenticeship of your son, who returns you no thanks for\nit.[149] You have yourself told me of his bad disposition, and how\nwrongly he has answered you when you have tried to give him good\nadvice. Latterly he has not ventured to do so, since I read him a\nlecture, and threatened that I would help to send him to the House of\nCorrection. I fear he will be a bad son to you.' Upon this she gave\nfree vent to her tears, and begged that if I obtained my liberty I\nwould not abandon her. This I promised, so far as lay in my power;\nfor I could not know what my circumstances might be.\n\n [148] In the margin is added: 'The woman who attended on me\n received eight rix-dollars monthly.'\n\n [149] In the margin: 'She had him learn wood-carving.'\n\nIn this way some days elapsed, and Jonatha and Ole knew not what the\nissue might be.\n\nOn May 19, at six o'clock in the morning, Ole knocked softly at my\nouter door. Jonatha went to it. Ole said softly, 'The King is already\ngone; he left at about four o'clock.' I know not if his hope was\ngreat; at any rate it did not last long. Jonatha told me Ole's news.\nI wished the King's Majesty a prosperous journey (I knew already what\norder he had given), and it seemed to me from her countenance she was\nto some extent contented. At about eight o'clock Totzloff came up to\nme and informed me that the Lord Chancellor Count Allefeldt had sent\nthe prison governor a royal order that I was to be released from my\nimprisonment, and that I could leave when I pleased. (This order was\nsigned by the King's Majesty the day before his Majesty started.)\n\nHis Excellency had accompanied the King. Totzloff asked whether I\nwished him to lock the doors, as I was now free. I replied, 'So long\nas I remain within the doors of my prison, I am not free. I will\nmoreover leave properly. Lock the door and enquire what my sister's\ndaughter, Lady Anna Catharina Lindenow, says, whether his\nExcellency[E68] sent any message to her (as he promised) before he\nleft. When Totzloff was gone, I said to Jonatha, 'Now, in Jesus'\nname, this very evening I shall leave. Gather your things together,\nand pack them up, and I will do the same with mine; they shall remain\nhere till I can have them fetched.' She was somewhat startled, but\nnot cast down. She thanked God with me, and when the doors were\nunlocked at noon and I dined, she laughed at Ole, who was greatly\ndepressed. I told her that Ole might well sigh, for that he would now\nhave to eat his cabbage without bacon.\n\n [E68] The Excellency alluded to is Ulrik Frederik Gyldenlove, a\n natural son of Frederik III. Anna Catharina Lindenow was daughter\n of Leonora's sister, Elizabeth Augusta, who married Hans Lindenow.\n\nTotzloff brought me word from my sister's daughter that his\nExcellency had sent to her to say that she was free to accompany me\nfrom the tower, if she chose. It was therefore settled that she was\nto come for me late the same evening.\n\nThe prison governor was in a great hurry to get rid of me, and sent\nthe tower-warder to me towards evening, to enquire whether I would\nnot go. I sent word that it was still too light (there would probably\nbe some curious people who had a desire to see me).\n\nThrough a good friend I made enquiry of her Majesty the Queen,\nwhether I might be allowed the favour of offering my humble\nsubmission to her Majesty (I could go into the Queen's apartment\nthrough the secret passage, so that no one could see me). Her Majesty\nsent me word in reply that she might not speak with me.\n\nAt about ten o'clock in the evening, the prison governor opened the\ndoor for my sister's daughter. (I had not seen him for two years.) He\nsaid, 'Well, shall we part now?' I answered, 'Yes, the time is now\ncome.' Then he gave me his hand, and said 'Ade!' (Adieu). I answered\nin the same manner, and my niece laughed heartily.\n\nSoon after the prison governor had gone, I and my sister's daughter\nleft the tower. Her Majesty the Queen thought to see me as I came\nout, and was standing on her balcony, but it was rather dark;\nmoreover I had a black veil over my face. The palace-square, as far\nas the bridge and further, was full of people, so that we could\nscarcely press through to the coach.\n\nThe time of my imprisonment was twenty-one years, nine months, and\neleven days.\n\nKing Frederick III. ordered my imprisonment on August 8, A.D. 1663;\nKing Christian V. gave me my liberty on May 18, 1685. God bless my\nmost gracious King with all royal blessing, and give his Majesty\nhealth and add many years to his life.\n\nThis is finished in my prison.\n\nOn May 19, at ten o'clock in the evening, I left my prison. To God be\nhonour and praise. He graciously vouchsafed that I should recognise\nHis divine benefits, and never forget to record them with gratitude.\n\nDear children! This is the greatest part of the events worth\nmentioning which occurred to me within the doors of my prison. I live\nnow in the hope that it may please God and the King's Majesty that I\nmay myself show you this record. God in His mercy grant it.\n\n1685. Written at Husum[E69] June 2, where I am awaiting the return of\nthe King's Majesty from Norway:\n\n [E69] This Husum is a village just outside Copenhagen, where\n Leonora remained for some months before she went to Maribo, as is\n proved by a letter from her dated Husum, September 18, 1685. Of\n course the last paragraphs must have been added after she left her\n prison, and the passage 'This is finished in my prison' refers, at\n any rate, only to what precedes.\n\nA.D. 1683. New Year's Day. To Myself.\n\n    Men say that Fortune is a rare and precious thing,\n    And they would fain that Power should homage to her bring.\n    Yet Power herself is blind and ofttimes falleth low,\n    Rarely to rise again, wherefore may Heaven know.\n    To-day with humorous wiles she holds her sovereign sway,\n    And could one only trust her, there might be goodly prey.\n    Yet is she like to Fortune, changeful the course she flies,\n    And both, oh earthly pilgrim, are but vain fraud and lies.\n    The former is but frail, the other strives with care,\n    And both alas! are subject to many a plot and snare.\n    Thou hast laid hold on Fortune with an exultant mind,\n    Affixed perhaps to-morrow the fatal _mis_ we find;\n    Then does thy courage fail, this prefix saddens thee,\n    Wert thou thyself Goliath or twice as brave as he.\n    And thou who art so small--already grey with care--\n    Thou know'st not whether evil this year thy lot may share.\n    For Fortune frolics ever, now under, now above,\n    Emerging here and there her varied powers to prove.\n    All that is earthly comes and vanishes again,\n    Therefore I cling to that which will for aye remain.\n\nOn March 14, 1683, I wrote the following:--\n\n    True is the sentence we are sometimes told:\n    A friend is worth far more than bags of gold.\n    Yet would I gladly ask, where do we find\n    A friend so virtuous that he is well inclined\n    To help another in his need and gloom\n    Without a thought of recompense to come?\n    Naught is there new in this, for selfish care\n    To every child of Eve has proved a snare.\n    Each generation hears the last complain,\n    And each repeats the same sad tale again;--\n    That the oppressed by the wayside may lie,\n    When naught is gained but God's approving eye.\n\n    See, at Bethesda's pool, how once there came\n    The halting impotent, some help to claim\n    Among those thousands. Each of pity free,\n    Had no hand for him in his misery\n    To bring him to the angel-troubled stream.\n    Near his last breath did the poor sufferer seem,\n    Weary and penniless; when One alone\n    Who without money works His wise own\n    Will, turned where the helpless suppliant lay,\n    And gently bade him rise and go his way.\n\n    Children of grief, rejoice, do not despair;\n    This Helper still is here and still will care\n    What He in mercy wills. He soothes our pain,\n    And He will help, asking for naught again.\n    And in due time He will with gracious hand\n    Unloose thy prison bars and iron band.\n\nA.D. 1684. The first day. To Peder Jensen Totzloff.\n\n    Welcome, thou New Year's day, altho' thou dost belong\n    To those by Brahe reckoned the evil days among,\n    Declaring that whatever may on this day begin\n    Can never prosper rightly, nor true success can win.\n    Now I will only ask if from to-day I strive\n    The evil to avoid and henceforth good to live,\n    Will this not bring success? Why should a purpose fail,\n    Altho' on this day made? why should it not prevail?\n    Oh Brahe, I believe, when we aright begin,\n    To-day or when it be, and God's good favour win,\n    The issue must be well, and all that matters here\n    Is to commend our ways to our Redeemer dear.\n\n    Begin with Jesus Christ this as all other days.\n    Pray that thy plans may meet with the Almighty's praise,\n    So may'st thou happy be, and naught that man can do\n    Can hinder thy designs, unless God wills it so!\n    May a rich meed of blessing be on thy head bestow'd,\n    And the Lord Jesus Christ protect thee on thy road\n    With arms of grace. Such is my wish for thee,\n    Based on the love of God; sure, that He answers me.\n\n\nLONDON: PRINTED BY\nSPOTTISWOODE AND CO., NEW-STREET SQUARE\nAND PARLIAMENT STREET\n\n\n\n\n       *       *       *       *       *\n\n\n\n\nTranscriber's note:\n\nThe following corrections were made:\n\np. 53: length the good-for-nothing[good-for nothing] fellow came down,\n       and\n\np. 55: there for ten days[25] a letter from Gul...[Gl...] which he\n\np. 56: patacoon[patacon] to those who were to restrain her, saying,\n\np. 59: came to see her, no one in consequence[consequenec] consoled\n       her,\n\np. 61: When the lawyer had said that they[t hey] had now taken\n\np. 64: lose in Dan...[Den...].\n\np. 67: It was necessary[neccessary] to descend the rampart into the\n\np. 92: he persuaded[pursuaded] me to undertake the English journey,\n\np. 106: with my attendant. I answered nothing else than[then] that\n\np. 114: silk camisole[camisolle], in the foot of my stockings there\n        were\n\np. 132: Castle[Cstale], I had sent a good round present for those in\n\np. 135: sad day, and I begged them, for Jesus'[Jesu's] sake, that\n\np. 137: decree? I only beg for Jesus'[Jesu's] sake that what I say\n\np. 172: might easily injure herself with one.'[[76]]\n\np. 174: Synge'[[E31]]:--\n\np. 230: of listening to reason, for she at once exclaimed 'Ach[!]\n\np. 239: Karen, Nils'[Nil's] daughter, left me one evening in 1669,\n\np. 241: and the Frenchman[Frenchmen] was conveyed to the Dark Church,\n\np. 241: through Uldrich[Udrich] Christian Gyldenlove. Gyldenlove\n\np. 246: her word moreover, and I so arranged it[at] six weeks\n\np. 259: In the same year, 1671, Karen, Nils'[Nil's] daughter, left\n\np. 264: silent, not if I were standing before the King's\n        bailiff![?][']\n\np. 268: in the time of Karen, Nils'[Nil's] daughter. Chresten, who\n\np. 272: In the same year Karen, Nils'[Nil's] daughter, left me for\n\np. 276: and a half after her marriage, and that suddenly,\n        suspicion[suspipicion]\n\np. 300: Supper in the Queen's church[[E65]]. Once, when she came\n\np. 311: [60] In[in] the margin is added: 'The sorrow manifested by\n        many would far\n\np. 311: [117] In the margin is added: '1666. While Karen, Nils'[Nil's]\n        daughter, waited\n\np. 311: Nils'[Nil's] daughter. When anything gave her satisfaction,\n        she would take\n\np. 311: to set Copenhagen[Copenagen] on fire in divers places, and\n        also the\n\np. 311: Autobiography[Autobiograpy] of Leonora as 'notre vieillard;'\n        he was a faithful\n\np. 311: which placed it at the disposal of Hannibal\n        Sehested[Schested] when he\n\np. 311: [E38] 'Anno 1666, soon after Karen, Nils'[Nil's] daughter,\n        came to me,\n\np. 311: [E51] Hannibal Sehested[Schested] was dead already in 1666,\n        as Leonora\n\np. 311: disposed to Hannibal Sehested[Schested].\n\np. 311: entitled 'Martilogium (for martyrologium[matyrologium]) der\n        Heiligen' (Strasburg\n\n\n\n***","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nJust and Unjust Wars\n\nFifth Edition\n\nJust and Unjust Wars\n\nA Moral Argument with Historical Illustrations\n\nFifth Edition\n\nMichael Walzer\n\nA Member of the Perseus Books Group\n\nNew York\n\nCopyright \u00a9 1977 by Basic Books\n\nCopyright \u00a9 2015 by Basic Books for Preface to the Fifth Edition and Postscript\n\nPublished by Basic Books,\n\nA Member of the Perseus Books Group\n\nAll rights reserved. Printed in the United States of America. No part of this book may be reproduced in any manner whatsoever without written permission except in the case of brief quotations embodied in critical articles and reviews. For information, address Basic Books, 250 West 57th Street, New York, NY 10107.\n\nBooks published by Basic Books are available at special discounts for bulk purchases in the United States by corporations, institutions, and other organizations. For more information, please contact the Special Markets Department at the Perseus Books Group, 2300 Chestnut Street, Suite 200, Philadelphia, PA 19103, or call (800) 810-4145, ext. 5000, or e-mail special.markets@perseusbooks.com.\n\nDesigned by Jack Lenzo\n\nLibrary of Congress Cataloging-in-Publication Data\n\nWalzer, Michael.\n\nJust and unjust wars : a moral argument with historical illustrations \/ Michael Walzer. \u2014 Fifth edition.\n\npages cm\n\nIncludes bibliographical references and index.\n\nISBN 978-0-465-05270-7 (e-book) 1. War. 2. War\u2014Moral and ethical aspects. 3. Just war doctrine. I. Title.\n\nU21.2.W345 2015\n\n172'.42\u2014dc23\n\n2015010999\n\n10 9 8 7 6 5 4 3 2 1\n\nAux martyrs de l'Holocauste\n\nAux r\u00e9volt\u00e9s des Ghettos\n\nAux partisans de for\u00eats\n\nAux insurg\u00e9s des camps\n\nAux combatants de la r\u00e9sistance\n\nAux soldats des forces alli\u00e9es\n\nAux sauveteurs de fr\u00e8res en peril\n\nAux vaillants de l'immigration clandestine\n\nA l'\u00e9ternit\u00e9\n\n\u2014Inscription at Yad Vashem Memorial, Jerusalem\nContents\n\nPreface to the Fifth Edition\n\nPreface to the First Edition\n\nAcknowledgments\n\nPart One The Moral Reality of War\n\n1 Against \"Realism\"\n\nThe Realist Argument\n\nThe Melian Dialogue\n\nStrategy and Morality\n\nHistorical Relativism\n\nThree Accounts of Agincourt\n\n2 The Crime of War\n\nThe Logic of War\n\nThe Argument of Karl von Clausewitz\n\nThe Limit of Consent\n\nThe Tyranny of War\n\nGeneral Sherman and the Burning of Atlanta\n\n3 The Rules of War\n\nThe Moral Equality of Soldiers\n\nThe Case of Hitler's Generals\n\nTwo Sorts of Rules\n\nThe War Convention\n\nThe Example of Surrender\n\n Part Two The Theory of Aggression\n\n4 Law and Order in International Society\n\nAggression\n\nThe Rights of Political Communities\n\nThe Case of Alsace-Lorraine\n\nThe Legalist Paradigm\n\nUnavoidable Categories\n\nKarl Marx and the Franco-Prussian War\n\nThe Argument for Appeasement\n\nCzechoslovakia and the Munich Principle\n\nFinland\n\n5 Anticipations\n\nPreventive War and the Balance of Power\n\nThe War of the Spanish Succession\n\nPre-emptive Strikes\n\nThe Six Day War\n\n6 Interventions\n\nSelf-Determination and Self-Help\n\nThe Argument of John Stuart Mill\n\nSecession\n\nThe Hungarian Revolution\n\nCivil War\n\nThe American War in Vietnam\n\nHumanitarian Intervention\n\nCuba, 1898, and Bangladesh, 1971\n\n7 War's Ends, and the Importance of Winning\n\nUnconditional Surrender\n\nAllied Policy in World War II\n\nJustice in Settlements\n\nThe Korean War\n\nPart Three The War Convention\n\n8 War's Means and the Importance of Fighting Well\n\nUtility and Proportionality\n\nThe Argument of Henry Sidgwick\n\nHuman Rights\n\nThe Rape of the Italian Women\n\n9 Noncombatant Immunity and Military Necessity\n\nThe Status of Individuals\n\nNaked Soldiers\n\nThe Nature of Necessity (1)\n\nSubmarine Warfare: The Laconia Affair\n\nDouble Effect\n\nBombardment in Korea\n\nThe Bombing of Occupied France and the Vemork Raid\n\n10 War Against Civilians: Sieges and Blockades\n\nCoercion and Responsibility\n\nThe Siege of Jerusalem, 72 A.D.\n\nThe Right to Leave\n\nThe Siege of Leningrad\n\nTaking Aim and the Doctrine of Double Effect\n\nThe British Blockade of Germany\n\n11 Guerrilla War\n\nResistance to Military Occupation\n\nA Partisan Attack\n\nThe Rights of Guerrilla Fighters\n\nThe Rights of Civilian Supporters\n\nThe American \"Rules of Engagement\" in Vietnam\n\n12 Terrorism\n\nThe Political Code\n\nThe Russian Populists, the IRA, and the Stern Gang\n\nThe Vietcong Assassination Campaign\n\nViolence and Liberation\n\nJean-Paul Sartre and the Battle of Algiers\n\n13 Reprisals\n\nDeterrence Without Retribution\n\nThe FFI Prisoners at Annecy\n\nThe Problem of Peacetime Reprisals\n\nThe Attack on Khibye and the Beirut Raid\n\nPart Four Dilemmas of War\n\n14 Winning and Fighting Well\n\n\"Asinine Ethics\"\n\nChairman Mao and the Battle of the River Hung\n\nThe Sliding Scale and the Argument from Extremity\n\n15 Aggression and Neutrality\n\nThe Right to Be Neutral\n\nThe Nature of Necessity (2)\n\nThe Rape of Belgium\n\nThe Sliding Scale\n\nWinston Churchill and Norwegian Neutrality\n\n16 Supreme Emergency\n\nThe Nature of Necessity (3)\n\nOverriding the Rules of War\n\nThe Decision to Bomb German Cities\n\nThe Limits of Calculation\n\nHiroshima\n\n17 Nuclear Deterrence\n\nThe Problem of Immoral Threats\n\nLimited Nuclear War\n\nThe Argument of Paul Ramsey\n\nPart Five The Question of Responsibility\n\n18 The Crime of Aggression: Political Leaders and Citizens\n\nThe World of Officials\n\nNuremberg: \"The Ministries Case\"\n\nDemocratic Responsibilities\n\nThe American People and the War in Vietnam\n\n19 War Crimes: Soldiers and Their Officers\n\nIn the Heat of Battle\n\nTwo Accounts of Killing Prisoners\n\nSuperior Orders\n\nThe My Lai Massacre\n\nCommand Responsibility\n\nGeneral Bradley and the Bombing of St. L\u00f4\n\nThe Case of General Yamashita\n\nThe Nature of Necessity (4)\n\nThe Dishonoring of Arthur Harris\n\nConclusion\n\nAfterword: Nonviolence and the Theory of War\n\nPostscript: A Defense of Just War Theory\n\nNotes\n\nIndex\nPreface to the Fifth Edition\n\nThe book of Ecclesiastes ends with a complaint: \"Of making many books, there is no end.\" On the occasion of the fifth edition of Just and Unjust Wars, I can't complain about that; I am grateful for the re-making of this book. But it is also true that \"Of making many wars, there is no end,\" and this should be a universal human complaint. I am writing just as the United States is ending two of the longest wars in its history (in Afghanistan and Iraq)\u2014and beginning another (to degrade and defeat the Islamic State in Syria and, again, Iraq). Each new edition of Just and Unjust Wars has coincided with new wars and new questions about war. I wrote about humanitarian intervention in the preface to the third edition and about wars for regime change in the preface to the fourth.\n\nAs this latest edition is being prepared for publication, intense arguments are under way about asymmetric warfare\u2014arguments that overlap a great deal with central issues in this book. But asymmetric warfare has its own set of moral difficulties and immoral cruelties that require specific, detailed consideration and judgment. For reasons I will come to, judgment is especially important\u2014hence this newly prepared preface.\n\nMost of the wars of the past several decades have been asymmetric; notably, for American citizens, the war against the Taliban in Afghanistan and the second Iraq war (after the first few weeks, when the success of the invasion produced an unexpected insurgency)\u2014also the Israeli wars in Lebanon and Gaza, the Sri Lankan war against the Tamil rebels, and the Russian war in Chechnya. Asymmetric warfare isn't new; the guerrilla wars discussed in chapter 11, especially the Vietnam War, are earlier examples. Some writers believe that traditional just war theory, with its long-established rules about when to fight and how to fight, cannot deal with asymmetry. But if the theory \"worked,\" as I think it did, in helping Americans understand what was wrong in Vietnam, there is no reason why it can't work in other times and places. It is a good theory; we remain indebted to the Catholic theologians who invented it centuries ago.\n\nBut the theory was invented to deal with what we now call \"conventional\" warfare\u2014where two armies are engaged and each one is pretty much like the other in organization and armaments. If there are any rules that govern the fighting, it is easy to imagine that they will be the same for the two armies and for all their soldiers. Wars of that sort are\u2014we should be thankful\u2014rare these days. Iran and Iraq fought an old-fashioned war in the 1980s, and the 1991 Iraq war, against the conquest of Kuwait, was roughly conventional. We can all think of possible wars like those, but they are not likely to happen anytime soon.\n\nIn the more immediate conditions of asymmetry, by contrast, there is only one army, organized, armed, and disciplined by a modern state; its opponents are insurgents, less well-organized, less well-armed, and often without a coordinated system of military discipline or of military justice. Do the same rules apply to armies and insurgents? I want to say that they do, but that requires an argument.\n\nThe insurgents claim the prerogatives of weakness\u2014which follow, they say, from the conventional doctrines of military necessity and last resort applied to their special circumstances. So the old rules do apply, but given the circumstances, not in the same way to the two sides. The insurgents have to be able to hide from the army's overwhelming firepower, so they can't wear uniforms. They can't fight along a \"front\"; they have to be able to strike anytime, anywhere, so they can't always distinguish combatants from civilians. In any case, military targets are often too dangerous for them to attack; killing civilians, they argue, is often the only thing they can do, so it is their \"last resort\" even if they don't actually try anything else. And they can't build fortresses or military bases, for if they don't have a \"front,\" they also don't have a \"rear\" where they can assemble or regroup in safety. The only protection they have is the cover of their own people's homes and neighborhoods.\n\nFor the conventional army, insurgents claiming these prerogatives are a big problem. The army gets no circumstantial exemption from the old rules; it is expected above all to maintain the distinction between combatants and civilians, even if the insurgents deliberately blur the distinction. But how can its soldiers fight against enemies who hide among the civilian population without killing civilians?\n\nThe proportionality rule (see \"The Argument of Henry Sidgwick\" in chapter 8) is supposed to provide criteria for judging those deaths, but proportionality in warfare is not an exact science. Historically, in conventional wars, the rule has most often been read permissively: the value of the military target is so great that a fairly high number of civilian casualties (collateral damage, as it is called) is \"not disproportionate.\" Think about an Allied decision to bomb a German tank factory in World War II. The factory is located in a working-class neighborhood\u2014not for the sake of civilian cover but because that is where factories were built before workers had cars. Given the aiming devices available in 1943, any attack on the factory would kill many civilians. But the death of many civilians, indeed, of almost any number of civilians, is \"not disproportionate\" to the value of stopping the production of tanks for the German war effort. This same argument can be made, and was regularly made, in defense of attacks on considerably less valuable targets\u2014which is the reason for my skepticism about the proportionality rule in this book.\n\nIn recent asymmetric conflicts, by contrast, the rule has been read restrictively. Asymmetry makes for micro-battles, small-scale engagements, and in these engagements, given the overwhelming power of the army and the weakness of the insurgents, even a very small number of civilian casualties seems disproportionate to the military value of the target. For the target may be nothing more than a couple of insurgents firing from the roof of a small apartment building (in Afghanistan) or a single rocket launching crew in the parking lot of a hospital (in Gaza).\n\nThe army argues that it has to respond to the gunfire from the roof and to the rockets from the parking lot, and the civilians it kills, even if their number looks disproportionate, are the moral responsibility of the insurgents who have chosen to fight from civilian cover. The insurgents blame the army's soldiers, who are indeed the actual killers; at the same time the insurgents commonly benefit from the civilian deaths in the court of public opinion. The weak are even more attractive when they are also victims, and it is probably true that insurgent fighters don't only hide behind civilians; they also deliberately expose civilians to attack. But the other side may also benefit from civilian deaths. The army aims, it says, only at military targets, but the collateral damage from its attacks may serve to deter future insurgent war making or discourage civilian support for the insurgents. So perhaps the army isn't as careful as it might be to avoid collateral damage.\n\nSorting all this out doesn't require us to abandon or significantly revise just war theory. It does require that the theory be applied, as the insurgents say, with close attention to the circumstances of asymmetry. Close attention is also critical attention: it doesn't necessarily support the insurgents' claims, nor does it allow just any military response from the army. But before addressing these battlefield issues, I need to consider, briefly, how the battles begin.\n\nThe army is, as it were, already in power, serving a sovereign state whose political policies and moral authority are contested, rightly or wrongly. So it's the insurgents who usually start the war. Claiming the prerogatives that I have just described, they often attack civilian targets, inviting a military response, the more savage the better. But how was the decision to attack reached? The insurgents are militarily weak, but they are not always or everywhere politically weak. They might have launched a political campaign against the policies and authority of the state\u2014through repeated demonstrations, civil disobedience, a mass march, a general strike\u2014and there probably were people in the insurgent organization who argued for this policy: politics before war. If war is the continuation of politics by other means, as Clausewitz said, then politics should come first; military force is supposed to be a \"last resort,\" but even if that's not right, the use of force shouldn't come before or instead of political action (see my Arguing about War, chapters 6\u20137, for an argument about the metaphysical character of \"lastness\"). We should look for a political struggle in advance of any decision to go to war. It seems to me that the justice of the war, assuming for now the justice of the cause, hangs on this previous struggle. When war is not the continuation but the replacement of politics, we should probably call it unjust.\n\nI am not going to say anything more about the causes for which insurgents fight; the recent history of asymmetric war includes a range of violent struggles\u2014from national liberation to religious crusade. Most just war theorists would call the first of these just and the second unjust; that is my own view. But what is critically at issue in asymmetric warfare is less the cause than the conduct of the war. The focus on conduct derives from the extreme danger to which civilians are subjected, first by the insurgents and then by the army. How civilians are treated is a matter of jus in bello, justice in war, but it certainly reflects on the character of the insurgents and of the army (and of the state for which the army fights). A just cause can be undone if it is pursued in unjust ways. Or, better, when we condemn the conduct of an asymmetric war, we are also arguing that the cause, if it is just, must be pursued differently\u2014by a different kind of war or by a return to politics.\n\nNow, let's say, the fighting has begun, and the first attacks are aimed indiscriminately at civilians\u2014with an argument attached: we can't do anything else. This is the standard argument in defense of terrorism, and it is hard to sustain, even if we set aside the political struggle that has just been pre-empted. Not every military target is impregnable to the insurgents. There are always vulnerable targets that can be attacked, if there is a will to attack them\u2014as anyone knows who has toured the sprawling facilities of a modern army. Further evidence for this claim comes from inside the insurgent organization itself, where there are always dissidents who oppose attacks on civilians and argue for the possibility of a purely military campaign. They testify both to the strength of the conventional rules of war and to the weakness of the argument from necessity. I see no reason to give up on those rules; they apply also, as we will see, to the other side.\n\nThe insurgents can in fact distinguish soldiers and civilians among their enemies, and we should condemn every refusal or failure to do that. Their stronger argument is that they can't help their enemies make the same distinction\u2014they can't separate themselves from their own civilians. Theirs, they say, is a popular insurgency; they fight from among the people because they are the people and the people are the insurgents. They aren't going to march off to some distant battlefield where they are certain to be overwhelmed. They will fight from where they live, or from where their people live, with whom they are one. Their human shields are volunteers, and if the insurgents benefit from the death of those shields, that is, from civilian deaths, they are not responsible for them. If the enemy army can't separate the insurgents from the people, it should give up, for there is no way that it can fight justly.\n\nThat is the primary argument of the insurgents, and it is much better than a common secondary argument: that they are fighting from such densely populated areas that they can't distance their fighters from residential neighborhoods. In the Gaza war of 2014, Israel claimed that some 15 percent of the rockets fired by Hamas were fired from residential neighborhoods and from schools, mosques, and hospitals within those neighborhoods. But if 85 percent of the rockets were fired from uninhabited areas, we have to conclude that the other firings were deliberately sited to provoke counterfire that would kill and injure civilians. Clearly the provocation worked. Population density is a problem for the army; I don't think it is an excuse for the insurgents. If the excuse isn't persuasive in Gaza, with one of the highest ratios of people to space in the world, then it must be far less persuasive in places like Vietnam, Afghanistan, and Iraq.\n\nBut the insurgents' claim that they are fighting from where they live, that they are at one with their people, is more plausible\u2014though the claim obviously has different truth value in different countries, at different times. The closer the insurgents are to their people, the stronger the argument for oneness, the more likely it is that they won't be blamed for not wearing uniforms (think of the Minutemen in the American War of Independence) or for fighting from homes and neighborhoods. But when they fight that way, they cannot blame the army for the civilians it kills in response\u2014so long as the soldiers live by the rules that still apply to them.\n\nMany conventional wars have been fought in crowded urban neighborhoods; we shouldn't expect insurgents to miss the opportunities offered by cities and also by towns and villages. There is, however, a difference here: a conventional army, fighting its way through a city, will probably kill civilians, but these killings do not benefit the opposing army. In asymmetric warfare, the army's killings definitely benefit the insurgents, who are therefore liable to the charge, which I have already made, that they deliberately expose civilians to enemy fire. But suppose they have the support of many of the men and women they expose, from whose homes and neighborhoods they choose to fight. Then, indeed, the army responding to insurgent attacks will face a difficult and highly charged moral and political decision: how to deal with \"the people,\" that is, with unarmed but possibly hostile civilians, who are indistinguishable from the insurgents.\n\nHere is what I take to be the central issue in asymmetric warfare (it's also an issue in conventional wars, but asymmetry gives it special significance): How should the army fight when its fighting puts \"enemy\" civilians at risk? \"Enemy\" is in scare quotes because, while some of these civilians may well sympathize with or actively support the insurgents (as the insurgents claim), some of them do not; some of them just wish they were somewhere else. And there are always the children, one-third or more of the population, who aren't anyone's enemies. So the insurgents are fighting from among a mixed group of civilians; the army's soldiers are attacking. How should the soldiers plan, organize, and carry out their attack? The key moral issue can be specified more clearly: What risks should the army ask its soldiers to take in order to reduce the risks they impose on \"enemy\" civilians?\n\nIn a sense, the question is unfair. It isn't the soldiers' fault that they aren't fighting on a conventional battlefield against a uniformed enemy. The moral difficulties of asymmetric warfare are imposed on the soldiers, not chosen by them. Nonetheless, they are well-armed and well-trained, and they are backed up by all the resources of a high-tech military force; the civilians they encounter are unarmed and untrained, highly vulnerable, with no backup at all. And it is the soldiers who, whatever their own intentions, put those civilians at risk. I argue in this book, with examples from World War II, Korea, and Vietnam, that soldiers have to accept some risk (I don't attempt to say how much) in order to protect civilians from their own deadly fire. The revised version of the double effect doctrine that I describe in chapter 9, which sets up this question, has been widely discussed and adopted by many just war theorists. But the specific issue of risk taking didn't get much attention until the wars in Afghanistan and Gaza.\n\nThe moral burden I place on soldiers is unfair in another way. The insurgents don't take any risks to avoid putting civilians at risk; instead, they often take risks or, in the case of suicide bombers, they choose death precisely in order to kill and injure civilians. Soldiers are supposedly fighting to protect those civilians, so why make their fight harder and more dangerous? Why ask them to take risks to avoid killing \"enemy\" civilians whom their enemies are deliberately putting at risk? There is a pragmatic answer to this question: asymmetric war is also a political struggle\u2014a battle for civilian support, that is, for \"hearts and minds.\" The U.S. Army's revised rules of engagement for soldiers in Afghanistan, announced in 2010, were aimed above all at reducing civilian casualties. The rationale was very simple, and it fits many other asymmetric wars: you don't win hearts and minds by destroying bodies. But in Israel's Gaza wars, winning the hearts and minds of the inhabitants of the Gaza Strip was probably not a plausible war aim (though Israel certainly would have benefited if Gazans came to believe that the Israeli army was more committed to civilian well-being than the Hamas militants were). In any case, a similar pragmatic argument applies: it is global rather than local hearts and minds that are at issue now, but in international society as it is today the loss is equally damaging\u2014or even more damaging.\n\nPragmatic arguments, however, only hold if they work\u2014in this case if a significant reduction of collateral damage brings with it local support or global sympathy. The moral argument holds whether or not this happens: these people should not be killed, and these soldiers have an obligation to do everything they can to avoid killing them. It is possible, as some critics claim, that the effective meaning of this obligation is that the soldiers won't be able to win asymmetric wars. Asymmetry describes a struggle between a very strong and a very weak military force: a high-tech army against a low-tech insurgency. The difference in firepower is huge. And yet, throughout history, the army rarely defeats the insurgents; even when it doesn't lose the war, it doesn't actually win. The U.S. Army could not defeat the Vietcong in Vietnam; nor could the United States along with its NATO allies defeat the Taliban in Afghanistan; nor were we able to defeat either the Sunni or the Shiite militias in Iraq. The Israelis did not lose the wars in Lebanon and Gaza, but they can't be said to have won. It is indeed possible to win\u2014by giving up on hearts and minds and on any semblance of moral decency and simply killing and killing until the insurgents' civilian cover is literally gone. The Russians in Chechnya and the government of Sri Lanka in its war against the Tamil rebels demonstrate that these wars can be won by a modern army. But can they be won by an army committed to just rules of engagement?\n\nThe answer to this question depends not only on the commitment and competence of the army but also on the moral\/political judgments of everyone else, locally and globally. So it is important to condemn military actions that break the rules, even if these actions don't reach to massive attacks on the civilian population. The army must not look to benefit from the collateral damage it inflicts in the course of legitimate attacks on military targets; it must take positive measures to limit civilian injury. And these positive measures include soldiers accepting risk, to some degree, in order to minimize the risks they impose on civilians. One form of risk taking has been fairly common in both America's and Israel's wars: warning of attacks to come, so that civilians can flee\u2014and insurgents can prepare, if that is possible, for the attack. It is a good thing to warn civilians of impending danger, but warnings are not sufficient. As American soldiers learned in Vietnam, many people don't leave: they are caring for elderly parents or sick children; they are afraid that their homes will be looted; they have no safe place to go.\n\nThere are, then, other things that must be done\u2014and not done. Consider my earlier example of Taliban insurgents firing at American soldiers from the roof of a small apartment building in an Afghan town. The soldiers don't know who is in the building. They could simply pull back and call in an air strike\u2014taking no risks themselves but radically endangering any civilians in the building. The 2010 rules of engagement rightly rule this out, so the soldiers have (in this simplified example) only three options: they can try to get someone into the building to see if civilians are living there, or they can try to get soldiers onto an adjacent roof, so that they can fire directly at the insurgents. Both these actions require the soldiers to accept (some degree of) risk. Or they can withdraw and wait for another encounter with the Taliban insurgents. No army likes to leave the battlefield to the enemy, but if the officer in the field thinks that the risks of the other options are too great, that is the right thing to do.\n\nBut what if there are civilians on the roof, dragged up there by force or willingly standing with the insurgents? And what if the army unit's withdrawal would leave other units, fighting nearby, at risk? Then, even if soldiers succeed in getting onto adjacent roofs, civilians will be killed or injured. Who is responsible for those deaths? If the numbers seem disproportionate to the military value of the building, as they probably will, given the restrictive understanding of proportionality, the army will be blamed. I believe, by contrast, that whenever soldiers have accepted risks in order to minimize the injuries they inflict on civilians, the blame should fall elsewhere. Calling in an air strike would probably be a war crime; killing civilians in a firefight in the circumstances I have just described is not.\n\nSince attacks from the air, from planes or drones, involve no risk for the attacking forces, it might be argued that our judgments is these cases can only be shaped by calculations of proportionality. That may be true, and then we will need to find our way to calculations that are neither too permissive nor too restrictive. But successful air attacks, aimed at legitimate targets, depend heavily on information from the ground, and the collection of information is a dangerous business. Too often, attacks have been launched without sufficient knowledge about the targets or with knowledge provided by unreliable informants, who are often pursuing private vendettas. Many civilians die in attacks of that sort, which should always be condemned. But when the intelligence work is seriously undertaken and its risks accepted, and when civilians are killed because they are being used as cover or deliberately exposed, the army can rightly claim that it has done the best it could in the circumstances of asymmetry.\n\nIt is important to get these judgments right because, as I've already said, asymmetric wars are also political struggles. The insurgents, the soldiers, and the endangered civilians are not the only people involved; all the rest of us are involved. In a sense, this is true also in conventional wars. The point of writing a book like Just and Unjust Wars is to facilitate the judgments that citizens have to make about the wars their countries fight. But in asymmetric warfare, the responsibility to judge the war, to join the arguments about how it is being fought, extends more widely. The world's judgments are important, and if the \"world\" gets things right, the war will probably end with justice done.\n\nThe insurgents should be condemned when they attack civilians and when they deliberately put civilians at risk; they should be praised and supported when they struggle, in the circumstances of asymmetry, to fight justly. The army should be condemned when it fails to do everything it can do, in the circumstances of asymmetry, to avoid killing civilians, and it should be praised and supported when it lives by the moral rules of engagement. Strong judgments of this sort will promote good endings, and I suspect that in the long run the anticipation of strong judgments will make asymmetric wars less likely.\n\nMichael Walzer\n\nPrinceton, New Jersey\n\nMay 2015\nPreface to the First Edition\n\nI did not begin by thinking about war in general, but about particular wars, above all about the American intervention in Vietnam. Nor did I begin as a philosopher, but as a political activist and a partisan. Certainly, political and moral philosophy ought to help us at those difficult times when we choose sides and make commitments. But it does so only indirectly. We are not usually philosophical in moments of crisis; most often, there is no time. War especially imposes an urgency that is probably incompatible with philosophy as a serious enterprise. The philosopher is like Wordsworth's poet who reflects in tranquility upon past experience (or other people's experience), thinking about political and moral choices already made. And yet these choices are made in philosophical terms, available because of previous reflection. It was, for example a matter of great importance to all of us in the American anti-war movement of the late 1960s and early 1970s that we found a moral doctrine ready at hand, a connected set of names and concepts that we all knew\u2014and that everyone else knew. Our anger and indignation were shaped by the words available to express them, and the words were at the tips of our tongues even though we had never before explored their meanings and connections. When we talked about aggression and neutrality, the rights of prisoners of war and civilians, atrocities and war crimes, we were drawing upon the work of many generations of men and women, most of whom we had never heard of. We would be better off if we did not need a vocabulary like that, but given that we need it, we must be grateful that we have it. Without this vocabulary, we could not have thought about the Vietnam War as we did, let alone have communicated our thoughts to other people.\n\nNo doubt we used the available words freely and often carelessly. Sometimes this was due to the excitement of the moment and the pressures of partisanship, but it also had a more serious cause. We suffered from an education which taught us that these words had no proper descriptive use and no objective meaning. Moral discourse was excluded from the world of science, even of social science. It expressed feelings, not perceptions, and there was no reason for the expression of feelings to be precise. Or rather, any precision it achieved had an entirely subjective reference: it was the domain of the poet and the literary critic. I don't need to rehearse this point of view (I shall criticize it in detail later on), though it's less prevalent now than it once was. What is crucial is that we disputed it, knowingly or unknowingly, every time we criticized American conduct in Vietnam. For our criticisms had the form at least of reports on the real world, not merely on the state of our own tempers. They required evidence; they pressed us, however trained we were in the loose use of moral language, toward analysis and investigation. Even the most skeptical among us came to see that these criticisms could be true (or false).\n\nIn those years of angry controversy, I promised myself that one day I would try to set out the moral argument about war in a quiet and reflective way. I still want to defend (most of) the particular arguments that underlay our opposition to the American war in Vietnam, but also and more importantly I want to defend the business of arguing, as we did and as most people do, in moral terms. Hence this book, which may be taken as an apology for our occasional carelessness and a vindication of our fundamental enterprise.\n\nNow, the language with which we argue about war and justice is similar to the language of international law. But this is not a book about the positive laws of war. There are many such books, and I have often drawn upon them. Legal treatises do not, however, provide a fully plausible or coherent account of our moral arguments, and the two most common approaches to the law reflected in the treatises are both in need of extra-legal supplement. First of all, legal positivism, which generated major scholarly works in the late nineteenth and early twentieth centuries, has become in the age of the United Nations increasingly uninteresting. The UN Charter was supposed to be the constitution of a new world, but, for reasons that have often been discussed, things have turned out differently. To dwell at length upon the precise meaning of the Charter is today a kind of utopian quibbling. And because the UN sometimes pretends that it already is what it has barely begun to be, its decrees do not command intellectual or moral respect\u2014except among the positivist lawyers whose business it is to interpret them. The lawyers have constructed a paper world, which fails at crucial points to correspond to the world the rest of us still live in.\n\nThe second approach to the law is oriented in terms of policy goals. Its advocates respond to the poverty of the contemporary international regime by imputing purposes to that regime\u2014the achievement of some sort of \"world order\"\u2014and then reinterpreting the law to fit those purposes. In effect, they substitute utilitarian argument for legal analysis. That substitution is certainly not uninteresting, but it requires a philosophical defense. For the customs and conventions, the treaties and charters that constitute the laws of international society do not invite interpretation in terms of a single purpose or set of purposes. Nor are the judgments they require always explicable from a utilitarian standpoint. Policy-oriented lawyers are in fact moral and political philosophers, and it would be best if they presented themselves that way. Or, alternatively, they are would-be legislators, not jurists or students of the law. They are committed, or most of them are committed, to restructuring international society\u2014a worthwhile task\u2014but they are not committed to expounding its present structure.\n\nMy own task is different. I want to account for the ways in which men and women who are not lawyers but simple citizens (and sometimes soldiers) argue about war, and to expound the terms we commonly use. I am concerned precisely with the present structure of the moral world. My starting point is the fact that we do argue, often to different purposes, to be sure, but in mutually comprehensible fashion: else there would be no point in arguing. We justify our conduct; we judge the conduct of others. Though these justifications and judgments cannot be studied like the records of a criminal court, they are nevertheless a legitimate subject of study. Upon examination they reveal, I believe, a comprehensive view of war as a human activity and a more or less systematic moral doctrine, which sometimes, but not always, overlaps with established legal doctrine.\n\nIn fact, the vocabulary overlaps more than the arguments do. Hence I must say something about my own use of language. I shall always refer to laws of international society (as these appear in legal handbooks and military manuals) as positive laws. For the rest, when I talk of law, I am referring to the moral law, to those general principles that we commonly acknowledge, even when we can't or won't live up to them. When I talk of the rules of war, I am referring to the more particular code that governs our judgments of combat behavior, and that is only partially articulated in the Hague and Geneva conventions. And when I talk of crimes, I am describing violations of the general principles or of the particular code: so men and women can be called criminals even when they cannot be charged before a legal tribunal. Since positive international law is radically incomplete, it is always possible to interpret it in the light of moral principles and to refer to the results as \"positive law.\" Perhaps that is what has to be done in order to flesh out the legal system and render it more attractive than it presently is. But it is not what I have done here. Throughout the book, I treat words like aggression, neutrality, surrender, civilian, reprisal, and so on, as if they were terms in a moral vocabulary\u2014which they are, and always have been, though most recently their analysis and refinement have been almost entirely the work of lawyers.\n\nI want to recapture the just war for political and moral theory. My own work, then, looks back to that religious tradition within which Western politics and morality were first given shape, to the books of writers like Maimonides, Aquinas, Vitoria, and Suarez\u2014and then to the books of writers like Hugo Grotius, who took over the tradition and began to work it into secular form. But I have not attempted a history of just war theory, and I quote the classical texts only occasionally, for the sake of some particularly illuminating or forceful argument. I refer more often to contemporary philosophers and theologians (and soldiers and statesmen), for my main concern is not with the making of the moral world but with its present character.\n\nPerhaps the most problematic feature of my exposition is the use of the plural pronouns: we, our, ourselves, us. I have already demonstrated the ambiguity of those words by using them in two ways: to describe that group of Americans who condemned the Vietnam War, and to describe that much larger group who understood the condemnation (whether or not they agreed with it). I shall limit myself henceforth to the larger group. That its members share a common morality is the critical assumption of this book. In my first chapter I try to make a case for that assumption. But it's only a case, it's not conclusive. Someone can always ask, \"What is this morality of yours?\" That is a more radical question, however, than the questioner may realize, for it excludes him not only from the comfortable world of moral agreement, but also from the wider world of agreement and disagreement, justification and criticism. The moral world of war is shared not because we arrive at the same conclusions as to whose fight is just and whose unjust, but because we acknowledge the same difficulties on the way to our conclusions, face the same problems, talk the same language. It's not easy to opt out, and only the wicked and the simple make the attempt.\n\nI am not going to expound morality from the ground up. Were I to begin with the foundations, I would probably never get beyond them; in any case, I am by no means sure what the foundations are. The substructure of the ethical world is a matter of deep and apparently unending controversy. Meanwhile, however, we are living in the superstructure. The building is large, its construction elaborate and confusing. But here I can offer some guidance: a tour of the rooms, so to speak, a discussion of architectural principles. This is a book of practical morality. The study of judgments and justifications in the real world moves us closer, perhaps, to the most profound questions of moral philosophy, but it does not require a direct engagement with those questions. Indeed, philosophers who seek such an engagement often miss the immediacies of political and moral controversy and provide little help to men and women faced with hard choices. For the moment, at least, practical morality is detached from its foundations, and we must act as if that separation were a possible (since it is an actual) condition of moral life.\n\nBut that's not to suggest that we can do nothing more than describe the judgments and justifications that people commonly put forward. We can analyze these moral claims, seek out their coherence, lay bare the principles that they exemplify. We can reveal commitments that go deeper than partisan allegiance and the urgencies of battle; for it is a matter of evidence, not a pious wish, that there are such commitments. And then we can expose the hypocrisy of soldiers and statesmen who publicly acknowledge these commitments while seeking in fact only their own advantage. The exposure of hypocrisy is certainly the most ordinary, and it may also be the most important, form of moral criticism. We are rarely called upon to invent new ethical principles; if we did that, our criticism would not be comprehensible to the people whose behavior we wanted to condemn. Rather, we hold such people to their own principles, though we may draw these out and arrange them in ways they had not thought of before.\n\nThere is a particular arrangement, a particular view of the moral world, that seems to me the best one. I want to suggest that the arguments we make about war are most fully understood (though other understandings are possible) as efforts to recognize and respect the rights of individual and associated men and women. The morality I shall expound is in its philosophical form a doctrine of human rights, though I shall say nothing here of the ideas of personality, action, and intention that this doctrine probably presupposes. Considerations of utility play into the structure at many points, but they cannot account for it as a whole. Their part is subsidiary to that of rights; it is constrained by rights. That is above all true of the classical forms of military maximization: the religious crusade, the proletarian revolution, the \"war to end war.\" But it's true also, as I will try to show, of the more immediate pressures of \"military necessity.\" At every point, the judgments we make (the lies we tell) are best accounted for if we regard life and liberty as something like absolute values and then try to understand the moral and political processes through which these values are challenged and defended.\n\nThe proper method of practical morality is casuistic in character. Since I am concerned with actual judgments and justifications, I shall turn regularly to historical cases. My argument moves through the cases, and I have often foregone a systematic presentation for the sake of the nuances and details of historical reality. At the same time, the cases are necessarily sketched in outline form. In order to make them exemplary, I have had to abridge their ambiguities. In doing that, I have tried to be accurate and fair, but the cases are often controversial and no doubt I have sometimes failed. Readers upset by my failures might usefully treat the cases as if they were hypothetical\u2014invented rather than researched\u2014though it is important to my own sense of my enterprise that I am reporting on experiences that men and women have really had and on arguments that they have really made. In choosing experiences and arguments for discussion, I have relied heavily on World War II in Europe, the first war of which I have memories and the paradigm, for me, of a justified struggle. For the rest, I have tried to pick out the obvious cases: those that have figured largely in the literature of war and those that play a part in contemporary controversies.\n\nThe structure of the book is explained in the second and third chapters, which introduce the main argument. Here I only want to say that my presentation of the moral theory of war is focused on the tensions within the theory that make it problematic and that make choice in wartime difficult and painful. The tensions are summed up in the dilemma of winning and fighting well. This is the military form of the means\/ends problem, the central issue in political ethics. I address it directly, and resolve or fail to resolve it, in Part Four; and the resolution, if it works, must be relevant also to the choices faced in politics generally. For war is the hardest place: if comprehensive and consistent moral judgments are possible there, they are possible everywhere.\n\nCambridge, Massachusetts, 1977\nAcknowledgments\n\nIn writing about war, I have had the support of many allies, institutional and personal. I began my research during the academic year 1971\u201372, while working at the Center for Advanced Study in the Behavioral Sciences in Stanford, California. I wrote a version of the preface and of chapter 1 at Mishkenot Sha'ananim (Peaceful Habitations) in Jerusalem, Israel, in the summer of 1974\u2014a visit made possible by the Jerusalem Foundation; the bulk of the book was completed in 1975\u201376, while I was a Guggenheim Fellow.\n\nFor almost a decade, I went to school with the members of the Society for Ethical and Legal Philosophy, and while none of them are responsible for any of the arguments in this book, they have collectively had a great deal to do with the writing of it. I am especially grateful to Judith Jarvis Thompson, who read the entire manuscript and made many valuable suggestions. With Robert Nozick I have quarreled amicably about some of the hardest issues in the theory of war and his arguments, hypothetical cases, queries, and proposals helped me shape my own presentation.\n\nMy friend and colleague Robert Amdur read most of the chapters and he often forced me to think about them again. Marvin Kohl and Judith Walzer read portions of the manuscript; their comments on matters of style and substance have often been incorporated into my pages. I am grateful also to Philip Green, Yehuda Melzer, Miles Morgan, and John Schrecker.\n\nDuring a quarter at Stanford University and for several years at Harvard, I taught a course on the just war, and learned while I was teaching\u2014from colleagues and students alike. I will always be glad of the cooling skepticism of Stanley Hoffmann and Judith Shklar. I also benefited from the comments and criticisms of Charles Bahmueller, Donald Goldstein, Miles Kahler, Sanford Levinson, Dan Little, Gerald McElroy, and David Pollack.\n\nMartin Kessler of Basic Books conceived this book almost before I did, and assisted and encouraged me at every stage of the writing of it.\n\nWhen I was almost finished, Betty Butterfield undertook to type the final draft and set an astonishing pace, both for herself and for me; without her, the completion of the book would have taken much longer than it did.\n\nAn early version of chapter 12, on terrorism, appeared in The New Republic in 1975. In chapters 4 and 16 I have drawn upon arguments first developed in Philosophy and Public Affairs in 1972. In chapters 14 and 15, I have used portions of an article published in 1974 in the Israeli philosophical quarterly Iyyun. I am grateful to the editors of the three journals for permission to reprint these materials.\n\nI am grateful to the various publishers who have kindly permitted me to reprint material which first appeared under their auspices:\n\nRolf Hochhuth, \"Little London Theater of the World\/Garden,\" lines 38\u201340, in Soldiers: An Obituary for Geneva. Copyright \u00a9 1968 by Grove Press, Inc. Reprinted by permission of Grove Press, Inc.\n\nRandall Jarrell, \"The Death of the Ball Turret Gunner,\" line 1, copyright \u00a9 1945 by Randall Jarrell. Renewed copyright \u00a9 1972 by Mrs. Randall Jarrell; and \"The Range in the Desert,\" lines 21\u201324, copyright \u00a9 1947 by Randall Jarrell. Renewed copyright \u00a9 1974 by Mrs. Randall Jarrell. Both appeared in The Complete Poems. Reprinted by permission of Farrar, Straus & Giroux, Inc.\n\nStanley Kunitz, \"Foreign Affairs,\" lines 10\u201317, in Selected Poems. 1928\u20131958. Copyright \u00a9 1958 by Stanley Kunitz. This poem originally appeared in The New Yorker. Reprinted by permission of Little, Brown and Company in association with the Atlantic Monthly Press.\n\nWilfred Owen, \"Anthem for Doomed Youth,\" line 1, and \"A Terre,\" line 6, in The Collected Poems of Wilfred Owen, edited by C. Day Lewis. Reprinted by permission of the Owen Estate and Chatto and Windus Ltd. and New Directions Publishing Corporation.\n\nGillo Pontecoro, The Battle of Algiers, edited and with an introduction by PierNico Solinas. Scene 68, pages 79\u201380. Reprinted by permission of Charles Scribner's Sons.\n\nLouis Simpson, \"The Ash and the Oak\" in Good News of Death and Other Poems. Poets of Today II. Copyright \u00a9 1955 by Louis Simpson. Reprinted by permission of Charles Scribner's Sons.\nPart One\n\nThe Moral Reality of War\n\nAgainst \"Realism\"\n\nFor as long as men and women have talked about war, they have talked about it in terms of right and wrong. And for almost as long, some among them have derided such talk, called it a charade, insisted that war lies beyond (or beneath) moral judgment. War is a world apart, where life itself is at stake, where human nature is reduced to its elemental forms, where self-interest and necessity prevail. Here men and women do what they must to save themselves and their communities, and morality and law have no place. Inter arma silent leges: in time of war the law is silent.\n\nSometimes this silence is extended to other forms of competitive activity, as in the popular proverb, \"All's fair in love and war.\" That means that anything goes\u2014any kind of deceit in love, any kind of violence in war. We can neither praise nor blame; there is nothing to say. And yet we are rarely silent. The language we use to talk about love and war is so rich with moral meaning that it could hardly have been developed except through centuries of argument. Faithfulness, devotion, chastity, shame, adultery, seduction, betrayal; aggression, self-defense, appeasement, cruelty, ruthlessness, atrocity, massacre\u2014all these words are judgments, and judging is as common a human activity as loving or fighting.\n\nIt is true, however, that we often lack the courage of our judgments, and especially so in the case of military conflict. The moral posture of mankind is not well represented by that popular proverb about love and war. We would do better to mark a contrast rather than a similarity: before Venus, censorious; before Mars, timid. Not that we don't justify or condemn particular attacks, but we do so hesitantly and uncertainly (or loudly and recklessly), as if we were not sure that our judgments reach to the reality of war.\n\nThe Realist Argument\n\nRealism is the issue. The defenders of silent leges claim to have discovered an awful truth: what we conventionally call inhumanity is simply humanity under pressure. War strips away our civilized adornments and reveals our nakedness. They describe that nakedness for us, not without a certain relish: fearful, self-concerned, driven, murderous. They aren't wrong in any simple sense. The words are sometimes descriptive. Paradoxically, the description is often a kind of apology: yes, our soldiers committed atrocities in the course of the battle, but that's what war does to people, that's what war is like. The proverb, all's fair, is invoked in defense of conduct that appears to be unfair. And one urges silence on the law when one is engaged in activities that would otherwise be called unlawful. So there are arguments here that will enter into my own argument: justifications and excuses, references to necessity and duress, that we can recognize as forms of moral discourse and that have or don't have force in particular cases. But there is also a general account of war as a realm of necessity and duress, the purpose of which is to make discourse about particular cases appear to be idle chatter, a mask of noise with which we conceal, even from ourselves, the awful truth. It is that general account that I have to challenge before I can begin my own work, and I want to challenge it at its source and in its most compelling form, as it is put forward by the historian Thucydides and the philosopher Thomas Hobbes. These two men, separated by 2,000 years, are collaborators of a kind, for Hobbes translated Thucydides' History of the Peloponnesian War and then generalized its argument in his own Leviathan. It is not my purpose here to write a full philosophical response to Thucydides and Hobbes. I wish only to suggest, first by argument and then by example, that the judgment of war and of wartime conduct is a serious enterprise.\n\nThe Melian Dialogue\n\nThe dialogue between the Athenian generals Cleomedes and Tisias and the magistrates of the island state of Melos is one of the high points of Thucydides' History and the climax of his realism. Melos was a Spartan colony, and its people had \"therefore refused to be subject, as the rest of the islands were, unto the Athenians; but rested at first neutral; and afterwards, when the Athenians put them to it by wasting of their lands, they entered into open war.\" This is a classic account of aggression, for to commit aggression is simply to \"put people to it\" as Thucydides describes. But such a description, he seems to say, is merely external; he wants to show us the inner meaning of war. His spokesmen are the two Athenian generals, who demand a parley and then speak as generals have rarely done in military history. Let us have no fine words about justice, they say. We for our part will not pretend that, having defeated the Persians, our empire is deserved; you must not claim that having done no injury to the Athenian people, you have a right to be let alone. We will talk instead of what is feasible and what is necessary. For this is what war is really like: \"they that have odds of power exact as much as they can, and the weak yield to such conditions as they can get.\"\n\nIt is not only the Melians here who bear the burdens of necessity. The Athenians are driven, too; they must expand their empire, Cleomedes and Tisias believe, or lose what they already have. The neutrality of Melos \"will be an argument of our weakness, and your hatred of our power, among those we have rule over.\" It will inspire rebellion throughout the islands, wherever men and women are \"offended with the necessity of subjection\"\u2014and what subject is not offended, eager for freedom, resentful of his conquerors? When the Athenian generals say that men \"will everywhere reign over such as they be too strong for,\" they are not only describing the desire for glory and command, but also the more narrow necessity of inter-state politics: reign or be subject. If they do not conquer when they can, they only reveal weakness and invite attack; and so, \"by a necessity of nature\" (a phrase Hobbes later made his own), they conquer when they can.\n\nThe Melians, on the other hand, are too weak to conquer. They face a harsher necessity: yield or be destroyed. \"For you have not in hand a match of valor upon equal terms . . . but rather a consultation upon your safety. . . .\" The rulers of Melos, however, value freedom above safety: \"If you then to retain your command, and your vassals to get loose from you, will undergo the utmost danger: would it not in us, that be already free, be great baseness and cowardice, if we should not encounter anything whatsoever rather than suffer ourselves to be brought into bondage?\" Though they know that it will be a \"hard matter\" to stand against the power and fortune of Athens, \"nevertheless we believe that, for fortune, we shall be nothing inferior, as having the gods on our side, because we stand innocent against men unjust.\" And as for power, they hope for assistance from the Spartans, \"who are of necessity obliged, if for no other cause, yet for consanguinity's sake and for their own honor to defend us.\" But the gods, too, reign where they can, reply the Athenian generals, and consanguinity and honor have nothing to do with necessity. The Spartans will (necessarily) think only of themselves: \"most apparently of all men, they hold for honorable that which pleaseth and for just that which profiteth.\"\n\nSo the argument ended. The magistrates refused to surrender; the Athenians laid siege to their city; the Spartans sent no help. Finally, after some months of fighting, in the winter of 416 B.C., Melos was betrayed by several of its citizens. When further resistance seemed impossible, the Melians \"yielded themselves to the discretion of the Athenians: who slew all the men of military age, made slaves of the women and children; and inhabited the place with a colony sent thither afterwards of 500 men of their own.\"\n\nThe dialogue between the generals and the magistrates is a literary and philosophical construction of Thucydides. The magistrates speak as they well might have done, but their conventional piety and heroism is only a foil to what the classical critic Dionysius calls the \"depraved shrewdness\" of the Athenian generals. It is the generals who have often seemed unbelievable. Their words, writes Dionysius, \"were appropriate to oriental monarchs . . . but unfit to be spoken by Athenians. . . .\"a Perhaps Thucydides means us to notice the unfitness, not so much of the words but of the policies they were used to defend, and thinks we might have missed it had he permitted the generals to speak as they probably in fact spoke, weaving \"fair pretenses\" over their vile actions. We are to understand that Athens is no longer itself. Cleomedes and Tisias do not represent that noble people who fought the Persians in the name of freedom and whose politics and culture, as Dionysius says, \"exercised such a humanizing influence on everyday life.\" They represent instead the imperial decadence of the city state. It is not that they are war criminals in the modern sense; that idea is alien to Thucydides. But they embody a certain loss of ethical balance, of restraint and moderation. Their statesmanship is flawed, and their \"realistic\" speeches provide an ironic contrast to the blindness and arrogance with which the Athenians only a few months later launched the disastrous expedition to Sicily. The History, on this view, is a tragedy and Athens itself the tragic hero. Thucydides has given us a morality play in the Greek style. We can glimpse his meaning in Euripides' The Trojan Women, written in the immediate aftermath of the conquest of Melos and undoubtedly intended to suggest the human significance of slaughter and slavery\u2014and to predict a divine retribution:\n\nHow ye are blind\n\nYe treaders down of cities, ye that cast\n\nTemples to desolation, and lay waste\n\nTombs, the untrodden sanctuaries where lie\n\nThe ancient dead; yourselves so soon to die!\n\nBut Thucydides seems in fact to be making a rather different, and a more secular, statement than this quotation suggests, and not about Athens so much as about war itself. He probably did not mean the harshness of the Athenian generals to be taken as a sign of depravity, but rather as a sign of impatience, toughmindedness, honesty\u2014qualities of mind not inappropriate in military commanders. He is arguing, as Werner Jaeger has said, that \"the principle of force forms a realm of its own, with laws of its own,\" distinct and separate from the laws of moral life. This is certainly the way Hobbes read Thucydides, and it is the reading with which we must come to grips. For if the realm of force is indeed distinct and if this is an accurate account of its laws, then one could no more criticize the Athenians for their wartime policies than one could criticize a stone for falling downwards. The slaughter of the Melians is explained by reference to the circumstances of war and the necessities of nature; and again, there is nothing to say. Or rather, one can say anything, call necessity cruel and war hellish; but while these statements may be true in their own terms, they do not touch the political realities of the case or help us understand the Athenian decision.\n\nIt is important to stress, however, that Thucydides has told us nothing at all about the Athenian decision. And if we place ourselves, not in the council room at Melos where a cruel policy was being expounded, but in the assembly at Athens where that policy was first adopted, the argument of the generals has a very different ring. In the Greek as in the English language, the word necessity \"doubles the parts of indispensable and inevitable.\" At Melos, Cleomedes and Tisias mixed the two of these, stressing the last. In the assembly they could have argued only about the first, claiming, I suppose, that the destruction of Melos was necessary (indispensable) for the preservation of the empire. But this claim is rhetorical in two senses. First, it evades the moral question of whether the preservation of the empire was itself necessary. There were some Athenians, at least, who had doubts about that, and more who doubted that the empire had to be a uniform system of domination and subjection (as the policy adopted for Melos suggested). Secondly, it exaggerates the knowledge and foresight of the generals. They are not saying with certainty that Athens will fall unless Melos is destroyed; their argument has to do with probabilities and risks. And such arguments are always arguable. Would the destruction of Melos really reduce Athenian risks? Are there alternative policies? What are the likely costs of this one? Would it be right? What would other people think of Athens if it were carried out?\n\nOnce the debate begins, all sorts of moral and strategic questions are likely to come up. And for the participants in the debate, the outcome is not going to be determined \"by a necessity of nature,\" but by the opinions they hold or come to hold as a result of the arguments they hear and then by the decisions they freely make, individually and collectively. Afterwards, the generals claim that a certain decision was inevitable; and that, presumably, is what Thucydides wants us to believe. But the claim can only be made afterwards, for inevitability here is mediated by a process of political deliberation, and Thucydides could not know what was inevitable until that process had been completed. Judgments of necessity in this sense are always retrospective in character\u2014the work of historians, not historical actors.\n\nNow, the moral point of view derives its legitimacy from the perspective of the actor. When we make moral judgments, we try to recapture that perspective. We reiterate the decision-making process, or we rehearse our own future decisions, asking what we would have done (or what we would do) in similar circumstances. The Athenian generals recognize the importance of such questions, for they defend their policy certain \"that you likewise, and others that should have the same power which we have, would do the same.\" But that is a dubious knowledge, especially so once we realize that the \"Melian decree\" was sharply opposed in the Athenian assembly. Our standpoint is that of citizens debating the decree. What should we do?\n\nWe have no account of the Athenian decision to attack Melos or of the decision (which may have been taken at the same time) to kill and enslave its people. Plutarch claims that it was Alcibiades, chief architect of the Sicilian expedition, who was \"the principal cause of the slaughter . . . having spoken in favor of the decree.\" He played the part of Cleon in the debate that Thucydides does record, that occurred some years earlier, over the fate of Mytilene. It is worth glancing back at that earlier argument. Mytilene had been an ally of Athens from the time of the Persian War; it was never a subject city in any formal way, but bound by treaty to the Athenian cause. In 428, it rebelled and formed an alliance with the Spartans. After considerable fighting, the city was captured by Athenian forces, and the assembly determined \"to put to death . . . all the men of Mytilene that were of age, and to make slaves of the women and children: laying to their charge the revolt itself, in that they revolted not being in subjection as others were. . . .\" But the following day the citizens \"felt a kind of repentance . . . and began to consider what a great and cruel decree it was, that not the authors only, but that the whole city should be destroyed.\" It is this second debate that Thucydides has recorded, or some part of it, giving us two speeches, that of Cleon upholding the original decree and that of Diodotus urging its revocation. Cleon argues largely in terms of collective guilt and retributive justice; Diodotus offers a critique of the deterrent effects of capital punishment. The assembly accepts Diodotus' position, convinced apparently that the destruction of Mytilene would not uphold the force of treaties or ensure the stability of the empire. It is the appeal to interest that triumphs\u2014as has often been pointed out\u2014though it should be remembered that the occasion for the appeal was the repentance of the citizens. Moral anxiety, not political calculation, leads them to worry about the effectiveness of their decree.\n\nIn the debate over Melos, the positions must have been reversed. Now there was no retributivist argument to make, for the Melians had done Athens no injury. Alcibiades probably talked like Thucydides' generals, though with the all-important difference I have already noted. When he told his fellow citizens that the decree was necessary, he didn't mean that it was ordained by the laws that govern the realm of force; he meant merely that it was needed (in his view) to reduce the risks of rebellion among the subject cities of the Athenian empire. And his opponents probably argued, like the Melians, that the decree was dishonorable and unjust and would more likely excite resentment than fear throughout the islands, that Melos did not threaten Athens in any way, and that other policies would serve Athenian interests and Athenian self-esteem. Perhaps they also reminded the citizens of their repentance in the case of Mytilene and urged them once again to avoid the cruelty of massacre and enslavement. How Alcibiades won out, and how close the vote was, we don't know. But there is no reason to think that the decision was predetermined and debate of no avail: no more with Melos than with Mytilene. Stand in imagination in the Athenian assembly, and one can still feel a sense of freedom.\n\nBut the realism of the Athenian generals has a further thrust. It is not only a denial of the freedom that makes moral decision possible; it is a denial also of the meaningfulness of moral argument. The second claim is closely related to the first. If we must act in accordance with our interests, driven by our fears of one another, then talk about justice cannot possibly be anything more than talk. It refers to no purposes that we can make our own and to no goals that we can share with others. That is why the Athenian generals could have woven \"fair pretenses\" as easily as the Melian magistrates; in discourse of this sort anything can be said. The words have no clear references, no certain definitions, no logical entailments. They are, as Hobbes writes in Leviathan, \"ever used with relation to the person that useth them,\" and they express that person's appetites and fears and nothing else. It is only \"most apparent\" in the Spartans, but true for everyone, that \"they hold for honorable that which pleaseth them and for just that which profiteth.\" Or, as Hobbes later explained, the names of the virtues and vices are of \"uncertain signification.\"\n\nFor one calleth wisdom, what another calleth fear; and one cruelty what another justice; one prodigality, what another magnanimity . . . etc. And therefore such names can never be true grounds of any ratiocination.\n\n\"Never\"\u2014until the sovereign, who is also the supreme linguistic authority, fixes the meaning of the moral vocabulary; but in the state of war, \"never\" without qualification, because in that state, by definition, no sovereign rules. In fact, even in civil society, the sovereign does not entirely succeed in bringing certainty into the world of virtue and vice. Hence moral discourse is always suspect, and war is only an extreme case of the anarchy of moral meanings. It is generally true, but especially so in time of violent conflict, that we can understand what other people are saying only if we see through their \"fair pretenses\" and translate moral talk into the harder currency of interest talk. When the Melians insist that their cause is just, they are saying only that they don't want to be subject; and had the generals claimed that Athens deserved its empire, they would simply have been expressing the lust for conquest or the fear of overthrow.\n\nThis is a powerful argument because it plays upon the common experience of moral disagreement\u2014painful, sustained, exasperating, and endless. For all its realism, however, it fails to get at the realities of that experience or to explain its character. We can see this clearly, I think, if we look again at the argument over the Mytilene decree. Hobbes may well have had this debate in mind when he wrote, \"and one [calleth] cruelty what another justice. . . .\" The Athenians repented of their cruelty, writes Thucydides, while Cleon told them that they had not been cruel at all but justly severe. Yet this was in no sense a disagreement over the meaning of words. Had there been no common meanings, there could have been no debate at all. The cruelty of the Athenians consisted in seeking to punish not only the authors of the rebellion but others as well, and Cleon agreed that that would indeed be cruel. He then went on to argue, as he had to do given his position, that in Mytilene there were no \"others.\" \"Let not the fault be laid upon a few, and the people absolved. For they have all alike taken arms against us. . . .\"\n\nI cannot pursue the argument further, since Thucydides doesn't, but there is an obvious rejoinder to Cleon, having to do with the status of the women and children of Mytilene. This might involve the deployment of additional moral terms (innocence, for example); but it would not hang\u2014any more than the argument about cruelty and justice hangs\u2014on idiosyncratic definitions. In fact, definitions are not at issue here, but descriptions and interpretations. The Athenians shared a moral vocabulary, shared it with the people of Mytilene and Melos; and allowing for cultural differences, they share it with us too. They had no difficulty, and we have none, in understanding the claim of the Melian magistrates that the invasion of their island was unjust. It is in applying the agreed-upon words to actual cases that we come to disagree. These disagreements are in part generated and always compounded by antagonistic interests and mutual fears. But they have other causes, too, which help to explain the complex and disparate ways in which men and women (even when they have similar interests and no reason to fear one another) position themselves in the moral world. There are, first of all, serious difficulties of perception and information (in war and politics generally), and so controversies arise over \"the facts of the case.\" There are sharp disparities in the weight we attach even to values we share, as there are in the actions we are ready to condone when these values are threatened. There are conflicting commitments and obligations that force us into violent antagonism even when we see the point of one another's positions. All this is real enough, and common enough: it makes morality into a world of good-faith quarrels as well as a world of ideology and verbal manipulation.\n\nIn any case, the possibilities for manipulation are limited. Whether or not people speak in good faith, they cannot say just anything they please. Moral talk is coercive; one thing leads to another. Perhaps that's why the Athenian generals did not want to begin. A war called unjust is not, to paraphrase Hobbes, a war misliked; it is a war misliked for particular reasons, and anyone making the charge is required to provide particular sorts of evidence. Similarly, if I claim that I am fighting justly, I must also claim that I was attacked (\"put to it,\" as the Melians were), or threatened with attack, or that I am coming to the aid of a victim of someone else's attack. And each of these claims has its own entailments, leading me deeper and deeper into a world of discourse where, though I can go on talking indefinitely, I am severely constrained in what I can say. I must say this or that, and at many points in a long argument this or that will be true or false. We don't have to translate moral talk into interest talk in order to understand it; morality refers in its own way to the real world.\n\nLet us consider a Hobbist example. In chapter XXI of Leviathan, Hobbes urges that we make allowance for the \"natural timorousness\" of mankind. \"When armies fight, there is on one side, or both, a running away; yet when they do it not out of treachery, but fear, they are not esteemed to do it unjustly, but dishonorably.\" Now, judgments are called for here: we are to distinguish cowards from traitors. If these are words of \"inconstant signification,\" the task is impossible and absurd. Every traitor would plead natural timorousness, and we would accept the plea or not depending on whether the soldier was a friend or an enemy, an obstacle to our advancement or an ally and supporter. I suppose we sometimes do behave that way, but it is not the case (nor does Hobbes, when it comes to cases, suppose that it is) that the judgments we make can only be understood in these terms. When we charge a man with treason, we have to tell a very special kind of story about him, and we have to provide concrete evidence that the story is true. If we call him a traitor when we cannot tell that story, we are not using words inconstantly, we are simply lying.\n\nStrategy and Morality\n\nMorality and justice are talked about in much the same way as military strategy. Strategy is the other language of war, and while it is commonly said to be free from the difficulties of moral discourse, its use is equally problematic. Though generals agree on the meaning of strategic terms\u2014entrapment, retreat, flanking maneuver, concentration of forces, and so on\u2014they nevertheless disagree about strategically appropriate courses of action. They argue about what ought to be done. After the battle, they disagree about what happened, and if they were defeated, they argue about who was to blame. Strategy, like morality, is a language of justification.b Every confused and cowardly commander describes his hesitations and panics as part of an elaborate plan; the strategic vocabulary is as available to him as it is to a competent commander. But that is not to say that its terms are meaningless. It would be a great triumph for the incompetent if they were, for we would then have no way to talk about incompetence. No doubt, \"one calleth retreat what another calleth strategic deployment. . . .\" But we do know the difference between these two, and though the facts of the case may be difficult to collect and interpret, we are nevertheless able to make critical judgments.\n\nSimilarly, we can make moral judgments: moral concepts and strategic concepts reflect the real world in the same way. They are not merely normative terms, telling soldiers (who often don't listen) what to do. They are descriptive terms, and without them we would have no coherent way of talking about war. Here are soldiers moving away from the scene of a battle, marching over the same ground they marched over yesterday, but fewer now, less eager, many without weapons, many wounded: we call this a retreat. Here are soldiers lining up the inhabitants of a peasant village, men, women, and children, and shooting them down: we call this a massacre.\n\nIt is only when their substantive content is fairly clear that moral and strategic terms can be used imperatively, and the wisdom they embody expressed in the form of rules. Never refuse quarter to a soldier trying to surrender. Never advance with your flanks unprotected. One might construct out of such commands a moral or a strategic war plan, and then it would be important to notice whether or not the actual conduct of the war conformed to the plan. We can assume that it would not. War is recalcitrant to this sort of theoretical control\u2014a quality it shares with every other human activity, but which it seems to possess to an especially intense degree. In The Charterhouse of Parma, Stendhal provides a description of the battle of Waterloo that is intended to mock the very idea of a strategic plan. It is an account of combat as chaos, therefore not an account at all but a denial, so to speak, that combat is accountable. It should be read alongside some strategic analysis of Waterloo like that of Major General Fuller, who views the battle as an organized series of maneuvers and counter-\u00admaneuvers. The strategist is not unaware of confusion and disorder in the field; nor is he entirely unwilling to see these as aspects of war itself, the natural effects of the stress of battle. But he sees them also as matters of command responsibility, failures of discipline or control. He suggests that strategic imperatives have been ignored; he looks for lessons to be learned.\n\nThe moral theorist is in the same position. He too must come to grips with the fact that his rules are often violated or ignored\u2014and with the deeper realization that, to men at war, the rules often don't seem relevant to the extremity of their situation. But however he does this, he does not surrender his sense of war as a human action, purposive and premeditated, for whose effects someone is responsible. Confronted with the many crimes committed in the course of a war, or with the crime of aggressive war itself, he searches for human agents. Nor is he alone in this search. It is one of the most important features of war, distinguishing it from the other scourges of mankind, that the men and women caught up in it are not only victims, they are also participants. All of us are inclined to hold them responsible for what they do (though we may recognize the plea of duress in particular cases). Reiterated over time, our arguments and judgments shape what I want to call the moral reality of war\u2014that is, all those experiences of which moral language is descriptive or within which it is necessarily employed.\n\nIt is important to stress that the moral reality of war is not fixed by the actual activities of soldiers but by the opinions of mankind. That means, in part, that it is fixed by the activity of philosophers, lawyers, and publicists of all sorts. But these people don't work in isolation from the experience of combat, and their views have value only insofar as they give shape and structure to that experience in ways that are plausible to the rest of us. We often say, for example, that in time of war soldiers and statesmen must make agonizing decisions. The pain is real enough, but it is not one of the natural effects of combat. Agony is not like Hobbist fear; it is entirely the product of our moral views, and it is common in war only insofar as those views are common. It was not some unusual Athenian who \"repented\" of the decision to kill the men of Mytilene, but the citizens generally. They repented, and they were able to understand one another's repentance, because they shared a sense of what cruelty meant. It is by the assignment of such meanings that we make war what it is\u2014which is to say that it could be (and it probably has been) something different.\n\nWhat of a soldier or statesman who does not feel the agony? We say of him that he is morally ignorant or morally insensitive, much as we might say of a general who experienced no difficulty making a (really) difficult decision that he did not understand the strategic realities of his own position or that he was reckless and insensible of danger. And we might go on to argue, in the case of the general, that such a man has no business fighting or leading others in battle, that he ought to know that his army's right flank, say, is vulnerable, and ought to worry about the danger and take steps to avoid it. Once again, the case is the same with moral decisions: soldiers and statesmen ought to know the dangers of cruelty and injustice and worry about them and take steps to avoid them.\n\nHistorical Relativism\n\nAgainst this view, however, Hobbist relativism is often given a social or historical form: moral and strategic knowledge, it is said, changes over time or varies among political communities, and so what appears to me as ignorance may look like understanding to someone else. Now, change and variation are certainly real enough, and they make for a tale that is complex in the telling. But the importance of that tale for ordinary moral life and, above all, for the judgment of moral conduct is easily exaggerated. Between radically separate and dissimilar cultures, one can expect to find radical dichotomies in perception and understanding. No doubt the moral reality of war is not the same for us as it was for Genghis Khan; nor is the strategic reality. But even fundamental social and political transformations within a particular culture may well leave the moral world intact or at least sufficiently whole so that we can still be said to share it with our ancestors. It is rare indeed that we do not share it with our contemporaries, and by and large we learn how to act among our contemporaries by studying the actions of those who have preceded us. The assumption of that study is that they saw the world much as we do. That is not always true, but it is true enough of the time to give stability and coherence to our moral lives (and to our military lives). Even when world views and high ideals have been abandoned\u2014as the glorification of aristocratic chivalry was abandoned in early modern times\u2014notions about right conduct are remarkably persistent: the military code survives the death of warrior idealism. I shall say more about this survival later on, but I can demonstrate it now in a general way by looking at an example from feudal Europe, an age in some ways more distant from us than Greece of the city states, but with which we nevertheless share moral and strategic perceptions.\n\nThree Accounts of Agincourt\n\nActually, the sharing of strategic perceptions is in this case the more dubious of the two. Those French knights, so many of whom died at Agincourt, had notions about combat very different from our own. Modern critics have still felt able to criticize their \"fanatical adherence to the old method of fighting\" (King Henry, after all, fought differently) and even to offer practical suggestions: the French attack, writes Oman, \"should have been accompanied by a turning movement around the woods . . .\" Had he not been \"overconfident,\" the French commander would have seen the advantages of the move. We can talk in a similar way about the crucial moral decision that Henry made toward the end of the battle, when the English thought their victory secure. They had taken many prisoners, who were loosely assembled behind the lines. Suddenly, a French attack aimed at the supply tents far in the rear seemed to threaten a renewal of the fighting. Here is Holinshed's sixteenth-century account of the incident (virtually copied from an earlier chronicle):\n\n. . . certain Frenchmen on horseback . . . to the number of six hundred horsemen, which were the first that fled, hearing that the English tents and pavilions were a good way distant from the army, without any sufficient guard to defend the same . . . entered upon the king's camp and there . . . robbed the tents, broke up chests, and carried away caskets and slew such servants as they found to make any resistance. . . . But when the outcry of the lackeys and boys which ran away for fear of the Frenchmen . . . came to the king's ears, he doubting lest his enemies should gather together again, and begin a new field; and mistrusting further that the prisoners would be an aid to his enemies . . . contrary to his accustomed gentleness, commanded by sound of trumpet that every man . . . should incontinently slay his prisoner.\n\nThe moral character of the command is suggested by the words \"accustomed gentleness\" and \"incontinently.\" It involved a shattering of personal and conventional restraints (the latter well-established by 1415), and Holinshed goes to some lengths to explain and excuse it, stressing the king's fear that the prisoners his forces held were about to rejoin the fighting. Shakespeare, whose Henry V closely follows Holinshed, goes further, emphasizing the slaying of the English servants by the French and omitting the chronicler's assertion that only those who resisted were killed:\n\nFluellen. Kill the [b]oys and the baggage! 'Tis expressly against the law of arms. 'Tis as arrant a piece of knavery, mark you now, as can be offert.\n\nAt the same time, however, he cannot resist an ironical comment:\n\nGower . . . they have burned and carried away all that was in the king's tent, wherefore the king most worthily hath caused every soldier to cut his prisoner's throat. O, 'tis a gallant king!\n\nA century and a half later, David Hume gives a similar account, without the irony, stressing instead the king's eventual cancellation of his order:\n\n. . . some gentlemen of Picardy . . . had fallen upon the English baggage, and were doing execution on the unarmed followers of the camp, who fled before them. Henry, seeing the enemy on all sides of him, began to entertain apprehensions from his prisoners; and he thought it necessary to issue a general order for putting them to death; but on discovering the truth, he stopped the slaughter, and was still able to save a great number.\n\nHere the moral meaning is caught in the tension between \"necessary\" and \"slaughter.\" Since slaughter is the killing of men as if they were animals\u2014it \"makes a massacre,\" wrote the poet Dryden, \"what was a war\"\u2014it cannot often be called necessary. If the prisoners were so easy to kill, they were probably not dangerous enough to warrant the killing. When he grasped the actual situation, Henry, who was (so Hume wants us to believe) a moral man, called off the executions.\n\nFrench chroniclers and historians write of the event in much the same way. It is from them that we learn that many of the English knights refused to kill their prisoners\u2014not, chiefly, out of humanity, rather for the sake of the ransom they expected; but also \"thinking of the dishonor that the horrible executions would reflect on themselves.\" English writers have focused more, and more worriedly, on the command of the king; he was, after all, their king. In the later nineteenth century, at about the same time as the rules of war with respect to prisoners were being codified, their criticism grew increasingly sharp: \"a brutal butchery,\" \"cold-blooded wholesale murder.\" Hume would not have said that, but the difference between that and what he did say is marginal, not a matter of moral or linguistic transformation.\n\nTo judge Henry ourselves we would need a more circumstantial account of the battle than I can provide here. Even given that account, our opinions might differ, depending on the allowance we were willing to make for the stress and excitement of battle. But this is a clear example, of a situation common in both strategy and morality, where our sharpest disagreements are structured and organized by our underlying agreements, by the meanings we share. For Holinshed, Shakespeare, and Hume\u2014\u00adtraditional chronicler, Renaissance playwright, and Enlightenment historian\u2014and for us too, Henry's command belongs to a category of military acts that requires scrutiny and judgment. It is as a matter of fact morally problematic, because it accepts the risks of cruelty and injustice. In exactly the same way, we might regard the battle plan of the French commander as strategically problematic, because it accepted the risks of a frontal assault on a prepared position. And, again, a general who did not recognize these risks is properly said to be ignorant of morality or strategy.\n\nIn moral life, ignorance isn't all that common; dishonesty is far more so. Even those soldiers and statesmen who don't feel the agony of a problematic decision generally know that they should feel it. Harry Truman's flat statement that he never lost a night's sleep over his decision to drop the atomic bomb on Hiroshima is not the sort of thing political leaders often say. They usually find it preferable to stress the painfulness of decision-\u00admaking; it is one of the burdens of office, and it is best if the burdens appear to be borne. I suspect that many officeholders even experience pain simply because they are expected to. If they don't, they lie about it. The clearest evidence for the stability of our values over time is the unchanging character of the lies soldiers and statesmen tell. They lie in order to justify themselves, and so they describe for us the lineaments of justice. Wherever we find hypocrisy, we also find moral knowledge. The hypocrite is like that Russian general in Solzhenitsyn's August 1914, whose elaborate battle reports barely concealed his total inability to control or direct the battle. He knew at least that there was a story to tell, a set of names to attach to things and happenings, so he tried to tell the story and attach the names. His effort was not mere mimicry; it was, so to speak, the tribute that incompetence pays to understanding. The case is the same in moral life: there really is a story to tell, a way of talking about wars and battles that the rest of us recognize as morally appropriate. I don't mean that particular decisions are necessarily right or wrong, or simply right or wrong, only that there is a way of seeing the world so that moral decision-making makes sense. The hypocrite knows that this is true, though he may actually see the world differently.\n\nHypocrisy is rife in wartime discourse, because it is especially impor\u00adtant at such a time to appear to be in the right. It is not only that the moral stakes are high; the hypocrite may not understand that; more crucially, his actions will be judged by other people, who are not hypocrites, and whose judgments will affect their policies toward him. There would be no point to hypocrisy if this were not so, just as there would be no point to lying in a world where no one told the truth. The hypocrite presumes on the moral understanding of the rest of us, and we have no choice, I think, except to take his assertions seriously and put them to the test of moral realism. He pretends to think and act as the rest of us expect him to do. He tells us that he is fighting according to the moral war plan: he does not aim at civilians, he grants quarter to soldiers trying to surrender, he never tortures prisoners, and so on. These claims are true or false, and though it is not easy to judge them (nor is the war plan really so simple), it is important to make the effort. Indeed, if we call ourselves moral men and women, we must make the effort, and the evidence is that we regularly do so. If we had all become realists like the Athenian generals or like Hobbists in a state of war, there would be an end alike to both morality and hypocrisy. We would simply tell one another, brutally and directly, what we wanted to do or have done. But the truth is that one of the things most of us want, even in war, is to act or to seem to act morally. And we want that, most simply, because we know what morality means (at least, we know what it is generally thought to mean).\n\nIt is that meaning that I want to explore in this book\u2014not so much its general character, but its detailed application to the conduct of war. I am going to assume throughout that we really do act within a moral world; that particular decisions really are difficult, problematic, agonizing, and that this has to do with the structure of that world; that language reflects the moral world and gives us access to it; and finally that our understanding of the moral vocabulary is sufficiently common and stable so that shared judgments are possible. Perhaps there are other worlds to whose inhabitants the arguments I am going to make would seem incomprehensible and bizarre. But no such people are likely to read this book. And if my own readers find my arguments incomprehensible and bizarre, that will not be because of the impossibility of moral discourse or the inconstant signification of the words I use, but because of my own failure to grasp and expound our common morality.\n\n Even oriental monarchs are not quite so toughminded as the Athenian generals. According to Herodotus, when Xerxes first disclosed his plans for an invasion of Greece, he spoke in more conventional terms: \"I will bridge the Hellespont and march an army through Europe into Greece, and punish the Athenians for the outrage they committed upon my father and upon us.\" (The Histories, Book 7, trans. Aubrey de Selincourt.) The reference is to the burning of Sardis, which we may take as the pretext for the Persian invasion. The example bears out Francis Bacon's assertion that \"there is that justice imprinted in the nature of men that they enter not upon wars (whereof so many calamities do ensue) but upon some, at least specious, grounds and quarrels.\" (Essay 29, \"Of the True Greatness of Kingdoms and Estates.\")\n\n Hence we can \"unmask\" strategic discourse just as Thucydides did with moral discourse. Imagine that the two Athenian generals, after their dialogue with the Melians, return to their camp to plan the coming battle. The senior in command speaks first: \"Don't give me any fine talk about the need to concentrate our forces or the importance of strategic surprise. We'll simply call for a frontal assault; the men will organize themselves as best they can; things are going to be confused anyway. I need a quick victory here, so that I can return to Athens covered with glory before the debate on the Sicilian campaign begins. We'll have to accept some risks; but that doesn't matter since the risks will be yours, not mine. If we are beaten, I'll contrive to blame you. That's what war is like.\" Why is strategy the language of hard-headed men? One sees through it so easily. . . .\n\nThe Crime of War\n\nThe moral reality of war is divided into two parts. War is always judged twice, first with reference to the reasons states have for fighting, secondly with reference to the means they adopt. The first kind of judgment is adjectival in character: we say that a particular war is just or unjust. The second is adverbial: we say that the war is being fought justly or unjustly. Medieval writers made the difference a matter of prepositions, distinguishing jus ad bellum, the justice of war, from jus in bello, justice in war. These grammatical distinctions point to deep issues. Jus ad bellum requires us to make judgments about aggression and self-defense; jus in bello about the observance or violation of the customary and positive rules of engagement. The two sorts of judgment are logically independent. It is perfectly possible for a just war to be fought unjustly and for an unjust war to be fought in strict accordance with the rules. But this independence, though our views of particular wars often conform to its terms, is nevertheless puzzling. It is a crime to commit aggression, but aggressive war is a rule-governed activity. It is right to resist aggression, but the resistance is subject to moral (and legal) restraint. The dualism of jus ad bellum and jus in bello is at the heart of all that is most problematic in the moral reality of war.\n\nIt is my purpose to see war whole, but since its dualism is the essential feature of its wholeness, I must begin by accounting for the parts. In this chapter, I want to suggest what we mean when we say that it is a crime to begin a war, and in the next I will try to explain why it is that there are rules of engagement that apply even to soldiers whose wars are criminal. This chapter introduces Part Two, where I will examine in detail the nature of the crime, describe the appropriate forms of resistance, and consider the ends that soldiers and statesmen may legitimately seek in fighting just wars. The next chapter introduces Part Three, where I will discuss the legitimate means of warfare, the substantive rules, and show how these rules apply in combat conditions and how they are modified by \"military necessity.\" Only then will it be possible to confront the tension between ends and means, jus ad bellum and jus in bello.\n\nI am not sure whether the moral reality of war is wholly coherent, but for the moment I need not say anything about that. It's enough that it has a recognizable and relatively stable shape, that its parts are connected and disconnected in recognizable and relatively stable ways. We have made it so, not arbitrarily, but for good reasons. It reflects our understanding of states and soldiers, the protagonists of war, and of combat, its central experience. The terms of that understanding are my immediate subject matter. They are simultaneously the historical product of and the necessary condition for the critical judgments that we make every day; they fix the nature of war as a moral (and an immoral) enterprise.\n\nThe Logic of War\n\nWhy is it wrong to begin a war? We know the answer all too well. People get killed, and often in large numbers. War is hell. But it is necessary to say more than that, for our ideas about war in general and about the conduct of soldiers depend very much on how people get killed and on who those people are. Then, perhaps, the best way to describe the crime of war is simply to say that there are no limits at either of these points: people are killed with every conceivable brutality, and all sorts of people, without distinction of age or sex or moral condition, are killed. This view of war is brilliantly summed up in the first chapter of Karl von Clausewitz's On War, and though there is no evidence that Clausewitz thought war a crime, he has certainly led other people to think so. It is his early definitions (rather than his later qualifications) that have shaped the ideas of his successors, and so it is worth considering them in some detail.\n\nThe Argument of Karl von Clausewitz\n\n\"War is an act of force,\" Clausewitz writes, \". . . which theoretically can have no limits.\" The idea of war carries with it for him the idea of limitlessness, whatever actual restraints are observed in this or that society. If we imagine a war fought, as it were, in a social vacuum, unaffected by \"accidental\" factors, it would be fought with no restraint at all in the weapons used, the tactics adopted, the people attacked, or anywhere else. For military conduct knows no intrinsic limits; nor is it possible to refine our notions of war so as to incorporate those extrinsic moral codes that Clausewitz sometimes calls \"philanthropic.\" \"We can never introduce a modifying principle into the philosophy of war without committing an absurdity.\" The more extreme the battle is, then, the more general and intense the violence employed on one side and the other, the closer to war in the conceptual sense (\"absolute war\") it is. And there can be no imaginable act of violence, however treacherous or cruel, that falls outside of war, that is not-war, for the logic of war simply is a steady thrust toward moral extremity. That is why it is so awful (though Clausewitz does not tell us this) to set the process going: the aggressor is responsible for all the consequences of the fighting he begins. In particular cases, it may not be possible to know these consequences in advance, but they are always potentially terrible. \"When you resorted to force,\" General Eisenhower once said, \". . . you didn't know where you were going. . . . If you got deeper and deeper, there was just no limit except . . . the limitations of force itself.\"\n\nThe logic of war, according to Clausewitz, works in this way: \"each of the adversaries forces the hand of the other.\" What results is a \"reciprocal action,\" a continuous escalation, in which neither side is guilty even if it acts first, since every act can be called and almost certainly is pre-emptive. \"War tends toward the utmost exertion of forces,\" and that means toward increasing ruthlessness, since \"the ruthless user of force who shrinks from no amount of bloodshed must gain an advantage if his opponent does not do the same.\" And so his opponent, driven by what Thucydides and Hobbes call \"a necessity of nature,\" does the same, matching the ruthlessness of the other side whenever he can. But this description, though it is a useful account of how escalation works, is open to the criticism that I have already made. As soon as we focus on some concrete case of military and moral decision-making, we enter a world that is governed not by abstract tendencies but by human choice. The actual pressures toward escalation are greater here, less there, rarely so overwhelming as to leave no room for maneuver. Wars no doubt are often escalated, but they are also (sometimes) fought at fairly steady levels of violence and brutality, and these levels are (sometimes) fairly low.\n\nClausewitz grants this, though without surrendering his commitment to the absolute. War, he writes, \"may be a thing which is sometimes war in a greater, sometimes in a lesser degree.\" And again, \"There can be wars of all degrees of importance and energy, from a war of extermination down to a mere state of armed observation.\" Somewhere between these two, I suppose, we begin to say, all's fair, anything goes, and so on. When we talk that way, we are not referring to the general limitlessness of war, but to particular escalations, particular acts of force. No one has ever experienced \"absolute war.\" In this or that struggle, we endure (or commit) this or that brutality, which can always be described in concrete terms. It is the same with hell: I cannot conceptualize infinite pain without thinking of whips and scorpions, hot irons, other people. Now, what is it that we think about when we say, war is hell? What aspects of warfare lead us to regard its initiation as a criminal act?\n\nThe same questions can be introduced in another way. War is not usefully described as an act of force without some specification of the context in which the act takes place and from which it derives its meaning. Here the case is the same as with other human activities (politics and commerce, for example): it's not what people do, the physical motions they go through, that are crucial, but the institutions, practices, conventions that they make. Hence the social and historical conditions that \"modify\" war are not to be considered as accidental or external to war itself, for war is a social creation. At particular points in time, it takes shape in particular ways, and sometimes at least in ways that resist the \"utmost exertion of forces.\" What is war and what is not-war are in fact something that people decide (I don't mean by taking a vote). As both anthropological and historical accounts suggest, they can decide, and in a considerable variety of cultural settings they have decided, that war is limited war\u2014that is, they have built certain notions about who can fight, what tactics are acceptable, when battle has to be broken off, and what prerogatives go with victory into the idea of war itself.a Limited war is always specific to a time and place, but so is every escalation, including the escalation beyond which war is hell.\n\nThe Limit of Consent\n\nSome wars are not hell, and it will be best to begin with them. The first and most obvious example is the competitive struggle of aristocratic young men, a tournament on a larger scale and with no presiding officer in the stands. Examples can be found in Africa, ancient Greece, Japan, and feudal Europe. Here is a \"contention by arms\" that has often captured the imagination, not only of children, but also of romantic adults. John Ruskin made it his own ideal: \"creative or foundational war is that in which the natural restlessness and love of contest are disciplined, by consent, into modes of beautiful\u2014though it may be fatal\u2014play. . . .\" Creative war may not be terribly bloody, but that is not the crucial thing about it. I have read accounts of tournaments that make them sound brutal enough, but no such account would lead anyone to say that it was a crime to organize a tournament. What rules out such a claim, I think, is Ruskin's phrase \"by consent.\" His beautiful aristocrats do what they choose to do, and that is why no poet ever described their deaths in terms comparable to those of Wilfred Owen writing of infantrymen in World War I:\n\nWhat passing-bells for these who die as cattle?\n\n\"To the youths who voluntarily adopt it as their profession,\" writes Ruskin, \"war] has always been a grand pastime. . . .\" We take their choice as a sign that what they are choosing cannot be awful, even if it looks that way to us. Perhaps they ennoble the brutal melee; perhaps not; but if this kind of war were hellish, these well-born young men would be doing something else.[b\n\nA similar argument can be made whenever fighting is voluntary. Nor does it matter a great deal if the men involved don't choose to fight, so long as they can choose to break off fighting without dire consequences. In certain primitive societies, whole age cohorts of young males go off to battle; individuals cannot avoid combat without exposing themselves to dishonor and ostracism. But there is no effective social pressure or military discipline on the battlefield itself. And then there takes place, as Hobbes says, \"on both sides a running away.\" When running away is acceptable, as it often is in primitive warfare, battles will obviously be short and casualties few. There is nothing that resembles \"the utmost exertion of forces.\" Those men who don't run away, but stand and fight, do so not because of the necessities of their case, but freely, as a matter of choice. They seek out the excitement of battle, perhaps because they enjoy it, and their subsequent fate, even if it is very painful, can't be called unjust.\n\nThe case of mercenaries and professional soldiers is more complex and needs to be examined with some care. In Renaissance Italy, wars were fought by mercenary soldiers recruited by the great condottieri, partly as a business venture, partly as a political speculation. City-states and principalities had to rely on such men because the political culture of the time did not allow for effective coercion. There were no conscript armies. The result was warfare of a very limited sort, since recruits were expensive and each army represented a considerable capital investment. Battle became a matter largely of tactical maneuver; physical confrontation was rare; relatively few soldiers were killed. Wars had to be won, as two of the condottieri wrote, \"rather by industry and cunning than by actual clash of arms.\" Thus the great defeat of the Florentines at Zagonara: \"no deaths occurred [in the battle],\" Machiavelli tells us, \"except those of Lodovico degli Obizi and two of his men, who, having fallen from their horses, were drowned in the mud.\" But, once again, I don't want to stress the limited character of the fighting but something prior to that, from which the limits follow: a certain sort of freedom in choosing war. Mercenary soldiers signed up on terms, and if they could not actually choose their campaigns and tactics, they could to some degree fix the cost of their services and so condition the choices of their leaders. Given that freedom, they might have fought very bloody battles and the spectacle would not lead us to say that war was a crime. A fight between mercenary armies is undoubtedly a bad way of settling political disputes, but we judge it bad for the sake of the people whose fate is being settled, not for the sake of the soldiers themselves.\n\nOur judgments are very different, however, if the mercenary armies are recruited (as they most often are) from among desperately impoverished men, who can find no other way of feeding themselves and their families except by signing up. Ruskin makes this point well when he tells his aristocratic warriors: \"Remember, whatever virtue and goodliness there may be in this game of war, rightly played, there is none when you . . . play it with a multitude of small human pawns . . . [when you] urge your peasant millions into gladiatorial war. . . .\" Then battle becomes a \"circus of slaughter\" in the midst of which no consensual discipline is possible, and those who die do so without ever having had a chance to live in another way. Hell is the right name for the risks they never chose and the agony and death they endure; the men responsible for that agony are rightly called criminals.\n\nMercenaries are professional soldiers who sell their services on the open market, but there are other professionals who serve only their own prince or people and, though they may earn their bread by soldiering, disdain the name of mercenary. \"We're either officers who serve their Tsar and country,\" says Prince Andrey in War and Peace, \"and rejoice in the success and grieve at the defeat of the common cause, or we're hirelings who have no interest in our master's business.\" The distinction is too gross; in fact there are intermediate positions; but the more a soldier fights because he is committed to a \"common cause,\" the more likely we are to regard it as a crime to force him to fight. We assume that his commitment is to the safety of his country, that he fights only when it is threatened, and that then he has to fight (he has been \"put to it\"): it is his duty and not a free choice. He is like a doctor who risks his life during an epidemic, using professional skills he chose to acquire but whose acquisition is not a sign that he hopes for epidemics. On the other hand, professional soldiers are sometimes exactly like those aristocratic warriors who relish battle, driven more by a lust for victory than by patriotic conviction, and then we may well be unmoved by their deaths. At least we will not say, they would not want us to say, what Owen says of his comrades in the trenches, that \"one dies of war like any old disease.\" They died instead of their own free will.\n\nWar is hell whenever men are forced to fight, whenever the limit of consent is breached. That means, of course, that it is hell most of the time; throughout most of recorded history, there have been political organizations capable of marshalling armies and driving soldiers into battle. It is the absence of political discipline or its ineffectiveness in detail that opens the way for \"creative war.\" The examples I have given are best understood as limiting cases, establishing the boundaries of hell. We ourselves are old inhabitants\u2014even if we live in democratic states where the government that decides to fight or not to fight is popularly elected. For I am not considering now the legitimacy of that government. Nor am I immediately interested in the willingness of a potential soldier to vote for a war he has been led to believe is necessary or to volunteer for it. What is important here is the extent to which war (as a profession) or combat (at this or that moment in time) is a personal choice that the soldier makes on his own and for essentially private reasons. That kind of choosing effectively disappears as soon as fighting becomes a legal obligation and a patriotic duty. Then \"the waste of the life of the combatants is one which,\" as the philosopher T. H. Green has written, \"the power of the state compels. This is equally true whether the army is raised by voluntary enlistment or by conscription.\"c For the state decrees that an army of a certain size be raised, and it sets out to find the necessary men, using all the techniques of coercion and persuasion at its disposal. And the men it finds, precisely because they go to war under constraint or as a matter of conscience, can no longer moderate their battles; the battles are no longer theirs. They are political instruments, they obey orders, and the practice of war is shaped at a higher level. Perhaps they really are obligated to obey orders in this or that case, but war is radically changed by the fact that they do so generally. The change is best represented for the modern period (though there are historical analogues) by the effects of conscription. \"Hitherto soldiers had been costly, now they were cheap; battles had been avoided, now they were sought, and however heavy were the losses, they could rapidly be made good by the muster-roll.\"\n\nNapoleon is said to have boasted to Metternich that he could afford to lose 30,000 men a month. Perhaps he could have lost that many and still have maintained political support at home. But he could not have done so, I think, had he had to ask the men he was about to \"lose.\" Soldiers might agree to such losses in a war forced upon them by the enemy, a war of national defense, but not in the sorts of wars that Napoleon fought. The need to seek their consent (whatever the form in which it was sought and given or not given) would surely limit the occasions of war, and if there were any chance at all of reciprocity from the other side, it would limit its means too. This is the sort of consent I have in mind. Political self-determination is not, judging from twentieth-century history, an adequate substitute, though it isn't easy to think of one that would be better. In any case, it is when individual consent fails that \"acts of force\" lose whatever appeal they previously had and become the constant object of moral condemnation. And after that, war also tends to escalate in its means, not necessarily beyond all limits, but certainly beyond those limits that ordinary humanity, as free of political loyalty as of political constraint, would establish if it could.\n\nThe Tyranny of War\n\nWar is most often a form of tyranny. It is best described by paraphrasing Trotsky's aphorism about the dialectic: \"You may not be interested in war, but war is interested in you.\" The stakes are high, and the interest that military organizations take in an individual who would prefer to be somewhere else, doing something else, is frightening indeed. Hence the peculiar horror of war: it is a social practice in which force is used by and against men as loyal or constrained members of states and not as individuals who choose their own enterprises and activities. When we say, war is hell, it is the victims of the fighting that we have in mind. In fact, then, war is the very opposite of hell in the theological sense, and is hellish only when the opposition is strict. For in hell, presumably, only those people suffer who deserve to suffer, who have chosen activities for which punishment is the appropriate divine response, knowing that this is so. But the greater number by far of those who suffer in war have made no comparable choice.\n\nI do not mean to call them \"innocent.\" That word has come to have a special meaning in our moral discourse. It doesn't refer there to the participants but to the bystanders of battle, and so the class of innocent men and women is only a subset (though it is often a frighteningly large subset) of all those in whom war takes an interest without asking their consent. The rules of war by and large protect only the subset, for reasons I will have to consider later on. But war is hell even when the rules are observed, even when only soldiers are killed and civilians are consistently spared. Surely no experience of modern warfare has etched its horror so deeply in our minds as the fighting in the trenches of World War I\u2014and in the trenches civilian lives were rarely at risk. The distinction of combatants and bystanders is enormously important in the theory of war, but our first and most fundamental moral judgment does not depend upon it. For in one sense at least, soldiers in battle and nonparticipating civilians are not so different: the soldiers would almost certainly be nonparticipants if they could.\n\nThe tyranny of war is often described as if war itself were the tyrant, a natural force like flood or famine or, personified, a brutal giant stalking his human prey, as in these lines from a poem by Thomas Sackville:\n\nLastly stood War, in glittering arms y-clad,\n\nWith visage grim, stern looks, and blackly hued;\n\nIn his right hand a naked sword he had\n\nThat to the hilts was all with blood embrued,\n\nAnd in his left (that kings and kingdoms rued)\n\nFamine and fire he held, and therewithal\n\nHe razed towns, and threw down towers and all.\n\nHere is the Grim Reaper in uniform, armed with a sword instead of a scythe. The poetic image enters also into moral and political thought, but only, I think, as a kind of ideology, obscuring our critical judgment. For it is a piece of mystification to represent tyrannical power as an abstract force. In battle as in politics, tyranny is always a relation among persons or groups of persons. The tyranny of war is a peculiarly complex relation because coercion is common on both sides. Sometimes, however, it is possible to distinguish the sides and to identify the statesmen and soldiers who first took the naked sword to hand. Wars are not self-starting. They may \"break out,\" like an accidental fire, under conditions difficult to analyze and where the attribution of responsibility seems impossible. But usually they are more like arson than accident: war has human agents as well as human victims.\n\nThose agents, when we can identify them, are properly called criminals. Their moral character is determined by the moral reality of the activity they force others to engage in (whether or not they engage in it themselves). They are responsible for the pain and death that follow from their decisions, or at least for the pain and death of all those persons who do not choose war as a personal enterprise. In contemporary international law, their crime is called aggression, and I will consider it later on under that name. But we can understand it initially as the exercise of tyrannical power, first over their own people and then, through the mediation of the opposing state's recruitment and conscription offices, over the people they have attacked. Now, tyranny of this sort rarely encounters domestic resistance. Sometimes the war is opposed by local political forces, but the opposition almost never extends to the actual exercise of military power. Though mutinies are common in the long history of war, they are more like peasant jacqueries, quickly and bloodily suppressed, than revolutionary struggles. Most often, real opposition comes only from the enemy. It is the men and women on the other side who are most likely to recognize and resent the tyranny of war; and whenever they do that, the contest takes on a new significance.\n\nWhen soldiers believe themselves to be fighting against aggression, war is no longer a condition to be endured. It is a crime they can resist\u2014though they must suffer its effects in order to resist it\u2014and they can hope for a victory that is something more than an escape from the immediate brutality of battle. The experience of war as hell generates what might be called a higher ambition: one doesn't aim to settle with the enemy but to defeat and punish him and, if not to abolish the tyranny of war, at least to reduce the probability of future oppression. And once one is fighting for purposes of this sort, it becomes terribly important to win. The conviction that victory is morally critical plays an important part in the so-called \"logic of war.\" We don't call war hell because it is fought without restraint. It is more nearly right to say that, when certain restraints are passed, the hellishness of war drives us to break with every remaining restraint in order to win. Here is the ultimate tyranny: those who resist aggression are forced to imitate, and perhaps even to exceed, the brutality of the aggressor.\n\nGeneral Sherman and the Burning of Atlanta\n\nWe are now in a position to understand what Sherman had in mind when he first announced that war is hell. He wasn't merely describing the awfulness of the experience, nor was he denying the possibility of moral judgment. He made such judgments freely, and he surely thought of himself as a righteous soldier. His maxim sums up, with admirable brevity, a whole way of thinking about war\u2014a one-sided and partial way of thinking, I shall argue, but powerful nonetheless. In his view, war is entirely and singularly the crime of those who begin it, and soldiers resisting aggression (or rebellion) can never be blamed for anything they do that brings victory closer. The sentence War is hell is doctrine, not description: it is a moral argument, an attempt at self-justification. Sherman was claiming to be innocent of all those actions (though they were his own actions) for which he was so severely attacked: the bombardment of Atlanta, the forced evacuation of its inhabitants and the burning of the city, the march through Georgia. When he issued the order for the evacuation and burning of Atlanta, the city's aldermen and the Confederate commander, General Hood, protested his plans: \"And now, sir,\" wrote Hood, \"permit me to say that the unprecedented measure you propose transcends, in studied and ingenious cruelty, all acts ever before brought to my attention in the dark history of war.\" Sherman replied that war is indeed dark. \"War is cruelty and you cannot refine it.\" And therefore, he went on, \"those who brought war into our country deserve all the curses and maledictions a people can pour out.\" But he himself deserves no curses at all. \"I know I had no hand in making this war.\" He is only fighting it, not by choice but because he has to. He has been forced to use force, and the burning of Atlanta (so that the city could not again serve as a military depot for Confederate forces) is simply one more example of that use, one of the entailments of war. It is cruel, no doubt, but the cruelty isn't his own; it belongs, so to speak, to the men of the Confederacy: \"You who, in the midst of peace and prosperity, have plunged a nation into war. . . . \" The Confederate leaders can easily restore peace by yielding obedience to federal law, but he can do so only by military action.\n\nSherman's argument expresses the anger that is commonly directed against those who begin a war and inflict its tyrannies on the rest of us. We disagree, of course, when it comes to giving the tyrants a proper name. But that disagreement is intense and heated only because we agree on the moral stakes. What is at issue is responsibility for death and destruction, and Sherman is by no means the only general to take a lively interest in such matters. Nor is he the only general to think that if his cause is just he cannot be blamed for the death and destruction he spreads around him\u2014for war is hell.\n\nIt is the Clausewitzian idea of limitlessness that is at work here, and if that idea is right, there would indeed be no response to Sherman's argument. But the tyranny of war is no more limitless than is political tyranny. Just as we can charge a tyrant with particular crimes over and above the crime of ruling without consent, so we can recognize and condemn particular criminal acts within the hell of war. When we answer the question, \"Who started this war?\" we have not finished distributing responsibility for the suffering that soldiers inflict. There are further arguments to make. That's why General Sherman, though he insisted that the cruelty of war could not be refined, claimed nevertheless to be refining it. \"God will judge . . . ,\" he wrote, \"whether it be more humane to fight with a town full of women [and children] at our back or to remove them in time to places of safety among their own friends and people.\" This is another kind of justification; and whether or not it is made in good faith, it suggests (what is certainly true) that Sherman had some responsibility for the people of Atlanta, even though he did not begin the war of which they were victims. When we focus exclusively on the fact of aggression, we are likely to lose sight of that responsibility and to talk as if there were only one morally relevant decision to be made in the course of a war: to attack or not to attack (to resist or not to resist). Sherman wants to judge war only at its outermost boundaries. But there is a great deal to be said about its interior regions, as he himself admits. Even in hell, it is possible to be more or less humane, to fight with or without restraint. We must try to understand how this can be so.\n\n This, of course, is exactly what Clausewitz wants to deny. In technical terms, he is arguing that war is never an activity constituted by its rules. War is never like a duel. The social practise of duelling includes and accounts for only those acts of violence specified in the rulebook or the customary code. If I wound my opponent, shoot his second, and then beat him to death with a stick, I am not duelling with him; I am murdering him. But similar brutalities in war, though they violate the rules, are still regarded as acts of war (war crimes). Hence there is a formal or linguistic sense in which military action is limitless, and this has undoubtedly influenced our understanding of such action. At the same time, however, \"war\" and related words are at least sometimes used in a more restrictive sense, as in the famous speech of Sir Henry Campbell-Bannerman, one of the leaders of the Liberal Party in Britain during the Boer War: \"When is war not war? When it is fought by methods of barbarism. . . .\" We do still refer to the Boer War, but the argument is not idiosyncratic. I will provide other examples later on.\n\n We can glimpse the mood of the happy warrior in a letter that Rupert Brooke wrote to a friend at the very beginning of World War I, before he knew what it would be like: \"Come and die. It'll be great fun.\" (Quoted in Malcolm Cowley, A Second Flowering, New York, 1974, p. 6.)\n\n Green is arguing against the proposition I have hitherto maintained: that no wrong is done in war if \"the persons killed are voluntary combatants.\" He denies this on the grounds that a soldier's life is not merely his own. \"The individual's right to life is but the other side of the right which society has in his living.\" But that, it seems to me, is only true in certain sorts of societies; it is hardly an argument that could have been made to a feudal warrior. Green goes on to argue, more plausibly, that in his own society it makes little sense to talk of soldiers fighting voluntarily: war is now a state action. The chapter on \"The Right of the State over the Individual in War\" in Green's Principles of Political Obligation, provides an especially clear description of the ways in which moral responsibility is mediated in the modern state; I have relied on it often in this and later chapters.\n\nThe Rules of War\n\nThe Moral Equality of Soldiers\n\nAmong soldiers who choose to fight, restraints of various sorts arise easily and, one might say, naturally, the product of mutual respect and recognition. The stories of chivalric knights are for the most part stories, but there can be no doubt that a military code was widely shared in the later Middle Ages and sometimes honored. The code was designed for the convenience of the aristocratic warriors, but it also reflected their sense of themselves as persons of a certain sort, engaged in activities that were freely chosen. Chivalry marked off knights from mere ruffians and bandits and also from peasant soldiers who bore arms as a necessity. I suppose that it survives today: some sense of military honor is still the creed of the professional soldier, the sociological if not the lineal descendent of the feudal knight. But notions of honor and chivalry seem to play only a small part in contemporary combat. In the literature of war, the contrast between \"then and now\" is commonly made\u2014not very accurately, but with a certain truth, as in this poem by Louis Simpson:\n\nAt Malplaquet and Waterloo\n\nThey were polite and proud,\n\nThey primed their guns with billets-doux\n\nAnd, as they fired, bowed.\n\nAt Appomattox too, it seems\n\nSome things were understood . . .\n\nBut at Verdun and at Bastogne\n\nThere was a great recoil,\n\nThe blood was bitter to the bone\n\nThe trigger to the soul. . . .\n\nChivalry, it is often said, was the victim of democratic revolution and of revolutionary war: popular passion overcame aristocratic honor. That draws the line before Waterloo and Appomattox, though still not quite correctly. It is the success of coercion that makes war ugly. Democracy is a factor only insofar as it increases the legitimacy of the state and then the effectiveness of its coercive power, not because the people in arms are a bloodthirsty mob fired by political zeal and committed to total war (in contrast to their officers, who would fight with decorum if they could). It is not what the people do when they enter the arena of battle that turns war into a \"circus of slaughter,\" but, as I have already argued, the mere fact that they are there. Soldiers died by the thousands at Verdun and the Somme simply because they were available, their lives nationalized, as it were, by the modern state. They didn't choose to throw themselves at barbed wire and machine guns in fits of patriotic enthusiasm. The blood is bitter to their bones, too; they, too, would fight with decorum if they could. Their patriotism is, of course, a partial explanation of their availability. The discipline of the state is not merely imposed on them; it is also a discipline they accept, thinking that they have to for the sake of their families and their country. But the common features of contemporary combat: hatred for the enemy, impatience with all restraint, zeal for victory\u2014these are the products of war itself whenever masses of men have to be mobilized for battle. They are as much the contribution of modern warfare to democratic politics as of democracy to war.\n\nIn any case, the death of chivalry is not the end of moral judgment. We still hold soldiers to certain standards, even though they fight unwillingly\u2014in fact, precisely because we assume that they all fight unwillingly. The military code is reconstructed under the conditions of modern warfare so that it comes to rest not on aristocratic freedom but on military servitude. Sometimes freedom and servitude co-exist, and then we can study the difference between them in clinical detail. Whenever the game of war is revived, the elaborate courtesies of the chivalric age are revived with it\u2014as among aviators in World War I, for example, who imagined themselves (and who have survived in the popular imagination) as airborne knights. Compared to the serfs on the ground, these were aristocrats indeed: they fought in accordance with a strict code of conduct that they invented themselves. There was thralldom in the trenches, however, and mutual recognition took a very different form. Briefly, on Christmas Day 1914, German and French troops came together, drank and sang together, in the no-man's land between their lines. But such moments are rare in recent history, and they are not occasions for moral invention. The modern rules of war depend upon an abstract rather than a practical fellowship.\n\nSoldiers cannot endure modern warfare for long without blaming someone for their pain and suffering. While it may be an example of what Marxists call \"false consciousness\" that they do not blame the ruling class of their own or of the enemy country, the fact is that their condemnation focuses most immediately on the men with whom they are engaged. The level of hatred is high in the trenches. That is why enemy wounded are often left to die and prisoners are killed\u2014like murderers lynched by vigilantes\u2014as if the soldiers on the other side were personally responsible for the war. At the same time, however, we know that they are not responsible. Hatred is interrupted or overridden by a more reflective understanding, which one finds expressed again and again in letters and war memoirs. It is the sense that the enemy soldier, though his war may well be criminal, is nevertheless as blameless as oneself. Armed, he is an enemy; but he isn't my enemy in any specific sense; the war itself isn't a relation between persons but between political entities and their human instruments. These human instruments are not comrades-in-arms in the old style, members of the fellowship of warriors; they are \"poor sods, just like me,\" trapped in a war they didn't make. I find in them my moral equals. That is not to say simply that I acknowledge their humanity, for it is not the recognition of fellow men that explains the rules of war; criminals are men too. It is precisely the recognition of men who are not criminals.\n\nThey can try to kill me, and I can try to kill them. But it is wrong to cut the throats of their wounded or to shoot them down when they are trying to surrender. These judgments are clear enough, I think, and they suggest that war is still, somehow, a rule-governed activity, a world of permissions and prohibitions\u2014a moral world, therefore, in the midst of hell. Though there is no license for war-makers, there is a license for soldiers, and they hold it without regard to which side they are on; it is the first and most important of their war rights. They are entitled to kill, not anyone, but men whom we know to be victims. We could hardly understand such a title if we did not recognize that they are victims too. Hence the moral reality of war can be summed up in this way: when soldiers fight freely, choosing one another as enemies and designing their own battles, their war is not a crime; when they fight without freedom, their war is not their crime. In both cases, military conduct is governed by rules; but in the first the rules rest on mutuality and consent, in the second on a shared servitude. The first case raises no difficulties; the second is more problematic. We can best explore its problems, I think, if we turn from the trenches and the front lines to the general staff at the rear, and from the war against the Kaiser to the war against Hitler\u2014for at that level and in that struggle, the recognition of \"men who are not criminals\" is hard indeed.\n\nThe Case of Hitler's Generals\n\nIn 1942, General von Arnim was captured in North Africa, and it was proposed by members of Dwight Eisenhower's staff that the American commander \"should observe the custom of by-gone days\" and permit von Arnim to visit him before he was sent into captivity. Historically, such visits were not merely matters of courtesy; they were occasions for the reaffirmation of the military code. Thus General von Ravenstein, captured by the British that same year, reports: \"I was taken to see . . . Auchinleck himself in his office. He shook hands with me and said: 'I know you well by name. You and your division have fought with chivalry.'\" Eisenhower, however, refused to allow the visit. In his memoirs, he explained his reasons:\n\nThe custom had its origin in the fact that the mercenary soldiers of old had no real enmity toward their opponents. Both sides fought for love of a fight, out of a sense of duty or, more probably, for money. . . . The tradition that all professional soldiers are comrades in arms has . . . persisted to this day. For me, World War II was far too personal a thing to entertain such feelings. Daily as it progressed there grew within me the conviction that, as never before . . . the forces that stood for human good and men's rights were . . . confronted by a completely evil conspiracy with which no compromise could be tolerated.\n\nOn this view, it doesn't matter whether or not von Arnim had fought well; his crime was to have fought at all. And similarly, it may not matter how General Eisenhower fights. Against an evil conspiracy, what is crucial is to win. Chivalry loses its rationale, and there are no limits left except \"the limitations of force itself.\"\n\nThat was Sherman's view too, but it does not account for the judgments that we make of his conduct, or of Eisenhower's, or even of von Arnim's and von Ravenstein's. Consider now the better-known case of Erwin Rommel: he, too, was one of Hitler's generals, and it is hard to imagine that he could have escaped the moral infamy of the war he fought. Yet he was, so we are told by one biographer after another, an honorable man. \"While many of his colleagues and peers in the German army surrendered their honor by collusion with the iniquities of Nazism, Rommel was never defiled.\" He concentrated, like the professional he was, on \"the soldier's task of fighting.\" And when he fought, he maintained the rules of war. He fought a bad war well, not only militarily but also morally. \"It was Rommel who burned the Commando Order issued by Hitler on 28 October 1942, which laid down that all enemy soldiers encountered behind the German line were to be killed at once. . . .\" He was one of Hitler's generals, but he did not shoot prisoners. Is such a man a comrade? Can one treat him with courtesy, can one shake his hand? These are the fine points of moral conduct; I do not know how they might be resolved, though I am sympathetic with Eisenhower's resolution. But I am sure, nevertheless, that Rommel should be praised for burning the Commando Order, and everyone who writes about these matters seems equally sure, and that implies something very important about the nature of war.\n\nIt would be very odd to praise Rommel for not killing prisoners unless we simultaneously refused to blame him for Hitler's aggressive wars. For otherwise he is simply a criminal, and all the fighting he does is murder or attempted murder, whether he aims at soldiers in battle or at prisoners or at civilians. The chief British prosecutor at Nuremberg put this argument into the language of international law when he said, \"The killing of combatants is justifiable . . . only where the war itself is legal. But where the war is illegal . . . there is nothing to justify the killing and these murders are not to be distinguished from those of any other lawless robber bands.\" And then Rommel's case would be exactly like that of a man who invades someone else's home and kills only some of the inhabitants, sparing the children, say, or an aged grandmother: a murderer, no doubt, though not one without a drop of human kindness. But we don't view Rommel that way: why not? The reason has to do with the distinction of jus ad bellum and jus in bello. We draw a line between the war itself, for which soldiers are not responsible, and the conduct of the war, for which they are responsible, at least within their own sphere of activity. Generals may well straddle the line, but that only suggests that we know pretty well where it should be drawn. We draw it by recognizing the nature of political obedience. Rommel was a servant, not a ruler, of the German state; he did not choose the wars he fought but, like Prince Andrey, served his \"Tsar and country.\" We still have misgivings in his case, and will continue to have them, for he was more than just unlucky in his \"Tsar and country.\" But by and large we don't blame a soldier, even a general, who fights for his own government. He is not a member of a robber band, a willful wrongdoer, but a loyal and obedient subject and citizen, acting sometimes at great personal risk in a way he thinks is right. We allow him to say what an English soldier says in Shakespeare's Henry V: \"We know enough if we know we are the king's men. Our obedience to the king wipes the crime of it out of us.\" Not that his obedience can never be criminal; for when he violates the rules of war, superior orders are no defense. The atrocities that he commits are his own; the war is not. It is conceived, both in international law and in ordinary moral judgment, as the king's business\u2014a matter of state policy, not of individual volition, except when the individual is the king.\n\nIt might, however, be thought a matter of individual volition whether particular men join the army and participate in the war. Catholic writers have long argued that they ought not to volunteer, ought not to serve at all, if they know the war to be unjust. But the knowledge required by Catholic doctrine is hard to come by; and in case of doubt, argues the best of the Schoolmen, Francisco de Vitoria, subjects must fight\u2014the guilt falling, as in Henry V, on their leaders. Vitoria's argument suggests how firmly political life is set, even in the pre-modern state, against the very idea of volition in time of war. \"A prince is not able,\" he writes, \"and ought not always to render reasons for the war to his subjects, and if the subjects cannot serve in the war except they are first satisfied of its justice, the state would fall into grave peril. . . .\" Today, of course, most princes work hard to satisfy their subjects of the justice of their wars; they \"render reasons,\" though not always honest ones. It takes courage to doubt these reasons, or to doubt them in public; and so long as they are only doubted, most men will be persuaded (by arguments something like Vitoria's) to fight. Their routine habits of law-abidingness, their fear, their patriotism, their moral investment in the state, all favor that course. Or, alternatively, they are so terribly young when the disciplinary system of the state catches them up and sends them into war that they can hardly be said to make a moral decision at all:\n\nFrom my mother's sleep I fell into the State.\n\nAnd then how can we blame them for (what we perceive to be) the wrongful character of their war?a\n\nSoldiers are not, however, entirely without volition. Their will is independent and effective only within a limited sphere, and for most of them that sphere is narrow. But except in extreme cases, it never completely disappears. And at those moments in the course of the fighting when they must choose, like Rommel, to kill prisoners or let them live, they are not mere victims or servants bound to obedience; they are responsible for what they do. We shall have to qualify that responsibility when we come to consider it in detail, for war is still hell, and hell is a tyranny where soldiers are subject to all sorts of duress. But the judgments we actually make of their conduct demonstrate, I think, that within that tyranny we have carved out a constitutional regime: even the pawns of war have rights and obligations.\n\nDuring the past hundred years, these rights and obligations have been specified in treaties and agreements, written into international law. The very states that enlist the pawns of war have stipulated the moral character of their mutual slaughter. Initially, this stipulation was not based upon any notion of the equality of soldiers but upon the equality of sovereign states, which claimed for themselves the same right to fight (right to make war) that individual soldiers more obviously possess. The argument that I have made on behalf of soldiers was first made on behalf of states\u2014or rather on behalf of their leaders, who, we were told, are never willful criminals, whatever the character of the wars they begin, but statesmen serving the national interest as best they can. When I discuss the theory of aggression and of responsibility for aggression, I will have to explain why that is an inadequate description of what statesmen do. For now, it is enough to say that this view of sovereignty and political leadership, which was never in accord with ordinary moral judgment, has also lost its legal standing, replaced in the years since World War I by the formal designation of war-making as a criminal activity. However, the rules of engagement have not been replaced but expanded and elaborated, so that we now have both a ban on war and a code of military conduct. The dualism of our moral perceptions is established in the law.\n\nWar is a \"legal condition which equally permits two or more groups to carry on a conflict by armed force.\" It is also, and for our purposes more importantly, a moral condition, involving the same permissiveness, not in fact at the level of sovereign states, but at the level of armies and individual soldiers. Without the equal right to kill, war as a rule-governed activity would disappear and be replaced by crime and punishment, by evil conspiracies and military law enforcement. That disappearance seems to be heralded by the United Nations Charter, where the word \"war\" does not appear but only \"aggression,\" \"self-defense,\" \"international enforcement,\" and so on. But even the UN's \"police action\" in Korea was still a war, since the soldiers who fought in it were moral equals even if the states were not. The rules of war were as relevant there as in any other \"conflict by armed force,\" and they were equally relevant to the aggressor, the victim, and the police.\n\nTwo Sorts of Rules\n\nThe rules of war consist of two clusters of prohibitions attached to the central principle that soldiers have an equal right to kill. The first cluster specifies when and how they can kill, the second whom they can kill. My chief concern is with the second, for there the formulation and reformulation of the rules reach to one of the hardest questions in the theory of war\u2014that is, how those victims of war who can be attacked and killed are to be distinguished from those who cannot. I don't believe that this question must be answered in this or that specific way if war is to be a moral condition. It is necessary, however, that at any particular moment there be an answer. War is distinguishable from murder and massacre only when restrictions are established on the reach of battle.\n\nThe first set of rules does not involve any such fundamental issue. Rules specifying how and when soldiers can be killed are by no means unimportant, and yet the morality of war would not be radically transformed were they to be abolished altogether. Consider, for example, those battles described by anthropologists in which warriors fight with bows and unfeathered arrows. The arrows fly less accurately than they would if they were feathered; they can be dodged; few men are killed. It is clearly a good rule, then, that arrows not be feathered, and we may fairly condemn the warrior who first arms himself with the superior and forbidden weapon and hits his enemy. Yet the man he kills was liable to be killed in any case, and a collective (intertribal) decision to fight with feathered arrows would not violate any basic moral principle. The case is the same with all other rules of this kind: that soldiers be preceded into battle by a herald carrying a red flag, that fighting always be broken off at sunset, that ambushes and surprise attacks be prohibited, and so on. Any rule that limits the intensity and duration of combat or the suffering of soldiers is to be welcomed, but none of these restraints seem crucial to the idea of war as a moral condition. They are circumstantial in the literal sense of that word, highly particularized and local to a specific time and place. Even if in practice they endure for many years, they are always susceptible to the transformations brought about by social change, technological innovation, and foreign conquest.b\n\nThe second set of rules does not seem similarly susceptible. At least, the general structure of its provisions seems to persist without reference to social systems and technologies\u2014as if the rules involved were (as I think they are) more closely connected to universal notions of right and wrong. Their tendency is to set certain classes of people outside the permissible range of warfare, so that killing any of their members is not a legitimate act of war but a crime. Though their details vary from place to place, these rules point toward the general conception of war as a combat between combatants, a conception that turns up again and again in anthropological and historical accounts. It is most dramatically exemplified when war is actually a combat between military champions, as among many primitive peoples, or in the Greek epics, or in the biblical tale of David and Goliath. \"Let no man's heart fail within him,\" says David, \"thy servant will go and fight this Philistine.\" Once such a contest has been agreed upon, soldiers themselves are protected from the hell of war. In the Middle Ages, single combat was advocated for precisely this reason: \"Better for one to fall than the whole army.\" More often, however, protection has been offered only to those people who are not trained and prepared for war, who do not fight or cannot: women and children, priests, old men, the members of neutral tribes, cities, or states, wounded or captured soldiers.c What all these groups have in common is that they are not currently engaged in the business of war. Depending on one's social or cultural perspective, killing them may appear wanton, unchivalrous, dishonorable, brutal, or murderous. But it is very likely that some general principle is at work in all these judgments, connecting immunity from attack with military disengagement. Any satisfactory account of the moral reality of war must specify that principle and say something about its force. I shall attempt to do both these things later on.\n\nThe historical specifications of the principle are, however, conventional in character, and the war rights and obligations of soldiers follow from the conventions and not (directly) from the principle, whatever its force. Once again, war is a social creation. The rules actually observed or violated in this or that time and place are necessarily a complex product, mediated by cultural and religious norms, social structures, formal and informal bargaining between belligerent powers, and so on. Hence, the details of noncombatant immunity are likely to seem as arbitrary as the rules that determine when battles should start and stop or what weapons may be used. They are more important by far, but similarly subject to social revision. Exactly like law in domestic society, they will often represent an incomplete or distorted embodiment of the relevant moral principle. They are subject, then, to philosophical criticism. Indeed, criticism is a crucial part of the historical process through which the rules are made. We might say that war is a philosophical creation. Long before philosophers are satisfied with it, however, soldiers are bound by its canons. And they are equally bound, because of their own equality, and without reference to the content or the incompleteness of the canons.\n\nThe War Convention\n\nI propose to call the set of articulated norms, customs, professional codes, legal precepts, religious and philosophical principles, and reciprocal arrangements that shape our judgments of military conduct the war convention. It is important to stress that it is our judgments that are at issue here, not conduct itself. We cannot get at the substance of the convention by studying combat behavior, any more than we can understand the norms of friendship by studying the way friends actually treat one another. The norms are apparent, instead, in the expectations friends have, the complaints they make, the hypocrisies they adopt. So it is with war: relations between combatants have a normative structure that is revealed in what they say (and what the rest of us say) rather than in what they do\u2014though no doubt what they do, as with friends, is affected by what they say. Harsh words are the immediate sanctions of the war convention, sometimes accompanied or followed by military attacks, economic blockades, reprisals, war crimes trials, and so on. But neither the words nor the actions have any single authoritative source; and, finally, it is the words that are decisive\u2014the \"judgment of history,\" as it is called, which means the judgment of men and women arguing until some rough consensus is reached.\n\nThe terms of our judgments are most explicitly set forth in positive international law: the work of politicians and lawyers acting as representatives of sovereign states, and then of jurists codifying their agreements and searching out the rationale that underlies them. But international law arises out of a radically decentralized legislative system, cumbrous, unresponsive, and without a parallel judicial system to establish the specific details of the legal code. For that reason, the legal handbooks are not the only place to find the war convention, and its actual existence is demonstrated not by the existence of the handbooks but by the moral arguments that everywhere accompany the practice of war. The common law of combat is developed through a kind of practical casuistry. Hence the method of this book: we look to the lawyers for general formulas, but to historical cases and actual debates for those particular judgments that both reflect the war convention and constitute its vital force. I don't mean to suggest that our judgments, even over time, have an unambiguous collective form. Nor, however, are they idiosyncratic and private in character. They are socially patterned, and the patterning is religious, cultural, and political, as well as legal. The task of the moral theorist is to study the pattern as a whole, reaching for its deepest reasons.\n\nAmong professional soldiers, the war convention often finds advocates of a special kind. Though chivalry is dead and fighting unfree, professional soldiers remain sensitive (or some of them do) to those limits and restraints that distinguish their life's work from mere butchery. No doubt, they know with General Sherman that war is butchery, but they are likely to believe that it is also, simultaneously, something else. That is why army and navy officers, defending a long tradition, will often protest commands of their civilian superiors that would require them to violate the rules of war and turn them into mere instruments for killing. The protests are mostly unavailing\u2014for instruments, after all, they are\u2014but within their own sphere of decision, they often find ways to defend the rules. And even when they don't do that, their doubts at the time and justifications after the fact are an important guide to the substance of the rules. Sometimes, at least, it matters to soldiers just whom they kill.\n\nThe war convention as we know it today has been expounded, debated, criticized, and revised over a period of many centuries. Yet it remains one of the more imperfect of human artifacts: recognizably something that men have made, but not something that they have made freely or well. It is necessarily imperfect, I think, quite aside from the frailties of humankind, because it is adapted to the practice of modern war. It sets the terms of a moral condition that comes into existence only when armies of victims meet (just as the chivalric code sets the terms of a moral condition that comes into existence only when there are armies of free men). The convention accepts that victimization or at least assumes it, and starts from there. That is why it is often described as a program for the toleration of war, when what is needed is a program for its abolition. One does not abolish war by fighting it well; nor does fighting it well make it tolerable. War is hell, as I have already said, even when the rules are strictly observed. Just for that reason, we are sometimes made angry by the very idea of rules or cynical about their meaning. They only serve as Prince Andrey says in that impassioned outburst that evidently also expresses Tolstoy's conviction to make us forget that war is \"the vilest thing in life. . . .\"\n\nAnd what is war, what is needed for success in war, what are the morals of the military world? The object of warfare is murder; the means employed in warfare\u2014spying, treachery, and the encouragement of it, the ruin of a country, the plunder of its inhabitants . . . trickery and lying, which are called military strategy; the morals of the military class\u2014absence of all independence, that is, discipline, idleness, ignorance, cruelty, debauchery, and drunkenness.\n\nAnd yet, even people who believe all this are capable of being outraged by particular acts of cruelty and barbarism. War is so awful that it makes us cynical about the possibility of restraint, and then it is so much worse that it makes us indignant at the absence of restraint. Our cynicism testifies to the defectiveness of the war convention, and our indignation to its reality and strength.\n\nThe Example of Surrender\n\nAnomalous the convention often is, but binding nonetheless. Consider for a moment the common practice of surrendering, the detailed features of which are conventionally (and in our own time, legally) established. A soldier who surrenders enters into an agreement with his captors: he will stop fighting if they will accord him what the legal handbooks call \"benevolent quarantine.\" Since it is usually made under extreme duress, this is an agreement that would have no moral consequences at all in time of peace. In war it does have consequences. The captured soldier acquires rights and obligations specified by the convention, and these are binding without regard to the possible criminality of his captors or to the justice or urgency of the cause for which he has been fighting. Prisoners of war have a right to try to escape\u2014they cannot be punished for the attempt\u2014but if they kill a guard in order to escape, the killing is not an act of war; it is murder. For they committed themselves to stop fighting, gave up their right to kill, when they surrendered.\n\nIt is not easy to see all this as the simple assertion of a moral principle. It is the work of men and women (with moral principles in mind) adapting to the realities of war, making arrangements, striking bargains. No doubt, the bargain is generally useful to captives and captors alike, but it is not necessarily useful in every case to either of them or to mankind as a whole. If our purpose in this particular war is to win as soon as possible, the spectacle of a prison camp must seem strange indeed. Here are soldiers making themselves at home, settling in for the duration, dropping out of the war before it is over, and bound not to renew the fighting, even if they can (through sabotage, harassment, or whatever), because they promised at the point of a gun not to do so. Surely these are promises that can sometimes be broken. Yet prisoners are not invited to calculate the relative utilities of keeping or of breaking them. The war convention is written in absolutist terms: one violates its provisions at one's moral, as at one's physical peril. But what is the force of these provisions? They derive ultimately from principles that I will take up later on, which explain the meaning of quarter, disengagement, and immunity. They derive immediately and specifically from the consensual process itself. The rules of war, alien as they often are to our sense of what is best, are made obligatory by the general consent of mankind.\n\nNow that, too, is a consent given under a kind of duress. Only because there is no escape from hell, it might be said, have we labored to create a world of rules within it. But let us imagine an escape attempt, a liberation struggle, a \"war to end war.\" Surely it would be foolish then to fight according to the rules. The all-important task would be to win. But it is always important to win, for victory can always be described as an escape from hell. Even the victory of an aggressor, after all, ends the war. Hence the long history of impatience with the war convention. That history is nicely summed up in a letter written in 1880 by the Prussian chief of staff, General von Moltke, to protest the Declaration of St. Petersburg (an early effort to codify the rules of war): \"The greatest kindness in war,\" wrote von Moltke, \"is to bring it to a speedy conclusion. It should be allowable, with that view, to employ all means save those that are absolutely objectionable.\" Von Moltke stops short of a total denial of the war convention; he recognizes absolute prohibitions of some unspecified sort. Almost everyone does. But why stop short if that means falling short of the \"greatest kindness\"? This is the form of the most common argument in the \u00adtheory of war and of the most common moral dilemma in its practice. The war convention is found to stand in the way of victory and, it is usually said, a lasting peace. Must its provisions, must this particular provision be obeyed? When victory means the defeat of aggression, the question is not only important; it is painfully difficult. We want to have it both ways: moral decency in battle and victory in war; constitutionalism in hell and ourselves outside.\n\n But these young men, Robert Nozick argues, \"are certainly not encouraged to think for themselves by the practice of absolving them of all responsibility for their actions within the rules of war.\" That is right; they are not. But we cannot blame them in order to encourage the others unless they are actually blameworthy. Nozick insists that they are: \"It is a soldier's responsibility to determine if his side's cause is just. . . . \" The conventional refusal to impose that responsibility flatly and across the board is \"morally elitist.\" (Anarchy, State, and Utopia, New York, 1974, p. 100.) But it isn't elitist merely to recognize the existence of authority structures and socialization processes in the political community, and it may be morally insensitive not to. I do agree with Nozick that \"some bucks stop with each of us.\" A great deal of this book is concerned with trying to say which ones those are.\n\n They are also susceptible to the kind of reciprocal violation legitimized by the doctrine of reprisal: violated by one side, they can be violated by the other. But this does not seem to be true of the other sort of rules, described below. See the discussion of reprisals in chapter 13.\n\n The lists are often more specific and more picturesque than this, reflecting the character of a particular culture. Here is an example from an ancient Indian text, according to which the following groups of people are not to be subjected to the exigencies of battle: \"Those who look on without taking part, those afflicted with grief . . . those who are asleep, thirsty, or fatigued or are walking along the road, or have a task on hand unfinished, or who are proficient in fine art.\" (S. V. Viswanatha, International Law in Ancient India, Bombay, 1925, p. 156.)\nPart Two\n\nThe Theory of Aggression\n\nLaw and Order in International Society\n\nAggression\n\nAggression is the name we give to the crime of war. We know the crime because of our knowledge of the peace it interrupts\u2014not the mere absence of fighting, but peace-with-rights, a condition of liberty and security that can exist only in the absence of aggression itself. The wrong the aggressor commits is to force men and women to risk their lives for the sake of their rights. It is to confront them with the choice: your rights or (some of) your lives! Groups of citizens respond in different ways to that choice, sometimes surrendering, sometimes fighting, depending on the moral and material condition of their state and army. But they are always justified in fighting; and in most cases, given that harsh choice, fighting is the morally preferred response. The justification and the preference are very important: they account for the most remarkable features of the concept of aggression and for the special place it has in the theory of war.\n\nAggression is remarkable because it is the only crime that states can commit against other states: everything else is, as it were, a misdemeanor. There is a strange poverty in the language of international law. The equivalents of domestic assault, armed robbery, extortion, assault with intent to kill, murder in all its degrees, have but one name. Every violation of the territorial integrity or political sovereignty of an independent state is called aggression. It is as if we were to brand as murder all attacks on a man's person, all attempts to coerce him, all invasions of his home. This refusal of differentiation makes it difficult to mark off the relative seriousness of aggressive acts\u2014to distinguish, for example, the seizure of a piece of land or the imposition of a satellite regime from conquest itself, the destruction of a state's independence (a crime for which Abba Eban, Israel's foreign minister in 1967, suggested the name \"policide\"). But there is a reason for the refusal. All aggressive acts have one thing in common: they justify forceful resistance, and force cannot be used between nations, as it often can between persons, without putting life itself at risk. Whatever limits we place on the means and range of warfare, fighting a limited war is not like hitting somebody. Aggression opens the gates of hell. Shakespeare's Henry V makes the point exactly:\n\nFor never two such kingdoms did contend\n\nWithout much fall of blood, whose guiltless drops\n\nAre every one a woe, a sore complaint\n\n'Gainst him whose wrongs gives edge unto the swords\n\nThat makes such waste in brief mortality.\n\nAt the same time, aggression unresisted is aggression still, though there is no \"fall of blood\" at all. In domestic society, a robber who gets what he wants without killing anyone is obviously less guilty, that is, guilty of a lesser crime, than if he commits murder. Assuming that the robber is prepared to kill, we allow the behavior of his victim to determine his guilt. We don't do this in the case of aggression. Consider, for example, the German seizures of Czechoslovakia and Poland in 1939. The Czechs did not resist; they lost their independence through extortion rather than war; no Czech citizens died fighting the German invaders. The Poles chose to fight, and many were killed in the war that followed. But if the conquest of Czechoslovakia was a lesser crime, we have no name for it. At Nuremberg, the Nazi leadership was charged with aggression in both cases and found guilty in both. Once again, there is a reason for this identity of treatment. We judge the Germans guilty of aggression in Czechoslovakia, I think, because of our profound conviction that they ought to have been resisted\u2014though not necessarily by their abandoned victim, standing alone.\n\nThe state that does resist, whose soldiers risk their lives and die, does so because its leaders and people think that they should or that they have to fight back. Aggression is morally as well as physically coercive, and that is one of the most important things about it. \"A conqueror,\" writes Clausewitz, \"is always a lover of peace (as Bonaparte always asserted of himself); he would like to make his entry into our state unopposed; in order to prevent this, we must choose war. . . . \" If ordinary men and women did not ordinarily accept that imperative, aggression would not seem to us so serious a crime. If they accepted it in certain sorts of cases, but not in others, the single concept would begin to break down, and we would eventually have a list of crimes more or less like the domestic list. The challenge of the streets, \"Your money or your life!\" is easy to answer: I surrender my money and so I save myself from being murdered and the thief from being a murderer. But we apparently don't want the challenge of aggression answered in the same way; even when it is, we don't diminish the guilt of the aggressor. He has violated rights to which we attach enormous importance. Indeed, we are inclined to think that the failure to defend those rights is never due to a sense of their unimportance, nor even to a belief (as in the street-challenge case) that they are, after all, worth less than life itself, but only to a stark conviction that the defense is hopeless. Aggression is a singular and undifferentiated crime because, in all its forms, it challenges rights that are worth dying for.\n\nThe Rights of Political Communities\n\nThe rights in question are summed up in the lawbooks as territorial integrity and political sovereignty. The two belong to states, but they derive ultimately from the rights of individuals, and from them they take their force. \"The duties and rights of states are nothing more than the duties and rights of the men who compose them.\" That is the view of a conventional British lawyer, for whom states are neither organic wholes nor mystical unions. And it is the correct view. When states are attacked, it is their members who are challenged, not only in their lives, but also in the sum of things they value most, including the political association they have made. We recognize and explain this challenge by referring to their rights. If they were not morally entitled to choose their form of government and shape the policies that shape their lives, external coercion would not be a crime; nor could it so easily be said that they had been forced to resist in self-defense. Individual rights (to life and liberty) underlie the most important judgments that we make about war. How these rights are themselves founded I cannot try to explain here. It is enough to say that they are somehow entailed by our sense of what it means to be a human being. If they are not natural, then we have invented them, but natural or invented, they are a palpable feature of our moral world. States' rights are simply their collective form. The process of collectivization is a complex one. No doubt, some of the immediate force of individuality is lost in its course; it is best understood, nevertheless, as it has commonly been understood since the seventeenth century, in terms of social contract theory. Hence it is a moral process, which justifies some claims to territory and sovereignty and invalidates others.\n\nThe rights of states rest on the consent of their members. But this is consent of a special sort. State rights are not constituted through a series of transfers from individual men and women to the sovereign or through a series of exchanges among individuals. What actually happens is harder to describe. Over a long period of time, shared experiences and cooperative activity of many different kinds shape a common life. \"Contract\" is a metaphor for a process of association and mutuality, the ongoing character of which the state claims to protect against external encroachment. The protection extends not only to the lives and liberties of individuals but also to their shared life and liberty, the independent community they have made, for which individuals are sometimes sacrificed. The moral standing of any particular state depends upon the reality of the common life it protects and the extent to which the sacrifices required by that protection are willingly accepted and thought worthwhile. If no common life exists, or if the state doesn't defend the common life that does exist, its own defense may have no moral justification. But most states do stand guard over the community of their citizens, at least to some degree: that is why we assume the justice of their defensive wars. And given a genuine \"contract,\" it makes sense to say that territorial integrity and political sovereignty can be defended in exactly the same way as individual life and liberty.a\n\nIt might also be said that a people can defend its country in the same way as men and women can defend their homes, for the country is collectively as the homes are privately owned. The right to territory might be derived, that is, from the individual right to property. But the ownership of vast reaches of land is highly problematic, I think, unless it can be tied in some plausible way to the requirements of national survival and political independence. And these two seem by themselves to generate territorial rights that have little to do with ownership in the strict sense. The case is probably the same with the smaller properties of domestic society. A man has certain rights in his home, for example, even if he does not own it, because neither his life nor his liberty is secure unless there exists some physical space within which he is safe from intrusion. Similarly again, the right of a nation or people not to be invaded derives from the common life its members have made on their piece of land\u2014it had to be made somewhere\u2014and not from the legal title they hold or don't hold. But these matters will become clearer if we look at an example of disputed territory.\n\nThe Case of Alsace-Lorraine\n\nIn 1870, both France and the new Germany claimed these two provinces. Both claims were, as such things go, well founded. The Germans based themselves on ancient precedents (the lands had been part of the Holy Roman Empire before their conquest by Louis XIV) and on cultural and linguistic kinship; the French on two centuries of possession and effective government. How does one establish ownership in such a case? There is, I think, a prior question having to do with political allegiance, not with legal titles at all. What do the inhabitants want? The land follows the people. The decision as to whose sovereignty was legitimate (and therefore as to whose military presence constituted aggression) belonged by right to the men and women who lived on the land in dispute. Not simply to those who owned the land: the decision belonged to the landless, to town dwellers and factory workers as well, by virtue of the common life they had made. The great majority of these people were apparently loyal to France, and that should have settled the matter. Even if we imagine all the inhabitants of Alsace-Lorraine to be tenants of the Prussian king, the king's seizure of his own land would still have been a violation of their territorial integrity and, through the mediation of their loyalty, of France's too. For tenantry determines only where rents should go; the people themselves must decide where their taxes and conscripts should go.\n\nBut the issue was not settled in this way. After the Franco-Prussian war, the two provinces (actually, all of Alsace and a portion of Lorraine) were annexed by Germany, the French conceding German rights in the peace treaty of 1871. During the next several decades, the question was frequently asked, whether a French attack aimed at regaining the lost lands would be justified. One of the issues here is that of the moral standing of a peace treaty signed, as most peace treaties are signed, under duress, but I shall not focus on that. The more important issue relates to the endurance of rights over time. Here the appropriate argument was put forward by the English philosopher Henry Sidgwick in 1891. Sidgwick's sympathies were with the French, and he was inclined to regard the peace as a \"temporary suspension of hostilities, terminable at any time by the wronged state. . . .\" But he added a crucial qualification:\n\nWe must . . . recognize that by this temporary submission of the vanquished . . . a new political order is initiated, which, though originally without a moral basis, may in time acquire such a basis, from a change in the sentiments of the inhabitants of the territory transferred; since it is always possible that through the effects of time and habit and mild government\u2014and perhaps through the voluntary exile of those who feel the old patriotism most keenly\u2014the majority of the transferred population may cease to desire reunion. . . . When this change has taken place, the moral effect of the unjust transfer must be regarded as obliterated; so that any attempt to recover the transferred territory becomes itself an aggression. . . .\n\nLegal titles may endure forever, periodically revived and reasserted as in the dynastic politics of the Middle Ages. But moral rights are subject to the vicissitudes of the common life.\n\nTerritorial integrity, then, does not derive from property; it is simply something different. The two are joined, perhaps, in socialist states where the land is nationalized and the people are said to own it. Then if their country is attacked, it is not merely their homeland that is in danger but their collective property\u2014though I suspect that the first danger is more deeply felt than the second. Nationalization is a secondary process; it assumes the prior existence of a nation. And territorial integrity is a function of national existence, not of nationalization (any more than of private ownership). It is the coming together of a people that establishes the integrity of a territory. Only then can a boundary be drawn, the crossing of which is plausibly called aggression. It hardly matters if the territory belongs to someone else, unless that ownership is expressed in residence and common use.\n\nThis argument suggests a way of thinking about the great difficulties posed by forcible settlement and colonization. When barbarian tribes crossed the borders of the Roman Empire, driven by conquerors from the east or north, they asked for land to settle on and threatened war if they didn't get it. Was this aggression? Given the character of the Roman Empire, the question may sound foolish, but it has arisen many times since, and often in imperial settings. When land is in fact empty and available, the answer must be that it is not aggression. But what if the land is not actually empty but, as Thomas Hobbes says in Leviathan, \"not sufficiently inhabited\"? Hobbes goes on to argue that in such a case, the would-be settlers must \"not exterminate those they find there but constrain them to inhabit closer together.\" That constraint is not aggression, so long as the lives of the original settlers are not threatened. For the settlers are doing what they must do to preserve their own lives, and \"he that shall oppose himself against [that], for things superfluous, is guilty of the war that thereupon is to follow.\" It is not the settlers who are guilty of aggression, according to Hobbes, but those natives who won't move over and make room. There are clearly serious problems here. But I would suggest that Hobbes is right to set aside any consideration of territorial integrity-as-ownership and to focus instead on life. It must be added, however, that what is at stake is not only the lives of individuals but also the common life that they have made. It is for the sake of this common life that we assign a certain presumptive value to the boundaries that mark off a people's territory and to the state that defends it.\n\nNow, the boundaries that exist at any moment in time are likely to be arbitrary, poorly drawn, the products of ancient wars. The mapmakers are likely to have been ignorant, drunken, or corrupt. Nevertheless, these lines establish a habitable world. Within that world, men and women (let us assume) are safe from attack; once the lines are crossed, safety is gone. I don't want to suggest that every boundary dispute is a reason for war. Sometimes adjustments should be accepted and territories shaped so far as possible to the actual needs of nations. Good borders make good neighbors. But once an invasion has been threatened or has actually begun, it may be necessary to defend a bad border simply because there is no other. We shall see this reason at work in the minds of the leaders of Finland in 1939: they might have accepted Russian demands had they felt certain that there would be an end to them. But there is no certainty this side of the border, any more than there is safety this side of the threshold, once a criminal has entered the house. It is only common sense, then, to attach great importance to boundaries. Rights in the world have value only if they also have dimension.\n\nThe Legalist Paradigm\n\nIf states actually do possess rights more or less as individuals do, then it is possible to imagine a society among them more or less like the society of individuals. The comparison of international to civil order is crucial to the theory of aggression. I have already been making it regularly. Every reference to aggression as the international equivalent of armed robbery or murder, and every comparison of home and country or of personal liberty and political independence, relies upon what is called the domestic analogy. Our primary perceptions and judgments of aggression are the products of analogical reasoning. When the analogy is made explicit, as it often is among the lawyers, the world of states takes on the shape of a political society the character of which is entirely accessible through such notions as crime and punishment, self-defense, law enforcement, and so on.\n\nThese notions, I should stress, are not incompatible with the fact that international society as it exists today is a radically imperfect structure. As we experience it, that society might be likened to a defective building, founded on rights; its superstructure raised, like that of the state itself, through political conflict, cooperative activity, and commercial exchange; the whole thing shaky and unstable because it lacks the rivets of authority. It is like domestic society in that men and women live at peace within it (sometimes), determining the conditions of their own existence, negotiating and bargaining with their neighbors. It is unlike domestic society in that every conflict threatens the structure as a whole with collapse. Aggression challenges it directly and is much more dangerous than domestic crime, because there are no policemen. But that only means that the \"citizens\" of international society must rely on themselves and on one another. Police powers are distributed among all the members. And these members have not done enough in the exercise of their powers if they merely contain the aggression or bring it to a speedy end\u2014as if the police should stop a murderer after he has killed only one or two people and send him on his way. The rights of the member states must be vindicated, for it is only by virtue of those rights that there is a society at all. If they cannot be upheld (at least sometimes), international society collapses into a state of war or is transformed into a universal tyranny.\n\nFrom this picture, two presumptions follow. The first, which I have already pointed out, is the presumption in favor of military resistance once aggression has begun. Resistance is important so that rights can be maintained and future aggressors deterred. The theory of aggression restates the old doctrine of the just war: it explains when fighting is a crime and when it is permissible, perhaps even morally desirable.b The victim of aggression fights in self-defense, but he isn't only defending himself, for aggression is a crime against society as a whole. He fights in its name and not only in his own. Other states can rightfully join the victim's resistance; their war has the same character as his own, which is to say, they are entitled not only to repel the attack but also to punish it. All resistance is also law enforcement. Hence the second presumption: when fighting breaks out, there must always be some state against which the law can and should be enforced. Someone must be responsible, for someone decided to break the peace of the society of states. No war, as medieval theologians explained, can be just on both sides.\n\nThere are, however, wars that are just on neither side, because the idea of justice doesn't pertain to them or because the antagonists are both aggressors, fighting for territory or power where they have no right. The first case I have already alluded to in discussing the voluntary combat of aristocratic warriors. It is sufficiently rare in human history that nothing more need be said about it here. The second case is illustrated by those wars that Marxists call \"imperialist,\" which are not fought between conquerors and victims but between conquerors and conquerors, each side seeking dominion over the other or the two of them competing to dominate some third party. Thus Lenin's description of the struggles between \"have\" and \"have-not\" nations in early twentieth-century Europe: \". . . picture to yourselves a slave-owner who owned 100 slaves warring against a slave-owner who owned 200 slaves for a more 'just' distribution of slaves. Clearly, the application of the term 'defensive' war in such a case . . . would be sheer deception. . . . \" But it is important to stress that we can penetrate the deception only insofar as we can ourselves distinguish justice and injustice: the theory of imperialist war presupposes the theory of aggression. If one insists that all wars on all sides are acts of conquest or attempted conquest, or that all states at all times would conquer if they could, then the argument for justice is defeated before it begins and the moral judgments we actually make are derided as fantasies. Consider the following passage from Edmund Wilson's book on the American Civil War:\n\nI think that it is a serious deficiency on the part of historians . . . that they so rarely interest themselves in biological and zoological phenomena. In a recent . . . film showing life at the bottom of the sea, a primitive organism called a sea slug is seen gobbling up small organisms through a large orifice at one end of its body; confronted with another sea slug of an only slightly lesser size, it ingurgitates that, too. Now the wars fought by human beings are stimulated as a rule . . . by the same instincts as the voracity of the sea slug.\n\nThere are no doubt wars to which that image might be fit, though it is not a terribly useful image with which to approach the Civil War. Nor does it account for our ordinary experience of international society. Not all states are sea-slug states, gobbling up their neighbors. They are always groups of men and women who would live if they could in peaceful enjoyment of their rights and who have chosen political leaders who represent that desire. The deepest purpose of the state is not ingestion but defense, and the least that can be said is that many actual states serve that purpose. When their territory is attacked or their sovereignty challenged, it makes sense to look for an aggressor and not merely for a natural predator. Hence we need a theory of aggression rather than a zoological account.\n\nThe theory of aggression first takes shape under the aegis of the domestic analogy. I am going to call that primary form of the theory the legalist paradigm, since it consistently reflects the conventions of law and order. It does not necessarily reflect the arguments of the lawyers, though legal as well as moral debate has its starting point here. Later on, I will suggest that our judgments about the justice and injustice of particular wars are not entirely determined by the paradigm. The complex realities of international society drive us toward a revisionist perspective, and the revisions will be significant ones. But the paradigm must first be viewed in its unrevised form; it is our baseline, our model, the fundamental structure for the moral comprehension of war. We begin with the familiar world of individuals and rights, or crimes and punishments. The theory of aggression can then be summed up in six propositions.\n\n1. There exists an international society of independent states. States are the members of this society, not private men and women. In the absence of an universal state, men and women are protected and their interests represented only by their own governments. Though states are founded for the sake of life and liberty, they cannot be challenged in the name of life and liberty by any other states. Hence the principle of non-\u00adintervention, which I will analyze later on. The rights of private persons can be recognized in international society, as in the UN Charter of Human Rights, but they cannot be enforced without calling into question the dominant values of that society: the survival and independence of the separate political communities.\n\n2. This international society has a law that establishes the rights of its members\u2014above all, the rights of territorial integrity and political sovereignty. Once again, these two rest ultimately on the right of men and women to build a common life and to risk their individual lives only when they freely choose to do so. But the relevant law refers only to states, and its details are fixed by the intercourse of states, through complex processes of conflict and consent. Since these processes are continuous, international society has no natural shape; nor are rights within it ever finally or exactly determined. At any given moment, however, one can distinguish the territory of one people from that of another and say something about the scope and limits of sovereignty.\n\n3. Any use of force or imminent threat of force by one state against the political sovereignty or territorial integrity of another constitutes aggression and is a criminal act. As with domestic crime, the argument here focuses narrowly on actual or imminent boundary crossings: invasions and physical assaults. Otherwise, it is feared, the notion of resistance to aggression would have no determinate meaning. A state cannot be said to be forced to fight unless the necessity is both obvious and urgent.\n\n4. Aggression justifies two kinds of violent response: a war of self-\u00addefense by the victim and a war of law enforcement by the victim and any other member of international society. Anyone can come to the aid of a victim, use necessary force against an aggressor, and even make whatever is the international equivalent of a \"citizen's arrest.\" As in domestic society, the obligations of bystanders are not easy to make out, but it is the tendency of the theory to undermine the right of neutrality and to require widespread participation in the business of law enforcement. In the Korean War, this participation was authorized by the United Nations, but even in such cases the actual decision to join the fighting remains a unilateral one, best understood by analogy to the decision of a private citizen who rushes to help a man or woman attacked on the street.\n\n5. Nothing but aggression can justify war. The central purpose of the theory is to limit the occasions for war. \"There is a single and only just cause for commencing a war,\" wrote Vitoria, \"namely, a wrong received.\" There must actually have been a wrong, and it must actually have been received (or its receipt must be, as it were, only minutes away). Nothing else warrants the use of force in international society\u2014above all, not any difference of religion or politics. Domestic heresy and injustice are never actionable in the world of states: hence, again, the principle of nonintervention.\n\n6. Once the aggressor state has been militarily repulsed, it can also be punished. The conception of just war as an act of punishment is very old, though neither the procedures nor the forms of punishment have ever been firmly established in customary or positive international law. Nor are its purposes entirely clear: to exact retribution, to deter other states, to restrain or reform this one? All three figure largely in the literature, though it is probably fair to say that deterrence and restraint are most commonly accepted. When people talk of fighting a war against war, this is usually what they have in mind. The domestic maxim is, punish crime to prevent violence; its international analogue is, punish aggression to prevent war. Whether the state as a whole or only particular persons are the proper objects of punishment is a harder question, for reasons I will consider later on. But the implication of the paradigm is clear: if states are members of international society, the subjects of rights, they must also be (somehow) the objects of punishment.\n\nUnavoidable Categories\n\nThese propositions shape the judgments we make when wars break out. They constitute a powerful theory, coherent and economic, and they have dominated our moral consciousness for a long time. I am not concerned to trace their history here, but it is worth emphasizing that they remained dominant even during the eighteenth and nineteenth centuries, when lawyers and statesmen regularly argued that war-making was the natural prerogative of sovereign states, not subject to legal or moral judgment. States went to war for \"reasons of state,\" and these reasons were said to have a privileged character, such that they needed only to be alluded to, not even expounded, in order to terminate all argument. The common assumption in the legal literature of the time (roughly from the age of Vattel to that of Oppenheim) is that states always have, like Hobbist individuals, a right to fight. The analogy is not from domestic to international society, but from the state of nature to international anarchy. But this view never seized the popular imagination. \"The idea of war and the launching of it,\" writes the foremost historian of the theory of aggression, \"were for the ordinary man and for public opinion always loaded with moral significance, demanding full approval if waged with right and condemnation and punishment if without. . . .\" The significance ordinary men attached was exactly of the sort I have been describing: they drew the terrifying experience of war, as Otto von Bismarck once complained, back to the familiar ground of everyday life. \"Public opinion,\" Bismarck wrote, \"is only too ready to consider political relations and events in the light of those of civil law and private persons generally. . . . [This] shows a complete lack of understanding of political matters.\"\n\nI am inclined to think that it shows a deep understanding of political matters, though not always in its applications a knowledgeable or sophisticated understanding. Public opinion tends to focus on the concrete reality of war and on the moral meaning of killing and being killed. It addresses the questions that ordinary men cannot avoid: should we support this war? should we fight in it? Bismarck works from a more distant perspective, turning the people who ask such questions into pawns in the high game of real-politik. But ultimately the questions are insistent and the distant perspective untenable. Until wars are really fought with pawns, inanimate objects and not human beings, warfare cannot be isolated from moral life. We can get a clear view of the necessary links by reflecting on the work of one of Bismarck's contemporaries and on one of the wars at which the German chancellor connived.\n\nKarl Marx and the Franco-Prussian War\n\nLike Bismarck, Marx had a different way of understanding political matters. He regarded war not merely as the continuation but as the necessary and inevitable continuation of politics, and he described particular wars in terms of a world historical scheme. He had no commitment to the existing political order, nor to the territorial integrity or political sovereignty of established states. The violation of these \"rights\" raised no moral problem for him; he did not seek the punishment of aggressors; he sought only those outcomes that, without reference to the theory of aggression, advanced the cause of proletarian revolution. It is entirely characteristic of Marx's general views that he should have hoped for a Prussian victory in 1870 because it would lead to German unification and ease the course of socialist organization in the new Reich and because it would establish the dominance of the German over the French working class.\n\nThe French need a drubbing [he wrote in a letter to Engels]. If the Prussians are victorious, then the centralization of state power will be favorable to the centralization of the working class. German preponderance will shift the center of the working class movement in Western Europe from France to Germany and . . . the German working class is theoretically and organizationally superior to that of France. The superiority of the Germans over the French . . . would mean at the same time the superiority of our theory over Proudhon's, etc.\n\nBut this was not a view that Marx could defend in public, not only because its publication would embarrass him among his French comrades, but for reasons that go directly to the nature of our moral life. Even the most advanced members of the German working class would not be willing to kill French workers for the sake of German unity or to risk their own lives merely in order to enhance the power of their party (or of Marx's theory!) within the ranks of international socialism. Marx's argument was not, in the most literal sense of the word, a possible account of the decision to fight or of the judgment that the war the Germans fought was, at least initially, a just war. If we are to understand that judgment, we would do better to begin with the simplistic assertion of a British member of the General Council of the International: \"The French,\" said John Weston, \"had invaded first.\"\n\nWe know now that Bismarck worked hard and with all his usual ruthlessness to bring about that invasion. The diplomatic crisis that preceded the war was largely of his contrivance. Nothing that he did, however, can plausibly be said to have threatened the territorial integrity or political sovereignty of France; nothing that he did forced the French to fight. He merely exploited the arrogance and stupidity of Napoleon III and his entourage and succeeded in putting the French in the wrong; it was the tribute he paid to the public opinion he deplored. Hence it has never been necessary to correct the argument of John Weston or of those members of the German Social Democratic Workers' Party who declared in July 1870 that it was Napoleon who had \"frivolously\" destroyed the peace of Europe: \"The German nation . . . is the victim of aggression. Therefore . . . with great regret, [we] must accept the defensive war as a necessary evil.\" The \"First Address\" of the International on the Franco-Prussian War, drafted by Marx on behalf of the General Council, took the same view: \"On the German side, the war is a war of defense\" (though Marx went on to ask, \"Who put Germany to the necessity of defending herself?\" and to hint at the true character of Bismarckian politics). French workers were called upon to oppose the war and to drive the Bonapartists from power; German workers were urged to join the war, but in such a manner as to maintain \"its strictly defensive character.\"\n\nSome six weeks later, the war of defense was over, Germany was triumphant at Sedan, Bonaparte a prisoner, his empire overthrown. But the fighting continued, for the chief war aim of the German government was not resistance but expansion: the annexation of Alsace-Lorraine. In the \"Second Address\" of the International, Marx accurately described the war after Sedan as an act of aggression against the people of the two provinces and against the territorial integrity of France. He did not believe that either the German workers or the new French republic would be capable of punishing that aggression in the near future, but he looked for punishment nonetheless: \"History will measure its retribution, not by the extent of the square miles conquered from France, but by the intensity of the crime of reviving, in the second half of the nineteen century, the policy of conquest.\" What is striking here is that Marx has enlisted history not in the service of the proletarian revolution but in the service of conventional morality. Indeed, he invokes the example of the Prussian struggle against the first Napoleon after Tilset and so suggests that the retribution he has in mind will take the form of a future French attack on the German Reich, a war of exactly the sort that Henry Sidgwick also thought justified by the German \"policy of conquest.\" But whatever Marx's program, it is clear that he is working within the terms set by the theory of aggression. When he is forced to confront the actualities of war and to describe in public the possible shape of a socialist foreign policy, he falls back upon the domestic analogy and the legalist paradigm in their most literal forms. Indeed, he argued in the \"First Address\" that it was the task of socialists \"to vindicate the simple laws of morals and justice, which ought to govern the relations of private individuals, as the rules paramount of the intercourse of nations.\"\n\nIs this Marxist doctrine? I am not sure. It has little in common with Marx's philosophic pronouncements on morality and little in common with the reflections on international politics that fill his letters. But Marx was not only a philosopher and a letter-writer; he was also a political leader and the spokesman of a mass movement. In these latter roles, his world-historical view of the significance of war was less important than the particular judgments he was called upon to make. And once he was committed to judgment, there was a certain inevitability to the categories of the theory of aggression. It was not a question of adjusting himself to what is sometimes condescendingly called the \"level of consciousness\" of his audience, but of speaking directly to the moral experience of its members. Sometimes, perhaps, a new philosophy or religion can reshape that experience, but this was not the effect of Marxism, at least not with regard to international warfare. Marx simply took the theory of aggression seriously, and so he placed himself in the front ranks of those ordinary men and women about whom Bismarck complained, who judged political events in the light of domestic morality.\n\nThe Argument for Appeasement\n\nThe war of 1870 is a hard case because, with the exception of those French liberals and socialists who challenged Bonaparte and those German social-democrats who condemned the annexation of Alsace-Lorraine, none of its participants are very attractive. The moral issues are muddy, and it would not be difficult to argue that the struggle was in fact an aggressive war on both sides, rather than on each in succession. But the issues are not always muddy; history provides wonderfully clear examples of aggression. The historical study of war virtually begins with such an example (with which I also began): the Athenian attack on Melos. But the easy cases raise problems of their own, or rather, one characteristic problem. Aggression most often takes the form of an attack by a powerful state upon a weak one (that is why it is so readily recognizable). Resistance seems imprudent, even hopeless. Many lives will be lost, and to what end? Even here, however, our moral preference holds. We not only justify resistance; we call it heroic; we do not measure the value of justice, apparently, in terms of lives lost. And yet such measurements can never be entirely irrelevant: who would want to be ruled by political leaders who paid them no mind? So justice and prudence stand in an uneasy relation to one another. Later on, I will describe various ways in which the argument for justice incorporates prudential considerations. But now it is important to stress that the legalist paradigm tends in a radical way to exclude them.\n\nThe paradigm as a whole is commonly defended in utilitarian terms: resistance to aggression is necessary to deter future aggressors. But in the context of international politics, an alternative utilitarian argument is almost always available. This is the argument for appeasement, which suggests that giving in to aggressors is the only way of avoiding war. In domestic society, too, we sometimes choose appeasement, negotiating with kidnappers or extortionists, for example, when the costs of refusal or resistance are greater than we can bear. But we feel badly in such cases, not only because we have failed to serve the larger communal purpose of deterrence, but also and more immediately because we have yielded to coercion and injustice. We feel badly even though all that we have yielded is money, whereas in international society appeasement is hardly possible unless we are willing to surrender values far more important. And yet the costs of war are such that the argument for surrender can often be put very strongly. Appeasement is a bad word in our moral vocabulary, but the argument is not morally obtuse. It represents the most significant challenge to what I have been calling the presumption in favor of resistance, and I want now to examine it in some detail.\n\nCzechoslovakia and the Munich Principle\n\nThe defense of appeasement in 1938 sometimes involved the claim that the Sudeten Germans were, after all, entitled to self-determination. But that is a claim that might have been met through some sort of autonomy within the Czech state or through boundary changes considerably less drastic than those that Hitler demanded at Munich. In fact, Hitler's goals reached far beyond the vindication of a right, and Chamberlain and Daladier knew this, or should have known it, and surrendered anyway. It was the fear of war rather than any view of justice that explains their actions. This fear was given theoretical expression in a very intelligent little book, published in 1939 by the English Catholic writer Gerald Vann. Vann's argument is the only attempt that I have come across to apply just war theory directly to the problem of appeasement, and for that reason I shall look at it closely. He defends what might be called the \"Munich principle\":\n\nIf a nation finds itself called upon to defend another nation which is unjustly attacked and to which it is bound by treaty, then it is bound to fulfill its obligations. . . . It may, however, be its right, and even its duty, to try to persuade the victim of aggression to avoid the ultimate evil of a general conflict by agreeing to terms less favorable than those which it can claim in justice . . . provided always that such a surrender of rights would not mean in fact a surrender once and for all to the rule of violence.\n\nThe \"duty\" here is simply \"seek peace\"\u2014Hobbes' first law of nature and presumably near the top of Catholic lists as well, though Vann's phrase \"the ultimate evil of a general conflict\" suggests that it is nearer to the top than in fact it is. In just-war doctrine, as in the legalist paradigm, the triumph of aggression is a greater evil. But it is certainly a duty to avoid violence if one possibly can; this is a duty that the rulers of states owe to their own people and to others as well, and it may override obligations established by international treaties and conventions. But the argument requires the limiting clause at the end, which I would have thought applicable in September 1938. That clause is worth examining, since its purpose is obviously to tell us when to appease and when not.\n\nImagine a state whose government strives to press its boundaries or its sphere of influence outward, a little bit here, a little bit there, continually over a period of time\u2014not quite Edmund Wilson's sea-slug state, something nearer to a conventional \"great power.\" Certainly the people against whom the pressure is being brought have a right to resist; Allied states and possibly other states as well ought to support their resistance. But appeasement, by the victim or the others, would not necessarily be immoral\u2014this is Vann's argument\u2014and there might even be a duty to seek peace at the expense of justice. Appeasement would involve a surrender to violence, but given a conventional power, it would not or might not involve absolute subjection to the \"rule of violence.\" I take it that absolute subjection is what Vann means by \"once and for all.\" He cannot mean \"forever,\" for governments fall, states decay, people rebel; we know nothing about forever. \"Rule of violence\" is a more difficult term. Vann can hardly set the limit of appeasement at the point where it means yielding to greater physical force; that is always what it means. As a moral limit, the phrase must point to something more unusual and more frightening: the rule of men committed to the continual use of violence, to a policy of genocide, terrorism, and enslavement. Then appeasement would be, quite simply, a failure to resist evil in the world.\n\nNow that is exactly what the Munich agreement was. Vann's argument, once we have understood its terms, undermines his own case. For there can be no doubt that Nazism represented the rule of violence, and that its true character was sufficiently known at the time. And there can be no doubt that Czechoslovakia was surrendered to Nazism in 1938; the remnants of its territory and sovereignty could not be defended\u2014at least not by the Czechs\u2014and that, too, was known at the time. But it remains a question whether Vann's argument might not apply to other cases. I will skip the Polish war, for the Poles were confronted again by Nazi aggression and had, no doubt, learned from the Czech experience. But the situation of Finland a few months later was different. There the \"Munich principle\" was urged by all of Finland's friends and by many Finns as well. It did not seem to them, despite the Czech experience, that an acceptance of Russian terms in the late fall of 1939 would have been \"a surrender once and for all to the rule of violence.\"\n\nFinland\n\nStalin's Russia was not a conventional great power, but its behavior in the months before the Finnish war was very much in the style of traditionalist power politics. It sought to expand at the expense of the Finns, but the demands it made were moderate, closely linked to questions of military security, without revolutionary implications. What was at issue, Stalin insisted, was nothing more than the defense of Leningrad, which was then within artillery range of the Finnish border (he did not fear a Finnish attack but a German attack from Finnish territory). \"Since we cannot move Leningrad,\" he said, \"we must move the border.\" The Russians offered to yield more land (though less valuable land) than they sought to take over, and that offer gave the negotiations at least something of the character of an exchange between sovereign states. At an early point in the talks, Marshal Mannerheim, who had no illusions about Soviet policy, strongly recommended making the deal. It was more dangerous for Finland than for Russia for the Finns to be so close to Leningrad. Stalin may well have intended an eventual annexation of Finland, or its transformation into a communist state, but that was not apparent at the time. Most Finns thought the danger, though serious enough, was something less than that. They feared further encroachments and pressures of a more ordinary kind. Hence the Finnish case offers a useful test of the \"Munich principle.\" Should Finland have agreed to terms less favorable than it could justly claim in order to avoid the carnage of war? Should its allies have pressed such terms upon it?\n\nThe first question cannot be answered flatly either way; the choice belongs to the Finns. But the rest of us have an interest, and it is impor\u00adtant to try to understand the moral satisfaction with which their decision to fight was greeted throughout the world. I am not referring here to the excitement that always attends the beginnings of a war and that rarely lasts for long, but rather to the sense that the Finnish decision was exemplary (as the British, French, and Czech decision to surrender, greeted with an uneasy combination of relief and shame, was not). There is, of course, a natural sympathy for the underdog in any competition, including war, and a hope that he can pull off an unexpected victory. But in the case of war, this is specifically a moral sympathy and a moral hope. It has to do with the perception that underdogs are also (usually) victims or potential victims: their struggle is right. Even if national survival is not at stake\u2014as in fact it was, for the Finns, once the war began\u2014we hope for the defeat of the aggressor in much the same way as we hope for the defeat of a neighborhood bully, even if he is not a murderer. Our common values are confirmed and enhanced by the struggle; whereas appeasement, even when it is the better part of wisdom, diminishes those values and leaves us all impoverished.\n\nOur values would also have been diminished, however, had Stalin quickly overwhelmed the Finns and then treated them as the Athenians did the Melians. But that suggests less the desirability of surrender than the critical importance of collective security and resistance. Had Sweden, for example, been publicly committed to send troops to fight with the Finns, there would probably never have been a Russian attack. And the British and French plans to come to Finland's aid, inept and self-serving as these were, probably played a decisive part, along with the early and unexpected victories of the Finnish army, in persuading the Russians to seek a negotiated settlement. The new borders established in March 1940 were far worse than those that had been offered to Finland four months earlier; thousands of Finnish soldiers (and a greater number of Russians) were dead; hundreds of thousands of Finnish civilians were driven from their homes. But against all this must be set the vindication of Finnish independence. I don't know how one strikes the balance, still less how one might have done so in 1939 when vindication seemed an unlikely or at best a chancy prospect. Nor can its value be measured even now; it involves national pride and self-respect as much as freedom in policy-making (which no state possesses absolutely and Finland, since 1940, to a lesser degree than many). If the Finnish war is commonly thought to have been worthwhile, it is because independence is not a value that can easily be traded off.c\n\nThe \"Munich principle\" would concede the loss or erosion of independence for the sake of the survival of individual men and women. It points toward a certain sort of international society, founded not on the defense of rights but on the adjustment to power. No doubt there is realism in this view. But the Finnish example suggests that there is also realism in the alternative view, and in a twofold sense. First, the rights are real, even to the people who must die to defend them; and second, the defense is (sometimes) possible. I don't want to argue that appeasement can never be justified, only to point to the great importance we collectively attach to the values the aggressor attacks. These values are summed up in the existence of states like Finland\u2014indeed, of many such states. The theory of aggression presupposes our commitment to a pluralist world, and that commitment is also the inner meaning of the presumption in favor of resistance. We want to live in an international society where communities of men and women freely shape their separate destinies. But that society is never fully realized; it is never safe; it must always be defended. The Finnish war is a paradigmatic example of the necessary defense. That is why, for all the complexity of the diplomatic maneuvering that preceded the war, the actual fighting has about it a great moral simplicity.\n\nThe defense of rights is a reason for fighting. I want now to stress again, and finally, that it is the only reason. The legalist paradigm rules out every other sort of war. Preventive wars, commercial wars, wars of expansion and conquest, religious crusades, revolutionary wars, military interventions\u2014all these are barred and barred absolutely, in much the same way as their domestic equivalents are ruled out in municipal law. Or, to turn the argument around once more, all these constitute aggressive acts on the part of whoever begins them and justify forceful resistance, as their equivalents would in the homes and streets of domestic society.\n\nBut this is not yet a complete characterization of the morality of war. Though the domestic analogy is an intellectual tool of critical importance, it doesn't offer an entirely accurate picture of international society. States are not in fact like individuals (because they are collections of individuals), and the relations among states are not like the private dealings of men and women (because they are not framed in the same way by authoritative law). These differences are not unknown or obscure. I have been ignoring them only for the sake of analytical clarity. I have wanted to argue that as an account of our moral judgments, the domestic analogy and the legalist paradigm possess great explanatory power. The account is still incomplete, however, and I must look now at a series of issues and historical cases that suggest the need for revision. I cannot exhaust the range of possible revision, for our moral judgments are enormously subtle and complex. But the major points at which the argument for justice requires the amendment of the paradigm are clear enough; they have long been the focus of legal and moral debate.\n\n The question of when territory and sovereignty can rightly be defended is closely connected to the question of when individual citizens have an obligation to join the defense. Both hang on issues in social contract theory. I have discussed the second question at length in my book Obligations: Essays on Disobedience, War, and Citizenship (Cambridge, Mass., 1970). See especially \"The Obligation to Die for the State\" and \"Political Alienation and Military Service.\" But neither in that book nor in this one do I deal in any detail with the problem of national minorities\u2014groups of people who do not fully join (or do not join at all) in the contract that constitutes the nation. The radical mistreatment of such people may justify military intervention (see chapter 6). Short of that, however, the presence of national minorities within the borders of a nation-state does not affect the argument about aggression and self-defense.\n\n I shall say nothing here of the argument for nonviolent resistance to aggression, according to which fighting is neither desirable nor necessary. This argument has not figured much in the development of the conventional view. Indeed, it poses a radical challenge to the conventions: if aggression can be resisted, and at least sometimes successfully resisted, without war, it may be a less serious crime than has commonly been supposed. I will take up this possibility and its moral implications in the Afterword.\n\n It is probably less important, then, that these calculations be rightly made (since we cannot be sure what that would mean) than that they be made by the right people. One might usefully compare the decisions of the Melians and the Finns in this regard. Melos was an oligarchy, and its leaders, who wanted to fight, refused to allow the Athenian generals to address a popular assembly. Presumably they feared that the people would refuse to risk their lives and their city for the oligarchs. Finland was a democracy; its people knew the exact nature of the Russian demands; and the government's decision to fight apparently had overwhelming popular support. It would fit well with the rest of the theory of aggression if the Finns were again taken as exemplary: the decision to reject appeasement is best made by the men and women who will have to endure the war that follows (or by their representatives). This says nothing, of course, about the arguments one might want to make in the popular assembly: these might well be prudential and cautionary rather than defiant and heroic.\n\nAnticipations\n\nThe first questions asked when states go to war are also the easiest to answer: who started the shooting? who sent troops across the border? These are questions of fact, not of judgment, and if the answers are disputed, it is only because of the lies that governments tell. The lies don't, in any case, detain us long; the truth comes out soon enough. Governments lie so as to absolve themselves from the charge of aggression. But it is not on the answers to questions such as these that our final judgments about aggression depend. There are further arguments to make, justifications to offer, lies to tell, before the moral issue is directly confronted. For aggression often begins without shots being fired or borders crossed.\n\nBoth individuals and states can rightfully defend themselves against violence that is imminent but not actual; they can fire the first shots if they know themselves about to be attacked. This is a right recognized in domestic law and also in the legalist paradigm for international society. In most legal accounts, however, it is severely restricted. Indeed, once one has stated the restrictions, it is no longer clear whether the right has any substance at all. Thus the argument of Secretary of State Daniel Webster in the Caroline case of 1842 (the details of which need not concern us here): in order to justify pre-emptive violence, Webster wrote, there must be shown \"a necessity of self-defense . . . instant, overwhelming, leaving no choice of means, and no moment for deliberation.\" That would permit us to do little more than respond to an attack once we had seen it coming but before we had felt its impact. Pre-emption on this view is like a reflex action, a throwing up of one's arms at the very last minute. But it hardly requires much of a \"showing\" to justify a movement of that sort. Even the most presumptuous aggressor is not likely to insist, as a matter of right, that his victims stand still until he lands the first blow. Webster's formula seems to be the favored one among students of international law, but I don't believe that it addresses itself usefully to the experience of imminent war. There is often plenty of time for deliberation, agonizing hours, days, even weeks of deliberation, when one doubts that war can be avoided and wonders whether or not to strike first. The debate is couched, I suppose, in strategic more than in moral terms. But the decision is judged morally, and the expectation of that judgment, of the effects it will have in allied and neutral states and among one's own people, is itself a strategic factor. So it is important to get the terms of the judgment right, and that requires some revision of the legalist paradigm. For the paradigm is more restrictive than the judgments we actually make. We are disposed to sympathize with potential victims even before they confront an instant and overwhelming necessity.\n\nImagine a spectrum of anticipation: at one end is Webster's reflex, necessary and determined; at the other end is preventive war, an attack that responds to a distant danger, a matter of foresight and free choice. I want to begin at the far end of the spectrum, where danger is a matter of judgment and political decision is unconstrained, and then edge my way along to the point where we currently draw the line between justified and unjustified attacks. What is involved at that point is something very different from Webster's reflex; it is still possible to make choices, to begin the fighting or to arm oneself and wait. Hence the decision to begin at least resembles the decision to fight a preventive war, and it is important to distinguish the criteria by which it is defended from those that were once thought to justify prevention. Why not draw the line at the far end of the spectrum? The reasons are central to an understanding of the position we now hold.\n\nPreventive War and the Balance of Power\n\nPreventive war presupposes some standard against which danger is to be measured. That standard does not exist, as it were, on the ground; it has nothing to do with the immediate security of boundaries. It exists in the mind's eye, in the idea of a balance of power, probably the dominant idea in international politics from the seventeenth century to the present day. A preventive war is a war fought to maintain the balance, to stop what is thought to be an even distribution of power from shifting into a relation of dominance and inferiority. The balance is often talked about as if it were the key to peace among states. But it cannot be that, else it would not need to be defended so often by force of arms. \"The balance of power, the pride of modern policy . . . invented to preserve the general peace as well as the freedom of Europe,\" wrote Edmund Burke in 1760, \"has only preserved its liberty. It has been the origin of innumerable and fruitless wars.\" In fact, of course, the wars to which Burke is referring are easily numbered. Whether or not they were fruitless depends upon how one views the connection between preventive war and the preservation of liberty. Eighteenth-\u00adcentury British statesmen and their intellectual supporters obviously thought the connection very close. A radically unbalanced system, they recognized, would more likely make for peace, but they were \"alarmed by the danger of universal monarchy.\"a When they went to war on behalf of the balance, they thought they were defending, not national interest alone, but an international order that made liberty possible throughout Europe.\n\nThat is the classic argument for prevention. It requires of the rulers of states, as Francis Bacon had argued a century earlier, that they \"keep due sentinel, that none of their neighbors do overgrow so (by increase of territory, by embracing of trade, by approaches, or the like) as they become more able to annoy them, than they were.\" And if their neighbors do \"overgrow,\" then they must be fought, sooner rather than later, and without waiting for the first blow. \"Neither is the opinion of some of the Schoolmen to be received: that a war cannot justly be made, but upon a precedent injury or provocation. For there is no question, but a just fear of an imminent danger, though no blow be given, is a lawful cause of war.\" Imminence here is not a matter of hours or days. The sentinels stare into temporal as well as geographic distance as they watch the growth of their neighbor's power. They will fear that growth as soon as it tips or seems likely to tip the balance. War is justified (as in Hobbes' philosophy) by fear alone and not by anything other states actually do or any signs they give of their malign intentions. Prudent rulers assume malign intentions.\n\nThe argument is utilitarian in form; it can be summed up in two propositions: (1) that the balance of power actually does preserve the liberties of Europe (perhaps also the happiness of Europeans) and is therefore worth defending even at some cost, and (2) that to fight early, before the balance tips in any decisive way, greatly reduces the cost of the defense, while waiting doesn't mean avoiding war (unless one also gives up liberty) but only fighting on a larger scale and at worse odds. The argument is plausible enough, but it is possible to imagine a second-level utilitarian response: (3) that the acceptance of propositions (1) and (2) is dangerous (not useful) and certain to lead to \"innumerable and fruitless wars\" whenever shifts in power relations occur; but increments and losses of power are a constant feature of international politics, and perfect equilibrium, like perfect security, is a utopian dream; therefore it is best to fall back upon the legalist paradigm or some similar rule and wait until the overgrowth of power is put to some overbearing use. This is also plausible enough, but it is important to stress that the position to which we are asked to fall back is not a prepared position, that is, it does not itself rest on any utilitarian calculation. Given the radical uncertainties of power politics, there probably is no practical way of making out that position\u2014deciding when to fight and when not\u2014on utilitarian principles. Think of what one would have to know to perform the calculations, of the experiments one would have to conduct, the wars one would have to fight\u2014and leave unfought! In any case, we mark off moral lines on the anticipation spectrum in an entirely different way.\n\nIt isn't really prudent to assume the malign intent of one's neighbors; it is merely cynical, an example of the worldly wisdom which no one lives by or could live by. We need to make judgments about our neighbor's intentions, and if such judgments are to be possible we must stipulate certain acts or sets of acts that will count as evidence of malignity. These stipulations are not arbitrary; they are generated, I think, when we reflect upon what it means to be threatened. Not merely to be afraid, though rational men and women may well respond fearfully to a genuine threat, and their subjective experience is not an unimportant part of the argument for anticipation. But we also need an objective standard, as Bacon's phrase \"just fear\" suggests. That standard must refer to the threatening acts of some neighboring state, for (leaving aside the dangers of natural disaster) I can only be threatened by someone who is threatening me, where \"threaten\" means what the dictionary says it means: \"to hold out or offer (some injury) by way of a threat, to declare one's intention of inflicting injury.\" It is with some such notion as this that we must judge the wars fought for the sake of the balance of power. Consider, then, the Spanish Succession, regarded in the eighteenth century as a paradigmatic case for preventive war, and yet, I think, a negative example of threatening behavior.\n\nThe War of the Spanish Succession\n\nWriting in the 1750s, the Swiss jurist Vattel suggested the following criteria for legitimate prevention: \"Whenever a state has given signs of injustice, rapacity, pride, ambition, or of an imperious thirst of rule, it becomes a suspicious neighbor to be guarded against: and at a juncture when it is on the point of receiving a formidable augmentation of power, securities may be asked, and on its making any difficulty to give them, its designs may be prevented by force of arms.\" These criteria were formulated with explicit reference to the events of 1700 and 1701, when the King of Spain, last of his line, lay ill and dying. Long before those years, Louis XIV had given Europe evident signs of injustice, rapacity, pride, and so on. His foreign policy was openly expansionist and aggressive (which is not to say that justifications were not offered, ancient claims and titles uncovered, for every intended territorial acquisition). In 1700, he seemed about to receive a \"formidable augmentation of power\"\u2014his grandson, the Duke of Anjou, was offered the Spanish throne. With his usual arrogance, Louis refused to provide any assurances or guarantees to his fellow monarchs. Most importantly, he refused to bar Anjou from the French succession, thus holding open the possibility of a unified and powerful Franco-\u00adSpanish state. And then, an alliance of European powers, led by Great Britain, went to war against what they assumed was Louis' \"design\" to dominate Europe. Having drawn his criteria so closely to his case, however, Vattel concludes on a sobering note: \"it has since appeared that the policy [of the Allies] was too suspicious.\" That is wisdom after the fact, of course, but still wisdom, and one would expect some effort to restate the criteria in its light.\n\nThe mere augmentation of power, it seems to me, cannot be a warrant for war or even the beginning of warrant, and for much the same reason that Bacon's commercial expansion (\"embracing of trade\") is also and even more obviously insufficient. For both of these suggest developments that may not be politically designed at all and hence cannot be taken as evidence of intent. As Vattel says, Anjou had been invited to his throne \"by the [Spanish] nation, conformably to the will of its last sovereign\"\u2014that is, though there can be no question here of democratic decision-\u00admaking, he had been invited for Spanish and not for French reasons. \"Have not these two Realms,\" asked Jonathan Swift in a pamphlet opposing the British war, \"their separate maxims of Policy? . . .\" Nor is Louis' refusal to make promises relating to some future time to be taken as evidence of design\u2014only, perhaps, of hope. If Anjou's succession made immediately for a closer alliance between Spain and France, the appropriate answer would seem to have been a closer alliance between Britain and Austria. Then one could wait and judge anew the intentions of Louis.\n\nBut there is a deeper issue here. When we stipulate threatening acts, we are looking not only for indications of intent, but also for rights of response. To characterize certain acts as threats is to characterize them in a moral way, and in a way that makes a military response morally comprehensible. The utilitarian arguments for prevention don't do that, not because the ways they generate are too frequent, but because they are too common in another sense: too ordinary. Like Clausewitz's description of war as the continuation of policy by other means, they radically underestimate the importance of the shift from diplomacy to force. They don't recognize the problem that killing and being killed poses. Perhaps the recognition depends upon a certain way of valuing human life, which was not the way of eighteenth-century statesmen. (How many of the British soldiers who shipped to the continent with Marlborough ever returned? Did anyone bother to count?) But the point is an important one anyway, for it suggests why people have come to feel uneasy about preventive war. We don't want to fight until we are threatened, because only then can we rightly fight. It is a question of moral security. That is why Vattel's concluding remark about the War of the Spanish Succession, and Burke's general argument about the fruitlessness of such wars, is so worrying. It is inevitable, of course, that political calculations will sometimes go wrong; so will moral choices; there is no such thing as perfect security. But there is a great difference, nonetheless, between killing and being killed by soldiers who can plausibly be described as the present instruments of an aggressive intention, and killing and being killed by soldiers who may or may not represent a distant danger to our country. In the first case, we confront an army recognizably hostile, ready for war, fixed in a posture of attack. In the second, the hostility is prospective and imaginary, and it will always be a charge against us that we have made war upon soldiers who were themselves engaged in entirely legitimate (non-threatening) activities. Hence the moral necessity of rejecting any attack that is merely preventive in character, that does not wait upon and respond to the willful acts of an adversary.\n\nPre-emptive Strikes\n\nNow, what acts are to count, what acts do count as threats sufficiently serious to justify war? It is not possible to put together a list, because state action, like human action generally, takes on significance from its context. But there are some negative points worth making. The boastful ranting to which political leaders are often prone isn't in itself threatening; injury must be \"offered\" in some material sense as well. Nor does the kind of military preparation that is a feature of the classic arms race count as a threat, unless it violates some formally or tacitly agreed-upon limit. What the lawyers call \"hostile acts short of war,\" even if these involve violence, are not too quickly to be taken as signs of an intent to make war; they may represent an essay in restraint, an offer to quarrel within limits. Finally, provocations are not the same as threats. \"Injury and provocation\" are commonly linked by Scholastic writers as the two causes of just war. But the Schoolmen were too accepting of contemporary notions about the honor of states and, more importantly, of sovereigns. The moral significance of such ideas is dubious at best. Insults are not occasions for wars, any more than they are (these days) occasions for duels.\n\nFor the rest, military alliances, mobilizations, troop movements, border incursions, naval blockades\u2014all these, with or without verbal menace, sometimes count and sometimes do not count as sufficient indications of hostile intent. But it is, at least, these sorts of actions with which we are concerned. We move along the anticipation spectrum in search, as it were, of enemies: not possible or potential enemies, not merely present ill-\u00adwishers, but states and nations that are already, to use a phrase I shall use again with reference to the distinction of combatants and noncombatants, engaged in harming us (and who have already harmed us, by their threats, even if they have not yet inflicted any physical injury). And this search, though it carries us beyond preventive war, clearly brings us up short of Webster's pre-emption. The line between legitimate and illegitimate first strikes is not going to be drawn at the point of imminent attack but at the point of sufficient threat. That phrase is necessarily vague. I mean it to cover three things: a manifest intent to injure, a degree of active preparation that makes that intent a positive danger, and a general situation in which waiting, or doing anything other than fighting, greatly magnifies the risk. The argument may be made more clear if I compare these criteria to Vattel's. Instead of previous signs of rapacity and ambition, current and particular signs are required; instead of an \"augmentation of power,\" actual preparation for war; instead of the refusal of future securities, the intensification of present dangers. Preventive war looks to the past and future, Webster's reflex action to the immediate moment, while the idea of being under a threat focuses on what we had best call simply the present. I cannot specify a time span; it is a span within which one can still make choices, and within which it is possible to feel straitened.\n\nWhat such a time is like is best revealed concretely. We can study it in the three weeks that preceded the Six Day War of 1967. Here is a case as crucial for an understanding of anticipation in the twentieth century as the War of the Spanish Succession was for the eighteenth, and one suggesting that the shift from dynastic to national politics, the costs of which have so often been stressed, has also brought some moral gains. For nations, especially democratic nations, are less likely to fight preventive wars than dynasties are.\n\nThe Six Day War\n\nActual fighting between Israel and Egypt began on June 5, 1967, with an Israeli first strike. In the early hours of the war, the Israelis did not acknowledge that they had sought the advantages of surprise, but the deception was not maintained. In fact, they believed themselves justified in attacking first by the dramatic events of the previous weeks. So we must focus on those events and their moral significance. It would be possible, of course, to look further back still, to the whole course of the Arab-\u00adJewish conflict in the Middle East. Wars undoubtedly have long political and moral pre-histories. But anticipation needs to be understood within a narrower frame. The Egyptians believed that the founding of Israel in 1948 had been unjust, that the state had no rightful existence, and hence that it could be attacked at any time. It follows from this that Israel had no right of anticipation since it had no right of self-defense. But self-defense seems the primary and indisputable right of any political community, merely because it is there and whatever the circumstances under which it achieved statehood.b Perhaps this is why the Egyptians fell back in their more formal arguments upon the claim that a state of war already existed between Egypt and Israel and that this condition justified the military moves they undertook in May 1967. But the same condition would justify Israel's first strike. It is best to assume, I think, that the existing cease-fire between the two countries was at least a near-peace and that the outbreak of the war requires a moral explanation\u2014the burden falling on the Israelis, who began the fighting.\n\nThe crisis apparently had its origins in reports, circulated by Soviet officials in mid-May, that Israel was massing its forces on the Syrian border. The falsity of these reports was almost immediately vouched for by United Nations observers on the scene. Nevertheless, on May 14, the Egyptian government put its armed forces on \"maximum alert\" and began a major buildup of its troops in the Sinai. Four days later, Egypt expelled the United Nations Emergency Force from the Sinai and the Gaza Strip; its withdrawal began immediately, though I do not think that its title had been intended to suggest that it would depart so quickly in event of emergency. The Egyptian military buildup continued, and on May 22, President Nasser announced that the Straits of Tiran would henceforth be closed to Israeli shipping.\n\nIn the aftermath of the Suez War of 1956, the Straits had been recognized by the world community as an international waterway. That meant that their closing would constitute a casus belli, and the Israelis had stated at that time, and on many occasions since, that they would so regard it. The war might then be dated from May 22, and the Israeli attack of June 5 described simply as its first military incident: wars often begin before the fighting of them does. But the fact is that after May 22, the Israeli cabinet was still debating whether or not to go to war. And, in any case, the actual initiation of violence is a crucial moral event. If it can sometimes be justified by reference to previous events, it nevertheless has to be justified. In a major speech on May 29, Nasser made that justification much easier by announcing that if war came, the Egyptian goal would be nothing less than the destruction of Israel. On May 30, King Hussein of Jordan flew to Cairo to sign a treaty placing the Jordanian army under Egyptian command in event of war, thus associating himself with the Egyptian purpose. Syria already had agreed to such an arrangement, and several days later Iraq joined the alliance. The Israelis struck on the day after the Iraqi announcement.\n\nFor all the excitement and fear that their actions generated, it is unlikely that the Egyptians intended to begin the war themselves. After the fighting was over, Israel published documents, captured in its course, that included plans for an invasion of the Negev; but these were probably plans for a counter-attack, once an Israeli offensive had spent itself in the Sinai, or for a first strike at some later time. Nasser would almost certainly have regarded it as a great victory if he could have closed the Straits and maintained his army on Israel's borders without war. Indeed, it would have been a great victory, not only because of the economic blockade it would have established, but also because of the strain it would have placed on the Israeli defense system. \"There was a basic asymmetry in the structure of forces: the Egyptians could deploy . . . their large army of long-term regulars on the Israeli border and keep it there indefinitely; the Israelis could only counter their deployment by mobilizing reserve formations, and reservists could not be kept in uniform for very long. . . . Egypt could therefore stay on the defensive while Israel would have to attack unless the crisis was defused diplomatically.\" Would have to attack: the necessity cannot be called instant and overwhelming; nor, however, would an Israeli decision to allow Nasser his victory have meant nothing more than a shift in the balance of power posing possible dangers at some future time. It would have opened Israel to attack at any time. It would have represented a drastic erosion of Israeli security such as only a determined enemy would hope to bring about.\n\nThe initial Israeli response was not similarly determined but, for domestic political reasons having to do in part with the democratic character of the state, hesitant and confused. Israel's leaders sought a political resolution of the crisis\u2014the opening of the Straits and a demobilization of forces on both sides\u2014which they did not have the political strength or support to effect. A flurry of diplomatic activity ensued, serving only to reveal what might have been predicted in advance: the unwillingness of the Western powers to pressure or coerce the Egyptians. One always wants to see diplomacy tried before the resort to war, so that we are sure that war is the last resort. But it would be difficult in this case to make an argument for its necessity. Day by day, diplomatic efforts seemed only to intensify Israel's isolation.\n\nMeanwhile, \"an intense fear spread in the country.\" The extraordinary Israeli triumph, once fighting began, makes it difficult to recall the preceding weeks of anxiety. Egypt was in the grip of a war fever, familiar enough from European history, a celebration in advance of expected victories. The Israeli mood was very different, suggesting what it means to live under threat: rumors of coming disasters were endlessly repeated; frightened men and women raided food shops, buying up their entire stock, despite government announcements that there were ample reserves; thousands of graves were dug in the military cemeteries; Israel's political and military leaders lived on the edge of nervous exhaustion. I have already argued that fear by itself establishes no right of anticipation. But Israeli anxiety during those weeks seems an almost classical example of \"just fear\"\u2014first, because Israel really was in danger (as foreign observers readily agreed), and second, because it was Nasser's intention to put it in danger. He said this often enough, but it is also and more importantly true that his military moves served no other, more limited goal.\n\nThe Israeli first strike is, I think, a clear case of legitimate anticipation. To say that, however, is to suggest a major revision of the legalist paradigm. For it means that aggression can be made out not only in the absence of a military attack or invasion but in the (probable) absence of any immediate intention to launch such an attack or invasion. The general formula must go something like this: states may use military force in the face of threats of war, whenever the failure to do so would seriously risk their territorial integrity or political independence. Under such circumstances it can fairly be said that they have been forced to fight and that they are the victims of aggression. Since there are no police upon whom they can call, the moment at which states are forced to fight probably comes sooner than it would for individuals in a settled domestic society. But if we imagine an unstable society, like the \"wild west\" of American fiction, the analogy can be restated: a state under threat is like an individual hunted by an enemy who has announced his intention of killing or injuring him. Surely such a person may surprise his hunter, if he is able to do so.\n\nThe formula is permissive, but it implies restrictions that can usefully be unpacked only with reference to particular cases. It is obvious, for example, that measures short of war are preferable to war itself whenever they hold out the hope of similar or nearly similar effectiveness. But what those measures might be, or how long they must be tried, cannot be a matter of a priori stipulation. In the case of the Six Day War, the \"asymmetry in the structure of forces\" set a time limit on diplomatic efforts that would have no relevance to conflicts involving other sorts of states and armies. A general rule containing words like \"seriously\" opens a broad path for human judgment\u2014which it is, no doubt, the purpose of the legalist paradigm to narrow or block altogether. But it is a fact of our moral life that political leaders make such judgments, and that once they are made the rest of us do not uniformly condemn them. Rather, we weigh and evaluate their actions on the basis of criteria like those I have tried to describe. When we do that we are acknowledging that there are threats with which no nation can be expected to live. And that acknowledgment is an impor\u00adtant part of our understanding of aggression.\n\n The line is from David Hume's essay \"Of the Balance of Power,\" where Hume describes three British wars on behalf of the balance as having been \"begun with justice, and even, perhaps, from necessity.\" I would have considered his argument at length had I found it possible to place it within his philosophy. But in his Enquiry Concerning the Principles of Morals (Section III, Part I), Hume writes: \"The rage and violence of public war: what is it but a suspension of justice among the warring parties, who perceive that this virtue is now no longer of any use or advantage to them?\" Nor is it possible, according to Hume, that this suspension itself be just or unjust; it is entirely a matter of necessity, as in the (Hobbist) state of nature where individuals \"consult the dictates of self-preservation alone.\" That standards of justice exist alongside the pressures of necessity is a discovery of the Essays. This is another example, perhaps, of the impossibility of carrying over certain philosophical positions into ordinary moral discourse. In any case, the three wars Hume discusses were none of them necessary to the preservation of Britain. He may have thought them just because he thought the balance generally useful.\n\n The only limitation on this right has to do with internal, not external legitimacy: a state (or government) established against the will of its own people, ruling violently, may well forfeit its right to defend itself even against a foreign invasion. I will take up some of the issues raised by this possibility in the next chapter.\n\nInterventions\n\nThe principle that states should never intervene in the domestic affairs of other states follows readily from the legalist paradigm and, less readily and more ambiguously, from those conceptions of life and liberty that underlie the paradigm and make it plausible. But these same conceptions seem also to require that we sometimes disregard the principle; and what might be called the rules of disregard, rather than the principle itself, have been the focus of moral interest and argument. No state can admit to fighting an aggressive war and then defend its actions. But intervention is differently understood. The word is not defined as a criminal activity, and though the practice of intervening often threatens the territorial integrity and political independence of invaded states, it can sometimes be justified. It is more important to stress at the outset, however, that it always has to be justified. The burden of proof falls on any political leader who tries to shape the domestic arrangements or alter the conditions of life in a foreign country. And when the attempt is made with armed force, the burden is especially heavy\u2014not only because of the coercion and ravages that military intervention inevitably brings, but also because it is thought that the citizens of a sovereign state have a right, insofar as they are to be coerced and ravaged at all, to suffer only at one another's hands.\n\nSelf-Determination and Self-Help\n\nThe Argument of John Stuart Mill\n\nThese citizens are the members, it is presumed, of a single political community, entitled collectively to determine their own affairs. The precise nature of this right is nicely worked out by John Stuart Mill in a short article published in the same year as the treatise On Liberty (1859) and especially useful to us because the individual\/community analogy was very much in Mill's mind as he wrote. We are to treat states as self-\u00addetermining communities, he argues, whether or not their internal political arrangements are free, whether or not the citizens choose their government and openly debate the policies carried out in their name. For self-\u00addetermination and political freedom are not equivalent terms. The first is the more inclusive idea; it describes not only a particular institutional arrangement but also the process by which a community arrives at that arrangement\u2014or does not. A state is self-determining even if its citizens struggle and fail to establish free institutions, but it has been deprived of self-determination if such institutions are established by an intrusive neighbor. The members of a political community must seek their own freedom, just as the individual must cultivate his own virtue. They cannot be set free, as he cannot be made virtuous, by any external force. Indeed, political freedom depends upon the existence of individual virtue, and this the armies of another state are most unlikely to produce\u2014unless, perhaps, they inspire an active resistance and set in motion a self-\u00addetermining politics. Self-\u00addetermination is the school in which virtue is learned (or not) and liberty is won (or not). Mill recognizes that a people who have had the \"misfortune\" to be ruled by a tyrannical government are peculiarly disadvantaged: they have never had a chance to develop \"the virtues needful for maintaining freedom.\" But he insists nevertheless on the stern doctrine of self-help. \"It is during an arduous struggle to become free by their own efforts that these virtues have the best chance of springing up.\"\n\nThough Mill's argument can be cast in utilitarian terms, the harshness of his conclusions suggests that this is not its most appropriate form. The Millian view of self-determination seems to make utilitarian calculation unnecessary, or at least subsidiary to an understanding of communal liberty. He doesn't believe that intervention fails more often than not to serve the purposes of liberty; he believes that, given what liberty is, it necessarily fails. The (internal) freedom of a political community can be won only by the members of that community. The argument is similar to that implied in the well-known Marxist maxim, \"The liberation of the working class can come only through the workers themselves.\" As that maxim, one would think, rules out any substitution of vanguard elitism for working class democracy, so Mill's argument rules out any substitution of foreign intervention for internal struggle.\n\nSelf-determination, then, is the right of a people \"to become free by their own efforts\" if they can, and nonintervention is the principle guaranteeing that their success will not be impeded or their failure prevented by the intrusions of an alien power. It has to be stressed that there is no right to be protected against the consequences of domestic failure, even against a bloody repression. Mill generally writes as if he believes that citizens get the government they deserve, or, at least, the government for which they are \"fit.\" And \"the only test . . . of a people's having become fit for popular institutions is that they, or a sufficient portion of them to prevail in the contest, are willing to brave labor and danger for their liberation.\" No one can, and no one should, do it for them. Mill takes a very cool view of political conflict, and if many rebellious citizens, proud and full of hope in their own efforts, have endorsed that view, many others have not. There is no shortage of revolutionaries who have sought, pleaded for, even demanded outside help. A recent American commentator, eager to be helpful, has argued that Mill's position involves \"a kind of Darwinian definition [The Origin of the Species was also published in 1859] of self-determination as survival of the fittest within the national boundaries, even if fittest means most adept in the use of force.\" That last phrase is unfair, for it was precisely Mill's point that force could not prevail, unless it were reinforced from the outside, over a people ready \"to brave labor and danger.\" For the rest, the charge is probably true, but it is difficult to see what conclusions follow from it. It is possible to intervene domestically in the \"Darwinian\" struggle because the intervention is continuous and sustained over time. But foreign intervention, if it is a brief affair, cannot shirt the domestic balance of power in any decisive way toward the forces of freedom, while if it is prolonged or intermittently resumed, it will itself pose the greatest possible threat to the success of those forces.\n\nThe case may be different when what is at issue is not intervention at all but conquest. Military defeat and governmental collapse may so shock a social system as to open the way for a radical renovation of its political arrangements. This seems to be what happened in Germany and Japan after World War II, and these examples are so important that I will have to consider later on how it is that rights of conquest and renovation might arise. But they clearly don't arise in every case of domestic tyranny. It is not true, then, that intervention is justified whenever revolution is; for revolutionary activity is an exercise in self-determination, while foreign interference denies to a people those political capacities that only such exercise can bring.\n\nThese are the truths expressed by the legal doctrine of sovereignty, which defines the liberty of states as their independence from foreign control and coercion. In fact, of course, not every independent state is free, but the recognition of sovereignty is the only way we have of establishing an arena within which freedom can be fought for and (sometimes) won. It is this arena and the activities that go on within it that we want to protect, and we protect them, much as we protect individual integrity, by marking out boundaries that cannot be crossed, rights that cannot be violated. As with individuals, so with sovereign states: there are things that we cannot do to them, even for their own ostensible good.\n\nAnd yet the ban on boundary crossings is not absolute\u2014in part because of the arbitrary and accidental character of state boundaries, in part because of the ambiguous relation of the political community or communities within those boundaries to the government that defends them. Despite Mill's very general account of self-determination, it isn't always clear when a community is in fact self-determining, when it qualifies, so to speak, for nonintervention. No doubt there are similar problems with individual persons, but these are, I think, less severe and, in any case, they are handled within the structures of domestic law.a In international society, the law provides no authoritative verdicts. Hence, the ban on boundary crossings is subject to unilateral suspension, specifically with reference to three sorts of cases where it does not seem to serve the purposes for which it was established:\n\n  * when a particular set of boundaries clearly contains two or more political communities, one of which is already engaged in a large-scale military struggle for independence, that is, when what is at issue is secession or \"national liberation\";\n  * when the boundaries have already been crossed by the armies of a foreign power, even if the crossing has been called for by one of the parties in a civil war, that is, when what is at issue is counter-\u00adintervention; and\n  * when the violation of human rights within a set of boundaries is so terrible that it makes talk of community or self-determination or \"arduous struggle\" seem cynical and irrelevant, that is, in cases of enslavement or massacre.\n\nThe arguments that are made on behalf of intervention in each of these cases constitute the second, third, and fourth revisions of the legalist paradigm. They open the way for just wars that are not fought in self-\u00addefense or against aggression in the strict sense. But they need to be worked out with great care. Given the readiness of states to invade one another, revisionism is a risky business.\n\nMill discusses only the first two of these cases, secession and counter-\u00adintervention, though the last was not unknown even in 1859. It is worth pointing out that he does not regard them as exceptions to the nonintervention principle, but rather as negative demonstrations of its reasons. Where these reasons don't apply, the principle loses its force. It would be more exact, from Mill's standpoint, to formulate the relevant principle in this way: always act so as to recognize and uphold communal autonomy. Nonintervention is most often entailed by that recognition, but not always, and then we must prove our commitment to autonomy in some other way, perhaps even by sending troops across an international frontier. But the morally exact principle is also very dangerous, and Mill's account of the argument is not at this point an account of what is actually said in everyday moral discourse. We need to establish a kind of a priori respect for state boundaries; they are, as I have argued before, the only boundaries communities ever have. And that is why intervention is always justified as if it were an exception to a general rule, made necessary by the urgency or extremity of a particular case. The second, third, and fourth revisions have something of the form of stereotyped excuses. Interventions are so often undertaken for \"reasons of state\" that have nothing to do with self-determination that we have become skeptical of every claim to defend the autonomy of alien communities. Hence the special burden of proof with which I began, more onerous than any we impose on individuals or governments pleading self-defense: intervening states must demonstrate that their own case is radically different from what we take to be the general run of cases, where the liberty or prospective liberty of citizens is best served if foreigners offer them only moral support. And that is how I shall characterize Mill's argument (though he characterizes it differently) that Great Britain ought to have intervened in defense of the Hungarian Revolution of 1848 and 1849.\n\nSecession\n\nThe Hungarian Revolution\n\nFor many years before 1848, Hungary had been a part of the Hapsburg Empire. Formally an independent kingdom, with a Diet of its own, it was effectively ruled by the German authorities in Vienna. The sudden collapse of those authorities during the March Days\u2014symbolized by the fall of Metternich\u2014opened the way for liberal nationalists in Budapest. They formed a government and demanded home rule within the Empire; they were not yet secessionists. Their demand was initially accepted, but controversy developed over the issues that have always plagued federalist schemes: the control of tax revenue, the command of the army. As soon as \"order\" was restored in Vienna, efforts began to reassert the centralist character of the regime, and these soon took the familiar form of military repression. An imperial army invaded Hungary, and the nationalists fought back. The Hungarians were now rebels or insurgents; they quickly established what international lawyers call their belligerent rights by defeating the Austrians and taking control of much of old Hungary. In the course of the war, the new government shifted leftwards; in April 1849, a republic was proclaimed under the presidency of Lajos Kossuth.\n\nThe revolution might be described, in contemporary terms, as a war of national liberation, except that the boundaries of old Hungary included a very large Slavic population, and the Hungarian revolutionaries seem to have been as hostile to Croat and Slovene nationalism as the Austrians were to their own claims for communal autonomy. But this is a difficulty that I am going to set aside, for it did not appear as such at the time; it did not enter into the moral reflections of liberal observers like Mill. The Hungarian Revolution was greeted with enthusiasm by such men, especially in France, Britain, and the United States, and its emissaries were eagerly received. Governmental response was different, in part because nonintervention was the general rule to which all three governments subscribed, in part because the first two were also committed to the European balance of power and therefore to the integrity of Austria. In London, Palmerston was formal and cold: \"The British government has no knowledge of Hungary except as one of the component parts of the Austrian Empire.\" The Hungarians sought only diplomatic recognition, not military intervention, but any British dealings with the new government would have been regarded by the Austrian regime as an interference in its internal affairs. Recognition, moreover, had commercial consequences that might have engaged the British more closely on the side of Hungary, for the revolutionaries hoped to purchase military supplies on the London market. Despite this, the establishment of formal ties, once the Hungarians had demonstrated that \"a sufficient portion of them\" were committed to independence and willing to fight for it, would not have been difficult to justify in Millian terms. There can be no doubt of the existence (though there was a reason to doubt the extent) of the Hungarian political community; it was one of the oldest nations in Europe, and its recognition as a sovereign state would not have violated the moral rights of the Austrian people. Military supply to insurgent armies is indeed a complex issue, and I will come back to it with reference to another case, but none of the complexities are apparent here. Soon enough, however, the Hungarians needed far more than guns and ammunition.\n\nIn the summer of 1849, the Austrian emperor asked for the help of Tsar Nicholas I, and Hungary was invaded by a Russian army. Writing ten years later, Mill argued that the British should have responded to this intervention with an intervention of their own.\n\nIt might not have been right for England (even apart from the question of prudence) to have taken part with Hungary in its noble struggle against Austria; although the Austrian government in Hungary was in some sense a foreign yoke. But when, the Hungarians having shown themselves likely to prevail in this struggle, the Russian despot interposed, and joining his force to that of Austria, delivered back the Hungarians, bound hand and foot, to their exasperated oppressors, it would have been an honorable and virtuous act on the part of England to have declared that this should not be, and that if Russia gave assistance to the wrong side, England would aid the right.\n\nThe qualification \"in some sense a foreign yoke\" with regard to Austrian rule in Hungary is curious, for whatever its meaning, it must also qualify the nobility and rightness of the Hungarian struggle for independence. Since Mill does not intend the latter qualification, we need not take the former seriously. The clear tendency of his argument is to justify assistance to a secessionist movement at the same time as it justifies counter-\u00adintervention\u2014indeed, to assimilate the one to the other. In both cases, the rule against interference is suspended because a foreign power, morally if not legally alien, is already interfering in the \"domestic\" affairs, that is, in the self-determinations of a political community.\n\nMill is right, however, to suggest that the issue is easier when the initial interference involves the crossing of a recognized frontier. The problem with a secessionist movement is that one cannot be sure that it in fact represents a distinct community until it has rallied its own people and made some headway in the \"arduous struggle\" for freedom. The mere appeal to the principle of self-determination isn't enough; evidence must be provided that a community actually exists whose members are committed to independence and ready and able to determine the conditions of their own existence.b Hence the need for political or military struggle sustained over time. Mill's argument doesn't cover inarticulate and unrepresented peoples, or fledgling movements, or risings quickly suppressed. But imagine a small nation successfully mobilized to resist a colonial power but slowly being ground down in the unequal struggle: Mill would not insist, I think, that neighboring states stand by and watch its inevitable defeat. His argument justifies military action against imperial or colonial repression as well as against foreign intervention. Only domestic tyrants are safe, for it is not our purpose in international society (nor, Mill argues, is it possible) to establish liberal or democratic communities, but only independent ones. When it is required for the sake of independence, military action is \"honorable and virtuous,\" though not always \"prudent.\" I should add that the argument also applies to satellite regimes and great powers: designed for the first Russian intervention in Hungary (1849), it precisely fits the second (1956).\n\nBut the relation between virtue and prudence in such cases is not easy to make out. Mill's meaning is clear enough: to threaten war with Russia might have been dangerous to Britain and hence inconsistent \"with the regard which every nation is bound to pay to its own safety.\" Now, whether or not it actually was dangerous was surely for the British to decide, and we would judge them harshly only if the risks they declined to run were very slight indeed. Even if counter-intervention is \"honorable and virtuous,\" it is not morally required, precisely because of the dangers it involves. But one can make much more of prudence than this. Palmerston was concerned with the safety of Europe, not only of England, when he decided to stand by the Austrian empire. It is perfectly possible to concede the justice of the Millian position, and yet opt for nonintervention on what are currently called \"world order\" principles. So justice and prudence are (with a certain worldly relish) set in opposition to one another in a way that Mill never imagined they could be. He thought, naively perhaps, that the world would be more orderly if none of its political communities were oppressed by foreign rule. He even hoped that Britain would one day be powerful enough, and have the necessary \"spirit and courage,\" to insist \"that not a gun [should] be fired in Europe by the soldiers of one Power against the revolted subjects of another,\" and to put itself \"at the head of an alliance of free peoples. . . .\" Today, I suppose, the United States has succeeded to those old-fashioned liberal pretensions, though in 1956 its leaders, like Palmerston in 1849, thought it imprudent to enforce them.\n\nIt might also be said that the United States had (and has) no right to enforce them, given the self-serving ways in which its government defines freedom and intervention in other parts of the world. Mill's England was hardly in a better position. Had Palmerston contemplated a military move on behalf of the Hungarians, Count Schwarzenberg, Metternich's successor, was prepared to remind him of \"unhappy Ireland.\" \"Wherever revolt breaks out within the vast limits of the British Empire,\" Schwarzenberg wrote to the Austrian ambassador in London, \"the English government always knows how to maintain the authority of the law . . . even at the price of torrents of blood. It is not for us,\" he went on, \"to blame her.\" He sought only reciprocity, and that kind of reciprocity among great powers is undoubtedly the very essence of prudence.\n\nTo set prudence and justice so radically at odds, however, is to misconstrue the argument for justice. A state contemplating intervention or counter-intervention will for prudential reasons weigh the dangers to itself, but it must also, and for moral reasons, weigh the dangers its action will impose on the people it is designed to benefit and on all other people who may be affected. An intervention is not just if it subjects third parties to terrible risks: the subjection cancels the justice. If Palmerston was right in believing that the defeat of Austria would shatter the peace of Europe, a British intervention ensuring that defeat would not have been \"honorable and virtuous\" (however noble the Hungarian struggle). And clearly, an American threat of atomic war in 1956 would have been morally as well as politically irresponsible. Thus far prudence can be, and has to be, accommodated within the argument for justice. But it should be said that this deference to third party rights is not at the same time a deference to the local political interests of the great powers. Nor does it involve the acceptance of a Schwarzenbergian reciprocity. Britain's recognition of Austria's imperial claims does not entitle it to a similar recognition. The prudential acceptance of a Russian sphere of influence in Eastern Europe does not entitle the United States to a free hand in its own sphere. Against national liberation and counter-intervention, there are no prescriptive rights.\n\nCivil War\n\nIf we describe the Hungarian Revolution as Mill did, assuming that Palmerston was wrong, ignoring the claims of Croats and Slovenes, it is virtually a paradigm case for intervention. It is also, so described, a historically exceptional, indeed, it is now a hypothetical case. For these circumstances don't often arise in history: a national liberation movement unambiguously embodying the claims of a single, unified political community; capable at least initially of sustaining itself on the battlefield; challenged by an unambiguously foreign power; whose intervention can however be deterred or defeated without risking a general war. More often history presents a tangle of parties and factions, each claiming to speak for an entire community, fighting with one another, drawing outside powers into the struggle in secret, or at least unacknowledged, ways. Civil war poses hard problems, not because the Millian standard is unclear\u2014it would require a strict standoffishness\u2014but because it can be and routinely is violated by degrees. Then it becomes very difficult to fix the point at which a direct and open use of force can plausibly be called a counter-\u00adintervention. And it is difficult also to calculate the effects of such a use of force on the already distressed inhabitants of the divided state and on the whole range of possible third parties.\n\nIn such cases, the lawyers commonly apply a qualified version of the self-help test. They permit assistance to the established government\u2014it is after all, the official representative of communal autonomy in international society\u2014so long as it faces nothing more than internal dissension, rebellion, and insurgency. But as soon as the insurgents establish control over some substantial portion of the territory and population of the state, they acquire belligerent rights and an equality of status with the government. Then the lawyers enjoin a strict neutrality. Now, neutrality is conventionally regarded as an optative condition, a matter of choice, not of duty. So it is with regard to wars between states, but in civil wars there seem to be very good (Millian) reasons for making it obligatory. For once a community is effectively divided, foreign powers can hardly serve the cause of self-determination by acting militarily within its borders. The argument has been succinctly put by Montague Bernard, whose Oxford lecture \"On the Principle of Non-intervention\" ranks in importance with Mill's essay: \"Of two things, one: the interference in the case supposed either turns the balance, or it does not. In the latter event, it misses its aim; in the former, it gives the superiority to the side which would not have been uppermost without it and establishes a sovereign, or a form of government, which the nation, if left to itself, would not have chosen.\"\n\nAs soon as one outside power violates the norms of neutrality and nonintervention, however, the way is open for other powers to do so. Indeed, it may seem shameful not to repeat the violation\u2014as in the case of the Spanish Civil War, where the noninterventionist policies of Britain, France, and the United States did not open the way for a local decision, but simply allowed the Germans and Italians to \"turn the balance.\" Some military response is probably required at such moments if the values of independence and community are to be sustained. But though that response upholds values shared throughout international society, it cannot accurately be described as law enforcement. Its character is not readily explicable within the terms of the legalist paradigm. For counter-\u00adintervention in civil wars does not aim at punishing or even, necessarily, at restraining the intervening states. It aims instead at holding the circle, preserving the balance, restoring some degree of integrity to the local struggle. It is as if a policeman, instead of breaking up a fight between two people, should stop anyone else from interfering or, if he cannot do that, should give proportional assistance to the disadvantaged party. He would have to have some notions about the value of the fight, and given the ordinary conditions of domestic society, those would be strange notions for him to have. But in the world of states they are entirely appropriate; they set the standards by which we judge between actual and pretended counter-interventions.\n\nThe American War in Vietnam\n\nI doubt that it is possible to tell the story of Vietnam in a way that will command general agreement. The official American version\u2014that the struggle began with a North Vietnamese invasion of the South, to which the United States responded in accordance with its treaty obligations\u2014\u00adfollows the legalist paradigm closely, but is on its surface unbelievable. Fortunately, it seems to be accepted by virtually no one and need not detain us here. I want to pursue a more sophisticated version of the American defense, which concedes the existence of a civil war and describes the U.S. role, first, as assistance to a legitimate government, and secondly, as counter-intervention, a response to covert military moves by the North Vietnamese regime. The crucial terms here are \"legitimate\" and \"response.\" The first suggests that the government on behalf of which our counter-intervention was undertaken had a local status, a political presence independent of ourselves, and hence that it could conceivably win the civil war if no external force was brought to bear. The second suggests that our own military operations followed upon and balanced those of another power, in accordance with the argument I have put forward. Both these suggestions are false, but they point to the peculiarly confined character of counter-intervention and indicate what one has to say (at least) when one joins in the civil wars of other states.\n\nThe Geneva Agreement of 1954, ending the first Vietnamese war, established a temporary frontier between the North and the South, and two temporary governments on either side of the line, pending elections scheduled for 1956. When the South Vietnamese government refused to permit these elections, it clearly lost whatever legitimacy was conferred by the agreements. But I shall not dwell on this loss, nor on the fact that some sixty states nevertheless recognized the sovereignty of the new regime in the South and opened embassies in Saigon. I doubt that foreign states, whether they act independently or collectively, sign treaties or send ambassadors, can establish or disestablish the legitimacy of a government. What is crucial is the standing of that government with its own people. Had the new regime been able to rally support at home, Vietnam today would have joined the divided states of Germany (until 1990) and Korea, and Geneva 1954 would be remembered only as the setting for another cold war partition. But what is the test of popular support in a country where democracy is unknown and elections are routinely managed? The test, for governments as for insurgents, is self-help. That doesn't mean that foreign states cannot provide assistance. One assumes the legitimacy of new regimes; there is, so to speak, a period of grace, a time to build support. But that time was ill-used in South Vietnam, and the continuing dependence of the new regime on the United States is damning evidence against it. Its urgent call for military intervention in the early 1960s is more damning evidence still. One must ask of President Diem a question first posed by Montague Bernard: \"How can he impersonate [represent] his people who is begging the assistance of a foreign power in order to reduce them to obedience?\" Indeed, it was never a successful impersonation.\n\nThe argument might be put more narrowly: a government that receives economic and technical aid, military supply, strategic and tactical advice, and is still unable to reduce its subjects to obedience, is clearly an illegitimate government. Whether legitimacy is defined sociologically or morally, such a government fails to meet the most minimal standards. One wonders how it survives at all. It must be the case that it survives because of the outside help it receives and for no other, no local reasons. The Saigon regime was so much an American creature that the U.S. government's claim to be committed to it and obligated to ensure its survival is hard to understand. It is as if our right hand were committed to our left. There is no independent moral or political agent on the other side of the bond and hence no genuine bond at all. Obligations to one's creatures (except insofar as they pertain to the personal safety of individuals) are as insignificant politically as obligations to oneself are insignificant morally. When the U.S. did intervene militarily in Vietnam, then, it acted not to fulfill commitments to another state, but to pursue policies of its own contrivance.\n\nAgainst all this, it is argued that the popular base of the South Vietnamese government was undermined by a systematic campaign of subversion, terrorism, and guerrilla war, largely directed and supplied from the North. That there was such a campaign, and that the North was involved in it, is clearly true, though the extent and timing of the involvement are very much in dispute. If one were writing a legal brief, these matters would be critically important, for the American claim is that the North Vietnamese were illegally supporting a local insurgency, with both men and material, at a time when the U.S. was still providing only economic assistance and military supply to a legitimate government. But that claim, whatever its legal force, somehow misses the moral reality of the Vietnamese case. It would be better to say that the U.S. was literally propping up a government\u2014and shortly a series of governments\u2014without a local political base, while the North Vietnamese were assisting an insurgent movement with deep roots in the countryside. We were far more vital to the government than they were to the insurgents. Indeed, it was the weakness of the government, its inability to help itself even against its internal enemies, that forced the steady escalation of American involvement. And that fact must raise the most serious questions about the American defense: for counter-\u00adintervention is morally possible only on behalf of a government (or a movement, party, or whatever) that has already passed the self-help test.\n\nI can say very little here about the reasons for insurgent strength in the countryside. Why were the communists able, and the government unable, to \"impersonate\" Vietnamese nationalism? The character and scope of the American presence probably had a great deal to do with this. Nationalism is not easily represented by a regime as dependent as Saigon was on foreign support. It is also important that North Vietnamese moves did not similarly brand those they benefited as foreign agents. In nations divided as Vietnam was, infiltration across the dividing line is not necessarily regarded as outside interference by the men and women on the other side. The Korean War might look very different than it does if the Northerners had not marched in strength across the 38th parallel, but had made covert contact, instead, with a Southern rebellion. In contrast to Vietnam, however, there was no rebellion\u2014and there was considerable support for the government\u2014in South Korea. These cold war dividing lines have the usual significance of an international border only insofar as they mark off, or come in time to mark off, two political communities within each of which individual citizens feel some local loyalty. Had South Vietnam taken shape in this way, American military activity, in the face of large-scale Northern connivance at terrorism and guerrilla war, might have qualified as counter-intervention. At least, the name would have been an arguable one. As it is, it is not.\n\nIt remains an issue whether the American counter-intervention, had it been such, could rightly have assumed the size and scope of the war we eventually fought. Some notion of symmetry is relevant here, though it cannot be fixed absolutely in arithmetic terms. When a state sets out to maintain or restore the integrity of a local struggle, its military activity should be roughly equivalent to that of the other intervening states. Counter-intervention is a balancing act. I have made this point before, but it is worth emphasizing, for it reflects a deep truth about the meaning of responsiveness: the goal of counter-intervention is not to win the war. That this is not an esoteric or obscure truth is suggested by President Kennedy's well-known description of the Vietnam War. \"In the final analysis,\" Kennedy said, \"it is their war. They are the ones who have to win it or lose it. We can help them, we can give them equipment, we can send our men out there as advisors, but they have to win it\u2014the people of Vietnam against the Communists. . . .\" Though this view was reiterated by later American leaders, it is not, unhappily, a definitive exposition of American policy. In fact, the United States failed in the most dramatic way to respect the character and dimensions of the Vietnamese civil war, and we failed because we could not win the war as long as it retained that character and was fought within those dimensions. Searching for a level of conflict at which our technological superiority could be brought to bear, we steadily escalated the struggle, until finally it was an American war, fought for American purposes, in someone else's country.\n\nHumanitarian Intervention\n\nA legitimate government is one that can fight its own internal wars. And external assistance in those wars is rightly called counter-intervention only when it balances, and does no more than balance, the prior intervention of another power, making it possible once again for the local forces to win or lose on their own. The outcome of civil wars should reflect not the relative strength of the intervening states, but the local alignment of forces. There is another sort of case, however, where we don't look for outcomes of that sort, where we don't want the local balance to prevail. If the dominant forces within a state are engaged in massive violations of human rights, the appeal to self-determination in the Millian sense of self-help is not very attractive. That appeal has to do with the freedom of the community taken as a whole; it has no force when what is at stake is the bare survival or the minimal liberty of (some substantial number of) its members. Against the enslavement or massacre of political opponents, national minorities, and religious sects, there may well be no help unless help comes from outside. And when a government turns savagely upon its own people, we must doubt the very existence of a political community to which the idea of self-determination might apply.\n\nExamples are not hard to find; it is their plenitude that is embarrassing. The list of oppressive governments, the list of massacred peoples, is frighteningly long. Though an event like the Nazi holocaust is without precedent in human history, murder on a smaller scale is so common as to be almost ordinary. On the other hand\u2014or perhaps for this very reason\u2014clear examples of what is called \"humanitarian intervention\" are very rare. Indeed, I have not found any, but only mixed cases where the humanitarian motive is one among several. States don't send their soldiers into other states, it seems, only in order to save lives. The lives of foreigners don't weigh that heavily in the scales of domestic decision-making. So we shall have to consider the moral significance of mixed motives.c It is not necessarily an argument against humanitarian intervention that it is, at best, partially humanitarian, but it is a reason to be skeptical and to look closely at the other parts.\n\nCuba, 1898, and Bangladesh, 1971\n\nBoth these cases might be taken up under the headings of national liberation and counter-intervention. But they each have a further significance because of the atrocities committed by the Spanish and the Pakistani governments. The brutal work of the Spaniards is easier to talk about, for it fell short of systematic massacre. Fighting against a Cuban insurgent army that lived off the land and apparently had large-scale peasant support, the Spaniards first worked out the policy of forced resettlement. They called it, without euphemism, la reconcentraci\u00f3n. General Weyler's proclamation required that:\n\nAll inhabitants of rural areas or areas outside the lines of fortified towns will be concentrated within the towns occupied by troops at the end of eight days. All individuals who disobey or who are found outside the prescribed areas will be considered as rebels and judged as such.\n\nI will ask later on whether \"concentration\" in itself is a criminal policy. The immediate crime of the Spaniards was to enforce the policy with so little regard for the health of the people involved that thousands of them suffered and died. Their lives and deaths were widely publicized in the United States, not only in the yellow press, and undoubtedly figured in the minds of many Americans as the major justification for the war against Spain. Thus the Congressional resolution of April 20, 1898: \"Whereas the abhorrent conditions which have existed for more than three years in the island of Cuba, so near our own borders, have shocked the moral sense of the people of the United States. . . .\" But there were other reasons for going to war.\n\nThe chief of these were economic and strategic in character, having to do, first, with American investment in Cuban sugar, a matter of interest to a section of the financial community; and second, with the sea approaches to the Panamanian Isthmus where the canal would one day be, a matter of interest to the intellectuals and politicians who championed the cause of American expansion. Cuba was a minor element in the plans of men like Mahan and Adams, Roosevelt and Lodge, who were more concerned with the Pacific Ocean than the Caribbean Sea. But the canal that would connect the two gave it a certain strategic value, and the war to win it was worthwhile insofar as it accustomed Americans to imperialist adventures (and led also to the conquest of the Phillipines). By and large, the historical debate over the causes of the war has focused on the different forms of economic and political imperialism, the search for markets and investment opportunities, the pursuit of \"national power for its own sake.\" It's worth remembering, however, that the war was also supported by anti-\u00adimperialist politicians\u2014or rather, that Cuban freedom was supported and then, in consequence of Spanish brutality, the humanitarian intervention of American military forces. The war we actually fought, however, and the intervention urged by populists and radical Democrats were two rather different things.\n\nThe Cuban insurgents made three requests of the United States: that we recognize their provisional government as the legitimate government of Cuba, that we provide their army with military supplies, and that American warships blockade the Cuban coast and cut off the supplies of the Spanish army. Given such help, it was said, the insurgent forces would grow, the Spaniards could not long hold out, and the Cubans would be left to reconstruct their country (with American help) and manage their own affairs. This was also the program of American radicals. But President McKinley and his advisors did not believe the Cubans capable of managing their own affairs, or they feared a radical reconstruction. In any case, the U.S. intervened without recognizing the insurgents, invaded the island, and quickly defeated and replaced the Spanish forces. The victory undoubtedly had humane effects. Though the American military effort was remarkably inefficient, the war was short and added little to the miseries of the civilian population. Relief operations, also remarkably inefficient at first, began as soon as the battles were won. In his standard account of the war, Admiral Chadwick boasts of its relative bloodlessness: \"War of itself,\" he writes, \"cannot be the great evil; the evil is in the horrors, many of which are not necessarily concomitant. . . . The war now beginning between the United States and Spain was one in which these greater horrors were largely to be absent.\" The horrors were indeed absent; far more so, at least, than in the long years of the Cuban Insurrection. But the invasion of Cuba, the three years of military occupation, the eventual granting of a drastically limited independence (under the provisions of the Platt Amendment) go a long way toward explaining the skepticism with which America's professions of humane concern have conventionally been regarded. The entire course of action, from 1898 to 1902, might be taken as an example of benevolent imperialism, given the \"piratical times,\" but it is not an example of humanitarian intervention.\n\nThe judgments we make in cases such as this don't hang on the fact that considerations other than humanity figured in the government's plans, or even on the fact that humanity was not the chief consideration. I don't know if it ever is, and measurement is especially difficult in a liberal democracy where the mixed motives of the government reflect the pluralism of the society. Nor is it a question of benevolent outcomes. As a result of the American victory, the reconcentrados were able to return to their homes. But they would have been able to do that had the United States entered the war on the side of the Spaniards and, together with them, decisively defeated the Cuban insurgents. \"Concentration\" was a war policy and would have ended with the war, whatever the war's end. The crucial question is a different one. Humanitarian intervention involves military action on behalf of oppressed people, and it requires that the intervening state enter, to some degree, into the purposes of those people. It need not set itself to achieve those purposes, but it also cannot stand in the way of their achievement. The people are oppressed, presumably, because they sought some end\u2014religious toleration, national freedom, or whatever\u2014unacceptable to their oppressors. One cannot intervene on their behalf and against their ends. I don't want to argue that the purposes of the oppressed are necessarily just or that one need accept them in their entirety. But it does seem that a greater attention is due them than the United States was prepared to pay in 1898.\n\nThis regard for the purposes of the oppressed directly parallels the respect for local autonomy that is a necessary feature of counter-\u00adintervention. The two revisionist principles reflect a common commitment: that intervention be as much like nonintervention as possible. In the one case, the goal is balance; in the other, it is rescue. In neither case, and certainly not in secessions and national liberation struggles, can the intervening state rightly claim any political prerogatives for itself. And whenever it makes such claims (as the United States did when it occupied Cuba and again when it imposed the Platt Amendment), we suspect that political power was its purpose from the start.\n\nThe Indian invasion of East Pakistan (Bangladesh) in 1971 is a better example of humanitarian intervention\u2014not because of the singularity or purity of the government's motives, but because its various motives converged on a single course of action that was also the course of action called for by the Bengalis. This convergence explains why the Indians were in and out of the country so quickly, defeating the Pakistani army but not replacing it, and imposing no political controls on the emergent state of Bangladesh. No doubt, strategic as well as moral interests underlay this policy: Pakistan, India's old enemy, was significantly weakened, while India itself avoided becoming responsible for a desperately poor nation whose internal politics was likely to be unstable and volatile for a long time to come. But the intervention qualifies as humanitarian because it was a rescue, strictly and narrowly defined. So circumstances sometimes make saints of us all.\n\nI shall not say very much about Pakistani oppression in Bengal. The tale is a terrible one and by now fairly well documented. Faced with a movement for autonomy in what was then its eastern province, the government of Pakistan, in March 1971, literally turned an army loose on its own people\u2014or rather, a Punjabi army loose on the Bengali people, for the unity of east and west was already a broken thing. The resulting massacre only completed the break and made it irreparable. The army was not entirely without direction; its officers carried \"death lists\" on which appeared the names of the political, cultural, and intellectual leaders of Bengal. There was also a systematic effort to slaughter the followers of these people: university students, political activists, and so on. Beyond these groups, the soldiers ranged freely, burning, raping, killing. Millions of Bengalis fled into India, and their arrival, destitute, hungry, and with incredible stories to tell, established the moral foundation of the later Indian attack. \"It is idle to argue in such cases that the duty of the neighboring people is to look on quietly.\" Months of diplomatic maneuvering followed, but during that time, the Indians were already assisting Bengali guerrillas and offering sanctuary not only to refugees but also to fighting men and women. The two-week war of December 1971 apparently began with a Pakistani air strike, but the Indian invasion required no such prior attack; it was justified on other grounds.\n\nThe strength of the Bengali guerrillas and their achievements between March and December are matters of some dispute; so is their role in the two-week war. Clearly, however, it was not the purpose of the Indian invasion to open the way for the Bengali struggle; nor does the strength or weakness of the guerrillas affect our view of the invasion. When a people are being massacred, we don't require that they pass the test of self-help before coming to their aid. It is their very incapacity that brings us in. The purpose of the Indian army, then, was to defeat the Pakistani forces and drive them out of Bangladesh, that is, to win the war. The purpose was different from that of a counter-intervention, and for an important moral reason. People who initiate massacres lose their right to participate in the normal (even in the normally violent) processes of domestic self-\u00addetermination. Their military defeat is morally necessary.\n\nGovernments and armies engaged in massacres are readily identified as criminal governments and armies (they are guilty, under the Nuremberg code of \"crimes against humanity\"). Hence humanitarian intervention comes much closer than any other kind of intervention to what we commonly regard, in domestic society, as law enforcement and police work. At the same time, however, it requires the crossing of an international frontier, and such crossings are ruled out by the legalist paradigm\u2014unless they are authorized, I suppose, by the society of nations. In the cases I have considered, the law is unilaterally enforced; the police are self-appointed. Now, unilateralism has always prevailed in the international arena, but we worry about it more when what is involved is a response to domestic violence rather than to foreign aggression. We worry that, under the cover of humanitarianism, states will come to coerce and dominate their neighbors; once again, it is not hard to find examples. Hence many lawyers prefer to stick to the paradigm. That doesn't require them, on their view, to deny the (occasional) need for intervention. They merely deny legal recognition to that need. Humanitarian intervention \"belongs in the realm not of law but of moral choice, which nations, like individuals must sometimes make. . . .\" But that is only a plausible formulation if one doesn't stop with it, as lawyers are likely to do. For moral choices are not simply made; they are also judged, and so there must be criteria for judgment. If these are not provided by the law, or if legal provision runs out at some point, they are nevertheless contained in our common morality, which doesn't run out, and which still needs to be explicated after the lawyers have finished.\n\nMorality, at least, is not a bar to unilateral action, so long as there is no immediate alternative available. There was none in the Bengali case. No doubt, the massacres were a matter of universal interest, but only India interested itself in them. The case was formally carried to the United Nations, but no action followed. Nor is it clear to me that action undertaken by the UN, or by a coalition of powers, would necessarily have had a moral quality superior to that of the Indian attack. What one looks for in numbers is detachment from particularist views and consensus on moral rules. And for that, there is at present no institutional appeal; one appeals to humanity as a whole. States don't lose their particularist character merely by acting together. If governments have mixed motives, so do coalitions of governments. Some goals, perhaps, are cancelled out by the political bargaining that constitutes the coalition, but others are super-added; and the resulting mix is as accidental with reference to the moral issue as are the political interests and ideologies of a single state.\n\nHumanitarian intervention is justified when it is a response (with reasonable expectations of success) to acts \"that shock the moral conscience of mankind.\" The old-fashioned language seems to me exactly right. It is not the conscience of political leaders that one refers to in such cases. They have other things to worry about and may well be required to repress their normal feelings of indignation and outrage. The reference is to the moral convictions of ordinary men and women, acquired in the course of their everyday activities. And given that one can make a persuasive argument in terms of those convictions, I don't think that there is any moral reason to adopt that posture of passivity that might be called waiting for the UN (waiting for the universal state, waiting for the messiah . . . ).\n\nSuppose . . . that a great power decided that the only way it could continue to control a satellite state was to wipe out the satellite's entire population and recolonize the area with \"reliable\" people. Suppose the satellite government agreed to this measure and established the necessary mass extermination apparatus. . . . Would the rest of the members of the U.N. be compelled to stand by and watch this operation merely because [the] requisite decision of U.N. organs was blocked and the operation did not involve an \"armed attack\" on any [member state]? . . .\n\nThe question is rhetorical. Any state capable of stopping the slaughter has a right, at least, to try to do so. The legalist paradigm indeed rules out such efforts, but that only suggests that the paradigm, unrevised, cannot account for the moral realities of military intervention.\n\nThe second, third, and fourth revisions of the paradigm have this form: states can be invaded and wars justly begun to assist secessionist movements (once they have demonstrated their representative character), to balance the prior interventions of other powers, and to rescue peoples threatened with massacre. In each of these cases we permit or, after the fact, we praise or don't condemn these violations of the formal rules of sovereignty, because they uphold the values of individual life and communal liberty of which sovereignty itself is merely an expression. The formula is, once again, permissive, but I have tried in my discussion of particular cases to indicate that the actual requirements of just interventions are constraining indeed. And the revisions must be understood to include the constraints. Since the constraints are often ignored, it is sometimes argued that it would be best to insist on an absolute rule of nonintervention (as it would be best to insist on an absolute rule of a nonanticipation). But the absolute rule will also be ignored, and we will then have no standards by which to judge what happens next. In fact, we do have standards, which I have tried to map out. They reflect deep and valuable, though in their applications difficult and problematic, commitments to human rights.\n\n The domestic analogy suggests that the most obvious way of not qualifying for nonintervention is to be incompetent (childish, imbecilic, and so on). Mill believed that there were incompetent peoples, barbarians, in whose interest it was to be conquered and held in subjection by foreigners. \"Barbarians have no rights as a nation [i.e., as a political community]. . . .\" Hence utilitarian principles apply to them, and imperial bureaucrats legitimately work for their moral improvement. It is interesting to note a similar view among the Marxists, who also justified conquest and imperial rule at certain stages of historical development. (See Shlomo Avineri, ed., Karl Marx on Colonialism and Modernization, New York, 1969.) Whatever plausibility such arguments had in the nineteenth century, they have none today. International society can no longer be divided into civilized and barbarian halves; any line drawn on developmental principles leaves barbarians on both sides. I shall therefore assume that the self-help test applies equally to all peoples.\n\n There is a further issue here, having to do with the natural resources that are sometimes at stake in secessionist struggles. I have argued that \"the land follows the people\" (chapter 4). But the will and capacity of the people for self-\u00addetermination may not establish a right to secede if the secession would remove not only land but also vitally needed fuel and mineral resources from some larger political community. The Katangan controversy of the early 1960s suggests the possible difficulties of such cases\u2014and invites us to worry also about the motives of intervening states. But what was missing in Katanga was a genuine national movement capable, on its own, of \"arduous struggle.\" (See Conor C. O'Brien, To Katanga and Back, New York, 1962.) Given the existence of such a movement, I would be inclined to support secession. It would then be necessary, however, to raise more general questions about distributive justice in international society.\n\n The case is different, obviously, when the lives at stake are those of fellow nationals. Interventions designed to rescue citizens threatened with death in a foreign country have conventionally been called humanitarian, and there is no reason to deny them that name when life and death are really at issue. The Israeli raid on Entebbe airport in Uganda (July 4, 1976) seems likely to become a classic case. Here there is, or ought to be, no question of mixed motives: the only purpose is to rescue these people toward whom the intervening power has a special commitment.\n\nWar's Ends, and the Importance of Winning\n\nWhat may be called the modernist view of war is grimly summed up in a poem by Randall Jarrell:\n\nProfits and death grow marginal:\n\nOnly the mourning and the mourned recall\n\nThe wars we lose, the wars we win;\n\nAnd the world is\u2014what it has been.\n\nWar kills; that is all it does; even its economic causes are not reflected in its outcomes; and the soldiers who die are, in the contemporary phrase, wasted. Jarrell speaks in the name of those wasted men, of comrades already dead and of others who know they will soon be killed. And theirs is an authoritative perspective: there have been so many of them. When soldiers die in small numbers, in encompassable battles, they can attribute some meaning to their deaths. Sacrifice and heroism are conceivable notions. But the slaughter of modern warfare overwhelms their capacity for moral understanding; cynicism is their last resort. It is not, however, our last resort, or the most important form of our perceptions of the war in which Jarrell fought. Indeed, most of his fellow survivors would still want to affirm that the world is different, and better, for the Allied victory and the defeat of the Nazi regime. And theirs, too, is an authoritative perspective: there are so many of them. In an age when human sensibility is finely tuned to all the nuances of despair, it still seems important to say of those who die in war that they did not die in vain. And when we can't say that, or think we can't, we mix our mourning with anger. We search for guilty men. We are still committed to a moral world.\n\nWhat does it mean not to have died in vain? There must be purposes that are worth dying for, outcomes for which soldiers' lives are not too high a price. The idea of a just war requires the same assumption. A just war is one that it is morally urgent to win, and a soldier who dies in a just war does not die in vain. Critical values are at stake: political independence, communal liberty, human life. Other means failing (an important qualification), wars to defend these values are justified. The deaths that occur in their course, on both sides, are morally comprehensible\u2014which is not to say that they are not also the products of military stupidity and bureaucratic snafu: soldiers die senselessly even in wars that are not senseless.\n\nBut if it is sometimes urgent to win, it is not always clear what winning is. On the conventional military view, the only true aim in war is \"the destruction of the enemy's main forces on the battlefield.\" Clausewitz speaks of \"the overthrow of the enemy.\" But many wars end without any such dramatic ending, and many war aims can be achieved well short of destruction and overthrow. We need to seek the legitimate ends of war, the goals that can rightly be aimed at. These will also be the limits of a just war. Once they are won, or once they are within political reach, the fighting should stop. Soldiers killed beyond that point die needlessly, and to force them to fight and possibly to die is a crime akin to that of aggression itself. It is commonly said of just war theory, however, that it does not in fact draw this line at any point short of destruction and overthrow, that the most extreme military argument and the \"moralist\" argument coincide in requiring that war be fought to its ultimate end. In the aftermath of World War II, a group of writers appeared who insisted that the pursuit of justice was deeply implicated in the horrors of twentieth-century war. They called themselves \"realists,\" and I shall use that name, though these were not in fact followers of Thucydides and Hobbes. Their argument was less general and ultimately less subversive of conventional morality. Just wars turn into crusades, they claimed, and then the statesmen and soldiers who fight them seek the only victory appropriate to their cause: total victory, unconditional surrender. They fight too brutally and too long. They sow justice and reap death. It is a powerful argument, though I shall want to suggest with reference both to the conduct of war and to the purposes for which it is fought that it makes no sense except as a moral argument. The remedy the realists proposed was to give up justice and aim at more modest outcomes. The remedy I want to propose instead is to understand better the justice at which we cannot help aiming.\n\nUnconditional Surrender\n\nAllied Policy in World War II\n\nThe realist position might be summed up in this way. It is a feature of democratic or liberal culture that peace is conceived as a normative condition. Wars can only be fought, then, if some \"universal moral principle\" requires it: the preservation of peace, the survival of democracy, and so on. And once war begins, this principle must be vindicated absolutely; nothing less than total victory will justify the resort to the \"evil instrument\" of military force. The threat to peace or democracy must be completely destroyed. \"Democratic cultures,\" as Kecskemeti has written in his well-known book on surrender, \"are profoundly unwarlike: to them, war can be justified only if it is waged to eliminate war. . . . This crusading ideology . . . is reflected in the conviction that hostilities cannot be brought to an end before the evil enemy system has been eradicated.\" The locus classicus of this ideology is the thought of Woodrow Wilson, and its most important material expression is the Allied demand for unconditional surrender in World War II.\n\nWhat is objectionable about democratic idealism, as the realists describe it, is that it sets goals that cannot possibly be reached, for which soldiers can only die in vain. This is a moral objection, and an important one if soldiers have in fact been asked to die for such purposes as \"the eradication of evil.\" Their most heroic efforts, after all, can only bring a particular war to an end; they cannot end war. They can save democracy from a particular threat, but they cannot make the world safe for democracy. But I am inclined to think that the significance of these Wilsonian slogans has been much overestimated in the realist literature. By the time Wilson brought the United States into World War I, the fighting had already been carried well beyond the limits of justice and reason. The worst of those \"injuries . . . to the structure of human society which a century will not efface\" had already been inflicted, and the men responsible were not innocent Americans but the tough-minded statesmen and soldiers of Britain, France, and Germany. Wilson's Fourteen Points made possible a German surrender on terms that fell far short of the war aims of Lloyd George and Clemenceau. Indeed, it was the German charge that these terms had not been honored in the actual peace settlement (which was true) that led the Allies to insist on unconditional surrender the second time around. \"No such arguments will be admitted by us as were used by Germany after the last war,\" Churchill told the House of Commons in February 1944. \"The policy of unconditional surrender,\" writes Kecskemeti, \"represents a studied contrast with President Wilson's political conduct of the war in 1918.\" But if that is true, it isn't easy to see how both Wilsonian and anti-Wilsonian policies, surrender on terms and unconditional surrender, can be attributed to \"the traditional moralistic all-or-nothing American approach to the problem of war and peace.\"\n\nFor all his idealism, Wilson fought a limited war; his ideals set the limits. (Whether these were the right limits or not is another question.) Nor was World War II an unlimited war, despite the refusal of the Allies to offer terms. The demand for unconditional surrender, Churchill assured the Commons, \"does not mean that we] are entitled to behave in a barbarous manner, nor that [we] wish to blot out Germany from among the nations of Europe.\" What it does mean, he went on, is that \"if we are bound, we are bound by our own consciences to civilization. We are not bound to the Germans as the result of a bargain struck.\" It would have been more precise had he said that the Allies were not bound to the German government, for the German people, the greater number of them, at any rate, must be included under the rubric of \"civilization.\" They were entitled to the protection of civilized norms and could never have been entirely at the mercy of their conquerors. There is really no such thing (in the moral world) as the unconditional surrender of a nation, for conditions inhere in the very idea of international relations, as they do in the idea of human relations\u2014and they are roughly the same in each. Even domestic criminals, with whom the authorities don't usually negotiate, never surrender unconditionally. If they cannot stipulate conditions above those established in the law, it is nevertheless true that the law recognizes rights\u2014the right not to be tortured, for example\u2014which are theirs as human beings and as citizens, whatever their crimes. Nations have similar rights in international society, above all the right not to be \"blotted out,\" deprived forever of sovereignty and freedom.[a\n\nConcretely, the policy of unconditional surrender involved two commitments: first, that the Allies would not negotiate with Nazi leaders, would have no dealings with them of any sort, \"except to instruct them about the details of orderly capitulation\"; secondly, that no German government would be recognized as legitimate and authoritative until the Allies had won the war, occupied Germany, and established a new regime. Given the character of the existing German government, these commitments do not seem to me to represent an excessive idealism. But they do suggest the outer limit of what can legitimately be sought in war. The outer limit is the conquest and political reconstruction of the enemy state, and only against an enemy like Nazism can it possibly be right to reach that far. In his lectures on American diplomacy, George Kennan suggests that unconditional surrender should not have been talked about, but he nevertheless agrees \"that Hitler was a man with whom a compromise peace was impracticable and unthinkable. . . .\" That is, one might say, a realistic moral judgment. It recognizes, without explicitly affirming, the evil of the Nazi regime, and it rightly places Nazism outside the (moral) world of bargaining and accommodation. We can understand the right of conquest and reconstruction only with such an example. The right does not arise in every war; it did not arise, I think, in the war against Japan. It exists only in cases where the criminality of the aggressor state threatens those deep values that political independence and territorial integrity merely stand for in the international order, and when the threat is in no sense accidental or transitory but is inherent in the very nature of the regime.\n\nOne must be careful here; it is at this point that just wars come nearest to crusades. A crusade is a war fought for religious or ideological purposes. It aims not at defense or law enforcement, but at the creation of new political orders and at mass conversions. It is the international equivalent of religious persecution and political repression, and it is obviously ruled out by the argument for justice. Yet the very existence of Nazism tempts us, as it tempted General Eisenhower, to imagine World War II as a \"crusade in Europe.\" So we must draw the line between just wars and crusades as clearly as we can. Consider the following argument of a nineteenth-\u00adcentury English jurist:\n\nThe first limitation of the general right, incident to every state, of adopting whatever form of government . . . [it] may please is this:\n\nNo state has a right to establish a form of government which is built upon professed principles of hostility to the government of other nations.\n\nThis is to draw the line very dangerously, for it suggests that we might make war against governments whose \"professions\" we have some reason to dislike or fear. But professions are not to the point. We have no clear knowledge as to when these are likely to be acted out and when they are not. No single form of government seems particularly prone to aggression. It is certainly not the case, as many nineteenth-century liberals imagined, that authoritarian states are more likely to make war than democracies are: the history of democratic regimes, beginning with Athens, offers no evidence of this. Nor is hostility to governments relevant here, except insofar as these represent the self-determining activities of nations. The Nazis were at war with nations, not governments alone; they were not merely professedly but actively hostile to the very existence of entire peoples. And it is only in response to hostility of this sort that the rights of conquest and political reconstruction come into existence.\n\nBut suppose the German people had risen against Nazism, as they rose against the Kaiser in 1918, and themselves created a new regime. The Allies were apparently committed not to deal even with a revolutionary German government. \"To the morally oriented Allies,\" writes Kecskemeti, \"any abatement from the strict rules of unconditionality meant that some element of the evil past would survive after the loser's surrender and make their victory meaningless.\" In fact, there was another, and a more realistic, motive for strictness: mutual distrust among Hitler's enemies, the needs of coalition politics. The Western powers and the Russians could agree on nothing except an absolute rule. Justice points the other way, for reasons closely akin to those that mark out and drastically limit the practice of intervention. Had the Germans themselves undertaken to destroy Nazism, there would have been every reason to help them and no need for an external reconstruction of their polity. A German revolution would have made the conquest of Germany morally unnecessary. But there was no revolution and painfully little resistance to Nazi rule. Politically significant opposition developed only within the ruling cadre itself, and only in the latter days of a losing war: thus the coup d'etat attempted by the German generals in July 1944. In peacetime, such an attempt would count as an act of self-determination, and if it were successful, other states would have no choice but to deal with the new government. Given a war such as the Nazis fought, and in which the generals were deeply implicated, the case is harder. I am inclined to think that by 1944 the Allies had a right to expect, and to impose, a more thoroughgoing renovation of German political life. Even the generals would have had to surrender unconditionally (as some of them, at least, were ready to do).\n\nUnconditional surrender is rightly regarded as a punitive policy. It is important to see exactly in what sense this is so. The policy would have penalized the German people only insofar as it declared their political liberty temporarily forfeit and subjected them to a military occupation. Pending the establishment of a post-Nazi and an anti-Nazi regime, the Germans were to be placed in political tutelage: it is a consequence of their failure to overthrow Hitler themselves, the chief of the ways in which they were collectively held responsible for the injuries he and his followers caused to other nations. The forfeiture of independence, however, entails no further loss of rights; the punishment was limited and temporary; it assumed, as Churchill said, the continued existence of a German nation. But the Allies also aimed at more particular and far-reaching punishments. They refused to compromise with the Nazi regime because they planned to put its leading members on trial for their lives. To wage war with such a goal in mind, Kecskemeti argues, is to succumb to \"the pedagogic fallacy,\" that is, to try to build a peaceful post-war world \"on the undying memory of a just chastisement.\" But that cannot be done because deterrence doesn't work in international as it does in domestic society: the number of actors is far smaller; their deeds are not stereotyped and reiterated; the lessons of punishment are interpreted very differently by those who administer and by those who receive them; and in any case, they soon become irrelevant as circumstances change. Now, \"just chastisement\" is exactly what the legalist paradigm would require, and Kecskemeti's criticism points toward the need for further revision. But he argues only that deterrence is ineffective, and his argument, while it is plausible enough, is by no means certainly true. I want to suggest instead that the special character of international society makes the full measure of domestic law enforcement morally infeasible, and at the same time that the special character of Nazism in fact required the \"chastisement\" of the leading Nazis.\n\nWhat is special about international society is the collective character of its members. Each decision-maker stands for an entire community of men and women; the impact of his aggressive and defensive wars is felt over a wide geographic and political range. War affects more people than domestic crime and punishment, and it is the rights of those people that force us to limit its purposes. We might consider a new version of the domestic analogy, oriented toward collective rather than individual action: the attack of one state upon another is more like a feudal raid than a criminal assault (even when it is, literally, a criminal assault). It resembles a feud more than a mugging, not only because there are no commonly accepted police, but also because the rituals of punishment will more probably extend than cut off the violence. Short of the most severe and extraordinary measures\u2014extermination, exile, political dismemberment\u2014an enemy state, like an aristocratic clan, and unlike a common criminal, cannot be entirely deprived of the power of renewed activity. But such measures can never be defended, and so enemy states must be treated, morally as well as strategically, as future partners in some sort of international order.\n\nStability among states, as among aristocratic factions and families, rests upon certain patterns of accommodation and restraint, which statesmen and soldiers would do well not to disrupt. But these patterns are not simply diplomatic artifacts; they have a moral dimension. They depend upon mutual understandings; they are comprehensible only within a world of shared values. Nazism was a conscious and willful challenge to the very existence of such a world: a program of extermination, exile, and political dismemberment. In a sense, aggression was the least of Hitler's crimes. It is not quite right, then, to describe the conquest and occupation of Germany and the trial of Nazi leaders as so many (unavailing) efforts to deter future aggressors. They are better understood as the expressions of a collective abhorrence, a reaffirmation of our own deepest values. And it is right to say, as many people said at the time, that the war against Nazism had to end with such a reaffirmation if it was to end meaningfully at all.\n\nJustice in Settlements\n\nThe policy of unconditional surrender, directed at the government but not the people of Germany, was an appropriate response to Nazism. But it isn't always appropriate. Doing justice, in the legalist sense, isn't always the right thing to do. (I have already argued that it cannot be the goal of counter-interventions.) The cardinal mistake of the realists is to suppose that if one fights for \"universal moral principles,\" one must always fight in the same way, as if universal principles did not have concrete and diverse applications. We need, then, to look at a case where limited aims were set, not by the requirements of a realistic analysis\u2014for realism imposes no moral requirements; aggressors can be realists, too\u2014but by the argument for justice.\n\nThe Korean War\n\nThe American war in Korea was officially described as a \"police action.\" We had come to the aid of a state defending itself against a fullscale invasion, committed ourselves to the hard work of international law enforcement. The United Nations' authorization enhanced our commitment, but its terms were in fact shaped unilaterally. Once again, we were at war with aggression itself as well as with a particular foe. Now, what were the war aims of the U.S. government? One would expect that American democracy, slow to anger but terrible in its righteous wrath, should have aimed at the total eradication of the North Korean regime. In fact, our initial aims were limited in character. In the Senate debate over President Truman's decision to rush American troops into battle, it was stated repeatedly that our sole purpose was to drive the North Koreans back to the partition line and to restore the status quo ante bellum. Senator Flanders insisted that the President \"would not be within his rights in pursuing the Korean forces . . . north of the 38th parallel.\" Senator Lucas, a spokesman for the administration, \"wholeheartedly agreed.\" The debate focused on constitutional issues; there had been no declaration of war and so the President's \"rights\" were limited. At the same time, the Senate did not want to declare war and enlarge those rights; its members were satisfied with what might be called a conservative war. \"The acquisitive state,\" writes Liddell Hart, \"inherently unsatisfied, needs to gain victory in order to gain its object. . . . The conservative state can attain its object . . . by foiling the other side's bid for victory.\"\n\nThat was the American goal until we ourselves, in the immediate aftermath of MacArthur's triumph at Inchon, crossed the 38th parallel. The decision to cross is not at all easy to figure out, but it seems to be an example of military hubris far more than of democratic idealism. Its larger political and moral implications do not seem to have been thought about much at the time; the move was defended mostly in tactical terms. To halt at the old line, it was said, would have surrendered the military initiative to the enemy and allowed him to rebuild his army for a new offensive. \"The aggressor's forces should not be permitted to take refuge behind an imaginary line,\" Ambassador Austin told the UN, \"because that would recreate the threat to the peace. . . .\" I will leave aside the odd notion that the 38th parallel was an imaginary line (how then did we recognize the initial aggression?). It is not implausible to suggest that the North Koreans had no right to a military sanctuary and that attacks across the 38th parallel with the limited purpose of preventing their regroupment might be justified. In responding to an armed invasion, one can legitimately aim not merely at a successful resistance but also at some reasonable security against future attack. But when we crossed the old line, we also took on a more radical purpose. Now it was the American goal, sanctioned, again, by the UN, to unify Korea by force of arms and create a new (democratic) government. And that required not limited attacks within the borders of North Korea, but the conquest of the entire country. The question is whether wars against aggression necessarily generate such far-reaching and exalted goals. Is this what justice requires?\n\nIf it is, we would have done well to settle for something less. But it would be strange for Americans to answer that question in the affirmative, since we had formally branded the North Korean attempt to unify the country by force a criminal aggression. Secretary of State Acheson seems to have felt the difficulty when he told the Senate (during the MacArthur hearings) that unification had never been our military objective. We aimed only \"to round up the people that were putting on the aggression.\" That would have created a political vacuum in the North, he went on, and Korea would then have been unified, not through force, but \"through elections and that sort of thing. . . .\" Disingenuous as this is, it nevertheless is indicative of what the argument for justice requires. Defending the morality of American policy, Acheson is forced to insist on the limited character of our military effort and to deny that it ever was a crusade against communism. He did believe, however, that the success of our police action required something very like the conquest of North Korea.\n\nClearly, the analogy in his mind was with domestic law enforcement, where one doesn't simply stop the criminal activity and restore the status quo ante; one also \"rounds up\" the criminals and holds them for trial and punishment. But this feature of the domestic model (and hence of the legalist paradigm) is not easily carried over into the international arena. For the roundup of the aggressors will most often require a military conquest, and conquest has effects that reach far beyond the people who are rounded up. It prolongs a war in which large numbers of innocent men and women are virtually certain to die, and it puts a whole nation, as we have seen, under political tutelage. It does this even if its methods are democratic (\"free elections and that sort of thing\"), because it replaces a regime which the people of the conquered nation had not themselves sought to replace\u2014indeed, for which they had recently fought and died. Unless the activities of that regime are a standing affront to the conscience of mankind, its destruction is not a legitimate military goal. And however grim a picture one paints, the North Korean regime was not such an affront; its policies were more like those of Bismarck's than of Hitler's Germany. Its leaders may well have been guilty of criminal aggression, but their physical capture and punishment seem at most the marginal benefit of a certain sort of military victory, not a reason for seeking such a victory.\n\nThe argument at this point might be put in terms of proportionality, a doctrine often said to fix firm limits to the length of wars and the shape of settlements. In this instance, we would have to balance the costs of continued fighting against the value of punishing the aggressors. Given our present knowledge of the Chinese invasion and its consequences, we can say that the costs were disproportionate (and the aggressors never punished). But even without such knowledge, a strong case might have been made that Acheson's \"round-up\" did not warrant its likely price. On the other hand, it is characteristic of arguments of this sort that an equally strong case could have been made on the other side, simply by enlarging our conception of the purposes of the war. Proportionality is a matter of adjusting means to ends, but as the Israeli philosopher Yehuda Melzer has pointed out, there is an overwhelming tendency in wartime to adjust ends to means instead, that is, to redefine initially narrow goals in order to fit the available military forces and technologies. Perhaps the conquest of North Korea could not be defended as a means of punishing aggressors; it might nonetheless have been defended as a means of doing that and simultaneously abolishing a border that could only be (in fact has been) the focus of future tension\u2014hence, avoiding wars to come. It is necessary in such arguments to hold ends constant, but how does one do that? In practice, the inflation of ends is probably inevitable unless it is barred by considerations of justice itself.\n\nNow justice in settlements is a complex notion, but it has a certain minimal content which seems to have been understood well enough by America's leaders at the beginning of the struggle. Once that minimal content has been realized, it is the rights of the people of the enemy country that rule out further fighting, whatever its added value.b These rights were no doubt badly represented by the North Korean regime, but that in itself is not, as we have seen, a sufficient reason for a war of conquest and reconstruction. It was the crime of the aggressor to challenge individual and communal rights, and states responding to aggression must not repeat the challenge once basic values have been upheld.\n\nI can now restate the fifth revision of the legalist paradigm. Because of the collective character of states, the domestic conventions of capture and punishment do not readily fit the requirements of international society. They are unlikely to have significant deterrent effects; they are very likely to extend rather than restrict the number of people exposed to coercion and risk; and they require acts of conquest that can only be aimed at entire political communities. Except when they are directed against Nazi-like states, just wars are conservative in character; it cannot be their purpose, as it is the purpose of domestic police work, to stamp out illegal violence, but only to cope with particular violent acts. Hence the rights and limits fixed by the argument for justice: resistance, restoration, reasonable prevention. I am afraid that these are not as constraining as they may sound. It will often require a fairly decisive military defeat to persuade aggressor states that they cannot succeed in their conquests. They would not have begun the fighting, obviously, unless their leaders had high hopes. And further military action may be necessary before a peace settlement can be worked out that provides even minimal security for the victim: disengagement, demilitarization, arms control, external arbitration, and so on.c Some combination of these, appropriate to the circumstances of a particular case, constitutes a legitimate war aim. If this falls short of the \"punishment of aggression,\" it has to be said that military defeat is always punishing and that the preventive measures I have listed are also penalties, indeed, collective penalties, insofar as they involve a certain derogation of state sovereignty.\n\n\"The object in war is a better state of peace.\" And better, within the confines of the argument for justice, means more secure than the status quo ante bellum, less vulnerable to territorial expansion, safer for ordinary men and women and for their domestic self-determinations. The key words are all relative in character: not invulnerable, but less vulnerable; not safe, but safer. Just wars are limited wars; there are moral reasons for the statesmen and soldiers who fight them to be prudent and realistic. Overreaching is common in war, however, and has many causes; I do not want to deny that a certain characteristic distortion of the argument for justice is one among them. Democratic idealism in the debased forms of self-righteousness and zeal sometimes prolongs wars, but so does aristocratic pride, military hubris, religious and political intolerance. A few sentences from David Hume's essay \"On the Balance of Power\" suggest that we should add to the list the \"obstinacy and passion\" with which even sophisticated statesmen, like those of eighteenth-century Britain, defend the balance:\n\nThe same peace which was afterwards made at Ryswick in 1697 was offered so early as the year ninety-two; that concluded at Utrecht in 1712 might have been finished on as good conditions . . . in the year eight; and we might have given at Frankfurt in 1743 the same terms which we were glad to receive at Aix-la-Chappelle in the year forty-\u00adeight. Above half of our wars with France . . . are owing more to our own imprudent vehemence than to the ambition of our neighbor.\n\nThe realists have (unrealistically) looked for a single enemy; in fact, they have more than they can handle without the support of a fully developed moral doctrine.\n\nIn the heated debates over America's Korean war, those political and military figures favoring the expansion of the conflict frequently cited the maxim: in war there is no substitute for victory. The idea, it should be said, is more readily traceable to Clausewitz than to Woodrow Wilson; it is anyway a silly idea, since it offers no definition of victory. In the case at hand, that word was presumably meant to describe a condition in which the enemy was utterly broken, without further resources. Given that meaning, it can safely be said that the maxim is historically as well as morally false. Nor is its falsehood an esoteric doctrine; it was widely accepted among American leaders in the early 1950s, and the government was able to sustain, through a difficult time, its search for a substitute. But the maxim is right in another sense. In a just war, its goals properly limited, there is indeed nothing like winning. There are alternative outcomes, of course, but these are accepted only at some cost to basic human values. And that means that there are sometimes moral reasons for prolonging a war. Consider those long months when the Korean negotiations were stalemated over the issue of the forcible repatriation of prisoners. The American negotiators insisted on the principle of free choice, lest the peace be as coercive as war itself, and accepted the continuation of the fighting rather than yield on that point. They were probably right, though it is difficult at this distance to weigh the values involved\u2014and here the doctrine of proportionality is surely relevant. In any case, it follows from the argument for justice that wars can end too soon. There is always a humanitarian impulse to stop the fighting, and attempts are often made by the great powers (or the United Nations) to impose a cease-fire. But it isn't always true that such cease-fires serve the purposes of humanity. Unless they create a \"better state of peace,\" they may simply fix the conditions under which the fighting will be resumed, at a later time and with a new intensity. Or they may confirm a loss of values the avoidance of which was worth a war.\n\nThe theory of ends in war is shaped by the same rights that justify the fighting in the first place\u2014most importantly, by the right of nations, even of enemy nations, to continued national existence and, except in extreme circumstances, to the political prerogatives of nationality. The theory incorporates arguments for prudence and realism; it is an effective bar to total war; and it is, I think, harmonious with other features of jus ad bellum. But the case is different with the theory of means, to which I now must turn. Here there appear to be tensions and even contradictions that are internal to the argument for justice. It is with reference to the conduct of war and not to the end for which it is fought that the urgent need to do justice seems sometimes to lead statesmen and soldiers to act unjustly, that is, to fight without restraint and with a crusading zeal.\n\nOnce we have agreed upon the character of aggression, and of those threats of war that constitute aggression, and of those acts of colonial oppression and foreign interference that justify interventions and counter-interventions, we have also made it possible to identify enemies in the world: governments and armies that can rightly be (and perhaps should be) resisted. The wars that result from this resistance are the responsibility of those governments and armies; the hell of war is their crime. And if it isn't always true that their leaders ought to be punished for their crimes, it is vitally important that they not be allowed to benefit from them. If they can rightly be resisted, they should also be successfully resisted. Hence the temptation to fight by any means\u2014which brings us up against what I have described in Part One as the fundamental dualism of our conception of war. For the rules of encounter take no cognizance whatever of the relative guilt of governments and armies. The theory of jus in bello, though it, too, is founded on the rights of life and liberty, stands independently of and apart from the theory of aggression. The limits it imposes are imposed equally and indifferently on aggressors and their adversaries. And the acceptance of these limits\u2014moderation in battle\u2014may well make it difficult to achieve the ends of war, even if these are moderate ends. Can the rules, then, be set aside for the sake of a just cause? I shall try to answer that question, or to suggest some ways in which it might be answered, but only after examining in detail the nature and practical workings of the rules themselves.\n\n It was once argued by jurists and philosophers that conquerors had a right to kill or enslave the citizens of a conquered state. Against this view, in the name of natural law or human rights, Montesquieu and Rousseau claimed that the conqueror's prerogatives extended only to the state, not to the individual men and women who composed it. \"The state is the association of men, not the men themselves; the citizen may perish and the man remain.\" (The Spirit of the Laws, X.3.) \"Sometimes it is possible to kill the State without killing a single one of its members; and war gives no right which is not necessary to the gaining of its object.\" (The Social Contract, I.4.) But this is still too permissive a view, for the rights of individuals include the right of political association, and if the citizen is killed or the state destroyed, something of the man dies too. Even the destruction of a particular regime is only defensible, as I will argue, in exceptional circumstances.\n\n Or it is the rights of one's own people. Consider the classic discussion of proportionality in war in Shakespeare's Troilus and Cressida (II.2). Hector and Troilus are debating the surrender of Helen:\n\nHector. Brother, she is not worth what she doth cost\n\nThe keeping.\n\nTroilus. What's aught but as 'tis valued?\n\nHector. But value dwells not in particular will.\n\nIt holds his estimate and dignity\n\nAs well wherein 'tis precious of itself\n\nAs in the prizer. 'Tis mad idolatry\n\nTo make the service greater than the god.\n\nTroilus quickly switches the argument from Helen herself to the honor of the Trojan warriors, and so wins the debate, for the value of honor seems indeed to dwell in particular wills. The move is typical, and it can be countered only with a moral claim: that the Trojan warriors have no right to put a whole city at risk for the sake of their own honor. It is not that the sacrifice is greater than the god, but that the men, women, and children likely to be sacrificed are not necessarily believers in the god and don't share in the worship.\n\n The list can be extended to include the temporary occupation of enemy territory, pending a peace settlement or for some period of time stipulated in the settlement. It does not include annexation, even as a measure of security against further attack. This is so partly for reasons that Marx suggests in his \"Second Address\" (with reference to Alsace-Lorraine): \"If limits are to be fixed by military interests, there will be no end to claims, because every military line is necessarily faulty, and may be improved by annexing some outlying territory; and moreover, they can never be fixed finally and fairly because they always must be imposed by the conqueror upon the conquered and consequently carry within them the seeds of fresh wars.\" It is true, however, that some lines are more \"faulty\" than others and that one can make out both plausible and implausible versions of the argument Marx is opposing. A stronger case against annexation, I should think, rests on the rights of the inhabitants of the annexed land.\nPart Three\n\nThe War Convention\n\nWar's Means and the Importance of Fighting Well\n\nThe purpose of the war convention is to establish the duties of belligerent states, of army commanders, and of individual soldiers with reference to the conduct of hostilities. I have already argued that these duties are precisely the same for states and soldiers fighting wars of aggression and wars of defense. In our judgments of the fighting, we abstract from all consideration of the justice of the cause. We do this because the moral status of individual soldiers on both sides is very much the same: they are led to fight by their loyalty to their own states and by their lawful obedience. They are most likely to believe that their wars are just, and while the basis of that belief is not necessarily rational inquiry but, more often, a kind of unquestioning acceptance of official propaganda, nevertheless they are not criminals; they face one another as moral equals.\n\nThe domestic analogy is of little help here. War as an activity (the conduct rather than the initiation of the fighting) has no equivalent in a settled civil society. It is not like an armed robbery, for example, even when its ends are similar in kind. Indeed, it is the contrast rather than the correspondence that illuminates the war convention. The contrast is readily explicated; we have only to think about the following sorts of cases. (1) In the course of a bank robbery, a thief shoots a guard reaching for his gun. The thief is guilty of murder, even if he claims that he acted in self-defense. Since he had no right to rob the bank, he also had no right to defend himself against the bank's defenders. He is no less guilty for killing the guard than he would be for killing an unarmed bystander\u2014a customer, say, depositing his money. The thief's associates might praise him for the first killing, which was in their terms necessary, and condemn him for the second, which was wanton and dangerous. But we won't judge him in that way, because the idea of necessity doesn't apply to criminal activity: it was not necessary to rob the bank in the first place.\n\nNow, aggression is also a criminal activity, but our view of its participants is very different: (2) In the course of an aggressive war, a soldier shoots another soldier, a member of the enemy army defending his homeland. Assuming a conventional firefight, this is not called murder; nor is the soldier regarded after the war as a murderer, even by his former enemies. The case is in fact no different from what it would be if the second soldier shot the first. Neither man is a criminal, and so both can be said to act in self-defense. We call them murderers only when they take aim at noncombatants, innocent bystanders (civilians), wounded or disarmed soldiers. If they shoot men trying to surrender or join in the massacre of the inhabitants of a captured town, we have (or ought to have) no hesitation in condemning them. But so long as they fight in accordance with the rules of war, no condemnation is possible.\n\nThe crucial point is that there are rules of war, though there are no rules of robbery (or of rape or murder). The moral equality of the battlefield distinguishes combat from domestic crime. If we are to judge what goes on in the course of a battle, then, \"we must treat both combatants,\" as Henry Sidgwick has written, \"on the assumption that each believes himself in the right.\" And we must ask \"how the duties of a belligerent, fighting in the name of justice, and under the restraints of morality, are to be determined.\" Or, more directly: without reference to the justice of their cause, how can soldiers fight justly?\n\nUtility and Proportionality\n\nThe Argument of Henry Sidgwick\n\nSidgwick answers this question with a twofold rule that neatly sums up the most common utilitarian view of the war convention. In the conduct of hostilities, it is not permissible to do \"any mischief which does not tend materially to the end [of victory], nor any mischief of which the conduciveness to the end is slight in comparison with the amount of the mischief.\" What is being prohibited here is excessive harm. Two criteria are proposed for the determination of excess. The first is that of victory itself, or what is usually called military necessity. The second depends upon some notion of proportionality: we are to weigh \"the mischief done,\" which presumably means not only the immediate harm to individuals but also any injury to the permanent interest of mankind, against the contribution that mischief makes to the end of victory.\n\nThe argument as stated, however, sets the interests of individuals and of mankind at a lesser value than the victory that is being sought. Any act of force that contributes in a significant way to winning the war is likely to be called permissible; any officer who asserts the \"conduciveness\" of the attack he is planning is likely to have his way. Once again, proportionality turns out to be a hard criterion to apply, for there is no ready way to establish an independent or stable view of the values against which the destruction of war is to be measured. Our moral judgments (if Sidgwick is right) wait upon purely military considerations and will rarely be sustained in the face of an analysis of battle conditions or campaign strategy by a qualified professional. It would be difficult to condemn soldiers for anything they did in the course of a battle or a war that they honestly believed, and had good reason to believe, was necessary, or important, or simply useful in determining the outcome. Sidgwick apparently thought this conclusion inescapable, once we agree to make no judgment as to the relative utility of different outcomes. For then we must grant that soldiers are entitled to try to win the wars they are entitled to fight. That means that they can do what they must to win; they can do their utmost, so long as what they do is actually related to winning. Indeed, they should do their utmost, so as to end the fighting as quickly as possible. The rules of war rule out only purposeless or wanton violence.\n\nThat is not, however, a small achievement. If it were made effective in practice, it would eliminate a great deal of the cruelty of war. For it has to be said of many of the people who die in the course of a war, soldiers as well as civilians, that their deaths do not \"tend materially to the end [of victory]\" or that the contribution they make to that end is \"slight\" indeed. These deaths are nothing more than the inevitable consequence of putting deadly weapons into the hands of undisciplined soldiers, and armed men into the hands of stupid or fanatical generals. Every military history is a tale of violence and destruction out of all relation to the requirements of combat: massacres on the one hand and, on the other, ill-planned and wasteful battles that are little better than massacres.\n\nSidgwick's twofold rule seeks to impose an economy of force. It requires discipline and calculation. Any intelligent military strategy, of course, imposes the same requirements. On Sidgwick's view, a good general is a moral man. He keeps his soldiers in check, keyed for battle, so that they don't run amuck among civilians; he sends them to fight only after having thought through a battle plan, and his plan is aimed at winning as quickly and as cheaply as possible. He is like General Roberts at the battle of Paardeberg (in the Boer War), who called off the frontal assaults on the Boer trenches ordered by Kitchener, his second in command, saying that the loss of life \"did not appear . . . to be warranted by the exigencies of the situation.\" A simple decision, though not as common in war as one might expect. I don't know if it was made out of any deep concern for human life; perhaps Roberts was thinking only of his honor as a general (who does not send his men to be slaughtered), or perhaps he was worried about the capacity of the troops to renew the fighting on the following day. It was in any case exactly the sort of decision that Sidgwick would require.\n\nBut though the limits of utility and proportionality are very impor\u00adtant, they do not exhaust the war convention; indeed, they don't explain the most critical of the judgments we make of soldiers and their generals. If they did, moral life in wartime would be a great deal easier than it is. The war convention invites soldiers to calculate costs and benefits only up to a point, and at that point it establishes a series of clearcut rules\u2014moral fortifications, so to speak, that can be stormed only at great moral cost. Nor can a soldier justify his violation of the rules by referring to the necessities of his combat situation or by arguing that nothing else but what he did would have contributed significantly to victory. Soldiers who reason in that way can never violate Sidgwick's limits, since all that Sidgwick requires is that soldiers . . . reason in that way. But justifications of this kind are not acceptable, or not always acceptable, either in law or morality. They have been \"generally rejected,\" according to the U.S. Army's handbook of military law, \" . . . for acts forbidden by the customary and conventional laws of war, inasmuch as [these laws] have been developed and framed with consideration for the concept of military necessity.\" Now, what sorts of acts are these, and what are the grounds for forbidding them, if Sidgwick's criteria don't apply? I will have to explain later on how \"military necessity\" is taken into account in framing the prohibitions; I am concerned now with their general character.\n\nBelligerent armies are entitled to try to win their wars, but they are not entitled to do anything that is or seems to them necessary to win. They are subject to a set of restrictions that rest in part on the agreements of states but that also have an independent foundation in moral principle. I don't think that these restrictions have ever been expounded in utilitarian fashion, though it is no doubt a good thing that they be expounded and that military conduct be shaped to their requirements. When we abstract from the utility of particular outcomes, focus exclusively on jus in bello, utilitarian calculations are radically constrained. It might be said that if every war in a series extending indefinitely into the future were to be fought with no other limits than those proposed by Sidgwick, the consequences for mankind would be worse than if every war in that same series were fought within limits fixed by some additional set of prohibitions.a But saying that does not suggest which prohibitions are the right ones. And any effort to figure out the right ones by calculating the likely effects over time of fighting wars in certain ways (an enormously difficult task) is sure to run up against unconstrained utilitarian arguments: that victory here and now will end the series of wars, or reduce the probability of future fighting, or avoid immediate and horrifying consequences. Hence anything should be permitted that is useful and proportionate to the victory being sought. Utilitarianism is obviously most effective when it points to outcomes about which we have (relatively) clear ideas. For that reason, it is more likely to tell us that the rules of war should be overridden in this or that case than it is to tell us what the rules are\u2014beyond Sidgwick's minimum injunctions which can't and don't ever have to be overridden.\n\nUntil the constraints are lifted and the substantial effects of victory and defeat are weighed in the balance, utilitarianism provides only a general endorsement of the war convention (the twofold rule and any others commonly accepted); after that, it is unlikely to specify rules at all but only particular courses of action. When to lift the constraints is one of the hardest questions in the theory of war. I will try to answer it in Part Four, and I will describe at that time the positive role of utilitarian calculation: to mark out those special cases where victory is so important or defeat so frightening that it is morally, as well as militarily, necessary to override the rules of war. But such an argument is not possible until we have recognized rules beyond Sidgwick's and understood their moral force.\n\nMeanwhile, it is worth dwelling for a moment on the precise nature of the general endorsement. The utility of fighting limited wars is of two sorts. It has to do not only with reducing the total amount of suffering, but also with holding open the possibility of peace and the resumption of pre-war activities. For if we are (at least formally) indifferent as to which side wins, we must assume that these activities will in fact be resumed and with the same or similar actors. It is important, then, to make sure that victory is also in some sense and for some period of time a settlement among the belligerents. And if that is to be possible, the war must be fought, as Sidgwick says, so as to avoid \"the danger of provoking reprisals and of causing bitterness that will long outlast\" the fighting. The bitterness that Sidgwick has in mind might, of course, be the consequence of an outcome thought to be unjust (like the annexation of Alsace-Lorraine in 1871), but it may also result from military conduct thought to be unnecessary, brutal or unfair, or simply \"against the rules.\" So long as defeat follows from what are widely regarded as legitimate acts of war, it is at least possible that it will leave behind no festering resentment, no sense of scores unsettled, no deeply felt need for individual or collective revenge. (The government or officers' corps of the defeated state may have reasons of its own to encourage such feelings, but that is another matter.) An analogy might be drawn, once again, with a family feud, its origin long forgotten, its justice no longer at issue. A feud of this sort may be carried on for many years, marked by the occasional killing of a father or a grown-up son, an uncle or a nephew, first of one family, then of the other. So long as nothing more happens, the possibility of reconciliation remains open. But if someone in a fit of anger or passion, or even by accident or mistake, kills a woman or a child, the result may well be a massacre or a series of massacres, not stopping until one of the families is wiped out or driven away. The case is at least similar to intermittent war among states. Some limits must be commonly accepted, and more or less consistently maintained, if there is ever to be a peace short of the complete submission of one of the belligerents.\n\nIt is probably true that any limits will be useful here, so long as they are in fact commonly accepted. But no limit is accepted simply because it is thought that it will be useful. The war convention must first be morally plausible to large numbers of men and women; it must correspond to our sense of what is right. Only then will we recognize it as a serious obstacle to this or that military decision, and only then can we debate its utility in this or that particular case. For otherwise we would not know which obstacle out of the infinite number that are conceivable, and the very large number that are historically recorded, is to be the subject of our debates. With regard to the rules of war, utilitarianism lacks creative power. Beyond the minimal limits of \"conduciveness\" and proportionality, it simply confirms our customs and conventions, whatever they are, or it suggests that they be overridden; but it does not provide us with customs and conventions. For that, we must turn again to a theory of rights.\n\nHuman Rights\n\nThe Rape of the Italian Women\n\nThe importance of rights may best be suggested if we look at a historical example placed, as it were, on the margin of Sidgwick's argument. Consider, then, the case of the Moroccan soldiers fighting with Free French forces in Italy in 1943. These were mercenary troops who fought on terms, and the terms included license to rape and plunder in enemy territory. (Italy was enemy territory until the Badoglio regime joined the war against Germany in October, 1943; I don't know if the license was then withdrawn; if so, the withdrawal seems to have been ineffective.) A large number of women were raped; we know the number, roughly, because the Italian government later offered them a modest pension. Now, the argument for giving soldiers privileges of this sort is a utilitarian one. It was made long ago by Vitoria in the course of a discussion of the right of sack: it is not unlawful to put a city to sack, he says, if it is \"necessary for the conduct of the war . . . as a spur to the courage of the troops.\" If this argument were applied to the case at hand, Sidgwick might respond that \"necessary\" is probably the wrong word here and that the contribution of rape and plunder to military victory is \"slight\" in comparison with the harm caused to the women involved. That is not an unpersuasive response, but it is not entirely convincing either, and it hardly gets at the root of our condemnation of rape.\n\nWhat is it we object to in the license given those Moroccan soldiers? Surely our judgment does not hang on the fact that rape is only a trivial or inefficient \"spur\" to masculine courage (if it is a spur at all: I doubt that brave men are the most likely rapists). Rape is a crime, in war as in peace, because it violates the rights of the woman who is attacked. To offer her as bait to a mercenary soldier is to treat her as if she were not a person at all but a mere object, a prize or trophy of war. It is the recognition of her personality that shapes our judgment.b And this is true even in the absence of a philosophical conception of human rights, as the following passage from the Book of Deuteronomy\u2014the first attempt that I have found to regulate the wartime treatment of women\u2014clearly indicates:\n\nWhen thou goest forth to battle against thine enemies, and the Lord thy God deliverest them into thy hands, and thou carriest them away captive, and seest among the captives a woman of goodly form, and thou hast a desire unto her, and wouldst take her to thee to wife; then thou shalt bring her home to thy house . . . and she shall . . . bewail her father and mother a full month; and after that thou mayest go in unto her, and be her husband, and she shall be thy wife. And . . . if thou have no delight in her, then thou shalt let her go whither she will; but thou shalt not sell her . . . for money, thou shalt not deal with her as a slave . . .\n\nThis falls far short of contemporary views, though I expect it would be as difficult to enforce today as it was in the time of the Judean kings. Whatever theological or sociological account of the rule is appropriate, it is clear that what is at work here is a conception of the captive woman as a person who must be respected, despite her capture; hence the month of mourning before she is sexually used, the requirement of marriage, the ban on slavery. She has lost some of her rights, we might say, but not all of them. Our own war convention requires a similar understanding. Both the prohibitions that are covered by Sidgwick's twofold rule and those that lie beyond it are properly conceptualized in terms of rights. The rules of \"fighting well\" are simply a series of recognitions of men and women who have a moral standing independent of and resistant to the exigencies of war.\n\nA legitimate act of war is one that does not violate the rights of the people against whom it is directed. It is, once again, life and liberty that are at issue, though we are now concerned with these two as they are individually rather than collectively possessed. I can sum up their substance in terms I have used before: no one can be forced to fight or to risk his life, no one can be threatened with war or warred against, unless through some act of his own he has surrendered or lost his rights. This fundamental principle underlies and shapes the judgments we make of wartime conduct. It is only inadequately expressed in positive international law, but the prohibitions established there have this principle as their source. Lawyers sometimes talk as if the legal rules were simply humanitarian in character, as if the ban on rape or on the deliberate killing of civilians were nothing more than a piece of kindness. But when soldiers respect these bans, they are not acting kindly or gently or magnanimously; they are acting justly. If they are humanitarian soldiers, they may indeed do more than is required of them\u2014sharing their food with civilians, for example, rather than merely not raping or killing them. But the ban on rape and murder is a matter of right. The law recognizes this right, specifies, limits, and sometimes distorts it, but doesn't establish it. And we can recognize it ourselves, and sometimes do, even in the absence of legal recognition.\n\nStates exist to defend the rights of their members, but it is a difficulty in the theory of war that the collective defense of rights renders them individually problematic. The immediate problem is that the soldiers who do the fighting, though they can rarely be said to have chosen to fight, lose the rights they are supposedly defending. They gain war rights as combatants and potential prisoners, but they can now be attacked and killed at will by their enemies. Simply by fighting, whatever their private hopes and intentions, they have lost their title to life and liberty, and they have lost it even though, unlike aggressor states, they have committed no crime. \"Soldiers are made to be killed,\" as Napoleon once said; that is why war is hell.c But even if we take our standpoint in hell, we can still say that no one else is made to be killed. This distinction is the basis of the rules of war.\n\nEveryone else retains his rights, and states remain committed, and entitled, to defend these rights whether their wars are aggressive or not. But now they do this not by fighting but by entering into agreements among themselves (which fix the details of noncombatant immunity), by observing these agreements and expecting reciprocal observance, and by threatening to punish military leaders or individual soldiers who violate them. This last point is crucial for an understanding of the war convention. Even an aggressor state can rightly punish war criminals\u2014enemy soldiers, for example, who rape or kill civilians. The rules of war apply with equal force to aggressors and their adversaries. And we can now see that it is not merely the moral equality of soldiers that requires this mutual submission; it is also the rights of civilians. Soldiers fighting for an aggressor state are not themselves criminals: hence their war rights are the same as those of their opponents. Soldiers fighting against an aggressor state have no license to become criminals: hence they are subject to the same restraints as their opponents. The enforcement of these restraints is one of the forms of law enforcement in international society, and the law can be enforced even by criminal states against \"policemen\" who deliberately kill innocent bystanders. For these bystanders do not forfeit their rights when their states wrongly go to war. An army warring against aggression can violate the territorial integrity and political sovereignty of the aggressor state, but its soldiers cannot violate the life and liberty of enemy civilians.\n\nThe war convention rests first on a certain view of combatants, which stipulates their battlefield equality. But it rests more deeply on a certain view of noncombatants, which holds that they are men and women with rights and that they cannot be used for some military purpose, even if it is a legitimate purpose. At this point, the argument is not entirely dissimilar from that which obtains in domestic society, where a man fighting in self-defense, for example, is barred from attacking or injuring innocent bystanders or third parties. He can attack only his attackers. In domestic society, however, it is relatively easy to distinguish bystanders and third parties, whereas in international society, because of the collectivist character of states and armies, the distinction is harder to make. Indeed, it is often said that it cannot be made at all, for soldiers are only coerced civilians, and civilians are willing supporters of their armies in the field. And then it cannot be what is due to the victims but only what is necessary for the battle that determines our judgments of wartime conduct. Here is the critical test, then, for anyone who argues that the rules of war are grounded in a theory of rights: to make the combatant\/noncombatant distinction plausible in terms of the theory, that is, to provide a detailed account of the history of individual rights under the conditions of war and battle\u2014how they are retained, lost, exchanged (for war rights) and recovered. That is my purpose in the chapters that follow.\n\n The alternative utilitarian argument is that of General von Moltke: additional prohibitions merely drag out the fighting, while \"the greatest kindness in war is to bring it to a speedy conclusion.\" But if we imagine a series of wars, this argument probably won't work. At any given level of restraint, let's say, a war will take so many months. If one of the belligerents breaks the rules, it might end more quickly, but only if the other side fails or is unable to reciprocate. If both sides fight at a lower level of restraint, the way may be shorter or longer; there isn't going to be any general rule. And if restraints have broken down in one war, they are unlikely to be maintained in the next, so any immediate benefits probably won't show up in the balance over time.\n\n In a powerful essay entitled \"Human Personality,\" Simone Weil has attacked this way of talking about what we can and cannot do to other people. Rights talk, she claims, turns \"what should have been a cry of protest from the depth of the heart . . . into a shrill nagging of claims and counter-claims. . . .\" And she applies her argument to a case very much like ours: \"if a young girl is being forced into a brothel she will not talk about her rights. In such a situation, the word would sound ludicrously inadequate.\" (Selected Essays: 1934\u20131943, ed. Richard Rees, London, 1962, p. 21.) Weil would have us refer ourselves instead to some notion of the sacred, of the image of God in man. Perhaps some such ultimate reference is necessary, but I think she is wrong in her claim about the \"sound\" of rights talk. In fact, arguments about human rights have played a significant part in the struggle against oppression, including the sexual oppression of women.\n\n In quoting this sentence I do not mean to endorse the military nihilism it represents. Napoleon, especially in his later years, was given to statements of this sort, and they are not uncommon in the literature on war. One writer claims that they illustrate a quality of leadership that he calls \"robustness.\" Napoleon's exclamation, \"I do not care a fig for the lives of a million men,\" is, he says, an extreme example of robustness. One could think of better names. (Alfred H. Burne, The Art of War on Land, London, 1944, p. 8.)\n\nNoncombatant Immunity and Military Necessity\n\nThe Status of Individuals\n\nThe first principle of the war convention is that, once war has begun, soldiers are subject to attack at any time (unless they are wounded or captured). And the first criticism of the convention is that this principle is unfair; it is an example of class legislation. It does not take into account that few soldiers are wholeheartedly committed to the business of fighting. Most of them do not identify themselves as warriors; at least, that is not their only or their chief identity; nor is fighting their chosen occupation. Nor, again, do they spend most of their time fighting; they neglect war whenever they can. I want to turn now to a recurrent incident in military history in which soldiers, simply by not fighting, appear to regain their right to life. In fact, they do not regain it, but the appearance will help us understand the grounds on which the right is held, and the facts of the case will clarify the meaning of its forfeiture.\n\nNaked Soldiers\n\nThe same tale appears again and again in war memoirs and in letters from the front. It has this general form: a soldier on patrol or on sniper duty catches an enemy soldier unaware, holds him in his gunsight, easy to kill, and then must decide whether to shoot him or let the opportunity pass. There is at such moments a great reluctance to shoot\u2014not always for moral reasons, but for reasons that are relevant nonetheless to the moral argument I want to make. No doubt, a deep psychological uneasiness about killing plays a part in these cases. This uneasiness, in fact, has been offered as a general explanation of the reluctance of soldiers to fight at all. In the course of a study of combat behavior in World War II, S. L. A. Marshall discovered that the great majority of men on the front line never fired their guns. He thought this the result above all of their civilian upbringing, of the powerful inhibitions acquired in its course against deliberately injuring another human being. But in the cases I shall list, this inhibition does not seem a critical factor. None of the five soldiers who wrote the accounts was a \"non-firer,\" nor, so far as I can tell, were the other men who figure importantly in their stories. Moreover, they give reasons for not killing or for hesitating to kill, and this the soldiers interviewed by Marshall were rarely able to do.\n\n1) I have taken the first case from a letter written by the poet Wilfred Owen to his brother in England on May 14, 1917.\n\nWhen we were marching along a sunken road, we got the wind up once. We knew we must have passed the German outposts somewhere on our left rear. All at once, the cry rang down, \"Line the bank.\" There was a tremendous scurry of fixing bayonets, tugging of breech covers, and opening pouches, but when we peeped over, behold a solitary German, haring along toward us, with his head down and his arms stretched in front of him, as if he were going to take a high dive through the earth (which I have no doubt he would like to have done). Nobody offered to shoot him, he looked too funny. . . .\n\nPerhaps everyone was waiting for an order to shoot, but Owen's meaning is undoubtedly that no one wanted to shoot. A soldier who looks funny is not at that moment a military threat; he is not a fighting man but simply a man, and one does not kill men. In this case, indeed, it would have been superfluous to do so: the comical German was soon taken prisoner. But that is not always possible, as the remaining cases suggest, and the reluctance or refusal to kill has nothing to do with the existence of a military alternative. There is always a nonmilitary alternative.\n\n2) In his autobiography Good-bye to All That, Robert Graves recalls the only time that he \"refrained from shooting a German\" who was neither wounded nor a prisoner.\n\nWhile sniping from a knoll in the support line, where we had a concealed loop-hole, I saw a German, about seven hundred yards away, through my telescopic sights. He was taking a bath in the German third line. I disliked the idea of shooting a naked man, so I handed the rifle to the sergeant with me. \"Here, take this. You're a better shot than I am.\" He got him; but I had not stayed to watch.\n\nI hesitate to say that what is involved here is a moral feeling, certainly not a moral feeling that is conceived to extend across class lines. But even if we describe it as the disdain of an officer and a gentleman for conduct that appears to be unmanly or unheroic, Graves' \"dislike\" still depends upon a morally important recognition. A naked man, like a funny man, is not a soldier. And what if the obedient and presumably unfeeling sergeant had not been with him?\n\n3) During the Spanish Civil War, George Orwell had a similar experience as a sniper working from a forward position in the republican lines. It would probably never have occurred to Orwell to hand his gun down the hierarchy of ranks; in any case, his was an anarchist battalion, and there was no hierarchy.\n\nAt this moment a man, presumably carrying a message to an officer, jumped out of the trench and ran along the top of the parapet in full view. He was half-dressed and was holding up his trousers with both hands as he ran. I refrained from shooting at him. It is true that I am a poor shot and unlikely to hit a running man at a hundred yards. . . . Still, I did not shoot partly because of that detail about the trousers. I had come here to shoot at \"Fascists;\" but a man who is holding up his trousers isn't a \"Fascist,\" he is visibly a fellow-creature, similar to yourself, and you don't feel like shooting at him.\n\nOrwell says, \"you don't feel like\" rather than \"you should not,\" and the difference between these two is important. But the fundamental recognition is the same as in the other cases and more fully articulated. Moreover, Orwell tells us that this \"is the kind of thing that happens all the time in wars,\" though with what evidence he says that, and whether he means that one doesn't feel like shooting or that one doesn't shoot \"all the time,\" I don't know.\n\n4) Raleigh Trevelyan, a British soldier in World War II, has published a \"diary of Anzio\" in which he recounts the following episode.\n\nThere was a wonderfully vulgar sunrise. Everything was the color of pink geraniums, and birds were singing. We felt like Noah must have done when he saw his rainbow. Suddenly Viner pointed across the stretch of scrubby heath. An individual, dressed in German uniform, was wandering like a sleep-walker across our line of fire. It was clear that for the moment he had forgotten war and\u2014as we had been doing\u2014was reveling in the promise of warmth and spring. \"Shall I bump him off?\" asked Viner, without a note of expression in his voice. I had to decide quickly. \"No,\" I replied, \"just scare him away.\"\n\nHere, as in the Orwell passage, the crucial feature is the discovery of a man \"similar to yourself,\" doing \"as we had been doing.\" Of course, two soldiers shooting at one another are quite precisely similar; one is doing what the other is doing, and both are engaged in what can be called a peculiarly human activity. But the sense of being a \"fellow-creature\" depends for obvious reasons upon a different sort of identity, one that is entirely dissociated from anything threatening. The fellowship of spring (reveling in the sun) is a good example, though even that is not untouched by the pressures of \"military necessity.\"\n\nOnly Sergeant Chesteron didn't laugh. He said that we should have killed the fellow, since his friends would now be told precisely where our trenches were.\n\nSergeants seem to bear much of the burden of war.\n\n5) The most reflective of the accounts I have found is by an Italian soldier who fought the Austrians in World War I: Emilio Lussu, later a socialist leader and anti-fascist exile. Lussu, then a lieutenant, together with a corporal, had moved during the night into a position overlooking the Austrian trenches. He watched the Austrians having morning coffee and felt a kind of amazement, as if he had not expected to find anything human in the enemy lines.\n\nThose strongly defended trenches, which we had attacked so many times without success had ended by seeming to us inanimate, like desolate buildings uninhabited by men, the refuge only of mysterious and terrible beings of whom we knew nothing. Now they were showing themselves to us as they really were, men and soldiers like us, in uniform like us, moving about, talking, and drinking coffee, just as our own comrades behind us were doing at that moment.\n\nA young officer appears and Lussu takes aim at him; then the Austrian lights a cigarette and Lussu pauses. \"This cigarette formed an invisible link between us. No sooner did I see its smoke than I wanted a cigarette myself. . . .\" Behind perfect cover, he has time to think about his decision. He felt the war justified, \"a hard necessity.\" He recognized that he had obligations to the men under his command. \"I knew it was my duty to fire.\" And yet he did not. He hesitated, he writes, because the Austrian officer was so entirely oblivious to the danger that threatened him.\n\nI reasoned like this: To lead a hundred, even a thousand, men against another hundred, or thousand, was one thing; but to detach one man from the rest and say to him, as it were: \"Don't move, I'm going to shoot you. I'm going to kill you\"\u2014that was different. . . . To fight is one thing, but to kill a man is another. And to kill him like that is to murder him.\n\nLussu, like Graves, turned to his corporal but (perhaps because he was a socialist) with a question, not an order. \"Look here\u2014I'm not going to fire on a man alone, like that. Will you?\" . . . \"No, I won't either.\" Here the line has been clearly drawn between the member of an army who makes war together with his comrades and the individual who stands alone. Lussu objected to stalking a human prey. What else, however, does a sniper do?\n\nIt is not against the rules of war as we currently understand them to kill soldiers who look funny, who are taking a bath, holding up their pants, reveling in the sun, smoking a cigarette. The refusal of these five men, nevertheless, seems to go to the heart of the war convention. For what does it mean to say that someone has a right to life? To say that is to recognize a fellow creature, who is not threatening me, whose activities have the savor of peace and camaraderie, whose person is as valuable as my own. An enemy has to be described differently, and though the stereotypes through which he is seen are often grotesque, they have a certain truth. He alienates himself from me when he tries to kill me, and from our common humanity. But the alienation is temporary, the humanity imminent. It is restored, as it were, by the prosaic acts that break down the stereotypes in each of the five stories. Because he is funny, naked, and so on, my enemy is changed, as Lussu says, into a man. \"A man!\"\n\nThe case might be different if we imagine this man to be a wholehearted soldier. In his bath, smoking his morning cigarette, he is thinking only of the coming battle and of how many of his enemies he will kill. He is engaged in war-making just as I am engaged in writing this book; he thinks about it all the time or at the oddest moments. But this is an unlikely picture of an ordinary soldier. War is not in fact his enterprise, but rather surviving this battle, avoiding the next. Mostly, he hides, is frightened, doesn't fire, prays for a minor wound, a voyage home, a long rest. And when we see him at rest, we assume that he is thinking of home and peace, as we would be. If that is so, how can it be justified to kill him? Yet it is justified, as most of the soldiers in the five stories understand. Their refusals seem, even to them, to fly in the face of military duty. Rooted in a moral recognition, they are nevertheless more passionate than principled decisions. They are acts of kindness, and insofar as they entail any danger at all or lower minutely the odds for victory later, they may be likened to superogatory acts. Not that they involve doing more than is morally required; they involve doing less than is permitted.\n\nThe standards of permissibility rest on the rights of individuals, but they are not precisely defined by those rights. For definition is a complex process, historical as well as theoretical in character, and conditioned in a significant way by the pressure of military necessity. It is time now to try to see what that pressure can and cannot do, and the \"naked soldier\" cases provide a useful instance. In the nineteenth century, an effort was made to protect one type of \"naked soldier\": the man on guard duty outside his post or at the edge of his lines. The reasons given for singling out this lone figure are similar to those expressed in the five stories. \"No other term than murder,\" wrote an English student of war, \"expresses the killing of a lone sentry by a pot shot at long range. It [is] like shooting a partridge sitting.\" The same idea is obviously at work in the code of military conduct that Francis Lieber drafted for the Union Army in the American Civil War: \"Outposts, sentinels, pickets are not to be fired upon, except to drive them in. . . .\" Now, a war is easily imaginable in which this idea was extended, so that only soldiers actually fighting, hundreds against hundreds, thousands against thousands, as Lussu says, could be attacked. Such a war would be constituted as a series of set battles, formally or informally announced in advance, and broken off in some clear fashion. The pursuit of a defeated army could be allowed, so neither side need be denied the possibility of a decisive victory. But perpetual harassment, sniping, ambush, surprise attack\u2014all these would be ruled out. Wars have indeed been fought in this way, but the arrangements have never been stable, because they give a systematic advantage to the army that is larger and better equipped. It is the weaker side that persistently refuses to fix any limits on the vulnerability of enemy soldiers (the extreme form of this refusal is guerrilla war), pleading military necessity. What does this mean?\n\nThe Nature of Necessity (1)\n\nThe plea takes a standard form. This or that course of action, it is said, \"is necessary to compel the submission of the enemy with the least possible expenditure of time, life, and money.\" That is the core of what the Germans call kriegsraison, reason of war. The doctrine justifies not only whatever is necessary to win the war, but also whatever is necessary to reduce the risks of losing, or simply to reduce losses or the likelihood of losses in the course of the war. In fact, it is not about necessity at all; it is a way of speaking in code, or a hyperbolical way of speaking, about probability and risk. Even if one grants the right of states and armies and individual soldiers to reduce their risks, a particular course of action would be necessary to that end only if no other course improved the odds of battle at all. But there will always be a range of tactical and strategic options that conceivably could improve the odds. There will be choices to make, and these are moral as well as military choices. Some of them are permitted and some ruled out by the war convention. If the convention did not discriminate in this way, it would have little impact upon the actual fighting of wars and battles; it would simply be a code of expediency\u2014which is what Sidgwick's twofold rule is likely to come to, under the pressure of actual warfare.\n\n\"Reason of war\" can only justify the killing of people we already have reason to think are liable to be killed. What is involved here is not so much a calculation of probability and risk as a reflection on the status of the men and women whose lives are at stake. The case of the \"naked soldier\" is resolved in this way: soldiers as a class are set apart from the world of peaceful activity; they are trained to fight, provided with weapons, required to fight on command. No doubt, they do not always fight; nor is war their personal enterprise. But it is the enterprise of their class, and this fact radically distinguishes the individual soldier from the civilians he leaves behind.a If he is warned that he is always in danger, it is not so great a disruption of his life as it would be in the case of the civilian. Indeed, to warn the civilian is in effect to force him to fight, but the soldier has already been forced to fight. That is, he has joined the army because he thinks his country must be defended, or he has been conscripted. It is important to stress, however, that he has not been forced to fight by a direct attack upon his person; that would repeat the crime of aggression at the level of the individual. He can be personally attacked only because he already is a fighter. He has been made into a dangerous man, and though his options may have been few, it is nevertheless accurate to say that he has allowed himself to be made into a dangerous man. For that reason, he finds himself endangered. The actual risks he lives with may be reduced or heightened: here notions of military necessity, and also of kindness and magnanimity, have free play. But the risks can be raised to their highest pitch without violating his rights.\n\nIt is harder to understand the extension of combatant status beyond the class of soldiers, though in modern war this has been common enough. The development of military technology, it might be said, has dictated it, for war today is as much an economic as a military activity. Vast numbers of workers must be mobilized before an army can even appear in the field; and once they are engaged, soldiers are radically dependent on a continuing stream of equipment, fuel, ammunition, food, and so on. It is a great temptation, then, to attack the enemy army behind its own lines, especially if the battle itself is not going well. But to attack behind the lines is to make war against people who are at least nominally civilians. How can this be justified? Here again, the judgments we make depend upon our understanding of the men and women involved. We try to draw a line between those who have lost their rights because of their warlike activities and those who have not. On the one side are a class of people, loosely called \"munitions workers,\" who make weapons for the army or whose work directly contributes to the business of war. On the other side are all those people who, in the words of the British philosopher G. E. M. Anscombe, \"are not fighting and are not engaged in supplying those who are with the means of fighting.\"\n\nThe relevant distinction is not between those who work for the war effort and those who do not, but between those who make what soldiers need to fight and those who make what they need to live, like all the rest of us. When it is militarily necessary, workers in a tank factory can be attacked and killed, but not workers in a food processing plant. The former are assimilated to the class of soldiers\u2014partially assimilated, I should say, because these are not armed men, ready to fight, and so they can be attacked only in their factory (not in their homes), when they are actually engaged in activities threatening and harmful to their enemies. The latter, even if they process nothing but army rations, are not similarly engaged. They are like workers manufacturing medical supplies, or clothing, or anything else that would be needed, in one form or another, in peacetime as well as war. An army, to be sure, has an enormous belly, and it must be fed if it is to fight. But it is not its belly but its arms that make it an army. Those men and women who supply its belly are doing nothing peculiarly warlike. Hence their immunity from attack: they are assimilated to the rest of the civilian population. We call them innocent people, a term of art which means that they have done nothing, and are doing nothing, that entails the loss of their rights.\n\nThis is a plausible line, I think, though it may be too finely drawn. What is more important is that it is drawn under pressure. We begin with the distinction between soldiers engaged in combat and soldiers at rest; then we shift to the distinction between soldiers as a class and civilians; and then we concede this or that group of civilians as the processes of economic mobilization establish its direct contribution to the business of fighting. Once the contribution has been plainly established, only \"military necessity\" can determine whether the civilians involved are attacked or not. They ought not to be attacked if their activities can be stopped, or their products seized or destroyed, in some other way and without significant risk. The laws of war have regularly recognized this obligation. Under the naval code, for example, merchant seamen on ships carrying military supplies were once regarded as civilians who had, despite the work they were doing, a right not to be attacked, for it was possible (and it sometimes still is) to seize their ships without shooting at them. But whenever seizure without shooting ceases to be possible, the obligation ceases also and the right lapses. It is not a retained but a war right, and rests only on the agreement of states and on the doctrine of military necessity. The history of submarine warfare nicely illustrates this process, through which groups of civilians are, as it were, incorporated into hell. It will also enable me to suggest the point at which it becomes morally necessary to resist the incorporation.\n\nSubmarine Warfare: The Laconia Affair\n\nNaval warfare has traditionally been the most gentlemanly form of fighting, possibly because so many gentlemen went into the navy, but also and more importantly because of the nature of the sea as a battlefield. The only comparable land environment is the desert; these two have in common the absence or relative absence of civilian inhabitants. Hence battle is especially pure, a combat between combatants, with no one else involved\u2014just what we intuitively want war to be. The purity is marred, however, by the fact that the sea is extensively used for transport. Warships encounter merchant ships. The rules governing this encounter are, or were, fairly elaborate. Worked out before the invention of the submarine, they bear the marks of their technological as well as their moral assumptions. A merchant ship carrying military supplies could lawfully be stopped on the high seas, boarded, seized, and brought into port by a prize crew. If the merchant seamen resisted this process at any stage, whatever force was necessary to overcome the resistance was also lawful. If they submitted peacefully, no force could be used against them. If it was impossible to bring the ship into port, it could be sunk, \"subject to the absolute duty of providing for the safety of the crew, passengers, and papers.\" Most often, this was done by taking all three on board the warship. The crew and passengers were then to be regarded not as prisoners of war, for their encounter with the warship was not a battle, but as civilian internees.\n\nNow, in World War I, submarine commanders (and the state officials who commanded them) openly refused to act in accordance with this \"absolute duty,\" pleading military necessity. They could not surface before firing their torpedoes, for their ships were lightly armed above decks and highly vulnerable to ramming; they could not provide prize crews from their own small number, unless they, too, were to return to port; nor could they take merchant seamen on board, for there was no room. Hence their policy was to \"sink on sight,\" though they did accept some responsibility to assist survivors after the ship was down. \"Sink on sight\" was especially the policy of the German government. The only alternative, its defenders have argued, was not to use submarines at all, or to use them ineffectively, which would have conceded control of the sea to the British navy. After the war was over, perhaps because the Germans lost it, the traditional rules were reaffirmed. The London Naval Protocol of 1936, ratified by all the major participants in the last and the next great war (by the Germans in 1939), explicitly provided that \"in their action with regard to merchant ships, submarines must conform to the rules of international law to which surface ships are subject.\" This is still the \"binding rule,\" according to respected authorities on naval law, though anyone who defends the rule must do so \"notwithstanding the experience of the Second World War.\"\n\nWe can best gain access to this experience by turning immediately to the famous \"Laconia order\" issued by Admiral Doenitz of the German U-boat command in 1942. Doenitz required not only that submarines strike without warning, but also that they do nothing whatsoever to help the crew members of a sunken ship: \"All attempts to rescue the crews of sunken ships should cease, including picking up men from the sea, righting capsized lifeboats, and supplying food and water.\" This order provoked great indignation at the time, and after the war its promulgation was among the crimes with which Doenitz was charged at Nuremberg. But the judges refused to convict on this charge. I want to look closely at the reasons for their decision. Since their language is obscure, however, I shall also ask what their reasons might have been and what reasons we might have for requiring or not requiring rescue at sea.\n\nThe issue clearly was rescue and nothing else; despite the \"binding rule\" of international law, the policy of \"sink on sight\" was not challenged by the court. The judges apparently decided that the distinction between merchant ships and warships no longer made much sense.\n\nShortly after the outbreak of the war, the British Admiralty . . . armed its merchant vessels, in many cases convoyed them with armed escort, gave orders to send position reports upon sighting submarines, thus integrating merchant vessels into the warning system of naval intelligence. On October 1, 1939, the Admiralty announced [that] British merchant ships had been ordered to ram U-boats if possible.\n\nAt this point, the court seemed to reason, merchant seamen had been conscripted for military service; hence it was permissible to attack them by surprise exactly as if they were soldiers. But this argument, by itself, is not a very good one. For if the conscription of merchant seamen was a response to illegitimate submarine attacks (or even to the strong probability of such attacks), it cannot be invoked to justify those same attacks. It must be the case that the \"sink on sight\" policy was justified in the first place. The invention of the submarine had made it \"necessary.\" The old rules were morally if not legally suspended because supply by sea\u2014a military enterprise whose participants had always been liable to attack\u2014had ceased now to be subject to nonviolent interdiction.\n\nThe \"Laconia order\" reached much further than this, however, for it suggested that seamen helpless in the sea, unlike wounded soldiers on land, need not be helped once the battle was over. Doenitz's argument was that the battle, in fact, was never over until the submarine was safe in its home port. The sinking of a merchant vessel was only the first blow of a long and tense struggle. Radar and the airplane had turned the wide seas into a single battlefield, and unless the submarine immediately began evasive maneuvers, it was or might be in great trouble. Seamen had once been better off than soldiers, a privileged class of near-combatants treated as if they were civilians; now, suddenly, they were worse off.\n\nHere again is the argument from military necessity, and again we can see that it is above all an argument about risk. The lives of the submarine crew would be endangered, Doenitz claimed, and the probability of detection and attack increased by this or that extent, if they attempted to rescue their victims. Now, this is clearly not always the case: in his account of the destruction of an Allied convoy in the Arctic Sea, David Irving describes a number of incidents in which German submarines surfaced and offered assistance to merchant seamen in lifeboats without increasing their own risks.\n\nLieutenant-Commander Teichert's U-456 . . . had fired the striking torpedoes. Teichert took his submarine alongside the lifeboats and ordered the Master, Captain Strand, to come aboard; he was taken prisoner. The seamen were asked whether they had sufficient water and they were handed tinned meat and bread by the submarine officers. They were told that they would be picked up by destroyers a few days later.\n\nThis occurred only a few months before Doenitz's order prohibited such assistance, and under conditions which made it perfectly safe. Convoy PQ 17 had dispersed, abandoned by its escorts; it was no longer in any sense a fighting force; the Germans controlled the air as well as the sea. The battle was clearly over, and military necessity could hardly have justified a refusal to help. I should think that if such a refusal, under similar circumstances, could be attributed to the \"Laconia order,\" Doenitz would indeed be guilty of a war crime. But nothing like this was demonstrated at Nuremberg.\n\nNor, however, did the court openly adopt the argument from military necessity: that under different circumstances the refusal to help was justified by the risks it entailed. Instead, the judges reaffirmed the binding rule. \"If the Commander cannot rescue,\" they argued, \"then . . . he cannot sink a merchant vessel. . . .\" But they did not enforce the rule and punish Doenitz. Admiral Nimitz of the U.S. Navy, called to testify by Doenitz's attorney, had told them that \"U.S. submarines [generally] did not rescue enemy survivors if by so doing the vessels were exposed to unnecessary or additional risk.\" British policy had been similar. In view of this, the judges declared that \"the sentence of Doenitz is not assessed on the ground of his breaches of the international law of submarine warfare.\" They did not accept the argument of the defense attorneys that the law had effectively been rewritten by informal collusion among the belligerents. But they apparently felt that this collusion did make the law unenforceable (or at least unenforceable against only one of the parties to its violation)\u2014a proper judicial decision, but one that leaves open the moral question.\n\nIn fact, Doenitz and his Allied counterparts had reasons for the policy they adopted, and these reasons fit roughly into the framework of the war convention. Wounded or helpless combatants are no longer subject to attack; in that sense they have regained their right to life. But they are not entitled to assistance so long as the battle continues and the victory of their enemies is uncertain. What is decisive here is not military necessity but the assimilation of merchant seamen to the class of combatants. Soldiers need not risk their lives for the sake of their enemies, for both they and their enemies have exposed themselves to the coerciveness of war. There are some people, however, who are safe against that coerciveness, or who ought to be safeguarded against it, and these people also have a part in the Laconia affair.\n\nThe Laconia was a liner carrying 268 British servicemen and their families, returning home from pre-war stations in the Middle East, and 1,800 Italian prisoners of war. It was torpedoed and sunk off the west coast of Africa by a U-boat whose commander did not know who its passengers were (liners were used extensively by the Allies as troopships). When Doenitz learned of the sinking, and of the identity of the people in the water, he ordered a massive rescue effort involving, initially, a number of other submarines. Italian warships were also asked to hurry to the scene, and the U-boat commander responsible for the sinking radioed in English a general call for help. But the submarines were instead attacked by several Allied planes whose pilots presumably did not know what was going on in the seas below or did not believe what they were told. The confusion is typical enough in time of war: ignorance on all sides, compounded by mutual fear and suspicion.\n\nIn fact, the planes did little damage, but Doenitz's response was harsh. He directed the German commanders to confine their rescue efforts to the Italian prisoners; the British soldiers and their families were to be set adrift. It was this spectacle of women and children abandoned at sea, and the subsequent order that seemed to require its repetition, that was widely thought to be outrageous\u2014and rightly so, it seems to me, even though \"unrestricted\" submarine warfare was by then commonly accepted. For we draw a circle of rights around civilians, and soldiers are supposed to accept (some) risks in order to save civilian lives. It is not a question of going out of their way or of being, or not being, good Samaritans. They are the ones who endanger civilian lives in the first place, and even if they do this in the course of legitimate military operations, they must still make some positive effort to restrict the range of the damage they do. This indeed was Doenitz's own position before the Allied attack, a position he maintained despite criticism from other members of the German High Command: \"I cannot put these people into the water. I shall carry on [the rescue effort].\" It is not kindness that is involved here, but duty, and it is in terms of that duty that we judge the \"Laconia order.\" A rescue effort undertaken for the sake of noncombatants can be broken off temporarily because of an attack, but it cannot be called off in advance of any attack merely because an attack may occur (or recur). For one attack at least has already occurred and put innocent people in danger of death. Now they must be helped.\n\nDouble Effect\n\nThe second principle of the war convention is that noncombatants cannot be attacked at any time. They can never be the objects or the targets of military activity. But as the Laconia affair suggests, noncombatants are often endangered not because anyone sets out to attack them, but only because of their proximity to a battle that is being fought against someone else. I have tried to argue that what is then required is not that the battle be stopped, but that some degree of care be taken not to harm civilians\u2014which means, very simply, that we recognize their rights as best we can within the context of war. But what degree of care should be taken? And at what cost to the individual soldiers who are involved? The laws of war say nothing about such matters; they leave the cruelest decisions to be made by the men on the spot with reference only to their ordinary moral notions or the military traditions of the army in which they serve. Occasionally one of these soldiers will write about his own decisions, and that can be like a light going on in a dark place. Here is an incident from Frank Richards' memoir of the First World War, one of the few accounts by a man from the ranks.\n\nWhen bombing dug-outs or cellars, it was always wise to throw the bombs into them first and have a look around them after. But we had to be very careful in this village as there were civilians in some of the cellars. We shouted down to them to make sure. Another man and I shouted down one cellar twice and receiving no reply were just about to pull the pins out of our bombs when we heard a woman's voice and a young lady came up the cellar steps. . . . She and the members of her family . . . had not left [the cellar] for some days. They guessed an attack was being made and when we first shouted down had been too frightened to answer. If the young lady had not cried out when she did, we would have innocently murdered them all.\n\nInnocently murdered, because they had shouted first; but if they had not shouted, and then killed the French family, it would have been, Richards believed, murder simply. And yet he was accepting a certain risk in shouting, for had there been German soldiers in the cellar, they might have scrambled out, firing as they came. It would have been more prudent to throw the bombs without warning, which means that military necessity would have justified him in doing so. Indeed, he would have been justified on other grounds, too, as we shall see. And yet he shouted.\n\nThe moral doctrine most often invoked in such cases is the principle of double effect. First worked out by Catholic casuists in the Middle Ages, double effect is a complex notion, but it is at the same time closely related to our ordinary ways of thinking about moral life. I have often found it being used in military and political debates. Officers will tend to speak in its terms, knowingly or unknowingly, whenever the activity they are planning is likely to injure noncombatants. Catholic writers themselves frequently use military examples; it is one of their purposes to suggest what we ought to think when \"a soldier in firing at the enemy foresees that he will shoot some civilians who are nearby.\" Such foresight is common enough in war; soldiers could probably not fight at all, except in the desert and at sea, without endangering nearby civilians. And yet it is not proximity but only some contribution to the fighting that makes a civilian liable to attack. Double effect is a way of reconciling the absolute prohibition against attacking noncombatants with the legitimate conduct of military activity. I shall want to argue, following the example of Frank Richards, that the reconciliation comes too easily, but first we must see exactly how it is worked out.\n\nThe argument goes this way: it is permitted to perform an act likely to have evil consequences (the killing of noncombatants) provided the following four conditions hold.\n\n  1. The act is good in itself or at least indifferent, which means, for our purposes, that it is a legitimate act of war.\n  2. The direct effect is morally acceptable\u2014the destruction of military supplies, for example, or the killing of enemy soldiers.\n  3. The intention of the actor is good, that is, he aims only at the acceptable effect; the evil is not one of his ends, nor is it a means to his ends.\n  4. The good effect is sufficiently good to compensate for allowing the evil effect; it must be justifiable under Sidgwick's proportionality rule.\n\nThe burden of the argument is carried by the third clause. The \"good\" and evil effects that come together, the killing of soldiers and nearby civilians, are to be defended only insofar as they are the product of a single intention, directed at the first and not the second. The argument suggests the great importance of taking aim in wartime, and it correctly restricts the targets at which one can aim. But we have to worry, I think, about all those unintended but foreseeable deaths, for their number can be large; and subject only to the proportionality rule\u2014a weak constraint\u2014double effect provides a blanket justification. The principle for that reason invites an angry or a cynical response: what different does it make whether civilian deaths are a direct or an indirect effect of my actions? It can hardly matter to the dead civilians, and if I know in advance that I am likely to kill so many innocent people and go ahead anyway, how can I be blameless?\n\nWe can ask the question in a more concrete way. Would Frank Richards have been blameless if he had thrown his bombs without warning? The principle of double effect would have permitted him to do so. He was engaged in a legitimate military activity, for many cellars were in fact being used by enemy soldiers. The effects of making \"bomb without warning\" his general policy would have been to reduce the risks of his being killed or disabled and to speed up the capture of the village, and these are \"good\" effects. Moreover, they were clearly the only ones he intended; civilian deaths would have served no purpose of his own. And finally, over an extended period of time, the proportions would probably have worked out favorably or at least not unfavorably; the mischief done would, let us assume, be balanced by the contribution to victory. And yet Richards was surely doing the right thing when he shouted his warning. He was acting as a moral man ought to act; he is not an example of fighting heroically, above and beyond the call of duty, but simply of fighting well. It is what we expect of soldiers. Before trying to state that expectation more precisely, however, I want to see how it works in more complex combat situations.\n\nBombardment in Korea\n\nI am going to follow here a British journalist's account of the way the American army waged war in Korea. Whether it is an entirely just account I do not know, but I am more interested in the moral issues it raises than in its historical accuracy. This, then, was a \"typical\" encounter on the road to Pyongyang. A battalion of American troops advanced slowly, without opposition, under the shadow of low hills. \"We were well into the valley now, half-way down the straight . . . strung out along the open road, when it came, the harsh stutter of automatic fire sputtering the dust around us.\" The troops stopped and dove for cover. Three tanks moved up, \"pounding their shells into the . . . hillside and shattering the air with their machine guns. It was impossible in this remarkable inferno of sound to detect the enemy, or to assess his fire.\" Within fifteen minutes, several fighter planes arrived, \"diving down upon the hillside with their rockets.\" This is the new technique of warfare, writes the British journalist, \"born of immense productive and material might\": \"the cautious advance, the enemy small arms fire, the halt, the close support air strike, artillery, the cautious advance, and so on.\" It is designed to save the lives of soldiers, and it may or may not have that effect. \"It is certain that it kills civilian men, women, and children, indiscriminately and in great numbers, and destroys all that they have.\"\n\nNow there is another way to fight, though it is only open to soldiers who have had a \"soldierly\" training and who are not \"roadbound\" in their habits. A patrol can be sent forward to outflank the enemy position. In the end, it often comes to that anyway, as it did in this case, for the tanks and planes failed to hit the North Korean machine gunners. \"At last, after more than an hour . . . a platoon from Baker Company began working their way through the scrub just under the ridge of the hill.\" But the first reliance was always on bombardment. \"Every enemy shot released a deluge of destruction.\" And the bombardment had, or sometimes had, its characteristic double effect: enemy soldiers were killed, and so were any civilians who happened to be nearby. It was not the intention of the officers who called in the artillery and planes to kill civilians; they were acting out of a concern for their own men. And that is a legitimate concern. No one would want to be commanded in wartime by an officer who did not value the lives of his soldiers. But he must also value civilian lives, and so must his soldiers. He cannot save them, because they cannot save themselves, by killing innocent people. It is not just that they can't kill a lot of innocent people. Even if the proportions work out favorably, in particular cases or over a period of time, we would still want to say, I think, that the patrol must be sent out, the risk accepted, before the big guns are brought to bear. The soldiers sent on patrol can plausibly argue that they never chose to make war in Korea; they are soldiers nevertheless; there are obligations that go with their war rights, and the first of these is the obligation to attend to the rights of civilians\u2014more precisely, of those civilians whose lives they themselves endanger.\n\nThe principle of double effect, then, stands in need of correction. Double effect is defensible, I want to argue, only when the two outcomes are the product of a double intention: first, that the \"good\" be achieved; second, that the foreseeable evil be reduced as far as possible. So the third of the conditions listed above can be restated:\n\n  1. The intention of the actor is good, that is, he aims narrowly at the acceptable effect; the evil effect is not one of his ends, nor is it a means to his ends, and, aware of the evil involved, he seeks to minimize it, accepting costs to himself.\n\nSimply not to intend the death of civilians is too easy; most often, under battle conditions, the intentions of soldiers are focused narrowly on the enemy. What we look for in such cases is some sign of a positive commitment to save civilian lives. Not merely to apply the proportionality rule and kill no more civilians than is militarily necessary\u2014that rule applies to soldiers as well; no one can be killed for trivial purposes. Civilians have a right to something more. And if saving civilian lives means risking soldier's lives, the risk must be accepted. But there is a limit to the risks that we require. These are, after all, unintended deaths and legitimate military operations, and the absolute rule against attacking civilians does not apply. War necessarily places civilians in danger; that is another aspect of its hellishness. We can only ask soldiers to minimize the dangers they impose.\n\nExactly how far they must go in doing that is hard to say, and for that reason it may seem odd to claim that civilians have rights in such matters. What can this mean? Do civilians have a right not only not to be attacked but also not to be put at risk to such and such a degree, so that imposing a one-in-ten chance of death on them is justified, while imposing a three-in-ten chance is unjustified? In fact, the degree of risk that is permissible is going to vary with the nature of the target, the urgency of the moment, the available technology, and so on. It is best, I think, to say simply that civilians have a right that \"due care\" be taken.b The case is the same in domestic society: when the gas company works on the lines that run under my street, I have a right that its workmen observe very strict safety standards. But if the work is urgently required by the imminent danger of an explosion on a neighboring street, the standards may be relaxed and my rights not violated. Now, military necessity works exactly like civil emergency, except that in war the standards with which we are familiar in domestic society are always relaxed. That is not to say, however, that there are no standards at all, and no rights involved. Whenever there is likely to be a second effect, a second intention is morally required. We can move some way toward defining the limits of that second intention if we consider two more wartime examples.\n\nThe Bombing of Occupied France and the Vemork Raid\n\nDuring World War II, the Free French air force carried out bombing raids against military targets in occupied France. Inevitably, their bombs killed Frenchmen working (under coercion) for the German war effort; inevitably too, they killed Frenchmen who simply happened to live in the vicinity of the factories under attack. This posed a cruel dilemma for the pilots, which they resolved not by giving up the raids or asking someone else to carry them out, but by accepting greater risks for themselves. \"It was . . . this persistent question of bombing France itself,\" says Pierre Mendes-France, who served in the air force after his escape from a German prison, \"which led us to specialize more and more in precision bombing\u2014that is, flying at a very low altitude. It was more risky, but it also permitted greater precision. . . .\" The same factories, of course, could have been (perhaps should have been) attacked by squads of partisans or commandos carrying explosives; their aim would have been perfect, not merely more precise, and no civilians except those working in the factories would have been endangered. But such raids would have been extremely dangerous and the chances of success, and especially of reiterated success, very slim. Risks of that sort were more than the French expected, even of their own soldiers. The limits of risk are fixed, then, roughly at that point where any further risk-taking would almost certainly doom the military venture or make it so costly that it could not be repeated.\n\nThere is obviously leeway for military judgment here: strategists and planners will for reasons of their own weigh the importance of their target against the importance of their soldiers' lives. But even if the target is very important, and the number of innocent people threatened relatively small, they must risk soldiers before they kill civilians. Consider, for example, the one case I have found from the Second World War where a commando raid was tried instead of an air attack. In 1943, the heavy water plant at Vemork in occupied Norway was destroyed by Norwegian commandos operating on behalf of the British S.O.E. (Special Operations Executive). It was vitally important to stop the production of heavy water so as to delay the development of an atomic bomb by German scientists. British and Norwegian officials debated whether to make the attempt from the air or on the ground and chose the latter approach because it was less likely to injure civilians. But it was very dangerous for the commandos. The first attempt failed, and thirty-four men were killed in its course; the second attempt, by a smaller number of men, succeeded without casualties\u2014to the surprise of everyone involved, including the commandos. It was possible to accept such risks for a single operation that would not, it was thought, have to be repeated. For a \"battle\" that extended over time, consisting of many separate incidents, it would not have been possible.\n\nLater in the war, after production was resumed at Vemork and security considerably tightened, the plant was bombed from the air by American planes. The bombing was successful, but it resulted in the deaths of twenty-two Norwegian civilians. At this point, double effect seems to work, justifying the air attack. Indeed, in its unrevised form it would have worked sooner. The importance of the military aim and the actual casualty figures (foreseeable in advance, let us assume) would have justified a bombing raid in the first place. But the special value we attach to civilian lives precluded it.\n\nNow, the same value attaches to the lives of German as to those of French or Norwegian civilians. There are, of course, additional moral as well as emotional reasons for paying that respect and accepting its costs in the case of one's own people or one's allies (and it is no accident that my two examples involve attacks on occupied territory). Soldiers have direct obligations to the civilians they leave behind, which have to do with the very purpose of soldiering and with their own political allegiance. But the structure of rights stands independently of political allegiance; it establishes obligations that are owed, so to speak, to humanity itself and to particular human beings and not merely to one's fellow citizens. The rights of German civilians\u2014who did no fighting and were not engaged in supplying the armed forces with the means of fighting\u2014were no different from those of their French counterparts, just as the war rights of German soldiers were no different from those of French soldiers, whatever we think of their war.\n\nThe case of occupied France (or Norway) is, however, complex in another way. Even if the French pilots had reduced their risks and flown at high altitudes, we would not hold them solely responsible for the additional civilian deaths they caused. They would have shared that responsibility with the Germans\u2014in part because the Germans had attacked and conquered France, but also (and more importantly for our immediate purposes) because they had mobilized the French economy for their own strategic ends, forcing French workers to serve the German war machine, turning French factories into legitimate military targets, and putting the adjacent residential areas in danger. The question of direct and indirect effect is complicated by the question of coercion. When we judge the unintended killing of civilians, we need to know how those civilians came to be in a battle zone in the first place. This is, perhaps, only another way of asking who put them at risk and what positive efforts were made to save them. But it raises issues that I have not yet addressed and that are most dramatically visible when we turn to another, and a much older, kind of warfare.\n\n In his moving account of the French defeat in 1940, Marc Bloch has criticized this distinction: \"Confronted by the nation's peril and by the duties that it lays on every citizen, all adults are equal and only a curiously warped mind would claim for any of them the privilege of immunity. What, after all, is a 'civilian' in time of war? He is nothing more than a man whose weight of years, whose health, whose profession . . . prevents him from bearing arms effectively. . . . Why should [these factors] confer on him the right to escape from the common danger?\" (Strange Defeat, trans. Gerard Hopkins, New York, 1968, p. 130.) But the theoretical problem is not to describe how immunity is gained, but how it is lost. We are all immune to start with; our right not to be attacked is a feature of normal human relationships. That right is lost by those who bear arms \"effectively\" because they pose a danger to other people. It is retained by those who don't bear arms at all.\n\n Since judgments of \"due care\" involve calculations of relative value, urgency, and so on, it has to be said that utilitarian arguments and rights arguments (relative at least to indirect effects) are not wholly distinct. Nevertheless, the calculations required by the proportionality principle and those required by \"due care\" are not the same. Even after the highest possible standards of care have been accepted, the probable civilian losses may still be disproportionate to the value of the target; then the attack must be called off. Or, more often, military planners may decide that the losses entailed by the attack, even if it is carried out at minimal risk to the attackers, are not disproportionate to the value of the target: then \"due care\" is an additional requirement.\n\nWar Against Civilians: Sieges and Blockades\n\nSiege is the oldest form of total war. Its long history suggests that neither technological advance nor democratic revolution are the crucial factors pushing warfare beyond the combatant population. Civilians have been attacked along with soldiers, or in order to get at soldiers, as often in ancient as in modern times. Such attacks are likely whenever an army seeks what might be called civilian shelter and fights from behind the battlements or from within the buildings of a city, or whenever the inhabitants of a threatened city seek the most immediate form of military protection and agree to be garrisoned. Then, locked into the narrow circle of the walls, civilians and soldiers are exposed to the same risks. Proximity and scarcity make them equally vulnerable. Or perhaps not equally so: in this kind of war, once combat begins, noncombatants are more likely to be killed. The soldiers fight from protected positions, and the civilians, who don't fight at all, are quickly made over (in a phrase I have taken from the military literature) into \"useless mouths.\" Fed last, and only with the army's surplus, they die first. More civilians died in the siege of Leningrad than in the modernist infernos of Hamburg, Dresden, Tokyo, Hiroshima, and Nagasaki, taken together. They probably died more painfully, too, even if in old-fashioned ways. Diaries and memoirs of twentieth-\u00adcentury sieges are entirely familiar to anyone who has read, for example, Josephus' harrowing history of the Roman siege of Jerusalem. And the moral issues raised by Josephus are familiar to anyone who has thought about twentieth-\u00adcentury war.\n\nCoercion and Responsibility\n\nThe Siege of Jerusalem 72 A.D.\n\nCollective starvation is a bitter fate: parents and children, friends and lovers must watch one another die, and the dying is terribly drawn out, physically and morally destructive long before it is over. Though it sounds like the end of the world, the following passage from Josephus refers to a time relatively early in the Roman siege.\n\nThe restraint of liberty to pass in and out of the city took from the Jews all hope of safety, and the famine now increasing consumed whole households and families; and the houses were full of dead women and infants; and the streets filled with the dead bodies of old men. And the young men, swollen like dead men's shadows, walked in the market place and fell down dead where it happened. And now the multitude of dead bodies was so great that they that were alive could not bury them; nor cared they for burying them, being now uncertain what should betide themselves. And many endeavoring to bury others fell down themselves dead upon them. . . . And many being yet alive went unto their graves and there died. Yet for all this calamity was there no weeping nor lamentation, for famine overcame all affections. And they who were yet living, without tears beheld those who being dead were now at rest before them. There was no noise heard within the city. . . .\n\nThis is not a firsthand account; Josephus was outside the walls, with the Roman army. According to other writers, it is the women who last longest in sieges, the young men who soonest fall into that deadly lethargy that precedes actual death. But the account is accurate enough: that is what a siege is like. Moreover, that is what it is meant to be like. When a city is encircled and deprived of food, it is not the expectation of the attackers that the garrison will hold out until individual soldiers, like Josephus' old men, drop dead in the streets. The death of the ordinary inhabitants of the city is expected to force the hand of the civilian or military leadership. The goal is surrender; the means is not the defeat of the enemy army, but the fearful spectacle of the civilian dead.\n\nThe principle of double effect, however it is expounded, provides no justification here. These are intentional deaths. And yet siege warfare is not ruled out by the laws of war. \"The propriety of attempting to reduce [a city] by starvation is not questioned.\" If there is a general rule that civilian deaths must not be aimed at, the siege is a great exception\u2014and the sort of exception that seems, if it is morally warranted, to shatter the rule itself. We must consider why it has been made. How can it be thought right to lock civilians up in the death trap of an encircled city?\n\nThe obvious answer is simply that the capture of cities is often an important military objective\u2014in the age of the city-state, it was the ultimate objective\u2014and, frontal assault failing, the siege is the only remaining means to success. In fact, however, it is not even necessary that a frontal assault fail before a siege is thought justifiable. Sitting and waiting is far less costly to the besieging army than attacking, and such calculations are permitted (as we have seen) by the principle of military necessity. But this argument is not the most interesting defense of siege warfare and not the one, I think, with which commanders themselves have assuaged their consciences. Josephus suggests the alternative. Titus, he tells us, lamented the deaths of so many Jerusalemites, \"and, lifting up his hands to heaven . . . called God to witness, that it was not his doing.\" Whose doing was it?\n\nAfter Titus himself, there are only two candidates: the political or military leaders of the city, who have refused to surrender on terms and forced the inhabitants to fight; or the inhabitants themselves, who have acquiesced in that refusal and agreed, as it were, to run the risks of war. Titus implicitly, and Josephus explicitly, opts for the first of these possibilities. Jerusalem, they argue, has been seized by the fanatical Zealots, who have imposed the war upon the mass of moderate Jews, ready otherwise to surrender. There is perhaps a measure of truth in this view, but it is not a satisfactory argument. It makes Titus himself into an impersonal agent of destruction, set off by the obstinacy of others, without plans and purposes of his own. And it suggests that cities (and why not countries?) that do not surrender are justly exposed to total war. Neither of these is a plausible proposition. Even if we reject them both, however, the attribution of responsibility in siege warfare is a complex business. This complexity helps explain though I shall argue that it does not justify the peculiar status of sieges in the laws of war. It also leads us to see that there are moral questions that must be answered before the principle of double effect comes into play. How did those civilians come to be so near the battlefield, where they are now (intentionally or incidentally) killed? Are they there by choice? Or have they been forced into their encounter with war and death?\n\nA city can indeed be defended against the will of its citizens\u2014by an army, beaten in the field, that retreats within its walls; by an alien garrison, serving the strategic interests of a distant commander; by militant, politically powerful minorities of one or another sort. If they were competent casuists, the leaders of any of these groups might reason in the following way: \"We know that civilians will die as a result of our decision to fight here rather than somewhere else. But we will not do the killing, and the deaths will not in any way benefit us. They are not our purpose, nor a part of our purpose, nor a means to our purpose. By collecting and rationing food, we will do all we can to save civilian lives. Those who die are not our responsibility.\" Clearly, such leaders cannot be condemned under the principle of double effect. But they can be condemned nevertheless\u2014so long as the inhabitants of the city decline to be defended. There are many examples of this sort of thing in medieval history: burghers eager to surrender, aristocratic warriors committed (not to the burghers) to continue the fight. In such cases, the warriors surely bear some responsibility for burgher deaths. They are agents of coercion within the city, as the besieging army is without, and the civilians are trapped between the two. But such cases are rare today, as they were in classical times. Political integration and civic discipline make for cities whose inhabitants expect to be defended and are prepared, morally if not always materially, to endure the burdens of a siege. Consent clears the defenders, and only consent can do so.\n\nWhat of the attackers? I assume that they offer surrender on terms; that is simply the collective equivalent of quarter and should always be available. But surrender is refused. There are then two military options. First, the strongholds of the city can be bombarded and the walls stormed. No doubt, civilians will die, but for these deaths the attacking soldiers can rightly say that they are not to blame. Though they do the killing, these deaths are, in an important sense, not their \"doing.\" The attackers are cleared by the refusal of surrender, which is an acceptance of the risks of war (or, moral responsibility is shifted onto the defending army, which has made surrender impossible). But this argument applies only to those deaths that are in fact incidental to legitimate military operations. The refusal of surrender does not turn the civilians into direct objects of attack. They have not thereby joined the war, though some of them may subsequently be mobilized for warlike activities within the city. They are simply in their \"proper and permanent abode,\" and their status as citizens of a besieged city is no different from their status as citizens of a country at war. If they can be killed, who cannot be? But then it would appear that the second military option is ruled out: the city cannot be surrounded, cut off, its people systematically starved.\n\nThe lawyers have drawn the line differently, though they, too, acknowledge that questions of coercion and consent precede questions of direct and indirect effect. Consider the following case from Machiavelli's Art of War:\n\nAlexander the Great, anxious to conquer Leucadia, first made himself master of the neighboring towns and turned all the inhabitants into Leucadia; at last the town was so full of people that he immediately reduced it by famine.\n\nMachiavelli was enthusiastic about this strategy, but it never became accepted military practice. Moreover, it is not accepted even if the purpose of the forced evacuation is more benign than Alexander's: simply to clear the suburbs for military operations, say, or to drive away people whom the besieging army cannot afford to feed. Had Alexander acted from such motives, and then taken Leucadia by storm, the incidental death of any of the evacuees would still be his special responsibility, since he had forcibly exposed them to the risks of war.\n\nThe legal norm is the status quo. The commander of the besieging army is not conceived to be, and does not think himself to be, responsible for those people who have always lived in the city\u2014who are there, so to speak, naturally\u2014nor for those who are there voluntarily, who sought the protection of city walls, driven only by the general fear of war. He is in the clear with regard to these people, however horribly they die, however much to his purpose it is that they die horribly, because he did not force them into their death place. He did not push them through the gates of the city before he locked them in. This is, I suppose, an understandable way of drawing the line, but it does not seem to me the right way. The hard question is whether the line can be drawn differently without ruling out sieges altogether. In the long history of siege warfare, this question has a specific form: should civilians be allowed to leave the city, saving themselves from starvation and relieving pressure on the collective food supply, after it has been invested? More generally, isn't locking them into the besieged city morally the same as driving them in? And if it is, shouldn't they be let out, so that those that remain, to fight and starve, can really be said to have chosen to remain? During the siege of Jerusalem, Titus ordered that any Jews who fled the city were to be crucified. It is the one point in his narrative where Josephus feels the need to apologize for his new master. But I want to turn now to a modern example, for these questions were directly addressed by the Nuremberg courts after World War II.\n\nThe Right to Leave\n\nThe Siege of Leningrad\n\nWhen its last road and rail links to the east were cut by advancing German forces, on September 8, 1941, Leningrad held over three million people, of whom about 200,000 were soldiers. This was roughly the peacetime population of the city. About half a million people had been evacuated before the siege began, but the number had been made up by refugees from the Baltic states, the Karelian Isthmus, and Leningrad's western and southern suburbs. These people ought to have been moved on, and the evacuation of the city itself speeded up; the Soviet authorities were frighteningly inefficient. But evacuation is always a difficult political issue. To organize it early and on a large scale seems defeatist; it is a way of acknowledging that the army won't be able to hold a line in front of the city. Moreover, it requires a massive effort at a time, it is usually said, when resources and manpower should be concentrated on military defense. And even when the danger is imminent, it is likely to encounter civilian resistance. Politics makes for two sorts of resistance: from those who hope to welcome the enemy and profit from his victory, and from those who are unwilling to \"desert\" the patriotic struggle. Inevitably, the very authorities organizing the evacuation are also conducting a propaganda campaign that makes desertion seem dishonorable. But the greater resistance is nonpolitical in character, deeply rooted in feelings of place and kin: the unwillingness to leave one's home, to separate from friends and family, to become a refugee.\n\nFor all these reasons, the large proportion of Leningraders trapped in the city after September 8 is not unusual in the history of sieges. Nor were they trapped absolutely. The Germans were never able to link up with Finnish forces either on the western or eastern shores of Lake Lagoda, and so there remained an evacuation route to the interior of Russia, at first by boat across the lake, and then as the waters froze, progressively by foot, sled, and truck. Until large-scale convoys could be organized (in January 1942), however, only a slow trickle of people were able to escape. A more immediate escape route was available\u2014through the German lines. For the siege was maintained along a wide arc south of the city, many miles long and in places thinly held. It was possible for civilians on foot to filter through the lines and, as desperation grew within the city, thousands attempted to do so. The German command responded to these attempts with an order, first announced on September 18, and then repeated two months later, to stop the escapes at all costs. Artillery was to be used \"to prevent any such attempt at the greatest possible distance from our own lines by opening fire as early as possible, so that the infantry is spared . . . shooting at civilians.\" I have not been able to find any account of how many civilians died as a direct or indirect result of this order; nor do I know whether or not infantrymen actually opened fire. But if we assume that the German effort was at least partially successful, many would-be escapees, hearing of the shelling or the shooting, must have remained in the city. And there many of them died. Before the siege ended in 1943, more than a million civilians were dead of starvation and disease.\n\nAt Nuremberg, Field Marshal von Leeb, who commanded Army Group North from June to December 1941, and who was therefore responsible for the first months of siege operations, was formally charged with war crimes because of the order of September 18. Von Leeb claimed in defense that what he had done was customary practice in wartime, and the judges, after consulting the legal handbooks, were led to agree. They cited Professor Hyde, an American authority on international law: \"It is said that if the commander of a besieged place expels the noncombatants, in order to lessen the number of those who consume his stock of provisions, it is lawful, though an extreme measure, to drive them back so as to hasten the surrender.\" No effort was made to distinguish \"expelled\" civilians from those leaving voluntarily, and probably the distinction is not relevant to the guilt or innocence of von Leeb. The benefit to the besieged army would be the same in either case. The laws of war permit the attackers to bar the benefit if they can. \"We might wish the law were otherwise,\" said the judges, \"but we must administer it as we find it.\" Von Leeb was acquitted.\n\nThe judges could have found cases in which civilians were allowed to leave besieged cities. During the Franco-Prussian War, the Swiss managed to arrange for a limited evacuation of civilians from Strasbourg. The American commander permitted civilians to leave Santiago before ordering the bombardment of that city in 1898. The Japanese offered free exit for noncombatants trapped in Port Arthur in 1905, but the offer was declined by the Russian authorities. These were all cases, however, in which the attacking army expected to carry the city by storm, and its commanders were willing to make a humanitarian gesture\u2014they would not have said that they were recognizing noncombatant rights\u2014that would cost them nothing. But when the defenders are to be waited out, subjected to slow starvation, the precedents are different. The siege of Plevna in the Russo-\u00adTurkish war of 1877 is more typical.\n\nWhen Osman Pasha's food supplies began to fail, he turned out the old men and women who were in the town and demanded free passage for them to Sofia or Rakhovo. General Gourko [the Russian commander] refused and sent them back.\n\nAnd the student of international law who cites this case, then comments: \"He could not do otherwise without detriment to his plans.\" Field Marshal von Leeb might have recalled the shining example of General Gourko.\n\nThe argument that needs to be made against both Gourko and von Leeb is suggested by the terms of the German order of September 18. Suppose that large numbers of Russian civilians, convinced that they would die if they returned to Leningrad, had persisted in the face of artillery fire and advanced on the German lines. Would the infantry have shot them down? Its officers were apparently uncertain. That sort of thing was the work of special \"death squads,\" not of ordinary soldiers, even in Hitler's army. Surely there would have been some reluctance, and even some refusals; and surely it would have been right to refuse. Or, suppose that these same refugees were not killed, but rounded up and imprisoned. Would it have been acceptable under the laws of war to inform the commander of the besieged city that they would be held without food, systematically starved, until he surrendered? No doubt, the judges would have found this unacceptable (even though they sometimes recognized the right to kill hostages). They would not have questioned the responsibility of von Leeb for these people whom he had, in my alternative case, actually locked up. But how is the siege of a city different?\n\nThe inhabitants of a city, though they have freely chosen to live within its walls, have not chosen to live under siege. The siege itself is an act of coercion, a violation of the status quo, and I cannot see how the commander of the besieging army can escape responsibility for its effects. He has no right to wage total war, even if civilians and soldiers within the city are politically united in refusing surrender. The systematic starvation of civilians under siege is one of those military acts which \"though permissible by custom, is a glaring violation of the principle by which custom professes to be governed.\"\n\nThe only justifiable practice, I think, is indicated in the Talmudic law of sieges, summed up by the philosopher Maimonides in the twelfth century (whose version is cited by Grotius in the seventeenth): \"When siege is laid to a city for the purpose of capture, it may not be surrounded on all four sides, but only on three, in order to give an opportunity for escape to those who would flee to save their lives. . . .\" But this seems hopelessly na\u00efve. How is it possible to \"surround\" a city on three sides? Such a sentence, it might be said, could only appear in the literature of a people who had neither a state nor an army of their own. It is an argument offered not from any military perspective, but from a refugee perspective. It makes, however, the crucial point: that in the direness of a siege, people have a right to be refugees. And then it has to be said that the besieging army has a responsibility to open, if it possibly can, a path for their flight.\n\nIn practice, many men and women will refuse to leave. Though I have described civilians under siege as people in a trap, hostage-like, life in the city is not like life in a prison camp; it is both much worse and much better. There is, for one thing, important work to do, and there are shared reasons for doing it. Besieged cities are arenas for a collective heroism, and even after ordinary love of place gives out, the emotional life of the threatened city makes departure difficult, at least for some of the citizens. Civilians performing essential services for the army will not, of course, be permitted to leave; they are in effect conscripted. Along with the civilian heroes of the siege, they are henceforth legitimate objects of military attack. The offer of free exit turns all those people who choose to remain in the city, or who are forced to remain, even if they are still in their \"proper and permanent abode,\" into something like a garrison: they have yielded their civilian rights. It is another example of the coerciveness of war that men and women must, in this case, leave their homes to maintain their immunity. But that is not a judgment on the siege commander. When he opens his lines to civilian refugees, he is reducing the immediate coerciveness of his own activity, and having done that he probably has a right to carry on that activity (assuming that it has some significant military purpose). The offer of free exit clears him of responsibility for civilian deaths.\n\nAt this point, the argument needs to be made more general. I have been suggesting that when we judge those forms of warfare that closely involve the civilian population, like sieges (and, as we shall see, guerrilla war), the issue of coercion and consent takes precedence over the issue of direction and indirection. We want to know how civilians came to be in militarily exposed positions: what force was used against them, what choices they freely made. There are a wide range of possibilities:\n\n  1. that they are coerced by their ostensible defenders, who must then share responsibility for the resulting deaths, even though they do no killing themselves;\n  2. that they consent to be defended, and so clear the military commander of the defending army;\n  3. that they are coerced by their attackers, driven into an exposed position and killed, in which case it doesn't matter whether the killing is a direct effect or a side effect of the attack, for it is a crime either way;\n  4. that they are attacked but not coerced, attacked in their \"natural\" place, and then the principle of double effect comes into play and siege by starvation is morally unacceptable; and\n  5. that they are offered free exit by their attackers, after which those that remain can justifiably be killed, directly or indirectly.\n\nThe last two of these are the most important, though I will want to qualify them later on. They require a clearcut reversal of contemporary law as stated or restated at Nuremberg, so as to establish and give substance to a principle that is, I think, commonly accepted: that soldiers are under an obligation to help civilians leave the scene of a battle. In the case of a siege, I want to say, it is only when they fulfill this obligation that the battle itself is morally possible.\n\nBut is it still militarily possible? Once free exit has been offered, and been accepted by significant numbers of people, the besieging army is placed under a certain handicap. The city's food supply will now last so much longer. It is precisely this handicap that siege commanders have in the past refused to accept. I don't see, however, that it is different in kind from other handicaps imposed by the war convention. It doesn't make siege operations entirely impractical, only somewhat more difficult\u2014given the ruthlessness of the modern state, one has to say, marginally more difficult; for the presence of large numbers of civilians in a besieged city is unlikely to be allowed to interfere with the provisioning of the army; and, as the Leningrad example suggests, the death of large numbers of civilians is unlikely to be allowed to interfere with the defense of the city. In Leningrad, soldiers did not starve, though civilians died of hunger. On the other hand, civilians were evacuated from Leningrad, once Lake Lagoda had frozen solid, and food supplies were brought in. In different circumstances, free exit might make a greater military difference, forcing a frontal assault on the city (because the besieging army may also have supply problems) or a major prolongation of the siege. But these are acceptable consequences, and they are only \"detrimental\" to the plans of the siege commander if he has not planned for them in advance. In any case, if he wants (as he probably will want) to lift his hands to heaven and say of the civilians he kills, \"It's not my doing,\" he has no choice but to offer them the chance to leave.\n\nTaking Aim and the Doctrine of Double Effect\n\nThe issue is more difficult, however, when a whole country is subjected to siege conditions, when an invading army sets about systematically to destroy crops and food supplies, for example, or when a naval blockade cuts off vitally needed imports. Here free exit is not a plausible possibility (mass migration would be necessary), and the question of responsibility takes on a somewhat different form. Once again, it should be stressed that the struggle to secure and deny supplies is a common feature of ancient as well as modern warfare. It was the subject of legislation long before the modern laws of war were worked out. The Deuteronomic code, for example, explicitly bans the cutting down of fruit trees: \"Only the trees of which thou knowest that they are not trees for food, them thou mayest destroy and cut down, that thou mayest build bulwarks against the city. . . .\" But few armies seem ever to have respected the ban. It was apparently unknown in Greece; during the Peloponnesian War, the destruction of olive groves was virtually the first act of an invading army; judging from Caesar's Gallic Wars, the Romans fought in the same way. In early modern times, long before the scientific destruction of crops became possible, the doctrine of strategic devastation was a kind of conventional wisdom among military commanders. \"The Palatinate was wasted [in the Thirty Years War] in order that the imperial armies should be denied the military produce of the country; Marlborough destroyed the farms and crops of Bavaria for a similar purpose [in The War of the Spanish Succession]. . . .\" The Shenandoah Valley was laid waste in the American Civil War; and the burning of farms on Sherman's march through Georgia had, among other purposes, the strategic goal of starving the Confederate army. In our own time, and with a more advanced technology, vast sections of Vietnam were subjected to a similar destruction.\n\nThe contemporary laws of war require that such efforts be directed, whatever their indirect effects, only against the armed forces of the enemy. Civilians in a city have been thought a legitimate target, civilians at large not so: they are, though in vast numbers, only the incidental victims of strategic devastation. The allowable military purpose here is to make the provisioning of the enemy army impossible, and when generals have exceeded that purpose\u2014attempting, like General Sherman, to end the war by \"punishing\" the civilian population\u2014they have been commonly condemned. Why this is so I am not sure, though why it should be so is easier to make out. The impossibility of free exit rules out any direct attack on the civilian population.\n\nThis is not, however, much protection for civilians, since military supplies cannot be destroyed without first destroying civilian supplies. The morally desirable rule is stated by Spaight: \"If under such peculiar conditions as existed in the Confederate States and in South Africa [during the Boer War] . . . the enemy depends for his supplies on the surplus of cereals, etc., held by the noncombatant population, then a commander is justified by the necessity of war in destroying or seizing that surplus.\" But it is not the case that the army lives off the civilian surplus; more likely, civilians are forced to make do with what is left after the army has been fed. Hence, strategic devastation is not aimed, and cannot be, at \"military produce,\" but at food supplies generally. And civilians suffer long before soldiers feel the pinch. But who is it who inflicts this suffering, the army that destroys food stocks or the army that seizes what remains for itself? This question is taken up in the British government's official history of World War I.\n\nThe British Blockade of Germany\n\nIn its origins, a blockade was simply a naval siege, an \"investment by sea,\" barring all ships from entering or leaving the blockaded area (usually a major port) and cutting off, so far as possible, all supplies. It was not thought legally or morally justifiable, however, to extend this interdiction to the trade of an entire country. Most nineteenth-century commentators shared the view that the economic life of an enemy country could never be a legitimate military objective. The denial of military supplies was, of course, permissible, and given the possibility of stopping and searching ships on the high seas, elaborate rules were developed for the regulation of wartime trade. Lists of goods qualified as \"contraband\" and liable to confiscation were regularly published by belligerent powers. Though these lists tended to get longer and more inclusive, the laws of naval warfare stipulated the existence of a category of \"conditional contraband\" (commonly thought to include foodstuffs and medical supplies) that could not be seized unless it was known to be destined for military use. The relevant principle here was an extension of the combatant\/noncombatant distinction. \"The seizure of articles of commerce becomes illegitimate so soon as it ceases to aim at enfeebling the naval and military resources of the [enemy] country and puts immediate pressure on the civilian population.\"\n\nIn the course of the First World War, these rules were undermined in two ways; first, by extending the notion of blockade, and then by assuming the military utility of all conditional contraband. The result was full-scale economic warfare, a struggle over supply analogous in its purposes and effects to strategic devastation. The Germans fought this war with the submarine; the British, who controlled at least the surface of the sea, used conventional naval forces, blockading the entire German coast. In this case, conventional forces won the day. The convoy system eventually overcame the submarine threat, whereas the blockade, according to Liddell Hart, was a decisive factor in Germany's defeat. \"The spectre of slow enfeeblement ending in eventual collapse,\" he argues, drove the High Command to undertake its disastrous offensive of 1918. More immediate, and less military consequences can also be traced to the blockade. The \"slow enfeeblement\" of a country unhappily entails the actual deaths of individual citizens. Though civilians did not starve to death in Germany during the last years of the war, mass malnutrition greatly heightened the normal effects of disease. Statistical studies carried out after the war indicate that some half million civilian deaths, directly attributable to diseases such as influenza and typhus, in fact resulted from the deprivations imposed by the British blockade.\n\nBritish officials defended the blockade in legal terms by calling it a reprisal for German submarine warfare. More important for our purposes, however, is their consistent denial that the interdiction of supply was aimed at German civilians. The Cabinet had planned only a \"limited economic war,\" directed, as the official history has it, \"against the armed forces of the enemy.\" But the German government maintained its resistance \"by interposing the German people between the armies and the economic weapons that had been leveled against them and by making the civil populace bear the suffering inflicted.\" The sentence invites ridicule, and yet it is hard to imagine any other defense of the naval blockade (or of strategic devastation in land warfare). The passive form of the verb \"inflicted\" carries the argument. Who did the inflicting? Not the British, though they stopped ships and confiscated cargoes; they took aim at the German army and sought only military ends. And then, the official historian suggests, the Germans themselves pushed civilians into the front line of the economic war\u2014it is as if they had driven them into the forward trenches at the Battle of the Somme\u2014where the British could not help but kill them in the course of legitimate military operations.\n\nIf we are to pursue this argument, we will have to assume what seems unlikely: that the British did not in fact aim at the benefits they won from the slow starvation of German civilians. Given that fortunate blindness, the claim that Britain be acquitted of those civilian deaths is at least interesting, though finally unacceptable. It is interesting, first of all, that the official British historian makes the claim in this complex form rather than simply asserting a war right (as in the siege cases) to starve civilians. And it is interesting, secondly, because the acquittal of the British depends so radically on the indictment of the Germans. Without \"interposition,\" the British have no case, for the revised principle of double effect bars the strategy they adopted.\n\nIt is, of course, false to say that the German government \"interposed\" the civilian population between the blockade and the army. The civilians were where they had always been. If they stood behind the army in the national food line, that is where they had always stood. The army's prior claim to resources was not invented in order to cope with the exigencies of the blockade. Moreover, that claim was probably accepted by the great majority of the German people, at least until the very last months of the war. When the British took aim at the enemy army, therefore, they were aiming through the civilian population, knowing that the civilians were there and that they were in their normal place, \"their proper and permanent abode.\" In relation to the German army, they were placed in exactly the same way as British civilians in relation to their own army. It may be that the British did not intend to kill them; killing them wasn't (if we take the official history seriously) a means to the end set by the Cabinet. But if the success of the British strategy did not depend upon civilian deaths, it nevertheless required that nothing at all be done to avoid those deaths. Civilians had to be hit before soldiers could be hit, and this kind of attack is morally unacceptable. A soldier must take careful aim at his military target and away from nonmilitary targets. He can only shoot if he has a reasonably clear shot; he can only attack if a direct attack is possible. He can risk incidental deaths, but he cannot kill civilians simply because he finds them between himself and his enemies.a\n\nThis principle rules out the extended form of the naval blockade and every sort of strategic devastation, except in cases where adequate provision can be made, and is made, for noncombatants. It is not a principle that has been commonly accepted in war, at least not by the combatants. But it is consistent, I think, with other parts of the war convention, and it has gradually won acceptance, for political as much as moral reasons, with reference to a very important form of contemporary warfare. The systematic destruction of crops and food supplies is a frequent strategy in anti-guerrilla struggles, and since the governments engaged in such struggles generally claim sovereignty over the territory and population involved, they have been inclined to accept responsibility for feeding civilians (which is not to say that civilians have always been fed). Just what this involves I will consider in the next chapter. I have been arguing here that even enemy civilians, over whom sovereignty is not claimed, are the responsibility of attacking armies, whenever those armies adopt strategies that put civilians at risk.\n\n It remains true, however, that the issue of \"interposition\" (or coercion) has to be resolved first. Consider an example from the Franco-Prussian War of 1870: during the siege of Paris, the French used irregular forces behind enemy lines to attack trains carrying military supplies to the German army. The Germans responded by placing civilian hostages on the trains. Now it was no longer possible to get a \"clear shot\" at what was still a legitimate military target. But the civilians on the trains were not in their normal place; they had been radically coerced; and responsibility for their deaths, even if these deaths were actually inflicted by the French, lay with the German commanders. On this point, see Robert Nozick's discussion of \"innocent shields of threats\" in Anarchy, State and Utopia, p. 35.\n\nGuerrilla War\n\nResistance to Military Occupation\n\nA Partisan Attack\n\nSurprise is the essential feature of guerrilla war; thus the ambush is the classic guerrilla tactic. It is also, of course, a tactic in conventional war; the concealment and camouflage that it involves, though they were once repugnant to officers and gentlemen, have long been regarded as legitimate forms of combat. But there is one kind of ambush that is not legitimate in conventional war and that places in sharp focus the moral difficulties guerrillas and their enemies regularly encounter. This is the ambush prepared behind political or moral rather than natural cover. An example is provided by Captain Helmut Tausend, of the German Army, in Marcel Ophuls' documentary film The Sorrow and the Pity. Tausend tells of a platoon of soldiers on a march through the French countryside during the years of the German occupation. They passed a group of young men, French peasants, or so it seemed, digging potatoes. But these were not in fact peasants; they were members of the Resistance. As the Germans marched by, the \"peasants\" dropped their shovels, picked up guns hidden in the field, and opened fire. Fourteen of the soldiers were hit. Years later, their captain was still indignant. \"You call that 'partisan' resistance? I don't. Partisans for me are men that can be identified, men who wear a special armband or a cap, something with which to recognize them. What happened in that potato field was murder.\"\n\nThe captain's argument about armbands and caps is simply a citation from the international law of war, from the Hague and Geneva conventions, and I shall have more to say about it later on. It is important to stress first that the partisans had here taken on a double disguise. They were disguised as peaceful peasants and also as Frenchmen, that is, citizens of a state that had surrendered, for whom the war was over (just as guerrillas in a revolutionary struggle disguise themselves as unarmed civilians and also as loyal citizens of a state that is not at war at all). It was because of this second disguise that the ambush was so perfect. The Germans thought they were in a rear area, not at the front, and so they were not battle-ready; they were not preceded by a scouting party; they were not suspicious of the young men in the field. The surprise achieved by the partisans was of a kind virtually impossible in actual combat. It derived from what might be called the protective coloration of national surrender, and its effect was obviously to erode the moral and legal understandings upon which surrender rests.\n\nSurrender is an explicit agreement and exchange: the individual soldier promises to stop fighting in exchange for benevolent quarantine for the duration of the war; a government promises that its citizens will stop fighting in exchange for the restoration of ordinary public life. The precise conditions of \"benevolent quarantine\" and \"public life\" are specified in the law books; I need not go into them here. The obligations of individuals are also specified: they may try to escape from the prison camp or to flee occupied territory, and if they succeed in their escape or flight, they are free to fight again; they have regained their war rights. But they may not resist their quarantine or occupation. If a prisoner kills a guard in the course of his escape, the act is murder; if the citizens of a defeated country attack the occupation authorities, the act has, or once had, an even grimmer name: it is, or was, \"war treason\" (or \"war rebellion\"), a breaking of political faith, punishable, like the ordinary treason of rebels and spies, by death.\n\nBut \"traitor\" does not seem the right name for those French partisans. Indeed, it is precisely their experience, and that of other guerrilla fighters in World War II, that has led to the virtual disappearance of \"war treason\" from the law books and of the idea of breaking faith from our moral discussions of wartime resistance (and of peacetime rebellion also, when it is directed against alien or colonial rule). We tend to deny, today, that individuals are automatically subsumed by the decisions of their government or the fate of its armies. We have come to understand the moral commitment they may feel to defend their homeland and their political community, even after the war is officially over. A prisoner of war, after all, knows that the fighting will go on despite his own capture; his government is in place, his country is still being defended. But after national surrender the case is different, and if there are still values worth defending, no one can defend them except ordinary men and women, citizens with no political or legal standing. I suppose it is some general sense that there are such values, or often are, that leads us to grant these men and women a kind of moral authority.\n\nBut though this grant reflects new and valuable democratic sensibilities, it also raises serious questions. For if citizens of a defeated state still have a right to fight, what is the meaning of surrender? And what obligations can be imposed on conquering armies? There can be no ordinary public life in occupied territory if the occupation authorities are subject to attack at any time and at the hands of any citizen. And ordinary life is a value, too. It is what most of the citizens of a defeated country most ardently hope for. The heroes of the resistance put it in jeopardy, and we must weigh the risks they impose on others in order to understand the risks they must accept themselves. Moreover, if the authorities actually do aim at the restoration of everyday peacefulness, they seem entitled to enjoy the security they provide; and then they must also be entitled to regard armed resistance as a criminal activity. So the story with which I began might end this way (in the film it has no end): the surviving soldiers rally and fight back; some of the partisans are captured, tried as murderers, condemned, and executed. We would not, I think, add those executions to the list of Nazi war crimes. At the same time, we would not join in the condemnation.\n\nSo the situation can be summed up: resistance is legitimate, and the punishment of resistance is legitimate. That may seem like a simple standoff and an abdication of ethical judgment. It is actually a precise reflection of the moral realities of military defeat. I want to stress again that our understanding of these realities has nothing to do with our view of the two sides. We can deplore the resistance, without calling the partisans traitors; we can hate the occupation, without calling the execution of the partisans a crime. If we alter the story or add to it, of course, the case is changed. If the occupation authorities do not live up to their obligations under the surrender agreement, they lose their entitlements. And once the guerrilla struggle has reached a certain point of seriousness and intensity, we may decide that the war has effectively been renewed, notice has been given, the front has been re-established (even if it is not a line), and soldiers no longer have a right to be surprised even by a surprise attack. Then guerrillas captured by the authorities must be treated as prisoners of war\u2014provided, that is, they have themselves fought in accordance with the war convention.\n\nBut guerrillas don't fight that way. Their struggle is subversive not merely with reference to the occupation or to their own government, but with reference to the war convention itself. Wearing peasant clothes and hiding among the civilian population, they challenge the most fundamental principle of the rules of war. For it is the purpose of those rules to specify for each individual a single identity; he must be either a soldier or a civilian. The British Manual of Military Law makes the point with special clarity: \"Both these classes have distinct privileges, duties, and disabilities. . . . An individual must definitely choose to belong to one class or the other, and shall not be permitted to enjoy the privileges of both; in particular . . . an individual [shall] not be allowed to kill or wound members of the army of the opposed nation and subsequently, if captured or in danger of life, pretend to be a peaceful citizen.\" That is what guerrillas do, however, or sometimes do. So we can imagine another conclusion to the story of the partisan attack. The partisans successfully disengage, disperse to their homes, and go about their ordinary business. When German troops come to the village that night, they cannot distinguish the guerrilla fighters from any other of the villagers. What do they do then? If, through searches and interrogations\u2014police, not soldier's, work\u2014they seize one of the partisans, should they treat him as a captured criminal or a prisoner of war (leaving aside now the problems of surrender and resistance)? And if they seize no one, can they punish the whole village? If the partisans don't maintain the distinction of soldiers and civilians, why should they?\n\nThe Rights of Guerrilla Fighters\n\nAs this example suggests, the guerrillas don't subvert the war convention by themselves attacking civilians; at least, it is not a necessary feature of their struggle that they do that. Instead, they invite their enemies to do it. By refusing to accept a single identity, they seek to make it impossible for their enemies to accord to combatants and noncombatants their \"distinct privileges . . . and disabilities.\" The political creed of the guerrillas is essentially a defense of this refusal. The people, they say, are no longer being defended by an army; the only army in the field is the army of the oppressors; the people are defending themselves. Guerrilla war is \"people's war,\" a special form of the lev\u00e9e en masse, authorized from below. \"The war of liberation,\" according to a pamphlet of the Vietnamese National Liberation Front, \"is fought by the people themselves; the entire people . . . are the driving force. . . . Not only the peasants in their rural areas, but the workers and laborers in the city, along with intellectuals, students, and businessmen have gone to fight the enemy.\" And the NLF drove the point home by naming its paramilitary forces Dan Quan, literally, civilian soldiers. The guerrilla's self-image is not of a solitary fighter hiding among the people, but of a whole people mobilized for war, himself a loyal member, one among many. If you want to fight against us, the guerrillas say, you are going to have to fight civilians, for you are not at war with an army but with a nation. Therefore, you should not fight at all, and if you do, you are the barbarians, killing women and children.\n\nIn fact, the guerrillas mobilize only a small part of the nation\u2014a very small part, when they first begin their attacks. They depend upon the counter-attacks of their enemies to mobilize the rest. Their strategy is framed in terms of the war convention: they seek to place the onus of indiscriminate warfare on the opposing army. The guerrillas themselves have to discriminate, if only to prove that they are really soldiers (and not enemies) of the people. It is also and perhaps more importantly true that it is relatively easy for them to make the relevant discriminations. I don't mean that guerrillas never engage in terrorist campaigns (even against their fellow countrymen) or that they never take hostages or burn villages. They do all those things, though they generally do less of them than the anti-guerrilla forces. For the guerrillas know who their enemies are, and they know where they are. They fight in small groups, with small arms, at close quarters\u2014and the soldiers they fight against wear uniforms. Even when they kill civilians, they are able to make distinctions: they aim at well-known officials, notorious collaborators, and so on. If the \"entire people\" are not really the \"driving force,\" they are also not the objects of guerrilla attack.\n\nFor this reason, guerrilla leaders and publicists are able to stress the moral quality not only of the goals they seek but also of the means they employ. Consider for a moment Mao Tse-tung's famous \"Eight Points for Attention.\" Mao is by no means committed to the notion of noncombatant immunity (as we shall see), but he writes as if, in the China of the warlords and the Kuomintang, only the communists respect the lives and property of the people. The \"Eight Points\" are meant to mark off the guerrillas first of all from their predecessors, the bandits of traditional China, and then from their present enemies, who ravage the countryside. They suggest how the military virtues can be radically simplified for a democratic age.\n\n  1. Speak politely.\n  2. Pay fairly for what you buy.\n  3. Return everything you borrow.\n  4. Pay for anything you damage.\n  5. Do not hit or swear at people.\n  6. Do not damage crops.\n  7. Do not take liberties with women.\n  8. Do not ill-treat captives.\n\nThe last of these is particularly problematic, for in the conditions of guerrilla war it must often involve releasing prisoners, something most guerrillas are no doubt loath to do. Yet it is at least sometimes done, as an account of the Cuban revolution, originally published in the Marine Corps Gazette, suggests:\n\nThat same evening, I watched the surrender of hundreds of Batistianos from a small-town garrison. They were gathered within a hollow square of rebel Tommy-gunners and harangued by Raul Castro:\n\n\"We hope that you will stay with us and fight against the master who so ill-used you. If you decide to refuse this invitation\u2014and I am not going to repeat it\u2014you will be delivered to the custody of the Cuban Red Cross tomorrow. Once you are under Batista's orders again, we hope that you will not take arms against us. But, if you do, remember this:\n\n\"We took you this time. We can take you again. And when we do, we will not frighten or torture or kill you. . . . If you are captured a second time or even a third . . . we will again return you exactly as we are doing now.\"\n\nEven when guerrillas behave this way, however, it is not clear that they are themselves entitled to prisoner of war status when captured, or that they have any war rights at all. For if they don't make war on noncombatants, it also appears that they don't make war on soldiers: \"What happened in that potato field was murder.\" They attack stealthily, deviously, without warning, and in disguise. They violate the implicit trust upon which the war convention rests: soldiers must feel safe among civilians if civilians are ever to be safe from soldiers. It is not the case, as Mao once suggested, that guerrillas are to civilians as fish to the ocean. The actual relation is rather of fish to other fish, and the guerrillas are as likely to appear among the minnows as among the sharks.\n\nThat, at least, is the paradigmatic form of guerrilla war. I should add that it is not the form such war always or necessarily takes. The discipline and mobility required of guerrilla fighters often preclude a domestic retreat. Their main forces commonly operate out of base camps located in remote areas of the country. And, curiously enough, as the guerrilla units grow larger and more stable, their members are likely to put on uniforms. Tito's partisans in Yugoslavia, for example, wore distinctive dress, and this was apparently no disadvantage in the kind of war they fought. All the evidence suggests that quite apart from the rules of war, guerrillas, like other soldiers, prefer to wear uniforms; it enhances their sense of membership and solidarity. In any case, soldiers attacked by a guerrilla main force know who their enemies are as soon as the attack begins; ambushed by uniformed men, they would know no sooner. When the guerrillas \"melt away\" after such an attack, they more often disappear into jungles or mountains than into villages, a retreat that raises no moral problems. Battles of this sort can readily be assimilated to the irregular combat of army units like Wingate's \"Chindits\" or \"Merrill's Marauders\" in World War II. But this is not what most people have in mind when they talk about guerrilla war. The paradigm worked out by guerrilla publicists (together with their enemies) focuses precisely on what is morally difficult about guerrilla war\u2014and also, as we shall see, about anti-guerrilla war. In order to deal with these difficulties, I shall simply accept the paradigm and treat guerrillas as they ask to be treated, as fish among the ocean's fish. What then are their war rights?\n\nThe legal rules are simple and clearcut, though not without their own problems. To be eligible for the war rights of soldiers, guerrilla fighters must wear \"a fixed distinctive sign visible at a distance\" and must \"carry their arms openly.\" It is possible to worry at length about the precise meaning of distinctiveness, fixity, and openness, but I do not think we would learn a great deal by doing so. In fact, these requirements are often suspended, particularly in the interesting case of a popular rising to repel invasion or resist foreign tyranny. When the people rise en masse, they are not required to put on uniforms. Nor will they carry arms openly, if they fight, as they usually do, from ambush: hiding themselves, they can hardly be expected to display their weapons. Francis Lieber, in one of the earliest legal studies of guerrilla war, cites the case of the Greek rebellion against Turkey, where the Turkish government killed or enslaved all prisoners: \"But I take it,\" he writes, \"that a civilized government would not have allowed the fact that the Greeks . . . carried on mountain guerrilla [war] to influence its conduct toward prisoners.\"\n\nThe key moral issue, which the law gets at only imperfectly, does not have to do with distinctive dress or visible weapons, but with the use of civilian clothing as a ruse and a disguise.a The French partisan attack perfectly illustrates this, and it has to be said, I think, that the killing of those German soldiers was more like assassination than war. That is not because of the surprise, simply, but because of the kind and degree of deceit involved: the same sort of deceit that is involved when a public official or party leader is shot down by some political enemy who has taken on the appearance of a friend and supporter or of a harmless passer-by. Now it may be the case\u2014I am more than open to this suggestion\u2014that the German army in France had attacked civilians in ways that justified the assassination of individual soldiers, just as it may be the case that the public official or party leader is a brutal tyrant who deserves to die. But assassins cannot claim the protection of the rules of war; they are engaged in a different activity. Most of the other enterprises for which guerrillas require civilian disguise are also \"different.\" These include all the possible varieties of espionage and sabotage; they can best be understood by comparing them to acts carried out behind enemy lines by the secret agents of conventional armies. It is widely agreed that such agents possess no war rights, even if their cause is just. They know the risks their efforts entail, and I see no reason to describe the risks of guerrillas engaged in similar projects any differently. Guerrilla leaders claim war rights for all their followers, but it makes sense to distinguish, if this is possible, between those guerrillas who use civilian dress as a ruse and those who depend upon camouflage, the cover of darkness, tactical surprise, and so on.\n\nThe issues posed by the guerrilla war paradigm, however, are not resolved by this distinction. For guerrillas don't merely fight as civilians; they fight among civilians, and this in two senses. First, their day-to-day existence is much more closely connected with the day-to-day existence of the people around them than is ever the case with conventional armies. They live with the people they claim to defend, whereas conventional troops are usually billeted with civilians only after the war or the battle is over. And second, they fight where they live; their military positions are not bases, posts, camps, forts, or strongholds, but villages. Hence they are radically dependent on the villagers, even when they don't succeed in mobilizing them for \"people's war.\" Now, every army depends upon the civilian population of its home country for supplies, recruits, and political support. But this dependence is usually indirect, mediated by the bureaucratic apparatus of the state or the exchange system of the economy. So food is passed from the farmer to the marketing co-op, to the food processing plant, to the trucking company, to the army commissary. But in guerrilla war, the dependence is immediate: the farmer hands the food to the guerrilla, and whether it is received as a tax or paid for in accordance with Mao's Second Point for Attention, the relation between the two men is face-to-face. Similarly, an ordinary citizen may vote for a political party that in turn supports the war effort and whose leaders are called in for military briefings. But in guerrilla war, the support a civilian provides is far more direct. He doesn't need to be briefed; he already knows the most important military secret; he knows who the guerrillas are. If he doesn't keep this information to himself, the guerrillas are lost.\n\nTheir enemies say that the guerrillas rely on terror to win the support or at least the silence of the villagers. But it seems more likely that when they have significant popular support (which they don't always have), they have it for other reasons. \"Violence may explain the cooperation of a few individuals,\" writes an American student of the Vietnamese war, \"but it cannot explain the cooperation of a whole social class [the peasantry].\" If the killing of civilians were sufficient to win civilian support, the guerrillas would always be at a disadvantage, for their enemies possess far more fire power than they do. But killing will work against the killer \"unless he has already pre-empted a large part of the population and then limits his acts of violence to a sharply defined minority.\" When the guerrillas succeed, then, in fighting among the people, it is best to assume that they have some serious political support among the people. The people, or some of them, are complicitous in guerrilla war, and the war would be impossible without their complicity. That doesn't mean that they seek out opportunities to help. Even when he sympathizes with the goal of the guerrillas, we can assume that the average civilian would rather vote for them than hide them in his house. But guerrilla war makes for enforced intimacies, and the people are drawn into it in a new way even though the services they provide are nothing more than functional equivalents of the services civilians have always provided for soldiers. For the intimacy is itself an additional service, which has no functional equivalent. Whereas soldiers are supposed to protect the civilians who stand behind them, guerrillas are protected by the civilians among whom they stand.\n\nBut the fact that they accept this protection, and depend upon it, doesn't seem to me to deprive the guerrillas of their war rights. Indeed, it is more plausible to make exactly the opposite argument: that the war rights the people would have were they to rise en masse are passed on to the irregular fighters they support and protect\u2014assuming that the support, at least, is voluntary. For soldiers acquire war rights not as individual warriors but as political instruments, servants of a community that in turn provides services for its soldiers. Guerrillas take on a similar identity whenever they stand in a similar or equivalent relationship, that is, whenever the people are helpful and complicitous in the ways I have described. When the people do not provide this recognition and support, guerrillas acquire no war rights, and their enemies may rightly treat them when captured as \"bandits\" or criminals. But any significant degree of popular support entitles the guerrillas to the benevolent quarantine customarily offered prisoners of war (unless they are guilty of specific acts of assassination or sabotage, for which soldiers, too, can be punished).b\n\nThis argument clearly establishes the rights of the guerrillas; it raises the most serious questions, however, about the rights of the people; and these are the crucial questions of guerrilla war. The intimacies of the struggle expose the people in a new way to the risks of combat. In practice, the nature of this exposure, and its degree, are going to be determined by the government and its allies. So the burdens of decision are shifted by the guerrillas onto their enemies. It is their enemies who must weigh (as we must) the moral significance of the popular support the guerrillas both enjoy and exploit. One can hardly fight against men and women who themselves fight among civilians without endangering civilian lives. Have these civilians forfeited their immunity? Or do they, despite their wartime complicity, still have rights vis-\u00e0-vis the anti-guerrilla forces?\n\nThe Rights of Civilian Supporters\n\nIf civilians had no rights at all, or were thought to have none, it would be a small benefit to hide among them. In a sense, then, the advantages the guerrillas seek depend upon the scruples of their enemies\u2014though there are other advantages to be had if their enemies are unscrupulous: that is why anti-guerrilla warfare is so difficult. I shall want to argue that these scruples in fact have a moral basis, but it is worth suggesting first that they also have a strategic basis. It is always in the interest of the anti-guerrilla forces to insist upon the soldier\/civilian distinction, even when the guerrillas act (as they always will if they can) so as to blur the line. All the handbooks on \"counter-insurgency\" make the same argument: what is necessary is to isolate the guerrillas from the civilian population, to cut them off from their protection and at the same time to shield civilians from the fighting. The last point is more important in guerrilla than in conventional war, for in conventional war one assumes the hostility of \"enemy civilians,\" while in a guerrilla struggle one must seek their sympathy and support. Guerrilla war is a political, even an ideological conflict. \"Our kingdoms lay in each man's mind,\" wrote T. E. Lawrence of the Arab guerrillas he led in World War I. \"A province would be won when we had taught the civilians in it to die for our ideal of freedom.\" And it can be won back only if those same civilians are taught to live for some counter-ideal (or in the case of a military occupation, to acquiesce in the re-establishment of order and ordinary life). That is what is meant when it is said that the battle is for the \"hearts and minds\" of the people. And one cannot triumph in such a battle by treating the people as so many enemies to be attacked and killed along with the guerrillas who live among them.\n\nBut what if the guerrillas cannot be isolated from the people? What if the lev\u00e9e en masse is a reality and not merely a piece of propaganda? Characteristically, the military handbooks neither pose nor answer such questions. There is, however, a moral argument to be made if this point is reached: the anti-guerrilla war can then no longer be fought\u2014and not just because, from a strategic point of view, it can no longer be won. It cannot be fought because it is no longer an anti-guerrilla but an anti-social war, a war against an entire people, in which no distinctions would be possible in the actual fighting. But this is the limiting case of guerrilla war. In fact, the rights of the people come into play earlier on, and I must try now to give them some plausible definition.\n\nConsider again the case of the partisan attack in occupied France. If, after the ambush, the partisans hide in a nearby peasant village, what are the rights of the peasants among whom they hide? German soldiers arrive that night, let's say, seeking the men and women directly involved or implicated in the ambush and looking also for some way of preventing future attacks. The civilians they encounter are hostile, but that doesn't make them enemies in the sense of the war convention, for they don't actually resist the efforts of the soldiers. They behave exactly as citizens sometimes do in the face of police interrogations: they are passive, blank, evasive. We must imagine a domestic state of emergency and ask how the police might legitimately respond to such hostility. Soldiers can do no more when what they are doing is police work; for the status of the hostile civilians is no different. Interrogations, searches, seizures of property, curfews\u2014all these seem to be commonly accepted (I will not try to explain why); but not the torture of suspects or the taking of hostages or the internment of men and women who are or might be innocent. Civilians still have rights in such circumstances. If their liberty can be temporarily abridged in a variety of ways, it is not entirely forfeit; nor are their lives at risk. The argument would be much harder, however, had the troops been ambushed as they marched through the village itself, shot at from the cover of peasant homes and barns. To understand what happens then, we must look at another historical example.\n\nThe American \"Rules of Engagement\" in Vietnam\n\nHere is a typical incident of the Vietnam War. \"An American unit moving along Route 18 [in Long An province] received small arms fire from a village, and in reply the tactical commander called for artillery and air strikes on the village itself, resulting in heavy civilian casualties and extensive physical destruction.\" Something like this must have happened hundreds, even thousands of times. The bombing and strafing of peasant villages was a common tactic of the American forces. It is a matter of special interest to us that it was permitted by the U.S. Army's \"rules of engagement,\" worked out, so it was said, to isolate the guerrillas and minimize civilian casualties.\n\nThe attack on the village near Route 18 looks as if it was intended to minimize only army casualties. It looks like another instance of a practice I have already examined: the indiscriminate use of modern fire power to save soldiers from trouble and risk. But in this case, the trouble and risk are of a sort very different from anything encountered on the front line of a conventional war. It is most unlikely that an army patrol moving into the village would have been able to locate and destroy an enemy position. The soldiers would have found . . . a village, its population sullen and silent, the guerrilla fighters hiding, the guerrilla \"fortifications\" indistinguishable from the homes and shelters of the villagers. They might have drawn hostile fire; more likely, they would have lost men to mines and booby traps, the exact location of which everyone in the village knew and no one would reveal. Under such circumstances, it was not difficult for soldiers to convince themselves that the village was a military stronghold and a legitimate target. And if it was known to be a stronghold, surely it could be attacked, like any other enemy position, even before hostile fire was encountered. In fact, this became American policy quite early in the war: villages from which hostile fire might reasonably be expected were shelled and bombed before soldiers moved in and even if no movement was planned. But then how does one minimize civilian casualties, let alone win over the civilian population? It was to answer this question that the rules of engagement were developed.\n\nThe crucial point of the rules, as they are described by the journalist Jonathan Schell, was that civilians were to be given warning in advance of the destruction of their villages, so that they could break with the guerrillas, expel them, or leave themselves. The goal was to force the separation of combatants and noncombatants, and the means was terror. Enormous risk was attached to complicity in guerrilla war, but this was a risk that could only be imposed on whole villages; no further differentiation was possible. It is not the case that civilians were held hostage for the activities of the guerrillas. Rather, they were held responsible for their own activity, even when this activity was not overtly military. The fact that the activity sometimes was overtly military, that ten-year-old children threw hand grenades at American soldiers (the incidence of such attacks was probably exaggerated by the soldiers, in part to justify their own conduct toward civilians) blurs the nature of this responsibility. But it has to be stressed that a village was regarded as hostile not because its women and children were prepared to fight, but because they were not prepared to deny material support to the guerrillas or to reveal their whereabouts or the location of their mines and booby traps.\n\nThese were the rules of engagement: (1) A village could be bombed or shelled without warning if American troops had received fire from within it. The villagers were presumed able to prevent the use of their village as a fire base, and whether or not they actually were able, they certainly knew in advance whether it would be so used. In any case, the shooting itself was a warning, since return fire was to be expected\u2014though it is unlikely that the villagers expected the response to be as disproportionate as it usually was, until the pattern had become familiar. (2) Any village known to be hostile could be bombed or shelled if its inhabitants were warned in advance, either by the dropping of leaflets or by helicopter loudspeaker. These warnings were of two sorts: sometimes they were specific in character, delivered immediately before an attack, so that the villagers only just had time to leave (and then the guerrillas could leave with them), or they were general, describing the attack that might come if the villagers did not expel the guerrillas.\n\nThe U.S. Marines will not hesitate to destroy immediately any village or hamlet harboring the Vietcong. . . . The choice is yours. If you refuse to let the Vietcong use your villages and hamlets as their battlefield, your homes and your lives will be saved.\n\nAnd if not, not. Despite the emphasis on choice, this is not quite a liberal pronouncement, for the choice in question is very much a collective one. Exodus, of course, remained an individual option: people could move out of villages where the Vietcong had established itself, taking refuge with relatives in other villages, or in the cities, or in government-run camps. Most often, however, they did this only after the bombing had begun, either because they did not understand the warnings, or did not believe them, or simply hoped desperately that their own homes would somehow be spared. Hence it was sometimes thought humane to dispense with choice altogether and forcibly to deport villagers from areas that were considered under enemy control. Then the third rule of engagement went into effect. (3) Once the civilian population had been moved out, the village and surrounding country might be declared a \"free fire zone\" that could be bombed and shelled at will. It was assumed that anyone still living in the area was a guerrilla or a \"hardened\" guerrilla supporter. Deportation had stripped away civilian cover as defoliation stripped away natural cover, and left the enemy exposed.\n\nIn considering these rules, the first thing to note is that they were radically ineffective. \"My investigation disclosed,\" writes Schell, \"that the procedures for applying these restraints were modified or twisted or ignored to such an extent that in practice the restraints evaporated entirely. . . .\" Often, in fact, no warning was given, or the leaflets were of little help to villagers who could not read, or the forcible evacuation left large numbers of civilians behind, or no adequate provision was made for the deported families and they drifted back to their homes and farms. None of this, of course, would reflect on the value of the rules themselves, unless the ineffectiveness were somehow intrinsic to them or to the situation in which they were applied. This was clearly the case in Vietnam. For where the guerrillas have significant popular support and have established a political apparatus in the villages, it is unrealistic to think that the villagers will or can expel them. This has nothing to do with the virtues of guerrilla rule: it would have been equally unrealistic to think that German workers, though their homes were bombed and their families killed, would overthrow the Nazis. Hence the only protection the rules provide is in advising or enforcing the departure not of guerrillas from peaceful villages but of civilians from what is likely to become a battlefield.\n\nNow, in a conventional war, removing civilians from a battlefield is clearly a good thing to do; positive international law requires it wherever possible. Similarly in the case of a besieged city: civilians must be allowed to leave; and if they refuse (so I have argued), they can be attacked along with the defending soldiers. But a battlefield and a city are determinate areas, and a battle and a siege are, usually, of limited duration. Civilians move out; then they move back. Guerrilla war is likely to be very different. The battlefield extends over much of the country and the struggle is, as Mao has written, \"protracted.\" Here the proper analogy is not to the siege of a city but to the blockade or strategic devastation of a much wider area. The policy underlying the American rules of engagement actually envisaged the uprooting and resettlement of a very substantial part of the rural population of Vietnam: millions of men, women, and children. But that is an incredible task, and, leaving aside for the moment the likely criminality of the project, there was never more than a pretense that sufficient resources would be made available to accomplish it. It was inevitable then, and it was known to be inevitable, that civilians would be living in the villages that were shelled and bombed.\n\nWhat happened is quickly described:\n\nIn August 1967, during Operation Benton, the \"pacification\" camps became so full that Army units were ordered not to \"generate\" any more refugees. The Army complied. But search and destroy operations continued. Only now the peasants were not warned before an air-strike was called on their village. They were killed in their villages because there was no room for them in the swamped pacification camps.\n\nI should add that this sort of thing doesn't always happen, even in anti-\u00adguerrilla war\u2014though the policy of forced resettlement or \"concentration,\" from its origins in the Cuban Insurgency and the Boer War, has rarely been carried out in a humane manner or with adequate resources. But one can find counter-examples. In Malaya, in the early 1950s, where the guerrillas had the support of only a relatively small part of the rural population, a limited resettlement (to new villages, not concentration camps) seems to have worked. At any rate, it has been said that after the fighting was over, few of the resettled villagers wanted to return to their former homes. That is not a sufficient criterion of moral success, but it is one sign of a permissible program. Since governments are generally thought to be entitled to resettle (relatively small numbers of) their own citizens for the sake of some commonly accepted social purpose, the policy cannot be ruled out altogether in time of guerrilla war. But unless the numbers are restricted, it will be difficult to make the case for common acceptance. And here, as in peacetime, there is some requirement to provide adequate economic support and comparable living space. In Vietnam, that was never possible. The scope of the war was too wide; new villages could not be built; the camps were dismal; and hundreds of thousands of displaced peasants crowded into the cities, forming there a new lumpen proletariat, miserable, sick, jobless, or quickly exploited in ill-paid and menial jobs or as servants, prostitutes, and so on.\n\nEven had all this worked, in the limited sense that civilian deaths had been avoided, the rules of engagement and the policy they embodied could hardly be defended. It seems to violate even the principle of proportionality\u2014which is by no means easy to do, as we have seen again and again, since the values against which destruction and suffering are to be measured are so readily inflated. But in this case, the argument is clear, for the defense of resettlement comes down finally to a claim something like that made by an American officer with reference to the town of Ben Tre: we had to destroy the town in order to save it. In order to save Vietnam, we had to destroy the rural culture and the village society of the Vietnamese. Surely the equation does not work and the policy cannot be approved, at least in the context of the Vietnamese struggle itself. (One can always shift, I suppose, to the higher mathematics of international statecraft.)\n\nBut the rules of engagement raise a more interesting question. Suppose that civilians, duly warned, not only refuse to expel the guerrillas but also refuse to leave themselves. Can they be attacked and killed, as the rules imply? What are their rights? They can certainly be exposed to risks, for battles are likely to be fought in their villages. And the risks they must live with will be considerably greater than those of conventional combat. The increased risk results from the intimacies I have already described; I would suggest now that it is the only result of those intimacies, at least in the moral realm. It is serious enough. Anti-guerrilla war is a terrible strain on conventional troops, and even if they are both disciplined and careful, as they should be, civilians are certain to die at their hands. A soldier who, once he is engaged, simply fires at every male villager between the ages of fifteen and fifty (say) is probably justified in doing so, as he would not be in an ordinary firefight. The innocent deaths that result from this kind of fighting are the responsibility of the guerrillas and their civilian supporters; the soldiers are cleared by the doctrine of double effect. It has to be stressed, however, that the supporters themselves, so long as they give only political support, are not legitimate targets, either as a group or as distinguishable individuals. Conceivably, some of them can be charged with complicity (not in guerrilla war generally but) in particular acts of assassination and sabotage. But charges of that sort must be proved before some sort of judicial tribunal. So far as combat goes, these people cannot be shot on sight, when no firefight is in progress; nor can their villages be attacked merely because they might be used as firebases or because it is expected that they will be used; nor can they be randomly bombed and shelled, even after warning has been given.\n\nThe American rules have only the appearance of recognizing and attending to the combatant\/noncombatant distinction. In fact, they set up a new distinction: between loyal and disloyal, or friendly and hostile noncombatants. The same dichotomy can be seen at work in the claims American soldiers made about the villages they attacked: \"This place is almost entire V.C. controlled, or pro-V.C.\" \"We consider just about everyone here to be a hard-core V.C., or at least some kind of supporter.\" It is not the military activities of the villagers that are being stressed in statements of this sort, but their political allegiance. Even with reference to that, the statements are palpably false, since at least some of the villagers are children who cannot be said to have any allegiance at all. In any case, as I have already argued in the example of the villagers of occupied France, political hostility does not make people enemies in the sense of the war convention. (If it did, there would be no civilian immunity at all, except when wars were fought in neutral countries.) They have done nothing to forfeit their right to life, and that right must be respected as best it can be in the course of attacks against the irregular fighters the villagers both resemble and harbor.\n\nIt is important to say something now about the possible shape of those attacks, though I cannot talk about them like a military strategist; I can only report on some of the things that strategists say. Bombing and shelling from a distance have undoubtedly been defended in terms of military necessity. But that is as bad an argument strategically as it is morally. For there are other and more effective ways of fighting. Thus a British expert on counter-insurgency writes that the use of \"heavily armed helicopters\" against peasant villages \"can only be justified if the campaign has deteriorated to the extent where it is virtually indistinguishable from conventional war.\" I doubt that it can be justified even then, but I want to stress again what this expert has grasped: that counter-insurgency requires a strategy and tactics of discrimination. Guerrillas can be defeated (and, similarly, they can win) only at close quarters. With regard to peasant villages, this suggests two different sorts of campaigns, both of which have been extensively discussed in the literature. In areas of \"low intensity operations,\" the villages must be occupied by small units specially trained for the political and police work necessary to seek out guerrilla supporters and informants. In areas where the guerrillas are effectively in control and the fighting intense, the villages must be encircled and entered in force. Bernard Fall has reported in some detail on a French attack of this sort in Vietnam in the 1950s. What is involved here is an effort to bring numbers, expertise, and technology directly to bear, forcing the guerrillas to give battle in a situation where fire can be relatively precise, or driving them into a surrounding net of soldiers. If the soldiers are properly prepared and equipped, they need not accept unbearable risks in fighting of this sort, and they need not inflict indiscriminate destruction. As Fall points out, a very considerable number of men are required for this strategy: \"No sealing off of an enemy force could be successful unless the proportion of attackers to defenders was 15 to 1 or even 20 to 1, for the enemy had in its favor an intimate knowledge of the terrain, the advantages of defensive organization, and the sympathy of the population.\" But these proportions are frequently achieved in guerrilla war, and the \"surround and storm\" strategy would be eminently feasible were it not for a second and more serious difficulty.\n\nSince the villages are not (or should not be) destroyed when they are stormed, and since the villagers are not resettled, it is always possible for the guerrillas to return once the specially assembled task force has moved on. Success requires that the military operation be followed by a political campaign\u2014and this neither the French in Vietnam nor the Americans who followed them were able to mount in any serious fashion. The decision to destroy villages from a distance was a consequence of this failure, which is not at all the same thing as the \"deterioration\" of guerrilla into conventional war.\n\nAt some point in the military progress of the rebellion, or in the decline of the political capacity of the government that opposes it, it may well become impossible to fight the guerrillas at close quarters. There aren't enough men or, more likely, the government, though it can win particular battles, has no staying power. As soon as the fighting is over, the villagers welcome back the insurgent forces. Now the government (and its foreign allies) face what is in effect, or rather what has become, a people's war. This honorific name can be applied, however, only after the guerrilla movement has won very substantial popular support. It is by no means true all the time. One need only study Che Guevara's abortive campaign in the jungles of Bolivia to realize how easy it is to destroy a guerrilla band that has no popular support at all. From there, one might trace a continuum of increasing difficulty: at some point along that continuum, guerrilla fighters acquire war rights, and at some further point, the right of the government to continue the struggle must be called into question.\n\nThis last is not a point which soldiers are likely to recognize or acknowledge. For it is an axiom of the war convention (and a qualification on the rules of war) that if attack is morally possible, counter-attack cannot be ruled out. It cannot be the case that guerrillas can hug the civilian population and make themselves invulnerable. But if it is always morally possible to fight, it is not always possible to do whatever is required to win. In any struggle, conventional or unconventional, the rules of war may at some point become a hindrance to the victory of one side or another. If they could then be set aside, however, they would have no value at all. It is precisely then that the restraints they impose are most important. We can see this clearly in the Vietnam case. The alternative strategies I have briefly outlined were conceivably a way of winning (as the British won in Malaya) until the guerrillas consolidated their political base in the villages. That victory effectively ended the war. It is not, I suppose, a victory that can be distinguished in any definitive fashion from the political and military struggle that preceded it. But one can say with some assurance that it has occurred whenever ordinary soldiers (who are not moral monsters and would fight by the rules if they could) become convinced that old men and women and children are their enemies. For after that, it is unlikely that the war can be fought except by setting out systematically to kill civilians or to destroy their society and culture.\n\nI am inclined to say more than this. In the theory of war, as we have seen, considerations of jus ad bellum and jus in bello are logically independent, and the judgments we make in terms of one and the other are not necessarily the same. But here they come together. The war cannot be won, and it should not be won. It cannot be won, because the only available strategy involves a war against civilians; and it should not be won, because the degree of civilian support that rules out alternative strategies also makes the guerrillas the legitimate rulers of the country. The struggle against them is an unjust struggle as well as one that can only be carried on unjustly. Fought by foreigners, it is a war of aggression; if by a local regime alone, it is an act of tyranny. The position of the anti-guerrilla forces has become doubly untenable.\n\n The case is the same with the wearing of civilian clothing as with the wearing of enemy uniforms. In his memoir of the Boer War, Deneys Reitz reports that Boer guerrillas sometimes wore uniforms taken from British soldiers. Lord Kitchener, the British commander, warned that anyone captured in khaki would be shot, and a considerable number of prisoners were later executed. While he insists that \"none of us ever wore captured uniforms with the deliberate intention of decoying the enemy, but only out of sheer necessity,\" Reitz nevertheless justified Kitchener's order by telling of an incident in which two British soldiers were killed when they hesitated to shoot at guerrillas dressed in khaki. (Commando, London, 1932, p. 247.)\n\n The argument I am making here parallels that made by lawyers with reference to \"belligerent recognition.\" At what point, they have asked, should a group of rebels (or secessionists) be recognized as a belligerent power and granted those war rights which customarily belong only to established governments? The answer has usually been that the recognition follows upon the establishment of a secure territorial base by the rebels. For then they actually function like a government, taking on responsibility for the people who live on the land they control. But this assumes a conventional or near-conventional war. In the case of a guerrilla struggle, we may have to describe the appropriate relation between the rebels and the people differently: it is not when the guerrillas look after the people that they acquire war rights, but when the people \"look after\" the guerrillas.\n\nTerrorism\n\nThe Political Code\n\nThe word \"terrorism\" is used most often to describe revolutionary violence. That is a small victory for the champions of order, among whom the uses of terror are by no means unknown. The systematic terrorizing of whole populations is a strategy of both conventional and guerrilla war, and of established governments as well as radical movements. Its purpose is to destroy the morale of a nation or a class, to undercut its solidarity; its method is the random murder of innocent people. Randomness is the crucial feature of terrorist activity. If one wishes fear to spread and intensify over time, it is not desirable to kill specific people identified in some particular way with a regime, a party, or a policy. Death must come by chance to individual Frenchmen, or Germans, to Irish Protestants, or Jews, simply because they are Frenchmen or Germans, Protestants or Jews, until they feel themselves fatally exposed and demand that their governments negotiate for their safety.\n\nIn war, terrorism is a way of avoiding engagement with the enemy army. It represents an extreme form of the strategy of the \"indirect approach.\" It is so indirect that many soldiers have refused to call it war at all. This is a matter as much of professional pride as of moral judgment. Consider the statement of a British admiral in World War II, protesting the terror bombing of German cities: \"We are a hopelessly unmilitary nation to imagine that we [can] win the war by bombing German women and children instead of defeating their army and navy.\" The key word here is unmilitary. The admiral rightly sees terrorism as a civilian strategy. One might say that it represents the continuation of war by political means. Terrorizing ordinary men and women is first of all the work of domestic tyranny, as Aristotle wrote: \"The first aim and end [of tyrants] is to break the spirit of their subjects.\" The British described the \"aim and end\" of terror bombing in the same way: what they sought was the destruction of civilian morale.\n\nTyrants taught the method to soldiers, and soldiers to modern revolutionaries. That is a crude history; I offer it only in order to make a more precise historical point: that terrorism in the strict sense, the random murder of innocent people, emerged as a strategy of revolutionary struggle only in the period after World War II, that is, only after it had become a feature of conventional war. In both cases, in war and revolution, a kind of warrior honor stood in the way of this development, especially among professional officers and \"professional revolutionaries.\" The increasing use of terror by far left and ultranationalist movements represents the breakdown of a political code first worked out in the second half of the nineteenth century and roughly analogous to the laws of war worked out at the same time. Adherence to this code did not prevent revolutionary militants from being called terrorists, but in fact the violence they committed bore little resemblance to contemporary terrorism. It was not random murder but assassination, and it involved the drawing of a line that we will have little difficulty recognizing as the political parallel of the line that marks off combatants from noncombatants.\n\nThe Russian Populists, the IRA, and the Stern Gang\n\nI can best describe the revolutionary \"code of honor\" by giving some examples of so-called terrorists who acted or tried to act in accordance with its norms. I have chosen three historical cases. The first will be readily recognizable, for Albert Camus made it the basis of his play The Just Assassins.\n\n1) In the early twentieth century, a group of Russian revolutionaries decided to kill a Tsarist official, the Grand Duke Sergei, a man personally involved in the repression of radical activity. They planned to blow him up in his carriage, and on the appointed day one of their number was in place along the Grand Duke's usual route. As the carriage drew near, the young revolutionary, a bomb hidden under his coat, noticed that his victim was not alone; on his lap he held two small children. The would-be assassin looked, hesitated, then walked quickly away. He would wait for another occasion. Camus has one of his comrades say, accepting this decision: \"Even in destruction, there's a right way and a wrong way\u2014and there are limits.\"\n\n2) During the years 1938\u201339, the Irish Republican Army waged a bombing campaign in Britain. In the course of this campaign, a republican militant was ordered to carry a pre-set time bomb to a Coventry power station. He traveled by bicycle, the bomb in his basket, took a wrong turn, and got lost in a maze of streets. As the time for the explosion drew near, he panicked, dropped his bike, and ran off. The bomb exploded, killing five passers-by. No one in the IRA (as it was then) thought this a victory for the cause; the men immediately involved were horrified. The campaign had been carefully planned, according to a recent historian, so as to avoid the killing of innocent bystanders.\n\n3) In November 1944, Lord Moyne, British Minister of State in the Middle East, was assassinated in Cairo by two members of the Stern Gang, a right-wing Zionist group. The two assassins were caught, minutes later, by an Egyptian policeman. One of them described the capture at his trial: \"We were being followed by the constable on his motorcycle. My comrade was behind me. I saw the constable approach him. . . . I would have been able to kill the constable easily, but I contented myself with . . . shooting several times into the air. I saw my comrade fall off his bicycle. The constable was almost upon him. Again, I could have eliminated the constable with a single bullet, but I did not. Then I was caught.\"\n\nWhat is common to these cases is a moral distinction, drawn by the \"terrorists,\" between people who can and people who cannot be killed. The first category is not composed of men and women bearing arms, immediately threatening by virtue of their military training and commitment. It is composed instead of officials, the political agents of regimes thought to be oppressive. Such people, of course, are protected by the war convention and by positive international law. Characteristically (and not foolishly), lawyers have frowned on assassination, and political officials have been assigned to the class of nonmilitary persons, who are never the legitimate objects of attack. But this assignment only partially represents our common moral judgments. For we judge the assassin by his victim, and when the victim is Hitler-like in character, we are likely to praise the assassin's work, though we still do not call him a soldier. The second category is less problematic: ordinary citizens, not engaged in political harming\u2014that is, in administering or enforcing laws thought to be unjust\u2014are immune from attack whether or not they support those laws. Thus the aristocratic children, the Coventry pedestrians, even the Egyptian policeman (who had nothing to do with British imperialism in Palestine)\u2014these people are like civilians in wartime. They are innocent politically as civilians are innocent militarily. It is precisely these people, however, that contemporary terrorists try to kill.\n\nThe war convention and the political code are structurally similar, and the distinction between officials and citizens parallels that between soldiers and civilians (though the two are not the same). What lies behind them both, I think, and lends them plausibility, is the moral difference between aiming and not aiming\u2014or, more accurately, between aiming at particular people because of things they have done or are doing, and aiming at whole groups of people, indiscriminately, because of who they are. The first kind of aiming is appropriate to a limited struggle directed against regimes and policies. The second reaches beyond all limits; it is infinitely threatening to whole peoples, whose individual members are systematically exposed to violent death at any and every moment in the course of their (largely innocuous) lives. A bomb planted on a streetcorner, hidden in a bus station, thrown into a caf\u00e9 or pub\u2014this is aimless killing, except that the victims are likely to share what they cannot avoid, a collective identity. Since some of these victims must be immune from attack (unless liability follows from original sin), any code that directs and controls the fire of political militants is going to be at least minimally appealing. It is so much of an advance over the willful randomness of terrorist attacks. One might even feel easier about killing officials than about killing soldiers, since the state rarely conscripts its political, as it does its military, agents; they have chosen officialdom as a career.\n\nSoldiers and officials are, however, different in another respect. The threatening character of the soldier's activities is a matter of fact; the unjust or oppressive character of the official's activities is a matter of political judgment. For this reason, the political code has never attained to the same status as the war convention. Nor can assassins claim any rights, even on the basis of the strictest adherence to its principles. In the eyes of those of us whose judgments of oppression and injustice differ from their own, political assassins are simply murderers, exactly like the killers of ordinary citizens. The case is not the same with soldiers, who are not judged politically at all and who are called murderers only when they kill noncombatants. Political killing imposes risks quite unlike those of combat, risks whose character is best revealed by the fact that there is no such thing as benevolent quarantine for the duration of the political struggle. Thus the young Russian revolutionary, who eventually killed the Grand Duke, was tried and executed for murder, as were the Stern Gang assassins of Lord Moyne. All three were treated exactly like the IRA militants, also captured, who were held responsible for the deaths of ordinary citizens. That treatment seems to me appropriate, even if we share the political judgments of the men involved and defend their resort to violence. On the other hand, even if we do not share their judgments, these men are entitled to a kind of moral respect not due to terrorists, because they set limits to their actions.\n\nThe Vietcong Assassination Campaign\n\nThe precise limits are hard to define, as in the case of noncombatant immunity. But we can perhaps move toward a definition by looking at a guerrilla war in which officials were attacked on a large scale. Beginning at some point in the late 1950s, the NLF waged a campaign aimed at destroying the governmental structure of the South Vietnamese countryside. Between 1960 and 1965, some 7,500 village and district officials were assassinated by Vietcong militants. An American student of the Vietcong, describing these officials as the \"natural leaders\" of Vietnamese society, argues that \"by any definition this NLF action . . . amounts to genocide.\" This assumes that all Vietnam's natural leaders were government officials (but then, who was leading the NLF?) and hence that government officials were literally indispensable to national existence. Since these assumptions are not remotely plausible, it has to be said that \"by any definition\" the killing of leaders is not the same as the destruction of entire peoples. Terrorism may foreshadow genocide, but assassination does not.\n\nOn the other hand, the NLF campaign did press against the limits of the notion of officialdom as I have been using it. The Front tended to include among officials anyone who was paid by the government, even if the work he was doing\u2014as a public health officer, for example\u2014had nothing to do with the particular policies the NLF opposed. And it tended to assimilate into officialdom people like priests and landowners who used their nongovernmental authority in specific ways on behalf of the government. They did not kill anyone, apparently, just because he was a priest or a landowner; the assassination campaign was planned with considerable attention to the details of individual action, and a concerted effort was made \"to ensure that there were no unexplained killings.\" Still, the range of vulnerability was widened in disturbing ways.\n\nOne might argue, I suppose, that any official is by definition engaged in the political efforts of the (putatively) unjust regime, just as any soldier, whether he is actually fighting or not, is engaged in the war effort. But the variety of activities sponsored and paid for by the modern state is extraordinary, and it seems intemperate and extravagant to make all such activities into occasions for assassination. Assuming that the regime is in fact oppressive, one should look for agents of oppression and not simply for government agents. As for private persons, they seem to me immune entirely. They are subject, of course, to the conventional forms of social and political pressure (which are conventionally intensified in guerrilla wars) but not to political violence. Here the case is the same with citizens as with civilians: if their support for the government or the war were allowable as a reason for killing them, the line that marks off immune from vulnerable persons would quickly disappear. It is worth stressing that political assassins generally don't want that line to disappear; they have reasons for taking careful aim and avoiding indiscriminate murder. \"We were told,\" a Vietcong guerrilla reported to his American captors, \"that in Singapore the rebels on certain days would dynamite every 67th streetcar . . . the next day it might be every 30th, and so on; but that this hardened the hearts of the people against the rebels because so many people died needlessly.\"\n\nI have avoided noticing until now that most political militants don't regard themselves as assassins at all but rather as executioners. They are engaged, or so they regularly claim, in a revolutionary version of vigilante justice. This suggests another reason for killing only some officials and not others, but it is entirely a self-description. Vigilantes in the usual sense apply conventional conceptions of criminality, though in a rough and ready way. Revolutionaries champion a new conception, about which there is unlikely to be wide agreement. They hold that officials are vulnerable because or insofar as they are actually guilty of \"crimes against the people.\" The more impersonal truth is that they are vulnerable, or more vulnerable than ordinary citizens, simply because their activities are open to such descriptions. The exercise of political power is a dangerous business. Saying this, I do not mean to defend assassination. It is most often a vile politics, as vigilante justice is most often a bad kind of law enforcement; its agents are usually gangsters, and sometimes madmen, in political dress. And yet \"just assassinations\" are at least possible, and men and women who aim at that kind of killing and renounce every other kind need to be marked off from those who kill at random\u2014not as doers of justice, necessarily, for one can disagree about that, but as revolutionaries with honor. They do not want the revolution, as one of Camus' characters says, \"to be loathed by the whole human race.\"\n\nHowever the political code is specified, terrorism is the deliberate violation of its norms. For ordinary citizens are killed and no defense is offered\u2014none could be offered\u2014in terms of their individual activities. The names and occupations of the dead are not known in advance; they are killed simply to deliver a message of fear to others like themselves. What is the content of the message? I suppose it could be anything at all; but in practice terrorism, because it is directed against entire peoples or classes, tends to communicate the most extreme and brutal intentions\u2014above all, the tyrannical repression, removal, or mass murder of the population under attack. Hence contemporary terrorist campaigns are most often focused on people whose national existence has been radically devalued: the Protestants of Northern Ireland, the Jews of Israel, and so on. The campaign announces the devaluation. That is why the people under attack are so unlikely to believe that compromise is possible with their enemies. In war, terrorism is associated with the demand for unconditional surrender and, in similar fashion, tends to rule out any sort of compromise settlement.\n\nIn its modern manifestations, terror is the totalitarian form of war and politics. It shatters the war convention and the political code. It breaks across moral limits beyond which no further limitation seems possible, for within the categories of civilian and citizen, there isn't any smaller group for which immunity might be claimed (except children; but I don't think children can be called \"immune\" if their parents are attacked and killed). Terrorists anyway make no such claim; they kill anybody. Despite this, terrorism has been defended, not only by the terrorists themselves, but also by philosophical apologists writing on their behalf. The political defenses mostly parallel those that are offered whenever soldiers attack civilians. They represent one or another version of the argument from military necessity.a It is said, for example, that there is no alternative to terrorist activity if oppressed peoples are to be liberated. And it is said, further, that this has always been so: terrorism is the only means and so it is the ordinary means of destroying oppressive regimes and founding new nations. The cases I have already worked through suggest the falsity of these assertions. Those who make them, I think, have lost their grip on the historical past; they suffer from a malign forgetfulness, erasing all moral distinctions along with the men and women who painfully worked them out.\n\nViolence and Liberation\n\nJean-Paul Sartre and the Battle of Algiers\n\nBut there is another argument which, because of the currency it has gained, must be taken up here, even though it has no immediate analogue in wartime debates. It has been put forward in its starkest form by Sartre in a justification of FLN terrorism in Algeria, published as a preface to Franz Fanon's The Wretched of the Earth. The summary lines of Sartre's argument are these:\n\nTo shoot down a European is to kill two birds with one stone, to destroy an oppressor and the man he oppresses at the same time: there remains a dead man and a free man.\n\nIn his usual fashion, with a certain zest for Hegelian melodrama, Sartre is here describing what he takes to be an act of psychological liberation. Only when the slave turns on his master, physically confronts him and kills him, does he create himself as a free human being. The master dies; the slave is reborn. Even if this were a believable picture of the terrorist act, the argument is not persuasive; it is open to two obvious and crippling questions. First, is the one-to-one relation necessary? Did it take one dead European to make one free Algerian? If so, there were not enough Europeans living in Algeria; more would have had to be brought over if the Algerian people were to free themselves by Sartrean means. If not, it must follow that someone else besides the man-who-kills can be liberated. . . . How? By watching? By reading about the murder in the newspaper? It is hard to see how vicarious experience can play an important part in a process of personal liberation (as described by an existentialist philosopher).\n\nThe second question raises more familiar issues: will any European do? Unless Sartre thinks all Europeans, including children, are oppressors, he cannot believe that. But if it is only liberating to attack and kill an agent of oppression, we are back with the political code. From Sartre's perspective, that cannot be right, since the men and women he is defending had explicitly rejected that code. They killed Europeans at random, as in the well-known scene from the (historically accurate) film The Battle of Algiers, in which a bomb is set off in a milk bar where French teenagers are drinking and dancing.\n\nMILK BAR. EXPLOSION. OUTSIDE. DAY.\n\nThe jukebox is flung into the middle of the street. There is blood, strips of flesh, material . . . the white smoke and shouts, weeping, hysterical girls' screams. One of them no longer has an arm and runs around howling despairingly; it is impossible to control her. . . . The sound of sirens is heard. . . . The ambulances arrive . . .\n\nSuch an event is not easily reconstructed as an existentialist encounter between masters and slaves.\n\nCertainly, there are historical moments when armed struggle is necessary for the sake of human freedom. But if dignity and self-respect are to be the outcomes of that struggle, it cannot consist of terrorist attacks upon children. One can argue that such attacks are the inevitable products of oppression, and in a sense, I suppose, that is right. Hatred, fear, and the lust for domination are the psychological marks of oppressed and oppressor alike, and their acting out, on either side, can be said to be radically determined. The mark of a revolutionary struggle against oppression, however, is not this incapacitating rage and random violence, but restraint and self-control. The revolutionary reveals his freedom in the same way as he earns it, by directly confronting his enemies and refraining from attacks on anyone else. It was not only to save the innocent that revolutionary militants worked out the distinction between officials and ordinary citizens, but also to save themselves from killing the innocent. Whatever its strategic value, the political code is intrinsically connected to psychological liberation. Among men and women trapped in a bloody struggle, it is the key to self-respect. The same thing can be said of the war convention: in the context of a terrible coerciveness, soldiers most clearly assert their freedom when they obey the moral law.\n\n Among revolutionaries as among government officials, this argument often slides from an analysis of particular cases of duress and necessity (which are rarely convincing) to the general claim that war is hell and anything goes. General Sherman's view is upheld, for example, by the Italian leftist Franco Solinas, who wrote the screenplay for Pontecorvo's The Battle of Algiers and defended the terrorism of the Algerian FLN: \"For centuries they've tried to prove that war is fair play, like duels, but war isn't and therefore any method used to fight it is good. . . . It's not a question of ethics or fair play. What we must attack is war itself and the situations that lead to it.\" (The Battle of Algiers, edited and translated by PierNico Solinas, New York, 1973, pp. 195\u201396.) Compare the same argument made by American officials in defense of the bombing of Hiroshima, chapter 16.\n\nReprisals\n\nDeterrence Without Retribution\n\nWhen the British imposed their blockade of Germany in 1916, they called it a reprisal; when the Germans began the systematic bombing of London in 1940, they defended themselves in the same way. No part of the war convention is so open to abuse, is so openly abused, as the doctrine of reprisals. For the doctrine is, or once was thought to be, permissive with regard to all the rest of the convention. It legitimates actions otherwise criminal, if these actions are undertaken in response to crimes previously committed by the enemy. \"Reprisals,\" writes a pacifist critic of the rules of war, \"mean doing what you think wrong on the plea that someone else did it first.\" And, he goes on, someone else will always do it first. Hence reprisals create a chain of wrongdoing at the end of which every responsible actor can point to some other actor and say \"tu quoque.\"\n\nIt is the explicit purpose of reprisals, however, to break off the chain, to stop the wrongdoing here, with this final act. Sometimes\u2014though it has to be said, not often\u2014that purpose is realized. I want to begin with a case in which it was realized, so that we can at least make sense of what was for many years the conventional opinion\u2014as stated, for example, by a nineteenth-\u00adcentury French lawyer: \"Reprisals are a means of preventing war from becoming entirely barbarous.\"\n\nThe FFI Prisoners at Annecy\n\nIn the summer of 1944, much of France was a battleground. Allied armies were fighting in Normandy; partisan groups, organized now into the French Forces of the Interior and in touch with both the Allies and the Gaullist Provisional Government in Algeria, operated on a large scale in many parts of the country. They wore insignias of battle; they bore their arms openly. It is clear that the 1940 armistice had effectively been voided, and the military struggle resumed. Nevertheless, the German authorities continued to treat captured partisans as war traitors or war rebels, subject to summary execution. On the day after the Allied landings, for example, fifteen partisans captured at Caen were immediately shot. And the executions continued, as the pace of the fighting increased, during the next months. The FFI complained of these executions to the Provisional Government, which in turn sent a formal protest to the Germans. Since they did not recognize the Government, the Germans refused to accept the protest. In their note, the French had threatened reprisals against German prisoners. The continued killing did not, however, elicit any such response\u2014perhaps because troops directly subject to the Provisional Government, recruited outside occupied France, were regularly accorded prisoner-\u00adof-war status by the Germans.\n\nIn August 1944, large numbers of German soldiers in Southern France began surrendering to partisan groups, and the FFI leadership was suddenly in a position to carry out the Government's threat. \"When . . . it became known that the Germans . . . had executed 80 French prisoners, and that further executions were imminent, the FFI command at Annecy decided that 80 of the prisoners in its] hands would in turn be shot.\" At this point, the Red Cross intervened, won a postponement of the executions, and sought from the Germans an agreement henceforth to treat captured partisans as prisoners of war. The partisans waited six days and then, the Germans not replying, the 80 prisoners were shot.[a The effects of the reprisal are not easy to make out, for the German army was hard-pressed, and many other factors must have figured in its decisions. It is apparently true, however, that no partisans were executed after the Annecy shootings.\n\nNow in one sense, this case is easy to judge: the Geneva Convention of 1929, which the French had signed and the FFI itself reaffirmed, explicitly barred reprisals against prisoners of war. No other group of innocent men and women was granted a similar immunity; prisoners were singled out because of the contract implied by surrender, in which they are promised life and benevolent quarantine. Killing them would be a breach of faith as well as a violation of the positive laws of war. But I shall not focus on this exception to the general rule of reprisals, for it does not open up the larger question, whether the deliberate killing of innocent men and women should ever be declared lawful or morally justified. And I doubt very much that we will want to say, in answer to that question, that some innocent people can be killed and others not. The case of the FFI prisoners is useful because it provides a classic example of reprisal, and one in which our sympathies are likely to be engaged, at least initially, on the side of the \"reprisers.\"\n\nReprisals of this sort have as their purpose the enforcement of the war convention. In international society, as in Locke's state of nature, every individual member (every belligerent power) claims the right to enforce the law. The content of this right is the same as it is in domestic society: it is first of all a right of retribution, to punish guilty men and women; it is secondly a right of deterrence, to protect oneself and others against criminal activity. In domestic society, these two most often go together. Criminal activity is deterred by punishing or threatening to punish guilty individuals. That, at least, is the commonly accepted doctrine. In international society, however, and especially in wartime, the two rights are not equally enforceable. It is often impossible to get at guilty individuals, but it's always possible to prevent or try to prevent further criminal activity by responding in kind as the French partisans did, that is, by \"punishing\" innocent people. The result might be described as a one-sided sort of law enforcement: deterrence without retribution.\n\nIt might also be described as a prime example of radical utilitarianism\u2014indeed, of a utilitarianism so radical that utilitarian philosophers have been concerned to deny its existence. Yet it is common enough in the theory as well as in the practice of war. One of the criticisms most frequently leveled against utilitarianism is that its calculations would under certain circumstances require the authorities to \"punish\" an innocent person (to kill or imprison him, under cover of punishment). The usual response has been to adjust the calculations so that they yield different and more conventionally acceptable results. But in the history of international law and in debates over wartime behavior, the effort at adjustment has mostly been foregone. Reprisals have been defended, with admirable directness, on strictly utilitarian grounds. Under the special conditions of combat, at least, utilitarian calculations have indeed required the \"punishing\" of innocent people. The political or military leaders of belligerent powers have commonly invoked the requirement, claiming that no other means were available to check the criminal excesses of their opponents. And detached observers, students of the law, and venerable doctors have generally accepted this as a possible argument \"in extreme cases\" (the cases, of course, are often disputed). Hence it is a \"principle of war law,\" according to a leading authority: \"For every offense punish someone; the guilty, if possible, but someone.\"\n\nThis is not an attractive principle, and it would not be accurate to explain the traditional acceptance of reprisals by reference to it alone. In wartime, after all, innocent people are often attacked and killed in the name of utility, in order, it is said, to shorten the war, save lives, and so on. But such attacks don't have the same status as reprisals. It is not their utility, assuming now that they are in fact useful, that makes reprisals different, but some other quality. This quality is misunderstood, I think, by those writers who describe reprisal as the most primitive feature of the war convention, a survival of the ancient lex talionis. For the talion is a return of evil for evil, and what is crucial about reprisal is precisely that evil, though it may be repeated, is not returned. The new crime has a new victim, who is not the original criminal though he probably has the same nationality. The particular choice is (so far as utility goes) quite impersonal; in this sense, reprisal is chillingly modern. Something, however, of the talion survives: not the idea of return, but the idea of response. Reprisal is characterized by a certain posture of looking back, acting after, which implies a willingness not to act at all, to abide by some set of restraints. \"They did it first.\" This sentence carries a moral argument. I do not believe that it is a very strong argument or one that will take us far. But it serves to mark off reprisal from other, equally useful violations of the war convention. There is no right to commit crimes in order to shorten a war, but there is a right, so it was once thought, to commit crimes (or rather, acts that would otherwise be called crimes) in order to cope with the previous criminal activity of one's enemies.\n\nThe backward-looking character of reprisals is confirmed by the rule of proportionality that restrains them. The rule is quite different and far more precise than that which figures, for example, in the doctrine of double effect. The partisan commanders at Annecy acted in strict accordance with its provisions when they decided to kill 80 Germans in response to the killing of 80 Frenchmen. Reprisals are limited with reference to previous crimes, not with reference to the crimes they are designed to deter (not with reference to their effects or their hoped-for effects). This point has sometimes been disputed by writers committed to utilitarian modes of thought. Thus McDougal and Feliciano argue, in characteristic style, \"that the kind and amount of permissible . . . violence is that which is reasonably designed so to affect the enemy's expectations about the costs and gains of reiteration or continuation of his initial criminal act as to induce the termination of and future abstention from such act.\" They admit that the amount of violence, so determined, may be greater than that originally inflicted by the enemy. In the Annecy case, it might well have been less: the shooting of 40 Germans, or 20, or 10, might have had the same effect as the shooting of 80. But however the calculations work out, this kind of forward-looking proportionality has never been accepted either by the general run of theorists writing about war or by ordinary practitioners. During World War II, to be sure, the Germans often responded to partisan activity in the occupied states of Europe by shooting ten hostages for every German killed. This proportion may have reflected a peculiar notion about the relative value of German lives, or it may have been \"reasonably designed so to affect the enemy's expectations, etc.\" In any case, the practice was universally condemned.\n\nIt was condemned, of course, not only because of the actual disproportion involved, but also because the previous partisan activity was in many cases not thought to violate the war convention. Hence the German response was simply utilitarian deterrence, not law enforcement. It is another feature of the backward-looking character of reprisals that the acts to which they respond must be crimes, violations of the recognized rules of war. Moreover, the rules must be commonly recognized, on both sides of the battleline, if the special character of reprisals is to be maintained. When the British army resorted to reprisals during the War of 1812, an opposition member of the House of Commons, who thought such conduct barbarous, asked why His Majesty's soldiers didn't scalp their captives when they fought with the American Indians or enslave them in their wars with the Barbary corsairs. I suppose the answer is that scalping and enslavement were not thought illegitimate by the Indians and the corsairs. And so the imitation of these practices by the British would not have been understood as law enforcement (nor would it have had any deterrent effect); it would only have confirmed their enemies' notions of appropriate wartime behavior. Reprisals may involve deterrence without retribution, but this must nevertheless be a reactive deterrence, and what it reacts to is a violation of the war convention. If there is no convention, there can be no reprisal.\n\nAt the same time, we are uneasy about reprisals precisely because there is a convention, and one that categorically rules out the acts that reprisal usually requires. If it is wrong, and for the deepest reasons, to kill innocent people, how can it be right to kill them? In treatises on international law, the defense of reprisal is always qualified, first by a great show of reluctance and anxiety, and secondly by some words about the extremity of the case. It is not easy to know what this last qualification means, however, and it appears in fact that any violation of the rules is sufficiently \"extreme\" to justify a proportionate response. Backward-looking proportionality is a genuine limit: it would have barred, for example, the two so-called reprisals with which I began this chapter. But extremity is not a limit at all. It is certainly not true that reprisals are undertaken only when the enemy's crimes pose a drastic danger to the war effort as a whole or to the cause for which the war is being fought. For the purpose of reprisal is not to win the war or prevent the defeat of the cause, but simply to enforce the rules. Perhaps the meaning of the appeal to extremity is like that of the show of reluctance: both suggest a view of reprisal as a last resort. In practice, again, the only action required before one reaches this last resort is a formal protest, such as the French delivered to the Germans in 1944, and a threat to respond in kind if this or that criminal activity is continued. But one might require much more than that, both in the way of law enforcement and in the way of military action. The FFI might, for example, have announced that they would treat German soldiers involved in the execution of captured partisans as war criminals; they might even have begun to publish the names of those who would be accused. Given the military situation of the German army in 1944, such an announcement could well have had a significant effect. Or the partisans might have attempted to raid the prisons or camps where their comrades were being held. Such raids were not impossible, though they would have involved risks entirely absent when one shoots down captured soldiers.\n\nIf the notion of last resort were taken seriously, it would limit reprisal in a radical way. But suppose that the partisans had issued the announcement and undertaken the raids without stopping the German executions. Would they then have been justified in shooting their prisoners? \"A reckless enemy often leaves his opponent no other means of securing himself against the repetition of barbarous outrage.\" But the truth is that there are always other means, more or less dangerous, more or less effective. To argue against the executions isn't to deny the partisans a last resort. It is only to say, for example, that military raids are their last resort. If the raids fail, they can only be tried again; there is nothing more to be done. (Reprisals might fail, too\u2014they usually do\u2014and what comes after that?) This is the conclusion that I want to defend, and I will defend it, once again, by reflecting on the status and character of the German prisoners.\n\nWho are these men? Once they were soldiers; now they are disarmed and helpless. Perhaps some of them are war criminals; perhaps some of them were involved in the murder of captured partisans. Then, surely, they should be put on trial, not shot out of hand. We will want to hear the evidence against them and make sure that we punish the right ones. Only a trial can signal our own commitment to the rules of war. But here, let us assume, are ordinary prisoners who neither made nor carried out criminal decisions. Their day-to-day activities were very much like those of their enemies. How can they be shot out of hand, treated more cruelly than we would treat suspected criminals? It seems incredible that some number of them should be arbitrarily separated from the rest and then killed, simply so that we can announce their deaths, and all this for the sake of justice! Killing them would be murder: the name is exact, no matter what crimes we hope to avoid by becoming murderers. For these men are not mere material out of whose lives we can fashion a deterrent strategy. Even as prisoners, or precisely as prisoners, they have rights against us.\n\nThe current thrust of international law is to condemn reprisals against innocent people, and for essentially the reasons that I have suggested: the helplessness of the victims rules them out as objects of military attack, and their noninvolvement in criminal activity rules them out as objects of retributive violence. The Geneva Convention of 1929, as we have seen, declared prisoners immune; the 1949 Conventions did the same for wounded, sick, and shipwrecked members of the armed forces and for civilian persons in occupied territory. This last provision effectively bars the killing of hostages, the paradigm case of using innocent people for one's own military purposes. The only class of disengaged men and women against whom reprisals are still legally defensible is the civilian population of the enemy country. Its members can still be held hostage, though only at a distance, for the good behavior of their government and army. It has been argued that this way of judging reprisals is a logical extension of the general principle \"that persons whose usefulness as bases of enemy power is precluded . . . by belligerent control or capture cease to be legitimate objects of violence.\" But this is to misstate the general principle. It would allow not only reprisals but also first strikes against enemy civilians. However peaceful their pursuits, after all, these civilians remain a \"significant base of enemy power,\" providing political and economic support to the armed forces. Even children are not \"precluded\" from serving that power: they will grow up to be soldiers, munitions workers, and so on. Yet such people are protected by the war convention; they are admitted, along with prisoners and wounded soldiers, to the class of the innocent. The underlying purpose of recent developments in the law is not to extend a general principle, which is already (in principle) fully extended, but to prohibit its violation in the special circumstances once thought to justify reprisals. And if there are good reasons for doing that, there would seem to be no good reasons for drawing the line as it has currently been drawn.b\n\nSo the necessary judgment is readily summed up: we must condemn all reprisals against innocent people, whether these people are \"subject to belligerent control\" or not. This is to set radical limits to a practice that once was commonly defended, and not with casual or inconsequential arguments. But I don't want to claim that those old arguments have no force at all. They correctly point to a certain moral difference between the initial crime and the reprisal-response. From a position of great detachment, these two may seem to constitute a vicious circle\u2014and a circle fully accounted for by the pious maxim that \"violence breeds violence.\" The maxim, however, is sometimes wrong and, what is more important, it fails to distinguish violence that is responsive and restrained from violence that is neither. Stand beside the French commanders at Annecy and the circle looks different. German guilt in this case is greater than that of the French, because the Germans acted first, breaking the conventional rules for some military advantage; the French reacted, repeating the violations for the declared purpose of re-establishing the rules. I don't know how to measure the difference between them; perhaps it isn't great; but it is worth stressing that there is a difference, even as we give their crimes a common name.\n\nWith regard to the most important of the rules of war, the violation of the rules for the sake of law enforcement is ruled out. The doctrine of reprisal, then, refers only to the lesser parts of the war convention, where the rights of the innocent are not at stake. Consider, for example, the ban on the use of poison gas. Winston Churchill was entirely justified when he warned the German government, early in World War II, that the use of gas by its armies would bring an immediate Allied reprisal. For soldiers have only a war right, and no more basic right, to be attacked with certain weapons and not with others. The rule about poison gas is legally established, but it is not morally required. Hence, when it is violated, parallel and proportionate violations, narrowly aimed at re-establishing the rule and at no larger military purpose, are morally permissible. They are permissible because the people against whom they are directed are already the legitimate objects of military attack. The case is the same with all those informal agreements and reciprocal arrangements that limit the extent and intensity of warfare. Here the threat of reprisal is the major means of enforcement, and there is no reason to hesitate about making the threat or carrying it out. It might be argued that when restraints of this sort are violated, they simply disappear, and then there is no reason to limit one's own violations by attending to the proportionality rule. But that is true only if reprisal fails to restore the old limits. One must aim first at restoration: in that sense, we still use reprisals as a bar to the barbarism of war.\n\nThe Problem of Peacetime Reprisals\n\nBut all this assumes that warfare of the ordinary sort is already in progress. What is at issue is the mode or means of attack. In the case of peacetime reprisals, what is at issue is the attack itself. It has come from across the border: a raid of one sort or another. The victim state responds with a second raid, which isn't aimed at re-affirming the rules of war but at re-\u00adestablishing the broken peace. The crime that is repeated is the act of force, the violation of sovereignty. It will be called aggression and justified as self-defense\u2014talked about, that is, in the language of jus ad bellum\u2014but it remains a \"military measure short of war\" as long as the restraints appropriate to reprisals, established by the theory of jus in bello, are maintained. And so it is best discussed here, with reference to those restraints.\n\nThe Attack on Khibye and the Beirut Raid\n\nThe term \"peacetime reprisals\" is not entirely accurate. The legal handbooks divide their subject into \"war\" and \"peace,\" but much of history is a demi-monde that neither word adequately describes. It is to this demi-monde that reprisals most commonly pertain; they are a form of action appropriate to periods of insurgency, border strife, cease-fire, and armistice. Now it is a feature of such periods that acts of force are not always acts of state in any simple sense. They are not the work of recognized officials and of soldiers acting on official orders, but (often) of guerrilla bands and terrorist organizations\u2014tolerated, perhaps patronized by the officials, but not directly subject to their control. Thus Israel, since its founding in 1948, has repeatedly been attacked by Palestinian guerrillas and terrorists operating out of the neighboring Arab states but not formally affiliated with their armies. In response to these attacks, the Israeli authorities have tried over the years virtually every conceivable form of counter-attack\u2014testing out, as it were, the politics and morality of reprisal. It is a grim and unusual history, providing the theorist with all the examples he could want (and more). And if it doesn't suggest that peacetime reprisals make for peace, it also doesn't point to any alternative response to illegitimate attacks.\n\nMost of the Palestinian raids have been the work of terrorists, not guerrillas; that is, following the argument of the last two chapters, they have been directed randomly against civilian targets: against farmers working near the border, buses on country roads, village schools and houses, and so on. Hence there is no question about their illegitimacy, whatever one thinks of the larger Arab-Israeli conflict. Nor can there be any question that the Israelis have a right to respond in some way. The right exists in the case of any across-the-border raid, but it is especially clear when the raid is aimed at civilians, who can offer no immediate resistance. Nevertheless, particular Israeli responses have indeed been questionable, for it is a hard matter to know what to do in such cases. Terrorists harbored by neighboring states with which one is not openly at war do not provide an easy target. Any military response will be marked by a kind of asymmetry characteristic of peacetime reprisal: the initial foray is unofficial; the counter-attack is the act of a sovereign state, challenging the sovereignty of another state. How do we judge such challenges? What are the rules that govern peacetime reprisals?\n\nThe first rule is a familiar one. Though the terrorist raid is aimed at civilians, the reprisal must not be so aimed. Moreover, the \"reprisers\" must take care that civilians are not the incidental victims of their attack. With regard to its conduct, peacetime reprisal is exactly like war itself, and so certain of our judgments are obvious enough. Consider, for example, the Israeli raid on Khibye:\n\nFollowing the killing of a woman and her two children in a village near Lod Airport, the Israelis launched a night attack against the Jordanian village of Khibye on 14 October 1953. . . . [They] fought their way into the village, rounded up the inhabitants, and blew up forty-\u00adfive houses. Not all the houses were cleared beforehand, and more than forty villagers were buried under the rubble. . . . The brutality of the raid led to sharp protests in Israel and abroad. . . .\n\nThese killings probably cannot be called \"unintended,\" and it certainly cannot be said that due care was taken to avoid them; so the protests were justified; the killings were criminal. But what if no civilians had died, or, as in most on-the-ground Israeli reprisals, only a small number, killed in the course of a firefight with Jordanian regulars? What are we to say of the raid itself, of the Jordanian soldiers killed in its course (who had no part in the murder of Israeli civilians), of the houses destroyed? This is not a standard military operation, though it is the most common form of peacetime reprisal. Its purpose is coercive: to force the officials of a neighboring state to keep the peace and to repress guerrillas and terrorists on their own side of the border. But it is not directly or continuously coercive; otherwise it would require a full-scale invasion. Reprisals have the form of a warning: if our villages are attacked, yours will also be attacked. Hence they must always respond to previous raids. And they are governed, after the rule of noncombatant immunity, by the rule of backward-looking proportionality. Though life cannot be balanced against life, the second raid must be similar in character and scope to the first.\n\nI am inclined to defend counter-attacks of this sort, when these two restraints are accepted. The defense, I should stress, doesn't depend in any way upon the notions of extremity or last resort. In peacetime, war is the last resort (and a long series of terrorist raids might justify a war, if no other means seemed likely to end the series). Reprisal is a first resort to force, once diplomacy has proven ineffective. It is, again, a \"military measure short of war,\" an alternative to war, and that description is an important argument in its favor. But the general argument remains a difficult one, as we can see if we turn to another historical example, where (in contrast to Khibye) the rules of immunity and proportionality were scrupulously respected.\n\nIn 1968, the focus of Palestinian terrorism shifted from Israel itself to the Israeli national airline and its passengers. On December 26 of that year, two terrorists attacked an Israeli plane preparing for takeoff at Athens Airport. Some 50 people were aboard at the time and, although only one was killed, it was clearly the purpose of the terrorists to kill as many as possible. They aimed their guns at the windows of the plane, at seat level. The two men were captured by Athenian police, and it was discovered that they were members of the Popular Front for the Liberation of Palestine, an organization with headquarters in Beirut. They were traveling on Lebanese documents. Repeatedly over the previous months, Israel had warned the Lebanese government that it could not \"escape responsibility\" for its support of groups like the PFLP. Now the Israelis undertook a dramatic reprisal.\n\nTwo days after the Athens attack, Israeli commandos landed by helicopter at Beirut Airport and destroyed 13 planes belonging to civilian airlines licensed in Lebanon. According to an Israeli news release, the commandos \"at great risk to themselves . . . exercised the strictest precautions to prevent civilian casualties. The planes were emptied of passengers and ground crews, and people in the vicinity were led away to safety.\" Whatever the extent of the risks involved, no one was killed; Lebanese authorities later claimed that two Israeli soldiers were wounded during the attack. From a military point of view, the raid was a spectacular success\u2014and, I think, from a moral point of view too. It was clearly responsive to the incident at Athens; it was parallel and proportionate in its means (for one can destroy a great deal of property in answer to the destruction of human life); and it was carried out so as to avoid civilian deaths.\n\nDespite all this, the Beirut raid was much criticized at the time (and condemned at the UN)\u2014above all, because of the seriousness of the attack upon Lebanese sovereignty. It is the attack upon Jordanian sovereignty that would stand out in the Khibye case, too, had civilian lives been spared. The killing of civilians is an affront to humanity, but attacks on military installations and the destruction of civilian property pose a more narrow and direct challenge to the state. Indeed, that is the purpose of the attacks; and the vulnerability of soldiers, on the one hand, and of airplanes, boats, buildings, and so on, on the other, hangs on the vulnerability of the sovereign state. Soldiers are vulnerable, if the state is, because they are the visible symbols and the active agents of its authority. And civilian property is vulnerable because the innocence of its owners extends only to their persons, not (or not necessarily) to their possessions. The value we attach to human life is such that rights to life are forfeit only when particular men and women are actually engaged in war-making or national defense. But the lesser value of property is such that property rights are forfeit whenever the state that protects property, and taxes it, is itself subject to attack. Individuals can be taxed without becoming legitimate targets, but property, or certain sorts of property, may be a legitimate target even if its owners are not.c But this argument hangs on the liability of the state, and that remains a matter of dispute.\n\nThe Israeli argument followed the pattern of positive law (or at least of positive law before the era of the UN). Israel insisted that the Lebanese government had an obligation to prevent the use of its territory as a base for terrorist raids. No one seems to deny the reality of the obligation, but it was argued on behalf of the Lebanese (though not by them) that the government in Beirut was in fact incapable of honoring it. Events since 1968 may seem to have borne out that claim, and if it is right, the Israeli attack would be difficult to defend. It is surely wrong to destroy the property of innocent people so as to bring pressure on other people who are in any case unable to act differently from the way they are acting. But one should never be too quick to deny the competence of an established government, for a certain loss of sovereignty is the legal and moral result of political powerlessness. If a government literally cannot control the inhabitants of the territory over which it supposedly presides, or police its borders, and if other countries suffer because of this incapacity, then surrogate controlling and policing are clearly permissible. And these may well go beyond the limits commonly accepted for reprisal raids. At this point, reprisal is like retributive punishment in domestic society: as punishment assumes moral agency, so reprisal assumes political responsibility. Both assumptions are worth holding onto, for as long as possible.\n\nThe critical question is whether one sovereign state can be forced by another to fulfill its obligations. It is the official position of the UN that this kind of law enforcement, even when it is restrained by the rules of war, is illegal. This position rests not only on the general claim of the UN to declare the (positive) law, but also on its readiness and ability, at least some of the time, to enforce the law itself. But the world organization was clearly not ready or able to enforce the law in 1968; nor has it been ready or able to do so at any time since. Nor is there any evidence that individual members of the UN, however they vote on ritual occasions, are prepared to renounce reprisals when the lives of their own citizens are at stake. Reprisals are clearly sanctioned by the practice of nations, and the (moral) reason behind the practice seems as strong as ever. Nothing the UN has actually done, no effects it can presently have, suggests a centralization of legal or moral authority in international life.d\n\nBut the sheer unreality of the UN position doesn't by itself establish the legitimacy of peacetime reprisals. In his edition of Kelsen's Principles of International Law, Robert Tucker has insisted that anyone defending reprisals must show \"that more often than not the independent use of force by states has served the purposes of law. . . .\" This is to shift the ground from the effectiveness of the UN to the utility of reprisal itself and to invite a historical examination the results of which are not likely to favor the \"reprisers\" in any decisive way. But the ground of reprisal is not its overall effectiveness. It is the right, in the difficult conditions of the demi-monde, to seek certain effects. So long as the conditions exist, the right must also exist, even if those same conditions (as in Locke's state of nature) make it unlikely that rightful action will have entirely satisfactory consequences. If, in a particular case, reprisal is certain to fail, then obviously it should not be tried. But whenever there is some substantial chance of success, it is the legitimate resort of a victim state; for no state can be required passively to endure attacks upon its citizens.\n\nReprisal is a practice carried over from the war convention to the world of \"peacetime,\" because it provides an appropriately limited form of military action. It is better, I think, to defend the limits than to try to abolish the practice. Soldiers engaged in a reprisal raid will cross over an international boundary, but they will quickly cross back; they will act destructively, but only up to a point; they will violate sovereignty, but they will also respect it. And finally, they will attend to the rights of innocent people. Reprisals are always limited responses to particular transgressions: crimes against the rules of war, small-scale breaches of the peace. Though they have often been used, they cannot rightly be used, as a cover for invasions or interventions or assaults upon innocent life. It may be that there are moments of extremity and crisis when state's rights and human rights have to be violated; but such moments are not generated by the particular crimes of our enemies, and the violations are not usefully called reprisals. None of the cases of reprisal that I have come across in the lawbooks and the military histories are extreme cases in any meaningful sense of that term. Nor does the war convention provide for extreme cases. Extremity lies, so to speak, beyond the reach of conventional provision. I will consider its character and provenance in Part Four of this book. The analysis of reprisals concludes the discussion of the ordinary means of war. I must turn now to those extraordinary means that the moral urgency of our ends seems sometimes to require.\n\n I have never understood why, in cases like this one, the men are not simply hidden away when their deaths are announced. Why must they actually be killed? Since deceit of various sorts is accepted under the war convention, it certainly should not be ruled out here. But I have been unable to find any case in which such a ruse was tried.\n\n It is not difficult, however, to account for the present legal situation. The threat to take reprisals against enemy civilians is a crucial feature of the contemporary system of nuclear deterrence, and statesmen and soldiers are not prepared solemnly to denounce that system. Moreover, though nuclear deterrence rests only on threats, and the acts threatened are of such a nature that moral men and women might well refuse at the final moment to carry them out, no one is prepared in advance to admit to inhibitions. \"Any act of cruelty to the innocent,\" wrote an American jurist of the pre-atomic age, \"any act, especially, by which noncombat\u00adants are made to feel the stress of war, is what brave men shrink from, although they may feel obliged to threaten it.\" (T. D. Woolsey, Introduction to the Study of International Law, New York, 1908, p. 211.) But can they threaten it effectively if it is known in advance that they will shrink from acting? I will take up the problems of nuclear deterrence in chapter 17.\n\n This is probably what the lawyers have in mind when they argue that, in cases of reprisal, the private citizen \"is held to be identified with his state.\" The identification is by no means total; it does not obliterate personal rights. Nor, I think, does the effect extend to private homes, which seem to share in the innocence of their inhabitants (unless they have been used as terrorist bases).\n\n With regard to the routine UN condemnations of Israeli reprisals, Richard Falk has written: \"One may argue against the fairness of such constraints upon Israel's discretion in these circumstances, but it is essentially an extra-legal appeal as the organs of the UN have the procedural capacity to authorize or prohibit specific uses of force, and it is the exercise of this capacity that most clearly distinguishes what is 'legal' from what is 'illegal' . . . in international society.\" I am not sure that any legislative body, domestic or international, can abolish self-help unless it provides alternative means of help, but I will leave such matters to the lawyers. Assuming Falk is right, it must be said that the extra-legal appeal is a moral appeal the success of which probably will and certainly should undermine the newly enacted \"law.\" See \"International Law and the US Role in Vietnam: A Response,\" in Falk, ed., The Vietnam War and International Law, Princeton, 1968, p. 493.\nPart Four\n\nDilemmas of War\n\nWinning and Fighting Well\n\n\"Asinine Ethics\"\n\nChairman Mao and the Battle of the River Hung\n\nIn the year 638 B.C., during the period of China's history known as the Spring and Autumn Era, the two feudal states of Sung and Ch'u fought a battle at the Hung River in central China. The army of Sung, led by its ruler Duke Hsiang, was drawn up in battle formation on the river's northern bank; the Ch'u army had to ford the stream. When its soldiers were halfway across, one of Hsiang's ministers came to him and said, \"They are many, and we are few. Pray let us attack them before they are all crossed over.\" The Duke refused. When the enemy army had reached the northern bank but had not yet re-formed its lines, the minister again asked leave to begin the fight; again the Duke refused. Only after the Ch'u soldiers were properly marshaled did he signal the attack. And then, in the ensuing battle, the Duke himself was wounded and his army put to flight. According to the chronicles, the people of Sung blamed their ruler for the defeat, but he said, \"The superior man does not inflict a second wound, and does not take prisoner anyone of grey hairs. When the ancients had their armies in the field, they would not attack an enemy when he was in a defile; and though I am but the poor representative of a fallen dynasty, I will not sound my drums to attack an unformed host.\"\n\nThis is the code of a feudal warrior, an obscure warrior in this case until Mao Tse-tung drew his story out of the chronicles in order to make a modern point. \"We are not the Duke of Sung,\" he declared in one of his lectures On Protracted War (1938), \"and we have no use for his asinine ethics.\" Mao's lecture was an innovative discussion of guerrilla tactics. His argument against the Duke of Sung, however, was familiar enough, and to Chinese as well as Western readers. It is an argument common among practical men, like Hsiang's minister, to whom winning is always more important than aristocratic honor. But it enters significantly into the theory of war only when winning is seen to be morally important, that is, only when the outcome of the struggle is conceived in terms of justice. Some 200 years after the battle at the River Hung, more than two millennia before the communist revolution, the philosopher Mo Tzu perfectly described Mao's case, as he himself must understand it.\n\nSuppose there is a country which is being persecuted and oppressed by its rulers, and a Sage . . . in order to rid the world of this pest raises an army and sets out to punish the evil-doers. If, when he has won a victory, he conforms to the doctrine of the Confucians, he will issue an order to his troops saying, \"Fugitives are not to be pursued, an enemy who has lost his helmet is not to be shot at; if a chariot overturns, you are to help the occupants to right it\"\u2014if this is done, the violent and the disorderly will escape with their lives and the world will not be rid of its pest.\n\nMo Tzu believed in the doctrine of Righteous War. Mao Tse-tung has introduced into China the western theory of the just war. No doubt, there are fine points of difference between these two ideas, which I cannot pursue here. But they are not different in any major way. They set up the tension between winning and fighting well in similar fashion, and for Mo Tzu and Chairman Mao they point to the same resolution: the feudal rules for fighting well are simply cast aside. The tension is overcome as soon as it is recognized. That doesn't mean that there are no rules of engagement at all; I have already cited Mao's \"Eight Points for Attention,\" which recapitulate in democratic style the old chivalric code. But for Mao himself the \"Eight Points\" apparently reflect only the utilitarian requirements of guerrilla war, and they cannot stand against the higher utility of winning\u2014which he is likely to describe in extravagant terms, a combination of Wilsonian idealism and Marxist apocalypse: \"The aim of war is to eliminate war. . . . Mankind's era of wars will be brought to an end by our own efforts, and beyond doubt the war we wage is part of the final battle.\" And in the final battle, no one will insist upon the \"Eight Points.\" Exceptions will readily be made whenever the conflict seems critical. Consider, for example, the last of the Eight: \"Do not ill-treat captives.\" Mao has also argued that guerrilla bands on the move cannot take prisoners. \"It is best first to require the prisoners to hand over their weapons and then to disperse them or execute them.\" Since prisoners are not conceived as men-with-rights, the choice between dispersal and execution is purely tactical, and to insist in all cases upon the rule against ill treatment would presumably be an example of \"asinine ethics.\"\n\nNor were rights thought to be at stake in the old warrior codes. Duke Hsiang believed it unworthy and demeaning to strike a wounded soldier or attack an unformed host. Combat was only possible between peers; otherwise war would not be an occasion for the display of aristocratic virtue. It is not hard to understand why anyone convinced of the moral urgency of victory would be impatient with such notions. Of what use is the (undoubted) virtue of the Duke of Sung if the world is ruled by violence and aggression? Indeed, a war in which the Duke's virtue was more important than a military triumph would seem to be a very unimportant war. Thus the argument of Hsiang's minister after the defeat of the Sung army: \"If we grudge a second wound, it would be better not to wound at all. If we would spare the grey-haired, we had better submit to the enemy.\" Either fight all-out or not at all. This argument is often said to be typical of American thought, but in fact it is universal in the history of war. Once soldiers are actually engaged, and especially if they are engaged in a Righteous War or a just war, a steady pressure builds up against the war convention and in favor of particular violations of its rules. And then, more often than the belligerent powers are prepared to admit\u2014itself a matter of interest\u2014the rules are broken. They are not broken for the sake of military necessity alone. That argument justifies too much, and it does so without reference to the cause for which the war is being fought. The rules are broken for the sake of the cause. It is with some version of the argument for justice that the violations are defended.\n\nOn this view, the rules have no standing in any war that is worth fighting. They are at most \"rules of thumb,\" general precepts of honor (or utility) to be observed only until observing them comes into conflict with the requirements of victory. But this is to misunderstand the status of the war convention. If we consider noncombatant immunity rather than warrior honor, and the protection of human rights rather than the expediencies of guerrilla war\u2014that is, if we attend to what is really fundamental in the rules of war\u2014the conflict between winning and fighting well is not so easily resolved. If we recognize, for example, that the protection afforded by the \"Eight Points\" is morally required, and that men and women are rightly indignant if they are robbed and ravaged by guerrilla bands, then Mao's rules take on a greater significance than their author attributes to them. They cannot simply be set aside; nor can they be balanced, in utilitarian fashion, against this or that desirable outcome. For the rights of innocent people have the same moral effectiveness in the face of just as in the face of unjust soldiers.\n\nAnd yet the case for breaking the rules and violating those rights is made sufficiently often, and by soldiers and statesmen who cannot always be called wicked, so that we have to assume that it isn't pointless. Anyway, we know its point all too well. We know how high the stakes sometimes are in war and how urgent victory can be. \"For there are peoples,\" as Simone Weil has written, \"[who] have never recovered after having once been conquered.\" The very existence of a community may be at stake, and then how can we fail to consider possible outcomes in judging the course of the fighting? At this point if at no other, the restraint on utilitarian calculation must be lifted. Even if we are inclined to lift it, however, we cannot forget that the rights violated for the sake of victory are genuine rights, deeply founded and in principle inviolable. And there is nothing asinine about this principle: the very lives of men and women are at stake. So the theory of war, when it is fully understood, poses a dilemma, which every theorist (though not, fortunately, every soldier) must resolve as best he can. And no resolution is serious unless it recognizes the force of both jus ad bellum and jus in bello.\n\nThe Sliding Scale and the Argument from Extremity\n\nThe immediate issue is whether we should discriminate between soldiers fighting a just war and soldiers fighting an unjust war. It is, of course, those who claim membership in the first group who raise the issue, making what might be called an appeal against combatant equality. Though such appeals are particular in character, they have a general form. They all involve the claim that the equality I have been defending is merely conventional and that the truth about war rights is best expressed in terms of a sliding scale: the more justice, the more right. Something like this appears to be what the philosopher John Rawls has in mind when he says, \"Even in a just war, certain forms of violence are strictly inadmissible; and when a country's right to war is questionable and uncertain, the constraints on the means it can use are all the more severe. Acts permissible in a war of legitimate self-defense, when these are necessary, may be flatly excluded in a more doubtful situation.\" The greater the justice of my cause, the more rules I can violate for the sake of the cause\u2014though some rules are always inviolable. The same argument can be put in terms of outcomes: the greater the injustice likely to result from my defeat, the more rules I can violate in order to avoid defeat\u2014though some rules, and so on. The value of this position is that it grants the existence of rights (of some sort) while still opening the way for soldiers resisting aggression to do (some of) the things they believe necessary for victory. It allows the justice of one's cause to make a difference in the way one fights. Exactly how much of a difference is allowed, however, is radically unclear, and so is the status of the men and women who are now drawn into the hell of war so that justice can triumph. The practical effects of the argument are probably more far-reaching than its proponents would like, but I will say nothing about these effects until I can look at a number of historical cases. First, however, something more must be said about the structure of the argument.\n\nAccording to the war convention as I have described it, there is no range of actions, over which the sliding scale might move, between legitimate combat and inadmissible violence. There is only a line, not entirely distinct but meant simply to mark off the one from the other. Given this view, the argument quoted from Rawls might be taken to mean that borderline cases should be decided systematically against that country whose \"right to war is questionable\" or even that the military and political leaders of that country should keep some distance away from the border, never doubling the doubtfulness of their cause with the doubtfulness of their methods. This last would simply be a plea for scrupulousness, which is always a good thing. But there is another meaning that can be drawn out of Rawls' argument (though I don't think it is his own meaning): that the class of \"strictly inadmissible\" acts should be kept very small, and space should be opened up within the rules of war where the sliding scale might be applied. The effect of sliding the scale to point x within this space, it should be said, is not to remove all restraints on military action up to that point, but rather to leave only the restraints of usefulness and proportionality. The sliding scale makes way for those utilitarian calculations that rules and rights are intended to bar. It creates a new class of generally inadmissible acts and of quasi-rights, subject to piecemeal erosion by soldiers whose cause is just\u2014or by soldiers who believe that their cause is just. And so it enables those soldiers to do terrible things and to defend in their own consciences and among their associates and followers the terrible things they do.\n\nNow, the extreme form of the sliding-scale argument is the claim that soldiers fighting a just war can do anything at all that is useful in the fighting. This effectively annuls the war convention and denies or suspends the rights that the convention was designed to protect. The war rights of the just are total, and any blame their actions entail falls upon the leaders of the other side. General Sherman took this view of war, as we have seen, and I have called it the \"war is hell\" doctrine. It is not so much a resolution of the tension between winning and fighting well as a denial of its moral significance. The only kind of justice that matters is jus ad bellum. Beyond that there are only such considerations as rational men will always attend to: they will not waste their substance in useless killing of the innocent, though they will kill them readily enough if victory seems to require it. It may be that this is what the sliding scale comes to in any case, but its advocates at least claim to recognize the existence of rules and rights, and so their argument requires a separate analysis.\n\nThe only alternative to the sliding scale, it is often said, is a position of moral absolutism. To resist the slide, one must hold that the rules of war are a series of categorical and unqualified prohibitions, and that they can never rightly be violated even in order to defeat aggression. But that is a hard line to take, and especially so in the modern age, when aggression has assumed such frightening forms. Perhaps the Duke of Sung was right not to break the warrior code for the sake of his dynasty. But if what is being defended is the state itself and the political community it protects and the lives and liberties of the members of that community. . . . Fiat justicia ruat coelum, do justice even if the heavens fall, is not for most people a plausible moral doctrine.\n\nThere is an alternative doctrine that stops just short of absolutism and that I shall try to defend in the chapters that follow. It might be summed up in the maxim: do justice unless the heavens are (really) about to fall. This is the utilitarianism of extremity, for it concedes that in certain very special cases, though never as a matter of course even in just wars, the only restraints upon military action are those of usefulness and proportionality. Throughout my discussion of the rules of war, I have been resisting this view and denying its force. I have argued, for example, against the notion that civilians can be locked into a besieged city or reprisals taken against innocent people \"in extreme cases.\" For the idea of extremity has no place in the making of the war convention\u2014or if it is said that combat is always extreme, then the idea is naturalized within the convention. The rules are adjusted to the everyday extremities of war; no further adjustment is possible if we are to have any rules at all, and if we are to attend to the rights of the innocent. But now the question is not one of rule-\u00admaking, but of rule-breaking. We know the form and substance of the moral code; we must decide, at a moment of desperation and looming disaster, whether to live (and perhaps to die) by its rules.\n\nThe sliding scale erodes the convention bit by bit, and so it eases the way for the decision-maker who believes himself \"forced\" to violate human rights. The argument from extremity permits (or requires) a more sudden breach of the convention, but only after holding out for a long time against the process of erosion. The reasons for holding out have to do with the nature of the rights at issue and the status of the men and women who hold them. These rights, I shall argue, cannot be eroded or undercut; nothing diminishes them; they are still standing at the very moment they are overridden: that is why they have to be overridden. Hence breaking the rules is always a hard matter, and the soldier or statesman who does so must be prepared to accept the moral consequences and the burden of guilt that his action entails. At the same time, it may well be that he has no choice but to break the rules: he confronts at last what can meaningfully be called necessity.\n\nThe tension between the rules of war and the theory of aggression, between jus in bello and jus ad bellum, can be dealt with in four different ways:\n\n  1. the war convention is simply set aside (derided as \"asinine ethics\") under the pressure of utilitarian argument;\n  2. the convention yields slowly to the moral urgency of the cause: the rights of the righteous are enhanced, and those of their enemies devalued;\n  3. the convention holds and rights are strictly respected, whatever the consequences; and\n  4. the convention is overridden, but only in the face of an imminent catastrophe.\n\nThe second and fourth of these are the most interesting and the most important. They explain how it is that morally serious men and women, who have some sense of what rights are, come nevertheless to violate the rules of war, escalate its brutality and extend its tyranny. The fourth seems to me the right argument. It provides the best account of the two kinds of justice and most fully recognizes the force of each. I shall focus on it in the chapters that follow, but try at the same time to suggest the inadequacies and dangers of the sliding scale. I will look first at a number of cases involving the practice of neutrality, perhaps the most disputed feature of the war convention. Since neutral rights constitute a kind of noncombatant immunity, they might have been taken up earlier on. The disputes they have generated, however, raise questions less about the content than about the force and endurance of rights in war. How long must one wait before breaking the rules? The answer I want to defend is best expressed by reversing Chairman Mao's dictum: with reference to our own conventions, and until the very last minute, we are all the Duke of Sung.\n\nAggression and Neutrality\n\nThe doctrine of neutrality has a twofold form, which is best expressed (and which is conventionally expressed) in the language of rights. States possess, first, a right to be neutral, which is simply an aspect of their sovereignty. In any prospective or on-going conflict between two other states, they are free to opt for what might be called the condition of \"thirdness.\" And if they do that, they then possess neutral rights, specified at great length in positive international law. As with the war convention generally, the initial right and the subsequent rights exist without reference to the moral character of the belligerent powers or to the probable outcome of the war. The more convinced we are, however, that one of the belligerents is an aggressor or that the outcome is going to be disastrous, the more likely we are to deny the very possibility of noninvolvement. How can the rest of us respect its right to stand and watch if, by violating that right, we might avert the destruction?\n\nThese questions have been posed with a special insistence in the years since World War II, but in fact the argument implicit in them is an old one. Consider, for example, a British proclamation issued in 1793: the political and military policies of the revolutionary government of France, it was said, involved \"all the surrounding powers in one common danger . . . giving them the right . . . imposing on them the duty, to stop the progress of an evil which exists only by the successive violation of all law and property. . . .\" The practical consequence of this sort of thing is obvious. If states don't do their duty, they can be forced to do it. One asserts the urgency of the struggle, and one erodes or denies the right to be neutral, in order to pave the way for the violation of neutral rights. The history of neutrality provides many examples of such violations, defended with some version of the argument from extremity or with the sliding scale, and I shall refer to that history in order to analyze those defenses. But first I must say something about the nature of neutrality itself and its place in the war convention.\n\nThe Right to Be Neutral\n\nNeutrality is a collective and voluntary form of noncombatancy. It is collective in that its benefits obtain for all the members of a political community without reference to the status of individuals. Soldiers and civilians are alike protected, so long as their state is \"not engaged in war-making.\" The rights of disengagement distribute equally to all citizens. Neutrality is voluntaristic in that it can be assumed at will by any state with regard to a war or a prospective war between any other states. Individuals can be conscripted, but states cannot. They may ask that other powers formally acknowledge their neutrality, but the condition is unilaterally assumed and the acknowledgment unnecessary. The \"scrap of paper\" that Germany brushed aside when it invaded Belgium in 1914 did not establish Belgian neutrality; the Belgians themselves did that. And had the Germans formally renounced their guarantee or waited for its expiration, their invasion would still have been the crime it was said to be at the time. It would have been a crime, that is, as long as the Belgians not only claimed the rights but also observed the duties of a neutral state.\n\nThese duties can be summed up very simply, although international law on this subject is elaborate and detailed: they require a strict impartiality toward the belligerents, without reference to the justice of their cause or to any sentiments of neighborliness, cultural affinity, or ideological agreement. It is not only fighting on one or another side that is prohibited, but every sort of official discrimination. This rule is very strict; if it is violated, neutral rights are forfeit, and the neutral state is subject to reprisals from whichever belligerent is injured by the violations. The rule applies, however, only to state action. Private citizens remain free to choose sides in a variety of ways, to campaign politically, raise money, even raise volunteers (though they cannot launch forays across the border). What is more important, normal patterns of trade may be maintained with both belligerents. Hence the neutrality of any given state is likely to be more helpful to one side than to the other. So far as the warring powers are concerned, neutrality is rarely a matter of equal benefit, for neither the balance of private sympathy and effort nor the balance of trade is likely to be even between them.a But neither can complain of the unofficial help the other receives. This is a help that cannot be helped; it derives from the very existence of the neutral state, its geography, economy, language, religion, and so on, and could only be interdicted by the most rigorous coercion of its citizens. But the neutral state is not required to coerce its own citizens. So long as it takes no positive action to help one side or the other, it has fulfilled its duty not to get involved, and then it is automatically entitled to the full enjoyment of its right not to get involved.\n\nThe moral basis of the right is not entirely clear, however, in large part because its domestic analogue is so unappealing. In both political and moral life, the \"neuter\" is not a person one instinctively likes. Perhaps he has a right to avoid if he can the quarrels of his neighbors, but what about their troubles? We have to ask again: can he stand and watch a neighbor being assaulted on the street? Might not the neighbor say at such a time, \"You're either for me or against me\"? As a revolutionary slogan, that sentence suggests, perhaps, an unwarranted pressure and a threat of retaliations to come. But in the case at hand, its message is simpler and less objectionable. Surely a strict neutrality here, a refusal to discriminate in any way in favor of the victim, would be disquieting and strange. Neighbors are not mere spectators, studying one another's misfortunes from some great distance. The social life they share entails a degree of mutual concern. On the other hand, if I am obligated to be \"for\" my neighbor, I am not obligated to rush to his rescue\u2014first, because that may not be an effective way of being for him; and second, because it may be disastrous for me. I have a right to weigh the risks of joining the battle. But let's assume that the risks are minor: there are a large number of us watching, and I can count on the support of the others if I take the lead; or there is a policeman around the corner, and I can count on him to take the lead. Then I have no right to be neutral, and any efforts on my part to escape, make excuses, bury my head in the sand, are sure to be thought reprehensible.\n\nBut the right of a state is different, and not only because there is no policeman around the corner. For there may well be a majority of states and an overwhelming predominance of force at least potentially available on behalf of a state under attack, thought to be the victim of aggression. All that stands in the way of mobilizing this force, it may be, is the war convention and the right of neutrality. Even in such a case, the right holds, because risk in war is very different from what it is in domestic fighting. Years ago, John Westlake argued that \"neutrality is not morally justifiable unless intervention in the war is unlikely to promote justice or could do so only at a ruinous cost to the neutral.\" Ruination is to be avoided, but is this only the ruination of states? When a state joins a war, it risks its survival to this or that degree, depending on the nature of the conflict, the power of its allies, and the readiness and fighting capacity of its army; and these risks may be acceptable or not. But at the same time, it condemns an indefinite number of its citizens to certain death. It does this, to be sure, without knowing which citizens those are. But the decision itself is irrevocable: once fighting begins, it is certain that soldiers (and probably civilians, too) will die. The right of neutrality follows from this fact. Like other provisions of the war convention, it represents a limit on the coerciveness of war. At least this group of men and women, citizens of the neutral state, who do not choose to risk their lives, will be protected from having to do so.\n\nBut why should these men and women be immune and free when so many others are driven into battle? In what possible way are they entitled to their neutrality? The question is especially important if we imagine a situation where a particular state's decision to be neutral means that more people will be killed than would be killed if it joined the war, for the participation of its armies might turn the tide and shorten the fighting by so many weeks or months. But the leaders of such a state are not required to calculate as if every human life carried the same moral weight for every decision-maker at every moment in time. Their people's lives are not international resources to be distributed in war so as to balance the risks or reduce the losses of other people. These are innocent lives. With reference to the soldiers of the neutral state, that means only that they have not yet been attacked and forced to fight. Still, they are disengaged, and no one has a right to challenge their disengagement. Perhaps that disengagement is a matter of luck; it is often, in cases of successful neutrality, a matter of geography. But people are entitled to their good fortune in such matters, as states are, or are presumed to be, entitled to their geographic locations.b\n\nSo neutral citizens are immune from attack; the coerciveness of war can never willfully be extended beyond the limits fixed by the material causes of the conflict and the military organization of the states involved. The leaders of a neutral state are entitled to maintain that immunity; indeed, they may be bound to do so, given the consequences of its loss for their fellow citizens. The same solidarity that makes noninvolvement at home morally questionable may well make it obligatory in the international arena: this group of men and women must save one another's lives first. They cannot do this by killing other people, unless those others are attacking them. The rules of neutrality suggest, however, that they can do it by allowing other people to die rather than dying themselves. If they have incurred obligations toward some of those people\u2014for the sake, perhaps, of collective security\u2014then, of course, they cannot allow them to die; otherwise, the right holds, even if its assertion seems ignoble.\n\nBut there is one sort of case in which this right might be denied. Imagine (what is easily imaginable) that some great power launches a campaign of conquest, aimed not merely at this or that state but at some larger ideological or imperial goal. Why should such a campaign be resisted only by its first victims, when in fact many other states will be threatened if the initial resistance fails? Or consider the common argument that aggression anywhere threatens everyone. Aggression is like crime: if one does not stamp it out, it will spread. Then again, there is no reason for the immediate victims to fight alone. They are fighting on behalf of future victims, that is, of all other states, and the others will reap the benefits of their fighting and dying. How can they stand aside? President Wilson took this position in his war message of April 2, 1917: \"Neutrality is no longer feasible or desirable when the peace of the world is involved and the freedom of its peoples.\" He presumably meant morally feasible, since a practical alternative to war, namely continued neutrality, clearly existed. The argument against that alternative must go something like this. If one imagines a particular aggressor moving on from one triumph to another, or if one imagines a radical increase in the incidence of aggression as a result of this particular triumph, then it has to be said that peace and freedom are in general danger. And then continued neutrality is not morally feasible; for while a neutral state has or may have a right to let others die in quarrels of their own, it cannot let them die on its behalf. Any danger that is shared by all the members of international society is morally coercive, even if it is not yet materially present, for all of them.\n\nThis argument, however, rests uneasily on \"imaginings\" about which there is no general agreement and which often look painfully implausible after the fact. It seems very strange today, for example, that any conceivable outcome of World War I could have been thought to pose a universal threat to peace and freedom (or a greater threat than was posed by the actual outcome). And this is so even if one grants that the war began with an act or a series of acts of aggression. The mere recognition of a criminal attack, without some profoundly pessimistic or, as in this case, highly extravagant view of its likely consequences, does not require the leaders of a neutral state to draw President Wilson's conclusions. They can always refuse to do so, imagining in their turn that their own country and the whole world are in no real danger. That is a unilateral view of the situation, to be sure, and one can argue (as I would often be inclined to do) with the leaders who put it forward. But they and their people are entitled to act on it. That is the real right of neutrality.\n\nThe Nature of Necessity (2)\n\nAt this point, however, the crucial moral decision may not lie with the neutral state. The belligerents also have a choice: to respect neutral rights or not. Violations of those rights are usually thought to be an especially bad kind of aggression\u2014on the principle, I suppose, that it is worse to strike out at uninvolved states than at states with which one has been quarreling. Unless we take a rather permissive view of the initial resort to violence, this seems a dubious principle. On the other hand, attacks on neutrals are usually an especially clear kind of aggression, whereas responsibility for the war itself may be difficult to assess. When armies move across the frontier of a state that has maintained a strict impartiality, we have little difficulty in recognizing the move as a criminal act. Violations short of armed attack are harder to recognize but almost equally reprehensible, for they invite and justify military responses from the other side. If neutrality collapses and the war is extended to new territory and people, the crime is that of the first violator (assuming a proportionate response from the second).\n\nBut what if neutrality is violated for a good cause: for the sake of national survival and the defeat of aggression; or, more largely, for the sake of \"civilization as we know it\" or the \"peace and freedom\" of the whole world? Here is the paradigmatic form of the collision between jus ad bellum and jus in bello. The belligerent power believes itself pressed by the exigencies of a just war. The neutral state is firm in its rights: its citizens are not bound to sacrifice themselves to someone else's exigencies. The belligerent power talks of the vital importance of the ends for which it is fighting; the neutral state invokes the rules of war. Neither side is entirely convincing, though in particular cases we must choose between them. I have tried to make the strongest possible case for neutral rights. Their violation almost certainly entails the killing (or the causing to be killed) of innocent people, and so it is not a casual matter even when the end in view is very important. Indeed, we are likely to recognize good men fighting for important ends by their reluctance to invade neutral states and force their citizens to fight. The value of that reluctance will be apparent if we look at two cases in which neutral rights were wrongly violated: first, on the plea of necessity, and second, with the argument more justice, more right. The first is the most famous violation of neutrality since the Athenian attack on Melos, and I have given it the name originally assigned in wartime propaganda.\n\nThe Rape of Belgium\n\nThe German attack on Belgium in August 1914 is unusual in that it was openly and honestly described by the Germans themselves as a violation of neutral rights. The speech of Chancellor von Bethmann Hollweg to the Reichstag on August 4 deserves to be remembered.\n\nGentlemen, we are now in a state of necessity, and necessity knows no law. Our troops have already entered Belgian territory.\n\nGentlemen, that is a breach of international law. It is true that the French government declared at Brussels that France would respect Belgian neutrality as long as her adversary respected it. We know, however, that France stood ready for an invasion. France could wait, we could not. A French attack on our flank on the lower Rhine might have been disastrous. Thus we were forced to ignore the rightful protests of the Government of Belgium. The wrong\u2014I speak openly\u2014the wrong we thereby commit we will try to make good as soon as our military aims have been attained.\n\nHe who is menaced as we are and is fighting for his highest possession can only consider how he is to hack his way through (durch-haven).\n\nThis is frank talk, though it is not quite like the \"frankness\" of the Athenian generals at Melos. For the chancellor does not step outside the moral world when he defends the German invasion. He grants that a wrong has been done, and he promises to make it good after the fighting is over. That promise was not taken seriously by the Belgians. Their neutrality having been violated and their borders crossed, they had no reason to expect anything good from the invaders; nor did they believe that their independence would be respected. They chose to resist the invasion, and once their soldiers were fighting and dying, it is hard to see how the wrong the Germans had done could ever be made good.\n\nThe force of von Bethmann Hollweg's argument lies not in the promise of reparation, but in the plea of necessity. This will be a useful occasion to consider again what the plea might mean\u2014and to suggest that here, as in military history generally, it means a great deal less than it appears to do. We can see clearly in the chancellor's speech the two levels at which the concept works. First, there is the instrumental or strategic level: the attack on Belgium was necessary, it is being argued, if German defeat was to be avoided. But that is an improbable argument. The attack had long seemed to the General Staff the most expedient way of striking a hard blow against the French and winning a quick victory in the west (before Germany was fully engaged with the Russians on the eastern front). By no means, however, was it the only way of defending German territory. A French invasion along the lower Rhine, after all, could only outflank the German army if the Germans were mobilized for action further north (along the Belgium frontier). The chancellor's actual claim was that the odds of victory would be improved and German lives saved if the Belgians were sacrificed. But that expectation, which turned out to be wrong, had nothing to do with necessity.\n\nThe second level of the argument is moral: not only is the attack necessary to win, but winning itself is necessary, since Germany is fighting for its \"highest possession.\" I don't know what von Bethmann Hollweg thought Germany's highest possession was. Perhaps he had in mind some notion of honor or military glory, which could only be upheld by victory over the nation's enemies. But honor and glory belong to the realm of freedom, not necessity. We are likely to think that Germany's victory was morally necessary (essential, required) only of its survival as an independent nation or the very lives of its people were at stake. And on the best construction of the German cause, that was certainly not the case; what was at stake was Alsace-Lorraine, Germany's African colonies, and so on. So the argument fails on both levels. It would have to succeed on both, I think, before the violation of Belgian neutrality could be defended.\n\nThe German chancellor puts forward exactly the sort of argument that would be appropriate at a time of genuine extremity. He rejects every kind of deceitfulness. He does not pretend that the Belgians have failed in their duty of impartiality. He does not claim that the French have already violated Belgian neutrality or even that they are threatening to do so. He does not argue that Belgium cannot rightly stand aside in the presence of (French) aggression. He recognizes the force of the war convention and hence of the right of neutrality, and he makes the case for overriding that right. He wants to override it, however, not at the last minute but at the very first, and not when Germany's survival is in danger but when the dangers are of a more ordinary kind. So his is not a plausible case; its structure is right, but not its content. Nor was it thought plausible at the time. The German invasion was almost universally condemned (by many Germans, too). It was an important reason for the determination and high morale with which Britain entered the war and for the sympathy with which the Allied cause was viewed in other neutral countries\u2014the United States, above all. Even Lenin, who led the leftist opposition to the war, thought the defense of Belgium a reason to fight: \"Let us suppose that all the states interested in the observation of international treaties declared war on Germany, with the demand for the liberation and indemnification of Belgium. In such a case, the sympathies of Socialists would, of course, be on the side of Germany's enemies.\" But, he went on, that is not what the war is really about. He was right; the war as a whole does not lend itself to an easy description in terms of justice and injustice. But the attack on Belgium does. We must turn now, and at much greater length, to a harder case.\n\nThe Sliding Scale\n\nWinston Churchill and Norwegian Neutrality\n\nThe day after Britain and France declared war on Germany in 1939, King Haakon VII formally proclaimed Norway's neutrality. The policy of the king and his government was not founded on political or ideological indifference. \"We never had neutrality of thought in Norway,\" the Foreign Minister wrote, \"and I never wanted it.\" Norway's political and cultural ties were with the Allies, and there seems no reason to doubt what historians of the period tell us: \"The Norwegians firmly believed in the high ideals of democracy, individual freedom, and international justice.\" They were not, however, prepared to fight for those ideals. The war was a struggle among the great powers of Europe, and Norway was very much a small power, traditionally disengaged from European machtpolitik, and now virtually disarmed. Whatever the moral importance of the issues over which the war was being fought, the Norwegian government could hardly intervene in any decisive way. Nor could it intervene at all without accepting great risks. Its first task was to make sure that Norway was still intact and its citizens alive at the end.\n\nWith this purpose in mind, the government adopted a strict policy of \"neutrality in deed.\" On balance, this policy favored the Germans, even though most of Norway's normal trade was with the Allied powers, especially Britain. For the Germans depended on Norway for a very large part of their iron ore supply. The ore was mined at Gallivare in northern Sweden, and during the summer months it was shipped out of the Swedish town of Lulea on the Baltic Sea. But in the winter, the Baltic froze; then the ore was moved by rail to Narvik on the Norwegian coast, the nearest warm-water port. There German ships picked it up and carried it down the coast, keeping within Norwegian territorial waters so as to avoid the British navy. The German ore supply was thus protected by Norwegian (and Swedish) neutrality, and for this reason the invasion of Norway was no part of Hitler's original strategic plan. Instead, \"[he] emphasized repeatedly that in his opinion the most desirable attitude for Norway as well as for the rest of Scandinavia would be one of complete neutrality.\"\n\nThe British view was very different. During the long months of the \"phony war,\" Scandinavian neutrality was a constant topic of Cabinet discussion. Winston Churchill, then First Lord of the Admiralty, proposed one plan after another to interdict the shipments of iron ore. Here was a chance, he argued, here was the only chance, to strike a quick blow against Germany. Instead of waiting for a German attack in France and the Low Countries, the Allies could force Hitler to disperse his armies and to fight\u2014Churchill never doubted that the Germans would fight for their ore supply\u2014in a part of the world where the strength of the British navy could most effectively be brought to bear. The French were also disinclined to wait for an attack on their own soil. Sir Edward Spears writes of Prime Minister Daladier that \"his views on military matters were confined to keeping warlike operations as remote from France as possible.\" The Norwegian prime minister no doubt had a parallel idea in mind. But there is this difference: the war which the Norwegians wished to see fought in France, and which the French were ready to fight in Norway, was France's and not Norway's war. Churchill confronted the same difficulty; Norwegian neutrality was a bar to each of his plans. It was only a moral and legal bar, perhaps, for he did not expect the Norwegians to fight very hard for their neutrality, but it was an important bar nonetheless, since the British were inclined to distinguish themselves from their enemies by their respect for international law and justice. \"All the cards are against us in playing with these neutrals,\" General Ironside, Chief of the Imperial General Staff, confided to his diary. \"Germany does not mean to respect them if it so suits her and we must respect them.\" The case was especially difficult because it did in fact suit the Germans, but not the British, to respect Norway's neutral rights.\n\nThe Russo-Finnish war opened a new possibility for Allied strategists (and moralists). The League of Nations, which had said nothing about the German attack on Poland, now condemned the Russians for waging an aggressive war. Churchill, who \"sympathized ardently with the Finns,\" proposed to send troops to Finland in fulfillment of Britain's obligations under the Covenant\u2014and to send them via Narvik, Gallivare, and Lulea. Under the plan drawn up by the General Staff, only a battalion of soldiers would actually have reached Finland, while three divisions would have guarded the \"lines of communication\" across Norway and Sweden, not only stopping the shipments of iron ore, but seizing it at its source and digging in for an expected German response in the spring. It was a bold plan which would almost certainly have led to a German invasion of Sweden and Norway and to large-scale military operations in the two countries. \"We have more to gain than to lose,\" Churchill argued, \"by a German attack on Norway.\" One immediately wants to ask whether the Norwegians had more to gain than to lose. Apparently they did not think so, for they rejected repeated requests that they permit the free passage of British troops. The Cabinet decided in favor of the expedition anyway, but the instructions prepared for its commander would have allowed him to proceed only in the face of \"token opposition.\" General Ironside worried that the political will necessary for success did not exist. \"We must . . . remain quite cynical about anything except stopping the iron ore.\" The Cabinet seems to have been cynical enough about its Finnish cover. As it turned out, however, the members were unwilling to do without it, and when the Finns sued for peace in March 1940, the plan was shelved.\n\nChurchill now pressed a more modest proposal. He urged the mining of Norwegian territorial waters, so as to force German merchant ships out into the Atlantic where the British navy could capture or sink them. It was a proposal he had made immediately after the war began and that he brought forward whenever his larger plans seemed in danger. Even this \"genteel little act of bellicosity,\" however, encountered opposition. Though the Cabinet seemed favorable to Churchill's original presentation (in September 1939), \"the Foreign Office arguments about neutrality were weighty, and I could not prevail. I continued . . . to press my point by every means and on all occasions.\" It is interesting to note, as Liddell Hart does, that a similar project had been brought forward in 1918 and rejected by the Commander-in-Chief, Lord Beatty. \"[He] said it would be most repugnant to the officers and men in the Grand Fleet to steam in overwhelming strength into the waters of a small but high-spirited people and coerce them. If the Norwegians resisted, as they probably would, blood would be shed; this, said the Commander-in-Chief, 'would constitute a crime as bad as any that the Germans had committed elsewhere.'\" The words have a somewhat archaic ring (and it should be said that Beatty's last line, repeated in 1939\u201340, would not have been true), but many Englishmen still felt a similar repugnance. These were more likely to be professional diplomats and soldiers than civilian politicians. General Ironside, for example, not always the cynic he pretended to be, wrote in his diary that the mining of Norwegian waters, though it could be described as \"a reprisal for the way Germany had treated neutral ships . . . may well start off some form of totalitarian war.\"\n\nChurchill presumably believed that Britain was in for that kind of war anyway, given the political character of its enemy. He defended his proposal with a moral argument focusing on the nature and long-term goals of the Nazi regime. It is not merely that he did not sympathize with Beatty's repugnance; he told the Cabinet that such feelings courted disaster, not for Britain alone but for all Europe.\n\nWe are fighting to re-establish the reign of law and to protect the liberties of small countries. Our defeat would mean an age of barbaric violence, and would be fatal, not only to ourselves, but to the independent life of every small country in Europe. Acting in the name of the Covenant, and as virtual mandatories of the League and all it stands for, we have a right, indeed are bound in duty, to abrogate for a space some of the conventions of the very laws we seek to consolidate and reaffirm. Small nations must not tie our hands when we are fighting for their rights and freedom. The letter of the law must not in supreme emergency obstruct those who are charged with its protection and enforcement. It would not be right or rational that the aggressive Power should gain one set of advantages by tearing up all laws, and another set by sheltering behind the innate respect for law of its opponents. Humanity, rather than legality, must be our guide.\n\nThis is a powerful argument, though its rhetoric is sometimes misleading; it requires close examination. I want to begin by accepting Churchill's description of the British as defenders of the rule of law. (Indeed, they vindicated their claim to that title by refusing for months to adopt his proposals.) It may even be accurate to talk of Britain as the \"virtual mandatory\" of the League of Nations, so long as one understands that phrase to mean that it was not the actual mandatory; the British decision to invade Norwegian waters was as unilateral as was Norway's decision to stay out of the war. The problem lies in the consequences Churchill believes to follow from the justice of Britain's cause.\n\nHe puts forward a version of what I have called the sliding scale argument: the greater the justice of one's cause, the more rights one has in battle.c But Churchill pretends that these are rights against the Germans. The British, he says, are entitled to violate those legal conventions behind which Germany is sheltering. Legal conventions, however, have (or sometimes have) their moral reasons. The purpose of the laws of neutrality is not primarily to protect belligerent powers but to save the lives of neutral citizens. It was in fact the Norwegians who were sheltered by the \"letter of the law\"; the Germans were only its secondary beneficiaries. This ordering suggests the crucial difficulty with the sliding scale. However much the rights of the British are enhanced by the justice of their cause, they can hardly acquire a title to kill Norwegians or to put their lives at risk unless Norwegian rights are somehow simultaneously diminished. The sliding scale argument presupposes and requires some such symmetry, but I do not see how it can be generated. It is not enough to argue that the just side can do more. Something must be said about the objects as well as the subjects of this military doing. Who is being done to? In this case, the objects are Norwegian citizens, who are in no sense responsible for the war into which they are to be dragged. They have not challenged the rule of law or the peace of Europe. How have they become liable to attack?\n\nThere is an implicit answer to this question in Churchill's Cabinet memorandum. He obviously believes that the Norwegians ought to be involved in the struggle against Germany, not only because their involvement would be good for Britain, but also because, if Britain and France were forced into a \"shameful peace,\" they would certainly be among the \"next victims.\" Neutral rights fade away, he argues, when brought up against aggression and illegal violence on the one hand and legitimate resistance on the other. Or at least, they fade away whenever the aggressor poses a general threat: to the rule of law, the independence of small nations, and so on. Britain is fighting on behalf of Germany's future victims, and they must sacrifice their rights rather than hinder the struggle. Taken as moral exhortation, this seems to me, in the circumstances of 1939\u201340, entirely justified. But it remains a question whether the sacrifice is to be required because the Norwegians recognize the German threat or because the British do. Churchill is repeating Wilson's argument of 1917: neutrality is not morally feasible. But this is a dangerous argument when made not by the leader of a neutral state but by a leader of one of the belligerents. It is not a question now of the voluntary surrender of neutral rights, but of their \"abrogation for a time.\" And even that phrase is a euphemism. Since human life is at stake, the abrogation is not temporary, unless Churchill plans to raise the dead after the war is over.\n\nIn most wars, it can plausibly be said that one side fights justly, or probably does, or fights with greater justice than the other, and in all these cases the enemy against which it fights may well pose a general threat. The right of third parties to be neutral is a moral entitlement to ignore those distinctions and to recognize or not to recognize that threat. It may well be that they have to fight if they do recognize a danger to themselves, but they cannot rightly be forced to fight if they do not. They may be morally blind, or obtuse, or selfish, but these faults do not turn them into the resources of the righteous. This is, however, exactly the effect of Churchill's argument: the sliding scale is a way of transferring the rights of third parties to the citizens and soldiers of a state whose war is, or is said to be, just.\n\nBut there is another argument in Churchill's memorandum which does not require the application of the sliding scale; it is most clearly suggested by the phrase \"supreme emergency.\" In an emergency, neutral rights can be overridden, and when we override them we make no claim that they have been diminished, weakened, or lost. They have to be overridden, as I have already said, precisely because they are still there, in full force, obstacles to some great (necessary) triumph for mankind. To British strategists, Norwegian neutrality was an obstacle of just this sort. It appears now that they greatly exaggerated the effects they could have had on Germany's war effort by cutting off the ore shipments. But their estimates were honestly made, and they were shared by Hitler himself. \"We can under no circumstances afford to lose the Swedish ore,\" he told General Falkenhurst in February 1940. \"If we do, we will soon have to wage war with wooden sticks.\" That attractive prospect must have weighed heavily with the British Cabinet. They had available to them a simple utilitarian argument, backed up by a theory of justice, for violating Norway's neutral rights: the violations were militarily necessary to defeat Nazism, and it was morally essential that Nazism be defeated.\n\nHere again is the two-level argument, and in this case the argument works on the second level: the moral necessity is clear (I will try to explain why this is so in the next chapter). That is why we are likely to be far more sympathetic to Churchill's than to von Bethmann Hollweg's position. But the instrumental or strategic claim is as questionable in the Norwegian as in the Belgian example. The Allied armies had not yet fought a single battle; the force of the German blitzkrieg had not yet been felt in the West; the military significance of the airplane was not yet understood. The British still had full confidence in the Royal Navy. The First Lord of the Admiralty certainly had such confidence: all his Norwegian plans depended upon naval power. Only a Churchill, having called the situation at the beginning of 1940 a \"supreme emergency,\" could still find words to describe Britain's danger six months later. The truth is that when the British finally decided \"to sail in overwhelming strength into the waters of a small but high-spirited people and coerce them,\" they were not thinking of avoiding defeat but (like the Germans in 1914) of winning a quick victory.\n\nSo the British move is another example of overriding at the first minute rather than the last. We judge it less harshly than the German attack on Belgium, not only because of what we know of the character of the Nazi regime, but also because we look back on the events of the next months which so quickly brought Britain to the brink of national disaster. But it has to be stressed again that Churchill had no foresight of that disaster. To understand and weigh the actions he advocated, we must stand beside him in those early months of the war and try to think as he did. Then the question is simply this: can one do anything, violating the rights of the innocent, in order to defeat Nazism? I am going to argue that one can indeed do what is necessary, but the violation of Norwegian neutrality was not necessary in April 1940; it was only a piece of expediency. Can one then reduce the risks of fighting Nazism, at the expense of the innocent? Surely one cannot do that, however just the struggle. Churchill's argument hangs on the reality and the extremity of the crisis, but here (in his own view) there was no crisis. The \"phony war\" was not yet a supreme emergency. The emergency came on unexpectedly, as emergencies are likely to do, its dangers first revealed by the fighting in Norway.\n\nThe final British decision was made late in March, and the Leads were mined on April 8. The next day, the Germans invaded Norway. Eluding the British navy, they landed troops all along the coast, even as far north as Narvik. It was a response not so much to the actual laying of the mines as to the months of plans, arguments, and hesitations, none of which were concealed from Hitler's agents and strategic analysts. It was also the response Churchill had expected and hoped for, though it came too soon and with complete surprise. The Norwegians fought bravely and briefly; the British were tragically unready to defend the country they had made vulnerable to attack. There were a number of counter-landings by British troops; Narvik was captured and held for a short time; but the navy was ineffective against the German airforce, and Churchill, still First Lord of the Admiralty, presided over a series of humiliating evacuations. Germany's ore supply was safe for the duration of the war, as it would have been had Norway's neutrality been respected. Norway was an occupied country, with a fascist government; many of its soldiers were dead; the \"phony war\" was over.\n\nAt Nuremberg in 1945, German leaders were charged with having planned and carried out an aggressive war against Norway. Liddell Hart finds it \"hard to understand how the British and French governments had the face to approve . . . this charge.\" His indignation derives from his belief that neutral rights are equally invulnerable to the claims of just and unjust belligerents. So they are, and it would have been better if after the war the British had acknowledged that the mining of the Leads had been a breach of international law and that the Germans were entitled, if not to invade and conquer Norway, at least to respond in some military way. I do not want to deny the anomaly of the argument that Hitler's Germany could have any rights at all in its wars of conquest. German entitlements, however, came by way of Norwegian rights, and so long as one recognizes the practice of neutrality, there is no way around them. In a supreme emergency, indeed, it may be necessary \"to hack one's way through,\" but it is no virtue to be too eager to do that or to do it too soon, for it is not the opposing army that is hacked through in such a case, but innocent men and women, whose rights are intact, whose lives are at stake.\n\n Neutral states have sometimes sought a more perfect neutrality by embargoing all trade with belligerent powers. But this does not seem a plausible course. For if the normal balance of trade favors one belligerent, a total embargo is likely to favor the other. There is no zero point; the status quo ante bellum seems the only reasonable norm.\n\n But this argument doesn't seem to work with reference to the property and prosperity (rather than the lives) of the citizens. If a state can discriminate economically against an aggressor, even if the costs to itself are considerable, it seems bound to do so, unless the discrimination is likely to involve it in the fighting. Aggressor states, of course, have a right to respond to discriminatory measures, by force if necessary. But they won't always be in a position to respond, and if they are not, the measures may be morally required. When the League of Nations invoked economic sanctions against Italy in the Ethiopian War of 1936, it made the requirement legal as well. But I should think that the moral obligation would have held had there been only an Ethiopian appeal and no League resolution. In any case, the example suggests the relative status of property rights in the theory of war.\n\n Hugo Grotius, who generally favors the sliding scale, is particularly clear on the question of neutrality: \"From what has been said we can understand how it is permissible for one who is waging a just war to take possession of a place situated in a country free from hostilities.\" He sets three conditions, the first of which does not quite fit the Norwegian case: \"that there is not an imaginary but a real danger that the enemy will seize the place and cause irreparable damage.\" But Churchill might have argued that the Germans enjoyed all the benefits of seizure without the effort. See Of the Law of War and Peace, Book II, Chapter ii, Section x.\n\nSupreme Emergency\n\nThe Nature of Necessity (3)\n\nEveryone's troubles make a crisis. \"Emergency\" and \"crisis\" are cant words, used to prepare our minds for acts of brutality. And yet there are such things as critical moments in the lives of men and women and in the history of states. Certainly, war is such a time: every war is an emergency, every battle a possible turning point. Fear and hysteria are always latent in combat, often real, and they press us toward fearful measures and criminal behavior. The war convention is a bar to such measures, not always effective, but there nevertheless. In principle at least, as we have seen, it resists the ordinary crises of military life. Churchill's description of Britain's predicament in 1939 as a \"supreme emergency\" was a piece of rhetorical heightening designed to overcome that resistance. But the phrase also contains an argument: that there is a fear beyond the ordinary fearfulness (and the frantic opportunism) of war, and a danger to which that fear corresponds, and that this fear and danger may well require exactly those measures that the war convention bars. Now, a great deal is at stake here, both for the men and women driven to adopt such measures and for their victims, so we must attend carefully to the implicit argument of \"supreme emergency.\"\n\nThough its use is often ideological, the meaning of the phrase is a matter of common sense. It is defined by two criteria, which correspond to the two levels on which the concept of necessity works: the first has to do with the imminence of the danger and the second with its nature. The two criteria must both be applied. Neither one by itself is sufficient as an account of extremity or as a defense of the extraordinary measures extremity is thought to require. Close but not serious, serious but not close\u2014neither one makes for a supreme emergency. But since people at war can rarely agree on the seriousness of the dangers they face (or pose for one another), the idea of closeness is sometimes made to do the job alone. Then we are offered what might best be called the back-to-the-wall argument: that when conventional means of resistance are hopeless or worn out, anything goes (anything that is \"necessary\" to win). Thus British Prime Minister Stanley Baldwin, writing in 1932 about the dangers of terror bombing:\n\nWill any form of prohibition of bombing, whether by convention, treaty, agreement, or anything you like, be effective in war? Frankly, I doubt it, and in doubting it, I make no reflection on the good faith of either ourselves or any other country. If a man has a potential weapon and has his back to the wall and is going to be killed, he will use that weapon, whatever it is and whatever undertaking he has given about it.\n\nThe first thing that has to be said about this statement is that Baldwin does not mean his domestic analogy to be applied literally. Soldiers and statesmen commonly say that their backs are to the wall whenever military defeat seems imminent, and Baldwin is endorsing this view of extremity. The analogy is from survival at home to victory in the international sphere. Baldwin claims that people will necessarily (inevitably) adopt extreme measures if such measures are necessary (essential) either to escape death or to avoid military defeat. But the argument is wrong at both ends. It is simply not the case that individuals will always strike out at innocent men and women rather than accept risks for themselves. We even say, very often, that it is their duty to accept risks (and perhaps to die); and here as in moral life generally, \"ought\" implies \"can.\" We make the demand knowing that it is possible for people to live up to it. Can we make the same demand on political leaders, acting not for themselves but for their countrymen? That will depend upon the dangers their countrymen face. What is it that defeat entails? Is it some minor territorial adjustment, a loss of face (for the leaders), the payment of heavy indemnities, political reconstruction of this or that sort, the surrender of national independence, the exile or murder of millions of people? In such cases, one's back is always to the wall, but the dangers one confronts take very different forms, and the different forms make a difference.\n\nIf we are to adopt or defend the adoption of extreme measures, the danger must be of an unusual and horrifying kind. Such descriptions, I suppose, are common enough in time of war. One's enemies are often thought to be\u2014at least they are often said to be\u2014unusual and horrifying. Soldiers are encouraged to fight fiercely if they believe that they are fighting for the survival of their country and their families, that freedom, justice, civilization itself are at risk. But this sort of thing is only sometimes plausible to the detached observer, and one suspects that its propagandistic character is also understood by many of the participants. War is not always a struggle over ultimate values, where the victory of one side would be a human disaster for the other. It is necessary to be skeptical about such matters, to cultivate a wary disbelief of wartime rhetoric, and then to search for some touchstone against which arguments about extremity might be judged. We need to make a map of human crises and to mark off the regions of desperation and disaster. These and only these constitute the realm of necessity, truly understood. Once again, I am going to use the experience of World War II in Europe to suggest at least the rough contours of the map. For Nazism lies at the outer limits of exigency, at a point where we are likely to find ourselves united in fear and abhorrence.\n\nThat is what I am going to assume, at any rate, on behalf of all those people who believed at the time and still believe a third of a century later that Nazism was an ultimate threat to everything decent in our lives, an ideology and a practice of domination so murderous, so degrading even to those who might survive, that the consequences of its final victory were literally beyond calculation, immeasurably awful. We see it\u2014and I don't use the phrase lightly\u2014as evil objectified in the world, and in a form so potent and apparent that there could never have been anything to do but fight against it. I obviously cannot offer an account of Nazism in these pages. But such an account is hardly necessary. It is enough to point to the historical experience of Nazi rule. Here was a threat to human values so radical that its imminence would surely constitute a supreme emergency; and this example can help us understand why lesser threats might not do so.\n\nIn order to get the map right, however, we must imagine a Nazi-like danger somewhat different from the one the Nazis actually posed. When Churchill said that a German victory in World War II \"would be fatal, not only to ourselves, but to the independent life of every small country in Europe,\" he was speaking the exact truth. The danger was a general one. But suppose it had existed for Britain alone. Can a supreme emergency be constituted by a particular threat\u2014by a threat of enslavement or extermination directed against a single nation? Can soldiers and statesmen override the rights of innocent people for the sake of their own political community? I am inclined to answer this question affirmatively, though not without hesitation and worry. What choice do they have? They might sacrifice themselves in order to uphold the moral law, but they cannot sacrifice their countrymen. Faced with some ultimate horror, their options exhausted, they will do what they must to save their own people. That is not to say that their decision is inevitable (I have no way of knowing that), but the sense of obligation and of moral urgency they are likely to feel at such a time is so overwhelming that a different outcome is hard to imagine.\n\nStill, the question is difficult, as its domestic analogue suggests. Despite Baldwin, it is not usually said of individuals in domestic society that they necessarily will or that they morally can strike out at innocent people, even in the supreme emergency of self-defense. They can only attack their attackers. But communities, in emergencies, seem to have different and larger prerogatives. I am not sure that I can account for the difference, without ascribing to communal life a kind of transcendence that I don't believe it to have. Perhaps it is only a matter of arithmetic: individuals cannot kill other individuals to save themselves, but to save a nation we can violate the rights of a determinate but smaller number of people. But then large nations and small ones would have different entitlements in such cases, and I doubt very much that this is true. We might better say that it is possible to live in a world where individuals are sometimes murdered, but a world where entire peoples are enslaved or massacred is literally unbearable. For the survival and freedom of political communities\u2014whose members share a way of life, developed by their ancestors, to be passed on to their children\u2014are the highest values of international society. Nazism challenged these values on a grand scale, but challenges more narrowly conceived, if they are of the same kind, have similar moral consequences. They bring us under the rule of necessity (and necessity knows no rules).\n\nI want to stress again, however, that the mere recognition of such a threat is not itself coercive; it neither compels nor permits attacks on the innocent, so long as other means of fighting and winning are available. Danger makes only half the argument; imminence makes the other half. Now let us consider a time when the two halves came together: the terrible two years that followed the defeat of France, from the summer of 1940 to the summer of 1942, when Hitler's armies were everywhere triumphant.\n\nOverriding the Rules of War\n\nThe Decision to Bomb German Cities\n\nThere have been few decisions more important than this one in the history of warfare. As a direct result of the adoption of a policy of terror bombing by the leaders of Britain, some 300,000 Germans, most of them civilians, were killed and another 780,000 seriously injured. No doubt, these figures are low when compared to the results of Nazi genocide; but they were, after all, the work of men and women at war with Nazism, who hated everything it stood for and who were not supposed to imitate its effects, even at lagging rates. And the British policy had further consequences: it was the crucial precedent for the fire-bombing of Tokyo and other Japanese cities and then for Harry Truman's decision to drop atomic bombs on Hiroshima and Nagasaki. The civilian death toll from Allied terrorism in World War II must have exceeded half a million men, women, and children. How could the initial choice of this ultimate weapon ever have been defended?\n\nThe history is a complex one, and it has already been the subject of several monographic analyses. I can review it only briefly, attending especially to the arguments put forward at the time by Churchill and other British leaders, and always remembering what sort of a time it was. The decision to bomb cities was made late in 1940. A directive issued in June of that year had \"specifically laid down that targets had to be identified and aimed at. Indiscriminate bombing was forbidden.\" In November, after the German raid on Coventry, \"Bomber Command was instructed simply to aim at the center of a city.\" What had once been called indiscriminate bombing (and commonly condemned) was now required, and by early 1942, aiming at military or industrial targets was barred: \"the aiming points are to be the built-up areas, not, for instance, the dockyards or aircraft factories.\" The purpose of the raids was explicitly declared to be the destruction of civilian morale. Following the famous minute of Lord Cherwell in 1942, the means to this demoralization were specified: working-\u00adclass residential areas were the prime targets. Cherwell thought it possible to render a third of the German population homeless by 1943.\n\nBefore Cherwell provided his \"scientific\" rationale for the bombing, a number of reasons had already been offered for the British decision. From the beginning, the attacks were defended as reprisals for the German blitz. This is a very problematic defense, even if we leave aside the difficulties of the doctrine of reprisals (which I have already canvassed). First of all, it appears possible, as one scholar has recently argued, that Churchill deliberately provoked the German attacks on London\u2014by bombing Berlin\u2014in order to relieve pressure on R.A.F. installations, until then the major Luftwaffe target. Nor was it Churchill's purpose, once the blitz began, to deter the German attacks or to establish a policy of mutual restraint.\n\nWe ask no favor of the enemy. We seek from them no compunction. On the contrary, if tonight the people of London were asked to cast their votes whether a convention should be entered into to stop the bombing of all cities, the overwhelming majority would cry, \"No, we will mete out to the Germans the measure, and more than the measure, that they have meted out to us.\"\n\nNeedless to say, the people of London were not in fact asked to vote on such a convention. Churchill assumed that the bombing of German cities was necessary to their morale and that they wanted to hear (what he told them in a radio broadcast of 1941) that the British air force was making \"the German people taste and gulp each month a sharper dose of the miseries they have showered upon mankind.\" This argument has been accepted by many historians: there was \"a popular clamor\" for revenge, one of them writes, which Churchill had to satisfy if he was to maintain a fighting spirit among his own people. It is especially interesting to note, then, that a 1941 opinion poll showed that \"the most determined demand for [reprisal raids] came from Cumberland, Westmoreland, and the North Riding of Yorkshire, rural areas barely touched by bombing, where some three-quarters of the population wanted them. In central London, conversely, the proportion was only 45 percent.\" Men and women who had experienced terror bombing were less likely to support Churchill's policy than those who had not\u2014a heartening statistic, and one which suggests that the morale of the British people (or perhaps better, their conventional morality) allowed for political leadership of a different sort than Churchill provided. The news that Germany was being bombed was certainly glad tidings in Britain; but as late as 1944, according to other opinion surveys, the overwhelming majority of Britishers still believed that the raids were directed solely against military targets. Presumably, that is what they wanted to believe; there was by then quite a bit of evidence to the contrary. But that says something, again, about the character of British morale. (It should also be said that the campaign against terror bombing, run largely by pacifists, attracted very little popular support.)\n\nReprisal was a bad argument; revenge was a worse one. We must concentrate now on the military justifications for terror bombing, which were presumably paramount in Churchill's mind, whatever he said on the radio. I can discuss these only in a general way. There was a great deal of dispute at the time, some of it technical, some of it moral in character. The calculations of the Cherwell minute, for example, were sharply attacked by a group of scientists whose opposition to terrorism may well have had moral grounds, but whose position, to the best of my knowledge, was never stated in moral terms. Explicit moral disagreement developed most importantly among the professional soldiers involved in the decision-\u00admaking process. These disagreements are described, in characteristic fashion, by a strategic analyst and historian who has studied the British escalation: \"The . . . debate had been beclouded by emotion on one side of the argument, on the part of those who as a matter of moral principle objected to making war on civilians.\" The focus of these objections seems to have been some version of the doctrine of double effect. (The arguments had, to the mind of the strategic analyst, \"a curiously scholastic flavor.\") At the height of the blitz, many British officers still felt strongly that their own air attacks should be aimed to minimize civilian casualties. They did not want to imitate Hitler, but to differentiate themselves from him. Even officers who accepted the desirability of killing civilians still sought to maintain their professional honor: such deaths, they insisted, were desirable \"only insofar as [they] remained a by-product of the primary intention to hit a military target. . . .\" A tendentious argument, no doubt, yet one that would drastically have limited the British offensive against cities. But all such proposals ran up against the operational limits of the bomber technology then available.\n\nEarly in the war, it became clear that British bombers could fly effectively only at night and, given the navigational devices with which they were equipped, that they could reasonably aim at no target smaller than a fairly large city. A study made in 1941 indicated that of those planes that actually succeeded in attacking their target (about two-thirds of the attacking force), only one-third dropped their bombs within five miles of the point aimed at. Once this was known, it would seem dishonest to claim that the intended target was, say, this aircraft factory and that the indiscriminate destruction around it was only an unintended, if foreseeable, consequence of the justified attempt to stop the production of planes. What was really unintended but foreseeable was that the factory itself would probably escape harm. If any sort of strategic bombing offensive was to be maintained, one would have to plan for the destruction that one could and did cause. Lord Cherwell's minute was an effort at such planning. In fact, of course, navigational devices were rapidly improved as the war went on, and the bombing of specific military targets was an important part of Britain's total air offensive, receiving top priority at times (before the June 1944 invasion of France, for example) and cutting into the resources allowed for attacks on cities. Today many experts believe that the war might have ended sooner had there been a greater concentration of air power against targets such as the German oil refineries. But the decision to bomb cities was made at a time when victory was not in sight and the specter of defeat ever present. And it was made when no other decision seemed possible if there was to be any sort of military offensive against Nazi Germany.\n\nBomber Command was the only offensive weapon available to the British in those frightening years, and I expect there is some truth to the notion that it was used simply because it was there. \"It was the only force in the West,\" writes Arthur Harris, chief of Bomber Command from early 1942 until the end of the war, \"which could take offensive action . . . against Germany, our only means of getting at the enemy in a way that would hurt at all.\" Offensive action could have been postponed until (or in hope of) some more favorable time. That is what the war convention would require, and there was also considerable military pressure for postponement. Harris was hard-pressed to keep his Command together in the face of repeated calls for tactical air support\u2014which would have been coordinated with ground action largely defensive in character, since the German armies were still advancing everywhere. Sometimes, in his memoirs, he sounds like a bureaucrat defending his function and his office, but obviously he was also defending a certain conception of how the war might best be fought. He did not believe that the weapons he commanded should be used because he commanded them. He believed that the tactical use of bombers could not stop Hitler and that the destruction of cities could. Later in the war, he argued that only the destruction of cities could bring the fighting to a quick conclusion. The first of these arguments, at least, deserves a careful examination. It was apparently accepted by the Prime Minister. \"The bombers alone,\" Churchill had said as early as September 1940, \"provide the means of victory.\"\n\nThe bombers alone\u2014that poses the issue very starkly, and perhaps wrongly, given the disputes over strategy to which I have already referred. Churchill's statement suggested a certainty to which neither he nor anyone else had any right. But the issue can be put so as to accommodate a degree of skepticism and to permit even the most sophisticated among us to indulge in a common and a morally important fantasy: suppose that I sat in the seat of power and had to decide whether to use Bomber Command (in the only way that it could be used systematically and effectively) against cities. Suppose further that unless the bombers were used in this way, the probability that Germany would eventually be defeated would be radically reduced. It makes no sense at this point to quantify the probabilities; I have no clear notion what they actually were or even how they might be calculated given our present knowledge; nor am I sure how different figures, unless they were very different, would affect the moral argument. But it does seem to me that the more certain a German victory appeared to be in the absence of a bomber offensive, the more justifiable was the decision to launch the offensive. It is not just that such a victory was frightening, but also that it seemed in those years very close; it is not just that it was close, but also that it was so frightening. Here was a supreme emergency, where one might well be required to override the rights of innocent people and shatter the war convention.\n\nGiven the view of Nazism that I am assuming, the issue takes this form: should I wager this determinate crime (the killing of innocent people) against that immeasurable evil (a Nazi triumph)? Obviously, if there is some other way of avoiding the evil or even a reasonable chance of another way, I must wager differently or elsewhere. But I can never hope to be sure; a wager is not an experiment. Even if I wager and win, it is still possible that I was wrong, that my crime was unnecessary to victory. But I can argue that I studied the case as closely as I was able, took the best advice I could find, sought out available alternatives. And if all this is true, and my perception of evil and imminent danger not hysterical or self-\u00adserving, then surely I must wager. There is no option; the risk otherwise is too great. My own action is determinate, of course, only as to its direct consequences, while the rule that bars such acts is founded on a conception of rights that transcends all immediate considerations. It arises out of our common history; it holds the key to our common future. But I dare to say that our history will be nullified and our future condemned unless I accept the burdens of criminality here and now.\n\nThis is not an easy argument to make, and yet we must resist every effort to make it easier. Many people undoubtedly found some comfort in the fact that the cities being bombed were German and some of the victims Nazis. In effect, they applied the sliding scale and denied or diminished the rights of German civilians so as to deny or diminish the horror of their deaths. This is a tempting procedure, as we can see most clearly if we consider again the bombing of occupied France. Allied fliers killed many Frenchmen, but they did so while bombing what were (or were thought to be) military targets. They did not deliberately aim at the \"built-up areas\" of French cities. Suppose such a policy had been proposed. I am sure that we would all find the wager more difficult to undertake and defend if, through some strange combination of circumstances, it required the deliberate slaughter of Frenchmen. For we had special commitments to the French; we were fighting on their behalf (and sometimes the bombers were flown by French pilots). But the status of the civilians in the two cases is no different. The theory that distinguishes combatants from noncombatants does not distinguish Allied from enemy noncombatants, at least not with regard to the question of their murder. I suppose it makes sense to say that there were more people in German than in French cities who were responsible (in some fashion) for the evil of Nazism, and we may well be reluctant to extend to them the full range of civilian rights. But even if that reluctance is justified, there is no way for the bombers to search out the right people. And for all the others, terrorism only reiterates the tyranny that the Nazis had already established. It assimilates ordinary men and women to their government as if the two really made a totality, and it judges them in a totalitarian way. If one is forced to bomb cities, it seems to me, it is best to acknowledge that one has also been forced to kill the innocent.\n\nOnce again, however, I want to set radical limits to the notion of necessity even as I have myself been using it. For the truth is that the supreme emergency passed long before the British bombing reached its crescendo. The greater number by far of the German civilians killed by terror bombing were killed without moral (and probably also without military) reason. The decisive point was made by Churchill in July of 1942:\n\nIn the days when we were fighting alone, we answered the question: \"How are you going to win the war?\" by saying: \"We will shatter Germany by bombing.\" Since then the enormous injuries inflicted on the German Army and manpower by the Russians, and the accession of the manpower and munitions of the United States, have rendered other possibilities open.\n\nSurely, then, it was time to stop the bombing of cities and to aim, tactically and strategically, only at legitimate military targets. But that was not Churchill's view: \"All the same, it would be a mistake to cast aside our original thought . . . that the severe, ruthless bombing of Germany on an ever-increasing scale will not only cripple her war effort . . . but will create conditions intolerable to the mass of the German population.\" So the raids continued, culminating in the spring of 1945\u2014when the war was virtually won\u2014in a savage attack on the city of Dresden in which something like 100,000 people were killed. Only then did Churchill have second thoughts. \"It seems to me that the moment has come when the question of bombing German cities simply for the sake of increasing the terror, though under other pretexts, should be reviewed. . . . The destruction of Dresden remains a serious query against the conduct of Allied bombing.\" Indeed it does, but so does the destruction of Hamburg and Berlin and all the other cities attacked simply for the sake of terror.\n\nThe argument used between 1942 and 1945 in defense of terror bombing was utilitarian in character, its emphasis not on victory itself but on the time and price of victory. The city raids, it was claimed by men such as Harris, would end the war sooner than it would otherwise end and, despite the large number of civilian casualties they inflicted, at a lower cost in human life. Assuming this claim to be true (I have already indicated that precisely opposite claims are made by some historians and strategists), it is nevertheless not sufficient to justify the bombing. It is not sufficient, I think, even if we do nothing more than calculate utilities. For such calculations need not be concerned only with the preservation of life. There is much else that we might plausibly want to preserve: the quality of our lives, for example, our civilization and morality, our collective abhorrence of murder, even when it seems, as it always does, to serve some purpose. Then the deliberate slaughter of innocent men and women cannot be justified simply because it saves the lives of other men and women. I suppose it is possible to imagine situations where that last assertion might prove problematic, from a utilitarian perspective, where the number of people involved is small, the proportions are right, the events hidden from the public eye, and so on. Philosophers delight in inventing such cases in order to test out our moral doctrines. But their inventions are somehow put out of our minds by the sheer scale of the calculations necessary in World War II. To kill 278,966 civilians (the number is made up) in order to avoid the deaths of an unknown but probably larger number of civilians and soldiers is surely a fantastic, godlike, frightening, and horrendous act.a\n\nI have said that such acts can probably be ruled out on utilitarian grounds, but it is also true that utilitarianism as it is commonly understood, indeed, as Sidgwick himself understands it, encourages the bizarre accounting that makes them (morally) possible. We can recognize their horror only when we have acknowledged the personality and value of the men and women we destroy in committing them. It is the acknowledgment of rights that puts a stop to such calculations and forces us to realize that the destruction of the innocent, whatever its purposes, is a kind of blasphemy against our deepest moral commitments. (This is true even in a supreme emergency, when we cannot do anything else.) But I want to look at one more case before concluding my argument\u2014a case where the utilitarian accounting, however bizarre, seemed so radically clear-cut to the decision-makers as to leave them, they thought, no choice but to attack the innocent.\n\nThe Limits of Calculation\n\nHiroshima\n\n\"They all accepted the 'assignment' and produced The Bomb,\" Dwight Macdonald wrote in August 1945 of the atomic scientists. \"Why?\" It is an important question, but Macdonald poses it badly and then gives the wrong answer. \"Because they thought of themselves as specialists, technicians, and not as complete men.\" In fact, they did not accept the assignment; they sought it out, taking the initiative, urging upon President Roosevelt the critical importance of an American effort to match the work being done in Nazi Germany. And they did this precisely because they were \"complete men,\" many of them European refugees, with an acute sense of what a Nazi victory would mean for their native lands and for all mankind. They were driven by a deep moral anxiety, not (or not most crucially) by any kind of scientific fascination; they were certainly not servile technicians. On the other hand, they were men and women without political power or following, and once their own work was done, they could not control its use. The discovery in November 1944 that German scientists had made little progress ended their own supreme emergency, but it did not end the program they had helped to launch. \"If I had known that the Germans would not succeed in constructing the atom bomb,\" Albert Einstein said, \"I would never have lifted a finger.\" By the time he found that out, however, the scientists had largely finished their work; now indeed technicians were in charge, and the politicians in charge of them. And in the event, the bomb was not used against Germany (or to deter its use by Hitler, which is what men like Einstein had in mind), but against the Japanese, who had never posed such a threat to peace and freedom as the Nazis had.b\n\nStill, it was an important feature of the American decision that the President and his advisors believed the Japanese to be fighting an aggressive war and, moreover, to be fighting it unjustly. Thus Truman's address to the American people on August 12, 1945:\n\nWe have used [the bomb] against those who attacked us without warning at Pearl Harbor, against those who have starved and beaten and executed American prisoners of war, against those who have abandoned all pretense of obeying international laws of warfare. We have used it in order to shorten the agony of war . . .\n\nHere again, the sliding scale is being used to open the way for utilitarian calculations. The Japanese have forfeited (some of) their rights, and so they cannot complain about Hiroshima so long as the destruction of the city actually does, or could reasonably be expected to, shorten the agony of war. But had the Japanese exploded an atomic bomb over an American city, killing tens of thousands of civilians and thereby shortening the agony of war, the action would clearly have been a crime, one more for Truman's list. This distinction is only plausible, however, if one renders a judgment not only against the leaders of Japan but also against the ordinary people of Hiroshima and insists at the same time that no similar judgment is possible against the people of San Francisco, say, or Denver. I can find, as I have said before, no way of defending such a procedure. How did the people of Hiroshima forfeit their rights? Perhaps their taxes paid for some of the ships and planes used in the attack on Pearl Harbor; perhaps they sent their sons into the navy and air force with prayers for their success; perhaps they celebrated the actual event, after being told that their country had won a great victory in the face of an imminent American threat. Surely there is nothing here that makes these people liable to direct attack. (It is worth noting, though the fact is not relevant in judging the Hiroshima decision, that the raid on Pearl Harbor was directed entirely against naval and army installations: only a few stray bombs fell on the city of Honolulu.)\n\nBut if Truman's argument on August 12 was weak, there was a worse one underlying it. He did not intend to apply the sliding scale with any precision, for he seems to have believed that, given Japanese aggression, the Americans could do anything at all to win (and shorten the agony of war). Along with most of his advisors, he accepted the \"war is hell\" doctrine; it is a constant allusion in defenses of the Hiroshima decision. Thus Henry Stimson:\n\nAs I look back over the five years of my service as Secretary of War, I see too many stern and heartrending decisions to be willing to pretend that war is anything else but what it is. The face of war is the face of death; death is an inevitable part of every order that a wartime leader gives.\n\nAnd James Byrnes, Truman's friend and his Secretary of State:\n\n. . . war remains what General Sherman said it was.\n\nAnd Arthur Compton, chief scientific advisor to the government:\n\nWhen one thinks of the mounted archers of Ghengiz Khan . . . the Thirty Years War . . . the millions of Chinese who died during the Japanese invasion . . . the mass destruction of western Russia . . . one realizes that in whatever manner it is fought, war is precisely what General Sherman called it.\n\nAnd Truman himself:\n\nLet us not become so preoccupied with weapons that we lose sight of the fact that war itself is the real villain.\n\nWar itself is to blame, but also the men who begin it . . . while those who fight justly merely participate in the hell of war, choicelessly, and there are no moral decisions for which they can be called to account. This is not, or not necessarily, an immoral doctrine, but it is radically one-sided; it evades the tension between jus ad bellum and jus in bello; it undercuts the need for hard judgments; it relaxes our sense of moral restraint. When he was choosing a target for the first bomb, Truman reports, he asked Stimson which Japanese cities were \"devoted exclusively to war production.\" The question was reflexive; Truman did not want to violate the \"laws of war.\" But it wasn't serious. Which American cities were devoted exclusively to war production? It is possible to ask such questions only when the answer doesn't matter. If war is hell however it is fought, then what difference can it make how we fight it? And if war itself is the villain, then what risks do we run (aside from the strategic risks) when we make decisions? The Japanese, who began the war, can also end it; only they can end it, and all we can do is fight it, enduring what Truman called \"the daily tragedy of bitter war.\" I don't doubt that that was really Truman's view; it was not a matter of convenience but of conviction. But it is a distorted view. It mistakes the actual hellishness of war, which is particular in character and open to precise definition, for the limitless pains of religious mythology. The pains of war are limitless only if we make them so\u2014only if we move, as Truman did, beyond the limits that we and others have established. Sometimes, I think, we have to do that, but not all the time. Now we must ask whether it was necessary to do it in 1945.\n\nThe only possible defense of the Hiroshima attack is a utilitarian calculation made without the sliding scale, a calculation made, then, where there was no room for it, a claim to override the rules of war and the rights of Japanese civilians. I want to state this argument as strongly as I can. In 1945, American policy was fixed on the demand for the unconditional surrender of Japan. The Japanese had by that time lost the war, but they were by no means ready to accept this demand. The leaders of their armed forces expected an invasion of the Japanese main islands and were preparing for a last-ditch resistance. They had over two million soldiers available for the fighting, and they believed that they could make the invasion so costly that the Americans would agree to a negotiated peace. Truman's military advisors also believed that the costs would be high, though the public record does not show that they ever recommended negotiations. They thought that the war might continue late into 1946 and that there would be as many as a million additional American casualties. Japanese losses would be much higher. The capture of Okinawa in a battle lasting from April to June of 1945 had cost almost 80,000 American casualties, while virtually the entire Japanese garrison of 120,000 men had been killed (only 10,600 prisoners were taken). If the main islands were defended with a similar ferocity, hundreds of thousands, perhaps millions, of Japanese soldiers would die. Meanwhile, the fighting would continue in China and in Manchuria, where a Russian attack was soon due. And the bombing of Japan would also continue, and perhaps intensify, with casualty rates no different from those anticipated from the atomic attack. For the Americans had adopted in Japan the British policy of terrorism: a massive incendiary raid on Tokyo early in March 1945 had set off a firestorm and killed an estimated 100,000 people. Against all this was set, in the minds of American decision-makers, the impact of the atomic bomb\u2014not materially more damaging but psychologically more frightening, and holding out the promise, perhaps, of a quick end to the war. \"To avert a vast, indefinite butchery . . . at the cost of a few explosions,\" wrote Churchill in support of Truman's decision, \"seemed, after all our toils and perils, a miracle of deliverance.\"\n\n\"A vast indefinite butchery\" involving quite probably the deaths of several million people: surely this is a great evil, and if it was imminent, one could reasonably argue that extreme measures might be warranted to avert it. Secretary of War Stimson thought it was the sort of case I have already described, where one had to wager; there was no option. \"No man, in our position and subject to our responsibilities, holding in his hand a weapon of such possibilities for . . . saving those lives, could have failed to use it.\" This is by no means an incomprehensible or, on the surface at least, an outrageous argument. But it is not the same as the argument I suggested in the case of Britain in 1940. It does not have the form: if we don't do x (bomb cities), they will do y (win the war, establish tyrannical rule, slaughter their opponents). What Stimson argued is very different. Given the actual policy of the U.S. government, it amounts to this: if we don't do x, we will do y. The two atomic bombs caused \"many casualties,\" James Byrnes admitted, \"but not nearly so many as there would have been had our air force continued to drop incendiary bombs on Japan's cities.\" Our purpose, then, was not to avert a \"butchery\" that someone else was threatening, but one that we were threatening, and had already begun to carry out. Now, what great evil, what supreme emergency, justified the incendiary attacks on Japanese cities?\n\nEven if we had been fighting in strict accordance with the war convention, the continuation of the struggle was not something forced upon us. It had to do with our war aims. The military estimate of casualties was based not only on the belief that the Japanese would fight almost to the last man, but also on the assumption that the Americans would accept nothing less than unconditional surrender. The war aims of the American government required either an invasion of the main islands, with enormous losses of American and Japanese soldiers and of Japanese civilians trapped in the war zones, or the use of the atomic bomb. Given that choice, one might well reconsider those aims. Even if we assume that unconditional surrender was morally desirable because of the character of Japanese militarism, it might still be morally undesirable because of the human costs it entailed. But I would suggest a stronger argument than this. The Japanese case is sufficiently different from the German so that unconditional surrender should never have been asked. Japan's rulers were engaged in a more ordinary sort of military expansion, and all that was morally required was that they be defeated, not that they be conquered and totally overthrown. Some restraint upon their war-making power might be justified, but their domestic authority was a matter of concern only to the Japanese people. In any case, if killing millions (or many thousands) of men and women was militarily necessary for their conquest and overthrow, then it was morally necessary\u2014in order not to kill those people\u2014to settle for something less. I have made this argument before (in chapter 7); here is a further example of its practical application. If people have a right not to be forced to fight, they also have a right not to be forced to continue fighting beyond the point when the war might justly be concluded. Beyond that point, there can be no supreme emergencies, no arguments about military necessity, no cost-accounting in human lives. To press the war further than that is to re-commit the crime of aggression. In the summer of 1945, the victorious Americans owed the Japanese people an experiment in negotiation. To use the atomic bomb, to kill and terrorize civilians, without even attempting such an experiment, was a double crime.\n\nThese, then, are the limits of the realm of necessity. Utilitarian calculation can force us to violate the rules of war only when we are face-to-face not merely with defeat but with a defeat likely to bring disaster to a political community. But these calculations have no similar effects when what is at stake is only the speed or the scope of victory. They are relevant only to the conflict between winning and fighting well, not to the internal problems of combat itself. Whenever that conflict is absent, calculation is stopped short by the rules of war and the rights they are designed to protect. Confronted by those rights, we are not to calculate consequences, or figure relative risks, or compute probable casualties, but simply to stop short and turn aside.\n\n George Orwell has suggested an alternative utilitarian rationale for the bombing of German cities. In a column written for the leftist journal Tribune in 1944, he argued that the bombing brought the true character of contemporary combat home to all those people who supported the war, even enjoyed it, only because they never felt its effects. It shattered \"the immunity of civilians, one of the things that have made war possible,\" and so it made war less likely in the future. See The Collected Essays, Journalism and Letters of George Orwell, ed. Sonia Orwell and Ian Angus, New York, 1968, Vol. 3, pp. 151\u2013152. Orwell assumes that civilians had really been immune in the past, which is false. In any case, I doubt that his argument would lead anyone to begin bombing cities. It is an apology after the fact, and not a convincing one.\n\n In his novel The New Men, C. P. Snow describes the discussions among atomic scientists as to whether or not the bomb should be used. Some of them, his narrator says, answered that question with \"an absolute no,\" feeling that if the weapon were used to kill hundreds of thousands of innocent people, \"neither science nor the civilization of which science is bone and fibre, would be free from guilt again.\" But the more common view was the one I have been defending: \"Many, probably the majority, gave a conditional no with much the same feeling behind it; but if there were no other way of saving the war against Hitler, they would be prepared to drop the bomb.\" The New Men, New York, 1954, p. 177 (Snow's emphasis).\n\nNuclear Deterrence\n\nThe Problem of Immoral Threats\n\nTruman used the atomic bomb to end a war that seemed to him limitless in its horrors. And then, for a few minutes or hours in August 1945, the people of Hiroshima endured a war that actually was limitless in its horrors. \"In this last great action of the Second War War,\" wrote Stimson, \"we were given final proof that war is death.\" Final proof is exactly the wrong phrase, for war had never been like that before. A new kind of war was born at Hiroshima, and what we were given was a first glimpse of its deadliness. Though fewer people were killed than in the fire-bombing of Tokyo, they were killed with monstrous ease. One plane, one bomb: with such a weapon the 350 planes that raided Tokyo would virtually have wiped out human life on the Japanese islands. Atomic war was death indeed, indiscriminate and total, and after Hiroshima, the first task of political leaders everywhere was to prevent its recurrence.\n\nThe means they adopted is the promise of reprisal in kind. Against the threat of an immoral attack, they have put the threat of an immoral response. This is the basic form of nuclear deterrence. In international as in domestic society, deterrence works by calling up dramatic images of human pain. \"In the groves of their academy,\" wrote Edmund Burke of the liberal theorists of crime and punishment, \"at the end of every vista, you see nothing but the gallows.\" The description is uncomplimentary, for Burke believed that domestic peace must rest upon some other foundation. But there is this much to be said for the gallows: in principle, at least, only guilty men need fear the death it brings. About the theorists of deterrence, however, it must be said, \"In the groves of their academy, at the end of every vista, you see nothing but the mushroom cloud\"\u2014and the cloud symbolizes indiscriminate slaughter, the killing of the innocent (as in Hiroshima) on a massive scale. No doubt, the threat of such slaughter, if it is believed, makes nuclear attack a radically undesirable policy. Doubled by a potential enemy, the threat produces a \"balance of terror.\" Both sides are so terrified that no further terrorism is necessary. But is the threat itself morally permissible?\n\nThe question is a difficult one. It has generated in the years since Hiroshima a significant body of literature exploring the relation between nuclear deterrence and just war. This has been the work mostly of theologians and philosophers, but some of the strategists of deterrence have also been involved; they worry about the act of terrorizing much as conventional soldiers worry about the act of killing. I cannot review this literature here, though I shall draw upon it freely. The argument against deterrence is familiar enough. Anyone committed to the distinction between combatants and noncombatants is bound to be appalled by the specter of destruction evoked, and purposely evoked, in deterrence theory. \"How can a nation live with its conscience,\" John Bennett has asked, \"and know that it is preparing to kill twenty million children in another nation if the worst should come to the worst?\" And yet, we have lived with that knowledge, and with our consciences too, for several decades now. How have we managed? The reason for our acceptance of deterrent strategy, most people would say, is that preparing to kill, even threatening to kill, is not at all the same thing as killing. Indeed it is not, but it is frighteningly close\u2014else deterrence wouldn't \"work\"\u2014and it is in the nature of that closeness that the moral problem lies.\n\nThe problem is often misdescribed\u2014as in the following analogy for nuclear deterrence first suggested by Paul Ramsey and frequently repeated since:\n\nSuppose that one Labor Day weekend no one was killed or maimed on the highways; and that the reason for the remarkable restraint placed on the recklessness of automobile drivers was that suddenly everyone of them discovered he was driving with a baby tied to his front bumper! That would be no way to regulate traffic even if it succeeds in regulating it perfectly, since such a system makes innocent human lives the direct object of attack and uses them as a mere means for restraining the drivers of automobiles.\n\nNo one, of course, has ever proposed regulating traffic in this ingenious way, while the strategy of deterrence was adopted with virtually no opposition at all. That contrast should alert us to what is wrong with Ramsey's analogy. Though deterrence turns American and Russian civilians into mere means for the prevention of war, it does so without restraining any of us in any way. Ramsey reproduces the strategy of the German officers during the Franco-Prussian War who forced civilians to ride on military trains in order to deter saboteurs. By contrast with those civilians, however, we are hostages who lead normal lives. It is in the nature of the new technology that we can be threatened without being held captive. That is why deterrence, while in principle so frightening, is so easy to live with. It cannot be condemned for anything it does to its hostages. It is so far from killing them that it does not even injure or confine them; it involves no direct or physical violation of their rights. Those critics of deterrence who are also committed consequentialists have had to imagine psychic injuries. Thus Eric Fromm, writing in 1960: \"To live for any length of time under the constant threat of destruction creates certain psychological effects in most human beings\u2014fright, hostility, callousness . . . and a resulting indifference to all the values we cherish. Such conditions will transform us into barbarians. . . .\" But I don't know of any evidence that bears out either the assertion or the prediction; surely we are no more barbarians now than we were in 1945. In fact, for most people, the threat of destruction, though constant, is invisible and unnoticed. We have come to live with it casually\u2014as Ramsey's babies, traumatized for life in all probability, could never do, and as hostages in conventional wars have never done.\n\nIf deterrence were more painful, we might have found other means of avoiding nuclear war\u2014or we might not have avoided it. If we had to keep millions of people under restraint in order to maintain the balance of terror, or if we had to kill millions of people (periodically) in order to convince our adversaries of our credibility, deterrence would not be accepted for long. The strategy works because it is easy. Indeed, it is easy in a double sense: not only don't we do anything to other people, we also don't believe that we will ever have to do anything. The secret of nuclear deterrence is that it is a kind of bluff. Perhaps we are only bluffing ourselves, refusing to acknowledge the real terrors of a precarious and temporary balance. But no account of our experience is accurate which fails to recognize that, for all its ghastly potential, deterrence has so far been a bloodless strategy.\n\nSo far as consequences go, then, deterrence and mass murder are very far apart. Their closeness is a matter of moral posture and intention. Once again, Ramsey's analogy misses the point. His babies are not really the \"direct object of attack,\" for whatever happens on that Labor Day weekend, no one will deliberately set out to kill them. But deterrence depends upon a readiness to do exactly that. It is as if the state should seek to prevent murder by threatening to kill the family and friends of every murderer\u2014a domestic version of the policy of \"massive retaliation.\" Surely that would be a repugnant policy. We would not admire the police officials who designed it or those pledged to carry it out, even if they never actually killed anybody. I don't want to say that such people would necessarily be transformed into barbarians; they might well have a heightened sense of how awful murder is and a heightened desire to avoid it; they might loathe the work they were pledged to do and fervently hope that they never had to do it. Nevertheless, the enterprise is immoral. The immorality lies in the threat itself, not in its present or even its likely consequences. Similarly with nuclear deterrence: it is our own intentions that we have to worry about and the potential (since there are no actual) victims of those intensions. Here Ramsey has put the case very well: \"Whatever is wrong to do is wrong to threaten, if the latter means 'mean to do.' . . . If counter-\u00adpopulation warfare is murder, then counter-population deterrent threats are murderous.\" No doubt, killing millions of innocent people is worse than threatening to kill them. It is also true that no one wants to kill them, and it may well be true that no one expects to do so. Nevertheless, we intend the killings under certain circumstances. That is the stated policy of our government; and thousands of men, trained in the techniques of mass destruction and drilled in instant obedience, stand ready to carry it out. And from the perspective of morality, the readiness is all. We can translate it into degrees of danger, high and low, and worry about the risks we are imposing on innocent people, but the risks depend on the readiness. What we condemn in our own government, as in the police in my domestic analogy, is the commitment to murder.a\n\nBut this analogy, too, can be questioned. We don't prevent murder any more than we control traffic in these bizarre and inhuman ways. But we do deter or seek to deter our nuclear adversaries. Perhaps deterrence is different because of the danger its advocates claim to avoid. Traffic deaths and occasional murders, however much we deplore them, do not threaten our common liberties or our collective survival. Deterrence, so we have been told, guards us against a double danger: first, of atomic blackmail and foreign domination; and second, of nuclear destruction. The two go together, since if we did not fear the blackmail, we might adopt a policy of appeasement or surrender and so avoid the destruction. Deterrence theory was worked out at the height of the cold war between the United States and the Soviet Union, and those who worked it out were concerned above all with the political uses of violence\u2014which are not relevant in either the traffic or police analogies. Underlying the American doctrine, there seemed to lurk some version of the slogan \"Better dead than Red\" (I don't know the Russian parallel). Now that is not really a believable slogan; it is hard to imagine that a nuclear holocaust was really thought preferable to the expansion of Soviet power. What made deterrence attractive was that it seemed capable of avoiding both.\n\nWe need not dwell on the nature of the Soviet regime in order to understand the virtues of this argument. Deterrence theory doesn't depend upon a view of Stalinism as a great evil (though that is a highly plausible view) in the same way that my argument about terror bombing depended upon an assertion about the evils of Nazism. It requires only that we see appeasement or surrender to involve a loss of values central to our existence as an independent nation-state. For it is not tolerable that advances in technology should put our nation, or any nation, at the mercy of a great power willing to menace the world or to press its authority outwards in the shadow of an implicit threat. The case here is very different from that which arises commonly in war, where our adherence to the war convention puts us, or would put us, at a disadvantage vis-\u00e0-vis them. For disadvantages of that sort are partial and relative; various counter-\u00admeasures and compensating steps are always available. But in the nuclear case, the disadvantage is absolute. Against an enemy actually willing to use the bomb, self-defense is impossible, and it makes sense to say that the only compensating step is the (immoral) threat to respond in kind. No country capable of making such a threat is likely to refuse to make it. What is not tolerable won't be tolerated. Hence any state confronted by a nuclear adversary (it makes little difference what the adversary relationship is like or what ideological forms it assumes), and capable of developing its own bomb, is likely to do so, seeking safety in a balance of terror.b Mutual disarmament would clearly be a preferable alternative, but it is an alternative available only to the two countries working closely together, whereas deterrence is the likely choice of either one of them alone. They will worry about one another's readiness to attack; they will each assume their own commitment to resist; and they will realize that the greatest danger of such a confrontation would not be the defeat of one side or the other but the total destruction of both\u2014and possibly of everyone else too. This in fact is the danger that has faced mankind since 1945, and our understanding of nuclear deterrence must be worked out with reference to its scope and imminence. Supreme emergency has become a permanent condition. Deterrence is a way of coping with that condition, and though it is a bad way, there may well be no other that is practical in a world of sovereign and suspicious states. We threaten evil in order not to do it, and the doing of it would be so terrible that the threat seems in comparison to be morally defensible.\n\nLimited Nuclear War\n\nIf the bomb were ever used, deterrence would have failed. It is a feature of massive retaliation that while there is or may be some rational purpose in threatening it, there could be none in carrying it out. Were our \"bluff\" ever to be called and our population centers suddenly attacked, the resulting war could not (in any usual sense of the word) be won. We could only drag our enemies after us into the abyss. The use of our deterrent capacity would be an act of pure destructiveness. For this reason, massive retaliation, if not literally unthinkable, has always seemed undo-able, and this is a source of considerable anxiety for military strategists. Deterrence only works, they argue, if each side believes that the other might actually carry out its threat. But would we carry it out? George Kennan has recently given what must be the moral response:\n\nLet us suppose there were to be a nuclear attack of some sort on this country and millions of people were killed and injured. Let us further suppose that we had the ability to retaliate against the urban centers of the country that had attacked us. Would you want to do that? I wouldn't . . . I have no sympathy with the man who demands an eye for an eye in a nuclear attack.\n\nA humane position\u2014though one that should probably be whispered, rather than published, if the balance of terror is to be sustained. But the argument might look very different if the original attack or the planned response avoided cities and people. If a limited nuclear war were possible, wouldn't it also be do-able? And might not the balance of terror then be re-established on the basis of threats that were neither immoral nor unconvincing?\n\nOver a brief timespan, in the late 1950s and the early 1960s, these questions were answered with an extraordinary outpouring of strategic arguments and speculations, overlapping in important ways with the moralizing literature I described earlier. For the debate among the strategists focused on the attempt (though this was rarely made explicit) to fit nuclear war into the structure of the war convention, to apply the argument for justice as if this sort of conflict were like any other sort. The attempt involved, first, a defense of the use of tactical nuclear weapons in deterring and, if that failed, in resisting conventional or small-scale nuclear attacks; and it involved, secondly, the development of a \"counter-force\" strategy directed at the enemy's military installations and also at major economic targets (but not at entire cities). These two had a similar purpose. By holding out the promise of a limited nuclear war, they made it possible to imagine actually fighting such a war\u2014they made it possible to imagine winning it\u2014and so they strengthened the intention that lay behind the deterrent threat. They transformed the \"bluff\" into a plausible option.\n\nUntil the late 1950s, the tendency of most people was to regard the atomic bomb and its thermonuclear successors as forbidden weapons. They were treated on analogy with poison gas, though the prohibition on their use was never legally established. \"Ban the bomb\" was everyone's policy, and deterrence was simply a practical way of enforcing the ban. But now the strategists suggested (rightly) that the crucial distinction in the theory and practice of war was not between prohibited and acceptable weapons but between prohibited and acceptable targets. Massive retaliation was painful and difficult to contemplate because it was modeled on Hiroshima; the people we were planning to kill were innocent, militarily uninvolved, as removed from and ignorant of the weapons with which their leaders threatened us as we were of the weapons with which our leaders threatened them. But this objection would disappear if we could deter our adversaries by threatening a limited and morally acceptable destruction. Indeed, it might disappear so entirely that we would be tempted to give up deterrence and initiate the destruction ourselves whenever it seemed to our advantage to do so. This was certainly the tendency of much strategic argument, and several writers painted rather attractive pictures of limited nuclear war. Henry Kissinger likened it to war at sea\u2014the very best kind of war, since no one lives in the sea. \"The proper analogy . . . is not traditional land warfare, but naval strategy, in which self-contained, highly mobile] units with great fire power gradually gain the upper hand by destroying their enemy counterparts without physically occupying territory or establishing a front line.\" The only difficulty is that Kissinger imagined fighting a war like that in Europe.[c\n\nTactical and counter-force warfare meets the formal requirements of jus in bello, and it was seized upon eagerly by certain moral theorists. That is not to say, however, that it makes moral sense. There remains the possibility that the new technology of war simply doesn't fit and cannot be made to fit within the old limits. This proposition can be defended in two different ways. The first is to argue that the collateral damage likely to be caused even by a \"legitimate\" use of nuclear weapons is so great that it would violate both of the proportionality limits fixed by the theory of war: the number of people killed in the war as a whole would not be warranted by the goals of the war\u2014particularly since the dead would include many if not most of the people for whose defense the war was being fought; and the number of people killed in individual actions would be disproportionate (under the doctrine of double effect) to the value of the military targets directly attacked. \"The disproportion between the cost of such hostilities and the results they could achieve,\" wrote Raymond Aron, thinking of a limited nuclear war in Europe, \"would be colossal.\" It would be colossal even if the formal limits on targeting were in fact observed. But the second argument against limited nuclear war is that these limits would almost certainly not be observed.\n\nAt this point, of course, one can only guess at the possible shape and course of the battles; there is no history to study. Neither moralists nor strategists can refer to cases; instead they design scenarios. The scene is empty; one can fill it in very different ways, and it is not impossible to imagine that limits might be maintained even after nuclear weapons had been used in battle. The prospect that they would be maintained and the war extended over time is so frightening to those countries on whose soil such wars are likely to be fought that they have generally opposed the new strategies and insisted upon the threat of massive retaliation. Thus, as Andr\u00e9 Beaufre has written, \"Europeans would prefer to risk general war in an attempt to avoid war altogether rather than have Europe become the theater of operations for limited war.\" In fact, however, the risks of escalation will be great whatever limits are adopted, simply because of the immense destructive power of the weapons involved. Or rather, there are two possibilities: either nuclear weapons will be held at such low levels that they won't be significantly different from or of greater military utility than conventional explosives, in which case there is no reason to use them at all; or their very use will obliterate the distinction between targets. Once a bomb has been aimed at a military target but has, as a side effect, destroyed a city, the logic of deterrence will require the other side to aim at a city (for the sake of its seriousness and credibility). It is not necessarily the case that every war would become a total war, but the danger of escalation is so great as to preclude the first use of nuclear weapons\u2014except by someone willing to face their final use. \"Who would even launch such hostilities,\" Aron has asked, \"unless he was determined to persist to the bitter end?\" But such a determination is not imaginable in a sane human being, let alone in a political leader responsible for the safety of his own people; it would involve nothing less than national suicide.\n\nThese two factors, the extent even of limited destruction and the dangers of escalation, seem to rule out any sort of nuclear war between the great powers. They probably rule out large-scale conventional war, too, including the particular conventional war about which the strategists of the 1950s and 1960s were most concerned: a Russian invasion of western Europe. \"The spectacle of a large Soviet field army crashing across the line into western Europe in the hope and expectation that nuclear weapons would not be used against it\u2014thereby putting itself and the USSR totally at risk while leaving the choice of weapons to us\u2014would seem to be hardly worth a second thought. . . .\" It is important to stress that the bar lies in the totality of the risk: not in the possibility of what the strategists called a \"flexible response,\" finely adjusted to the scope of the attack, but in the stark reality of ultimate horror should the adjustments fail. It may well be that \"flexible response\" enhanced the value of a counter-population deterrent by making it possible to reach that final point in \"easy\" stages, but it is also and more importantly true that we have never begun the staged escalation and are never likely to begin it, because of what lies at the end. Hence the persistence of counter-population deterrence, and hence also the virtual end of the strategic debate, which petered out in the middle 1960s. At that point, I think, it became clear that given the existence of large numbers of nuclear weapons and their relative invulnerability, and barring major technological breakthroughs, any imaginable strategy is likely to deter a \"central war\" between the great powers. The strategists helped us to understand this, but once it was understood it became unnecessary to adopt any of their strategies\u2014or at least, any particular one of them. We continue to live, then, with the paradox that pre-existed the debate: nuclear weapons are politically and militarily unusable only because and insofar as we can plausibly threaten to use them in some ultimate way. And it is immoral to make threats of that kind.\n\nThe Argument of Paul Ramsey\n\nBefore deciding (or refusing) to live with this paradox, I want to consider in some detail the work of the Protestant theologian Paul Ramsey, who has over a period of years argued that there exists a justifiable deterrent strategy. From the beginning of the moral and strategic debates, Ramsey has been a sharp opponent of the advocates of counter-city deterrence and also of those of its critics who think that it is the only form of deterrence and therefore opt for nuclear disarmament. He has condemned both these groups for the all-or-nothing character of their thinking: either total and immoral destruction or a kind of \"pacifistic\" inertia. He argues that these twin perspectives conform to the traditional American view of war as an all-out conflict, which must therefore be avoided whenever possible. Ramsey himself, I think, is a Protestant soldier in a different tradition; he would have Americans gird themselves for a long, continuous struggle with the forces of evil.\n\nNow if there is to be a justified deterrent strategy, there must be a justified form of nuclear war, and Ramsey has conscientiously argued \"the case for making just war possible\" in the modern age. He takes a lively and well-informed interest in the strategic debates and has at various times defended the use of tactical nuclear weapons against invading armies and of strategic weapons against nuclear installations, conventional military bases, and isolated economic objectives. Even these targets are only \"conditionally\" permissible, since the proportionality rule would have to be applied in each case, and Ramsey does not believe that its standards will always be met. Like everyone (or almost everyone) who writes about these matters, he has no zest for nuclear combat; his main interest is in deterrence. But he needs at least the possibility of legitimate warfare if he is to maintain a deterrent posture without making immoral threats. That is his central purpose, and the effort to achieve it involves him in a highly sophisticated application of just war theory to the problems of nuclear strategy. In the best sense of the word, Ramsey is engaged with the realities of his world. But the realities in this case are intractable, and his way around them is finally too complex and too devious to provide a plausible account of our moral judgments. He multiplies distinctions like a Ptolemaic astronomer with his epicycles and comes very close at the end to what G. E. M. Anscombe has called \"double-think about double effect.\" But his work is important; it suggests the outer limits of the just war and the dangers of trying to extend those limits.\n\nRamsey's central claim is that it is possible to prevent nuclear attack without threatening to bomb cities in response. He believes that \"the collateral civilian damage that would result from counter-force warfare in its maximum form\" would be sufficient to deter potential aggressors. Since the civilians likely to die in such a war would be the incidental victims of legitimate military strikes, the threat of counter-force warfare plus collateral damage is also morally superior to deterrence in its present form. These are not hostages whom we intend to murder (under certain circumstances). Nor are we planning their deaths; we are only pointing out to our possible enemies the unavoidable consequences even of a war justly fought\u2014which is, we could honestly say were we to adopt Ramsey's proposal, the only sort of war we were preparing to fight. Collateral damage is simply a fortunate feature of nuclear warfare; it serves no military purpose, and we would avoid it if we could, though it is clearly a good thing that we cannot. And since the damage is justifiable in prospect, it is also justifiable here and now to call that prospect to mind for the sake of its deterrent effects.\n\nBut there are two problems with this argument. First, the danger of collateral damage is unlikely to work as a deterrent unless the damage expected is radically disproportionate to the ends of the war or the value of this or that military target. Hence Ramsey is driven to argue that \"the threat of something disproportionate is not always a disproportionate threat.\" What that means is this: proportionality in combat is measured, let's say, against the value of a particular missile base, while proportionality in deterrence is measured against the value of world peace. So the damage may not be justifiable in prospect (under the doctrine of double effect), and yet the threat of such damage may still be morally permitted. Perhaps that argument is right, but I should stress that its result is to void the proportionality rule. Now there is no limit on the number of people whose deaths we can threaten, so long as those deaths are to be caused \"collaterally\" and not by taking direct aim. As we have seen before, the idea of proportionality, once it is worked on a bit, tends to fade away. And then the entire burden of Ramsey's argument falls on the idea of death by indirection. That is indeed an important idea, central to the permissions and restraints of conventional war. But its standing is undermined here by the fact that Ramsey relies so heavily on the deaths he supposedly doesn't intend. He wants, like other deterrent theorists, to prevent nuclear attack by threatening to kill very large numbers of innocent civilians, but unlike other deterrent theorists, he expects to kill these people without aiming at them. That may be a matter of some moral significance, but it does not seem significant enough to serve as the cornerstone of a justified deterrent. If counter-force warfare had no collateral effects, or had minor and controllable effects, then it could play no part in Ramsey's strategy. Given the effects it does have and the central part it is assigned, the word \"collateral\" seems to have lost much of its meaning. Surely anyone designing such a strategy must accept moral responsibility for the effects on which he is so radically dependent.\n\nBut we have not yet seen the whole of Ramsey's design, for he doesn't pull back from the hardest questions. What if the likely collateral damage of a just nuclear war isn't great enough to deter a would-be aggressor? What if the aggressor threatens a counter-city strike? Surrender would be intolerable, and yet we cannot ourselves threaten mass murder in response. Fortunately (again), we don't have to. \"We do not need . . . to threaten that we will use [nuclear weapons] in case of attack,\" Bernard Brodie has written. \"We do not need to threaten anything. Their being there is quite enough.\" So it is, too, according to Ramsey, with counter-city strikes: the mere possession of nuclear weapons constitutes an implicit threat which no one actually has to make. If the immorality lies in uttering the threat, then it may in practice be avoided\u2014though one may wonder at the ease of this solution. Nuclear weapons, Ramsey writes, have a certain inherent ambiguity: \"they may be used either against strategic forces or against centers of population,\" and that means that \"apart from intention, their capacity to deter cannot be removed from them. . . . No matter how often we declare, and quite sincerely declare, that our targets are an enemy's forces, he can never be quite certain that in the fury or the fog of war his cities may not be destroyed.\" Now, the possession of conventional weapons is both innocent and ambiguous in exactly the way Ramsey suggests. The fact that I am holding a sword or a rifle doesn't mean that I am going to use it against innocent people, though it is quite effective against them; it has the same \"dual use\" that Ramsey has discovered in nuclear weapons. But the bomb is different. In a sense, as Beaufre has said, it isn't designed for war at all. It is designed to kill whole populations, and its deterrent value depends upon that fact (whether the killing is direct or indirect). It serves the purpose of preventing war only by virtue of the implicit threat it poses, and we possess it for the sake of that purpose. And men and women are responsible for the threats they live by, even if they don't speak them out loud.\n\nRamsey presses on. Perhaps the mere possession of nuclear weapons won't be enough to deter some reckless aggressor. Then, he suggests, we must distinguish \"between the appearance and the actuality of being . . . committed to go to city exchanges. . . . In that case, only the appearance should be cultivated.\" I am not sure exactly what that means, and Ramsey (for once) seems reluctant to say, but presumably it would allow us to hint at the possibility of massive retaliation without actually planning for it or intending to carry it out. Thus we are offered a continuum of increasing moral danger along which four points are marked out: the articulated prospect of collateral (and disproportionate) civilian deaths; the implicit threat of counter-city strikes; the \"cultivated\" appearance of a commitment to counter-city strikes; and the actual commitment. These may well be distinct points, in the sense that one can imagine policies focused around each of them, and these would be different policies. But I am inclined to doubt that the differences make a difference. To rule out the last for moral reasons, while permitting the first three, can only make people cynical about one's moral reasons. Ramsey aims to clear our intentions without prohibiting those policies that he believes necessary (and that probably are necessary under present conditions) for the dual prevention of war and conquest. But the unavoidable truth is that all these policies rest ultimately on immoral threats. Unless we give up nuclear deterrence, we cannot give up such threats, and it is best if we straightforwardly acknowledge what it is we are doing.\n\nThe real ambiguity of nuclear deterrence lies in the fact that no one, including ourselves, can be sure that we will ever carry out the threats we make. In a sense, all we ever do is \"cultivate the appearance.\" We strain for credibility, but what we are putatively planning and intending remains incredible. As I have already suggested, that helps make deterrence psychologically bearable, and perhaps also it makes a deterrent posture marginally better from a moral standpoint. But at the same time, the reason for our hesitancy and self-doubt is the monstrous immorality that our policy contemplates, an immorality we can never hope to square with our understanding of justice in war. Nuclear weapons explode the theory of just war. They are the first of mankind's technological innovations that are simply not encompassable within the familiar moral world. Or rather, our familiar notions about jus in bello require us to condemn even the threat to use them. And yet there are other notions, also familiar, having to do with aggression and the right of self-defense, that seem to require exactly that threat. So we move uneasily beyond the limits of justice for the sake of justice (and of peace).\n\nAccording to Ramsey, this is a dangerous move. For if we \"become convinced,\" he writes, \"that in the matter of deterrence a number of things are wicked which are not,\" then, seeing no way of avoiding wickedness, we will \"set no limits on it.\" Once again, this argument is precisely right with reference to conventional warfare; it catches the central error of what I have called the \"war is hell\" doctrine. But it is persuasive in the case of nuclear warfare only if one can describe plausible and morally significant limits, and that Ramsey has not done; nor have the strategists of \"flexible response\" been able to do it. All their arguments depend upon the ultimate wickedness of counter-city strikes. The pretense that this is not so carries with it dangers of its own. To draw insignificant lines, to maintain the formal categories of double effect, collateral damage, noncombatant immunity, and so on, when so little moral content remains is to corrupt the argument for justice as a whole and to render it suspect even in those areas of military life to which it properly pertains. And those areas are wide. Nuclear deterrence marks their outer limits, forcing us to contemplate wars that can never be fought. Within those limits there are wars that can and will and perhaps even should be fought, and to which the old rules apply with all their force. The specter of a nuclear holocaust does not invite us to act wickedly in conventional wars. Indeed, it probably is a deterrent there, too; it is hard to imagine a repetition of Dresden or Tokyo in a conventional war between nuclear powers. For destruction on such a scale would invite a nuclear response and a drastic and unacceptable escalation of the struggle.\n\nNuclear war is and will remain morally unacceptable, and there is no case for its rehabilitation. Because it is unacceptable, we must seek out ways to prevent it, and because deterrence is a bad way, we must seek out others. It is not my purpose here to suggest what the alternatives might look like. I have been more concerned to acknowledge that deterrence itself, for all its criminality, falls or may fall for the moment under the standard of necessity. But as with terror bombing, so here with the threat of terrorism: supreme emergency is never a stable position. The realm of necessity is subject to historical change. And, what is more important, we are under an obligation to seize upon opportunities of escape, even to take risks for the sake of such opportunities. So the readiness to murder is balanced, or should be, by the readiness not to murder, not to threaten murder, as soon as alternative ways to peace can be found.\n\n Would it make any difference if this commitment were mechanically fixed? Suppose we set up a computer which would automatically respond to any enemy attack by releasing our missiles. Then we informed our potential enemies that if they attacked our cities, theirs would be attacked. And they would be responsible for both attacks, we might say, since in the interval between the two, no political decision, no act of the will, would be possible on our side. I don't want to comment on the possible effectiveness (or the dangers) of such an arrangement. But it is worth insisting that it would not solve the moral problem. The men and women who designed the computer program or the political leaders who ordered them to do so would be responsible for the second attack, for they would have planned it and organized it and intended that it should occur (under certain conditions).\n\n This is obviously the grim logic of nuclear proliferation. So far as the moral question goes, each new balance of terror created by proliferation is exactly like the first one, justified (or not) in the same way. But the creation of regional balances may well have general effects upon the stability of the great power equilibrium, thereby introducing new moral considerations that I cannot take up here.\n\n Kissinger later moved away from these views, and they have pretty much dropped out of the strategic debates. But this picture of limited nuclear war is worked out in graphic detail in a novel by Joe Haldeman (The Forever War, New York, 1974), where the fighting goes on not at sea but in outer space. Many of the strategic speculations of the 1950s and 1960s have ended up as science fiction. Does this mean that the strategists had too much imagination or that the authors of science fiction have too little?\nPart Five\n\nThe Question of Responsibility\n\nThe Crime of Aggression: Political Leaders and Citizens\n\nThe assignment of responsibility is the critical test of the argument for justice. For if war is fought not under the aegis of necessity but, most often, of freedom, then soldiers and statesmen have to make choices that are sometimes moral choices. And if they do that, it must be possible to single them out for praise and blame. If there are recognizable war crimes, there must be recognizable criminals. If there is such a thing as aggression, there must be aggressors. It is not the case that for every violation of human rights in wartime we can name a guilty person or group of persons. The conditions of war supply a plethora of excuses: fear, coercion, ignorance, even madness. But the theory of justice should point us to the men and women from whom we can rightly demand an accounting, and it should shape and control the judgments we make of the excuses they offer (or that are offered on their behalf). It does not point to people by their proper names, of course, but by their offices and circumstances. We learn the names (sometimes) only as we work our way through cases, attending to the details of moral and military action. Insofar as we name the right names, or at least insofar as our assignments and judgments are in accordance with the actual experience of war, sensitive to all its painfulness, the argument for justice is greatly strengthened. There can be no justice in war if there are not, ultimately, responsible men and women.\n\nThe question here is of moral responsibility; we are concerned with the blameworthiness of individuals, not their legal guilt or innocence. Much of the debate about aggression and war crimes, however, has focused on the latter issue, not the former. And as we read through these arguments, or listen to them, it often seems that what is being said is this: that if an individual is not legally liable for some particular act or omission but, as it were, merely immoral, not much can usefully be said about his guilt. For legal liability is a matter of definite rules, well-known procedures, and authoritative judges, while morality is nothing more than endless talk, where every talker has an equal right to his opinions. Consider, for example, the view of a contemporary law professor who believes that the \"essentials\" of \"the question of war crimes\" can be set forth \"with tolerable clarity and brevity,\" so long as one caveat is accepted: \"I shall make no attempt to say what is immoral\u2014not because I believe morality unimportant, but because my views on it are entitled to no more weight than Jane Fonda's or Richard M. Nixon's, or yours.\" Of course, morality is unimportant if all opinions are equal, because then no particular opinion has any force. Moral authority is no doubt different from legal authority; it is earned in different ways; but Professor Bishop is wrong to think that it doesn't exist. It has to do with the capacity to evoke commonly accepted principles in persuasive ways and to apply them to particular cases. No one can argue about justice and war, as I have been doing, without striving for an authoritative voice and laying claim to a certain \"weightiness.\"\n\nMoral argument is especially important in wartime because\u2014as I have said before, and as Bishop's \"brevity\" makes clear\u2014the laws of war are radically incomplete. Authoritative judges are rarely called to the business of judging. Indeed, there are often prudential reasons for not calling them, for even well-wrought judicial decisions are likely at certain moments in the history of international society to be understood only as acts of cruelty and vengeance. Trials like those that took place at Nuremberg after World War II seem to me both defensible and necessary; the law must provide some recourse when our deepest moral values are savagely attacked. But such trials by no means exhaust the field of judgment. We have more to do in these matters, and it is my purpose to do it here: to point at criminals and possible criminals across the whole range of wartime activity, though not to suggest, except tangentially, how we should deal with such people. What is crucial is that they can be pointed at; we know where to look for them, if we are ready to look.\n\nThe World of Officials\n\nI will begin with the assignments and judgments that are required by the crime of war itself. That is to begin with politics rather than combat, civilians rather than soldiers, for aggression is first of all the work of political leaders. We must (naively) imagine them sitting around the elegant table of an old-fashioned chancellery or in the electronic fastness of a modern command room plotting illegitimate attacks, conquests, interventions. No doubt it is not always like that, though recent history provides ample evidence of direct and open criminal planning. \"Statesmen\" are more devious, aiming at war only indirectly, like Bismarck in 1870, and taking a very complicated view of their own efforts. Then it is not easy, perhaps, to mark out aggressors, though I think we should start with the assumption that it is always possible. The men and women who lead their people into war owe them and us an accounting. For every person who is killed, every drop of blood that falls is\n\n. . . . a sore complaint\n\nGainst him whose wrongs gives edge unto the swords.\n\nListening to the excuses and lies, and also to the true accounts, of political leaders, we search for the \"wrongs\" that lie behind the fighting and are its moral cause.\n\nThe lawyers have not always encouraged this search. Until 1945, at least, they have held that \"acts of state\" cannot be the crimes of individual persons. The legal reasons for this denial lie in the theory of sovereignty, as it was once understood. Sovereign states by definition know no superiors, it was argued, and accept no external judgments: hence there is no way to prove the criminality of acts imputed to the state, that is, carried out by recognized authorities in the course of their official duties (unless domestic law provides procedures for bringing such proof to bear). This argument is without moral effect, however, for in this regard states were never morally but only legally sovereign. All of us are capable of judging the acts of political leaders, and we commonly do so. Nor does legal sovereignty any longer provide protection against external judgments. Here Nuremberg is the decisive precedent.\n\nBut there is another, more informal version of the \"act of state\" doctrine, which refers not to the sovereignty of the political community but to the representativeness of its leaders. We are often urged not to condemn the acts of statesmen, or not to be too quick to condemn them, since, after all, these people are not acting selfishly or for private reasons. They are, as Townsend Hoopes wrote of America's leaders during the Vietnam War, \"struggling in good conscience . . . to serve the broad national interest according to their lights.\" They are acting for the sake of other people and in their name. The same assertion can be made on behalf of military officers, except when the crimes they commit are passionate or selfish. It might be made, too, on behalf of revolutionary militants who kill innocent people for the sake of the cause (not because of any personal grudge), even though the cause has no official but only a putative connection to the national interest. These are leaders, too; they may have risen to their \"offices\" by means not all that different from those adopted by more conventional officials, and they can sometimes say that acts of the movement or the revolution are as representative as acts of state. If this argument is acceptable in the case of statesmen and officers, I can see no reason to reject it in the case of revolutionaries. But it is a bad argument in all these cases, for it is false to suggest that representative functions are morally risk-free. They are instead peculiarly risky, precisely because statesmen, officers, and revolutionaries act for other people and with wide-ranging effects. They act sometimes so as to endanger the people they represent, sometimes so as to endanger the rest of us; they can hardly complain if we hold them subject to moral judgment.\n\nPolitical power is a good that people seek. They aspire to office, connive at control and leadership, compete for positions from which they can do evil as well as good. If they hope to be praised for the good they do, they cannot escape blame for the evil. Still, blame is always resented, even when we may think it well-deserved, and it is important to try to say why this is so. Moral criticism goes very deep; it calls into question a leader's good faith and his personal rectitude. Since political leaders are rarely cynical about their work, and can never afford to appear to be cynical, they take such criticism seriously and dislike it intensely. Disagreement they can accept (if they are democratic leaders), but not accusations of criminality. Indeed, they are likely to treat all moral criticism as an illegitimate displacement of political controversy. I suppose they are right to recognize that morality is often a mask for politics. The case is the same with the law. Legal accusation can be a very powerful form of political attack, but though it is often used in that way, and often degraded in the use, it remains true nevertheless that political leaders are bound by the legal code and can rightly be charged and punished for criminal acts. Similarly with the moral code: though the terms of praise and blame are universally available and often misused, the code is still binding, and praise and blame are at least sometimes appropriate. The misuse of law and morality is common in wartime, and so we have to be careful not only in punishing political leaders for the wars they wage but also in stigmatizing them. They have no a priori claim to escape the stigma of aggression, however, when they violate the rights of another people and force its soldiers to fight.\n\nActs of state are also acts of particular persons, and when they take the form of aggressive war, particular persons are criminally responsible. Just who those persons are, and how many they are, is not always apparent. But it makes sense to begin with the head of state (or the effective head) and the men and women immediately around him, who actually control the government and make key decisions. Their accountability is clear, like that of the commanders of a military campaign for the strategy and tactics they adopt, for they are the source rather than the recipients of superior orders. When they defend themselves, they don't look up the political hierarchy, but across the battleline: they blame their opponents for forcing them to fight. They point to the intricate complexity of the pre-war maneuvering and to the extravagant demands and harassing actions of their adversaries. They have long stories to tell:\n\nWho first attacked? Who turned the other cheek?\n\nAggression perpetrated is as soon\n\nDenied, and insult rubbed into the injury\n\nBy cunning agents trained in these affairs,\n\nWith whom it's touch-and-go, don't tread-on-me,\n\nI-dare-you-to, keep-off, and kiss-my-hand.\n\nTempers could sharpen knives, and do; we live\n\nIn states provocative.\n\nIn order to work our way through the claims and counter-claims, we need a theory such as I have attempted to set forth in Part Two of this book. Often enough, despite the cunning agents, the theory is readily applied. It is worth setting down some of the cases about which we have, I think, no doubts: the German attack on Belgium in 1914, the Italian conquest of Ethiopia, the Japanese attack on China, the German and Italian interventions in Spain, the Russian invasion of Finland, the Nazi conquests of Czechoslovakia, Poland, Denmark, Belgium, and Holland, the Russian invasions of Hungary and Czechoslovakia, the Egyptian challenge to Israel in 1967, and so on\u2014the twentieth century makes for easy listing. I have argued that the American war in Vietnam belongs to the same series. Sometimes, no doubt, the going is more muddy; political leaders are not always in control of their own provocations, and wars do break out without anyone planning or intending to violate anyone else's rights. But insofar as we can recognize aggression, there should be little difficulty in blaming heads of state. The hard and interesting problems arise when we ask how responsibility for aggression is diffused through a political system.\n\nAt Nuremberg, the crime of aggression (\"crime against peace\") was said to involve \"the planning, preparation, initiation, and waging of [aggressive] war.\" These four activities were distinguished from the planning and preparation of particular military campaigns and from the actual fighting of the war, which were (rightly) held to be noncriminal in character. Now, \"planning, preparation, initiation, and waging\" would appear to be the work of a fairly large number of people. But in fact the courts restricted the range of accountability so that convictions were obtained only against those officials who were part of \"Hitler's inner circle of advisors\" or who played such a major role in the making or execution of policy that their protests and refusals would have had a significant impact. Persons lower down the bureaucratic hierarchy, though their contribution was cumulatively significant, were not held individually responsible. It is not at all clear, however, just where we should draw that line; nor is it clear that we ought to assign blame in the same way as we assign legal culpability. The best way to deal with these issues is to turn immediately to a critical case.\n\nNuremberg: \"The Ministries Case\"\n\nIn an important article on responsibility for crimes of war, Sanford Levinson has analyzed the Nuremberg verdicts, focusing especially on the trial of Ernst von Weizsaecker, who was State Secretary of the German Foreign Ministry from 1938 to 1943, second only to von Ribbentrop (one of the \"inner circle\") in the foreign policy hierarchy. I want to follow Levinson's account, and then draw some conclusions from it. Von Weizsaecker was charged with crimes against the peace and initially convicted, but the conviction was reversed upon review. His defense emphasized two points: first, that he took no part in actual policy planning, and secondly, that within the Foreign Ministry he opposed Nazi aggression; he was also involved, at least marginally, in underground opposition to Hitler's regime. The review court accepted this defense, emphasizing its second part: von Weizsaecker's diplomatic activity, which \"aided and abetted\" German war plans, was so important that it would have been held against him had he not criticized Hitler's policies within his ministry and passed information to more active opponents outside. Thus the line of criminal responsibility was drawn so as to include officials like von Weizsaecker, while he himself was acquitted because, though he clearly played a part in \"preparing\" an aggressive war, he also \"opposed and objected to\" that war.\n\nThe prosecution argued the insufficiency of this opposition: since he knew of plans for aggression, it was said, he had a positive duty to reveal those plans to the potential victims. But the court rejected this argument because of the risks such action would have entailed and also because it might have led to greater German losses on the battlefield.\n\nOne may quarrel with, and oppose to the point of violence and assassination, a tyrant whose programs mean the ruin of one's country. But the time has not yet arrived when any man would view with satisfaction the ruin of his own people and the loss of its young manhood. To apply any other standard of conduct is to set up a test that has never yet been suggested as proper and which, assuredly, we are not prepared to accept as either wise or good.\n\nThis is too strong, I think, for it is obviously not a question of \"viewing with satisfaction\" the battle losses of one's own side. One might be greatly saddened by them and still feel it morally right to protect the innocent people of the victim state. And surely we would think it both wise and good, indeed heroic, had some German opponent of Hitler warned the Danes or the Belgians or the Russians of the coming attacks. But there is probably no legal or moral obligation to act in this way. Not only the risk but also the inner pain that a man might feel at such a time is more than we require. On the other hand, von Weizsaecker's alternative actions, though they satisfied the judges, may have amounted to less than we require. For he continued to serve the regime whose policies he disapproved; he did not resign.\n\nThe issue of resignation came up more directly in connection with charges that von Weizsaecker was guilty of war crimes and crimes against humanity, the latter relating to the extermination of the Jews. Here, too, he argued \"that minimal participation should be negated by the fact that he opposed what was being done.\" But in this case, intra-office opposition was not deemed sufficient. The SS had formally requested the Foreign Ministry's opinions in regard to its policy on the Jewish question. And von Weizsaecker, though he knew what that policy was, had voiced no objections. Apparently he thought his silence the price of his office, and he wanted to retain his office so that he \"might be in a position to initiate or aid in attempts to negotiate peace\" and so that he might continue to pass on information to Hitler's underground opponents. But the court held that \"One cannot give consent to . . . the commission of murder because by so doing he hopes eventually to be able to rid society of the chief murderer. The first is a crime of imminent actuality while the second is but a future hope.\" The court did not believe that failure to resign was itself a matter of criminal liability. While it might be true that no \"decent man could continue to hold office under a regime which carried out . . . wholesale barbarities of this kind,\" indecency is not a crime. But to hold office and keep silent was a punishable offense, and von Weizsaecker was sentenced to seven years in prison.\n\nNow, the criteria of \"significant contribution\" or the possibility of \"significant protest\" seem entirely appropriate in deciding upon trial and punishment. The standards of blame, however, are much more strict: we need to say more about indecency. If von Weizsaecker was bound to resign in protest, I don't see why lesser officials with similar knowledge were not similarly bound. In the United States during the Vietnam years, only a very small number of foreign policy officials resigned, most of them holding low-level positions, but those resignations were morally heartening (to those of us, at least, who knew their reasons) in a way which suggests that they should have been imitated. The courage required to resign in Germany in the late 1930s or early 1940s was far greater than that required in the U.S. three decades later, where opposition to the war was public and vociferous. But it was not a death-defying courage that was necessary even in Germany, but something less, well within the reach of ordinary people. Many officials who failed to resign offered excuses for not doing so, which suggests that they recognized the imperative, however dimly. These excuses were mostly like von Weizsaecker's, focused on distant goods. But there were also men who remained in office in order to engage, often at great personal risk, in concrete and immediate acts of benevolence or sabotage. The most extraordinary of these was the SS lieutenant Kurt Gerstein, whose case has been carefully documented by Saul Friedlander.\n\nGerstein represented the type of man who, by virtue of his deepest convictions, disavowed the Nazi regime, even hated it inwardly, but collaborated with it in order to combat it from within and to prevent worse things from happening.\n\nI cannot retell Gerstein's story here; it is enough to say that it demonstrates that it was possible to live a moral life even in the SS, though at a cost in personal agony (Gerstein eventually committed suicide) which we can expect few people to pay. Resignation is much easier, and sometimes, I think, we must take it as the minimal sign of moral decency.\n\nVon Weizsaecker's case invites us to reflect on one further problem. The State Secretary was a diplomat who carried out negotiations with foreign countries under instructions from his superiors. But he was also an advisor to those superiors; his own views were frequently requested. Now advisors are in a curious position with regard to both legal and moral judgment. Their most important advice is often given orally, whispered in the ruler's ear. What is written down may be incomplete, tailored to the requirements of bureaucratic correspondence. We miss the nuances and qualifications, the subtle signs of doubt, the private emphases and hesitations. If sufficient documentation is available, we may go ahead and make judgments anyway. It's certainly not the case that only \"line\" and never \"staff\" officials can be held responsible for decisions made. But whispering in the ruler's ear is problematic; it is easier to suggest what should be said than what we should do if we suspect that it hasn't been said.\n\nWhat von Weizsaecker said was probably insufficient, for according to his own account he urged nothing more than the likelihood of German defeat; his opposition to Hitler's policies were always expressed in expediential terms. Perhaps those were the only terms likely to be effective in Germany during those years. That is probably true in other cases, too, even with governments less openly committed to a program of conquest. But it is often important to use the language of morality, if only to break through the forms of euphemism and silence with which officials conceal even from themselves the extent and nature of the crimes they are committing. Sometimes the best way for an advisor to say no is simply to give an accurate name to the policy he is being asked to approve. This point is beautifully made in a speech in Shakespeare's King John. With hints and indirection, John had ordered the murder of his nephew Arthur, Duke of Brittany. Later he came to regret the murder and turned on his courtier, Hubert de Burgh, who had carried it out.\n\nHadst thou but shook thy head or made a pause\n\nWhen I spoke darkly what I purposed,\n\nOr turned an eye of doubt upon my face,\n\nAs bid me tell my tale in express words,\n\nDeep shame had struck me dumb, made me break off . . .\n\nBut thou dist understand me by my signs,\n\nAnd didst in signs again parley with sin;\n\nYea, without stop, didst let thy heart consent,\n\nAnd consequently thy rude hand to act,\n\nThe deed which both our tongues held vile to name.\n\nThe speech is hypocritical, but it captures the common quality of bureaucratic acquiescence, and it suggests very forcefully that advisors and agents, when they have the opportunity, must speak out \"in express words,\" using the moral language that we all know. They may be judged insufficiently tough or hard-headed if they talk that way. But to be \"tough\" enough to carry out policies that are literally unmentionable is either to be very cowardly or very wicked.\n\nDemocratic Responsibilities\n\nWhat about the rest of us\u2014citizens, let's say, of a state engaged in an aggressive war? Collective responsibility is a hard notion, though it is worth stressing at once that we have fewer problems with collective punishment. Resistance to aggression is itself \"punishing\" to the aggressor state and is often described in those terms. With reference to the actual fighting, as I have already argued, civilians on both sides are innocent, equally innocent, and never legitimate military targets. They are, however, political and economic targets once the war is over; that is, they are the victims of military occupation, political reconstruction, and the exaction of reparative payments. We may take the last of these as the clearest and simplest case of collective punishment. Reparations are surely due the victims of aggressive war, and they can hardly be collected only from those members of the defeated state who were active supporters of the aggression. Instead, the costs are distributed through the tax system, and through the economic system generally, among all the citizens, often over a period of time extending to generations that had nothing to do with the war at all. In this sense, citizenship is a common destiny, and no one, not even its opponents (unless they become political refugees, which has its costs, too) can escape the effects of a bad regime, an ambitious or fanatic leadership, or an overreaching nationalism. But if men and women must accept this destiny, they can sometimes do so with a good conscience, for the acceptance says nothing about their individual responsibility. The distribution of costs is not the distribution of guilt.\n\nAt least one writer has tried to argue that political destiny is a kind of guilt: existential, unavoidable, frightening. For the soldier or citizen of a state at war, writes J. Glenn Gray in his philosophical memoir of World War II, is the member of a \"coarse, vulgar, heedless, and violent\" community and, willy-nilly, a participant in an enterprise \"whose spirit is to win at any cost.\" He cannot cut himself loose.\n\nHe is bound to reflect that his nation has given him refuge and sustenance, provided him with whatever education and property he calls his own. He belongs and will always belong to it in some sense no matter where he goes or how hard he seeks to alter his inheritance. The crimes, therefore, that his nation or one of its units commits cannot be indifferent to him. He shares the guilt as he shares the satisfaction in the generous deeds and worthy products of nation or army. Even if he did not consciously will them and was unable to prevent them, he cannot wholly escape responsibility for collective deeds.\n\nMaybe; but it is not an easy move from \"the ache of guilt,\" which Gray almost lovingly describes, to hard talk about responsibility. It might be better to say of loyal citizens who watch their government or army (or their comrades in battle) doing terrible things that they feel or should feel ashamed rather than responsible\u2014unless they actually are responsible by virtue of their particular participation or acquiescence. Shame is the tribute we pay to the inheritance that Gray describes. \"A burning sense of shame at the deeds of his government and the acts of horror committed by German soldiers and police was the mark of a conscientious German at the close of the war.\" That is exactly right, but we won't ourselves blame that conscientious German or call him responsible; nor need he blame himself unless there was something he should have done, and could do, in the face of the horror.\n\nPerhaps it can always be said of such a person that he could have done more than he did do. Certainly conscientious men and women are likely to believe that of themselves; it is a sign of their conscientiousness.\n\nOn this or that occasion he has been silent when he should have spoken out. In his own smaller or larger circle of influence he has not made his whole weight felt. Had he brought forth the civil courage to protest in time, some particular act of injustice might have been avoided.\n\nSuch reflections are endless and endlessly dispiriting; they lead Gray to argue that behind collective responsibility there lies \"meta-physical guilt,\" which derives from \"our failure as human beings to live in accordance with our potentialities and our vision of the good.\" But some of us, surely, fail more dismally than others; and it is necessary, with all due caution and humility, to mark out standards by which we can measure the respective failures. Gray suggests the right standard, though he goes on very quickly to insist that we can never apply it to anyone but ourselves. But that kind of self-regard is not possible in politics and morality. Judging ourselves, we necessarily judge other people, with whom we share a common life. And how is it possible to criticize and blame our leaders, as we sometimes must do, without involving their enthusiastic followers (our fellow citizens)? Though responsibility is always personal and particular, moral life is always collective in character.\n\nThis is Gray's principle, which I mean to adopt and expound: \"The greater the possibility of free action in the communal sphere, the greater the degree of guilt for evil deeds done in the name of everyone.\" The principle invites us to focus our attention on democratic rather than authoritarian regimes. Not that free action is impossible even in the worst of authoritarian regimes; at the very least, people can resign, withdraw, flee. But in democracies there are opportunities for positive response, and we need to ask to what extent these opportunities fix our obligations, when evil deeds are committed in our name.\n\nThe American People and the War in Vietnam\n\nIf the argument in chapters 6 and 11 is right, the American war in Vietnam was, first of all, an unjustified intervention, and it was, secondly, carried on in so brutal a manner that even had it initially been defensible, it would have to be condemned, not in this or that aspect but generally. I am not going to re-argue that description, but assume it, so that we can look closely at the responsibility of democratic citizens\u2014and at a particular set of democratic citizens, namely, ourselves.\n\nDemocracy is a way of distributing responsibility (just as monarchy is a way of refusing to distribute it). But that doesn't mean that all adult citizens share equally in the blame we assign for aggressive war. Our actual assignments will vary a great deal, depending on the precise nature of the democratic order, the place of a particular person in that order, and the pattern of his own political activities. Even in a perfect democracy, it cannot be said that every citizen is the author of every state policy, though every one of them can rightly be called to account. Imagine, for example, a small community where all the citizens are fully and accurately informed about public business, where all of them participate, argue, vote on matters of communal interest, and where they all take turns holding public office. Now this community, let us say, initiates and wages an unjust war against its neighbors\u2014for the sake of some economic advantage, perhaps, or out of zeal to spread its (admirable) political system. There is no question of self-defense; no one has attacked it or is planning to do so. Who is responsible for this war? Surely all those men and women who voted for it and who cooperated in planning, initiating, and waging it. The soldiers who do the actual fighting are not responsible as soldiers; but as citizens, they are, assuming that they were old enough to have shared in the decision to fight.a All of them are guilty of the crime of aggressive war and of no lesser charge, and we would not hesitate in such a case to blame them publicly. Nor would it make any difference whether their motive was economic selfishness or a political zeal that appeared to them entirely disinterested. Either way, the blood of their victims would complain against them.\n\nThose who voted against the war or who refused to cooperate in the waging of it could not be blamed. But what would we think of a group of citizens that didn't vote? Had they voted, let's say, the war might have been avoided, but they were lazy, didn't care, or were afraid to come down on one side or the other of a hotly disputed issue. The day of the crucial decision was a day off from work; they spent it in their gardens. I am inclined to say that they are blameworthy, though they are not guilty of aggressive war. Surely those of their fellow citizens who went to the assembly and opposed the war can blame them for their indifference and inaction. This seems a clear counter-example to Gray's assertion that \"No citizen of a free land can justly accuse his neighbor . . . of not having done as much as he should to prevent the state of war or the commission of this or that state crime. But each can . . . accuse himself. . . .\" In a perfect democracy, we would know a great deal about one another's duties, and just accusations would not be impossible.\n\nImagine now that the minority of citizens that was defeated could have won (and prevented the war) if instead of merely voting, they had held meetings outside the assembly, marched and demonstrated, organized for a second vote. Let's assume that none of this would have been terribly dangerous to them, but they chose not to take these measures because their opposition to the war wasn't all that strong; they thought it unjust but were not horrified by the prospect; they hoped for a quick victory; and so on. Then they are blameworthy, too, though to a lesser degree than those slothful citizens who did not even bother to go to the assembly.\n\nThese last two examples resemble the good samaritan cases in domestic society, where we commonly say that if it is possible to do good, without risk or great cost, one ought to do good. But when the issue is war, the obligation is stronger, for it is not a question of doing good, but of preventing serious harm, and harm that will be done in the name of my own political community\u2014hence, in some sense, in my own name. Here, assuming still that the community is a perfect democracy, it looks as if a citizen is blameless only if he takes back his name. I don't think this means that he must become a revolutionary or an exile, actually renouncing his citizenship or loyalty. But he must do all he can, short of accepting frightening risks, to prevent or stop the war. He must withdraw his name from this act (the war policy) though not necessarily from every communal action, for he may still value, as he probably should, the democracy he and his fellow citizens have achieved. This, then, is the meaning of Gray's maxim: the more one can do, the more one has to do.\n\nWe can now drop the myth of perfection and paint a more realistic picture. The state that goes to war is, like our own, an enormous state, governed at a great distance from its ordinary citizens by powerful and often arrogant officials. These officials, or at least the leading among them, are chosen through democratic elections, but at the time of the choice very little is known about their programs and commitments. Political participation is occasional, intermittent, limited in its effects, and it is mediated by a system for the distribution of news which is partially controlled by those distant officials and which in any case allows for considerable distortions. It may be that a politics of this sort is the best we can hope for (though I don't believe that) once the political community reaches a certain size. Anyway, it is no longer as easy to impose responsibility as it is in a perfect democracy. One doesn't want to regard those distant officials as if they were kings, but for certain sorts of state action, secretly prepared or suddenly launched, they bear a kind of regal responsibility.\n\nWhen a state like this commits itself to a campaign of aggression, its citizens (or many of them) are likely to go along, as Americans did during the Vietnam War, arguing that the way may after all be just; that it is not possible for them to be sure whether it is just or not; that their leaders know best and tell them this or that, which sounds plausible enough; and that nothing they can do will make much difference anyway. These are not immoral arguments, though they reflect badly on the society within which they are made. And they can, no doubt, be made too quickly by citizens seeking to avoid the difficulties that might follow if they thought about the war for themselves. These people are or may be blameworthy, not for aggressive war, but for bad faith as citizens. But that is a hard charge to make, for citizenship plays such a small part in their everyday lives. \"Free action in the communal sphere\" is a possibility for men and women in such a state only in the formal sense that serious governmental restraint, actual repression, doesn't exist. Perhaps it should also be said that the \"communal sphere\" doesn't exist, for it is only the day-by-day assumption of responsibility that creates that sphere and gives it meaning. Even patriotic excitement, war fever, among such people is probably best understood as a reflex of distance, a desperate identification, stimulated, it may be, by a false account of what is going on. One might say of them what one says of soldiers in combat, that they are not to blame for the war, since it is not their war.b\n\nBut as an account of all the citizens, even in such a state, this is certainly exaggerated. For there exists a group of more knowledgeable men and women, members of what political scientists call the foreign policy elites, who are not so radically distanced from the national leadership; and some subset of these people, together with others in touch with them, is likely to form an \"opposition\" or perhaps even a movement of opposition to the war. It would seem possible to regard the entire group of knowledgeable people as at least potentially blameworthy if that war is aggressive and unless they join the opposition. To say that is to presume upon the knowledge they have and their private sense of political possibility. But if we turn to an actual case of imperfect democracy, like the United States in the late 1960s and early 1970s, the presumption doesn't seem unwarranted. Surely there was knowledge and opportunity enough among the country's elites, the national and local leaders of its political parties, its religious establishments, its corporate hierarchies, and perhaps above all its intellectual teachers and spokesmen\u2014the men and women whom Noam Chomsky has named, in tribute to the role they play in contemporary government, \"the new mandarins.\" Surely many of these people were morally complicitous in our Vietnam aggression. I suppose one can also say of them what many of them have said of themselves: that they were simply mistaken in their judgments of the war, failed to realize this or that, thought that was true when it was not, or hoped for this result which never came about. In moral life generally, one makes allowances for false beliefs, misinformation, and honest mistakes. But there comes a time in any tale of aggression and atrocity when such allowances can no longer be made. I cannot mark out that time here; nor am I interested in pointing at particular people or certain that I can do so. I only want to insist that there are responsible people even when, under the conditions of imperfect democracy, moral accounting is difficult and imprecise.\n\nThe real moral burden of the American war fell on that subset of men and women whose knowledge and sense of possibility were made manifest by their oppositional activity. They were the ones most likely to reproach themselves and one another, continually asking whether they were doing enough to stop the fighting, devoting enough time and energy, working hard enough, working as effectively as they could. For most of their fellow citizens, anxious, apathetic, and alienated, the war was merely an ugly or an exciting spectacle (until they were forced to join it). For the dissidents, it was a kind of moral torture\u2014self-torture, as Gray describes it, though they also tortured one another, wastefully, in savage internecine conflicts over what was to be done. And this self-torture bred a kind of self-righteousness vis-\u00e0-vis the others, an endemic failing on the Left, though understandable enough under conditions of aggressive war and mass acquiescence. The expression of that self-righteousness, however, is not a useful way to get one's fellow citizens to think seriously about the war or to join the opposition: nor was it useful in this case. It is not easy to know what course of action might serve these purposes. Politics is difficult at such a time. But there is intellectual work to do that is less difficult: one must describe as graphically as one can the moral reality of war, talk about what it means to force people to fight, analyze the nature of democratic responsibilities. These, at least, are encompassable tasks, and they are morally required of the men and women who are trained to perform them. Nor is it dangerous to perform them, in a democratic state, waging war in a distant country. And the citizens of such a state have time to listen and reflect; they, too, are in no immediate danger. War imposes harsher burdens than any these people have to bear\u2014as we shall see when we consider, finally, the moral life of men at arms.\n\n Why aren't they responsible as soldiers? If they are morally bound to vote against the war, why aren't they also bound to refuse to fight? The answer is that they vote as individuals, each one deciding for himself, but they fight as members of the political community, the collective decision having already been made, subject to all the moral and material pressures that I described in chapter 3. They act very well if they refuse to fight, and we should honor those\u2014they are likely to be few\u2014who have the self-certainty and courage to stand against their fellows. I have argued elsewhere that democracies ought to respect such people and ought certainly to tolerate their refusals. (See the essay on \"Conscientious Objection\" in Obligations.) That doesn't mean, however, that the others can be called criminals. Patriotism may be the last refuge of scoundrels, but it is also the ordinary refuge of ordinary men and women, and it requires of us another sort of toleration. But we should expect opponents of the war to refuse to become officers or officials, even if they feel bound to share combat risks with their countrymen.\n\n But see the note in Anne Frank's Diary: \"I don't believe that only governments and capitalists are guilty of aggression. Oh no, the little man is just as keen on it, for otherwise the people of the world would have risen in revolt long ago.\" I'm sure she is right about the keenness, and I don't want to excuse it. But we don't, for all that, call the little men war criminals, and I am trying to explain why we don't. (The Diary of a Young Girl, trans. B. M. Mooyaart-Doubleday, New York, 1953, p. 201.)\n\nWar Crimes: Soldiers and Their Officers\n\nWe are concerned now with the conduct of war and not its overall justice. For soldiers, as I have already argued, are not responsible for the overall justice of the wars they fight; their responsibility is limited by the range of their own activity and authority. Within that range, however, it is real enough, and it frequently comes into question. \"There wasn't a single soldier,\" says an Israeli officer who fought in the Six Day War, \"who didn't at some stage have to decide, to choose, to make a moral decision . . . quick and modern though [the war] was, the soldier was not turned into a mere technician. He had to make decisions that were of real significance.\" And when faced with decisions of that sort, soldiers have clear obligations. They are bound to apply the criteria of usefulness and proportionality until they come up against the basic rights of the people they are threatening to kill or injure, and then they are bound not to kill or injure them. But judgments about usefulness and proportionality are very difficult for soldiers in the field. It is the doctrine of rights that makes the most effective limit on military activity, and it does so precisely because it rules out calculation and establishes hard and fast standards. Hence in my initial cases I will focus on specific violations of rights and on the defenses that soldiers commonly offer for these violations. The defenses are basically of two sorts. The first refers to the heatedness of battle and the passion or frenzy it engenders. The second refers to the disciplinary system of the army and the obedience it requires. These are serious defenses; they suggest the loss of self that is involved in warfare, and they remind us that most soldiers most of the time have not chosen the combat and discipline they endure. Where is their freedom and responsibility?\n\nBut there is a related issue that I must consider before trying to mark out the realm of freedom from the coercions and hysteria of war. The war convention requires soldiers to accept personal risks rather than kill innocent people. This requirement takes different forms in different combat situations, and I have already discussed these in considerable detail; my concern now is with the requirement itself. The rule is absolute: self-\u00adpreservation in the face of the enemy is not an excuse for violations of the rules of war. Soldiers, it might be said, stand to civilians like the crew of a liner to its passengers. They must risk their own lives for the sake of the others. No doubt this is easy to say, less easy to do. But if the rule is absolute, the risks are not; it is a question of degree; the crucial point is that soldiers cannot enhance their own security at the expense of innocent men and women.a This might be called an obligation of soldiering as an office, but it is a hard question whether one can rightly be said to assume such obligations when one comes into the office as unwillingly as most soldiers do. Imagine a liner manned by kidnapped sailors: would the members of such a crew be bound, as the ship was sinking, to see to the safety of the passengers before seeing to their own?\n\nI am not sure how to answer that question, but there is a crucial difference between the work of coerced crew members and that of military conscripts: the first group is not in the business of sinking ships, the second is. Conscripts impose risks on innocent people; they are themselves the immediate source of the danger and they are its effective cause. And so it is not a question of saving themselves, letting others die, but of killing others in order to improve their own odds. Now that they cannot do, because that no man can do. Their obligation isn't in practice mediated by the office of soldiering. It arises directly from the activity in which they are engaged, whether that activity is voluntary or not, or at least it arises so long as we regard soldiers as moral agents and even if we regard them as coerced moral agents. They are not mere instruments; they do not stand to the army as their weapons do to them. It is precisely because they do (sometimes) choose to kill or not, to impose risks or accept them, that we require them to choose in a certain way. That requirement shapes the whole pattern of their rights and duties in combat. And when they break out of that pattern, it is a matter of some significance that they don't by and large deny the requirement. They claim, instead, that they literally were not able to fulfill it; that they were not at the moment of their \"crime,\" moral agents at all.\n\nIn the Heat of Battle\n\nTwo Accounts of Killing Prisoners\n\nIn his fine memoir of World War I, Guy Chapman tells the following story. After a minor but bloody advance from one line of trenches to the next, he encountered one of his fellow officers, his face \"slack and haggard, but not from weariness.\" Chapman asked him what was wrong.\n\n\"Oh, I don't know. Nothing. . . . At least. . . . Look here, we took a lot of prisoners in those trenches yesterday morning. Just as we got into their line, an officer came out of a dugout. He'd got one hand above his head, and a pair of fieldglasses in the other. He held the glasses out to S________, . . . and said, 'Here you are, sergeant, I surrender.' S________ said, 'Thank you, sir,' and took the glasses with his left hand. At the same moment, he tucked the butt of his rifle under his arm and shot the officer straight through his head. What the hell ought I to do?\"\n\n\"I don't see that you can do anything,\" I answered slowly. \"What can you do? Besides I don't see that S________'s really to blame. He must have been half mad with excitement by the time he got into that trench. I don't suppose he ever thought what he was doing. If you start a man killing, you can't turn him off like an engine. After all, he is a good man. He was probably half off his head.\"\n\n\"It wasn't only him. Another did exactly the same thing.\"\n\n\"Anyhow, it's too late to do anything now. I suppose you ought to have shot both on the spot. The best thing now is to forget it.\"\n\nThat sort of thing happens often in war, and it is commonly excused. Chapman's argument makes some sense: it is, in effect, a plea of temporary insanity. It suggests a kind of killing frenzy that begins in combat and ends in murder, the line between the two being lost to the mind of the individual soldier. Or it suggests a frenzy of fear such that the soldier cannot recognize the moment when he is no longer in danger. He is not, indeed, a machine that can just be turned off, and it would be inhumanly righteous not to look with sympathy on his plight. And yet, if it is true that enemy soldiers are often killed trying to surrender, it is also true that a relatively small number of men do the \"extra\" killing. The rest seem ready enough to stop as soon as they can, whatever the state of mind they had worked themselves into during the battle itself. This fact is morally decisive, for it suggests a common acknowledgment of the right to quarter, and it proves that the right can in fact be recognized, since it often is, even in the chaos of combat. It is simply not true of soldiers, as one philosopher has recently written, that \"war . . . in some important ways makes psychopaths of them all.\" The argument has to be more particular than that. When we make allowances for what individual soldiers do \"in the heat of battle,\" it must be because of some knowledge we have that distinguishes these soldiers from the others or their circumstances from the usual ones. Perhaps they have encountered enemy troops who feigned surrender in order to kill their captors: then the war rights of other troops are made problematic in a new way, for one cannot be sure when killing is \"extra.\" Or perhaps they have been under some special strain or have been fighting too long and are near to nervous exhaustion. But there is no general rule that requires us to make allowances, and sometimes, at least, soldiers should be censured or punished for killings that take place after the battle is over (though summary execution is probably not the best form of punishment). They should certainly never be encouraged to believe that a total lack of restraint can be excused merely by reference to the passions that cause it.\n\nThere are officers, however, who encourage exactly that belief, not out of compassion but calculation, not because of the heat of the battle but in order to raise the temperature of men in combat. In his novel The Thin Red Line, one of the best accounts of jungle fighting in World War II, James Jones tells of another incident of \"extra\" killing. He describes a new army unit, its members un-blooded and without confidence in their ability to fight. After a hard march through the jungle, they come upon a Japanese position from the rear. There is a brief and savage fight. At a certain point, Japanese soldiers start trying to surrender, but some of the Americans cannot or will not stop the killing. Even after the firefight is definitely over, those Japanese who have succeeded in surrendering are brutally treated\u2014by men, so Jones wants to suggest, who are caught up in a kind of intoxication, their inhibitions suddenly gone. The commanding officer watches all this and does nothing. \"He did not want to jeopardize the new toughness of spirit that had come over the men after achieving success here. That spirit was more important than whether or not a few Jap soldiers got kicked around or killed.\"\n\nI suppose that soldiers must be \"men of spirit,\" like Plato's guardians, but Jones' colonel has mistaken the nature of their spiritedness. It is almost certainly true that they fight best when they are most disciplined, when they are most in control of themselves and committed to the restraints appropriate to their trade. \"Extra\" killing is less a sign of toughness than of hysteria, and hysteria is the wrong kind of spiritedness. But even if the colonel's calculations were correct, he would still be bound to stop the killing if he could, for he cannot train and toughen his men at the expense of Japanese prisoners. He is also bound to act so as to prevent such killings in the future. This is a crucial aspect of what is called \"command responsibility,\" and I will take it up in detail later on. It is important to stress now that it is a large responsibility; for the general policy of the army, expressed through its officers, the climate they create by their day-to-day actions, has far more to do with the incidence of \"extra\" killing than does the intensity of the actual fighting. But this doesn't mean that individual soldiers must be excused; indeed, it suggests once again that heatedness isn't the issue, but murderousness; and for their own murderousness individuals are always responsible, even when under the conditions of military discipline they are not exclusively so.\n\nIt is a feature of criminal responsibility that it can be distributed without being divided. We can, that is, blame more than one person for a particular act without splitting up the blame we assign. When soldiers are shot trying to surrender, the men who do the actual shooting are fully responsible for what they do, unless we recognize particular extenuating circumstances; at the same time, the officer who tolerates and encourages the murders is also fully responsible, if it lay within his power to prevent them. Perhaps we blame the officer more, for his coolness, but I have tried to suggest that combat soldiers, too, should be held to high standards in such matters (and they will surely want their enemies held to high standards). The case looks very different, however, when combatants are actually ordered to take no prisoners or to kill the ones they take or to turn their guns on enemy civilians. Then it is not their own murderousness that is at issue but that of their officers; they can act morally only by disobeying their orders. In such a case, we are likely to divide as well as distribute responsibility: we regard soldiers under orders as men whose acts are not entirely their own and whose liability for what they do is somehow diminished.\n\nSuperior Orders\n\nThe My Lai Massacre\n\nThe incident is infamous and hardly needs retelling. A company of American soldiers entered a Vietnamese village where they expected to encounter enemy combatants, found only civilians, old men, women, and children, and began to kill them, shooting them singly or collecting them in groups, ignoring their obvious helplessness and their pleas for mercy, not stopping until they had murdered between four and five hundred people. Now, it has been argued on behalf of these soldiers that they acted, not in the heat of battle (since there was no battle) but in the context of a brutal and brutalizing war which was in fact, if only unofficially, a war against the Vietnamese people as a whole. In this war, the argument goes on, they had been encouraged to kill without making careful discriminations\u2014encouraged to do so by their own officers and driven to do so by their enemies, who fought and hid among the civilian population. These statements are true, or partly true; and yet massacre is radically different from guerrilla war, even from a guerrilla war brutally fought, and there is considerable evidence that the soldiers at My Lai knew the difference. For while some of them joined in the murders readily enough, as if eager to kill without risk, there were a few who refused to fire their guns and others who had to be ordered to fire two or three times before they could bring themselves to do so. Others simply ran away; one man shot himself in the foot so as to escape the scene; a junior officer tried heroically to stop the massacre, standing between the Vietnamese villagers and his fellow Americans. Many of his fellows, we know, were sick and guilt-ridden in the days that followed. This was not a fearful and frenzied extension of combat, but \"free\" and systematic slaughter, and those men that participated in it can hardly say that they were caught in the grip of war. They can say, however, that they were following orders, caught in the grip of the United States Army.\n\nThe orders of Captain Medina, the company commander, had in fact been ambiguous; at least, the men who heard them could not agree afterwards as to whether or not they had been told to \"waste\" the inhabitants of My Lai. He is quoted as having told his company to leave nothing living behind them and to take no prisoners: \"They're all V.C.'s, now go and get them.\" But he is also said to have ordered only the killing of \"enemies,\" and when asked, \"Who is the enemy?\" to have offered the following definition (in the words of one of the soldiers): \"anybody that was running from us, hiding from us, or who appeared to us to be the enemy. If a man was running, shoot him; sometimes even if a woman with a rifle was running, shoot her.\" That is a very bad definition, but it isn't morally insane; barring a loose interpretation of the \"appearance\" of enmity, it would have excluded most of the people killed at My Lai. Lieutenant Calley, who actually led the unit that entered the village, gave far more specific orders, commanding his men to kill helpless civilians who were neither running nor hiding, let alone carrying rifles, and repeating the command again and again when they hesitated to obey.b The army's judicial system singled him out for blame and punishment, though he claimed he was only doing what Medina had ordered him to do. The enlisted men who did what Calley ordered them to do were never charged.\n\nIt must be a great relief to follow orders. \"Becoming a soldier,\" writes J. Glenn Gray, \"was like escaping from one's own shadow.\" The world of war is frightening; decisions are difficult; and it is comforting to slough off responsibility and simply do what one is told. Gray reports soldiers insisting on this special kind of freedom: \"When I raised my right hand and took the [army oath], I freed myself of the consequences for what I do. I'll do what they tell me and nobody can blame me.\" Army training encourages this view, even though soldiers are also informed that they must refuse \"unlawful\" orders. No military force can function effectively without routine obedience, and it is the routine that is stressed. Soldiers are taught to obey even petty and foolish commands. The teaching process has the form of an endless drill, aimed at breaking down their individual thoughtfulness, resistance, hostility, and waywardness. But there is some ultimate humanity that cannot be broken down, the disappearance of which we will not accept. In his play The Measures Taken, Bertolt Brecht describes militant communists as \"blank pages on which the Revolution writes its instructions.\" I suppose there are many drill sergeants who dream of a similar blankness. But the description is a false one and the dream a fantasy. It is not that soldiers don't sometimes obey as if they were morally blank. What is crucial is that the rest of us hold them responsible for what they do. Despite their oath, we blame them for the crimes that follow from \"unlawful\" or immoral obedience.\n\nSoldiers can never be transformed into mere instruments of war. The trigger is always part of the gun, not part of the man. If they are not machines that can just be turned off, they are also not machines that can just be turned on. Trained to obey \"without hesitation,\" they remain nevertheless capable of hesitating. I have already cited examples of refusal, delay, doubt, and anguish at My Lai. These are internal confirmations of our external judgments. No doubt we can make these judgments too quickly, without hesitations and doubts of our own, paying too little attention to the harshness of battle and the discipline of the army. But it is a mistake to treat soldiers as if they were automatons who make no judgments at all. Instead, we must look closely at the particular features of their situation and try to understand what it might mean, in these circumstances, at this moment, to accept or defy a military command.\n\nThe defense of superior orders breaks down into two more specific arguments: the claim of ignorance and the claim of duress. These two are standard legal and moral claims, and they seem to function in war very much as they do in domestic society. It is not the case, then, as has often been argued, that when we judge soldiers we must balance the necessities of military discipline (that obedience be quick and unquestioning) against the requirements of humanity (that innocent people be protected). Rather, we view discipline as one of the conditions of wartime activity, and we take its particular features into account in determining individual responsibility. We do not excuse individuals in order to maintain or strengthen the disciplinary system. The army may cover up the crimes of soldiers or seek to limit liability for them with that end (or that pretended end) in view, but such efforts do not represent the delicate working out of a conception of justice. What justice requires is, first of all, that we commit ourselves to the defense of rights and, second, that we attend carefully to the particular defenses of men who are charged with violating rights.\n\nIgnorance is the common lot of the common soldier, and it makes an easy defense, especially when calculations of usefulness and proportionality are called for. The soldier can plausibly say that he does not know and cannot know whether the campaign in which he is engaged is really required for the sake of victory, or whether it has been designed so as to hold unintended civilian deaths within acceptable limits. From his narrow and confined vantage point, even direct violations of human rights\u2014as in the conduct of a siege, for example, or in the strategy of an anti-guerrilla campaign\u2014may be unseen and unseeable. Nor is he bound to seek out information; the moral life of a combat soldier is not a research assignment. We might say that he stands to his campaigns as to his wars: he is not responsible for their overall justice. When war is fought at a distance, he may not be responsible even for the innocent people he himself kills. Artillery men and pilots are often kept in ignorance of the targets at which their fire is directed. If they ask questions, they are routinely assured that the targets are \"legitimate military objectives.\" Perhaps they should always be skeptical, but I don't think we blame them if they accept the assurances of their commanders. We blame instead the far-seeing commanders. As the example of My Lai suggests, however, the ignorance of common soldiers has its limits. The soldiers in the Vietnamese village could hardly have doubted the innocence of the people they were ordered to kill. It is in such a situation that we want them to disobey: when they receive orders which, as the army judge said at the Calley trial, \"a man of ordinary sense and understanding would, under the circumstances, know to be unlawful.\"\n\nNow, this implies an understanding not only of the circumstances but also of the law, and it was argued at Nuremberg and has been argued since that the laws of war are so vague, uncertain, and incoherent that they can never require disobedience. Indeed, the state of the positive law is not very good, especially where it relates to the exigencies of combat. But the prohibition against massacre is plain enough, and I think it is fair to say that common soldiers have been charged and convicted only for the knowing murder of innocent people: shipwrecked survivors struggling in the water, for example, or prisoners of war, or helpless civilians. Nor is it a question here only of the law, for these are acts that not only \"violate unchallenged rules of warfare,\" as the British field manual of 1944 states, but that also \"outrage the general sentiments of humanity.\" Ordinary moral sense and understanding rule out killings like those at My Lai. One of the soldiers there remembers thinking to himself that the slaughter was \"just like a Nazi-type thing.\" That judgment is precisely right, and there is nothing in our conventional morality that renders it doubtful.\n\nBut the excuse of duress may hold even in a case like this, if the order to kill is backed up by a threat of execution. I have argued that soldiers in combat cannot plead self-preservation when they violate the rules of war. For the dangers of enemy fire are simply the risks of the activity in which they are engaged, and they have no right to reduce those risks at the expense of other people who are not engaged. But a threat of death directed not at soldiers in general but at a particular soldier\u2014a threat, as the lawyers say, \"imminent, real, and inevitable\"\u2014alters the case, lifting it out of the context of combat and war risk. Now it becomes like those domestic crimes in which one man forces another, under threat of immediate death, to kill a third. The act is clearly murder, but we are likely to think that the man in the middle is not the murderer. Or, if we do think him a murderer, we are likely to accept the excuse of duress. Surely someone who refuses to kill at such a time, and dies instead, is not just doing his duty; he is acting heroically. Gray provides a paradigmatic example:\n\nIn the Netherlands, the Dutch tell of a German soldier who was a member of an execution squad ordered to shoot innocent hostages. Suddenly he stepped out of rank and refused to participate in the execution. On the spot he was charged with treason by the officer in charge and was placed with the hostages, where he was promptly executed by his comrades.\n\nHere is a man of extraordinary nobility, but what are we to say of his (former) comrades? That they are committing murder when they fire their guns, and that they are not responsible for the murder they commit. The officer in charge is responsible, and those among his superiors who decided on the policy of killing hostages. Responsibility passes over the heads of the members of the firing squad, not because of their oaths, not because of their orders, but because of the direct threat that drives them to act as they do.\n\nWar is a world of duress, of threat and counter-threat, so we must be clear about those cases in which duress does, and those in which it does not count as an excuse for conduct we would otherwise condemn. Soldiers are conscripted and forced to fight, but conscription by itself does not force them to kill innocent people. Soldiers are attacked and forced to fight, but neither aggression nor enemy onslaught forces them to kill innocent people. Conscription and attack bring them up against serious risks and hard choices. But constricted and frightening as their situation is, we still say that they choose freely and are responsible for what they do. Only a man with a gun at his head is not responsible.\n\nBut superior orders are not always enforced at the point of a gun. Army discipline in the actual context of war is often a great deal more haphazard than the firing squad example suggests. \"It is a great boon of frontline positions,\" writes Gray, \"that . . . disobedience is frequently possible, since supervision is not very exact where danger of death is present.\" And in rear areas as well as at the front, there are ways of responding to an order short of obeying it: postponement, evasion, deliberate misunderstanding, loose construction, overly literal construction, and so on. One can ignore an immoral command or answer it with questions or protests; and sometimes even an overt refusal only invites reprimand, demotion, or detention; there is no risk of death. Whenever these possibilities are open, moral men will seize upon them. The law seems to require a similar readiness, for it is a legal principle that duress excuses only if the harm the individual soldier inflicts is not disproportionate to the harm with which he is threatened. He is not excused for the murder of innocent people by the threat of demotion.\n\nIt has to be said, however, that officers are far more capable than enlisted men of weighing the dangers they face. Telford Taylor has described the case of Colonel William Peters, an officer in the Confederate Army during the American Civil War, who refused a direct order to burn the town of Chambersburg, Pennsylvania. Peters was relieved of his command and placed under arrest, but he was never brought before a court martial. We may admire his courage, but if he anticipated that his superiors would (\"prudently,\" as another Confederate officer said) avoid a trial, his decision was relatively easy. The decision of an ordinary soldier, who may well be subject to summary justice and who knows little of the temper of his more distant superiors, is much harder. At My Lai, those men who refused to fire never suffered for their refusal and apparently did not expect to suffer; and that suggests that we must blame the others for their obedience. In more ambiguous cases, the duress of superior orders, though it is not \"imminent, real, and inevitable\" and cannot count as a defense, is commonly regarded as an extenuating factor. That seems the right attitude to take, but I want to stress once again that when we take it we are not making concessions to the need for discipline, but simply recognizing the plight of the common soldier.\n\nThere is another reason for extenuation, unmentioned in the legal literature, but prominent in moral accounts of disobedience. The path that I have marked out as the right one is often a very lonely path. Here, too, the case of the German soldier who broke ranks with his fellow executioners and was promptly executed by them is unusual and extreme. But even when a soldier's doubts and anxieties are widely shared, they are still the subject of private brooding, not of public discussion. And when he acts, he acts alone, with no assurance that his comrades will support him. Civil protest and disobedience usually arise out of a community of values. But the army is an organization, not a community, and the communion of ordinary soldiers is shaped by the character and purposes of the organization, not by their private commitments. Theirs is the rough solidarity of men who face a common enemy and endure a common discipline. On both sides of a war, unity is reflexive, not intentional or premeditated. To disobey is to breach that elemental accord, to claim a moral separateness (or a moral superiority), to challenge one's fellows, perhaps even to intensify the dangers they face. \"This is what is most difficult,\" wrote a French soldier who went to Algeria and then refused to fight, \"being cut off from the fraternity, being locked up in a monologue, being incomprehensible.\"\n\nNow, incomprehensible is perhaps too strong a word, for a man appeals at such a time to common moral standards. But in the context of a military organization, that appeal will often go unheard, and so it involves a risk that may well be greater than that of punishment: the risk of a profound and morally disturbing isolation. This is not to say that one can join in a massacre for the sake of togetherness. But it suggests that moral life is rooted in a kind of association that military discipline precludes or temporarily cuts off, and that fact, too, must be taken into account in the judgments we make. It must be taken into account especially in the case of common soldiers, for officers are more free in their associations and more involved in discussions about policy and strategy. They have a say in the shape and character of the organization over which they preside. Hence, again, the critical importance of command responsibility.\n\nCommand Responsibility\n\nBeing an officer is not at all like being a common soldier. Rank is something men compete for, aspire to, glory in, and so even when officers were initially conscripted, we need not worry about holding them rigidly to the duties of their office. For rank can be avoided even when service cannot. Junior officers are killed at a high rate in combat, but still there are soldiers who want to be officers. It is a question of the pleasures of command; there is nothing quite like it (so I am told) in civilian life. The other side of pleasure, however, is responsibility. Officers take on immense responsibilities, again unlike anything in civilian life, for they have in their control the means of death and destruction. The higher their rank, the greater the reach of their command, the larger their responsibilities. They plan and organize campaigns; they decide on strategy and tactics; they choose to fight here rather than there; they order men into battle. Always, they must aim at victory and attend to the needs of their own soldiers. But they have at the same time a higher duty: \"The soldier, be he friend or foe,\" wrote Douglas MacArthur when he confirmed the death sentence of General Yamashita, \"is charged with the protection of the weak and unarmed. It is the very essence and reason of his being . . . [a] sacred trust.\" Precisely because he himself, gun in hand, artillery and bombers at his call, poses a threat to the weak and unarmed, he must take steps to shield them. He must fight with restraint, accepting risks, mindful of the rights of the innocent.\n\nThat obviously means that he cannot order massacres; nor can he terrorize civilians with bombardment or bombing, or uproot whole populations in order to create \"free-fire zones,\" or take reprisals against prisoners, or threaten to kill hostages. But it means more than that. Military commanders have two further and morally crucial responsibilities. First, in planning their campaigns, they must take positive steps to limit even unintended civilian deaths (and they must make sure that the numbers killed are not disproportionate to the military benefits they expect). Here the laws of war are of little help; no officer is going to be criminally charged for killing too many people if he does not actually massacre them. But the moral responsibility is clear, and it cannot be located anywhere else than in the office of commander. The campaign belongs to the commander as it does not belong to the ordinary combatants; he has access to all available information and also to the means of generating more information; he has (or ought to have) an overview of the sum of actions and effects that he is ordering and hoping for. If, then, the conditions set by the doctrine of double effect are not met, we should not hesitate to hold him accountable for the failure. Second, military commanders, in organizing their forces, must take positive steps to enforce the war convention and hold the men under their command to its standards. They must see to their training in this regard, issue clear orders, establish inspection procedures, and assure the punishment of individual soldiers and subordinate officers who kill or injure innocent people. If a great deal of such killing and injuring takes place, they are presumptively responsible, for we assume that it lay within their power to prevent it. Given what actually happens in war, military commanders have a great deal to answer for.\n\nGeneral Bradley and the Bombing of St. L\u00f4\n\nIn July 1944, Omar Bradley, in command of American forces in Normandy, was engaged in planning a breakout from the invasion beachheads established the month before. The plan that he worked out, code-named COBRA and approved by Generals Montgomery and Eisenhower, called for the carpet bombing of an area three and a half miles wide, one and a half deep, along the P\u00e9riers road outside the town of St. L\u00f4. \"Air bombing, we calculated, would either destroy or stun the enemy in the carpet\" and so permit a quick advance. But it also posed a moral problem, which Bradley discusses in his autobiography. On July 20, he described the coming attack to some American newsmen:\n\nThe correspondents listened quietly to the outline of our plan, craned their necks as I pointed to the carpet and . . . tallied the air strength that had been assigned to us. At the close of the briefing, one of the newsmen asked if we would forewarn the French living within bounds of the carpet. I shook my head as if to escape the necessity for saying no. If we were to tip our hand to the French, we would also show it to the Germans. . . . The success of COBRA hung upon surprise; it was essential we have surprise even if it meant the slaughter of innocents as well.\n\nBombing of this sort, along the line of battle and in close support of combat troops, is permitted by positive international law. Even indiscriminate fire is permitted within the actual combat zone. Civilians are thought to be forewarned by the proximity of the fighting. But as the correspondent's question suggests, this does not resolve the moral issue. We still want to know what positive measures might have been taken to avoid \"the slaughter of innocents\" or reduce the damage done. It is important to insist on such measures because, as this example clearly shows, the proportionality rule often has no inhibitory effects at all. Even if a large number of civilians lived in those five square miles near St. L\u00f4, and even if all of them were likely to die, it would seem a small price to pay for a breakout that might well signal the end of the war. To say that, however, is not to say that those innocent lives are forfeit, for there may be ways of saving them short of calling off the attack. Perhaps civilians all along the battlefront could have been warned (without giving up surprise in a particular sector). Perhaps the attack could have been redirected through some less populated area (even at greater risk to the soldiers involved). Perhaps the planes, flying low, could have aimed at specific enemy targets, or artillery have been used instead (since shells could then be aimed more precisely than bombs), or paratroops dropped or patrols sent forward to seize important positions in advance of the main attack. I am in no position to recommend any of these courses of action, although, in the event, any of the last of them might have been preferable, even from a military point of view. For the bombs missed the carpet and killed or wounded several hundred American soldiers. How many French civilians were killed or wounded Bradley does not say.\n\nHowever many civilians died, it cannot be said that their deaths were intentional. On the other hand, unless Bradley worked his way through the sorts of possibilities I have listed, it also cannot be said that he intended not to kill them. I have already explained why that negative intention ought to be required from soldiers; it is the domestic equivalent of what the lawyers call \"due care\" in domestic society. With reference to specific and small-scale military actions (like the bombing of cellars described by Frank Richards), the people required to take care are common soldiers and their immediate superiors. In cases such as the COBRA campaign, the relevant individuals stand higher in the hierarchy; it is on General Bradley that we rightly focus our attention, and on his superiors. Once again, I have to say that I cannot specify the precise point at which the requirements of \"due care\" have been met. How much attention is required? How much risk must be accepted? The line isn't clear. But it is clear enough that most campaigns are planned and carried out well below the line; and one can blame commanders who don't make minimal efforts, even if one doesn't know exactly what a maximum effort would entail.\n\nThe Case of General Yamashita\n\nThe same problem of specifying standards comes up when one considers the responsibility of commanders for the actions of their subordinates. They are bound, as I have said, to enforce the war convention. But even the best possible system of enforcement doesn't preclude particular violations. It proves itself the best possible system by seizing upon these in a systematic way and by punishing the individuals who commit them so as to deter the others. It is only if there is a massive breakdown of this disciplinary system that we demand an accounting from the officers who preside over it. This, in effect, is the demand formally made upon General Yamashita by an American military commission in the aftermath of the Philippine campaign in 1945. It was said of Yamashita that he was responsible for a large number of specified acts of violence and murder inflicted upon unarmed civilians and prisoners of war. That these acts had in fact been committed by Japanese soldiers no one denied. On the other hand, no evidence was presented to show that Yamashita had ordered the violence and murder nor even that he had known about any of the specified acts. His responsibility lay in his failure \"to discharge his duty as commander to control the operations of the members of his command, permitting them to commit brutal atrocities. . . .\" Defending himself, Yamashita claimed that he had been entirely unable to exercise control over his troops: the successful American invasion had disrupted his communication and command structure, leaving him in effective charge only of the troops whom he personally led, in retreat, into the mountains of northern Luzon; and these troops had committed no atrocities. The commission refused to accept this defense and sentenced Yamashita to death. His appeal was carried to the U.S. Supreme Court, which declined to review the case, despite memorable dissents by Justices Murphy and Rutledge. Yamashita was executed on February 22, 1946.\n\nThere are two ways of describing the standard to which Yamashita was held by the commission and the Court majority. The defense lawyers argued that the standard was one of strict liability, radically inappropriate in cases of criminal justice. That is to say, Yamashita was convicted without reference to any acts he committed or even to any omissions that he might have avoided. He was convicted of having held an office, because of the duties said to inhere in that office, even though the duties were in fact undo-able under the conditions in which he found himself. Justice Murphy went further: the duties were undo-able because of the conditions that the American army had created.\n\n. . . read against the background of military events in the Philippines subsequent to October 9, 1944, these charges amount to this: \"We, the victorious American forces, have done everything possible to destroy and disorganize your lines of communication, your effective control of your personnel, your ability to wage war. In these respects we have succeeded. . . . And now we charge and condemn you for having been inefficient in maintaining control of your troops during the period when we were so effectively besieging and eliminating your forces and blocking your ability to maintain effective command.\"\n\nThis is probably an accurate description of the facts of the case. Not only was Yamashita unable to do the things that commanders should do, but if we push the argument back, he was in no sense the author of the conditions which made those things impossible. I should add, however, that the other judges did not believe, or did not admit, that they were enforcing the principle of strict liability. According to Chief Justice Stone, the question was \"whether the law of war imposes on an army commander a duty to take such appropriate measures as are within his power to control the troops under his command. . . .\" It is easy to answer that question affirmatively, but not at all easy to say what measures are \"appropriate\" under the adverse conditions of combat, disorganization, and defeat.\n\nOne wants to set the standards very high, and the argument for strict liability is utilitarian in character: holding officers automatically responsible for massive violations of the rules of war forces them to do everything they can to avoid such violations, without forcing us to specify what they ought to do. But there are two problems with this. First of all, we don't really want commanders to do everything they can, for that requirement, taken literally, would leave them little time to do anything else. This point is never as telling in their case as it is in the case of political leaders and domestic crime: we don't require our leaders to do everything they can (but only to take \"appropriate measures\") to prevent robbery and murder, for they have other things to do. But they, presumably, have not armed and trained the people who commit robbery and murder, and these people are not directly in their charge. The case of military commanders is different; hence we must expect them to devote a great deal of time and attention to the discipline and control of the men with guns they have turned loose in the world. But still, not all their time and attention, not all the resources at their command.\n\nThe second argument against strict liability in criminal cases is a more familiar one. Even doing \"everything\" is not the same as doing it successfully. All we can require is serious efforts of specific sorts; we cannot require success, since the conditions of warfare are such that success isn't always possible. And the impossibility of success is necessarily an excuse\u2014given serious effort, an entirely satisfactory excuse\u2014for failure. To refuse to accept the excuse is to refuse to regard the defendant as a moral agent: for it is in the nature of moral agents (of human beings) that their best efforts sometimes fail. The refusal disregards the defendant's humanity, makes him into an example, pour encourager les autres; and that we have no right to do to anyone.\n\nThese two arguments seem to me right, and they exonerate General Yamashita, but they also leave us with no clear standards at all. In fact, there is no philosophical or theoretical way of fixing such standards. That is also true with regard to the planning and organization of military campaigns. There is no sure rule against which to measure the conduct of General Bradley. The discussion of double effect in chapters 9 and 10 pointed only in a fairly crude way toward the sorts of considerations that are relevant when we make judgments about such matters. The appropriate standards can emerge only through a long process of casuistic reasoning, that is, by attending to one case after another, morally or legally. The chief failure of the military commission and the Supreme Court in 1945, aside from the fact that they failed to do justice to General Yamashita, is that they made no contribution to this process. They did not specify the measures that Yamashita might have taken; they did not suggest what degree of disorganization might serve as a limit on command responsibility. Only by making such specifications, again and again, can we draw the lines that the war convention requires.\n\nWe can say more than this, I think, if we turn back briefly to the My Lai case. The evidence brought forward at the trial of Lieutenant Calley and the materials collected by newsmen carrying on their own investigations of the massacre clearly suggest the responsibility of officers superior to both Calley and Medina. The strategy of the American war in Vietnam, as I have already argued, tended to put civilians at risk in unacceptable ways, and ordinary soldiers could hardly ignore the implications of that strategy. My Lai was itself in a free-fire zone, routinely shelled and bombed. \"If you can shoot artillery . . . in there every night,\" one soldier asked, \"how can the people in there be worth so much?\" In effect, soldiers were taught that civilian lives were not worth much, and there seems to have been little effort to counteract that teaching except by the most formal and perfunctory instruction in the rules of war. If we are fully to assign blame for the massacre, then, there are a large number of officers whom we would have to condemn. I cannot put together a list here, and I doubt that all of them could have been or ought to have been legally charged and tried\u2014though this might have been a useful occasion to apply, and improve upon, the Yamashita precedent. But that many officers are morally chargeable seems certain, and their blameworthiness is not less than that of the men who did the actual killing. Indeed, there is this difference between them: in the case of the ordinary soldiers, the burden of proof lies with us. As in any murder case, we must prove their knowing and willful participation. But the officers are presumptively guilty; the burden of proof, if they would demonstrate their innocence, lies with them. And until we find some way of imposing that burden, we shall not have done all that we can do in defense of the \"weak and unarmed,\" the innocent victims of war.\n\nThe Nature of Necessity (4)\n\nI have left the hardest question for last. What are we to say about those military commanders (or political leaders) who override the rules of war and kill innocent people in a \"supreme emergency\"? Surely we want to be led at such a time by men and women ready to do what has to be done\u2014what is necessary; for it is only here that necessity, in its true sense, comes into the theory of war. On the other hand, we cannot ignore or forget what it is they do. The deliberate killing of the innocent is murder. Sometimes, in conditions of extremity (which I have tried to define and delimit), commanders must commit murder or they must order others to commit it. And then they are murderers, though in a good cause. In domestic society, and particularly in the context of revolutionary politics, we say of such people that they have dirty hands. I have argued elsewhere that men and women with dirty hands, though it may be the case that they had acted well and done what their office required, must nonetheless bear a burden of responsibility and guilt. They have killed unjustly, let us say, for the sake of justice itself, but justice itself requires that unjust killing be condemned. There is obviously no question here of legal punishment, but of some other way of assigning and enforcing blame. What way, however, is radically unclear. The available answers are all likely to make us uneasy. The nature of that uneasiness will be apparent if we turn again to the case of British terror bombing in World War II.\n\nThe Dishonoring of Arthur Harris\n\n\"He will perhaps go down in history as a giant among the leaders of men. He gave Bomber Command the courage to surmount its ordeals. . . .\" So writes the historian Noble Frankland about Arthur Harris, who directed the strategic bombing of Germany from February 1942 until the end of the war. Harris was, as we have seen, the determined advocate of terrorism, resisting every attempt to use his planes for other purposes. Now, terror bombing is a criminal activity, and after the immediate threat posed by Hitler's early victories had passed, it was an entirely indefensible activity. Hence Harris' case isn't really an example of the dirty hands problem. He and Churchill, who was ultimately responsible for military policy, faced no moral dilemma: they should simply have stopped the bombing campaign. But we can take it as an example, nonetheless, for it apparently had that form in the minds of British leaders, even of Churchill himself at the end. That is why Harris, though of course criminal charges were never brought against him, was not treated after the war as a giant among the leaders of men.\n\nHe had done what his government thought necessary, but what he had done was ugly, and there seems to have been a conscious decision not to celebrate the exploits of Bomber Command or to honor its leader. \"From this work,\" writes Angus Calder, \"Churchill and his colleagues at last recoiled. After the strategic air offensive officially ended in mid-April [1945], Bomber Command was slighted and snubbed; and Harris, unlike other well-known commanders, was not rewarded with a peerage.\" In such circumstances, not to honor was to dishonor, and that is exactly how Harris regarded the government's action (or omission). He waited a while for his reward and then, resentfully, left England for his native Rhodesia. The men he led were similarly treated, though the snub was not so personal. In Westminster Abbey, there is a plaque honoring those pilots of Fighter Command who died during the war, listing them all by name. But the bomber pilots, though they suffered far heavier casualties, have no plaque; their names are unrecorded. It is as if the British had taken to heart Rolf Hochhuth's question:\n\nIs a pilot who bombs\n\npopulation centers under orders\n\nstill to be called a soldier?\n\nAll this makes a point, though it does so indirectly and in so equivocal a fashion that we cannot but notice its moral awkwardness. Harris and his men have a legitimate complaint: they did what they were told to do and what their leaders thought was necessary and right, but they are dishonored for doing it, and it is suddenly suggested (what else can the dishonor mean?) that what was necessary and right was also wrong. Harris felt that he was being made a scapegoat, and it is surely true that if blame is to be distributed for the bombing, Churchill deserves a full share. But Churchill's success in dissociating himself from the policy of terrorism is not of great importance; there is always a remedy for that in retrospective criticism. What is important is that his dissociation was part of a national dissociation\u2014a deliberate policy that has moral significance and value.\n\nAnd yet, the policy seems cruel. Stated in general terms, it amounts to this: that a nation fighting a just war, when it is desperate and survival itself is at risk, must use unscrupulous or morally ignorant soldiers; and as soon as their usefulness is past, it must disown them. I would rather say something else: that decent men and women, hard-pressed in war, must sometimes do terrible things, and then they themselves have to look for some way to reaffirm the values they have overthrown. But the first statement is probably the more realistic one. For it is very rare, as Machiavelli wrote in his Discourses, \"that a good man should be found willing to employ wicked means,\" even when such means are morally required. And then we must look for people who are not good, and use them, and dishonor them. Perhaps there is some better way of doing that than the way Churchill chose. It would have been better if he had explained to his countrymen the moral costs of their survival and if he had praised the courage and endurance of the fliers of Bomber Command even while insisting that it was not possible to take pride in what they had done (an impossibility that many of them must have felt). But Churchill did not do that; he never admitted that the bombing constituted a wrong. In the absence of such an admission, the refusal to honor Harris at least went some small distance toward re-establishing a commitment to the rules of war and the rights they protect. And that, I think, is the deepest meaning of all assignments of responsibility.\n\nConclusion\n\nThe world of necessity is generated by a conflict between collective survival and human rights. We find ourselves in that world less often than we think, certainly less often than we say; but whenever we are there, we experience the ultimate tyranny of war\u2014and also, it might be argued, the ultimate incoherence of the theory of war. In a troubling essay entitled \"War and Massacre,\" Thomas Nagel has described our situation at such a time in terms of a conflict between utilitarian and absolutist modes of thought: we know that there are some outcomes that must be avoided at all costs, and we know that there are some costs that can never rightly be paid. We must face the possibility, Nagel argues, \"that these two forms of moral intuition are not capable of being brought together into a single, coherent moral system, and that the world can present us with situations in which there is no honorable or moral course for a man to take, no course free of guilt and responsibility for evil.\" I have tried to avoid the stark indeterminacy of that description by suggesting that political leaders can hardly help but choose the utilitarian side of the dilemma. That is what they are there for. They must opt for collective survival and override those rights that have suddenly loomed as obstacles to survival. But I don't want to say, any more than Nagel does, that they are free of guilt when they do that. Were there no guilt involved, the decisions they make would be less agonizing than they are. And they can only prove their honor by accepting responsibility for those decisions and by living out the agony. A moral theory that made their life easier, or that concealed their dilemma from the rest of us, might achieve greater coherence, but it would miss or it would repress the reality of war.\n\nIt is sometimes said that the dilemma ought to be concealed, that we should draw the veil (as Churchill tried to do) over the crimes that soldiers and statesmen cannot avoid. Or, we should avert our eyes\u2014for the sake of our innocence, I suppose, and the moral certainties. But that is a dangerous business; having looked away, how will we know when to look back? Soon we will avert our eyes from everything that happens in wars and battles, condemning nothing, like the second monkey in the Japanese statue, who sees no evil. And yet there is plenty to see. Soldiers and statesmen live mostly on this side of the ultimate crises of collective survival; the greater number by far of the crimes they commit can neither be defended nor excused. They are simply crimes. Someone must try to see them clearly and describe them \"in express words.\" Even the murders called necessary must be similarly described; it doubles the crime to look away, for then we are not able to fix the limits of necessity, or remember the victims, or make our own (awkward) judgments of the people who kill in our name.\n\nMostly morality is tested only by the ordinary pressures of military conflict. Mostly it is possible, even when it isn't easy, to live by the requirements of justice. And mostly the judgments we make of what soldiers and statesmen do are singular and clearcut; with whatever hesitations, we say yes or no, we say right or wrong. But in supreme emergencies our judgments are doubled, reflecting the dualist character of the theory of war and the deeper complexity of our moral realism; we say yes and no, right and wrong. That dualism makes us uneasy; the world of war is not a fully comprehensible, let alone a morally satisfactory place. And yet it cannot be escaped, short of a universal order in which the existence of nations and peoples could never be threatened. There is every reason to work for such an order. The difficulty is that we sometimes have no choice but to fight for it.\n\n Telford Taylor suggests a possible exception to this rule, citing a hypothetical case which has often been discussed in the legal literature. A small detachment of troops on a special mission or cut off from its main force takes prisoners \"under such circumstances that men cannot be spared to guard them . . . and that to take them along would greatly endanger the success of the mission or the safety of the unit.\" The prisoners are likely to be killed, Taylor says, in accordance with the principle of military necessity. (Nuremberg and Vietnam, New York, 1970, p. 36.) But if it is only the safety of the unit that is in question (its mission may already have been accomplished), the proper appeal would be to self-\u00adpreservation. The argument from necessity has not, despite Taylor, been accepted by legal writers; the argument from self-preservation has won greater support. In his military code for the Union Army, for example, Francis Lieber writes that \"a commander is permitted to direct his troops to give no quarter . . . when his own salvation makes it impossible to cumber himself with prisoners.\" (Taylor, p. 36n.) But surely in such a case the prisoners should be disarmed and then released. Even if it is \"impossible\" to take them along, it is not impossible to set them free. There may be risks in doing that, but these are exactly the sorts of risks soldiers must accept. The risks involved in leaving wounded men behind are of the same sort, but that is not a satisfactory reason for killing them. For a useful discussion of these issues, see Marshall Cohen, \"Morality and the Laws of War,\" in Held, Morgenbesser, and Nagel, eds., Philosophy, Morality, and International Affairs, New York, 1974, pp. 76\u201378.\n\n It may be useful to suggest the sorts of commands that should be issued at such a time. Here is an account of an Israeli unit entering Nablus during the Six Day War: \"The battalion CO got on the field telephone to my company and said, 'Don't touch the civilians . . . don't fire until you're fired at and don't touch the civilians. Look, you've been warned. Their blood be on your heads.' In just those words. The boys in the company kept talking about it afterwards. . . . They kept repeating the words. . . . 'Their blood be on your heads.'\" The Seventh Day: Soldiers Talk About the Six Day War, London, 1970, p. 132.\nAfterword: Nonviolence and the Theory of War\n\nThe dream of a war to end war, the myth of Armageddon (the last battle), the vision of the lion lying down with the lamb\u2014all these point toward an age definitively peaceful, a distant age that lies across some unknown time-break, without armed struggle and systematic killing. It will not come, so we have been told, until the forces of evil have been decisively defeated and mankind freed forever from the lust for conquest and domination. In our myths and visions, the end of war is also the end of secular history. Those of us trapped within that history, who see no end to it, have no choice but to fight on, defending the values to which we are committed, unless or until some alternative means of defense can be found. The only alternative is nonviolent defense, \"war without weapons,\" as it has been called by its advocates, who seek to adjust our dreams to our realities. They claim that we can uphold the values of communal life and liberty without fighting and killing, and this claim raises important questions (secular and practical questions) about the theory of war and the argument for justice. To treat them as they deserve would require another book; I can offer only a brief essay, a partial and tentative analysis of the ways in which nonviolence relates, first, to the doctrine of aggression, and then to the rules of war.\n\nNonviolent defense differs from conventional strategies in that it concedes the overrunning of the country that is being defended. It establishes no obstacles capable of stopping a military advance or preventing a military occupation. \"Although minor delaying actions against the incursions of foreign troops and functionaries may be possible,\" writes Gene Sharp, \"civilian defense . . . does not attempt to halt such entry, and cannot successfully do so.\" That is a radical concession, and I don't think that any government has ever made it willingly. Nonviolence has been practiced (in the face of an invasion) only after violence, or the threat of violence, has failed. Then its protagonists aim to deny the victorious army the fruits of its victory through a systematic policy of civilian resistance and noncooperation: they call upon the conquered people to make themselves ungovernable. I want to stress that it is not war but civilian resistance that has usually been regarded as a last resort, because war holds out at least the possibility of avoiding the occupation that evokes or requires the resistance. But we might reverse this ordering were we to decide that resistance is as likely to end the occupation as military action is to prevent it, and at a much lower cost in human lives. There is as yet no evidence that that proposition is true, \"no cases in which . . . civilian defense has caused an invader to withdraw.\" But no nonviolent struggle has ever been undertaken by a people trained in advance in its methods and prepared (as soldiers are in the case of war) to accept its costs. So it might be true; and if it is, we should have to regard aggression very differently from the way we do at present.\n\nIt might be said that nonviolence abolishes aggressive war simply by virtue of the refusal to engage the aggressor militarily. Invasion is not morally coercive in the ways I described in chapter 4, men and women cannot be forced to fight, if they have come to believe that they can defend their country in some other way, without killing and being killed. And if there really is some other way, at least potentially effective, then the aggressor cannot be charged with forcing them to fight. Nonviolence de-escalates the conflict and diminishes its criminality. By adopting the methods of disobedience, noncooperation, boycott, and general strike, the citizens of the invaded country transform aggressive war into a political struggle. They treat the aggressor in effect as a domestic tyrant or usurper, and they turn his soldiers into policemen. If the invader accepts this role, and if he responds to the resistance he encounters with curfews, fines, jail sentences, and nothing more, the prospect is opened up of a long-term struggle, not without its difficulties and painfulness for civilians, but far less destructive than even a short war, and winnable (we are assuming) by those same civilians. Allied states would have no reason to intervene militarily in such a struggle; which is a good thing, since if they too were committed to nonviolent defense, they would have no means of intervening. But they could bring moral and perhaps also economic pressure to bear against the invaders.\n\nThis, then, would be the position of the invaders: they would hold the country they had \"attacked,\" could establish military bases wherever they pleased, and enjoy whatever strategic benefits these yielded them (vis-\u00e0-vis other countries, presumably). But their logistics problems would be severe, for unless they brought along their own personnel, they could not depend upon the local transportation or communication systems. And since they could hardly bring along an entire workforce, they would have great difficulty exploiting the natural resources and the industrial productivity of the invaded country. Hence the economic costs of the occupation would be high. The political costs might well be higher. Everywhere their soldiers would encounter sullen, resentful, withdrawn, and noncooperative civilians. Though these civilians would never take up arms, they would rally, demonstrate, and strike; and the soldiers would have to respond, coercively, like the hated instruments of a tyrannical regime. Their military \u00e9lan might well fade, their morale erode, under the strains of civilian hostility and of an on-going struggle in which they never experienced the release of an open fight. Eventually, perhaps, the occupation would become untenable, and the invaders would simply leave; they would have won and then lost a \"war without weapons.\"\n\nThis is an attractive, even though it is not a millennial, picture. Indeed, it is attractive precisely because it is not millennial, but conceivable in the world we know. It is only just conceivable, however; for the success I have described is possible only if the invaders are committed to the war convention\u2014and they won't always be committed. While nonviolence by itself replaces aggressive war with political struggle, it cannot by itself determine the means of struggle. The invading army can always adopt the common methods of domestic tyrants, which go well beyond curfews, fines, and jail sentences; and its leaders, though they are soldiers, may well be tempted to do that for the sake of a quick \"victory.\" Tyrants will not, of course, lay siege to their own cities or bomb or bombard them; nor will invaders who encounter no armed opposition. But there are other, probably more efficient, ways of terrorizing a people whose country one controls, and of breaking their resistance. In his \"Reflections on Gandhi,\" George Orwell points out the importance of exemplary leadership and wide publicity in a nonviolent campaign and wonders whether such a campaign would even be possible in a totalitarian state. \"It is difficult to see how Gandhi's methods could be applied in a country where opponents of the regime disappear in the middle of the night and are never heard from again.\" Nor would civilian resistance work well against invaders who sent out squads of soldiers to kill civilian leaders, who arrested and tortured suspects, established concentration camps, and exiled large numbers of people from areas where the resistance was strong to distant and desolate parts of the country. Nonviolent defense is no defense at all against tyrants or conquerors ready to adopt such measures. Gandhi demonstrated this truth, I think by the perverse advice he gave to the Jews of Germany: that they should commit suicide rather than fight back against Nazi tyranny. Here nonviolence, under extreme conditions, collapses into violence directed at oneself rather than at one's murderers, though why it should take that direction I cannot understand.\n\nIf one faces an enemy like the Nazis, and if armed resistance is impossible, it is virtually certain that the men and women of the occupied country\u2014those who have been marked out for survival, at any rate, and perhaps even those who have been marked out for death\u2014will yield to their new masters and obey their decrees. The country will grow silent. Resistance will be a matter of individual heroism or of the heroism of small groups, but not of collective struggle.\n\nThe success of nonviolent resistance requires that soldiers (or their officers or political leaders) refuse at some early point, before civilian endurance is exhausted, to carry out or support a terrorist policy. As in guerrilla war, the strategy is to force the invading army to bear the onus of civilian deaths. But here the onus is to be made especially clear (especially unbearable) by the dramatic absence of any armed struggle in which civilians might be collusive. They will be hostile, certainly, but no soldiers will die at their hands or at the hands of partisans who have their secret support. And yet, if their resistance is to be broken decisively and quickly, the soldiers will have to be prepared to kill them. Since they are not always prepared to do that, or since their officers are not always sure that they will do it again and again, as might be necessary, civilian defense has had a certain limited effectiveness\u2014not in expelling an invading army, but in preventing the attainment of particular goals set by its leaders. As Liddell Hart has argued, however, these effects have only been possible\n\nagainst opponents whose code of morality was fundamentally similar [to that of the civilian defenders], and whose ruthlessness was thereby restrained. It is very doubtful whether non-violent resistance would have availed against a Tartar conqueror in the past, or against a Stalin in more recent times. The only impression it seems to have made on Hitler was to excite his impulse to trample on what, to his mind, was contemptible weakness\u2014although there is evidence that it did embarrass many of his generals, brought up in a better code . . .\n\nIf one could count on that \"better code\" and look forward to a nonviolent test of wills\u2014civilian solidarity against military discipline\u2014there would, I think, be no reason to fight: political struggle is better than fighting, even when victory is uncertain. For victory in war is also uncertain; and here it might be said, as it cannot easily be said in the case of war, that the citizens of the occupied country will win if they deserve to win. As in the domestic struggle against tyranny (so long as the struggle doesn't degenerate into massacre), we judge them by their capacity for self-help, that is, by their collective determination to defend their liberty.\n\nWhen one cannot count on the moral code, nonviolence is either a disguised form of surrender or a minimalist way of upholding communal values after a military defeat. I don't want to underestimate the importance of the second of these. Though civilian resistance evokes no moral recognition among the invading soldiers, it can still be important for its practitioners. It expresses the communal will to survive; and though the expression is brief, as in Czechoslovakia in 1968, it is likely to be long remembered. The heroism of civilians is even more heartening than that of soldiers. On the other hand, one should not expect much more from civilians confronted with a terrorist or potentially terrorist army than brief or sporadic resistance. It is easy to say that \"Non-violent action is not a course for cowards. It requires the ability and determination to sustain the battle whatever the price in suffering. . . .\" But this sort of exhortation is no more attractive than that of a general telling his soldiers to fight to the last man. Indeed, I prefer the exhortation of the general, since he at least addresses himself to a limited number of men, not to an entire population. The case is similar with guerrilla war, which has this advantage over civilian resistance: it recapitulates the military situation where only a relatively few people are asked \"to sustain the battle\"\u2014though the others will suffer too, as we have seen, unless the opposing army fights in accordance with the war convention.\n\nThe comparison with guerrilla war is worth pursuing further. In an armed insurrection, the coercing and killing of civilians by enemy soldiers has the effect of mobilizing other civilians and bringing them into the insurgent camp. The indiscriminate violence of their opponents is one of the major sources of guerrilla recruitment. Nonviolent resistance, on the other hand, is possible on a significant scale only if civilians are already mobilized and prepared to act together. The resistance is simply the physical expression of that mobilization, directly, in the streets, or indirectly, through economic slowdowns and political passivity. Now the coercion and killing of civilians is likely to break the solidarity of the resistance, spreading terror through the country and eventually producing a dulled acquiescence. At the same time, it may demoralize the soldiers who are called upon to do what appears to them\u2014if it appears to them\u2014indecent work, and it may undercut support for the occupation among the friends and relatives of those soldiers. Guerrilla war can produce a similar demoralization, but the effect is compounded by the fear soldiers must feel in the face of the hostile men and women among whom they are forced to fight (and die). In the case of nonviolent defense, there will be no fear; there will only be disgust and shame. The success of the defense is entirely dependent upon the moral convictions and sensibilities of the enemy soldiers.\n\nNonviolent defense depends upon noncombatant immunity. For this reason, it is no service to the cause to ridicule the rules of war or to insist (as Tolstoy did) that violence is always and necessarily unrestrained. When one wages a \"war without weapons,\" one appeals for restraint from men with weapons. It is not likely that these men, soldiers subject to military discipline, are going to be converted to the creed of nonviolence. Nor is it critical to the success of the \"war\" that they be converted, but only that they be held to their own putative standards. The appeal that is made to them takes this form: \"You cannot shoot at me, because I am not shooting at you; nor am I going to shoot at you. I am your enemy and will remain so as long as you occupy my country. But I am a noncombatant enemy, and you must coerce and control me, if you can, without violence.\" The appeal simply restates the argument about civilian rights and soldierly duties that underlies the war convention and provides its substance. And this suggests that the transformation of war into a political struggle has as its prior condition the restraint of war as a military struggle. If we are to aim at the transformation, as we should, we must begin by insisting upon the rules of war and by holding soldiers rigidly to the norms they set. The restraint of war is the beginning of peace.\nPostscript: A Defense of Just War Theory\n\nIn the years since this book was first published, just war theory has become a minor academic industry; it has proved particularly engaging for contemporary philosophers, though political theorists and legal scholars are also working on the difficult issues of morality and war. The literature seems vast, given what writing on this subject was like in the middle years of the last century. Vietnam is the major reason for the surge of interest, but interest has been sustained by America's subsequent wars, which have been greater in number and longer in duration than anyone expected.\n\nMany recent books and articles are critical of what is called standard (or even orthodox) just war theory, which this book is often taken to represent. I have read some of this literature, but mostly I have listened to the critical arguments at lectures and academic conferences in the United States and abroad\u2014and sometimes I have responded to them. I want to respond here, not to particular philosophers but to the cohort of recent academic writers about war who believe that the distinctiveness of just war theory is unnecessary; war's dilemmas are in no way exceptional; they do not differ from the moral dilemmas of everyday life, and they can be dealt with by the familiar methods of analytic philosophy. What is at issue, in part, is the very subject of the theory. What is just war theory about?\n\nThere are at least two answers to this question, one of which I want to defend as strongly as I can. The first answer is that just war theory is about war, and the second answer is that just war theory is about moral philosophy. The difference is most simply a matter of focus. Toward what issues, what hard questions, what specific circumstances, is the theorist's attention directed? But we might also think of it in a backward looking way. What was the theorist reading before she began writing? And since I bear some small responsibility for the academic industry, I will begin autobiographically, with my own reading matter.\n\nBefore I wrote Just and Unjust Wars, I read some key texts in Catholic moral theology and early international law\u2014Augustine, Aquinas, and Vitoria; Grotius and Puffendorf, and a few others. I read a handful of \u00adnineteenth- and twentieth-century legal textbooks and a couple of contemporary theorists, like Paul Ramsey, the most important Protestant writer on war in the 1950s and 1960s. But the greater part by far of my reading was not in theory at all but in military history, both academic and popular, and then in the memoir literature produced by soldiers of different ranks (preferably the lower ranks: junior officers and foot soldiers, who make the toughest moral decisions on the battlefield); and then in wartime journalism and commentary (especially about Vietnam, the immediate occasion of my own writing). Finally I read many of the novels and poems that deal with the experience of fighting and the company of soldiers. The nontheoretical genres, and the books and articles they include, seemed to me the critically necessary material for my project\u2014partly because I had never been a soldier myself and I needed to learn as much as I could about the experience of war. But I also focused on histories, memoirs, essays, novels, and poems because I wanted the moral arguments of my book to ring true to their authors\u2014and to the men and women about whom they were writing.\n\nLooking over the recent flood of books and articles about justice and injustice in war, I sense that many authors are not reading the way I did. They are preoccupied with the academic literature about moral philosophy and just war theory. They are reading the journals, not the journalists; they are reading each other. This is a common academic practice, but it has always seemed to me problematic\u2014especially when the subject is politics and war.\n\nAfter reading each other, these theorists argue with each other (and sometimes with the rest of us), disagreeing about significant theoretical points and about fine points, too. Some of the disagreements are about ethical issues like self-defense and responsibility\u2014issues that arise not only in wartime but also in civil society in time of peace and in many ordinary domestic contexts. Many of these theorists take the view that issues of this sort can be delineated most clearly and addressed most conclusively in contexts far removed from war and even in hypothetical and elaborately constructed cases that have no historical or practical reference at all. So they have no need to read, say, military history; the debate is focused elsewhere, and all that is necessary is to read the works of the other participants in the debate.\n\nI don't want to deny the possible usefulness of this sort of philosophical labor. Issues that arise in war, but also in other real and imagined contexts, can certainly be addressed, illuminated, and perhaps even resolved, in the other contexts. But I worry that the illuminations and resolutions won't ring true to the people I have always tried to address, for whom war is a primary subject and a personal experience. But how can that be? Surely if we have figured out what personal responsibility (for example) means\u2014in the way that many contemporary philosophers figure things out, by abstracting from particular cases, by inventing examples that test every possible definition, by calling each other to account with increasingly refined examples\u2014then we also know what personal responsibility means in war. There is no point in reading historical analyses of military decisions, or subjective accounts of decision making in the field, or fictional narratives about combat, since none of these are designed for philosophical purposes. What counts is the cleverness of the design and the questions that it highlights and helps us answer.\n\nFor these theorists wars and battles are like street crimes and marital disputes in civil society; they involve the same kind of moral dilemmas. They are \"cases\" to which theorists need to apply the rules of everyday morality. In order to figure out the rules, they may start from the cases, playing with them, changing their details, even inventing possible and impossible variations that test our understanding of the rules as they are, or as they might be, or should be. But these theorists have no commitment to the actual cases until they know the applicable rules, and hypothetical cases will do just as well in figuring things out\u2014perhaps better, since they impose no reality constraints on their designers. Still, all the cases, real and hypothetical, are in some sense familiar, that is, they can be imagined to arise, however improbable they are, in everyday life.\n\nI want to argue, against this view, that wars and battles are not \"cases\" to which the law and morality of everyday life can be applied; by definition, they don't take place in civil society. War is a long-standing human practice (however uncomfortable we are with it), which represents a radical break with our ordinary social activities. The practice of war has been argued about and reflected on over many centuries, and it has its own law and even its own morality\u2014which have been produced through the adaptation of ordinary law and morality to the peculiar circumstances of war. If we want to understand why that adaptation was necessary and what it has produced, we need to turn first to war itself. We need to understand what wars and battles are, how they have been experienced over the years, and how their moral and legal rules have been worked out. That's the point of reading military history and soldiers' memoirs. Only then can we turn to particular wars, particular battles, and particular incidents in battles; only then do we have cases to which we can apply the rules and come to grips with the peculiar tensions to which these applications are subject. And then, finally, we will be in a position to argue for or against revisions of the rules. We can learn a lot about the rules and the tensions and the possibly necessary revisions by reading what earlier international lawyers and just war theorists have written, but some of their key arguments will seem strange or incomprehensible unless we begin with the literature of war itself.\n\nIt is especially strange that just war theory, in its standard form, requires us to judge the conduct of a war independently of our judgment of its character, so that what soldiers are permitted to do or barred from doing in battle doesn't depend on whether their war is just or not. I am going to focus here on this strange requirement, which lies at the center of contemporary philosophical debates. What the required independence of the two judgments means is that we grant soldiers on both sides, whether their cause is just or unjust, an equal right to fire their guns, so long as they aim only at each other and not at innocent civilians. We treat soldiers on the battlefield as moral equals. Many (it may be most) of the philosophers working on these issues today are highly critical of this equality, and of the separation of ad bellum justice from in bello justice. They think it is obviously wrong to judge soldiers by how well they fight, without reference to the rightness or wrongness of the war they are fighting. They want soldiers fighting a just war to be able to do things, like firing their guns, that soldiers fighting an unjust war are barred from doing. And their argument makes a lot of sense if, as I wrote about one of them, we imagine war to be a peacetime activity. Indeed, standard just war theory is untenable if we take wars and battles to be like street crimes and marital disputes.\n\nWe need an example here, not hypothetical, so let's compare aggressive war to a bank robbery in, say, Philadelphia, the city of brotherly love. Let Philadelphia represent a peaceful civil society. We would certainly not judge the conduct of the robbery independently of the wrongfulness of robbing banks, as if the wrong didn't make a difference. It does make a difference. If there is a shoot-out between the bank robber and a bank guard, it can't be the case that the two of them have an equal right to shoot so long as neither of them aims at innocent bystanders. Though they are both subject to moral constraints, the guard has rights that the robber obviously doesn't have. So, by analogy, shouldn't we say that the just warrior, defending his country, has rights that the unjust warrior, invading the country, doesn't have?\n\nThis is the central challenge (though there are others, related to this one) posed by many contemporary moral philosophers to the standard or orthodox theory of just war\u2014the theory that I have tried to develop and defend. According to the standard theory, aggressive war is indeed a crime, but it isn't the crime of the ordinary soldiers who fight it. The criminals are the men and women, mostly men, the political and military leaders, who consult together and decide, let's say, to attack a neighboring country. The Nuremberg tribunal got it right, then, when it indicted the heads of the Nazi party, state, and army, and allowed ordinary German soldiers to go home. But how can these soldiers be guiltless, who marched into Poland, Russia, Belgium, France, and so many other countries? To answer that question (which I have now put in the strongest possible way), we must focus on what actually happens in the world of war. For this is a world where life is radically unlike life in Philadelphia or in any peaceful civil society, even one beset by armed and possibly violent bank robbers.\n\nWhat is special, what is peculiar, about war? I have a short list of features that moral and political theorists, philosophers too, should attend to\u2014a list that could be expanded. It is designed only to serve the immediate purposes of my argument here.\n\nFirst, the circumstances of war are intensely coercive, and they are coercive in ways that are probably not equaled anywhere else. Slavery and imprisonment are highly coercive institutions, and conscription for military service is sometimes compared to them (by its opponents). The comparison can be useful in political debates, though it is greatly exaggerated. But it is hard to exaggerate the coerciveness of the battlefield, where life is always at risk and soldiers are compelled to act in ways that have no precedent in their own, or in any, civilian experience. Command decisions are also subject to the coerciveness of war\u2014hence the claim of \"military necessity.\" Moral theorists who try to set limits to that claim, as I tried to do in Just and Unjust Wars, must acknowledge its possible force. In the heat of battle, officers are driven to do cruel things, which they would never imagine doing in domestic society, by the belief that they have no alternative or, rather, that the only alternative is defeat\u2014which they take to mean subjection and possibly death, not so much for themselves as for their country and their fellow citizens. If we want, sometimes, to challenge that belief, we must understand the circumstances that produce it.\n\nSome of the legal and moral institutions of war acknowledge and even legitimize its coerciveness. Consider the prisoner of war convention (I will repeat here an argument I make in chapter 3), which makes the practice of individual or group surrender on the battlefield possible. Surrender is an implicit agreement: the surrendering soldier, threatened with death, agrees to give up fighting, and his captors agree, whether they think his war just or unjust, to provide him with what lawyers call \"benevolent quarantine for the duration of the war.\" But this is an agreement that the captive has made under extreme duress, and according to the ordinary law and morality of domestic life, it cannot have any binding force. And yet we recognize and accept its binding force, so that soldiers who try to escape from a prisoner of war camp are treated as if they have broken their word and thereby acted contrary to the law and morality of war; they are subject to punishment, even to capital punishment. I suppose one could argue that in the \"original position,\" as it is described by John Rawls, all potential soldiers would agree to the prisoner of war convention and that this hypothetical consent gives it binding force. But then we would have to explain why many prisoners try to escape and rejoin the fighting and why we celebrate their efforts\u2014why they are commonly described as heroes in books and films when they might plausibly be described as men who have broken their (hypothetical or tacit) promise and put a useful and humane convention at risk.\n\nThe coerciveness of war explains the prisoner of war convention, for there is no other way to allow soldiers to stop fighting without being killed. But there is a second feature of war that explains why some soldiers violate the convention and why we call them heroes.\n\nSecond, war is an intensely collective and collectivizing experience. When political or moral theorists talk about war, and especially about just war, they commonly begin with the right of individual self-defense. But this is only a bare beginning, and as an introduction to the understanding of warfare, it is somewhat misleading. Wars are fought by individuals, indeed, but not by individuals who are principally engaged in defending themselves. They are members of a collective, to which they attach value, often great value, and they are engaged in a project that is not merely their own. Most often the collective is a state, but it can also be a militant organization that functions like a state: it recruits fighters for a cause, trains and organizes them, and sends them into battle. The cause of many militant organizations is to establish a state; the cause for which states organize armies is to defend their own existence and the common life and individual lives of their citizens.\n\nIt is a fact of our moral history that many individuals are willing to risk their lives for these causes. This is not easy to understand. If states exist primarily to defend life, then how can life be sacrificed to defend the state? That is the question posed most clearly in the political theory of Thomas Hobbes, to which he offered no satisfactory answer. And in fact the defense of the state (and the pursuit of the cause of any militant organization) not only requires the defenders to put their own lives at risk but also requires them to put the lives of many other people at risk\u2014the fighters they oppose, most obviously, but also the civilians they and their opponents claim to be defending. Some philosophers, working upward, so to speak, from individual self-defense, doubt that any of this can be justified. The value of my life, or yours, may justify violent self-defense, but it is hard for these philosophers to understand how any collective could have that kind of value.\n\nThey have not, however, succeeded in convincing the rest of us. Patriotism and loyalty are, no doubt, often misguided, but they shouldn't be incomprehensible. Collectives like the state (or the army of the state) are indeed instrumental; they have no intrinsic value. But they make possible, and then they defend, another collective, a community whose existence is of centrally important value to (most of) its members. This value has many sources: history and memory, traditions of belief and practice, a culture of political engagement, the continuity of families, the sense of place, the immediacy of a way of life. When these seem to be in danger, many of us will risk our lives in their defense. Even in ordinary times, we are, all of us, collectivists of some sort\u2014on behalf of our families, or religious communities, or nations, or nation-states. But this is a fairly weak collectivism, which only sometimes wins out over individual self-interest. The acceptance of martyrdom in a time of religious persecution is the chief example of this kind of victory. Persecution creates conditions something like those experienced in a nation under attack, when the sense of danger intensifies our collectivist sensibilities, our patriotism and loyalty. And then the defense of the common life\u2014and of the necessary agencies of that defense, like states and armies\u2014regularly trumps the defense of the self. That is why captured soldiers will sometimes try to escape from the prisoner of war camp: they want to resume the fighting (and they are willing to accept the risks of doing that) because they think that the victory of their nation or the success of their cause is critically important.\n\nThe intensified collectivism of war also intensifies the coerciveness of war. In some wars, soldiers fight because they have been impressed or conscripted. But when the common life is in danger or when people think it is in danger, citizens will rush to enlist in the army. We call them volunteers, but they are probably acting under very strong social pressure and also under the pressure of their own consciences\u2014where conscience means what the word suggests: the knowledge they share with God, in the original religious formulation, or the knowledge they share with other members of their community, in secular understandings. In wartime, and commonly in both the religious and secular versions, what young men and women know is that they ought to volunteer (conscientious objection is exceptional even when it is permitted). This is collectivized knowledge, and it provides a powerful push toward what is only in a highly qualified sense a \"voluntary\" enlistment.\n\nOnce you are in the army, you are a member of a very strong collective of combatants, and all the people you leave behind are members of another strong collective, the civilian population. These memberships are matters of life and death\u2014hence they are \"strong\" in ways that no other memberships, no peacetime memberships, can ever be. I may be a committed professional, a lawyer, doctor, or teacher; I may be a devout member of a religious community, or a political party activist, or a class-conscious worker, but none of these affiliations, though they are undoubtedly very important, is collectivized in as radical a way as combatants and noncombatants are, where being a member determines whether you can, or cannot, be targeted and killed. Perhaps there are soldiers who, given the morality of everyday life, don't deserve to be targeted (they are against the war; they shoot their guns in the air), perhaps there are civilians who do deserve to be targeted (they are fierce and uncompromising hawks). In the circumstances of war, we cannot make these distinctions.\n\nThe moral equality of soldiers finds its parallel in the moral equality of civilians. Individuals are incorporated into both these collectives without regard to their personal moral standing. By contrast, in peacetime civil society, life and death decisions are made, in hospitals and courtrooms, for example, with careful attention to individual cases. Soldiers may receive that kind of attention after the war, if they are tried for war crimes. But when the two armies are engaged, and civilians are radically at risk, individual attentiveness isn't possible. We fight with soldiers; we don't fight with civilians.\n\nThe collectivism of war extends, so to speak, all the way down. It would be wrong to think of a battle as a series of encounters between individual soldiers. War is often chaotic, and soldiers are sometimes cut off from their units, forced to fight on their own, thinking at that moment only of their own survival. But battles commonly are collective engagements, shaped by a strategic plan and then by the tactical decisions of field commanders, which are enforced by rigorous military discipline. Soldiers fight together, helping each other, hoping to survive but aiming at a local victory: a target destroyed, a hill captured, a road opened, an enemy battalion outflanked or surrounded. The local victory has value only as a part of some larger scheme, but if that value is real to the soldiers involved, it justifies the risks they take, and it may justify (as self-defense would not) the risks they impose on nearby civilians. Soldiers make individual decisions about the risks they take and the risks they impose, but they make these decisions in the context of the collectivism (and the coerciveness) of war.\n\nNo one reading the literature of war can miss the sense of its strangeness\u2014and its awfulness. To be forced to risk your life, again and again, for collective purposes: this is not what anyone wants for himself or for the people he cares about. He may think that it is important to fight (it sometimes is); he may even think that the cause is worth dying for (it sometimes is), but he would rather be doing something else. The sense that many soldiers have of being radically committed and of wishing so strongly that they weren't\u2014that too probably finds no easy equivalent in ordinary life. And this internally contradictory but emotionally powerful sense is common to soldiers on both sides of the battle. Indeed, soldiers on both sides recognize, if only intermittently, the feelings they share; they see themselves in the others.\n\nPerhaps one can construct hypothetical cases that reproduce this feature, or other features, of wartime experience, though I doubt that cases taken from civil society (the bank robber and the bank guard, for example) or any of the hypothetical and constructed cases (variations on the famous trolley car story, say) actually get close enough. And I worry that theorists who focus on these kinds of cases aren't thinking about war at all. They are not interested, or not sufficiently interested, in what actually happens on the battlefield and what it feels like to be there.\n\nThird, war is a world of radical and pervasive uncertainty. The outcome of skirmishes and battles is hard to predict; the life chances of any particular soldier change from minute to minute; the knowledge available to officers making decisions is painfully limited. These physical and factual uncertainties probably can't be matched in ordinary civilian life, and they exist alongside moral uncertainties that almost certainly can't be matched. Most often in civilian life we have a fairly good idea about what ought to be done or about who can tell us what ought to be done. Moral practice has a certain habitual quality, and moral and legal authority is routinized. But in the anarchic society of states, which is also the world of war, both morality and authority are radically contested.\n\nThis doesn't mean that individual soldiers and groups of soldiers won't be convinced of the justice of their cause; they will be convinced, or most of them will, with whatever lingering anxieties. The soldiers of country A will follow the lead of their parents and peers, of their teachers and preachers, and of their prime ministers and presidents (this is another example of the collectivizing tendency of warfare). The difficulty is that all these people, from parents to presidents, exist in two entirely separate sets. And the soldiers from country B, opponents of country A, will be equally convinced, and some of them similarly anxious; they will be following the same leads. And there is no one in the world, literally no one, to whom these two groups of soldiers can turn for impartial and authoritative guidance. Uncertainty exists, so to speak, at the highest level. In international society, there are virtually no cases where warring states, and the neutral states watching them, agree on a single moral, political, or military description of the war; nor is there any routine way of appealing from this or that description to some ultimate judge. Wars end, one way or another, but disagreement doesn't. The moral contests outlast the battles\u2014as we can see if we compare the history books produced for state schools in the victorious and in the defeated country.\n\nBut what if a particular soldier knows that the war his country is fighting is unjust? Or what if the rest of us know that and think he should know it too? How can he fight in such a war? How can anyone fighting unjustly claim a right to kill his opponents, who are fighting justly? These are rhetorical questions; they are commonly asked by philosophers who insist that jus ad bellum determines jus in bello\u2014or, at least, that the two can't be independent of each other\u2014and who further insist that they know which of the warring states has ad bellum justice on its side. Reflecting on the certainty that, it seems to me, underlies these questions, I am reminded of Oliver Cromwell's response to similar certainties among Puritan ministers during the English Revolution: \"Think ye in the bowels of Christ, that ye may be wrong!\" That's not where I think, but I take the point. Given the circumstances of war, soldiers have a right to be wrong, and they have a right to think that they may be wrong and to defer to the decisions of (let's say) their democratically elected leaders. They also have a right to think that though they should oppose the war as citizens, they are bound to fight it as soldiers\u2014because there are many states that would not survive for long in the world of war, in international society as it exists today, without a disciplined army. And, finally, soldiers have a right to refuse to fight in a war they believe to be unjust; in cases like the Nazi war effort with which I began this argument, refusal is certainly the best response. But it is an act of heroism, and it can't be morally required; unheroic conduct isn't criminal conduct.\n\nBut mostly soldiers sincerely believe that their war is just, for reasons that seem sufficient to them, and this belief gives a certain shape to their battles, which are fought between soldiers certain of their cause in a world where all causes are uncertain. I am not making a relativist argument here; I have argued over many years that (most) wars, on one side or the other, are objectively just or unjust. But this objectivity has no political or judicial embodiment. There is no agent of objectivity. And that is one reason for the deep principle of (standard, orthodox) just war theory: that soldiers have an equal right to fight, whether their cause is, or isn't, objectively just. Soldiers on both sides have exactly the same rights and obligations. If they are captured, they must be treated similarly and equally, that is, in accordance with the (morally strange) prisoner of war convention. And after the war, they should be encouraged and helped in exactly the same way to go home and resume their civilian existence.\n\nIn fact, all the special features of the world of war conspire to produce this principle of warrior equality, which obviously has no domestic equivalent (again: bank robbers and bank guards are not equals, even if the bank turns into a battlefield). The different forms of wartime coerciveness\u2014social pressure, conscription, army discipline, military necessity\u2014impact the soldiers on both sides in roughly the same way. The heightened sense of collective belonging and commitment is felt in similar ways by all of them. And they all live with the same uncertainties.\n\nIf we want to constrain the conduct of soldiers on the battlefield, we must recognize these similarities. We must insist that all soldiers, whether they think their cause is just or unjust, and whether we think their cause is just or unjust, have the same rights and, what is even more important, the same obligations. No group of soldiers, claiming to be just warriors, can arrogate to themselves rights they deny to others or claim exemption from everyone else's obligations\u2014for if that is allowed and justified, there will soon be no constraints at all.\n\nThe moral equality of soldiers is perhaps the strangest rule of war. But philosophers who deny its morality seem to me to miss the force of that preposition: \"rule of war.\" To understand the rule, you have to take an interest not only in moral theory, which accounts for the strangeness of the rule, but in war itself, which accounts for the existence of the rule.\nNotes\n\nPreface to the First Edition\n\n. The most concise and forceful exposition of these reasons is Stanley Hoffmann, \"International Law and the Control of Force,\" in The Relevance of International Law, ed. Karl Deutsch and Stanley Hoffmann (New York, 1971), pp. 34\u201366. Given the present state of the law, I have most often cited positivists of an earlier age, especially W. E. Hall, John Westlake, and J. M. Spaight.\n\n. The pioneering work of this sort is Myres S. McDougal and Florentino P. Feliciano, Law and Minimum World Public Order (New Haven, 1961).\n\n. For a useful study of these writers, see James Turner Johnson, Ideology, Reason, and the Limitation of War: Religious and Secular Concepts, 1200\u20131740 (Princeton, 1975).\n\n1 Against \"Realism\"\n\n. This and subsequent quotations are from Hobbes' Thucydides, ed. Richard Schlatter (New Brunswick, N.J., 1975), pp. 377\u201385 (The History of the Peloponnesian War, 5: 84\u2013116).\n\n. Dionysius of Halicarnassus, On Thucydides, trans. W. Kendrick Pritchett (Berkeley, 1975), pp. 31\u201333.\n\n. See F. M. Cornford, Thucydides Mythistoricus (London, 1907), esp. ch. XIII.\n\n. The Trojan Women, trans. Gilbert Murray (London, 1905), p. 16.\n\n. Werner Jaeger, Paideia: The Ideals of Greek Culture, trans. Gilbert Highet (New York, 1939), I, 402.\n\n. H. W. Fowler, A Dictionary of Modern English Usage, second ed., rev. Sir Ernest Gowers (New York, 1965), p. 168; cf. Jaeger, I, 397.\n\n. Plutarch's Lives, trans. John Dryden, rev. Arthur Hugh Clough (London, 1910), I, 303. Alcibiades also \"selected for himself one of the captive Melian women. . . . \"\n\n. Hobbes' Thucydides, pp. 194\u2013204 (The History of the Peloponnesian War, 3:36\u201349).\n\n. Thomas Hobbes, Leviathan, ch. IV.\n\n. The Charterhouse of Parma, I, chs. 3 and 4; J. F. C. Fuller, A Military History of the Western World (n.p., 1955), II, ch. 15.\n\n. C. W. C. Oman, The Art of War in the Middle Ages (Ithaca, N.Y., 1968), p. 137.\n\n. Raphael Holinshed, Chronicles of England, Scotland, and Ireland, excerpted in William Shakespeare, The Life of Henry V (Signet Classics, New York, 1965), p. 197.\n\n. Henry V, 4:7, ll. 1\u201311.\n\n. David Hume, The History of England (Boston, 1854), II, 358.\n\n. Ren\u00e9 de Belleval, Azincourt (Paris, 1865), pp. 105\u201306.\n\n. See the summary of opinions in J. H. Wylie, The Reign of Henry the Fifth (Cambridge, England, 1919), II, 171ff.\n\n. For an excellent and detailed account, which suggests that Henry's action cannot be defended, see John Keegan, The Face of Battle (New York, 1976), pp. 107\u201312.\n\n2 The Crime of War\n\n. Clausewitz should now be read in the new translation by Michael Howard and Peter Paret, On War (Princeton, 1976). But this book appeared after my own work was finished; I have quoted Clausewitz from a graceful, though abridged version by Edward M. Collins, War, Politics, and Power (Chicago, 1962), p. 65. Cf. Howard and Paret, p. 76.\n\n. Press conference, January 12, 1955.\n\n. Clausewitz, p. 64. Cf. Howard and Paret, pp. 75\u201376.\n\n. Clausewitz, pp. 72, 204. Cf. Howard and Paret, pp. 81, 581.\n\n. John Ruskin, The Crown of Wild Olive: Four Lectures on Industry and War (New York, 1874), pp. 90\u201391.\n\n. Wilfred Owen, \"Anthem for Doomed Youth,\" in Collected Poems, ed. C. Day Lewis (New York, 1965), p. 44.\n\n. Thomas Hobbes, Leviathan, ch. XXI. For a description of primitive warfare of this sort, see Robert Gardner and Karl G. Heider, Gardens of War: Life and Death in the New Guinea Stone Age (New York, 1968), ch. 6.\n\n. Quoted in J. F. C. Fuller, The Conduct of War, 1789\u20131961 (n.p., 1968), p. 16.\n\n. Machiavelli, History of Florence (New York, 1960), Bk. IV, ch. I, p. 164.\n\n. Ruskin, p. 92.\n\n. War and Peace, trans. Constance Garnett (New York, n.d.), Part Two, III, p. 111.\n\n. \"A Terre,\" Collected Poems, p. 64.\n\n. Fuller, Conduct of War, p. 35.\n\n. Thomas Sackville, Earl of Dorset, \"The Induction,\" Works, ed. R. W. Sackville-\u00adWest (London, 1859), p. 115.\n\n. This and the following quotations are from William Tecumseh Sherman, Memoirs (New York, 1875), pp. 119\u201320.\n\n3 The Rules of War\n\n. Louis Simpson, \"The Ash and the Oak,\" Good News of Death and Other Poems, in Poets of Today II (New York, 1955), p. 162.\n\n. See for example, Fuller, Conduct of War, ch. II (\"The Rebirth of Total War\").\n\n. Edward Rickenbacker's Fighting the Flying Circus (New York, 1919) is a lively (and typical) account of the chivalry of the air. In 1918, Rickenbacker wrote in his flight diary: \"Resolved today that . . . I will never shoot at a Hun who is at a disadvantage . . .\" (p. 338). For a general account, see Frederick Oughton, The Aces (New York, 1960).\n\n. Quoted in Desmond Young, Rommel: The Desert Fox (New York, 1958), p. 137.\n\n. Eisenhower, Crusade in Europe (New York, 1948), pp. 156\u201357.\n\n. Ronald Lewin, Rommel as Military Commander (New York, 1970), pp. 294, 311. See also Young, pp. 130\u201332.\n\n. Quoted in Robert W. Tucker, The Law of War and Neutrality at Sea (Washington, D.C. 1957), p. 6n. Tucker's discussion of the legal issues is very useful; see also H. Lauterpacht, \"The Limits of the Operation of the Law of War,\" in 30 British Yearbook of International Law (1953).\n\n. Henry V, 4:1, ll. 132\u201335.\n\n. Francisco de Vitoria, De Indis et De Iure Belli Relationes, ed. Ernest Nys (Washington, D.C., 1917): On the Law of War, trans. John Pawley Bate, p. 176.\n\n. Randall Jarrell, \"The Death of the Ball Turret Gunner,\" in The Complete Poems (New York, 1969), p. 144.\n\n. See below, ch. 18. For a historical account of these issues, see C. A. Pompe, Aggressive War: An International Crime (The Hague, 1953).\n\n. Quincy Wright, A Study of War (Chicago, 1942), I, 8.\n\n. Gardner and Heider, Gardens of War, p. 139.\n\n. First Samuel, 17:32.\n\n. Johan Huizinga, Homo Ludens (Boston, 1955), p. 92.\n\n. War and Peace, Part Ten, XXV, p. 725.\n\n. For a discussion of this agreement, see my essay \"Prisoners of War: Does the Fight Continue After the Battle?\" in Obligations: Essays on Disobedience, War and Citizenship (Cambridge, Mass., 1970).\n\n. Moltke in Seinen Briefen (Berlin, 1902), p. 253. The letter is addressed to J. C. Bluntschli, a noted scholar of international law.\n\n4 Law and Order in International Society\n\n. Henry V, 1:2, ll. 24\u201328.\n\n. The judges distinguished \"aggressive acts\" from \"aggressive wars,\" but then used the first of these as the generic term: see Nazi Conspiracy and Aggression: Opinion and Judgment (Washington, D.C., 1947), p. 16.\n\n. Quoted in Michael Howard, \"War as an Instrument of Policy,\" in Herbert Butterfield and Martin Wight, eds., Diplomatic Investigations (Cambridge, Mass., 1966), p. 199. Cf. On War, trans. Howard and Paret, p. 370.\n\n. John Westlake, Collected Papers, ed. L. Oppenheim (Cambridge, England, 1914), p. 78.\n\n. See Ruth Putnam, Alsace and Lorraine from Caesar to Kaiser: 58 B.C.\u20131871 A.D. (New York, 1915).\n\n. Henry Sidgwick, The Elements of Politics (London, 1891), pp. 268, 287.\n\n. Leviathan, ch. 30.\n\n. Leviathan, ch. 15.\n\n. For a critique of this analogy, see the two essays by Hedley Bull, \"Society and Anarchy in International Relations,\" and \"The Grotian Conception of International Society,\" in Diplomatic Investigations, chs. 2 and 3.\n\n. See Vitoria, On the Law of War, p. 177.\n\n. Lenin, Socialism and War (London, 1940), pp. 10\u201311.\n\n. Edmund Wilson, Patriotic Gore (New York, 1966), p. xi.\n\n. It is worth noting that the United Nations' recently adopted definition of aggression closely follows the paradigm: see the Report of the Special Committee on the Question of Defining Aggression (1974), General Assembly Official Records, 29th session, supplement no. 19 (A\/9619), pp. 10\u201313. The definition is reprinted and analyzed in Yehuda Melzer, Concepts of Just War (Leyden, 1975), pp. 26ff.\n\n. On the Law of War, p. 170.\n\n. See L. Oppenheim, International Law, vol. II, War and Neutrality (London, 1906), pp. 55ff.\n\n. C. A. Pompe, Aggressive War, p. 152.\n\n. Quoted in Pompe, p. 152.\n\n. Quoted in Franz Mehring, Karl Marx, trans. Edward Fitzgerald (Ann Arbor, 1962), p. 438.\n\n. Minutes of the General Council of the First International: 1870\u20131871 (Moscow, n.d.), p. 57.\n\n. Roger Morgan, The German Social-Democrats and the First International: 1864\u20131872 (Cambridge, England, 1965), p. 206.\n\n. \"First Address of the General Council of the International Working Men's Association on the Franco-Prussian War,\" in Marx and Engels, Selected Works (Moscow, 1951), I, 443.\n\n. \"Second Address . . . ,\" Selected Works, I, 449 (Marx's italics).\n\n. Selected Works, I, 441.\n\n. See the arguments made by Churchill at the time: The Gathering Storm (New York, 1961), chs. 17 and 18; also Martin Gilbert and Richard Gott, The Appeasers (London, 1963). For a recent scholarly reappraisal somewhat more sympathetic to Chamberlain, see Keith Robbins, Munich: 1938 (London, 1968).\n\n. Gerald Vann, Morality and War (London, 1939).\n\n. Max Jakobson, The Diplomacy of the Winter War (Cambridge, Mass., 1961), p. 117.\n\n. Jacobson reports an admission by the Swedish prime minister that had Sweden been publicly committed to assist Finland in the autumn of 1939, the Soviet Union would probably not have attacked (p. 237).\n\n5 Anticipations\n\n. D. W. Bowett, Self-Defense in International Law (New York, 1958), p. 59. My own position has been influenced by Julius Stone's critique of the legalist argument: Aggression and World Order (Berkeley, 1968).\n\n. Quoted from the Annual Register, in H. Butterfield, \"The Balance of Power,\" Diplomatic Investigations, pp. 144\u201345.\n\n. Francis Bacon, Essays (\"Of Empire\"); see also his treatise Considerations Touching a War with Spain (1624), in The Works of Francis Bacon, ed. James Spedding et al. (London, 1874), XIV, pp. 469\u2013505.\n\n. Oxford English Dictionary, \"threaten.\"\n\n. M. D. Vattel, The Law of Nations (Northampton, Mass., 1805), Bk. III, ch. III, paras. 42\u201344, pp. 357\u201378. Cf. John Westlake, Chapters on the Principles of International Law (Cambridge, England, 1894), p. 120.\n\n. Jonathan Swift, The Conduct of the Allies and of the Late Ministry in Beginning and Carrying on the Present War (1711), in Prose Works, ed. Temple Scott (London, 1901), V, 116.\n\n. As late as the eighteenth century, Vattel still argued that a prince \"has a right to demand, even by force of arms, the reparation of an insult.\" Law of Nations, Bk. II, ch. IV, para. 48, p. 216.\n\n. Compare the argument of Hugo Grotius: \"The danger . . . must be immediate and imminent in point of time. I admit, to be sure, that if the assailant seizes weapons in such a way that his intent to kill is manifest, the crime can be forestalled; for in morals as in material things a point is not to be found which does not have a certain breadth.\" The Law of War and Peace, trans. Francis W. Kelsey (Indianapolis, n.d.), Bk. II, ch. I, section V, p. 173.\n\n. Walter Laquer, The Road to War: The Origin and Aftermath of the Arab-\u00adIsraeli Conflict, 1967\u20138 (Baltimore, 1969), p. 110.\n\n. Edward Luttwak and Dan Horowitz, The Israeli Army (New York, 1975), p. 212.\n\n. Luttwak and Horowitz, p. 224.\n\n6 Interventions\n\n. \"A Few Words on Non-Intervention\" in J. S. Mill, Dissertations and Discussions (New York, 1873), III, 238\u201363.\n\n. See Irving Howe, ed., The Basic Writings of Trotsky (New York, 1963), p. 397.\n\n. John Norton Moore, \"International Law and the United States' Role in Vietnam: A Reply,\" in R. Falk, ed., The Vietnam War and International Law (Princeton, 1968), p. 431. Moore addresses himself specifically to the argument of W. E. Hall, International Law (5th ed., Oxford, 1994), pp. 289\u201390, but Hall follows Mill closely.\n\n. For a brief survey, see Jean Sigmann, 1848: The Romantic and Democratic Revolutions in Europe, trans. L. F. Edwards (New York, 1973), ch. 10.\n\n. Charles Sproxton, Palmerston and the Hungarian Revolution (Cambridge, 1919), p. 48.\n\n. \"Non-Intervention,\" pp. 261\u201362.\n\n. See S. French and A. Gutman, \"The Principle of National Self-\u00addetermination,\" in Held, Morgenbesser, and Nagel, eds., Philosophy, Morality, and International Affairs (New York, 1974), pp. 138\u201353.\n\n. This is the general position of R. J. Vincent, Nonintervention and World Order (Princeton, 1974), esp. ch. 9.\n\n. Sproxton, p. 109.\n\n. See, for example, Hall, International Law, p. 293.\n\n. \"On the Principle of Non-Intervention\" (Oxford, 1860), p. 21.\n\n. See Hugh Thomas, The Spanish Civil War (New York, 1961), chs. 31, 40, 48, 58; Norman J. Padelford, International Law and Diplomacy in the Spanish Civil Strife (New York, 1939) is an incredibly na\u00efve defense of the nonintervention agreements.\n\n. A useful statement of this position can be found in the essay by John Norton Moore already cited; see note 3 above. For an example of the official view, see Leonard Meeker, \"Vietnam and the International Law of Self-Defense\" in the same volume, pp. 318\u201332.\n\n. I shall follow the account of G. M. Kahin and John W. Lewis, The United States in Vietnam (New York, 1967).\n\n. \"On the Principle of Non-Intervention,\" p. 16.\n\n. See Gregory Henderson, Korea: The Politics of the Vortex (Cambridge, Mass., 1968), ch. 6.\n\n. Kahin and Lewis, p. 146.\n\n. Ellery C. Stowell suggests some possible examples in Intervention in International Law (Washington, D.C., 1921), ch. II. For contemporary legal views (and newer examples), see Richard Lillich, ed., Humanitarian Intervention and the United Nations (Charlottesville, Va., 1973).\n\n. Quoted in Philip S. Foner, The Spanish-Cuban-American War and the Birth of American Imperialism (New York, 1972), I, 111.\n\n. Quoted in Stowell, p. 122n.\n\n. See, for example, Julius W. Pratt, Expansionists of 1898 (Baltimore, 1936) and Walter La Feber, The New Empire: An Interpretation of American Expansion (Ithaca, 1963); also Foner, I, ch. XIV.\n\n. Foner, I, ch. XIII.\n\n. F. E. Chadwick, The Relations of the United States and Spain: Diplomacy (New York, 1909), pp. 586\u201387. These lines are the epigraph to Walter Millis' account of the war: The Martial Spirit (n.p., 1931).\n\n. Millis, p. 404; it should be noted that Millis also writes of the American decision to go to war: \"Seldom can history have recorded a plainer case of military aggression . . .\" (p. 160).\n\n. For a contemporary account by a British journalist, see David Loshak, Pakistan Crisis (London, 1971).\n\n. John Westlake, International Law, vol. I, Peace (2nd ed., Cambridge 1910), pp. 319\u201320.\n\n. Thomas M. Franck and Nigel S. Rodley, \"After Bangladesh: The Law of Humanitarian Intervention by Military Force,\" 67 American Journal of International Law 304 (1973).\n\n. Julius Stone, Aggression and World Order, pp. 99.\n\n7 War's Ends, and the Importance of Winning\n\n. \"The Range in the Desert,\" The Complete Poems, p. 176.\n\n. B. H. Liddell Hart, Strategy (2nd rev. ed., New York, 1974), p. 339: Liddell Hart himself holds a different, and a much more sophisticated, position.\n\n. War, Politics and Power, p. 233; cf. the new translation of Howard and Paret, p. 595.\n\n. The work of Reinhold Niebuhr was the major inspiration of this group, Hans Morganthau its most systematic theorist. For works more immediately relevant to my purposes in this chapter, see George Kennan, American Diplomacy: 1900\u20131950 (Chicago, 1951); John W. Spanier, The Truman-MacArthur Controversy and the Korean War (Cambridge, Mass., 1959); Paul Kecskemeti, Strategic Surrender: The Politics of Victory and Defeat (New York, 1964). For a useful critique of the \"realists,\" see Charles Frankel, Morality and U.S. Foreign Policy, Foreign Policy Association Headline Series, no. 224 (1975).\n\n. Spanier, p. 5.\n\n. Kecskemeti, pp. 25\u201326.\n\n. On the connection between Wilson's \"world view\" and his desire for a compromise peace, see N. Gordon Levin, Jr., Woodrow Wilson and World Politics: America's Response to War and Revolution (New York, 1970), pp. 43, 52ff.\n\n. The Hinge of Fate (New York, 1962), p. 600.\n\n. Kecskemeti, pp. 217, 241.\n\n. Hinge of Fate, p. 600; see also Churchill's cabinet memorandum of January 14, 1944, p. 599.\n\n. American Diplomacy, pp. 87\u201388.\n\n. Robert Phillimore, Commentaries upon International Law (Philadelphia, 1854), I, 315.\n\n. Kecskemeti, p. 219.\n\n. See Raymond G. O'Connor, Diplomacy for Victory: FDR and Unconditional Surrender (New York, 1971).\n\n. Kecskemeti, p. 240.\n\n. For a general view of punishment as public condemnation, see \"The Expressive Function of Punishment,\" in Joel Feinberg, Doing and Deserving (Princeton, 1970), ch. 5.\n\n. Glen D. Paige, The Korean Decision (New York, 1968), pp. 218\u201319.\n\n. Strategy, p. 355.\n\n. Quoted in Spanier, p. 88.\n\n. Quoted in David Rees, Korea: The Limited War (Baltimore, 1970), p. 101.\n\n. Concepts of Just War, pp. 170\u201371.\n\n. Liddell Hart, Strategy, p. 338.\n\n. Hume, Theory of Politics, ed. Frederick Watkins (Edinburgh, 1951), pp. 190\u201391.\n\n8 War's Means and the Importance of Fighting Well\n\n. Elements of Politics, pp. 253\u201354.\n\n. Elements of Politics, p. 254; for a contemporary statement from a roughly similar point of view, see R. B. Brandt, \"Utilitarianism and the Rules of War,\" 1 Philosophy and Public Affairs 145\u201365 (1972).\n\n. Byron Farwell, The Great Anglo-Boer War (New York, 1976), p. 209.\n\n. The Law of Land Welfare, U.S. Department of the Army Field Manual FM 27\u201310 (1956), para. 3. See the discussion of this provision in Telford Taylor, Nuremberg and Vietnam (Chicago, 1970), pp. 34\u201336, and Marshall Cohen, \"Morality and the Laws of War,\" Philosophy, Morality, and International Affairs, pp. 72ff.\n\n. Elements of Politics, p. 264.\n\n. For an example of the \"morality\" of the feud, see Margaret Hasluck, \"The Albanian Blood Feud,\" in Paul Bohannan, Law and Warfare: Studies in the Anthropology of Conflict (New York, 1967), pp. 381\u2013408.\n\n. The story is told in Ignazio Silone, \"Reflections on the Welfare State,\" 8 Dissent 189 (1961); De Sica's film Two Women is based on an incident from this period in Italian history.\n\n. On the Law of War, pp. 184\u201385.\n\n. Deuteronomy 21:10\u201314. This passage is ignored in Susan Brownmiller's analysis of the \"true Hebraic concept . . . of rape\" in Against Our Will: Men, Women, and Rape (New York, 1975), pp. 19\u201323.\n\n. See, for example, McDougal and Feliciano, Law and Minimum World Public Order, p. 42 and passim.\n\n9 Noncombatant Immunity and Military Necessity\n\n. S. L. A. Marshall, Men Against Fire (New York, 1966), chs. 5 and 6.\n\n. Wilfred Owen, Collected Letters, ed. Harold Owen and John Bell (London, 1967), p. 458 (14 May 1917).\n\n. Good-bye to All That (rev. ed., New York, 1957), p. 132.\n\n. The Collected Essays, Journalism and Letters of George Orwell, ed. Sonia Orwell and Ian Angus (New York, 1968), II, 254.\n\n. The Fortress: A Diary of Anzio and After (Hammondsworth, 1958), p. 21.\n\n. Sardinian Brigade: A Memoir of World War I, trans. Marion Rawson (New York, 1970), pp. 166\u201371.\n\n. Archibald Forbes, quoted in J. M. Spaight, War Rights on Land (London, 1911), p. 104.\n\n. Instructions for the Government of Armies of the United States in the Field, General Orders 100, April, 1863 (Washington, 1898), Article 69.\n\n. M. Greenspan, The Modern Law of Land Warfare (Berkeley, 1959), pp. 313\u201314.\n\n. G. E. M. Anscombe, Mr. Truman's Degree (privately printed, 1958), p. 7; see also \"War and Murder\" in Nuclear Weapons and Christian Conscience, ed. Walter Stein (London, 1963).\n\n. See Sir Frederick Smith, The Destruction of Merchant Ships under International Law (London, 1917) and Tucker, Law of War and Neutrality at Sea.\n\n. H. A. Smith, Law and Custom of the Sea (London, 1950), p. 123.\n\n. Tucker, p. 72.\n\n. Tucker, p. 67.\n\n. Doenitz, Memoirs: Ten Years and Twenty Days, trans. K. H. Stevens (London, 1959), p. 261.\n\n. The Destruction of Convoy PQ 17 (New York, n.d.), p. 157; for other examples, see pp. 145, 192\u201393.\n\n. Nazi Conspiracy and Aggression: Opinion and Judgment, p. 140.\n\n. Doenitz, Memoirs, p. 259.\n\n. Old Soldiers Never Die (New York, 1966), p. 198.\n\n. Kenneth Dougherty, General Ethics: An Introduction to the Basic Principles of the Moral Life According to St. Thomas Aquinas (Peekskill, N.Y., 1959), p. 64.\n\n. Dougherty, pp. 65\u201366; cf. John C. Ford, S. J. \"The Morality of Obliteration Bombing,\" in War and Morality, ed. Richard Wasserstrom (Belmont, Calif., 1970). I cannot make any effort here to review the philosophical controversies over double effect. Dougherty provides a (very simple) text book description, Ford a careful (and courageous) application.\n\n. For a philosophical version of the argument that it cannot make a difference whether the killing of innocent people is direct or indirect, see Jonathan Bennett, \"Whatever the Consequences,\" Ethics, ed. Judith Jarvis Thomson and Gerald Dworkin (New York, 1968).\n\n. Reginald Thompson, Cry Korea (London, 1951), pp. 54, 142\u201343.\n\n. I have been helped in thinking about these questions by Charles Fried's discussion of \"Imposing Risks on Others,\" An Anatomy of Values: Problems of Personal and Social Choice (Cambridge, Mass., 1970), ch. XI.\n\n. Quoted from the published text of Marcel Ophuls' documentary film, The Sorrow and the Pity (New York, 1972), p. 131.\n\n. Thomas Gallagher, Assault in Norway (New York, 1975), pp. 19\u201320, 50.\n\n10 War Against Civilians: Sieges and Blockades\n\n. The Works of Josephus, trans. Tho. Lodge (London, 1620): The Wars of the Jews, Bk. VI, ch. XIV, p. 721.\n\n. See, for example, Elena Skrjabina's remarkable memoir, Siege and Survival: The Odyssey of a Leningrader (Carbonville, Ill., 1971).\n\n. Charles Chaney Hyde, International Law (2nd rev. ed., Boston, 1945), III, 1802.\n\n. The Works, p. 722.\n\n. M. H. Keen, The Laws of War in the Late Middle Ages (London, 1965), p. 128, for an account of aristocratic obligations in such cases.\n\n. The Art of War, trans. Ellis Fameworth, rev. with an intro. by Neal Wood (Indianapolis, 1965), p. 193.\n\n. Spaight's discussion is the best: War Rights, pp. 174ff.\n\n. The Works, p. 718.\n\n. I shall follow the account of Leon Goure, The Siege of Leningrad (Stanford, 1962).\n\n. Goure, p. 141; Trials of War Criminals before the Nuremberg Military Tribunals (Washington, D.C., 1950), XI, 563.\n\n. The citation is from Hyde, International Law, III, 1802\u201303.\n\n. Spaight, pp. 174ff.\n\n. Spaight, pp. 177\u201378.\n\n. Hall, International Law, p. 398.\n\n. The Code of Maimonides: Book Fourteen: The Book of Judges, trans. Abraham M. Hershman (New Haven, 1949), p. 222; Grotius, Law of War and Peace, Bk. III, ch. XI, section xiv, pp. 739\u201340.\n\n. See Skrjabina, Siege and Survival, \"Leningrad.\"\n\n. Deuteronomy 20:20.\n\n. Hobbes' Thucydides, pp. 123\u201324 (2:19\u201320); War Commentaries of Caesar, trans. Rex Warner (New York, 1960), pp. 70, 96 (Gallic Wars 3:3, 5:1).\n\n. A. C. Bell, A History of the Blockade of Germany (London, 1937), pp. 213\u201314.\n\n. Spaight, p. 138.\n\n. Hall, International Law, p. 656.\n\n. B. H. Liddell Hart, The Real War: 1914\u20131918 (Boston, 1964), p. 473.\n\n. The studies were carried out by German statisticians, but the results are accepted by Bell. He is a little reluctant, however, to regard these results as a sign of the \"success\" of the British blockade: see p. 673.\n\n. Bell, p. 117. Cf. the same argument made by a French historian, Louis Gui\u00adchard, The Naval Blockade: 1914\u20131918, trans. Christopher R. Turner (New York, 1930), p. 304.\n\n11 Guerrilla War\n\n. The Sorrow and the Pity, pp. 113\u201314.\n\n. For a useful survey of the legal situation, see Gerhard von Glahn, The Occupation of Enemy Territory (Minneapolis, 1957).\n\n. See, for example, W. F. Ford, \"Resistance Movements and International Law,\" 7\u20138 International Review of the Red Cross (1967\u201368) and G. I. A. D. Draper, \"The Status of Combatants and the Question of Guerrilla War,\" 45 British Yearbook of International Law (1971).\n\n. Quoted in Draper, p. 188.\n\n. Quoted in Douglas Pike, Viet Cong (Cambridge, Mass., 1968), p. 242.\n\n. Mao Tse-tung, Selected Military Writings (Peking, 1966), p. 343.\n\n. Dickey Chapelle, \"How Castro Won,\" in The Guerrilla\u2014And How to Fight Him: Selections from the Marine Corps Gazette, ed. T. N. Greene (New York, 1965), p. 223.\n\n. Draper, p. 203.\n\n. See Michael Calvert, Chindits: Long Range Penetration (New York, 1973).\n\n. Draper, pp. 202\u201304.\n\n. Guerrilla Parties Considered with Reference to the Laws and Usages of War (New York, 1862). Lieber wrote this pamphlet at the request of General Halleck.\n\n. Jeffrey Race, War Comes to Long An (Berkeley, 1972), pp. 196\u201397.\n\n. See The Guerrilla\u2014And How to Fight Him; John McCuen, The Art of Counter-\u00adRevolutionary War (London, 1966); Frank Kitson, Low Intensity Operations: Subversion, Insurgency, and Peacekeeping (Harrisburg, 1971).\n\n. Seven Pillars of Wisdom (New York, 1936), Bk. III, ch. 33, p. 196.\n\n. For a graphic description of soldiers going beyond these limits, see Victor Kolpacoff's novel of the Vietnam War, The Prisoners of Quai Dong (New York, 1967).\n\n. Race, p. 233.\n\n. Jonathan Schell, The Military Half (New York, 1968), pp. 14ff.\n\n. For an account of forcible deportation, see Jonathan Schell, The Village of Ben Suc (New York, 1967).\n\n. The Military Half, p. 151.\n\n. Orville and Jonathan Schell, letter to The New York Times, Nov. 26, 1969; quoted in Noam Chomsky, At War with Asia (New York, 1970), pp. 292\u201393.\n\n. See the description of the camps that the British set up for Boer farmers: Farwell, Anglo-Boer War, chs. 40, 41.\n\n. Sir Robert Thompson, Defeating Communist Insurgency (New York, 1966), p. 125.\n\n. Don Oberdorfer, Tet (New York, 1972), p. 202.\n\n. Schell, The Military Half, pp. 96, 159.\n\n. Kitson, p. 138.\n\n. Street Without Joy (New York, 1972), ch. 7.\n\n. See the account of Regis Debray, Che's Guerrilla War, trans. Rosemary Sheed (Hammondsworth, 1975).\n\n12 Terrorism\n\n. But Liddell Hart, the foremost strategist of the \"indirect approach,\" has consistently opposed terrorist tactics: see, for example, Strategy, pp. 349\u201350 (on terror bombing).\n\n. Rear Admiral L. H. K. Hamilton, quoted in Irving, Destruction of Convoy PQ 17, p. 44.\n\n. Politics, trans. Ernest Barker (Oxford, 1948), p. 288 (1314a).\n\n. The Just Assassins, in Caligula and Three Other Plays, trans. Stuart Gilbert (New York, 1958), p. 258. The actual historical incident is described in Roland Gaucher, The Terrorists: From Tsarist Russia to the OAS (London, 1965), pp. 49, 50n.\n\n. J. Bowyer Bell, The Secret Army: A History of the IRA (Cambridge, Mass., 1974), pp. 161\u201362.\n\n. Gerold Frank, The Deed (New York, 1963), pp. 248\u201349.\n\n. James E. Bond, The Rules of Riot: Internal Conflict and the Law of War (\u00adPrinceton, 1974), pp. 89\u201390.\n\n. Pike, Viet Cong, p. 248.\n\n. Race, War Comes to Long An, p. 83, which suggests that it was precisely the best public health officers, teachers, and so on who were attacked\u2014because they constituted a possible anti-communist leadership.\n\n. Pike, p. 250.\n\n. Pike, p. 251.\n\n. The argument, I suppose, goes back to Machiavelli, though most of his descriptions of the necessary violence of founders and reformers have to do with the killing of particular people, members of the old ruling class: see The Prince, ch. VIII and Discourses, I:9 for examples.\n\n. The Wretched of the Earth, trans. Constance Farrington (New York, n.d.), pp. 18\u201319.\n\n. Gillo Pontecorvo's The Battle of Algiers, ed. Piernico Solinas (New York, 1973), pp. 79\u201380.\n\n13 Reprisals\n\n. G. Lowes Dickinson, War: Its Nature, Cause, and Cure (London, 1923), p. 15.\n\n. H. Brocher, \"Les principes naturels du droit de la guerre,\" 5 Revue de droit international et de legislation compar\u00e9e 349 (1873).\n\n. Robert B. Asprey, War in the Shadows: The Guerrilla in History (New York, 1975), I, 478.\n\n. Frits Kalshoven, Belligerent Reprisals (Leyden, 1971), pp. 193\u2013200.\n\n. Kalshoven, pp. 78ff.\n\n. See, for example, the essays of H. J. McCloskey and T. L. S. Sprigge in Contemporary Utilitarianism, ed. Michael D. Bayles (Garden City, N.Y., 1968).\n\n. Spaight, War Rights, p. 120.\n\n. Spaight, p. 462.\n\n. McDougal and Feliciano, Law and Minimum World Public Order, p. 682.\n\n. See Robert Katz, Death in Rome (New York, 1967) for an account of one of the most brutal Nazi reprisals.\n\n. Spaight, p. 463n.\n\n. Greenspan is typical: \"Only in exceedingly grave cases should there be resort to reprisals.\" Modern Law of Land Warfare, p. 411.\n\n. Lieber, Instructions, Article 27 (emphasis added).\n\n. Kalshoven, pp. 263ff.\n\n. McDougal and Feliciano, p. 684.\n\n. Churchill, The Grand Alliance (New York, 1962), p. 359. A distinction similar to the one I am defending here is suggested by Westlake: \". . . the laws of war are too deeply rooted in humanity and morality to be discussed on the footing of contract alone, except it may be some parts of no great importance which convention might have settled otherwise than it has.\" International Law, II, 126.\n\n. See Kalshoven on \"non-belligerent reprisals,\" pp. 287ff.\n\n. Luttwak and Horowitz, The Israeli Army, p. 110.\n\n. For accounts and evaluations of the raid, see Richard Falk, \"The Beirut Raid and the International Law of Reprisal,\" 63 American Journal of International Law (1969) and Yehuda Blum, \"The Beirut Raid and the International Double Standard: A Reply to Professor Falk,\" 64 A.J.I.L. (1970).\n\n. See the general condemnation voted by the Security Council on April 9, 1964, cited in Sydney D. Bailey, Prohibitions and Restraints in War (London, 1972), p. 55.\n\n. Hans Kelson, Principles of International Law, 2nd ed., rev. Robert W. Tucker (New York, 1967), p. 87.\n\n14 Winning and Fighting Well\n\n. The Chinese Classics, trans. and ed. James Legge, vol. V: The Ch'un Ts'ew with The Tso Chuen (Oxford, 1893), p. 183.\n\n. Military Writings, p. 240.\n\n. Quoted in Arthur Waley, Three Ways of Thought in Ancient China (Garden City, New York, n.d.), p. 131.\n\n. Military Writings, pp. 81, 223\u201324.\n\n. Basic Tactics (New York, 1966), p. 98.\n\n. The Chinese Classics, V, 183.\n\n. The Need for Roots, trans. Arthur Wills (Boston, 1955), p. 159.\n\n. A Theory of Justice (Cambridge, Mass., 1971), p. 379. Compare Vitoria: \". . . whatever is done in right of war receives the construction most favorable to the claims of those engaged in a just war.\" On the Law of War, p. 180.\n\n. This seems to be G. E. M. Anscombe's position in the two essays already cited: Mr. Truman's Degree and \"War and Murder.\"\n\n. For a discussion of what it means to \"override\" a moral principle, see Robert Nozick, \"Moral Complications and Moral Structures,\" 13 Natural Law Forum 34\u201335 and notes (1968).\n\n15 Aggression and Neutrality\n\n. Philip C. Jessup, Neutrality: Its History, Economics, and Law (New York, 1936), IV, 80 (emphasis added).\n\n. W. E. Hall, The Rights and Duties of Neutrals (London, 1874) is the best account of the laws of neutrality.\n\n. Westlake, International Law, II, 162.\n\n. The speech is reprinted in The Theory and Practice of Neutrality in the Twentieth Century, ed. Roderick Ogley (New York, 1970), p. 83.\n\n. Theory and Practice of Neutrality, p. 74.\n\n. Liddell Hart, The Real War, pp. 46\u201347.\n\n. For an example of the American response, see James M. Beck, The Evidence in the Case: A Discussion of the Moral Responsibility for the War of 1914 (New York, 1915), esp. ch. IX.\n\n. Socialism and War, p. 15.\n\n. Nils Oervik, The Decline of Neutrality: 1914\u20131941 (Oslo, 1953), p. 241.\n\n. Oervik, p. 223.\n\n. Churchill, The Gathering Storm (New York, 1961), Bk. II, ch. 9.\n\n. Assignment to Catastrophe (New York, 1954), I, 71\u201372.\n\n. Time Unguarded: The Ironside Diaries 1937\u20131940, ed. Roderick Macleod and Denis Kelly (New York, 1962), p. 211.\n\n. Ironside Diaries, p. 185.\n\n. Ironside Diaries, p. 216.\n\n. History of the Second World War (New York, 1971), p. 53.\n\n. Ironside Diaries, p. 238.\n\n. The Gathering Storm, p. 488.\n\n. Oervik, p. 237.\n\n. For an account of the campaign, see J. L. Moulton, A Study of Warfare in Three Dimensions: The Norwegian Campaign of 1940 (Athens, Ohio, 1967).\n\n. History of the Second World War, p. 59. Cf. General Ironside's entry for February 14, 1940: \"Winston is now pressing for his laying of mines in neutral Norwegian waters as the only means of forcing the Germans to violate Scandinavia and so give us a chance of getting into Narvik.\" The Ironside Diaries, p. 222.\n\n16 Supreme Emergency\n\n. Quoted in George Quester, Deterrence Before Hiroshima (New York, 1966), p. 67.\n\n. See J. Glenn Gray, The Warriors: Reflections on Men in Battle (New York, 1967), ch. 5: \"Images of the Enemy.\"\n\n. But the claim that one can never kill an innocent person abstracts from questions of coercion and consent: see the examples cited in chapter 10.\n\n. See Quester, Deterrence and F. M. Sallagar, The Road to Total War: Escalation in World War II (Rand Corporation Report, 1969); also the official history by Sir Charles Webster and Noble Frankland, The Strategic Air Offensive Against Germany (London, 1961).\n\n. Noble Frankland, Bomber Offensive: The Devastation of Europe (New York, 1970), p. 41.\n\n. The Story of the Cherwell minute is told, most unsympathetically, in C. P. Snow, Science and Government (New York, 1962).\n\n. Quester, pp. 117\u201318.\n\n. Quoted in Quester, p. 141.\n\n. Quoted in Angus Calder, The People's War: 1939\u20131945 (New York, 1969), p. 491.\n\n. Calder, p. 229; the same poll is cited by Vera Brittain, a courageous opponent of British bombing policy: Humiliation with Honor (New York, 1943), p. 91.\n\n. \". . . it was not [Cherwell's] ruthlessness that worried us most, it was his calculations.\" Snow, Science and Government, p. 48. Cf. P. M. S. Blackett's post-war critique of the bombing, worked out in narrowly strategic terms: Fear, War, and the Bomb (New York, 1949), ch. 2.\n\n. Sallagar, p. 127.\n\n. Sallagar, p. 128.\n\n. Frankland, Bomber Offensive, pp. 38\u201339.\n\n. Frankland, Bomber Offensive, p. 134.\n\n. Sir Arthur Harris, Bomber Offensive (London, 1947), p. 74.\n\n. Calder, p. 229.\n\n. The Hinge of Fate, p. 770.\n\n. For a detailed account of this attack, see David Irving, The Destruction of Dresden (New York, 1963).\n\n. Quoted in Quester, p. 156.\n\n. Memoirs of a Revolutionist (New York, 1957), p. 178.\n\n. Robert C. Batchelder, The Irreversible Decision: 1939\u20131950 (New York, 1965), p. 38. Batchelder's is the best historical account of the decision to drop the bomb, and the only one that treats the moral issues in a systematic way.\n\n. A. Russell Buchanan, The United States and World War II (New York, 1964), I, 75.\n\n. \"The Decision to Use the Atomic Bomb,\" Harper's Magazine (February 1947), repr. in The Atomic Bomb: The Great Decision, ed. Paul R. Baker (New York, 1968), p. 21.\n\n. Speaking Frankly (New York, 1947), p. 261.\n\n. Atomic Quest (New York, 1956), p. 247.\n\n. Mr. Citizen (New York, 1960), p. 267. I owe this group of quotations to Gerald McElroy.\n\n. Batchelder, p. 159.\n\n. Batchelder, p. 149.\n\n. Triumph and Tragedy (New York, 1962), p. 639.\n\n. \"The Decision to Use the Bomb,\" p. 21.\n\n. Speaking Frankly, p. 264.\n\n. The case would be even worse if the bomb were used for political rather than military reasons (with the Russians rather than the Japanese in mind): on this point, see the careful analysis of Martin J. Sherwin, A World Destroyed: The Atomic Bomb and the Grand Alliance (New York, 1975).\n\n17 Nuclear Deterrence\n\n. \"The Decision to Use the Bomb,\" in The Atomic Bomb, ed. Baker, p. 21.\n\n. Reflections on the Revolution in France (Everyman's Library, London, 1910), p. 75.\n\n. See, for example, Nuclear Weapons and Christian Conscience, ed. Stein; Nuclear Weapons and the Conflict of Conscience, ed. John C. Bennett (New York, 1962); The Moral Dilemma of Nuclear Weapons, ed. William Clancy (New York, 1961); Morality and Modern Warfare, ed. William J. Nagle (Baltimore, 1960).\n\n. \"Moral Urgencies in the Nuclear Context,\" in Nuclear Weapons and the Conflict of Conscience, p. 101.\n\n. The Just War: Force and Political Responsibility (New York, 1968), p. 171.\n\n. \"Explorations into the Unilateral Disarmament Position,\" in Nuclear Weapons and the Conflict of Conscience, p. 130.\n\n. See the novel Fail-Safe by Eugene Burdick and Harvey Wheeler (New York, 1962) for a possible scenario.\n\n. Paul Ramsey, \"A Political Ethics Context for Strategic Thinking,\" in Strategic Thinking and Its Moral Implications, ed. Morton A. Kaplan (Chicago, 1973), pp. 134\u201335.\n\n. George Urban, \"A Conversation with George F. Kennan,\" 47 Encounter 3:37 (September 1976).\n\n. For a review and critique of this literature, see Philip Green, Deadly Logic: The Theory of Nuclear Deterrence (Columbus, Ohio, 1966).\n\n. Nuclear Weapons and Foreign Policy (New York, 1957), p. 180.\n\n. On War, trans. Terence Kilmartin (New York, 1968), p. 138.\n\n. See the article \"Warfare, Conduct of\" in the Encyclopaedia Britannica (15th ed., Chicago, 1975), Macropaedia, Vol. 19, p. 509.\n\n. On War, p. 138.\n\n. Bernard Brodie, War and Politics (New York, 1973), p. 404 (author's emphasis).\n\n. The bulk of Ramsey's articles, papers, and pamphlets are collected in his book The Just War; see also his earlier work War and the Christian Conscience: How Shall Modern War Be Justly Conducted? (Durham, 1961).\n\n. \"War and Murder,\" p. 57.\n\n. The Just War, p. 252; see also p. 320.\n\n. The Just War, p. 303.\n\n. War and Politics, p. 404.\n\n. The Just War, p. 253 (author's emphasis); see also p. 328.\n\n. \"Warfare,\" p. 568.\n\n. The Just War, p. 254; see also pp. 333ff.\n\n. The Just War, p. 364. Ramsey is paraphrasing Anscombe's critique of pacifism: see \"War and Murder,\" p. 56.\n\n18 The Crime of Aggression: Political Leaders and Citizens\n\n. Joseph W. Bishop, Jr., \"The Question of War Crimes,\" 54 Commentary 6:85 (December 1972).\n\n. See the suggestion of Sanford Levinson, \"Responsibility for Crimes of War,\" 2 Philosophy and Public Affairs 270ff. (1973).\n\n. For a useful account of this doctrine, tracing it back to the jurisprudence of John Austin, see Stanley Paulson, \"Classical Legal Positivism at Nuremberg,\" 4 Philosophy and Public Affairs 132\u201358 (1975).\n\n. Quoted in Noam Chomsky, At War with Asia, p. 310.\n\n. Stanley Kunitz, \"Foreign Affairs,\" in Selected Poems: 1928\u20131959 (Boston, 1958), p. 23.\n\n. Trials of War Criminals Before the Nuremberg Military Tribunals, vol. 11 (1950), pp. 488\u201389; see the discussion in Levinson, pp. 253ff., and in Greenspan, Modern Law of Land Warfare, pp. 449\u201350.\n\n. Trials of War Criminals, vol. 14 (n.d.), p. 383; see Levinson, p. 263.\n\n. Trials of War Criminals, vol. 14, p. 472; see Levinson, p. 264.\n\n. For a discussion of the Vietnam cases, see Edward Weisband and Thomas M. Franck, Resignation in Protest (New York, 1976).\n\n. Kurt Gerstein: The Ambiguity of Good, trans. Charles Fullman (New York, 1969).\n\n. Trials of War Criminals, vol. 14, p. 346.\n\n. King John 4:2, ll. 231\u201341.\n\n. For the contemporary law of reparations, see Greenspan, pp. 309\u201310, 592\u201393.\n\n. The Warriors, pp. 196\u201397.\n\n. The Warriors, p. 198.\n\n. The Warriors, p. 199.\n\n. In thinking about these issues, I have been greatly helped by the essays in Joel Feinberg's Doing and Deserving.\n\n. The Warriors, p. 199.\n\n. See Richard A. Falk, \"The Circle of Responsibility,\" in Crimes of War, ed. Falk, G. Kolko, and R. J. Lifton (New York, 1971), p. 230: \"The circle of responsibility is drawn around all who have or should have knowledge of the illegal and immoral character of the war.\"\n\n. American Power and the New Mandarins (New York, 1969).\n\n19 War Crimes: Soldiers and Their Officers\n\n. The Seventh Day: Soldiers Talk About the Six Day War (London, 1970), p. 126.\n\n. I owe this point to Dan Little.\n\n. Guy Chapman, A Passionate Prodigality (New York, 1966), pp. 99\u2013100.\n\n. Richard Wasserstrom, \"The Responsibility of the Individual for War Crimes,\" in Philosophy, Morality, and International Affairs, p. 62.\n\n. The Thin Red Line (New York, 1964), pp. 271\u201378.\n\n. On the difficulties of surrender in the midst of a modern battle, see John Keegan, The Face of Battle, p. 322.\n\n. See the discussion of this point by Samuel David Resnick, Moral Responsibility and Democratic Theory, unpublished Ph.D. dissertation (Harvard University, 1972).\n\n. Seymour Hersh, My Lai 4: A Report on the Massacre and its Aftermath (New York, 1970); see also David Cooper, \"Responsibility and the 'System,'\" in Individual and Collective Responsibility: The Massacre at My Lai, ed. Peter French (Cambridge, Mass., 1972), pp. 83\u2013100.\n\n. Hersh, p. 42.\n\n. The Warriors, p. 181.\n\n. The Measures Taken, in The Jewish Wife and Other Short Plays, trans. Eric Bentley (New York, 1965), p. 82.\n\n. The best account of the present legal situation is Yoram Dinstein, The Defense of Obedience to Superior Orders in International Law (Leiden, 1965).\n\n. McDougal and Feliciano, Law and Minimum World Public Order, p. 690.\n\n. Quoted in Kurt Baier's analysis of the Calley trial, \"Guilt and Responsibility,\" Individual and Collective Responsibility, p. 42.\n\n. See Wasserstrom, \"The Responsibility of the Individual.\"\n\n. Quoted in Telford Taylor, Nuremberg and Vietnam, p. 49.\n\n. The Warriors, pp. 185\u201386.\n\n. The Warriors, p. 189.\n\n. McDougal and Feliciano, pp. 693\u201394 and notes.\n\n. Nuremberg and Vietnam, p. 55n.\n\n. Jean Le Meur, \"The Story of a Responsible Act,\" in Political Man and Social Man, ed. Robert Paul Wolff (New York, 1964), p. 204.\n\n. Quoted in A. J. Barker, Yamashita (New York, 1973), pp. 157\u201358.\n\n. Omar N. Bradley, A Soldier's Story (New York, 1964), pp. 343\u201344.\n\n. For the relevant law, see Greenspan, Modern Law of Land Warfare, pp. 332ff.\n\n. See Fried, Anatomy of Values, pp. 194\u201399.\n\n. I shall follow the account of A. Frank Reel, The Case of General Yamashita (Chicago, 1949).\n\n. Reel, p. 280: the appendix of this book reprints the Supreme Court decision.\n\n. On strict liability, see Feinberg, Doing and Deserving, pp. 223ff.\n\n. Hersh, p. 11.\n\n. \"Political Action: The Problem of Dirty Hands,\" 2 Philosophy and Public Affairs (1973), pp. 160\u201380.\n\n. Frankland, Bomber Offensive, p. 159.\n\n. Calder, The People's War, p. 565; Irving, Destruction of Dresden, pp. 250\u201357.\n\n. Soldiers: An Obituary for Geneva, trans. Robert David MacDonald (New York, 1968), p. 192.\n\n. The Discourses, Bk. I, ch. XVIII.\n\n. 1 Philosophy and Public Affairs (1972), p. 143.\n\nAfterword: Nonviolence and the Theory of War\n\n. Exploring Nonviolent Alternatives (Boston, 1971), p. 93; cf. Anders Boserup and Andrew Mack, War Without Weapons: Non-Violence in National Defense (New York, 1975), p. 135.\n\n. Sharp, p. 52.\n\n. But an enemy state might threaten to bomb rather than invade; on this possibility, see Adam Roberts, \"Civilian Defense Strategy,\" in Civilian Resistance as a National Defense, ed. Roberts (Hammondsworth, 1969), pp. 268\u201372.\n\n. Collected Essays, Journalism, and Letters, vol. 4, p. 469.\n\n. Louis Fischer, Gandhi and Stalin, quoted in Orwell's \"Reflections,\" p. 468.\n\n. \"Lessons from Resistance Movements\u2014Guerrilla and Non-Violent,\" in Civilian Resistance, p. 240.\n\n. For a brief account of Czech resistance, see Boserup and Mack, pp. 102\u201316.\n\n. Sharp, p. 66; but he believes that the degree and extent of suffering will be \"vastly smaller\" than in regular warfare (p. 65).\nIndex\n\nAcheson, Dean, \u2013\n\nact of state doctrine, \u2013\n\nadvisors, responsibility of, \u2013, \u2013\n\nAfghanistan, xix\n\naggression, \u2013, ,\n\nas act of state, \u2013\n\ncrime of, \u2013,\n\ndefinition of, \u2013\n\ndomestic analogy and, \u2013,\n\nneutrality and, \u2013, \u2013\n\nnonviolent defense against, \u2013\n\nresponsibility for, \u2013\n\ntheory of, \u2013, , \u2013\n\nthreats and, \u2013, , ,\n\nAgincourt, battle of, \u2013\n\nAlcibiades, \u2013\n\nAlexander the Great,\n\nAlgiers, battle of, \u2013\n\nAlsace-Lorraine, \u2013\n\nambush, , ,\n\nin guerrilla war, \u2013, \u2013,\n\nAnjou, Duke of, \u2013\n\nannexation, , , ,\n\nAnscombe, G. E. M., ,\n\nappeasement, \u2013, \u2013\n\nAquinas, St. Thomas, xxvi,\n\naristocratic soldiers, \u2013, \u2013\n\nAristotle,\n\nArnim, Hans-J\u00fcrgen von, \u2013\n\nAron, Raymond, \u2013\n\nassassination, \u2013, \u2013\n\nasymmetric warfare, xiii\u2013xiv\n\nnoncombatant casualties in, xviii\u2013xxii\n\nproportionality rule in, xv\n\nrisk taking in, xix\u2013xxii\n\nAthens, \u2013\n\nAtlanta, burning of, \u2013\n\natomic bomb\n\nSee Hiroshima, nuclear deterrence\n\nAustin, Warren,\n\nAustrian Empire, \u2013\n\nBacon, Francis, , \u2013\n\nbalance of power, \u2013, , ,\n\nbalance of terror, \u2013, \u2013\n\nBaldwin, Stanley,\n\nBangladesh. See India's intervention in East Pakistan\n\nbarbarians, , ,\n\nBeatty, Admiral David, \u2013\n\nBeaufre, Andr\u00e9, ,\n\nBeirut raid, \u2013\n\nBelgium\n\nGerman \"rape of,\" , \u2013\n\nneutrality of,\n\nbelligerent rights, , ,\n\nbenevolent quarantine, \u2013, , \u2013,\n\nBennett, John,\n\nBernard, Montague, ,\n\nBethmann Hollweg, Theobald von, \u2013\n\nBishop, Joseph W., Jr.,\n\nBismarck, Otto von, \u2013,\n\nBloch, Marc,\n\nblockades, ,\n\nof Germany by Great Britain, \u2013,\n\nmoral responsibility in, \u2013\n\nin Spanish American War,\n\nBoer War, ,\n\nBomber Command (R.A.F.), \u2013, \u2013\n\nbombing\n\nterror, \u2013, , \u2013\n\nSee also specific locations\n\nboundaries, \u2013, \u2013, \u2013,\n\nBradley, General Omar, \u2013,\n\nBrecht, Bertolt,\n\nBritain. See Great Britain\n\nBrodie, Bernard,\n\nBrooke, Rupert,\n\nBurke, Edmund, , \u2013,\n\nByrnes, James, ,\n\nCalder, Angus,\n\nCalley, Lieutenant William, \u2013, ,\n\nCampbell-Bannerman, Henry,\n\nCamus, Albert, \u2013\n\nCastro, Raul,\n\nChadwick, Admiral F. E.,\n\nChapman, Guy, \u2013\n\nCherwell, Lord (F. A. Lindemann), \u2013\n\nChina, , \u2013\n\n\"Chindits,\"\n\nchivalry, \u2013\n\nChomsky, Noam,\n\nChurchill, Winston,\n\natomic bomb and, \u2013\n\non bombing German cities, , , , \u2013\n\non German victory, \u2013\n\nNorway's neutrality and, \u2013\n\nreprisal warning,\n\non supreme emergency, \u2013, ,\n\nunconditional surrender and,\n\ncitizens, ,\n\nSee also democratic citizens; noncombatants\n\ncivil war, , , \u2013\n\nnonintervention in, , \u2013\n\nself-help in, ,\n\nSpanish, \u2013\n\nof U.S., \u2013, ,\n\ncivilians, ,\n\nSee also noncombatant immunity; rights\n\nClausewitz, Karl von, xvi, \u2013, ,\n\nCleon, ,\n\nCOBRA, \u2013\n\ncode of honor, \u2013\n\ncoercion,\n\nof conscripts, \u2013, , ,\n\nresponsibility and, , \u2013\n\nin sieges, \u2013,\n\nof war, \u2013,\n\ncollateral damage, \u2013\n\nSee also double effect, noncombatant casualties; noncombatant immunity\n\ncollective freedom, \u2013, \u2013, \u2013\n\ncollective punishment, \u2013\n\ncollective rights, , ,\n\ncollective security, ,\n\ncollectivism, of war, \u2013\n\ncolonization, , , , \u2013\n\ncombatant\/noncombatant distinction, \u2013, \u2013, , \u2013\n\nSee also innocence; noncombatant immunity; rights\n\ncommand responsibility, \u2013, \u2013\n\n\"Commando Order,\"\n\ncommandos, , \u2013\n\ncommunity\n\npolitical, \u2013\n\nof soldiers, \u2013\n\nCompton, Arthur,\n\nconquest, \u2013, , , ,\n\nconscientious objection, \u2013, ,\n\nconscripts, , ,\n\ncoercion of, \u2013, , ,\n\nconsent\n\nin sieges, \u2013,\n\nin social contract, \u2013\n\nin surrender,\n\nin war generally, \u2013\n\ncontraband,\n\ncontract,\n\ncounter-city warfare, , \u2013\n\ncounter-force warfare, \u2013\n\ncounter-intervention\n\nin Hungarian Revolution, \u2013\n\nself-help and,\n\nin Vietnam War,\n\ncrime,\n\nof aggression, \u2013,\n\nSee also war crimes\n\nCromwell, Oliver,\n\ncrusade, \u2013,\n\nCuba\n\ninsurrection against Spain, \u2013\n\nrevolution in,\n\nCzechoslovakia\n\n\"Munich principle\" and, \u2013\n\nRussian invasion of, \u2013,\n\nin World War II, , \u2013, \u2013\n\nDavid and Goliath,\n\nDeclaration of St. Petersburg,\n\nDeladier, Edouard,\n\ndemocracy, , , , , \u2013\n\nchivalry and, \u2013\n\nmoral responsibility and, \u2013\n\nunconditional surrender and,\n\ndemocratic citizens\n\nconscientious objection among, \u2013\n\nmoral responsibility of, \u2013\n\nsoldiers compared to,\n\ndeterrence, , , \u2013, \u2013\n\nSee also nuclear deterrence\n\nDeuteronomy, \u2013,\n\nDiodotus,\n\nDionysius of Halicarnassus, \u2013\n\ndiplomacy, ,\n\nbefore Six Day War,\n\nDoenitz, Admiral Karl, \u2013\n\ndomestic analogy, \u2013, \u2013\n\naggression and, \u2013,\n\nappeasement and, \u2013\n\narmed robbery, \u2013\n\nof deterrence, -272\n\nof due care, \u2013\n\nfamily feuds, \u2013\n\nlaw enforcement and, , \u2013\n\non neutrality, \u2013\n\nself-defense and, \u2013\n\ndouble effect, doctrine of, xix,\n\nintention in, , \u2013\n\nlimited nuclear war and, \u2013\n\nnoncombatant immunity and, \u2013, \u2013\n\nproportionality rule in, , \u2013\n\nrevision of,\n\nrisk in, \u2013, \u2013\n\nsieges and, \u2013\n\nDresden, firebombing of,\n\nDryden, John,\n\ndue care, \u2013, , \u2013\n\nduels, \u2013\n\nduress, \u2013, , \u2013,\n\nEast Pakistan. See India's intervention in East Pakistan\n\nEban, Abba,\n\nEcclesiastes, xiii\n\nEgypt, Six Day War and, \u2013\n\nEinstein, Albert,\n\nEisenhower, General Dwight D., \u2013, ,\n\nends of war, , \u2013, , , ,\n\nenemies, ,\n\nidentification of, \u2013, \u2013\n\nSee also insurgents\n\nequality\n\nSee soldiers, moral equality of\n\nescalation, \u2013, ,\n\nin limited nuclear war, \u2013\n\nespionage,\n\nEthiopia War (1936), , \u2013\n\nEuripides,\n\nextremity\n\nutilitarianism of, \u2013, \u2013\n\nFalk, Richard,\n\nFalkenhurst, General Nikolaus von,\n\nFall, Bernard,\n\nFanon, Franz,\n\nFeliciano, Florentino P.,\n\nfeuds, , \u2013\n\nFinland,\n\nindependence of, \u2013\n\n\"Munich principle\" and, \u2013\n\nRusso-Finnish War, \u2013,\n\nFlanders, Ralph, \u2013\n\nFLN (Algeria), \u2013\n\nforced resettlement, ,\n\nin Vietnam War, \u2013\n\nFrance, \u2013,\n\nAlsace-Lorraine and, \u2013\n\nbombing of St. L\u00f4, \u2013\n\nGerman occupation of, \u2013, \u2013, \u2013,\n\nreprisals in, \u2013, \u2013,\n\nSpanish Civil War and, \u2013\n\nWar of Spanish Succession and, \u2013\n\nWorld War II Allied bombing in, \u2013,\n\nFranco-Prussian War, \u2013, ,\n\nFrank, Anne,\n\nFrankland, Noble, \u2013\n\nfreedom, \u2013, \u2013,\n\ncollective, \u2013, \u2013, \u2013\n\nmoral responsibility and, \u2013, , ,\n\nrules of war and,\n\nSee also coercion; consent; duress; tyranny\n\nFrench Forces of the Interior, \u2013, \u2013,\n\nFriedlander, Saul,\n\nFromm, Erich,\n\nFuller, Major General J. F. C.,\n\nGandhi, Mohandas, \u2013\n\nGaza war, xvii\u2013xix\n\nGeneva Agreement (1954),\n\nGeneva Conventions (1929 and 1949), , \u2013\n\nGenghis Khan,\n\nGermany, \u2013, ,\n\nAlsace-Lorraine and, \u2013\n\nBelgium neutrality and, , \u2013\n\nFrance occupation by, \u2013, \u2013, \u2013,\n\nFranco-Prussian War and, \u2013,\n\nGreat Britain blockade of, \u2013,\n\nGreat Britain bombing of cities in, xv, \u2013, \u2013\n\nNorway's neutrality and, \u2013\n\nSpanish Civil War and, \u2013\n\nunconditional surrender for,\n\nGerstein, Kurt,\n\nGraves, Robert,\n\nGray, J. Glenn, \u2013, \u2013, ,\n\nGreat Britain, \u2013\n\nGerman city bombing by, xv, \u2013, \u2013\n\nGermany blockade by, \u2013,\n\nHungarian Revolution and, \u2013\n\nLondon Naval Protocol of, , , \u2013,\n\nNorway's neutrality and, \u2013\n\nR.A.F. of, \u2013, \u2013\n\nRusso-Finnish War and, \u2013\n\nSpanish Civil War and, \u2013\n\nWar of Spanish Succession and, \u2013\n\nGreece, \u2013, \u2013,\n\nGreen, T. H.,\n\nGrotius, Hugo, xxvi, ,\n\nguerrilla war,\n\nambush in, \u2013, \u2013,\n\nanti-guerrilla warfare, \u2013, \u2013\n\nfighters' rights in, \u2013\n\nideology in, , \u2013\n\ninternational law and, \u2013,\n\nnoncombatant immunity in, \u2013\n\nnonviolent defense and, \u2013\n\nas \"people's war,\" \u2013,\n\nprisoners of war in, , \u2013, \u2013,\n\nsabotage in, \u2013,\n\nwinning in, \u2013\n\nGuevara, Che,\n\nguilt,\n\nreparations and, \u2013\n\nshame and, \u2013\n\nHaakon, VII (king of Norway),\n\nHaldeman, Joe,\n\nHarris, Air Marshal Arthur, , , \u2013\n\nHenry V, of England, \u2013, ,\n\nHerodotus,\n\nHiroshima, , , , , \u2013\n\nhistorical relativism, \u2013\n\nHitler, Adolf, , ,\n\nadvisors of, \u2013,\n\natomic bomb and,\n\nCommando Order from,\n\nNorway and, \u2013\n\nHobbes, Thomas, , \u2013, , ,\n\nHochhuth, Rolf,\n\nHolinshed, Raphael,\n\nhonor, \u2013, \u2013\n\nHood, General John,\n\nHoopes, Townsend,\n\nhostages, , \u2013,\n\nHsiang, Duke of Sung, \u2013\n\nhumanitarian intervention, ,\n\nin Cuba's insurrection against Spain, \u2013\n\nIndia in East Pakistan, \u2013\n\njudgment in, \u2013\n\nlegalist paradigm revision for, \u2013\n\nmotives in, , \u2013\n\nHume, David, , ,\n\nHungarian Revolution (1848\u20131849)\n\nGreat Britain and, \u2013\n\nJohn Stuart Mill on, \u2013\n\nprudence related to, \u2013\n\nRussia and, \u2013\n\nHussein, King of Jordan,\n\nHyde, Charles Chaney,\n\nhypocrisy, xxvii, \u2013,\n\nimperialism, , , , \u2013, \u2013\n\nIndia's intervention in East Pakistan, \u2013\n\nindividual rights, \u2013, \u2013, ,\n\ninnocence\n\ndefined, ,\n\nterrorism and, , , \u2013,\n\ninsurgents, \u2013\n\nin asymmetrical warfare, xiv\n\nneutrality related to,\n\nnoncombatant casualties and, xvii\u2013xviii\n\npolitics of, xvi\n\nproportionality rule and, xv\u2013xvi\n\nself-help for, ,\n\nintelligence, xxi\u2013xxii\n\nintention, \u2013\n\nin double effect, \u2013\n\nin nuclear deterrence, , \u2013\n\npreventive war and, \u2013\n\ninternational law,\n\nguerrilla war and, \u2013,\n\nkilling and, \u2013\n\nneutrality and, \u2013,\n\npreventive war and,\n\nreprisal and, \u2013\n\nwar convention and,\n\nInternational on the Franco-Prussian War, \u2013\n\ninternational society\n\naggression and, \u2013, \u2013\n\ncollective character of,\n\nindependent states as,\n\npunishment from, \u2013\n\nrights of members, \u2013\n\nstability in, \u2013\n\nintervention,\n\ncounter-intervention, \u2013, \u2013\n\nhumanitarian, , \u2013\n\nSee also nonintervention; self-determination\n\nIraq,\n\nIrish Republican Army, ,\n\nIronside, General Edmund, \u2013\n\nIrving, David, \u2013\n\nIsrael\n\nBeirut raid by, \u2013\n\nEntebbe airport raid by,\n\nGaza war and, xvii\u2013xix\n\nKhibye raid by, \u2013\n\nPalestinian raids on, \u2013\n\nSix Day War of, \u2013, ,\n\nItalian-Ethiopian War, , \u2013\n\nItaly, , \u2013\n\nRenaissance, \u2013\n\nwomen's rape in, \u2013\n\nJaeger, Werner,\n\nJapan\n\nin Russo-Japanese War,\n\nTokyo firebombing, , \u2013,\n\nunconditional surrender and, \u2013\n\nin World War II, , , , , \u2013\n\nJarrell, Randall,\n\nJerusalem, siege of, \u2013,\n\nJews\n\nin Jerusalem siege, \u2013,\n\nNazism and, \u2013,\n\nSee also Israel\n\nJones, James,\n\nJordan,\n\nJosephus, \u2013,\n\njustice,\n\nlegalist paradigm of, \u2013\n\nMarx on, \u2013\n\nmeaning of, \u2013\n\nprudence and, \u2013\n\nrealists' critique of, \u2013\n\nresponsibility and, \u2013\n\nin settlements, \u2013,\n\ntensions in the theory of, \u2013,\n\nvigilante, \u2013\n\nof war and in war (jus ad bellum and jus in bello) distinguished, \u2013, \u2013, ,\n\nSee also aggression; rights; war convention\n\nKatangan secession,\n\nKecskemeti, Paul, \u2013, \u2013\n\nKelsen, Hans,\n\nKennan, George, ,\n\nKennedy, John,\n\nKhibye raid, \u2013\n\nkilling, \u2013, , \u2013\n\ndomestic analogy on, \u2013\n\ninternational law and, \u2013\n\nmurder compared to, \u2013, \u2013, , \u2013, \u2013, , , \u2013,\n\nstandards for, \u2013\n\nSee also massacres; noncombatant immunity\n\nKissinger, Henry,\n\nKitchener, General H. H.,\n\nKorean War, \u2013\n\nbombardment in, \u2013\n\nlaw enforcement in, \u2013\n\nnegotiations in,\n\nproportionality rule in, \u2013\n\npunishment in, \u2013\n\nrights in, \u2013\n\nU.S. ends in, \u2013\n\nKossuth, Lajos,\n\nLaconia Affair, \u2013\n\nlanguage\n\nof morality, xxiii\u2013xxvii,\n\nof positive laws, xxv\u2013xxvi\n\nof strategy, \u2013\n\nabout war, xxiii\u2013xxiv\n\nlast resort, xiv, xvi, , , \u2013, ,\n\nlaw enforcement, ,\n\ndomestic analogy and, , \u2013\n\nreprisals as, \u2013, \u2013\n\nLawrence, T. E.,\n\nLeague of Nations, ,\n\nLebanon, xiii, xx,\n\nLeeb, Field Marshal R. von, \u2013\n\nlegal positivism, xxiv\u2013xxvi, , , , , ,\n\nUN and, \u2013\n\nlegalist paradigm,\n\nrevision of, \u2013, , \u2013, \u2013\n\nstated, \u2013\n\nlegitimacy, \u2013, , , \u2013\n\nin acts of war, \u2013, , \u2013, , ,\n\nof Vietnam government, \u2013\n\nLenin, V. I., , \u2013\n\nLeningrad, siege of, , \u2013,\n\nLevinson, Sanford,\n\nLex talionis,\n\nLiddell Hart, B. H., , , , , \u2013\n\nLieber, Francis, , ,\n\nlimited nuclear war\n\ncollateral damage in, \u2013\n\ncounter-force warfare and, \u2013\n\ndouble effect and, \u2013\n\nescalation in, \u2013\n\nflexible response and, ,\n\nmassive retaliation in, \u2013\n\nnaval warfare compared to,\n\nnuclear deterrence and, \u2013\n\nPaul Ramsey on, \u2013\n\nlimited war, \u2013, , , , ,\n\nLocke, John,\n\nLondon Naval Protocol (1936), , , \u2013,\n\nLouis XIV, of France, \u2013\n\nLucas, Scott,\n\nLussu, Emilio, \u2013\n\nMacArthur, General Douglas,\n\nMacdonald, Dwight,\n\nMachiavelli, Nicolo, , ,\n\nMaimonides, Moses,\n\nMalaya,\n\nMannerheim, Marshal K. G.,\n\nMao Tse-tung, \"Eight Points for Attention,\" , \u2013\n\nMarshall, S. L. A.,\n\nMarx, Karl, \u2013,\n\nMarxism, , , \u2013\n\nmassacres, \u2013\n\nSee also India's intervention in East Pakistan; My Lai massacre\n\nmassive retaliation, , \u2013,\n\nMcDougal, Myres, S.,\n\nMcKinley, William,\n\nMedina, Captain Ernest, \u2013,\n\nMelos, \u2013,\n\nMelzer, Yehuda,\n\nMendes-France, Pierre,\n\nmercenaries, \u2013, ,\n\nmerchant seamen, \u2013\n\n\"Merrill's Marauders,\"\n\nMetternich, K. von, ,\n\nMill, John Stuart, \u2013\n\n\"Ministries Case,\" \u2013\n\nMo Tzu,\n\nMoltke, General H. J. von, \u2013,\n\nMontesquieu, Baron de,\n\nmoral absolutism, \u2013\n\nmoral reality of war, defined,\n\nmoral relativism, \u2013\n\nmoral responsibility. See responsibility, moral\n\nmorality, ,\n\nhypocrisy and, xxvii, \u2013\n\nlanguage of, xxiii\u2013xxvii,\n\npolitics and, \u2013\n\npractical, xxvii, \u2013\n\nstrategy and, \u2013\n\nMoroccan soldiers, \u2013\n\nMoyne, Lord (W. E. Guinness), ,\n\n\"Munich principle,\" \u2013\n\nmurder, , ,\n\nkilling compared to, \u2013, \u2013, , \u2013, \u2013, , , \u2013,\n\nMurphy, Frank, \u2013\n\nMy Lai massacre, \u2013, \u2013\n\nMytilene, \u2013\n\nNagel, Thomas,\n\nNapoleon Bonaparte, ,\n\nNapoleon III, of France,\n\nNasser, Gamal Abdel, \u2013\n\nNational Liberation Front (Vietnam), ,\n\nnational minorities, \u2013\n\nnaval warfare\n\nlimited nuclear war compared to,\n\nmerchant seamen in, \u2013\n\nnecessity in, \u2013\n\nnoncombatant immunity in, \u2013\n\nNorway's neutrality and, \u2013, \u2013\n\nrisk in, \u2013\n\nsink on sight in, \u2013\n\nSee also London Naval Protocol\n\nNazism\n\nJews and, \u2013,\n\nsupreme emergency of, \u2013, \u2013\n\nas ultimate threat, \u2013\n\nunconditional surrender and, \u2013\n\nnecessity,\n\nclaim of, criticized, \u2013\n\nas defense of terrorism, \u2013, , \u2013\n\nmilitary, defined, \u2013\n\nmoral, in India's intervention in East Pakistan, \u2013\n\nnature of, \u2013, \u2013, \u2013\n\nin naval warfare, \u2013\n\nresponsibility and, , \u2013\n\nrules of war and, \u2013\n\nsupreme emergency and, \u2013\n\nin Thucydides, \u2013\n\nwar convention and, \u2013, \u2013\n\nneutrality, , , ,\n\naggression and, \u2013, \u2013\n\ndomestic analogy on, \u2013\n\ninternational law and, \u2013,\n\nof Norway, \u2013\n\nproperty and, \u2013\n\nright of, \u2013\n\nsliding scale of, \u2013\n\nsupreme emergency and, \u2013\n\nviolations of, \u2013\n\nNgo Dinh Diem,\n\nNicholas I, of Russia, \u2013\n\nNimitz, Admiral Chester,\n\nnoncombatant immunity, ,\n\ndouble effect and, \u2013, \u2013\n\nin guerrilla war, \u2013\n\nin naval warfare, \u2013\n\nin nonviolent defense,\n\nin nuclear deterrence,\n\nreprisal and, \u2013\n\nself-defense and,\n\nterrorism and, \u2013, \u2013\n\nin war convention, xiv\u2013xxi, , \u2013, \u2013, \u2013\n\nwho qualifies for, , , \u2013, , \u2013\n\nnoncombatants, \u2013, , \u2013\n\nnonintervention,\n\nin civil war, , \u2013\n\nMill's defense of, \u2013\n\nnonviolent defense,\n\ncommunal values in, \u2013\n\nguerrilla war and, \u2013\n\nnoncombatant immunity in,\n\npolitics of, \u2013,\n\nwar convention in,\n\nNorway\n\nneutrality of, \u2013\n\nVemork raid and, \u2013\n\nNozick, Robert,\n\nnuclear deterrence\n\nimmoral threats and, \u2013\n\nintention in, , \u2013\n\nlimited nuclear war and, \u2013\n\nnoncombatant immunity in,\n\nproportionality rule in,\n\nreprisal in, \u2013, \u2013\n\nsupreme emergency and, \u2013\n\nwar convention and,\n\nSee also limited nuclear war\n\nNuremberg Trials, , , ,\n\nDoenitz in, \u2013\n\n\"Ministries Case,\" \u2013\n\nvon Leeb, \u2013\n\nobedience, , \u2013\n\noccupation, \u2013\n\nFrench resistance to German, \u2013\n\nnonviolent resistance to, \u2013\n\nofficers,\n\ncommand responsibility of, \u2013, \u2013\n\ndue care by, \u2013\n\nstrict liability of, \u2013\n\nsuperior orders, \u2013\n\nofficials\n\nadvisors to, \u2013\n\nheads of state, , \u2013\n\nmoral responsibility of, \u2013\n\nresignation by, \u2013\n\nsabotage by, \u2013\n\nsilence of,\n\nOkinawa, battle of,\n\nOman, C. W. C., \u2013\n\nOphuls, Marcel, \u2013\n\nOppenheim, L.,\n\nOrwell, George, \u2013,\n\nOwen, Wilfred, , ,\n\nPaardeberg, battle of,\n\nPakistan. See India's intervention in East Pakistan\n\nPalestinians, xvii\u2013xix, \u2013\n\nPalmerston, Henry John Temple, , \u2013\n\nPanama canal, \u2013\n\nParis, siege of,\n\npeace\n\nas end of war, , ,\n\nas normative condition,\n\npeacetime reprisals, \u2013\n\nPearl Harbor,\n\nPeloponnesian War, \u2013\n\n\"people's war,\" , \u2013,\n\nPeters, Colonel William,\n\nPhillipines, The, \u2013, \u2013\n\nPlato,\n\nPlatt Amendment, \u2013\n\nPlevna, siege of,\n\nPlutarch,\n\npoison gas, ,\n\nPoland, in World War II, , , \u2013\n\npolitical assassination, \u2013\n\npolitical community, rights of, \u2013\n\nPopular Front for the Liberation of Palestine,\n\nPort Arthur,\n\npositive laws, xxv\u2013xxvi, , , \u2013,\n\npre-emptive strikes, \u2013, \u2013\n\npreventive war\n\nbalance of power and, \u2013\n\ncriteria for, \u2013\n\nself-preservation in, \u2013\n\nsuspicion and, \u2013\n\nprisoners of war,\n\nin guerrilla war, , \u2013,\n\nkilling of, at Agincourt, \u2013\n\nkilling of, in heat of battle, \u2013\n\nin reprisals, \u2013, \u2013,\n\nrights of, , \u2013,\n\nprofessional soldiers, \u2013,\n\nproportionality rule, xvi, xxi\u2013xxii, ,\n\nin doctrine of double effect, , \u2013\n\nin Korean War, \u2013\n\nin nuclear deterrence,\n\nin reprisals, \u2013, \u2013\n\nin war convention, xv, \u2013\n\nprudence, \u2013\n\npunishment\n\ncollective, \u2013\n\nby international society, \u2013\n\njust war as, \u2013\n\nin Korean War, \u2013\n\nreprisal and, \u2013,\n\nunconditional surrender as, \u2013\n\nquarter, in surrender, , , , \u2013\n\nR.A.F. See Bomber Command\n\nRamsey, Paul, \u2013, \u2013,\n\nrandomness, of terrorism, \u2013, ,\n\nrape, \u2013\n\nRavenstein, General Johann von,\n\nRawls, John, ,\n\nrealism, \u2013, , ,\n\nlimited nuclear war and, \u2013,\n\nmoral argument in, \u2013\n\nrelativism and, \u2013\n\nSee also Melos\n\nrealists, \u2013, , \u2013, ,\n\njustice critique of, \u2013\n\non winning, \u2013\n\nrefugees, \u2013\n\nReitz, Deneys,\n\nrelativism\n\nhistorical, \u2013\n\nmoral, \u2013\n\nRenaissance Italy, \u2013\n\nreparations, , , \u2013\n\nreprisals, \u2013\n\nbelligerent, rationale of,\n\ninternational law and, \u2013\n\nas law enforcement, \u2013, \u2013\n\nin nuclear deterrence, \u2013, \u2013\n\npeacetime, \u2013\n\nprisoners of war in, \u2013, \u2013,\n\nproportionality rule in, \u2013, \u2013\n\npunishment and, \u2013,\n\nvictims of, \u2013, \u2013\n\nresignation, \u2013\n\nresistance, See nonviolent defense\n\nresponsibility, legal, ,\n\nfor aggression, \u2013\n\nof officers (command responsibility), , , \u2013\n\nof officials, \u2013\n\nof soldiers, \u2013\n\nSee also international law\n\nresponsibility, moral,\n\nin blockades, \u2013\n\ncollective, \u2013\n\nof democratic citizens, \u2013\n\nfreedom and, \u2013, , ,\n\nguilt and, \u2013\n\nresignation and, \u2013\n\nin sieges, \u2013\n\nof soldiers, xix\u2013xxi, , \u2013, , \u2013\n\nin supreme emergency, \u2013\n\nfor war crimes, , , \u2013, , \u2013,\n\nretribution, \u2013\n\nSee also reprisals\n\nrevolution, \u2013\n\nin Cuba,\n\nFranco-Prussian War and, \u2013\n\nSee also Hungarian Revolution\n\nRichards, Frank, \u2013,\n\nrights, xxviii\n\naggression and, \u2013\n\nbelligerent, , ,\n\nof civilians, ,\n\ncollective, , ,\n\nconquest, \u2013\n\nof guerrilla fighters, \u2013\n\nindividual, \u2013, \u2013, ,\n\nin international society, \u2013\n\nto life and liberty, \u2013\n\nof merchant seamen, \u2013\n\nof neutrality, \u2013\n\noverriding of, \u2013, , \u2013, \u2013\n\npolitical community, \u2013\n\nof prisoners of war, , \u2013,\n\nof self-defense, , \u2013, \u2013\n\nof soldiers, \u2013\n\nto territorial integrity, \u2013\n\nof women, \u2013\n\nSee also noncombatant immunity; prisoners of war; sovereignty; state rights\n\nrisks, xx\u2013xxii, ,\n\nin doctrine of double effect, \u2013, \u2013\n\nin naval warfare, \u2013\n\nof soldiers, \u2013\n\nRoberts, General Frederick Sleigh,\n\nRoman Empire, , \u2013, \u2013\n\nRommel, General Erwin, \u2013\n\nRoosevelt, Franklin,\n\nRousseau, Jean-Jacques,\n\n\"rules of engagement\"\n\nin Afghanistan, xix\n\nin Vietnam War, \u2013\n\nrules of war, xxvi\n\nextremity in, \u2013\n\nmoral absolutism in, \u2013\n\nnecessity and, \u2013\n\nnoncombatants in, \u2013\n\noverriding of, \u2013\n\nsieges and,\n\ntwo sorts of, \u2013\n\nviolation of, \u2013\n\nSee also utilitarianism\n\nRuskin, John, ,\n\nRussia\n\nCzechoslovakia's invasion by, \u2013,\n\nFinland and, \u2013,\n\nHungarian revolution and, \u2013\n\nLeningrad, siege of, , \u2013,\n\nNicholas I of, \u2013\n\nnuclear deterrence and, ,\n\nRusso-Finnish War, \u2013,\n\nRusso-Japanese War,\n\nRusso-Turkish War,\n\nRutledge, Wiley Blount,\n\nsabotage,\n\nin guerrilla war, \u2013,\n\nby officials, \u2013\n\nsack, \u2013\n\nSackville, Thomas,\n\nSartre, Jean-Paul, \u2013\n\nSchell, Jonathan, \u2013\n\nSchwarzenberg, Count,\n\nsecession\n\nboundaries and,\n\nHungarian Revolution as, \u2013\n\nnatural resources and,\n\nself-determination in, \u2013\n\nSedan, battle of, \u2013\n\nself-defense, ,\n\ndomestic analogy and, \u2013\n\njustification of, \u2013\n\nnoncombatant immunity and,\n\npeacetime reprisals as,\n\npermissible acts in,\n\nright of, , \u2013, \u2013\n\nsupreme emergency and,\n\nself-determination,\n\nJohn Stuart Mill's defense of, \u2013\n\npolitical,\n\nin secession, \u2013\n\nself-help in, \u2013\n\nvirtues in,\n\nself-help,\n\nin civil war, ,\n\ncounter-intervention and,\n\nfor insurgents, ,\n\nself-determination and, \u2013\n\nself-preservation, \u2013,\n\nself-respect, \u2013, \u2013\n\nSergei, Grand Duke, \u2013,\n\nsettlements, justice in, \u2013\n\nstatus quo ante and, \u2013\n\nSee also Korean War\n\nShakespeare, William, \u2013, , , , \u2013\n\nshame, \u2013\n\nSharp, Gene, \u2013\n\nSherman, General William Tecumseh,\n\n\"war is hell\" doctrine, \u2013, ,\n\nSidgwick, Henry, ,\n\nutilitarianism of, \u2013\n\nsieges\n\ncoercion in, \u2013,\n\nconsent in, \u2013,\n\ndouble effect and, \u2013\n\nof Jerusalem, \u2013,\n\nof Leningrad, , \u2013,\n\nmoral responsibility in, \u2013\n\nof Plevna,\n\nrefugees from, \u2013\n\nSee also blockades\n\nSimpson, Louis, \u2013\n\nSix Day War,\n\ndiplomacy before,\n\nEgypt and, \u2013\n\nIsraeli pre-emption in, \u2013, \u2013\n\nself-defense and, ,\n\nsliding scale, \u2013, \u2013\n\nsnipers, \u2013,\n\nSnow, C. P.,\n\nsocial contract, \u2013\n\nsocialism, \u2013, \u2013\n\nMarxism, , , \u2013\n\nsoldiers\n\naristocratic, \u2013, \u2013\n\ncitizens compared to,\n\ncommunity of, \u2013\n\nconscripts, \u2013, , , , , ,\n\ndeaths of, \u2013\n\ndecision-making by,\n\nlegal responsibility of, \u2013\n\nmercenaries, \u2013, ,\n\nmoral equality of, \u2013\n\nmoral responsibility of, xix\u2013xxi, , \u2013, , \u2013\n\nMoroccan, \u2013\n\nnaked, \u2013\n\nobedience of, , \u2013\n\nprofessional, \u2013,\n\nrights of, \u2013\n\nuniforms and, \u2013, \u2013\n\nSolinas, Franco,\n\nSolzhenitsyn, Alexander,\n\nsovereignty, \u2013\n\ndefinition,\n\nresponsibility and,\n\nright of neutrality and,\n\nself-determination and,\n\nterritorial integrity and, \u2013\n\nSoviet Union, ,\n\nSee also Russia\n\nSpaight, J. M.,\n\nSpain\n\nCivil War, \u2013\n\nCuba's insurrection against, \u2013\n\nSpanish Civil War, \u2013\n\nSpanish Succession, War of, \u2013\n\nSpanish-American War, \u2013\n\nSparta, \u2013, \u2013\n\nSpears, Edward,\n\nSt. L\u00f4, bombing of, \u2013\n\nStalin, Joseph,\n\nStalinism,\n\nStendhal (Marie-Henri Beyle),\n\nStern Gang, ,\n\nStimson, Henry, \u2013, ,\n\nStone, Harlan Fiske,\n\nStraits of Tiran, \u2013\n\nStrasbourg,\n\nstrategic devastation, \u2013\n\nstrategy\n\nincompetence in,\n\nlanguage of, \u2013\n\nmorality and, \u2013\n\nstrict liability, \u2013\n\nSuarez, Francisco, xxvi\n\nsubmarines, \u2013,\n\nSuez War, \u2013\n\nsuperior orders, \u2013\n\nSupreme Court, U.S., \u2013\n\nsupreme emergency,\n\nChurchill on, \u2013, ,\n\ndefined, \u2013\n\nmoral responsibility in, \u2013\n\nof Nazism, \u2013, \u2013\n\nnecessity and, \u2013\n\nneutrality and, \u2013\n\nnuclear deterrence and, \u2013\n\nutilitarianism and,\n\nSee also Hiroshima\n\nsurrender,\n\nof individual soldiers, , \u2013, , , \u2013\n\nnational, \u2013\n\nquarter in, , , , \u2013\n\nin sieges, \u2013\n\nin war convention, \u2013\n\nSee also unconditional surrender\n\nSweden, , \u2013\n\nSwift, Jonathan,\n\nSwiss,\n\nTalmud, law of sieges,\n\nTausend, Captain Helmut, \u2013\n\nTaylor, Telford, ,\n\nterritorial integrity, \u2013\n\nterror bombing, \u2013, , \u2013\n\nterrorism, xvii\n\nassassination compared to, \u2013\n\nin battle of Algiers, \u2013\n\ncode of honor and, \u2013\n\ninnocence and, , , \u2013,\n\nmilitary necessity, claim of, \u2013, , \u2013\n\nnoncombatant immunity and, \u2013, \u2013\n\nrandomness of, \u2013, ,\n\nunconditional surrender and,\n\nThirty Years War, ,\n\nthreats\n\naggression and, \u2013, , ,\n\nimmoral, \u2013\n\nin preventive war, \u2013\n\nbefore Six Day War, \u2013\n\nultimate, \u2013\n\nThucydides, \u2013\n\nTilset, battle of,\n\nTitus, emperor of Rome,\n\nTokyo, firebombing of, , \u2013,\n\nTolstoy, Leo, , , ,\n\ntraitors, \u2013,\n\ntrench warfare, , , , ,\n\nTrevelyan, Raleigh,\n\nTrotsky, Leon,\n\nTruman, Harry,\n\nHiroshima decision by, , \u2013\n\nKorean War and, \u2013\n\nTucker, Robert W.,\n\nTurkey,\n\nRusso-Turkish War,\n\ntyranny of war, \u2013, \u2013\n\nUN. See United Nations\n\nunconditional surrender\n\nappropriateness of, \u2013\n\nChurchill and,\n\nJapan and, \u2013\n\nNazism and, \u2013\n\nas punishment, \u2013\n\nrights in, \u2013\n\nterrorism and,\n\nWilson, Woodrow, and, \u2013\n\nuniforms, \u2013, \u2013\n\nUnited Nations (UN), xxiv\u2013xxv, , \u2013\n\nIndia's intervention in East Pakistan and, \u2013\n\nUnited States (U.S.)\n\nCivil War of, \u2013, ,\n\nCuban Insurrection and, \u2013\n\nKorean War and, \u2013\n\nSpanish Civil War and, \u2013\n\nSpanish-American War, \u2013\n\nVietnam War and, \u2013\n\nYamashita and, \u2013\n\nutilitarianism\n\natomic bomb and, , \u2013\n\nof extremity, \u2013, \u2013\n\npreventive war and, ,\n\nrape, utilitarian defense of, \u2013\n\nreprisal and, \u2013\n\nSidgwick on, \u2013\n\nsliding scale of, \u2013\n\nin strict liability,\n\nsupreme emergency and,\n\nterror bombing and,\n\nin war convention, \u2013\n\nVann, Gerald, \u2013\n\nVattel, Emmerich de, \u2013\n\nVemork raid, \u2013\n\nvictory. See winning\n\nVietnam, \u2013\n\nVietnam War, xxiii\u2013xxiv, , ,\n\ncounter-intervention in,\n\nforced resettlement in, \u2013\n\nMy Lai massacre, \u2013, \u2013\n\nnoncombatant warning in, \u2013\n\nNorth Vietnam insurgents in, \u2013\n\nresponsibility for, \u2013\n\n\"rules of engagement\" in, \u2013\n\nU.S. intervention and, \u2013\n\nvigilante justice, \u2013\n\nVitoria, Francisco de, , , ,\n\nwar\n\ncivil, \u2013, , , \u2013,\n\ncoercion of, \u2013,\n\ncollectivism of, \u2013\n\ndefinition of,\n\nguerrilla, \u2013, , , \u2013\n\nimperialist, ,\n\nlimited nuclear, \u2013\n\nlogic of, \u2013\n\n\"people's,\" , \u2013,\n\npreventive, \u2013\n\nreasons for, , \u2013, , \u2013,\n\nrevolutionary, \u2013, \u2013, ,\n\nas state right,\n\nuncertainties of, \u2013\n\nuniqueness of, \u2013\n\nvalues in, \u2013\n\nwar convention, \u2013, ,\n\nargument against, \u2013\n\natomic bomb and, \u2013\n\ncynicism about, \u2013, \u2013\n\ndefinition of,\n\ndouble effect in, xix, \u2013\n\nguerrilla war and, , \u2013,\n\ninsurgents in, xiv\u2013xv, xviii\n\ninternational law and,\n\njustifications in, \u2013\n\nlimited nuclear war and, \u2013,\n\non naval warfare, \u2013\n\nnecessity and, \u2013, \u2013\n\nnoncombatant immunity in, xiv\u2013xxi, , \u2013, \u2013, \u2013\n\nin nonviolent defense,\n\nnuclear deterrence and,\n\noverriding of, \u2013\n\nprohibitions in, \u2013\n\nproportionality rule in, xv, \u2013\n\nreprisals and, , , \u2013, ,\n\nsurrender in, \u2013\n\ntension with theory of aggression, \u2013\n\nutilitarianism in, \u2013\n\nSee also rules of war\n\nwar crimes, , \u2013\n\nNuremberg Trials, , , \u2013, , \u2013,\n\n\"war is hell\" doctrine, \u2013,\n\nWar of 1812, \u2013\n\nwar rebellion,\n\nwar treason,\n\nwarfare\n\nasymmetric, xiii\u2013xxii\n\ncounter-city, , \u2013\n\ncounter-force, \u2013\n\nlimited, \u2013, , , , ,\n\nsiege, \u2013,\n\nsubmarine, \u2013,\n\ntrench, , , , , \u2013, , \u2013\n\nSee also naval warfare\n\nWaterloo, battle of, , \u2013\n\nWebster, Daniel, \u2013\n\nWeil, Simone, ,\n\nWeizsaecker, Ernst von, \u2013\n\nWestlake, John,\n\nWeston, John,\n\nWeyler y Nicolau, General Valeriano,\n\nWilson, Edmund,\n\nWilson, Woodrow, \u2013, ,\n\nwinning, xx, \u2013,\n\nethics against, \u2013\n\nin guerrilla war, \u2013\n\nof Korean War, \u2013\n\nmeaning of,\n\nrealists on, \u2013\n\nsacrifices for, \u2013\n\nSee also settlements, justice in\n\nwomen, \u2013\n\nas noncombatants,\n\nXerxes,\n\nYamashita, General Tomoyuki, , \u2013\n\nYugoslav partisans,\n\nZagonara, battle of, \n\n## Contents\n\n  1. Contents\n  2. Preface to the Fifth Edition\n  3. Preface to the First Edition\n  4. Acknowledgments\n  5. Part One: The Moral Reality of War\n  6. 1 Against \"Realism\"\n  7. The Realist Argument\n  8. The Melian Dialogue\n  9. Strategy and Morality\n  10. Historical Relativism\n  11. Three Accounts of Agincourt\n  12. 2 The Crime of War\n  13. The Logic of War\n  14. The Argument of Karl von Clausewitz\n  15. The Limit of Consent\n  16. The Tyranny of War\n  17. General Sherman and the Burning of Atlanta\n  18. 3 The Rules of War\n  19. The Case of Hitler's Generals\n  20. Two Sorts of Rules\n  21. The War Convention\n  22. The Example of Surrender\n  23. Part Two: The Theory of Aggression\n  24. 4 Law and Order in International Society\n  25. The Rights of Political Communities\n  26. The Case of Alsace-Lorraine\n  27. The Legalist Paradigm\n  28. Unavoidable Categories\n  29. Karl Marx and the Franco-Prussian War\n  30. The Argument for Appeasement\n  31. Czechoslovakia and the Munich Principle\n  32. Finland\n  33. 5 Anticipations\n  34. Preventive War and the Balance of Power\n  35. The War of the Spanish Succession\n  36. Pre-emptive Strikes\n  37. The Six Day War\n  38. 6 Interventions\n  39. Self-Determination and Self-Help\n  40. Secession\n  41. Civil War\n  42. The American War in Vietnam\n  43. Humanitarian Intervention\n  44. Cuba, 1898, and Bangladesh, 1971\n  45. 7 War's Ends, and the Importance of Winning\n  46. Unconditional Surrender\n  47. Justice in Settlements\n  48. The Korean War\n  49. Part Three: The War Convention\n  50. 8 War's Means and the Importance of Fighting Well\n  51. Utility and Proportionality\n  52. Human Rights\n  53. 9 Noncombatant Immunity\n  54. Naked Soldiers\n  55. The Nature of Necessity (1)\n  56. Submarine Warfare: The Laconia Affair\n  57. Double Effect\n  58. Bombardment in Korea\n  59. The Bombing of Occupied France and the Vemork Raid\n  60. 10 War Against Civilians: Sieges and Blockades\n  61. Coercion and Responsibility\n  62. The Right to Leave\n  63. Taking Aim and the Doctrine of Double Effect\n  64. The British Blockade of Germany\n  65. 11 Guerrilla War\n  66. The Rights of Guerrilla Fighters\n  67. The Rights of Civilian Supporters\n  68. The American \"Rules of Engagement\" in Vietnam\n  69. 12 Terrorism\n  70. The Russian Populists, the IRA, and the Stern Gang\n  71. The Vietcong Assassination Campaign\n  72. Violence and Liberation\n  73. 13 Reprisals\n  74. The FFI Prisoners at Annecy\n  75. The Problem of Peacetime Reprisals\n  76. The Attack on Khibye and the Beirut Raid\n  77. Part Four: Dilemmas of War\n  78. 14 Winning and Fighting Well\n  79. The Sliding Scale and the Argument from Extremity\n  80. 15 Aggression and Neutrality\n  81. The Right to Be Neutral\n  82. The Nature of Necessity (2)\n  83. The Rape of Belgium\n  84. The Sliding Scale\n  85. 16 Supreme Emergency\n  86. Overriding the Rules of War\n  87. The Limits of Calculation\n  88. 17 Nuclear Deterrence\n  89. Limited Nuclear War\n  90. The Argument of Paul Ramsey\n  91. Part Five: The Question of Responsibility\n  92. 18 The Crime of Aggression: Political Leaders and Citizens\n  93. The World of Officials\n  94. Nuremberg: \"The Ministries Case\"\n  95. Democratic Responsibilities\n  96. The American People and the War in Vietnam\n  97. 19 War Crimes: Soldiers and Their Officers\n  98. In the Heat of Battle\n  99. Superior Orders\n  100. Command Responsibility\n  101. General Bradley and the Bombing of St. L\u00f4\n  102. The Case of General Yamashita\n  103. The Nature of Necessity (4)\n  104. The Dishonoring of Arthur Harris\n  105. Conclusion\n  106. Afterword: Nonviolence and the Theory of War\n  107. Postscript: A Defense of Just War Theory\n  108. Notes\n  109. Index\n\n## Landmarks\n\n  1. Cover\n  2. Table of Contents\n  3. Preface\n  4. Start of Content\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n**Making Your Small Farm Profitable**\n\n# Making Your Small Farm Profitable\n\nRon Macher\n\nForeword by Howard W. \"Bud\" Kerr, Jr.\n\nThe mission of Storey Publishing is to serve our customers by publishing practical information that encourages personal independence in harmony with the environment.\n\nEdited by Deborah Burns and Marie Salter\n\nCover design by Meredith Maker\n\nCover photograph \u00a9 by Larry Lefever\/Grant Heilman Photography, Inc.\n\nText design and production by Erin Lincourt\n\nPhotographs by Larry Lefever\/Grant Heilman Photography, Inc., except those on page 232 by Nick DeCandia; 98 and 150 by Jane Grushow\/Grant Heilman Photography, Inc.; 50, 74, 184 by Grant Heilman\/Grant Heilman Photography, Inc.; 42 by Joel Sartore\/Grant Heilman Photography, Inc.; 16 by Kristi Ann Gilman-Miller.\n\nLine drawings by Chuck Galey, except those on pages 12, 56, 82, 161 (bottom) by Cathy Baker; 161 (top) by Brigita Fuhrmann; 161 (middle) by Millie Holderread; 54 and 55 by Alison Kolesar; 31, 39, 78, 83, 95, 125, 130, 136, 145, 148, and 187 by Elayne Sears; 162 by Becky Turner.\n\nIndexed by Nan Badgett\/Word\u2022a\u2022bil\u2022i\u2022ty\n\n\u00a9 **1999 by Margaret Radcliffe**\n\nAll rights reserved. No part of this book may be reproduced without written permission from the publisher, except by a reviewer who may quote brief passages or reproduce illustrations in a review with appropriate credits; nor may any part of this book be reproduced, stored in a retrieval system, or transmitted in any form or by any means\u2014electronic, mechanical, photocopying, recording, or other\u2014without written permission from the publisher.\n\nThe information in this book is true and complete to the best of our knowledge. All recommendations are made without guarantee on the part of the author or Storey Publishing. The author and publisher disclaim any liability in connection with the use of this information. For additional information please contact Storey Publishing, 210 MASS MoCA Way North Adams, MA 01247.\n\nStorey books are available for special premium and promotional uses and for customized editions. For further information, please call 1-800-793-9396.\n\nPrinted in United States by Versa Press  \n20 19 18 17 16 15 14\n\n**Library of Congress Cataloging-in-Publication Data**\n\nMacher, Ron.\n\nMaking your small farm profitable \/ Ron Macher.\n\np. cm.\n\nIncludes bibliographical references and index  \nISBN 978-1-58017-161-8  \n1. Farm management. 2. New agricultural enterprises. I. Title.\n\nS561. M24 1999\n\n630\u2032.68\u2014dc21 99-16219\n\nCIP\n\n## Dedication\n\nTo my wife, Joanne,  \nand to my children, Jean and Jeff\n\n## Contents\n\nForeword\n\nPreface\n\nAcknowledgments\n\nI. Getting Started\n\n1. Deciding to Farm\n\nWhat Is a Small Farm?\n\nBecoming a Farmer\n\nThe Farming Life\n\nAgripreneurship\n\n2. Starting a Farm\n\nStarting a Farm Plan\n\nEvaluating Your Resources\n\nGetting Help\n\nII. Farming\n\n3. Some Principles of Good Farming\n\n4. A Living, Healthy Soil\n\nThe Physical Nature of Soil\n\nWorking the Soil\n\nOur Living Soil\n\nSoil and Plant Roots\n\nCrop Rotation\n\nCover Crops and Green Manure\n\nNitrogen and Legumes\n\nFeeding Livestock in Rotations\n\n5. Weatherproofing Your Farm\n\nThe Effects of Climate\n\nWater\n\nAltering Your Farm Environment\n\nExtending the Season\n\nCaring for Livestock in Winter\n\nIII. Planning and Marketing\n\n6. Your Goals and Farm Planning\n\nPlanning\n\nSetting Goals\n\nDeveloping Your Farm Plan\n\n7. Marketing\n\nEight Steps to Identifying the Market\n\nNiche Marketing\n\nAll About Niche Markets\n\nAdd-on Value\n\nTwelve Ways to Sell Your Products\n\nPricing Your Product\n\nAdvertising\n\nYour Farm as a Destination\n\n8. Selecting Your Enterprise\n\nTypes of Enterprises\n\nEnterprise Cost Analysis\n\nDiversity\n\nSustainability\n\nIv. management\n\n9. Machinery\n\nAcquiring Machinery\n\nDetermining Equipment Size\n\nVariables to Consider\n\nEconomics of Machinery\n\nBuying Equipment\n\n10. Farm Management\n\nRecipe for Success\n\nKnowledge Is Power\n\nMaking Choices\n\nManagement Tools\n\nManaging Labor\n\nPlanning for Farm Efficiency\n\n11. Where We Are Going\n\nLand and Farms\n\nIndustrialization of Agriculture\n\nFarms and People\n\nSuccess on Sustainable Farms\n\nMy Vision\n\nAppendix A: Metric Conversions Chart\n\nAppendix B: Resource Lists\n\nBooks\n\nBook Sources\n\nPeriodicals\n\nInternet Sites\n\nUniversity Sources\n\nFederal Agencies\n\nMap Sources\n\nResource Lists\n\nSeed and Plant Catalogs\n\nSupplies\n\nAssociations and Organizations\n\nIndex\n\n## Foreword\n\nThe agricultural industry is as old as America.\n\nGenerations before Columbus discovered the New World, natives of the Western Hemisphere grew maize, squash, and root crops. Our ancestors tilled the soil for subsistence and later embraced farming as a vocation. Over the years, because of scientific breakthroughs, new technology, and improved systems, the number of people employed in farming has declined; still, the business of farming remains vital to our well-being as a people and a nation.\n\nMost of America's nearly two million farms are considered \"small,\" with seven out of ten grossing less than $50,000 a year. Despite their preponderance, operators of small farms have often felt neglected by our national farm programs. Sources of advice for farmers starting out have about dried up, with agricultural county agents admitting that they have time to service only full-time farmers\u2014a group whose numbers are declining. The state departments of agriculture have marketing advice aplenty but are of little help to newcomers asking questions about credit, cropping recommendations, and cultural information. Needed are individual human beings whose hands are on the rural pulse and who have lots of information in their heads, but it remains to be seen who will train them or who will pay them. People have been spoiled for more than a century, recipients of free advice from the government, but that advice is gone now and won't be coming back.\n\nA big establishment and greater sales volume do not guarantee a corresponding increase in profits. Likewise, buying land and calling that land a farm cannot ensure that your investment will be profitable. The days of starting out on a few dollars are over, and farming is now complex. People in the business are specialists, and many even have training in business theory. Such is farming in 1999, and so it will be in the years to come.\n\nToday, as well as tomorrow, the most important piece of farm equipment is knowledge. Understanding complex situations or agribusiness production and marketing problems will be paramount to staying in business year after year. In this era of high technology, vacillating consumer wants and tastes, and shifting market conditions, farm managers and agricultural entrepreneurs are seeking help to cope with these and other situations. In the past\u2014indeed, too frequently\u2014people would \"shut the gate after the horse was out of the barn\" and then go look for the horse. Now, the preferred method is to have a plan of action in place before going to the barn\u2014knowledge is the key.\n\nIt starts with you. You must decide what is good for you, your family, and the farm business before purchasing resources or even planting a seed. Anyone can own a farm and call themselves a farmer, but to become profitable you must acquire and apply business skills.\n\n_Making Your Small Farm Profitable_ is your road map for planning a successful journey into the vast and diverse landscape of agriculture in the years ahead. Ron Macher is a veteran farmer, a true friend of mine and yours, and a huge advocate of small farms. Ron has \"been there and done that\" for agriculture for many decades. He writes from the heart for the sole purpose of instructing and guiding novice entrepreneurs, wanna-bes, and tried and true \"dirt under the fingernails\" farm people. Ron, like me, is close to the earth\u2014we are both proven small farm operators. We have been close friends and colleagues championing America's small farm community for many years. (Ron was in the private sector, while I served the public via the U.S. Department of Agriculture.)\n\nIn the early 1980s we witnessed the tragic loss of many family farms. Ron did not stand idly by as this occurred; rather, he launched a new publication aimed at Midwest farm families. The fledgling farm magazine _Missouri Farm_ grew in popularity, and later the magazine's name was changed to _Small Farm Today_ \u2014now the industry bellwether.\n\nOver the years, with the advent of new crops and technology, Ron found that his working hours were increasingly devoted to being an executive, not a farmer. This role brought a new kind of obligation, and Ron soon felt the kind of responsibility he has today; he got a glimpse of the complicated future he and others would face in industrialized agriculture. Already, he was greatly disturbed by the failure of government, both federal and state, to provide the information that small farm operators required. He took it upon himself to meet that need and began editing and publishing his magazine, now in circulation for more than 15 years.\n\n_Making Your Small Farm Profitable_ is the outgrowth of Ron's vast experience in farming and his knowledge of journalism. Over the years, he learned how to share this experience and delegate the details of his business to others. Written for people who want to be profitable regardless of their farm status, novice or veteran, full-time or part-time, this book is your guide from the earliest stages of farming, when all things and topics are nebulous, to a genuine high point in farming\u2014when you make a bank deposit, confirming that your small farm _is_ profitable.\n\nHoward W. \"Bud\" Kerr, Jr.  \nFormer Director, Office of Small-Scale Agriculture,  \nU.S. Department of Agriculture\n\n## Preface\n\nWhen you are disking a field to plant corn, the sun is shining, and the earth smells fresh, you are probably not thinking about whether that process will make you money. Farmers invariably find production agriculture more fun than the business side of farming, which involves heady subjects like marketing, sales, and cost evaluations. Still, to become a successful agripreneur, you'll need to learn to enjoy the business side of farming as well. It is my sincere hope that this book will help beginning and established farmers make their farms profitable and their livelihoods satisfying, moving us forward to a more consistently sustainable agriculture and ultimately fostering sustainable communities.\n\n### What Is Agripreneurship?\n\nAgripreneurship is the profitable marriage of agriculture and entrepreneurship\u2014more plainly, turning your farm into a business. Most farmers regard agriculture as a combination of philosophy and lifestyle, so in its broadest sense agripreneurship binds together philosophy, lifestyle, and business, yielding ideals that give you purpose and goals to strive for. Agripreneurship is a mental attitude that can give you the strength and motivation to break from tradition.\n\n### About Sustainable Agriculture\n\nSustainable agriculture is a site-specific, whole-farm approach to agriculture. Land, people, goals, capital, crops, and livestock are managed to yield the best possible results on the farm. Not tied to any one model of agriculture, sustainable agriculture strives to reduce costs and increase the efficiency of the family farm.\n\nBy reducing inputs, sustainable agriculture encourages conservation and multiple uses of resources. It promotes diversity, using multiple species and natural methods to recycle matter and nutrients to maintain the land's productivity, now and in the future. Sustainable agriculture encourages local food production, providing food to society at a reasonable cost while supporting the farmer with an acceptable level of income. Finally, sustainable agriculture fosters a diverse and sustainable farming community as well as a sustainable society, drawing together farmers, lenders, consumers, and institutions in cooperative partnerships.\n\n### A Sustainable Community\n\nWhen considered strictly from an agricultural standpoint, we quickly recognize that society comprises four major groups: farmers, lenders, consumers, and institutions.\n\n  * Farmers are those who provide food and fiber to our nation. Their farms provide a reserve of natural habitat for wildlife, supply oxygen, and act as filters for our watersheds. Farmers instill in their children a strong work ethic and a good set of values.\n\n  * Lenders are important suppliers of start-up capital for companies of all sorts. In agriculture, lenders supply monies to agripreneurs and processing facilities, allowing farmers to sell their products directly at retail prices.\n\n  * Consumers are those who buy the products that individual farmers and institutions produce. Consumers provide farmers with the income they need to continue their operations.\n\n  * Institutions include universities, government agencies, and businesses. Institutions create consumer goods, add value to products, and perform research.\n\nThese four groups are the cornerstones of our society, the community of America. To be a sustainable community, the community requires sustainable agriculture. Respect and cooperation among these groups promotes just that.\n\n### Why Do You Need This Book?\n\n_Making Your Small Farm Profitable_ provides a blueprint for farmers. In one volume, I try to tie together the whole picture of farming, show new farmers what they can do and how to choose where to begin, and share my keys to success. I have been a farmer for 34 years and have made a lot of mistakes in that time; some people call this \"experience.\" My fifteen years as a publisher have exposed me to thousands of farm failures and farm success stories. This book should keep you from making some of the mistakes I have made and seen others make\u2014mistakes that can potentially cost thousands of dollars. All of this useful knowledge will be yours with a modest investment of reading time and by practicing the thinking process detailed here.\n\n### About This Book\n\nBack in the first rural crisis in 1984, I started a magazine for small farmers (now _Small Farm Today_ ), because information they needed was not readily available. Recently, I decided to write a book about farming, because the information for new farmers and farmers trying to survive in this age of industrial agriculture is not readily available. There are many good books about individual species of livestock, many useful gardening books, many great books about the reasons for and philosophy of farming, but few books that tie all these aspects together or that help the reader choose where to start.\n\nThis book takes a how-to approach\u2014it does not specifically detail how to raise, say, cattle or pumpkins, but rather how to farm successfully and profitably. It is a whole-farm planning approach that ties together outside and on-farm resources with your personal, family, and farm goals. It's about the principles that make your thinking process work like a well-oiled machine.\n\nThroughout the book, you'll find called out in the text pearls of wisdom, guiding principles that, if followed, will help make your farming life more manageable and productive. We begin with a discussion of the basics, information and techniques that will assist you to get started or that will help improve your existing operation. Many of these are things I wish I had known years ago when I was starting out.\n\nLater chapters discuss principles of good farming, resources (location, soil, water, and climate among them), and basic farming methods to help you determine what and how you want to farm.\n\nThen we help you refine your goals, following that up with a discussion of marketing, enterprise choices, machinery, and management. In the final chapters, we tie it all together and send you out as an agripreneur.\n\nYou _can_ make money on your farm. You _can_ enjoy it. Read on to find out how.\n\n## **Acknowledgments**\n\nMy special thanks to my wife, Joanne, for the use of half her kitchen table for a year, and for her tolerance of the many piles of books and papers that covered it. I also thank Deborah Burns and Marie Salter of Storey Books, for their good ideas and professionalism. To the readership of _Small Farm Today_ magazine\u2014you asked for a book on making a small farm pay\u2014here it is.\n\nFinally, I am grateful for the help and assistance of Paul Berg, editor at _Small Farm Today_. Writers draw strength and inspiration from countless sources, and many people influenced what I have written. In particular, Paul was the prodder who helped me create the most useful book I could. His editing skills and suggestions enabled me to say what I wanted in the best possible manner. My thanks to Paul for all the hours and effort he expended in helping me get this book completed.\n\n###### Part i\n\n## Getting Started\n\n###### Chapter One\n\n## Deciding to Farm\n\nWhy do people farm?\n\nFarming can be rewarding. There is nothing like the feeling that comes from walking through a still-foggy field in the morning to find newborn triplets from your favorite ewe, or seeing a foot-long ear on your latest strain of open-pollinated corn, or just looking across your land at sunset and saying, \"This belongs to me!\"\n\nFarming can be frustrating. Why did that chain choose to break on your planter right in the middle of the farthest field at 3 P.M., forcing you to have to go to town to get another, only for it to be dark when you got back? Why did it rain for three straight weeks while you were trying to plant your vegetables, not raining a drop since? A man once told me, \"There's no way in hell I'd have a job where my income depended on whether or not it rained!\"\n\nFarming can be tough. The number of farmers in the United States has shrunk from 6.5 million in the 1930s to only 2 million in the 1990s. Less than 12 percent of all farmers make a sustainable living wage from the farm, and we are losing almost one hundred farms per day. Agricultural experts (economists in particular) have been predicting the demise of small farms for the past 50 years. \"Look before you leap\" certainly seems like good advice if you are thinking about becoming a farmer.\n\n_Today's small farms range from quarter-acre specialty herb farms to many-acred traditional and alternative crop and livestock farms that sell value-added products directly to the consumer. Can you visualize your farm_?\n\nThe total number of farms in the United States has declined from a high approaching 70,000 in the 1930s to 20,000 today.\n\nSo why do people farm?\n\nThe reasons that people begin to farm are as varied as the snowflakes that are just now beginning to fall on my newly budding trees. Some farmers are sons and daughters following in their parents' footsteps. Some have fond memories of spending summers on the farm of a grandfather or uncle. Some are influenced (as I was) by an elderly neighbor talking about his life on the farm.\n\nThe Rural Sociology Department at the University of Missouri did a study in the 1950s, asking why people chose to farm. Large and small farmers alike were questioned. Here were the first three reasons:\n\n  1. \"I like to work outdoors.\"\n\n  2. \"It's a good place to raise my children.\"\n\n  3. \"I'll always have a place to live and food to eat.\" (Not necessarily true, if you have a mortgage.)\n\nFourteenth of the fifteen reasons listed was \"It provides a good income,\" and fifteenth was \"I don't know how to do anything else.\" Obviously, lifestyle was a big reason that people farmed, if income was so far down on the list.\n\nAlmost 40 years after the University of Missouri study, the DuPont Company hired Doanes Agricultural Service to do a study on why farmers farm and what farming practices they used, among other such questions.\n\nIn that study, conducted in 1996, the first three answers were still lifestyle choices, but \"good income\" moved up to seventh place. Income seems to be more important to people now than it was four decades ago, but lifestyle is still the principal reason someone farms.\n\nLater in this chapter, I will explain the importance of farm income for small farmers, and why it should be your number one goal; the reason will probably surprise you. But even though you need a good income to be a farmer, that is probably not _why_ you want to be a farmer.\n\nThis is also true of me. The reasons I farm all involve the independent lifestyle, self-reliance, being in harmony with nature, and the feelings of goodness that come from my land. Self-reliance and a love of the outdoors are an ingrained part of our American heritage, and farming is a way to celebrate that heritage. Henry Ford said it best: \"The further removed we are from the land, the greater our insecurity.\"\n\n### What Is a Small Farm?\n\nIf you are planning to farm, you are almost certainly going to start with a small farm. You will be in good company\u201479 percent of all the farms in the United States are small farms, according to the 1997 Census of Agriculture. There is also good news for small farmers\u2014while the total number of farms has decreased, the number of small farms is actually increasing at a rate of about 2 percent per year, and this trend is predicted to continue for at least the next 10 to 20 years.\n\nHow much land do you have to have to be a small farm? The answer is up to you. I define a small farm as any farm that has 179 acres or fewer, or that has a gross income of $50,000 per year or less. This seems to be a reasonable compromise between acreage and income\u2014but not everyone agrees. As you read this book, don't get hung up on numbers and definitions. There are hundreds of government bureaucrats and university professors making $100,000 a year arguing about what's a large farm and what's a small farm. To make the matter worse, many people confuse small farms, family farms, and sustainable agriculture. The U.S. Department of Agriculture has no official definition of farm sizes. Meanwhile, according to the National Small Farm Commission, a small farm is any farm grossing $250,000 per year or less. This is 94 percent of all the farms in the United States, and includes what I would call midsize farms.\n\n# Terms\n\nThe confusion between small farms, family farms, and sustainable agriculture is somewhat understandable, because all three share similar characteristics.\n\n  * A _small farm_ is any farm that comprises 179 acres or less, or that grosses $50,000 or less per year. Small farms are usually family farms but may or may not be sustainable.\n\n  * A _family farm_ is any size farm\u2014small, medium, or large\u2014in which family members supply the majority of needed farm labor. A family farm is not necessarily sustainable.\n\n  * _Sustainable agriculture_ is an economically viable, environmentally sound, and socially acceptable system of agriculture that may be used on any size farm.\n\nThere even seems to be some argument as to what constitutes a farmer. The first definition of a farmer in Webster's _New_ _World Dictionary_ is \"a person who earns his living by farming; especially, one who manages or operates a farm.\" While Webster was content to define a farm and a farmer using only words, the rest of the world seems to prefer definitions involving numbers. The Usda has changed the definition of a farm nine times. In the 1950s and 1960s, a farm was any place smaller than 10 acres but selling $250 or more of agricultural products per year, or any place of 10 acres or more from which $50 of agricultural products were sold per year.\n\nCurrently, the Usda defines you as a farmer if you sell $1,000 worth of agricultural products per year. In 1996, the U.S. Bureau of the Census proposed changing that figure to $10,000 (presumably to save itself some counting). There was an outcry from universities, because the sizes of their federally funded programs are determined by the number of farmers living in their state; in the end, the $1,000 figure has prevailed.\n\nMy own definition of a farmer is quite simple: If you think you're a farmer, you are a farmer. Any size acreage, from a small garden to 3,000 acres, where a person or family tries to make a living (or part of a living) from the land is a farm. A small farm is any farm that is small, regardless of cash inflow or outgo. The \"Biggest Little Farm in America\" grosses $238,000 from \u00bd acre of gourmet vegetables. I know several owners of 80-acre farms who gross $50,000 to $60,000 per year, with a final (net) profit of $25,000 to $30,000. On research plots on my 80-acre farm, with a combination of vegetables and livestock, we have grossed $3.00 per square foot, and netted $1.32 per square foot. If we expanded these practices to a full acre (43,560 square feet), this would yield a gross profit of $130,680 and a net profit of $57,500.\n\nThe definitions of farms and farmers probably always will be contested by a variety of university, corporate, government, and individual entities. If we are to progress as farmers, these definitions are not necessarily important. We must simply know who we are, what we are about, and where we are going. Agricultural problems are solved by solutions, not definitions.\n\n# How The IRS Defines a Farmer\n\nThe U.S. Internal Revenue Service (Irs) has its own definition of a farmer. Because farmers, unlike hobbyists, are allowed tax reductions, the Irs defines them rather strictly. The short explanation (available in \"Farmer's Tax Guide 225\") is that you may file as a farmer by March 1 if at least two-thirds of your gross income is from farming. Further, your farming activities are presumed not to be a hobby if profits result in any three of five consecutive tax years, ending with the tax year in question\u2014with exceptions for raising horses. Obviously, the Irs is not interested in unsuccessful farmers.\n\nA more detailed explanation is available in publication N-1000, \"Farmers for Tax Purposes\" (italics are mine):\n\nWho is a farmer: _All individuals, partnerships, or corporations that cultivate, operate, or manage farms for gain or profit, either as owners or tenants, are regarded as farmers_. To be classified as a farmer, a taxpayer must meet the two-fold test of participation to a significant degree in the growing process and assumption of a substantial risk of loss from that process. Thus, a taxpayer who gets rent based on farm production is a farmer. But if the rent is fixed, he isn't a farmer unless he materially participates in the farm's operation or management. A taxpayer is engaged in the business of farming if he belongs to a partnership so engaged.... _Farmers include persons engaged in oyster farming, the raising of bees, breeding and raising chinchillas, mink, foxes, and other furbearing animals_. Feedlot operators have been held to be farmers with respect to stock they own, as well as with respect to stock handled for customers. _But a taxpayer engaged in forestry or the raising of timber is not engaged in farming. Neither is a taxpayer who sells Christmas trees grown, without planting or cultivation, on land he owns_.\n\nWell, that really clears things up! This is why I call myself an agripreneur, instead of a farmer.\n\n### Becoming a Farmer\n\nFarming, in some respects, is easy; making a living from a farm is hard. In addition to being a farmer, I want to help you become an agripreneur.\n\n_Agripreneur_ is a term I coined in 1987 to describe the readership of my magazine, _Small Farm Today_. An entrepreneur, according to Webster's dictionary, is someone who runs a business at his\/her own financial risk\u2014a middleman. An agripreneur is someone who runs an agricultural business\u2014farming in particular\u2014at his or her own risk.\n\nI will discuss this in more detail in later chapters, but for now, ask yourself some simple questions:\n\n  * \"How much money do I need in order to live comfortably and support my family? $15,000? $20,000? $50,000?\"\n\n  * \"How long will it take to achieve this level of income, and will I be happy when I achieve it?\"\n\n  * \"Do I want to farm part time? Full time? Start part time and grow from there?\"\n\n  * \"How much money do I already have in order to start farming, and how much do I wish to borrow?\"\n\n  * \"What skills and resources do I have? Am I good with livestock, or can I learn to be? Am I good with machinery, or maybe carpentry\u2014to build hog houses or poultry houses? What skills do I have that I really enjoy? What skills am I weak in or do I really dislike?\"\n\n# Self \u2013 Evaluation\n\nBecoming an agripreneur involves changing the way you think about farming. When you decide to start a farm, you need to ask how farming will satisfy both your monetary needs and your personal goals, now and in the future. This is a question that you will want to reevaluate frequently as you continue in farming.\n\nThese are not questions you will know all the answers to now, but you need to keep them in mind as we proceed.\n\nDo not be critical of your skills at this point. There are many people entering farming today with no farming background. These people have good business and computer skills\u2014which can be a farming asset\u2014but don't have any of the basic day-today knowledge to run a farm, from milking a cow to planting corn. If you are one of these people, don't worry; you can learn.\n\n#### Full-Time Farming: Pros and Cons\n\nIf you want to start out as a full-time farmer and have no experience, you have chosen a difficult road; you will have to do all your learning while you are striving to make enough money to satisfy your personal goals. It is even harder if you took out a mortgage and a big loan; on a full-time farm, your business, your home, and your lifestyle are all tied together in one big package.\n\nOn the other hand, in full-time farming you are your own boss and set your own hours. You are the Ceo of planning, developing, and determining your destiny. It is highly rewarding to lay out a plan and guide it to completion, correcting your failures and overcoming obstacles. No amount of money can give you the feeling you get in the springtime from watching the birth of a scruffy bright-eyed calf or a litter of tiny piglets, as your farm renews and repopulates itself in its never-ending cycle of life.\n\n#### Part-Time Farming: Pros and Cons\n\nPart-time farming allows you to have your cake and eat it too. Combining farming with a town job or seasonal off-farm labor can be difficult to manage, juggling the demands of both workplaces, but you'll have more financial safety. It is certainly a safer transition for new farmers; their farming mistakes will not have the capability to cost them their entire income. An off-farm job provides the financial security and the borrowing power of a steady paycheck, and reduces the risks involved in a weather-dependent business. You can experience the rewards of farming and learn as you go without the stress of a \"root-hog-or-die\" existence.\n\n#### Gaining Knowledge\n\nI was lucky enough to have some knowledge when I started. I grew up in a rural area, although my father did not farm. I worked on several farms nearby and learned skills as varied as driving a team of mules to harvest a potato crop, raking and baling hay, and milking a ten-cow herd by hand for grade C milk. If you are new to farming and you know any farmers, talk to them and do chores with them whenever you get the chance.\n\nPeople who grew up on farms do not always discuss their finances in depth, nor do they necessarily know why they do things the way they do. They may simply be carrying out the instructions they learned about _how_ to do it, without understanding the principles of _why_ they do it. To be a successful agripreneur, you must understand your operations completely, from the how to the why.\n\n### The Farming Life\n\nPrinciple: _Farming goals must be family goals. To move to a farm, the entire family needs to be prepared for what life will be like_.\n\nAlthough life on the farm is much easier than it was in the past\u2014due to the advent of electricity, county water, and the automobile\u2014a farm is still a rural environment, and when it gets dark, there are no streetlights. Dust and mud are \"up close and personal\" every day and every season. Dirt and gravel roads are harder to manage than concrete, and no city crew comes out to clean snow off your street. You will probably be far away from medical help, a fire department, and the police. Your children will have fewer playmates, and visiting the neighbors may require hopping into a car.\n\nOn a farm, you'll have to deal with nature. Nature is cyclical, slow, and unpredictable. After a beef cow is bred, she is pregnant for 9 months, and then you must wait another 6 months before you can sell her feeder calf to get back any income. Many field crops take 6 months to a year from planting to harvest to income. The weather can act up at any time, drowning your crops or freezing your livestock.\n\nIf you are new to farming, all of this will require some adjustment. Fortunately, there are genuine advantages to a rural life. People in a rural environment learn to become more independent\u2014to rely on their own inner strength. Family members who work together and share will learn to depend on each other much more than they would in an urban environment, where everyone is going in different directions.\n\nThe most important thing that agriculture has furnished this country with is not food or fiber, but, rather, a set of children with a work ethic and a good set of values. Doing farm chores that must be done\u2014it is not acceptable to say, \"Oh, I'll feed the goats next Saturday\"\u2014gives children a basis for a work ethic that will continue throughout their lives. These values carry over into the rural community, creating a quality of life that we all want\u2014and need.\n\nChildren who grow up on a farm learn responsibility and reliability.\n\nOne of the biggest differences between a town business and a farm is that at the end of the day, you leave a town business and go home. With a farm, your business, your home, your family, and your lifestyle are all linked. Dinner-table discussions on a farm usually include a lot more \"business talk\" than do dinner-table discussions in town. And when your home and farm are at the same location and your mortgage entangles both, if your farm is not profitable, you can lose both your home and your business in one fell swoop.\n\nOn the bright side, watching the sun set after a hard day's work is much more satisfying from your porch than it is from rush-hour traffic. And to start the day's work, all you have to do is step outside instead of stuffing your briefcase and jumping into your car.\n\nNone of the obstacles is hard to overcome. The important thing is for the entire family to be prepared for a lifestyle change. Husband, wife, and children must know why they are making this choice and what part they will contribute to the whole.\n\n### Agripreneurship\n\n_Sustainable farming_ is the current catchphrase in agricultural circles. Sustainable farming simply means farming that sustains itself\u2014a continuous cycle that does not wear out the land or the farmer, replenishes the livestock and crops, and enables the family to continue farming. There are many elements to sustainability, such as diversity and \"low input\"\u2014but the most important principle of sustainability is to be profitable. Memorize this sentence: \"To be sustainable, it must be profitable.\"\n\nThis is important for the survival of your farm and is part of my definition of agripreneurship. Apripreneurs must have a positive attitude and practice a sustainable type of farming that satisfies both personal and family goals. Because agripreneurship involves sustainability, and sustainability requires profitability, we will make this the guiding principle of agripreneurship.\n\n#### A Profitable Farm\n\nPrinciple: _Your farm must be profitable_.\n\nMany people believe the goal of being profitable is to get wads of money so they can spend it on what they want. These people pursue high-paying jobs and put in endless hours of work to get rich, so they can be happy.\n\nThe reason to make your farm profitable is simple: As long as you take in more money than you spend, your farm can improve in fertility and has a potential for sustainability. A farm can generate large numbers of bills\u2014your goal is to be able to pay them.\n\nIf your farm has to be subsidized with off-farm income, it is not sustainable in the long run. Eventually you'll deplete your savings, or you will retire from your job. Even part-time farms should be sustainable. Make your farm pay for itself.\n\nIn today's small-farm market, your only real financial security lies in your ability to sell yourself and your products. You will be responsible for your own success or failure. In conventional agriculture, a farmer simply raises the crop and then hauls it to the middleman to sell. The middleman passes on the crop to the retailer and eventually to the consumer. To be an agripreneur, you must be your own middleman, selling your product directly to the consumer.\n\n#### The Big Picture\n\nPrinciple: _You must look at the whole picture_.\n\nAgripreneurship involves examining your operation as a whole. Its parts should support each other, and you need to be aware of how efficiently each part is succeeding\u2014or failing.\n\nIn a factory job, workers simply screw part A into part B and never see the whole. In most office jobs, each worker gets a piece of a project, or a single account out of several. An agripreneur must be aware of each piece, each account, and how they are linked. Your farm is the entire factory, the whole company, and you need to know how all of it works.\n\nTo see the whole picture, you must look beyond your farm, too. Is your product something you can sell in your area? Are there \"new\" products that would be appreciated in the local community? How does what you raise affect neighboring farms? How does what they raise affect you? What will be the overall effects in the local region?\n\n#### Planning\n\nPrinciple: _Think. Think. Think_.\n\nAgripreneurship is the \"thinking person's agriculture.\" You must plan carefully, for both your farm and your family. Your plan should include time for rest and recreation, because this renews your spirit and gives you time to think about your farm and learn what others do. Farming is physical by nature, but it is not how hard you work, but how smart you work. The difference between a conventional farmer and an agripreneur is that the agripreneur does more thinking and less doing. If you wear yourself out physically and mentally, you will accomplish nothing. Learn to enjoy the trip, as well as the destination.\n\nIn agripreneurship, your reasoning process, backed by reading and research, enables you to make decisions that affect or contribute to all facets of your farm and your personal and family goals. A thinking person evaluates all of his or her options and tries to keep an open mind. My favorite maxim is, \"Just because it's new doesn't make it better, and just because it's old doesn't make it obsolete.\" An agripreneur will take the most useful of the new with the best of the old and apply it to his or her own farm in an individual style. To discover what will work and what won't requires research, experimentation, and thought.\n\n###### Food For Thought\n\nThere are other principles to both sustainability and agripreneurship, but this will get you started. Remember that agripreneurship\u2014and farming\u2014is a never-ending process of discovery. Your farm will evolve into a unique entity and meet your personal goals, family goals, and farm goals. Once you grasp and apply these principles, you can become independent, self-sufficient, successful, and happy. Just think of this as your agricultural self-help book.\n\n###### Chapter Two\n\n## Starting a Farm\n\nThere are many different types of farms, from large monoculture farms that produce a single product to general diversified farms that raise a little bit of everything (see chart for examples). Farms can range in size from thousands of acres down to \u00bd acre or even smaller. A shiitake mushroom farmer in southern Missouri got her start with eight mushroom logs. Now she farms thousands of logs.\n\nThe important thing to consider when choosing a farm type is to find something you like to raise that is compatible with your climate and that can be marketed for a profit. We will discuss this further after you have done some farm planning.\n\n### Starting a Farm Plan\n\nWhether you are buying a new farm or jump-starting an old farm, you must have a plan. To begin your farm planning, first assess your available resources, both personal and property.\n\nIf you are new to farming, there is a definite advantage to deciding what you will farm (vegetables, grains, dairy cattle, ostrich,\n\n_Farming furnishes children with a work ethic, a good set of values, and a true sense of family_\n\n###### Some Types Of Farms\n\nFarm Type | Product\/Characteristics  \n---|---  \nTraditional crop | Field crops such as corn, soybeans, wheat, and milo  \nCrop and livestock | May use rough ground for pasturing livestock and the rest of the property for traditional field crops  \nSpecialty crop | Herbs, cut or dried flowers (outdoors or in greenhouses), industrial field crops such as guayule (a rubber substitute) and kenaf (used as fiber by the paper industry), one or several varieties of vegetables, fruits (from kiwis to apple orchards), and berries; fruits and vegetables can be marketed as either fresh produce or value-added crops (canned tomatoes or dried blueberries or jam, for example)  \n_Unusual crop_  \nButterflies | Live butterflies, for use at weddings and parties  \nBees and insects | Rent insects out for pollinating crops  \n_Aquaculture_  \nWorms | Worms for fish bait  \nGoldfish and tropical fish | Fish for the aquarium trade  \nFee fishing | Consumers pay a daily fee or a per-pound fee for fish they catch on the farm  \nFood fish | Catfish, trout, and other fish are raised for sale to stores, restaurants, and consumers  \n_Exotic animals_  \nElk, bison, deer, mountain sheep, donkeys, ratites (e.g., ostriches, emus, and rheas) | Raised for meat, hides, and specialty by-products such as emu oil or velvet from elk antlers (used medicinally)\n\n# What Is Added Value?\n\nValue is added when a raw commodity is changed in a way that makes it more convenient to market and gives it a longer marketing life. Examples of value-added crops include cornmeal, jams and jellies, and tomato juice.\n\nfor example) and how you will market it (roadside stand, farmers' market, sale barn, value added) before you purchase the land. On the other hand, it is difficult to find the perfect farm that has everything you want at a price you can afford. So whether it is a new farm or an old one, you need to evaluate the property. By _property_ , I mean not just the land you own, but also what you will place on it.\n\n### Evaluating Your Resources\n\nMake a big resource list (see box on page 20)\u2014try to include on it everything you can think of. A list of what you know and what you need to know is the first step to a successful farm. If you know what you have available, it will make it easier to work out what is required for whatever enterprise you choose.\n\nFarm planning is like a road map: You must know where you're going, or it will be awfully hard to get there. In other words, you must know how many dollars you need to live, both now and in the future, and how many potential dollars a farm operation can expect. It is obviously not possible to come up with exact numbers, but you must have some realistic figures in mind. You began evaluating your capital and skills in chapter 1, in the Self-Evaluation section, but let's consider them more closely now.\n\n# Resource List\n\nFor each point, consider what you have available and what you will need to achieve your goals.\n\n  * Capital (money)\n\n  * Skills\n\n  * Labor available (time, people)\n\n  * Land\n\n  * Soils\n\n  * Water\n\n  * Location\/access\n\n  * Climate\n\n  * Equipment\n\n  * Marketing needs of the area\n\n#### Capital\n\nIn chapter 1, you evaluated what income level would be comfortable for you and your family. You also began to consider how much money you would need to start farming, or to change your operation. Now let's think about the amount of money you currently have. Ask yourself these questions:\n\n  * Do I have any savings?\n\n  * How much am I willing to risk on this new farming project?\n\n  * How much can I borrow\u2014and how much am I willing to borrow?\n\n  * Can I rent land with or without the option to buy later to maximize my limited capital? Should I rent or should I buy?\n\n  * Do neighbors or nearby equipment dealers have machinery I can rent, rather than buy?\n\n  * Can I buy a cheap \"fixer-upper\" and use my knowledge and time to multiply some sweat into cash?\n\nThese options all arise from the direct monies you have available. Consider other means to gain what you need as well, such as barter. Barter can be expressed in many ways. You could farm a piece of land on shares\u2014two-thirds to the owner, one-third to you. You can trade crops and livestock back and forth between nearby farms, or offer them in trade for a fence, or hay baling, or some plywood for a sheep shelter. Your skills\u2014or just some honest sweat helping another farmer\u2014are also tradable. Only your imagination limits the capital you can get.\n\n#### Skills\n\nIn chapter 1, you began a list of your personal resources, including the skills you have available. By listing the skills you have, then evaluating those you will need for your farming operation, you can determine what you will need to learn, or what you need someone else to do in your place.\n\nThe only way to acquire all the basic skills you need to be a farmer is to farm. Although you can (and should) read a book to learn the theory of how to plow a field, in reality there are far too many variables to explain. If you are completely new to farming, it may be wise to work for another farmer for a while, helping him in return for being taught some basic how-tos. Workshops, seminars, classes, and conferences are other places to learn at least the rudiments of many skills. If you cannot find anyone to learn from, read about it, then just jump in and do it. If you're patient, eventually you'll get it right\u2014as they say, \"Experience is what we learn from our mistakes.\"\n\nKeep in mind that agriculture can be dangerous, especially when lacking basic livestock- and machinery-handling skills. Don't get injured before you really get going just because of pride. Make sure you have a good idea of what you are doing, and make sure you have the proper tools to do it.\n\nMany old farmers would be glad to teach you their skills. Talk to them about volunteering in exchange for instruction, at sale barns, at feed and supply stores, or at their farms.\n\n# Skills a Farmer Needs\n\nFollowing are lists of skills that are needed for some basic types of farming. Of course, you might research what is needed for the operation you are planning. For example, elk farming, although similar to cattle farming, has very different requirements for each skill, and knowledge of collecting velvet (a product of the antlers) will also be necessary.\n\nSome of these may not be as simple as they appear. Moving cattle, for example, requires knowledge of the best places to stand to appear in the animals' fields of vision without alarming them, and what types of motion and movement work best.\n\nField-crop skills\n\nSoils\n\nPlowing, disking, harrowing\n\nPlanting, cultivation, harvesting\n\nSkills for each piece of equipment used\n\nWeather\/season knowledge\n\nCrop knowledge\n\nPest control, weed control, fertilization\n\nSeed saving\/preparation\n\nIrrigation\n\nMarketing\n\nVegetable-crop skills\n\nSoils\n\nCrop knowledge\n\nField preparation\n\nPlanting, seeds, transplants\n\nVertical growing, trellis (e.g., peas), cages (e.g., tomatoes)\n\nRaised-bed growing\n\nFlat-field growing\n\nMulching, cultivation\n\nHarvesting\n\n\u2022 Hand\n\n\u2022 Mechanical\n\nWeather\/season knowledge\n\nUse of season extenders\n\nPest control, weed control, fertilization\n\nSeed saving\/preparation\n\nIrrigation\n\nMarketing\n\nVine- and bramble-crop skills\n\nSoils\n\nPlanting\n\nPropagation\n\nIdentifying new wood, pruning, trellising\n\nCultivation\n\nHarvesting\n\n\u2022 Hand\n\n\u2022 Mechanical\n\nAdding value\n\nWinter preparation\n\nWeather\/season knowledge\n\nCrop knowledge\n\nPest control, weed control, fertilization\n\nIrrigation\n\nMarketing\n\nCattle skills\n\nApplying Id tags\n\nCastration\n\nBreed knowledge, breeding\n\nDehorning (removal of horns), hoof trimming\n\nMedication application, shots with needle, worming\n\nDelivering calves\n\nMilking cows\n\nHandling\/moving cattle\n\nRestraining large animals\n\nBehavioral knowledge\/control for:\n\n\u2022 Aggression\n\n\u2022 Birthing\n\n\u2022 Sickness\n\nShelter needs\n\nFeeding\/water\n\nManure handling\n\nFencing\n\nMarketing\n\nHog skills\n\nBreed knowledge, breeding\n\nFlushing\n\nApplying Id tags\n\nCastration, ringing\n\nRestraint for health care\n\nBirthing of litters\n\nHandling\/moving hogs\n\nRestraining large animals\n\nBehavioral knowledge\/control for:\n\n\u2022 Aggression (boars and sows)\n\n\u2022 Birthing\n\n\u2022 Sickness Shelter needs Feeding\/water\n\nManure handling\n\nFencing\n\nMarketing\n\nSheep skills\n\nBreed knowledge, breeding\n\nFlushing\n\nApplying Id tags\n\nTagging (removal of wool around udder), docking (removal of tails)\n\nMedication application\n\n\u2022 Shots with needle\n\n\u2022 Worming\n\nDelivering multiple lambs\n\nShearing\n\nHandling\/moving sheep\n\nBehavioral knowledge\/control for:\n\n\u2022 Aggression (rams)\n\n\u2022 Birthing\n\n\u2022 Sickness\n\nShelter needs\n\nFeeding\/water\n\nManure handling\n\nFencing\n\nMarketing\n\nUseful hand-tool skills\n\nWoodworking\n\nConcrete working\n\nFencing\n\nElectrical\n\nChain-saw operation\n\nRototiller usage\n\nHand mowers\n\n#### Labor\n\nYou must assess your available labor, in terms both of yourself and of your family. Will it be just you working, or will you have some combination of family members? If you have an idea of what combination of crops and livestock you plan to raise, figure out whether you have enough labor to take care of these enterprises. Remember, as we discussed in chapter 1, your financial goals must be your whole family's goals in order to be successful.\n\nFarm labor is generally measured by the number of 10-hour work days required under average conditions to take care of a certain number of acres or so many head of livestock (see labor charts, pages 26 and 27). Six 10-hour days per week for 50 weeks allows about 3,000 hours of labor available per year for one full-time farmer. In general, increasing the number of labor hours means increasing income, but in the case of a thinking agripreneur, this is not always so.\n\n# Family Labor\n\nFamily labor makes use of each family member's skills in a way that yields the greatest benefit to the farm. Each of us is better at some things than others, and tasks should be divided accordingly. Try to avoid strict rules and stereotypes: If a woman is better at lambing ewes, and a man is better at keeping books, do it that way. Even small children\u2014with patience, instruction, and careful supervision on your part\u2014can be helpful. Small children can handle baby chicks, fill feeders or water pans (a little at a time), pull weeds, or pick small fruits once you show them how. Give older children more responsibility, and listen to their ideas. If you want children to develop a strong work ethic while learning useful skills, they must first enjoy their work. You must teach them how. (See chapter 10 for more on family labor.)\n\nUsing the amount of hours listed in the labor charts, it seems that one person could therefore take care of 428 beef cattle (7 hours per cow), 750 sheep (4 hours per ewe), or 7,500 laying chickens (40 hours per 100 hens)\u2014although this would not allow time for building or equipment maintenance or marketing. A better time allowance (though still far from ideal) on a diversified family farm might be 20 acres of corn (152 hours), 20 acres of grass\/legume hay (218 hours), 420 acres of pasture (29.4 hours), 100 beef cattle (700 hours per year), 100 sheep (400 hours per year), and 2,000 laying hens (800 hours per year). This would total 2,300 hours per year, leaving 700 hours to use on maintenance, making hay, marketing, or a part-time job.\n\nWhen allotting time for labor, remember to allow for the proper distribution of labor throughout the year, so that you are not crowded for time during critical months. Field corn, for instance, requires 7.6 hours per acre per year\u2014but 1.1 hours of that are in December to March, 3 hours (planting) are in April and May, 1 hour (cultivation) is in June and July, 0.8 hour is in August and September, and 1.7 hours (harvest) are in October and November. The heaviest months are April through May and October through November. In contrast, in a two-litter-per-sow Butcher Hog system, with litters farrowed on March 1 and September 1, 16 hours are required in December to March, 8 hours in April to July, 4 hours in August, 8 hours in September, and 4 hours in October and November. Note the increased labor hours at farrowing time.\n\nBecause your goals should be your family's goals, it is important to set aside time for family in your farming life. Don't let your work schedule get so full that you neglect other obligations, such as time for church, vacations, or children's sports events. Remember, too, that labor is not accomplished solely on the farm. Allow time for education: Attend seminars and conferences. Nonfarm activities give you personal contacts with different viewpoints and may inspire ideas about how to improve the management of your own business.\n\n# Labor Charts\n\nThese charts give an estimate of time required for all direct labor involved in the normal care of crops and livestock: planting and harvesting, feeding, milking, cleaning stalls and pens, castration, docking, assisting in birth, and normal health care such as shots and vaccinations. They do not include establishment of a product (start-up times), machinery and building maintenance, cleaning bins and other general farm chores, adding value, or marketing of the product. The labor hours are based on the types of equipment a small farmer could be expected to own (a three-plow tractor, for example). These times are approximate and should be used only as guidelines in comparison to your actual records.\n\n###### Total Hours Of Labor Per Year To Produce The Crop At Average Efficiency From Field Preparation To Harvest\n\nData from _Farm Business Planning Guide for Organization_ #6500, University of Missouri Cooperative Extension, Columbia, Mo, 1965; _Selected Fruit and Vegetable Planning Budgets_ , Ec 959, by Charles D. DeCourley and Kevin C. Moore, Department of Agricultural Economics, University of Missouri\u2014Columbia, 1987; and _Enterprise Budgets: Northeast Oklahoma 1985_ , prepared by Bill Burton, Oklahoma State University Cooperative Extension, Claremore, Ok, 1985.\n\n# Total Hours Of Labor Per Year To Raise Animals At Average Efficiency\n\nAnimal Unit | Hours\/Animal Unit  \n---|---  \nButcher hogs: 1 sow (2 litters\/year) | 40  \nFeeder pigs: 1 sow (2 litters\/year) | 22  \nFeeder pig finishing: 100 pigs | 80  \nSheep: 1 ewe | 4.0  \nHoneybees: 1 hive | 6.2  \nDairy goats: 1 doe | 1.2  \nChickens: 100 laying hens | 40  \nChickens: 500 broilers | 9  \nTurkeys: 100 birds | 10  \nDairy cattle: 1 cow  \nFluid milk market | 80  \nManufacturing milk market | 85  \nReplacement heifer to 2 years old | 20  \nBeef cattle: 1 cow and calf  \nStocker calf | 7  \nSteer calf |   \nWintered only | 3.5  \nWintered and grazed | 4  \nFinished immediately | 5  \nWintered and finished | 6  \nWintered, grazed, and finished | 8  \nHeifer calf  \nFinished immediately | 4  \nWintered and finished | 6  \nYearling steer  \nFinished immediately | 5  \nWintered and finished | 6  \nPlain wintered and short fed | 5\n\nData from _Farm Business Planning Guide for Organization_ #6500, University of Missouri Cooperative Extension, Columbia, Mo, 1965, and _Enterprise Budgets: Northeast Oklahoma 1985_ , prepared by Bill Burton, Oklahoma State University Cooperative Extension, Claremore, Ok, 1985.\n\n# Terms\n\n**_Finished immediately_** Fed only grain after weaning\n\n**_Heifer calf_** Female who has not yet given birth\n\n**_Steer calf_** Castrated bull calf\n\n**_Stocker calf_** Calf sold at weaning (6 to 7 months of age)\n\n**_Wintered only_** Weaned from mother and fed hay or grain over winter\n\n**_Wintered and finished_** Fed hay and grain until spring, then given all the grain the animal desires to finish\n\n**_Wintered and grazed_** Fed hay or grain over winter, then grazed on spring grass until sold (usually in autumn)\n\n**_Wintered, grazed, and finished_** Fed hay and grain after weaning, grazed from spring to autumn, then finished on more grain until marketed\n\n**_Yearling steer_** Steer about 1 year old; a \"short\" yearling is just under a year, a \"long\" yearling is just over a year.\n\n#### Land\n\nQuestions to begin assessment of the property include:\n\n  * What size is my farm?\n\n  * How much of the land is producing timber or brush?\n\n  * How much is suitable only for pasture and how much is suitable for grain crops?\n\nThis can be further divided into small-grain-crop (wheat, oats) and large-grain-crop (corn, soybeans) suitability, based on soil types. Large grains usually require more fertile soils.\n\nWhen buying property, remember that real estate is a piece of land with a set of rights that may or may not be included. Make sure you have water rights and mineral rights, and are aware of easements and any other intrusions on the property. Check at the Abstract Office or Title Company in your town to see what rights have been sold.\n\nCondition of land, soil type, and climate are the main factors in determining what crops will grow on an area of land. Although it is possible, in fact, to grow almost any crop anywhere, the cost to grow some crops in some areas may not be acceptable. Grain and vegetable crops do best in deep, fertile soils and level fields. Fruit crops do best where there are no extremes of temperature. They need well-drained soil, but do not require a level topography. The condition of a land is also known as its capability.\n\n#### Classifying Land\n\nLand is generally classified into eight Capability Classes. Each class allows for certain types of usage and treatment. Classes are based on slope of land, available water capacity, and soil drainage. Classes I to Iv are suited for cultivation. Classes V to Viii may be pasture, woodland, or wildlife areas. These are general conditions, and the land may be capable of alternative uses with proper treatment. For example, a hillside may be used for crops if water is prevented from running straight down the slope and eroding the soil. Terracing (a series of steps or ridges made from soil running at right angles to the direction of slope), contour strip-cropping (alternating strips of six to twelve rows of two different crops, forming bands following the contours of the land), and crop rotation (alternating crops by year or season to provide better soil and more ground cover) can make a hillside viable for crops. Your Soil Conservation Department can provide you with maps of land capability. Check with your university extension office or your state Department of Agriculture.\n\n**Class I land** , the best possible, generally has a slope of 0 to 2 percent, with deep, fertile soil that does not erode easily. It holds water well, has a good supply of plant nutrients, and is free of rocks. It is good for any use, from cultivated crops to pasture, from woodland to wildlife.\n\n**Class Ii land**, usually with a slope of 2 to 5 percent, is almost as good, but some conservation practices may be necessary to keep it productive, due to erosion, wetness, or lack of water retention.\n\n**Class Iii land**, with a 5 to 10 percent slope, may require intensive erosion control if regularly cultivated. It may be excessively wet in spring or poorly aerated.\n\n**Class Iv land** is still suitable for cultivation, but only for limited periods, and may be better used as pasture. It usually has a slope of 10 to 15 percent.\n\nThe other four classes are not generally suitable for cultivation. **Class V land** , with a slope of 15 to 20 percent, is limited to plants that can cope with extremely wet, poorly aerated, rocky soils. **Class Vi land** is suitable for pastureland or woodland, with limited conservation practices. It may have shallow soil, steep slopes, high erosion, or lots of rocks. **Class Vii land** is good for wildlife, but may have severe limits for pasture or woods. **Class Viii land** won't grow much of anything.\n\n#### Soils\n\nPrinciple: _Natural fertility and slope of land are critical on small acreages_.\n\nThere are more than 18,000 different soils in the United States. They vary from shallow to deep, from clay to sand to loam, from well drained to wet. Each type of soil is suitable for particular kinds of crops or land usage. For example, the Clinton-Boone-Lindley areas in Wisconsin, Minnesota, Iowa, Illinois, and Missouri vary from gentle to sharply rolling hills. Most of this region is uncleared woodland or pastureland. Cultivated areas are small, and crops depend on slope. This is not a good soil type on which to plant 400 acres of corn, but it might make excellent orchard land.\n\nThere will be more on soils in chapter 4, but you need to know the natural fertility of your farm soils and roughly what condition they are in currently. For any farm of less than 10 acres, the natural fertility of the farm is very important, because the physical size of the land will restrict the choice of enterprises. For instance, a cow-calf operation, which takes 3 acres to run a cow-calf pair, will not allow for much volume of income. The poorer the soils, the more area it would take; in some areas of the West, it takes 10 acres to run a cow-calf unit. Ten acres of vegetables or U-pick berries, however, would produce a large volume of income\u2014but these require adequate water. Each enterprise must be evaluated in the light of the current resources and conditions.\n\nA good place to start your soil research is with soil maps of your area. Contact your university extension office or state Department of Agriculture to find the nearest map source, or consult the appendix for the national offices.\n\n#### Water\n\nBecause just one cow drinks 50 to 90 gallons of water a day, you can see how important water is to the farm. If you're going to raise crops and livestock, you need lots of water. We will cover this in more detail in chapter 5, but when you are thinking about buying your farm, find out what water resources you can expect to have. And be sure to ask these questions:\n\nOne cow drinks 50 to 90 gallons of water per day.\n\n  * Does the property have a creek or river running through it?\n\n  * Is this water source unpolluted?\n\n  * Where does the water come from?\n\n  * Does it start on my farm?\n\n  * Does the property have any ponds (or \"tanks,\" as they are called in Texas)?\n\n  * Can cattle, sheep, or horses water from these ponds, or are they too shallow, with too much mud and vegetation around the edge?\n\nWater plays a definite role in what crops can be grown. Small-grain crops like wheat are generally grown in areas of 15 to 25 inches of rainfall per year. Large-seed grains, like corn, are grown in areas having at least 20 inches of rain from April to September and a 150-day growing season with hot days and nights.\n\n###### Water Requirements Of Crops\n\nCrop | Lb Water per Lb Dry Matter (average)  \n---|---  \nAlfalfa | 831  \nBarley | 534  \nClover, red | 789  \nClover, sweet | 770  \nCorn | 368  \nCotton | 646  \nMilo | 328  \nOats | 597  \nPotatoes | 636  \nRye | 685  \nSorghum | 322  \nWheat | 513\n\nData was gathered in Colorado. This table is for general comparisons only; measurements vary by region and climatological conditions. One Eastern U.S. study, for example, found that corn required only 271 pounds of water per pound of dry matter.\n\nLook into how water is supplied by the county, too. Many rural areas have what is known as _rural water_ , which means a large number of rural residents are served by one drilled well with treated water. Check your water rights (the rights to access water flowing across and underneath your land) at the county courthouse. Out West in arid regions, if you do not secure your water rights with the property, you may not even have drinking water. Don't just assume you can use the water on your land because it was that way where you grew up.\n\nIf the property has a well, find out all you can about the amount of water the previous owners used\u2014and how they used it. Will it fit your needs? Check with the State Geological Survey for information on the well's water flow and quantity (gallons per minute). Ask about reliability during drought or normal dry weather. Water quality is also important. Have the water tested for mineral content and pollution. Taste the water. Is it high in iron or sulfur, or otherwise unpleasant to taste or smell?\n\nWells must reach a permeable stratum where water flows freely, allowing easy access from the surface. The stratum can vary widely in depth in different regions.\n\n#### Location\/Access\n\nDifferent areas of the United States are noted for different crops and livestock due to their topography (mountains, rivers, plains, for example), soil types, and climate. Kansas is known for wheat production, and Texas and Missouri for cattle. (Texas and Missouri have the largest number of small farms in the United States.)\n\nChoosing the part of the country where you want to live will influence the type of farming you can do. You need to match the type of farming you are interested in with the soils and climate of the area. Look at the growing conditions of each crop and animal you are interested in, and determine if they are suitable for an area.\n\nOther factors, such as livestock availability, must also be planned for; if you want to raise hogs in the western mountains, for instance, breeding stock will be many, many miles away. Distance from markets must also be considered. To direct-market your produce, you should be within 40 miles of a fairly large population. This is also important if you plan to sell some of your products through restaurants and grocery stores, for example.\n\nAre you considering a U-pick operation or some form of agritourism? In order to sell produce on-farm or through a roadside stand, you need to be near some well-traveled roads, perhaps on the way to a tourist site, such as a boating lake. Ask yourself how many roads lead to your farm. Are they blacktop? Are you on the blacktop or a major highway or are you back in the \"boonies\"?\n\nEasy access is important for hauling your crops and livestock to town, even if you don't have a U-pick operation. How close are you to a large-population town where you can purchase all the products and input items you need on the farm? Access to the farm is important both for visitors and for your family. (See chapter 7 for more information.)\n\n#### Climate\n\nClimate relates to your farm location. The Usda has issued a map of climate zones for growing plants. Zone designations are based on the minimum weather temperatures in that region, which determine the general hardiness of plants able to grow there. There are eleven zones for the United States.\n\nIn general, the lower the zone number, the hardier your plants must be to do well. In Zone 3, the first zone in the forty-eight contiguous states, plants must be able to endure \u201340 degrees Fahrenheit\n\nHardiness zones in the United States\n\nin order to survive year-round, while in Zone 6, in the Midwest and West, the mercury never goes below \u201310 degrees Fahrenheit. Plants in Zone 9 (Florida, southern Texas, southern Arizona, and California) never have to deal with temperatures lower than 30 degrees Fahrenheit. Many plants are rated by zone or hardiness level. To be safe, choose plants that are rated one zone below the one in which you live.\n\nNo matter where you live, microclimates can enable you to grow plants that do not normally live in the area, making a successful niche market for your produce. Microclimates can be created through the use of trees or other windbreaks, from natural depressions or rises in the land, or from man-made weather breaks, such as buildings. You can always create your own artificial microclimate through such means as greenhouses and indoor plant farming. We will discuss this more in chapter 5.\n\n##### Equipment\n\nOnce you have assessed your land, consider the equipment needed for your various crop and livestock enterprises. Machinery and high-quality hand tools are expensive to buy and costly to maintain and repair.\n\nMachinery is a tool to help you do more work faster than you could by hand. If you don't use the time saved to do more profitable things on your farm (like planning or marketing), your machinery is just an expense you don't really need. Machinery should enable you to produce the volume of business necessary to make a profit with as little labor as possible.\n\nRemember that the farm has many free energy and power sources, starting with the sun, which can be utilized to reduce your machinery, labor, and resource needs. For example, crop rotations\u2014using plants themselves to nourish the soil\u2014improve your yields without requiring purchase of energy-consuming commercial fertilizers, and thus make your operation more sustainable.\n\nIn addition to machinery, equipment includes the physical plant of the property\u2014buildings and fencing. With careful placement of\n\n# Free Energy Sources\n\n## Sun\n\n  * Photosynthesis from the sun produces crops and forages.\n\n  * Sunshine generates heat, which may be stored by greenhouses or mulches.\n\n  * Sunshine may be used for water collection through condensation.\n\n  * Sunlight can sanitize soil, killing microorganisms such as bacteria and molds.\n\n  * The sun provides essential nutrients like vitamin D.\n\n## Water\n\n  * Running water may generate power.\n\n  * Rain (and snow) provides free water for crops and livestock. Collect it for future use.\n\n## Soil Enhancers\n\n  * Nitrogen fixation from leguminous plants eliminates adding nitrogen.\n\n  * Crop rotations provide elements to other crops, replacing applications of fertilizer.\n\n  * Organic matter from plants (green manure, pasture, crop waste) is fuel for microbial activity in your soil, which helps plants get the nutrients they need.\n\n  * Worms and various fungi\/bacteria will migrate to and breed in healthy soil. These help aerate the land and provide essential minerals to the plants.\n\n  * Azotobacters are a form of free-living bacteria that can fix the nitrogen of the air into the soil. They will also be found in soil with a high organic content.\n\n## Energy from Livestock\n\n  * Livestock will naturally plow a pasture or garden as they graze (especially hogs).\n\n  * Manure from animals in either raw or compost form can be used to enrich the soil.\n\n## Other Free Energy Sources\n\n  * Wind may be used to generate power.\n\n  * Trees can cool livestock in their shade.\n\n  * Windbreaks prevent water and heat loss.\n\n  * Trees may provide a fuel or building source.\n\n  * Pasture and crop stubble can be used to feed animals.\n\npasture fencing, you can give different nutritional levels of forage to different-size animals or animals in different phases of production\u2014like nursing cows or bred ewes. (It takes a higher level of nutrition for a female nursing and raising offspring than it does to maintain a pregnant animal.) For example, in general, legumes (clover, lespedeza, alfalfa) have a higher protein content and relative feed value (Rfv) than grasses. To flush ewes (increasing ovulation), place them on lespedeza or alfalfa pasture. You can also get similar results with a 5- to 6-inch-tall grass pasture; even though the grass has a lower Rfv, quantity can balance its lower quality. In the same vein, multiple-birth animals (sheep, hogs) benefit from weight gain from an increasing level of nutrition, which makes them ovulate more to produce more offspring. The right type of fencing allows animals to be easily moved to where they need to be according to their nutritional needs.\n\nHow do you determine your machinery needs and costs? Again, analyze your crop and livestock needs. We cover equipment and machinery in more detail in chapter 9.\n\nThe females of multiple-birth animals such as sheep and hogs should graze in areas with high-quality forage to satisfy the nutritional needs of pregnancy and lactation.\n\n### Getting Help\n\nIf you are new to farming, you may not be able to answer all of the questions that arise from your farm plan. By the time you finish reading this book, you will be able to answer more of them, but it is important for you to learn from a variety of sources. Go to your local library and do some research on enterprises that interest you. Talk to farmers you know, or go to a farmers' market in the area you want to move to and talk to the vendors there about what they grow and what works for them. Visit your local university extension office and get advice on crops and livestock that will work in your area. If they have a expert on direct marketing or adding value, plan on frequent consultations with him or her.\n\nAs you learn from all of these people, remember that everyone has answers, but they are not always the right answers for you or your farm\u2014always get second, third, and fourth opinions. If someone says you can't do \"X\" on your farm, you will have to decide how important it is to you to do \"X,\" and if it is important, research further to find out whether there is a way. I will speak more about resources for learning later on.\n\n###### Food For Thought\n\nBefore you decide what to farm, read all that you can on as many different topics of interest as possible. Knowing where you are going is the first step to getting there. Starting a farm may seem like an impossible task, but if you have faith in yourself and your family and you do your research carefully, you will succeed in getting the farm\u2014and lifestyle\u2014you want.\n\n###### Part Ii\n\n## Farming\n\n###### Chapter Three\n\n## Some Principles of Good Farming\n\nIn addition to deciding where and what to farm, you'll need some basic guidelines on farming itself. Although how-to methods will be discussed later, you must learn basic principles to apply to your farm. We have already covered some principles. I would like to repeat them, just so you will be able to look at them as a group:\n\n  * Plan your farm and your goals carefully.\n\n  * Farm goals should be family goals.\n\n  * Look at the whole picture.\n\n  * A farmer should learn and grow through reading and meetings.\n\n  * Read, research, and experiment.\n\n  * Think.\n\n  * Old is not always obsolete. New is not always better.\n\n  * Natural fertility and slope of land are critical.\n\n  * To be sustainable, a farm must be profitable.\n\n  * A farm _must_ be profitable.\n\nNow, let's continue with some other important principles.\n\n_Farm goals should be family goals. Your children learn responsibility by being involved in the farm goals_.\n\n**Principle :** _To be sustainable, a farm must be environmentally sound and socially acceptable_.\n\nFor your farm to be sustainable, you must develop cropping and livestock systems that are environmentally sound and socially acceptable. For instance, plowing up and down the hill so your soil washes out onto a public road is not environmentally sound, because you lose a lot of healthy soil. Nor is it socially acceptable, because everybody pays for the cost of cleanup. As another example, huge feedlots of large numbers of concentrated animals are encountering more and more opposition today, some even from neighboring farmers. Lack of social acceptance will prevent this type of farm from being sustainable.\n\n**Principle :** _Avoid debt_.\n\nDealing with bankers is like selling through a middleman. The farmer makes less profit because he has to pay interest and principal. Start-up debt is okay if you can pay at least 20 percent (preferably 50 percent) on the land and stretch the payments out as low and as long as possible with no prepayment penalty when you have a good year. For all other projects, _grow_ into the enterprise rather than _borrowing_ into it. The principle is to avoid debt as much as you can and to make payments in line with farm production ups and downs.\n\n**Principle :** _Keep costs down_.\n\nWhenever you try something new in the way of crops or livestock, do it on a small scale and grow into it while learning. It could save you lots of money.\n\n**Principle :** _Try for low inputs_.\n\nLow inputs may improve your soil and make your operation more sustainable, in addition to saving money. Of course, the less money you spend, the more you have to work with.\n\n**Principle :** _Do things on time_.\n\nAccomplishing tasks on time is an important principle of farming that requires your labor and machinery requirements to match. Fields are best sized to what you can do in one day. You can drive only one tractor or tiller at a time. Timing is important because of the seasonal and cyclical nature of farming. If you want your cows to calve close together, the bull must be with your cows for only 60 days, or two heat cycles. Close calving means more attention on your part, and a more uniform-sized calf crop for selling purposes.\n\n**Principle :** _Plan your farm to minimize your work_.\n\nWork can be minimized by planning your farm layout wisely. Run a travel lane with access to crop fields and pastures down the middle of your farm, to allow for efficient movement of livestock and machinery; create square or rectangular fields to maximize the efficiency of your machinery; and plan placement of sheds and gardens for best access from the house and fields to save time and energy.\n\nIf you are raising herd-type animals (for example, cattle, sheep, elk), maintain several animals whenever possible. Herds of animals are more content than are one or two animals, who are always wanting to rejoin the main herd. Uniform bunches of animals enable you to creep-feed calves on higher-quality forage than is needed for the mother cows. Remember that culling is important.\n\n**Principle :** _Develop a system of production that balances farm resources and available labor_.\n\nSmall acreages lend themselves to good timing. Due to their small size, you can reach locations quickly. With low numbers of livestock, tasks do not take long to perform. This often allows you to beat the weather or use it to your advantage.\n\n**Principle :** _Keep good records_.\n\nIt's essential to keep careful financial and performance records and a diary. One of the most important things you need to know is the cost of production for each enterprise. It is nice to know ewe #29 had triplets for the last three years. It is nicer to know that it cost \"X\" dollars of feed for the ewe and \"Y\" dollars for the lambs, and the cost of the pasture, hay, equipment, and so on, allotted to her was so many cents per pound, because now you know what price you need to sell at to gain a profit.\n\n**Principle :** _Learn basic veterinary skills and tasks_.\n\nWhether new to the farm or established, good how-to skills are important. Vet calls today are easily $25 just for the trip, plus labor and medicine, so it pays to learn how to dock sheep tails, castrate calves, and deliver babies, for example.\n\n**Principle :** _Learn carpentry, electrical, and machinery repair skills_.\n\nCarpentry, electrical, and machinery skills all reduce costs. If you do not have these skills, some courses could be helpful.\n\n**Principle :** _Learn stockman skills, and keep gentle livestock._\n\nStockman skills are also important, but you may not be able to learn them in school. Talk to someone in the business and see what he knows, or read books and magazines. A basic example is knowing where to stand when herding cattle\u2014standing in a place in their field of view saves time in herding and reduces the animals' stress level.\n\nResearch has shown that gentle livestock reproduce and grow faster and better than do wild, nervous animals. But there are other reasons that gentle livestock are important. If you have a pet cow or ewe, for instance, one that will always come to a feed bucket when you call, that animal will help you move your livestock from one pasture to the next or into the working corral area with a minimum of trouble. The pet will also bring the herd in for a close visual inspection for new babies, bad eyes, sickness, or other problems. Being around your animals, patting and scratching the pets, and observing the others all help keep them gentle. Using a treat like cattle cubes or an ear of corn also speeds along the process.\n\n**Principle :** _Take good care of your buildings, machinery, and livestock_.\n\nBuildings and machinery that are taken care of will always cost less to maintain and will rarely need replacement. Obey your engine care instructions on machinery, and frequently inspect your buildings and machinery for wear, weather damage, and general condition.\n\nLivestock also respond to better treatment. Healthy, stress-free livestock will gain weight more quickly, stay in better condition, and have better performance in birthing and raising young. Learn everything you can to keep your livestock healthy and unstressed, and make sure they have adequate shelter, feed, and water.\n\n**Principle :** _Have a good water system, and save every drop of water that falls on your farm_.\n\nA good water system for livestock is essential. Again, it is better if your livestock can go to the water, instead of you hauling it to them. I have found it tiring to dip water out of a pond and haul it to the hog water tank on another part of the farm. Your own pond or lake with a submersible pump and plastic pipe makes life more pleasant for you and your livestock.\n\nWater can be stored in ponds and in the soil itself. For instance, if your soil is 4 to 5 percent organic matter, it can absorb 4 to 6 inches of rain per hour without erosion, which would cause runoff. If your organic matter is only about 2 percent, your soil can absorb only \u00bd to 1\u00bd inches of rain per hour. (See chapter 5 for more on water.)\n\n**Principle :** _Maintain or improve the soil fertility_.\n\nMaintaining soil fertility is a process of thinking about the \"why\" of the way you do things on your farm. Several processes improve and help fertility, like composting on either large or small scale, animal manures, and green manures for cover crops.\n\nAnimal manures can be good for the soil. One of the best ways to spread manure is to let the animals do it. Hauling piles of manure from animals standing in a muddy barn lot requires that you spend time and labor forking out the manure, plus fuel and expensive machinery (the manure spreader) to spread it. If you keep your animals on pasture instead, they will spread it across the field as they defecate while they graze. This natural method lets your animals do the work for you.\n\n**Principle :** _Let the animals do as much feed harvesting on their own as possible_.\n\nIt is much easier to stockpile fields of grass for winter feed and drive the livestock to that field one time than to haul hay daily. To put up hay for winter forage requires time, money, and equipment. You may not be able to avoid hay altogether, but the cost savings when you can are money in your pocket.\n\n**Principle :** _Use crop rotations._\n\nCrop rotations are vital to a sustainable farming system. They enhance soil fertility and help control plant diseases, weeds, and erosion. Soil fertility in a rotation is maintained by the growing of a legume (like clover) or sod crop (grass) to provide nitrogen fixing and buildup of humus. Crop rotations also lend themselves to livestock usage; the legumes and sod crops in the rotations may be used for grazing and for hay.\n\n**Principle :** _Have 2 years' worth of hay and grain in storage_.\n\nThis is a goal you should try to achieve for a number of reasons. The obvious one is weather. You never know when the winter will be longer, or the summer hotter and dryer, and you will need extra feed to supplement your livestock on pasture. If you are having a weather problem, chances are your neighbors are too, and feed prices will be high. When you have your own feed, you are more likely make it through to rain. If not, you will be selling your livestock on a depressed market. Fortunately, droughts and bad winters do not happen too often, but it is always best to be prepared.\n\nHere are some principles we will cover in future chapters:\n\n  * To be sustainable, a farm must be diversified.\n\n  * A farmer must be skilled at buying and marketing.\n\n  * Plan to have products to sell year-round.\n\n  * Plan to have produce available when others do not, or have unique products.\n\nThese will be covered in more detail in chapters 8 and , but the basic idea is to learn to market, and to have something to sell at all times. Your bills are monthly, not seasonal, so you should have something to sell year-round. A diversified farm is protected from weather conditions or price variabilities that hit one crop or animal but not another. Markets vary at different times: When hog prices are low, sheep prices may be high. You should also diversify your marketing methods, so you are not dependent on only one type of income. You also need to set and control your prices, rather than just taking the price someone offers.\n\nAnd finally, some advice for you to practice on yourself:\n\n  * Be disciplined.\n\n  * Don't procrastinate.\n\n  * Practice scheduled, efficient, and productive work habits.\n\n  * Keep a positive attitude.\n\n  * Be happy.\n\n###### Food For Thought\n\nLearning the basic principles of farming will provide you with the tools needed to run a successful operation. Though all of the principles in this chapter are important, the most important ones are the last few mentioned. A positive, happy attitude and a disciplined approach to problem solving will make your goals easier to obtain.\n\n###### Chapter Four\n\n## A Living, Healthy Soil\n\nMost farmers of any type will say it takes 5 years to learn how to farm a specific piece of ground on a particular farm. Most business people will tell you the learning curve for a new start-up business is 10 to 12 years. To be organically certified, your land must not have had chemical fertilizers or weed and pest sprays used on it for 3 years or more, to give the soil a chance to renew itself and return to life.\n\nAll these statistics are based on the assumption that you can survive in your business long enough to reach a mature level. All businesses, farming included, need to be least-cost producers in order to survive the ups and downs of the marketplace. To work at least cost, to get the most out of your land, to be sustainable and profitable, you must have good soil.\n\n_A living, healthy soil is the foundation for your farm's productivity and success_.\n\nIt is best, of course, if your farm begins with good soil. If it does not, though, don't be discouraged. There are techniques that will increase your soil's fertility, including crop rotation, application of organic matter, and careful choice of crops. The 4-acre Student Garden in Santa Cruz, California, started as a hillside with a clay soil that would barely grow weeds. Within 3 years, using organic methods, the students created an excellent fertile soil. Note, though, that improving soil is not a quick fix. You can count on a minimum of two years, and probably more, before your soil is what you want it to be.\n\n**Principle** : _The foundation of your farm and your most important production tool is a living, healthy soil_.\n\nHere we will go into some technical detail, because good soil is the foundation of a good farm\u2014and to attain good soil, you must understand the basics of soil fertility. Even experienced farmers may not have a good grasp of soil mechanics, because they have only learned to add N-P-K (nitrogen, phosphorus, potassium) at the right time to make their corn and soybean rotation work. Good soil has a much greater complexity than that, an interaction among your plants and livestock, insects and worms, microorganisms, and soil types.\n\n### The Physical Nature of Soil\n\nSoil is formed by water and wind erosion, and temperature variances, which crack large rocks into smaller rocks (in Missouri, sometimes not small enough), then down into particles that combine with decomposing organic matter (dead and decaying plants and animals).\n\nNature forms soil in layers. The top layer is the highest in organic matter and, hence, the most fertile. The top layer is usually 6 to 7 inches in depth. This layer is what is normally plowed for planting. The next layer is the subsoil and is obviously weathered. It has little organic matter. The third layer, the substratum, is more or less the parent rock\/material from which the particles for soil are formed.\n\n#### Soil Types\n\nBecause soils contain particles of varying size and materials, and organic content also varies in different areas, there is a wide range of soils. Three principle types are recognized: sandy, clay, and loam soils.\n\nAs mentioned earlier, there are more than 18,000 different soils in the United States. Every state (and every farm) contains many different soils. These soils are members of eighteen different orders of soil. Orders that cover a broad range of the United States include the ultisols, the mollisols, and the alfisols. _Ultisols_ are a southeastern soil, highly weathered, with subsurface clay. The clay stores water and nutrients for plants to use. _Mollisols_ are primarily a midwestern soil, with a fertile surface layer of high organic content. They are excellent for corn, soybeans, and wheat. _Alfisols_ are productive soils of the mideastern states. They require careful management.\n\n#### Soil Content\n\nThe orders and suborders define the soil types, but what you really need to know is what your soil contains. Soil types are classified according to content: sand, clay, loam, and organic matter.\n\n##### Sandy Soils\n\nSandy soils are up to 70 percent sand by weight. They are known as \"light\" soils, or soils that drain readily, because sand particles are fairly large and irregular in shape, and do not press together tightly, thus allowing water to make its way among the particles. Sandy soils are usually lower in nutrient content than the other soil types, as the water washes the nutrients out. Organic matter can do wonders for sandy soils, but they still require special management, as we will discuss later. Sandy soils come in different classes, such as _sand_ and _loamy sand_.\n\n##### Clay Soils\n\nClay soils contain at least 35 percent clay, and usually 40 percent or above. Class names are _clay_ and _sandy clay_. Clay soils are made up of flat particles that can pack tightly together, making for poor drainage and aeration. Clay soils are sticky when wet and hard when dry, and if cultivated at the wrong time will give you a summer of hard, lumpy soil to work with. If you plow clay soils when they are too wet, they will puddle or run together and be like brick when dry.\n\n##### Loam Soils\n\nLoam soils are a mixture of 45 percent sand, 40 percent silt, and 15 percent clay particles. When sand is the dominant particle, it is called _sandy loam_ ; likewise with silt loam and clay loams.\n\nLoamy soils are what you want, because they combine the best characteristics of clay and sand soils. They have good aeration and drainage, but retain good water-holding capacity, which preserves nutrients and prevents nutrient leaching.\n\nSee the resource list in the appendix for contacts for aerial photos and topographic and soil maps for your area.\n\n### Working the Soil\n\nLike animals, plants, and other living organisms, soils respond to good care and suffer under bad handling.\n\nWorking soil when too wet destroys structure tilth and harms beneficial microbial life. Test your soil before you work it.\n\nTo test your soil to see whether it is of the right moisture to work with, make a loose ball of dirt and drop it from about waist high. If it breaks apart, your soil is ready to work. If it does not break, then wait a half day or a day with no rain before you start working the ground.\n\nIf you are working your ground with a tractor and there is water in the plow sole, _stop_ \u2014it is too wet. If the plow furrows glisten in the sun, or look slick rather than crumbly, the ground is too wet\u2014stop plowing. When turning over the furrow slice, it should be crumbly and falling apart.\n\n### Our Living Soil\n\nLiving soil is much more than just the different kinds of mineral particles that hold the plants upright. It is teeming with life throughout its strata, from bacteria to worms. Most of your soil's organisms inhabit the top 6 or 7 inches, the topsoil. These organisms include insects and other \"bugs,\" earthworms, bacteria, and fungi.\n\nBillions of soil organisms exist in your soil, and many will work to improve soil fertility.\n\n#### Earthworms\n\nThe earthworm is the largest of the soil's creatures, and performs a multitude of tasks. Burrowing lets air into the soil, as well as water. Worms move surface organic matter underground into their burrows. This is one way they fertilize your soil. The other way is a special talent of worms: When they eat organic matter and it passes through their body, the result is worm manure\u2014worm castings\u2014which contain 5 times more nitrogen, 7 times more phosphorus, 3 times more magnesium, 11 times more potassium, and 1\u00bd times more calcium than the surrounding soil.\n\nNight crawlers ( _Lumbricus terrestris_ ) are not native to the United States, but they have adapted to life in cultivated fields better than have native earthworms.\n\nUsda researcher Henry Hopp conducted many extensive studies on worms in the 1940s, and noted that worms move the soil by two methods. In loose soils, they slip through the holes; in tight soils, they simply eat their way through. While worms deposit some of their castings on the surface, they mainly deposit them underground, which fertilizes your soil. In Australia and New Zealand, in areas where worms were not natural to the soils, pasture production increased 30 to 50 percent with the introduction of earthworms.\n\nThe best way to maintain a high adult worm population is with mulch or a cover crop. Worms are most active in the spring and fall and go into a kind of hibernation when it is too dry or too cold. A cover crop growing in the fall keeps the ground temperature from dropping quickly and killing your worms. The cover crop lets the soil freeze slowly, allowing the worms to move deeper and thus survive. The larger the adult worm population you can maintain, the more young you will have and the more fertility your soil will have.\n\n#### Bacteria\n\nThe earthworm is one of the largest inhabitants of the soil community, but there are billions of organisms in each gram of healthy topsoil. You may have up to 2 tons of bacteria per acre. Bacteria are either aerobic (they need air to live; these are the most beneficial ones) or anaerobic (these do not need air to live).\n\nBacteria in your soil make nitrogen and sulfur compounds into a form usable by plants. This is important because nitrogen is often the limiting factor in plant growth. Soil may contain only small amounts of nitrogen in forms that plants can utilize, and some plants are heavy feeders of nitrogen. There are several bacteria that can make soil minerals available for plants.\n\n_Azotobacter_ is a free-living microorganism that fixes nitrogen from the soil and air into its body tissue. Found naturally in soils, azotobacters can be encouraged to multiply by the addition of organic matter, humus, and humic acids to the soil.\n\nA nitrogen-fixing bacterium called _Rhizobium_ occurs naturally on leguminous plants like peas, beans, and clover. This bacterium takes nitrogen from the air and converts it to a form that plants can utilize. The population of _Rhizobia_ increases as the amount of organic matter and humus in the soil increases. It is also possible to buy _Rhizobia_ in small packets from many companies in order to inoculate seeds. This may be necessary if you have no _Rhizobia_ in your soil, which is possible if no nitrogen-fixing crops have ever grown there. We'll discuss nitrogen fixing later in this chapter.\n\nAnother important bacterium is the _actinomycete_. Even though actinomycetes are bacteria, they also share some similarities with fungi. They act as decomposers of organic matter and facilitators in humus formation. Actinomycetes can be found everywhere, but are particularly rich in sod soils and compost. To be most effective, they need a neutral pH of 6.0 to 7.5.\n\n#### Fungi\n\nThese organisms include mold, yeasts, and mushrooms. They get their nutrients from decomposing organic material or live organic material. Although some fungi are harmful, what we are concerned with here are beneficial fungi, such as _mycorrhiza_ , a mycelium fungus that has a mutually beneficial relationship with plant roots. Plant roots provide carbohydrates to sustain mycorrhiza, and the fungus helps the plant absorb nutrients and water. About 80 percent of the agricultural plants grown have a mycorrhizal root association. Mycorrhiza can be enhanced with increased organic matter and minimal disturbance of the soil.\n\n### Soil and Plant Roots\n\nAs we have seen, your soil is both a physical structure and a group of living organisms. Like livestock, your soil needs food, water, and air to reproduce and function.\n\nPlant roots help supply your living soil's needs, and connect the plant to it. They remove nutrients and water from the soil to feed the plant, but they also feed the soil. Plant roots are always getting rid of dead tissue, which makes excellent food for soil microorganisms. Many food transactions occur right around the plant roots; this area is known as the _rhizosphere_.\n\nPlant roots can be up to three times the mass of the aboveground plant.\n\nThe roots of plants are typically two to three times the mass of the aboveground plant yield and cover a large area. H. Dittmer, writing in the _American Journal of Botany_ , 1937\u201338, found that a single rye plant had a root length of 377 miles. The root hairs numbered 14.5 billion. The surface area was more than   of an acre. Combined, the roots and root hairs had a length of 6,990 miles, with a combined surface area of 63,784 square feet\u2014close to 1.5 acres.\n\nAs already mentioned, most of your soil's organisms are found in the upper 6 or 7 inches\u2014the plow layer, or topsoil. Although most of the plant's roots are in this layer, many plants send roots 5 or 6 feet deep in search of nutrients. These deep roots help provide a structure to the soil, preventing erosion and increasing the soil's capacity to hold water. To achieve deep roots, the soil must contain good amounts of nitrogen.\n\nA loose soil with good tilth or structure lets air reach to the roots, which are then able to work better. Tight, waterlogged soils have no air, and plants in them soon yellow and die.\n\nSoil is a loosely connected body of particles of irregular shapes and sizes. The spaces between these particles form cavities, some large and some too small to be seen by the naked eye. Small spaces allow for capillary action by the soil, similar to a paper towel soaking up water. The various cavities, or pores, are filled with water and air, both of which are essential to plant growth. We will talk more about water in chapter 5.\n\nContinuous cropping results in rapid loss of soil organic matter. Alternating shallow- and deep-rooted plants like clover and alfalfa provides better drainage from channels left by dead roots. Deep-rooted plants bring up minerals from the subsoil. A rotation with a sod crop maintains the organic matter supply and furnishes the raw food for soil bacteria. By keeping a crop on the land, most of the toxic nutrient leaching is minimized. Crop rotation improves yields as well as crop quality.\n\n### Crop Rotation\n\nCrop rotation is the process of planting a different crop after each previous crop, which allows the different plants to take advantage of nutrients the previous plants didn't use, and to put different nutrients into the soil to avoid depletion of overall nutrients. For example, corn uses nitrogen; soybeans replace it. Following corn with soybeans avoids nitrogen depletion.\n\nCrop rotation dates back to Roman times. Farmers in those days began rotations to replenish the land instead of using up its fertility and abandoning the field.\n\nA rotation should be planned so that you have the greatest possible value of salable or usable crop material during a period of years. When planning a rotation, also consider labor needs and soil fertility. Ideally, a rotation will help spread out your labor needs, because you have a diversity of crops ready to harvest at different times of the year.\n\nCrops are divided into three classes: grain crops, like wheat and barley; grass crops, including sods and legumes used for pasture\/hay; and cultivated crops, like corn and soybeans. You can substitute crops within types, depending on weather conditions and year. For instance, you can plant barley instead of wheat; both are small grains. Ideally, farm fields would be square, but this often does not happen in real life, so just try to keep plots all the same size\u2014that is, if you plant 10 acres of corn, follow it with a full 10 acres of soybeans, then 10 acres of hay, and so on.\n\n#### Advantages of Rotation\n\nThere are many advantages to crop rotation. The biggest one is building and maintaining organic matter and putting nitrogen back in your soil by plowing under immature forage crops, primarily legumes. (Legumes are plants that fix nitrogen in the soil.) Crop rotation lets you grow a soil crop or legume crop on all the fields of your farm.\n\nHere are some other advantages:\n\n  * Rotations of different crops provide varying root systems, some deep, some shallow, which bring different crop nutrients to the plow layer for use by the next planting.\n\n  * Rotations improve drainage tilth and water-holding capacity of the soil, which also reduces erosion.\n\n  * By alternating crops on the same fields, you use a natural method of breaking up insect pest and disease cycles, and this also helps in eliminating weed species.\n\n  * Rotation hosts beneficial microbiological life, which discourages disease; improves microbiological activity, which helps plants absorb nutrients better; and creates an inhospitable soil environment for many soilborne diseases.\n\n  * By rotating crops, a farmer's labor load is spread throughout the season, making for more timely operations.\n\n  * Rotation or diversification of your crops provides protection against total crop and economic failure, and provides year-round distribution of labor.\n\n  * Rotations cut costs and time by reducing purchased fertilizer and allowing easier tillage because of improved tilth.\n\n  * On the average, rotating your crops will give you a 10 percent increase in yields; continual planting of the same crop results in mineral depletion. Crop rotation allows you to make money while you build your soil.\n\n#### Short-Term Rotation\n\nRotation can be either short term or long term. A short-term rotation will take place in 1 year or less. An example is fall-planted wheat overseeded in spring with red clover. To protect the wheat, the clover should be overseeded on a snow cover or on frozen ground, preferably both. This can be done by hand, by powered seeders mounted on tractors, pickup trucks, or all-terrain vehicles, and even by plane! After the grain is harvested in July, the clover grows through the wheat stubble and can be grazed in the fall or incorporated by plowing or disking before another fall grain crop is planted.\n\n# Seven Advantages Of Crop Rotation\n\n  1. Keeps soil in suitable physical condition\n\n  2. Helps maintain the supply of organic matter and nitrogen in the soil\n\n  3. Is a practical means of using farm manure\n\n  4. Keeps soil occupied with crops\n\n  5. Changes location of feeding range of roots\n\n  6. Counteracts possible development of toxic substances\n\n  7. Improves crop quality\n\nFrom Clyde E. Leighty, \"Soils and Men,\" _Usda y earbook of Agriculture, 1938_ (Washington, Dc: U.S. Department of Agriculture, 1938), 410.\n\nAnother short-term rotation might be rye and hairy vetch seeded in standing corn. After the corn is picked, the rye and vetch grow through the winter and are then plowed under in the spring for a green-manure crop, after which corn, beans, or milo may be planted. A short rotation for vegetables might be broccoli followed by buckwheat, then perhaps rye or turnips.\n\nMinor rotations are another type of short-term rotation. (Both major and minor rotations can be used on the same farm.) A minor rotation is when a small area is set aside for a short-term use\u2014a truck garden, a hog pasture, a lambing area, for example. The minor rotation can be part of the major rotation, or it can be temporarily or permanently set aside and fenced. A more permanent setup of a minor rotation allows you to add feeding and watering systems.\n\nAn example of a minor rotation might be hog pasture, potatoes, then truck garden or alfalfa. The hogs turn up the soil for potato planting and contribute manure to the soil fertility, as does a legume like alfalfa. A late summer\/fall garden of crops such as beets, cabbage, collards, lettuce, and spinach will allow you some late sales, along with extra greens at the table.\n\nIn this example of short-term crop rotation, corn is followed by rye and hairy vetch, which are then plowed under before corn is planted again.\n\nMinor crop rotation in this example includes hogs in early spring, potatoes in spring and summer, and vegetables and alfalfa in fall.\n\n# Factors In The Maintenance Of Soil Fertility\n\n  * Cover crop\n\n  * Crop residues\n\n  * Crop rotation\n\n  * Drainage\n\n  * Farm manures\n\n  * Fertilization\n\n  * Irrigation\n\n  * Liming\n\n  * Organic matter\n\n  * Tillage\n\nFrom Clyde E. Leighty, \"Soils and Men,\" _Usda y earbook of Agriculture, 1938_ (Washington, Dc: U.S. Department of Agriculture, 1938), 409.\n\n#### Long-Term Crop Rotations\n\nLong-term rotations are usually 2 to 5 years. They are more complicated than short-term rotations, but usually not more difficult. They often include hay and sod crops, which are utilized by livestock. A popular eighteenth- and nineteenth-century rotation was the Norfolk rotation: root crop\/barley\/legume\/wheat. The legume allowed introduction of livestock such as sheep or cattle into the rotation, by using the legume for grazing or hay.\n\nSome common 3-year rotations are corn\/rye\/clover or corn\/barley\/clover. An example of a 5-year rotation is:\n\nA common 5-year rotation in the southern United States is given below (left), with a variation at right.\n\n#### Planning a Rotation\n\nEvery farm has its own set of management and climatic constraints to deal with, but there are some basic rules of thumb to rotations, as mentioned by Nicolas Lampkin in _Organic Farming_. The following ideas are adapted from his rotation designs for England, but are applied to circumstances in the United States:\n\n  * Alternate deep-rooted plants (e.g., corn) with shallow-rooted plants (e.g., cabbage) to improve soil structure and thus drainage.\n\n  * Alternate between plants having a high biomass of roots (legumes such as red clover and orchard grass) with crops with a low biomass root system, like corn and soybeans.\n\n  * Nitrogen-fixing crops like soybeans should be followed by nitrogen-using crops like corn.\n\n  * Keep the soil covered with crops as much of the time as possible to prevent erosion and reduce weeds (see box).\n\n  * Some crops grow extremely fast, like sun hemp, buckwheat, radishes, and corn. They should be alternated with crops that grow slowly, such as winter wheat and red clover. Slow-growing crops are more susceptible to weed pressure and should follow weed-suppressing crops like winter rye, which has an _alleopathic effect_ , that is, an ability to suppress weed germination and growth.\n\n# Keep It Covered\n\nI sow rye and hairy vetch in the standing corn in August. When the corn is harvested, there is a ground cover until spring plowing. The soil is uncovered only about 30 days in the spring after the corn is planted and between then and when it is about knee-high in mid-June. The corn planted in mid-May will be shading out weeds, slowing rainfall with its leaves, and holding the soil with its roots. The rye that was plowed under just before planting the corn sops up any leftover nitrogen from the previous year's crop and makes it available to the next crop. The hairy vetch, a legume, puts up to 100 pounds of nitrogen back into the soil. The rye and vetch together make a huge amount of biomass to turn under, improving the soil tilth and increasing microbiological activity of the soils.\n\n  * Alternate from leaf to straw crops to help with weed suppression. Mechanical cultivation reduces weeds in row crops, while straw crops shade out and steal nutrients and water from weeds, thereby stunting the weeds' growth.\n\n  * Alternate between fall and spring plantings of crops. This spreads out your workload, reduces weather risk, and helps suppress weeds that germinate at different times of the year.\n\n  * Balance your rotation between the cash and non\u2013immediate cash crops, because, as I've said before, it has to be profitable to be sustainable. This system makes it more profitable because it spreads risk and allows for cash crops (corn, wheat) and crops that build the soil (legumes, grasses, cover crops).\n\n### Cover Crops and Green Manure\n\nCover crops are used to prevent erosion, shade out weeds, and protect the soil from freezing and thawing. They also may reduce use of herbicides and pesticides by providing a haven for beneficial insects, as well as breaking the disease cycles of a monoculture system. By improving the health and microbiological activity of the soil, they also improve crop yield.\n\nCover crops are not necessarily incorporated into the soil, although they may be used as green manure (see below). Using a legume like red clover or hairy vetch in your field-crop or market-garden rotations gives you a plant useful for animal grazing or as hay, plus the benefits to the rotation of soil cover and nitrogen fixation. Red clover provides 2 to 3 tons of dry matter (the weight of forage after drying) and 70 to 150 pounds of nitrogen per acre. Hairy vetch supplies 3 to 6 tons of dry matter and 40 to 150 pounds of nitrogen.\n\n#### Green Manure\n\nA green-manure crop is a cover crop that is incorporated into the soil, even if it was not planted for that purpose, such as weeds growing on a flooded bottom field. Green-manure crops are turned under for the purpose of adding organic matter and\/or nitrogen to the soil. This reduces your fertilizer costs, as the cover crop will furnish enough nitrogen for the following crop.\n\nFor every foot of green-manure crop you turn under, you put about 1 ton of organic matter back into your soil. This is because your green manure is just as big below the ground (roots) as it is above. About half\u20141,000 pounds\u2014of this tonnage is lost almost immediately through evaporation. Thus, green manuring is used to maintain rather than increase organic matter.\n\n#### Legumes or Non-Legumes?\n\nGreen-manure crops and cover crops can be legumes (see next section) or non-legumes. As green manure, legumes add organic matter and nitrogen. Non-legumes add only organic matter. Volume or bulk is the most important goal in supplying organic matter to the soil and is more easily achieved by crops such as rye. If the crop is too mature when turned under, much of the available nitrogen may be used up in allowing bacteria to decay the crop. But with legumes, there is enough nitrogen to decay the crop and have nitrogen available for the next crop.\n\n### Nitrogen and Legumes\n\nNutrients move slowly in the soil. Roots usually grow toward the nutrients when they are activated by nitrogen and phosphorus (nutrients) already present in the soil. This makes nitrogen a necessary component of plant growth.\n\nLegumes are crops, such as soybeans and hairy vetch, that \"fix\" nitrogen (that is, move nitrogen from the air or the soil to storage nodules in the plant), allowing increased plant growth and deeper roots. They have a _Rhizobia-_ bacteria connection that allows legumes to take nitrogen from the air and store it in nodules on the plant root. This nitrogen is then released when the crop is plowed under and decomposes. You may need to supply the initial _Rhizobia_ by applying an inoculant consisting of _Rhizobia_ bacteria to the seeds to start the process.\n\nThe nitrogen in the legumes you turn under comes from both the nitrogen they drew up from the soil and the nitrogen they fixed from the air. Roughly one-third of the nitrogen comes from the soil and two-thirds comes from the air. If you sow a combination, say a (non-legume) rye and hairy vetch (a legume), the rye helps capture what nitrogen the vetch missed in the soil and uses it to grow lots of organic matter. While the rye uses the free nitrogen, the vetch continues to fix more nitrogen from the soil into a plant-usable form. The rye will hold the nitrogen inside its slender green stems until both the leguminous and non-leguminous plants are turned under, furnishing enough nitrogen for the next crop.\n\nLegumes take nitrogen from the air and store it in their roots. When the plant is plowed under and begins to decompose, the nitrogen is released into the soil.\n\nThe amount of nitrogen is determined by the time of cutting\u2014the more time a plant has to gather nitrogen in an unstressed growth period, the greater the yield. Dr. C. J. Willard, an Ohio expert on legumes, noted that sweet clover in the Midwest cut the final week of April contained 124 pounds of nitrogen, while that turned under the last week in May contained 160 pounds.\n\nThe best time to incorporate (plow under) a legume is in early bloom, when it is most succulent. Controlling the growth stages of legumes and non-legumes to keep them consistent can be accomplished by mowing or grazing.\n\nIn southern areas of the United States, where a green manure is plowed under, it may have to be followed by a fall crop like rye to hold the nutrients until the new crop is ready to be planted in spring. This is because milder winters promote continued soil activity, which uses up the available nutrients. Also, many regions of the South suffer more soil leaching, allowing rain to drain away the necessary nutrients. The rye will prevent this leaching and reduce erosion.\n\n# Nitrogen Fixation\n\n  1. Use the correct rhizobial inoculant for the legume you are growing. Ask your local seed dealer for the correct inoculant.\n\n  2. Soils need iron, sulfur, and molybdenum for nitrogen fixation to function properly.\n\n  3. A green-manure crop plowed under causes a large increase in microbiological activity. Soil bacteria reproduce themselves and double their population in as few as 7 days.\n\n  4. Weather affects the nitrogen release from green-manure crops. Warmer temperatures and field capacity of soil water at approximately 60 percent work the best for nitrogen release.\n\n  5. Most soil bacteria need a pH between 6.0 and 8.0 to perform well.\n\n# Best Cover Crops\n\nLeguminous cover crops take nitrogen from the air and put it in the soil. They will also often attract beneficial insects.\n\n  * Clover, Berseem _(Trifolium alexandrinum)_\n\n  * Clover, crimson _(Trifolium incarnatum)_\n\n  * Clover, red _(Trifolium pratense)_\n\n  * Clover, subterranean _(T. subterraneum)_\n\n  * Clover, sweet (white and yellow) _(Melilotus alba; M. officinalis)_\n\n  * Clover, white _(Trifolium repens)_ Ladino ( _T. repens_ forma _lodigense_ )\n\n  * Cowpeas _(Vigna unguiculata)_\n\n  * Field peas ( _Pisum sativum_ var. _arvense_ )\n\n  * Medic _(Medicago)_\n\n  * Vetch, hairy _(Vicia villosa)_\n\n  * Vetch, woolly-pod _(Vicia dasycarpa)_\n\nHere are some non-leguminous crops that are compatible with legumes. These furnish large amounts of biomass and increase nutrient-holding capacity when planted as a mix with legumes.\n\n  * Barley _(Hordeum)_\n\n  * Buckwheat _(Fagopyrum esculentum)_\n\n  * Oats _(Avena)_\n\n  * Rye, winter _(Secale cereale)_\n\n  * Ryegrass, annual _(Lolium multiflorum)_\n\n  * Sorghum\/Sudangrass hybrids _(Sorghum bicolor_ x _S. bicolor_ var. _sudanese)_\n\n  * Wheat _(Triticum)_\n\n### Feeding Livestock in Rotations\n\nA crop rotation with legumes and non-legumes provides your farm with the opportunity to increase organic matter and soil fertility. The main purpose of pasture is for livestock feed\u2014either by grazing animals directly on the pasture or by cutting and storing it for later feeding. Some of your fields will probably be in permanent pasture and some will be mowed for hay.\n\nMowing or grazing a plant causes some roots to die. Roots continually grow and die throughout the season. As stated earlier, the root tonnage per acre often exceeds the plant yield threefold, returning organic matter back to the soil.\n\nThe time needed for a pasture to recover after mowing or grazing depends on several factors, like fertility, soil, and moisture, but mostly on the degree of defoliation (what percentage of the plant was removed). The higher the percent removed, the longer the recovery or regrowth period. Mowing removes plant growth uniformly; grazing does not.\n\n#### Mowing\n\nIf pasture grass reaches 12 inches or taller, it should be harvested for hay. At early bloom, non-legume plants have reached their maximum growth. For legumes, the point where they have the most nitrogen is early bloom. This also produces the highest-quality hay, if you intend to mow it. With legumes, yield goes up but quality goes down after this stage of growth.\n\n#### Grazing\n\nJim Gerrish, an intensive-grazing expert at the University of Missouri, says that in general, animals should be turned into pastures when grasses are 6 to 10 inches in height and grazed to about half the current height. This avoids cropping the lower, growing stage of the grass. In other words, \"Graze half and leave half.\"\n\n#### Management-Intensive Grazing\n\nThere are several types of grazing methods you can use, but management-intensive grazing seems to be the best method now available. Basically, management-intensive grazing is having a minimum of sixteen grazing paddocks on which cattle are moved every one or two days, with each paddock receiving as long a rest period as possible.\n\nIn this system, grass is a crop and cattle, sheep, or goats are your combines. You \"harvest\" your grass when ready, just as you would corn or beans. With a standard rotation, you use only 35 to 40 percent of the forage available. Management-intensive grazing uses 80 to 85 percent of the available forage.\n\nIn New Zealand, farmers using this system are getting 700 pounds of beef per acre without irrigation or fertilization. On irrigated alfalfa, close to 2,000 pounds of beef per acre has been attained. The upper limits of this method of grazing will be determined by forage varieties, weather, and your management, combined with cattle genetics.\n\nA good rule of thumb is not to have any cattle on any paddock more than 4 days. You may have to shift your livestock at different rates, depending on the time of year. In Missouri, fescue needs 60 to 90 days rest in July and August, but 30 days would be too much in May, when grass is growing rapidly.\n\nIn a standard four-pasture rotation, the forage is used 25 percent of the time and rested 75 percent of the time. In a sixteen-paddock management-intensive grazing system, the forage is used only 6 percent of the time and rested 94 percent of the time.\n\n#### Pastures for Erosion Control\n\nPastures will do more than provide food for your livestock. Sod pastures, which provide a thick, even sod covering with the grasses interlocked, are important in controlling erosion. Kentucky bluegrass and perennial ryegrass grow into sod pastures. In contrast, many forages (e.g., alfalfa and warm-season grasses) grow in bunches. The better the sod, the more erosion is controlled\u2014which keeps the soil and nutrients you want to have on your land.\n\nWeaver and Noll, in 1935, tested an upland silt loam soil with a 10 percent slope and grass 2 inches high. Under those conditions, excellent sod in a 1-inch rain in 30 minutes had only 12 percent runoff and no erosion. Fair pasture had 55 percent runoff with some erosion. Poor pasture had 75 percent runoff\u2014a soil loss of 4.6 tons per acre, with the only difference being the sod under the grass!\n\nIn _Quality Pasture_ , Allan Nation notes, \"Usda studies in 1930 showed that rotating cropland through sod-farming pastures for three to five years increased usable rainfall in Iowa by 6.4 inches a year over continuous corn. Grass crops absorb 87.4 percent of the rainfall versus 69.4 percent for a field of corn.\" In other words, good pasture increases the amount of water you can save for your farm\u2014which leads us to the next chapter.\n\n###### Food For Thought\n\nThe soil is the foundation for your farm. Good soil allows you to create an abundance of nutritious crops and healthy livestock. Take care when choosing crops and livestock, and apply management practices that will allow your farm to maintain or increase its soil fertility.\n\n## Chapter Five  \nWeatherproofing Your Farm\n\nThe climate and weather of your particular region are natural resources. Although we seldom think about it, our soils and their fertility are a direct result of climate, because climate determines which plants will grow and which animals will inhabit the area. Weathering of rock, combined with the dead plants and animals, makes up the nutritional basis or fertility of your soil.\n\n### The Effects of Climate\n\nIt is important to match the climate with the type of farming you wish to do. Although you can grow just about anything you want wherever and whenever you want, it may not be cost-effective. For instance, growing catfish in Alaska would be too expensive because water temperature needs to be 65 degrees Fahrenheit for the catfish to begin eating and at least 80 degrees for quick, efficient growth. The cost of constantly heating a catfish tank or pond would be prohibitive.\n\n_Planting a shelter of trees for your farm will provide microclimates that enable your crops and livestock to be more productive and protected from weather extremes_.\n\nLook for crops and livestock that thrive in the climate and weather of your region. Consider the crop's or livestock's home climate. If a breed of cattle or sheep has been raised in an arid region of the country, it may not do well if it is raised in a humid, wet area, which might cause parasite problems. Consider what is already being raised in your area. If you are new to farming, ask older farmers in the region what crops they grow now and what crops used to be grown in the area.\n\n#### Temperature\n\nVarying temperatures determine what plants grow best in what region, inspiring nicknames such as the Corn Belt and the Cotton Belt.\n\nLow temperatures are not necessarily bad; many plants, bulbs, and seeds need a period of exposure to cold in order for them to germinate and grow properly. Snow is an excellent insulator and is very important in producing a good wheat crop.\n\nPlants' life cycles are tied to the seasons and to temperature changes. Annual plants live only 1 year, producing seed, then dying when the temperature drops. Perennials die back to the ground after a hard freeze and survive over winter to grow again next year.\n\n### Water\n\nWater (or the lack of it) affects everything on your farm\u2014you, your livestock, your plants, and your soil. Living creatures need water for digestion and secretion, to regulate body temperature and feed tissues, to serve as a lubricant for joints and muscles, and to protect embryos in the womb.\n\n#### Water and Livestock\n\nAn embryo is 90 percent water. A newborn calf is 75 to 80 percent water. An adult cow is 50 to 60 percent water. Animals can lose all their fat and half their protein and still survive, but the loss of one-tenth of their body's water means death. You can quickly see how important water is to your farm.\n\nCatch every drop of water that falls on your farm through cropping and tillage methods that slow down evaporation and water flow.\n\nAnimals may need large quantities of water. An adult human normally requires 5 to 6 pints of water daily. A mature cow requires 50 to 90 gallons per day. This does not mean you need a lake for each animal. Adequate forage reduces the need for free-standing water. Sheep can go weeks without drinking and may thrive without freestanding water if they have good pasture. The nutrient value of crops is also affected by the amount of moisture available to them.\n\n#### Water and Plants\n\nIf soil is low in water, plants will find it difficult to grow or to perform all the functions that water facilitates, like transfer of nutrients through roots to other plant parts. Minerals in the soil can be absorbed by plants only when the minerals are in solution. Water is necessary for photosynthesis, the process by which plants extract energy from the sun. Without water there would be no microorganisms in the soil. Some of these organisms are good, like teramycin (used as antibiotics), and some are bad, like apple scab fungus, whose spores are loosened by large raindrops, but they all require water.\n\nWater Needs Vary\n\nThe amount of rainfall is a major factor in determining which plants grow best in your region. Xerophytes are plants that can grow with limited water. Xerophytes may be succulent or nonsucculent. Succulents are plants such as cactus and milkweed that contain water storage cells to sustain them.\n\nWestern regions of the United States are known for their grass production, as many grasses can tolerate low natural rainfall. Gama grasses, curly mesquite, saltbush, and shrub oak all have xero-graphic (arid-tolerant) modifications. These plants grow quickly in the spring or following a rainy season. The shoots mature and then die, but the roots may survive for a year with no rain. The dried tops will furnish food for livestock. In southwestern Texas, sheep and cattle are grazed year-round on these grasses, which cure on the stem. We cannot change the weather, but we _can_ take advantage of it.\n\nHydrophytes, in contrast to xerophytes, are plants that grow in or near water, such as water lilies. Mesophytes fit between the two extremes, and they include most crops commonly grown. They also include both warm-season grasses (big bluestem, for example) and cool-season grasses, such as fescue.\n\nLike most plants, corn needs water most when it is growing rapidly, such as when the silk is being fertilized to form the corn seed (kernels).\n\nPlants lose immense amounts of water through transpiration (evaporation). The rate of transpiration increases with temperature and light intensity. In addition, as plants grow they gain more leaf surface, which increases their rate of transpiration. Actively growing plants contain more water than they do solids. For every pound of plant dry matter, there are 5 to 10 pounds of water\u2014but to produce that pound of dry matter requires several hundred pounds of water. The most efficient plants (pineapple, cactus, milo, gama grass, for example) use 250 to 300 pounds of water to produce 1 pound of dry matter. Less efficient plants (pumpkin, squash, cucumber, for example) may use 1,000 pounds of water to produce 1 pound of dry matter. Corn may transpire up to 2 quarts of water per plant per day, or 300,000 gallons of water per acre. Aquatic plants merely need to absorb the water\u2014about 10 pounds per pound of growth. Which plants you choose will depend on your climate.\n\n# Climate And Livestock\n\nClimate and weather affect livestock both directly and indirectly:\n\n  1. The climate can cause species to evolve with specific adaptations to that temperature. Some breeds of cattle, like Santa Gertrudis, were developed for hot-weather areas and have shaker muscles that help them get rid of flies, which are more numerous in hot areas. Some woolly animals, like alpacas, will find it difficult to cope with high temperatures, but do well in colder regions.\n\n  2. Indirectly, weather affects the vegetative growth in a region, which in turn makes some breeds more adaptive to that region. Irish Dexter cattle, for instance, were developed in mountainous areas having short growing seasons and sparse vegetation; they are a small animal with a dual purpose for meat and milk. Just as there is an inherent fertility in soils, livestock has an inherent capacity to produce the best under whatever are their optimal climate conditions.\n\nMost plants need water when their growth is rapid. For instance, corn needs water when the corn is silking the ear and being fertilized by the tassel (the male part of the plant) to form the corn seed (the kernels). Insufficient water at this time will cause ears to fill unevenly with kernels. Tree fruits and berries are another example: their water requirements are highest just before maturity, when they increase greatly in size. This means that you must keep track not only of how much rain falls in your area, but also of when it falls.\n\n#### Climate and Parasites\n\nParasites and diseases thrive in warm, humid weather. Tomato blight, for instance, can attack tomatoes if the weather is hot and humid, causing leaves to yellow and fall off, reducing the plant to survival mode or death\u2014and this leads to a decrease in production.\n\nWheat harvest in the Midwest is associated with the hot, dry days of July and August, but if the weather is hot and humid, a disease called rust may cut yield in half. The wind can carry rust spores from mature wheat in Canada all the way to Texas, affecting plants along the way. When it rains, spores are deposited on newly sprouted small grains.\n\nWeather and climate also influence the degree of parasites in livestock. Roundworms, for instance, need moisture and rainfall for the larvae to be able to infect hogs and turkeys.\n\nRain is the main cause of worm problems in sheep and goats, particularly when combined with warm temperatures. Worms are not as much of a problem in the drier western regions and the cold northern regions as they are in the South. There, the warm, humid weather allows worm parasites in sheep to flourish.\n\n### Altering Your Farm Environment\n\nSo far, we have looked at principles of weather and climate and how they affect our crops and livestock and the choices we might make regarding what we will raise. We know that we cannot control the weather, so it is important to find ways to adapt to our conditions by protecting against the elements and conserving water. Cooperating with seasonal weather cycles will also reduce the need for purchased feedstuffs and eliminate birthing in bad weather. For example, I plan all my calves to birth in April and May, when the weather is settling and the grass is growing.\n\n#### Providing Shelter\n\nAnimals on pasture will use shelter only when needed, but it is good to have in rough weather\u2014particularly for young livestock.\n\nIf you are on a small acreage and have no access to land with previously built shelters or naturally occurring livestock protection, you will have to build some sort of shelter. In my opinion, the best animal shelters are three-sided, open to the south, well bedded with straw, and portable. Some people even build shelters from hay bales. Smaller, portable shelters allow for multiple uses for crops and livestock, and will change as your farming operation evolves. A shelter under 40 inches high is best for small livestock\u2014sheep, hogs, and poultry\u2014to prevent rain and snow blow-in.\n\nMultiuse barns should be avoided. They are harder to adapt, are fixed in place, and with multiple livestock types in one building, can increase disease problems. If your building cannot be portable or you already have one in place, and you use the building for birthing animals, do not use it again for birthing for 6 months or more. This will help to prevent disease.\n\nDifferent animals need different kinds of shelter at different times of production. For instance, a thick cedar grove is excellent for calving or lambing. Cedars have vast amounts of fallen needles for bedding, and usually do not have low-hanging branches in a thick stand. They shed snow and rain in all but the heaviest downpours, and sheep like to be underneath them. I rotate pastures so that when it is time to lamb, the pasture with my cedar grove has grass that has not been grazed to the ground when I need it for nursing mothers.\n\n# The Best Shelters\n\nThe best animal shelters are:\n\n  * Three-sided\n\n  * Open to the south\n\n  * Well bedded with straw\n\n  * Portable\n\n  * For small livestock, less than 40 inches high\n\nThe best animal shelters are portable, three-sided, open to the south, and well bedded with straw.\n\nMost livestock are pretty good mothers and do better on their own, rather than being confined to a small farm lot with a high potential for disease. They do especially well birthing on grass pastures when the grass is really growing (April\/May for the Midwest) or whenever the temperature is in the 60s in your area. Sunlight and fresh air go a long way toward keeping your livestock healthy.\n\n##### Predator-Proof Poultry Shelters\n\nIf you have predator problems, a fenced-in shelter to which you lead the livestock at night is essential. Poultry, in particular, need a place where they will be protected from coyotes, owls, raccoons, opossums, cats, and other predators. This may be a large poultry house to which chickens are returned each night, or a small portable coop such as a Smedley unit (normally for hogs).\n\nA popular idea today is the chicken tractor, where poultry are kept in a portable 10-by-12-foot cage covered with chicken wire, with half of it covered in aluminum roofing. These open-bottomed shelters allow you to move your chickens daily through your garden or fields to eat insects, forage for weeds, and incorporate their droppings in the soil without damaging your plants.\n\n##### Sheds\n\nYou will also need shelter for machinery. If your machinery is left out in the rain and sun, you are just throwing your money away. Rust and weather wear shorten the life of tools more quickly than does anything else. A machine shed where you can keep your tools and machinery dry, store repair equipment, and have a place to work on your tools will enhance your machinery, save you money, and improve your quality of life.\n\nFinally, I also have a shed with a raised floor where I store my feeds, conveniently located near my livestock. It is carefully sealed to keep out rodents\u2014and it does so most of the time.\n\nA chicken tractor is a win-win situation. It lets you move your chickens to a different spot in your gardens or fields every day, where they can devour insects and weeds and add their droppings to the soil.\n\n#### Conserving Water\n\nThere are two ways to provide water for your farm. The first is irrigation (drip or standard), which waters crops when they need it. Irrigation facilitates practices such as foliar feeding (spraying a liquid fertilizer on the plant leaves) and incorporating fertilizer into the water. The drawback is the expense\u2014the cost of irrigation equipment and the cost of the water. To provide water 12 inches deep on 1 acre (43,560 square feet) requires 325,850 gallons of water (this is called 1 acre foot of water). One acre inch of water equals 27,154 gallons.\n\nThe other method of providing water for your farm is simply to conserve every drop of rain that falls on it. This has the advantage of being sustainable and cheap. Even if you plan to do some irrigation, conserve water to keep your costs to a minimum. Here are some management strategies to save water:\n\n  * Make sure your soil is worm-friendly (see Earthworms on page 56), which helps hold water in the soil.\n\n  * Make sure your land has crops (including pasture and cover crops) on it during rainy seasons, to prevent runoff and to hold water.\n\n  * Plan your field rows to retain water rather than lose it\u2014this means that if you plant on hillsides, for example, always terrace your rows to prevent runoff.\n\n  * Always provide a place for water flow to accumulate, rather than leaving your farm.\n\n##### Ponds\n\nEven farms with the best soil will have rainfall that causes runoff. Ponds are an effective method of storing water for later use. Ponds\u2014or tanks, as they are called in the western states\u2014can be used for many purposes: crop irrigation, watering livestock, home water supply, fire safety, erosion prevention, wildlife habitat, swimming, and fishing. A pond should be fenced to water livestock below the dam and keep the water pure and clean for human use.\n\nIf stocked with fish, the pond can add income to your farm through fee fishing. You can also consider aquaculture in on-farm ponds. Freshwater shrimp can be raised in ponds as far north as Kentucky, and crayfish even farther north. I raised cat-fish for many years, with an on-farm fish-processing plant (consisting of a room with two sinks and a freezer). I sold the fish to local grocery stores.\n\n##### Cover Crops\n\nWe briefly discussed growing cover crops in chapter 4, but I would like to focus on the weatherproofing benefits of this practice. Cover crops slow down both wind and water erosion, as the thickly sown plants form many little surface dams to slow water and soil movement. The root systems do the same below the surface. With slower movement, the water has a chance to soak in, allowing plants to have an increased supply of water later in the season. Cover crops also stop nutrient leaking. Winter rye, for instance, sops up excess nitrogen in the soil, while hairy vetch is good at hanging onto phosphorus. The plants release these nutrients when they are plowed or disked under as green manure.\n\n##### Timing\n\nWhen you plant your crops is critical in regard to saving moisture. Every time you work the soil for planting or cultivation, you do two things. First, you lose some moisture from the soil. Second, stirring the soil increases the microbial action within it, burning up nutrients and organic matter.\n\nThis means that green-manure crops need to be turned under and left to rot for about 2 weeks before you plant corn. The green-manure crop from last fall is a storage base that holds plant nutrients in place over the winter. They become available to your new crop when you turn them under. However, turning them under does cause loss of moisture. It is important to turn under fall crops early in the spring to keep them from using the moisture needed for your new crop year.\n\n#### Creating a Microclimate\n\nYou cannot change the weather, but you can modify its effects by creating microclimates. A beneficial microclimate is an area that is protected from the worst effects of wind, weather, and temperature. It often tends to stay frost-free when surrounding areas are not, or it may offer protection from a hot sun in July.\n\nExamine your farmland to see if you have any natural microclimates. These will be sheltered areas, possibly in a depression where the wind does not blow (although depressions often suffer from frost problems) or on a low hillside that is not as affected by wind or frost. If possible, walk over your property on a frosty morning in late fall or early spring and see if any areas are less affected\u2014or not affected at all. These will be good places to utilize. You may wish to modify them further, to enhance the effect.\n\nA little record keeping goes a long way. Buy a soil thermometer and test soil daily. Common plants such as lilac enter the first leaf and first flower stages of growth at specific temperatures and weather conditions. Checking temperatures and dates, and comparing these over the years, will make your farm more successful.\n\nIf there are no natural microclimates, create your own. In planning a microclimate, the best place to start is with a wind-break, which I will discuss in more detail in the following section. The windbreak controls a large microclimate, which sets the stage for a more subtle microclimate in your garden or planting area. Keep the ground protected, for example, with plastic or straw, to create a warmer environment for seedlings, hold extra water, and reduce weeds. (I tend to avoid plastic because it creates a mess when it breaks down; straw, on the other hand, breaks down into extra organic matter.)\n\nAir flows just like water and settles in low spots. A raised bed might be just high enough to keep plants away from frost. Raised beds also provide warmer soil temperatures in the spring, sometimes by as much as 5 to 10 degrees.\n\nIt is also possible to create a microclimate for an individual plant by use of season extenders or similar tools. Research by D. R. Paterson and D. R. Earhart at the Texas Agricultural Experiment Station reported that when tomatoes were grown in black plastic with cages and then the bottoms of the cages were wrapped with common roofing paper, losses due to wind and hail decreased by 46 percent. After 5 weeks, there was a 62 percent increase in growth, with 86 percent more marketable fruit and a 49 percent increase in total yield. This data may not apply to all tomato varieties, but the method is worth a try if you have a wind problem early in the season.\n\n#### Windbreaks and Sustainability\n\n**Principle :** _Plant or create windbreaks to prevent heat and moisture loss_.\n\nWind can be a problem, as it leaches heat and moisture from crops and livestock. Therefore, crops and livestock protected from wind use moisture and nutrients more efficiently. One way to remove wind as a problem is through a windbreak.\n\nWindbreaks reduce wind velocity, slow wind erosion, and create microclimates. Soil particles move when wind speeds are 13 miles per hour, 1 foot above the ground. When wind flow removes fine soil particles from one field, organic matter and nutrients go with it, so you have to increase fertilizer input or accept lower economic production in your crops.\n\nDecide where to plant your windbreaks based on the direction of the prevailing winds. In most areas of the United States, you'll need to plant windbreaks to the north and west of the area you want to protect. There is more information on height, length, and density of windbreaks in the section that follows. In general, though, windbreaks should be at least 100 feet longer than the area you want to protect, to prevent wind from whipping around the windbreak edges. For home protection, windbreaks should be 100 to 150 feet long, located no more than 300 feet away from the house, yard, and outbuildings.\n\nA wind velocity of 5 miles per hour is slowed to \u00bd mile per hour in the lee of a windbreak\u2014a 90 percent decrease. A 30-mile-per-hour wind will be reduced by 50 percent, to 15 miles per hour, by a windbreak. The lee can be a distance of up to thirty times the height of the trees (see next section).\n\nUniversity research shows that heat energy savings of up to 40 percent are possible when you use windbreaks. One study showed that a house with a constant temperature of 70 degrees Fahrenheit protected by a windbreak requires 23 percent less fuel than a house exposed to the full sweep of the wind. Permanent windbreaks on 40-acre fields at the University of Nebraska's Mead Research Station increased soybean yields by 18 percent, corn yields by 20 percent, and wheat yields by 22 percent.\n\nWindbreaks are a worthwhile investment. James R. Brandle, Bruce B. Johnson, and Terry Akeson, in a University of Nebraska study, found that \"a full windbreak occupying 5 percent of the field is economically viable\" and will \"more than compensate for the cost of establishing the windbreak and the loss of output from acres taken out of production.\"\n\nWindbreaks slow wind erosion, reduce water evaporation, and moderate temperatures to protect your crops and livestock.\n\n# Thirteen Benefits Of Windbreaks\n\n  1. Increase crop yields as well as hay and pasture yields\n\n  2. Reduce soil erosion\n\n  3. Trap winter moisture for spring crops\n\n  4. Reduce water and nutrients needed by crops\n\n  5. Moderate temperature and humidity\n\n  6. Reduce crop input costs\n\n  7. Provide valuable wildlife habitat, such as nesting places for birds, which feed on insects in the fields\n\n  8. Reduce heating costs 10 to 40 percent when placed around the farmstead\n\n  9. Provide tree products, nut products, and suitable areas to grow high-dollar medicinal herbs like goldenseal, ginseng, and brambles (e.g., blackberries and raspberries)\n\n  10. Produce firewood and fenceposts\n\n  11. Provide as much as 50 percent savings in feed costs for livestock\n\n  12. Reduce wind damage to crop plants\n\n  13. Spread snow evenly in a field\n\n##### Kinds of Windbreaks\n\nTo determine the best kinds of trees for windbreaks, consult your local university extension office or state Department of Agriculture. You will choose trees based on their lifespan, density, growth patterns (for example, evergreens do not shed needles, and thus provide more protection later in the year), and height. Trees or shrubs can be planted in single rows or in mixed groups.\n\n**Height**. The height of a windbreak determines the protected area. A rough formula predicts wind speed reductions in an area of 2 to 5 times the height of the windbreak on the windward side and up to 30 times the height on the leeward side. This means that with 30-foot trees, the protected zone spans 60 to 150 feet on the side the wind is coming from and up to 900 feet on the side away from the wind.\n\n**Length**. The length of the windbreak determines the amount of area receiving protection. According to windbreak expert James Brandle, the maximum efficiency for windbreaks requires that the length be 10 times the height.\n\n**Density**. Varying the density of the windbreak can influence what you do with it. For instance, 25 to 35 percent density is best for even spreading of snow across a wheat field but will not control soil erosion as well as a 40 to 60 percent density. Evergreen trees, among them Eastern Red Cedar, are fairly dense. They are good choices for windbreaks because they don't lose their needles in wintertime, unlike deciduous trees (broadleafs) such as oak.\n\nThe main advantages of tree windbreaks are their height (30 to 50 feet) and longevity (50 years). Trees are not the only wind-break materials, however. Perennial grasses and legumes can also be planted in strips. As well, small grains like rye, wheat, and oats can be planted in strips to slow wind erosion, conserve moisture, and provide a suitable microclimate for vegetables that need more wind protection, such as onions and early-planted cole crops. Corn rows planted in a field of soybeans or wheat grass in wheat fields are also beneficial. Two rows of corn are usually alternated with sixteen rows of soybeans or four to sixteen rows of corn and soybeans are planted in alternate strips.\n\nWheat grass is planted in double rows spaced 36 inches apart, every 48 feet across the wheat field; wheat grass grows about 5 feet tall. Annual plant windbreaks can be placed around an area perpendicular to the prevailing wind direction, just as with trees. In most areas of the United States, this means placing the windbreak on the north and west sides of the area to be protected. If there is no predominant wind direction, Laurie Hodges and James R. Brandle recommend planting annual windbreaks with the rows closer together, following the land contours or in a serpentine pattern, to slow the winds and protect the other plants.\n\nWindbreaks are not made solely of trees. For example, you can use corn as a windbreak to provide a beneficial microclimate for your soybeans, increasing their yield.\n\nStrips of small grains can also be used as lures to keep insects away from your cash crop. Insects prefer the small grains and will remain in them, rather than eating other crops. It is suggested that the strips of small grains be 45 to 60 feet apart in your field and of a width appropriate for your equipment.\n\n### Extending the Season\n\nSeason extenders are a way to get crops growing early or to lengthen their growing period late in the year, to allow you to have crops when other farmers do not. This is an easy way to reduce competition and increase your profits. Greenhouses, cloches, tunnels, and floating row covers can all be used to extend your growing season.\n\n# Solar Growing\n\nAccording to Shane Smith, author of the _Bountiful Solar Greenhouse_ , food production in greenhouses actually had its beginnings in a medical prescription for the Roman emperor Tiberius Caesar. He was told by his doctor to eat a fresh cucumber each day\u2014even in winter\u2014so his workers created a pit in the earth and covered it with a transparent stone. Manure may also have been used for its heat-producing qualities.\n\nShane Smith is a founder of the Cheyenne Community Greenhouse in Wyoming. Cheyenne is a cold place to live, and it ranks fourth highest in the nation for yearly average wind speed. The Greenhouse, a community project, is 5,000 square feet and is 100 percent passive solar heated. Two hundred 55-gallon black-painted drums containing water store solar heat for the greenhouse. The north, east, west, and roof are heavily insulated. The Greenhouse has never required back-up heating, even during the coldest weather.\n\nAnna Edey's year-round solar greenhouse, Solviva (Swedish for \"sun life\"), is located in Martha's Vineyard, a small island off Cape Cod, Massachusetts. Solviva houses both crops and livestock, and uses only the sun and animals' body heat for warmth. Anna has 100 chickens laying eggs in the greenhouse to supplement the 80 to 100 pounds of salad greens that are produced every week. She grosses $70,000 to $100,000 per year; the 104-by-28-foot structure cost only $30,000 to build in 1983.\n\nYou can grow lots of bedding plants in a small greenhouse\u2014up to 5,000 plants have been grown in a 16-by-16-foot space. These two examples of permanent freestanding greenhouses have production figures of 3 to 4 pounds of food crops per square foot. Your efficiency will improve as you learn, so don't be afraid to start.\n\nIf these examples have you interested in greenhouses, but you're not sure about the investment, there are lots of ways to extend your growing season cheaply, and to get a feel for solar growing:\n\n  * If you have a south-facing house, barn, or shed, just lean a window, a sliding glass door, or a piece of plexiglass against the building. Cover the ends with plastic or straw bales, and you have a miniature greenhouse for early greens or starting plants for transplanting.\n\n  * Lay out several straw bales in parallel rows. Cover the rows with window panes, or 2-by-2- or 2-by-4-inch wood frames covered with plastic. Placing milk jugs painted black and filled with water between the bales will help dispense heat at night when things cool down. Bales placed crossways at the ends of the two rows of bales will hold in even more heat at night, and provide excellent protection against the wind.\n\n  * A cold frame is simply a wood frame covered with a window pane. These mini-greenhouses are easily transportable.\n\n  * Milk jugs make great tiny greenhouses to start plants in the field early in the season. You can start sweet potato slips in April under milk jugs. Just cut off the bottom of the jug and place it over the plant, pushing it far enough into the soil to hold against winds.\n\n  * Raised-bed gardens with hoops of Pvc pipe covered with plastic make good season extenders. Raised beds are large frames of wood (or tire rubber), usually placed with rebar rods driven beside them for structure, then filled with soil. A typical raised bed might measure 5 by 30 feet. The hoops and plastic can also be used on flat-ground gardens.\n\nI have a closed room in my machine barn in which I start tomatoes, peppers, melons, and herbs early. I set hooks in the ceiling and string fluorescent lights about 4 inches above a series of folding tables. ( _Note:_ Fluorescent lights must be replaced at least every other year, as their intensity does fade.) I plan planting dates to allow seeds to sprout and grow to about \u00bd to 1 inch 2 weeks before frost-free dates in spring. Once they are growing well, I move them outside during the day, placing them in a ring of square hay bales to protect them from the sun. This hardens them off (adjusts them to outdoor temperatures) and prepares them for planting. On the first frost-free date, I transplant them.\n\n# Growing Potatoes\n\nPlant potatoes as early as possible in the spring. Hill them when they are 3 or 4 inches tall. (To hill a potato, mound the earth around it until the hill is 8 or 9 inches high and 12 inches wide at the base.) Then spread straw heavily on top, which not only conserves moisture already in the ground, but also soaks up any new rain and virtually eliminates weed problems.\n\nSeason extenders will also save moisture. Covering garden beds with black plastic or mulch early in spring will hold in moisture enough for you to grow a good tomato crop with little or no added water. Mulch such as straw keeps the ground moist through the hottest summer months.\n\n### Caring for Livestock in Winter\n\n**Principle :** _Let your animals learn to cope with winter_.\n\nYour livestock may need better shelter in winter to protect them from the wind. Overall, though, the best thing you can do for your animals is to let them learn to cope. In northern areas, cattle will learn to graze through snow and can use the snow as a water source.\n\nWe are apt to do too many things for our animals that are not conducive to developing good, hardy breeding stock. Stock that can survive on the land on its own will minimize our input in time and supplies\u2014allowing our labor to be used elsewhere and saving money.\n\n#### Providing Water\n\nSheep obviously need some water, but can usually do quite well on range or pasture with what nature provides. I have seen my own Katahdin Hair sheep go without water for up to a week in the winter, even though it was available to them.\n\nCattle, hogs, rabbits, poultry, and horses require a source of freestanding water to drink. This means that you must check your water source each morning and break any surface ice so the water can be reached.\n\nIf you have only a few head of livestock, the best way to water them in winter is with buckets of warm water. If your livestock numbers are large and spread out, an automatic energy-free waterer saves both time and water. You still need to visit your livestock every day, though, to check that the device is working.\n\nHealthy livestock can easily withstand subzero temperatures if they are relatively dry and protected from the wind. In this example, the hills serve as a windbreak.\n\n#### Providing Feed\n\nLivestock require more feed in winter, as pasture grasses decline and their nutritive value decreases. Plan to store crops such as ear corn and hay for your winter feeding. You might also investigate cold-hardy forage crops that will allow your stock to graze into late winter\u2014or all winter, in the South.\n\n###### Food For Thought\n\nSustainable agriculture is an ongoing discovery of how things work and interact on your farm. Saving every drop of water that falls on your land involves using tillage methods like fall-sown cover crops, windbreaks to slow moisture depletion, and terraces and dense wind-breaks to distribute water and snow. By seeing the interconnectedness of how one thing affects another, we can put together a series of practices that will make a farm sustainable and successful.\n\n###### Part Iii\n\n## Planning and Marketing\n\n###### Chapter Six\n\n## Your Goals and Farm Planning\n\nWhy do you need goals for your life and your business? The answer is simple: If you don't know where you are going, how will you know when you get there? The more goals you write down, the better your chances of success.\n\nGoals vary. There are short-term, intermediate, and long-term goals. There are personal goals, family goals, business goals, and farm goals. A family goal might be to develop a system of farming that allows you to spend more time with your children. A personal goal might be to have enough farm income to quit your town job and farm full time. A business goal might be to achieve a 20 percent return on your total investment. A farm goal might be to develop a system of farming that is economically viable, socially acceptable, and environmentally sound.\n\nAs you can see, goals can overlap, and different goal aspects can be present in a single goal. The important thing is to figure out what your goals are, so you can plan your farm.\n\n_Diversified crops give you a multitude of markets\u2014and marketing options\u2014throughout the year_.\n\n### Planning\n\nA young couple, Earnie and Martha Bohner of Persimmon Hill Berry Farm, spent 3 years accumulating budgets and information on various crops and livestock before they chose the enterprises they thought suited their soils, location, and monetary goals.\n\nAfter determining what they wanted to grow, they established a 10-year plan, a 5-year plan, a 3-year plan, and a 1-year plan, broken down into what to do every 2 weeks.\n\nIn the process of studying various enterprises, they attended conferences and seminars, bought books, and talked to farmers. This intense study of the crops they were interested in exposed them to different tillage methods, irrigation methods, and marketing methods. Some of the knowledge that they gained from farmers, instead of from a book, changed their goals or the order in which they chose to accomplish their goals\u2014some would take longer and some could be achieved more quickly. The Bohners eventually decided to start with U-pick berries and shiitake mushrooms. (More information on their operation can be found in chapter 7.)\n\nEarnie and Martha worked full-time jobs while accumulating their information. Will you need 3 years of study to put together a perfect plan? That will depend on your available time and persistence for the project. You might need only 6 months to a year\u2014but remember, \"Nobody plans to fail, but many fail to plan.\"\n\n**Principle :** _Planning and flexibility are two keys to farm success_.\n\nIn addition to planning, you must be willing to be flexible and shrewd. Take advantage of new ideas, new markets, and any opportunities that arise. Right after college I was a foreman on a sod-laying crew, and the bluegrass sod we were cutting belonged to an elderly farmer who used to tell me about the Depression. He said fat hogs were selling for 3 cents per pound. One day there was a really bad ice storm. He loaded his hogs carefully and hauled them 125 miles to the St. Louis market. Because of the storm, there were too few hogs there to satisfy the buyers that day, so his hogs brought 5 cents a pound. He took advantage of the weather to increase his profits.\n\nMonetary cycles and cash-flow shortages can make any enterprise\u2014no matter how much you love it\u2014into a living hell, draining both your enthusiasm and your pocketbook. A sustainable, diversified farming operation gives you the best chance of earning a living from the farm and still having a life. Yes, I said the best chance\u2014no guarantees. The best and most frugal farmer in the world cannot survive unless he or she can pay the bills, good weather or bad. Planning and flexibility will allow for a steady cash flow.\n\n#### A Planning Example\n\nLet's take some space on your farm for a planning example, without considering budgets or marketing. You have decided to plant some vegetables.\n\n**Slope**. First, consider the land itself. Most vegetables need a warm environment in order to prosper. A gentle slope is a good place to plant. Why? Well, the sun is the primary source of heat for soil. The heat is greater in the summer when the sun's rays hit the soil directly; in wintertime, the sun's rays hit the soil at a slant, because the sun is lower in the sky as a result of the tilt of the earth. But on a slope, the sun is able to make direct contact with the soil for a longer period of time due to the angle of the land. This means it will warm up and dry out sooner in the spring, and thus can be planted earlier.\n\nIn general, in the northern hemisphere, trees grow on northern slopes and grasses on southern slopes. Southern slopes tend to be warmer and drier than northern slopes.\n\n**Moisture**. The angle of slope of a piece of property can modify the climatic conditions on that land. On steep slopes, more rain runs off instead of penetrating the soil. Thus, the soil is drier and provides less water for your crops. On sloping ground, there is more moisture toward the bottom of the slope. Your choice of vegetables or field crops should take advantage of this moisture.\n\n**Soil**. You will also need to consider the soil of the slope. Sandy soils are porous and warm up quickly in spring. Clay and clay loam soils hold water and dry slowly in spring. Evaporation of water from the soil also reduces the direct effect of the sun's rays.\n\n**Planting**. Now consider how you will plant your early-vegetable crop. If this slope is more than 2 percent, you want to run your vegetable rows perpendicular to and around the slope, each row of plants forming a dam to catch water and slow the erosion of soil. If the slope is greater than 5 percent, different growing methods must be applied, such as strip-cropping (see box).\n\n**Management**. You'll also need to consider the effects of your other farm operations. For example, running livestock on the slope the year before you plant, or feeding hay to stock and concentrating the manure on the slope, may furnish all the fertilizer you need\u2014virtually free of charge.\n\nThis may seem like a lot of decisions to make just to plant some vegetables, but things will get easier. Once it becomes a habit for you to consider all your options, some decisions become almost instinctive.\n\n# Strip-Cropping\n\nStrip-cropping is used on land that is moderately sloping. Crops are planted in alternating strips, each strip being from 4 to 12 rows wide, horizontally across the slope. The idea of alternating strips is to allow for a thicker crop (hay, for example) to prevent water flow from eroding the slope, while still utilizing the slope for more high-value crops.\n\nVegetables, for instance, are surrounded by bare ground or mulch, so they need a thick crop alternating with them. The alternating strips might be corn and soybeans on a gentler slope, corn and hay on a steeper slope. Crops are rotated in the strips: for example, 2 years of corn, then the corn strip is sown to a hay crop, then the hay strip is plowed under, furnishing organic matter and soil fertility for the next corn crop. The hay strips may be grazed, using electric fence to keep sheep or cattle out of the corn. (See chapter 4 for more on crop rotations.)\n\n### Setting Goals\n\nFollowing is a list of sample goals. If these goals do not seem to fit your plan, make your own. For instance, the goals for a fee-hunting ranch would certainly different from those I have presented here. Your own list may be much longer or shorter, but the more effort you put into it, the greater your chances of success.\n\n#### Long-Term Goals\n\nLong-term goals, 10 years or more, should be written down and reviewed monthly to see whether you are proceeding in the right direction or if you have veered off track. As time goes on and your experiences or markets alter your plans, you may change your goals. Be sure to write down your altered goals and spend some time planning how accomplishing the new goal will differ from what you had previously planned.\n\nWrite down broad-scope goals as long-term goals. Some examples:\n\n**_Goal 1_. We want to farm**. This is a good broad goal, but if it is your starting point, you have a lot of research to do.\n\n**_Goal 2_. We want a sustainable, diversified, profitable farm because it gives us the best chance for success**. All of these are necessary for success, but the research still needs to be done.\n\n**_Goal 3_. We want a pasture-based farm; or we want a diversified crop and livestock farm; or we want to raise purebred animals**. Your research and needs will help you choose a direction before you decide on specific crops or livestock. The process of establishing a successful, respected farm takes several years.\n\n**_Goal 4_. We are from the city, so I will work part time for a farmer, maybe on weekends, to learn basic skills and ask questions, questions, questions, so that I can gather an idea of what I want to do. I will work for free if necessary to achieve this goal**. Everyone should have some method(s) of gaining knowledge as a long-term goal. You\n\nshould be learning about farming from your farm for the rest of your life. Sustainable agriculture is an ongoing process of discovery.\n\n**_Goal 5_. We\u2014the family unit, wife and children, my brother and I, my neighbor and I\u2014will be in total agreement about our monetary and labor goals**. It is great to say, \"Let's raise blueberries.\" But who does the planting, the pruning, the marketing? Will these chores be split? How? If the unit cannot agree at the outset, the project is doomed to failure. This is a long-term goal because it is broad in scope. Saying \"Sheila will plant the blueberries this spring\" is a short-term goal, but planning the family's overall direction for an operation is long term.\n\n**_Goal 6_. We will reduce input costs. We want a type of farming where crops and livestock complement one another, and return a greater profit to us while improving our soil, our net worth, and giving us a higher quality of life**. Reducing input costs probably involves either improving soil fertility (healthier soil requires fewer inputs) or increasing diversity. Soil fertility is a long-term goal (see Goal 16). Using diversity\u2014a combination of crops and livestock that complement each other\u2014can, with careful planning, reduce costs. For example, hogs eating your own homegrown corn reduces feed expenditures. Improving soil and net worth and improving quality of life should be separate goals. As individual goals, some of these might be achievable in a shorter time frame.\n\n_Goal 7_. We want to sell products that fit the organic market niche.\n\nThis is a more specific goal that may be achievable in a shorter time\u2014but until you know what you are doing, treat it as a long-term goal. What does it take to be certified organic? Will crop yields be lower, the same, or higher? Remember that being organic does not make your farm sustainable, and vice versa.\n\n**_Goal 8._ We will try any new project on a limited basis and keep good records. I will not bet the farm on hearsay or hype from universities, newspapers, magazines, and so on. I will first test it on my farm and also talk to other farmers who have done it**. Once again, the breadth of scope makes this a long-term goal. Any individual project to which you apply this principle will be a short- or intermediate-term goal. If the project works, it may revert again to a long-term goal. Often, it is worthwhile to write down what may seem like plain common sense as a goal. This allows you to come back to it and think about it periodically.\n\n**_Goal 9_. We will buy only that machinery that we absolutely need**. Machinery depreciates from the day you buy it, and when it gets rusty it really goes down in value. However, if you have good tool skills, used machinery and tools can be a really good buy.\n\n**_Goal 10_. We will keep good records and know our cost of production per bushel, per plant, per animal, per acre**. Records are a dayto-day short-term goal, an intermediate goal, and a long-term goal. To be really valuable, records must be kept on a long-term basis and evaluated periodically. Eventually, they will allow you to make profitable long-term decisions, based on the production capability of your farm. If you do not know your cost of production and you market directly to the consumer, how will you know what to charge?\n\n**_Goal 11_. We will use 5-year averages for budgets with a cash flow of 120 percent**. Say you are selling fresh brown eggs to neighbors, market customers, and a local restaurant. If your expenses are averaging $100 per week (you will average out your expenses over a 5-year period, adjusting for any unforeseen long-term changes, such as switching from grain-fed to range-fed), then according to your goal, your projected revenues should be $120 per week (120 percent of $100). The extra $20 is your profit, to apply to living expenses, savings, farm upgrades, and so on.\n\n**_Goal 12_. We will reduce or eliminate as many risks as possible in our operation**. Agriculture has major problems or risks that affect profitability, such as weather, prices, government regulations, disease (plant and animal), weeds, and poor soil fertility. If 20 percent of your income derives from overcoming these obstacles and 80 percent comes from your value-added direct marketing efforts, you have reduced your risk in farming. Value adding is important.\n\n**_Goal 13_. We will continue to educate ourselves every year**. Make an effort to attend and participate in conferences and seminars every year. Read as much as possible on research for your particular crops and livestock. Use this new information to update, change, or eliminate some goals after test trials on your farm.\n\n**_Goal 14_. We will direct-market as much farm production as possible**. Selling retail brings higher profits and more customer contact. Selling direct also gives you the chance to sell other products you raise to customers.\n\n**_Goal 15_. We will have a marketing plan with sales every month of the year**. Sales are essential every month, because you have bills every month. This is a short-term, intermediate, and long-term goal. Even if it is ultimately achieved, periodic reevaluation is required.\n\n**_Goal 16_. I will ceaselessly strive to build soil fertility by using cover crops, green manures, livestock, strip-cropping, and other soil-conserving tillage methods**. This is a lifetime goal. Improving soil fertility is a long, slow process, because every crop you raise on the land uses soil nutrients. You must carefully balance your choices of crops, livestock, and management techniques to improve soil fertility. You will probably need a lot of trial and error to learn how your soil reacts over the years.\n\n#### Medium or Intermediate-Term Goals\n\nThese goals are what you will seek to accomplish in 3 to 5 years. They must be specific tasks, and will often be broken down from long-term goals. Some medium goals:\n\n**_Goal 1_. I will purchase breeding stock, machinery, buildings, and so on as low-interest loans with 3- to 5-year terms**. Operating loans and lines of credit are for items to be used up and paid for in 1 year, such as seed, fertilizer, and veterinary supplies. A good ratio for the bank is 2:1\u2014in other words, $2 worth of assets for every $1 of liabilities (debt). Of course, it is always better to avoid any debt (see Goal 3).\n\n**_Goal 2_. I want to plant blueberries and sell them at the farmers' market and to local stores**. Keep in mind that it will take at least 3 years for any production and 5 to 6 years for full production. It would be wise to consider single-season vegetable and\/or field crops to give you income while your blueberries are maturing. This will also allow you to develop some loyal customers at farmers' markets and stores in advance, so you will have a developed market for your berries.\n\nAlways confirm market interest for a crop before you plant. Because there are no immediate returns on your planting, these types of crops will need 3- to 5-year loans, not operating loans, unless you can get a revolving line of credit. Revolving credit pays up and pays down, usually for a specific amount, say $10,000 to $40,000.\n\n**_Goal 3_. I will avoid short-term debt like the plague**. With a good ratio of 2:1, you can weather some tough financial storms. If you do whatever it takes to make the payment on your long-term loan at the ratio of 2:1, you usually have about 50 percent equity in your operation. When short-term debt piles up, it gets harder and harder to get out from under. This is also a long-term goal, because you will always want to avoid debt.\n\n**_Goal 4_. As I add new enterprises to my goal planning in the next couple of years, I will compare \"apples\" to \"apples\" by taking a look at the gross income per acre minus the direct cash costs per acre, which equals the gross margin on profit per acre or per head of livestock**. By leaving out fixed costs and mortgage payments, for example, you analyze the direct profitability of the enterprise on its own merits\u2014or lack thereof. Looking at income and expenses for 3 to 5 years will give you a good idea of what you can do and how much it will cost.\n\n**_Goal 5_. I will set in operation some risk management for my farm**. What kind of risks will you have in, say, a U-pick operation or a breeding-stock operation? What happens if you get sick and cannot work? Do you have a reserve in case of drought or flood? Can you buy crop or livestock insurance?\n\nUse life and health insurance to your advantage. In 1999, the government began gross revenue insurance, which covers almost all crops and livestock. If things go awry, you are guaranteed a certain gross revenue per acre or a certain yield (bushels per acre), all for a 20 percent higher premium.\n\nYou can control costs and increase output for increased gross income, but you have no control over the government and politicians and the laws they might pass that will affect your bottom line. Make a plan to deal with unexpected changes.\n\n**_Goal 6_. We want to convert our cattle operation to a management-intensive grazing operation**. You've estimated that it will require 2 years to build fence and lay water lines. How will this affect the cash flow? Will it generate more income the first year or second?\n\n#### Short-Term Goals\n\nYou will seek to achieve short-term goals within 1 to 2 years' time:\n\n**_Goal 1_. We have existing portable hog houses and will purchase feeder pigs to sell as whole-hog sausage in the fall**. Any short-term goal will probably spawn other short-term goals. In this case, for instance, decisions need to be made on numbers and types of pigs, feed, and marketing. Other goals prompted by this one might include \"We will buy 4 crossbred (Yorkshire x Hampshire) feeder pigs from Bob Smith in late August,\" or \"We will mix the sausage at 80 percent lean and 20 percent fat, and sell at a retail price of $2.50 per pound (120 percent of expenses, from the long-term goals).\"\n\n**_Goal 2_. We will build two portable chicken tractors and raise about 200 birds for sale as meat in about 8 weeks**. Chicken tractors are portable cages that allow chickens to graze in your garden between the rows and fertilize it, without eating your plants (see page 83).\n\n**_Goal 3_. We will buy ready-to-lay pullets and start an egg-laying operation**. Long-term and intermediate goals are often loosely defined or even philosophical, but short-term goals require that specific decisions be made immediately. How many pullets? What feed? What production level will be managed for? (Seventy to 80 percent is a good starting level.) What will we charge for a dozen eggs?\n\n**_Goal 4_. We will figure in time for rest and recreation this fall**. Vacations, entertainment, and rest are necessary to renew your body and spirit; they also allow some family time away from the concerns of the farm. Make sure that rest and recreation are always on your list of goals.\n\n#### Achieving Your Goals\n\nSetting your long-term, intermediate, and short-term goals is like using a map, and the goals are your destination, depending on where you are at on the road. You must have a plan. Some people say, \"I have a plan in my head\" or \"Planning is boring. I want to raise ostrich now!\" These people will fail without an excessive amount of luck. Planning is the easiest and least risky way to make your small farm successful and profitable.\n\nI cannot count the number of times we at _Small Farm Today_ have received calls from farmers who say something like, \"Well, I just finished growing my specialty popcorn. How do I market it?\" We try to suggest possible marketing avenues, but if they have not been planned in advance, the farmer's family will likely be eating a lot of popcorn, or whatever else they raised without planning on how to dispose of it.\n\nSuccessful farmers write down their goals. They will know how to raise their product, what their cost of production is, and, with careful planning, where they are going and how to get there.\n\n### Developing Your Farm Plan\n\nYour farm plan should be like a small business plan. Consult with a local Small Business Development office for information on forming a business plan. You will need this both for loans and for your own planning purposes.\n\nA business plan explains how you will achieve revenues from sales and how you will spend expense money. In particular, it demonstrates how you will repay loans and interest. It also shows how your business fits into the marketplace, examines who your customers are, how you will price your product, and what future developments may occur. A good plan accounts for variables that may affect income. Farms have some variables to consider that are not typical of other businesses:\n\n**Weather**. Weather will always be your first and biggest problem, but with cover crops, buildup of organic matter, and some limited drip irrigation or other limited water usage, you have eliminated all but the temperature aspects and catastrophic problems like floods and tornadoes. Although you may not have specifics on weather in your plan, you need to consider its effects and devise ways to counter them (add-on value, for example).\n\n**Weed, disease, and insect problems**. These are the target of the production strategy of your business plan. You should employ a production method that lowers purchased inputs, such as chemical pesticides. Weed, disease, and insect problems can best be handled by building the organic matter and fertility of your soil. Building soil fertility must be one of your broad, long-term goals. The healthier your soil, the fewer problems you will have with weeds, disease, and insects. Any problem that can be cured with a \"spray-to-death\" program can be cured in another way that is less expensive and less harmful to the farmer and the environment. And if you plan to be organic, another cure _must_ be found.\n\n**Management**. Japanese farmer Masanobu Fukuoka calls his technique the do-nothing method. What he really means is to manage your farm in a manner that reduces costs, making it sustainable because it becomes profitable. Jim Gerrish, grassman, and Allan Nation, editor of the _Stockman Grass Farmer_ newspaper, changed the term \"intensive grazing\" to Mig\u2014management-intensive grazing. They were underscoring the importance of management and the thinking process in farm success.\n\nFancy equipment or high-priced show livestock does not necessarily make your farm profitable. On my own farm of 80 acres, I have one old tractor, a tiller, and a few implements. I grow about 5 acres of open-pollinated corn as feed for my sausage hogs (bought cheap as feeder pigs) and poultry. This is primarily because these animals are not as good at converting pasture to profits as are my Katahdin Hair sheep, which I have more of and which use more grass\u2014an annual renewal source. These are not show sheep, but sheep bred for my farm conditions. Pasturing poultry and hogs can certainly reduce your feed costs and give you a better, healthier product to sell, but poultry and hogs are simple, one-stomach animals. Sheep and cattle, in contrast, are ruminants (they have four stomachs), with the ability to consume larger amounts of roughage. They can be raised to a marketable weight on forage alone. Low inputs, efficient livestock, and direct sales keep my operation profitable.\n\n**Government regulations**. Like it or not, you must be involved in meetings about legislation. The general population is four to five generations removed from the farm and this includes our legislators, yet they pass laws about an industry with which they are unfamiliar. It is very important that legislation that affects direct marketing be addressed so the small farmer has a chance to compete with the big boys.\n\nFarmers are less than 2 percent of the total population. We feed the other 98 percent. Every farmer must be a spokesperson for our industry. You cannot live in the woods doing your own thing, so to speak, because you can be legislated out of business if you are not aware of what is going on.\n\n**Adding value**. Value-added products include jams, cuts of meat, sausage, rubs, shampoos, craft projects, and much more.\n\n# True Story\n\nMasanobu Fukuoka is a Japanese farmer famous for his natural farming methods, which he described in his books _One Straw Revolution_ and _Natural Farming_. He grows rice and winter grain on a d-acre field, and has harvested 22 bushels each of grain and rice from that field\u2014without any inputs at all. Fukuoka's philosophy translates into a thinking-type agriculture. American agriculture is steeped in tradition, but not in thinking about _how_ and _why_ we do the things we do.\n\nFukuoka, born in 1914, categorizes farming into three basic types:\n\n**_Mahayana_ natural farming** is perfection\u2014the farming of Nature. Natural farming follows the Buddhist philosophy of _mu_ , or nothingness, by obeying a \"do-nothing nature.\" Fukuoka says Nature has everything; no matter how man struggles, he will never be more than a small, imperfect part of its totality. This notion of farming is more of a philosophical ideal than a farming method attainable by man.\n\n**_Hinayana_ natural farming** tries to remove human knowledge and action and utilizes the pure forces of Nature. It is the natural farming method that results when man works with Nature; this is the method that Fukuoka uses. This farming method takes advantage of natural occurrences. It is a \"do-nothing\" philosophy still, but with allowances for the necessary intrusions of man, such as planting. A typical application of _hinayana_ in the United States is overseeding red clover in winter wheat. The clover sprouts in the spring, but its growth is held back by the shade of the faster-growing wheat. When the wheat is harvested in July, the clover is exposed to the sun and grows rapidly, providing a cover crop. Nature determines the interaction of the crops; man assists.\n\nFukuoka's natural farming has five major principles: no tillage, no fertilizer, no pesticides, no weeding, and no pruning.\n\n**Scientific farming** uses human knowledge and action in an effort to establish a superior way of farming. Fukuoka believes scientific farming is limited to short-term goals; its achievements may be better in some ways (e.g., yields) but are far inferior in all other ways, and will lead eventually to failure. In comparison, natural farming is total and comprehensive. Fukuoka describes American organic farming as just another type of scientific farming. He says scientific farming is judged on a scientific basis; natural farming should be judged on philosophical grounds.\n\nNot all of Fukuoka's ideas are applicable to American farming, but he is always thought-provoking.\n\nAdding value reduces risk by turning a basic product into a product with additional profit and a longer shelf life. Weather, price, government regulations, and weed, disease, and insect problems are the main obstacles to small farm profits. If the majority of your income comes from a value-added product, you have eliminated most of the farming risk.\n\nValue-added products allow for a drastic reduction in volume, with little change in gross revenues. Most people with value-added products have excess field production (more crops than can be turned into product), which can be sacrificed without loss to the product. The add-on price also protects the gross revenues. It only takes 5 bushels of corn sold as cornmeal to equal the gross revenues of 100 bushels of corn sold at the elevator. (See chapter 7 for more on add-on value.)\n\n**Marketing**. When success and profit are your broad, long-term goals, direct marketing becomes very important. I will discuss marketing and adding value in the next chapter.\n\n###### Food For Thought\n\nIt is my hope that as you read this book, you will commit the basic truths of each chapter to memory to be used every time you make a farm decision or define or redefine your goals. The most fundamental of these practices are to clarify and prioritize your goals, keep them uppermost in your mind, and plan your enterprise with them in mind.\n\n###### Chapter Seven\n\n## Marketing\n\nWhat are some differences between the marketing strategies used in traditional agriculture and those used in alternative agriculture?\n\nTraditional agriculture grows the crop or livestock and then looks for someone to buy it. Alternative agriculture looks for the market first and then grows what the market wants. By combining this with added value and direct marketing at retail prices, farmers move upward in the marketing process, for much greater profits.\n\nTraditional agriculture depends on \"wholesale\" prices at elevators and sale barns. However, a farm that sells surplus products above what the family eats is a small business, and small businesses succeed by practicing retail rather than wholesale marketing. Direct marketing is the profit equalizer for family farms.\n\nTraditional agriculture simply sells the commodity that is grown. Alternative agriculture will seek to add value to the product by turning it into a retail consumer item such as jam, sausage, or a wool sweater. By adding value-added products to direct marketing, farmers gain additional profit. It is important to remember that in direct marketing, you are marketing yourself as well as your product.\n\n**Principle :** _A farmer must be skilled at buying and marketing_.\n\n_Marketing begins by reaching the consumer. Each fall this roadside display of pumpkins attracts the eye of tourists, photographers, and pumpkin buyers as they travel U.S. Route 7 in southwestern Vermont_.\n\n### Eight Steps to Identifying the Market\n\nHow do we determine what the market or consumers want?\n\n**Step 1**. Obtain maps of your state and county.\n\n**Step 2**. Locate your farm on the maps and draw circles with your farm at the center with a radius of 25, 50, and 100 miles.\n\n**Step 3**. Count how many towns, cities, or population centers are within those ranges. Most of your farm's customers will be in the 25- to 50-mile range.\n\nWith your farm at the center, draw circles at a radius of 25, 50, and 100 miles.\n\n**Step 4**. Using your maps, add up the number of people present in your area. Your local agriculture statistics office (check with your local extension office or State Department of Agriculture) may help you compile these numbers.\n\n**Step 5**. Find out how many alternative agriculture farmers are in the circled areas of the maps, what they are growing, and what they are doing with their produce. If there are already ten strawberry farmers with 10 acres each, the market is probably already saturated for U-pick strawberries\u2014but it may not be if you are willing to process the berries into jams, jellies, or pies. Remember, too, that there will never be an oversupply of the \"best\"\u2014successful marketing is selling quality. You may have to do some driving and checking of local newspapers, farmers' markets, and your local Extension and agriculture statistics office to get this information.\n\n**Step 6**. Go to your local grocery store or visit several of them in the population centers. What vegetables and meat products are already available in these stores? Can the stores purchase these items locally or do they import them?\n\n**Step 7**. After gathering the information, determine what crops and livestock are missing that might have sales potential. Check which products are imported from other states and ask yourself why. Hardly any states are self-sufficient in food production; most grow around 30 percent of what they eat. Growing produce locally helps improve the regional rural economy and is desired by consumers. The market is there for the taking.\n\n**Principle :** _Plan to have produce available when others do not, or offer unique products_.\n\n# True Story\n\nDwight James, a banker-turned-farmer from Alabama, discovered that his state was a pumpkin-deficient one. In other words, the pumpkins being purchased in Alabama (population: 4.3 million) were coming from Tennessee and New Mexico. The market was there, but Alabama farmers were not taking advantage of it. By competing with imported pumpkins, Dwight could save transportation costs and damages, and rely on \"homegrown\" civic pride to increase sales. Since this discovery, Dwight is now the president of a sixty-member pumpkin association. Their promotions in the state of Alabama have resulted in the marketing of many varieties of pumpkins for different uses. Because many of these people had never grown pumpkins before, it is clear they are risk takers and price makers\u2014not price takers\u2014which is what agripreneurship is all about.\n\n**Step 8**. Find out who your customers are. If you have any ethnic groups in your area, investigate growing specialty produce and marketing methods that appeal to them. For instance, certain Vietnamese ethnic groups prefer live black chickens. There are certainly a wide variety of oriental vegetables to grow\u2014or Russian, Amerind, French, Arabic, Mexican.... Check farmers' markets and local ethnic stores or associations to talk to people about what they are interested in. Get specific ideas of how they want the product to appear. A satisfied customer is a return customer.\n\n### Niche Marketing\n\n**Principle :** _Raise crops and livestock for niche markets_.\n\nSmall farmers should raise crops and livestock only for niche markets. This means relying on alternative crops and livestock, or alternative marketing, rather than the traditional types the universities promote. While universities will claim their research is size-neutral, in the real world, that simply is not true. For example, if your university or seed dealer comes up with a new variety of corn that yields an additional 10 bushels to the acre, who benefits the most? The farmer with 50 acres of corn or the farmer with 2,000 acres of corn? The man growing 50 acres of corn gets a 500-bushel increase in his crop. The man with 2,000 acres gets a 20,000-bushel increase, which enables him to purchase new technology at a faster rate than can the 50-acre farmer. The 2,000-acre farmer also has a lower per-unit cost of production and more acres to spread his equipment over.\n\nYou might well say, \"I have only five acres. I cannot compete, why do it?\" Well, you _can_ compete, but you need to raise different crops, or the same crops for different reasons. Here's an example.\n\nSelling corn will ordinarily bring you about $2 per bushel. A bushel of shelled corn is 56 pounds, so this amounts to about 3.6 cents per pound. On the other hand, say you take the same corn and add value to it by processing it into cornmeal. You will make $82.88\n\n# True Story\n\nCurtis Bennett, in Mexico, Missouri, had no land and no machinery. He had been working for other farmers, but wanted an operation of his own. He wished to continue growing the crops he was familiar with, but to make more money and improve his quality of life. He met a farmer with land to lend at the 1994 National Small Farm Trade Show and Conference. Once he had the land, Curtis decided the organic market had excellent potential, and started growing organic corn and tofu soybeans.\n\nHe spends about $18 per acre for organic fish emulsion foliar fertilizer for his corn and $12 per acre on his soybeans. He plants the corn following the soybeans so he does not have to buy nitrogen for the corn. He sows about 25 pounds per acre of hairy vetch between August 15 and September 15, which provides an additional 100 pounds of nitrogen for the following year. Sowing in annual rye also helps as a green-manure crop and as a method of weed control.\n\nYields on the Bennett farm have varied from 33 to 50 bushels for soybeans and been about 115 bushels for corn. These are very respectable yields for low inputs. Curtis's methods have resulted in soybeans that tested at 44 percent protein, a very high-quality bean. The organic industry requires a minimum of 38 percent protein, and pays a premium price for higher-quality beans. Quality tofu soybeans sell for about four times the price of regular soybeans.\n\nLike many farmers, Curtis is always busy. He is a member of the Heartland Organic Marketing Cooperative in Iowa, and has served on its board of directors. The cooperative direct-markets its soybeans, bypassing distributors. About 90 percent of Curtis's beans are sold domestically, and about 10 percent go overseas to Japanese buyers. Most of his corn goes to California for taco chips, burrito wraps, and other natural foods.\n\nCurtis started with nothing but an idea and his own motivation, and now is well on his way to success. Finding the right alternative market and growing the product for it provided him with the key to increased profit.\n\nper bushel ($1.48 per pound) in gross profit. You will have to cover the cost of bags to put the cornmeal in\u2014about 15 cents each for a plastic-lined cloth bag. This means 56 one-pound bags would cost $8.40. This $8.40 will have to be subtracted from the amount you make from selling the cornmeal. Subtracting the $8.40 for bags and the $2.00 opportunity cost ( _opportunity cost_ is the price of the original opportunity) by selling corn by the bushel leaves a profit of $72.48, which will pay your labor, management, and marketing costs. A good goal is to have 50 percent of your income as net profit\u2014in this case, about $36.25 per bushel.\n\nIf each acre yields 100 bushels, selling corn at $2 per bushel will net you $50. Selling it as cornmeal will produce a net income of about $3,625 per acre\u2014including payment for labor and management. By selling the corn as cornmeal\u2014or growing high-dollar, high-yield varieties, or selling \"Indian corn\" for decorative purposes, or any other number of alternative marketing methods\u2014you can vastly increase your level of profit. Of course, selling 5,600 bags of cornmeal will require much more marketing effort than would hauling the corn to the nearest elevator and unloading it.\n\n#### Farming Methods and Marketing\n\nHow you choose to raise your crops and livestock can influence your marketing choices. Obviously, planting standard commodity crops on a large scale does not lend itself to direct marketing. Even variations of commodity crops (popcorn, white corn, tofu soybeans, for example) usually require contracting with someone to buy the product, rather than direct marketing. This is also true of large-scale operations involving vegetables, medicinal herbs, and industrial crops. Many farming methods, however, lead directly to niche markets. For example, choosing to raise purebred livestock, show-ring livestock, ethnic produce, or your own strains of open-pollinated plants creates a market niche for you. Some varieties of alternative livestock have built-in markets, such as velvet from elk or deer.\n\nIf you raise your products by organic or natural means, you will have an excellent marketing approach for consumers. Many people nowadays want to know that their food was raised outdoors in humane conditions, without herbicides, pesticides, antibiotics, growth hormones, or other chemicals. So consider how you will raise your crops and livestock when you are choosing your niche markets.\n\n### All About Niche Markets\n\nSo exactly what is a niche market? Niche markets are of two types. Sometimes a niche market means a different type of crop or livestock than the two traditional triumvirates of corn, wheat, and beans and cattle, hogs, and sheep. Cut flowers, for example, are a different type of crop that can net as much as $15,000 per acre. Or it can mean produce is marketed differently from the way traditional crops and livestock are, such as at farmers' markets or at a U-pick farm.\n\nHere are some other characteristics of niche markets.\n\n**1. Small size**. A niche market is small and may be easily saturated. Careful research is required before you choose what to raise. You must assess both the current and potential demand for your product. Because of limited demand, you may choose to raise small quantities of several different crops.\n\n# Farming By The Square Foot\n\nIf you are a traditional farmer with a large acreage, you need to set aside a small area to practice intensive growing for direct marketing. Forget about bushels, acres, or thousands of livestock. Instead, think of your farm in terms of square feet. There are 43,560 square feet in 1 acre. If you get 20 cents per pound for the product and produce 1 pound per square foot, the result is $8,712 per acre. The potential of netting $1 per square foot or $43,560 net profit from 1 acre is very real (remember, you are selling direct). The catch is that 1 pound of produce per square foot equals 21.7 tons of produce per acre; you'll need 400 to 1,000 customers to consume that much produce.\n\n**2. Add-on value**. Niche markets are high-yield, high-dollar, high add-on-value types of crops and livestock. You add on value to a raw commodity by turning it into a different product for a different market, such as selling painted gourds, or marketing freezer beef or whole hog sausage instead of delivering livestock to the sale barn. Niche markets frequently lend themselves to processing-manufacturing, which supplies jobs for the rural community as well as for the farmers. For example, many farmers who started U-pick berry operations find themselves making jams and jellies out of surplus berries or berries not good enough for U-pick sales.\n\n# True Story\n\nEarnie Bohner's U-pick blueberry operation in Lampe, Missouri, yields about 10,000 pounds per acre and he sells them for $1.35 per pound, or $13,500 per acre. Earnie and his wife, Martha, joined forces with a chef and his wife to develop new products, such as blueberry barbecue sauce, shiitake mushroom sauce, dried shiitakes, and blazons. (Blazons are dried blueberries that sell for $10 per pound\u2014which considerably increases the gross profit on the same per-acre yield.) By putting his berries into small jars as jam, featuring his Persimmon Hill Berry Farm label, for $3.50 per jar, the same 10,000 pounds per acre yields $98,000 in gross revenues.\n\n  1. **Lack of mechanization**. Niche crops are not easily mechanized. The \"Suits,\" otherwise known as speculators and Corporate America, need absolute control of their crops, with mechanization potential for planting on large areas. Consistent production is hard to maintain without mechanization, and unmechanized crops require large numbers of hired help for large scales of production. If a crop takes lots of stoop labor, Corporate America will be less likely to invest in it. A warning: If you see corporations become interested in a particular crop or livestock, get out of it and take your lumps before you are wiped out in the market crash. On a positive note, if you market your production in a way that Corporate America cannot, such as by processing or manufacturing a homemade recipe, you can prosper even with corporate competition.\n\n  2. **Limited interest**. Niche markets are markets often overlooked by mainstream and Corporate America. They may be too small, or require too much labor or management, or they may have been abandoned due to market crashes or lack of profit. You can succeed by finding a different way to market a product that others believe to have limited potential. It is also possible to enter a market during a \"crash\" period and acquire your stock at a low price. For example, hogs were selling for as low as 18 cents per pound in 1998. Anyone wanting to acquire breeding stock for a hog sausage niche market operation could have bought several hogs, started a breeding operation, and turned some of them into sausage at a selling price of $2.50 per pound (which equals a gross profit of about $2.00 per pound).\n\n  3. **Controlled volume**. A niche market can start with small sales and grow into the volume that you desire. In contrast, corporations like high-volume marketing of products, which yields a lower return but is justified by the volume of sales.\n\n  4. **Low capital risk**. A niche market can usually be gotten into and out of without a large amount of capital expenditures at risk. Niche markets can be long lasting or short lived, which is why you need to be able to get in and out of production. As an example, let's say your niche market is raising purebred hogs on pasture with low-cost farrowing huts to sell to other breeders. If the bottom were to drop out of the hog market, it would be fairly easy to cull your herd to a minimum and convert the rest of the pasture to crops or some other livestock. If you raise confinement hogs in expensive facilities, you are stuck with raising hogs whether or not you profit. These expensive facilities are practically impossible to convert to anything else.\n\n  5. **Diversification**. Niche markets, because of their small size and short duration, almost require that you be diversified. When you sell one animal or one plant, you are not diversified and therefore not sustainable. If you raise small amounts of different produce, it will be easier to sell. For instance, selling 25 pounds of artichokes to a customer is difficult. But you might be able to sell that customer 3 pounds of artichokes, 10 pounds of tomatoes, a 6-pound roast from your Dexter cattle, two dozen eggs, and 5 pounds of apples.\n\n# True Story\n\nChuck DeCourley, of Columbia, Missouri, was promotions director for _Small Farm Today_ magazine. He wanted to start farming but didn't want to spend tremendous amounts of money. After surveying the market, he found that ratite (ostrich, emu, and rhea) prices had dropped following the speculator boom. According to newspaper articles, many farmers, particularly in Texas and Oklahoma, had given up when the \"breeder's market\" collapsed. Some farmers released their stock into the wild, some sold it at dirt-cheap prices, and a few just killed all their birds. But, as they say, one man's misfortune is another man's fortune.\n\nChuck purchased emus at low prices and started breeding them. Once he and a friend had enough birds, they started culling them for slaughter. They have marketed emu meat sticks ($1.39), emu sausage ($6.00 for 1 pound), and seven kinds of emu oil products, including a pain rub ($7.00 for 4 ounces). They also sell carved eggs for $150 and up. Chuck is successfully making money off livestock that are being turned loose in some states just to get rid of them. He has since left his magazine career to pursue farming full time.\n\n**8. Direct contact with consumers**. To be a niche product marketer, you must become a salesperson. If you do not believe in your product 100 percent, your customers won't believe in it. If you do not want to deal with the public, you might be better off joining a cooperative marketing effort with other farmers.\n\n### Add-on Value\n\n**Principle :** _Add-on value can increase both gross income and net profit_.\n\nAdd-on value is a simple concept. It means that you have processed a raw commodity to create a new product.\n\n**Example 1**. Berries can be sold as jams and jellies. The jams and jellies add value because they sell at a higher price per unit, but from the same acreage and same yield. Because they can be stored and sold throughout the year, they will produce a larger gross income over a longer period than will fresh berries, which can only be sold in-season.\n\n**Example 2**. When you sell hogs to the packers, you are selling hogs as a raw commodity. If you turn them into sausage, you are selling a finished product\u2014an add-on value\u2014direct to the consumer.\n\nFresh berries are perishable, but jams and jellies can be sold throughout the year.\n\n**Example 3**. Elk are a good add-on value animal. Their antlers are harvested yearly for velvet or craft projects, their meat is sold at a premium, and their hides are turned into handbags, boots, and wallets.\n\n**Example 4**. Ratites are used for meat, emu oil products, carved eggs, \"leather\" (from the legs and hides), and feather craft projects.\n\n# More Add-On Value Ideas\n\n  * **Fruits**. Jams, jellies, barbecue sauces, vinegars, dried fruits, chocolate-dipped fruits, wines, pies, juices, ice cream, crushed ice drinks.\n\n  * **Meats**. Cuts of meat, sausage, hams, meat sticks, prepared meals (e.g,. chicken cordon bleu, enchiladas, turkey, and stuffing).\n\n  * **Grains**. Flour, breads, doughnuts, recipe kits, soy nuts.\n\n  * **Fiber**. Yarns, clothing, felted products, small figures, stuffed animals.\n\n  * **Decorative crafts**. Christmas or autumn wreaths or swags, dried flower arrangements, cornhusk dolls, painted gourds, gourd birdhouses, painted or carved eggs, scarecrows, feather arrangements, potpourri, herbal gift baskets, pressed flower pictures.\n\n  * **Vegetables\/mushrooms\/herbs**. Garlic braids, mushroom logs, potted plants, dried herbs, pepper wreaths, seed packets and grow-it-yourself kits, recipes with appropriate accompanying dried herbs and mushrooms, herbal dips, seasoning mixtures, relishes, mustards, canned vegetables, salsas, popcorn.\n\n  * **Health and beauty products**. Shampoos, soaps, lotions, foot baths, herbal teas.\n\n  * **Other ideas**. Oils, leather products, honey, candles, T-shirts with farm logo.\n\nTo add value, processing or manufacturing is usually, but not always, necessary. Craft projects, for instance, are generally done by hand at a table, rather than in a Usda kitchen. Processing can be something as simple as cracking and removing the shells of walnuts or pecans so the \"meat\" is exposed. Each step in processing adds time or effort, but will result in a higher-value end product. For example, whole walnuts sell for only 10 to 50 cents per pound, while walnut meats sell for $7 per pound. Specific varieties of hand-cracked walnuts yield 25 to 30 percent nut meats. There are machines to crack walnuts, which decrease time and labor, but they also decrease quality. Machine-cracked walnuts yield only about 7 percent nut meats.\n\nProcessing changes the form of a raw commodity. There are several reasons for processing, as follows.\n\n  1. **Ease of storage**. It is much easier, for example, to fit packages of sausage and pork chops into a freezer than a whole hog carcass.\n\n  2. **Convenience**. Most consumers would rather buy shelled pecans or boneless chicken than go to the effort of shelling or boning on their own\u2014if the price is right.\n\n  3. **Longevity**. Canned fruits and vegetables will easily outlast their fresh counterparts, as do jams, jellies, and dried fruits. Most meats can be stored safely for 3 to 6 months in the freezer, but meat begins to lose quality after that. Cured meats can be frozen for only 1 or 2 months.\n\n**Principle :** _Plan to have products to sell year-round_.\n\n#### Processing Food\n\nYou need no special permits for craft projects, and will not normally require much processing equipment. If you sell value-added food, however, you are probably going to need an approved kitchen with appropriate permits to manufacture it in. You must check health regulations\u2014local, state, and Usda. call your local health inspector and Department of Agriculture for more information. University extension offices often have information on adding value and kitchen requirements.\n\nIf you need a processing kitchen, your first step should be to find out what is available in your area. When starting, it is usually better to rent a facility from someone else, both to save money and to get an idea of what you will need. Check out churches, schools, restaurants, and nearby value-added farm businesses for a kitchen you could rent that will fulfill your needs. Your local Health Department or extension service may be able to give you some leads.\n\nYou will also need information on how to do the processing and labeling. The state agriculture department or extension may have a value-added program to help with labels. If you sell across state lines, there will also be federal guidelines that must be satisfied. Talk to other added-value farmers and call businesses that have similar products in grocery stores to get information on companies that supply labels, cans, bottles, shipping packages, and other processing supplies.\n\nWith meat, processing often runs afoul of Usda regulations. There are practically no small-scale processing plants for chickens, for example. One way around this is to sell the animal \"live,\" then slaughter it for the customer. Check with your state's Department of Agriculture for rules and maximum numbers allowed. (In Missouri, each farm is allowed to sell 20,000 live chickens per year without a Usda inspection.)\n\nIn general, you can butcher fish, chickens, and rabbits at home. To sell pork, beef, lamb, or cabrito (goat), the meat must go through a state-inspected or Usda plant, but you can still sell direct if you sell the animals live.\n\nProcessing equipment for meat animals can be purchased from a variety of sources. Processing equipment might include fruit and wine presses, grinding mills for grain, poultry shears, knives, food choppers, and sausage stuffers, plus bags, egg cartons, and shrink wrappers. A number of companies carry rabbit- and poultry-processing equipment. (See the appendix for more information.)\n\nFarmers today are lucky to make a 5 percent return on their farms when selling raw products like corn. Food processors consistently receive a 20 to 25 percent return on their investment. If you have done your research and priced your products correctly, addon value will increase not just gross income, but net profits as well.\n\n### Twelve Ways to Sell Your Products\n\n**Principle :** _Direct marketing is the profit equalizer for small family farms_.\n\nThere are a multitude of methods by which you can sell what your farm produces. Let's take a quick look at some of these. The first eight listed here are direct-marketing opportunities; the others may involve selling to retail outlets.\n\n#### Selling to Friends and Neighbors\n\nThis is the best place for farmers new to direct marketing to start. You and your spouse should compile a list of about 100 people you know through work, clubs, church, and so on. Include people like your banker, your barber, your letter carrier, and relatives. You will use this list for contacts, and, if you continue to sell this way, as a mailing list to let people know when your products will be available.\n\nThis is an easy way to start direct sales, but it does have some drawbacks. It requires a lot of work to contact people to sell your products. Also, because these are your friends and neighbors, they may expect to get a lower price as a friendly bonus. They may also be slow to pay. You must remind them that for you to stay in business and your farm to be sustainable, you must make a profit. A good business deal is good for both parties.\n\nFriends and neighbors may be more critical of your product. You must explain exactly what they are getting. Customers not familiar with farming frequently assume a 250-pound hog will produce 250 pounds of edible meat. Instead, a 250-pound hog will probably produce 150 to 170 pounds of cut and wrapped meat, including chops, hams, and sausage. If you explain to your customers what to expect before they buy the meat, it is more likely that you will keep a satisfied customer. Remember, an unsatisfied friend or neighbor can cause a lot of problems.\n\nOn my farm, I sell hogs at 80 cents a pound liveweight, which means about $100 for half of a hog, with the customer paying for the processing. Processing costs run between $35 and $45, depending on how the customer wants his meat cut and whether he wants any cured meat. Alternatively, I sell 80 percent lean sausage at a flat $2.50 per pound, and get about 100 pounds of sausage from a 225-to 250-pound hog. I sold all my pork this year in the midst of 10-cent hog prices at the local sale barn. Customers were willing to pay premium prices for quality meat. My pork is fresher and leaner, and my hogs were raised on pasture and not given any antibiotics or growth hormones. My pork comes from happy hogs\u2014and my customers know it.\n\nFarmers' markets provide a lot of customers in one place, which works well if you have a product they want.\n\n#### Farmers' Markets\n\nFarmers' markets are perfect for direct marketing. Consider selling at any farmers' markets within 1 hour's driving time. Your state Department of Agriculture should have a list of markets.\n\nThe biggest advantage of selling at a farmers' market is that you'll find lots of consumers in one place. It is much easier to sell a little bit of produce to a lot of customers than a lot of produce to a few customers. If you have something new and different from the rest of the market, provide taste samples to the customers. Recipes featuring your produce are also a good marketing strategy.\n\nThe major disadvantages to farmers' markets are cost and time. Time spent at the market is time not spent on the farm. When you figure all your expenses\u2014production costs, booth fees, labor, and transportation\u2014selling here may not be cost-effective. Count your pennies carefully.\n\nCheck out the market the season before you intend to sell there. See what the farmers are raising, and what they are not. Is there anything missing\u2014yellow or purple snap beans, or heirloom tomatoes such as 'Brandywine'? Talk to the farmers and customers to see what is needed that you could raise. This is a good place to start.\n\nIf there is no farmers' market in your area, consider starting one. Evaluate the customer potential, determine the exhibitor (farmer) potential, and design a charter\u2014all these are very important. Decide fees, dates, and hours, and investigate insurance and location. This entails a lot of work and requires community support. If you do start a farmer's market, holding special events at it\u2014bake sales, festivals, concerts\u2014can be an additional way to draw revenue. Finally, a farmers' market will require the same thing you do\u2014advertising to let people know it is there.\n\n#### Roadside Stands\n\nThese will usually be on or near your farm and can vary from a pickup bed, to a temporary shelter, to a shop. If your own property is off the beaten track, inquire at area businesses such as discount stores and gas stations about setting up a temporary stand on their property, or pool your produce with a farmer on a good road. Make your stand attractive, with bright signs and clear information.\n\nRoadside stands bring customers to you at little expense. If you do well, you can develop this into a tidy little on-farm shop. With some advertising, it could attract people from miles away. Unfortunately, time is again a negative factor. Time spent in the stand may be worth less than at a farmers' market (think in terms of customer dollars per hour), unless you are on a well-traveled road with an easy exit. You will have to decide what the stand's hours will be. Will it be open daily, three times a week, or only on weekends? Will it be open all day, or just around rush hour, when customers are heading home? In some areas, the farmer leaves a can for money and lets customers help themselves. I'm not trusting enough for this approach, although it might be a good way of disposing of excess produce. Check insurance requirements to operate a roadside stand.\n\n#### Community-Supported Agriculture (Csa)\n\nThis is a subscription service where the farmer signs up customers for a monthly or seasonal fee and agrees to deliver to them a percentage of the farm output for a season. Csas are more than just a way to sell your produce: You form a definite relationship with your customer, and your customer shares in the production risk of your farm. By receiving the yearly fees up front, the farmer avoids borrowing money and paying interest, and has a paid market before he or she plants anything. The customers know how and where their food is grown, and receive fresh produce at a good price. Customers will also get a good lesson in the vagaries of weather!\n\nWorking a Csa is relatively simple. Most Csas have a starting base of about thirty customers. The customer will pay, say, $300 for 7 months, and in turn receives a weekly delivery of his percentage of the produce\u2014probably about 10 pounds per week. Customers may be allowed to work on the farm in peak labor times to reduce their cost. Some farms have customers pick up their produce on-farm, with a \"rollover\" table where a customer can remove produce he or she does not want and substitute something else. Some farms provide a yearly list of what they are planting; others seek input from the customers as to their preferences.\n\nThe main disadvantages of a Csa are determining a fair price and finding customers to sign up for it. Many people are afraid to commit money before getting any product. Most (but not all) successful Csas are near metropolitan areas with a large customer base. You don't want to start a Csa until you have a good knowledge of what you can raise each year, and what it costs to raise it.\n\n# True Story\n\nSam and Elizabeth Smith grew organic produce for local markets and restaurants for many years on their Williamstown, Massachusetts, farm. They decided to turn their farm into a Csa because they wanted a closer tie with the local community. Csa members actually become part of the farm. The Smiths have apprentices from all over the country who live on the farm from spring until harvest to learn about organic farming and the Csa experience. The apprentices not only help tend the crops and animals but they also help pick the vegetables when members come to do their pickup. Members choose certain crops, such as strawberries, raspberries, cherry tomatoes, snap peas, string beans, kale, herbs, and flowers.\n\nAs separate items, the Smiths' Caretaker Farm raises organic lamb, pork, and beef. Honey and eggs are available for sale. A local baker has joined the operation and provides fresh healthy breads, rolls, and special desserts, as well as baking classes. The Smiths don't grow corn, but it's available to members thanks to a participating neighboring farm.\n\nCaretaker Farm has become a thriving community of members who value local, fresh, organic produce and enjoy their weekly visits to the farm to pick up their share. A steering committee helps manage the membership administration. There are seasonal festivals, such as planting and harvesting potatoes and harvesting pumpkins, with square dances and potluck dinners. There is a pond where members can swim; there is a children's garden and classes for children in making various craft projects.\n\n#### Catalog Sales\n\nValue-added products are required for catalog sales, which can be a natural outgrowth from a mailing list of customers. Starting a catalog requires a lot of thought and careful market research. Printing, packaging, and mailing the catalog\u2014that's the easy part. The hard part is juggling supply and demand, making sure you have the necessary supplies on hand without spending too much money or creating a surplus. If you don't have what the customer wants when he wants it, you have lost a sale and possibly a customer. If you have thousands of unsold jelly jars, you have lost a business.\n\nI recommend catalog sales only for those with several years of experience in both farming and direct marketing of value-added products. A catalog requires a large volume of production. Factoring in the weather, amounts of production, and projected customer returns calls for knowledge and experience. It might be wise to have another farmer close by who can supply you with extra produce if you run short\u2014but it must be of a quality comparable to yours.\n\n#### Shows and Fairs\n\nA booth at trade shows and community events can promote your farm and increase your sales. These booths are a good way to tap into local markets, appealing both to brand-new customers and to people who have already heard about your Aunt Sally's Special Mustard Recipe but have never tried it. You might consider offering some kind of exhibition in trade for the booth\u2014cooking, milking a goat, or demonstrating a craft (soapmaking, spinning, felting, for example). Have cards or flyers available with maps to show interested people how to find your farm.\n\nCheck with the chambers of commerce in nearby communities, and also with your local extension office, to see what events are available to you. Don't look just at traditional farm and craft shows. Consider everything, from ballooning events to food fairs, from music festivals to church gatherings, from association conferences to Living History events. If no such events exist in your area, consider hosting one yourself.\n\nThe next step up from local events is a booth at a major farm and craft show or a state fair. You must be careful about costs. Remember that you have to recover the booth fee (usually $50 for small shows and $250 or more for larger shows) before you make a profit. If you are in another city, you may have to recover transportation, lodging, and meal costs as well. I recommend attending a major show the year before you exhibit at it to assess the customer potential. Talk to entrepreneurs marketing similar products and ask what they think of the show. In the end, though, finding out if you will have enough customers to justify the expense can be done only by trying a show or two and keeping good records. Be warned that some shows will be great, outdoing all expectations\u2014but some will be duds.\n\nThe \"major leagues\" of shows are the food festivals and gourmet food fairs, where booths can cost $2,000 or more, plus fines if you don't do exactly as instructed. These are not for faint-of-heart, low-volume producers, or those without a good cash reserve. You must be professional, knowledgeable, competent, and know your costs of production to the tenth of a cent. If a buyer asks what it costs you to deliver 100 six-packs of jam to 20 different stores, you have to be able to say, \"Yes, I can do that for Xx dollars and cents per case.\"\n\n#### U-Pick Farms\n\nU-picks are definitely a \"people-person\" business. All sizes, shapes, and ages of people will come to your farm, full of all kinds of problems, stories, and misconceptions about farmers. They will gladly share their experiences while you try to deal with five customers waiting to check out, a small child trampling plants in the field, and cars blocking the parking lot. If you are a \"people person,\" though, and believe you can sell your visitors products that are wholesome while providing them a fun experience, this can be a good way to market.\n\nSome families make it an annual event to pick their own produce.\n\nU-picks do away with the expense and time of harvesting your crop and hauling the produce to town. (Some farms pick part of their crop and sell it to hurried visitors at a higher price.) Once customers are on the farm, you have a great opportunity to sell them other products, too.\n\nThere are many other cost considerations, however. You need to check into the requirements for U-picks before you start: insurance, parking, bathroom facilities, a shady rest area, and so on. For information, talk to insurance companies, other U-pick farmers, and your local extension office. Check county regulations, as well.\n\nThink, too, about the problems inherent in letting large numbers of people onto your farm\u2014consider fence placement and security, for starters. Once again, it is helpful to do your research, and consult with other U-pick farmers and your extension service. You'll need to offer pointers to inexperienced pickers to reduce damage to your crops. There will always be some free \"field tastings,\" but discourage massive consumption. Require parental supervision of children\u2014you are not a baby-sitter (unless you want to set this up as an extra perk for a fee). Some farms require phone-in appointments, while others just have drive-bys. Phoning allows control of customer numbers with production, but requires more advertising until you acquire a regular clientele.\n\nU-picks used to sell only by the container, but most now choose to sell by weight. Thus, accurate scales are essential.\n\nIt is best to have a large population center within 45 minutes' driving time, unless you have some innovative ways to draw people to your property. Earnie Bohner, at Persimmon Hill Farm in Lampe, Missouri, operates a U-pick that is well over an hour away from any major city. He's on a road that leads to a camp, however, and many parents driving down to see their kids stop by. He offers homemade muffins and fresh berry smoothies, plus his unique addon-value berry and mushroom products (dried berries, mushroom sauce, and berry barbecue sauces). His special products and fun farm serve as a tourist magnet. He even has tour buses come. There is more information on attracting customers in the advertising section later in this chapter.\n\n#### Food Circles\n\nA food circle is sort of a cross between a Csa subscription farm and a cooperative. A limited group of farmers and others in town pool their produce\u2014usually 1 farmer for each 10 nonfarm families is a good ratio. Members may pay a fee to join, and some circles require a deposit. A tightly structured food circle will have a distribution site (perhaps more than one) where produce is delivered weekly. Members buy the produce at the distribution site. A member who does not buy enough during a certain period (monthly or seasonally) may have to pay a fine. A loosely structured food circle may just have people get together on an informal basis to trade or barter their goods.\n\nThe major disadvantage of a food circle is that lots of people must participate in starting it up and keeping it going. A farmer will find a lot of his or her time spent on organizational matters. It does have an advantage over cooperatives in that a set volume of produce is not needed from a farmer, and an advantage over Csas in that the customer does not have to contribute as much money up front; this makes it easier to attract customers. Farmers may not be free to set the price on their produce, however. Most food circles set a consistent price for all like produce, combine it, and return the farmer a percentage of money equal to the percentage he contributed.\n\n#### Grocery and Health-Food Stores\n\nThese can be good outlets for your products, but you will be selling them wholesale, so you must move more product to make up for your reduced income and increased time and travel. While helpful, any wholesale selling should be a small portion of your total sales volume.\n\n# True Story\n\nI was interested in getting into an alternative agriculture venture that would give me more return on my dollar than cattle. I decided to investigate raising catfish. I did my research carefully on raising them, then looked into marketing. I inquired at grocery stores in the nearest large city (about 30 minutes away) to see if they would have an interest in purchasing catfish. They did. I then checked with the Health Department to find out how I would have to arrange the fish for sale. I was told to take them to the stores in a cooler with a layer of ice, then a layer of fish.\n\nWhen I finally had fish to market, I took them to town, arranged as the Health Department specified. I visited the first three stores that had indicated interest and every one of them turned me down. When the fourth store said no, I asked the manager why he did not want the fish. He said he could not use fish in ice\u2014they had to be in bags. I checked with the Health Department again and was told that bags were fine, as long as they had a hole in the bottom and ice around them. Once I changed to this, all of the stores bought my fish. The lesson here is, ask questions to find out what the customer wants.\n\nStores may be hard to break into if they do not already carry local produce. Try to guarantee them certain amounts, packed the way they prefer\u2014they may be more amenable. Of course, make sure you will be able to supply the amounts you promise. Consignment sales may be a way to get your products into the store. These also have the advantage of giving you a higher profit, but you will have to spend more time in arranging displays and accounting.\n\n#### Restaurants\n\nThis again is not a retail market, but you may get prices slightly better than wholesale. Target specialty restaurants with staff chefs: They will be more interested in buying fresh produce. They also tend to be very particular. It is best to talk to the chefs before you grow, to find out what they want and what quantities they will need. Bring complimentary samples so they can experiment. Some chefs can be a bit temperamental. Tactfully encourage their purchases with one-sheet newsletters of information and quotes from food fairs and gourmet magazines. If you find a chef receptive to trying your produce, grow one or two new herbs or vegetables each season and give them as samples.\n\nSome more tips on selling to restaurants:\n\n  * Make sure anything you supply to a restaurant is clean and attractive. Dirty produce is a quick way to lose a sale.\n\n  * Restaurants should be in a tourist area or close to a large population, or their volume may not be worth your time and effort.\n\n  * Try to have several restaurants as customers, and have produce that is used in a wide variety of dishes. If your sales are based on produce used only in the chef's house special, or in a single salad, you will be in trouble if that dish changes.\n\n  * Arrange delivery dates. If the chef can't make his specialty when he wants to, you may lose him as a customer. Leave your number with the restaurant for rush orders.\n\n  * Above all, make sure to supply the quantities a restaurant requests. If the chef asked for ten cases, don't bring six. Never promise what you can't deliver.\n\nIt is important to evaluate the volume of business available for each of your products. When I started my aquaculture business, I was selling trout and catfish to restaurants. I found though, that I was selling ten catfish for every trout, because there were only enough upscale restaurants in my area to use 25 pounds of trout per week. In contrast, I sold 50 to 100 pounds of catfish per week to several \"home-style\" restaurants, plus these fish sold much better in the grocery stores. I soon dropped the trout and raised only catfish.\n\n#### Market Pools and Groups\n\nMarket pools are often retail. They are simply a group of farmers in an area who agree to refer sales to each other and promote each other's businesses. A farmer may refer a customer to someone else in the pool if he or she is currently out of what the customer wants or if the customer is looking for something just a little bit different. This system also allows the customer to have more choice. To go a step farther, the farmers may band together to sell their stock as a group at the sale barn, or to a company looking for larger numbers than a single farmer can provide. If this is done on a regular basis, it would be better to form a cooperative.\n\nMarket groups are usually regional or national associations that try to put together buyers and farmers. There are a number of these groups, particularly in the herb and flower businesses. The advantage is that the farmer does not have to do sales. The disadvantage is the commission that most of these groups charge. To find these groups, start by talking to some of the herb and flower associations and magazines listed in the appendix.\n\n#### Cooperatives\n\nCooperatives are another way of getting someone else to do the marketing. These are especially good for farmers who dislike the effort of direct marketing or who have no \"people skills.\" Cooperatives work by gathering together a group of small producers who are all raising the same product(s). They pool their produce to provide the volume required by major buyers. The selling is done by a co-op salesperson, freeing up the farmers for more production time. The disadvantage is that farmers will not get the retail price for their produce (although they may get a small premium for quality), and part of their profits are eaten up by co-op fees (including the salary of the salesperson). A member may also be in trouble if the co-op has contracts for X amount of produce and his or her yield is poor that year. Depending on co-op rules, there may be fines or other penalties.\n\nThere are all types of cooperatives. A co-op may be as small as eight to ten ranchers pooling calves to get a bunch of 100 for sale as first-time calvers, or it could be a group of farmers specializing in heirloom tomatoes for delis. Co-ops can also be quite big, such as the sugar beet cooperatives in the Dakotas, where farmers who join sign strict production contracts.\n\nInvestigate cooperatives carefully before joining. If you want to start one, do your research first, and talk to lots of other cooperatives. Even loose cooperatives need rules and regulations for members to abide by. It must be determined what percentage of sales goes to co-op salespeople and advertising, how administrative duties will be carried out, and how many growers will be allowed in the co-op. The new \"New Generation\" cooperatives set strict limits for numbers of members, differentiating them from older co-op models such as the Missouri Farmers' Association and rural electric co-ops. In a small co-op, make sure the other members are all people you trust and are willing to do business with.\n\n### Pricing Your Product\n\nThe price you put on your crops, livestock, and value-added products is what determines your income. People new to direct marketing often underprice their products and thus cheat themselves. You must keep in mind that you have a quality product (if you don't, you need to work on it until you do) and that you deserve compensation for it. You should be reimbursed for your direct expenses in raising it, the cost of your land and tools, your time in labor and management, and any costs involved in adding value and marketing it.\n\nAccurate farm records are essential. You must know your costs of production before you can be reimbursed for them. If you do not know your costs of production, including labor, you cannot know the profit margin necessary to stay in business. As someone new to farming, your production records will not be as accurate as they will after 5 years of recording expenses, experiencing weather variations, and selling. Remember to adjust your prices as you gain experience.\n\nCosts of production for crops include:\n\n  * Seed or plants\n\n  * Purchased fertilizers (including the costs of transportation for driving back and forth from your local horse stable with \"free\" fertilizer)\n\n  * Any additional equipment you need for the garden and plants (pots, starter soil, fluorescent lights, cloches)\n\n  * Gasoline\n\n  * Depreciation of machinery\n\n  * Labor\n\nIf you grow four major crops, split the dollars spent on fuel, repairs, and maintenance among them.\n\nCosts of production for livestock include:\n\n  * Purchasing the stock\n\n  * Feed\n\n  * Veterinarian visits\n\n  * Medicine\n\n  * Fencing and shelter\n\n  * Labor\n\nA meticulous accounting will include such things as the miles you drive to and from the sale barn. Time spent selling door to door or at farmers' markets should be included, as well as any advertising dollars you spend.\n\nOnce you have allowed for expenses, decide what profit to allow yourself. This is where you need to check your competitor's prices. What are similar products selling for at farmers' markets? in grocery stores? at nearby roadside stands? Is your product different\u2014perhaps of more quality, all natural, or an heirloom variety? If so, price it accordingly. The customer will not think it is quality if you price it too cheaply. If you're not sure people will pay a high price for your product, give out samples to encourage interest. I used to sell 'Red Cup', an heirloom stuffing tomato, at a premium price to a small deli. The deli, in turn, sold it as a special, stuffed with tuna salad. Any variety of produce not found in stores should be sold at a higher price due to its rarity. This applies to meat and value-added products as well.\n\nYou may be tempted to underprice your produce to make some quick cash. This is a mistake. If, for example, you were to undercut your fellow farmers at a farmers' market, you would make them angry. Also, new customers will expect to see these prices every time, and you will get a reputation for low prices rather than for quality. Finally, it will cost you money in the long run. A drop in price requires a much greater increase in sales volume for the same net profit. By the way, if you have extra produce at the end of the day, take it home\u2014your chickens and hogs will appreciate the treat.\n\nAs time goes by, you may want to consider raising your prices. In general, for a moderate price increase, determine if the total dollar increase in gross revenues is higher than losing the gross income of 10 percent of your customers (at least in the short term). Remember, it is better to raise the price in small increments over a long period of time than to have a big jump in a short period.\n\n# Increasing Prices\n\nLet's say you are selling shares in a garden Csa for $100 and have ten regular customers, giving you $1,000 in gross income. If you raised your prices to $125, you can count on losing 10 percent of your customers\u2014in this case, one customer. This would leave you with nine customers, each paying $125, or $1,125\u2014worth the price increase.\n\n### Advertising\n\n**Principle :** _Advertising is necessary to help your customers find you_.\n\nAdvertising is a great way to gather customers and increase profit at your farm, and it doesn't have to cost buckets of money. The first and best way to advertise is word of mouth. Tell your friends and neighbors to spread the word. Tell your barber or banker about your produce. Give out free samples at association, community, and church events. Make sure you include directions to your farm via business cards, brochures, maps, and recipe cards with your address included. Pool your resources with fellow farmers, and offer directions to each other's operations. Give talks or slide shows about your operation to local business associations, garden societies, 4-H and other children's clubs, churches, and schools\u2014this is an excellent way to get free publicity. By creating a network of friends, relatives, neighbors, businesses, clubs, and community, you will enhance your business and your relationships.\n\nThe next step in advertising is flyers and notices. Place them on community bulletin boards, such as at hardware and grocery stores and local restaurants, and at service places such as barber-shops and dentists' and doctors' offices. Local motels may place your literature on their display rack to show their lodgers things to do while in town. If you have seasonal or one-time events at your farm, you can always submit these as calendar items to your chamber of commerce, plus to local newspapers, magazines, and radio stations. Remember to allow lead time for publications before an event\u2014some magazines may require two or more months' notice. Check with your extension office and state Department of Agriculture about any advertising opportunities they offer.\n\n# Keep Signs Simple\n\nWith roadside stands and farmers' markets, your advertising is often just words on posters or a wipe-off board. Make your signs attractive and colorful.\n\nRoadside signs must be simple. A rule of thumb is to have 1 inch of letters for every 10 miles per hour a car will be traveling. On a 40-mile-per-hour road, make your letters 4 inches tall. Put the signs far enough before the turnoff that cars can slow down in order to turn. I recommend placing a sign on roads approaching your farm from each direction and another sign at the turnoff. Put the signs about g mile from your farm, clearly indicating the remaining distance.\n\nMake signs in colors that stand out (yellow is not a good choice). On a colored background (such as neon pink), use only black lettering. The more professional and attractive the sign, the more business you will get.\n\nMake sure your roadside signs are visible and easy to read.\n\nBuying classified and display advertisements in ad gazettes, newspapers, and magazines may be useful, but it can also be expensive. This is even more true of radio and television. Instead, consider trying to get them to do a feature story on you. It is great publicity at no cost. Kelly Klober, direct marketer and author of _A Guide to Raising Pigs_ , recommends sending a box of samples to local radio or television personalities to promote your farm.\n\nKlober also recommends maintaining a mailing list of customers and sending them word of developments on your farm: new produce or livestock, baby animals, start of the season, special opportunities. When you have 200 names, investigate acquiring a bulk mailing permit at the post office. Make sure all the local media are on your mailing list. Send them regular press releases or a newsletter, or even just a postcard. Press releases should be limited to a single page, double-spaced.\n\n# Getting The Most From Your Ad\n\nWhen you buy advertising in newspapers and magazines, there are some rules to consider. A classified ad should have a catchy headline (Fresh Wholesome Vegetables, or Lean Tasty Beef) and contact information. If you are providing a service such as delivery, list it.\n\nDisplay advertising should have a catchy headline, a brief description of the product, contact information, and a picture\u2014either a photograph or a line drawing\u2014to get the customer's attention. Don't overload your ad with words or fill all the available space: White space will also draw the customer's eye.\n\nGood placement also enhances an ad's sales potential. The right-hand outer column is the best place to be. Talk to the publication's salesperson about preferred placement. There may be a charge for the best position, or you may get it free as a bonus if you buy multiple ads.\n\nTalk about prepayment and multiple-time discounts, and anything else you can do to bring the price down. Newspapers have fairly set prices, but magazines and ad gazettes may have some bargaining room.\n\nWeb sites are an up-and-coming sales technique. You may be able to arrange for notice on your community's site, or you may want your own Web page. Check out different companies for prices and services. It is best not to be buried too far in another company's Web site (e.g., _http:\/\/www.freshlamb.com\/_ is a better site choice than _http:\/\/www.somecompanies\/htolg\/findit\/services\/farms\/freshlamb.com_ ). Do not spend too much money on this method until you are sure of its effectiveness. On-line catalog sales may be something to consider as you expand.\n\n### Your Farm as a Destination\n\nThere are other ways to attract people to your farm, either by offering additional activities on the grounds or by providing services that keep customers interested in coming back. Activities might include fee hunting or fishing, swimming and boating, a bed-and-breakfast, ice skating, a petting zoo, a restaurant or caf\u00e9, a craft shop or art gallery (featuring works of local\/regional artists), and a children's play area (hay-bale mazes are popular).\n\nYou can encourage community and farm associations to hold events at your farm. This is another opportunity to sell your products and develop more names for your mailing list. Anything from a balloon race start (but consider the effect on your livestock) to a conservation wildlife day may help draw customers. You might sponsor your own Farm Day with fun activities to attract townsfolk, or a Field Day with an explanation of your profitable farming methods to attract fellow farmers. You may want to have a value-added workshop or class, where you charge people to come to your farm. This allows you to market your skills as well as your product. If you develop a farm kitchen, rent it to other value-added agripreneurs.\n\nTalk to schools, daycare centers, senior citizens' homes, travel agencies (touring companies), and clubs about doing tours of your farm. You can either conduct these free as promotional events or charge a fee per person or busload. Senior citizens enjoy visiting farms for reasons of nostalgia. They will all have a \"snakes in the blackberries\" story and a \"Remember when....\" A few busloads of senior citizens at a flat fee per person (say $2), plus additional sales, will enhance your profit margin. Visitors will also be ripe for future sales such as Christmas gift packs, and will share their great experience with their friends and relatives. Make sure they leave the farm with a brochure or catalog.\n\nSeasonal attractions such as hayrides and bonfires are another way to get people to come out. Decorate your farm for Halloween or Christmas. Sponsor a School Day through local elementary schools, where children get a small pumpkin free. (They'll encourage their parents to bring them back to the farm to buy bigger pumpkins and other sale items.) Have a pumpkin-carving contest (entry fee: just the cost of the pumpkin), with space donated for advertising through the local paper. Emphasizing a family farm experience helps make these visits an annual event, at the least. Sponsor a pumpkin or Christmas tree fund-raiser through a children's organization (4-H, Boy or Girl Scouts, for example) or a church or service group. Sell value-added holiday corn decorations or wreaths. Have an egg hunt at Easter, with sales of value-added decorated eggs or rabbits as pets. While you have the visitors on the farm, tell them about other opportunities they will have, such as buying fresh sweet corn in the spring. Keep a sign-up sheet on hand to expand your mailing list. You can also acquire addresses off checks.\n\nA welcoming farm on a well-traveled road will be popular with locals and tourists alike.\n\nOf course, to get people to come back, you will have to offer attractive services. A graveled parking lot and either gravel or concrete walkways are a big plus. If your fields are not close to the parking, offer wagon or cart rides out and back. Put straw between rows on a U-pick farm, or even plant grass for a nice walkway. Strategically located trash cans are essential for your own benefit. Shaded rest or picnic areas, hot or chilled food and drinks (pies, bread, muffins, cider, fruit shakes, hot chocolate), nice rest rooms, soda machines, shopping bags, all increase your customer base and profit. Your operation needs to have an attractive appearance.\n\nHere are a few more tips to keep in mind:\n\n  * Anyone manning the cashbox should be friendly, knowledgeable, and clean.\n\n  * Your produce should be attractive and clearly labeled with prices and types.\n\n  * Free samples always boost your sales, but be careful not to give away too many if you have large numbers of visitors.\n\n  * Accepting credit cards will almost always increase sales.\n\n  * Consider an 800 number for value-added sales.\n\n  * Always keep an eye out for what attracts you when you visit other operations and events. Innovation and an attractive operation will bring in customers\u2014and keep them coming back.\n\n###### Food For Thought\n\nMany people consider marketing to be the hardest part of their operation. It's not easy\u2014but direct marketing is what will improve your profit margin. Start planning your marketing approach now, including market identification, niche development, add-on value, and advertising, and your operation will prosper.\n\n###### Chapter Eight\n\n## Selecting Your Enterprise\n\nOnce you have set your goals, it is time to select your enterprise or enterprises. To do this, you'll have to research your farm and the enterprises you're interested in, then determine what you'll do and how you'll do it. Let's look at the factors you should consider:\n\n  * **What can your farm support?** Soil test your farm soil to determine what crops will grow best with its natural inherent fertility. Blueberries, for instance, like acid soil. Legumes such as clover and alfalfa prefer sweeter soils. Consider slopes and types of growth already present. A hilly area is usually more suited to livestock than to crops. Wooded areas might be left for tree crops.\n\n_Heirloom apples are in high demand today due to their superior flavor. They are just one of many possible enterprises you can choose to pursue_.\n\n  * **How will you water your crops and livestock?** You can plan flood irrigation, soaker hoses, drip irrigation, sprinklers, or a bucket. Consider your water sources in terms of access and cost. If you choose the bucket, your water source had better be close to the crop or a water trough. You may have a well, a spring, ponds, a rural water district, or heavy rains. Make sure you save all the water that falls on the farm. (See chapter 5 for more on water.)\n\n  * **What are the costs and returns of your enterprise?** Before you begin, plot a cash-flow budget for the season, including harvest and storage costs. Make both short- and long-term projections. Consider legal requirements, equipment and labor, transportation, facilities, and start-up costs to acquire seed and livestock. Returns will depend on your production levels and marketing methods. Consider how much of a particular crop to grow, and whether you will keep it on a small scale or expand in the future.\n\n  * **Are there permits required or special regulations for your product?** Many states have special requirements for shipping nursery plants or reptiles in and out of the state. Others require a permit for certain operations, such as a release from the Conservation Department to raise deer or a permit for fee hunting for game birds. Processing kitchens must require fulfilling regulations of city and\/or county health codes. Remember to add to your start-up costs fees for permits and money for equipment to satisfy regulations.\n\n  * **You will need to keep records**. Keep track of expenditures, livestock breeding schedules, crop yields, weather, and other markers you can use to improve your farm and choose future enterprises. Will you need any special records, like sales tax records and bloodline papers? Consider the time and cost of these records. To keep purebred stock, for example, you will probably want it registered, and may have to pay both for registration and to be a member of the registration association.\n\n  * **Can you plan in some quick returns for start-up?** Start-up capital is important. Once you know how much capital you need or can afford, determine how soon you can get a return on your investment. As I have said before, it is important to look at not just what you can raise, but how you will market it. Radishes are the fastest crop I know of. It takes 10 to 20 pounds of seed per acre to yield 900 to 2,000 pounds of radishes in only 21 days. This is all good, but can you market this many radishes from 1 acre\u2014much less 5 acres or more? Ready-to-lay chicken pullets can be laying inside of a week of delivery. Snap beans take 50 to 60 days, while blueberries take 3 to 5 years before in full production. You need to consider a mix of short-and long-term investments.\n\n  * **Must you have general farm liability?** Will the enterprise need liability insurance? U-picks and roadside stands require insurance for injury to customers. You may want to check into crop protection insurance, although it is usually harder to secure at a reasonable cost for alternative crops.\n\n  * **When will you start?** Start-up time is extremely important. In traditional agriculture, you can look at data from the starting year forward and determine whether the farm will make it, depending on how many good years of livestock and crop prices followed the start-up year, and the debt involved. The wrong year to start or expand caused great financial stress to farmers in the drought years of 1953 and 1954, and during the rural crises of the 1980s, for example. Although alternative agriculture is not quite as tied to the climate as is traditional farming, choosing when to start is still important (see box, page 154). This is as true of farmers changing their operation as it is of those new to farming.\n\nA new farmer I knew tried to start a vegetable operation in July. That particular year the weather was drier than normal, and with no practical way to get water, the start-up operation faltered. He become discouraged and quit farming. Patience goes a long way in any business\u2014it is important not to rush your start. Research carefully and choose your time, when the season and weather are appropriate.\n\n# Starting Out\n\nWhen should you start farming? The best time to start is the late summer\u2014say, August or September before the growing season you want to be producing income in. If you cannot do this for some reason, grow something in your backyard or in a vacant lot, but gain experience. Working on someone else's farm is a good idea.\n\nAssuming you can start in late fall of the season before you want or need the income, what do you do?\n\n  * Lay out a work plan for everything you have to do.\n\n  * You should already have the basic tools necessary for the coming year. Get them to your farm, and grease, oil, and otherwise prepare them for use now and storage over the winter. Buy whatever equipment you still need in the fall or before the spring season, when it costs more and you don't have time.\n\n  * Plan your marketing program. How much will you have to sell? How many customers will you need? Will you have to develop a market? Is the market already there?\n\n  * Lay out fields and buildings.\n\n  * Build whatever fencing and buildings you will need.\n\n  * Take soil tests.\n\n  * Gather as much free fertilizer as you can: Clean out barns, gather leaves and manure, start a compost heap, and spread it on crop ground.\n\n  * Plant fall cover crops so you have green-manure crops to turn under in the spring.\n\n  * Buy livestock as needed. Breed your animals to give birth in early spring.\n\n  * Order seeds and plants in January.\n\n  * Thirty days before the frost-free date (usually in April), start crops that require extra germination time (peppers, tomatoes, gourds, melons) under fluorescents lighting in a warm building. After about 2 weeks, take these transplants outside to a sheltered spot to harden them off, returning them inside at night.\n\n  * Plant everything on or after the frost-free date.\n\n  * Think about the big picture all the time.\n\n  * **How does the new enterprise fit with the rest of your farm projects?** You must consider what your enterprise will bring to the mix of crops and livestock that you already grow or want to grow. Rotating crops and livestock on your land will keep it healthy and rich. If customers will drive up to 50 miles for fresh berries, but will travel only 2 miles for snap beans, you may need berries on your farm in order to sell snap beans. Crops like pumpkins and Indian corn provide seasonal income, but are heavy feeders and will deplete soil nutrients. Season extenders like greenhouses enable you to have early and late crops, but they take up space and money. Tunnels or cloches may be an inexpensive alternative, but may also require more labor and management.\n\nChoose crops and livestock that support each other and reduce the costs of their partner. A simple example is hogs and corn. Train the hogs to graze the cornstalks during late growth and turn the animals onto the field after harvest. Feed the corn to the hogs throughout the year, and the hogs will turn under the pasture and fertilize it.\n\n  * **What type of production method works best and is most profitable for this enterprise?** Consider conventional, sustainable, natural, and organic methods. Crops can be grown in greenhouses, raised beds, rows, or trellises. Livestock can be raised on feedlots or pastures. You may want to practice management-intensive grazing (see chapter 4). Carefully examine the costs of production, labor, yield, and market potential to determine how to proceed.\n\n  * **Will the enterprise provide seasonal or monthly income or some combination of the two?** If it provides a short-term income during the year, combine it with other enterprises whose incomes mature at different times. Consider adding value to extend your enterprise's income period.\n\n# Farming Methods\n\n  * _Conventional farming:_ High-volume farming with a low margin of profit; uses any means possible to increase yields.\n\n  * _Sustainable farming:_ Low-input farming aimed at direct markets. Sustainable farming is environmentally sound, socially acceptable, and economically viable.\n\n  * _Natural farming:_ Natural farming avoids artificial inputs (antibiotics, chemical pesticides) and may be aimed at either commodity or direct markets; may be sustainable.\n\n  * _Organic farming:_ Certified organic farmers tend to have a low volume of production to be sold at premium prices to direct or niche markets. They avoid all synthetic pesticides and additives; may be sustainable.\n\n###### Comparison of Farming Types\n\n  * **What are the labor requirements of your new enterprise?** For example, berries require large amounts of hand labor. If the enterprise is constructed as a U-pick operation, though, with the customer doing the majority of the harvest labor, or as a Csa (community-supported agriculture; see pages 132\u2013134), where shareholders share the risk and provide some labor on the farm, your labor will be reduced. Consider all of the labor for all of your crops and livestock, and leave some time for yourself and your family.\n\n  * **Who will furnish the labor for your enterprise?** It may be you alone, your family, or hired labor, seasonal or full time. Labor costs in time and money must be considered for planting, harvesting, and marketing, to determine how many people are needed for a particular crop or livestock.\n\n  * **What will be the machinery requirement for this enterprise?** Consider your alternatives in terms of both time saved and investment costs. Will a $500 rototiller do the job or will it take a $3,000 to $5,000 used tractor? Consider renting, leasing, and buying options. Check with your neighbors to see what they have available and what you can barter for. If you need specialized planting equipment or harvesting equipment, like a potato digger or a cherry tree shaker, make sure you can get it. There is harvest machinery for just about anything you want, but do you have the volume necessary to make machine-harvest a viable option?\n\n  * **What kind of industry support is available?** How accessible are the plants, seeds, and breeding stock for your enterprises? You may also need research or management support. Look for production budgets, and check for university or extension personnel with knowledge of your crop or animal. Check with area farmers and associations for expert consultants. If no research has been done on an alternative crop, start small and grow into it.\n\n  * **How will you market your products?** You can sell wholesale, at a lower profit, or retail at a higher profit. You can add on value to a raw commodity to increase your profit and market time, at the cost of more work and investment. Contract selling locks in sales price, but many clauses for grades and quality are subject to discount. For the most part, you are better off doing it all on your own, and carrying all the risk but taking all the profit. Nobody can watch all the eggs in your basket better than you.\n\n  * **Where will you market your products?** Much as your farm risk is reduced by having a variety of crops and livestock, your profit risk is reduced by having a variety of markets. A farmers' market is a great place to sell produce, but you have to haul it to town. U-picks get the customers to provide part of the labor but require good liability insurance. Roadside stands provide good sales if they are located on a well-traveled hardtop road with room for parking. Route sales work well if you can combine the trip with some other project to lower your costs. At one time, when I had 200 laying hens, I delivered eggs to businesses along the road on the way to get feed every Friday morning. (See chapter 7 for more marketing information.)\n\n  * **Is there a market already established or will you have to create your own?** Even if there is a farmers' market in your area, it may not allow sales of your product. Some markets do not permit meat products or craft projects to be sold. Local stores may be able to buy from local farmers or their buying may be tied to an out-of-state headquarters.\n\n  * **How will you advertise?** You must advertise to let people know who you are and what you have for sale. You can use word of mouth, newspapers, magazines, radio, television, or direct mail. (See chapter 7 for more information on advertising.)\n\n  * **What level of experience do you have for this enterprise?** Are you willing to learn? Raising elk is similar to raising cattle, and could be an easy change. If you have never planted a crop, it is better to start with a garden than with 5 acres. If you have not practiced direct marketing, consider how much you like dealing with the public, and how good you could be as a salesperson. Which family members are best suited to deal with the public? The key is to be always friendly, be willing to learn from your customers, and never be afraid to ask questions.\n\n  * **How will you price your product?** You can sell by volume, by weight, by the head, or by the pound. The final price must include the percentage of profit you need plus all of your costs, including labor. Consider what the market will stand when planning your price. Can you sell your product above, below, or the same as store price? Do you have any special features to raise the price, such as organically grown, higher quality, or uniqueness? If you are the only person in your area with Christmas trees, you have a mini-monopoly and can sell at a premium price. As the only person selling okra, you can sell it at a premium price if there is a demand for it. If no one is familiar with okra, you will have to create a market before it will sell well.\n\n  * **How much competition is there for sales of your product?** Look at other farms engaged in your enterprise within 25, 50, and 100 miles. Look at how they are marketing it and what volume of sales they have. Can you offer something they don't? If you and forty other farmers are selling a standard breed of tomato at midseason at a local farmers' market, count on prices no better than wholesale.\n\n  * **How do you feel about your enterprise?** There's an old saying among cattle buyers: \"Bought right is half sold.\" The same is true in picking a business enterprise. Never choose an enterprise just because it will be profitable. You have to like what you're doing; you'll be more successful if you enjoy your work.\n\nThe items I have listed here are all questions you must ask yourself about any new enterprise. However, they are not the only questions. In researching your choice, ask yourself whatever questions arise from its unique circumstances. Keep in mind its requirements for planting, growing, harvesting, and marketing. By approaching the planning in a systematic way, you will find out not just about this one enterprise but also how it compares to other enterprises. This way, you'll be able to have the most profitable mix on your farm.\n\n### Types of Enterprises\n\nWhat you can raise makes an almost infinite list. It expands even more when you factor in how you are marketing your enterprise. I want to cover a few enterprise choices, along with some of their advantages and disadvantages and their potential markets.\n\n  * **Cattle**. Cattle are normally raised for beef or dairy. Both require large volume, unless you can develop direct markets. _Consider_ small breeds such as Dexter cattle and Miniature Herefords, less common breeds such as Murray Grey and Belted Galloways, intensive grazing, raising organically or naturally, selling beef halves or quarters.\n\n  * **Hogs**. To be successful, aim for the purebred or value-added markets. _Consider_ rare breeds such as Mulefoot and Tamworth, raising on pasture, organic or natural, selling meat cuts or sausage.\n\n  * **Sheep**. Traditionally, sheep have been raised for wool, but there are increasing meat and dairy markets. The wool market is not profitable unless you add value through spinning or weaving. Some sheep are being raised exclusively for meat, such as Katahdin Hair sheep. There is some trophy hunting being done with breeds like Barbados Blackbelly. Dairy sheep like East Friesian are being used for cheesemaking. _Consider_ rare breeds such as Navajo-Churro and Miniature Babydoll Southdowns, \"new\" breeds such as Dorper (a meat sheep), intensive grazing, selling meat cuts or sausage, seasonal lamb markets.\n\n  * **Goats**. Meat and dairy goats are common. Some fiber goats, such as angora, are prized for spinning and weaving. _Consider_ ethnic markets for cabrito (goat meat), crossbreeding with Boer goats (a prized but expensive meat goat from South Africa), goat soaps, miniature breeds such as Nigerian Dwarf.\n\nDairy goat\n\n  * **Rabbits**. Rabbits are raised for meat and pelts. Small and dwarf breeds are sold as pets. Some breeds, such as angora, are raised for fiber for yarns. Deborah Lemmer, a canny marketer from Shaggy Shagbark Acres in Michigan, likes to demonstrate spinning directly from a rabbit sitting quietly in her lap. _Consider_ rabbit jerky and sausage, restaurant sales.\n\nDucks\n\n  * **Poultry**. Poultry includes chickens, turkeys, ducks, guineas, geese, and peafowl. There is a large chicken fancier market. You can sell purebred poultry, chicks, eggs, or meat. One farmer in Canada raises chickens bred exclusively for fly-tying feathers. _Consider_ rare breeds such as Dominique chickens and Khaki Campbell ducks, pastured poultry, egg ornaments, ethnic markets for black-feathered breeds, organic eggs.\n\nTurkey\n\n  * **Elk, deer, and bison**. These are gaining acceptance in the meat market for being low in fat and high in protein. All require a large initial investment in fencing and permits. Elk are now considered livestock in many states. Velvet from elk and deer antlers is a high-ticket item used for medicinal purposes. The purebred market is doing well selling breeding stock. Hides are also a good seller. _Consider_ jerky and sausage, restaurant sales, trophy hunting.\n\n  * **Ratites**. The bottom dropped out of the speculation market on ostriches, emus, and rheas a few years ago, but they still have sales potential. Meat on all of them is low in fat and high in nutrients. Hides and eggs have value-added potential; ostrich feathers and emu oil are also possible products. Initial investment is fairly low, but it takes effort to build a market. _Consider_ sales to gourmet restaurants, emu oil pain rub, decorations with hide and feathers, carved or painted eggs.\n\n  * **Game birds**. Pheasants and quail can be raised for breeding, meat, or hunting. Initial investment is fairly low, but permits are needed and markets must be sought. _Consider_ sales to restaurants and upscale grocery stores, selling birds to hunters or providing hunts on your property, catalog sales of chicks.\n\nLlama\n\n  * **Camelids**. Llamas and alpacas are used primarily for fiber and breeding markets. Llamas are gaining acceptance as pack animals and guard animals for sheep flocks. There is really no meat market for camelids. Alpacas can be a fairly hefty investment, but items made from alpaca wool sell at a premium. Heat can be a problem. _Consider_ Alpaca yarn, llama sweaters, blankets, guided trips in National Forests with llamas.\n\nAlpaca\n\n  * **Small mammals and reptiles**. Small mammals include African Pygmy hedgehogs, Sugar gliders, chinchillas, and gerbils for the pet market. Costs are low and space requirements are few, but permits may be required. Locating a market is the hardest part. _Consider_ pet shows and sales, pet stores, local fairs.\n\n# A Fish Story\n\nThere are twenty-eight major fish-producing countries. On a worldwide basis, at least ninety species of fish, thirteen species of shrimp, prawns, and crayfish, and a wide variety of shellfish and marine plants are produced using aquaculture techniques. Many developing countries are turning to fish farming to satisfy their protein needs, increase their self-dependence, and reduce risks from the dwindling ocean fish supply.\n\nAccording to the Global Aquaculture Association, as reported in the January\/February 1998 issue of _Fish Farming News,_ trade in seafood products now generates $50 billion annually. There is an increasing demand among Americans for fish products. Shrimp accounts for about 25 percent of the fresh and frozen seafood consumed in the United States\u2014and farm-raised shrimp makes up nearly one-third of the total shrimp supply. Population growth is expected to increase shrimp demand by about 2 to 3 percent per year. Currently, the Usda spends only about   as many dollars on aquaculture as it does on crop research, but that is increasing.\n\nThe potential of aquaculture for a farming enterprise or home consumption is tremendous. I raised catfish in the 1980s in cages 36 inches in diameter by 4 feet deep. My already existing ponds were too deep to seine; the cages made them usable. It was fairly easy to produce 400 to 600 pounds of catfish in one of these cages in just 20 or 21 weeks, with feed efficiency of 1\u00bd to 2 pounds of feed per pound of gain. A 5 percent death loss and a 55 percent dress-out (the percentage of usable meat after the removal of skin and viscera) grosses about $500 per cage, or could furnish about 210 pounds of fresh fish for your freezer. I experimented the first year, then built a 10-foot by 6-foot fish-processing plant complete with a used 6-foot by 6-foot walk-in cooler, and butchered my own fish as well as neighboring farmers' fish.\n\n  * **Aquaculture**. Catfish, trout, and bass can be raised for breeding stock, meat, or fee fishing. Meat fish such as tilapia are used in recirculating systems. Exotic small fish can be raised for the pet market. Costs can range from very low to extremely high. Farm-raised catfish and trout have good grocery and restaurant sales potential, although a small processing kitchen will be necessary. _Consider_ restaurant and grocery store sales, fee fishing, sales at farmers' markets\u2014live (if allowed) or packed in ice\u2014sales through Csas (see chapter 7).\n\n  * **Vermiculture (worms)**. Selling worms for bait or for farm fields has a lucrative market. Some are grown and sold in compost kits for city dwellers. Low cost, but it takes work to keep selling them. _Consider_ selling with \"compost kits\" at environmental shows and Earth Day fairs. Sell kits to local schools.\n\n  * **Bees, butterflies, and other insects**. Bees can be sold for breeding or rented for pollinating crops. Their primary market is value-added products from pollen, wax, and honey. Butterflies for pollinating and release at events such as weddings are gaining in popularity. Beneficial insects are also gaining acceptance as chemical use declines. These are all management-intensive, and market potential may be small. _Consider_ grocery store honey sales through a grocery store or Csa (see chapter 7), trading pollination for a crop share.\n\n  * **Dogs**. Dogs are raised for pets, breeding stock, livestock guarding, herding, and hunting. The market is well established and steady. There are hundreds of breeds to choose from, and investment cost is low. If training is required, dogs are obviously management-intensive. _Consider_ training dogs, stock dog clinics and shows, dog shows.\n\n  * **Equines**. Horses, donkeys, and mules are not inexpensive. Draft breeds are gaining in interest, but require a lot of time and effort to raise and train. Riding stock will not sell easily unless you are well known, although you might consider trail riding or establishing a stable. Donkeys are being used as livestock guardians. _Consider_ breeding, racing, draft animal schools, Plow Days (a field day of draft-animal demonstrations to gather customers).\n\n  * **Pasture, hay, and cover crops**. These are the perfect way to combine crops and livestock. Except for baled hay, their market potential is not high, but they are necessary components in the interaction of different enterprises on your farm. _Consider_ hay trade to neighbors for crops, hay-bale mazes at a U-pick, miniature bales (6 inches long) for sale at country craft events.\n\n  * **Traditional grains**. It is hard to profit from field corn, wheat, and soybeans without adding value. There is excellent potential for open-pollinated corn and unique soybeans, such as tofu. Open-pollinated corn is desired for its high feed value (usually more nutritious than hybrid corn) and its open-pollinated qualities, specifically, the ability to save seed from it for replanting. Some breeds also have sales potential for decorative purposes (\"Indian\" corns and Cherokee Blue, for example). Less traditional grains such as barley, oats, and sunflowers may have market potential, but should be considered mostly for on-farm use. _Consider_ cornstalks and miniature corn for Halloween, chocolate-covered soybeans, organic markets, seed sales, cornhusk- and wheat-weaving craft projects, flours, and breads.\n\n  * **Vegetables and melons**. The traditional crops for direct marketing. A huge variety of both open-pollinated and hybrid varieties are available. Invest the effort to have them available before or after the regular season to increase sales. _Consider_ rare and exotic varieties with a unique appearance or flavor, ethnic specialties, \"baby\" vegetables, pepper sauces and wreaths, tomato sauces.\n\n  * **Herbs and medicinal herbs**. Herbs are intensive crops with increasing market potential. Find buyers for medicinal herbs before raising them. High-dollar herbs, such as ginseng and echinacea, require a big investment in care. Herbs' best potential is probably on a small scale, as accompaniment to value-added items and farmers' market sales. _Consider_ value-added soup mixes, vinegars, essential oils, and soaps, or sell fresh specialty herbs like lemongrass to specialty restaurants.\n\n  * **Mushrooms**. Mushrooms are easy to raise but require a few years before realizing much income. To raise them on a large scale may require special facilities and a good source of wood. Sales of specialty varieties to restaurants, grocery stores, and consumers is lucrative, as is selling mushroom-growing kits\u2014prepared logs inoculated with mushroom spawn. _Consider_ value-added items like dried mushrooms and soup mixes.\n\n  * **Gourds and pumpkins**. These are good seasonal items. It is best to add value to them for marketing. _Consider_ carved pumpkins, specialty miniature and white pumpkins, painted gourds, pumpkin pie (usually made from butternut squash), specialty gourds such as luffa and dipper gourds.\n\n  * **Industrial crops**. Crops such as crambe, meadowfoam, guayule, and jojoba are used for nonfood industrial applications, such as lubricants and plastics. These often have excellent potential for profits, but require an enormous investment in time, labor, and learning, without an obvious market readily available. I do not consider them sustainable at the current time, in that you cannot use them on your own farm, and you are at the mercy of a very small number of buyers for market price and availability. (If only one company will buy a crop and it goes belly-up, you could be in trouble.)\n\n  * **Small fruits**. Strawberries, brambles (e.g., raspberries), ribes (e.g., gooseberries), and other berries have great potential for U-pick and value-added markets. Planting varieties that mature at different times of the season will ensure a steady flow of customers. Grapes have value-added potential as juice, wines, and flavoring. Unusual varieties can attract interested customers. _Consider_ : U-picks, farmers' markets, grocery stores, health stores, syrups, dried berries, chocolate-covered berries, jams and jellies, fruit drinks.\n\n  * **Fruit and nut trees**. Orchard fruits like apples, pears, peaches, and cherries, and nuts such as hazelnuts, walnuts, and pecans do well at U-pick farms and farmers' markets, and may have grocery potential. For fruits, particularly apples, try to have varieties with different flavors and that mature at different times. _Consider_ value-added potential, or a \"fruit and nut subscription service,\" where a basket is mailed to the customer two or three times a year.\n\n  * **Flowers, cut and dried**. The cut-flower market is growing (pardon the pun), and there are certainly a multitude of varieties. With a greenhouse and a careful mix of maturity times and types, sales can continue year-round. Combine it with a nursery tree operation. There are good value-added markets (get-well baskets, dried arrangements) and also seasonal potential (Valentine's Day, Easter). _Consider_ craft show sales, grocery stores and farmers' markets, flower shops, providing arrangements to upscale banquet halls.\n\n  * **Timber**. Tree crops can be raised for timber, firewood, or nursery stock. Sell nursery stock for windbreaks or wildlife shelter. When planted carefully and allowed to grow to maximum potential, tree crops can be highly profitable. One good choice is a quick-growing variety, such as _Pawlonia_ or bamboo. Christmas trees also have excellent seasonal sales potential. Think before you clear an area of trees\u2014you may want to call your extension service or Forestry Department to get an evaluation.\n\nChristmas trees can provide income outside the traditional growing season. Note, too, that not all farm crops need to be edible.\n\n  * **Other alternatives**. There are many other alternative crops, from fava beans to shrimp, and from lions to sweet sorghum. The rarer it is, the harder it may be to locate a market\u2014but then competition is reduced, too. Try to pick crops and livestock that will be easy to raise, can be marketed in two or three different ways, and support each other.\n\n### Enterprise Cost Analysis\n\nFollowing are a list of budgets compiled from university, government, and private sources. These budgets are intended to help you in the process of selecting an enterprise. Keep in mind that you need to select those enterprises that match your sales, labor, and capital resources.\n\nThese budgets are good for planning, and for comparing \"apples to apples\" as to which enterprises are the highest grossing or most labor intensive. They also allow you to get a basic feel for investment and the labor involved. Let's analyze the budget below for a sow and sixteen pigs, direct-market as sausage, to see what information we can gather.\n\n# Warning!\n\nThe following budgets are not complete and may not even be accurate in your area of the country. Feed costs, for instance, may be 20 to 30 percent higher in some parts of the nation. Demand for machinery may be different in one region from another. Some of the vegetables and livestock I mention may not be prevalent enough for you to buy breeding stock in your area or purchase seed to plant. Start-up depreciation, interest, and tax costs are not included, and labor costs are not removed from the net profit. These budgets do not usually take alternative or added-value marketing methods into account. Thus, it is possible to get much more money out of an enterprise than is shown.\n\nFirst, consider feed costs. The $1,116.00 feed costs divided by the gross income of the fed hogs, $4,687.50, means your feed costs are 24 percent of the gross income. Can you grow the feed or will you have to buy it? Will changing feed prices affect your project? In this case, prices will not have much effect, because the percentage of feed costs to gross income is relatively low and your profit margin is high.\n\nWhat about labor? Half the labor, 20 hours, went to producing the product (hogs for sausage) and half the labor went to marketing the product (sausage). If we divide the net profit, $4039.50, by 40 hours, we find you have an income of $100.99 per hour for your labor expended on hogs.\n\nLeaving out weekends and holidays, farmers need about 300 ten-hour days of labor to be fully employed, or 3,000 hours per year. (Yes, I know it's not fair that farmers have to work 10-hour days instead of 8, but that's part of the joy of being a farmer.) So using these figures, you would know that if you had six sows, ninety market hogs, and six cull sows, they would take 240 hours (6 sows \u00d7 40 hours) of labor for your hog project per year, leaving 2,760 hours for crops or other livestock projects. Dollars earned per hour is one way to measure profitability.\n\nAnother way to look at and compare profitability is net profit divided by gross income. In this budget, that equals 78 percent. A high percentage like this means that operating costs are low, as compared to projects like beef cows and stock calves, where the percentage is 61 percent.\n\nRemember always that profitability depends on how well you market your product. Just selling retail instead of wholesale can make some projects profitable. In this hog example, it was profitable even with hogs selling for 10 cents per pound on the commodity market, because the product was not sold as a hog, but instead was direct-marketed straight to the consumer as a different product, whole hog sausage.\n\n### Profitability Comparison of Enterprises\n\n#### Livestock\n\nBeef cow\/stocker calf  \n---  \nIncome  \n500-lb. calf \u00d7 .95 (calf crop) \u00d7 .84 \u00d7 $0.80 | $319.20  \nOne cull cow: 1000 lbs. \u00d7 .16 $0.40 | $64.00  \nTotal gross income: | $383.20  \nExpenses  \nFeed: hay 1.5 tons | $75.00  \nPasture 3 tons | $38.00  \nProtein and salt | $5.00  \nOther var. costs | $30.00  \nNet Profit | $235.20  \nLabor | 7 Hours  \nBison (1 cow)  \n---  \nBased on 15 cows and 1 bull  \nGross income | $1,686.00  \nExpenses  \nFeed | $486.00  \nOther var. costs | $155.00  \nNet profit | $1,045.00  \nLabor | 10 hours  \nBobwhite quail (meat birds)  \n---  \nPurchased as day old, sold at 13 weeks, 3 groups of 1,000 birds, 15% death loss  \nGross income 2,550 birds x $2.80 | $6,757.00  \nExpenses  \nFeed 9,112 lbs. | $1,457.00  \nOther var. costs | $1,680.00  \nNet profit | $3,620.00  \nLabor | 220 hours  \nChickens, broilers  \n---  \n1000 chicks, 10% mortality, 8\u20139 weeks to market. 900 chicks after mortality \u00d7 4 lbs = 3,600 lbs.  \nIncome  \n3600 lbs. \u00d7 $2 | $7,200.00>  \nExpenses  \nFeed: 10,800 lbs. (3 lbs. per lb. sold) @ $0.14 | $1,512.00  \nOther var. costs | $1,000.00  \nNet profit | $4,688.00  \nLabor | 500 hours  \nChickens, purebred  \n---  \n20 hens with production 60% (220 eggs per hen)  \n220 eggs \u00d7 .60 (hatching rate) | 132 eggs per hen  \n132 eggs \u00d7 20 hens | 2,640 chicks  \nIncome  \n2640 chicks \u00d7 $3 | $7,920.00  \nExpenses  \nFeed | 280.00  \nOther var. costs | 200.00  \nNet profit | $7,440.00  \nLabor | 40 hours  \nDairy goat  \n---  \nDoes averaging two kids  \nIncome  \nMilk sales, culls, and replacements | $586.00  \nExpenses  \nFeed | $234.00  \nOther var. costs | $127.50  \nNet profit | $224.50  \nLabor | 22 hours  \nEarthworms  \n---  \nProduction 4 pounds per square foot of worm bed  \nIncome  \n840 pints (100 worms to a pint) | $840.00  \nExpenses  \nVariable costs | $80.00  \nNet profit | $760.00  \nLabor | 112 hours  \nElk  \n---  \nBased on 25 cows and 2 bulls in the breeding herd  \nIncome  \nBreeding stock and velvet | $4,144.00  \nExpenses  \nFeed | $429.00  \nOther var. cost | $149.00  \nNet profit | $3,566.00  \nLabor | 9 hours  \nFallow deer (1 doe)  \n---  \nBased on 100 does and 5 bucks  \nIncome  \nVenison, breeding stock, hides | $253.00  \nExpenses  \nFeed | $59.00  \nOther var. costs | $63.00  \nNet profit | $131.00  \nLabor | 3 hours  \nHogs, feeder pig purchased  \n---  \n40-lb. pig fed to 250 pounds. Direct market as sausage ($2.50\/lb.)  \nIncome  \n100 pounds sausage \u00d7 $2.50 | $250.00  \nExpenses  \nFeed | $48.00  \nPig cost | $32.00  \nNet profit | $170.00  \nLabor | 8 hours  \nHogs, sow and 16 pigs raised (2 litters per year)  \n---  \nDirect Market as Half Carcasses  \nIncome  \n15 market hogs \u00d7 250 lbs. \u00d7 $0.80 | $3,000.00  \n1 cull sow (400 lbs. \u00d7 .50 \u00d7 $2.50) | $500.00  \nTotal gross income | $3,500.00  \nExpenses  \nFeed | $1,116.00  \nOther var. costs | 32.00  \nNet profit | $2,352.00  \nLabor | 20 hours  \nHogs, sow and 16 pigs raised (2 litters per year)  \n---  \nDirect Market as Sausage ($2.50 per pound)  \nIncome  \n15 market hogs \u00d7 250 lbs. \u00d7 .50 \u00d7 $2.50 | $4,687.50  \n1 cull sow (400 lbs. \u00d7 .50 \u00d7 $2.50) | $500.00  \nTotal gross income | $5,187.50  \nExpenses  \nFeed | $1,116.00  \nOther var. costs | 32.00  \nNet profit | $4,039.50  \nLabor | 20 hours  \nMarket labor | 20 hours  \nHoneybees  \n---  \nIncome  \n60 pounds honey per hive \u00d7 $ | $180.00  \nExpenses  \nVariable costs | $90.00  \nNet profit | $90.00  \nLabor 11 | hours  \nPheasant  \n---  \nPurchased 200 day-olds, sold at 20 weeks for flight birds  \nIncome  \n80 roasters\/80 hens | $1,120.00  \nExpenses  \nFeed | $423.00  \nChick ($0.90) | $180.00  \nOther var. costs | $205.00  \nNet profit | $312.00  \nLabor | 120 hours  \nRabbits (fryers)  \n---  \n20 does, 2 bucks, 5 litters per year. Sell 5 lbs. at 10 weeks  \nGross income 3,500 lbs. \u00d7 $0.80 | $2,800.00  \nExpenses  \nFeed 5 tons | $1,400.00  \nOther var. costs | $129.00  \nNet profit | $1,271.00  \nLabor | 200 hours  \nSheep  \n---  \nOne ewe spring lamb born March, April, and May. Lambs 150% lamb crop, 20% replacements  \nIncome  \n1.50 \u00d7 120 lbs. \u00d7 $0.80 | $144.00  \n150 lbs. cull ewe \u00d7 $0.20 = 30 lbs. \u00d7 $0.20 | $6.00  \nWool: 10 lbs. \u00d7 $1.50 (added-value wool) | $15.00  \nGross income | $165.00  \nExpenses  \nFeed 500 lbs. hay \u00d7 $.05\/lb. | $25.00  \nGrain 100 lbs. ewe and lamb \u00d7 $.035\/lb. | $3.50  \nFair pasture .75 acres | $11.25  \nWormers, salt, minerals | $5.00  \nOther var. costs | $30.00  \nNet profit | $90.25  \nLabor | 6 hours  \nSheep, feeder lamb  \n---  \nBought 70 pounds, fattened to 120 pounds  \nIncome  \nLamb 120 lbs. \u00d7 $0.75 | $90.00  \nExpenses  \nFeeder lamb 70 lbs. \u00d7 $0.78 | $54.60  \nFeed | $14.00  \nVariable costs | $7.38  \nNet profit | $14.02  \nLabor | 1 hour  \nSheep, meat type  \n---  \nOne ewe spring lamb born March, April, May, 20% replacements  \nIncome  \nLambs 1.75 lamb crop \u00d7 120 \u00d7 $0.70 | $147.00  \nExpenses  \n150 \u00d7 .20 = 30 lbs. \u00d7 $0.20 | $6.00  \nFeed | $44.75  \nOther var. costs | $30.00  \nNet profit | $66.25  \nLabor | 6 hours  \nSheep, milk  \n---  \nMilking 160 days\/year (lambing 1.5 times\/year) 1.65 lambs per ewe  \nIncome  \n160 lbs. milk | $104.00  \n95 lbs. lamb | $128.25  \nWool and hides | $10.00  \nCull ewes and ram | $7.50  \nGross income | $249.75  \nExpenses  \nFeed | $85.45  \nOther var. costs | $20.86  \nNet profit | $143.44  \nLabor | 15 hours per ewe\n\n#### Crops\n\nAsparagus  \n---  \nAverage production 1, 2, and 3 years  \nGross income\/acre | $3,333.00  \nTotal var. costs | $1,210.00  \nNet profit | $2,123.00  \nLabor | 24 hours  \nBell pepper  \n---  \nGross income\/acre (9,960 lbs.) | $3,357.00  \nTotal var. costs | $2,334.00  \nNet profit | $1,023.00  \nLabor | 185 hours  \nBlackberries  \n---  \nYears 2\u201315  \nGross income\/acre (8,400 lbs.) | $6,300.00  \nTotal var. costs | $1,454.00  \nNet profit | $4,846.00  \nLabor | 210 hours  \nBlueberries  \n---  \nYears 3\u201315  \nGross income\/acre (4,000 lbs.) | $6,400.00  \nTotal var. costs | $1,200.00  \nNet profit | $5,200.00  \nLabor | 200 hours  \nBroccoli  \n---  \nGross income\/acre (8,250 lbs.) | $4,169.00  \nTotal var. costs | $1,002.00  \nNet profit | $2,136.00  \nLabor | 136 hours  \nCabbage  \n---  \nGross income\/acre (16,250 lbs.) | $1,746.00  \nTotal var. costs | $1,897.00  \nNet profit | $\u2013151.00  \nLabor | 115 hours  \nCantaloupe  \n---  \nGross income\/acre (10,000 lbs.) | $1,955.00  \nTotal var. costs | $1,056.00  \nNet profit | $896.00  \nLabor | 108 hours  \nCucumber  \n---  \nGross income\/acre (12,500 lbs.) | $2,150.00  \nTotal var. costs | $1,255.00  \nNet profit | $895.00  \nLabor | 153 hours  \nEggplant  \n---  \nGross income\/acre (16,500 lbs.) | $3,973.00  \nTotal var. costs | $2,118.00  \nNet profit | $1,855.00  \nLabor | 207 hours  \nLima beans  \n---  \nGross income\/acre (3,200 lbs.) | $1,600.00  \nTotal var. costs | $985.00  \nNet profit | $615.00  \nLabor | 128 hours  \nOkra  \n---  \nGross income\/acre (9,000 lbs.) | $6,160.00  \nTotal var. costs | $2,197.00  \nNet profit | $3,963.00  \nLabor | 301 hours  \nOnion  \n---  \nGross income\/acre (1,500 sacks 50 lbs. apiece) | $10,500.00  \nTotal var. costs | $5,992.00  \nNet profit | $4,508.00  \nLabor | 20 hours  \nPotatoes  \n---  \nGross income\/acre (15,000 lbs.) | $2,150.00  \nTotal var. costs | $1,086.00  \nNet profit | $1,064.00  \nLabor | 105 hours  \nPumpkin  \n---  \nGross income\/acre (28,000 lbs.) | $2,800.00  \nTotal var. costs | $1,419.00  \nNet profit | $1,381.00  \nLabor | 25 hours  \nRaspberries (Red Heritage fall berries)  \n---  \nYears 2\u201315  \nGross income\/acre (7,500 lbs.) | $6,375.00  \nTotal var. costs | $1,192.00  \nNet profit | $5,183.00  \nLabor | 111 hours  \nSnap beans  \n---  \nGross income\/acre (4,200 lbs.) | $2,163.00  \nTotal var. costs | $1,002.00  \nNet profit | $1,162.00  \nLabor | 109 hours  \nSpinach  \n---  \nGross income\/acre (6,000 lbs.) | $2,220.00  \nTotal var. costs | $1,031.00  \nNet profit | $1,189.00  \nLabor | 103 hours  \nStrawberries  \n---  \nYears 2\u20134  \nGross income\/acre (6,900 lbs.) | $4,116.00  \nTotal var. costs | $1,466.00  \nNet profit | $2,650.00  \nLabor | 96 hours  \nSummer squash  \n---  \nGross income\/acre (13,860 lbs.) | $3,811.00  \nTotal var. costs | $1,311.00  \nNet profit | $2,500.00  \nLabor | 140 hours  \nSweet corn  \n---  \nGross income\/acre (900 dozen) | $1,363.00  \nTotal var. costs | $705.00  \nNet profit | $658.00  \nLabor | 71 hours  \nSweet potatoes  \n---  \nGross income\/acre (15,000 lbs.) | $3,350.00  \nTotal var. costs | $1,570.00  \nNet profit | $1,780.00  \nLabor | 115 hours  \nTomatoes  \n---  \nGross income\/acre (17,000 lbs.) | $1,360.00  \nTotal var. costs | $623.00  \nNet profit | $737.00  \nLabor | 76 hours  \nWatermelon  \n---  \nGross income\/acre (20,000 lbs.) | $7,686.00  \nTotal var. costs | $3,101.00  \nNet profit | $4,585.00  \nLabor | 306 hours\n\n### Diversity\n\n**Principle :** _A diversified farm has the best chance of success_.\n\nDiversity in enterprises provides you with economic stability. By diversity I mean raising both crops and livestock, and more than one kind of each. If you have crops and livestock that you can sell at different times of the year, and have something else to sell when the market for any particular product drops, you will have a steadier income.\n\nEnterprises that feed into one another strengthen this bond. Use rabbit waste to raise worms, or set up a recirculating water system to feed fish waste to plants. Running and feeding livestock on a plot that you will raise vegetables on next year can provide all or part of the fertility needed for your vegetable crops. Rotating your crops and using cover crops in between can also provide some grazing and possibly hay for your livestock. By using crop ground or vegetable ground for livestock in winter and crops in spring and summer, you can triple your gross per square foot.\n\nThis same multiplication factor works if you can graze cattle, sheep, and goats on the same pasture. This, however, is highly management-intensive and should be done only after you have gained some experience in intensively grazing one species.\n\nA diversified, sustainable farm will give you the best chance to succeed in agriculture. Keep diversity in mind when choosing your enterprises.\n\n### Sustainability\n\n**Principle :** _To be sustainable, farm enterprises must be profitable, environmentally sound, and socially acceptable_.\n\nSustainability on the farm means that it is profitable\u2014an unprofitable farm cannot sustain its financial operations. If it is unprofitable, it must be subsidized with off-farm income. This is fine for getting started, and certainly cheaper than a bank loan. In the future, however, it should stand on its own\u2014if the farm is not profitable, you will not be able to realize your lifestyle or monetary goals.\n\nThe enterprises that make up your farm must be environmentally sound and socially acceptable to be sustainable. Sustainable crops and livestock can adjust to and thrive on a low-input-type system. My Katahdin Hair sheep are a breed noted for their natural parasite resistance (lower worming expense). I lamb them on pasture in May when grass is up and growing. Katahdins have few lambing problems, and are good mothers and milkers. Under my feeding and grazing system\u2014all grass, except when breeding them or during lambing season\u2014I have little management or cost inputs. They provide me with steady income, year after year, at little cost to me, my farm, or the environment.\n\nI believe free-range poultry and pasture hogs will once again become the norm in agricultural production due to their sustainable properties\u2014low inputs, easy management, and environmental soundness. The other reason that they will do well is that today's consumer wants to know that his or her food is safe. Consumers are interested in who grows their food and how it is grown. Last, they are interested in quality. Consider the sustainability of any enterprises you choose, and how you can increase your sustainability over time.\n\n###### Food For Thought\n\nNow that you have this chapter under your belt, you should be able to pick the combination of livestock and crop enterprises that matches your farm, labor, and capital resources. If you choose carefully, you are on the road to success.\n\n###### Part Iv\n\n## Management\n\n###### Chapter Nine\n\n## Machinery\n\nPrinciple: _Machinery and tools should save time or reduce the need for additional labor. Otherwise, they are a waste of money_.\n\nMachinery, by my definition, is any piece of equipment or tool\u2014including hand tools\u2014that makes the job easier, faster, and better. A hoe, a wheelhoe, a two-wheel tiller, and a tractor are all machinery, each with advantages and disadvantages.\n\nThe purpose of any tool or piece of machinery is to save time by reducing labor. This increases the work that can be accomplished and decreases the cost of doing the job. To decide if a tool or piece of machinery is worthwhile, its cost must be balanced against the time saved and what you can do with that time. If a piece of machinery will not save you time or reduce the need for additional labor, don't buy it. If a tool saves time, but you don't use the time on other projects, that tool is unnecessary.\n\nAs I am a four-shot-a-day diabetic with three congestive heart failures, I use machinery because it saves the parts of my body that still work. I am also interested in producing a surplus of my crops and livestock so I have plenty to sell to my customers in a timely manner.\n\n_Small farmers should not dismiss buying a larger used tractor if it will fit their operation and if the price fits their budget_.\n\n# Good Health\n\nI am talking about myself, so let me pursue a tangent for a moment and make you aware of the good physical wellness you need to farm, whether or not you use machinery. Many people come into farming after years of sitting at a desk and getting little exercise. This is especially true of people just moving to the country as retirees, but may be valid even for young people who have simply tired of city life. (A recent study of the _Small Farm Today_ readership found that 47 percent had been farming less than 5 years.) The equipment recommendations I make are for younger, reasonably physically fit adults. If you are not younger or physically fit, don't give up, though. Go on a diet, work into things slowly, and use equipment to enhance the physical strength you do have.\n\nRemember, \"Where there's a will, there's a way.\" I farmed full time for 30 years, and even with the challenges I mentioned above, I still farm 80 acres, and still do 80 percent of my own farming chores. (I hire out the rest.) I have sheep, hogs, and poultry for livestock, raise open-pollinated corn, and grow a market garden. The point is, if you want to do it, go for it\u2014just use your common sense and, if need be, hire neighbors or young people for the really heavy work. I know an elderly retired farmer who farms stock tanks full of soil about waist high. He makes them accessible to his golf cart so he can travel about and tend the plants himself. This is healthy for him and healthy for his community. You can do it if you want to.\n\n### Acquiring Machinery\n\nThe first decision to make about machinery is the level of use for which you need it. People who are growing their own food on a small scale, for instance, can get by with very little equipment\u2014there are many ways to prepare the ground with hand tools for planting, rather than by using a tractor or a big rototiller. Mulching the ground the year before it is needed (softening the soil, killing the vegetation) and then opening furrows in the mulch was a technique practiced by the writer Ruth Stout in her no-work gardens, and she successfully produced large harvests of vegetables. Those who advocate raised-bed gardening may use a spade or a special tine tool to loosen the beds. Before you can make choices, you must know what your options are. Let's discuss the varieties of machinery available, starting with the original supplier of horsepower.\n\n#### Draft Animals\n\nDraft animals were the original \"machinery\" used on farms to save time and effort. Although most farmers have replaced them with tractors and combines, they are still a viable tool today. Sustainable agriculture is very broad in its definition, and there is more than one model for farming success. The Amish, for example, are a successful agrarian society who use draft animals. Draft horses and other draft animals can be very useful on a small farm and may be more economical for some jobs than a tractor. The primary argument for tractors is that they do the work faster and at less cost. A tractor can plow 12 acres in a day, while four horses would take 3 days. But if you only have 2 acres of fields, two horses can plow the entire plot in a single day, and a tractor will not save that much time.\n\nCosts for draft horses have remained relatively fixed as time goes by\u2014feed costs have increased some, but not exponentially; horse-drawn equipment is rare, but relatively cheap (although it is increasing in cost as more farmers shift to draft animals); and horses are much cheaper than new tractors and similar machinery. Horses also reproduce themselves, ensuring a continuation of available equipment.\n\nDraft animals may provide an economic power source for your small farm if you are willing to take the time to learn how to use them from an experienced teamster.\n\nHowever, if you decide on draft animals as your power source, there is a definite learning curve. Here are some tips to get you started:\n\n  * Read all you can about the subject.\n\n  * Enlist the help of an experienced horseman or oxen handler (drover) to help you pick out your animals and equipment.\n\n  * Get older, experienced animals until you develop your skills.\n\nIt is not easy to drive a 2,000-pound horse with 12-inch dinner-plate hoofs in a straight line 40 inches wide between two rows of corn\u2014and plowing is even harder on you and the horse than is cultivating.\n\nDraft animals must be approached just as other methods of power are\u2014by evaluating cost of the power source (draft animals) and cost of the equipment. When choosing between a tractor and a draft animal, considerations include size of operation, ease of use, whether the animals will be bred and sold, and relative costs. For 200 acres of crops, I recommend a tractor. For 2 to 80 acres, draft animals are an option. There is one other thing to consider: Do you enjoy being around draft animals, working with them, and raising them? If not, a tractor is better for you. If you do, some of the other considerations become less important, and you can find a way to make it work.\n\n#### Tractors\n\nA tractor is considered the usual indispensable tool for farms today. Tractors are used for tilling, planting, harvesting, hauling manure and supplies, plowing snow, and leveling roads. They replaced draft animals on most farms, and are the obvious choice for those without the time or temperament to deal with living creatures as tools. Tractors range from little more than lawn mowers, to garden tractors (up to about 20 horsepower), to behemoths of more than 100 horsepower. There are also specialty tractors, with narrow bodies or high clearance for use in orchard or vegetable beds.\n\n# The Amish\n\nThe Amish society is more than 300 years old, according to Donald B. Kraybill in his book _The Riddle of the Amish Culture_. He notes that from a small group of 5,000 in 1900, they have blossomed to more than 100,000 in North America today. The question Kraybill asks is, How do they manage to flourish in the midst of industrialization?\n\nThe typical Amish farm family has one or two driving horses and six to eight horses or mules for fieldwork. Families who no longer live on the farm have one or two horses for transportation. Amish are not required to own a horse but it is the assumed mode of transportation, as cars are forbidden.\n\nThe horse, according to Kraybill, represents key values involving tradition, time, nature, and sacrifice. Horses make for a slower pace that conforms to nature: Horses don't have headlights, for example, so they can't be used in the fields at night. Daily travel is limited to about 25 miles, and the size of Amish farms is typically about 50 acres. The horse also links the Amish with the natural rhythm of the seasons. And the horse keeps the Amish out of cities.\n\nThe Amish succeed by having low input costs, minimal needs, and a strong support network of family, friends, and neighbors. They produce quality goods in traditional ways, and avoid many of the costs of an industrial society. They maintain high soil fertility by mostly natural means, and often utilize the crops themselves or use direct markets. They believe in planning crops to have work throughout the year. The Amish have proved that horses can be cost-competitive with modern machinery, but it takes knowledge, understanding, and a love of animals to succeed.\n\nThe key to choosing a tractor for your property is to decide what you can afford for the time saved, what you will be doing with it, and what implements will be attached. On most small farms, there will be only one tractor, so it must be suited for everything from plowing to hauling a wagonload of firewood.\n\n#### What Will You Do with It?\n\nConsider the size of your fields and the turning radius of the tractor. If you have 5 acres, a 60-horsepower tractor is unnecessary. If you have 40 acres of vegetables, you will probably want a fairly substantial tractor. Most small farms could use a 30- to 50-horsepower tractor with a three-point hitch. Don't overlook a high-horsepower used tractor, though, if the price is right. Jobs come up for which you would be glad to have the extra power. For safety's sake, get a wide-front-end tractor. They are much more stable and safer for all types of farm jobs.\n\n##### Implements\n\nTractor implements are rated by category. Category I (20 \u2013 45 horsepower) are usually best for small operations, although Category Ii (50\u2013100 horsepower) are sometimes appropriate. Those in Category Ii are usually too expensive and too unwieldy. Tillage tools include moldboard plows, chisel plows, disks, harrows, and mulchers. Planting implements include drills, corn planters, and broadcast seeders. Cultivation tools include cultivators and rotary hoes. Flame weeders, blades, loaders, rakes, balers, mowers\u2014such as brush hogs\u2014sleds, bale carriers, manure spreaders, and trailers are other specialized implements. There are also pull-type combines and corn pickers. If you buy a tractor, make sure you get the best use from it by adding appropriate implements.\n\n#### Hand Tools\n\nBefore you consider a tractor (or a draft animal), look at less expensive alternatives. The first option for farming is hand tools, including hoes, shovels and spades, trowels, transplanters, seeders, drills, rakes, forks, scythes and sickles, shears, calf hooks, field knives, wheel hoes, wheel cultivators and blades, flame weeders, spreaders, wheelbarrows, and wagons. Steve Salt, a vegetable farmer and frequent contributor to _Small Farm Today_ magazine, praises hand tools for their low cost, durability, minimal environmental\n\n###### Implements and Performance\n\nimpact, maneuverability, and precision. They give farmers a closer perspective on their crops and soil, too, he says.\n\nExamine hand tools in terms of the amount of soil they will work and how much time it will take. A wheel hoe may break the soil faster than will a hand-held hoe, but it may not work in some tough, heavy clay soils. Choose tools carefully. Some hoes are sturdier, and a wide variety of them have specialized functions. A sharp-bladed heavy field hoe allows you to clean a field of weeds much quicker and for less effort than does a dull sheet-metal garden hoe. If\n\n# True Story\n\nSteve Salt lives near Yarrow in northern Missouri. Steve, his wife, and children own 147 acres of land and produce vegetables, herbs, and small fruits on about 7 acres and also produce four or five acres of sweet sorghum. Steve trades labor on a neighbor's crop for the use of the neighbor's processing equipment and boiling pans for the sorghum. The neighbor and he both sell the sorghum when it's ready.\n\nSteve is a diversified market gardener, using mostly hand tools (and some small powered tools) to manage produce for a myriad of niche markets. His main crops are sweet sorghum and ethnic and heirloom vegetables. Steve's sorghum is picked by hand or cut with cane knives by family and crews of neighbors, who receive sorghum syrup in trade. He grows about 300 varieties of vegetables and about 150 species of plants. He sells most of his crops at farmers' markets. Sixty different kinds of tomatoes and sixty kinds of peppers are popular with his Hispanic customers, as well as Mexican herbs and plain old head cabbage. Chinese greens are favorites with his Asian customers. His small fruits are enjoyed by everyone.\n\nAn interesting sideline for Steve is that he boards about eighty head of rodeo stock (horses) on the rest of his farm. He feeds and cares for them and calls the owners for any vet care or illness problems. He is also finishing a two-volume book on ethnic vegetables.\n\nyou only have a small garden though, you may not wish to invest $40 in a quality field hoe. Most of the time, you will want quality and durability in your tools, so examine them carefully before you buy. Look in the resource listings in the appendix for suggestions.\n\n#### Small Motors\n\nThe next step up from hand tools is a two-wheeled tractor and a walk-behind tiller. These hand-guided machines usually range from 5 to 20 horsepower. They start at about $250 for tilling; most are $600 to $800. At higher prices, they allow power take-off-driven additions such as snowblowers and mowers. These are usually an excellent investment for a small-scale market gardener.\n\n#### Processing Equipment\n\nProcessing equipment requires careful consideration before you invest in it. Most added-value processing will involve large stock, poultry and rabbits, or crops. For large stock\u2014beef, hogs, sheep, goats, bison, elk\u2014processing must be done in a county-inspected or Usda- inspected plant if you plan to sell the meat to the general public. Processing at home is not possible. Crop processing (jams, jellies, bread, for example) involves establishing a Usda-approved kitchen, as discussed in chapter 7.\n\nProcessing of poultry\u2014chickens, turkeys, ducks, emus\u2014as well as game birds and rabbits, can be done at home. The amounts allowed and sale conditions vary from state to state, but usually several thousand animals may be slaughtered\u2014call your county and state health departments for more information. For poultry butchering, you will usually need:\n\n  * a killing cone to hold the chicken down while you cut its throat to prevent wing flapping from bruising meat;\n\n  * a rack on which to hang killed birds to drain blood;\n\n  * a stainless-steel table on which to eviscerate birds;\n\n  * a scalder to loosen feathers;\n\n  * a plucker to remove feathers;\n\n  * tanks full of ice water to cool off birds; and\n\n  * a cooler in which to store birds.\n\nMost home operations stop here; customers will pick up the birds in coolers of ice to take them home for freezing or eating fresh. If you start selling to stores, you will probably need a wrapping machine and appropriate containers. Rabbits and fish will require some, but not all, of the equipment listed for poultry. Some suppliers for this home equipment are listed in the back of this book.\n\nProcessing may or may not increase your profits. If you add value to your product, but you increase your labor costs 10 times, or your container costs increase fivefold, you may not actually increase profits. The cost of buying processing equipment or establishing a kitchen will also reduce your profits for a time. The question is whether you will process enough units of volume to increase your profits. Establish break-even costs before investing in the equipment. Will you be able to sell the processed product at a price that will cover your processing costs, or does that put you out of the competitive market? My best suggestion is to start by helping someone do his processing. Decide the best way to do it, then buy your equipment.\n\n#### Fencing\n\nYou may not think of fencing as machinery, but netting and fencing in the right places will certainly ease your burden of labor. Fencing requires careful thought about placement, along with quality materials, correctly used. Poorly constructed fences call for constant repair and livestock roundups. Good fences will pay for themselves in time saved in protecting, managing, and moving livestock, and in allowing the best use of every acre of the farm. Good fences demonstrate pride of ownership, aid land steward-ship, and increase the overall value of your farm.\n\nWhen doing initial planning, make sure your perimeter fence (on the property lines) is able to hold any type of livestock, from hogs to cattle. You need to keep your livestock in and your adjoining neighbors' livestock out. As the old saying goes, \"Good fences make good neighbors.\"\n\nInside your property, make use of lanes to ease movement of livestock from one end of the farm to the other and from field to field. Lanes should be big enough to graze and drive equipment through, rather than narrow, eroded dirt trails. Build a fence wherever possible in such a way that you can mow with a tractor on both sides of it. This allows easy access for maintenance and care of the fence. Make use of temporary inside fencing for seasonal pastures and crop rotations. Electrical fencing, especially the new high-tensile wire and fast-pulse chargers, is economical and more valuable as temporary fencing around grain crops or for intensive grazing.\n\nGet the best fencing and materials you can for durability and economy. At most farms, this will usually mean woven wire or three or four strands of barbed wire on the perimeter, and barbed wire fences on the interior. Elk farms require heavy, 8-foot-tall fencing. Horse farms often use wood or vinyl fencing. For small stock, fences built with hog panels are good. Chicken wire may be necessary to contain poultry.\n\n#### Other Machinery\n\nAdditional machinery for specialized uses includes hay balers, hay wrappers, and combines. There are specialized tools as well, ranging from sawmills, to tree shakers, to cotton gins. Most of these are too expensive and too specialized to justify purchase for a small operation. Rent or borrow from a neighbor if you can.\n\nKeep in mind that some types of structures cross the line into tools and machinery. Loading chutes greatly ease the task of transferring livestock. Squeezes, headgates, catch crates, and corrals serve the same purpose. To decrease your labor and costs, examine each type of livestock and crop, and consider what will work best to increase their quality and decrease your time invested.\n\n# Building a Barbed Wire Fence\n\n  1. Mow or clean the area where the fence is to be built.\n\n  2. Set a corner post at each end of the fence.\n\n  3. Stretch one barbed wire between these two corner posts to establish and line up your fence line. Leave in place; it helps keep out dogs and coyotes. Staple 2 inches above the ground line.\n\n  4. Make a common post-brace assembly at each end of the fence.\n\n  5. Install a line brace each 660 feet on flat terrain, or more if you have hilly terrain to work with.\n\n  6. Mark and set or drive line posts 12 to 16 feet apart.\n\n  7. Stretch the top barbed wire and staple the wire snugly, but do not drive staples so tight that staples cut into the wire.\n\n  8. Stretch three to six lower wires to proper tension and staple.\n\n## Materials for a Double-Span End Brace\n\n## Installing Braces\n\n  1. Attach braces using the dowel-pin method as shown below.\n\n    1. Notch the post to fit the brace.\n\n    2. Drill \u215c-inch holes 2 inches deep in post and brace.\n\n    3. Drive a \u215c-inch steel pin in the ends of each brace.\n\n  2. Wrap the ends of braces with several strands of No. 11 galvanized wire and twist tight. This adds strength and keeps the brace from splitting.\n\n  3. Insert brace pins in holes and assemble the braces.\n\n  4. Fasten tension wires.\n\n    1. Use four strands (2 loops) of No. 9 smooth galvanized wire.\n\n    2. Tie wire 4 inches from top and bottom of posts.\n\n    3. Staple wire to each post with three 1g-inch staples.\n\n    4. Twist wire tight with a short stick or board; leave stick in place for later tightening.\n\n## Wire Stretching and Stapling\n\n  1. Stretch woven wire with a fence stretcher or tractor until tension curve is half its original depth.\n\n  2. Use fence clamps (two 2 \u00d7 6s bolted together) on all woven wire fences. Attach parallel to and between two wire stays.\n\n  3. Attach stretcher chains so there is an equal number of line wires below and above line of pull. One stretcher is sufficient on 26- to 32-inch fences; higher fences require two stretchers.\n\n  4. Staple fence to posts on ridges and depressions before the stretcher is released.\n\n  5. Cut wire, remove the two vertical or stay wires, then tie and staple to end or corner posts.\n\n  6. Use 1-in. staples for hardwood, 1g-in. staples for softwood.\n\n  7. Drive staples in line posts diagonally with the grain\u2014loose enough to allow wire to slip through staples. Staple wire securely to corner, end, and brace posts.\n\nFrom Carl Scheneman and Albert Hagan, _Good Fences for Your Farm_ , University of Missouri Extension Circular 667, 1956.\n\n#### Computers\n\nThe last piece of machinery I would like to discuss is the newest tool for farms\u2014the computer. I recommend investing in this tool for a variety of reasons. A computer is excellent for storing budgets and income data, figuring taxes, and managing records. There are also farm programs for specific purposes, from keeping track of field yields to managing ostrich operations. E-mail can be a great way to communicate with other farmers and gather information from bulletin boards, such as Sheep-L and Graze-L. Last, with Internet access, you can download all sorts of useful information, from grazing tips, to okra recipes, to site plans.\n\nComputers do not have to be expensive. For most farm uses, you won't need a top-of-the-line model with snazzy graphics. A simple computer that can handle a word processor and an income program will do most of the functions you need. Look at used computers, or low-end models. As always, consider cost and benefits carefully before making a purchase.\n\n### Determining Equipment Size\n\nThe old adage \"Make it yourself, wear it out, use it up, or make it do\" is a principle that applies to any small farmer, and to the tools and equipment on his farm. The small farmer's most limiting factors for success are time and capital. Machinery frequently requires large amounts of both. Fixed costs on equipment that is used infrequently is an expense that small farmers do not need and cannot afford if their operations are to be economically viable and thus sustainable. So the question you must ask is not \"What do I want?\" but, instead, \"What do I actually need?\"\n\nFarmer Steve Salt offers these guidelines for tool and equipment usage. Human-powered tools\u2014hand hoes, wheel hoes, push plows\u2014are inexpensive and reliable for use on up to about a half acre of crops. Five- to 20-horsepower two-wheel walk-behind tractors can handle from \u00bd to 2 acres. These could handle even 10 acres, if you custom-hire the heavy jobs like plowing, or you could rent a tractor and plow for the time you need it. If you go beyond the \u00bd- to 10-acre market-garden range and get into 10 to 80 acres or more, a combination of a two-wheel walk-behind tractor and a four-wheel ride-on tractor can be justified, especially on livestock and crop farms. You simply cannot move a 1,500-pound round bale with a two-wheel tractor, nor can you plow, cultivate, and harvest 40 acres of field crops in a timely and profitable manner.\n\nOn my own 80-acre farm, I use a combination of one 8-horsepower Bcs tiller and one 23.5-horsepower Ford tractor, and have a neighbor haul my share of big round bales, which he bales. Another neighbor picks my corn with his harvester, although I hand-pick some of it early in the season for my hogs.\n\n### Variables to Consider\n\nBefore pondering costs and ownership, carefully consider these issues:\n\n  * **Rental or leasing**. Is this possible when we need the machinery, or is there a high demand at that time? How far do you have to move the machinery to get to your farm? Is delivery included in rental?\n\n  * **Neighbors**. Can you exchange machines with a neighbor, or barter for the use of what you need? This could be as simple as, \"I will bale your hay if you combine my corn.\" Transportation costs must be figured if you are trading with someone other than a neighbor whose land adjoins yours.\n\n  * **Hiring**. Is there someone in the area who has the machinery needed whom you can hire? My neighbor bales my hay in trade for a percentage of the hay crop. If you hire people you are unfamiliar with, get references\u2014are they reliable and available for your crop?\n\n  * **Business potential**. If you do not have the acres necessary to justify the cost of the machine, but there is a need for it in your area, can you buy it to do custom work for your neighbors (and yourself)? Some farmers purchase machinery together, but to avoid problems, I recommend avoiding this unless it is done as a cooperative, with all arrangements among parties worked out on paper in advance.\n\n  * **Used equipment**. Compare new prices and used prices for the same piece of machinery. Is new necessary? If the machinery you need is a low-usage item, used equipment might be your best buy. Many farmers today are using tractors 30 and 40 years old, and some tools that are even older\u2014perhaps adapted from horse-drawn implements. You might question how tractors can even last 20 years. They can because they are tough and well made. If they are well cared for and maintained properly, your machinery should last your lifetime and more. If you buy used, examine the equipment carefully, and, if possible, operate it to get a feel for it. For a tractor, consider its horsepower and the types of equipment you will be hooking up to it.\n\n  * **Scale of use**. This is really a part of your thinking process about what crops to grow. Look at machine costs as a percentage of the total costs and net profit, according to the crop grown. If you only have one cherry tree and 5 acres of tomatoes, consider machinery for your tomatoes before you worry about the cherry tree.\n\n  * **Time**. Don't forget the importance of time invested. How does it affect the price you get for your crop?\n\n### Economics of Machinery\n\nPrinciple: _Buy the machinery you need at the best price possible, and make it last by faithful maintenance_.\n\nThere is a simple key to buying machinery: Buy only what you absolutely need at the best price possible, and make it last by cleaning, adjusting, and using it properly\u2014and storing it away from the effects of weather. If you do not know how, acquire an operator's manual or consult other farmers who are using the same type of equipment you are interested in.\n\nMachinery cost is a fixed cost of production, and according to _Used Farm Equipment_ , Northeast Regional Agricultural Engineering Services at Cornell University, its cost makes it important to \"minimize machinery costs for a level of productivity and efficiency that will maximize profits.\"\n\n#### Machinery Costs\n\nMachinery costs are of two types: ownership and operating costs. You may own a combine, but perhaps you use it three months of the year, at harvest, because it is no good for any other jobs on the farm. A tractor, in comparison, may be used every day, especially if you have a diversified livestock and crop farm.\n\n##### Ownership Costs\n\nOwnership costs (also called fixed costs), which are about the same amount every year, include the following:\n\n  * **Depreciation**. The machinery decreases in value no matter how much or little you use it. (This amount also needs to be set aside each year to buy new machinery when the one you have wears out.)\n\n  * **Taxes**. Fees imposed by the government include property tax (personal and real estate), sales tax, income tax, Social Security, and possibly others.\n\n  * **Insurance on facilities and equipment**. Insurance is a form of risk management. The risks of fire, theft, flood, and so on could put you out of business if you're not insured.\n\n  * **Interest** , if you take out a loan to buy the machinery. Interest costs must be evaluated carefully before purchase\u2014does the benefit of the equipment justify the added expense?\n\n  * **Storage facilities**. If do not already have a shed to store your machinery, you may have to build one. Caring for your tools by keeping them out of the weather is one of the best ways to save on future costs.\n\n_Note_ : These ownership costs will accrue even if your equipment never leaves the machine shed!\n\n# Tips For Storing Equipment\n\nHere are some tips from Harold Tucker, the lubricants technical director at Phillips 66 Company, a division of Phillips Petroleum (Bartlesville, Ok), on the proper methods of winterizing machinery.\n\nPostseason Shut-Down\n\n  * Clean the engine compartment and the outside of the machinery using high-pressure washing equipment, if necessary. When mixed with oil, corrosive material in dirt, such as acids and ash from fertilizer and pesticides, can damage an engine.\n\n  * Remove residual crops from all farm equipment. Stray pieces of corn or wheat, for example, contain moisture that can corrode metal components.\n\n  * Overhaul, change oil, and lubricate all machinery as soon as possible after the harvest. Drain the oil to remove contaminants, then refill with new oil for long-term storage. Change all filters.\n\n  * Observe used oil carefully. Note unusual colors or thickness (viscosity). This could mean a cross-contamination of fluids.\n\n  * Grease press wheels and clutch parts on planters and fittings on tractors and combines.\n\n  * Remove rust, and repaint equipment. Keep machinery rust-free.\n\n  * Remove batteries, clean contact points with baking soda, and reinstall the batteries. Then apply a coating of grease on the terminals to prevent corrosion.\n\nThe Northeast Regional Agricultural Engineering Service predicts ownership costs of a new tractor at about 13 percent of its cost and 16 percent of the cost for used tractors, based on 8 percent interest. Ownership cost increases one-half of 1 percent for every 1 percent increase in the interest rate. Other machinery such as a plow or a planter has an ownership cost of about 14 percent for new equipment and 17 percent for used equipment.\n\n##### Operating Costs\n\nThe other machinery cost is operating costs. Operating costs\u2014also known as variable costs\u2014are the cost of using a piece of machinery. Variable costs include fuel, oil, lubrication, filters, repairs, and maintenance. For hand tools, this might be as simple as repairing a handle or sharpening the tool. Sharpening qualifies as maintenance because it makes the tool work efficiently and it lengthens the tool's useful life. Fuel, either gas or diesel, is a cost every time you turn the key and use the tractor. The same can be said for oil, filters, grease, belts, and other machine parts.\n\nWhen considering the economics of machinery, consider the costs in various ways. First, the actual purchase price: How will you pay for it? (Will you need a loan?) Then consider the fixed and variable costs. Use them to figure your costs per hour and costs peracre. The fewer the acres, the higher will be the cost per acre. Smaller acreage usually means adjusting to less expensive tools to maximize profitability.\n\nUsing the figures from the Equipment Costs chart for a $13,000 tractor used for 600 hours per year, the cost per acre on an 80-acre farm is $60.68. This is before taxes ($120) and labor ($10 per hour \u00d7 600 hours = $6,000) are added in; these increase the final figure to $137.18 per acre! This is obviously very high. However, if you had a down payment or a trade-in, this figure might realistically drop to $50 to $60 per acre. Also, most tractors will run way beyond the 10-year life expectancy most people use for their figuring. My first tractor was a 1948 8N Ford with a 7-foot sicklebar mower, purchased for $800 in 1965. It was 17 years old when I bought it, and I used it until 1975, at which time I sold it for $1,000, without the mower. It is still being used by its third owner.\n\nAnother good way to compare costs is to get a custom rate sheet from your local extension office. For instance, in Missouri we have a custom rate for moldboard plowing ranging from $7 to $15 per acre. Disking runs from $4 to $10 per acre. Cultivating is $3.50 to $7 per acre. Rotary hoe rates are $3 to $5.50 per acre. Corn planting is $5 to $12 per acre. Picking corn in the ear is $17 to $25 per acre. Once you have the custom rate sheet and can compare costs, it is relatively easy to make an economic decision based just on cost.\n\nOf course, more than just cost is involved if you rent or get your farming done on a custom basis. How timely are the custom operations in your area? Can you get references? Some co-ops have small windows of opportunity for harvests such as sweet corn. Weather delays and breakdowns can make ownership seem cheap in comparison to losing your crop. Finally, like everything else in this book, it is up to you to weigh the pros and cons and make the best decision you can live with, labor-wise and money-wise.\n\n  * Run each piece of equipment at low speed for up to 30 minutes periodically throughout the winter to lubricate gears and internal components, unless the equipment has been thoroughly winterized.\n\n  * Use quality lubricants from a reputable supplier.\n\nPreseason Start-Up\n\n  * Check hoses and connections for leaks before operating equipment. The most common areas for leaks include oil pan seals, transmission connections, and the rear and front main seals.\n\n  * Visually inspect engine oil paths by looking for wet spots on the hoses. Do not run your hands along the hoses during inspection. Engine fluids are under high pressure and a leak may force oil under the skin. With the eyes alone, look at the bottoms of fittings and hoses where fluid will accumulate and drip. In addition, machinery that has been sitting in the same place for a long time can leave easy-to-identify oil spots on the ground.\n\n  * Drain winter oil and change filters. Change the transmission fluid and antifreeze, if needed.\n\n  * Invest in a scheduled equipment preventive maintenance program. Set regular cycles to change oil and filters. During the growing season, preventive maintenance is one element of farming that operators can control.\n\n  * Use oil and hydraulic fluid analyses to help prolong equipment life through early detection of abnormal wear.\n\n# Equipment Costs\n\nHere are some formulas to help you figure your costs for machinery. They do not include property taxes or labor of the operator.\n\n## Fixed Costs\n\n  1. Annual depreciation = new cost \u00f7 years of useful life\n\n  2. Average annual insurance = new cost x g x insurance rate\n\nInsurance rate: nonmotorized at .012 ($12\/$100 valuation)\n\nmotorized at 0.010 ($10\/$1000 valuation)\n\n  3. Annual housing = square feet storage \u00d7 cost per square foot\n\nCost per square foot = new cost per square foot \u00f7 years of life + repairs as percent of new cost per square foot\n\nRepairs as percent of new cost per square foot = 1.5 percent\n\n  4. Average annual interest rate = new cost \u00d7 \u00bd \u00d7 interest rate\n\n  5. Hourly fixed costs = (A + B + C + D) \u00f7 hours per year\n\n## Variable Costs\n\n  6. Hourly repair costs = new cost \u00d7 repairs as percent of new cost \u00f7 hours of service life\n\n  7. Hourly fuel costs = gallons fuel per hour \u00d7 cost per gallon fuel\n\n  8. Hourly lube costs = hourly fuel costs \u00d7 10 percent\n\n  9. Hourly variable costs = F + G + H\n\n  10. Total hourly costs = E + I\n\n## Costs per Acre\n\n  11. Yearly variable costs = I \u00d7 hours per year\n\n  12. Yearly fixed costs = A + B + C + D\n\n  13. Costs per acre = (K + L) \u00f7 acres used\n\nIn the following example, a tractor costs $13,000 new, with a life expectancy of 6,000 hours and 600 hours annual use on an 80-acre farm. It requires 100 square feet of storage space. The storage space costs $3 per square foot new and is depreciated over 30 years. The annual interest rate is 12 percent. Repairs are 90 percent of new cost. Fuel consumption is 2.31 gallons\/hour and costs $1 gallon.\n\n  1. Annual depreciation = 13,000 \u00f7 10 = $1,300.00\n\n  2. Average annual insurance = (13,000 \u00f7 2) \u00d7 0.01 = $65.00\n\n  3. Annual housing = 100 \u00d7 0.145 = $14.50 Cost per square foot = (3 \u00f7 30) + (3 \u00d7 0.015)\n\n  4. Average annual interest = (13,000 \u00f7 2) \u00d7 0.12 = $780.00\n\n  5. Hourly fixed costs = (1,300.00 + 65.00 + 14.50 + 780.00) \u00f7 600 = $3.60\n\n  6. Repairs = 13,000 \u00d7 (0.90 \u00f7 6000) = $1.95\n\n  7. Fuel = 2.31 \u00d7 1.00 = $2.31\n\n  8. Lube = 2.31 \u00d7 0.10 = $0.23\n\n  9. Hourly variable costs = 1.95 + 2.31 + 0.23 = $4.49\n\n  10. Total cost hour = 3.60 + 4.49 = $8.09\n\n  11. Yearly variable costs = 4.49 \u00d7 600 = $2,694.00\n\n  12. Yearly fixed costs = A + B + C + D = $2,160.00\n\n  13. Costs per acre = (2,694 + 2,160) \u00f7 80 = $60.68\n\nAdapted from Charles DeCourley and Kevin Moore, Ec 959: _Selected Fruit and Vegetable Planning Budgets_ (University of Missouri\u2014Columbia, Department of Agricultural Economics, 1987).\n\n# What Kind Of Equipment Do i Need?\n\nDepending on the size of your operation, you may need only some of this equipment. There may also be specialized pieces necessary for particular crops or livestock. These lists are merely to give you an idea of what you'll need.\n\nTruck Farm (Small-Scale Vegetable Farm)\n\nCrop and Livestock Farm Hay Farm\n\nHay Farm\n\n### Buying Equipment\n\nTo buy equipment, you must first know where to find it. Tractor dealerships usually have both new and used equipment and may or may not have some type of warranty period. The second place to look is in your local newspapers or farm magazines for equipment for sale by owner. Auctions are another good place to look, although the enthusiasm of competitive bidders may drive the price beyond its real worth. Know ballpark values before you go to an auction.\n\nBuying used machinery is one way to keep your costs down, though you must be sure you are making a wise investment. First, consider the equipment's age and usage. A tractor's useful life is about 12,000 hours. If you use a tractor 600 hours per year, the useful life is 20 years. Wendell Bower, in his book _Modern Concepts of Farm Machinery Management_ , says that the first year of life for a new tractor is the most expensive; each year, operational costs get less until wear and tear causes repair costs to increase. At this time, Bower suggests trading tractors, because your average use-per-hour costs will increase. This normally occurs at about 10 years of age, or 6,000 to 7,000 hours of use.\n\nNext, figure the costs per hour and the costs per acre to see if this is a good investment. You should be able to figure these out quickly with a pocket calculator. It's not a bargain if it costs too much to use!\n\nHere are some tips on purchasing:\n\n  * Do not get hung up on a particular brand. Do consider, however, local availability of parts, or you may have a long wait while hard-to-find parts are shipped to your farm.\n\n  * Talk to fellow farmers to find reputable dealers and auctioneers with a good reputation. When buying, investigate why the equipment is being sold, and see if there are any maintenance records.\n\n  * Check out the equipment carefully for rust and signs of major accidents. Look for added welds or fresh paint that might disguise these problems. On tractors, examine the tires for wear. Inspect engines carefully, and examine filter condition. You may be able to have analysis of oils and lubricants done at a local lab\u2014check with your extension office.\n\n  * Consider comfort and safety. How comfortable are the seat and shocks? How easy are the controls to reach and operate? On a tractor, does it have a rollover bar?\n\n  * If possible, try out the implement or drive a tractor around before buying it. Make sure everything is working.\n\n# Author's Note\n\nThe following tables are outdated, inaccurate, and price varies from one region to another. So why include them? It is extremely difficult to find comparison data for different equipment. These tables will help you begin to compare the costs, usage rates, and lifespan of equipment you might consider for your farm. I have always found these tables useful; I believe you will, too.\n\n###### Estimated Machinery Costs\n\n###### Estimated Variable Costs of Equipment (per acre)\n\n###### Food For Thought\n\nAs you seek out and decide on tools and machinery, remember that the best tool is your own mind. Without it, all of your machinery is just lumps of metal and wood.\n\n###### Chapter Ten\n\n## Farm Management\n\nAll of the previous chapters have been about management in one way or another. Now I want you to think of management not as how it applies to any individual part of your operation but as a tool in its own right. Management is what will ultimately provide you with profit\u2014and success.\n\nSuccessful farm management is two things:\n\n  1. Farming is taking all you can from the soil so you can sell the surplus to make a profit.\n\n  2. Farming is also putting back into the soil all you can so you can maintain and increase its fertility.\n\nThis give and take must be in balance for your farm to be sustainable in the long run\u2014and management is what determines the balance.\n\n_Dairy farms are labor intensive and require astute observation of animal behavior. Grass-based dairies and seasonal dairying reduce labor requirements tremendously while still providing adequate income_.\n\nFor instance, I have a small, 900-square-foot hog lot on a very gentle south slope (great for my early garden). There is a portable hog house at one end of the lot. The first time I used the lot, it was solid, tough fescue pasture, where domesticated vegetables would quickly be cornered and disposed of by weeds. The three 40-pound feeder pigs I bought plowed up the fescue free of charge, ate lots of grubs and weeds, and generally made the land ready for the vegetable crop that followed.\n\nI have continued this program because it works well. Each pig furnishes an average of 8 pounds of manure per day, from its 40-pound beginning weight to its 225-pound butchering weight, according to the _Swine Handbook of Housing and Equipment_. Hog manure on a per-ton basis contains 13.8 pounds of nitrogen, 4.6 pounds of phosphorus, and 9 pounds of potassium. Eight pounds of manure per day times three pigs equals 24 pounds of manure a day times a 120-day feeding period equals 2,880 pounds of manure, or about 39 pounds of nitrogen, 13 pounds of phosphorus, and 25 pounds of potassium for my vegetable crop. To help with this fertility program, I sow annual ryegrass and hairy vetch in the lot after the pigs are sold as whole hog sausage for $2.50 per pound. This green-manure crop is then tilled under about two weeks before I want to plant vegetables. Every foot of growth I plow down contributes about 2,000 pounds of organic matter.\n\nNutrient and usage cycles can be repeated over and over. The hogs could also be run back on cornfields if you wish, or you might substitute poultry or sheep for the livestock portion of this cycle. The only limit is your imagination.\n\nLivestock that is sold off the farm will take with it some of the farm's nutrients, but in my example above, because I purchased the pigs, I also brought in nutrients. If you sell all your hay and do not feed it to livestock on your farm, you are depleting your soil fertility. The best program is to bale your hay and feed it back to sheep and cattle on your hay field and let these animals recycle their nutrients in the form of manure. None of these cycles is ever 100 percent effective, or at least is not practical to be so, but you should strive for 100 percent. If you do not reach this lofty goal, regroup, plan, and go forward again. As they say, \"Practice makes perfect.\"\n\n### Recipe for Success\n\nAll through this book I have talked about the principles of good farming. These principles are the same whether you live in New Mexico, Missouri, or Alaska. Principles of good farming work anywhere in the world.\n\nSpecifics of farming are a different matter\u2014what works for your neighbor may not work for you. Everybody seems to want a recipe for success\u2014Do steps 1, 2, and 3, and Bingo! Success! In farming however, there is no single recipe, because sustainable agriculture is site-specific. In other words, it is what works on your farm with your soil types, your management techniques, your capital, and your monetary requirements. You are the best person to put together the package of principles that works for you. For instance, using farming or tillage practices that conserve soil and improve fertility through the use of cover crops is an important principle. In the north-central sections of the United States, rye and hairy vetch might be your cover crops; in the South, cowpeas might be a better choice. The principle is the same, but the crops change\u2014and, of course, your management of these different crops will be different.\n\nThe most important management principle is, \"Look at the big picture.\" How does everything on your farm fit together to make your operation smooth-running and profitable?\n\n#### Take Stock of Resources\n\nTo get a handle on the big picture on your particular farm, you will need to look at specifics. In chapter 1 you asked yourself, \"How many acres of tillable ground do I have? How many acres of pasture or timber ground? What farming skills do I have? What equipment do I have?\"\n\nIn other words, take stock of all your existing resources. In the early 1940s, the Missouri Extension Service had a program called Balanced Farming that expressed just the idea I'm trying to get across here. Unfortunately, \"progress\" caused the service to drop the program. It was solid farming principle then\u2014and it is still appropriate today. In fact, it is what sustainable farm management is all about. Each resource should be managed to maximize its best use and best profit.\n\n#### Practice Balanced Farming\n\nBalanced farming is a process of planning and putting together a combination of measures to give optimum (not necessarily maximum) net farm income, to provide conservation and improvement of soil fertility, and other resources. The resources to be combined are land, labor, capital, and management.\n\nThe key word is \"balanced.\" One full-time person expends 300 ten-hour days in labor or 3,000 hours per year. If you lay out a farm plan that requires 4,000 hours of labor to accomplish, your labor requirements will be unbalanced if you have only one full-time worker (yourself). When your labor requirements are too high, your production rates (bushels\/acre, pigs\/litter, and so on) are frequently so low that the additional acres or numbers of livestock won't give you any substantial income growth. On the other hand, if your farm plan requires only 2,000 hours of labor, you will probably get a higher rate of production, because things will be done in a timely manner. The total production possible will be reduced, however, because you will not be fully employed. A combination of crops and livestock keeps you fully employed because the livestock fills the seasonal (wintertime) lull in the crop-production cycles.\n\nIf your farming operation is balanced, you will have time for field days, reading research papers, attending workshops, and learning from other resources that can give you the knowledge to improve your operational procedures and your net income at virtually no cost. For example, make sure your labor time is fully employed through the combined use of crops and livestock. Maybe you have discovered through the reading of research papers that side-dressing corn at 12 to 18 inches tall with nitrogen is more efficient, costs less, and is better for plant growth than applying nitrogen at planting time, which risks loosing the nitrogen through leaching before the corn plant is big enough to use it. Knowledge is an on-farm resource supplied by you, the operator.\n\n#### Optimize Resources\n\nBesides land, labor, capital, and management balancing, balanced farming optimizes the use of your land resources. If you have rolling, steep timber ground, it can be competitive with other crops like corn if it furnishes lumber for building or fence posts and uses the natural inherent fertility of the soils present to do so. Good, level, well-drained soils are optimized by growing field crops like corn, soybeans, vegetables, and herbs. Rolling pasture ground might best be kept in pasture most of the time and rotated with other crops so you might grow 1 year of corn, 2 years of hay, and 2 years of pasture, or it might be in permanent pasture. The balance comes from using the natural inherent soil type already there.\n\nJust because a farm is in timber does not mean it is necessary to bulldoze the trees and plant it in corn, or spray it to death and plant in fescue, like much of the timber ground in Missouri was treated in the 1960s. Timber can produce posts, firewood, wildlife, light grazing, and lumber for building and pens. In my area, taking trees off the steep hills exposes them to erosion, and the hills are difficult to work on with machinery. The best management practice may be to leave the ground in timber\u2014its best use\u2014and, if managed correctly, its best profit. Returns per acre can compare favorably with crops and pasture. If the trees are already there, management is all that is needed to have a resource that is capable of producing a nice income with little start-up capital.\n\nAs another management example, I once bought a little creek-bottom farm that had been timbered for logs, but the woodsmen left all the treetops lying on the ground. I could have complained about the expense of clearing it, or tried to work around it, but instead I viewed it as a resource. It made perfect firewood. I cut and sold 100 cords of firewood from those tops at $75 a cord, and only lacked $600 to make the first year's payment of $8,100 on the property.\n\n### Knowledge Is Power\n\nPrinciple: _Knowledge is your most important resource_.\n\nKnowledge is your best on-farm resource. What you have in your head and what you can do with your hands is what will direct your management decisions\u2014and no banker or government directive can take it from you. The more you know, the better your decision making will be.\n\nTo utilize knowledge, however, you must have information. I cannot emphasize this enough\u2014you must read, read, Read! Attend seminars and workshops, and talk to other farmers, especially those trying new ideas. Profit from the mistakes others have made. There is no point in reinventing the wheel.\n\nFrequent used-book stores (and look in the agriculture, nature, and gardening sections). Check your local library. Look through university libraries for research papers and books from 1910 to 1960. Extension bulletins from this period cover diversified family farms, where cover crops, rotations, and multiple livestock species were once common. Get catalogs from used-book sources advertised in farming magazines. Subscribe to those magazines that cover topics you are interested in. Look for book publishers (such as Storey Books) that put out useful materials. (See the listings in the appendix for more sources.)\n\nThe most productive use of any farmer's time is spent in planning, figuring costs, and marketing. You are not lazy when you are planning\u2014you are working smarter, rather than harder. A little modern management applied to time-tested information will help make your farm a success.\n\n### Making Choices\n\nYou must always make choices. Pastures mowed in June and July remove the majority of early and late weeds before they go to seed. Multiple-species grazing\u2014say cattle, sheep, and goats\u2014is not easy, but will accomplish the same purpose as mowing, minus the fuel and labor costs. Sheep will eat 90 percent of the common weeds, which generally are high in trace minerals important for animal health and reproduction.\n\nOnce you have chosen your crops and livestock, marketing decisions must be made (if they were not made first). The ability to sell yourself and your farm is what will provide financial security.\n\n### Management Tools\n\nLet me now take a couple of topics from earlier chapters, and explain how they fit into farm management.\n\n#### Crop Diversity as a Management Tool\n\nTraditional agriculture, as they say, is the only business in the world that buys retail, sells wholesale, and pays the freight both ways. This is not the way to succeed. Today's farmer must be a producer, a marketer, and a salesperson. Part of management is finding a successful balance between production and marketing, and also allotting time for family life, so you do not become bogged down with work, feel discouraged, and quit farming.\n\nSustainable agriculture and direct marketing are both tools of management. Sustainable small-farm management is based on diversification. Many modern farms are monocultures (all corn, all hogs, all soybeans). They are usually that way because producers find out what crop they can grow best on their land with the least management. Switching this type of thinking to an alternative crop does not improve it. When exotic animals came on the farm scene and farmers started raising emu and elk, for instance, they claimed they were diversified. They were not. When all you raise is one animal, you are not diversified, even if the animal is exotic. When you raise only one animal on your farm, you are extremely vulnerable to price fluctuations and labor extremes.\n\nMonocultures cause other problems, too. Continued use of one crop, especially if it is a heavy feeder (using large amounts of N-P-K) like corn, depletes soil fertility. Just changing crops is not a rotation. A true crop rotation alternates heavy- and light-feeding crops to complement their nutrient requirements and increase soil fertility.\n\n#### Crop Rotations as a Management Tool\n\nRotations increase the total yield of crops over the years of the rotation. Rotations improve soil by the use of soil-building legume crops; they also help even out the distribution of both labor and machinery use. For instance, a small-grain crop (wheat, oats) eliminates the need for plowing if it follows a row crop like corn or soybeans. Legume hays or pasture seeding following small grains eliminates plowing and sometimes even disking. Normal rotations of 3 to 5 years usually consist of a row crop, a small-grain crop, and a sod crop. Different parts of the farm may require managing more than one type of rotation because of different soil types or varied terrain. Rotations, according to Nicolas Lampkin in _Organic Farming_ , have to maintain soil fertility, organic matter levels, and structure, while ensuring that sufficient nutrients, especially nitrogen, are available and that nutrient losses are minimized. Rotations are the main means of reducing weed, disease, and pest problems by achieving crop diversity both in space and in time.\n\n# A Sample Rotation\n\nVegetables can also be combined with traditional field crops. For instance, you might plant corn, followed by potatoes, followed by a small grain, followed by a hay crop. I plant corn in early May (in Missouri). In early August, I sow winter rye and hairy vetch, so the corn ground is partially exposed for only about 75 days, from May 15 to August 1, to keep down erosion and reduce nutrient leaching. The rye and vetch cover the ground the same year the corn is planted and can furnish hay, seed, grain crop, or green manure plowed under in the second year. I will also have legume grasses and hay available for livestock grazing, which contributes fertility and income to the rotation plan. The rye sops up any nitrogen left from the corn crop and holds it in the rye plant to be used the next year. The hairy vetch, a legume, furnishes nitrogen, which is useful for, say, a tomato crop that follows the corn crop. On or before May 10, I will mow the rye and vetch and plant caged tomatoes in the resulting mulch. The mulch helps control weeds, and the alleopathic tendencies (that is, the ability to prevent germination or growth of another plant) of winter rye also help with weed suppression. The rye and vetch mulch holds soil moisture and slows erosion.\n\nAlthough I have mainly discussed traditional crops and livestock in rotations in chapter 4, the principle of rotation works equally well with vegetables and livestock. In general, potatoes, sweet corn, broccoli, and strawberries are big feeders of nitrogen, while beans, peas, pumpkins, and lettuce are low users of nitrogen. Of course, big feeders will have to be followed in the rotation by a legume, while low users can be followed by a non-leguminous crop. ( _The Knotts Handbook for Vegetable Growers_ , by Oscar Lorenz and Donald Maynard, is the old standby for complete lists of all sorts on the vegetables I talk about here.)\n\nIn this simple rotation, peas and beans provide nitrogen for the following year's sweet corn, while the ryegrass protects the soil from erosion and provides green manure (organic matter).\n\nDick Raymond, author of _Joy of Gardening_ , has a 2-year rotation that tills under only crops grown on the plot. He uses no leaves, no mulch, no compost, no manure, and no fertilizer. The first year he grows a green-manure crop of peas followed by snap beans followed by annual ryegrass. The second year, he plants a crop of sweet corn (a heavy nitrogen feeder) followed by annual ryegrass.\n\nThis rotation alternates nitrogen-fixing crops (peas and beans) with a nitrogen-demanding crop, sweet corn. The annual ryegrass keeps the soil covered most of the time, thereby reducing erosion and nutrient leaching, especially in winter. The winter cover also preserves the earthworm population. The alternating depth of root systems\u20146 feet deep for sweet corn and 2 feet deep for peas and beans\u2014brings different soil nutrients into the system. The different root biomasses (ryegrass is dense; peas, beans, and sweet corn are lighter) furnish earthworms and other soil organisms with material to live on. The peas are a spring crop, the beans and corn summer crops, and the annual ryegrass is a fall crop. This spreads out the workload, and the different germination times help with weed control. Last, three of the four crops involved in this rotation can be used as edible or cash crops.\n\nLivestock are needed in a balanced rotation of crops and are important in maintaining soil fertility. Cattle, sheep, and goats in particular can convert low-grade roughage like weeds or cornstalks into salable meat products, while contributing to soil fertility\u2014and they do all the work of spreading the manure. Cattle, sheep, and goats also utilize rough ground that is too erosion-prone to be anything but pasture.\n\n### Managing Labor\n\nLivestock spreads your labor into use for the entire year, making you fully employed on your farm. There will be some overlap of crop and livestock chores, but having livestock helps even out the  peaks and valleys of labor use. This is just as important for good management as is practicing value-added methods to spread your marketing throughout the year.\n\nFor example, on my farm I maintain a Katahdin Hair sheep flock that I lamb on the grass pastures in May and June. Not lambing in cold weather saves me labor and vet bills, while using hair sheep allows me to avoid shearing and other labor normally associated with wool. Although there is some overlap with corn planting, the sheep pretty much do their thing at a time of year when weather and grass conditions are good. I usually buy feeder pigs to be sold as sausage in August or September to utilize early corn and surplus corn. This eliminates the labor of marketing the corn. If I let the pigs graze in part of the cornfield, I eliminate the harvesting of that area. The pigs are slaughtered in December or January, to eliminate feeding during the worst of the winter. Sales of their sausage and ear corn for seed give me some winter income. The lambs are sold for breeding stock or meat from August through April, and their cycle starts over. Basically, this schedule allows me to avoid major labor in the cold winter months, while still getting some winter income through value-added sales. Also, by choosing and breeding for animals that can care for themselves, I eliminate unnecessary labor involved in livestock health care.\n\nLabor is a limiting factor in farm management. No matter how much you love it, you can be efficient at only so much. All your own family's labor should be utilized fully before you think about hiring help, because this labor does not require a cash outlay, as an employee would. In raising vegetables, however, peak harvest labor may require hired help in order to have the best-quality product for sale.\n\n#### Using Family Labor\n\nIf you want your children to love the farm and maybe even go into business with you, the farm has to be a positive experience for them. This means thinking carefully before exposing them to some animals or equipment. Integrating your children into the farm chores, though, will help them to learn responsibility, timeliness, and a work ethic. Unlike many \"in-town\" responsibilities, the animals must be fed\/watered\/milked on time, or serious consequences occur.\n\nTeach your children safety in everything they do on the farm whether it is working with livestock, machinery, or around the house and garden. Whenever you give a child a chore, explain not only how to do it but also why he or she is doing it. Emphasize doing a chore correctly, rather than demanding speed.\n\nThe Amish do not have central heat. Because the heat is in just one room, the family congregates to that room except at bedtime. This brings the family together, and children learn many things by listening and seeing the interaction and relationship of their parents and siblings. I don't advocate turning off the heat in your house\u2014but allowing full family discussions on the farm at dinnertime, or afterward, in the living room, is certainly beneficial. Children learn by example. You are the example. If you get angry and beat on your livestock, they probably will also. If you drive your machinery fast and do not regularly grease and oil it, they will probably do the same.\n\nYoung children, age 5 or 6, can do simple chores like collecting chicken eggs. Children this age need adult supervision at all times, and their size in relationship to the animals' size must be taken into account. An old hen will sit on her eggs to protect them, especially if she is broody. A nest box is at about eyeball level with a young child and the hen can threaten eyes by pecking. One way of protecting your child is to give her a small trash can lid, held up like a shield. The child can then push the hen aside and get the eggs and, at most, will get a harmless peck on the hand. Of course, you need to show the child how to do this, explaining why the hen does what she does, so your child will not only understand that the hen is just protecting her property (the eggs), but will also build confidence to do this.\n\nChildren's muscles are developed enough to carry, say, a 5-gallon water bucket that is half full when they are 9 or 10 years old. Do not expect them to work as hard as you do, nor as long. Adult supervision is necessary until they have the confidence and judgment to do the chore on their own. Be sure to praise your children every chance you get. It builds self-esteem and confidence. It makes them want to do more.\n\nExtra care must be taken with larger animals. Small children look like dogs or predators to animals like sheep and, again, the mothers will always want to protect their young. I didn't allow my children around hogs and cattle until they were in their teens. This should also be true of any engine-powered farm equipment. Too many accidents have happened to children on and near tractors and mowers. Wait until they are old enough to understand the dangers.\n\nMake sure to teach older children the full spectrum of farming activities. Take them to the farmers' market and let them run the cash register. Take them to the processing plant to hear the instructions you give. Your children need to learn that successful farming includes marketing as well as growing. Let them have a small plot to develop their own crop or raise an animal, then sell it along with yours. The 4-H program encourages this, although it usually works with more traditional crops and livestock. While I disapprove of some of the overly competitive aspects of some programs, it is an excellent way for your children to interact with other farm families. If your child does his or her research carefully, you may end up with a new profitable crop for your farm.\n\n#### Hired Labor\n\nHiring labor for the farm is often difficult. Employ outside help only when you have to, and then get the best available. You must be prepared to pay as much as a job of comparable responsibility in town would pay\u2014unless it is an apprenticeship, where the person is working for experience. When hiring, make sure you see a r\u00e9sum\u00e9 and interview a person carefully about experience, initiative, and responsibility. Establish a written contract, so both people know what they are getting.\n\nIf you must hire outside labor, you'll have to consult a tax accountant, because you may need to withhold taxes and Social Security from their paychecks. You may also have to pay unemployment insurance and worker's compensation insurance. Document employee work with performance reviews. If you need to fire a bad employee, you will need documentation of his infractions to avoid complaints or a lawsuit.\n\nTo find hired help, first talk to fellow farmers, who may have suggestions based on whom they have worked with. Older children from nearby families who are looking for some extra cash may be willing to do jobs on a part-time or seasonal basis. Check community bulletin boards at local restaurants or stores for specialized help, such as baling hay. A classified ad in the local paper may also be an option.\n\nOnce you have good employees, you want to make sure they'll stay. Your interaction with any employee is important for a good working relationship. Your job is to make the person proud and happy to work for you. Do not treat people as just another expense. Be clear with your directions, and show them what to do. Be fair and honest all the time. Incentives or bonus plans for certain levels of production instill pride of ownership in hired help. Extra money may help on the employees' home front when they have to work late for some reason\u2014an employee's family has to be happy with the job, too, or you will not have an employee for long.\n\n### Planning for Farm Efficiency\n\nFarmstead arrangement is very important to efficient labor use, especially for building locations. According to _The Farm Management Handbook_ , every 100 feet of unnecessary distance between the house and farm adds up to 14 miles of travel a year for each daily round trip. If you have a barn about 1,000 feet from your house and can move it 500 feet closer, you will save about 700 miles of walking each year.\n\nEvery step you don't take is time that can be used productively elsewhere on your farm.\n\nThe size and shape of your fields also affect labor. The smaller the field, the more times you have to turn the tractor when tilling the field, and the less time you spend doing productive labor. The larger the field, however, the less important this savings becomes. Obviously, a garden plot would be better to till by hand or with a two-wheel tractor rather than with your 45-horsepower tractor. You also have to consider leaving enough \"waste\" space between your plot and the fences to allow for turning your equipment. There is no optimal size or shape, except what will work best on your farm. So when planning your fields and pastures, look at your maps carefully, and plot the best placements of access lanes, buildings, pastures, gardens, and fields. Keep walking distances and future uses (rotations, new crops, for example) in mind while planning. No matter how carefully you plan, some new circumstance will change it\u2014but if you have a good plan, adjustments will be less costly in time, labor, and money.\n\n#### Building Reserves\n\nReserves are another method of securing financial safety: They will help you through unsteady markets or shaky financial periods. To build reserves requires careful management.\n\n#### Money\n\nThe most obvious reserve to build is that of money. I assume most of you have discovered that money not put in a savings account is usually spent. Setting aside a small amount each month for savings will be a great help. Also, when you buy machinery, it will depreciate each year. _Depreciation_ is the cost spread over the life of the equipment. For example, consider a tiller that costs $1,000 and has an expected lifespan of 10 years. The depreciation equals cost divided by expected lifespan ($1,000 \u00f7 10), in this case $100 per year. Set aside an amount equal to the depreciation, so you will have most of the cash needed to replace it when it wears out. If your markets are uneven throughout the year, you will need reserves for slower periods. It is always better to skimp a bit in summer than to have to do without in winter.\n\nHaving extra monies reserved will also let you take advantage of sales or advertising opportunities without interrupting your present cash flow. It can get you through droughts and floods, and allow vacations and seminar attendance. A small fund to increase your knowledge through book purchases, magazine subscriptions, and conference attendance will help ensure your continued success.\n\n#### Crops and Livestock\n\nThe most obvious resource to reserve other than cash is your crops and livestock. Finding a way to preserve your crops for winter feed will save you money. By saving ear corn, baling hay, and storing potatoes, you are making your crops work through additional seasons. A good hay year should result in extra hay in the barn for a reserve against a bad hay year. Adding value is another way to build reserves, by ensuring that you have products to sell throughout the year.\n\nYou can build reserves in livestock by saving back females, which reduces your replacement costs. You save back females by keeping your best young ewe stock each year, while culling the rest for sale. All ram lambs must be sold. Rams should be purchased every year or alternated to prevent inbred genetic problems. This practice will ultimately improve the quality and reproductive performance of the livestock, as you select for the best stock for your farm and management methods. This is another kind of reserve, a reserve of quality.\n\nQuality reserves are also possible with other aspects of your farm. Build your soils with cover crops and careful rotation of crops and livestock\u2014this will result in a reserve of nutrients for future crops. Quality soil is the reserve that forms the basis of farm sustainability.\n\nReserves can also involve how you plan the use of your farm, with resources both large and small. Timber growing on your farm will always be a good resource for the future, whether in sales or in firewood. Salvaging equipment and structures for other uses is an excellent way to save money. Wire can always be conserved. On my farm, I tie small bits of wire to fencing so I always have a supply on hand. I use it for tying together fences, fixing machinery, tying down water troughs, closing gates, and a multitude of other uses. Lumber and fencing can always be used for repairing similar structures, or save them for future projects.\n\nMake a list of how many different ways you can build reserves on your farm. Examine each crop and type of livestock, your land, your machinery, and structures. Decide what is best to do, and plan how to do it. With careful management, you will weather any storm.\n\n###### Food For Thought\n\nDiversity, rotations, and reserves are all different aspects of farm management, as is the information throughout all of these chapters. Management covers production cost, production expenses, labor, machinery labor and use, land use, soil maintenance, crops, livestock, budgets, diminishing physical output, increasing mental effort, and maximizing economic returns. To sum up farm management is to say that it is all about how you think about the various aspects of farming in relationship to one another. Your management will determine your success.\n\n###### Chapter Eleven\n\n## Where We Are Going\n\nMy grandmother always used to say, \"If you don't know where you're going, how will you know when you get there?\" To really know where we are going, first we must know where we have been. The history of agriculture does much to explain how we got to where we are today, and from this, we can determine where we want to be.\n\n### Land and Farms\n\nIn the early days of agriculture in this country, it was common practice just to move to a different farm when the soil got poor on the land you were farming. Land had little monetary value and was used to encourage people to homestead various areas where railroads and other businesses wished to have customers.\n\nThen dust storms and the Depression hit; Americans became concerned about their food supply and the conservation of our soil. \"Land\u2014they're not making any more of it!\" was a common quote.\n\n_Direct marketing whole-hog sausage enables you to capture a niche market that is far more profitable than hauling your hogs to the sale barn and saying, \"What will you give me for them?\"_\n\nWith the increasing use of the automobile after World War Ii, rapid movement became available from cities to the country, and vice versa. People were interested in independence, self-sufficiency, and the good life. A little piece of land and a house outside the city was the American dream. Factories and businesses followed this path to give their employees a taste of country life, and farmland started disappearing. Later, as land prices rose and the best return on money was from developments, ranging from housing to shopping malls to recreation centers, farmland began vanishing, to be replaced by concrete and asphalt. Today, during every single minute of every single day, we lose 2 acres of farmland to development.\n\nThroughout the 1940s and 1950s, a number of writers came forth and exposed various agricultural policies and practices. Around 1943, Louis Bromfield, author of _Malabar Farm_ , was warning of future hunger caused by disappearance of farmers. A New York City newspaper headline read, \"City Facing First Famine in Our History.\" Edward Faulkner's _Plowman's Folly_ questioned the use of the plow due to its destruction of the soil. Arthur Moore, in _The Farmer and the Rest of Us_ , noted that of the 6 million farms in the United States in 1945, one-third of them averaged 20 acres in size and sent about $100 worth of food to the marketplace in a year. They produced about 3 percent of the production. The 3 million top-producing farms grew 90 percent of our farm commodities. I include these comments and figures for two reasons: First, many of the ideas from the 1940s and 1950s have become the sustainable agriculture ideas of the 1990s and the new millennium. Second, we must recognize that farmers and agriculture are disappearing from the American countryside and mind. But there is good news! Though the number of farms has decreased in recent years, the number of small farmers is increasing and will likely continue to do so. In the 1990s, we have about 1.9 million farms; 75 percent of them are small farmers, with 178 acres or fewer, who produce 20 percent of our farm commodities.\n\nGiven society's choice of where its food is grown and how it is grown, this could well be more than 50 percent in a few short years. Local farm production leads to stable local communities and citizen participation in local government. Small farms furnish a set of children with a work ethic and a good set of values. Isn't this the kind of nation we want to live in? We need more farmers growing more crops for different reasons, not fewer farmers going to town for more financially successful jobs.\n\nHistorical geographer Paul B. Frederic notes that situation is of critical importance in terms of farm characteristics, as relative location determines how the use of a particular site changes through time. Changing proximity to markets leads to change in farm functions. According to _Farming in the Midwest 1840\u20131900_ , edited by James Whitaker, in most cases, farms that retain their pioneer atmosphere are those most remote from population centers. This is still true today, as cattle is ranched in the traditional manner in the ranges of the West, and Appalachian farmers continue their traditional subsistence farming. Population centers tend to support smaller farms with direct-marketed products\u2014the antithesis of the modern \"traditional\" farm. Small farm agripreneurs need to be located within 40 to 50 miles of a city for the best marketing results.\n\n### Industrialization of Agriculture\n\nOriginally, farms were small, and mostly fed just the families who lived on them. With the growth of cities, some farms started producing food and fiber for urban dwellers. As time went by, farms grew large and mechanization came on the scene in the form of reapers, binders, corn pickers, and so on, up to the modern-day combine. John Deere developed the first steel plows, and tractors eventually replaced the horse on farms as the main power source.\n\nBy the 1950s, agriculture had become fully mechanized. Hybrid corn was the rage, and farmers began to use chemicals to reduce labor costs. Farms became more specialized, and diversification and its benefits gave way to new agricultural thinking. \"Fencerow to fencerow\" and \"Bigger is better\" became the watchwords of the young tigers of the 1970s.\n\nThe rural crisis of the 1980s brought the agricultural dream crashing down. Out-of-control land prices, combined with a drop in commodity prices, killed a lot of farms. Suicides increased, along with divorces; dreams were destroyed and families were torn apart. The rural crisis brought about a change in agriculture. Farmers began to question the land-grant institutions and they began to question the way they farmed and why. Farmers began to really study farming, and, in particular, marketing. The survivors of the '80s made it either because of excellent equity positions or because they were diversified farms that developed niche markets for traditional or nontraditional crops and livestock, or a combination of both.\n\nThe rural crisis was followed by the consolidation of agricultural processing\u2014now, for example, four companies process 87 percent of all the beef in the United States. And other commodities have similar numbers. The same consolidation has happened with the chemical companies, which bought out the seed companies, giving them virtually complete control of major crops like corn, wheat, and beans, from production through processing and marketing. Also, in the 1990s, seed companies began charging a technology fee in addition to the cost of the seed. Recently, the Usda and Delta Land and Pine Company developed a \"terminator\" gene that makes saving seed impossible, because it is genetically programmed not to germinate the second year. This has received so much bad publicity that Monsanto (which bought out Delta Land and Pine) chose not to proceed with it, but they have continued to develop other varieties of terminator genes. Other companies are also developing terminator technology. The problem with terminator technology (other than the risk of genetic contamination of your open-pollinated crops) is that it forces you to buy from the seed companies every year, rather than being able to save your own seed.\n\nAgain, the good news is that even as farm numbers shrink and major companies take over more of the market, the number of small farms is increasing. Small farms of 178 acres or fewer continue to gain in numbers at the rate of 2 percent a year and will continue to do so for at least the next 10 years. I firmly believe that this rate of increase will become even higher as more farmers opt out of traditional marketing and into direct marketing, either by themselves or as part of a cooperative.\n\n### Farms and People\n\nTo begin with, everyone was a farmer. Many of our early politicians were farmers, and some, like George Washington and Thomas Jefferson, had extensive landholdings in which they experimented with crops and practiced growing legumes for cover crops and soil improvement. As people moved to the cities, the percentage of farmers started to shrink. The average person today is four or five generations removed from the farm, and the people making agriculture decisions have little or no experience with farming.\n\nIf we want agriculture to evolve into a more sustainable pattern, the four major groups of importance to agriculture\u2014farmers, lenders, consumers, and institutions\u2014must be convinced at the same time that these changes are beneficial. This will require educating these people as to what farming is about, and why sustainable practices will ultimately benefit everyone\u2014and make them more money.\n\n### Success on Sustainable Farms\n\nDespite the somewhat gloomy note struck by my brief history of agriculture, there are more opportunities to start and succeed on a farm today than ever before. Common sense, along with appropriate technology and direct marketing, has put the small-farm dream in reach of anyone who wants to try to achieve it. It can even be done without first going into debt up to your ears to get started.\n\nThe whole purpose of this book is to give the principles of good farming, which will work on any type of farm and in any state. The way you picture your farm and its goals and relationships is the Big Picture principle, the balance between your crops and livestock, both locally and globally. The smaller the acreage you have, the more important your level of soil fertility, so that you can produce the necessary volume to be profitable.\n\nEstablishing your goals and working to achieve them will create a successful small farm. The most important thing to remember is that the goals must be the entire family's goals. Direct marketing is mandatory for success. To gain the volume needed for success in your monetary goals, you must receive retail prices for most, if not all, of your production. As I write this sentence (November 1998), I will make $100 to $130 per head in net profit on my hogs by selling them as sausage rather than the current market price of 18 cents per pound from the packers. You can do the same.\n\nFarm planning is a road map of how to reach your destination. Farm planning is a principle. Selecting your enterprise correctly to match your farm, your resources, and your management level are important to success. Spend your time analyzing budgets and profit potential.\n\nMachinery and hand tools are labor-saving devices to enable you to do more in less time or to do it in a timely manner. Sweet corn, for instance, has a one- to two-day window of harvest for the picture-perfect ear. Planting every 2 weeks spreads your marketing and marketing risk of timeliness.\n\nFarm management is the process of taking all the principles in all the chapters and making them work together. When you do that, your farming operation will run smoothly, like a well-oiled machine.\n\nI'm sure that as you go along, you will find more principles that are specific to your farm and soil type, but what we have discussed are the basics; with these few, you can start and you can succeed!\n\n#### First Steps\n\nI have met many people during my years as publisher of _Small Farm Today_ magazine who say they want to farm, but it seems they never start. Starting a project like farming, especially if you have no experience, can be quite frightening, but also challenging. You must see yourself as an agripreneur.\n\n**Experiment**. If you are already farming, but want to move to more-profitable alternatives, set aside a small area to work with and start experimenting. If you are new to farming, start your new farm now, even if it is in the backyard of your subdivision. Start researching, start plotting a business plan, start visiting farmers' markets and talking to fellow farmers. Grow into farming, but do not borrow yourself into farming.\n\n**Rent land**. There are lots of ways to start\u2014if you don't have any land, rent some. Apprenticing on a working farm is another way to gain experience, with part of your salary being used to save for some land. Use of a vacant lot in the city can often be had merely for clearing it. You can then grow produce on it rent-free.\n\n**Learn as you grow**. You can grow enough to feed one adult for one year on 1,000 square feet. Do it yourself\u2014it's good practice. Sell some of your produce to your neighbors. Keep records, look at your soil, and examine how it works. You can accumulate tools and equipment, paying cash and storing them until you get your land if you have a plan.\n\n**Always watch your bottom line**. If you cannot make a profit on 1 acre, having 10 acres or 80 will only make the project 10 to 80 times worse. Thinking equals profits. The more you think about farming and study farming, the more opportunities you will see for reducing your costs, increasing your profits, and becoming sustainable. If you simply reduce your expenses 10 percent, you increase your net profit 40 percent. Food for thought: If your produce or meat is fresher and better than store-bought, why should you sell it for 20 percent less than a store? If you work hard to produce a high-quality, \"best-of-the-best\" product, you deserve to get paid for it. Price your products at what they are worth.\n\nWhen should you start farming? If you are talking about traditional agriculture\u2014corn, wheat, and soybeans, or cattle, hogs and sheep\u2014the anwer is to start at the bottom of the price cycle or seasonal low. All commodities prices run in cycles between low supply\/high demand\/high prices and high supply\/low demand\/low prices. Crop cycles depend on whether there are any surplus or deficits in previous years. Cattle cycles are normally long, whereas hogs and sheep\u2014multibirth animals with short gestation periods\u2014are normally shorter. It should be noted, however, that environmentally controlled housing, especially for poultry and hogs, has played havoc with seasonal cycles for livestock. Further consolidation of agriculture has essentially caused the marketing system for commodities to fail because there is in effect no supply and demand essential for price discovery. Do not embark on this type of farming (traditional agriculture with conventional markets) unless you want to be a contract grower making no management decisions and shouldering a large portion of the financial risk of farming, which is Land, Labor, and Machinery. However, I would encourage you to start farming using the methods I talked about in this book!\n\n#### Be a Spokesperson\n\nMuch as I would like to see us have 6.5 million farmers again, we have only 1.9 million. Two percent of the population feeds the other 98 percent. To make the changes I have talked about, to keep your farm on an even keel, you must communicate your needs to others. Because the politicians who make the laws and the people who vote on them are four or five generations removed from the farm, it is up to you to educate them. Every  man, woman, and child involved in agriculture must be a spokesperson for the agriculture industry. In that role you can reaffirm the farm and food connection, and at the same time build relationships with your customers.\n\n### My Vision\n\nThroughout this book I have talked about the contributions that small sustainable farms make to the community and to our society. I also believe we must have a sustainable society to have a sustainable agriculture. Just reading the words seems to give me roots in something permanent\u2014roots in a better place to live and grow.\n\nWe easily have room for many more small farms in the United States. This is one of the few bright places in American agriculture because small farms are increasing. In the future, I see small sustainable farms as being the predominant type of localized agriculture. More small farms will be involved in processing their own products, and there will be many regional products like Ozark wines, and Boone County hams, and Washington apples. The regionality will be significant as we shift to a more seasonal diet that goes along with a decentralized agriculture. With more people connected to agriculture, we will have a better understanding of our world, and will understand that what we do today has consequences for our children and their children.\n\nSmall farms provide more time for interaction with family and neighbors. The neighborhood will be very broad, as computers and the Internet help small farmers share and gain knowledge with each other around the world. In the future, I believe we will spend much more time thinking, rather than doing, and as a result, we will become more at peace with one another and the world in general.\n\nEverything in life is dependent on our soils, the sun, and water. When larger numbers of people are involved with the soil and the basics of life, food, shelter, and clothing, the world will become a better place, because more people will appreciate and understand how we get these basic tenets and how important they are. In today's world, this is easy to forget, because we are a mobile society that can move away from the consequences of our actions; if the soil is poor, plow up another field, or sell the farm and get another. In the future, we will not have that luxury; we must improve what we have. A small farm must be fertile to provide for the family's lifestyle and livelihood.\n\nSmall farms hold the key to reconnect people with the land and the food they must have to sustain life. Small farms are an ongoing process of discovery about life and people. Small farms are the wave of the future, leading to a better world.\n\nIt is now your chance to join this movement. You now have all the building blocks\u2014the principles of a successful small farm. All you have to do is start.\n\n###### Happy and profitable farming!\n\n_Note_ : If you have comments or questions, I invite you to contact me in care of the publisher, Storey Publishing.\n\n###### Appendix a\n\n## Metric Conversions Chart\n\n###### Appendix b\n\n## Resource Lists\n\nIn this appendix, I have provided a list of resources that I have found valuable in small farming endeavors. First, though, I wish to promote two resources without which you would not be holding this book:\n\n  * Storey Publishing, 800-793-9396, www.storey.com. Storey publishes a wide variety of farming books, including this one. I particularly recommend the Storey's Guide to Raising farm animals series.\n\n  * Small Farm Today magazine, 800-633-2535, www.smallfarmtoday.com. Bimonthly. I started this how-to magazine of alternative and traditional crops, livestock, and direct marketing in 1984 to provide information to small farmers and small acreage landowners. It led eventually to this book.\n\n### Books\n\nThere are a multitude of useful books; those listed here are ones I use frequently. The catch is that many of them are out of print. Check with your local bookstore to see if you can order them. And check your local and university libraries, and visit lots of used bookstores\u2014you may come across a treasure.\n\n#### Alternative Livestock\n\nBenyon, Peter H., and John E. Cooper, et al., Manual of Exotic Pets (Ames, Ia: Iowa State University Press, 1991).\n\nGuide to exotic pets\u2014mammals, birds, and reptiles.\n\nBirutta, Gale, Storey's Guide to Raising Llamas (North Adams, Ma: Storey Publishing, 1997).\n\nIn-depth llama guide.\n\nJohnson, Jim, James Harvey Johnson, and Stanley T. Weiner, Husbandry and Medical Management of Ostriches, Emus, and Rheas (College Station, Tx: Wildlife and Exotic Animal Consultants, 1992).\n\nShort but detailed guide to ratites.\n\nKyle, Russell, A Feast in the Wild (Oxford, Uk: Kudu Publishing, 1987). A look at the potential food use of several exotic species. von Kerckerinck, Josef, Deer Farming in North America (Rhinebeck, Ny: Phanter Press, 1987).\n\nExcellent guide to raising fallow deer.\n\nYerex, David, and Ian Spiers, Modern Deer Farm Management (Carterton, New Zealand: Ampersand Publishing, 1987).\n\nGood deer farming guide.\n\n#### Aquaculture\n\nBrown, E.E., and J.B. Gratzek, Fish Farming Handbook (Westport, Ct: Avi Publishing, 1980).\n\nRaising food, bait, and tropical fish.\n\nHuner, Jay V., and E. Evan Brown, eds., Crustacean and Mollusk Aquaculture in the United States (Westport, Ct: Avi Publishing, 1985).\n\nRaising crawfish, shrimp, clams, and more.\n\n\u2014\u2014\u2014, ed., Freshwater Crayfish Aquaculture (Binghamton, Ny: Food Products Press, 1998).\n\n#### Cattle\n\nMorrison, Frank B., Feeds and Feeding, 22nd ed. (Morrison, Ia: The Morrison Publishing, 1959).\n\nThe bible of livestock feed.\n\nSalatin, Joel, Salad Bar Beef (Swoope, Va: Polyface Farms, 1995).\n\nHow to profit from a small beef cattle operation.\n\nSmith Thomas, Heather, Storey's Guide to Raising Beef Cattle (North Adams, Ma: Storey Publishing, 1998).\n\nHandling, breeding and care of beef cattle.\n\nvan Loon, Dirk, The Family Cow (North Adams, Ma: Storey Publishing, 1976). The basics of a family cow.\n\n#### Draft Animals\n\nMiller, Lynn R., Work Horse Handbook (Sisters, Or: Farmers Book Service, 1985).\n\nGuide to using draft horses.\n\nTelleen, Maurice, The Draft Horse Primer (Waverly, Ia: Draft Horse Journal, 1977).\n\nSelection, care, and use of work horses and mules.\n\n#### Earthworms\n\nBarrett, Thomas J., Harnessing the Earthworm (Eagle River, Wi: Shields Publications, 1947).\n\nThe original book on earthworm culture.\n\nErnst, David, The Farmer's Earthworm Handbook (Brookfield, Wi: Lessiter Publications, 1995).\n\nManaging earthworms to improve soils.\n\nGaddie, Ronald E. Sr., and Donald E. Douglas, Earthworms for Ecology and Profit, Vol. 1 (Ontario, Ca: Bookworm, 1976).\n\nEarthworm farming.\n\nLee, K.E., Earthworms: Their Ecology and Relationships with Soil and Land Use (Orlando, Fl: Academic Press, 1985).\n\nA technical guide to worms and soils.\n\nMinnich, Jerry, The Earthworm Book (Emmaus, Pa: Rodale, 1977).\n\nHow to raise and use earthworms.\n\n#### Field Crops\n\nAlternative Field Crops Manual (Madison: University of Wisconsin-Extension, Madison\/University of Minnesota Center for Alternative Plant and Animal Products, 1992).\n\nA collection of papers on alternative crops, from adzuki beans to wild rice.\n\nLampkin, Nicolas, Organic Farming (Ipswich, Uk: Farming Press Books, 1990).\n\nOrganic farming in Europe.\n\nMartin, John H., Warren H. Leonard, and David L. Stamp, Principles of Field Crop Production, 3rd ed. (New York: Macmillan, 1976).\n\nBasics of crop production.\n\nRobinson, Raoul A., Return to Resistance (Davis, Ca: agAccess, 1996).\n\nBreeding crops to reduce pesticide dependence.\n\nSpecialty and Minor Crops Handbook, 2nd ed. (Oakland: University of California Danr, 1997).\n\nProfiles of 63 specialty and minor crops.\n\n#### Gamebirds\n\nMullin, John, Game Bird Propagation 5th ed. (Goose Lake, Ia: Wildlife Harvest Publications, inc., 1994).\n\nBreeding gamebirds.\n\nWoodard, Allen, Pran Vohra, and Vern Denton, Game Bird Breeders Handbook (Blaine, Wa: Hancock House, 1993).\n\nRaising pheasant, partridge, and quail.\n\n#### Gardening\n\nBartholomew, Mel, Cash from Square Foot Gardening (North Adams, Ma: Storey Publishing, 1985).\n\nMore information on square foot gardening.\n\n\u2014\u2014\u2014, Square Foot Gardening (Emmaus, Pa: Rodale Press, 1981).\n\nPlanning a garden in square feet.\n\nColeman, Elliot, The New Organic Grower, 2nd ed. (White River Junction, Vt: Chelsea Green, 1995).\n\nOrganic tools and techniques for home and market gardeners.\n\nPoincelot, Raymond P., No-Dig, No-Weed Gardening (Emmaus, Pa: Rodale Press, 1986).\n\nOrganic gardening without tilling.\n\n#### Greenhouses and Solar Gardening\n\nEdey, Anna, Solviva (Martha's Vineyard, Ma: Trailblazer Press, 1985).\n\nUsing a biodynamic solar greenhouse for saving money on crops and livestock.\n\nPoisson, Leandre, and Gretchen Vogel Poisson, Solar Gardening (White River Junction, Vt: Chelsea Green Publishing, 1994).\n\nGrow vegetables year-round with \"mini-greenhouses.\"\n\nSmith, Shane, The Bountiful Solar Greenhouse (Santa Fe, Nm: John Muir Publications, 1982).\n\nGrowing food year-round with solar greenhouses.\n\n#### Herbs and Flowers\n\nByczynski, Lynn, The Flower Farmer (White River Junction, Vt: Chelsea Green Publishing, 1997).\n\nGuide to raising and selling organic cut flowers.\n\nMiller, Richard Alan, The Potential of Herbs as a Cash Crop, 2nd ed. (Metairie, La: Acres Usa, 1985).\n\nThe classic guide to the herb business.\n\nShores, Sandy, Growing and Selling Fresh-Cut Herbs (North Adams, Ma: Storey Publishing, 1999).\n\nExcellent handbook for herbal agripreneurs.\n\nStevens, Alan, Field-Grown Cut Flowers (Edgerton, Wi: Avatar's World, 1997).\n\nProduction guide for fresh and dried cut flowers.\n\nSturdivant, Lee, Herbs for Sale (Friday Harbor, Wa: San Juan Naturals, 1994).\n\nGrowing and marketing herbs.\n\n\u2014\u2014\u2014, and Tim Blakley, Medicinal Herbs in the Garden, Field, and Marketplace (Friday Harbor, Wa: San Juan Naturals, 1998).\n\nGuide to establishing a medicinal herb business.\n\n#### History\n\nCochrane, Willard W., The Development of American Agriculture: A Historical Analysis (Minneapolis: University of Minnesota Press, 1981).\n\nA look at how we got to where we are in agriculture.\n\n#### Homesteading\n\nEmery, Carla, The Encyclopedia of Country Living, 9th ed. (Seattle, Wa: Sasquatch Books, 1994).\n\nThe best general information book I have found, from winemaking to meat drying.\n\nWigginton, Eliot, The Foxfire Book, vol. 1 (Garden City, Ny: Anchor Press, 1972).\n\nHomecraft how-to. Nine other volumes followed this one.\n\n#### Honeybees\n\nBonney, Richard E., Beekeeping: A Practical Guide (North Adams, Ma: Storey Publishing, 1993).\n\nAcquiring and managing bees.\n\nMorse, Roger A., The New Complete Guide to Beekeeping (Woodstock, Vt: Countryman Press, 1994).\n\nStarting and maintaining bees.\n\n#### Inspirational\n\nBerry, Wendell, The Gift of Good Land (Sisters, Or: Farmer's Book Service, 1981).\n\nMore cultural and agricultural essays.\n\n\u2014\u2014\u2014, The Unsettling of America (New York: Avon Books, 1978).\n\nEssays on the importance of sustainable agriculture.\n\nLogsdon, Gene, You Can Go Home Again (White River Junction, Vt: Chelsea Green, 1998).\n\nMore wonderful inspiration.\n\n#### Machinery and Computers\n\nBowman, Greg, ed., Steel in the Field (Burlington, Vt: Sustainable Agriculture Network, 1997).\n\nAn excellent guide to weed management tools.\n\nCampidonica, Mark, How to Find Agricultural Information on the Internet (Oakland: University of California Danr, 1997).\n\nGood introduction to finding farming resources on the Net.\n\nFarm Conveniences and How to Use Them (New York: Lyons Press, 1999).\n\nLabor-saving devices from the 1800s.\n\n#### Marketing Methods\n\nAbleman, Michael, On Good Land (San Francisco: Chronicle Books, 1998).\n\nStory of a small community farm in California.\n\nCopeland, John D., Recreational Access to Private Lands, 2nd ed. (Fayetteville, Ar: National Center for Agricultural Law Research and Information, 1998).\n\nLegal advice on establishing on-farm activities.\n\nDoane, D. Howard, Vertical Farm Diversification (Norman: University of Oklahoma Press, 1950).\n\nTaking farms beyond raw material production.\n\nGroh, Trauger, and Steven McFadden, Farms of Tomorrow Revisited (White River Junction, Vt: Chelsea Green Publishing, 1997).\n\nA blueprint for Community Supported Agriculture farms.\n\nRogak, Lisa, The Complete Country Business Guide (Grafton, Nh: William Hills Publishing, 1998).\n\nIntroduction to starting a rural business.\n\n#### Pastures and Cover Crops\n\nHughes, H.D., Maurice E. Heath, and Darrel S. Metcalfe, Forages, 2nd ed. (Ames: Iowa State University Press, 1962).\n\nExcellent guide to forages. The 5th edition (1995) is a two-volume set.\n\nManaging Cover Crops Profitably, 2nd ed. (Burlington, Vt: Sustainable Agriculture Network, 1998).\n\nAn excellent guide to choosing and growing cover crops.\n\nMurphy, Bill, Greener Pastures on Your Side of the Fence (Colchester, Vt: Arriba Publishing, 1987).\n\nUsing the Voisin system of grazing management to improve pasture productivity.\n\nNation, Allan, Quality Pasture (Jackson, Ms: Green Park Press, 1995).\n\nCreating, managing, and profiting from quality pasture.\n\nSmith, Burt, Pingsun Leung, and George Love, Intensive Grazing Management (Kamuela, Hi: The Graziers Hui, 1986).\n\nManaging forage and animals for profit.\n\n#### Poultry\n\nThe American Standard of Perfection (Troy, Ny: American Poultry Association, Inc., 1993).\n\nA complete description of all recognized breeds and varieties of domestic poultry, periodically revised.\n\nDamerow, Gail, Storey's Guide to Raising Chickens (North Adams, Ma: Storey Publishing, 1995).\n\nAn in-depth guide.\n\nHastings Belshaw, R.H., Guinea Fowl of the World (Northamptonshire, Uk: Nimrod Book Services, 1985).\n\nGuide to breeding guineas.\n\nLee, Andy, and Pat Foreman, Chicken Tractor, Straw Bale Edition (Columbus, Nc: Good Earth Publications, 1998).\n\nPortable chicken pens in the garden.\n\nLevi, Wendell Mitchell, The Pigeon (Sumter, Sc: Levi Publishing Co., Inc., 1957).\n\nComplete information on raising pigeons.\n\nSalatin, Joel, Pastured Poultry Profits (Swoope, Va: Polyface Farms, 1993).\n\nRotational grazing chickens in portable cages.\n\nSchwanz, Lee, ed., The Family Poultry Flock (Brookfield, Wi: Farmer's Digest, Inc., 1981).\n\nBasics of choosing and raising poultry.\n\nThear, Katie, Free-range Poultry (Ipswich, Uk: Farming Press Books, 1990).\n\nGuide to grazing chickens free-range.\n\n#### Shelter and Fencing\n\nBurch, Monte, How to Build Small Barns and Outbuildings (North Adams, Ma: Storey Publishing, 1992).\n\nFundamentals of general construction.\n\nDamerow, Gail, Fences for Pasture and Garden (North Adams, Ma: Storey Publishing, 1992).\n\nSelecting, planning, and building fences.\n\nMerrilees, Dong, Ralph Wolfe, and E. Loveday, Low-Cost Pole Building Construction (North Adams, Ma: Storey Publishing, 1980).\n\nHow to build a small home, barn or other pole structure.\n\nSmall-Acreage Farming\n\nAngier, Bradford, One Acre and Security (New York: Random House, 1972).\n\nGeneral information on starting a farm.\n\nBromfield, Louis, From My Experience (New York: Harper & Brothers, 1955).\n\nMore about Malabar.\n\n\u2014\u2014\u2014, Malabar Farm (New York: Ballantine, 1947, 1970).\n\nThe author returns to organic farming and tells about it.\n\nLogsdon, Gene, The Contrary Farmer (White River Junction, Vt: Chelsea Green, 1993).\n\nGreat inspiration and useful advice.\n\nMiller, Ralph C. and Lynn R. Miller, eds. Ten Acres Enough: The Small Farm Dream Is Possible (Sisters, Or: Farmer's Book Service, 1981).\n\nA reprint of the 1864 classic, with updated essays. Part inspiration, part useful advice.\n\nOlson, Michael, MetroFarm (Santa Cruz, Ca: Ts books, 1994).\n\nAnother good guide to small parcel success.\n\nSalatin, Joel, You Can Farm (Swoope, Va: Polyface Farms, 1998).\n\nThe author explains his system for success.\n\nSmart, Charles Allen, Rfd (athens: Ohio University Press, 1998).\n\nCity dweller turns farmer during the Depression.\n\nSmith, Miranda, ed., The Real Dirt (Burlington, Vt: Northeast Region Sare, 1994).\n\nOrganic and low-input practices in the Northeast.\n\nWhateley, Booker T., How to Make $100,000 Farming 25 Acres (Chillicothe, Il: American Botanist, 1996).\n\nThe classic on small farm success.\n\nThe Yearbooks of Agriculture, 1910\u20131962, 1976. (Washington, Dc: Us Department of Agriculture).\n\nThese books have all sorts of useful information. In particular, I recommend 1938: Soils and Men; 1941: Climate and Man; 1948: Grass; 1949: Trees; 1955: Water; 1957: Soil; and 1976: Living on a Few Acres.\n\n#### Small Fruits and Tree Crops\n\nHarlan, Michael, and Linda Harlan, Growing Profits (Citrus Heights, Ca: Moneta Publications, 1997).\n\nStarting and operating a backyard nursery.\n\nOtto, Stella, The Backyard Berry Book (Maple City, Mi: OttoGraphics, 1995).\n\nHow to grow berries.\n\n\u2014\u2014\u2014, The Backyard Orchardist (Maple City, Mi: OttoGraphics, 1993).\n\nGrowing fruit trees in the home garden.\n\nWampler, Ralph L., and James E. Motes, Pick-Your-Own Farming (Norman: University of Oklahoma Press, 1984).\n\nGrowing crops for U-pick farms.\n\n#### Small Stock\n\nCheeke, Peter R., Nephi M. Patton, and George S. Templeton, Rabbit Production, 5th ed. (Danville, Il: Interstate Printers and Publishers, Inc., 1982).\n\nExcellent guide to raising rabbits.\n\nDrummond, Susan Black, Angora Goats the Northern Way (Freeport, Mi: Stoney Lonesone Farm, 1985).\n\nRaising and utilizing angora goats.\n\nKlober, Kelly, Storey's Guide to Raising Pigs (North Adams, Ma: Storey Publishing, 1997).\n\nAn excellent source for small-scale pig raising.\n\nKruesi, William K., The Sheep Raiser's Manual (Charlotte, Vt: Williamson Publishing, 1985).\n\nGood guide to raising sheep.\n\nMacKenzie, David, Goat Husbandry, 4th ed. (London: Faber & Faber Limited, 1980).\n\nGuide to raising goats.\n\nThornton, Keith, Outdoor Pig Production (Ipswich, Uk: Farming Press Books, 1990).\n\nBasics of raising pigs.\n\n#### Soils\n\nAlbrecht, William A., The Albrecht Papers, Vol. 1, (Metairie, La: Acres Usa, 1975).\n\nClassic research on sustainable soils and their importance. Three more volumes follow.\n\nBuckman, Harry O., and Nyle C. Brady, The Nature and Properties of Soils, 7th ed. (New York: Macmillan Company, 1969).\n\nAn excellent guide to soils and soil management.\n\nFaulkner, Edward H., Plowman's Folly (New York: Grosset & Dunlap, 1943).\n\nAlternative cultivation methods.\n\nGershuny, Grace, Start with the Soil (Emmaus, Pa: Rodale, 1993).\n\nOrganic gardener's guide to improving soil.\n\n\u2014\u2014\u2014, and Joseph Smillie. The Soul of Soil, 3rd ed. (White River Junction, Vt: Chelsea Green, 1996).\n\nA guide to ecological soil management.\n\nKinsey, Neal, and Charles Walters, Hands-On Agronomy (Metairie, La: Acres Usa, 1993).\n\nBuilding and maintaining the soil.\n\nWalters, Charles Jr., and C.J. Fenzau, An Acres Usa primer (Metairie, La: Acres Usa, 1979).\n\nPrinciples of soil care from an unusual perspective.\n\n#### Sustainable Practices\n\nAltieri, Miguel A., Agroecology (Boulder, Co: Westview Press, 1987).\n\nA scientific justification of alternative agriculture.\n\nAvory, Allan, Holistic Resource Management (Covelo, Ca: Island Press, 1988).\n\nComprehensive system planning.\n\nEdwards, Clive, et al., eds. Sustainable Agricultural Systems (Ankeny, Ia: Soil and Water Conservation Society, 1990).\n\nPractical research on sustainable agriculture.\n\nFukuoka, Masanobu, The Natural Way of Farming (New York: Japan Publications, Inc., 1985).\n\n\"Do-nothing\" organic farming in Japan.\n\nJackson, Wes, et al., eds., Meeting the Expectations of the Land (Berkeley, Ca: North Point Press, 1984).\n\nEssays on sustainable agriculture.\n\n\u2014\u2014\u2014, New Roots for Agriculture (Lincoln: University of Nebraska Press, 1985).\n\nReasons for sustainable agriculture and how to practice it.\n\nNational Research Council, Alternative Agriculture (Washington, Dc: National Academy Press, 1989)\n\nAlternative farming benefits and case studies.\n\n#### Vegetables\n\nDamerow, Gail, The Perfect Pumpkin (North Adams, Ma: Storey Publishing, 1997).\n\nGrowing and using pumpkins.\n\nDeppe, Carol, Breed Your Own Vegetable Varieties (Boston: Little, Brown and Company, 1993).\n\nPlant breeding for the home gardener.\n\nDeWitt, Dave, and Paul W. Bosland, The Pepper Garden (Berkeley, Ca: Ten Speed Press, 1993).\n\nHow to grow peppers.\n\nJeavons, John, How to Grow More Vegetables 5th ed. (Berkeley, Ca: Ten Speed Press, 1995).\n\nSustainable biointensive organic horticulture.\n\nLorenz, Oscar A., and Donald N. Maynard, Knott's Handbook for Vegetable Growers, 3rd ed. (New York: John Wiley & Sons, 1988).\n\nThe essential authority on production of vegetables.\n\nStamets, Paul, Growing Gourmet and Medicinal Mushrooms (Berkeley, Ca: Ten Speed Press, 1993).\n\nA bible of mushrooms.\n\nWeaver, William Woys, Heirloom Vegetable Gardening (New York: Henry Holt and Company, 1997).\n\nPlanting, growing, and saving seeds.\n\nWilbur, Charles W., How to Grow World Record Tomatoes (Metairie, La: Acres Usa, 1998).\n\nGrowing huge organic tomato plants.\n\n### Book Sources\n\n###### Acres Usa\n\n800-355-5313\n\nwww.acresusa.com\n\nPublishes and carries a variety of farming books.\n\n###### Chelsea Green Publishing Company\n\n800-639-4099\n\nwww.chelseagreen.com\n\nPublishes and carries a variety of farming books.\n\n###### Country Store Books\n\n800-633-2535\n\nwww.smallfarmtoday.com\/countrystorebks.html\n\nCarries a variety of farming books.\n\n###### Countryman Press\n\n800-245-4151\n\nwww.countrymanpress.com\n\nFocuses on the outdoors and gardening.\n\n###### Hancock House Publishers\n\n800-938-1114\n\nwww.hancockhouse.com\n\nPublishes aviculture (bird) books.\n\n###### Storey Publishing\n\n800-793-9396\n\nwww.storey.com\n\nPublishes a wide variety of farming books.\n\n###### Sustainable Agriculture Research and Education\n\n202-720-5384\n\nwww.sare.org\n\nPublishes several sustainable farming and gardening books.\n\n###### Ten Speed Press\n\n800-841-2665\n\nwww.tenspeedpress.com\n\nPublishes a variety of garden books.\n\n### Periodicals\n\n#### Magazines and Newspapers\n\nMagazines about specific crops or animals offer the best opportunity to further your knowledge. They not only have up-to-date information on what you wish to raise, but they also contain display advertisements and breeder's directories that list places to buy what you need and information contacts.\n\n###### Acres Usa\n\n800-355-5313\n\nwww.acresusa.com\n\nEco-agriculture information, ranging from standard to extreme; monthly\n\n###### Alpacas Magazine\n\nAlpaca Owners & Breeders Association\n\n615-834-4195\n\nwww.alpacainfo.com\n\nInformation on alpacas; quarterly\n\n###### American Bee Journal\n\n217-847-3324\n\nwww.americanbeejournal.com\n\nInformation on honeybees\n\n###### Animal Finders' Guide\n\n812-898-2678\n\nwww.animalfindersguide.com\n\nListings of rare and exotic animals for sale; 18 issues per year\n\n###### Back Home\n\n800-992-2546\n\nwww.backhomemagazine.com\n\nSustainable living\n\n###### Bee Culture\n\n800-289-7668\n\nwww.beeculture.com\n\nAmerican beekeeping; monthly\n\n###### Biodynamics\n\nBiodynamic Farming and Gardening Association\n\n888-516-7797\n\nwww.biodynamics.com\n\nInformation on the biodynamic (organic) method of agriculture; quarterly\n\n###### The Brayer\n\nAmerican Donkey & Mule Society, Inc.\n\n972-219-0781\n\nwww.lovelongears.com\/brayer\n\nDonkey and mule information; bimonthly\n\n###### Country Smallholding\n\n+44-18-5843-8839\n\nwww.countrysmallholding.com\n\nSmall farming in England; monthly\n\n###### Countryside & Small Stock Journal\n\n800-551-5691\n\nwww.countrysidemag.com\n\nHomesteading; bimonthly\n\n###### Domestic Rabbits\n\nAmerican Rabbit Breeders Association, Inc.\n\n309-664-7500\n\nwww.arba.net\n\nNews from the American Rabbit Breeders Association; bimonthly\n\n###### Draft Horse Journal\n\n319-352-4046\n\nwww.drafthorsejournal.net\n\nDraft horse news. Quarterly.\n\n###### Fish Farming News\n\n800-989-5253\n\nwww.fish-news.com\/ffn.htm\n\nInformation on the aquaculture business; 7 issues a year\n\n###### Goat Rancher\n\n888-562-9529\n\nwww.goatrancher.com\n\nInformation on meat goats; monthly\n\n###### The Maine Organic Farmer & Gardener\n\nMaine Organic Farmers and Gardeners Association\n\n207-568-4142\n\nwww.mofga.org\n\nOrganic agriculture; quarterly\n\n###### Mother Earth News\n\nOgden Publications\n\n800-234-3368\n\nwww.motherearthnews.com\n\nI am actually recommending the old issues, pre-1986, for useful information on small farm homesteading. The later issues are more environmentally focused and have less information applicable to farmers.\n\n###### Mules and More\n\n573-646-3934\n\nwww.mulesandmore.com\n\nInformation on mules; monthly\n\n###### The Natural Farmer\n\nNortheast Organic Farmers Association\n\nwww.nofa.org\/tnf\/index.php\n\nOrganic farming information; quarterly\n\n###### North American Elk\n\nNorth American Elk Breeders Association\n\n402-756-3355\n\nwww.naelk.org\n\nElk information; bimonthly\n\n###### Organic Gardening\n\nwww.organicgardening.com\n\nGrowing chemical-free food and flowers; bimonthly\n\n###### Poultry Press\n\n765-827-0932\n\nwww.poultrypress.com\n\nPromotes standardbred poultry; monthly\n\n###### Ranch & Rural Living\n\n325-655-4434\n\nwww.ranchmagazine.com\n\nFocuses on sheep and goats\n\n###### Rare Breeds Journal\n\n308-665-1431\n\nwww.rarebreedsjournal.com\n\nExotic and minor breeds of animals for the pet industry\n\n###### Rural Heritage\n\n319-362-3027\n\nwww.ruralheritage.com\n\nFocuses on farming and logging with draft animals; bimonthly\n\n###### Rural Property Bulletin\n\n888-327-6289\n\nwww.landandfarm.com\n\nRural property sale listings\n\n###### Small Farmer's Journal\n\n800-876-2893\n\nwww.smallfarmersjournal.com\n\nSmall farming, focusing on draft animals; quarterly\n\n###### Small Farm Today\n\n800-633-2535\n\nwww.smallfarmtoday.com\n\nHow-to information on alternative and traditional crops, livestock, and direct marketing; bimonthly\n\n###### The Stockman Grass Farmer\n\n800-748-9808\n\nwww.stockmangrassfarmer.net\n\nInformation on intensive rotational grazing; monthly\n\n###### Western Mule\n\n417-859-6853\n\nwww.westernmulemagazine.com\n\nInformation on mules; monthly\n\n#### Newsletters\n\nNewsletters are another source of information, ranging from the magazine-like Growing for Market to the university-published Sustainable Agriculture.\n\n###### Ag News & Views\n\nThe Samuel Roberts Noble Foundation\n\n580-223-5810\n\nwww.noble.org\/Ag\/news_views\n\nGeneral farm information; monthly\n\n###### American Livestock Breeds Conservancy News\n\n919-542-5704\n\nwww.albc-usa.org\n\nInformation on rare breeds; bimonthly\n\n###### Center for Rural Affairs\n\n402-687-2100\n\nwww.cfra.org\n\n###### The Cut Flower Quarterly\n\nAssociation of Specialty Cut Flower Growers\n\n440-774-2887\n\nwww.ascfg.org\n\nField and greenhouse cut-flower information; quarterly\n\n###### Growing for Market\n\n800-307-8949\n\nwww.growingformarket.com\n\nNews and ideas for market gardeners; monthly\n\n###### HortIdeas\n\n<http:\/\/users.mikrotec.com\/~gwill\/index.html>\n\nReports the latest research and tools for gardeners; bimonthly\n\n###### Leopold Letter\n\nLeopold Center for Sustainable Agriculture\n\n515-294-3711\n\nwww.leopold.iastate.edu\n\nNews and Research on sustainable agriculture; quarterly\n\n###### Sustainable Agriculture\n\nMinnesota Institute for Sustainable Agriculture\n\n800-909-6472\n\nwww.misa.umn.edu\n\nSustainable agriculture news; bimonthly\n\n### Web Sites\n\nBesides those listed below, there are many more Web sites that have good information, so do some keyword searches using topics of interest (e.g., sustainable, buffalo, organic, small farm). Use links from the sites you find to explore more options.\n\n###### Department of Animal Science\n\nOklahoma State University\n\nwww.ansi.okstate.edu\n\n###### Urban Agriculture Notes\n\nCity Farmer\n\nwww.cityfarmer.org\n\nUrban farming information\n\n###### The CyberFarm\n\nOhio State University Extension\n\n<http:\/\/extension.osu.edu>\n\nSmall farm information\n\n###### Forage Information System\n\n<http:\/\/forages.oregonstate.edu>\n\nForage information\n\n###### Greenweb\n\n<http:\/\/boldweb.com\/gw>\n\nGardening information\n\n###### Research & Extension\n\nLangston University\n\nwww.luresext.edu\n\nAquaculture and goat information\n\n###### MachineFinder\n\nwww.machinefinder.com\n\nSearch for used equipment on-line\n\n###### Minnesota Institute for Sustainable Agriculture\n\nwww.misa.umn.edu\n\nSustainable agriculture information\n\n###### Educational Materials\n\nNorth Dakota State University Cooperative Extension\n\nwww.ag.ndsu.edu\/pubs\n\nFarming information\n\n###### Openair Market Network\n\nwww.openair.org\n\nFarmers' market information\n\n###### Ostriches On Line\n\nwww.ostrich.com\n\nIncludes on-line newsletter\n\nIn addition to the Internet, there are also newsgroups and mailing lists. Newsgroup availability varies from server to server, so I recommend just looking at your available list for farm-related groups. Mailing lists are basically digests of messages sent to you via e-mail.\n\n### University Sources\n\nConsult your local extension office and university agriculture department and ask about any sustainable, value-added, or small farm programs it has available. Check if there are any extension specialists in topics you are investigating. My list here is of universities and departments that I feel are promoting small farm issues or have useful small farm information.\n\n###### California Small Farm Center\n\nUniversity of California, Davis\n\n530-752-8136\n\nwww.sfc.ucdavis.edu\n\nAgriculture and Natural Resources\n\n###### University of California\n\n510-987-0107\n\nwww.ucanr.org\n\nColorado Cooperative Extension\n\n###### Colorado State University\n\n970-491-6281\n\nwww.ext.colostate.edu\n\nIdaho College of Agricultural Life Sciences\n\n###### University of Idaho\n\n208-885-6681\n\nwww.ag.uidaho.edu\n\nNew-farmer packets\n\n###### Indiana Media Distribution Center\n\nPurdue University\n\n888-398-4636\n\nwww.ag.purdue.edu\/agcomm\n\nIowa Leopold Center for Sustainable Agriculture\n\n###### Iowa State University\n\n515-294-3711\n\nwww.leopold.iastate.edu\n\nLouisiana AgCenter Publications\n\n###### Louisiana State University\n\n225-578-4161\n\nwww.lsvagcenter.com\n\nMaryland Cooperative Extension Publications\n\n###### University of Maryland\n\n301-405-2436\n\n<http:\/\/pubs.agnr.umd.edu>\n\nMichigan Educational Materials Distribution Center\n\n###### Michigan State University Extension\n\n517-353-6740\n\nwww.emdc.msue.msu.edu\n\nMissouri Missouri Alternative Center\n\n###### University of Missouri\n\n573-882-1905\n\n<http:\/\/agebb.missouri.edu\/mac>\n\nInformation packets\n\n###### Montana Extension Publications\n\nMontana State University\n\n406-994-1750\n\nwww.msuextension.org\n\nPennsylvania Agricultural Alternative Publications\n\n###### Penn State Cooperative Extension\n\n<http:\/\/agalternatives.aers.psu.edu>\n\nFact sheets with budgets\n\n###### West Virginia Extension Service\n\nWest Virginia University\n\n304-293-5691\n\n<http:\/\/www.wvu.edu\/>~exten\/\n\nWisconsin University of Wisconsin Extension\n\n608-262-4387\n\nwww.uwex.edu\/topics\/publications\n\n### Federal Agencies\n\nIn addition to the agencies below, I recommend contacting your state Department of Agriculture to find out what offices and grants could aid your farm. For example, in Missouri we have the AgriMissouri program for adding value and labeling. Your State Department of Agriculture is also your best source for legal information.\n\n###### Alternative Farming Systems Information Center\n\nNational Agricultural Library Room\n\n301-504-6559\n\n<http:\/\/afsic.nal.usda.gov>\n\nPublishes information on farming alternatives\n\n###### Family & Small Farms\n\nNational Institute of Food and Agriculture\n\n202-720-4423\n\nwww.nifa.usda.gov\/familysmall farms.cfm\n\nPublications, fact sheets\n\n###### National Sustainable Agriculture Information Services\u2014Attra\n\nAppropriate Technology Transfer for Rural Areas (Attra)\n\n800-346-9140\n\nwww.attra.org\n\nProvides free information on sustainable agriculture topics.\n\n###### Small Business Center\n\nU.S. Chamber of Commerce\n\nwww.uschamber.com\/sb\n\nHas addresses of Small Business Development Centers and other programs.\n\n###### Sustainable Agriculture Research and Education (Sare)\n\n202-720-5384\n\nwww.sare.org\n\nResearch and grants on sustainable agriculture; four regional U.S. offices\n\n### Map Sources\n\nThere are several sources for maps of your property. Cartographic maps and platte books may be purchased from your local Usda natural Resources Conservation Service (Nrcs) agency or may be found commercially. Aerial photographic maps may be purchased from your local Farm Service Agency. Soil maps may be found at a local or state soil survey office. If you cannot find where these are located, here are national offices:\n\n###### Aerial Photography Field Office\n\nFarm Service Agency\n\n801-844-2900\n\nwww.fsa.usda.gov\n\nNational Cartography and Geospatial Center Library National Resource Conservation Service\n\n817-509-3200\n\nwww.ncgc.nrcs.usda.gov\n\nNational Soil Survey Center\n\n###### Natural Resources Conservation Service\n\n402-437-5499\n\n<http:\/\/soils.usda.gov\/about\/nssc>\n\n### Resource Lists\n\n###### The Herbal Green Pages\n\nHerb Growing and Marketing Network\n\n717-393-3295\n\nwww.herbworld.com\n\nMore than 6,000 herb-related businesses, associations, and suppliers listed\n\n###### Organic Pages Online\n\nOrganic Trade Association\n\n413-376-1213\n\nwww.theorganicpages.com\n\nMore than 1,000 organic growers, associations, suppliers, and certification groups listed\n\n###### Family & Small Farms\n\nNational Institute of Food and Agriculture\n\n202-720-4423\n\nwww.nifa.usda.gov\/familysmallfarms.cfm\n\nUseful lists of small farm programs and resources by state\n\n### Seed and Plant Catalogs\n\nThere are hundreds of seed companies. These are some I like.\n\n###### Abundant Life Seed Foundation\n\nSaginaw, Oregon\n\n541-767-9606\n\nwww.abundantlifeseeds.com\n\nOpen-pollinated, rare, and heirloom seeds\n\n###### Albert Lea Seed House\n\nAlbert Lea, Minnesota\n\n800-352-5247\n\nwww.alseed.com\n\nField crop and pasture seed\n\n###### Bountiful Gardens\n\nWillits, California\n\n707-459-6410\n\nwww.bountifulgardens.org\n\nOrganic vegetables, cover crops, and information\n\n###### E & R Seed\n\nMonroe, Indiana\n\n260-692-6827\n\nVegetables, flowers, and supplies\n\n###### Fedco Seeds\n\nWaterville, Maine\n\n207-873-7333\n\nwww.fedcoseeds.com\n\nVegetables, herbs, flowers, cover crops; tree and bulb catalog also available\n\n###### Filaree Farm\n\nOkanogan, Washington\n\n509-422-6940\n\nwww.filareefarm.com\n\nGarlic and supplies\n\n###### Fungi Perfecti Llc\n\nOlympia, Washington\n\n800-780-9126\n\nwww.fungi.com\n\nGourmet and medicinal mushrooms and supplies\n\n###### Irish Eyes\u2014Garden City Seeds\n\nThorp, Washington\n\n509-933-7150\n\nwww.irisheyesgardenseeds.com\n\nVegetables and supplies\n\n###### Harris Seeds\n\nRochester, New York\n\n800-544-7938\n\nwww.harrisseeds.com\n\nVegetables and flowers\n\n###### Johnny's Selected Seeds\n\nWinslow, Maine\n\n877-564-6697\n\nwww.johnnyseeds.com\n\nVegetables for market growers\n\n###### Jordan Seeds Inc.\n\nWoodbury, Minnesota\n\n651-738-3422\n\nwww.jordanseeds.com\n\nVegetables\n\n###### J.W. Jung Seed Co.\n\nRandolph, Wisconsin\n\n800-297-3123\n\nwww.jungseed.com\n\nVegetables, flowers, and berries\n\n###### Otis Twilley Seed Company\n\nHodges, South Carolina\n\n800-622-7333\n\nwww.twilleyseed.com\n\nVegetables and flowers\n\n###### Peaceful Valley Farm & Garden Supply\n\nGrass Valley, California\n\n888-784-1722\n\nwww.groworganic.com\n\nOrganic farming tools and supplies\n\nRenee's Garden\n\nFelton, California\n\n888-880-7228\n\nwww.reneesgarden.com\n\nFlowers, herbs, and vegetables\n\n###### R.H. Shumway's\n\nRandolph, Wisconsin\n\n800-342-9461\n\nwww.rhshumway.com\n\nVegetables and more\n\n###### Richters Herbs\n\nGoodwood, Ontario\n\n905-640-6677\n\nwww.richters.com\n\nHerbs and books\n\n###### Ronniger's Potato Farm\n\nAustin, Colorado\n\n877-204-8704\n\nwww.ronnigers.com\n\nRupp Seed\n\n###### Wauseon, Ohio\n\n800-700-1199\n\nwww.ruppseeds.com\n\nVegetables, pumpkins, and watermelons\n\n###### Sandhill Preservation Center\n\nCalamus, Iowa\n\n563-246-2299\n\nwww.sandhillpreservation.com\n\nAll heirloom, untreated; farmer owned, heirloom vegetables and poultry\n\n###### Seed Savers Exchange\n\nDecorah, Iowa\n\n563-382-5990\n\nwww.seedsavers.org\n\nFarmer seed exchange catalogs and yearbook\n\n###### Seeds of Change\n\nEl Guique, New Mexico\n\n888-762-7333\n\nwww.seedsofchange.com\n\nOrganic vegetables, herbs, and supplies\n\n###### Southern Exposure Seed Exchange\n\nMineral, Virginia\n\n540-894-9480\n\nwww.southernexposure.com\n\nFarmer owned; vegetables\n\n###### Stark Brothers\n\nLouisiana, Missouri\n\n800-325-4180\n\nwww.starkbros.com\n\nFruit trees and peppers only\n\n###### Territorial Seed Co.\n\nCottage Grove, Oregon\n\n800-626-0866\n\nwww.territorial-seed.com\n\nTomato Growers Supply Co.\n\n###### Fort Myers, Florida\n\n888-478-7333\n\nwww.tomatogrowers.com\n\nTomatoes and peppers only\n\n###### Totally Tomatoes\n\nRandolph, Wisconsin\n\n800-345-5977\n\nwww.totallytomato.com\n\nTomatoes and peppers only\n\n### Supplies\n\n###### A.M. Leonard, Inc.\n\nPiqua, Ohio\n\n800-543-8955\n\nwww.amleo.com\n\nGarden supplies\n\n###### Agronics\n\nAlbuquerque, New Mexico\n\n866-510-5795\n\nwww.agronicsinc.com\n\nSoil mineralization supplies\n\n###### CropKing Inc.\n\nWadsworth, Ohio\n\n800-321-5656\n\nwww.cropking.com\n\nCumberland General Store\n\n###### Alpharetta, Georgia\n\n800-334-4640\n\nwww.cumberlandgeneral.com\n\nHomesteading equipment\n\n###### Gardens Alive!\n\nLawrenceburg, Indiana\n\n513-354-1482\n\nwww.gardensalive.com\n\nOrganic fertilizers and supplies\n\n###### Gardener's Supply Company\n\nBurlington, Vermont\n\n888-833-1412\n\nwww.gardeners.com\n\nGarden supplies\n\n###### Gempler's\n\nMadison, Wisconsin\n\n800-382-8473\n\nwww.gemplers.com\n\nSupplies of all sorts, including Ipm\n\n###### Harmony Farm Supply & Nursery\n\nSebastopol, California\n\n707-823-9125\n\nwww.harmonyfarm.com\n\nOrganic crop and garden supplies\n\n###### Hummert International\n\nEarth City, Missouri\n\n800-325-3055\n\nwww.hummerts.com\n\nHorticulture supplies\n\n###### Lehman's\n\nKidron, Ohio\n\n888-438-5346\n\nwww.lehmans.com\n\nHomesteading equipment\n\n###### MidWest Plan Service\n\nIowa State University\n\nAmes, Iowa\n\n800-562-3618\n\nwww.mwps.org\n\nNasco\n\n###### Fort Atkinson, Wisconsin\n\n800-558-9595\n\nwww.enasco.com\/farmandranch\n\nFarm and ranch supplies\n\n###### Ohio Earth Foods, Inc.\n\nHartville, Ohio\n\n330-877-9356\n\nwww.ohioearthfood.com\n\nNatural fertilizer and supplies\n\n###### Peaceful Valley Farm Supply\n\nGrass Valley, California\n\n888-784-1722\n\nwww.groworganic.com\n\nNatural supplies, including irrigation\n\n###### Planet Natural\n\nBozeman, Montana\n\n800-289-6656\n\nwww.planetnatural.com\n\nOrganic and natural garden supplies\n\n###### Prohoe\n\nMunden, Kansas\n\n800-536-5450\n\nwww.prohoe.com\n\nQuality hoes\n\n###### Smith & Hawken Ltd.\n\nPueblo, Colorado\n\n800-940-1170\n\nwww.smithandhawken.com\n\nGarden supplies\n\n###### Veldsma & Sons, Inc.\n\nStockbridge, Georgia\n\n800-458-7919\n\nwww.veldsma.com\n\nSupplies for Christmas-tree growers and pumpkin marketers\n\n###### Wood-Mizer\n\nIndianapolis, Indiana\n\n800-553-0182\n\nwww.woodmizer.com\n\nPortable sawmills\n\n###### Worm's Way\n\nBloomington, Indiana\n\n800-247-9676\n\nwww.wormsway.com\n\nGarden supplies for urban farming\n\n### Associations and Organizations\n\nI am not listing very many associations, as there is one for every breed of animals and variety of crop under the sun. I have mostly picked generalized organizations that apply nationwide. For more information on associations and organizations in your area, talk to fellow farmers and the staff at your local extension office. Attend meetings and shows of local groups to increase your knowledge.\n\n###### American Farmland Trust\n\n202-331-7300\n\nwww.farmland.org\n\nOrganization dedicated to saving farmland\n\n###### American Herb Association\n\n530-265-9552\n\nwww.ahaherb.com\n\nHerb information\n\n###### American Livestock Breeds Conservancy\n\n919-542-5704\n\nwww.albc-usa.org\n\nInformation on rare and endangered breeds of livestock\n\n###### American Pastured Poultry Producers Association\n\n888-662-7772\n\nwww.apppa.org\n\nInformation on pastured poultry\n\n###### Biodynamic Farming and Gardening Association\n\n888-516-7797\n\nwww.biodynamics.com\n\nEcological farm management\n\n###### Center for Rural Affairs\n\n402-687-2100\n\nwww.cfra.org\n\nInformation on national events affecting family farmers\n\n###### Herb Growing and Marketing Network\n\n717-393-3295\n\nwww.herbworld.com\n\nHerb information\n\n###### Herb Research Foundation\n\n303-449-2265\n\nwww.herbs.org\n\nHerb research information\n\n###### Kansas Rural Center\n\n785-873-3431\n\nwww.kansasruralcenter.org\n\nSustainable farming information\n\n###### Kerr Center for Sustainable Agriculture\n\n918-647-9123\n\nwww.kerrcenter.com\n\nInformation and packets on sustainable agriculture topics\n\n###### Maine Organic Farmers and Gardeners Association\n\n207-568-4142\n\nwww.mofga.org\n\nOrganic information and factsheets\n\n###### Michael Fields Agricultural Institute\n\n262-642-3303\n\nwww.michaelfieldsaginst.org\n\nSustainable farm promotion\n\n###### Michigan Integrated Food and Farming Systems\n\n517-432-0712\n\nwww.miffs.org\n\nFarmer support\n\n###### National Christmas Tree Association\n\n636-449-5070\n\nwww.christmastree.org\n\nRepresents both farmers and retailers of Christmas trees\n\n###### Northeast Organic Farming Association\n\n203-888-5146\n\nwww.nofa.org\/tnf\/index.php\n\nOrganic information\n\n###### Practical Farmers of Iowa\n\n515-232-5661\n\nwww.practicalfarmers.org\n\nResearch for family farmers\n\n###### Samuel Roberts Noble Foundation\n\n580-223-5810\n\nwww.noble.org\n\nGeneral farm information\n\n###### Southern Sustainable Agriculture Working Group\n\n479-422-5831\n\nwww.ssawg.org\n\nSustainable Farming Association of Minnesota\n\n866-760-8732\n\nwww.sfa-mn.org\n\nSustainable farm information and newsletter\n\n###### Virginia Biological Farmers Association\n\n540-463-6363\n\nwww.vabf.org\n\nInformation on organic farming and newsletter\n\n## Index\n\nNote: Page numbers in _italic_ indicate illustrations; those in **boldface** indicate charts.\n\n**A**\n\nAccess, to farm, 34\u201335\n\nActinomycete,\n\nAdvertising, 144\u201347, _, _,\n\nAgripreneurship\n\ndefined, xi\n\npersonal qualities for,\n\nprinciples of, 13\u201315, 43\u201349, 213\u201315\n\nself-evaluation for, 9\u201311\n\nAlfisols,\n\nAmish farmers,\n\nAquaculture, , 163\u201364\n\nAssociations, for farmers, 267\u201368\n\n_Azotobacter_ ,\n\n**B**\n\nBacteria, in soil, __ ,\n\nBalance, in farming, 216\u201318\n\nBarns,\n\nBarter,\n\nBig picture principle, , 215\u201316\n\nBison,\n\nBooks, recommended, 244\u201354\n\nBudgets, sample, 168\u201370, **170\u201380**\n\nBuildings, care of,\n\nBusiness plan, 110\u201313\n\n**C**\n\nCamelids, , __\n\nCapability classes, 29\u201330\n\nCapital, , 20\u201321\n\nCatalog sales, ,\n\nCattle\n\ndefinitions for,\n\ntypes of,\n\nwater requirements for, , __ , ,\n\nClimate\n\neffects on farm type, 75\u201376\n\nevaluating, 35\u201337, __\n\nlivestock and,\n\nparasites and,\n\nCold frames,\n\nCommunity events, as markets, 134\u201335\n\nCommunity-Supported Agriculture (CSA), 132\u201334,\n\nCompetition,\n\nComputers,\n\nCooperatives,\n\nCorn\n\nwater requirements for, __ , 79\u201380\n\nas windbreak, , __\n\nCost analysis, 168\u201370, **170\u201380**\n\nCosts\n\ncontrolling,\n\nof equipment, 201\u20137, **209\u201311**\n\nplanning and, 152\u201353\n\npricing products and, 142\u201344\n\nCover crops, 66\u201370, ,\n\nCrop rotation\n\nadvantages of, , 60\u201362\n\ndefined, ,\n\nlong-term, 64\u201365, **, **\n\nas management tool, 220\u201323, __\n\nnecessity of,\n\nfor pasture, 70\u201372, __ ,\n\nplanning, 65\u201366,\n\nshort-term, 61\u201362, __\n\nCrops. _See also specific types_\n\nchoosing, , 151\u201360\n\nclasses of,\n\nproduction costs for,\n\nprofitability, **176\u201380**\n\nreserves of,\n\ntemperature and,\n\ntypes of, 165\u201368\n\nwater for, **** , 77\u201380, 150\u201351\n\nCSA, 132\u201334,\n\n**D**\n\nDebt, , 106\u20137\n\nDeer,\n\nDiseases\n\nfarm plan and,\n\nwater and,\n\nDiversity, in farming\n\nas management tool, 219\u201320\n\nniche marketing and, 123\u201324\n\nprofitability and, , , 180\u201381\n\nDogs, as enterprise,\n\n**E**\n\nEarthworms, __ ,\n\nEfficiency, , 227\u201329, __\n\nElk,\n\nEnergy sources,\n\nEnvironment, protecting, , 181\u201382\n\nEquines,\n\nEquipment. _See_ Machinery and equipment\n\nErosion control, pastures for,\n\nExperience, . _See also_ Knowledge\n\n**F**\n\nFairs, as markets, 134\u201335\n\nFamily\n\nlabor from, , 224\u201326\n\nlifestyle of, 11\u201313\n\nFarmers. _See also_ Agripreneurship\n\ndefined, 6\u20138\n\nnumber of,\n\nFarmers' markets, __ ,\n\nFarming. _See also_ Agripreneurship\n\nbalanced, , 216\u201318\n\nfuture of, 241\u201342\n\nhistory of, 233\u201337\n\nlifestyle, 11\u201313\n\nproduction types, , 155\u201356\n\nFarm plan, 17\u201319, 110\u201313\n\nFarms\n\nfamily, defined,\n\nnumber of, **** , ,\n\nreasons for owning, 4\u20135\n\nsmall, defined, 4\u20136\n\nas tourist destinations, 147\u201349, __\n\ntypes of, **** , 135\u201337, __\n\nFeed. _See also_ Pastures\n\nharvesting,\n\nstoring, 48\u201349,\n\nwinter,\n\nFences, 194\u201397, _, _\n\nFlexibility, need for, 100\u2013101\n\nFlowers,\n\nFood circles, 137\u201338\n\nFood processing, 127\u201329\n\nFruits, ,\n\nFukuoka, Masanobu,\n\nFungi, in soil, __ , 57\u201358\n\n**G**\n\nGame birds,\n\nGoals\n\nachieving,\n\nlong-term, 103\u20136\n\nmedium, 106\u20138\n\nneed for,\n\nshort-term, 108\u20139\n\nGoats, , __\n\nGovernment agencies, 261\u201362\n\nGovernment regulations, ,\n\nGrains,\n\nGrasses, perennial,\n\nGrazing, 71\u201372, __\n\nGreenhouses, 92\u201393\n\nGreen manure, 66\u201370,\n\nGroceries, as markets, 138\u201339\n\n**H**\n\nHairy vetch, , ,\n\nHand tools, 190\u201393\n\nHardiness zones, , __\n\nHealth, farming and,\n\nHealth-food stores, as markets, 138\u201339\n\nHerbs,\n\nHogs,\n\nHydrophytes,\n\n**I**\n\nInsects\n\ndeterring, with wind break,\n\nas enterprise,\n\nfarm plan and,\n\nInsurance, ,\n\nIrrigation,\n\n**K**\n\nKnowledge, gaining\n\nas goal, 103\u20134,\n\nimportance of, ,\n\nas resource, 218\u201319\n\n**L**\n\nLabor\n\nfamily, , 224\u201326\n\nmanaging, 223\u201327\n\nrequirements for, 24\u201325, **\u2013**,\n\nLand, evaluating, 28\u201330\n\nLegumes\n\ndefined,\n\nas green manure or cover crops, 67\u201369, __\n\nmowing,\n\nnitrogen and, 68\u201369, __\n\nas windbreaks,\n\nLifestyle, farming, 11\u201313\n\nLivestock. _See also specific types_\n\ncare of, , 94\u201396, __\n\nchoosing, 151\u201360\n\nclimate and,\n\nas energy source, , 187\u201388, __ , , ****\n\nfeed for, 48\u201349,\n\nfeeding in rotations, 70\u201372, __\n\ngentle, 46\u201347\n\nparasites in,\n\nproduction costs for,\n\nprofitability of, **170\u201376**\n\nreserves of,\n\nshelter for, 81\u201383, _, _\n\ntypes of, 160\u201365\n\nwater for, 76\u201377, , 150\u201351\n\nLocation, of farm, 34\u201335\n\nLow inputs, 44\u201345\n\n**M**\n\nMachinery and equipment\n\nacquisition options, 199\u2013200\n\nbuying, 207\u20138\n\ncare of,\n\ncosts of, 201\u20137, **209\u201311**\n\ndefined,\n\ngoals for,\n\nrequirements for, 37\u201339, , 186\u201387, 198\u201399,\n\nshelters for, ,\n\ntypes of, 187\u201398, __ , ****\n\nwinterizing, 202\u20133\n\nMammals, small,\n\nManagement\n\nfarm plan and, 110\u201311\n\nof labor, 223\u201327\n\nprinciples of, 213\u201315,\n\ntools for, 219\u201323\n\nManure, , 66\u201370, ,\n\nMaps, sources for,\n\nMarket groups,\n\nMarketing strategies\n\ndirect sales, 129\u201338\n\nmarket identification, 116\u201318, __\n\nniche marketing, 118\u201324\n\nplanning, ,\n\nretail sales, 140\u201341\n\ntraditional vs. alternative,\n\nwholesale sales, 138\u201340\n\nMesophytes,\n\nMetric converstion chart, ****\n\nMicroclimates, creating, , 86\u201387\n\nMollisols,\n\nMoney, reserves of,\n\nMotors, small,\n\nMowing, pasture,\n\nMycorrhiza,\n\n**N**\n\nNight crawlers _(Lumbricus terrestris),_\n\nNitrogen, 68\u201369\n\n**O**\n\nOpportunity cost, defined,\n\nOrganic certification, ,\n\nOrganizations, for farmers, 267\u201368\n\n**P**\n\nParasites, water and,\n\nPastures\n\nas enterprise,\n\nfor erosion control,\n\nrotations of, 70\u201372, __ ,\n\nPeriodicals, recommended, 254\u201358\n\nPermits,\n\nPlanning. _See also_ Farm plan; Resource evaluation\n\ncrop rotation, 65\u201366\n\nenterprises, 151\u201360\n\nexamples of, 100\u2013102\n\nfarm type, 17\u201319, ****\n\nneed for,\n\nas principle of agripreneurship,\n\nPonds, 84\u201385\n\nPotatoes,\n\nPoultry\n\nshelters for, 82\u201383, __\n\ntypes of, , __\n\nPredators, shelters from, 82\u201383, __\n\nPricing, 142\u201344,\n\nProduction methods, , , 155\u201356\n\nProfitability\n\nattaining, 239\u201340\n\nas goal, 104\u20138\n\nplanning and, , 152\u201353,\n\nas principle of agripreneurship, 13\u201314\n\nsample budgets for, 168\u201370, **170\u201380**\n\nsustainability and,\n\nProperty, evaluating, 28\u201330\n\n**R**\n\nRabbits,\n\nRainfall. _See also_ Water\n\namount of, , __\n\nconserving, 84\u201385\n\nRaised beds, ,\n\nRatites,\n\nRecord keeping\n\nas goal,\n\nmicroclimates and,\n\nplanning and,\n\npricing products and, 142\u201343\n\nas principle of agripreneurship, 45\u201346\n\nRed clover,\n\nReptiles, as enterprise,\n\nResearch, , ,\n\nReserves, building, 229\u201331\n\nResource evaluation\n\ncapital, , 20\u201321\n\nclimate, 35\u201337, __\n\nequipment, 37\u201339,\n\nlabor, 24\u201325, **26\u201327**\n\nland, 28\u201330\n\nlocation\/access, 34\u201335\n\nskills, 9\u201310, , **22\u201323**\n\nsoil, 30\u201331\n\nwater, 32\u201333, _, _\n\nResource lists, 262\u201363\n\nResource management, 215\u201318\n\nRestaurants, as markets, 139\u201340\n\n_Rhizobia_ , ,\n\nRisk management, 105\u20136,\n\nRoadside stands, 131\u201332, ,\n\nRoute sales,\n\nRye, , ,\n\n**S**\n\nSeasons, extending, 92\u201394\n\nSeed catalogs, 263\u201365\n\nSheep\n\ntypes of,\n\nwater requirements for,\n\nShelters\n\nfor livestock, 81\u201383, _, _\n\nfor machinery and equipment, ,\n\nSigns, roadside, , __\n\nSkills\n\nevaluating, 9\u201310, , **22\u201323**\n\nnecessary, 46\u201347\n\nSocial acceptance, of farming, , 181\u201382\n\nSoil\n\nevaluating, 30\u201331, ,\n\norganisms in, 55\u201358, _, _,\n\nphysical nature of, 52\u201354\n\nplant roots and, 58\u201359, __\n\nworking with, 54\u201355, __\n\nSoil enhancers,\n\nSoil fertility\n\nas goal, , ,\n\nmaintaining,\n\nas principle of agripreneurship, 47\u201348\n\nSolar energy,\n\nSpecial events, as advertising, 147\u201349\n\nStrip-cropping,\n\nSupply sources, 265\u201367\n\nSustainable agriculture\n\ndefined, xi\u2013xii, ,\n\nvs. other methods,\n\nprinciples of, , , , 181\u201382, 213\u201315\n\n**T**\n\nTanks, 84\u201385\n\nTemperature, crop choices and,\n\nTimber,\n\nTiming, , , 153\u201354,\n\nTools. _See_ Machinery and equipment\n\nTractors, 188\u201390,\n\nTrade shows, as markets, 134\u201335\n\nTrees, as windbreaks, 89\u201390\n\n**U**\n\nUltisols,\n\nUniversity sources, 260\u201361\n\nU-pick farms, 135\u201337, __ , ,\n\n**V**\n\nValue-added products\n\nbenefits of,\n\ndefined,\n\nequipment for, 193\u201394\n\nin farm plan, 111\u201313\n\nmarketing, , ,\n\nproducing, 127\u201329\n\ntypes of, 125\u201326, __\n\nVegetables, 165\u201366\n\nVermiculture,\n\nVeterinarians,\n\n**W**\n\nWater\n\nconserving, , 84\u201385, ,\n\nas energy source,\n\nevaluating, 31\u201333, **** , _, _, 151\u201352\n\nfor livestock, 76\u201377,\n\nfor plants, 77\u201380\n\nWeather, farm plan and, . _See also_ Climate\n\nWeb sites\n\nadvertising with,\n\nas resources, 258\u201359\n\nWeeds, farm plan and,\n\nWindbreaks\n\nbenefits of, , 87\u201389, __\n\ntypes of, 89\u201391\n\nWinter\n\nlivestock in, 94\u201396, __\n\nmachinery and equipment in, 202\u20133\n\nWorms\n\nas enterprise,\n\nin soil, , __\n\n**X**\n\nXerophytes, \n\n## Other Storey Titles You Will Enjoy\n\nChicken Coops, by Judy Pangman.\n\nA collection of hen hideaways to spark your imagination and inspire you to begin building.\n\n180 pages. Paper. Isbn 978-1-58017-627-9. Hardcover. Isbn 978-1-58017-631-6.\n\nEquipping Your Horse Farm, by Cherry Hill and Richard Klimesh.\n\nA guide to every aspect of selecting, maintaining, and operating essential equipment for your horse operation.\n\n192 pages. Paper. Isbn 978-1-58017-843-3. Hardcover. Isbn 978-1-58017-844-0.\n\nThe Gardener's A-Z Guide to Growing Organic Food, by Tanya L. K. Denckla.\n\nAn invaluable resource about growing, harvesting, and storing for 765 varieties of vegetables, fruits, herbs, and nuts.\n\n496 pages. Paper. Isbn 978-1-58017-370-4.\n\nGrass-Fed Cattle, by Julius Ruechel.\n\nThe first complete manual in raising, caring for, and marketing grass-fed cattle.\n\n384 pages. Paper. Isbn 978-1-58017-605-7.\n\nSmall-Scale Livestock Farming, by Carol Ekarius.\n\nA natural, organic approach to livestock management to produce healthier animals, reduce feed and health care costs, and maximize profit.\n\n224 pages. Paper. Isbn 978-1-58017-162-5.\n\nStarting & Running Your Own Small Farm Business, by Sarah Beth Aubrey.\n\nA business-savvy reference that covers everything from writing a business plan and applying for loans to marketing your farm-fresh goods.\n\n176 pages. Paper. Isbn 978-1-58017-697-2.\n\nStorey's Barn Guide to Sheep.\n\nStep-by-step visuals for all aspects of sheep care in a handy, hanging format.\n\n96 pages. Paper with concealed wire-o binding. Isbn 978-1-58017-849-5.\n\nThe Vegetable Gardener's Bible, by Edward C. Smith.\n\nA reinvention of vegetable gardening that shows how to have your most successful garden ever.\n\n320 pages. Paper. Isbn 978-1-58017-212-7.\n\nThe Weather-Resilient Garden, by Charles W. G. Smith.\n\nA defensive approach to landscaping, plus emergency and long-term solutions for reviving weather-damaged landscapes.\n\n416 pages. Paper. Isbn 978-1-58017-516-6.\n\nThese and other books from Storey Publishing are available wherever quality books are sold or by calling 1-800-441-5700. Visit us at www.storey.com.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n# **COLOMBIA**\n\nANDREW DIER\n\n## **Contents**\n\n**Index**\n\n**List of Maps**\n\n**Discover Colombia**\n\n**Bogot\u00e1**\n\n**Cartagena and the Caribbean Coast**\n\n**Boyac\u00e1 and the Santanderes**\n\n**Medell\u00edn and the Coffee Region**\n\n**Cali and Southwest Colombia**\n\n**The Pacific Coast**\n\n**San Andr\u00e9s and Providencia**\n\n**The Amazon and Los Llanos**\n\n**Background**\n\n**Essentials**\n\n**Resources**\n\n  |    \n---|---\n\n## [**DISCOVER  \nColombia**](toc.html#ch_00)\n\nPlanning Your Trip\n\nIF YOU HAVE...\n\nThe Best of Colombia\n\nJET-SETTING\n\nMUSIC AND DANCE FESTIVALS\n\nMOUNTAIN HIGHS\n\nThe Wild Coasts\n\nColonial Towns and Countryside\n\nIdyllic colonial towns, fast-paced cities, stunning archaeological sites, jaw-dropping scenery, and secluded beaches should be enough to put Colombia on your must-see list. But what really seals the deal is the contagious _alegr\u00eda_ (happiness) of the people you will meet along the way.\n\nBogot\u00e1 and Medell\u00edn offer all the architecture, culture, restaurants, and nightlife that you would expect in any major world city. For urbanites, these are destinations in their own right. But they are also excellent bases to enjoy creature comforts while organizing trips throughout the country.\n\n  |    \n---|---\n\nCartagena and the towns of Villa de Leyva and Barichara will transport you back to the 18th century, when citizens were ruled by a king in Madrid. Cartagena never fails to seduce those who stroll along its narrow, cobblestone streets adorned by bougainvillea cascading from balconies above.\n\nThe Andes Mountains combined with Colombia's tropical location mean that every possible natural setting is within reach. The Sierra Nevada del Cocuy, virtually unknown outside of Colombia, offers incredible trekking amid glaciers and snowcapped peaks. Coffee farms are nestled in verdant valleys abundant with orchids. Los Llanos, Colombia's eastern plains, and the Amazon basin are tropical wonderlands, with innumerable opportunities for nature and wildlife viewing.\n\nEven the beaches here are varied. The Caribbean Coast features gems like Parque Nacional Natural Tayrona, where glacier-fed streams flow from snowcapped mountains into the Caribbean Sea. The Pacific Coast offers solitude and a chance to spot humpback whales breaching. And, far from the mainland in the Caribbean, the islands of San Andr\u00e9s and Providencia are a sultry respite from the rush of city living.\n\nColombia is one of the most biodiverse countries on the planet. Its people are just as diverse. Beyond differences in language, dialects, and accents, you can tell where someone is from by the songs that they sing, the instruments that they play, and the dances to which they move. In Cali, salsa, with its the fancy footwork, color, and brass instruments, is nothing less than an obsession. In the Llanos, _joropo_ is a tribute to Colombian cowboy tradition.\n\nA remarkable transformation has occurred in this corner of South America. And now is a great time to experience this change. Colombia has laid out its welcome mat and beckons: _\u00a1Bienvenidos!_\n\n  |    \n---|---\n\n### **Planning Your Trip**\n\n#### **Where to Go**\n\n##### **Bogot\u00e1**\n\nAgainst the backdrop of the Andes Mountains, the country's cool capital is a **cosmopolitan melting pot.** It's a city of stunning colonial and modern **architecture,** art and culture, **glitzy shopping, five-star dining,** and **euphoric nightlife.**\n\n##### **Cartagena and the Caribbean Coast**\n\nThe Caribbean coastline runs the gamut from the eerie **desert landscapes** of **La Guajira** in the far north to the **untamed jungles** near **Capurgan\u00e1** along the Panamanian border. In between are the tropical jungles and mystical **Ciudad Perdida** of the **Sierra Nevada de Santa Marta,** as well as Cartagena, the seductive **colonial jewel** of the Caribbean.\n\n##### **Boyac\u00e1 and the Santanderes**\n\nCradle of Colombian independence, the departments of Santander and Boyac\u00e1 are graced with **stunning countryside,** from the awe-inspiring **Ca\u00f1\u00f3n del Chicamocha** to the snowcapped peaks of the **Sierra Nevada del Cocuy. San Gil** is the outdoor adventure capital, while nearby **Barichara** is one of the most beautiful **colonial pueblos** in the country. The sacred **Laguna Iguaque** and the nearby town of **Villa de Leyva,** with its serene whitewashed buildings and cobblestone streets, are truly picturesque.\n\n**IF YOU HAVE...**\n\n\u2022 **ONE WEEK:** Visit Medell\u00edn, stay at a coffee farm, and fly to Cartagena.\n\n\u2022 **TWO WEEKS:** Add the Caribbean Coast, then Bogot\u00e1 and Villa de Leyva.\n\n\u2022 **THREE WEEKS:** Add Cali, Popay\u00e1n, and Parque Nacional Natural Isla Gorgona.\n\n\u2022 **FOUR WEEKS:** Add an excursion to the Amazon or Los Llanos.\n\n##### **Medell\u00edn and the Coffee Region**\n\nAmbitious Medell\u00edn is known for its **temperate climate** and **fun nightlife.** For a break from the city, the **Reserva Natural R\u00edo Claro** makes a fantastic midweek distraction. Photogenic Paisa pueblos abound, with **Jard\u00edn, Jeric\u00f3, Salamina,** and **Salento** some of the most colorful. Stay at one of countless **coffee haciendas** in the lush rolling hills. The landscape is dotted with towering wax palms and brightly colored _barranquero_ birds. The snow-covered volcanic peaks of **Parque Nacional Natural Los Nevados** beckon mountain climbers.\n\n##### **Cali and Southwest Colombia**\n\nColombia's third-largest city is a joyous one of music and dance. When the sun goes down it's hard to resist Cali's hypnotic salsa rhythms. To the west, beyond the **endless sugarcane fields** of the **Valle de Cauca,** stands the White City of **Popay\u00e1n,** a **historic colonial city** of presidents and poets. It makes a great base from which to explore the _p\u00e1ramos_ of the **Parque Nacional Natural Purac\u00e9** and, beyond that, the mysterious archaeological sites of **Tierradentro** and **San Agust\u00edn.** Under the looming shadow of **Volc\u00e1n Galeras, Pasto** is known for its raucous Carnaval de Negros y Blancos in January. In the **bucolic countryside** south toward Ecuador are **mountains** and **emerald-green lakes.**\n\nIglesia La Ermita, Cali\n\n##### **The Pacific Coast**\n\nThe Pacific is Colombia's wild coast, where the thick jungles of **Choc\u00f3** meet the beaches and endless ocean at wonderfully remote **Bah\u00eda Solano** and **Nuqu\u00ed.** Warm Pacific waters are a playground for **humpback whales** that spend August through October here. Sea turtles are return visitors, too. **Parque Nacional Natural Isla Gorgona** is simply spectacular, an island home to endemic species such as the blue anole lizard. Serious divers will want to make the journey to **Santuario de Flora y Fauna Malpelo,** where schools of hammerhead sharks slowly circle.\n\n##### **San Andr\u00e9s and Providencia**\n\nThe paradises of English-speaking San Andr\u00e9s and Providencia offer everything you'd expect from a Caribbean island vacation. **Fantastic diving** will keep you occupied for days off of sunny San Andr\u00e9s. The daily routine of lounging on **remote beaches,** eating fresh seafood, lazing in hammocks, and **stargazing** on the beach in perfect Providencia will have you hooked.\n\n##### **The Amazon and Los Llanos**\n\nThe Amazon **rainforest** is the lungs of the world. Visit an eco-lodge on the **R\u00edo Yavar\u00ed,** where you can take canoe rides above the treetops in the flooded jungle. Observe birds and pink dolphins by day and look for alligators as darkness falls. Spend a couple of days in the Ticuna village of **San Mart\u00edn** and enjoy the blissfully car-free hamlet of **Puerto Nari\u00f1o.** In Los Llanos, take in the astonishing wildlife at **Hacienda La Aurora.** Bathe in the **multicolored waters** of natural wonder Ca\u00f1o Cristales.\n\nthe Colombian Amazon\n\n#### **When to Go**\n\nBecause Colombia straddles the equator, the temperatures and length of days are nearly constant year-round. There are, however, distinct dry and rainy seasons. Throughout most of the country, **December through February** and **July through August** are considered **_verano_ (dry season). _Invierno_ (rainy season)** is usually between **April and May** and again between **September and November.**\n\nIn San Andr\u00e9s and Providencia, June through November is rainy and from February through April it's drier. In the Amazon the drier months are between June and September and the rainy season is December through May. It's worth a visit during either season. In the Pacific coast region, it rains year-round.\n\n**High tourist seasons** run from **mid-December through mid-January,** during **Easter week (Semana Santa)** and, to a lesser extent, school vacations from **June to August.** During high season, hotel rates and airline ticket prices soar. Colombians from the interior flock to the Caribbean coast during the New Year's holidays, creating a party atmosphere. In contrast, Bogot\u00e1 becomes a ghost town during the major holidays. Hotels and flights can also get booked up on the 10 or so _puentes_ (long weekends) of the year.\n\nMany of the major **festivals and celebrations** take place between December and February: the Feria de Cali, the Carnaval de Negros y Blancos in Pasto, Hay Festival in Cartagena, and the Carnaval de Barranquilla. **Easter week** celebrations are big in colonial cities such as Popay\u00e1n, Momp\u00f3x, Pamplona, and Tunja, while during that time every two years Bogot\u00e1 puts on the Festival Iberoamericano de Teatro. **Humpback whales** make their appearance off the Pacific coast from **July to October.**\n\nCa\u00f1o Cristales in Los Llanos\n\n#### **Before You Go**\n\n##### **Passports and Visas**\n\nTravelers to Colombia who intend to visit as tourists for a period of under 90 days will need only to present a **valid passport** upon entry in the country. You may be asked to show **proof of a return ticket.** Tell the immigration officer if you intend to stay up to 90 days, otherwise they will probably give you a stamp permitting a stay of 60 days. Language schools and universities will be able to assist those who may require a year-long **student visa.**\n\n##### **Vaccinations**\n\nThere are **no obligatory vaccination requirements** for visiting Colombia. However, proof of the **yellow fever vaccine** may be requested upon arrival at the Parque Nacional Natural Tayrona or at the Leticia airport in the Amazon. This vaccination can be obtained at Red Cross clinics throughout the country. If you are traveling onward to countries such as Brazil, Ecuador, or Peru, you may have to provide proof of the vaccine upon entry to those countries.\n\nThe Centers for Disease Control and Prevention (CDC) recommends travelers to have all of the **basic vaccinations** updated. In addition, for most travelers to Colombia, the CDC recommends the **hepatitis A** and **typhoid** vaccinations. **Hepatitis B, rabies,** and **yellow fever** vaccinations are recommended for some travelers. If you plan to go to the Amazon region, **antimalarial drugs** may be recommended.\n\n##### **Transportation**\n\nMost travelers arrive by plane to Colombia, with the vast majority arriving at the **Aeropuerto Internacional El Dorado** in Bogot\u00e1. There are numerous nonstop international flights into Bogot\u00e1 from the eastern seaboard of the United States and one flight from Toronto. From Florida and New York there are **nonstop flights** to **Cartagena** and **Barranquilla.** There are flights from Florida to **Medell\u00edn, Armenia,** and **Cali.**\n\nThere are **overland border entries** from **Venezuela** (into C\u00facuta) and **Ecuador** (to Ipiales) and **by boat** from **Peru** or **Brazil** to the Amazonian port of Leticia and from **Panama** to Capurgan\u00e1 or Cartagena.\n\n**Intra-country flights** are easy, safe, increasingly more economical, and, above all, quick. Taking the **bus** to just about anywhere in the country is an inexpensive, popular, and slower option. **Renting a car** is a viable option in the **coffee region** where roads are good. In the **major cities,** there are extensive **rapid bus networks,** and in **Medell\u00edn** there is a clean and efficient **Metro. Private buses** and **taxis** are ubiquitous in cities, although cabs should be ordered by phone. The best way to see the sights of most cities is usually **on foot.**\n\n##### **What to Take**\n\nFor those interested in jungle exploration, **waterproof hiking boots** and possibly some **collapsible trekking poles** are musts. If going to the Amazon, Capurgan\u00e1, or to the Pacific coast, a **waterproof camera bag** and **silica gel** may prevent the heartache of a ruined camera. For caving, visiting the tombs of Tierradentro, and for finding your way at night, a **small flashlight** will be of great use and comfort. To spot humpbacks, anacondas, and birds, **binoculars** will be great to have. If you plan on spending much time on the coast, bringing your own **snorkeling gear** is a good idea.\n\nsunset in Puerto Nari\u00f1o, the Amazon\n\nTo protect against the sun, pack a **wide-brimmed hat** ; against the rain, a **lightweight rain jacket** and **compact umbrella** ; against the cold, a sweatshirt and lightweight sweaters; against mosquitoes, lightweight and light-colored long-sleeved shirts and some strong repellent. For long bus rides, earplugs, eye masks, and **luggage locks** will make the trip more relaxing. Finally, a **Latin American Spanish dictionary** will help you get your point across and make friends.\n\nCasual attire is fine at most restaurants, theaters, and religious venues. Some restaurants in Bogot\u00e1 and Cartagena may expect more of an effort. In large cities, you'll want to dress to impress in bars and clubs. Shorts are generally frowned upon in interior cities.\n\n### **The Best of Colombia**\n\nThere's not one clear-cut, common way to visit uncommon Colombia. For a first-time visit, a tour of the coffee region and Cartagena on the Caribbean coast will be a beautiful and easy week-long introduction to this fascinating country. In two weeks, you can squeeze in Medell\u00edn, a Paisa pueblo or two, the sublime colonial town of Villa de Leyva in Boyac\u00e1, and cap it all off with a weekend in fast-paced Bogot\u00e1. With a third week, you can explore southwest Colombia and salsa the night away in Cali.\n\n##### **Medell\u00edn and the Coffee Region**\n\n###### **DAY 1**\n\nArrive in the evening at the Aeropuerto Internacional Jos\u00e9 Mar\u00eda C\u00f3rdova in Rionegro, outside of Medell\u00edn. Make the one-hour trip via cab or bus into town. Get settled at the no-nonsense **Hotel Ibis,** across from the **Museo de Arte Moderno de Medell\u00edn,** or at the friendly **Urban Buddha** hostel in the leafy Laureles neighborhood.\n\nHead to the always-lively Parque Lleras area of the Poblado neighborhood. Familiarize yourself with Colombian cuisine at **Mondongo's,** then have a beer at the coolest corner store in town, **El Social Tienda Mixta.**\n\nMedell\u00edn at night\n\n###### **DAY 2**\n\nDiscover downtown Medell\u00edn by taking a ride on the Metro. Here you can check out the finest art museum in the region, the **Museo de Antioquia,** and have your picture taken in front of your favorite rotund Fernando Botero sculpture in the adjacent plaza.\n\nHop on the Metro again to see symbols of the new Medell\u00edn: the Metrocable gondola network and the **Biblioteca Espa\u00f1a,** a boldly designed public library built on the side of a mountain. From there, transfer once more to another Metrocable line to the **Parque Arv\u00ed,** a huge recreational area.\n\nHead back to your hotel and freshen up before checking out a salsa or tango bar after dinner.\n\n###### **DAY 3**\n\nTake the three-hour bus ride through the southern Antioquian countryside to the picture-perfect Paisa town of **Jard\u00edn.** Hang out with the locals in the sublime **Parque Principal,** a park bursting with flowers. Explore the neo-gothic **Bas\u00edlica Menor de la Inmaculada Concepci\u00f3n.** Nurse a beer or sip a hot _tinto_ at one of the park's _tiendas_ (shops).\n\nRelax at the low-key hostel **Casa Selva y Caf\u00e9,** a pleasant walk away from the town center. Birding and nature enthusiasts will want to stay at **La Esperanza.**\n\n###### **DAY 4**\n\nSet off for the coffee region by heading to **Manizales** in the morning on a five-hour bus ride. Once in town, have a coffee under the shadow of the remarkable **El Cable** tower, a gondola system that once transported coffee over the mountains to the R\u00edo Magdalena.\n\nCheck in to a **coffee farm** in the valleys near Chinchin\u00e1 in the late afternoon. The folks from **Hacienda Venecia** or **Hacienda Guayabal,** only about a half hour away, can pick you up in town.\n\nverdant valleys in the coffee region\n\n###### **DAY 5**\n\nTake a tour of a coffee farm today, and admire the orderly rows of deep green coffee plants adorned with bright red beans. Cap it off with a cup of 100 percent Colombian\u2014served black. It's called _tinto._ In the afternoon, take a bus to one of the region's cutest pueblos, **Salento,** a five-hour trip.\n\nStay at the bright orange **Tralala** hostel and have dinner at wonderful **La Eliana.**\n\n###### **DAY 6**\n\nWalk through pasture land and tropical forest of the Valle de Cocora to the **Reserva Acaime,** where you can watch the hummingbirds flit about at the feeders while you warm up with a _tinto_ (coffee). Then head back down through a wonderland of 60-meter-high (200-foot-high) wax palms, Colombia's national tree, in the Valle de Cocora.\n\nSpend the night again in Salento. Before you retire for the night, stroll the atmospheric **Calle Real.**\n\n**Jet-Setting**\n\nParque Nacional Natural Tayrona\n\nIf you don't have the time for a long overland trip, consider flying to these destinations from the major cities.\n\nGetaways close to airports in larger cities such as Santa Marta, Neiva, Monter\u00eda, and Yopal are served by various major airlines. Check smaller airlines such as **Satena** (Colombian toll-free tel. 01\/800-091-2034, www.satena.com) or **Aerol\u00ednea de Antioquia** (Colombian toll-free tel. 01\/800-051-4232, www.ada-aero.com) for the smaller destinations.\n\n**FROM BOGOT\u00c1**\n\n\u2022 **Hacienda La Aurora:** See wildlife like never before at this cattle ranch and nature reserve in the Llanos. (Airport: Yopal)\n\n\u2022 **Ca\u00f1o Cristales:** Discover the secret of the multicolored waters at this serene national park in the southern Llanos. (Airport: La Macarena)\n\n\u2022 **San Agust\u00edn:** You'll be blown away by Colombia's best-preserved archaeological site, nestled in the rolling mountains south of Neiva. (Airport: Neiva)\n\n\u2022 **Parque Nacional Natural Tayrona:** Enjoy a quick beach break at the spectacular PNN Tayrona. Be sure to save some time for a jungle hike to **El Pueblito,** remnants of a former Tayrona settlement high in the mountains. (Airport: Santa Marta)\n\n**FROM MEDELL\u00cdN**\n\n\u2022 **Pacific Coast:** Breathtaking sunsets, remote beaches, and the chance to see visiting humpback whales make for a memorable getaway from Medell\u00edn. (Airport: Bah\u00eda Solano or Nuqu\u00ed)\n\n\u2022 **Capurgan\u00e1:** On the Panamanian border, Capurgan\u00e1 is a Darien Gap outpost, home to tropical rainforest and wild beach landscapes. (Airport: Acand\u00ed or Capurgan\u00e1)\n\n\u2022 **Reserva Natural Viento Solar:** Rejuvenate at this nature reserve set in tropical dry forest along a strip of forgotten coast north of Monter\u00eda. (Airport: Monter\u00eda or Tol\u00fa)\n\n**FROM CALI**\n\n\u2022 **Parque Nacional Natural Isla Gorgona:** Colombia's version of the Galapagos Islands was also once its Alcatraz. You'll have this incredible island park almost all to yourself. (Airport: Guapi)\n\n##### **Cartagena and the Caribbean Coast**\n\n###### **DAY 1**\n\nFrom nearby Pereira, fly to **Cartagena,** 1.5 hours away. Once you land and change into the airy attire standard for the sultry city, get to know the area by taking a stroll on the massive ramparts protecting the city.\n\nHave dinner at **La Vitrola,** a Cartagena classic. Spend the night at the **Hotel Sofitel Legend Santa Clara** or the **Blue House** hostel, a good budget option.\n\n###### **DAY 2**\n\nWalk the Old City streets, getting lost and found again as you amble from the divine **Plaza de Bol\u00edvar** to the **Plaza Santo Domingo** to **Las Murallas,** the city's walls. Be sure to check out the impressive **Castillo de San Felipe** in the late afternoon.\n\nBougainvillea spills onto the streets of Cartagena's Old City.\n\nDon't miss out on the old-style fun at **Caf\u00e9 Havana,** and be sure to try one of their famous rum drinks. Go for pizza afterward at nearby **Pavia** before returning to your hotel for the night.\n\n###### **DAY 3**\n\nFor a change of pace, take a cab or bus to **Bocagrande,** Cartagena's version of Miami Beach. A walk along the bay in the Castillo Grande district is a fine way to pass the day.\n\nSpend some time in the hip and happening area of **Getseman\u00ed,** a neighborhood of tapas bars and watering holes. For inventive local cuisine, try **La Cocina de Pepina.**\n\n###### **DAYS 4-6**\n\nFrom the Muelle Tur\u00edstico in town, take a boat to the beaches of **Islas del Rosario,** the area's finest beaches, and spend a couple of nights at an island hotel. It's worth splurging for the **Coralina Isla Boutique.**\n\nReturn to Cartagena in the late afternoon and take one last walk on the walls, enjoying a cocktail at **Caf\u00e9 del Mar** and dinner at **La Cevicher\u00eda.**\n\n###### **EXCURSION TO LA GUAJIRA**\n\nIf you have 4-5 extra days, consider taking a trip to **La Guajira,** a desert peninsula that is home to the Way\u00fau people.\n\nStart at the beach in **Palomino** or in **Riohacha,** the departmental capital of La Guajira, and join up with an organized tour group. After a dusty ride past countless cacti and lonely goats, you'll arrive at **Cabo de la Vela,** where you can take a dip in the Caribbean Sea or try your luck wind- or kite-surfing.\n\nThe next stop is **Punta Gallinas,** the northernmost point of South America. Spend a day or two on a photo safari of the unusual landscape of desert dunes that drop dramatically into the sea.\n\nTake a canoe trip to explore the mangroves, then share a huge, freshly prepared lobster with a friend. If there's time, check out the **Santuario de Fauna y Flora Los Flamencos** to the southwest for an early morning or late afternoon canoe ride in search of flamingos.\n\n##### **Bogot\u00e1 and Boyac\u00e1**\n\n###### **DAY 1**\n\nFly into **Bogot\u00e1.** Set in the Andes at an elevation of 2,625 meters (8,612 feet), the Colombian capital city can be especially cool, so pack some layers and an umbrella. In the late afternoon, wander the historic **Candelaria** district and visit the world-famous **Museo del Oro.** Stay at the **Casa Platypus** downtown.\n\nthe Plaza de Bol\u00edvar in Bogot\u00e1\n\n###### **DAY 2**\n\nTake a bus or hire a car for the 3.5-hour trip to the low-key pueblo of **Villa de Leyva,** one of Colombia's best preserved colonial towns, in the department of Boyac\u00e1.\n\nEnjoy the unique atmosphere in Villa de Leyva by walking its stone streets. Check out the woolen _ruanas_ (ponchos) at **Alieth Tejido Artesanal,** and if you have time, check out the **Convento del Santo Ecce Homo** in the desert nearby.\n\nStay at the **Casa Viena Hostel** or splurge at the **Hotel Plaza Mayor,** where the views are great.\n\n###### **DAY 3**\n\nVisit the **Santuario Flora y Fauna Iguaque** just outside of town and hike to the mist-shrouded **Laguna Iguaque** for some morning exhilaration. Relax in Villa de Leyva for the evening.\n\n###### **DAYS 4-5**\n\nReturn to Bogot\u00e1 and, if it's a weekend day, go to the top of the **Torre Colpatria** for an incredible 360-degree view of the massive city. Spend a night in the **Zona Rosa** and splurge on a meal at **Andr\u00e9s Carne de Res,** Colombia's most famous restaurant. Here, the line between dining and rumba gets blurred at around 8 o'clock.\n\n**Music And Dance Festivals**\n\ndancers getting ready for Carnaval in Barranquilla\n\nGet your groove on at these colorful music and dance festivals.\n\n**CALI**\n\nCale\u00f1os boast that their city is the world capital of salsa, and there's no denying that it's an integral part of daily life in Cali. The last week of the year is the **Feria de Cali,** a week-long event of salsa concerts, parties, and pageantry that takes over the city.\n\nOther festivals worth checking out are August's **Festival Mundial de Salsa,** which showcases the glitz and frenetic footwork of the dance, and the **Festival Petronio \u00c1lvarez,** a September celebration of Pacific Coast music and culture.\n\n**VILLAVICENCIO**\n\nOn the Llanos, the great eastern plains of Colombia, cowhands work on cattle ranches during the day. At night, they get out their harps and jam a Llanero form of waltz called _joropo._ During the **Torneo Internacional del Joropo** in June, musicians and dancers from across the Llanos converge on Villavicencio, participating in open-air concerts and competitions. Cowboys show their stuff in Llanero rodeos during the week-long festival.\n\n**MEDELL\u00cdN**\n\nTango has a long history in Medell\u00edn. The **Festival Internacional de Tango** is held each year in June, offering four days of free concerts and dance performances across the city.\n\n**BARRANQUILLA**\n\nColombia's favorite festival is the **Carnaval de Barranquilla,** held each year in February. _Cumbia_ , an intriguing mix of indigenous, African, and Spanish musical styles, takes center stage at this multi-day event of parades, concerts, and parties.\n\n**BOGOT\u00c1**\n\nTypical of the way this metropolis rolls, Bogot\u00e1 doesn't have just one music celebration. From July to November, the action takes place in the city's largest park, the Parque Sim\u00f3n Bol\u00edvar, during the **Festivales al Parque** series of festivals: Salsa al Parque, Jazz al Parque, Opera al Parque, and the thumping Rock al Parque. Best of all, it's free.\n\nIf the next morning is a Sunday, enjoy the city's **Ciclov\u00eda** by renting a bike and joining the thousands of Bogotanos hitting the streets for a little exercise. If it's not a Sunday, you can still stroll the streets.\n\n##### **San Agust\u00edn, Popay\u00e1n, and Cali**\n\n###### **DAYS 1-2**\n\nFly into Neiva and immediately head south to **San Agust\u00edn,** the most important archaeological site in Colombia. It's the country's version of Easter Island, set amid lush countryside in the Cordillera Central of the Andes. Take your time visiting the park: Two days should do it.\n\npre-Columbian statue in San Agust\u00edn\n\nStay within walking distance of the park at one of the cute hotels dotting the countryside.\n\n###### **DAY 3**\n\nFlag down a bus headed westward and check out the country's other archaeological site. **Tierradentro** is a series of underground burial tombs spread atop hills in the lush Valle de Cauca countryside. It's a scenic place, and you should take your time and walk the sites at a leisurely pace.\n\nStay in the village of San Andr\u00e9s de Pisimbal\u00e1 at the **La Portada** hotel.\n\n###### **DAY 4**\n\nTake the four-hour scenic bus ride to the historic White City of **Popay\u00e1n.** Wander the streets and linger in the beautiful **Parque Caldas.**\n\nStay at the **Hostel Caracol,** where friendly staff will let you take the hostel dog for a late-afternoon walk around town. Before retiring, settle into a booth at **El Sotare\u00f1o** and listen to some tango music over a cold beer.\n\n###### **DAY 5**\n\nTake the easy two-hour bus ride to Cali. In the late afternoon, enjoy the atmosphere of the **Parque San Antonio,** the best place for people-watching or enjoying the sunset.\n\nHave dinner in one of the cozy restaurants in the sloping San Antonio neighborhood, and stay there, too, at the **Ruta Sur** hostel or the **San Antonio Hotel Boutique.** Later, check out a _salsateca_ (dance club) for a truly authentic Cali experience. Get a good night's rest to prepare for your flight home tomorrow.\n\n##### **Excursions and Side Trips**\n\n###### **SAN ANDR\u00c9S AND PROVIDENCIA**\n\nIf you're looking to get away from it all, go to the island of Providencia, part of the San Andr\u00e9s Archipelago, off the coast of Nicaragua. Allot at least four days for some solid beach relaxation time.\n\nFly into San Andr\u00e9s. To get to Providencia from there, it's just a short flight. Once in Providencia, stay at the **Hotel Sirius,** where dive experts can take you to the reefs for a few underwater adventures. Add 1-2 days in San Andr\u00e9s if you're into snorkeling, diving, and drinking coco locos.\n\niguana in San Andr\u00e9s\n\n###### **THE AMAZON**\n\nThis quick but meaningful Amazon adventure will require at least four days. Leticia is the gateway to the Colombian Amazon. It's a two-hour flight from Bogot\u00e1. At the **Reserva Natural Tanimboca,** you can stay in a treehouse in the jungle, just minutes from town.\n\nFor the next couple of days, take a boat up the world's most powerful river and visit the Ticuna community of **San Mart\u00edn.** Continue onward to the decidedly eco-friendly **Puerto Nari\u00f1o.**\n\nAlternatively, you can head straight to the eco-lodges along the **R\u00edo Yavar\u00ed,** in Brazil, where you can take day and nighttime safaris, discovering the abundant life of the Amazonian rainforest and river.\n\n**Mountain Highs**\n\nColombia is a great place to conquer the longest continental mountain range in the world, the famed **Andes Mountains.** Extending from Chile and Argentina northward to Colombia and Venezuela, the Andes split into three chains at the Colombia-Ecuador border.\n\nThe highest mountains in Colombia are within about 40 kilometers of the palm-lined beaches of the Parque Nacional Natural Tayrona. This is the **Sierra Nevada de Santa Marta,** a mountain chain independent of the Andes, which comprises the world's highest coastal mountains.\n\n\u2022 **Ciudad Perdida:** Take the famed multi-day jungle hike to archeological site Ciudad Perdida (Lost City), high in the mountains of the Sierra Nevada de Santa Marta.\n\n\u2022 **Sierra Nevada del Cocuy:** Dramatic snowcapped mountains, valleys filled with armies of _frailej\u00f3n_ plants, and crystalline mountain lakes await at the stunning Sierra Nevada del Cocuy. You can spend two or three days day-hiking through the Parque Nacional Natural Cocuy, or the more adventurous can organize a six-day tour.\n\n\u2022 **Parque Nacional Natural Los Nevados:** Parque Nacional Natural Los Nevados, in the Cordillera Central, offers hikers of all abilities the opportunity to explore misty cloud forests and get glimpses of snow-covered volcanoes. Take a day tour to the park from Manizales, a one- or three-day trek toward Laguna del Ot\u00fan from Pereira, or a challenging multi-day trek from Salento to the Nevado del Tolima. Keep your eyes peeled for the iconic Andean condor.\n\n\u2022 **Parque Nacional Natural Purac\u00e9:** In Parque Nacional Natural Purac\u00e9, ambitious hikers can get up at dawn, hike through the tropical forest to the top of Volc\u00e1n del Purac\u00e9, and be back in Popay\u00e1n for dinner.\n\n\u2022 **Parque Municipal Natural Planes de San Rafael and Parque Nacional Natural Tatam\u00e1:** In the Cordillera Occidental, check out the lesser-visited Parque Municipal Natural Planes de San Rafael, a former cattle ranch that has been converted into a nature reserve. Beyond that, there's Parque Nacional Natural Tatam\u00e1, where you can see the Pacific Ocean beyond the carpet of green of the Choc\u00f3 rainforest.\n\nthe Ciudad Perdida, in the Sierra Nevada de Santa Marta\n\n\u2022 **Parque Natural Chicaque and Parque Nacional Natural Chingaza:** Within minutes of busy Bogot\u00e1 are various mountain adventures fit for day trips. Parque Natural Chicaque is a private park south of the city. Start your hike in the cold cloud forest, and within minutes the climate and natural surroundings have morphed into tropical hot country. PNN Chingaza is a serene national park of _p\u00e1ramos_ and mountain lakes, and is the source of water for eight million thirsty Bogotanos.\n\n### **The Wild Coasts**\n\nIf you have the time and an adventurous spirit, check out some of Colombia's wildest stretches of coastline, from the rocky Pacific to the Darien to the dry tropical forests of C\u00f3rdoba. Try this itinerary in September or October when the whales are frolicking in the Pacific, the waters are calm in the Caribbean, and the tourists have gone back to work and school.\n\n##### **Caribbean Coast**\n\nThis week-long itinerary takes you to lesser known points along the Caribbean coast. Be prepared to take several modes of transportation in order to get around. If you don't have a full week to spend here, prioritize a visit to Capurgan\u00e1 or Reserva Natural Viento Solar.\n\n###### **DAY 1**\n\nFrom Medell\u00edn, take the hour-long flight to the Darien Gap community of **Capurgan\u00e1.** On the Caribbean side of the isthmus, a horse-drawn taxi will take you to your jungle lodge. Stay at friendly **Caba\u00f1as Darius** or at the honeymoon-worthy **Bah\u00eda Lodge.**\n\n###### **DAY 2**\n\nWalk through the jungle, over the mountain, to the sleepy community of **Sapzurro,** paying no attention to the cacophony of annoyed howler monkeys. Continue on to Panama if you'd like: It's just a 20-minute hike away, on the other side of a steep and muddy hill. At **La Miel** on the Panama side of the border, take a dip in the calm waters.\n\npoison dart frog, Capurgan\u00e1\n\n###### **DAY 3**\n\nSpend the day diving or snorkeling around the reefs off the coastline of Capurgan\u00e1. Enjoy dinner on the beach at **Donde Josefina.**\n\nIf you have more time, contact **Posada del Gecko** to arrange a side trip to Panama's San Blas islands, where you can visit a Guna indigenous village.\n\n###### **DAY 4**\n\nTake a boat to the port of Turbo (or fly to Monter\u00eda) and hop on a bus to **Reserva Natural Viento Solar.** It's a long day of interesting traveling from the Darien, usually involving several modes of transportation. Expect to be on the road and sea for about eight hours.\n\nThis nature reserve is in **tropical dry forest** on the C\u00f3rdoba coast. Take a relaxing walk on the beach in the afternoon.\n\n###### **DAY 5**\n\nWalk through the forest surrounding Reserva Natural Viento Solar and look for impossibly cute _osos perezosos_ (sloths). Snooze in a hammock, then **kayak** in the calm waters before the sun goes down. Spend your last night at the reserve in an open-air **thatched roof hut,** lulled to sleep by the gentle breeze and rolling waves in the distance.\n\n###### **DAY 6**\n\nMake your way to the beach community of **Tol\u00fa,** with a detour to **San Antero.** Have a spectacular meal at **Pesecar,** with an even more spectacular view of Bah\u00eda de Cispat\u00e1 and the mangroves beyond.\n\nTake a mangrove cruise and visit the **Asocaiman** turtle and alligator nature refuge run by local fishers. Continue onwards to Tol\u00fa, where you can take an evening _bici-taxi_ (pedicab) tour of this charming fishing town.\n\n###### **DAY 7**\n\nIf you have more time, set sail to the **Islas de San Bernardo** and spend a night or two at a rustic or extremely swank hotel\u2014your choice! If you're there during the week, you might have your own private island. Otherwise, catch a flight at the Aeropuerto Golfo de Morrosquillo back to Medell\u00edn.\n\n##### **Pacific Coast**\n\nThe Pacific coast is different from the Caribbean side. There are few roads, and the main mode of transportation is by boat. The best way to visit the Pacific is to pick a spot and limit your time to that area. If you make your accommodations arrangements beforehand, your hostel or lodge will pick you up from the airport. Your lodge can also arrange **humpback whale-watching** trips from July to October. Seeing these great creatures is a highlight of any visit to the Pacific.\n\n###### **BAH\u00cdA SOLANO**\n\nThis town at the fringe of the jungle is a great base for any activity. Go diving or sport-fishing, or hike in the jungle or along the beach to crystal-clear swimming holes and waterfalls like the **Cascada Chocolatal.** Or hang out in orderly and walkable Bah\u00eda Solano and experience city life, Pacific style.\n\nSpend a few nights at one of Bah\u00eda Solano's hotels, like **Posada Tur\u00edstica Rocas de Cabo Marzo** or **Posada Tur\u00edstica Hostal del Mar.** The flight to Bah\u00eda Solano from Medell\u00edn takes one hour. You can even walk from the airport to your hotel.\n\n###### **EL VALLE**\n\nEl Valle boasts broad beaches and the fantastic **Estaci\u00f3n Septiembre Sea Turtle Hatchery,** where newborn sea turtles are born and released into the turbulent waters of the Pacific Ocean, as well as unforgettable sunsets over **Playa Almejal.**\n\nsunset on Playa Almejal, El Valle\n\nStay the night at one of the several fantastic beachside lodges and hostels of El Valle, or get to know the cultural life of the people of the Pacific by staying at one of the _posadas nativas_ (guesthouses owned and operated by locals) here, such as **Villa Maga** or **El Nativo.**\n\nGet to El Valle by flying into the Bah\u00eda Solano airport. El Valle is home to one of the few roads in this area, so you can hop on a _colectivo_ to reach your lodge.\n\n###### **NUQU\u00cd**\n\nFive-star eco-lodges, as well as a few economical options, abound on the coastline near Nuqu\u00ed. If you stay at one of the eco-lodges, like **El Cant\u00edl, Morromico,** or **Luna de Miel,** you'll quickly become accustomed to being pampered. Their all-inclusive packages include great seafood dinners, access to remote beaches, guided nature hikes (in search of poisonous frogs!), and day trips to Afro-Colombian villages.\n\nThe flight into Nuqu\u00ed from Medell\u00edn takes 45 minutes.\n\n###### **PARQUE NACIONAL NATURAL UTR\u00cdA**\n\nThis beautiful **national park** between Nuqu\u00ed and Bah\u00eda Solano is the perfect option for those who want to be surrounded by nature. Take a hike, discover a remote beach, or go swimming. At night, you can hunt for glow-in-the-dark mushrooms, then fall asleep to the sounds of the rainforest.\n\nParque Nacional Utr\u00eda on the Pacific coast\n\nYou can reach PNN Utr\u00eda from either the Bah\u00eda Solano or Nuqu\u00ed airport. Park staff will pick you up.\n\n### **Colonial Towns and Countryside**\n\nGorgeous countryside, historic pueblos and cities, and outdoor adventures: Get a taste of what Bogot\u00e1, Boyac\u00e1, and Santander have to offer.\n\n##### **Days 1-3**\n\nSpend a couple of days in **Bogot\u00e1** wandering the **Candelaria,** the capital city's _centro hist\u00f3rico,_ then visit the **Quinta de Bol\u00edvar,** Sim\u00f3n Bol\u00edvar's old country home. Don't miss the **Cerro de Monserrate,** a pilgrimage site with unsurpassed views of the metropolis. Hike up, then take a ride on the gondola or tram back down.\n\nLearn about Colombia's past from the time of the Muiscas to its shaky years as an independent nation in the city's excellent museums, like the extensive art galleries of the **Colecci\u00f3n de Arte del Banco de la Rep\u00fablica** and the mesmerizing **Museo del Oro.**\n\nCatedral Primada, La Candelaria district of Bogot\u00e1\n\n##### **Days 4-5**\n\nAfter setting off from Bogot\u00e1, make your way to **Tunja,** a three-hour bus ride. Tunja is a city of spectacular **colonial churches.** Spend a few hours checking them out, then head on to the nearby colonial town of **Villa de Leyva,** one of the country's most beautiful and well-preserved pueblos, just 45 minutes away.\n\nVilla de Leyva's charm is in its quiet atmosphere and lovely whitewashed colonial architecture. The **Plaza Mayor** is the top place to experience both. In the countryside nearby, check out the **Santuario Fauna y Flora Iguaque** for a half-day hike to Laguna Iguaque, which was sacred to the Muiscas. In the adjacent arid deserts, visit the lovely **Convento del Santo Ecce Homo.**\n\n##### **Day 6**\n\nTake a bus to **R\u00e1quira** and spend the day visiting the **Convento de la Candelaria,** just outside of town. Tour the complex, then head back to town to shop for handicrafts, like the city's famous clay pots.\n\n##### **Day 7**\n\nToday is a travel day. Return to Tunja to catch a bus bound for **Barichara,** which is Villa de Leyva's rival for most beautiful pueblo. Judge for yourself as you walk the stone streets of this old tobacco town in the department of Santander.\n\nStroll the famous **Camino Real** to the indigenous village of **Guane** and return to Barichara in time for the spectacular sunset. Stay at the **Color de Hormiga Hostel** and wake up to the chirping of cheerful, amazingly colorful birds.\n\n##### **Day 8**\n\nBarichara makes a great base for all sorts of outdoor adventures in and around San Gil. Spend a day hiking to waterfalls, rafting, splashing in swimming holes, or caving. Have dinner at **Gringo Mike's** in San Gil. Stay in Barichara unless you want to be in the middle of the action of San Gil.\n\n##### **Days 9-11**\n\nHead north to **Bucaramanga.** On the way there, visit the **Ca\u00f1\u00f3n del Chicamocha** and be blown away by the views. It's a 1.5-hour trip to the canyon.\n\nvivid green hills near Bucaramanga\n\nOnce you make it to Bucaramanga, chat up other world travelers in town at the **Kasa Guane.** If charm is what you seek, head to the colonial district of nearby Gir\u00f3n, checking in at the **Gir\u00f3n Chill Out Hotel Boutique** and strolling the town's old streets in the evening.\n\nOn your last morning in the area, cap things off by paragliding at **Mesa de Ruitoque** outside of Bucaramanga. If soaring above the green valleys is too much action, visit the beautiful **Jard\u00edn Bot\u00e1nico Eloy Valenzuela** in **Floridablanca** and munch on a sweet _oblea_ (wafer) in the town center afterward.\n\nFrom the Bucaramanga airport, catch a flight back to Bogot\u00e1.\n\n##### **Excursion to El Cocuy**\n\nGot more time and need some mountains to conquer? You can get to the **Sierra Nevada del Cocuy** by land directly from Tunja or from Bucaramanga, going through the highland university town of **Pamplona.** Add at least four more days for this option.\n\n## **BOGOT\u00c1**\n\nHIGHLIGHTS\n\nHISTORY\n\nPLANNING YOUR TIME\n\nSAFETY\n\nORIENTATION\n\nSights\n\nLA CANDELARIA\n\nAVENIDA JIM\u00c9NEZ\n\nCENTRO INTERNACIONAL\n\nWESTERN BOGOT\u00c1\n\nNORTHERN BOGOT\u00c1\n\nSOUTHERN BOGOT\u00c1\n\nEntertainment and Events\n\nNIGHTLIFE\n\nPERFORMING ARTS\n\nFESTIVALS AND EVENTS\n\nShopping\n\nHANDICRAFTS\n\nCLOTHES AND ACCESSORIES\n\nANTIQUES\n\nFLOWERS AND MARKETS\n\nJEWELRY\n\nBOOKS\n\nSHOPPING MALLS\n\nSports and Recreation\n\nBIKING\n\nRUNNING\n\nHIKING\n\nSOCCER\n\nTOURS\n\nAccommodations\n\nLA CANDELARIA\n\nCENTRO INTERNACIONAL\n\nWESTERN BOGOT\u00c1\n\nCHAPINERO\n\nNORTHERN BOGOT\u00c1\n\nFood\n\nLA CANDELARIA\n\nAVENIDA JIM\u00c9NEZ\n\nCENTRO INTERNATIONAL\n\nCHAPINERO\n\nNORTHERN BOGOT\u00c1\n\nInformation and Services\n\nVISITOR INFORMATION\n\nTELEPHONES\n\nPOST OFFICES AND COURIER SERVICES\n\nINTERNET CAF\u00c9S\n\nNEWSPAPERS AND MAGAZINES\n\nSPANISH LANGUAGE COURSES\n\nMONEY\n\nVISAS AND OFFICIALDOM\n\nHEALTH\n\nLAUNDRY\n\nGetting There\n\nBY AIR\n\nBY BUS\n\nGetting Around\n\nTRANSMILENIO\n\nPRIVATE BUSES\n\nTAXIS\n\nWALKING\n\nBIKING\n\nCAR RENTAL\n\nVicinity of Bogot\u00e1\n\nTENJO\n\nZIPAQUIR\u00c1 AND VICINITY\n\nLAGUNA DE GUATAVITA AND VICINITY\n\nPARQUE NACIONAL NATURAL CHINGAZA\n\nSUESCA\n\nSOUTH AND WEST OF BOGOT\u00c1\n\nBusy Bogot\u00e1 is Colombia's cool capital\u2014and not just in terms of its famously chilly nights. A few years ago, visitors would arrive at the El Dorado airport and spend two days maximum in the Andean metropolis before taking the next flight to Cartagena. Now people are staying awhile, and it's easy to see why.\n\n**HIGHLIGHTS**\n\nLOOK FOR M TO FIND RECOMMENDED SIGHTS, ACTIVITIES, DINING, AND LODGING.\n\nM **Plaza de Bol\u00edvar:** Colombia's most important and most photographed plaza is named for Sim\u00f3n Bol\u00edvar, the man who gave the country independence (click here).\n\nM **Iglesia Museo Santa Clara:** This stunning colonial-era church is decorated in the Mudejar style (click here).\n\nM **Manzana Cultural:** Colombia's tumultuous history has given rise to some noteworthy creative expression that is on display in the art museums of the city's cultural block (click here).\n\nM **Museo del Oro:** Anthropology, history, and art combine in this extraordinary presentation of pre-Columbian gold artifacts (click here).\n\nM **Cerro de Monserrate:** The views atop this hill are incredible both by day and by night (click here).\n\nM **Jard\u00edn Bot\u00e1nico:** Countless shades of green are on display in this lovely park minutes from downtown (click here).\n\nM **Ciclov\u00eda:** When a city can get a quarter of its population to get out and ride a bike on a Sunday, you know it's doing something right (click here).\n\nM **Nemoc\u00f3n:** The plaza and streets of this little-visited salt-mining town are full of charm (click here).\n\nM **Laguna de Guatavita:** This sacred lake is the source of the El Dorado myth (click here).\n\nM **Parque Natural Chicaque:** Minutes from La Candelaria, the cloud forests of this park seem miles away from everything (click here).\n\nThere is the Museo del Oro, of course, undoubtedly one of the best museums in Latin America. There are precious few reminders of the Muisca settlement of Bacat\u00e1 in this vast concrete jungle of today, but this museum is a stellar tribute to a people who all but disappeared within decades of the Spanish conquest.\n\nThen there is the living museum that is the historic district, La Candelaria. Every street block has its unique story to tell: the flower vase that changed history, the loyal companion who saved the Liberator's neck, the generosity of a famous painter. Colonial churches surprise with their quiet, steadfast beauty, and grandiose buildings along the Avenida Jim\u00e9nez stand as testament to the aspirations of the \"Athens of South America.\" Red buses, glitzy shopping areas, and stunning libraries set in manicured parks are proof that Bogot\u00e1 can, with a little investment and good government, overcome the formidable challenges of its recent past.\n\nA melting pot of nearly eight million, Bogot\u00e1 is home to Colombians from every corner of the country who come to study, seek opportunity, or crave the freedom and anonymity that this sprawling city of eight million offers. It shouldn't come as a surprise that it is the country's culinary and cultural capital as well. This is the place to enjoy nouvelle Colombian cuisine, with flavors from the two coasts at a host of innovative restaurants. It's the place where there is always something going on\u2014a massive theater festival, a symphony concert, a dance marathon courtesy of a big-name DJ, a gallery opening\u2014it's just a matter of finding out when and where. Bogotanos' reputation for being gloomy and cerebral is unfair. You only need to experience the sheer _alegr\u00eda_ (joyfulness) of Andr\u00e9s Carne de Res one weekend night for proof.\n\nWhen the sensory overload and intensity of this over-caffeinated city becomes too much, the _p\u00e1ramos_ (highland moors), cloud forests, and mountain lakes of extraordinary natural parks beckon. Parque Nacional Natural Chingaza, Parque Natural Chicaque, and Laguna de Guatavita are all only about an hour away.\n\n#### **HISTORY**\n\nAs early as AD 300, the Muisca people settled along the Cordillera Oriental (Eastern Mountain Range) of the Andes Mountains, forming a loose confederation. Bacat\u00e1 (now Bogot\u00e1) was the seat of the Zipa, head of the southern confederation. The Muiscas had an agricultural economy but also extracted salt and emeralds, wove fine textiles, and actively traded for cotton, shells, and gold with other indigenous peoples. The names of many of their settlements\u2014Ch\u00eda, Suba, Engativ\u00e1\u2014survive, though no physical traces remain.\n\nLured by tales of riches, three European armies converged on Muisca territory in 1538. An army headed by Spanish conquistador Gonzalo Jim\u00e9nez de Quesada arrived from Santa Marta. Another army, headed by Spaniard Sebasti\u00e1n de Belalc\u00e1zar, arrived from the south. A third army, led by German expeditionary Nikolaus Federmann, arrived from present-day Venezuela.\n\nBy the time Federmann and Belalc\u00e1zar arrived, Jim\u00e9nez de Quesada had plundered the Muisca lands and had founded, in August 1538, a settlement that he named Santa Fe de Bogot\u00e1 del Nuevo Reino de Granada de las Indias del Mar. In the late 17th century, the population was less than 15,000 inhabitants. European diseases had almost completely wiped out the Muisca population. Marriages between Muiscas and the Spanish formed the _mestizo_ base of the city.\n\nThe city was the seat of the first provisional government established after Colombia's declaration of independence in 1810. In 1819, the name of the city was changed to Bogot\u00e1, and it became capital of the newly formed Gran Colombia. The city was not connected by railroad to the outside world until the end of the 19th century\u2014and then only to Girardot, a port on the R\u00edo Magdalena.\n\nThe early decades of the 20th century were a period of growth and prosperity. The post-war period was a time of rapid, haphazard development that saw the establishment of many new industries. Much of the growth was unplanned, and sprawling slums developed, especially in the south of the city.\n\nBy the 1990s, Bogot\u00e1 had become synonymous with poverty, crime, and urban sprawl. A series of mayors, including Enrique Pe\u00f1alosa and Antanas Mockus, transformed the city. Pe\u00f1alosa undertook large projects such as the TransMilenio rapid bus system, reclaimed public space, and invested heavily in education and basic services. Mockus worked to improve security and increase civic consciousness. Between 1995 and 2003, the city transformed itself.\n\nDespite all its challenges Bogot\u00e1 continues to be the economic, cultural, and educational powerhouse of Colombia. The city is a magnet for people from all over Colombia and, in recent years, even from abroad. Today Bogot\u00e1 ranks as the fifth largest city in South America.\n\n#### **PLANNING YOUR TIME**\n\nAt the minimum, give Bogot\u00e1 two days. In that short time span, you can cover La Candelaria, head up to Monserrate, discover the Museo del Oro, and enjoy some good meals in the Zona T, Zona G, or the Macarena.\n\nWith about five days you can explore neighborhoods like the Macarena, check out the botanical gardens, or make a day trip to the Parque Natural Chicaque or to the Laguna de Guatavita. If you're here over a Sunday, you'll absolutely have to head out to the Ciclov\u00eda.\n\nIf you are staying in Colombia for 10 days, you can try a city-country combo by adding Villa de Leyva. Or make it a city-coast combo, adding a Caribbean Coast destination such as Cartagena or Santa Marta.\n\nMany museums are closed on either Monday or Tuesday. The Museo del Oro is closed Mondays and the art museums of the Manzana Cultural are closed Tuesdays. During the end-of-year holidays and Holy Week (Semana Santa), Bogot\u00e1 becomes a ghost town as locals head for the countryside, the coast, or abroad. There is very little traffic at those times, but many restaurants are closed and nightspots are empty, especially around Christmas. Bogot\u00e1 is a particularly dull place to be on New Year's Eve. Semana Santa is perhaps less lonely and can be a good time to visit, especially when the biennial theater festival is on. On long weekends, many Bogotanos skip town; those from the provinces come for a visit.\n\n#### **SAFETY**\n\nBogot\u00e1 is much safer than it once was, but it is no Copenhagen. The best advice is to, as Colombians would say, _\"no dar papaya.\"_ Literally, that translates to \"don't give any papayas.\" Don't hand someone the opportunity to take advantage of you.\n\nWhile strolling in La Candelaria, keep a watchful eye on cameras and other gadgets. Better yet, leave valuables\u2014including passports\u2014locked away in the hotel safe if possible. Private security guards and police now regularly patrol La Candelaria at night, although it may feel a little spooky after 10 or 11 at night.\n\nYou will often come across homeless people or those who claim to be displaced. Most\u2014but not all\u2014of these people are harmless. While ample social services do exist in the city, many of the city's destitute do not have the wherewithal to access them. When street people ask for money, you may want to have some spare change, a bottle of water, or leftover food to give out (but only if you do not feel threatened in any way).\n\nTraveling by the city's SITP buses is safe and comfortable. The red TransMilenio buses can get crowded, so be aware of pickpockets. Private buses and _colectivos_ are less safe and drivers can be reckless.\n\nBogot\u00e1 has had a serious problem with taxi crime, commonly known as _paseo milonario._ But recent technological advances have made a noticeable dent in these crimes. Tappsi, a popular and free smartphone app, is indispensable. With this app, you can request a cab, find out the name of the driver, and have your trip tracked by a friend. Alternatively, you can order a cab over the phone. Avoid hailing cabs off the street, particularly when you are alone, when it is late at night, and near nightclubs and upscale dining areas.\n\nIf you are heading out for a night on the town, do not accept drinks from strangers. Leave credit\/debit cards, your passport, and expensive cell phones at home.\n\nDuring an emergency, call 123 from any phone.\n\n#### **ORIENTATION**\n\nSprawling Bogot\u00e1 covers some 1,776 square kilometers (686 square miles), filling a large part of the _altiplano_ (high plateau) or savannah of Bogot\u00e1. In all likelihood, much of your time will be spent along the corridor that is the **Carrera** or **Avenida 7** (most often called the **S\u00e9ptima** ). The S\u00e9ptima extends, parallel to the eastern mountains, from the Plaza de Bol\u00edvar in La Candelaria through the Centro Internacional, Chapinero, and northern neighborhoods to Usaqu\u00e9n and beyond.\n\n**La Candelaria** is the oldest part of town, dating to the 16th century. With the Plaza de Bol\u00edvar at its heart, it is a neighborhood full of historic buildings, interesting museums, and hostels.\n\nAdjacent to La Candelaria is **Avenida Jim\u00e9nez,** also known as the \"Eje Ambiental.\" This is a pedestrian street that is shared with TransMilenio. In addition to being the home of the Museo del Oro, colonial churches, the Quinta de Bol\u00edvar, and Monserrate, the area is also known for its grand early-20th-century architecture.\n\nnorthern Bogot\u00e1\n\nFarther north is the **Centro Internacional.** Major banks have their headquarters in this part of town, and two major museums\u2014the Museo de Arte Moderno de Bogot\u00e1 and the Museo Nacional\u2014are two of the major tourist attractions in the neighborhood.\n\nJust above the bullfighting ring and the iconic Torres del Parque complex is the quirky neighborhood of the **Macarena,** full of art galleries and cozy restaurants. The popular Parque Nacional marks the end of this area that is often considered **downtown.**\n\nThe Distrito Capital of Bogot\u00e1 comprises 20 _localidades_ (official neighborhoods), each with its own local mayor and neighborhood council. **Chapinero** is one of the largest ones along the Carrera 7 (S\u00e9ptima) corridor. It extends to Calle 100, but most people consider Chapinero to include the neighborhoods from around Calle 45 to about Calle 72. Below the S\u00e9ptima is a gritty commerce center that is also considered the center of gay nightlife. There are no major sights of interest in Chapinero.\n\nChauffeured SUVs whizzing by and bodyguards lingering about on the street are tell-tale signs that you have arrived in the swanky northern neighborhoods. The **Zona G,** between Calles 69 and 70 above the S\u00e9ptima; the **Zona Rosa,** between Calles 81 and 85 and Carreras 11 and 15; and the **Parque de la 93** area, between Calles 91 and 94 and also between Carreras 11 and 15, are home to excellent restaurants, famous nightspots, glitzy malls, and fancy hotels. It is the center of hedonism in Bogot\u00e1. Finally, above the S\u00e9ptima between Calles 120 and 125 is **Usaqu\u00e9n,** a sleepy pueblo that has been swallowed by big Bogot\u00e1. Usaqu\u00e9n is becoming a trendy restaurant area and is also known for its Sunday flea market.\n\nIf you look at a map of Bogot\u00e1 you will realize that this corridor from La Candelaria to Usaqu\u00e9n is a tiny sliver of this massive city. West of the S\u00e9ptima and in the center of Bogot\u00e1 is the Parque Sim\u00f3n Bol\u00edvar, along with the Jard\u00edn Bot\u00e1nico and the Biblioteca Virgilio Barco. These are wonderful green spaces worth checking out on sunny days. These sights are not far from the Avenida El Dorado (Calle 26), which connects the El Dorado airport with downtown. In addition to its new TransMilenio line, this nicely designed thoroughfare is lined with hotels, shopping centers, and the fortress-like U.S. Embassy.\n\n**Southern Bogot\u00e1** includes massive working-class and poor neighborhoods. Sights are few and far between. The Santuario del 20 de Julio and Paloqueamo market are worth visiting and are just a few minutes south of the Plaza de Bol\u00edvar. In the huge _localidad_ of Kennedy (named in honor of President John F. Kennedy, who visited the area while announcing infrastructure aid in 1961) is the fantastic Biblioteca Tintal public library. The Teatro Mayor Julio Mario Santo Domingo is in the _localidad_ of Suba in the northwest of the city. Its stunning theater hosts concerts and dance performances from internationally renowned artists.\n\nBogot\u00e1 street addresses are generally easy to figure out. _Calles_ (streets) run east-west (perpendicular to the mountains), while _carreras_ go north-south (parallel to the mountains). For example, the Museo del Oro address is Calle 16 No. 5-41. This means it is on Calle 16, 41 meters from Carrera 5. The Centro Andino shopping mall is at Carrera 11 No. 82-71, or on Carrera 11, 71 meters from Calle 82. The higher the number of the _calle_ goes, the farther north you are. Similarly, the higher the number of the _carrera_ , the farther west you go.\n\nPerhaps because the _calle_ and _carrera_ system was a little too logical, the city planners have also created _avenidas_ (avenues), _diagonales,_ and _transversales._ Both _diagonales_ and _transversales_ are streets on the diagonal. To add to the fun, some _calles_ are also called _avenida calles,_ and likewise there are some called _avenida carrera._ Just ignore the _avenida_ part of the name. Avenida Calle 26 is also known as the Avenida El Dorado. Carrera 30 (which goes past the Estadio El Camp\u00edn) is also known as the Avenida Quito or NQS. Lastly, there are some streets that are called _bis,_ as in Calle 70A _bis_ or Carrera 13 _bis_. It's like an extra little street. Finally, addresses in the south of Bogot\u00e1 have _sur_ (south) in their address. The address for the 20 de Julio shrine is Calle 27 Sur No. 5A-27.\n\n### **Sights**\n\nEverything you need to see in Bogot\u00e1 is downtown, from La Candelaria to the Centro Internacional. Most museums have at least limited English explanations, and some have English-language tours. Photography is allowed at most sights, although the military police guarding the Casa de Nari\u00f1o are sensitive about photography. Some churches and shopping centers may prohibit you from taking photos.\n\n#### **LA CANDELARIA**\n\nLa Candelaria is a living museum. It is a reminder of Spanish power and ambition in the New World; a tribute to the yearning for freedom embodied by Colombia's founding fathers; and a reflection on the tenacity of the independent Colombian republic to persevere in the face of adversity. La Candelaria is a bustling place and has been for centuries. These days, university students, government bureaucrats, tourists, and old-timers who have lived in the area for decades pass each other along the narrow streets and frequent the same caf\u00e9s.\n\nYou could spend a couple of days admiring the colonial churches and exploring the many museums in the area, but if you don't have that much time, three or four hours will give you a good sense of the area and its significance. All of the sights in La Candelaria are easily and best visited on foot. Areas above the Chorro de Quevedo (toward the eastern mountains), as well as some parts to the west, bordering the Avenida Caracas, can be a little sketchy and should be avoided.\n\n##### M **Plaza de Bol\u00edvar**\n\nEvery respectable Colombian city has a Plaza de Bol\u00edvar, but none have quite the history as this one. Between Carreras 7-8 and Calles 10-11, the **Plaza de Bol\u00edvar** is the natural starting point for any tour of La Candelaria. Originally known as the Plaza Mayor, the plaza has had several reincarnations during its history. In colonial times, it was where the Friday market took place. It was also the setting for executions, including that of independence heroine Policarpa Salavarrieta (whose picture graces the $10,000 peso bill). Following the death of Sim\u00f3n Bol\u00edvar, Congress renamed the plaza in his honor in 1846. A diminutive statue of the \"Liberator,\" the first of many Bol\u00edvar statues in the world, stands in the middle of the plaza. Today the plaza is home to political demonstrations, inauguration ceremonies for the Bogot\u00e1 mayor, and concerts.\n\n###### **CATEDRAL PRIMADA AND CAPILLA SAGRARIO**\n\nThe neoclassical facade of the **Catedral Primada** (mass noon Tues.-Sat., 10:30am, noon, and 1:30pm Sun.) dominates the plaza. It was built in 1807, and this is the fourth cathedral built on that same site. The tombs of Gonzalo Jim\u00e9nez de Quesada, founder of Bogot\u00e1, and independence figure Antonio Nari\u00f1o are in a side chapel on the right.\n\nNext door to the cathedral is the **Capilla El Sagrario** (Cra. 7 No. 10-40, mass 7:30am and 5pm Mon.-Fri., 5pm Sun.). This chapel was built much earlier than the cathedral, in the 1600s, and is considered to be an excellent example of Santa Fe (as Bogot\u00e1 was known) architecture. The interior is decorated with a Mudejar or Moorish-style vaulted wooden ceiling. Along the sides of the cross-shaped chapel are several large works depicting biblical scenes by Colombian baroque painter Gregorio V\u00e1squez de Arce y Ceballos. A ceremony was held here to honor the army and Sim\u00f3n Bol\u00edvar following their decisive victory over the Spaniards at the Battle of Boyac\u00e1 in 1819.\n\nthe Presidential Guard\n\n###### **CASA DEL FLORERO**\n\nAcross Calle 10 on the northeast corner of the plaza is the **Casa del Florero** (Cra. 7 No. 11-28, tel. 1\/334-4150, 9am-5pm Tues.-Fri., 10am-4pm Sat.-Sun, COP$3,000), also known as the **Museo del 20 de Julio** or **Museo de la Independencia.** This small house used to be a general store run by a Spaniard, Jos\u00e9 Gonz\u00e1lez-Llorente. The story goes that his refusal to lend a vase to a pair of Creoles sparked the ire of either incredibly sensitive or cunning locals, who launched a protest during the busy market day against Spanish rule. Historians today dispute much of the tale, but the shattered remains of that colorful vase are exhibited today in the museum. Maybe the most interesting exhibit in the museum is a room that shows the transformation of the Plaza de Bol\u00edvar over time, with raw footage of two of the most traumatic events in recent Colombian history: the Bogotazo riots following the assassination of Jorge Eli\u00e9cer Gait\u00e1n in 1948 and the siege of the Palacio de Justicia following a takeover by the M-19 guerrilla group in 1985. A free guided tour in English is given every Wednesday at 3pm.\n\n###### **GOVERNMENT BUILDINGS**\n\nThe newest building on the plaza, completed in 1991, is the **Palacio de Justicia** on the north side. Housing the Supreme Court and other high courts, this building replaced the previous one, which was destroyed following the tragic events of 1985. (That building had replaced a previous justice building that was burned to the ground during the Bogotazo.) On November 6, 1985, M-19 guerrillas stormed the building, perhaps in cahoots with infamous drug kingpin Pablo Escobar, killing several justices and holding some 350 people in the building hostage. After hours of stand-off, the military counterattacked, coordinating their assault from the Casa del Florero. The fight concluded the next day with the building engulfed in flames, result of a military rocket. More than 100 people were killed. Controversy remains even today about the tragedy and the government's actions. Several victims\u2014mostly workers in the cafeteria\u2014were seen being escorted to safety, never to be found again. Five years later the M-19 demobilized, becoming a political movement. Today, it is telling that there is not even a plaque mentioning the tragedy. Nevertheless, clearly some wounds have healed: former M-19 guerrilla Gustavo Petro was elected mayor in 2011, with his office ( _alcald\u00eda_ ) in the **Palacio Li\u00e9viano** on the west side of the plaza.\n\nOn the south side of the square is the neoclassical **Capitolio Nacional,** home of the bicameral Colombian Congress. Designed by architect Thomas Reed, the Capitolio took over 70 years to build, finally being completed in 1926. Gargoyles keep watch atop the building behind the Ionic columns of the front. For about two months in 2009 the entire facade was covered with 1,300 massive ants, a project of Colombian artist Rafael G\u00f3mezbarros. The work was a commentary on forced displacement resulting from Colombia's armed conflict.\n\n##### **West of the Plaza**\n\n###### **ESCUELA DE ARTES Y OFICIOS DE SANTO DOMINGO**\n\nOne of the best trade schools in Latin America for woodworking, embroidery, silversmithing, and leatherworking is the **Escuela de Artes y Oficios de Santo Domingo** (Cl. 10 No. 8-73, tel. 1\/282-0534, www.eaosd.org, 9am-5pm Mon.-Fri., free). Attracting students and teachers from around the world, this school is supported by the Fundaci\u00f3n Mario Santo Domingo. A brief tour of the school is possible (call ahead to arrange). You will notice a warm and collegial atmosphere at the school, where some 600 students are enrolled. If you are staying in Bogot\u00e1 for a while, you can inquire about taking a class. The school is housed in two lovely colonial buildings from the 1600s that are connected by a courtyard. A store\u2014which could be mistaken for a small design museum\u2014sells a limited number of items made by students. Many more are sold at the annual Feria de Artesan\u00edas in December.\n\n###### **IGLESIA DE LA CONCEPCI\u00d3N**\n\nThe **Iglesia de la Concepci\u00f3n** (Cl.10 No. 9-50, 7am-5pm daily) was completed in 1595, making it the second oldest church in the city. Along with a convent, it used to take up an entire block of old Santa Fe. The convent (which no longer exists) was built for the daughters and granddaughters of the conquistadors. The spectacular geometric designs on the ceiling and the polychromatic presbytery are among the most striking aspects of the church. If you pop in, you will no doubt see many faithful\u2014most of humble means\u2014in the pews, in silent meditation. This city block is called the Calle del Divorcio. This refers to a nearby residence for separated or single women who were not allowed into convents and could not live in their family's home.\n\nFarther down the street beyond the Iglesia de la Concepci\u00f3n is the historic labyrinthine artisans market known as the **Pasaje Rivas.**\n\n###### **MUSEO DE LA POLIC\u00cdA NACIONAL**\n\nThe grandiose Palacio de la Polic\u00eda, built in the early 20th century, was once the headquarters for the national police and today is home to the **Museo de la Polic\u00eda Nacional** (Cl. 9 No. 9-27, tel. 1\/233-5911, 8am-noon and 2pm-5pm Tues.-Sun., free). Obligatory tours are given by knowledgeable and friendly cadets who are fulfilling their one-year public service obligation. The museum does have its fair share of guns, but there are also exhibits on different technologies employed by police in pursuit of the bad guys, along with tributes to police dogs. If you go up to the rooftop, you can get a unique view of the city. In the streets around the museum are dozens of shops selling police and military uniforms. Here you can pick up an official \"Polic\u00eda\" baseball cap, but it wouldn't be a good idea to wear it while in Colombia.\n\n###### M **IGLESIA MUSEO SANTA CLARA**\n\nIt is easy to pass by the stone exterior of the **Iglesia Museo Santa Clara** (Cra. 8 No. 8-91, tel. 1\/337-6262, www.museoiglesiasantaclara.gov.co, 9am-5pm Tues.-Fri., 10am-4pm Sat.-Sun., adults COP$3,000), but that would be a shame, as this is one of the most beautiful sights in Bogot\u00e1. Once part of a convent, the little church is an extraordinary example of Mudejar style in Santa Fe. This convent for barefoot Franciscan nuns known as the Clarisas was completed in 1647. It originally housed 12 nuns, who were descendants of conquistadors, and 12 Creole maidens. Perhaps the most stunning aspect design-wise can be admired by craning your neck and looking up: The single nave is beautifully illuminated by hundreds of golden floral motifs. The church is now strictly a museum; it often hosts edgy contemporary art exhibitions. Admission is free on Sundays.\n\nGolden floral motifs adorn the nave at the Iglesia Museo Santa Clara.\n\n##### **South of the Plaza**\n\n###### **CASA DE NARI\u00d1O**\n\nYou can have your picture taken with members of the Presidential Guard (they don't mind) at the gates of the neoclassical **Casa de Nari\u00f1o** (Cra. 8 No. 6-26, www.presidencia.gov.co), home to Colombia's presidents. As is suggested by its name, the presidential palace stands on the site of the birth house of Antonio Nari\u00f1o, who was one of the early voices for independence in New Granada, which was how the Spaniards named the territory. In 1906 Nari\u00f1o's house was razed to make way for the first presidential palace, which was designed by the same French architect who designed the Palacio Li\u00e9vano on the Plaza de Bol\u00edvar. The palace has served as home for Colombian presidents off and on since 1886. Minutes after the 2002 inauguration of President \u00c1lvaro Uribe, the exterior of the palace was slightly damaged by missiles fired from FARC guerrillas. Several missiles landed on humble homes in slums nearby, killing 13.\n\nTours are given of the Casa de Nari\u00f1o, but you must make a reservation several days in advance. For more information on taking the tour visit the website. Even if you don't visit the interior of the palace, you can watch the changing of the Presidential Guard on Wednesday, Friday, and Sunday at 4pm.\n\nAlso on the grounds of the Casa de Nari\u00f1o is the oldest **astronomical observatory** in the New World. This was the initiative of famed botanist and scientist Jos\u00e9 Celestino Mutis. It was completed in 1803. You can inquire about tours conducted by the Universidad Nacional at the Claustro de San Agust\u00edn.\n\n###### **IGLESIA AND CLAUSTRO DE SAN AGUST\u00cdN**\n\nFacing the palace, the **Iglesia de San Agust\u00edn** (Cra. 7 No. 7-13, 9am-5pm Mon.-Sun.) was part of the first Augustinian monastery in the Spanish New World, completed in 1668. The Franciscans and Dominicans beat the Augustinians to the punch in Santa Fe, relegating them to the far extremes of Santa Fe. It is a three-nave temple, which distinguished it from other churches at the time. San Agust\u00edn has seen its share of drama over the years. An earthquake destroyed the two towers in 1785 (they rebuilt just one). In 1861 in the midst of liberal reforms, the government took control of the church from the Augustinians. The next year the church was the scene of a presidential coup attempt during the Battle of San Agust\u00edn, as Conservatives attacked Liberals who were holed up in the church and adjacent monastery (which no longer stands). The church suffered damage yet again during the Bogotazo riots. The **Claustro de San Agust\u00edn** (Cra. 8 No. 7-21, tel. 1\/342-2340, 9am-5pm Mon.-Sat., 9am-4pm Sun., free) didn't serve long as a seminary, and in fact was used as a garrison in which Antonio Nari\u00f1o was imprisoned. During the Bogotazo rampage in 1948, international delegates in town for the 9th Pan-American Conference sought shelter from the mayhem there. Today this beautiful cloister is run by the Universidad Nacional, which puts on temporary art exhibits and hosts educational activities.\n\n###### **MUSEO ARQUEOL\u00d3GICO**\n\nThe **Museo Arqueol\u00f3gico** (Cra. 6 No. 7-43, tel. 1\/243-0465, www.musarq.org.co, 8:30am-5pm Mon.-Fri., 9am-4pm Sat., COP$3,000) holds an extensive and nicely presented collection of ceramic work of pre-Columbian indigenous peoples. In addition there is a room on colonial-era decorative arts, in acknowledgement of the history of this 17th-century home of a Spanish marquis. A small caf\u00e9 adjoins the museum.\n\n##### **East of the Plaza**\n\n###### **MANZANA JESU\u00cdTICA**\n\nThree important colonial buildings make up the **Manzana Jesu\u00edtica** (Jesuit Block). In the early 17th century, the Compa\u00f1\u00eda de Jes\u00fas, a group of Jesuit priests arriving from Cartagena, was given permission by the Spanish ruling authority to build a church and school on the southeastern side of the Plaza Mayor (later to become the Plaza de Bol\u00edvar). As part of its commitment to social justice and to education, the cloister of the **Colegio Mayor de San Bartolom\u00e9** (Cra. 7 No. 9-96, tel. 1\/44-2530, closed to the public) was founded in 1604. The facade of the school was completed in the early 20th century and is considered an excellent example of Republican architecture. It has been in operation continuously since that year and is the oldest school in Colombia. Important figures in the Colombian independence struggle, such as Antonio Nari\u00f1o and Francisco de Paula Santander were students at the school. **Iglesia de San Ignacio,** a church dedicated to the founder of the Jesuit order, was completed in 1643. The church has undergone a massive renovation for years, with the large cupola being restored, the roof above the nave being redone (it was on the verge of collapse), and meticulous restoration of the baroque interior, which includes paintings by many famous painters from the colonial era.\n\nWell worth a visit, the **Museo Colonial** (Cra. 6 No. 9-77, tel. 1\/341-6017, www.museocolonial.gov.co, 9am-5pm Tues.-Fri., 10am-4pm Sat.-Sun., COP$3,000) showcases a fine collection of art and religious artifacts from the colonial era, including the largest collection of works by Gregorio V\u00e1squez de Arce y Ceballos. On the bottom floor is an exhibit that explores life in colonial times. The museum courtyard is quiet and green. Admission is free on Sundays.\n\n###### **TEATRO COL\u00d3N**\n\nInspired by the Teatro Santi Giovanni e Paolo in Venice, the **Teatro Col\u00f3n** (Cl. 10 No. 5-62, tel. 1\/284-7420) was designed by Pietro Cantini to commemorate the 400th anniversary of Christopher Columbus's 1492 landing in the New World. Tours of the theater have not been offered during the long restoration of the theater, but you can call or stop by and inquire about these. The best way of visiting the theater, of course, is to see a performance there.\n\n###### **PALACIO DE SAN CARLOS**\n\nToday housing the Ministry of Foreign Relations, the colonial-era **Palacio de San Carlos** (Cl. 10 No. 5-51, closed to the public) was the home of Colombian presidents from 1825 until 1908. During the Bol\u00edvar dictatorship and the turbulent Gran Colombia period, Bol\u00edvar's companion Manuela S\u00e1enz earned the nickname \"Liberator of the Liberator\" for helping him escape through a palace window\u2014saving him from an 1828 assassination attempt. A plaque marking the exact spot draws the curiosity of passersby today.\n\n###### **MUSEUMS**\n\nThe **Museo de Trajes Regionales** (Cl. 10 No. 6-18, tel. 1\/341-0403, www.museodetrajesregionales.com, 9am-4pm Mon.-Fri., 9am-2pm Sat., COP$3,000), which showcases traditional costumes from the different regions of Colombia, is best known for being the home of Manuela S\u00e1enz, Sim\u00f3n Bol\u00edvar's companion. The museum is next door to the **Plaza de Cuervo,** a tropical patio in the middle of historic Bogot\u00e1. Behind the elegant palm trees is the house where Antonio Nari\u00f1o is said to have translated the Declaration of the Rights of Man from French into Spanish in 1793. After making about 100 copies of it for distribution to rouse the masses, he became nervous and started to furiously destroy them. (He got busted by the Spanish authorities anyway.)\n\nThe **Museo Militar** (Cl. 10 No. 4-92, tel. 1\/281-3086, 9am-4pm Tues.-Fri., 10am-4pm Sat.-Sun., free, must present identification) is in a 17th-century house that was home to independence hero Capt. Antonio Ricaurte. Dozens of mannequins dressed in Colombian military uniforms keep you company as you wander the corridors of this museum. One room is dedicated to Colombia's participation in the Korean War. Over 4,300 Colombians fought in the war waged nearly 15,000 kilometers away, with 163 losing their lives. Colombia was the only country in Latin America to send troops in support of the United Nations\/United States coalition. Two patios are filled with cannons, tanks, and fighter jets.\n\nThe **Museo de Bogot\u00e1** (Cra. 4 No. 10-18, tel. 1\/352-1864, www.museodebogota.gov.co, 9am-5:30pm Mon.-Fri., 10am-4:30pm Sat.-Sun., free) may be of special interest to city planner types. A permanent exhibition examines the development of Bogot\u00e1 through the years, and temporary shows have highlighted photography, historic figures in the city, and profiles of neighborhoods in the metropolis.\n\n##### M **Manzana Cultural**\n\nThe **Manzana Cultural** (Cl. 11 No. 4-41) of the Banco de la Rep\u00fablica is a \"Cultural Block\" (not Cultural Apple) that comprises the Biblioteca Luis \u00c1ngel Arango, the library's concert hall, the Museo Botero, the Museo de Arte, the Colecci\u00f3n de Arte del Banco de la Rep\u00fablica, and the Casa de la Moneda. Without a doubt it is one of the most important addresses for visual arts in Colombia\u2014and a required stop on any visit to Bogot\u00e1.\n\n###### **BIBLIOTECA LUIS \u00c1NGEL ARANGO**\n\nThe **Biblioteca Luis \u00c1ngel Arango** (Cl. 11 No. 4-14, tel. 1\/343-1224, www.banrepcultural.org, 8am-8pm Mon.-Sat., 8am-4pm Sun.) is reportedly one of the busiest libraries in the world, with over 5,000 visitors each day. Part of the same complex and located behind the library, the **Casa Republicana** (8am-8pm Mon.-Sat. and 8am-4pm Sun., free) often hosts temporary art exhibits. There is also a beautiful chamber music concert hall in the large complex.\n\n###### **COLECCI\u00d3N DE ARTE DEL BANCO DE LA REP\u00daBLICA**\n\nWith 14 galleries highlighting Colombian art from the 17th century to present day, the **Colecci\u00f3n de Arte del Banco de la Rep\u00fablica** (Cl. 11 No. 4-41, tel. 1\/343-1316, www.banrepcultural.org, 9am-7pm Mon. and Wed.-Sat., 10am-5pm Sun., free) is an excellent opportunity to discover Colombian art. Look for the series of \"dead nuns.\" It was customary to paint nuns twice in their lifetimes: once when they entered the convent and once more moments after passing away. The nuns from this particular series lived at the nearby convent of the Iglesia de la Concepci\u00f3n.\n\nAnother highlight is the spectacular\u2014if a tad on the gaudy side\u2014 _La Lechuga_ monstrance (a monstrance is a receptacle to hold the Host). It's called _La Lechuga,_ meaning lettuce, because of its 1,486 sparkling emeralds, but it is also adorned by hundreds of diamonds, rubies, amethysts, and pearls. The Spaniard who created this extraordinary piece charged the Jesuits the equivalent of a cool US$2 million when he finished it in 1707. Hidden away in a vault for over 200 years, it was acquired by the Banco de la Rep\u00fablica in 1987 for US$3.5 million.\n\ncourtyard at the Casa de la Moneda\n\nNineteenth-century landscapes, portraits by impressionist and Bogot\u00e1 native Andr\u00e9s Santa Mar\u00eda, and works from an array of well-known Colombian artists from the 20th century (including Alejandro Obreg\u00f3n, Eduardo Ram\u00edrez, Guillermo Wiedemann, and Luis Caballero) are other museum highlights. Free guided tours in Spanish are offered several times a day.\n\n###### **MUSEO DE ARTE DEL BANCO DE LA REP\u00daBLICA**\n\nBehind the Colecci\u00f3n de Arte, in a sleek modern \"white box,\" is the **Museo de Arte del Banco de la Rep\u00fablica** (Cl. 11 No. 4-21, tel. 1\/343-1212, www.banrepcultural.org, 9am-7pm Mon. and Wed.-Sat., 10am-5pm Sun., free), which hosts temporary exhibits and has one floor dedicated to 20th-century Latin American and European art from the Banco de la Rep\u00fablica collection. On the bottom floor is the **Parqueadero** (2pm-7pm Wed.-Mon.)\u2014the \"Parking Lot\"\u2014a sort of laboratory on contemporary art.\n\n###### **MUSEO BOTERO**\n\nIn the **Museo Botero** (Cl. 11 No. 4-41, tel. 1\/343-1212, www.banrepcultural.org, 9am-7pm Mon. and Wed.-Sat., 10am-5pm Sun., free) there are still lifes, portrayals of everyday life in Colombian pueblos, and social commentaries by the most accomplished Colombian artist, Medell\u00edn-born Fernando Botero. In addition to paintings of corpulent Colombians, there are bronze and marble sculptures of chubby cats and pudgy birds. One side of the lovely colonial house, which surrounds a sublime courtyard, displays the artist's exceptional personal collection of European and American art, including works by Picasso and Dali. All of these were donated by the _maestro_ to the Banco de la Rep\u00fablica so that Colombians of all backgrounds could appreciate and enjoy them without paying a peso\u2014an extraordinary opportunity. Once the home of archbishops during the colonial era, the building was set ablaze during the 1948 disturbances of the Bogotazo. It has been painstakingly recreated.\n\n###### **CASA DE LA MONEDA**\n\nConnected to the Museo Botero and the Colecci\u00f3n de Arte by patios and a Botero gift shop, the **Casa de la Moneda** (Cl. 11 No. 4-93, tel. 1\/343-1212, www.banrepcultural.org, 9am-7pm Mon. and Wed.-Sat., 10am-5pm Sun., free) was where the New World's first gold coins were produced starting in the early 17th century. The museum's **Colecci\u00f3n Numism\u00e1tica** shows the history of the Nueva Granada mint.\n\n###### **CENTRO CULTURAL GABRIEL GARC\u00cdA M\u00c1RQUEZ**\n\nDesigned by Rogelio Salmona, the **Centro Cultural Gabriel Garc\u00eda M\u00e1rquez** (Cl. 11 No. 5-60, tel. 1\/283-2200, www.fce.com.co, 9am-7pm Mon.-Sat., 10:30am-5pm Sun., free) was a gift from the Mexican government in honor of the 1982 Nobel Prize winner for literature, Colombian Gabriel Garc\u00eda M\u00e1rquez. Gabo, as he is called, has lived in Mexico since the 1960s. On the main level, where you can enjoy a nice sunset view of the cathedral, is a bookstore with an ample selection of books on Colombia. Next to the Juan Valdez Caf\u00e9 below is a space where photography and art exhibits are often shown.\n\n#### **AVENIDA JIM\u00c9NEZ**\n\nThe Avenida Jim\u00e9nez used to be the R\u00edo San Francisco and the extreme northern boundary of Santa Fe. For the architectural enthusiast, there are several gems on this street that stand in tribute of the city's inflated view of itself during the first half of the 20th century. Most of these historic buildings can only be enjoyed from the exterior. In 2000, in an effort to reinvent the historic Avenida Jim\u00e9nez, architect Rogelio Salmona created the **Eje Ambiental** (Environmental Corridor), which extends from the Universidad de los Andes campus to the Avenida Caracas. Vehicular traffic is banned from the street except for the red buses of the TransMilenio. Ample pedestrian space has made this a pleasant place for a stroll.\n\nIn 2012, the city created a pedestrian zone from the Plaza de Bol\u00edvar to the Calle 26. This busy commercial area is now a fun way to check out the city's core, do a little shopping, sightseeing, and people-watching.\n\n##### **Historic Architecture**\n\nYou don't have to be an expert on architecture to appreciate the many impressive buildings lining the entire length of the Avenida Jim\u00e9nez. Most of these gems were built in the early 20th century. To the west side of the S\u00e9ptima (Carrera 7) are: the neoclassical **Palacio de San Francisco** (Av. Jim\u00e9nez No. 7-56), prior home to the Cundinamarca departmental government; the **Edificio L\u00f3pez** (Av. Jim\u00e9nez No. 7-65), which was built by the same construction firm that built the Chrysler building in New York; and the modernist **Edificio Camacho,** farther down and on the right.\n\nIt was on the southwest corner of the S\u00e9ptima and Avenida Jim\u00e9nez that populist Liberal Party presidential candidate Jorge Eli\u00e9cer Gait\u00e1n was assassinated on April 9, 1948, which sparked the tragic Bogotazo riots. Up to 3,000 were killed. This precipitated the bloody period of La Violencia that swept the country. At McDonald's, a plaque and flowers mark the spot where the tragedy took place. A young Gabriel Garc\u00eda M\u00e1rquez, then a law student at the Universidad Nacional, lived near the Palacio de San Francisco at that time, and with his building in flames, he and his brother rushed back\u2014to save his typewriter.\n\nOn the eastern side of the S\u00e9ptima, notable buildings include the modernist **Banco de la Rep\u00fablica** (Cra. 7 No. 14-78); the **Universidad de Nuestra Se\u00f1ora del Rosario** (Cl. 12C No. 6-25), founded in 1653, which is housed in a colonial building that was originally a monastery; the **Edificio Monserrate** (Av. Jim\u00e9nez No. 4-49), which was home to _El Espectador_ newspaper; the fabulous restored **Hotel Continental** (Av. Jim\u00e9nez No. 4-19), once the most exclusive hotel in town; the neoclassical **Academia Colombiana de Historia** (Cl. 10 No. 9-95); the 17th-century **Iglesia and Convento de las Aguas** (Cra. 2 No. 18A-58), where Artesan\u00edas de Colombia has a store; and finally (at the end of the Eje Ambiental) the campus of the **Universidad de Los Andes,** one of the top universities in Latin America, with several stunning new buildings. Los Andes has around 25,000 students.\n\n##### **Churches**\n\nTypical of most all colonial-era churches, the **Iglesia de San Francisco** (Cl. 16 No. 7-35, 6:30am-8pm Mon.-Fri., 6:30am-12:40pm and 4:30pm-8pm Sat.-Sun.) looks somber from the outside, but inside it's decorated by a fantastic golden altar, considered a masterwork of American baroque. This is the oldest of all the churches in the city, built by the Franciscans in 1557. The church is often full of working-class faithful. Adjacent to the San Francisco is the **Iglesia de Veracruz** (Cl. 16 No. 7-19), which is where several independence figures, executed by the Spaniards, are laid to rest.\n\nThe third church in this row is called **Iglesia La Tercera** (Cl. 16 No. 7-35, 7am-6pm Mon.-Fri., 11am-1pm Sat.-Sun.), and it is one of the jewels of colonial churches in Bogot\u00e1. It was built in the late 18th century, about 50 years before Colombian independence. Architecturally, the main highlight is its barrel-vaulted ceiling decorated with geometric designs and altarpieces made of cedar and walnut. Unlike other churches, the interior is not covered with gold leaf.\n\n##### M **Museo del Oro**\n\nSome visitors come to Bogot\u00e1 specifically to see the world-renowned **Museo del Oro** (Cra. 6 No. 15-88, tel. 1\/343-2233, www.banrep.gov.co, 9am-6pm Tues.-Sat., 10am-4pm Sun., COP$3,000). During your museum experience, you will see just a fraction of the thousands of treasures of the Banco de la Rep\u00fablica, Colombia's central bank, since its first acquisition in 1939. The museum tells the story of how\u2014and why\u2014the native peoples of Colombia created such incredibly detailed and surprisingly modern designs of gold jewelry and religious objects.\n\nWhat is astonishing about the collection is the sophistication of the work. It is almost all smelted, with Muisca and Sin\u00fa peoples employing a \"lost wax\" technique, with various metals being purposefully alloyed. Here, rather than large, hammered pieces, as in countries like Peru, you will see intricately crafted and designed jewelry.\n\nOne of the highlights, without a doubt, is the golden raft created by local Muisca people. The raft portrays the ritual of El Dorado, \"the Golden One.\" Another piece to look for is the collection's first acquisition, the Quimbaya P\u00f3poro. This was used during religious ceremonies. The unforgettable offering room is filled with golden treasures. English explanations are good throughout the museum (so is the audio tour). Just beyond the gift shop is a very popular restaurant that specializes in Colombian and Mediterranean cuisine. If possible try to avoid visiting the museum on weekends, when crowds soar, especially on Sunday, when admission is free.\n\n##### **Museo de la Esmeralda**\n\nOn the 23rd floor of the Avianca building is the **Museo de la Esmeralda** (Cl. 16 No. 6-66, tel. 1\/286-4259, www.museodelaesmeralda.com.co, 10am-6pm Mon.-Sat., COP$5,000). The museum has an impressive recreation of an emerald mine and then several examples of different emeralds from Colombia and elsewhere. Guides, fluent in Spanish and English, will make sure you know that the best emeralds do\u2014without a doubt\u2014come from Colombia, primarily from the Muzo mines in the Boyac\u00e1 department. Although there is no pressure to do so, you can purchase all different classes of emeralds, and their jewelers can transform the emeralds you choose into rings or earrings within a day. Even if you are not interested in purchasing an emerald it is fun to check out the gems under a magnifying glass, as you learn why some emeralds are much more precious than others. The museum also has a small store on the main floor of the building that sometimes has discounted coupons for museum entry. Security at the Avianca building is stringent, and you will need to bring a photocopy of your passport and produce a telephone number of your hotel for entry.\n\n##### **Quinta de Bol\u00edvar**\n\nThe **Quinta de Bol\u00edvar** (Cl. 20 No. 2-91 Este, tel. 1\/336-6410\/19, 9am-5pm Tues.-Sat., 10am-4pm Sun., COP$3,000) is a lovely country estate that was presented by Francisco de Paula Santander, Vice President of the Rep\u00fablica de Gran Colombia, as a gift to Sim\u00f3n Bol\u00edvar in 1820. El Libertador was president of Colombia from 1819 to 1830. Bol\u00edvar stayed there during his brief and sporadic visits to Bogot\u00e1, a city he did not like. He spent approximately 432 nights there, give or take. Built in 1800, it is a beautiful example of a late colonial-era house. Furnished with period pieces and set in a beautiful garden under cypress and walnut trees, it is one of the most popular touristic sights in the city. On Wednesdays there are guided tours in English at 11am (if there is a group of at least three). Reserve your spot the day before. Each day there are Spanish-language tours at 11am and 2pm, if you'd like to practice your _espa\u00f1ol._ An audio tour is available for just COP$1,000, but the narrators are a bit long-winded. Admission is free on Sundays. It is just a five-minute walk uphill from the Quinta to Monserrate.\n\n##### M **Cerro de Monserrate**\n\nRiding or hiking up to the top of this mountain, the **Cerro de Monserrate,** and taking in the views of the city by day or by night is a memorable one. To get to the top, take a funicular tramway (7:45am-11:45am Mon.-Sat., 6am-6pm Sun., daytime round-trip COP$15,400, nighttime round-trip COP$17,000) or the _telef\u00e9rico_ (cable car, noon-midnight Mon.-Sat., 9:30am-6:30pm Sun., daytime round-trip COP$15,400, nighttime round-trip COP$17,000).\n\nYou can also hike to the top, which, due to large crowds on weekends and holidays, is a good plan for a weekday morning. The path is open 5am-4pm Wednesday-Monday. There is no charge to make the somewhat challenging ascent on foot. Those over 75 years old, under a meter tall, or very pregnant are supposedly prohibited from making the climb, but this doesn't appear to be enforced. Going at a fast clip, the walk will take under 45 minutes. If you do decide to walk up, bring plenty of sun protection.\n\nCerro de Monserrate, as seen from the Quinta de Bol\u00edvar\n\nIn the past there have been reports of bandits lingering in the woods along the path, but the security situation has vastly improved. Bored police cadets are stationed at three or four points along the trail until 4pm, and when there are no police there are plenty of vendors selling refreshments or several others huffing and puffing going up or leisurely coming down. If you feel as if you have done your exercise for the day, you can purchase a one-way ticket at the top to ride the funicular or tramway back down for under COP$8,000.\n\nFor the faithful, the white chapel atop, the **Santuario de Monserrate,** may be the goal of this hike. It is not of interest architecturally speaking, and it has been destroyed and rebuilt several times since the 1600s, but it is the highest church around, at about 3,152 (10,341 feet) above sea level. Inside, a 17th-century sculpture of the Fallen Christ of Monserrate attracts many believers. Some pilgrims climb the hill on their knees during Holy Week, believing that the Fallen Christ grants miracles to those who do so.\n\nThere are two pricey restaurants on the top of the mountain\u2014a romantic setting for marriage proposals and a favorite spot for locals to bring visitors. These are French-Colombian **Restaurante Casa San Isidro** (tel. 1\/281-9270, www.restaurantecasasanisidro.com, noon-midnight Mon.-Sat., COP$30,000) and **Restaurante Casa Santa Clara** (tel. 1\/281-9309, www.restaurantecasasantaclara.com, noon-6pm Tues.-Sat., COP$25,000), which serves mostly Colombian fare.\n\nTo the south of Monserrate rises the **Cerro de Guadalupe,** with a large statue of the virgin. It can only be accessed by road and was, until recently, unsafe to visit. If you would like to visit (the views are about the same as from Monserrate), take a microbus on Sunday from the intersection of Calle 6 and Avenida Caracas. As you enter the ticket office at the base of Monserrate, you may see an old photograph of a tightrope walker crossing the 890 meters from Monserrate to Guadalupe blindfolded. This stunt was performed by Canadian daredevil Harry Warner in 1895.\n\n#### **CENTRO INTERNACIONAL**\n\n##### **Museo de Arte Moderno de Bogot\u00e1**\n\nAcross from the Parque de la Independencia on Avenida 26 is the **Museo de Arte Moderno de Bogot\u00e1** (Cl. 24 No. 6-00, tel. 1\/286-0466, www.mambogota.com, 10am-6pm Tues.-Sat., noon-5pm Sun., COP$4,000). It often puts on interesting exhibitions highlighting Colombian and Latin American artists. The cinema shows independent films and documentaries. Nicknamed MAMBO, it is another creation by the late architect Rogelio Salmona.\n\n##### **Torre Colpatria Observation Deck**\n\nThe **Torre Colpatria Observation Deck** (Cra. 7 No. 24-82, tel. 1\/283-6665, 6pm-8pm Fri., 11am-8pm Sat., 11am-5pm Sun., COP$4,000) offers unparalleled 360-degree views of Bogot\u00e1. The vista of the city from the Colpatria bank tower is arguably superior to that of Monserrate. At 48 floors, the building remains Colombia's tallest. At night the tower goes into disco mode, as it decks out in colorful lights.\n\n##### **Parque de la Independencia**\n\nThe **Parque de la Independencia,** long a favorite for young lovers and those seeking a pleasant stroll under the towering eucalyptus and wax palm trees, was created in 1910 in celebration of Colombia's 100-year anniversary of independence from Spain. The Quiosco de la Luz houses a tourist information center (Punto de Informaci\u00f3n Tur\u00edstica, or PIT). The park is undergoing a major expansion with the construction of a **Parque del Bicentenario** (it originally was to be completed in 2010). This exciting project will bring greenspace above the TransMilenio line on Calle 26.\n\n###### **PLANETARIO DE BOGOT\u00c1**\n\nOn the north side of the park is the modernist **Planetario de Bogot\u00e1** (Cl. 26B No. 5-93, tel. 1\/281-4150, www.idartes.gov.co, 10am-5pm Tues.-Sun., COP$3,000-10,000), which was built in the late 1960s and houses an air and space museum, **Museo del Espacio.** A statue of Copernicus stands outside.\n\n###### **PLAZA DE SANTAMAR\u00cdA**\n\nNext to the planetarium is the former Plaza de Toros de Santamar\u00eda, now renamed **Plaza de Santamar\u00eda.** The neo-Mudejar brick arena was built in the 1930s by a Spanish architect and was modeled after bullfighting rings in Madrid. Less messy events such as meditation sessions and even diving exhibitions have taken plaza in the plaza since Mayor Gustavo Petro banned bullfighting in 2012.\n\n###### **TORRES DEL PARQUE**\n\nAbout 100 steps up from the bullfighting ring and planetarium are the iconic **Torres del Parque.** These three brick apartment buildings, running parallel to the eastern mountains, were designed in the 1960s by Rogelio Salmona, the most accomplished architect from Bogot\u00e1 during the late 20th century. The buildings are perfectly integrated with the Parque de la Independencia and the Plaza de Santamar\u00eda. French-born Salmona studied with Le Corbusier and was awarded the Alvar Aalto Prize in 2003 for his lifetime achievements. Public space takes up almost three-fourths of the area in the towers complex, and art galleries, caf\u00e9s, and bodegas are nice places to linger on a chilly day.\n\nThe Torres del Parque were designed by Rogelio Salmona.\n\n##### **The Macarena**\n\nJust above the Torres del Parque, the laid-back **Macarena neighborhood** (also known as Zona M) is known for its art galleries and cozy restaurants. The adjective \"bohemian\" is frequently thrown about to characterize the barrio, which steeply slopes up from the Carrera 5 to the Circunvalar ring road. It is indeed an \"artsy\" place\u2014that is most evident by the handful of galleries lining the east side of Carrera 5. While you may have to ring a doorbell to enter, gallery staff are more than happy for you to come in and check out what's on. The gallery **Valenzuela Kennler** (Cra. 5 No. 26-28, tel. 1\/243-7752, www.vkgaleria.com, 10am-6pm Mon.-Fri., 11am-5pm Sat.) features experimental artists and, often, video art. **Alonso Garc\u00e9s** (Cra. 5 No. 26-92, tel. 1\/337-5827, www.alonsogarcesgaleria.com, 10am-1pm and 2pm-6pm Mon.-Fri., 10am-2pm Sat.) features work by major contemporary Colombian artists. It also has a nice bookstore. **NC Arte** (Cra. 5 No. 26-76, tel. 1\/282-1474, www.ncearte.org, 10am-6pm Tues.-Sat.) is a newcomer with cool installations. The mix of intellectuals, artists, students, leftists, hipsters, and foreigners makes this neighborhood like no other in Bogot\u00e1.\n\n##### **Museo Nacional**\n\nThe **Museo Nacional** (Cra. 7 No. 28-66, tel. 1\/381-6470, www.museonacional.gov.co, 10am-6pm Tues.-Sat., 10am-5pm Sun., free) was designed by English architect Thomas Reed (who also designed the Capitolio Nacional) in the late 1800s to serve as the penitentiary for Cundinamarca, which was at that time one of nine states of the United States of Colombia. This prison was a cross-shaped panopticon, with a central tower from which guards could monitor prisoners housed in the three wings. It was in the late 1940s that the prison was converted into a museum. The permanent collection examines the history of Colombia from pre-Columbian cultures to the 20th century. On the top floor is a nice introduction to Colombian art. The museum often holds temporary exhibits on the ground floor. There is a pretty good museum store, and Juan Valdez Caf\u00e9 brews coffee in the lovely courtyard.\n\n##### **Parque Central Bavaria**\n\nBelow Carrera 13 is **Parque Central Bavaria** (Cra. 13 No. 28A-21), a large urban renovation project on the first site of the Bavaria brewery. The Bavaria Kopp's Deutsche Bierbrauerei was founded by German immigrant Leo Kopp and his four brothers. Bavaria is one of the few surviving\u2014and thriving\u2014businesses from the 19th century in Colombia. Its beers include \u00c1guila, Club Colombia, Coste\u00f1a, P\u00f3ker, and Pilsen. The brewery moved from this location in the 1980s and only two of the original brewery's buildings remain today, now home to several restaurants.\n\n##### **Parque Nacional**\n\nA center of activity on the weekends, the **Parque Nacional** (between Cras. 5-7 and Clls. 35-39) is the largest park in downtown Bogot\u00e1 and is the second oldest one in the city. The park is set between a lovely English Tudor-style neighborhood called La Merced and, to the north, the Universidad Javeriana, which was founded by the Jesuits. On Sundays and holidays when there is Ciclov\u00eda, free aerobics classes draw huge crowds in the park. In addition there are numerous fields and courts to practice sports, including several clay tennis courts. On the northwest corner of the park is a whimsical sculpture by Enrique Grau called _Rita 5:30._\n\n##### **Cementerio Central**\n\nThe most important cemetery in Colombia is the **Cementerio Central** (Cra. 20 No. 24-80, tel. 1\/269-3141, 9am-4pm daily), where prominent political, cultural, and business figures rest. Before the cemetery was built in 1830, distinguished persons were buried in churches following Spanish tradition. Francisco de Paula Santander, who is known as Colombia's Thomas Jefferson; Gustavo Rojas Pinilla, a military dictator from the 1950s; Luis Carlos Gal\u00e1n, a liberal presidential candidate who was assassinated under orders of Pablo Escobar in 1989; and Leo Kopp, the German founder of the Bavaria brewery, are all buried here. Some people pray at Kopp's tomb, asking for wishes. There is also a part of the cemetery where thousands of victims from the Bogotazo riots from April 1948 are buried, many of them chillingly listed as \"N. N.\" (\"no name\").\n\nImmediately west of the cemetery is a remarkable art installation called _Auras Anonimas_ by Colombian artist Beatriz Gonz\u00e1lez. An abandoned columbarium (structure to keep ashes) is covered with around 9,000 primitive black and white paintings of people carrying away the dead. It is a powerful reflection on the violence and death in Colombia.\n\n###### **CENTRO DE MEMORIA, PAZ Y RECONCILIACI\u00d3N**\n\nA memorial to victims of violence associated with the armed conflict is adjacent to the Cementerio Central. The **Centro de Memoria, Paz y Reconciliaci\u00f3n** (Cra. 19B No. 24-86, <http:\/\/centromemoria.gov.co>, 11am-1pm and 2pm-4pm Mon., 8am-10am, 11am-1pm, and 2pm-4pm Tues.-Fri., free) is one of the first memorials to victims of violence in Colombia\u2014an important milestone.\n\n###### **CEMENTERIO BRIT\u00c1NICO**\n\nNeighboring Cementerio Central is the **Cementerio Brit\u00e1nico** (English Cemetery, Cl. 26 No. 22-75, tel. 1\/334-0057), which was donated by the city to the British government in 1829 in recognition of help provided by the British Foreign Legion during the war of independence. Since then it has been the main burial ground for the city's Protestants. A fence at the back of the cemetery was made with the barrels of the legionnaires' bayonets. It is a green, peaceful place\u2014just knock at the door and the family of caretakers will show you in.\n\n###### **PARQUE RENACAMIENTO**\n\nThe **Parque Renacamiento,** just west of the cemeteries, opened in 2000 and is noteworthy for its bronze sculpture _Man on a Horse,_ donated by Fernando Botero.\n\nsoaring wax palms at the Jard\u00edn Bot\u00e1nico\n\n#### **WESTERN BOGOT\u00c1**\n\n##### **Parque Sim\u00f3n Bol\u00edvar**\n\nNicknamed the city's lungs, when it was built in the late 1960s the **Parque Sim\u00f3n Bol\u00edvar** (between Clls. 53-63 and Cras. 48-68, 6am-6pm daily) was in the countryside. Now, it's almost exactly in the middle of the city. Two popes have celebrated mass there: Pope Paul VI in 1968 and Pope John Paul II in 1986. The park is an excellent place for watching ordinary Bogotanos at play, especially on the weekends. Numerous festivals and concerts take place here. There are more than 16 kilometers of trails in the park. In August, traditionally the windiest month, thousands of families try their luck catching a breeze for their colorful kites.\n\n##### **Biblioteca Virgilio Barco**\n\nOpen since 2001, the stunning **Biblioteca Virgilio Barco** (Av. Cra. 60 No. 57-60, tel. 1\/315-8890, 2pm-8pm Mon., 8am-8pm Tues.-Sat., 9:30am-5:30pm Sun.), across the street from the Parque Sim\u00f3n Bol\u00edvar, is yet another project designed by Rogelio Salmona and is one of four fantastic library-parks in the city created by Mayor Enrique Pe\u00f1alosa. The purpose of these mega libraries is to provide citizens access to books, Internet, and cultural\/educational opportunities in a peaceful environment. While relatively plentiful in northern neighborhoods, green spaces\u2014even trees\u2014are few and far between in the massive lower-income neighborhoods. A bike path ( _cicloruta_ ) surrounds the park and is popular with young inline skaters. The well-maintained grounds are a playground for the young, the old, and the canine.\n\n##### M **Jard\u00edn Bot\u00e1nico**\n\nColombia is one of the most biodiverse countries on the planet, and the **Jard\u00edn Bot\u00e1nico** (Av. Cl. 63 No. 68-95, tel. 1\/437-7060, 8am-5pm Tues.-Fri., 9am-5pm Sat.-Sun., COP$2,700) does an excellent job of showing off that diversity. It won't be hard to find the Colombian national tree, the towering wax palm. And inside the greenhouse, be on the lookout for the Bogot\u00e1 orchid\u2014yes, Bogot\u00e1 has its own official orchid! The gardens take you on a tour of the many different climates in the country\u2014from the _p\u00e1ramos_ (highland moors) to cloud forests to tropical jungles. Feel free to stray from the paths and get closer. The garden has its own farm and composting station that you can wander about as well. One of the perks of working there is getting fresh organic vegetables! Run by the city, the botanical garden also runs a community garden project in neighborhoods and carries out educational projects across the city.\n\n##### **Other Nearby Attractions**\n\nFor the romantic ones, a stroll around the **Parque de los Novios** (Lovers Park, Cl. 63 No. 45-10, 6am-6pm daily) might be just the thing for a sunny afternoon. In addition to renting an aquatic bicycle you can also check out the motocross track. The highlight at the **Museo de los Ni\u00f1os Colsubsidio** (Av. Cra. 60 No. 63-27, tel. 1\/225-7587, 9am-4pm Mon.-Sat., COP$5,500) is an old Avianca Boeing 727 jet that kids can explore. Also for the kids is the **Salitre M\u00e1gico** (www.salitremagico.com.co, 10am-6pm Wed.-Sun., COP$20,000-50,000) amusement park. There you can ride the _rueda de Chicago_ (Ferris wheel) or the _monta\u00f1a rusa_ (roller coaster).\n\n#### **NORTHERN BOGOT\u00c1**\n\nNorthern Bogot\u00e1 does not have many tourist sights. But the shopping and restaurant areas might be nice to stroll around on an afternoon.\n\nThe **Zona Rosa** (between Clls. 81-85 and Cras. 11-15), an area of shopping, dining, and nightlife, is a tribute to hedonism. Well-known Colombian designers, such as Sylvia Tcherassi, Lina Cantillo, and Ricardo Pava, have boutiques here, catering to the Colombian jet-set. The **Centro Andino, Atlantis Plaza,** and **El Retiro** \u2014the holy trinity of shopping malls\u2014never seem to go out of fashion. On weekend evenings the entire area buzzes with activity and anticipation. Calle 82 and Carrera 13 form a T\u2014hence the moniker **Zona T** \u2014and are pedestrian streets lined with restaurants and happening watering holes. This is where Bogot\u00e1 comes alive at night.\n\nThe **Parque de la 93** (between Cras. 11A-13 and Clls. 93A-B) is a manicured park surrounded by restaurants. Workers from the area stroll the park on their lunch hour. Sometimes there are big screens set up with bean bags strewn about for people to watch soccer matches. At night it is a popular dining area, but not nearly as rowdy as the Zona T.\n\nThe **Parque Chic\u00f3** (Cra. 7 No. 93-01) is a quiet spot in the north on an old hacienda from the colonial era. The **Museo** (10am-5pm Mon.-Fri., 8am-noon Sat., COP$2,500) has a small collection of pre-Columbian art, religious art, and decorative objects from around the world.\n\nOnce upon a time, charming **Usaqu\u00e9n** was its own distinct pueblo. Now, not even at the fringes of big Bogot\u00e1, miraculously somehow Usaqu\u00e9n has retained much of its colonial charm. It has become a dining and drinking hot spot with many restaurants and bars around the main square. On Sundays the neighborhood comes alive during its popular **flea market** (Cra. 5 at Cl. 119B).\n\n#### **SOUTHERN BOGOT\u00c1**\n\nIf you mention going to southern Bogot\u00e1 for sightseeing, Bogotanos may give you a baffled look. El Sur, the South, is synonymous with poverty and violence for many. Barrios lack green spaces, and neighborhoods are almost across the board ugly. This is where the housekeepers and the drivers for wealthy families in the north live. They can earn in a month a little more than what some from Bogot\u00e1 society spend on a dinner in the Zona G on a Friday night.\n\nBut it is not necessarily a place full of despair. The middle class is growing; young people are earning college decrees; the city government is investing in TransMilenio lines, parks, and libraries; and the private sector is building malls. This is evident in all the massive _localidades_ (official neighborhoods) in the south: Bosa, Ciudad Bol\u00edvar, Kennedy, Los M\u00e1rtires, and Soacha, which is its own municipality.\n\n##### **Divino Ni\u00f1o**\n\nThe **Templo del 20 de Julio** (Cl. 27 Sur No. 5A-27, tel. 1\/372-5555) is one of the most important pilgrimage sites in Colombia, set in the working class neighborhood of 20 de Julio, just a couple of blocks from the new TransMilenio Portal del 20 de Julio station. July 20, 1810, is the day celebrated as Colombia's independence day. On Sundays and religious holidays, hundreds of thousands of faithful come to pray to the **Divino Ni\u00f1o,** a small statue of a smiling young Jesus in his pink robe, which is kept in a chapel behind the church. Around 30 masses are held between 5am and 7pm on Sundays to meet the extraordinary demand. It is indeed a colorful sight. Built by Salesian priests in 1942, the church provides groceries to poor families in the neighborhood who sign up for it. Nearby the church and plaza are shops selling Divino Ni\u00f1o statues and keychains.\n\n##### **Biblioteca Tintal**\n\nThe **Biblioteca Tintal** (Av. Ciudad de Cali No. 6C-09, 2pm-8pm Mon., 8am-8pm Tues.-Sat., 9:30am-5:30pm Sun.) is a beautiful, modern public library built on the site of an unused trash recycling facility. Inaugurated in 2001, it was one of many projects conceived and built by Mayor Pe\u00f1alosa. The library is easily accessed by TransMilenio (Biblioteca Tintal station). The trip there will take you through the large southwestern _localidad_ of Kennedy.\n\n### **Entertainment and Events**\n\nBogot\u00e1 is, without a doubt, the cultural capital of Colombia. In 2012, the city was named a UNESCO City of Music. Because it's the Colombian melting pot, all regions of Colombia are represented in their musical traditions here. Merengue, salsa, _cumbia_ (traditional Caribbean music), _vallenato_ (love ballads accompanied by accordion): You name it, you can hear it.\n\nThe city also attracts international artists, who perform at some of the city's spectacular theaters, such as the gorgeous **Teatro Mayor Julio Mario Santo Domingo** in Suba, the architectural gem of the Sala de Conciertos Luis \u00c1ngel Arango for chamber music, and the iconic Teatro Jorge Eli\u00e9cer Gait\u00e1n downtown. Pop icons such as Paul McCartney and Lady Gaga perform before the masses at the Estadio El Camp\u00edn soccer stadium, and nearby in the Parque Sim\u00f3n Bol\u00edvar, the city puts on several \"al Parque\" free music festivals each year. Rock al Parque is the best known, but there is also Jazz al Parque, Opera al Parque, and even Gospel al Parque.\n\nthe stunning Teatro Mayor Julio Mario Santo Domingo\n\n#### **NIGHTLIFE**\n\nThe capital city is also Colombia's nightlife capital. The Zona Rosa may still dominate the nightlife landscape, but downtown hasn't completely surrendered. La Candelaria has its share of long-standing smaller bars, catering to Bogotanos and visitors alike; the non-SUV crowd hangs out in Macarena hideaways; while the party till dawn crowd throngs the nearby nightclub Radio Berl\u00edn. Gay bars and caf\u00e9s thrive in Chapinero, with massive Theatr\u00f3n, as it has for over a decade, reigning as the club with something for everyone.\n\nMost of the nightspots are, like everything else, located along the Carrera 7. To get the latest on nightlife, and find out about parties, check out Vive In (www.vive.in) or Plan B (www.planb.com.co). Many electronic music parties, attracting big name DJs, take place outside of the city towards Ch\u00eda, and often the best way to find out about them is by stumbling upon posters on streetlight posts.\n\n##### **Bars and Clubs**\n\nThanks to **Bogota Beer Company** (tel. 1\/742-9292, www.bogotabeercompany.com), a successful chain of pubs with several locations throughout the city, sipping on a Candelaria artisan beer has become trendy in Bogot\u00e1. They also serve decent burgers. Try one of the northern locations (Cl. 85 No. 13-06 or Cra. 11A No. 93A-94). Sitting on the terrace and listening to rock at the always-packed **Pub Bogot\u00e1** (Cra. 12A No. 83-48, tel. 1\/691-8711, www.thepub.com.co, noon-close daily, no cover), you're in a strategic position to watch people cruising the Zona T.\n\nMost clubs have a cover between COP$10,000 and 30,000 (rarely). Covers usually include a _consumible_ (complimentary drink). You can usually try to negotiate with the bouncer on the cover, especially if you're with a group. Finally, it's always a good idea to head out on the town with lots of smaller bills. Sometimes bartenders suffer from forgetfulness and fail to return your change. Tips are not expected at bars.\n\nIt doesn't look like much from the outside, but inside **El Coq** (Cl. 84 No. 14-02, tel. 1\/611-2496, hours vary Wed.-Sat., cover COP$20,000), a relaxed and groovy bar in the Zona Rosa, it is pretty stylish. Also in the Zona Rosa, **Armando Records** (Cl. 85 No. 14-46, www.armandorecords.org, hours vary Tues.-Sat., cover COP$15,000) attracts a slightly grungy but cool crowd. The terrace is a fun (but sometimes cold) spot. Live bands and well-known international DJs regularly play at Armando. **La Villa** (Cra. 14A No. 83-56, hours vary Tues.-Sat., cover COP$15,000) hosts the popular Gringo Tuesdays parties, but has all kinds of themed parties catering to locals and visitors alike.\n\nFor fans of _vallenato_ (love ballads accompanied by accordion), the old-school **Rinc\u00f3n Rafael Ricardo** (Cl. 85 No. 14-55, tel. 1\/530-2118, hours vary Thurs.-Sat., no cover) and flashy **Matildelina** (Cl. 81 No. 11-34, tel. 1\/805-2933, 9pm-3am Thurs.-Sat., COP$20,000 cover) are the places to go in the Zona Rosa. Live bands from the Caribbean Coast perform regularly on the big stage at Matildelina, warming up the crowd. **Salom\u00e9 Pagana** (Cra. 14A No. 82-16, tel. 1\/221-1895 or 1\/218-4076, 6pm-2:30am Tues. and Thurs.-Sat., cover COP$15,000) is a Zona Rosa staple that is your salsa and _cubana_ headquarters, often hosting well-known singers and bands.\n\n**C\u00e9ntrico** (Cra. 7 No. 32-16, 41 floor, hours vary Wed.-Sat., cover COP$20,000) is a hot bar-restaurant where you can sip your cocktails while overlooking the city from the 41st floor. It is pretty fancy, so dress to impress.\n\n**Trampa de Vallenato Galer\u00edas** (Cl. 53 No. 27A-31, no phone, 5pm-3am Thurs.-Sat.) may have the warm authenticity that you have been craving. This _vallenato_ club is regularly voted as the top in the city. To hear _cubana_ and salsa music, you can pop into the charming little downtown bar **Son Salom\u00e9** (Cra. 7 No. 40-31, 2nd floor, tel. 1\/285-0547, hours vary daily) for a drink or two to unwind.\n\nOn Fridays it's often rock that the students, hipsters, and visiting foreigners groove to at classic **Candelario Bar** (Cra. 5 No. 3-14, tel. 1\/342-3742, 9pm-3am Fri.-Sat.), but don't be surprised to hear electronic, reggae, or Latin beats. It also serves lunch during the week. **Quiebra de Canto** (Cra. 5 No. 17-76, tel. 1\/243-1630, 6:30pm-3am Wed.-Sat., cover COP$10,000) is a classic haunt where jazz, funk, and salsa are often the order of the night. Wednesdays are especially popular in the two-floor joint. A different vibe can be found at the **Viejo Almac\u00e9n** (Cl. 15 No. 4-30, 6pm-2am Wed.-Sun.), a tango bar named after the famous Viejo Almac\u00e9n in Buenos Aires.\n\nSo, it's 6am and you still need to dance? Near the bullfighting ring and in a basement, **Radio Berl\u00edn** (Cra. 6 No. 26-57, 9pm-5am Fri.-Sat., COP$20,000 cover) is almost too cool for school. It's occasionally open on Thursdays. If you're looking for a late-night groove, often featuring international DJs, this is your place.\n\nIn the Macarena, cool **Baum** (Cl. 33 No. 6-24, cell tel. 316\/494-3799, 10pm-5am Fri.-Sat., COP$15,000) attracts a fun crowd and often hosts international DJs.\n\n##### **LGBT**\n\nBogot\u00e1 is not lacking in gay nightlife spots. At last count there were over 100 gay establishments in the city. This is the place, after all, where many gay Colombians gravitate to so that they can escape endless questions from relatives about when they are going to get married. This isn't a gay bar town, as most people skip that step and head straight to the clubs.\n\nOn Wednesday nights the place to go is **Cav\u00fa Club** (Cra. 15 No. 88-71, tel. 1\/249-9987, www.cavuclub.com, 9pm-3am Wed. and some weekend nights, cover COP$15,000). Here the music is _m\u00fasica pa' planchar_ (music to iron by), and there is usually a performance by a drag queen, such as regular La Lupe. At reliable **Blues Bar** (Cl. 86A No. 13A-30, tel. 1\/616-7126, 9pm-3am Thurs.-Sat., cover COP$15,000) you can drink and listen to cool music as you warm up around the bonfire in the patio.\n\nAs far as clubbing goes, **Theatr\u00f3n** (Cl. 58 No. 10-32, tel. 1\/235-6879, www.theatrondepelicula.com, 9pm-3am Fri.-Sat., cover COP$20,000) is a humongous disco in Chapinero. Theatr\u00f3n is one-stop shopping for the gay crowd. It has no fewer than nine dance floors, featuring different types of music, including reggaet\u00f3n, _vallenato_ , pop, house, and trance. In the main room there is usually a drag show or contest at around midnight on Saturdays. There are few bars and clubs specifically for women, although at Theatr\u00f3n they won't feel like second-class citizens. Theatr\u00f3n occasionally puts on special parties for women. All the major electronic music clubs are gay-friendly. Salsa and _vallenato_ clubs\u2014not so much.\n\n#### **PERFORMING ARTS**\n\n##### **Classical Music and Opera**\n\nYou may not think of classical music when you think Bogot\u00e1, or South America for that matter, but the city is home to two excellent orchestras and an opera, and hosts talented performers year round. As is the case for most concerts and events in Bogot\u00e1, purchasing tickets in advance from Tu Boleta (tel. 1\/593-6300, www.tuboleta.com) is the most convenient option.\n\nThe publicly financed **Orquesta Filarm\u00f3nica de Bogot\u00e1** (www.filarmonicabogota.gov.co) and the **Sinf\u00f3nica Nacional de Colombia** (www.asociacion-sinfonica.org) are the two main orchestras in town and the most important ones in the country. The _filarm\u00f3nica_ performs on the Universidad Nacional campus at the **Auditorio Leon de Greiff** (Cra. 45 No. 26-85, www.divulgacion.unal.edu.co); the **Auditorio Fabio Lozano** (Cra. 4 No. 22-61, tel. 1\/242-7030, ext. 1905) at the Universidad Jorge Tadeo; and occasionally at the **Teatro Jorge Eli\u00e9cer Gait\u00e1n** (Cra. 7 No. 22-47, tel. 1\/379-5750, www.teatrojorgeeliecer.gov.co). The Auditorio Le\u00f3n de Greiff is hard to miss: There is a huge iconic stencil of revolutionary Che Guevara on its exterior. There is often an international guest soloist at these concerts. Although tickets are available at the _taquillas_ (ticket offices) at these theaters a few hours before performance time, it is recommended to purchase tickets, which are usually embarrassingly inexpensive (usually COP$20,000-40,000), in advance at a Tu Boleta outlet (such as in Centro Andino or El Retiro).\n\nThe _sinf\u00f3nica_ performs at the same theaters, as well as the **Colsubisidio Auditorium** (Cl. 26 No. 25-40, tel. 1\/343-2673) and at the spectacular **Teatro Mayor Julio Mario Santo Domingo** (Av. Cl. 170 No. 67-51, tel. 1\/377-9840, www.teatromayor.com), which has two concert halls. This theater, public library, and cultural center in the working class _localidad_ of Suba is worth a visit regardless of whether there is a performance on. The prominent Santo Domingo family donated nearly US$31 million for the construction of this beautiful center.\n\nAt the **Sala de Conciertos Luis \u00c1ngel Arango** (Cl. 11 No. 4-14, tel. 1\/381-2929, www.banrepcultural.org\/musica, ticket office 1pm-8pm Mon.-Fri.) in La Candelaria, chamber music concerts featuring acclaimed international artists are regularly held in its spectacular, modernist theater in the Biblioteca Luis \u00c1ngel Arango. An added bonus: free _tinto_ or _arom\u00e1tica_ (herbal tea) at intermission.\n\nThe **\u00d3pera de Colombia** (tel. 1\/608-8752 or 1\/608-2860, www.operadecolombia.com), one of few opera companies in South America, is highly regarded. They perform classic operas during their season, which usually extends from August to October. The **Teatro Jorge Eli\u00e9cer Gait\u00e1n** (Cra. 7 No. 22-47, tel. 1\/379-5750, ext. 213, www.teatrojorgeeliecer.gov.co) and the **Teatro Cafam de Bellas Artes** (Av. Cra. 68 No. 90-88, tel. 1\/644-4900, www.teatrodebellasartesdebogota.com) host the soirees.\n\n##### **Theater**\n\nThe largest theater company, the **Teatro Nacional** (Cl. 71 No. 10-25, tel. 1\/217-4577, www.teatronacional.com.co), has three different theaters and performances take place just about every day. Their main theater is named in honor of Fanny Mikey, an Argentinian actress who moved to Bogot\u00e1 and started its famed theater festival.\n\n**Teatro Libre** (Cl. 62 No. 9A-65, tel. 1\/542-1559, www.teatrolibre.org) has its main location in Chapinero and another in Candelaria (Cl. 12B No. 2-44, tel. 1\/281-3516). In its 40-plus years of existence, its repertoire has included mostly classic theater as well as works by Colombian playwrights. **Casa Ensamble** (Cra. 24 No. 41-69, tel. 1\/368-9268, www.casaensamble.com), in the cute neighborhood of La Soledad, is an alternative performance space, with avant-garde plays such as _T\u00edteres Pornos (Porno Puppets)_. The theater, which sometimes feels more like a cabaret, is a project of well-known Colombian actress Alejandra Borerro. Source of neighborhood pride, the **Fundaci\u00f3n Gilberto Alzate Avenda\u00f1o** (Cl. 10 No. 3-16, tel. 1\/282-9491, www.fgaa.gov.co), in La Candelaria, puts on theater and music performances featuring local talent year round in addition to art exhibits. Many of their events are free of charge.\n\n##### **Film**\n\nMost movie theaters in Bogot\u00e1, as in the rest of the country, are located inside big shopping malls. In the north that means Centro Andino, Atlantis Plaza, and Centro Comercial Granahorrar, the latter also showing more independent flicks and hosting film festivals. A couple of small cinemas in the north specialize in independent films: **Cineman\u00eda** (Cra. 14 No. 93A-85, tel. 1\/621-0122, www.cinemania.com.co), near the Parque de la 93, and **Cinema Para\u00edso Caf\u00e9 + Bar** (Cra. 6 No. 120A-56, tel. 1\/215-5316, www.cinemaparaiso.com.co). Downtown, go to Calle 24 at the **Cine Colombia Embajador** (Cl. 24 No. 6-01, tel. 1\/404-2463, www.cinecolombia.com) for the usual Hollywood releases. It's across from the **Museo de Arte Moderno de Bogot\u00e1** (MAMBO, Cl. 24 No. 6-00, tel. 1\/286-0466, www.mambogota.com). The museum often shows foreign, classic, and art films. It's best to go in person to get the schedule.\n\n#### **FESTIVALS AND EVENTS**\n\nWhile Bogot\u00e1 lacks celebrations that unite the whole city, such as the Carnaval de Barranquilla or the Feria de Cali, a number of annual festivals and events have their followers.\n\n##### **Festival Iberoamericano de Teatro**\n\nEvery two years during Easter week, theater and dance take over the city during the **Festival Iberoamericano de Teatro** (www.festivaldeteatro.com.co). Attracting more than 100 prestigious international troupes and companies and over 170 representing Colombia, this festival is a living tribute to Fanny Mikey, an Argentinian actress who adopted Colombia as her home. She started the biennial affair in 1988. Known for her bright red hair and distinctive smile, she passed away in 2008. With over 800 performances in the span of two weeks, it is one of the largest such theater festivals in the world. There are always theater groups from English-speaking countries, and there are typically many circus and dance performances. To take a break from the show, you can always hang out at the Carpa Cabaret at night, where you can drink and dance alongside actors from across the globe. Besides performances in theaters, there is an impressive series of free performances in parks and plazas in neighborhoods across the city and workshops for acting students.\n\n##### **ArtBo**\n\nMore than 50 art galleries representing 400 artists from the Americas converge on Bogot\u00e1 each November during **ArtBo** (www.artboonline.com), the **Feria Internacional de Arte de Bogot\u00e1,** one of the top contemporary art fairs in Latin America. It is held each year at the Corferias fairground (Cra. 40 No. 22C-67, www.corferias.com). One space is dedicated to young, emerging artists.\n\n##### **Feria de Artesan\u00edas**\n\nIn December, and just in time for Christmas, the Corferias fairground (Cra. 40 No. 22C-67, www.corferias.com) is the setting for the fantastic\u2014if overwhelming\u2014 **Feria de Artesan\u00edas** (www.expoartesanias.com). During two weeks, artisans come from across Colombia to showcase and sell their handicrafts. Many artisans, particularly indigenous peoples and Afro-Colombians from rural areas, have their trip to Bogot\u00e1 sponsored by Artesan\u00edas de Colombia, the event's organizer. You will find that one day will not be enough to see\u2014and buy\u2014everything. The fair is also a great place for yummy Colombian snacks like _patacones_ (fried plantains).\n\n##### **Music Festivals**\n\nIn this city of music, free music festivals take place at the Parque Sim\u00f3n Bol\u00edvar during the latter half of the year. The series began in the mid-1990s and has grown in popularity ever since. The most famous outdoor music festival is by far the festival **Rock al Parque** (www.rockalparque.gov.co, July), the largest free outdoor rock festival in Latin America. The 2012 edition attracted 120,000 rockers. Variation include **Salsa al Parque** (Aug.), **\u00d3pera al Parque** (Aug.), **Jazz al Parque** (Sept.), **Hip Hop al Parque** (Oct.) and **Colombia al Parque** (Nov.). Find schedule information online (www.culturarecreacionydeporte.gov.co).\n\nInternational and national jazz and Latin jazz artists perform annually at the long-running **Festival Internacional de Jazz de Bogot\u00e1.** Most concerts are held at the **Teatro Libre** (Cl. 62 No. 9-65, tel. 1\/217-1988, www.teatrolibre.org) in Chapinero. This usually takes place in early September, with tickets available at all Tu Boleta stands.\n\n##### **Races**\n\nIf the altitude doesn't make you huff and puff along the streets of La Candelaria, maybe you would be up for the challenge of a running race in Bogot\u00e1. The **Media Marat\u00f3n** is the city's biggest race, attracting runners from around the world. It usually takes place in August, starting at the Plaza de Bol\u00edvar and ending in the Parque Sim\u00f3n Bol\u00edvar. Nike sponsors its **We Run 10K** each year in October. The most unusual race of all takes place in December during the **Ascenso Torre Colpatria** race. That's when runners ascend a stairwell 48 floors to the top of the Colpatria building downtown.\n\n### **Shopping**\n\n#### **HANDICRAFTS**\n\nMarkets selling Colombian handicrafts will likely find you before you find them. The **Pasaje Rivas** (between Cras. 9-10 and Clls. 10-11) dates to the late 19th century. This traditional shopping corridor\u2014one of the few remaining\u2014makes for a fun detour. The passages are so narrow that it is impossible to not interact with the carpenters selling their furniture and women peddling their hand-woven baskets and curios.\n\nThe Pasaje Rivas is great for atmosphere, but if you are looking for high quality, visit one of the **Artesan\u00edas de Colombia** (www.artesaniasdecolombia.com.co) stores. The same people who put on the amazing Feria de Artesan\u00edas every December have two stores in Bogot\u00e1. The most picturesque, by far, is at a stunningly white colonial church, the **Iglesia Las Aguas** (Cra. 2 No. 18A-58, tel. 1\/284-3095, 9am-6pm Mon.-Fri., 10am-noon Sat.). You can also visit the Chic\u00f3 location (Cl. 86A No. 13A-10, tel. 1\/691-7149, 10am-7pm Mon.-Sat.).\n\nPasaje Rivas is a traditional shopping corridor.\n\n**El Balay** (Cra. 15 No. 75-75, tel. 1\/347-1462, 9:30am-7pm Mon.-Sat.) is another option in northern Bogot\u00e1. While they have their share of trinkets, you might find a nice hammock or _chamba_ (casserole dish).\n\n#### **CLOTHES AND ACCESSORIES**\n\nLooking for a cool T-shirt? Check out **America del Sur** (Cl. 85 No. 12-83, www.americadelsur.com.co, 11am-7:30pm Mon.-Sat.), which has mostly Colombia-themed shirts, or **BrincaBrinca** (Cra. 14 No. 85-26, tel. 1\/530-1136, www.brincabrinca.com, 10am-7pm Mon.-Sat.). **Cyclus** (Cra. 7 with Cl. 54, east side, tel. 1\/249-720, www.cyclus.com.co, 10am-7pm Mon.-Sat.) is a unique store that makes all sorts of messenger bags, backpacks, and wallets out of recycled tires. The slogan of this environmentally friendly boutique is appropriately \"It's a round trip.\"\n\n#### **ANTIQUES**\n\nOne street near the Zona Rosa is dedicated almost exclusively to antiques. Nicknamed the **Calle de los Anticuarios** (Cl. 79A between Cras. 7-9), this pleasant one-way street, nice for a mid-morning stroll, is lined by a handful of antique shops as well as some restaurants and, at its top, the Iglesia Santa Mar\u00eda de los \u00c1ngeles, a popular choice for weekend weddings.\n\nProminent on the street are: **Cinco en Punto** (Cl. 79B No. 8-31, tel. 1\/248-9798, 10am-6pm Mon.-Sat.), offering a range of curios from vases to furniture; **Anticuario Novecento** (Cl. 79B No. 7-60, tel. 1\/606-8616, www.anticuarionovecento.com, 10am-6:30pm Mon.-Sat.), with a wide collection that includes religious art from colonial Colombia along with Baccarat crystal from the 1930s; and **Bol\u00edvar Old Prints** (Cl. 79B No. 7-46, tel. 1\/695-5006, www.bolivaroldprints.com, 10:30am-6pm Mon.-Sat.), which specializes in old maps from Latin America and is owned by a French expat. The website of **Asociaci\u00f3n de Anticuarios de Colombia** (Cl. 79B No. 8-49, tel. 1\/248-5756, www.asociacionanticuariosdecolombia.com) has a more complete listing of shops as well as an interesting page regarding Colombian heritage pieces that are in peril of disappearing through illegal sales and transport abroad.\n\nA smaller antique area is in Chapinero on the Carrera 9 from Calle 60 to Calle 63. Check out **Librer\u00eda Errata** (Cra. 9 No. 61-16, tel. 1\/249-6234, www.libreriaerrata.com, 10am-6:30pm Mon.-Sat.) for old books and **Ayer & Co.** (Cl. 62 No. 9-11, tel. 1\/219-9789, 10am-1:30pm and 2:30pm-6:30pm Mon.-Sat.), which sells _de todo un poco_ (a little of everything).\n\n#### **FLOWERS AND MARKETS**\n\nThe flower market at the **Parque El Virrey** (Cl. 86 at Cra. 15, daily) is always colorful, and, if you are the bargaining type, you might enjoy purchasing some flowers, if only to enjoy for a couple of days. A nearby florist, **Flor Expres** (Cra. 13A No. 86A-49, tel. 1\/691-7335, 9am-7pm Mon.-Fri., 9am-5pm Sat.), is also good, with many unusual varieties and orchids, but there's no haggling involved. The king of all flower markets remains **Paloquemao** (Av. 19 No. 25-02, 3am-noon Mon.-Fri.) downtown, but to see it at its most vibrant, you have to get there really early. Best days: Friday and Sunday. Worst day: Monday. There is much more to Paloquemao, with all those exotic Colombian fruits, vegetables, meats, and more. There is a TransMilenio station nearby (Paloquemao station), but it's best to cab it if you go early in the morning.\n\nTwo popular flea markets take place every Sunday. The **Mercado de Pulgas San Alejo** (Cra. 7 No. 24-70, 9am-5pm Sun.) takes place in front of the Torre Colpatria. Uptown in Usaqu\u00e9n is the **Mercado De Las Pulgas Toldos De San Pelayo** (Cra. 7B No. 124-77, 8am-5pm Sun.). The crowds that go to these are worlds apart!\n\n#### **JEWELRY**\n\nColombia is one of the top emerald-producing countries in the world, boasting three major mining areas, mostly located in the Boyac\u00e1 department. Bogot\u00e1 is probably the best place in the country to pick up one of those gems, but it would be wise to walk into jewelry stores armed with knowledge about how you can tell what is a good gem, an idea about prices, etc. Downtown, check out the **Museo Internacional de la Esmeralda** (Cl. 16 No. 6-66, tel. 1\/286-4268, 10am-6pm Mon.-Sat.); **Joyeria Relojeria Museum** (Emerald Trade Center, tel. 1\/342-2957, 9am-7pm Mon.-Fri., 9am-5pm Sat.); or the many stores on the block of Carrera 6 between Calles 10 and 13. You can also try your luck wheeling and dealing with the men milling about on the Jim\u00e9nez just below the S\u00e9ptima. But there, you're on your own.\n\nTwo top-end jewelers at the Centro Andino are **Li\u00e9vano** (Centro Andino Local 157, tel. 1\/616-8608, 10:45am-7:45pm Mon.-Sat.) and **Bauer** (Centro Andino, tel. 1\/478-5454, 11am-7pm Mon.-Sat.).\n\nSpecializing in Colombian gold is internationally recognized **Galeria & Museo Cano** (Cra. 13 No. 27-97, Torre B, Int. 1-19, tel. 1\/336-3255, www.galeriacano.com.co, 9am-7pm Mon.-Fri., 9am-5pm Sat.).\n\n#### **BOOKS**\n\nFor English books, travel guides, newspapers, and magazines, **Author's** (Cl. 70 No. 5-23, tel. 1\/217-7788, 10am-8pm Mon.-Sat., 10am-6pm Sun.) is the best and perhaps only place in town. Author's also has a large selection of children's books. In the north, try **Librer\u00eda Lerner** (Cl. 92 No. 15-23, tel. 1\/636-4295, www.librerialerner.com.co, 9am-7pm Mon.-Sat.), a great place to find Colombian literature, or **Librer\u00eda Central** (Cl. 94 No. 13-92, tel. 1\/622-7423, 10am-6pm Mon.-Sat.), which also has some English- and German-language books.\n\n#### **SHOPPING MALLS**\n\nColombia is mall crazy, and Bogot\u00e1, with over 20 of them, is the capital of this infatuation. Symbolic of its growing middle class, glitzy shopping malls have popped up literally all across the city. The latest and largest is **Tit\u00e1n Plaza** (Cra. 72 No. 80-94, Cl. 80 at Av. Boyac\u00e1, www.titanplaza.com), proudly home of Colombia's first Gap.\n\nIn northern Bogot\u00e1, **Centro Andino** (Cra. 11 No. 82-71, hours vary daily, www.centroandino.com.co) was a big deal when it opened in 1993, being the first high-end shopping center. The German school, the longstanding Colegio Andino, was razed to make way for it. Colombian men's clothes brand **Arturo Calle, Bosi** for shoes and leather, high-end jeweler **Bauer,** and **L.A. Cano** (specializing in handicrafts) are a few of the many stores in the mall. There is a **Cine Colombia** movie theater as well. The food court on the top floor provides an unusual vista of the Zona Rosa, and caf\u00e9s on the terrace below are popular for late afternoon _onces_ (tea time).\n\nNext door to Andino is glitzy **El Retiro** (Cl. 81 No. 11-94, www.elretirobogota.com, hours vary daily), home of the **Plaza de Andr\u00e9s** restaurant; **Mercedes Salazar,** with whimsical jewelry; and **Mundo \u00danico,** which sells skimpy men's underwear. The TurisBog sightseeing bus stops in front of the mall on Calle 81.\n\nFinally, the **Atlantis Plaza** (Cl. 81 No. 13-05, www.atlantisplaza.com, hours vary daily) has a **Cinemark** movie theater, the swimwear shop **Onda de Mar,** and restaurants such as **Crepes & Waffles** and **Hard Rock Caf\u00e9.**\n\n### **Sports and Recreation**\n\n#### **BIKING**\n\n##### M **Ciclov\u00eda**\n\nThe **Ciclov\u00eda** is one of the best things about Bogot\u00e1. No wonder it has been copied in cities around the world\u2014from all across Colombia to New York to Brussels. Every Sunday and on holidays (two times at night, even) about 121 kilometers of Bogot\u00e1 streets are closed to vehicular traffic so that cyclists, joggers, dog walkers, skaters, and people-watchers can claim the streets. The Ciclov\u00eda started small in the 1970s as a neighborhood initiative. Today it is an institution, and really one of the few spaces in which people of all classes in Bogot\u00e1 mix. On particularly sunny days, over two million people have been estimated to have participated in the Ciclov\u00eda. That's the equivalent of the entire population of Houston, Texas, out on a bike! Always be prepared for sun, cold, and rain.\n\nWhile popular with joggers and others, it may be more enjoyable on a bike, especially because you can cover a lot more of the city pedaling rather than walking. The Ciclov\u00eda on the Avenida S\u00e9ptima and on the Carrera 15 are two of the most popular routes, but those are just a fraction of the possibilities. You can go for miles and miles. In fact, this may be a chance to explore parts of the city that you would have never considered before.\n\nThere is no need to take a guided group tour, as the Ciclov\u00eda is easy to figure out. If you ever get lost, you can always ask the helpful Ciclov\u00eda staff, patrolling the routes. Or just ask one of the hundreds of thousands of others out for some fresh air which way to go. Bring money with you so you can grab a freshly squeezed orange juice along the way. Bike repair stations are located on all routes. Keep an eye on the time, as you don't want to be far from your hotel when the cars come roaring back at the strike of 2pm.\n\n##### **Ciclopaseo de los Mi\u00e9rcoles**\n\nFast becoming an in-the-know institution is this group of over a hundred cyclists of all ages and abilities that gets together every other Wednesday night for a nighttime ride along the _ciclorutas_ (bike paths) and streets of Bogot\u00e1. The **Ciclopaseo de los Mi\u00e9rcoles** has been going strong for about seven years. The group meets at bike shop **Welcome** (Cl. 96 No. 10-57, tel. 1\/256-0915) at 7pm. Find out about the next ride on Twitter (@elciclopaseo) or on Facebook. There is no charge.\n\nThe Sunday Ciclov\u00eda is a Bogot\u00e1 institution.\n\n##### **Bike Rentals**\n\nMany bike shops have begun to rent bikes specifically for the Ciclov\u00eda. Try **Pure Bike Shop** (Cra. 13 No. 78-47, tel. 1\/476-5058, www.purebikeshop.com, daily rental COP$45,000), **Eco Byke** (cell tel. 311\/519-2332), or **Bogot\u00e1 Bike Tours** (Cra. 3 No. 12-72, tel. 1\/281-9924, www.bogotabiketours.com). Many shops offer group bike tours.\n\n#### **RUNNING**\n\nTraffic in the city makes it tough to find pleasant places to run, but there are a few. The **Parque Nacional** (between Clls. 36-39 and Cras. 7-5), along with the cute English-style Merced neighborhood next to it, is not a bad place for a short morning jog downtown. The **Parkway** (Av. 24 between Clls. 45 and 34) is another option. This is a lovely strip of green in a quiet part of the Teusaquillo neighborhood. Of course the **Parque Bol\u00edvar** is also very popular, as is the park of the **Biblioteca Virgilio Barco.** In the north try the **Parque El Virrey** (Cras. 8-15 near Cl. 87). One side of the canal has a bike path, the other a foot path. In this _play_ (fashionable) area, you may want to make sure your workout outfit is perfectly color-coordinated. Be careful crossing Carrera 11.\n\n#### **HIKING**\n\nThe mountains surrounding the city are just too inviting to not explore. There are more mountain paths to conquer besides the one to the top of the **Cerro de Monserrate.**\n\nAnother hike in the north that is wildly popular is at the **Quebrada La Vieja,** a path that is operated by the Acueducto. It is open Monday through Saturday 5am-9:30am. On Sundays and holidays it is closed. It may be tricky to find the path at first. It is on the east (mountain side) of the Circunvalar at Calle 71. Most people walk up along the right side of Calle 72 and go through the tunnel under the Circunvalar to reach the path. The hike takes about two hours total.\n\n**Amigos de la Monta\u00f1a** (www.amigosdelamontana.org) helps to maintain the Quebrada La Vieja. They also arrange group walks. Another group, **Camino Bogot\u00e1** (www.caminobogota.wordpress.com), regularly organizes hikes in the mountains around the city. Most of these excursions are accompanied by police officers. A third organization, **Caminar Colombia** (tel. 1\/366-3059 or 1\/241-0065, www.caminarcolombia.com), offers \"ecological walks,\" usually on Sunday. These are usually outside of, but not far from, Bogot\u00e1. These walks cost around COP$40,000 including transportation. On the day of the hike, the group usually meets at 6:30am at the Los Heroes shopping center (Cra. 19A No. 78-85). The TransMilenio station there is called Los Heroes.\n\nflying kites at the Biblioteca Virgilio Barco\n\n#### **SOCCER**\n\nThere are three professional _f\u00fatbol_ clubs in Bogot\u00e1. **Santa Fe** (www.independientesantafe.com) is known as \"Expreso Rojo\" (the Red Express). **Millonarios** (www.millonarios.com.co) has, along with Am\u00e9rica of Cali, won the most national titles. Their color is blue. Both compete at the **Estadio El Camp\u00edn** (Cra. 30 at Cl. 57). There is a TransMilenio station in front of the stadium (Camp\u00edn station). The big match in town is Santa Fe versus Millonarios, and it can get quite heated in the stands. Note that you're not allowed to bring in belts or sharp objects to the matches.\n\nThe green team, **La Equidad** (www.equidadclubdeportivo.com), is the third club in the city. They are affiliated with La Equidad insurance company. Their modern, covered stadium is in the south of the city.\n\nTickets for all matches can be purchased at Tu Boleta (www.tuboleta.com, tel. 1\/593-6300) or Ticket Express (www.ticketexpress.com, tel. 1\/609-1111) outlets. You can also go to each team's ticket offices: **La Tienda Roja** (Cl. 64A No. 50B-08) for Santa Fe or **La Tienda Azul** (Centro Comercial Gran Estaci\u00f3n, Local 2-50, Av. Cl. 26 No. 62-47) for Millonarios. The most sought-after seats in the house at the Estadio El Camp\u00edn are _platea occidental alta_ and _baja._ Ticket prices range between COP$20,000 and COP$70,000.\n\nThere are two soccer seasons each year. One goes from February to June and the second from June to December. Almost all the matches are on Wednesdays and on weekends.\n\nColombia hosted the Under 20 Soccer World Cup in 2011, which was a source of pride for the country, with the championship match (Brazil defeated Portugal) played in Bogot\u00e1. When the Colombian national team is playing a match, traffic magically disappears on the city's streets!\n\n#### **TOURS**\n\n##### **Walking Tours**\n\nA free walking tour of La Candelaria is given, in English, every Tuesday and Thursday at 10am and 2pm starting at the tourist information office on the southwest corner of the Plaza de Bol\u00edvar. Spanish tours are offered every day. Stop by or call (tel. 1\/283-7115) a day before to reserve your place.\n\nArchitecture buffs might be interested in taking a Rogelio Salmona walking tour, exploring some of the architect's most celebrated works. These are organized by the **Fundaci\u00f3n Rogelio Salmona** (Cra. 6 No. 26-85, Piso 20, tel. 1\/283-6413, www.fundacionrogeliosalmona.org). There are three different tours: Centro Hist\u00f3rico, the Centro Internacional, and the Biblioteca Virgilio Barco area. Tours last about three hours, each with an expert guide. English-speaking guides can be arranged. The tours are quite pricey, at COP$180,000 for two people. Prices lower substantially if you latch onto a group of at least 12 (COP$45,000).\n\n##### **Bus Tours**\n\n**TurisBog** (Parque Central Bavaria, Local 120, Manzana 2, tel. 1\/336-8805, www.turisbog.com, adults COP$63,000) is a hop-on, hop-off bus tour that began operation in late 2012. The green double-decker bus makes seven stops: the Maloka science museum in Salitre, the Jard\u00edn Bot\u00e1nico, the Parque de la 93, El Retiro mall, Monserrate, the Parque Central Bavaria in the Centro Internacional, and the Corferias fairgrounds near the Calle 26. Included with the purchase of your ticket is a GPS-activated audio guide in both Spanish and English, a guided walking tour of La Candelaria, and discounts to several restaurants and attractions.\n\n##### **Bike Tours**\n\nIf you'd like the camaraderie of a group of other visitors as you get to know the city and get in a little exercise, try one of the many excursions offered by **Bogot\u00e1 Bike Tours** (Cra. 3 No. 12-72, tel. 1\/281-9924, www.bogotabiketours.com). They offer bike tours around the city and also many other walking tours, such as an unusual graffiti tour.\n\n##### **Tours Outside of Bogot\u00e1**\n\nMany hotels can arrange tours to attractions such as Zipaquir\u00e1 and Laguna de Guatavita, or visits to these can be made via public transportation or by hiring a driver for the day. An agency that specializes in daily Catedral de Sal tours is www.tourcatedraldesal.com. These cost COP$96,000 per person.\n\nThere are some extraordinary national natural parks ( _parques nacional natural,_ or PNN) quite close to Bogot\u00e1, making for excellent day hikes. Sometimes these are a bit more difficult to organize without transportation and not being familiar with the area. **Aventureros** (Cra. 15 No.79-70, tel. 1\/467-3837, www.aventureros.co) organizes mountain bike trips outside of Bogot\u00e1, for instance to the Desierto de Tatacoita near Nemoc\u00f3n. **Ecoglobal Expeditions** (tel. 1\/579-3402, www.ecoglobalexpeditions.com) organizes excursions to multiple destinations throughout Colombia, including the famous Ca\u00f1o Cristales and hikes in El Cocuy. They also can organize day trips to parks nearby Bogot\u00e1, such as the Parque Natural Nacional Sumapaz, containing the world's largest _p\u00e1ramo_ (highland moor), and the PNN Chingaza. **Colombia Oculta** (tel. 1\/630-3172, ext. 112, cell tel. 311\/239-7809, www.colombiaoculta.org) is a similar organization, with similar destinations.\n\n### **Accommodations**\n\nAs tourism has grown in Bogot\u00e1, so too have the number of accommodations options. This is evident in the Centro Hist\u00f3rico, with dozens of hostels catering to backpackers, and also in the north, with five-star hotels changing the landscape in upscale shopping and dining areas. Thus room rates tend to climb as you go from south to north, with Chapinero appropriately offering the most in-between options. Weekend rates are often less expensive in the larger hotels that cater to business people.\n\nWhile it is probably more desirable to stay along the Carrera 7 corridor, other parts of town may be more convenient depending on your length of visit or budget. If you want to be close to the airport, many hotels, several quite new, line Calle 26 (Avenida El Dorado) in western Bogot\u00e1. This part of town is known for steel and glass, not colonial charm. There are few interesting restaurant options within walking distance at night; however, it is quite accessible to downtown during the day thanks to the new TransMilenio line (15-20 minutes) and at night by taxi. Besides being close to the airport, hotels in this area are close to the Corferias fairgrounds, the Parque Sim\u00f3n Bol\u00edvar area, and the U.S. Embassy.\n\nNorth of Calle 100 there are many hotel options in what feels like suburbia. You might find some good deals online there, but you will be quite far from downtown attractions.\n\nHotel rates sometimes automatically include sales tax of 10 percent (IVA). Most hotels include free wireless Internet and breakfast (although the quality of breakfast will vary). While all the fancy hotels and backpacker places have English speaking staff\u2014at least at the front desk\u2014smaller hotels may not. Note that room rates usually depend on the number of persons, not necessarily on the size of the room.\n\nExcept for some international chains and upper-end hotels, most hotels will not have heating or air conditioning in their rooms. You'll have to make do with extra blankets and body heat on those chilly Bogot\u00e1 nights.\n\nA final word: _Moteles_ are always, _residencias_ are usually, and _hospedajes_ are sometimes Colombian love hotels.\n\n#### **LA CANDELARIA**\n\nTravelers on a budget will find plentiful, friendly options in La Canderlaria close to all the important sights. Yet it still might feel a little desolate late at night, especially on holidays when the university students are gone and Bogotanos skip town.\n\n##### **Under COP$70,000**\n\nSleek **Explora Hostels** (Cl. 12C No. 3-19, tel. 1\/282-9320, www.explorahostels.com, COP$22,000 dorm, COP$50,000 d) is small with minimalist decor. There is not much common space, so you might need earplugs at night if your neighbors are in party mode (unless you join them). **Cranky Croc** (Cl. 12D No. 3-46, tel. 1\/342-2438, www.crankycroc.com, COP$23,000 dorms, COP$70,000 d) is big and airy with wood floors throughout, and always has excursions and activities on offer for its guests. Private security guards make this street feel safe after dark.\n\n**La Vieja Suiza** (Cl. 12 No. 3-07, tel. 1\/286-9695, www.laviejasuiza.com, COP$60,000 d) is a cozy and quiet place, run by two Swiss guys, that is connected to their bakery. It's nice to be awoken by the aroma of freshly baked (Swiss) bread.\n\n##### **COP$70,000-200,000**\n\n**Platypus Hostel** (Cl. 12F No. 2-43, tel. 1\/352-0127, www.platypusbogota.com, COP$22,000 dorm, COP$100,000 d) is the pioneer backpacker lodge in La Candelaria. Although somewhat worn, it is a welcoming place, where you can mix with other travelers. There are three houses in the Platypus kingdom, all on the same street. The main one, where you check in, is livelier. Platypus is just off the Eje Ambiental (Av. Jim\u00e9nez).\n\nThe folks at Platypus now have gone upscale with the newish M **Casa Platypus** (Cra. 3 No. 12F-28, tel. 1\/281-1801, www.casaplatypus.com, COP$40,000 dorm, COP$150,000 d). It is comfortable, sparkling clean, and friendly. The rooftop terrace is an excellent place to unwind with a glass of wine after a day hitting the streets.\n\nM **Masaya Intercultural** (Cra. 2 No. 12-48, tel. 1\/747-1848, www.masaya-experience.com, COP$22,000 dorm, COP$100,000 d) near LaSalle University offers different accommodation options depending on your budget or style, from luxurious private rooms to bunk beds. Tourists stay at this newish hostel range from backpackers to budget travelers. Staff are super friendly. Guests and students conglomerate by the bar\/restaurant area in front.\n\nThe **Abad\u00eda Colonial** (Cl. 11 No. 2-32, tel. 1\/341-1884, www.abadiacolonial.com, COP$145,000 s, COP$200,000 d), an Italian-run midrange option, is surprisingly quiet in back around the interior patio. The restaurant specializes in\u2014surprise\u2014Italian cuisine.\n\n##### **Over COP$200,000**\n\nM **Hotel de la \u00d3pera** (Cl. 10 No. 5-72, tel. 1\/336-2066, www.hotelopera.com.co, COP$330,000 d) still reigns as the luxury place to stay in La Candelaria. One republican-style house and one colonial house have been converted into this hotel. There are two restaurants, including one on the rooftop that has one of the best views downtown. The hotel also offers a spa.\n\n**Casa Deco** (Cl. 12C No. 2-36 tel. 1\/283-7032, www.hotelcasadeco.com, COP$230,000 d) is a nicely refurbished art deco building with 21 well-appointed although somewhat chilly rooms (all named by the color of their interior). The terrace is an excellent place for relaxing on a late afternoon.\n\n#### **CENTRO INTERNACIONAL**\n\nThis part of town, once a modern and upscale commercial district, is on the rebound. The new TransMilenio line that has opened on the S\u00e9ptima has been a major factor in transforming the area into a walkable and well-situated place to stay while discovering the city. There are, however, still few hotel options.\n\n##### **COP$70,000-200,000**\n\nIn the heart of the Centro Internacional, few surprises are in store at **Ibis Museo** (Transversal 6 No. 27-85, tel. 1\/381-4666, www.ibishotel.com, COP$120,000). Across from the Museo Nacional, the hotel has 200 smallish rooms and a 24-hour restaurant (breakfast not included). Right on the S\u00e9ptima, this French economy hotel chain is a nice place to be on Ciclov\u00eda Sundays.\n\nThe 850-room **Crown Plaza Tequendama** (Cra. 10 No. 27-51, tel. 1\/382-0300, www.cptequendama.com.co, COP$180,000 d) was the most exclusive address in Bogot\u00e1 for many years. Charles de Gaulle even stayed there. It retains its elegance of yesteryear, with shoe-shiners in the lobby, several restaurants and caf\u00e9s on-site, and smartly dressed bellboys. Rooms are comfortable, and the location is agreeable to taking in the sights downtown. Security is tight here.\n\n#### **WESTERN BOGOT\u00c1**\n\nIf you are only passing through and would like to be close to the airport, you may consider staying in one of the many hotels along the Avenida Calle 26 (Avenida El Dorado). With the new TransMilenio line on the El Dorado, it is easy to hop on one of the red buses and spend the day visiting the major sights in the Centro Hist\u00f3rico. There's not much in the way of charm in this part of town. There is, however, a mall: **Gran Estaci\u00f3n** (Av. El Dorado No. 62-47). Plus, along the Avenida and the TransMilenio line is a nice bike route and jogging path.\n\n##### **COP$70,000-200,000**\n\n**Aloft Hotel** (Av. Cl. 26 No. 9-32, tel. 1\/741-7070, www.starwoodhotels.com\/aloft hotels, COP$169,999 d), a member of the Starwood Hotel Group, is smartly decorated, modern, and within five minutes of the airport. There are 142 rooms in this property. It is also within about a five-minute walk of the Portal El Dorado TransMilenio station.\n\n##### **Over COP$200,000**\n\nThe **Marriott** (Av. Cl. 26 No. 69B-53, tel. 1\/485-1111, www.marriott.com, COP$400,000) is luxurious and is close to the Salitre business and shopping area. The two restaurants\u2014one a cool sushi bar and the other serving Italian fare\u2014are excellent. The hotel is within walking distance of a TransMilenio station, and it takes under 12 minutes to get to the airports. Open since in 2009, it is one of the first luxury international hotels to arrive in Bogot\u00e1 in recent years.\n\n#### **CHAPINERO**\n\nHalfway between downtown and the Zona Rosa, the area of Chapinero between Calle 53 and Calle 72 offers quite a few midrange accommodation options. Chapinero Alto, to the east of the Avenida S\u00e9ptima, is a quiet and leafy middle-class neighborhood. Below the S\u00e9ptima (to the west), it's gritty. During weekdays Chapinero bustles with merchants and students, and on weekend nights the area is transformed into a mostly gay nightlife area. Hotels in Chapinero are about a 10-minute cab ride from the upscale restaurant areas in the north.\n\n##### **COP$70,000-200,000**\n\nThe Viaggio chain has nine reasonably priced, furnished apartment buildings in Bogot\u00e1. **Viaggio 6.1.7.** (Cl. 61 No. 7-18, tel. 1\/744-9999, www.viaggio.com.co, COP$163,000 d) is a high-rise centrally located on the S\u00e9ptima. Rooms have tiny kitchenettes, but breakfast is included in the price. You can rent rooms on a daily, weekly, or monthly basis.\n\nClassical music fills **6 Suites** (Cra. 3B No. 64A-06, tel. 1\/752-9484, www.6suiteshotel.com, COP$150,000 d), which has exactly that in a small house. Some packages include dinner. There is a Saturday vegetable and fruit market in a small park next to the house, as well as a round-the-clock police station.\n\nModerately priced, clean, and centrally located, the **Abitare 56 Hotel** (Cl. 56 No. 7-79, tel. 1\/248-0600, www.abitare56.com, COP$113,000 d) has 28 rooms and is a great deal.\n\n**Casona del Patio** (Cra. 8 No. 69-24, tel. 1\/212-8805, www.lacasonadelpatio.net, COP$135,000 d) is in an English Tudor-style home, and its 24 rooms are reasonably priced despite its location near the Zona G. It has all wood floors, and there is private security on the street at night. With 10 rooms, the **Matisse Hotel** (Cl. 67 No. 6-55, tel. 1\/212-0177, www.matissehotel.com, COP$180,000 d) is just minutes away from the Zona G in an English Tudor-style house above the S\u00e9ptima.\n\nTwo hotels in Chapinero exclusively market to gay and lesbian clientele. **High Park Suites** (Cra. 4 No. 58-58, tel. 1\/249-5149, contacto@highparksuites.com, COP$191,000 d) has four spacious rooms and is in Chapinero Alto. If you are planning on taking taxis everywhere you go, the location is perfectly fine. However, if you'd like to walk, it is on the east side of the Carrera 5 speedway\u2014crossing the street there can be like crossing the Indianapolis 500. Warhol-mad **San Sebastian** (Cl. 62 No. 9-49, tel. 1\/540-4643, www.hbsansebastian.com, COP$180,000 d) is on the other side of the S\u00e9ptima, not far from gay mecca Theatr\u00f3n and the gay-friendly gym and supermarket.\n\n##### **Over COP$200,000**\n\nIn the leafy Chapinero Alto neighborhood, **The Book Hotel** (Cra. 5 No. 57-79, tel. 1\/704-2454, www.thebookhotel.co, COP$273,000 d), which markets itself as gay-friendly, offers very comfortable and modern rooms, and the moderately priced adjacent restaurant with a terrace is popular with locals on their lunch hour.\n\n#### **NORTHERN BOGOT\u00c1**\n\nUptown is a good option if comfort trumps budget and you want to be close to loads of excellent restaurants.\n\n##### **Under COP$70,000**\n\nA good deal within easy walking distance of all the restaurants and stores of the Zona Rosa is **Chapinorte Bogot\u00e1 Guesthouse** (Cl. 79 No. 14-59, tel. 1\/256-2152, www.chapinortehostelbogota.com, COP$64,000 s with shared bath). It's a real bargain for the north.\n\n##### **COP$70,000-200,000**\n\nOn a quiet street in a wealthy neighborhood minutes from the Zona Rosa, **Retiro 84** (Cl. 84 No. 9-95, tel. 1\/616-1501 www.retiro84.com, COP$173,000 d) has 16 rooms. The breakfast area is not very happening, but the hotel is comfortable and is reasonably priced for this high-rent part of town. It's popular with business travelers in town for longer stays. Near the Atlantis Plaza shopping mall in the Zona Rosa, **Hotel Saint Simon** (Cra. 14 No. 81-34, tel. 1\/621-8188, www.hotelsaintsimonbogota.com, COP$180,000 d) is a good value. It is a fairly nondescript brick hotel with about 60 carpeted but well-maintained rooms.\n\n##### **Over COP$200,000**\n\n**BH** (www.bhhoteles.com) is a relatively new Colombian chain of hotels, mostly catering to business travelers. All are comfortable with minimalist design. They operate several hotels in Bogot\u00e1. **BH Tempo** (Cra. 7 No. 65-01, tel. 1\/742-4095, COP$220,000 d) has 63 rooms and is close to the Zona G. **BH Quinta** (Cra. 5 No. 74-52, tel. 1\/742-4908, COP$280,000 d) is in an English Tudor-style house on the busy Carrera Quinta. The most expensive of their hotels is the **BH Retiro** (Cl. 80 No. 10-11, tel. 1\/756-3177, COP$370,000), overlooking a park. It's a five-minute walk to the Centro Andino. Farther north still is **BH Parque 93** (Cra. 14 No. 93A-69, tel. 1\/743-2820, COP$220,000).\n\nIf you'd like to be in the middle of the action in the Zona Rosa, a few comfortable options are around or below the US$150 per night range. Near the Atlantis Plaza shopping mall is the **GHL Hotel Hamilton** (Cra. 14 No. 81-20, tel. 1\/621-5455, www.ghlhoteles.com, COP$290,000 d) of the GHL hotel chain. Cool M **84 DC** (Cl. 84 No. 9-67, tel. 1\/487-0909, www.84dc.com.co, COP$248,000 d) blends in well in this upscale neighborhood. It has 24 spacious, modern rooms. Guests have some privileges at the nearby Bodytech gym.\n\n**B3** (Cra. 15 No. 88-36, tel. 1\/593-4490, www.hotelesb3.com, COP$200,000 d) is one of the most striking hotels in town, due to its wonderful living facade of plants. The lobby area is a lively place in the early evening, when guests munch on tapas and sip cocktails at the bar.\n\nSpanish midrange hotel chain M **NH Bogot\u00e1 93** (Cl. 93 No. 12-41, tel. 1\/589-7744, COP$240,000) is an unpretentious entry in Bogot\u00e1. It offers 137 smart rooms, a nice rooftop terrace, and a small gym.\n\nFinally, the **Hilton** (Cra. 7 No. 72-41, tel. 1\/600-6100, www.hilton.com, COP$250,000 d) is back in Bogot\u00e1, after having abandoned its location downtown during harsher times. This time the Hilton is in a slick black high-rise on the S\u00e9ptima near Calle 72. If you book early enough, you can get a good deal on rooms.\n\nAlthough it's right next door to Andr\u00e9s D.C. and the rest of the Zona Rosa revelry, you'd never know it in your quiet, comfortable room at the **Boh\u00e8me Royal** (Cl. 82 No. 12-35, tel. 1\/618-0168, www.hotelesroyal.com, COP$310,000 d). You can use the gym at its partner hotel, the Andino Royal on Calle 85. Stately **Hotel Morrison** (Cl. 84 Bis No. 13-54, tel. 1\/622-3111, www.morrisonhotel.com, COP$300,000) overlooks a nicely manicured park in the Zona Rosa. It offers spacious rooms and has a \"New York style\" restaurant. Guests enjoy privileges at the Spinning Center gym across the street. You may want to avoid rooms on the south side of the hotel, where you might feel the pulsating beats from nearby discos.\n\nAt **B.O.G.** (Cra. 11 No. 86-74, tel. 1\/639-9999, www.boghotel.com, COP$420,000 d), every detail of the hotel has been thought out. An extraordinary giant photograph of an emerald in the gym area downstairs will inspire you to buy one. The most widely heralded chef in Colombia, Leonor Espinosa, has a nouvelle Colombian cuisine restaurant, and the rooftop pool is luxurious. Just a few blocks away, M **Cit\u00e9** (Cra. 15 No. 88-10, tel. 1\/646-7777, www.citehotel.com, COP$400,000 d) may lack some of the finer touches of B.O.G., but its location right on the El Virrey park could not be better. The terrace of the restaurant is a popular place for Sunday brunch. They have a rooftop pool and provide bikes for guests to use on the Ciclov\u00eda, which passes by in front every Sunday and holiday.\n\n**Estelar** (Cl. 93 No. 11-19, tel. 1\/511-1555, www.hotelesestelar.com, COP$430,000 d) has a fabulous rooftop bar and pool, and you will be sure to get a good night's sleep thanks in part to the soundproof windows. It's close to the Parque de la 93. The discreet **Hotel Port\u00f3n Bogot\u00e1** (Cl. 84 No. 7-55, tel. 1\/616-6611, www.hotelportonbogota.com.co, COP$400,000 d) prides itself on its tight security, making it a favorite of visiting diplomats. It has an elegant old-school feel, especially in the restaurant and lounge area, where they light the three fireplaces every evening at 7. Port\u00f3n guests have unlimited access to the very close Bodytech gym.\n\nIf you have real money to burn, try the **JW Marriott Hotel Bogot\u00e1** (Cl. 73 No. 8-60, tel. 1\/481-6000, www.marriott.com\/bogjw, COP$600,000 d). The bar at this 245-room hotel can make more than 70 types of martinis\u2014enough said. Of course, you don't have to shell out 600,000 pesos to saddle up at the bar and sip one of those martinis.\n\n### **Food**\n\nBogot\u00e1 is the best city in the country when it comes to dining, with more than its fair share of excellent restaurants. Unfortunately many of these places charge Manhattan prices.\n\nA 10 percent tip is usually included in the price, but it is a requirement for the server to ask you if you'd like the _servicio incluido._ You can say no, but that would be considered harsh. If you are truly impressed with the service, you can always leave a little additional on the table.\n\nThe Bogot\u00e1 dish par excellence is _ajiaco._ This is a hearty potato and chicken soup, seasoned with the secret herb _guascas._ (Oops\u2014there goes the secret.) On some dreary days, there can be nothing better. Heated debate can arise about what else to include in the soup. A small piece of corn on the cob usually goes in, as does a dollop of cream, but capers and avocado slices are controversial additions.\n\nReservations are helpful on weekend evenings, especially in the Zona G. Restaurant staff will be more than happy to order a cab for you by phone. That is a very good idea, especially at night.\n\nTap water in Bogot\u00e1\u2014 _de la llave_ \u2014is perfectly fine and good tasting. Besides, if you ever order a fresh lemonade or fruit juice, you're getting _agua de Bogot\u00e1_ anyhow.\n\n#### **LA CANDELARIA**\n\n##### **Caf\u00e9s, Bakeries, and Quick Bites**\n\nHere in coffee country, there is no shortage of tucked away caf\u00e9s where you can sip a _tinto,_ but if you want something more, it's best to stick to the pros. That means **Juan Valdez Caf\u00e9** (Centro Cultural Gabriel Garc\u00eda M\u00e1rquez) and **Oma** (Museo Arqueol\u00f3gico). These chains are all over Bogot\u00e1 and indeed Colombia, and have a strong following. Their success is one reason perhaps that Starbucks has not yet ventured into Colombia.\n\nIf you'd like to stay away from the chains, check out the M **Caf\u00e9 de la Pe\u00f1a\/Pasteler\u00eda Francesa** (Cra. 3 No. 9-66, tel. 1\/336-7488, www.cafepasteleria.com, 8am-8pm Mon.-Sat., 9am-6pm Sun.). This bakery\/caf\u00e9 is where locals pick up their daily baguette. They serve quiches and light lunches with a few Colombian touches as well. Inside it has a quiet, homey feeling.\n\nThe classic place for a _tamal_ and a hot chocolate, the **Puerta Falsa** (Cl. 11 No. 6-50, tel. 1\/286-5091, 7am-10pm daily) claims to be one of the oldest operating restaurants in Bogot\u00e1, having opened in 1816.\n\n##### **Colombian and Fusion**\n\n**Capital Cocina y Caf\u00e9** (Cl. 10 No. 2-99, tel. 1\/342-0426, 9:30am-9pm Mon.-Fri., 2pm-9pm Sat., COP$17,000) is your hip little spot on the corner. Hearty lunches are reasonably priced, and the restaurant is vegetarian-friendly. Breakfast is served until noon (just on weekdays), and it's a cozy place for a nightcap. Using Andean ingredients, such as healthy quinoa, one block over is the tiny M **Quinoa y Amaranto** (Cl. 11 No. 2-95, tel. 1\/565-9982, 8am-4pm Mon., 8am-9pm Tues.-Fri., 8am-5pm Sat.), a warm little place that is a haven for vegetarians downtown. With lunches for around COP$11,000 it's an herbivore bang for your buck.\n\nAn always popular Colombian seafood joint specializing in fried fish and _cazuelas_ (seafood stews) is **Pescadero la Subienda** (Cra. 6 No. 10-27, tel. 1\/284-9816, noon-6pm Mon.-Sat., COP$20,000). **El Olivar** (Cra. 6 No. 10-40, tel. 1\/283-2847, 7:30am-5pm Mon.-Thurs., 7:30am-10pm Fri., 11am-4pm Sat., COP$25,000), across the street, is a more upscale fusion place\u2014and the prices reflect that. Popular at lunchtime with bureaucrats, it serves hearty soups, such as _cazuelas_ based in coconut milk, and also Mediterranean cuisine. Finally, **Mar\u00eda Tomasa Caribbean Cuisine** (Cra. 6 No. 10-82, tel. 1\/744-9097, 9am-4:30pm Mon.-Sat., COP$25,000) is a cheerful place with a Coste\u00f1o feel. Seafood dishes abound, but there are also the customary _arepa de huevo_ (egg fried in corn meal) and juices that you can only get on the coast\u2014like _n\u00edspero_ (sapodilla) juice.\n\n##### **International**\n\n**La Manzana** (Cl. 11 No. 4-93, tel. 1\/284-5335, 9am-7pm Mon. and Wed.-Sat., 10am-5pm Sun.) is inside the Banco de la Rep\u00fablica art complex. It's a quiet environment, overlooking a modern fountain and magnolia trees in the courtyard between the museums. This restaurant specializes in Mediterranean cuisine and pastas. They have some pretty good desserts, too.\n\nIf mushrooms are your thing, head to **Merlin Caf\u00e9 Galer\u00eda Restaurante** (Cra. 2 No. 12-84, tel. 1\/284-9707, noon-1am Mon.-Sat., COP$20,000). They promise the best mushrooms in Bogot\u00e1. It's a funky place. Just across the way is candlelit **El Gato Gris** (Cra. 1 No. 13-12, tel. 1\/342-1716, www.gatogris.com, 9am-midnight Mon.-Thurs., 9am-3am Fri.-Sat.). The menu has a wide range of fare, including steaks and pastas, and they are proud of their cr\u00eapes as well. Both of these are close to the Chorro de Quevedo.\n\nTwo classy international cuisine restaurants have been a part of the Candelaria scene for many years now\u2014meaning they are doing something right. **Bonaparte** (Cra. 8 No. 11-19, tel. 1\/283-8788, noon-4:30pm Mon.-Sat., COP$30,000) is an authentic French bistro. It's known for its cr\u00eapes as well as heartier dishes such as beef Roquefort. Bonaparte remains popular with gossiping politicians and court justices. **Mi Viejo** (Cl. 11 No. 5-41, tel. 1\/566-6128, noon-5pm Mon.-Sun., COP$30,000) was the first Argentinian restaurant in La Candelaria, and it has a loyal following. Paradise for beef-eaters, this friendly spot has, as would be expected, an extensive Argentinian wine selection.\n\nCajun food is the thing at **La Condesa Irina Lazaar** (Cra. 6 No. 10-19, tel. 1\/283-1573, lunch Mon.-Sat., COP$35,000). This American-run spot is very easy to miss, but if you are in the mood for pork chops or perhaps crab cakes, this is the place. There are only six tables, so it is best to reserve in advance. If you persuade him, the friendly owner might consider opening the restaurant for you for dinner.\n\nNear loads of backpacker hostels, the **Crazy Mongolian** (Cl. 12D No. 3-77, 12:30pm-9:30pm Mon.-Sat., COP$12,000) lets you choose the ingredients in generous portions of Mongolian barbecue.\n\nThe food at the **Mirador** restaurant (Cl. 10 No. 5-22, tel. 1\/336-2066, noon-10 pm daily, COP$35,000) on the top of the Hotel de la \u00d3pera may get mixed reviews, but the views? Above the red roofs and church steeples of La Candelaria, they are incredible.\n\n#### **AVENIDA JIM\u00c9NEZ**\n\n##### **Caf\u00e9s, Bakeries, and Snacks**\n\nFor a sandwich or coffee on the go check out tiny **La Jamoner\u00eda Sandwich Gourmet** (Cl. 12C No. 6A-36, tel. 1\/283-0361, 7am-6pm Mon.-Fri., 7am-2pm Sat.). If you're in one of those moods, you can order a \"Cheese Lonely\" sandwich for about COP$7,000.\n\nBrush shoulders with the locals at **La Gran Parilla Santa Fe** (Av. Jim\u00e9nez No. 5-65, tel. 1\/334-4745, 11:30am-5:30pm Mon.-Sat., COP$15,000), a no-surprises and budget-friendly Colombian restaurant downtown. They do the _comida t\u00edpica_ (Colombian fare) thing specializing in grilled meats. **Pasteler\u00eda La Florida** (Cra. 7 No. 21-46, 8:30am-9pm daily, COP$10,000) is a Cachaco institution, sort of a Colombian greasy spoon diner.\n\n##### **Colombian and Fusion**\n\nFor a hearty meal of _mamona_ (grilled meat), run, don't walk, to M **Capachos Asadero** (Cl. 18 No. 4-68, tel. 1\/243-4607, www.asaderocapachos.com, 11:30am-3:30pm Tues.-Thurs., 11:30am-5pm Fri.-Sun.), an authentic _llanero_ (cowboy) restaurant. For under COP$20,000 you get a healthy portion of grilled meat tenderly cooked for several hours, fried yucca, and a _maduro_ (fried plaintain). Goes down well with a beer. On weekends they have live music and dance performances. It's open every day at lunch.\n\nFor a taste of the Colombian Pacific, you can't beat the seafood lunch places on Carrera 4 at Calle 20. There are several of them, all serving about the same thing. Try **Sabores del Mar** (corner of Cl. 20 and Cra. 4, lunch Mon.-Sat., COP$15,000) or **Sabores del Pac\u00edfico** (Cr. 4 No. 20-29, lunch Mon.-Sat, COP$15,000). If you're looking for cheap fried fish or other typical dishes in a place oozing with character, try the **Mercado de las Nieves** (Cl. 19 No. 8-62, no phone, lunch Mon.-Sat., COP$12,000). The passage that connects Calle 19 with Calle 20 is filled with mom-and-pop restaurants serving mostly fish dishes. (A sign marks the place as Pasaje La Macarena, but few call it that.)\n\nIs it time for a chain? You can't beat always-reliable M **Crepes & Waffles** (Av. Jim\u00e9nez No. 4-55, tel. 1\/676-7600, www.crepesywaffles.com.co, noon-8:30pm Mon.-Sat., noon-5pm Sun.), found all over Colombia. Fill up on a _cr\u00eape de sal_ (savory cr\u00eape) for around COP$15,000, but save room for the scrumptious deserts (such as a mini-waffle with Nutella and vanilla ice cream). Also, if you have been searching the world over for a cr\u00eape with tofu in it, the cr\u00eape Gandhi awaits. This particular location is in the easy-on-the-eyes Monserrate building on the Eje Ambiental. Other popular locations are on the Zona T, Parque de la 93, and at the airport, where you can get a healthy breakfast before that morning flight.\n\n#### **CENTRO INTERNATIONAL**\n\n##### **Caf\u00e9s and Snacks**\n\n**Andante Ma Non Troppo** (Cra. 5 No. 15-21, tel. 1\/341-7658, COP$12,000) is a long-running caf\u00e9 that also serves breakfast, sandwiches, and salads.\n\n##### **Colombian and Fusion**\n\nIn the Centro Tequendama, a popular spot for Cali food is **Fulanitos** (Cra. 13 No. 27-00, Local 101, tel. 1\/281-7913, noon-5pm Mon.-Fri, COP$15,000). Another inviting place with a set lunch menu of Colombian favorites is **Ruta** (Cl. 37 No. 13A-26, tel. 1\/751-9239, noon-10pm Mon.-Sat., COP$12,000). At **Los Cauchos** (Cl. 26B No. 3A-20, tel. 1\/243-4059, noon-10pm Mon.-Sat., COP$25,000), preparing excellent Colombian food is a family affair. They have been around in the Macarena since 1976. Check out their plate of the day: Monday it's _ajiaco_ (chicken and potato soup) and on Friday it's _puchero,_ a hearty plate of chicken, pork, beef, potatoes, yuca, and corn in a tomato-onion sauce.\n\n**Leo Cocina y Cava** (Cl. 27B No. 6-75, tel. 1\/286-7091, noon-midnight Mon.-Sat., COP$45,000), by internationally acclaimed chef Leonora Espinosa, has been featured by _Cond\u00e9 Nast_ magazine as one of the tops\u2014in the world. Her empire has expanded northward with her new sleek and savvy restaurant, **La Leo** (Cra. 11 No. 86-74 at the B.O.G. hotel, tel. 1\/639-9999, 6am-10am, noon-3pm, and 7pm-11:30pm Mon.-Sat., COP$40,000), which some say is more style than substance.\n\n##### **International**\n\nThe Macarena is a veritable United Nations of cuisine. On a quiet street behind the Museo Nacional, M **Donostia** (Cl. 29 Bis No. 5-84, tel. 1\/287-3943, noon-4pm Mon., noon-4 and 7pm-11pm Tues.-Sat., COP$30,000) is a swanky place for Spanish-Colombian cuisine. You'll want to linger on the colorful sofas. **Al\u00f4 Brasil** (Cra. 4 No. 26B-88, tel. 1\/337-6015, noon-3pm and 6-10pm Tues.-Wed., noon-3pm and 6pm-midnight Thurs.-Fri., noon-midnight Sat., 1pm-5pm Sun., COP$22,000) serves the famous _feijoada brasileira,_ a stew of beans with beef and pork. The ladies at **Agave Azul** (Cra. 3A No. 26B-52, tel. 1\/560-2702, www.restauranteagaveazul.blogspot.com, 1pm-10pm Sat., noon-3pm Sun., COP$70,000 set menu) have a passion for Mexican food. Tequila is an important part of the equation at this creative place\u2014you were warned. Reservations are necessary.\n\nIn the Parque Bavaria the Mexican place **San Lorenzo** (Cra. 13 No. 28A-21, tel. 1\/288-8731, noon.-4pm Mon.-Fri., COP$23,000) packs in the banking crowd at lunchtime on the fourth floor of the old Bavaria brewery.\n\n#### **CHAPINERO**\n\n##### **Caf\u00e9s and Snacks**\n\n**Pan de Nobles** (Cra. 9 No. 60-82, tel. 1\/606-7262, 7am-8pm Mon.-Sat., 9am-6pm Sun., COP$10,000) started out as a whole-wheat bread bakery. Today it has expanded into a healthy and vegetarian empire. There is a sit-down restaurant upstairs with a set menu. Downstairs on the corner is a \"vegetarian express\" popular with university students, who chow down on veggie burgers between classes. And the bakery still sells unusual breads, such as feijoa bread and soy arepas (cornmeal cakes).\n\nA typical bakery\/caf\u00e9 that has packed them in for breakfast and lunchtime since 1943 is **San Marcos** (Cra. 13 No. 40-36, 6am-8:30pm Mon.-Sat., 7:30am-7pm Sun.). They specialize in lasagnas and pastas.\n\n##### **Colombian**\n\nIt is hard to find a seat, especially on the terrace, at **Canela y Candil** (Cra. 8 No. 56-32, tel. 1\/479-3245, lunch Mon.-Fri., COP$12,000). **Las Margaritas** (Cl. 62 No. 7-77, tel. 1\/249-9468, noon-4:30pm Mon.-Fri., 8am-6:30pm Sat.-Sun.) has been around for over a century. Try the _puchero,_ a meaty stew, on Thursday. The _ajiaco_ (chicken and potato soup) is also good. For the best of original coastal cuisine, try friendly M **MiniMal** (Cra. 4A No. 57-52, tel. 1\/347-5464, www.mini-mal.org, 12:30pm-10pm Mon.-Wed., 12:30pm-11pm Thurs.-Sat., COP$25,000). The stingray _cazuela_ (stew) is one of the more exotic items on the menu. There's also a funky gift shop where you can get one-of-a-kind, 100 percent Colombian handicrafts.\n\n##### **International**\n\nHipster-ish **Salvo Patria** (Cra. 54 No. 4A-13, tel. 1\/702-6367, noon-10pm Mon.-Sat., COP$22,000) is a happening place. There's a variety of interesting appetizers, sandwiches, meaty main courses, and vegetarian options. You'll be tempted to try a carafe of gin _lulada_ (a drink made with the juice of a _lulo_ , a type of orange).\n\nTo go old school in Chapinero Alto, there are two options. First, **Giuseppe Verdi** (Cl. 58 No. 5-35, tel. 1\/211-5508, noon-11pm Mon.-Sat., noon-9pm Sun., COP$20,000) has been around forever, serving typical Italian dishes. They have added a small terrace caf\u00e9 for more informal meals or a glass of wine. **La Poularde** (Cra. 4 No. 54-56, tel. 1\/249-6156, noon-3pm and 7pm-11pm Mon.-Sat., 12:30pm-4pm Sun., COP$25,000) serves very traditional French dishes, such as escargot and cr\u00eapes suzette. There's a 60 percent chance of Edith Piaf songs being played while you're here.\n\n#### **NORTHERN BOGOT\u00c1**\n\n##### **Caf\u00e9s and Quick Bites**\n\nWhen it comes to _onces_ (Colombian tea time), M **Myriam Camhi** (Cl. 81 No. 8-08, tel. 1\/345-1819, 7am-8pm Mon.-Sun.) takes the cake. Just take a look at the decadent desserts on display. The Napoleon de Arequipe and chocolate flan are favorites. While it is known for sweet indulgence, the extensive lunch menu is also nice, with many lighter dishes such as wraps and a pretty good salad bar. On Fridays Myriam serves _ajiaco_ (chicken and potato soup). With a more local feel, **Brot Caf\u00e9** (Cl. 81 No. 7-93, tel. 1\/347-6916, 7:30am-7pm daily) has a fiercely loyal following for breakfast and during afternoon _onces._ They are famous for their freshly baked chocolate baguettes. It's also open for lunch. Surrounded by green on their terrace it's easy to forget the chaos of the city.\n\nM **Siuka** (Cl. 79A No. 8-82, tel. 1\/248-3765, 9am-7pm Mon.-Sat., 11am-5pm Sun.) has got it all. Well, according to its name at least. _Siuka_ means \"everything\" in Chibcha, the language of the Muiscas. It's hard to find any fault at all with this airy, friendly, and minimalist newcomer. Suika adjoins the good **Los S\u00e1nduches de Sr. Ostia** high-class sandwich joint (Cl. 79A No. 8-82, tel. 1\/248-3311). Instead of french fries, you can get spicy carrots! (It's by the same people as Donostia downtown.)\n\n**Nick's** (Cra. 9 No. 79A-28, tel. 1\/321-4108, 11am-10pm Mon.-Sat.) is a quaint place specializing in sandwiches that is popular with hipsters and advertising types. If you arrive on bike, Nick will give you a discount. **Do\u00f1a Dicha** (Cra. 11 No. 78-78, tel. 1\/629-7452, 10am-7pm Mon.-Sat.) bakes delicious bread and is a nice place for a cappuccino (always served with a spoon dipped in chocolate). Best to go there during non-rush hour, as the traffic on the Carrera 11 might spoil the mood. M **Brown** (Calle 77A No. 12-26, tel. 1\/248-0409, www.brownesunareposteria.com, 11am-6pm Mon.-Fri., 11am-4pm Sat.) is cute as a button and has a light lunch menu (half a sandwich, soup, salad, and drink for COP$16,500) and pretty good brownies. It's a nice place to get stuck during the rain. Remember to request your coffee to be strong if that's the way you like it.\n\n**Diletto Caf\u00e9** (Cra. 9 No. 80-45, tel. 1\/317-7383, 7:30am-8:30pm Mon.-Fri., 9am-8pm Sat., 11am-7pm Sun.) often seems to host caffeinated business meetings\u2014and they also can fry eggs. For under COP$10,000 you can have scrambled eggs, a croissant, and a cappuccino for breakfast. **Pan Pa' Ya** (Cl. 82 No. 8-85, www.panpaya.com.co, 7am-9pm daily) is an inexpensive bakery that everyone loves. There are many other locations, even in Weston, Florida! Plan your visit strategically: _Almojabanas_ (cheese rolls) come out of the oven at 8:30am and 6:30pm; _palitos de queso_ (cheese sticks) at 9:30am and 4:30pm.\n\n##### **Colombian**\n\nAt M **Fulanitos 81** (Cl. 81 No. 10-56, tel. 1\/622-2175, lunch daily, COP$20,000) there is always a line at lunchtime. This is Cali cuisine at its best. Try the _chuletas de cerdo_ (pork chops) or _sancocho_ (soup), and have a refreshing _lulada_ (a drink made with the juice of a _lulo_ , a type of orange), of course.\n\nIf you want a good introduction to Colombian delicacies in an elegant atmosphere that doesn't feel like the Carnaval de Barranquilla, the **Casa Club Colombia** (Cl. 82 No. 9-11, tel. 1\/744-9077, 8am-1am, COP$25,000) is an excellent choice. In a lovely house where the fireplace is always lit, _bandeja paisa_ (dish of beans, various meats, yuca, and potatoes) and all your favorites from all corners of the country are on the menu.\n\n**Andr\u00e9s Carne de Res** is the required stop for all visitors to Colombia. It's sort of like a shrine, but one where you can have mojitos, _patacones_ (fried plantains), and a menu full of other typical Colombian dishes. This is one of the first places around that embraced Colombian culture and cuisine with gusto, convinced that it is something to be proud of. And it has worked. The food is good and the atmosphere fantastic. The original, and to many, the best M **Andr\u00e9s** (Cl. 3 No. 11A-56, tel. 1\/863-7880, noon-3am Thurs.-Sat., noon-11pm Sun., COP$30,000) is in Ch\u00eda, about 45 minutes away. This house seems to go on forever, and on weekend nights the slide from dinner to rumba is a slow but definitive one. **Andr\u00e9s D.C.** (tel. 1\/863-7880, noon-midnight Sun.-Wed., noon-3am Thurs.-Sat., COP$35,000) is for an urban Andr\u00e9s experience\u2014for you and about 1,199 others. Look for the windmills next to the El Retiro mall. M **La Plaza de Andr\u00e9s** is in the mall itself (8am-10pm daily, COP$25,000). The colorful Plaza is more reasonably priced and is not the full-on rumba experience, although it does have its personality.\n\nWelcome to Santa Marta\u2014in Bogot\u00e1, that is. The M **Gaira Cumbia Caf\u00e9** (Cra. 13 No. 96-11A, tel. 1\/746-2696, www.gairacafe.com, 9am-10pm Mon.-Wed., 9am-2am Thurs., 9am-3am Fri.-Sat., 9am-6pm Sun., COP$25,000) specializes in Caribbean cuisine. This is a special place, a tribute to music and family. It is run by Guillermo Vives and his mom. Guillo, as he is known, is a talented musician, and is the brother of Carlos Vives, the multiple Grammy Award-winning _vallenato_ singer. The food is quite good, and it is popular at lunchtime. At night, Guillo and other musicians regularly perform, to the delight of the whiskey-drinking crowd. On weekends they have special activities for children in the morning. Some just go for the rumba on weekend nights, for which there is a cover.\n\n##### **International**\n\n**Bagatelle** (Cl. 82 No. 9-09, tel. 1\/621-2614, 7am-10pm Mon.-Sat., 8am-5pm Sun., COP$20,000) is a long-standing French p\u00e2tisserie and restaurant. Their terrace, under the trees, is a nice place for lunch or to sip on a caf\u00e9 au lait and nibble at a delicious bread pudding.\n\nFrench restaurant **Criteri\u00f3n** (Cl. 69A No. 5-75, tel. 1\/310-1377, noon-4pm and 7pm-11pm Mon.-Sat., 9am-1pm and 7pm-11pm Sun., COP$40,000) is the standard bearer when it comes to haute cuisine in Bogot\u00e1. It is the creation of the Rausch brothers, who are among the top chefs in Bogot\u00e1.\n\n**Harry's Bakery** (Cra. 6 No. 69A-24, tel. 1\/321-3940, noon-11pm Mon.-Sat., noon-6pm Sun., COP$20,000) is headed by another highly regarded chef, Harry Sass\u00f3n. It serves sandwiches, burgers, popcorn shrimp, and other diner fare. Harry's mom is in charge of the decadent desserts.\n\nClassy **Astrid & Gaston** (Cra. 7 No. 67-64, tel. 1\/211-1400, www.astridygastonbogota.com, 12:30pm-3:30pm and 7:30pm-11:30pm Mon.-Sat., bar open until 3am Fri.-Sat., COP$35,000), direct from Lima, and stylish **Rafael** (Cl. 70 No. 4-65, tel. 1\/255-4138, 12:30pm-3pm and 7:30pm-11pm Mon.-Sat., COP$30,000) are rivals for the top Peruvian food in town. At Astrid, try an only-in-Colombia coca pisco sour.\n\nThe little block of restaurants on Carrera 13 between Calles 85 and 86 has some great options. The vast menu of generous designer burgers at M **Agad\u00f3n Burger Bar** (Cra. 13 No. 85-75, tel. 1\/255-4138, noon-10pm Mon.-Wed., noon-midnight Thurs.-Sat., noon-4:30pm Sun., COP$20,000) will leave you satisfied. It's run by a pair of Israelis, and they even have a couple of vegetarian burgers on the menu. **Casa** (Cra. 13 No. 85-24, tel. 1\/236-3755, noon-midnight daily, COP$22,000), in a cozy house with fireplaces and cool artwork on the walls, serves food meant to be shared. You're surrounded by _palma bobas_ and other tropical plants in their patio. Across the street is elegant **La Brasserie** (Cra. 13 No. 85-35, tel. 1\/257-6402, noon-midnight Mon.-Sat., noon-6pm Sun., COP$32,000). Oh, yes, and the Clintons have dined here.\n\n**Opa** (Cra. 14 No. 90-80, tel. 1\/218-9682, noon-8pm daily, COP$12,000) is a cheap and good gyros joint popular with the work crowd on dull Calle 90. **Gyros y Kebabs** (Cra. 13 No. 82-28, tel. 1\/635-9324, noon-11pm daily, COP$15,000) does standard Lebanese food very well. It's cozy to sit by the bread oven and watch the bread guy slide pita bread in and back out, piping hot. Yes, they have an afternoon happy hour.\n\nIn Usaqu\u00e9n try **Abasto** (Cra. 6 No. 119B-52, tel. 1\/215-1286, <http:\/\/abasto.com.co>, 7am-10pm Mon.-Thurs., 7am-11pm Fri., 9am-10:30pm Sat., 9am-5pm Sun., COP$22,000), where they use only the freshest and often organic ingredients.\n\n##### **Asian**\n\nHaving had just a tiny Asian immigration, unlike Peru, Panama, and Brazil, Colombia doesn't have dazzling Asian cuisine, but you'll probably be pleasantly surprised at the offer in Bogot\u00e1. The pan-Asian cuisine chain M **Wok** (Quinta Camacho, Cra. 9 No. 69A-63, tel. 1\/212-0167, www.wok.com.co, noon-11pm Mon.-Sat., noon-8pm Sun., COP$23,000) is hard to beat. A second location is in the Museo Nacional (Cra. 6 Bis No. 29-07, tel. 1\/287-3194). The menu is astoundingly extensive, inventive, and fresh. Hearty fish soups and curries based in coconut milk will warm you up, and there are numerous vegetarian dishes, such as a Vietnamese-inspired grilled tofu sandwich. The quality of their sushi is also good. Wok is an environmentally and socially responsible company, working with family farmers and fishers in small communities throughout Colombia. Don't let the fact that it is a chain dissuade you.\n\nExcellent service awaits at sleek **Watakushi** (Cra. 12 No. 83-17, tel. 1\/744-9097, noon-3pm and 6pm-11pm Mon.-Thurs., noon-11pm Fri.-Sat., noon-5pm Sun., COP$35,000), one of the many restaurants operated by local restaurant wizard Leo Katz. It is on the Zona T. **Sushi Gozen** (Cl. 94 No. 14-11, tel. 1\/257-0282, noon-3pm and 6:30pm-midnight Mon.-Fri., noon-midnight Sat., COP$25,000) is a Colombian-Japanese restaurant popular with Japanese business people (a good sign) that has much more than just sushi. Finally, **Arigato** (Cl. 76 No. 12-22, tel. 1\/248-0764, 11am-9pm Mon.-Sat., COP$20,000) is a family-run Japanese restaurant. The fresh fish is flown in regularly from the Pacific Coast.\n\nThere are few Indian restaurants in Bogot\u00e1, but of those, **Flor de Loto** (Cl. 90 No. 17-31, tel. 1\/617-0142, noon-3 p.m. and 6pm-9pm Mon.-Sat., COP$24,000) is probably the best. The head chef is originally from the Punjab. The restaurant surrounds a peaceful garden. It's cash-only here.\n\n##### **Italian**\n\nItalian restaurants are in abundance in Bogot\u00e1. At M **La Divina Comedia** (Cl. 71 No. 5-93, tel. 1\/317-6987, noon-4pm and 7pm-11pm Mon.-Sat., COP$25,000) go for the divine _tortellata_ (a mix of stuffed pastas). At unpretentious **Trattor\u00eda San Giorgio** (Cl. 81 No. 8-81, tel. 1\/212-3962, noon-10:30pm Mon.-Sat., noon-6pm Sun., COP$22,000), Italian regulars are often found sipping wine and enjoying a multi-course meal. M **DiLucca** (Cra.13 No. 85-32, tel. 1\/257-4269, noon-midnight daily, COP$25,000) is consistently good, with both pastas and pizzas, and they deliver. The atmosphere is rather lively inside.\n\nM **Julia** (Cra. 5 No. 69A-19, tel. 1\/348-2835, noon-11pm daily, COP$25,000) serves the best pizza in town. Theirs is the irregular-shaped and paper-thin Roman crust. They also have a couple of non-cheese pizzas, which is unusual for Bogot\u00e1. It is astounding to see what the chefs manage to come up with in such a tiny\u2014and infernally hot\u2014kitchen. As no reservations are accepted in this tiny restaurant, it's best to get there on the early side. Second best pizza in town? That would be **Archie's** (Cl. 82 No. 13-07, tel. 1\/610-9162, www.archiespizza.com, breakfast to late night daily, COP$22,000). This chain is all over Bogot\u00e1 and in many cities countrywide. Archie's delivers.\n\n### **Information and Services**\n\n#### **VISITOR INFORMATION**\n\nFor information on all things Bogot\u00e1, go to the **Punto de Informaci\u00f3n Tur\u00edstica** (PIT). In La Candelaria there is a PIT on the southwest corner of the Plaza de Bol\u00edvar (tel. 1\/283-7115, daily). Other locations include the Quiosco de la Luz in the Parque de la Independencia (Cra. 7 at Cl. 26, tel. 1\/284-2664, 9am-5pm Mon.-Sat., 10am-4pm Sun.), both airport terminals, both main and south bus terminals, and in the north at the Centro Comercial Granahorrar on Calle 72. The attendants will bend over backwards to help you out any way they can, providing maps and tips.\n\n#### **TELEPHONES**\n\nThe telephone code for Bogot\u00e1 and many surrounding towns is 1. From abroad, dial 57 for Colombia, then 1 for Bogot\u00e1, followed by the 7-digit number. To call a cell phone from a landline, first dial 03 and then the 10-digit number. To do the reverse, call 03-1 (the 1 for Bogot\u00e1). Prepaid cell phones or SIM cards can be purchased at any Claro or Movistar store. When in the city, there is no need to dial the 1 before the landline number.\n\n##### **Emergency Numbers**\n\nFor emergencies, just remember 1-2-3. The single emergency hotline is 123. While some operators may speak English, that is probably unlikely. You should provide the neighborhood you are in and a precise street number.\n\nU.S. citizens who have health, safety, or legal emergencies can contact the **U.S. Embassy** at 1\/275-2000.\n\n#### **POST OFFICES AND COURIER SERVICES**\n\nHere are some of the main offices of 4-72, the national post office, in Bogot\u00e1: Centro Internacional (Cra. 7 No. 27-54, tel. 1\/245-4015), Chapinero (Cl. 67 No. 8-39, tel. 1\/248-7810), and Chic\u00f3 (Car. 15 No. 85-61, tel. 1\/621-9508). The office hours are the same at all locations (8am-5pm Mon.-Fri., 9am-1pm Sat.). Private courier services are the way most people send correspondence domestically. Two major companies are Servientrega and Deprisa. **Servientrega** is in the Centro Internacional (Mailboxes Cra. 13 No. 32-16) and in Chapinero (Mailboxes Cl. 67 No. 7-28). **Deprisa** is operated by Avianca. There are many offices throughout the city, including in the north at the Centro Comercial Granahorrar (Cl. 73 No. 9-42), near Andino (Cr. 11 No. 81-17), and near the Parque El Virrey (Cra. 15 No. 88-53). Downtown they can be found at Calle 13 No. 7-09. Hours of these offices are typically 8am-5pm Monday-Friday and 9am-1pm Saturday.\n\n#### **INTERNET CAF\u00c9S**\n\nInternet caf\u00e9s are plentiful, especially in La Candelaria area and Chapinero, although you may have some difficulty locating caf\u00e9s in the wealthier residential neighborhoods of Chic\u00f3 and Rosales.\n\n#### **NEWSPAPERS AND MAGAZINES**\n\nThe **_City Paper_** is a free monthly newspaper in English with information on events, interesting profiles, and essays. It is generally distributed to hotels, restaurants, and caf\u00e9s during the first two weeks of the month. Two other freebies, **_ADN_** and **_Metro,_** both in Spanish, are newspapers that are distributed on street corners in the mornings. Another fun publication is the hip and free bimonthly **_Cartel Urbano,_** which examines Bogot\u00e1 cultural life. **_GO_** is a monthly publication on things going on in the city. **_El Tiempo_** and **_El Espectador_** are the two main newspapers in town, and are good sources for information. **_Semana_** is considered the best news magazine, and is published weekly.\n\n#### **SPANISH LANGUAGE COURSES**\n\nThe Spanish spoken in Bogot\u00e1 is considered neutral and clear, compared to accents you may have heard in the Caribbean, Spain, or Argentina. Therefore Bogot\u00e1 is an excellent place to study Spanish, if you have some time to invest. The best schools are operated by the major universities in town. These include the **Universidad Externado** (Centro de Espa\u00f1ol para Extranjeros, CEPEX, Cra. 1A No. 12-53, tel. 1\/353-7000 or 1\/342-0288, www.uexternado.edu.co\/cepex), the **Universidad Nacional** (Edificio 229, Torre Sur, Primer Nivel, tel. 1\/316-5000), and the **Universidad Javeriana Centro Latinoamericano** (Transv. 4 No. 42-00 Piso 6, tel. 1\/320-8320, www.javeriana.edu.co\/centrolatino).\n\n#### **MONEY**\n\nATMs are everywhere throughout the city, and this is probably your best option to get Colombian pesos. Transaction fees vary. Some ATMs on the streets are closed at night. Be discreet and cautious when taking out money.\n\nTo change money, try **New York Money** at the Centro Andino mall (tel. 1\/616-8946), Atlantis Plaza (tel. 1\/530-7432), or at the Centro Comercial Granahorrar shopping center (tel. 1\/212-2123). They are open also on Sundays and holidays. Note that you'll need to show your passport to change money.\n\n#### **VISAS AND OFFICIALDOM**\n\nIf you need to stay beyond the 60 or 90 days allowed to visitors from the United States, Canada, Australia, New Zealand, and most European countries, you will need to go to **Migraci\u00f3n Colombia** (Cl. 100 No. 11B-27, tel. 1\/595-4331). It is best to go there a few days before your current visa expires.\n\n#### **HEALTH**\n\n##### **Altitude**\n\nAt 2,580 meters, Bogot\u00e1 is the third highest capital city in the world (behind La Paz, Bolivia, and Quito, Ecuador). It is common to feel short of breath and fatigued during the first two days at the higher altitude. Other symptoms of altitude sickness include headache and nausea. Take it easy for those first few days in Bogot\u00e1 and avoid caffeine and alcohol. If you are sensitive to high altitude, see a doctor, who can prescribe medication to mitigate the effects of high altitude, before your trip. If you feel symptoms such as fever or gradual loss of consciousness, see a doctor immediately.\n\nAlso, keep in mind that, being so high up, you are also that much closer to the sun. When it is sunny those rays are deceivingly potent.\n\n##### **Hospitals, Clinics, and Pharmacies**\n\nBogot\u00e1 has excellent physicians and hospitals. Two of the best hospitals are the **Fundaci\u00f3n Santa Fe** (Cl. 119 No. 7-75, www.fsfb.org.co, emergency tel. 1\/629-0477, tel. 1\/603-0303) and the **Cl\u00ednica del Country** (Cra. 16 No. 82-57, tel. 1\/530-1350, www.clinicadelcountry.com). The **Fundaci\u00f3n Shaio** (Diag. 110 No. 53-67, emergency tel. 271-4050, tel. 1\/624-3211) hospital specializes in cardiology. For sexual and reproductive health matters, **Profamilia** (Cl. 34 No. 14-52, tel. 1\/339-0900, www.profamilia.org.co), a member of the International Planned Parenthood Federation, offers clinical services. It is steps away from the Profamilia TransMilenio station on the Avenida Caracas.\n\nMom-and-pop pharmacies are all over the city, and sometimes these can be less stringent about requiring physical prescriptions. The Supermarket chain **Carulla** (www.carulla.com) usually has an in-store pharmacy, and the Venezuelan chain **Farmatodo** (tel. 1\/743-2100, www.farmatodo.com.co) has around 30 stores in Bogot\u00e1; some of them are open 24 hours a day.\n\n##### **Dental Services**\n\nDental care is excellent in Bogot\u00e1. Americans are known to come to Colombia specifically for dental treatments and surgeries. **Marl\u00f3n Becerra** (Cl. 91 No. 15-15, tel. 1\/746-1111) has several offices in Bogot\u00e1.\n\n#### **LAUNDRY**\n\nWash and dry services that charge by the pound or kilo are plentiful in La Candelaria and in Chapinero. This service is often called _lavander\u00eda,_ as opposed to dry cleaning ( _lavado en seco_ ). In La Candelaria two such services are: **Limpia Seco Sarita** (Cra. 3 No. 10-69, tel. 1\/233-9980) and **Extra-R\u00e1pido** (Cl. 12 No. 2-62, tel. 1\/282-1002). In Chapinero there is **Lava Seco** (Cra. 9 No. 61-03, tel. 1\/255-2582), another **Lava Seco** (Cl. 66 No. 8-20, tel. 1\/249-7072), and **Lavander\u00eda San \u00c1ngel** (Cl. 69 No. 11A-47, tel. 1\/255-8116). A good dry cleaning service is **Classic** (Cra. 13A No. 86A-13, tel. 1\/622-8759).\n\n### **Getting There**\n\n#### **BY AIR**\n\nThe **Aeropuerto Internacional El Dorado** (BOG, Cl. 26 No. 103-09. tel. 1\/266-2000, www.elnuevodorado.com) is undergoing a massive expansion. The international terminal will finally be connected with the Puente Aereo (Avianca domestic terminal).\n\nYou will need to show your luggage receipts before passing through customs. A customs agent will take one copy of the customs declaration and then you may be required to put both checked and carry-on bags through a scanner. They are mostly looking for weapons, cash, and fruits and vegetables.\n\nThere are money exchange offices and ATMs just outside of the customs area.\n\nIf you are not being picked up by a hotel shuttle or friend, it is imperative to use the official taxi services available outside the arrivals area.\n\n#### **BY BUS**\n\nBogot\u00e1 has three bus terminals. These are the **Terminal del Sur,** the **Portal del Norte,** and the main bus station, the **Terminal de Transportes** (Diagonal 23 No. 69-60, tel. 1\/423-3630, www.terminaldetransporte.gov.co) in Salitre.\n\nYou can catch a bus to just about anywhere at the Terminal de Transportes. The terminal is well organized and clean and is divided into three \"modules,\" each generally corresponding to a different direction: Module 1 is south, Module 2 is east\/west, and Module 3 is north.\n\nThere are two other modules, 4 and 5, corresponding to long-distance taxi services and to arrivals. All modules are located in the same building. Each module has an information booth at the entrance with an attendant who can point you in the right direction.\n\nThe terminal has plenty of fast food restaurants, ATMs, a pharmacy, and a Dunkin' Donuts every 50 meters. You can stow your bags in a locker or check them in a storage area. Wireless Internet is available in some areas of the building, and if not, there are Internet and telephone services. To pass the time, you can win big at the many casinos in the terminal or enjoy some peace at the second-floor chapel.\n\nIn the arrivals module, there is a tourist information office (PIT), where the helpful attendants can give you a map of the city and assist you in getting to your hotel. There is also an organized and safe taxi service and plenty of public transportation options available.\n\nDuring the Christmas and Easter holidays, the bus terminal is a busy place with crowds and packed buses. This is also true on _puentes_ (long weekends).\n\nThe terminal website is not bad, with a map of the modules, information on bus companies with links to their websites, timetables, and price information. Prices listed online are comparable to the prices found at the terminal.\n\nCheck prices with a couple of companies, as levels of comfort can vary. Some companies even offer Wi-Fi in their buses, and most show loud, violent movies for your enjoyment.\n\nThe **Portal del Norte** (Autopista Norte with Cl. 174), part of the TransMilenio station of the same name, may be more convenient if you are traveling to nearby destinations. As you exit northbound TransMilenio buses, there are well-marked exits to platforms for different nearby destinations. The area for Zipaquir\u00e1 (shortened to Zipa on signage) and Ch\u00eda is at the far left of the platform. Destinations such as Laguna de Guatavita (COP$7,400), Sesquil\u00e9 (COP$5,500), Suesca (COP$6,000), and Tenjo (COP$3,600) are straight ahead. You pay the bus driver directly for these trips.\n\nTransMilenio bus along Avenida Jim\u00e9nez\n\nMeanwhile, in front of the \u00c9xito supermarket\/store on the east side of the Autopista is where buses going a little farther on to places such as Nemoc\u00f3n, Villa de Leyva, Tunja, and Bucaramanga pick up passengers. It is not nearly as organized as at the main terminal, but it all somehow manages to work out. It's best to catch these buses during the daytime.\n\nThe **Terminal del Sur** (Autopista Sur with Cra. 72D) is near the Portal Sur of TransMilenio. This station serves locations in the south of Cundinamarca, such as Tequendama, and farther south to Girardot, Ibagu\u00e9, Neiva, Popay\u00e1n, Armenia, Cali, and all the way to Mocoa in Putumayo.\n\n### **Getting Around**\n\n#### **TRANSMILENIO**\n\nWhile it has become the public transport system that Bogotanos seem to love to hate, the red buses of TransMilenio have clearly transformed the city. This dedicated mass transit system began rolling along the Avenida Caracas in 2000 near the end of the Enrique Pe\u00f1alosa mayorship. Today, the Caracas line moves more passengers than most subway lines. One of the great characteristics of the TransMilenio project is that, in addition to buses, a requirement has been made to create wide sidewalks, pedestrian bridges, and bike lanes alongside the bus lines. The system has been lauded internationally and copied by several cities as a very cost effective mass transit system. However, lack of investment in new stations and buses has meant that the buses are overcrowded.\n\nHow do you determine which bus to take? It's not that easy at first. The stations are divided into 12 zones. For example, the Museo del Oro is classified as Zona J and Calle 85 is Zona B. The first step may be to figure out the zone you are going to and the zone you are starting out from. Because there are many express lines, or lines that skip certain stations, figuring out the system map may give you a migraine. If you are not up for the challenge, just ask one of the attendants at the station.\n\nYou are most likely to use the Caracas line to go north and south, with buses that spur off to the Las Aguas station as a convenient way of getting to Centro Hist\u00f3rico attractions. In the opposite direction, that same line\u2014bound for the Portal del Norte\u2014will get you close to the Zona Rosa (Calle 85). The line that goes along the Avenida 30 stops near the Parque Sim\u00f3n Bol\u00edvar, the Estadio El Camp\u00edn, the Universidad Nacional, and Paloquemao. The new line that goes to the Portal Eldorado stops at the cemeteries (Centro Memoria), the fairgrounds (Corferias), and close to the U.S. Embassy (Gobernaci\u00f3n). Unfortunately, TransMilenio is not an option to reach the airport.\n\nThe system operates 5am-midnight Monday through Saturday and 6am-11pm on Sunday and holidays. Fares are a little more expensive during rush hours (COP$1,700 as opposed to COP$1,400). Also, Bogotanos are usually in a hurry to get where they're going. So the \"local\" routes\u2014often called \"Ruta F\u00e1cil\"\u2014may be less crowded. Keep your wallet in your front pocket and watch your things, especially during rush hour.\n\nIn 2012, Bogot\u00e1 began replacing private buses with the city SITP buses (www.sitp.gov.co). SITP buses stop only at designated stops. You must purchase a refillable card to ride, which can be found at numerous locations. This is the best way to travel between La Candelaria and the north.\n\n#### **PRIVATE BUSES**\n\nThey are intimidating at first, but sometimes private buses are the only way to go. The good (and bad) thing about buses is that, although they are not supposed to, they will stop just about anywhere, even in the middle of the street. In both big buses and _colectivos_ (minivans), you pay the driver upon entry. It's best to have small bills and change. Exit buses at the back door. You can use the button to alert the driver to stop. Sometimes buses don't come to a complete stop (especially for young men)\u2014they just slow down. Take precautions when exiting any bus that hasn't stopped completely. Heading downtown from the Zona Rosa or Chapinero areas, look for buses that say Normand\u00eda on them.\n\n#### **TAXIS**\n\nIt's estimated that over a million people take a cab each day in the city. There are around 50,000 taxis (mostly yellow Hyundai vehicles) circulating the streets of Bogot\u00e1, so you will rarely have a hard time locating one. However, it's important to always order cabs by phone or the smartphone app Tappsi for safety.\n\nA trip from the Zona Rosa area to La Candelaria costs around COP$15,000. A _taximetro_ calculates units, which determine the price. The rates are listed on a _tarjet\u00f3n_ (large card) with the driver's information. That card is always supposed to be visible. There are special surcharges for cab services ordered by phone, for nighttime, and for going to the airport. Taxi drivers do not expect tips, but you can always round up the fare if you'd like. During the end-of-year holidays, drivers may ask for a holiday tip.\n\n#### **WALKING**\n\nYou get a real feeling for the city\u2014the good, the bad, and the ugly\u2014by walking its streets. All areas from the historic district through to the Centro Internacional and Macarena are accessible on foot, and walking is often the best way to get around. The same is true for upscale shopping and residential areas to the north. All of these areas are safe to walk around, but it's never a good idea to advertise your tourist status with bulky cameras and backpacks. Old, leafy neighborhoods like Teusaquillo-La Soledad are nice to wander around during the daytime as well. The worst thing about walking in the city is dealing with drivers. Generally speaking, they have very little respect for pedestrians or cyclists. Look for stoplights at intersections to help you safely cross streets. Also, note that when traffic lights turn yellow, that means green. Finally, keep in mind that, when there is no traffic, there are apparently also no speed limits.\n\n#### **BIKING**\n\nBogot\u00e1 has a huge bike path network, one of the most extensive in Latin America. These are called the _ciclorutas_. For those tired of getting stuck in traffic or dealing with buses, biking it to work has become a nice alternative. But you really have to keep your wits about you. Crossing the street can be tricky, as drivers don't hold a lot of respect for bikers, unfortunately. In addition, most bike paths follow alongside busy thoroughfares like the Carrera 11, so, during rush hours especially, you could be inhaling polluted air. It's nicer on weekends or late at night. If in town for just a limited time, you may prefer to make your bike experience a stress-free Ciclov\u00eda one (Sundays and holidays), rather than having to deal with the bike paths. A map of the 344-kilometer bike lane network is available at www.movilidadbogota.gov.co. As part of the public spaces along TransMilenio routes there are always _ciclorutas_ , such as along Calle 26 (Avenida El Dorado). Another long stretch from downtown to the World Trade Center on Calle 100 goes along the Carrera 11. It's strongly recommended to wear a helmet, and using a bike lock is a good idea.\n\n#### **CAR RENTAL**\n\nWith more than a million aggressive drivers on the clogged streets of Bogot\u00e1, renting a vehicle is a horrible idea for visitors. However, if you are planning to travel to places nearby Bogot\u00e1 like Villa de Leyva or would like to take your time touring parks or villages, it might be an option. **National** (Cra. 7 No. 145-71, www.nationalcolombia.com), **Avis** (Av. 19 No. 123-52 Local 2, tel. 1\/629-1722, www.avis.com), and **Hertz** (Av. Caracas No. 28A-17, tel. 1\/327-6700, www.rentacarcolombia.co) have offices in Bogot\u00e1. Your driver's license is accepted here in Colombia.\n\n### **Vicinity of Bogot\u00e1**\n\n#### **TENJO**\n\nThe town of Tenjo is charming. If you would like to visit a pueblo in the green pastureland outside of Bogot\u00e1, this makes a fine day trip. The plaza is shady and compact with a colonial church on one side. People keep warm wrapped in their heavy woolen _ruanas_ (ponchos) as they relax on park benches, and people sell _obleas_ (wafers) and _almojabanas_ (cheese rolls) to passersby.\n\nJust outside of town are two recommended restaurants: **La Granja** (Km. 12 V\u00eda Siberia-Tenjo, tel. 1\/864-6148, www.lagranjatenjo.com, COP$25,000), which has a petting farm, and nearby **Viveros Tirr\u00e1** (Km. 11 V\u00eda Siberia-Tenjo, cell tel. 312\/397-9940, www.restauranteviverostirra.com, COP$25,000). Both of these are popular with families on the weekends.\n\n#### **ZIPAQUIR\u00c1 AND VICINITY**\n\nA favorite day trip for visitors to Bogot\u00e1 is the city of **Zipaquir\u00e1** (pop. 112,000). About an hour's drive from Bogot\u00e1, Zipaquir\u00e1 is known for its Catedral de Sal\u2014a cathedral built in a salt mine. Zipaquir\u00e1 is named for the Muisca leader of the Bacat\u00e1 confederation\u2014the Zipa. The Muisca settlement was very close to the mines, and they traded salt for other commodities with other indigenous groups.\n\nThe **Catedral de Sal** (tel. 1\/852-3010, www.catedraldesal.gov.co, 9am-5:30pm daily, COP$20,000) is part of the Parque del Sal and the top \"Wonders of Colombia\" as voted by Colombians. The original cathedral was built by miners in 1951, but due to safety concerns, a new and larger cathedral was built and opened in 1995. The cathedral is indeed an impressive feat of engineering. Tours are obligatory, but you can stray from your group. The tours go past the Stations of the Cross and finally there is a massive cross in the cavernous sanctuary. Masses actually do take place here in the depths on Sundays, and they attract many faithful. Other features include a museum, a rock-climbing wall, and a children's 3-D film, which you could probably skip.\n\nThe picturesque main plaza in Zipaquir\u00e1, with palm trees rising against a backdrop of green mountains, is always the center of activity in town. Here locals gather to gossip, get their shoes shined, or munch on an _oblea_ (wafer) oozing with caramel. Dominating the plaza is a cathedral designed by Friar Domingo de Petr\u00e9s, who also designed the Bogot\u00e1 and Santa Fe de Antioquia cathedrals. Construction began in 1805; 111 years later, in 1916, it was completed and dedicated.\n\nOn the main road into town from Bogot\u00e1, there are several grilled meat-type restaurants, with teenage boys furiously waving red flags to attract customers. A local favorite for a hearty lunch of grilled fish, chicken, or vegetables is **Casa Nnova** (Cra. 10 at Cl. 3), across from the fancy La Cascada restaurant. A good option for overnight accommodations is **Hotel Cacique Real** (Cl. 6 No. 2-36, tel. 1\/851-0209, www.hotelcaciquereal.com, COP$88,000 d).\n\nZipaquir\u00e1 is an easy day trip. On weekends, families and tourists alike take the Turistren from the Usaqu\u00e9n station. Bands play Colombian _papayera_ music as you slowly chug through the savannah of Bogot\u00e1 on this three-hour trip. The train leaves in the morning and returns in the late afternoon, giving you more than enough time to visit the salt mines. You could also (and probably will want to) just take the train one way, returning by bus or taxi back to Bogot\u00e1. Otherwise, buses depart the Portal del Norte TransMilenio station every 10 minutes or so, all day long, and the fare is inexpensive. If arriving via TransMilenio, when exiting the bus take a left and you will see signs pointing the way for \"Zipa\" buses. The attendants can also direct you. You'll pay the bus driver directly. The trip takes about 45 minutes. You can either walk or take a short taxi ride to the Parque del Sal.\n\n##### M **Nemoc\u00f3n**\n\nWith only 10,000 residents, the sleepy pueblo of **Nemoc\u00f3n** (www.nemocon-cundinamarca.gov.co) is 15 kilometers from Zipaquir\u00e1 and just 65 kilometers from Bogot\u00e1. It is a cute, compact colonial-era town, also home to salt mines, but it does not attract nearly the same number of visitors that Zipaquir\u00e1 does. That's part of its allure. In pre-Columbian times, this was also a Muisca settlement devoted to salt extraction.\n\nOn the plaza, the church is set against a backdrop of eucalyptus-covered hills, obviously some sort of reforestation effort. There is a small salt museum on the corner, and students will be happy to give you a tour in Spanish. About a 10-minute walk toward the hills are the **mines** (9am-5pm daily, COP$17,000). Tours take about 90 minutes. The beautifully renovated section of the mines that you visit is no longer used for salt extraction. In the depths of the mines you will see all types of stalactites and stalagmites. The pools where salt and water were mixed to pump out the salt are a highlight. The reflection of the illuminated vaults on the surface of the pools, combined with the cool lighting, is amazing. You can take some fun photos inside. Once you get back out into daylight, there are simple restaurants with names like the **Venado de Oro** (the Golden Deer) on the plaza or **La Casa de la Gallina** (the Hen House) on Calle 2 (No. 4-24). There's not much in the way of accommodations in Nemoc\u00f3n, but most visitors make it a day trip. Not far from Nemoc\u00f3n is a desert microclimate called the **Desierto de Tatacoita.** It's a popular mountain-biking area.\n\nthe sleepy pueblo of Nemoc\u00f3n\n\n#### **LAGUNA DE GUATAVITA AND VICINITY**\n\n##### M **Laguna de Guatavita**\n\nThe El Dorado myth, which became an obsession for gold-thirsty Europeans in the New World, is based on a Muisca Indian ritual that took place here in this perfectly round mountain lake.\n\nFollowing the death of the Muisca _cacique_ (chief), a nephew would be chosen to succeed him. The day of the ceremony, the nephew would be sequestered in a cave. Then, stripped naked and covered with mud and gold dust, he would be rowed to the center of the sacred lake with incense and music filling the air. Once there, gold, silver, emeralds, and other tributes were tossed into the cold waters, and the cacique would dive in. Incidentally, the gold used in these ceremonies mostly came from outside Muisca territories. This was an area rich in emeralds, salt, and corn\u2014gold, not so much.\n\nPart of what historians know about the ceremony was confirmed with the finding in 1856 of the miniature golden raft depicting the ceremony. This piece was not found in Guatavita, but rather in a cave close to Bogot\u00e1. That raft, of course, is one of the main displays at the Museo del Oro.\n\nFrom the European perspective, there must have been a profound sense of disappointment once they realized that the \"city of gold\" just didn't exist. It was the promise of wealth beyond their wildest dreams that had sustained them as they made their arduous trek through the swamps of the muggy R\u00edo Magdalena valley, swatting away mosquitoes night after night as they desperately tried to get some rest. When they arrived in Guatavita, they drained the lake, at least three times, to see what could be found at the bottom. A giant cut in the lake can be still seen today.\n\nLaguna de Guatavita\n\nAfter years of neglect, today **Laguna de Guatavita** (9am-4pm Tues.-Sat., COP$13,600) is being given the respect it deserves. An environmental agency maintains the park and, in order to preserve it, has forbidden direct access to the lake. The lake is much better appreciated from above on the well-maintained path along the top of the crater. To reduce the impact on the fragile environment, the path does not go all the way around. On the weekends, you must join a tour group to see the lake. These leave every half hour. Guides are knowledgeable and passionate about their work. English tours are possible, especially for larger groups, but those should be reserved in advance. During the week you can amble along the path at your own pace.\n\nWhile much of the brick path is flat, there is a fairly steep climb, making it difficult for those with physical limitations. The entire tour takes less than an hour. At the end of the walk, you can walk or hop on a minibus to the entrance of the park. When Monday is a holiday, the park is open Wednesday-Monday.\n\n###### **GETTING THERE**\n\nGetting to the Laguna de Guatavita via public transportation is a little tricky, but doable. At the TransMilenio Portal del Norte station, take a bus bound for the town of Sesquil\u00e9. On the main square in front of the church you can usually find taxis that will take you to the Laguna de Guatavita and back (one way around COP$25,000). One (and just one) _colectivo_ (minivan) by Cootranscovadonga leaves at 8:45am on weekends and holidays for the park, returning at around 4pm, making for a very long day. In addition, several _colectivos_ leave all day long from Sesquil\u00e9 bound for El Hato\/Rancher\u00eda\/El Uval that can drop you off about a two-kilometer hike to the park.\n\nMany visitors opt to hire a driver for the day to make the trip to Guatavita and back. This varies in price and it would be wise to check with a few drivers and bargain. Make sure to specify if there will be additional stops (for a coffee break or for a lakeside lunch). Hotels, hostels and travel agencies in Bogot\u00e1 can also arrange this trip for you.\n\nAnother option is driving. Once you get out of Bogot\u00e1, it is fairly stress-free for Colombian standards, as the road is good (mostly four-lane) and with decent signage. Take the Autopista Norte towards Tunja and take the exit to Sesquil\u00e9. Past the town a dirt road leads up to the park. If you get lost, locals along the road will help you find your way. You can park right at the park entrance.\n\nAn excellent place on the way for a mid-morning coffee and arepa is **Carajillo Restaurante** (Km. 41).\n\n##### **Nueva Guatavita**\n\nThe actual town of Guatavita no longer exists. Back in the 1960s, when this kind of thing could be done without cries from environmentalists or community activists, the town was flooded in order to build the **Emblase de Tomin\u00e9,** a large reservoir. They moved the town a little bit inland, calling it Nueva Guatavita. All the buildings here are painted white, as if they were built in the colonial period, but the streets are California-wide. There are two small museums by the water. The **Museo de Arte Religioso de Guatavita La Antigua** (COP$1,000) displays some of the relics from the church salvaged before the great flood. There is a short presentation about the flood and some photos of the submerged town. You used to be able to see the top of an obelisk from the town cemetery, but either a boat ran over it or it just crumbled into the depths. Nearby is a small indigenous culture museum. A few restaurants in town serve fried trout and other local specialties.\n\n##### **Emblase de Tomin\u00e9**\n\nBetween Nueva Guatavita and the Laguna de Guatavita alongside the Tomin\u00e9 reservoir are a handful of restaurants, hotels, and marinas.\n\n**La Juanita** (cell tel. 310\/213-5793, felipespath@gmail.com, www.lajuanitaguatavita.com\/place) is wonderfully crunchy-granola, in a quiet place within walking distance of the reservoir and about halfway between the Laguna de Guatavita and Nueva Guatavita. They grow their own vegetables, have yoga classes, and offer pottery making and horseback-riding excursions. You can even walk from there to the Laguna de Guatavita.\n\nOverlooking the water, family-run **Los Pinos** (Km. 11 V\u00eda Sesquil\u00e9-Guatavita, cell tel. 310\/777-6631, www.lospinos.com.co, weekends and holidays only) is a fine place for an afternoon lunch after a day at the lake. Grilled fish and barbecued pork ribs are their specialties, although they can accommodate vegetarians. You can rent a Sunfish sailboat for excursions on the water.\n\nLiterally next door to the restaurant, the **Club Marina de Guatavita** (cell tel. 312\/592-7468) rents sailboats (Sunfish COP$60,000 per day). You can also rent windsurfing equipment and take classes. You can water-ski for about COP$40,000 for a half-hour trip (including wetsuit). Canoes can be rented for only COP$10,000 per hour. There is also camping available next to the water, which costs COP$25,000. There are five very basic cabins for rent.\n\n#### **PARQUE NACIONAL NATURAL CHINGAZA**\n\nThe **Parque Nacional Natural Chingaza** extends over 76,000 hectares (188,000 acres) in Cundinamarca and Meta and makes for an excellent day trip of hiking among armies of _frailejones_ plants through the melancholy and misty _p\u00e1ramo_ (highland moor). One of the better hikes is a 3.5-hour one that takes you to the **Lagunas de Siecha,** which include the three mountain lakes: Suram\u00e9rica, Siecha, and Guasca. Along with Laguna de Guatavita, these were also sacred Muisca lakes.\n\nThe park limits the number of visitors, so it is best to request an entry permit in advance. Hire an experienced guide (COP$50,000\/day), as trails are often not obvious. The guides are generally very knowledgeable and friendly. However, they may not speak English. To arrange a visit, call **Parques Nacionales** (tel. 1\/353-2400, www.parquesnacionales.gov.co) and request permission to visit PNN Chingaza. They will ask you to send the names of the members of your group in an email and confirm the reservation. They will also provide contact information for the local association of guides. You can save a lot of money, and help the local economy, if you do it this way rather than through an organized private tour.\n\n_frailejones_ at the Parque Nacional Natural Chingaza\n\nPacking rubber boots is a must, as you will be hiking along really muddy paths. Sneakers just won't do. A light raincoat or windbreaker and sweatshirt are essential, as well as a packed lunch, snacks, and water. The hike to the _lagunas_ takes about 3.5 hours.\n\nOther excursions within the Parque Nacional Natural Chingaza can be made from different entry points.\n\nIn the town of **Guasca** you can relax, eat, and stay at the **Posada Caf\u00e9 La Huerta** (cell tel. 315\/742-0999, www.cafelahuerta.com). They make great American cornbread. If you decide to stay with them for a weekend, they can arrange your transportation and visit to PNN Chingaza.\n\nDown the road a ways from Posada Caf\u00e9 La Huerta is the abandoned colonial-era **Capilla de Siecha,** a picturesque white chapel in the middle of farmland. It is guarded by many sheep and some tiny dogs, and you can buy a ticket for entrance from an elderly farmer.\n\nFrom Bogot\u00e1, take a bus to Guasca and from there take a _buseta_ (minivan) towards Paso Hondo. The _buseta_ will leave you at the intersection of the road that leads to the park. It is about a 90-minute walk to the park entrance from there. These leave at 6:30am, 7:30am, and 9:30am and so forth. You should leave Bogot\u00e1 by around 7am to make the 9:30am _buseta_. Buses to Guasca leave from the Portal del Norte bus terminal as well as from an informal bus pickup area between Calles 72 and 73 at Carrera 14. It may sound iffy, but if you go there, you can ask any taxi or bus driver where to find the right bus.\n\nIf you travel with private transportation, it is possible to drive closer to the park, but only if you have a four-wheel-drive vehicle.\n\n#### **SUESCA**\n\nIt's all about rock climbing in Suesca. On the weekends, Bogotanos converge on this little Cundinamarca town and head to the _rocas._ Most climbing takes place along cliffs just behind the town\u2014some up to 250 meters high\u2014parallel to some old train tracks and the R\u00edo Bogot\u00e1. It's a beautiful setting, and the fresh smell of eucalyptus trees and mountain mist add to the feeling of the place.\n\nSeveral outdoors shops in Suesca rent equipment and organize rock-climbing classes and excursions. **Explora Suesca** (cell tel. 311\/249-3491 or 317\/516-2414, www.explorasuesca.com) rents out bikes in addition to rock-climbing gear and classes. **Monodedo** (Cra. 16 No. 82-22, tel. 1\/616-3467, cell tel. 316\/266-9399, www.mondodedo.com) has an office in Bogot\u00e1. Expect to pay around COP$60,000 for a three-hour rock-climbing excursion with a guide.\n\nMany folks like making Suesca a camping weekend. The most popular place for this is **Campo Base** (cell tel. 320\/241-9976 or 321\/415-3930), right across from the rocks. They've got hot water, a place for cooking, and they rent out tents.\n\nAt the other end\u2014way other end\u2014of the spectrum is **Casa Lila** (cell tel. 320\/204-8262 or 300\/835-9472, patriciavalenciaturismo@gmail.com, COP$220,000). This luxurious bed-and-breakfast with nine rooms is so cozy, with fireplaces all around and its own restaurant, that it may be hard to leave. It's right next to an old train station at kilometer 3. In between those two is **El Hostal Vivac** (cell tel. 312\/539-5408, www.elvivachostal.com, shared room COP$25,000, private room COP$65,000).\n\nAfter all that rock climbing, it's time for some Thai food. Check out **Restaurante Vamonos Pa'l Monte** for Phuket vegetables or pad Thai. It's right at the entrance of the _rocas_. The other really popular place is **Rica Pizza** (cell tel. 312\/379-3610), on the main road, also near the entrance to the park. They serve more than pizza.\n\nBuses from Bogot\u00e1 regularly serve Suesca from the Portal del Norte station. You can also contact one of the tour companies based in Bogot\u00e1 that specialize in rock climbing, who can arrange for transportation for a day trip.\n\n#### **SOUTH AND WEST OF BOGOT\u00c1**\n\n##### M **Parque Natural Chicaque**\n\nIt's hard to believe that such natural beauty is so close and accessible to Bogot\u00e1. You can basically take TransMilenio to this natural, private park and be walking amid the cloud forest within an hour.\n\n**Parque Natural Chicaque** (tel. 1\/368-3114\/18, www.chicaque.com) is a little-known and underappreciated gem in Bogot\u00e1. This private park offers 18 kilometers of excellent paths that meander through virgin cloud forest. It's more than likely that it is only silence that you hear as you explore the park, save for the occasional bird or rustle of leaves. It's got a pretty good website with detailed information on prices and services (in English as well).\n\nHorseback riding is possible, and you can climb a towering oak tree (and spend the night there for COP$110,000). There are two good restaurants, one at the entrance of the park and one below in the lodge. Rooms at the lodge are very simple, designed for families (one adult COP$86,000 including all meals). There are a few cabins with fireplaces for more privacy (COP$250,000 for two people and all meals), and camping (COP$50,000 including all meals) is also available.\n\nGetting to the park is fairly easy. You can take the TransMilenio to the Portal del Sur station and take a park bus from there. This bus leaves at 8am, 10am, 1:30pm, and 4:30pm. There is a fee of COP$6,000 for this trip. The rendezvous point is the fast food **Restaurante Choribroaster** (Cra. 72D No. 57J-03). It is near a large Jumbo store.\n\n##### **Salto del Tequendama**\n\nWest of Chicaque, on the road towards Mesitas del Colegio, is the **Salto del Tequendama** (Tequendama Waterfall). Back in the early part of the 20th century, this was a lovely place to visit, and the falls are certainly dramatic. What was once an elegant hotel overlooks the falls. It has very recently opened its doors once again, this time serving as an exhibition space. The problem with any visit to Tequendama is the horrible stench coming from the R\u00edo Bogot\u00e1, which you must drive along, into which much of the city's waste is poured. It is a shame.\n\nvista from the Parque Natural Chicaque\n\n##### **Parque de Orquideas del Tequendama**\n\nThe **Parque de Orquideas del Tequendama** (Km. 19 V\u00eda Bogot\u00e1-Mesitas del Colegio, cell tel. 300\/464-5960, www.orquideasdeltequendama.com, 10am-5pm Sat.-Sun., COP$7,000) has been described as _\"un jard\u00edn de los dioses\"_ (\"a garden of the gods\"). Within minutes of chilly Salto de Tequendama, you descend toward the R\u00edo Magdalena valley and it is suddenly\u2014and violently\u2014hot. At this farm, Omar Chaparro has around 6,000 varieties of orchids on display, including delicate miniature orchids, fragrant orchids, and many orchids that are nearly extinct. You'll see the official orchids of Bogot\u00e1 and Colombia there, too. A true expert and enthusiast on orchids, Omar has a seed bank with the ultimate goal of propagating orchid species and planting them in their natural habitats throughout Colombia so that they will survive for another generation. Guided tours, unfortunately only in Spanish, are interesting, as Omar and his staff educate guests on the flowers. There is a little open-air restaurant that serves lunch on Sundays, with, of course, colorful views. You can visit during the week if you make a reservation in advance.\n\nMidway between the falls and the orchid farm is the **Zool\u00f3gico Santa Cruz,** a zoo in which jaguars native to Colombia are sadly confined to small pens.\n\nPublic transportation is available from the Portal del Sur bus station to both the Salto del Tequendama and the orchid farm.\n\n##### **Bogot\u00e1 to Melgar**\n\nHeading from Bogot\u00e1 towards the R\u00edo Magdalena valley, you'll pass several _tierra caliente_ (hot country) resort towns, such as **Anapoima, Girardot,** and\u2014just across the border into the Tolima department\u2014 **Melgar** and **Carmen de Apical\u00e1.** Many wealthy Bogotano families have second homes in this part of Cundinamarca, while Melgar, the \"city of swimming pools,\" caters to a more middle-class clientele. These towns are easily reached by bus from Bogot\u00e1.\n\nParque de Orqu\u00eddeas del Tequendama\n\nSeveral websites (www.fincasenarriendo.com or www.alquilerdefincasenmelgar.net) rent houses in this part of the country. Searching under _alquilar_ (to rent) _, fincas_ (farms), and _casas_ (houses) will provide you with several options.\n\nDuring World War II, **Fusagasug\u00e1,** between Bogot\u00e1 and Melgar, had an internment facility for Germans, Japanese, and Italians at the Hotel La Sabaneta. Most of the more than 400 people there had moved to Colombia decades prior to the war and were forced to leave their families and stay in the hotel under the watchful eyes of the Colombian police. There is but a faint reminder of the facility, with just a part of the hotel facade still standing.\n\nA landmark just before Melgar that is an obsession of Colombian kids is the **Nariz del Diablo** (Devil's Nose), a rock formation that juts out along that windy road.\n\n##### **Bogot\u00e1 to Honda**\n\n**Guaduas** is home to one of the few heroines in the Colombian independence movement, Policarpa (endearingly known as La Pola) Salavarrieta. A thatched roof house, the **Casa de la Pola,** is now a small museum dedicated to her life and is located just off the main plaza. Near a statue of La Pola, vendors at a small outdoor market sell things like hammocks and goat's milk. The surrounding blocks of the town have several colonial buildings set along cobblestone streets. A popular place for pastries is **Nectar,** right on the square, and **La Pesebrera Gourmet** restaurant, in a colonial house nearby the plaza, is a fun place for a hearty lunch; it doubles as a tourist information center.\n\nThere is not much in the way of tourist attractions in this hot country town; however, its plaza is a divine place for a cool beer under the shade of massive ceiba trees. In the town of Villeta, the old two-lane road (through Facatativ\u00e1) and the newer road (which leads to Calle 80 in Bogot\u00e1) meet. Buses from Villeta to Bogot\u00e1 are frequent and cost around COP$12,500.\n\nPuente Navarro, Honda\n\nIn **Sasaima** , towards Facatativ\u00e1, is a lovely old country farm\/vacation home built by a Swiss architect in the 1940s that has been converted into a hotel\/spa. It's called **El Refugio** (tel. 1\/243-3620\/25, www.elrefugiohotelspa.com), and a refuge in the lush Colombian countryside it certainly is. Rooms are comfortable and the restaurant serves nice meals. It's a popular place for city slickers to get away from the honking horns of Bogot\u00e1.\n\n##### **Honda**\n\nThe steamy town of Honda, known as the City of Bridges, rests on the banks of the R\u00edo Magdalena, almost exactly halfway between Bogot\u00e1 and Medell\u00edn. It was the country's first and most important interior port city. The R\u00edo Magdalena has shaped its history, and was the reason for its rise to importance. From the 16th century until the mid-20th century, the R\u00edo Magdalena was the main transportation route connecting Bogot\u00e1 to the Caribbean coast and the rest of the world.\n\nHonda was founded in 1539. As a port, Honda was a place where quinine, coffee, lumber, and slaves were loaded and unloaded along the banks of the river.\n\nFrom this city of 29 bridges, steamships would ply the route toward the coast. In 1919, the Barranquilla-based SCADTA airlines became the very first airline of the Americas, bringing seaplane service to Honda. The river served as an airstrip. It must have been quite a sight to see these planes on the muddy Magdalena. SCADTA would later become Avianca Airlines.\n\nThe steamships and seaplanes no longer make their appearances in Honda today. But if you head down to the river's edge, you can ask a local angler to give you a quick jaunt along the river in his boat. Be sure to walk across the bright yellow **Puente Navarro,** a pedestrian bridge built by the San Francisco Bridge Company in 1898. Check out the **Museo del R\u00edo Magdalena** (Cl. 10 No. 9-01, tel. 8\/251-0129) on the way.\n\nHonda and its sleepy streets makes for a nice stopover en route between Bogot\u00e1 and Medell\u00edn. A popular weekend destination for Bogotanos in need of _tierra caliente_ (hot country) relaxation, the town has some good hotel options. For pampering try the **Posada Las Trampas** (Cra. 10A No. 11-05, tel. 8\/251-7415, www.posadalastrampas.com). For a friendly welcome, stay at the **Casa Belle Epoque** (Cl. 12 No. 12A-21, tel. 8\/251-1176, www.casabelleepoque.com), a moderately priced hotel popular with international travelers.\n\n##### **Tobia**\n\nThe one and only game in town in Tobia, about 2.5 hours from Bogot\u00e1, is adventure sports. Specifically that includes white-water rafting on the **R\u00edo Negro,** ziplines, rock climbing, and horseback riding. The scenery is quite beautiful as you enter town through a valley surrounded by cliffs. The town itself is nothing special.\n\nIt will not be hard to find tour companies to help you organize your adventures. Touts are everywhere selling packages. Here are some of the prices: rafting (COP$30,000), horseback riding (COP$30,000), and canopy tours (COP$40,000). **EcoAndes** (tel. 1\/803-1130 or 1\/252-6529, www.ecoandes.net) is a good option for a tour operator. **Los Tobianos** (cell tel. 314\/397-0360, www.lostobianos.com) and **Dosis Verde** (tel. 1\/232-3735 or 1\/492-9329, www.dosisverde.com) are others.\n\nThe largest of the basic hotels in town is **La Gaitana** (tel. 1\/631-0461, cell tel. 313\/466-9092, www.lagaitana.com). On the other side of the river is **Hotel San Juanito.**\n\nTobia is easily reached by public transportation, costing around COP$16,000 from the Terminal de Buses. It is also a pretty easy drive if you have your own car. There are some expensive tolls, however.\n\nAn excellent stop-off between El Vino and La Vega is **La Vara** restaurant, overlooking a verdant valley. You can enjoy an arepa (cornmeal cake) and chocolate or a heartier, meatier lunch.\n\nBuses depart all day long for both of these \"hot country\" destinations. It takes 2.5-3 hours to get there from the Terminal de Transporte.\n\n## **CARTAGENA AND THE CARIBBEAN COAST**\n\nHIGHLIGHTS\n\nPLANNING YOUR TIME\n\nCartagena\n\nSIGHTS\n\nENTERTAINMENT AND EVENTS\n\nSHOPPING\n\nSPORTS AND RECREATION\n\nACCOMMODATIONS\n\nFOOD\n\nINFORMATION AND SERVICES\n\nGETTING THERE\n\nGETTING AROUND\n\nBOCACHICA\n\nPLAYA BLANCA AND ISLAS DEL ROSARIO\n\nMOMPOX\n\nBarranquilla\n\nSIGHTS\n\nFESTIVALS AND EVENTS\n\nACCOMMODATIONS\n\nFOOD\n\nGETTING THERE\n\nGETTING AROUND\n\nPUERTO COLOMBIA\n\nSanta Marta\n\nORIENTATION\n\nSIGHTS\n\nSHOPPING\n\nRECREATION\n\nACCOMMODATIONS\n\nFOOD\n\nINFORMATION AND SERVICES\n\nGETTING THERE AND AROUND\n\nTAGANGA\n\nM MINCA\n\nM PARQUE NACIONAL NATURAL TAYRONA\n\nPNN TAYRONA TO PALOMINO\n\nPALOMINO\n\nPARQUE NACIONAL NATURAL SIERRA NEVADA DE SANTA MARTA\n\nLa Guajira\n\nRIOHACHA\n\nSANTUARIO DE FAUNA Y FLORA LOS FLAMENCOS\n\nM ALTA GUAJIRA\n\nCABO DE LA VELA\n\nPUNTA GALLINAS\n\nPARQUE NACIONAL NATURAL MACUIRA\n\nWestern Caribbean Coast\n\nGOLFO DE MORROSQUILLO\n\nSAN ANTERO\n\nMONTER\u00cdA\n\nM CAPURGAN\u00c1 AND SAPZURRO\n\nColombia's Caribbean coast extends 1,760 kilometers (1,095 miles), from Venezuela to Panama, and is longer than California's coastline. The coastal area varies dramatically, with an astonishing array of landscapes: desolate deserts, snowcapped mountains, lowland swamps, dry savannahs, and rainforest. This region has so much to offer, it could easily be your only destination in Colombia.\n\n**HIGHLIGHTS**\n\nLOOK FOR M TO FIND RECOMMENDED SIGHTS, ACTIVITIES, DINING, AND LODGING.\n\nM **Cartagena's Old City:** Lose yourself in the romance of Cartagena's narrow streets and plazas, and cap off your day sipping a mojito atop the massive fortified walls (click here).\n\nM **Carnaval de Barranquilla:** Madcap and euphoric, this is Colombia's most famous celebration. Put on a costume and dance your way down the parade route (click here).\n\nM **Minca:** Take a break from the beach and chill in this refreshing town set in the foothills of the spectacular Sierra Nevada de Santa Marta mountains (click here).\n\nM **Parque Nacional Natural Tayrona:** Mountains meet jungles meet beaches at this popular national park near Santa Marta (click here).\n\nM **Ciudad Perdida Trek:** Climb the thousand-plus stone steps through the cloud forest to the mystical Lost City, the most important settlement of the Tayrona civilization (click here).\n\nM **Alta Guajira:** Be mesmerized by the stark beauty of desert landscapes, get to know Way\u00fau culture, and dine on fresh lobster at the top of South America (click here).\n\nM **Capurgan\u00e1 and Sapzurro:** Walk barefoot along deserted beaches, trek through dense rainforest among colorful frogs and howling monkeys, or cool off in a crystalline brook (click here).\n\nCartagena is a majestic walled city full of magnificent churches and palaces, picturesque balcony-lined streets, and romantic plazas. Colombia's colonial past lives on in Mompox, a once-thriving port on the R\u00edo Magdalena where it feels as if time has stopped. The old city of Santa Marta has positioned itself as a great base from which to explore the beaches and mountains of the north-central coast.\n\nBeach options abound here. The most famous are at Parque Nacional Natural Tayrona: glimmering golden sand beaches with the jungle backdrop of the Sierra Nevada. Islands in the Parque Nacional Natural Corales del Rosario y San Bernardo, between Tol\u00fa and Cartagena, beckon visitors with their white sandy beaches and five-star hotels.\n\nThere are many options for nature lovers. Minca, a small town located on the slopes of the Sierra Nevada not far from Santa Marta, offers unparalleled bird-watching opportunities. In the jungles that envelop Capurgan\u00e1 and Sapzurro, you can go bird-watching, listen to the cries of howler monkeys, and count the colorful frogs you encounter along the many jungle paths in the area. Offshore, dive with the occasional sea turtle and observe myriad marine life in nearby waters.\n\nAdventurous types can hike up to Ciudad Perdida in the Sierra Nevada. The views of the Lost City, with its eerie, beautiful terraces set atop the mountain, are simply unforgettable. Travel up the Guajira Peninsula, home of the Way\u00fau, Colombia's largest indigenous community. Here you'll find Cabo de la Vela, where the desert meets the sea, and stark, magnificent Punta Gallinas, the northernmost point in South America, where windswept dunes drop dramatically into Caribbean waters.\n\nThe Caribbean coast is vibrant with music and dance. It is the birthplace of _vallenato_ (love ballads accompanied by accordion), which has European, African, and Amerindian roots. The coast is also home to many musical strains and dances with strong African rhythms, such as _cumbia,_ a melodious traditional music once danced by African slaves, and _mapal\u00e9_ and _champeta,_ a more recent urban music born in Cartagena. The Carnaval de Baranquilla, declared a Masterpiece of the Intangible Heritage of Humanity by UNESCO, offers an unparalleled introduction to Caribbean music and folklore. Any time of year, Barranquilla's modern, interactive Museo del Caribe is an excellent overview of the people of the Caribbean and their very vibrant culture.\n\n#### **PLANNING YOUR TIME**\n\nThere are a lot of sights to see in Cartagena. Take some time to wander the streets of the Old City and soak up the beauty and atmosphere. Two days will suffice, but three days is ideal. The coastal island of Bar\u00fa or the Islas del Rosario archipelago can be done in an easy day or overnight trip.\n\nFrom Cartagena, there are many possible excursions. Getting to the riverside town of Mompox involves five hours of travel by road and river, so staying two nights there is necessary. The seaside towns of Cove\u00f1as and Tol\u00fa are easy excursions with direct bus links from Cartagena. Barranquilla is an easy day or overnight trip.\n\nSanta Marta is an excellent base for exploring the northern coast. If you prefer to be on the sea, the laid-back seaside village of Taganga, only a short ride from Santa Marta, could also be your base for exploration of the north.\n\nSanta Marta offers many excursions. Spending a few days in the tranquil Sierra Nevada town of Minca is a welcome escape from the heat and an excellent base to explore the nearby mountains. The Parque Nacional Natural Tayrona is a possible day trip from Santa Marta or Taganga, but you will probably want to stay at least two nights to explore its beaches and jungles. The seaside resort of Palomino is just a 1.5-hour ride from Santa Marta. All of these towns make fine starting points for the four- to five-day trek to Ciudad Perdida, a must for any backpacker.\n\nMost visitors to the Guajira Peninsula join an organized tour from Riohacha. From Riohacha it is possible to visit Cabo de la Vela via a bus to Uribia and then a ride in the back of a local passenger truck, but it will take several hours. Most tourists visit Cabo de la Vela as part of a tour of La Guajira. There is no transportation north of Cabo de la Vela, so you will need to join a tour to get to magnificent Punta Gallinas or to Parque Nacional Natural Macuira. Do not drive though the Guajira Peninsula without a guide: It is dangerous.\n\nTo the southwest of Cartagena, you'll find coastal communities Tol\u00fa and Cove\u00f1as and the untamed coastline of C\u00f3rdoba, all easily accessed by bus from Cartagena or Monter\u00eda. Medell\u00edn is the best gateway to the villages of Capurgan\u00e1 and Sapzurro on the Darien Gap. You could happily spend up to five days in this area.\n\n### **Cartagena**\n\nCartagena is unforgettable and magical, a highlight of any trip to Colombia. The main attraction is the Old City (Cartagena's _centro hist\u00f3rico_ ), which includes two districts: the Walled City (Ciudad Amurallada) and burgeoning Getseman\u00ed. The magnificent walls of Ciudad Amurallada enclose narrow streets lined with magnificent _casas altas_ (two-story houses that were home to wealthy merchants) with bougainvillea spilling out over their balconies. Getseman\u00ed is an old working-class colonial neighborhood (also once enclosed by a wall) that today lures visitors with its many lodging, dining, and nightlife options. Near the Old City is the magnificent Castillo de San Felipe, one of the most impressive Spanish fortifications in the New World.\n\nOutside the Old City, Cartagena is a large, poor, and sprawling city of almost one million people. If you want to get a sense of how big it is, visit La Popa for incredible views of the city and the bay. To get to know the real Cartagena first-hand, visit the frenetic Mercado de Bazurto and then, for contrast, take a stroll through high-end Bocagrande.\n\nIf you're looking to play at the beach, escape to the Islas del Rosario, as the beaches of Cartagena are unappealing.\n\n##### **History**\n\nCartagena de Indias was founded in 1533 by Spanish conquistador Pedro de Heredia on a small Carib indigenous settlement. The city owes its glory to its strategic geographic position and easily defended harbor. Heavily armed convoys of galleons sailed once a year from Spain to the New World, transporting agricultural and manufactured goods; on the way back they took on silver and gold from Peru and Mexico. The convoy's treasures were stored in Cartagena until the sail back to Spain.\n\nCartagena's Old City\n\nThe city proper was constructed at the north end of the large harbor, on a marshy island separated from the mainland. During the 16th century, the city was sacked by pirates numerous times, most notably by Sir Francis Drake in 1568. With the construction of fortifications, the city was spared major attacks. Castillo de San Felipe was built on the mainland to protect from an overland invasion. Pairs of forts were constructed at various passage points in the harbor to stop intruders. The construction of the fortifications took almost two centuries and was completed by mid-18th century.\n\nWith the construction of the Canal del Dique connecting Cartagena to the R\u00edo Magdalena in 1650, the city became the main entry port to Nueva Granada. The city also prospered as one of the main slave ports in Spanish America. It is estimated that more than one million slaves passed through the city. The Spanish crown forbade the enslavement of indigenous people, so plantation and mine owners bought African slaves, transported from Santo Domingo or West Africa. Conditions on and after the trip were horrendous. Slaves often escaped and created free communities known as _palenques,_ such as the town of San Basilio, south of Cartagena.\n\nDuring the fight for independence from Spain, Cartagena sided with the revolutionary movement. After Caracas, it was the second city in Nueva Granada to set up an independent junta, and it formally declared independence in 1812. In 1815, it was recaptured by the Spanish under the command of Pablo Murillo, after a siege that lasted more than three months. Cartagena was retaken by revolutionaries in 1821.\n\nDuring the 19th century, Cartagena no longer enjoyed the activity and status it had as one of Spain's main ports. In the last decades of the century, Barranquilla eclipsed Cartagena as the main center of economic activity on the Caribbean coast. The economic decline had one good side effect: preserving the colonial past. The Old City remained largely intact through the 20th century, prompting UNESCO to declare it a World Heritage Site.\n\nDuring the 20th century, Cartagena became a major industrial center and domestic tourist destination. In the past two decades, the city has received an estimated 100,000 displaced people, most fleeing violence in the Atlantic coast. The city has been unable to cope with this influx, and the result is a vast belt of shantytowns.\n\nCartagena remained relatively peaceful even during the worst periods of violence in the 1980s and 1990s, attracting tourists from across Colombia. In the past decade, Cartagena has become a major international tourist destination, with a proliferation of chic hotels in the Old City, glitzy Miami-style condominiums and hotels in Bocagrande, and new resorts along the coast north of the city. It boasts the Hay literary festival, a classical music festival, and an acclaimed film festival that attracts visitors from the world over.\n\n##### **Safety**\n\nAll in all, the city is a safe place; however, the general precautions for all Colombian cities apply to Cartagena as well. Be careful while walking on the walls after dark. Police have an iffy reputation here, and have been known to stop and frisk young non-Colombian men in the evening, purportedly out of suspicion of drug possession, but in actuality they are looking for money. Some of the poor fishing villages on the islands nearby Cartagena are not safe to visit, at least not alone.\n\n##### **Orientation**\n\nCartagena is located on the Caribbean coast of Colombia about 130 kilometers (80 miles) southwest of Barranquilla and over 1,100 kilometers (680 miles) north of Bogot\u00e1. The city is at the north end of a large bay with the same name.\n\nThe focal point of Cartagena is the **Old City,** known as the **_centro hist\u00f3rico,_** or simply El Centro. The Old City is the original Spanish settlement completely enclosed by a massive stone wall. The Walled City is set out in a fairly regular grid with numerous plazas. The streets here have names that change from block to block; no one really knows them or uses them. Find your way by identifying the main squares\u2014Torre de los Coches, Plaza de la Aduana, Plaza de Santo Domingo, Plaza de Bol\u00edvar, and Plaza Fernandez de Madrid\u2014and making your way from one to the other. It takes some practice, but walking the charming streets of the Old City is a pleasure.\n\nHistorically, the main entrance to the Walled City was the gate where the Torre del Reloj (clock tower) now stands. Just south is the Bah\u00eda de las \u00c1nimas and the Muelle de los Pegasos, from which point tourist boats to the Islas del Rosario depart.\n\nDuring colonial times, the poorer district of **Getseman\u00ed** was a separate island connected by bridge to the Old City. In the late 19th and early 20th centuries, the mangroves and marshes were filled in.\n\nSoutheast of Getseman\u00ed is **La Matuna,** a busy commercial center full of 20th-century high-rises. To the east is the massive Castillo de San Felipe and farther on is La Popa, the only significant hill in Cartagena. East of La Popa is Mercado de Bazurto, Cartagena's huge central market, which is on Avenida Pedro de Heredia, the main road that leads to the bus terminal and out of the city.\n\nJust north of the Old City is the 19th-century district of **El Cabrero,** where the villa of former president Rafael N\u00fa\u00f1ez, today the Casa Museo Rafael N\u00fa\u00f1ez, is located. Farther north are the residential neighborhoods of Marbella and Crespo, where the airport is. About two kilometers farther north, next to the fishing village of La Boquilla, is the seaside development of Las Am\u00e9ricas.\n\nSouth of the Old City is **Bocagrande,** with its many high-rise hotels and residential buildings. It was first developed in the 1960s and 1970s as a domestic tourist destination. It is the stomping ground of Cartagena's rich residents. Since around 2005, however, there has been a spurt of construction of high-end, Miami Beach-inspired white residential skyscrapers. The main attractions here are the beaches. They are very popular on weekends. (There's even a gay-ish beach at the very end known as Hollywood.) But they're just OK. The sand is gray, and there's a constant stream of persistent vendors and masseuses.\n\ntypical street in Cartagena\n\nThe peninsula of Castillo Grande on the bay side of Bocagrande is an upscale residential neighborhood. The sidewalk that wraps around it makes for a pleasant stroll or place to jog.\n\n#### **SIGHTS**\n\n##### M **Old City**\n\n###### **LAS MURALLAS**\n\nReferring to Cartagena's **_murallas_** (walls), Colombians endearingly call the city \"El Corralito de Piedra\" (little stone corral). These walls are one of the most salient features of the city. After Drake sacked the city in 1568, the Spanish started fortifying access to the bay and the perimeter wall around the city. The effort took almost two centuries to complete. The walls that can be seen today are mostly from the 17th and 18th centuries.\n\nThe most impressive part of the wall is the stretch that runs parallel to the sea. This includes three _baluartes_ (bulwarks, or ramparts) where Spaniards stood ready to defend the city from attack. The massive **Baluartes de San Lucas y de Santa Catalina,** built in the very north of the city to repel attacks from land, are known as Las Tenazas because they are shaped like pincers. When the sea started depositing sediments and expanding the seashore, thus enabling the enemy to maneuver south along the wall, the Spanish built a spike to halt them. This defensive structure, known as El Espig\u00f3n de la Tenaza, is now home to the **Museo de las Fortificaciones** (Baluarte de Santa Catalina, tel. 5\/656-0591, www.fortificacionesdecartagena.com, 8am-6pm daily, COP$7,000). At the westernmost tip of the walls, facing the sea, is the equally impressive **Baluarte Santo Domingo,** now home to Caf\u00e9 del Mar. At the southern tip of the segment of walls facing the sea, next to the Plaza Santa Teresa, are the **Baluartes de San Ignacio y de San Francisco Javier,** also home to a pleasant outdoor bar.\n\nA _paseo_ (walk) on the walls is the quintessential Cartagena late-afternoon experience, enjoyed by international visitors, Colombian honeymooners, and Cartagenan high school students still in their school uniforms. The best time for this promenade is around 5pm. In the evenings, vacationers head to the handful of bars for a pre- or post-dinner drink. Avoid strolling the wall late at night, especially alone.\n\n###### **CLAUSTRO AND IGLESIA SAN PEDRO CLAVER**\n\nThe **Claustro San Pedro Claver** (Plaza de San Pedro Claver No. 30-01, tel. 5\/664-4991, 8am-5:30pm Mon.-Fri., 8am-4:30pm Sat.-Sun., COP$9,000) is an old Jesuit monastery, now museum, where Pedro Claver served as a priest. He is known for his compassion towards newly arrived African slaves, and is said to have baptized thousands of them. For his dedication to slaves, the priest was the first person to be canonized in the New World. The museum has relics and art from the colonial era and sometimes hosts temporary exhibitions. You can visit the small quarters where San Pedro Claver lived and climb up to the choir balcony of the Iglesia de San Pedro Claver. The cloister has a three-story courtyard brimming with flowers and trees.\n\nAdjacent to the monastery is the **Iglesia de San Pedro Claver** (Plaza de San Pedro Claver No. 30-01, tel. 5\/664-4991, masses 6:45am and 6pm Mon.-Sat., 7am, 10am, noon, and 6pm Sun.), which is adorned by a beautiful marble altar and is the final resting place for San Pedro Claver.\n\n###### **MUSEO DE ARTE MODERNO DE CARTAGENA DE INDIAS**\n\nCartagena's main art museum, the **Museo de Arte Moderno de Cartagena de Indias** (Cl. 30 No. 4-08, Plaza de San Pedro Claver, tel. 5\/664-5815, 9am-noon and 3pm-6pm Mon.-Fri., 10am-1pm Sat., COP$5,000, Tues. free) is on the square in front of the San Pedro Claver plaza, which is filled with several metallic sculptures by Cartagenero Edgardo Carmona that depict quotidian scenes of Cartagena life. The museum has a small permanent collection of works from 20th-century Colombian artists, including native sons Alejandro Obreg\u00f3n and Enrique Grau. The museum is in the old Customs House.\n\n###### **MUSEO NAVAL DEL CARIBE**\n\nAdjoining the rear of the Iglesia de San Pedro Claver is the **Museo Naval del Caribe** (Cl. 31 No. 3-62, Cl. San Juan de Dios, tel. 5\/664-2440, www.museonavaldelcaribe.com, 10am-5:30pm daily, COP$7,000). This museum provides a history lesson of the earliest indigenous dwellers who lived in the area, continuing through the Spanish conquest and including a bit about the many (mostly English) pirates who tried to steal the Spaniards' gold loot, which they had absconded with from all across South America. The second floor has a lot of replicas of grand ships from the period and a history of the Colombian navy (you may be surprised to learn of its participation in the Korean War). There are few explanations in English. Part of the building dates to the 17th century and was a Jesuit convent; after they were expelled by the king, it was converted into a hospital.\n\n###### **PLAZA DE BOL\u00cdVAR**\n\nOne of the city's most pleasant squares is the **Plaza de Bol\u00edvar,** once used for bullfights. It was transformed into a park in the late 19th century. With a statue of Sim\u00f3n Bol\u00edvar in the center, nearly always with a disrespectful pigeon resting upon his head, this shady spot is very inviting.\n\n###### **CATEDRAL BAS\u00cdLICA MENOR**\n\nDiagonal to the Plaza de Bol\u00edvar is the built-to-last **Catedral Bas\u00edlica Menor** (tel. 5\/664-7283, masses 10am-noon Mon.-Sat., 8am, noon, and 7pm Sun., COP$13,000, free during masses). It was built in the 16th century and doubled as a fortress. It was attacked by Sir Francis Drake in 1586. The facade, along with most of the interior, has been stripped of the Italianate stucco exterior that was added in the 20th century and restored to its former austere stone look. The cathedral's pale orange belltower, which dates to the early 20th century, can be seen across the Old City. Stroll the ornate cathedral during or before a mass to visit for free. Audio guides are available 8am-6pm daily.\n\n###### **PALACIO DE LA INQUISICI\u00d3N**\n\nOn the south side of the plaza is the **Palacio de la Inquisici\u00f3n** (Cl. 34 No. 3-11, Plaza de Bol\u00edvar, tel. 5\/664-4570, 9am-6pm Mon.-Sat., 10am-4pm Sun., COP$15,000). This remarkable 18th-century construction, one of the finest examples of colonial architecture in Cartagena standing today, was the headquarters of the Spanish Inquisition in Cartagena. In this building was housed the Tribunal del Santo Oficio, whose purpose was to exert control over the Indians, mestizos, and African slaves not only in Nueva Granada but also in New World colonies in Central America, the Caribbean, and Venezuela. The tribunal was active from 1610 until the late 17th century. There were two other tribunals in the New World: one in Lima and another in Mexico City.\n\nThe first floor of the building is a museum displaying the weapons of torture employed by authorities as part of the Inquisition. In Cartagena as elsewhere, the most common punishable crime was \"witchcraft,\" and hundreds of supposed heretics (indigenous people were excluded from punishment) were condemned here. On the second floor are exhibition spaces dedicated to the restoration of the building and to the history of Cartagena. Most explanations are written in Spanish; you may decide to hire one of the English-speaking guides (COP$35,000 for a group up to five persons). On your way out, take a right and then another right onto the Calle de la Inquisici\u00f3n and look for a small window on the palace wall. This was a secret spot where citizens of colonial Cartagena could anonymously report others for various and sundry heresies.\n\n###### **MUSEO DEL ORO ZEN\u00da**\n\nThe **Museo del Oro Zen\u00fa** (Cra. 4 No. 33-26, Plaza de Bol\u00edvar, tel. 5\/660-0778, 10am-1pm and 3pm-7pm Tues.-Fri., 10am-1pm and 2pm-5pm Sat., 11am-4pm Sun., free), on the north side of the Plaza de Bol\u00edvar, exhibits gold jewelry and funerary objects from the Zen\u00fa indigenous people, who were the original dwellers of the R\u00edo Magdalena area and R\u00edo Sin\u00fa valley, to the southwest of Cartagena. It has excellent Spanish and English explanations. A smaller version of the Museo del Oro in Bogot\u00e1, this museum has a regional focus and is one of the few tributes to indigenous culture today in Cartagena.\n\n###### **CASA MUSEO RAFAEL N\u00da\u00d1EZ**\n\nJust beyond the wall in the Cabrero neighborhood is the **Casa Museo Rafael N\u00fa\u00f1ez** (Calle Real del Cabrero, tel. 5\/664-5305, 9am-5:30pm Tues.-Sun., COP$10,000). This is the house of Rafael N\u00fa\u00f1ez, the four-time former president of Colombia, author of the 1886 Colombian constitution, and author of the 11 verses of Colombia's national anthem. N\u00fa\u00f1ez governed Colombia from this, his coastal home. The museum, which underwent a renovation in 2013, has memorabilia from his political life and is a beautiful example of 19th-century Cartagena architecture. N\u00fa\u00f1ez was born and died in Cartagena, and he lies at rest in the Ermita de Nuestra Se\u00f1ora de las Mercedes across the street.\n\n##### **Castillo de San Felipe**\n\nThe largest Spanish fort in the New World, the magnificent **Castillo de San Felipe** (Cerro de San L\u00e1zaro, east of Old City, 8am-6pm daily, COP$17,000) must have given pirates pause as they contemplated an attack on the city. It was built atop the Cerro de San L\u00e1zaro outside of the Walled City to repel attacks by land. Construction was begun in 1639 and completed over a century later. It was never captured. Tunnels enabled soldiers to quickly move about without being noticed, and cells housed the occasional unlucky prisoner.\n\nToday, visitors ramble through tunnels and secret passageways (a flashlight will come in quite handy), and views from the highest points of the fort are magnificent.\n\nThe monolithic Castillo de San Felipe is one of the most impressive Spanish forts in the New World.\n\nThe best time to visit the fort is in the late afternoon, when the intense sun abates. Audio tours (COP$10,000) are available. For many, the view of the fort from a distance suffices, especially at nighttime when it is lit up. If you want to do things up for a special celebration, you can rent out the entire _castillo_ (fireworks are an additional charge!). Contact the Sociedad de Mejoras P\u00fablicas de Cartagena (Castillo de San Felipe de Barajas, tel. 5\/656-0590, www.fortificacionesdecartagena.com) for more information.\n\nTo get there, take a bus from Avenida Santander (COP$1,500). You can also walk from the Old City, though maneuvering through and around traffic is not fun. A taxi will cost COP$5,000.\n\n##### **La Popa**\n\n**La Popa** is a 150-meter-high (500-foot-high) hill east of the Castillo de San Felipe, so named because of its resemblance to a ship's stern ( _popa_ in Spanish). La Popa is home to the **Convento Nuestra Se\u00f1ora de la Candelaria** (Cra. 20A 29D-16, tel. 5\/666-0976, 9am-5:30pm daily, COP$5,000), which was built by Augustinian monks, reportedly on a pagan site of worship. The monastery has a lovely courtyard, a small chapel where faithful pray to the Virgen de la Candelaria, and memorabilia from Pope John Paul II's visit to the monastery in 1986. Most tourists come here for the views over Cartagena and out to the sea. You can take a taxi there from the Old City for COP$50,000-60,000 round-trip. Arrange the price in advance and make sure the driver will wait for you there.\n\n##### **Sightseeing Tours**\n\nThe folks at **This Is Cartagena** (Av. Centenario No. 30-42, tel. 5\/660-0969, www.ticartagena.com, 9am-6pm Mon.-Fri., COP$115,000) offer two tours of the city. One is a full city tour (about four hours) that hits all the major sights, from La Popa to the beaches of Bocagrande. It's the comfortable way to see the city: in an air-conditioned vehicle. Entries to all the sights are included in the tour price. The second tour is a walking tour of the Old City, in which you'll learn a lot about this fascinating city, its history, architecture, and people. This Is Cartagena offers several other tours, including day trips to the Islas de Rosario, a photography tour, and even a \"historic drinking tour.\"\n\n**Plaza to Plaza Walking Tour**\n\nthe Iglesia Santo Toribio, one of Cartagena's beautiful colonial churches\n\nThe best way to get to know Cartagena is to go for a morning stroll, finding your way from plaza to plaza, and even getting lost a couple of times.\n\nStart at the **Plaza de los Coches** (opposite Getseman\u00ed), once the main entry point to the city. It is easily identifiable by the iconic 19th-century **Torre del Reloj** (clock tower) that tops the entrance though the wall. Inside stands a statue of Cartagena's founder Pedro de Heredia. During the colonial period, this plaza was the site of the city's slave market. During much of the 20th century it bustled with commercial activity. Today, the plaza is filled with watering holes catering to visitors.\n\nImmediately to the southwest is the large triangular **Plaza de la Aduana,** once the seat of power in colonial Cartagena. It is surrounded by stately colonial mansions. A statue of Christopher Columbus presides in the center. It's also got a fair share of ATMs and is where the main tourist office is located.\n\nAdjacent to the southeast is the **Plaza de San Pedro,** a small square located in front of the towering **Iglesia de San Pedro Claver** and attached convent where Saint Peter Claver, a Jesuit monk dedicated to succoring African slaves, ministered.\n\nWalking two blocks north on Calle de San Pedro you'll arrive at the city's heart, the leafy **Plaza de Bol\u00edvar,** a shady park with benches, fountains, and a statue of Sim\u00f3n Bol\u00edvar in the middle. It is surrounded by some of the most important buildings of the city, including the **Catedral Bas\u00edlica Menor** and the **Palacio de la Inquisici\u00f3n.**\n\nOn Calle de los Santos de Piedra and Calle de Nuestra Se\u00f1ora del Rosario is the **Plaza de Santo Domingo,** heart of the former upper class quarter. You will notice many superb two-story _casas altas_ built by rich merchants. The plaza is dominated by the austere Iglesia de Santo Domingo. A rotund nude bronze sculpture by Fernando Botero, live musical performances, and many outdoor caf\u00e9s liven up the popular plaza.\n\nNext is the large, green **Plaza Fern\u00e1ndez de Madrid** in the historic working class San Diego district. On one side is the charming **Iglesia Santo Toribio** with its magnificent wooden ceiling and cannonball damage (compliments of English pirate Edward Vernon).\n\nOn Calle Cochera del Hobo is the tiny **Plaza San Diego,** which is surrounded by inviting restaurants. It's also where you can join the locals who gather around Do\u00f1a Dora's food stall. This is the place to sample a _carima\u00f1ola_ and _arepa de huevo,_ two deep-fried and totally delicious treats.\n\nEnd your roaming at the **Plaza de las B\u00f3vedas** at the extreme northwest of the city. Once the location of a military storehouse, this is where you can load up on handicrafts at the Galer\u00eda de las B\u00f3vedas.\n\n#### **ENTERTAINMENT AND EVENTS**\n\n##### **Nightlife**\n\n**La Esquina Sandiegana** (corner of Cl. del Sant\u00edsimo and Cl. de los P\u00fantales, 5pm-2am Sun.-Thurs., 5pm-3am Fri.-Sat., no cover) is a locals' place, where the music is salsa and the drink is beer. Its walls are decorated with salsa posters, album covers, and photographs of salsa greats. It's a hole-in-the-wall bar in the San Diego neighborhood.\n\n**Bazurto Social Club** (Av. del Centenario Cra. 9 No. 30-42, tel. 5\/664-3124, www.bazurtosocialclub.com, 7pm-3am Thurs.-Sat., cover varies) is an always-lively restaurant-bar, popular with Colombians and international visitors alike. Get a taste for Afro-Colombian _champeta_ beats, as live acts, including the Bazurto All Stars, often perform here. They also serve food, such as shrimp empanadas and paella. The house drink is the fruity rum _machacos._\n\nThe former site of a convent, **El Coro** (Cl. del Torno No. 39-29, tel. 5\/650-4700, 5pm-2am Sun.-Thurs., 5pm-3am Fri.-Sat., no cover), at the Hotel Santa Clara, is an inviting, if posh, spot for an after-dinner drink. From Wednesday to Sunday, live music, with a nod to Havana, is on offer until late. How many places can you enjoy Latin jazz while downing mojitos in a former convent?\n\n**Caf\u00e9 Havana** (intersection of Cl. de la Media Luna and Cl. del Guerrero, cell tel. 314\/556-3905 or 310\/610-2324, www.cafehavanacartagena.com, 8:30pm-4am Thurs.-Sat., cover varies) famously got the endorsement of former Secretary of State Hillary Clinton on her trip to Colombia in 2012. It's a place for rum drinks and dancing. Caf\u00e9 Havana is open Sundays when the following Monday is a holiday.\n\n**Donde Fidel** (Plaza de los Coches, tel. 5\/664-3127, noon-2am Sun.-Thurs., noon-3am Fri.-Sat., no cover) is a tiny salsa-lovers' spot where the action spills out onto the plaza in front. Good times and cold beer can be found here.\n\nYou've walked atop the massive walls of the Old City, and now it's time for some drinks. A sundown drink, with the Caribbean breeze kissing your face on the _murallas_ (walls), is a Cartagena experience that shouldn't be missed. There are three options. First, the **Baluarte Tasca-Bar** (Cl. San Juan de Dios, tel. 5\/660-0468, www.baluartesanfranciscojavier.com, 5:30pm-2am daily, no cover) is an open-air restaurant-bar at the northwesternmost corner of the wall, across from the Plaza de Santa Teresa. It's chilled out here, not trendy (but the drink prices are on the steep side: COP$24,000 for a margarita). The most happening spot would be, without a doubt, the **Caf\u00e9 del Mar** (tel. 5\/664-6513, 5pm-3am daily, no cover), on the wall near the Plaza de Santo Domingo entrance. Here the music is loungey and electronic, and it stays busy until late. It's spread out atop the wall, making it hard to mingle with others or even to carry on much of a conversation. But the drinks will quench your thirst and the music is seductive. The third wall option is to go rogue: hang out on the wall, drink an \u00c1guila beer sold by a roaming vendor, and listen to the music emanating from Caf\u00e9 del Mar.\n\nWednesdays are the new Saturdays in Cartagena. In Getseman\u00ed, **Visa por un Sue\u00f1o** (Media Luna Hostel, Cl. de la Media Luna No. 10-46, tel. 5\/664-0639, 9pm-3am Wed., COP$10,000) is a weekly party held on the rooftop of the Media Luna Hostel that has become Cartagena's most famous soiree, when backpackers mix it up with Colombians. Get there early\u2014this party gets packed.\n\n##### **Festivals and Events**\n\nCartagena feels like a celebration all the time, but especially November to February, when an array of cultural events are featured.\n\n###### **HAY FESTIVAL**\n\n**Hay Festival** (www.hayfestival.com) is an important international festival that began in Wales nearly 30 years ago. It celebrates literature, music, environmental awareness, and community and is held in various cities across the world, including in Cartagena in late January. Bill Clinton has called it the \"Woodstock of the mind.\" In addition to talks and concerts, the festival holds educational programs for youth in the neighborhoods of Cartagena. It also provides free or discounted tickets to students. Most of the events take place in the **Teatro Heredia** (Cl. de la Chicher\u00eda No. 38-10, tel. 5\/664-6023 or 5\/664-9631). While the festival's name is pronounced as the English \"hay,\" in Colombia it's often pronounced as the Spanish _\"hay\"_ (\"ai\"). Hay Festival is thus a double entendre: _hay festival_ in Spanish means yes, there is a festival!\n\n###### **FESTIVAL INTERNACIONAL DE M\u00daSICA**\n\nOver the course of a week in early to mid-January, the churches, plazas, and theaters of the Walled City become the setting for classical music concerts by musicians from all over the world during the **Festival Internacional de M\u00fasica** (International Music Festival, www.cartagenamusicfestival.com, tickets www.tuboleta.com). Most concerts sell out far in advance, but if you can't get tickets, you might be able to catch a free performance in one of the churches or plazas in the Old City.\n\n###### **FESTIVAL INTERNACIONAL DE CINE DE CARTAGENA DE INDIAS**\n\nIf you're in town during late February and are looking for an excuse to escape the heat, here it is: the **Festival Internacional de Cine de Cartagena de Indias** (International Film Festival, tel. 5\/664-2345, www.ficcifestival.com). A tradition since the 1960s, this week-long film festival has an interesting program of documentaries, Colombian films, and shorts; a series of roundtable discussions with prominent actors and directors; and educational activities in neighborhoods throughout the city. The venues include historic buildings and plazas.\n\n###### **CONCURSO NACIONAL DE BELLEZA**\n\nBeauty contests, and especially the **Concurso Nacional de Belleza** (Miss Colombia Pageant, tel. 5\/660-0779, www.srtacolombia.org), are a big deal in Colombia. The coronation of Se\u00f1orita Colombia takes place every November and is the highlight of Cartagena's Independence Day celebrations. Aspirers for the title represent each of the departments of the country, in addition to some cities. Ladies from the Valle del Cauca and Atl\u00e1ntico have won the most titles (10 each) since the pageant began in the 1930s. In 2001, the first Miss Colombia of Afro-Colombian heritage was chosen: Vanessa Mendoza, who represented the Choc\u00f3 department. Tickets to the main events\u2014the swimsuit competition at the Cartagena Hilton and the coronation at the Centro de Convenciones\u2014are hard to come by but not impossible to purchase.\n\n###### **ELECTRONIC MUSIC FESTIVALS**\n\nIn early January every year, especially on the first weekend after New Year's Day, one or two big beachside electronic music festivals take place. The most well-known and regular festival is **Ultra Mar,** but there are others. These parties are often the main reason the under-35 crowd from Bogot\u00e1, Medell\u00edn, and Cali converges on the city during the New Year's holidays. Tickets and information can be found at Tu Boleta (www.tuboleta.com). Double-check before heading out to one of these events, as sometimes the location changes at the last minute.\n\n#### **SHOPPING**\n\n##### **Shopping Centers and Malls**\n\nThe bazaar-like **Centro Comercial Getseman\u00ed** (Cl. 30 No. 8B-74, tel. 5\/664-2508, hours vary daily) shopping center doesn't really cater to tourists. It's made up of hundreds of small mom-and-pop kiosks that sell just about anything: computer supplies, notebooks, beauty supplies, handicrafts, and knickknacks. You can probably get your nails done here as well.\n\n**Centro Comercial Caribe Plaza** (Pie de la Popa, Cl. 29D No. 22-108, tel. 5\/669-2332, 10am-8pm Mon.-Thurs., 10am-9pm Fri.-Sun.) is an upscale modern mall near the Castillo de San Felipe with numerous clothing and shoe stores, movie theaters, and a food court.\n\nfresh fish at the Mercado de Bazurto\n\n###### **MERCADO DE BAZURTO**\n\nDefinitely not for the faint of heart, a visit to the sprawling, grimy **Mercado de Bazurto** (Av. Pedro Heredia, 5am-4pm daily) is the best way to connect with the real Cartagena. On the periphery of the market be sure to peruse the seafood area, where women sell the catch of the day to restaurant owners. Be amazed at all the different kinds of fruit on offer. To get to the market, take a bus from Avenida Santander (COP$1,500) or a taxi (COP$7,000 from the Old City).\n\nAnother way of visiting the market is the **Mercado de Bazurto Tour** (tel. 5\/660-1492, cell tel. 315\/655-4120, cevicheria@hotmail.com, COP$250,000 pp), organized by Jorge Escand\u00f3n, the owner of La Cevicher\u00eda and Bazurto Social Club. On the tour, you'll learn about the ingredients that make Caribbean cuisine special, particularly the seafood and exotic fruits. You'll also meet the vendors who have worked their entire lives behind a stall at the market. Afterwards you'll head to a beach house and have a gourmet lunch featuring lobster and other delicacies prepared by Jorge and his staff.\n\n##### **Handicrafts**\n\nThe most historic place to pick up some Colombian handicrafts is at **Las B\u00f3vedas** (extreme northeastern corner of the wall, 9am-6pm daily). Once a military storehouse, today it's the place to buy multicolored hammocks and all kinds of Colombian _artesan\u00edas_ of varying quality.\n\nFor high quality handicrafts you're better off going to **Artesan\u00edas de Colombia** (Centro de Convenciones, Local 5, tel. 5\/660-9615, 10am-7pm Mon.-Sat.). This is a government entity whose mission it is to promote Colombian handicrafts and craftspeople. This store sells handicrafts from across the country but specializes in masks from the Carnaval de Barranquilla, woven _mochilas_ (handbags) from indigenous groups in the Sierra Nevada, and the colorful embroidery of _molas_ from indigenous groups in the Darien Gap region near Panama.\n\n##### **Jewelry**\n\nColombian emeralds are considered to be some of the finest in the world. Many jewelers in the Centro Hist\u00f3rico sell emerald jewelry and can custom make jewelry for you. One of the most highly regarded jewelers is **Galer\u00eda Cano** (Plaza Bol\u00edvar No. 33-20, Local 679, tel. 5\/664-7078, www.galeriacano.com.co, 9am-7pm Mon.-Fri., 10am-7pm Sat.). They specialize in gold, silver, and emerald jewelry. Cano has other locations in the airport and at the Hotel Santa Clara, as well as in Bogot\u00e1.\n\n##### **Cigars**\n\nIn Cartagena, touches of Cuba are found everywhere: mojitos, music, and, too, the cigars. At **La Cava del Puro** (Cl. de las Damas No. 3-106, tel. 5\/664-9482, www.lacavadelpuro.com, 9am-8pm Mon.-Sat., 10am-8pm Sun.) they don't sell just any old stogie; here the cigars come from Havana and are of the best quality. Smoking is not only permitted here, but in fact promoted. Sometimes a little whiskey is even served to perusing clients. Note that cigars with labels that say \"Hecho in Cuba\" may be confiscated by customs agents upon arrival in the United States. Cigars that come from the Barichara area in the Colombian department of Santander are quite good and much cheaper, and you can take them across borders with no questions asked.\n\n##### **Clothing**\n\nAlong Calle Santo Domingo there are several boutiques of top Colombian designers. Bogotana **Bettina Spitz** (Cl. de la Mantilla No. 3-37, tel. 5\/660-2160, www.bettinaspitz.com, 11am-1pm and 2pm-8pm daily) sells casual, beach, and formal clothes for women, as well as an array of accessories, shoes, and some men's items. **Jon Sonen** (Cl. Ricaurte No. 31-56, tel. 5\/664-1092 or 5\/660-4682, www.jonsonen.com, 10am-8pm Sun.-Thurs., 10am-9pm Fri.-Sat.) is a Colombian label specializing in menswear, with stores throughout the country.\n\nYou'll notice that guayabera shirts are what men wear around Cartagena, to restaurants and events. It's possible to spend a fortune on them, but if you want to blend in without busting your budget, go to **Arte y Creaciones** (Cl. Don Sancho No. 36-94, cell tel. 320\/583-9091, noon-8pm Mon.-Fri., 10am-8pm Sat.-Sun.), where a cotton guayabera will run you about COP$55,000. Another option for cheap guayaberas is **Centro de Artesan\u00edas de San Jacinto** (Cl. de la Iglesia No. 35-59, tel. 5\/660-1574, 9am-8:30pm daily).\n\n##### **Books**\n\n**Abaco Libros** (Cl. 36 No. 3-86, Cl. de la Mantilla, tel. 5\/664-8290, 9am-9pm Mon.-Sat., 3pm-9pm Sun.) is a small book shop\/caf\u00e9 with a variety of books on Cartagena, top Colombian novels, and a selection of magazines, classics, and best sellers in English. **Librer\u00eda Nacional** (Cl. Segunda de Badillo No. 36-27, tel. 5\/664-1448, 8:30am-12:30pm and 2pm-6:30pm Mon.-Fri., 8:30am-5pm Sat.) is a chain bookstore with shelves full of Colombian and Spanish-language books, but not much in the way of books in English.\n\n#### **SPORTS AND RECREATION**\n\nIf you're looking for a place to jog or stroll, the bayside path along the peninsula of **Castillo Grande** (Cra. 5 from Clls. 6-10 and along Cl. 6 from Cras. 6-14) in the Bocagrande sector is very nice, particularly in the late afternoon. From here you'll have great views of the Cartagena port and La Popa in the far distance. Forming an L shape, the path is about two kilometers long.\n\n##### **Biking**\n\nThe best time to explore Cartagena by bike is early on a Sunday morning or on a Sunday or Monday evening when there is little activity and no traffic in the Old City. **Pato Bikes** (Cl. de la Media Luna, Cra. 8B No. 25-110, tel. 5\/664-0639, cell tel. 301\/423-9996, 9pm-9pm daily, COP$20,000) and **Bike Route** (Callej\u00f3n de los Estribos No. 2-78, cell tel. 318\/456-1392, 10am-11pm daily, COP$5,000\/hour, COP$30,000\/day) are both located in Getseman\u00ed and rent bikes. In addition to bike rental, **Bicitour Getseman\u00ed** (Cl. Carretero, Getseman\u00ed, cell tel. 300\/357-1825, COP$5,000\/2-hour rental) offers guided tours of the Old City on two wheels. Many hostels and some hotels also have bicycles for hire.\n\n##### **Diving**\n\n**Diving Planet** (Cl. Estanco del Aguardiente No. 5-94, tel. 5\/664-2171, www.divingplanet.org, 8am-7pm Mon.-Sat.) offers classes and diving excursions to some 25 locations throughout the Parque Nacional Natural Corales del Rosario y San Bernardo. A one-day mini-course with two immersions costs COP$305,000. A day trip with two immersions for those who are certified divers costs COP$290,000. If you pay in advance on their webpage or pay in cash once in Cartagena you'll receive a discount. Multiple day PADI certification courses are also available. Some of these plans include an overnight on the white beaches of the Islas del Rosario. Snorkeling excursions are also available (COP$190,000).\n\n##### **Yoga**\n\n**Santuario del Yoga** (Cl. El Estanco del Aguardiente No. 5, tel. 5\/668-5338, cell tel. 313\/649-3133, 8:30am-5pm daily, COP$15,000 per class) offers yoga classes in a small studio in the Walled City and also classes on the beach on occasion. Some instructors are bilingual.\n\n#### **ACCOMMODATIONS**\n\nCartagena remains the top tourist destination in Colombia (and is second only to Bogot\u00e1 in terms of international arrivals), and the crowds keep coming. Hotel options have flourished since 2000. The Old City and Getseman\u00ed have become a favorite location for high-end boutique hotels. At the other end of the spectrum, hostels have begun to appear in these same neighborhoods. Finding a good midrange option, however, is a challenge.\n\nA short bus or taxi ride away from colonial Cartagena is its version of Miami Beach: Bocagrande. Large high-rise hotels facing the waters of the Caribbean are the norm. This area is popular with Colombian tourists. There are some midrange and budget options here, though not with a view to the sea.\n\nFor those interested in beaches, the Las Am\u00e9ricas area two kilometers past the airport is home to many new, large high-rise hotels. Finally, there are lodging options in the Islas de Rosario. Spending a night there may be more satisfying, albeit much more expensive, than a day trip.\n\nPeak tourist seasons in Cartagena are during the last week of November (which is when the city celebrates its independence from Spain and hosts the Miss Colombia beauty pageant), from mid-December to mid-January, Semana Santa (Holy Week) in March or April, June-July during school vacations, and during any of the long weekends when Monday is a holiday (check a Colombian calendar). Cartagena is at its liveliest (and more fun) when the out-of-towners converge on it, particularly during the end-of-year holidays when it's two to three weeks of celebration. But finding a place to stay may be difficult, as room rates spike.\n\n##### **Old City and Getseman\u00ed**\n\n###### **UNDER COP$70,000**\n\nThe Shangri-La of backpacker hostels in Cartagena is the famous **Media Luna** (Cl. de la Media Luna No. 10-46, tel. 5\/664-3423, www.medialunahostel.com, COP$37,000 dorm). Located on the edge of Getseman\u00ed, it's a high-energy kind of place with multicultural socializing (and flirting) centered on the small pool in the courtyard. If you're looking to break out of your shell, this may be the place. It has a capacity of over 100 with just a couple of private rooms (book early for those). There's a burrito place attached to the Luna, they organize lots of activities, and bikes are available to rent. Then there's the bar: On Wednesday nights, turn up on the early side (around 9pm) to squeeze in at their famous Visa por un Sue\u00f1o party. This hostel isn't the best place for travelers over the age of 28.\n\nIn the Old City, the Uruguayan hostel chain **El Viajero** (Cl. Siete Infantes No. 9-45, tel. 5\/660-2598, www.elviajerohostels.com, COP$30,000 dorm, COP$75,000 pp d) has air-conditioned dorm rooms of various sizes and a handful of private rooms, which are located across the street in a more subdued environment. A decent breakfast is included in the room rate.\n\nLow-key hostel option M **Blue House** (Plaza Fern\u00e1ndez Madrid, corner of Cl. del Curato and Cra. 7 No. 38-08, tel. 5\/668-6501, www.bluehouseht.com, COP$35,000 dorm, COP$150,000 d) is within the Walled City on a relatively quiet side street just a few blocks from the Plaza Santo Domingo. It's got just one dorm room and two private rooms. You'll have no problem finding the hostel\u2014it's true blue.\n\n###### **OVER COP$200,000**\n\nThe two classic upmarket hotels in the Walled City are the Santa Teresa and the Santa Clara. The **Hotel Charleston Santa Teresa** (Cra. 3A No. 31-23, tel. 5\/664-9494, www.hotelcharlestonsantateresa.com, COP$718,000 d) was originally built as a convent for Clarisa nuns. Post-independence it served many different purposes: headquarters for the police, a jail, a pasta factory. In the 1980s it was finally converted into a hotel. There are two wings to this historic hotel, a colonial one and a modern wing. The two inner courtyards are lovely, and you'll be astounded by the floral displays. Concierges will help arrange any excursion you'd like. Amenities such as four restaurants, a rooftop pool, a spa, and gym ensure a relaxing stay. It's steps away from the wall.\n\nThe other old city classic is in San Diego. The **Hotel Sofitel Legend Santa Clara** (Cl. del Torno 39-29, tel. 5\/650-4700, www.sofitel.com, COP$747,000 d) is a 122-room hotel synonymous with class and luxury. The stunning colonial courtyard alone is worth taking a peek, even if you're not a guest. The Santa Clara originally served as a convent.\n\nFrom the rooftop terrace of M **Hotel LM** (Cl. de la Mantilla No. 3-56, tel. 5\/664-9100, www.hotel-lm.com, COP$800,000 d) guests enjoy spectacular views of the rooftops of old Cartagena, including the Iglesia San Pedro Claver. This luxury hotel has seven spacious rooms and an \"interactive\" kitchen where guests order in advance and can even participate in food preparation.\n\n**Casa Lola** (Cl. del Guerrero No. 29-108, tel. 5\/664-1538, www.casalola.com.co, COP$374,000 d) is designed and managed by a Spanish couple who were some of the first hoteliers to take a chance on Getseman\u00ed. The hotel, spread over two buildings (one colonial and one republican-era), has 10 smartly designed rooms.\n\nIt's not hard to determine how the owners decided on **Makondo Hotel Boutique** (Cl. del Curato No. 38-161, tel. 5\/660-0823, www.hotelmakondo.com, COP$210,000 d) as the name for their small hotel: It's next door to Gabriel Garc\u00eda M\u00e1rquez's house and named for the fictitious Colombian pueblo portrayed in the author's _One Hundred Years of Solitude._ This small hotel is boutique-lite, with less exorbitant prices. It's got 10 rooms, some rather small. It's within a stone's throw of several good restaurants.\n\n##### **Bocagrande**\n\n###### **OVER COP$200,000**\n\nSeveral mediocre hotels along Carrera 3 in Bocagrande fit in the category of economy hotels. For a little more, but for less than most hotels in Cartagena, the **Hotel San Pietro** (Cra. 3 No. 4-101, tel. 5\/665-2369, www.pietro.com, COP$228,000 d), located on the same stretch, is more comfortable than its neighbors. It has 35 rooms of different sizes, a cute reading room with books you can check out, and a rooftop terrace with a hot tub (though it can only be used during the day, under the blazing sun). The owners also have an adjacent Italian restaurant.\n\nThe old classic in town is the **Hotel Caribe** (Cra. 1 No. 2-87, Bocagrande, tel. 5\/650-1160, www.hotelcaribe.com, COP$350,000 d). These swanky digs, comprising three large buildings, are next to the beach (they have a beach club exclusively for guests). Most visitors, however, seem to prefer to lounge by the pools and drink a fruity cocktail.\n\nFor a no-surprises brand-name hotel experience, the **Hilton Cartagena** (Av. Almirante Brion, El Laguito, tel. 5\/665-0660, www.hilton.com, COP$411,000 d) won't fail you. It's at the tip of Bocagrande, isolated from the crowds, and has multiple pools and restaurants as well as a gym. Guests have to pay extra for wireless Internet access, though.\n\n#### **FOOD**\n\nSeafood reigns supreme in Cartagena cuisine. Popular fish are _pargo rojo_ (red snapper), _corvina_ (sea bass), _dorado_ (mahi mahi), and _sierra_ (swordfish). Avoid _mero_ (grouper) as it is threatened in the Caribbean waters. Shellfish include _langosta_ (lobster), _langostinos_ (prawns), and _chipi chipis_ (tiny clams). These main dishes are often accompanied with delicious coconut rice and _patacones_ (fried plantains).\n\nThough many restaurants in the Walled City sport Manhattan prices, an inexpensive meal is not impossible to find. There are still a few mom-and-pop restaurants featuring set (cheap!) lunches for locals who would balk at paying over COP$10,000 for their midday meal.\n\n##### **Old City**\n\nTransport yourself to the Havana of yesteryear at M **La Vitrola** (Cl. Baloco No. 2-01, tel. 5\/664-8243, noon-3pm and 7pm-midnight daily, COP$35,000), an always elegant, always packed restaurant that specializes in Caribbean seafood, such as their popular tuna steak with avocado and mango, as well as pasta dishes. Immaculately dressed bartenders are a blur of constant motion as they perform their nightly mojito ritual: plucking mint leaves, crushing them with sugar in tall glasses, pouring in soda and rum, squeezing in some fresh lime juice, then giving the concoction a few vigorous shakes. La Vitrola is pricey, but the atmosphere, with live Cuban music in the evenings, makes it worthwhile.\n\nServing up the best, freshest ceviche in town is **La Cevicher\u00eda** (Cl. Stuart No. 7-14, tel. 5\/660-1492, noon-11pm Mon.-Sat., noon-10pm Sun., COP$28,000). It's got a creative menu, featuring ceviche with mango and ceviche with coconut and lime juice, and outdoor seating on a quiet street.\n\nTastefully decorated with a lovely garden area, upper-crust **Restaurante FM** (Cl. 2 de Badillo No. 36-151, tel. 5\/664-7973, noon-3pm and 7pm-11:30pm daily, COP$38,000) is named for its owner, Francisco Montoya, who has created a menu that features Caribbean and Mediterranean dishes.\n\n**El Balc\u00f3n** (Cl. Tumbamuertos No. 38-85, cell tel. 300\/336-3876, www.elbalconcartagena.com, noon-midnight daily, COP$22,000) is a friendly place with a view in Plaza San Diego. Get here in the early evening and enjoy a sundowner cocktail as you listen to lounge music or have a light meal like a refreshing gazpacho or their shrimp \"sexviche.\" Casual and cute, **Collage Charladero** (Cl. Roman No. 5-47, tel. 5\/660-7626, noon-midnight Mon.-Sat., COP$22,000) serves sandwiches, burgers, falafels, fresh juices (watermelon with lime and mint), and refreshing sangria in a clean and cool environment close to all the historic sights.\n\nThe **Enoteca** (Cl. San Juan de Dios No. 3-39, tel. 5\/664-3806, www.enoteca.com.co, noon-11:30pm daily, COP$30,000) never seems to lose its popularity. This institution is best known for its pizzas, professional service, and nice atmosphere, although its pastas are overpriced. While the interior patio decorated with fountains and twinkling lights is certainly atmospheric, you can also dine in their wine cellar room near the front, where the air conditioner always hums.\n\nFor a little curry with your shrimp, try **Ganesha** (Cl. de las Bovedas No. 39-91, tel. 5\/660-9165, www.ganesharestaurante.com, noon-3pm and 6:30pm-11pm Tues.-Sun., COP$24,000), an authentic Indian restaurant with an extensive menu with many vegetarian options.\n\n**La Cocina de Carmela** (across from Librer\u00eda Nacional, Cl. Segunda de Badillo No. 36-50, cell tel. 301\/348-7881, 11:30am-11pm Mon.-Sat., COP$12,000) is an unpretentious bargain spot where Colombian and international dishes (served buffet style) are on offer at lunchtime. At night it's \u00e0 la carte, specializing in seafood and pasta dishes.\n\n**Crepes & Waffles** (Cl. Baloco Edificio Pi\u00f1eres, Local 1, tel. 5\/664-6062, www.crepesywaffles.com.co, noon-10:30pm Mon.-Thurs., noon-11:30pm Fri.-Sat., 8am-10:30pm Sun.) is a wildly successful and reliable Colombian family-style chain that specializes in savory and sweet cr\u00eapes and just sweet waffles. With restaurants as far away as Spain, the restaurant has a progressive policy of hiring women who are heads of their households. Besides healthy and quick meals, Crepes is a good place for an ice cream break on a muggy Cartagena afternoon.\n\nIn 1965, Dora Gav\u00edria starting selling her _fritos_ (fried snacks) to locals and students in order to support her family of five children. Today she stays in the kitchen mostly, letting her adult children run her stand, but everyone still calls it **Do\u00f1a Dora** (Plaza San Diego, 4pm-10pm daily). There's always a crowd gathered around the small food stall taking turns dabbing a little more hot sauce on their _arepa de huevo_ (egg fried in corn meal) _, carima\u00f1olas_ (meat-stuffed yuca fritters), and empanadas. Beer is the perfect companion for her _fritos._ As you are strolling the narrow streets of the Old City, look for street corner vendors of _agua de coco_ (coconut water). This natural sports drink is sold in the actual coconut\u2014just add a straw. It's an unbeatable thirst quencher on hot days. The going price for coconut water is COP$2,000.\n\n##### **Getseman\u00ed and Manga**\n\nThe Plaza de la Trinidad in Getseman\u00ed is the heart of the neighborhood. It's home to some swanky spots perfect for a couple of drinks or a meal. A little bit of Barcelona can be found there at the **Champagner\u00eda del Mediterraneo** (Plaza de la Trinidad, tel. 5\/646-3576, 11am-midnight Wed.-Mon., COP$24,000), where Spanish wines accompany Serrano ham sandwiches. **Demente** (Plaza de la Trinidad, cell tel. 311\/831-9839, www.demente.com.co, 4pm-2am Mon.-Sat., COP$22,000) is an ultra-cool tapas bar on a competing corner across the plaza that also specializes in cocktails and tapas. It's an open-air spot with a retractable roof, where the music is funky, the cocktails are fine, and the cigars are Cuban. It's a fun place for an evening of tapas and drinks.\n\nCheck out **Di Silvio** (Cl. de la Sierpe No. 9A-08, tel. 5\/660-2205, 6:30pm-11:30pm Tues.-Sun., COP$24,000), an upscale Italian restaurant just off of the Plaza de la Trinidad. M **Pavia** (Cl. Guerrero 29-75, tel. 5\/664-3308, 6pm-midnight Mon.-Sun., COP$15,000) is a funky little pizza and pasta joint that is run by a musician and artist from Italy.\n\nVIPs such as President Santos have been known to sample the authentic Italian dishes at **I Balconi** (Cl. del Guerrero No. 29-146, cell tel. 311\/392-0936, www.ibalconi.com, noon-10pm Sun.-Thurs., noon-midnight Fri.-Sat., COP$18,000). It gets boisterous here as the evenings wear on and the wine flows. It's above Caf\u00e9 Havana. Ask for a table on one of the balconies so you can enjoy the street life from on high.\n\nAt M **La Cocina de Pepina** (Callej\u00f3n Vargas, Cl. 25 No. 9A-06, Local 2, tel. 5\/664-2944, noon-4pm and 6pm-10pm Tues.-Sat., noon-4pm Sun.-Mon., COP$25,000), typical dishes from across the Caribbean coast are thoughtfully reinvented. It's a cozy place in an alleyway near the Calle del Arsenal. Make a reservation for dinner.\n\n**Marea by Rausch** (Centro de Convenciones, Cra. 8, tel. 5\/654-4205, www.mareabyrausch.com, noon-3pm and 7pm-10pm Tues.-Sat., 4pm-10pm Sun., COP$45,000) is an ultra-chic seafood restaurant that is the brainchild of the Rausches, two brother chefs from Bogot\u00e1. Specialties include a tuna tartar and prawns in a coconut and saffron sauce. This restaurant has excellent views of the bay and the Torre del Reloj.\n\nThe food at the **Club de Pesca** (Fuerte San Sebasti\u00e1n del Pastelilo, Manga, tel. 5\/660-4594, noon-11pm daily, COP$55,000) is overpriced and overrated, but the view is unsurpassable. This Cartagena classic is in the old San Sebasti\u00e1n del Pastelillo fort with magnificent views to the bay. It's a favorite spot for wedding banquets, and some guests arrive at the fort in yachts. On the menu try the _jaiba gratinada,_ which is a crab au gratin.\n\n##### **Bocagrande**\n\nElegant **Arabe Internacional** (Cra. 3 No. 8-83, tel. 5\/665-4365, www.restaurantearabeinternacional.com, noon-3:30pm and 7pm-10pm Mon.-Fri., noon-10pm Sat.-Sun., COP$25,000) has been serving authentic Middle Eastern cuisine since 1965. It's a popular place for the Cartagena business crowd.\n\nIf you just want a hearty, authentic Colombian meal without the bells and whistles, head to **Mac Dugan's** (Av. San Mart\u00edn No. 9-42, tel. 5\/665-5101, 11am-9pm daily, COP$12,000), a family-run restaurant in Bocagrande.\n\n#### **INFORMATION AND SERVICES**\n\nIn addition to locations at the airport and at the cruise ship port, there are city-run **tourist information kiosks** near the Torre del Reloj (no phone, 9am-noon and 1pm-6pm Mon.-Sat., 9am-5pm Sun.) and an air-conditioned main office in the historic **Casa de Marquez Plaza de la Aduana** (tel. 5\/660-1583, 9am-noon and 1pm-6pm Mon.-Sat., 9am-5pm Sun.). The website for the **Cartagena Tourism Board** is www.cartagenadeindias.travel.\n\nIn case of an **emergency** , call the police at 123, 112, or 5\/628-4748. For medical emergencies call tel. 5\/667-5244.\n\nAlthough postcards are relatively easy to purchase, sending them is a different matter. Mailing postcards and letters is not common here and not that easy. The post office, **4-72,** has a branch in the Walled City (Cl. de la Moneda No. 7-94, tel. 5\/670-0102, 8am-noon and 2pm-5:30pm Mon.-Fri., 9am-12:30pm Sat.). It costs COP$2,000 for postage to the United States and Canada. This branch has a small exhibition space with old postage stamps on display.\n\n##### **Volunteering**\n\nThe **Fundaci\u00f3n Juan Felipe G\u00f3mez** (Cl. 31 No. 91-80, Ternera, tel. 5\/661-0937, www.juanfe.org) was the inspiration of Catalina Escobar, a Colombian businesswoman. As a volunteer in a maternity clinic in Cartagena, she was holding a tiny infant, born to a teenage mother, who died in her arms, all because the mother didn't have the most basic financial resources to get proper care for her son. Within days, Escobar's own son died in a tragic death. Driven by grief and the desire to help young women to lead healthy and happy lives, she founded this organization. At the Fundaci\u00f3n Juan Felipe G\u00f3mez, named for the child she lost, young women (often pregnant or new mothers) take classes designed to provide them with workforce skills. Because of Escobar's efforts, she was nominated a CNN Hero of the Year in 2012. Anybody with skill or interest in nutrition, computers, fashion, or teaching English can get involved. Both short- and long-term volunteers are welcomed. If you'd like to visit the center, call in advance to arrange a tour. You can take a public bus or a 20-minute taxi to the _fundaci\u00f3n_.\n\n#### **GETTING THERE**\n\nCartagena's **Aeropuerto Internacional Rafael N\u00fa\u00f1ez** (CTG, tel. 5\/656-9202, www.sacsa.com.co) is located to the east of the city, about a 12-minute cab ride from Cartagena.\n\n**JetBlue** (www.jetblue.com) operates three flights a week between New York-JFK and Cartagena. Nonstop from Florida, **Spirit Airlines** (www.spirit.com) has a flight from Fort Lauderdale, and **Avianca** (Col. toll-free tel. 01\/800-095-3434, www.avianca.com) has one out of Miami. **Copa** (tel. 5\/665-8495, www.copaair.com) serves Cartagena from its hub in Panama City, Panama. The main national carriers, Avianca and **LAN Colombia** (Col. toll-free tel. 01\/800-094-9490, www.lan.com), operate many flights each day to Cartagena from various Colombian cities. **Viva Colombia** (tel. 5\/642-4989, www.vivacolombia.co) often offers inexpensive fares between Cartagena and Medell\u00edn, Bogot\u00e1, Cali, and Pereira. **ADA** (Col. toll-free tel. 01\/800-051-4232, www.ada-aero.com) serves the city with flights from Medell\u00edn, Monter\u00eda, and C\u00facuta. **Easy Fly** (tel. 5\/693-0400, www.easyfly.co) has a nonstop from Bucaramanga.\n\nRegular bus service connects Cartagena with all major cities and all coastal cities. The **Terminal de Transportes** (Diag. 56N. 57-236, tel. 5\/663-0454) is a long 20- to 30-minute cab ride from the _centro hist\u00f3rico._ Expect to pay about COP$20,000 for the trip.\n\n#### **GETTING AROUND**\n\nThe Old City, Getseman\u00ed, and Bocagrande are very walkable. For short hops, there are cabs. Taxis here do not have meters, so it's quite possible you won't get the local rate. Before hopping in a cab, ask a local or two how much you should pay. From the Old City to Bocagrande, expect to pay around COP$6,000. A ride to the airport will cost COP$10,000, and a trip to Las Am\u00e9ricas will go for COP$12,000. Tipping is not customary for cabbies. Although an additional cost is added if you use a phone service for a cab, added costs for night pickups, air conditioning, and traveling on holidays are not permitted, and you can protest those.\n\nIt may seem overwhelming at first, but taking a public bus is a cheap way to get from point A to point B\u2014and you'll hardly ever be overcharged. To hop on a bus to Bocagrande from the Old City, walk down to Avenida Santander along the sea and flag down just about any bus you see (or look for a sign in the window that reads \"Bocagrande\"). The ride will set you back COP$1,500. When you want out, yell _\"\u00a1Parada!\"_ On the main road just to the east of the walls, you'll see a nonstop parade of buses loading and unloading. From here you can go to the Castillo de San Felipe, to the Mercado de Bazurto, or to the bus terminal, for the same low price of COP$1,500.\n\n#### **BOCACHICA**\n\nBocachica, which means \"Small Mouth,\" is one of two entrances to the Bah\u00eda de Cartagena. It is at the southern end of the bay. The other, much wider entrance, is Bocagrande (\"Big Mouth\"), in the northern part of the bay near Cartagena. In 1640, when three galleons sank at Bocagrande and blocked that passage, the Spaniards decided to fortify the more easily defensible Bocachica.\n\nThe **Fuerte de San Fernando** and **Bater\u00eda de San Jos\u00e9** are two forts constructed at either side of Bocachica that were the first line of defense of the bay. The Fuerte de San Fernando, at the southern tip of the island of Tierrabomba, is a particularly impressive example of 18th-century military architecture. It is very well preserved and you can still see the barracks, kitchen, storerooms, and chapel enclosed within the massive fortifications. The low-lying Bater\u00eda de San Jos\u00e9 is a much more modest affair. The only way to get to Bocachica is by _lanchas_ that depart from the Muelle de los Pegasos, the tourist port in Cartagena near the Torre del Reloj (COP$7,000). The 45-minute trip through the bay provides interesting views of Cartagena and the port. In Bocachica, there are a few small restaurants where you can eat fried fish, coconut rice, and _patacones_ (fried plantains) and drink a cold beer.\n\n#### **PLAYA BLANCA AND ISLAS DEL ROSARIO**\n\nSouth of Cartagena is the elongated island of **Bar\u00fa** , which is separated from the mainland by the Canal del Dique, a manmade waterway built in 1650 to connect Cartagena with the R\u00edo Magdalena. On Bar\u00fa lies **Playa Blanca,** a Caribbean paradise of idyllic, white-sand beaches bordering warm waters of dozens of shades of blue. West of Bar\u00fa, and about 25 kilometers southwest of Cartagena, is the archipelago **Islas del Rosario,** part of the much larger **Parque Nacional Natural Corales del Rosario y San Bernardo**. On Bar\u00fa and the Islas del Rosario, traditional Afro-Colombian communities with rich cultural heritages coexist with the vacation houses of Colombia's rich and famous.\n\nBar\u00fa is home to several beautiful beaches. With the exception of Playa Blanca, most of these are inaccessible to the general public. The 25 small coral islands of the Islas del Rosario are a marine wonderland. The once-spectacular coral reefs off of Bar\u00fa and surrounding the archipelago have been badly damaged by the increased flow of fresh water from the Canal del Dique, which has been dredged in recent years.\n\nA trip to Playa Blanca and the Islas del Rosario affords the chance to blissfully bask in the sun and splash about in the ocean. However, be aware that the standard tours, such as those operated by **Optitours** (Av. Santander No. 46-94, Cartagena, tel. 5\/666-5957, cell tel. 300\/394-4848, www.opitours.com, COP$50,000-70,000) can often be crowded on weekends. Most tours involve cruising down the Bah\u00eda de Cartagena, passing through the Strait of Bocachica, and heading to the **Oceanario Islas del Rosario** (www.oceanariocolombia.com, 10am-3pm Tues.-Sun., COP$25,000) on the island of San Mart\u00edn de Pajarales, part of the Islas del Rosario. Very popular with Colombian families on vacation, the aquarium features a dolphin show. You can swim in the water and rest in the shade if you decide not to patronize the aquarium. Then the boats head off to Playa Blanca, where you can buy lunch. The insistent vendors at Playa Blanca can make for an unpleasant experience, but you can try telling them _\"no, gracias.\"_\n\nIf you are willing to pay more, there are more upscale day tours to the _islas_. One is a day trip (COP$173,000) to the luxurious beachside **Hotel San Pedro de Majagua** (Cartagena office: Cl. del Torno No. 39-29, tel. 5\/650-4460, <http:\/\/nuevo.hotelmajagua.com>, COP$400,000 d) on Isla Grande. They take care of transportation from the Muelle de Marina Santa Cruz in Manga (boats leave at 9am daily). The price also includes a seafood lunch and a visit to the Oceanario Islas del Rosario, whether you want to go there or not. You can also spend the night at this comfortable hotel (COP$350,000 d including transportation).\n\nAnother recommended day tour is the **This Is Cartagena** (Av. Centenario No. 30-42, tel. 5\/660-0969, cell tel. 317\/259-3773, www.thisiscartagena.com, 9am-6pm Mon.-Fri., yacht tour COP$200,000) tour to the islands. It's a more relaxed experience, as groups are rarely larger than eight persons.\n\nIf you are willing to splurge on an overnight stay, upscale hotel options include **Coralina Isla Boutique** (cell tel. 310\/764-8835, COP$577,000 d) or **Agua Azul Beach Resort** (cell tel. 320\/680-2134 or 314\/504-3540, COP$1,300,000 d high season, COP$700,000 d low season). For a special occasion, you can rent the luxurious houseboat run by the travel agency Aviatur. Their **Casa Navegante** (tel. 1\/587-5181, www.aviaturecoturismo.com, COP$1,200,000) is moored on the beautiful Bah\u00eda Chol\u00f3n in the Islas de Rosario.\n\n#### **MOMPOX**\n\nThis town, founded in 1540 on the eastern edge of a large island between two branches of the R\u00edo Magdalena (the Brazo de Loba and the Brazo Mompox), was an opulent center of trade, connecting the interior of the country with Cartagena during the colonial era. But then the mighty river changed its course in the late 18th century. Mompox's importance steadily declined, never to return.\n\nMompox is what Cartagena looked like before it became a tourist destination, and it's hard to deny the melancholic charm this oppressively hot town retains even today. The attraction here is strolling the wide streets, admiring the magnificent whitewashed houses decorated with intricate iron latticework, and watching the river flow by. In 1995, because of its architectural importance, it was declared a UNESCO World Heritage Site.\n\nThe R\u00edo Magdalena runs through Mompox.\n\nThe town is spread out along the river. It does not have a central plaza, but three squares, each with a church, facing the river. It is believed that each of these squares is on the location of a former indigenous settlement. From south to north these are: **Plaza de Santa B\u00e1rbara, Plaza de la Concepci\u00f3n** (also known as Plaza Mayor), and **Plaza de San Francisco.** Three main streets run parallel to the river: Calle de la Albarrada (which corresponds to Carrera 1) facing the river; the Calle Real del Medio, Mompox's main street, one block west of the river; and the Calle de Atr\u00e1s (literally, the \"street behind\").\n\nThere are two historical churches worth visiting in Mompox. However, they are only regularly open during mass times. Nonetheless, the Casa Amarilla can call the church to request that someone open up the doors so that you can take a quick peek. The **Iglesia de Santa B\u00e1rbara** (Cl. de la Albarrada and Cl. 14, mass 4pm Sun.), built in 1630, is well worth a visit. The facade is painted a striking yellow, with colorful floral decorations. It has an unusual baroque octagonal tower with a balcony wrapping around it. Inside, it has a magnificent gilded altar. Another noteworthy church is the **Iglesia de San Agust\u00edn** (Cl. Real del Medio and Cl. 17, masses 7pm daily, with additional 9am mass on Sun.), which houses the Santo Sepulcro, a gilded reproduction of Christ's tomb, which is carried through the streets during Semana Santa. The only museum in town is the **Museo Cultural de Arte Religioso** (Cl. Real del Medio No. 17-07, tel. 5\/685-6074, 9am-noon and 3pm-4pm Tues. and Thurs.-Fri., 9am-noon Sat.-Mon., COP$2,000), which has displays of gold- and silverwork from the colonial era. Mompox silver- and goldsmiths made a name for themselves with their intricate filigree jewelry. Another interesting sight is the **Piedra de Bol\u00edvar** (Cl. de la Albarrada and Cl. 17), a monument facing the river with a stone slab that lists all the visits Sim\u00f3n Bol\u00edvar made to Mompox. Finally, Mompox's atmospheric 19th-century **Cementerio Municipal** (Cl. 18 and Cra. 4, 8am-noon and 2-5pm daily, free) is well worth a detour.\n\nthe ornate Iglesia de Santa B\u00e1rbara, Momp\u00f3x\n\n##### **Festivals and Events**\n\n**Semana Santa,** or Holy Week (Easter), which is held during late March or April, is the most important celebration in Mompox, when visitors from all over Colombia converge on the town to watch its religious processions and attend concerts. You'll have to book months in advance to get a hotel room during that time.\n\n##### **Shopping**\n\nMompox is famous for its intricate gold filigree jewelry. Look for the **Joyer\u00eda Filimompox** (Cl. 23 No. 3-23, tel. 5\/685-6604 or 313\/548-2322), where the staff will explain their craft to you during your visit to their workshop. They accept credit cards. At the **Escuela Taller de Artes y Oficios de Santa Cruz de Mompox** (Claustro de San Agust\u00edn, Cl. 16 No. 1A-57, tel. 5\/685-5204), young people learn traditional handicrafts. Visitors are welcome to drop by and watch these artisans at work. Inside, there's an interior courtyard, an inviting place to linger for a while.\n\n##### **Accommodations and Food**\n\n**Bioma Hotel Boutique** (Cl. Real del Medio No. 18-59, tel. 5\/685-6733, cell tel. 315\/308-6365, www.bioma.co, COP$190,000 d) may be one of the most comfortable options in town, as it offers 12 air-conditioned rooms, a dipping pool, and good food. The M **Casa Amarilla** (Cl. de la Albarrada No. 13-59, tel. 5\/685-6326, cell tel. 301\/362-7065, www.lacasaamaraillamompos.com, COP$25,000 dorm, COP$100,000 d), owned by a British travel writer, is another excellent choice, with accommodations for the backpacker as well as private rooms for those seeking more comfort. After a careful restoration, the Casa Amarilla opened a luxury colonial house called the **Casa de la Concepci\u00f3n** (Cl. de la Albarrada No. 13-59, tel. 5\/685-6326, cell tel. 301\/362-7065, www.lacasaamaraillamompos.com, COP$1,500,000 house rental) in 2013. It has four bedrooms and two interior patio gardens, and the second story balcony has a fine view to the plaza below.\n\nHotels in Mompox are the best options food-wise, but be sure to confirm with them before arriving. During off-season many do not have cooks on call. Otherwise, the **Comedor Coste\u00f1o** (Cl. de la Albarrada No. 18-45, tel. 5\/685-5263, 7am-5pm daily) is a restaurant that serves _comida t\u00edpica_ (Colombian fare) overlooking the Magdalena. On the renovated Plaza de la Concepci\u00f3n there are some open-air caf\u00e9s that serve snacks and drinks. This is a nice weekend night gathering area. Plaza Santo Domingo (Cl. 18 and Cra. 3) has **food stalls** serving pizza and other fast food.\n\n##### **Getting There**\n\nMost visitors arrive in Mompox from Cartagena, and there are several ways to make the journey.\n\nThere is one direct bus that leaves from the Terminal de Transportes in Cartagena (Diagonal 57 No. 24-236, tel. 5\/663-0454) at 6:30am. The ride takes eight hours and costs COP$50,000. More comfortable is a door-to-door service (COP$75,000) with a company like **Toto Express** (cell tel. 310\/707-0838), which takes six hours. The fastest way involves a van, a boat, and a taxi: take a van (COP$40,000 pp, 3.5 hours) from outside the Terminal de Transportes in Cartagena to Magangu\u00e9, a port on the Magdalena; there hop on a _chalupa_ boat that will take you to a spot called Bodega de Mompox (COP$7,000, 30 mins.); and from there take a shared taxi or _mototaxi_ (COP$15,000, 30 minutes) to Mompox.\n\n### **Barranquilla**\n\nColombia's fourth largest city (pop. 1.6 million) is known for its busy port and for the bacchanalian Carnaval de Barranquilla, designated a World Masterpiece of the Oral and Intangible Heritage of Humanity by UNESCO. This, the most famous celebration in Colombia, is a time of music, dancing in the streets, and revelry. It lasts only about four days, but the city starts readying for it days (if not weeks) in advance.\n\nDuring the rest of the year there's not a whole lot to lure the visitor to Barranquilla. It is not a colonial city, but vestiges of its early 20th century importance can be seen in its El Prado district.\n\n#### **SIGHTS**\n\nTwo museums give the visitor a good insight into Barranquilla's people and culture. The first, **Casa del Carnaval** (Cra. 54 No. 49B-39, tel. 5\/370-5437 or 5\/379-6621, 9am-5pm Tues.-Thurs., 9am-6pm Sat.-Sun., COP$5,000), is _carnaval_ headquarters, and its Sala Carnaval Elsa Caridi provides an interactive introduction to the annual event. After a visit here, you'll come to understand the many different components of the celebration, like the different musical styles: _cumbia, mapal\u00e9, chand\u00e9,_ and _son_. While at first blush it may seem that the Carnaval de Baranquilla is just a big party, there is more to it than meets the eye. Behind every costume, parade, and dance there is a story. Knowledgeable guides will share this story and their genuine enthusiasm for the festival at this well-done museum.\n\nThe second of Barranquilla's top two museums is **Museo del Caribe** (Cl. 36 No. 46-66, tel. 5\/372-0582, www.culturacaribe.org, 8am-5pm Tues.-Fri., 9am-6pm Sat.-Sun., COP$10,000), one of the finest museums in the country, with a focus on Coste\u00f1o (Caribbean coast) culture. There is a room dedicated to Gabriel Garc\u00eda M\u00e1rquez, which may be hard to understand if you are not familiar with many of his works or if your Spanish isn't perfect; slide shows on ecosystems from the Caribbean region; and exhibits on the people of the Caribbean, including the many different indigenous tribes who live there. Of particular interest is a room that examines immigration to the region, from African slaves to \"Turcos,\" meaning those mostly coming from Syria, Lebanon, and Palestine. The museum has a restaurant with reasonably priced meals and a cute, tiny caf\u00e9 on the plaza in front. Both of these keep regular museum hours.\n\nThe old downtown of the city is very real, in its rundown and dirty state. The **Paseo Bol\u00edvar** (Cl. 34 between Cras. 38-45) is the main drag downtown. It's lined with discount shops and charming used-book stands, where you can often find some Colombian classics\u2014even in English\u2014if you look hard enough. There's always a crowd at the newspaper kiosks, which seems like a scene from a different era. On the restored **Plaza San Nicol\u00e1s** (between Clls. 32-33 and Cras. 41-42) is the neo-gothic **Iglesia San Nicol\u00e1s Tolentino** (Cra. 42 No. 33-45, tel. 5\/340-2247), which took about 300 years to build.\n\nNot about roses and chocolates, the **Museo Rom\u00e1ntico** (Cra. 54 No. 59-199, tel. 5\/344-4591, 9am-11:30am and 2:30pm-5:30pm Mon.-Fri., COP$5,000) is really a history museum of Barranquilla with artwork, old _carnaval_ costumes, missives signed by Sim\u00f3n Bol\u00edvar, and a typewriter used by Gabriel Garc\u00eda M\u00e1rquez. It was once the majestic home of Jewish immigrants who arrived in Colombia at the turn of the 20th century. It is run by an elderly historian and his wife.\n\nPop singer Shakira has won two Grammys and countless other awards, and is Barranquilla's favorite daughter. To honor her, the people of the city put a statue of her stroking a guitar in a prominent place: in front of the **Estadio Metropolitano Roberto Mel\u00e9ndez** (Aves. Circunvalar Alberto Pumarejo and Murillo).\n\n#### **FESTIVALS AND EVENTS**\n\n##### M **Carnaval de Barranquilla**\n\nFor most Colombians, Barranquilla is synonymous with _carnaval_ , and they boast that the Carnaval de Barranquilla is the world's biggest after Rio, although folks from New Orleans may balk at this claim. In Colombia, this is really the only place where the bacchanal is celebrated, although some Caribbean cities and even Bogot\u00e1 make an effort.\n\nDuring the four days prior to Ash Wednesday, in late February or early March, the **Carnaval de Barranquilla** (www.carnavaldebarranquilla.org) is full of Coste\u00f1o pageantry: costumes, music, dance, parades, and whiskey.\n\nOfficially, Carnaval gets going on Saturday, but on the Friday night before, **La Guacherna** is held. One event of the night is the Desfile Gay, which is when outrageously costumed men dressed in drag parade down a street to the hoots and hollers of thousands of bystanders.\n\nSaturday is the main event. That's the day of the **Batalla de las Flores** (Battle of the Flowers) parade. It's when floats carrying beauty queens and dancers and thousands in _comparsas_ (groups) in elaborate costumes make their way down the Calle 40 under the sizzling Barranquilla sun. This event dates back to 1903, when the celebration was begun as a celebration of the end of the Guerra de Mil D\u00edas (Thousand Days' War). Participation in the parades is serious business here, involving planning, practice, money, and, sometimes, connections. However, there is one _comparsa_ during the Batalla de las Flores in which just about anyone can participate, and it's one of the most popular. That's the _comparsa_ of \"Disfrazate como Quieras\"\u2014go however you like. Anybody in a costume, from the silly to the sexy, can join. To participate, visit the web page (www.disfrazatecomoquieras.com).\n\nOn Sunday, during the Gran Parada de Tradici\u00f3n y Folclor, groups of dancers perform on the Calle 40 to the typical, hypnotic music of _carnaval_ \u2014a mix of African, indigenous, and European sounds. On Monday there is another parade, the Gran Parade de Comparsas, and, starting in the late afternoon, a massive concert attracting more than 30 musical groups. These compete for the award of Congo del Oro. On Tuesday, after four days of music and dancing, things wind down with the parade Joselito Se Va con las Cenizas. This is when Joselito, a fictitious Barranquillero, dies after four days of rumba, and his body is carried through the streets as bystanders weep. On Wednesday, Barranquilleros call in sick.\n\nYou can watch all the action of the parades from the _palcos_ (bleachers) that line Calle 40. Keep in mind that the parades take place in the middle of the day, meaning lots of sun and heat. Tickets for the _palcos_ can be ordered online at Tu Boleta (www.tuboleta.com).\n\nWith regard to _carnaval_ , it's said that _\"quien lo vive, es quien lo goza\"_ (\"whoever experiences it is who enjoys it\"). But to do that, it's crucial to get those hotel and flight reservations early.\n\n#### **ACCOMMODATIONS**\n\nAs a business destination, Barranquilla has a number of hotel options, although hostels are almost nonexistent. Rates significantly drop on weekends.\n\n**Meeting Point Hostel** (Cra. 61 No. 68-100, tel. 5\/318-2599, www.themeetingpoint.hostel.com, COP$60,000 d, COP$25,000 dorm) is the only hostel catering to international backpackers in Barranquilla. It's run by an Italian-Colombian family. If it feels like you're staying in their house, that's because it is their house, down to kids on the sofa playing video games. The neighborhood is quiet and green, and about a 15-minute walk from the El Prado area.\n\n**Country International Hotel** (Cra. 52 No. 75-30, tel. 5\/369-5900, ext. 120, www.countryinthotel.com, COP$238,000 d) has a nice pool and comfortable rooms. It's located in a good area. **Hotel Estelar Alto Prado** (Cl. 76 No. 56-29, tel. 5\/336-0000, COP$292,000 d) is a modern, ultra comfortable, and stylish address in Barranquilla.\n\nM **Hotel El Prado** (Cra. 54 No. 70-10, tel. 5\/369-7777, www.hotelelpradosa.com, COP$236,000 d) debuted in 1930, and for decades, before the relatively recent boom in luxury cookie-cutter hotels, it was the luxury address in town. It's got 200 rooms, a massive boiler room, and a fab pool to lounge around drinking a cocktail. Non-guests can spend an afternoon getting pampered here for only COP$38,000, a price that includes lunch and all the poolside lounging you need.\n\n#### **FOOD**\n\n**El Huerto** (Cra. 52 No. 70-139, tel. 5\/368-7171, 8am-7:30pm Mon.-Sat., 10am-3pm Sun., COP$12,000) has been serving the vegetarian minority of Barranquilla since 1986. They have a set lunch menu every day and sell baked goods to go as well.\n\nBarranquilla's many residents of Lebanese and Syrian descent have a few Middle Eastern restaurants to choose from. One of the best is **Arabe Gourmet** (Cra. 49C No. 76-181, tel. 5\/360-5930, 11am-10pm daily, COP$25,000).\n\nThe M **Restaurante Bar La Cueva** (Cra. 43 No. 59-03, tel. 5\/340-9813, noon-3pm and 6pm-10pm Mon.-Thurs., noon-3pm and 6pm-1am Fri.-Sat., COP$25,000) has history and lots of character. It was the hangout of Gabriel Garc\u00eda M\u00e1rquez and artists such as Alejandro Obreg\u00f3n in the 1960s. Elephant tracks, memorabilia, and photos make it seem like a museum, but it is still a restaurant, and a popular one at that. The specialty here is seafood. There's live music on Friday and Saturday evenings. Be sure to check out the Obreg\u00f3n work _La Mulata de Obreg\u00f3n,_ complete with a bullet hole thanks to a drunken friend of the artist.\n\nFor steak lovers, the top two options in Barranquilla are **La Bonga del Sinu** (Cra. 53 No. 82-10, tel. 5\/358-5035, 11:30am-10pm Mon.-Wed., 11:30am-11pm Thurs.-Sat., 11:30am-9pm Sun., COP$25,000) and **Buffalo Grill** (Cra. 51B No. 79-97, Local 2, tel. 5\/378-6519, noon-11pm Mon.-Thurs., noon-midnight Fri.-Sat., noon-10pm Sun., COP$25,000).\n\nThanks to its location in a strip mall, you may not have soaring expectations for **Restaurante Plaza 52** (Cra. 52 No. 72-114, Local C9, tel. 5\/358-1806, 10am-8pm daily, COP$6,500 set lunch). But the lunchtime crowds (and lines) give it away. They serve great down-home food at rock-bottom prices.\n\nBarranquilla's version of street food can be found at the popular, competing **food stands** at the intersection of Carrera 52 and Calle 71. There's always a crowd composed of construction workers and office types hanging out here.\n\n#### **GETTING THERE**\n\nFrom Barranquilla's **Aeropuerto Internacional Ernesto Cortissoz** (Soledad, www.baq.aero), there is excellent air connection with all major cities in Colombia and a handful of nonstop international flights as well. The airport is south of the city in Soledad. Taxis cost about COP$20,000 from downtown to the airport.\n\n**Avianca** (Centro Comercial Gran Centro Cra. 53 No. 68-242, tel. 5\/360-7007, www.avianca.com, 8am-noon and 2pm-6pm Mon.-Fri., 9am-12:30pm Sat.) flies nonstop from Miami, Bogot\u00e1, Cali, and Medell\u00edn. **LAN** (Centro Comercial Buenavista Cra. 53 No. 98-99, Col. toll-free tel. 01\/800-094-9490, 10am-8pm Mon.-Fri., 10am-noon and 1pm-7pm Sat., 10am-noon and 1pm-5pm Sun.) has flights to Bogot\u00e1. On **Copa** (Col. toll-free tel. 01\/800-011-2600, www.copair.com) there are nonstop flights to San Andr\u00e9s and to Panama City.\n\n**Viva Colombia** (tel. 5\/319-7989, www.vivacolombia.com.co) flies from Medell\u00edn, **Easy Fly** (tel. 5\/385-0676, www.easyfly.com.co) has nonstop flights from Bucaramanga and Valledupar, and **ADA** (Col. toll-free tel. 01\/800-051-4232, www.ada-aero.com) flies from Monter\u00eda.\n\nThere is regular bus service to all points in Colombia from the **Terminal Metropolitana de Transportes** (Km. 1.5 Prolongaci\u00f3n Cl. Murillo, tel. 5\/323-0034, www.ttbag.com.co). Fast van service is also available to Cartagena (COP$17,000) and Santa Marta (COP$17,000) from the **Berlinastur terminal** in town (Cl. 96 No. 46-36, tel. 5\/385-0030, www.berlinastur.com).\n\n#### **GETTING AROUND**\n\nBarranquilla is not much of a walking city, with taxi cabs the most convenient way to get around. Unfortunately the cabs are not metered, so you might be charged a little more than locals if you're unfamiliar with rates. In town, you should never pay more than COP$13,000 to get from point A to point B. For those with smartphones, the app **Tappsi** enables you to order a cab from anywhere in the city, receive the license plate number and name of the driver, and send this information to a friend, for security purposes.\n\n**Transmetro** (www.transmetro.gov.co, 5am-11pm Mon.-Sat., 6am-10pm Sun., COP$1,400) is the Barranquilla version of the TransMilenio rapid bus system in Bogot\u00e1, and it runs along two main avenues: Avenida Murillo (also known as Calle 45) and Avenida Olaya Herrera (also known as Carrera 46). There are stations in front of the Museo del Caribe, the cathedral, and the stadium, and you can take the bus downtown from there.\n\n#### **PUERTO COLOMBIA**\n\nAbout a 40-minute bus ride (COP$2,000) outside of Barranquilla is Puerto Colombia, which was once Colombia's most important port.\n\nThe pier, the main attraction in town, was built at the turn of the 20th century. At the time, it was one of the longest piers in the world. The pier was severely damaged in major storms in 2009 and 2013, and there are huge gaps in the pier today. Today you can walk the pier and ask local fishers about their catch. Military personnel stand guard, meanwhile, in order to prohibit smuggling of illegal contraband. There are plans to refurbish the pier and to create an artificial beach, in hope of attracting weekend day-trippers and bringing back some of the town's former glory. Part of that effort included the construction of a boardwalk along the water.\n\nThe old train station has been converted into a cultural center by the **Fundaci\u00f3n Puerto Colombia** (town center, tel. 5\/309-6120, fundacionpuertocoombia@gmail.com, 8:30am-12:30pm and 3pm-7pm Mon.-Fri., 9am-1pm Sat., 4pm-6pm Sun., free), and there is often an art exhibit going on here.\n\nThere are several seafood restaurants around the pier area. The most famous, perhaps, is **Mi Viejo Muelle** (Cl. 2 No. 3-175, tel. 5\/309-6727, noon-8pm daily, COP$22,000). It's got a nice large deck with a view, and old photos of Puerto Colombia decorate the walls.\n\n### **Santa Marta**\n\nSanta Marta is coming into its own as a major tourist destination on the Caribbean coast. In addition to its charming historic district, great hotel options, and restaurants, Santa Marta offers an excellent base from which to explore the Sierra Nevada and the deserts of La Guajira.\n\nSanta Marta was the first permanent Spanish settlement in colonial Colombia, and remained relatively rural until the second half of the 20th century, when it became a major domestic tourist destination. In the mid-1970s, treasure hunters discovered Ciudad Perdida high in the Sierra. Ciudad Perdida was one of the most important settlements of the Tayrona indigenous people. The National Institute of Anthropology carefully excavated the site, opening it to tourism in the 1980s.\n\nIn recent years, the old historic downtown, long neglected as development had moved to the Rodadero district, has seen a renaissance. It has become a destination of its own and a staging point for visits to the unspoiled beaches of Parque Nacional Natural Tayrona and hiking trips to Ciudad Perdida.\n\n#### **ORIENTATION**\n\nThe _centro hist\u00f3rico_ extends from the busy Calle 22 (Avenida Santa Rita) in the west to the Avenida del Ferrocarril in the east, toward the seaside village of Taganga. And from south to north, the borders are the same Avenida del Ferrocarril in the south to the _malec\u00f3n_ (Carrera 1C\/Avenida Rodrigo de Bastidas). The focal point of the Centro is the Parque de los Novios (Carreras 2A-3 and Calles 19-20). This is a lovely park with pedestrian streets (Calle 19 and Carrera 3) intersecting on its eastern side. Most sights are within a smaller range of streets from Carrera 5 (Camp Serrano) to the water and between Calles 20 and 14.\n\nThe Rodadero district is a mini-Miami with condos and hotels lining the beach. It's west of downtown Santa Marta, just around the bend on the main highway. There are constant bus links all day between the two.\n\n#### **SIGHTS**\n\n##### **Centro Hist\u00f3rico**\n\nThe compact _centro hist\u00f3rico_ in Santa Marta in itself feels like a living museum, with its mix of colonial and republican-era architecture. All major sights, save for the Quinta de San Pedro Alejandrino, are located here and can be visited in one day.\n\nThe **Parque de los Novios** (Cras. 2A-3 and Clls. 19-20) is a symbol of the city's rejuvenation. The pedestrian streets (except for one) around the plaza have a lot to do with it. Today it's a most pleasant place for a stroll and a meal. Restaurants have cropped up along the park's periphery. On the Calle 20 side of the park is the grandiose neoclassical **Palacio de Justicia.** Visitors can only admire it from the outside, as the building is not open to the public.\n\nThe **Catedral de Santa Marta** or **Bas\u00edlica Menor** (Cr. 5 No. 16-30, tel. 5\/421-2434, masses at noon and 6pm Mon.-Sat., 7am, 10am, noon, and 6pm Sun.) took around 30 years to build and was completed in 1794, toward the end of Spanish reign in Nueva Granada, as colonial Colombia was called. It is one of the oldest cathedrals in Latin America. The city's founder, Rodrigo de Bastidas, is buried there, and Sim\u00f3n Bol\u00edvar laid in rest there before his body was moved to Caracas.\n\nThe **Plaza de Bol\u00edvar** (Cras. 1-2 and Clls. 14-15) has a statue of Sim\u00f3n Bol\u00edvar on horseback, ready to destroy the oppressors. The **Banco de la Rep\u00fablica** (Cl. 14 No. 1C-37, tel. 5\/421-0251, www.banrep.gov.co, 8:30am-6pm Mon.-Fri., 9am-1pm Sat., free) often has art exhibits and also has a public library on the third floor (a quiet place to read or work).\n\nThe **Museo de Oro Tairona** (Cl. 14 No. 1C-37, tel. 5\/421-0251, www.banrepcultural.org, 8:30am-6pm Mon.-Fri., 9am-1pm Sat., free) is in the historic **Casa de la Aduana,** perhaps the oldest customs house in the Americas, dating back to 1531. A smaller version of the famous Museo del Oro in Bogot\u00e1, this focuses on the Tayrona people, who were the native settlers of the region and forebears of the Kogis, Arhuacos, Kankuamos, and Wiwas who live in the Sierra Nevada. There are ceramic and gold artifacts on display, and a description of the Ciudad Perdida archaeological site. A visit to this museum may enrich your hike up to the Lost City, as you'll have a better understanding of the people who once inhabited it.\n\nAlong the waterfront is the **Paseo de Bastidas** (Cra. 1) boardwalk. A sunset walk along the pier that extends from the boardwalk is a daily Santa Marta ritual. Great views are to be had here of the port on the right, the Isla Morro in the sea, and the gorgeous sailboats on the left docked at the Marina Santa Marta.\n\n##### **Quinta de San Pedro Alejandrino**\n\nThe Liberator, Sim\u00f3n Bol\u00edvar, spent his final days in Santa Marta, passing away at the age of 47 at the **Quinta de San Pedro Alejandrino** (Mamatoco, tel. 5\/433-2995, www.museobolivariano.org.co, 9am-6pm daily, COP$12,000). This country estate is now a museum where visitors can see the bedroom in which Bol\u00edvar died in 1830. A modern wing houses two art galleries. Young guides will offer to take you around the complex for a small tip of about COP$2,000, but you're probably better off on your own. The _quinta_ is set in a manicured botanical garden. There is a small snack bar and gift shop on the grounds.\n\n#### **SHOPPING**\n\nThe shop **Santa Marca** (Cl. 17 No. 2-45, tel. 5\/423-5862, www.santamarca.co, 8am-8pm Mon.-Sat.) has gifts and souvenirs made by local creative types. The **Centro Comercial Arrecife** (Cra. 4 No. 11A-119, tel. 5\/422-8873, 9am-9pm daily) is a large shopping mall in the Rodadero area; it has a food court, a Carulla supermarket, a movie theater, and a selection of mostly Colombian brand clothing shops.\n\nLocals converge on Santa Marta's Paseo de Bastidas to enjoy the sunset.\n\n#### **RECREATION**\n\n**Deva Yoga Studio** (Cra. 21 No. 15-18, tel. 5\/431-0354, www.newfuturesociety.org, classes at 9am and 6:30pm Mon.-Fri., 9am Sat., COP$20,000\/class) offers hatha and Tibetan yoga classes and meditation classes. It is affiliated with the New Future Society International, an international yoga organization.\n\nThe tour agency **Aventura Sierra Nevada** (Restaurante Marisol, Cra. 3A No. 16-30, cell tel. 311\/216-5419, www.aventurasierranevada.com, 10am-9pm daily) organizes activities along the Caribbean coast, from kite surfing courses to bike tours of various lengths, hikes to Tayronaka, a trip to the Yumake nature reserve, and inner tube trips down the R\u00edo Don Diego.\n\n#### **ACCOMMODATIONS**\n\nThe Centro Hist\u00f3rico of Santa Marta was considered to be crumbling, desolate, and even dangerous before the early 2000s, but gentrification has taken hold and today it's gone boutique. In addition to posh boutique hotels, there are comfortable hostels. The Rodadero has plenty of high-rise hotels, most of which are all-inclusive. The village of Taganga is also very close to the city, and some prefer to stay at this relaxed beach village and visit Santa Marta for the day or head to restaurants in the Centro Hist\u00f3rico in the evenings.\n\n##### **Under COP$70,000**\n\nM **Aluna** (Cl. 21 No. 5-27, tel. 5\/432-4916, www.alunahotel.com, COP$30,000 dorm with fan, COP$100,000 d with a\/c) gets just about everything right. This hostel has a mix of dorm rooms and private rooms, all immaculate, each with private bath. There's plenty of space to hang out over the three floors of the hostel. There is a small shaded interior courtyard and an excellent top floor terrace that provides vantage points over the city and to the Sierra Nevada in the distance. The hostel restaurant serves gazpacho, falafel, and, for breakfast, those banana pancakes that every traveler craves. It's run by a friendly Dubliner, Patrick.\n\nThe Quinta de San Pedro Alejandrino is where Sim\u00f3n Bol\u00edvar spent his final days.\n\nFun and high energy: That's **La Brisa Loca** (Cl. 14 No. 3-58, tel. 5\/431-6121, www.labrisaloca.com, COP$28,000 dorm, COP$90,000 d), a revamped mansion turned hostel that has earned its spot as a backpacker favorite. There's a small pool in the main courtyard with private rooms and dormitories surrounding it over three floors. On the top floor is an excellent bar that, thanks to its daily drink specials, gets jammed with backpackers and locals alike. On match days, it's soccer enthusiasts who cram the bar. In addition to on-site parties, the hostel organizes a wide array of outdoor adventures to the Sierra Nevada and rents out paddleboards for days at the beach.\n\n**Drop Bear Hostel** (Cra. 21 No. 20-36, tel. 5\/435-8034, www.dropbearhostel.com, COP$18,000 hammock, COP$25,000 dorm, COP$65,000 d) is a funky newcomer to the Santa Marta hostel scene. It's in a huge house that was built by a drug trafficker in the 1980s. Each night the Australian and New Zealander owners give tours of the house, including secret tunnels and nooks where money was stashed away. The bar is aptly named the Cartel Bar. Rooms are massive, and perhaps the nicest feature at Drop Bear is the big pool, which is the main gathering area at this sociable place.\n\n##### **COP$70,000-200,000**\n\nThe M **Hotel del Parque** (Cl. 19 No. 4-45, tel. 5\/420-7508, COP$90,000 d) on the pleasant pedestrian Calle 19 is a gem of a find. Supremely low-key, this hotel has got just a handful of air-conditioned rooms, is very well maintained, and, best of all, it's fairly priced. There's complimentary coffee, but no free breakfast; however, there are many options within walking distance nearby.\n\n##### **Over COP$200,000**\n\nA Spanish couple has developed a small empire of boutique lodging and dining options in old Santa Marta. **La Casa del Farol** (Cl. 18 No. 3-115, tel. 5\/423-1572, www.lacasadelfarol.com, COP$317,000 d) was their first, and one of the first boutique hotels in the city. It's in an 18th-century house and has six rooms that are named for different cities of the world. From the tiny wading pool on the rooftop you get a nice view of the city.\n\n**La Casa del Agua** (Cl. 18 No. 4-09, tel. 5\/423-1572, www.lacasadelagua.com.co, COP$225,000 d) is across the street from La Casa del Farol. It has four rooms of varying sizes and styles. Try for one with a balcony. The small pool downstairs is a welcome sight after a day out and about in the heat.\n\nFor some boutique pampering, try the 10-room **Casa de Isabella** (Callej\u00f3n del R\u00edo, Cra. 2 No. 19-20, tel. 5\/431-2082, cell tel. 301\/466-5656, www.casaisabella.com, COP$250,000 d). It's a tastefully revamped republican-era house with nods to both colonial and republican styles, and it surrounds a tamarind tree that's over 200 years old. The suites on top have fantastic private terraces and hot tubs.\n\nIf you're more interested in a beach holiday, consider the beachfront **Tamac\u00e1 Beach Resort Hotel** (Cra. 2 No. 11A-98, tel. 5\/422-7015, www.tamaca.com.co, COP$315,000 d) in Rodadero. It has 81 rooms with all the usual amenities and a fantastic pool area that overlooks the water. The hotel has two towers, with the beachside tower preferred by most.\n\n#### **FOOD**\n\nFoodies from Europe and North America have converged on Santa Marta, making their dreams of opening up a restaurant become reality. And Samarians and travelers alike thank them.\n\nThere ought to be more restaurants like M **Marisol** (Cl. 19 No. 3-56, tel. 5\/420-6511, www.marisolsantamarta.com, 8am-11pm daily, COP$18,000), an unpretentious spot that serves up healthy meals that are not too expensive, as well as deliciously fresh juices. It's run by a man from Cali who ran a successful restaurant in Berlin for many years. The location on the pedestrian Calle 19 is particularly peaceful, and you can grab a seat outside at night if you want. Sandwiches, pastas, and salads appear on the lunch and dinner menus. You can also grab a late breakfast here and chat with the servers if they're not busy.\n\nFor pizza the Sicilian way, look no further than **La Pizzeria Italia Gourmet** (Cra. 3A No. 16-24, tel. 5\/422-7329, 5pm-11pm Mon.-Sat., COP$30,000) on the cute pedestrian alley, the Callej\u00f3n del Correo. It's next door to the excellent **Donde L'Italiano** (Cra. 3A No. 16-26, cell tel. 316\/429-1131, 5pm-11pm Mon.-Sat., COP$30,000) a cheerful restaurant where even the wallflower _pasta arrabiata_ tastes exquisite. Go for a table in the courtyard in back.\n\nAn American-run place that delivers quality meals and drinks that will remind you of home is **El Frescanjero** (Cra. 7A No. 19-63, tel. 5\/422-0379, 11am-2:30pm and 5:30pm-10:30pm Tues.-Sat., COP$20,000). They've got the most eclectic menu on the coast with Japanese chicken with ginger, vegetarian tacos, po'boys, and a platter of German sausage that automatically comes with a beer. And oh, the cocktail specials!\n\nA locally run spot for a non-fussy, home-cooked meal is **Mandragora** (Cl. 20 No. 6-54, tel. 5\/421-9392, cell tel. 301\/400-3442, 11:30am-3pm Mon.-Sat., COP$12,000). Each day there is a set menu with fresh fish or typical Colombian dishes, such as _bandeja paisa_ (dish of beans, various meats, yuca, and potatoes) on the menu.\n\nFor North Americans who've been on the road for a while, it's a treat to stumble upon M **Agave Azul** (Cl. 14 No. 3-58, tel. 5\/431-6121, noon-10pm Tues.-Fri., 5pm-11pm Sat., COP$25,000). It's a Tex-Mex place, complete with margaritas, burritos, and nachos, in a space awash in warm colors. It's run by the same two American brothers who have M **Ouzo** (Cra. 3 No. 19-29, tel. 5\/423-0658, 6pm-11pm Mon.-Sat., COP$25,000). Specializing in Mediterranean fare, Ouzo has brought some class to the park. Sizzling seafood or pasta accompanied by a glass of white wine will work out just fine for you. You can dine alfresco by candlelight if you wish, although you may be bothered by street vendors or beggars. It's not uncommon for visitors to hit both of these excellent restaurants in the same day.\n\nThe fab **Restaurante Panamerican** (Cl. 18 No. 1C-10, tel. 5\/421-2901, noon-3pm and 6pm-10pm Mon.-Sat., noon-4pm Sun., COP$25,000) hasn't changed in decades\u2014and why should it? While its heyday is long past, the Panamerican has an enormous menu specializing in steaks and seafood, but you can also just order a martini and enjoy the retro-chic atmosphere.\n\nThe way Samarios talk, it would seem that M **Donde Chucho** (Cl. 6 at Cra. 3, tel. 5\/422-1752, 11:30am-10:30pm daily, COP$25,000) is the only restaurant in Santa Marta and that it has been serving its mixed seafood platters since Bol\u00edvar was in town. It's in fast-moving Rodadero, and the open-air place is filled with photos of Chucho, the owner and chef, posing with Colombian beauty queens, sports stars, and the odd politician. It's hard to believe that Chucho began this seafood empire (he owns four restaurants now) with a humble wooden ceviche stand in Rodadero in the 1990s.\n\nThe terrace of **Pepe Mar** (Cra. 1 No. 6-05, Rodadero, tel. 5\/422-2503, noon-10pm Tues.-Thurs. and Sun., until midnight Fri.-Sat., COP$26,000) is the best place for people-watching in all of Rodadero. Try the fried red snapper with coconut rice and, of course, a _patac\u00f3n_ (fried plantain) or two.\n\n#### **INFORMATION AND SERVICES**\n\nIn addition to a stand at the airport, there is a **PIT** (Punto de Informaci\u00f3n Tur\u00edstica, Cra. 1 No. 10A-12, tel. 5\/438-2587, 9am-noon and 2pm-6pm Mon.-Fri., 9am-1pm Sat.) tourist information booth along the waterfront.\n\n**Lavanderia Para\u00edso** (Cl. 22 No. 2A-46, tel. 5\/431-2466, cell tel. 315\/681-1651, 9am-6pm Mon.-Fri., 9am-1pm Sat.) will wash your clothes and have them ready for you to pick up by the next day. With a little charm, you might be able to persuade them to have your things ready before 5pm the same day.\n\n#### **GETTING THERE AND AROUND**\n\nSanta Marta is easily accessed by air and by land from all major cities in Colombia.\n\nThe **Aeropuerto Internacional Sim\u00f3n Bol\u00edvar** (tel. 5\/422-4604 or 5\/422-4490) is 16 kilometers (10 miles) west of the _centro hist\u00f3rico._ Domestic carriers **Avianca** (www.avianca.com), **LAN Colombia** (www.lan.com), **Viva Colombia** (www.vivacolombia.com.co), and **Easy Fly** (www.easyfly.com.co) connect Santa Marta with the major cities of Colombia. Copa has nonstop flights to its hub in Panama City, Panama.\n\nThere is hourly bus service to Cartagena, Barranquilla, and Riohacha. Many buses to nearby coastal destinations leave from the market area in the _centro hist\u00f3rico._ Long-haul buses for destinations such as Bogot\u00e1, Medell\u00edn, and Bucaramanga depart from the Terminal de Transportes (Cl. 41 No. 31-17, tel. 5\/430-2040) outside of town.\n\nTaxis to Taganga cost around COP$8,000, and _colectivo_ buses are around COP$1,200. These can be found on the waterfront near the Parque Bol\u00edvar, along Carrera 5, or at the market at Carrera 11 and Calle 11.\n\nThe best way to get around the _centro hist\u00f3rico_ is on foot.\n\n#### **TAGANGA**\n\nThis popular beachside community is only about a 20-minute ride through the desert to the northeast from Santa Marta and is actually considered part of the city. In the 1970s, this sleepy fishing village was discovered by hippie-types looking for an escape from urban life, and Taganga evolved into a mecca for backpackers. On any given Saturday along the bayside promenade, you'll brush shoulders with a truly mixed lot of humanity: Colombian families, diving fanatics, traveling musicians, and general sunseekers of all ages and nationalities. It's as close as Colombia gets to Venice Beach.\n\n##### **Recreation**\n\n###### **DIVING AND SNORKELING**\n\nThe warm waters (24-28\u00b0C\/75-82\u00b0F) off of Taganga provide some good diving and snorkeling opportunities. Diving excursions take you into the waters off of the Parque Nacional Natural Tayrona to the northeast, the Isla Morro off the coast of Santa Marta, or to a shipwreck near the beaches of Rodadero. The best months for diving here are between July and September.\n\nRun by a Paisa couple, **Tayrona Dive Center** (Cra. 1C No. 18A-22, tel. 5\/421-5349, cell tel. 318\/305-9589, www.tayronadivecenter.com, 8am-noon and 2pm-6pm daily) is a very organized agency that offers PADI certification courses (COP$650,000) over a period of three days with six dives each day, a one-day mini-course (COP$160,000), and diving excursions for those with experience. They also have a hotel (Cra. 1C No. 18A-22, tel. 5\/421-5349, cell tel. 318\/305-9589, www.tayronadivecenter.com, COP$40,000 pp) with eight rooms, five with views of the water. Rooms have a safe deposit box and big refrigerators. The hotel is exclusively for divers during high season.\n\n**Oceano Scuba** (Cra. 2 No. 17-46, tel. 5\/421-9004, cell tel. 316\/534-1834, www.oceanoscuba.com.co, 8am-noon and 2pm-6pm daily) offers the whole array of diving activities, from one-day dives for certified divers (COP$110,000) to a one-day beginner's course (COP$180,000) to an Open Water PADI certification course (COP$600,000) that lasts three days. Night dives (COP$80,000), during which you might come across eels hunting the waters, and snorkeling (COP$50,000) are also on offer.\n\n###### **BIKING AND TREKKING**\n\nBiking and hiking trips are the specialty of **Elemento Outdoor Adventure** (Cl. 18 No. 3-31, tel. 5\/421-0870, cell tel. 310\/605-0929, www.elementooutdoor.com, 8am-noon and 2pm-6pm Mon.-Fri., 9am-4pm Sat.-Sun.). Elemento offers a range of mountain bike adventures, including a one-day downhill trip from Los Pinos to Minca (COP$140,000) with swimming hole stops along the way. There are also multi-day adventures in the area. For less adrenaline-pumping days, Elemento also offers visits to eco-farms, nature reserves, and indigenous communities.\n\nThe fishing village of Taganga has become a backpacker mecca.\n\n###### **BEACHES**\n\nThere is a popular beach in front of the La Ballena Azul hotel, but the best beaches are a quick boat ride away. Beaches along the coast from Taganga to the **Parque Nacional Natural Tayrona** can be visited by boat. It costs about COP$40,000 round-trip to go to Tayrona from Taganga, but you can negotiate that price, especially if you are in a group. Although all visitors to the park are supposed to pay an entrance fee (and it is steep for non-Colombians), some boat captains will take you to beaches where no park employees will charge you for park entrance, which park officials rightly do not condone. During the windy months of December-February, boat transportation can be rough and dangerous.\n\n**Playa Grande** is probably the best beach to visit, and it is one of the closest to Taganga. It costs COP$10,000 round-trip to get there by boat. To arrange for boat transportation to any of these beaches, just head to the beach in front of the promenade or at the La Ballena Azul. There are always boaters hanging about waiting for customers.\n\n##### **Accommodations**\n\nOwned by Olga from Bogot\u00e1, **Hostal Pelikan** (Cra. 2 No. 17-04, tel. 5\/421-9057, cell tel. 316\/756-1312, www.hostalpelikan.com, COP$25,000 dorm, COP$65,000 d) is a decent place to stay, with a nice terrace area for your morning coffee. Breakfast (with fruit!) is an additional fee.\n\nThe wooden cabins at **La Casa del Profe** (Cl. 21 No. 5A-36, cell tel. 311\/882-8912, COP$50,000 pp) dot a mountain's edge and offer great views of the bay. There are eight rooms here, and guests may use the kitchen. It's about a 15-minute hike to the action in Taganga.\n\nM **La Ballena Azul** (Cra. 1 at Cl. 18, tel. 5\/421-9009, www.hotelballenaazul.com, COP$173,000 d) is a Taganga classic, started by a Frenchwoman years ago. It's still in the family, and they still serve cr\u00eapes in their restaurant. Ballena Azul is on the beach, with the best location in town\u2014in the center of activity.\n\nProbably the most luxurious option in Taganga is the **Hotel Bah\u00eda Taganga** (Cl. 4 No. 1B-35, tel. 5\/421-0653, cell tel. 310\/216-9120, www.hotelbahiataganga.com, COP$235,000 d). It's on the eastern side of the bay. Head to the pool in the late afternoon and watch the sun slip behind the mountains.\n\n##### **Food**\n\nOn the outdoor terrace of M **Bitacora** (Cra. 1 No. 17-13, tel. 5\/421-9482, 9am-11:30pm, COP$18,000) diners get front seat views to the boardwalk. Bitacora specializes in fresh, Taganga seafood, but there are also many vegetarian options as well as pastas. If you want to cool off with a coconut lemonade, this is the spot.\n\nM **Babaganoush Restaurante y Bar** (Cra. 1C No. 18-22, 3rd floor above Taganga Dive Center, cell tel. 318\/868-1476, 1pm-11:30pm Wed.-Mon., COP$20,000) is an excellent Dutch-run restaurant and bar with amazing views. It's a true crowd pleaser with a diverse menu of falafels, seafood, steak, and even a shout-out to Southeast Asia. But surprisingly, you won't find any babaganoush! Go in the evening for the atmosphere, drinks, and sunsets. The daily happy hour is hard to pass up.\n\n##### **Getting There and Around**\n\nTaganga is easily reached from Santa Marta and from points east, such as the Parque Nacional Natural Tayrona and Palomino. Minibuses and buses ply that route daily. Minibuses from the _centro hist\u00f3rico_ of Santa Marta depart from the market area on Carrera 5 and also from Carrera 1 near the Parque Sim\u00f3n Bol\u00edvar. The trip costs about COP$1,500. Taxis from Santa Marta cost around COP$12,000, more from the bus terminal or airport. Once in Taganga you can walk everywhere you need to go.\n\n#### M **MINCA**\n\nIf you've had your fill of beaches or the seductive Caribbean cities, maybe it's time for an altitude adjustment. Artists, nature lovers, coffee farmers, and transplanted urbanites in the village of Minca (pop. 500) look down upon their neighbors in nearby Santa Marta\u2014literally. At elevation of 660 meters, midway up the Sierra, you get a bird's-eye view of Santa Marta, just 45 minutes away. You also get a great bird's-eye view of birds, especially higher up at the edge of the Parque Nacional Sierra Nevada de Santa Marta. The blissful routine of mountain hikes, dips in invigorating swimming holes, and sunset ogling may make you want to linger here.\n\n##### **Recreation**\n\n###### **HIKING**\n\nMinca is a paradise for those with a pair of hiking boots and a backpack slung over their shoulder. Hostels and hotels can point you in the right direction to several gentle hikes along tranquil mountain roads and paths to swimming holes of either freezing cold or wonderfully refreshing pure water, depending on the thickness of your skin. Three popular walks with swimming holes are within easy walking distance from Minca: Balneario Las Piedras (45 mins.), Pozo Azul (1 hr.), and the Cascadas Marinka (1 hr.).\n\nPeaceful and cool Minca is a nice break from the beach.\n\nFor a challenge, try the three-hour hike (one way) to the Los Pinos hostel at an elevation of 1,600 meters (5,250 feet). From there, or nearby, you can often get a fanastic glimpse of the snowcovered Pico Crist\u00f3bal Col\u00f3n and Pico Bol\u00edvar, the highest mountain peaks in the country.\n\nThe Sierra Nevada de Santa Marta is a renowned coffee-growing region. The **Finca La Victoria** (no phone, 9am-4pm daily, COP$5,000) is a family-run coffee farm that you can visit for a small fee. It is between Pozo Azul and Los Pinos, about a one-hour walk from town.\n\nIn and around Minca there are no safety issues, and you can set off and up the mountain on your own without a guide.\n\n###### **BIRD-WATCHING**\n\nHigh into the Sierra Nevada, at an elevation of around 2,400 meters (7,875 feet), the **Reserva El Dorado** (www.proaves.org) is one of the finest bird-watching reserves in the country. For reservations (COP$569,000 3 nights all incl.), contact **EcoTurs** in Bogot\u00e1 (Cra. 20 No. 36-61, tel. 1\/287-6592, info@ecoturs.org) or **Aviatur** (tel. 1\/587-5181, www.aviaturecoturismo.com) in Bogot\u00e1. The accommodations are excellent, with 10 spacious rooms, great food, and, crucially, hot showers. The area is home to 19 endemic species, including the Santa Marta antpitta, Santa Marta parakeet, Santa Marta bush tyrant, blossom crown, and screech owls. Anybody can stay at El Dorado, even the non-birding crowd.\n\n##### **Acccommodations**\n\n###### **UNDER COP$70,000**\n\nM **Oscar's Hostal Finca La Fortuna** (400 m from town entrance near casino area, cell tel. 313\/534-4500, <http:\/\/hotelfincalafortuna.blogspot.com>, COP$20,000 hammock, dorm COP$25,000, COP$40,000 d) consists of simple and naturally luxurious cabins for a small capacity of guests dramatically set on a bluff with extraordinary views of Santa Marta and the surrounding countryside. Oscar's is completely off-grid, with solar panels providing electricity and rainwater collection and a well for all water use. Much of the land that you see from here (some 70 acres that is now a sea of trees) is owned by a man named Oscar, who is on a mission to undo some of the damage humans have done to the mountains of the Sierra Nevada through cattle ranching. The sunsets here, particularly from mid-June until mid-December, are \"living art,\" as Oscar puts it. One- to two-day mule tours in the Sierra Nevada, with Oscar, can be arranged for COP$90,000 per day, per person. Reasonably priced breakfasts include homemade granola (COP$4,000), and other meals can be arranged as well.\n\nM **Rancho de la Luna** (300 m from town entrance near casino area, tel. 5\/422-3160, cell tel. 317\/249-7127, www.ranchodelalunaenminca.com, COP$70,000), in the countryside outside of Minca near Oscar's Hostal, is a guesthouse with two lodges with basic but very comfortable facilities (and great views of Santa Marta). But the real selling point is their wellness program: healthy food, massages, and yoga classes. These have additional costs, although there are packages that include yoga, a massage, meals, and two nights' accommodations for COP$200,000 pp.\n\nM **El Mirador Hotel** (200 m from town entrance, cell tel. 311\/671-3456 or 318\/368-1611, www.miradorminca.wordpress.com, COP$25,000 dorm, COP$45,000 d) is an enchanting hostel with a great view, warm hosts, and delicious meals. The hostel has three rooms, two private rooms and a dorm room with three beds. It's set in a lush garden, and the lovely dining area is open air. The restaurant is open to the public nightly and meals cost around COP$20,000.\n\nMany travelers head up the mountain to spend some time at sociable **Hostal Los Pinos** (near Campano, cell tel. 313\/587-7677 or 321\/898-0641, lospinoshostal@yahoo.com, COP$20,000 dorm, COP$60,000 d). Hostal Los Pinos is on a mountain ridge (1,400 meters\/4,600 feet) where the views are unbelievable. When it's clear, you can glimpse the snowcapped mountains of the Sierra Nevada. This is a fun spot, with lots of hanging around. There are some nice walks you can take from here, as well as high-adrenaline downhill bike rides, and hikes to waterfalls hidden in the mountains. And occasional paintball duels on-site.\n\n###### **COP$70,000-200,000**\n\nOperated by ProAves, the **Hotel Minca** (near town entrance, tel. 5\/421-9958, cell tel. 317\/437-3078, www.hotelminca.com, COP$135,000 d), once a convent, is one of the first hotels in Minca. There are 13 spacious rooms in this old-fashioned building with broad verandas with hammocks. There's a nature path on the grounds, and numerous hummingbird feeders along the open-air dining area ensure that you'll have a breakfast-time show.\n\n**Hostal Casa Loma** (50 m uphill from the church, cell tel. 313\/808-6134 or 321\/224-6632, www.casalomaminca.com, COP$20,000 dorm, COP$80,000 d) is on a hilltop with a truly amazing vantage point over Santa Marta. It's a friendly place, where delicious food (often vegetarian) is served, and you mix and mingle with other travelers. Cabins farther on the hillside are quieter than the rooms near the main social area. Camping (COP$15,000) is also available. To get to Hostal Casa Loma, you must climb up a winding path just behind the town church. Casa Loma offers yoga classes (COP$20,000) on their forest terrace, massages (COP$55,000), and shows two films a week in their outdoor forest cinema.\n\n**Hostal Palo Alto** (near Reserva El Dorado, cell tel. 300\/642-1741 or 312\/677-1403, www.tangaratours.co, tangaratours@gmail.com, COP$80,000 pp) is a mountain paradise, up high at an elevation of 1,700 meters, where you don't have to be a bird enthusiast to enjoy the crisp mountain air and natural beauty of the sierra\u2014but if you are, this a great place to be. It's near the El Dorado bird-watching reserve. Accommodations here are basic, but comfortable.\n\n##### **Food**\n\nThere are a handful of good eateries in Minca, mostly catering to international travelers. Hotels and hostels are always a reliable and reasonably priced option for guests and non-guests alike. They are open every day with lunch hours generally noon-3pm and dinner 6pm-10pm. Main dishes rarely cost more than COP$20,000. Standouts include **Hostal Casa Loma** (50 m above the church, cell tel. 313\/808-6134 or 321\/224-6632, www.casalomaminca.com), **El Mirador Hotel** (200 m from town entrance, cell tel. 311\/671-3456 or 318\/368-1611, www.miradorminca.wordpress.com), and **Rancho de la Luna** (300 m from town entrance near casino area, tel. 5\/422-3160, cell tel. 317\/249-7127, www.ranchodelalunaenminca.com).\n\nTowards the church from the town entrance is **Hola from La Sierra Caf\u00e9** (town center, cell tel. 310\/703-2870, holafromlasierracafe@gmail.com, 9am-9pm Wed.-Sun.). This friendly hippie-ish spot serves light and healthy meals, including breakfasts (pancakes!). They bake bread daily and also sell locally produced organic coffee and other products.\n\n**Bururake Fusion** (town center, noon-3pm and 7pm-10pm Wed.-Sun., COP$18,000) has a daily menu and offers a little of everything: hamburgers, pastas, vegetarian dishes, and refreshing fruit juices. If you have the munchies after that morning hike, look for **Empanadas Don Luis** (no phone, 9am-7pm daily). They're the best.\n\nDining at M **Ei Mox Muica** (300 m from town entrance, cell tel. 311\/699-6718, 10am-9pm daily, COP$18,000) is like being invited to a friend's house. They only have two tables, candlelit at night, and these overlook a lush garden. The menu includes a variety of pastas, salads, cr\u00eapes, and wines. Andrea's specialty is cooking while Andr\u00e9s is a woodcarver. With his father, accomplished painter and sculptor Manuel Bohorquez, he organizes woodcarving and ceramics workshops for visiting artists. Contact Andr\u00e9s in advance for information at andresescultor@hotmail.com.\n\n##### **Information and Services**\n\nBring plenty of cash with you to Minca: There are no ATMs here. There is a small **tourist information stand** (near police station, cell tel. 317\/308-5270, 10am-6pm daily). It's run by the tour agency **Jungle Joe's** (www.junglejoeminca.com).\n\n##### **Getting There and Around**\n\nMinca is easily reached from Santa Marta. _Taxis colectivos_ (shared taxis) depart on a regular basis from the market at Calle 11 and Carrera 12. These cost COP$7,000. Private taxis from the airport cost around COP$50,000 and taxis from the Centro cost COP$40,000.\n\n#### M **PARQUE NACIONAL NATURAL TAYRONA**\n\nPerhaps the best known national park in Colombia, the **Parque Nacional Natural Tayrona** (PNN Tayrona, 34 km northeast of Santa Marta on the Troncal del Caribe highway, tel. 5\/421-1732, www.aviaturecoturismo.com or www.parquesnacionales.gov.co, 8am-5pm daily, COP$37,500 non-Colombian, COP$14,000 Colombian resident, COP$7,500 children, COP$7,500 students under age 25 with a valid ID) encompasses gorgeous beaches, tropical rainforests, and archaeological sites.\n\nThe park extends over 12,000 hectares (30,000 acres) of land from the edge of Taganga to the southwest to the R\u00edo Piedras on the east. The southern border of the park is the Troncal del Caribe highway and to the north is the Caribbean Sea. To the east and south of the PNN Tayrona is the PNN Sierra Nevada de Santa Marta, a much larger national park.\n\nThe frequently tempestuous waters of the PNN Tayrona provide dramatic scenery, with palms growing atop massive island boulders, waves crashing up against them. There are more than 30 golden sand beaches in the park that are set dramatically against a seemingly vertical wall of jungle. Although you can't see them from the park, the snow-covered peaks of the Sierra Nevada de Santa Marta mountains are only 42 kilometers from the coast.\n\nwild horses on the beach at the Parque Nacional Natural Tayrona\n\nThe park includes significant extensions of highly endangered dry tropical forests, mostly in the western section of the park. You will notice that these forests are much less dense than the humid tropical forests. At higher elevations you will see magnificent cloud forests. In addition to beaches, the coast includes marine estuaries and mangroves. The park includes streams with chilly waters that flow from high in the sierra: In the western part of the park, many of these run dry during the dry season, while in the eastern sector they have water year-round.\n\nThe forest in Parque Nacional Natural Tayrona is alive with plant and animal life. Over 1,300 plant, 396 bird, and 99 mammal species have been identified here. Four species of monkeys live in the park, and they can often be spotted. Five species of wild cats have been identified in the park. These are the margay, jaguar, ocelot, panther, and jaguarundi. Their numbers are few and these great cats are expert at hiding in the jungle: Don't count on stumbling across them during your visit! Other mammals include sloths, anteaters, armadillos, deer, and 40 types of bats. Birds include migratory and resident species, including the rare blue-billed curassow (locally called El Paujil), a threatened bird that lives in the cloud forest.\n\n###### **PLANNING YOUR TIME**\n\nThere are two rainy seasons: April-June and September-November, with the latter more intense. During these times, trails can be extremely muddy. If at all possible, avoid visiting the PNN Tayrona during the high seasons mid-December through mid-January and Semana Santa, and to a lesser extent during the Colombian summer school holidays from mid-June until mid-July. During holidays the park is swarmed with visitors. Long holiday weekends ( _puentes_ ) are also quite busy here, regular weekends less so, but during the week is by far the best. While many visit the park on day-trips from Santa Marta, spending one or two nights in the park is recommended, even though accommodations and food are expensive.\n\n##### **Recreation**\n\n###### **BEACHES**\n\nThe beaches in Parque Nacional Natural Tayrona are spectacular, but while the water may appear inviting, currents are deceivingly strong, and, despite the warnings posted on the beach, many people have drowned here. Of the park's 34 beaches, there are only 6 where you are allowed to swim. There are no lifeguards on duty in the park, and no specific hours for swimming. The best swimming beach is **La Piscina,** which is between the beaches of Arrecifes and Cabo San Juan (where you can also take a dip). To the west of Cabo San Juan is a clothing-optional beach. La Piscina is an inviting cove with crystal-clear waters. A natural rock barrier in the water keeps the waters always calm. It's a 20-minute walk west from Arrecifes.\n\nSome of the other beaches open to swimming are in the less-visited western part of the park. **Playa Neguanje** is accessed by car or taxi (COP$15,000 from Santa Marta) through the Zalangana entrance (12 km northeast of Santa Marta). **Playa del Muerto** (Playa Cristal) is another recommended beach in the same area. It is over 20 kilometers from the entrance to the beach. You can visit some of the beaches on the western end of the park all the way to Cabo San Juan by boat from Taganga, but park staff prefer for visitors to enter the park by land. The waters can also be quite rough, especially between December and February.\n\n###### **HIKING**\n\nA highlight of any visit to the PNN Tayrona is the trek up to **El Pueblito** (also called Chairama, 3 km, 1.5 hrs. one way), which consists of ruins of what was an important Tayrona settlement. (Unless, that is, you have already visited the more impressive Ciudad Perdida site.) Here there are well-preserved remnants of terraces, and a small Kogi community still lives near the site. The somewhat challenging path through the tropical jungle is steep and the stone steps can be slippery, but it's well worth it. Hikers can go up to El Pueblito without a guide. El Pueblito is usually accessed from within Tayrona by walking west along the beach from the Arrecifes area. It can also be accessed from the main highway, the Troncal del Caribe.\n\nAt about 24 kilometers northeast of Santa Marta, ask to be let off at the Calabazo entrance to the park. It's about a 2.5-hour trek from there. It's not necessary, but if you'd like you can hire a guide to lead you to El Pueblito. Inquire about this when you check in.\n\n##### **Accommodations and Food**\n\nThere are numerous lodging options in the PNN Tayrona for every budget, from the high-end Ecohabs to camping. The travel agency **Aviatur** (Bogot\u00e1 office Av. 19 No. 4-62, tel. 1\/587-5181 or 1\/587-5182, www.aviaturecoturismo.com) manages most all lodging facilities in the park. Neither the Ecohabs nor the cabanas could be considered a bargain, but the Ecohabs, where you can awake to a beautiful view of the sea, are indeed special, and worth one or two nights. The cabanas are set back from the beach but are quite comfortable too.\n\nThe **Ecohabs** (COP$448,000 pp all meals incl.), in the Ca\u00f1averal sector, consist of 14 private _boh\u00edos_ (thatched-roof cabins) that, from a distance, look like giant nests amidst the trees. Really they are modeled after the thatched-roof houses of the Tayrona people. They sleep 2-4 persons. There are two floors to the Ecohabs. On the first floor is the bathroom and an open-air social area. On the second floor is the bedroom. A flashlight is necessary if you need to go to the bathroom in the middle of the night, as you have to go outside and downstairs. This is inconvenient for some.\n\nIn nearby Arrecifes, there are six two-story **cabanas** (12 rooms, COP$365,650 pp all meals incl.) with a capacity of four persons each. These are like jungle duplexes. The two units are divided by thin walls. There is also a **hammock area** (COP$23,000 pp) in Arrecifes with a capacity of 60 hammocks. For those choosing this option, there are lockers and you can lock things in the safety box at the lobby area.\n\nThere are **campgrounds** (COP$15,000 pp) in both Ca\u00f1averal and Arrecifes and also at Cabo San Juan, which is a 15-minute walk to the west from the beach at La Piscina. It tends to get very crowded, bordering on unpleasant, during long weekends and holidays.\n\nSafety boxes are included in all rooms, and can be provided to campers as well. That said, some prefer to leave a bag or valuables at a trusted hotel in Santa Marta.\n\nThe park has two restaurants in Ca\u00f1averal, close to the Ecohabs, and in Arrecifes near the cabanas. They are open every day 7am-9pm, and the specialty here is fresh seafood. Expect to pay around COP$30,000 for a lunch or dinner entr\u00e9e. There are some snack bars in the park as well.\n\n##### **Getting There and Around**\n\nFrom Santa Marta you can take any bus eastbound along the Troncal del Caribe to the main entrance (Zaino Gate). _Colectivo_ buses can be caught at the intersection of Carrera 11 and Calle 11 (the market) in Santa Marta, and the trip takes about an hour and costs under COP$5,000. You can also take a cab for about COP$60,000.\n\nThere is usually an extremely thorough inspection of backpacks and bags upon entering the park. Visitors are not allowed to bring in plastic bags (to protect sea turtles) and no alcohol (although it is served at restaurants and snack bars in the park). Be sure to bring bug repellent, a flashlight, and good hiking boots.\n\nAt the administrative offices, visitors pay entrance fees. These fees are not included in the Aviatur package prices. Although technically park visitors are supposed to carry proof of a yellow fever vaccination, this is rarely, if ever, checked.\n\nIt's about four kilometers from the offices to Ca\u00f1averal, and vans make this route on an ongoing basis (COP$2,000). From there it is a sweaty 45 more minutes on foot through the jungle to Arrecifes. Mules can be hired to carry your bags, or you can rent a horse for COP$20,000.\n\n#### **PNN TAYRONA TO PALOMINO**\n\nIf you prefer less civilization and more tranquility, the coast between PNN Tayrona and Palomino has some interesting places to hang your _sombrero vueltiao_ (Colombian hat) for a few days. Between these two very popular tourist destinations, there are beaches that are rather overlooked by the masses. Day trips to PNN Tayrona can be easily coordinated from this area.\n\nImpossibly placed upon giant beach boulders, the guesthouse **Finca Barlovento** (Playa Los Naranjos to the east of PNN Tayrona, Bogot\u00e1 tel. 1\/325-6998, www.fincabarloventosantamarta.com, COP$400,000 d incl. 2 meals) is located between the sea and the R\u00edo Piedras. It's an amazing place to stay, and the food's good, too. There are just three rooms and one more luxurious cabin here. Jungle excursions can be arranged by the hotel, but you'll want to enjoy some beachside afternoons. You'll have the beach to yourself.\n\nSelf-described as \"chilled out,\" **Coste\u00f1o Beach Surf Camp and Ecolodge** (Playa Los Naranjos, cell tel. 310\/368-1191, www.costenosurf.com, COP$30,000 dorm, COP$80,000 d) gets exceptionally high marks from its guests for its laid-backness. And while you don't have to be a surfer or skater to fit in here, it doesn't hurt either. Boards are rented for COP$30,000 per day, and classes cost COP$25,000. Coste\u00f1o Beach has both dorm and private accommodations on this former coconut farm. It's also solar powered.\n\n**Playa Koralia** (48 km east of Santa Marta, cell tel. 310\/642-2574 or cell tel. 317\/510-2289, www.koralia.com, COP$209,000 d) is a beach-chic hotel on the beach just east of the Parque Nacional Natural Tayrona. It's rustic: There is no electricity. Candlelit meals (always vegetarian-friendly), star-gazing, an evening drink by the bonfire, and a little spa time are Playa Koralia's recipe for peace.\n\n#### **PALOMINO**\n\nSwaying coconut palms and uncrowded beaches: That's what the Caribbean is all about, isn't it? And at Palomino that's exactly what you get. This town on the Troncal del Caribe is just across the border in the La Guajira department. And it's become quite a popular destination, particularly with backpackers. Caribbean currents can be frustratingly strong here, but the cool waters of the nearby R\u00edo Palomino flowing down from the Sierra Nevada de Santa Marta are always refreshing and much more hospitable towards visitors. Palomino is a good stop to make between Santa Marta and desert adventures in the Alta Guajira.\n\n##### **Recreation**\n\nRecreational activities in and around Palomino include easy day trip walks to the R\u00edo Palomino, about an hour away, where there is also tubing. This river forms the eastern border of the town. Also in the area are the fantastic jungle waterfalls at **Quebrada Valencia** (between PNN Tayrona and Palomino). All hotels and hostels organize these easy excursions. Samarian families visit these swimming holes to cool off on the weekends.\n\n**Chajaka** (office on the south side of the main coastal highway, cell tel. 313\/583-3288) offers interesting day trips or multi-day hiking trips (COP$120,000) into the Sierra Nevada to visit Kogui communities and experience the jungle.\n\n**Shivalila Yoga** (La Sirena hostel, cell tel. 321\/450-7359, yogashivalila@gmail.com, COP$15,000 class) offers yoga classes at the crunchy La Sirena hostel on the beach. Inquire at the hostel for the weekly schedule.\n\n##### **Accommodations and Food**\n\nPalomino is well on its way to dethroning Taganga as the deluxe backpacker resort. That's thanks largely to one famous hostel: **The Dreamer** (Palomino, cell tel. 300\/609-7229, www.thedreamerhostel.com, COP$29,000 dorm, COP$110,000 d). This is by far the most social option this side of Santa Marta. Dorm and private accommodations are in _malokas_ (cabins) surrounding an always-happening pool area and outdoor snack bar.\n\nNext door to The Dreamer is **Caba\u00f1as San Sebasti\u00e1n** (Palomino, cell tel. 300\/432-7170 or cell tell. 310\/775-4630, www.sansebastianpalomino.com.co, COP$30,000 pp). It consists of two cabins and three rooms for rent, just a few meters from the beach.\n\nDon't miss the excellent juice stand at the beach in front of The Dreamer, where friendly Alejandro even has his own organic chocolate for sale.\n\nAt the **Hotel Hukumeizi** (Palomino, cell tel. 315\/354-7871 or 317\/566-7922, www.turismoguajira.com, info@hukumeizi.com, COP$250,000 pp all meals) there are 16 cute, round _boh\u00edos_ (bungalows) with a restaurant in the center. If you go during the week, you'll probably have the place to yourselves, but service may be less attentive. From here it's about a 15-minute walk along the beach to the R\u00edo San Salvador and about an hour from the R\u00edo Palomino.\n\nM **El Matuy (Donde Tuchi)** (Palomino, cell tel. 315\/751-8456, www.elmatuy.com, COP$180,000 pp including meals) is a privately owned nature reserve with 10 cabins amid the palms and no electricity (this means candlelight evenings and no credit card machine). Hotel staff can help organize horseback riding or other activities. Food is varied, yet portions are not extremely generous.\n\nOn the Troncal del Caribe, the **Hostal Mochileros Culturart** (Cl. 1B No. 4-25, cell tel. 312\/626-6934) is a cultural center for Palomino, where there are often musical performances and other events in the evenings. There is a restaurant here with fine veggie burgers and refreshing juices, and, as the name suggests, there are rooms available at this hostel.\n\n##### **Getting There and Around**\n\nThere is regular bus transportation along the Troncal del Caribe between Santa Marta and Riohacha. Take a bus bound for Palomino from Santa Marta at the market on Carrera 11 and Calle 11. It's about a two-hour trip. It costs COP$10,000. On the highway where the bus drops you off, there are young men on motorbikes who will take you to your hotel (COP$3,000) on the beach.\n\n#### **PARQUE NACIONAL NATURAL SIERRA NEVADA DE SANTA MARTA**\n\nEncompassing almost the entire Sierra Nevada mountain range is the **Parque Nacional Natural Sierra Nevada de Santa Marta** (www.parquesnacionales.gov.co). This park has a total area of 383,000 hectares (945,000 acres), making it one of the larger parks in Colombia.\n\nThe main attraction is the **Ciudad Perdida (Lost City)** , the most important archaeological site of the Tayrona, the pre-Columbian civilization that inhabited the Sierra Nevada. The Tayrona had a highly urbanized society, with towns that included temples and ceremonial plazas built on stone terraces. There are an estimated 200 Tayrona sites, but Ciudad Perdida is the largest and best known. Many of these towns, including Pueblito (in the Parque Nacional Natural Tayrona), were occupied at the time of the Spanish conquest. Today, an estimated 30,000 indigenous people who are descendants of the Tayronas, including the Kogis, Arhuacos, Kankuamos, and Wiwas, live on the slopes and valleys of the Sierra Nevada de Santa Marta. These people believe that the Sierra Nevada is the center of the universe and that the mountain's health controls the entire Earth's well-being. Many areas of the sierra are sacred sites to these people and are barred to outsiders.\n\nThe Sierra Nevada de Santa Marta mountain range is best described as a giant pyramid, which is bordered on the north by the Caribbean and on the southeast and southwest by the plains of northern Colombia. Although some believe that the range is a distant extension of the Cordillera Oriental (Eastern Mountain Range) of the Andes, most geologists believe it is a completely independent mountain system.\n\nIt is the world's highest coastal mountain range, with the twin peaks of **Pico Crist\u00f3bal Col\u00f3n** and **Pico Bol\u00edvar** (the two are called **Chinund\u00faa** by indigenous groups in the area) reaching 5,776 meters (18,950 feet; Col\u00f3n is said to be slightly higher than Bol\u00edvar) but located only 42 kilometers from the sea. Pico Crist\u00f3bal Col\u00f3n is the world's fifth most prominent mountain after Mount Everest (Nepal\/Tibet, China), Mount Aconcagua (Argentina), Mount McKinley (U.S.), and Mount Kilimanjaro (Tanzania). In addition, there are seven other snow-covered peaks that surpass 5,000 meters: Simonds, La Reina, Ojeda, Los Nevaditos, El Guardi\u00e1n, Tulio Ospina, and Codazzi. Treks to these peaks used to be possible from the northern side of the mountains, starting at the Arhuaco indigenous village of Nabusimake (Cesar), but are no longer permitted by the indigenous communities.\n\nThe PNN Sierra Nevada de Santa Marta encompasses the entire mountain range above 600 meters (16,400 feet). In addition, a small segment of the park east of the PNN Tayrona, from the R\u00edo Don Diego to the R\u00edo Palomino, extends to sea level. This means that the park encompasses the entire range of tropical ecosystems in Colombia, from low-lying tropical forests (sea level to 1,000 meters), cloud forests (1,000-2,300 meters), high mountain Andean forest (2,300-3,500 meters), _p\u00e1ramo_ (highland moor, 3,500-4,500 meters), super _p\u00e1ramo_ (4,500-5,000 meters), and glaciers (above 5,000 meters). However, because access to the upper reaches of the park is limited, what visitors will most be able to appreciate is low-lying tropical and cloud forest.\n\nThe isolation of the range has made it an island of biodiversity, with many plant and animal species found nowhere else. The Sierra Nevada de Santa Marta is home to 187 mammal species, including giant anteaters, spider monkeys, peccaries, tree rats, jaguars, and pumas. There are 46 species of amphibians and reptiles, including several that live above 3,000 meters that are found nowhere else on the planet. There are an astonishing 628 bird species, including the Andean condor, blue-knobbed curassow, sapphire-bellied hummingbird, and black-solitary eagle, as well as many endemic species. There are at least 71 species of migratory birds that travel between Colombia and North America.\n\nthe mystical Ciudad Perdida\n\n##### M **Ciudad Perdida Trek**\n\nA highlight for many visitors to Colombia is the four- to six-day, 52-kilometer (32-mile) round-trip trek to the **Ciudad Perdida (Lost City)** in the Sierra Nevada mountains of the Caribbean coast. The Ciudad Perdida is within the confines of the Parque Nacional Natural Sierra Nevada de Santa Marta.\n\nThe Ciudad Perdida, called Teyuna by local indigenous tribes and Buritaca 200 by archaeologists, was a settlement of the Tayrona, forebears of the people who inhabit the Sierra Nevada today. It was probably built starting around AD 700, at least 600 years before Machu Picchu. There is some disagreement as to when it was abandoned: There is evidence of human settlement until the 16th century. The site was visited in the early 1970s by _guaqueros_ (treasure hunters) who pillaged the site. News of its discovery in 1976 marked one of the most important archaeological events of recent years. From 1976 to 1982, archaeologists from the Colombian National Institute of History and Anthropology painstakingly restored the site.\n\nSpread over some 35 hectares (86 acres), the settlement consists of 169 circular terraces atop a mountain in the middle of dense cloud forest. Archaeologists believe that this sophisticated terrace system was created in part to control the flow of water in this area known for torrential rainfall for much of the year.\n\nPlazas, temples, and dwellings for tribal leaders were built on the terraces in addition to an estimated 1,000 _boh\u00edos_ (traditional thatched roof huts), which housed between 1,400 and 3,000 people. A fire was always at the center of the _boh\u00edo,_ and there was a domestic area where food and water were stored and cooking took place, as well as an artisan area for goldsmithing.\n\nSurrounding the Ciudad Perdida were farms of coca, tobacco, pumpkin, and fruit trees. The city was connected to other settlements via an intricate system of mostly stone paths.\n\nThe hike is, by and large, uphill, as you reach an elevation of 1,100 meters (3,600 feet). There are nearly 20 river crossings to be made. Towards the end of the third day, you will climb about 1,200 often treacherously slippery stone steps until you reach the spectacular terraces of Ciudad Perdida. For many this sight makes all the sweat, fatigue, and mosquito bites worthwhile.\n\nThere is one set fee (COP$600,000) for the trek. This does not change, no matter if you're making the trek in three, four, or five days. If you are in very good shape and prefer taking the Ciudad Perdida express route, the trek can be done in three nights and four days. This requires six hours hiking per day and rising early. For some, the long nights of card playing at campsites can get old quick; others enjoy the camaraderie with hikers from all over the world. In case of an emergency on the mountain, a burro or helicopter will be sent to retrieve the hiker, for a fee.\n\nIt's important to only go with a reputable tour company, such as **Magic Tour** (Cl. 16 No. 4-41, Santa Marta, tel. 5\/421-5820; Cl. 14 No. 1B-50, Taganga, tel. 5\/421-9429; www.magictourcolombia.com) or **TurCol** (Cl. 13 No. 3-13, Centro Comercial San Francisco Plaza, Local 115, Santa Marta; Cl. 19 No. 5-40, Taganga, tel. 5\/421-2556; www.turcol.com or www.buritaca200.com). A third option is **Baquianos Tour** (Cl. 10C No. 1C-59, Santa Marta, tel. 5\/431-9667, www.lostcitybaquianos.com). Your tour company will provide food (advise in advance if you have special dietary needs or wants), hammocks or cots with mosquito netting, and mules to carry up supplies.\n\nYou'll need to bring a small to medium-sized backpack, enough to carry a few days of clothes, good hiking boots with strong ankle support, sandals for stream crossings (keeping your boots dry), long pants, mosquito repellent, water purifying tablets, sunscreen, cash for refreshments to purchase along the way, a small towel, toilet paper, hand sanitizer, flashlight, sealable bags to keep things dry, light rain jacket, and a water container. If you have them, trekking poles may be a nice addition. Sleeping bags may provide more comfort at night but aren't necessary.\n\nFrom mid-December through mid-January you'll have plenty of company along the way: It's high tourist season. Other high seasons are during Semana Santa and June-July. The wettest months tend to be April-May and September-November. Expect a daily downpour and doable, but sometimes rather treacherous, river crossings during those times of the year. When it's raining or has been raining, the trek is more challenging. On the plus side, there are usually fewer crowds on the mountain at that time.\n\nCampsites along the way turn into backpacker villages during high seasons, but they never turn in to rowdy scenes by any means. Upon arrival the routine is fairly standard. You'll often be able to cool off in the pristine waters of nearby swimming holes, have dinner, and hit the hammocks. Sleeping in hammocks can be uncomfortable for those not used to them. Earplugs come in handy for light sleepers.\n\nThe trek begins at the settlement of Mamey on the R\u00edo Buritaca. Along the way you will no doubt come in contact with Kogi people who live in the Sierra Nevada, and will pass through the village of Mutanyi. These lands are theirs and visitors are encouraged to refrain from taking photographs of them without prior permission. One of the most popular tourist activities in Colombia, the trek is considered quite safe.\n\n##### **Estaci\u00f3n San Lorenzo**\n\nDeeper and higher into the Sierra Nevada de Santa Marta from the town of Minca and set amid pristine cloud forest is the **Estaci\u00f3n San Lorenzo** (Santa Marta parks office, Cl. 17 No. 4-06, tel. 5\/423-0752 or 5\/421-3805, sierranevada@parquesnacionales.gov.co, 8am-noon and 2pm-6pm Mon.-Fri.; Bogot\u00e1 parks office, tel. 1\/353-2400, www.parquesnacionales.gov.co), part of the Parque Nacional Natural Sierra Nevada de Santa Marta. At this spot at 2,200 meters (7,200 feet) in the Sierra Nevada, visitors can make guided walks through the cloud forest to see birds and enjoy views of the snow-covered peaks in the park. The two cabins (COP$25,000-35,000 pp) of San Lorenzo are wonderfully isolated amid the forest.\n\nA stay here can be an inexpensive way to get to know the Sierra Nevada, but there are many hoops you must go through to arrange it. First you must contact the parks office, preferably in Santa Marta, to inquire about availability. Then you will be required to make a _consignaci\u00f3n_ (deposit) to their bank account to cover the cost of your stay. This involves filling out a deposit form and standing in line at their bank (Banco de Bogot\u00e1).\n\nThe easiest way to get to San Lorenzo is by _mototaxi_ from Minca, and this can cost around COP$50,000. Alternatively, and if you are traveling in a small group, it may make sense to arrange transportation with an SUV, which can cost up to COP$250,000.\n\nAccommodations are basic here, and there are just two cabins, each with three rooms holding six beds each, for a total capacity of 36. No meals are provided here, but there is a rustic kitchen facility with basic utensils and cooking implements. You will have to provide your own propane gas canister (readily available in Santa Marta) and all your food for cooking. The temperature drops significantly in the evening, thus it is important to bring warm clothes. There is a fireplace in both cabins.\n\n### **La Guajira**\n\nThe vast Guajira Peninsula has some of the most rugged, beautiful landscapes in Colombia. It is home to the Way\u00fau indigenous people, who have maintained their independent way of life through centuries. Though many Way\u00fau now live in cities and towns, their traditional _rancher\u00edas_ (settlements) dot the desert. With the growth of tourism, many have set up lodging using traditional _rancher\u00eda_ houses made out of _yotojoro,_ the dried hearts of cactus plants.\n\nThe Colombian side of the peninsula (Venezuela shares the other part) can be divided into three sections. The Baja Guajira (Lower Guajira), near the Sierra Nevada de Santa Marta, is fertile agricultural and cattle-ranching land. The much more arid middle swath, with the departmental capital of Riohacha and the unlovely towns of Uribia and Maicao, is home to the majority of the population. The Alta Guajira (Upper Guajira), from Cabo de la Vela to Punta Gallinas, is sparsely populated and has some truly otherworldly landscapes. Focus your visit on this last part.\n\nLack of infrastructure, especially in the north, makes visiting the Guajira a challenge, so most people opt for organized tours. Though it is possible to get to Cabo de la Vela on public transportation, do not travel elsewhere in the Alta Guajira without a dependable local guide. The roads are unmarked tracks in the sand, and getting lost is inevitable. More urgently, in this somewhat lawless place, where the Colombian government has limited authority, there are unscrupulous people ready to prey on unsuspecting visitors.\n\nDuring the rainy months September to November, it can be difficult to travel through the desert, which can become muddy to the point of impassable.\n\n##### **History**\n\nSpanish navigator Juan de la Cosa, who was a member of Columbus's first three voyages, disembarked in Cabo de la Vela in 1499, making the Guajira Peninsula one of the first places visited by Europeans in South America. It was not until 1535 that explorer Fernando de Enciso founded a settlement near Cabo de la Vela, which became a center of pearl extraction. This early settlement was relocated to Riohacha, which was founded in 1544. The traditional Way\u00fau inhabitants of the peninsula put up strong resistance to Spanish advances. During the 19th and 20th centuries, the peninsula was used primarily as a smuggling route.\n\nFor better or for worse, the fortunes of the Guajira changed in 1975 when the Colombian government entered into an agreement with oil giant Exxon to develop the Cerrej\u00f3n open-pit coal mine 80 kilometers (50 miles) southeast of Riohacha. This project involved the construction of Puerto Bol\u00edvar, a coal port located in Bah\u00eda Portete, and of a railway to transport the coal. Production started in 1985. Coal has since become one of Colombia's main exports. The mine has generated more than US$2 billion in royalties for the Colombian government. Little of this wealth has trickled down to the people of La Guajira. It is the fourth poorest department in Colombia.\n\n#### **RIOHACHA**\n\nCalled S\u00fcchiimma in the Way\u00fau language, meaning \"city of the river,\" Riohacha (pop. 231,000) is La Guajira's slow-paced departmental capital. It is one of the oldest cities in Colombia. Not a tourist destination itself, Riohacha is an excellent base from which to launch tours of Alta Guajira.\n\n##### **Sights**\n\nAlong Calle 1, also known as Avenida Marina, is the **Paseo de la Playa** boardwalk, where locals and tourists take their evening strolls and take in the sea air. Along the way are kiosks where vendors sell ceviches and Way\u00fau women set up their brightly colored _mochilas_ (traditional woven handbags) on the sidewalk, nonchalantly waiting for customers. The kilometer-long pier known as the **Muelle Tur\u00edstico** is another favorite place for a walk. There are no railings, so maybe those who have had a couple of drinks should sit this one out.\n\n**Parque Padilla** (Cl. 2 and Cra. 7), Riohacha's shady main plaza, is named after favorite son Admiral Jos\u00e9 Prudencio Padilla, who was the most prominent Afro-Colombian commander in the revolutionary wars. To one side of the plaza is the only remnant of colonial Riohacha, the heavily reconstructed **Catedral de Nuestra Se\u00f1ora de los Remedios** (Cl. 2 No. 7-13, tel. 5\/727-2442, masses 6:30am and 7pm Mon.-Sat., 7am, 11am, and 7pm Sun.), originally erected in the 16th century. The remains of Padilla repose there.\n\nthe endless deserts of La Guajira\n\nOn the Riohacha-Maicao road just outside of town is the **Sendero Eco-Cultural El Ri\u00edto** (no phone, 7am-6pm daily), a pleasant nature walk where you can observe birds that was completed in 2013. This is a great place to stretch your legs before or after a long and bumpy ride through the desert.\n\n##### **Accommodations and Food**\n\nRiohacha isn't known for fantastic lodging or dining, but then again you will probably be spending limited time here.\n\nThe **Hotel Castillo del Mar** (Cra. 9A No. 15-352, tel. 5\/727-5043, cell tel. 316\/525-1295, hotelcastillodelmar@gmail.com, COP$25,000 dorm, COP$100,000 d) is about a 10-minute walk from the boardwalk in a quiet residential area. It's the choice of backpackers. There are 11 rooms, some with air conditioning, some without. It could be more comfortable, but for the price it's a bargain.\n\nCloser to downtown, the **Hotel Arimaca** (Cl. 1 Av. La Marina No. 8-75, tel. 5\/727-3515, www.hotelarimaca.com, COP$130,000 d) will do for a night. It has 50 rooms and caters to the business crowd during the week. **Oceano** (Cra. 15 No. 9-21, tel. 5\/728-1108, www.oceanohotel.co, COP$140,000 d) has 15 rooms and is a couple of blocks from the boardwalk.\n\n**Taroa Lifestyle Hotel** (Cl. 1 No. 4-77, tel. 5\/729-1122, www.taroahotel.com, COP$200,000) is the first so-called \"Way\u00fau lifestyle hotel\" in Colombia, and it opened in July 2013. It is only about half an hour to the northeast. It's on the beach and has 46 rooms.\n\nThere are several seafood restaurants along Avenida Marina. Try the **Casa del Marisco** (Cl. 1 4-43, tel. 5\/728-3445, 10am-10pm daily, COP$25,000), a restaurant that serves an array of fresh seafood dishes and pastas.\n\nFor for a wider range of dishes, try **Saz\u00f3n Internacional** (Cl. 1 No. 3-57, tel. 5\/728-0415, noon-10pm Tues.-Sun.). For something quick go to the food court of the modern **Centro Comercial Suchiimma** (Cl. 15 No. 8-56, 10am-9pm daily). There are some vegetarian options available here, and at the Jumbo store, you can pick up all the provisions you need for a long ride through the desert.\n\n##### **Getting There and Around**\n\n**Aeropuerto Almirante Padilla** (Cl. 29B No. 15-217, tel. 5\/727-3854) is five minutes north of town. There are only one or two flights per day from Bogot\u00e1 to Riohacha on **Avianca** (ticket office Cl. 7 No. 7-04, tel. 5\/727-3624, www.avianca.com, 8am-6pm Mon.-Fri., 9am-1pm Sat.). On Tuesdays and Saturdays, **Tiara Air Aruba** (Cl. 2 No. 6-64, Local 2, tel. 5\/727-3737, www.tiara-air.com, 8am-noon and 2pm-6pm Mon.-Fri.) operates flights between Riohacha and Aruba, with connections to and from Fort Lauderdale.\n\nThe **bus station** (Av. El Progreso and Cl. 11) has frequent services to Maicao, Santa Marta, and Barranquilla. There are also services to Valledupar, Cartagena, and Bogot\u00e1. Shared taxis to Uribia, where you can pick up trucks to Cabo de la Vela, leave from Calle 15 and Carrera 1.\n\nThe center of town, along the boardwalk, is easily walkable.\n\n###### **BORDER CROSSING**\n\nFor stays under 90 days, U.S., Canadian, and most European citizens do not require a visa to enter Venezuela. You may be required to show proof of a hotel reservation and proof of an air ticket departing from Venezuela, and your passport must not expire within six months of entry into Venezuela. There is a Venezuelan consulate in Riohacha for further queries: **Consulado de Venezuela** (Cra. 7 No. 3-08, Edificio El Ejecutivo, Piso 2, tel. 5\/727-4076, 8am-noon and 2pm-5pm Mon.-Thurs., 8am-1pm Fri.). There is an entry point to Venezuela at the town of Maicao.\n\n#### **SANTUARIO DE FAUNA Y FLORA LOS FLAMENCOS**\n\nAny day you see a pink flamingo is a good day. That's enough of a reason to visit the **Santuario de Fauna y Flora Los Flamencos** (Camarones, cell tel. 301\/675-3862 or 313\/514-0366, ecoturismosantuario@gmail.com, www.parquesnacionales.gov.co, free). This park is 25 kilometers southwest of Riohacha and is home to thousands of _Phoenicopterus ruber ruber_ or American flamingos. The 700-hectare sanctuary, which is part of the national park system, encompasses a magnificent coastal estuary where the flamingo fish for shrimp in the shallow waters.\n\nTo get there, you can take a bus from Riohacha going towards Camarones or points southwest. From Camarones take a _mototaxi_ to the entrance of the park (COP$2,000). To see the flamingos up close, take a _chalupa_ (wooden boat) onto the lagoon (COP$15,000 for two people). To avoid the glaring sun, visit the park early in the morning or late in the afternoon.\n\nAt the **Centro de Visitantes Los Mangles** (cell tel. 301\/675-3862 or 313\/514-0366, ecoturismosantuario@gmail.com, COP$100,000 meals and tours) there are five tiny yet spic and span cabins that can be rented, and there is an area for sleeping in _chinchoros_ (large hammocks; COP$20,000). These are located between the bay and the sea. In addition to the flamingos, the beach at the park is quite nice and you can take an excursion through the mangroves. It's a charming place, and is a good option if you'd prefer not to stay in Riohacha.\n\n#### M **ALTA GUAJIRA**\n\nThe **Alta Guajira** (Upper Guajira) comprises the entire peninsula east of Cabo de la Vela and Uribia. It is very sparsely populated: The three largest settlements, where most of the tourism infrastructure is located, are Cabo de la Vela, Punta Gallinas at the very northern tip, and Nazareth, in the northeast. The terrain has a striking ochre color, with rocky and sandy patches. The vegetation is mostly shrubs and cacti. The Caribbean coast here is broken by three large bays with stunning turquoise and aquamarine waters: Bah\u00eda Portete, Bah\u00eda Honda, and Bah\u00eda Hondita. The last of these is easily accessible to tourists in day trips from Punta Gallinas. There are a few low mountain ranges, including the Serran\u00eda de la Macuira (864 meters\/2,835 feet), located in the extreme northeastern corner of the peninsula, but overall the terrain is low and slightly undulated.\n\nThe only destination in the Alta Guajira that is accessible by public transportation is Cabo de la Vela. Though it is possible to contract transportation by land or sea to Punta Gallinas, most visitors opt to visit the region in an easily organized tour.\n\nThe sands of Alta Guajira are a favorite location for raucous 4x4 races and competitions. A large annual event is the **Rally Aventura Guajira** (www.ruedalibrexcolombia.net), which takes place in August, when more than 200 vehicles trek across the desert from Riohacha to Cabo de la Vela. If this isn't your bag you may want to double-check your dates of travel to make sure they don't coincide with the event.\n\n##### **Tours**\n\nThe standard Alta Guajira tour involves going to Cabo de la Vela on day one, with a stop at the now-abandoned salt mines of the Salinas de Manaure (not worth a visit), spending the night at Cabo de la Vela, continuing on the next day to Punta Gallinas, and returning to Riohacha on day three (COP$340,000-380,000 per person, including food and lodging). A longer tour involves two additional nights in Nazareth to visit the Parque Natural Nacional Macuira (COP$800,000-880,000 per person, including food and lodging).\n\nMake sure to check how many people are in your SUV, as there have been reports of tour operators who cram seven people into a vehicle, making for an uncomfortable ride. Tour guides generally have a limited grasp of English.\n\nThe most comfortable option by far is to rent an SUV with a driver for your own party. This costs around COP$400,000 per day, not including food and lodging.\n\nThe desert countryside seems endless and is beautiful in its own desolate way. You'll be amazed at how these drivers know which way to go, as there are no road signs, only cacti and occasional goats. Every once in a while, you will have to pay \"tolls\" to Way\u00fau children who have set up quasi road blocks. To gain their permission to cross, drivers hand over crackers, cookies, or candy.\n\n###### **TOUR COMPANIES**\n\nTour companies offer package tours or private vehicles. A highly recommended tour company is **Expedici\u00f3n Guajira** (Cl. 2 No. 5-06, tel. 5\/727-2336, cell tel. 311\/439-4677 or 301\/464-2758, franklin_penalver@yahoo.com), managed by Franklin Penalver. His guides, of Way\u00fau origin, know the area like the back of their hands.\n\n**Solera Travels** (Cra. 9A No. 15-352, cell tel. 316\/525-1295, gerenciacomercial@soleratravels.com) has an office at the Castillo de Mar hostel and sells all the regular tour packages. **Kaishi** (Plaza Principal, Uribia, cell tel. 311\/429-6315, www.kaishitravel.com) is another agency that has a good reputation.\n\n#### **CABO DE LA VELA**\n\n**Cabo de la Vela** (known as Jepira in the Way\u00fau language), 180 kilometers north of Riohacha, is a small Way\u00fau fishing village spread along the Caribbean Sea. It comes as a shock\u2014a pleasant one\u2014after several hours driving through the arid landscape to finally arrive at the waters of the Caribbean. Here the beaches are nice, the views otherworldly, and the atmosphere peaceful. The smooth waters and ample winds provide near perfect conditions for windsurfing and kite surfing, and Cabo de la Vela has become a destination for these sports.\n\nThere are several pleasant excursions near town, and if you take an organized package tour, all of these should be included in the price. One is to **El Faro,** a lighthouse on a high promontory with spectacular nearly 360-degree views of the surrounding ocean. Another is to the **Ojo del Agua,** a small but pleasant beach near a freshwater spring. Farther afield is the **Pil\u00f3n de Az\u00facar,** a high hill that affords incredible views to the surrounding region. Nearby is the **Playa de Pil\u00f3n,** a beautiful ochre-colored beach. To the west is the **Jepirachi Wind Farm,** the first of its kind of Colombia; the turbines make for a somewhat surreal sight in the midst of this barren territory. Organized tours include stops at these spots. If you are on your own, hotels can organize these excursions for you. Trips to these sights in SUVs cost around COP$15,000 per person, round-trip, with a minimum of two persons.\n\nCabo de la Vela, La Guajira\n\n##### **Recreation**\n\nCabo de la Vela is an excellent place to practice or learn how to kite or windsurf. **Eoletto** (Rancher\u00eda Utta, cell tel. 321\/468-0105 or 314\/851-6216, www.windsurfingcolombia.com) is a **windsurfing and kitesurfing** school run by Etto from Germany. An eight-hour windsurfing course costs COP$380,000; kitesurfing is COP$900,000. Rentals are available for COP$50,000 per hour.\n\n##### **Accommodations**\n\nFamily-run guesthouses are plentiful in Cabo de la Vela, and are quite rudimentary. Freshwater is always scarce here in the desert, so long showers are not an option. Floors are usually sandy, and electricity is limited. The street in Cabo de la Vela along the sea is lined with about 15 guesthouses. There are a few lodgings outside of town towards El Faro, such as Rancheria Utta.\n\nThe M **Rancheria Utta** (300 m northwest of town, V\u00eda al Faro, cell tel. 312\/687-8237 or 313\/817-8076, www.rancheriautta.com, COP$15,000 hammock, COP$35,000 bed pp) is a nice place to stay. It is just far enough from the village of Cabo de la Vela that you can experience the magical ambience of being far away from civilization. Cabins are simple with walls made from the hearts of _yotojoro_ (cacti). That is a traditional form of construction in the desert. There are 11 _caba\u00f1as_ with a total of 35 beds and plenty of inviting _chinchorros_ (hammocks) for lazing about. Being on the beach at night, looking up at the stars and listening to the sound of gentle waves breaking nearby, is unforgettable. A pleasant restaurant at the hotel serves breakfast (COP$6,000), lunch (COP$15,000), and dinner (COP$15,000). The fare is mostly seafood (lobster is a favorite but will cost extra), but they can accommodate vegetarians.\n\nThe **Hospedaje Jarrinapi** (cell tel. 310\/366-4245 or 311\/683-4281, www.jarrinapi.com, COP$12,000 hammock, COP$35,000 pp) is a large hotel with 19 _yotojoro_ cabins for a total capacity of 60 people. This hotel has electricity 24 hours a day and has a nice, orderly kitchen and restaurant. **Posada Pujur\u00fa** (cell tel. 300\/279-5048, <http:\/\/posadapujuru.blogspot.com>, COP$20,000 hammock, COP$50,000 bed) has 14 rooms and a space for hammocks. Pujur\u00fa is next to a kite-surfing school.\n\n##### **Getting There**\n\nFrom Riohacha, shared taxis ply the route to Uribia (COP$15,000, 1 hr.). Catch these at the intersection of Calle 15 and Carrera 5. Ask the driver to drop you off at the spot where passenger trucks depart for Cabo de la Vela and intermediate _rancher\u00edas_. Trucks make this route, from Uribia to Cabo de la Vela (COP$15,000, 2 hrs. or more). These trucks depart until 2pm every day. The uncomfortable ride on a bench in the back of the truck can take several hours, depending on how many stops are made, but this is the Guajira way to go.\n\n#### **PUNTA GALLINAS**\n\n**Punta Gallinas** is a settlement on a small peninsula jutting into the Caribbean at the very northernmost tip of the South American continent. It is home to about 100 Way\u00fau who claim this beautiful spot as their ancestral land. The landscape here is a symphony of oranges, ochres, and browns, dotted with cactus and shrubs. The peninsula is hemmed in to the south by Bah\u00eda Hondita, a large bay with bright aquamarine waters and thin clusters of mangroves, and to the north by the deep blue Caribbean.\n\nActivities in and around Punta Gallinas include a visit to the _faro_ (lighthouse), which is the northernmost tip of South America; canoe rides in Bah\u00eda Hondita to see flamingos and mangroves; or visits to two spectacular beaches. These are the remote and unspoiled beaches at **Dunas de Taroa** (Taroa dunes), where windswept and towering sand dunes drop abruptly some 30 meters into the sea, and at **Punta Aguja,** at the southwest tip of the peninsula of Punta Gallinas. These excursions are included in tour prices. If you are on your own, hotels charge around COP$20,000 per person to see the dunes (five-person minimum), COP$150,000 for a group boat ride on the Bah\u00eda Hondita to spy on flamingos, and COP$20,000 per person to go to Punta Aguja (five-person minimum).\n\nThere are two good lodging options in Punta Gallinas, both with splendid views of the Bah\u00eda Hondita. M **Hospedaje Luzmila** (Punta Gallinas, cell tel. 312\/626-8121 or 312\/647-9881, COP$20,000 hammock, COP$30,000 bed pp) has 10 _caba\u00f1as_ with 20 beds and is spread out alongside the bay. Breakfast (COP$5,000), lunch (COP$15,000), and dinner (COP$15,000) are served in the restaurant. Lobster dishes cost extra.\n\nM **Donde Alexandra** (Punta Gallinas, cell tel. 313\/512-7830 or 318\/760-8501, COP$12,000-20,000 hammock, COP$30,000 bed pp) has 10 rooms and 25 beds. Meals are not included in the prices but are usually around COP$6,000 for breakfast, COP$15,000 for lunch, and COP$15,000 for dinner, unless you order lobster (COP$45,000). Donde Alexandra has sweeping vistas of the bay and beyond from the restaurant area.\n\n##### **Getting There**\n\nAlthough most travelers visit Punta Gallinas on an organized tour, it is possible to travel on your own from the Puerto de Pescadores at Puerto Bol\u00edvar on a _lancha_ (boat) arranged by Hospedaje Luzmilla or Donde Alexandra (COP$100,000 pp round-trip, minimum 5 passengers). In the rainy season, from September to November, this may be the only option.\n\n#### **PARQUE NACIONAL NATURAL MACUIRA**\n\nThe remote **Parque Nacional Natural Macuira** (PNN Macuira, 260 km northeast of Riohacha, cell tel. 311\/688-2362, macuira@parquesnacionales.gov.co, 8am-6pm, free) covers an area of 25,000 hectares and encompasses the entire Serran\u00eda de la Macuira (Macuira Mountain Range), an isolated mountainous outcrop at the northeastern tip of the Guajira Peninsula. These mountains are a biological island in the middle of the surrounding desert. The Macuira range, which is 35 kilometers (22 miles) long and 864 meters (2,835 feet) at its highest point (Cerro Palua), captures moisture-laden winds from the sea that nourish a unique low-elevation tropical cloud forest teeming with ferns, orchids, bromeliads, and moss. At lower elevations, there are tropical dry forests. The park includes 350 species of plants, 140 species of birds (17 endemic), and more than 20 species of mammals. The park is within the large Alta Guajira Way\u00fau Reservation. In the lower parts of the range, within the park, live many Way\u00fau families who raise goats and grow corn and other subsistence crops.\n\nThe northernmost point in South America is Punta Gallinas.\n\nThe gateway to the park is the Way\u00fau town of **Nazareth**. Visitors are required to register at the PNN Macuira Park Office, where they must hire an authorized guide (a good idea anyhow, since the trails are not marked). There are various hikes varying 1.5-6 hours, and the cost of a guided walk per group of eight ranges COP$35,000-45,000. Day hikes meander along river beds through tropical dry forest, leading to different destinations. One leads to the **Arewol\u00fc Sand Dunes,** which are, surprisingly, located in the midst of the forest. Another takes you to the **Shipano\u00fc pools**. A third hike goes to **Cerro Tojoro,** which affords beautiful views of the coast and the mountain range. There is also a hike along the entire length of the range. If you can, plan to hike early in the morning to increase your chances of seeing birds and other wildlife. Unfortunately for visitors, hiking to the Macuira's unusual cloud forest is not permitted by the Way\u00fau.\n\nThere are bare-bones lodging options in town. One suitable option is **Hospedaje V\u00eda Manaure** (entrance to town, cell tel. 314\/552-5513, COP$15,000 hammock), which has a capacity of 50.\n\nThere is no public transportation to Nazareth.\n\n### **Western Caribbean Coast**\n\nThe portion of the Caribbean coast that stretches west from Cartagena, down to the Golfo de Urab\u00e1, and then juts north through the Darien Gap and the border with Panama is less visited and less familiar than the coastal region to the east. That may be just the ticket for those yearning to explore the undiscovered and escape the crowds.\n\nThe Golfo de Morrosquillo is home to the beach towns of Tol\u00fa and Cove\u00f1as, which have long been popular family getaways. Just north of the gulf, the Islas de San Bernardo are where you'll find the world's most densely populated island, tiny Santa Cruz del Islote. Other nearby attractions include the inland town of San Antero, known throughout Colombia for its Festival del Burro, and, farther southwest, the quiet and peaceful retreat Reserva Natural Viento Solar in the community of R\u00edo Cedro. This is a place to truly disconnect from the world, and it's only accessible by motorbike via a dirt path.\n\nAcross the Golfo de Urab\u00e1, the tropical rainforests of the Darien provide an exuberant backdrop to the seaside towns of Capurgan\u00e1 and Sapzurro, which are accessible only by boat or plane. The diving sites off the coast and toward Panama's San Blas Islands is superb.\n\nHowever, what most visitors remember and cherish about a visit to this remote part of Colombia is being submerged in truly wild nature.\n\n#### **GOLFO DE MORROSQUILLO**\n\nLargely unknown to international visitors, the twin beach communities of **Tol\u00fa** and **Cove\u00f1as** on the Golfo de Morrosquillo are popular with vacationing Colombian families from Medell\u00edn and Monter\u00eda. If you can afford it, there are several resort islands off the Caribbean coast, such as Tintip\u00e1n and M\u00facura, where you can practically have the beach to yourself, especially during the week.\n\nWhile Cove\u00f1as is mostly a long line of beachfront hotels (and an important Ecopetrol oil pipeline), neighboring Tol\u00fa to the east has an easy if run-down charm to it. Here you can get around on foot or by _bici-taxi_ (bicycle cabs). On weekend evenings, action is centered in the main plaza, but on the weekend, it shifts to the boardwalk, where vendors sell ceviche, bars blast music, and kids play in the water.\n\nOne of the main weekend activities here is to take a day-trip tour of the **Islas de San Bernardo**. These always include a look at **Santa Cruz del Islote,** the most densely populated island in the world, with one person for every 10 square meters, and a stop at **Isla Murica** or **Isla Palma** for lunch and a swim. At Isla Palma, a resort run by all-inclusive operator Decameron, international tourists may be put off by the confined dolphins and other animals. Contact **Mundo Mar** (Av. 1 No. 14-40, tel. 5\/288-4431, www.mundomar.com.co). Tour operators sell this package for around COP$57,000. They are located in the hotels along the boardwalk. They can also arrange trips to the Islas del Rosario closer to Cartagena.\n\nBeaches in Tol\u00fa aren't great; beaches at Playa El Franc\u00e9s or to the west of Cove\u00f1as Punta Bol\u00edvar are nicer. Before a weekday visit, ask at your hotel about the security situation at these beaches. They are somewhat remote.\n\n##### **Accommodations**\n\nM **Villa Babilla** (Cl. 20 No. 3-40, Tol\u00fa, tel. 5\/288-6124, www.villababillahostel.com, COP$70,000) is the best place to stay in Tol\u00fa by a long shot. Rooms have no TV or air conditioning. You can cook your own meals here. There is an Olimpica grocery store a couple blocks away.\n\nThe **Camino Verde** (tel. 5\/288-5132, www.vacacionescaminoverde.com, COP$160,000 d, COP$240,000 d w\/meals) hotel is on the Playa El Franc\u00e9s, a few kilometers east of Tol\u00fa, and is a peaceful spot to relax, especially during the week.\n\nM **Punta Norte** (Tintip\u00e1n, cell tel. 310\/655-4851, www.hotelpuntanorte.com, COP$220,000 pp) is run by a friendly Uruguayan (Punta Norte is most often referred to as Donde El Uruguayo, or \"Where's the Uruguayan?\") and his artist wife. This all-inclusive hotel is on the tiny island of Tintip\u00e1n. Rooms are simple and the lobsters are huge! Days are spent lounging on beaches or discovering nearby islands. Bring plenty of insect repellent should you decide to go to this remote island paradise.\n\nFor white sandy beaches, warm aquamarine waters, and the occasional calorie-loaded cocktail, go to the luxury resort of **Punta Faro** (Bogot\u00e1 tel. 1\/616-3136, www.puntafaro.com, COP$535,000 pp high season), an island resort in the Islas de San Bernardo. High season prices are almost double that of off season rates. There are 45 rooms on the island, ensuring that, while you won't have the island all to yourself, you won't be packed in like sardines. Most guests take a hotel boat from Cartagena, but you can also arrive via Tol\u00fa.\n\n##### **Getting There and Around**\n\nThere is frequent bus service from both Tol\u00fa and Cove\u00f1as to Cartagena (3 hrs., COP$30,000) and Monter\u00eda (2.5 hrs., COP$25,000). In addition, the airline **ADA** (Col. toll-free tel. 01\/800-051-4232, www.ada-aero.com) flies from Medell\u00edn to Tol\u00fa, making this a quick, easy, and often inexpensive beach destination. Getting around Tol\u00fa? Take a _bici-taxi_ (bicycle cab).\n\n#### **SAN ANTERO**\n\nThis seaside town's claim to fame is the annual **Festival del Burro,** one of those many only-in-Colombia celebrations. There are burro races, burro costume contests (in 2013 the winning burro was dressed as the newly announced pope, who beat out a Shakira burro and a Transmiburro, a four-legged version of Bogot\u00e1's TransMilenio bus system), and parades. They even have a modern arena that hosts all the fun. The origin of the festival is a religious one. During Semana Santa, an effigy of Judas would ride into town on a burro, and would afterwards be burned for having betrayed Christ. The festival evolved over time to be more about burros and less about Judas.\n\nAn unexpected find of undisturbed mangroves and forest awaits at the **Bah\u00eda Cispat\u00e1** nearby. An interesting project by **Asocaiman** (caimanmiranda@hotmail.com) helps in the protection and propagation of alligators and turtles. These creatures were once hunted for their meat and eggs, but today, former local hunters have been trained on the importance of protecting these species. They now work for the animals' protection. You can visit the refuge (tips are encouraged) and take a short tour led by one of the former hunters. They offer different boat tours of the mangroves, which range in cost COP$15,000-45,000. A locally run and quite friendly hotel, the **Mangle Colora'o** (Vereda Amaya, tel. 4\/811-0722, cell tel. 301\/203-7071, COP$35,000 pp), is just across the street.\n\nSan Antero borders the water and has some nice beaches at Playa Blanca. Here there is a long string of waterfront hotels popular with weekenders from Monter\u00eda and Medell\u00edn. During the week, it's very quiet. The **Cispat\u00e1 Marina Hotel** (tel. 4\/811-0197 or 4\/811-0887, www.cispata.com, COP$123,000 pp) has an enviable location overlooking the the Bah\u00eda Cispat\u00e1 and, on the other side, the beaches of Playa Blanca. The hotel comprises 16 cute, red-roofed _caba\u00f1as_ as well as smaller apartments. In addition to the bay and the sea, the hotel also has a large pool.\n\nGo eat at M **Pesecar** (Bah\u00eda Cispat\u00e1, cell tel. 312\/651-2651, 7am-9pm daily). It's worth the trip to San Antero just for lunch\u2014fresh, very fresh, seafood, at this restaurant with an unbeatable bayside location.\n\nNo trip to the Caribbean coast is complete without a visit to a mud volcano. In San Antero there is a large one, where you'll be able to enjoy the therapeutic properties of the mud without bumping up against anyone. Laugh therapy is one of the many treatments available.\n\nalligators at the Asocaiman conservation center, Bah\u00eda Cispat\u00e1\n\n##### **Reserva Natural Viento Solar**\n\nIt's hard to find a place more peaceful than **Reserva Natural Viento Solar** (village of R\u00edo Cedro, cell tel. 311\/312-2473, www.vientosolar.org, students and backpackers COP$20,000 no meals, COP$130,000 pp all meals incl.). This private natural reserve composed of tropical dry forest is on a mostly undeveloped C\u00f3rdoba coastline near the settlement of R\u00edo Cedro, southwest of San Antero.\n\nAt this reserve extending over 200 hectares (500 acres) along the Caribbean coast, activities include kayaking, nature walks, bird-watching, swimming, and yoga. Gentle _osos perezosos_ (sloths) reside in this undisturbed reserve, and you may also see howler monkeys, boa constrictors, and iguanas, as well as many species of birds. The reserve is run by a charismatic Paisa woman, Elena Posada, who is affectionately known as La Mona.\n\nReserva Natural Viento Solar is accessed through the town of Lorica, a fishing town on the banks of the R\u00edo Sin\u00fa. It's famous for its waterfront market. From the Tol\u00fa-Monter\u00eda highway, you can catch a shared taxi (COP$15,000) that will take you to the coastal hamlet of San Bernardo del Viento. From there, Viento Solar will arrange for a _mototaxi_ (COP$10,000) to take you the rest of the way to the reserve.\n\n#### **MONTER\u00cdA**\n\nThe center of Monter\u00eda, Colombia's cattle-ranching capital (pop. 409,000), has recently been given a facelift, and in the late afternoon or early evening, it's a pleasant place for a stroll. The Plaza de Bol\u00edvar is gorgeous and the spectacularly white **Catedral de San Jeronimo** stands prominently facing it. The **Banco de la Rep\u00fablica** (Cra. 3 No. 28-59, tel. 4\/782-3382) may have an art exhibit to check out. For a pleasant walk or jog, head to the **Parque Lineal Ronda del Sin\u00fa** between the muddy R\u00edo Sin\u00fa and the Avenida Primera. This lovely park under the shade of tall trees and with a view to the river has bike and jogging paths, playgrounds, workout stations, an amphitheater, and juice stalls. This is one of the nicest urban parks in the country.\n\nReserva Natural Viento Solar\n\nThe coolest thing about Monter\u00eda is the ingenious (and eco-friendly) system to cross the R\u00edo Sin\u00fa: the **_planchones._** These are small ferries attached to a cable that crosses the river as the captain rows passengers across. It only costs COP$400, making it one of the top cheap thrills in all of Colombia. Even though it takes under five minutes to cross, and there's not much need to get to the other side, the _planchones_ themselves almost make Monter\u00eda worth a trip.\n\n##### **Accommodations and Food**\n\nHotels tend to be overpriced in Monter\u00eda. The **Hotel Casa Real** (Cl. 29 No. 6-26, tel. 4\/782-4004, www.hotelcasarealmonteria.net, COP$173,000 d) is a few blocks from the Avenida Primera and is also close to a police station. Many rooms are tiny with no windows.\n\nWhen in Monter\u00eda, one eats beef. The famous restaurants are on the outskirts of town and are large outdoor cowboy-ish places. While you could order grilled chicken, when in Monter\u00eda, order a thick, juicy steak. **Bonga del Sin\u00fa** (Km. 5 V\u00eda Ceret\u00e9, tel. 5\/786-0085, www.labongadelsinu.com, noon-11pm Mon.-Sat. and noon-9pm Sun., COP$22,000) doesn't disappoint its hungry patrons. Along the Avenida Primera are several bars and outdoor restaurants, and the volume kicks up a notch or two on Saturday nights. In a nod to the significant Arab immigration in the area, Monter\u00eda has a couple of spots where kibbe trumps steaks. Try **Farah Delicias Arabes** (Cra. 6 No. 60-42, tel. 4\/789-9680, www.farahdeliciasarabes.com, 4:30pm-10:30pm daily, COP$15,000), an authentic Lebanese restaurant. There is also a **Juan Valdez Caf\u00e9** (Cl. 44 No. 10-139, tel. 4\/785-1607) at the Alamedas del Sin\u00fa shopping mall. Get your macchiato fix here because you're a long way from another decent cup of joe!\n\n#### M **CAPURGAN\u00c1 AND SAPZURRO**\n\nThe sparsely populated **Darien Gap** , a 160-kilometer-long (100-mile-long) and 50-kilometer-wide (30-mile-wide) stretch of mountainous jungle and swamp extending from Panama to Colombia, has long captured the imagination of adventurers. The two crescent-shaped villages of Capurgan\u00e1 and Sapzurro, built between the sea and the interior mountains of the small and low Darien Mountain Range, are within howling distance of the Panamanian border, but they seem far, far away from anything else. Capurgan\u00e1 and Sapzurro are on the eastern edges of the Darien Gap, a stretch of land that connects Central America (via Panama) with South America (via Colombia). The stretch of tropical rainforest here is the only interruption in the famous Pan-American Highway, which extends from Alaska to Patagonia.\n\nWith its absence of roads and the cover provided by the jungle's canopy, the entire Darien region has been a major corridor for the trafficking of illegal drugs from Colombia into Central America, which has also meant that there has been a heavy presence of both guerrillas and paramilitaries. Capurgan\u00e1 and Sapzurro suffered greatly from drug-related violence during the 1990s and early 2000s. Thanks in part to a strong military presence, safety in the area has vastly improved. Though drug trafficking continues deep in the jungle, kidnappings and violent skirmishes don't affect locals or visitors.\n\nthe wild, rocky coast of Carpurgan\u00e1\n\nWhile much of the Colombian Darien is lowland and swamp, as it is part of the R\u00edo Atrato basin, near the border with Panama, the terrain is mountainous and covered in tropical jungle. Within minutes of leaving your hotel you'll be surrounded by the sounds of the jungle, accompanied only by the occasional bright green and black speckled toad and maybe a band of howler monkeys.\n\nHere in the Colombian Darien, the majority of the population is Afro-Colombian. Capurgan\u00e1 is the larger village of the two, though both are tiny. There are no cars in either village. Get around on foot, by bike, or by boat.\n\nTo get here, you have to either take a flight from Medell\u00edn (to either the Capurgan\u00e1 or Acand\u00ed airport) or take a _lancha_ (from Turbo or Acand\u00ed). Should the seas be too rough or if a general strike shuts down everything (as when we were there), you may just be stuck in the jungle for a couple days more.\n\n##### **Recreation**\n\n###### **HIKING**\n\nThere are several jungle walks to make around Capurgan\u00e1. These take you through dense jungle overflowing with tropical vegetation and home to howler monkeys, birds, colorful frogs, and snakes. While the walks are short and fairly straightforward, you may want to ask at your hotel or hostel for a guide, especially for the walk between Capurgan\u00e1 and Sapzurro. Guides cost about COP$10,000. Wear hiking boots (waterproof if possible) and a swimsuit underneath your clothes for dips in the water off of Sapzurro or in freshwater swimming holes, and set off in the morning hours to avoid trying to navigate your way in the late afternoon.\n\nAn easy walk to make, without the need of a guide, is to **La Coquerita** (20-minute walk north from town, cell tel. 311\/824-8022, COP$2,000), a delightfully ramshackle waterside hangout where you can have a refreshing coco-lemonade, maybe some guacamole and _patacones_ (fried plantains), and take a dip in the refreshing freshwater or saltwater pools. There are also some handicrafts on sale here. To get there, walk along the Playa Caleta beach just north of the port, passing in front of the Hotel Almar. Continue along the jungle path that hugs the coastline. La Coquerita is under a kilometer from town, and the path is well-marked. Look out for the black and fluorescent green frogs along the way, but don't touch them; they're poisonous.\n\nThere are two ways to go to the idyllic hamlet of **Sapzurro:** by boat or on foot. The path to Sapzurro leads you through the exuberant rainforest to a lookout point and then down directly to the Sapzurro beach. The hike takes two hours.\n\nTo set off for Sapzurro, start at the soccer field, on the southern end of town, and ask the way. Midway up the uphill path is a shack that is the home of a man who claims to protect the jungle, Once you find him, you know you're on the right track. He expects those who pass through to pay him about COP$1,000. At the top of the mountain there is a nice overlook with views of Capurgan\u00e1 and the coastline. The hike is not difficult, but the path can get muddy and slippery in places. Wear hiking boots and pick up a walking stick along the way to help you manage on the steep parts.\n\na frog on the path to Sapzurro\n\nOnce in Sapzurro, you're a short hike (15 minutes) up to the border with **Panama** and the village of **La Miel.** This easy walk begins on the same street as Caba\u00f1as Uvali and the Reserva Natural Tacarcun\u00e1. The border crossing is at the top of a steep hill with embedded steps. You'll need to show a passport to cross over to Panama. There is not much to the community of La Miel. It has a small military outpost, many young children running around, and a pleasant beach where you can swim and have a seafood lunch or drink.\n\nAnother walk to make is to the **El Cielo** waterfall **,** a 50-minute walk (about 3 km) through the jungle. It's easy to make and is flat, although you'll have to make around a dozen shallow stream crossings. Bring a bathing suit to cool off in the swimming holes you'll encounter. To get to heavenly El Cielo, set out on the road that runs parallel to the airstrip. Ask locals for directions.\n\nIt is possible to walk between Capurgan\u00e1 and El Aguacate, but the path, along the shore, is rocky and a bit treacherous.\n\n###### **DIVING AND SNORKELING**\n\nAs you'd expect, the warm, turquoise waters off the coast of Capurgan\u00e1 and all the way up to San Blas in Panama make for fantastic diving, and there are over 30 diving sites to choose from. The best time for underwater exploration is from May to November. During those months, visibility is exceptional with hardly any waves around the diving spots. There are coral walls, reef rocks, and caves to explore close to the coastline.\n\n**Dive and Green Diving Center** (facing the port, cell tel. 311\/578-4021 or 316\/781-6255, www.diveandgreen.com, 7:30am-12:30pm and 2pm-6pm daily) is the best place to organize a diving trip (for certified divers an excursion costs COP$190,000) or to take a PADI certification course (5 days, COP$820,000) with a bilingual instructor. For these packages it is best to pay in cash. Credit card transactions will have an additional fee. For those interested in snorkeling, they can help make arrangements for you, though they don't themselves lead snorkeling trips. Dive and Green offers all the equipment you need. If you are on the fence about whether diving is for you, they offer a Discover Scuba Diving day for COP$150,000. Dive and Green has accommodations: four rooms in a house adjacent to their offices. These cost COP$25,000 per person. Although in town, it's facing the water, guaranteeing a pleasant evening breeze.\n\n##### **Accommodations**\n\nThere are a surprising number of excellent and inexpensive accommodations options in both Capurgan\u00e1 and Sapzurro. While there are a few large, all-inclusive hotels with welcome drinks and the works, the most interesting and comfortable options are the smaller guesthouses and hostels. Nearly all hotels are owned and operated by out-of-towners.\n\n###### **CAPURGAN\u00c1**\n\nMany hotels and hostels are near the _muelle_ (port) in Capurgan\u00e1. Here you have the advantage of being in or near the hub of activity. Many visitors stay at one of the few all-inclusive hotels in Capurgan\u00e1, but those options have zero charm.\n\nThe M **Posada del Gecko** (Centro, cell tel. 314\/525-6037, www.posadadelgecko.com, posadadelgecko@hotmail.com, COP$20,000 dorm, COP$35,000 d) is the best place to stay in town. It's run by an Italian-Colombian couple and offers both dorm and private rooms spread over two houses, with a capacity of 28 persons. In between is a spacious open-air garden ideal for lounging in a hammock or the hot tub. Enjoy a good Italian dinner by candlelight from the restaurant (7:30pm-11pm daily). It's open to non-guests as well, but it's best to go by in advance and make a reservation. The hotel organizes three-day excursions to the San Blas Islands (Panama), in which you visit a Guna indigenous community, frolic on pristine white-sand beaches, and snorkel.\n\nAlthough there are no sandy beaches there, outside of the center of Capurgan\u00e1 in the Playa Roca area, about a 15-minute walk or horse ride away, are several excellent guesthouses amid the trees. At night you'll need no air conditioner, and in the morning you may awake to birdsong.\n\nOne of the perks of staying at welcoming **Caba\u00f1as El Tuc\u00e1n** (Playa Roca, www.cabanatucancapurgana.com, COP$65,000), run by a friendly Bogotana-Italian couple, is that they make their own pasta and are good cooks. This house in the jungle is clean and comfortable, and the prices of their two spacious rooms are reasonable. Right across the path from El Tuc\u00e1n is M **Caba\u00f1as Darius** (Playa Roca, cell tel. 314\/622-5638, www.cdarius.blogspot.com, capurga05@gmail.com, COP$85,000 pp incl. 2 meals), another Colombian-international endeavor. It's a very nice guesthouse in the trees. Rooms are spacious and clean, and it's cool enough at night that you won't miss air conditioning. Balconies and hammocks provide lounging space, but the top selling point is the warm hospitality and Nery's unbelievable cooking. A third option in the same area is **Hotel Los Robles** (Playa Roca, cell tel. 314\/632-8408 or 314\/632-8428, www.capurganalosrobles.es.tl, COP$85,000 d, COP$70,000 pp). This lodge has quite the entrance\u2014a winding path lined by bright fuchsia ginger flowers. _Caracol\u00ed_ and _higuer\u00f3n_ trees provide shade and a home for birds. There are 12 rooms in two houses.\n\nThe most low-key place to stay in the area is Playa Aguacate. A German has carved a little paradise out of the jungle, and, once there, you won't want to leave. It's popular with honeymooners and those celebrating special occasions. Simple and comfortable cabins, each with a sea view, make up the M **Bah\u00eda Lodge** (Playa Aguacate, cell tel. 314\/812-2727, www.bahia-lodge.com, COP$190,000 pp 2 meals per day). Over the hill from is the lodge is **Hotel Las Ceibas** (Bah\u00eda Aguacate, cell tel. 313\/695-6392, Medell\u00edn tel. 4\/331-7440, www.hotellasceibas.com.co). It has three rooms in two houses set amid gardens, and the rooms have pleasant balconies with hammocks for late afternoon relaxation.\n\n###### **SAPZURRO**\n\nIn Sapzurro, there are also quite a few options. The only drawback is that there are fewer restaurant options. Most hotels offer meals, though, and are usually the best bet at any rate. **Caba\u00f1as Uvali** (cell tel. 314\/624-1325, COP$40,000 pp) is a friendly, clean, and straightforward little place in town. It's about a five-minute walk to the beach. **La Posada Hostal & Camping** (www.sapzurrolaposada.com, cell tel. 312\/662-7599 or 310\/410-2245, COP$65,000 pp d) has one luxury apartment, with a view, a dorm-style room, and a large space for camping under the big mango tree. They have a little tiki bar over the water, which can be set up for your romantic Sapzurro dinner.\n\nThe M **Resort Para\u00edso Sapzurro** (cell tel. 313\/685-9862, www.paraiso-sapzurro-colombia.com, COP$10,000 dorm, COP$50,000 pp d) has basic beachfront cabins. Free avocadoes and mangoes is not a bad perk. This hostel is more commonly known as \"Donde El Chileno,\" after the Chilean owner. The hostel can organize oversea journeys directly to Cartagena.\n\nThe M **Sapzurro Reserva Natural Tacarcuna** (on the path to La Miel, cell tel. 314\/622-3149, COP$40,000 pp) is a special place for anyone interested in the flora and fauna of the region. Owners Martha and Fabio completed a botanical garden of native species with nature trails and a butterfly farm in 2013, and their other passion is birds. Behind their house up the mountain into the jungle, you can grab your binoculars and wait and watch for birds to make their appearances. Four different types of toucans, cuckoos, parakeets, owls, antipittas, tanagers, and many other species can be seen here. Throngs of migratory birds arrive between August and November each year. The two cabins available are cute, spic-and-span, and of course surrounded by nature. Don't confuse this natural reserve with the all-inclusive Tacurcuna Hotel near the Capurgan\u00e1 airport. They can organize a number of nature hikes in the area and near Acand\u00ed.\n\n##### **Food**\n\nHotels are usually the best options for food in the villages. M **Donde Josefina** (Playa Caleta, cell tel. 316\/779-7760, COP$30,000, noon-9pm daily) remains the top restaurant for a delicious, gourmet seafood dinner, right on the beach in the heart of Capurgan\u00e1. Dining on lobster in a coconut and garlic sauce under the swaying branches of a palm tree: That sounds like a vacation! A decent bakery overlooking the soccer field in Capurgan\u00e1 serves breakfast. In Sapzurro, the best place for a meal is at **Do\u00f1a Triny** (Hostal Do\u00f1a Triny, cell tel. 312\/751-8626 or 313\/725-8362, noon-10pm daily).\n\n##### **Information and Services**\n\nBring extra cash to Capurgan\u00e1: There are no ATMs, and credit cards are not accepted in most establishments. To avoid bringing wads of pesos, many hotels will allow (or require) you to make a _consignaci\u00f3n_ (deposit) to their bank account in advance. That can usually be done from any city in Colombia. The nearest bank, **Banco Agrario** (Cl. Las Flores with Cl. Consistorial, tel. 4\/682-8229, 8am-11:30am and 2pm-4:30pm Mon.-Fri.) is in Acand\u00ed, a half-hour boat ride away to the south from Capurgan\u00e1. To be on the safe side, bring along some extra cash.\n\n##### **Getting There and Around**\n\n**Aerol\u00ednea de Antioquia** (ADA, tel. 4\/444-4232, www.ada.com.co) serves both Capurgan\u00e1 and Acand\u00ed from the Aeropuerto Olaya Herera in Medell\u00edn. Acand\u00ed is the municipality to which the village of Capurgan\u00e1 is linked. It is south of Capurgan\u00e1 on the Darien. There is one flight per day on ADA Monday through Saturday to Acand\u00ed. Direct flights to Capurgan\u00e1 were temporarily suspended at the time of writing.\n\nBoats are a good way to get around Carpurgan\u00e1.\n\nTo get from Acand\u00ed to Capurgan\u00e1, you'll have to take a horse from the airport (seriously) to the docks (a 15-minute trip), at which point you'll take a 30-minute long _lancha_ (boat) ride onwards to Capurgan\u00e1 (COP$17,000). There is always a _lancha_ at 1pm daily. The seas can be rough at times, so, always try to get a seat in back. You may want to keep your camera or other electronics in their cases so they won't be exposed to seawater. Demand a life vest. The return trip from Capurgan\u00e1 to Acand\u00ed leaves at 7:30am daily.\n\nAll boats arrive at the _muelle_ (docks) in Capurgan\u00e1, which are in the middle of town. Most hotels are within walking distance, although some, like Caba\u00f1as Darius, are a bit of a walk. Try to find someone with a horse to take you there (about COP$10,000). Hotels in Aguacate and Sapzurro are reachable only by taking another _lancha_ from the docks in Capurgan\u00e1, about a 20-minute ride. Those hotels will arrange your transportation from Capurgan\u00e1 in advance.\n\nThere are bus links from Medell\u00edn (8 hours), Monter\u00eda (4 hours), and from cities across the Caribbean coast to the rough and tumble coastal port city of Turbo on the Golfo de Urab\u00e1. As an alternative to taking a flight to either Capurgan\u00e1 or to Acand\u00ed, you can take a 2.5- to 3-hour boat ride from Turbo. These usually depart at 8am, costing about COP$60,000. The early morning departure means that you will probably have to spend the previous night in Turbo. That's not ideal, but if you must, most tourists agree that **Residencias La Florida** (Cra. 13 No. 99A-56, tel. 4\/827-3531, COP$30,000 d) is an all right accommodations option, close to the port, and the hotel staff is quite helpful arranging your onward transportation.\n\nDo not plan to take a boat from Turbo to Capurgan\u00e1 from December to March. The 2.5-hour journey can be awful during this time of high winds, and the waves can be unrelenting. If you are unlucky enough to be in the front of the boat, you will step off the boat with, at the very least, a painfully sore back. This is likewise true for the trip from Acand\u00ed to Capurgan\u00e1, although it is a much shorter ride.\n\nReturning to the mainland from Capurgan\u00e1, it's always best to reserve a day in advance for _lanchas_ bound for Acand\u00ed or Turbo, especially during peak tourist times. Your hotel should be able to do this for you, but just in case, you can call Sara at the _muelle_ (314\/614-0704).\n\nFrom Panama, you can take a flight from Panama City to Puerto Obald\u00eda. From there, boats frequently make the journey onward to Capurgan\u00e1. It's just a 30-45 minute ride and costs about US$15. You may be required to show proof of yellow fever vaccination to enter Colombia.\n\nIf traveling onward to Panama, you must go to the Colombian **Ministerio de Relaciones Exteriores** (Cl. del Comercio, cell tel. 311\/746-6234, 8am-5pm Mon.-Fri., 9am-4pm Sat.) the day before for an exit stamp.\n\n## **BOYAC\u00c1 AND THE SANTANDERES**\n\nHISTORY\n\nPLANNING YOUR TIME\n\nHIGHLIGHTS\n\nBoyac\u00e1\n\nM VILLA DE LEYVA\n\nR\u00c1QUIRA\n\nTUNJA\n\nTUNJA TO LAGO TOTA\n\nSIERRA NEVADA DEL COCUY\n\nM PARQUE NACIONAL NATURAL EL COCUY\n\nSantander\n\nBUCARAMANGA\n\nM CA\u00d1\u00d3N DEL CHICAMOCHA\n\nSAN GIL\n\nM BARICHARA\n\nNorte de Santander\n\nPAMPLONA\n\nC\u00daCUTA\n\nLocated north of Bogot\u00e1, the mountainous departments of Boyac\u00e1, Santander, and Norte de Santander (these last two are known collectively as the Santanderes) are rich in history, natural beauty, and outdoor activities. The countryside is dotted with historic colonial towns, including two of the most beautiful and well-preserved in Colombia: Villa de Leyva and Barichara. The scenery of the region runs the gamut from the desert landscape near Villa de Leyva to the bucolic rolling hills and pastures of agriculturally rich Boyac\u00e1, and from the awe-inspiring R\u00edo Chicamocha canyon to the dramatic snowcapped peaks of the Sierra Nevada del Cocuy (Cocuy Range).\n\nOutdoor activities are the draw here, like trekking in the Sierra Nevada del Cocuy and white-water rafting, caving, and paragliding near San Gil. Except for the frenetic and modern Bucaramanga, stoic Tunja, and the border city of C\u00facuta, a refreshingly slow pace prevails. The pueblos of Boyac\u00e1 are easily accessed from Bogot\u00e1 and can even be visited on a long weekend. It will take a little more time to discover Santander, located between Bogot\u00e1 and the Caribbean coast. Although most people only stop in the sultry city of C\u00facuta on their way to Venezuela or on a visa run, it is a pleasant surprise. The historic pueblo of Pamplona is the most chilled-out place in all of Norte de Santander.\n\n#### **HISTORY**\n\nBefore the Spanish conquest, Boyac\u00e1 was part of the Muisca heartland. Hunza, where present day Tunja is located, was the seat of the Zaque, one of the Muisca leaders. The Sun Temple, one of the Muiscas' sacred sites, was in Sogamoso, northeast of Tunja.\n\nBoyac\u00e1 and the Santanderes played a major role in the struggle for independence. In 1811, Boyac\u00e1 became the seat of the Provincias Unidas de la Nueva Granada (United Provinces of New Granada), the first republican independent government. It was in Boyac\u00e1 in 1819 that the two decisive battles of independence were fought: the Batalla del Pantano de Vargas (Battle of the Vargas Swamp) and the Batalla del Puente de Boyac\u00e1 (Battle of the Bridge of Boyac\u00e1). These battles marked the end of Spanish domination in Colombia.\n\nSantander was one of the more dynamic regions in 19th-century Colombia, with an export economy based on the cultivation of quinine, coffee, cocoa, and tobacco. In the early 20th century, Norte de Santander became the first major coffee-producing region in Colombia.\n\nThe mid-20th-century fighting between Liberals and Conservatives was particularly acute in Santander and Norte de Santander. In 1960, the Ejercito de Liberaci\u00f3n Nacional (National Liberation Army) or ELN guerrilla group was born in rural Santander.\n\nThe region has experienced steady economic growth since the early 2000s. Bucaramanga, the capital of Santander, has become a prosperous center of manufacturing and services. C\u00facuta, in the neighboring Norte de Santander department, is a center of commerce whose fortunes are linked to Venezuela's.\n\nWhile poverty is widespread in the Boyac\u00e1 countryside, the area is an important agricultural center and supplies Bogot\u00e1 with much of its food. The departmental capital of Tunja has also become a major center of learning: It is home to 10 universities.\n\n#### **PLANNING YOUR TIME**\n\nThere are three main draws in Boyac\u00e1 and Santander: the lovely colonial town of Villa de Leyva, the snowcapped wonderland of the Sierra Nevada del Cocuy, and, in Santander, the action-packed area around San Gil, including the nearby town of Barichara.\n\nVilla de Leyva can be visited in a short two-day excursion from Bogot\u00e1, but you could easily spend a couple more relaxing days seeing all the sights, including a hike to Laguna Iguaque. Add on a day to visit the churches of Tunja, but be sure to confirm hours beforehand. To further explore Boyac\u00e1, extend your visit for a couple of days to the area around Sogamoso, particularly the postcard-perfect towns of Iza and Mongu\u00ed and Lago de Tota. There are good public transportation links throughout Boyac\u00e1, but this is also a fairly easy place to drive.\n\nGetting to the Sierra Nevada del Cocuy is a schlep (11 hours by bus from Bogot\u00e1), so a trip there requires a minimum of 4-5 days to make it worthwhile. To just do day hikes into the park, base yourself in either G\u00fcic\u00e1n or El Cocuy, or nearer to the park in one of several lodges. To do the six-day circuit around the park, plan on 10 days so as to include a day or two of acclimatization before you embark. This is a remote area and there are fewer public transportation options. Buses depart for the area from Tunja. Roads are for the most part in good shape.\n\n**HIGHLIGHTS**\n\nLOOK FOR M TO FIND RECOMMENDED SIGHTS, ACTIVITIES, DINING, AND LODGING.\n\nM **Villa de Leyva:** One of Colombia's most visited and beloved colonial pueblos, relaxed Villa de Leyva is just a couple of hours away from Bogot\u00e1 (click here).\n\nM **Santuario Flora y Fauna Iguaque:** Hike through Andean forest and mysterious _p\u00e1ramo_ to a sacred Muisca lake (click here).\n\nM **Tunja's Historic Churches:** Glimpse the splendor of Tunja's colonial past in its beautiful churches (click here).\n\nM **Parque Nacional Natural El Cocuy:** Stunning scenery greets you at every turn in this remote park of snowcapped peaks (click here).\n\nM **Ca\u00f1\u00f3n del Chicamocha:** Experience the blue skies and deep canyons at this photogenic park not far from Bucaramanga and San Gil (click here).\n\nM **Paragliding near San Gil:** Soar through the air with the greatest of ease in Colombia's recreational capital (click here).\n\nM **Barichara:** Decompress and rejuvenate in one of the most beautiful pueblos in the country (click here).\n\nM **Camino Real:** Follow in the footsteps of native Guane indigenous traders and Spanish colonists on this meandering path through the Santander countryside (click here).\n\nSan Gil and Barichara have a lot to offer, so plan on spending at least three days. Barichara is a much more beautiful base for exploring the region, but San Gil is home to the main adventure sport tour operators.\n\nThere are good public transportation links between Bucaramanga and San Gil and between San Gil and Barichara. However, getting from Bucaramanga and San Gil to Tunja or Villa de Leyva is not fun, as the highway is often saturated with big trucks and buses. On holidays it can be difficult to get a seat on a bus out of or to San Gil, an intermediary stop between Bucaramanga and Bogot\u00e1.\n\nBetween Villa de Leyva and San Gil it is 6-7 hours, with a change of bus in Barbosa and\/or Tunja. Even though San Gil and the Sierra Nevada del Cocuy are only 75 kilometers as the crow flies, to get from one to the other, you must transfer buses in Tunja, requiring more than 15 hours on various buses.\n\nThere's frequent bus service between Bucaramanga and Pamplona\/C\u00facuta.\n\n### **Boyac\u00e1**\n\nTo the northeast of Bogot\u00e1 and the department of Cundinamarca, the Boyac\u00e1 department is a mostly rural agriculture-oriented area of bucolic highlands, home to campesinos (peasants) often dressed in their warm woolen _ruanas_ (capes) as they tend to their dairy cows and potato crops. Boyac\u00e1 is also known for its role in Colombian history: The capital city of Tunja was effectively the runner-up to Bogot\u00e1 when the Spaniards sought a capital city for their New World territory of Nueva Granada. Its colonial-era importance can be seen today in the number of impressive churches that stand in its historic center. Nearby, Villa de Leyva has the perfect combination of colonial charm, good hotels and restaurants, attractions, and fantastic weather. Boyacenses are known for their politeness, shyness, and honesty, and will often address you not with the formal _usted_ but rather with the super-deferential _sumerc\u00e9,_ a term that is derived from the old Spanish _su merced_ (literally, \"your mercy\").\n\n#### M **VILLA DE LEYVA**\n\nThis enchanting colonial pueblo is set in an arid valley (Valle de Saquencip\u00e1) and has been a major tourist destination for decades. The population triples on weekends, when city folk from Bogot\u00e1 converge on the town. The surrounding desert scenery, a palette of ever-changing pastels, is gorgeous, the typically sunny weather is never too hot nor too cool, and the town's architecture of preserved whitewashed houses along stone streets is charming.\n\nThe influx of visitors every weekend doesn't diminish the appeal of VDL, as it's known. A surprising number of activities and attractions are in reach, including paleontological and archaeological sites and outdoor activities such as biking and hiking. The nearby Santuario Flora y Fauna Iguaque is one of the most accessible national parks in the country, and you need only a decent pair of boots to hike to its sacred lakes. Villa de Leyva is also a good base from which to explore the Boyac\u00e1 countryside and towns such as R\u00e1quira.\n\n##### **Sights**\n\n###### **PLAZA MAYOR**\n\nVilla de Leyva's **Plaza Mayor** is one of the most photographed locations in Colombia. The town's main square, the largest plaza in the country, is indeed photogenic, but it can be a frustrating task capturing it all in one take: At 14,000 square meters (3.5 acres), it's big. In the middle of the square is a Mudejar-style well, the Ara Sagrada, that was the source of water for the townspeople in colonial times. On the western side of the square is the **Iglesia Parroquial** (Cra. 9 No. 12-68, 8am-noon and 2pm-6pm Tues.-Sat., 8am-noon Sun.), made out of stone, adobe, and wood, which was built in the 17th century. It features a large golden _retablo_ altar.\n\nOn the western side of the plaza is the quirky **Casa Museo Luis Alberto Acu\u00f1a** (Cra. 10 No. 12-83, tel. 8\/732-0422, www.museoacuna.com.co, 9am-6pm daily). In addition to cubist-influenced paintings of pre-Hispanic indigenous culture, rooms are filled with the artists' private art collection and antiques. The courtyard holds some wood sculptures of the artists. The museum also has a pint-sized gift shop. Acu\u00f1a was instrumental in the restoration of and preservation of colonial architecture in Villa de Leyva.\n\nOne of the oldest houses in Villa de Leyva, and best preserved, is the **Casa Juan de Castellanos** (Cra. 9 No. 13-11) on the northeast corner of the Plaza Mayor. It is so well preserved, in fact, that it today serves as the main office of the city government. The house is not officially open to the public, but you can take a peek. The house belonged to Spaniard Juan de Castellanos, who came to the New World as a soldier. He was an important chronicler of the time.\n\nAcross from the Casa Juan de Castellanos is the historic **Casa del Primer Congreso de lasProvincias Unidas de la Nueva Granada** (Cra. 9 No. 13-04). Restored by artist Luis Alberto Acu\u00f1a in the 1950s, this is where the era of the Patria Boba, as it would later (and derisively) be known, was begun. The **Casa Real F\u00e1brica de Licores** (Cl. 13 No. 8-03) was the first official distillery in Nueva Granada. After standing in ruins for decades, the house was restored in the 1950s.\n\na charming cobblestone street in Villa de Leyva\n\nThe **Museo El Carmen de Arte Religioso** (Cl. 14 No. 10-04, 10am-1pm and 2pm-5pm Sat., Sun., and holidays, COP$2,500) presents paintings, crucifixes, manuscripts, and religious figures from the colonial era. The museum is on the southwest corner of the grassy **Plazoleta de la Carmen.** The complex (which dates to around 1850) also includes a monastery and convent.\n\nThe **Casa Museo Antonio Nari\u00f1o** (Cra. 9 No. 10-25, 9am-noon and 2pm-5pm Thurs.-Tues., tel. 8\/732-0342, free) is a house in which independence figure Antonio Nari\u00f1o lived and died. It was built in the 17th century, and the museum displays some of his manuscripts as well as items from everyday life in the 19th century, such as a giant mortar used to mill corn. The short and sweet museum often puts on temporary art exhibits, which may have a small admission charge.\n\n###### **OTHER SIGHTS**\n\nOn the **Plaza Ricaurte,** the 19th-century **Convento de San Agust\u00edn** today houses the highly respected **Instituto Humboldt** (Cra. 8 No. 15-98, tel. 8\/732-0791, www.humboldt.org.co, free), a research institute dedicated to conservation and environmental education. A small room on threatened animal species in Colombia is open to the public, and a tour is given on Fridays (3pm-5pm). The Humboldt occasionally hosts cultural events.\n\nThe **Casa Museo Capit\u00e1n Antonio Ricaurte** (Cl. 15 No. 8-17, 9am-noon and 2pm-5pm Tues.-Sun., no phone, free) is in the small house where this independence figure was born in 1786. He is a hero of the Colombian air force, and one room is filled with uniforms and memorabilia of that military branch. He died heroically, sacrificing his life by blowing himself up with a cache of gunpowder so that it would not land in the hands of Spaniards.\n\n##### **Entertainment and Events**\n\nThe most popular place to hang out in the evenings is on the Plaza Mayor, where the thing to do is buy a couple of beers and watch the world go by. But there are other watering holes in town. **La Cava de Don Fernando** (Cra. 10 No. 12-03, tel. 8\/732-0073) is a spot for a cocktail where the music is generally, but not always, rock.\n\nFor a movie night, head to the **Cine Club Casa Quintero** (Casa Quintero, 2nd fl., tel. 8\/732-1801, Thurs.-Sun. evenings).\n\nThe dark, crystal-clear skies above Villa de Leyva make for great stargazing. In February each year the town hosts the **Festival de Astron\u00f3mica de Villa de Leyva** (www.astroasasac.com), during which all are invited to view the stars from powerful telescopes in the Plaza Mayor.\n\n##### **Shopping**\n\nIn Villa de Leyva as in the rest of Boyac\u00e1, the special handicraft is woolen goods. But the town is home to many creative types, and small jewelers, galleries, and handicraft shops are found on every street.\n\n**Alieth Tejido Artesanal** (Cl. 13 No. 7-89, tel. 8\/732-1672, www.alieth.8m.com) is an association of about 35 women who weave woolen sweaters, _ruanas_ (ponchos), _mochilas_ (backpacks), gloves, scarves, and some colorful, slightly psychedelic bags. A tour, the \"Ruta de la Lana,\" can be taken to nearby farms to learn about the process from sheep to sweater. It costs COP$48,000 per person, lasts for about five hours, and snacks and a souvenir are included. Alieth Ort\u00edz, the head of this interesting program, requests reservations be made a few days in advance so that they can organize things with the artisans. Another excellent store to browse wool items is **Creaciones Dora** (Cra. 10 No. 10-02, 9am-7pm Mon.-Fri., 9am-9pm Sat.).\n\nBoyac\u00e1 is known for its woven woolen scarves and _ruanas._\n\n**La Libelula** (Cra. 9 No. 14-35, tel. 8\/732-0040, 10am-7pm daily) specializes in leather: handbags, belts, and accessories. The mysterious shop **Misterio** (Cl. 14 No. 9-85, tel. 8\/732-0418, 10am-8pm daily) sells colorful scarves, handmade jewelry, and semiprecious stones like quartz, amethyst, and emeralds. Coal mining is a major industry in rugged Boyac\u00e1; **Arte al Carb\u00f3n** (Cl. 15 No. 9-46, www.artealcarbon.galeon.com, 9am-7pm Mon.-Sat.) sells jewelry made out of coal by women from mining communities.\n\n##### **Recreation**\n\n###### **HIKING**\n\nClose to town, sporty locals regularly take a brisk morning hike up to the **Santo,** a statue on the eastern side of Villa de Leyva. The walk takes about an hour in total, and it is a steep climb. From the statue of the saint you can get a good view of the town and will better appreciate the scale of the fantastic Plaza Mayor. To get to the path, walk east along Calle 11 to the tennis court and track\/soccer field. This is to the north of the Hotel Duruelo. The path entrance is marked and it practically leads straight up. The rocks are covered, unfortunately, with religious graffiti. It's best to make the climb early in the morning, before the midday heat envelops the valley. Although the view is nice, you'll be better off leaving your camera at your hotel, not necessarily due to safety reasons, but rather because you may not want to be loaded down with things as you climb. At parts you may be on all fours!\n\n###### **TOURS**\n\nTour companies in Villa de Leyva mostly specialize in adventure activities nearby. The best outfit is **Colombian Highlands** (Renacer, Av. Cra. 10 No. 21, tel. 8\/732-1201, www.colombianhighlands.com), the same folks running the Renacer hostel. **Armonita Tours** (Av. Perimetral No. 8-08, tel. 1\/643-3883, www.amonitatour.com) can take you to the Santa Sof\u00eda area for waterfall rappelling (COP$85,000), caving (COP$75,000), and the Paso de \u00c1ngel hike (COP$75,000). They also can arrange horseback riding trips to the Pozos Azules, and they rent bikes. In addition, they have day-trip packages to places like Museo El F\u00f3sil de Monquir\u00e1 and the Convento del Santo Ecce Homo. These cost about COP$60,000 and include transportation and entry to the attractions.\n\n###### **BIKE RENTAL**\n\nYou'll need plenty of sunscreen and water, but renting a bike to see the sights in the valley near Villa de Leyva is a great way to spend a day and get some good exercise as you huff and puff up that hill to the Convento del Santo Ecce Homo.\n\nMountain bikes, not necessarily of the highest quality, are readily available for rent in Villa de Leyva. These usually all go for about COP$20,000 for a half day (until 1 or 2pm). **Sentimiento Natura** (Cl. 8 No. 9-47, cell tel. 321\/217-2455) rents bikes, usually from a house that is conveniently located near the road through the valley. **Bici Motos** (Transversal 10 No. 7A-10, Barrio Los Olivos, cell tel. 321\/225-5769) is also on the way, and the friendly owner lives above the bike shop. That means you can get there on the early side (6am-7am) if you want to pick up a bike. (He doesn't mind.) He charges only COP$10,000 for a five-hour rental.\n\n##### **Accommodations**\n\nVilla de Leyva gets hopping on weekends and on holidays, and hotel rates bump up accordingly. Rates are even higher during the Christmas holidays through the second week of January and during Holy Week. During the week you will have your choice of hotels and ought to try to negotiate a better price. Many will provide additional discounts if you pay in cash.\n\nThere aren't many midrange hotel options in town, as most hotels target the luxury hotel market. A number of hotels that call themselves boutique have sprung up in recent years just outside of the town or farther in the valley. They may be comfortable but they may lack in the charm department. Hostels in Villa de Leyva are reliably friendly options, and staff are chock full of knowledge and tips on where to go and what to do.\n\n###### **UNDER COP$70,000**\n\nM **Renacer** (Av. Cra. 10 No. 21, tel. 8\/732-1201, www.colombianhighlands.com, COP$35,000 shared room, COP$65,000 d) is the best-known hostel in town and is popular for good reasons. Its location is a hike away from town (about a 15-minute walk), but guests will be reimbursed for their taxi ride upon arrival. It is set amidst green at the foot of a mountain. Facilities are well kept and there is ample open-air common space. There are seven rooms and cabanas for varying numbers of guests, some with private bathrooms. There is also a place for those arriving in campers or vans. The on-site restaurant, **Pekish,** is excellent, with options to please anyone, such as falafels (COP$12,000) and Vietnamese spring rolls (COP$10,000). In addition to the hostel, Renacer, through **Colombian Highlands** (www.colombianhighlands.com), arranges outdoor expeditions to nearby attractions and can even assist in excursions outside of the Villa de Leyva area. They have very good information on how to hike or bike the area solo. This is an excellent place to swap travel tips with backpackers from around the world.\n\nRun by an Austrian-Colombian couple, the M **Casa Viena Hostel** (Cra. 10 No. 19-114, Sector de la Banadera, tel. 8\/732-0711, www.hostel-villadeleyva.com, COP$17,000 s, COP$40,000 d) is a quiet and relaxed guesthouse on the same road as Renacer (just before it). It has just three rooms and the owners live in the same house. You may bump into them going to the shared bathrooms. They opened a new farmhouse in 2013 called **Casa Puente Piedra** (COP$25,000 pp) that is even more tranquil. It is within walking distance of the Santuario Flora y Fauna Iguaque. They have excellent mountain bikes for rental. These cost COP$25,000 for a half day.\n\nA low-key and lesser-known hostel option is **Hostal Rana** (Cl. 10A No. 10-31, tel. 8\/732-0330, www.hostalrana.com or www.learnspanishinvilladeleyva.com, COP$20,000 dorm, COP$40,000 d). It opened in 2010 and has one dorm room and four private rooms. Rooms are clean and beds are firm. There is a small kitchen for use. One problem is that the tiny camping area in the courtyard area is sort of icky. They can arrange for Spanish lessons here.\n\nThe no-frills **Hospedaje Los Balcones de la Plaza** (Cl. 13 No. 9-94, cell tel. 314\/360-8568, COP$45,000 pp w\/shared bath) has about four rooms and occupies a corner of real estate overlooking the Plaza Mayor. The views from your balcony can't be beat. They don't provide wireless Internet, but the town government does on the Plaza Mayor. Here there is no common area, but the plaza is as common as it gets.\n\nIf you ask locals for a less expensive hotel option, many will tell you to check out **Hospeder\u00eda Don Paulino** (Cl. 14 No. 7-46, tel. 8\/732-1227, COP$35,000 s, COP$65,000 d). It's not a fancy place by any means, but the price can't be beat! The 16 rooms all have wireless Internet and TV, but the ones on the first floor are a little on the stuffy and small side. From the second floor balcony overlooking the outdoor patio you can enjoy the sunset. There's no breakfast included but each morning they do provide coffee.\n\n###### **COP$70,000-200,000**\n\nFamily-run **Hospeder\u00eda La Roca** (Cl. 13 No. 9-54, COP$160,000 d) has been a cheapie quietly overlooking the Plaza Mayor for years, but it's no longer a budget option. More than 20 rooms with high ceilings surround two interior courtyards that are filled with greenery. Try for one on the second floor with a squint of a view of the mountains. Around the corner is the welcoming **Posada de Los \u00c1ngeles** (Cra. 10 No. 13-94, tel. 8\/732-0562, COP$110,000 d), a lovely option overlooking the Plazoleta de Carmen. Some rooms have balconies overlooking the church. Take your American-style breakfast in the cheerfully painted patio filled with potted plants and flowers. No wireless Internet.\n\nFriendly and colorful **Sol de la Villa** (Cra. 8A No. 12-28, tel. 8\/732-0224, COP$120,000) has 30-some comfortable rooms and an excellent \"in town\" location. Walls are a little on the thin side, as are curtains. Rooms towards the back and upstairs are best. A nice service they provide, one that all environmentally aware hotels ought to offer, is free filtered water, so there's no need to buy plastic bottles of water each day. The inviting M **Hospeder\u00eda El Marqu\u00e9s de San Jorge** (Cl. 14 No. 9-20, tel. 8\/732-0240, www.hospederiaelmarquesdesanjorge.com, COP$130,000-200,000 d) is just a block from the Plaza Mayor, has two interior patios that are filled with greenery, and has clean and comfortable modern rooms (despite having been around since 1972). It's a bargain compared to other luxury hotels in town.\n\n###### **OVER COP$200,000**\n\nThe location of the M **Hotel Plaza Mayor** (Cra. 10 No. 12-31, tel. 8\/732-0425, www.hotelplazamayor.com.co, COP$306,000 d), with a bird's-eye view of the Plaza Mayor from its western side, is unrivaled. The hotel's terrace is a great place to watch goings-on in the plaza and to take a photo of the cathedral bathed in a golden light in late afternoon. Rooms are spacious, some have a fireplace, and all are tastefully decorated. Breakfast is served in the pleasant courtyard.\n\nTwo other upscale options face parks. On the cute Parque Nari\u00f1o, the elegant **Hotel La Posada de San Antonio** (Cra. 8 No. 11-80, tel. 8\/732-0538, www.hotellaposadadesanantonio.com, COP$295,000 d) is lavishly decorated and has a pleasant restaurant area, a cozy reading room, a pool, an art gallery, a billiards room, and even a small chapel. It was originally a wealthy family's home built in 1845. On the Plaza de Ricaurte, the **Hotel Plazuela de San Agust\u00edn** (Cl. 15 No. 8-65, Plaza de Ricaurte, tel. 8\/732-2175, www.hotelplazuela.com, COP$300,000 d) is a cozy hotel with enormous (yet carpeted) rooms. Mornings get off to a nice start with breakfast in the courtyard, near a fountain. The hotel is two blocks from the Plaza Mayor. They have another hotel in the countryside towards Santa Sof\u00eda.\n\n##### **Food**\n\n###### **CAF\u00c9S, BAKERIES, AND QUICK BITES**\n\nThe many caf\u00e9s and bakeries in town have fiercely loyal clienteles. Bakery **Pan T\u00edpica** (Cl. 11 No. 11A-64) is an old local favorite and specializes in _mogolla batida_ (whipped bread). **Panader\u00eda San Francisco** (Cl. 10, 8am-7pm) is famous for its _galletas de maiz_ (corn cookies). The owners of **Pasteler\u00eda Francesa** (Cl. 10 No. 6-05, 9am-7pm Thurs.-Mon.), a French bakery, often skip town, so don't be disappointed if it is closed, foiling your plan to sip a caf\u00e9 au lait accompanied by a _pain au chocolat_ (chocolate croissant). **Panader\u00eda Do\u00f1a Aleja** (Cl. 14 No. 9-21, 8am-8pm Mon.-Sat., 9am-4pm Sun.) is known for its _mogollas_ (rolls).\n\nAt **Sybarita Caffe** (Cra. 9 No. 11-88, cell tel. 316\/481-1872, 8am-8pm daily) the owners are on a mission to bring coffee appreciation to the masses. Even if you are one of those \"coffee is coffee\" people, once you try one of their coffees (from the southern Colombian highland departments of Cauca or Nari\u00f1o), you may just be jolted out of your slumber. That's some good coffee! If you want your coffee from the Coffee Region, Quind\u00edo to be specific, then **Caf\u00e9 Los Gallos** (Cra. 8 No. 12-96, cell tel. 300\/659-9511, 9am-8pm daily) is your place. This sweet place is filled with rooster paraphernalia; it's named after the family name.\n\n**Gelateria Pizzer\u00eda Santa Lucia** (Cra. 10 No. 10-27, cell tel. 314\/305-8150, 11am-9:30pm) serves homemade ice cream and yogurts. It's all natural, and the pizza's not bad either. If you just want a thin crust pizza without a big production and expense try **Crepes Pizza y Algo M\u00e1s** (Cra. 9 11-80, cell tel. 313\/854-2051, www.crepespizza.blogspot.com, 6pm-10pm Mon.-Thurs., 1pm-10pm Fri.-Sun., COP$12,000).\n\n**Merengues y Besitos** (Cra. 9 No. 11-84, cell tel. 312\/394-3601, 10am-7pm daily) has very sweet sweets wrapped in colorful packaging.\n\n###### **COLOMBIAN**\n\nThe **Albahaca Restaurante-Bar Viejoteca** (Cra. 8A No. 13-46, cell tel. 313\/844-6613, 10am-9pm daily) is a favorite for visitors for two reasons: the lovely ambience, especially in the evening, and for its non-outrageous prices! Their top dishes include _cuchuco de trigo con espinazo de cerdo_ (buckwheat soup with pork back, COP$17,000) and grilled trout in _uchuva_ (Peruvian groundcherry) sauce (COP$18,000). Ask for a table in the garden or by the fireplace. The word _viejoteca_ is in the name because the owners like oldies music.\n\n**MiCocina** (Cl. 13 No. 8-45, tel. 8\/732-1676, www.restaurantemicocina.com, noon-10pm daily, COP$25,000), where there is a cooking school within the restaurant, has earned a name for itself as an ever-so-slightly upscale restaurant serving the best of Colombian cuisine. After a _calentado bogotano,_ a beloved hangover cure made with fried eggs and potatoes, save room for the cheese ice cream from Paipa. It's mostly Colombian meat-based dishes here, but they offer a few vegetarian plates.\n\nLocals tend to steer clear of the overpriced restaurants on the Plaza Mayor. When it comes to _comida,_ it's got to be _buena, mucha, y barrata_ (good, plentiful, and cheap). Close to the Terminal de Transportes, but not too close, **Los Kioscos de los Caciques** (Cra. 9 No. 9-05, cell tel. 311\/475-8681, noon-3pm and 6pm-8pm daily, COP$6,000) specializes in filling local dishes such as _mazamorra chiquita_ (beef stew with potatoes, corn, and other vegetables) and _cuchuco con espinazo_ (stew with a base of pork spine and potatoes). You can also order from the menu. It's an atmospheric place, where you dine in thatched kiosks. At the Saturday market, those in the know go to **Donde Salvador** (between Clls. 12-13 and Cras. 5-6, Plaza de Mercado) for _mute rostro de cordero,_ a hearty corn-based soup with lamb. You can also, of course, pick up plenty of cheap and fresh fruit. **La Parilla** (Cra. 9 No. 9-17, 7am-9pm daily, set lunch COP$5,000) is an everyman kind of place. At the plaza, **Estar de la Villa** (Cl. 13 No. 8-58, tel. 8\/732-0251, 10am-9pm daily, COP$8,000) is always packed, often with employees from some of the fancier restaurants nearby.\n\n**Traveling Taste Buds**\n\nHot, sweet, and gooey\u2014the arepas of Tinjac\u00e1 are worth both the calories and the trip.\n\nForget about counting calories as you try these local specialties near Villa de Leyva.\n\n**WINE**\n\nVilla de Leyva is one of a handful of areas in Colombia where wine is produced. Take a tour of **Vi\u00f1edo Aim Karim** (Km. 10 V\u00eda Santa Sof\u00eda, cell tel. 317\/518-2746, www.marquesvl.com, 10am-5pm, COP$5,000) and try their Marqu\u00e9s de la Villa wine. Their sauvignon blanc won an award in Brussels in 2011.\n\n**SAUSAGE**\n\nAbout 25 kilometers west of Villa de Leyva, the town of Sutamarch\u00e1n is famous for its spicy _longaniza_ sausage. The best place to sample this is at **La Fogata** (tel. 8\/725-1249, www.longanizasutamarchan.com). It's on the main road on the left as you go toward R\u00e1quira.\n\n**AREPAS**\n\nMost visitors to Colombia develop a love or hate relationship with arepas, corn-based pancakes that accompany just about every meal. Every region has their own distinct type of arepa, and every Colombian believes that theirs is superior to the rest. It would be hard to find anyone who could resist the famed _arepa quesuda_ from the town of **Tinjac\u00e1** about 18 kilometers southwest of Villa de Leyva. Meaning \"sweating arepa,\" _arepa quesudas_ are two small arepas with sweet, melted cheese in the middle. They're a big mess to eat, but they're so good.\n\n**JAM**\n\nTinjac\u00e1 is also known for its delicious jams made by **El Robledal** (Vereda Santa B\u00e1rbara, cell tel. 310\/226-5299, www.elrobledal.co). Check out their exotic fruit jams such as _uchuva, lulo,_ and rhubarb. Their products can also be found in Villa de Leyva at the Savia restaurant in the Casa Quintero on the Plaza Mayor.\n\n**BROILED HEN**\n\nS\u00e1chica is an orderly, quiet town just outside of Villa de Leyva on the way toward Tunja. Here, the local specialty is broiled hen. Try it at **La Candelaria** (Cl. 3 No. 2-48, cell tel. 311\/845-7786).\n\n###### **INTERNATIONAL**\n\n**Mercado Municipal** (Cra. 8 No. 12-25, tel. 8\/732-0229, noon-3pm Mon.-Thurs., noon-3pm and 8-midnight Fri.-Sun. and holidays, COP$22,000) has without a doubt one of the coolest settings in VDL. It is in a courtyard (which was once the third patio of a parsonage) filled with herb gardens in which a traditional Mexican barbecue wood-burning oven is built into the ground. In it they cook their famous barbecued goat. International dishes on the menu include pastas and several vegetarian offerings. It's open for breakfast on the weekends and there is a nice bakery in front. The set lunch special is a very good deal.\n\nAuthentic French food can be found in Villa de Leyva! That would be at **Chez Remy** (Cra. 9 No. 13-25, tel. 311\/848-5000, noon-10pm Fri.-Sat., noon-4pm Sun., COP$24,000). The French-inspired dishes include a _quenelle de mar_ (COP$28,000) that combines a myriad of tastes from the faraway sea: salmon, hake, shrimp, and lobster. But on chilly nights, the French onion soup (COP$9,000) really hits the spot.\n\nThe **Casa Quintero** on the corner of the Plaza Mayor has several restaurants in a sort of fancy food court setting. There is a little something for everyone here, including a Lebanese restaurant, a surprisingly filling arepa joint, and a pizza place. If you are in the mood for Mexican, try **La Bonita,** run by the same people as the Mercado Municipal, where you can sample delicious dishes such as _lomo a la tampiquena_ (COP$33,000), which includes grilled baby beef, a chicken flauta, a quesadilla, and rice with beans. Or go for a barbecued pork taco (COP$26,000).\n\nWhile you await your rosemary, veggie, _higo_ (fig), or barbecue burger (COP$8,500) and refreshing basil lemonade at **Vastago** (Cl. 13 No. 8-43, 9am-8pm daily, COP$17,000), you can check out the little shops and stands in a sort of arcade that has several small shops (ceramics, jewelry) and vendors (old Colombian magazines, antiques). There's a cupcake caf\u00e9 in back if you want something sweet afterwards.\n\n**La Ricotta** (Cra. 10 No. 11-49, tel. 8\/732-1042, noon-10pm Fri.-Sun., COP$16,000) makes its own pastas and is a reasonably priced Italian cuisine option.\n\n###### **VEGETARIAN**\n\n**Casa Salud Natural** (Cl. 12 No. 10-74, no phone, noon-9pm daily, COP$14,000) is a mostly lunch place where you get a set meal of a soup and a vegetable protein. A pricey vegetarian\/organic restaurant, **Savia** (Casa Quintero, Plaza Mayor, tel. 8\/732-1778, noon-9pm Thurs.-Mon., COP$25,000) has an extensive menu and also sells locally produced jams and other items for sale in their _tienda_ (store).\n\n##### **Information and Services**\n\nThe Villa de Leyva **tourist office** (corner Cra. 9 and Cl. 13, off of Plaza Mayor, tel. 8\/732-0232, 8am-12:30pm and 2pm-6pm Mon.-Sat., 9am-1pm and 3pm-6pm Sun.) has free tourist maps and brochures. There are several ATMs in Villa de Leyva, particularly along the southern end of the Plaza Mayor. Internet caf\u00e9s are also numerous.\n\nAn efficient and inexpensive laundry service in town near the bus terminal is **Lava Express** (Cra. 8 No. 8-21, cell tel. 320\/856-1865, 8am-noon and 2pm-7pm Mon.-Fri., 8am-7pm Sat.). They can provide rush service and pick up and return your items to your hotel.\n\nIn case of an emergency contact the **Polic\u00eda Nacional** (tel. 8\/732-1412 or 8\/732-0391) or the **Hospital San Francisco** (tel. 8\/732-0516 or 8\/732-0244).\n\n##### **Getting There and Around**\n\nWith a recently expanded four-lane highway that bypasses Tunja, Villa de Leyva is easily accessible by private car or by public bus from Bogot\u00e1, as well as from Tunja. It isn't a crazy idea to rent a car in Bogot\u00e1 and drive to Villa de Leyva. That gives you a lot of flexibility to be able to drive around the countryside and visit enchanting pueblos to your heart's content! Nearly all hotels have parking lots.\n\nThere are a few direct buses to Villa de Leyva from Bogot\u00e1 (COP$20,000); however, often it is quicker and easier to take a bus from the Terminal Norte to Tunja and then transfer to a _buseta_ (small bus) onward towards Villa de Leyva. These leave around every 15 minutes from the Terminal Villa de Leyva, not the main bus terminal. The last bus leaves Tunja at 7pm. This trip costs just COP$6,000.\n\nVice versa, the last bus bound for Tunja departs the **Terminal de Transportes** (Cra. 9 between Clls. 11-12) in Villa de Leyva at 6pm. It takes about 45 minutes.\n\nReturning to Bogot\u00e1, several companies offer two daily buses that depart between 5 and 6am and again at around 1pm. There are many more options on Saturdays, Sundays, and Monday holidays. These tend to leave in the late afternoon at around 3pm.\n\nTo get to Villa de Leyva from Bucaramanga or San Gil in Santander, you'll have to hop on a bus to Tunja. This highway that extends from Bogot\u00e1 to Venezuela is a busy one, and the journey can take five or six hours.\n\nOnce in Villa de Leyva, it is easy (and more importantly, a pleasure) to walk everywhere. A few streets around the Plaza Mayor, including the main drag, the Calle 13, are pedestrian only. Even on non-pedestrian streets it's hard for vehicles to zoom along.\n\n##### **Vicinity of Villa de Leyva**\n\n###### M **SANTUARIO FLORA Y FAUNA IGUAQUE**\n\nOne of the country's most accessible national parks is about 13 kilometers from Villa de Leyva. The **Santuario Flora y Fauna Iguaque** (www.parquesnacionales.gov.co, COP$37,500 non-Colombians, COP$14,000 residents in Colombia, COP$7,500 children and students) is an excellent place to experience the unique landscape of the Andean _p\u00e1ramo_ (highland moor) as well as dry tropical forest. The protected area extends for some 6,750 hectares. It is also a park of several _lagunas_ (mountain lakes). The Laguna Iguaque in particular is known as a sacred lake for the Muisca Indians who predominated in the area. According to their beliefs, the goddess Bachu\u00e9 was born out of the blue-green waters of this lake, giving birth to humanity.\n\nMost day-trippers based in Villa de Leyva visit the park to make the climb up to the Laguna Iguaque. The climb, which takes you through three ecosystems\u2014Andean forest, sub- _p\u00e1ramo_ , and _p\u00e1ramo_ \u2014begins at the Centro Administrativo Carrizal at an elevation of 2,800 meters (9,185 feet) and ends 4.6 kilometers (2.6 miles) later at the Laguna Iguaque (3,650 meters\/11,975 feet). The enjoyable hike takes about 3-4 hours to make. Along the way you may be able to spot different species of birds and perhaps some deer or foxes. At the mist-shrouded Laguna Iguaque, you'll be surrounded by hundreds of _frailejones,_ an unusual cactus-like plant found only in this special ecosystem.\n\nIt is best to make the hike during the week, as the trails get crowded on weekends. You do not need a guide for the hike to the Laguna Iguaque. During particularly dry spells the threat of forest fires forces the park to forbid entry to visitors. That is most likely to occur in January or August. Ask beforehand at your hostel or hotel to find out if the park is open to visitors.\n\nIf you are interested in exploring other paths in the park, consider overnighting at the **Centro de Visitantes Furachiogua,** the park's basic accommodations facilities (catering mostly to student groups). Seven rooms have 6-8 beds each (COP$38,000 pp), and the restaurant is open to day-trippers as well. This is about 700 meters beyond the Centro Administrativo Carrizal. There are camping facilities near the cabins (COP$10,000 pp). To inquire about accommodations or to make a reservation contact the community organization **Naturar-Iguaque** (cell tel. 312\/585-9892 or 318\/595-5643, naturariguaque@yahoo.es). A guided walk to the Laguna Iguaque costs COP$80,000 for a group of 1-6.\n\nBuses serving the town of Arcabuco from the bus station in Villa de Leyva can stop at the Casa de Piedra (8 km from Villa de Leyva). The first bus leaves at 6am with another departing at 10am. From there it's about three kilometers\/two miles (about an hour's walk) to the east to the Centro Administrativo Carrizal visitors center.\n\n###### **PALEONTOLOGICAL SITES**\n\nDuring the Cretaceous period (66-145 million years ago), the area around Villa de Leyva was submerged in an inland sea. Some of the marine species that lived in the area included the pliosaurus, plesiosaurus, and ichthyosaurus.\n\nTowards the end of this period, many species became extinct. Simultaneously the Andes mountains were created when the earth shifted. As the waters gave way to mountains, the bones of these species became imbedded in rock, guaranteeing their preservation. Today there are a handful of paleontological sites worth visiting, where you can view fossils of parts of massive dinosaurs to small ammonites, of which there are thousands. Excavations continue throughout the valley.\n\nIn 1977, locals made a fantastic discovery: a distant relative of carnivorous marine reptiles from the pliosaurus family, to be classified as a _Kronosaurus boyacensis Hampe._ It roamed this part of the earth some 110 million years ago. The first ever find of this species in the world can be seen, imbedded in the earth extending for about 10 meters, in the location of its discovery at the **Museo El F\u00f3sil de Monquir\u00e1** (Km. 4 V\u00eda Santa Sof\u00eda, Vereda Monquir\u00e1, www.museoelfosil.com, COP$6,000). Guides give a brief tour of the museum, which has hundreds of other animal and plant fossils on display. This is a major tourist sight, and there are souvenir shops and juice stands nearby.\n\nAcross the street from Museo El F\u00f3sil de Monquir\u00e1 is the **Centro de Investigaciones Paleontol\u00f3gicas** (cell tel. 314\/219-2904, www.centropaleo.com, 9am-noon and 2pm-5pm Mon., Wed., and Thurs., 8am-5pm Fri.-Sun., COP$8,000), which opened in late 2012. On view here are parts of a _Platypterygius boyacensis,_ as well as a _Callawayasaurus colombiensis,_ which were all found nearby. While the center is a museum as well, the main focus of this nonprofit organization is research. Technicians in lab coats and white gloves carefully work behind the glass preparing and preserving fossils (Fri.-Sun.). An informative 20-minute tour of the center in Spanish is included.\n\n**Gondava** (Km. 6 V\u00eda Santa Sof\u00eda, www.granvalle.com, 9am-5pm Tues.-Sun., COP$13,000) is more about fun than paleontology. This park is geared toward kids, taking visitors back in time, around 100 million years ago, when Earth was the domain of giant dinosaurs. This park has to-scale replicas of dozens of terrestrial and aquatic dinosaurs. The largest is the _brachiosaurus,_ which stands 14 meters (46 feet) high. Other park attractions include a playground, labyrinth, and a 3-D movie theater.\n\nOn the northeastern edge of town is the **Museo Paleontol\u00f3gico de Villa de Leyva** (Cra. 9 No. 11-42, tel. 8\/732-0466, www.paleontologico.unal.edu.co, 9am-noon and 2pm-5pm Tues.-Sat., 9am-3pm Sun., COP$3,000). Run by the Universidad Nacional, this museum provides an introduction to the fossils that have been found in the area, which date back 110-130 millions of years ago. On display are ammonites, which have become a symbol for the area, and other prehistoric animals that roamed the area. In addition, the museum has an arboretum with gardens of palms, oaks, and an Andean forest. This is behind the museum. It is about a 15-minute walk from the Plaza Mayor to the museum. On weekday mornings it can be a zoo with school groups being herded through.\n\n###### **CONVENTO DEL SANTO ECCE HOMO**\n\nSet idyllically atop a hill overlooking the town, the **Convento del Santo Ecce Homo** monastery (Cra. 6A No. 51A-78, tel. 1\/288-6373, 9am-5pm Tues.-Sun., COP$5,000) was founded by the Dominicans on Palm Sunday in 1620. The monks were evicted on several occasions from this, their beautiful home. During the struggle for independence it was taken over by rebel troops under the command of a French general in 1816. In a matter of weeks the Spaniards seized it. Following definitive independence, President Santander, not a huge fan of the church, annexed the convent and ordered it to be used as a school. It was finally recuperated by the Dominicans in 1868.\n\nThe monastery is a delight to visit. Believers may find inspiration here; non-believers will appreciate the quiet beauty of its setting. In addition to the church, there are two striking small baroque chapels, plus a museum. Part of the museum is dedicated to indigenous cultures in the area. A monk's cell, library, and dining hall area provide a glimpse into monastery life. Surrounded by stone columns, the courtyard is awash in a rainbow of colors, with flowers always in bloom. Across the street from the monastery is a guesthouse and camping area open to the public.\n\nFor those feeling energetic, the trip up to the monastery makes for a great bike ride from Villa de Leyva. Be sure to get an early start and stop off at the **Cabanita Roja** (Vereda Barbilla, 1 km before the monastery, cell tel. 321\/211-9653) for a pick-me-up snack to give you the stamina to conquer that last long hill.\n\n###### **HIKING**\n\nA popular hike for those seeking a thrill is the **Paso de \u00c1ngel** near the town of **Santa Sof\u00eda.** It's about six kilometers from town. The thrill comes when you walk along a narrow precipice between two canyons. You can take public transportation to Santa Sof\u00eda and walk to the Paso de Angel. **Colombian Highlands** (Renacer, Av. Cra. 10 No. 21, tel. 8\/732-1201, cell tel. 311\/308-3739, www.colombiahighlands.com) offers tours to the area and has maps if you'd like to do this on your own. You can take public transportation to Santa Sof\u00eda and walk to the Paso de Angel.\n\nThe popular **La Periquera Waterfalls** can be reached by taking a _colectivo_ (small bus) bound for Gachantiva and getting off at El Uvalito. There are several falls, but the most impressive has a drop of 15 meters. The entrance to the falls is about 11 kilometers (7 miles) from town. This trip can be easily made on bike as well. A restaurant at the entrance serves snacks.\n\n###### **GETTING THERE AND AROUND**\n\nThe environs of Villa de Leyva can be visited on bike, by taxi, or by public transportation. **CoomultransVilla** has hourly buses in the mornings, more or less, from the Terminal de Transportes in Villa de Leyva to Santa Sof\u00eda (COP$2,500) departing at 6:45am, 8am, 9am, and 10am. They can let you off within easy walking distance of all the sights (Museo El F\u00f3sil de Monquir\u00e1, Convento del Santo Ecce Homo, etc.), as well as Santa Sof\u00eda if you are interested in the Paso de \u00c1ngel walk. Check the opening hours of the sights you'd like to visit so that you won't have to wait around to enter. There are also buses in the afternoon. The last bus departing Santa Sof\u00eda bound for Villa de Leyva leaves at around 4. You'll have to be on the lookout for it and flag it down. This company also can take you to the Periquera waterfalls (take the Gachantiva bus), fairly close to Iguaque (Arcabuco bus), and to R\u00e1quira. It's easy and cheap. It's best to confirm all the bus schedules in advance.\n\n#### **R\u00c1QUIRA**\n\nThis town, 28 kilometers (17 miles) away from Villa de Leyva, is synonymous with _artesan\u00edas_ (handicrafts). The main drag is lined with colorful shops, where in a one-stop shopping frenzy you can pick up **handicrafts** of every size and shape and from all across the country (and even from China): hammocks, _mochilas_ (hats or handbags), and row after row of trinkets.\n\nIt's easy to be turned off by the trinket shopping scene, but there is some authenticity to be found in this little town. R\u00e1quira is the capital of Colombian **ceramics** and has been since before the arrival of the Spaniards. In fact, it is said that the name R\u00e1quira means \"city of clay pots\" in the Chibcha language of the Muiscas, who lived in the area. All of those reddish flower pots and planters you may have seen throughout the country most likely came from here.\n\nSeveral shops in town specialize in pottery, and behind the shops you will often see trucks being carefully loaded up with pottery as they fulfill orders from across Colombia and beyond. One big shop within two blocks of the pleasant Parque Principal (Cras. 3-4 and Clls. 3-4) where you can peruse aisles upon aisles of the pottery is **Todo R\u00e1quira** (Cra. 5 No. 3A-05, tel. 8\/735-7000, www.todoraquira.com, 9am-6pm daily). The front of the store is filled with a variety of handicrafts, but if you meander to the back, you'll see their pottery factory and can check out bowls, flowerpots, and other items. Many designs are elegant in their simplicity, like the many square planters. It is said that some 500 families in the area make their living harvesting the clay in nearby areas or firing the pottery in their own workshops. A dwindling number of women in the area do things the old-fashioned way\u2014with their hands. They make mostly decorative items like candlestick holders and clay hen piggybanks. These aren't perfect, but they have so much character.\n\nR\u00e1quira is Colombia's ceramics capital.\n\nIf you'd like to observe the ceramic-making process you can visit the _taller_ (workshop) of **Isaias Valero** (cell tel. 310\/774-5287, COP$5,000 suggested donation). You can watch him at work, and he can show you the multi-day steps that go into to the process. If he is there, Isaias will gladly welcome your visit. To get to the workshop, walk up about 68 steps on the right just before the Casa de la Cultura.\n\n##### **Convento de la Candelaria**\n\nOne of the oldest monasteries in Latin America, and one that is still in use today, is the **Convento de la Candelaria** near R\u00e1quira. A pair of Augustinian missionaries arrived in this desert area in 1588 with the mission of bringing Christianity to the native Muisca people. They lived in caves (which you can visit) until the monastery was constructed.\n\nThe complex includes two cloisters, which hold a chapel and a museum. The museum is a hodgepodge of religious art and objects, examples of technological advances through the years\u2014from a _reloj borracho_ (drunken clock) to an early Apple computer\u2014and a display on the Colombian saint Ezequiel Moreno y D\u00edaz, who is said to have healed cancer victims.\n\nYou may see a handful of young soon-to-be priests doing upkeep, often softly singing, in one of the colorful courtyards. They live in one part of the monastery with older monks in another.\n\nAdjacent to the monastery is a modern hotel, **Posada San Agust\u00edn** (vocaciones@agustinosrecoletos.com.co, COP$168,000 d), which often hosts yoga and meditation retreats. Rooms are immaculately clean and completely free from clutter. Some even have hot tubs. At nighttime you can sit around a fire in the common area and sip hot, spiced wine. Meals are served in the restaurant, and they can also prepare vegetarian food. It's a quiet and peaceful place and there are some walks you can do nearby, one leading to a waterfall (a 1.5-hour walk).\n\n##### **Getting There**\n\nBuses from Villa de Leyva to R\u00e1quira depart from the Terminal de Transportes, leaving between 7am and 8:30am. You may, however, prefer to hire a cab for the day, especially if you're traveling in a small group. Cab drivers typically charge COP$80,000 (for the car) to visit the Convento de la Candelaria and R\u00e1quira, for example, with a couple of stops along the way. They will _esperar_ (wait) for you to visit each stop. Be specific about where you want to go and the price at the beginning (put things in writing) to avoid unpleasant surprises later. If you are in a group you may want to upgrade to a minivan. Contact **Transportes G &J** (cell tel. 313\/259-0589 or 311\/746-8434) for a minivan.\n\nIf you'd like to visit the monastery, you can rent a taxi from R\u00e1quira, which will cost about COP$20,000-30,000 round-trip. The trip takes about 15 minutes each way to make. (The taxi driver will wait for you while you visit it.) You can also take public transportation from the Parque Principal, and the bus can leave you at the intersection with a dirt road that leads to the monastery. Buses (COP$2,000) are infrequent, however. You can also walk to the monastery from town. It's a pretty seven-kilometer (four-mile) walk through the hilly countryside to reach the monastery from there.\n\n#### **TUNJA**\n\nThis university town (pop. 178,000), home to the Universidad de Boyac\u00e1, boasts some spectacular churches. Make sure you arrive during church visiting hours, as the city does not have much else to offer. As there are frequent bus connections with Bogot\u00e1 and Santander, Tunja is a good base from which to explore Boyac\u00e1.\n\n##### **Sights**\n\nEverything you need to see in Tunja is located in its _centro hist\u00f3rico._\n\n###### M **HISTORIC CHURCHES**\n\nTunja is a city of churches, with over a dozen that date to colonial times. Hours of visitation can be irregular, but they are always open for mass, which is a good time to take a look. Most churches celebrate mass at about 7am and 6pm daily, with more frequent masses on Sundays. There tend to be more churches open for visitation in the mornings (8am-11:30am) than in the afternoons.\n\nOn the eastern side of the **Plaza de Bol\u00edvar** (Cl. 19 at Cra. 9), **Catedral Santiago de Tunja** (Cra. 9 at Cl. 19) is a 16th-century construction, originally built out of wood and earthen _tapia pisada,_ which is an adobe technique. It was the first cathedral to be built in Nueva Granada. It has three naves, four side chapels, and two front chapels.\n\n**Santa Clara La Real** (Cra. 7 No. 19-58, Cl. 21 No. 11-31, tel. 8\/742-5659 or 8\/742-3194, 8am-11:30am and 3pm-4:30pm Mon.-Fri., 8am-11:30am Sat., masses 7am and 5pm Mon.-Sat., 7am, 11am, and 5pm Sun.) was built between 1571 and 1574 and was the first Clarisa convent in Nueva Granada. It has one nave with spectacular gold decorations adoring its presbytery with golden garlands, grapes, pineapples (which were a sacred indigenous symbol), pelicans, an anthropomorphic sun, and other symbols of nature. Also look for the seal of Tunja, the double-headed eagle, modeled on the seal of Emperor Charles V, who gave the city its charter. In the choir is the tiny cell where Madre Josefa del Castillo lived for over 50 years in the late 17th and early 18th centuries. From there she wrote two books and several poems, with themes of sexual repression and mystical descriptions of heaven and hell. Near her cell are some frescoes made with coal, an abundant resource in the area. The adjacent Convento Santa Clara La Real was undergoing a painstakingly careful restoration at the time of writing.\n\nThe sky-blue interior of the **Iglesia de Santa B\u00e1rbara** (Cra. 11 No. 16-62, between Clls. 16-17, tel. 8\/742-3021, 8:30am-12:30pm and 2pm-6pm daily, masses 5:30pm and 6pm Mon.-Fri., 7am, 9am, 10am, and 11am Sat., noon, 5pm, 6pm, and 7pm Sun.) and Mudejar ceiling designs make this one of the prettiest churches in Tunja. The single-nave structure, with two chapels making the form of a cross, was completed in 1599. When it was built, it was raised at the edge of Tunja, near an indigenous settlement.\n\nBuilt in the 1570s, the **Templo de Santo Domingo de Guzm\u00e1n** (Cra. 11 No. 19-55, tel. 8\/742-4725, 8am-11:30am Mon.-Fri., 7am and 6pm masses Mon.-Fri., 7am and 6pm masses Sat., 7am, 10am, noon, and 6pm masses Sun.) is one of the most elaborately decorated churches in Colombia. Visitors have been known to audibly gasp at their first sight of the spectacular Capilla del Rosario, a chapel constructed of wood painted in red and gold-plated floral designs. It's often dubbed the Sistine Chapel of baroque art in Latin America. Figures of El Nazareno and El Jud\u00edo Errante are part of the collection of paintings and woodcarvings in this church with several chapels. If you have time to visit just one church in Tunja, make it this one.\n\nThe **Claustro de San Agust\u00edn** (Cra. 8 No. 23-08, tel. 8\/742-2311, ext. 8306, www.banrepcultural.org\/tunja, 8:30am-6pm Mon.-Fri., 9am-1pm Sat., free) dates to the late 16th century. It served as an Augustinian convent until 1821, when it was taken over by the government. The friars were sent to another convent, and the building would become the home of the Colegio de Boyac\u00e1 and later transferred to the Universidad de Boyac\u00e1. Adorning the corridors around the patio are several murals dating back to the colonial era. The _claustro_ (cloister) is administered by the Banco de la Rep\u00fablica, and they often hold cultural events here. You can settle down with a book or work on your computer in the pleasant reading rooms.\n\nOther religious sights worth visiting include the 17th-century **Iglesia San Ignacio** (Cra. 10 No. 18-41, tel. 8\/742-6611, 8am-noon and 2pm-5pm Wed.-Sat.), which now serves as a theater, and the **Templo y Convento San Francisco** (Cra. 10 No. 22-32, tel. 8\/742-3194, 10:30am-12:30pm and 3pm-5:30pm daily, 7am, 11am, noon, and 7pm mass Mon.-Fri., 11am, noon, 6pm, and 7pm mass Sat., 8am, 10am, 11am, noon, 5pm, 6pm, and 7pm mass Sun.), one of the oldest churches and monasteries in Tunja. It was an important base for evangelization of nearby indigenous communities.\n\na decorative church ceiling\n\n###### **MUSEUMS**\n\nThe Mudejar-Andalusian style **Casa del Fundador Gonzalo Su\u00e1rez Rend\u00f3n** (Cra. 9 No. 19-68, Plaza de Bol\u00edvar, tel. 8\/742-3272, 8am-noon and 2pm-6pm daily, COP$3,000) was built in the middle of the 16th century. The most remarkable aspects of the house are the frescoes of mythological creatures, human figures, exotic animals, and plants. These whimsical paintings date from the 17th century, although not much else is known about them.\n\n**Casa del Escribano del Rey Don Juan de Vargas** (Cl. 20 No. 8-52, tel. 8\/74-26611, 9am-noon and 2pm-5pm Tues.-Fri., 9am-noon and 2pm-4pm Sat.-Sun., COP$2,000) was owned by an important person in colonial Tunja, the scribe to the king. The scribe's jurisdiction covered all of present-day Boyac\u00e1, Santander, Norte de Santander, and parts of Venezuela and Cundinamarca. Student guides will give you a thorough tour of the museum. The house showcases furniture and other examples of colonial life, but the highlight of this Andalusian-style house has to be the unusual painted ceilings portraying exotic animals and mythological creatures, similar to the frescoes that can be found in the Casa del Fundador.\n\nThe childhood home of former president Gustavo Rojas Pinilla is now a museum: **Museo Hist\u00f3rico Casa Cultural Gustavo Rojas Pinilla** (Cl. 17 No. 10-63, tel. 8\/742-6814, 8:30am-noon and 2pm-6pm Mon.-Fri.). Rojas, after seizing power in 1953, became the only dictator that Colombia has ever had. Upstairs are two exhibition spaces, one with memorabilia of Rojas and the other with portraits of 12 presidents that hailed from Boyac\u00e1. Despite his anti-democratic credentials, Rojas is revered in Tunja as the man who brought the mid-20th-century violence between Liberals and Conservatives to an end.\n\n##### **Shopping**\n\nSmall **7 Kuero's** (Cl. 23 No. 9-90, tel. 8\/743-7328, josekueros@hotmail.com, 9am-5pm Mon.-Sat.) is a shop specializing in leather goods. Next to the cathedral, the **Sombrer\u00eda Richard** (Cr. 9 No. 19-06, tel. 8\/747-1276, daily) is an old-school hat shop selling hats that seem to be from the 1950s.\n\n**Unicentro** (Av. Universitaria No. 39-77, tel. 8\/745-4108, 8am-11pm daily) is Tunja's mall, holding a movie theater, food court, and a _plazoleta de caf\u00e9s,_ an area with many different caf\u00e9s. It is located in the modern neighborhood of La Pradera, to the north of downtown.\n\n##### **Accommodations**\n\nMost overnight visitors to Tunja stay in the decent hotels in the _centro hist\u00f3rico_ within easy walking distance of the Plaza de Bol\u00edvar and sights of interest.\n\nTwo blocks from the Plaza de Bol\u00edvar is M **Hotel Casa Real** (Cl. 19 No. 7-65, tel. 8\/743-1764, www.hotelcasarealtunja.com, COP$58,000 s, COP$72,000 d), which is a colonial-style house with 10 rooms surrounding a divine courtyard. That's where a very nice breakfast is served for an additional cost in the morning. You can order your breakfast the night before and even request it to be delivered to your room. Rooms are tastefully decorated and comfortable. Prices here are astoundingly low. The courtyard walls are decorated with lovely tile paintings depicting Boyac\u00e1 country scenes. The artist, **Adriano Guio** (cell tel. 314\/319-0822, aguio1@hotmail.com), has a studio near the town of Nobsa.\n\nAcross the street from Hotel Casa Real is the **Hotel Ocet\u00e1** (Cl. 19 No. 7-64, tel. 8\/742-2886, www.hotelocetatunja.com, COP$90,000 d). It opened in 2012 and is clean and functional; beds are firm.\n\nWith the same owners as Hotel Casa Real, **Hotel Alicante** (Cra. 8 No. 19-15, tel. 8\/744-9967, www.hotelalicantetunja.com, COP$72,000 d) caters to business clientele. This small hotel may not have the charm of Casa Real, but it is clean and matter of fact.\n\nThe fancy hotel in town is, as it has been for decades, the **Hotel Hunza** (Cl. 21A No. 10-66, tel. 8\/742-4111, www.hotelhunza.com, COP$228,000 d). It's got luxurious king-size beds and card keys to get in. Amenities include a decent sized indoor pool and a steam room. Its neighbor is the Iglesia Santo Domingo, which makes for a strange view. The hotel is a popular place for wedding banquets. There is a lively bar near the entrance, but it shouldn't keep you up at night.\n\n##### **Food**\n\n_Comida t\u00edpica_ (Colombian fare) rules the day in this city lacking in restaurant options. For a really local, greasy-spoon-type experience, try **Restaurante Maizal** (Cra. 9 No. 20-30, tel. 8\/742-5876, 7am-8:45pm Mon.-Sat., 9am-4:45pm Sun., COP$12,000). It has been serving _sancocho_ (beef stews), _mondongo_ (tripe stew), and _ajiaco_ (chicken and potato soup) to Tunja for over 50 years. Another old-timer is **El Bodeg\u00f3n Express** (Cra. 10 No. 18-45, cell tel. 321\/221-4460, 8am-4pm Mon.-Sat., COP$10,000). It's next to the Iglesia San Ignacio. It specializes in trout dishes and _cocido boyacense_ (COP$6,000), which has a variety of meats and some of the unusual tubers from the area, such as _cubios, ibias,_ and _rubas._\n\nM **Empanadas de Mi Tierra** (Cra. 10 No. 17-67, cell tel. 320\/414-0857, 10am-7pm daily) is a fast food joint that has 15 types of empanadas, with varieties such as Mexican, Asian, vegetarian, and cheese with quail's eggs. And you can douse them with many types of sauces. The empanadas go down well with a cool _avena cubana,_ a creamy and cold oatmeal drink.\n\n**Pizza Nostra** (Cl. 19 No. 10-63, tel. 8\/740-2040, 11am-8pm daily, COP$18,000) has a few locations in and around town. The most famous one is at the Pozo de Donado (tel. 8\/740-4200, 11am-11pm daily), a small park and Muisca archaeological site surrounding a lake.\n\nIt's a tradition in Tunja to while away the hours in caf\u00e9s. It must be the chilly weather. While the actual coffee around town may disappoint, the atmosphere, with groups of retirees dressed in suits brushing shoulders with bevies of college students, does not. Put some _aguardiente_ (anise-flavored liquor) in your coffee and enjoy the great view from the second floor of **Caf\u00e9 Tabore** (Cl. 19 No. 9-57, tel. 8\/742-2048, 7am-8pm daily), overlooking the Plaza de Bol\u00edvar. The lively Pasaje Vargas on the west side of the plaza is lined with several caf\u00e9s. Try **Capital Caf\u00e9** (7am-7pm daily), which is at the entrance of the _pasaje_ (passage) close to the Plaza de Bol\u00edvar. **Caf\u00e9 San Rafael** (Cra. 11 No. 18-35, no phone, 8:30am-8pm Mon.-Sat.) is a tad more elegant than the rest.\n\n##### **Getting There and Around**\n\nSituated 150 kilometers northeast of Bogot\u00e1 and 21 kilometers southeast of Villa de Leyva, Tunja is easy to get to by car or by bus. Buses to Bogot\u00e1, other towns in Boyac\u00e1, and to all major cities in Colombia depart from the **Terminal de Transportes** (Cra. 7 No. 16-40). Buses from Bogot\u00e1 cost about COP$18,000 and from Villa de Leyva are COP$6,000. The best way to get around the _centro hist\u00f3rico_ is on foot.\n\n##### **Puente de Boyac\u00e1**\n\nThis war memorial on the road between Bogot\u00e1 and Tunja celebrates a decisive battle, the **Batalla del Puente de Boyac\u00e1,** that effectively ended Spanish control of Nueva Granada. At this site today there are several memorials and statues, including the Plaza de Banderas, where flags from all the departments of Colombia fly. There is also a sculpture of Gen. Francisco Paula de Santander and a large sculpture of Gen. Sim\u00f3n Bol\u00edvar surrounded by angels representing the South American countries that he liberated (Bolivia, Colombia, Ecuador, Peru, and Venezuela). There is a small bridge on the memorial grounds, but this is from the 1930s; the original Puente de Boyac\u00e1 long gone.\n\nSantander and Bol\u00edvar achieved immortality as heroes of Colombian independence for their victory here. After defeating the Spaniards at the Batalla del Pantano de Vargas on July 25, 1819, revolutionary troops under their command marched toward Bogot\u00e1. South of Tunja, they engaged with the main Spanish army, defeating it decisively on August 7 at the Batalla del Puente de Boyac\u00e1. The engagement was a small affair with fewer than 3,000 men on each side, with about 100 royalists and only 13 rebels losing their lives. Bol\u00edvar marched onward to Bogot\u00e1, which he took without a fight, ushering in independence.\n\nAt 6pm every day there is a short flag-lowering ceremony. You can have your picture taken with Colombian soldiers. Buses passing through between Bogot\u00e1 and Tunja can drop you off here, or you can contract a taxi from Tunja.\n\n#### **TUNJA TO LAGO TOTA**\n\n##### **Paipa**\n\nThe town of Paipa (pop. 32,000) is known for two things: its thermal baths and the Lanceros monument.\n\n###### **COMPLEJO TERMAL**\n\nThe **Complejo Termal** (Km. 4 V\u00eda Pantano de Vargas, tel. 8\/785-0068, www.termalespaipa.co, 6am-10pm daily), the hot springs, is the biggest attraction in town. There are two parts to the hot springs complex; the **Parque Acu\u00e1tico** (COP$13,000) has three thermal water pools and pools for kids, and the **Centro de Hidroterapia** (COP$41,000) offers six activities for 20 minutes each, including hydro-massage, saunas, and mudbaths. On the weekends the park is packed, especially the Parque Acu\u00e1tico, which teems with families, so it's highly recommended to go on a much calmer weekday. To do the Centro de Hidroterapia circuit, call in advance to make a reservation. Slots are available daily at 7am, 10am, 1pm, 4pm, and 7pm. A limited number of people, around 16, are allowed during each session.\n\nBetween Paipa and the Complejo Termal is the manmade **Lago Sochagota.** The pedestrian path around the lake attracts locals and visitors alike, especially in the late afternoon. This is a nice thing to do after a good soak at the baths.\n\n###### **LOS LANCEROS**\n\nOn the site of the **Batalla del Pantano de Vargas** (Battle of the Vargas Swamp) stands Colombia's largest sculpture, **_Los Lanceros_** (9 km south of Paipa on Paipa-Pantano de Vargas road, free). The massive monument was designed by Colombian sculptor Rodrigo Arenas Betancourt and built in commemoration of 150 years of Colombian independence. Bronze sculptures show the 14 _lanceros_ (lancers on horseback) charging into battle, fists clenched in the air, with fear and defiance depicted on their faces. Above them is an odd triangular concrete slab that points into the heavens. It is 36 steps up to the platform of the monument, the age of Sim\u00f3n Bol\u00edvar on that fateful day. The monument generates strong opinions, and there is little gray area: You either love it or detest it.\n\nThe Batalla del Pantano de Vargas was a decisive battle during Sim\u00f3n Bol\u00edvar's independence march on Bogot\u00e1 in 1819. After crossing the Llanos from Venezuela and climbing up the Andes via the P\u00e1ramo de Pisba, the revolutionary army under Bol\u00edvar engaged with a contingent of Spanish troops at the Pantano de Vargas on July 25, 1819. Exhausted after their long slog over the Cordillera Oriental mountains, the revolutionary troops were nearly defeated. However, a charge by 14 armed horsemen led by Juan Jos\u00e9 Rend\u00f3n saved the day. Soldiers from the British Foreign Legion, under the command of Irishman James Rooke, also played a decisive role in this battle. The royalists lost 500 men in the battle, while 350 revolutionaries perished.\n\nAcross from the monument is the **Casa Museo Comunitario Juan Vargas** (COP$1,000), a small museum mostly about the military campaigns of Sim\u00f3n Bol\u00edvar. It was in this house that Juan Vargas, his wife, and their 12 children were executed by the Spaniards for supporting the rebel troops. Oddly, there is not much information on the battle that occurred in the swamp across the street, but it's worth a quick look anyhow.\n\nthe memorial of the Batalla del Pantano de Vargas, _Los Lanceros_\n\n###### **ACCOMMODATIONS AND FOOD**\n\nWhile there are some inexpensive hotels in town, it's much more pleasant to stay outside of town near the Complejo Termal, even if it means splurging somewhat. Midweek rates at these fancy hotels drop substantially.\n\nThe M **Hacienda El Salitre** (Km. 3 V\u00eda Paipa-Toca, tel. 8\/785-1510, www.haciendadelsalitre.com, COP$350,000 d) is set in the countryside under towering eucalyptus trees, and you'll pass grazing cows to get there. At the hotel, go for one of the rooms with a thermal bathtub. Each day you'll be treated to a thermal bath three times (staff come in and change the water each time). Rooms are cozy, warm, and spacious, but not quite luxurious. The hotel has a very nice restaurant with outdoor seating, plus a caf\u00e9 and a bar. Even if you are just passing through, the restaurant, with its lovely setting, is the best around. Here you can get a massage, and can take a horse out for a trot to a nearby lake. From Sunday to Friday there is a 30 percent discount. It's a big wedding banquet and honeymoon location on the weekends. The hacienda served as a barracks during the Pantano de Vargas battle in 1819.\n\nOverlooking Lago Sochagota is the **Estelar Paipa Hotel y Centro de Convenciones** (tel. 8\/785-0944, www.hotelesestelar.com, COP$286,000 d). This upmarket chain hotel is modern, service oriented, and well maintained. On site is a spa with thermal baths, the main attraction, and there are other activities on-site to keep you busy, such as a pool, tennis court, golf course, and horseback riding. With over 100 rooms, it is a popular place for large groups. Rates here don't substantially drop during the week, but it may be worth asking.\n\n###### **GETTING THERE AND AROUND**\n\nPaipa is 40 kilometers (25 miles) northeast of Tunja, and frequent buses make this route. The half-hour journey from Tunja costs COP$5,000. From Paipa, taxis to the Complejo Termal area cost about COP$7,500. Buses from Sogamoso depart from the intersection of Carrera 14 and Calle 16 and cost about COP$4,000. The trip takes about 45 minutes.\n\n##### **Sogamoso**\n\nThis city of over 100,000 habitants is about 75 kilometers (46 miles) east of Tunja and is known for being an important pre-Hispanic Muisca center. It was known as Suamoxi. It's a city of little charm; however, the Museo Arqueol\u00f3gico de Sogamoso is worth a stop.\n\n###### **SIGHTS**\n\nRun by the Universidad Pedag\u00f3gica y Tecnol\u00f3gica de Colombia (UPTC) university in Tunja, the **Museo Arqueol\u00f3gico de Sogamoso** (Cl. 9A No. 6-45, tel. 8\/770-3122, 9am-noon and 2pm-5pm Mon.-Sat., 9am-3pm Sun. and holidays, COP$5,000) has an extensive collection of artifacts of the Muisca civilization, the main indigenous group of Colombia. Muiscas lived in the area that is today the departments Boyac\u00e1, Santander, and Cundinamarca. Suamoxi was the seat of power for a confederation led by the Iraca. The Bacat\u00e1 confederation (near Bogot\u00e1) and the Hunza (Tunja) confederation, led by the Zaque, were the most powerful of the Muisca confederations. The most memorable sight on the museum grounds is the fantastic Templo del Sol, a re-creation of a Muisca temple that was burnt to the ground by the Spaniards in the late 16th century. The museum is worth visiting, even though the exhibition spaces are drab and the sequence of exhibits does not flow very lucidly. That is a shame because there is an interesting history to tell and the collection is impressive. If you have the time and speak Spanish, hire a guide. Inquire at the ticket office. Look for the exhibit on _ocarinas,_ which are whistles, usually ceramic, that are often zoomorphic in form. Also see the stunning black-and-white geometric designs of _torteros,_ which are spindles used in spinning yarn, as well as remarkably well-preserved red-and-white ceramic vessels and urns.\n\n###### **ACCOMMODATIONS AND FOOD**\n\nThe M **Hotel Finca San Pedro** (Km. 2 V\u00eda Aquitania, tel. 8\/770-4222, www.fincasanpedro.galeon.com, COP$25,000 dorm, COP$80,000 d) is a lush hostel set among lovely gardens with fruit trees, vegetables, and flowers. (It's odd that there's no fruit at breakfast!) It's a popular place with international backpackers visiting Boyac\u00e1. The owner's son gives yoga classes on occasion. This friendly spot, a five-minute cab ride from Sogamoso (COP$4,000), is the best option in the area.\n\nIt is said that Bol\u00edvar stayed at the **Hacienda Suescun** (Km. 4 V\u00eda Sogamoso-Tibasosa, tel. 8\/779-3333, cell tel. 312\/596-4506, www.haciendasuescunhotel.com, COP$164,000 d) before he headed off to face the Spaniards at the decisive Batalla del Puente de Boyac\u00e1. This _hacienda,_ surrounded by tall trees covered with Spanish moss dangling towards the ground, has 18 rooms. It's north of Sogamoso. They have horses that can be taken out for a ride in the countryside.\n\n###### **GETTING THERE**\n\nThere are easy bus connections between both Bogot\u00e1 (3 hrs., COP$23,000) and Tunja to Sogamoso (1 hr., COP$15,000). The **Terminal de Transportes** (Cra. 17 between Clls. 11-11A, tel. 8\/770-330) is downtown. Many buses connect Sogamoso with Paipa, Iza, Mongu\u00ed, and Aquitania. Sogamoso is also a gateway to Los Llanos, with frequent bus service between Sogamoso and Yopal. This is a good, less expensive option for traveling to the Hacienda La Aurora south of Yopal. The trip between Sogamoso and Yopal takes about four hours to make and costs about COP$25,000.\n\n##### **Mongu\u00ed**\n\nThe chilly (average temperature is 13\u00b0C\/55\u00b0F) highland colonial town of Mongu\u00ed was founded in 1601 and was a strategic town for the Spaniards, as it was located between Tunja and the vast Llanos, the eastern plains. It has been designated as one of the most beautiful towns in Boyac\u00e1. Its narrow cobblestone streets are lined with white and green houses, many well over a couple of centuries old. It's in a valley below the highland moor of **P\u00e1ramo de Ocet\u00e1,** dubbed the most beautiful _p\u00e1ramo_ in the world. You can decide for yourself by hiking among its armies of _frailej\u00f3n_ plants, mountain lakes, and enormous boulders. Hotel staff can contact a knowledgeable local guide (around COP$50,000) to accompany you through this unusual landscape.\n\n###### **SIGHTS**\n\nThree colonial constructions in Mongu\u00ed have been declared national monuments. The stone **Bas\u00edlica y Convento de Nuestra Se\u00f1ora de Mongu\u00ed** stand on the Plaza de Bol\u00edvar. The Franciscan convent today houses a Museo de Arte Religioso highlighting the work of the famous 17th-century Colombian baroque painter Gregorio V\u00e1squez de Arce y Ceballos. Other historic buildings are the **Capilla de San Antonio de Padua,** which was the town's first church, and the photogenic stone bridge, the **Puente de Calicanto.**\n\nToday Mongu\u00ed is almost as famous for its soccer ball-making industry as for its colonial beauty. Around 70 percent of the town works in about 20 small factories in this industry, which has been around since the 1950s. They churn out some 30,000 balls each month. (More balls are produced during World Cup years as demand tends to spike.) You can pick up a \"Made in Mongu\u00ed\" soccer ball at the shop **Balones Hurtado** (Cl. 7 No. 3-60, tel. 8\/778-2021, www.baloneshurtado.com). Their slogan is \"more than a ball...inspiration for your feet!\"\n\n###### **ACCOMMODATIONS AND FOOD**\n\nThere are not many accommodations options in and around town, but one of the best is **La Casona de San Francisco de Asis** (Cra. 4A No. 3-41, tel. 8\/778-2498, COP$40,000 pp d). Rooms have a view over the R\u00edo Morro canyon, and the hotel is quite tidy. The restaurant, which has been in service for over two decades, is also one of the best in town. The restaurant specializes in _cocido boyacense,_ which has a variety of meats and some of the unusual tubers from the area, such as _cubios, ibias,_ and _rubas._\n\nThe **Calicanto Real** (cell tel. 311\/811-1519, juliosaenz66@hotmail.com, COP$25,000 pp) is an old house with five rooms located on the other side of the Puente de Calicanto. It was once owned by a wealthy emerald miner, was abandoned for several years, and now it is has been refurbished as a hotel. Rooms are spacious with nice views and have a lot of character, but the beds are soft. Adjacent to the hotel is a tavern filled with decorations like cowboy hats, animal heads, and an homage to Mongu\u00ed's most famous poet: Mora S\u00e1enz R\u00f3mulo, also known as \"El Indio Romulo.\" Born in the 1930s, the charismatic poet became famous nationwide for defining a genre of campesino poetry, commenting on social ills using the language of rural folk. He recorded several albums of poetry, was awarded dozens of medals for his contributions to cultural life, and even served as mayor of his hometown.\n\nThe **Hospedaje el Rinc\u00f3n de Duzgua** (Vereda Duzgua, tel. 8\/778-2130) is a cabin outside of town, and it's a good place for exploring the P\u00e1ramo de Ocet\u00e1.\n\n###### **GETTING THERE**\n\nThere are two roads between Sogamoso and Mongu\u00ed. The old but scenic route is partly unpaved and winds through eucalyptus forests and the pueblo of Morca. It's about 20 kilometers (12 miles) and makes an excellent bike ride. On the new road, a bus ride to Mongu\u00ed costs about COP$4,000 and takes about 45 minutes. The bus leaves from the intersection of Carrera 14 and Calle 16.\n\n##### **Iza**\n\nSerene and sleepy Iza (pop. 2,081), 14 kilometers (9 miles) southwest of Sogamoso on the way to Lago Tota, is a charming pueblo. There are not heaps of activities to do here, and there is no huge attraction except for possibly the nearby hot springs. This well-preserved town, originally a Muisca settlement before the conquest, is nestled in a valley of green pastures and is surrounded by low mountains. Iza's a good place to walk about, take in some fresh air, exchange pleasantries with locals (and cows) surprised by your presence, and escape from the tourist trail.\n\ntraffic in Iza\n\nThe thermal baths are just outside of town (you can even walk there) at **Termales Erika** (Vereda Aguacaliente, tel. 8\/779-0038, 7am-6pm daily, COP$8,000). They are closed on Mondays for cleaning, thus Tuesday is the best day for a soak!\n\nIn Iza, the local specialty is cakes, pies, and other sugary sweets. Bakers constantly swat away bees at their stands on the shady Parque Principal as they await customers on the weekends. A good place to eat something other than sweets is **La Casona Parrilla Bar** (cell tel. 320\/222-6293, 1pm-10pm Sat.-Sun., COP$12,000). It specializes in grilled meats.\n\nTo get to Iza from Sogamoso, take a bus (COP$3,000) that leaves from the intersection of Carrera 11 with Calle 8. This is known as the Puente de Pesca.\n\n##### **Lago Tota**\n\nOne of the most popular destinations in Boyac\u00e1 is the Lago Tota, Colombia's largest lake. The views are spectacular here with mountains, valleys, and fields surrounding the lake. It measures 47 kilometers (29 miles) in perimeter. The main town on the lake is Aquitania; however, most visitors choose to stay at one of the cozy lakeside lodges nearby. A day trip from Sogamoso to Playa Blanca, a chilly lakeside beach, can also be arranged. But to truly relax, plan to stay the night so that you can enjoy watching the sun slowly slip away in the distance at sunset and relax by the fireplace with a glass of wine (bring your own) as the night wears on. Biking, walks, and boat rides to one of the handful of uninhabited islands on the lake are other activities you may enjoy. If you have a mountain bike, a nice ride is along the western side of the lake, along a mostly dirt road.\n\nThe lake and surrounding countryside, a patchwork of fields of green onions and potatoes, is beautiful, without a doubt. However, the lake is in peril. The dumping of fertilizers and pesticides from lakeside farms has been the primary reason that the Lago de Tota, a lake that provides drinking water for hundreds of thousands of people, has been declared one of the top five most threatened wetlands in the world by the World Wetlands Network. There are other culprits as well: large caged trout farms, the use of lake water at a nearby steel plant, and the most recent threat, oil exploration in the area by a large French oil company.\n\n###### **RECREATION**\n\n**Playa Blanca** (COP$3,500 entry) is the lake's beach, and a strange scene often awaits you there. Boys playing soccer on the white sand, university students from Bogot\u00e1 hanging out drinking beer, and teenage boys in swimsuits alongside _abuelas_ (grandmothers) bundled up in their wool _ruanas_ (ponchos) watching the proceedings. Besides sampling one of 16 fresh trout dishes on offer at the restaurant (open 8am-5pm daily), other activities at the beach include taking a tour of the lake (COP$6,000) and horseback riding.\n\n###### **ACCOMMODATIONS**\n\nA handful of inviting lodge-type hotels are around the lake to the north and west of the lakeside town of **Aquitania.** Bargains can be had during the week, when you will have the lodge (if not the lake) blissfully to yourself. The area caters to weekenders from Bogot\u00e1.\n\nThe Decameron all-inclusive hotel chain has agreements with three hotels in the area. The M **Decameron Refugio Santa In\u00e9s** (Km. 29 V\u00eda Sogamoso-Aquitania, tel. 8\/772-8860, cell tel. 313\/261-2429, santaineshotel@gmail.com, COP$99,000 pp d) is a comfortable lodge-type hotel with 13 rooms and two cabins. Wood ceilings and floors add to the atmosphere. Set on the eastern side of the lake, the hotel's terrace is an ideal vantage point to watch the sunset. Beds are very comfortable, there is wireless Internet access, and breakfast is included. The restaurant offers other meals as well. Hotel staff can arrange for walks to a _p\u00e1ramo,_ and horseback riding and taking a boat around the lake are other activities on offer. This is the nicest Decameron option. **Hotel El Camino Real** (Km. 20 V\u00eda Sogamoso-Aquitania, tel. 8\/770-0684, mauriciofigueroa@yahoo.com, www.decameron.com, COP$84,000 pp d) and **Refugio Rancho Tota** (Km. 21 V\u00eda Sogamoso-Aquitania, tel. 8\/770-8083, www.hotelranchotota.com, COP$80,000 pp d) are the other two. These have similar pricing and similar facilities, and both have small spas.\n\nserene countryside around Colombia's largest lake, Lago Tota\n\nFor charm, and a room with a view, there are two longstanding stone lodge options. **Rocas Lindas** (cell tel. 310\/349-1107, www.hotelrocaslindas.wordpress.com, COP$85,000 pp d) is a cozy lodge with 10 rooms and one cabin. There's no wireless Internet here, and this hotel could use some upgrading. M **Pozo Azul** (Bogot\u00e1 tel. 1\/620-6257, cell tel. 320\/384-1000, www.hotelrefugiopozoazul.com, COP$196,000 d) was one of the first nice hotels on the lake, and it still oozes charm. When you walk in you'll often see guests gathered by a circular fireplace in the lobby area. The hotel has 15 rooms and two _caba\u00f1as_. It is on an inlet of the lake. Some beds are on the soft side, and you have to descend 80 steep steps to get from the parking lot to the lodge. That could be difficult for those with physical limitations, and it will leave all but Olympic athletes out of breath when they finally reach the hotel.\n\n##### **Getting There**\n\nBuses that go directly to Playa Blanca, via Iza, depart Sogamoso from the intersection of Carrera 11 with Calle 8 (Puente de Pesca). Otherwise, take any bus bound for the town of Aquitania (Plaza Principal). From the market, four blocks away, take another bus to Playa Blanca. It takes about an hour to get to the lake, and the bus costs about COP$4,000. Taxis are also available.\n\n#### **SIERRA NEVADA DEL COCUY**\n\nThe Sierra Nevada del Cocuy, the highest mountains within the Cordillera Oriental (Eastern Range) of the Andes mountain chain, is 260 kilometers (162 miles) northeast of Bogot\u00e1 in northern Boyac\u00e1. The entire mountain range is contained within and protected by the **Parque Nacional Natural El Cocuy,** the country's fifth largest national park. With its 11 jagged snowcapped peaks, massive glacier-formed valleys, extensive _p\u00e1ramos_ (highland moors) studded with exotic _frailej\u00f3n_ plants, and stunning crystalline mountain lakes, streams, and waterfalls, it is one of the most beautiful places in Colombia. The sierra appeals to serious mountaineers and rock climbers, but it is also a place that nature-lovers with little experience and no gear can explore by doing easily organized day hikes.\n\nthe beach at Playa Blanca on Lago Tota\n\n###### **PLANNING YOUR TIME**\n\nGetting to the Sierra Nevada del Cocuy entails a long, grueling trip, albeit through the beautiful, verdant countryside of Boyac\u00e1. Ideally you would want to spend at least four days there, taking in the spectacular mountain landscapes.\n\nThe park has three sectors: the Northern, Central and Southern Sectors, each with many options for day hikes, more strenuous ascents to the snowcapped peaks, or highly technical rock-climbing expeditions. There is also a spectacular six-day trek along a valley between the two main ridges of the sierra. It is not a highly technical trek but requires good high-altitude conditioning. For many visitors, this is the main reason to visit the sierra.\n\nThe towns of El Cocuy and G\u00fcic\u00e1n are convenient arrival and departure points to visit the area. In both you can find basic tourist services, tour operators and guides, and stores to stock up on food, though not trekking equipment (though this can be rented from local tour operators). Both have a few interesting sights and are departure points for day hikes. El Cocuy is better located to access the Southern Sector of the park and G\u00fcic\u00e1n the Northern Sector. However, since both of these towns are around 20 kilometers (12 miles) from the park and there is limited public transportation, a good option is to base yourself nearer to the park edge in one of several pleasant lodges or campsites. You could easily spend a few days in each one of the three sectors, setting off on beautiful day hikes.\n\nThe only way to do the six-day hike around the park is with an organized tour, as the trails are not marked. If you are planning to do this trek, you may want to arrive a few days earlier to do some high altitude acclimatization hikes. Many peaks are above 5,000 meters (16,000 feet) high.\n\nThe only dependable time to visit the Sierra Nevada del Cocuy is from December to March, during the _verano_ (main dry season) in the Cordillera Oriental. At other times, there may be permanent cloud cover and much rain. High season, when Colombian visitors flock to the mountains, is from mid-December to mid-January, and again in Holy Week (late March or April). So, if you have the flexibility, visit in early December or from late January through early March.\n\nThe best available topographical maps of the Sierra Nevada de Cocuy, which might be helpful in planning your visit, can be viewed and downloaded online (www.nevados.org).\n\n##### **El Cocuy**\n\nEl Cocuy is a charming colonial town nestled in the lower folds of the Sierra Nevada del Cocuy at an altitude of 2,750 meters. The town is meticulously kept up, with whitewashed houses painted with a band of aquamarine blue. The only real sight to check out is in the Parque Principal, where there is a large diorama of the Sierra Nevada del Cocuy. This will allow you to understand the broken mountain geography, with its multitude of snowcapped peaks, lakes, and valleys. In the town there are decent accommodations, a few tour operators, and some stores to stock up for a visit to the park, though no specialized mountaineering stores.\n\n###### **RECREATION**\n\nFor a spectacular panoramic view of the entire sierra, take a hike to **Cerro Mahoma** (Mahoma Hill), to the west of the town of El Cocuy. It is a strenuous six- to seven-hour excursion often used by people who are acclimatizing to high altitude before trekking in the Sierra Nevada del Cocuy. The trailhead is outside of town on the road that leads to the town of Chita. As the trail is not marked and splits several times, it is better to go with a guide. For an experienced local guide, contact the local guide association, **ASEGUICOC** (Asociaci\u00f3n de Prestadores de Servicios Ecotur\u00edsticos de G\u00fcic\u00e1n y El Cocuy, cell tel. 311\/557-7893, aseguicoc@gmail.com).\n\n###### **ACCOMMODATIONS AND FOOD**\n\n**Hotel la Posada del Molino** (Cra. 3 No. 7-51, tel. 8\/789-0377, www.elcocuycasamuseo.blogspot.com, COP$20,000 pp d) is a friendly guesthouse. Rooms in this old house are set around two colorful interior patios. The house has a little history to it, as well. Apparently during the deadly feuds between G\u00fcic\u00e1n and El Cocuy (G\u00fcic\u00e1n was conservative and El Cocuy was liberal), the famous Virgen Morenita image was taken from its shrine in G\u00fcic\u00e1n and hidden away in the house where the hotel is located. You can see the room that hid this secret.\n\n**Casa Mu\u00f1oz** (Cra. 5 No. 7-28, tel. 8\/789-0328, www.hotelcasamunoz.com, COP$20,000 pp d) has a great location overlooking the main plaza in town. It offers a restaurant in the patio on the main floor. Rooms are fine, though somewhat small, with firm beds and wooden floors.\n\nHotels, like the Casa Mu\u00f1oz, generally offer the best food, but don't expect to be gobsmacked come dinnertime. Vegetarians may want to travel with a can of emergency lentils to hand over to kitchen staff to warm up for you.\n\n###### **INFORMATION AND SERVICES**\n\nAt the offices of the **Parque Nacional Natural El Cocuy** (Cl. 5 No. 4-22, tel. 8\/789-0359, 7am-noon and 1pm-4:45pm daily, cocuy@parquesnacionales.gov.co, COP$37,500 non-Colombians, COP$14,000 residents, COP$7,500 children\/students), you can obtain a park entry permit and general information.\n\nThere is an **ATM** at the Banco Agrario at Carrera 4 and Calle 8.\n\n###### **GETTING THERE**\n\nThe towns of El Cocuy and G\u00fcic\u00e1n are served from Bogot\u00e1 by three bus companies. The trip takes 11 hours, stops at El Cocuy, and terminates at G\u00fcic\u00e1n. The most comfortable option is with the bus company **Libertadores** (COP$50,000), which operates a big bus, leaving Bogot\u00e1 at 8:30pm. The return trip departs El Cocuy at 7:30pm. Bus line **Fundadores** (COP$45,000) has two buses, leaving Bogot\u00e1 at 5am and 4:30pm, returning from El Cocuy at 7:30am and 8:30pm. **Concord** (COP$45,000) also has two services, leaving Bogot\u00e1 at 3am and 5pm and leaving El Cocuy for Bogot\u00e1 at 5:30am and 7:30pm.\n\n##### **G\u00fcic\u00e1n**\n\nLong before its foundation in 1822, G\u00fcic\u00e1n was a place of significance for the U'wa indigenous people. The U'wa fiercely resisted the Spanish conquest, and, rather than submit to domination, their chief G\u00fcic\u00e1ny led the people to mass suicide off a nearby cliff known as El Pe\u00f1\u00f3n de los Muertos. This little-known act of defiance is the New World's equivalent of the Masada mass suicide in ancient Judea.\n\nThe town, damaged by fires and civil war, is a mix of modern and old buildings, without much charm. However, it is a convenient starting point to visit the Northern Sector of the park. It has good accommodations, several tour operators, and some interesting sights and is the starting point for numerous beautiful day hikes.\n\nFolks in G\u00fcic\u00e1n resent that the national park carries the \"El Cocuy\" name. They feel that this natural wonder is just as much theirs as it is their rivals in the town of El Cocuy. You can score points with them by referring to the park as Parque Nacional Natural El G\u00fcic\u00e1n.\n\n###### **SIGHTS**\n\nThe main sight in town is the image of the Virgen Morenita de G\u00fcic\u00e1n, located in the **Iglesia de Nuestra Se\u00f1ora de la Candelaria** (Parque Principal). This image of the virgin, with strong indigenous traits, appeared to the survivors of the U'wa mass suicide and ushered in their conversion to Christianity.\n\nAt the entrance to the town on the road from El Cocuy is the **Monumento a la Dignidad de la Raza U'wa** (Monument to U'Wa Dignity), a large statue that depicts the culture and history of the U'Wa people. It was designed by a local artist with input from the community.\n\n###### **RECREATION**\n\nThere are several pleasant day hikes to be done from G\u00fcic\u00e1n. A mildly strenuous three-kilometer (two-mile), two-hour round-trip hike takes you to the base of the **Pe\u00f1\u00f3n de los Muertos** (3,800 meters\/12,500 feet), site of the U'wa mass suicide. The 300-meter cliff is imposing, and the thought of hundreds of people jumping off in defiance will send shivers through your body. From the Parque Principal, follow the road east towards the Vereda San Juan. Several signs indicate the way, so you will not need a guide.\n\nA longer and more strenuous 11-kilometer (7-mile), six-hour hike leads northeast along the **Sendero del Mosco** (Mosco Trail) up the R\u00edo Cardenillo, passing sheer cliffs to a spot called Parada de Romero, which is the initial (or ending, depending on which way you go) segment of the six-day circuit around the Sierra Nevada del Cocuy. The hike ends at an altitude of 3,800 meters and is a good acclimatization walk. The trailhead is off the road that leads from El G\u00fcic\u00e1n to the Parque Nacional Natural El Cocuy. As the trail is not marked, it is better to take a guide. Contact the association of local guides, **ASEGUICOC** (Asociaci\u00f3n de Prestadores de Servicios Ecotur\u00edsticos de G\u00fcic\u00e1n y El Cocuy, cell tel. 311\/557-7893, aseguicoc@gmail.com).\n\n###### **ACCOMMODATIONS**\n\nThe **Brisas del Nevado** (Cra. 5 No. 4-57, cell tel. 310\/629-9001, <http:\/\/brisasdelnevado.com>, COP$35,000 pp) has the best accommodations and restaurant in town. Four rooms in the original house sleep 2-4 persons each. Outside is a nicer cabin with two rooms. The only problem is it is located next to a _tejo_ bar. _Tejo_ is a Colombian sport heavily associated with drinking.\n\n**El Eden** (Tr. 2 No. 9-58 Urbanizaci\u00f3n Villa Nevada, cell tel. 311\/808-8334, luishernandonc@hotmail.com, COP$30,000 pp) is in a residential neighborhood about 10 minutes up from the main plaza. It's a friendly place with lots of rooms, and you can use their kitchen. Rabbits and parakeets are caged in the garden below.\n\n**Hotel Guaicani** (Cl. 5 No. 6-20, cell tel. 312\/524-3449, guaicany@hotmail.com, COP$20,000 pp d) is not wonderful, but it does offer trekking services and equipment rental.\n\nJust outside of town is the **Hotel Ecol\u00f3gico El Nevado** (road to El Cocuy, cell tel. 320\/808-5256 or 310\/806-2149, www.hoteleconevado.jimdo.com, COP$60,000 d), in a spacious and green setting. There are two parts to the hotel: the original quaint farmhouse with an interior patio and a modern wing. The farmhouse has loads more character, but the modern wing is, well, modern.\n\n###### **INFORMATION AND SERVICES**\n\nAt the offices of the **Parque Natural Nacional El Cocuy** (Tr. 4 No. 6-60, 7am-noon and 1pm-4:45pm daily) you can obtain a park entry permit and general information.\n\n###### **GETTING THERE**\n\nG\u00fcic\u00e1n is served from Bogot\u00e1 by three bus companies. The trip takes 11 hours and terminates at G\u00fcic\u00e1n, with a stop at El Cocuy. The most comfortable option is with company **Libertadores** (COP$50,000), which operates a big bus, leaving Bogot\u00e1 at 8:30pm and returning from G\u00fcic\u00e1n at 7pm. **Fundadores** (COP$45,000) has two buses, leaving Bogot\u00e1 at 5am and 4:30pm and departing G\u00fcic\u00e1n at 7am and 8pm. **Concord** (COP$45,000) also has two services, leaving Bogot\u00e1 at 3am and 5pm and returning to Bogot\u00e1 from G\u00fcic\u00e1n at 5am and 7pm. If you are in a hurry, you can take **Cootransdatil** to Soat\u00e1 (COP$15,000) at 7am, 11am, or 2pm. From Soat\u00e1 there are frequent departures for Duitama and Bogot\u00e1.\n\n#### M **PARQUE NACIONAL NATURAL EL COCUY**\n\nLocated about 20 kilometers (12 miles) east of the towns of El Cocuy and G\u00fcic\u00e1n, the **Parque Nacional Natural El Cocuy** (tel. 8\/789-0359, cocuy@parquesnacionales.gov.co, COP$37,500 for non-Colombian visitors, COP$14,000 Colombians and residents, COP$7,500 children\/students) covers an area of 306,000 hectares (760,000 acres) spanning the departments of Boyac\u00e1, Arauca, and Casanare.\n\nThe Sierra Nevada del Cocuy, consisting of two parallel ranges 30 kilometers long with 11 peaks higher than 5,000 meters, is the centerpiece of the park. However, the park extends far north and east from the sierra and includes extensive tracts of temperate and tropical forests. It also includes 92,000 hectares (230,000 acres) of U'wa indigenous _resguardos_ (reservations), which are not open to tourism.\n\nThe Sierra Nevada del Cocuy is home to the largest expanse of glaciers in Colombia, extending 16 square kilometers (6 square miles). What are usually referred to as _nevados_ (snowcapped mountains) are in fact glacier-capped mountains. The highest peak is **Ritacuba Blanco** (5,380 meters\/17,650 feet). Other notable glacier-capped peaks are **Ritacuba Negro** (5,350 meters, 17,550 feet), **San Pabl\u00edn Norte** (5,200 meters, 17,060 feet), **C\u00f3ncavo** (5,200 meters\/17,060 feet), and **Pan de Az\u00facar** (5,100 meters, 16,730 feet). One of the most striking peaks in the Sierra Nevada del Cocuy is the **P\u00falpito del Diablo** or Devil's Pulpit (5,100 meters, 16,730 feet), a massive rectangular flat-top rock formation. (A side note: Be sure to refer to it as the P\u00falpito del Diablo (Devil's Pulpit) and not Pulp\u00edto del Diablo, which means the Devil's Little Octopus.) Of these, Ritacuba Blanco, C\u00f3ncavo, and Pan de Az\u00facar can be ascended by anyone in good physical shape and do not require mountain climbing skills.\n\nUnfortunately, all the glaciers in Colombia, including those of the Sierra Nevada del Cocuy, are rapidly melting due to global warming. A 2013 report by the Colombian Hydrological, Meteorological, and Environmental Studies Institute (IDEAM) forecasts that, by 2030, all the glaciers in Colombia will have disappeared.\n\nAt the base of the peaks are numerous glacier-formed valleys supporting _p\u00e1ramos,_ unique tropical high altitude ecosystems of the Andes. The _p\u00e1ramos_ are covered with beautiful _frailejones,_ plants that have imposing tall trunks and thick yellow-greenish leaves. Other _p\u00e1ramo_ vegetation includes shrubs, grasses and _cojines_ (cushion plants).\n\nErwin Krauss, a Colombian of German descent was the first modern explorer of the sierra in the 1930s. In the 1960s and 1970s, Colombian and European expeditions climbed most of the peaks. During the 1980s and 1990s, there was significant ELN and FARC presence and tourism all but disappeared.\n\nIn the past decade, the army has reestablished control of the area around the Sierra Nevada de Cocuy, and tourists have started to come back. In the 2012-2013 season, there were an estimated 9,000 visitors. The Colombian Park Service has been scrambling to deal with the influx of visitors.\n\nEntry permits (which include entry fees) are required and can be easily obtained at the park offices in El Cocuy or G\u00fcic\u00e1n. In peak season from mid-December to mid-January and during Easter week, it is better to obtain the permit several weeks in advance through the Park Service in Bogot\u00e1. Call (tel. 1\/353-2400) or email (ecoturismo@parquesnacionales.gov.co) with the names of visitors, passport numbers, and expected dates of your arrival. The Park Service will provide instructions for paying and will send the permit by email.\n\nParque Nacional Natural El Cocuy\n\n##### **Hiking**\n\nThere are three separate sectors where you can do spectacular one- to two-day hikes into the park. Each of these sectors has lodges and camping grounds that serve food and make convenient starting points for these hikes. You can get to any of these lodges on the morning _lechero_ (milk truck). Every morning the milk man collects fresh milk from family dairy farms throughout the countryside.\n\n###### **SOUTHERN SECTOR**\n\nThis sector is accessed by a road from the **Alto de la Cueva,** a stop on the _lechero_ route. There are two good lodging options in this area, and they serve as points of reference: **Caba\u00f1as Guicany,** a lodge at Alto de la Cueva outside the park, and **Caba\u00f1a Sisuma,** a lodge 10 kilometers (6 miles) from Alto de la Cueva inside the park.\n\nOne popular day hike in the Southern Sector is up to **Lagunillas** through a wide glacier-formed valley strewn with different types of _frailejones_ and passing four large lakes. From Alto de la Cueva it is a six- to seven-hour round-trip hike to an altitude of 4,300 meters (14,100 feet). From the Caba\u00f1a Sisuma lodge within the park, it is a four- to five-hour round-trip hike. You do not need a guide to do this excursion.\n\nA strenuous hike takes you to the **Pan de Az\u00facar.** From the Caba\u00f1a Sisuma lodge it is a six-hour round trip hike to the border of the glacier that covers Pan de Az\u00facar or 10 hours round-trip to the top of the glacier. Along the way you will pass the **P\u00falpito del Diablo** (Devil's Pulpit), a stunning, huge rectangular flat-topped rock formation. From the top of Pan de Az\u00facar there are spectacular views of the Laguna Grande de la Sierra, P\u00falpito del Diablo, and C\u00f3ncavo peaks. A guide is required for this hike.\n\n###### **CENTRAL SECTOR**\n\nThe starting point for visits to the Central Sector is **Hacienda La Esperanza,** which is a stop on the daily _lechero._ From there, it is a strenuous six-hour round-trip hike to **the Laguna Grande de la Sierra,** a beautiful lake nestled between the C\u00f3ncavo and P\u00falpito del Diablo peaks. A guide is not necessary for this hike. By camping at the lake, it is possible to ascend to the **C\u00f3ncavo** (5,200 meters\/17,060 feet), **Concavito** (5,100 meters, 16,730 feet), or **Toti** (4,900 meters\/16,075 feet) peak. Each ascent involves a strenuous four- to five-hour round-trip hike and should be done with a guide. From the Laguna Grande de la Sierra, it is also possible to reach Caba\u00f1a Sisuma, in the Southern Sector, in nine hours. A guide is necessary as this trail is not well marked.\n\n###### **NORTHERN SECTOR**\n\nThe starting point for hikes in the Northern Sector is **Caba\u00f1as Kanwara.** A short and mildly strenuous three- to four-hour round-trip hike takes you to the **Alto Cimiento del Padre,** a mountain pass at 4,200 meters (13,800 feet). This hike offers spectacular views of Ritacuba Negro peak. This hike does not require a guide.\n\nCaba\u00f1as Kanwara is also the starting point for hikes to the gently sloping **Ritacuba Blanco,** the highest peak in the Sierra Nevada del Cocuy. The ascent to the top can be done in one grueling 9- to 10-hour excursion, leaving at 2am or 3am in order to reach the peak in the morning when conditions are best for climbing on the glacier. Most people, however, split the trek into two, camping at the Playitas camp spot halfway up. A guide is necessary for this trek.\n\n###### **SIX-DAY CIRCUIT**\n\nAn unforgettable experience is to do the six-day trek through the glacier-formed valleys lying between the two north-south ranges of mountains. Along the whole trip you will have glacier-capped mountains on both sides. There are a few mountain passes, but generally the altitude is 4,000-4,500 meters (13,100-14,800 feet). You do not need to be an expert mountaineer, but in addition to being in good physical condition, you need to be acclimatized to the altitude. A few days of day treks before doing the circuit may be required. Do not attempt this trek without a knowledgeable guide, as it's easy to get lost in this treacherous landscape. The basic tour, which involves carrying all your own gear, will cost on average COP$700,000 per person. Don't pay less than that because it means the operator is skimping on the guide's salary. High-end tours, with porters and a cook, will cost COP$1,500,000 per person.\n\n###### **TOUR OPERATORS AND GUIDES**\n\nWhether you decide to do a couple of day hikes or the six-day trek, securing a reliable, professional guide will greatly increase your enjoyment. For day hikes, contact the local guide association **Asociaci\u00f3n de Prestadores de Servicios Ecotur\u00edsticos de G\u00fcic\u00e1n y El Cocuy** (ASEGUICOC, cell tel. 311\/557-7893, aseguicoc@gmail.com). For day hikes, expect to pay about COP$80,000-100,000. If you ascend to the top of a glacier, the daily rate goes up to COP$130,000-150,000 and includes necessary gear. One highly knowledgeable guide is **Julio Su\u00e1rez** (cell tel. 311\/509-4413, ucumary13@gmail.com).\n\nOne of the leading trekking operators in the Sierra Nevada del Cocuy is **Colombia Trek** (Cra. 4 No. 6-50, G\u00fcic\u00e1n, cell tel. 320\/339-3839, www.colombiatrek.com), run by knowledgeable veteran Rodrigo Arias. It is one of the few operators offering English-speaking guides, and it is highly recommended.\n\nAnother tour company based in El Cocuy is **Servicios Ecotur\u00edsticos G\u00fcic\u00e1ny** (Cra. 5 at Cl. 9, El Cocuy, cell tel. 310\/566-7554), run by Juan Carlos Carre\u00f1o, son of the owner of Caba\u00f1as G\u00fcic\u00e1ny.\n\nAvoid horseback rides through the park. Horses and cattle have caused significant damage to the flora of the park, and both are officially illegal. Unfortunately, many lodge owners do not agree with this environmental policy and refuse to adhere to it.\n\n##### **Accommodations**\n\nWhile not luxurious by any means, the lodging options in and around the park are just what you'd expect and want in this mountain environment. And the owners are all quite attentive and extremely friendly. Plan to spend some time hanging out in and around your hotel. It's nice to explore the countryside and meet locals.\n\n###### **SOUTHERN SECTOR**\n\nThe best located accommodation in the Southern Sector is M **Caba\u00f1a Sisuma** (cell tel. 311\/236-4275 or 311\/255-1034, aseguicoc@gmail.com, COP$35,000 pp), a cozy cabin inside the park in the Lagunillas sector run by the local tour guide association ASEGUICOC. It has six rooms, good food, and fireplaces to keep one warm. It is a two-hour hike into the park from the Alto de la Cueva, a stop on the daily _lechero._\n\nAnother pleasant and comfortable option is M **Caba\u00f1as G\u00fcic\u00e1ny** (Alto de la Cueva, cell tel. 310\/566-7554, cab_guaicany@yahoo.es or guaicany@hotmail.com, COP$50,000 pp with meals, COP$30,000 without meals, COP$10,000 pp camping), owned by old timer Eudoro Carre\u00f1o. The _lechero_ can drop you off at the lodge. It's rustic and the owner is a delight to chat with over a hot _tinto_ in his rustic kitchen.\n\n###### **CENTRAL SECTOR**\n\nM **Hacienda La Esperanza** (cell tel. 310\/209-9812, haciendalaesperanza@gmail.com, COP$50,000 d), a working farm on the edge of the park, provides accommodations in a rustic farmhouse oozing with character. The family running the hotel is very hospitable, and the host is a trained chef who enjoys pampering his guests. Nothing beats hanging out by the fireplace in the late afternoon drinking something hot after a day of mountain climbing. The _lechero_ makes a stop at this hacienda.\n\n###### **NORTHERN SECTOR**\n\nThe most conveniently located place to stay in this sector is M **Caba\u00f1as Kanwara** (cell tel. 311\/231-6004 or 311\/237-2260, infokanwara@gmail.com, COP$35,000 pp). This lodge of cute wooden A-frame houses has a great location and serves good food, too. To get there, you must get off the _lechero_ at Hacienda Ritacuba and walk 90 minutes towards the park.\n\n##### **Getting There and Around**\n\nFrom El Cocuy and G\u00fcic\u00e1n to the three park sectors there are three transportation possibilities: hiking 4-5 hours uphill from these towns to the park, taking an express service costing COP$80,000-100,000 (ouch!), or riding an early morning _lechero_ (milk truck). This is a working truck that picks up milk along a predetermined route. Merchandise and passengers share the back of the truck, which is covered with canvas. Don't expect any comforts, but expect to have some good tales to tell. The _lechero_ leaves G\u00fcic\u00e1n from Carrera 5 and Calle 6 every morning at 5:30am and stops at El Cocuy around 6am. Around 7:30am it arrives at Alto de la Cueva, where you can get off to visit the Southern Sector. Around 9am it pulls right up to Hacienda La Esperanza in the Central Sector. Around 10:30am it reaches Hacienda Ritacuba, from where you can walk up to Caba\u00f1as Kanwara in the Northern Sector.\n\n### **Santander**\n\nBeautiful, lush scenery, a delightful climate, well-preserved colonial pueblos, and friendly, outgoing people\u2014this is the Santander department. Located in northeast Colombia, Santander lies to the north of Boyac\u00e1 and southwest of Norte de Santander. Bucaramanga is the modern capital city, but you'll probably be drawn to the countryside. San Gil and the Ca\u00f1\u00f3n del Chicamocha will keep you busy with a smorgasbord of outdoor adventures, while nearby Barichara will seduce you with its tranquil ambiance.\n\n#### **BUCARAMANGA**\n\nThe Ciudad Bonita (Beautiful City) is the capital of the department of Santander. Bucaramanga is a busy and growing city with a young and vibrant population and an agreeable climate where the flowers are always in bloom. Its central location makes for a strategic launching point for visits to the Santander countryside and is a midway point between Bogot\u00e1 and Santa Marta on the Caribbean coast as well as C\u00facuta in the far east. Including neighborhoods that are an extension of Bucaramanga (Floridablanca, Gir\u00f3n, and Piedecuesta), the population exceeds a million.\n\n###### **ORIENTATION**\n\nMost of your time will probably be spent in Cabecera (the upscale shopping and residential area), in the city center (between Cras. 9-17 and Cl. 45 and Av. Quebrada Seca), and in nearby municipalities such as Gir\u00f3n and Floridablanca.\n\n_Carreras_ (avenues) run north to south, increasing in number from west to east. The main _carreras_ are 15, 27, and 33. _Calles_ (streets) run east to west and increase in number from north to south.\n\n##### **Sights**\n\nBucaramanga's main sights are contained within the walkable city center. If you're staying in the Cabecera neighborhood it's a long, hot walk to the city center, so you're better off taking a cab.\n\nBucaramanga prides itself on its parks, and one of the most famous is the **Parque Garc\u00eda Rovira** (Cras. 10-11 and Clls. 36-37). Filled with towering palms, it doesn't provide much shade, but with the pale yellow and white 19th-century **Catedral San Laureano** (Cra. 12 No. 36-08) standing prominently on the park's eastern side, it is rather photogenic. On the west side of the park is Bucaramanga's oldest church, the **Capilla de los Dolores** (Cra. 10 No. 36-08). This unassuming, white-washed structure dates back to 1748 and no longer has a religious mission. It's generally not open to the public. Across from it is **La Casa del Libro Total** (Cl. 35 No. 9-81, tel. 7\/630-3389, www.lacasadellibrotal.com, 8am-10pm Mon.-Fri.), a newish cultural center that (oddly) has a number of bank offices and at the same time exhibition spaces (air-conditioned) for interesting art exhibits. There is also a small library, and a caf\u00e9 serves free coffee.\n\nThe Libertador, Sim\u00f3n Bol\u00edvar, stayed in his friend Juan Eloy Valenzuela's house, now known as the **Museo Casa de Bol\u00edvar** (Cl. 37 No. 12-15, tel. 7\/630-4258, 8am-noon and 2pm-6pm Mon.-Fri., 8am-noon Sat., COP$2,000) for about 70 days in 1828 while he awaited news from the Convenci\u00f3n de Oca\u00f1a. (Things went badly at that convention, with a rift between Bol\u00edvar and Santander growing wider, and the end result was Bol\u00edvar's self-declaration as dictator.) The museum has personal belongs of the Liberator, an original diary from the first Expedici\u00f3n Bot\u00e1nica led by Jos\u00e9 Celestino Mutis, an original shield of the Estados Unidos de Colombia, and an exhibit on the Guane indigenous people from the area.\n\nAcross the street from the Museo Casa de Bol\u00edvar, the **Casa de la Cultura** (Cl. 37 No. 12-46, hours vary) hosts occasional art exhibitions and other events. The restaurant on the first floor is packed at lunchtime.\n\nFive or six blocks to the east is the **Parque Santander** (Cras. 19-20 and Clls. 35-36). It's a lively park in the middle of the hustle and bustle of modern Bucaramanga. Hare Krishnas beat drums, unimpressed skateboarders show off, and dozens others look on. The Romanesque Revival **Catedral de la Sagrada Familia** (Cl. 36 No. 19-56) took over a hundred years to complete. It was finished in 1865. Some of the most striking features inside include the many stained glass windows. The church, with twin towers and statues of the Virgin Mary, the baby Jesus, and Joseph in between, looks particularly grandiose at night when it is lit.\n\nThe **Museo de Arte Moderno de Bucaramanga** (Cl. 37 No. 26-16, tel. 7\/645-0483, www.museodeartemodernodebucaramanga.blogspot.com, 8:30am-noon and 2pm-5:30pm Mon.-Fri., 8am-noon Sat., COP$2,000) is worth checking out, but it's only open when there is an exhibit. The **Centro Cultural Posada Tres Culturas** (Cl. 37 No. 24-62, tel. 7\/683-9142, www.librostresculturas.com, 9am-noon and 2pm-7pm Mon.-Sat.) is near the museum and often has events going on. It has a nice art bookstore.\n\nThe **Parque San P\u00edo** (between Cras. 33-35 and Clls. 45-46) is a vibrant greenspace near the Cabecera neighborhood. At the western end stands the Fernando Botero sculpture _Mujer de Pies Desnuda._ On the opposite end is the modern **Iglesia San P\u00edo** (Cra. 36 No. 45-51), where there are paintings on permanent display by local artist Oscar Rodr\u00edguez Naranjo. Farther up is the **Museo Guane** at the **Universidad Aut\u00f3noma de Bucaramanga** (UNAB, Av. 42 No. 48-11, tel. 7\/643-6111), which has a collection of over 600 ceramic pieces (figures, ceremonial and daily vessels, shell necklaces, stone utensils, and ceramic spindle whorls) found near Bucaramanga. Some 90 pieces are on display in a small lobby area. Nobody seems to know where the exhibition space is, and you'll have to ask around. You may have to climb around lounging students to even get a look at the collection.\n\n##### **Nightlife**\n\nThe nightlife scene in Bucaramanga? Maybe exuberant is the right word to describe it. Most bars and clubs are open Thursday through Saturday, closing at 2am or 3am.\n\n**Caf\u00e9 Con Verso** (Cl. 44 No. 28-63, tel. 7\/647-1486, 4:30pm-late Mon.-Sat.) is a pleasant caf\u00e9 with occasional live music and film nights. **La Birrer\u00eda Pub & Grill** (Cra. 36 No. 43-46, tel. 7\/657-7675, noon-midnight Sun.-Thurs., Fri.-Sat. noon-2am) serves sports bar-type food (although there are some healthy selections) and beer. It's open-air and waitstaff are very attentive. This is the place to watch big _f\u00fatbol_ matches. **El Garaje** (Cl. 48 No. 33-39, tel. 7\/657-4768) is more about burgers and beer.\n\n**La Esquinita de los Recuerdos** (Cl. 22 No. 25-55, tel. 7\/632-0640 or 7\/645-6861, hours vary Tues.-Sat.) is a beloved bar and a good place to have a beer while listening to old (Latino style) favorites. The bar itself is an oldie, more or less, having been around since 1965. **Cali Son** (Cl. 33 No. 31-33, no phone) is one of the top salsa bars in Bucaramanga.\n\nAs you might imagine from its name, **Dash** (Cl. 52 No. 34-27, cell tel. 315\/624-6905) is a high-energy club popular with the college crowd.\n\n##### **Shopping**\n\nBucaramanga is well known in Colombia for its leather shoes, handbags, and wallets. **Nora Lozza First Class** (Centro Comercial El Cacique, Tr. 93 No. 34-99, Local 113, 10am-8pm daily) is a well-known designer of leather bags and accessories made in Bucaramanga. There are stores in El Cacique and several other shopping malls.\n\n**Latin Lover** (Cra. 35 No. 44-41, tel. 7\/695-1369, 9am-noon and 2pm-8pm Mon.-Sat.), created by a pair of Bucaramanga hipsters, sells groovy and original Latino-chic T-shirts. They cost around COP$60,000.\n\nFor handicrafts, check out the woven items, including handicrafts made from leaves of the _fique_ palm tree, and other accessories at **Luz y Vida** (Centro Comercial Cuarta Etapa, 4th floor, Local 402\/9, tel. 7\/673-0680, cell tel. 317\/316-4487, www.artesaniasluzyvida.webnode.com.co, 10am-8pm Mon.-Sat., 10am-5pm Sun.). This is an association of women heads of household who have been forcibly displaced from their homes.\n\n##### **Accommodations**\n\n###### **UNDER COP$70,000**\n\nIt's not just backpackers who flock to the M **Kasa Guane** (Cl. 49 No. 28-1, tel. 7\/657-6960, www.kasaguane.com, COP$25,000 dorm, COP$80,000 d). This busy yet friendly place with both dorms and private rooms hosts activities, provides tons of insider information and tips, and is in a great location in Cabecera. The guys here will get you hooked up with paragliding and give you expert insider tips on all the Bucaramanga party spots. On weekends the top floor bar gets lively.\n\n**Nest Fly Site Hostel** (Km. 2 V\u00eda Mesa de Ruitoque, cell tel. 312\/0432-6266, www.colombiaparagliding.com, COP$25,000 dorm, COP$60,000 d) is the place to stay if you're interested in paragliding. It's right next door to the fly site. It's a quiet and cute place, 20 minutes away from the bustle of Buca. Nest is run by the same people as Colombia Paragliding and the Kasa Guane hostel. It is near the Ruitoque town next to the Las \u00c1guilas launching pad for most paragliding flights near Bucaramanga.\n\n###### **COP$70,000-200,000**\n\n**Antigua Bel\u00e9n Bed and Breakfast** (Cra. 31 No. 17-22, tel. 7\/634-9860, www.hotelantiquabelen.com, COP$133,000 d with a\/c) has 13 rooms in a modern house full of antiques. It's located in a rather dull part of town not terribly close to nor too far from anything. Breakfast is served in a pleasant patio in the back.\n\nThe **Hostal UNAB** (Av. 42 No. 48-160, tel. 7\/643-6111, ext. 652, COP$154,000 d) has just four comfortable rooms and a restaurant on-site. Right across the street from the university, it might be an odd place to be for a visit to Buca. But it is extremely low-key. It's about a 15-minute walk down to the Parque San P\u00edo from here.\n\n###### **OVER COP$200,000**\n\n**Hotel Guane** (Cl. 34 No. 22-72, tel. 7\/634-7014, www.hotelguane.com, COP$206,000 d) is a mid-sized hotel with 40 air-conditioned rooms, a pool, and two restaurants. Cheesy decor. **La Serran\u00eda** (Cl. 33 No. 30-26, tel. 7\/691-7535, www.laserraniahotel.com, COP$250,000 d) has about 50 new and minimalist-style rooms along with a rooftop pool and restaurant. It's overpriced. **Ciudad Bonita** (Cl. 35 No. 22-01, tel. 7\/635-0101, www.hotelciudadbonita.com.co, COP$260,000 d) is the fancy hotel in town. It has 70 rooms, two restaurants, a caf\u00e9, a pool, gym, sauna, and there's live music Thursday, Friday, and Saturday evenings. The area it's in is not a pleasant place to walk around day or night.\n\n##### **Food**\n\nWant to eat like a local? Look for these Santanderean specialties: _cabrito con pepitoria_ (goat fricassee), _carne oreada_ (dried meat), and _mute santandereano_ (a corn-based meaty stew). And don't forget the ants: fried big bottom ants or _hormigas culonas._ These queen ants are harvested throughout Santander, typically after Semana Santa. After months of hibernation, on one prickly hot day, the queens leave their colony. That's when they are caught. They are later toasted. Eating ants has always been popular and dates back hundreds of years to the Guane culture. Wealthy Santandereanos used to be embarrassed to admit any fondness for the creepy-crawlers, but that's changed, and in fact the ants are showing up more and more on the plates of diners on a quest for the exotic.\n\nM **Santanero Colonial** (Cl. 41 No. 10-54, tel. 7\/696-0538, 7:30am-4pm Mon.-Thurs., 7:30am-late Fri., COP$15,000) is one of the top choices for government bureaucrats on lunch break. There is always a set lunch menu (plus \u00e0 la carte), and frequently you'll have to wait a bit to be seated. Tables are set up around a pleasant sunny patio. It is behind the Gobernaci\u00f3n building. **La Aldaba** (Cl. 37 No. 12-32, tel. 7\/642-4062, noon-3pm Mon.-Sat.) is a popular place for an inexpensive lunch. They always have a lunch special that features trout, chicken, or beef for under COP$12,000.\n\nOne of Bucaramanga's favorite restaurants is M **El Viejo Chiflas** (Cra. 33 No. 34-10, tel. 7\/632-0640, 9am-midnight Mon.-Wed., Thurs.-Sun. 24 hours, COP$23,000). The atmosphere here is cowboy style with wooden tables and interiors, and the menu features local specialties, such as goat and the Santander classic _carne oreada._ And there's always an arepa (cornmeal cake) with your meal. Portions can be huge. **Los Tejaditos** (Cl. 34 No. 27-82, tel. 7\/634-6028, www.restaurantelostejaditos.com, 11am-10pm Tues.-Sat., 11am-5pm Sun., COP$23,000) is an old-style restaurant with a popular special menu at lunchtime. **Mercag\u00e1n** (Cra. 33 No. 42-12, tel. 7\/632-4949, www.mercaganparrilla.com, 11am-6pm Mon. and Thurs., 11am-11pm Tues.-Wed. and Fri.-Sat., 11am-4pm Sun., COP$25,000) is a legendary steakhouse in Bucaramanga that has multiple locations, including in many shopping malls.\n\nFor a break from the _comida t\u00edpica_ (Colombian fare) thing, there are a few options in Buca. **Radha Govinda's** (Cra. 34 No. 51-95, tel. 7\/643-3382, lunch Mon.-Sat.) is a vegetarian option. It's Hare Krishna-run and is on a quiet street in Cabecera. The **Embajada China** (Cl. 49 No. 32-27, tel. 7\/647-1931, 10am-10pm daily, COP$15,000) is run by a Chinese family, and they serve generous portions. It's in Cabecera near the Kasa Guane. Stir-fries, salads, and pastas are on the menu at **Tavolo Gourmet** (Cra. 35 No. 48-84, tel. 7\/643-7461, www.tavologourmet.com, 11am-10pm Tues.-Sun., COP$18,000). It's a bright and airy place in a fancy neighborhood. The wildly popular Colombian chain **Crepes & Waffles** (Centro Comercial La Florida, Cl. 31 No. 26A-19, 3rd floor, Local 3090, tel. 7\/632-1345, 11:45am-9:30pm Mon.-Sat., 11:45am-8:30pm Sun., COP$17,000) is a welcome sight.\n\nWe'll admit that the cr\u00eapes at M **Cr\u00eapes D'Or** (Cl. 46 No. 34-28, tel. 7\/657-4770, 3:30pm-10pm Mon.-Sat., COP$13,000) are nothing to write home about, although they're fine and fairly priced. What makes this unpretentious family-run spot a delight is its setting overlooking the Parque San P\u00edo. Imagine an outdoor terrace where you look out onto park goers, dogwalkers, and joggers and not onto a steady stream of traffic!\n\n**Kirama** (Cl. 49 No. 33-37, tel. 7\/657-6989, 6am-9pm daily) is a wildly popular spot for breakfast on the run, like an _arepa boyacense_ (cornmeal cake stuffed with cheese). Do not be confused by Karima, which is immediately next to it. Kirama says Karima came later. **Pan Pa Ya** (Cl. 49 No. 28-38, tel. 7\/685-2001, 8am-10pm Mon.-Sat, 9am-noon and 5pm-8pm Sun.) is in all the major cities of Colombia, and it's always a reliable place for a decent cup of coffee, pastries, and inexpensive breakfasts (eggs, fresh fruit).\n\nThe **Mercado P\u00fablico** (between Cras. 15-16 and Clls. 33-34, daily) downtown is a great place to wander about. On multiple floors, you can do some cheap shopping, walking swiftly past the meat section, get a cheap meal on the fourth floor, and enjoy some pretty nice views of the city as well. On the top floor they sell loads of Piedecuesta cigars for cheap as well as baskets, herbs, and _artesan\u00edas_ (handicrafts). Several stands sell juices and lunches. Some stalls even sell bull's eyes, if you're feeling adventurous.\n\n##### **Information and Services**\n\nThe **tourist office** (Cl. 30 No. 26-117, tel. 7\/634-1132) is parkside at the Parque de los Ni\u00f1os. Police can be reached by dialing 123, the **Hospital Universitario Gonz\u00e1lez Valencia** (Cra. 33 No. 28-126) by calling tel. 7\/634-6110.\n\n##### **Getting There and Around**\n\nThe **Aeropuerto Internacional de Palo Negro** (V\u00eda Lebrija), Bucaramanga's airport, is 25 kilometers (15 miles) west of town. **Avianca** (Cl. 52 No. 35A-10, tel. 7\/657-3888, www.avianca.com, 8am-6pm Mon.-Fri., 8am-1pm Sat.), **EasyFly** (tel. 7\/697-0333, www.easyfly.com.co), **VivaColombia** (tel. 1\/489-7989, www.vivacolombia.com.co), and **LAN** (Col. toll-free tel. 01\/800-094-9490, www.lan.com) all serve the city. Taxis to and from the airport to Bucaramanga cost COP$32,000.\n\nFrequent bus service is offered between Bucaramanga and all major cities nationwide as well as small locales in Santander. The **Terminal de Transportes** (Km. 2 Tr. Metropolitana, tel. 7\/637-1000, www.terminalbucaramanga.com) is modern, clean, and open-air. It is off of Calle 70 on the way towards Gir\u00f3n.\n\nThe **MetroL\u00ednea** (www.metrolinea.gov.co) is the Bucaramanga version of the Bogot\u00e1 TransMilenio. These green buses are clean and efficient, and the system covers just about the entire city, although it can be difficult to figure out. Maps of the system are hard to come by, obligating you to ask fellow travelers for information. You can purchase cards for the regular buses (ones that do not have dedicated lanes) at kiosks on the streets.\n\nTaxis are plentiful in Bucaramanga. To order one call **Radio Taxis Libres** (tel. 7\/634-8888) or **Taxmovil** (tel. 7\/633-9090).\n\nThe city center area is easily visited on foot, although there is a lot of traffic. The Cabecera area is also more or less walkable, especially in the evenings when traffic calms down.\n\n##### **Mesa de Ruitoque**\n\nA surprisingly quiet and rural area to the southeast of Bucaramanga and Floridablanca, Mesa de Ruitoque sits on a plateau and is perfect for paragliding.\n\n###### **PARAGLIDING**\n\nThe Bucaramanga area is a great place to get over that fear and fly your first tandem paragliding flight, or to take a 10-day course, and the plateau of Mesa de Ruitoque, just 10 minutes from downtown, is where to go.\n\nThe area is blessed with 350 flyable days per year, meaning more air time and less waiting around. **Colombia Paragliding** (www.colombiaparagliding.com) offers tandem flights of different durations from the **Voladero Las \u00c1guilas** (Km. 2 V\u00eda Ruitoque, tel. 7\/678-6257, www.voladerolasaguilas.com.co) launch point just outside of town near Ruitoque. The Kasa Guane hostel can arrange transportation for you in their van. Instructors are all certified.\n\nA 10-minute flight costs COP$50,000, 20 minutes is $90,000, and a 30-minute flight costs $120,000. Winds are best at this fly site in the afternoon, from noon until 4pm, and the site itself opens each day at 10am. A 10-day certification course is offered by Colombia Paragliding at the \u00c1guilas site with additional flight time in Chicamocha. It costs COP$2,300,000 including transportation, meals, and lodging.\n\nThe views are quite spectacular from above Bucaramanga. (Bring your camera!) The best way for pilots to judge the winds is by observing the _chulos_ (black vultures) as they fly and glide high above. At the fly site there is also a snack bar. The place gets crazy crowded on weekends and on holidays. For a fee of COP$25,000 you can get a DVD of your flight.\n\n##### **Floridablanca**\n\nThere aren't loads of reasons to make a special trip to Floridablanca, which has evolved to become essentially a southeastern suburb of Bucaramanga, five kilometers away, but a good one is to take a bite out of one of their famous _obleas,_ crisp paper-thin wafers filled with gooey and delicious _arequipe_ (caramel spread). **Obleas Floridablanca** (Cra. 7 No. 5-54, tel. 7\/648-5819, 10am-8pm daily) is the most famous _oblea_ factory of them all. They've been around since 1949 and as you can imagine have their share of loyal customers. There are around 30 types of _obleas_ you can order, although there really is no need to go beyond the classic _oblea_ with just _arequipe._ The names of the _obleas_ are whimsical. Two of the more popular ones are the _amor eterno_ (eternal love), which has _arequipe,_ cheese, and blackberry jam, and the _noviazgo_ (courtship), which has _arequipe_ and cheese. They also have do-it-yourself _oblea_ kits that you can take back home with you.\n\na resident yellow-footed tortoise at the Jard\u00edn Bot\u00e1nico Eloy Valenzuela\n\nThe **Museo Arqueol\u00f3gico Regional Guane** (Casa de la Cultura, Cra. 7 No. 4-35, tel. 7\/619-8181, 8am-noon and 2pm-6pm Mon.-Fri., COP$1,000) is in need of some love, but the collection of ceramics on display is impressive and extensive. In the courtyard are some pre-Hispanic designs that were found on a large boulder.\n\nAlong the manicured lawns of the **Jard\u00edn Bot\u00e1nico Eloy Valenzuela** (tel. 7\/634-6100, 8am-5pm daily, COP$4,000) you can wander along paths (sharing them with turtles) and view enormous ceibas and other trees. If you look closely in the tops of trees you might even see some sloths. The R\u00edo Fr\u00edo (it's a stream, really) flows through the gardens. The gardens were revamped and reloaded in 2012. Security is present in the park, and there are always visitors, but be on your guard at some of the far reaches of the gardens. It's not terribly easy, but you can get to the gardens by way of MetroL\u00ednea from Bucaramanga.\n\n##### **Gir\u00f3n**\n\nWhile Bucaramanga is a pulsating tribute to the economic growth of modern Colombia, nearby Gir\u00f3n, under 15 minutes and 12 kilometers (7 miles) away to the west, is a living reminder of the colonial past, at least in the town's historic center: The population of Gir\u00f3n is around 150,000 today! It's somewhat surprising to see, in this age of television, Internet, and shopping malls, how the plaza is still the main meeting place in Gir\u00f3n, as it has been since the 17th century. Locals take pride in their town, and winners of the best facades, doors, and windows contest proudly display their plaques in front of their white-washed _tapia pisada_ (adobe) homes.\n\nWhile it is easy to visit at any time, on weekends Gir\u00f3n has a festive air to it as city folk from Bucaramanga and other day-trip visitors stroll the town's cobblestone streets. There isn't much in the way of tourist sights here, but be sure to visit **Bas\u00edlica Menor San Juan Bautista** and the **Parque las Nieves,** and walk along the _malec\u00f3n_ (wharf).\n\nYou might want to consider making Gir\u00f3n your base, instead of Bucaramanga. It's got charm, it's quiet, and traveling back and forth to the metropolis is not an issue. The trip takes under 15 minutes, and taxis will cost only around COP$10,000. You can also get to Gir\u00f3n on the MetroL\u00ednea system.\n\nM **Gir\u00f3n Chill Out Hotel Boutique** (Cra. 25 No. 32-06, tel. 7\/646-1119, www.gironnchillout.com, COP$144,000 d) is run by an Italian couple (hence the Italian flag), and while the name suggests an Ibiza atmosphere, the hotel is in a remodeled colonial house. It's quiet, cute, and homey, and they serve authentic Italian food. **Las Nieves** (Cl. 30 No. 25-71, tel. 7\/681-2951, www.hotellasnievesgiron.com, COP$116,000 d with a\/c, COP$82,000 d with fan) is right on the plaza, and the best rooms, although at the same time the most used rooms, are the six that have a balcony overlooking the plaza. The interior of the hotel is full of palm trees and greenery, and the owner's dog will let you pet him\/her. There are about 30 rooms, many full of twin beds. It's OK.\n\n**La Casona** (Cl. 28 No. 28-09, tel. 7\/646-7195, www.lacasona-restaurante.com, noon-8pm Tues.-Sun., COP$16,000) is a spiffy old place, and they have a fun _onces_ (tea time) menu for late afternoon tea, Colombian style: You get a _tamal_ (tamale), cheese, breads, and hot chocolate.\n\n#### M **CA\u00d1\u00d3N DEL CHICAMOCHA**\n\nWith an absolutely spectacular location above the Ca\u00f1\u00f3n del Chicamocha, the privately run **Parque Nacional del Chicamocha** (PANACHI, 54 km\/34 mi south of Bucaramanga, tel. 7\/639-4444, www.parquenacionaldelchicamocha.com, 9am-6pm Tues. and Thurs., 9am-7pm Fri.-Sun., COP$40,000 admission plus round-trip gondola ride) is a mostly cheesy amusement park geared towards Colombian families, but the views? Insert your favorite superlative here. This privately run park has several attractions, like an ostrich farm, areas that celebrate Santander culture and traditions, extreme sports, and soon a water park, but most people opt for the cable car ride (6.3 kilometers\/3.9 miles) across the canyon over the R\u00edo Chicamocha. For locals and for tourists who are opposed to backtracking, this is not just a nice excursion, it's a means of transportation. On weekends, holidays, and during the Christmas and Easter holidays, from Bucaramanga you can take a PANACHI bus (cell tel. 316\/696-3780) to the park, as a day trip. To get there using public transportation, buses leaving from the Bucaramanga bus terminal bound for San Gil can drop you off at the park. (There are always buses passing the park between San Gil and Bucaramanga, and you can flag them down and hop on.) If you happen to be in that part of town, buses also depart for San Gil from the Papi Quiero Pi\u00f1a store in Floridablanca. By the way, the vistas from the road that hugs the canyon high above the R\u00edo Chicamocha make that right-hand side window seat from Bucaramanga worth fighting for.\n\na gondola ride across the Ca\u00f1\u00f3n del Chicamocha\n\n##### **Mesa de los Santos**\n\nThe village of Los Santos is on the other side of the canyon from the PANACHI. In this area, known as Mesa de los Santos, there are a number of country homes for weekenders from Bogot\u00e1, a few places to stay, and numerous outdoorsy things to do, like visit El Duende waterfall. It ranks second in the world for annual number of tremors at 390 per month (a rate of about one every two hours).\n\nThe **Refugio La Roca** (Km. 22 in La Mojarra, cell tel. 312\/333-1480, www.refugiolaroca.blogspot.com, COP$60,000 d shared bath) is a kind of hippy-ish pad for \"rock climbers, walkers, artists, and other dreamers.\" Their specialty is rock climbing and it's got quite a jaw-dropping view, overlooking the Chicamocha canyon. Here you can camp for COP$10,000 per night (they also have tents and sleeping bags for rent). There is also one dorm room with four beds for COP$80,000.\n\nAt the other end of the spectrum is **El Roble** hotel (after La Mesa toll booth, Mesa de los Santos, tel. 1\/232-8595, www.cafemesa.com, COP$398,000 d). The rate includes a tour of the on-site coffee plantation. This lovely getaway is on a large certified organic coffee farm, set underneath towering oak trees, and teeming with over 100 species of birds. **Ecoposada Vina de Aldana** (cell tel. 317\/270-5077 or 300\/438-5522, ecoposada.dealdana@gmail.com, COP$80,000 d) is a guesthouse with 20 rooms at a vineyard, where you can take walks, view birds, and take a tour of the vineyard.\n\n#### **SAN GIL**\n\nRafting, paragliding, caving, mountain biking, canyoning, hiking, birding, and rappelling are all within reach in San Gil, Colombia's outdoor adventure capital. This spry city (pop. 43,000) on the steep banks of the R\u00edo Fonce is 95 kilometers (59 miles) southwest of Bucaramanga. It caters to international tourists. Even if your idea of \"adventurous\" is merely being in Colombia, the breathtaking Santander scenery of canyons, rivers, waterfalls, and mountains is more than enough reason to warrant a visit.\n\nDuring the late 19th and early 20th centuries, San Gil and nearby towns built their prosperity on quinine, coffee, cocoa, and tobacco cultivation. Today, old tile-roofed hangars to dry tobacco, known as _caneys,_ still dot the landscape. Today it is a peaceful place to visit and a top tourist destination. San Gil can feel claustrophobic, but accommodations in San Gil are plentiful, comfortable, and inexpensive, and it's got Gringo Mike's.\n\nThere's no need to stay in bustling San Gil in order to enjoy the many outdoor activities that the area offers. You can easily organize rafting trips or paragliding adventures from quieter and more charming towns such as Barichara (20 kilometers\/12 miles north).\n\n##### **Sights**\n\nOn the R\u00edo Fonce about a 15-minute walk from town is the **Jard\u00edn Bot\u00e1nico El Gallineral** (8am-5pm daily, COP$8,000). This used to be just a park, but it has been given a fussy makeover to apparently make it more appealing to tourists, and now it is a botanical garden. There are cute stalls selling handicrafts, sweets, and coffee along the park's orderly paths. It is a pretty place, and a late-afternoon walk among the towering trees is a nice plan. A restaurant on-site is open for lunch.\n\n##### **Shopping**\n\nThe best place in San Gil to browse handicrafts is at **Corporaci\u00f3n Patrimonio Guane, Agata y Yarigui** (Cra. 10 No. 9-67, cell tel. 313\/892-9681).\n\n##### **Recreation**\n\n###### **RAFTING AND KAYAKING**\n\nThree rivers near San Gil offer some excellent rafting adventures. The **R\u00edo Fonce,** whose banks the town stands on, is the closest and one of the best suited for rafting. It's a category II-III. It's fine year-round, although in March and April the water level is higher. A rafting trip on the Fonce costs about COP$30,000 for a 90-minute trip. The **R\u00edo Su\u00e1rez** is a category III-V river. The starting point is about an hour's drive towards Bogot\u00e1. You'll definitely get wet on this one. The trip leaves at 10am, returning at 4pm, and costs COP$125,000. You're on the water for about 2.5 hours. The third river is the **R\u00edo Chicamocha,** but many consider the previous rivers to be the best.\n\nThe R\u00edo Fonce flows through San Gil.\n\n**Colombia Rafting Expeditions** (Cra. 10 No. 7-83, tel. 7\/724-5800, www.colombiarafting.com) is considered the best rafting company in town. They focus exclusively on river activities. The walls of their small office are covered with diplomas and certificates earned by their team of experienced guides. They can organize trips down all the rivers, determining which one is right for you based upon on skill level, your sense of adventure, and water levels. They also do kayak trips. This company takes safety concerns very seriously and conducts safety training exercises in English.\n\nThree-day kayaking courses (four hours per day starting at 8am) are offered by Colombia Rafting. These cost COP$400,000 and take place on the R\u00edo Fonce. They can also arrange for kayaking on the R\u00edo Chicamocha. They also rent out kayaks to those who can demonstrate their level of experience. You can also try your hydrospeeding (riverboarding) skills on the R\u00edo Fonce. That costs COP$45,000.\n\n###### M **PARAGLIDING**\n\nThere are two main paragliding areas near San Gil. One is the spectacular **Ca\u00f1\u00f3n del Chicamocha** and the other is 16 kilometers (10 miles) away at **Las Vueltas** in Curit\u00ed. Tandem paragliding trips over the Chicamocha, of about a 45-minute duration, cost around COP$170,000. That price includes transportation to and from the landing and pickup sites. Chicamocha paragliding flights take place in the mornings. At the windy Las Vueltas location, it will cost you about COP$60,000. Those flights are held in the afternoon. As far as courses go, **Colombian Paragliding** (cell tel. 312\/432-6266, www.colombiaparagliding.com) has the best reputation. They are based near Bucaramanga.\n\n###### **WATERFALLS AND RAPPELLING**\n\nThe **Cascadas de Juan Cur\u00ed** (road to Charal\u00e1, COP$7,000) are quite close to San Gil and easily reached on public transportation or by bike. These falls, about 18 meters high, are privately owned by two neighbors who are fierce rivals! For a more rustic climb through the jungle to reach your refreshing goal, go to the second entrance (Donde Efigenia). It's about a 15-minute hike, and it can be treacherous at points. Wear some shoes you don't mind getting muddy and wet. And bring a bathing suit to cool off in one of the pools. You can camp there as well. It's a nice excursion. If you don't want to sweat and struggle at all, take the first entrance. To test your rappelling skills here, contact **P\u00e1ramo Extremo** (Cra. 4 No. 4-57, tel. 7\/725-8944, www.paramosantanderextremo.com). They can organize an excursion, with all the safety equipment and an experienced guide, for COP$45,000.\n\n###### **CAVING**\n\nSeveral caves around San Gil make for good exploring. The **Cueva Indio** is one of the most popular. It's filled with bats, and you don't really have to do much bending over to explore. It is near the town of P\u00e1ramo, just beyond the Cascadas de Juan Cur\u00ed. An excursion including equipment and a guide costs COP$25,000, but that doesn't include transportation. Contact **P\u00e1ramo Extremo** (Cra. 4 No. 4-57, tel. 7\/725-8944, www.paramosantanderextremo.com) in the town of P\u00e1ramo.\n\nThe **Cueva Vaca,** near Curit\u00ed, is the most challenging of the caves in the area. You will be in water and mud the entire time you are underground, and at one point you'll have to swim underwater to get through to the next cave. It's action packed and there are some tight squeezes as well, but the adventure is worth it. There are lots of stalactites and stalagmites and bats to see. It costs COP$25,000 plus about COP$3,000 in bus transportation. **Colombia Rafting** (Cra. 10 No. 7-83, cell tel. 311\/283-8647, www.colombiarafting.com) or other outfitters can organize a trip here.\n\nThe Cascadas de Juan Cur\u00ed are a great day trip from San Gil.\n\nThe **La Antigua** cave is on the road towards Barichara. **El Dorado Hostel** (Cl. 2 No. 8-55, tel. 7\/723-7588, www.eldoradohostel.com) organizes an extreme trip that includes the cave plus canyoning, rappelling, and two waterfall descents. All that adventure during just five hours! This trip costs COP$80,000 including transportation.\n\nThe Medell\u00edn-based outfit **Expedici\u00f3n Adventure** (cell tel. 314\/258-9499, www.expedicionadventure.blogspot.com, expedicionadventure@gmail.com) specializes in unique 3- to 20-day caving trips to mostly unexplored and unspoiled areas in Santander. The starting point is usually in Barbosa, a town between Tunja and Barichara.\n\n###### **BIKING**\n\n**Colombian Bike Junkies** (Cl. 12, No. 8-35, cell tel. 316\/327-6101 or 313\/411-5332, www.colombianbikejunkies.com), run by a pair from Seattle, Washington, and the United Kingdom, organizes downhill day-trip rides, crazy canyon adventures, multi-activity combos, and multi-day adventures. One day trip starts at 2,000 meters on the top of the Ca\u00f1\u00f3n del Chicamocha, going, down, down, down through beautiful countryside to the ghost town of Jordan. After a swim and lunch, there is yet one more downhill trip near Curit\u00ed. Some 50 kilometers (30 miles) of downhill riding! All on top-of-the-line mountain bikes. If you want to rent a cheap-o bike for the day, go to **Bicicleter\u00eda El Ring** (Cl. 7 No. 10-14, tel. 7\/724-3189).\n\n###### **SWIMMING**\n\nOn weekends and on holidays, families head to swimming holes to splash about. The atmosphere is joyous, and there's usually music and plenty of food and drink as well. (A little trash, too, unfortunately.) **Pozo Azul** is about five minutes by bus or taxi from San Gil (or a 20-minute walk). **Pescaderito** is in Curit\u00ed, about a 40-minute bus ride away, and there are five swimming holes in which to cool off. During the week it's quieter.\n\n###### **TOURS**\n\nYour hostel or hotel can organize any activity you are interested in doing, but in case you'd like to shop around, contact the following companies. **Planeta Azul** (Parque El Gallineral, tel. 7\/724-0000, www.planetaazulcolombia.com) is an agency that organizes rafting trips as well as a whole host of other activities, like bungee jumping (COP$46,000), caving (COP$40,000), rappelling (COP$40,000), paragliding (COP$60,000), and horseback riding (COP$95,000) to keep you stimulated. **Aventura Total** (Cl. 7 No. 10-27, tel. 7\/723-8888, www.aventuratotal.com.co) has a good reputation as well. They offer all-inclusive packages that include rafting, caving, and other activities as well as hotel accommodations. Aventura Total often organizes activities for large school groups. **Nativox** (Cra. 11 No. 7-14 Malec\u00f3n, tel. 7\/723-9999, www.nativoxsangil.com) is similar to the previous two.\n\n##### **Accommodations**\n\nGood and affordable lodging options are plentiful in San Gil. However, for more space, fresh air, or for more luxury, consider staying in Barichara, Mesa de los Santos, or Pinchote, all close by.\n\nOne of the first hostels in town catering to international backpackers, **Macondo Hostal** (Cra. 8 No. 10-35, tel. 7\/724-8001, www.macondohostel.com, COP$18,000 dorm, COP$55,000 d w\/bath) remains an excellent choice. Clean dorm rooms, popular with backpackers, and private rooms, popular with older travelers, quickly fill up\u2014make a reservation in advance! They have a hot tub, small garden, and hammocks for post-adventure relaxing. Staff are extremely knowledgeable, helpful, and great at organizing rafting, paragliding, and all other outdoor activities in the region. On the corner next door is the **Hostal Colombo Ingl\u00e9s** (Cra. 8 No. 9-133, tel. 7\/724-3787, www.hostelcolomboingles.com, COP$18,000 dorm, COP$70,000 d), which opened in late 2013. It's small and the staff are friendly. There is no English connection here; it's just a name.\n\nWelcome to _Sam_ Gil. Native entrepreneur Sam has two lodging options in the center of town. Super-clean **Sam's VIP Hotel** (Cr. 10 No. 12-33, tel. 7\/724-2746, www.samshostel.com, COP$17,000 dorm, COP$70,000 d) has a great location overlooking the plaza. The terrace is a great place for hanging out in the evenings. Plus there's a teeny pool and a sauna. His second place, with 11 rooms, is **La Mansion de Sam Hotel Boutique** (Cl. 12 No. 8-71, tel. 7\/724-6044, <http:\/\/hotelmansionsangil.com>, COP$70,000-100,000 d) which is set in an old house just a block from the main plaza. Sam has an inviting pub that specializes in steaks, ribs, and beer. Colorful artwork by local artist \"Rosenkranz\" adorns the walls of the rooms. La Mansion has a lot of character, cool decoration, and big rooms, some with balconies. The only drawback is that cars sometimes park in the interior patio.\n\nThe multi-story **Hostel Santander Alem\u00e1n** (Cra. 10 No. 15-07, tel. 7\/724-0329, www.hostelsantanderalemantv.com, COP$20,000 dorm, COP$60,000 d) is half a block from the local bus station. It's very clean. Friendly folks, but there's nothing German about it.\n\nIf you'd like to get away from the backpacker scene but still pay close to backpacker prices, there are three clean cheapies that may fit the bill. If you stay at the **Hotel Capri** (Cl. 10 No. 9-31, tel. 7\/724-4218, hotelcaprisangil@yahoo.es, COP$40,000), get a room on the third floor overlooking the street. **The Hotel Abril** (Cl. 8 at Cra. 10, tel. 7\/724-8795, hotelabrilsangilss@yahoo.es, COP$45,000 pp) has 28 rooms, good fans, wireless Internet, and hot water. With just six rooms, M **Posada Familiar** (Cra. 10 No. 8-55, tel. 7\/724-8136, COP$45,000 pp high season) makes you feel at home. The patio is filled with flowers and plants, you can cook in the kitchen, and the owner is extremely nice.\n\nTo enjoy the peace of the countryside and charm of a colonial town but still be within easy striking distance of San Gil restaurants and activities, consider staying in the hamlet of **Pinchote**. M **Hotel Boutique Wassiki** (Km. 3 V\u00eda San Gil-Bogot\u00e1, tel. 7\/724-8386, www.wassiki.com, COP$164,000-227,000 d) is an excellent upscale hotel and offers well-appointed and airy rooms, comfortable common areas, a beautiful dining room, lots of hammocks, and a pool. It's got a fine view of the valley below and is within walking distance of the idyllic Plaza Principal of Pinchote.\n\n##### **Food**\n\nMostly Tex-Mex M **Gringo Mike's** (Cl. 12 No. 8-35, tel. 7\/724-1695, 8am-11:45am and 5pm-10pm daily, COP$18,000) is paradise for Americans who have been on the road a while. Guac and chips, barbecue burgers, black bean burgers, Philly cheese steak sandwiches, burritos, and even breakfast burritos. It's a bummer it isn't open for lunch, though. Wait staff are on the nonchalant side, and the place is full of tourists, but who cares? The margaritas are perfect!\n\nTo brush elbows with the locals try **Rogelia** (Cra. 10 No. 8-09, tel. 7\/724-0823, 7am-7:30pm daily, COP$12,000) or **Man\u00e1** (Cra. 10 No. 9-42, lunch daily, COP$12,000), which is a popular place for lunch, and inexpensive, too. Try the grilled chicken stuffed with ham and cheese, but don't expect gourmet nor charm.\n\nThe best aspect about the **Gallineral Restaurante** (Parque Gallineral, cell tel. 300\/565-2653, 8am-5pm daily, COP$20,000) is its lush setting.\n\nOn a second-floor open-air terrace, **La Terraza de Sevilla Video-Bar** (Cl. 10 No. 9-09, tel. 7\/724-3422, 8am-2am Mon.-Sat., COP$12,000) specializes in grilled hamburgers and hot dogs. It's the place to drink beer and watch soccer. For a coffee or drink and a friendly atmosphere, **La Habana** (Cra. 9 No. 11-68, tel. 7\/724-6279, 9am-midnight Mon.-Thurs., 9am-2am Fri.-Sat., 4pm-midnight Sun.) is a good choice.\n\nThe market is small, and the atmosphere is peaceful. As you walk through the stalls, you may only hear the hushed tones of the vendors. For a huge fresh fruit juice or just plain fruit in the morning, this is the place to go.\n\n##### **Getting There and Around**\n\nThe main **Terminal de Transportes** (V\u00eda al Socorro, tel. 7\/724-5858) for buses to the major cities, such as Bucaramanga, Tunja, and Bogot\u00e1, is five minutes out of town, on the other side of the river. The journey to Bucaramanga by bus takes 2.5 hours and costs COP$15,000. Traveling to Tunja by bus will take 3-5 hours, to Bogot\u00e1 7, and to Santa Marta or Medell\u00edn about 12 (v\u00eda Bucaramanga). Taxis to and from the main bus terminal to the town center cost COP$3,200.\n\nA smaller bus terminal for nearby towns is on Carrera 15 at Calle 11. It serves towns such as Barichara, Charal\u00e1, Curit\u00ed, and Pescadero. It doesn't have an official name, but some refer to it as the Mini Terminal. Buses to Barichara depart every half hour 6am-6:30pm and cost COP$3,800.\n\n#### M **BARICHARA**\n\nIn 1975, when it was declared a national monument, Barichara (pop. 8,000) was named the most beautiful pueblo in Colombia. Despite its popularity with weekenders and a steady stream of international visitors, it hasn't lost its charm. This old tobacco town of sloping cobblestoned streets and white-washed colonial-era homes is permanently blessed with bright blue skies and warm temperatures. Located 20 kilometers (12 miles) northwest of San Gil, the town is on a plateau that overlooks the R\u00edo Su\u00e1rez. Don't skimp on your time here.\n\nBarichara is one of the country's most beautiful pueblos.\n\n##### **Sights**\n\nOn the serene Parque Principal is the **Templo de la Inmaculada Concepci\u00f3n,** with two grandiose towers that soar 22 meters into the air. When lit up at night, the sandstone church is particularly striking. The church was completed around 1780. On the west side of the park is the mayor's office. Next to it is the Casa de Cultura.\n\nUp the picturesque Calle 6, at the top of the hill is the **Capilla de Santa B\u00e1rbara,** a Romanesque-style church that is a popular place for weddings. There is a cheesy sculpture garden, **Parque de las Artes,** at the edge of the R\u00edo Su\u00e1rez canyon.\n\nThere are two other colonial churches to see, the **Capilla de Jes\u00fas** (Cra. 7 at Cl. 3), next to the cemetery, and the **Capilla de San Antonio** (Cra. 4 at Cl. 5). All around town you'll see houses and walls that utilize the _tapia pisada_ adobe technique, and often, on these brilliantly white walls you'll see a small patch of the mud interior left exposed on purpose, to show passersby that it's not just a modern brick construction but real _tapia pisada._\n\nBarichara is the birthplace of Pres. Aquileo Parra G\u00f3mez, who was the 11th president of the Estados Unidos de Colombia. His childhood home, **Casa Aquileo Parra G\u00f3mez** (Cl. 6 at Cra. 2), has been extremely well preserved and is an excellent example of typical 19th-century Barichara architecture. The site is also a handicraft workshop for the elderly, who make woven bags and other items out of the natural fiber _fique,_ which are sold for a pittance. It is an excellent social program, and they seem to have a good time. They are there Monday through Thursday.\n\nthe Camino Real between Barichara and Guane\n\n###### M **CAMINO REAL**\n\nA must-do activity in Barichara is to take the 5.3-kilometer (3.3-mile) Camino Real path to the pueblo of Guane. It's a lovely path that zigzags down from the plateau of Barichara through farmland, affording nice views of the countryside and an excellent opportunity to burn off a few vacation calories. Parts of the path are lined with stone walls that have been there for centuries.\n\nBefore the conquest, indigenous tribes throughout what is now Colombia traded crops and goods with each other utilizing an extensive network of footpaths. These trails meandered through the countryside of present-day Santander, Boyac\u00e1, Norte de Santander, Cundinamarca, and beyond. During Spanish rule, the paths continued to be a major means of communication between colonial towns, and the networks became known as Caminos Reales.\n\nIn the late 19th century, a German, Geo von Legerke, restored the Barichara-Guane Camino Real and built a stone bridge across the R\u00edo Su\u00e1rez in order to improve transportation to the mighty R\u00edo Magdalena.\n\nThe hike down takes two hours, and you don't need a guide: It's well marked, well trodden, and safe. To get to the trailhead, walk east along Carrera 10 to the Piedra de Bol\u00edvar, where you'll see the stone path leading down towards the valley.\n\nIn Guane you can check out the small **Museo Isaias Ardila D\u00edaz** (Parque Principal, hours vary), which has three rooms, one on paleontology (fossils), the next on archaeology (mummy), and a third on colonial life in rural Santander. _Sabaj\u00f3n,_ which is the Colombian version of eggnog, is the sweet specialty in Guane, and it is sold in various shops around the park.\n\nIf you are not up for the hike a (cute) bus departs the Parque Principal in Barichara at 6am, 9:30am, 11:30am, 2:30pm, and 5:30pm (it returns 30 minutes later from Guane).\n\n##### **Festivals and Events**\n\nLittle Barichara proudly hosts two annual film festivals. The **Festival Internacional de Cine de Barichara** (www.ficba.com.co) takes place in June, and the **Festival de Cine Verde** (www.festiver.org), an environmentally themed festival, is held every September.\n\nCobblestone streets in the colonial village of Guane\n\n##### **Shopping**\n\nBarichara has always been a magnet for artists and craftspeople, and many have shops in town.\n\nThe **Fundaci\u00f3n Escuela Taller Barichara** (Cra. 5 No. 4-26, tel. 7\/726-7577, www.tallerdeoficiosbarichara.com, 8am-7pm Mon.-Thurs., 8am-10pm Fri.-Sat., 8am-4pm Sun.) is a gallery, museum, school, shop, and restaurant, all wrapped up in one. Occasional photography and painting exhibitions are held at this lovely cultural center, decorative objectives traditional from the area are always on display, ceramics and other items made by students are for sale, and anyone can take a month-long or longer course here. They offer dozens for free, and the Cruces restaurant is the best restaurant in town (it's open on weekends and holidays).\n\nAn interesting stop to make is at the **Taller de Papel de Fique** (Cl. 6 No. 2-68, no phone, 8am-3pm Mon.-Thurs.). At this workshop, craftspeople make beautiful paper out of the natural fiber of _fique._ On sale in their small store are cards, stationery, and handicrafts, all produced using that natural fiber. They are also now experimenting with other paper made from pineapple leaves. Short tours explaining the paper-making process are given, and for this there is a small charge.\n\nOne of the best-known ceramic artists in town is **Jimena Rueda** (Cra. 5 No. 2-01, cell tel. 314\/400-5071). In addition to browsing her work, ask about the famous rustic handmade pottery of the Guane people. There is only one person who knows and uses this technique: **Ana Felisa Alquichire.** Do\u00f1a Ana Felisa has been declared a living national cultural treasure by the Colombian presidency.\n\n**Galer\u00eda Anil** (Cl. 6 No. 10-46, cell tel. 311\/470-1175) is the studio for local artists Jasm\u00edn and Carlos.\n\n##### **Accommodations**\n\nWith its growth in popularity, accommodations options to fit all budgets and styles have popped up in Barichara.\n\nBackpackers and budget travelers have several options in Barichara. The M **Color de Hormiga Hostel** (Cl. 6 No. 5-35, cell tel. 315\/297-1621, <http:\/\/colordehormiga.com\/hostel.html>, COP$45,000 d) used to house teachers from a neighboring school. It's decorated with institutional furniture that was left behind and kept the groovy tiled floors as they are. Funky! There are seven small rooms for one or two people, each with its own private bath. The kitchen is open for use by guests. The M **Reserva Natural** (Vereda San Jos\u00e9 Alto, cell tel. 315\/297-1621, COP$70,000 pp d), from the same owner as the Color de Hormiga Hostel, is a step up, with more luxury, more solitude, and a crazy bird show every morning while you have a healthy breakfast. Birds representing all colors of the rainbow appear like clockwork every morning to munch on pieces of banana and papaya to the delight of guests enjoying their breakfasts. This is about a 10-minute walk from town. The staff is incredibly friendly.\n\nThe **Tinto Hostel** (Cl. 6 No. 2-61, Bloque E Casa 1, tel. 7\/726-7725, www.hosteltintobarichara.com) is a friendly hostel. It's in a weird, mostly residential cul de sac, about a 10-minute downhill walk from town. It seems farther away than it actually is. They are helpful with organizing outdoor adventures in the San Gil area. The funky hostel award in Barichara goes to **Casa Bak\u00fa** (Cl. 5 No. 9-69, cell tel. 301\/419-2136, bakuhostal@hotmail.com). Baku, as in the capital of Azerbaijan. Like it or not, it's a social place. It's tiny. The common area, with a little homage to Bob Marley, is basically a garden with a bar and chairs. That's where they serve breakfast, which is included.\n\nOn the edge of town past the hospital, **Artepolis** (Cra. 2 at Cl. 2, cell tel. 300\/203-4531) was getting going when we arrived. It is a Frenchman's idea of creating a space for creative people to come and find creative inspiration in the marvelous setting of Barichara. Its formal and serious sounding name is the Centro Internacional de Encuentro y Formaci\u00f3n para el Arte y Cultura.\n\nAhh, the boutique hotel. Barichara didn't have them before; now it does! **La Nube** (Cl. 7 No. 7-39, tel. 7\/726-7161, www.lanubeposada.com, COP$330,000 d) was boutique before that word entered the Colombian hotel lexicon. It's still a comfortable choice. It has seven rooms and a good restaurant (breakfast not included), and the patio is a nice place for relaxing to the soothing sound of a fountain. **Achiotte Hotel Boutique** (Cl. 5 No. 3-52, tel. 7\/726-7512, COP$220,000 d) is a well-done, quiet hotel, with large rooms and common spaces filled with bamboo, flowers, and trees. You can shower here in the open air (nobody will see you!). At present the hotel has about nine rooms, which makes it really feel boutique. Along with a pool, there are plans to add several more rooms, which may change the feeling.\n\nM **El Cogollo** (Cra. 11 No. 7-37, cell tel. 311\/202-4391, www.baricharacogollo.com, COP$280,000) is a very comfortable boutique-type hotel with eight rooms. The hotel uses construction materials and techniques from the earth, such as _tapia pisada, bahreque, adobe,_ and stone. They operate the travel agency **Barichara Travel** (www.baricharaguanecito.com).\n\nFinally, there is the peaceful **Posada Sue\u00f1os de Antonio** (Cra. 9 No. 4-25, tel. 7\/726-7793, www.suenosdeantonio.com, COP$120,000 d), with five spacious rooms surrounding an interior patio.\n\n##### **Food**\n\nThe **Restaurante y Caf\u00e9 Las Cruces** (Cra. 5 No. 4-26, tel. 7\/726-7577, www.tallerdeoficiosbarichara.com, Fri.-Sun. and daily during high season, COP$28,000) is considered the top restaurant in Barichara. It's in the patio of the Fundaci\u00f3n Escuela Taller Barichara. **Plenilunio Caf\u00e9** (Cl. 6 No. 7-74, tel. 7\/726-7485, 6:30pm-10pm daily, COP$22,000) serves mostly Italian food but also has backpacker favorites like veggie burgers. There are just a handful of tables in this cozy spot, and most all of them are occupied by content foreign visitors on balmy Barichara evenings.\n\n**La Puerta** (Cl. 6 No. 8-51, tel. 7\/726-7649, www.baricharalapuerta.com, lunch and dinner daily high season, COP$22,000) is a beautiful place, candlelit at night. They serve tasty pastas and use local, organic ingredients when possible. **Casta\u00f1etos** (Cl. 6 No. 10-43, tel. 7\/726-7765, 6:30pm-10pm daily high season, COP$24,000) is the best pizzeria in town.\n\nAt the other end of town, near the canyon, is **Al Cuoco** (Cra. 4 No. 3B-15, cell tel. 312\/527-3628, noon-9pm or 10pm daily, COP$22,000), an Italian place run by a Roman. They make their own pasta.\n\nLocals throng to **El Balc\u00f3n de Mi Pueblo** (Cl. 7 No. 5-62, cell tel. 318\/280-2980, noon-5pm daily, COP$12,000) because they serve good, meaty Colombian food ( _cabro, carne oreada, churrasco_ ) without serving up Bogot\u00e1 prices! It's a cute place, up on the second floor. Another favorite is the lunch-only option **Misif\u00fa** (Cra. 6 No. 6 31, tel. 7\/726-7321, noon-6pm daily, COP$12,000). Their specialty is the local specialty _carne oreada,_ a dry and toothsome steak, reminiscent of beef jerky.\n\nFor a coffee or some of their world-famous (or at least pueblo-famous) _galletas de cuajada_ (cheese cookies), head to **Panader\u00eda Barichara** (Cl. 5 No. 5-33, tel. 7\/726-7688, 7am-1pm and 2pm-8pm daily). They've been around since 1954.\n\nThe best nightlife in town? Head to the **Mirador** bar on the west side of town overlooking the R\u00edo Su\u00e1rez at around 5:30pm. Free sunsets are included with the price of your \u00c1guila beer!\n\n##### **Getting There**\n\nMost visitors arrive either in their own transportation or by bus to Barichara. Buses from Bogot\u00e1 depart from the Autopista Norte Station. The journey to San Gil takes six hours or more, and you'll have to transfer in San Gil. It costs COP$30,000. From Bucaramanga, from the Piedecuesta terminal, a bus leaves at 4:45pm Monday-Friday. On Saturday the bus departs at 9am and on Sunday at 7:30pm. It takes three hours and costs COP$15,000. _Busetas_ leave San Gil every half hour from the Terminal de Transportes Monday-Sunday starting at 6:10am, with the last bus departing at 8:15pm. The 20-kilometer (12-mile) journey takes 45 minutes.\n\n### **Norte de Santander**\n\nThis department in the northeast of the country borders Venezuela to the east and Santander to the south. The two main places of interest are Pamplona and C\u00facuta, two very different cities. Pamplona is a charming and cool highland town that was important during the colonial era, though much of its colonial architecture has disappeared due to earthquakes and the march toward progress. To the north, the departmental capital of C\u00facuta is a large, boiling hot commercial city and gateway to Venezuela. Both cities are easily accessed by road from Bucaramanga. The southernmost area of Norte de Santander and the northernmost area of Catatumbo have been plagued with guerrilla and paramilitary activity in recent years and are best avoided.\n\n#### **PAMPLONA**\n\nThis historic and charming colonial town is a refreshing change from the _calor_ (heat) of C\u00facuta and Bucaramanga, set in a lush, agriculturally rich valley at 2,300 meters (7,500 feet). In addition to colonial remnants like the Casa de las Tres Mar\u00edas (now Museo de Arte Moderno Eduardo Ram\u00edrez Villamizar), Pamplona is known for being the home of abstract expressionist artist Eduardo Ram\u00edrez, and for being a surprisingly lively college town, home to the Universidad de Pamplona and thousands of students.\n\n##### **Sights**\n\nPamplona has its share of museums, and they are all easily visited in a day on foot. The best museum here is the **Museo de Arte Moderno Eduardo Ram\u00edrez Villamizar** (Cl. 5 No. 5-75, tel. 7\/568-2999, www.mamramirezvillamizar.com, 9am-noon and 2pm-5pm Tues.-Sun., COP$3,000), prominently located on the Parque Agueda Gallardo. This museum, in a lovingly restored 16th-century house, features the work of this modernist sculptor and painter and also puts on temporary shows of modern and contemporary Colombian artists. In the courtyard, surrounding a magnolia tree, are many Ram\u00edrez sculptures. Born in Pamplona in 1922, Ram\u00edrez passed away in Bogot\u00e1 in 2004.\n\nAround the corner is the **Museo Arquidiocesano de Arte Religioso** (Cra. 5 No. 4-87, tel. 7\/568-2816, 10am-noon and 3pm-5pm Wed.-Mon., COP$2,000). It houses oil paintings from masters such as Gregorio Arce y Ceballos and others, wood carvings dating back to the 17th century, and silver and gold ceremonial items.\n\nThe most interesting churches to check out include the imposing **Catedral Santa Clara** (Cl. 6 between Cras. 5-6), which dates back to 1584, and the **Ermita del Se\u00f1or del Humilladero** (Cl. 2 between Cras. 7-8), which is next to the cemetery, filled with above-ground tombs. It is famous for its realistic carving Cristo del Humilladero.\n\nThe **Casa Mercado** (Cl. 6 between Cras. 4-5) stands on the previous location of a Jesuit college; this covered market was built in 1920. The **Museo Casa Colonial** (Cl. 6 No. 2-56, tel. 7\/568-2043, www.casacolonialpamplona.com, 8am-noon and 2pm-6pm Mon.-Fri., free) packs quite a punch in its 17th-century abode. It includes exhibits on some of the native cultures from the area, touches on the independence movement and struggles of the early Colombian republic, and takes the visitor through to the 20th century.\n\nFinally, the small **Museo Casa Anzo\u00e1tegui** (Cra. 6 No. 7-48, 9am-noon and 2pm-5:30pm Mon.-Sat., COP$1,000) examines the life of General Jos\u00e9 Antonio Anzo\u00e1tgui and the fight for independence from Spain. It was in this house that this war hero died in 1819. He was the head of Bol\u00edvar's honor guard and was promoted to general following the Batalla del Puente de Boyac\u00e1.\n\n##### **Accommodations and Food**\n\nM **Hostal 1549** (Cl. 8B No. 8-64, Calle los Miserables, tel. 7\/568-0451, www.1549hostal.com, COP$130,000) has a big problem: Your room is so cozy that it will take an effort to get out and explore the town. Seven spacious rooms have big, comfortable beds, and many have fireplaces. The hotel has an adjacent restaurant where breakfast is served. At night locals gather to drink, but they are usually shown the door by 11pm.\n\nSomewhat quirky, the **Hotel Ursua** (Cl. 5 No. 5-67, tel. 7\/568-2470, COP$40,000 d) has a fantastic location right on the main park, and rooms, with beds that will do, come in all shapes and sizes. The restaurant serves inexpensive breakfasts and lunches.\n\nOnce home to a scribe to the Spanish authorities in the 18th century, **El Solar** (Cl. 5 No. 8-10, tel. 7\/568-2010, www.elsolarhotel.com, COP$110,000 d) is one of the most popular accommodation and restaurant options in Pamplona. It has 10 rooms, and 21 beds. Breakfast at their restaurant is included, as is wireless Internet.\n\n**Pierro's Pizza** (Cra. 5 No. 8B-67, tel. 7\/568-0160, 5pm-11pm daily, COP$20,000) is the most popular place for pizza (as well as a whole host of other favorites), and it's run by an Italian. Other favorites include **Sal y Pimienta** (Cra. 5A No. 8B-66, cell tel. 301\/496-2464, 5pm-10pm daily, COP$18,000), and **Restaurante P\u00edoko** (Cl. 5 No. 5-49 tel. 7\/568-3031, 7am-9pm, COP$18,000) specializes in trout dishes.\n\nVegetarians will want to hunt for the hard-to-find Hare Krishna-run **Majesvara** (Cra. 3B No. 1C-26, cell 310\/267-9307, 11am-2pm and 6pm-9pm Mon.-Sat., COP$8,000). If you find it try their set lunch or dinner menu.\n\nEvery town should have a place like **Stanco La Rokola** (Cl. 9 No. 5-23 Plazuela Almeida). It's a tiny nook on the Plazuela Almeida, where for about COP$2,000 you can order a shot of tequila, vodka, or rum, and then be on your merry way.\n\n##### **Getting There**\n\nPamplona is reached by bus from C\u00facuta. These leave on an hourly basis, and the two-hour trip costs about COP$10,000. You can also take a bus to Pamplona from Bucaramanga. It costs about COP$28,000 and takes under five hours. Pamplona's bus station is the spic and span **Terminal de Transportes** (Barrio El Camell\u00f3n), about a 10-minute walk from the town center.\n\n#### **C\u00daCUTA**\n\nMidway between Bogot\u00e1 and Caracas, the sizzling capital (pop. 637,000) of the Norte de Santander department straddles the border with Venezuela. Streets are lined with vendors selling cheap Venezuelan gasoline. A favorite and cheap beer served in restaurants and bars is Polar, straight from Venezuela. There is a constant stream of traffic crossing the border on the Puente Internacional in both directions. Venezuelans used to come to C\u00facuta for shopping, but now it is the Colombians who are going east to load up on goods. Nonetheless, there is always a Venezuelan presence in C\u00facuta, especially on weekends and holidays.\n\nIn recent years, the city has seen a large influx of persons fleeing violence in other parts of Norte de Santander and Arauca. During the 1990s, there was a bloody turf war between paramilitaries and leftist guerrillas. In 2008, in response to months of simmering tension, pop singer Juanes organized _Paz sin Fronteras_ (Peace Without Borders), a concert on the border between Colombia and Venezuela, to the delight of nearly 300,000 fans. Today the city is considered a safe place to visit.\n\nMany foreign travelers in C\u00facuta are there either to cross over into Venezuela so that they can extend their visit in Colombia on their tourist visa, or they are on their way to Caracas. Despite its reputation as a hot, uninteresting city, C\u00facuta is a pleasant place to explore for a couple of days.\n\nThe downtown area of the City of Trees is quite walkable, with trees lining its broad streets. There are a handful of republican period churches and buildings worth a look. And at night, particularly on weekends, restaurants, caf\u00e9s, and bars in the Caobos district are pleasant gathering places for Cucute\u00f1os of all ages.\n\nThe main tourist attraction is just outside of the city in Villa del Rosario, on the way toward Venezuela. That's where Colombia's Thomas Jefferson, General Francisco de Paula Santander was born, and also is where the first constitution for Gran Colombia was drafted. Sim\u00f3n Bol\u00edvar officially became the country's president here.\n\n###### **ORIENTATION**\n\nThe western boundary of C\u00facuta is basically the parched R\u00edo Pamplonita and, before that, the Avenida Libertadores. A lot of the big restaurants and nightlife spots are located here, and this is where the Ciclov\u00eda is held on Sundays. The Diagonal Santander represents the boundary of the downtown towards the north. It links the Terminal de Transportes in the northwest of the city to the stadium and finally merges into the Autopista Internacional (the road that leads to San Antonio, Venezuela) in the east. All the decent hotels and most sights of interest in C\u00facuta are located in the downtown area between Avenida 0 to the east and Avenida 6 to the west and Calle 8 to the north and Calle 13 to the south. Ventura Plaza Centro Comercial is a good landmark to remember. It is between Avenida 0 and the Diagonal Santander. The Aeropuerto Camilo Daza is in the northwest of the city.\n\n##### **Sights**\n\n###### **CITY CENTER**\n\nAll the sights in downtown C\u00facuta can easily be visited on foot. There is not a tourism culture here, so obtaining basic information such as opening hours and telephone numbers is not always easy. At the beautifully restored **Biblioteca P\u00fablica Julio P\u00e9rez Ferrero** (Av. 1 No. 12-35, Barrio La Playa, tel. 7\/595-5384, www.bibliocucuta.org, 8am-noon and 2pm-6pm Mon.-Fri., 9am-noon Sat.) there is always something going on: photography exhibits on old C\u00facuta, classes and workshops, concerts. The library building, declared a national monument, originally served as a hospital in the late 18th century. The library also operates the **Museo Centenario** (Cl. 14 No. 1-03, tel. 7\/595-5384, www.museocentenario.com, 8am-noon and 2pm-6pm Mon.-Fri., 9am-noon Sat.), which is often the host of art exhibitions and events. It is a newish addition to the cultural scene. It is open only when there is something going on.\n\nThe **Palacio de Gobierno Departamental** (Clls. 13-14 and Avs. 4-5, not open to the public) is noteworthy for its republican, neoclassical architecture. **Torre del Reloj** (Cl. 13 and Avs. 3-4) houses a clock that plays the national anthem on its bells at noon each day. The bells came from Italy in the early 19th century. You can ask to check it out. From the top, if you can squeeze past the bells you can get a nice view of the city with the **Monumento Cristo Rey** (Av. 4 at Cl. 19) in the distance. The Torre del Reloj is part of the **Casa de la Cultura** complex. It also hosts art exhibitions. The neoclassical **Catedral San Jos\u00e9** (Av. 5 No. 10-53) faces the shady **Parque Santander** (Avs. 5-6 and Clls. 10-11). Finally, be sure to walk along Calle 10 or Calle 11. Canned music blares from loudspeakers on the streets all day long.\n\nThe **Banco de la Rep\u00fablica** (Diagonal Santander No. 3E-38, tel. 7\/575-0131, www.banrepcultural.org\/cucuta, 8am-11:30am, 2pm-6pm Mon.-Fri.) always has an art exhibition on view, usually featuring a Colombian artist. Concerts are also held at their theater. It is just outside of the city center.\n\nParque Gran Colombiano\n\n###### **PARQUE GRAN COLOMBIANO**\n\nThe major historical site in C\u00facuta is actually seven kilometers away in **Villa del Rosario** on the road towards San Antonio del T\u00e1chira, Venezuela. The **Parque Gran Colombiano** is in the middle of the busy highway that leads to the Venezuelan border. Although much of the park is green space where couples kiss under palm trees and others jog or walk their dogs, the most important historical sites within the park are the **Casa Natal del General Santander** (Km. 6 Autopista Internacional, tel. 7\/570-0265, 8am-11am and 2pm-5pm Tues.-Sat., 9am-5pm Sun. and holidays, free) and the ruins of the **Templo del Congreso.** The museum tells the story of General Francisco de Paula Santander and is set in his childhood home. The Templo del Congreso is where Gran Colombia's Constitution of 1821 was drafted and where Sim\u00f3n Bol\u00edvar was sworn in as president (and Santander as vice president). The church was badly damaged in the C\u00facuta Earthquake of 1875, and only the dome, in a different style altogether, was rebuilt. It's unavoidable that there is a bronze statue of Bol\u00edvar inside the ruins. The congress met for over a month to draft the constitution, and on their breaks, they would rest under the shade of a huge tamarind tree. It's still there, right in front of the church.\n\nAcross the highway is the **Casa de la Bagatela,** which was the seat of the executive branch of Gran Colombia. It was named La Bagatela in honor of independence figure Antonio Nari\u00f1o, who penned a revolutionary paper in Bogot\u00e1 by that same name. Don't bother scurrying across the highway to the Casa de la Bagatela, as there's nothing much to see.\n\nYou can take a shared taxi or a _buseta_ (small bus) bound for San Antonio del T\u00e1chira at the Ventura Plaza mall. These cost around COP$2,500. Ask to be dropped off at the Parque Gran Colombiano. Private taxis cost about COP$7,500 from downtown C\u00facuta.\n\n##### **Nightlife**\n\nThere are three trendy nightlife areas: the lovely and leafy Caobos neighborhood, the _malec\u00f3n_ (wharf) along the R\u00edo Pamplonita, and in the Centro Comercial Bol\u00edvar.\n\nIn Caobos, in the newly branded Zona E (because many spots are along the Avenida 1E) there are dozens of pub-like places where you can have a bite and\/or have a brew outside on the terrace. It's not a bad atmosphere. Pubs are all more or less the same, but be on the lookout for these popular spots: **American Pub Radio** (Av. 1E No. 16-20, tel. 7\/594-8398, www.americanpubradio.com); **Saxo Pub** (Cl. 16 No. 1E-13 tel. 7\/571-4270, www.saxopub.com, 4pm-2am daily); and **British Pub** (Cl. 17 No. 1E-05, 5:30pm-2am Tues.-Thurs., 5:30pm-3am Fri. and Sat.).\n\nTo the east and along the banks of the R\u00edo Pamplonita, many of the restaurants in town evolve into party places as the night wears on. Another very C\u00facute\u00f1o way to party is to grab your friends and pick up some booze, drive down to the Avenida de los Libertadores and park, and pump up the _vallenatos_ (ballads accompanied by accordions) or, occasionally, electronica. To the north the nearby **Centro Comercial Bol\u00edvar** has a number of bars, including a couple of gay clubs, as well as clubs catering to salsa aficionados.\n\n##### **Recreation**\n\nOn Sunday morning, head to the Avenida de los Libertadores\/Paseo de los Proceres\/ _malec\u00f3n_ along the R\u00edo Pamplonita. You can work up a sweat with the locals here as they ride their bikes and jog during their mini **Ciclov\u00eda** (7am-1pm). Bike rentals aren't available, but you can always jog or people-watch.\n\nTo arrange an organized tour of points of interest in Norte de Santander contact **Crischarol Tours** (Cl. 13 No. 5-60, tel. 7\/572-0407, www.crischaroltours.blogspot.com). They offer day-trip tours to Pamplona.\n\n##### **Accommodations**\n\nAstonishingly, C\u00facuta, the sixth largest city in Colombia, did not have any international or even national chain hotels until Holiday Inn announced its arrival in late 2014. With 98 rooms, and located across from the Ventura Plaza mall, it is a much needed addition. Most downtown hotels have seemingly been around forever, the kind where you are handed the remote control in its plastic cover when you check in, and staff are dressed in outdated uniforms.\n\nOutside of town, in Villa del Rosario, there are other hotel options. These are popular places on the weekends. Continuing onward into San Antonio, hotel options are abysmal. They are cheap, but abysmal.\n\nIt once probably seemed very flashy, but retro **Hotel Tonchal\u00e1** (Av. 0 at Cl. 10, tel. 7\/575-6444, www.hoteltonchala.com, COP$209,0900 d) has kept up with the times by updating the rooms (there are about 100 of them). The hotel has a pool, gym, and sauna. It's within easy walking distance of Ventura Plaza Centro Comercial.\n\nStaff dressed in bright pink uniforms at the **Hotel Arizona Suites** (Av. 0 No. 7-62, tel. 7\/573-1884, www.hotelarizonasuites.com, COP$245,000) are so exceptionally friendly, even the most persnickety of guests will find it hard to lodge any complaint. Rooms are fine, a little on the small side, and definitely overpriced. The restaurant overlooks a small pool, and the spa area was renovated in 2012. The location is OK, not great, near two busy streets and about a 10-minute walk to sights downtown.\n\nIf you would rather be surrounded by greenery than concrete you might want to consider the **Hotel Villa Antigua** (Autopista San Antonio-Villa del Rosario, tel. 7\/570-0399, www.hotelvillantigua.amawebs.com, COP$120,000 d). It is geared mostly towards Colombian families and groups as a weekend place to kick back and drink by the large pool. During the week you'll likely have the place to yourself. They have cabanas of several sizes and then spacious regular hotel rooms. Breakfast is served in an outdoor restaurant overlooking the pool, but it smells like either gas or strong floor wax. Wireless Internet is available in the lobby. The Parque Gran Colombiano is just across the street, but you must be very careful crossing the street. This is the main drag to San Antonio, and cars and buses absolutely zoom by. At the park you can go for a morning or late afternoon walk. If you want to go into town during the day or at night, it is not an issue: Taxis are cheap and it takes about 15 minutes.\n\n##### **Food**\n\nThe most typical food from C\u00facuta includes _hayacas cucutenas,_ similar to tamales; _mute,_ a meaty stew; and _pastel de garbanzo_ (a fried garbanzo bean pastry).\n\nThe M **Embajada Antioque\u00f1a** (Cl. 6 No. 3-48, tel. 7\/571-7673, 8am-10pm daily, COP$20,000) has hearty breakfasts (meaty _caldos_ or broths), lunches, and dinners. It's been around for about 40 years, and the atmosphere, with tango music and Colombian classics, is quite nice. Try the baby beef or their _t\u00edpica_ plate with ground beef chorizo, _chichar\u00f3n_ (sausage), rice, and arepa, or at lunchtime the COP$8,000 set lunch. **La Mazorca** (Av. 4 No. 9-23, tel. 7\/571-1800, 7am-8pm daily, COP$18,000) is a popular chain restaurant serving the gamut of Colombian fare.\n\nFor a welcome break from meat eating, try M **Champi\u00f1on** (Cl. 10 No. 0-05, tel. 7\/571-1561, 8am-8:30pm Mon.-Sat., COP$12,000). This, C\u00facuta's best and biggest vegetarian restaurant, opens in the morning for coffee and pastries, then offers a generous set lunch. Off the menu you can order veggie burgers, salads, and pastas. **Aceituna** (Av. 0 No. 13-135, tel. 7\/583-7464, 10:30am-10pm Mon.-Sat., COP$18,000) is a long-standing Lebanese food restaurant run by a Lebanese-Colombian family.\n\nVegetarians, almost always prohibited from the joys of Colombian street food delights, will rejoice at the sight of _pasteles de garbanzo_ purveyor **100% Garbanzo** (Cl. 11 No. 2-83, cell tel. 313\/498-1712, 8am-noon and 2pm-6pm Mon.-Fri.). This snack bar is a great place to sample the bean pastry.\n\nWhen darkness falls in C\u00facuta, especially on the weekends, the energy shifts to the _malec\u00f3n_ area, about a 10-minute cab ride from downtown. That's where many of the big restaurants are located. **Londeros Sur** (Av. Libertadores No. 0E-60B, tel. 7\/583-3335, www.restaurantelonderos.com, COP$25,000) is famous around town for its Argentinian steaks; **Rodizio** (Av. Libertadores No. 10-121, tel. 7\/575-1719, www.rodizio.com.co, COP$30,000) for its Brazilian-style grilled meats; and **Rodeo** (Av. Libertadores No. 16-38, www.rodeogourmet.com, COP$25,000) is another very popular carnivorous option. **Balc\u00f3n Paisa** (Av. Los Libertadores No. 6-40, tel. 7\/575-0244, COP$22,000) is one of the most popular restaurants on the _malec\u00f3n_ and is huge. In addition to a vast menu of Colombian dishes, they often feature live shows. Late at night it becomes more rumba than restaurant. And for a departure, **La Gran Muralla** (Av. Libertadores No. 10-84, tel. 7\/575-3946, COP$18,000) is one of the best-known Chinese places in town.\n\nThe mall, the **Ventura Plaza Centro Comercial** (between Clls. 10 and 11 at Diagonal Santander, www.venturaplaza.com.co) is usually a safe bet for a bite of fast food, if you have run out of ideas. The food court area is open until 10:30pm.\n\n##### **Information and Services**\n\nThe **tourist office** (Cl. 10 No. 0-30, no phone, 8am-noon and 2pm-6pm Mon.-Fri., 8am-noon Sat.) has lots of brochures and maps for not only C\u00facuta but also the rest of the Norte de Santander department. Staff can provide some good tips on visiting natural attractions in the area.\n\nIn case of an emergency, contact the **Polic\u00eda Nacional** (tel. 7\/576-0622, or 123). A major hospital in town is the **Hospital Erasmo Meoz** (Av. 11E with Cl. 4N Guaimaral, tel. 7\/574-6888).\n\n##### **Getting There and Around**\n\nThe **Terminal de Transportes** (Avs. 7-8 between Clls. 1-2) in C\u00facuta is dirty, chaotic, and generally unpleasant, and is probably the main reason so many arrive in C\u00facuta with a poor impression of the city. Try to get your ticket in advance or at least find out the schedule so you don't have to be there longer than necessary. If you are waiting for a bus, it's best to wait outside on the curb, far from the claustrophobic station. There are numerous buses to Pamplona and other towns near C\u00facuta. The bus to Bogot\u00e1 costs around COP$70,000 and the trip takes 14 hours. To Bucaramanga it costs COP$30,000 and takes six hours, and to San Gil it costs about COP$45,000 and takes eight hours.\n\nThe airport, **Aeropuerto Camilo Daza** (Km. 5 Autopista Panamericana), is a 15-minute cab ride from downtown. **Avianca** (Cl. 13 No. 5-22, tel. 7\/571-3877, www.avianca.com, 8am-noon and 2pm-6pm Mon.-Fri., 8am-noon Sat.) flies nonstop between C\u00facuta and Bogot\u00e1 and Medell\u00edn; **EasyFly** (tel. 7\/595-5005, www.easyfly.com.co) connects C\u00facuta with Bucaramanga and Medell\u00edn; and **LAN** (Col. toll-free tel. 01\/800-094-9490, www.lan.com.co) flies to Bogot\u00e1.\n\nIf you are staying downtown, all attractions are within relatively easy walking distance. It's always a good policy to call a cab beforehand, and always after dark. Two reputable services are **Taxis Radio Taxi Cone** (tel. 7\/582-1666) and **Radio Taxi Radio** (tel. 7\/583-6828).\n\n###### **CROSSING INTO VENEZUELA**\n\nCrossing the border from C\u00facuta into San Antonio del T\u00e1chira on the Venezuelan side is easy. _Busetas_ (shared taxis) depart from in front of the Ventura Plaza mall. Expect to pay about COP$2,500 for a bus ticket. Taxis will cost upwards of COP$20,000. Be sure to get off before the bridge so you can get your passport stamped by immigration officials. Visas are not required for North Americans or European Union citizens. Hotels in San Antonio del T\u00e1chira are absolutely dismal, so it's far better to either stay in C\u00facuta or continue onwards to Caracas immediately. For more information on visas you can visit the **Consulado Venezuelano** (Av. Aeropuerto Camilo Daza, Sector Corral de Piedra, Zona Industrial, Cl. 17 Esquina, tel. 7\/579-1954 or 7\/579-1951, 8am-10am and 2pm-3pm Mon.-Thurs., 8am-10am Fri.).\n\n## **MEDELL\u00cdN AND THE COFFEE REGION**\n\nHIGHLIGHTS\n\nHISTORY\n\nPLANNING YOUR TIME\n\nMedell\u00edn\n\nSIGHTS\n\nENTERTAINMENT AND EVENTS\n\nSPORTS AND RECREATION\n\nSHOPPING\n\nACCOMMODATIONS\n\nFOOD\n\nINFORMATION AND SERVICES\n\nGETTING THERE AND AROUND\n\nNorthern and Eastern Antioquia\n\nSANTA FE DE ANTIOQUIA\n\nMAGDALENA MEDIO\n\nGUATAPE\n\nSouthern Antioquia\n\nM JARD\u00cdN\n\nJERIC\u00d3\n\nThe Coffee Region\n\nMANIZALES\n\nCOFFEE FARMS\n\nM SALAMINA\n\nM PARQUE NACIONAL NATURAL LOS NEVADOS\n\nARMENIA\n\nVICINITY OF ARMENIA\n\nSALENTO\n\nM VALLE DE COCORA\n\nFILANDIA\n\nPEREIRA AND VICINITY\n\nVALLE DEL R\u00cdO OT\u00daN\n\nSANTA ROSA DE CABAL\n\nBELALC\u00c1ZAR\n\nSANTUARIO\n\nPARQUE MUNICIPAL NATURAL PLANES DE SAN RAFAEL\n\nIBAGU\u00c9\n\nMedell\u00edn is one of the most dynamic cities of Colombia, with a vibrant cultural scene and nightlife. However, the hallmarks of this region are the coffee plantations and tiny Paisa pueblos. This verdant countryside, with its exuberant vegetation and many shades of green, is simply spectacular.\n\n**HIGHLIGHTS**\n\nLOOK FOR M TO FIND RECOMMENDED SIGHTS, ACTIVITIES, DINING, AND LODGING.\n\nM **Museo de Antioquia:** The galleries of this art museum, a fabulous art deco building from the early 20th century, are filled with works from the best Colombian artists spanning nearly four centuries. And the terrace caf\u00e9 is the best place for people-watching in the Centro (click here).\n\nM **Reserva Natural R\u00edo Claro:** A jungle paradise in the Magdalena Medio is set along a canyon that was only relatively recently discovered. This private natural reserve offers caving, river rafting, and a surplus of peace (click here).\n\nM **Jard\u00edn:** Just a few hours south of busy Medell\u00edn, the pace of life slows to a crawl in this picture-perfect coffee town of brightly colored Paisa homes. Jard\u00edn is surrounded by lush green mountains full of recreational opportunities (click here).\n\nM **Salamina:** Life goes on much as it always has in this rather remote Paisa town known for its superb architecture and warm hospitality of its people (click here).\n\nM **Parque Nacional Natural Los Nevados:** Dozens of hikes leading through tropical jungles afford fantastic views of snowcapped volcanoes and mountain lakes in this easily accessed national park (click here).\n\nM **Museo del Oro Quimbaya:** This fantastic gold museum pays tribute to the original settlers of what is now the fertile coffee region: the indigenous Quimbaya people (click here).\n\nM **Jard\u00edn Bot\u00e1nico del Quind\u00edo:** This is the best botanical garden in the region. You take a guided tour through a tropical forest full of _guadua_ forests and dozens of species of birds. (click here).\n\nM **Valle de Cocora:** One of the most dramatic and photographed scenes in Colombia is this valley filled with towering wax palms, Colombia's national tree. Nearby Salento offers coffee plantations and a typical Paisa pueblo experience (click here).\n\nJard\u00edn, a town in southern Antioquia\n\nMedell\u00edn, the surrounding department of Antioquia, and the coffee region departments of Caldas, Risaralda, and Quind\u00edo comprise the central, mountainous section of Colombia, covering the Cordillera Central (Central Range) and Cordillera Occidental (Western Range) north of Cali. The mountains then flatten out into the Caribbean coastal lowlands. Antioquia and the coffee region lie on some of the most beautiful mountain landscapes in Colombia.\n\n#### **HISTORY**\n\nDespite its inaccessible terrain, Antioquia was an important province of colonial Nueva Granada due to its abundant gold deposits. It attracted settlers who panned the rivers or cultivated food for the mining camps. Santa Fe de Antioquia, founded in 1541, was the main colonial settlement.\n\nAfter independence, the province continued to prosper, even attracting foreign investment in gold mining. Demographic pressure triggered a southward migration known as the _colonizaci\u00f3n antioque\u00f1a._ Waves of settlement brought Paisa families to unoccupied lands in the south of Antioquia, the coffee region, and the northern part of the Valle del Cauca.\n\nDuring the early part of the 20th century, coffee was a major source of prosperity in Antioquia. Medell\u00edn grew rapidly and became the industrial powerhouse of Colombia. The last decades of the 20th century were difficult times for Antioquia, with the triple scourge of drug trafficking, paramilitary armies, and guerrillas. In the past decade, the government has made huge strides in bringing back the rule of law, and today Antioquia is one of the safest and most prosperous regions in Colombia.\n\n#### **PLANNING YOUR TIME**\n\nWeather-wise, anytime of the year is a good time to visit Medell\u00edn and the coffee region. There's a reason why they call Medell\u00edn the \"City of Eternal Spring\": The entire region has a temperate climate.\n\nMedell\u00edn completely empties during the end of the year holidays, from December 15 to January 15, and also during Semana Santa (Easter week). This is peak time for pueblos and coffee region haciendas. During school vacations (June-July), natural parks and reserves and coffee haciendas get busy again.\n\nGive Medell\u00edn three days. During that amount of time, you can experience \"old Medell\u00edn\" sights in the Centro, such as the Museo de Antioquia, as well as check out the modern Medell\u00edn icons that are the subject of great pride: the Metrocable, Biblioteca Espa\u00f1a, the caf\u00e9 culture of cool El Poblado, and the Parque Explora. Consider spending a weekend in Medell\u00edn, when hotel rates drop, and especially if you're interested in checking out the city's nightlife scene. In a week you can add one or two other destinations in Antioquia, such as the Reserva Natural R\u00edo Claro or one of the picture-perfect Paisa pueblos, such as Jard\u00edn or Jeric\u00f3. These are within about a three-hour bus ride from the Antioquian capital of Medell\u00edn.\n\nMany visitors visit the gorgeous colonial town of Santa Fe de Antioquia to the north of Medell\u00edn as a day trip, but it's better to spend one night there, in order to enjoy strolling its quaint streets after the sun has gone down. On the banks of the mighty R\u00edo Cauca, Santa Fe is one of the hottest towns in the region. Guatape, with its famous rock, El Pe\u00f1ol, makes for a nice overnight on the way to or back from the R\u00edo Claro reserve, where two nights are necessary. These three destinations are popular on weekends and holidays. The stunning natural beauty of R\u00edo Claro is best enjoyed during the week.\n\nTo the south of Medell\u00edn are two picture-perfect Paisa pueblos: Jard\u00edn and Jeric\u00f3. A couple of days in one of those should be enough. You can continue from there southward into the coffee region on winding country roads.\n\nMedell\u00edn is not a base for visiting the coffee region: The cities of Armenia, Manizales, and Pereira are. Pereira has good air connections, while the Manizales airport is often shrouded in fog.\n\nOne of the joys of this region is to book a few days at a coffee hacienda. Many tour operators will pack your days with day-trip activities. Resist! As it gets dark at 6pm every day, it would be a shame to miss spending some daylight hours strolling the grounds of the _finca_ (farm), lazing in a hammock or rocking chair, or doing nothing at all.\n\nYou'll need a week or more to decompress on the coffee farm and see the region's top sights: the Valle de Cocora, the Jard\u00edn Bot\u00e1nico del Quind\u00edo, and Museo del Oro Quimbaya near Armenia, and one or two of the national and regional parks. There are very good transportation links between the three major cities and Salento. Roads are generally excellent. While renting a car in Colombia is not often the best option, here it makes sense.\n\nThough you can enjoy this magnificent region just by taking a bus ride from one place to the next, there are a few spots that warrant special attention. East of Medell\u00edn toward the R\u00edo Magdalena lays the Reserva Natural R\u00edo Claro, a lush, deep canyon. The Parque Nacional Natural Los Nevados and surrounding regional parks (such as Parque Regional Natural Ucumar\u00ed) offer innumerable opportunities for day hikes and longer treks. A bit off the beaten track, Parque Municipal Natural Planes de San Rafael offers beautiful hikes in the less-visited Cordillera Occidental.\n\n### **Medell\u00edn**\n\nWhile Medell\u00edn is the country's second in terms of population and importance, perpetually behind Bogot\u00e1, this city of around 2.7 million is the only Colombian metropolis with an urban train system. It's the first city with a cable car transportation system.\n\nThe first settlement in the region, near the present-day Poblado sector, was established in 1616. Medell\u00edn proper was founded in 1675, and was designated the capital of Antioquia in 1826.\n\nIn the 1980s, Pablo Escobar, born in nearby Rionegro, established a cocaine-trafficking empire based Medell\u00edn. In its heyday, the Medell\u00edn Cartel controlled 80 percent of the world's cocaine trade. When President Virgilio Barco cracked down on the cartel in the late 1980s, Escobar declared war on the government. He assassinated judges and political leaders, set off car bombs to intimidate public opinion, and paid a bounty for every policeman that was murdered in Medell\u00edn\u2014a total of 657. In 1991, Medell\u00edn had a homicide rate of 380 per 100,000 inhabitants, the highest such rate on record anywhere in the world. In 1993, Escobar was killed while on the run from the law.\n\nDuring the 1990s, leftist guerrillas gained strength in the poor _comunas,_ or sectors, of Medell\u00edn, waging a vicious turf war with paramilitaries. At the turn of the century, the homicide rate was 160 per 100,000 inhabitants, making Medell\u00edn still one of the most dangerous places on Earth.\n\nShortly after assuming power in 2002, President \u00c1lvaro Uribe launched Operaci\u00f3n Orion to wrest the poor _comunas_ of Medell\u00edn from the leftist guerrillas, and violence decreased notably. By 2005, homicides were still high by international standards but a fraction of what they had been a decade before. Under the leadership of Mayor Sergio Fajardo, elected in 2004, and his successors Alonso Salazar and An\u00edbal Gaviria, Medell\u00edn has undergone an extraordinary transformation. In partnership with the private sector, the city has invested heavily in public works, including a new cable car transportation system, museums, and libraries. In recent years, the city has become a major tourist destination and has attracted significant foreign investment.\n\nMedell\u00edn boasts a network of spectacular public libraries, greenspaces and plazas, and its train and cable car system. Beyond the Centro, the neighborhood of Laureles and the municipality of Envigado have their own distinct identity and atmosphere.\n\n##### **Orientation**\n\nMedell\u00edn is in the Valle de Aburr\u00e1, with the trickling and polluted R\u00edo Medell\u00edn dividing the city into east and west. Both the Metro (Line A) and the Avenida Regional or Autopista Sur, a busy expressway, run parallel to the river.\n\nThe main neighborhoods are **El Poblado,** including the mini-hood of Provenza, the **Centro,** and the **Carabobo Norte** area (often referred to as **Universidades** ).\n\nThe El Poblado area is full of great restaurants, bars, hostels, hotels, and glitzy shopping malls. It is also full of tall brick high-rise apartment buildings, home to the well-to-do. If you arrive in Medell\u00edn from the Rionegro airport, you will descend the hill into the valley and land more or less in El Poblado. Luxury hotels and malls line the Avenida El Poblado. The Parque Lleras is the center of the Provenza neighborhood, a small, leafy, and very hip part of El Poblado. It is on the eastern side of the Avenida El Poblado. To the west, down the Calle 10, is the Parque del Poblado, and a few blocks farther is the El Poblado Metro station.\n\nAcross the river from El Poblado is the Terminal del Sur bus station and the Aeropuerto Olaya Herrera. The Cerro Nutibara (Pueblito Paisa) is north of the airport. Northwest of the Cerro Nutibara is the quiet neighborhood of Laureles, and farther west is the stadium area. The B line of the Metro connects the Centro with the stadium area.\n\nBetween El Poblado and the Centro is the Barrio Colombia and an industrial area known appropriately enough as Industriales. This is an up-and-coming area with new hotels and high-rises being built in what is known as the Ciudad del R\u00edo, where the Museo de Arte Moderno de Medell\u00edn is located. Barrio Colombia is also home to many nightspots.\n\nThe heart of the Centro is the Plaza Botero and Parque Berr\u00edo Metro station area. The Centro is between El Poblado to the south and Carabobo Norte to the north. The Avenida El Poblado, also known as Carrera 43A, connects El Poblado with El Centro. The Metro does as well.\n\nThe northern neighborhoods of Medell\u00edn are massive. To the west of the R\u00edo Medell\u00edn there are few places of interest. The Cerro Volador is a hill and landmark in the northwest of the city. To the east of the river is the Aranjuez neighborhood. The Acevedo Metro station is where you can pick up the Metrocable to the Biblioteca Espa\u00f1a and Parque Arv\u00ed.\n\nOn the other side of town, in the far south, are the municipalities of Envigado and Itag\u00fci. Envigado has some great restaurants, a busy main plaza, and the Parque El Salado. The Avenida El Poblado connects El Poblado with Envigado. Itag\u00fci is an industrial town with little of interest for the tourist except for bars and clubs, many open until the wee hours.\n\n##### **Safety**\n\nAs in other cities in Colombia, it is best to avoid hailing taxis on the street, particularly at night. Instead, have your hotel, or the restaurant or shop you're at, call a taxi for you. Be careful at clubs and bars, and don't accept drinks from strangers.\n\nNeighborhoods such as La Provenza and Laureles are safe to walk about day or night. The Centro should be avoided after dark. It is not a good idea to take a carefree stroll in the northern or western neighborhoods or, especially, in the _comunas_ (city sectors) on the surrounding hills, but specific sights mentioned can be visited. Not only are they clean and efficient, but the Metro, Metrocable, and Metropl\u00fas are safe.\n\n#### **SIGHTS**\n\nTo see Medell\u00edn in full motion, visit the Centro during the week. On Saturdays, it's quieter, although the Peatonal Carabobo bustles with activity. On Sundays downtown is almost deserted except for tourists and street people. Most visits to the Centro start from the Parque Berr\u00edo Metro station. The main sights can easily be seen on foot and in a few hours.\n\n##### **Centro**\n\n###### M **MUSEO DE ANTIOQUIA**\n\nThe **Museo de Antioquia** (Cra. 52 No. 52-43, tel. 4\/251-3636, www.museoantioquia.org.co, 10am-5:30pm Mon.-Sat., 10am-4:30pm Sun., COP$10,000) is one of the top art museums in the country, with an extensive permanent collection of works from Colombian artists from the 19th century to modern times. Look for the iconic painting _Horizontes_ ( _Horizons_ ) by Francisco A. Cano. It's a painting that depicts the _colonizaci\u00f3n antioque\u00f1a,_ when, for a variety of reasons, families from Antioquia headed south to settle in what is now known as the coffee region. In the contemporary art rooms, you'll see _Horizontes_ (1997) by Carlos Uribe. This painting presents the same bucolic scene, except this time, 84 years later, in the background, a plane is seen spraying pesticides over the countryside, in an attempt to kill coca and marijuana crops. Native son Fernando Botero has donated several of his works, over 100 of them, to the museum. There is also a small room on a series of works by Luis Caballero. The museum is in an architectural gem, an art deco-style building from the 1930s. It originally served as the Palacio Municipal. There is a free guided tour at 2pm every day.\n\n##### **Carabobo Norte**\n\nOne of the country's top universities is the **Universidad de Antioquia** (Cl. 67 No. 53-108, tel. 4\/263-0011, www.udea.edu.co). The university was founded in 1803; however, it has only been at its current location in the Ciudad Universitaria, a few blocks west of the Centro, since 1968. There are over 37,000 students enrolled here, and the campus, full of plazas and public art, has a vibrant student energy In addition to a busy calendar of cultural events, the university has an excellent museum, **MUA** (Cl. 67 No. 53-108, Bloque 15, tel. 4\/219-5180, www.udea.edu.co, 8am-5:45pm Mon.-Fri., 9am-12:45pm Sat., free), that features changing contemporary art exhibits and a permanent natural history exhibit. Access to the campus for all visitors is at the Porter\u00eda del Ferrocarril entrance. Visitors must present a photo ID.\n\nAdjacent to the university near the Metro station, the **Parque Explora** (Cra. 52 No. 73-75, tel. 4\/516-8300, www.parqueexplora.org, 8:30am-5:30pm Tues.-Fri., 10am-6:30pm Sat.-Sun, COP$18,000) is one of the most iconic buildings of modern Medell\u00edn. It's a series of four futuristic red boxes. The ticket office closes 90 minutes before closing time. The **aquarium** (9am-5pm Tues.-Fri., included in admission cost) is the highlight of Explora. This is one of the largest aquariums in Latin America. Check out the tanks of Colombian marine creatures, including life in Colombian rivers such as the Amazon and Orinoco. Look for the giant piraruc\u00fa, an endangered fish that lives in the Amazon River and in its tributaries. The **Planetario Medell\u00edn** (Cra. 52 No. 71-117, tel. 4\/516-8300, www.planetariomedellin.org, 8am-5pm Tues.-Wed., 8am-7pm Thurs.-Fri., 10am-6pm Sat.-Sun., COP$12,000) is across the street from the Parque Explora.\n\nIf you feel like a walk in the park, the **Jard\u00edn Bot\u00e1nico de Medell\u00edn** (Cra. 52 No. 73-298, tel. 4\/444-5500, www.jbmed.org, 9am-5pm daily, free) is the place for you. It is across the street from Parque Explora. The highlight here is the Orquiderama, an open-air wood lattice-like structure where events are held. The botanical gardens are more akin to a tropical city park. It's a popular hangout for students, who will giggle as they say hello to you in English. In addition to the fantastic setting and good food of **In Situ** (inside the gardens, tel. 4\/460-7007, www.botanicomedellin.org, noon-3pm Mon., noon-3pm and 7pm-10pm Tues.-Sat., COP$25,000), there are caf\u00e9s and a nice gift shop.\n\nBehind the botanical gardens is the **Esquina de las Mujeres** (Cra. 51 at Cl. 73), a small public space with busts of accomplished women from Medell\u00edn and Antioquia from the colonial era to the present day. They represent many walks of life: indigenous women, activists, social workers, and artists who all made a contribution to society. Not many locals know about this homage, which was unveiled in 2007.\n\nPresidents, artists, and writers rest in the **Museo Cementerio de San Pedro** (Cra. 51 No. 68-68, tel. 4\/516-7650, www.cementeriosanpedro.org.co, 8am-5:30pm daily, free). Marble statues and elaborate tombs pay tribute to influential Antioque\u00f1os from the 19th century onward, but the reminders of the city's recent turbulent past may strike you as more interesting. One plot, near the tomb of Fidel Cano, founder of the once influential _El Espectador_ newspaper, contains the tombs of several members of drug kingpin Pablo Escobar's associates and guards. (He is buried in a cemetery in the neighboring town of Itag\u00fci.) Some tombs have stickers identifying allegiance to one of Medell\u00edn's soccer clubs, others have touching handwritten notes from wives and children left behind. There is a free tour of the cemetery on Sundays (2pm). To see the cemetery in a different light, check it out on a full moon evening (7pm-9pm). It's open to the public then and there are usually free concerts and other cultural activities going on. Check the cemetery's up-to-date webpage for a complete schedule of activities.\n\n##### **Cerro Nutibara**\n\nTo see an authentic Paisa pueblo, go to Jard\u00edn, Jeric\u00f3, or Salamina. They're just a couple of hours away and are as real as you can get. Can't do that? Then go to the **Pueblito Paisa** (Cl. 30A No. 55-64, tel. 4\/235-6476, 5am-midnight daily, free), atop the Cerro de Nutibara, one of two hills that interrupt the flat landscape of the Valle de Aburr\u00ed. Here, at this rather cheesy celebration of Paisa culture, you'll be greeted by smiling folks decked out in traditional costume. There is also a small **Museo de la Ciudad.** Plenty of food and handicrafts are on sale here. Also on the hill are a sculpture park and an amphitheater. You could go just for the views: From this high point there's a good view of Colombia's second city. The hill is also a popular place for an early morning jog. The Pueblito Paisa is lit with thousands of multi-colored lights at Christmastime.\n\n##### **Northern Medell\u00edn**\n\nThe **Casa Gardeliana** (Cra. 45 No. 76-50, tel. 4\/213-5965, Barrio Manrique, 9am-5pm Mon.-Sat. 10am-4pm Sun., free) has wall-to-wall tango memorabilia and some cool souvenirs. Plus a seat from the ill-fated airplane that crashed in the city and killed tango icon Carlos Gardel. Unfortunately, the museum does not tell the story of tango in Medell\u00edn in a clear manner, but it is just steps from the Metropl\u00fas Manrique station. If you are visiting downtown or the Cementerio de San Pedro, from there you can easily hitch a ride on the Metropl\u00fas.\n\nIn the neighboring barrio of Aranjuez is the **Casa Museo Pedro Nel G\u00f3mez** (Cra. 51B No. 85-24, Barrio Aranjuez, tel. 4\/233-2633, free). This delightful museum houses an extensive collection of the painter's works, including several murals for which he is best known. Much of his work portrays the plight of campesinos (rural peasants), workers, and indigenous people. His house, now the museum, was designed by G\u00f3mez, and the location of it was chosen by his Italian-born wife. The hills overlooking the city here reminded her of Florence, somehow. In the new wing of the museum there is a small public library. The courtyard holds a snack bar-caf\u00e9. The museum is not easy to get to, and you will probably have to take a cab there. Many visitors combine this visit with a trip to the nearby Casa Gardeliana.\n\n###### **BIBLIOTECA ESPA\u00d1A**\n\nWhen this public library was opened in the low-income neighborhood of Santo Domingo, King Juan Carlos came from Madrid for the ceremony. Spain, after all, helped to fund the project. It's one of many newly created _biblioteca parques_ (public library parks) in Medell\u00edn. More than a place for books, these library parks have become community centers and sources of pride in neighborhoods that continue to struggle with poverty and violence. The **Biblioteca Espa\u00f1a** (Cra. 33B No.107A-100, tel. 4\/385-7531, www.reddebibliotecas.org, 8am-7pm Mon.-Sat. and 11am-5pm Sun.) is the most famous and most visited of the 24 public libraries in the city, and, like each of them, it is stunning. The library resembles giant boulders clinging to the edge of the mountainside. It was designed by architect and Barranquilla native Giancarlo Mazzanti, who won a prize for this work at the VI Bienal Iberoamericana de Arquitectura y Urbanismo in Lisbon in 2008.\n\n**Downtown Medell\u00edn Walking Tour**\n\n**PLAZA BOTERO TO THE PLAZA DE LOS PIES DESCALZOS**\n\nMedell\u00edn's brash downtown is a compact history tour comprising stoic remnants from the colonial era, brick and mortar evidence of Medell\u00edn's rising as Colombia's most important industrial center in the early 20th century, and the vibrant public spaces, modern transportation systems, and futuristic architecture showing this proud city's 21st-century optimism.\n\nBegin the tour at the Parque Berr\u00edo Metro station and walk five minutes north to the Plaza Botero.\n\n**PLAZA BOTERO**\n\nMost visits downtown begin under the shadows of the **Palacio de la Cultura Rafael Uribe Uribe** (Cra. 51 No. 52-03, 8am-5pm Mon.-Fri., 8am-2pm Sat., free), an occasional host of art exhibits. The **Plaza Botero** (in front of the Museo de Antioquia, Cra. 52 No. 52-43) gets its name for its 23 corpulent bronze sculptures by Fernando Botero. Passersby often pose in front of the sculptures, such as _La Mano (The Hand)_ and _Eva (Eve)_ for a quick snapshot. One of the most prolific, and by far the best known, of contemporary Colombian artists, Botero donated these sculptures to his hometown of Medell\u00edn. His paintings and sculptures of rotund people often portray campesino (rural) life, but many of them are also commentaries on the violence in Colombia.\n\n**PEATONAL CARABOBO**\n\nTo the south, the **Peatonal Carabobo** is a pedestrian walkway that extends for eight blocks. Lined with shoe shops, five-and-dime stores, and snack bars, it's busy, loud, and colorful. (Although there is usually a police presence, be sure to watch your stuff!)\n\nOn the right-hand side is Medell\u00edn's oldest church, the brilliantly white **Iglesia de la Vera Cruz** (Cl. 51 No. 52-38, tel. 4\/512-5095), which dates back to 1682. It is often filled with working-class faithful, sitting or standing in meditation and prayer. It's a refuge of quiet in this busy commercial area. The only other living testament to Spanish rule in Medell\u00edn is the white-washed **Bas\u00edlica Nuestra Se\u00f1ora de la Candelaria** on the Parque Berr\u00edo a few blocks away.\n\nThe Belgian architect who designed the grandiose **Palacio Nacional** (Cra. 52 No. 48-45, tel. 4\/513-4422) in the 1920s probably never expected that it would, over time, become the domain of around 400 tennis shoes and jeans vendors. It was originally built to house governmental offices, and today, when you walk through the corridors of this historic building, all you'll hear is the chorus of _\"a la orden\"_ (\"at your service!\") from hopeful shop attendants. Towards the end of Peatonal Carabobo is **Donde Ram\u00f3n,** a small kiosk in the middle of the walkway, jam-packed with antique objects like brass horse stirrups or old _carrieles_ (leather handbags) from Jeric\u00f3.\n\n**PARQUE DE LAS LUCES**\n\nAfter years of abandonment and urban decay, in 2005 the artificial forest of the **Parque de las Luces** or **Plaza Cisneros** (Cl. 44 at Cra. 52) was opened in an effort to rejuvenate the area. The park, consisting of 300 illuminated posts, looks somewhat odd during the day but is spectacular at night when it shines. Check it out by car at night as it is not safe to roam about after dark. On the east side of the plaza are two historic early-20th-century brick buildings: the **Edificio Carr\u00e9** and **Edificio V\u00e1squez** (Cl. 44B No. 52-17, tel. 4\/514-8200). When they were built they were the tallest buildings in Medell\u00edn. These buildings were once important warehouse facilities during the industrial boom of the early 20th century. The plaza used to be the home of the main marketplace. On the western side of the plaza is the **Biblioteca EPM** (Cra. 54 No. 44-48, tel. 4\/380-7516, 8:30am-5:30pm Mon.-Sat.), a stunning public library sponsored by the electric company EPM (Empresas P\u00fablicas de Medell\u00edn), built in 2005. In addition to reading rooms, there are occasional exhibitions and cultural events held at the library.\n\nAcross from the Parque de las Luces on the southern side of Calle 44 is the **Estaci\u00f3n Ferrocarril** (Cra. 52 No. 43-31, tel. 4\/381-0733), the old main train station. There's not much to see here, except for a train engine and forgotten old tracks.\n\n**PLAZA MAYOR**\n\nTo the west of the Estaci\u00f3n Ferrocarril is the **Centro Administrativo La Alpujarra** (Cl. 44 No. 52-165), which houses the Departamento de Antioquia government offices. The sculpture **_Homenaje a la Raza,_** by Rodrigo Arenas Betancur, stands in the middle of the large intermediary plaza. Just beyond is the **Plaza de la Libertad** (Cra. 55 between Clls. 42-44), a complex of modern office space and interesting public space complete with urban gardens.\n\nCross the pedestrian bridge over the lanes of the Metropl\u00fas bus station. Metropl\u00fas is the latest addition to Medell\u00edn's transportation network. It debuted in 2013. Here is the **Plaza Mayor** (Cl. 41 No. 55-80, www.plazamayor.com.co), the city's preeminent convention and event venue; it has a fair share of nice restaurants. The **Teatro Metropolitano** (Cl. 41 No. 57-30, tel. 4\/232-2858, www.teatrometropolitano.com), built from 20th-century brick, hosts concerts.\n\nFinally, the **Plaza de los Pies Descalzos** (Cra. 58 No. 42-125) is a plaza filled with a _guadua_ (Colombian bamboo) forest and fountains, where you can take off your shoes and play. It's surrounded by eateries on one side and the massive **Museo del Agua** (Cra. 57 No. 42-139, tel. 4\/380-6954, 8am-6pm Tues.-Fri., 10am-7pm Sat.-Sun., COP$4,000) on the other. In the distance is a long-standing Medell\u00edn architectural icon: the **Edificio Inteligente** (Cra. 58 No. 42-125, tel. 4\/380-4411). Built in the late 1950s, it has served as the headquarters of EPM, the utility company. To find out what's so smart about it, you can take a free tour of the building. They are offered Monday-Friday. Call in advance to reserve.\n\nGetting to Santo Domingo is an attraction in itself. The neighborhood is connected to the metropolis by the Metrocable cable car system. Take the Metro towards Niqu\u00eda station and transfer to the Metrocable at Acevedo. The Santo Domingo station is the third and final stop.\n\nWhen the Metrocable K line was opened in 2004, it was the first of its kind in the world: a gondola-like public transport system with a socio-economic purpose, connected to a metro. The system, consisting of gondolas, has eliminated eternal climbs up and down the mountain for low-income residents.\n\n###### **PARQUE ARV\u00cd**\n\nFor some fresh, and oftentimes crisp, country air, a visit to the **Parque Arv\u00ed** (Santa Elena, tel. 4\/444-2979, www.parquearvi.org, 9am-5pm Tues.-Sun., free), covering 16,000 hectares (40,000 acres) of nature, hits the spot after a few days of urban exploring.\n\nHighlights in the park include the seven well-marked nature paths, which meander through cloud forests thick with pine and eucalyptus trees, over brooks, along ancient indigenous paths, to mountain lakes and lookout points with spectacular views of the Valle de Aburra and Medell\u00edn below. Most paths, with a distance of under three kilometers (two miles), are not strenuous whatsoever. There is a longer path of over 10 kilometers (six miles) that is excellent for biking. Ask at the information booth upon arrival at the Arv\u00ed Metrocable station for suggestions on walks to make. There are often free guided nature walks as well.\n\n**Getting Up the Hill**\n\nDuring the late 1990s to 2000s, thousands of families from rural areas in Antioquia, C\u00f3rdoba, and Choc\u00f3 were forced to leave their homes due to violence. Moving to Medell\u00edn to start a new life, many arrived in the low-income neighborhoods along the steep slopes of the mountains surrounding the city. But here, where many live in meager brick homes covered with corrugated zinc roofs secured only by large stones, horrific violence has followed them. First it was turf wars between guerrillas and paramilitaries in the early 2000s. Today the violence is caused by drug-trafficking gangs with links to former paramilitaries. This wave of violence has given birth to a new phenomenon: intra-urban displacement, during which families have been displaced within the city due to urban violence. For many, this is the second displacement that their families have had to endure.\n\nCity leaders have sought to improve the quality of life in the _comunas_ in a variety of innovative ways. Two lines of the Metrocable gondola system have made a huge difference in allowing residents to travel to work or school in the city without having to walk up and down the mountainside. Spectacular modern public libraries have been built in many low-income communities, providing a safe and pleasant space to study, read, and connect to the Internet. These have developed into important cultural centers, with an active schedule of films, children's activities, and other cultural activities. New homes have been built and donated to 200-300 displaced families in the neighborhood, with funds from the national government under President Juan Manuel Santos.\n\nIn 2012, the city debuted its latest project, this time aimed at improving life in the Comuna 13, the most notorious of the _comunas_ in the entire city. This time the project involved the creation of open-air escalators in this neighborhood. These are a series of six dual, interconnected escalators that extend down the slopes for some 384 meters (1,260 feet). The system operates from early in the morning until about 10 at night. They are monitored by city employees, and their use is free. It is the first time in the world escalators have been used in order to improve the lives of the less fortunate.\n\nThe escalators have made a difference in the lives of Comuna 13 residents, although there are some who believe that the money spent on the project (around US$6 million) could have been better used otherwise. There have been alarming reports as well that some gangs have been intimidating residents by charging them to use the escalators, under the threat of dire consequences.\n\nDespite the high levels of violence affecting residents (never foreign tourists), the escalators have become a tourist attraction, and even appear in the city's tourism promotional materials. Celebrities and dignitaries from President Juan Manuel Santos to French fashion designer Francois Girbaud have taken a ride on the escalators.\n\nIt is indeed a strange kind of tourism, with which some may feel uncomfortable. However, if you would like to see this escalator project, you certainly can. Go during the day, and you must not wander the streets of the Comuna 13. Never remain in the neighborhood after dark. To get there, take the Metro L\u00ednea B to San Javier station. As you depart the station, in front are _colectivos_ (small buses) that regularly transport passengers to the Comuna 13. It's about a 15-minute trip and costs under COP$1,500. Ask anyone which bus to take, and let the bus driver know that you'd like to go to the _escaleras el\u00e9ctricas._\n\nFrom San Javier, there is also a Metrocable line (L\u00ednea J) that has three stops and travels to the top at La Aurora.\n\nA trip to the Parque Arv\u00ed in northern Medell\u00edn is an excellent break from the city.\n\nOther recreational activities are on offer in the different _nucleos_ (nuclei) of the park. Understanding the nuclei and layout of the park can be confusing. Staff at the information booth at the entrance will provide you with a map and assist you in planning your visit. At the Nucleo Comfenalco, you can rent a paddleboat on the Piedras Blancas reservoir, and there is a small hotel, as well as a butterfly pavilion. In the Nucleo Mazo there is a market.\n\nTo explore the park on bike, you can rent a bike for free with the city's Encicla program by showing an ID card. These can be rented at any of the several Encicla stations in the park, but you will be required to renew the rental at any station if you have been cycling for more than an hour. Encicla staff can provide you with a map and recommendations.\n\nThe park is a nice day trip to make, and you might consider the getting there the best part about the excursion. To get there from the city, take the Metro to the Acevedo station in the north of the city (L\u00ednea A towards Niqu\u00eda). From there, transfer to the Metrocable (L\u00ednea K) to the Santo Domingo station. From there you must transfer to the Parque Arv\u00ed line (L\u00ednea L, COP$4,200 one way), which has an additional cost.\n\nThe temperature can drop substantially and abruptly in the park. Pack along a light sweater and a lightweight rainproof jacket. Try to get an early start so that you can enjoy the park without rushing. There are various snack bars and restaurants throughout the park.\n\n##### **Southern Medell\u00edn**\n\nThe Ciudad del R\u00edo area, between downtown and El Poblado near a Metro line, has developed into an up-and-coming neighborhood largely due to the arrival of the **Museo de Arte Moderno de Medell\u00edn** (MAMM, Cra. 44 No. 19A-100, tel. 4\/444-2622, www.elmamm.org, 9am-5:30pm Tues.-Fri., 10am-5:30pm Sat., 10am-5pm Sun., COP$8,000). The coolest part about the MAMM is its location in an old warehouse, typical of the Barrio Colombia area. This was the home of Talleres Robledo, a steel mill that began operations in the 1930s. Exhibitions (usually two at a time) are hit or miss. The museum store, the _tienda,_ is an excellent place to pick up a whimsical Medell\u00edn souvenir. Many items, like T-shirts, notepads, or cute doo-dads, are the product of local creative minds.\n\nThe **Parque El Salado** (Vereda El Vallano, Envigado, tel. 4\/270-3132, www.parqueelsalado.gov.co, 9am-5pm Tues.-Sun., COP$3,000), a municipal park covering 17 hectares (42 acres) in Envigado, has trails and activities, such as a zipline, and is a good place to get some fresh air. On weekends it gets packed with families on a _paseo de olla._ Literally a soup-pot excursion, _paseo de olla_ usually means _sancocho,_ a hearty beef stew. Essential gear for a day out at the park includes giant aluminum pots for slowly heating a _sancocho_ over a campfire. There is plenty of fresh air at this park, but from afar it may appear that there is a forest fire in the picnic area with all of the campfires. Getting to the park is easy using public transportation. From the Envigado Metro station look for a green bus with a sign that says Parque El Salado. It's about a 20-minute ride up towards the mountains.\n\n#### **ENTERTAINMENT AND EVENTS**\n\n##### **Nightlife**\n\nSince 1969, **El Social Tienda Mixta** (Cra. 35 No. 8A-8, tel. 4\/311-5567) has been selling the basics to local residents (soap, sugar, coffee); it's only been a recent phenomenon that it's now the hippest place to be seen at night, when it is converted into the most popular bar in Provenza! It's so popular on weekend evenings, you can forget about finding a vacant plastic chair.\n\nWant to check out the nightlife with other party people? That's the idea behind the **Pub Crawl Medell\u00edn** (cell tel. 300\/764-6145, pubcrawlmed@gmail.com, Sat. evenings, COP$30,000). In this night of shenanigans, revelers (groups of about 12) get together, then hit several bars (enjoying courtesy shots along the way), and then wind up the night dancing to the beats at a popular dance club. Each Saturday the group explores different nightspots.\n\nEvery Thursday evening, the Medell\u00edn microbrewery **3 Cordilleras** (Cl. 30 No. 44-176, tel. 4\/444-2337, www.3cordilleras.com, 5:30pm-9pm Thurs., COP$20,000) offers a tour of their brewery, during which you learn about the beer-making process. At the end of the tour, the grand finale is tasting several of their artisan beers and friendly socializing. On the final Thursday of each month, after the tour there is live music and beer.\n\n**Calle Nueve** (Cl. 9 No. 43B-75, tel. 4\/266-4852, 6pm-2am Mon.-Sat., no cover), in a nondescript white house, is a hipster's paradise in El Poblado. Music varies wildly from salsa to house to folk. The dim lighting and well-worn couches provide the perfect chilled-out atmosphere.\n\n###### **SALSA, TANGO, AND JAZZ**\n\nMedell\u00edn is no Cali, but salsa has its aficionados here. If the musical genres _son, la charanga, el guaguanco,_ and _la timba_ don't mean anything to you now, they might after a night at **Son Havana** (Cra. 73 No. 44-56, tel. 4\/412-9644, www.sonhavana.com, 8pm-3am Wed.-Sat., cover Sat. COP$8,000) often has live performances. Nearby is **El T\u00edbiri** (Cra. 70 at Cl. 44B, hours vary Wed.-Sat.), an underground salsa joint on Carrera 70, which is hugely popular on the weekends. They say the walls sweat here, as after 10pm it gets packed with revelers, many of whom are university students. Friday nights are big at El T\u00edbiri.\n\nThe downtown **Sal\u00f3n M\u00e1laga** (Cra. 51 No. 45-80, tel. 4\/231-2658, www.salonmalaga.com, 9am-11pm daily, no cover)\u2014boy, has it got character. It's filled with old jukeboxes and memorabilia, and has its clientele who come in for a _tinto_ (coffee) or beer during the day. The Saturday tango show at 5:30pm and oldies event on Sunday afternoons are especially popular with locals and travelers alike, but a stop here is a fine idea anytime.\n\nNear the Parque de la Periodista, a major weekend hangout for the grungy set, there are some small bars big on personality. Tuesday nights are bordering on legendary at **Eslab\u00f3n Prendido** (Cl. 53 No. 42-55, tel. 4\/239-3400, 3pm-11pm Tues.-Sat., cover varies), a hole-in-the-wall salsa place that really packs them in! **El Acontista** (Cl. 53 No. 43-81, tel. 4\/512-3052, noon-10pm Mon.-Thurs., noon-midnight Fri.-Sat.) is an excellent jazz club downtown. It's got a bookstore on the second floor and live music on Monday and Saturday evenings. They've got great food, too, making it an excellent stop after a day visiting the Centro.\n\nAn authentic tango spot in Envigado is **Bar Atlenal** (Cl. 38 Sur No. 37-3, tel. 4\/276-5971, 3pm-2am daily). Friday night is the best time to go to see a tango performance, but to listen to some tango music from the juke box and have a beer, go any day of the week. It's an institution, with more than six decades of history. Allegiance to the soccer club Atl\u00e9tico Nacional is evident on the walls of the bar. Included is an homage to star player Andr\u00e9s Escobar. Also in Envigado is **La Venta de Dulcinea Caf\u00e9 Cultural** (Cl. 35 Sur No. 43-36, tel. 4\/276-0208, www.laventadedulcinea.jimdo.com, 2pm-11pm Mon.-Sat.), where salsa, _milonga,_ and tango nights are often held. Check the webpage for a schedule.\n\n###### **DANCE CLUBS**\n\nFamous **Mango's** (Cra. 42 No. 67A-151, tel. 4\/277-6123, 5pm-6am daily, no cover), decked out like a Wild West saloon, is a festive club popular with foreigners and locals alike, and gets going late. **Jes\u00fas Dulce M\u00edo\u2014Mil Juguetes** (Cra. 38 No. 19-255, Km. 2 V\u00eda Las Palmas, tel. 4\/266-6020, www.fondadulcejesusmio.com, 7pm-3am Tues.-Sat., COP$10,000 cover) is a popular club near El Poblado. Wednesday is karaoke night.\n\nIf you go to **Fahrenheit** (Cra. 42 No. 79-125, Itag\u00fci, tel. 4\/354-6203 www.discotecafahrenheit.com, 10pm-6am Thurs.-Sat., cover varies) you should dress to impress. It's a late-night place in the neighboring town of Itag\u00fci. Thursdays are electronica nights, while Saturdays are for crossover, a mix of popular music with Latin tunes. Guys should expect to pay around COP$25,000 for cover, ladies _nada._\n\n###### **GAY BARS AND CLUBS**\n\nThere is a lively gay and youthful nightlife scene in Medell\u00edn. **Donde Aquellos** (Cra. 38 No. 9A-26, tel. 4\/312-2041, cell tel. 313\/624-1485, 4:30pm-2am daily) is an easy-going kind of place near the Parque Lleras in El Poblado. This friendly bar is a good place for a terrace drink. **Culture Club** (Cra. 43F No. 18-158, hours vary Thurs.-Sat., cover varies) is the hottest dance club and gets hopping at around midnight on weekends. It's a fashionable place, with chandeliers and red velvet.\n\n##### **Cinema and Theaters**\n\n**Otraparte** (Cra. 43A No. 27A Sur-11, tel. 4\/448-2404, www.otraparte.org, 8am-8pm Mon.-Fri., 9am-5pm Sat.-Sun.) is a cultural center that offers a dynamic program of free concerts, films, book launches, and even free yoga classes in Envigado.\n\n##### **Festivals and Events**\n\n**Festicamara** (www.festicamara.com), an international chamber music festival, is held in March or April each year, with concerts across the city at venues like the fabulous **Orquiderama** in the Jard\u00edn Bot\u00e1nico.\n\nThe **Festival Internacional de Tango** (www.festivaldetangomedellin.com) takes place each year during the last week of June, commemorating the anniversary of the death of Carlos Gardel. This festival, and in fact the perseverance of tango culture in Medell\u00edn, is largely due to one man's passion and efforts. Argentine Leonardo Nieto visited Medell\u00edn in the 1960s, primarily to get to know this city where tango icon Carlos Gardel died in an airplane crash. He fell in love with the city, stayed, and created the Casa Gardelina and the Festival Internacional de Tango. During this festival, tango concerts and events take place across the city, in nightclubs, theaters, parks, and on street blocks.\n\n**International Day of Laziness**\n\nPaisas are known throughout Colombia to be some of the most hard-working and driven people in the country. The Medell\u00edn Metro, routine 7am business meetings, the orderly pueblos in the Antioquian countryside, and even former president \u00c1lvaro Uribe, a native Paisa, are examples of this industriousness. Uribe's famous words upon taking office in 2002 were _\"trabajar, trabajar, trabajar\"_ (\"work, work, work\"). Laziness is quiet simply anathema to Paisas.\n\nBut you can't be productive _all_ the time. The people of **Itag\u00fci,** an industrial town bordering Medell\u00edn, have taken that to heart. In fact, on one day each year they not only take it easy, they embrace and celebrate the virtues of slothfulness during their **D\u00eda Internacional de la Pereza** (International Day of Laziness) celebrations. On that day in August, residents rise at the leisurely hour of 10am, put out their hammocks and beds in front of their houses, and laze the day away, sometimes still in their pajamas. The day's events include a bed (on wheels) race and general goofing off. Ironically, most of the action (or inaction) of that day takes place in the Itag\u00fci Parque del Obrero (Worker's Park).\n\nSince 1991, Medell\u00edn has hosted an impressive **Festival Internacional de Poes\u00eda de Medell\u00edn** (www.festivaldepoesiademedellin.org), which routinely attracts poets from dozens of countries, who share their work in more than 100 venues across the city. It's held in early July.\n\nAs the leading textile manufacturing center in Colombia, Medell\u00edn is the obvious choice for the most important fashion event in the country: **Colombiamoda** (<http:\/\/colombiamoda.inexmoda.org.co>). It attracts designers and fashionistas from across the globe, and during this week, the Plaza Mayor becomes a fabulous model-fest.\n\nThe **Feria de las Flores** (www.feriadelasfloresmedellin.gov.co) is the most important festival of the year in Medell\u00edn, and is when the city is at its most colorful. It's a week-long celebration of Paisa culture, with horseback parades, concerts, and the highlight, the Desfile de los Silleteros. That is when flower farmers from Santa Elena show off incredibly elaborate flower arrangements in a parade through the city streets. The festival takes place in July or August each year, and it's a great time to visit the city.\n\nAt Christmastime, Medell\u00edn sparkles with light, every night. It all begins at midnight on December 1, during the **Alborada.** That's when the sights and sounds of fireworks and firecrackers envelop the entire Valle de Aburr\u00e1. On December 7, the **Alumbrado Navide\u00f1o,** the city's Christmas light display, begins. The Cerro de Nutibara and the R\u00edo Medell\u00edn, along with other city sites, are illuminated with 14.5 million multi-colored lights. Sponsored by the electric company, it's an incredible sight to behold.\n\n#### **SPORTS AND RECREATION**\n\n##### **Biking**\n\nThe Medell\u00edn **Ciclov\u00eda** (8am-1pm Sun.) has many routes, including along the Avenida El Poblado and along the R\u00edo Medell\u00edn. On Tuesday and Thursday evenings there is a **Ciclov\u00eda Nocturna** (8pm-10pm) on two routes: along the river and around the stadium area. Not to be outdone by their neighbor, the cities of Envigado and Itag\u00fci also have a Ciclov\u00eda on Sundays.\n\n**Encicla** (office in \u00c9xito Colombia store, Cl. 48D No. 66-61, tel. 4\/436-6271, cell tel. 310\/390-9314, 8am-6pm daily, www.encicla.gov.co) is Medell\u00edn's bike share program. It's free to use for those over the age of 18. Nice routes are along the Carrera 70 and around the universities. Visitors must register in person at the Encicla office, and must show their passport. Encicla bike paths follow Carreras 65 and 70, connecting with the Estadio and Universidad Metro stations. In **Parque Arv\u00ed** (Santa Elena, www.parquearvi.org) there are six **Encicla en el Parque stations.** At this park, the registration procedure is less involved.\n\n**Bike Rent** (Cra. 35 No. 7-14, cell tel. 310\/448-3731, www.bikerent.com.co, 9am-7pm Mon., Wed., and Fri.-Sat., 9am-10pm Tues. and Thurs., 8am-1pm Sun., COP$25,000 half day, COP$35,000 full day) rents good bikes cheaply, and the prices decrease as the number of hours you rent them increases. They also have information on routes and suggestions at this convenient Provenza location.\n\n**Ciclo Barranquero** (tel. 4\/538-0699, cell tel. 314\/806-5892, ciclobarranquero@gmail.com, COP$70,000-90,000 pp) organizes interesting day-trip bike rides for all levels of cyclists in the city and beyond, such as in the nearby pueblo of Santa Elena and in Guatape. Bikes and necessary equipment are included in the price, but transportation to the meeting point is not. **Ecoturismo Arewaro** (Cra. 72A No. 30A-21, tel. 4\/444-2573, cell tel. 300\/652-4327, www.ecoturismoarewaro.com, COP$46,000 pp) organizes day-trip walks and bike trips in parks and pueblos near Medell\u00edn. _Arewaro_ means \"gathering of friends\" in the Way\u00fau language.\n\n##### **Yoga and Gyms**\n\n**Atman Yoga** (Tr. 37 No. 72-84, tel. 4\/311-1132, www.atman-yoga.org, donations requested) holds free yoga classes every day at its main location in Laureles. They are quite popular, especially in the evenings. There are also classes at the Jard\u00edn Bot\u00e1nico on Tuesdays and Thursdays at 5:30pm and at other times during the week. In El Poblado there are several yoga studios, such as **108 Yoga** (Cl. 5 Sur No. 30-72, tel. 4\/266-7232) and **Sati Yoga Y Meditaci\u00f3n** (Cl. 10 No. 36-14, 2nd floor, tel. 4\/352-4143, cell tel. 315\/499-6625).\n\nThe gym **El Molino** (Cra. 79 No. 37-55, tel. 4\/411-2714, 5am-10pm Mon.-Thurs., 5am-8pm Fri., 8am-3pm Sat., 9am-2pm Sun., COP$35,000 weekly pass) in Laureles has a large weight room, cardio machines, and a pool and sauna, and holds various classes. You may not think of schlepping all the way to Envigado to get in a workout, but at multi-level **Dinamo Fitness** (Tr. 29S No. 32B-126, tel. 4\/334-1512, www.dinamofitness.com, 5am-10pm Mon.-Thurs., 5am-2pm Sat., 8am-2pm Sun.) the price is right. A two-day pass costs COP$22,000, a ticket for six entries is COP$60,000, and a ticket for 12 entries is COP$120,000.\n\nMany parks in Medell\u00edn have **outdoor gyms** where you can get in a free workout. These are especially popular with young men who work out on the _barras_ (pull-up bars). One such gym is in the Provenza barrio (Cra. 37 at Cl. 8); another is in the Ciudad del R\u00edo behind the Museo de Arte Moderno de Medell\u00edn (Cl. 20 at Cra. 45).\n\n##### **Paragliding**\n\nFor incredible views of both the verdant Antioquian countryside and the metropolis in the distance, check out a paragliding adventure organized by the **Aeroclub San Felix** (Km. 6 V\u00eda San Pedro de los Milagros, tel. 4\/388-1077, www.parapenteencolombia.com, 20-min. flight COP$80,000, complete course COP$1,600,000). Bus transportation to the town of San Felix is available from the Portal del Norte bus station, and the Aeroclub San Felix can also arrange transportation from El Poblado for an additional cost.\n\n##### **Centro Deportivo Atanasio Girardot**\n\nAt the **Centro Deportivo Atanasio Girardot** (Cras. 70-74 and Clls. 48-50, tel. 4\/369-9000, pool tel. 4\/430-1330, ext. 146), in addition to soccer matches and the occasional big-name concert, there are running tracks, basketball courts, and swimming pools free of charge and open to the public, including visitors. To use the pools all you need is to present an ID and bring a bathing cap. They sell them on-site for under COP$20,000. The Olympic-sized pool is only open to those with a membership in the swimming league, but there are four others available to the public. The nice blue track is often packed with Paisas getting in their morning sweat. It's open to the public during the morning hours, and usually on Tuesdays and Thursday evenings it is also open 8pm-10pm. There are also jazzercise, yoga, rumba, and aerobics classes given several times a day starting at 6am. Just show up and join the fun. Call (tel. 4\/369-9000, ext. 118) for class schedule information. The complex is accessible from the Estadio Metro station.\n\nthe Estadio Atanasio Girardot complex\n\n##### **Soccer**\n\nMedell\u00edn has two professional teams, and Envigado has one. By far the most famous team, with followers across the country, is **Atl\u00edtico Nacional** (www.atlnacional.com.co). Nacional, wearing the green and white of the Antioquian flag, has been playing since 1947. It's one of the most successful teams in Colombia and has won the top division 11 times. Nacional defeated Santa Fe from Bogot\u00e1 for the 2013 Liga Postob\u00f3n championship, which was a very big deal. Nacional is wildly popular with young men and boys in Medell\u00edn, Antioquia, and beyond. The cheap seats at Nacional games are always packed with kids from the barrios. The other team in town is **Deportivo Independiente Medell\u00edn** (www.dalerojo.net). This is the oldest club in Colombia and was originally called Medell\u00edn Foot Ball Club when it was established in 1913. Their colors are red and blue. Both teams play at the **Estadio Atanasio Girardot** (Cl. 48 No. 73-10, www.inder.gov.co). Tickets can be purchased at **Ticket Factory Express** (tel. 4\/444-4446, www.ticketexpress.com.co).\n\n##### **Tours**\n\n**Turibus** (www.turibuscolombia.com, 9am-7:40pm daily, COP$28,000 24-hour pass) operates a hop-on, hop-off service that has seven stops in the city, including the Plaza Botero and the Cerro Nutibara\/Pueblito Paisa. They also offer tours to other parts of the Antioquia department, such as to Jeric\u00f3 and Guatape.\n\nWhile some may find it unseemly to go on a tour of Pablo Escobar's Medell\u00edn, others find it fascinating. During the three-hour **Pablo Escobar Tour** (cell tel. 317\/489-2629, www.paisaroad.com, 10am daily, COP$35,000) offered by Paisa Road, you'll see where the world's most notorious drug baron grew up, learn about the violent world of the cartels, and visit his tomb. The meeting point for the tours (offered in English) is at the Black Sheep and Casa Kiwi hostels in the Provenza area of El Poblado. If you'd like to visit Escobar's grave independently, you can take the Metro to the Sabaneta station. The **Parque Jardines Montesacro** (Cra. 42 No. 25-51, Autopista Sur Itag\u00fci, tel. 4\/374-1111, 9am-5pm daily, free) is within walking distance from there.\n\n#### **SHOPPING**\n\nThe Provenza neighborhood and the area around Parque Lleras are home to several boutique clothing and accessories shops. **Santa Fe** (Cl. 43 No. 7 Sur-107, www.centrocomercialsantafe.com) is the largest and flashiest of Medell\u00edn's malls. It's on the Avenida El Poblado.\n\n#### **ACCOMMODATIONS**\n\nAccommodation options to fit every budget and taste are plentiful in Medell\u00edn. El Poblado has the most options, with luxury hotels along the Avenida El Poblado and hostels and boutiques in the walkable Provenza area, close to a smorgasbord of restaurants and bars and close-ish to the El Poblado Metro station. Laureles is a quiet and green residential area with a growing number of fine options for those wanting an escape from the madding crowd. There aren't many reasons anyone would want to stay in the Centro, an area of town that feels unsafe at night. As is the case in cities in the interior of the country, on weekends Medell\u00edn empties somewhat, hotel prices fall, and vacancies increase.\n\n##### **Under COP$70,000**\n\nDozens of hostels cater to more than the traditional twenty-something backpacker. Today these economical options offer comfortable private rooms for those who are looking for a friendly environment and keeping an eye on their expenses.\n\nThe **Casa Kiwi** (Cra. 36 No. 7-10, tel. 4\/268-2668, www.casakiwihostel.com, COP$20,000 dorm, COP$60,000 d) is a backpacker's institution in El Poblado. A cold and _\u00fcber_ -cool atmosphere pervades the place at times, though, and it does have the reputation of being a party pad (loud). It's got a capacity of 60, with an array of private rooms and dorm options, plus a nice sundeck above. Nonguests can visit on Friday evenings, when there's live music on the deck.\n\nWith just one dorm room and two private rooms (with a shared bath), **La Miscelanea Hostel** (Cra. 35 No. 7-86, tel. 4\/311-8635, www.lamiscelanea.co, COP$20,000 pp) is a low-key and relaxed option run by a friendly Paisa couple. This hostel in Provenza started as a restaurant\/bar, and they still have occasional live music performances in their bar area. There's lots of vegetarian fare on offer, and it's open to the public. The _tienda_ (store) has funky things made by Medell\u00edn's creative folk.\n\nM **Urban Buddha** (Circular 73A No. 38-55, tel. 4\/413-9322, www.buddhahostel.com, COP$20,000 dorm, COP$60,000 d) is a friendly place to stay in the leafy neighborhood of Laureles. The garden out back is a peaceful urban refuge and nice place to study one's Spanish or chat with other world travelers. The Spanish owners of this hostel also run the **Secret Buddha Hostel** (Cl. 94 B Sur No. 51-121, La Estrella, tel. 4\/279-5152, cell tel. 312\/892-6521, www.buddhahostel.com, COP$25,000 dorm, COP$65,000 d) in the Estrella municipality, outside of Medell\u00edn. It's green and quiet out there, but you can still head into town easily on the Metro.\n\n##### **COP$70,000-200,000**\n\nIn contrast to many Colombian cities, Medell\u00edn has a fair variety of midrange hotels. Chain hotels tend to be best, holding few surprises.\n\nThe M **Casa Hotel Asturias Medell\u00edn** (Circular 4 No. 73-124, tel. 4\/260-2872, COP$89,000 d) is on a delightful corner of the tree-lined and quiet Laureles neighborhood. That's the big selling point for this small hotel. Rooms are modern and comfortable, although not terribly huge. It's a good deal.\n\nWell located in the Provenza neighborhood, **Acqua Hotel Express** (Cra. 35 No. 7-47, tel. 4\/448-0482, cell tel. 320\/788-4424, www.hotelacqua.com, COP$131,000 d) is a good value. Its 43 rooms are spic and span and comfortable.\n\nFrench budget chain M **Hotel Ibis** (Cl. 20 No. 44-16, tel.4\/444-1554, www.ibis.com, COP$99,000 d) has modern rooms with comfortable beds at great rates, and is located in an interesting area in the Ciudad del R\u00edo, across the street from the Museo de Arte Moderno de Medell\u00edn. There's no gym, but the neighborhood is quiet, making it a decent place for a morning jog. The best views are on the hotel's south side. The hotel restaurant offers buffet meals for an additional price. On the weekends it's very quiet, and room rates drop to an unbelievable COP$79,000.\n\nLocated across from the Atanasio Girardot sports complex, the **Hotel Tryp Medell\u00edn** (Cl. 50 No. 70-24, tel. 4\/604-0686, www.tryphotels.com, COP$146,000 d) has 140 large, comfortable (if spartan) rooms and an excellent rooftop terrace with a whirlpool and steam room. Guests have access to an extremely loud gym on the lobby floor. Restaurants are nonexistent in this area, except for street food, and hotel room service is iffy.\n\nThe **Hotel BH El Poblado** (Cra. 43 No. 9 Sur 35, tel. 4\/604-3534, www.bhhoteles.com, COP$170,000 d) is across from the enormous Centro Comercial Santa Fe. This Colombian chain hotel with 70 rooms has huge, comfortable beds and modern rooms, and despite its location on a major street (Av. El Poblado), it's not that noisy. An included breakfast buffet is served in a pleasant open-air terrace. It's also got the world's tiniest hotel gym with about three cardio machines.\n\nIt's got a boring and frankly ugly location, but the standard-to-the-core **GHL Comfort Hotel San Diego** (Cl. 31, No. 43-90, www.ghlhoteles.com, COP$124,000 d) offers good prices and the staff is attentive. A mediocre breakfast is served on the top-floor terrace (featuring an excellent view), and amenities include a sauna and small gym. It's close to a couple of malls and is between the Centro and El Poblado on a main road. The Ciclov\u00eda passes by in front on Sundays, making it a snap to get out and move.\n\nHard-core city people will be the ones interested in staying in the Centro. The **Hotel Nutibara Conference Center** (Cl. 52A No. 50-46, tel. 4\/511-5111, www.hotelnutibara.com, COP$132,000 d) is the best choice. It's a faded, grand old hotel located steps from the Museo de Antioquia. With wide corridors and huge rooms with parquet floors, it retains mid-20th-century elegance and personality.\n\n#### **FOOD**\n\nThe revitalization and resurgence of Medell\u00edn that began in the early 2000s has also led to culinary revolution, with countless new dining options popping up throughout the city. The best neighborhoods for dining are Provenza and El Poblado and the Zona M in Envigado.\n\n##### **Colombian**\n\n**Mondongo's** (Cra. 70 No. C3-43, tel. 4\/411-3434, www.mondongos.com.co, 11:30am-9:30pm daily, COP$20,000) is a well-known and popular place for typical Colombian food and for drinks with friends. _Mondongo_ is a tripe stew, a Colombian comfort food, In addition to the Carrera 70 location there is another Mondongo's on the busy Calle 10 in El Poblado (Cl. 10 No. 38-38, tel. 4\/312-2346) that is a popular drinking hole as well. They've even got a location in Miami.\n\nAnother popular place on the Carrera 70 strip is **La Tienda** (Cra. 70 Circular 3-28, tel. 4\/260-6783, 10am-2am daily). It's a festive restaurant that morphs into a late-night drinking place as Medell\u00edn evenings wear on. Their _bandeja paisa_ is famous. It's a signature Antioquian dish that includes beans, rice, sausages, and pork rinds.\n\nAlong the Avenida Las Palmas above El Poblado are several large and famous grilled meat and _comida t\u00edpica_ restaurants. They are especially popular on weekend afternoons. **Hato Viejo** (Cl. 16 No. 28-60, Av. Las Plamas, tel. 4\/268-5412 or 4\/268-6811, noon-11pm daily, COP$25,000) is a popular place for a weekend lunch with the gang. On Friday nights they have live music. **San Carb\u00f3n** (Cl. 15A No. 30-80, tel. 4\/311-7602, www.sancarbon.com.co, noon-10pm weekdays, noon-2am on weekends, COP$29,000) often has live music Wednesday-Sunday. Specialties include barbecue pork ribs and pepper steak.\n\nThe Provenza area has a number of cute and original Colombian specialty restaurants. **Cazuelas de Mi Tierra** (Cra. 37 No. 8A-116, tel. 4\/448-6810, www.cazuelasdemitierra.com, 8am-5pm Mon.-Wed., 8am-7pm Thurs.-Sat., 10am-4pm Sun., COP$20,000) has a special each day and always plenty of hangover-combating creamy _cazuelas_ (stews).\n\n**Mi Bu\u00f1uelo** (Cl. 8 No. 35-33, tel. 4\/311-5370, 6:30am-8pm Mon.-Sat., 6:30am-3pm Sun.), meanwhile, is a tribute to those unassuming, perfectly round, fried balls of dough, _bu\u00f1uelos._ **Arepitas Pa' Papa** (Cra. 34 No. 7-73, tel. 4\/352-2455, 11am-2:30pm and 6pm-10pm Mon.-Sat., COP$15,000) lets you create an arepa (cornmeal cake) with your favorite toppings.\n\nM **Queareparaenamorarte** (tel. 4\/542-0011, cell tel. 316\/741-4458, 12:30pm-8:30pm Mon.-Wed., 12:30pm-10:30pm Thurs.-Sat., 12:30pm-7pm Sun., COP$25,000) is not your typical _comida t\u00edpica_ restaurant. Juilian, owner, chef, and expert on Colombian cuisine, has traveled the country over and has brought the secrets back from grandmothers' kitchens from the Amazon to Santa Marta.\n\n##### **Fusion**\n\nM **In Situ** (Jard\u00edn Bot\u00e1nico, tel. 4\/460-7007, www.botanicomedellin.org, noon-3pm Mon., noon-3pm and 7pm-10pm Tues.-Sat., noon-4pm Sun., COP$30,000) may have the nicest view of any eatery in Medell\u00edn. It's surrounded by a million shades of green on the grounds of the Jard\u00edn Bot\u00e1nico. It's an elegant place for a lunch, but if you've been sweating it visiting the city, you may feel out of place among the sharply dressed business and society crowds. In Situ has an interesting menu with items such as apple sea bass (COP$30,000) and beef medallions in a coffee sauce with a plantain puree (COP$29,000).\n\nNext to the Museo de Arte Moderno de Medell\u00edn is hip **Bonuar** (Cra. 44 No. 19A-100, tel. 4\/235 3577, www.bonuar.com, 10am-7pm Tues.-Fri., 11am-6pm Sat., noon-4pm Sun. holidays, COP$22,000), where the burgers (including a Portobello and lentil version) are famous, but so is the brunch. It's a cool place with a nice outdoor seating area. During weekdays go in the evening when it's livelier.\n\nA classic, old-school restaurant is **La Provincia** (Cl. 4 Sur No. 43A-179, tel. 4\/311-9630, www.restaurantelaprovincia.com, noon-3pm and 7pm-midnight Mon.-Sat., COP$28,000). It is a fusion of Mediterranean cuisine (lots of seafood) with Colombian flair. Reserve a table on the romantic patio out back if you can. Try the exotic grilled fish fillet in a peanut sauce with green papaya strips.\n\n**El Herbario** (Cra. 43D No. 10-30, tel. 4\/311-2537, www.elherbario.com, noon-3pm and 7pm-11pm daily, COP$24,000) has an inventive menu with items such as lemongrass tuna, turmeric prawns, and artichoke risotto. Spacious and minimalistic, it can feel a little like eating in a warehouse, though. The attached store sells exotic jams and chutneys and the like.\n\n##### **American**\n\nChef Ricardo Ram\u00edrez studied culinary arts in New Orleans, came back to Colombia, and immediately went to work designing the menu for the Cajun restaurant **Stella** (Cra. 44A No. 30 Sur-7, tel. 4\/448-4640, 11am-11pm Tues.-Sat., 11am-3pm Sun., COP$22,000). He's got things right\u2014there are po'boys, muffaletta sandwiches, catfish, jambalaya, and even alligator sausage. (They get the alligators from a farm near Monter\u00eda.) The non-reptile crowd can try the vegetarian \u00e9touff\u00e9e. Sunday brunches are often accompanied by live jazz music.\n\nThe most innovative restaurant to come Medell\u00edn's way in a long time is M **Aloha Bar & BBQ** (Cra. 37A No. 8A-70, tel. 4\/444-1148, 11am-11pm Mon.-Thurs., 11am-4am Fri.-Sat., 10am-9pm Sun., COP$25,000). Run by a Hawaiian couple, this is the place for pork sliders, teriyaki ribs, and cole slaw. It's open way late on the weekends, great for a late night nosh after hitting the Parque Lleras bars.\n\n##### **Spanish**\n\nCozy and chic Spanish restaurant **El Barral** (Cl. 30 Sur No. 43A-38, tel. 4\/276-1212, noon-10pm Mon.-Sat., COP$30,000) specializes in paella, tapas, and sangria, and does them well.\n\n##### **Steak**\n\nWith Colombian newspapers plastered on the walls displaying headlines of yesteryear, the Argentinian steakhouse **Lucio Carb\u00f3n y Vino** (Cra. 44A No. 30 Sur-40, Envigado, tel. 4\/334-4003, noon-midnight Mon.-Sat., COP$32,000) specializes in grilled steak, paired with a nice Malbec.\n\n##### **British**\n\n**Cockers Greasy Spoon** (Cl. 7 No. 35-56, Provenza, cell tel. 301\/520-2668, 9am-2:30pm Tues.-Sat.) knows how to fry up an authentic British breakfast, like baked beans and bacon on toast. They make their own sausage, as well as most everything else on the menu, but the blokes who run the place admit that they don't lay the eggs. For lunch, try the fish and chips. It's a house specialty.\n\n##### **Thai**\n\nAuthentic Asian restaurants are few and far between. **Royal Thai** (Cra. 8A No. 37A-05, tel. 4\/354-2843, www.royalthaicolombia.com, COP$27,000) gets mixed reviews, and it's expensive, but hey, it's Thai.\n\n##### **Indian**\n\n**Naan** (Cra. 35 No. 7-75, tel. 4\/312-6285, COP$22,000) is a small and trendy Indian place in the Provenza area.\n\n##### **Middle Eastern**\n\nThere are a few good Lebanese and Arab food options in El Poblado and in Laureles. At **Tabun** (Cra. 33 No. 7-99, tel. 4\/311-8209, www.eltabun.com, noon-10pm Mon.-Thurs., noon-11pm Fri.-Sat., noon-9pm Sun., COP$22,000), in addition to usual Arab fare, they also have a few Indian dishes. Plus belly dancers on weekends!\n\n**Fenicia** (Cra. 73 No. C2-41, Av. Jard\u00edn, tel. 4\/413-8566, www.feniciacomidaarabe.com, noon-8pm Mon. Thurs., noon-9pm Fri.-Sat., noon-4pm Sun., COP$15,000) is an authentic Lebanese restaurant run by a family who immigrated to Colombia years ago.\n\n##### **Italian**\n\nAt **Crispino** (Circular 1A No. 74-04, Laureles, tel. 4\/413-3266, noon-11pm Mon.-Thurs., noon-midnight Fri.-Sat., noon-5pm Sun., COP$20,000), owner Salvatore, direct from Naples, offers authentic Italian cuisine.\n\nWhereas most restaurants in El Poblado face rather busy and noisy streets, **Toscano** (Cl. 8A No. 34-20, tel. 4\/311-3094, cell tel. 314\/739-6316, 10:30am-11pm Tues.-Sun., COP$13,000 lunch set menu, COP$20,000) is on a quiet street largely isolated from the neighboring riffraff. It's a delight to sit outside and have a pasta dish with a glass of wine.\n\n##### **Vegetarian**\n\nMost restaurants except the hard-core Colombian _parilla_ -type places now offer at least one lonely vegetarian dish on their menu. No need to pity the herbivore any longer. In Provenza, make a beeline for the cool atmosphere and fantastic vegetarian food at two-story M **Verdeo** (Cra. 35 No. 8A-3, tel. 4\/444-0934, www.ricoverdeo.com, noon-10pm Mon.-Wed., noon-11pm Thurs.-Sat., noon-4pm Sun., COP$18,000). This vegetarian haven discreetly set at the end of a street in the Provenza neighborhood could be the best vegetarian restaurant in Colombia. Veggie burgers go down well with an artisanal beer, but there are also Asian and Italian inspired \u00e0 la carte options. Lunch menus are inventive, and are a bargain. Downstairs is more atmospheric, but if you dine upstairs, you can look out over an urban _guadua_ (bamboo) forest. An organic market, **Ceres** (Cra. 35 No. 8A-3, www.ceresmercadoorganico.com) is on the second floor of Verdeo.\n\n**Lenteja Express** (Cra. 35 No. 8A-76, tel. 4\/311-0186, cell tel. 310\/879-9136, 11am-9pm Mon.-Sat.) specializes in veggie burgers: chickpea burgers, lentil burgers, Mexican burgers.\n\nIn the Laureles area, **Pan y Vida Caf\u00e9** (Cra. 51D No. 67-30 Policlinica, tel. 4\/583-8386, 7am-7pm Mon.-Wed., 7am-10pm Thurs.-Fri., 7am-4pm Sat., COP$12,000) serves healthy meals, often featuring the Andean super grain quinoa and organic vegetables. At lunchtime they offer a choice of two set meals. Outdoor seating is a fine idea on a pretty day. Pan y Vida is open in the mornings for coffee, juices, and pastries.\n\n##### **Caf\u00e9s and Quick Bites**\n\n**Fellini** (Plaza Mayor, Cl. 41 No. 55-80, Local 105, tel. 4\/444-5064, www.fellini.com.co, noon-8pm Mon.-Fri., noon-4pm Sat., COP$15,000) specializes in burgers, but they also serve sandwiches, salads, and pastas. Plan to eat here after your long day of sightseeing downtown.\n\nThe **Juan Valdez Caf\u00e9** (Cra. 37A No. 8A-74, 10am-9pm daily), atop the Parque Lleras, is a point of reference for the area and the place to meet up with someone for a cappuccino. It's popular with travelers and locals alike.\n\nIf you're feeling decadent, as in you'd like your latte in an actual coffee cup and served to you at a table, try **Pergamino Caf\u00e9** (Cra. 37 No. 8A-37, tel. 4\/268-6444, 10am-9pm Mon.-Fri., 11am-9pm Sat). It's on a quietish street in the Provenza area.\n\n**Manzzino** (Circular 72 No. 38-44, tel. 4\/580-7000, 10am-9pm Mon.-Sat., 10am-6pm Sun.) will quickly become your fave Uruguayan neighborhood bakery caf\u00e9 in all of Medell\u00edn, hands down. They've got apple pies, scrumptious almond cakes, quiches, and sandwiches, and you can enjoy them on a delightful terrace as you watch neighborhood folks go about their business in _tranquilo_ (peaceful) and delightful Laureles.\n\nFour in the morning and you've got the munchies? Join the legion of taxi drivers, college kids, and miscellaneous night owls at **Trigo Laurel** (Circula 1A No. 70-06, tel. 4\/250-4943, 24 hours daily). It never closes, not on New Year's not on the 20 de Julio, when Colombians celebrate their independence. They specialize in baked goods, but they also serve cheap lunches. It's on a quiet corner of Carrera 70.\n\nFresh juices are the specialty at **Cosechas Express** (Cl. 10 No. 35-25, tel. 4\/266-9139, 8am-6:30pm Mon.-Fri., 9am-4pm Sat.). There's an infinite number of possibilities here, as you can mix and match.\n\n#### **INFORMATION AND SERVICES**\n\n##### **Spanish-Language Courses**\n\n**Universidad EAFIT Centro de Idiomas** (Cra. 49 No. 7S-50, Edificio 31, Oficina 201, tel. 4\/261-9500, ext. 9439 or ext. 9669, COP$880,000 38-hr. course) offers intensive (20 hours per week) and semi-intensive (10 hours per week) Spanish classes. Their courses are certified by the Spanish Cervantes Institute. The **Universidad de Antioquia** (Cra. 52 No. 50-13, Edificio Suramericana, tel. 4\/219-8332, ext. 9003, www.idiomasudea.net) offers personal language instruction at a price of COP$40,000 per hour. Group classes can also be arranged.\n\nA language exchange called \"The Lab\" takes place every Wednesday at **Buena Vista Bar** (Cra. 37 No. 8A-83, cell tel. 313\/788-7440, 7pm-1:30am Wed.). At this friendly gathering of Colombians, travelers, and expats, you can mingle with others and brush up on (or show off) your Spanish, Portuguese, French, Italian, German, or English skills. Afterwards, enjoy an international fiesta featuring salsa music on the bottom floor and international beats on the upstairs terrace of this cool space in the Parque Lleras area.\n\n##### **Tourist Information**\n\nMedell\u00edn produces the most comprehensive tourist information of any city in Colombia. In addition to tourist information booths at the bus terminals and airports, there is a large office at the **Plaza Mayor** (Cl. 41 No. 55-80, tel. 4\/261-7277, 8am-noon and 2pm-6pm Mon.-Sat.). The tourism office webpage (www.medellin.travel) is up to date with information on what's going on in the city.\n\nThe main newspaper in Medell\u00edn is **_El Colombiano_** (www.elcolombiano.com). Other online resources for events and activities in Medell\u00edn and in the area are **Medell\u00edn Living** (www.medellinliving.com), a blog site run by expats; **Medell\u00edn Style** (www.medellinstyle.com), which has information on DJ events in town; **Plan B** (www.planb.com.co); ticket outlet **Tu Boleta** (www.tuboleta.com); and **Guia Gay Colombia** (www.guiagaycolombia.com), for information on gay and lesbian nightlife.\n\n#### **GETTING THERE AND AROUND**\n\nIt's a snap getting to centrally located Medell\u00edn from just about anywhere in Colombia, and from Florida. And once in the Antioquian capital, getting around is pretty easy, too.\n\n##### **By Air**\n\nThere are nonstop flights from Medell\u00edn's **Aeropuerto Internacional Jos\u00e9 Mar\u00eda C\u00f3rdova** (MDE, Rionegro, tel. 4\/402-5110 or 4\/562-2885) to all major cities in Colombia. The airport is simply referred to as \"Rionegro.\" Avianca, LAN, and Viva Colombia operate domestic flights. Internationally, Avianca serves Miami, Panama City, Lima, and Madrid; American Airlines has a nonstop flight to Miami; Spirit and JetBlue to Fort Lauderdale; AeroGal and LAN to Quito; Copa to Caracas and Panama City; and Insel Air to Cura\u00e7ao.\n\nThe Rionegro airport is about 45 minutes (35 km\/22 miles) from downtown, depending on traffic. Taxis cost around COP$60,000 between the city and the airport.\n\nAlternatively, there are _busetas,_ small buses, that leave the airport bound for the San Diego neighborhood, which is convenient to El Poblado. These can be found as you exit the terminal towards the right. Upon arrival in Medell\u00edn, there are taxis on standby. An organized and legitimate group of young people will help place your bags in the cab, even though you may not want or need this service. They expect a COP$1,000-2,000 tip.\n\nTraveling to the airport, there are buses (Conbuses, tel. 4\/231-9681) that depart from a side street just behind the Hotel Nutibara (Cl. 52A No. 50-46, tel. 4\/511-5111) in the Centro. These depart from about 4:30am until 7:20pm every day, and the trip costs COP$8,000. The buses are hard to miss: They're green and white with the word _aeropuerto_ printed in all caps on the front window.\n\n**Aeropuerto Olaya Herrera** (AOH, Cra. 65A No. 13-157, tel. 4\/403-6781, www.aeropuertoolayaherrera.gov.co) is the super-convenient in-town airport. **EasyFly** serves Monter\u00eda, C\u00facuta, Bucaramanga, Apartad\u00f3, and Quibd\u00f3; **Satena** serves Bogot\u00e1, Quibd\u00f3, Apartad\u00f3, Bah\u00eda Solano, and Nuqui. **ADA** (Aerol\u00edneas Antioque\u00f1as) serves a whole slew of cities throughout Colombia, especially western Colombia. The Olaya Herrera terminal was built in the 1930s and is an architectural gem.\n\n##### **Intercity Buses**\n\nMedell\u00edn has two bus terminals: the Sur and the Norte. The **Terminal del Sur** (Cra. 65 No. 8B-91, tel. 4\/444-8020 or 4\/361-1186) is across from the Aeropuerto Olaya Herrera, and it serves destinations in southern Antioquia and the coffee region. The **Terminal del Norte** (Cra. 64C No. 78-580, tel. 4\/444-8020 or 4\/230-9595) is connected to the Caribe Metro station. It serves Santa Fe de Antioquia and Guatape, the Caribbean Coast, and Bogot\u00e1.\n\n##### **Metro**\n\nMedell\u00edn's **Metro** (tel. 4\/444-9598, www.metrodemedellin.gov.co) is the only urban train system in the country. It's a safe and super-clean system of two lines: L\u00ednea A, which runs from Niqu\u00eda (north) to La Estrella (south), and L\u00ednea B, from San Antonio in the Centro west to San Javier. The Metro line A is useful for traveling between El Centro, El Poblado, and Envigado. Metro line B has a stop at the stadium. The current Metro fare is COP$1,800; however if you think you may use the Metro, Metrocable, and Metropl\u00fas system on a regular basis, consider purchasing a refillable Tarjeta C\u00edvica card that is valid on all three transportation networks. The cost per ride with the Tarjeta C\u00edvica modestly drops to COP$1,600. The card can be purchased at Metro ticket booths.\n\nthe Medell\u00edn Metro\n\n##### **Metrocable**\n\nThe Metrocable public transportation system, consisting of gondola _(telef\u00e9rico)_ lines, was inaugurated in 2004 and consists of three lines, with two under construction. It has been internationally lauded as an innovative approach to solving the particular transportation needs of the isolated and poor _comunas_ (residential sectors), built on mountainsides of the city. The three Metrocable lines are: L\u00ednea J from the San Javier Metro station to La Aurora in the west, L\u00ednea K from the Acevedo Metro station in the north to Santo Domingo, and L\u00ednea L from Santo Domingo to the Parque Arv\u00ed. The Metrocable runs 9am-10pm daily. The Metrocable L\u00ednea L from Santo Domingo to the Parque Arv\u00ed operates 9am-6pm Tuesday-Sunday. When Monday is a holiday, the L\u00ednea L runs that day and does not operate the next day, Tuesday.\n\n##### **Metropl\u00fas Rapid Bus**\n\nThe first line of the **Metropl\u00fas** (www.metroplus.gov.co) rapid bus system, with dedicated bus stops similar to those of the TransMilenio in Bogot\u00e1, debuted in 2013. There are two Metropl\u00fas lines: L\u00ednea 1 and L\u00ednea 2. L\u00ednea 1 connects the working-class neighborhood of Arjuanez in the north with the Universidad de Medell\u00edn in the southwest. L\u00ednea 2 connects the same two sectors but passes through the Centro and Plaza Mayor area. To access the system, you have to use the Tarjeta C\u00edvica, which can be purchased at any Metro station.\n\n##### **Taxis**\n\nTaxis are plentiful in Medell\u00edn. Order them over the phone when possible. A few taxi companies have easy-to-remember numbers: tel. 4\/444-4444, tel. 4\/335-3535, and tel. 4\/211-1111. Friendly **Miguel Espinosa** (cell tel. 311\/378-3565) is a cabbie based around the Laureles area. You can also order reliable cabs using the smartphone app Tappsi.\n\n### **Northern and Eastern Antioquia**\n\n#### **SANTA FE DE ANTIOQUIA**\n\nLiving and breathing colonial charm, this pueblo 80 kilometers (50 miles) northwest of Medell\u00edn is the best of Antioquia. The historic center of the town is compact, with landmarks of plazas, parks, and churches. Santa Fe was founded in 1541 by Jorge Robledo, a ruthless conquistador. An important center for gold mining, Santa Fe was capital of Antioquia until 1823, when it lost that title to Medell\u00edn. On the banks of the R\u00edo Cauca, its proximity to Medell\u00edn makes it an easy trip for those interested in seeing a colonial-era jewel of a pueblo.\n\nWith the average temperature a sizzling 27\u00b0C (81\u00b0F), it can be a challenge to fully enjoy strolling the lovely streets of the pueblo during the heat of the day. If you can, plan to go for the night (one weekday night will do), arriving in late afternoon. That's the nicest time to stroll the streets.\n\n##### **Sights**\n\nThe town's narrow stone streets are adorned with charming plazas and parks and five historic churches. It's a delight to stroll the town in the late afternoon, after the heat of the day has subsided. Churches and historic buildings in Santa Fe are often built in the typical _calicanto_ style, a mix of brick and stone construction materials. Historic colonial churches, with majestic facades, often face parks and are illuminated at night.\n\nThe \"grandmother\" of churches in Antioquia, the **Templo de Santa B\u00e1rbara** (Cl. 8 at Cra. 8, masses 7am and 6pm Mon.-Sat. 6am and 6pm Sun.), with its many baroque elements, was built towards the end of the 18th century. Next to it, in what was a Jesuit college, is the **Museo de Arte Religioso** (Cl. 11 No. 8-12, tel. 4\/311-3808, 9am-5pm Sat.-Sun., COP$2,000), a museum that highlights paintings, sculptures, and gold and silver pieces from the Spanish New World colonies.\n\nA nicely presented museum housed in a colonial-style house, the **Museo Juan del Corral** (Cl. 11 No. 9-77, tel. 4\/853-4605, www.museojuandelcorral.com, 9am-noon and 2pm-5:30pm Mon.-Fri., 10am-5pm Sat.-Sun., free) has exhibits on the history of Santa Fe, including historical items from 1813 when Antioquia was declared free. The museum also puts on temporary exhibits of contemporary Colombian artists, and other cultural events are held here.\n\nSix kilometers (four miles) outside of town, on an old road that leads to the town of Sopetr\u00e1n on the other side of the R\u00edo Cauca, is an architectural wonder, the **Puente de Occidente,** a suspension bridge made of iron and steel. It was built towards the end of the 19th century by Jos\u00e9 Mar\u00eda Villa, an engineer who studied in New Jersey and worked on the Brooklyn Bridge. It's a narrow bridge and has been closed to vehicular traffic, for the most part. _Mototaxis_ can take you there from town, across the bridge, and back for COP$15,000. The bridge is easily reached by bike as well.\n\n##### **Festivals and Events**\n\nThe big event in Santa Fe is the week-long **Festival de Cine de Antioquia** (www.festicineantioquia.com), a film festival held each year in early December. There is usually an international director or actor who is the guest of honor. Some free showings are held outdoors in the town's plazas and parks.\n\n##### **Accommodations**\n\nMedell\u00edn families converge on Santa Fe en masse on weekends, and for many the draw is to lounge by the pool at one of the hotels lining the main road leading into town. Hotels in town, however, have more charm. Hotel prices can drop substantially during the week.\n\nThe historic town of Santa Fe de Antioquia makes for a pleasant overnight trip from Medell\u00edn.\n\nIn town, the M **Hotel Mariscal Robledo** (Cl. 10 No. 9-70, tel. 4\/853-1111, cell tel. 313\/760-0099, www.hotelmariscalrobledo.com, COP$120,000-170,000 d) is far and away the most comfortable hotel, and one oozing with personality. Antiques, especially with a cinematic theme, decorate the lobby and common areas. Rooms on the second floor, which have not been given a 21st-century makeover, are nonetheless comfortable, and have far more character. The pool area is luxurious.\n\nOn the boutique side, the **Hotel Casa Tenerife** (Cra. 8 No. 9-50, tel. 4\/853-2261, www.hotelcasatenerife.com, COP$162,000 d) has 12 rooms, is tastefully decorated, and has a nice pool and courtyard area. It often caters to couples celebrating romantic getaways, with such details as rose petals on the bed.\n\nOn the Plaza Mayor are two options. The family-run **Hotel Caser\u00f3n Plaza** (Cl. 9-41, Plaza Mayor, tel. 4\/853-2040, www.hotelcaseronplaza.com.co, COP$145,000-208,000 d) has an excellent location but is overpriced for what you get. Some of the 33 rooms have air conditioning, which is a plus in Santa Fe. There is also a small pool in back, another plus. **Hostal de la Plaza Mayor** (Cra. 9 No. 9-59, tel. 4\/255-7427, cell tel. 311\/396-5628, <http:\/\/hostalplazamayorsantafedeantioquia.blogspot.com>, COP$50,000 d) is the budget option in town. Staff are friendly, but it's a little run down.\n\n##### **Food**\n\nThere are few places in Colombia where one can dine to the soft tones of classical or jazz music. The **Restaurante Bar La Comedia** (Parque Santa Barbara, tel. 4\/853-1243, noon-3pm and 6pm-10pm Wed.-Sun., COP$18,000) is one such place. Light dishes, sandwiches, and cr\u00eapes dominate the small menu, and this is also an option for late afternoon _onces,_ tea time. It's diagonal to the Santa B\u00e1rbara church. **Restaurante Port\u00f3n del Parque** (Cl. 10 No. 11-03, tel. 4\/853-3207, noon-8pm Sun.-Thurs., noon-9:30pm Sat.-Sun., COP$20,000) is lavishly decorated with portraits and paintings by owner Olga Cecilia. In addition to typical Paisa specialties (lunch specials during the week go for under COP$10,000), the extensive menu offers seafood and international cuisine. Finally, the restaurant at the **Hotel Mariscal Robledo** (Cl. 10 No. 9-70, tel. 4\/853-1111, cell tel. 313\/760-0099, www.hotelmariscalrobledo.com, 8am-3pm and 7pm-10pm daily, COP$25,000) is always a good choice.\n\n##### **Recreation**\n\n**Naturaventura** (Hotel Mariscal Robledo, Cl. 10 No. 9-70, tel. 4\/853-1946, naturaventura1@hotmail.com) organizes nature walks, bike trips, horseback riding, and rafting excursions. For horseback riding, contact **Gu\u00edas Turantioquia** (tel. 4\/853-1148), which organizes day-trip horseback riding tours in and around Santa Fe.\n\n##### **Shopping**\n\nSpaniards were once attracted to Santa Fe because of its gold. Today it is famous for its intricate filigree jewelry. To peruse some, visit **ORFOA** (Cl. 9 No. 6-02, tel. 4\/853-2880, 9am-noon and 2pm-6pm daily) or **Dulces & Artesan\u00edas Clavellina** (Hotel Mariscal Robledo, Cl. 10 No. 9-70, tel. 4\/853-2195, 9am-noon and 2pm-6pm daily). Clavellina is the symbolic flower of Santa Fe.\n\n**Guarnieler\u00eda y Marroquiner\u00eda** (Cl. 10 No. 7-66, cell tel. 314\/847-8354, noon-7pm Mon.-Fri., 10am-7pm Sat.-Sun.) sells authentic Jeric\u00f3 _carrieles_ (shoulder bags used by Paisa cowhands) and other locally made leather handicrafts. **La Casa Solariega** (Cl. de la Amargura No. 8-09, tel. 4\/853-1530, 9am-noon and 2pm-6pm daily) has an eclectic collection of handicrafts, paintings, and antiques in a typical Santa Fe house.\n\n##### **Information and Services**\n\nA **tourist information office** on the Plaza Mayor (Cra. 9 and Cl. 9, tel. 4\/853-1022) has maps and hotel information.\n\n##### **Getting There and Around**\n\nThere is regular bus service, several times a day, from the **Terminal de Transportes del Norte** (Cra. 64 No. 78-344, tel. 4\/267-7075, www.terminalesmmedellin.com) in Medell\u00edn to Santa Fe. The journey takes two hours and takes you through a feat of modern engineering: the **T\u00fanel Fernando G\u00f3mez Mart\u00ednez,** the longest tunnel in South America. To return, walk a couple of blocks to the Medell\u00edn-Turbo highway near the market at Carrera 10 and flag down passing buses. Most of them are going to Medell\u00edn. The trip costs under COP$10,000.\n\n#### **MAGDALENA MEDIO**\n\n##### M **Reserva Natural R\u00edo Claro**\n\nA visit to the spectacular, privately run **Reserva Natural R\u00edo Claro** (Medell\u00edn office tel. 4\/268-8855, cell tel. 311\/354-0119, www.rioclaroelrefugio.com) is a highlight for anyone visiting Colombia. In the steamy and remote Magdalena Medio region of Antioquia, the reserve encompasses 450 hectares (1,100 acres) along the R\u00edo Claro canyon, a babbling, crystal-clear river. This reserve is a place to enjoy the unspoiled beauty of the river and its jungle and to disconnect from the hectic pace of urban life.\n\nThe story behind the park begins with an oft-repeated tale about a pesky jaguar. It seems that the cat was blamed for killing some livestock of a campesino in the area. In a quest to track down the guilty party (the jaguar got away unharmed), the farmer followed its tracks through the jungle, over several days, and to a spectacular canyon. When Juan Guillermo Garc\u00e9s heard about the astonishing discovery, he had to see this undiscovered territory for himself. Garc\u00e9s immediately knew that this was a special place, and he made a commitment to purchase the land to protect it from development, including a highway that was to pass through this pristine land.\n\nOn weekends, R\u00edo Claro receives many visitors. In addition to those staying at the reserve, many day visitors spend the afternoon at R\u00edo Claro. Don't go on a Saturday, Sunday, or holiday if you seek a peaceful commune with nature. If you visit the reserve midweek, you'll most likely have the place practically to yourself, which is heavenly.\n\n###### **RECREATION**\n\nGuides don't speak English, generally. There are two must-do activities at the reserve. The first is an easy rafting trip down the river (COP$20,000), during which you can see the karstic jungle, in which trees grow atop rocks. This excursion takes about two hours. The second must-do activity is a combination swim\/hike trip to the **Caverna de los Gu\u00e1charos** (COP$15,000 pp). This guided walk has its challenging moments: wading across the swiftly flowing river, making your way through the dark, dark cavern, climbing out of the cavern, and then making your way back across the river. _Gu\u00e1charo_ birds (oilbirds), living inside the cavern, act like they own the place (the cavern is, after all, named for them). They don't like it when human intruders invade their space, and they'll let you know that with their screeching. The cavern is made of marble; its stalactites and stalagmites are impressive. Waterproof shoes with good traction are recommended, as you'll be wading in water most of the time. Also, it's nice to have a headlamp so that you'll have hands free. You can take your camera, but at a certain point it will need to be kept in a water repellent bag, which the guide will have. If you're up for both trips, go on the cavern tour in the morning and go rafting in the late afternoon.\n\nReserva Natural R\u00edo Claro\n\nOther activities at the reserve include rock-climbing, a zip line, hanging out on the marble beach, self-guided nature walks, and tubing. These are all arranged by R\u00edo Claro staff.\n\n###### **ACCOMMODATIONS**\n\nThe reserve has a variety of accommodations options. Contact the R\u00edo Claro office (tel. 4\/268-8855, cell tel. 311\/354-0119, www.rioclaroelrefugio.com) for all reservations and information. The **Hotel El Refugio** (COP$80,000) is above the reception and dining area, and is a comfortable all-wooden lodge construction. The best and most isolated is at the far end near the canyon, a 15-minute walk from the main reception area in the **Caba\u00f1as El Refugio** (COP$95,000-140,000 pp), where rooms are quite spectacular and open-air. You'll sleep well here with the sounds of the rushing water to lull you asleep. Rooms are completely open, but there are no problems with mosquitoes.\n\nThe **Hotel R\u00edo Claro** (COP$95,000 pp) is across the highway from the rest of the reserve but still along the river, and it has a big pool. These are small concrete bungalows. The hotel is popular with student groups. All meals are included in the room rates. Tell staff when you make your reservation if you have any dietary needs or special requests, like fresh fruit.\n\n###### **GETTING THERE**\n\nThe reserve is easily reached by bus from Medell\u00edn. All buses between Medell\u00edn and Bogot\u00e1 pass in front of the R\u00edo Claro entrance, where there is a small security booth. From Medell\u00edn, it takes around three hours, costing around COP$20,000. Be sure to tell the driver you'd like to be dropped off at the _\"entrada de la Reserva R\u00edo Claro.\"_ (\"the entrance to the R\u00edo Claro Reserve\").\n\n##### **Hacienda Napoles**\n\nThe **Hacienda Napoles** (Puerto Triunfo, cell tel. 314\/892-2307, www.haciendanapoles.com, 9am-6pm Tues.-Sun., COP$32,000) was a vacation home for Pablo Escobar, complete with an airstrip and exotic animals, including quite a few hippos, who apparently adapted nicely to the muggy climes of the R\u00edo Magdalena area. Today Hacienda Napoles is a theme park with giant dinosaur sculptures, some of which were built by Escobar for his children; two water parks (additional fees); hippopotami, zebras, and ostriches; an Africa museum; the remnants of Pablo Escobar's country house (now a museum); a collection of old cars that were destroyed following his death; and his private airplane landing strip.\n\nAvoid the oppressive heat and intense sun of midday (and the crowds on weekends) by visiting early on a weekday morning. The park can easily be visited from R\u00edo Claro, which is about an hour away. When Monday is a holiday, the park closes on Tuesday rather than on Monday.\n\n#### **GUATAPE**\n\nThe stone monolith La Piedra dominates the landscape here, but the Guatape area is more than just a big rock: It's a weekend playground chock full of recreational activities that keep the crowds from Medell\u00edn busy.\n\n##### **Sights**\n\nGuatape is a resort town. Aside from La Piedra, it's known for its _z\u00f3calos,_ colorful designs of the friezes on the lower part of houses. Many of these honor the traditions of townspeople, such as farming and fishing, others have sheep or other animals, and still others hot rods or Pink Panther. A particularly colorful street is the **Calle del Recuerdos** near the Parque Principal.\n\nOn a serene mountainside near Guatape, beyond El Encuentro hostel on the same road is the **Monasterio Santa Mar\u00eda de la Epifan\u00eda** (www.monjesbenedictinosguatape.org), home to around 30 Benedictine monks. Guests, up to eight at a time, are welcome to stay. Every day of the week at the 5:15pm _visperas_ (vespers) service, the public is invited to hear the monks sing Gregorian chants.\n\nCheck out the **Iglesia de Piedra** in the town of El Pe\u00f1ol, a modern construction that resembles La Piedra, which is quite a strange sight.\n\n##### **La Piedra Pe\u00f1ol**\n\nKnown simply as La Piedra, **La Piedra Pe\u00f1ol** (8am-6pm daily, COP$10,000) is a giant rock monolith that soars 200 meters (650 feet) into the sky from the scenic and meandering Embalse Pe\u00f1ol-Guatape, an important reservoir covering some 64 square kilometers (25 square miles) that is an important producer of hydro-electric energy for the country. There's been quite a rivalry between the towns of El Pe\u00f1ol and Guatape over the years, over which town can claim La Piedra for their own. It is located between the two, a tad closer to the Guatape side. Things digressed to a point where folks from Guatape began to paint their town's name in large letters on one prominent side of the rock. People from El Pe\u00f1ol were not amused, and this giant marking of territory was halted by authorities. Today all that remains of that brouhaha is what appear to be the letters \"GI.\"\n\nThe 360-degree views from the top of La Piedra over the Guatape reservoir and Antioqiuan countryside are worth the toil of climbing up over 600 steps, in a ramshackle brick and concrete stairwell that is stuck to the rock, to the top. To celebrate your feat, you can have a drink at one of the snack bars there.\n\nIn front of La Piedra, there is a statue of the man who first climbed the monolith in 1954. Inspired by a priest, Luis Villegas L\u00f3pez and two friends took five days to slowly climb up cracks in the rock. They had to deal with a beehive and a rainstorm along the way, adding to the challenge. It's one of the top tourist attractions in Antioquia. From the bottom of the rock, look up and notice the hundreds of bromeliads growing along the sides of it.\n\nLa Piedra can be visited several ways. You can walk from Guatape, which takes 45 minutes. (Sunscreen and water are essential.) You can bike it, although the road that winds its way up to the rock entrance is quite steep. You can take a _mototaxi_ from your hotel (COP$10,000), or you can hop on a Jeep from the Parque Principal (between Cras. 28-29 and Clls. 31-32) in Guatape. It's best to make your visit during the early morning hours or late in the afternoon due to potent sun rays.\n\nThe town is surrounded by a large reservoir operated by EPM, the Medell\u00edn utility company. The reservoir was built in phases during the 1970s and was not without controversy, as the flooding of the area began without the full consent of the inhabitants. Finally all families were resettled by EPM by 1979, and the town of El Pe\u00f1ol gradually became covered with rising waters, with only a church steeple remaining as a reminder of the town's past.\n\n##### **Tours**\n\nA popular excursion is to take a **boat tour** with brothers Luis and Rodolfo Londo\u00f1o (cell tel. 312\/794-7150 or 312\/236-5783, COP$50,000-100,000 per boat) to some of the islands of the reservoir. A standard stop on the tour is to (or rather, above) the submerged town of Viejo Pe\u00f1ol. It was flooded on purpose during the construction of the reservoir and nearby dam in 1978. The only real remnant of the town is a large cross rising out of the water. A small historical museum displays old photos and historical memorabilia from the old town on the waterside. These tours typically last 45 minutes to 1.5 hours.\n\n##### **Accommodations**\n\nDuring the week, prices drop significantly at most hotels, especially if you pay in cash.\n\nM **El Encuentro** (tel. 4\/861-1374, cell tel. 311\/619-6199, www.hostalelencuentro.com, COP$20,000 dorm, COP$65,000 d), a 12-minute walk up from town, remains one of the best options in Guatape. Run by a friendly Californian named Greg, the hostel is on the Guatape-Pe\u00f1ol reservoir, and rooms are spread throughout two houses, with about 10 rooms in total. Most of these are private rooms, and some have a shared bath. A larger apartment and two dorms are available here, as well as a place for tents down by the lake. The staff at El Encuentro can organize a plethora of outdoorsy things to do: downhill mountain bike rides on their excellent bikes, hikes, and jumping off of bridges. Spanish classes can be arranged, and you can study your verbs on the nice deck.\n\nIn town is the newer **Tomate Caf\u00e9 Hostel** (Cl. 30 No. 28-120, tel. 4\/861-1100, cell tel. 312\/216-1199, www.tomatecafehostel.com, COP$18,000 dorm, COP$40,000 d). It's run by a Paisa family and has four small private rooms and two dorm rooms. A strong cup of coffee is always on offer here, as well as healthy and vegetarian food in their restaurant. It's next to a disco, so on weekend nights it can get thumping. You were warned!\n\nAt M **Mi Casa Guatape** (tel. 4\/861-0632, cell tel. 301\/457-5726, www.micasaguatape.com, COP$20,000 dorm, COP$60,000 d) guests wake up, step outside with a cup of coffee in hand, and greet their neighbor, La Piedra, with a warm _Buenos d\u00edas._ You can't get much closer to that big rock than from this small English-Colombian hostel. The hostel has five private rooms and one four-bed dorm as well as two kitchens for use. When not outdoors climbing La Piedra or taking out their kayak for a spin, guests can laze in hammocks on the deck, watch movies, or bond with the owners' sweet dog. Mi Casa works closely with **Adventure Activities** (cell tel. 301\/411-4442), just next door, a group that organizes an intense-rock climbing excursion up one of dozens of routes up La Piedra (COP$60,000, 7 hours), as well as other outings. Owner Sean takes guests on a waterfall hike (6 km\/4 miles round-trip, COP$15,000, 4 hours). It's easy to go into town from the hostel by catching a ride with a passing Jeep or with Mi Casa's preferred _mototaxi_ driver. Mi Casa is about three kilometers before Guatape on the main road (25-min. walk or COP$1,500 taxi ride) and is across the street from landmark El Estadero La Mona.\n\nThere are a couple of upscale hotels in town. **Hotel Portobello** (Cl. 32 No. 28-29, tel. 4\/861-0016, cell tel. 312\/783-4050, www.hotelportobeloguatape.com, COP$215,000 d) has 16 rooms, and most of them have a view of the lake. You can obtain a 25 percent discount during the week if you pay in cash.\n\n##### **Food**\n\nFish like massive carp and trout from the reservoir are the specialty in Guatape. Reliable fish and Colombian cuisine restaurants include **La Fogata** (Cra. 30 No. 31-32, tel. 4\/861-1040, cell tel. 314\/740-7282, 8am-8pm daily), on the waterfront.\n\nPizza and pasta are on the menu at **Rafaelos** (below Hotel Portobello on the waterfront, tel. 4\/861-0016, cell tel. 310\/200-9020, 11am-11pm Wed.-Sun.).\n\nCraving a curry? M **Donde Sam** (El Pe\u00f1ol, near church, tel. 4\/851-5401, cell tel. 320\/667-5870, 11am-11pm daily, COP$15,000) is worth the trip. Owner and chef Sam, from Agra, and his Colombian wife, Lina, serve up authentic Indian dishes (as well as other international cuisine). Lunches, like curry vegetables or chicken, are accompanied by a soup and salad. It's livelier at night, and sometimes they put on mood-setting music in an attempt to transport the crowd to Asia.\n\nSometimes exceptional hospitality can give one a sugar headache. That's what happens at Gloria Elena's generous candy tastings at **Dulces de Guatape** (Cl. 29 No. 23C-32, Barrio Villa del Carmen, tel. 4\/861-0724, 7am-6pm). At this small candy factory, they make all kinds of sweets, many with _arequipe_ (caramel) and some with fruits like the tart _uchuva_ and guava as well as some chocolate bonbons that have peanuts and almonds.\n\n##### **Getting There and Around**\n\nThere is frequent bus service from Medell\u00edn's north terminal to Guatape. The trip takes about two hours and costs COP$12,000. Buses depart Guatape at a bus terminal that was completed in 2013 on the waterfront. It's just one block from the main plaza. Buses returning to Medell\u00edn often fill up in a hurry on Sundays, especially during holidays. If you are relying on public transportation, book your return bus trip early. The last bus for Medell\u00edn departs at 6pm. The return fare is also COP$12,000.\n\n### **Southern Antioquia**\n\n#### M **JARD\u00cdN**\n\nSometimes place names fit perfectly. Such is the case in the picture-perfect Antioquian town of Jard\u00edn. The main park gushes year-round with trees and flowers always in bloom, and the streets are corridors of color as well, with brightly painted houses one after another.\n\nFor many years this town has been a favorite country getaway for Paisas from Medell\u00edn. It's becoming popular with international travelers, too, but still, if you arrive during the week, you'll feel like you've stumbled upon something special. On weekends, and especially on holidays, a festive atmosphere fills the air, and the Plaza Principal buzzes with activity.\n\nWhile the main selling points of Jard\u00edn are its good looks, nearby tropical forests and cloud forests, home to natural attractions such as the Caverna El Esplendor and the ProAves bird-watching reserve, provide good excuses for lacing up those hiking boots.\n\n##### **Sights**\n\nThe **Parque Principal** is the center of life in Jard\u00edn. It's full of colorful wooden chairs, eight flower gardens, a handful of tall trees that provide welcome shade, and a constant cast of characters passing through, hanging out, or sipping a coffee. Prominent on the east side of the park is the neo-gothic cathedral the **Bas\u00edlicaMenor de la Inmaculada Concepci\u00f3n** (Cra. 3 No. 10-71, mass daily 11am), a 20th-century construction, the striking interior of which is painted shades of turquoise.\n\nthe Parque Principal in pretty Jard\u00edn\n\nThe **Museo Clara Rojas** (Cra. 5 No. 9-31, tel. 6\/845-5652, <http:\/\/mcrpjardin.blogspot.com>, 8am-noon and 2pm-6pm, COP$2,000) has 19th-century period furniture and relics from the _colonizaci\u00f3n antioque\u00f1a,_ as well as a small collection of religious art, including a painting of Jesus as a child surrounded by lambs with medals hanging around their necks. The town's tourism office is behind the museum, operating the same hours as the museum.\n\n##### **Recreation**\n\nWalking around Jard\u00edn is a pleasant way to get to know the town and surrounding mountains. Setting out for a walk towards the surrounding western mountains, to the Alto de las Flores or Salto del \u00c1ngel, makes for a great morning. If you lose your way, ask for directions. On the east side of town, there is a charming path, the **Camino Herrera,** which leads to the Casa de los Fundadores. In that area are several coffee plantations.\n\nJard\u00edn has not one, but two mini chairlifts in town. The **Cable Aereo** (8am-6pm daily, COP$5,000 round-trip) goes up to the Cristo Rey hill. The other, more rustic **La Garrucha** (8am-6pm daily, COP$4,000), goes across town. Although these are popular with tourists, they were built with a purpose in mind: so that rural farmers would have an easier way to bring their coffee and other crops to market.\n\n**Condor de los Andes** (tel. 4\/845-5374, cell tel. 311\/746-1985, condordelosandes@colombia.com) is a tourism operator that organizes walks, paragliding, and waterfall rappelling. Their most popular activity is a day-long **rappelling** adventure to the **Caverna El Esplendor.** This cavern in the jungle outside of town is reached on foot (about a 1.5-hour walk). Once there you rappel down a 50-meter-high (164-foot-high) waterfall into the cavern. Transportation and lunch are included in the price (COP$95,000 pp), and they usually depart Jard\u00edn at around 8am, returning by 4pm. The group also offers paragliding (COP$75,000, 25 mins.) and rappelling at the 53-meter-high (174-foot-high) **Cascada Escalera** (COP$55,000). Condor de los Andes has a small hostel (COP$35,000 pp) five blocks from the Parque Principal.\n\nThose with an inner cowboy may want to take a horseback tour to the **Salto del \u00c1ngel** waterfall. Contact **John Jairo** (cell tel. 312\/825-4524) to reserve your spot.\n\nThe mountains that envelop most of Jard\u00edn are protected lands encompassing some 28,000 hectares (69,000 acres). This area is called the **Reserva Cuchilla Jard\u00edn Tamesis.** Within the reserve are caverns, waterfalls, caves, and nature paths. The **park office** (Alcald\u00eda building, 2nd floor, Cra. 3 No. 10-10, tel. 4\/845-5668, cell tel. 321\/758-7534, dmicuchilla@corantioquia.gov.co) offers free guided walks to these natural attractions.\n\n###### **BIRDING**\n\nColombia's premier bird-watching and conservation group, **ProAves** (www.proaves.org), operates a bird-watching park, the **Reserva Natural de las Loro Orejiamarillo,** within Reserva Cuchilla Jard\u00edn Tamesis. This is where the yellow ear parrot _(Ognorhynchus icterotis)_ can be seen, an endangered species in Colombia. They make their nests in the majestic _palma de cera_ (wax palm) trees. Another exotic bird to look for is the _colibr\u00ed de frontino (Coeligena orina),_ a species of hummingbird. In addition to birdlife, there have been spottings of pumas, the _oso de anteojos_ (an Andean bear), and deer.\n\nIt is ideal to get an early start to view the parrots\u2014as early as 5am. As the elevation is fairly high, the temperatures dip as low as 4\u00b0C (39\u00b0F). Rubber boots and warm clothing are essential.\n\nTo coordinate a visit to the bird-watching park, contact **EcoTurs** (Cra. 20 No. 36-61, Bogot\u00e1, tel. 1\/287-6592, www.ecoturs.org) in advance as staff are not always at the site. This tour agency manages visits to this and all of the other ProAves reserves across the country. In Jard\u00edn, contact Joana Villa (cell tel. 312\/867-1740), or contact Angela G\u00f3mez in Bogot\u00e1 (cell tel. 313\/852-9158) for more information about this park in Antioquia. There is a COP$15,000 entrance fee per person for Reserva Natural de las Loro Orejiamarillo; a guide service costs COP$50,000 per group; and round-trip jeep transportation along rugged mountain roads to the reserve is a whopping COP$240,000.\n\n**Devils of Riosucio**\n\nEvery two years in January, in the sleepy coffee- and plantain-growing town Riosucio in northern Caldas near the Antioquian town of Jard\u00edn, residents (and a growing number of visitors) go to the devil during the revelry of the **Carnaval de Riosucio**. This festival, one of the most beloved in the region, has an interesting story. It began out of a plea made by local priests for two feuding pueblos of Riosucio\u2014the gold-mining village of Quiebralomo and La Monta\u00f1a, home to a large indigenous population\u2014to get along. In 1847 both communities were nudged to participate in that year's Three Kings Day commemoration and to set aside their differences, temporarily at least. If they didn't come together that year in peace, they would invite the wrath of the devil.\n\nOver time, it was that last bit that resonated with the townspeople. From that year onward, groups of families, friends, and neighbors would get together and create elaborate floats and costumes, seemingly in homage to the devil over this five-day celebration. The festival is run by the Rep\u00fablica del Carnaval, which reigns over the town during that time, and the culmination of the event is the ceremonial burning of an effigy of the devil. The festival gets going on the first Friday of January with the most colorful activities taking place on Sunday.\n\n**La Esperanza** (cell tel. 312\/837-0782, COP$180,000 pp all meals incl.) is a private nature reserve run by an American, Doug Knapp, set on a mountain ridge 15 minutes from town. Sunrises, with a view to Jard\u00edn, and sunsets, looking out towards the mountains of Los Farallones del Citar\u00f3, can't be beat. A Jack of many trades, birder Knapp built three comfortable cabins complete with siesta-friendly decks and natural light pouring through the windows. He's also carved out some forest paths that meander through the property. Oh, and he cooks, too.\n\nAt La Esperanza, you don't have to go far to catch a glimpse of some spectacular birds. Knapp's colleagues have documented the presence of eight endemic birds, including the Parker's antbird, the whiskered wren, the Colombian chacalaca, and the yellow-headed manakin. More than 365 species are estimated to live in the Jard\u00edn area.\n\n##### **Accommodations**\n\nIf you are planning to visit Jard\u00edn on a _puente_ (long weekend) or during holidays, you will need to make a reservation at a hotel well in advance. On regular weekends, there is usually no problem in finding a hotel, although the best options do tend to fill up. During the week, the town is yours, and prices drop substantially (especially if you plan to pay in cash).\n\nAlthough Jard\u00edn has plenty of reasonably priced hotel options, there is just one hostel. M **Casa Selva y Caf\u00e9** (Casa del Lago Vereda La Salada, tel. 4\/845-5430, cell tel. 318\/518-7171, www.hostalselvaycafe.com, COP$25,000 dorm, COP$50,000-90,000 d) is a cozy countryside spot, about a 12-minute walk away from the hustle and bustle of Jard\u00edn city life. Back behind a little pond surrounded with flowers and fruit trees with pastureland and mountains behind, it is pure peace here. Alexandra, the owner, is a yoga teacher and gives classes on-site every Tuesday and Thursday, which are free for hostel guests. For nonguests the classes cost COP$20,000. The two private rooms and two dorms are spacious and clean with high ceilings and hardwood floors.\n\n**Hotel Casa Grande** (Cl. 8 No. 4-33, tel. 4\/845-5487, cell tel. 311\/340-2207, www.hotelcasagrande.co, COP$30,000 pp) features 12 rooms, which have a capacity of 2-5 persons each. Most rooms have 2-4 beds to accommodate families. Breakfast is included in the price, and dinner can be arranged at the hotel as well. The friendly owner, a Jard\u00edn native, can supply tourist information for the area.\n\n**Hotel Valdivia Plaza** (Parque Principal, next to the Museo Clara Rojas, tel. 4\/845-5055, cell tel. 316\/528-1047, COP$58,000 d) has 20 rooms and is clean, but isn't bursting with personality. Splurge for a room with a private balcony overlooking the park.\n\n**Hotel Jard\u00edn** (Cra. 3 No. 9-14, cell tel. 310\/380-6724, COP$40,000 pp) has 11 spacious and modern apartments with a capacity of 4-8 persons each. It's a bargain. This is the most colorful house in a most colorful town, with orange, yellow, red, and blue balconies, doors, and trim. The house was restored in 2012.\n\nA comfortable, if conservative, choice is **Comfenalco Hotel Hacienda Baland\u00fa** (Km. 1 V\u00eda Jard\u00edn Riosucio, tel. 4\/845-5561, COP$158,000 d), a hotel with all the extras: restaurant, sauna, and swimming pool. It's a tranquil 15- to 20-minute walk from town.\n\n##### **Food**\n\nThe soft glow of candlelight at M **Caf\u00e9 Europa** (Cl. 8 No. 4-02, cell tel. 312\/230-2842, 11am-3pm and 6pm-10pm Wed.-Mon.) is hard to resist for weekenders. This corner restaurant run by a German photographer and travel writer serves nice pizzas. Order a bottle of French wine (COP$35,000) and settle in. There's no rush in Jard\u00edn. The menu at **Pastelatte** (Cra. 4 No. 8-45, cell tel. 301\/482-3908, noon-8:30pm Wed.-Mon., COP$14,000) features cr\u00eapes, cheesecakes, coffee, sandwiches, and pastas, and service is speedy and always with a smile. **Zodiaco** (off the main plaza in front of the Hotel El Dorado, tel. 4\/845-5615, 8am-11pm daily, COP$15,000) is a _comida t\u00edpica_ (Colombian fare) restaurant, but it's a couple of notches fancier than most restaurants in town.\n\nAt **Las Margaritas** (Cra. 3 No. 9-68, tel. 4\/845-6651, 7am-9pm daily, COP$15,000), the specialty is _pollo a la Margarita_ (chicken fried with a Parmesan cheese breading). This back-to-Paisa-basics place is good for a hearty breakfast. Vegetarians will appreciate a generous morning serving of _calentado_ (beans and rice). If you want to add some juice (not a part of the typical Paisa breakfast), there is a juice stall two doors down from Las Margaritas, as well as fruit vendors in the park. The _tienda_ (store) next door often has fresh Colombian pastries, such as _almojabanas_ (cheese rolls) and _pandebonos_ (delicious pastries made of yuca flour and cheese).\n\nIt's a weekend ritual in Jard\u00edn: spend the afternoon with family and friends at one of the _trucheras_ (trout farms). One of the largest and best known of these is **La Trucher\u00eda** (Km. 5 V\u00eda Riosucio, tel. 4\/845-5159, noon-6pm daily, COP$18,000). Trout is served infinite ways here: _a la mostaza_ (mustard), with fine herbs, and stuffed with vegetables, to name a few. And what better way to round out the day than with a rousing paintball game!\n\nThe **Caf\u00e9 de los Andes** (Parque Principal, Cra. 5 No. 9-73, tel. 4\/845-6239, 8:30am-9pm Thurs.-Mon., 9:30am-8pm Tues.-Wed.) is the brew of choice from Jard\u00edn, and their caf\u00e9 on the terrace of the Casa del Caf\u00e9 is the finest spot in Jard\u00edn for a caffeine jolt. Go for an espresso; other coffee drinks are disappointingly weak. It's in the Casa del Caf\u00e9; if you go upstairs you might see the bean-to-bag process in action.\n\n**Dulces del Jard\u00edn** (Cl. 13 No. 5-47, tel. 4\/845-6584, www.dulcesdejardin.com, 8am-6pm Mon.-Sat.) is the candy-maker in town. In addition to _arequipe_ (caramel) sweets, they make all-natural jams and fruit spreads (COP$6,000) from pineapple, coconut, and papaya.\n\n**Siglo XXI** (Cra. 6 No. 9-18, no phone, noon-8pm daily) is a hole in the wall where you can have a beer and brush shoulders with locals. The walls of this pub are decorated with faded photos of the town and of local _futbolistas_ (soccer players).\n\n##### **Getting There and Around**\n\nThe bus company **Transportes Suroeste Antioque\u00f1o** (tel. 4\/352-9049, COP$18,000) leaves Medell\u00edn each day bound for Jard\u00edn, leaving from the Terminal de Transportes Sur.\n\nThere is also one bus at 6:30am that leaves for Manizales to the south in the coffee region from Jard\u00edn. This route goes through the town of Riosucio.\n\n#### **JERIC\u00d3**\n\nJeric\u00f3 and Jard\u00edn are two (colorfully painted) peas in the same pod. Both are fiercely traditional Paisa pueblos, and they won't change for anybody. Although it is the closer of the two to Medell\u00edn, Jeric\u00f3 feels more remote, and less visited, and therein lies its charm. Set on a gentle slope of a mountain overlooking a valley dotted with cattle ranches and farms of coffee, tomato, plantain, and, cardamom, Jeric\u00f3 still is very much a Paisa cowboy outpost. Colombians know Jeric\u00f3 for two very different reasons. The first is its unique handicraft, the _carriel,_ a shoulder bag made out of leather and cowhide that is a symbol of Paisa cowboy culture. The second is its homegrown saint, Laura Montoya, who was canonized in 2013.\n\nJeric\u00f3 is a pleasant place to hang one's (cowboy) hat for a night, and its sleepy streets lined with brightly colored wooden balconies and doors are a playground for shutterbugs.\n\n##### **Sights**\n\nThe **Catedral de Nuestra Se\u00f1ora de las Mercedes** (Cl. 7 No. 4-34, Plaza de Bol\u00edvar, tel. 4\/852-3494) is a brick construction that towers over the Parque Reyes. The cathedral is where Laura Montoya was officially declared a saint (Colombia's first) during a ceremony in May 2013. Born into poverty in 1874, Montoya was raised by her grandmother. She began her adulthood as a teacher but later decided to enter religious service. Montoya set out on a lengthy missionary mission into the jungle to witness to indigenous people. She later started a religious order that focused on marginalized peoples that has since spread to many countries. Two miracles are attributed to Montoya. At the entrance to the cathedral, there is a bronze statue of the saint alongside an indigenous child, representing Montoya's devotion to assisting impoverished communities in remote areas.\n\nBelow the cathedral is the **Museo de Arte Religioso** (Cl. 7 No. 4-34, Plaza de Bol\u00edvar, tel. 4\/852-3494, 8:30am-noon and 1:30pm-6pm Mon.-Fri., 8:30am-6pm Sat., 9am-noon and 1:30pm-5:30pm Sun., COP$2,000), in which religious art and ceremonial items from the colonial period onwards are on display. Often the museum hosts temporary art exhibitions.\n\nthe colorful Paisa town of Jeric\u00f3\n\nThe best museum in town, and probably the best outside of Medell\u00edn, is the **Museo de Jeric\u00f3 Antioquia** (Cl. 7 between Cras. 5-6, tel. 4\/852-4045, cell tel. 311\/628-8325, 8am-noon and 1:30pm-6pm Mon.-Fri., 9am-5pm Sat.-Sun., COP$2,000). This museum has several rooms, with some dedicated to archaeology (ceramics and other items from the Ember\u00e1 indigenous group of western Colombia) and the rest to contemporary art from Antioquian artists. The museum also shows films during the week, and concerts featuring a range of musical genres are held on the last weekend of each month.\n\nOn the Morro El Salvador, or Cerro de Cristo Rey (called this because of the white statue of Christ on a pedestal), four blocks from the plaza, is the **Jard\u00edn Bot\u00e1nico Los Balsos,** a small botanical garden. For COP$8,000 round-trip you can take a _telef\u00e9rico_ (gondola) to the **Parque Las Nubes,** a park on a hill overlooking Jeric\u00f3. The park is also known as the Parque Los Venados (Deer Park) for its many four-legged residents. Some short paths lead to a waterfall and to a grotto, and the views of the town and Antioquian countryside are sweeping. Neither the garden nor the park has an entry fee, and they are open to the public daily from sunrise to sunset.\n\n##### **Shopping**\n\nTo find your very own _carriel_ shoulder bag or other leather souvenir from Jeric\u00f3, walk down Carrera 5. On the righthand side of the street are a couple of classic shops like **Guarnieleria Jeric\u00f3** (Cra. 5 No. 5-35, tel. 4\/852-3370, 9am-6pm daily) and **Taller de Guarnieler\u00eda & Talabarter\u00eda** (Cra. 5 No. 5-03, tel. 4\/852-3128, cell tel. 311\/716-9895, 9am-6pm daily). The classic _carriel_ goes for about COP$130,000. Oddly, you probably won't see many people other than tourists actually using these unique handbags. _Carrieles_ were used by _arrieros_ (Paisa cowboys) for their horseback trips around Tierra Paisa. These bags are accordion-like, with several divisions in them for carrying items like money, a lock of hair, a knife, or a candle. Some suspect that the name _carriel_ is derived from the English \"carry all,\" while others say it comes from the French _cartier,_ or handbag.\n\nOn the same street is a sweets store, **Delicias del Cardamomo** (Cra. 5 No. 2-128, tel. 4\/852-5289, 9am-6pm daily) that sells cardamom candies, cardamom cookies, and plain old cardamom seeds. Cardamom is a relatively important crop in Jeric\u00f3.\n\n##### **Accommodations**\n\nAs in all Colombian pueblos, room rates drop during the week.\n\nThe best value in town is the **Casa Grande** (Cl. 7 No. 5-54, tel. 4\/852-3229, cell tel. 311\/329-2144, www.hotelcasagrande.freshcreator.com, COP$40,000 pp). It's a nicely renovated old house with 15 simple rooms. Rooms facing the street are preferable. **Hotel Port\u00f3n Plaza** (Cl. 7 No. 3-25, tel. 4\/852-3009, www.hotelportonplaza.com, COP$35,000 pp) runs a close second, although it's much larger. It is just off of the plaza. Ask for room 209 for a good view, or a second-floor room with a view over the street.\n\n##### **Food**\n\nOn the Parque Principal there are quite a few restaurant and caf\u00e9 options along its east side. On late afternoons, the entire length of one side of the plaza is full of folks enjoying a _tinto_ (coffee) and watching the comings and goings of townspeople milling about the plaza. A meal with a view is the selling point of **El Balc\u00f3n Restaurante** (Parque Principal, Cra. 4 No. 6-26, tel. 4\/852-3191, cell tel. 311\/784-4419, 8am-9pm daily, COP$15,000). From its perch on a balcony, you have front row seats to the action below in the plaza and a nice vista of the mountains in the distance. The Colombian dishes are filling.\n\n**Mandala** (Cl. 7 No. 5-55, tel. 4\/852-3331, 7am-10pm daily, COP$15,000) is as funky as it gets in Jerico. This restaurant, which serves everything from paella to Colombian comfort food, is also a hangout spot where live music is sometimes on offer. **Casa Gourmet** (Cl. 7 No. 3-16, tel. 4\/852-4323, 11am-10pm daily, COP$15,000), across from the Hotel Port\u00f3n Plaza, serves fast food like pizza, burgers, and cr\u00eapes. \"Gourmet\" is a bit of a stretch, but if you're looking for a quick meal that strays from the rigid _comida t\u00edpica_ fare, this will do.\n\n##### **Information and Services**\n\nThere is a small **tourist booth** (tel. 4\/852-852-3101, cell tel. 321\/612-3743, www.jericoturistico.com, 9am-6pm Thurs.-Tues.) across from the Catedral de Nuestra Se\u00f1ora de las Mercedes. You can get information on hotels, restaurants, and things to do. They also sell sweets, like _crema de solteras,_ a typical sweet from Jeric\u00f3, and cardamom candies, as well as handicrafts. The tourist office organizes walking tours of the town in Spanish on weekends. Inquire at the office or contact guide Maribel (cell tel. 313\/672-0199).\n\n##### **Getting There and Around**\n\nThere is regular bus service to Jeric\u00f3 from the Terminal del Sur in Medell\u00edn (COP$20,000), starting at 5am and going until 6pm. Buses arrive and depart from the **Parque Principal.** To get to Jard\u00edn from Jeric\u00f3, you must take a _chiva_ (rural bus) to the town of Andes or to the town of Pe\u00f1alisa and transfer there to Jard\u00edn.\n\n### **The Coffee Region**\n\nBlessed by lush, tropical vegetation, meticulously manicured countryside dotted with beautiful haciendas and towns, spring-like weather, and a backdrop of massive, snowcapped mountains, Colombia's coffee region is almost Eden. Nature here is a thousand shades of green: bright green bamboo groves, emerald colored forests with spots of white _yarumo_ trees, dark green coffee groves, and green-blue mountains in the distance punctuated by brightly colored flowers and polychromatic butterflies and birds.\n\nThough the main cities and many towns lack charm, dozens of well-preserved villages offer colorful balcony-clad buildings. Life in most of these towns remains untouched by tourism. A visit on market day, with bustling streets jammed with Jeeps, burdened people, and goods, is a memorable one.\n\nAnd then there is coffee. It is true that Brazil and Vietnam are the world's top coffee producers, but arabica beans are grown throughout Colombia. Coffee grown in some parts of the country (such as Cauca and Nari\u00f1o) is considered superior to that from this region, but here, more than anywhere else, coffee is an inseparable part of Paisa identity. While the extent of land devoted to coffee farming is diminishing, the numbers are still impressive: The department of Caldas contains over 80,000 hectares (200,000 acres) of coffee farms; Risaralda, 52,000 hectares (128,000 acres); and Quind\u00edo, 30,000 hectares (74,000 acres). Visit a coffee farm to understand the laborious production process or\u2014even better\u2014stay overnight.\n\n##### **History**\n\nIn pre-Columbian times, this region was inhabited by the Quimbaya people. In 1537, Spanish conquistador Sebasti\u00e1n de Belalc\u00e1zar conquered the region as he moved north from Ecuador towards the central Muisca region. Due to the sparse indigenous population and lack of precious metals, the region, which was governed from faraway Popay\u00e1n, was largely uninhabited during most of the colonial period.\n\n**The Birth of the Coffee Economy**\n\nDuring the 19th century, demographic pressures spurred settlers from the northwestern province of Antioquia to migrate south, giving origin to what is known as the _colonizaci\u00f3n antioque\u00f1a._ For this reason, the coffee region is akin to Antioquia, with similar dialect, cuisine, and architecture. As the settlers made their way south, they founded towns and started farms: Salamina was established in 1825, Manizales in 1849, Filandia in 1878, and Armenia in 1889.\n\nThe region prospered enormously throughout the 20th century due to ideal conditions for producing coffee. The Colombian National Coffee Federation, owner of the Juan Valdez brand, provided technical assistance, developed infrastructure, and helped stabilize prices. High international coffee prices during the 1980s and 1990s made the region one of the most prosperous areas in the country.\n\nThe fall of global coffee prices in the past decade has forced the region to reinvent itself. Rather than produce a low-value commodity, many farmers have invested in producing high-quality strains that fetch much higher prices. Growers have also diversified, planting other crops, such as plantains, often interspersed through coffee plantations. Finally, agro- and eco-tourism has provided a much-needed new source of revenue.\n\n##### **Planning Your Time**\n\nIt's hard to go wrong as a tourist in the coffee region. No matter your starting point or home base, an immersion in coffee culture is easy, nearby, and rewarding. If you can, plan to spend about five days in this most pleasant part of Colombia. In that time you can stay at a coffee hacienda, visit a natural park, and tour a picture-perfect pueblo.\n\nHowever, if time is short, a quick visit can be equally as rewarding. With easy transportation links to the major cities of the region and excellent tourism infrastructure to meet all budget needs, Salento has the trifecta of coffee region attractions: It's a cute pueblo, coffee farms are within minutes of the main plaza, and jungle hikes that lead through tropical forest to the Valle de Cocora are easy to organize. The town gets packed with visitors on weekends and during holidays, resulting in a more festive atmosphere, but also traffic jams.\n\nAnother option is to stay a couple of days at a hacienda. You can leisurely explore the farms and countryside, relax, and possibly go for a day trip or two to a nearby attraction. Many haciendas are high-end, like Hacienda Bambusa, Finca Villa Nora, and Hacienda San Jos\u00e9. However, budget travelers can also enjoy the unique atmosphere of hacienda life at Hacienda Guayabal outside of Manizales, Villa Martha near Pereira, and Finca El Ocaso in Salento. Meanwhile, Hacienda Venecia has something for travelers of all budgets.\n\nFor bird-watchers, the lush region offers many parks and gardens to marvel at the hundreds of species in the area. The Reserva del R\u00edo Blanco near Manizales, Jard\u00edn Bot\u00e1nico del Quind\u00edo near Armenia, and the Santuario de Flora y Fauna Ot\u00fan-Quimbaya near Pereira are within minutes of the city, guided walks are regularly offered, and birdlife is abundant. Outside of metropolitan areas, the Parque Municipal Natural Planes de San Rafael, which adjoins the Parque Nacional Natural Tatam\u00e1, is less known, but is a natural paradise.\n\nDay trips to natural parks, including the Parque Nacional Natural Los Nevados, are easily organized. PNN Los Nevados is home to _p\u00e1ramos_ (highland moors), lunar landscapes, and snowcapped volcanoes. It can be accessed from many points, and it can even be visited by car. Multiple day treks offer challenges.\n\nIf there were a Cute Pueblo Region of Colombia, this might be it. While you'll have more flexibility driving your own vehicle, it's easy to check out a pueblo or two from the region's major cities traveling by public transportation. A night or two is enough to experience the village life.\n\n#### **MANIZALES**\n\nThe capital of the Caldas department, Manizales (pop. 393,200) is the region's mountain city. Instead of developing in a lowland valley like Armenia or Pereira, Manizales is set atop meandering mountain ridges. This location means that getting around town involves huffing and puffing up and down hills on foot, enduring rollercoaster-like bus or taxi rides along curvy roads, or taking the scenic route on the city's expanding Cable Aereo cable car network.\n\nAbove the coffee farms below at a higher altitude of 2,160 meters (7,085 feet), in Manizales good views abound. And any self-respecting sight around town has got to have a _mirador_ (scenic lookout). On a clear day you can see the peaks of the Parque Nacional Natural Los Nevados in the distance, and this city serves as an excellent base to discover that rugged park of snowcapped volcanoes.\n\nVisitors will keep busy here with many easily organized day-trip possibilities to coffee farms and natural parks. While in other cities dusk is a down time, when visitors return to their hotels or go online, in Manizales this is a good time to head out: to soak in a hot spring, to stroll the promenade along the Avenida 12 de Octubre and await a sunset, or to hang out in the Juan Valdez Caf\u00e9 by the Torre del Cable.\n\n###### **ORIENTATION**\n\nThe two main drags in Manizales\u2014Avenida Santander (Carrera 23) and the Paralela (Carrera 25)\u2014will take you to where you want to go in town. They connect the Zona Rosa\/El Cable area with downtown and with the Avenida 12 de October, which leads to Chipre and the Monumento a los Colonizadores.\n\n##### **Sights**\n\nAs in all the major cities in the region, the real sights are in the tropical jungles. Manizale\u00f1os can be envied for having tropical forests at their backdoor. Here you can visit a park, feel like you're far away in the jungles of Colombia, but still be within the city limits. Birds, such as the colorful _barranquero_ (blue-crowned motmot; the unofficial bird of Manizales) and many varieties of trees and flowers abound. Just outside of Manizales are coffee haciendas, parks, and gardens, all easily visited from the city. In town, one day should suffice to see the urban attractions.\n\n###### **CENTRO**\n\nDowntown Manizales is bustling with activity during weekdays but vacates in a hurry in the evenings. There aren't many sights of interest, save for some republican period architecture and some noteworthy churches. Go on a weekend day if you can.\n\nThe **Plaza de Bol\u00edvar** (Cras. 21-22) holds an odd sculpture to honor Sim\u00f3n Bol\u00edvar created by Antioque\u00f1o Rodrigo Arenas Betancourt. It's known as the _Condor-Bol\u00edvar,_ portraying the Liberator with a body of a condor, the national bird.\n\nFacing the plaza, the neo-gothic **Catedral Bas\u00edlica de Manizales** (Cra. 22 No. 22-15, tel. 6\/883-1880, open until 6:30pm daily) is imposing. Construction began in the late 1920s and was completed in 1936. It replaced the previous cathedral on the same spot, which had been damaged by earthquakes and had to be demolished. For 360-degree views of Manizales and beyond, climb around 500 steps up the spiral Corredor Polaco (the Polish corridor) in Colombia's tallest church tower. To climb the tower, you'll need a guide (COP$7,000). The tower is open 9am-noon and 2pm-5pm Thursday-Sunday.\n\nManizales is Colombia's coffee capital.\n\nNot as grandiose as the cathedral a couple of blocks away, the interior of the **Inmaculada Concepci\u00f3n** (Cl. 30 at Cra. 22, in the Parque Caldas, tel. 6\/883-5474, 7am-noon and 2pm-6:30pm daily) is much more beautiful. The neo-gothic-style church, completed in 1909, was built with _bahareque_ and _guadua,_ natural materials used in construction across Colombia. The wooden rib vaulted ceiling is made of cedar, as are the columns and pews.\n\n###### **CHIPRE**\n\nThis neighborhood to the west of downtown is known for its views and sunsets. Manizale\u00f1os like to boast how Chilean poet Pablo Neruda, when strolling on the promenade in Chipre along the Avenida 12 de Octubre, marveled at this \"sunset factory.\" Atop the futuristic lookout tower, **La Torre al Cielo** (10am-7pm Tues.-Sun., COP$2,000) you're guaranteed a nice vista.\n\nAt the far end of the walkway is the **Monumento a los Colonizadores** (Av. 12 de Octubre and Cra. 9, tel. 6\/872-0420, ext. 22, 10am-6pm daily, free), designed by Luis Guillermo Vallejo. This monument honors the courage and sacrifice of Antioquian colonizers who settled the city, and it depicts an Antioquian family on horseback and on foot forging ahead, with cattle in tow, to this part of Colombia during the _colonizaci\u00f3n antioque\u00f1a,_ when hundreds of families migrated from Medell\u00edn to settle farms in the coffee region area. Manizales residents played a role in building the monument by donating keys and the like to be melted and used in its construction. The sculpture stands atop 20 tons of _piedra de mani_ (peanut stone), the namesake for the city. It was inaugurated in 2001. To get to this part of the city, look for a bus with a \"Chipre\" sign along Avenida Santander (Cra. 23).\n\nEl Cable is a remnant of a gondola system that transported coffee over the Central Cordillera Mountains.\n\n###### **EL CABLE**\n\nOne of the city's icons is a soaring wooden tower known as **El Cable** (52 meters\/171 feet tall) that once supported the unusual gondola system that transported coffee (10 tons per hour!), other materials, and sometimes people from Manizales over the Central Cordillera. The system reached elevations of 3,700 meters (12,100 feet) and descended to the town of Mariquita (495 meters\/1,625 feet). From there the coffee would be transported overland to Honda on the banks of the R\u00edo Magdalena. The rest of the journey in Colombia would take the coffee north to the port city of Barranquilla, where it would be transferred to big boats bound for North America and Europe. This system was developed in the 1920s and would last until the early 1960s.\n\nOf the 376 towers that supported the line, this particular tower, which was in the town of Herveo, was the only one built out of wood (all the others were made of iron). They were all supposed to be made of iron, but the boat carrying one of them from Europe to Colombia was sunk by a German submarine in the Atlantic Ocean during World War I. An English engineer in Colombia, who also designed the neat **Estaci\u00f3n del Cable,** the adjacent station for the cable transport system, designed this tower using wood found locally. The Estaci\u00f3n del Cable, a historic building, appropriately today houses the architecture department for the Universidad Nacional.\n\n###### **RESERVA DEL R\u00cdO BLANCO**\n\nAn array of birds, some found only in Colombia, can be spotted during an excursion to the **Reserva del R\u00edo Blanco** (Vereda Las Palomas, tel. 6\/870-3810, cell tel. 310\/422-1883, www.fundegar.com, 6am-6pm daily), only three kilometers (two miles) outside of Manizales. Most hostels can arrange this day trip, charging COP$20,000 per person for a guided group hike to strategic places in the jungle apt for spotting birds. Toucans and antpittas can often be seen, but over 362 species have been identified throughout the reserve. From August until March the area receives many migratory birds from North America. Private and longer bird-watching tours are also possible.\n\nAn _oso andino_ (Andean bear) that lived in captivity for most of his life was adopted and now lives in an enclosed field in the reserve near the cabins. It was decided that he would not be able to be released back into the wild.\n\nThere are comfortable lodging options available in the park in two **cabins** (tel. 6\/870-3810, cell tel. 310\/422-1883, US$55 pp all meals) that have a total of eight rooms. These are specifically for bird-watchers. Birding guides are offered for the additional cost of US$60 for a Spanish-speaking guide and US$90 for a guide who speaks English. The porch of the lodge is an excellent place to commune with hummingbirds, who are regular customers at the hummingbird feeders.\n\nTaxi transportation to the reserve costs COP$25,000 from Manizales.\n\n###### **RECINTO DEL PENSAMIENTO**\n\nFor a walk in the park, the **Recinto del Pensamiento** (Km. 11 V\u00eda al Magdalena, tel. 6\/874-4157, www.recintodelpensamiento.com, COP$3,000) is a tranquil green space with guided nature walks, a chair lift (COP$17,000), a two-hectare orchid forest with 12,000 orchids, and a butterfly farm. The centerpiece of the park is the **Pabell\u00f3n de Madera,** a large open-air event space made of _guadua_ (a type of bamboo) built by a renowned Colombian architect, Sim\u00f3n V\u00e9lez.\n\n###### **ECOPARQUE LOS YARUMOS**\n\nOn undisturbed mountainsides throughout Colombia, you have undoubtedly noticed the silvery-white leaves of the _yarumo blanco_ tree. Should you get a closer look, you'll see that the leaves of this tree are actually green; it's a fuzzy layer on them that makes them appear white. The **Ecoparque Los Yarumos** (Cl. 61B No. 15A-01, Barrio Toscana, tel. 6\/872-0420 ext. 22, 9am-6pm Tues.-Sun., free) is named for those deceptive trees. A few _yarumos_ can be seen in the more than 50 hectares (125 acres) of cloud forest jungle that makes up this park. The park has nature paths, a lookout tower, and activities such as jungle zip lines. To get to the park on public transportation, you can take a bus from the Manizales city center bound for Minitas. It's about a 7- to 10-minute walk from the bus stop to the park entrance. Taxis are also cheap, costing under COP$4,000 from the El Cable area.\n\n###### **HOT SPRINGS**\n\nNear Manizales are two popular _termales_ (hot springs). Bring your own towel and sandals for both, and visit on a weekday if you want to avoid the crowds.\n\nNear a river, **Tierra Viva** (Km. 2 V\u00eda Enea-Gallinazo, tel. 6\/874-3089, www.termalestierraviva.com, 9am-midnight Mon.-Thurs., 9am-1am Fri.-Sun., COP$12,000-14,000) is closest to Manizales and less expensive. It consists of one pool, some bare-bones changing rooms, and a snack bar.\n\nConsidered the better and hotter of the springs, **Termales Oto\u00f1o** (Km. 5 V\u00eda Antigua El Nevado del Ruiz, tel. 6\/874-0280, <http:\/\/termaleselotono.com>, 7am-midnight daily, day pass COP$25,000, COP$107,000-356,000 d) is a hot spring hotel complex a 25-minute ride to the southeast of the city. It's larger, with four pools (though two of these are reserved exclusively for hotel guests).\n\n##### **Festivals and Events**\n\nThe **Plaza de Toros** (Cra. 27 10A-07, tel. 6\/883-8124, www.cormanizales.com), or bullfighting ring, takes center stage every year at Manizales's biggest bacchanal, the **Feria de Manizales** (www.feriademanizales.gov.co), a celebration of the city's founding. During the festivities there are also concerts, a ballad festival (Festival de Trova), and a Miss Coffee beauty pageant. This citywide party is held in early January.\n\nThe theater festival in Bogot\u00e1, held every two years, is the most important theatrical event in Colombia. In second place is the annual **Festival Internacional de Teatro de Manizales** (www.festivaldemanizales.com). It's always held during the first week of September and attracts theater troupes primarily from the Americas. In addition to performances in theaters throughout the city, free performances are given in parks and plazas, and an educational program for aspiring young actors takes place in schools.\n\n##### **Shopping**\n\nTwo popular shopping malls with inexpensive food options, movie theaters, and numerous clothing and other shops are **Fundadores** (Cl. 33B No. 20-03, tel. 6\/889-4318, www.centrocomercialfundadores.com, 9am-9pm daily) and **Cable Plaza** (Cra. 23 No. 65-11, tel. 6\/875-6595, 9am-9pm daily).\n\nFor handicrafts, peruse the locally made wares such as woodcarvings and woven items at the **Artesan\u00edas de Caldas** (Plaza de Bol\u00edvar, Gobernaci\u00f3n de Caldas, Cra. 21 and Clls. 22-23, tel. 6\/873-5001, www.artesaniasdecaldas.com, 8am-noon and 2pm-6pm Mon.-Sat.) downtown.\n\n##### **Sports and Recreation**\n\nThe **Ciclov\u00eda** in Manizales takes place every Sunday from 8am to noon along the Avenida Santander (Carrera 23) and other main streets. The last Thursday of each month a **Ciclov\u00eda Nocturna** (7pm-10pm) is held. It's the nighttime version of the Ciclov\u00eda.\n\n**Once Caldas** (www.oncecaldas.com.co) is the Manizales soccer club, and their stadium, the **Estadio Palogrande** (Cra. 25 No. 65-00) is in the Zona Rosa, within walking distance from many hostels and hotels. Once Caldas won the Copa Libertadores de America in 2004, defeating Boca Juniors from Buenos Aires, which was a big deal around these parts. Tickets can be purchased in the Cable Plaza Mall (Cra. 23 No. 65-11, tel. 6\/875-6595, 9am-9pm daily).\n\n##### **Accommodations**\n\nThe El Cable area in Manizales is the best place to stay, due to a large number of lodging options and its proximity to restaurants and shopping centers. It's a quiet neighborhood, bustling with international visitors, particularly on Calle 66. There is little traffic on streets here, which is nice; however, that often means that vehicles of all sorts zoom by at high speeds.\n\nOne of the long-standing budget accommodations in Manizales is **Mountain Hostels** (Cl. 66 No. 23B-91, tel. 6\/887-4736, cell tel. 300\/521-6120, www.mountainhostels.com.co, COP$22,000 dorm, COP$60,000 d). Spread over two houses, it has a variety of room types and a small restaurant where you can order a healthy breakfast. Rooms aren't fancy, however.\n\n**Hostal Kumanday** (Cl. 66 No. 23B-40, tel. 6\/887-2682, cell tel. 315\/590-7294, www.kumanday.com, COP$25,000 dorm, COP$40,000 pp d) is a quiet and clean option on the same street as Mountain Hostels. There are 10 rooms and one small dorm room, and all options include breakfast (but no fruit). Staff are a little shy. Kumanday has its own, highly recommended, tour agency that specializes in hiking in and around the Parque Nacional Natural Los Nevados. They also offer a downhill mountain-bike trip (COP$125,000 day trip, 3-person minimum) in the park.\n\n**Casa Lassio Hostal** (Cl. 66 No. 23B-56, tel. 6\/887-6056, cell tel. 310\/443-8917, COP$23,000 dorm, COP$35,000 private room pp) has six rooms, and they also organize bike tours.\n\nThe Colombian chain Estelar (www.hotelesestelar.com) has three hotels in the Manizales area. They are among the top hotels in the city, and all have spacious and clean rooms, as well as at least one room that is accessible for people with disabilities. Weekend rates tend to be significantly lower than during the week. There are few reasons for wanting to stay in downtown Manizales, but if you do, **Hotel Estelar Las Colinas** (Cra. 22 No. 20-20, tel. 6\/884-2009, COP$188,080 d) is the best option (but only on weekends, when traffic, noise, and general urban stress is manageable). The hotel's 60-some rooms are large and clean, but the restaurant and bar area is a little gloomy. A breakfast buffet is available for an additional cost.\n\nThe M **Estelar El Cable** (Cra. 23C No. 64A-60, tel. 6\/887-9690, COP$294,000 d) has 46 rooms over nine floors and is the upscale option in the El Cable\/Zona Rosa area. Breakfast and a light dinner are often included in room rates. Rooms are spacious and clean, with pressed wood floors. A small gym offers modern cardio machines.\n\nIf you prefer birds and trees, check out Estelar's 32-room hotel at the **Recinto del Pensamiento** (Km. 11 V\u00eda al Magdalena, tel. 6\/889-7077, COP$210,000 d). Outside the city, surrounded by nature, this hotel feels a little isolated. It's a popular place for business conferences and events during the week. Rooms are spacious.\n\n##### **Food**\n\nFor excellent down-home, regional cuisine, head to **Don Juaco** (Cl. 65 No. 23A-44, tel. 6\/885-0610, cell tel. 310\/830-2218, noon-10pm daily, COP$15,000), which has been serving contented diners for decades. Try the Paisa hamburger: a hamburger sandwiched in between two arepas (cornmeal cakes). Enjoy it or the popular set lunch meals on the pleasant terrace.\n\nFor delectable grilled meat dishes, **Palogrande** (Cra. 23C No. 64-18, tel. 6\/885-3177, 11am-10pm daily, COP$25,000) is the place you want. Located on a quiet street, it's a rather elegant, open-air place with a nice atmosphere.\n\nFor Italian fare, there are two decent options. Try **Spago** (Cl. 59 No. 24A-10, Local 1, tel. 6\/885-3328, cell tel. 321\/712-3860, noon-3pm and 6pm-10pm Mon.-Sat., noon-3pm Sun., COP$25,000), which is one of the upmarket restaurants in town, with tasty thin-crust pizzas. **Il Forno** (Cra. 23 No. 73-86, tel. 6\/886-8515, noon-10pm Mon.-Sat., noon-3pm Sun., COP$22,000) is a family-style chain restaurant with a great view of the city.\n\nVegetarian restaurants exist in Manizales, but it's difficult to track them down. **Orellana** (Cl. 50 No. 26-40, tel. 6\/885-3907, noon-3pm Mon.-Sat., COP$10,000) serves healthy set lunches and is located in the Versalles neighborhood. It's near the supermarket Confamiliares de la 50. **Rushi** (Cra. 23C No. 62-73, tel. 6\/881-0326, cell tel. 310\/538-8387, 11am-9pm Mon.-Sat., 11am-3pm Sun., COP$12,000) is a vegetarian restaurant close to the Zona Rosa area. It offers set lunches and \u00e1 la carte dishes such as paella and vegetarian fried rice.\n\nIn Colombia, you are never far from a bakery selling sweets. In Manizales there is one bakery where you can have your sweets without the guilt. At **Cero Az\u00facar** (Cra. 23C No. 62-42, tel. 6\/881-0625, 8:30am-8pm daily), all their cakes, cookies, and ice creams are made with the natural sweetener stevia.\n\nA fantastic place for a late afternoon cappuccino and snack is M **Juan Valdez Caf\u00e9** (Cra. 23B No. 64-55, tel. 6\/885-9172, 10am-9pm daily). Yes, it's a chain, and there's one in any self-respecting mall in Colombia. But this one is different: Locals proudly boast that it is the largest Juan Valdez on the planet. But what truly sets this one apart is its great location, under the shadow of the huge wooden tower that once supported the coffee cable car line that ran from Manizales to Mariquita.\n\n##### **Information and Services**\n\nA **Punto de Informaci\u00f3n Tur\u00edstica** (Cra. 22 at Cl. 31, tel. 6\/873-3901, 7am-7pm daily) can be of assistance in organizing excursions to parks and coffee farms throughout Caldas. The staff are eager to help. A small **tourist office** is also located in the main hall of the Terminal de Transportes (Cra. 43 No. 65-100).\n\nIn case of an emergency, Manizales has a single emergency line: 123. To speak directly with the police, call 112.\n\n##### **Getting There and Around**\n\n**Avianca** (Cra. 23 No. 62-16, Local 110, tel. 6\/886-3137, www.avianca.com, 8am-6:30pm Mon.-Fri., 8am-1pm Sat.) and **ADA** (airport tel. 6\/874-6332, 5:45am-5:45pm Mon.-Fri., 8am-5:30pm Sat., 3pm-5:30pm Sun.) serve **Aeropuerto La Nubia** (tel. 6\/874-5451), about 10 kilometers (six miles) southeast of downtown. But the runway is often shrouded in clouds. Because of this, the airport is closed 35 percent of the time and always at night.\n\nThere is regular and speedy **bus service** to both Armenia (COP$17,500, 2 hours) and Pereira (COP$11,000, 70 mins.). **Arauca** (www.empresaarauca.com.co) runs buses to Pereira every 15 minutes and is considered a good company. Buses bound for Cali and Medell\u00edn cost around COP$35,000-40,000 and take five hours each. Buses to Bogot\u00e1 cost COP$47,000 and take about nine hours.\n\nThe **Terminal de Transportes** (Cra. 43 No. 65-100, tel. 6\/878-5641, www.terminaldemanizales.com) is spacious, orderly, and clean. From there it is about a 15-minute taxi ride to the Zona Rosa area. The terminal adjoins the cable car **Cable Aereo** station. The cable car route transports passengers from the terminal (Estaci\u00f3n Cambulos) to the Fundadores station (Cra. 23 between Clls. 31-32) in the Centro.\n\nPublic transportation is not the most organized in Manizales. Private buses are easy to use, but you'll probably have to ask someone for help to determine which bus to flag down. To get downtown from the Zona Rosa, take a bus bound for Chipre.\n\nTaxicabs are also plentiful. **Guillermo Ort\u00edz** (cell tel. 313\/766-8376) is highly recommended. Reliable cab companies include **Taxi La Feria** (tel. 6\/884-8888), **Taxi Ya** (tel. 6\/878-0000), and **Taxi Express** (tel. 6\/880-2000). Many hotels and hostels routinely work with one or two specific taxi drivers, but if not, they will always order a cab for you.\n\nWalking from the Zona Rosa area, along the Avenida Santander, to downtown will take about 45 minutes.\n\n#### **COFFEE FARMS**\n\nThe Caldas countryside is home to coffee haciendas, large and small. Hacienda Venecia and Hacienda Guayabal are two of the most highly recommended for coffee tours, as well as overnight stays. They are located near Chinchin\u00e1, only a 30-minute drive from Manizales.\n\n##### **Hacienda Venecia**\n\nOne of the most well-known, organized, and most visited coffee farms is **Hacienda Venecia** (Vereda El Rosario, V\u00eda a Chinchin\u00e1, cell tel. 320\/636-5719, www.haciendavenecia.com, coffee tour COP$35,000, COP$70,000-250,000 d). This large working coffee plantation has been in the same family for four generations, and their coffee was the first in Colombia to receive UTZ certification for sustainable farming, in 2002. The farm is set far from the highway, providing a peaceful atmosphere. When you're there, you're surrounded by coffee plants growing everywhere you look.\n\nIf you are day-tripping, organizing an excursion to the farm for a coffee tour is easy from Manizales. In fact, you won't have to do much at all except inform the hostel or hotel where you are staying that you'd like to go. The Hacienda Venecia makes a daily pickup at the main hostels in town at around 9am.\n\nTours are given at 9:30am with an additional tour offered at 2:30pm, depending on demand. The 2.5-hour tour begins with a comprehensive presentation of coffee-growing in Colombia and in the world, the many different aromas of coffee, and how to differentiate between a good bean and a bad bean. (And you'll be offered a knock-your-socks-off espresso to boot.) Later, the tour heads outside through the plantation, where you'll see coffee plants at all stages in the growing process. You'll also be able to observe the soaking and drying process. At the end, in the lovely original hacienda house, it's time to roast some beans and drink another freshly roasted cup of Venecia coffee. A typical lunch, such as _ajiaco_ (chicken and potato soup), is offered as well (COP$10,000) at the end of the tour. A farm tour by Jeep and private tours (both COP$50,000) can also be arranged.\n\nThere are lodging options suitable for all budgets at Venecia. In the hostel, often a lively center of activity, accommodations are basic (and bathrooms are shared) but comfortable. For more luxury and seclusion, you may want to stay at the old hacienda house.\n\nOther activities at the farm include horseback riding (for an additional fee) or walks around the plantation on your own. This is particularly pleasant to do early in the morning, when birds (more than 116 species!) are chirping.\n\n##### **Hacienda Guayabal**\n\nNear Venecia, **Hacienda Guayabal** (Km. 3 V\u00eda Chinchin\u00e1-Pereira, cell tel. 314\/772-4895, www.haciendaguayabal.com, tour COP$30,000, COP$95,000 pp all meals) has a dramatic setting, with mountains and valleys covered in coffee crops and _guadua_ (bamboo) completely enveloping the hacienda. This is indeed a coffee farm, and one of the pioneers in coffee farm tourism, but equally interesting is to take a nature walk through the _guaduales_ (forests of Colombia's bamboo) that always spring up along water sources. This hacienda has been in Do\u00f1a Mar\u00eda Teresa's family for around 50 years.\n\nIf you come, you might as well stay, so that you can enjoy the peace and warm hospitality of this special place. While accommodations in the six rooms are not luxurious, they are more than adequate. Meals are delicious, one of the things for which Guayabal is known. Tours around the _finca_ (farm) take about two hours, and you learn about the coffee process as you maneuver along the rows of orderly coffee plants. In addition, you can hike up to a spectacular lookout on a mountainside for breathtaking views of the hills, the valleys, the forests, and the farms all around. Near the guesthouse is a hut made from _guadua_ with recycled floor tiles that has a small coffee bar where you can have a cup of coffee and wait for birds of every color and shape to fly up to nibble on a piece of banana. Tranquility is the watchword here; it's no wonder Guayabal is occasionally host to meditation retreats.\n\n#### M **SALAMINA**\n\nDesignated as one of Colombia's most beautiful pueblos, Salamina features history, beauty, personality, and spectacular countryside; yet, for the most part, it remains off of most tourists' radar. When you visit this historic town, you'll feel as if you have stumbled upon a hidden gem. The historic center of Salamina is marked by colorful and well-preserved two-story houses with their stunning woodwork, doors, and balconies. Salamina is often called the _pueblo madre_ (mother town), as it was one of the first settlements of the Antioquian colonization. It's older than Manizales.\n\n##### **Sights**\n\nThe **Plaza de Bol\u00edvar** (Clls. 4-5 between Cras. 6-7) is the center of activity in Salamina. It's an attractive plaza with a gazebo and large fountain brought over from Germany. Carried by mules over the mountains from the coast, it took a year to arrive, in several pieces, to its final destination. The **Bas\u00edlica Menor La Inmaculada Concepci\u00f3n** (Cl. 4 between Cras. 6-7) has an unusual architecture. The single nave worship hall is rectangular and flat with wooden beams and no columns. The church was designed by an English architect, who is said to have modeled it on the First Temple in ancient Jerusalem.\n\nThe **Casa Rodrigo Jim\u00e9nez Mej\u00eda** (Cl. 4 and Cra. 6) is the most photographed house in Salamina. The colors of this exceptionally preserved house were chosen in an interesting way. An owner of the house called kids from the town to gather in the plaza and to give the owner their proposal on what colors to use for the house's exterior. The winner was a four-year-old girl, who chose bright orange, yellow, and green.\n\nThe **Casa de la Cultura** (Cra. 6 No. 6-06, tel. 6\/859-5016, 8am-noon and 2pm-5pm Mon.-Fri., free) displays photos of old Salamina. It's often a hub of activity. It's also known as the Casa del Diablo, and a jovial devil wood-carving above the door greets visitors as they enter.\n\nFor decades, the **Cementerio San Esteban** (Cra. 3 between Clls. 2-4, no phone), the town cemetery, was divided into three sections: one for the rich, one for the poor, and another for so-called \"N.N.\" bodies (non-identified corpses, or \"no names\"). A wall was built to divide the rich from the poor, but it was knocked down at the behest of a priest in 1976. A skull and crossbones is displayed over the cemetery entrance. There is a small neo-gothic style chapel (open occasionally) on the grounds.\n\nIn the village of **San F\u00e9lix,** 30 kilometers (19 miles) east of Salamina, you can hike through serene countryside and admire a forest of _palmas de cera_ (wax palms) from the hills above. Afterwards, on the village plaza, ask at the stores for a refreshing _helado de salpic\u00f3n_ (ice cream made from chunks of fresh fruit in frozen watermelon juice). A bus makes the round-trip (COP$10,000 each way) to San Felix twice a day, once in the early morning and again in the afternoon. It leaves from the plaza.\n\n##### **Festivals and Events**\n\nSalamina's **Semana Santa** (Holy Week) celebrations, which fall in either March or April, are not that well known, but it is, nonetheless, a great time to get to know this cute town. Orchids and other flowers adorn the balconies of houses, adding even more color. In addition, free classical, religious, and jazz music concerts are held in churches, plazas, and even the cemetery.\n\na friendly devil in Salamina\n\nSan Felix is known for its **Exposici\u00f3n de Ganado Normando** in July, when local farmers show off their best Norman cows, with various competitions. It's an important event for ranchers throughout the region, and also a chance to see an authentic display of Paisa culture.\n\nHalloween is a big deal in Salamina. Here it's called the **Tarde de Mar\u00eda La Parda,** named after a local woman who is said to have sold her soul to the devil in order to obtain riches. Her ghost supposedly causes mischief in the countryside every now and then. Events for Tarde de Mar\u00eda La Parda take place in Plaza de Bol\u00edvar, and there are costume parties at night on that day.\n\nDecember 7 is a special day\u2014or rather, night\u2014to be in Salamina. That's when the lights are turned off in Salamina, and the streets and balconies are illuminated with handmade lanterns made by locals. This beautiful celebration is called the **Noche de las Luces,** a night to stroll the streets and enjoy the special atmosphere. Locals greet each other serving sweets, snacks, or drinks. Music fills the air and the evening culminates in a fireworks show.\n\n##### **Accommodations**\n\nThe best place to stay in Salamina by a long shot is the M **Casa de Lola Garcia** (Cl. 6 No. 7-54, tel. 6\/859-5919, www.lacasadelolagarcia.com, COP$120,000 d), which opened its doors in 2012. The dream of a native Salamine\u00f1o, musician Mauricio Cardona Garc\u00eda, the carefully restored house was once the home of his grandmother, Lola Garc\u00eda. Rooms are spacious and comfortable. If you provide Mauricio with some notice, meals at the hotel can be arranged.\n\nTwo other hotels in town, while not fancy, will do the trick if you're sticking to a budget. **Hospedaje Casa Real** (Cra. 6A No. 5-33, tel. 6\/859-6355, cell tel. 311\/784-2364, www.hospedajecasareal.wix.com, COP$50,000-80,000 d) has 24 rooms and is around the corner from the Plaza de Bol\u00edvar. The owners also have a _finca_ with lodging facilities in the countryside. **Hotel Colonial** (Cl. 5 No. 6-74, tel. 6\/859-5078, cell tel. 314\/627-9124, hotelcolonial2011@hotmail.com, COP$35,000-50,000 d) is right on the square and has a variety of room options. In both of these hotels, ask to see the rooms before you check in, as their characteristics vary.\n\na campesino near Salamina\n\nBased in Manizales, the travel agency **Rosa de los Vientos** (Centro Comercial Parque Caldas Nivel 2, Local PB45, tel. 6\/883-5940, www.turismorosadelosvientos.com) can arrange a home stay (COP$25,000-35,000 pp) in one of the many historic homes in Salamina. The owner, Jackeline Rend\u00f3n, is from Salamina, knows the owners well, and will match you with a good fit.\n\n##### **Food**\n\nPopular and atmospheric **Tierra Paisa** (no phone, 8am-9pm daily, COP$7,000), below the Hotel Colonial on the park, serves typical Colombian food, like _bandeja paisa_ (a quintessential Paisa dish of beans, various meats, yuca, and potatoes), at incredibly low prices.\n\nYou can't leave Salamina without trying their specialties. One is _macana,_ a hot drink made of milk, ground up cookies, cinnamon, and sugar. The other is _huevos al vapor,_ a boiled egg that is methodically steamed using giant coffee urns and served in a coffee cup. For the quintessential Salamina breakfast, go for both.\n\n##### **Getting There and Around**\n\nThere is frequent **shared taxi service** to Salamina from Manizales, costing (COP$11,000). These depart from the Terminal de Transportes (Cra. 43 No. 65-100) in Manizales.\n\nOne **bus** leaves Medell\u00edn at 7am daily bound for Salamina and other communities in the area. It departs from the **Terminal del Sur** (Cra. 65 No. 8B-91, tel. 4\/444-8020 or 4\/361-1186). The trip takes 4-6 hours on rural roads.\n\n#### M **PARQUE NACIONAL NATURAL LOS NEVADOS**\n\nThis national park covers 583 square kilometers (225 miles) of rugged terrain along the Central Cordillera between the cities of Manizales to the north, Ibagu\u00e9 to the southeast, and Pereira to the northwest. Whether you do a day trip or a multi-day trek, a visit to **Parque Nacional Natural Los Nevados** (www.parquesnacionales.gov.co) allows you to enjoy first-hand the stark beauty of the upper reaches of the Andes, far above the forest line, with its intriguing vegetation and fauna. Within the park are three snowcapped volcanoes, **Nevado del Ruiz** (5,325 meters\/17,470 feet), **Nevado del Tolima** (5,215 meters\/17,110 feet), and **Nevado Santa Isabel** (4,950 meters\/16,240 feet), as well as myriad lakes, such as the **Laguna del Ot\u00fan.**\n\nThis rugged landscape was formed by volcanic activity and later sculpted by huge masses of glaciers. At their maximum extension, these glaciers covered an area of 860 square kilometers (332 square miles). They began to recede 14,000 years ago and, according to a 2013 study by the Colombian Institute of Hydrology, Meteorology, and Environmental Studies (IDEAM), will completely disappear by 2030.\n\nMost of the park consists of _p\u00e1ramo,_ a unique tropical high altitude ecosystem, and super _p\u00e1ramo,_ rocky terrain above the _p\u00e1ramo_ and below the snow line. _P\u00e1ramo_ is a highland tropical ecosystem that thrives where UV radiation is higher, oxygen is scarcer, and where temperatures vary considerably from daytime to nighttime, when the mercury falls below freezing. It is the kingdom of the eerily beautiful _frailejones,_ plants with statuesque tall trunks and thick yellow-greenish leaves. Other _p\u00e1ramo_ vegetation includes shrubs, grasses, and cushion plants ( _cojines_ ). The super _p\u00e1ramo_ has a stark, moonlike landscape, with occasional dunes of volcanic ash. Though it's largely denuded of vegetation, bright yellow plants called _litamo real_ and orange moss provide splashes of color. On a clear day, the views from the _p\u00e1ramo_ or super _p\u00e1ramo_ of the snowcapped volcanoes and lakes are simply stunning.\n\nThe black and white Andean condor, _vultur gryphus,_ with its wingspan of up to three meters (10 feet), can sometimes be spotted gliding along the high cliffs in the park. While it is estimated that there are over 10,000 of the birds on the continent (mostly in Argentina), there are few remaining in Colombia. Some estimates report that by the mid-1980s, there were no more than 15 left in Colombia, due in large part to poaching by cattle ranchers. In an effort to boost their numbers in Colombia, a reintroduction program was initiated in the park (and in other parts of the country) in the 1990s in conjunction with the San Diego Zoo, where newborns were hatched. Today it is estimated that there are 200-300 condors soaring above Colombia's Andean highlands. Numbers of the endangered birds in Los Nevados range 8-15. Other fauna includes spectacled bears _(oso de anteojos),_ tapirs, weasels, squirrels, bats, and many species of birds.\n\nThe Nevado del Tolima and Nevado del Ruiz volcanoes are considered active, with the Ruiz presenting more activity. In 1986 it erupted, melting the glacier, which in turn created a massive mudslide that engulfed the town of Armero, burying an estimated 20,000 of the town's 29,000 residents.\n\n###### **ORIENTATION**\n\nThe Northern Sector of the park includes the Nevado del Ruiz, with its three craters (Arenales, La Pira\u00f1a, and La Olleta), and extends south to the extinct Cisne volcano and Laguna Verde. This part can be accessed by vehicle.\n\nThe Southern Sector includes everything from the Nevado Santa Isabel south to the Quind\u00edo peak, as well as Nevado del Tolima. This area has fewer visitors as access is only by foot or by horse, from Manizales, Pereira, Salento, or Ibagu\u00e9.\n\n##### **The Northern Sector**\n\nThe **Northern Sector** (turnoff to Las Brisas entry point at Km. 43 V\u00eda Manizales-Honda, tel. 6\/887-1611, www.parquesnacionales.gov.co, 8am-2pm daily high season, COP$43,500 for nonresidents, COP$28,500 for residents, COP$24,500 for students with valid student ID, COP$5,000 per vehicle) is the most visited part of the park. Until recently, day trippers could drive from Manizales right up to El Refugio, a camp at the base of the Ruiz, and climb up to the main Arenales crater (5,325 meters\/17,470 feet) in a strenuous three-hour hike. The Cisne visitors center provided lodging in this sector of the park and allowed easy access to the Nevado Santa Isabel and Laguna Verde.\n\nthe lunar-like landscape of the Parque Nacional Natural Los Nevados\n\nDue to increased activity at Ruiz, the entire Northern Sector was closed from March 2012 through early 2013. In January 2013 a small area, from the Las Brisas entry station to the Valle de las Tumbas (also known as Valle del Silencio) was reopened to visitors in organized tours and private vehicles. El Refugio at the base of the Ruiz, the Chalet Arenales camping site, and the Cisne visitors center are off limits. At the time of writing, there were no plans to reopen these facilities.\n\nIf you don't have a vehicle, the only way to visit this part of the park is on an organized day tour from Manizales. These tours leave at 6am, drive to Las Brisas park entry station, and continue on to the Valle de las Tumbas, making stops along the way to gaze at the landscape, particularly the Nevado del Ruiz and La Olleta crater (weather permitting), and to view birds and vegetation. On the way back to Manizales, the tour stops for an hour at the rundown Termales de Ruiz outside the park for a quick soak in warm sulfur-laden waters. The tours return to Manizales by 5pm. This experience will be unsatisfying for people who want to move their legs. **Ecosistemas** (Cra. 21 No. 23-21, Manizales, tel. 6\/880-8300, www.ecosistemastravel.com) offers this day tour for COP$100,000 for residents and COP$120,000 for nonresidents.\n\nIf you have a vehicle (a car with 4WD is not necessary) you can drive the Brisas-Valle de las Tumbas segment but you will be required to take a guide (included in the entry price) in your vehicle.\n\nAnother possibility to view Nevado del Ruiz without entering the park is to take an early morning milk truck _(lechero)_ leaving Manizales at 5am to El Sif\u00f3n, northeast of the park. The trip affords beautiful views of the _p\u00e1ramo_ and the Ruiz in the distance and a chance to share a ride with farmers from the region. At El Sif\u00f3n you can catch breakfast and walk back along the road to be picked up by the truck as it returns to Manizales, where you will arrive around noon. All the major hostels in Manizales know about how to organize this excursion.\n\n##### **The Southern Sector**\n\nThe **Southern Sector** offers numerous trekking opportunities, several of which can be done without a guide. There are no official entry stations and permits are not required, but be prepared to pay an entry fee if you bump into a ranger. If you want to be meticulous, you can pay in advance by contacting the Parques Nacionales office in Bogot\u00e1 (tel. 1\/353-2400, www.parquesnacionales.gov.co), sending the names of the visitors to the park, depositing the entry fee at a bank, and receiving a permit by email. However, this is a huge hassle and very few people do it.\n\n##### **Nevado Santa Isabel Trek**\n\nA spectacular day trek from Manizales is up to the snow line of the **Nevado Santa Isabel.** It is a long day trip, starting with a bumpy 50-kilometer (31-mile) drive to the border of the park at Conejeras and then a three-hour (5.5-kilometer\/3.4-mile) hike up the canyon of the R\u00edo Campo Alegre and then to the snow line. This hike requires good physical condition as it takes you from an elevation of 4,000 meters (13,100 feet) up to 4,750 meters (15,600 feet) through _p\u00e1ramo_ and super _p\u00e1ramo_. More serious mountaineers can extend the trek to the summit of the Nevado Santa Isabel (4,950 meters\/16,240 feet) by camping past Conejeras and doing an early morning ascent to the top. At sunrise, the views onto the surrounding high mountain landscape, with the Nevado del Ruiz and Nevado del Tolima in the background, are magnificent. The ascent to the top requires specialized gear.\n\nThere is no public transportation to Conejeras and the trails are not clearly marked, so an organized tour from Manizales is the way to go. A recommended tour operator is **Kumanday** (Cl. 66 No. 23B-40, Manizales, tel. 6\/887-2682, cell tel. 315\/590-7294, kumandaycolombia@gmail.com, www.kumanday.com). The folks at **Mountain Hostels** (Cl. 66 No. 23B-137, Manizales, tel. 6\/887-4736, www.mountainhostels.com.co) can also help you organize this trek.\n\n##### **Laguna del Ot\u00fan Trek**\n\nA popular three-day trek from Pereira is to the **Laguna del Ot\u00fan.** The starting point is El Cedral, a _vereda_ (village) 21 kilometers (13 miles) east of Pereira at an altitude of 2,100 meters (6,900 feet). The end point of the trek at the lake is at 3,950 meters (13,000 feet). This 19-kilometer (12-mile) hike provides an incredible close-up view of the transitions from humid tropical forest to higher altitude tropical forests and the _p\u00e1ramo._ The trek follows the valley of the crystalline R\u00edo Ot\u00fan, first through the Parque Regional Natural Ucumar\u00ed and then into the Parque Nacional Natural Los Nevados. It's not too strenuous. Most trekkers split the climb into two segments, camping at El Bosque or Jord\u00edn on the way up and spending one night at the Laguna del Ot\u00fan. The return hike can be done in one day. The path is easy to follow, though quite rocky and muddy. A guide is not necessary.\n\nThe only accommodation along this route is at the **Centro de Visitantes La Pastora** (6 km\/4 mi from El Cedral toward Laguna del Ot\u00fan, no phone, cell tel. 312\/200-7711, COP$22,000 pp) in the Parque Regional Natural Ucumar\u00ed. The dormitory-style rooms are clean and comfortable in this cozy lodge. Meals (COP$6,000-9,000) by the fireplace are excellent. It is possible to buy snacks along the way, but there is no food at the _laguna,_ so bring cooking equipment and food along with tents and sleeping bags.\n\nTo get to El Cedral from Pereira, take a _chiva_ (rural bus) offered by **Transportes Florida** (tel. 6\/331-0488, COP$5,000, 2 hrs.), which departs from Calle 12 and Carrera 9 in Pereira. On weekdays, the bus departs at 7am, 9am, and 3pm. On weekends there is an additional bus at noon. The buses return from El Cedral approximately at 11am, 2pm, and 5pm.\n\nThe Laguna del Ot\u00fan can also be visited on an organized tour in a long day trip from Pereira. This involves leaving Pereira at 5am and driving 88 kilometers (55 miles) to Potos\u00ed (3,930 meters\/12,895 feet) near the park border and then hiking two hours to the lake. This is not a strenuous walk. A recommended tour operator in Pereira for this excursion is **Cattleya Ser** (Cl. 99 No. 14-78, La Florida, cell tel. 314\/642-6691 or 311\/380-8126, www.cattleyaser.com.co). The **Kolibr\u00ed Hostel** (tel. 6\/331-3955, cell tel. 321\/646-9275, www.kolibrihostel.com) in Pereira also offers guided treks to the laguna.\n\n##### **Nevado del Tolima and Paramillo del Quind\u00edo Treks**\n\nThere are two ways to reach the classically cone-shaped Nevado del Tolima (5,215 meters\/17,110 feet). The somewhat easier and more scenic route is from Vereda del Cocora near Salento, which takes four days. A more strenuous route is up from El Silencio, near Ibagu\u00e9, which can be done in two days.\n\nFrom **Vereda del Cocora** (2,200 meters\/7,215 feet), you hike 7-8 hours (13.5 kilometers\/8.4 miles) through the Valle del Cocora, up the R\u00edo Quind\u00edo canyon, through the P\u00e1ramo Romerales to the Finca La Primavera at 3,680 meters (12,075 feet). There you spend the night (COP$10,000 pp) and take a simple meal. On the second day you hike 6-7 hours (12 kilometers\/7.5 miles) to a campsite at 4,400 meters (14,450 feet) near the edge of the super _p\u00e1ramo_. On the third day, you depart the campsite at 2am and climb a further 8 kilometers (5 miles) to reach the rim of the Tolima crater at 7 or 8am, when there are incredible views to the Quind\u00edo, Santa Isabel, Cisne, and Ruiz peaks. That evening you sleep again at the Finca La Primavera and return to Vereda del Cocora on the following day. The path is not clearly marked and it is easy to lose your way (there's a reason why one part is called the Valle de los Perdidos or Valley of the Lost!), so it is best go with a guide. The ascent to the glacier requires specialized gear.\n\nFrom Ibagu\u00e9 the starting point for the trek to Nevado del Tolima is **El Silencio,** a small _vereda_ (settlement) 28 kilometers (17 miles) north of the city at the end of the beautiful R\u00edo Combeima river canyon. From El Silencio, you'll walk 2.5 kilometers (1.5 miles) along a mountain path to **El Rancho Tolima Termales** (tel. 8\/266-2152, cell tel. 310\/817-2526, www.ranchotolimatermales.com, 24 hours daily, COP$5,000) hot springs. From there, there are several routes up to the top of Tolima. The most direct route is via La Cueva. It is a strenuous six- to eight-hour (15-kilometer\/9-mile) hike up dense tropical forest, _p\u00e1ramo,_ and super _p\u00e1ramo_ to **Latas,** an unmarked camp spot near some large rusting metal sheets. Water is available nearby.\n\nFrom Latas, the ascent up the glacier to the rim of the volcano takes 3-4 hours and requires crampons and an ice axe. The return trip takes 5-6 hours, not including a soothing dip at the _termales._ Unless you are an experienced mountaineer, a guide is necessary.\n\nAnother less traveled but beautiful hike is to the **Paramillo del Quind\u00edo** (4,750 meters\/15,585 feet), an extinct volcano that once was covered by a glacier. The 17-kilometer (10.5-mile) ascent from La Primavera Farm takes eight hours and can be done in one long day. Alternatively, you can split the hike in two, camping so as to arrive at the top of the crater in the early morning when visibility is best. There are spectacular views of the Tolima, Santa Isabel, and Ruiz volcanoes. This is a strenuous but not technically difficult climb.\n\nRecommended guides for the Tolima and Quind\u00edo treks are **P\u00e1ramo Trek** in Salento (cell tel. 311\/745-3761, paramotrek@gmail.com) and **Truman David Alfonso Bejarano** in Ibagu\u00e9 (cell tel. 315\/292-7395, trumandavid01@gmail.com), who can organize excursions from Salento or Ibagu\u00e9. His blog (www.truman-adventure.blogspot.com) has detailed information about the various possible routes up to Nevado del Tolima.\n\n##### **Mountain Biking**\n\nThere are many possible mountain-bike trips through the park, ranging from the easy (all downhill) to the fairly strenuous. **Kumanday** (Cl. 66 No. 23B-40, Manizales, tel. 6\/887-2682, cell tel. 315\/590-7294, kumandaycolombia@gmail.com, www.kumanday.com) offers several excursions.\n\n#### **ARMENIA**\n\nThe defining moment for Colombia's Ciudad Milagro (Miracle City) arrived uninvited on the afternoon of January 25, 1999, when an earthquake registering 6.4 on the Richter scale shook the city. One thousand people lost their lives, nearly half the city became instantly displaced, and thousands of nearby coffee farms were destroyed. The miracle of this coffee region city can be seen in how it rapidly rebuilt and began to thrive once more.\n\nAs is the case with sister cities Pereira and Manizales, Armenia was settled in the late 19th century by Antioquian colonizers. The city is not a tourist destination itself, but you'll be astonished to see, within just a few blocks of the city center, a sea of green coffee farms. That lush countryside is the real attraction.\n\nThe city was founded in 1889 and initially named Villa Holgu\u00edn to honor then-president Carlos Holgu\u00edn Mallarino. It is widely believed that the city was renamed Armenia to honor victims of the 1894-1896 Hamidian massacres of ethnic Armenians living in the Ottoman Empire.\n\n###### **ORIENTATION**\n\nArmenia is a small city by Colombian standards, home to 294,000 residents. Although there is not much to see or do downtown, it is a compact area, and it's easy to get around on foot. The northern areas of the city are where the hotels, malls, and restaurants are to be found. That part of town, around the Hotel Armenia, is also walkable.\n\n_Carreras_ run north-south and _calles_ east-west. Main drags include Carreras 14 (Avenida Bol\u00edvar), 18, and 19, as well as the Avenida Centenario, which runs parallel to the R\u00edo Quind\u00edo on the eastern side of the city. Carrera 14 is pedestrian-only downtown.\n\n##### **Sights**\n\nStanding in downtown Armenia's **Plaza de Bol\u00edvar** (between Cras. 12-13 and Clls. 20-21) is a sculpture of Sim\u00f3n Bol\u00edvar (northern side of the plaza) and the love-it-or-hate-it **Monumento al Esfuerzo,** by Rodrigo Arenas Betancourt, built in the 1960s. This sculpture stands in remembrance of the sacrifices made and hardships faced by Antioquian settlers who arrived in the area seeking opportunity. The modern **Catedral de la Inmaculada Concepci\u00f3n** (Cra. 12 between Clls. 20-21, hours vary), completed in 1972, is a concrete, triangular-shaped building that replaced the previous cathedral, which had stood since 1927. The plaza is on the stark side.\n\nA stroll down the **pedestrian street** from the Plaza de Bol\u00edvar to the Parque Sucre is a pleasant way to see the modern downtown at its busiest.\n\n###### M **MUSEO DEL ORO QUIMBAYA**\n\nEven if you have visited the world-famous Museo del Oro in Bogot\u00e1, it is worth the trek to Armenia just to visit the **Museo del Oro Quimbaya** (Av. Bol\u00edvar No. 40N-80, tel. 6\/749-8433, www.banrepcultural.org, 10am-5pm Tues.-Sun., free) on the outskirts of town. In contrast to the Gold Museum in Bogot\u00e1, this museum, designed by famed architect Rogelio Salmona, focuses exclusively on the Quimbaya nation, which predominated in the coffee region before the Spanish conquest. Much of the museum is devoted to ceramic and gold decorative and ceremonial items that were found in the area. Excellent explanations in English provide interesting background information on the history, ways of life, and traditions of the Quimbaya people.\n\n##### **Festivals and Events**\n\nArmenios celebrate their city's founding in October with their **Fiestas Cuyabras** or Fiestas de Armenia. (People from Armenia are also called Cuyabras, after a bush that produced pumpkin-like fruit that was once widespread in the region.) City parks and plazas are the stages for cultural events, a beauty pageant, and a fun Yipao (Jeep) parade. These U.S. military Jeeps (called Jeep Willys), a symbol of the region, began arriving in Colombia around 1946, after World War II.\n\n##### **Recreation**\n\nSeveral city parks and plazas are great places to enjoy the delicious Armenia climate. These include the **Parque de la Vida** (Cra. 13 at Cl. 8N) and the **Parque Sucre** (Cra. 13 at Cl. 13) downtown, which is adjacent to a delightful pedestrian street. Locals and visitors gather in the late afternoon at the **Caf\u00e9 Quind\u00edo** in the park for _onces_ (tea time).\n\nThe **Parque El Bosque** (Cl. 21 No. 22-23) is a green space that has a bust of Abraham Lincoln that was donated to the city by the Armenian community of Fresno, California, as a way of expressing their gratitude for naming the city in solidarity with the decimated Armenian nation in the early 20th century. The bullfighting ring is in this park.\n\n**Globos Colombia** (cell tel. 320\/667-7818, www.globoscolombia.com, COP$390,000) offers commanding views of the Zona Cafetera from a hot-air balloon. Flights usually depart 6am-6:30am and last 45 minutes. A hearty Paisa breakfast is included in the tour. Flights depart from nearby Armenia as well as near Pereira, about an hour's drive away.\n\n##### **Shopping**\n\nThe **Centro Comercial Unicentro** (Cra. 14 No. 6-02, tel. 6\/731-2667, 8am-9pm daily) along the Avenida Bol\u00edvar has the usual array of Colombian mall stores, fast food joints, an \u00c9xito department store, a movie theater, several ATMs, and food and coffee courts (with spectacular views of the bucolic valley). **El Portal del Quind\u00edo** (Av. Bol\u00edvar 19N No. 46-057, www.elportaldelquindio.com, 8am-9pm daily) is down the road from Unicentro and offers similar shops.\n\n##### **Accommodations**\n\nThe **Casa Quimbaya** (Cl. 16N No. 14-92, tel. 6\/732-3086, cell tel. 312\/590-0066, www.casaquimbaya.com, COP$20,000 dorm, COP$60,000 d) is the budget hostel option in Armenia. It is near the Universidad del Quind\u00edo. There are two dorm rooms and four private rooms. It's in an ordinary-looking house on a quiet street, very close to the action of Carrera 14.\n\nA midrange option downtown is **Casa Hotel del Parque** (Cra. 14 No. 12-26, tel. 6\/731-3166, www.casahoteldelparque.com, COP$99,000 d). It has five rooms and a great location on the Parque Sucre and the pedestrian street.\n\nThe **Armenia Hotel** (Av. Bol\u00edvar and Cl. 8N, tel. 6\/746-0099, cell tel. 320\/696-9111, www.armeniahotel.com.co, COP$220,000 d) has long been considered the most elegant place to stay in town. However, it has lost some of its panache over the years. It's got 129 rooms on nine floors, and a big atrium smells of eucalyptus emanating from the steam room. Rooms and bathrooms are spacious, the beds are fine, and you could go to town raiding the mini-bar. A spa, pool, and small gym are available on the premises, and guests also have privileges at a gym four blocks away.\n\nThe first U.S. hotel chain will soon be arriving in Armenia in the form of **Best Western Plus Mocawa.** It will have 97 rooms over 16 floors, a gym and spa, and a coffee bar in the lobby. Its location on the Avenida Bol\u00edvar is close to malls. It is scheduled to open in late 2013.\n\n##### **Food**\n\nArmenia's gastronomic center is uptown along Avenida Bol\u00edvar.\n\nIn the Centro, **Lucerna** (Cl. 20 No. 14-40, tel. 6\/741-1005, 9am-7:30pm Mon.-Sat., 11am-6:30pm Sun.) is a classic. This retro-looking _sal\u00f3n de t\u00e9_ (tearoom) is always packed; you can order a meal or snack.\n\nIf you've got an appetite, head to **La Fogata** (Av. Bol\u00edvar No. 14N-39, tel. 6\/749-5980, www.lafogata.com.co, noon-midnight daily, COP$28,000), a classic in Armenia. Their filet mignon is known as the best in town. Several Peruvian dishes make the menu interesting, and the loungey Caf\u00e9 La Fogata is a popular place for a cocktail after work or before a night out dancing. Nearby, the **Caf\u00e9 Quind\u00edo Gourmet** (Parque de la Vida, Cra. 14\/Av. Bol\u00edvar 7N, tel. 6\/745-4478, www.cafequindio.com.co, 11am-9:30pm Mon.-Sat., noon-4:30pm Sun.) serves coffee drinks from Armenia's preferred coffee brand, Caf\u00e9 Quind\u00edo, and more substantial meals such as pastas and sandwiches.\n\nPractically hidden behind a residential complex near the Armenia Hotel, **Zaki** (Cra. 13 No. 8N-39, Edificio Bamb\u00fa, Local 104, tel. 6\/745-1220, noon-3pm and 6pm-10pm Mon.-Thurs., noon-3pm and 6pm-11pm Fri.-Sat., COP$20,000) is a sushi joint that also serves a mish-mash of other Asian-inspired dishes. It is one of a handful of restaurants and bars in this part of town that cater to Armenia yuppies.\n\n**Natural Food Plaza** (Cra. 14 No. 4-51, tel. 6\/745-1597, 7:30am-6pm Mon.-Thurs., 7:30am-4pm Fri. and Sun., COP$10,000) always has a set lunch, but you can also order Paisa dishes, like tamales and _bandeja paisa_ \u2014the quintessential Paisa dish of beans, various meats, yuca, and potatoes\u2014reinvented vegetarian-style, all to the soothing sounds of elevator music.\n\nMany of the best-known restaurants in Armenia are on the outskirts of town, like **El Roble** (Km. 12 V\u00eda Armenia Pereira, tel. 6\/740-5120, 6:30am-9pm daily, COP$15,000), which is a sprawling 100 percent Colombian cuisine family-style restaurant.\n\n##### **Information and Services**\n\n**Tourist offices** are located at the bus station (Cl. 35 No. 20-68) and in the **Edificio de la Gobernaci\u00f3n** (Plaza de Bol\u00edvar, tel. 6\/741-7700, 8am-noon and 2pm-6pm Mon.-Sat.).\n\n##### **Getting There and Around**\n\nMajor airlines **Avianca** (Centro Comercial Portal del Quind\u00edo, Av. Bol\u00edvar No. 19N-46, 2nd floor, tel. 6\/734-5205, www.avianca.com, 10:30am-1:30pm and 2:30pm-7:30pm Mon.-Sat., 11am-1:30pm and 2:30pm-7pm Sun.) and **LAN** (Col. toll-free tel. 01\/800-094-9490, www.lan.com) as well as smaller Colombian carriers **EasyFly** (tel. 6\/747-9031, www.easyfly.com.co) and **Aerol\u00edneas de Antioquia** (Col. toll-free tel. 01\/800-051-4232 www.ada-aero.com) serve the **Aeropuerto Internacional El Eden** (Km. 10 V\u00eda La Tebaida, tel. 6\/747-9400). **Spirit Air** (www.spirit.com) has two weekly nonstop flights from Fort Lauderdale.\n\nThe bus terminal, the **Terminal de Transportes** (Cl. 35 No. 20-68, tel. 6\/747-3355, www.terminalarmenia.com) is just south of downtown, about 13 blocks from the Plaza de Bol\u00edvar. There is frequent service to Pereira (1 hour, COP$8,000), Salento (1 hour, COP$4,000), and Manizales (4 hours, COP$17,000). Buses bound for Medell\u00edn (6.5 hours, COP$38,000) leave all day from before dawn to around midnight. While short-distance buses depart until about 10pm or later, it's better to travel earlier if possible out of safety reasons and in order to enjoy the scenery along the way.\n\nThe rapid bus system in Armenia is called the **Tinto** (www.tinto.com.co), after the ubiquitous little coffees. A line on Avenida Bol\u00edvar connects the northern part of the city with downtown. The website can be confusing, so it's best to ask someone how to get around.\n\n#### **VICINITY OF ARMENIA**\n\n##### M **Jard\u00edn Bot\u00e1nico del Quind\u00edo**\n\nJust 10 minutes outside of town, the well-run **Jard\u00edn Bot\u00e1nico del Quind\u00edo** (Km. 3 V\u00eda al Valle, tel. 6\/742-7254, cell tel. 310\/835-0236, www.jardinbotanicoquindio.org, 9am-4pm daily, English tour COP$30,000) is home to hundreds of tree and plant species, many of which are threatened. Knowledgeable volunteer guides, who are usually college students, lead visitors on a mandatory 2.5-hour tour along jungle paths, stopping every so often to point out flora that you would have overlooked had you walked through on your own. That might strike you as a major time commitment, but it really doesn't seem like it. In addition to palms (which aren't technically trees) and _guadua_ (which is actually related to grass), look out for _matapalos,_ a tree that wraps itself around other trees, strangling them as they fight for sunlight. It's been lovingly nicknamed the _abrazo de la suegra_ (mother-in-law's hug).\n\nIn Colombia where there is tropical forest, there will be birds. The gardens are no exception, and they are home to at least 119 species. The birds are at their most active early in the morning. Some of the commonly seen species include tanagers, toucans, owls, woodpeckers, the multi-colored _torito cabecirrojo_ (red-headed barbet), and iconic _barranqueros_ or _barranquillos_ (blue-crowned motmots). These birds make their nests in the earth. Rodent residents who frequently make cameo appearances are _ardillas_ (squirrels) and cute _guatines_ (Central American agoutis). By far the most photographed sector of the park is the _mariposario_ (enclosed butterfly garden) in the shape of a giant butterfly, home to thousands. This is an interactive experience, in which visitors are encouraged to coax the insects to light on their fingers, arms, and shoulders. Butterflies are livelier when the sun is out.\n\nJard\u00edn Bot\u00e1nico del Quind\u00edo\n\nGuides are volunteers, and although the entry price is steep, it's good form to tip the guides after the tour. Call in advance to inquire about English-speaking tours.\n\nIt's easy to get to the park using public transportation from Armenia. Just look for a bus from the Plaza de Bol\u00edvar or along Avenida Bol\u00edvar that says \"Jard\u00edn Bot\u00e1nico.\"\n\n##### **Theme Parks**\n\nThese parks are always mobbed with Colombian families on weekends and holidays.\n\n**RECUCA** (Km. 5 V\u00eda La Y-Barcelona, Vereda Callelarga, tel. 6\/749-8525, www.recuca.com, 9am-3pm daily, tour with lunch COP$30,000) is a theme park, but one without rollercoasters or water rides. RECUCA stands for Recorrido de la Cultura Cafetera (Coffee Culture Experience). Upon arrival at the _finca_ (farm), you'll be greeted by smiling employees dressed in traditional bean-picking garb. Then you'll explore a coffee farm, lend a hand by picking some ripe beans, and learn about the whole process. After that, you'll enjoy a big Paisa lunch (beans and rice for herbivores). If you prefer, you can just take part in a coffee-tasting session (COP$11,000). You can get to RECUCA by taking a bus bound for Barcelona from the Terminal de Transportes in Armenia (Cl. 35 No. 20-68). The bus drops you off at the park entrance. From there it is a 30-minute walk, or the guard at the entrance can order a Jeep for you (COP$5,000).\n\nThe **Parque Nacional del Caf\u00e9** (Km. 6 V\u00eda Montenegro-Pueblo Tapao, tel. 6\/741-7417, www.parquenacionaldelcafe.com, 9am-6pm daily, COP$22,000-55,000) is near the town of **Montenegro,** 12.5 kilometers (8 miles) west of Armenia. While part of the park is devoted to telling the story of coffee production in Colombia, it's mostly an amusement park with rollercoasters, a chair-lift ride, horseback rides, a coffee show, a water park, and other attractions.\n\n**PANACA** (Km. 7 V\u00eda Vereda Kerman, tel. 6\/758-2830, cell tel. 313\/721-9211, www.panaca.com.co, 9am-6pm daily, COP$30,000-60,000) is an agricultural-themed amusement park near the town of **Quimbaya,** where visitors can see and interact with all types of farm animals and watch the occasional pig race.\n\n##### **Festivals and Events**\n\nIn June or sometimes July, **Calarc\u00e1** puts on an event to honor what made the coffee region what it is today. A number of the usual festival events take place during the **Fiesta Nacional del Caf\u00e9** (www.calarca.net), but it's the **Desfile de Yipao** that steals the show. That's when Jeep Willys\u2014U.S. military Jeeps from World War II and the Korean War that were sold to farmers in the coffee region\u2014are laden down with people, animals, and furniture, and go on parade. There are competitions (essentially Willy beauty pageants) and a contest in which the Jeep Willys are loaded down with 1,800 kilos of cargo and race forward on two wheels only. The ubiquitous Jeep Willy has become a symbol of the region.\n\n##### **Accommodations and Food**\n\nWith more than a century's experience growing coffee, the **Hacienda Combia** (Km. 4 V\u00eda al Valle-Vereda La Bella, tel. 6\/746-8472, cell tel. 314\/682-5395, www.combia.com.co, COP$153,000 d) produces Caf\u00e9 Inspiraci\u00f3n, their brand of high-quality coffee. It is operated by the same owners as the Hacienda San Jos\u00e9 near Pereira. It has around 30 rooms and an infinity pool that has a fantastic view, and there are coffee tours available through the nearby fields. This hotel is not far from a highway, but you can easily block out reminders of suburbia by focusing on the fertile lands that surround you and are home to colorful birds. Its proximity to the airport (airport pickups can be arranged) and easy access make it popular for events with Colombian businesses. It also attracts foreign embassy staff living in Bogot\u00e1.\n\n**Bakkho** (Cl. 41 No. 27-56, Calarc\u00e1, tel. 6\/743-3331, www.bakkho.com, noon-9pm Tues., noon-10pm Wed.-Sun., COP$35,000) is considered to be the top restaurant in Quind\u00edo. Here, presentation and ambience is everything, and it's no surprise that this is where locals come to celebrate special occasions. Fare is international with many seafood dishes. Bakkho has various locations in the area, but the Calarc\u00e1 location is the original.\n\nSurrounded by 160 hectares of pineapple, cacao, banana, and citrus crops, it's hard to imagine a more relaxing place than sublime M **Hacienda Bambusa** (off V\u00eda Calarc\u00e1-Caicedonia south of Armenia, tel. 6\/740-4935, cell tel. 321\/313-7315, www.haciendabambusa.com, COP$160,000 pp). Its isolation is a selling point, as you feel far from everything, providing the perfect environment to disconnect. The house and much of the furniture are made of _guadua_ and other traditional materials, and rooms are luxurious and tastefully decorated. There are only seven rooms, each with a private balcony or terrace. The views from those balconies are spectacular, with endless farms punctuated by _guadua_ forests and mountains in the distance. Meals are prepared by the acclaimed Bakkho restaurant. It is isolated here, and it would be a shame to spend the day rushing around the area sightseeing. Here at the farm there are cacao tours to take, horses to ride, birds to watch, and massages to be enjoyed. The Armenia airport is about 40 minutes away, and taking a cab from there costs COP$45,000, although the hotel can arrange all your transportation. Bambusa offers a range of packages, some including activities and excursions, transportation to and from the airport, and all meals. They do not bump up prices during high season (nor reduce prices during low season).\n\nNear the town of Quimbaya and off the road 800 meters past some ordinary looking houses and apartments, a spectacularly well-maintained red-and-white-painted hacienda awaits: M **Finca Villa Nora** (tel. 6\/741-5472, cell tel. 310\/422-6335, www.quindiofincavillanora.com, COP$180,000 d with 2 meals). It's a 120-year-old house that is charming and full of character. It's built in the typical Paisa style. Amid fruit trees, flowers, coffee fields, and a huge ficus tree, at Villa Nora the air is pure, sunsets lovely, and drinks on the verandah not a bad idea. There are only seven rooms at this quiet refuge.\n\n#### **SALENTO**\n\nOn the western edge of the Parque Nacional Natural Los Nevados, the pueblo of Salento (pop. 7,000) is one-stop shopping for those seeking a quintessential coffee region experience. The town, an enchanting pueblo, home to coffee growers and cowboys, is adorned with the trademark colorful balconies and facades of Paisa architecture. It was one of the first settlements in the region during the 19th-century Antioquian colonization. In the nearby countryside, coffee farms dominate the landscape. Here you can be a Juan Valdez, the iconic personification of Colombian coffee, during a coffee tour in which you harvest coffee beans, learn about the bean-to-bag process, and sip the freshest coffee you've ever tasted.\n\nWithin minutes of town is the Valle de Cocora, where you can play tree tag in forests of _palma de cera_ (wax palm, the Colombian national tree), the skyscrapers of the palm family. Some of these can reach up to 60 meters (200 feet) high. For a more challenging hike continue on to the Reserva Acaime, a private nature reserve of tropical forest, babbling brooks, and not a few hummingbirds. From here adventurers can ascend into the _p\u00e1ramos_ (highland moors) and, eventually, the snowcapped mountains of the Parque Natural Nacional Los Nevados.\n\nSalento is easily accessed from between both Armenia and Pereira, and in the town and nearby countryside there are hostels, hotels, and good restaurants. It is a popular tourist destination, so if you'd like to experience Salento without the crowds, go during the week.\n\n##### **Sights**\n\nThe **Plaza de Bol\u00edvar** or **Plaza Principal** is the center of town and center of activity. The festive pedestrian **Calle Real** (between Cl. 1 and Cl. 5) is the most photogenic street in Salento. It is lined with restaurants and souvenir shops painted in a rainbow of colors. It starts at the Plaza Principal and leads up to the **Alto de la Cruz Mirador** (scenic lookout atop the Calle Real). At the cross you can get a great bird's-eye view of the Calle Real and Salento. Farther on is another lookout with views over the surrounding jungles and valleys. But it's really about atmosphere in this Quind\u00edo town.\n\n##### **Coffee Tours**\n\nIn the outskirts of Salento, an excellent place to learn about the coffee process from seed to cup is the **Finca El Ocaso** (V\u00eda Salento-Vereda Palestina, cell tel. 310\/451-7329, cafeelocaso@hotmail.com, www.fincaelocasosalento.com, 8:30am-4:30pm daily, tour COP$8,000). This family-run farm with some 12 hectares (30 acres) of coffee crops produces coffee that has several international certifications, such as the German UTZ and the Rainforest Alliance. Elevation here is around 1,780 meters, a good altitude to grow coffee. Gregarious Don Elias and his wife, Gloria Luz, run the farm, and they enjoy showing their farm to visitors. It's a fairly interactive tour, lasting around 40 minutes, in which you plant a coffee seed, strap a basket to your hip to harvest some ripe, red beans, and grind the coffee pulp. Then, of course, you get to try a freshly roasted cup at the end. The _finca_ (farm) also has three cozy rooms (COP$35,000-100,000), decorated with period furniture, available for rent in the traditional coffee plantation house. You can also rent the whole house (COP$420,000). If you'd like a tour in English, it's best to give the owners some advance notice. It's about an hour-long walk from town, or you can hire a Jeep Willy.\n\nYou can also check out the **Finca Don Eduardo coffee tour** (Plantation House, Alto de Coronel, Cl. 7 No. 1-04, cell tel. 316\/285-2603, COP$20,000). There are two daily, at 9am and 3pm. This organic _finca_ is run by the folks from Plantation House.\n\n##### **Recreation**\n\nFor the real Paisa experience, **horseback riding** is a good way to enjoy the countryside around Salento. In the Plaza Principal there are usually horses at the ready, especially on weekends. One popular excursion is to some nearby waterfalls. **Don \u00c1lvaro** (cell tel. 311\/375-1534, 3-hr. trip COP$40,000 pp) treats his horses well and is considered the best guide for this activity.\n\nSalento, along with the neighboring countryside, is a nice place for a **bike ride.** Most hostels can arrange bike rental. Additionally, **CicloSalento** (near Plantation Hostel, Alto de Coronel, Cl. 7 No. 1-04, cell tel. 318\/872-9714, COP$10,000\/hr., COP$35,000\/day) rents out good quality mountain bikes with helmets. Caution: The winding road leading into town from the Valle de Cocora does not have a shoulder for bikes. Vehicles tend to speed along this road, making this a dangerous stretch for cyclists and pedestrians.\n\n##### **Accommodations**\n\nAs Salento has grown in popularity, with Paisa weekenders and international travelers, excellent accommodations options (from backpacker lodges to coffee plantations and camping options) to fit all budget types have similarly grown. Although the number of higher-end hotels in town is growing, it is often the case that small hostels and nearby coffee farms will suit your needs just fine.\n\nOne of the best hostels in the area is M **Tralala** (Cra. 7 No. 6-45, cell tel. 314\/850-5543, www.hostaltralalasalento.com, COP$18,000 dorm, COP$45,000 d). It's hard to miss this in-town option: It's a two-story white house with bright orange wooden trim. At Tralala there are only seven rooms, including a dormitory that sleeps six, making for a chilled-out environment for the guests. Run by a Dutchman, the hostel is spic and span and tastefully decorated. Its minimalist style provides a nice vacation for the eyes. Staff are friendly and knowledgeable, and the kitchen area is a pleasant area to hang out and chitchat with others. There's a sun deck and garden area in case relaxation is needed.\n\nLondoner Tim was one of the first to help transform Salento from a sleepy Paisa pueblo into one of Colombia's top tourist destinations. His **Plantation House** (Alto de Coronel, Cl. 7 No. 1-04, cell tel. 316\/285-2603, www.theplantationhousesalento.com, COP$22,000 dorm, COP$55,000 d), with 24 rooms total, remains one of the top places to get to know Salento and the surrounding areas. Catering to international visitors, this hostel has two houses, one of which is over 100 years old. It's quiet and green around the hostel, and, though you'll be bound to meet other travelers like yourself, there is plenty of space to find a little solitude. Plantation House can organize bike excursions, horseback riding, and hikes to the Valle de Cocora. The owners of the Plantation House have their own organic coffee farm, **Finca Don Eduardo** (Alto de Coronel, Cl. 7 No. 1-04, cell tel. 316\/285-2603, COP$15,000 dorm, COP$35,000 d), about 15 minutes outside of town, which has one private room and one dormitory. This coffee plantation is over 80 years old and set amid lush, rolling hills. It is an environmentally friendly hostel: Solar panels enable guests to have a hot shower, and a rainwater collection system provides that water.\n\nAnother excellent hostel-type option is **La Serrana Eco-farm and Hostel** (Km. 1.5 V\u00eda Palestina Finca, cell tel. 316\/296-1890, www.laserrana.com.co, COP$22,000 dorm, COP$55,000 d). It's situated on a bluff with lovely views of coffee farms in every direction. The nine rooms, of various types and sizes, are comfortable, and there is also a women-only dorm room. Camping is also available for COP$12,000. It's a peaceful place where you can enjoy sunrises and sunsets, go for a walk into town, or just hang out. La Serrana is best known for its delicious (and nutritious) family-style dinners and other meals. Vegetarians always have options, and the cooks make an effort to buy local, fresh food. La Serrana has another, smaller, lodging option, M **Las Camelias** (Km. 1.5 V\u00eda Palestina Finca, cell tel. 316\/296-1890, www.laserrana.com.co, COP$70,000 d), a colonial-style house you can see from the hostel. This is geared for couples who want a little more privacy\u2014there are only three rooms. Rooms, drenched with natural light, are spacious, with hardwood floors and fireplaces. Common space is ample with large windows, and there is a kitchen for guest use. From La Serrana it is a short distance to the Finca Ocasa coffee farm.\n\nCentrally located M **Hostal Ciudad de Segorbe** (Cl. 5 No. 4-06, tel. 6\/759-3794, www.hostalciudaddesegorbe.com, COP$85,000 d) is a bed and breakfast run by a Colombian and Spanish pair. The renovated house is over 100 years old and is built in the traditional Paisa style. The hostel's eight rooms have high wooden ceilings with gorgeous original geometric designs and small balconies. One room is equipped for guests with disabilities. Pictures of Spanish towns like Segorbe, the hometown of one of the owners, decorate the walls. Excellent service is provided to guests, such as transportation assistance and help with organizing sightseeing activities. There are plans to add more rooms to the hotel in the adjacent lot, which may make it less of an intimate stay, but it's still a good bet.\n\n###### **CAMPING**\n\nFour kilometers outside of Salento, on the banks of the R\u00edo Quind\u00edo, is **Camping Monteroca** (Valle del R\u00edo Quind\u00edo, cell tel. 315\/413-6862, www.campingmonteroca.com, COP$70,000 cabin, COP$15,000 tent), a sprawling campground catering mostly to Colombian weekenders. The camp has 11 cabins, one of which is called the Hippie Hilton, and several of them have awesome waterbeds. There is a lot of space for tents here, as well. Monteroca has a restaurant and two bars. Recreational activities such as horseback riding (COP$12,000 per hour), a three-hour hike to nearby waterfalls (COP$25,000), and yoga classes are on offer as well. To get there from Salento, take a Jeep bound for Las Veredas. They leave every 15 minutes from the Plaza Principal during weekends.\n\n##### **Food**\n\nSalento offers varied restaurant options, not just the standard _comida t\u00edpica_ fare.\n\n**Mojiter\u00eda** (Cl. 4 No. 5-54, cell tel. 310\/409-2331, 2pm-11pm daily, COP$18,000) is a lively spot where you can grab a quick bite (appetizers, salads, soups, and pastas) or try one or two of the many mojitos on offer. At night it takes on a bar atmosphere.\n\nIt's a real treat to discover a restaurant like M **La Eliana** (Cra. 2 No. 6-45, cell tel. 314\/660-5987, 10am-9pm daily, COP$20,000), where great service, a cozy atmosphere, and fantastic food are the norm. This Spanish-run spot a few blocks from the center of town is the only place in this part of the woods where you can find curry dishes and gourmet pizzas on the menu. And try as they might, the friendly cocker spaniels aren't allowed to mingle with diners.\n\nOne of the best regional food restaurants is **Camino Real Parrilla Bar** (Cra. 6A No. 1-35, cell tel. 314\/864-2587, 10am-midnight Sun.-Thurs., 10am-2am Fri.-Sat., COP$18,000). After a grueling climb to the Mirador, this popular place at the top of Calle Real makes for a great stop. The restaurant has outdoor seating and a huge, fairly varied menu with grilled meats, a few salads, and lots of _trucha_ (trout). At night it's an _aguardiente_ **,** a popular local drinking hangout.\n\n**Alegra** (Cra. 6 No. 2-52, cell tel. 301\/462-4458, 2pm-8:20pm Mon., Tues., Thurs., and Fri., 12:30pm-8:20pm Sat.-Sun., COP$18,000) is a cute place about a block from the ruckus of Calle Real. Here you can enjoy cilantro pesto pasta, a veggie burger, and a glass of wine as you listen to jazz in the background. It's run by a friendly woman from Bogot\u00e1.\n\n**Brunch** (Cl. 6 No. 3-25, cell tel. 311\/757-8082, 6:30am-9:30pm daily, COP$15,000), a hip little joint with graffiti and messages from hundreds of visitors from around the globe decorating the walls, is another restaurant with the international traveler in mind. They do serve brunch, but also breakfast, lunch, and dinner. The menu seems aimed squarely at Americans: Buffalo wings, Philly cheese steak sandwiches, black bean burgers, and peanut butter brownies. Menu items are assigned whimsical names like Wax Palm Pancakes. **Beta Town** (Cl. 7 No. 3-45, cell tel. 321\/218-7043, 6pm-midnight daily) is a popular place for burgers, beer, and hanging out. They've even got a _tejo_ field, where you can try your luck at this only-in-Colombia sport.\n\n##### **Information and Services**\n\nHostels usually provide the best tourist information, but there is a city-run tourist kiosk, the **Punto de Informaci\u00f3n Tur\u00edstica** (10am-5pm Wed.-Mon.), in front of the Alcald\u00eda (city offices) in the Plaza Principal.\n\n##### **Getting There and Around**\n\nThere is frequent bus service from Pereira, Armenia, and other cities to Salento. The last bus from Armenia leaves at 8pm (under COP$4,000). From Pereira, there are four direct buses each weekday, costing under COP$6,000. There is more frequent service on weekends. As Salento is well established on the tourist route, thieves are known to prey on foreigners on late-evening buses traveling from Pereira to Salento. Keep a vigilant eye on your possessions.\n\nBuses to Armenia (every 20 mins., COP$4,000) and Pereira (COP$6,000) depart from the the intersection of Carrera 2 at Calle 5, with the last bus departure at 6pm daily. For Filandia you have to first go to Armenia.\n\n#### M **VALLE DE COCORA**\n\nThe main attraction for most visitors to Salento is seeing the _palmas de cera_ (wax palms) that shoot up towards the sky in the Valle de Cocora. These are some of the tallest palms in the world, reaching 50-60 meters high (200 feet), and they can live over 100 years. They have beautiful, smooth, cylindrical trunks with dark rings. In 1985, they were declared to be the national tree of Colombia.\n\nThe Valle de Cocora is a 15-kilometer (9-mile) section of the lower R\u00edo Quind\u00edo valley. Much of it has been turned into pastureland, but, thankfully, the palms have been preserved. The palms look particularly stunning in the denuded pastureland.\n\nthe picturesque Valle de Cocora and its famous wax palms\n\nThe gateway to the valley is the **Vereda de Cocora** , a stretch of restaurants specializing in trout. This _vereda_ (settlement) is a major domestic tourist destination and can get incredibly crowded during holidays and weekends. Most Colombian tourists come for a late lunch and take a stroll along the main road behind the _vereda_ to view the palms. However, there are two much more rewarding excursions: a leisurely 90-minute walk to **La Monta\u00f1a,** a ranger station for the local environmental agency, or a more intense five-hour loop through the valley and up the R\u00edo Quind\u00edo and back via La Monta\u00f1a.\n\nFor the La Monta\u00f1a hike, continue down the main road beyond the Vereda de Cocoa and pass through the gate of a private farm on the right where a signpost reads \"FCA. EL BOSQUE 7.6 KM.\" Follow a path that meanders six kilometers (four miles) up hills converted to pastureland and then along a ridge on one side of the Valle de Cocora to the ranger station. The views from the path onto the wax palms in the valley and mountainside are stunning. If you opt for the longer hike, that same spectacular scenery comes at the end of a walk that saves the valley of the palms for last, like a delicious dessert.\n\nFor the longer hike, take a right through a gate painted blue after the last building in the Vereda de Cocora. After walking about four kilometers (2.5 miles) through pasture, you'll enter the dense forest. The path crisscrosses the trickling R\u00edo Quind\u00edo. After three kilometers (two miles) you reach the **Reserva Acaime** (cell tel. 321\/636-2818 or 320\/788-1981, COP$4,000), a private reserve created to preserve the surrounding cloud forest. With the entrance fee, you can enjoy a complimentary cup of hot chocolate, _agua de panela_ (a hot sugary drink), or coffee and watch throngs of hummingbirds of several varieties fly up to feeders. It's quite a show. You can also stay at Acaime, either in private rooms or a large dormitory (COP$40,000 pp including all meals).\n\nFrom Reserva Acaime, you backtrack a kilometer and then climb a steep path to La Monta\u00f1a. The last leg of the hike, back from La Monta\u00f1a to Vereda del Cocora, provides the best and most memorable photo opportunities of the valley and hundreds upon hundreds of wax palms. Make sure your camera batteries are charged, as you'll want to take many pictures. The entire loop takes 4-5 hours. You may want to wear rubber or waterproof boots, as the path along the R\u00edo Quind\u00edo is muddy. There is no need for a guide.\n\nTo get to Vereda del Cocora from Salento, take a Jeep Willy (COP$3,500), which leave the Plaza Pincipal at 6am, 7:30am, 9:30am, and 11:30am each day.\n\n##### **Trek to Finca La Primavera**\n\nIf you would like to do a longer expedition, you can extend the Valle de Cocora hike beyond Reserva Acaime to the **P\u00e1ramo de Romerales** on the border of the Parque Nacional Natural Los Nevados and to **Finca La Primavera,** a working farm located at an altitude of 3,680 meters (12,075 feet). This excursion allows you to enjoy the transition from cloud forest to _p\u00e1ramo_ (highland moor) but requires two days. The path from Acaime continues to **Estrella de Agua,** a research station, through the P\u00e1ramo de Romerales and finally Finca Primavera, where you can bunk for COP$10,000 per person and arrange for meals. The entire hike from Vereda de Cocora to Primavera is 13.5 kilometers (8.5 miles) and takes 7-8 hours. This trek does not require camping gear; all you need is food for snacking. However, it is quite cold at Finca La Primavera, and a sleeping bag makes a difference. Hiring a guide is a good idea as the trail is poorly marked.\n\nFrom Finca La Primavera, you can continue into the **Parque Nacional Natural Los Nevados,** hiking to the Nevado del Tolima (5,215 meters\/17,110 feet) or the less traveled Paramillo del Quind\u00edo (4,750 meters\/15,585 feet).\n\nRecommended guides are **P\u00e1ramo Trek** (cell tel. 311\/745-3761, paramotrek@gmail.com) in Salento and **Truman David Alfonso Bejarano** (cell tel. 315\/292-7395, trumandavid01@gmail.com) in Ibagu\u00e9, who can organize excursions from Salento or Ibagu\u00e9.\n\n#### **FILANDIA**\n\nTo visit an authentic coffee town without the tourists, head to Filandia (pop. 12,000), a cute pueblo between Armenia and Pereira. Sights are few; this town is about atmosphere. The name Filandia has nothing to do with the Nordic country of Finland (which in Spanish is Finlandia).\n\nThe focal point on the **Parque Central** (between Cras. 4-5 and Clls. 6-7) is the church, the **Templo Mar\u00eda Inmaculada** (Cra. 7), which was built in the early 20th century. From the plaza explore the charming streets of the town, including the **Calle del Tiempo Detenido** (Cl. 7 between Cras. 5-6) and the **Calle del Empedrado,** two streets of two-story houses made of _bahareque_ (a natural material) adorned by colorful doors and windows. Stop by the town's oldest construction, the **Droguer\u00eda Bristol** (Cra. 6 No. 5-63) along the way. A nice view of the countryside can be had near the _cl\u00ednica mental_ (mental hospital; Cra. 8 No. 7-55). On the street pick up one of Filandia's famous reed baskets.\n\nOn the road towards Quimbaya, just outside of town, is the wooden **Ecoparque Mirador de las Colinas Iluminadas** (10am-7pm Mon.-Fri., 10am-9pm Sat.-Sun., COP$3,000), which looks like a wooden spaceship. From the top are nice views of the countryside, and inside, looking down, is a strange mosaic of a giant blue butterfly. On the way towards the _mirador_ , pop into one of the many handicraft shops, where you can browse baskets until the cows come home.\n\n##### **Accommodations and Food**\n\nAccommodations and restaurants in Filandia are limited. In a traditional Paisa house, the **Hostal La Posada del Compadre** (Cra. 6 No. 8-06, tel. 6\/758-3054, cell tel. 313\/335-9771, www.laposadadelcompadre.com, COP$60,000 d) offers a handful of rooms and ample outdoor hangout space. Rooms are large, beds are adequate, breakfast is included, and the prices are reasonable.\n\nWith a prime location on the main square, the **Hostal Tibouchina** (Cl. 6A No. 5-05, tel. 6\/758-2646, COP$40,000 pp d) has seven rooms and pleasant common areas, including a large kitchen. It's on the second floor above a caf\u00e9\/bar. The interior rooms, which lack windows, are on the stuffy side. On the other hand, if you get one of the rooms facing the street, you may hear music and the goings-on in the plaza until the wee hours on weekends.\n\nThe M **Hostal Colina de Lluvia** (Cra. 4 No. 5-15, COP$25,000 dorm, COP$60,000 d private) opened in mid-2013 and is easily the best place to stay in Filandia. Tastefully decorated rooms are spic and span with comfortable beds, and there is a small garden patio.\n\nCandlelit tables, lounge music, and art on the walls\u2014you won't believe your eyes when you see M **Helena Adentro** (Cra. 7 No. 8-01, cell tel. 312\/873-9825, noon-2am weekends). Started by a New Zealander and a Paisa, it's by far the coolest spot in Filandia, Quind\u00edo, and perhaps this side of Medell\u00edn. Cured meats and goat cheeses come from local farmers, along with the coffee. They have their own brand of coffee but also serve coffee from other regions of Colombia, using different brewing techniques. Locals keep coming back for the inventive libations here, such as the house cocktail, the Adentro Helena (aguardiente, _lulo_ juice, and lime).\n\nThe popular place for a cappuccino is **Jahn Caf\u00e9** (Cl. 6 No. 5-45, 7:30am-midnight Mon.-Fri., 7:30am-2am Sat.-Sun.).\n\n##### **Information and Services**\n\nThe Filandia tourist office is in the **Casa del Artesano** (Cra. 5 at Cl. 7, 2nd floor, tel. 6\/758-2172, 7am-noon and 1:30pm-4:30pm Mon.-Fri.).\n\n##### **Getting There and Around**\n\nBuses to Filandia leave from Armenia (COP$4,000, every 20 minutes) and Pereira (COP$5,000, hourly) all day long until around 8pm. These circulate the town picking up passengers, especially on Carrera 7.\n\n#### **PEREIRA AND VICINITY**\n\nIn Pereira (pop. 465,000), the capital of the Risaralda department, you can get your boots muddy, see exotic birds, and experience the tropical Andean forest during the day, then later enjoy a good meal out or hit the town until late. Pereira is perfectly situated for day trips to the countryside, be it elegant haciendas or natural parks such as Santuario de Flora y Fauna Ot\u00fan-Quimbaya. In town, you can easily see the major sights of interest in one day: the spectacular cathedral, the Plaza de Bol\u00edvar with its statue of a nude Sim\u00f3n Bol\u00edvar on horseback charging ahead, and the Museo de Arte de Pereira. Manizales, Armenia, and Salento are within one hour driving or riding of Pereira.\n\n##### **Sights**\n\nDowntown, in the **Plaza de Bol\u00edvar** (Clls. 19-20 and Cras. 7-8) stands a bronze sculpture by Rodrigo Arenas Betancourt depicting the Liberator Sim\u00f3n Bol\u00edvar on horseback charging ahead to fight the Spaniards\u2014naked. Facing the plaza is the **Nuestra Se\u00f1ora de la Pobreza Catedral** (Cl. 20 No. 7-30, tel. 6\/335-6545, masses every hour 6am-noon and 5pm Mon.-Sat., 6am-noon and 5pm-8pm Sun.). The cathedral was originally built in 1890, using industrial-era building techniques, and was damaged by an earthquake, needing to be almost completely reconstructed. It was rebuilt with a wooden ceiling and supports made from cumin laurel, a tree native to Colombia that is now endangered.\n\nThe **Museo de Arte de Pereira** (Av. Las Am\u00e9ricas No. 19-88, tel. 6\/317-2828, www.museodeartedepereira.com, 9am-noon and 2pm-6pm Tues.-Fri., 10am-5pm Sat.-Sun., free) is one of the best art museums in the region, and deserving of a visit. It features temporary exhibitions of contemporary Latin American artists. It's south of downtown.\n\nIn the **Parque Olaya Herrera** (between Cras. 13-14 and Clls. 19-23) is the well-preserved **Antigua Estaci\u00f3n del Tren,** a photogenic old train station. There is a Megabus station in the park, and the park is a nice place for a morning jog.\n\nThe **Zool\u00f3gico Mateca\u00f1a** (Av. 30 de Agosto, tel. 6\/314-2636, www.zoopereira.org, 9am-6pm daily, COP$15,000), located within screeching, howling, and roaring distance of the airport, has an extensive section on Colombian animals. It's better than many zoos in Colombia; however, big cats, including Colombian jaguars, have little space to move about.\n\nThe **Viaducto C\u00e9sar Gav\u00edria Trujillo** is a modern cable bridge that connects Pereira with its industrial neighbor of Dosquebradas. It's a point of reference and source of city pride. It is named for former president C\u00e9sar Gav\u00edria, who is from Pereira and who served as president during the early 1990s.\n\n##### **Recreation**\n\nFor bike tours and rentals, contact **Retro Ciclas** (cell tel. 310\/540-7327 or 312\/437-4882, www.mtbtourscolombia.com). One of the more popular tours is a trip to the village of Estaci\u00f3n Pereira (COP$86,000), where in the town you'll take two different and exciting means of transportation: a _brujita,_ a motorcycle-powered cart that zooms along old train tracks, and later a _garrucha,_ which is a gondola-like metallic basket that transports passengers over the R\u00edo Cauca. Another trip on offer is along the R\u00edo Ot\u00fan (COP$80,000) to the Santuario de Flora y Fauna Ot\u00fan-Quimbaya.\n\n##### **Accommodations**\n\nIn town, it's best to stay in the Circunvalar area. The Centro has options, but prices are comparable to hotels on the Circunvalar, where you can walk without much concern at night and there are plenty of restaurants and nightlife spots nearby.\n\nThe M **Kolibr\u00ed Hostel** (Cl. No. 16-35, tel. 6\/331-3955, cell tel. 321\/646-9275, www.kolibrihostel.com, COP$22,000 dorm, COP$60,000 d) is a welcome newcomer to Pereira, filling a void of budget accommodations for international clientele near the Circunvalar. In addition to a mix of private rooms and dorms, Kolibr\u00ed has two long-stay apartments. It's run by a Dutch-Colombian couple who have traveled extensively in the area to some off-the-map places, and they offer tours, such as to the village of Estaci\u00f3n Pereira, to the Santuario de Flora y Fauna Ot\u00fan-Quimbaya, and an interesting orchid tour. Bars, restaurants, and malls are within walking distance from the hostel. Great breakfasts are offered on the deck, where they also have a barbecue grill.\n\nThe **Hotel Movich** (Cra. 13 No. 15-73, tel. 6\/311-3300, COP$249,000 d) is a good option if you like comfort, don't want any surprises, and want the conveniences that the Circunvalar offers. It's across the street from the Iglesia de Carmen. The pool (usually open until 9pm) and gym (open 24 hours) are quite nice. A massive breakfast buffet is included in the room rate.\n\nA restful sleep is assured at the **Hotel Don Alfonso** (Cra. 13 No. 12-37, tel. 6\/333-0909, www.donalfonsohotel.com, COP$240,000 d), a small boutique-style hotel on the main nightlife and shopping drag of the Avenida Circunvalar. It has 11 comfortable air-conditioned rooms, each with inviting beds covered by quilts.\n\n###### **HACIENDAS**\n\nWithin minutes of the Pereira's bright lights are some gorgeous and luxurious hacienda hotels. Some of them are popular places for special events, such as weddings on the weekend and corporate seminars during the week.\n\n**Hacienda Malabar** (Km. 7 V\u00eda a Cerritos, Entrada 6, tel. 6\/337-9206, www.hotelmalabar.com, COP$257,500 d) is an authentic hacienda with seven rooms, ample gardens to wander, and a pool. The wooden ceilings with their geometric designs and tile floors with Spanish Mudejar designs throughout the house are spectacular.\n\nM **Villa Martha** (Km. 9 V\u00eda a Marsella, 1 km from the main road, tel. 6\/322-9994, cell tel. 310\/421-5920, www.fincavillamartha.com, COP$75,000 pp with meals) offers the most affordable coffee farm experience. Here you can kick back and relax, take a dip in a pool that has \"Villa Martha\" written on the bottom, stroll the countryside, and take a tour of the coffee plantation. It's not luxurious, but the warm hospitality of the _finca_ (farm) owners, Martha and her husband, Rafael, more than compensates. Rooms by the pool are nicer. Villa Martha doesn't allow non-guests to visit for the day.\n\nM **Castilla Casa de Huespedes** (Km. 10 V\u00eda a Cerritos, tel. 6\/337-9045, cell tel. 315\/499-9545, www.haciendacastilla.com, COP$281,000 d), built in the 19th century, is set amid fruit trees and has a pool to boot. The nine rooms are lovely, and staff are friendly. Once there, at this serene spot in the countryside, you'll probably not even be aware of the fact that a fried chicken restaurant is just around the corner along the highway! They make their own jam here, and a majestic cedar tree near the pool area looks even more regal when illuminated at night.\n\nThe **Hacienda San Jos\u00e9** (Km. 4 V\u00eda Pereira-Cerritos, Entrada 16, Cadena El Tigre, tel. 6\/313-2612, www.haciendahotelsanjose.com, COP$275,000-310,000 d) was built in 1888 and has been in the Jaramillo family for generations. It's in the countryside, and the entrance to it, lined with palms, is a dramatic one. The home is spectacular, and the lovely wooden floors make a satisfying creak when you step on the planks. Service is impeccable and the restaurant, excellent. The grounds make for a nice late afternoon stroll, and you can admire an enormous and regal old _sam\u00e1n_ tree, well into its second century of life, as you dine alfresco. Living Trips (www.livingtrips.com) manages this hotel, and they can arrange day trip excursions for you. The restaurant is open to the public, and members of the public can also come for the day and enjoy the pool. It is almost always booked on weekends and during holidays. This gorgeous hacienda is a particularly popular place for weddings on weekends. The Mateca\u00f1a airport is only 10 minutes away.\n\nLuxury hotel **Sazagua** (Km. 7 V\u00eda Cerritos, Entrada 4, tel. 6\/337-9895, www.sazagua.com, COP$446,000 d) is not technically a hacienda, as it is located in a country club type environment. Here attention to detail reigns. The 10 rooms are impeccable, the common space is inviting, the gardens are perfectly manicured (surrounded by elegant heliconia flowers, birds, and the occasional iguana), and you can lounge by the pool or enjoy a massage at the spa. Nonguests can enjoy the spa facilities for a separate charge. It's not a traditional hacienda, but it sure feels good there.\n\n##### **Food**\n\nPereira makes it easy for visitors, gastronomically speaking. All the top restaurants are in more or less the same area, along and nearby the Avenida Circunvalar.\n\n**Mama Flor** (Cl. 11 No. 15-12, tel. 6\/335-4713, noon-10pm Mon.-Sat., noon-5pm Sun., COP$15,000) is a cute joint catering to meat-lovers with mostly open-air seating and old photographs of Pereira decorating the walls. It's mostly a lunch place. Grilled beef options like tri-tip (in Colombia called _punta de anca_ ) are menu favorites.\n\nThe specialty at somewhat swanky **Mediterraneo** (Av. Circunvalar No. 4-47, tel. 6\/331-0397, noon-2am Mon.-Thurs., noon-3am Sat., noon-1am Sun., COP$25,000) is seafood, but the menu is varied. The lighting is a little on the dark side, but the restaurant is all open-air.\n\n**Ambar Diego Panesso** (Cra. 17 No. 9-50, tel. 6\/344-7444, noon-3pm and 6pm-10pm Mon.-Sat., COP$30,000) serves up elaborate dishes like portobello mushrooms stuffed with apple puree and bacon bits to the Pereira elite. While vegetarian dishes are mostly nonexistent on the menu, the kitchen will gladly take on the challenge and whip up a pasta dish for you. It's in the upscale Pinares neighborhood. Another restaurant on the elegant side is **El Mirador** (Av. Circunvalar at Cl. 4, Colina, tel. 6\/331-2141, noon-2am Mon.-Sat., COP$30,000), a steakhouse that has an incredible view of the city. There's an extensive list of Argentinian wines.\n\nThe menu at **El Meson Espa\u00f1ol** (Cl. 14 No. 25-57, tel. 6\/321-5636, noon-3pm and 7pm-midnight daily, COP$22,000) runs the gamut from paellas (the house specialty) to pad Thai.\n\nDine under a giant Italian flag at the open-air **Portobello** (Cra. 15 No. 11-55, tel. 6\/325-0802, noon-3pm and 6pm-10pm Mon.-Wed., noon-3pm and 6pm-11pm Thurs.-Sat., noon-9pm Sun., COP$25,000), the top address for pasta and other Italian favorites, where pizzas are cooked in a wood-burning oven.\n\n**Diego Parrilla** (Cl. 10B No. 15-09, tel. 6\/333-8503, cell tel. 317\/402-1980, noon-11pm Mon.-Sat daily, COP$30,000) is a popular steakhouse. It's in the pleasant Los Alpes area, not far from the Circunvalar.\n\nFor a hearty Colombian meal, like a big bowl of _ajiaco_ (a filling potato-based stew), or to hang out and have a couple of beers at night, **La Ruana** (Av. Circunvalar No. 12-08, tel. 6\/325-0115, 8am-2am Mon.-Sat., 8am-10pm Sun., COP$20,000) is the place to go on the Avenida Circunvalar.\n\n**Bermeo** (tel. 6\/333-0909, noon-3pm and 6pm-11pm daily, COP$25,000), which specializes in international cuisine, adjoins Hotel Don Alfonso. While they don't have much in the way of veggie options, they can accommodate vegetarians.\n\nTwo always reliable and reasonably priced Colombian chain restaurants that have healthy options on their menus hold prime space in the **Centro Comercial Parque Arboleda** (Circunvalar No. 5-20, www.parquearboleda.com). **Archie's** (tel. 6\/317-0600, www.archiespizza.com, 8am-11pm daily, COP$22,000) has great pizzas (try the thin-crust _pizzas r\u00fasticas_ ) and salads. Its location on the top floor, with a breezy terrace, is a cool one. They also deliver. Directly below on the ground floor, meals at **Crepes & Waffles** (tel. 6\/331-5189, www.crepesywaffles.com.co, noon-9pm Sun.-Wed., noon-10pm Thurs.-Sat., COP$25,000) serves both savory and sweet cr\u00eapes, including many vegetarian options. Desserts, such as mini-waffles with chocolate sauce and vanilla ice cream, are hard to resist.\n\n##### **Information and Services**\n\nIn Pereira, you can call 123 for any type of emergency.\n\nThere is a small **tourist information booth** in the lobby of the Centro Cultural Lucy Tejada (Cl. 10 No. 16-60, tel. 6\/311-6544, www.pereiraculturayturismo.gov.co, 8am-noon and 2pm-6:30pm Mon.-Fri.).\n\n##### **Getting There and Around**\n\nExcellent bus connections are available between Pereira and most major cities. The **Terminal de Transportes de Pereira** (Cl. 17 No. 23-157, tel. 6\/315-2323, www.terminaldepereira.com) is relatively close to the Avenida Circunvalar area. It is clean.\n\nThe articulated bus rapid transit system, the **Megabus** (tel. 6\/335-1010), has three routes and connects with 28 intra-city buses. It's not terribly convenient for those staying near the Avenida Circunvalar, unfortunately.\n\nThe **Aeropuerto Mateca\u00f1a** (Av. 30 de Agosto, tel. 6\/314-2765) is pint-sized, and about a 10-minute ride east from the city. It's quite convenient for those planning on staying at one of the several haciendas in that area. The two largest Colombian airliners, **Avianca** (Av. Circunvalar No. 8B-23, tel. 6\/333-0990, www.avianca.com, 8am-1pm and 2:30pm-6pm Mon.-Fri., 8am-noon Sat.) and **LAN** (Cl. 19 No. 8-34, Local 102, 8am-6pm Mon.-Fri., 9am-1pm Sat.) serve Pereira. Panamanian-based **Copa** (Av. Circunvalar No. 8B-51, Edificio Bancafe, Local 103, Col. toll-free tel. 01\/800-011-2600, www.copaair.com, 8am-6pm Mon.-Fri., 9am-noon Sat.) has nonstop flights to Panama City five days a week. Budget **Viva Colombia** (www.vivacolombia.co) flies nonstop between Pereira and the Caribbean cities of Cartagena (4 weekly flights) and Santa Marta (3 weekly flights).\n\nFor those looking to rent a car, **Hertz** (airport tel. 6\/314-2678, www.hertz.com, 8am-6pm Mon.-Fri., 8am-noon Sat.) has an office at the airport.\n\n#### **VALLE DEL R\u00cdO OT\u00daN**\n\nA visit to the Valle del R\u00edo Ot\u00fan (R\u00edo Ot\u00fan Valley) between Pereira and the Laguna del Ot\u00fan is an interesting, highly enjoyable, and easy to organize introduction to Andean cloud forests. There are many possibilities for visiting the valley, from day trips out of Pereira to multi-day excursions, with very pleasant lodging facilities in the Santuario de Flora y Fauna Ot\u00fan-Quimbaya and Parque Regional Natural Ucumar\u00ed.\n\nThe R\u00edo Ot\u00fan flows 78 kilometers (48 miles) from the Laguna del Ot\u00fan to the R\u00edo Cauca and is the main source of water for Pereira. The conservation of the upper segment of the river, from Pereira to the Laguna del Ot\u00fan, has been a success story, thanks to reforestation land protection efforts.\n\n###### **PLANNING YOUR TIME**\n\nThe Santuario de Flora y Fauna Ot\u00fan-Quimbaya is 14.4 kilometers (9 miles) southeast of Pereira along the R\u00edo Ot\u00fan. The Parque Regional Natural Ucumar\u00ed is 6.6 kilometers (4 miles) upriver.\n\nYou can do day trips out of Pereira to either, but don't try to do both in one day. Santuario de Flora y Fauna Ot\u00fan-Quimbaya is easily accessible by public transportation, and the main nature trails can be visited in one day. However, getting to Parque Regional Natural Ucumar\u00ed involves public transportation and a two-hour hike. It can be visited on a long day trip but it is much preferable to spend a night or two at the comfortable Pastora visitor center in the midst of the Andean forest. You can combine a visit to both, visiting Santuario de Flora y Fauna Ot\u00fan-Quimbaya and then spending a day or two in Parque Regional Natural Ucumar\u00ed.\n\nDecember and July-August are drier months, and considered the best time for a hike to the Laguna del Ot\u00fan. However, during mid-December through mid-January and Semana Santa (Holy Week) in March or April the trails can be packed with hikers, as this is high season for Colombians. The hike through Parque Regional Natural Ucumar\u00ed to the Laguna del Ot\u00fan is very popular then, and there can be over a hundred hikers camping each night at that mountain lake.\n\nthe R\u00edo Ot\u00fan\n\n##### **Santuario de Fauna y Flora Ot\u00fan-Quimbaya**\n\nThe **Santuario de Fauna y Flora Ot\u00fan-Quimbaya** (Km. 4.5 V\u00eda Florida-El Cedral, Vereda La Suiza, cell tel. 313\/695-4305, www.parquesnacionales.gov.co, COP$5,000) covers 489 hectares (1,208 acres) of highly biodiverse Andean tropical forest at altitudes between 1,750 and 2,250 meters (5,740 and 7,380 feet). The vegetation is exuberant, and there are animal-viewing opportunities. The park is home to more than 200 species of birds, including endangered multicolored tanagers and the large _pava caucana_ (Cauca guan). And, although you may not see them, you'll definitely hear the _mono aulladores_ (howler monkeys). They make quite a brouhaha.\n\nThe main activities at the park are guided walks along three nature paths led by knowledgeable and enthusiastic guides from a local community ecotourism organization, the **Asociaci\u00f3n Comunitaria Yarumo Blanco** (cell tel. 312\/200-7711, yarumoblanco2009@hotmail.com). The tours cost COP$35,000 for one path, COP$70,000 for two, and COP$80,000 for all three (per group of any size). Visitors can also rent mountain bikes (COP$10,000 all day) and ride along the main road bordering the crystalline R\u00edo Ot\u00fan.\n\nThe visitors center, also run by the association, offers simple but comfortable lodging (COP$32,000-42,000 pp) and meals (COP$6,000-9,000).\n\nTo get to Ot\u00fan-Quimbaya from Pereira, take a bus operated by **Transportes Florida** (tel. 6\/331-0488, COP$4,000, 90 mins.) from Calle 12 and Carrera 9 in Pereira. On weekdays, the bus departs at 7am, 9am, and 3pm. On weekends there is an additional bus at noon. The buses return from Ot\u00fan-Quimbaya at approximately 9:30am, 11:30am, and 5:30pm.\n\n##### **Parque Regional Natural Ucumar\u00ed**\n\nThe **Parque Regional Natural Ucumar\u00ed** is 6.6 kilometers (4 miles) southwest of Santuario Flora y Fauna Ot\u00fan-Quimbaya. This regional park covers an area of 3,986 hectares (9,850 acres) of Andean tropical forest at altitudes between 1,800 and 2,600 meters (5,900 and 8,500 feet). The main path follows the R\u00edo Ot\u00fan through lush cloud forests, with waterfalls feeding into the river. The park is a wonderful place to view nature, with more than 185 species of birds.\n\nThe starting point of the main path is **El Cedral,** a small _vereda_ (settlement) southwest of Santuario Flora y Fauna Ot\u00fan-Quimbaya. The path is a well-trod one (by humans and horses) and is often muddy and rocky. It is best to take rubber or waterproof boots. It takes about 2.5 hours to climb to the main La Pastora visitors center, six kilometers from El Cedral.\n\nIf you are on a day trip from Pereira, you can have lunch at the visitors center (COP$9,000 pp) and set off on one of three nature hikes before returning to El Cedral to catch the last bus at 5pm. Even better, you can spend a night or two in the clean and very cozy dormitory-style rooms (COP$22,000 pp). As the temperature begins to drop in the late afternoon, you can sit by the lodge's fireplace and sip hot chocolate and eat cheese (as is the custom in Colombia). There is no electricity, making a stay at La Pastora truly restful. To make a reservation, contact the ecotourism organization **Fecomar** (cell tel. 312\/200-7711, fecomar.anp@hotmail.com, www.fecomar.com.co). They can arrange horses, if you would rather ride than hike up.\n\nPast La Pastora, the path continues 13 kilometers to the **Laguna del Ot\u00fan** in the **Parque Nacional Natural Los Nevados.**\n\nTo get to El Cedral take the **Transportes Florida** bus (tel. 6\/331-0488, COP$5,000, 2 hrs.) from Calle 12 and Carrera 9 in Pereira. On weekdays, the bus departs at 7am, 9am, and 3pm. On weekends there is an additional bus at noon. The buses return from Santuario Flora y Fauna Ot\u00fan-Quimbaya at approximately at 9am, 11am, and 5pm.\n\n#### **SANTA ROSA DE CABAL**\n\nThe dusty town of Santa Rosa de Cabal means one thing to most Colombians: _termales_ (hot springs). To get to the most well-known springs you'll have to pass through Santa Rosa de Cabal. In town, at the **Parque las Araucarias** (between Cras. 14-15 and Clls. 12-13), the main square, there are juices to be drank, _chorizo santarosano_ sausages to be eaten (a specialty here), handicrafts to be bought, and people to be watched. The other point of interest is the **Santuario La Milagrosa** (Cl. 7 at Cra. 14, tel. 6\/368-5201 or 6\/368-5168), a generally drab modern church with a fantastic stained glass window.\n\n##### **Hot Springs**\n\nThere are two _termales_ (hot springs) near Santa Rosa de Cabal: **Termales de Santa Rosa de Cabal** (Km. 9 V\u00eda Termales, tel. 6\/364-5500, www.termales.com.co, 9am-11:30pm daily, COP$14,000-31,000) and **Termales San Vicente** (18 km east of Santa Rosa de Cabal, tel. 6\/333-3433, www.sanvicente.com.co, 8am-midnight daily, COP$19,000-60,000). Both are wildly popular with Colombian families on weekends and holidays, and are quieter during the week. Both springs offer transportation from Pereira, and entry fees drop during the week.\n\nBuilt in 1945, the Termales de Santa Rosa de Cabal hot springs are closer to Santa Rosa de Cabal. There are two areas in the complex. The first area, on the left as you enter the park, was recently built and is called the **Termales Balneario.** These consist of three large pools for adults and one for children. This area is the most popular for day-trip visitors.\n\nThe oldest part of the complex, called **Termales de Hotel** (hours vary Mon.-Fri., COP$24,000), is farther on at the base of some spectacular waterfalls of cool and pure mountain waters. The highest waterfall drops some 175 meters (575 feet). If you choose to stay the night, there are three options. **La Caba\u00f1a** (COP$183,000 pp, meals incl.) is the newest and most comfortable place to stay and has 17 rooms. La Caba\u00f1a guests are allowed entry to the Termales Balneario, the Termales de Hotel, and their own small private pool. The advantage of staying at one of the hotels is that you can enjoy full use of the pools from 6am on, before the day-trip crowd begins arriving at 9am.\n\nAt both Termales Balneario and Termales de Hotel, there are additional activities on offer, such as a guided nature walk (COP$14,000) to some waterfalls\u2014wear shoes with traction for this, the path is slippery\u2014and spa treatments such as massages (COP$60,000, 45 mins.) and other services in a shabby-looking spa area.\n\nThe San Vicente hot springs are more remote, but the scenery of rolling hills, mountains in the distance, and farms is enchanting. Particularly scenic are the _pozos de amor,_ small natural pools the size of whirlpools that perfectly fit two. The all-inclusive _pasadia_ (day pass) option with transportation costs COP$60,000. Buses leave Pereira at 8am, returning at 5pm. San Vicente also offers accommodations, with cabins and a small hotel. A cabin for two people costs COP$173,000 per person without meals.\n\n##### **Accommodations and Food**\n\nThe bucolic countryside outside of Santa Rosa is home to many roadside, family-style restaurants. The best two are run by the same owner. On the road towards the Termales de Santa Rosa, **Mamatina** (Km. 1 V\u00eda Termales, La Leona, tel. 6\/363-4899, 9am-10pm daily, COP$18,000) specializes in trout covered with sausage, _sancocho_ (a meaty stew), and grilled meats. They can also accommodate vegetarians, usually a nutritious meal of beans and rice. Next door to the restaurant is their hotel by the same name (tel. 6\/363-4899), which offers clean and comfortable rooms ranging in price from COP$40,000 per person to COP$140,000 for the suite with a hot tub. Horseback riding and walks through the countryside can be arranged here.\n\nOn the way towards Termales de San Vicente is their other, newer hotel. The M **Hospedaje Don Lolo** (Km. 5 V\u00eda Termales San Vicente, cell tel. 316\/698-6797, hospedajedonlol@gmail.com, COP$50,000 pp d) is a hotel on a farm with cows, pigs, fish, horses, and dogs. If you're interested, you can lend a hand milking a cow or two. Some walks through the countryside are options as well, such as to an old Indian cemetery, to a big waterfall, and through jungle to see birds and butterflies. The countryside views and fresh air are delightful. If you're lucky, you may be able to see the Nevado del Ruiz in the distance in the early morning. The **Don Lolo** (Km. 5 V\u00eda Termales San Vicente, cell tel. 316\/698-6797) restaurant just down the road has a lot of personality and is a popular stopping off point going to or returning from the Termales de San Vicente.\n\n#### **BELALC\u00c1ZAR**\n\nThe coffee and plantain town of Belalc\u00e1zar rests impossibly on a ridge, with fantastic views of the Valle de Cauca on one side and the Valle del R\u00edo Risaralda on the other. Besides the incredible views, Belalc\u00e1zar, an agricultural town off the tourist map, offers the visitor pure coffee region authenticity. Belalc\u00e1zar has a distinct architecture, with its houses covered with colorful zinc sheets to protect against strong winds.\n\nsunny street in Belalc\u00e1zar\n\nBuilt in 1954 in hopes of preventing further bloodshed during the bloody Violencia period, the 45.5-meter-high (149-foot-high) **Monumento a Cristo Rey** (Km. 1 V\u00eda Pereira-Belalc\u00e1zar, COP$3,000) has become the symbol of this town. To get to the top of the statue of Jesus, you'll have to climb 154 steps. From atop, on a clear day, you can see six Colombian departments: Caldas, Risaralda, Quind\u00edo, Valle del Cauca, Tolima, and Choc\u00f3; and both Central and Occidental mountain ranges. The other attraction in town is the **Eco Parque La Estampilla hike** (1.5 km, open daylight hours, free). It's on the northeast side of town, a 10-minute walk from the **Parque Bol\u00edvar** (Clls. 15-16 and Cras. 4-5), in which you can wander a winding path through forests of _guadua_ (bamboo).\n\nThe best time to check out Belalc\u00e1zar life at its most vibrant is on market day\u2014Saturday\u2014when farmers from the countryside converge into town to sell coffee beans, plantains, pineapples, and other crops. On this day the Parque Bol\u00edvar buzzes with activity as Jeep Willys, packed with farmers and market-goers, come and go all day long. It's quite a carnival atmosphere.\n\nThe best hotel in town is the **Hotel Balc\u00f3n Colonial** (Cra. 4A No. 12-10, tel. 6\/860-2433, COP$35,000 d). It's clean, cool, and basic, having just nine rooms.\n\n##### **Getting There**\n\nIt is 45 kilometers (29 miles) from Pereira to Belalc\u00e1zar. Buses leave from the Pereira Terminal de Transportes (Cl. 17 No. 23-157, Pereira, tel. 6\/315-2323). **Flota Occidental** (tel. 6\/321-1655) is the bus company that serves Belalc\u00e1zar (COP$6,000, 1.5 hrs.).\n\n#### **SANTUARIO**\n\nIt's worth the arduous journey to this remote village on a mountaintop in the Cordillera Occidental (Western Mountains) just to take a photo of its famous **Calle Real,** dotted with stately Paisa houses that have been done up in a rainbow of colors. Calle Real is one of the most photographed streets in Colombia.\n\nOn Saturdays, campesinos come into town to sell their coffee, cacao, sugarcane, and other crops. There is so much activity on market day in the **Plaza de Bol\u00edvar** (between Clls. 6-7 and Cras. 5-6) that you'll be tempted to find a front row seat in a caf\u00e9 and take it all in: produce and coffee being unloaded and loaded, Jeep Willys filled with standing-room-only passengers arriving and departing, farmers drinking beer in taverns, women selling sweets in the park, and children being children. Many farmers, money in hand, whoop it up in town and stay the night.\n\nAlthough Santuario is indeed picture-perfect, it's far better to continue on to the **Parque Nacional Natural Tatam\u00e1** than to spend the night in Santuario, even if you are not interested in doing any hiking or trekking. Hotels in town are not recommended.\n\nThere is regular bus transportation from the Terminal de Transportes de Pereira (Cl. 17 No. 23-157, Pereira, tel. 6\/315-2323) to Santuario. **Flota Occidental** (tel. 6\/321-1655) makes this two-hour trip (COP$6,000) three times a day: 6:45am, noon, and 5:20pm.\n\n#### **PARQUE MUNICIPAL NATURAL PLANES DE SAN RAFAEL**\n\nLocated in the remote, little visited Cordillera Occidental (Western Mountains), the Tatam\u00e1 Massif contains one of the world's few remaining pristine _p\u00e1ramos_ (highland moors). The topography of the mountain range is very broken, especially the jagged **Cerro Tatam\u00e1** (4,250 meters\/13,945 feet), which is the highest point in the Cordillera Occidental. The range is highly biodiverse, with an estimated 564 species of orchids and 402 species of birds. It is also home to pumas, jaguars, and _osos anteojos,_ the only breed of bear in Colombia. The central part of the massif is protected by the 15,900-hectare (39,300-acre) Parque Natural Nacional Tatam\u00e1.\n\nAccess to Tatam\u00e1 is through the Parque Municipal Natural Planes de San Rafael, which acts as a buffer zone on the eastern side of Tatam\u00e1 near the town of Santuario, but which is an attraction in itself.\n\nThe **Parque Municipal Natural Planes de San Rafael** (10 km from Santuario, cell tel 311\/719-1717, <http:\/\/planesdesanrafael.blogspot.com>) covers an area of 11,796 hectares (29,149 acres) of cloud forest between the altitudes of 2,000 and 2,600 meters, with significant patches of primary growth. The main activities are nature walks conducted by friendly and knowledgeable guides of a local community organization, the **Asociaci\u00f3n de Gu\u00edas e Interpretes Ambientales (GAIA),** many of whom got their start through participation in groups of youth bird-watchers.\n\nWithin the park there are four main paths. The shortest, called the **Lluvia de Semillas** (Rainfall of Seeds), allows visitors to see a forest in recuperation. It is a one-kilometer (0.6-mile) loop through land that was once used for cattle grazing and, over the past 15 years, has been slowly returning to a forest. The 9.6-kilometer (6-mile) round-trip **Cascadas** trail is a strenuous path to the border of the Parque Nacional Natural Tatam\u00e1 at an elevation of 2,600 meters. It crisscrosses the R\u00edo San Rafael and culminates at a group of waterfalls. Along the way you can see a great variety of birds and large patches of primary forest. The hike takes 3.5 hours up and 2.5 hours down. The 12-kilometer (7.5-mile) **Quebrada Risaralda** hike takes six hours and can be combined into a loop with the Cascadas hike. Finally, the **Laguna Encantada** path is a nine-kilometer (5.5-mile) circuit that takes five hours and is especially good for bird-watching, with the possibility of viewing many hummingbirds. The best time for these hikes is early in the morning. Costs for these excursions are COP$25,000-35,000 per group of any size.\n\n##### **Parque Nacional Natural Tatam\u00e1**\n\nFrom Parque Municipal Natural Planes de San Rafael it is also possible to organize excursions into the **Parque Nacional Natural Tatam\u00e1** (tel. 6\/368-7964, www.parquesnacionales.gov.co), located at the highest point of the Cordillera Occidental between the departments of Choc\u00f3, Risaralda, and Valle del Cauca. Though it is not officially open to ecotourism, the folks at GAIA can organize an excursion that requires at least two nights of camping. The first day involves a 12-kilometer (7.5-mile), eight-hour hike to a campground at 3,200 meters. The following day you explore the upper reaches of the Tatam\u00e1, with the unusual shrub-covered _p\u00e1ramo_ (high tropical mountain ecosystem), craggy outcrops, and deep gorges. From the top you can see the Choc\u00f3 lowlands. You return by nightfall to camp and return to Parque Municipal Natural Planes de San Rafael on the following day.\n\n##### **Accommodations**\n\nThe clean, comfortable, and quite cozy lodge (COP$20,000 pp) at the Parque Municipal Natural Planes de San Rafael visitors center accommodates 40 people. Meals (COP$6,000-9,000) are nothing short of delicious. To reserve lodging, contact the ecotourism organization **FECOMAR** (cell tel. 312\/200-7711, fecomar.anp@hotmail.com, www.fecomar.com.co) or call the park administrator (cell tel. 311\/719-1717).\n\n##### **Getting There**\n\nTo get to the Parque Municipal Natural Planes de San Rafael visitors center and lodge, pick up one of the Jeeps that leave from the main square in Santuario each day at 7am and 3pm. A taxi can also take you for COP$20,000.\n\n#### **IBAGU\u00c9**\n\nIbagu\u00e9 (pop. 593,000) is the capital city of the Tolima department. It is a hot city of traffic jams that does not seduce many travelers. It is, however, a city of music and hosts several music festivals each year. It's hard to believe, but just to the west of Ibagu\u00e9, a city in one of the most important tropical-fruit-producing regions in Colombia, looms the Nevado del Tolima, a snowcapped volcano in the eastern side of the Parque Nacional Natural Los Nevados. Ibagu\u00e9 is a great launch pad for the challenge of ascending to that mountain.\n\n##### **Sights**\n\nShould you have some time to spend in Ibagu\u00e9, a handful of sights are worth taking a look. Shade is not an issue at the **Plaza de Bol\u00edvar** (between Cras. 2-3 and Clls. 9-10): It's full of huge, centuries-old trees.\n\nStaff at the tourist kiosk in the plaza in front of the **Catedral Inmaculada Concepci\u00f3n** (Cl. 10 No. 1-129, tel. 8\/263-3451, 7am-noon and 2:30pm-7:30pm daily) can suggest hiking excursions and tips on trekking up to the Nevado del Tolima. The **Banco de la Rep\u00fablica** (Cra. 3A No. 11-26, tel. 8\/263-0721, 8:30am-6pm Mon.-Fri., 8:30am-1pm Sat., free) always has an exhibit in its one exhibition hall on the second floor. It's just off of **Calle Peatonal** (Cra. 3 between Clls. 10-15), which is refreshingly free from cars and motorbikes.\n\nThe **Conservatorio del Tolima** (Cra. 1 between Clls. 9-10, tel. 8\/826-1852, www.conservatoriodeltolima.edu.co, open only for concerts) is perhaps Ibagu\u00e9's claim to fame and the reason it calls itself the \"Capital Musical de Colombia.\" A few bars from the Colombian national anthem are painted on the exterior of the yellow republican-era building.\n\nMusic festivals take place in the city year-round, including the **Festival Nacional de la Musica Colombiana** (www.fundacionmusicaldecolombia.com, Mar.), the **Festival de Jazz** (May), and the **Festival Folcl\u00f3rico Colombiano** (www.festivalfolclorico.com, June or July).\n\nThe **Museo de Arte del Tolima** (Cra. 7 No. 5-93, tel. 8\/273-2840, www.museodeartedeltolima.org, 10am-6pm daily, COP$3,000) is a small museum with a permanent collection and temporary exhibition space dedicated to contemporary Colombian artists. A leafy park in this pleasant part of town is the **Parque Centenario** (Cra. 6 between Clls. 8-10). Along with usual park goings-on, cultural events are often held in an amphitheater.\n\n##### **Accommodations and Food**\n\nThe **Hotel Ambeima** (Cra. 3 No. 13-32, tel. 8\/263-4300, www.hotelambeima.com, COP$89,000 d) is downtown on the Calle Peatonal and is a decent midrange choice. The best hotel in town is the **Hotel Estelar Altamira** (Cra. 1A No. 45-50, tel. 8\/266-6111, COP$203,000 d). There are no hostels that cater to international backpackers.\n\nThe trendy dining and drinking spot in Ibagu\u00e9 is 15 stories high, with a superb bird's-eye view of the city and the Parque Nacional Natural Los Nevados. **Altavista** (Cra. 2 at Cl. 11, tel. 8\/277-1381, noon-midnight Mon.-Wed., noon-3am Thurs.-Sat., noon-4pm Sun., COP$25,000) has a little bit of everything: Asian-inspired dishes, tapas, and even several vegetarian options. The decor and atmosphere are all South Beach. As the sun slips behind the mountains in the distance, bartenders swing into full motion as Altavista turns into a _play_ (fashionable) nightclub scene.\n\n##### **Getting There and Around**\n\nThe **Terminal de Transportes** (Cra. 2 No. 20-86, tel. 8\/261-8122, www.terminalibague.com) is in a rough part of town, so you should take a cab to and from the bus station. For the most part, Ibagu\u00e9 is not a walkable city, except in the center of town. From the Ibagu\u00e9 airport, **Aeropuerto Nacional Perales** (V\u00eda al Aeropuerto, tel. 8\/267-6096), there are nonstop flights to Bogot\u00e1 and Medell\u00edn.\n\n## **CALI AND SOUTHWEST COLOMBIA**\n\nHIGHLIGHTS\n\nPLANNING YOUR TIME\n\nCali\n\nSIGHTS\n\nENTERTAINMENT AND EVENTS\n\nSHOPPING\n\nSPORTS AND RECREATION\n\nACCOMMODATIONS\n\nFOOD\n\nINFORMATION AND SERVICES\n\nGETTING THERE\n\nGETTING AROUND\n\nRESERVA NATURAL ANAHUAC\n\nHACIENDA PARA\u00cdSO AND MUSEO DE LA CA\u00d1A DE AZ\u00daCAR\n\nBuga\n\nSIGHTS\n\nACCOMMODATIONS AND FOOD\n\nGETTING THERE\n\nVICINITY OF BUGA\n\nROLDANILLO\n\nPasto\n\nSIGHTS\n\nENTERTAINMENT AND EVENTS\n\nSHOPPING\n\nRECREATION\n\nACCOMMODATIONS\n\nFOOD\n\nINFORMATION AND SERVICES\n\nGETTING THERE AND AROUND\n\nM LAGUNA LA COCHA\n\nM LAGUNA VERDE\n\nVOLC\u00c1N CUMBAL\n\nCAF\u00c9 DE ALB\u00c1N\n\nIPIALES\n\nPopay\u00e1n\n\nSIGHTS\n\nENTERTAINMENT AND EVENTS\n\nRECREATION\n\nACCOMMODATIONS\n\nFOOD\n\nGETTING THERE AND AROUND\n\nVICINITY OF POPAY\u00c1N\n\nM TIERRADENTRO\n\nM SAN AGUST\u00cdN\n\nThe distances between the cities in the southwest of Colombia are not great as the crow (or in this case, the condor) flies. However, Cali, Popay\u00e1n, Pasto, and their environs are worlds apart. Cali is a city that melds its colonial past with visions of a modern future. In Pasto, the largest city in the Nari\u00f1o department, Volc\u00e1n Galeras looms in the distance. Popay\u00e1n is the White City, historic home of presidents and poets.\n\n**HIGHLIGHTS**\n\nLOOK FOR M TO FIND RECOMMENDED SIGHTS, ACTIVITIES, DINING, AND LODGING.\n\nM **Salsa in Cali:** There is no better place to practice your moves than at one of Cali's countless _salsatecas_ (salsa clubs). But even if you have two left feet, you can still get into the spirit by taking in a salsa show, going to Salsa al Parque, or enjoying the one-of-a-kind atmosphere at an old-school salsa bar (click here).\n\nM **Laguna La Cocha:** Take a boat ride from the colorful fishing village of Encano to the tiny tropical rainforest Isla de la Corota, home to Colombia's smallest national park (click here).\n\nM **Laguna Verde:** The emerald-green waters of this crater lake in the Nudo de los Pastos mountains on the border with Ecuador are a fantastic sight to behold (click here).\n\nM **_Centro Hist\u00f3rico_ in Popay\u00e1n:** Explore volcanoes, coffee farms, indigenous markets, or hot springs by day, but save some energy and time to amble the streets of the White City's beautiful historic center after the sun goes down (click here).\n\nM **Tierradentro:** Dozens of elaborate burial chambers lie beneath the hilltops in this important archaeological site in Cauca (click here).\n\nM **San Agust\u00edn:** A visit to this incredible archaeological site, coupled with nature hikes or rafting adventures, is a highlight of any visit to Colombia. This is Colombia's Easter Island (click here).\n\nIn Cali, Colombia's third largest city, vestiges of the colonial past endure in the historic center. Its centerpiece is the superbly maintained 16th-century Iglesia La Merced. But Cali is best known for its warm people and its hot salsa dancing. Across the city, visitors will see Cale\u00f1os at play: hanging out in the Parque San Antonio, strolling along La Loma de la Cruz artisan market, and dancing at Salsa al Parque, a free outdoor salsa party that's held once a month. This is the place to take a salsa lesson or two, practice your fancy footwork on the dance floor, or settle back in a booth at a popular _salsateca,_ enjoying the music and one-of-a-kind atmosphere.\n\nThe Valle de Cauca, just a couple hours' ride from the departmental capital, offers some unsung destinations. As you set off from the city, the flat countryside is dominated by fields of sugarcane, interrupted only by occasional _sam\u00e1n_ trees, which look like they belong in Africa. Every once in a while, a verdant island stands out in the middle of the cane fields\u2014a hacienda, or country estate. Buga, a Catholic pilgrimage site, is perfectly situated near the weekend playground of Lagos Calima. In the north of the department is the sleepy town of Roldanillo, which was put on the map thanks to its famous son, abstract expressionist artist Omar Rayo. For adventurous types, this cute town is about a different art: the art of flying. Each year hundreds catch a breeze as they paraglide over the valley.\n\nAlthough many travelers make a quick detour on their way to or from Ecuador to admire the neo-gothic gem of Las Lajas church, set impossibly in a narrow canyon near Ipiales, few spend the time to get to know the department of Nari\u00f1o. Yet it is here that every mountain you ascend, every river you traverse, and every turn you make reveals even more jaw-dropping scenery. Jagged mountains, turquoise lakes, and sleeping volcanoes provide dramatic backdrops to the neat patchwork of potato crops that sustain many families in this agricultural region. Volc\u00e1n Galeras is a sometimes ominous presence in Pasto's skyline, it doesn't stop revelers from living it up during the Carnaval de Negros y Blancos in January. Not far from the city is Laguna La Cocha, with fishing communities of brightly colored clapboard houses and wooden fishing boats.\n\nUpstream along the R\u00edo Cauca from the Valle de Cauca is the \"White City\" of Popay\u00e1n, home to the Universidad del Cauca. Students fill the city's caf\u00e9s, bars, and its stately main plaza, the Parque Caldas. Nearby are several interesting day trips, from organic coffee farms to hot springs to indigenous markets. In the national park of Purac\u00e9, intrepid hikers trek to the rim of a snow-covered active volcano. Farther on are two of Colombia's most important and beautiful archaeological sites: Tierradentro, where dozens of ancient underground burial chambers are preserved, and San Agust\u00edn, where hundreds of well-preserved stone statues of deities and animals stand in silence.\n\n#### **PLANNING YOUR TIME**\n\nThree days is just enough time to get a feel for the Cali way. Try visiting the city over part of the weekend so that you can check out a salsa show or visit a _salsateca_ (salsa club). Other sights of interest in Valle de Cauca can be visited out of Cali or perhaps from Buga. Roldanillo makes a nice side trip between Cali and the Coffee Region near Armenia.\n\nIt's an easy trip between Cali and Popay\u00e1n, about three hours by minivan on a good road. If you'd like to check out sights near Popay\u00e1n, such as the Guambiano town of Silvia, or hike to the Purac\u00e9 volcano, plan for at least 3-4 days in the capital of Cauca. A circuit tour of the sights of Tierradentro, the Tatacoa desert, and San Agust\u00edn can be done out of Popay\u00e1n. For that you'll need five days.\n\nIt is often slow going on the Pan-American Highway between Ipiales, through Pasto, to Popay\u00e1n. (But the scenery? Simply incredible.) While Ipiales is not a great base for sightseeing, the larger city of Pasto is. You can happily spend a few days seeing the city and taking some day trips such as La Cocha or the Laguna Verde. The strenuous climb up Volc\u00e1n Cumbal takes planning but is worth it.\n\nSome of the big goings in the region are the Semana Santa processions in Popay\u00e1n (Easter week), the Feria de Cali (last week of December), and the Carnaval de Negros y Blancos (January 4-7) in Pasto.\n\n### **Cali**\n\nCali's relaxed place is evident everywhere you go in this diverse city of three million. It may be the infernal heat slowing everyone down, without regard to _tinto_ (coffee) intake. Here in the Valle del Cauca, where sugarcane fields go on forever, you're far removed from uptight Bogot\u00e1 and over-ambitious Medell\u00edn. In a swipe at those two cities, locals like to say that, _\"Cali es Cali y lo dem\u00e1s es loma_ \" (\"Cali is Cali but those other cities are just hills\").\n\nWhile the colonial churches of La Merced and San Antonio are some of the loveliest you will see in Colombia, Cali is not a city packed with must-see sights. Yet tourists keep falling in love with the \"Sultan of the Valley.\" Is it that late refreshing afternoon breeze? The people? Or is it the salsa?\n\nCali is, indeed, all about salsa. That's pretty clear once you hop in any cab in the city. The music is turned on, and there's an 80 percent chance you'll hear the driver singing along to the hypnotic rhythms of conga drums, bongos, and timbales. At the bus terminal, _muchachos_ working for the bus companie break into some fancy footwork as they wait for customers. In hotels, housekeeping staff walk the corridors cheerfully singing.\n\nCali is the world's salsa capital. Incorporating a little salsa into your visit to Cali is a necessity if you want to get to know this seductive city, its people, and their ways.\n\n##### **History**\n\nPresent-day Valle del Cauca was settled by the Calima people as early as 1200 BC. The first Europeans to arrive were soldiers under the command of Sebasti\u00e1n de Belalc\u00e1zar. This Spanish conquistador served under Francisco Pizarro in the conquest of Peru, but then decided to seek his own fame by conquering most of present-day Ecuador and southwestern Colombia. In 1536, he founded Santiago de Cali, slightly north of its current location. During the colonial period and most of the 19th century, Cali was a small agricultural settlement, surrounded by haciendas, with a significant slave population.\n\nThe construction of railway links spurred economic development and in the 20th century, Valle de Cauca became sugarcane country. Cali developed rapidly, becoming a center of manufacturing. In 1971, the city hosted the Pan-American Games. That was probably Cali's heyday, before it started a long process of decay.\n\nDuring the 1980s and 1990s, Cali became a global drug trafficking center and seat of the eponymous drug cartel. It is widely believed that Cali Cartel kingpins Gilberto and Miguel Rodr\u00edguez Orejuela provided key support to government authorities in their war against the Medell\u00edn Cartel, which ultimately resulted in the assassination of Pablo Escobar and breakup of his empire. For a time in the 1990s, the Cali Cartel ruled the global cocaine business unchallenged.\n\nAs a result of drug-related violence, Cali's civic leaders fled, taking investment and business elsewhere. During the 1990s and early 21st century, the city had a double scourge of urban violence and guerrilla intimidation. The FARC (Fuerzas Armadas Revolucionarias de Colombia; Revolutionary Armed Forces of Colombia) occupied much of the surrounding mountainous areas.\n\nCurrently Cali is living a renaissance of sorts, undertaking major infrastructure projects, such as the MIO mass transit system, and is beginning to attract new investments. And in 2013, Cali was the proud host of the World Games, an international sports competition.\n\n##### **Orientation**\n\nNearly all of the city's tourist attractions are located in the **Centro,** in three areas in particular: **La Merced, La Ermita,** and **Plaza Cayzedo\/Iglesia San Francisco.** You can visit all these areas on foot in one day. The Centro is brimming with activity during weekdays. Avoid the midday heat by planning your visit in the morning or late afternoon. Avoid lingering downtown after dark.\n\nthe towering palms of the Plaza Cayzedo\n\nInviting neighborhoods such as **San Antonio, Granada,** and **El Pe\u00f1on** may lack attractions but are a delight to get to know. Within a 10- to 15-minute cab ride from the Centro, these tree-lined barrios are filled with restaurants, shops, and hotels. Late-night activity on weekends tends to shift to northern areas (not far from Granada) such as Menga and Yumbo.\n\nYou may have difficulty figuring out the lay of the land in Cali, as it is not as straightforward as other cities in Colombia. The R\u00edo Cali (actually more of a stream), the Tres Cruces hill, the Torre de Cali, and the Intercontinental Cali are well-known points of reference.\n\n##### **Climate**\n\nThe average temperature in _caliente_ Cali is about 24\u00b0C (75\u00b0F); however, the average daily high is a sizzling 30\u00b0C (86\u00b0F). But 4:30pm-6:30pm, when the sun begins its descent over the western Cordillera and into the Pacific, the drop in temperature (to around 19\u00b0C\/66\u00b0F), softening of the sun's rays, and a gentle breeze combine to make the weather absolutely _delicioso._ In residential areas like San Antonio, the perfect temps are a magnet drawing people outside. Plan your day accordingly: Do all your necessary emailing and online time-wasting during the heat of the day or later at night, and make that late afternoon stroll an event.\n\n##### **Safety**\n\nIf you ask a taxi driver in Cali if a certain barrio (neighborhood) is _seguro_ (safe), you may get a response along the lines of, _\"\u00a1Es seguro que te roban!\"_ (\"It's sure that you'll be robbed!\"). But don't worry\u2014that's a common joke. Follow the general precautions of any large Colombian city, especially regarding the use of taxis and precautions to take in nightspots, and you'll be fine.\n\nEven though pickpockets are about, walking by day is fine in Centro (downtown), as it is always bustling with people. But be alert near sights such as Iglesia La Ermita. Walking within neighborhoods is generally safer than walking between them, such as between the Centro and Granada or San Antonio. MIO (the bus system) is a safe option.\n\nThere are community police stations (Centros de Atenci\u00f3n Inmediata or CAI) in every neighborhood and often in parks. There is a CAI in the Parque de San Antonio, not far from the church. The national toll-free hotline for any emergency is 123. The police have an additional number, 112.\n\n#### **SIGHTS**\n\nAll the major historical and tourist sights in Cali are found in the Centro, or downtown. The best time to visit churches is during mass; otherwise there is a good chance they'll be closed.\n\n##### **La Merced Complex**\n\nOn June 25, 1536, the city of Santiago de Cali was founded by Sebasti\u00e1n de Belalc\u00e1zar. He changed his mind and moved the city shortly thereafter to its present location, and a mass was held to celebrate the foundation of the city. It was on this site that the **Iglesia La Merced** (Cra. 4 at Cl. 7, tel. 2\/889-2309, 6:30am-10am and 4pm-7pm daily, masses at 7am and 6pm Mon.-Sat., 9am and 6pm Sun.) was built, sometime around 1545. The oldest church in Cali, it is a lovely example of typical colonial construction of the time, with its thick, whitewashed walls. The church, in the shape of a cross, has a single nave with red wooden beams. The only extravagance to be seen is the golden baroque altar with a statue of the Virgen de las Mercedes, who is the patron saint of Cali. It stands out against the austerity of this house of worship.\n\nAdjacent to the church is the **Museo de Arte Colonial y Religioso La Merced** (Cra. 4 No. 6-117, tel. 2\/888-0646, cell tel. 312\/731-5948, 9am-noon and 2pm-5pm Mon.-Fri., 9am-2pm Sat., museocolonial.lamerced@gmail.com, COP$4,000 adults, COP$3,000 students, COP$2,000 children), a small museum containing Quite\u00f1o school paintings, silver objects, statues, and religious items from the colonial period. Guides, usually college students, will be happy to show you around for a small fee. The museum is located in the church's chapels of the Virgen de la Merced and Virgen de los Remedios.\n\nThe well-done **Museo Arqueol\u00f3gico La Merced** (Cra. 4 No. 6-59, tel. 2\/885-4665, 9am-1pm and 2pm-6pm Mon.-Sat., museolamerced@une.net.co, COP$4,000 adults, COP$2,000 children) is housed in part of the Augustinian convent and has two exhibition rooms highlighting ceramics from native cultures of the region: Tolima, Quimbaya, Calima, Tierradentro, San Agust\u00edn, Tumaco, and Nari\u00f1o.\n\n###### **MUSEO DEL ORO CALIMA**\n\nWithin the Banco de la Rep\u00fablica building across the street from Iglesia La Merced is the **Museo del Oro Calima** (Cl. 7 No. 4-69, tel. 2\/684-7754, www.banrepcultural.org\/cali\/museodelorocalima, 9am-5pm Tues.-Fri., 10am-5pm Sat., free). Cali's gold museum has a collection of more than 600 ornamental gold and utilitarian ceramics, attributed to the ancient Calima people, that have been unearthed northwest of present-day Cali.\n\nThe **Sala de Exposiciones de la Banco de la Rep\u00fablica** (Cl. 7 No. 4-69, tel. 2\/684-7751, 9am-5:30pm Mon.-Fri., free) is in the same building as the Museo del Oro, and it often hosts traveling exhibits from the bank's art collection as well as special exhibits by regional artists. It's worth a peek.\n\n###### **IGLESIA LA ERMITA**\n\nNear the R\u00edo Cali is one of the city's most iconic landmarks: the mini, neo-gothic **Iglesia La Ermita** (Cra. 13 at Cl. 1, no phone, masses at 7am and 5pm Mon.-Fri., 10am and 5pm Sat.-Sun.). Originally built in the 16th century, the church was destroyed by earthquakes in 1787 and in 1925. There was not much left after the 18th-century tremor, except for the painting of the Se\u00f1or de la Ca\u00f1a, another demonstration of how important sugarcane has been to the people of the Valle del Cauca. Its survival was attributed to a miracle. The three-nave church has an Italian marble altar and many stained-glass windows. The current building was completed in the 1940s, and its design is said to have been modeled after the towering Ulm Minster in Germany.\n\n###### **PLAZOLETA JAIRO VARELA AND BULEVAR DEL R\u00cdO**\n\nMegaprojects to revitalize the deteriorating downtown began in 2010. In 2013 the **Plazoleta Jairo Varela** and **Bulevar del R\u00edo** were completed. The Plazoleta Jairo Varela pays homage to a beloved salsa singer and founder of the Grupo Niche. Jairo Varela passed away in Cali in 2012. It is a large outdoor cultural space to be used for concerts and other cultural activities, located near the CAM office tower (Av. 2N between Clls. 10 and 11), which is the main municipal government building. Along the river, the pedestrian walkway Bulevar del R\u00edo was also unveiled. It extends from Calle 5 to La Ermita, and within weeks of its opening became an instant hit with weekend joggers. These new projects provide an exciting contrast to some of the grandiose relics of the 19th century, such as the **Teatro Jorge Isaacs** (Cra. 3 at Cl. 12) and **Puente Ortiz** (Cra. 1 at Cl. 12).\n\n###### **PLAZA CAYZEDO AND AROUND**\n\nThe dozens of majestic wax palms in the **Plaza Cayzedo** (Cras. 4-5 between Clls. 11-12) create a green oasis in the middle of gritty downtown Cali. Like the Plaza de Bol\u00edvar in Bogot\u00e1, Plaza Cayzedo was known as the Plaza Mayor during the colonial era, and was often the scene of lively markets that were held after mass at the San Pedro cathedral. It was also a place of open-air concerts by military bands and the occasional bullfight. In 1913, about a century following his execution, it was renamed to honor the most famous independence figure from Cali, Joaqu\u00edn de Cayzedo y Cuero. Passing through the park on the brick walkways you'll encounter a colorful cross-section of Cale\u00f1os, from university students to shoe-shiners to dapper older men watching the world go by from the comfort of a park bench.\n\nDating to the turn of the 19th century, the brilliantly white neoclassical-style **Catedral San Pedro** (Cl. 11 No. 5-35, tel. 2\/881-1378, masses at 9am, 10am, 11am, and noon Mon.-Fri., 9am and 5pm Sun.) on the southern corner of the Plaza Cayzedo has been rebuilt several times due to destruction caused by earthquakes. The most stunning building on the plaza, however, is the French neoclassical gem the **Palacio Nacional** (Cra. 4 No. 12-04), also known as the Palacio de Justicia, on the eastern side of the Plaza de Cayzedo. Completed in the early 1930s, it houses various judicial bodies of the Valle de Cauca departmental government. The Palacio Nacional is not open to the public.\n\nThe distinctive red-brick church complex of the **Iglesia San Francisco** (Cl. 10 No. 6-00, tel. 2\/884-2457, masses at 7am and 5pm Mon.-Sat., 9am and 6pm Sun.) includes the **Capilla de la Inmaculada,** the **Convento de San Joaqu\u00edn,** and the **Torre Mud\u00e9jar,** all built between the 17th and 19th centuries by Franciscans. The architectural star here is the Torre Mud\u00e9jar, a four-story bell tower 23 meters (75 feet) high. It is divided into four red-brick sections, each level with a different geometric design. It is considered one of the best examples of neo-Mudejar design in the Americas. The architect of the tower was supposedly a Moor who had fled Spanish authorities, seeking refuge in the convent. In return for the free lodging, he designed the bell tower.\n\n##### **Outside the Centro**\n\n###### **MUSEO LA TERTULIA**\n\nOne of the best art museums in the country is Cali's **Museo La Tertulia** (Av. Colombia No. 5 Oeste-105, tel. 2\/893-2939, www.museolatertulia.com, 10am-6pm Tues.-Sat., 2pm-6pm Sun., COP$4,000 adults, COP$2,000 students, free Sun.). Museum galleries highlight contemporary Colombian artists such as Beatriz Gonz\u00e1lez, Hugo Zapata, Omar Rayo, and many more. Built in the 1960s, Museo La Tertulia is perhaps the most important cultural center in Cali. The word _tertulia_ refers to a social gathering for talking and sharing ideas about culture, art, and other themes. A cinema (showings usually 7pm and 9:15pm Tues. and Sat., 4pm and 7pm Sun., COP$5,000) shows art films and hosts festivals such as EuroCine. A concert hall offers chamber music concerts and poetry readings, and the lush grounds house an amphitheater. The museum caf\u00e9 on the terrace is a popular meeting place in the evenings.\n\nthe Torre Mud\u00e9jar in downtown Cali\n\n###### **ZOOL\u00d3GICO DE CALI**\n\nThe **Zool\u00f3gico de Cali** (Cra. 2 Oeste and Cl. 14, tel. 2\/488-0888, 9am-4:30pm daily, COP$14,000 adults, COP$9,000 children) is, by far, the best zoo in the country, although that may not say very much. The landscape itself would make a lovely and tranquil botanical garden. Its location straddling the R\u00edo Cali and the great diversity in flora (providing welcome shade) make it a pleasant place to spend a few hours in the morning or late afternoon. Most of the zoo is dedicated to Colombian species. The condor exhibit may be disturbing to some visitors, as these large birds are confined in relatively small cages. American flamingos frolic nearby. Beware of swarms of families on weekends. The zoo is accessible by MIO by taking the A02 bus from the San Bosco station.\n\n###### **IGLESIA AND PARQUE DE SAN ANTONIO**\n\nThe small white **Iglesia de San Antonio** (Cl. 1 Oeste at Cra. 10, tel. 2\/893-7185), similar to Iglesia La Merced in the Centro, is beautiful in its simplicity. Its adobe walls are white, accented by the wooden beams above. The brick entrance and bell tower are also striking. The church was built in the mid-18th century, because the residents in the growing area of San Antonio wanted to worship in their own neighborhood instead of having to schlep down the hill to pray at another church. If the main doors are closed (which is often the case as the church is only open for mass), head to the back left of the church and ring (once) at the door. Clarisa nuns from the convent there may give you a quick tour. A cautionary note: Do not express any disappointment to them that this church is _just_ from 1747, unlike Iglesia La Merced (which was built about two hundred years before). They are sensitive about that! On weekends couples exchange vows at this picturesque spot.\n\nOnce you've made it to the top of the hill, plan to stick around awhile: **Parque San Antonio** is the place to experience San Antonio life. You'll see hipsters with their dogs mingling with other canine lovers, couples sipping a cold beer and enjoying the view as the sun goes down, and intellectual types engrossed in books. On one side of the park are small unpretentious restaurants, bars, and ice cream shops.\n\n###### **LA LOMA DE LA CRUZ**\n\nWhile technically it's known as an _artesan\u00edas_ (handicrafts) market, **La Loma de la Cruz** (Cl. 5 between Cras. 14-16, 9am-10pm daily) has such a pleasant atmosphere, especially in the early evening from Thursday through Sunday, that it is more of a tourist attraction than a shopping experience\u2014plus, it's free. The entrance at the bottom of the _loma,_ or hill, is hard to miss with the man-made waterfall cascading down and a sign that reads \"Loma de la Cruz.\" More than 20 kiosks sell typical Colombian handicrafts, such as hand-woven handbags and the like. After perusing the handicrafts and other items for sale, you can often enjoy concerts featuring Andean music, poetry readings, dance performances, or open-air films in the small amphitheater. You can walk to the hill from the San Antonio area.\n\n###### **EL CERRO DE CRISTO REY**\n\nThe statue of Christ on **El Cerro de Cristo Rey** (south of the Cerro de las Tres Cruces in the Los Andes neighborhood) stands 26 meters (85 feet). It was created by an Italian sculptor to celebrate 50 years of peace following the Guerra de Mil D\u00edas (the Thousand Days' War) over the turn of the 20th century. That civil war claimed around 100,000 lives in Colombia and Panama. While there is a path to the top, it is best to take a taxi (COP$45,000). Along the way you can check out the sculpture on the side of the mountain called _El Lamento de la Pacha Mama,_ which is a tribute to indigenous peoples, and munch on an _empanada_ at a roadside stall. Don't make the Cristo Rey excursion after dark.\n\n#### **ENTERTAINMENT AND EVENTS**\n\n##### M **Salsa**\n\nIn Cali, there are many ways you can get a taste of (and very likely get hooked on) salsa.\n\n###### **SALSA SHOWS**\n\nA good way to observe the incredibly intricate and fast footwork of Cale\u00f1o salsa dancers is to go to a performance of one of the big salsa shows in town. In this cabaret-style environment, you'll be amazed at the talent and exhausted by the high energy of these dancers, often ranging in age from 4 to 40.\n\nThe most famous show of them all is **Delirio** (Parque del Amor, Cl. 69 No. 4N-88, tel. 2\/893-7610, www.delirio.com.co, COP$120,00), a sort of Cirque du Soleil\u2014\u00e0 la Cali. This group, combining dance, music, and circus, has delighted audiences all over the world. Performances go from about 8pm to well after midnight. The group is constantly updating their shows, creating segments on different themes that inspire them, such as La Mar\u00eda and even Michael Jackson tributes. During intermission, spectators are invited to dance on the stage, but there's no pressure. Performances are usually held the last Friday of every month, and at other times during the Feria de Cali. Minors under the age of 18 are not allowed inside the big tent.\n\nOther popular ongoing shows include **Ens\u00e1lsate** (cell tel. 316\/480-7822, www.ensalsate.co, COP$90,000) which takes place the second week of each month and during the Feria de Cali at the Sal\u00f3n Ritz of the Hotel Dann Carlton (Cra. 2 No. 1-60, tel. 2\/893-3000). It's a three-act show with a mix of music and dance with salsa, music from the Caribbean, tango, and even some hip-hop added to the mix. Tickets can be obtained at Tu Boleta (www.tuboleta.com).\n\n###### **SALSA AL PARQUE**\n\nOn the last Saturday night of the month, go to **Salsa al Parque** (Parque de los Estudiantes), also called the Parque de Santa Librada or, most commonly, the **Parque de Jovita** (Cl. 5 at Cra. 22, fculturalatina@gmail.com, 4pm-midnight, free). This friendly and open-air freebie is known as an _audici\u00f3n,_ which is a chance for salsa enthusiasts called _coleccionistas,_ who collect salsa albums, to play their favorites for the crowd. Young and old alike, from all walks of life, gather at the park for these events, with the common denominator being a love of salsa. The event is organized by the Fundaci\u00f3n Cultural Nuestra Cosa Latina. They usually sell CDs to defray their costs. There's nothing like it!\n\n###### **SALSATECAS**\n\nNo matter where you go on a Saturday night, there's a good chance that you'll hear some salsa. But there are some places\u2014 _salsatecas_ \u2014where it's all about salsa and nothing more. Most are open from Wednesday until Sunday, closing at around 2am. Big _salsatecas,_ like Tin Tin Deo, will have a cover.\n\nSurrounding the Parque de la Alameda are a handful of _salsatecas,_ many of them quite old-school. These are excellent places to soak up the atmosphere, enjoy the music, nurse your drink, and watch the locals dance. **El Habanero Club** (Cl. 7A No. 23A-01, tel. 2\/557-5829, www.bolerohabanero.co, hours vary Wed.-Sat.) is one such place. At this fabulous small club specializing in _musica Antillana_ (music from the Caribbean), you may feel like you've stepped into 1940s Cuba when you open the door. International visitors are welcome here, but plan on staying awhile: the owners don't like it when gringos nurse one beer and leave. **Libaniel** (Cl. 7A No. 23-68, tel. 2\/557-5157), on the opposite corner from El Habanero, is one of those established clubs where the \"new\" waiters have only been working there for 10 years. On Fridays and Saturdays it's \"Salsa y m\u00e1s,\" and a good time to check it out. They even serve tamales to satisfy your midnight cravings. A third club on the park is another famous name, the **Port\u00f3n Caldense** (Cra. 23B No. 7-32, tel. 2\/557-7616, www.portoncaldense.com). It's larger than the other places previously mentioned, and similarly full of atmosphere. On Thursdays they switch gears with a tango night.\n\nThe **Casa Latina** (Cl. 7 No. 27-38, tel. 2\/556-6549, garylatina1@gmail.com, no cover) is a welcoming bar with tons of personality that draws in the salsa aficionados of Cali. Owner Gary Dom\u00ednguez has theme nights, usually on Saturdays, celebrating a star salsa performer. Salsa memorabilia cover the walls, and the DJ booth that Gary mans while drinking a beer is jam-packed with records. How he knows what is where is a mystery\u2014he must have a system. You may want to contact the bar in advance to see if they have any special events coming up.\n\nIn the San Fernando neighborhood along Calle 5 are some of the best-known big salsa clubs in Cali. **Tin Tin Deo** (Cl. 5 No. 38-71B, tel. 2\/514-1537, www.tintindeo.com, 7:30pm-1am Thurs., 7:30pm-3am Fri.-Sat., cover COP$10,000-COP$15,000) is a requirement on a Thursday night. At Tin Tin Deo they also dance _chichoky,_ which is a new style of Cali salsa that incorporates African rhythms. Saturday night is also big here, when the music is _pachanguero,_ which is sort of \"party music,\" Cali style. Tin Tin Deo is the new _chico_ on the block\u2014it's only been around since 1985, started by some friends from the Universidad del Valle. It's always a good bet.\n\n**Conga** (Cl. 5 No. 30-17, tel. 2\/556-5608, 9pm-3am Thurs.-Sat., cover COP$15,000) is more of an insider's place. Fridays and Saturdays are _viejoteca_ nights, which are oldies nights, with Sundays geared more toward younger folk. Finally **Zaperoco Bar** (Av. 5N No. 16-46, tel. 2\/661-2040, www.zaperocobar.com, 8pm-3am Thurs.-Sat., cover COP$20,000) confidently calls itself the best rumba in Cali. It regularly hosts live acts featuring salsa, music from the Pacific, and Cuban _son_ music. The long lineup of _orquestas_ (bands), is planned far in advance. The service isn't as friendly as at other bars.\n\n##### **Other Nightlife**\n\n**La Topa Tolondra** (Cl. 5 No. 13-27, cell tel. 314\/664-1470, 6pm-1am Wed.-Thurs., 6pm-3am Fri.-Sat.) is a cool little spot near La Loma de la Cruz, where you can have a beer with local bohemians. On the cusp between El Pe\u00f1on and San Antonio is the extremely mysterious and too cool **Greco Bar** (Cra. 4 No. 2-116, cell tel. 312\/854-0151, greco613-@hotmail.com, 5pm-midnight Wed.-Sat). It's in an old house, where the lighting is way dim\u2014if you don't watch your step you'll twist your ankle fumbling down the stairs. From the outside it looks sort of abandoned, but the hipsters shooting the breeze in the hammocks out front give it away. Overlooking the park in San Antonio is **Atahualpa** (Cra. 10 No. 1-15 Oeste, tel. 2\/893-7206, 1pm-midnight Thurs.-Sat.). On the first floor they sell handicrafts and on the terrace you can have a beer or two. This is a fantastic perch from which to watch San Antonio come alive as night falls. Meanwhile, the jet-setters of Cali hang out in El Pe\u00f1on on Avenida 2 Oeste, a strip of fashionable restaurant\/bars popular with the after-work crowd. Terrace seating is hard to come by on Thursday and Friday evenings.\n\n###### **ELECTRONIC**\n\nCali has a fairly heavy electronic music scene. Clubs are not huge draws in town, but parties or raves are. Usually between June and August each year, the **Black and White Sensation Party** (www.blackandwhitesensation.com), with a big-time international and national DJ lineup, has grown to become one of the most famous rave-type parties in the country. To find out about upcoming parties, visit www.tuboleta.com or consult the Cali section of www.planb.com.co. Also be on the lookout for flyers in trendy places in Granada.\n\nBig-time electronic music clubs, the places that stay open late, tend to congregate in the Yumbo and Menga areas to the north of Granada. **El\u00edptica** (Cra. 38 No. 7-161, cell tel. 311\/342-2294, 10pm-4am Thurs., 10pm-6am Fri.-Sat., COP$20,000) is an electronic music club that occasionally hosts international DJs.\n\n###### **LGBT**\n\nGay clubs in Cali are friendly and mixed with women and men. They tend to get hopping at around 11pm and stay open until 3am. **Queens Bar** (Av. 9A No. 15-07, tel. 2\/396-5338, www.queensbarcali.com, cover COP$15,000) is one of the hottest gay clubs of the moment, with three different dance floors each with its own type of music. It's located in Granada.\n\n**Lulu Latino** (Km. 2 V\u00eda Yumbo, cover COP$15,000) is the old standard as far as gay clubs go. There are often _barra libre_ (open bar) promotions. Go for the rum instead of poor quality vodka.\n\n###### **JAZZ AND ROCK**\n\nNear the R\u00edo Cali, **Duke's Bar** (Av. 4 Oeste No. 1-66, cell tel. 301\/418-6618, 6pm-1am Wed.-Thurs., 6pm-2am Fri.-Sat.) is the place for a change of rhythm. Here it's Louis Armstrong and John Coltrane who rule. Duke's also serves meals. For all your rock needs\u2014hard rock, glam rock, Spanish rock, you name it\u2014head to **Rock City Bar** (Cl. 5 No. 12-57, 4pm-3am Wed.-Sat., COP$6,000 cover weekends).\n\n###### **TANGO**\n\nTango lovers (of which there are many in Colombia) may feel outnumbered in this salsa town, but not at **La Matraca** (Cra. 11 No. 22-80, tel. 2\/885-7113, www.lamatracacali.com, 6pm-2:30am Fri.-Sat., 3pm-11pm Sun.). On the Parque Obrero, La Matraca used to be a corner shop where you could buy staples like rice and potatoes and hear tango from the owner's collection. No more potatoes here; it's just music, dancing, and drinks. It tends to be happening on Sundays after 3pm. This neighborhood is a little rough around the edges, so it's recommended to order a cab there and back.\n\n###### **LOUNGES**\n\nThe jet-set crowd of Cali hangs out in El Pe\u00f1on on Avenida 2 Oeste, a strip of fashionable restaurant\/bars popular with the after-work crowd. Terrace seating is hard to come by on Thursday and Friday evenings. In the same area but with a totally different, much more relaxed atmosphere is **Malec\u00f3n Cubar** (Cl. 1 Oeste No. 1-32, tel. 2\/892-2977, 3pm-2am Tues.-Sat.). Here you can groove to Caribbean sounds, drink a mojito or two, and have a meal. Drinks are a little pricey.\n\n##### **Cinema**\n\nAll the major malls have movie theaters. **Cine Colombia** operates several of those cinemas, including the one at Chipchape (Av. 6N No. 37N-25, tel. 2\/644-2463, www.cinecolombia.com). **Museo La Tertulia** (Cra. 1 Oeste No. 5-105, tel. 2\/893-2941, www.museolatertulia.com) shows artsy flicks. Alternative cultural space **Lugar a Dudas** (Cl. 15 N No. 8N-41, www.lugaradudas.org, tel. 2\/668-2335) also shows independent films, usually on Tuesdays and Saturdays at 7pm.\n\nThere are two beautiful historic theaters built in the early 20th century in the Centro: **Teatro Jorge Isaacs** (Cra. 3 No. 12-28, tel. 2\/880-9027) and the **Teatro Municipal Enrique Buenaventura** (Cra. 5 No. 6-64, tel. 2\/81-3131, www.teatromunicipal.gov.co.) Contact the theaters directly or visit www.tuboleta.com for information on upcoming performances.\n\n##### **Festivals and Events**\n\nThe **Festival Petronio \u00c1lvarez** (www.festivalpetronioalvarez.com, mid-Aug.) is a series of outdoor concerts that celebrates Afro-Colombian music and culture. The vibrations of the drums and good vibe of the crowd at this annual festival may intoxicate you! It's named for a famous musician from the Buenaventura area on the Pacific Coast.\n\nThe **Festival Mundial de Salsa** (www.cali.gov.co, mid-Sept.) began in 2006. It is a fiercely competitive dance competition, attracting thousands of salsa dancers of all ages from around the world. The finals are usually held in the Plaza de Toros. For tickets go to www.colboletos.com.\n\nCale\u00f1os love their _feria_ (fair). During the last week of the year when other cities become virtual ghost towns, the opposite occurs in Cali: It becomes Colombia's party central during the beloved **Feria de Cali** (www.feriadecali.com). Occurring between Christmas and New Year's, the _feria_ is a celebration that crosses barriers of class and age, with parades, concerts, beauty pageants, _cabalgatas_ (horseback processions), and plenty of dancing and drinking. In addition to official activities, parties and other events take over the entire city during this week.\n\n#### **SHOPPING**\n\nIf you are hunting for some Colombian handicrafts to take back home, try **La Cale\u00f1ita** (Cra. 24 No. 8-53, tel. 2\/556-1172, www.lacalenita.com, 9am-6pm Mon.-Sat.). This store is all of Colombia under one roof, and it has a range of merchandise, from high-quality jewelry and woven items to general Colombian kitsch.\n\nMalls and more malls: That's how Cali shops. There's open-air **Chipichape** (Cl. 38N 6N-35, tel. 2\/659-2199, www.chipichape.com.co, 6am-midnight daily), not far from Granada, as well as smaller **Centenario** (Av. 4N No. 7N-46, tel. 2\/683-9604, www.centenariocc.com, 8am-11:45pm daily), which has glorious air conditioning. To the south are newer malls like **Jard\u00edn Plaza** (Cra. 98 No. 16-200, tel. 2\/324-7222, www.jardinplaza.com, 8am-11:30pm daily), which is also open-air. It's near the Universidades MIO station. Each of these _centro comerciales_ (malls), or CCs, has all the big-name Colombian brands (for example, Velez shoes, La Riviera cosmetics, Totto backpacks, and Arturo Calle menswear) and expensive international stores such as Adidas, Esprit, and Lacoste. Each also has small handicrafts stores or kiosks. Food courts have a variety of options, including fast food places with fun names like Mr. Arepa and Patacontodo, serving up Colombian delights.\n\n#### **SPORTS AND RECREATION**\n\n##### **El Cerro de las Tres Cruces**\n\nA weekend ritual for many Cale\u00f1os is to hike up **El Cerro de las Tres Cruces** (Three Cross Hill), west of the Santa Monica neighborhood. The climb will get your blood pumping, and at the top and along the way, you'll have some good views of Cali, especially early in the day. The ascent will take about an hour. At the top, if you still feel energetic, you can join Cali's fit and fabulous as they work out in the makeshift outdoor gym next to the crosses. It's quite a scene up there. Bring a little cash with you to enjoy a freshly squeezed orange juice.\n\nmaking the hike to the top of El Cerro de las Tres Cruces\n\nThere is a story behind the hill's namesake crosses. According to legend, in 1837 two friars decided that they had had enough of the prostitution, plagues, fires, dengue fever, and famine in Cali, and placed the blame squarely on the Buziraco, a demon who, after having been expelled from Cartagena, made his way to Cali to this hilltop. The first cross that was set on the hill was destroyed by an earthquake (Buziraco's fault), so in 1938 it was decided to build three concrete crosses. They have withstood the test of time so far, and there have been no further reports of Buziraco's antics.\n\nThe hardest part about the walk is figuring out where to start it. Various paths lead to the top\u2014not far from Granada\u2014in the Altos de Normand\u00eda neighborhood or in Juanamb\u00fa. If you can find your way to Avenida 10 Oeste at Calle 12N, you will be close to the path and can ask anyone you come across for directions. If you're not going with someone who knows how to get there, take a cab and request to be dropped off close to the _sendero al Cerro de las Tres Cruces_ (path to Three Crosses Hill).\n\nIt's recommended to take this popular hike on weekend mornings only, when you are assured of being in good company (hundreds of others), although there is usually always a police presence along the well-marked and well-trodden path. Parts of the path are quite steep, and you may need to climb up on all fours at some points. Therefore, don't bring items that you don't need, so you can have your hands free. And bring water.\n\n##### **Tours**\n\n**TOVYT Tours** (Av. 8N No. 15A Norte-40, tel. 2\/370-6800, www.tovyt.com), with an office in Granada, can organize private tours and outings if you have a minimum of four people in your group. These include trips to the Hacienda El Para\u00edso, bird-watching tours to the Kilometer 18 area west of the city, and excursions to San Cipriano.\n\n**The Little Witches of San Cipriano**\n\nA popular day trip from Cali is to take a ride on the _brujitas,_ literally \"little witches,\" of San Cipriano. Train tracks run through the two villages of C\u00f3rdoba and San Cipriano, although trains rarely pass by. Local entrepreneurs saw an opportunity here in providing a quick transportation alternative to walking the train tracks for residents to get from one village to the other. They created a wooden cart transportation system that was set upon the rails. It was originally propelled manually using long sticks. The drivers resembled witches on broomsticks flying by, hence the name _brujitas._ Nowadays, passengers\u2014up to about 10\u2014zip by, as the _brujitas_ are powered by motorbikes (less charming but more adventurous). The teenage drivers like to go fast, so hold on and, if you see a train coming towards you, get ready to jump off. It's about a 30-minute journey from C\u00f3rdoba to San Cipriano.\n\nUpon arrival in San Cipriano, you can have a hearty seafood lunch or continue on just beyond the village to a protected area (admission COP$2,000), where you can wander down a path that leads to a refreshing swimming hole in the R\u00edo San Cipriano. Inner tubes can be rented, and you can opt to float back to C\u00f3rdoba.\n\nA spin on the _brujita_ costs about COP$8,500 round-trip for tourists, less for locals. However, touts may try to charge you up to COP$80,000 for the ride. Pay for the trip only after you have returned to C\u00f3rdoba.\n\nThe experience is best enjoyed in a group. These are often organized by hostels in Cali. If you would like to go on your own, though, you can take any bus bound for Buenaventura and ask the driver to let you off at C\u00f3rdoba. Buses depart Cali from the Terminal para Buenaventura in the southwest of the city. Look for the metallic black sculpture of the _mariamulata_ bird (the local name for a great-tailed grackle) by renowned artist Enrique Grau (Cl. 7 Oeste No. 3-03). The ride costs about COP$18,000. Be sure to make a pineapple pit stop at **Pi\u00f1as del 44.** It's at kilometer 44 on the highway, and buses will often take a break here.\n\nSan Cipriano isn't the only place in Colombia with _brujitas._ Visitors to the Coffee Region can take a ride in the village of Estaci\u00f3n Pereira along the R\u00edo Cauca near Pereira.\n\n**Rioja Travel** (tel. 2\/660-7092, www.riojatravel.com) offers many of the same day trips as TOVYT Tours. The agency can arrange an outing for a minimum of two people. A day trip of six hours to Lago Calima costs COP$206,000 per person; a horseback tour along the R\u00edo Pance costs COP$84,000 per person.\n\n**FHURE Travel** (Cra. 94A No. 45-90, tel. 2\/372-4092, www.fhuretravel.com) offers day trips to the Parque Nacional Natural Los Farallones near the town of Pance in the south of the city. This excursion costs about COP$56,000 per person.\n\n##### **Bird-Watching**\n\nColombia is home to over 1,900 species of birds, 76 of which are endemic. To get a glimpse of this celebrated diversity in the Cali area, check out a bird-watching excursion offered by the organization **Mapalina** (cell tel. 318\/627-7062 or 316\/805-2117, www.mapalina.com, from US$100). Associated with the American Birding Association, Mapalina is a birding organization led by the nonprofit Asociaci\u00f3n R\u00edo Cali, and it organizes birding field trips year-round to a half dozen locations near Cali, from tropical San Cipriano to the west to the Laguna Sonso in the northeast near Buga. Some of the birds you might see include the multicolored tanager, gold-ringed tanager, banded ground-cuckoo, long-wattled umbrellabird, and the unusual Andean cock-of-the-rock. Trips, almost always with an English-speaking guide, are tailor-made according to the wishes of the visitors, and can be arranged for groups of 1-10 persons. From October to April there are usually more birds in the area, as many migratory species arrive from the north. Each year in December the organization participates in a day-long bird census in a cloud forest area near Kilometer 18, west of Cali, in collaboration with the Red Nacional de Observadores de Aves de Colombia bird-watching network. Anyone is invited to participate in the census.\n\n##### **Gyms, Spas, and Pools**\n\nFor a day of health and pampering, head to the **Intercontinental Cali** (Av. Colombia No. 2-72, tel. 2\/882-3225, COP$50,000 day pass), where you can work out at the hotel's gym, splash about in the big pool, and rejuvenate at the spa. If you're in San Antonio and would like to spin, go to the minuscule **Centro de Acondicionamiento F\u00edsico Fernando Ben\u00edtez** (Cl. 3 No. 5-22, cell tel. 316\/281-4143, www.acondicionamientofisicofb.com, 6am-8pm Mon.-Sat., COP$15,000 day pass), which allows visitors to drop in and take a spinning class.\n\n**Acuaparque de la Ca\u00f1a** (Cra. 8 No. 39-01, tel. 2\/438-4812, www.acuaparquecali.com, 9am-6pm daily, COP$12,500 adults, COP$10,000 children) is an insanely popular place to get wet on the weekends. It's quieter during the week. It's not really a place for serious training, more of a place to cool off and people-watch. It's also got several big water flume rides. You can take MIO buses P52C, P40A, or P24B to the park.\n\n##### **Biking**\n\nSunday is bike day. In Cali this recreation initiative is called **Ciclovida** (8am-1pm Sun.). Vehicular traffic is closed for some 30 kilometers with a main route extending from Calle 9 at Carretera 66 Sur to Calle 70 at Carretera 1N. It's next to impossible to find bikes to rent, but you can always jog!\n\n##### **Spectator Sports**\n\nThere are two professional soccer teams in Cali: **Deportivo** (www.deportivocali.com.co) and **Am\u00e9rica** (www.america.com.co). Deportivo, whose colors are green and white, is one of the most successful teams in the country. It is also the only team with its own stadium, the **Estadio Deportivo Cali** (Km. 8 V\u00eda Cali-Palmira, tel. 2\/688-0808). It's the largest soccer stadium in the country, with a capacity of over 50,000.\n\nAm\u00e9rica was linked to the Cali Cartel through the now extradited brothers, who were affiliated with the club for some 15 years, Gilberto and Miguel Rodr\u00edguez Orejuela. The fact that their mascot is a red devil with pitchfork in hand is coincidental, although fits with Am\u00e9rica's reputation! Am\u00e9rica plays at the **Pascual Guerrero Stadium** (Cra. 36 No. 5B-32, tel. 2\/556-6678), which is where the 1971 Panamerican Games were held. It got a big facelift just in time for the 2011 Under-20 World Cup.\n\nThe big match in town is the Clasico del Valle de Cauca between Deportivo and Am\u00e9rica. Although Deportivo leads that series, Am\u00e9rica holds the most national titles, with 13, while Deportivo has 8. Go to www.tuboleta.com for tickets and information on soccer matches.\n\n#### **ACCOMMODATIONS**\n\nGranada, El Pe\u00f1on, and San Antonio are considered the best places to stay in Cali. These neighborhoods are safe, walkable, and offer diverse dining options. There are numerous hotels in the Centro within a few blocks of all the main tourist sights, but it's not a desirable place to be at night. The top-end hotels all provide air conditioning in their rooms; budget accommodations probably will not.\n\n##### **Granada**\n\nLeafy streets and proximity to restaurants and nightspots make the trendy barrio (neighborhood) of Granada a nice place to stay while in Cali. There are options for all budgets and styles. New sidewalks, refurbished streets, and speed humps began to appear in much of the area in 2013, making the area more pedestrian-friendly. The area is fine to walk about in the evening hours, but on weekday nights it feels deserted.\n\n###### **UNDER COP$70,000**\n\nThe hostel scene in Granada offers several options, and Granada runs neck and neck with San Antonio for backpacker business.\n\nAfter a few days at the M **Iguana Hostel** (Av.\/Cl. 9N No. 22N-22, tel. 2\/660-8937, www.iguana.com.co, COP$18,000 dorm, COP$45,000 d w\/bath) you'll think of the staff as new friends, such is the hospitality at this long-running hostel. Free salsa classes take place just about every evening, and the staff are full of nuggets of information, from ideas for day-trip excursions, to where to go to hear music on a Monday night, to what pizza place delivers late. The hostel is located on a steep, quiet street within easy walking distance to restaurants. There are a couple of patios to hang out in, and a fierce black and white cat guards the front gate at night. Iguana is one of the oldest hostels in Cali, but the accommodations are still fresh.\n\n**Pelican Larry's** (Cl. 20N No. 6AN-44, tel. 2\/420-3955, www.hostelpelicanlarry.com, COP$18,000 dorms, COP$50,000 d) has dorm rooms, private rooms with shared bath, and one private room with its own bathroom. This crash pad is close to the action of Calle Sexta.\n\n###### **COP$70,000-200,000**\n\nFrom the terrace on the 10th floor at **Aqua Hotel** (Av. 8N No. 10-91, tel. 2\/667-2388, www.aquagranada.com, COP$120,000 d), you get a great view. On one side, the Iglesia Ermita peeks through two nondescript towers. You can also see the Torre de Cali skyscraper. On the other side look for Tres Cruces, Granada, and the shenanigans taking place atop the NOW Hotel. Breakfast is served in your spacious room, which has a functional kitchenette and enormous refrigerator. Aqua is on the always-busy Avenida Octava, but traffic noise does not seem to be an issue.\n\nGranada is full of restaurants and watering holes, but the Calle Novena is not a main thoroughfare, making it a pleasant and mostly peaceful place to stay. The **Hotel Plaza Mayor** (Av. 9AN No. 14-70, tel. 2\/667-0303, www.hotelplazamayorcali.com, COP$140,000 d low season, COP$175,000 d high season) is your basic nothing-special hotel, but it's a good value for Granada. Many of the rooms have kitchenettes.\n\n###### **OVER COP$200,000**\n\nOpened in January 2014, **Marriott Cali** (Av. 8N 9-64, tel. 1\/485-1111, www.marriott.com) is the latest luxury international hotel chain to arrive in Cali, on a busy thoroughfare near the Granada neighborhood. It offers 170 rooms, a pool, restaurants, and a gym.\n\nCatering mostly to the Bogot\u00e1 jet-set, **NOW Hotel** (Av. 9AN No. 10N-74, tel. 2\/488-9797, www.nowhotel.com.co, COP$297,000 d) is an island of South Beach in the middle of Granada. It has 19 luxurious rooms with interiors decorated by Bo Concept. Large beds, balconies, and DirecTV are some of the perks here. There are two restaurants in the hotel, and there are certainly worse places to be at sunset than the rooftop terrace bar. The hotel bar is a popular place for a drink, and sometimes there are parties here, especially on weekends. If you need peace and quiet for a good night's sleep, this isn't the best option.\n\nThe Torre de Cali, the tallest skyscraper in the city, is a classic point of reference, and is home to the **Hotel Plaza Torre de Cali** (Av. de las Am\u00e9ricas No. 18N-26, tel. 2\/683-3535, www.hoteltorredecali.com, COP$246,000 d), with rooms on floors 11-20. The views from the southeast side of the hotel towards the valley are unbeatable. The hotel has spruced itself up, hoping to compete with the top hotels in the city. It is in the Versailles area, about a 15-minute walk to the Zona Rosa area of Granada. Avoid walking around this part of town late at night.\n\n##### **El Pe\u00f1on**\n\nThis tidy, upscale, and somewhat boring neighborhood is home to upscale restaurants and hotels. It is sort of an island between San Antonio and (across the R\u00edo Cali) Centenario and Granada. Unfortunately, crossing the river is a frightful experience, with pedestrian crosswalks nonexistent; you'll probably want to take a cab. To hike up the hill to San Antonio, you will also have to scamper across the Carrera 4, where cars zoom by at high speeds.\n\n###### **COP$70,000-200,000**\n\nIf you can name your favorite salsa singer with no hesitation, the place for you might be the **Posada Salsa Boutique** (Cl. 4 Oeste No. 3A-39, tel. 2\/376-0866, www.posadasalsa.com, COP$70,000 s, COP$110,000 d). Each of the six small rooms in this hotel that opened in 2010 has a different theme in homage to the great salsa stars, like the Cecilia Cruz room and the H\u00e9ctor Lavoe room. You can request your favorite one. Salsa classes are offered, as you'd expect, but the best part about this hotel is its location on a quiet, tree-lined street.\n\nAlso on Calle 4 Oeste, **Aloja** (Cl. 4 Oeste No. 3A-50, tel. 2\/557-0933, www.aloja.com.co, COP$120,000 apt.) has only three furnished apartments for rent, but if one is available, it's a great bargain. The three-story building is above an excellent ice cream shop, which sweetens the deal, and is a two-minute walk from a variety of neighborhood restaurants. Apartments are equipped with everything you'll probably need and come with a high-tech security system.\n\n**El Pe\u00f1on Hotel** (Cl. 1 Oeste No. 2-61, tel. 2\/893-3625, hotelelpenon@hotmail.com, COP$156,000 d) offers rooms large enough for two people to do jumping jacks in and is reasonably priced, especially for this part of town. Rooms are carpeted.\n\n###### **OVER COP$200,000**\n\nMaybe it doesn't have the same cachet as it once did during its 60-odd years in Cali, but the M **Intercontinental Cali** (Av. Colombia No. 2-72, tel. 2\/882-3225, www.ihg.com, COP$385,000 d), run by the Colombian Estelar group, retains its old-school elegance with all the amenities you'd expect: huge comfortable beds, a spa, multiple restaurants and bars, and a check-in counter that must be 20 meters long. You can walk from the Intercontinental Cali to restaurants in El Pe\u00f1on, but most guests prefer to cab it.\n\n##### **San Antonio**\n\nThis neighborhood of artists and hipsters has options for all budgets, with tons of hotel and restaurant options, and also has that charm you might be seeking. All the action is centered on the Parque San Antonio, which is one of the best places to be when the sun sets. You can walk from San Antonio to the historic center in the daytime, about a 20-minute walk, if you're up for it.\n\n###### **UNDER COP$70,000**\n\nIn an old house with lots of charm, M **La Casa Caf\u00e9** (Cra. 6 No. 2-13, tel. 2\/893-7011, www.lacasacafecali.blogspot.com, COP$15,000 dorm, COP$30,000 d) is a friendly spot with an ideal location on a quiet street in San Antonio. Upstairs there are four rooms\u2014one dormitory and three private rooms with shared bathrooms\u2014and there is a small kitchen for guest use. Downstairs is a caf\u00e9 and reading area where local bands play sometimes.\n\nThe Uruguayan hostel chain **El Viajero** (Cra. 5 No. 4-56, tel. 2\/893-8342, www.hostelsincali.com, COP$20,000 dorm, COP$37,000 d) arrived in Cali in mid-2013. This location offers the backpacking crowd a pool, a bar, free salsa classes in their dance studio, a basic breakfast, and the occasional circus performance.\n\n###### **COP$70,000-200,000**\n\nThe M **San Antonio Hotel Boutique** (Cra. 6 No. 2-51, tel. 2\/524-6364, www.hotelboutiquesanantonio.com, COP$179,000 d) is the best option in San Antonio if personalized attention and comfort are your priorities. Two of the hotel's 10 rooms have their own terrace, but anyone can enjoy the rooftop terrace. Rooms have air conditioning and comfortable beds. Guests can make free calls to the United States and Canada.\n\nOpen since October 2012, comfy M **Ruta Sur** (Cra. 9 No. 2-41, tel. 2\/893-6946, www.hostalrutasurcali.blogspot.com, COP$25,000 dorm, COP$85,000 private d) offers two small dormitory rooms with triple bunkbeds (!), along with four rather luxurious private rooms (with bathrooms) where weary flashpackers (backpackers who like their comfort every once in a while) can get a sound rest 1after days on the road. At night you can laze in a hammock under the stars in an open-air interior patio. Here there are no salsa classes or bar tours on offer: It's all about comfort and rest.\n\nSan Antonio is Cali's most charming neighborhood.\n\n#### **FOOD**\n\nCali lacks the international cuisine options that are found in Bogot\u00e1, but a handful of creative fusion restaurants help to fill in that void. Cali society frequents the most highly regarded restaurants in the posh neighborhoods of El Pe\u00f1on and Granada. Prices for a night out are on the uptick in San Antonio, as yuppies and visitors alike crowd restaurants after soaking in the scene at the park. It's rather lonesome after dark in the Centro, but there are numerous options for a typical _valledecaucana_ lunch following a morning hitting the pavement as you visit Cali's historical sights. Nothing accompanies a hearty meal in the hot city like a cool _lulada_ drink made from the _lulo_ fruit (a variety of orange). Tap water is fine to drink in Cali.\n\n##### **Centro**\n\n###### **CAF\u00c9S, BAKERIES, AND QUICK BITES**\n\nVegetarians may be surprised to stumble across vegetarian options downtown. For a pastry and a coffee, or for a vegetarian buffet lunch, check out **Nutricentro** (Cra. 5 No. 7-40, tel. 2\/895-9777, 10am-6pm Mon.-Sat.). It's a cheerful place near the Merced sights.\n\n###### **COLOMBIAN AND FUSION**\n\nReasonable lunch options abound in the Centro. On weekends and evenings, options are fewer and less popular.\n\nJust across the street from Iglesia La Merced is the colonial building that houses the Sociedad de Mejoras\u00fablicas de Cali, an organization in support of public works. Discreetly facing an interior patio is M **Mi Valle del Cauca** (Sociedad de Mejoras P\u00fablicas, Cra. 4 No. 6-76, tel. 2\/883-6309, 11am-3pm Mon.-Fri., set lunch COP$12,000), an excellent lunch spot. It's popular with professionals on their lunch break. Try the tortilla soup and fish dishes.\n\nThere's no need to feel embarrassed about dining in a parking lot. Despite its less-than-picturesque location, **DA'gusto** (Cra. 4 No. 9-49, tel. 2\/881-8697, 6am-5pm Mon.-Sun., COP$8,000) packs in the crowds day in and day out at lunchtime. Service is quick and the action in the kitchen is fast, yet friendly. _Sancocho,_ a meaty stew, is a favorite, as is fried fish. Set lunches are served with juice, salad, and lots of carbs: potatoes, yuca, plantains, and rice. You can also order dishes such as beans and rice off the menu.\n\nAt the lunch counter of tiny and tidy **Rikus** (Cra. 4 No. 12-56, tel. 2\/896-0795, 10am-5pm Mon.-Sat., COP$6,000) you can eat cheap Colombian fare and watch the staff furiously stir, mix, pour, and serve.\n\n##### **Granada**\n\n###### **CAF\u00c9S, BAKERIES, AND QUICK BITES**\n\nA top contender in the city for the crown of best _pandebono,_ a delicious pastry made of yuca flour and cheese, is **Kuty Panader\u00eda** (Av. 6N No. 27N-03, tel. 2\/661-1465, www.panaderiakuty.com, 6am-9pm daily). Kuty also serves other fast food fare all day long. For heaps of old-style character, nothing beats M **Miami** (Av. 6 No. 16N-98, tel. 2\/396-5998, 7am-8pm daily). Here, right on Avenida Sexta, you can fuel up on freshly baked croissants and a _caf\u00e9 con leche_ (coffee with milk) and be endlessly entertained by the hustle and bustle of one of Cali's busiest streets. Miami even has Wi-Fi.\n\nIf you're looking to get your coffee _fuerte_ (strong), go to **Juan Valdez** (Av. 9N No. 17-11, tel. 2\/660-7337, www.juanvaldezcoffee.com, 10am-9pm Mon.-Thurs., 10am-midnight Fri.-Sat., 11am-9pm Sun.). This shop, the Colombian version of Starbucks, is on a fancy corner in Granada and has a wonderful terrace under the shade of acacia trees, providing the perfect setting for writing in that journal of yours or browsing on your tablet.\n\n###### **COLOMBIAN AND FUSION**\n\n**Ringlete** (Cl. 15AN No. 9N-31, tel. 2\/660-1540, www.ringlete.com, noon-3pm and 6:30pm-10pm Mon.-Sat., noon-4:30pm Sun., COP$24,000) features _nueva cocina Vallecaucana_ (new Valle de Cauca cuisine) in a brightly decorated restaurant in Granada. Check out the pork chops, shrimp ceviche, or seafood _cazuela_ (stew).\n\nM **Carambolo** (Cl. 14N No. 9N-18, tel. 2\/667-5656, noon-midnight Mon.-Sat., noon-5pm Sun., COP$30,000) has a creative menu combining Mediterranean and Colombian flavors. Many dishes in this restaurant are whimsically named, such as the Shakira (stuffed eggplant combining Middle Eastern and Caribbean flavors) and the Celia Cruz\u2014shrimp coated in coconut and _chontaduro_ (fruit from a type of palm tree) and served in a _maracuya_ (passion fruit) sauce. It's a popular place with Cali society and out-of-towners.\n\n**Platillos Voladores** (Av. 3N No. 7-19, tel. 2\/668-7750, www.platillosvoladores.com, COP$35,000, noon-3pm and 7pm-11pm Mon.-Sat.), run by chef Vicky Acosta, is consistently rated as one of Cali's top restaurants. Fusion is the watchword here: Thailand, Lebanon, Italy, France, and the Colombian Pacific are among the cuisines that Acosta works with. Menu favorites include a fresh fillet of fish from the Pacific in a caramelized _chontaduro_ (a starchy fruit that grows on a variety of palm) and garlic sauce (COP$43,000), quinoa stir-fry (COP$26,000), and an exotic and spicy ostrich carpaccio (COP$23,000). Reservations are recommended.\n\n###### **TEX-MEX**\n\n**D'Toluca** (Cl. 17N No. 8N-46, tel. 2\/668-9372, 11:30am-3pm and 5:30pm-11:30pm Mon.-Thurs. and Sun., 11:30am-3pm and 5:30pm-1am Fri.-Sat., COP$15,000) serves up reliable Tex-Mex cuisine and often has lunch or drink specials.\n\n###### **CR\u00caPES**\n\nAlthough it's a chain present in all big malls in Colombia, **Crepes & Waffles** (Av. 6AN No. 24N-70, tel. 2\/485-4474, www.crepesywaffles.com.co, noon-10pm Sun.-Thurs., noon-11pm Fri.-Sat., COP$20,000) is a reliable friend. There are dozens of savory cr\u00eapes, all consistently good. What's important is to save room for dessert. This particular location on the Avenida Sexta is big and breezy and is between Granada and Chipichape.\n\ncaf\u00e9 in Granada\n\n##### **El Pe\u00f1on**\n\n###### **CAF\u00c9S, BAKERIES, AND QUICK BITES**\n\nDelicious homemade ice cream awaits you at **Calathea** (Cl. 4 Oeste No. 3A-50, tel. 2\/371-0188, 11am-7pm Mon.-Sat., noon-6pm Sun.). Exotic flavors you can't get back home are constantly being invented. There's coconut-lemon, strawberries with red wine, and the native fruit _araz\u00e1_ with mint.\n\n###### **COLOMBIAN AND FUSION**\n\nOpen-air M **Panero** (Cl. 3 Oeste No. 3A-18, tel. 2\/892-3333, 9:30am-7pm Mon.-Sat., COP$12,000) serves delicious vegetarian lunches from a set menu. If you miss lunch, grab a veggie burger or try one of the many tempting desserts. Panero is located on a sublime tree-lined street and its terrace is an agreeable place to be for lunch or for a late afternoon tea.\n\n###### **INTERNATIONAL**\n\nAlain from Alsace came to Cali, loved it, and opened **Petite France** (Cra. 3A Oeste No. 3-53, tel. 2\/893-3079, noon-2:30pm and 6:30pm-10:30pm Mon.-Sat., COP$24,000). This French restaurant in El Pe\u00f1on is highly recommended by locals. Save room for the delicious tarte flamb\u00e9.\n\nThe terrace at **La Pizzeria** (Intercontinental Hotel, Av. Colombia No. 2-72, tel. 2\/886-1010, noon-11:30pm Mon.-Sun., COP$25,000) at the Intercontinental Hotel is almost always full at night, which is a good sign. The pizza is deemed by many to be the best in town. If you want to enjoy your pie and the pleasing temps of a Cali evening by dining alfresco, get there early.\n\n##### **San Antonio**\n\n###### **CAF\u00c9S, BAKERIES, AND QUICK BITES**\n\nWeary travelers and San Antonians alike flock to M **Macondo Postres y Caf\u00e9** (Cra. 6 No. 3-03, tel. 2\/893-1570, www.macondocafe.blogspot.com, 11:30am-11pm Mon.-Thurs., 11:30am-midnight Fri.-Sat., 4:30pm-11pm Sun.) at all hours of the day. It's one of the best places to hang out in San Antonio when you're in no rush to run to the Centro. If you're in the mood for an artisan beer or a decent coffee while you depart to Wi-Fi land, or you want a hearty meal, including the much sought-after veggie burger, Macondo is your neighborhood place. At night they show artsy movies, and there are often live jazz and blues bands performing on Saturday evenings.\n\n###### **COLOMBIAN AND FUSION**\n\nIn a city with a surprising number of vegetarian restaurants, or at least veg-friendly options, M **El Buen Alimento** (Cra. 2 No. 4-53, tel. 2\/375-5738, 11:30am-10pm Tues.-Thurs., 11:30am-11pm Fri.-Sat., 11:30am-9pm Sun., COP$15,000) gives Panero in El Pe\u00f1on a run for its money for the title of veggie champion. There's always a set menu option for lunch, including fresh juice, soup, and the main course, but you can also order \u00e0 la carte. Veggie burgers, pastas, and vegetarian tamales\u2014an unimaginable option for most Colombians\u2014are plentiful. This cheerful spot in San Antonio with bright decor is even popular with devout meat-eaters.\n\nThe inspiration for **Ojo de Perro Azul** (Cra. 9 No. 1-27, tel. 2\/893-6956, 5pm-10pm Tues.-Thurs., 5pm-2am Fri.-Sat., 3pm-9pm Sun., COP$15,000) was a street dog named Juancho, who had one blue eye. Ojo de Perro Azul is a laid-back place that oozes San Antonio style. The food is OK here, but the ambiance is what makes it enjoyable. It's a good place to go later in the evening, have a nibble, and have cocktails while you enjoy the music. If you're asking yourself, \"Shouldn't it be Perro de Ojo Azul?,\" you'd be correct. The owners have a sense of humor. Sticking with the color theme, there is M **El Pargo Rojo** (Cr. 9 No. 2-09, tel. 2\/893-6087, 8am-3pm Mon.-Sun., COP$18,000), or The Red Snapper. Fresh fried sea bass and _cazuelas_ (seafood stews) from the waters of the Pacific are specialties at this San Antonio favorite. In the morning you can try an _arepa de huevo_ (egg fried in corn meal), if you are feeling low on cholesterol. El Pargo Rojo is not open for dinner.\n\nGoing back to blue, **Azul** (Cra. 9 No. 4-02B, tel. 2\/893-6057, noon-3pm and 6pm-11pm Mon.-Fri., 6pm-11pm Sat., COP$25,000) is a fusion-style restaurant, with Mediterranean, Colombian, Middle Eastern, and Asian flavors represented on the menu by chef Martha Izquierdo. Ask for the _clandestinos_ \u2014dishes that don't appear on the menu. With colorful decor, Azul is a good choice for dinner.\n\n#### **INFORMATION AND SERVICES**\n\n##### **Tourist Information**\n\nThe **Oficina de Turismo Municipal de Cali** (Cra. 5 No. 6-05, Oficina 102, tel. 2\/885-8855, ext. 122, 8am-noon and 1:30pm-5pm Mon.-Sat.) has maps and brochures on Cali and environs. Friendly young police cadets work the booth, but their English skills are minimal.\n\nThe online entertainment and events guide **Plan B** (www.planb.com.co) has a section on Cali. Ticket companies **Tu Boleta** (www.tuboleta.com) and **Colboletos** (www.colboletos.com) are also good resources.\n\n##### **Money**\n\nATMs are not as easy to come by in San Antonio compared to other neighborhoods in the city, but you can always count on them in shopping malls. To receive a wire transfer, **Western Union** (www.westernunion.com) has several offices in Cali. Check the website for locations. **Titan Intercontinental** (Cl. 11 No. 4-48, tel. 2\/898-0898, www.titan.com.co, 8am-5pm Mon.-Fri., 9am-1pm Sat.), a currency exchange office, is located in the Centro. Banks do not normally change money.\n\n##### **Spanish-Language Classes**\n\nAll the major universities in Cali offer Spanish programs for foreigners. Check the **Universidad Javeriana Programa de Espa\u00f1ol Funcional para Extranjeros** (Cl. 18 No. 118-250, tel. 2\/321-8200, ext. 510, www.javierianacali.edu.co) and **Universidad Santiago de Cali Programa de Lenguas Extranjeras** (Cl. 5 at Cra. 62, tel. 2\/518-3000, ext. 411, www.usc.edu.co\/idiomas).\n\n##### **Communications**\n\n###### **INTERNET ACCESS**\n\nWireless Internet availability is more the norm than the exception at restaurants, caf\u00e9s, and big shopping malls. **Juan Valdez** (Av. 9N No. 17-11, tel. 2\/660-7337, www.juanvaldezcoffee.com, 10am-9pm Mon.-Thurs., 10am-midnight Fri.-Sat., 11am-9pm Sun.) in Granada is a good place to get connected. Small Internet caf\u00e9s, open until about 8pm, are plentiful too. There you can surf the Internet for hours and pay next to nothing.\n\n###### **MAIL SERVICES**\n\nThe post office, **4-72,** has an office at the Chipichape shopping center (Av. 6 No. 35-47, Bodega 4, Local 426, tel. 2\/379-7164, www.4-72.com.co, 9am-1pm and 2pm-6pm Mon.-Sat.). Other courier services such as **Servientrega** (Av. 8N No. 14N-10, Local 3, tel. 2\/660-3384, 8:30am-12:30pm and 1:30pm-6pm Mon.-Sat.) have many locations in the city.\n\n###### **TELEPHONE**\n\nThe city telephone code for Cali is 2, but you'll only need to use it if you're calling Cali from a different part of the country, or from abroad. It's generally easy to find people selling use of their cell phones, called _minutos_ (minutes), for cheap on the street downtown and in _tiendas_ (stores) elsewhere. Cell phone numbers must have ten digits. To report any emergency, dial 123.\n\n###### **NEWSPAPERS**\n\nThe main daily newspaper in town is **_El Pa\u00eds_** (www.elpais.com.co), although **_El Tiempo_** is also frequently available in drugstores, bookstores, and malls. The free monthly **_Cali Cultural_** (www.calicultural.net) has an extensive listing of cultural events in the city.\n\n##### **Health Services**\n\nCali has excellent medical facilities. The **Fundaci\u00f3n Valle del Lili** (Av. Sim\u00f3n Bol\u00edvar Cra. 98 No. 18-49, tel. 2\/331-9090, appointment hotline tel. 2\/680-5757, www.valledellili.org) and the **Centro M\u00e9dico Imbanaco** (Cra. 38A No. 5A-100, tel. 2\/682-1000, appointment hotline tel. 2\/685-1000, www.imbanaco.com) are two of the top hospitals in the country.\n\nA recommended dental office is **Orthofami** (Cl. 19 No. 4-55, tel. 2\/373-4447, www.orthofami.com, 8am-6pm Mon.-Fri., 9am-1pm Sat.). The **PROFAMILIA Clinic** (Cl. 23N No. 3N-40, tel. 2\/661-8032, 7am-5pm Mon.-Fri., 8am-noon Sat.), in the Versailles neighborhood, offers a wide range of sexual and reproductive health services and products at low cost for women and men.\n\n#### **GETTING THERE**\n\n##### **By Air**\n\nThe Cali airport, the **Aeropuerto Internacional Alfonso Bonilla Aragon** (Palmira, tel. 2\/280-1515, www.aerocali.com.co) is under an hour's drive away. A taxi ride from San Antonio to the airport will cost around COP$50,000. Minibuses from the airport to the city are usually at the ready just outside of the departure hall. They cost about COP$12,000 and take you to the Terminal de Transportes de Cali, the Cali bus station (Cl. 30N 2AN-29).\n\nThere is free wireless Internet in some areas of the airport departure hall. In the terminal, restaurants serve the usual breakfast fare; you can also get refreshed before your flight with your last _lulada_ (a drink made with the juice of a _lulo_ , a type of orange).\n\nAll major Colombian airlines and some international carriers serve Cali. **Avianca** (Av. 6A No. 31N-11, Local 3, tel. 2\/660-7028, www.avianca.com, 8am-6pm Mon.-Fri., 9am-1pm Sat.) has numerous flights between Cali, Bogot\u00e1, and Medell\u00edn throughout the day. It also operates nonstop flights to Cartagena, Barranquilla, Pasto, and Tumaco. Internationally, Avianca flies nonstop between Cali and Madrid and Cali and Miami, and through Medell\u00edn to New York's JFK airport.\n\nOwned by the military, **Satena** (Centro Comercial Paseo de la Quinta, Cl. 5 No. 46-83, Local 213, tel. 2\/554-6919, www.satena.com) serves more exotic destinations, such as Guapi, the gateway to the island of Gorgona; Puerto Asis, bordering Ecuador in the Putumayo department; and Quibd\u00f3, the capital city of the Choc\u00f3 department.\n\n**Copa Airlines** (Centro Comercial Chipichape, Cl. 3N No. 6N-35, Local 318, Col. toll-free tel. 01\/800-011-2600, www.copaair.com, 8am-7pm Mon.-Fri., 8am-5pm Sat., 9am-4pm Sun.) flies from Cali to Panama City, Panama, as well as to Bogot\u00e1 and to the island of San Andr\u00e9s in the Caribbean. **LAN Colombia** (Cl. 25N No. 6B-36, Col. toll-free tel. 01\/800-094-9490, www.lan.com, 8am-6pm Mon.-Fri., 9am-1pm Sat.) flies nonstop to Bogot\u00e1 and Quito, Ecuador. Discount airliner **Viva Colombia** (Cali call center tel. 2\/380-8989, www.vivacolombia.co) offers daily flights between Cali and Bogot\u00e1, Medell\u00edn, Cartagena, and Santa Marta. **American Airlines** (Intercontinental Cali Hotel, Av. Colombia No. 2-72, Local 6, tel. 9\/800-052-2555, www.aa.com, 8am-6pm Mon.-Fri., 9am-1pm Sat.) is the only U.S.-based airline with service to Cali. They operate daily flights out of Miami. Charter flights are offered on **TAC** (Centro Comercial Colon Plaza, Cra. 1 No. 61A-30, Local 6, tel. 2\/439-4084) from Cali to Guapi, El Charco, and Timbiqu\u00ed.\n\nEcuadorean carrier **TAME** (Cra. 4A 12-41, Oficina 118, Edificio Seguros Bol\u00edvar, tel. 2\/888-9101, www.tame.com.ec, 8am-noon and 2pm-6pm Mon.-Fri., 9am-noon Sat.-Sun.) flies between popular vacation spot Esmeraldas on the Pacific Coast and Cali. **Aerogal** (www.aerogal.com.co) flies to Guayaquil.\n\n##### **By Bus**\n\nThe organized and bustling **Terminal de Transportes** (Cl. 30N No. 2AN-29, tel. 2\/668-3655, www.terminalcali.com) in Cali is a relatively quick taxi ride away from Granada. A small information booth is located in the center of it all. Attendants will be able to give you a rough idea of bus fare prices, help you organize day-trip travel, and even provide you with a Cali tourist map if you are just arriving. There is an efficient taxi stand at the main entrance (Puerta 3). If you are taking MIO to the terminal, the nearest station is Las Am\u00e9ricas, about two blocks away.\n\n#### **GETTING AROUND**\n\n##### **By Taxi**\n\nYellow taxis are plentiful in Cali, and you will need to travel by cab often to get around, especially at night. Taxi drivers are generally helpful, chatty, and honest. However, it is always advisable to order a cab over the phone or using the smartphone app Tappsi. Staff at hotels, restaurants, and (sometimes) clubs should be happy to do this for you. **Taxi Express** (tel. 2\/555-5555) has a phone number you won't forget. Another service is **Taxis y Autos de Cali** (tel. 2\/664-0000). Make sure they turn on the _taximetro_ (meter) when you get started, and note that there may be small nighttime travel or telephone surcharges added on. _Taxistas_ (cab drivers) don't expect tips. And they rarely have change for large bills.\n\nA cab from Cali to the airport will cost around COP$50,000 and take under an hour. From Granada expect to pay about COP$5,000 to get to the bus terminal or to Chipichape and about COP$6,500 to get to San Antonio.\n\n##### **By Bus**\n\nWith 82 routes and a total of 711 buses at last count, the Masivo Integrado de Occidente or **MIO** (www.mio.com.co or www.metrocali.gov.co) is Cali's public transport system. It comprises several dedicated bus lanes with stations (similar to that of the TransMilenio network in Bogot\u00e1), as well as _alimentadores_ \u2014feeder buses that connect with the articulated MIO network at various points. MIO can take you just about anywhere in the city, but the system map is not easy to figure out.\n\nMIO is useful if you are staying in or near Granada or plan to visit the Centro or sights in the south such as shopping malls or the Universidad del Valle campus, or if you plan on going to Pance. The bright blue buses are immaculately maintained and quite safe, too. Note that if you would like to ride one of the _alimentadores_ you will have to present a MIO card on-board. Those have to be bought at the MIO stations. MIO runs 5am-11pm Monday-Saturday and 6am-10pm Sundays and holidays. The current fare for a single trip to any point in the city is COP$1,600.\n\nThe first MIO buses began to operate in 2008, but private bus companies, often using old and polluting buses, were still in operation. In a noble effort to provide some order to the transportation chaos in the city, in late 2012 Cali mayor Rodrigo Guerrero implemented a gradual elimination of those private bus companies, replacing them with city-run MIO buses in order to create a single public bus system. Powerful bus company owners fought him every step of the way. The mayor faced bus strikes, hunger strikes, roadblocks, lawsuits, and threats, but he persevered. It got so nasty at one point, the mayor's transportation secretary had to seek refuge abroad due to threats. If Guerrero is successful, Cali will be the first city in Colombia with a single bus system covering almost the entire city and even extending it to the airport in Palmira. They are getting there: MIO coverage has reached 87 percent.\n\n##### **By Car and Motorcycle**\n\nThe roads in the Valle de Cauca region are generally of high quality and the terrain is flat. So, while it's not a fantastic idea in the city, renting a car or motorbike for excursions outside of Cali (to Valle haciendas, Buga, Lago Calima, and Roldanillo) may be an option. It is not difficult, even, to drive from Cali to Armenia, for example, if you are ready to move on to the Coffee Region. Driving to Popay\u00e1n or to Pasto along the Pan-American Highway is more taxing because of winding, two-lane roads. The same goes for travel westward to Buenaventura.\n\n**Hertz** (Av. Colombia No. 1-14, El Pe\u00f1on, tel. 2\/892-0437, www.rentacarcolombia.co, 8am-noon and 2pm-6pm Mon.-Fri., 8am-3pm Sat.; airport tel. 2\/666-3283, 7am-5pm Mon.-Sat.) has two offices in Cali. For motorcycle rental and tours, get in touch with Mike, the Danish owner of **Motolombia** (tel. 2\/396-3949, www.motolombia.com). Though it's based in Cali, Motolombia can also hook you up with bikes in Bogot\u00e1 and Medell\u00edn.\n\n#### **RESERVA NATURAL ANAHUAC**\n\nOn lazy weekend days, Cale\u00f1os descend en masse on the R\u00edo Pance, south of the city, to splash around in the cool water. Near the town of Pance, at the edge of the Parque Nacional Natural los Farallones, is **Reserva Natural Anahuac** (1 km before Pance on the Cali-Pance road, tel. 2\/331-4828, www.reservanaturalanahuac.com.co, 7:30am-9:30pm daily, COP$5,000), a well-maintained private nature reserve along the R\u00edo Pance that is very accessible. It has a pleasant path that meanders through _guadua_ (a type of bamboo) forests, a fishing pond, a restaurant, and, if you would like to stay the night, some cabins (COP$22,000 pp) and a camping area. There is no need for a guide to make this excursion and it may be just the ticket on a hot day.\n\nTo get to the park, you can take MIO to the Universidades station in the south of Cali and transfer to an _alimentador_ (feeder) MIO bus, the A-19, towards Pance. This ride on MIO is relaxing and pleasant as you pass through Cali suburbia and gawk at the astonishingly fierce-looking Cordillera Occidental mountains in the distance. At the intersection of the Autopista Sur with the V\u00eda Pance is the huge, open-air **La Boquer\u00eda Restaurant** (Cl. 18 No. 122-100, tel. 2\/555-2199, 8am-9pm daily, COP$25,000), which serves hearty Colombian fare. Upon arrival at Anahuac, take care crossing the street: Cars and buses have little regard for pedestrians. Try to beat the crowds by returning to Cali during the early afternoon. Otherwise, finding transportation can be difficult.\n\n#### **HACIENDA PARA\u00cdSO AND MUSEO DE LA CA\u00d1A DE AZ\u00daCAR**\n\nThe **Hacienda Para\u00edso** (14 km from the Pan-American Highway, El Cerrito, tel. 2\/514-6848, ext. 105, haciendaparaiso@inciva.gov.co, www.inciva.org, 9:30am-4:30pm Tues.-Sun., COP$5,000) is an estate near the town of Cerrito that is famous for being the home of famed Colombian poet Jorge Isaacs. His romantic novel _La Mar\u00eda_ is a love story about two adolescents\u2014Efra\u00edn and Mar\u00eda\u2014set against the backdrop of an idyllic hacienda in the Valle de Cauca during the 19th century. The house is frankly rather underwhelming, but the setting amid sugarcane plantations is beautiful. The museum has period furniture and objects and some background on the famous book.\n\nNearby is the **Hacienda Piedechinche and Museo de la Ca\u00f1a de Az\u00facar** (El Cerrito, tel. 2\/550-6076, www.museocanadeazucar.com, 9am-4pm daily, COP$5,000). This hacienda is owned by the Providencia sugar company. Here you amble through some lush and immaculately maintained grounds, seeing different examples of sugarcane mills from sugar-producing regions in Colombia. You can also visit the 18th-century house where the hacienda owners lived.\n\nHacienda Para\u00edso\n\nIf you'd like to spend the night in the country, you can stay at the **Hotel Piedemonte** (cell tel. 316\/555-1462, haciendaparaiso@inciva.gov.co, COP$50,000 pp d). It's on the Hacienda Para\u00edso grounds and has 20 chalets as well as a restaurant. During the week when there are fewer crowds at the hacienda, it is a peaceful place, and guests can lounge by natural pools and perhaps read _La Mar\u00eda,_ or go for a horseback ride.\n\nIf you do not have your own transportation, you can hire a cab for the day from Cali to the area and back, although the cost-benefit ratio may not work out. Another option is to take a bus from the Cali bus terminal (Cl. 30N No. 2AN-29) to the village of Amaime for about COP$5,000. The bus company **COODETRANS Palmira** can take you there. There are always taxis waiting at the bus stop for the arrival of buses. From here take a cab to visit both haciendas. You can negotiate with the driver to take you to both haciendas and then return you to catch a minibus back to Cali from Palmira (COP$50,000) or take you directly back to Cali (COP$80,000). The driver can drop you off in Palmira, about an hour from Cali, where transportation to Cali is easy to acquire. Buses from Buga toward Palmira can also drop you off at the roadside in the village of Amaime, a few kilometers from the sights.\n\n### **Buga**\n\nThe city of Guadalajara de Buga (pop. 115,000), founded in 1555, was one of the first cities established by the Spaniards in New Granada. It is best known as a place of pilgrimage. More than a million Colombian faithful come each year to pray at the Bas\u00edlica Se\u00f1or de los Milagros. Buga may not be chock full of attractions, but it is an excellent launching point from which to discover many lesser-known sights and breathe the fresh air of the Valle de Cauca. Plus, it's a friendly kind of place.\n\n#### **SIGHTS**\n\nThe star attraction in town for religious pilgrims is the **Bas\u00edlica Se\u00f1or de los Milagros** (Cra. 14 No. 3-62, tel. 2\/228-2823, www.milagrosdebuga.com, 5:30am-7:30pm daily). Built in the early 20th century, this pink church is not of architectural significance: It's known for its \"Cristo Negro\" or Se\u00f1or de los Milagros\u2014a charred woodcarving of Christ that is displayed in a chapel behind the altar.\n\nPraying to the \"Black Christ\" is believed to provide miracles, and the story behind this icon is one of generosity, devotion, and miracles. In colonial times an indigenous woman who had converted to Christianity saved for years to purchase a crucifix. One day a man crossed her path crying because he'd go to jail if he didn't pay a debt. The woman showed her generosity to by giving him all the money she had saved. Months later, she noticed a small crucifix floating down the river towards her. She picked it up and made an altar to pray to it. The crucifix grew in size, prompting her and others to believe it had miraculous powers. After years of deterioration, the church decided to burn it and replace it with another, but it never burned, remaining charred and black\u2014another miracle.\n\nThe shops lining the Avenida del Milagroso, a pedestrian walkway leading to the church, sell all manner of basilica-related trinkets. Each September 14 there is a large procession in and around town featuring the Se\u00f1or de los Milagros. It's accompanied by special masses and ceremonies in the church.\n\nBuga is a major pilgrimage destination.\n\nThe **Parque Cabal** (between Clls. 6-7 and Cras. 14-15) is the center of this slow-paced city. Old-timers drink their _tinto_ (small cups of black coffee) in corner caf\u00e9s in the late afternoon, engrossed in political conversations with their friends, while lottery vendors circulate among the tables hoping to sell a couple of tickets. On the corner of Calle 6 and Carrera 15 is the **Catedral de San Pedro** (8am-noon and 2pm-6pm Mon.-Fri.), a beautifully preserved, three-nave church that was originally built in the 16th century. It is a couple of blocks west of the park.\n\n#### **ACCOMMODATIONS AND FOOD**\n\nThe best part about the small M **Buga Hostel** (Cra. 13 No. 4-83, tel. 2\/236-7752, www.bugahostel.com, COP$16,000 dorm, COP$35,000-COP$45,000 d), aside from its friendly owners, is its **Holy Water Ale Caf\u00e9,** where you can saddle up to the bar, try one of their home brews on tap, and chow down on sourdough pizza or the best black-bean burger this side of Austin. It's open daily for lunch and dinner. The hostel has one large dorm room and two private rooms. The hostel has a guide on staff who organizes excursions to the Sonso and Yotoco natural reserves and other sights in the Valle de Cauca. These nature hikes are open to anyone and are reasonably priced.\n\nThe **Hotel Guadalajara** (Cl. 1 No. 13-33, tel. 2\/236-2611, www.hotelguadalajara.com.co, COP$200,000 d) is retro without trying to be. It's just three blocks away from Bas\u00edlica Se\u00f1or de los Milagros and has been around for 50 years. The pool is a good place to cool off. The hotel often has special weekend rates.\n\n#### **GETTING THERE**\n\nBuses depart Cali for Buga all day long. Tickets cost around COP$8,000, and the journey through the sugarcane plantations of the Valle takes under two hours. Buses will often leave you at the main highway, and from there you'll have to walk about 20 minutes or take a cab into town. Cabs are always at the ready to meet arriving buses, and this is easy to do. The pleasant open-air Buga bus station, modern and clean, is a straightforward 15-minute walk from the Buga Hostel.\n\n#### **VICINITY OF BUGA**\n\n##### **Natural Reserves**\n\nThere are three natural parks close to Buga where you can enjoy a day of hiking, canoeing, and bird-watching. Run by **INCIVA** (tel. 2\/514-6848, www.inciva.org), an agency that promotes cultural and ecological points of interest in the Valle de Cauca department, the **Parque Natural Regional El V\u00ednculo** (El V\u00ednculo, cell tel. 321\/831-4775, pnrelvinculo@insciva.gov.co, 8am-noon and 1pm-5pm Mon.-Fri., COP$3,000) is a center for natural exploration and education. Here you can take a short guided nature walk through the dry tropical forest for a small fee. Contact INCIVA in advance to set this up. Previously a cattle ranch, the land was donated to INCIVA by its owner in 1969 to help preserve and protect threatened wildlife and ecological diversity. You can also visit the park on weekends with a prior reservation. Parque Natural Regional El V\u00ednculo is to the south of Buga towards Palmira, only about three kilometers away.\n\nAt the **Reserva Natural Laguna de Sonso** (tel. 2\/228-1922, www.lagunasonso.tripod.com, COP$5,000) you may see many different species of waterfowl and other animals in this wetland reserve on the eastern banks of the R\u00edo Cauca to the southwest of Buga. It is a mostly marshy park, and a walk through it takes about an hour. Buses from the Buga bus station (Cl. 4 before the main highway) going towards Darien or Cali can take you there for around COP$2,000, although you will need to walk one kilometer to get to the park entrance. If you see a fisherman once you arrive at the lake, you can ask him to take you around in his canoe, for a small fee.\n\nMuseo Rayo\n\nA third option is the **Reserva Natural Bosque de Yotoco** (Km. 18 Carretera Buga-Madronal, tel. 2\/228-1922, 7:30am-6pm daily, COP$5,000), where a couple of hour-long guided walks\u2014El Corb\u00f3n and El Cedro\u2014along the R\u00edo Yotoco can be organized. This is the only way to visit the park. Contact expert guide Valent\u00edn (cell tel. 315\/471-4758) to set up a tour. You can get a good view of Lago Calima from the park. This park is also accessible by bus (COP$5,000). Take one headed to Darien and ask the bus driver to let you off at Yotoco.\n\n##### **Lago Calima**\n\nThe **Lago Calima** is the largest artificial lake in Colombia and is a favorite spot for Cale\u00f1os seeking weekend rejuvenation. With winds reaching 43 knots (46 mph), it's a great place for kite-surfing and windsurfing. But other sports, such as waterskiing, boating, and swimming, are also popular. Colombian weekenders often stay in lake houses with their friends and family, but there are a handful of standard hotels if you don't have the option of renting a house. If you are planning a weekend jaunt to this area, you may want to have your own transportation.\n\nDarien is the \"town\" around Lake Calima. It is a cute pueblo with a few restaurants, stores, and other services. Its **Museo Arqueol\u00f3gico Calima** (Cl. 10 No. 12-50, tel. 2\/253-3121, www.calimadarien.com, 8am-noon and 1pm-5pm Tues.-Fri., 10am-6pm Sat.-Sun., COP$3,000) has a collection of ceramics dating to 8,000 BC, and it includes artifacts from the Ilama, Yotoco, and Sonso cultures.\n\nOne of the main draws of Lago Calima is its wind- and kite-surfing opportunities. **Kite Colombia** (cell tel. 317\/821-4889, www.kitecolombia.com, US$500 for a 5-day course including lodging, or US$55 for a 1-day introduction course) offers a range of kite-surfing courses and can also arrange accommodations in its lakeside hostel (US$15 dorm, US$22 d). In the mornings, try some Spanish lessons (US$200 for 20 hours).\n\n#### **ROLDANILLO**\n\nAgainst a backdrop of mountains and the vast valley to its east, Roldanillo (pop. 35,000) is on the map for two reasons: It was the home of modernist artist Omar Rayo, and it's the undisputed paragliding capital of Colombia. This town is just two hours from Buga by bus, far away from the well-trodden tourist route.\n\nThe **Museo Rayo** (Cl. 8 No. 8-53, tel. 2\/229-8623, www.museorayo.co, 9am-6pm daily, COP$5,000) is the only real sight in town, and it is well worth a visit. Omar Rayo, who was part of the Op Art, or Optical Art, movement, is known for bold and abstract paintings that often appear three-dimensional. Dedicated mostly to Rayo's paintings, drawings, and sculptures, the museum comprises five octagonal exhibition spaces, each one with a different theme. Temporary exhibits showcase other renowned Colombian artists. Rayo passed away in Roldanillo in 2010.\n\nIf you have always wanted to feel what it's like to be a condor soaring over the sugarcane fields below, **Cloud Base Colombia** (Cra. 9 No. 8-71, tel. 2\/229-9106, www.cloudbasecolombia.com) has a solution for you. This tourist agency, started in 2011 by a pair of German and Swiss paragliding fanatics, can arrange for paragliding and hang gliding lessons and excursions. If the idea of leaping off a cliff into the bright blue skies of the valley doesn't appeal to you, they can come up with some other ideas for you to pass the time. You can think of them as the Roldanillo tourist information center.\n\nThe optimal time to fly is between late December and late March, although anytime is fine. Roldanillo regularly hosts big-time paragliding competitions, such as the Paragliding World Cup Semifinal in January 2013.\n\nThe best option in town is **Casa Vieja** (Cra. 9 No. 8-71, tel. 2\/229-9106, cell tel. 312\/808-8841, COP$35,000 s, COP$50,000 d). It's a small hostel with a cute garden run by the guys at Cloud Base Colombia. They have 12 rooms and they know how to grill. **Balcones del Parque** (Cl. 8 No. 6-86, tel. 2\/259-5151, celdosman@hotmail.com, COP$25,000 dorm, COP$70,000 d) could be very nice, as its location, overlooking the park, is fantastic.\n\n### **Pasto**\n\nMany people overlook unassuming Pasto, a city of more than 400,000 people. It's seen by backpackers as a place to spend the night on the way between Ecuador and Popay\u00e1n, not a destination in its own right. But Pasto, along with the stunning Nari\u00f1o countryside, deserves your time and attention.\n\nWith church steeples rising out of its colonial center, Pasto is set in the verdant Valle de Atriz with the deceivingly gentle, sloping Volc\u00e1n Galeras watching over it. The valley is a rich agricultural region where potatoes are king. This city even has a potato named after it\u2014the _papa pastusa_ \u2014which is sold in every supermarket in Colombia. Pasto, the \"Ciudad Sorpresa,\" indeed may surprise you with a number of museums and sights that will keep you intellectually stimulated for more than a couple days. Of particular interest is the extraordinary handicraft technique called _barniz de Pasto,_ as well as some incredible wood carvings, an influence of Quite\u00f1o culture.\n\nWonderful day trips can be made from the city to La Cocha and to Laguna Verde to the south, and an overnight trip to climb Volc\u00e1n Cumbal is a good plan for the adventurous.\n\nDue to its rugged terrain and poorly patrolled border with Ecuador, Nari\u00f1o, the department of which Pasto is capital, became a major corridor for drug and guerrilla activity. The areas described in this guide are safe, however.\n\n#### **SIGHTS**\n\n##### **Historic Churches**\n\nThe most important and only colonial-era church in the city is the **Iglesia de San Juan Bautista** (Cl. 18A No. 25-17, tel. 2\/723-5440, 7:30am-11am Sun.-Fri., 3:30pm-6:30pm Sat.), in the heart of the Centro on the **Plaza Nari\u00f1o.** The original construction was built in the 16th century, but an earthquake demolished that, and in 1669 the current church was built. The interior has outstanding geometric Mudejar designs on the ceiling and around the presbytery.\n\nStatues of angels set atop the twin towers of the **Iglesia de Cristo Rey** (corner Cl. 20 and Cra. 24) beckon from blocks away. This is a stunning gothic revival church. The sanctuary is lined by 19 woodcarvings created by famous local sculptor Alfonso Zambrano and Ecuadorian woodcarvers. Above, light streams through enormous stained glass windows, creating a mystical environment.\n\nThe **Catedral de Pasto** (Cra. 26 No. 17-23, tel. 2\/723-3328, 7am-11am Sun.-Fri., 3pm-7pm Sat.) was being renovated at the time of the writing of this guide. It was built in 1920. The stately red-brick church is composed of three naves.\n\n##### **Museo Taller Alfonso Zambrano Pay\u00e1n**\n\nIn a colonial-style house built in the 1960s, the **Museo Taller Alfonso Zambrano Pay\u00e1n** (Cl. 20 No. 29-78, tel. 2\/731-2837, hernandozambrano@gmail.com, 8am-noon and 2pm-4pm Mon.-Sat., free) is a tribute to Alfonso Zambrano, one of Pasto's most famous sons. Zambrano is best known in Pasto for having carved extraordinary designs for carnival floats. Each year an award was given for the best design, and he won it over and over again. Zambrano won so often that he was prohibited at a certain point from participating in the competition due to his extraordinary skill. Woodcarving and painting continue to take place at this combination workshop\/museum. And it remains a family affair: It is now operated by Zambrano's daughter, and the artist's grandsons can regularly be found at work carving designs out of cedar. The museum's small collection of pre-Columbian ceramics and Quite\u00f1o school paintings is impressive.\n\n##### **Museo Taminango de Artes y Tradiciones Populares de Nari\u00f1o**\n\nSet in a colonial house built in the early 17th century (said to be the oldest house still standing in Pasto), the **Museo Taminango de Artes y Tradiciones Populares de Nari\u00f1o** (Cl. 13 No. 27-67, tel. 2\/723-5539, 8am-noon and 2pm-6pm Mon.-Fri., 9am-1pm Sat., COP$2,000) is a museum dedicated to handicrafts. An American missionary, Catalina Morgan, who arrived in Pasto in 1934, set out to preserve the house, which had fallen into disrepair. She raised enough money to restore it and to convert it, eventually, into a cultural center. It was opened as a museum in 1989. The museum presents traditional handicrafts from Nari\u00f1o, including explanations of the _barniz de Pasto_ technique. This technique, developed by indigenous groups, uses the leaves and fruits of the _mopa mopa_ bush, from which a resin is extracted. It is dyed in different colors using vegetables dyes, and thin sheets of it are applied to decorate wooden boxes and other objects. Adjacent to the museum is a small handicrafts store.\n\nPlaza Nari\u00f1o, Pasto's main plaza\n\n##### **Museo del Oro Nari\u00f1o**\n\nThe **Museo del Oro Nari\u00f1o** (Cl. 19 No. 21-27, tel. 2\/721-9100, ext. 2624, www.banrepcultural.org, 10am-5pm Tues.-Sat., free), in the Banco de la Rep\u00fablica building facing the Plaza Carnaval, is a small but good collection of pre-Columbian ceramics from the Nari\u00f1o altiplano (high plains) and Pacific coast. The Pasto indigenous group populated the area around Pasto and Ipiales (south of Pasto) and parts of Ecuador. The predominant group in the area was the Quillacingas, who had arrived from the Caribbean region and who were fierce warriors.\n\nOf particular interest in the museum are the stunning _discos giratorios_ (metallic discs plated with gold and copper designs) that were presumed to have been used\u2014specifically, spun\u2014in hypnotic religious ceremonies. In the same building you can often find temporary exhibits featuring Colombian artists.\n\n##### **Museo del Carnaval**\n\nTo learn all about the city's _carneval_ celebration, go to the **Museo del Carnaval** (Cl. 19 at Cra. 42, 8am-11:30am and 2pm-5:30pm Mon.-Fri., free). This museum, located in a former slaughterhouse, shows off what Pastuosos are most proud of: their famous Carnaval de Negros y Blancos in early January. A guide will show you the colorful floats, costumes, and masks from the annual celebration and can tell you everything you ever wanted to know about it. Visiting the museum is the next best thing to being a part of the craziness in January.\n\n#### **ENTERTAINMENT AND EVENTS**\n\n##### **Nightlife**\n\nPasto is a party town during the Carnaval de Negros y Blancos, but you can still find good times if you're not visiting in January. Thanks to its being home to the Universidad de Nari\u00f1o, there are a few bars and _pe\u00f1as_ (Andean music bars) that you might want to check out.\n\nTry some _hervidos_ (hot alcoholic fruit drinks) at **Mestizo Pe\u00f1a Bar** (Cl. 18 No. 27-67, tel. 2\/723-7754, 6pm-1am Thurs.-Sat.) and **Embrujo Andino Pe\u00f1a Bar** (Cra. 23 No. 19-58, cell tel. 313\/604-4935, www.embrujoandinobar.tk). Both are popular _pe\u00f1as,_ often showcasing live music.\n\nAs far as bars go, **Volcaf\u00e9** (Cl. 18 No. 24-29, tel. 2\/722-4301, 8am-1am Mon.-Sat.) on the Plaza Nari\u00f1o is a good place for a drink in the late afternoon as you overlook the goings-on the plaza. **Cola de Gallo** (Cl. 18 No. 27-47, tel. 2\/722-6194, 3pm-1am Mon.-Sat.) serves its own _caf\u00e9_ (coffee), which is distributed nationally by Juan Valdez, and is a relaxed place for a drink later on in the evening.\n\n##### **Cinema**\n\nTwo shopping centers have movie theaters that show mostly Hollywood blockbusters. **Royal Films** is in the \u00c9xito shopping center (Cra. 22D No. 2-56), and **Valle de Autriz** is in the shopping mall of the same name (Cra. 42 No. 18A-94, tel. 2\/731-6129, www.cinemasvalledeatriz.com.co). Movies cost around COP$5,000.\n\n##### **Festivals and Events**\n\n###### **CARNAVAL DE NEGROS Y BLANCOS**\n\nJanuary 2-6 every year, the population of sleepy Pasto explodes as up to 300,000 visitors from Colombia and beyond converge on the city during the **Carnaval de Negros y Blancos** (www.carnavaldepasto.org, Jan.), which is recognized as a world heritage tradition by UNESCO. The celebration actually gets going on December 28, the \"day of the innocents.\" That's a day of purification and celebration of the natural beauty of the area; many hop on their bikes on this day, cruising down the main parade route. On December 31 a parade pokes fun at politicians and other unpopular figures from the previous year (and sometimes effigies of them are burned).\n\nOn January 2 the parade of the colonies takes place, a celebration of cultures from the Nari\u00f1o department, which includes a horseback procession. January 3 is a day of celebration just for children.\n\nJanuary 4 celebrates the arrival of the Castaneda family, who arrived in Pasto (from the Putumayo department), were welcomed with open arms, and worked to help it grow. This marks the symbolic beginning of the _negros y blancos_. January 5 is the day of the _negros,_ when revelers paint themselves or others black. This day is a celebration of diversity. Finally the festival concludes with the day of the _blancos,_ when fantastic floats slowly make their way along the city streets, and hundreds of thousands of onlookers are busy pelting others with white powder (a Colombian version of Holi, the Indian festival of colors).\n\nIn addition to the parades, there are concerts featuring Andean and other styles of music, and presentations of elaborately costumed stilt walkers and dancers. The _carnaval_ spills over one more day to January 7, when the Cuy Festival is held, but you have to be fond of roasted guinea pig to truly enjoy this event!. The main parade route departs the stadium and ends in the Plaza de Carnaval in the Centro.\n\nFor the _carnaval_ , locals strategically seek out the best spot to watch the parades, with some getting out as early as 6am. The parades\u2014and the drinking\u2014start at 9am on the dot. Airlines add dozens of flights during this time to Pasto, but it's still best to make plans several months in advance if you want to be a part of the fun.\n\n#### **SHOPPING**\n\nFine handicrafts from the region can be picked up at **Obando Barniz de Pasto** (Cra. 25 No. 13-4, tel. 2\/722-0363, 9am-6pm Mon.-Fri.), a business that has been in existence since the 1850s, which specializes in _barniz de Pasto._ **Artesan\u00edas de Colombia** (Cra. 27 No. 12-89, tel. 2\/729-9433, 8am-noon and 2pm-6pm Mon.-Fri.), adjacent to the Museo Taminango, has a wide range of local handicrafts, of the highest quality, including wooden jewelry boxes employing the _enchapado en tamo_ (straw marquetry) technique, in which they are decorated with thin strands of barley and wheat stems. Artesan\u00edas de Colombia is much more than a store, as it conducts numerous training programs for local artisans. On the main plaza is the combination casino\/handicraft store **Shirakaba** (Cl. 18 No. 24-69, tel. 2\/723-9890, 8am-6pm Mon.-Sat).\n\n#### **RECREATION**\n\n##### **Santuario de Fauna y Flora Galeras**\n\nThe Quillacinga people called it Urcunina, the Mountain of Fire, and when the Spaniards arrived, they renamed it **Volc\u00e1n Galeras,** because the gently sloping volcano resembled the sails of ships in the Mediterranean Sea. Today, Pasto residents call it a menace, as this volcano sits right above the city, only eight kilometers away on its western side. Galeras has been one of the most active volcanoes in Colombia, and indeed the world, in recent years. There have been minor eruptions almost every year over the past decade, with the last reported in 2013. In 1993, during an international meeting of volcanologists, six scientists in the crater of the volcano were killed when it erupted.\n\nThe **Santuario de Fauna y Flora Galeras** (tel. 2\/732-0493, www.parquesnacionales.gov.co) is part of the national park system, but much of the park has been closed for many years due to the volcano's activity.\n\nTwo hikes can be done, one outside the park and one inside. One hike is from the neighborhood of Anganoy to the park ranger's station, the Caba\u00f1a de Control\u2014Urcunina. This is a leisurely walk along a slowly sloping road and takes about an hour, and it requires no guide. It starts from the neighborhood of Anganoy (Nido de Aguilas). Once there, look for the road that leads up to the park. To get to Anganoy, either take a cab (COP$4,000) or take the C7 SIT bus.\n\nThe second hike, which takes you into the park, is the 5.7-kilometer-long (3.5-mile-long) San Felipe-Laguna Telpis hike up the northern side of the volcano to the Laguna Telpis, a mountain lake. The hike begins in the village of San Felipe near the town of Yacuanquer. This hike takes three hours to the top, and at the Telpis ranger station you must pay the park entrance fee of COP$2,000. To get to the starting point you can take a taxi (COP$4,000) from Pasto.\n\nBefore setting off on any hike near Galeras, call the ranger's station (tel. 2\/732-0493) to ask about the status of the trails.\n\n#### **ACCOMMODATIONS**\n\nM **La Maison del Ejecutivo** (Cl. 19 No. 37-16B, tel. 2\/731-0043, www.lamaisondelejecutivo.com, COP$120,000 d) is a cozy place to stay in a peaceful neighborhood a short taxi ride from the Plaza Nari\u00f1o. Rooms are quite comfortable. What sets this place apart, though, is the excellent service. The French owner, Patrice, has lived in Pasto with his Colombian wife for many years. He is quite knowledgeable about tourist attractions and can give some expert travel tips. Breakfasts are generous and healthy, a rarity for hotels in Colombia. In fact, you may want to call ahead and see if you can go there for breakfast even if you're not staying there. Dinner is also served for those who wish, and given the options in town, that might be a good idea.\n\nThe **Juan Sebasti\u00e1n Hotel** (Cra. 29 No. 20-18, tel. 2\/731-0983, COP$100,000 d) opened in 2011, is centrally located, and is a good value. It has 35 rooms, offers wireless Internet, and includes a very basic breakfast.\n\n**Fernando Plaza** (Cl. 20 No. 21B-16, tel. 2\/729-1432, www.hotelfernandoplaza.com, COP$124,000 d) is a business hotel in a somewhat quieter part of the Centro a few blocks from the Iglesia de Cristo Rey. It offers immaculate and comfortable rooms.\n\nStarted by an Australian many years ago but having been under other management for quite some time, the **Koala Inn** (Cl. 18 No. 22-37, tel. 2\/722-1101, COP$28,000 s, COP$45,000 d) remains the only backpacker lodge in Pasto, and, although staff are pleasant, this dingy hostel just barely cuts it.\n\n#### **FOOD**\n\nPasto is Colombia's _cuy_ capital. _Cuy_ is guinea pig meat, which is prepared by slowly barbecuing the meat for about an hour. The delicacy appears on many menus around town, but it's more of a weekend or special occasion dish, so you may have to seek it out.\n\nEateries popular with locals in the Centro include **Restaurante Chipichape** (Cra. 28 No. 18-78, tel. 2\/722-8992, 8am-10pm Mon.-Sat., 8am-5pm Sun., COP$6,500) and **Picanter\u00eda Ipiales** (Cl. 19 No. 23-37, tel. 2\/723-0393, 9:30am-8:30pm Mon.-Sat., 11am-6pm Sun., COP$7,000). Similar to each other, both specialize in _lapingacho_ plates. _Lapingachos_ are a type of cheese-filled potato cake, made from _papa pastusa (a local potato variety),_ and the meals usually contain beef, pork, or chicken surrounded by about five of these cakes. There is always some roasted _capia_ (corn) on the side.\n\nCafeteria, bakery, and pizzeria **La Merced** (Cra. 22 No. 17-37, tel. 2\/723-8830, 7am-10pm daily) has something for everyone, from typical Colombian fare to seafood to pizzas. It also has an array of sweets and baked goods.\n\nThe only strictly vegetarian restaurant is **Pan Integral** (Cra. 29 No. 20-34, no phone, 9am-6pm Mon.-Sat., COP$4,500), where a set lunch menu goes for only COP$4,500, but a nicer option is **Huerta del Chef** (Cra. 36 No. 18-114, cell tel. 301\/447-0350, 7am-2:30pm Mon.-Fri., 7am-1pm Sun., COP$12,000).\n\nThe best pizza place in town is **Alina Pizza Gourmet** (Cl. 20 No. 38-07, tel. 2\/731-3565, 6pm-10pm daily, COP$25,000), a hip and overpriced joint on the Avenida Los Estudiantes. It serves only pizza\u2014including many vegetarian options\u2014and drinks. The walls are decorated with thousands of photos and the atmosphere is friendly.\n\nFor hanging out, try **Caf\u00e9 La Catedral** (Cra. 26 No. 16-37, tel. 2\/729-8584, www.cafelacatedral.com, 8:30am-12:30pm and 3pm-8:30pm Mon.-Fri., 10am-12:30pm and 3pm-8:30pm Sat.), where you can have a coffee or light meal and watch the action in the plaza. It's a shame it isn't open on Sunday morning, though, as it's nearly impossible to find a decent cup of coffee at that time.\n\n#### **INFORMATION AND SERVICES**\n\nThe **Pasto tourist office** (Cl. 18 No. 25-25, tel. 2\/723-4962, 8am-noon and 2pm-6pm Mon.-Sat.) is right off the Plaza Nari\u00f1o and across from the Iglesia de San Juan Bautista. Here you can purchase a city map (COP$2,000), pick up materials on nearby attractions, and purchase some _artesan\u00edas._ The timid staff may not volunteer information, so you may have to be persistent.\n\n#### **GETTING THERE AND AROUND**\n\nThe **bus terminal** (Cra. 6 No. 16D-50, tel. 2\/730-8955, www.terminaldepasto.com.co), about a 10-minute taxi ride from the Centro, is well-organized, with shops, cafeterias, and ATMs. From the bus terminal, _colectivos_ (small, fast buses) make the journey to and from Ipiales all day long and cost around COP$7,000. Get a window seat if you make this trip along the Pan-American Highway: The scenery is breathtaking and the hairpin curves thrilling. It is a two-lane road, so accidents and road construction can cause delays. Buses going to Bogot\u00e1 take around 20 hours and cost COP$80,000, while buses from Popay\u00e1n take about six hours, costing COP$30,000.\n\n**Avianca** (Cl. 19 No. 25-77, tel. 2\/723-2320, www.avianca.com, 8am-6:30pm Mon.-Fri., 8am-1pm Sat.) serves the **Aeropuerto Antonio Nari\u00f1o** (Chachagui, tel. 2\/232-8141 or 2\/732-8064), which is about 35 kilometers north of the city. The landing strip is dramatically set upon a plateau 50 meters above rich agricultural land, with mountains all around. Landing here in August when the winds blow can be alternately exciting and terrifying. There is regular flight service to Bogot\u00e1 and Cali on Avianca and **Satena** (Cl. 18 No. 27-74, tel. 2\/722-0623 or 2\/733-4266, www.satena.com). A taxi into town will cost around COP$20,000. Shared _colectivos_ can sometimes be found for less.\n\nIn Pasto, transportation alternatives are pretty straightforward and organized. Taxis cost COP$3,500 to anywhere in the city. As a matter of precaution, after dark order taxis by phone.\n\nThe bus system is quite good, as private bus companies have been by and large replaced by the public **SIT** network (www.ciudadsorpresa.com.co). Bus fares currently stand at COP$1,100. Unfortunately there are few signs pointing out bus stops, so you will need to ask which bus to take and from where to catch it. The vast majority of tourist sights are in the Centro, and are best visited on foot if possible.\n\n#### M **LAGUNA LA COCHA**\n\nOnly about 45 minutes outside of Pasto is **Laguna La Cocha** and the smallest park in the national park system, the **Santuario de Fauna y Flora Isla de la Corota.** This excursion is a delight. Minibuses from Pasto will take you to the lakeside fishing village of Encano, on the shores of La Cocha, and is home to about 200 families. The cheerfully painted wooden A-frames, the flowerboxes, and the colorful _lanchas_ (wooden boats) waiting at the ready will remind you of someplace\u2014but probably not Colombia! There are maybe a dozen restaurants in the Encano, all serving La Cocha trout, dozens of different ways.\n\nFrom the village you can hire a boat to take you to the sanctuary on tiny **Isla de la Corota** , not far away (about a 10-minute ride). This excursion costs about COP$20,000 per boat, as the boat's owner will wait for you and take you back to the mainland. On the island, you'll have to pay an entry fee (COP$1,000) and sign in at the ranger's office. From there you'll walk through the virgin rainforest on a wooden walkway. Although the vegetation is tropical with 500 species of plants, including ferns, bromeliads, orchids, lichen, and _siete cueros_ trees, the climate is actually quite cool. It's about 2,800 meters (9,200 feet) above sea level. The highest points on the island have similar vegetation to that of _p\u00e1ramos_ (highland moors). It's nice to go on a weekday when there are few visitors, so that you can enjoy the wonderful peace that the island brings. It's a lovely excursion, one that won't take long: the island covers only about 16 hectares (40 acres) of land.\n\nSurrounding the lake are more than 50 private natural reserves managed by local farmers through the Asociaci\u00f3n de Desarollo Campesino (www.adc.org.co). Many of these offer accommodations for visitors. One such reserve is **El Encanto Andino** (Vereda Santa Teresita, cell tel. 321\/263-2663, www.lagunacocha.blogspot.com, COP$100,000 d incl. all meals, transportation COP$25,000 pp). Here you can take walks through the jungle, visit an orchid farm, and do some bird-watching, among other activities. Food is produced at the reserve, and it is all organic. It is difficult to get there, but the pristine environment and unique experience may make it worth the trouble. Contact the hotel for transportation assistance.\n\nEasier to get to and more luxurious is the Swiss chalet-like **Hotel Sindamonoy** (tel. 2\/721-8222, cell tel. 314\/863-5186, www.hotelsindamanoy.com, COP$176,000 d) overlooking the lake. It has 23 large rooms, most with a lake view. If you are just coming for the day, you can take a boat to the hotel and have lunch at the restaurant, the best on the lake.\n\nTo get to Encano, take a _colectivo_ (a small minivan) from Pasto. They leave from in front of the hospital (along Avenida Colombia) facing the big Alkosto store (not from the bus terminal). Expect to wait about 20 minutes for the car to fill up with passengers. It is a 45-minute drive and costs COP$4,000.\n\n#### M **LAGUNA VERDE**\n\nThe hike up to the sulfurous **Laguna Verde** (3,800 meters\/12,500 feet), a dazzling, emerald green crater lake on the north side of the dormant **Volc\u00e1n Azufral,** is easy to make from Pasto. Laguna Negra is a smaller neighboring lake. A sacred site for the Pasto indigenous people, the volcano is part of the Nudo de los Pastos mountain range, which serves as a natural border between Colombia and Ecuador. The vegetation in the _p\u00e1ramo_ (highland moor) is sparse, with low shrubs, wildflowers, moss, and lichen. There is little fauna to be seen, except for the occasional _gavilan_ (vulture) gliding through the air. From here on a clear day you can see as far as the Galeras volcano in Pasto.\n\nfishing community at Laguna La Cocha\n\nThe Nudo de los Pastos is where the great Andes mountain range coming north from Ecuador splits into two ranges in Colombia: the Cordillera Central (Central Mountain Range) and the Cordillera Occidental (Western Mountain Range). The Cordillera Central continues northwards through Pasto, Cali, much of the Coffee Region, Medell\u00edn, and eventually into the Bol\u00edvar department (province) near the Caribbean. The Cordillera Occidental rises between the Pacific Ocean and the R\u00edo Cauca and continues through the departments of Choc\u00f3 and Antioquia to the Gulf of Urab\u00e1 on the Caribbean coast.\n\nThe **Reserva Natural Azufral** is managed by a community organization, the **Asociaci\u00f3n Azufral los Andariegos T\u00faquerres** (Cra. 6A No. 16D-50, tel. 2\/730-8955, cell tel. 316\/713-3823). From the park ranger's cabin, it is a six-kilometer hike to Laguna Verde and Laguna Negra. Hikers are requested to register at the cabin and pay a small entry fee (COP$2,000). It is an easy, gradual ascent as it follows a dirt road all the way up the mountain. This can take 3-5 hours. It is hard to get lost, especially on weekends and holidays when there are many fellow hikers.\n\nWeather can change on a dime, temperatures can dramatically drop, and the winds can be fierce. Wear warm clothing, bring along a waterproof windbreaker and drinking water, and pick up some fruit from the T\u00faquerres market to keep you going. Economical breakfasts, lunches, and a hot _agua de panela_ (hot drink made from raw brown sugar) or _tinto_ (black coffee) can be had at the ranger's cabin. Call in advance (cell tel. 316\/713-3823) to arrange for meals. The best time to make the trek is July-October or January-April when the weather is drier.\n\nview from the path to Laguna Verde\n\n**T\u00faquerres** , the nearest town to the hike's starting point, is not beautiful, but if you make the hike on a Thursday, be sure to catch the open-air **Santamar\u00eda Market** (6am-2pm). There's the usual cornucopia of fruits and vegetables, including some crazy-looking potatoes. Stray dogs hang out in the freshly slaughtered meat section. The call of _\"a mil, a mil, a mil\"_ (\"for one thousand, for one thousand\") rings across the market, as increasingly anxious vendors try to sell their vegetables (and rabbits and chickens) to a dwindling number of potential customers.\n\n##### **Guides and Tour Agencies**\n\nContracting a guide is not necessary for this hike, especially on weekends when there are many making the climb. Nevertheless, knowledgeable guides are available from the community-based **Asociaci\u00f3n Azufral los Andariegos T\u00faquerres** (T\u00faquerres office: Cra. 13 No. 19-26, tel. 2\/728-0586), the group that manages the park. You can request a guide in advance by calling Jorge Noguera, head of the association (cell tel. 316\/713-3823, COP$30,000 group). If you'd prefer an organized tour from Pasto and would like to avoid having to deal with public transportation, contact Jaime L\u00f3pez at **Viajes Cielo & Tierra** (cell tel. 317\/437-7436, www.turismopasto.com, COP$140,000-COP$200,000). This group can provide a bilingual guide for the excursion, which leaves Pasto (they can pick you up at your hotel) bright and early at 5am. Viajes Cielo & Tierra offers many other outdoor adventures and multi-day tours in the Nari\u00f1o department.\n\n##### **Getting There**\n\nIf you are making this trip independently, there is regular _colectivo_ service from the Pasto bus terminal (Cra. 6A No.16D-50, tel. 2\/730-8955) bound for T\u00faquerres starting at 5am. The trip will cost about COP$8,000 and takes two hours to make. Plan to leave Pasto no later than 8am, however.\n\nFrom T\u00faquerres it is about 15 kilometers to the hike's starting point (the park ranger's cabin), just past the San Roque village. _Colectivos_ (COP$1,000) leave from the Parque Bol\u00edvar in the morning to San Roque. (The park ranger cabin is 500 meters from there.) This is the cheapest option. However, this service is infrequent (ask a local about the bus upon arrival). Private taxis (COP$20,000-COP$40,000 round-trip) can be contracted from T\u00faquerres to take you all the way to the park ranger's cabin and pick you up at a designated time afterwards, but it is recommended to contact Jorge Noguera, from the **Asociaci\u00f3n Azufral los Andariegos T\u00faquerres** (cell tel. 316\/713-3823, COP$30,000), in advance to coordinate this transportation, as the association is always an honest broker. The trip up takes you past idyllic farmhouses and fields of potato crops.\n\nbuckets of potatoes at the Santamar\u00eda Market in T\u00faquerres\n\n#### **VOLC\u00c1N CUMBAL**\n\nAt 4,764 meters (15,630 feet) high, the snow-dusted **Volc\u00e1n Cumbal** is the highest volcano in the Nudo de los Pastos mountain range in southern Colombia. It is 15 kilometers northwest of Cumbal (pop. 20,000), a laid-back and orderly town of broad streets.\n\nThe volcano has not seen any activity since 1930, but the Colombian Geological Service upped its level of threat from green to yellow in 2012. Indigenous people used to extract sulphur rock and snow and ice for sale in the market at Ipiales, and to make _helado de paila,_ an ice cream made in large copper bowls.\n\nTo hike to the top of Volc\u00e1n Cumbal, you must spend the night in the town of Cumbal and get an early morning start. It is a strenuous hike: five to six hours up and three back. The volcano has two peaks; most visitors climb the southern one. From the top, there are spectacular views of the mountainous surrounding countryside extending into Ecuador.\n\nThe path to the volcano begins in the settlement called La Origa, which is 20 minutes from town. From there it is a tough hike up. The path is not clearly marked, and it could be easy to get lost, particularly when fog rolls in over the mountains, so be sure to get a guide. A recommended guide is Fidencio Cuaical of **Turicumbes** (cell tel. 310\/513-7234, turicumbes@hotmail.com, COP$60,000), a group of young, rural guides, based out of Cumbal.\n\nIf you prefer an organized excursion with transportation from Pasto, you can arrange it with **Viajes Cielo & Tierra** (tel. 2\/731-3314, cell tel. 317\/437-7436, www.turismopasto.com, turismocieloytierra@hotmail.com).\n\nThe best time of year to make the climb is during the dry months between July and September.\n\n##### **Accommodations**\n\nThe **Hotel Para\u00edso Real** (Cl. 17 at Cra. 9, cell tel. 314\/640-6046, COP$25,000 d) is really the only lodging available in Cumbal. Rooms are clean but quite basic. Surprisingly, there is wireless Internet, and the hotel adjoins a small bakery. Adjacent to the hotel is a basic restaurant where they serve trout fished from the sacred Laguna Cumbal, about six kilometers from town.\n\n##### **Getting There**\n\nTo get to Cumbal from Pasto, you must take a bus to Ipiales first. This costs around COP$10,000, and the trip takes two hours. At the Ipiales terminal, transfer to a minivan bound for Cumbal; the trip takes about an hour (COP$6,000).\n\n#### **CAF\u00c9 DE ALB\u00c1N**\n\nFor an unforgettable experience in a Nari\u00f1o coffee town, you can't beat the homestay program of **Caf\u00e9 de Alb\u00e1n** (cell tel. 314\/632-3765, cafealban@gmail.com or edgarivanpa@gmail.com, www.cafealban.org, COP$60,000 pp per day) in the town of San Jos\u00e9 de Alb\u00e1n, which is 68 kilometers northeast of Pasto, an agro-tourism project started by an American. For a few days you can live with a local family and learn the ins and outs of the coffee growing, harvesting, and roasting process. Near the town are markets to visit, hikes to hike, and opportunities to learn about handicrafts from the region. You can also volunteer by teaching English at Caf\u00e9 de Alb\u00e1n's school.\n\n#### **IPIALES**\n\nBorder towns are rarely beauties, and Ipiales, on the Pan-American Highway on the other side of Ecuador, is no exception. However, Ipiales is home to the stunning neo-gothic Santuario Nuestra Se\u00f1ora de las Lajas, a major pilgrimage site that is worth a look.\n\nFor meals and accommodations, the area around the Plaza La Pola is where to look. With hookahs on the tables and dishes such as prawn fettuccini on the menu, **La Terraza** (Cra. 6 No. 13-17, tel. 2\/775-7677, 10am-9pm Mon.-Sat., 10am-3pm Sun., COP$18,000) might seem a little ambitious for Ipiales. It is on the second floor overlooking the Plaza La Pola. Also on the menu are shwarmas and burritos. **Los Andes** (Cra. 5 No. 14-44, tel. 2\/773-4338, www.hotellosandes.com, COP$90,000 d) is one of the few good hotel options in town. Carpeted rooms are fine (there are 33) and they even have a gym.\n\n##### **Las Lajas**\n\nThe most famous neo-gothic construction in all of Colombia, and one of the most visited pilgrimage sites for Catholics, is the **Santuario Nuestra Se\u00f1ora de las Lajas** (tel. 2\/775-4462, 7am-6pm daily), seven kilometers from the city. It is truly a photogenic church built impossibly in the middle of a gorge and over a river. The basilica is packed with Colombian and Ecuadorean tourists on Sundays and religious holidays and, to a lesser extent, on Saturdays. Minivans head towards the basilica all day long from the bus terminal, and it costs about COP$2,000. You will need to walk down a long walkway to the cathedral. It's lined with knickknack shops and diners where you can grab a _quimbolito_ pastry and a sweet, weak coffee to warm up, usually served by a lady in a _ruana_ (poncho). Along the walkway you'll notice thousands plaques of prayers and thanksgiving. The pews are always packed for masses at the cathedral, and a museum (entry COP$2,000) has old photographs of prior cathedral constructions. Families often bring picnic lunches and watch their kids run around by the river, the trickling R\u00edo Gu\u00e1itara. Hungry street dogs wait for scraps. The bridge leading to the cathedral is a favorite spot for photos.\n\nFor much of its history, the cathedral has been under construction. First built as a small chapel in the 18th century, its latest and stunning neo-gothic incarnation took about 33 years (until 1949) to complete. It was designed by an architect from Pasto and built by an Ecuadorean.\n\nThe sanctuary is a place of miracles, according to believers. An indigenous girl and her mother were traveling home and, during a storm, sought refuge from the rain under the protection of rocks. The girl, Rosa, who was deaf and dumb, discovered the illuminated image of Mary on the side of the canyon after a lightning bolt struck, and she astonished her mother by later uttering the words _\"Mamita, la mestiza me llama\"_ (\"Mommy, she is calling me\").\n\nAfter seeing the cathedral, the walk back up is steep. There are plans to build a cable car transportation system to make it all easier and make this a bigger tourism draw.\n\nllama at Las Lajas\n\n##### **Crossing the Border**\n\nThe border with Ecuador is at the town of Rumichaca about three kilometers south of the center of Ipiales. Minibuses depart from the center all day long and from the bus terminal as well. These cost about COP$1,500, or you can take a cab, which will cost COP$6,000. Leaving Colombia, you will need to fill out an entry form at the Ecuadorean side of the bridge and request either a 30- or 90-day-long permission. The border is open 5am-10pm.\n\n##### **Getting There**\n\nThere is frequent _colectivo_ (small bus) service between Pasto and Ipiales along the Pan-American Highway. It's a beautiful, if slow-going, ride, but remember that you'll have to stay awake to enjoy the awe-inspiring scenery of mountains, valleys, and rivers. The road is two-lane and traffic is heavy, especially with big trucks. If there is a wreck on it, this 2.5-hour journey can take much longer. The trip costs about COP$8,000.\n\nBuses to Popay\u00e1n take about eight hours and cost COP$32,000; to Cali the journey takes 11 hours and costs COP$46,000. You can take a 23-hour bus ride to Bogot\u00e1 for COP$107,000, but airliner **Satena** (Cra. 7 No. 16-49, tel. 2\/725-6085, www.satena.com, 9am-6pm Mon.-Sat.) has a morning flight to the capital a few times a week from the tiny Ipiales airport.\n\n### **Popay\u00e1n**\n\nThe temperate capital of the Cauca department, Popay\u00e1n is the White City along the banks of the R\u00edo Cauca between the Cordilleras Central and Occidental (Central and Western Mountain Ranges). It is a dignified city, proud of its place in history as the home of presidents, poets, and priests. It retains its colonial charm despite earthquakes and modernization. Religion retains its importance in the lives of its people; during the annual Holy Week celebrations the entire city takes part in solemn processions through the streets. Idyllic churches and museums are the main places of interest in Popay\u00e1n, but lingering in the Parque Caldas on a sunny afternoon or strolling the lonely streets on a Sunday evening may be what you remember most.\n\nPopay\u00e1n is also a great base from which to explore sights nearby. Just outside of town is the Guambarino indigenous town of Silvia, famous for its colorful Tuesday market. In Coconuco, you can take a dip in the hot springs, and Parque Nacional Purac\u00e9 is a nearby national park where you can hike to the rim of a volcano, the Volc\u00e1n Purac\u00e9.\n\nFarther afield in Cauca is the archaeological site of Tierradentro, and beyond that, in the Huila department, is San Agust\u00edn. These two sights can be combined in a circuit trip in three or four days from Popay\u00e1n, although many tourists choose one or the other.\n\n#### **SIGHTS**\n\n##### M **Centro Hist\u00f3rico**\n\nThe **Parque Caldas** (Clls. 4-5 and Cras. 6-7) in the center of Popay\u00e1n is a lovely pedestrian square and the city's main point of reference. It's a fantastic place to have a coffee (there's a Juan Valdez Caf\u00e9 on the square) or just hang out, day or night. Your discovery of Popay\u00e1n begins here.\n\n###### **MUSEUMS**\n\nThe **Casa Museo Edgar Negret and Museo Iberoamericano de Arte Moderno de Popay\u00e1n** (MIAMP, Cl. 5 No. 10-23, tel. 2\/824-4546, www.museonegret.wordpress.com, 8am-noon and 2pm-6pm Wed.-Mon., COP$2,500) is in the home of Edgar Negret, a Colombian artist best known for massive abstract iron sculptures that adorn public spaces in cities throughout Colombia and in museums throughout the world. Negret donated this 18th-century house to the city in an effort to promote its rebirth following the devastating 1983 quake.\n\nAn oft-talked-about piece at the museum is a model of Negret's proposal for a monument to Sim\u00f3n Bol\u00edvar in Bogot\u00e1 that ended up being too abstract for the Bogotanos.\n\nThe museum was the scene of excitement when, in 2011, an etching by Pablo Picasso was stolen from Negret's private collection. Worth over US$65,000, it was recovered when the thief tried to sell it a few months later. Negret passed away in Bogot\u00e1 in 2012 on his 92nd birthday.\n\nThe **Museo de Historia Natural de la Universidad del Cauca** (Cra. 2 No. 1A-25, tel. 2\/820-9861, 9am-noon and 2pm-5pm daily, COP$3,000) was founded in 1936. This expansive museum on the edge of the historic center is considered the best natural history museum in Colombia and highlights the astounding variety of species, both plant and animal, that are found in Colombia. A guide will show you through.\n\n**Museo Nacional Guillermo Valencia** (Cra. 6 No. 2-69, tel. 2\/820-6160, 10am-noon and 2pm-5pm Tues.-Sun., COP$2,000) is an 18th-century house near the Puente del Humilladero that was the home of Popay\u00e1n poet Guillermo Valencia. His son, Guillermo Leon Valencia, was president in the 1960s. The museum may not be of great interest to foreign visitors, but the house is undeniably a beauty. The **Puente Humilladero** and **Puente de la Custodia** are historic bridges next to the Museo Casa Valencia. The Puente de la Custodia dates to the 18th century.\n\nThe **Museo de Arte Religioso** (Cl. 4 No. 4-56, tel. 2\/824-2759, 9am-12:30pm and 2pm-6pm Mon.-Fri, 9am-2pm Sat., COP$5,000), run by the Arquidiocesis de Popay\u00e1n, has 10 rooms of religious art from the colonial era in an 18th-century neoclassical house covering Quite\u00f1o, Popay\u00e1n, and Spanish styles. You'll probably be guided through by a police cadet.\n\n###### **CHURCHES**\n\nThere are several colonial churches dating from the 17th to 18th centuries to visit in Popay\u00e1n. Most of them have been restored following earthquakes over the years. The **Iglesia San Francisco** (Cl. 4 and Cra. 9, tel. 2\/824-0160) is one of the most beautiful churches and dates to the late 18th century. You can ask at the church to see the mummies that were found here following the earthquake. The **Iglesia La Ermita** (Cl. 5 and Cra. 2, tel. 2\/820-9725) is older, dating to the 16th century. It has some fine woodcarvings and paintings. The **Iglesia Santo Domingo** (Cl. 4 and Cra. 5, tel. 2\/824-0536) is where the Good Friday procession begins every year. The neoclassical **cathedral** (Cl. 5 and Cra. 6, tel. 2\/824-1710) on the Parque Caldas was completed in the early 20th century. The cathedral's official name is **Catedral Bas\u00edlica de Nuestra Se\u00f1ora de la Asunci\u00f3n de Popay\u00e1n,** but it is always referred to as _\"la catedral.\"_\n\n##### **Cerro El Morro del Tulc\u00e1n**\n\nFor a quick early morning or afternoon walk and some nice views of the city, check out the **Cerro El Morro del Tulc\u00e1n,** a hill to the northeast of Popay\u00e1n. A statue of the city's founder, Sebasti\u00e1n de Belalc\u00e1zar, stands on horseback on top of the hill. It is thought that this hill is actually a man-made pyramid built by pre-Columbian peoples. There have been reports of bandits at the hill, so do not go very late in the afternoon.\n\nA corny handicraft market, the **Pueblito Patojo** (9am-6pm daily), adjoins the Cerro el Morro. Also nearby is the **Cerro de las Tres Cruces**.\n\n#### **ENTERTAINMENT AND EVENTS**\n\n##### **Nightlife**\n\nFor unsurpassed old-school, Popay\u00e1n-style atmosphere, **El Sotare\u00f1o** (Cra. 6 No.8-05, no phone, 8pm-midnight daily) can't be beat. It's a cozy mom-and-pop place where the pop plays old vinyl tunes (lots of tango) from his collection and patrons of all ages fill in the booths. For nightlife of a different sort, head to the drinking hall alongside university students at the **Campanario** shopping mall (Cra. 9 No. 24AN-21, 5pm-9pm daily), on the outskirts of town.\n\n##### **Festivals and Events**\n\nThe most important religious site in Colombia during **Semana Santa** (Holy Week) is Popay\u00e1n. During Easter week, solemn processions take place on the streets of the center, a tradition that has been fulfilled every year in the White City since 1566. This is the only time of the year that mummies discovered in the Iglesia San Francisco are displayed.\n\nthe White City of Popay\u00e1n\n\n**Congreso Nacional Gastron\u00f3mico** (www.gastronomicopopayan.org) is an annual food festival held in early September each year, mostly at the Hotel Dann Monasterio. You may be surprised to learn that Popay\u00e1n is one of UNESCO's Cities of Gastronomy, along with Cheng Du, China; Ostersund, Sweden; and Jeonju, North Korea.\n\n#### **RECREATION**\n\n##### **Guided Tours**\n\nMarcela from **Finca Alas y Raices** (cell tel. 320\/742-3470, www.alasyraices.org) conducts walking tours of Popay\u00e1n (COP$60,000) as well as day trips to Silvia and multi-day trips to San Agust\u00edn. Lorena Torres of **Kolumbien Linda Tours** (Cl. 3 No. 25-35, cell tel. 314\/892-6699, www.kolumbien-linda-tours.com) specializes in German and English tours to San Agust\u00edn from Popay\u00e1n (COP$250,000). **Luna Paz Tour** (Cra. 11 No. 4-85, tel. 2\/821-9595, www.lunapaztour.jimdo.com) organizes many interesting tours, such as to Tierradentro (COP$354,000 pp) and to Purac\u00e9 (COP$120,000 pp).\n\nThe staff at **Hosteltrail** (Cra. 11 No. 4-16, tel. 2\/831-7871, www.hosteltrailpopayan.com) and **Hostel Caracol** (Cl. 4 No. 2-21, tel. 2\/820-7335, www.hostelcaracol.com) can arrange outings, such as a day trip to the Coconuco hot springs with an exhilarating bike ride back; a trip to the Silvia market on Tuesday; tours of the Finca Alas y Raices coffee farm; and trips to **Mam\u00e1 Lombriz** (cell tel. 316\/482-8655, www.mamalombriz.com), a farm on the outskirts of town where worms work!\n\n#### **ACCOMMODATIONS**\n\nPopay\u00e1n has a fair number of accommodations options, and hostels are some of the better ones. Rooms are in short supply during Holy Week and during the Congreso Nacional Gastron\u00f3mico.\n\n###### **UNDER COP$70,000**\n\nRelaxed is the best word to describe the M **Hostel Caracol** (Cl. 4 No. 2-21, tel. 2\/820-7335, www.hostelcaracol.com, COP$48,000 d shared bath). That atmosphere has a lot to do with its easy-going staff. They are full of great suggestions on how to make the most of your visit, including how to do their \"Circuito Sur,\" a trip that includes San Agust\u00edn, Tatacoa desert, and Tierradentro. There is free use of computers and wireless Internet. If you feel like mingling with other travelers or locals, check out their small caf\u00e9. M **Hosteltrail** (Cra. 11 No. 4-16, tel. 2\/831-7871, www.hosteltrailpopayan.com, COP$45,000 d shared bath, COP$55,000 d private bath), on the other side of the park and owned by the same Scottish couple, is a sociable place. Hosteltrail is about a 15-minute walk from the bus station.\n\n**Parklife** (Cl. 5 No. 6-19, cell tel. 300\/249-6240, www.parklifehostel.com, COP$18,000 dorm, COP$45,000 d) has the nicest location in Popay\u00e1n. Literally right next to the _catedral_ and overlooking the Parque Caldas, it can't get much better. This is a lively and bright Spanish-Romanian-Irish-run hostel, with a pair of private \"rooms with a view\" that overlook the park and two dorm rooms with 8-10 beds each.\n\n###### **COP$70,000-200,000**\n\n**La Casa de Mina** (Cl. 3 No. 2-37, cell tel. 310\/494-4082, www.lacasademima.com, COP$150,000 d) is a quiet and cozy bed and breakfast a few blocks from the Parque Caldas. Seven rooms overlook three courtyards. The owner, Do\u00f1a Olga, lives here as well, and will make you feel right at home. **Hotel La Plazuela** (Cl. 5 No. 8-13, tel. 2\/824-1084, www.hotellaplazuela.com.co, COP$110,000 d) is a colonial-style house with a lovely large interior courtyard. The rooms are not as great as the setting, which is hard to beat. The hotel restaurant is popular with city government and university employees on their lunch break.\n\n###### **OVER COP$200,000**\n\nThe finest option in town, although falling short of five stars, is the classic M **Hotel Dann Monasterio** (Cl. 4 No. 10-14, tel. 2\/824-2191, www.hotelesdann.com, COP$362,000 d). It's housed in an old monastery overlooking a serene interior courtyard where you can have a coffee. It has a pool out back amid spacious, well-kept grounds.\n\n#### **FOOD**\n\nFive-course meals are standard at the **Restaurante del Hotel Camino Real** (Cl. 5 No. 5-59, tel. 2\/824-3595, www.hotelcaminoreal.com.co, noon-3pm and 6pm-9:30pm, COP$25,000). The cuisine here is mostly French, but also on the menu are the empanadas that made Popay\u00e1n famous: potato- and peanut-filled _empanadas de pipi\u00e1n._ It can get busy here around lunchtime.\n\n**La Semilla Escondida** (Cl. 5 No. 2-28, tel. 2\/820-6437, noon-3pm and 6pm-10pm Mon.-Sat.) serves healthy lunches and an economical set lunch for just COP$8,000 in a peaceful corner of town. Go for the cr\u00eapes\u2014the owner is French, after all.\n\n**Wipala** (Cra. 2 No. 2-38, tel. 2\/823-3141, wipalacolectivocultural@gmail.com, 10am-9pm daily, COP$15,000) is a live music venue\/gallery\/restaurant surrounding a verdant patio. In the evenings it's a cool place to visit and mingle with locals as well as visitors and expats. Wipala serves some unique juices, some with coca leaf. There's also a veggie burger made out of beets, which tastes much better than it sounds. Wednesday nights are curry nights, a tradition started by the folks at Hosteltrail.\n\nIf you want Mexican food, go to a restaurant owned by a Mexican. **Tequila's** (Cl. 5 No. 9-25, tel. 2\/822-2150, 5pm-9pm Wed.-Sun., COP$15,000) is a great little cantina where you can delight in the dynamic duo of enchiladas and micheladas (beer cocktail). The owners are a Popay\u00e1nensa-Mexican couple who lived in Long Island for many years.\n\nAbove all, the M **Restaurante Italiano** (Cl. 4 No. 8-83, tel. 2\/824-0607, 11am-10pm daily, COP$20,000) is consistent. Run by a Swiss woman, it's been in Popay\u00e1n for years, stays open late, and serves mega portions of pasta dishes and also pizzas. Load up on your carbs here before your Purac\u00e9 hike. During the day the set lunch\u2014usually Colombian fare\u2014is hard to beat.\n\n**Capriccio Caf\u00e9 & Te** (Cl. 5 No. 5-63, tel. 2\/832-3053) looks and feels like a caf\u00e9 should. It's a nice place to hang out, maybe have a pastry, but you may want to stick with the pros at **Juan Valdez** (Parque Caldas, Cra. 7 No. 4-36, tel. 2\/839-5332) if you want a good brew. Se\u00f1or Valdez has quite the location on the western side of the park.\n\n**Cuarsenor Caf\u00e9** (Cra. 6 No. 3-85, tel. 2\/834-4040, COP$6,000) is hopping all day long, packed mainly with local workers. It's a bargain breakfast spot, but it also serves hearty lunches with trout or grilled chicken. There's an adjoining bakery. **Fruti Jugos** (Cl. 5 No. 7-66, tel. 2\/839-5826) serves gigantic, fresh Colombian juices out of plastic pitchers. So refreshing. You can even order aphrodisiac juices.\n\n#### **GETTING THERE AND AROUND**\n\nThere are frequent buses to Cali (COP$15,000, 4 hours) and Pasto (5 hours, COP$25,000) from the **Terminal de Transportes** (Tr. 9 No. 4N-125, Oficina 201, tel. 2\/823-1817, www.terminalpopayan.com), a modern bus station about a 15-minute walk from downtown. The **Aeropuerto Guillermo Leon Valencia** is nearby, and Popay\u00e1n is served by **Avianca** (Cra. 5 No. 3-85, tel. 2\/824-4505, www.avianca.com, 8am-noon and 2pm-6pm Mon.-Fri., 9am-1pm Sat.).\n\n#### **VICINITY OF POPAY\u00c1N**\n\n##### **Finca Alas y Raices**\n\n**Finca Alas y Raices** (Timbio, cell tel. 320\/742-3470, www.alasyraices.org, COP$50,000 tour and lunch pp) is an organic coffee farm run by a Colombian and Swiss pair. They welcome visitors for tours of their farm and explanations of the coffee-producing process.\n\n##### **Market at Silvia**\n\nThe **Market at Silvia** (Silvia, Tues. 5am-2pm) is a popular day trip destination from Popay\u00e1n. It is about an hour's bus ride (COP$6,000) away. On market days starting at dawn, Guambiano indigenous people converge on the market from nearby communities to buy and sell fruits, vegetables, and textiles. There are few handicrafts to purchase. The market and the people are photogenic; however, if you would like to take close-up photos of people, it's imperative to ask for permission first. The market occurs rain or shine. To get to the market, take a bus bound for Silvia from the Terminal de Transportes. They leave every 20 minutes or so, and the trip takes about an hour.\n\n##### **Termales de Coconuco**\n\nFor a dip in some _termales_ (hot springs), **Coconuco** (cell tel. 310\/543-7172, www.termalesaguatibia.com) is the place. Nestled among the hills are two springs, one _tibia_ (cool-ish) and one _herviendo_ (boiling hot), if sulfurous. Most of the action, especially on the weekend, takes place at the latter, as Colombian families while away the day (into evening) there. It's a gregarious scene and inevitably someone will share a box of _aguardiente,_ the anise-flavored liquor of choice here. But it's a calm environment during the week. The _tibia_ spring is more natural with a mud pool and an awesome water slide. It costs COP$10,000 to enter and the hot springs cost COP$4,000. The town of Coconuco is an indigenous community about 30 kilometers outside of Popay\u00e1n, and a bus ride there from the terminal (Trans. 9 No. 4N-125) to town takes about one hour, costing COP$4,000. From the town you have to walk about four kilometers along a well-marked road. On clear days you can get some good views of the Purac\u00e9 volcano.\n\n##### **Parque Nacional Natural Purac\u00e9**\n\nThe **Parque Nacional Natural Purac\u00e9** (COP$20,000 non-Colombians, COP$8,500 Colombian residents, COP$4,000 children) is a national park covering some 83,000 hectares (205,100 acres) that includes two important mountainous formations within the Cordillera Central (Central Mountain Range): the Serran\u00eda de los Coconucos (Coconucos Range) and the Masizo Colombiano (Colombian Massif). It is a region of immense environmental importance that was declared a UNESCO Biosphere Reserve in 1979.\n\nThe Serran\u00eda is a six-kilometer-long chain of volcanoes, including the snow-covered Pan de Az\u00facar (5,000 meters\/16,400 feet), Coconuco (4,600 meters\/15,100 feet), Purac\u00e9 (4,580 meters\/15,000 feet), and Sotar\u00e1 (4,400 meters\/14,400 feet). Purac\u00e9 and Sotar\u00e1 are currently active. The Masizo Colombiano is a mountainous formation where five major Colombian rivers are born: R\u00edo Magdalena and R\u00edo Cauca, which flow into the Caribbean; the R\u00edo Pat\u00eda, which flows into the Pacific; and the R\u00edo Caquet\u00e1 and R\u00edo Putumayo, which are tributaries of the Amazon.\n\nThe park includes over 30 _lagunas_ (mountain lakes), including the Laguna del Magdalena, which is the source of the R\u00edo Magdalena. Most of the park lies at an altitude greater than 2,600 meters (8,500 feet).\n\nThe primary attraction at this park is the fairly strenuous climb to the summit of the **Volc\u00e1n Purac\u00e9** (7 km) in the northern part of the park. The ascent, which requires about five hours up and three down, takes you through high mountain tropical jungle and then _p\u00e1ramo,_ a unique high mountain Andean ecosystem.\n\nGuides are available at the Pilimbal\u00e1 office, and it's a good idea to hire one. Call the Popay\u00e1n parks office (tel. 2\/823-1223) or the Pilimbal\u00e1 office in the park (tel. 8\/521-2578 or 8\/521-2579) in advance to arrange for a guide. Guides usually charge around COP$35,000 for the hike up to the crater.\n\nHigh season is from mid-December to mid-January, as well as during Semana Santa (Easter week) and school vacations from mid-June to mid-July. There are cabins in the park (COP$25,500-COP$35,000 pp high season), and camping is also available (COP$9,500 pp). To inquire about a reservation, contact the Pilimbal\u00e1 station (tel. 8\/521-2578 or 8\/521-2579). A restaurant here serves all three meals for under COP$5,000 each. If you would like to request a vegetarian meal or have other dietary needs, phone the restaurant in advance (tel. 8\/521-2577); ask for Se\u00f1ora Feliza.\n\n###### **GETTING THERE AND AROUND**\n\nThe main Pilimbal\u00e1 park ranger's station is 44 kilometers east of Popay\u00e1n. From Popay\u00e1n, buses depart the Terminal de Transportes (COP$5,000) bound for the community of La Plata. Buses generally leave at 6:30am, but there are often delays. The trip takes approximately 1.5 hours. Get off at the Cruce de la Mina, also known as El Crucero, and from there walk about 800 meters towards the left to the ranger's station. There are return buses coming from the opposite direction until around 5:30pm.\n\n#### M **TIERRADENTRO**\n\nFrom AD 500 to 900, the area of Tierradentro was settled by an agricultural society that dug magnificent decorated underground tombs, produced large stone statues, and built oval buildings on artificial terraces. As in the case of San Agust\u00edn, these people disappeared without a trace. We do not even know what they called themselves. By the time of the Spanish conquest, the area was inhabited by the Paez, a Chibcha-speaking people who still inhabit the area and who are organized in _cabildos_ (indigenous ruling bodies) that are recognized by the Colombian government.\n\nthe well-preserved burial chambers of Tierradentro Archeological Park\n\n**Tierradentro** is the site of a major indigenous necropolis that includes monumental funeral statues and hypogea (underground burial chambers). These chambers, some 12 meters wide, are decorated with intricate red and black anthropomorphic and zoomorphic geometric designs, some of which are in relief. This archaeological park was declared a UNESCO World Heritage Site in 1995.\n\nThese awe-inspiring burial chambers reveal the existence of a rich and complex society that devoted significant time and effort to preparing the way to the afterlife. There is also an intriguing symmetry to be found between these ornate underground chambers and the houses of the living above ground. Though there are some similarities between the statues and tombs of San Agust\u00edn, it is believed that these were two distinct cultures.\n\n##### **Sights**\n\n###### **TIERRADENTRO ARCHEOLOGICAL PARK**\n\n**Tierradentro Archeological Park** (Inza, 91 km east of Popay\u00e1n, www.tierradentro.info, 8am-4pm daily, COP$20,000) is a circuit of five sites set on four hills _(altos)_. If you are staying at La Portada in San Andr\u00e9s de Pisimbal\u00e1, you can avoid the walk down to the main entrance of the park (30 minutes) and take the path that leads straight from the hostel to the Alto de San Andr\u00e9s, one of the five sites, and continue onward towards Alto de Aguacate and the museum and park office.\n\nIt's better to start at the museum, so you can get a good introduction to the park before you tackle the sites. The sites are best visited in a counterclockwise manner. There are two museums across the street from each other. On the Aguacate side is the archaeological museum, and on the other side across from the ticket booth is the ethnographic museum. After checking out the museums and purchasing your ticket, you can continue to **Alto de Segovia,** which has some of the most impressive burial chambers. They are all very well preserved. Beyond **Alto del Duende** is **El Tabl\u00f3n,** which has large stone statues similar to those of San Agust\u00edn. You will follow the road to get there.\n\nstone carving at Tierradentro Archeological Park\n\nAfter crossing through the town, just next to La Portada is the path to the **Alto de San Andr\u00e9s.** Parts of the trail\u2014especially between Alto de San Andr\u00e9s and Alto de Aguacate\u2014are poorly marked. When in doubt, keep going left and upwards\u2014and don't panic. You will eventually reach the top of the mountain ridge. Along the way you may run across _campesinos_ (farmers) tending to their coffee and banana plantations who can point you in the right direction. Getting to the **Alto de Aguacate** is a steep climb, but the views of the countryside are breathtaking. The entire circuit can be done in a day\u2014a very long day. Your admission ticket is good for two days, so you could go at a slower pace and visit half the first day and the remaining half on the second day. If you are in a hurry, check out the most impressive site, the Alto de Segovia. You can also inquire about a horseback tour of the park. La Portada hotel can arrange that for you.\n\nA good flashlight is a must to be able to get a good look at the interior paintings of the tombs and some of the elaborate artwork decorating them. Rubber boots could be useful if you are there during the rainy months of March-June or September-November. Also, be sure to bring water and snacks. It's a tough six-hour walk from La Portada to Alto de San Andr\u00e9s, Alto del Aguacate, and down to the museum and park entrance. And that's just a little more than halfway. Near the museums you can get refreshed with a juice, water, or homemade ice cream at one of the little stores or nearby hostels.\n\nIn Alto de Segovia and Alto del Duende, a park guard will open up some of the tombs and let you in and back out. The steps can be quite steep and obviously on the dark side.\n\n###### **BIBLIOTECA P\u00daBLICA LA CASA DEL PUEBLO DE GUANACAS**\n\nAfter Tierradentro, you can visit the **Biblioteca P\u00fablica La Casa del Pueblo de Guanacas** (8am-6pm daily), an award-winning public library in the town of Guanacas. It's made of _guadua,_ a native species of bamboo, and was the idea of a couple of young locals, who presented the idea of a community library to the architecture department of the Universidad Javeriana in Bogot\u00e1. Sim\u00f3n Hosie, an award-winning architect who involves local communities intensely in the design process, took on the project, the Japanese Embassy provided funding, and residents of the village, both young and old, participated in its construction. The library opened in 2004.\n\nTo get there, take a bus toward Popay\u00e1n from the Tierradentro site and ask to get off at Guanacas. It's an easy walk down to the library.\n\n##### **Accommodations**\n\nBasic accommodations are plentiful near the official park entrance and museum, and are all located on the same road. **Hospedaje Pisimbal\u00e1** (cell tel. 311\/605-4835 or 321\/263-2334, COP$20,000 pp) is a neat and comfortable option, with a cool thatched roof gazebo in front. There's a good restaurant as well. **Lucerno** (no tel., COP$15,000 pp) is bare-bones, and the water may be cold, but the owners of this hostel, Secundino and Carmelita, are extremely friendly. It was one of the first such hostels around, and was given its name by a Swiss visitor from Lucerne. It has nine rooms. **El Refugio** (tel. 2\/825-2904, hotelalbergueelrefugio@gmail.com, COP$60,000 d) is the largest and most luxurious option. It's even got a pool.\n\nM **La Portada** (cell tel. 311\/601-7884, laportadatierradentro@hotmail.com, www.laportadahotel.com, COP$30,000 d w\/private bath, COP$20,000 s) is the best option. In the town of San Andr\u00e9s de Pisimbal\u00e1, it is very well-run and orderly, and it's made out of _guadua,_ a type of bamboo (even the shower curtain rods). Leonardo, the owner, is friendly and full of good information. His wife is an excellent cook, too.\n\n##### **Getting There**\n\nThe trip to Tierradentro from Popay\u00e1n will take you through gorgeous countryside of farms, villages, and gentle mountains shrouded in mist.\n\nOne bus a day leaves from Popay\u00e1n directly to San Andr\u00e9s de Pisimbal\u00e1. It leaves at 10:30am and takes four hours. From San Andr\u00e9s de Pisimbal\u00e1 to Popay\u00e1n there is one direct bus that departs at 6am.\n\nAnother option is to take one of the buses bound for Inz\u00e1 and get off at El Cruce, which is a short walk from the museum area. These leave at 5:30am, 8:30am, 10:30am, 1pm, and 3pm. It is a five-hour trip and costs COP$20,000.\n\nFrom San Agust\u00edn you can take the 6am bus bound for Bogot\u00e1, then transfer in Garz\u00f3n to the bus towards La Plata. From there you can take the 10:30am bus to San Andr\u00e9s de Pisimbal\u00e1. That trip takes about three hours. Going from Pisimbal\u00e1 to San Agust\u00edn you can take a bus at 6am. It arrives in the early afternoon. Public transportation between San Agust\u00edn and Tierradentro will require various transfers.\n\n#### M **SAN AGUST\u00cdN**\n\nThe small colonial town of San Agust\u00edn, nestled within the folds of the southern Colombian Andes, would probably be an attractive destination in its own right. At an elevation of 1,800 meters (5,900 feet), it is set in a place of enormous natural beauty and has wonderful spring-like weather. However, its fame comes from its location near the largest pre-Columbian archaeological site south of Central America and north of Per\u00fa. From approximately AD 100-800, this region was home to an indigenous culture that produced spectacular monumental funeral statues hewn out of volcanic rock. Researchers do not know what these people called themselves, and so, lacking a better name, they have been labeled the San Agust\u00edn Culture.\n\nTo visit the main archaeological sites near San Agust\u00edn, you will need about two days. However, it makes no sense to rush your stay here, as it is an incredibly pleasant and peaceful place to visit, with several options for hiking and rafting amidst spectacular mountain landscapes.\n\n###### **HISTORY**\n\nThe area around San Agust\u00edn was occupied as early as 3300 BC. Starting in the 1st century AD, the people of San Agust\u00edn created hundreds of monumental funeral stone statues set on large platforms. Very little is known about these people, except that they lived from agriculture and formed fairly compact settlements. By AD 800, this society had mysteriously vanished and other indigenous peoples coming from the Amazon basin occupied the area. Today, the inhabitants are predominantly mestizo.\n\nSpanish chroniclers of the 16th century mention the statues, as did Colombian naturalist Francisco Jos\u00e9 de Caldas in the early 19th century. However, it was not until 1913 that German ethnographer Konrad Theodor Preuss conducted the first systematic excavation. When he was finished, he carted off (probably illegally) 35 large statues that are now at the Ethnographic Museum in Dahlem on the outskirts of Berlin. Dav\u00edd Dellenback, American-born researcher and long-time resident of San Agust\u00edn, discovered on a 1992 trip to Berlin that only three of these statues are on exhibit, while the rest were piled away haphazardly in a storehouse. He found no trace of the numerous ceramic pieces that Preuss took and which were probably destroyed during the Allied bombings of World War II. Dellenback and his wife, Martha Gil, have promoted a petition, signed by more than 1,800 residents of San Agust\u00edn and nearby towns, for the German authorities to return the statues to their rightful home. Dellenback and Gil coauthored a comprehensive guide to San Agust\u00edn called _The Statues of the Pueblo Escultor,_ which is available in shops in San Agust\u00edn.\n\nSan Agust\u00edn is Colombia's most important archaeological site.\n\n##### **Sights**\n\nThe extremely well-maintained **Parque Arqueol\u00f3gico de San Agust\u00edn** (8am-4pm daily, COP$20,000) covers 80 hectares (200 acres) of what was one of the most important ritual areas of the San Agust\u00edn Culture. It was first established in 1937 and was declared a UNESCO World Heritage Site in 1995. The park contains over 130 statues with striking human and animal-like features, as well as carved tombs and monumental stone tables or dolmens. Your ticket also valid for Parque Arqueol\u00f3gico Alto de los \u00cddolos.\n\nNear the entrance to the park is the **Museo Arqueol\u00f3gico** , which contains smaller statues, pottery, tools, and jewelry, as well as explanations on the San Agust\u00edn culture. At the time of writing, the museum was undergoing renovations. Most of the monumental stone objects are concentrated on four funeral hills designated Mesitas A, B, C, and D. The pleasant **Bosque de las Estatuas** contains a path that meanders among statues relocated from other locations in the San Agust\u00edn area. The unusual **Fuente de Lavapi\u00e9s** contains bas-reliefs of human and animal figures sculpted on the rocky bed of a stream. This is the only non-funeral site in the park. On a hill behind the _fuente_ is the Alto de Lavapi\u00e9s, where excavations have shown human presence dating back to 3300 BC. The park is easy to navigate: plan on a couple of hours to stroll leisurely and absorb the beauty. The park is two kilometers (1.2 miles) west of town and can be reached on foot or by bus.\n\nThere are four smaller, yet definitely worthwhile sights, near the town of San Agust\u00edn: **La Chaquira** , a large bas-relief sculpted in a rock face overlooking the spectacular R\u00edo Magdalena gorge; **El Purutal,** which contains two magnificent, very well-conserved statues with original bright pigment covering; and **La Pelota** and **El Tabl\u00f3n** , each with additional funeral monuments.\n\nEight kilometers (five miles) southwest of the town of Isnos on the other side of the Magdalena, the **Parque Arqueol\u00f3gico Alto de los \u00cddolos** (8am-5pm daily, COP$20,000) is the second largest archaeological park. It includes an anthropomorphic statue that measures 4.3 meters (14 feet), along with large sarcophagi. A ticket here is also valid for Parque Arqueol\u00f3gico Nacional de San Agust\u00edn. The much smaller **Parque Arqueol\u00f3gico Alto de Las Piedras** (8am-5pm daily, free), six kilometers (four miles) north of Isnos on the road that leads to the Salto de Borodones waterfall, has statues and tombs with original pigments.\n\nAlso in the vicinity of San Agust\u00edn are several natural sights. The entire upper gorge of the **R\u00edo Magdalena** is majestic. At **El Estrecho,** the river spurts thought a 2.2-meter (7-foot) rocky funnel. On the Isnos side are two waterfalls, the 400-meter (1,300-foot) **Salto de Borodones** and 200-meter (650-foot) **Salto de Morti\u00f1o**.\n\nArmed with a map and a willingness to ask directions, it is possible to visit all the sights on the San Agust\u00edn side by foot. This can make for a wonderful escape. However, many tourists opt for half-day or full-day horseback tours, which can cover several of these sights, including the Parque Arqueol\u00f3gico Alto de \u00cddolos. Make sure to book reliable guides that take good care of their horses through your hotel. Rates depend on the length of ride and number of people. Visiting the sites on the Isnos side by public transportation can be difficult; most people opt for day-long jeep tours that cover the Parque Arqueol\u00f3gico Alto de los \u00cddolos, Parque Alto de Las Piedras, El Estrecho, and the waterfalls.\n\n##### **Recreation**\n\nThe upper R\u00edo Magdalena offers some of the best white-water river rafting in Colombia, set within spectacular mountain landscapes. **Magdalena Rafting** (www.magdalenarafting.com) offers numerous options, both for novices and experienced rafters, starting from short 90-minute to day-long trips.\n\nUntil recently, hiking in the Masizo Colombiano was no-go because of security. Now it is possible to ascend to the spectacular upper reaches of these mountains. One easily organized trip is to the **Laguna del Magdalena,** birthplace of the R\u00edo Magdalena, located in the P\u00e1ramo de la Papas at 3,327 meters (10,915 feet). The trip takes 3-4 days and can be done on foot or by horse.\n\n##### **Accommodations**\n\nThere are several cute and friendly accommodations options in San Agust\u00edn, and they can each provide expert advice on the area, arrange recreational activities such as horseback riding and rafting, and assist with travel needs.\n\nM **El Maco** (tel. 8\/837-3437, cell tel. 320\/375-5982, www.elmaco.ch, COP$16,000-COP$50,000 pp) is set among the hills about a 20-minute walk from town. Accommodations consist of seven small cabins, a tipi, a chalet, and an indigenous-style _maloka_ (cabin) distributed throughout some pleasant gardens, home to dogs and chickens. They can organize some great horseback riding trips to archaeological spots. El Maco is run by Rene, who moved to the area from Switzerland in 1994. **La Casa de Francois** (tel. 8\/837-3847, cell tel. 314\/358-2930, www.lacasadefrancois.com, COP$18,000 dorm, COP$50,000 d) offers cabins, two dorm rooms with 10 beds total, and lots of hammocks to laze about after a hard day's work of archaeological exploration. The restaurant is quite good here, and reasonably priced, and they bake their own bread. It's 10 minutes from town.\n\nColombian-run **Huaka Yo** (Bogot\u00e1 tel. 1\/489-9269, cell tel. 320\/846-9763, www.huakayo.com, COP$116,000 d high season) is just 200 meters from the archaeological park. It consists of a large house that has 12 rooms and six loft spaces, each with its own private bath. Huaka Yo is a five-minute bus ride from town or a 20-minute walk. They prefer that you make a deposit to their bank account to hold reservations, which means less cash you have to carry with you. The **Casa del Japones** (Cl. 8 No. 12-83, cell tel. 312\/525-9552, COP$12,000 pp) is a quiet and small hostel (only eight rooms) in a cozy country house outside of town. It's an economical option, and vegetarian food is on offer. The banana pancakes exceed one's expectations.\n\nhorses at the ready, San Agust\u00edn\n\n**Alto de los Andaquies Hostal** (cell tel. 312\/444-7368 or 316\/635-6006, www.andaquies.com, COP$30,000 pp) has 15 rooms with hot water spread over three rustic houses just one kilometer from both the town and archaeological park. They produce coffee on this lush farm and other crops such as plantains, bananas, and yucca. They organize two- to four-day tours of all the major sights (COP$250,000 pp d all-inclusive).\n\n##### **Food**\n\nMost restaurants are on the main drag in town, the Calle 5. **Donde Richar** (Cl. 5 No. 23-45, cell tel. 312\/432-6399, noon-7pm daily, COP$23,000) is famous for its grilled meats. **Tomate** (Cl. 5 No. 16-04, cell tel. 314\/265-5527, 8am-3pm Thurs.-Mon., COP$8,000) is San Agust\u00edn's veggie headquarters, serving breakfast and a daily special. For pastas and pizzas go to Ugo's place, **Restaurante Italiano** (Vereda El Tabl\u00f3n, cell tel. 314\/375-8086, 6pm-9:30pm Thurs.-Sun., COP$15,000). It's a bit outside of town, but worth the trip.\n\n##### **Getting There and Around**\n\nBuses from Bogot\u00e1 go through Neiva. Buses from Neiva usually stop in the town of Pitalito, 138 km south, and from there you have to take a shared taxi to San Agust\u00edn (COP$5,000), about 45 minutes away.\n\nFrom Popay\u00e1n there are about four buses per day to Pitalito, and you'll have to get off beforehand at San Agust\u00edn (COP$35,000, 7 hours). This journey takes you through the spectacular scenery of Parque Nacional Natural Purac\u00e9.\n\nFrom Tierradentro, take a bus to La Plata (COP$12,000, 2.5 hours) and change there for Pitalito (COP$22,000, 2.5 hours), where you can hop on a shared taxi for San Agust\u00edn (COP$5,000, 45 mins.).\n\nUpon arrival in Tierradentro, you may be left at El Cruce. From there it's about a COP$5,000 truck ride into town. Once you step off in El Cruce, you may be swarmed by insistent touts who earn a commission at hotels for scoring some guests.\n\nBus and taxi companies have their offices at the corner of Calle 3 and Carrera 10 in the town of Tierradentro. In town, buses shuttle folks about town on a regular basis, although you can usually get where you want to go by walking.\n\n**The False Indian Ambassador**\n\nIn December 1962, the high society of the dusty provincial capital of Neiva was delighted by the unannounced visit of the Indian ambassador to Colombia, Shari Lacshama Dharhamdhah. He was on a private visit to Neiva, planning to travel onward to see the archaeological sites of San Agust\u00edn. But some things about him were odd. For example, he arrived by train and without clothes befitting a high ranking diplomat. He explained that his limousine had broken down on the trip from Bogot\u00e1 and that his luggage would be forwarded to Neiva. Yet, he looked Indian, spoke garbled Spanish, practiced yoga, and was a vegetarian.\n\nNews of his arrival was conveyed by a passenger sitting next to him on the train and spread like wildfire throughout Neiva, a city where nothing eventful ever happened. He was lodged in the presidential suite of Hotel Neiva Plaza (the city's most elegant hotel), presented with new clothes from Neiva's finest haberdashery, and wined and dined by leading politicians, business people, and socialites for five days.\n\nAlas, it turned out that the ambassador was not really an international diplomat. Colombia did not even have diplomatic relations with India at that time. The man's real name was Jaime Torres and he was a high school teacher in the nearby city of Ibagu\u00e9. A former schoolmate identified him and gave away his secret. He was briefly detained, but, as he had not committed any crime, he was freed. Not much is known about Torres's whereabouts after this incident, except that he eventually moved to New Haven, Connecticut.\n\nThe crazy story was made into a comedic film, _El Embajador de la India,_ in 1987.\n\n##### **Desierto de Tatacoa**\n\nLocated next to the R\u00edo Magdalena and within view of the snowcapped Nevado del Huila, the 330-square-kilometer (127-square-mile) **Desierto de Tatacoa** makes for an unusual day trip. Technically it's not a desert, just a semi-arid zone with dry tropical forest, but it feels and looks like one.\n\nA visit to Tatacoa includes a stop at the historic town of **Villavieja,** with its 17th-century **Capilla de Santa Barbara** (Plaza Principal Villavieja, restricted hours), a church founded by the Jesuits in honor of the indigenous cacique Tocaya, who was killed by the Spaniards. The **Museo Paleontol\u00f3gico** (Cl. 3 No. 3-05, 7:30am-1pm Mon.-Fri., 7am-6pm Sat.-Sun., COP$2,000) provides a sample of the many fossils, dating from 3.8 million years ago, that have been found in the desert.\n\nThe dry conditions at Tatacoa make for ideal stargazing. The **Observatorio Astron\u00f3mico Tatacoa** (Vereda El Cuzco, tel. 8\/879-7584, cell tel. 310\/465-6765, www.tatacoa-astronomia.com, 6:30am-9pm, COP$5,000), near town, provides telescopes for visitors to scan the sky in the evening hours. Camping is permitted next to the observatory. Each year in June-July, usually over a long weekend, hard-core and novice astronomers head to the desert for the annual Star Party event. There is a restaurant next to the observatory that serves mostly grilled goat or mutton.\n\n###### **TOURS**\n\n**Chaska Tours** (Hostal El Maco, Neiva, tel. 8\/837-3437, cell tel. 311\/271-4802, info@chaskatours.net, COP$100,000 pp) organizes day trip tours from San Agust\u00edn to Tatacoa. There is an option to spend the night in the desert.\n\n##### **Neiva**\n\nThe extremely hot (average highs 32-35\u00b0C\/90-95\u00b0F) capital of the Huila department (province), Neiva (pop. 340,000) has few attractions for tourists. However, if you do happen to stop here, a nice place for a stroll and sitting in an outdoor caf\u00e9 is the **Malec\u00f3n R\u00edo Magdalena** (Av. Circunvalar from Cacica Gaitana monument to the Caracoli docks). This is the boardwalk along the R\u00edo Magdalena.\n\nNeiva really gets going during the **Fiestas de San Juan y San Pablo** (June 15-June 30). The highlight is the Reinado del Bambuco, a beauty queen pageant named after _bambuco,_ a traditional music of the Colombian Andes. Entrants, in flowery and frilly costumes, must dance the Sanjuanero, a famous _bambuco,_ and other folk songs.\n\n###### **ACCOMMODATIONS**\n\nCatering to business clientele, the **Hotel Neiva Plaza** (Cl. 7 No. 4-62, tel. 8\/871-0806, www.hotelneivaplaza.com, COP$189,000 s, COP$243,000 d) is in front of the Parque Santander. It has 86 spacious rooms, a gym, and a pool. It's the reliable (and overpriced) choice in Neiva.\n\nIn the Tatacoa desert, and within walking distance of the Observatorio Astron\u00f3mico Tatacoa, **Posada Elvira Clever** (Vereda El Cusco, cell tel. 312\/559-8576, COP$20,000 pp) has two rooms available in her small and simple zinc-roof house. There is no electricity here. In Villavieja, **La Casona** (Cl. 3 No. 3-60, tel. 8\/879-7636, cell tel. 320\/243-9705, <http:\/\/hotellacasonavillavieja.blogspot.com>, COP$50,000 d) is a small hotel in an old house located on the main park, with friendly owners. Sim\u00f3n Bol\u00edvar is said to have stayed here.\n\n###### **GETTING THERE AND AROUND**\n\nIn addition to connections with the major cities of Colombia, from the **Terminal de Transportes de Neiva** (Tr. 5 No. 53-12, tel. 8\/873-1232) in the south of the city, you can catch a minibus to Villavieja (COP$7,000, 1 hr.) and to San Agust\u00edn (COP$25,000, 4 hrs.). The **Aeropuerto Benito Salas** (Cra. 6 No. 32-45, tel. 8\/875-8198) is in the north on the road towards Tatacoa. If you'd like to explore the region with your own wheels you can rent a vehicle at **ANT Rent A Car** (Av. 26 No. 5-12, tel. 8\/872-2859, ant.rentacar@hotmail.com). It also has chauffeured cars available.\n\nIn Villavieja, _mototaxis_ regularly transport visitors to the desert sights. These cost about COP$15,000 per person.\n\n## **THE PACIFIC COAST**\n\nHIGHLIGHTS\n\nHISTORY\n\nSAFETY\n\nPLANNING YOUR TIME\n\nChoc\u00f3\n\nBAH\u00cdA SOLANO\n\nEL VALLE\n\nM PARQUE NACIONAL NATURAL UTR\u00cdA\n\nNUQU\u00cd\n\nINFORMATION AND SERVICES\n\nGETTING THERE AND AROUND\n\nSouth Pacific Coast\n\nBUENAVENTURA\n\nTUMACO\n\nM PARQUE NACIONAL NATURAL ISLA GORGONA\n\nM SANTUARIO DE FLORA Y FAUNA MALPELO\n\nColombia is the only country in South America with coastline on both the Atlantic and the Pacific Oceans. While the Caribbean coast is developed and populated, Colombia's Pacific coast is wild, remote, and mysterious. For the few who venture to this little-visited area, the beaches, jungles, and people are simply unforgettable.\n\n**HIGHLIGHTS**\n\nLOOK FOR M TO FIND RECOMMENDED SIGHTS, ACTIVITIES, DINING, AND LODGING.\n\nM **Estaci\u00f3n Septiembre Sea Turtle Hatchery and Release Program:** Witnessing a release of valiant baby sea turtles fearlessly scampering across the sands into the crashing waves of the Pacific Ocean is an unforgettable experience (click here).\n\nM **Parque Nacional Natural Utr\u00eda:** Jungle walks, swimming, and whale-watching are the order of the day when you stay at this beautiful national park near Nuqu\u00ed (click here).\n\nM **Parque Nacional Natural Isla Gorgona:** Nature has reclaimed this island, which was once Colombia's Alcatraz. Walk through the jungle to Isla Gorgona's beautiful secluded beaches (click here).\n\nM **Santuario de Fauna y Flora Malpelo:** There's incredible diving among hammerhead sharks around this tiny rocky island almost 500 kilometers from the Colombian coast (click here).\n\nThe Pacific coast of Colombia extends for 1,392 kilometers (865 miles), just under the length of the California coast. The departments of Choc\u00f3, Valle de Cauca, Cauca, and Nari\u00f1o each share real estate on the Pacific, with Choc\u00f3 boasting the largest stretch of coast. The tiny islands of Gorgona and Malpelo, both national parks, are 35 kilometers (22 miles) off the coast of Guapi and 490 kilometers (305 miles) from Buenaventura, respectively.\n\nThe Pacific coast is populated by indigenous peoples, primarily the Ember\u00e1, who live mainly in small riverside settlements in the interior of Choc\u00f3; _colonos_ or \"colonists\" who have arrived from Antioquia for generations; and Afro-Colombians, descendants of African slaves who make up the majority of the population (over 80 percent).\n\nIt is a sparsely populated region, but there are three major cities: Buenaventura (pop. 363,000), Colombia's most important port; Tumaco (pop. 188,000), bordering Ecuador; and Quibd\u00f3 (pop. 162,000), the interior capital of the department of Choc\u00f3. Those cities are of far less interest to tourists than the coastal towns of Nuqu\u00ed and Bah\u00eda Solano, where both sea and land provide countless opportunities to appreciate the natural world.\n\nThere are four national parks or protected areas on the Pacific coast. These include the Parque Nacional Natural (PNN) Utr\u00eda, halfway between Bah\u00eda Solano and Nuqu\u00ed; the less visited PNN Sanquianga near Tumaco; the PNN Isla Gorgona, a spectacular island park that was once a prison; and the Santuario de Fauna y Flora (SFF) Malpelo, an internationally known wildlife preserve that offers superb diving.\n\n#### **HISTORY**\n\nIn pre-Columbian times, the region was inhabited by indigenous groups, notably the Tumaco-La Tolita people, who produced stunning goldwork. During the colonial era and into the early 20th century, the entire Pacific coast corridor was governed (or rather, not governed at all) from the city of Popay\u00e1n. The main economic activity was gold mining, mostly in the Pat\u00eda river valley in Nari\u00f1o and the Choc\u00f3 region. Afro-Colombian people, now more than 80 percent of the region's population, are descendants of slaves who were forced to work in these mines and in the haciendas of the Cauca river valley.\n\nThese slaves began arriving in New Granada from different parts of Africa starting in the 17th century. Most arrived in Cartagena and were immediately separated from their families, sold, and transported across the colony. Some slaves revolted or escaped from their owners and established communities in inaccessible areas. After the abolition of slavery in 1851, many former slaves migrated and settled along the rivers and coast of the Pacific. These communities survived intact through the 20th century but, in recent years, have been severely affected by the internal conflict. Many displaced Afro-Colombians from the Pacific have resettled in cities such as Cali.\n\nDue to a lack of transportation links, the Pacific region has always been isolated from the rest of Colombia. While rich in biodiversity and cultural identity, cities such as Buenaventura, Quibd\u00f3, and Tumaco are underdeveloped; plagued with inadequate government services, poor infrastructure, and lack of economic opportunity. Poverty rates in urban areas often exceed 70 percent. Outside the cities, inhabitants subsist on small scale farming, fishing, and illegal gold mining and forestry.\n\nOne emerging bright spot is ecotourism, especially around Bah\u00eda Solano and Nuqu\u00ed, where many enterprising Paisas from Antioquia have started tourist-related businesses. A more recent development has been the strengthening of local organizations, which are making strides in sharing the tourism peso with the broader community. The government has been promoting the region, especially for whale-watching.\n\n#### **SAFETY**\n\nThough most of the coastal settlements of interest to tourists are quite safe, parts of the Pacific, especially in the Nari\u00f1o department bordering Ecuador and the Choc\u00f3 department bordering Panama, remain major drug-trafficking corridors. The abrupt geography, dense jungle, easy water transportation routes, and lack of government border control have attracted drug traffickers, paramilitaries, and FARC guerrillas alike. With a strong and visible police and military presence, the Bah\u00eda Solano-Nuqu\u00ed area is considered safe. Those towns are OK to walk around at night. The interior jungles of the Pacific provide cover for drug traffickers and illegal groups. Ask hotel staff for updated security tips.\n\n#### **PLANNING YOUR TIME**\n\nIt requires more effort (and more money) to visit the Pacific coast than other parts of Colombia, so it would be a shame to just spend just a couple of days there. Plan for about five days, split across two locations. Visit between July and October, when humpback whales travel 8,000 kilometers from Antarctica to give birth to their young in the warm waters off of the Colombian coast.\n\nFy into Bah\u00eda Solano and fly out of Nuqu\u00ed, spending two or three days in each area, and staying within the Choc\u00f3 department. Time here can be complete beach relaxation, filled with activities such as jungle hikes and canoe trips, or a combination of both.\n\nThere is not much activity at night, save for some excellent stargazing on the beach, at least in northern coastal communities. This might us the perfect place to decompress and catch up on sleep.\n\nThree full days are sufficient to visit Parque Nacional Natural Isla Gorgona. If plan on diving or lessons, spend a full week.\n\nTourism picks up during humpback whale-watching season, especially during Colombian school vacations in August. The end-of-year holiday season is also a popular time, especially for Colombian families. Reserve far in advance to stay at some of the higher-end hotels during the holidays. Other times of the year are fairly quiet.\n\nRegardless of where you go, it will rain during your stay, maybe once or twice a day. Choc\u00f3 is one of the rainiest places on the planet.\n\n##### **What to Take**\n\nHot and humid weather the norm (the average temperature is 28\u00b0C\/82\u00b0F), so plan on getting wet each day\u2014either on purpose or by accident. Daytime activities nearly always involve water. For walks along rivers, sandals with traction or rubber boots are very useful (flip-flops not so much). Clothes and towels take days to dry, sometimes refusing to dry altogether. Bring only lightweight clothing, packing a few more T-shirts than you usually would. And have them ready to throw in the nearest washing machine when you arrive in the city.\n\nBug repellent is a good idea, although, thanks to a pleasant breeze most evenings, mosquitoes are not usually a problem. There's no need to take malaria pills or any extraordinary precautions. All hotels provide mosquito netting, usually covering all windows in addition to a mosquito net over your bed.\n\nWaste management is an issue for towns, hotels, and restaurants along the coast. Beer and soda in bottles is preferable to aluminum cans, as the glass is returnable. However, hydration is very important. Tap water isn't safe; hotels all provide filtered or boiled drinking water for guests. Bring a heavy-duty water bottle that can be refilled again and again. If you don't see filtered water readily available for guest use, don't be shy to ask kitchen staff to refill your bottle with _agua filtrada_ (filtered water). Bags _(bolsas)_ of water are usually sold in corner stores and are preferable to plastic bottles. There is an expanding landfill in the jungle between El Valle and Bah\u00eda Solano; however, you may want to spare the rainforest by taking accumulated plastic back with you for recycling in Medell\u00edn or another large city. This is a requirement on the pristine island of Gorgona.\n\nBinoculars are good for jungle outings and excellent for whale-watching. Take along only the most necessary electronics with you to the jungle. Extreme humidity, river crossings, sudden rain showers, and bumpy boat rides are not kind to laptops and cameras. Waterproof bags for cameras are essential, as well as sachets of silica gel to throw in camera cases.\n\nHere on the Equator, it gets dark at around 6pm every evening year-round. There is limited electricity in this part of Colombia, with generators usually cranking on for only a few hours at night. A flashlight is good for strolls on the beach or about town. A lack of nighttime entertainment makes this a great time for reading. Pack books and magazines rather than relying on electronic gadgets. A deck of cards can also provide some evening entertainment.\n\nBring cash. Credit cards are not accepted at hotels. At some of the top-end places, you can make a deposit into the hotel's bank account before arriving, limiting the amount of cash you carry.\n\n### **Choc\u00f3**\n\nYou fly into Choc\u00f3 over green mountains, part of the Cordillera Occidental, punctuated by orderly Antioquian pueblos and pastureland. But then, the jungle begins: thick, impenetrable tropical forest, of a thousand shades of green. This lowland tropical forest, with vegetation not unlike that found in the Amazon, begins to rise with undulating forested hills as you pass over the smaller Serran\u00eda del Baud\u00f3 mountain range, one of the rainiest places on Earth. If you look down you will notice the tops of these low mountains shrouded in clouds. Every once in a while a milk-chocolate brown river meanders its way through the jungle westward, and tiny Afro-Colombian and Ember\u00e1 communities of thatched or zinc-topped roofs spring up alongside them. Here rivers and streams are the only means of transportation, just as it has been for centuries. Just after the captain announces your \"initial descent,\" you suddenly find yourself above the turquoise coastal waters of the Pacific. Get a window seat.\n\nAlong the Pacific coast of the Choc\u00f3 department, there are four general areas that have decent tourism infrastructure: Bah\u00eda Solano, El Valle, the Parque Nacional Natural Utr\u00eda, and Nuqu\u00ed.\n\n#### **BAH\u00cdA SOLANO**\n\nWhen many visitors arrive at the tiny Bah\u00eda Solano airport, after they collect their bags they head straight to the village of El Valle about 22 kilometers (14 miles) away and to the hotels on the beaches of Playa Almejal. But Ciudad Mutis, as Bah\u00eda Solano is officially named (after the famed botanist), is actually a good base for your visit\u2014there's no need to rush off. It is one of Colombia's sport-fishing capitals, and excellent diving and whale-watching excursions can be arranged from here. The town, although the largest one in the Choc\u00f3 Pacific coast, is small, and everywhere is accessible on foot.\n\nNice jungle walks to swimming holes fed by crystalline freshwater waterfalls are within walking distance from the town, and depending on the tides, you can also walk to the beaches of **Punta Hu\u00edna** and **Playa Mecana**. These can also be easily reached by boat. During low tide, the bay becomes a soccer field; when the tide comes in, it's a place to cool off.\n\nBah\u00eda Solano\n\n##### **Recreation**\n\n###### **HIKING**\n\nThere are three easily done walks in or around Bah\u00eda Solano. **Punta Hu\u00edna, Playa Mecana,** and the **Cascadas Cocacola** can also be reached on foot, and are all under two hours walking from town. You may want to go with a guide (hotels can arrange this) at least the first time, and find out the day's tide information before heading out. If the tide has come in, you'll have to take a boat back to the town.\n\nThe **Virgen de la Loma** path is an easy, 30-minute climb up through lush vegetation to the top of a hill with a nice view of the bay. The entrance to the trail is well-marked and is only 20 meters from the Hostal del Mar. For this you'll need no guide. About two blocks from there on the west side of town is the trail to the **Cascada Choc\u00f3latal,** which takes you along a river to a roaring waterfall. Taking about 40 minutes or so, this hike may not necessarily require a local guide, but it may be helpful, especially so you'll have someone with trained eyes to point out the occasional colorful frog, lizard, bird, or humongous spider to you. Otherwise they are quite tricky to spot. You will have to crisscross the narrow, shallow creek several times, and it can be treacherous as the rocks are slippery. You'll need to have both hands free. Wear a bathing suit so you can frolic in the cool waters of the swimming hole at the five-meter waterfall. That's your refreshing reward!\n\nThe third hike is called the **Cascada del Aeropuerto.** This hike is right across the street from the airport and leads to a towering jungle waterfall. You probably won't need a guide, but it is not impossible to become lost. Follow the stream and note that it will eventually veer to the left. It's about a half hour walk. All of these paths are maintained by a group of community members with the help of high school students who have posted signs to point the way.\n\n###### **SPORT-FISHING**\n\nThe waters off of the Choc\u00f3 coast are excellent for sport fishing. The **Posada Tur\u00edstica Rocas de Cabo Marzo** (tel. 4\/682-7525, cell tel. 313\/681-4001, bahiatebada@hotmail.com, www.posadaturisticarocasdecabomarzo.com) regularly organizes fishing adventures in the area. Species that can be found in the waters here include marlins (blue, black, and striped) and sailfish. These are considered endangered and are caught and then released. Yellowfin tuna, red snapper, wahoo, and sierra are other fish that are caught and eaten. The best time of the year for fishing is March-June. Between October and December is also a good time to fish (catch and release) for marlin and sailfish, but this is a rainy season and the waters are rough. Rental of a boat and an eight-hour day excursion costs around COP$1,600,000, with a maximum of four fishers.\n\nRocas de Cabo Marzo organizes catch-and-release tournaments like the **Torneo Internacional de Pesca Deportiva.** The tournament takes place far from the coastline, as much of the Choc\u00f3 coast, especially around the Cabo Marzo area, is protected as a Zona Exclusiva de Pesca Artesanal del Choc\u00f3 (ZEPA). That means it is limited to only local fishers using their traditional fishing methods. This initiative is supported by Conservation International (www.conservation.org.co).\n\n###### **DIVING**\n\nDiving in the Pacific is much different from diving in the Caribbean. First of all, it's more expensive, due to the high price of gasoline. Secondly, the variety of fish is different: While in the Caribbean it is common to see colorful tropical fish, in the Pacific the fish are much larger. Both **Posada Tur\u00edstica Hostal del Mar** (tel. 4\/682-7415, cell tel. 314\/630-6723, hostaldelmarbahiasolano@yahoo.com) and **Posada Tur\u00edstica Rocas de Cabo Marzo** (tel. 4\/682-7525, cell tel. 313\/681-4001, bahiatebada@hotmail.com, www.posadaturisticarocasdecabomarzo.com) offer diving excursions. A popular trip is to the shipwrecked _Sebasti\u00e1n de Belalc\u00e1zar_ just to the northeast of Bah\u00eda Solano. Another good place to dive is around Cabo Marzo to the north near the Panamanian border. Rocas de Cabo Marzo charges COP$120,000 per person (without equipment) for a diving trip to that location. Double-check on the security situation before going to Cabo Marzo, as drug traffickers operate in the area.\n\n##### **Accommodations and Food**\n\nThere are a handful of quite good accommodations options located near the bay. They can each help organize whale-watching and other excursions for you. M **Posada Tur\u00edstica Hostal del Mar** (tel. 4\/682-7415, cell tel. 314\/630-6723, hostaldelmarbahiasolano@yahoo.com, COP$50,000 pp) is run by Rodrigo Fajardo and Estrella Rojas. They are pioneers in the area in terms of community organizing and ecotourism. There are four comfortable cabins tucked in their gardens filled with orchids, vegetation, chickens, and a lazy cat, Julia. Meals, often served with their homemade _aj\u00ed_ (hot sauce), are taken on a picnic table in the middle of this miniature tropical paradise. Rodrigo is a certified diving instructor, and you can learn that sport in the waters of the Pacific with him or go out for a trip if you already know what you're doing. A nearby shipwreck is a popular place for underwater exploration. Both he and Estrella know the area exceptionally well and can give you pointers on how to make the most of your stay and can coordinate day trips. Estrella has a small _tienda_ in the arrivals area of the airport, where she sells handicrafts made by locals.\n\nthe port of Bah\u00eda Solano\n\n**Biodiversity in Choc\u00f3**\n\nThis region is part of the **Choc\u00f3 Biogeogr\u00e1fico,** one of the most biodiverse regions in the world. It comprises a wide swath of rainforest extending from Panama to Ecuador, hemmed in by the Pacific Ocean to the west and by the Andes to the east. The Colombian part of this region includes an amazing 8,000 plant species, of which 2,000 are endemic. (In comparison, all of Canada, which is hundreds of times larger, has 3,270 plant species, of which 140 are endemic.) In addition there are 838 bird species, 261 amphibian species, 188 reptile species, and 180 mammal species. The relative isolation of this region has resulted in a high level of endemism of about 25 percent. It is one of the wettest places on Earth, with annual rainfall of 10,000 millimeters (33 feet!), with some places registering up to 20,000 millimeters (66 feet). Due to the high rainfall, this biodiversity hotspot boasts one of the most dense river networks in the world, with dozens of major arteries, such as the Baud\u00f3, San Juan, and Pat\u00eda. The Colombia Pacific is quite well preserved: An estimated 75 percent of the region is still covered with rainforest, and there are significant areas of mangroves.\n\nM **Posada Tur\u00edstica Rocas de Cabo Marzo** (tel. 4\/682-7525, cell tel. 313\/681-4001, bahiatebada@hotmail.com, www.posadaturisticarocasdecabomarzo.com, COP$110,000 pp d, all meals incl.) is a cozy lodge-like guesthouse with five rooms. Room rates include airport pickup and drop-off, all meals, and excursions such as jungle hikes and walks to nearby beaches. Their small hotel restaurant might just be the only place in the jungle where you'll find homemade pizza. It's open daily, but just for guests. In addition to fishing and diving expeditions, Rocas de Cabo Marzo organizes visits to the Estaci\u00f3n Septiembre turtle hatchery program (COP$60,000), a walk to the Playa and R\u00edo Mecana (COP$100,000 for 2 people), and a specialized expedition to see poisonous frogs (as well as howler monkeys and sloths) in the jungle with an indigenous guide.\n\nLess ecologically minded, but quite comfortable, is the **Hotel Yubarta** (tel. 4\/682-7509, cell tel. 314\/773-5066, COP$90,000 d). It is a multi-story building that offers 11 spacious air-conditioned rooms, a pleasant garden, and free _tinto_ (coffee) all day long. Meals can be taken at the rather swanky adjacent restaurant, **Do\u00f1a Aida** (7:30am-10am, noon-3pm, and 5:30pm-8:30pm Mon.-Sat., COP$18,000), which has a pleasant deck. The restaurant is open to the public.\n\n**El Mara\u00f1on** (Playa Mecana, cell tel. 320\/671-6163, COP$35,000 pp) is a locally owned beachside lodge with just five _caba\u00f1as_. It's a five-minute boat ride here from Bah\u00eda Solano (and you can walk there during low tide). It's easy to take a hike to the R\u00edo Mecana, and they organize fishing expeditions and whale-watching trips. **Choiban\u00e1 Casa** (cell tel. 310\/878-1214 or 312\/548-2969, www.choibana.com, COP$120,000 pp incl. meals and transportation) is a cute and colorfully painted wooden house on the beach past rowdy Playa Hu\u00edna. There are just five rooms, including one spectacular hut majestically set atop a rock. It's a 20-minute boat ride from Bah\u00eda Solano.\n\n#### **EL VALLE**\n\nThis fishing community to the south of Bah\u00eda Solano is authentic if grubby, with wooden houses lining dirt (often muddy) streets. However, just outside of town, about a 15-minute walk north is the **Playa Almejal,** a broad beach with hotels set back against the jungle. The beach is home to thousands of nervous little _cangrejos fantasmas_ (ghost crabs) scurrying about. The gray sandy beaches are often covered with driftwood and other debris. But during the nearly always spectacular sunsets, the pastels of the sky are perfectly reflected on the wet sands. It's a magical scene. The water is great for swimming, jumping in the waves, and surfing.\n\nPlaya Almejal\n\n##### **Recreation**\n\nThere are some pleasant excursions to make nearby Playa Almejal. The **Cascada El Tigre** can be reached by boat or by walking (4 hrs. one way, COP$40,000 guide), and this waterfall of cool water right on a secluded cove makes for an unforgettable shower\/massage. The **R\u00edo Tund\u00f3** just outside of El Valle is a great place for a jungle canoe or kayak trip. This excursion, which usually includes a two-hour hike in the verdant jungle, costs about COP$40,000 per person.\n\nPersuasive local guides regularly make the rounds at hotels offering day-trip excursions, but don't feel pressured to sign on to any trip immediately. Excursions can by pricey due to high gasoline prices. It may be worth comparing quotes with other guests first, and negotiating a better deal.\n\n##### M **Estaci\u00f3n Septiembre Sea Turtle Hatchery and Release Program**\n\nFrom August until December, female olive Ridley sea turtles return to the beaches of the area (to the same spot where they were born) to lay up to 80 eggs. Around 40-60 days later, these eggs hatch, and baby turtles are born. At the **Estaci\u00f3n Septiembre Sea Turtle Hatchery and Release Program** (Playa La Cuevita, 5 km south of El Valle, cell tel. 314\/677-2488 or 314\/675-3353), turtle eggs are collected from the beaches and protected from stray dogs, birds, and from humans. They remain in the sand until baby turtles are born. Then they are released into the ocean.\n\nThis program is run by the Fundaci\u00f3n Natura and administered by the community-based organization the **Fundaci\u00f3n Caguama.** ( _Caguama_ means sea turtle in the Ember\u00e1 language.)\n\nThere are two ways to see this important work at the Estaci\u00f3n Septiembre. You can visit for the day. (It's best to contact them in advance so they can coordinate your visit.) They request a small donation of perhaps COP$5,000. Or you can stay at one of the three simple rooms at the Estaci\u00f3n Septiembre (Fundaci\u00f3n Natura in Bogot\u00e1, Cra. 21 No. 39-43, tel. 1\/245-5700, COP$60,000 pp all meals incl.).\n\nA highlight of your visit to the Pacific coast might be witnessing a release of fearless newborn turtles as they scamper their way into the water. These take place at the Estaci\u00f3n Septiembre, especially during the month of September.\n\n##### **Festivals and Events**\n\nThere are two big annual events in Valle. During one week in July the entire community parties it up during the **Fiesta de la Virgen Carmen.** It's a week of street dances and carnival. In September is the **Festival de Viajeros Sin Maletas** (Travelers Without Luggage Festival), an environmental and conservation awareness-generation event supported by various community groups, nonprofit organizations, and international donors in celebration of the many migrating animals who visit the area each year. In addition to educational activities with children, there are theater and musical performances.\n\n##### **Accommodations and Food**\n\nOn Playa Almejal, the **Posada Don Ai** (Playa Almejal, cell tel. 314\/651-1160, COP$120,000 pp incl. meals) is a relaxed place with nine small cabins each with a porch and requisite hammock. Meals, usually fried fish, _patacones,_ rice, and salad, are included in the price of your stay and are served under a breezy thatched roof dining area overlooking the Pacific. It's quite a good value. Even if you are not staying there, you can stop by for a meal. Just beyond Don Ai's on the beach is M **El Almejal** (tel. 4\/412-5050, COP$170,000 pp). This eco-conscious lodge, with tastefully done and airy cabins, is a consistent favorite for those who want a little more comfort. Above in the jungle are the very top-end cabins. These are situated along a bird-watching trail (they organize early morning bird-watching walks here). El Amejal has its own organic garden and sea turtle hatchery program.\n\nThe third option is the most economical, and by far the funkiest. When you first walk up from the beach, barefoot and backpack on, you'll pass through several enormous boulders, and then the M **Humpback Turtle** (Playa Almejal, cell tel. 312\/756-3439, www.humpbackturtle.com, COP$15,000 camping pp, COP$25,000 dorm bed, COP$80,000 d) suddenly appears. Nestled at the edge of the jungle, this very colorful and rather primitive hostel, often called Donde Tyler, Donde Taylor, or Donde el Gringo Ese after the American owner Tyler, has a dorm room with six beds and four private rooms as well as a campsite. With an organic garden, composting, water conservation measures such as the use of dry toilets, and using plastic water bottles for construction materials, this is by far the most environmentally minded option in the area.\n\nBetween Playa Almejal and the town of El Valle are other options, including Villa Maga and El Nativo. Just across from the Almejal beach, these are both run by _nativos,_ as local Afro-Colombians identify themselves. **El Nativo** (cell tel. 311\/639-1015, nativo58@hotmail.com, COP$60,000 d incl. meals) has two simple cabins (four rooms in total) made from natural materials such as _guadua_ (bamboo) and palm leaves. They can organize day-trip excursions for you. Nearby is M **Villa Maga** (cell tel. 320\/777-4767, www.villamaga.net, COP$45,000 pp), with a similar setup as El Nativo, although with more natural light in the A-framed cabins and little more comfortable.\n\nHang loose and kick back at the Humpback Turtle hostel.\n\n**Do\u00f1a Rosal\u00eda** (El Valle, past Internet caf\u00e9, no phone) is the only real restaurant in El Valle, and it's a good one. Here you are served in the dining room of Do\u00f1a Rosal\u00eda's small, but immaculate, house. If you can't find it, just ask anybody around. Everybody knows Rosal\u00eda! If you have a special request\u2014if you'd like lentils or beans instead of fish, for instance\u2014let her know by dropping by beforehand. The hotels on Playa Almejal (Caba\u00f1as Punta Roca\/Do\u00f1a Betty, Don Ai, and El Almejal) all serve good seafood meals.\n\n_Tiendas_ (shops) in El Valle sell snacks and usually have seating if you want a cold drink or beer. A couple of bakeries offer freshly baked bread.\n\nAs for drinking spots, besides _tiendas_ in El Valle where you can drink beer with the locals, the best option by far is **El Mirador.** Hard to miss, it's the only multicolored bar set on a boulder on the beach. It's between Humpback Turtle and El Almejal. It's only open on Sunday afternoons, when it gets packed with mostly locals, but visitors are more than welcome.\n\n#### M **PARQUE NACIONAL NATURAL UTR\u00cdA**\n\nHalfway between Bah\u00eda Solano and Nuqu\u00ed, the **Parque Nacional Natural Utr\u00eda** (www.parquesnacionales.gov.co, 8am-5pm daily, park entrance COP$37,500 non-Colombians, COP$14,000 Colombians and residents, COP$7,500 students with ID) has a spectacular location on the edge of the jungle but close to some great beaches. It encompasses over 54,000 hectares (135,000 acres) of tropical forest, mangroves, and waters. Several nature paths await exploration, including a wooden bridge walkway over mangroves, and you can also walk to a nearby secluded beach with a guide and arrange to be picked up later. A well-marked path through the jungle leads to El Valle in about two hours. During whale-watching season, humpbacks have been known to swim into the narrow lagoon in front of the park cabins, providing exclusive shows for park guests.\n\nMost of the excursions require a guide. However, there are some short walks you can make near the Centro de Visitantes. One of the more unusual activities at the park is searching for glow-in-the-dark mushrooms in the evening along the nature trail, which you can do on your own. Another is a walk above the mangroves on an elevated wooden walkway.\n\nPark facilities are managed by the local community organization Mano Cambiada. The park has three beautiful wooden cabins (separated into three inviting private rooms, each with its own bathroom), with a total capacity of over 30, and there's an open-air restaurant near the cabins. Private rooms go for about COP$192,000 per person including all meals. There is one cabin with dormitory-style accommodations. A few days at this park during the week (or off-season) when it isn't crowded with groups could be wonderfully relaxing. If the park is crowded, however, it could become quite a social scene, especially during the long evenings. Round-trip transportation from either the Bah\u00eda Solano or Nuqu\u00ed airports costs COP$300,000 per person.\n\nOther excursions include whale-watching, from June until October (COP$87,000), a two-hour jungle hike to Playa Cocalito (COP$32,000), a boat trip to Playa Blanca for snorkeling (COP$32,000), and a boat ride to the Jurubir\u00e1 community, where residents live in simple wooden houses on stilts, including a walk to some hot springs (COP$93,000). Kayaks are available for free to those staying at the park.\n\nIf you are not staying at the park, it can be visited on a day trip from El Valle, Bah\u00eda Solano, or Nuqu\u00ed. But that excursion gets a failing grade from many, due to its high cost (upwards of COP$125,000 per person). The standard park visit includes a chat about the national park system, a short walk, snorkeling in the lagoon in front of the Centro de Visitantes, and a boat ride to Playa Blanca (part of the park), where you can snorkel and have a greasy lunch. Playa Blanca is a party place for locals on weekends and holidays.\n\n##### **Accommodations and Food**\n\nIt is far better to stay at the park, preferably during the week (and not during high season). Travel agency **Aviatur** (Av. 19 No. 4-62, Bogot\u00e1, tel. 1\/587-5181 or 1\/587-5182, www.aviaturecoturismo.com) offers package tours to PNN Utr\u00eda including two nights accommodations, all meals, and transportation to and from the Nuqu\u00ed airport. This costs COP$816,000 per person based on double accommodation. You can also contact **Mano Cambiada** (cell tel. 313\/759-6270 or 310\/348-6055, www.nuquipacifico.com).\n\n#### **NUQU\u00cd**\n\nThere is not much to the town of Nuqu\u00ed, except for some basic services and the airport. However, surrounding beaches and ecolodges are wonderful. Here you can head off to the jungle for hikes in search of colorful frogs, go on a whale-watching expedition, try your luck surfing, and row in dugout canoes along serpentine rivers that serve as highways between communities. Or you can blissfully relax in a hammock and while away the hours to the sound of crashing waves in the background. All hotels listed can keep you as busy or lazy as you want.\n\n##### **Accommodations**\n\nFrom locally owned accommodations in typical wooden Choc\u00f3 houses to cabins on the beach to luxurious secluded resorts, the beaches near Nuqu\u00ed have options for every budget. All hotels can organize hikes, whale-watching trips, diving excursions, and other activities. **El Cant\u00edl** (tel. 4\/448-0767, www.elcantil.com, COP$717,000 pp, 2 nights, all meals and transportation incl.) is perhaps the most widely known ecolodge in the area. Run by a couple from Medell\u00edn, El Cant\u00edl is composed of small but comfortable cabins nestled on the edge of the jungle\u2014view of the ocean included. It is about a 35-minute boat ride from Nuqu\u00ed. Whale-watching, diving, and surfing can also be arranged. Behind El Cant\u00edl, you can take a jungle hike and go on an exotic frog safari. Beautiful nearby beaches can be explored on your own, and you'll likely be alone on the sand. The fresh seafood at the hotel restaurant is both abundant and delicious. Electricity is powered by their own hydroelectric plant.\n\nIn solitary splendor, M **Morromico** (45 mins. north of Nuqu\u00ed, tel. 8\/521-4172 or 8\/522-4653, www.morromico.com, COP$200,000 pp incl. meals) is another upper-end resort, with only four rooms, a private beach, free sunsets, and freshly caught red snapper cooked in fresh coconut milk for lunch. What more could you ask for? This hotel is close to the Parque Nacional Natural Utr\u00eda.\n\n**Guachalito** is another favorite beach near Nuqu\u00ed. One of the better lodging options there is M **Luna de Miel** (tel. 4\/683-6152, cell tel. 314\/431-2125, lunademiel@hotmail.com, COP$150,000 pp incl. meals), a simple but very comfortable lodge of two cabins. It's a peaceful place.\n\nThere are more economical and authentic locally owned options in some of the villages. **Mano Cambiada** (cell tel. 313\/759-6270 or 310\/348-6055, www.nuquipacifico.com) can help organize a stay.\n\n#### **INFORMATION AND SERVICES**\n\nIn Bah\u00eda Solano there is an **ATM** at the **Banco Agrario.** (There is no ATM in El Valle, Utr\u00eda, or Nuqu\u00ed.) At the Banco Agrario (tel. 4\/682-7522, 8am-2pm Mon.-Fri.) in Bah\u00eda Solano, you can have money wired ( _giro_ ) to you or deposited directly into a hotel account ( _consignaci\u00f3n_ ). In Nuqu\u00ed, at **Super Giros** (tel. 4\/683-6067) you can also have money wired to you. In Nuqu\u00ed there are two banks: Banco Agrario and Bancolombia.\n\nOver 3,000 majestic humpback whales make their way to Colombia's Pacific coast from August until October each year.\n\nDo not expect wireless Internet, regular cell phone service, or 24-hour electricity at your hotel along the coast. There are Internet caf\u00e9s in the towns, such as Bah\u00eda Net in Bah\u00eda Solano, but connection speed is very slow.\n\nThere are **hospitals** in Bah\u00eda Solano (tel. 4\/682-7016 or 4\/682-7884) and in Nuqu\u00ed (tel. 4\/683-6003), as well as **pharmacies** in El Valle, Bah\u00eda Solano, and Nuqu\u00ed. There is a significant police and military presence in these places as well. If you have access to a phone, emergencies can be reported by calling 112.\n\n#### **GETTING THERE AND AROUND**\n\nTwo airports serve the central Choc\u00f3 coast: Bah\u00eda Solano and Nuqu\u00ed. Military-owned **Satena** (tel. 1\/423-8530, www.satena.com), **ADA** (tel. 4\/444-4232, www.ada-aero.com), and charter carrier **TAC** (tel. 4\/361-0945, www.taccolombia.com) serve both Bah\u00eda Solano and Nuqu\u00ed out of the Aeropuerto Olaya Herrera in Medell\u00edn. There are also flights out of the departmental capital of Quibd\u00f3. The trip takes under an hour. Another option is with the charter airline **Selvazul** (tel. 4\/362-2590, www.selvazul.net). From the Aeropuerto Olaya Herrera in Medell\u00edn this airline serves both Bah\u00eda Solano and Quibd\u00f3.\n\nWhen you arrive at the Bah\u00eda Solano or Nuqu\u00ed airport, most hotels will have arranged to pick you up at the airport if you have a reservation. This is the worry-free, recommended way to go. If you have not made a reservation, there are always some folks affiliated with hotels from the area at the airport to persuade visitors to stay with them.\n\nTo get to El Valle and Playa Almejal on your own from the Bah\u00eda Solano airport you can take a _mototaxi_ or truck for COP$10,000-20,000 per person depending on your negotiating skills. Much of the road to El Valle, through the jungle, has been paved. It costs much less (COP$2,000 on _mototaxi_ ) to get to Bah\u00eda Solano from the airport, since it's only two kilometers away.\n\nBoats leave Bah\u00eda Solano and El Valle for PNN Utr\u00eda, Nuqu\u00ed, and beaches beyond. As gasoline costs about three times as much as it does in the rest of Colombia, boat trips will seem expensive: at least COP$60,000 to PNN Utr\u00eda, COP$80,000 to Nuqu\u00ed. You can try to negotiate a better price, especially if you have a group going with you.\n\nIf comfort is an overrated commodity for you, travel between Buenaventura and Bah\u00eda Solano by boat can be arranged by hitching a ride on a cargo boat. These often depart on Fridays. Basic accommodations (bunk bed) and food are provided. The journey will take around 20 hours, costing COP$120,000. In Bah\u00eda Solano, head to the docks to inquire about these, or call Capitan C\u00e9sar (cell tel. 314\/686-3232) or the harbormaster (tel. 4\/682-7064).\n\nIf you want to go to the southern coastal city of Tumaco by sea, you will probably have to hop on a boat headed to Buenaventura first. Best to contact the harbormaster or go to the Bah\u00eda Solano port in person to find out about ship departures.\n\nVessels bound for Panama from Bah\u00eda Solano depart weekly. To find out about the next boat, go to Bah\u00eda Solano and ask around for El Profesor Justino; everyone knows who he is. The trip usually costs about COP$100,000 and takes about four hours. If leaving the country by boat, you will need to get an exit stamp at **Migraci\u00f3n Colombia** (www.migracioncolombia.gov.co). Request assistance from one of the local hotels in getting to Migraci\u00f3n Colombia.\n\n### **South Pacific Coast**\n\nThe beaches of the South Pacific coast of Colombia in and around the port cities of Buenaventura and Tumaco have not been on the radar screen of many international visitors, but they have drawn Colombian tourists for decades. Off the coast, you'll find the island oases of Parque National Natural Isla Gorgona and the Santuario de Flora y Fauna Malpelo, where nature rules the day.\n\n##### **Safety**\n\nBoth Buenaventura and Tumaco have been severely affected by turf battles involving rival drug trafficking gangs over the past few years. The beaches, however, are secluded and safe, and tourists are always welcomed with open arms. But do avoid the interior of the cities.\n\n#### **BUENAVENTURA**\n\nBuenaventura is the largest city on the coast, with a population of over 300,000, and it is Colombia's busiest port. The golden years of Buenaventura are long gone, but one of the remnants of its heydays is the elegant **Hotel Estelar Estaci\u00f3n** (Cl. 2 No. 6-8, tel. 2\/241-9512, www.sht.com.co, COP$250,000 d). It's near the Muelle Tur\u00edstico, the tourist port where you can have a generous seafood meal at restaurants like **Le\u00f1os y Mariscos** (Cl. 1 No. 5-08, tel. 2\/241-7000) and catch a boat to the nearby beaches of Juanchaco and Ladrilleros.\n\nThe gray sandy beaches of **Juanchaco** and **Ladrilleros** have their charm, and the surf's often up in Ladrilleros. From the beachside bluffs, during whale-watching season (June-November) you can spot the humpbacks frolicking in the waters.\n\n_Lanchas_ (boats) make the trip to Ladrilleros from the Muelle Tur\u00edstico in Buenaventura several times a day. **Astur\u00edas** (Muelle Tur\u00edstico Local No. 2, tel. 2\/240-4048, www.buceaencolombia.com) is one company that provides this service with boats at 10am, 1pm, and 4pm. Round-trip fares are around COP$55,000. It's a 45-minute ride to Juanchaco. Ladrilleros is a half-hour walk from Juanchaco. A 45-minute walk farther from Ladrilleros is the quiet La Barra beach. A popular hotel here, with a great view of the infinite Pacific, is the **Reserva Aguamarina** (tel. 2\/246-0285, www.reservaaguamarina.com). From here you can take a walk to waterfalls and discover other remote beaches.\n\n#### **TUMACO**\n\nIn the Nari\u00f1o department, Tumaco is the second largest port on the Pacific coast. It has a population of around 170,000 and sits on the coastline border with Ecuador. Tumaco is famous for its incredibly photogenic natural stone archway on El Morro beach. It's about 10 minutes from downtown, on the same island where the airport is located. A good place to stay is **Hotel Los Corales** (tel. 2\/727-2779, cell tel. 312\/841-1949, www.hotelloscorales.com, COP$100,000 d), a beach hotel that boasts it has hosted not one but two Colombian superstar singers: Juanes and Carlos Vives.\n\n**Bocagrande** island, home to a handful of hotels and restaurants and bars just outside of the city, is popular with local holidaymakers. A good place to stay is in one of the 25 _caba\u00f1as_ at **Hotel Las Lilianas** (cell tel. 310\/396-0906, COP$55,000 pp, meals included). For an extra charge of COP$20,000 the hotel arranges _lancha_ transportation from and to Tumaco. The trip takes about 25 minutes. The Bocagrande island inspired a famous Colombian _bolero_ song from the 1960s, _Noches de Bocagrande._ Pretty much every Colombian of a certain age knows that tune, although many mistakenly believe it is named for the Bocagrande area of Cartagena.\n\nTumaco is known for its excellent, fresh seafood, like _pescado sudado_ (fish stew) _, langostinos al coco_ (coconut shrimp), and ceviche. _Piangua_ (mangrove cockle), which grows in the mangroves, is a local delicacy, and is also threatened. A classic restaurant is **El Muelle** (Viaducto al Morro, tel. 2\/727-2383, COP$10,000-40,000).\n\n##### **Getting There and Around**\n\nFrom Cali there are regular buses that take you to Buenaventura. This three-hour trip costs about COP$20,000. **Satena** (Col. toll-free tel. 01\/800-091-2034, www.satena.com) flies to Buenaventura from Bogot\u00e1. While it is possible to take public transportation from Pasto (4 hrs.), that road has had security problems in the past. It's best to fly. **Avianca** (Col. toll-free tel. 01\/800-0953-3434, www.avianca.com) serves Tumaco from Bogot\u00e1 and Cali. Cargo boats can take you from Tumaco to San Lorenzo in Ecuador. Schedules vary, so it is best to go to the port directly and inquire about this option.\n\n#### M **PARQUE NACIONAL NATURAL ISLA GORGONA**\n\nA visit to the **Parque Nacional Natural Isla Gorgona** (www.parquesnacionales.gov.co) is an unforgettable one, no matter the time of year. This 9- by 2.6-kilometer (5.6- by 1.6-mile) island about 60 kilometers (37 miles) off the coast was thought to have been originally settled by Guna indigenous peoples who lived near present-day Panama. It was named by the Spaniards after the mythical female Greek monster, Gorgon, who, among other things, wore a belt made of snakes and even had them coming out of her hair. Spanish conquistador Francisco Pizarro landed on the island in 1527. He and his men were unhappy guests on the island. They didn't stay long: too many men were getting bitten by snakes. Following that, the island was mostly uninhabited.\n\nParque Nacional Natural Isla Gorgona\n\nCapuchin monkeys are cute\u2014from a distance.\n\nSurrounded by sharks and crawling with snakes, it served as a prison starting in 1959, primarily for those accused of atrocities during La Violencia, when Liberals and Conservatives fought each other in cities and towns across Colombia. Many say that the prison was modeled after the Alcatraz Penitentiary in California. Once the prison was closed in 1984, the island was converted into a national park.\n\nWater\u2014in the form of babbling brooks, trickling streams, roaring waterfalls, dewdrops on leaves, crashing waves, and gentle rain storms\u2014is the proof that this island is very much alive. Because of the 90 percent humidity, mist is often seen rising from the thick tropical jungle. It has its own permanent cloud lingering above its highest points.\n\n##### **Recreation**\n\n###### **HIKING**\n\nHiking excursions are included in the price of package tours with travel agency Aviatur. You are not allowed to stray far from the hotel area on your own, and all hikes must be made with a guide. Excursions can also be paid for \u00e0 la carte. They cost COP$10,000-30,000 each per person.\n\nA popular hike is a half-day trek around the edge of the island and through the jungle to Playa Palmeras, a beautiful beach on the southwestern side of the island that overlooks the smaller Isla Gorgonilla. On the way you'll have the possibility of seeing a variety of birds, snakes, monkeys, and amphibians. If you are very lucky, along the way may spot the stunning blue anole lizard, the only all-blue lizard in the world and one that exists only on this particular island. It has become threatened due to the presence of non-native iguanas. The beach is gorgeous and would be perfect except for all the flip-flops and plastic bottles\u2014flotsam that has made its way to the island shore from all around the world. On this hike a lunch is provided and the return is via boat.\n\nAnother walk takes you to Yundigua, an area of the park with excellent snorkeling. Return from this two-hour excursion is also by boat.\n\nYou can also take a two-hour walking tour to the site of the old prison, which is an eerie experience. Nature has all but reclaimed the prison, but you can almost sense the presence of ghosts lingering on. A small museum examines the brutal conditions of prison life on the island. It is open daily and is free.\n\n###### **DIVING AND WHALE-WATCHING**\n\nIsla Gorgona is an excellent place for diving. There is a dive center on the island, and courses for beginners are available. You can arrange for diving classes during a week spent at Isla Gorgona. Five excursions can be arranged to one of the seven main dive sites near the island. **Aviatur** (Av. 19 No. 4-62, Bogot\u00e1, tel. 1\/587-5181 or 1\/587-5182, www.aviaturecoturismo.com) offers a diving package that includes six dives and just one excursion on land that costs around COP$1,500,000 per person. A PADI certification course costs about COP$1,600,000 per person. Two extra dives cost COP$215,000.\n\nAnother option can be to contract a diving trip from an independent dive company. These depart from either Buenaventura or Guapi. **Arrecifes de Pac\u00edfico** (Cra. 38 No. 8A-17, Cali, tel. 2\/514-1691, cell tel. 321\/642-6015) organizes four-day\/three-night trips to Gorgona throughout the year. These typically depart by boat from Buenaventura. It's about an 11-hour trip to Isla Gorgona from there, and accommodations are on board the boat, which has a capacity of about 25 passengers. This trip, including six dives and one night dive with transportation, meals, and accommodations, costs around COP$1,390,000 per person (not including equipment). Diving spots include Tiburonera, Plaza de Toros, Monta\u00f1ita 1, Cazuelam, Parguera, El Viudo, and Remanso de la Parguera. For beginners, they recommend a short course in Cali before departing to Gorgona, where certification can be offered.\n\nGorgona is an excellent place for humpback whale-watching from July to October. If diving is too deep for you, within only about 20 meters of the island shore you can swim among thousands of colorful tropical fish and the occasional sea turtle. Sometimes you can hear humpback whales singing in the far-off distance.\n\n##### **Accommodations and Food**\n\nTravel agency **Aviatur** (Av. 19 No. 4-62, Bogot\u00e1, tel. 1\/587-5181 or 1\/587-5182, www.aviaturecoturismo.com) currently has a contract to operate ecotourism activities on the island. While you can visit Isla Gorgona for the day, the only way to stay on the island is to book a tour with Aviatur. A stay of three nights, including transportation from Guapi, all meals, and three excursions, costs COP$850,000 per person for two persons. During whale-watching season from mid-July through August, the package includes two whale-watching excursions and three on-land excursions and costs COP$1,002,162 per person based on double accommodations.\n\nOne area of the island has the park offices, a dozen comfortable cabins, and the pleasant open-air restaurant. All accommodations and meals (mostly seafood) are included in the package price. Call in advance if you have special dietary needs. At lunch, if it is sunny, you might be able to see dolphins frolic nearby. A band of white-headed temperamental capuchin monkeys hangs out near the settlement.\n\n##### **Information and Services**\n\nThere is a nurse's office equipped with vials of snake bite antivenin, should it come to that. But, since the park's opening, they have never had to administer those. (A worker was once bitten by a non-poisonous snake, however.) There is also a small store with handicrafts, souvenirs, and snacks. In the evenings there is even Internet service, but you may have to wait for one of the computers. Tap water is pure coming from the freshwater of the island, and it is then filtered. You can drink it from the tap in your cabin with no worries.\n\n**Handicrafts**\n\nThe main reason visitors pass through **Guapi,** an impoverished town of 30,000 and the largest town on the department of Cauca's coastline, is to catch a boat to Isla Gorgona.\n\nIf, on your way to or from Gorgona, you have some time in Guapi, be sure to check out the fantastic basketry and handicrafts at the **COOPMUJERES** handicraft store (Cra. 2 No. 10-39, cell tel. 311\/385-1014, guapicoopmujeres@yahoo.es, 8am-noon and 2pm-6pm Mon.-Sat., 8am-noon Sun.), across the street from the Hotel R\u00edo Guapi. This enterprise is completely run by Afro-Colombian women artisans, many of whom are heads of their households. They specialize in woven items such as hats and basketry, using natural fibers commonly found in surrounding jungles, such as _paja tetera, chocolatillo,_ and others. When the cooperative was started in 1992, the idea was met with much skepticism by men in Guapi, who doubted the women's ability to successfully run a business. Over 20 years later, it looks like they've proven the naysayers wrong.\n\nThis is one of the most ecological resorts you may ever visit. All electricity is provided by a small hydroelectric plant. For hot water, each cabin has solar power cells on its rooftop.\n\n##### **Getting There**\n\nThe easiest and best way to get to Isla Gorgona is to organize a _lancha_ (boat) with **Aviatur** (Av. 19 No. 4-62, Bogot\u00e1, tel. 1\/587-5181 or 1\/587-5182, www.aviaturecoturismo.com) from the coastal town of Guapi. These boats leave every Friday and Monday morning at around 11, about an hour after the arrival of the morning **Satena** (tel. 1\/423-8530, www.satena.com) flight from Cali. An Aviatur representative will meet you at the Guapi airport and take you over to their office\/embarkation point on a _mototaxi._ The trip to the island takes just under two hours, sometimes over rough seas. It costs COP$150,000 round-trip. If you are not able to take the Aviatur _lancha_ (if you arrive on a different day, for example), you will have to rent an entire _lancha_ from Guapi for about COP$500,000, which may not be so bad if you have others to share the cost.\n\nFrom Buenaventura, boats heading to Bocas de Satinga can drop you off at Isla Gorgona (COP$120,000 one way). If there are four or five passengers, the boat will detour and drop you off at the island. If there are fewer passengers, the boat will radio Gorgona and rendezvous with an Aviatur speedboat, which is an adventure in its own right. Contact Aviatur about this option.\n\n#### M **SANTUARIO DE FLORA Y FAUNA MALPELO**\n\nCovering around 900,000 hectares (2.2 million acres) of protected Pacific Ocean waters and the tiny **Isla Malpelo,** the **Santuario de Flora y Fauna Malpelo** (www.parquesnacionales.gov.co, COP$85,000-159,000) was established in 1995. The area was declared a Particularly Sensitive Sea Area by the International Maritime Organization in 2002 and a UNESCO World Heritage Site in 2006. The steep volcanic rock of Isla Malpelo is nearly 500 kilometers (300 miles) from the coast of Colombia. It is administered by **Parques Nacionales** (tel. 1\/353-2400, ext. 138, www.parquesnacionales.gov.co).\n\nIt is the largest no-fishing zone in the Eastern Tropical Pacific, thus making it one of the top places for diving in the world. There are 11 main dive sites, including the most important site, **La Nevera,** where it is common to see scores of hammerhead sharks. The deep waters surrounding the island are home to some of the most important coral formations in the Colombian Pacific. Mollusks and crustaceans, fish such as snapper, endangered _mero_ (grouper), large populations of hammerhead sharks, whale sharks, sun ray sharks, and manta rays are found in abundance in the sanctuary. It is one of the few places in the world where the short-nosed ragged-toothed shark, a deepwater shark, has been spotted. Inhospitable to much animal life, the island is home to crabs, lizards, and geckos. Among birds, the largest colony in the world of the Nazca booby is found on Malpelo.\n\nEven experienced divers will be blown away by the diversity of sea life in the Santuario de Flora y Fauna Malpelo.\n\nMalpelo is for experienced divers only. To get there you must coordinate with one of the following authorized diving tour groups:\n\n\u2022 Out of Buenaventura: **Embarcaciones Astur\u00edas** (tel. 2\/242-4620, cell tel. 313\/767-2864, barcoasturias@yahoo.com)\n\n\u2022 From Cali: **Pacific Diving** (tel. 2\/558-3903, info@cascoantiguocolombia.com, seawolfaboard@gmail.com) and **Arrecifes del Pac\u00edfico** (tel. 2\/514-1691, cell tel. 321\/642-6015, www.arrecifesdelpacifico.com)\n\n\u2022 Out of Panama City: **Coiba Dive Expeditions** (tel. 507\/232-0216, www.coibadiveexpeditions.com) and the German group **Inula UAA Adventures** (tel. 507\/667-95620, inuladiving@gmail.com, www.inula-diving.de)\n\nThe **Fundaci\u00f3n Malpelo** (www.fundacionmalpelo.org) is a nonprofit organization working to protect this sanctuary. The island is under constant threat from illegal fishing, particularly of hammerhead sharks. In 2012 it was estimated that 200 tons of fish were illegally caught in the Colombian Pacific, mostly by boats hailing from Costa Rica, Ecuador, and from Asian countries. During Holy Week, shark fin stew is sold in Buenaventura.\n\n## **SAN ANDR\u00c9S AND PROVIDENCIA**\n\nHIGHLIGHTS\n\nHISTORY\n\nTHE LAND\n\nPLANNING YOUR TIME\n\nSan Andr\u00e9s\n\nSIGHTS\n\nENTERTAINMENT AND EVENTS\n\nRECREATION\n\nACCOMMODATIONS\n\nFOOD\n\nINFORMATION AND SERVICES\n\nGETTING THERE AND AROUND\n\nProvidencia and Santa Catalina\n\nM PARQUE NACIONAL NATURAL OLD PROVIDENCE MCBEAN LAGOON\n\nENTERTAINMENT AND EVENTS\n\nRECREATION\n\nACCOMMODATIONS\n\nFOOD\n\nINFORMATION AND SERVICES\n\nGETTING THERE AND AROUND\n\nThe San Andr\u00e9s Archipelago is made up of seven atolls and three major islands: San Andr\u00e9s, Providencia, and Santa Catalina. San Andr\u00e9s is 775 kilometers (492 miles) northeast of the Colombian mainland and only 191 kilometers (119 miles) east of Nicaragua. The islands are fairly small: San Andr\u00e9s, the largest island, has an area of 26 square kilometers (10 square miles), Providencia just 17 square kilometers (6.5 square miles), and Santa Catalina, attached to Providencia by a photogenic pedestrian bridge, is 1 square kilometer (247 acres) in size.\n\n**HIGHLIGHTS**\n\nLOOK FOR M TO FIND RECOMMENDED SIGHTS, ACTIVITIES, DINING, AND LODGING.\n\nM **Spratt Bight Pathway:** Epicenter of all goings on in San Andr\u00e9s and against a backdrop of the idyllic Johnny Cay in the distance, this beachside promenade is part boardwalk\u2014and part catwalk (click here).\n\nM **Jard\u00edn Bot\u00e1nico:** This well-tended botanical garden sits on a bluff overlooking the turquoise sea. A walk among the native trees, plants, and flowers provides a pleasant break from the beach (click here).\n\nM **Snorkeling and Diving off of San Andr\u00e9s:** The dozens of dive sites among thriving coral formations and steep ocean walls off of San Andr\u00e9s can keep divers blissfully busy for days (click here).\n\nM **Parque Nacional Natural Old Providence McBean Lagoon:** Paddle through the mangrove lagoons and snorkel offshore among tropical fish at this small national park (click here).\n\nM **Beaches on Providencia:** Undertake the tough field work of determining your favorite palm-lined Providencia beach. Your investigations could take several days (click here).\n\nM **The Peak:** From The Peak, the highest point on Providencia, hikers enjoy fantastic peeks from the jungle toward the deep blue sea (click here).\n\nOnce serving as a base for notorious English pirate Henry Morgan, Providencia\u2014or Old Providence, as English-speaking locals call it\u2014and its tiny tag-along neighbor of Santa Catalina are places to experience how the Caribbean used to be before tourism developed. Here visitors enjoy small bungalow-style hotels and home-cooked Creole food. The beaches are pristine and secluded and the waters are an inviting turquoise. Seafood, particularly fresh crab, is always on the menu, accompanied by cold beer.\n\nMore developed San Andr\u00e9s is popular with rowdy Colombian vacationers escaping the chilly climes of the Andes. However, it has many of the same charms as Providencia. Sunbathing, snorkeling, diving, and relaxing are always the order of the day.\n\nOn both islands English and a Creole patois are spoken, in addition to Spanish.\n\n#### **HISTORY**\n\nLittle is known of the early history of San Andr\u00e9s, Providencia, and Santa Catalina. In pre-Columbian times, the Miskito people of Central America visited the islands but never settled there. In 1628, English privateers brought back information to England about the islands that led to the foundation, in 1631, of a Puritan colony on Providence Island, the English name for Providencia. This project was backed by the Providence Island Company, a joint stock company formed by prominent English Puritans who were also involved in establishing settlements in New England. The Providence settlement was contemporaneous with the Massachusetts Bay Colony. At the time, it was expected that this lush tropical Eden would be far more successful than the New England settlements. It was hoped that tobacco and cotton could be easily grown there. In 1631, 100 settlers arrived from England on the _Seaflower._\n\nThe project was short-lived. Providence Island Company denied the settlers land ownership and required significant contribution of manpower to build the island's defenses. Rather than establishing a self-sustaining agricultural community, the colonists imported slaves and established a plantation-based economy. Correspondence from that time reveals decidedly un-Puritanical activity, such as drinking and gambling. The death knell of the colony was the decision of Providence Island Company to obtain a privateering patent and engage in outright piracy. This enraged and provoked the Spanish, the dominant power in that part of the Americas.\n\nWary of having their New World gold stolen from them, the Spaniards cracked down, and an attack on the islands in 1641 put an end to the Puritan experiment. During the following half century, the island was fought over by Spain and England. Pirates such as Edward Mansvelt and Henry Morgan used Providencia as a base. Nominally under the Spanish crown, the island welcomed a small number of settlers from Britain and other Caribbean islands during the late 17th and 18th centuries. In 1821, the archipelago became part of the newly independent Republic of Gran Colombia.\n\nDuring the 19th century, another influx of immigrants from the British Caribbean included many former slaves, which led to the creation of the Raizal community. One settler who lived on Providence, Phillip Beekman Livingston, traveled to the United States and was ordained a Baptist minister, and he introduced that faith to the islands. He was also instrumental in freeing the islands' slaves, starting with his own, 17 years before Colombia abolished slavery in 1853. As a result of his work, the Baptist faith became a distinctive part the islands' culture.\n\nColombia exerted greater power over the islands in the early 20th century, delegating educational instruction to the Catholic Church and forbidding the use of English on official business. In 1953, dictator Gustavo Rojas Pinilla declared San Andr\u00e9s a free port. This led to a massive influx of outsiders, mostly Colombian duty-free tourists and settlers, but also a contingent of Middle Eastern merchants, who altered the face of San Andr\u00e9s forever. A dense shopping district sprouted up on the North End of San Andr\u00e9s after the declaration of free port. The English-speaking Raizal people became a minority on their own island and lost control of much land. Providence, which was not declared a free trade zone, was spared this onslaught.\n\nThe 1991 Colombian constitution gave the islands some autonomy and put an end to immigration from the mainland. Providencia enacted strict zoning and land ownership regulations that have preserved the island's Raizal identity. Both the Colombian and Nicaraguan governments have declared interest in opening these waters to oil exploration, prompting a grass roots \"Old Providence, not Oil Providence\" campaign.\n\nIn recent decades, Nicaragua has contested Colombian jurisdiction over the islands, renouncing the 1928 Esguerra-Barcenas treaty and filing a suit at the International Court of Justice. In 2001, the court reaffirmed Colombian sovereignty over the islands and atolls but left the maritime border up in the air. In 2012, the court decided that roughly 70,000 square kilometers of sea north and south of San Andr\u00e9s, which had previously been Colombian, were in fact Nicaraguan. Colombians and islanders were shocked, especially because of the loss of traditional fishing areas. Two large atolls became enclaves in the Nicaraguan maritime area. The court decision cannot be appealed, but Colombian president Juan Manuel Santos has declared that it will not abide by the decision until Nicaragua ensures the Raizal fishers have access to their traditional fishing areas.\n\niguana in San Andr\u00e9s\n\n#### **THE LAND**\n\nThe archipelago covers 280,000 square kilometers of marine area (it was 350,000 before the 2012 International Court decision). It includes three major islands and seven atolls, and well-preserved coral reefs, particularly the barrier reef surrounding Providence and Santa Catalina, home to more than 80 species of corals and 200 species of fish.\n\nThe islands were once covered by forest. Though much has been cleared, especially in San Andr\u00e9s, significant tracts of forest remain, with cedars, cotton trees, stinking toes, birch gums, and other indigenous trees. The abundance of fruit-bearing trees and plants includes breadfruit, tamarind, mango, and guava, though much of the fruit that is consumed on the islands is imported from Colombia and Central America. There are several large, well-preserved mangrove lagoons, notably the McBean Lagoon in Providencia.\n\nThe islands support a wide range of reptiles, including snakes, iguanas, geckos, and lizards, including the blue or green lizard. Other land animals include crabs, especially the black and shankey crabs, which effect massive migrations to and from the sea to spawn. Coralina (www.coralina.gov.co), the archipelago's environmental agency, recruits army personnel to block traffic on Providencia's roads to protect these migrating crabs. Four species of protected sea turtles nest here. Approximately 100 bird species have been identified on the islands, but only 18 are resident. The island's only non-human land mammals are bats. Dolphins and whales are sighted occasionally.\n\nDespite obvious environmental degradation, especially in San Andr\u00e9s, the archipelago remains one of the best preserved corners of the Caribbean. In 2000, the 300,000 square kilometers of the Seaflower Biosphere Reserve became part of UNESCO's \"Man and the Biosphere\" program, which aims to preserve both biological and ethnic diversity, combining conservation with sustainable use by local communities. The 2012 International Court decision transferred about 45 percent of the biosphere to Nicaragua. Islanders hope that Nicaragua will continue to preserve this priceless marine nature reserve.\n\n#### **PLANNING YOUR TIME**\n\nHigh tourist seasons on both islands are during the Christmas and New Year's holidays. It may be hard to find a hotel from mid-December until mid-January. During this time, as throngs of Colombian families and a growing number of Brazilians and Argentinians take over San Andr\u00e9s. Also popular are Easter week and school vacations, between mid-June and August. May and September are quiet. Because it's more difficult to reach, Providencia never feels crowded.\n\nThe average temperature is 27\u00b0C (81\u00b0F). During the dry season between January and April, water rationing can be necessary, especially in Providencia, where it rains as little as five days per month. The rainy season extends from June until November, when it can rain 20-24 days per month. October is the rainiest month and is also when hurricanes occasionally churn up the warm Caribbean waters. March and April are some of the best months for snorkeling and diving because the waters are calm. December and January are windy, making snorkeling and diving challenging. Strong winds can prompt airlines to cancel flights into and out of Providencia.\n\nSan Andr\u00e9s is a possible long weekend getaway from mainland Colombia. However, most opt to stay 5-7 days. Week-long all-inclusive plans are popular. A visit to Providencia from San Andr\u00e9s can be a budget buster, but it is well worth the expense if you are interested in getting away from it all. A jaunt to Providencia involves an extra flight, and hotels and restaurants are generally more expensive than in San Andr\u00e9s, which itself is already more expensive than the mainland. If you want to do some serious diving, plan for at least a week, say three days in San Andr\u00e9s and four days in Providencia.\n\n### **San Andr\u00e9s**\n\nSurrounded by a large barrier reef, San Andr\u00e9s is Colombia's Caribbean playground. Here the waters are of seven shades of blue, the sandy beaches are white, and coco locos, the official island cocktail, are always served. Days here are spent lazing on the beach, island hopping, snorkeling and diving, and enjoying fresh seafood. For many Colombians, the deals at the many duty-free stores are too good to pass up\u2014that's one reason why they visit the island in the first place.\n\nSan Andr\u00e9s has a population of about 75,000, about two-thirds of which are of mainland Colombian origin. The rest are English- and Creole-speaking Raizales, many of whom have origins as Jamaican slaves. There is also a community of \"Turcos\" or \"Arabes,\" whose roots can be traced to mostly Lebanon and Syria. Their presence on the island is not an insignificant one, as demonstrated by a brilliantly white modern mosque that stands prominently in the commercial center.\n\n##### **Orientation**\n\nThe island of San Andr\u00e9s resembles a seahorse floating gently eastward in the western Caribbean Sea. It is only about 13 kilometers (8 miles) long from top to bottom and 3 kilometers (2 miles) wide, and has a total area of 26 square kilometers (10 square miles). The Circunvalar ring road more or less circles the entire island.\n\nThe beach at Spratt Bight is full of activity.\n\nThe \"town\" of San Andr\u00e9s is usually called the Centro or the North End. It is in the snout of the seahorse, in the northeast. This is the center of activity and where the majority of the island's restaurants, hotels, and shops (nearly all of which are owned and operated by mainland Colombians) are found. About 1.6 kilometers (1 mile) of the main drag here, Avenida Colombia, is the _paseo peatonal_ or _malec\u00f3n,_ the Spratt Bight Pathway, a delightful pedestrian promenade along the Spratt Bight beach. About two kilometers northwest of the Centro is the airport. The west side is quieter, with a handful of points of interest, hotels, and restaurants. The coastline on the west side is all coral; there are no beaches. At the southernmost point of the island is the Hoyo Soplador blow hole. Continuing counterclockwise, the town of San Luis extends along the southeastern edge of the island. This area has some good beaches, hotels, and restaurants, and is much more laid-back than the Centro.\n\nThe middle part of the island, called **La Loma** (The Hill), is the highest point on the island. The main point of reference here is the stately white First Baptist Church. This is home to the largest community of Raizal people.\n\n#### **SIGHTS**\n\n##### M **Spratt Bight Pathway**\n\nFor many, their first stop in San Andr\u00e9s after checking in to their hotel is the **Spratt Bight Pathway** (Centro). This pedestrian walkway is the liveliest stretch on the island, lined with restaurants, hotels, and souvenir shops on one side. On the ocean side of the pathway is the island's most popular beach, **Spratt Bight,** which looks out toward the enticing Johnny Cay in the distance.\n\n##### **Casa Museo Isle\u00f1a**\n\n**Casa Museo Isle\u00f1a** (Km. 5 Av. Circunvalar, tel. 8\/512-3419, 8:30am-5pm daily, COP$8,000) is a reconstruction of a typical island wooden house that provides a glimpse into island life in the 19th century. After a required guided tour (15 minutes), your cheerful young guide will tell you \"now let's dance!\" Reggae dancing is a rather strange component of the museum experience, but then again, it's hard to say no. Those smiling guides are a persuasive lot.\n\n##### **Cueva de Morgan (Morgan's Cave)**\n\nIt would seem that all caves hidden along the coasts of San Andr\u00e9s and Providencia are reputed to hold hidden treasures stashed away by notorious pirates. On the western side of San Andr\u00e9s is **Cueva de Morgan** (Morgan's Cave, tel. 8\/513-2946, 9am-6pm daily, COP$10,000), a sort of theme park where Welsh privateer\/pirate Captain Henry Morgan allegedly stored some of his loot (but there's no evidence to prove this). There isn't much to see at the cave itself. That's why the park owners added on some reconstructions of traditional wooden island cabins that serve as mini-musuems on island culture and ways of life. You visit these on a guided tour that is included in the cost. One is an art gallery where local dancers often perform to calypso beats. All in all, it's a tourist trap.\n\n##### **Hoyo Soplador**\n\nAt the **Hoyo Soplador** on the island's southern tip, the attraction is a hole in the coral where, when the tide and winds are right, water sprays up, reaching heights of more than 10 meters. It can't compare to Old Faithful, but then again, can you order a coco loco in Yellowstone?\n\n##### M **Jard\u00edn Bot\u00e1nico**\n\nThe **Jard\u00edn Bot\u00e1nico** (V\u00eda Harmony Hill in front of Hotel Sol Caribe Campo, tel. 8\/513-3390, www.caribe.unal.edu.co, 8:30am-5pm Mon.-Fri., 10am-4pm Sat.-Sun., COP$5,000) is easily the most peaceful place on San Andr\u00e9s. In this lovely botanical garden run by the Universidad Nacional, you can stroll along several paths and view trees and plants that grow in San Andr\u00e9s. From the five-story lookout tower, you can take in an impressive view of the island and its barrier reefs. Guided tours, included in the price of admission, are technically required, but if you are in a hurry or arrive in the late afternoon, you can request to amble the trails unaccompanied.\n\n##### **First Baptist Church**\n\nThe white, clapboard **First Baptist Church** (La Loma, no phone, services 7:30pm Thurs. and 10:30am Sun., COP$3,000 donation requested) was built in 1844 and rebuilt before the turn of the 20th century using wood imported from Alabama. It was the first Baptist church established on the island. A guide will give you a little history of the church and allow you to climb up to the bell tower for a commanding view of the island. The Sunday worship service can last several hours. Church members dress up for services, and you'll often see a smattering of tourists in the balcony on Sundays. The church is an excellent place to hear gospel music.\n\n##### **Paradise Farm**\n\nJob Saas, a local Raizal man, operates **Paradise Farm** (Cove Seaside, Km. 11 Polly Higgs Rd., tel. 8\/513-0798, cell tel. 315\/770-3904, donations accepted). Saas decided to transform the former standard family farm into one with a focus on conservation and the environment. Here you can see animals, such as iguanas and turtles, and plants that are threatened due to overdevelopment on San Andr\u00e9s. Saas uses the same farming techniques that his family has used for decades. It is a great initiative on this island where environmental awareness is lacking. He welcomes visitors to the farm, and, if you are lucky, you can hear his band play.\n\nbreadfruit at Jard\u00edn Bot\u00e1nico\n\n##### **Big Pond**\n\nManaged by a Rastafarian community, the **Big Pond** (La Loma, no phone, no set visiting hours, donations requested) is a pond on the top of La Loma, home to a few domesticated alligators. When called, they will swim close to the shore, where they are fed a diet of white bread. The alligators live in harmony with turtles, and herons watch the action from a tree nearby. Upon arrival, ask for Fernando. There is no set entry fee.\n\n#### **ENTERTAINMENT AND EVENTS**\n\n##### **Nightlife**\n\nThe nightlife scene on San Andr\u00e9s is big and brash. The most famous nightspots are near Spratt Bight. Clubs generally open from Thursday to Saturday during off-season but every night during high season. Things get cranking around 10pm. The perennial top discos are **Coco Loco** (Av. Colombia, tel. 8\/513-1047), **Extasis** (Hotel Sol Caribe, tel. 8\/512-3043), and **Blue Deep** (Sunrise Hotel). **Aquarius** (Av. Colombia between Bah\u00eda Sardina and Hotel Ton\u00e9, tel. 8\/512-5933) is more of a bar scene during the afternoon and early evening, turning into a dancing spot later on. You'll feel like you're in Ibiza at the **Majia Restaurante Italiano y Cocktail Bar** (Av. Colombia, cell tel. 318\/860-5234, www.majiasanandres.com), where chill-out music and mojitos go along with a beachside view. Guest DJs usually spin on weekends.\n\n#### **RECREATION**\n\n##### **Beaches**\n\nSome of the best beaches on the island include **Spratt Bight,** near the Centro in front of the pedestrian walkway; **San Luis,** near Chammey Marina; **Cocoplum** ; **Bah\u00eda Sonora** (near Rocky Cay) beaches; and the **Parque Regional Johnny Cay.**\n\nOut of all of these, the Johnny Cay beaches are probably the best. **Johnny Cay** is the island that beckons off the Spratt Bight beach. During peak tourist seasons, on weekends, and on holidays, though, it gets crazy packed.\n\nTo get to Johnny Cay, you must take a boat, called a _lancha_ in Spanish, from Spratt Bight, It is a quick 15-minute ride there. There are always boats owned by individuals (not organized tour companies) at the ready at the Spratt Bight beach. To arrange a trip, your negotiating skills will be put to the test. Hiring an individual boat can cost up to COP$200,000. The inexpensive option is to take a day tour (COP$20,000). These leave from Spratt Bight by 9:30am every day of the year, and the boats return at around 4pm.\n\nIn the late afternoon, the island clears out, but you can stay until almost 6 when the last boats leave. It's nice to be one of the last visitors on the island as the sun begins its descent. There are no accommodations options on the island. While there, take a walk around the entire island, where flocks of birds are likely the only company you'll have. It takes about 15 minutes.\n\n##### M **Snorkeling and Diving**\n\nSan Andr\u00e9s is surrounded by a well-preserved coral reef teeming with marine life that makes it a diver's and snorkeler's paradise. On the eastern edge is the windward barrier, 15 kilometers long and 60-80 meters wide, with significant live coral communities. Beyond the reef, the shelf ends abruptly with a vertical wall that drops hundreds of meters. To the west, the windward barrier protects a large marine lagoon that has seagrass cover. The reef on the western, leeward side is a bit less well preserved due to tourism and boat traffic, but it also has beautiful patches of coral and significant marine life. In all, the waters surrounding San Andr\u00e9s include more than 40 species of corals and 131 species of fish. It is common to see large schools of brightly colored jacks, tangs, grunts, and snapper, as well as barracudas, groupers, and parrotfish. Other marine creatures include turtles, stingrays, moray eels, octopus, squid, and lobster.\n\na diving lesson\n\nA unique feature of San Andr\u00e9s is that the dives are very close to shore, which means a 10- to 30-minute boat ride maximum. The water is warm and has excellent visibility year round. The best conditions for diving are January to May, with stronger winds in June and July. Popular dive sites are: The Pyramids, a shallow 4-meter dive with striking anemones and fish; Nirvana, a reef at about 15 meters, teeming with marine life; Trampa Tortuga, a reef at about 15 meters with great visibility; and Blue Wall, on the eastern edge of the windward barrier, which starts at 6 meters and drops to 300 meters with magnificent corals and large tube sponges.\n\nMost dive operators also offer short (three hours) introductory courses for beginners, costing around COP$155,000 per person, which allow you to do an easy dive without being certified. There are also many opportunities to do full introductory and advanced courses with certification. A three-day Open Water Diver certification course typically costs around COP$800,000.\n\nRecommended diving operators on San Andr\u00e9s include **Banda** (Hotel Lord Pierre, tel. 8\/513-1080, www.bandadiveshop.com), **Blue Life** (Hotel Sunrise Beach, Local 112\/113, tel. 8\/512-5318, www.bluelifedive.com), **Sharky's Surf Shop** (Sunset Hotel, Km. 13 Carretera Circunvalar, tel. 8\/512-0651, www.sharkydiveshop.com), and **Karibik** (Av. Newball 1-248, Edificio Galeon, tel. 8\/512-0101, www.karibikdiver.com). Diving excursions typically include two dives and cost COP$170,000. Night diving trips can be arranged by most dive shops.\n\n##### **Other Water Sports**\n\nSamuel Raigosa, better known as Chamey, is the **kite-surfing** guru of San Andr\u00e9s, and those at **Chamey's N\u00e1utica** (Km. 4 V\u00eda San Luis, tel. 8\/513-2077, cell tel. 317\/752-4965) are experts on kite-surfing. A one-hour class costs COP$70,000.\n\nkite-boarding in San Andr\u00e9s\n\nThe group **Ecofiwi Turismo Ecol\u00f3gico** (V\u00eda San Luis, Mango Tree sector, tel. 8\/513-0565, cell tel. 316\/567-4988 or 316\/624-3396, 9am-4pm daily) offers **kayak tours** of the mangroves in the Old Point Regional Mangrove Park led by local guides. The kayaks are completely transparent, providing kayakers with up-close views of sea life such as upside down jellyfish, sea cucumbers, seagrass beds, and also birds such as frigatebirds, pelicans, herons, and migratory birds. Snorkeling is also part of the tour (equipment included). The two-hour tour costs COP$50,000, and that cost includes a snack of a crab empanada and a juice or something of the like, plus a CD of photos from the trip. There are no additional costs. The group also offers **artisanal fishing tours,** during which the visitor goes fishing with local Raizal fishers. A half-day fishing tour costs COP$200,000 and an entire day is COP$300,000.\n\nThe **West View** (tel. 8\/513-0341, cell tel. 312\/308-8942, 9am-6pm daily, COP$8,000) is a park with two components: on the water side there is a restaurant and features include a waterslide, a trampoline, snorkeling, and Aquanaut suits that you can rent to walk on the floor of the ocean. Across the road are some houses made entirely of coconuts, fruit orchards, dozens of lizards and turtles to gawk at, and a cave to enter. South of the West View is **La Piscina Natural** (cell tel. 318\/363-6014, COP$5,000), a low-key spot for snorkeling that attracts fewer crowds.\n\n##### **Tours**\n\nThe locally run **Coonative Brothers** (Spratt Bight Beach, tel. 8\/512-1923) company offers tours to some of the best known beaches and swimming spots. A standard day tour costs COP$20,000 and leaves at 9:30 every morning. That includes a 1.5-hour stop at **El Acuario\/La Piscina\/Haynes Cay,** where you can wade and swim in waters labeled \"seven shades of blue.\" Animal lovers may find the attraction of \"swimming with the manta rays\" to be disturbing. On busy tourist days, the manta rays are handled over and over again, being lifted out of the water for snapshots with smiling tourists. They are fed a steady diet of white sandwich bread. For the sake of the rays, it's best to avoid participating in this activity. The rest of the day is spent on **Johnny Cay,** where you can buy lunch and drinks, and rent snorkeling equipment. (If you go to Johnny Cay, don't pay for a guide. There is no need.). You can just show up at the beach at around 9am to join the tour.\n\nLocal boaters affiliated with Coonative Brothers also offer full-day tours with more stops, including a visit to the San Andr\u00e9s mangroves for COP$60,000 per person. There is a minimum of 10 passengers for these tours. If you would like to go out on a private trip to one of those locations, that can cost up to COP$200,000, depending on your negotiating skills. Inquire at the Coonative Brothers' beach kiosk.\n\nAnother option to get out on the water is to take a glass-bottom boat tour. During this tour, you make several stops to coral reefs, to sunken ships, and to exotic islands. You'll be able to get in the water and snorkel several times to observe sealife. For information regarding these tours contact **San Andr\u00e9s Unlimited** (Tom Hooker Road No. 8-75, South End, tel. 8\/513-0035, tel. 8\/513-0129, cell tel. 316\/889-8701, or 310\/625-2938). A 1-hour 45-minute tour costs around COP$45,000 per person.\n\n#### **ACCOMMODATIONS**\n\nOn this island where tourism is king, lodging options are plentiful, except during high season (mid-December to mid-January, Holy Week, and, to a lesser extent, during school vacations from June to July). Top-end hotels and low-end hostels are not as common as mid- to upper-range all-inclusive hotels. Colombian chain Decameron has five properties on the island and is building its largest hotel yet near the airport, expected to be ready in 2015. There are no familiar international hotel chains on the island.\n\nYou will probably want to stay on or near the beach on the eastern side of the island, including in quiet San Luis. From here, you can hop on public transportation or hail a cab if you want to go to town, or rent a motorbike, golf cart, jeep ( _mulita_ ), or bicycle. The busy downtown (North End) of San Andr\u00e9s can feel claustrophobic, but you'll always be within walking distance of restaurants and services, and you can often find some good deals in this area. Waterfront hotels here have pools, not beaches. Stay near the airport if you prefer more seclusion but still want to be close to the action in the city center.\n\nThe western side of the island has coral coastline instead of beaches, and the few hotels cater mostly to divers, so this side feels more isolated.\n\nIn the interior of the island are some _posadas nativas,_ guesthouses owned and operated by locals, many with deep roots on the island. Staying at a _posada nativa_ is the best way to get to know the local culture. For a list of guesthouse options, you can visit the webpage of the program of _posadas nativas_ sponsored by the Colombian Ministry of Tourism and Ministry of Rural Affairs: www.posadasturisticasdecolombia.com.\n\nIn the North End, boutique hotel M **Casa Harb** (Cl. 11 No. 10-83, tel. 8\/512-6348, www.casaharb.com, COP$800,000 d) is by far the most luxurious place to stay in San Andr\u00e9s. The five suites, the lobby, dining area, and spa area are thoughtfully decorated with fantastic art and furniture from Morocco to Malaysia, personally chosen by owner Jak Harb. The **Hostal Mar y Mar** (Av. Colombia No. 1-32, cell tel. 317\/401-6906, COP$150,000 d), in the same part of the island, is a squeaky clean and friendly little place that opened in 2012. There are only four rooms and meals are not included. It's two blocks from the beach. Noise from airplane take-offs may be a nuisance in the mornings for some.\n\nAlthough it may look retro-Miami Beach, brilliantly white **Casablanca** (Av. Colombia No. 3-59, tel. 8\/512-4115, www.hotelcasablancasanandres.com, COP$391,000 d low season, COP$659,000 d high season) is an upscale option facing the beachfront pedestrian walkway. Of the 91 spacious rooms it offers, 10 of them are _caba\u00f1as._ There is a small pool and, more importantly, a pool bar, **Coco's.** The hotel has three on-site restaurants.\n\nThe Decameron chain seems to be on a mission to take over San Andr\u00e9s. They currently operate five hotels on San Andr\u00e9s. Close to town is their boutique hotel, **Decameron Los Delfines** (Av. Colombia No. 1B-86, tel. 8\/512-7816, Bogot\u00e1 tel. 1\/628-0000, www.decameron.co, COP$310,000 d). It has 39 very comfortable rooms and a pool. As is the case with all Decameron hotels, all meals are included. Fortunately, you are permitted to dine at other Decameron hotels on the island. This hotel does not have a beach.\n\nThe small hotels in the busy downtown are far more reasonably priced than those with a view to the sea and are just a few blocks away. The most popular choice for backpackers is the five-floor **El Viajero** (Av. 20 de Julio No. 3A-122, tel. 8\/512-7497, www.elviajerohostels.com, COP$52,000 dorm, COP$132,000-200,000 d), which is part of a Uruguayan chain. It has several air-conditioned gender-separated dorms, as well as private rooms. The top floor bar serves cold beer and assorted rum drinks, and there are several common areas with wireless Internet and computers. A paltry breakfast is included, and a kitchen is provided for guests' use. Staff aren't overly friendly, but they can arrange excursions.\n\nIt's surprising just how peaceful the **Posada Mary May** (Av. 20 de Julio No. 3-74, tel. 8\/512-5669, COP$60,000-110,000 d) is. Every morning you can pick up a cup of coffee in the lovely courtyard that is shaded by a huge avocado tree. On the downside, beds (usually three per room) are on the soft side in the spacious rooms, wireless Internet is sporadic, and in general the place could use an update. Around the corner, **Cli's Place** (Av. 20 de Julio No. 3-47, tel. 8\/512-0591, luciamhj@hotmail.com, COP$160,000 d), owned by Cletotilde Henry, has four double rooms in the main house as well as a _caba\u00f1a_ that accommodates seven people. You will feel at home here, although the price is high. **Casa D'Lul\u00fa** (Av. Antioquia No. 2-18, tel. 8\/512-2919, laposadadelulu.sanandresyprovidencia@hotmail.com, COP$150,000 d) has 10 rooms and three large studio apartments.\n\nBrightly colored **Cocoplum Hotel** (V\u00eda San Luis No. 43-49, tel. 8\/513-2121, www.cocoplumhotel.com, COP$240,000-420,000 d) in the San Luis area has the most important feature for a beach hotel: It is actually on the beach, with rooms that are steps from the water. Rooms are comfortable and spacious with kitchens. The restaurant and bar face the water, and quite often the only thing you hear is the palm branches rustling in the wind. Food is not fantastic, so don't go for the all-meals-included plan.\n\n**Ground Road Native Place** (Circunvalar No. 54-88, before the health clinic, San Luis, tel. 8\/513-3887, cell tel. 313\/776-6036, edupeterson1@hotmail.com, COP$50,000 pp) is a small, comfortable _posada nativa_ with five spacious rooms with air conditioning and wireless Internet in San Luis. It's in the home of Edula and George Peterson and is just a three-minute walk to the beach.\n\nAn option in La Loma is **Coconut Paradise Lodge** (Claymount No. 50-05, La Loma, tel. 8\/513-2926, oldm26@hotmail.com, COP$50,000 pp with breakfast), a beautiful turn-of-the-20th century wooden home with six rooms. It's a great place to stay if you are interested in learning about Raizal culture. It's close to the botanical gardens and the San Luis beaches. Try for the top floor room, which has great views and a delicious breeze.\n\nOn the quiet west side of the island, for those interested in diving, M **Sunset Hotel** (Km. 13 Circunvalar, tel. 8\/513-0433, 0420, www.sunsethotelspa.com, COP$196,000 low season, COP$320,000 d high season) is a great option. It has 16 bright and basic rooms that surround a small pool. While there is no beach, the hotel's dive shop, Sharky's, offers diving lessons and organizes diving excursions. You can go snorkeling in the waters across the street. Week-long diving packages may be a good option. And as its name indicates, great sunsets are included at no extra cost. You can also rent bikes here.\n\n#### **FOOD**\n\nSeafood is on every menu in every restaurant in San Andr\u00e9s. Fish, lobster, crab, and conch are likely to come from the waters off of San Andr\u00e9s and Providencia. However, _langostinos_ (prawns) and _camarones_ (shrimp) often come from either the Pacific or from the Cartagena area on the mainland. A Caribbean specialty you'll likely find only on San Andr\u00e9s, Providencia, and Jamaica is the rundown or _rond\u00f3n._ It is a filling to-the-max stew that has fish or conch, pig's tail, dumplings, yuca, and other ingredients slowly cooked in coconut milk. All restaurants are beach casual, and most of the larger ones accept credit cards.\n\n##### **Caf\u00e9s, Bakeries, and Quick Bites**\n\nPart of the Casablanca Hotel, the groovy turquoise **Sea Watch Caf\u00e9** (Av. Colombia, 6am-11pm daily, COP$18,000) is as close as it comes in Colombia to a New York-style coffee shop. Here you can have a leisurely breakfast as you watch the tourists file by on the walkway out front. They also offer pizza, hamburgers, ceviche, pasta, and desserts.\n\nFrom the outside, M **Coffee Break** (Av. Colombia No. 3-59, in front of Parque de la Barracuda, tel. 8\/512-1275, www.coffeebreak.com.co, 7am-11pm daily) often appears empty or even closed. But when you go inside, it's almost always packed with visitors and locals alike sipping on Vietnamese coffee, munching on nachos, or smearing cream cheese on their toasted bagels. Customers here take their time (likely because of the air conditioning!). A bakery\/caf\u00e9 popular with locals is **Bread Fruit** (Av. Francisco Newball No. 4-169, outside the Sunrise Hotel, tel. 8\/512-6044, 7:30am-8:30pm Mon.-Sat.). It's named after the breadfruit tree, which is typical to the area. (There is no breadfruit on the menu.)\n\nFinally, Miss Carmen is a familiar face on the Spratt Bight walkway, where she has been selling her homemade empanadas, ceviche, and cakes for years. Her stand doesn't really have a name, but you can call it **La Mesa Grande de Carmen** (Av. Colombia pathway).\n\n##### **Seafood**\n\nNorth of downtown, the M **Fisherman's Place** (Cra. 9 No. 1-10 Spratt Bight, tel. 8\/512-2774, noon-4pm daily, COP$20,000) is a restaurant run by a cooperative of local fishers. It overlooks the water and is near the airport, and is always packed. Try the Rond\u00f3n T\u00edpico Especial. Ask anyone in town what's the best seafood place on the island and a solid majority will mention **La Regatta** (tel. 8\/512-0437, www.restaurantelaregatta.com, noon-11pm daily, COP$40,000) next to the Club N\u00e1utico. It is open-air and juts out onto the water. For a sampling of the finest of San Andr\u00e9s seafood, try their Fiesta N\u00e1utica, which has lobster tails, prawns, and crab. Just far enough for a little peace and quiet from the Centro, **Niko's Seafood Restaurant** (Av. Colombia No. 1-93, tel. 8\/512-7535, 11am-11pm daily, COP$30,000) is what a family-run seafood place should be: over the water, not fancy-schmancy, and no lounge music. Lobster is the house specialty.\n\n**Seafood, Sustainably**\n\nTo protect their sustainability, some seafood should be avoided during certain times of the year, and some should be avoided completely.\n\n_Langosta_ (lobster), _pargo_ (snapper), and _caracol pala_ (conch) are the three most fished species around San Andr\u00e9s and Providencia, and are very often found on restaurant menus. It is recommended to avoid ordering conch from June to October (but due to overfishing it's wise to avoid conch entirely), lobster from April until June, and _cangrejo negro_ (black crab) from April to July.\n\nOther threatened species include Atlantic blue fin tuna, tarpon, lebranche mullet, _robalo blanco_ (white sea bass), _mero guasa_ (goliath grouper), _cherna_ (Nassau grouper), and the masked hamlet, which is only found in the waters off Providencia.\n\nAlthough the namesake for M **Miss Celia** (Av. Newball and Av. Raizal, tel. 8\/513-1062, restaurantemisscelia@gmail.com, noon-10pm daily, COP$30,000) passed away not too long ago, the restaurant continues the Raizal cuisine tradition in this cute, colorful, and authentic spot. Located in front of the Club Na\u00fatico, Miss Celia is surrounded by gardens and flowers, and the sounds of reggae and other local music add to the atmosphere. They recommend to their foreign guests to only order _rond\u00f3n_ at lunchtime, as they may not be able to handle it at night (it's a heavy dish).\n\nM **Donde Francesca** (El Pirata Beach, San Luis, tel. 8\/513-0163, cell tel. 315\/770-1315, restaurantedondefrancesca@gmail.com, 7:30am-8:30pm daily, COP$35,000) is colorful and has great food. The menu includes _langosta tempura_ (tempura lobster, COP$50,000) and _pulpo reducci\u00f3n al bals\u00e1mico_ (balsamic octopus, COP$34,000). In-the-know locals make a weekly visit to M **Restaurante Lidia** (Ground Rd. No. 64-65, San Luis, tel. 8\/513-2192) a ritual. It's only open on Sundays and on holiday Mondays. This place gets great reviews from local foodies. Lidia's crab empanadas are recommended.\n\nThe **Restaurante Punta Sur** (Km. 15.8, South End, tel. 8\/513-0003, 10am-6pm daily, COP$30,000) is close to the Hoyo Soplador. Sitting on the terrace when the waves come crashing in, it feels like you might be taken out to sea. With a small pool and deck area, it's a nice place to enjoy an afternoon. This family restaurant is a great place for some fresh seafood or drinks. _Arroz con camarones_ (rice with shrimp) and grilled lobster are a couple of the more popular menu items.\n\n##### **International**\n\n**Margherita e Carbonara** (Av. Colombia No. 1-93, tel. 8\/512-1050, 11am-11pm daily, COP$30,000) gets packed at night during high season due to its prized location near the big hotels. It's a boisterous family-style place where the pastas aren't bad.\n\n**Majia Restaurante Italiano y Cocktail Bar** (Av. Colombia, cell tel. 318\/860-52344, www.majia.co, COP$25,000) is a good choice for some authentic Italian pastas. It's run by a couple from Florence. **Mr. Panino** (Edificio Bread Fruit Local 106-7, tel. 8\/512-3481, www.misterpaninosanandres.com, 10:30am-10pm Mon.-Sat., 11am-4pm Sun., COP$30,000) is a reliable, somewhat upscale Italian restaurant, popular at both lunch and dinner. It's nice to sit on the high wooden tables in the back. Try their _risotto con langostinos,_ a prawn risotto that's a generous plate to share.\n\nAlthough there is a strong Lebanese influence on the island, Middle Eastern food is hard to come by in San Andr\u00e9s. **Hansa Pier** (Av. Colombia next to Tres Casitas hotel, cell tel. 313\/758-4604, noon-10pm daily, COP$20,000) has a great waterfront location in town, but the falafels and shwarma are nothing to write home about.\n\nIt may have an unfortunate name, but the **Gourmet Shop Assho** (Av. Newball in front of Parque de la Barracuda, tel. 8\/512-9843, 12:30pm-midnight Mon.-Sat., 6pm-11:30pm Sun., COP$30,000) is an excellent choice for a break from the seafood platter. The salads, pasta, and other dishes are good, and on every table there is a big bottle of imported spicy Asian chili sauce. With gourmet food items and wine for sale along the walls, and thousands of empty wine bottles decorating the ceiling, it's a cozy place. For something quick, you can try the hole in the wall (literally) **Gourmet Shop To Go** (Av. Newball in front of Parque de la Barracuda, tel. 8\/512-9843, 11am-3pm daily, COP$15,000) in the same building.\n\n##### **Markets**\n\n**Super Todo** (Av. 20 de Julio No. 3-41, tel. 8\/512-6366, 8am-8pm daily) is the largest supermarket on San Andr\u00e9s. It's in the Centro.\n\n#### **INFORMATION AND SERVICES**\n\nA humongous **tourist office** (Av. Newball, 8am-noon and 2pm-6pm daily) was opened in 2012. It's located between downtown and San Luis, across from Club N\u00e1utico. Tourism bureau staff are on hand at a tourist information kiosk where Avenida Colombia intersects with Avenida 20 de Julio (8am-7pm daily).\n\n#### **GETTING THERE AND AROUND**\n\nSan Andr\u00e9s is served by all the major Colombian airlines. In addition to their counters at the airport, most of them have ticket offices in the Centro. Copa has nonstop flights to Panama City. Air Transat (www.airtransat.com) operates charter flights between Canada and San Andr\u00e9s. The **Aeropuerto Gustavo Rojas Pinillas** (Centro, tel. 8\/512-3415) is very close to many hotels. Cabs to the airport cost COP$10,000. Satena operates flights to Providencia from San Andr\u00e9s.\n\nPublic buses serve the entire island. These cost about COP$2,000 each way. To get to San Luis, you can take the bus from the Parque de la Barracuda just south of the Centro.\n\nRenting a car is possible, but it's not recommended because parking is scarce, distances are not far, and, more importantly, there are more fun options. Most visitors rent heavy-duty, gas-powered golf carts referred to as _mulas_ (literally, mules) for the day instead. **Millennium Rent A Car** (Av. Newball, in front of the Parque La Barracuda, tel. 8\/512-3114) rents golf carts (COP$70,000 day) and _mulas_ (COP$150,000 day). **Rent A Car Esmeralda** (Av. Colombia, in front of Buxo del Caribe, tel. 8\/512-8116 or 8\/512-1934) offers the same prices. Although you can rent both golf carts and _mulas_ for multiple days, their use is prohibited after 6pm.\n\nTo take a day tour of the entire island, you can rent a van with the professional and knowledgeable driver **Jos\u00e9 Figueroa** (cell tel. 316\/317-2020). Taxis are plentiful in the Centro.\n\nBikes can be rented at **Bicycle Rental Shop** (Cra. 1B, Sector Punta Hansa, in front of Edificio Hansa Reef, cell tel. 318\/328-1790 or 321\/242-9328, 8am-6pm daily, COP$45,000\/day).\n\n### **Providencia and Santa Catalina**\n\nSecluded palm-lined beaches, gorgeous turquoise Caribbean waters, mellow locals, fresh seafood, and rum drinks make it easy to become smitten with Providencia and tough to leave.\n\nLocated about 90 kilometers (56 miles) north of San Andr\u00e9s, these islands are the easygoing cousins of hyperactive San Andr\u00e9s. Of volcanic origin, Providencia and Santa Catalina are older islands than San Andr\u00e9s, and are smaller in area and population than it, having a total area of about 18 square kilometers (7 square miles) and a population of only 5,000. Only 300 people live on minuscule Santa Catalina, an island known as the \"Island of Treasures,\" which was once home to an English fort.\n\n##### **Orientation**\n\nThe two islands of Providencia and Santa Catalina combined are about seven kilometers long and four kilometers wide (four miles by 2.5 miles). The harbor area of Providencia is called Santa Isabel and is the center of island activity. Other settlements on the island are usually referred to by the names of their beaches or bays. The main ones are on the western side of the island: Manchineel Bay (Bah\u00eda Manzanillo), on the southern end, which has some excellent beaches; Southwest Bay (Bah\u00eda Suroeste); and Freshwater Bay (Aguadulce), home to many hotels and restaurants. A ring road encircles the entire island of Providencia.\n\n#### M **PARQUE NACIONAL NATURAL OLD PROVIDENCE MCBEAN LAGOON**\n\nThe **Parque Nacional Natural Old Providence McBean Lagoon** (office Jones Point, east of airport, tel. 8\/514-8885 or 8\/514-9003, www.parquesnacionales.gov.co, 8am-12:30pm and 2pm-6pm daily, COP$14,000 non-Colombians, COP$8,500 Colombians, COP$4,000 students) is a small national park on the northeast coast of the island. It occupies about 1,485 hectares\/3,670 acres (1,390 hectares\/3,435 acres of that is in the sea). Here you can observe five different ecosystems: coral reefs, sea grass beds, mangroves, dry tropical forests, and volcanic keys.\n\n**Crab Cay** (Cayo Cangrejo) is one of the main attractions of the park, and it's an easy place for some splashing about in the incredibly clear, warm waters. This is a great place for some easy snorkeling, and, in addition to tropical fish, you may see manta rays or sea turtles. A short five-minute nature path on this minuscule island takes you to the top of the island. A snack bar on Crab Cay sells water, soft drinks, beer, coco locos, and snacks like ceviche. They are there every day until around 1pm.\n\nBoat tours, organized by all hotels and dive shops, motor around the coast of Providencia, stopping at beaches and at Crab Cay for snorkeling or swimming. These tours depart the hotels at around 9am each morning and cost about COP$35,000 per person. Once you disembark at Crab Cay, you'll have to pay the park entry fee of COP$14,000. Following the stop at Crab Cay, the boats go to Southwest Bay for a seafood lunch, not included in the price of the tour.\n\nsailing race in Providencia\n\nOtherwise you can hire a boat for yourself at around COP$350,000 total. Upon arrival at the island, you'll be required to pay the park entry fee. All hotels can arrange this more exclusive option.\n\nThe park's **Iron Wood Hill Trail** is a three-kilometer (1.8-mile) round-trip nature trail along which you can explore the tropical dry forest landscape, and will see different types of lizards, birds, and flora. There are nice views from here of the coastline. You are required to go with a local guide arranged by the parks office (Jones Point, east of airport, tel. 8\/514-8885 or 8\/514-9003, www.parquesnacionales.gov.co, 8am-12:30pm and 2pm-6pm daily, COP$25,000 pp plus park entry fee).\n\nAn additional activity is to hire a **kayak** and row to Crab Cay or through the park's McBean Lagoon mangroves. Passing through the mangroves you'll enter the **Oyster's Creek Lagoon,** where you'll see several species of birds, like blue and white herons and pelicans, as well as crabs, fish, and some unusual jellyfish. This is an interesting trip. Try to go early in the morning or late in the afternoon, as the sun can be brutal. Kayaks can be rented at the Posada Coco Bay (Maracaibo sector on the northeastern side of the island, tel. 8\/514-8226, cell tel. 311\/804-0373, www.posadacocobay.com, COP$30,000). A kayak with a guide costs COP$50,000, and for snorkeling equipment tack on another COP$10,000.\n\n#### **ENTERTAINMENT AND EVENTS**\n\n##### **Nightlife**\n\nIn Manchineel Bay, follow the reggae music down the beach and you'll discover **Roland Roots Bar** (Manchineel Bay, tel. 8\/514-8417, hours vary), which will easily become one of your favorites. It is set back among coconut palms and is the perfect place to spend a lazy, sunny day in Providencia. Or go at night, when you can order your rum drink to go and walk to the beach and stargaze, or hang out by a bonfire. Roland's competition is **Richard's Place** on the beach in Southwest Bay. Both of these spots often light up bonfires on the beach on weekend nights. Both also serve food, like fried fish.\n\n##### **Festival del Chub**\n\nIn early January of each year, the Parque Nacional Natural Old Providence McBean organizes the **Festival del Chub** (tel. 8\/514-8885 or 8\/514-9003). Chub is a plentiful but not very popular fish for consumption. The purpose of the festival is to encourage fishers and consumers to choose chub instead of other fish like red snapper, the stocks of which have been depleted throughout the Caribbean. There is one area of the island, Rocky Point (Punta Rocosa), where chub is widely eaten, and that is where the festival is held. In addition to an all-chub seafood festival, where you can try chub burgers and chub ceviche, there is also a sailing race from Southwest Bay to Manzanillo. It's fun to hang out at **Roland Roots Bar** (Manchineel Bay, tel. 8\/514-8417) in the morning to watch the sailors ready their boats for the race. This colorful festival usually takes place on a Saturday.\n\n#### **RECREATION**\n\n##### M **Beaches**\n\nThe best beaches on Providencia can be found generally on the western side of the island. From Manchineel Bay (Bah\u00eda Manzanillo) on the southern end to Allan or Almond Bay in the northwest, they are each worth exploring, if you have the time. On these beaches, the waters are calm, the sand golden, and there's always a refreshing breeze.\n\n**Manchineel Bay,** home to Roland Roots Bar, is an exotic beach where you can relax under the shade of a palm tree. Be careful of falling coconuts. In **Southwest Bay (Suroeste),** there are a couple of hotels and restaurants nearby, and you can sometimes see horses cooling off in the water or people riding them along the shoreline. The beaches of **Freshwater Bay** are very convenient to several hotels and restaurants.\n\nbeach in Providencia\n\nThe beach at **Allan Bay** (or **Almond Bay** ) is more remote. It's notable for its large octopus sculpture on the side of the road (can't miss it) and nicely done walkway down to the beach from the ring road. The beach area is a public park, and there is a snack bar and stand where you can purchase handicrafts. You'll have to either drive to this beach or hitch a ride on a taxi.\n\nA couple of coves on Santa Catalina have some secluded beaches on the path to Morgan's Head, and there is decent snorkeling nearby.\n\n##### **Snorkeling and Diving**\n\nProvidencia, which is surrounded by a 32-kilometer-long large barrier reef, is a fantastic place to dive or to learn to dive. The water temperature is always warm, and water visibility is usually 25-35 meters (82-115 feet).\n\nPopular diving sites are **Felipe's Place,** made up of several ledges with significant coral and marine life; **Turtle Rock,** a large rock at 20 meters covered with black coral; **Tete's Place,** teeming with fish; **Confusion,** with corals and sponges at 20-40 meters; and **Nick's Place,** a deep crack in the island's shelf that starts at 18 meters and drops to 40 meters. Good snorkeling can be done near **Cayo Cangrejo,** at the small islands of **Basalt** and **Palm Cays,** and around **Morgan's Head** in Santa Catalina, among other places.\n\n**Enjoy the Reef** (Southwest Bay, cell tel. 312\/325-8207) has snorkeling and diving tours for very small groups, and snorkeling tours for children.\n\nThe **Hotel Sirius** (Southwest Bay, tel. 8\/514-8213, www.siriushotel.net) is serious about diving and offers a PADI certification (COP$850,000) that includes four immersions over open water during a period of five days. They also offer a mini-course (COP$185,000), which includes a double immersion excursion. Hotel Sirius's diving courses have an excellent reputation.\n\n##### **Hiking**\n\n###### M **THE PEAK**\n\n**The Peak** (El Pico) is the highest point (360 meters\/1,181 feet) on Providencia, and from this mountaintop the 360-degree views are stunning. This hike takes about 1.5 hours to the top and under an hour down. The path to The Peak begins in the middle of the island and meanders along relatively well-marked trails through tropical rainforest and tropical dry forest. You'll likely come across lizards, cotton trees, and maybe a friendly dog who will follow you up to the top and back.\n\nFrom the top you'll be able to see the barrier reef that extends for 32 kilometers off of the east coast of the island. This reef is the second longest in the Caribbean and is part of the Parque Nacional Natural Old Providence McBean Lagoon.\n\nTo get to the starting point, go to the Bottom House (Casa Baja) neighborhood in the southeastern corner of the island just to the east of Manchineel Bay. Although you may come across a sign pointing towards The Peak, roads are not marked very well, so you will probably have to ask for directions to get to the starting point.\n\nAt the beginning of the walk, follow a path straight ahead, veering towards the right, and five minutes later go towards the right before a two-story house. You'll then go left (not to the right of the concrete well). From here on, you will pass a small garden, then follow a rocky creek straight on, fording it back and forth several times. You'll go through a gate and eventually veer to the left as you begin climbing up the hill. After you cross over a wooden bridge the path becomes steep; hold on to the wooden handrails. Occasional signs identify some of the trees or fauna you might see along the way.\n\nDuring rainy seasons, the path can become muddy and slippery. Make sure to bring a bottle of water with you. Guides are not necessary for this walk, but it's not impossible to get lost. All hotels can contract a guide for you, and this usually costs around COP$50,000.\n\n###### **SANTA CATALINA**\n\nFrom atop Santa Catalina island, English colonists and privateers once ruled, keeping their eyes peeled for potential enemies\u2014usually the Spanish Armada or competing Dutch pirates. Today you can see some remains from 17th-century English rule at **Fort Warwick.** It is adjacent to a big rock called **Morgan's Head.** If you look hard enough, it resembles the head of Henry Morgan, the notorious Welsh pirate and admiral of the English Royal Navy who marauded the Spanish New World colonies during the mid-17th century Morgan captured Santa Catalina from the Spaniards in 1670. Morgan's Head is next to **Morgan's Cave,** where the pirate supposedly hid his loot. You can go snorkeling inside the cave along with the occasional shark. Crossing the bridge, particularly at night, you may be able to spot manta rays gracefully swimming about. Start this hike at the colorful pedestrian bridge that connects Providencia with Santa Catalina in the Santa Isabel area. Once on Santa Catalina, take a left and follow the path.\n\n##### **Tours**\n\n**Paradise Tours** (Freshwater Bay, tel. 8\/514-8283, cell tel. 311\/605-0750, paradisetourscontact@gmail.com) is your one-stop shop, offering tours around the island, snorkeling excursions, diving excursions, and fishing excursions. One of their popular tours is the **Reefs and Snorkeling Tour** (3-4 hours, min. 4 people, COP$85,000), during which you boat to coral reefs around the island, exploring the underwater cities that exist just below the surface. Snorkeling equipment on this tour is extra.\n\nA double immersion diving excursion offered by the same agency costs COP$140,000, not including diving equipment, and lasts four hours. A full-day trip to El Faro reef, nine kilometers off of the island, costs COP$110,000. It's an excellent place for snorkeling in warm, crystalline waters.\n\nOn land, Paradise Tours offers several hiking options, such as to The Peak, where you can see coral reefs in the not so far distance; to Manchineel Hill, where you might see wild orchids on your way; and to the Iron Wood Hill in the Parque Nacional Natural Old Providence McBean Lagoon. These cost COP$85,000.\n\nA popular excursion is to take a boat tour around the island. The tours, departing at around 9am and returning at 3pm, make several stops, including Crab Cay and Santa Catalina. Any hotel can assist you in arranging one, and the boats make the rounds to pick up tourists at various hotels. These tours cost around COP$35,000 per person and usually leave from Freshwater Bay. If you prefer, you can rent a boat for just yourself and your crew, but that will cost more, up to COP$350,000. But in this option, you can decide when and where to go.\n\n#### **ACCOMMODATIONS**\n\nLocated directly over the lapping waters on the eastern side of Providencia, M **Posada Coco Bay** (Maracaibo, tel. 8\/514-8903 or 8\/514-8226, posadacocobay@gmail.com, www.posadacocobay.com, COP$180,000 d) is a small guesthouse with five comfortable rooms, three of which are on the water side. The other two (more spacious) options are across the street. You can go snorkeling just outside the hotel, and you can rent kayaks here, but there is no beach. You will have to rent a golf cart or _mula_ to get to island restaurants and beaches.\n\nBy far the most luxurious option on Providencia is at **Deep Blue** (Maracaibo Bay, tel. 8\/514-8423, www.hoteldeepblue.com, COP$600,000 d). It has 13 luxurious rooms sloping up a hill. A deck with a small pool provides spectacular views of the water. There is no beach, and unless you plan on dining exclusively at their elegant restaurant, you will need to find transportation to get to other restaurants and beaches on the island.\n\nThe **Hotel Old Providence** (Santa Isabel, tel. 8\/514-8691 or 8\/514-8094, COP$100,000 d) is the only option in the \"town\" area of Santa Isabel. It's close to Santa Catalina and offers basic comfortable rooms with air conditioning. Breakfast is not provided.\n\nIf you'd like to stay in Santa Catalina, close to the colorful pedestrian bridge is the guesthouse **Posada Villa Santa Catalina** (Santa Catalina, tel. 8\/514-8398, cell tel. 311\/257-3054, villasdesantacatalina@yahoo.com, www.villasantacatalina.com, COP$50,000 pp d). It's a comfortable and clean option and has air conditioning in the room. A small beach is about a 10-minute walk away.\n\nSomewhat far from everything is the M **Posada Refugio de la Luna** (Bluff, eastern side, tel. 8\/514-8460, providenciarefugiodelaluna@gmail.com, COP$170,000 d), a guesthouse with just one very comfortable and spacious room. Carmeni, the owner, is a papier-m\u00e2ch\u00e9 artist and has her studio upstairs in the house.\n\nThe Colombian all-inclusive chain **Decameron** (Bogot\u00e1 office tel. 1\/219-3030, www.decameron.co) has an affiliation with four locally owned and operated guesthouses in Providencia. These are all about the same high quality: clean, comfortable, and with air conditioning. Decameron requires that you make all of your travel arrangements with them in a tourist package. In exchange for getting the rights to make reservations at these hotels Decameron helped rebuild these family-run guesthouses after Hurricane Beta damaged them and the island in 2005.\n\nThere are three Decameron affiliated hotels in Freshwater Bay. The least expensive option is simply called **Relax** (Freshwater Bay, tel. 8\/514-8087, COP$80,000 pp). It has a small pool, hot water, and eight rooms, and is near a couple of restaurants and stores. It is across the road from the beach. **Miss Elma** (Freshwater Bay, tel. 8\/514-8229, COP$180,000 d) has just four rooms and a restaurant on the beach, with each room overlooking the sea. **Hotel Posada del Mar** (Freshwater Bay, tel. 8\/514-8052, posadadelmar@latinmail.com, www.posadadelmarprovidencia.com, COP$190,000 pp d) is a 24-room hotel with air conditioning and a pool. Oddly, instead of a beach, a grassy lawn overlooks the water. Not all rooms have a sea view.\n\n**Caba\u00f1as Miss Mary** (Southwest Bay, tel. 8\/514-8454, hotelmissmary@yahoo.com, COP$180,000 d) is beachside in the southwest with five beach-view rooms and three others. The restaurant is pretty good.\n\nM **Hotel Sirius** (Southwest Bay, tel. 8\/514-8213, www.siriushotel.net, COP$250,000 d) is a beachside hotel that specializes in diving and snorkeling excursions. (But you don't have to be a diver to enjoy your stay here.) It offers some huge rooms, and the friendly manager will make every effort to ensure you have a pleasant stay in Providencia.\n\n#### **FOOD**\n\nProvidencia is practically synonymous with fresh Caribbean seafood. A Providencia specialty is black crab. These fast-moving crabs live on the interior mountains and descend to the sea en masse once a year to lay their eggs in April or May. Many restaurants in Providencia do not accept credit cards. Hotel restaurants are open every day, while others often close on Sundays.\n\nThe **Deep Blue Hotel Restaurant** (Maracaibo Bay, tel. 8\/514-8423, noon-3pm and 6pm-10pm daily, COP$35,000) is the most elegant and pricey restaurant on the island. However, menu items are innovative and beautifully presented, and the service is excellent. It's a perfect place for a romantic \"last night in Providencia\" meal, particularly under the stars on the dock. M **Caribbean Place** (Freshwater Bay, tel. 8\/514-8698, noon-3pm and 6pm-10pm Mon.-Sat., COP$25,000) is one of the best seafood places in Providencia. Try the delicious fish in ginger butter sauce or coconut shrimp, and for dessert, the coconut pie. Cheerfully decorated, it is a great choice for both lunch and dinner.\n\nThe Canadian owner of M **Caf\u00e9 Studio** (Southwest Bay on ring road, tel. 8\/514-9076, 11am-10pm Mon.-Sat., COP$25,000) is likely to be found in this excellent restaurant's busy kitchen. It is a favorite not only for lunch and dinner, but also for afternoon coffee and their trademark cappuccino pie. Caf\u00e9 Studio has a varied menu, with pastas, interesting seafood dishes, and salads.\n\nM **Old Providence Taste** (Old Town Bay, to the west of Santa Isabel, tel. 8\/514-9028, 11:30am-3pm Mon.-Sat., COP$18,000), on the beach to the west of Santa Isabel, is run by a local sustainable seafood and farming co-op. Each day they offer a different menu, depending on what fishers and farmers bring in. It's the best deal on the island. They can also organize visits to farms and excursions with local fishers.\n\nThe **Miss Mary Hotel** (Southwest Bay, tel. 8\/514-8454, noon-3pm and 6pm-9pm daily, COP$20,000) has an open-air restaurant overlooking the beach. It's a nice place for lunch.\n\nFor a pizza night, try **Blue Coral** (Freshwater Bay, tel. 8\/514-8718, 11am-3pm and 6pm-9pm Mon.-Sat., COP$20,000). Though not out of this world, the pizzas and pastas here can taste exotic after several days of seafood.\n\nFor a midafternoon ice cream fix head to **Donde Puchi** (Santa Isabel, hours vary). **Miss Lucy's** (Southwest Bay, on the ring road, no phone, open daily) is a general store, but they also serve inexpensive meals, including _rond\u00f3n._ It's a friendly, local hangout.\n\n**Kalaloo Point Caf\u00e9-Boutique** (near Halley View lookout, eastern side of the island, tel. 8\/514-8592) is a cute caf\u00e9 and shop in a wooden house where you can have a cup of coffee or cool off with a Frenchy's frozen fruit bar. In the store they sell tropical dresses by a Colombian designer and various knick-knacks. There's also a small library.\n\nA small **grocery store** (open-11pm daily) is in Freshwater Bay below the Hotel Pirata Morgan.\n\n#### **INFORMATION AND SERVICES**\n\nThere is a **tourist office** (Santa Isabel, tel. 8\/514-8054, ext. 12, www.providencia.gov.co, 8am-noon and 2pm-6pm Mon.-Fri.) in the town area near the port. They may be able to assist with accommodations, including _posadas nativas_ (guesthouses owned and operated by locals), and give you some maps. A bank, ATM, and Internet caf\u00e9 are in the town. Since 2013, there is free wireless Internet on the island.\n\nIn case of an emergency the police can be reached at 112 or 8\/514-8000. For medical emergencies, call 125.\n\n#### **GETTING THERE AND AROUND**\n\nThere are two ways to travel to Providencia: by plane or by fast catamaran boat service from San Andr\u00e9s. There are three daily flights on **Satena** (Centro Comercial New Point, Local 206, San Andr\u00e9s, tel. 8\/512-1403; Aeropuerto El Embrujo, Providencia, tel. 8\/514-9257, www.satena.com). Charter flights are usually organized by Decameron (Colombian toll-free tel. 01\/800-051-0765, www.decameron.co) from San Andr\u00e9s to Providencia. All flights are on small propeller planes, and there are strict weight limitations. Passengers are only allowed 10 kilograms (22 pounds) in their checked baggage, and each passenger is required to be weighed upon check in along with their carry-on bag, which makes for an amusing photo op. The average weight per passenger cannot exceed 80 kilograms (176 pounds), including luggage. The flight takes about 25 minutes. The airport in Providencia is called **Aeropuerto El Embrujo** (tel. 8\/514-8176, ext. 6528). It is on the northeast side of the island near the Parque Nacional Natural Old Providence McBean Lagoon.\n\nThe **Catamaran Sensation** (tel. 8\/512-5124, www.elsensation.com, COP$65,000 one-way) provides fast boat service (three hours) between San Andr\u00e9s and Providencia. It provides service on Sunday, Wednesday, and Friday during low season. There is greater frecuency during high season. Boats leave San Andr\u00e9s at 7:30am from the Casa de la Cultura near the Hotel Arena Blanca and leave Providencia from the docks in Santa Isabel at 3:30pm. The catamaran service, while cheaper than air travel, often gets ghastly reviews due to the rough seas and resulting seasickness among the passengers. When the winds are strong and the waters are choppy between the two islands, especially between June and July and again in December and January, the ride can be extremely rough, requiring boat attendants to constantly circulate among the passengers to distribute sea sickness bags. This is especially true on the San Andr\u00e9s to Providencia leg. Waters are normally calmer the other way around.\n\nTaxis are expensive in Providencia, costing around COP$20,000 no matter where you go. _Mototaxis_ (motorcycle taxis) are much cheaper and you can find them almost anywhere. You can also flag down passing vehicles and hitchhike (paying a small fee). As in San Andr\u00e9s, you can rent golf carts and _mulas_ (gasoline-powered golf carts) in Providencia. All hotels can arrange this for you. They cost around COP$120,000 for one day.\n\n## **THE AMAZON AND LOS LLANOS**\n\nHIGHLIGHTS\n\nPLANNING YOUR TIME\n\nThe Amazon\n\nLETICIA\n\nALONG THE R\u00cdO AMAZONAS\n\nM PUERTO NARI\u00d1O\n\nM R\u00cdO YAVAR\u00cd\n\nLos Llanos\n\nVILLAVICENCIO\n\nM CA\u00d1O CRISTALES\n\nM HACIENDA LA AURORA\n\nThe Amazon and Los Llanos cover the eastern two-thirds of the country, a vast territory with very little population. Topographically they are the same: low-lying undulating terrain that is periodically flooded. But because of soil and climate, they have evolved different vegetation: dense rainforest in the Amazon and lush tropical savannahs in Los Llanos. The main draw in both the Amazon and Los Llanos is the unique natural landscapes and the magnificent wildlife inhabiting them.\n\n**HIGHLIGHTS**\n\nLOOK FOR M TO FIND RECOMMENDED SIGHTS, ACTIVITIES, DINING, AND LODGING.\n\nM **San Mart\u00edn de Amacayacu:** Experience life in this Ticuna village in the brimming-with-vitality Parque National Natural Amacayacu (click here).\n\nM **Puerto Nari\u00f1o:** No freeways, no traffic jams, no honking horns: In this eco-minded indigenous town overlooking the R\u00edo Loretoyaco, life is peaceful and the air is always pure (click here).\n\nM **Lago Tarapoto:** Pink dolphins perform for you in their natural habitat, and you can finally overcome your long-held piranha-phobia by taking a dip in this serene lake surrounded by lush jungle near Puerto Nari\u00f1o (click here).\n\nM **R\u00edo Yavar\u00ed:** Spend a few days under the immense Amazon rainforest canopy at a spectacular eco-lodge (click here).\n\nM **Ca\u00f1o Cristales:** Nature shows its psychedelic side at this stream of vibrant colors in the vast Llanos (click here).\n\nM **Hacienda La Aurora:** Take a safari on horseback through this enormous cattle ranch cum nature reserve in the heart of the Llanos and be astounded by the abundant wildlife. If you're lucky(!), you may even come across an anaconda (click here).\n\nA trip to the Amazon is a highlight not only to any visit to Colombia, but a highlight in any person's life. The survival of this vast ecosystem, the preservation of which is by no means assured, is of great importance to humanity. Learning about its variety of plants and animals, how it acts to stabilize the world's climate, how indigenous people managed to make a home there for thousands of years without disturbing its balance, and how modern civilization is threatening to destroy it is fascinating. Long after an introduction to Amazonia, one can't help reflecting on its significance for all of humanity.\n\nThis vast terrain of undulating hills and savannahs, with large patches of forest, abounds with wildlife: _chig\u00fciros_ (capybaras), deer, armadillos, sloths, anteaters, monkeys, anacondas, and an infinity of birds. Sadly, advancing human settlement and hunting have decimated much of it, but at places like Hacienda La Aurora you can view this wondrous wildlife in all its glory.\n\nThe Llanos is synonymous with cattle ranching and the cowboy way of life. If you are not squeamish, viewing traditional cattle-ranching activities, such as herding and branding calves, as they have been done for centuries by _llaneros_ (plainsmen), is an essential Llanos experience. Finally, the Llanos is home to a natural wonder not to be found anywhere else in the world: the vivid red, purple, yellow, and green streams of Ca\u00f1o Cristales in the southern extreme of the remote Serran\u00eda de la Macarena.\n\n#### **PLANNING YOUR TIME**\n\nTraveling in the Amazon and Los Llanos entails long-distance travel, mostly point to point from Bogot\u00e1 by airplane, and is therefore more expensive. To visit the Amazon, at least five days are required, and more if you want to spend some time in a nature reserve in the rainforest. The destinations in Los Llanos\u2014Ca\u00f1o Cristales and Hacienda La Aurora\u2014could be done in three days, though ideally you would want to spend more time there.\n\nThough the Amazon rainforest covers about one-third of the country east of the Andes and south of the R\u00edo Guaviare, the only real option to visit it is from the Amazon port city of Leticia, which has a multitude of options and ecotourism operators. The rest of the Colombian Amazon simply does not have even the minimum infrastructure to accommodate an independent traveler, and the region may be unsafe.\n\nThe great eastern plains of Colombia, Los Llanos, which comprise a further third of the country east of the Andes and north of the R\u00edo Guaviare, are the least explored region of the country. The reason is simply a lack of infrastructure, along with, until recently, security concerns.\n\n### **The Amazon**\n\nCovering an expanse of 8.2 million square kilometers (3.2 million square miles), the Amazon rainforest is the largest humid tropical forest in the world. Rainforests are important because of the enormous biodiversity that they sustain. And among rainforests, the New World rainforests are the most biodiverse. In fact, the Amazon jungle is home to one-tenth of all species on Earth, though it occupies only 1.6 percent of the world's surface. It holds more than 40,000 plant species, 3,000 fish species, 1,300 bird species, 428 mammal species, and 380 reptile species. By contrast, all of Canada, which occupies a surface larger than the Amazon rainforest, has 3,270 plant species, 1,100 fish species, 838 bird species, 188 reptile species, and 180 mammal species. Rainforests are also important as the world's main \"lungs,\" sucking in vast amounts of carbon dioxide through photosynthesis. Their ongoing destruction means the loss of invaluable biodiversity and increased warming.\n\nThe formation of the Amazon basin started about 180 million years ago, in the Jurassic era, when the westerly drifting American Continental Plate (South America) collided with the Nazca Plate (under the Pacific Ocean), forming the Andes. Water flowing eastward down the mountains accumulated in a vast freshwater lake that was hemmed in on the east by old mountainous formations (now the Guyana and Brazilian highlands). Large amounts of sediments were deposited, forming the basis for the Amazon's undulated topography. Around 28 million years ago, the water broke through the eastern mountain barrier and started flowing east into the Atlantic, forming the Amazon drainage basin.\n\nThe Amazon River (R\u00edo Amazonas), which measures about 6,400 kilometers (4,000 miles) in length, is fed by more than 1,000 tributaries. Though Colombia only has 180 kilometers (112 miles) on the Amazon river itself, several of its major rivers originate and flow through the Colombian Amazon region into the mighty river, including the Putumayo and the Caquet\u00e1. It is estimated that one-fifth of all the water that runs off the Earth's surface flows through this basin. The gradient is very slight: Leticia, which is more than 2,000 kilometers (1,200 miles) from the mouth of the river, stands at an elevation of 96 meters (315 feet). During the annual flood, lasting from November to April, the river can rise up to 50 meters (165 feet), submerging large sections of the jungle. Average river velocity is 1.5 kilometers per hour (0.9 miles per hour), though it increases slightly with the flooding.\n\na rainy trip on the Amazon\n\nThe topography of the Amazon consists of two distinct but intermingled areas: _terra firme,_ the undulated lands that are above the highest flood point (which comprise two-thirds of the surface of the basin), and _varzea,_ floodplains along the main rivers, which can extend up to 50 kilometers (30 miles) from the river. _Varzea,_ rich in sediments transported by the rivers, is where most human activity is concentrated.\n\nThere are two distinct types of rivers in the Amazon region: the predominant white rivers, which carry sediments down from the Andes, and the black rivers, which originate in the Guyanese and Brazilian highlands that were long ago denuded of soil due to erosion. As these waters travel through the flooded forest, they pick up pigments that give them their characteristic black color. _Igap\u00f3_ is the name given to jungles flooded by black-water rivers. The largest of the black rivers, and the largest tributary of the Amazon, is the R\u00edo Negro, called R\u00edo Guain\u00eda in Colombia. It flows into the Amazon at Manaus, creating the extraordinary _encontro das aguas,_ where white and black waters flow side by side for several kilometers until they mix.\n\nThe forest itself has a complex, layered structure. Towering trees, held up by complex buttresses at the base of the trunks, soar 40 meters (130 feet) high, forming the jungle's canopy. Occasionally, trees known as _emergentes_ rise above the canopy to a height of 60 meters (200 feet). According to a Ticuna myth, a giant fallen ceiba tree is the origin of the Amazon River. The canopy, flooded by sunlight, is full of plant and animal life. If you don't suffer from vertigo, a climb up to the canopy is an unforgettable experience. Below the canopy, shade tolerant species of trees and plants comprise the underbrush ( _sotobosque_ ) and support many epiphytes (plants that live on others), such as orchids and bromeliads. Large networks of vines entangle the growth.\n\n**Explorers in the Amazon**\n\nThe first Europeans to travel to the Amazon were Spanish conquistadors Gonzalo Pizarro (half brother of Francisco Pizarro, the infamous conqueror of Peru) and Francisco de Orellana, who, in 1541, headed down the R\u00edo Napo in present-day Ecuador to search for the mythical \"Land of Cinnamon.\" Pizarro, frustrated, turned back after one year. Orellana followed the course of the Napo, eventually floating down the entire course of the Amazon to its mouth in the Atlantic. Reportedly, he was attacked by women warriors and hence the region came to be named after the Amazons of Greek mythology.\n\nDuring the colonial period, the Spaniards largely ignored the region as there were no ready sources of riches. French naturalist Charles Marie de la Condamine was the first European scientific explorer to visit the region. In 1743, he traversed the entire basin, discovering, among other things, quinine and latex (for rubber). Another notable explorer was Alexander von Humboldt, who visited the Casiquiare Canal, which links the R\u00edo Orinoco and R\u00edo Negro, in 1800. It is located in southern Venezuela.\n\nThe waters of the Amazon are home to more than 1,500 species of fish, including the endangered piraruc\u00fa, one of the largest freshwater fishes on Earth, and notorious meat-eating piranha. They are also home to dolphins, both pink and gray. Pink dolphins evolved separately and have horizontal neck mobility that allows them to navigate the flooded forest easily, while gray dolphins are distant relations of the seafaring kind. Other aquatic mammals include manatees and _nutrias_ (otters). There are dozens of species of turtles, alligators, lizards, snakes, and frogs. Land-faring mammals include deer, anteaters, armadillos, tapirs, jaguars, ocelots, and pumas, though sighting of these large cats is quite rare. The trees support sloths, squirrels, and many species of moneys and bats. With more than 3,000 species of birds, the Amazon is truly a bird-watcher's paradise. During the floods, a canoe ride through the partially submerged trees will allow you to spot a variety of birds, including herons, kingfishers, ducks, woodpeckers, oropendulas, kiskadees, and hawks. Finally, there are innumerable insects, including giant leaf-cutting ants, as well as centipedes and scorpions.\n\nTo truly get a sense of the place, you need to get into the jungle, either by doing a trek or taking canoe rides in the flooded jungle. Then, the small details that make up this wonderland will come into focus: a ray of sun shining through the canopy; a massive, 40-meter-high ceiba tree; a vine that has wound itself around a tree like a boa constrictor; an orange mushroom popping up from a fallen tree, accelerating its final stage of decay; a single bright blue butterfly that crosses your path momentarily and then flutters away; a leaf as big as your head floating down to the ground; a whimsical song from a bird somewhere above in the canopy.\n\n##### **History**\n\nDuring the 20th century, settlement has been mostly limited to a swath of jungle in the Caquet\u00e1 and Putumayo departments near the Andes. There, oil and plentiful land have attracted settlers from the interior of the country. However, the sheer inaccessibility of most of the Colombian jungle has spared the type of development seen in Brazil. During the drug wars of the 1990s and early 2000s, coca cultivation spread deeper into the jungle in the departments of Caquet\u00e1, Putumayo, Guaviare, and Vaup\u00e9s, bringing along the FARC, and Leticia became a center for drug trafficking. At present much of the Amazonian drug business appears to have shifted to the Peruvian side of the river.\n\n##### **Climate**\n\nIt is always muggy in the Amazon, and rarely is there a breeze to provide some relief to the heat. The border town of Leticia reports an average 85 percent humidity year-round with an average temperature of 25.8\u00b0C (78.4\u00b0F). The region has one dry season, between June and August, and one rainy season, between January and May. In August it can rain as little as 10 days per month. During the dry season, rivers shrink, creating beaches, and trees and shrubs appear in parts of the jungle that during the rainy season are hidden under water.\n\nDuring the rainy season, water falls from the skies and pours down from the Andes into the mighty river, and canoes become the only means of getting from point A to point B in the jungle. You can glide in canoes through the treetops, an unforgettable experience. Ponchos, rubber boots, and insect repellent are especially critical during the rainy season.\n\n##### **Environmental Threats**\n\nUnfortunately, this diverse ecosystem is under severe threat. Over the past 40 years, 20 percent of the Amazon jungle has been destroyed. If strong measures are not taken, half of what remains could be destroyed within the next few decades. The main causes of the destruction (in order of importance) are cattle ranching, agriculture, dams, and illegal mining. The main means for its destruction are roads. Without these, human encroachment is limited to the borders of navigable rivers. Voracious, short-sighted development in Brazil, where road development has been greatest, is the main cause of the destruction of this wonderland. Though the Brazilian authorities tout decreasing levels of deforestation, the roads crisscrossing the jungle have made irreparable damage inevitable. Significant deforestation has also occurred along the Andes piedmont, especially in the headwaters of the Caquet\u00e1 and Putumayo rivers in Colombia, where illegal coca cultivation has been one of the main culprits.\n\nThere is alarming evidence that, as deforestation progresses, the Amazon ecosystem is breaking down and will be unable to sustain itself. With deforestation comes lower evaporation and rainfall. As the forest dries up, it may become prone to fires (which it is not currently), changing the overall dynamics. The Amazon has not yet reached that scary \"tipping point\" after which it cannot sustain itself, but vastly reduced measured rainfall points in that direction.\n\nThough the Colombian section of the Amazon rainforest represents only 10 percent of the total, it is the best preserved, due to a dearth of roads, and also the most likely to be preserved thanks to enlightened policies. From 1986 to 1990, President Virgilio Barco transferred 163,000 square kilometers (63,000 square miles\u2014twice the surface of Austria or 15 percent of Colombia) to national parks and indigenous _resguardos_ (land collectively owned by indigenous groups) and protected areas. Predio Putumayo, the largest _resguardo,_ measures 59,000 square kilometers (23,000 square miles), the size of Costa Rica. Subsequent governments have continued to expand the protected areas, and now at least 65 percent of all the Colombia Amazon is a protected area, either through the system of national parks or through indigenous _resguardos._\n\nThe 1991 constitution enshrined significant rights for Colombia's indigenous peoples, adding further protections. Though the threat of illegal logging and mining is ever present, particularly due to the presence of valuable rare earth minerals, Colombia seems to have taken successful steps to preserve a large section of one of the world's most important ecosystems.\n\nIn 2013, the Colombian government took a positive step by more than doubling the size of its Parque Nacional Natural Serran\u00eda de Chiribiquete, in the Amazon departments of Caquet\u00e1 and Guaviare, to over 28,000 square kilometers (11,000 square miles). It is the largest national park in Colombia.\n\n**The Peruvian Amazon Company**\n\nThe Colombian section of the Amazon was largely untouched until the mid-19th century, when quinine and then rubber extraction attracted Colombian and Peruvian adventurers. Vast tracts of land with rubber trees and plentiful indigenous labor seemed like a perfect combination to make a fortune. In 1901, Julio C\u00e9sar Arana, a Peruvian _cauchero_ (rubber baron), founded the Casa Arana, a company that operated a ruthless system of rubber extraction based on torture and slavery. The company, later known as the Peruvian Amazon Company headquartered in London, operated out of La Chorrera on the R\u00edo Putumayo. A visiting American, W. E. Hardenburg, witnessed the horrors and in 1909 published a damning article in the British magazine _Truth._ This prompted the British government to order an inquiry, which uncovered the terrible conditions. In 1912, Parliament opened an investigation, which cleared the British Board of Directors of all responsibility in the atrocities. At the same time they determined that over 32,000 Huitoto people had been murdered or worked to death during a five-year period. Huitoto leaders estimate that over 80,000 were killed between 1912 and 1929.\n\nThe Peruvian Amazon Company was liquidated in 1916, but, incredibly, Arana continued operations through the 1930s. It was not until 2012 that the Colombian government formally apologized to the indigenous people for these atrocities in a letter by President Santos at a ceremony in La Chorrera commemorating the 100th anniversary of the genocide.\n\n#### **LETICIA**\n\nVisitors come to Leticia to experience the jungle. This border town of 40,000 doesn't have much in the way of charm, and it's clogged with buzzing motorbikes, but, alas, all is not lost here. It has improved since the 1970s and 1980s when it was synonymous with cocaine and exotic animal trafficking. During that anything-goes time, poor native villagers would catch monkeys (to be sent to labs and zoos) in exchange for clothes, and unknown and fierce-looking men would routinely zoom up and down the river in speedboats to unknown destinations.\n\nToday, without a doubt, ecotourism is the future for Leticia, and more and more Colombians and visitors from abroad are discovering the area, for better or for worse. There was a 300 percent increase in visitors to Leticia between 2002 and 2006.\n\nA handful of sights worth checking out in town and along the Kilometers road will keep you occupied for a couple of days, and there are comfortable accommodations options. But best of all, it is close to the jungle and the R\u00edo Amazonas is always at the ready to take you there.\n\nIt is possible to book package tours that include all the sights and take care of your accommodations. However, a growing number of visitors explore the jungle independently. Leticia is an excellent base for that.\n\n###### **ORIENTATION**\n\nLeticia, the capital city of the Amazonas department of Colombia, is the southernmost city in the country, and it sits on the northern side of the R\u00edo Amazonas at the convergence of Colombia with Brazil and Peru. It is 1,100 kilometers (700 miles) southeast of Bogot\u00e1. It borders the grubby Brazilian town of Tabatinga to the east, and Isla de Santa Rosa (Peru) is an island in the river to Leticia's south. Just to the north of the city the town abruptly ends and the rainforest takes over\u2014no suburbs here. The closest Colombian town of any significance is Puerto Nari\u00f1o, 87 kilometers (54 miles) to the northwest.\n\nLeticia is laid out on a grid that is easy to figure out. The airport is north of town on the Avenida V\u00e1squez Cobo, which turns into Carrera 10, one of the main drags in town. _Carreras_ run north-south with _calles_ going from east to west. The _malec\u00f3n,_ from where all boats depart, is on the eastern side of town at the end of Calle 8. Carretera Los Kil\u00f3metros, also called V\u00eda a Tarapaca, leads to Mundo Amaz\u00f3nico and the Reserva Natural Tanimboca. There are some Huitoto settlements beyond those attractions, and then the road abruptly stops, surrendering to the jungle.\n\nbreakdancers in Leticia\n\n##### **Sights**\n\nOccupying just one (air-conditioned!) room in the red **Banco de la Rep\u00fablica** building, the **Museo Etnogr\u00e1fico de Leticia** (Cra. 11 No. 9-43, tel. 8\/592-7783, 8:30am-6pm Mon.-Fri., 9am-1pm Sat., www.banrepcultural.org\/leticia, free) provides a good introduction to the traditions and ways of life of some of the main indigenous people who live in the Colombian Amazon region, including the Ticunas, the Huitotos, and the Yukunas. Colorful feather crowns made of _guacamaya_ (macaw) feathers and descriptions of _chagras_ (islands of small vegetable plots in the middle of the jungle) and _malocas_ (community houses) are part of the exhibit. Explanations are provided in both Spanish and English. Sometimes art exhibits and other events are held in the building as well.\n\nThe **Parque Santander** (between Cras. 10-11 and Clls. 10-11) is a quiet place where unoccupied locals go for a brief reprieve from the intense midday sun. That is, it's quiet until around 5:30 each evening, when thousands of _loros_ (parrots) gather in the trees. It's a cacophonous racket. At 6pm on the dot, the Colombian national anthem blares from the loudspeakers from the military base facing the park. People stop what they are doing and stand at attention. The birds, however, have no such respect. They won't quiet down for anyone! The _loros_ have not always been here and are not native to the area. Some say that eccentric American hotel owner, anaconda wrestler, and convicted drug trafficker Mike Tsalickis released them in the park during his final days in Leticia in the 1980s.\n\nThe **Muelle Tur\u00edstico** (Cra. 11 at Cl. 8), also known as the _malec\u00f3n,_ is a busy bus station on water. It once faced a broad channel into the Amazon, but the current has brought sediments, creating a new island in front called, paradoxically, Isla de la Fantas\u00eda (it's a grubby neighborhood). During dry season, the channel closes for navigation and passengers must tramp across the island to embark. The port is seedy and grimy but 100 percent authentic. It's a real clash of cultures here, as tourists await their river tours while villagers, hailing from the very places the tourists will visit, arrive in the city to stock up on supplies. For any trip along the Amazon, including to the Peruvian town of Santa Rosa, you'll leave from here.\n\nIf you are interested in learning more about some of the medicinal plants, fruits, and trees you will see in the Amazon, a visit to the **Mundo Amaz\u00f3nico** (Km. 7.7 V\u00eda a Tarapac\u00e1, tel. 8\/592-6087, www.mundoamazonico.com, 8am-2pm daily, four-trail tour COP$36,000, one-trail tour COP$10,000) is a must. In the park you can take a walk among exotic fruit trees (like _copoaz\u00fa_ ) found in the area, learn about indigenous farming techniques, see some medicinal plants found in the rainforest, and see unusual fish, reptiles, and amphibians in the aquarium and terrarium area. On this last tour you can observe the prehistoric-looking _mata-mata_ turtle. Each guided tour takes 30-45 minutes. However, if you are in a hurry, you can request to take just one tour or do a fast trek around the park. Rafael Clavijo, the owner of the park, is a dedicated environmentalist, is knowledgeable about the flora and fauna of the Amazon, and speaks English. It is easy to take public transportation to the park. Look for a green Kilometer 11 bus (not towards Lagos) departing from the Parque Orellana (Cra. 11 between Clls. 7-8) across from the Hotel Anaconda. The bus costs only COP$2,000. (The trip costs COP$15,000 by _moto-taxi._ ) Tell the bus driver you'd like to be dropped off at Mundo Amaz\u00f3nico. It is a pleasant 10- to 15-minute walk from the road to the park entrance. You can inquire about day-trip excursions on offer.\n\nThe **Reserva Natural Tanimboca** (Carretera Los Kil\u00f3metros\/V\u00eda a Tarapaca, office Cra. 10 No. 11-69, tel. 8\/592-7679, www.tanimboca.com) is a nature reserve and lodge and is the best place close to Leticia where you can gain a real appreciation for the Amazonian jungle. Once there you can marvel at the stunning _maloca_ (community house) that they have built, check out the serpentarium, take a jungle walk, kayak, and experience the jungle from above by canopying. They also have some truly spectacular accommodations options. Spend the night 12 meters (40 feet) high in one of their three treehouses in the _dosel_ (canopy). Up above it's just you in the canopy, and thousands upon thousands of chatty jungle creatures. Although you may not be interested in long-term rental, these small thatched houses are comfortable and come equipped with a toilet. (You can also sleep in a hammock in the _maloca._ ) Included in your treehouse stay is a nocturnal jungle walk. Local cuisine, mostly grilled fish, is served at their restaurant. Tanimboca also organizes excellent multi-day tours of the Amazon region. The reserve is on the Kilometers road, about 15 minutes from town. It is accessible by the Kilometers public bus, which leaves from across the Hotel Anaconda.\n\n**Border Disputes**\n\nDuring the 19th and early 20th centuries, the border between Ecuador, Peru, and Colombia was a matter of dispute. In 1922, Colombia and Peru signed the Salom\u00f3n-Lozano Treaty, settling their common border at the expense of Ecuador. In 1932, a group of Peruvian civilians and some soldiers occupied Leticia. It is not clear whether the Peruvian government supported this attack. The occupation of Leticia led to a war in which both countries scrambled to get troops to this remote area. In 1932, Peru took the remote town of Tarapac\u00e1. In 1933, Colombia sent a fleet up the Amazon (including two new warships purchased from France), retook Tarapac\u00e1, and captured the Peruvian town of G\u00fceppi. As troops from both countries were preparing for a major confrontation, the League of Nations brokered a truce on May 24, 1932. This was the first time that the League, precursor to the United Nations, actively intervened in a dispute between two countries. On June 19, Peru returned Leticia to Colombia.\n\nAcross the Peruvian border is privately owned **Reserva Natural Marash\u00e1** (Cra. 10 No. 7-55, tel. 8\/592-5622, www.reservamarasha.com, COP$225,000 pp, all meals and activities included). It offers a range of ecotourism activities and is 25 kilometers (15 miles) from Leticia, so it can be easily visited as a day trip. The standard day trip includes excursions in kayaks or canoes and various nature walks. Lunch and transportation from Leticia and back is also included. Seven cabins can lodge 2-15 people. Conveniently located near Leticia, Marash\u00e1 is a good alternative to more remote reserves in the R\u00edo Yavar\u00ed area.\n\n##### **Festivals and Events**\n\nFrom around the 15th of July to the 20th, the **Festival de Confraternidad Amaz\u00f3nica** has been going strong since 1987 and is a celebration of Amazonian culture and friendship between the three neighboring countries of Colombia, Brazil, and Peru.\n\nYou have to like events that celebrate fish, especially ugly three-meter-long ones! The **Festival Piraruc\u00fa de Oro** takes place over three days at the end of November and beginning of December. It is more formally known as the **Festival Internacional de M\u00fasica Popular Amazonense.** Named in honor of the enormous piraruc\u00fa river fish, this is actually a cultural festival with numerous musical and dance performances.\n\n##### **Shopping**\n\nThe **Mercado Municipal** (Cl. 8 at Cra. 12, 7am-3pm daily) is a good place to pick up rubber boots or other gear for any jungle trip. (Many reserves and travel agencies can either rent or loan you a pair, so find out beforehand.) As you're trying on rubber boots, make sure they are easy to take off. Always shake out your socks, shoes, and boots before putting them on in the jungle. In the food section of the market, you can pick up some fruit or a dirt cheap meal.\n\nThe **Museo Uirapuru** (Cl. 8 No. 10-35, tel. 8\/592-7056, 9am-noon and 3pm-7pm Mon.-Sat., 9am-noon Sun.) is more a handicraft store than museum, although you can take a look at various river creatures in aquariums and snakes in jars in the back. Traditional medicines are also sold here.\n\n##### **Recreation**\n\n###### **TOURS**\n\nTour companies based in Leticia or Bogot\u00e1 offer a wide range of tour packages, from day trips to week-long packages. Hotels and hostels can organize these packages as well, or at least refer you to a tour agency.\n\nOne-day tours leaving from Leticia are a popular option for exploring the Amazon. These tours hit all the major sights: Victoria Regia, Isla de los Micos, Macedonia, Puerto Alegr\u00eda, and Lago Tarapoto along the river from Leticia to Puerto Nari\u00f1o. These cost around COP$170,000 each, depending on how many people are in the tour group. Tours leave Leticia at about 7am, returning at 5:30pm. You can also do a half-day tour, which does not include Puerto Nari\u00f1o. This will cost around COP$60,000-80,000 depending on your negotiating skills.\n\nA three-day tour may be a good option if you have limited time and would rather not have to worry about organizing things on your own. Before booking a package tour, find out where you'll be overnighting. Some tours may have you spending several nights in Leticia. It's OK to spend a night or two in dusty Leticia, but to get a real taste of the Amazon you really need to get to the jungle!\n\n**Ecodestinos** (Cl. 8 No. 7-99, Local 2, Leticia, tel. 8\/592-4816; Cra. 70H No. 127A-72, Bogot\u00e1, tel. 1\/608-8031, www.ecodestinos.com.co) is affiliated with Aviatur, one of the top travel agencies in Colombia, popular with Colombian tourists. They offer various package tours of the Amazon. Their \"Amazonas Selva y R\u00edo\" tour starts at COP$598,000 per person and includes two nights accommodations in Leticia, all meals, and tours to Puerto Nari\u00f1o, Lago Tarapoto, Parque Amacayacu, and Ticuna villages. The Ecodestinos website has an extensive listing of their offers, and these can be booked online. There are also some day-trip excursions, such as kayak tours (COP$75,000 pp) near Leticia, for those who are traveling independently.\n\n**Yurupary Amazonastours** (Cl. 8 No. 7-26, tel. 8\/592-4743, www.hotelyurupary.com) is affiliated with the Hotel Yurupary. A four-day, three-night package including a stay at their lodge on the Peru side of the R\u00edo Yavar\u00ed starts at COP$640,000 per person. In addition, they offer full-day tours on the river that cost about COP$117,000 (for groups of four or more) and half-day tours (COP$99,000) by land to a Huitoto community that include a nature walk.\n\n**Tanimboca** (www.tanimboca.org, tel. 8\/592-7679) is affiliated with the Reserva Natural Tanimboca in the jungle just outside of Leticia. Package tours with Tanimboca include jungle walks, a couple of nights in their fabulous treehouses at their reserve, and overnight visits to Puerto Nari\u00f1o and to the Marash\u00e1 reserve in Peru. They offer mostly private or small group tours. For a stay of five days and four nights, including activities, expect to pay around COP$1,300,000 per person. They can also arrange private one-day tours on the river. Tanimboca is a highly recommended and reputable agency.\n\n##### **Accommodations**\n\nSurprisingly, there are very good accommodations options, for all budgets, in Leticia. All of these are owned and managed by Colombians from other parts of the country or by Europeans.\n\n**Apaporis Hostel** (Cra. 10 No. 6-17, cell tel. 312\/522-0446 or 311\/886-5996, COP$20,000 dorm, COP$35,000 d) is a small, sparkling clean hostel on a quiet street. It has one dormitory room, two private rooms, a kitchen, and a garden (including a small organic vegetable garden) out back. It's owned by Elizabeth, a young Colombian entrepreneur. The largest hostel you probably ever have seen just might be M **Mahatu Jungle Guesthouse** (Cl. 7 No. 1-40, tel. 8\/592-7384, cell tel. 311\/539-1265, www.mahatu.org, COP$25,000 dorm, COP$60,000 d), literally straddling the Brazilian border. It's run by the affable Gustavo Alvarado, and you feel like you're in the country here. It is a peaceful, green place with dorm accommodations and private rooms spread out on a huge property that has a pool and two lakes! You can even take out a paddleboat for a quick spin. There's no shortage of hammocks around here. It's about a 15-minute walk into town from the hostel.\n\nRun by a Swiss-Colombian couple, **La Jangada Hospedaje** (Cra. 9 No. 8-106, cell tel. 311\/498-5447, <http:\/\/la-jangada-hostel.minihostels.com>, COP$25,000 dorm, COP$70,000 d) is a friendly hostel option in town. They have an extensive program of day trips from which to choose.\n\n**Waira** (Cra. 10 No. 7-36, tel. 8\/592-4428, www.wairahotel.com.co, COP$148,000 d) is a midrange option that caters to Colombian tourists. It looks swanky from the outside, but its 41 rooms are on the small side. Wireless Internet and air conditioning are available. The **Hotel Anaconda** (Cra. 15 No. 93-75, tel. 8\/218-0125, www.hotelanaconda.com.co, COP$160,000 pp) was one of the first hotels in Leticia and has been in operation for years. The 50 air-conditioned rooms are large, there is wireless Internet in the lobby, and a restaurant is on-site. It's no longer the swank hotel it may have once been, but it's in the heart of town and they have a big pool and a poolside bar! If you're not staying here and want to cool off you can get a day pass that costs COP$12,000. Note that per person room rates decrease as the number of guests increases.\n\nAn excellent choice in Leticia is the friendly and professionally run M **Amazon B &B** (Cl. 12 No. 9-30, tel. 8\/592-4981, www.theamazonbb.com, COP$216,000 d). It's on a quiet street away from the bustle of the city but within easy walking distance to restaurants and services at the same time. It's a popular place for a good rest before and\/or after a few days in the jungle. There's no air conditioning in the six _caba\u00f1as_ (they have fans), but they are modern and tastefully decorated. Breakfast is included. They also offer Spanish classes and can arrange all sorts of excursions for you. **Hospedaje Los Delfines** (Cra. 9 No. 12-81, tel. 8\/592-7488, COP$70,000 d) has wireless Internet, but there's no air conditioning, and breakfast is not included.\n\nThe cr\u00e8me de la cr\u00e8me of hotels in Leticia is the all-inclusive **Decameron Decalodge Tikuna** (Cra. 11 No. 6-11, tel. 8\/592-6600, www.decameron.com, COP$520,000 d). It's a spacious place with a very good open-air restaurant (open to non-guests) where you dine under a gigantic green anaconda-like snake. There's even a vegetarian menu. Facilities are nice here: a swimming pool, a _maloca_ (community house), and tastefully done _caba\u00f1as_ complete with comfy beds and hammocks. The hotel organizes excursions and activities, so you don't have to plan anything! Internet is expensive here.\n\nThe **Omshanty Jungle Lodge** (Km. 11 V\u00eda Leticia-Tarapac\u00e1, cell tel. 311\/489-8985, www.omshanty.com, COP$15,000 dorm, COP$55,000 d) is north of Leticia and offers clean dorm-style accommodations as well as private rooms. It's really in the jungle! You can cook meals in their kitchen, and there are some small mom-and-pop restaurants across the street. Efficient and inexpensive public transportation is available. The lodge can also organize stays in nearby indigenous communities. The lodge is just before the Reserva Natural Tanimboca.\n\n##### **Food**\n\nLeticia is not a culinary capital, but it is the place to sample some unusual Amazonian dishes. The standard Amazon meal includes fried fish, cassava, rice, _patacones_ (fried plantains), and perhaps a small salad. Piraruc\u00fa is the king of fish around here. It is one of the largest fish in the world, reaching up to three meters (10 feet) long and 350 kilograms (770 pounds). This fish is threatened, and regional governments have banned its fishing and consumption from November to March. You may not see it in the wild, although it does pop up to the surface to breathe every 15 minutes. You have a reasonably good chance, however, of hearing it. It makes a deep bellowing sound that echoes across the river. A particular dish popular here is the _patarasca,_ which is two types of fish, usually _dorado_ and _pintado,_ grilled with herbs and vegetables in banana leaves. This is accompanied by a juice such as _copoaz\u00fa_ or the ever-popular Brazilian beer.\n\nM **El Cielo** (Cl. 7 No. 6-50, cell tel. 312\/351-0427, 4pm-11pm Mon., Wed., and Fri., 11am-5pm Sun., COP$20,000) has the most interesting menu in town: Amazon fusion. They make pizza dough out of cassava flour, and Amazonian ants and _mojojoy_ (worms) may appear as ingredients in some dishes on the menu. There are vegetarian dishes like cr\u00eapes and lasagnas as well, and the cocktails are fine.\n\nWith dusty handicrafts from the Amazon adorning its walls, **Tierras Amaz\u00f3nicas** (Cl. 8 No. 7-50, hours vary Tues.-Sun., COP$20,000) strives to be Leticia's version of the famous Andr\u00e9s Carne de Res in Bogot\u00e1. Some of the unusual dishes you can order here include _chicharr\u00f3n de piraruc\u00fa,_ which are sort of like fish nuggets, and piraruc\u00fa steamed in banana leaf. Big lemonades (to complement the big food portions) here hit the spot. There's not much for vegetarians here. Sometimes they have live music to satisfy both Brazilian and Colombian tastes.\n\n**El Abuelo** (Cra. 11 at Cl. 7, no phone, set lunches COP$12,000) is a popular place with locals. It serves up the usual seafood dishes. **El Sabor** (Cl. 8 No. 9-25, tel. 8\/592-4774, set lunch COP$12,000) has inexpensive and tasty set lunches.\n\nThe only place in town where you'll find burritos, cheesy cr\u00eapes, and other fast food is **Amektiar** (Cl. 9 No. 8-15, tel. 8\/592-6094, 4pm-midnight). Across from Parque Santander, **Casa del Pan** (Cl. 11 No. 10-20, 6:30am-11pm Mon.-Sat.) is Leticia's version of Starbucks, a place for breakfast and a carb fix after a day in the jungle. It also serves refreshing lemonades and juices. At **Barbacoas** (Cra. 10 No. 8-28, daily) you can pick up a coffee and light breakfast in the morning, and at night have a beer and watch locals play pool.\n\n**Supermercado Hiper Kosto** (Cl. 8 No. 9-31, tel. 8\/592-8067) is a very basic grocery store where you can stock up on jungle provisions.\n\n##### **Information and Services**\n\n###### **VISAS AND OFFICIALDOM**\n\nTo travel to Tabatinga, Brazil, or Santa Rosa, Peru, for the day, or for stops at Peruvian villages on the way to Puerto Nari\u00f1o, there is no need for immigration formalities, but it's a good idea to carry your passport with you just in case.\n\nIf you are traveling on to destinations in the interior of Brazil or Peru from Leticia, you must obtain an exit stamp at the Migraci\u00f3n Colombia office at Aeropuerto Internacional V\u00e1squez Cobo (3 km north of town, tel. 8\/592-4562). There is a **Migraci\u00f3n Colombia** (Cl. 9 No. 9-62, tel. 8\/592-6001) office in town, but it does not provide entry or exit stamps for visitors. This office mainly provides services for Colombian citizens or non-Colombian residents.\n\nOnce you get your passport stamped at the airport, if you're continuing on to Manaus, Brazil, you will need to present your papers at the Brazilian **Polic\u00eda Federal** (650 Av. Da Amizade, Tabatinga, 7am-noon and 2pm-6pm Mon.-Fri.), near the Tabatinga hospital. If continuing to Peru, get your Peruvian entry stamp at the police office in Isla de Santa Rosa, which is on the main path through town.\n\nAt the **Brazilian Consulate** (Cra. 10 No. 10-10, Piso 2, Leticia, 8am-2pm Mon.-Fri., tel. 8\/592-7530), you can obtain a visa for Brazil, which is necessary for U.S. and Canadian citizens. Again, this is only necessary if you are planning to stay overnight in Brazil. You must have a yellow fever vaccination card and an onward airline ticket ready to present. Processing time is two to three 2-3 days. For U.S. citizens, the visa costs a hefty COP$430,000.\n\nNo visa is required to visit Peru for under 90 days. The **Peruvian Consulate** (Cl. 11 No. 5-32, Leticia, tel. 8\/592-3947, www.embajadadelperu.org.co, 8am-2pm Mon.-Fri.) can assist with further information.\n\n###### **HEALTH AND MEDICAL SERVICES**\n\nThe Colombian health authorities recommend getting a yellow fever vaccination 10 days before arriving in the area. The World Health Organization yellow health card, nonetheless, is not regularly checked at the airport. Malaria is very rare, but to be on the safe side, consider taking anti-malarial pills starting before your visit and up until four weeks after departure from the Amazon area. These can easily be purchased in pharmacies across Colombia without prescription.\n\nThe major hospital in town is the **Hospital San Rafael** (Cra. 10 No. 13-78, tel. 8\/592-7074). The **Cl\u00ednica Amazonas** (Cra. 6 No. 6-05, tel. 8\/592-5579) is open 24 hours a day, as is the **IPS Ind\u00edgena Trapecio Amaz\u00f3nico** (Cra. 9 No. 9-62). A dental clinic in town is the **Centro Odontol\u00f3gico del Amazonas** (Cra. 10 No. 12-109, tel. 8\/592-5953).\n\nFor whatever ails you, go to **Productos Naturales del Trapecio Amaz\u00f3nico** (Cl. 8 No. 9-87, tel. 8\/592-4796, 9am-6pm Mon.-Sat.) and you're bound to find a remedy. Traditional remedies to cure a laundry list of maladies, from impotence to arthritis and obesity, are available, and the knowledgeable staff can suggest certain medicinal therapies for you.\n\n###### **EMERGENCIES**\n\nReport emergencies to the local **Polic\u00eda Nacional** (Cra. 11 No. 12-30, emergency line 112 or 8\/892-5060).\n\n###### **TOURIST INFORMATION**\n\nFor information on the area, the **Fondo de Promoci\u00f3n Ecotur\u00edstica del Amazonas** (Cl. 8 No. 9-75, tel. 8\/592-4162, www.fondodepromocionamazonas.com) may be of help. It is near the Museo Uirapuru.\n\n###### **MONEY**\n\nThere are several Colombian banks in Leticia with ATMs. These include: **Banco de Bogot\u00e1** (Cra. 10 No. 10-108), **BBVA** (Cl. 7 No. 10-12, tel. 8\/592-4975), **Banco Agrario** (Cl. 8 No. 10-66, tel. 8\/100-0000), and **Bancolombia** (Cra. 11 No. 9-52, tel. 8\/592-6067). Generally, banking hours are 8am-11:30am and 2pm-4pm on weekdays. Each of these has ATMs. This is the best (if not only) place in the region to get cash. As you venture further afield from Leticia, credit cards are rarely accepted and ATMs are nonexistent. Colombian currency is accepted in the entire Amazon region near Leticia, including in Brazil and Peru.\n\n###### **INTERNET ACCESS**\n\nLeticia has a few Internet cafes, but don't expect rapid connections. **Amazon Technology** (Cra. 10 No. 7-85, 8:30am-noon and 3pm-9pm Mon.-Sat., 3pm-9pm Sun.) is a comfortable place to check email.\n\n###### **LAUNDRY**\n\nThere is a **laundry service** (Cra. 10 No. 9-32) that will have your clothes ready (and dry!) within one day.\n\n##### **Getting There And Around**\n\nAll major national carriers serve the Leticia airport, **Aeropuerto Internacional General Alfredo V\u00e1squez Cobo** (3 km north of town). **Avianca** resumed service in November 2013 with flights from Bogot\u00e1. **Copa** (Cl. 7 No. 10-36, tel. 8\/592-7838, 8am-12:30pm Mon.-Fri., 9am-1pm Sat., www.copaair.com) and **LAN** (Colombian toll-free tel. 01\/800-094-9490, www.lan.com) offer daily flights from Bogot\u00e1. **Satena** (Cra. 11 No. 9-42, Local 1, cell tel. 312\/457-6291, Colombian toll-free tel. 01\/800-091-2034, www.satena.com) flies from Leticia to La Chorrera (with connections to Araracuara and San Vincente del Cagu\u00e1n), La Pedrera, and Tarapaca. Flights to those supremely exotic destinations are usually once a week. Viva Colombia will launch flights to Leticia (out of Medell\u00edn) in the near future. There are no international flights from Leticia. Taxis to and from the airport should cost under COP$10,000.\n\nAll river transportation to **Puerto Nari\u00f1o, Caballococha** (Peru), and to points in between departs from the **Muelle Tur\u00edstico** (Tourist Wharf) in Leticia. To Puerto Nari\u00f1o there are daily boats at 6am, 8am, 10am, and 2pm. The trip takes two hours and costs COP$29,000 per person. By paying extra, it is possible to arrange for these boats to leave you off at stops along the way to Puerto Nari\u00f1o, such as Isla de los Micos, and be picked up by a later boat. Tickets for these boats can be obtained at the port in the Malec\u00f3n Plaza Local 101. This little shopping area is on the lefthand side when facing the port. There are three agencies with offices there: **Transportes Amaz\u00f3nicos** (tel. 8\/592-5999, cell tel. 313\/347-8091), **L\u00edneas Amazonas II** (tel. 8\/592-6711, cell tel. 311\/532-0633), and **Expreso Unidos Tres Fronteras** (tel. 8\/592-4687, cell tel. 311\/452-6809). It's best to go in person to the offices. It's organized and straightforward.\n\nThe boats to and from **Manaus** depart from the main port in **Tabatinga**. Slow cargo boats depart frequently, take three days, and cost COP$210,000 if you sleep in a hammock or COP$1,100,000 for a two-person cabin with air conditioning. These usually leave on Wednesdays and Saturdays at around 2 or 3 in the afternoon. These prices include food on board. If you plan to sleep in a hammock (which you must purchase in town beforehand), try to get on board early (up to four hours) to stake out a good place. In fact, you can board the boat as soon as you buy a ticket, even a day before departure. Stock up on snacks and water, as the food, which is included in the price, is not that great. A good book and a deck of cards will come in handy. There aren't any mosquitoes to bother you while snoozing in your hammock at night, but every once in a while a flying beetle may annoy you. You will be searched, presumably for drugs, once you board the boat. The reverse journey can also be made (from Manaus to Leticia), but it will take six days.\n\nWeekly fast boats (30 hours) depart on Friday and cost COP$430,000. So as not to miss your boat, remember that Tabatinga is one hour later than Leticia. Be sure to get your departure stamp at the Leticia airport! You can also take a flight from Tabatinga to Manaus.\n\nTo Iquitos, **Transtur** (Rua Marechal Mallet No. 349, tel. 97\/8113-5239, iquitostours@hotmail.com) and **Golfinho** (Av. Marechal Mallet No. 306, tel. 97\/3412-3186) have offices in Tabatinga. Boats leave six days a week from Tabatinga, stopping at the Peruvian town of Santa Rosa, across from Leticia, for immigration purposes. The trip takes nine hours and costs COP$140,000. Departure time is 3:30am or 5:30am, depending on the boat company. There are also boats, usually twice in the afternoon, to the Peruvian town of **Caballococha,** where you can catch a flight to Iquitos.\n\nLeticia is a small town; you can walk everywhere you'd like to go. Some hotels have bikes you can rent to explore the town and beyond. _Moto-taxis_ (motorcycle taxis) and _motocarros_ (three-wheeled _moto-taxis_ ) are plentiful. Sights along Kilometer 11 can be reached by public bus or _moto-taxi._ Sights down the river can be reached by taking a _lancha_ from the port.\n\n#### **ALONG THE R\u00cdO AMAZONAS**\n\nSeveral sights provide excellent photo ops along the R\u00edo Amazonas westward from Leticia to the Parque Amacayacu. These are usually visited as part of a package one-day tour offered by all hotels and travel agencies. Most of these tours continue past Amacayacu to Puerto Nari\u00f1o and the Lago Tarapoto. That is the easiest way to visit them, but it's a rather \"touristy\" proposition.\n\nFor greater flexibility and to avoid feeling part of the herd you can also charter your own boat for around COP$300,000, but you'll have to be specific about where you want to go in order to negotiate a good price. Head to the _malec\u00f3n_ (wharf) and ask any of the boat captains lingering about for this option.\n\nWhen on the water, keep your eyes peeled for dolphins, both gray and pink, and pelicans above. Every once in a while, you'll pass a fisherman in a _peque peque_ dugout canoe, just barely above water. When it rains you'll see them furiously scooping out water with their hands so as not to sink!\n\n##### **Victoria Regia**\n\nSeven kilometers (four miles) and about a 15-minute boat ride west from Leticia, **Victoria Regia** (COP$5,000 admission) is a private reserve that is usually the first stop on the river. You can view large circular _Victoria amazonica_ lily pads and their lovely white lotus flowers floating atop the water. These are some of the largest water plants in the world, and the leaves can measure up to 1.5 meters (5 feet) in diameter, with roots extending 7 meters (23 feet) below the water's surface. They say that these plants are so strong they can support the weight of a one-year old child. It's not recommended, however, to test this theory on your offspring. You can also marvel at a magnificent old ceiba tree farther along on the park walkway. Ceibas are some of the tallest trees in the world. This is also your chance to get that quintessential Amazon photo with a colorful _guacamaya_ (macaw) or two on your shoulders.\n\n##### **Puerto Alegr\u00eda**\n\nOn the Peruvian side of the river farther along is the community of **Puerto Alegr\u00eda.** Here the attraction is exotic animals. When tourist boats show up at the community dozens of times each day, local women and children greet the tourists with all sorts of animals in hand: alligators, sloths, turtles. For a contribution you can be photographed holding several of the animals. Tourists are told that the animals are released into the wild after a period of time, but that seems hard to believe. Even if that is the case, they must have a hard time adjusting to their natural habitat after years in captivity, being held by humans day in and day out. You may find it disturbing how these creatures are used for human entertainment here.\n\n##### **Isla de los Micos**\n\nThe **Isla de los Micos** (Monkey Island) is the most popular tourist attraction on the river. It's about 40 kilometers (25 miles) west of Leticia. At this island, owned by the Colombian hotel chain Decameron, elevated walkways meander through the jungle, and with just a morsel of fruit in your hand, you'll make the monkeys go bananas. They'll proceed to climb all over you in hopes of a snack, as if you were a tree. The monkeys are not native to the island, rather they were brought there by controversial hotel owner and entrepreneur Mike Tsalickis in the 1970s. Up to 12,000 supposedly lived there at one point. Some even were used for medical experiments. It is a tourist trap, but it's hard to deny that kids love it.\n\nlily pads at Victoria Regia\n\n##### **Macedonia**\n\nThe Ticuna village of **Macedonia** is a regular stop for tourist boats. When visitors arrive, they are invited in to the _maloca_ (community house), where an authentic ceremonial dance is performed. Tourists are led onto the middle of the dance floor to the beat of a turtle shell drum. Around the _maloca_ you can peruse an array of handicrafts at stalls set up by local women. A specialty is _palo de sangre_ wood carvings. Although touristy, it is nice that the community manages all the activities here, and all the income goes directly to them.\n\n##### **Reserva Calanoa**\n\nTucked away on the bank of the Amazon, west of the Macedonia community and just before the community of Makagua and the Parque Nacional Natural Amacayacu, is M **Reserva Calanoa** (R\u00edo Amazonas, 60 km\/37 miles west of Leticia, cell tel. 311\/842-4392, www.calanoaamazonas.com, COP$230,000 pp). Most people speeding by in their boats do not notice it. That's OK, because here the emphasis is on appreciation of the natural surroundings, of indigenous cultures, and learning: about art, photography, cooking, and flora and fauna. Activities offered (most are included in the overall price) include day and nighttime canoe trips, bird- and dolphin-watching, and canoe trips, by day and by night. It is a project of a Canadian-Colombian couple, Diego and Marlene Samper. Diego is an accomplished photographer. Calanoa offers one-of-a-kind tours, for example an annual workshop on natural fibers in February. During that week, participants learn the art of weaving _palma de caran\u00e1_ leaves that form the stunning _maloca_ rooftops that you have seen at every stop in the Amazon.\n\nReserva Calanoa has four beautifully done cabins made of all natural materials, and a lovely dining and relaxing area overlooks the river. The lodgings are called the Hotel de Selva. There is a strong emphasis here on working with and supporting neighboring indigenous communities, such as in the settlement of Mocagua, where Calanoa is undertaking a mural painting project in order to preserve and celebrate indigenous culture.\n\nLife abounds in the Parque Nacional Natural Amacayacu.\n\n##### **Parque Nacional Natural Amacayacu**\n\nThe prime, unspoiled plot of land known as Parque Nacional Natural Amacayacu covers some 300,000 hectares (740,000 acres). It has its southern border on the banks of the Amazon between the R\u00edo Amacayacu (\"River of Hammocks\") and the R\u00edo Matamata and extends northward to the R\u00edo Cotuhe. It was declared a national park in 1975. The park is characterized by undulating hills, swamps, and an intricate network of streams. The highest point in the park reaches 200 meters (650 feet) above sea level. It is estimated that in the park there are more than 5,000 plant species, 150 mammal species (including pink dolphins, tapirs, jaguars, manatees, nutrias, and numerous primates), 500 species of birds, about 100 species of fish, and the list goes on. Resident animals such as squirrel monkeys, sloths, wild boars, and jaguars are hard to spot in the park, and in the jungle in general.\n\nEach year much of the park is flooded during the rainy months of April and May. In 2012 it was a particularly wet wet season, resulting in extensive damage to park structures. The park has since been closed to tourism, although the Ticuna settlement of San Mart\u00edn can be visited.\n\nIf you are traveling by boat up the spectacular and serpentine Amacayacu (this can apply to any jungle cruise you take in the region), insist that the captain completely cut the engine at least once or twice during the journey, so that you can enjoy the incredible sounds of the jungle. When you float along in silence, hearing nothing but the calls of distant monkeys, shrieks of birds, or the constant hum of legions of frogs and insects, it is a magical experience. It makes you think, that, despite the tsunami of evidence to the contrary, just maybe we can, for the first time in the history of humanity, turn things around and save this remarkable ecosystem. Boat drivers are usually in a hurry, so you'll have to ask them something like: _\"Podemos parar aqu\u00ed sin motor un minutico por favor?\"_ (\"Would it be possible to stop here without the motor for a moment, please?\").\n\nDeep within the Parque Amacayacu, a dedicated team of animal lovers is rehabilitating monkeys that have been rescued from poor conditions in captivity. **Fundaci\u00f3n Maikuchiga** (Leoncio S\u00e1nchez, cell tel. 313\/397-1981, www.maikuchiga.org) is a group that rescues and cares for dozens of primates, like woolly monkeys, red howlers, and brown capuchins, who have been injured, orphaned, or rescued from poor conditions in captivity in the Colombian Amazon. Dr. Sara Bennett is the \"mother of the monkeys\" and runs the show here. She has been in Colombia for many years, originally arriving to conduct research on Amazonian trees. One of her greatest accomplishments has been in convincing local tribes to no longer hunt woolly monkeys, in order to protect their survival. Her aim is to promote the protection and awareness of these species, and generally to promote conservation efforts. You can visit the foundation to get to know their work, and they are always in need of financial support. Maikuchiga can be reached on foot from San Mart\u00edn during dry months.\n\n###### M **SAN MART\u00cdN DE AMACAYACU**\n\nUp the R\u00edo Amacayacu, within the PNN Amacayacu, is the Ticuna community of **San Mart\u00edn.** The community has organized itself to receive tourists and offers walks, canoe rides, and other activities. Friendly and knowledgeable community elder Victor \u00c1ngel Pereira (cell tel. 310\/769-7305) will receive you and get you organized. Entrance to the community costs COP$5,000, and this is an interesting day-trip excursion from either Leticia or Puerto Nari\u00f1o. There is a handicrafts store where local girls sell beautiful handwoven _mochilas_ (handbags), bark scrolls from the _yanchama_ tree on which scenes of jungle animals are painted using all natural dyes, and jewelry. You can also do a homestay with a local family for only about COP$10,000 per night in a hammock.\n\nThe M **Casa de Gregorio** (cell tel. 310\/279-8147, heike_van_gils@hotmail.com) is a lodge run by a Ticuna-Dutch couple, Heike and Jos\u00e9 Gregorio. She arrived in San Mart\u00edn as a doctoral student in agriculture sciences at the Universidad Nacional in 2004, and he is a Ticuna community leader. Through their **Small World Foundation** (www.smallworldfoundation.org), they work to improve the lives of the residents of this indigenous community, by installing toilets, starting a kindergarten, and purchasing rainwater tanks. A stay at the Casa de Gregorio provides visitors with a unique opportunity to discover the jungle and get to know Ticuna culture. A new, deluxe cabin was finished in 2013, and that costs COP$120,000 for a double in a luxurious king bed. There are two other simple double rooms and a small cabin, with a total capacity of 10. Lodging for two costs COP$80,000, and there are additional costs for meals, the community entry fee of COP$5,000, and for guides. Although it is possible to come for a day trip, this is not a recommended option. To get a taste of village life, it's best to not rush things and stay at least three or four days. To get there, you can take a boat for about 1.5 hours from Leticia for COP$24,000. These depart at 8am, 10am, and 2pm. Ask to be dropped off at Bocana Amacayacu (not the Parque Amacayacu). The return trip costs COP$29,000. You will need to arrange with Casa de Gregorio transportation from Bocana Amacayacu to San Mart\u00edn. That costs COP$30,000. You can also take a _peque-peque_ canoe from Puerto Nari\u00f1o or walk from there to San Mart\u00edn. You can also walk from Puerto Nari\u00f1o in tours organized by various hotels and agencies there. That expedition (you'll need a guide; ask Heike) takes three hours.\n\n#### M **PUERTO NARI\u00d1O**\n\nWhen you disembark at the village of Puerto Nari\u00f1o, atop a sloping hill overlooking the R\u00edo Loretoyac\u00fa, you'll wonder: Where are the motorbikes? Here in idyllic Puerto Nari\u00f1o, there are no roads and no motorized vehicles whatsoever. Environmentally minded and forward-thinking town council members decided many years ago that they wanted Puerto Nari\u00f1o to chart a different path than almost all other towns in Colombia (and for that matter in the world), and for their efforts, this town was named the first tourism sustainable town in the country. Here \"roads\" are actually palm-lined sidewalks that connect all the neighborhoods of this community together. Puerto Nari\u00f1o is so peaceful, you'll probably want to linger a while.\n\nan artisan at the Ticuna community of San Mart\u00edn\n\n##### **Sights**\n\nTo get a bird's-eye view of Puerto Nari\u00f1o and the rivers and jungle beyond, climb up the steps to the **Mirador Nai-pata** (COP$7,000, closes at 6pm). You pick up an entry ticket in the adjacent house. The tall treehouse (which is what _nai-pata_ means in Ticuna) is the perfect place to be at dusk.\n\nThe **Centro de Interpretaci\u00f3n Ambiental Nat\u00fctama** (cell tel. 312\/410-1925, www.natutama.org, 8am-12:30pm and 2pm-5pm, donations encouraged) is run by the conservation and education nonprofit Nat\u00fctama. At their center, you can watch some excellent videos about two important river species: the pink dolphin and the manatee. While the pink dolphin is celebrated in indigenous mythology, the unfortunate manatee is not. Thus it has been hunted to the brink of extinction. The focus of this organization is conservation awareness among the community, and in large part due to their educational outreach activities, the number of manatees in the Puerto Nari\u00f1o area has grown from 11 in 2002 to 24 in 2012. They also sell handicrafts and T-shirts, the proceeds of which help them carry out their activities. Nat\u00fctama means, in Ticuna, the \"world below the water.\"\n\n##### **Recreation**\n\n###### M **LAGO TARAPOTO**\n\n**Lago El Correo** and **Lago Tarapoto** are about a 20-minute boat ride from Puerto Nari\u00f1o, and this area is a good place for dolphin spotting (both pink and gray), swimming, piranha fishing, and nature hikes. Lago Tarapoto is connected with the Amazon, so by swimming in its serene waters you can truthfully say that you swam in the Amazon. The Lago Tarapoto at 37 square kilometers (14 square miles) is much larger than the adjacent Lago El Correo, which is closer to Puerto Nari\u00f1o. There are several spots in this area where you can see _renacos,_ also known as _el arbol que camina_ (the tree that walks), a tree with a jumble of above-ground roots. To get to the lake you'll have to go with a guide on a boat. The tourist office in Puerto Nari\u00f1o or any hotel can help organize a visit to these lakes and surrounding flooded jungles. This excursion, pleasant to make in the late afternoon, will cost COP$50,000 per person.\n\na bird's-eye view of Puerto Nari\u00f1o\n\n##### **Festivals and Events**\n\nThe **Festival Aut\u00f3ctono de Danza, Murga y Cuento** takes place at the end of December through early January each year, and is a celebration of indigenous culture and identity. Each night the town gathers around the basketball court for evenings of storytelling, dance, and the requisite beauty pageant. Interestingly, an important component of the pageant is a demonstration of the girls' knowledge of their native tongue.\n\n##### **Accommodations and Food**\n\n**Hospedaje Wone** (Cra. 1 No. 4-14, cell tel. 314\/266-5496 or 320\/878-5785, wonenemico@hotmail.com, COP$20,000 d) is a pleasant, locally owned place, with potted plants and flowers throughout, that has three rooms. It's near the port. A bathroom is outside in the back. They lost everything recently during the disastrous 2012 rainy season. After rebuilding, they are crossing their fingers that nature will be on their side for a while.\n\nA cozy and inexpensive option in Puerto Nari\u00f1o is M **Malocas Napu** (Cl. 4 No. 5-72, cell. tel. 314\/437-6075 or 313\/800-2771, www.malocanapu.com, COP$35,000 d, pp). There are eight rooms here in two _malocas_ (community houses). Bathrooms are separate. They can take you on fun excursions to Lago Tarapoto, including a little piranha fishing on a dugout canoe, if you are so inclined, and also show you the flooded jungle. Napu works closely with the travel agency Ecodestinos.\n\nFriendly **Hotel Lomas del Paiy\u00fc** (Cl. 7 No. 2-26, cell tel. 313\/268-4400, www.hotellomasdelpaiyu.turismo.co, COP$50,000-70,000 d) offers 22 clean, if a little stuffy, rooms. The top end option in town is the **Hotel Casa Selva** (Cra. 2 No. 6-72, cell tel. 311\/280-7319, Bogot\u00e1 tel. 1\/657-1468, www.casaselvahotel.com, COP$150,000 d). The rooms are immaculate and it is undeniably comfortable, but the atmosphere is rather businesslike.\n\nIf you'd like to get away from the hustle and bustle of Puerto Nari\u00f1o but still be within walking distance of it, the M **Alto del \u00c1guila Caba\u00f1as del Fraile** (cell tel. 314\/234-7292, 314\/201-3154, or 311\/502-8592, altodelaguila@hotmail.com, COP$25,000 pp with private bath) is your best bet. It's actually quite fun, too, with resident monkeys monkeying about always making things interesting. Cabins are clean and cheerful, the kitchen area is a comfortable place to have a _tinto_ and read, and you can take a kayak out for a spin on the river for free. The owner is indeed a Franciscan missionary, a straight-talking one at that, and he's a joy to meet.\n\nThe best restaurant in town is the **Restaurante Las Margaritas.** They specialize in the usual fish dishes but can also whip up vegetarian fare. Another friendly spot is **Delicias Amaz\u00f3nicas Metane** (Cra. 2 between Clls. 6-7). Set lunches go down well with a _copoaz\u00fa_ juice as you watch everybody walk by on the pathway in front. The small grocery store **Mercaselva** (Cra. 2 and Cl. 5, 6:30am-8:30pm daily) also makes OK pizzas. Order yours a few hours in advance.\n\n##### **Information and Services**\n\nThe helpful **tourist office** (Cra. 1 at Cl. 5, Palacio Municipal, no phone, 7am-noon and 2pm-5:45pm Mon.-Fri.) can help you organize excursions with official tour guides. There are no ATMs in Puerto Nari\u00f1o. There are a couple of Internet caf\u00e9s, but the connections are very slow.\n\n##### **Getting There and Around**\n\nIt takes just under two hours on a public boat to make the 87-kilometer (54-mile) river journey from Leticia to Puerto Nari\u00f1o without stopping. Tickets (COP$24,000 one way) for this trip can be purchased at the Leticia _malec\u00f3n_. Look for the office at Malec\u00f3n Plaza Local 101, to the left of the _malec\u00f3n_. Three companies provide this service: **Transportes Amaz\u00f3nicos** (tel. 8\/592-5999), **L\u00edneas Amazonas II** (tel. 8\/592-6711), and **Expresos Unidos Tres Fronteras** (tel. 8\/592-4687). There are usually three boats per day starting at 8am, 10am, and 2pm.\n\nWhen leaving Puerto Nari\u00f1o bound for Leticia, make sure you reserve your spot a day or so in advance. You can do this at the office on stilts along the walkway to the docks. Boats leave Puerto Nari\u00f1o at 7:30am, 11am, 2pm, and 4pm. You can also take a boat to Caballococha, Peru, from Puerto Nari\u00f1o. Ask at the office about this option.\n\n#### M **R\u00cdO YAVAR\u00cd**\n\nThe **R\u00edo Yavar\u00ed** (Rio Javari in Brazil) begins in Peru and serves as a border between Brazil and Peru. Its waters flow some 1,050 kilometers (650 miles) before it meets the Amazon in Brazil. About a six-hour journey from Leticia by boat (three hours when the jungle is flooded), this part of the Amazon basin is unspoiled, isolated, and is home to two excellent private natural reserves where you will be immersed in the jungle. Spend at least three days or up to a week at one of the Yavar\u00ed nature reserves (they are both excellent) to gain a real appreciation for jungle life. The longer you stay the more wildlife you are apt to see: pink dolphins, alligators, snakes, and dozens of birds. Although this region is technically in Brazil, it very well may be one of the highlights of your trip to Colombia.\n\n##### **Reserva Natural Palmar\u00ed**\n\nLocated on a bluff overlooking the river, the **Reserva Natural Palmar\u00ed** (office Cra. 10 No. 93-72, Apt. 602, Bogot\u00e1, tel. 1\/610-3514, www.palmari.org or www.travesiassas.com, COP$256,000) is a pioneer in ecotourism in this part of the Amazon. Once you arrive, you will be paired with a guide who will accompany you throughout your stay. You won't be grouped together with others. Activities offered include jungle walks (including nighttime), treks, canopying, kayaking, canoe rides, and visits to nearby indigenous communities. Usually guests spend one night in the jungle. Near Reserva Natural Palmar\u00ed you can admire massive ceiba trees, also called _lupuna_ trees. These noble giants reach up to 70 meters (230 feet) high and have witnessed a lot in their over 400 years of life! Ticuna Indians believe that these trees are what started life and created the river.\n\nbeautiful mushroom in the jungle\n\nThe reserve, first and foremost, has a strong commitment to the environment and to the community. The rooftops are made from a durable, recycled material imported from Canada. That is because of the growing scarcity of the native palm trees. Palmar\u00ed works together on sustainable agriculture and ecotourism projects as well as environmental education with local communities, and has been instrumental in the construction of schools in several villages. _Mucho_ credit is due to gregarious Axel, the German-Colombian owner of the reserve, for being a forward-thinking eco-example.\n\nPalmar\u00ed has a range of accommodations options, and if you are traveling in a group, this is an excellent choice. Palmar\u00ed is a favorite for Colombian school field trips, but you'll be warned about that possibility when you make a reservation. You can sleep in a hammock, in a communal lodge, or in private rooms. Food is delicious and varied, and, yes, there is cold beer. You can also get online as well, although this might be the perfect time for an Internet diet. There are two great places at Palmar\u00ed to spend the late afternoon hours as you watch the sun go down: the lookout tower and the swing set. Although it is possible (and adventurous) to get to Palmar\u00ed on your own, they will arrange your transportation directly from Leticia.\n\nTo get to Palmari from Leticia, you can take public transportation along the river, which will require multiple transfers, or you can let Palmar\u00ed take care of everything and go direct (COP$150,000 pp one way).\n\nview from the lookout tower at the Reserva Natural Palmar\u00ed\n\n##### **Reserva Natural Heliconia**\n\nIn Brazil, 109 kilometers (68 miles) southeast of Leticia, the **Reserva Natural Heliconia** (office Cl. 13 No. 11-74, Leticia, tel. 8\/592-5773, cell tel. 311\/508-5666, www.amazonheliconia.com, COP$1,600,000 pp d for 4 days\/3 nights incl. transportation) is a fantastic lodge and nature reserve hidden in the dense Amazonian jungle on a tributary of the R\u00edo Yavar\u00ed, which flows into the R\u00edo Amazonas. And that makes it all the more exotic, like a bird of paradise flower (for which it is named) growing in the middle of a sea of green in the jungle. Ideally, plan on spending at least three nights here, as anything less than that will feel rushed.\n\nActivities here (included in the price except for canopying) include nature walks, bird-watching, pink dolphin-watching, canoeing above the inundated forest, fishing, and canopying. You'll usually have two outings each day, and sometimes in the evenings you can explore the jungle at night, panning the darkness with your flashlight as you look for red eyes of jungle beasts looking back at you. It's an unsettling feeling to be immersed in the pitch black jungle against a backdrop of chirping insects, frogs, and birds. Another nocturnal activity is to take a canoe ride in search of alligators (or, if there are no gators, simply enjoying the sounds of the jungle and the millions of stars above).\n\nThe comfortable cabins at this reserve are made of all natural materials. Cabins come in different sizes, such as for two guests or families, and they have an area for larger groups. At night you don't even realize there are others around, such is the privacy.\n\nA hospitable local Brazilian family takes care of all the day-to-day details. There's no Internet access or cell phone coverage, and electricity comes on only for a few hours in the evening. Meals are taken in an open-air thatched roof _maloca_ in the center. Food is basic and does not vary much, usually fried fish, rice, and plantains. The friendly kitchen staff can, however, accommodate vegetarians if provided with some advance notice.\n\n### **Los Llanos**\n\nThe vast plains of eastern Colombia known as Los Llanos or Llanos Orientales (Eastern Plains) are lush tropical grasslands teeming with wildlife. They comprise the lands west of the Andes and north of the Amazon rainforest, and extend well into Venezuela. While this part of Colombia comprises around 25 percent of the total area in the country, it is home to only about 3 percent of the population. The border between the Amazon and the Llanos is roughly at the R\u00edo Guaviare, a tributary of the R\u00edo Orinoco, but the transition is gradual.\n\nElevations in the Llanos rarely exceed 300 meters (1,000 feet), and the land gradually descends from the Andes piedmont in the west towards the R\u00edo Orinoco to the east. The plains are drained by a multitude of large rivers, such as the Guaviare, Vichada, and Meta Rivers, that flow down from the Andes. The savannahs are covered with long-stemmed and carpet grasses in the drier areas and swamp grasses in low-lying humid areas. There are also thick patches of forest throughout the plains and along the rivers (known as gallery forests).\n\nThe climate is marked by two clear seasons: the _invierno,_ or rainy season, which lasts from April to November, and the _verano,_ or dry season, from December to March. During the rainy season, the rivers overflow and large parts of the Llanos are flooded. During the dry season, the land becomes parched.\n\nThe Llanos are literally full of wildlife, with more than 100 species of mammals and 1,300 species of birds, including many migratory birds. Mammals include several species of deer and rabbit, anteaters, armadillos, tapirs, otters, jaguars, pumas, and _chig\u00fciros_ (capybaras), the world's largest rodent. The plains are also home to the giant anaconda and to one of the most endangered species on earth, the Orinoco crocodile, which reaches up to seven meters (23 feet) long. Unfortunately, human settlement and hunting have vastly decreased the wildlife.\n\n##### **History**\n\nAt the time of the Spanish conquest, the Llanos were inhabited by several indigenous people, including the Guahibos, Achaguas, and Jiraras. The first European to explore the region was German conquistador Nikolaus Federmann, who set off in 1538 from Venezuela and crossed these plains on his way to the Muisca highlands of El Dorado in the Cordillera Oriental (Eastern Range of the Andes). Gonzalo Jim\u00e9nez de Quesada, the founder of Bogot\u00e1, was granted dominion over a large part of the Llanos but took little interest due to the apparent lack of treasures and sparse population. Starting in the second half of the 16th century, Jesuits set up missions to convert the indigenous people and established large cattle ranches. During the wars of independence, Venezuelan _llaneros_ (plainsmen) were an important element in Bol\u00edvar's army and played a key role in the expedition that crossed the plains, climbed the Andes, and finally defeated the Spanish army at the Batalla de Puente de Boyac\u00e1 on August 7, 1819.\n\nThanks to its lush grasslands, historically the Llanos have been an important cattle-ranching region in Colombia. To this day, the Llanos are synonymous with cowboy culture. Most of human habitation takes place along a narrow fringe of land bordering the Andes, in cities such as Villavicencio and Yopal.\n\nIn the 1980s oil was discovered, first in the far eastern department of Arauca and then in Casanare, and today, the area is Colombia's most important oil-producing region, with several pipelines linking the oil fields to the Caribbean port of Cove\u00f1as, from where it is exported. The lucrative oil industry became a tempting target for guerrilla insurgencies operating in the area. In the 1980s, the guerrilla group ELN (the National Liberation Army) extorted large amounts of money from the oil companies and their contractors and launched a bombing campaign of the pipelines. The 780-kilometer (485-mile) Ca\u00f1o Lim\u00f3n-Cove\u00f1as pipeline, which transports oil from oil fields in Arauca owned by California-based Occidental Oil, was bombed an incredible 170 times in 2001 alone. With Plan Colombia came money to defend this infrastructure, and bombings had dropped to only 17 by 2004.\n\n_chig\u00fciros_ (capybaras)\n\nThe FARC (Fuerzas Armadas Revolucionarias de Colombia; Revolutionary Armed Forces of Colombia) has been present in the region since the 1960s. From 1998 to 2002 their position was strengthened when President Pastrana granted them as part of a peace process a large demilitarized zone the size of Switzerland in the Meta and Caquet\u00e1 departments. The FARC managed this zone like a mini-state, using it to grow and process coca, smuggle arms, and hold kidnap victims. They even built roads and a recreation center for FARC commanders.\n\nAs a result of the guerrilla presence, agricultural output from the Llanos was depressed for many years. With increased security in the mid-2000s, the Llanos emerged as an important region for big agriculture, with large multinationals such as Cargill setting up massive plantations of African palm, soy, maize, and rubber, especially in the northeast Vichada region. A source of concern to environmentalists is the fact that the Llanos, unlike the Amazon, does not have many protected areas and could succumb to unchecked development.\n\nToday the region is still suffering sporadic violence caused by illegal groups, especially in Arauca and parts of Caquet\u00e1. The current governor of Meta, elected in 2011, is Alan Jara, who was kidnapped by the FARC in 2001 and was held for nearly eight years.\n\n#### **VILLAVICENCIO**\n\nThe capital of the department of Meta is only 80 kilometers (50 miles) southeast of Bogot\u00e1 and is a gateway to the vast Llanos. Villavicencio does not have much in the way of tourist attractions; it does, however, attract Bogotanos en masse in search of sun and a swimming pool. It's also a base from which to visit Ca\u00f1o Cristales.\n\nCenters of activity (aside from shopping malls) in Villavicencio are parks like **Parque los Fundadores** (Av. 40 and V\u00eda a Bogot\u00e1), which has an enormous sculpture by Rodrigo Arenas Betancourt, and the **Plaza Los Libertadores** (Cl. 39 between Cras. 32-33), over which the cathedral, the late-19th-century **Catedral Metropolitana Nuestra Se\u00f1ora del Carmen,** stands. Near the Plaza Los Libertadores are some pleasant pedestrian side streets, the result of an urban renewal project.\n\n##### **Festivals and Events**\n\n_Joropo_ is the music and dance of the Llanos. With European origins (it is said that it is related to Spanish flamenco), it's a dance of fast and fancy footwork, accompanied by folk music that utilizes the harp and guitar and other instruments. The **Torneo Internacional del Joropo** is a showcase of this particularly Llanos art form, and the festival has taken place in Villavicencio since 1960. Over a thousand dance pairs bedecked in traditional costume, including many children, participate. In addition to the dancing, there are rodeo events and, of course, a beauty pageant. It's held over four days in late June.\n\n**Los Llanos for Beginners**\n\nFor a taste of the Llanos, head eastward from Villavicencio to **Lagos de Menegua** (17 km east of Puerto L\u00f3pez, cell tel. 315\/326-6068, tel. 1\/616-0439, www.lagosdemenegua.com, COP$154,000 pp d). This family-friendly resort set in the grassy hills of the Llanos has 24 rooms and a large swimming pool. Here you can take a horseback ride through the ranch, take a jungle walk, and see some of the wildlife that is abundant here: alligators, cute _chig\u00fciros_ (capybaras), monkeys, and many varieties of birds. While there are some walks you can make on your own, the resort offers guided walks on horseback (COP$35,000 pp), Jeep tours of the savannah (COP$25,000), and nighttime excursions to see alligators (COP$50,000 pp). _Colectivos_ bound for Puerto Gait\u00e1n leave from the bus stations in both Bogot\u00e1 (COP$37,000) and Villavicencio (COP$15,000). Tell the driver you'd like to be dropped off at kilometer 17 between Puerto L\u00f3pez and Puerto Gait\u00e1n. It's just past Puerto L\u00f3pez, where an obelisk marks the geographic center of Colombia, and the R\u00edo Meta.\n\nCowboy culture in the Llanos is not limited to men. Proof of that is the **Concurso Mundial de la Mujer Vaquera** (www.mundialmujervaquera.com), a competition in which cowgirls show their skills in a series of events held at the Parque Las Malocas in Villavicencio in late March every year. _Llaneras_ are well represented and get the support of the hometown crowd, but it is an international event: The 2013 winner hailed from Mexico.\n\nThe **Encuentro Mundial del Coleo** (www.mundialcoleo.com.co) is for men, and they demonstrate their _coleo_ skills. _Coleo_ is a _llanero_ cowboy technique to recapture stray bulls so that they can be branded. The cowboys do this by galloping on horseback and thrusting the bulls to the ground by pulling their tails. Animal rights sympathizers should skip this event! The competition takes place at the Parque Las Malocas in October of every year.\n\n##### **Accommodations and Food**\n\nThose en route to Ca\u00f1o Cristales in La Macarena may need to spend a night in Villavo, as locals call it. One of the best hotels in town is the M **GHL Hotel Villavicencio** (Cra. 39C No. 19C-15, www.ghlhoteles.com, COP$240,000 d). It is on the backside of a small shopping mall (Villa Centro) that has a Yumbo supermarket. Rooms are spacious, the restaurant is pretty good, and on the top floor is a pool, with a fantastic view of the Llanos.\n\nA budget option is **Mochilero's Hostel** (www.mochileroshostel.com, COP$24,000 dorm, COP$50,000 d), which opened in 2012 and has six rooms (with fan). It's close to shopping malls and restaurants.\n\nWhen it comes to local cuisine, _llaneros_ like their beef, and open-air steakhouses are, have always been, and always will be the rage in Villavo. The specialty here is _mamona,_ which is grilled veal. Ingredients of this dish aren't complex: a one-year-old calf, salt, and beer.\n\n**El Amarradero del Mico** (V\u00eda Vanguardia Restrepo, cell tel. 313\/829-6228, noon-midnight daily, COP$22,000) is a popular open-air restaurant on the outskirts of town featuring something for all manner of carnivores. For a departure from the local specialties, try **Pizza Nostra** (Av. 40 No. 25A-47, tel. 8\/668-4000, noon-11pm daily, COP$18,000) for pizzas, pastas, and hamburgers.\n\nAt the Unicentro mall there is a food court with a decent Italian restaurant and a Juan Valdez Caf\u00e9. **Oliva Mediterranea** (Unicentro shopping mall, Av. 40 No. 26C-10, 3rd floor, tel. 8\/668-2020, noon-9pm daily, COP$18,000) makes a noble effort to make you think you're not in a food court. On the menu are Italian dishes, pizzas, and salads.\n\n**La Estaca Club** (Cra. 40 No. 57, no phone, 6pm-2am Mon.-Wed., 6pm-3am Thurs.-Sat., 6pm-10pm Sun., COP$15,000) is a popular pub-like gathering place with a pleasant terrace where you can eat soul food such as buffalo wings and have a couple of beers.\n\n##### **Information and Services**\n\nThere are **tourist information stands** at the airport (Vereda Vanguardia v\u00eda Restrepo) and at the bus terminal (Cra. 1 No. 15-05) and one at Llanocentro shopping mall. They generally keep the hours of 8am-5pm daily.\n\n##### **Getting There and Around**\n\nThere's really no need to fly to Villavicencio from Bogot\u00e1, unless you seek comfort, are averse to the assertive motoring style of Colombian bus drivers, or are in a hurry, as there are **buses** (Flota la Macarena, tel. 1\/421-5556, www.flotalamacarena.com, 2 hrs., COP$20,000) between the two cities practically 24 hours a day. From Villavicencio's **Terminal de Transportes de Villavicencio** (Cra. 1 No. 15-05, Anillo Vial, no phone) there's no shortage of _colectivo_ (smaller bus) service to Bogot\u00e1, but expect at least another frustrating hour in the Bogot\u00e1 traffic during much of the day. _Colectivos_ going to Puerto L\u00f3pez cost about COP$14,000. It's a good road.\n\nAirlines **LAN** (Colombian toll-free tel. 01\/800-094-9490, www.lan.com) and **Satena** (Cra. 31 No. 39-37, 2nd floor, tel. 8\/662-1260, www.satena.com) operate flights to the **Aeropuerto Vanguardia** (Vereda Vanguardia V\u00eda Restrepo, tel. 8\/670-9610) in Villavicencio from Bogot\u00e1. It's about a 45-minute long flight. There are connection flights to La Macarena, usually via small charter planes.\n\n#### M **CA\u00d1O CRISTALES**\n\nTaking a flight (or two) to a remote corner of Colombia with a troubled past just to see some river algae may not, on the surface, sound like a wise investment of precious vacation time. But, here in the remote Llanos, you'll be rewarded as you trek through the stark lowland hills of the Serran\u00eda de la Macarena, with its unusual dry tropical vegetation, and behold the vibrant purple, fuchsia, goldenrod, and green _Macarenia clavigera_ plants swaying in the gushing streams of **Ca\u00f1o Cristales.**\n\nthe colorful pools of Ca\u00f1o Cristales\n\nThe Serran\u00eda de la Macarena (Macarena Range) is a 120-kilometer-long, 30-kilometer-wide (75-mile-long, 19-mile wide) mountain range 70 kilometers (43 miles) south of Villavicencio and 45 kilometers (28 miles) east of the Andes. The range, which is entirely contained within the 629,280-hectare (1.6 million-acre) **Parque Nacional Natural Sierra de la Macarena,** is the highly eroded remnant of mountains that once towered on the supercontinent of Panagea, before the South American plate separated from the African plate and drifted westward to its present location, crashing into the Nazca plate and creating the Andes. These ancient outcrops, which form the Guyana Shield, dot the northwest Amazon basin in Colombia, Venezuela, and Guyana.\n\nThe Macarena is also unique in that it is at the confluence of three highly distinct ecosystems: the Amazon to the south, the Llanos to the north, and the Andes mountain rainforests to the west. A large number of endemic plants evolved in this isolated mountain range, including the striking _Macarenia clavigera,_ which draws tourists from all over Colombia and from abroad.\n\nCa\u00f1o Cristales is a stream that flows from west to east in the very southern part of the sierra and flows into the R\u00edo Guayabero, a tributary of the R\u00edo Guaviare. In its upper reaches, it has three branches that join to form the Ca\u00f1o Cristales stream proper. The surrounding landscape is quite dry and rocky, covered with unusual _Vellozia macarenensis_ plants, which have evolved to survive the dry climate and bush fires. These plants produce beautiful white flowers that add to the beauty of the stark environs.\n\nThe small town of **La Macarena** on the R\u00edo Guayabero is the gateway to Ca\u00f1o Cristales. It was part of the demilitarized zone granted to the FARC as part of the 1998-2002 peace process. Evincing little respect for nature, the FARC destroyed the park facilities, chased away the staff, and built a road right through the park, sparking squatters to take illegal possession of lands within the park. The town of La Macarena is now home to a 4,000-strong army base, one of the largest in Colombia. The areas in the vicinity of the town, including all the sights described in this section, are safe to visit. Tourism has given this isolated community new life. A highly successful nature guide training program for high school students has given youths in the town the opportunity to gain a living through ecotourism. Six thousand visitors came to Ca\u00f1o Cristales in 2012, and each year more and more are coming.\n\nTo visit Ca\u00f1o Cristales, you need a minimum of one full day and two nights, but staying a day or two longer is definitely worthwhile. The best time of the year to visit is during the rainy season, from May until November, when you can marvel at the _Macarenia clavigera_ in bloom.\n\n##### **Recreation**\n\n###### **HIKING**\n\nAll excursions require a guide. Visitors are asked to avoid using sunscreen or mosquito repellent, as it can damage the _Macarenia clavigera._ Therefore, make sure to bring a wide-brimmed sun hat and wear lightweight, long-sleeve shirts and pants to protect your skin. Wear shoes that have good traction on slippery rocks and that you don't mind getting wet.\n\nThe most popular trek is a half-day excursion that involves a pleasant 20-minute boat ride up the R\u00edo Guayabero from the town of La Macarena, a six-kilometer (3.7-mile) truck ride, and a two-kilometer (1.2 mile) hike to Ca\u00f1o Cristales. There, you'll admire the multicolored stream as it gushes through pools and waterfalls. At the end you get to cool off in the Piscina del Turista, a natural pool, and have a waterside lunch.\n\nAnother longer, all-day excursion takes you to the upper part of Ca\u00f1o Cristales, where you'll visit the three different branches of the river. The vegetation here is much denser, and the _Macarenia clavigera_ also take on yellow and green hues. This excursion involves wading across the stream numerous times. If it has been raining, the level and force of the water increases considerably, making for an exhilarating experience. Depending on the level of the water and physical conditions of the trip, you may visit one, two, or all three of the branches.\n\nTwo other half-day excursions also involve a ride up the R\u00edo Guayabero to visit **Cristalitos,** a smaller stream that is Ca\u00f1o Cristales in miniature, or to **El Mirador,** a hike to the top of a hill that offers sweeping views of the R\u00edo Guayabero and surrounding countryside.\n\nAnother excursion, which can only be done in the dry months of December through April, is 20 kilometers (12 miles) up to the **Raudal del Guayabero** to admire the white-water rapids, ancient indigenous petroglyphs, and interesting rock formations known as **Ciudad de Piedra.**\n\n###### **TOURS**\n\nThe easiest way to visit Ca\u00f1o Cristales is to buy a packaged tour. Two well-reputed local operators, **Ecotur\u00edsmo Sierra de la Macarena** (Av. Alfonso L\u00f3pez No. 40-28, Villavicencio, tel. 8\/664-3364, cell tel. 314\/325-3522, www.ecoturismomacarena.com) and **Cristales Aventura Tours** (La Macarena, cell tel. 313\/294-9452, cristalesaventuratours@hotmail.com) offer two- to five-night packages from Villavicencio. Expect to pay around COP$1,300,000 per person for a four-day trip. This includes air transportation out of Villavicencio, accomodations in La Macarena, and tours to Ca\u00f1o Cristales. The tours include transportation from Villavicencio in small charter planes, local transportation, food, and accommodations. These operators hire guides from UNIGMA, an association of young, local guides that have received specialized training since high school. You can also buy these tours from tourist operators elsewhere in the country, but these will simply take a margin and send you with the two local operators.\n\nIt is quite feasible, and cheaper, to organize your Ca\u00f1o Cristales trip on your own by arranging air transportation, local transportation, accommodations, and food. For all hikes in and around Ca\u00f1o Cristales you are required to hire a guide from **UNIGMA** (cell tel. 320\/856-7571, guiasunigma@hotmail.com, <http:\/\/canocristalesguiasdeturismo.wordpress.com>, COP$100,000 per day for a group of up to 7 people).\n\n##### **Accommodations and Food**\n\nHotels and restaurants in La Macarena, all economically priced, are nothing special. M **Centro Vacacional Punto Verde** (Parque Principal, cell tel. 310\/341-8899, cvpuntoverde@hotmail.com, COP$40,000 pp incl. all meals) is the best option by far, with nine rooms spread out behind an ample and leafy common area, including a small pool. Breakfast is included, and the restaurant, the Punta Verde, is the best in town. It's open to hotel guests and non-guests. Let them know ahead of time if you require vegetarian meals.\n\n**Casa Hotel** (Parque Principal, cell tel. 313\/292-9925 or 314\/279-2764, COP$35,000 pp) has 21 rooms, some with air conditioning. **La Cascada** (Cl. 5 No. 7-35, tel. 8\/560-3132, cell tel. 313\/294-9452, cristalesaventuratours@hotmail.com, COP$80,000 d) has 32 smallish rooms over two floors.\n\n##### **Information**\n\nThe group of local guides, UNIGMA, operates a **tourist information office** at the tiny Aeropuerto Javier Nore\u00f1a Valencia (5-min. walk from Parque Principal). It's the most buzzing place in La Macarena and is open daily 8am-5pm.\n\n##### **Getting There and Around**\n\nTo **Aeropuerto Javier Nore\u00f1a Valencia** in La Macarena, there are flights from Bogot\u00e1 via **Satena** (tel. 1\/423-8530, Colombian toll-free tel. 01\/800-091-2034, www.satena.com) Sundays at 2:11pm and on Fridays at 10:54am. There are charter flights from Villavicencio on small charter planes (and, on occasion, old DC-3s). Contact **Ecoturismo Sierra de la Macarena** (Av. Alfonso L\u00f3pez No. 40-28, Villavicencio, tel. 8\/664-3364, www.ecoturismomacarena.com) or **Cristales Aventura Tours** to buy a seat on these flights. Travel by land to La Macarena, while theoretically possible, is not advisable due to safety concerns.\n\nLa Macarena is a small town and you can get everywhere on foot. Boats up the R\u00edo Guayabero to Ca\u00f1o Cristales, Cristalitos, and El Mirador will cost COP$60,000-70,000 per group of up to 10 people and COP$250,000 to the Raudal del Guayabero.\n\n#### M **HACIENDA LA AURORA**\n\n**Hacienda La Aurora** (near the town of Paz de Ariporo, Casanare, cell tel. 310\/580-5395, www.juansolito.com), 180 kilometers (112 miles) northwest of Yopal in the department of Casanare and about double that from Villavicencio, is in many ways a typical extensive cattle-ranching farm of the Llanos. It was once part of the immense 430,000-hectare (1.1 million-acre) Jesuit hacienda that was subdivided over time. The hacienda measures 17,000 hectares (42,000 acres) and has 6,000 head of cattle. However, in one aspect it differs radically from other haciendas in the Llanos: Since the Barrag\u00e1n family, the current owners, purchased it in the 1970s, they have not allowed any hunting of animals, even if that means if jaguars and pumas attack and eat their cattle. This prohibition on hunting comes from the family's deep-seated conviction that cattle ranching must be a sustainable activity and that it must coexist peaceful with the Llanos' abundant biodiversity.\n\nAs a result, the hacienda abounds with wildlife that mixes with cattle herds and is not afraid of humans. As you travel through the farm, on the back of a specially outfitted safari truck, on horseback, or on foot, there is wildlife everywhere you look: herds of absurdly cute looking _chig\u00fciros_ (the world's largest rodents, also known as capybaras), deer, foxes, anteaters, armadillos, sloths, monkeys, tortoises, and caimans. Even non-bird-watchers will be amazed by what they see: huge ungainly _garz\u00f3n soldados_ (jabirus), bright red ibises, families of borrowing owls, incredibly exotic plumed hoatzins, parakeets, macaws, and toucans. In the dry season, it is not unusual to see anacondas on the banks of the R\u00edo Ariporo or in watering holes. There are also water buffalo and herds of wild horses roaming about. Finally, pumas and jaguars can also be spotted (albeit through wildlife cameras installed in strategic points on the farm). There is most definitely nowhere in Colombia where you can see this much wildlife in its natural setting.\n\nHacienda La Aurora is also very much a working cattle farm. The herds are spread out throughout the hacienda. There is a main hacienda house, which is the center of operations. When cattle-ranching activities need to be done, such as taming horses, branding calves, or rounding up cattle for sale, _vaqueros_ (cowboys) move to one of several smaller peripheral camps called _fundos_ to do these activities. Viewing the traditional cattle-ranching ways of the Llanos is a fascinating part of a visit to La Aurora.\n\nHaciendas such as this used to be the norm in the Llanos. However, many have been subdivided into smaller, more modern ranches with fenced pastures. Others have been converted into large African palm, corn, soy, or rubber plantations. La Aurora's administrators complain that it is increasingly difficult to find personnel willing to do what they call the \"work of the Llano\"\u2014cattle ranching the traditional way\u2014as wages in the oil industry are much better and the work is less taxing.\n\ndouble-striped thick knee bird\n\nTo fully enjoy a visit to Hacienda La Aurora, stay at least two full days. Accommodations at La Aurora are at the Ecolodge Juan Solito, which borders the ranch. Visits are easier in the dry season (Nov.-Mar.), when getting around the farm is not difficult and wildlife congregates around the water holes. However, a visit in the rainy season is also interesting. The ground often gets drenched, but is not impossible to visit the ranch and spot animals. Also, in the rainy season it is possible to do boat excursions.\n\n##### **Recreation**\n\nAll visitors pay a fixed price of COP$120,000 per person per day for all activities on the ranch. In return, all activities are tailored to your wishes and you are accompanied by a dedicated guide. Easily arranged activities include photo safaris on the back of a truck specially outfitted with benches, visits to view the traditional cattle-ranching activities of the hacienda, excursions on horseback, nature hikes, and canoe trips up the R\u00edo Ariporo (only in the rainy season). You can also swim in the Ariparo, though you must be careful of stingrays, electric eels, piranhas (a minor threat), and anacondas, but they are on the riverbank. Maybe just skip the swimming!\n\nIf you have a particular interest, such as specialized bird-watching or photographing specific wildlife, these activities can be arranged at no additional cost.\n\n##### **Accommodations and Food**\n\nLodging and food is provided at the **Ecolodge Juan Solito** (near Paz de Ariporo, Casanare, cell tel. 320\/342-6409, www.juansolito.com, COP$180,000 pp including all meals) just across the R\u00edo Ariporo on the south side of the Hacienda La Aurora. Juan Solito is owned by incredibly friendly and attentive Nelson Barrag\u00e1n, of the family that owns the hacienda. There are seven simple and comfortable rooms at the lodge. Meals are taken with the lodge staff on long benches in a thatched-roof dining room overlooking the river. Vegetarian fare can be arranged with advance notice. Sometimes, when there is a large group of guests, local musicians and dancers perform traditional _joropo_ music and dance, with Nelson playing the harp.\n\n##### **Getting There**\n\nThis Llanos adventure begins in **Yopal** , the orderly and pleasant capital city of Casanare. Due to the presence of large multinational oil companies in the area, there are several flights every day between Bogot\u00e1 and the Yopal airport, **Aeropuerto Alcaravan-EYP** (Cl. 40 No. 19-20, tel. 8\/635-8352). **LAN** (tel. 01\/800-094-9490, www.lan.com) and **Avianca** (Cl. 10 No. 22-22, tel. 8\/634-8406, www.avianca.com, 8am-noon and 2pm-6pm Mon.-Fri., 8:30am-12:30pm Sat.) offer various nonstops between the two cities. **EasyFly** (Col. toll-free tel. 01\/800-012-3279, www.easyfly.com.co) serves both Bogot\u00e1 and Bucaramanga from Yopal.\n\nFrom Yopal, the easiest way to get to La Aurora is by contracting transportation directly with them. For COP$380,000 (each way) they will pick you up (maximum four passengers) at the Yopal airport or bus station and take you directly to the farm.\n\nIf you'd like to save some money and don't mind a little adventure, you can take public transportation to the farm entrance. From the **Terminal de Transportes de Yopal** (Cra. 23 between Clls. 25-26) take a bus to Paz de Ariparo (COP$17,000, 1.25 hours; frequent departures). From there, take a slow bus to Monta\u00f1as de Totumo (COP$22,000; 6am, 12:30pm, and 3pm). The 12:30 bus continues on along a road that passes in front of the entrance of Ecolodge Juan Solito. Tell the bus driver to leave you at \"Finca La Vigia,\" where there is a sign that says \"Reserva Casanare.\" Confirm with the ecolodge for a pickup at Monta\u00f1as de Totumo (the 6am and 3pm buses) or at the entrance to the farm (12:30pm bus).\n\n## **BACKGROUND**\n\nThe Land\n\nGEOGRAPHY\n\nCLIMATE\n\nFlora and Fauna\n\nRAINFOREST\n\nCLOUD FOREST\n\n_P\u00c1RAMOS_\n\nTROPICAL DRY FORESTS\n\nTROPICAL GRASSLANDS\n\nHistory\n\nBEFORE COLUMBUS\n\nTHE SPANISH CONQUEST (1499-1550)\n\nCOLONIAL NUEVA GRANADA (1550-1810)\n\nTHE STRUGGLES FOR INDEPENDENCE (1810-1821)\n\nGRAN COLOMBIA: A FLAWED UNION (1821-1830)\n\nCIVIL WARS AND CONSTITUTIONS (1830-1902)\n\nPEACE AND REFORM (1902-1946)\n\nLA VIOLENCIA (1946-1953)\n\nDICTATORSHIP (1953-1957)\n\nTHE NATIONAL FRONT (1957-1974)\n\nUNDER SIEGE (1974-1991)\n\nCOLOMBIA ON THE BRINK (1992-2002)\n\nREGAINING ITS FOOTING (2002-PRESENT)\n\nGovernment and Economy\n\nGOVERNMENT\n\nECONOMY\n\nPeople and Culture\n\nDEMOGRAPHY\n\nRELIGION\n\nLANGUAGE\n\n### **The Land**\n\nColombia covers a land area of 1.14 million square kilometers (440,000 square miles), roughly the size of Texas and California combined, making it the fourth largest South American country in area after Brazil, Argentina, and Peru. It is located in the northwest corner of South America, with seacoast on both the Pacific and the Atlantic, and bordering Venezuela, Brazil, Peru, Ecuador, and Panama. The Amazonian departments of Putumayo, Caquet\u00e1, Amazonas, and Vaup\u00e1s in the south of the country straddle the Equator.\n\nFor a country of its size, it has an astonishing variety of landscapes, including the dense rainforests of the Amazon and the Pacific Coast, the vast grassland plains of the Llanos, the lofty Andes Mountains, and the Caribbean islands of San Andr\u00e9s and Providencia. Colombia's mountainous regions themselves hold a succession of vertically layered landscapes: tropical rainforests at their base, followed by cloud forests at higher elevations, topped by the unique tropical high mountain _p\u00e1ramo_ (highland moor) above 3,500 meters. The country boasts several peaks higher than 5,000 meters, including Nevado del Ruiz (5,325 meters\/17,470 feet) and Pico Crist\u00f3bal Col\u00f3n (5,776 meters\/18,950 feet).\n\n#### **GEOGRAPHY**\n\n##### **Regi\u00f3n Andina**\n\nThis central part of Colombia is dominated by the Andes mountain range. This region, which is referred to as the Regi\u00f3n Andina or simply _el interior_ (the interior) is the heartland of the country. It covers roughly 25 percent of the surface of the country and is home to 60 percent of the Colombian population.\n\nThe Andes mountain range, 8,000 kilometers (5,000 miles) long, runs the entire length of South America. The Andes are relatively young mountains, and some of the loftiest in the world after the Himalayas, resulting from the collision of the westward-moving South American plate with the Nazca and Antarctic plates starting 145 million years ago. The heavier Nazca and Antarctic plates to the west subducted under the lighter and more rigid South American plate, propelling it upwards and forming the Andes. In Colombia, as a result of a complex pattern of tectonic collisions, three parallel ranges were formed. At the Masizo Colombiano (Colombian Massif), a mountain range 175 kilometers north of the border with Ecuador, the Andes split into the Cordillera Occidental (Western Range), Cordillera Central (Central Range), and Cordillera Oriental (Eastern Range).\n\nThe Cordillera Occidental is the lowest and least populated of the three ranges. It runs roughly 750 kilometers parallel to the Pacific Coast and ends 150 kilometers from the Caribbean Sea. Its highest point is the Cerro de Tatam\u00e1 (4,250 meters\/13,945 feet). Of Colombia's three ranges, it has the least human intervention and is home to some of the world's only pristine high mountain _p\u00e1ramos_ ecosystems, notably that covering the Cerro de Tatam\u00e1.\n\nThe Central Cordillera is the highest of the three ranges and is the continuation in Colombia of the main Andes range. It runs roughly 800 kilometers and tapers off in the northern Caribbean plains, 200 kilometers form the Caribbean coast. Like the Andes in Ecuador, it is dotted with volcanoes. North of the Masizo Colombiano is the Serran\u00eda de los Coconucos (Coconucos Range), a range of 15 volcanoes including the Volc\u00e1n del Purac\u00e9 (Purac\u00e9 Volcano, 4,580 meters\/15,025 feet). Farther north, the Cordillera Central reaches its maximum elevation at the massive Nevado del Huila (5,750 meters\/15,585 feet). Farther north is a large complex formed by the Nevado del Tolima (5,215 meters\/17,110 feet), Nevado Santa Isabel (4,950 meters\/16,240 feet), and Nevado del Ruiz (5,325 meters\/17,470 feet). In its northern part, the Cordillera Central broadens to form the uneven highland that comprises the mountainous heartland of Antioquia with Medell\u00edn as its capital.\n\nSeveral of the volcanoes of the Cordillera Central have seen recent activity, notably Volc\u00e1n Galeras (4,276 meters\/14,029 feet), near the southern city of Pasto, which last erupted in 2005, forcing evacuation of nearby settlements. Volc\u00e1n Galeras is currently closed to visitors because of the threat of volcanic activity. In 1985, the Nevado del Ruiz erupted unexpectedly, creating a landslide that engulfed the town of Armero, killing more than 20,000 people. Since 2012, the Nevado del Ruiz has seen some activity, which has restricted access to the northern part of the Parque Nacional Natural Los Nevados.\n\nThe Cordillera Oriental, which like the Cordillera Occidental is non-volcanic, extends more than 1,100 kilometers to the border with Venezuela. The range broadens to form a broad high plateau called the Altiplano Cundiboyacense, which extends 200 kilometers north of Bogot\u00e1. This is an area of broad valleys with the extremely rich soil of sedimentary deposits. It is not one continuous plane, but rather a series of valleys. The Sabana de Bogot\u00e1, or Bogot\u00e1 High Plateau, where Bogot\u00e1 is located, is one particularly broad valley. North of the altiplano is the soaring Sierra Nevada del Cocuy, a mountain range with 11 glacier-covered peaks, including Ritacuba Blanco (5,380 meters\/17,650 feet). North of El Cocuy, the Cordillera Oriental loses altitude and splits in two: A smaller western segment forms the Serran\u00eda de Perij\u00e1 on the border between Colombia and Venezuela, and a larger branch continues into Venezuela to form the Venezuelan Andes.\n\nColombia is a mountainous country.\n\nThe 1,500-kilometer-long R\u00edo Magdalena flows along a broad valley that separates the Cordillera Central and the Cordillera Oriental, making it the main commercial waterway of Colombia. Due to heavy sedimentation, it is now only navigable when waters rise during the rainy seasons in the central part of the country (Apr.-May and Oct.-Nov.). The R\u00edo Cauca, which flows parallel to the Magdalena along the much narrower valley between the Cordillera Central and the Cordillera Occidental, is the main tributary of the Magdalena. They join in northern Colombia and flow into the Caribbean.\n\nAndean Colombia is a seismically volatile area, and the country has suffered some major earthquakes in the past. The most deadly measured 7.5 on the Richter Scale and occurred in C\u00facuta in 1875. It killed 10,000 and completely destroyed the city. In recent years, around 600 were killed in the Pacific port city of Tumaco during a quake and tsunami in 1979; 300 perished in the 1983 Holy Week earthquake in Pop\u00e1yan; and over 1,100 died in the Armenia quake of 1999.\n\n##### **Caribe**\n\nColombia's Caribbean Coast runs 1,760 kilometers from the border of Panama to Venezuela, just longer than the California coast. However, the term Caribe or Regi\u00f3n Caribe refers to much more than the narrow strip of coast; it encompasses basically all of Colombia north of the Andes, including a vast area of plains. This region covers 15 percent of the surface of Colombia and is home to 20 percent of the population.\n\nThe terrain is mostly low-lying and undulating. Near the border with Panama, the land is covered by dense tropical forests, similar to those of the Pacific Coast. Farther east is the Golfo de Urab\u00e1 (Gulf of Urab\u00e1), a large, shallow bay. Between the Golfo de Urab\u00e1 and Cartagena is the Golfo de Morrosquillo, a broad inlet that is 50 kilometers wide. Off the shore of the Golfo de Morrosquillo are two small archipelagos, the Islas de San Bernardo (San Bernardo Islands) and the Islas del Rosario (Rosario Islands), with beautiful coral reefs. Inland to the south is a large area of savannahs in the departments of C\u00f3rdoba and Sucre largely devoted to cattle ranching. This area was once covered by dry tropical forests, which have been largely felled.\n\nThe bay of Cartagena, farther east, is a magnificent deep bay that caught the attention of the Spaniards early on. To the southeast of Cartagena is the lower valley of the Magdalena and Cauca rivers, a vast expanse of low-lying lagoons and lands prone to seasonal flooding. The R\u00edo Magdalena flows into the Caribbean east of Cartagena at the port city of Barranquilla. Farther to the east along the coast is a major mountain range, the Sierra Nevada de Santa Marta. It was formed by the collision of the South American plate and the Caribbean plate to the north and is entirely independent of the Andes. This mountain is home to Colombia's two highest peaks, the twin Pico Crist\u00f3bal Col\u00f3n and Pico Bol\u00edvar (5,776 meters\/18,950 feet), and is considered the highest coastal mountain range in the world. The Sierra Nevada de Santa Marta contains the same range of vertically layered landscapes as the Andes, from low-lying tropical forest through cloud forest, Andean forests, _p\u00e1ramo,_ and glaciers. There are eight peaks with elevations greater than 5,000 meters.\n\nNortheast of the Sierra Nevada de Santa Marta is La Guajira peninsula, an arid peninsula jutting into the Caribbean. Punta Gallinas, at the tip of La Guajira, is the northernmost point in South America. There are a few low-lying mountain ranges in La Guajira, such as the Serran\u00eda de la Macuira (864 meters\/2,835 feet), which is covered with rainforest. The Sierra Nevada de Santa Marta and Serran\u00eda de la Macuira are biological islands, and their upper reaches are home to numerous endemic species that evolved in isolation.\n\n##### **Pac\u00edfico**\n\nThe Pacific Coast of Colombia extends 1,329 kilometers from Ecuador to Panama, about the same length as the coast of California. The term Pac\u00edfico, as it relates to Colombia, designates all the land\u2014jungle to be more accurate\u2014that lies between the Pacific Ocean and the Cordillera Occidental. This region covers 6 percent of Colombia and is home to about 2 percent of the population.\n\nThe topography of this region is mostly flat, with the low-lying coastal Serran\u00eda del Baud\u00f3 (1,810 meters\/5,940 feet) providing a mountainous backdrop to the coastal plain and forming an inland basin that is drained by the mighty R\u00edo Atrato, which flows northwards into the Caribbean Sea. The coast has a number of bays and inlets, notably the Ensenada de Utr\u00eda (Utr\u00eda Inlet), visited by humpback whales traveling every winter from the Antarctic Sea to give birth in the warm waters of the Colombian Pacific. South of Buenaventura, the coast has extensive mangroves, much of which are well-preserved. Offshore are two islands: Isla Gorgona is 35 kilometers off the coast on the continental shelf, and tiny Malpelo is 490 kilometers off the coast. Both of these islands are likely of volcanic origin.\n\nThe Colombian Pacific region is one of the wettest places on Earth, with average annual rainfall of 10,000 millimeters (33 feet). Due to the enormous amount of precipitation, the region has a dense river network with dozens of major arteries, such as the R\u00edo Baud\u00f3, R\u00edo San Juan, and R\u00edo Pat\u00eda.\n\n##### **Los Llanos**\n\nThe Llanos, Colombia's vast eastern plains, cover an area of 250,000 square kilometers, roughly 25 percent of Colombia's territory. The plains are hemmed in to the west by Cordillera Oriental and to the south by the Amazon rainforest, and extend far into Venezuela. Though the transition between the Amazon and the Llanos is gradual, the R\u00edo Guaviare, which flows from west to east at a longitude that is roughly midway between the northern and southern tips of Colombia, is considered the demarcation line between these two areas. The Llanos are home to about 1.5 million inhabitants, or about 3 percent of the population, making it the region with the second lowest population density\u2014after the Amazon region\u2014in the country.\n\nAfter the genesis of the Andes, water flowing eastward down the mountains accumulated in a vast freshwater lake that was confined on the east by old mountainous formations (now the Guyana and Brazilian highlands). Large amounts of sediments were deposited, forming the basis for the Llanos' undulating topography. Near the Andes, elevations can reach 300 meters and, moving east, slowly decrease in altitude until they reach the north-flowing R\u00edo Orinoco, which forms the border between Colombia and Venezuela.\n\nThe only significant mountain range in Los Llanos is the Serran\u00eda de la Macarena (Macarena Range), a 120-kilometer-long, 30-kilometer-wide range that is 45 kilometers east of the Andes just of the R\u00edo Guayabero, a tributary of the R\u00edo Guaviare. This range is part of the Guyana Shield complex of ancient, highly eroded remnants of mountains that existed long before the formation of the Andes.\n\nThe Llanos are drained by a multitude of large rivers, such as the R\u00edo Guaviare, R\u00edo Vichada, and R\u00edo Meta, which flow down from the Andes and meander towards the east. All the rivers of the Llanos are tributaries of the Orinoco\u2014for this reason, this region is also often called La Orinoqu\u00eda.\n\n##### **Amazon**\n\nThe Amazon region of Colombia comprises 400,000 million kilometers, or roughly 35 percent of Colombia's territory, including all the territory east of the Andes and south of the R\u00edo Guaviare. The Colombian portion covers only 10 percent of the entire Amazon drainage basin. Total population in the Amazon region is 1.1 million, or 2 percent of the country's population. It is the most sparsely populated area in the country.\n\nLike the Llanos, the Amazon has an undulating terrain, interrupted occasionally with ancient, low-lying mountainous formations of the Guyana Shield, such as the Serran\u00eda de Chiribiquete, a series of highly eroded tabletop mountains. The Amazon consists of two distinct but intermingled areas: _terra firme_ , the undulated lands that are above the highest flood point, and _varzea_ , floodplains along the main rivers, which can extend 50 kilometers from the river.\n\nThere are two types of rivers in the Amazon: the predominant white rivers, which carry sediments down from the Andes; and the black rivers, which originate within the rainforest in the Guyana Shield formations that were long ago denuded of soil due to erosion. As these waters travel though the flooded forest, they pick up pigments that give them their characteristic black color. _Igapo_ is the name given to jungles flooded by black water rivers. Most of the rivers of the Colombian Amazon are white, such as the massive R\u00edo Putumayo, R\u00edo Caquet\u00e1, R\u00edo Apaporis, and R\u00edo Vaup\u00e9s, all of which are more than 1,000 kilometers long. The main black river in Colombia is the R\u00edo Guain\u00eda, which does originate in the Andes, and which is the headwater of the largest black river of the Amazon\u2014the R\u00edo Negro\u2014which flows into the milky Amazon at Manaus, in Brazil.\n\n#### **CLIMATE**\n\nColombia has a typically tropical climate, with no change of seasons. Climate is related primarily to elevation, and there are defined annual precipitation patterns.\n\nIn the mountainous areas, temperature decreases approximately six degrees Celsius per every 1,000 meters of elevation (three degrees Fahrenheit per every 1,000 feet). The common designations for the altitudinal zones are as follows: _tierra caliente_ (hot lands) is anywhere below 1,000 meters of elevation; _tierra templada_ (temperate lands) is anywhere between 1,000 and 2000 meters; and _tierra fr\u00eda_ (cold land) is anywhere above 2,000 meters. Roughly 80 percent of the country is _tierra caliente_ , 10 percent is _tierra templada,_ and 7 percent is _tierra fr\u00eda_.\n\nCartagena, which is at sea level, has an average temperature of 27.5\u00b0C (81.5\u00b0F); Medell\u00edn, which is at 1,600 meters (5,250 feet), has an average temperature of 22\u00b0C (71.5\u00b0F); and the capital city of Bogot\u00e1, which is built at 2,625 meters (8,612 feet), has an average temperature of 13.5\u00b0C (56\u00b0F).\n\nPrecipitation patterns vary throughout the country. In the Andean region, there are generally two periods of _verano_ (dry season, literally \"summer\"), from December to March and from June to September, and two periods of _invierno_ (rainy season, literally \"winter\"), in April and May and from October to November. In the Caribbean Coast, the dry period is from December to April and the rainy season is from May to November. In the Pacific it rains almost the entire year, but there is a slight dry spell from December to March. In the Llanos, there are two very marked seasons: a very dry _verano_ from November to March and a very wet _invierno_ from April to October. In the Amazon, it rains almost the entire year, but there is a slight dry spell from August to October.\n\nExtreme weather in Colombia is rare, but the country is susceptible to weather phenomena such as El Ni\u00f1o or La Ni\u00f1a, when temperatures in the Pacific Ocean rise or fall, respectively. More than 400 people died during devastating flooding and mudslides that occurred throughout much of the country during a particularly strong Ni\u00f1a in 2010-2011.\n\nSan Andr\u00e9s and Providencia are occasionally, and the Caribbean mainland of Colombia rarely, in the path of Atlantic hurricanes August through October. The last storm of significance was Hurricane Beta in 2005. It caused considerable damage in Providencia.\n\n### **Flora and Fauna**\n\nWhen it comes to biodiversity, Colombia is a place of superlatives. Though representing only 0.2 percent of the Earth's surface, it is home to about 10 percent of all the species in the world. The country has an estimated 55,000 plant species, including 3,500 species of orchids. Only Brazil, with seven times the land surface, has as many plant species. Colombia is the country with the greatest number of bird species in the world\u2014about 1,800. It's also home to about 3,200 fish, 750 amphibian species, 500 reptile, and 450 mammal species. No wonder Colombia was designated as one of 17 so-called megadiverse countries, a select club of countries that are home to an outsized proportion of the world's biodiversity. Other megadiverse countries include Australia, Brazil, China, Democratic Republic of Congo, Indonesia, Madagascar, Mexico, the United States, and South Africa.\n\nThis enormous biodiversity is the result of Colombia's location in the tropics, where year-round sunlight and high precipitation are conducive to plant growth, plus the country's mountainous topography with numerous climatic zones and microclimates that have created biological islands where species have evolved in relative isolation. Furthermore, the recent Ice Ages were not as severe in this part of the world and as a result many ancient species were preserved. Finally, Colombia's location on the crossroads between Central and South America has further enriched the country's biodiversity.\n\n#### **RAINFOREST**\n\nRainforests are among the most complex ecosystems on Earth. They have a layered structure with towering trees that soar 30-40 meters high to form the forest's canopy. Some of the most common rainforest trees are the ceiba, mahogany, myrtle, laurel, acacia, and rubber trees. Occasionally, particularly high trees known as _emergentes_ pierce the canopy, reaching as high up as 60 meters (200 feet). Below the canopy is the _sotobosque_ , a middle layer of smaller trees and palms that vie for the sunlight filtering in through the canopy. In the canopy and _sotobosque_ there are many epiphytes (plants such as orchids and bromeliads) that have adapted to live on top of trees so as to be nearer to the sunlight. Near the ground live plants that require little sunlight, including ferns, grasses, and many types of fungi. The two main rainforests in Colombia, the Amazon and the Choc\u00f3, have the same layered structure, though they have some differences in their flora and fauna.\n\n**Colombian Fruits**\n\nColombia is a land bursting with exotic fruit. Sold from the back of pickup trucks by farmers on the roadside, overflowing at stalls in colorful markets in every town and village, lined up in neat rows in the produce section at fancy grocery stores and at juice stands\u2014just about anywhere you go, delicious fruit is in reach.\n\nYou know pineapple, papaya, mangos, and bananas, but be sure to try these tropical delights that you may not have encountered outside of Colombia.\n\n\u2022 **_Pitahaya_** (dragon fruit): Looking like a yellow grenade, _pitahayas_ have a sweet white meat inside.\n\n\u2022 **_Guan\u00e1bana_** (soursop): By far the most weird-looking fruit, soursop resemble prehistoric dinosaur eggs. Inside the large green spiky fruit is a milky and slimy flesh. _Guan\u00e1bana_ is great in juices and desserts.\n\n\u2022 **_Granadilla:_** Crack open this orangey-yellow fruit and slurp down the slimy gray contents, seeds and all. It's delicious.\n\n\u2022 **_Higo_** (prickly pear): This green fruit comes from cactus plants and has sweet, if tough, orange-colored meat.\n\n\u2022 **_Chirimoya_** (cherimoya): This green fruit that resembles a smooth artichoke is covered with a smooth, silky skin and filled with delectable, sweet pulp.\n\n\u2022 **_N\u00edspero_** (sapodilla): A fruit with a deep brown color that tastes like a prepared sweet.\n\n\u2022 **_Mangostino_** (mangosteen): Crack open a deep purple mangosteen and enjoy the sweet segments inside. They're full of antioxidants.\n\n\u2022 **_Uchuva_** (Cape gooseberry): Known in English as Cape gooseberries, these tart yellow berries are a cousin of the tomato and are tasty on their own or in salads, but are often used in jams and sweets.\n\n\u2022 **_Mamoncillo:_** Tough-skinned grapes (don't eat the skin), mamoncillos are usually only sold at street markets.\n\nThe Amazon Rainforest is home to a large number of vertebrates. Over millennia, a large number of canopy-dwelling species evolved. Monkeys, such as the large and extremely agile spider monkey, the woolly monkey, and the howler monkey, evolved prehensile tails that allowed them to move easily from branch to branch. Anteaters, such as the tamandua and the _oso mielero_ (giant anteater), and the incredibly cute _kinkaj\u00fa_ (kinkajou) also developed prehensile tails. Other inhabitants of the canopy include sloths, such as the adorable-looking three-toed sloth, whose strategy is not agility but passivity: It eats tree vegetation and is covered with algae which gradually turn it green to allow for good camouflage. The canopy is also home to myriad bats and many birds, including exotic eagles, curassows, toucans, woodpeckers, cotingas, and macaws.\n\nNotable is the majestic harpy eagle, with powerful claws and the ability to fly unencumbered through the canopy. It preys on monkeys and sloths, which it kills with the force of its claws. The _tigrillo_ (tiger cat) is a small and extremely endangered species. It has a long tail that helps with its balance as it moves from tree to tree.\n\nOn the ground, large vertebrates include the extremely endangered tapir, an ancient mammal species that can grow two meters long (over six feet) and weigh 300 kilograms (660 pounds). It is equally at ease on land as in the water. Other land mammals include the giant armadillo, giant anteater, deer, and boars, such as the _sa\u00edno_ and _pecar\u00ed._ Smaller mammals include the _guat\u00edn_ and _borugo,_ both rodents. These animals are often prey to the puma and jaguar, both of which inhabit the Amazon but are difficult to observe in the wild.\n\nThe rivers of the Amazon are home to more than 1,500 species of fish, including endangered _piraruc\u00fa,_ one of the largest freshwater fishes on Earth. There are also dolphins, both pink and gray. The former evolved separately from the ocean-going dolphins when the Amazon was an inland sea. The Amazonian gray dolphins are sea dolphins that adapted to living in freshwater. Other aquatic mammals include the highly endangered manatee and otters.\n\nThe Choc\u00f3 Rainforest is particularly rich in palms, of which 120 species have been identified. In fact, it is sometimes referred to as the \"Land of the Palms.\" The forest also abounds in cycads, ancient plants that have a stout trunk and crowns of hard, stiff leaves. Choc\u00f3 is also notable for more than 40 species of brightly colored poisonous frogs, known locally as _ranas kokois._ These small frogs are covered with a deadly poison and have evolved stunning coloring, from bright orange to red, gold, and blue. They are active in the day and therefore relatively easy to spot. Of Colombia's 1,800 species of birds, more than 1,000 have been identified in the Choc\u00f3, including a large number of hummingbirds. Offshore, the Pacific welcomes the annual migration of Antarctic humpback whales. The beaches of the Pacific Coast are popular nesting areas for sea turtles, in particular the _tortuga golfina_ (olive ridley) and _tortuga carey_ (hawksbill) sea turtles. On the island of Gorgona 35 kilometers from the mainland, an unusual\u2014and quite stunning\u2014species of lizard is the blue anole lizard. Highly threatened, it is found only on Gorgona.\n\n#### **CLOUD FOREST**\n\nRainforests that grow at higher altitudes on the flanks of the Andes are known as montane rainforests or cloud forests because they are often enveloped in mist that results from the condensation of warm air against chillier mountain currents. Unlike the lowland rainforest, cloud forests only have two layers, the canopy and ground layer. Generally, the vegetation is less dense than the lowland rainforest. However, it is home to many palms, ferns, and epiphytes, particularly orchids.\n\nThe type of cloud forest vegetation is dictated by altitude. _Selva subandina_ (sub-Andean forest) vegetation grows between the altitudes of 1,000-2,300 meters (3,300-7,500 feet), where temperature varies 16-23\u00b0C (61-73\u00b0F). Plant species include the distinctive Dr. Seuss-like white _yarumo_ with its oversized leaves, as well as cedar, oak, and mahogany trees. Many palms grow here, including the svelte wax palm and _tagua_ , which produces a nut that resembles ivory. Ferns include the striking _palma boba_ or tree fern. Colombia's premier crop, coffee, is grown at this elevation.\n\nAt elevations between 2,300-3,600 meters (7,500-12,000 feet), the vegetation is described as _selva Andina_ (Andean forest). This vegetation is even less dense and at higher elevations the trees are smaller. The vegetation bears some resemblance to landscapes in the northern and southern hemispheres. _Selva Andina_ includes many oak, _encenillo, sietecuero_ (glory bush), and pine trees.\n\nMammals include the spectacled or Andean bear, the only species of bear in South America, the mountain (or woolly) tapir, anteaters, armadillos, sloths, boars, foxes, and _olingos,_ a small arboreal carnivore of the raccoon family. In 2013, the _olinguito_ (small _olingo_ ), an incredibly cute-looking animal, was declared a new species. Other unusual animals include the slow-moving _guagua loba_ and _guat\u00edn,_ both of which are rodents. In addition, numerous species of monkeys inhabit the cloud forest, including noisy troops of howler monkeys. Birds include many types of _barranqueros_ (motmots), including the spectacular blue-crowned motmot. Other common birds include _t\u00e1ngaras_ (tanagers), woodpeckers, warblers, parrots, owls, and ducks, including the beautiful white and black torrent duck.\n\n#### **P\u00c1RAMOS**\n\n_P\u00e1ramos_ are a unique tropical highland ecosystem that thrives above 3,500 meters (11,500 feet), where UV radiation is higher, oxygen is scarcer, and where temperatures vary from minus 2 to 10 degrees Celsius (28-50\u00b0F). Due to frequent mist and precipitation, _p\u00e1ramos_ are often saturated with water and have many lakes. They are true \"water factories\" that provide water to many of Colombia's cities, notably Bogot\u00e1. Though _p\u00e1ramos_ exist throughout the New World tropics, most are located in Colombia. The Parque Nacional Natural Sumapaz, south of Bogot\u00e1, is the world's largest _p\u00e1ramo_.\n\n_P\u00e1ramo_ vegetation includes more than 50 species of _frailej\u00f3n_ (genus _Espeletia_ ), eerily beautiful plants that have imposing tall trunks and thick yellow-greenish leaves. Other _p\u00e1ramo_ vegetation includes shrubs, grasses, and _cojines_ (cushion plants). Mammals include the spectacled bear, _p\u00e1ramo_ tapir, weasels, squirrels, and bats. The _p\u00e1ramo_ is the realm of the majestic black and white Andean condor, which has a wingspan of up to three meters (10 feet). The condor, whose numbers had declined to the point of extinction, are found in the national parks of the Sierra Nevada de Santa Marta, Sierra Nevada del Cocuy, and Los Nevados. The _p\u00e1ramo_ lakes welcome many types of ducks, including the Andean duck, as well as smaller birds.\n\n#### **TROPICAL DRY FORESTS**\n\nTropical dry forests exist in areas where there is a prolonged dry season. The vegetation includes deciduous trees that lose their leaves during the dry season, allowing them to conserve water. Trees on moister sites and those with access to ground water tend to be evergreen. Before Columbus, this ecosystem covered much of the Colombian Caribbean coast. However, much of it was cut down for cattle ranching. Pockets still exist east of the Golfo de Morrosquillo and at the base of the Sierra Nevada de Santa Marta. Tropical dry forests are the most endangered tropical ecosystem in the world.\n\nThis unusual _frailej\u00f3n,_ a type of cactus, grows in Colombia's _p\u00e1ramos._\n\nhawk\n\nThough less biologically diverse than rainforests, tropical dry forests are home to a wide variety of wildlife. They were once the stomping ground of the now highly endangered _marimonda_ , or white-fronted spider monkey.\n\n#### **TROPICAL GRASSLANDS**\n\nLos Llanos (The Plains) of Colombia are covered with lush tropical grasslands. Vegetation includes long-stemmed and carpet grasses in the drier areas and swamp grasses in low-lying humid areas. There are also thick patches of forest throughout the plains and along the rivers (known as gallery forests). These plains are teeming with wildlife, including deer, anteaters, armadillos, tapirs, otters, jaguars, pumas, and _chig\u00fciros_ (also known as capybaras), the world's largest rodent. The Llanos are also home to the giant anaconda and to one of the most endangered species on Earth, the Orinoco crocodile, which reaches up to seven meters (23 feet) long.\n\n### **History**\n\n#### **BEFORE COLUMBUS**\n\nLocated at the juncture between Central and South America, what is now Colombia was a necessary transit point for the migration of people who settled South America. However, as these peoples left few physical traces of their passage, little is known of them. The oldest human objects found in Colombia, utensils discovered near Bogot\u00e1, are dated from 14,000 BC. With the expansion of agriculture and sedentary life throughout the territory of present-day Colombia around 1000 BC, various indigenous cultures started producing stunning ceramic and gold work, as well as some monumental remains. These remains provide rich material evidence of their development. Nonetheless, there are significant gaps in the understanding of the history of these early peoples.\n\nFrom around 700 BC, the area of San Agust\u00edn, near the origin of the R\u00edo Magdalena in southern Colombia, was settled by people who practiced agriculture and produced pottery. Starting in the 1st century AD, the people of San Agust\u00edn created hundreds of monumental stone statues set on large platforms, which comprise the largest pre-Columbian archaeological site extant between Mesoamerica and Peru. By AD 800, this society had disappeared.\n\nIn the northwestern plains of Colombia, south of present-day Cartagena, starting in the 1st century AD, the Sin\u00fa people constructed a large complex of mounds in the shape of fish bones that regulated flooding, allowing cultivation in flooded and dry seasons. During rainy seasons, the water flooded the lower cavities, allowing for cultivation on the mounds; during dry season, cultivation took place in the cavities that had been enriched by the flood waters. These monumental formations are still visible from the sky. By the time of the Spanish conquest, these people no longer inhabited the area.\n\nFrom AD 500 to 900, the area of Tierradentro, west of San Agust\u00edn, was settled by an agricultural society that dug magnificent decorated underground tombs, produced large stone statues, and built oval-shaped buildings on artificial terraces. As in the case of the San Agust\u00edn and the Sin\u00fa people, it is not known what happened to these people.\n\nAt the time of the conquest, present-day Colombia was populated by a large number of distinct agricultural societies that often maintained peaceful trading relations among themselves. The two largest groups were the Muisca people, who lived in the altiplano (highlands) of the Cordillera Oriental, and the Tayrona, who lived on the slopes of the Sierra Nevada de Santa Marta. Other groups included the Quimbaya, who settled the area of the present-day Coffee Region; the Calima, in present-day Valle del Cauca; and the Nari\u00f1os, in the mountainous areas of southwest Colombia.\n\nThese indigenous societies were mostly organized at the village level with loose association with other villages. Only the Muisca and the Tayrona had a more developed political organization. Though these were all agricultural societies, they also engaged in hunting, fishing, and mining and produced sophisticated ceramics and gold work. Each group specialized in what their environment had to offer and engaged in overland trade. For example, the Muiscas produced textiles and salt, which they traded for gold, cotton, tobacco, and shells with other groups.\n\nThe Muiscas, a Chibcha-speaking people, were the largest group, with an estimated 600,000 inhabitants at the time of the Spanish conquest. They settled the Cordillera Oriental in AD 300 and occupied a large territory that comprises most of the highland areas of the present-day departments of Cundinamarca and Boyac\u00e1. At the time of the conquest, they were organized into two large confederations: one in the south headed by the Zipa, whose capital was Bacat\u00e1 near present-day Bogot\u00e1; and another headed by the Zaque, whose capital was at Hunza, the location of present-day Tunja. The Muiscas had a highly homogeneous culture, skilled in weaving, ceramics, and gold work. Their cosmography placed significant importance on high Andean lakes, several of which were sacred, including Guatavita, Siecha, and Iguaque.\n\nThe Tayrona, who settled the slopes of the Sierra Nevada de Santa Marta, were also a Chibcha-speaking people. They had a more urban society, with towns that included temples and ceremonial plazas built on stone terraces, and practiced farming on terraces carved out of the mountains. There are an estimated 200 Tayrona sites, of which Ciudad Perdida (Lost City), built at 1,100 meters in the Sierra Nevada de Santa Marta, is the largest and best known. Many of these towns, including El Pueblito in the Parque Nacional Natural Tayrona, were occupied at the time of the Spanish conquest. The Kogis, Arhuacos, Kankuamos, and Wiwas, current inhabitants of the sierra, are their descendants and consider many places in the sierra sacred.\n\n#### **THE SPANISH CONQUEST (1499-1550)**\n\nAs elsewhere in the New World, the arrival of Europeans was an unmitigated disaster for the Native American societies. Though there were pockets of resistance, on the whole the indigenous people were unable to push back the small number of armed Spanish conquistadors. Harsh conditions after the conquest and the spread of European diseases, such as measles and smallpox, to which the indigenous people had no immunity, killed off millions of indigenous people. The Spanish conquest of present-day Colombia took about 50 years and was largely completed by the 1550s.\n\nIn 1499, the first European set foot on present-day Colombia in the northern Guajira peninsula. In 1510, a first, unsuccessful colony was established in the Gulf of Urab\u00e1 near the current border with Panama. In 1526, the Spanish established Santa Marta, their first permanent foothold, from where they tried, unsuccessfully, to subdue the Tayronas. In 1533, they established Cartagena, which was to become a major colonial port.\n\nIn 1536, Gonzalo Jim\u00e9nez de Quesada set off south from Santa Marta to conquer the fabled lands of El Dorado in the Andean heartland. After a year of grueling travel up the swampy R\u00edo Magdalena valley, 200 surviving members of Jim\u00e9nez de Quesada's 800 original troops arrived in the Muisca lands near present-day Bogot\u00e1. After a short interlude of courteous relations, the Spaniards' greed led them to obliterate the Muisca towns and temples. They found significant amounts of gold, especially in the town of Hunza, but they were, by and large, disappointed. In 1538, Jim\u00e9nez de Quesada founded Santa Fe de Bogot\u00e1 as the capital of this new territory, which he called Nueva Granada\u2014New Granada\u2014after his birthplace.\n\nSebasti\u00e1n de Belalc\u00e1zar, a lieutenant of Francisco Pizarro, led a second major expedition that arrived in the Muisca lands from the south. Having conquered the Inca city of Quito, Belalc\u00e1zar and his army traveled north, conquering a vast swath of land from present-day Ecuador to the _s\u00e1bana_ (high plateau) of Bogot\u00e1. Along the way, he founded several cities, including Popay\u00e1n and Cali in 1536. He arrived shortly after Quesada had founded Bogot\u00e1. Incredibly, a third conquistador, the German Nikolaus Federmann, arrived in Bogot\u00e1 at the same time, having traveled from Venezuela via the Llanos. Rather than fight for supremacy, the three conquistadors decided to take their rival claims to arbitration at the Spanish Court. In an unexpected turn of events, none of the three obtained title to the Muisca lands. When Bogot\u00e1 became the administrative capital of New Granada, they came under the sway of the Spanish Crown. Other expeditions swept across the Caribbean coast, through current-day Antioquia and the Santanderes.\n\n#### **COLONIAL NUEVA GRANADA (1550-1810)**\n\nFor most of its colonial history, Nueva Granada, as colonial Colombia was called, was an appendage of the Viceroyalty of Peru. In 1717, Spain decided to establish a viceroyalty in Nueva Granada but changed its mind six years later because the benefits did not justify the cost. In 1739, the viceroyalty was reestablished, with Santa Fe de Bogot\u00e1 as its capital. It was an unwieldy territory, encompassing present-day Colombia, Venezuela, Ecuador, and Panama. To make it more manageable, Venezuela and Panama were ruled by captain-generals and Ecuador by a president. At the local level, the viceroyalty was divided into _provincias_ (provinces), each with a local assembly called a _cabildo_.\n\nSettlement in Nueva Granada occurred primarily in three areas: where there were significant indigenous populations to exploit, as in the case of Tunja in the former Muisca territory; where there were gold deposits, as in Cauca, Antioquia, and Santander; and along trade routes, for example at Honda and Momp\u00f3x on the R\u00edo Magdalena. Cartagena was the main port of call for the biennial convoys of gold and silver sent to Spain. Bogot\u00e1 lived off of the official bureaucracy and sustained a fair number of artisans. Present-day Antioquia and Santander supported small-scale farming to provide provisions to the gold mining camps. Nueva Granada was one of the least economically dynamic of Spain's New World possessions. The mountainous topography and high transportation costs meant that agricultural production was primarily for local consumption and gold was the only significant export.\n\nColonial society was composed of a small Spanish and Creole (descendants of Spanish settlers) elite governing a large mestizo (mixed indigenous-white) population. The Spanish had initially preserved indigenous communal lands known as _resguardos_ , but the demographic collapse of the native population and intermarriage meant that, unlike in Peru or Mexico, there were relatively few people who were fully indigenous. There were also black slaves who were forced mostly to work in the mines and haciendas (plantations). Society was overwhelmingly Catholic and Spanish-speaking.\n\nCulturally, Nueva Granada was also somewhat of a backwater. Though there was a modest flourishing of the arts, Bogot\u00e1 could not compete with the magnificent architectural and artistic production of Quito, Lima, or Mexico City. The only truly notable event of learning that took place was the late 18th-century Expedici\u00f3n Bot\u00e1nica (Botanical Expedition), headed by Spanish naturalist Jos\u00e9 Celestino Mutis, the personal doctor to one of the viceroys. The aim of the expedition was to survey all the species of Nueva Granada\u2014that was a rather tall order given that Colombia is home to 10 percent of the world's species! However, the expedition did some remarkable research and produced beautiful prints of the fauna and flora.\n\n**Colombia's National Parks**\n\nFrom undisturbed coral reefs to the Amazonian jungle to snow-covered mountain ranges, Colombia's national park system is indeed a treasure and making the effort to visit them is worthwhile for any visitor. The country's system of natural parks and protected areas covers more than 14 million hectares: around 13.4 percent of the country! It includes 42 Parques Nacionales Naturales (National Natural Parks), which are major areas of ecological interest that have remained largely untouched by human intervention; and 10 Santuarios de Flora y Fauna (Flora and Fauna Sanctuaries), areas that are devoted to the preservation of specific ecosystems. Of the 42 parks, 24 are open for tourism. The rest are officially off limits, mostly due to security reasons or in order to respect the territory of indigenous communities.\n\nIn 1960, PNN Cueva de los Gu\u00e1charos, in the southwest, was the first park to be established. The number of parks steadily increased, especially from 1986 to 1990 when President Virgilio Barco doubled the extension of parks from roughly 5 million hectares to 10 million hectares. In the past few years, the government has again been increasing the number and extension of parks. In 2013 President Juan Manuel Santos doubled the size of the PNN Serran\u00eda de Chiribiquete to its present 2.8 million hectares, or three times the size of Yellowstone National Park.\n\nCharged with the considerable task of administering this huge system are a mere 430 rangers: roughly one person for every 33,000 hectares. Rangers face a great challenge in protecting the parks against threats related to human encroachment, particularly cattle ranching and the planting of illicit crops. There are other threats as well, such as illegal mining and logging. Paradoxically, what has preserved many of the parks until now has been the lack of security due to Colombia's internal conflict. As security conditions improve, there will be increasing pressure on these natural habitats. The Parks Service is actively engaging with the communities that live near the parks and is transferring the operation of much of the ecotourism infrastructure to community-based organizations as part of an effort to enlist local communities in the preservation of the land.\n\nEntry permits and entry fees are only required in a handful of highly visited parks, such as PNN Tayrona, PNN Gorgona, PNN Cocuy, and PNN Los Nevados. At these, you will automatically be charged if you book lodging in advance, or if not, upon arrival. If you want to be meticulous, you can obtain the entry permit and pay entry fees in advance by contacting the **Parques Nacionales** (tel. 1\/353-2400, www.parquesnacionales.gov.co) in Bogot\u00e1.\n\nThe late colonial period saw unrest in Nueva Granada. Starting in 1781, a revolt known as the Rebeli\u00f3n de los Comuneros took place in the province of Socorro (north of Bogot\u00e1) in present-day Santander as a result of an attempt by colonial authorities to levy higher taxes. It was not an anti-royalist movement, however, as its slogan indicates: _\u00a1Viva el Rey, Muera el Mal Gobierno!_ (\"Long live the king, down with bad government!\"). Rather it was a protest against unfair taxes, not much different from the Boston Tea Party. However, it gave the Spanish government a fright. A rebel army, led by Jos\u00e9 Antonio Gal\u00e1n, marched on Bogot\u00e1. Negotiations put an end to the assault, and later the authorities ruthlessly persecuted the leaders of the revolt.\n\n#### **THE STRUGGLES FOR INDEPENDENCE (1810-1821)**\n\nThough there was some ill feeling against the colonial government, as the Rebeli\u00f3n de los Comuneros attests, as well as rivalry between the Spanish- and American-born elites, it was an external event, the Napoleonic invasion of Spain, that set off the chain of events that led to independence of Nueva Granada and the rest of the Spanish dominion in the New World.\n\nIn 1808, Napoleon invaded Spain, took King Ferdinand VII prisoner, and tried to impose his own brother, Joseph, as king of Spain. The Spaniards revolted, establishing a Central Junta in Seville to govern during the king's temporary absence from power. Faced with the issue of whether to recognize the new Central Junta in Spain, the colonial elites decided to take matters in their own hands and establish juntas of their own. The first such junta in Nueva Granada was established in Caracas in April 1810. Cartagena followed suit in May and Bogot\u00e1 on July 20, 1810. According to popular myth, the revolt in Bogot\u00e1 was the result of the failure of a prominent Spaniard merchant to lend a flower vase to a pair of Creoles.\n\nThough they pledged alliance to Ferdinand VII, once the local elites had tasted power, there was no going back. Spanish authorities were expelled, and in 1811, a government of sorts, under the loose mantle of the Provincias Unidas de Nueva Granada (United Provinces of New Granada), was established with its capital at Tunja. Bogot\u00e1 and the adjoining province of Cundinamarca stayed aloof from the confederation, arguing that it was too weak to resist the Spanish. Subsequently, various provinces of Nueva Granada declared outright independence, starting with Venezuela and Cartagena in 1811 and Cundinamarca in 1813.\n\nSeveral cities remained loyal to the crown, namely Santa Marta and deeply conservative Pasto in the south. From 1812-1814 there was a senseless civil war between the Provincias Unidas and Cundinamarca\u2014that is why this period is called the Patria Boba, or Foolish Fatherland. Ultimately, the Provincias Unidas prevailed with the help of a young Venezuelan captain by the name of Sim\u00f3n Bol\u00edvar.\n\nAfter the restoration of Ferdinand VII, Spain attempted to retake its wayward colonies, with a military expedition and reign of terror known as the Reconquista\u2014the Reconquest. The Spanish forces took Cartagena by siege in 1815 and took control of Bogot\u00e1 in May 1816. However, in 1819, a revolutionary army composed of Venezuelans, Nueva Granadans, and European mercenaries headed by Bol\u00edvar arrived across the Llanos from Venezuela and decisively defeated the Spanish army in the Batalla del Puente de Boyac\u00e1\u2014the Battle of the Boyac\u00e1 Bridge\u2014on August 7, near Tunja. The rest of the country fell quickly to the revolutionary army. With support from Nueva Granada, Bol\u00edvar defeated the Spanish in Venezuela in 1821. Panama, which had remained under Spanish control, declared independence in 1821. Finally, Bol\u00edvar dispatched Antonio Jos\u00e9 de Sucre to take Quito in 1822, bringing an end to the Spanish rule of Nueva Granada.\n\n#### **GRAN COLOMBIA: A FLAWED UNION (1821-1830)**\n\nShortly after the Battle of Boyac\u00e1, the Congress of Angostura, a city on the R\u00edo Orinoco in Venezuela, proclaimed the union of Nueva Granada, Venezuela, and Ecuador under the name of the Rep\u00fablica de Colombia. Historians refer to this entity as Gran Colombia. In 1821, while the fight for independence was still raging in parts of Venezuela and Ecuador, a constitutional congress met in C\u00facuta. An ongoing debate about whether a centralist or federalist scheme was preferable resulted in a curious compromise: the Rep\u00fablica de Colombia assumed a highly centralist form, considered necessary to finish the battle for independence, but left the issue of federalism open to review after 10 years. The document was generally liberal, enshrining individual liberties and providing for the manumission of slaves, meaning that the children of slaves were born free.\n\nBol\u00edvar, who was born in Venezuela, was named president. Francisco de Paula Santander, who was born near C\u00facuta in Nueva Granada, was named vice president. Santander had fought alongside Bol\u00edvar in the battles for independence of Nueva Granada and was seen as an able administrator. While Bol\u00edvar continued south to liberate Ecuador and Peru, Santander assumed the reins of power in Bogot\u00e1. He charted a generally liberal course, instituting public education and a curriculum that included avant garde thinkers such as Jeremy Bentham. However, the highly centralist structure was unsavory to elites in Venezuela and Ecuador, who disliked rule from Bogot\u00e1. Shortly after the Congress of C\u00facuta, revolt broke out in Venezuela and Ecuador. In 1826, Bol\u00edvar returned from Bolivia and Peru, hoping for the adoption in Gran Colombia of the Bolivian Constitution, an unusual document he drafted that called for a presidency for life.\n\nThere had been a growing distance between Bol\u00edvar and Santander: Bol\u00edvar saw Santander as an overzealous Liberal reformer while Santander disliked Bol\u00edvar's authoritarian tendencies. In 1828, after a failed constitutional congress that met in Oca\u00f1a in eastern Colombia, Bol\u00edvar assumed dictatorial powers. He rolled back many of Santander's liberal reforms. In September 1828 there was an attempt on Bol\u00edvar's life in Bogot\u00e1. This was famously foiled by his companion, Manuela S\u00e1enz. The last years of Gran Colombia were marked by revolts in various parts of the country and a war with Peru. In 1830, a further constitutional assembly was convened in Bogot\u00e1, but by that point Gran Colombia had ceased to exist: Venezuela and Ecuador had seceded. In March 1830, a physically ill Bol\u00edvar decided to leave for voluntary exile in Europe and died on his way in Santa Marta.\n\n#### **CIVIL WARS AND CONSTITUTIONS (1830-1902)**\n\nAfter the separation of Venezuela and Ecuador, what is now Colombia adopted the name Rep\u00fablica de Nueva Granada. In 1832, it adopted a new constitution that corrected many of the errors of the excessively centralist constitution of Gran Colombia. There was a semblance of stability with the orderly succession of elected presidents. The elimination of some monasteries in Pasto sparked a short civil conflict known as the Guerra de los Supremos, which lasted 1839-1842. During this war, Conservative and Liberal factions coalesced for the first time, establishing the foundation of Colombia's two-party system. Generally, the Conservative Party supported the Catholic church, favored centralization, and followed the ideas of Bol\u00edvar. The Liberal Party supported federalism and free trade and identified with the ideas of Santander.\n\nThe country's rugged topography meant that Nueva Granada was not very integrated into the world economy. Gold, extracted mostly in Antioquia, was the main export. Most of the country eked out its subsistence from agriculture, with trade restricted within regions. This period saw some economic development, such as steam navigation on the Magdalena and Cauca rivers, and a contract for the construction of the trans-isthmanian railroad in Panama, which had yet to secede.\n\nMid-century saw the rise of a new class of leaders who had grown up wholly under republican governments. They ushered in a period of liberal reform. In 1851, Congress abolished slavery. In 1853, a new constitution established universal male suffrage, religious tolerance, and direct election of provincial governors. The government reduced tariffs and Nueva Granada experienced a short export-oriented tobacco boom.\n\nConflicts between radical reformers within the Liberal Party, moderates, and Conservatives led to unrest in various provinces. In 1859, discontented Liberals under Tom\u00e1s Cipriano de Mosquera revolted, leading to generalized civil war in which the Liberals were ultimately victorious. Once in power, they pushed radical reform. Mosquera expropriated all non-religious church property, partly in vengeance against church support of the Conservatives in the previous civil war.\n\nThe 1863 constitution was one of the world's most audacious federalist experiments. The country was renamed the Estados Unidos de Colombia (United States of Colombia), comprising nine states. The president had a two-year term and was not immediately re-electable. All powers that were not explicitly assigned to the central government were the responsibility of the states. Many of the states engaged in true progressive policies, such as establishing public education and promoting the construction of railroads. This period coincided with agricultural booms in quinine, cotton, and indigo that, for the first time, brought limited prosperity. This period saw the establishment of the Universidad Nacional (National University) and the country's first bank.\n\na plaque in Momp\u00f3x, denoting Sim\u00f3n Bol\u00edvar's presence\n\nIn 1880 and then in 1884, a coalition of Conservatives and moderate Liberals, who were dissatisfied with the radical policies, elected Rafael N\u00fa\u00f1ez as president. N\u00fa\u00f1ez tried to strengthen the power of the central government, sparking a Liberal revolt. The Conservatives were ultimately victorious and, in 1886, enacted a new centralist constitution that lasted through most of the 20th century. The country was rechristened Rep\u00fablica de Colombia, the name it has conserved since then. During the period 1886 through 1904, known as the Regeneraci\u00f3n, the Conservative Party held sway, rolling back many of the previous reforms, especially anticlerical measures and unrestricted male suffrage. The Liberal Party, excluded from power, revolted in 1899. The ensuing Guerra de los Mil D\u00edas (Thousand Days' War), which raged through 1902, was a terribly bloody conflict. It is not clear how many died in the war, but some historians put the figure as high as 100,000, or an incredible 2.5 percent of the country's population of four million at the time.\n\nOne year after the end of the war, Panama seceded. During the late 19th century, there had been resentment in Panama about the distribution of revenues from the transit trade that mostly were sent to Bogot\u00e1. However, in 1902 the local Panamanian elites had become alarmed at the lackadaisical attitude of the government in Bogot\u00e1 regarding the construction of an interoceanic canal. After the failure of the French to build a canal, Colombia had entered into negotiations with the United States. In the closing days of the Guerra de los Mil D\u00edas, Colombia and the United States signed the Hay-Terran Treaty, which called for the construction of the canal, surrendering control over a strip of land on either side of the canal to the United States. The Americans threatened that if the treaty were not ratified, they would dig the canal in Nicaragua. Arguing that the treaty undermined Colombian sovereignty, the congress in Bogot\u00e1 unanimously rejected the treaty with the United States in August 1903. That was a big mistake: A few months later, Panama seceded with the support of the United States.\n\n#### **PEACE AND REFORM (1902-1946)**\n\nUnder the leadership of moderate Conservative Rafael Reyes, who was president 1904-1909, Colombia entered a period of peace and stability. Reyes focused on creating a professional, nonpartisan army. He gave representation to Liberals in government, enacted a protective tariff to spur domestic industry, and pushed public works. During his administration, Bogot\u00e1 was finally connected by railway to the R\u00edo Magdalena. He reestablished relations with the United States, signing a treaty that provided Colombia with an indemnity for the loss of Panama. During the 1920s and 1930s, Colombia was governed by a succession of Conservative Party presidents. Though there was often electoral fraud, constitutional reform that guaranteed minority representation ensured peace.\n\nExpanding world demand for coffee spurred production across Colombia, especially in southern Antioquia and what is now known as the Coffee Region, creating a new class of independent farmers. Improved transportation, especially the completion of the railways from Cali to Buenaventura on the Pacific Coast and from Medell\u00edn to the R\u00edo Magdalena, was key to the growth of coffee exports. In the Magdalena Medio region and in Norte de Santander, U.S. companies explored and started producing petroleum. Medell\u00edn became a center of textile manufacturing. With its broken geography, air transportation developed rapidly. The Sociedad Colombo Alemana de Transportes A\u00e9reos (Colombian German Air Transportation Society) or SCADTA, the predecessor of Avianca, was founded in Barranquilla in 1919, and is reputedly the second oldest commercial aviation company in the world (the oldest is KLM).\n\nIn 1930, a split Conservative ticket allowed the Liberals to win the elections. After being out of power for 50 years, the Liberal Party was happy to regain control of the state apparatus. This led to strife with Conservatives long accustomed to power\u2014presaging the intense interparty violence that was to erupt 14 years later.\n\nFrom 1932-1933, Colombia and Peru fought a brief war in the Amazon over the control of the port city of Leticia. The League of Nations brokered a truce, the first time that this body, which was a precursor to the United Nations, actively intervened in a dispute between two countries.\n\nStarting in 1934, Liberal president Alfonso L\u00f3pez Pumarejo undertook major social and labor reforms, with some similarities to Roosevelt's New Deal. His policies included agrarian reform, encouragement and protection of labor unions, and increased spending on education. He reduced the Catholic church's sway over education and eliminated the literacy requirement for male voters. Many of these reforms simply returned the country to policies that had been enacted by Liberals in the 1850s, 80 years prior. In opposition to these policies, a new radical right, with a confrontational style and strains of fascism and anti-Semitism, arose under the leadership of Laureano G\u00f3mez.\n\nDuring World War II, Colombia closely allied itself with the United States and eventually declared war on the Axis powers in retaliation for German attacks on Colombian merchant ships in the Caribbean Sea. The government concentrated those of German-descent in a hotel in Fusagasug\u00e1 near Bogot\u00e1 and removed all German influence from SCADTA.\n\n#### **LA VIOLENCIA (1946-1953)**\n\nIn the 1946 elections, the Liberal Party split its ticket between establishment-backed Gabriel Turbay and newcomer Jorge Eli\u00e9cer Gait\u00e1n. Gait\u00e1n was a self-made man who had scaled the ladders of power within the Liberal Party despite the opposition of the traditional Liberal elite. He had a vaguely populist platform and much charisma. The moderate Conservative Mariano Ospina won a plurality of votes and was elected to the presidency. As in 1930, the transfer of power from Liberals to Conservatives and bureaucratic re-accommodation led to outbursts of violence.\n\nOn April 9, 1948, a deranged youth killed former presidential candidate Gait\u00e1n as he left his office in downtown Bogot\u00e1. His assassination sparked riots and bloodshed throughout the country, with severe destruction in the capital. The disturbance in Bogot\u00e1, known as El Bogotazo, occurred during the 9th Inter-American Conference, which had brought together leaders from all over the hemisphere. Young Fidel Castro happened to be in Bogot\u00e1 that day, though he had no part in the upheaval.\n\nThe assassination of Gait\u00e1n fueled the violence that had started in 1946. Over the course of 10 years, an estimated 100,000-200,000 people died in what was laconically labeled La Violencia (The Violence). This conflict was comparable in destruction of human life with the Guerra de los Mil D\u00edas, the last civil war of the 19th century. The killing took place throughout the country, often in small towns and rural areas. Mostly it involved loyalists of the predominant party settling scores or intimidating members of the opposite party to extract land or economic gain. In some cases, the violence was sheer banditry. Numerous, horrific mass murders took place. The police often took sides with the Conservatives or simply turned a blind eye. In response, some Liberals resorted to armed resistance, giving birth to Colombia's first guerrilla armies. The Liberal Party boycotted the 1950 elections, and radical Conservative Laureano G\u00f3mez was elected president. His government pursued authoritarian and highly partisan policies, further exacerbating the violence.\n\n#### **DICTATORSHIP (1953-1957)**\n\nIn 1953, with the purported aim of bringing an end to fighting between Liberals and Conservatives, the Colombian army, under the command of General Gustavo Rojas Pinilla, staged a coup. Rojas was able to reduce, but not halt, the violence, by curtailing police support of the Conservatives and by negotiating an amnesty with Liberal guerrillas. In 1954, Rojas was elected for a four-term period by a hand-picked assembly. Incidentally, it was this, non-democratically elected assembly that finally got around to extending suffrage to women, making Colombia one of the last countries in Latin America to do so. Rojas tried to build a populist regime with the support of organized labor, modeled after Per\u00f3n in Argentina. His daughter, Mar\u00eda Eugenia Rojas, though no Evita, was put in charge of social welfare programs. Though a majority of Colombians supported him at first, his repressive policies and press censorship ended up alienating the political elites.\n\n#### **THE NATIONAL FRONT (1957-1974)**\n\nIn May 1957, under the leadership of Liberals and Conservatives, the country went on an extended general strike to oppose the dictatorship. Remarkably, Rojas voluntarily surrendered power and went into exile in Spain. As a way to put an end to La Violencia, Liberal and Conservative Party leaders proposed alternating presidential power for four consecutive terms while divvying up the bureaucracy on a 50-50 basis. The proposal, labeled the National Front, was ratified by a nationwide referendum and was in effect 1958-1974.\n\nThe National Front dramatically reduced the level of violence. After years of fighting, both factions were ready to give up their arms. During this period, thanks to competent economic management, the economy prospered and incomes rose. The government adopted import substitution policies that gave rise to a number of new industries, including automobiles.\n\nBy institutionalizing the power of the two traditional parties, the National Front had the unintended consequence of squeezing out other political movements, especially from the left. As a result, during the 1960s a number of leftist guerrilla groups appeared. Some were simply the continuation, under a new name, of the Liberal guerrillas formed during La Violencia. The Fuerzas Armadas Revolucionarias de Colombia (FARC) was a rural, peasant-based group espousing Soviet Marxism. The Ej\u00e9rcito de Liberaci\u00f3n Nacional (ELN) was a smaller group inspired by the Cuban revolution. The even smaller Ej\u00e9rcito Popular de Liberaci\u00f3n (EPL) was a Maoist-inspired group. The Movimiento 19 de Abril (M-19) was a more urban group formed by middle-class intellectuals after alleged electoral fraud deprived the populist ANAPO Party (Alianza Nacional Popular; created by ex-dictator Rojas) of power. During the 1970s and 1980s, the M-19 staged flashy coups, such as stealing Bol\u00edvar's sword (and promising to return it once the revolution had been achieved) in 1974 and seizing control of the Dominican Republic Embassy in Bogot\u00e1 in 1980.\n\n#### **UNDER SIEGE (1974-1991)**\n\n##### **The Drug Trade and the Rise of Illegal Armed Groups**\n\nDue to its relative proximity to the United States, treacherous geography, and weak government institutions, Colombia has been an ideal place for cultivation, production, and shipment of illegal drugs, primarily to the United States. During the 1970s, Colombia experienced a short-lived marijuana boom centered around the Sierra Nevada de Santa Marta. Eradication efforts by Colombian authorities and competition from homegrown marijuana produced in the U.S. quickly brought this boom to an end.\n\nDuring the late 1970s, cocaine replaced marijuana as the main illegal drug. Though most of the coca cultivation at the time was in Peru and Bolivia, Colombian drug dealers based in Medell\u00edn started the business of picking up coca paste in Peru and Bolivia, processing it into cocaine in Colombia, and exporting the drug to the United States, where they even controlled some distribution at the local level. At its heyday in the mid-1980s, Pablo Escobar's Medell\u00edn Cartel controlled 80 percent of the world's cocaine trade. The rival Cali Cartel, controlled by the Rodr\u00edguez brothers, emerged in the 1980s and started to contest the supremacy of the Medell\u00edn Cartel, leading to a bloody feud.\n\nDuring the 1980s and 1990s, coca cultivation shifted from Peru and Bolivia to Colombia, mainly to the Amazon regions of Putumayo, Caquet\u00e1, Meta, and Guaviare. Initially, leftist guerrillas such as the FARC protected the fields from the authorities in return for payment from the cartels. Eventually, they started processing and trafficking the drugs themselves. Though the guerrillas had other sources of income, such as kidnapping and extortion, especially of oil companies operating in the Llanos, the drug trade was a key factor in their growth. With these sources of income, they no longer needed popular support and morphed into criminal organizations. By the mid-1980s, the FARC had grown into a 4,000-strong army that controlled large portions of territory, especially in the south of the country.\n\nDuring the 1980s and 1990s, the price of land was depressed as a result of the threat from the guerrillas. Using their vast wealth and power of intimidation, drug traffickers purchased vast swaths of land, mostly in the Caribbean coast of Colombia, at bargain prices. To defend their properties from extortion, they allied themselves with traditional landowners to create paramilitary groups. These groups often operated with the direct or tacit support of the army.\n\nColombian campesinos (small farmers), caught in the middle of the conflict between guerrillas and paramilitaries, suffered disproportionately. They were accused by both guerrillas and paramilitaries of sympathizing with the enemy, and the government was not there to protect them. The paramilitaries were particularly ruthless, often ordering entire villages to abandon their lands or massacring them. The conflict between guerrillas and paramilitaries is at the source of the mass displacement of people in Colombia. According to the Office of the United Nations High Commissioner for Refugees, the number of displaced people in Colombia ranges 3.9-5.3 million, making it the country with the most internal refugees in the world.\n\n##### **Peace Negotiations with the FARC and M-19**\n\nIn 1982, President Belisario Betancur was elected with the promise of negotiating peace with the guerrillas. The negotiations with the guerrillas got nowhere, but the FARC did establish a political party, the Uni\u00f3n Patri\u00f3tica (UP), which successfully participated in the 1986 presidential elections and 1988 local elections, managing to win some mayoralties. The paramilitaries and local elites did not want the political arm of the FARC to wield local power. As a result, the UP was subjected to a brutal persecution by the paramilitaries, who killed more than 1,000 party members. In the midst of this violence, Colombia suffered one of its worst natural disasters: the eruption of the Nevado del Ruiz in November 1985, which produced a massive mudslide that engulfed the town of Armero, killing more than 20,000 people.\n\nIn 1985, the M-19 brazenly seized the Palacio de Justicia in Bogot\u00e1. The Colombian army responded with a heavy hand, and in the ensuing battle, half of Colombia's Supreme Court justices had been killed. Many people, including many cafeteria employees, disappeared in the army take-over, and there is speculation that they were executed and buried in a mass grave in the south of Bogot\u00e1. Weakened by this fiasco, leaders of the M-19 took up President Virgilio Barco's offer to negotiate peace. The government set down clear rules, including a ceasefire on the part of the M-19, before talks could proceed. Unlike the FARC, the M-19 was still an ideological movement. The leaders of the M-19 saw that by participating in civil life they could probably gain more than by fighting. And they were right: In 2011 the people of Bogot\u00e1 elected Gustavo Petro, a former M-19 guerrilla, as their mayor. On March 19, 1990, Barco and the M-19's young leader, Carlos Pizarro, signed a peace agreement, the only major successful peace agreement to date between the authorities and a major guerrilla group.\n\n##### **The Rise and Fall of the Medell\u00edn Cartel**\n\nInitially, the Colombian establishment turned a blind eye to the rise of the drug cartels and even took a favorable view of the paramilitaries, who were seen as an antidote to the scourge of the guerrillas. For a time, Escobar was active in politics and cultivated a Robin Hood image, funding public works such as parks and housing projects. Rather than stick to his business, as the Cali Cartel did, Escobar started to threaten any official who tried to check his power. In 1984, he had Rodrigo Lara Bonilla, the Minister of Justice, assassinated. When the government subsequently cracked down, Escobar declared outright war. He assassinated judges and political leaders, set off car bombs to intimidate public opinion, and paid a reward for every policeman that was murdered in Medell\u00edn\u2014a total of 657. To kill an enemy, he planted a bomb in an Avianca flight from Bogot\u00e1 to Cali, killing all passengers on board. The Medell\u00edn Cartel planted dozens of massive bombs in Bogot\u00e1 and throughout the country, terrorizing the country's population. The cartel is allegedly responsible for the assassination of three presidential candidates in 1990: Luis Carlos Gal\u00e1n, the staunchly anti-mafia candidate of the Liberal Party; Carlos Pizarro, the candidate of the newly demobilized M-19; and Bernardo Jaramillo, candidate of the Uni\u00f3n Patri\u00f3tica.\n\nThere was really only one thing that Escobar feared\u2014extradition to the United States. Through bribery and intimidation, he managed to get extradition outlawed, and he negotiated a lopsided deal with the government of C\u00e9sar Gaviria: In return for his surrender, he was allowed to control the jail where he was locked up. From the luxurious confines of La Catedral, as the prison was named, he continued to run his empire. In 1992 there was an outcry when it became known that he had interrogated and executed enemies within the jail. When he got wind that the government planned to transfer him to another prison, he fled. In December 1993, government intelligence intercepted a phone call he made to this family, located him in Medell\u00edn, and killed him on a rooftop as he attempted to flee. It is widely believed that the Cali Cartel actively aided the authorities in the manhunt.\n\n##### **A New Constitution**\n\nThe 1990s started on a positive footing with the enactment of a new constitution in 1991. The Constitutional Assembly that drafted the charter was drawn from all segments of the political spectrum, including the recently demobilized M-19. The new constitution was very progressive, devolving considerable power to local communities and recognizing the rights of indigenous and Afro-Colombian communities to govern their communities and ancestral lands. The charter created a powerful new Constitutional Court, which has become a stalwart defender of basic rights, as well as an independent accusatory justice system, headed by a powerful attorney general, which was created to reduce impunity.\n\n#### **COLOMBIA ON THE BRINK (1992-2002)**\n\n##### **The Unchecked Growth of New Cartels, Paramilitaries, and Guerrillas**\n\nDrug cultivation and production increased significantly during the 1990s. The overall land dedicated to coca cultivation rose from 60,000 hectares in 1992 to 165,000 hectares in 2002. As a result of the government's successful crackdown first on the Medell\u00edn Cartel and then on the Cali Cartel, drug production split into smaller, more nimble criminal organizations. During the 1990s, the paramilitaries became stand-alone organizations that engaged in drug trafficking, expanding to more than 30,000 men in 2002. They created a national structure called the Autodefensas Unidas de Colombia, or AUC, under the leadership of Carlos Casta\u00f1o. The AUC coordinated activities with local military commanders and committed atrocious crimes, often massacring scores of so-called sympathizers of guerrillas.\n\nAt the same time, the guerrillas expanded significantly during the 1990s. Strengthened by hefty revenues from kidnapping, extortion, and drug trafficking, they grew to more than 50,000 mostly peasant fighters in 2002. Their strategy was dictated primarily by military and economic considerations and they had little to no public support. At their heyday, the FARC covered the entire country, attacking military garrisons and even threatening major urban centers such as Cali. They performed increasingly large operations, such as attacking Mit\u00fa, the capital of the department of Vaup\u00e9s, in 1998 or kidnapping 12 members of the Assembly of Valle del Cauca in Cali in 2002. The FARC commanders moved around the countryside unchecked. In the territories they controlled, they ruled over civilians, often committing heinous crimes. In 2002, they attacked a church in the town of Bojay\u00e1 in Choc\u00f3, killing more than 100 unarmed civilians, including many children, who had sought refuge there.\n\n##### **Plan Colombia**\n\nThe increasing growth of drug exports from Colombia to the United States in the 1990s became a source of concern for the U.S. government. From 1994 to 1998, the U.S. was reluctant to provide support to Colombia because the president at the time, Ernesto Samper, was tainted by accusations of having received campaign money from drug traffickers, and because of evidence about human rights abuses by the Colombian army. When Andr\u00e9s Pastrana was elected president in 1998, the Colombian and U.S. administrations designed a strategy to curb drug production and counteract the insurgency called Plan Colombia. This strategy had both military and social components, and was to be financed jointly by the U.S. and Colombia. Ultimately, the United States provided Colombia, which was becoming one of its strongest and most loyal of allies in Latin America, with more than US$7 billion, heavily weighted towards military aid, especially for training and for providing aerial mobility to Colombian troops. While the impact of Plan Colombia was not immediately visible, over time it changed the balance of power in favor of the government, allowing the Colombian army to regain the upper arm in the following years.\n\n##### **Flawed Peace Negotiations with the FARC**\n\nPresident Pastrana embarked on what is now widely believed to have been an ill-conceived, hurried peace process with the FARC. He had met Manuel Marulanda, the head of the FARC, before his inauguration in 1998 and was convinced that he could bring about a quick peace. Without a clear framework, in November 1998 he acceded to the FARC's request to grant them a demilitarized zone the size of Switzerland in the eastern departments of Meta and Caquet\u00e1. In hindsight, it seems clear that the FARC had no interest or need to negotiate as they were at the peak of their military power. Rather, the FARC commanders saw the grant of the demilitarized zone as an opportunity to strengthen their organization.\n\nFrom the beginning, it became clear that the FARC did not take the peace process seriously. Marulanda failed to show up at the inaugural ceremony of the peace process, leaving a forlorn Pastrana sitting alone on the stage next to a now famous _silla vacilla_ (empty seat). They ran the demilitarized zone as a mini-state, nicknamed Farclandia, using it to smuggle arms, hold kidnap prisoners, and process cocaine. During the peace negotiations, the FARC continued their attacks on the military and civilians. In February 2002, after the FARC kidnapped Eduardo Gechem, senator and president of the Senate Peace Commission, Pastrana declared the end of this ill-advised demilitarized zone and sent in the Colombian Army.\n\nprotest for peace in Bogot\u00e1\n\n##### **A Failed State?**\n\nIn 2002, the Colombian Army was battling more than 50,000 guerrillas and 30,000 paramilitaries, with an estimated 6,000 child soldiers. The insurgents controlled approximately 75 percent of the country's territory. An estimated 100,000 antipersonnel mines covered 30 of 32 departments. More than 2.5 million people had been internally displaced between 1985 and 2003, with 300,000 people displaced in 2002 alone. Not surprisingly, prestigious publications such as _Foreign Policy_ described Colombia at the time as failed state.\n\n#### **REGAINING ITS FOOTING (2002-PRESENT)**\n\n##### **\u00c1lvaro Uribe's Assault on the Guerrillas**\n\nIn the 2002 elections, fed-up Colombians overwhelmingly elected \u00c1lvaro Uribe, a former governor of Antioquia who promised to take the fight to the guerrillas. Uribe had a real grudge against the FARC, who had assassinated his father. The FARC were not fans of his, either. In a brazen show of defiance, during Uribe's inauguration ceremony in Bogot\u00e1 on August 7, 2002, the guerrilla group fired various rockets aimed at the presidential palace, during a post-swearing in reception. Several rockets struck the exterior of the palace, causing minor damage (attendees were unaware of the attack), but many more fell on the humble dwellings in barrios nearby, killing 21.\n\nDuring his first term, Uribe embarked on a policy of Seguridad Democr\u00e1tica, or Democratic Security, based on strengthening the army, eradicating illicit crops to deprive the guerrillas of revenues, and creating a controversial network of civilian collaborators who were paid for providing tips that led to successful operations against the insurgents. The government increased military expenditure and decreed taxes on the rich totaling US$4 billion to finance the cost of the war. Colombian military personnel grew from 300,000 in 2002 to 400,000 in 2007.\n\nFrom 2002 to 2003, the army evicted the FARC from the central part of the country around Bogot\u00e1 and Medell\u00edn, although that did not prevent them from causing terror in the cities. In February 2003, a car bomb attributed to the FARC exploded in the parking lot of the exclusive social club El Nogal, killing more than 30 people\u2014mostly employees. From 2004 to 2006, the army pressed the FARC in its stronghold in the southern part of the country. Aerial spraying of coca crops brought down cultivated areas from 165,000 hectares in 2002 to 76,000 in 2006.\n\nIn 2006, Uribe was reelected by a landslide, after Congress amended the constitution to allow for immediate presidential reelection. There is clear evidence that the government effectively bribed two congressmen whose vote was necessary for passage of the measure. Uribe interpreted the election results as a mandate to continue single-mindedly pursuing the guerrillas. The FARC came under severe stress, with thousands of guerrillas deserting and, for the first time, subjected to effective strikes against top commanders. No longer safe in their traditional jungle strongholds in Colombia, many FARC operatives crossed the border into Venezuela and Ecuador, causing tension between Colombia and the governments of those countries.\n\nIn early 2008, the Colombian military bombed and killed leading FARC commander Ra\u00fal Reyes in a camp in Ecuador, causing a diplomatic crisis with that country. Later that year, the military executed Operaci\u00f3n Jaque (Operation Checkmate), a dramatic rescue operation in which they duped the FARC into handing over their most important hostages. The hostages released included three U.S. defense contractors and Ingrid Betancur, a French-Colombian independent presidential candidate who was kidnapped by the FARC during the 2002 presidential election as she proceeded by land, against the advice of the military, towards the capital of the former FARC demilitarized zone. In 2008, Manuel Marulanda, founder of the FARC, died a natural death. At that time, it was estimated that the FARC forces had plummeted to about 9,000 fighters, half of what they had been eight years before.\n\nThe Colombian army has been implicated in serious human rights abuses. Pressure from top brass to show results in the war against the guerrillas and the possibility of obtaining extended vacation time led several garrisons to execute civilians and present them as guerrillas killed in combat. In 2008, it was discovered that numerous young poor men from the city of Soacha, duped by false promises of work, had been taken to rural areas, assassinated by the army, and presented as guerrillas killed in anti-insurgency operations. This macabre episode\u2014referred to as the scandal of _falsos positives_ (false positives)\u2014was done under the watch of Minister of Defense Juan Manuel Santos, who was later elected president of Colombia.\n\n##### **Peace Process with the AUC**\n\nFrom 2003 to 2008, the Uribe government pursued a controversial peace process with the right wing paramilitaries, the Autodefensas Unidas de Colombia. As part of that process, an estimated 28,000 paramilitary fighters demobilized, including most of the high level commanders. In 2005, the Colombian Congress passed a Justice and Peace Law to provide a legal framework for the process. Unlike previous peace laws that simply granted an amnesty to the insurgents, this law provided for reduced sentences for paramilitaries who had committed serious crimes in exchange for full confessions and reparation of victims. Domestic and international observers were extremely skeptical about the process, worrying that the paramilitaries would use their power to pressure for lenient terms. These misgivings were justified by evidence that they used their power of coercion to influence the results of the 2006 parliamentary elections, a scandal referred to as _parapol\u00edtica_. Many congresspersons, including a first cousin of Uribe, ended up in prison.\n\nIt soon became clear that the paramilitary commanders were not sincere in their commitment to peace. Many refused to confess crimes and transferred their assets to front men. Covertly, they continued their drug-trafficking operations. The government placed scant importance on the truth and reparation elements of the Justice and Peace Law, severely underfunding the effort to redress crimes committed against more than 150,000 victims who had signed up as part of the process. Through 2008, the paramilitaries had confessed to a mere 2,700 crimes, a fraction of the estimated total, and refused to hand over assets. Fed up with their lack of cooperation, in 2008 Uribe extradited 14 top ranking paramilitary commanders to the United States, where they were likely to face long sentences. However, the extradition severely hampered the effort to obtain truth and reparation for the victims of their crimes.\n\nThe difficulty in redressing the crimes against victims has been further hampered by the growth of the dozens of small _bacrim (bandas criminals,_ or illegal armed groups) who have taken territorial control of former paramilitary areas, intimidating victims who have returned to their rightful lands under the peace process. Many of these _bacrim_ inherited the structures of the former AUC groups and employed former paramilitaries.\n\n##### **Social and Economic Transformation**\n\nDuring the past decade, Colombia has made some remarkable strides in improving social and economic conditions. Due to improved security conditions, investment, both domestic and international has boomed, totaling almost US$80 billion from 2003 to 2012. Economic growth has averaged 4.7 percent 2002-2012, a significant increase over the prior decades. The number of people below poverty, as measured by the ability to buy a wide basket of basic goods and services, has declined from 59.7 percent in 2002 to 32.7 percent in 2012. In Colombia's 13 largest cities, which represent 45 percent of the population, poverty has fallen to 18.9 percent. In terms of basic needs, most urban areas are well served in terms of education, health, electricity, water, and sewage. However, there is a wide gap between the cities and rural areas, where 30 percent of the country's population lives. As of 2012, rural poverty stood at 46.8 percent. Though income inequality has been slowly falling, Colombia still has one of the most unequal distributions of income in the world.\n\n##### **Peace with the FARC?**\n\nIn the 2010 elections, Uribe's former minister of defense, Juan Manuel Santos, was elected by a large majority. Santos continued to pursue an aggressive strategy against the FARC. Army operations killed Alfonso Cano, the new leader of the FARC, as well as V\u00edctor Julio Su\u00e1rez Rojas, the guerrillas' military strategist. As evidenced in the diary of Dutch FARC member Tanya Nijmeijer, found by the Colombian Army after an attack on a rebel camp, morale within the FARC had sunk to an all-time low.\n\nAt the same time, Santos recognized the need to address non-military facets of the violence. In 2011, Congress passed a comprehensive Victims and Land Restitutions Law, to redress the weaknesses of Uribe's Justice and Peace Law. This law provides a comprehensive framework to redress the crimes committed against all victims of violence since 1985.\n\nAfter a year of secret negotiations, Santos announced the start of peace dialogues with the FARC in October 2012 in Cuba. These have proceeded at a slow pace, surprisingly without any halt to military actions. Former president Uribe and his allies have been very vocal against this initiative, claiming that a military defeat of the FARC is the only sensible path forward. At the time of writing, it was not clear whether the guerrillas were sincere in their desire for peace.\n\n### **Government and Economy**\n\n#### **GOVERNMENT**\n\nUnder the 1991 constitution, Colombia is organized as a republic, with three branches of power\u2014the executive, the legislative, and the judicial. The country is divided into 32 _departamentos_ (departments or provinces) and the Distrito Capital (Capital District), where Bogot\u00e1 is located. The departments are in turn divided into _municipios_ (municipalities). These _municipios_ include towns and rural areas.\n\nThe president of the republic, who is both head of state and head of government, is elected for a four-year term. With the exception of the military dictatorship of Gen. Gustavo Rojas Pinilla 1953-1957, presidents has been elected by the people since 1914. In 2005, then-president \u00c1lvaro Uribe succeeded in changing the constitution to allow for one immediate presidential re-election. In 2009, he attempted to get the constitution changed once more to allow for a second reelection but was thwarted by the powerful Constitutional Court, which decreed that this change would break the necessary checks and balances of the constitutional framework.\n\nPresidential elections will be held in May 2014 and 2018. If no candidate receives more than 50 percent of the votes, there will be a run-off election. Inauguration of the president takes place on August 7, the anniversary of the Batalla del Puente de Boyac\u00e1, which sealed Colombia's independence from Spain.\n\nThe legislative branch is made up of a bicameral legislature: the Senado (102 members) and the C\u00e1mara de Representantes (162 members). These representatives are elected every four years. Senators are voted for on a nationwide basis, while representatives are chosen for each department and the Distrito Capital. In addition, two seats in the Senate are reserved for indigenous representation. In the C\u00e1mara de Representantes, there are seats reserved for indigenous and Afro-Colombian communities as well as for Colombians who live abroad.\n\nAll Colombians over the age of 18\u2014with the exception of active duty military and police as well as those who are incarcerated\u2014have the right to vote in all elections. Women only gained the right to vote in 1954.\n\n##### **Political Parties**\n\nHistorically Colombia has had a two-party system: the Conservative Party and the Liberal Party. The Conservative Party has traditionally been aligned with the Catholic Church and has favored a more centralized government, and followed the ideas of Sim\u00f3n Bol\u00edvar. The Liberal Party favored a federal system of governing, has opposed church intervention in government affairs, and was aligned with the ideas of Gen. Francisco Paula Santander.\n\nThe hegemony of the two largest political parties came to a halt in the 2002 presidential election of rightist candidate \u00c1lvaro Uribe, who registered his own independent movement and then established a new party called El Partido de la Unidad. Since then, traditional parties have lost some influence. A third party, the Polo Democr\u00e1tico, became a relatively strong force in the early 2000s, capturing the mayorship of Bogot\u00e1, but has since faded, leaving no clear representative of the left.\n\nColombia is known for its coffee production.\n\nThe Partido Verde (Green Party) ran strong in the 2010 elections, with former mayor Antanas Mockus becoming the main rival against eventual winner Juan Manuel Santos. This party has little to do with the global Green Party movement. Political parties today have become personality oriented, and many candidates have been known to shop around for a party\u2014or create their own\u2014rather than adhere to the traditional parties.\n\n#### **ECONOMY**\n\nColombia has a thriving market economy based primarily on oil, mining, agriculture, and manufacturing. The country's GDP in 2013 was US$226 billion and per capita GDP was US$10,100, placing it as a middle-income country. Growth over the past decade has been a robust 4.7 percent. Inflation has averaged 3.8 percent in the past five years and unemployment has hovered around 10 percent.\n\nDuring the colonial period and up until the early 20th century, small-scale gold mining and subsistence agriculture were the mainstays of Colombia's economy. Starting in the 1920s, coffee production spread throughout the country and rapidly became Colombia's major export good. Coffee production is of the mild arabica variety and is produced at elevations of 1,000 to 1,900 meters, mostly by small farmers. During most of the 20th century, Colombia emphasized increasing the volume of production, using the Caf\u00e9 de Colombia name and mythical coffee farmer Juan Valdez and his donkey Paquita to brand it. A severe global slump in coffee prices during the past decade has led to a reassessment of this strategy and an increasing focus on specialty coffees. Today, coffee represents only 3 percent of all Colombian exports.\n\nColombia's wide range of climates, from hot on the coast to temperate in the mountains, means that the country produces a wide range of products. Until recently, sugar cane production, fresh flowers, and bananas were the only major export-driven agribusiness. However, improvements in security in recent years have resulted in a boom in large scale agricultural projects in palm oil, rubber, and soy. Cattle ranching occupies an estimated 25 percent of the country's land. Commercial forestry is relatively underdeveloped, though there is considerable illegal logging, especially on the Pacific Coast.\n\nIn recent decades, oil production and mining have become major economic activities. The main center of oil production is the Llanos, the eastern plains of Colombia, with oil pipelines extending from there over the Cordillera Oriental to Caribbean ports. Oil currently represents roughly half of all Colombian exports. There is also significant natural gas, which is used mostly for residential use. Large-scale mining has been focused on coal and nickel, with large deposits in the Caribbean coastal region. With the improvement of security conditions in the past decade, many international firms, such as Anglogold Ashanti, have requested concessions for large-scale gold mining, often with opposition from the community. Illegal gold mining, often conducted with large machinery, is a severe threat to fragile ecosystems, especially in the Pacific Coast rainforest.\n\nDuring the post-war period, Colombia pursued an import substitution policy, fostering the growth of domestic industries such as automobiles, appliances and petrochemical goods. Since the early 1990s, the government has been gradually opening the economy to foreign competition and tearing down tariffs. In recent years, the country has signed free trade agreements with the United States and the European Union. Today, the country has a fairly diversified industrial sector. The country is self-sufficient in energy, with hydropower supplying the bulk of electricity needs.\n\nUntil recently, tourism was minimal because of widespread insecurity and a negative image. Things started to change in the mid-2000s, and the annual number of international visitors has increased almost threefold from 600,000 in 2000 to 1.7 million in 2012. While Bogot\u00e1 and Cartagena still receive the bulk of visitors, almost the entire country has opened up for tourism, though there are still pockets of no-go zones. This boom in tourism has fostered a growth of community and ecotourism options, often with the support from government. The network of _posadas nativas_ (guesthouses owned and operated by locals) is one initiative to foment tourism at the community level, particularly among Afro-Colombians. In recent years, Parques Nacionales has transferred local operation of ecotourism facilities in the parks to community-based associations.\n\n### **People and Culture**\n\n#### **DEMOGRAPHY**\n\nColombia was estimated to have had a population of a little over 47 million in 2013 and has the third highest population in Latin America, behind Brazil and Mexico and slightly higher than Argentina. Around four million Colombians live outside of Colombia, mostly in the United States, Venezuela, Spain, and Ecuador. The population growth rate has fallen significantly in the past two decades and is now 1.1 percent. The population of the country is relatively young, with a median age of 28.6 years. Average life expectancy is 75 years.\n\nSixty percent of the Colombian population lives in the highland Andean interior of the country, where the three largest cities are located: Bogot\u00e1 (7.7 million), Medell\u00edn (3.4 million), and Cali (3.1 million).\n\nIt is increasingly an urban country, with around 75 percent of the population living in urban areas. This trend began during La Violencia and accelerated in the 1970s and 1980s. At least 3.9 million persons have been internally displaced due to the armed conflict in Colombia, leaving their homes in rural areas and seeking safety and economic opportunity in large cities.\n\nMost of the population (over 86 percent) is either mestizo (having both Amerindian and white ancestry) or white. People of African (10.4 percent) and indigenous or Amerindian (over 3.4 percent) origin make up the rest of the Colombian population. There is a tiny Romani or Roma population of well under 1 percent of the population, but nonetheless they are a protected group in the constitution.\n\n**Gay Rights in Colombia**\n\nIn a country still struggling with armed conflict and basic human rights, it might come as a surprise that gay and lesbian rights have not been pushed aside. Colombia has some of the most progressive laws regarding the rights of LGBT people in the western hemisphere. Since 2007, same-sex partners have enjoyed full civil union rights with a wide range of benefits, such as immigration, inheritance, and social security rights.\n\nHowever, when it comes to marriage, it's a little more complicated. In 2011, the top judicial body, the Colombian Constitutional Court, ordered Congress to regulate marriage rights by mid-2013. Congress dithered, leaving the issue up in the air. Since then, scores of same-sex couples have gotten married despite the vehement opposition of the country's Procurador (inspector general), although the country's powerful attorney general has sided with same-sex couples in this legal battle. This limbo has left a bad taste in the mouths of just about everyone: gay rights advocates, the Catholic Church, and powerful conservative politicians.\n\nThere are more than 80 indigenous groups, with some of the largest being the Way\u00fau, who make up the majority in the La Guajira department; the Nasa, from Cauca; the Ember\u00e1, who live in the isolated jungles of the Choc\u00f3 department, and the Pastos, in Nari\u00f1o. Departments in the Amazon region have the highest percentages of indigenous residents. In Vaup\u00e9s, for example, 66 percent of the population is of indigenous background. Many indigenous people live on _resguardos_ , areas that are collectively owned and administered by the communities.\n\nAfro-Colombians, descendants of slaves who arrived primarily via Spanish slave trade centers in the Caribbean, mostly live along both Pacific and Caribbean coasts and in the San Andr\u00e9s Archipelago. Choc\u00f3 has the highest percentage of Afro-Colombians (83 percent), followed by San Andr\u00e9s and Providencia (57 percent), Bol\u00edvar (28 percent), Valle del Cauca (22 percent), and Cauca (22 percent). Cali, Cartagena, and Buenaventura have particularly large Afro-Colombian populations. In the Americas, Colombia has the third highest number of citizens of African origin, behind Brazil and the United States.\n\nWhile Colombia has not attracted large numbers of immigrants, there have been periods in which the country opened its doors to newcomers. In the early 20th century, immigrants from the Middle East, specifically from Lebanon, Syria, and Palestine arrived, settling mostly along the Caribbean Coast, especially in the cities of Barranquilla, Santa Marta, Cartagena, and Maicao in La Guajira. From 1920 to 1950, a sizable number of Sephardic and Ashkenazi Jews immigrated. Colombia has not had a large immigration from Asia, although in the early 20th century there was a small immigration of Japanese to the Cali area.\n\n#### **RELIGION**\n\nOver 90 percent of Colombians identify as Roman Catholics, and it has been the dominant religion since the arrival of the Spaniards. The numbers of evangelical Christians, called simply _cristianos_ , continue to grow, and there are other Christian congregations, including Mormons and Jehovah's Witnesses, but their numbers are small. In San Andr\u00e9s and Providencia, the native Raizal population\u2014of African descent\u2014is mostly Baptist.\n\nThe Jewish community\u2014estimated at around 5,000 families\u2014is concentrated in the large cities, such as Bogot\u00e1, Medell\u00edn, Cali, and Barranquilla. There are significant Muslim communities, especially along the Caribbean Coast, and there are mosques in Barranquilla, Santa Marta, Valledupar, Maicao (La Guajira), San Andr\u00e9s, and Bogot\u00e1.\n\n**Happy Monday!**\n\nColombians enjoy a long list of holidays (over 20). With a few exceptions, such as the Independence celebrations on July 20 and August 7, Christmas, and New Year's Day, holidays are celebrated on the following Monday, creating a _puente_ (literally bridge, or three-day weekend).\n\nDuring Easter week and between Christmas Day and New Year's, interior cities such as Bogot\u00e1 and Medell\u00edn become ghost towns as locals head to the nearest beach or to the countryside. Conversely, beach resorts, natural reserves and parks, and pueblos fill up. Along with that, room rates and air tickets can increase substantially.\n\nThe following is a list of Colombian holidays, but be sure to check a Colombian calendar for precise dates. Holidays marked with an asterisk are always celebrated on the Monday following the date of the holiday.\n\n\u2022 A\u00f1o Nuevo (New Year's Day): January 1\n\n\u2022 D\u00eda de los Reyes Magos (Epiphany)*: January 6\n\n\u2022 D\u00eda de San Jos\u00e9 (Saint Joseph's Day)*: March 19\n\n\u2022 Jueves Santo (Maundy Thursday): Thursday before Easter Sunday\n\n\u2022 Viernes Santo (Good Friday): Friday before Easter Sunday\n\n\u2022 D\u00eda de Trabajo (International Workers' Day): May 1\n\n\u2022 Ascensi\u00f3n (Ascension)*: Six weeks and one day after Easter Sunday\n\n\u2022 Corpus Christi*: Nine weeks and one day after Easter Sunday\n\n\u2022 Sagrado Coraz\u00f3n (Sacred Heart)*: Ten weeks and one day after Easter Sunday\n\n\u2022 San Pedro y San Pablo (Saint Peter and Saint Paul)*: June 29\n\n\u2022 D\u00eda de la Independencia (Independence Day): July 20\n\n\u2022 Batalla de Boyac\u00e1 (Battle of Boyac\u00e1): August 7\n\n\u2022 La Asunci\u00f3n (Assumption of Mary)*: August 15\n\n\u2022 D\u00eda de la Raza (equivalent of Columbus Day)*: October 12\n\n\u2022 Todos Los Santos (All Saint's Day)*: November 1\n\n\u2022 D\u00eda de la Independencia de Cartagena (Cartagena Independence Day)*: November 11\n\n\u2022 La Inmaculada Concepci\u00f3n (Immaculate Conception): December 8\n\n\u2022 Navidad (Christmas): December 25\n\nSemana Santa\u2014Holy or Easter Week\u2014is the most important religious festival in the country, and Catholics in every village, town, and city commemorate the week with a series of processions and masses. The colonial cities of Popay\u00e1n, Mompox, Tunja, and Pamplona are known for their elaborate Semana Santa processions. Popay\u00e1n and Mompox in particular attract pilgrims and tourists from Colombia and beyond. In cities such as Bogot\u00e1, Cali, and Cartagena, there are multitudinous processions to mountaintop religious sites, such as Monserrate, the Cerro de la Cruz, and El Monasterio de la Popa, respectively.\n\n#### **LANGUAGE**\n\nSpanish is the official language in Colombia. In the San Andr\u00e9s Archipelago, English is spoken by Raizal natives who arrived from former English colonies after the abolition of slavery.\n\nAccording to the Ministry of Culture, there are at least 68 native languages, which are spoken by around 850,000 people. These include 65 indigenous languages, two Afro-Colombian languages, and Romani, which is spoken by the small Roma population.\n\nThree indigenous languages have over 50,000 speakers: Way\u00fau, primarily spoken in La Guajira; P\u00e1ez, primarily spoken in Cauca; and Ember\u00e1, primarily spoken in Choc\u00f3.\n\n## **ESSENTIALS**\n\nGetting There\n\nBY AIR\n\nBY CAR OR MOTORCYCLE\n\nBY BUS\n\nBY BOAT\n\nGetting Around\n\nBY AIR\n\nBY BUS\n\nBY CAR, MOTORCYCLE, OR BICYCLE\n\nBY BOAT\n\nVisas and Officialdom\n\nPASSPORTS AND VISAS\n\nCUSTOMS\n\nEMBASSIES AND CONSULATES\n\nTips for Travelers\n\nACCESS FOR TRAVELERS WITH DISABILITIES\n\nWOMEN TRAVELING ALONE\n\nGAY AND LESBIAN TRAVELERS\n\nHealth and Safety\n\nVACCINATIONS\n\nMEDICAL SERVICES\n\nDISEASES AND ILLNESSES\n\nCRIME\n\nInformation and Services\n\nMONEY\n\n### **Getting There**\n\n#### **BY AIR**\n\nMost visitors to Colombia arrive by air at the **Aeropuerto Internacional El Dorado** in Bogot\u00e1, with some carrying on from there to other destinations in the country. There are also nonstop international flights to the **Aeropuerto Internacional Jos\u00e9 Mar\u00eda C\u00f3rdova** in Medell\u00edn and to the airports in Cali, Cartagena, Barranquilla, and Armenia.\n\n##### **From North America**\n\n**Avianca** (www.avianca.com) has nonstop flights between Bogot\u00e1 and Miami, Fort Lauderdale, Orlando, Washington, and New York-JFK. From Miami there are also nonstops to Medell\u00edn, Cali, Barranquilla, and Cartagena.\n\n**American** (www.american.com) flies between Miami and Dallas and Bogot\u00e1; Miami and Medell\u00edn; and Cali and Medell\u00edn. **Delta** (www.delta.com) flies from Atlanta and New York-JFK to Bogot\u00e1. **United** (www.united.com) has flights from Newark and Houston to Bogot\u00e1.\n\n**Jet Blue** (www.jetblue.com) has service between Orlando and Fort Lauderdale and Bogot\u00e1, Cartagena and New York, and Medell\u00edn and Fort Lauderdale. **Spirit** (www.spirit.com) has flights from Fort Lauderdale to Bogot\u00e1, Medell\u00edn, Cartagena, and Armenia.\n\nthe airport shuttle in Acand\u00ed\n\n**Air Canada** (www.aircanada.com) operates nonstops from Toronto to Bogot\u00e1.\n\n##### **From Europe**\n\n**Avianca** (www.avianca.com) has service to Bogot\u00e1 and Medell\u00edn from Madrid and Barcelona. **Air France** (www.airfrance.com) flies from Paris to Bogot\u00e1. **Iberia** (www.iberia.com) serves Bogot\u00e1 from Madrid. **Lufthansa** (www.lufthansa.com) offers service between Bogot\u00e1 and Frankfurt.\n\n##### **From Latin America**\n\n**Avianca** (www.avianca.com) flies to many capitals in Latin America, including Buenos Aires, S\u00e3o Paulo, Rio de Janeiro, Valencia, Caracas, Lima, Santiago, and La Paz in South America; Canc\u00fan, Guatemala City, Mexico City, San Jos\u00e9, San Juan, San Salvador, and Panama City in Central America; and Havana, Santo Domingo, Punta Cana, Aruba, and Cura\u00e7ao in the Caribbean. Aerol\u00edneas Argentinas, AeroGal, Aeromexico, Air Insel, Conviasa, Copa, Cubana, LAN, Gol TAM, TACA, and Tiara Air Aruba also have connections to Colombia.\n\n#### **BY CAR OR MOTORCYCLE**\n\nA growing number of travelers drive into Colombia in their own car or with a rented vehicle. The most common point of entry is at the city of Ipiales on the Pan-American Highway: the Rumichaca border crossing with Ecuador at Ipiales (Tulc\u00e1n on the Ecuador side). This entry point is open 5am-10pm daily.\n\nOn the Venezuelan side, the border at C\u00facuta and San Antonio del T\u00e1chira is open 24 hours a day. Although there are other border crossings with Venezuela, this is the recommended overland point of entry.\n\n#### **BY BUS**\n\nFrequent buses that depart Quito bound for Cali (20 hours) or Bogot\u00e1 (30 hours). You can also take a taxi from the town of Tulc\u00e1n to the border at Ipiales and from there take an onward bus to Pasto, Popay\u00e1n, Cali, or beyond. In Quito contact **L\u00edneas de los Andes** (www.lineasdelosandes.com.co).\n\n#### **BY BOAT**\n\nIt is possible to enter the country from Panama, usually via the San Blas islands. **Blue Sailing** (U.S. tel. 203\/660-8654, www.bluesailing.net) offers sailboat trips between various points in Panama to Cartagena. The trip usually takes about 45 hours and costs around US$500. Sometimes, particularly during the windy season between November and March, boats stop in Sapzurro, Colombia, near the border. **San Blas Adventures** (contact@sanblasadventures.com, www.sanblasadventures.com) offers multi-day sailboat tours to the San Blas islands that usually depart from Cart\u00ed and end up in the Panamanian border village of La Miel. From there you can walk over the border to Sapzurro and take a _lancha_ (boat) from there on to Capurgan\u00e1. There are regular morning boats from Capurgan\u00e1 to both Acand\u00ed and Turbo. During the windy season, especially between December and February, this trip can be quite rough.\n\nIt is also possible to hitch a ride on a cargo boat from Ecuador to Tumaco or Buenaventura; however, service is irregular.\n\n### **Getting Around**\n\n#### **BY AIR**\n\nAir travel is an excellent, quick, and, thanks to the arrival of discount airliners such as VivaColombia, an often economical way to travel within Colombia. Flying is, without a doubt, the best option for those for whom the idea of 16 hours in a bus is not very appealing. Airlines have generally excellent track records and maintain modern fleets.\n\nBogot\u00e1 is the major hub in the country, with domestic **Avianca** (tel. 1\/401-3434, www.avianca.com) flights departing from the Puente Aereo (not from the main terminal of the adjacent international airport). All other domestic carriers: **LAN Colombia** (Colombian toll-free tel. 01\/800-094-9490, www.lan.com), **Viva Colombia** (tel. 1\/489-7989, www.vivacolombia.com.co), **EasyFly** (tel. 1\/414-8111, www.easyfly.com.co), **Satena** (Colombian toll-free tel. 01\/800-091-2034, www.satena.com), and **Copa** (tel. 01\/800-011-0808, www.copaair.com) do depart from the new domestic wing of the international airport.\n\nFor some destinations (namely Leticia in Amazonia), the Pacific Coast destinations of Bah\u00eda Solano and Nuqu\u00ed, La Macarena (Ca\u00f1o Cristales) in Los Llanos, and San Andr\u00e9s and Providencia in the Caribbean, the only viable way to get there is by air.\n\nIf you plan to fly to Caribbean destinations such as Cartagena, San Andr\u00e9s, Providencia, and Santa Marta during high tourist season, be sure to purchase your ticket well in advance, as seats quickly sell out and prices go through the roof. If your destination is Cartagena or Santa Marta, be sure to check fares to Barranquilla. These may be less expensive, and that city is only about an hour away. Similarly, if you plan to go to the Carnaval de Barranquilla in February, check fares to both Cartagena and Santa Marta. If you are flying to the Coffee Region, inquire about flights to Pereira, Armenia, and Manizales, as the distances between these cities are short. The Manizales airport, however, is often closed due to inclement weather.\n\nMedell\u00edn has two airports: **Aeropuerto Internacional Jos\u00e9 Mar\u00eda C\u00f3rdova** (in Rionegro) and **Aeropuerto Olaya Herrera.** All international flights and most large airplane flights depart from Rionegro, a town about an hour away from Medell\u00edn. The airport is simply referred to as \"Rionegro.\" **Satena** (Colombian toll-free tel. 01\/800-091-2034, www.satena.com) and **Aerol\u00edneas Antioque\u00f1as-ADA** (Colombian toll-free tel. 01\/800-051-4232, www.ada-aero.com) use the Olaya Herrera airport, which is conveniently located in town. This is a hub for flights to remote communities in the western and Pacific Region, including Acand\u00ed and Capurgan\u00e1 near the Panamanian border.\n\nThere are often strict weight restrictions for flights to Providencia from San Andr\u00e9s and generally on small planes, such as on the military-owned Satena airline.\n\n#### **BY BUS**\n\nThe vast majority of Colombians travel by bus. This is the money-saving choice and often the only option for getting to smaller communities. There are different types of buses, from large coaches for long-distance travel to _colectivos_ for shorter distances. _Colectivos_ (minivans) are often much quicker, although you won't have much leg room. There are also shared taxi options between many towns.\n\nWhen you arrive at a bus station with guidebook in hand and backpack on, you will be swarmed by touts barking out city names to you, desperately seeking your business on their bus. You can go with the flow and follow them, or, if you prefer a little more control and calm, you can instead walk past them to the ticket booths. Forge ahead and shake your head while saying _\"gracias.\"_\n\nBe alert and aware of your surroundings and of your possessions when you arrive at bus stations, are waiting in the bus terminal, and are onboard buses. Try to avoid flashing around expensive gadgets and cameras while onboard. During pit stops along the way, be sure to keep your valuables with you at all times. Don't feel obliged to tell anyone your life story, where you are off to, or where you are staying. At the same time, most Colombians are extremely friendly and are simply just curious about foreigners visiting their country.\n\nDuring most bus rides of more than a few hours' length, you will be subjected to violent films that the bus companies apparently find appropriate to be shown to small children. Earplugs, eyemasks, and even sleeping pills available at most pharmacies for those super-long journeys may come in handy, but make sure your possessions are well-guarded.\n\nBus drivers like to drive as fast as possible, and generally have few qualms about overtaking cars even on hairpin curves. You're better off not paying too much attention to the driving\u2014it will do you no good! Large buses tend to be safer than smaller ones, although they may not go as fast.\n\nDuring major holidays, purchase bus tickets in advance if you can, as buses can quickly fill up.\n\nBuses may be stopped by police, and you may be required to show or temporarily hand over your passport (keep it handy). Sometimes all the passengers may be asked to disembark from the bus so that the police can search it for illegal drugs or other contraband. Young males may be given a pat-down. Even if it annoys you, it is always best to keep one's cool and remain courteous with police officers who are just doing their job.\n\nWithin cities, traveling by bus is often the easiest way to get around. Many cities, such as Medell\u00edn, Cali, Armenia, Bucaramanga, Pereira, Barranquilla, and (soon) Cartagena, have adopted the Bogot\u00e1 rapid bus system model of the TransMilenio.\n\n#### **BY CAR, MOTORCYCLE, OR BICYCLE**\n\nDriving in Colombia is generally a poor idea for tourists. Roads are often in a poor state and are almost always just two lanes, speed limits and basic driving norms are not respected, driving through large towns and cities can be supremely stressful, signage is poor, sudden mudslides can close roads for hours on end during rainy seasons, and roads can be unsafe at night.\n\nOne exception is the Coffee Region. Here the roads are excellent, often four lanes, distances are short, and traffic is manageable. If you are planning on spending some time visiting coffee farms and idyllic towns, this might be a good option.\n\nAnother region where renting a car may make sense is in Boyac\u00e1. Here the countryside is beautiful and traffic is manageable.\n\nThere are car rental offices in all the major airports in the country. **Hertz** (tel. 1\/756-0600, www.rentacarcolombia.co) and the national **Colombia Car Rental** (U.S. tel. 913\/368-0091, www.colombiacarsrental.com) are two with various offices nationwide.\n\nTouring Colombia on motorcycle is an increasingly popular option. One of the best motorcycle travel agencies in the country is **Motolombia** (tel. 2\/392-9172, www.motolombia.com), based in Cali. Many motorcyclists take the Pan-American Highway through Colombia to Ecuador to continue on a once-in-a-lifetime South American bike tour.\n\nBicyclists will not get much respect on Colombian roads, and there are rarely any bike lanes of significance. In Santander and in Boyac\u00e1 the scenery is absolutely spectacular, but, especially in Santander, it is often quite mountainous. In the Valle de Cauca, around Buga and towards Roldanillo, the roads are good and flat! Staff at **Colombian Bike Junkies** (cell tel. 316\/327-6101, www.colombianbikejunkies.com), based in San Gil, are experts on biking throughout the country.\n\nEvery Sunday in cities across Colombia thousands of cyclists (joggers, skaters, and dog walkers, too) head to the city streets for some fresh air and exercise. This is the **Ciclov\u00eda** , an initiative that began in Bogot\u00e1, when city streets are closed to traffic. Except in Bogot\u00e1, it may be difficult to find a bike rental place, but you can still head out for a jog. _Ciclorutas_ (bike paths) are being built in the major cities as well, and Bogot\u00e1 has an extensive _cicloruta_ network. Again, cyclists don't get much respect from motorists, so be careful!\n\n#### **BY BOAT**\n\nIn some remote locations in Colombia the most common way to get around is by _lancha_ or boat. Many of the isolated villages and beaches and the Parque Nacional Natural Utr\u00eda along the Pacific Coast are accessed only by boat from either Bah\u00eda Solano or Nuqu\u00ed. The same goes for Isla Gorgona. To get to this island park, you normally have to take a boat from Guapi or from Buenaventura. All hotels or travel agencies can organize these trips for you.\n\nAlthough there are some flights from Medell\u00edn to the Darien Gap village of Capurgan\u00e1, it is often more convenient to either fly to the town of Acand\u00ed and take a boat onward to Capurgan\u00e1 or take a boat from the grubby town of Turbo. Waters can be rough, especially from November to March.\n\nIn the Amazon region, the only way to get from Leticia to attractions nearby, including Puerto Nari\u00f1o and the ecolodges on the R\u00edo Yavar\u00ed, is by a boat on the Amazon, which is a memorable experience. All boats leave from the _malec\u00f3n_ (wharf) in Leticia.\n\nThe island resorts off of the coast of Cartagena are accessed only by boat from the Muelle Tur\u00edstico near the Old City. If visiting Mompox from Cartagena, you'll need to take a ferry from the town of Magangu\u00e9 along the mighty R\u00edo Magdalena.\n\n### **Visas and Officialdom**\n\n#### **PASSPORTS AND VISAS**\n\nU.S. and Canadian citizens do not need a passport for visits to Colombia of under 90 days. Upon arrival, specify if you plan to spend more than 60 days in the country, as immigration officials may automatically stamp a 60-day permission (instead of 90). You could be asked to show a return ticket.\n\nThere is an exit tax of about US$66, divided into two categories (the Tasa Aeroportuaria and the Timbre Aeroportuario), that must be paid in cash in Colombian pesos or U.S. dollars upon departure from Colombian airports. Those who stay fewer than 30 days may be eligible for an exemption of one of the taxes. If you are exempt from one of the taxes, you may be sent at the airport to the airport authority office for an exemption stamp. Sometimes the taxes are included in the price of the airline ticket.\n\nTo renew a tourist visa, you must go to an office of **Migraci\u00f3n Colombia** (www.migracioncolombia.gov.co) to request an extension of another 90 days. Some travelers prefer to make a \"visa run\" to C\u00facuta, get their Colombian exit stamp, cross into Venezuela, and return the same day.\n\n#### **CUSTOMS**\n\nUpon arrival in Colombia, bags will be spot checked by customs authorities. Duty-free items up to a value of US$1,500 can be brought in to Colombia. Firearms are not allowed into the country, and many animal and vegetable products are not allowed. If you are carrying over US$10,000 in cash you must declare it.\n\nDeparting Colombia, expect thorough security checks with police looking for illegal drugs. They also may look for pre-Columbian art and exotic animals.\n\n#### **EMBASSIES AND CONSULATES**\n\nThe **United States Embassy** (Cl. 24 Bis No. 48-50, Gate One, tel. 1\/275-4900, <http:\/\/bogota.usembassy.gov>) is in Bogot\u00e1. In case of an emergency, contact the U.S. Citizen Services Hotline (tel. 1\/275-2000). Non-emergency calls are answered at the American Citizen Services Section from Monday through Thursday 2pm-4pm. To be informed of security developments or emergencies during your visit, you can enroll in the Smart Traveler Enrollment Program (STEP) on the U.S. Embassy website. In Barranquilla, there is a **Consular Agency Office** (Cl. 77B No. 57-141, Suite 511, tel. 5\/369-0419, 8am-noon Mon.-Fri.).\n\nThe **Canadian Embassy** (Cra. 7 No. 114-33, Piso 14, tel. (57-1) 657-9800, www.canadainternational.gc.ca\/colombia-colombie) is in Bogot\u00e1. There is a **Canadian Consular Office** (Bocagrande Edificio Centro Ejecutivo Oficina 1103, Cra. 3, No. 8-129, tel. 5\/665-5838) in Cartagena. For emergencies, Canadian citizens can call the emergency hotline in Canada collect (Can. tel. 613\/996-8885).\n\n### **Tips for Travelers**\n\n#### **ACCESS FOR TRAVELERS WITH DISABILITIES**\n\nOnly international and some national hotel chains offer rooms (usually just one or two) that are wheelchair accessible. Hostels and small hotels in secondary cities or towns will not. Airport and airline staff will usually bend over backwards to help those with disabilities, if you ask.\n\nGetting around cities and towns is complicated, as good sidewalks and ramps are the exception, not the rule. Motorists do not stop\u2014or even slow down\u2014for pedestrians.\n\n#### **WOMEN TRAVELING ALONE**\n\nAlong the Caribbean and Pacific coasts especially, women traveling alone should expect to be on the receiving end of flirts and various friendly offers by men and curiosity by everyone. In other parts of the country this is not as prevalent. Women should be extra cautious in taxis and buses. Always order taxis by phone and avoid taking them alone at night. While incidents are unlikely, it is not a fantastic idea to go out for a jog, a walk on a remote beach, or a hike through the jungle on your own. Walking about small towns at night alone may elicit looks or comments. Don't feel obliged to reveal your life story, where you are staying, or where you are going to inquisitive strangers. There have been incidents in the past with single women travelers in remote areas of La Guajira.\n\ngay pride flag in Bogot\u00e1's Plaza de Bol\u00edvar\n\n#### **GAY AND LESBIAN TRAVELERS**\n\nIn urban areas, especially in Bogot\u00e1, there is wide acceptance (or at least tolerance) of gays and lesbians, except for perhaps some of the poor neighborhoods. But public displays of affection between same-sex couples will generally get stares everywhere.\n\nThere is a huge gay community in Bogot\u00e1 (with the epicenter of gay life the Chapinero area) and gay nightlife scenes in all the other major cities. Bars and clubs are usually quite mixed with gay men and lesbians.\n\nIt may be more of an annoyance than anything else, but cab drivers will routinely ask foreign men what they think of Colombian women and will suggest that they should \"get\" one. (Some will offer to help.) A word or two about the beauty of women from Medell\u00edn or Cali is usually a good response.\n\nGay men should be cautious in nightclubs and on online dating service sites, as persuasive thieves may use their seduction skills to get the chance to steal stuff back at your hotel room! Always keep an eye on your drinks at nightclubs, don't accept drinks or sips from strangers, and don't take cabs off the street, especially in front of clubs late at night.\n\nSame-sex couples should not hesitate to insist on _matrimonial_ (double) beds at hotels. Most hotels in the cities and even in smaller towns and rural areas are becoming more clued in on this.\n\n### **Health and Safety**\n\n#### **VACCINATIONS**\n\nThere are no vaccination requirements for travel to Colombia. However, for certain regions, such as Amazonia and the Sierra Nevada de Santa Marta, it is officially recommended to carry proof of vaccination against yellow fever. You may be required to show this upon entry to the Parque Nacional Natural Tayrona. If traveling onwards to Brazil, you may be required to present proof of yellow fever vaccination. The yellow fever vaccination may not be recommended for persons who are HIV positive, for pregnant women, for children, or for others with weakened immune systems.\n\nThe Centers for Disease Control (CDC) recommends that travelers to Colombia get up-to-date on the following vaccines: measles-mumps-rubella (MMR), diphtheria-tetanus-pertussis, varicella (chickenpox), polio, and the yearly flu shot.\n\n#### **MEDICAL SERVICES**\n\nColombia has excellent hospitals in its major cities. Sixteen hospitals in Colombia (in Bogot\u00e1, Medell\u00edn, Bucaramanga, and Cali) were listed in the _Am\u00e9rica Econom\u00eda_ magazine listing of the top 40 hospitals of Latin America. Four hospitals were in the top 10. Those were the **Fundaci\u00f3n Santa Fe de Bogot\u00e1** (www.fsfb.org.co), the **Fundaci\u00f3n Valle del Lili** (www.valledellili.org) in Cali, the **Fundaci\u00f3n Cardioinfantil** (www.cardioinfantil.org) in Bogot\u00e1, and the **Fundaci\u00f3n Cardiovascular de Colombia** (www.fcv.org) in Floridablanca, near Bucaramanga. For sexual and reproductive health issues, **Profamilia** (www.profamilia.org.co) has a large network of clinics that provide walk-in and low-cost services throughout the country.\n\n**Aerosanidad SAS** (tel. 1\/439-7080, 24-hour hotline tel. 1\/266-2247 or tel. 1\/439-7080, www.aerosanidadsas.com) provides transportation services for ill or injured persons in remote locations of Colombia to medical facilities in the large cities.\n\nTravel insurance is a good idea to purchase before arriving in Colombia, especially if you plan on doing a lot of outdoor adventures. One recommended provider of travel insurance is **Assist Card** (www.assist-card.com). Before taking a paragliding ride or white-water rafting trip inquire to see whether insurance is included in the price of the trip\u2014it should be.\n\n#### **DISEASES AND ILLNESSES**\n\n##### **Malaria, Yellow Fever, and Dengue Fever**\n\nIn low-lying, tropical areas of Colombia, mosquito-borne illnesses such as malaria, dengue fever, and yellow fever are not uncommon.\n\nMalaria is a concern in the entire Amazon region and in lowland areas of Antioquia, Choc\u00f3, C\u00f3rdoba, Nari\u00f1o, and Bol\u00edvar. There is low to no malarial risk in Cartagena and in areas above 1,600 meters. The Colombian Ministry of Health estimates that there are around 63,000 annual cases of malaria in the country, 20 of which result in death. Most at risk are children under the age of 15. Malaria symptoms include fever, headache, chills, vomiting, fatigue, and difficulty breathing. Treatment involves the administration of various antimalarial drugs.\n\nYellow fever is another disease that is transmitted by mosquitoes. Early symptoms of yellow fever are similar to those of malaria and dengue fever: fever and chills, flu-like symptoms, and yellow-colored skin and eyes (jaundice). Every year there are around 20 cases of yellow fever reported in Colombia. It is most commonly contracted in low-lying areas, such as around the Sierra Nevada de Santa Marta, along the R\u00edo Magdalena, and in the Amazon region. There's no specific treatment for yellow fever, but it could involve blood transfusions, dialysis for kidney failure, and intravenous fluids.\n\nThe number of cases of dengue fever in Colombia has grown from 5.2 cases per 100,000 residents in the 1990s to around 18.1 cases per 100,000 in the 2000s. It is another mosquito-borne illness. The most common symptoms of dengue fever are fever; headaches; muscle, bone, and joint pain; and pain behind the eyes. It is fatal in less than 1 percent of the cases. Treatment usually involves rest and hydration and the administration of pain relievers for headache and muscle pain.\n\n###### **PREVENTION**\n\nUse mosquito nets over beds when visiting tropical areas of Colombia. Examine them well before using, and if you notice large holes in the nets request another one. Mosquitos tend to be at their worst at dawn, dusk, and in the evenings. Wear lightweight, long-sleeved, and light-colored shirts, long pants, and socks and keep some insect repellent handy.\n\nDEET is considered effective in preventing mosquito bites, but there are other, less-toxic alternatives.\n\nIf you go to the Amazon region, especially during rainy seasons, take an antimalarial prophylaxis starting 15 days before arrival, continuing 15 days after departing the region. According to the CDC, the recommended chemo prophylaxis for visitors to malarial regions of Colombia is atovaquone-proguanil, doxycycline, or mefloquine. These drugs are available at most pharmacies in Colombia with no prescription necessary.\n\n##### **Altitude Sickness**\n\nThe high altitudes of the Andes, including in Bogot\u00e1 (2,625 meters\/8,612 feet), can be a problem for some. If arriving directly in Bogot\u00e1, or if you are embarking on treks in the Sierra Nevada del Cocuy or in Los Nevados, where the highest peaks reach 5,300 meters (over 17,000 feet), for the first couple of days take it easy and avoid drinking alcohol. Make mountain ascents gradually if possible. You can also take the drug acetazolamide to help speed up your acclimatization. Drinking coca tea or chewing on coca leaves may help prevent _soroche,_ as altitude sickness is called in Colombia.\n\n##### **Traveler's Diarrhea**\n\nStomach flu or traveler's diarrhea is a common malady when traveling through Colombia. These are usually caused by food contamination resulting from the presence of E. coli bacteria. Undercooked meats, raw vegetables, dairy products, and ice are some of the main culprits. If you get a case of traveler's diarrhea, be sure to drink lots of clear liquids and perhaps an oral rehydration solution of salt, sugar, and water.\n\n##### **Tap Water**\n\nTap water is fine to drink in Colombia's major cities, but you should drink bottled, purified, or boiled water in the Amazon, in the Pacific coast, in the Darien Gap region, in La Guajira, and in San Andr\u00e9s and Providencia. If the idea of buying plastic bottles of water upsets you, look for _bolsitas_ (bags) of water. They come in a variety of sizes and use less plastic.\n\n#### **CRIME**\n\nColombia is safe to visit, and the majority of visitors have a wonderful experience in the country. But these remain uncertain times as guerrilla groups continue to fight the military, smaller groups of former paramilitaries ( _bacrim_ ) operate in cities and towns across the country as small drug lords, and dangerous gangs rule many urban areas. It isn't always safe all the time.\n\nThe threat of kidnapping no longer terrorizes Colombians as it did in the 1990s when Colombia earned the unwanted distinction of kidnapping capital of the world, but it continues to be a source of income for illegal armed groups.\n\nWhile kidnapping of foreigners has decreased dramatically, it still happens. In June 2013, a former U.S. Navy Seal, Kevin Scott Sutay, was kidnapped by the FARC as he was trekking (against the advice of the Colombian police and others) through the Amazonian rainforest towards Ecuador. In 2012, a pair of German brothers who were driving across Colombia in their four-wheel-drive vehicle were kidnapped in the Catatumbo region in the eastern department of Cesar by the ELN guerrilla group, who accused them of being spies. A Norwegian was kidnapped in 2013 as he was attempting to cross through the Darien Gap into Panama on foot. A Spanish couple, touring the deserts of La Guajira in their own vehicle, were kidnapped by common criminals near Cabo de la Vela in May 2013 and rescued by the police about a month later.\n\nColombian police\n\nThe areas where these foreigners were kidnapped, with the exception of Cabo de la Vela, are known to be volatile regions where tourists would be wise to contract local drivers. While it is a good idea to get informed on the security situation of areas in Colombia before traveling by checking the U.S. Embassy website (<http:\/\/bogota.usembassy.gov>), some areas are widely known to be iffy for visitors due to the presence of illegal and armed groups. These areas include much of the Amazon region, including the departments of Putumayo, Caquet\u00e1, Guaviare, Vaup\u00e9s, and Amazonas except for around Leticia and Puerto Nari\u00f1o; the southern Llanos (visiting Ca\u00f1o Cristales by plane is OK); the department of Arauca; the Catatumbo region of Cesar and Norte de Santander; most of the Choc\u00f3 department, with the exceptions of Quibd\u00f3 and the coastal tourist areas of Bah\u00eda Solano and Nuqu\u00ed; the Darien Gap region, including the Los Katios park (Capurgan\u00e1 and Sapzurro are considered safe); and parts of Cauca near Santander de Quilichao.\n\nAround tourist attractions in large cities, there is a strong police presence. This is especially true in the Old City of Cartagena and in the Centro Hist\u00f3rico of Bogot\u00e1. Street crime and homelessness are problems in poor neighborhoods and downtown areas of all the major cities after dark.\n\nParticularly in Bogot\u00e1, but in other large cities as well, countless locals and visitors alike have been victims of the taxi crime of _paseo milonario_ (the millionaire's ride). This occurs when you take a cab off the street, and within minutes the driver makes a sudden turn or invents an excuse that he needs to stop. At that point, usually two others will jump into the cab on either side of the victim and will threaten him or her at knife or gunpoint. The victim will then be driven to several ATMs in the city and forced to withdraw large amounts of cash. This crime tragically claimed the life of an American Drug Enforcement Agency agent in 2013 in Bogot\u00e1. Largely due to this crime, which soiled Colombia's reputation abroad, authorities have redoubled their efforts to prevent this crime and prosecute the perpetrators.\n\nPrevention is the key to avoiding becoming a victim. Always order a cab by phone or by the popular and free smartphone application Tappsi (available in most cities). When you order a cab, or have one ordered for you, be sure to jot down the _placas_ (license plate numbers). The operator will give these to you. When the cab arrives, confirm the _placas._ It is especially important to order a cab at night and when leaving upscale restaurants, shopping areas, and nightclubs.\n\nAnother crime to be aware of is poisoning. Poisoning most often occurs in nightclubs, when someone will either poison your drink or offer you a drink that has been poisoned. They will then easily persuade the victim to leave the club with them, and will force him or her to withdraw large sums of money from ATMs or will rob the victim of possessions. Although this is commonly called _burundanga_ poisoning, after a flower called the _borrachero_ whose seeds, when consumed, can render one helpless and in a zombie-like trance, the drug used by most criminals is not from the flower but a potent cocktail of drugs including the anti-anxiety drug lorazepam.\n\n### **Information and Services**\n\n#### **MONEY**\n\n##### **Currency**\n\nColombia's official currency is the peso, which is abbreviated as COP. Prices in Colombia are marked with a dollar sign, but remember that you're seeing the price in Colombian pesos. COP$1,000,000 isn't enough to buy a house in Colombia, but it will usually cover a few nights in a nice hotel!\n\nBills in Colombia are in denominations of $1,000, $2,000, $5,000, $10,000, $20,000, $50,000, and $100,000. Coins in Colombia got a makeover in 2012, so you may see two different versions of the same coin amount. Coins in Colombia are in denominations of $50, $100, $200, $500, and $1,000. The equivalent of cents is _centavos_ in Colombian Spanish.\n\nThe exchange rate for Colombian pesos fluctuates. The best way to make a quick (though imprecise) conversion to U.S. dollars is to take half of the amount and move the decimal three places to the left. Thus, think of COP$10,000 to be around US$5, and COP$20,000 to be US$10.\n\nMost banks in Colombia do not exchange money. For that, you'll have to go to a exchange bank, located in all major cities. There are money changers on the streets of Cartagena, but the street is not the best place for safe and honest transactions!\n\nTravelers checks are not worth the hassle of carrying around anymore, as they are hard to cash. Dollars are rarely accepted, save for high-end hotels or in San Andr\u00e9s. To have cash wired to you from abroad, look for a Western Union office. These are located only in major cities.\n\nCounterfeit bills are a problem in Colombia, and unsuspecting international visitors are often the recipient of them. Bar staff, taxi drivers, and street vendors are the most common culprits of this. It's good to always have a stash of small bills as a preventative measure. Tattered and torn bills will also be passed off to you, which could pose a problem. Try not to accept those.\n\n##### **Consignaciones**\n\n_Consignaciones_ (bank transfers) are a common way to pay for hotel reservations (especially in remote areas such as in Amazonia or in Choc\u00f3), tour packages or guides, or entry to national parks. Frankly it's usually a pain to make these deposits in person, as the world of banking can be confusing for non-Colombians. On the plus side, making a deposit directly into the hotel's bank account provides some peace of mind as it will diminish the need to carry with you large amounts of cash. To make a _consignaci\u00f3n_ you will need to know the recipient's bank account and whether that is a _corriente_ (checking) or _ahorros_ (savings) account, and you will need to show some identification and probably have to provide a fingerprint. Be sure to hold on to the receipt to be able to notify the recipient of your deposit.\n\n##### **ATMs**\n\nThe best way to get cash is to use your bank ATM card. These are almost universally accepted at _cajeros autom\u00e1ticos_ (ATMs) in the country. _Cajeros_ are almost everywhere except in the smallest of towns or in remote areas. Withdrawal fees are relatively expensive, although they vary. You can usually take out up to around COP$300,000-500,000 (the equivalent of around US$150-250) per transaction. Many banks place limits on how much one can withdraw in a day (COP$1,000,000).\n\n##### **Credit and Debit Cards**\n\nCredit and debit card use is becoming more and more prevalent in Colombia; however, online credit card transactions are still not so common except for the major airlines and some of the event ticket companies, such as www.tuboleta.com or www.colboletos.com. When you use your plastic, you will be asked if it's _credito_ (credit) or _debito_ (debit). If using a _tarjeta de credito_ (credit card) you will be asked something like, _\"\u00bfCuantas cuotas?\"_ or _\"\u00bfNumero de cuotas?\"_ (\"How many installments?\"). Most visitors prefer one _cuota_ ( _\"Una, por favor\"_ ). But you can have even your dinner bill paid in up to 24 installments! If using a _tarjeta de debito,_ you'll be asked if it is a _corriente_ (checking) or _ahorros_ (savings) account.\n\n##### **Tipping**\n\nIn most sit-down restaurants, a 10 percent service charge is automatically included in the bill. Wait staff are required to ask you, _\"\u00bfDesea incluir el servicio?\"_ (\"Would you like to include the service in the bill?\"). Many times restaurant staff neglect to ask tourists about the service inclusion. Of course if you find the service to be exceptional, you can leave a little extra in cash. Tipping is not expected in bars, nor is it expected at caf\u00e9s, although tip jars are becoming more common in larger cities.\n\nIt is not customary to tip taxi drivers. But if you feel the driver was a good one, driving safely and was honest, or if he or she made an additional stop for you, waited for you, or was just pleasant, you can always round up the bill (instead of COP$6,200 give the driver COP$7,000 and say _\"Qu\u00e9dese con las vueltas por favor\"_ (\"Keep the change\"). Note that sometimes a tip is already included in the fare for non-Colombian visitors!\n\nIn hotels, usually a tip of COP$5,000 will suffice for porters who help with luggage, unless you have lots of stuff. Tips are not expected, but are certainly welcome, for housekeeping staff.\n\n## **RESOURCES**\n\nSpanish Phrasebook\n\nPRONUNCIATION\n\nBASIC AND COURTEOUS EXPRESSIONS\n\nTERMS OF ADDRESS\n\nTRANSPORTATION\n\nACCOMMODATIONS\n\nFOOD\n\nSHOPPING\n\nHEALTH\n\nCOMMUNICATIONS\n\nAT THE BORDER\n\nAT THE GAS STATION\n\nVERBS\n\nNUMBERS\n\nTIME\n\nDAYS AND MONTHS\n\nSuggested Reading\n\nHISTORY\n\nTHE DRUG WAR AND ARMED CONFLICTS\n\nNATURAL HISTORY\n\nETHNOGRAPHY\n\nARCHITECTURE\n\nTRAVEL\n\nPHOTOGRAPHY AND ILLUSTRATED BOOKS\n\nFICTION\n\nInternet and Digital Resources\n\nACCOMMODATIONS\n\nBIRDING\n\nCARTAGENA\n\nECO-TOURISM\n\nEMBASSIES AND VISAS\n\nENTERTAINMENT, CULTURE, AND EVENTS\n\nHISTORY AND HUMAN RIGHTS ISSUES\n\nLANGUAGE COURSES\n\nMEDELL\u00cdN\n\nNEWS AND MEDIA\n\nTRANSPORTATION\n\nTRAVEL INFORMATION\n\nVOLUNTEERING\n\n### **Spanish Phrasebook**\n\nKnowing some Spanish is essential to visit Colombia, as relatively few people outside the major cities speak English. Colombian Spanish is said to be one of the clearest in Latin America. However, there are many regional differences.\n\nSpanish commonly uses 30 letters\u2014the familiar English 26, plus four straightforward additions: ch, ll, \u00f1, and rr, which are explained in \"Consonants,\" below.\n\n#### **PRONUNCIATION**\n\nOnce you learn them, Spanish pronunciation rules\u2014in contrast to English\u2014don't change. Spanish vowels generally sound softer than in English. ( _Note:_ The capitalized syllables below receive stronger accents.)\n\n##### **Vowels**\n\n**a** like ah, as in \"hah\": _agua_ AH-gooah (water), _pan_ PAHN (bread), and _casa_ CAH-sah (house)\n\n**e** like ay, as in \"may:\" _mesa_ MAY-sah (table), _tela_ TAY-lah (cloth), and _de_ DAY (of, from)\n\n**i** like ee, as in \"need\": _diez_ dee-AYZ (ten), _comida_ ko-MEE-dah (meal), and _fin_ FEEN (end)\n\n**o** like oh, as in \"go\": _peso_ PAY-soh (weight), _ocho_ OH-choh (eight), and _poco_ POH-koh (a bit)\n\n**u** like oo, as in \"cool\": _uno_ OO-noh (one), _cuarto_ KOOAHR-toh (room), and _usted_ oos-TAYD (you); when it follows a \"q\" the **u** is silent; when it follows an \"h\" or has an umlaut, it's pronounced like \"w\"\n\n##### **Consonants**\n\n**b, d, f, k, l, m, n, p, q, s, t, v, w, x, y, z, and ch** pronounced almost as in English; **h** occurs, but is silent\u2014not pronounced at all\n\n**c** like k as in \"keep\": _cuarto_ KOOAR-toh (room), casa KAH-sah (house); when it precedes \"e\" or \"i,\" pronounce **c** like s, as in \"sit\": _cerveza_ sayr-VAY-sah (beer), _encima_ ayn-SEE-mah (atop)\n\n**g** like g as in \"gift\" when it precedes \"a,\" \"o,\" \"u,\" or a consonant: _gato_ GAH-toh (cat), _hago_ AH-goh (I do, make); otherwise, pronounce **g** like h as in \"hat\": _giro_ HEE-roh (money order), _gente_ HAYN-tay (people)\n\n**j** like h, as in \"has\": _Jueves_ HOOAY-vays (Thursday), _mejor_ may-HOR (better)\n\n**ll** like y, as in \"yes\": _toalla_ toh-AH-yah (towel), _ellos_ AY-yohs (they, them)\n\n**\u00f1** like ny, as in \"canyon\": _a\u00f1o_ AH-nyo (year), _se\u00f1or_ SAY-nyor (Mr., sir)\n\n**r** is lightly trilled, with tongue at the roof of your mouth like a very light English d, as in \"ready\": _pero_ PAY-roh (but), _tres_ TRAYS (three), _cuatro_ KOOAH-troh (four)\n\n**rr** like a Spanish r, but with much more emphasis and trill. Let your tongue flap. Practice with _burro_ (donkey), _carretera_ (highway), and Carrillo (proper name), then really let go with _ferrocarril_ (railroad)\n\n_Note:_ The single small but common exception to all of the above is the pronunciation of Spanish **y** when it's being used as the Spanish word for \"and,\" as in \"Ron y Kathy.\" In such case, pronounce it like the English ee, as in \"keep\": Ron \"ee\" Kathy (Ron and Kathy).\n\n##### **Accent**\n\nThe rule for accents, the relative stress given to syllables within a given word, is straightforward. If a word ends in a vowel, an n, or an s, accent the next-to-last syllable; if not, accent the last syllable.\n\nPronounce _gracias_ GRAH-seeahs (thank you), _orden_ OHR-dayn (order), and _carretera_ kah-ray-TAY-rah (highway) with stress on the next-to-last syllable.\n\nOtherwise, accent the last syllable: _venir_ vay-NEER (to come), _ferrocarril_ fay-roh-cah-REEL (railroad), and _edad_ ay-DAHD (age).\n\n**Exceptions** to the accent rule are always marked with an accent sign: (\u00e1, \u00e9, \u00ed, \u00f3, or \u00fa), such as _tel\u00e9fono_ tay-LAY-foh-noh (telephone), _jab\u00f3n_ hah-BON (soap), and _r\u00e1pido_ RAH-pee-doh (rapid).\n\n#### **BASIC AND COURTEOUS EXPRESSIONS**\n\nColombians use many courteous formalities. Whenever approaching anyone for information or some other reason, do not forget the appropriate salutation\u2014good morning, good evening, etc. Standing alone, the greeting _hola_ (hello) can sound brusque.\n\n**Hello.** _Hola._\n\n**Good morning.** _Buenos d\u00edas._\n\n**Good afternoon.** _Buenas tardes._\n\n**Good evening.** _Buenas noches._\n\n**How are you?** Colombians have many ways of saying this: _\u00bfC\u00f3mo est\u00e1s\/como est\u00e1? \u00bfQu\u00e9 hubo\/Qu'hubo? \u00bfC\u00f3mo va\/vas? \u00bfQue tal?_\n\n**Very well, thank you.** _Muy bien, gracias._\n\n**Okay; good.** _Bien._\n\n**Not okay; bad.** _Mal._\n\n**So-so.** _M\u00e1s o menos._\n\n**And you?** _\u00bfY Usted?_\n\n**Thank you.** _Gracias._\n\n**Thank you very much.** _Muchas gracias._\n\n**You're very kind.** _Muy amable._\n\n**You're welcome.** _De nada._\n\n**Goodbye.** _Adi\u00f3s._\n\n**See you later.** _Hasta luego. Chao._\n\n**please** _por favor;_ (slang) _por fa_\n\n**yes** _s\u00ed_\n\n**no** _no_\n\n**I don't know.** _No s\u00e9._\n\n**Just a moment, please.** _Un momento, por favor._\n\n**Excuse me, please (when you're trying to get attention).** _Disculpe._\n\n**Excuse me (when you've made a mistake).** _Perd\u00f3n. Que pena._\n\n**I'm sorry.** _Lo siento._\n\n**Pleased to meet you.** _Mucho gusto._\n\n**How do you say . . . in Spanish?** _\u00bfC\u00f3mo se dice . . . en espa\u00f1ol?_\n\n**What is your name?** _\u00bfC\u00f3mo se llama (Usted)? \u00bfC\u00f3mo te llamas?_\n\n**Do you speak English?** _\u00bfHabla (Usted) ingl\u00e9s? \u00bfHablas ingl\u00e9s?_\n\n**Does anyone here speak English?** _\u00bfHay alguien que hable ingl\u00e9s?_\n\n**I don't speak Spanish well.** _No hablo bien el espa\u00f1ol._\n\n**Please speak more slowly.** _Por favor hable m\u00e1s despacio._\n\n**I don't understand.** _No entiendo._\n\n**Please write it down.** _Por favor escr\u00edbalo._\n\n**My name is . . .** _Me llamo . . . Mi nombre es..._\n\n**I would like . . .** _Quisiera . . . Quiero . . ._\n\n**Let's go to . . .** _Vamos a . . ._\n\n**That's fine.** _Est\u00e1 bien._\n\n**All right.** _Listo._\n\n**cool, awesome** _ch\u00e9vere, rico, super_\n\n**Oh my god!** _\u00a1Dios m\u00edo!_\n\n**That's crazy!** _\u00a1Qu\u00e9 locura!_\n\n**You're crazy!** _\u00a1Est\u00e1s loca\/o!_\n\n#### **TERMS OF ADDRESS**\n\nWhen in doubt, use the formal _Usted_ (you) as a form of address.\n\n**I** _yo_\n\n**you (formal)** _Usted_\n\n**you (familiar)** _t\u00fa_\n\n**he\/him** _\u00e9l_\n\n**she\/her** _ella_\n\n**we\/us** _nosotros_\n\n**you (plural)** _Ustedes_\n\n**they\/them** _ellas_ (all females); _ellos_ (all males or mixed gender)\n\n**Mr., sir** _se\u00f1or_\n\n**Mrs., madam** _se\u00f1ora_\n\n**miss, young lady** _se\u00f1orita_\n\n**wife** _esposa_\n\n**husband** _esposo_\n\n**friend** _amigo\/a_\n\n**girlfriend\/boyfriend** _novia_ (female); _novio_ (male)\n\n**partner** _pareja_\n\n**daughter; son** _hija; hijo_\n\n**brother; sister** _hermano; hermana_\n\n**mother; father** _madre; padre_\n\n**grandfather; grandmother** _abuelo; abuela_\n\n#### **TRANSPORTATION**\n\n**Where is . . . ?** _\u00bfD\u00f3nde est\u00e1 . . . ?_\n\n**How far is it to . . . ?** _\u00bfA cu\u00e1nto queda . . . ?_\n\n**from . . . to . . .** _de . . . a . . ._\n\n**How many blocks?** _\u00bfCu\u00e1ntas cuadras?_\n\n**Where (Which) is the way to . . . ?** _\u00bfCu\u00e1l es el camino a . . . ? \u00bfPor d\u00f3nde es...?_\n\n**bus station** _la terminal de buses\/terminal de transporte_\n\n**bus stop** _la parada_\n\n**Where is this bus going?** _\u00bfA d\u00f3nde va este b\u00fas?_\n\n**boat** _el barco, la lancha_\n\n**dock** _el muelle_\n\n**airport** _el aeropuerto_\n\n**I'd like a ticket to . . .** _Quisiera un pasaje a . . ._\n\n**roundtrip** _ida y vuelta_\n\n**reservation** _reserva_\n\n**baggage** _equipaje_\n\n**next flight** _el pr\u00f3ximo vuelo_\n\n**Stop here, please.** _Pare aqu\u00ed, por favor._\n\n**the entrance** _la entrada_\n\n**the exit** _la salida_\n\n**(very) near; far** _(muy) cerca; lejos_\n\n**to; toward** _a_\n\n**by; through** _por_\n\n**from** _de_\n\n**right** _la derecha_\n\n**left** _la izquierda_\n\n**straight ahead** _derecho_\n\n**in front** _en frente_\n\n**beside** _al lado_\n\n**behind** _atr\u00e1s_\n\n**corner** _la esquina_\n\n**stoplight** _la sem\u00e1foro_\n\n**turn** _una vuelta_\n\n**here** _aqu\u00ed_\n\n**somewhere around here** _por aqu\u00ed_\n\n**there** _all\u00ed_\n\n**somewhere around there** _por all\u00e1_\n\n**road** _camino_\n\n**street** _calle, carrera_\n\n**avenue** _avenida_\n\n**block** _la cuadra_\n\n**highway** _carretera_\n\n**kilometer** _kil\u00f3metro_\n\n**bridge; toll** _puente; peaje_\n\n**address** _direcci\u00f3n_\n\n**north; south** _norte; sur_\n\n**east; west** _oriente (este); occidente (oeste)_\n\n#### **ACCOMMODATIONS**\n\n**hotel** _hotel_\n\n**Is there a room available?** _\u00bfHay un cuarto disponible?_\n\n**May I (may we) see it?** _\u00bfPuedo (podemos) verlo?_\n\n**How much is it?** _\u00bfCu\u00e1nto cuesta?_\n\n**Is there something cheaper?** _\u00bfHay algo m\u00e1s econ\u00f3mico?_\n\n**single room** _un cuarto sencillo_\n\n**double room** _un cuarto doble_\n\n**double bed** _cama matrimonial_\n\n**single bed** _cama sencilla_\n\n**with private bath** _con ba\u00f1o propio_\n\n**television** _televisor_\n\n**window** _ventana_\n\n**view** _vista_\n\n**hot water** _agua caliente_\n\n**shower** _ducha_\n\n**towels** _toallas_\n\n**soap** _jab\u00f3n_\n\n**toilet paper** _papel higi\u00e9nico_\n\n**pillow** _almohada_\n\n**blanket** _cobija_\n\n**sheets** _s\u00e1banas_\n\n**air-conditioned** _aire acondicionado_\n\n**fan** _ventilador_\n\n**swimming pool** _piscina_\n\n**gym** _gimnasio_\n\n**bike** _bicicleta_\n\n**key** _llave_\n\n**suitcase** _maleta_\n\n**backpack** _mochila_\n\n**lock** _candado_\n\n**safe** _caja de seguridad_\n\n**manager** _gerente_\n\n**maid** _empleada_\n\n**clean** _limpio_\n\n**dirty** _sucio_\n\n**broken** _roto_\n\n**(not) included** _(no) incluido_\n\n#### **FOOD**\n\n**I'm hungry.** _Tengo hambre._\n\n**I'm thirsty.** _Tengo sed._\n\n**Table for two, please.** _Una mesa para dos, por favor._\n\n**menu** _carta_\n\n**order** _orden_\n\n**glass** _vaso_\n\n**glass of water** _vaso con agua_\n\n**fork** _tenedor_\n\n**knife** _cuchillo_\n\n**spoon** _cuchara_\n\n**napkin** _servilleta_\n\n**soft drink** _gaseosa_\n\n**coffee** _caf\u00e9, tinto_\n\n**tea** _t\u00e9_\n\n**drinking water** _agua potable_\n\n**bottled carbonated water** _agua con gas_\n\n**bottled uncarbonated water** _agua sin gas_\n\n**beer** _cerveza_\n\n**wine** _vino_\n\n**glass of wine** _copa de vino_\n\n**red wine** _vino tinto_\n\n**white wine** _vino blanco_\n\n**milk** _leche_\n\n**juice** _jugo_\n\n**cream** _crema_\n\n**sugar** _az\u00facar_\n\n**cheese** _queso_\n\n**breakfast** _desayuno_\n\n**lunch** _almuerzo_\n\n**daily lunch special** _men\u00fa del d\u00eda_\n\n**dinner** _comida_\n\n**the check** _la cuenta_\n\n**eggs** _huevos_\n\n**bread** _pan_\n\n**salad** _ensalada_\n\n**lettuce** _lechuga_\n\n**tomato** _tomate_\n\n**onion** _cebolla_\n\n**garlic** _ajo_\n\n**hot sauce** _aj\u00ed_\n\n**fruit** _fruta_\n\n**mango** _mango_\n\n**watermelon** _patilla_\n\n**papaya** _papaya_\n\n**banana** _banano_\n\n**apple** _manzana_\n\n**orange** _naranja_\n\n**lime** _lim\u00f3n_\n\n**passionfruit** _maracuy\u00e1_\n\n**guava** _guayaba_\n\n**grape** _uva_\n\n**fish** _pescado_\n\n**shellfish** _mariscos_\n\n**shrimp** _camarones_\n\n**(without) meat** _(sin) carne_\n\n**chicken** _pollo_\n\n**pork** _cerdo_\n\n**beef** _carne de res_\n\n**bacon; ham** _tocino; jam\u00f3n_\n\n**fried** _frito_\n\n**roasted** _asado_\n\n**Do you have vegetarian options?** _\u00bfTienen opciones vegetarianas?_\n\n**I'm vegetarian.** _Soy vegetarian(o)._\n\n**I don't eat . . .** _No como_ . . .\n\n**to share** para compartir\n\n**Check, please.** _La cuenta, por favor._\n\n**Is the service included?** _\u00bfEst\u00e1 incluido el servicio?_\n\n**tip** _propina_\n\n**large** _grande_\n\n**small** _peque\u00f1o_\n\n#### **SHOPPING**\n\n**cash** _efectivo_\n\n**money** _dinero_\n\n**credit card** _tarjeta de cr\u00e9dito_\n\n**debit card** _tarjeta de d\u00e9bito_\n\n**money exchange office** _casa de cambio_\n\n**What is the exchange rate?** _\u00bfCu\u00e1l es la tasa de cambio?_\n\n**How much is the commission?** _\u00bfCu\u00e1nto es la comisi\u00f3n?_\n\n**Do you accept credit cards?** _\u00bfAceptan tarjetas de cr\u00e9dito?_\n\n**credit card installments** _cuotas_\n\n**money order** _giro_\n\n**How much does it cost?** _\u00bfCu\u00e1nto cuesta?_\n\n**expensive** _caro_\n\n**cheap** _barato; econ\u00f3mico_\n\n**more** _m\u00e1s_\n\n**less** _menos_\n\n**a little** _un poco_\n\n**too much** _demasiado_\n\n**value added tax** _IVA_\n\n**discount** _descuento_\n\n#### **HEALTH**\n\n**Help me please.** _Ay\u00fademe por favor._\n\n**I am ill.** _Estoy enferma\/o._\n\n**Call a doctor.** _Llame un doctor._\n\n**Take me to . . .** _Ll\u00e9veme a . . ._\n\n**hospital** _hospital, cl\u00ednica_\n\n**drugstore** _farmacia_\n\n**pain** _dolor_\n\n**fever** _fiebre_\n\n**headache** _dolor de cabeza_\n\n**stomach ache** _dolor de est\u00f3mago_\n\n**burn** _quemadura_\n\n**cramp** _calambre_\n\n**nausea** _n\u00e1usea_\n\n**vomiting** _vomitar_\n\n**medicine** _medicina_\n\n**antibiotic** _antibi\u00f3tico_\n\n**pill** _pastilla, pepa_\n\n**aspirin** _aspirina_\n\n**ointment; cream** _ung\u00fcento; crema_\n\n**bandage (big)** _venda_\n\n**bandage (small)** _cura_\n\n**cotton** _algod\u00f3n_\n\n**sanitary napkin** _toalla sanitaria_\n\n**birth control pills** _pastillas anticonceptivas_\n\n**condoms** _condones_\n\n**toothbrush** _cepillo de dientes_\n\n**dental floss** _hilo dental_\n\n**toothpaste** _crema dental_\n\n**dentist** _dentista_\n\n**toothache** _dolor de muelas_\n\n**vaccination** _vacuna_\n\n#### **COMMUNICATIONS**\n\n**Wi-fi** _wifi_\n\n**cell phone** _celular_\n\n**username** _usuario_\n\n**password** _contrase\u00f1a_\n\n**laptop computer** _port\u00e1til_\n\n**prepaid cellphone** _celular prepago_\n\n**post office** _4-72_\n\n**phone call** _llamada_\n\n**letter** _carta_\n\n**stamp** _estampilla_\n\n**postcard** _postal_\n\n**package; box** _paquete; caja_\n\n#### **AT THE BORDER**\n\n**border** _frontera_\n\n**customs** _aduana_\n\n**immigration** _migraci\u00f3n_\n\n**inspection** _inspecci\u00f3n_\n\n**ID card** _c\u00e9dula_\n\n**passport** _pasaporte_\n\n**profession** _profesi\u00f3n_\n\n**vacation** _vacaciones_\n\n**I'm a tourist.** _Soy turista._\n\n**student** _estudiante_\n\n**marital status** _estado civil_\n\n**single** _soltero_\n\n**married; divorced** _casado; divorciado_\n\n**widowed** _viudado_\n\n**insurance** _seguro_\n\n**title** _t\u00edtulo_\n\n**driver's license** _pase de conducir_\n\n#### **AT THE GAS STATION**\n\n**gas station** _estaci\u00f3n de gasolina_\n\n**gasoline** _gasolina_\n\n**full, please** _lleno, por favor_\n\n**tire** _llanta_\n\n**air** _aire_\n\n**water** _agua_\n\n**oil (change)** _(cambio de) aceite_\n\n**My . . . doesn't work.** _Mi . . . no funciona._\n\n**battery** _bater\u00eda_\n\n**tow truck** _gr\u00faa_\n\n**repair shop** _taller_\n\n#### **VERBS**\n\nVerbs are the key to getting along in Spanish. They employ mostly predictable forms and come in three classes, which end in _ar, er,_ and _ir,_ respectively:\n\n**to buy** _comprar_\n\n**I buy, you (he, she, it) buys** _compro, compra_\n\n**we buy, you (they) buy** _compramos, compran_\n\n**to eat** _comer_\n\n**I eat, you (he, she, it) eats** _como, come_\n\n**we eat, you (they) eat** _comemos, comen_\n\n**to climb** _subir_\n\n**I climb, you (he, she, it) climbs** _subo, sube_\n\n**we climb, you (they) climb** _subimos, suben_\n\nHere are more (with irregularities indicated):\n\n**to do or make** _hacer_ (regular except for _hago,_ I do or make)\n\n**to go** _ir_ (very irregular: _voy, va, vamos, van_ )\n\n**to walk** _caminar_\n\n**to wait** _esperar_\n\n**to love** _amar_\n\n**to work** _trabajar_\n\n**to want** _querer_ (irregular: _quiero, quiere, queremos, quieren_ )\n\n**to need** _necesitar_\n\n**to read** _leer_\n\n**to write** _escribir_\n\n**to send** _enviar_\n\n**to repair** _reparar_\n\n**to wash** _lavar_\n\n**to stop** _parar_\n\n**to get off (the bus)** _bajar_\n\n**to arrive** _llegar_\n\n**to stay (remain)** _quedar_\n\n**to stay (lodge)** _hospedar_\n\n**to rent alquilar**\n\n**to leave** _salir_ (regular except for _salgo,_ I leave)\n\n**to look at** _mirar_\n\n**to look for** _buscar_\n\n**to give** _dar_ (regular except for _doy,_ I give)\n\n**to give (as a present or to order something)** _regalar_\n\n**to carry** _llevar_\n\n**to have** _tener_ (irregular: _tengo, tiene, tenemos, tienen_ )\n\n**to come** _venir_ (irregular: _vengo, viene, venimos, vienen_ )\n\nSpanish has two forms of \"to be\":\n\n**to be** _estar_ (regular except for _estoy,_ I am)\n\n**to be** _ser_ (very irregular: _soy, es, somos, son_ )\n\nUse _estar_ when speaking of location or a temporary state of being: \"I am at home.\" _\"Estoy en casa.\"_ \"I'm happy.\" _\"Estoy contenta\/o.\"_ Use _ser_ for a permanent state of being: \"I am a lawyer.\" _\"Soy abogada\/o.\"_\n\n#### **NUMBERS**\n\n**zero** _cero_\n\n**one** _uno_\n\n**two** _dos_\n\n**three** _tres_\n\n**four** _cuatro_\n\n**five** _cinco_\n\n**six** _seis_\n\n**seven** _siete_\n\n**eight** _ocho_\n\n**nine** _nueve_\n\n**10** _diez_\n\n**11** _once_\n\n**12** _doce_\n\n**13** _trece_\n\n**14** _catorce_\n\n**15** _quince_\n\n**16** _dieciseis_\n\n**17** _diecisiete_\n\n**18** _dieciocho_\n\n**19** _diecinueve_\n\n**20** _veinte_\n\n**21** _veinte y uno_ or _veintiuno_\n\n**30** _treinta_\n\n**40** _cuarenta_\n\n**50** _cincuenta_\n\n**60** _sesenta_\n\n**70** _setenta_\n\n**80** _ochenta_\n\n**90** _noventa_\n\n**100** _cien_\n\n**101** _ciento y uno_\n\n**200** _doscientos_\n\n**500** _quinientos_\n\n**1,000** _mil_\n\n**10,000** _diez mil_\n\n**100,000** _cien mil_\n\n**1,000,000** _mill\u00f3n_\n\n**one half** _medio_\n\n**one third** _un tercio_\n\n**one fourth** _un cuarto_\n\n#### **TIME**\n\n**What time is it?** _\u00bfQu\u00e9 hora es?_\n\n**It's one o'clock.** _Es la una._\n\n**It's three in the afternoon.** _Son las tres de la tarde._\n\n**It's 4 a.m.** _Son las cuatro de la ma\u00f1ana._\n\n**six-thirty** _seis y media_\n\n**quarter till eleven** _un cuarto para las once_\n\n**quarter past five** _las cinco y cuarto_\n\n**hour** _una hora_\n\n**late** _tarde_\n\n#### **DAYS AND MONTHS**\n\n**Monday** _lunes_\n\n**Tuesday** _martes_\n\n**Wednesday** _mi\u00e9rcoles_\n\n**Thursday** _jueves_\n\n**Friday** _viernes_\n\n**Saturday** _s\u00e1bado_\n\n**Sunday** _domingo_\n\n**today** _hoy_\n\n**tomorrow** _ma\u00f1ana_\n\n**yesterday** _ayer_\n\n**day before yesterday** _antier_\n\n**January** _enero_\n\n**February** _febrero_\n\n**March** _marzo_\n\n**April** _abril_\n\n**May** _mayo_\n\n**June** _junio_\n\n**July** _julio_\n\n**August** _agosto_\n\n**September** _septiembre_\n\n**October** _octubre_\n\n**November** _noviembre_\n\n**December** _diciembre_\n\n**week** _una semana_\n\n**month** _un mes_\n\n**after** _despu\u00e9s_\n\n**before** _antes_\n\n**holiday** _festivo_\n\n**long weekend** _puente_\n\n### **Suggested Reading**\n\n#### **HISTORY**\n\nBushnell, David. _The Making of Modern Colombia: A Nation in Spite of Itself._ Berkeley, CA: University of California Press, 1993. Mandatory reading for students of Colombian history. Bushnell, an American, is considered the \"Father of the Colombianists\".\n\nHemming, John. _The Search for El Dorado._ London: Joseph, 1978. Written by a former director of the Royal Geographical Society, this book explores the Spanish gold obsession in the New World. It's a great companion to any visit to the Gold Museum in Bogot\u00e1.\n\nLynch, John. _Sim\u00f3n Bol\u00edvar: A Life._ New Haven, CT: Yale University Press, 2007. This biography of the Liberator is considered one of the best ever written in English, and is the result of a lifetime of research by renowned English historian John Lynch.\n\nPalacios, Marco. _Between Legitimacy and Violence: A History of Colombia, 1875-2002._ Durham, NC: Duke University Press Books, 2006. Written by a Bogotano academic who was a former head of the Universidad Nacional, this book covers Colombia's economic, political, cultural, and social history from the late 19th century to the complexities of the late 20th century, and drug-related violence.\n\n#### **THE DRUG WAR AND ARMED CONFLICTS**\n\nBowden, Mark. _Killing Pablo: The Hunt for the World's Greatest Outlaw._ New York: Grove Press, 2001. This account of U.S. and Colombian efforts to halt drug trafficking and terrorism committed by drug lord Pablo Escobar was originally reported in a 31-part series in _The Philadelphia Inquirer._\n\nDudley, Steven. _Walking Ghosts: Murder and Guerrilla Politics in Colombia._ New York: Routledge Press, 2004. Essential reading for anyone interested in understanding the modern Colombian conflict, this book is written by an expert on investigating organized crime in the Americas.\n\nGonsalves, Marc, Tom Howes, Keith Stansell, and Gary Brozek. _Out of Captivity: Surviving 1,967 Days in the Colombian Jungle._ New York: Harper Collins, 2009. Accounts of three American military contractors who were held, along with former presidential candidate Ingrid Betancourt, by FARC guerrillas for over five years in the Colombian jungle.\n\nLeech, Garry. _Beyond Bogot\u00e1: Diary of a Drug War Journalist in Colombia._ Boston: Beacon Press, 2009. The basis for this book is the author's 11 hours spent as a hostage of the FARC.\n\nOtis, John. _Law of the Jungle: The Hunt for Colombian Guerrillas, American Hostages, and Buried Treasure._ New York: Harper, 2010. This is a thrilling account of the operation to rescue Ingrid Betancourt and American government contractors held by the FARC. It's been called a flip-side to _Out of Captivity._\n\n#### **NATURAL HISTORY**\n\nHilty, Steven L., William L. Brown, and Guy Tudor. _A Guide to the Birds of Colombia._ Princeton, NJ: Princeton University Press, 1986. This massive 996-page field guide to bird-rich Colombia is a must for any serious bird-watcher.\n\nMcMullan, Miles, Thomas M. Donegan, and Alonso Quevedo. _Field Guide to the Birds of Colombia._ Bogot\u00e1: Fundaci\u00f3n ProAves, 2010. This pocket-sized field guide published by ProAves, a respected bird conservation society, is a more manageable alternative to Hilty's guide.\n\n#### **ETHNOGRAPHY**\n\nDavis, Wade. _One River: Explorations and Discoveries in the Amazon Rain Forest._ New York: Simon & Schuster, 1997. From the author of _The Serpent and the Rainbow,_ this is a rich description of the peoples of the Amazonian rain forest, and the result of Davis' time in the country alongside famed explorer Richard Evan Schultes.\n\nReichel-Dolmatoff, Gerardo. _Colombia: Ancient Peoples & Places._ London: Thames and Hudson, 1965. A thorough anthropological investigation of the indigenous cultures across Colombia by an Austrian-born anthropologist who emigrated to Colombia during World War II.\n\n\\----. _The Shaman and the Jaguar: A Study of Narcotic Drugs Among the Indians of Colombia._ Philadelphia: Temple University Press, 1975. An examination of shamanic drug culture in Colombia, particularly among indigenous tribes from the Amazon jungle region.\n\n#### **ARCHITECTURE**\n\nEscovar, Alberto, Diego Obreg\u00f3n, and Rodolfo Segovia. _Gu\u00edas Elarqa de Arquitectura._ Bogot\u00e1: Ediciones Gamma, 2005. Useful guides for anyone wishing to learn more about the architecture of Bogot\u00e1, Cartagena, and Medell\u00edn.\n\n#### **TRAVEL**\n\nLamus, Mar\u00eda Cristina. _333 Sitios de Colombia Que Ver Antes de Morir._ Bogot\u00e1: Editorial Planeta Colombiana, 2010. Colombian version of _1,000 Places to See Before You Die_ (only available in Spanish).\n\nMann, Mark. _The Gringo Trail._ West Sussex: Summersdale Publishers, 2010. A darkly comic tale of backpacking around South America.\n\nNicholl, Charles. _The Fruit Palace._ New York: St. Martin's Press, 1994. A wild romp that follows the seedy cocaine trail from Bogot\u00e1 bars to Medill\u00edn to the Sierra Nevada and a fruit stand called the Fruit Palace during the wild 1980s. The English author was jailed in Colombia for drug smuggling as he conducted research for the book.\n\n#### **PHOTOGRAPHY AND ILLUSTRATED BOOKS**\n\nOften only available in Colombia, coffee table books by Colombian publishers Villegas Editores and the Banco de Occidente are gorgeous, well-done, and often in English. Save room in your suitcase for one or two.\n\nCobo Borda, Juan Gustavo, Gustavo Morales Lizcano, and C\u00e9sar David Mart\u00ednez. _Colombia en Flor._ Bogot\u00e1: Villegas Editores, 2009. This book features fantastic photographs of flowers you will see in Colombia.\n\nDavis, Wade and Richard Evans Schultes. _The Lost Amazon: The Photographic Journey of Richard Evans Schultes._ Bogot\u00e1: Villegas Editores, 2009. A fantastic journey deep into the Amazonian jungle by famed explorer Richard Evans Schultes.\n\nD\u00edaz, Hern\u00e1n. _Cartagena Forever._ Bogot\u00e1: Villegas Editores, 2002. A tiny little book of stunning black-and-white images of the Cartagena of yesteryear, by one of Colombia's most accomplished photographers.\n\nD\u00edaz, Merlano, Juan Manuel and Fernando Gast Harders. _El Choc\u00f3 Biogeogr\u00e1fico de Colombia._ Banco de Occidente Credencial Cali, 2009. A spectacular trip through the unique and biodiverse Choc\u00f3 region.\n\nFreeman, Benjamin and Murray Cooper. _Birds in Colombia._ Bogot\u00e1: Villegas Editores, 2011. Dazzling photographs of native bird species found in Colombia, a veritable birding paradise.\n\nHurtado Garc\u00eda, Andr\u00e9s. _Unseen Colombia._ Bogot\u00e1: Villegas Editores, 2004. Photos and descriptions of the many off-the-beaten-track destinations in the country.\n\nMonta\u00f1a, Antonio and Hans Doering. _The Taste of Colombia._ Bogot\u00e1: Villegas Editores, 1994. A thorough survey of Colombian cuisine by region, with recipes included.\n\nOrtiz Valdivieso, Pedro and C\u00e9sar David Mart\u00ednez. _Orqu\u00eddeas Especies de Colombia._ Bogot\u00e1: Villegas Editores, 2010. Jaw-dropping photos of orchids, from the unusual to the sublime, found in the forests of Colombia.\n\nRivera Ospina, David. _La Amazon\u00eda de Colombia._ Cali: Banco de Occidente Credencial, 2008. An excellent souvenir of your visit to the Amazon region.\n\n\\----. _La Orinoqu\u00eda de Colombia._ Cali: Banco de Occidente Credencial, 2005. One of the least visited areas of Colombia is the R\u00edo Orinoco basin in the Llanos and Amazon regions.\n\nVarious. _Colombia Natural Parks._ Bogot\u00e1: Villegas Editores, 2006. Gorgeous photos from all of Colombia's spectacular national parks.\n\nVillegas, Liliana. _Coffees of Colombia._ Bogot\u00e1: Villegas Editores, 2012. Everything you'd like to know about Colombian coffee in one charming and compact book.\n\nVillegas, Marcelo. _Guadua Arquitectura y Dise\u00f1o._ Bogot\u00e1: Villegas Editores, 2003. Profiles of minimalistic and modern constructions throughout Colombia all made from guadua.\n\n#### **FICTION**\n\nCaballero, Antonio. _Sin Remedio._ Bogot\u00e1: Alfaguara, 2006. A novel about a struggling poet in 1970s Bogot\u00e1 by one of Colombia's best-known columnists.\n\nEspinosa, Germ\u00e1n. _La Tejedora de Coronas._ Bogot\u00e1: Alfaguara, 2002. This novel contains the remembrances of Genovevea Alcocer, who, during the 18th century, leaves her native Cartagena to travel the world. Considered one of the most beautiful books of Latin American literature from the 20th century.\n\nFranco, Jorge. _Rosario Tijeras._ New York: Siete Cuentos, 1999. An acclaimed suspense novel that takes place in violent Medell\u00edn in the 1980s.\n\nGarc\u00eda M\u00e1rquez, Gabriel. _Innocent Er\u00e9ndira and Other Stories._ New York: Harper, 2005. A collection of short stories, including a memorable tale of poor Er\u00e9ndira and her awful grandmother.\n\n\\----. _Love in the Time of Cholera._ New York: Alfred A. Knopf, 1988. A Colombian love and lovesickness story set in a fictitious version of Cartagena.\n\n\\----. _One Hundred Years of Solitude._ New York: Harper, 2006. One of the Nobel Prize-winning author's classic novels, this book tells the story of the Buend\u00eda family from the fictitious town of Macondo.\n\nIsaacs, Jorge. _Mar\u00eda._ Rockville, Wildside Press, 2007. Considered one of the most important Latin American romantic novels of the 19th century, this love story takes place among the sugar cane fields near Cali.\n\nS\u00e1nchez Baute, Alonso. _L\u00edbranos del Bien._ Bogot\u00e1: Alfaguara, 2008. The incredible account of how two friends from the Caribbean coastal city of Valledupar became powerful adversaries.\n\nVallejo, Fernando. _Our Lady of the Assassins._ London: Serpent's Tail, 2001. A novel that takes place in Medell\u00edn, about a gay writer who returns to his violent hometown, where he falls in love with a young contract killer.\n\n### **Internet and Digital Resources**\n\n#### **ACCOMMODATIONS**\n\n**Hostel Trail**\n\n**www.hosteltrail.com**\n\nRun by a Scottish couple living in Popay\u00e1n, this is an excellent resource on hostels throughout South America.\n\n**Posadas Tur\u00edsticas de Colombia**\n\n**www.posadasturisticasdecolombia.gov.co**\n\nFind information on interesting accommodations alternatives, like home stays.\n\n#### **BIRDING**\n\n**ProAves**\n\n**www.proaves.org**\n\nExcellent website for the largest birding organization in the country.\n\n#### **CARTAGENA**\n\n**This is Cartagena**\n\n**www.tic.com**\n\nExperience Cartagena like a local.\n\n#### **ECO-TOURISM**\n\n**Parques Nacionales Naturales de Colombia**\n\n**www.parquesnacionales.gov.co**\n\nColombia's national parks website has information on all of the natural parks and protected areas in the country.\n\n**Aviatur Ecoturismo**\n\n**www.aviaturecoturismo.com**\n\nPackage tours of the Amazon, PNN Tayrona, PNN Isla Gorgona, and more are available from one of Colombia's most respected travel agencies.\n\n**Fundaci\u00f3n Malpelo**\n\n**www.fundacionmalpelo.org**\n\nThis nonprofit organization works to protect Colombia's vast maritime territory, including the Santuario de Flora y Fauna Malpelo.\n\n**Fundaci\u00f3n Natura**\n\n**www.natura.org.co**\n\nThe Fundaci\u00f3n Natura operates several interesting eco-tourism reserves in the country.\n\n#### **EMBASSIES AND VISAS**\n\n**U.S. Embassy in Colombia**\n\n**<http:\/\/bogota.usembassy.gov>**\n\nThe Citizen Services page often has security information for visitors, and is where you can register your visit in case of an emergency.\n\n**Colombian Ministry of Foreign Relations**\n\n**www.cancilleria.gov.co**\n\nOffers information on visas and other travel information.\n\n#### **ENTERTAINMENT, CULTURE, AND EVENTS**\n\n**Vive In**\n\n**www.vive.in**\n\nUpdated information on restaurants, entertainment, and cultural events in Bogot\u00e1.\n\n**Plan B**\n\n**www.planb.com.co**\n\nCompetitor of Vive In, Plan B offers information on what's going on in Bogot\u00e1, Medell\u00edn, and Cali.\n\n**Tu Boleta**\n\n**www.tuboleta.com**\n\nThe top event ticket distributor in the country, Tu Boleta is a good way to learn about concerts, theater, parties, and sporting events throughout Colombia.\n\n**Banco de la Rep\u00fablica**\n\n**www.banrepcultural.org**\n\nInformation on upcoming cultural activities sponsored by the Banco de la Rep\u00fablica in 28 cities in the country.\n\n#### **HISTORY AND HUMAN RIGHTS ISSUES**\n\n**CIA World Factbook Colombia**\n\n**www.cia.gov**\n\nBackground information on Colombia from those in the know.\n\n**Centro de Memoria Hist\u00f3rica**\n\n**www.centrodememoriahistorica.gov.co**\n\nExcellent website on the human toll of the Colombian conflict.\n\n**International Crisis Group**\n\n**www.crisisgroup.org**\n\nIn-depth analysis of the human rights situation in Colombia.\n\n**Colombia Diversa**\n\n**www.colombiadiversa.org**\n\nCovers LGBT rights in Colombia.\n\n#### **LANGUAGE COURSES**\n\n**Spanish in Colombia**\n\n**www.spanishincolombia.gov.co**\n\nOfficial government website on places to study Spanish in Colombia.\n\n#### **MEDELL\u00cdN**\n\n**Medell\u00edn Living**\n\n**www.medellinliving.com**\n\nThis website run by expats is an excellent purveyor of insider information on the City of Eternal Spring.\n\n#### **NEWS AND MEDIA**\n\n**_El Tiempo_**\n\n**www.eltiempo.com**\n\nEl Tiempo is the country's leading newspaper.\n\n**_El Espectador_**\n\n**www.elespectador.com.co**\n\nThis is Colombia's second national newspaper.\n\n**_Revista Semana_**\n\n**www.semana.com**\n\nSemana is the top news magazine in Colombia.\n\n**La Silla Vacia**\n\n**www.sillavacia.com**\n\nPolitical insiders dish about current events.\n\n**Colombia Reports**\n\n**<http:\/\/colombiareports.co>**\n\nColombian news in English.\n\n**_The City Paper Bogot\u00e1_**\n\n**www.thecitypaperbogota.com**\n\nWebsite of the capital city's English-language monthly.\n\n**Colombia Calling**\n\n**www.richardmccoll.com\/colombia-calling**\n\nWeekly online radio program on all things Colombia from an expat perspective.\n\n#### **TRANSPORTATION**\n\n**Moovit**\n\nThis app will help you figure out public transportation in Bogot\u00e1.\n\n**Tappsi**\n\nTo order a safe taxi in Colombia's large cities, first upload this excellent app.\n\n**SITP**\n\n**www.sitp.gov.co**\n\nThis is the official website of the ever-improving (yet confusing) public bus transportation system in Bogot\u00e1.\n\n#### **TRAVEL INFORMATION**\n\n**Colombia Travel**\n\n**www.colombia.travel**\n\nThis is the official travel information website of Proexport, Colombia's tourism and investment promotion agency.\n\n**Pueblos Patrimoniales**\n\n**www.pueblospatrimoniodecolombia.travel**\n\nFind a pueblo that suits your needs at this informative website.\n\n#### **VOLUNTEERING**\n\n**Conexi\u00f3n Colombia**\n\n**www.conexioncolombia.com**\n\nThis website is one-stop shopping for the nonprofit sector in Colombia. Find out how you can help, here.\n\n## **Index**\n\n### **A**\n\nAcademia Colombiana de Historia:\n\nAcu\u00f1a, Luis Alberto:\n\nAerol\u00ednea de Antioquia: ,\n\nAeropuerto Internacional El Dorado:\n\nAfro-Colombian community: , , , ,\n\nagricultural fairs:\n\nair travel: 457-458,\n\nAlborada:\n\nAlfonso Bejarano, Truman David: ,\n\nAlieth Tejido Artesanal: ,\n\nAllan Bay:\n\nalligators:\n\nAlmond Bay:\n\nAlonso Garc\u00e9s:\n\nAlta Guajira: ,\n\naltitude sickness:\n\nAlto Cimiento del Padre:\n\nAlto de Aguacate:\n\nAlto de la Cruz Mirador:\n\nAlto de la Cueva:\n\nAlto del Duende:\n\nAlto de San Andr\u00e9s:\n\nAlto de Segovia:\n\nAlumbrado Navide\u00f1o:\n\n\u00c1lvarez, Petronio:\n\n\u00c1lvaro, Don:\n\nAmazon rainforest: , , 399-419,\n\nAm\u00e9rica:\n\nAmigos de la Monta\u00f1a:\n\namusement parks:\n\nAnapoima:\n\nAndes Mountains: ,\n\nAndr\u00e9s Carne de Res: ,\n\nanimals: 433-437\n\nAntigua Estaci\u00f3n del Tren:\n\nantiques:\n\nAnzo\u00e1tgui, General Jos\u00e9 Antonio Anzo\u00e1tgui:\n\naquariums: ,\n\nArana, Julio C\u00e9sar:\n\narchaeological sights: Camino Real ; Museo Arqueol\u00f3gico (Bogot\u00e1) ; Museo Arqueol\u00f3gico Calima ; Museo Arqueol\u00f3gico de Sogamoso ; Museo Arqueol\u00f3gico La Merced ; Museo Arqueol\u00f3gico Regional Guane ; Museo de Jeric\u00f3 Antioquia ; Parque Nacional Natural Tayrona ; Raudal del Guayabero ; San Agust\u00edn , ; Tierradentro 345-348\n\narepas:\n\nArewol\u00fc Sand Dunes:\n\nArmenia: , 275-279\n\nArtBo:\n\nart galleries: ArtBo ; Banco de la Rep\u00fablica ; Casa Museo Luis Alberto Acu\u00f1a ; Macarena ; Manzana Cultural ; Museo Arquidiocesano de Arte Religioso ; Museo de Antioquia ; Museo de Arte Moderno de Bogot\u00e1 ; Museo de Arte Moderno de Bucaramanga ; Museo de Arte Moderno de Cartagena de Indias ; Museo de Arte Moderno de Medell\u00edn ; Museo de Arte Moderno Eduardo Ram\u00edrez Villamizar ; Museo de Arte Religioso de Guatavita La Antigua ; Museo de Arte Religioso (Jeric\u00f3) ; Museo de Arte Religioso (Popay\u00e1n) ; Museo de Arte Religioso (Santa Fe de Antiqua) ; Museo El Carmen de Arte Religioso\n\nAscenso Torre Colpatria:\n\nASEGUICOC: , ,\n\nAsocaiman: ,\n\nAsociaci\u00f3n Azufral los Andariegos T\u00faquerres: , ,\n\nAsociaci\u00f3n Comunitaria Yarumo Blanco:\n\nAsociaci\u00f3n de Gu\u00edas e Interpretes Ambientales (GAIA):\n\nastronomy: , ,\n\nAstur\u00edas:\n\nAtlantis Plaza:\n\nAtl\u00edtico Nacional:\n\nATMs:\n\nATVs:\n\nAutodefensas Unidas de Colombia:\n\nAvenida Jim\u00e9nez: , 44-47\n\n### **B**\n\nbackpacking: _see_ trekking\n\nBah\u00eda Cispat\u00e1:\n\nBah\u00eda de Cartagena:\n\nBah\u00eda Lodge: ,\n\nBah\u00eda Solano: , ,\n\nBah\u00eda Sonora:\n\nBalones Hurtado:\n\nBaluarte Santo Domingo:\n\nBaluartes de San Ignacio y de San Francisco Javier:\n\nBaluartes de San Lucas y de Santa Catalina:\n\nBanco de la Rep\u00fablica (Bogot\u00e1):\n\nBanco de la Rep\u00fablica (C\u00facuta):\n\nBanco de la Rep\u00fablica (Ibagu\u00e9):\n\nBanco de la Rep\u00fablica (Leticia):\n\nBanco de la Rep\u00fablica (Monter\u00eda):\n\nBanco de la Rep\u00fablica (Santa Marta):\n\nBanda:\n\nbanking: ,\n\nBarichara: , , , 207-212\n\nBarranquilla: , , 119-123\n\nBar\u00fa:\n\nBasalt:\n\nBatalla de las Flores:\n\nBatalla del Pantano de Vargas:\n\nBatalla del Puente de Boyac\u00e1:\n\nBater\u00eda de San Jos\u00e9:\n\nbaths, thermal: , , ,\n\nBavaria brewery:\n\nbeaches: Bah\u00eda Solano ; Cabo de la Vela ; Centro de Visitantes Los Mangles ; Darien Gap ; Dunas de Taroa ; Golfo de Morrosquillo ; Islas de Rosario , ; La Miel ; Nuqu\u00ed ; Ojo del Agua ; Palomino , ; Parque Nacional Natural Tayrona , , ; Playa Almejal ; Playa Blanca (Bar\u00fa) ; Playa Blanca (Lago Tota) ; Playa Blanca (Parque Nacional Natural Utr\u00eda) ; Playa de Pil\u00f3n ; Providencia , , , 390-391; Reserva Natural R\u00edo Claro ; San Andr\u00e9s , , , ; San Antero ; Santa Marta , , ; Sapzurro , ; South Pacific Coast , , ; Spratt Bight , ; Taganga ,\n\nbeauty contests:\n\nBelalc\u00e1zar:\n\nBennett, Dr. Sara:\n\nbest-of itinerary: 15-22\n\nBiblioteca EPM:\n\nBiblioteca Espa\u00f1a: ,\n\nBiblioteca Luis \u00c1ngel Arango:\n\nBiblioteca P\u00fablica La Casa del Pueblo de Guanacas:\n\nBiblioteca P\u00fablica Julio P\u00e9rez Ferrero:\n\nBiblioteca Tintal:\n\nBiblioteca Virgilio Barco: ,\n\nBig Pond:\n\nbiking: Bogot\u00e1 61-62; Cali ; Cartagena ; Cerro de Monserrate ; Convento del Santo Ecce Homo ; C\u00facuta ; Desierto de Tatacoita ; Manizales ; Medell\u00edn ; Parque Arv\u00ed ; Pereira ; Salento ; Santa Marta ; transportation ; Villa de Leyva\n\nbiodiversity:\n\nbird-watching: Amazon rainforest ; Buga preserves ; Cali ; coffee region ; El Encanto Andino ; El Roble ; Estaci\u00f3n San Lorenzo ; Hacienda Venecia ; Jard\u00edn ; Jard\u00edn Bot\u00e1nico del Quind\u00edo ; Lagos de Menegua ; Manizales ; Minca ; Parque Municipal Natural Planes de San Rafael ; Parque Nacional Natural Amacayacu ; Parque Nacional Natural Los Nevados ; Parque Nacional Natural Macuira ; Parque Nacional Natural Tayrona ; Reserva Calanoa ; Reserva del R\u00edo Blanco ; Reserva Natural Heliconia ; San Gil ; Santuario de Flora y Fauna Malpelo ; Santuario de Flora y Fauna Ot\u00fan-Quimbaya ; Santuario Flora y Fauna Iguaque ; Sapzurro Reserva Natural Tacarcuna ; Sendero Eco-Cultural El Ri\u00edto ; Sierra Nevada de Santa Marta\n\nBlack Christ:\n\nblowholes:\n\nBlue House: ,\n\nBlue Life:\n\nboating: ,\n\nBocachica:\n\nBocagrande: , , , ,\n\nBogot\u00e1: 29-92; accommodations 65-69; city layout 33-36; entertainment and events 54-58; fly-in destinations from ; food 69-75; highlights ; history ; information and services 76-78; itineraries , ; maps ; music and dance festivals , ; planning tips , ; safety ; shopping 58-61; sights 36-54; sports and recreation 61-65; tours 63-64; transportation 78-81; vicinity of Bogot\u00e1 82-92\n\nBogotazo riots: ,\n\nBol\u00edvar, Sim\u00f3n: independence movement ; _Los Lanceros_ ; Museo Casa de Bol\u00edvar ; Quinta de Bol\u00edvar ; Quinta de San Pedro Alejandrino\n\nborder disputes:\n\nBosque de las Estatuas:\n\nbotanical gardens: _see_ gardens\n\nBoyac\u00e1: , , 164-194\n\nBoyac\u00e1 and the Santanderes: , 160-219; Boyac\u00e1 164-194; highlights ; map ; Norte de Santander 212-219; planning tips ; Santander 194-212\n\nBrazil: ,\n\nbroiled hen:\n\nBucaramanga: ,\n\nBuenaventura: 367-368\n\nBuga: 325-328\n\nBulevar del R\u00edo:\n\nbullfighting:\n\nburro festival:\n\nbus travel: , ; _see also specific place_\n\nbutterflies:\n\n### **C**\n\nCaba\u00f1as Darius: ,\n\nCaba\u00f1as Guicany: ,\n\nCaba\u00f1a Sisuma: ,\n\nCaba\u00f1as Kanwara: ,\n\nCabanita Roja:\n\nCable Aereo:\n\nCabo de la Vela: ,\n\nCaf\u00e9 de Alb\u00e1n:\n\nCaf\u00e9 del Mar: ,\n\nCaf\u00e9 Havana: ,\n\nCaf\u00e9 Quind\u00edo:\n\nCalarc\u00e1:\n\nCali: , 302-323; accommodations 314-317; entertainment and events 308-311; fly-in destinations from ; food 317-320; itinerary ; map ; music and dance festivals , ; sights 305-308; sports and recreation 311-314\n\nCali and Southwest Colombia: , 298-353; Buga 325-328; Cali 302-323; highlights ; map ; Pasto 328-339; planning tips ; Popay\u00e1n 340-353\n\nCali Cartel:\n\nCalle del Empedrado:\n\nCalle del Recuerdos:\n\nCalle del Tiempo Detenido:\n\nCalle Peatonal:\n\nCalle Real (Salento): ,\n\nCalle Real (Santuario):\n\nCaminar Colombia:\n\nCamino Bogot\u00e1:\n\nCamino Herrera:\n\nCamino Real: , ,\n\nCampo Base:\n\nCandelaria district: , ,\n\nCa\u00f1o Cristales: , , 423-425\n\ncanoeing\/kayaking: Buga preserves ; Hacienda La Aurora ; Oyster's Creek Lagoon ; Parque Nacional Natural Utr\u00eda ; Reserva Calanoa ; Reserva Natural Heliconia ; Reserva Natural Marash\u00e1 ; Reserva Natural Palmar\u00ed ; Reserva Natural Viento Solar , ; R\u00edo Tund\u00f3 ; San Andr\u00e9s ; San Gil\n\nCa\u00f1\u00f3n del Chicamocha: , , , ,\n\ncanopy tours: Amazon rainforest ; Reserva Natural Heliconia ; Reserva Natural Palmar\u00ed ; Reserva Natural Tanimboca\n\nCapitolio Nacional:\n\nCapurgan\u00e1: , , , , , ,\n\nCarabobo Norte: ,\n\nCarmen de Apical\u00e1:\n\nCarnaval de Barranquilla: , , ,\n\nCarnaval de Negros y Blancos: ,\n\nCarnaval de Riosucio:\n\n_carriel_ bags:\n\nCartagena: , 97-119\n\nCartagena and the Caribbean Coast: ; Barranquilla 119-123; Cartagena 97-119; geography ; highlights ; itinerary , 24-25; La Guajira 142-149; map ; planning tips , ; Santa Marta 124-142; western Caribbean coast 150-159\n\nCartagena's Old City:\n\ncar travel: ,\n\nCasa Aquileo Parra G\u00f3mez:\n\nCasa de Gregorio:\n\nCasa de la Aduana:\n\nCasa de la Bagatela:\n\nCasa de la Cultura: , ,\n\nCasa de la Moneda:\n\nCasa de la Pola:\n\nCasa del Carnaval:\n\nCasa del Escribano del Rey Don Juan de Vargas:\n\nCasa del Florero:\n\nCasa del Fundador Gonzalo Su\u00e1rez Rend\u00f3n:\n\nCasa del Primer Congreso de las Provincias Unidas de la Nueva Granada: 164-165\n\nCasa de Nari\u00f1o:\n\nCasa Gardeliana:\n\nCasa Juan de Castellanos:\n\nCasa Lila:\n\nCasa Mercado:\n\nCasa Museo Antonio Nari\u00f1o:\n\nCasa Museo Capit\u00e1n Antonio Ricaurte:\n\nCasa Museo Comunitario Juan Vargas:\n\nCasa Museo Edgar Negret and Museo Iberoamericano de Arte Moderno de Popay\u00e1n\u2014MIAMP:\n\nCasa Museo Isle\u00f1a:\n\nCasa Museo Luis Alberto Acu\u00f1a:\n\nCasa Museo Pedro Nel G\u00f3mez:\n\nCasa Museo Rafael N\u00fa\u00f1ez:\n\nCasa Natal del General Santander:\n\nCasa Platypus: ,\n\nCasa Real F\u00e1brica de Licores:\n\nCasa Republicana:\n\nCasa Rodrigo Jim\u00e9nez Mej\u00eda:\n\nCasa Selva y Caf\u00e9: ,\n\nCasa Viena Hostel: ,\n\nCascada Choc\u00f3latal: ,\n\nCascada del Aeropuerto:\n\nCascada El Tigre:\n\nCascada Escalera:\n\nCascadas:\n\nCascadas Cocacola:\n\nCastillo de San Felipe: ,\n\nCastillo Grande:\n\ncattle ranching: , ,\n\nCattleya Ser:\n\nCaverna de los Gu\u00e1charos:\n\nCaverna El Esplendor:\n\ncaves: , , ,\n\nCayo Cangrejo:\n\nCementerio Brit\u00e1nico:\n\nCementerio Central:\n\nCementerio Municipal:\n\nCementerio San Esteban:\n\nCentral Sector, Parque Nacional Natural El Cocuy: ,\n\nCentro Administrativo La Alpujarra:\n\nCentro Andino:\n\nCentro, Cali:\n\nCentro Cultural Gabriel Garc\u00eda M\u00e1rquez:\n\nCentro Cultural Posada Tres Culturas:\n\nCentro de Hidroterapia:\n\nCentro de Interpretaci\u00f3n Ambiental Nat\u00fctama:\n\nCentro de Investigaciones Paleontol\u00f3gicas:\n\nCentro de Memoria, Paz y Reconciliaci\u00f3n:\n\nCentro Deportivo Atanasio Girardot:\n\nCentro de Visitantes Furachiogua:\n\nCentro de Visitantes La Pastora:\n\nCentro de Visitantes Los Mangles:\n\n_centro hist\u00f3rico_ (Popay\u00e1n): ,\n\n_centro hist\u00f3rico_ (Santa Marta):\n\nCentro Internacional: 47-51\n\nCentro, Manizales:\n\nCentro, Medell\u00edn:\n\nceramics:\n\nCerro de Guadalupe:\n\nCerro de las Tres Cruces:\n\nCerro de Monserrate: , , ,\n\nCerro El Morro del Tulc\u00e1n:\n\nCerro Mahoma:\n\nCerro Nutibara:\n\nCerro Tatam\u00e1:\n\nCerro Tojoro:\n\nChapinero:\n\nChapparo, Omar:\n\nChaska Tours:\n\nchildren's activities: Carnaval de Negros y Blancos ; coffee region theme parks ; Gondava ; Hacienda Napoles ; Monkey Island 411-412; Museo de los Ni\u00f1os Colsubsidio ; Parque Nacional del Chicamocha ; Puerto Alegr\u00eda ; Salitre M\u00e1gico ; Zool\u00f3gico de Cali ; Zool\u00f3gico Mateca\u00f1a\n\nChinund\u00faa:\n\nChipre:\n\nChoc\u00f3: , 358-367\n\nChoc\u00f3 Biogeogr\u00e1fico:\n\nchub fish:\n\nchurches: Barichara ; Bas\u00edlica Menor de la Inmaculada Concepci\u00f3n ; Bas\u00edlica Menor La Inmaculada Concepci\u00f3n ; Bas\u00edlica Menor San Juan Bautista ; Bas\u00edlica Menor, Santa Marta ; Bas\u00edlica Nuestra Se\u00f1ora de la Candelaria ; Bas\u00edlica Se\u00f1or de los Milagros ; Bas\u00edlica y Convento de Nuestra Se\u00f1ora de Mongu\u00ed ; Capilla de Jes\u00fas ; Capilla de la Inmaculada ; Capilla de los Dolores ; Capilla de San Antonio ; Capilla de San Antonio de Padua ; Capilla de Santa Barbara ; Capilla de Siecha ; Capilla El Sagrario ; Catedral Bas\u00edlica de Manizales ; Catedral Bas\u00edlica de Nuestra Se\u00f1ora de la Asunci\u00f3n de Popay\u00e1n ; Catedral Bas\u00edlica Menor ; Catedral de la Inmaculada Concepci\u00f3n ; Catedral de la Sagrada Familia ; Catedral de Nuestra Se\u00f1ora de las Mercedes ; Catedral de Nuestra Se\u00f1ora de los Remedios ; Catedral de Pasto ; Catedral de Sal ; Catedral de San Jeronimo ; Catedral de San Pedro ; Catedral de Santa Marta ; Catedral Inmaculada Concepci\u00f3n ; Catedral Metropolitana Nuestra Se\u00f1ora del Carmen ; Catedral Primada ; Catedral San Jos\u00e9 ; Catedral San Laureano ; Catedral San Pedro ; Catedral Santa Clara ; Catedral Santiago de Tunja ; Convento de la Candelaria ; Convento del Santo Ecce Homo ; First Baptist Church ; Iglesia and Convento de las Aguas ; Iglesia de Cristo Rey ; Iglesia de la Concepci\u00f3n ; Iglesia de la Vera Cruz ; Iglesia de Nuestra Se\u00f1ora de la Candelaria ; Iglesia de Piedra ; Iglesia de San Agust\u00edn , ; Iglesia de San Antonio ; Iglesia de San Francisco ; Iglesia de San Ignacio ; Iglesia de San Juan Bautista ; Iglesia de San Pedro Claver , ; Iglesia de Santa B\u00e1rbara (Mompox) ; Iglesia de Santa B\u00e1rbara (Tunja) ; Iglesia de Veracruz ; Iglesia La Ermita (Cali) ; Iglesia La Ermita (Popay\u00e1n) ; Iglesia La Merced ; Iglesia La Tercera ; Iglesia Museo Santa Clara , ; Iglesia Parroquial ; Iglesia San Francisco , ; Iglesia San Ignacio ; Iglesia San Nicol\u00e1s Tolentino ; Iglesia San P\u00edo ; Iglesia Santo Domingo ; Iglesia Santo Toribio ; Inmaculada Concepci\u00f3n ; Nuestra Se\u00f1ora de la Pobreza Catedral ; Parque Acu\u00e1tico ; Pasto ; Plaza Cayzedo ; Popay\u00e1n ; Santuario La Milagrosa ; Santuario Nuestra Se\u00f1ora de las Lajas ; Templo de Santa B\u00e1rbara ; Templo Mar\u00eda Inmaculada ; Tunja ,\n\nCiclopaseo de los Mi\u00e9rcoles:\n\nCicloSalento:\n\nCiclov\u00eda Bogot\u00e1: , ,\n\nCiclov\u00eda Cali:\n\nCiclov\u00eda Manizales:\n\nCiclov\u00eda Medell\u00edn:\n\ncigars:\n\nCity Center, C\u00facuta:\n\nCiudad de Piedra:\n\nCiudad Perdida: , ,\n\ncivil wars:\n\nClaustro de San Agust\u00edn: ,\n\nClaustro San Pedro Claver:\n\nClaver, Pedro:\n\nclimate: , 432-433\n\nclock tower:\n\nclothing:\n\nclothing-optional beaches:\n\nCloud Base Colombia:\n\ncloud forests: , , , 435-436\n\nClub Marina de Guatavita:\n\ncoal:\n\ncoastal itineraries: 24-26\n\ncocaine trafficking: ,\n\nCocoplum:\n\ncoffee farms: Belalc\u00e1zar ; Caf\u00e9 de Alb\u00e1n ; coffee region ; El Roble ; Finca Alas y Raices , ; Finca Don Eduardo , ; Finca La Victoria ; Hacienda Guayabal ; Hacienda Venecia ; industry statistics ; Manizales ; planning tips , ; RECUCA ; Salento ; Villa Martha\n\nColecci\u00f3n de Arte del Banco de la Rep\u00fablica: ,\n\nColecci\u00f3n Numism\u00e1tica:\n\nColegio Mayor de San Bartolom\u00e9:\n\nColombia al Parque:\n\nColombiamoda:\n\nColombian Highlands:\n\nColombia Paragliding: ,\n\nColombia Trek:\n\ncolonial history: 439-441\n\ncolonial towns: 27-28\n\nColor de Hormiga Hostel: ,\n\nComuna 13:\n\nConcavito:\n\nC\u00f3ncavo: ,\n\nConcurso Mundial de la Mujer Vaquera:\n\nConcurso Nacional de Belleza:\n\nCondor de los Andes:\n\ncondors:\n\nConfusion:\n\nCongreso Nacional Gastron\u00f3mico:\n\nConservatorio del Tolima:\n\n_consignaciones_ (bank transfers):\n\nconstitutional history: ,\n\nconsulates:\n\nConvento de la Candelaria: ,\n\nConvento del Santo Ecce Homo: , ,\n\nConvento de San Agust\u00edn:\n\nConvento de San Joaqu\u00edn:\n\nConvento Nuestra Se\u00f1ora de la Candelaria:\n\nCOODETRANS Palmira:\n\nCoonative Brothers:\n\nCOOPMUJERES:\n\nCoralina Isla Boutique: ,\n\nCordillera Central: ,\n\nCordillera Occidental: ,\n\nCove\u00f1as:\n\ncowgirls:\n\ncows, Norman:\n\nCrab Cay:\n\ncrafts: Barichara ; Bogot\u00e1 ; Bucaramanga ; Caf\u00e9 de Alb\u00e1n ; Cali ; _carriel_ bags , ; Cartagena ; Casa Aquileo Parra G\u00f3mez ; Ecoparque Mirador de las Colinas Iluminadas ; Escuela de Artes y Oficios de Santo Domingo ; Festival Aut\u00f3ctono de Danza, Murga y Cuento ; fiber art ; Guapi ; La Loma de la Cruz ; Macedonia ; Manizales ; Market at Silvia ; Museo de Trajes Regionales ; Museo Taminango de Artes y Tradiciones Populares de Nari\u00f1o ; Pasto ; Pueblito Paisa ; Pueblito Patojo ; R\u00e1quira ; San Gil ; San Mart\u00edn de Amacayacu ; Villa de Leyva\n\ncredit cards:\n\ncrime: 465-467\n\nCristales Aventura Tours:\n\nCristalitos:\n\nCristo Negro:\n\ncrocodiles:\n\nC\u00facuta: ,\n\nCueva de Morgan (Morgan's Cave): ,\n\nCueva Indio:\n\nCueva Vaca:\n\nculture: 454-456\n\ncurrency exchange:\n\ncustoms (immigration):\n\n### **D**\n\ndance festivals: general discussion ; Bogot\u00e1 57-58; Carnaval de Barranquilla , ; Festival Aut\u00f3ctono de Danza, Murga y Cuento ; Festival Iberoamericano de Teatro ; Festival Internacional de M\u00fasica Popular Amazonense ; Fiesta de la Virgen Carmen ; Torneo Internacional del Joropo\n\nDarien:\n\nDarien Gap:\n\ndebit cards:\n\ndemographics: 454-455\n\nDengue fever:\n\nDeportivo Cali:\n\nDeportivo Independiente Medell\u00edn:\n\ndeserts: ,\n\nDesfile de Yipao:\n\nDesierto de Tatacoa:\n\nDesierto de Tatacoita:\n\nD\u00eda Internacional de la Pereza:\n\ndictatorship:\n\ndisabilities, access for travelers with:\n\ndiseases:\n\ndiving: Bah\u00eda Solano , ; Capurgan\u00e1 ; Cartagena ; Isla Gorgona ; Providencia , , ; San Andr\u00e9s , , , ; Santuario de Flora y Fauna Malpelo ; Taganga\n\nDivino Ni\u00f1o:\n\ndolphins, river: ,\n\ndomestic flights:\n\nDominican monasteries:\n\nDonde Josefina: ,\n\nDonde Ram\u00f3n:\n\ndress, regional:\n\nDroguer\u00eda Bristol:\n\ndrug trafficking: , , , , ,\n\ndry tropical forests:\n\nDunas de Taroa:\n\n### **E**\n\nearthquakes:\n\nEaster week:\n\nEcodestinos:\n\nEcofiwi Turismo Ecol\u00f3gico:\n\neconomy: , 453-454\n\nEco Parque La Estampilla hike:\n\nEcoparque Los Yarumos:\n\nEcoparque Mirador de las Colinas Iluminadas:\n\nEcoposada Vina de Aldana:\n\nEcosistemas:\n\necotourism: ,\n\nEcotur\u00edsmo Sierra de la Macarena:\n\nEcuador: ,\n\nEdificio Camacho:\n\nEdificio Carr\u00e9:\n\nEdificio Inteligente:\n\nEdificio L\u00f3pez:\n\nEdificio Monserrate:\n\nEdificio V\u00e1squez:\n\nEje Ambiental:\n\nEl Acuario\/La Piscina\/Haynes Cay:\n\nEl Cable: ,\n\nEl Cabrero:\n\nEl Cant\u00edl: ,\n\nEl Cedral:\n\nEl Centro:\n\nEl Cerro de Cristo Rey:\n\nEl Cerro de las Tres Cruces:\n\nEl Cielo: ,\n\nEl Cocuy: ,\n\nEl Dorado myth:\n\nEl Encanto Andino:\n\nEl Estrecho:\n\nEl Faro:\n\nEl Mirador:\n\nEl Museo del Espacio:\n\nEl Nativo: ,\n\nEl Pe\u00f1on: ,\n\nEl Poblado:\n\nEl Pueblito: ,\n\nEl Purutal:\n\nEl Rancho Tolima Termales:\n\nEl Retiro:\n\nEl Roble:\n\nEl Robledal:\n\nEl Silencio:\n\nEl Social Tienda Mixta: ,\n\nEl Sotare\u00f1o: ,\n\nEl Tabl\u00f3n: ,\n\nEl Valle: , 361-364\n\nembassies:\n\nEmblase de Tomin\u00e9:\n\nemeralds: , , ,\n\nEncuentro Mundial del Coleo:\n\nEnjoy the Reef:\n\nenvironmental issues: Amazon deforestation , ; Centro de Interpretaci\u00f3n Ambiental Nat\u00fctama ; Festival de Cine Verde ; Festival de Viajeros Sin Maletas ; Fundaci\u00f3n Maikuchiga ; Hay Festival ; Instituto Humboldt ; Los Llanos ; Paradise Farm ; Puerto Nari\u00f1o ; Reserva Natural Palmar\u00ed ; sustainable seafood ,\n\nErmita del Se\u00f1or del Humilladero:\n\nescalators, Comuna 13:\n\nEscobar, Pablo: , ,\n\nEscuela de Artes y Oficios de Santo Domingo:\n\nEsquina de las Mujeres:\n\nEstaci\u00f3n del Cable:\n\nEstaci\u00f3n Ferrocarril:\n\nEstaci\u00f3n San Lorenzo:\n\nEstaci\u00f3n Septiembre Sea Turtle Hatchery and Release Program: , ,\n\nEstadio Atanasio Girardot:\n\nEstadio Deportivo Cali:\n\nEstadio El Camp\u00edn:\n\nEstadio Metropolitano Roberto Mel\u00e9ndez:\n\nEstadio Palogrande:\n\nEstrella de Agua:\n\nEuropean explorers:\n\nExplora Suesca:\n\nExposici\u00f3n de Ganado Normando:\n\n### **F**\n\nFARC Revolutionaries: Ca\u00f1o Cristales ; growth ; Los Llanos ; peace process , ,\n\nfauna: 433-437\n\nFecomar:\n\nFelipe's Place:\n\nFeria de Artesan\u00edas:\n\nFeria de Cali: ,\n\nFeria de las Flores:\n\nFeria de Manizales:\n\nFeria Internacional de Arte de Bogot\u00e1:\n\nFesticamara:\n\nFestival Aut\u00f3ctono de Danza, Murga y Cuento:\n\nFestival de Astron\u00f3mica de Villa de Leyva:\n\nFestival de Cine de Antioquia:\n\nFestival de Cine Verde:\n\nFestival de Confraternidad Amaz\u00f3nica:\n\nFestival de Jazz:\n\nFestival del Burro:\n\nFestival del Chub:\n\nFestival de Viajeros Sin Maletas:\n\nFestivales al Parque: ,\n\nFestival Folcl\u00f3rico Colombiano:\n\nFestival Iberoamericano de Teatro:\n\nFestival Internacional de Cine de Barichara:\n\nFestival Internacional de Cine de Cartagena de Indias:\n\nFestival Internacional de Jazz de Bogot\u00e1:\n\nFestival Internacional de M\u00fasica:\n\nFestival Internacional de M\u00fasica Popular Amazonense:\n\nFestival Internacional de Poes\u00eda de Medell\u00edn:\n\nFestival Internacional de Tango: ,\n\nFestival Mundial de Salsa: ,\n\nFestival Nacional de la Musica Colombiana:\n\nFestival Petronio \u00c1lvarez: ,\n\nFestival Piraruc\u00fa de Oro:\n\nfestivals: ; _see also specific place_\n\nFHURE Travel:\n\nFiesta de la Virgen Carmen:\n\nFiesta Nacional del Caf\u00e9:\n\nFiestas Cuyabras:\n\nFiestas de San Juan y San Pablo:\n\nFilandia: 286-287\n\nfilm festivals: , ,\n\nFinca Alas y Raices: ,\n\nFinca Don Eduardo coffee tour:\n\nFinca El Ocaso:\n\nFinca La Primavera:\n\nFinca La Victoria:\n\nfireworks:\n\nfish:\n\nfishing: Bah\u00eda Solano ,  Choc\u00f3 ; Lago Tarapoto ; Laguna La Cocha , ; Providencia ; Puerto Colombia ; Reserva Natural Heliconia ; San Andr\u00e9s ; Taganga\n\nfitness:\n\nflea markets:\n\nflora: 433-437\n\nFloridablanca: ,\n\nflower festival:\n\nflower markets:\n\nfolk festivals:\n\nfood: , ,\n\nfootwear:\n\nFort Warwick:\n\nfossils:\n\nFreshwater Bay: 390-391\n\nfruit:\n\nFuente de Lavapi\u00e9s:\n\nFuerte de San Fernando:\n\nFundaci\u00f3n Caguama:\n\nFundaci\u00f3n Maikuchiga:\n\nFundaci\u00f3n Malpelo:\n\nFundaci\u00f3n Puerto Colombia:\n\nFusagasug\u00e1:\n\n### **G**\n\nGardel, Carlos:\n\ngardens: Convento del Santo Ecce Homo ; Hacienda Piedechinche ; Jard\u00edn Bot\u00e1nico (Bogot\u00e1) , ; Jard\u00edn Bot\u00e1nico del Quind\u00edo , 279-280; Jard\u00edn Bot\u00e1nico de Medell\u00edn ; Jard\u00edn Bot\u00e1nico El Gallineral ; Jard\u00edn Bot\u00e1nico Eloy Valenzuela , ; Jard\u00edn Bot\u00e1nico Los Balsos ; Jard\u00edn Bot\u00e1nico San Andr\u00e9s , ; Mundo Amaz\u00f3nico ; Parque de Orquideas del Tequendama\n\ngay nightlife: , ,\n\ngay rights:\n\ngay travelers, tips for:\n\ngeography: 428-432\n\nGetseman\u00ed: , , ,\n\nGirardot:\n\nGir\u00f3n:\n\nGir\u00f3n Chill Out Hotel Boutique: ,\n\nglaciers: ,\n\nGlobos Colombia:\n\ngold: Mompox ; Museo Cultural de Arte Religioso ; Museo del Oro (Bogot\u00e1) ; Museo del Oro Calima ; Museo del Oro Nari\u00f1o ; Museo del Oro Quimbaya ; Museo del Oro Zen\u00fa ; Museo de Oro Tairona\n\nGolfo de Morrosquillo:\n\nGolfo de Urab\u00e1:\n\nGondava:\n\ngovernment: 452-453\n\ngovernment buildings:\n\nGranada: ,\n\nGran Colombia:\n\ngrasslands, tropical:\n\ngratuities:\n\nGringo Mike's: ,\n\nGuaduas:\n\nGuajira Peninsula:\n\nGuane: ,\n\nGuapi:\n\nGuasca:\n\nGuatape: 250-252\n\nguerilla activity: \u00c1lvaro Uribe ; Ca\u00f1o Cristales ; late 20th century ; Los Llanos ; Medell\u00edn , ; modern day 448-450; Norte de Santander\n\nG\u00fcic\u00e1n:\n\ngyms:\n\n### **H**\n\nHacienda Guayabal: ,\n\nHacienda La Aurora: , , , 425-427\n\nHacienda La Esperanza: ,\n\nHacienda Napoles:\n\nHacienda Para\u00edso:\n\nHacienda Piedechinche and Museo de la Ca\u00f1a de Az\u00facar:\n\nHacienda Venecia: ,\n\nHalloween:\n\nhammerhead sharks:\n\nhang gliding:\n\nHay Festival:\n\nhealthcare: , , , 464-465\n\nhens, broiled:\n\nhighland moors: ,\n\nhigh tourist season:\n\nhiking: Bah\u00eda Solano ; Bogot\u00e1 ; Buga preserves ; Caf\u00e9 de Alb\u00e1n ; Cali 311-312; Camino Real ; Ca\u00f1o Cristales ; Caverna de los Gu\u00e1charos ; Cerro de Monserrate ; Cerro El Morro del Tulc\u00e1n ; Ciudad Perdida Trek ; Darien Gap ; Eco Parque La Estampilla ; G\u00fcic\u00e1n ; Hacienda Guayabal ; Hacienda La Aurora ; Lago Tarapoto ; Laguna de Guatavita ; Laguna del Magdalena ; Laguna Verde ; Minca ; Nuqu\u00ed ; Palomino ; P\u00e1ramo de Ocet\u00e1 ; Parque Municipal Natural Planes de San Rafael 295-296; Parque Nacional Natural Chingaza ; Parque Nacional Natural El Cocuy , , 192-193; Parque Nacional Natural Isla Gorgona ; Parque Nacional Natural Los Nevados , , , ; Parque Nacional Natural Macuira ; Parque Nacional Natural Tayrona ; Parque Nacional Natural Utr\u00eda ; Providencia ; Reserva Natural Azufral ; Reserva Natural Palmar\u00ed ; Reserva Natural Viento Solar ; San F\u00e9lix ; San Gil ; Santa Catalina ; Santa Marta ; Santuario de Fauna y Flora Galeras ; Santuario Flora y Fauna Iguaque ; Sapzurro 154-155; Taganga ; Valle de Cocora ; Valle del R\u00edo Ot\u00fan 291-292; Villa de Leyva , ; Volc\u00e1n Cumbal\n\nHip Hop al Parque:\n\nhistory: , 437-452\n\nholidays:\n\n_Homenaje a la Raza:_\n\nhomicides:\n\nHonda: ,\n\nhorseback riding: Cali area ; El Matuy ; Emblase de Tomin\u00e9 ; Hacienda La Aurora ; Hacienda Venecia ; Lagos de Menegua ; Parque Natural Chicaque ; Playa Blanca ; Popay\u00e1n ; Pozo Azul ; Salento , ; San Agust\u00edn ; Santa Fe de Antiqua ; Tobia\n\nhorses, wild:\n\nHostel Caracol: ,\n\nhot-air ballooning:\n\nHotel Continental:\n\nHotel Estelar Estaci\u00f3n:\n\nHotel Ibis: ,\n\nHotel Plaza Mayor: ,\n\nHotel Sirius: , ,\n\nHotel Sofitel Legend Santa Clara: ,\n\nhot springs: Coconuco , ; Complejo Termal ; Santa Rosa de Cabal ; Termales Balneario ; Termales de Coconuco ; Termales de Hotel ; Termales de Santa Rosa de Cabal ; Termales Erika ; Termales Oto\u00f1o ; Termales San Vicente\n\nHoyo Soplador:\n\nhumpback whales: , , ,\n\n### **I**\n\nIbagu\u00e9: 296-297\n\nIguaque: ,\n\nillness:\n\nimmigration: ,\n\nINCIVA:\n\nindependence movement: general discussion ; Batalla del Puente de Boyac\u00e1 ; Cartagena ; Casa de la Pola ; _Los Lanceros_\n\nIndian ambassador:\n\nindigenous communities: history 437-438; Macedonia ; Museo Etnogr\u00e1fico de Leticia ; Peruvian Amazon Company ; Puerto Nari\u00f1o 414-416; Reserva Calanoa ; San Mart\u00edn de Amacayacu ; Taganga ; _see also specific place_\n\nInquisition:\n\nIn Situ: ,\n\nInstituto Humboldt:\n\nIpiales:\n\nIron Wood Hill Trail:\n\nIsaacs, Jorge:\n\nIsla de la Corota:\n\nIsla de los Micos:\n\nIsla Gorgona:\n\nIsla Malpelo:\n\nIsla Murica:\n\nIsla Palma:\n\nIslas del Rosario: , ,\n\nIslas de San Bernardo: ,\n\nItag\u00fci:\n\nitineraries: , 15-28\n\nIza:\n\n### **J**\n\njaguars:\n\nJairo, John:\n\njam:\n\nJard\u00edn: , , , 252-256\n\njazz: ,\n\nJazz al Parque:\n\nJeep tours:\n\nJepirachi Wind Farm:\n\nJeric\u00f3: , 256-258\n\nJesuit Block:\n\nJohnny Cay: ,\n\njoropo dance: ,\n\nJuanchaco:\n\njungle tours:\n\n### **KL**\n\nKaribik:\n\nKasa Guane: ,\n\nKite Colombia:\n\nkite surfing: Cabo de la Vela ; Lago Calima ; San Andr\u00e9s ; Santa Marta\n\nKumanday:\n\nLa Antigua:\n\nLa Caba\u00f1a:\n\nLa Candelaria: 36-44\n\nLa Casa del Libro Total:\n\nLa Cevicher\u00eda: ,\n\nLa Chaquira:\n\nLa Cocina de Pepina: ,\n\nLa Coquerita:\n\nLadrilleros:\n\nLa Eliana: ,\n\nLa Equidad:\n\nLa Esperanza: ,\n\nLa Fogata:\n\nLa Garrucha:\n\nLago Calima:\n\nLago El Correo:\n\nLagos de Menegua:\n\nLago Sochagota:\n\nLago Tarapoto: ,\n\nLago Tota:\n\nLa Guacherna:\n\nLa Guajira: , , 142-149\n\nLaguna de Guatavita: ,\n\nLaguna del Magdalena:\n\nLaguna del Ot\u00fan: , ,\n\nLaguna Encantada:\n\nLaguna Grande de la Sierra:\n\nLaguna Iguaque: , ,\n\nLaguna La Cocha: ,\n\nLagunas de Siecha:\n\nLaguna Verde: ,\n\nLagunillas:\n\nLa Juanita:\n\nLa Loma:\n\nLa Loma de la Cruz:\n\nLa Macarena:\n\nLa Matuna:\n\nLa Merced Complex:\n\nLa Miel: ,\n\nLa Monta\u00f1a:\n\nLa Nevera:\n\nlanguage:\n\nLa Pelota:\n\nLa Piedra Pe\u00f1ol:\n\nLa Piscina:\n\nLa Piscina Natural:\n\nLa Popa:\n\nLa Portada: ,\n\nLas Lajas:\n\n_Las Muralles_ (Cartagena city walls): ,\n\nLas Vueltas:\n\nLatas:\n\nLa Torre al Cielo:\n\nLa Violencia:\n\nLa Vitrola: ,\n\nLaziness Day, International:\n\nlegends: , ,\n\nLe\u00f1os y Mariscos:\n\nLeticia: 403-411\n\nLGBT nightlife: , ,\n\nLGBT rights:\n\nLGBT travelers, tips for:\n\nlibraries: Biblioteca EPM ; Biblioteca Espa\u00f1a ; Biblioteca P\u00fablica Julio P\u00e9rez Ferrero ; Biblioteca P\u00fablica La Casa del Pueblo de Guanacas ; Biblioteca Tintal ; Biblioteca Virgilio Barco ; Bogot\u00e1\n\nliterature: Hay Festival ; M\u00e1rquez, Gabriel Garc\u00eda ,\n\nLlanos Orientales:\n\nLluvia de Semillas:\n\n_Los Lanceros:_\n\nLos Llanos: , 420-427,\n\nLos Pinos:\n\nLost City (Ciudad Perdida):\n\nLuna de Miel: ,\n\n### **M**\n\nMacarena, Sierra de la:\n\nMacarena neighborhood:\n\nMacedonia:\n\nMagdalena Medio: 248-250\n\nmalaria:\n\nMalec\u00f3n R\u00edo Magdalena:\n\nMalpelo: 371-372\n\nMam\u00e1 Lombriz:\n\nmanatees:\n\nManchineel Bay:\n\nManga:\n\nmangroves: , ,\n\nManizales: , 260-267\n\nManzana Cultural: ,\n\nManzana Jesu\u00edtica:\n\nMapalina:\n\nmarijuana cultivation:\n\nMarket at Silvia:\n\nM\u00e1rquez, Gabriel Garc\u00eda: ,\n\nMedell\u00edn: , , , 225-246\n\nMedell\u00edn and the coffee region: , 220-297; coffee region 259-297; highlights ; history ; itinerary 15-16; maps ; Medell\u00edn 225-246; northern and eastern Antioqua 246-252; planning tips , ; southern Antioquia 252-258\n\nMedell\u00edn Cartel: ,\n\nMedia Marat\u00f3n:\n\nmedical care: , , , 464-465\n\nMelgar:\n\nMesa de los Santos:\n\nMesa de Ruitoque: ,\n\nMikey, Fanny:\n\nMillonarios:\n\nMinca: , 132-134\n\nmining: emeralds ; Nemoc\u00f3n ; salt\n\nMirador Nai-pata:\n\nMompox: 117-119\n\nMonasterio Santa Mar\u00eda de la Epifan\u00eda:\n\nMondongo's: ,\n\nmoney: ,\n\nMongu\u00ed:\n\nMonkey Island:\n\nMonodedo:\n\nmonstrance, _La Lechuga:_\n\nMontenegro:\n\nMonter\u00eda:\n\nMontoya, Laura:\n\nMonumento a Cristo Rey, Belalc\u00e1zar:\n\nMonumento a la Dignidad de la Raza U'wa:\n\nMonumento al Esfuerzo:\n\nMonumento a los Colonizadores:\n\nMonumento Cristo Rey, C\u00facuta:\n\nMorgan, Captain Henry:\n\nMorgan's Cave: ,\n\nMorgan's Head: ,\n\nMorromico: ,\n\nmotorcycling: ,\n\nmountain adventures:\n\nmountain biking: Desierto de Tatacoita ; Parque Nacional Natural Los Nevados ; San Gil , ; Taganga\n\nMountain Hostels:\n\nMUA:\n\nMuelle Tur\u00edstico: ,\n\nMuisca people: ,\n\nMundo Mar:\n\nmuseums: Armenia ; Barranquilla 119-120; Bogot\u00e1 , 39-40, , , 43-44, , , 49-50; Bucaramanga ; Cali , ; Cartagena , ; Colecci\u00f3n de Arte del Banco de la Rep\u00fablica , ; Guaduas ; Ibagu\u00e9 ; Medell\u00edn , , ; Jard\u00edn ; Jeric\u00f3 256-257; Mompox ; Museo de Antioquia , , ; Museo de Arte Moderno de Medell\u00edn , ; Museo del Oro (Bogot\u00e1) , , , ; Nueva Guatavita ; Paipa ; Pamplona ; Pasto 329-331; Pereira ; Popay\u00e1n ; Roldanillo ; San Agust\u00edn ; San Andr\u00e9s ; Santa Fe de Antioquia ; Santa Marta ; Sogamoso ; Tunja 178-179; Villa de Leyva , ,\n\n### **N**\n\nnational parks: general discussion ; Parque Nacional del Chicamocha ; Parque Nacional Natural Amacayacu ; Parque Nacional Natural Chingaza , ; Parque Nacional Natural Corales del Rosario y San Bernardo ; Parque Nacional Natural El Cocuy , 186-188, 190-194; Parque Nacional Natural Isla Gorgona , , ; Parque Nacional Natural Los Nevados , , ; Parque Nacional Natural Macuira ; Parque Nacional Natural Old Providence McBean Lagoon , ; Parque Nacional Natural Purac\u00e9 , , ; Parque Nacional Natural Sierra de la Macarena ; Parque Nacional Natural Sierra Nevada de Santa Marta ; Parque Nacional Natural Tatam\u00e1 , ; Parque Nacional Natural Tayrona , , , 134-137; Parque Nacional Natural Utr\u00eda , 364-365; Santuario de Flora y Fauna Malpelo\n\nNaturar-Iguaque:\n\nnaval museums:\n\nNazareth:\n\nNC Arte:\n\nNegret, Edgar:\n\nNeiva:\n\nNemoc\u00f3n: ,\n\nNevado del Ruiz:\n\nNevado del Tolima: ,\n\nNevado Santa Isabel: ,\n\nNick's Place:\n\nNoche de las Luces:\n\nNorman cows:\n\nNorte de Santander: 212-219\n\nnorthern and eastern Antioqua: 246-252\n\nnorthern Bogot\u00e1: ,\n\nNorthern Medell\u00edn:\n\nNudo de los Pastos: ,\n\nNueva Granada:\n\nNueva Guatavita:\n\nN\u00fa\u00f1ez, Rafael:\n\nNuqu\u00ed: , ,\n\n### **O**\n\n_obleas_ (wafers):\n\nObleas Floridablanca:\n\nobservatories: ,\n\nObservatorio Astron\u00f3mico Tatacoa:\n\nOceanario Islas del Rosario:\n\noff-road vehicles:\n\nOjo del Agua:\n\nOld City Cartagena: , , , ,\n\nOnce Caldas:\n\nopera:\n\n\u00f3pera al Parque:\n\norchids: , ,\n\nOrquiderama:\n\nOyster's Creek Lagoon:\n\n### **P**\n\nPacific coast: , 354-372; Choc\u00f3 358-367; geography ; highlights ; history ; itinerary 25-26; maps ; planning tips , 356-358; South Pacific coast 367-372\n\npacking tips:\n\nPaipa:\n\nPaisa culture: ,\n\nPalacio de Gobierno Departamental:\n\nPalacio de Justicia: ,\n\nPalacio de la Cultura Rafael Uribe Uribe:\n\nPalacio de la Inquisici\u00f3n: ,\n\nPalacio de San Carlos:\n\nPalacio de San Francisco:\n\nPalacio Li\u00e9viano:\n\nPalacio Nacional: ,\n\npaleontological sights: ,\n\n_palmas de cera_ (wax palms): , ,\n\nPalm Cays:\n\nPalomino: , 137-138\n\nPamplona: ,\n\nPANACA:\n\nPanama: ,\n\nPanama border crossing:\n\nPan de Az\u00facar: ,\n\nParadise Farm:\n\nparagliding: Condor de los Andes ; Medell\u00edn ; Roldanillo ; San Gil , ; Santander\n\nparamilitary activity: late 20th century ; Medell\u00edn ; modern day ; Norte de Santander ; Uribe's peace process\n\nParamillo del Quind\u00edo: ,\n\nP\u00e1ramo de Ocet\u00e1:\n\nP\u00e1ramo de Romerales:\n\nP\u00e1ramo Extremo:\n\n_p\u00e1ramos_ (highland moors): ,\n\nP\u00e1ramo Trek: ,\n\nparks: Ecoparque Los Yarumos ; Parque Arv\u00ed , , ; Parque Bol\u00edvar , ; Parque Cabal ; Parque Caldas , ; Parque Centenario ; Parque Central (Filandia) ; Parque Central Bavaria ; Parque Chic\u00f3 ; Parque de Jovita ; Parque de la  ; Parque de la Independencia ; Parque de las Luces 230-231; Parque de la Vida ; Parque del Bicentenario ; Parque de los Novios (Bogot\u00e1) ; Parque de los Novios (Santa Marta) ; Parque de Orquideas del Tequendama ; Parque El Bosque ; Parque El Salado ; Parque El Virrey ; Parque Explora ; Parque Garc\u00eda Rovira ; Parque Gran Colombiano ; Parque las Araucarias ; Parque las Nieves ; Parque Las Nubes ; Parque Lineal Ronda del Sin\u00fa ; Parque los Fundadores ; Parque Municipal Natural Planes de San Rafael , ; Parque Nacional: , ; Parque Nacional del Caf\u00e9 ; Parque Natural Chicaque , , ; Parque Natural Regional El V\u00ednculo ; Parque Olaya Herrera ; Parque Padilla ; Parque Principal, Jard\u00edn , ; Parque Regional Johnny Cay: ; Parque Regional Natural Ucumar\u00ed ; Parque Renacamiento ; Parque San Antonio , ; Parque San P\u00edo ; Parque Santander , , ; Parque Sim\u00f3n Bol\u00edvar ; Parque Sucre ; Recinto del Pensamiento ; Santuario de Fauna y Flora Isla de la Corota ; West View\n\nParkway:\n\nParque Acu\u00e1tico:\n\nParqueadero:\n\nParque Arqueol\u00f3gico Alto de Las Piedras:\n\nParque Arqueol\u00f3gico Alto de los \u00cddolos:\n\nParque Arqueol\u00f3gico de San Agust\u00edn:\n\nParque de las Artes:\n\nParques Nacionales: general information , ; _see also specific national park_\n\nparrots:\n\nPasaje Rivas: ,\n\nPascual Guerrero Stadium:\n\nPaseo Bol\u00edvar:\n\nPaseo de Bastidas:\n\nPaseo de la Playa:\n\nPaso de \u00c1ngel:\n\npassports: ,\n\nPasto: , 328-339\n\nPavia: ,\n\nPay\u00e1n, Zambrano:\n\nPeak, The: ,\n\nPeatonal Carabobo:\n\nPe\u00f1\u00f3n de los Muertos:\n\nPereira: 287-291\n\nperforming arts: Bogot\u00e1 ; Festival Iberoamericano de Teatro ; Festival Internacional de Teatro de Manizales\n\nPeruvian Amazon Company:\n\nPescaderito:\n\nPesecar: ,\n\nPico Bol\u00edvar:\n\nPico Crist\u00f3bal Col\u00f3n:\n\nPiedra de Bol\u00edvar:\n\npilgrimage sites: , ,\n\nPil\u00f3n de Az\u00facar:\n\nPi\u00f1as del 44:\n\npiracy: ,\n\n_planchones_ ferries:\n\nPlanetario de Bogot\u00e1:\n\nPlanetario Medell\u00edn:\n\nplane travel: 457-458,\n\nplanning tips: 10-12\n\nplants: 433-437\n\nPlaya Almejal: ,\n\nPlaya Blanca (Bar\u00fa):\n\nPlaya Blanca (Lago Tota):\n\nPlaya Blanca (Parque Nacional Natural Utr\u00eda):\n\nPlaya Cocalito:\n\nPlaya del Muerto:\n\nPlaya de Pil\u00f3n:\n\nPlaya Grande:\n\nPlaya Mecana: ,\n\nPlaya Neguanje:\n\nPlaza Botero:\n\nPlaza Cayzedo:\n\nPlaza de Bol\u00edvar (Armenia):\n\nPlaza de Bol\u00edvar (Bogot\u00e1): , ,\n\nPlaza de Bol\u00edvar (Cartagena): ,\n\nPlaza de Bol\u00edvar (Ibagu\u00e9):\n\nPlaza de Bol\u00edvar (Manizales):\n\nPlaza de Bol\u00edvar (Pereira):\n\nPlaza de Bol\u00edvar (Salamina):\n\nPlaza de Bol\u00edvar (Salento):\n\nPlaza de Bol\u00edvar (Santa Marta):\n\nPlaza de Bol\u00edvar (Tunja):\n\nPlaza de Cuervo:\n\nPlaza de la Aduana:\n\nPlaza de la Concepci\u00f3n:\n\nPlaza de la Libertad:\n\nPlaza de las B\u00f3vedas:\n\nPlaza de los Coches:\n\nPlaza de los Pies Descalzos: ,\n\nPlaza de San Francisco:\n\nPlaza de San Pedro:\n\nPlaza de Santa B\u00e1rbara:\n\nPlaza de Santamar\u00eda:\n\nPlaza de Santo Domingo:\n\nPlaza de Toros:\n\nPlaza Fern\u00e1ndez de Madrid:\n\nPlaza Los Libertadores:\n\nPlaza Mayor: , ,\n\nPlaza Nari\u00f1o:\n\nPlaza Ricaurte:\n\nPlaza San Diego:\n\nPlaza San Nicol\u00e1s:\n\nPlaza Santo Domingo: ,\n\nPlazoleta de la Carmen:\n\nPlazoleta Jairo Varela:\n\npoets: ,\n\npolice, national: ,\n\npolitics: 452-453\n\nPopay\u00e1n: , , 340-353\n\nPosada del Gecko: ,\n\nPosada San Agust\u00edn:\n\nPosada Tur\u00edstica Hostal del Mar: ,\n\nPosada Tur\u00edstica Rocas de Cabo Marzo: , , ,\n\nPozo Azul:\n\nprecipitation:\n\npre-Columbian history: 437-438\n\npresidential palace:\n\nprevention, disease:\n\nprisons:\n\nProAves:\n\nProvidencia: , , 388-395; beaches\n\nPueblito Paisa:\n\nPueblito Patojo:\n\nPuente de Boyac\u00e1:\n\nPuente de Calicanto:\n\nPuente de la Custodia:\n\nPuente de Occidente:\n\nPuente Humilladero:\n\nPuente Navarro:\n\nPuente Ortiz:\n\nPuerto Alegr\u00eda:\n\nPuerto Colombia:\n\nPuerto Nari\u00f1o: , , , 414-417\n\nP\u00falpito del Diablo: ,\n\nPunta Aguja:\n\nPunta Gallinas: ,\n\nPunta Hu\u00edna: ,\n\nPurac\u00e9 volcano:\n\n### **QR**\n\nQuebrada La Vieja:\n\nQuebrada Risaralda:\n\nQuebrada Valencia:\n\nQuimbaya nation:\n\nQuimbaya (town):\n\nQuinta de Bol\u00edvar: ,\n\nQuinta de San Pedro Alejandrino:\n\nrafting:\n\nrainfall:\n\nrainforests: , 433-435; _see also_ Amazon rainforest\n\nRally Aventura Guajira:\n\nrappelling: Caverna El Esplendor ; San Gil ,\n\nR\u00e1quira: ,\n\nRastafarian community:\n\nRaudal del Guayabero:\n\nRayo, Omar:\n\nRecinto del Pensamiento:\n\nRECUCA:\n\nRefugio la Roca:\n\nreggae dancing:\n\nRegi\u00f3n Andina:\n\nreligion:\n\nReserva Acaime: ,\n\nReserva Aguamarina:\n\nReserva Calanoa:\n\nReserva Cuchilla Jard\u00edn Tamesis:\n\nReserva del R\u00edo Blanco:\n\nReserva El Dorado:\n\nReserva Natural Anahuac:\n\nReserva Natural Azufral:\n\nReserva Natural Bosque de Yotoco:\n\nReserva Natural de las Loro Orejiamarillo:\n\nReserva Natural Heliconia:\n\nReserva Natural Laguna de Sonso:\n\nReserva Natural Marash\u00e1:\n\nReserva Natural Palmar\u00ed:\n\nReserva Natural R\u00edo Claro: , ,\n\nReserva Natural Tanimboca: ,\n\nReserva Natural Viento Solar: , ,\n\nRevolutionary Armed Forces of Colombia: _see_ FARC Revolutionaries\n\nRicaurte, Capit\u00e1n Antonio:\n\nR\u00edo Amazonas:\n\nR\u00edo Chicamocha:\n\nR\u00edo Claro canyon:\n\nR\u00edo Fonce:\n\nRiohacha: ,\n\nRioja Travel:\n\nR\u00edo Magdalena:\n\nR\u00edo Negro:\n\nR\u00edo Ot\u00fan Valley:\n\nR\u00edo Pance:\n\nR\u00edo Su\u00e1rez:\n\nRiosucio:\n\nR\u00edo Tund\u00f3:\n\nR\u00edo Yavar\u00ed: , , , 417-419\n\nRitacuba Blanco: ,\n\nRock al Parque:\n\nrock climbing: Parque Nacional Natural El Cocuy ; Refugio La Roca ; Suesca ; Tobia\n\nRojas Pinilla, Gustavo:\n\nRoldanillo:\n\nrubber industry:\n\nRuta Sur: ,\n\n### **S**\n\nSala de Exposiciones de la Banco de la Rep\u00fablica:\n\nSalamina: , , 268-270\n\nSalento: , , 281-284\n\nSalitre M\u00e1gico:\n\nSalmona, Rogelio:\n\nSalom\u00f3n-Lozano Treaty:\n\nSalsa al Parque: ,\n\nsalsa dancing: Cali , , 308-309, ; festivals ; Medell\u00edn\n\nsalt mining:\n\nSalto de Borodones:\n\nSalto del \u00c1ngel:\n\nSalto del Tequendama:\n\nSalto de Morti\u00f1o:\n\nSan Agust\u00edn: , , , , 348-352\n\nSan Andr\u00e9s: 378-388\n\nSan Andr\u00e9s and Providencia: , 373-395; geography ; highlights ; history 374-377; itinerary ; map ; planning tips ; Providencia and Santa Catalina 388-395; San Andr\u00e9s 378-388\n\nSan Andr\u00e9s Unlimited:\n\nSan Antero: ,\n\nSan Antonio: ,\n\nSan Antonio Hotel Boutique: ,\n\nSan Cipriano:\n\nSan F\u00e9lix:\n\nSan Gil: , ,\n\nSan Luis:\n\nSan Mart\u00edn de Amacayacu: , , ,\n\nSan Pabl\u00edn Norte:\n\nSanta Catalina: 388-395\n\nSanta Clara La Real:\n\nSanta Cruz del Islote:\n\nSanta Fe:\n\nSanta Fe de Antiqua:\n\nSantamar\u00eda Market:\n\nSanta Marta: 124-142\n\nSantanderes, the: , 194-219\n\nSanta Rosa de Cabal:\n\nSanta Sof\u00eda:\n\nSantuario:\n\nSantuario de Fauna y Flora Galeras:\n\nSantuario de Fauna y Flora Isla de la Corota:\n\nSantuario de Fauna y Flora Los Flamencos: ,\n\nSantuario de Flora y Fauna Malpelo: , ,\n\nSantuario de Flora y Fauna Ot\u00fan-Quimbaya: ,\n\nSantuario de Monserrate:\n\nSantuario Flora y Fauna Iguaque: , , ,\n\nSantuario La Milagrosa:\n\nSantuario Nuestra Se\u00f1ora de las Lajas:\n\nSapzurro: , , ,\n\nSapzurro Reserva Natural Tacarcuna:\n\nSatena: , ,\n\nseafood: ,\n\nseasons, best travel:\n\nsea turtles:\n\nSemana Santa (Mompox):\n\nSemana Santa (Popay\u00e1n):\n\nSemana Santa (San F\u00e9lix):\n\nSendero del Mosco:\n\nSendero Eco-Cultural El Ri\u00edto:\n\nSe\u00f1or de los Milagro:\n\nServicios Ecotur\u00edsticos G\u00fcic\u00e1ny:\n\nShakira:\n\nSharky's Surf Shop:\n\nShipano\u00fc pools:\n\nSierra de la Macarena:\n\nSierra Nevada del Cocuy: , , , 186-188,\n\nSierra Nevada de Santa Marta: , ,\n\nSmall World Foundation:\n\nsnorkeling: Capurgan\u00e1 ; La Piscina Natural ; Parque Nacional Natural Utr\u00eda ; Providencia ; San Andr\u00e9s , ; Santa Catalina ; Taganga\n\nsoccer: , , ,\n\nSogamoso:\n\nsouthern Antioquia: 252-258\n\nsouthern Bogot\u00e1: 52-54\n\nsouthern Medell\u00edn:\n\nsouth Pacific coast: 367-372\n\nSouthwest Bay (Suroeste):\n\nSouthwest Colombia: _see_ Cali and Southwest Colombia\n\nspace museum:\n\nSpanish conquest: 438-439\n\nSpanish language:\n\nSpanish language courses: ,\n\nspas: ; _see also specific place_\n\nsport-fishing: ; _see also specific place_\n\nSpratt Bight: ,\n\nSpratt Bight Pathway: ,\n\nstargazing: , , ,\n\nSu\u00e1rez, Julio:\n\nSuesca:\n\nsugar industry:\n\nsurfing: , ,\n\nswimming: , ,\n\n### **T**\n\nTaganga:\n\ntango: , , , ,\n\nTanimboca:\n\nTarde de Mar\u00eda La Parda:\n\ntaxi crime:\n\nTayrona: , 134-137\n\nTayrona people:\n\nTeatro Col\u00f3n:\n\nTeatro Heredia:\n\nTeatro Jorge Isaacs:\n\nTeatro Libre:\n\nTeatro Mayor Julio Mario Santo Domingo:\n\nTeatro Metropolitano:\n\ntemperatures: , 432-433\n\nTemplo del 20 de Julio:\n\nTemplo de la Inmaculada Concepci\u00f3n:\n\nTemplo del Congreso:\n\nTemplo de Santa B\u00e1rbara:\n\nTemplo de Santo Domingo de Guzm\u00e1n:\n\nTemplo Mar\u00eda Inmaculada:\n\nTemplo y Convento San Francisco:\n\nTenjo:\n\nTete's Place:\n\ntheme parks:\n\nthermal baths: , ,\n\nThis Is Cartagena:\n\nTicuna villages: ,\n\nTierradentro: , , , 345-348\n\nTierradentro Archeological Park:\n\nTierra Viva:\n\nTinjac\u00e1:\n\ntipping:\n\nTobia:\n\nTodo R\u00e1quira:\n\nTol\u00fa: ,\n\nTorneo Internacional del Joropo: ,\n\nTorneo Internacional de Pesca Deportiva:\n\nTorre Colpatria: ,\n\nTorre Colpatria Observation Deck:\n\nTorre del Reloj: ,\n\nTorre Mud\u00e9jar:\n\nTorres, Jaime:\n\nTorres del Parque:\n\nToti:\n\nTOVYT Tours:\n\ntrain rides:\n\nTralala: ,\n\ntransportation: , 457-461\n\nTransportes Florida: , ,\n\ntraveler's diarrhea:\n\ntrekking: Ciudad Perdida Trek ; Finca La Primavera ; Laguna del Magdalena ; Laguna del Ot\u00fan ; Parque Nacional Natural El Cocuy , , ; Parque Nacional Natural Los Nevados , ; Parque Nacional Natural Tatam\u00e1 ; Santuario Flora y Fauna Iguaque\n\ntropical dry forests: , , ,\n\ntropical grasslands:\n\nTsalickis, Mike:\n\nTumaco:\n\nTunja: , , 176-180\n\nT\u00faquerres:\n\nTuricumbes:\n\nTurtle Rock:\n\nturtles, sea:\n\n### **U**\n\nUltra Mar:\n\nUNESCO Biosphere Reserves:\n\nUNESCO City of Gastronomy:\n\nUNESCO City of Music:\n\nUNESCO World Heritage Sites: , ,\n\nUNESCO World Heritage traditions:\n\nUNIGMA:\n\nUniversidad Aut\u00f3noma de Bucaramanga:\n\nUniversidad de Antioquia:\n\nUniversidad de Los Andes:\n\nUniversidad de Nuestra Se\u00f1ora del Rosario:\n\nUniversidades:\n\nUrban Buddha: ,\n\nUribe, \u00c1lvaro:\n\nUsaqu\u00e9n:\n\nU'wa people:\n\n### **V**\n\nvaccinations: ,\n\nValenzuela Kennler:\n\nValero, Isaias:\n\nValle de Cauca: ,\n\nValle de Cocora: , 284-286\n\nValle del R\u00edo Ot\u00fan:\n\nVargas, Juan:\n\nvegetation: 433-437\n\nVenezuela border crossing: , ,\n\nVereda de Cocora: ,\n\nViaducto C\u00e9sar Gav\u00edria Trujillo:\n\nViajes Cielo & Tierra: ,\n\nVictoria Regia:\n\nVilla de Leyva: , , , , 164-172\n\nVilla del Rosario:\n\nVilla Maga: ,\n\nVillavicencio: , 421-423\n\nVillavieja:\n\nVi\u00f1edo Aim Karim:\n\nviolence, drug-related: ,\n\nVirgen de la Loma:\n\nvisas: ,\n\nvolcanoes: Parque Nacional Natural Los Nevados ; Voladero Las \u00c1guilas ; Volc\u00e1n Azufral ; Volc\u00e1n Cumbal ; Volc\u00e1n Galeras , ; Volc\u00e1n Purac\u00e9\n\nvolunteering:\n\n### **W**\n\nwaterfalls: Bah\u00eda Solano ; Cascada El Tigre ; Cascadas de Juan Cur\u00ed ; El Cielo ; El Duende ; Jard\u00edn , ; La Periquera ; Parque Las Nubes ; Parque Municipal Natural Planes de San Rafael ; Parque Nacional Natural El Cocuy ; Parque Nacional Natural Isla Gorgona ; Quebrada Valencia ; Villa de Leyva ; Reserva Aguamarina ; Reserva Cuchilla Jard\u00edn Tamesis ; R\u00edo Magdalena ; Termales de Hotel\n\nwater safety:\n\nwax palms: , ,\n\nWay\u00fau people:\n\nweather: ,\n\nwestern Bogot\u00e1: 51-52\n\nwestern Caribbean coast: 150-159\n\nWest View:\n\nwhale-watching: general discussion ; Bah\u00eda Solano ; best season ; Buenaventura ; Isla Gorgona ; Nuqu\u00ed ; Pacific Coast ; Parque Nacional Natural Utr\u00eda\n\nWhite City:\n\nwhite-water rafting: R\u00edo Magdalena ; San Gil ; Tobia\n\nwildlife: 433-437\n\nwildlife: Amazon rainforest ; Asocaiman ; Buga preserves ; Choc\u00f3 Biogeogr\u00e1fico ; Estaci\u00f3n Septiembre Sea Turtle Hatchery and Release Program ; Fundaci\u00f3n Maikuchiga ; Hacienda La Aurora ; La Esperanza ; Lagos de Menegua ; Los Llanos ; Parque Nacional Natural Amacayacu ; Parque Nacional Natural Macuira ; Parque Nacional Natural Tayrona ; Parque Regional Natural Ucumar\u00ed ; Reserva Acaime ; Reserva Calanoa ; Reserva del R\u00edo Blanco ; Reserva Natural de las Loro Orejiamarillo ; Reserva Natural Heliconia ; Reserva Natural Palmar\u00ed ; Reserva Natural R\u00edo Claro ; Reserva Natural Viento Solar ; Santuario de Fauna y Flora Galeras ; Santuario de Fauna y Flora Isla de la Corota ; Santuario de Fauna y Flora Los Flamencos ; Santuario de Flora y Fauna Malpelo 371-372; Santuario de Flora y Fauna Ot\u00fan-Quimbaya , ; Santuario Flora y Fauna Iguaque ; Sapzurro Reserva Natural Tacarcuna ; Sierra Nevada de Santa Marta ; Victoria Regia\n\nwindsurfing: ,\n\nwineries: ,\n\nwinter:\n\nwomen's crafts:\n\nwomen travelers:\n\n### **XYZ**\n\nyellow fever:\n\nyoga: Cartagena ; Medell\u00edn ; Palomino ; Reserva Natural Viento Solar ; Santa Marta\n\nYurupary Amazonastours:\n\nZipaquir\u00e1:\n\nZona G:\n\nZona M:\n\nZona Rosa: ,\n\nZona T:\n\nZool\u00f3gico de Cali:\n\nZool\u00f3gico Mateca\u00f1a:\n\nZool\u00f3gico Santa Cruz: \n\n## **List of Maps**\n\n**Front color map**\n\nColombia: 2-3\n\n**Discover Colombia**\n\nchapter divisions map:\n\n**Bogot\u00e1**\n\nBogot\u00e1:\n\nChapinero and Zona G:\n\nAvenida Jim\u00e9nez:\n\nCentro Internacional:\n\nNorthern Bogot\u00e1:\n\nVicinity of Bogot\u00e1:\n\n**Cartagena and the Caribbean Coast**\n\nCartagena and the Caribbean Coast:\n\nCartagena:\n\nOld City: 102-103\n\nBarranquilla:\n\nSanta Marta:\n\nLa Guajira:\n\n**Boyac\u00e1 and the Santanderes**\n\nBoyac\u00e1 and the Santanderes:\n\nTunja:\n\nParque Nacional Natural El Cocuy:\n\nBucaramanga:\n\nSan Gil and Vicinity:\n\nC\u00facuta:\n\n**Medell\u00edn and the Coffee Region**\n\nMedell\u00edn and the Coffee Region:\n\nEl Poblado:\n\nCentro:\n\nManizales:\n\nParque Nacional Natural Los Nevados:\n\nArmenia:\n\nPereira:\n\n**Cali and Southwest Colombia**\n\nCali and Southwest Colombia:\n\nCali:\n\nPasto:\n\nPopay\u00e1n:\n\n**The Pacific Coast**\n\nThe Pacific Coast:\n\n**San Andr\u00e9s and Providencia**\n\nSan Andr\u00e9s:\n\nProvidencia:\n\n**The Amazon and Los Llanos**\n\nThe Amazon and Los Llanos: \n\n## **Acknowledgments**\n\n_Mil gracias_ to: Dylan Misrachi, Alejandro Maldonado, Camilo Polo, Marcela Manrique, Marco Chiandetti, Alberto Dante, Eric S\u00e1nchez, Nery Luz Hoyos, Martha Rubio and Fabio Jim\u00e9nez, Rainbow Nelson, Elena Posada, Marcela S\u00e1nchez, Jay Speller, Alex Zuluaga, Jorge Barbosa, Paola Lenis, Johana Granados, Matthew Freiberg, everyone at Iguana Hostel in Cali, Los Pinos folks in Minca, Claudia of Hostal Ruta Sur in Cali, Heike van Gils, Victor Sepulveda, Jade Gosling, Felipe Escobar, Rodrigo Andr\u00e9s Fajardo, Tyler Stacy, Stefan Schnur, Tim Harbour, Tony Clark and Kim Macphee, Richard Emblin and Mar\u00eda Claudia Pe\u00f1a, Richard McColl, Diego Duarte, Alexa and Daniel from Kolibr\u00ed in Pereira, Don Ai in El Valle, Ryan H. from Medell\u00edn, Javier Gamba, Patrice Chambragne, Carolina, Julia, and Diana Barco, Paulino Gamboa, Martin Climent, Mauricio Quintero, Juan Guillermo Garc\u00e9s, Ximena Arosemena and everyone at the Reserva Natural R\u00edo Claro, Julio Su\u00e1rez, Greg Shiffer, Alexandra G\u00f3mez, Guillermo Valderrama, Luzlilian G\u00f3mez, Juanita Saenz, Marta Royo, Shaun from Macondo in San Gil, Douglass Knapp, Max Hartshorne, Mary Luz Parra, Diana and Pepe G\u00f3mez, Juan Pablo Echeverri of Hacienda Venecia, Irene Torres, Jorge Torres and the staff of Hacienda Guayabal, Daniel Maxian, Karla Miliani, Finca Villa Martha, Jackeline Rend\u00f3n, Mauricio Cardona, H\u00e9ctor Buitrago, Hemmo Misker, Fibas Jard\u00edn Bot\u00e1nico del Desierto--Villa de Leyva, Jorge Raul D\u00edaz, Nelson Barrag\u00e1n and everyone at Hacienda La Aurora, Goran Mihajlovic, Rafael Clavijo, Axel H. Antoine-Feill S., Kelly Brooks, Yaneth Caballero, Chris Hardyment, Alonso S\u00e1nchez, Gustavo Osorio, and Jorge Navas.\n\nThanks to everyone at Avalon Travel, especially Leah Gordon, Domini Dragoone, Mike Morgenfeld, and Grace Fujimoto. For their moral support, canines Tiga and Lumi, and finally, and most of all, to my partner Vio for his good sense, good humor, and incredible patience.\n\n**MOON COLOMBIA**\n\nAvalon Travel\n\na member of the Perseus Books Group\n\n1700 Fourth Street\n\nBerkeley, CA 94710, USA\n\nwww.moon.com\n\nEditor: Leah Gordon\n\nSeries Manager: Kathryn Ettinger\n\nCopy Editor: Deana Shields\n\nGraphics and Production Coordinator: Domini Dragoone\n\nCover Design: Faceout Studios, Charles Brock\n\nMoon Logo: Tim McGrath\n\nMap Editor: Mike Morgenfeld\n\nCartographer: Stephanie Poulain\n\nIndexer: Rachel Kuhn\n\neISBN-13: 978-1-61238-628-7  \nISBN-13: 978-1-61238-627-0\n\nISSN: 2334-0533\n\nPrinting History\n\n1st Edition \u2013 August 2014\n\n5 4 3 2 1\n\nText \u00a9 2014 by Andrew Dier.\n\nMaps \u00a9 2014 by Avalon Travel.\n\nAll rights reserved.\n\nSome photos and illustrations are used by permission and are the property of the original copyright owners.\n\nFront cover photo: bell tower of the Cathedral of Cartagena, \u00a9 Alfredo Maiquez\/Getty Images.\n\nTitle page photo: aerial view of Guatape Lake, \u00a9 Daniel Alvarez\/123RF.\n\nFront color photos: click here \u00a9 Jannis Werner\/123RF; click here, click here (top), click here, click here (top, bottom left), click here, click here, click here, click here, click here, click here, click here, click here, click here, click here \u00a9 Andrew Dier; click here (top left), click here (bottom left) \u00a9 Mariusz Prusaczyk\/123RF; click here (top right) \u00a9 Gregory Hills; click here (bottom), click here (bottom right), click here (bottom right), click here, click here, click here \u00a9 Jesse Kraft\/123RF; click here \u00a9 Lakasz Janyst\/123RF; click here, click here \u00a9 Rafcha\/123RF; click here \u00a9 Flaperval\/123RF; click here \u00a9 Alejandra Triana Munoz\/123RF.\n\nMoon Handbooks and the Moon logo are the property of Avalon Travel. All other marks and logos depicted are the property of the original owners. All rights reserved. No part of this book may be translated or reproduced in any form, except brief extracts by a reviewer for the purpose of a review, without written permission of the copyright owner.\n\nAll recommendations, including those for sights, activities, hotels, restaurants, and shops, are based on each author's individual judgment. We do not accept payment for inclusion in our travel guides, and our authors don't accept free goods or services in exchange for positive coverage.\n\nAlthough every effort was made to ensure that the information was correct at the time of going to press, the author and publisher do not assume and hereby disclaim any liability to any party for any loss or damage caused by errors, omissions, or any potential travel disruption due to labor or financial difficulty, whether such errors or omissions result from negligence, accident, or any other cause.\n\n**KEEPING CURRENT**\n\nIf you have a favorite gem you'd like to see included in the next edition, or see anything that needs updating, clarification, or correction, please drop us a line. Send your comments via email to feedback@moon.com, or use the address above.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzttcc b/data_all_eng_slimpj/shuffled/split2/finalzzttcc
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+{"text":"This was a random find whilst looking for a new book to read in Waterstones. The blurb sounded quite interesting and I decided to give it a try. I am so glad I did! It turns out it is part of a quintet series and I have just ordered the next one. L'Engle writes in a very accesible way and completely engulfs you in the story.\nHave you read this or any of her other works? The back of this copy has facts and questions and even an extract from another book. It's full of add ons.\nI loved Charles Wallace. He reminds me so much of one of the children I used to work with. I also loved the three old ladies, Mrs Who, Mrs Why and Mrs Which. Brilliant characters!\nMy favourite part was as they arrive on one of the planets and the whole world works like clockwork. I find alternate realities quite interesting.\nI'd love to hear what you thought of this if you've read it. Which character was your favourite or did you most relate to? What was your favourite part?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Every accommodation has a sense of discrete luxury, combining contemporary and Indian decor.\nIdeal for corporate , business and social events, with well-equipped state-of-the-art banquet facilities.\nBrimming with old world charm and the grandeur of a bygone era, The Hotel Park Elanza is a Chennai's Famous grand heritage hotel.\nOver its rich history, the hotel has hosted royalty, distinguished guests and celebrities, and continues to charm today, with its acclaimed Traditional Afternoon Tea and exceptional service.\nFeaturing magnificent views overlooking Chennai, Hotel Park Elanza, with dining on two levels, is open seven days a week. Hotel Park Elanza menu features traditional pizzas, pastas, seafood hotpot and Indian cuisine showcasing regional produce from Mountains.\nGuests can select from our Menu or take advantage and indulge in our special 2 Course Set Menu (Entree & Mains OR Mains & Dessert). Beverages based on consumption.\nA spa sanctuary to refresh you from the stresses of life and enliven your senses. We pride ourselves on our reputation for friendly, professional service with dedicated and highly qualified therapists and skin care products, all committed to your relaxation.\nOur expert Majestic meeting planners will be by your side throughout the entire process. A conference & events coordinator will act as your single point of contact, offering expert advice, exceeding your expectations and taking care of the little details so you can focus on achieving your goals.\nOur dedicated team will anticipate your needs with a range of meeting supplies provided on the day. Our Meeting Enhancements include: Delegate Amenities: Notepad, pens, water and a selection of mints.\nWe offer a range of spacious and comfortable guestrooms to help your delegates relax and re-energise at the end of a busy day. Each of our guestrooms features an ergonomic workspace with wireless internet access, LED television and stunning views of Chennai.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On February 25th of this year, our Open Caucus looked at Youth in the Economy. What we found is that youth are facing higher unemployment\/ underemployment; they are paying more for an education that is becoming less valued in the job market; and soaring home prices and student debts are forcing them to delay starting a family. When our witnesses were asked what young Canadians can do to change these circumstances, they answered with one voice- \"vote.\" Canadians 18-24 have the lowest voter turnout of any age demographic in the country. We need to increase the Canadian youth participation in the next federal election and beyond. We are proud to hold this meeting in conjunction with student representatives from Queen's University's Model Parliament. They feel it is important that all youth, not just those found on campuses around Canada, actively engage and speak out for their future.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Back in September last year, Sharp showed us what the TV industry has in store once 4K becomes the norm and people move on to bigger and better things: the 70-inch, Aquos 8K television. The display is about to launch in Europe, and, as expected, it carries a hefty price tag.\nAlready available in China and Japan, Sharp's LV-70X500E, to give it its full name, will arrive in European markets before the end of April\u2014a month later than originally planned\u2014and cost 11,999 Euros (around $13,779) with taxes included.\nSo, what do you get for all that cash? There's the 33.2 million pixels from its 7680 x 4320 resolution, HDR support through HLG and HDR10, a display with 1000 nits of peak brightness, local dimming over 216 individual zones, an 8 ms GtG response time, and 86-percent BT2020 color gamut coverage.\nThe TV also features eight HDMI ports\u2014 four HDMI 2.0 (with 4:4:4 chroma subsampling & HDCP 2.2) and four HDMI 1.4 inputs\u2014a 2.1 audio subsystem, a USB port for showing 4K video or 8K still images, and a LAN port for IP control.\nUnlike Japan, Europe has no experimental 8K broadcasts, so the TV is aimed at those in the medical sector who could use it for high-resolution scans, as well as designers, engineers, and other professionals.\nNo word when, or if, sharp plans to bring the Aquos 8K television to the US. With 4K technology only now starting to become more mainstream, it's unlikely that everyday consumers would rush out and buy such an expensive display.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The founder and past owner of Le Beau Caf\u00e9 & Fudgerie in Beausejour has re-invented herself. Corinne wants to welcome all her friends and former customers from Beausejour and surrounding areas to \"The Savvy Soaperie\" \u2013 the home of luxurious 100% natural handmade soap and skincare products.\nAs some of you may already know, Corinne started writing in her mom's cookbooks and trying recipes when she was only six. She has an extensive background in cooking, fudge making and is a qualified florist. She spent many years both in the floral industry and in the restaurant industry. Consequently, she is no stranger to the chemistry of plants and food. Her passion for soapmaking brings together her love of flowers, plants and cooking. As she explains, \"making soap is much like cooking and requires the effective blending of top quality ingredients to consistently get the right balance.\" In line with that, she has made the important choice of handcrafting her delightful products with the best ingredients that nature has to offer. She makes her handcrafted soaps with pure plant oils like organic sunflower oil & olive oil, golden cocoa butter, creamy raw shea butter, purifying clays, seeds, organic vegetables, locally grown sun-warmed herbs and generous amounts of pure essential oils such as lovely lavender, patchouli and geranium. Corinne is very proud of the fact that she stands among the \"rare soapmakers\" who DO NOT use any synthetic fragrances, synthetic colors, and additives. Her handmade soaps and skincare products are 100% natural.\nAfter taking numerous courses in soapmaking and aromatherapy in various parts of Canada, she carefully selected and refined every recipe for her soaps and body products to create the one-of-a-kind, \"skin-loving wonders\".\nBeing able to offer these extraordinary products to both new and existing valued customers fuels Corinne's entrepreneurial spirit. She encourages each one of you to engage in regaining your skin's terrain and natural flora by lathering yourself with her sensuous handcrafted soap and soaking up her ultra-nourishing skincare creations. Not only will they appeal to your senses \u2013 they will leave your skin soft, clean, healthy and naturally nourished.\nFor nurturing the mind, body & spirit, Corinne uses 100% pure essential oils in her Aromatherapy and Pure Soy Candles. She invites you to try these and indulge your soul!\nIn summary, what Corinne does at The Savvy Soaperie goes much further than skin deep. It's about contributing to your total health, reducing the amount of harsh chemicals being absorbed by your skin and entering your entire body (your skin is your largest organ). It's about treating your skin and the Earth equally well! When we don't use harmful chemicals on our bodies, they don't get into our skin, our water, our streams, our rivers and our lakes. Together we can be healthier and we can make a difference!\nFor further information please visit her website at www.savvysoaperie.com where you will find a lovely collection of handmade extra gentle and invigorating 100% natural soaps, specially formulated facial soaps, exfoliating soaps, skin-pampering bath delights, whipped shea body butters, aromatherapy, pure soy candles, gift cards, gifts and beautiful glazed ceramic soap dishes that Corinne also makes herself. You are sure to find something you love as well as a great variety of gift ideas for everyone!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzuhph b/data_all_eng_slimpj/shuffled/split2/finalzzuhph
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+{"text":"Pravasi Bhartiya Divas was celebrated in the Embassy of India on 09th January 2019 where Ambassador Sh. Vinay Mohan Kwatra interacted with the members of Indian community and felicitated the singer Mme. Raphaelle Brochet for singing Mahatma Gandhi's favourite Bhajan \"Vaishnav Jan To\" in French along with winners of \"Bharat ko Janiye Quiz\".","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"American, in a positive way.\nNot American enough, not European enough. He's just one Franzl.\nI'd be pissed off if you said \"very\"\nIt's a miracle he's still alive, considering his home state.\nA dying bread of American.\nAs much as Obama is American !!!\nAmerican? Yes, in a good way, of course.\nOklahoman? You don't fit there, dude.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What music program do I use when writing and producing music? And where can you get it?\nI sometimes get the question what music program I use when writing and producing music. Today, there are many different digital audio workstation (DAWs), all of which are working the same way, and it's simply a matter of finding one that you enjoy.\nCubase, ProTools, Logic, Garageband are some used to create music. I, after being faithful Steinberg Cubase for many years, have switched to Presonus Studio One. And I'm very pleased. The working method is wonderful, as many are \"drag&drop\" functions. It's an easy-to-understand interface and it's stable. It almost never crashes.\nAnd have you worked in Logic or Cubase before and are used to all the shortcuts, so there are complete settings so you can use your working method just as usual. The new version of Studio One also has very creative ways to write music and you may want to try different arrangements. A picture says more than 1000 words and a video says even more so you have a few minutes left so check out the video below that quickly describes Studio One.\nThe only thing I miss which is understandable when they are competitors is that I would like to be able to import my old songs I made in cubase to Studio One in an easy way. Today, it's a bit difficult to find out what synth and sound you used, etc.\nCopyright \u00a9 2019 BeeZed Productions. All rights reserved.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We are starting migration from rbx1-3a\/b-a9 to rbx1-5a\/b-vr and rbx1-4a\/b-vr. This should have no impact on your services, since it will be no-packet loss migration.\nWe will start with network (91.121.200.x) on Monday, 22:00 CEST.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Can a certificate chain be used for a hidden service's private key? It would be nice to be able to periodically re-sign a public key and upload it to the host. This would mean that in the case of the private key on the host being compromised, it is only valid for a short amount of time, due to the manual reauthorization process.\nHost: I am ONION_ADDRESS, my public key is HOST_PUBLIC_KEY.\nHSDir: I don't trust that public key.\nHost: Here is the signature from root key ROOT_PUBLIC_KEY: SIGNED_HOST_PUBLIC_KEY.\nHSDir: ROOT_PUBLIC_KEY is valid, and that signature is valid. Go ahead.\nHSDir: ROOT_PUBLIC_KEY is valid, but that signature has expired. Goodbye.\nThe ROOT_PRIVATE_KEY is stored on a machine under physical control of the hidden service operator. The remote host is considered the weakest point. Should the remote host be compromised, all that can be obtained is HOST_PRIVATE_KEY (and of course HOST_PUBLIC_KEY), and the SIGNED_HOST_PUBLIC_KEY. Due to the signature being created with an expiry date, the adversary can only masquerade as the hidden service for as long as SIGNED_HOST_PUBLIC_KEY remains valid. Should the hidden service operator become aware of the host becoming compromised, he would not renew the signature. Once the SIGNED_HOST_PUBLIC_KEY expires, due to the absence of a renewed signature, HOST_PUBLIC_KEY will not be trusted, resulting in the adversary's loss of control of the hidden service.\nThe purpose of this is to limit the impact which a compromise would have. Further, once SIGNED_HOST_PUBLIC_KEY expires, the hidden service operator can regain exclusive control of the .onion address \u2013 it would not be necessary for a new .onion to be created.\nAre there currently any methods (or any planned methods) which would allow for the host to be given partial\/temporary authority to serve as the hidden service?\nBrowse other questions tagged hidden-services security private-key certificate or ask your own question.\nFrom what depends .onion addresses - public or private key of the service?","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzwubv b/data_all_eng_slimpj/shuffled/split2/finalzzwubv
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+{"text":"Made with 100% Cottons front and back. I have Pre washed before I made it, To ensure this sham stays nice after Washing. Machine washed cold water and dried flat, you can air fluff it after its dry and its still awesome! I also used Double Batting ( 1 layer 100 cotton, 1 layer X-loft Poly ) for added textile and \"Puff\" and to really show off the quilting.I used Decorative Threads, In variegated variety of colors.\nThe color is Solid White Kona, with colored thread and a small multicolored trim.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Supreme Court has upheld lower court decisions finding a former head of the Public Security Intelligence Agency, Shigetake Ogata, guilty of defrauding real estate and cash from Chongryon, the group of pro-Pyongyang Korean residents.\nOgata, 79, was first found guilty and sentenced in the Tokyo District Court in 2009 before appealing the decision to the Tokyo High Court. In the top court ruling dated Monday, presiding Justice Katsumi Chiba upheld Ogata's sentence of two years and 10 months in prison, suspended for five years.\nThe court also upheld the guilty verdict of co-defendant Tadao Mitsui, 80, sentenced to three years in prison, suspended for five years.\nThe court agreed with findings in the original case that Ogata and Mitsui defrauded Chongryon of its central Tokyo head office building in 2007, when the group was looking to temporarily offload the property to prevent its seizure by a government-run debt collection agency.\nThe pair were found to have compelled Chongryon to transfer ownership of the property to a company led by Ogata while lying to the group that they had found a suitable buyer, and to have pocketed \u00a5484 million in cash, supposedly needed to court investors.\nOgata's and Mitsui's lawyers said the men were only trying to help Chongryon, also known as the General Association of Korean Residents in Japan, and did not intend to defraud the group.\nDisputes continue over the fate of the property, with Chongryon having filed an appeal earlier this month with the Supreme Court over a Tokyo High Court decision that allowed the sale of the property to Japanese real estate developer Marunaka Holdings.\nOgata served as head of the Public Security Intelligence Agency as well as heading the Sendai and Hiroshima high public prosecutor's offices, before retiring to become a lawyer in 1997.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"LAS VEGAS, Nevada -- Even after losing more than $3 million in a gambling frenzy over a three-year period in the late '90s, Daren Leverenz still thinks he can beat Las Vegas. He's even constructed a TV series around that belief, \"Man Vs. Vegas,\" which debuts Friday night at 10 on CMT (Cox cable channel 57).\nLeverenz is the star of this eight-show series, in which he attempts to win back the fortune he lost from 1997 to 1999. He has since repaired his finances and liquidated all of his assets to build a nest egg of about $1 million for his made-for-TV venture.\nIn the series Leverenz grabs tourists at random -- at ATMs and in elevators -- and hands them several thousands of dollars to play on his behalf.\n\"If they win, I give them some of the profits,\" Leverenz said during a phone interview from his office in L.A. on Monday. \"But they never lose their own money. I want them to share in the experience of being on this roller coaster.\"\nLeverenz, who made a fortune through investments in the dot.com market in the '90s, became a self-described adrenaline addict during his gambling forays. A birthday bash in which he turned $2,000 into $180,000 got him started; near the end he lost $250,000 in a single day the week Bellagio opened.\nThe 35-year-old resident of the San Fernando Valley town of Calabasas has filmed the series at the Plaza and Aladdin. His favorite forms of gambling are blackjack and baccarat, and he's learned to enjoy craps and poker. But he'll wager on any form of gambling -- even betting on NHL games -- and is unafraid of trying to beat the Vegas odds.\n\"It's all about attitude,\" he said. \"You have to always be positive.\"\nSounds good, but our money is on Vegas.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Crew neck sweatshirts heat pressed with Walleye designs. These sweatshirts are high quality 50\/50 cotton\/poly blend shirts custom heat pressed with great art images. Choose from many different Walleye Designs. Available in color and size choices, including extended sizes in some sweatshirts as special order.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Friday might be my favorite day of the week on this blog. I can always depend on amazing authors, illustrators, teachers, and librarians to finish my sentences. A huge THANK YOU to everyone who has participated over the past two years.\nThis week's special guest is Aaron Starmer. He dropped by to chat with me about The Riverman, storytelling, reading, school libraries, and my beach house. I wrote the words in red, and he wrote the words in black. Many thanks, Aaron!\nFiona Loomis is \"unknowable in the way that all girls are unknowable, but also in her own way.\" That's a line from the book and it sums up the hero Alistair's view of Fiona in the beginning. She's a mystery. But there's a problem with viewing people as mysteries. You tend to want to figure them out, rather than get to know them. And that's the wrong way to form a relationship, as I think Alistair discovers. Fiona is knowable. Like anyone, it takes time and effort to get to know her.\nI hope The Riverman surprises readers. I know a lot of people will be a bit befuddled when they finish it, but I urge them to look at it with fresh eyes. They don't have reread the thing, just ask themselves why the story was presented in the way it was. They might begin to connect more dots than they realize.\nThe Indubitable Dweeb is the name of my blog. I don't update it nearly as much as I should. There's some fun stuff buried in the archives from 2010 and 2011, but I've been too busy to keep at it. And the readership consists of about three people, so no one's really clamoring for more. The name comes from my first book (DWEEB) and from the fact that I am undoubtedly a dweeb. Indubitable is an underused word, in my opinion.\nAs a child I was a storyteller. It's always been an integral part of my life and I think people sometimes undervalue storytelling in their lives. That's not to say that everyone needs to be rabid readers of novels, but I think that the ability to enjoy and to tell a story is an essential one. It's more than recreation. It's how we understand other people. It's one of the ways we form empathy.\nWhenever I visit a school I tell a story. It's been 25 years since I was in middle school, but certain situations are universal so I like to air my adolescent laundry. Detail my embarrassments. My triumphs. The mundane situations. Middle school is a glorious and hideous time and everything in between. The message for kids shouldn't always be \"It gets better\" or \"Enjoy it while you can.\" They are in the middle of something significant. They should be in it, survive it, laugh at it. I am who I am because of those stories.\nReading is boring sometimes. It's a chore sometimes. But (and this is a rather large but) when you find a single book that grabs you, trust me there are countless other books that will grab you as well.\nSchool libraries are there to help you find that next book that will grab you. That's the job of librarians. To make sure you don't waste your time. If you've ever visited a new city, you know it helps to be guided by someone who lives there. Librarians live in the library (really they have murphy beds that fold down from the self-help section).\nMr. Schu, you should have asked me to dog-sit at your beach house for a month or two. I assume you have a beach house, and a dog, and the desire to give me a vacation, because I could use one. If I'm wrong, then you should have asked me if I'm grateful for the opportunity to share The Riverman with readers of your site. And the answer would be an indubitable YES.\nI am giving away one copy of The Riverman.\n1. It will run from 4\/11 to 11:59 p.m. on 4\/13.\nBorrow The Riverman from your school or public library. Whenever possible, please support independent bookshops.\nThis look like something my daughter and I will love! Looking forward to it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaaiyx b/data_all_eng_slimpj/shuffled/split2/finalzzzaaiyx
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@@ -0,0 +1,5 @@
+{"text":"Guitarist looking for new people and to have fun!\nPeople should get to know me because I'm genuine, caring And always up for a laugh. You never know until you try!\nSomeone who is genuine and fun, pretty and caring. I want to find someone on the same wavelength as Me and I can enjoy being myself around.\nfeatherboat hasn't asked any friends to write a recommendation yet.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Watch out, Cyber Monday. Green Monday is gaining traction with shoppers.\nConsumers flexed their thumb-shopping muscles on Dec. 12, Green Monday -- the second Monday in December that's become another big day for online holiday shopping, according to ChannelAdvisor. The company runs the e-commerce platforms of 3,000 retailers ranging from Staples (SPLS) and Bed Bath & Beyond (BBBY) to Urban Outfitters (URBN).\nOn Green Monday, e-commerce sales at ChannelAdvisor's retail clients grew 19% over 2010 -- not far behind Cyber Monday, when the firm's merchant customers saw online sales rise 25% from the year ago period.\nChannelAdvisor does not disclose total sales figures.\nThe sales gains on Green Monday are a sign of consumers' robust and consistent online shopping pattern this holiday season, ChannelAdvisor spokeswoman Sarah O'Dea tells DailyFinance.\n\"Last year, we saw dips and spikes in online shopping,\" she says. \"It's been more steady this year.\"\nAnother telling number: Online sales from Dec. 1 to Dec. 12 for ChannelAdvisor's retail clients surpassed what the company calls \"Cyber Five,\" the five days from Thanksgiving until Cyber Monday. O'Dea says this suggests that shoppers will continue to tap e-commerce for the remainder of the holiday season.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Conway Electric Exto+4 AC\/DC 8ft.\nAvoid the tippy-canoe tendencies of long narrow lightweight and dangerous power strips with the stability and power of the Exto+4. The square cast-aluminum housing won't slide or tip. It s designed to stay where you place it with grippy urethane feet underneath its heavy cast aluminum and stainless steel housing, which is nearly burn-proof. Plug all your goods into the four 15-amp rated electrically grounded outlets that are also internally tamper resistant helping prevent shock (especially around kids), one of the only extension cords with this feature. The 14-gauge ultra-flex wire is rated to 15 amps. You can even mount the Exto+4 to a wall or other surface using the self-tapping mounting holes on the underside. With the Exto+4you can finally give your electricity the love it deserves, though you might fall in love with it too.\nNon-slip, non-marking urethane feet keep the Exto in place on a table, counter or other hard surface.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ten-8 Fire Equipment will be debuting new brands and product at Fire Rescue East from January 18th through the 20th at the Ocean Center in Daytona Beach, FL. Explore fire equipment and apparatus solutions in 6,000 square feet of booth; booth #1101.\nThis year Ten-8 is showcasing six fire trucks and three ambulances in addition to an endless array of fire equipment! Best of all, Ten-8's vendors will be present in the booth to assist with finding your department the solutions you need or are looking for! Vendor partners include the following: Pierce Manufacturing, Inc.; Frontline Communications; Braun Ambulances; Osage Ambulances; Foundation Ambulance; MSA; Key Hose; Skeeter; and many more.\nIf you're into fire apparatus, be sure to check out two new brands of fire trucks now being carried exclusively by Ten-8 as well as three new Pierce products! We will count them down since there are so many.\nSkeeter is a Texas-based company specializing in brush trucks. Ten-8 began its exclusive partnership with Skeeter in August 2017.\nFrontline Communications is a Florida-based company specializing in command vehicles. Frontline is also a sister company to Pierce Manufacturing, Inc. Ten-8 began heavy promotion of these high-tech communication vehicles in October 2017.\nPierce Manufacturing, Inc.'s 110' Ascendant aerial platform! The 110' Ascendant platform will be a certain show stopper as it is a single-axle platform with a record-breaking reach of 110'!\nPierce's new Ford mini-pumper will also be on display. It may be a mini, but it packs a punch and gets the job done!\nPierce unveiled the new Ford Power Stroke this fall! Designed in partnership with Roush and Ford, the new engine is an economical choice for budget conscious departments who don't want to sacrifice performance of quality.\nFoundation Ambulance is making its Florida debut as Ten-8's official ambulance remount partner. Meet with the Foundation team and explore the demo remount in the booth!\nIn case you needed another reason to come to the Ten-8 booth, let us give you one more! You can not only view industry revolutionizing fire apparatus, but you can also test drive (or ride in) the new Pierce Ford Power Stroke! Request a demo from a Ten-8 or Pierce representative in the booth and drive the demo located outside of the convention center! When you drive or ride in the demo unit, you can enter to win a Cairns leather helmet!\nBeyond apparatus, Ten-8 will also be featuring some of the most popular equipment brands in the industry to include MSA, Fire-Dex, TFT, and Key Hose. Ten-8 is also excited to support its partnership with the Firefighter Cancer Support Network by unveiling a new decon kit to help protect firefighters from the cancer epidemic.\nBe sure to stop by Ten-8's booth to see new products, drive new apparatus, enter to win a leather helmet, and view the new decon kit!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Your free preview of The Air Current: Boeing and Embraer are at the finish line to unveil its commercial joint venture.\nAs Boeing digs into understanding what makes Embraer tick, they're finding that its structure (by design) makes it inherently difficult to pull apart.\nEntering her room first thing in the morning slightly bleary eyed, I lamented the new hamster nest-like pile of debris on her floor. But that's when it hit me.\nThe handover of the E190-E2 to Norway's Wider\u00f8e may very well be the final first delivery Embraer will make as an independent plane maker.\nBoeing and Embraer are at the finish line on a commercial aerospace tie up that will reshape the global industrial landscape and firmly establish a new superduopoly to challenge its newly-united European and Canadian rivals, according to two people familiar with both company's plans.\nThe joint venture, whose announcement is expected as early as Thursday or early next week, would establish its plans for moving forward with a strategic alliance, according to the people.\nWhile the final shape of the deal, however, is not known to The Air Current, Embraer's business jet unit will not be included along with its commercial regional and small single-aisle E-Jets, according to one of the people.\nThe agreement was to be announced as early as last week, according to the two people familiar with its plans and third who was briefed on the situation, but has been mired in stop-and-go negotiations on various points as stakeholders in the U.S. and Brazil weigh in on the shape of the future venture.\nBrazilian President Michael Temer was set meet late Tuesday with defense and security chiefs to review the deal, according to a Tuesday report from Bloomberg News. And Brazilian military officials testified to lawmakers in Brasilia its concerns over the deal had been allayed.\nAdditionally, newspaper Valor Economico reported late on Wednesday that Embraer Chief Executive Paulo Souza e Silva said the negotiations were in their final stages.\nA spokesman for Boeing declined comment. A spokesman for Embraer did not immediately respond to a request for comment.\nAn agreement would be a culmination of years of courtship between Embraer and Boeing. The relationship dates back to the 1990s when Embraer was a supplier on the McDonnell Douglas MD-11 program and has progressed through years of growing closeness on commercial technology testing and joint marketing of military hardware.\nAnnouncement of the pair's definitive intent to integrate would offer a significant strategic counterweight to the newly finalized the C Series Aircraft Limited Partnership \u2013 the union between Airbus and Bombardier's new single-aisle jet. That agreement was struck in October and closed on July 1. News of merger talks between Boeing and Embraer were first reported in December.\nIn the immediate term, such an announcement from Embraer and Boeing would be weighty counter-programming to its united rivals. Airbus plans to unveil a full rebranding of the Canadian 110 to 160-seat jets on July 10 in Toulouse, ahead of the Farnborough International Air Show outside of London.\nOver the coming years, a tie up between the U.S. \u2013 the world's largest economy, and Brazil \u2013 expected to be the fifth largest in the world by 2050 \u2013 establishes a new industrial beachhead to defend its market position as China rapidly rises in its global aerospace ambitions.\n**Annual memberships renew one year from the date of subscription at $199. Your monthly subscription gets you full access and will be automatically renewed each month unless you opt out. Once TAC is launched, you will receive more information on how to gain full access to the site.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Why does all my questions under Religion & Spirituality get taken down?\nPoll: how do you mentally and physically prepare to get back to work after holidays?\nThey're watching us aren't they?\nWhy doesn't Yahoo want us to edit our nicknames and pictures?\nWe ask more questions than answering them, has curiosity just gotten the better of us?\nA question is only for female users?\nUK Can someone please explain to me why certain journalists are making such a big deal about a Muslim winning GB Bake Off?\nName someone who`s made you laugh this past week?\nTell me if I did anything wrong ? I posted a picture of a 22 y\/o model from the web in the Beauty form and was reported .?\nAre you on your period right now?\nHow are some users able to change their usernames again?\nWhy do people like to report other people?\nCurrently, should anyone move your question without your permission?\nI know you know that I know that they know that we know that they know we all know, you know?\nCan I be your friend for the rest of your life?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Located on the site of the former Sacramento Bee parking garage, the 4-story apartment community will include studios, 1-, 2- and a few 3-bedroom units ranging in size from 450 to 1,300 square feet with 8,000 square feet of ground floor commercial space and 310 parking spaces (for visitors and tenants). Amenities will feature a resort-style pool, outdoor kitchens, state-of-art fitness center, bike lounge, pet spa, package lockers and more. The project's name pays homage to the McClatchy Company's (owners of the Sacramento Bee) long-term commitment and contribution to the Midtown Quarter.\nThe Midtown Quarter is comprised of three development projects by SKK Developments and partners; The Press, Q19 Apartments and 20PQR Townhomes. Together, the three projects total nearly 400 units and create the highest concentration of housing in Midtown Sacramento. The Midtown Quarters' proximity to employment, restaurants, bars and Midtown activity afford residents an increasingly urban and sustainable lifestyle.\nThe Press is anticipated to be completed Spring 2020.\nBuilt on a legacy of more than seven decades, the DeBartolo name is recognized as an icon in the real estate industry. Since our beginnings in 1944 when legendary entrepreneur Edward J. DeBartolo, Sr. pioneered the first shopping mall concept and developed some of the most well-known and nationally-recognized shopping landmarks, our legacy has been synonymous with success. Today, DeBartolo Development is one of the largest private real estate investment and development companies in the country and invests in real estate assets of all sizes and scopes, specializing in opportunistic acquisitions and market-driven, ground-up development of multifamily, hospitality, retail and mixed-use projects throughout the United States. For more information about DeBartolo Development, please visit our Website at www.debartolodevelopment.com.\nSKK Developments is a real estate firm that specializes in mixed-use and multi-family development projects in Midtown Sacramento. SKK Developments' past projects, such as the Fremont Building, 1801 L and L Street Lofts, have played an integral role in placemaking and often been earmarked as catalysts for the revitalization of Midtown Sacramento. Currently SKK Developments has nearly 500 units under construction within Sacramento's urban core and an additional 1,000 units in the planning and entitlements process. For more information please visit: www.skkdevelopments.com.\nThe mission of the Midtown Association (MA) is to create a center for culture, creativity and vibrancy in Sacramento's urban core. For more information about MA, call 916-442-1500, visit www.exploremidtown.org. or follow on social media \u2013 Facebook at www.facebook.com\/exploremidtown\/ and @ExploreMidtown on Instagram and Twitter.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzadkob b/data_all_eng_slimpj/shuffled/split2/finalzzzadkob
new file mode 100644
index 0000000000000000000000000000000000000000..d4d6aa8ead8f2055c9b07dd3e310f1a33f70545d
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzadkob
@@ -0,0 +1,5 @@
+{"text":"We provide a wide range of services for you and your family at all life stages.\nThe Practice has a strong commitment to preventative medicine and up to date clinical management.\nThe links below provide an overview of the most popular services we offer, but we are always happy to discuss any particular needs you may have.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A modern, acappella setting of the In Monte Oliveti text. (\"On Mount Olivet he prayed to the Father: Father if it is possible, let this cup pass from me. The spirit is indeed willing, but the flesh is weak; thy will be done.\") Thick, lush harmonies and rich colors will challenge and delight your advanced mixed-ensemble.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In the summer of 2015, VH1 aired Twinning, a competition between pairs of identical twins that tested their abilities when separated. The winners were Claire Buitendorp and Shawn Buitendorp, who are now using their supertwin powers to destroy a season of Project Runway.\nOkay, that's too strong, and they are bringing a lot of energy to this season, though in the same way that a tsunami brings someone a refill of their water. It's too much, too destructive, and so unnecessary.\nI am very much enjoying the rest of the cast and the work so far, but I am wondering what we're losing with the twin focus. It's not like they're entirely stunt casting; this profile of them explains what they offer, and is also more than the one-dimensional version of them we're getting on the show.\nAt least twice, Tim Gunn walked into the Project Runway workroom, which was strewn with oddly clean recycled material, and exclaimed \"Good heavens!\" Later, when one team of designers told him their team was named \"Ballin' on a Budget,\" his expressions suggested he'd just been hit in the face with a wad of lemon pulp.\nThat's about how I felt every time the editing turned its attention to a twin and her drama or tears or both. It's just so often that the show's attention turns to them and their uninteresting problems: with the challenge, with their models, with the show, with their self-doubt.\nClaire Hair helpfully explained that her twin Is Shawn Gone Already was struggling because she had to work on a team, use unconventional materials, and design high fashion for a \"larger\" woman, none of which she'd done before.\nThe twins took their team down with them, whirling around the workroom in a tornado of self-doubt and attention-seeking. Sentell's actual design, plastic wrapped around his model in such a way that I was pretty convinced I could do better with a trash bag and three minutes, did him no favors and he deserved to be sent home for it, but he also probably spent too much time helping his demanding teammates.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Stock Market Volatility is Back: Approaching a Decade of Lost Returns on Investments. The S&P 500 can fall another 42%.\nThe stock market is off to a horrible start for 2009. Many thought that things could not get worse than what we experienced in 2008. Yet market volatility, a sign of an unhealthy economy, is still with us and appearing again in a ferocious way. From January 4, 2008 to February 21, 2008 the S&P 500 was off by -9.2%. The S&P 500 during that same time for 2009 is now off by -14.75%. The issues we will now face are a continuing stream of declining earnings because of the pullback in consumption and the tightening up of the credit markets. It also doesn't help that consumers are psychologically more cautious because of what is going on.\nThe only sector that is not seeing double-digit losses is healthcare but even that, it is still down for the year. Of course, much of the downfall has occurred because of the pounding financials keep taking. The current level for financials is an adjusted 99.19. What was the level of financials back in the October 2007 peak? Try 475.86. What that means is the financial portion of the S & P 500 has fallen 79% in less than 2 years. Many other areas are also facing pain in the mean time. You don't lose $7 trillion in market cap and expect other sectors to be fine.\nYou need to remember how this ratio works. If earnings are increasing at the bottom and prices drop or stay steady, the P\/E will have a smaller nominal value. This is good. It means the price of the stock is cheaper. But what is occurring even though the S & P 500 is off 50.6% from its peak value is that earnings have been tanking as well. So that is why in 2008 you see the P\/E move up. This is somewhat counterintuitive but happens in most recessions. Take a look at the 2001 recession and the bubble peak. The P\/E shot up to over 45 before bottoming out at 19.29 in 2004.\nIf we are to believe this is one of the worst and deepest recessions since World War II, we can expect prices to continue to fall. Now let us go back to the P\/E chart above. During the 1940s the S&P 500 dropped to a P\/E of 5.9 in 1949. We are nowhere remotely close to that and until we see the P\/E at 10 or so, then we can assume a bottom (that is unless earnings start exploding to the upside which is not going to happen).\nI ran a quick analysis and the average quarter earnings since 1940 is 15.79. We are still above that. So there is much more correcting to do.\nSome are arguing that the fair value of the S&P 500 should be somewhere around 440 if we take a multiple of 15 which would be in line with historical P\/E ratios. That is a stunning number but makes sense. That means the S&P would need to fall an additional 330 points, a drop of 42% from where we are currently at. That is hard to imagine yet the math points us in that direction.\nme where a park my tiny money now.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Installation is pretty straightforward, it's much like a really big sticker. The challenge is to apply the Lamin-X so it doesn't leave any bubbles \u2013 they supply a water mister and a beveled edge so that you can push the air bubbles out, but it's pretty tricky and time-consuming to get it right. The good news is that some of the bubbles will work themselves out, so after about 30 minutes of carefully installing and pushing out bubbles, mine turned out really well. The main headlights are definitely harder to install than the fog lights given the headlights are bigger and more oddly shaped, whereas the fog lights are nice and small and a pretty regular shape (circle).\nI mainly installed these for looks to give my car a \"track day\" look, although I took off the headlight covers after about a week as it was just too aggressive looking and wasn't clean enough for me; however, I kept the fog light covers and really like the little bit of contrast they provide.\nWhat I didn't expect was the Lamin-X film does change the color of the light, giving it a yellow hue, as you can see in the photo above. This is great for the fog lights to give the car a neat look when the fogs are on, but another reason I was glad to take them off the headlights as it made everything look yellow at night and probably would look suspicious to cops to see a car with all yellow lights.\nYou can pick up a set for about $65 and install them yourself, so if you're looking for a cheap way to give your car a new look, definitely check it out!\nI think it looks cool but since you did the clear mod it kind of hid that. I agree it looks better without the yellow on the headlights. Maybe the clear mod and a blue tint would have looked better?\nNot bad. I went with the yellow bulb route. You have white HIDs normally?\nDid you remove your fog lights when applying or did you just stick the Lamin-x straight on the the fog lights while mounted. Thanks!\nI took the grilles off (which just pop out by pressing in tabs) but left the fog lights on the car and just washed them really clean.\nIt's a good idea that we use cover for the fog lights, I've tried it with the red lamin and it great! But I don't think we should use cover for headlight, because it will limit our sight. Anyway, thanks for your nice post!\nGreat post, it's really helpful for me. Thanks!\nI think I'll buy the smoke cover for my fog lights. Thanks so much for this recommendation!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Are Gym Gloves Good For Back and Biceps Workout?\nQuestion: I try to go to the gym three to four times per week; I do Crossfit and weightlifting and I want to know are gym gloves good for back and biceps workout? I've tried the rubbery gripads and and I get calluses when I go barehand gloves seem to be the answer. Which are the best weightlifting gloves 2017?\nThe short answer to your question is yes. Any time you can use some sort of hand protection you not only increase your confidence executing the exercise, but you protect your hands from developing calluses as you mentioned in your question. While you may find people who think that having calluses from working out is some sort of badge of honor or level of fitness, they are greatly mistaken. Callosity is actually a medical condition where the skin grows thicker and more tough. This is the body's natural response to excess pressure and friction placed on a specific area. The best weightlifting gloves 2017 are going to be those that you find most comfortable for your workout.\nYou mentioned a back and biceps workout. Let's assume you're doing Bicep Zottman Curls and dumbbells rows for your back. GymPaws\u00ae leather grip pads are going to be a barrier between the skin on your hands and the textured metal of the dumbbell you're holding. You don't feel the rough surface because of the leather front and slightly padded interior. Using GymPaws gym gloves also eases hand fatigue as your hand may start to feel sore far before the bicep brachioradials muscle is fatigued, or before you've even effected the larger rhomboid major or minor muscles of the back.\nIf you're still asking are gym gloves good for back and biceps workout all you have to do is research Amazon (http:\/\/www.amazon.com\/shops\/gympaws ) and read GymPaws Gym Gloves reviews to see which are the best weightlifting gloves 2017. Look for those customer reviews that mention specific exercises. Keep in mind that all of your purchases are protected by our 100% satisfaction guarantee whether you buy gym gloves online or you get them at your local Crossfit box.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On this great occasion, I would like to share about room decoration idea. Some days ago, we try to collected images for your need, we found these are awesome images. Okay, you can inspired by them. We got information from each image that we get, including set size and resolution.\nYou must click the picture to see the large or full size image. If you like and want to share you must click like\/share button, maybe you can help other people can visit here too. Here there are, you can see one of our room decoration idea gallery, there are many picture that you can surf, do not miss them.\nInterior designers concerned in a library interior design plan, ought to interview staff members of the library to realize perception into the wants of the staff plus the design of the workers work areas and furnishings similar to a employees desk. Work areas for staff must be ergonomic and purposeful. Staff ought to have the ability to easily exit from their desk if the theft alarm sounds as well. Another facet of library interior design is a theft detection system have to be installed in a strategic location. An necessary side of library interior design involves the collection of furnishings. Library interior design is considered one of the numerous niche areas of designers come across in their skilled careers. Library interior design or architecture requires detailed planning earlier than plans are drawn. Interior designers or architects can select normal furniture after which add a customized characteristic to it to suit the needs of the library design undertaking.\nThe color will immediately draw your eyes to that space and then you will discover the desk immediately. The inside designer will need name on all their supplier contacts to supply out the very best deal for the library in order to remain under budget. If the library has a set budget, the designer might want to pay shut attention to it when selecting furniture. The designer will need to incorporate areas for computer use as well. Commercial furnishings is manufactured to withstand heavy use. There are additionally industrial manufacturers that design furniture particularly for libraries and different commercial buildings. Library interior design involves designing for different age groups, people who are there for study or analysis purposes and individuals who simply need to loosen up with friends and browse the paper. Some individuals make a focal level by masking one wall in a daring shade of pink or vibrant, graphic wallpaper.\nIf in case you have a phenomenal piece that you really want people to note, make it your focal point. You'll be able to draw attention to a fantastic fireplace by decorating a mantel with daring accents of coloration. The inside designer should use their data of construction and design together with their marketing expertise to design a library that may usher in all ages and compete with the trendy bookstore\/coffee shop. This is when the interior designer should use their communication expertise to plan the design. Once the designer has obtained data from the workers, they should then prepare some drawings for workers to evaluation. Some areas the designer may give attention to are lighting, trueline woodworks acoustics and signage. Areas for youngsters ought to include fingers on play areas and quiet reading areas for fogeys and youngsters.\nThe feeling, when paired with the matching equipment which characteristic button particulars, will give the bedroom that man's contact that a lady can be comfortable to take pleasure in as effectively. Many bookstores characteristic nice comfortable chairs and espresso service engaging guests to stay longer. It is vital for designers concerned in library interior design to concentrate on the ambience that the library should convey to its guests. Interior designers can facilitate the planning course of if they are skilled in communication and information of the operate of a library. Everyone should easily access all services provided by the library. The furniture should be durable and comfy. The wants of the library patrons must be adhered to when planning the design. If you happen to inherited your grandmother's buffet desk that has been within the household for generations before her, design your room around it. Other elements involved within the design of the library are related to the specific procedures and policies in place by the library.\nIf you have any sort of questions pertaining to where and the best ways to make use of exotic paint colors, you could call us at our own page.\nBelow are 27 best pictures collection of room decoration idea photo in high resolution. Click the image for larger image size and more details.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This proposal was brought by the D66. GroenLinks, SP, ChristenUnie, Partij voor de Dieren, 50Plus, Kuzu\/Ozturk, Klein and ruling party PvdA all voted in favor, NOS reports. Other ruling party VVD voted against. According to the party, it would be too expensive and would put the power supply guarantee in danger.\nThere are currently 11 operating power plants in the Netherlands, five of which are from the 80's and will be closed down soon in any case. The proposal does not set a time limit on the government, according to NU. The Kamer only calls on the cabinet to start phasing out coal-fired power plant and draw up a plan for the sector to do so. The parliamentarians want this plan by 2016.\nEarlier this week the PvdA already called for coal plants to be closed in a legislative proposal drawn up in cooperation with GroenLinks. The call also sounded from 64 scientists in a letter to parliament.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Such companies can be used very advantageously for property management, especially in conjunction with the Cyprus' Double Tax Treaties and the EU Directives.\nThe structuring of ownership through a Cyprus Company can reduce capital gains, transfer or stamp duties, and inheritance taxes; and simplify a variety of other complex high-tax country issues.\nA Cyprus IBC can own Real Estate property both in Cyprus and in other countries including the beneficial shareholders' country of residence.\nThe Cyprus Company's beneficial shareholders that own the property can sell its shares instead of the property. The property's legal ownership has not changed, so all stamp duties and transfer duties in the property's jurisdiction are avoided.\nThe gain on the disposal of the Cyprus Company's shares is tax exempt in Cyprus (there may be a taxable gain only if the property is situated in Cyprus).\nThe use of Cyprus Entities is recommended especially for EU property where there are strict anti-avoidance measures in force (Portugal, Greece, and a number of other EU Member States). Indicatively, in Greece a special 3% annual tax on the property's ratable value is levied if the property is owned by a non-EU offshore entity!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"E-commerce is as old as the internet itself and with the rapid advancement in information technology; e-commerce is fast gaining importance in the retail world as well. It is expected that share of B2B e-commerce sales to grow from 9.9 percent in 2016 to 12.1 percent by 2020 and is forecasted to cross $1 trillion marks in near future.\nEven with the growing trend for online commerce, B2B e-commerce is still riddled with hurdles such as lack of expertise or complex business process. Yet the major drawback is the channel conflict in e-commerce. Channel conflict occurs when the manufacturer bypasses the distributor or retailer and sells their product directly to the consumer. Channel conflict may also occur within the sales department of retail or corporate as well. For example, a direct sales representative have to compete with the other sales channels such as mail campaign, telephone or online.\nWhat can be done to deal with channel conflict?\nIt's highly beneficial for the B2B commerce to have both the physical and online presence to flourish their business. However, this requires proper business strategy, sharing ideas and technology with channel partners and time for the long term success. The following practices might help the B2B e-commerce to mitigate the risk of both in-house and external channel conflict.\nLack of expertise is the biggest problem for the sale rep to fully take advantage of an e-commerce website. It is important to teach sales rep on how to avail the instant access to specs, pricing, and availability, comparisons, and data sheet online. Instant access to information will help the sales rep to guide the potential customer better.\nAdding a sales rep dashboard on the e-commerce website is the key to analyzing performance indicator such as quota, sales by customer, sales by product, customer order history and order status etc.\nThe addition of these features on e-commerce website will enable sales rep to make instantly and smoothly general a proposal for the client from their smart devices.\nAdding a workflow support from a request to a quote to a click to approve and order will enable the customers to instantly act on the quote.\nProviding sales rep with the instant chat function to talk to customers online will enable them to guide the customers towards a successful sale. Adding alerts or message whenever the customer has placed or not placed the order in a given period of time will also help sales rep to understand their customers better.\nThe best way to avoid channel conflict is to give the dealer or distributor the edge on price and undercutting on a price will position you as the direct competition for the dealer or the local store network.\nOffer your most trusted partners to promote their physical location on your e-commerce site. As many people still prefer to check the product in person before purchasing them even after their research online. This will help in promoting your dealers and distributors online and eventually result in increased sales. It can be a win-win situation for both manufacturers and distributors.\nShared platform means shared knowledge for both the manufacturers and distributors. By sharing data and collaborating with channel partners, it can aggregate datasets and provide new opportunities to arise. This can be beneficial for the entire sales ecosystem.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Our candidates have exceptional journeys. Read their inspiring success stories about the paths they've walked and how their SEO experience has made a difference.\n19 year old Joseph is the eldest of five siblings; he's a twin and has three young siblings - triplets. Joseph and his siblings currently live in London with his mum who is a recently graduated nurse. This year, he begins a four-year degree in Chemical Engineering following an elective gap-year which he took to improve his grades and secure his desired course at UCL.\nAt Greig City Academy, where he went to school, the average grades for students are low, but for his A-levels, Joseph was able to achieve 4As! He's interested in Trading, Investment Banking and Asset Management and hopes to forge a career in Investment Banking in the future.\nHow did you hear about SEO London and why did you apply?\nI remember the day I first heard about SEO. I met a cool guy at this Deloitte event we were both attending; I was in year 12 and he was in year 13. We started talking about careers and he mentioned SEO. That night, I remember going on his LinkedIN and scrolling through an endless list of experiences and achievements. That's when I told myself that I was going to achieve what HE had achieved.\nI'm proud to say that I too have had a lot of experiences with SEO. I've been on the SEO Scholars programme as well as the Academy. In fact, I've enjoyed three stints at Barclays for the past 3 years. I also had a stint at Latham and Watkins in the team that covered Private Equity. I learned two things; that law wasn't really for me, but perhaps at some point in the distant future, I'd like to do Private Equity. Knowing this early on is really important!\nThe formative moment was at my first assessment centre - it was for Citi, and I did it through SEO. I saw how friendly one of the candidates was with the SEO staff. So though the entire event was very professional, I saw a human element to how SEO related to the people it aimed to help. And that's the operative word - help. They're not here to be stern or to browbeat you.\nThough I didn't have expectations coming into the programme, this was one of the many surprises I've encountered. But quite apart from that, I think I just expected SEO to expand my horizons as I wanted to do something beyond just school and classwork.\nSo perhaps in that vein, the Scholars Programme delivered, and then some. For that I'm really grateful. It's been a really positive experience that's made a difference in my life.\nIn my gap year for example, I received a lot of support. SEO gave me motivation to really push myself and take that time to improve upon my grades. They also provided me with experiences to further enhance my prospects and gave me much needed exposure to the professional world.\nI can't stress how useful this has been. Through SEO I've met so many people and I've done so many things. I'm definitely more well-rounded as a result. But more than that, I've become a lot more self-aware. Through these experiences you pick up on a lot of unspoken things like the proper way to write a professional email - we weren't taught this at school. These soft-skills are typically distilled to public school students but not so much to state school students like me. I don't think the value of this is emphasised enough.\nFinally, through my experience I've learned many valuable lessons. In a nutshell; I learned that I can do it. I'm now so empowered to go out in the world and make an impact. As a 125% person who knows no limits, I believe in hard work but I also know that not always enough. You need opportunity and SEOs helps you get it.\nAdopted at just 20 days old by Italian parents following birth in Brazil, Marco, aged 25, grew up in Italy and lived there until 2010 before moving to England at age 19. After a period of youthful exuberance in London, he decided to reevaluate his options and improve his prospects. Whilst he'd been a strong academic performer with A*s at GCSE level, he couldn't attend university without Level 3 qualifications, so he decided to return to Italy and complete his A-Level equivalent. He'd always enjoyed learning so this seemed a logical thing to do.\nMarco cites going to university as one of his greatest achievements. It was the culmination of a mighty effort; he worked during the day to pay his way through school, and had to study through the night. This was tough and because of extenuating circumstances, he was unable to attain the grades he was capable of, but still, he managed to gain admittance into University or Westminster for a BSc in Economics.\nDetermined to prove himself, Marco began to aim high with the objective of making a name for himself in the corporate world. He currently has an offer from AON and has had multiple work experience stints at BP.\nThis entire period taught him about drive, motivation and ambition. He discovered that harnessing these internal forces can lead one to accomplish things way beyond the imagination. But also crucial is the help of others - he notes innumerable tips, suggestions and help were pivotal in his attainment of competitive internship and placement offers. Above all, this entire period has taught him how to be more responsible. The years leading up to the present were for Marco invaluable in terms of building his confidence and in teaching him the value of making mistakes; mistakes are valuable learning experiences and he acknowledges how drastically his life has changed from when he first came to the UK, aged 19, to now, at 25.\nI stumbled upon SEO rather fortuitously whilst I was browsing for work experience opportunities on the internet - I was going into second year at uni. I found a page advertising SEO Summer Launchpad and I decided to apply. For the time I've been on the programme, I've done the SEO Summer Launchpad through which I got a year-long placement scheme at BP.\nThroughout my interactions with SEO's dedicated team and in attending their programmes, my experiences have been exceeded, without a doubt. I came into it without much in the way of expectations but I would have never imagined having as many opportunities as I've had now.\nSuffice to say SEO has helped me a lot. I think the greatest impact has been academic and professional. To cite examples, I've received an offer from AON and I've done a year-long placement at BP. From a purely professional perspective, I now have a deep, almost intrinsic understanding of markets and it has allowed me to discover myself and what I enjoy. I also learned a great deal about my passion to interact with customers as well as my desire to work with mathematics. SEO allowed me to explore my interests in a professional setting to see what was really for me. Without SEO, I'm not sure I'd have discovered consulting, which I believe is what I'm most suited to. Personally speaking, the SEO programme helped me crystallize my values. It has also helped me think more seriously about giving back - it has amplified my desire to help others and it has given me the drive to take action and 'just do it'. But more fundamentally, I think the experiences I've had with SEO allowed me to find my niche and I now have a sense of direction.\nThere are plenty of takeaways and I think these will be useful to people who are thinking about joining SEO. Firstly, you have time. I think life can appear scary and because so many things compete for our time and attention, decisions can appear to have a finality which isn't reflected in reality. It's good to have an open mind and commit to finding what you want. Every experience is valuable, even if it teaches you about what you shouldn't do.\nMercedes is an Industrial Economics undergraduate at Nottingham. She lives in Cambridgeshire with her parents; her dad is a lawyer and her mum is an admin worker. She attended a good state school where academic excellence was valued. Alongside her parents' encouragement, she developed a desire to strive for improvement. This has materialised into a sense of determination and willpower which she shares with her peer group.\nLooking into the future, Mercedes sees herself working in financial markets. Aside from Banking, she's also looking at financial journalism. This stems from her passion for news and knowing what's going on in the world. She also hopes to set up a business in the future.\nI first heard about SEO at a women-only work experience programme with HSBC. An upcoming SEO event was brought up in conversation and it piqued my interest. Being from the countryside where Banking careers are rarely talked about, this was all new to me. After discovering a little bit more about SEO, I applied!\nFollowing acceptance to SEO, I did the First Year Fast Track, which I highly recommend. I got to hear from many different banks, some I'd never heard of. I had many conversations with people in the industry and through that I realised that I was suited to Markets. Aside from that, I also learned about what made each firm different, which helped massively with my cover letter. This worked in my favour because it showed that I'd done my homework!\nBeing female and black in a white-male dominated field can be difficult from an access perspective. The help and success I've had with SEO reassures me that SEO's work is breaking down those barriers and as such, I remain a staunch advocate. I'm always encouraging people to apply!\nPersonally, I had high expectations coming into the SEO programme because I knew what my friends had achieved through it. Yet they were exceeded! I was amazed at the help I got with my interviews. I received practice interview questions which really helped put my mind at ease. Suffice to say in spite of my high expectations, SEO still managed to over deliver!\nTruthfully, without SEO, I wouldn't be in the position that I'm in both professionally and personally. It's such a powerful network. Also, going through the SEO programme taught me that I have far more potential than I thought. I never expected to have an internship at Goldman Sachs.\nThat said, I want to make it clear that SEO works to provide opportunity. The onus is on you to seize and maximise it. For example, I did not go out at all during Fresher's. I did all my Spring applications whilst everyone was partying, but I didn't feel I was missing out because for me, my future was more important.\nBecause of that sacrifice, I got many interviews which through even more work, turned into offers. For my Goldman interview, I did lots of research; I knew their stock price, I knew the state of the markets - exchange rates for example - and I followed many different news stories. You have to do the work; SEO is there to ensure that you're rewarded!\nAdele is the first of 3 children born to two teachers. She attended a good state secondary school in France where she got on very well with her teachers and was pushed to apply to France's top university. She was successful in securing a place at The Sorbonne where she graduated with a Bachelor of Geography and a Bachelor of History. She then continued on to Sciences Po where she read a Master's in Public Law and Geography.\nAs she did her Master's, she recalls experiencing difficulties which prevented her from fully focussing on her future. At one point, she had to balance her studies alongside a 20-hour a week job. This proved challenging, but she was able to complete her studies and move to London where she started working as a financial head-hunter. This experience has since inspired her to go from being the recruiter to the one being recruited. She's currently undergoing a Master's in International Wealth management at ESCP in Paris, in the hope of attaining a graduate position in Asset Management at an Investment Bank.\nI first heard about SEO London through a friend whose story inspired me to apply. He hadn't completed any work experience in finance before joining SEO but with their help he managed to get a summer internship and a graduate position at Barclays in Sales and Trading. Knowing I was in a similar position, I decided to apply. For me, the opportunity to learn, to meet a diverse range of people sharing the same goals and aspirations drew me to SEO London.\nNow that I've gone through the programme, it's clear that my expectations have been exceeded. I loved the diverse range of presentations by great speakers. It's one thing to read about their experiences, but hearing from them is really powerful. I also felt that they were really trying to help us; there was a strong sense of community within the SEO network and to know that I'll be a part of this community gives me a sense of pride.\nDrishti is a second-year Economics student at UCL. Her family moved to the UK when she was ten. Her Dad's a Software Engineer and her mum teaches Hindi at a local private school. She has a younger brother who's pursuing Computer Science.\nHaving excelled in Maths at Dr Challoners School, where she completed her Sixth Form education, she decided to study Economics to pursue her love for maths with real world application. She's looking to secure a summer internship in Equity Research or Management Consultancy. Her ideal role would be one which combines knowledge of the market with a sales persona.\nAfter attending many events with the Economics and Finance society, I was approached by the Vice President of the society who noticed that I had a passion and enthusiasm for finance. He told me about SEO London and I applied for the First Year Fast Track programme which I was fortunate enough to be accepted on.\nI found it intense but rewarding at the same time. It was the first time I'd been involved with a diversity initiative and if I'm totally honest, I'd been oblivious to diversity issues regarding gender and ethnicity before this.\nMy experiences with SEO have definitely made a huge difference in my life. I realised my experiences to date had not been representative of the current issues and learned a lot about myself and what field is right for me.\nI'll also take away more contacts, a greater sense of perspective and openness. This specifically links to IBD - I now have a greater appreciation for what they do and why it's important.\nOxford Economics and Management undergraduate Haseem lives in London with his family. His parents both came over from Pakistan to settle in the UK and have lived in London ever since. His dad is a recently retired businessman who formerly ran a small photocopying business. His mum studied Economics, Maths and Statistics at the University of Peshawar and helped with the bookkeeping of the business. She is also a part time beautician. His brother runs a marketing consultancy, which he set up after completing a Business Management Degree at the University of East Anglia.\nBefore starting at Oxford, Haseem went to Ilford County grammar, a state school in Essex. In his first year at university, he secured 10 investment banking spring week offers. Now going into his second year, he holds an offer for a summer internship at Goldman Sachs on the derivatives desk in Private Wealth Management.\nHis immediate goal is to convert his summer internship into a full-time role. Looking further into the future, Haseem aims to work hard and set himself up for early retirement, whilst also building up his network and skill set in order to transition into Islamic Banking. As a devout Muslim, he wants to use his skills to give back to his religious community.\nI first heard about SEO through my sister who's at the LSE. The reason I applied was because I wanted to get exposure; that is, to meet with firms and their employees. I'd seen the firms that SEO worked with and I thought that working with them would be a great opportunity, so I signed up for the SEO scholars programme in Year 12.\nI also did the First Year Fast Track and what stood out for me immediately was how much help I received. I remember sending Kutchi, who is part of the SEO Investment Banking team, my Goldman cover letter which was in excess of the 300 word limit. She sent it back with 50 words cut out and I didn't even notice!\nAlongside my other experiences on the SEO programme, there's no question that my expectations have been exceeded. For example, Zubair, another member of the SEO team, really helped me with my interview preparation. For Morgan Stanley and BAML, 90% of the interview questions SEO asked me came up again!\nI was also impressed by the breadth of the network - SEO alumni are everywhere. Towards the end of my Goldman Spring Week, I had to choose two desks for my summer internship. It was an agonising decision - perhaps the hardest I've ever had to make - and I turned to SEO for advice. They put me in touch with an alumnus who gave me advice and this helped me make a decision. SEO does more than just help people get internships. Even when I'd got an offer, SEO were still looking after me to make sure I made the right choice. That definitely stood out!\nI feel that SEO has made a difference in my life. I think the SEO experience sets you on an altogether different path from the one that would have unfolded without it, and that's very powerful.\nAs for lessons I take from my SEO experience, I'd say I now have a higher estimation of my potential. There have been times when I've doubted myself. I would pursue goals and opportunities without fully backing myself. Yet as time went on, I started to get more and more confident, especially with the help I received from SEO.\nThe results I got confirmed that I had more capability than I previously thought. After getting interviewed by 10 different firms and getting 10 offers, I'm much more confident about how I come across in person.\nWhat did you do to get 10 Spring Week offers?\nI started by being super-organised. My aim was to submit applications as early as possible, so I created an excel spreadsheet with all the banks I wanted to apply to; the opening dates, the deadlines, the divisions, the motivational questions and the different online tests. I knew when the SEO sponsor firms were opening for application so I'd prepare my application and send them to Zubair or Kutchi for them to review. I also structured my cover letter for recycling: The bits about why I wanted to do banking and why I was an ideal candidate could be recycled.\nIt meant I only had to spend time on the section detailing why I wanted to work for that particular bank. For this, I researched their results; what their strengths are; their corporate responsibilities and how they linked to my values and anything in the news, especially if they'd been involved in a deal. Whilst all this took a lot of time, I didn't need to spend as much time as other people did writing them from scratch! I'd always send my cover letters to SEO for review and this prep meant that I could do an application in a few hours when it took others a day or more. I also made a point to do my applications before university because I had more time to apply. This worked out well because when I started university, I'd already done 11 out of 14 applications.\nAnother advantage I had is that I didn't do this alone. I took a gap year but some of my friends didn't. This meant I had a support system of people who'd gone through the process before. I worked on my applications with some of my friends who'd already succeeded and their advice proved invaluable. This combined with SEO's invaluable training and support helped me submit strong applications which resulted in the offers that I got. But I really want to emphasise that this takes work. SEO is there to ensure that certain barriers don't get in the way of you getting the results you deserve from the work you put in. But you have to put in the work!\nSecond-year Sheffield University Economics student Nathaniel is an inspiring and impressive SEO candidate. His story demonstrates what is achievable through hard work and determination.\nHe grew up in a single parent background, living with his mum and three of his siblings. He remembers moving around a lot and changing schools frequently. Very early on, he developed a desire to succeed; he got a part-time job in his teens to help out at home. With three spring weeks from Bank of America Merrill Lynch, Nomura and Blackrock under his belt, Nathaniel's experience shows how SEO can help those from under-represented backgrounds.\nI first heard about SEO London in a chance encounter with a member of an online forum just before starting university. I'd always wanted to work in finance as an analyst - I just wasn't sure about the area of finance. This guy had an impressive resume with many week-long work experience stints at some of the biggest and most prestigious investment banks. Having heard how SEO London had made all this possible, I immediately applied to join SEO because I wanted to emulate his success.\nWith SEO, I've done the First Year Fast Track as well as the recent SEO Banking Academy. I wasn't sure what to expect going into it, but the First Year Fast Track was especially beneficial; what I learned there filtered into my Spring Week success. That said, I put in a lot of work. I dedicated as much time as I could towards my applications. I shunned going out; I'd rush back from lectures to continue - I was just so determined. That's probably the most important ingredient: work ethic.\nAt the Fast-Track, I obtained first-hand accounts of working life in the financial industry. I gained a more intimate knowledge of the various roles in finance and specifically, in investment banking. I also attended multiple insight days at leading institutions and learned how to deliver presentations and stock pitches. But above all, I met really good people, many of whom have become my friends.\nAlso, my SEO experience has allowed me to expand my network and build an extensive network of contacts. And best of all? We've helped each other with applications, exchanging valuable tips and supporting each other throughout the journey.\nThis I feel has been critical in the success I've had. I feel that SEO has already made a huge difference in my life, especially from a professional standpoint as I'm now well on the path to a career in Investment Banking. I recognise that my background could have led my life along a trajectory quite unlike the one I'm currently on. SEO provided the opportunities which helped facilitate that shift.\nLooking ahead, I feel much more certain about my future; I have a sense of clarity which I did not possess prior to joining SEO London. More importantly, my success has given me confidence and self-belief which I'll take with me moving forward!\nAspiring Trader Rebecca is a University of Surrey Economics and Finance student. Her parents are in the medical industry - her dad is a doctor and her mum is a nurse. Her desire to go into Finance came from her brother - a Warwick mathematician who also wants to go into Investment Banking. She's always looked up to him and watching him excel academically motivated her to do the same.\nBorn in Scotland, Rebecca and her family moved around a lot, but she attended secondary school at Manchester High School for Girls. From there she moved to Saint John Payne School. For A-Level, she was at Chelmsford County High School for girls. She treasures her educational experience, having been to state, grammar and independent schools.\nShe currently has her sights set on becoming a trader at a bulge bracket Investment Bank with the intention of developing her career and rising through the ranks.\nI heard about SEO London through friends, just before I started university. I submitted my application for First Year Fast Track and got a call letting me know I'd been selected. In the same call they were kind enough to give me advice before my Nomura Spring Week interview and I was lucky enough to get it.\nThrough the SEO First Year Fast Track I got a place on the Women's Emerging Program at Nomura; it's a two-week long programme. I was assigned a desk with 5 screens and it was up to us to go around various desks and ask if we could shadow. We were given lots of exercises to do and one of them was a 10 minute stock-pitch! I also did a BP-IST insight day and prior to that, I'd had a week's summer experience with BP.\nSEO has really built my confidence. I used to be really shy and I'd never put my hand up to answer a question. I was encouraged to get more involved. I was pulled aside and reassured that this was a safe space. Now I feel like I can speak in front of people. I had to deliver a speech to become Head of Investment for the Economics Society - something I would have never done a year ago - and I actually got it!\nI think I've gained more of a passion to pursue a career within trading. I know that currently there are many initiatives to get more women into trading. I've received nothing but encouragement and support from SEO and even been introduced to traders from Citi.\nUsama Yusuf maximised the opportunities afforded by the SEO Scholars programme during his time at secondary school. As a result, he has secured a fully-funded university scholarship to study at University College London and has just completed the first year of his undergraduate studies.\nThe first time I came across SEO was at school when my school teacher told us about this programme called SEO Scholars. He mentioned that the benefits of the programme included mentoring, help with university applications, CVs and cover letters, possible university insight trips, and even the opportunity to gain some work experience. As a 15-year-old, who just wanted to do all he could to better his future, I was sold on the idea and applied.\nI can now safely say, upon reflection, that applying to SEO Scholars was one of the best decisions of my life. With the help and support of SEO, during my time on the Scholars programme I managed to complete work experience at various institutions; I did 3 days at Barclays, one week at Freshfields Bruckhaus Deringer and 4 weeks at Citi.\nOn top of the work experience, I went to corporate insight days at KPMG and Goldman Sachs. I went to university open days at Southampton and Cambridge and even had the opportunity to bowl with bankers from Goldman Sachs at a charity event. All of those experiences and all of those different learning opportunities were well worth the 30 minutes it took me to apply for the programme.\nIf it wasn't for the help SEO had given me over the years, and the opportunity to be able to dream big as a result of the different exposures I got whilst I was on the SEO scholars programme, I would have not gotten into UCL and I would have not have secured a full scholarship from Credit Suisse.\nSEO London offers fantastic programmes with fantastic opportunities and systems in place to ensure that you gain the most from your time on the all the different projects they offer. To top it off, it has amazing staff who care deeply about your success and your journey. I would recommend SEO London to all young people and would strongly encourage them to apply!\nHave you got a story you'd like to share?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"in a late 90's grey Geo Prism.\nand it is sitting on the little boy's lap.\nwhile staring out of the passenger side window.\nThe wires are exposed and the sun is rising.\nA light summer fog is in the air.\nIs the author an exposed wire fetishist?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Marvel comics have entertained generations of readers\u2014while influencing broad segments of popular culture with characters like The Human Torch, Captain America, The Incredible Hulk, The Fantastic Four, and Spider-Man\u2014for over 50 years. Daniels tells the Marvel story with grace and wit, and takes us behind the scenes to meet the comics' creative writers and artists. Illustrated with over 700 examples of cover art, story art, memorabilia and photographs, and 17 profiles of main Marvel super heroes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As a Santa Ana DUI lawyer, I go out of my way to insure you are protected in court and with DMV. However, you only have 10 days to report to DMV your arrest for DUI and to request a DMV hearing in order to protect your driving privileges. You must call us immediately after your release in order to give us a chance to help request your DMV administrative hearing before the 10-day deadline.\nMost DUI arrests happen during holidays, at checkpoints, and sporting events. DUI arrests in Santa Ana often happen in major streets such as Harbor BLVD, Main Street, Bristol Street, Edinger Avenue, Warner Avenue, First Street and other such larger streets. Larger streets are major arteries with lots of traffic, increasing the odds of finding a drunk driver. If you find yourself having had a few drinks, it's always best to call a ride service company or a taxi. However, if it's too late, please call us and we'll help mitigate your damages. It's a very tough proposition to go alone against the district attorney in DUI cases.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Want to become a .NET Developer with all the perks?\nValuable tips and tutorials about Microsoft's and other technology stacks.\nSpecial offers when new courses are released.\nI'll also share with you my journey from the humble beginning to where I am now, and what will happen next.\nBy teaching students practical programming skills, CodeStrengthen contributes to the improvement of life via career change and advancement. Whether you are looking for your first developer job, aiming for a higher salary or starting a side business, we provide the technical skills you need to be successful.\n\"Being a developer myself for more than seven years, I experience firsthand an amazing number of opportunities you will attract by knowing how to code! Through easy to follow video lessons, you will learn to develop, design, test and deploy real-world business applications. It is the best way to learn FAST and to achieve your goals quickly.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Thomson Reuters Corporation (TRI.TO) reported a profit for third quarter that dropped from the same period last year.\nThe company's profit totaled $291 million, or $0.37 per share. This compares with $348 million, or $0.46 per share, in last year's third quarter.\nExcluding items, Thomson Reuters Corporation reported adjusted earnings of $74 million or $0.11 per share for the period.\nThe company's revenue for the quarter rose 1.6% to $1.29 billion from $1.27 billion last year.\n-Earnings (Q3): $74 Mln. vs. $196 Mln. last year.\n-EPS (Q3): $0.11 vs. $0.27 last year.\n-Revenue (Q3): $1.29 Bln vs. $1.27 Bln last year.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Thank you for pledging to Vote 1 Sun this election and stop attacks on Queensland solar.\nCan you share this pledge with friends and family on Twitter or Facebook?\nThere are hundreds of thousands of Queensland solar supporters and owners - let's get them all on board this election!\nCan you pitch in to help us get mobile billboards roaming in key electorates before election day?\nI want to be sure that my green, solar power that I'm giving to the state power companies for no generation cost other than a small distribution cost is being used for others at the price I sell it at, not the price it is resold at.\nI think it would be fair if every kw produced is taken off our bills.\nAll around the world solar power has been taken up more rapidly than mobile phones, but those politicians living in the pockets of fossil fuel companies either don't notice or have their eyes firmly shut. We are not going to go away; we are the wave of the future.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Stereogum recently premiered Bayonne's official music video for \"Spectrolite.\" The song is said to be about \"a stone that his girlfriend at the time brought him back from Australia,\" which is reflected through the music video's travel theme brought to life with stop-motion animation by director Lauren Gregory. Read more about \"Spectrolite\" via the Stereogum premiere here. You can view the video for \"Spectrolite\" here.\n\"Spectrolite\" is currently on Spotify, as well iTunes and Amazon.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"PainCreamsDelivered.com was created in response to an unmet need in the healthcare industry. Patients across the nation are suffering from chronic pain, so they turn to dangerous, sedative, addictive oral pain medication. Most are unaware of alternative treatment options. We've partnered with compounding pharmacies and doctors around the United States to provide safe and effective topical pain creams to patients.\nOur pain creams are non-addictive, non-narcotic, and non-sedative. They offer direct and on-site pain relief. Because the creams are applied topically, there isn't the same risk of organ toxicity and gastrointestinal distress that patients experience with commonly prescribed oral medication.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I've had dental braces for about four months now and I wanted to take a scuba diving course in the summer. My dentist says I should not but I wanted a second opinion. Will it cause any problems with my brackets, will I need any special equipment, etc? Thanks.\nIf you are wearing rubber bands in your braces from jaw to jaw, i recommend taking them out for the dive. Wiggling your jaws to cope up with ear pressure may cause the rubber bands to snap and get lodged somewhere in your mouth or possibly swallowing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"The format is following the ideas of the highly successful Winter Olympic Games disciplines of ski and snowboard boardercross, where riders follow a set track with Obstacles they need to jump over. The whole competition is held in multiple elimination series, where only the best riders advance from round to round until the final.\nThe IKA and the World Sailing Youth Olympic Games working party have been working on the final details of the format since the confirmation of the discipline in 2016 to provide riders, coaches, national associations and event organisers will all necessary information.\nMore content and the final qualification event dates will be added as soon as tbey become available.\nThe final rulebook, including on-water refereeing, will be published shortly, and well before the next major event, the 2017 TT:R Slalom European Championships in Gizzeria, Italy.\nThe event is open to competitors from all over the world. Riders will compete in separate age disciplines (open men, open women, Boys U19, Girls U19).\nTo allow riders ideal preparation for the event, a coaching clinic will be held by multiple Kiteracing World Champion Steph Bridge at the same venue from 8 to 11 July 2017.\nWaarmee het low-wind (foilen) idee aan de kant geschoven is?\nDus Rob is een beetje slordig geweest, duidelijk.\nMaar heeft dit ook gevolgen voor 2020? Is dit dan ook het idee voor 2020 of gaan we toch naar een echte \"zeil\" benadering met driehoek en foils?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"After promising results with Parkinson's patients, researchers are trying the new therapeutic approach against Alzheimer's.\nThe U.S. Food and Drug Administration (FDA) has given the go-ahead for research trials to determine whether deep brain stimulation (DBS)\u2014a procedure involving the implantation in the brain of a pacemaker-like device\u2014can halt or slow the progression of Alzheimer's disease (AD). The technique has been used with success in the treatment of symptoms of Parkinson's disease (PD), and preliminary studies involving small groups of AD patients have been encouraging. Up to 50 people with early signs of AD will be recruited to participate in the AD trials, known as the ADvance Study, at seven different sites in the U.S. and Canada.\nDBS for AD patients involves the implantation of a battery pack underneath the skin of the chest which delivers low-grade electrical impulses through a subcutaneous wire to an electrode positioned in an important memory pathway in the brain called the fornix. In preliminary studies, electrical stimulation of the fornix appeared to increase neuronal activity in that region, and was linked to improved cognition and slower cognitive decline in many patients.\nThe ADvance Study is intended to add to scientific understanding of the precise mechanism by which electrical stimulation brings about brain changes. Half of the participants will be fitted with active pacemakers, and half with placebo pacemakers that provide no stimulation. Researchers will compare the two groups over a 12-month period, and then activate the DBS system in the placebo group. All participants will undergo periodic psychological, physiological, and cognitive assessments, and brain scans will be performed to measure glucose metabolism and changes in brain structure in memory regions.\nMany of the sites collaborating on the ADvance Study (see What You Can Do) are in the process of recruiting study participants. Applicants for the trial must be in good general health and diagnosed with early-stage AD, between 55 and 80 years of age, live at home, have been on a stable dose for at least two months of an FDA-approved drug for AD (Aricept, Razadyne, or Exelon), and have a reliable spouse, relative, or caregiver who agrees to cooperate with the study.\nTags add, alzheimer, aricept, brain, brain activity, cognitive decline, exelon, food, glucose, health, memory, metabolism, parkinson s disease, razadyne, skin, strength.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Universal Audio has long had a successful business selling hardware DSP effects, many of them carefully-modeling classic analog gear. These products use dedicated DSP hardware for number-crunching, requiring that you connect an extra box to your computer. UA has certainly had their loyalists, and for fans of the products, the dedicated gear is simply a convenient way to get all of these sound-processing goodies. But it's fair to ask the question, as many producers have who read this site, what's the advantage? Why not simply use native processing on your computer?\nApollo, UA's new hardware, answers that question more emphatically. By integrating the processing prowess of the UA platform into a high-quality audio interface, you can now add UA effects live, as you record and mix, with extreme low latencies. UA reports latencies below a couple of milliseconds. That's possible, theoretically, on a desktop computer, but not generally on a laptop and very often not with any real reliability. You can do it in a lab, but it's not something typical users see.\nSo, in one box, you effectively get your whole studio: the audio interface, the DSP power, and real low-latency sound processing. It's not the first audio interface with DSP, but it might be the most compelling case yet for why that combination make sense.\nAnd here's where things get interesting: via Thunderbolt, a single MacBook Air, costing just around $1000, could be your whole studio machine. And while Apollo runs a couple grand above that, that means the total price tag is stunningly low compared to what you'd pay just a short time ago.\nAll of this is academic until you actually have something to do with sound. So, UA is also expanding their developer platform to additional outside development; more on that soon.\nApollo isn't for everyone; obviously, some people won't like being tied to hardware, and native plug-ins do work for a lot of people. But it does solve problems for many potential producer customers by making something reliable, predictable, low-latency, extensible with lots of excellent processing tools, and all in one single-box solution.\nApollo will initially be Mac-only, but will come to Windows, too \u2013 and with more PCs supporting Thunderbolt in 2012, that means the MacBook is far from your only choice. So, you've got one add-on that's your interface, your pres, and your mix\/master\/effect toolbox.\nDedicated front-panel controls: preamp gain, channel selection, mic pad, +48V phantom power, low cut, monitor level, and dual headphone controls.\n4 digitally-controlled analog mic preamps, 8 balanced line inputs and outputs, dual front-panel JFET DIs, digitally-controlled analog monitor outputs, 8 channels of ADAT, 2 channels of S\/PDIF, word clock I\/O, FireWire 800 (standard), and a Thunderbolt expansion bay \u2014 making it a well-equipped centerpiece for the modern project studio.\nAnalog emulation plug-ins from Ampex, Lexicon, Manley, Neve, Roland, SSL, Studer, etc.\nThunderbolt will be available on a sold-separately Option Card; UA says it reduces latency and audio buffer sizes, improves high sample-rate performance, and allows greater UAD plug-in instances over FireWire.\nOf course, because Thunderbolt also connects to FireWire devices, you don't lose your FireWire investment. The only bad news is that you only get Thunderbolt here as an Option Card; I imagine we'll eventually see UA ship Thunderbolt connections standard.\nThere are both two-core and four-core versions, powered by Analog Devices SHARC processors, running an estimated street of US$1999 and $2499, respectively. Apollo's Thunderbolt Option Card will be shipping in the first half of 2012, with pricing TBD.\nWindows 7 summer; 10.6 and 10.7 Mac OS X when it ships.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For many years now, the state of Maine has been fertile ground for economic innovation. Public banking has been part of that story, and 2015 has featured passionate speeches, intelligent advocacy, and hard-hitting economic analysis. Randall Parr has been instrumental in keeping the movement alive and, importantly, in communicating goings-on in Maine with the rest of the nation.\nBack in 2013, Randall Parr published an OpEd in the Bangor Daily News analyzing the state's budgetary shortfalls. Parr traced the need for the legislature to raise taxes (hardly popular in a state that values fiscal responsibility) to the state's failure to create a public bank at that time. Parr used the example of the Bank of North Dakota to point out the potential of a public bank to deposit millions of dollars into government coffers.\nA public bank could have insulated Maine from reduced tax revenues during economic contractions, increasing liquidity, jobs and personal income, as the Bank of North Dakota has done for that state.\n. . . The 2013 \"Maine Treasurer's Cash Pool Report\" indicates millions of dollars were deposited by Maine in private banks and U.S. government securities during the fiscal year that ended June 30. If invested in a state bank, it could have created an equivalent amount of credit, which the \"money multiplier\" could have multiplied to billions of dollars.\nNow, in 2015, Maine has an opportunity to duplicate North Dakota's success. A bill to establish The Maine Street Bank was introduced in the Maine legislature on 13 January 2015. Titled An Act to Create a Public Bank, the bill proposes to create a public bank by 1 July 2017 for the state of Maine that is largely modeled after the nearly century old Bank of North Dakota. It would also authorize the formation of public banks in municipalities and cities around the state.\nRepresentative Dianne Russell, a former member of the Nation magazine's national \"Progressive Honor Roll\" who is in her fourth term in Maine's House, is the primary advocate for the bill, which she says has the potential to save working people from the tragedies of business closures and unemployment.\nEven though Maine is required to balance its budget by law, its latest budget again predicts shortfalls for 2016 and 2017, which means that to achieve a balanced budget Maine must either reduce spending or increase revenues [See p 21 Tables c1 and c2 2016-7 \"Budget Overview.pdf\".] This is the same situation that Maine faced at this time in 2013 and prior years. The 2008 recession created continuing deficits in each state without a state-owned bank.\nRecessions result in higher deficits because consumption and investment fall during recessions, generating less tax revenue. After 2008, U S state economic activity increased only in North Dakota, which has had budget surpluses each year since 2008, while all other states had shortfalls - because only North Dakota has a state-owned bank into which state funds are deposited by law. All other states are dependent on shenanigans of the national financial system, while North Dakota is shielded from it. Banks create money when they lend it, and extinguish money when loans are repaid. North Dakota creates its own money supply within its own borders, while Maine deposits state funds in private banks in France, England, Japan, Canada, and Wall Street. So Maine is helping foreign economies by exporting its money supply while North Dakota is boosting its own economy by lending within its state..\nLD24 gives Maine a chance to create a state-owned bank modeled on the Bank of North Dakota to create an economy which is independent of national recessions,. With a state-owned bank, Maine's budget would be more stable and would be independent of the ups and downs of the U S economy. Maine would no longer have to deprive its children of a decent education and deprive its citizens of decent health care or other needs to balance its budget.\nThe proposed budget for 2016-2017 plans to reduce the income tax rate for wealthiest taxpayers and continue lower brackets unchanged. Since wealthiest taxpayers spend far lower shares of their incomes, consumption will not increase due to tax cuts for the top tax bracket; thus economic activity in Maine will not increase; new jobs will not be created; incomes will not rise; prosperity will not result; and poverty will not be reduced. Cutting the top bracket will yield smaller state revenues and increase deficits.\nNorth Dakota's success is not due to its oil and gas production, which only began in 2010. In fact North Dakotas recent fossil energy boom has caused excess wear and tear on roads costing more to maintain, reduced its ability to transport farm produce due to oil and gas shipments, and suffered costly deadly accidents caused by exploding gas and oil. Other fossil energy states have faced deficits and recession like Maine. Economic activity fell in California, Texas, Oklahoma, Louisiana, and Alaska since 2008 while North Dakota activity grew. U S fossil fuel subsidies, including the depletion allowance, make Oil and Gas more profitable than renewable energy.\nLocal banks are more profitable in North Dakota than in Maine. Loan delinquencies are lower in North Dakota than Maine. Foreclosure rate is lower in North Dakota than Maine. Personal income is higher in North Dakota than Maine. Unemployment rate is lower in North Dakota than Maine. Upward mobility is higher in North Dakota than Maine. Education spending growth per student is higher in North Dakota than Maine where it is declining. North Dakota does not pay the debt service costs like Maine and its state bank returns about 18% on assets to its state treasury annually.\nIn a simulation, assuming 70% of all Maine cash deposited in a state-owned bank lent within Maine would create over 100K jobs, increase personal income over $4B, increase revenue over $200M, and eliminate debt service payments because the state bank could lend money for state projects for infrastructure and other needs. Details are in my book, Occupying a new Maine Economy, creating a State-owned bank. I provided copies of this book to the Governor and state legislators.\nIn his state address Maine's Governor set the goal of economic prosperity and ending poverty. Cutting the top income tax rate will lead to continuing budget deficits would increase, not end, poverty. It will put us deeper into the hole that Maine people now occupy. The Governor's strategy is a \"trickle-down\" approach that has failed repeatedly to revive the economy over the past 30 years and quadrupled the national debt between 1980-8.\nFor more information about Maine's public banking movement, visit the Maine Public Bank Coalition.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Anvil! The Story of Anvil: No, really: This actually is like a real-life Spinal Tap; it's the story of a Canadian metal band that sank into oblivion yet never quit \u2014 and still hasn't gave up hope. Director Sacha Gervasi is a fan who takes his subject seriously and turns out a fantastic film. Not rated. Bijou. See review this issue.\nCrank: High Voltage: Jason Statham reprises his role as a hitman who has to undertake stranger and stranger escapes in order to stay alive. Now, his heart's been replaced with a battery-powered thing that requires jolts of energy. But he wants his real heart back! R. 85 min. Movies 12.\nHangover, The: This summer's dirty-fun buzz movie stars Bradley Cooper, Zach Galifinakis and Ed Helms as three guys who have no idea what happened at the bachelor party last night. Where'd that guy's tooth go? Where'd the baby come from? Apparently it's a really good time finding out. Cinemark. VRC Stadium 15.\nHeckler's Night: Mock loudly and cheerfully as the Goat screens Commando. 7 pm Wednesday, June 10, Wandering Goat. Free.\nMelvin: Local filmmaker Henry Weintraub's latest work is a feature-length film about the titular kid (Leif Fuller), who rises from the grave to enlist a nerdy college kid (Patrick O'Driscoll) to take out the bullies who killed him. Bloody, funny and entirely Oregon-made. 11:30 pm Saturday, June 6, Bijou.\nMy Life in Ruins: The star of My Big Fat Greek Wedding, Nia Vardalos, returns with another Greek adventure; this time she's a professor who heads to Greece to work as a tour guide. Of course she meets a man there. Cinemark.\nNext Day Air: When a shipment of high-grade cocaine ends up in the hands of two small-time crooks, everything goes to hell. It's not the plot I want to see this for, though; it's Mos Def, who brightens up the trailers. With Donald Faison, Mike Epps and Debbie Allen. R. Movies 12.\nRocky Horror Picture Show, The: Do the time warp again! Catch the long-term 1970s camp cult classic fave. R. Midnight Saturday, June 6, Bijou.\nTwelve: Nikita Mikhalkov's Oscar-nominated (for Best Foreign Language Film) film is loosely based on Twelve Angry Men; here, 12 male jurors debate the case of an adopted Chechen teenager accused of murdering his Russian stepfather, in the process revealing their own personalities. \"A powerful new film inspired by a powerful older one,\" said Roger Ebert. PG13. 160 min. Bijou.\nDance Flick: How many genres are left for Hollywood to make spoof flicks of? I hope we're running out. PG13. VRC Stadium 15.\nUp: In the latest film from Pixar, a crotchety old balloon salesman sends his house into the sky (via balloons, of course) to escape from it all \u2014 only to find that he has an unwanted stowaway on his porch. The praise is already flowing. PG. Cinemark. VRC Stadium 15. See review this issue.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Fashion is an essential fraction of your existence. Whatsmode designs are amazing which you cannot overlook as it offers you with unique designs that add more charm to your personality. Though fashion is something which is ever altering and you need to transform with it too.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"98-ND-1 represents a major step forward, just as the 10 hybrids from 98-NP do. 98-ND-1 is yellow, about the size of danfordiae hort. So what makes it special? You can see the \u00c7at parent via the dotting near the fall's throat, along with its white patch. Interestingly none of the 98-NP hybrids show this characteristic (the dotting in particular also shows in 98-OO and 98-PR clones). 98-ND-1 also shows bicolour possibilities. I have been wanting good breeding stock from the \u00c7at hybrid to use with sxd clones -- wanting its genes in hopefully larger, more robust clones, etc. Time will tell whether this proves out for 98-ND-1. It's not large, but what's more important by far is the genetic variability it brings to the table. I'm expecting good things from this clone down the road.\nContrasting the future promises of 98-ND-1 I believe some of the 98-NP clones are ready for introducing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Strathroy, ON \u2013 The Strathroy Brewing Co. is now open in Strathroy-Caradoc, just west of London, ON, at 62 Albert St.\nIt was originally reported to be nearing its opening date in January 2013.\nStrathroy Brewing launches with its 5.5% ABV 1812 Independence Pale Ale, an unfiltered, bottle-conditioned Pale Ale that utilizes English and Belgian yeast strains.\n\"1812 Independence Pale Ale is dedicated to the strong Canadian spirit embodied by the peacemakers who protected our lands from invasion during the War of 1812 and preserved our independence from our American neighbours. Canadians from all walks of life tenaciously resisted the American invasion until peace was restored in 1815,\" writes Strathroy Brewing.\nThe brewery's a retail store is stocking 1812 Independence Pale Ale in 355mL bottles.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Today, we're reviewing the ten best water hoses.\nEven though simple, water hoses are functional household accessory that are perfect for refilling water reservoir tanks and watering flowers.\nIn combination with powerful nozzles, they are also perfect for cleaning vehicles, cleaning dirt and grime from wall and garage floors, and filling an emptying outdoors tubs and standalone pools on demand.\nIf you perform any of the highlighted tasks often and want a well-built water hose that lasts long, this article reviews the top 10 best water hoses in the market that will satisfy your needs.\nApex 8695-25 is a well-built 25-foot by 5\/8-inch hot water hose with a commercial-grade all-rubber construction. It is tough, holds hot water up to 180 degrees without melting or losing its shape, and has an advanced, flexible design that does not kink or break over the years.\nThis makes it one of the most suitable brands for cleaning grit and grime from garage floors and filling outdoor recreational accessories such as hot water tubs. Apex 8695-25 is hot water hose is durable and affordable.\nAt 50 inches long and 3\/4-inches thick, Gorilla Hose is an expandable hose made of durable high-strength fabric. The fabric is kink-proof. It is also durable and comes lined with a self-cleaning latex core that prevents build up of grit and growth of molds and mildew.\nGorilla Hose is lightweight, tangle-free, and has solid brass ends that do not leak once fitted to plastic and metallic faucets such as taps in the home. With each purchase, you get a 12-month manufacturer's warranty for defects.\nAs its name suggests, this 25-feet by 5\/8-inch NeverKink 8615-25 Series 2000 garden hose by Teknor Apex is an advanced watering accessory that does not kink, crack, nor tangle as many low-quality models do.\nIt is ultra-flexible, has Rigid Sleeves at its ends that prevent kinking on faucets, and high-strength and crush-proof aluminum couplings that last long. NeverKink 8615-25 Series 2000 is cheap, has an advanced Microshield against mildew and mold and has potent antimicrobial properties.\nDo you depend on water hoses to water your lawn and flowers, clean your garage, and clean your car in the evenings when you return home from work? Have you experimented with several cheap and high-end hoses with negative results?\nIf you are shopping for a new garden hose and want a long and durable model, NeverKink 8642-100 Series 3000 by Teknor Apex is one of the best around.\nIt is 100 feet by 5\/8-inches, has an extra-heavy-duty design that remains flexible down to 45 degrees, and high-strength aluminum couplings that are crush and leak-proof. It ships tangle and kink-proof, has rigid sleeves that protect it from kinking on home faucets, and has a well-made antibacterial Microshield that also prevents mildew and mold.\nWater Right PCH-050-MG-6PKRS is a lead-safe coil garden hose made of durable polyurethane. It is 50 inches long, 3\/8-inches thick, and has a striking olive green theme that users find interesting.\nThis hose is USA-made, 100% lead-free, has a UV stabilized NSF and FDA-approved design that never tarnishes, and has specially engineered strain reliefs that protect it from kinking and or losing its functionality in all weather conditions.\nOther desirable features are its 12-inch tails that ease installation on faucets, chrome-plated and machined brass fittings that lasts long, and phthalate and lead-free construction.\nEven though shorter than many high-end hoses in the market, Teknor Apex 828VR-25 AquaDrain is a flexible and functional model that is suitable for doing light watering and cleaning jobs.\nIt is flexible, kink-proof, and very easy to use. It is also light, tangle free, and made of lead, phthalate, and BPA-free components that do not affect health. You will enjoy using it.\nA recommended hose in top 10 best water hose in 2018 reviews, Orbit 27890 is a 25-foot coil hose with a six-pattern-spray nozzle that will satisfy your range of watering needs. You can use it for cleaning your car.\nYou can also use it to clean your walls and or floors without sacrificing performance. This hose is affordable and coils to a compact size for convenient storage.\nFor homeowners shopping for functional water hoses for regular usage, Gilmour 10 Series is a 1\/2 inch by 50 feet flexogen water hose with durable 8-ply construction with a burst strength or approximately 500 PSI. Its polished surface does not kink.\nIt resists mildew, stains, and minor abrasions, and has a Flow Guard Plus protective collar that resists kinks at faucets. As many hoses highlighted here, Gilmour 10 Series hose is cheap, crush resistant, and has Full-Flo machined metal couplings.\nA best seller online, this 15-foot by 5\/8-inch hose by Apex is a well-made watering accessory that will satisfy your light-duty gardening needs.\nIt is USA-made, made of reinforced 3-ply material, and has stylish yet crush-proof standard brass couplings.\nCamco 22853 is our pick for the best water hose in 2018. Its 20% thicker heavy-duty water hose is durable than many standard hoses in the market.\nIt is long (50 feet); lead, phthalate, and BPA-free; and has a CSA low lead content-certified design that does not affect health nor leave an unpalatable plastic taste in drinking water. It has a durable fitting and strain relief ends.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our slow cooker is an essential piece of kitchen equipment in our house. We use ours multiple times per week for a wide variety of meals and ingredient preparation.\nIn truth, a slow cooker doesn't so much save time but move it. It allows you to do the prep work for an evening meal earlier in the day.\nHere's an example. Let's say that we know that a given evening is going to be very busy. One child has soccer practice, another child has dance practice, and my wife has to stay late for an after-work meeting. Our solutions for dinner are now limited. We're either going to eat out, get take-out, or prepare something very quick and dirty at home.\nA slow cooker solves this problem. Whenever we arrive home, a good home-cooked meal made from basic ingredients is sitting there fully cooked, hot, and waiting for us. We just set the table and eat \u2013 it's far faster than even eating out.\nSince you're able to make the time transition described above, you've suddenly made it possible to cook at home when it previously didn't really work all that well.\nYou've either turned a meal eaten out to one eaten at home, or you've turned a prepackaged meal into a fresh, healthier, and probably tastier meal.\nIn either case, that's a net gain. The gain is clearer when compared to a meal eaten out, but it's still prevalent when compared to a prepackaged meal, as the costs on those are often quite high compared to what you get. Even if your costs are equivalent, the meal you've prepared is of better quality than the prepackaged one.\nFor your first slow cooker, I'd suggest picking up a very low-end cooker, just to see if you'll use it or not. A low-end slow cooker (or crock pot, as they're often labeled) usually just has a dial on the front that enables you to choose warm, low, or high for settings and is \"on\" whenever it's plugged in. Straightforward, indeed.\nIf you find you're using that one a lot, it's worthwhile to invest in one with some more sophisticated features. A timer is a very useful feature, as is a programmable slow cooker that allows you to have the slow cooker adjust from low to high at a designated time. This enables you to put in the ingredients, have the dish start a few hours after you leave, and kick up to high just before you're planning on returning, resulting in a perfect meal. This really adds to your flexibility with using this for family meals when you're out for the day. A good example of this type is the Hamilton-Beach 6 quart programmable model.\nThere's an absolute abundance of slow cooker recipes online. If you're just starting out, I suggest sticking to simpler recipes. One approach I often use is the \"five ingredient slow cooker meal.\" All you really do is combine five ingredients that you like and know will go well together, put the slow cooker on low for several hours, and then enjoy. The post linked there has several examples of this, but you almost can't mess this up as long as you stick to what seems good together to you.\nOnce you've tried some of these, start digging into the abundance of more detailed slow cooker recipes online. I've posted two different sets of slow cooker recipes over the years (this one and that one) and many of my Friday afternoon food posts are slow cooker recipes. Not only that, there are entire blogs and websites devoted to slow cookers, such as A Year of Slow Cooking.\nThe only real limit is your creativity (or perhaps ingredient availability in your area).","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Saturday was a good day for writing inspiration. I got to spend the day at The Library Station for the monthly ORA (Ozarks Romance Authors) meeting. Bryon stayed home and worked on his boat with Matt and, Grace entertained herself.\nThe great thing about ORA is it really helps keep me pumped for writing. Of course you couldn't tell that from the blog posts because I skipped a day. I did take my first thirty re-edited pages of my novel back and the group critiqued it. Two of the group had read the first draft when it was in first person and liked it better the old way. The rest of them didn't know any better. They all said they wanted to read more so that's a good sign I suppose.\nIt's staying in third person. I'm just not a first person fiction kind of girl. First person is so incredibly limiting. I like being able to read the minds of my characters.\nI made it through all of my first paper and ink edits. Now I have to go back and fix them all in Word. I swear, editing is more difficult than the writing. I say I made it through all of my edits, but Saturday one of the authors in the club took a bit of time to give me some advice and pointed out how many times I was using the passive tense. It was like opening the Pandora's Box of passive voice and she only read two pages.\nHere's the difference in passive and active voice straight from the one true source, Wikipedia: Caesar was stabbed by Brutus = passive voice. Brutus stabbed Caesar = active voice.\nSo now I have to go back and clean all of that up as well. It's going to be huge pain, but I'm doing it because I know she's right. She's published more than seventy novels. I even had one on my bookshelf that I haven't read yet. I took it, and she signed it for me. I want her to adopt me and be my mentor but it turns out prolific authors only get that way by \u2026 well, \u2026 writing. Not babysitting new would-be writers. I get that.\nI got a lot of good advice Saturday, and I registered for four writing classes that are coming up on Monday nights the last two weeks of March and the first two weeks of April. I used to say I'm as smart as I plan to be, but I think I'm ready to learn something new. Since I decided not to try to go to the RT Book Lovers Convention in Chicago, this will be the next best thing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"VIDEO: Young Heads: New Millennial Subcultures No. 1 \u2013 Directed by Max Cutting & Rich Luxton, this short captures an emerging community of young skaters at Mile End park which they recognise as their home.\nAmongst the eclectic selection of films at our 37th Screen Social event last week, we screened a series of fresh and nuanced shorts entitled Young Heads: New Millennial Subcultures. In partnership with the historic British brand Fred Perry, Tack Studio and Jocks & Nerds magazine teamed up to create a series of films that detail the worlds of three different millennial groups. Featuring Skaters, Junglists & Post-Punks \u2013 these films explore identity and belonging amongst the 'young heads' of society.\nFounded in 1952, Fred Perry and its laurel wreath logo are today synonymous with British Subculture. From the heady 60s mod scene to their current Subculture blog, Fred Perry as a brand has remained an iconic stalwart of British fashion.\nEmerging early in the 90's amidst the sounds of rave, hip-hop, reggae, dancehall \u2013 Jungle has been revitalised through a community of modern 'junglists' who stick to the original sounds of the genre.\nIn the third episode, Nottingham music band Kagoule talks about how their city defines the band's identity and sound as well as the mutual respect and beliefs local musicians share.\nFormed in 2011 and influenced by 90's alternative rock, Lucy Hatter, Lawrence English & Cai Burns try not to be categorised in a specific music genre even though their sound is often described as 'post-punk'.\nHighlighting the visual identity of the social groups, Max Cutting & Rich Luxton's documentary style also underscores the strong bonds defining these communities.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Senior I.T. Consultant with a versatile skill set, multiple professional certifications, numerous work performance awards, and more than 20 years of professional work experience.\noptimizing SQL Server Views, Stored Procedures, and Functions for maximum performance gains.\nwatching good movies and t.v. shows: Game of Thrones and Battlestar Galactica are two recent favorites.\ngardening, especially water gardens and growing carnivorous plants.\nplaying almost any sport. Favorites include snowboarding, soccer, swimming, cycling, paintball, volleyball, and wallyball.\nreading good books. Currently loving the Jack Reacher series!\nhiking and sitting by a campfire under a starry sky.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Arriving at a hut after a day's walk usually sparks a rush of activity, the details of which depend on the time of arrival and the weather. What follows is an overview of the routine with an early arrival on a sunny day. When arriving late the focus is food and sleeping!\nClaim a bunk, Sarah and I prefer bottom bunks, Sandie a top bunk. If there are few fellow trampers we can spread out, if the hut is full it's an exercise in containment, especially if wanting to keep track of your gear. Can't replace it if lost!\nHanging them out to dry on whatever suitable outpost is available: tree branches, bushes, rocks, walking poles, inside drying line or rack.\nTaking stuff out of the pack: food bag, sleeping gear, clothes bag.\nWriting our details in the hut book and reading others' posts, particularly from people we have met before (they're all speeding along ahead of us).\nChatting with other trampers \u2013 we've met some lovely people :).\nDivying the tasks for cooking, getting and filtering water and doing dishes.\nHaving a hot chocolate drink as dessert for extra calories; if the water is unfiltered it needs to be boiled for at least three minutes.\nCleaning teeth, using boiled or filtered water.\nGetting breakfast ready; for me that is soaking our delicious muesli \u2013 can only do this after dinner as we need all our pots for cooking.\nGetting snacks ready from the main food bag into my daily ration bag which I carry in my front \"balance bags\".\nChecking and bringing in washing; hanging inside or accept it will have to be attached to the outside of the pack the next day, to be air-dried that way.\nDepending on the quality of the hut, i.e. whether or not fly screens are on the windows, we go through a ritual of ridding the place of as many sandflies as possible.\nThen suddenly the light is fading, four or more hours have melted away and it is time to hit the sack\u2026. wake up time is 6 am for a 7\/7.30 am start of walking!\nRakaia seems to be the place.\nA long poem for a long and demanding day \u2013 things did change after this!\nThe story of river crossings to Top Wairoa Hut after a day's rain; two days after my tumble.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Dantian, dan t'ian, dan tien or tan t'ien is loosely translated as \"elixir field\", \"sea of qi\", or simply \"energy center\". Dantians are important focal points for meditative and exercise techniques such as qigong, martial arts such as t'ai chi ch'uan, and in traditional Chinese medicine.\nTaoist and Buddhist teachers often instruct their students to centre the mind in the navel or lower dantian. This is believed to aid control of thoughts and emotions. Acting from the dantian is considered to be related to higher states of awareness or samadhi.\nThe Taoist concept of dantians as energy centers is similar to the Indian yoga concept of chakras as key points where prana is stored (see also nadis). The major difference, however, is that Taoist dantians are the major energetic storage mechanisms whereas the yogic chakras are not so much storage centers, but energetic vortices which act as intake and output ports. Many traditions consider the dantians and the chakras to be separate, albeit cooperative energetic mechanisms.\nLower dantian (\u4e0b\u4e39\u7530, Xi\u00e0 D\u0101nti\u00e1n): below the navel (about three finger widths below and two finger widths behind the navel), which is also called \"the golden stove\" (\u91d1\u7089 pinyin: J\u012bn l\u00fa) or the namesake \"cinnabar field\" proper, where the process of developing the elixir by refining and purifying essence (jing) into vitality (qi) begins.\nUpper dantian (\u4e0a\u4e39\u7530, Sh\u00e0ng D\u0101nti\u00e1n): at the forehead between the eyebrows or third eye, which is also called \"the muddy pellet\", associated with the pineal gland. This cauldron is where Shen or spirit is refined into Wu Wei or emptiness.\nThe term dantian used by itself usually refers to the lower dantian, which is considered to be the foundation of rooted standing, breathing, and body awareness in qigong and martial arts. The lower dantian has been described to be \"like the root of the tree of life\".\nIn speaking of the lower of the three energy centers, the term dantian is often used interchangeably with the Japanese word hara (\u8179; Chinese: f\u00f9) which means simply \"belly\". In Chinese, Korean, and Japanese traditions, it is considered the physical center of gravity of the human body and is the seat of one's internal energy (qi). A master of calligraphy, swordsmanship, tea ceremony, martial arts, among other arts, is held in the Japanese tradition to be \"acting from the hara\".\nThe lower dantian corresponds to the yoga concept of the swadhisthana chakra. In yoga philosophy, it is thought to be the seat of prana that radiates outwards to the entire body.\nI added a small snippet about the Dantians to the image description. Due to character constants I could only add so much. However this should help shed a little-bit more light it. Your right though, the hara is called the seat of the chi, it's called the lower Dantian as well.\nHmmmnm \u2026 I thought the tan tian is actually the brain and the hara is the seat of chi.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"\"Sweet Bamboula\" Song by Bamboula 2000 from the New Society CD (2000). Featured photography by Gason Ayisyin (www.GasonAyisyin.com). Video by AR, Inc (asiarainey@gmail.com).\nPreserve and foster respect for New Orleans' indigenous traditions and historic places through cultural and educational programming.\nUndertake, support and promote research on Congo Square's history and significance in the cultural and economic development of New Orleans and the world.\nPromote public awareness of Congo Square's history and significance as well as vanguard its depiction in all print and electronic media.\nEnsure that the economic development of Congo Square results in the betterment of local African-American communities.\nProvide Congo Square Living Classroom Fieldtrips to children & schools.\nServe as a voice of consensus regarding all aspects of Congo Square on the input and direction of our local, national and international membership.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Putting a better spin on records. Sunrise Records is a leading Winnipeg retailer of music, film, games, and pop culture items. Founded in 1977, Sunrise has been connecting Winnipeg with music that has spanned vinyl, 8-track, cassette, CD and digital. While the industry changes, one constant remains, Sunrise Records is your destination for music. Visit us today at CF Polo Park.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The El Segundo Unified School District is committed to an educational program that recognizes the unique value, needs, and talents of individual students.Our Gifted and Talented Education (GATE) program is designed to provide access to appropriate and unique learning opportunities through a variety of strategies, including differentiated instruction, project-based learning, and special programs.\nThe ESUSD Board of Education recently approved revisions to our qualification requirements for GATE. Please refer to the Frequently Asked Questions and the Referral Letter below.\nOur District GATE Advisory Committee consists of parents, teachers, and administrators led by our GATE Coordinator, Grace Long.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It's that time of year. Everyone is looking to make changes to their diet, and US News and World Report published its Best Diet Rankings for 2017. And once again, the Paleo Diet ranked very poorly and in last place was Whole30. In first place is the DASH diet, originally developed to prevent and lower high blood pressure. I figured I'd add to previous rebuttals here and here with a shiny new version of the same old argument.\nI'm a dietitian and I love real food. I've seen people give up processed foods and it can change their life. It's amazing to me that there is still so much bias against giving up processed foods. The reality is, not many people can moderate when it comes to hyper-palatable foods. I'm sure it's pretty threatening to classically trained nutrition experts who still believe in low-fat dogma that the Whole30 book and Cookbook tops The New York Times Best Seller List. The Paleo Solution and blogs like Nom Nom Paleo are runaway successes. This is such a sharp departure from the norm. But maybe real food is what people need. And, given the great success of Paleo and Whole30, maybe it's what people want because it works.\nMaybe real food is what people want. They're tired of the low-fat dogma and processed foods.\nNever before have humans had to push away such an abundance of hyper-stimulating, highly processed, nutritionally void food-like products. These modern foods bypass our normal satiety cues, causing us to eat many more calories than we need for our sedentary lives. We're overfed, obese, exhausted and malnourished all at the same time. Maybe we need to eat less fillers and more real food. I'd like to take a moment to provide some counterpoints to show how eating a nutrient-dense, real food diet, can actually be better than diets that completely avoid meat, require drinking 2 shakes a day containing high-fructose corn syrup, allow you to skip eating real food to save your \"points\" for alcohol, and ones that allow only 500 \u2013 800 calories a day.\nThe reason these diets work is because they force people to invest in a complete dietary transformation. One of the criticisms in the US News Report is that \"most diabetes experts recommend a diet that includes whole grains, legumes and dairy products.\" But if you stop to think for a second, aren't many of those foods just adding more carbs with little other nutritional value? Also, aren't we failing diabetics with our advice? We don't need to drink orange juice and eat whole grain cereal with low-fat milk every morning. Beans are pretty difficult to digest and not a better source of protein than meat. Grains are not a better source of nutrients and fiber than roots and tubers. And while I see some benefit in high quality, full fat dairy in the form of yogurt or cheese, I also think it's a great exercise to eliminate it for 30-days to see how it makes you feel. As far as milk goes, well milk is great at putting weight ON mammals, that's what it's for. With 75% of the world's population lactose intolerant, and most of the nutritional benefits in milk coming from fat-soluble vitamins, it's time to reconsider recommending 3 cups of low-fat or skim milk each day. There are no risks in avoiding sugar, grains and dairy and in fact, giving up these foods can help prevent disease.\nBoth diets focus on fresh meat and seafood, vegetables (including roots and tubers), fruit and healthy fats. Both diets avoid grains, sugar, legumes and dairy, plus industrially processed vegetable oils like soybean and canola. The Whole30 is intended to be a 30-day nutrition challenge that is based on the Paleo template, but further excludes honey, maple, and \"Paleo treats.\" The Whole30 is designed to reset your metabolism and taste buds, break your sugar addiction, and get folks back in the kitchen cooking real food instead of relying on convenient but unhealthy processed foods. The Paleo Diet is more of a template for an overall lifestyle that usually means a 30-day \"squeaky-clean\" intro (similar to the Whole30), then an 80\/20 long term maintenance phase, meaning 80% Paleo and 20% healthy modern foods (some rice, cheese, yogurt, and the occasional treat) if tolerated. The Whole30 is not the Whole365. Most people who do a Whole30 end up also following a Paleo template of 80\/20 after their 30-days.\nLet's do a quick rundown and rebuttal of the assumptions that USNWR and many other popular press critics have about paleo-style diets.\nNobody is literally saying that domesticated cows were around during hunter gatherer times. Instead, these diet are simply looking back at what humans evolved to eat and focus on those foods as a template to for a more modern version of that diet. Yes, there is some evidence of a little grain consumption several tens of thousands of years ago, but generally humans evolved eating meat, roots and tubers and seasonal fruits, and that diet seems to work great for those who have tried it. Paleo is simply a template for eating a diet mimicking the hunter-gatherer way of eating.\nI showed my cookbooks to my nutrition professor while completing my RD, and her exact words were, \"Wow, you need to change the name of your book because this isn't paleo, this is really healthy!\" She admitted she had a bias against the word paleo, but once she understood that I was intelligent, very healthy, understood nutrient metabolism and also advocated for sustainability, her guard came down, and she was able to look at \"Paleo\" without cringing. Although she is a vegetarian for personal reasons, she said, \"If I ate meat, this is exactly how I would eat.\" So let's please remove our stereotypes and keep an open mind, as real scientists do.\nAssumption: We need to be eating grains to be healthy.\nSweet potatoes also have only 90mg of omega-6 per cup compared to the same amount of wheat cereal delivering 452mg. Grains are a big source of omega-6s, the kind of inflammatory fats that should be consumed less, not more. The foods with the highest omega 6's are vegetable oils (margarine, commercial salad dressings) and another huge sources is grains, yes, even whole grains.\nType 2 diabetes is a disease of carbohydrate intolerance. Sugar is addictive. You would never tell an alcoholic to just self moderate and hope for the best, so why would you tell someone with metabolic disorder from overconsumption of carbohydrates to moderate? They need to give up the sugar. Instead, we're telling them to eat less meat and more grains and legumes, and skim milk (more than 1\/2 the calories from skim milk come from sugar.) In fact, the American Dietetics Association lists beans as the ideal source of protein.\nThe RDA for carbohydrate is only set at the arbitrary value of 130g\/day in order to avoid ketosis, however they also say ketosis is not a danger (and this paper also backs that up). Ketosis is not ketoacidosis.\nNow, the Paleo and Whole30 plans aren't void of carbs, but people will naturally eat much less in the form of carbs if they're not consuming breakfast cereals, bagels, breads, pasta, and cake. And it turns out, when you eliminate these foods from your plate, you end up replacing them with nutrient dense foods that are difficult to over consume because they're not hyper-palatable. Nobody is going off the rails for saut\u00e9ed spinach, roasted salmon and water, but they DO line up never-ending bowls of pasta, bread sticks and wine.\nLet's get rid of our bias of low-carb diets and instead, take an honest look at the research proving that a lower-carb diet can actually prevent or reverse type 2 diabetes. The carbohydrate choices for Paleo and Whole30 are completely reasonable, nutrient-dense, and delicious without overstimulating our appetites causing us to over consume. This is why most people with blood sugar disregulation do very well on Paleo and Whole30.\nI challenge this on several levels. First of all, we need to be eating more protein and better protein. There's an assumption that all we need to eat is 0.8g\/kg per day. This is the absolute minimum to avoid disease, not the amount for optimal health. Also, this often gets translated to a baseline recommendation of 56 grams a day for men and 46 grams per day for women, numbers based on a \"reference\" man of 70kg (154lbs) and a \"reference\" woman at 57kg (125lbs). The average weights for Americans are much higher.\nThe Acceptable Macronutrient Distribution Range (AMDR) for protein is 10% \u2013 35% of calories from protein. On a 2,000 calorie diet, this means 20% of calories from protein is actually 100g\/day. That's double the RDA. According to the Dietary Reference Intakes by the Institute of Medicine, \"the current state of the literature does not permit any recommendation of the upper level for protein to be made on the basis of chronic disease risk.\" This means there is no worry for people to consume more than the upper limit, though it would be pretty difficult to consume 35% of your calories from protein (the upper end of the AMDR.) That would be 175g of protein on a 2,000 diet.\nProtein is the most satiating macronutrient. The more protein you consume, the more full you feel. Now, if you are trying to get more protein, here are the most efficient sources.\nLet's look at calories next. If we're trying to eat more protein but fewer calories, which sources of protein are the best choices? Let's look at how many calories you need to consume of these foods to get 30g of protein in a meal.\nThese two charts illustrate that animal protein is both the most efficient forms in terms of calories and grams of protein per serving. It should also be noted that plant-based proteins are not as easily digested and do not contain the full amino acid profile. Plus, beyond protein, animal flesh has a lot more vitamins and minerals than plant-based proteins, like heme-iron which is not available in plants.\nAdditionally, relying on food frequency questionnaires can hardly drive policy, because it's been proven that people lie when trying to recall what they ate. The truth is, when comparing someone on a vegetarian-type diet to a person on a Standard American Diet, you can't simply call out red meat as the driver of disease. There are many other lifestyle factors that typical vegetarians do differently than a typical American, like generally taking more care of themselves by doing yoga and other movement, not smoking, drinking less, etc. In this study that compared those who shop at health food stores (so, accounting for lifestyle) there was no different in mortality, heart disease or stroke found between meat eaters and vegetarians. Eating lots of fresh vegetables and living a health conscious life are both important, something central to the most vegetarians, Paleo and Whole30 followers.\nRed meat itself is not the villain, it's more likely to be the fries, 72oz soda, and deep fried apple pie that was consumed with it.\nThe other reason why people call meat \"bad\" is the assumption that it's better environmentally to eat only plants. Many parts of the world (such as much of Africa) are not well suited to cropping of wheat, corn, soy, and water-sucking plants like lettuce. Most of the world's landmass is not suitable for growing vegetables and grains, and instead is much better used as grazing land for animals. I got into this much deeper in this post, but the summary is that we're attacking meat as being an environmentally toxic and unhealthy way to get protein this is not necessarily the case. I do agree that we need to be eating fewer animals that eat grains (like CAFO chicken boneless skinless chicken breasts) but if we focus on animals that are converting grass (something we can not eat, on land that does not compete with humans for edible food) to flesh, then it's pretty clear that eating these animals is quite efficient for human nutrition AND it can improve the soil and the climate, helping to sequester carbon.\nI feel that most of the anti-meat energy comes from the guilt associated with killing an animal for our nourishment. Factory farming is not good for animal welfare or the environment, but to rule out all meat because of factory farming is like saying I'm not going to eat vegetables because I don't believe in GMOs. Don't hate the player, hate the game. It's not that hard to find good meat. You can get grass-fed beef at Walmart now.\nDairy is actually not the most efficient way to get calcium or vitamin D. US News actually has a post saying that you don't need dairy to get calcium highlighting sardines, vegetables, and nuts and seeds, all foods approved for Paleo and Whole30. Cod liver oil and salmon are fantastic sources of vitamin D, foods that are not allowed on plant-based diets. Plus, milk is fortified with vitamin D, so it's really no different from taking it in supplement form along with some calcium. So why attack the Whole30 as being deficient in these nutrients? Also, I'd like to point out that on the sample day breakdown of nutrients, USNWR scored the Whole30 diet as containing nearly 3 times the recommended intake of Vitamin D. I find it funny that there was no Vitamin D for the Paleo diet, considering that the two diets are nearly identical, yet the Vegan diet had B12 above the RDA listed in the USNWR. Was this in the form of a supplement? Without listing what foods were used to calculate the findings, it's not very transparent and leaves a questioner like me puzzled.\nAlso, I found it interesting to read this comment, \"Avoid diets that require a supplement \u2026 as this indicates the meal plan is lacking in nutritional value and the object of the diet creators is to use the diet for their own profit, one expert advised.\" This is completely made up information. On the health and nutrition page of the USNWR, it was acknowledged that \"A supplement isn't required, but the Whole30 founders write in their book that they 'believe many people would benefit' from high-quality fish oil, vitamin D3, magnesium and 'maybe some digestive help, like enzymes or probiotics.\" This recommendation is completely reasonable, yet not a requirement of the diet. Vegans on the other hand absolutely require the supplementation of B12 to avoid serious neurological damage. Vegans also need to supplement with DHA. And, isn't SlimFast based on supplements?\nHere are a few more nutrition criticisms from USNWR, \"While its focus on veggies and lean meat is admirable, experts couldn't get past the fact that entire food groups, like dairy and grains, are excluded on the Paleo diet. \"The risk of nutrient deficiency is real, unless the person takes a multivitamin,\" one panelist commented. Here, its rating lagged behind most other diets.\" And another, \"Experts worried about dieters missing out on key nutrients on the Paleo diet, given that it shirks entire food groups. Its rating classifies it as \"somewhat unsafe.\" It was among the poorest performers in this category.\" Again, what nutrients exactly can't we get from meat, seafood, vegetables, fruit and nuts?\nI'm completely perplexed by this one. In breakdown of nutrients on a Whole30 diet, the sodium was listed as 4,758 mg. Interestingly, the Paleo Diet came up as only having 726mcg. There was no place that I could find exactly which foods were used to compute the nutrition scores \u2013 a major flaw because there is no way I can verify the data. On a diet with no processed foods, how is it possible for someone who is basically cooking all their meals to have a sodium content of over TWO TEASPOONS of salt a day? Nobody adds this much salt to their roasted chicken and saut\u00e9ed kale. On the \"recipes\" tab of the USNWR review, there's a sample day listed and there is NO ADDED SALT at all. Now, humans need salt, and we all know that eating processed foods is where the majority of our sodium intake comes from, so on a diet void of processed food, the Whole30 and Paleo actually looks a whole lot like a \"low sodium\" diet to me \u2013 in fact the nutritional breakdown for Paleo shows it as having 1\/2 the sodium (only 726mg) of the halo'd DASH diet (at 1507mg and 2101mg), one that allows margarine, fruit juices, white pasta, and includes (gasp) RED MEAT.\nHigh Saturated Fat Leads to Insulin Resistance?\nI took a look at the Atkins reviews as well. I don't know how Atkins scored better than Paleo and Whole30, when they eliminate many more foods and include processed bars as part of the plan. I do appreciate the low-carb approach though, and I've seen many people do well on Atkins. Here's a quote: \"One panelist observed that the high level of saturated fat intake on the diet can hike the risk of insulin resistance, a hallmark of Type 2 diabetes.\" How is this possible when going low-carb is fantastic at reducing insulin resistance, and has been shown to reduce glucose, insulin, triglyceride, ApoB and saturated fat (especially palmitoleic acid) concentrations, reducing small dense LDL particle numbers, glycated haemoglobin (HbA1c) levels, blood pressure and body weight while increasing low HDL-cholesterol concentrations and reversing non-alcoholic fatty liver disease (NAFLD)?\nDiets that contain whole grains, low-fat dairy, legumes and sugar are healthier.\nI took a look at the breakfasts listed for the various diets. The breakfast listed for the Paleo Diet is not really what paleo folks eat. This was taken from Loren Cordain's book, who doesn't \"own\" Paleo. Nobody owns Paleo. That's the problem with people citing only his book as \"THE\" paleo book. His just happened to be the first book with that title. Most people on the Paleo Diet and Whole30 eat about 4-6oz protein per meal. So, I called the breakfast Whole30\/Paleo, since the example for a Paleo breakfast in USNWR is unrealistic. The breakfast shown is quite similar to what I would eat for breakfast. I should note that prior to adopting a low carb\/higher protein and fat diet, I ate very similar to DASH and was in metabolic syndrome but I was not overweight. I was a blood sugar mess though, and any of the other breakfasts listed here would certainly leave me starving by 9:30 and hypoglycemic.\nCompared to the Whole30\/Paleo breakfast, these other examples are low in fat, low in protein, and high in carbs. Eating this way in a fasting state (first thing in the morning) can set a person up for a blood sugar roller coaster. I was particularly unimpressed with the DASH diet (the #1 ranked diet.) Cereal, toast with margarine and orange juice? Who eats margarine still? I thought that left with the 1980's. And we all know orange juice is just sugar. There are way better sources of vitamin C than orange juice.\nDid you see the Macrobiotic diet? It starts the day with NO FAT at all (and most of these other diets are pretty low fat in my opinion.) The oatmeal has 12g of protein but at 64g carbs plus 2 slices of whole wheat bread adding another 8g of protein plus 26g carbs, and apple butter adding another 14g carbs, (and nothing else as far as vitamins or minerals,) that's a total of 104g carbs (more than I eat in a whole day) with 20g of incomplete protein and no fat to slow the digestion down. I don't understand how you're supposed to make it to lunch without passing out. I also really don't understand how all of this skim milk and toast (with margarine) is more nutrient-dense than the real food breakfast shown for Whole30\/Paleo.\nPeople like to criticize both the Paleo and Whole30 plans as being expensive because \"all that meat and produce can add up.\" Yet, when you actually compare healthy food to processed foods, real food is cheaper. Let's look at candy bars, which I don't think anyone is complaining are so expensive. The average price I found in the US is $1.24 for a standard Snickers bar. That's $0.66 per ounce.\nConsumer Reports purchased 300 packages of ground beef in 103 stores in 26 cities across the United States in 2015. They paid an average of $4.95 per pound for conventional beef (which, I would argue is better than a snickers bar) and an average $7.83 per pound for grass-fed organic beef. That's $0.39 and $0.49 per ounce, respectively. So, even grass-fed organic beef is $0.17 cheaper by weight than a snickers bar. Amazon sells a 6-pack of 5.5 ounce sliced Tofurky for $27.05, which equals a little over $13.00 per pound or $0.82 per ounce. Organic, grass-fed beef is $0.33 less per ounce than Tofurky. Organic vegetables like carrots, potatoes, and zucchini area all cheaper than candy bars and processed fake meat products.\nAdditionally, comparing the nutrition in fresh fruits, vegetables and nutrient-dense meat to processed food and meat-like products is definately not apples to apples. And the sustainability factor of these highly processed foods needs to be considered. If someone did a full life cycle assessment on Tofurky, looking at all of the inputs (water, fossil fuels, etc.) to make one floppy little piece of this fake meat, there's no way that it's more environmentally friendly than a cow that ate grass in a field.\nJenny Craig ranked #10 and SlimFast is #20, way above Paleo and Whole30. On SlimFast, you drink 2 of these shakes a day, plus eat 2 bars with equally disturbing ingredients. The snacks listed are a banana and apple (on an empty stomach, hello sugar rush!) and then one \"real\" meal. And this is healthier than eating real food?\nThe goal of the Paleo diet and the Whole30 are to help people eat food that is closer to its real form, to reset tastebuds and to stop eating junk food. It's not necessary to spend hours and hours in the kitchen either. A quick Paleo or Whole30 approved meal can be made by picking up a plain roasted chicken, grabbing a few carrots and hitting the salad bar. That's pretty easy. Also, there is no weighing and measuring your food, because the idea is that if you eat real food, your body will self-regulate. Shouldn't that be the goal of any weight loss program? Do we really want people reliant on bars and shakes as a long-term diet strategy?\nAlso, its super snarky to say that you can't eat out unless you \"eat in your neighbor's cave\". Lots of restaurants offer a piece of salmon over a green salad, saut\u00e9ed shrimp and vegetables, or a burger with no bun, baked potato and a side of fresh veggies. The reviewers really found that it's more difficult to eat Whole30 and Paleo at a restaurant than it is to order a vegan dish? How easy is it to order a Slimfast shake at a diner? The attitude given to Paleo and Whole30 by the US News team has palpable bias. Here's a quote from the Paleo review, \"This diet should go back where it came from.\" Nice one.\nHuman evolution has taught us a thing or two about which foods we can thrive on. Before the agricultural revolution, humans who didn't die from childbirth or accidents had remarkably good health. They didn't need to measure and weigh their food. They didn't need to drink two shakes and have two junky bars a day. They didn't eliminate meat. They didn't die of type 2 diabetes. They ate food closest to its natural form.\nPaleolithic nutrition improves plasma lipid concentrations of hypercholesterolemic adults to a greater extent than traditional heart-healthy dietary recommendations.\nBenefits of a Paleolithic diet with and without supervised exercise on fat mass, insulin sensitivity, and glycemic control: a randomized controlled trial in individuals with type 2 diabetes.\nCardiovascular, Metabolic Effects and Dietary Composition of Ad-Libitum Paleolithic vs. Australian Guide to Healthy Eating Diets: A 4-Week Randomised Trial.\nStrong and persistent effect on liver fat with a Paleolithic diet during a two-year intervention.\nPaleolithic nutrition for metabolic syndrome: systematic review and meta-analysis.\nA Palaeolithic diet improves glucose tolerance more than a Mediterranean-like diet in individuals with ischaemic heart disease.\nPalaeolithic diet decreases fasting plasma leptin concentrations more than a diabetes diet in patients with type 2 diabetes: a randomised cross-over trial.\n\"Even short-term consumption of a Paleolithic-type diet improved glucose control and lipid profiles in people with type 2 diabetes compared with a conventional diet containing moderate salt intake, low-fat dairy, whole grains and legumes.\" \u2013 Metabolic and physiologic effects from consuming a hunter-gatherer (Paleolithic)-type diet in type 2 diabetes.\nConsumers are getting smart. They no longer trust nutrition advice, because what they're being told doesn't work. Cholesterol and saturated fat are no longer supposed to be \"nutrients of concern,\" so why the continued recommendations of low-fat milk? Why the fear of Paleo and Whole30? Why the cynical comments? People want results. They want to eat real food. They want to feel full and feel good, and say goodbye to their sugar tooth, expanding waistband, and exhaustion. Why not admit that grains and legumes don't need to be part of a healthy diet? When 75% of the world's population is lactose intolerant, maybe it's time to stop recommending so much milk, which isn't even the best source of calcium to begin with. Maybe people need to completely eliminate processed food for a period instead of drinking shakes and eating bars.\nCountless people have found that when they eliminate grains, legumes, sugar and dairy for 30-days, it's the beginning to a whole new way of relating to food. After their 30-day intro, they're usually able to reincorporate a small amount of \"modern\" foods in moderation (like rice or full fat, plain yogurt, or the occasional treat or glass of wine) with no problem, while sticking to a general real food template as their base. Many people report that their blood sugar issues go away, food cravings disappear, acne improves, they sleep better, lose weight, perform better in the gym, and they just feel fantastic.\nIf we really knew so much about the right diets for everyone and how to get folks to stick to them, would we still have such a growing epidemic of diabetes and obesity today? Maybe it's possible that we are still just learning. Nutrition is a new and evolving science, and it's time to admit we've been wrong about things. Maybe it's time to open our minds a bit to new ways of thinking and be a bit more humble about our \"expertness\".\n20 Ways EAT Lancet's Global Diet is Wrongfully Vilifying Meat Film Update! Is a Meat Tax a Good Idea?\nIf I wasn't at work I would yell loudly and clap for a very long time. The Whole 30 and the Paleo diet have literally saved my life. I am still overweight now, but the weight is coming off in a healthy way that will last for the long term. So, it is a journey and will take quite some time for me because it was seriously bad before, but it won't be a yo-yo diet and this life is sustainable. I still made gf Christmas cookies and had my fair share, but that isn't every week. I do not get the argument that these diets are bad\u2026how can eating whole foods be bad? I don't get it. Thanks for standing up for the truth.\nAmazing post! Such a great explanation of what I've been trying to share with people. Thank you!!\nI am a registered dietitian and certified diabetes educator who has experienced firsthand the power of a whole foods diet changing my life and health. I 100% agree with the author and wish that the rest of the rest of the mainstream medical community would catch up already and realize the power of nutrition. Thank you for sharing this article.\nGreat article! You hit a home run with your thoughtful and well researched rebuttal of the USNWR article. \"Consumers are getting smart.\" Absolutely! Let's hope that more and more consumers vote for a whole and healthier way of eating with their purses.\nWow, at last we have a thorough, no-nonsense and comprehensive rebuttal to this debate that has been going on every year since the increase in popularity of ancestral diets like Paleo and Whole30. I'm so grateful for my colleague Diana Rodgers' hard work in putting this great article together so I have something credible to share with my clients (I'm a registered dietitian too!).\nAnd this is why the populations in the west are screwed! I'm in the UK and they still serve out-dated nutritional advice here to diabetics also.\nThe ingredient lists you shared for popular \"diet\" foods say it all. Those are not healthy, sustainable, or nourishing foods. Following a whole food diet is sometimes hard, but that doesn't mean it shouldn't be done. There is plenty of room for customization in these diet approaches- they are not one size fits all. Even the Washington Post included Whole30 in its dietitian-approved WaPo Food's 5 Diets project. Contributor and Registered Dietitian Ellie Krieger stated in an article that Whole30 could help people add \"structure and strategy\" to their day to day eating in hopes that participants can learn to transcend the diet mentality and land on a sustainable and satisfying way of eating. Thank you for your hard work and great information, Diana.\nI am also a registered dietitian, and I approve of Diana's message!\nI'm a registered dietitian and certified diabetes educator, and I couldn't agree more with this entire post. I regularly work with patients that have been failed by one of more of the high-scoring diets on the USNWR list. Thank you so much for this clear and concise rebuttal!\nSo well put and backed by credible and thorough. I am a RD and could not agree more with this whole post!\nI've often felt that steering people towards whole foods has been an uphill battle, as the amount of misinformation that must be addressed and countered is overwhelming. This article is one of the most logical, comprehensive, objective, evidence-based pieces that I've read in a very long time (if not ever) regarding the power that eating real, minimally-processed food can have on our health.\nYou've done us a great service, Diana!.\nWhat an excellent article! I think consumers are getting the message, and the industry-backed diets are scrambling. A whole-foods diet is such a powerful way to regain health. I hope it's just a matter of time before the rest of the health-care field catches on. From one Registered Dietitian to another\u2026 Thank you for your time and energy in putting this rebuttal together!\nI'm an RD and I LOVE this. Thank you for writing such a thoughtful, well-researched, and oh yeh, TRUE response to the disaster of diet-rankings that was just put out.\nI'm a real food RD and couldn't agree with this article more! There's no wonder why there is so much confusion in the nutrition industry. The food industry is doing everything it can to make a buck, despite the health of our country.\nSuch a great post! I am a first year school nurse and from day one have been feeling like ripping my hair out seeing what they are feeding our kids. Especially for breakfast!! It just makes me so angry and I feel as though there is nothing I can do. After all, I am only the school nurse, not the district's registered dietitian *insert eye roll*. One of the major contributors to our regional conference was a milk lobbyist and several RD's spouting a bunch of nutritional non-sense. I can only hope I'll be here long enough to maybe one day see a change in our school cafeterias. It's tough to sway opinion when you've only been taught one way your entire life.\nHi Brittany! First, I agree whole-heartedly with this article. Second, please try to find some compassion for your food services dietitian. Schools are required to follow federal guidelines in terms of what they serve. They don't have an option on much of it and generally do the best they can. The federal guidelines require milk, grains, and fruits for all breakfasts. Protein can be served but it is optional and not emphasized in the rules. That just shows you what we are up against (yes, I'm one of those school nutrition ogres too). We need to change the federal guidelines in order to change how school meals are served. It's a tall order and one your dietitian can't do on her own. Please give her some slack and a kind word next time you see her; we get beat up a lot for stuff we can't control.\nI am also a Registered Dietitian and Certified Diabetes Educator and am so grateful to Dianna for this well researched and comprehensive article!! It boggles my mind that a whole foods eating plan can be ranked less healthy than plans based on processed foods which contain a plethora of artificial ingredients including artificial sweeteners and high fructose corn syrup. We need to help people see past all of the marketing hype surrounding processed food products and appreciate the health benefits of a whole foods diet. JERF\u2026Just Eat Real Food!! One of the most important steps you can take towards better health! Thanks so much Dianna for helping to get the word out!\nAs an RD myself I could not agree more and I have seen first hand the difference eating real food makes.\nAbsolutely! I'm also a dietitian and agree wholeheartedly! Thanks for writing such a solid response!\nMic Drop! That is amazing!! Bravo!\nThank you \u2013 will share with all my clients.\nI am a dietetic intern and couldn't agree more. It frustrates me that even the curriculum we are taught includes this information. Thank you for writing this thorough excellent article! Cannot wait to also, be a real food RD.\nThanks so much for spreading the real food message. Keep up the good fight! As a fellow RD, I completely agree with your points here.\nI love this article and I love the number of dietitians showing their support. We have so much work to do and when we put it in basic common sense terms our patients and clients seem to understand how\/why paleo-ish is a good choice (usually better than other RDs and healthcare professionals). Keep up the good work.\nThose other \"Diets\" advertise in those magazines. No kidding they are ahead of Whole30 and Paleo.\nI don't think there is a marketing campaign for the DASH diet or Raw or Vegan. I think there's just a lot of assumptions and bias against meat and for grains.\nNo, but slim fast and atkins probably do. And if your health advice goes against what your ads preach, then you risk offending them and losing money.\nI have a question about the environmental impact of meat. I'm a farmer, growing primarily vegetables, and in a sense, I eat in a Paleo fashion, but with less meat. I typically try and eat meat every other day, and I prioritize locally sourced meat from here in New Mexico.\nI understand that oftentimes animals can use land we can't farm, but I've also heard that a lot of deforestation is caused by the beef industry, namely in South America. While the Walmart beef may be grassfed, I doubt those cows have been raised sustainably on appropriate land. Might they have been grain finished as well? I feel that, unless you really know the circumstances for your meat, you can't be confident that your environmental concerns are spoken for in buying it. What do you think?\nAnd I have a question I've never really answered for myself. I've always felt that Americans are overly fixated on meat, in part because we can get it so easily. But, back in the day, meat used to be difficult to attain, and without refrigeration, difficult to keep around. What do we know about how we used to consume meat in Paleolithic times? In my imagination, we ate good meat, but less of it because it was difficult to hunt.\nWe did not eat less meat as hunter gatherers than we do today from the evidence I've found. I reference some of these studies in my posts.\nMy husband and I did our first Whole 30 round in February of 2016. He had been diagnosed with Diabetes and told he would have to take medication which he refused. I have many health issues related to inflammation that have kept me from sustainable exercise programs and fitness activities. I was continuously gaining weight as was my husband. We were caught up in the convenience of processed foods. I worked with a nutritionist and the balanced plate method for food intake which per standards had me eating healthier, but weight loss was not happening. After 30 days of Whole 30, I had lost 24 lbs and felt better than I had felt in 20 years. My husband's bloodwork numbers were so good that the Dr. told him not only do you not need medication to control diabetes, but we will be taking you off blood pressure meds. He had lost 20 lbs. We continued with the introduction of some of the foods restricted with Whole 30 and I was surprised to find that dairy and wheat were primary causes of inflammation issues. We continued with Whole 30 and paleo recipes following our initial Whole 30 round and I have lost 67 lbs and he has lost 45. To start of the new year we are doing another Whole 30 round. We see this as a lifestyle change not a diet. Our doctors are very happy with the results and have vowed to learn more about Whole 30. Thank you for your article.\nI don't have any nutrition-related accolades, but I do have my experience. Not only has the ancestral lifestyle vastly improved my lipid and blood sugar profiles, it essentially cured the spirit-draining fibromyalgia I was plagued with for so many years. No more tender points, no more joint pain, no more cognitive fog or constant fatigue. I also have friends who have improved their autoimmune conditions with low carb Paleo. I fully believe that this diet, with its focus on nutrient-dense and efficient foods, will fit, at a minimum, everyone with an inflammatory condition, if not most everyone else.\nI made the switch a year ago to Paleo after 15+ years of severe digestive problems been diagosed with IBSD, Ulcerative Colitis, Stomach Ulcers, hiatal herna\u2026\u2026 I was on a roller coaster of medication (max'd out on 3 different ones daily, so I could eat and because I ate) and nothing and I mean nothing helped longer than 10 days. I grew up to believe eating everything unless you have an allergy was the best way to eat. I went to the dr's to get a refill and he had a fill in \u2013 wen went over my file why I was taking the meds etc he suggested Paleo\u2026.Now believe me I was on the fence I thought it was a fad diet and that wasn't my issue I just wanted to enjoy food again.\nNeedless to say after looking into for a month really digging in and finding out what made this different I figured I had nothing to lose I've tried everything else.\nFabulous article! I've tried so many diets and different ways of eating over the years, and Paleo\/Whole30 have been the only ones to truly make a difference in how I feel. I'm on my third Whole30 and convinced my 18-year-old son to do it with me. For years he's been having terrible sleep issues, digestive problems and overall just didn't feel well. I told him it would change his life. He started out VERY skeptical, but now just a little over a week into Whole30, he said he can't believe how good he feels, how much more energy he has, and how well he's been sleeping. It's amazing how your body resets when you eat REAL food!\nI agree with your facts and comments about the article! My boyfriend and I are hitting the No Sugar, clean, healthy 28-day challenge through his Crossfit gym. No I don't do Crossfit but walk and hit the gym twice a week. Since being with him, we've hit the Paleo diet hard and then fell off. This past Christmas we indulged (as I think almost everyone did!) and as soon as the challenge started our minds switched. I will say that not having sugar or alcohol has put a whole new spin on things. We love our craft beer and enjoy it immensely but taking it out and made us feel so much better! Granted we're still going through our detox period but I can't wait to see how we feel after 28 days! Thanks for focusing on the importance of eating naturally and staying strong. I have heart disease on both sides of the family, so for me it's utterly important to watch my cholesterol and Vit D levels! Thanks again!\nExcellent!!! I rarely comment on blog posts but this one deserved a shout out! I'm an RD and believe in real food\/Paleo\/Whole30 wholeheartededly. After being diagnosed with autoimmune disease, a Paleo diet has helped my body to heal and brought to light so many issues that I was having that I had no idea were related to the crap I was feeding my body. I feel healthier and stronger than ever. I'll definitely be sharing this over and over again!!!\nWhooping and hollering for JOY!!! This is the BEST article I have read about food and diet and filled with REAL EVIDENCE! I have been on a healing journey for nearly 20 years now and finally discovered I have an autoimmune disease and we began a healing autoimmune paleo diet lifestyle January 1st. My husband and I are both feeling healthier with each day! Thank you! Thank you! Thank you!\nDiana, very well written! Thank you for taking the time to write this post and for representing Real Food RD's in the best way!\nYES!!! A very well written, thorough and thoughtful post. I'm an Australian dietitian and whole-heartedly agree with everything written in this article. Thanks Diana.\nAbsolutely agree with eating clean and avoiding processed food, however, I think the promotion of meat and animal products especially bovine is what is harming our planet. More and more evidence showing how cattle and dairy production is causing catastrophic harm. It is good and well to promote self-care, but I think any sustainable message about food and food production should also involve consideration of whether that food production can be sustainable. If you must have cheese and yoghurts, avoid cows milk. Shop for goats and sheep milk cheeses!\nRhoni I will agree with your comment if you are referring to the current methods of factory farming meat and dairy, where animals are penned and large amounts of fossil fuels are used to grow grains, ship it to the animals and haul their waste away. That has to stop. However, pasture raised beef and dairy, using Allan Savory methods actually build soils, sequester carbon and reverse the damage done by factory farming and mono culture big ag. Those rich deep soils we used to have in the bread basket of America were formed by 60,000,000 bison and a multitude of other large grass eaters constantly being moved by predators. The grass lands and the grazing animals evolved together, building incredibly deep soils. We can duplicate this soil building system with portable fencing, good grass\/animal management and very minimal fossil fuel inputs. Plus the animals are a whole lot happier and healthier.\nAs a real food RD, I much appreciate Diana's work and agree 100%. Keep up the good work!\nAs a Registered Dietitian who helps people implement a paleo diet, I could not agree more strongly with this article. I've personally witnessed the incredible health benefits that it offers in hundreds of my clients. It's unfortunate that ill-informed articles like this still appear, but that should only encourage us to continue getting the word out. Lives are at stake. I applaud this excellent response, Diana!\nI am a registered dietitian and I feel that the Whole 30 shouldn't even be included in this list of \"best diets\" because it isn't a diet that is meant to be sustained year in and year out like the DASH diet. Sure, it is restrictive, but it is an elimination diet designed to last for only 30 days. Then the person should start adding back foods in a sequential manner to determine if they have any food sensitivities. I am finding that a lot of people have problems with gluten, dairy, sugar and certain grains-and the whole 30 is a great way for them to identify these issues, while also learning how to prepare whole foods. The Whole 30 is something that dietitians should have in their toolbox as another resource in helping reduce chronic disease and to support overall health. Thank you for posting such a great article!\nYou are an eloquent writer and a patient scholar: I admire your willingness to take on this idiocy and tear it apart word for word. Kind of the way we should be tearing apart our food, meaty morsel by meaty morsel.\nOne point I would clarify or add would be about vitamin A\u2026 as you know, sweet potatoes actually have beta-carotene and a fair number of us cannot convert that well, or at all, into vitamin A. Denise Minger wrote a great post about it here https:\/\/deniseminger.com\/2016\/10\/20\/why-do-some-people-do-well-as-vegans-and-vegetarians-clues-from-the-magical-world-of-genetics\/. Since reading her work, I've included scanning for vegan-tolerant genetic markers and have found it very interesting: happy vegetarians more often able to do the conversion, intentional meat eaters often unable to do so. I was happy I did the test in a young schizophrenic patient I have who was considering going vegan: strongest \"obligate carnivore\" genes I've seen and for that he was convinced to stay with a Paleo type diet.\nThanks for this column which I will share!\nyes, the nutrient database calls it \"vitamin A\" so I went with that, but I'm familiar with Denise's work and think she's great!\nThis is an excellent article, however, I have one qualm: you state that protein is the most satiating macronutrients and that is false. Fat is the most satiating macronutrients and fat most certainly should not be feared.\nThank you so much for addressing this issue. It literally makes me want to cry when I read in the media that processed JUNK is actually better than what nature has provided us. This was a very well written article that has hit on so many points of debate.\nWhat are your thoughts on the book, The China Study?\nI appreciated this article until you said that the risks of a vegan diet outweigh the benefits. As a vegan I can still appreciate the possible health aspects of other kinds of whole foods diets, but you seem to have published this not in defense of a paleo diet, but in offense to a vegan one. That study you cited regarding the risks of a vegan diet said that the conclusion was that a vegan diet was deficient in B12 \u2013 but in all other areas was sufficient. Call me daft, but that doesn't look like a study showing that the risks of a vegan diet outweigh the benefits. It's disappointing that you show concern for misinformation regarding the paleo diet but propagate the same misinformation about a vegan one. There is nothing wrong with having a bit of B12 fortified food to make up the deficit there, since the rest of the diet is more than sufficient. You said yourself, just above, in response to someone else, that correlation doesn't equal causation. And yet you do not eat grains based on their correlation with certain maladies of the body. I would like to point out that many people eat grains with enormous health benefits, but diabetics may not see these as they have a specific health condition. There is no single diet that benefits all. We all have different needs for our bodies \u2013 and at different life stages \u2013 that you cannot make the blanket statement that one diet is better than another. I agree with some of your points on paleo, even though I choose to follow a vegan diet, but it's disappointing that people keep flogging that old dead horse about the so-called \"risks\". I could show you a study citing the dangerously high levels of vitamin A that are prevalent in the diets of many paleo eaters and yet you would defend this based on what you know to be true. The inverse is also true for veganism. Like I said, different bodies, different needs, different tolerances for certain foods. No need to cherry-pick data.\nI don't eat grains because I have celiac disease, not because of correlations. Vitamin A is a concern in supplement form, not in real food form for healthy people. I'm working on a book to address your other points (literally, I'm not being sarcastic here \u2013 there's a lot to address about veganism and I'm writing a book on it).\nI'm also celiac so I totally understand where you're coming from in respect to a Paleo diet which omits grains, since there would be no risk of exposure when you're avoiding them altogether. I do avoid the obvious grains but choose to include others such as quinoa and buckwheat in my diet as I believe they are extremely healthful. I would be interested in your further points and will keep an eye out for your book, but having spent countless hours myself researching nutrition, I do feel confident in saying the data supports the healthfulness of a whole-food plant-based diet and it's role in reversing disease and illness. Of course I don't know everything, but off the bat I would probably say that data showing the risks or detriments of the diet would be coming from subjects who are not eating appropriately, or whose special\/specific requirements may mean that it is not the optimal diet for them. Not unhealthy, just not optimal for some. And then there are those vegans still dying of heart disease because they think if a potato chip doesn't have animal products it must be okay. Ludicrous, obviously, but you will get these type of people. I think a lot of concerns about the 'vegan' diet have come from broadly painting all vegans as following the same diet, and therefore getting mixed data. This is something I have come across before. A processed vegan diet is just as bad as the standard western diet in my opinion. To see the true health benefits you must be eating whole foods. So I suppose at least we agree on that point. Looking forward to your book though. Thanks for taking the time to reply.\nHumans need nutrients in meat. If you need to supplement a \"whole foods\" diet with nutrients, this is not a good diet. I realize there's a difference between eating twinkles and salad, and I do feel one could argue that a well planned vegan diet is better than a crappy version of a Western diet, but vegan is NOT optimal for humans. It's just not. We have no evidence of this. If you were to take a well planned vegan diet and then add some grass-fed beef, wild salmon, pasture-raised chicken eggs or other good meat, this is optimal. Working on a book about it now.\nAs an experienced Registered Dietitian, I can't believe any Dietitian would ever call a vegan diet unhealthy. That's a completely uneducated statement. Vegan's eat more fruits, vegetables, nuts and other healthy foods than any other diet. The vegans I know avoid fast food, processed foods, and refined sugar. They have lower rates of every chronic disease. It reduces inflammation, cholesterol, cancer risk and diabetes. You don't have to take a B12 supplement, most nondairy milks are fortified. Many people need B12 supplements for other reasons besides being vegan. Why do you feel so insecure about your diet choices that you feel another diet can't be healthy? Also you can't say eating red meat is good for the planet. Where's your research? It takes thousands of gallons of water for one pound of beef, not to mention the methane gas, runoff in our waters.\nHumans are omnivores \u2013 you were taught this in your RD training. Beef is the best source of iron and B12 \u2013 you were also taught this in your RD training. Saturated fat and dietary cholesterol are no longer \"nutrients of concern\" as per last year's updated guidelines. This isn't 1990 anymore. Try growing your own food and see if you can do it without any animal \"suffering\" because you can't. Have a great day.\nI enjoyed reading your site and my own experience contradicts some of what you have said; in particular that saturated fat is no longer a concern. I developed heart disease while following these diets. Yes I lost weight; yes, my blood work revealed improvement. At the same time, I found out, after an MI, that I had plaque build up in my coronary arteries. I did extensive testing, after the fact, of the causes of this problem. After the VAP Cholesterol test, I found high levels of lipoprotein(A). This lipoprotein was the culprit in my plaque buildup and is directly related to saturated, monounsaturated, and polyunsaturated fat consumption in my diet. After switching to a plant-based diet (based on Colin Caldwell's research), my lipo(A) numbers dropped dramatically. While my story may be unique, I do offer a word of caution to folks and encourage all of us to tread carefully.\nIt's telling that our pets are now at risk! \u2026there only needs to be the most cursory look at ingredients in our petfood. Diabetes should never happen to our pets, but they are eating what we are feeding them. Advertising for petfood includes hooks that catch our attention, including Pasta, cheese\u2026turkey with cranberry\u2026can you see how ridiculous this is?\u2026and if you can then why is this not ridiculous for us!\nI've read so many of these sites promoting Whole30, Paleo and the like, and followed these diets, in one form or another, for 20 years. My own story starts with the fact that I embraced these diets and later developed heart disease. I ate exactly what people said to eat: protein, some plant-based foods, and so-called \"healthy\" fats. My blood work appeared to prove the advocates of Whole30 etc right\u2026everything looked good. In fact, what was happening to my arteries and veins was a buildup of plaque, particularly in my coronary arteries. I had no idea. My blood work told the story that all was fine. The reality was quite different. After an MI, I had a heart cath and the truth was revealed; 90% blockage in one coronary artery. My advice to folks is simply this: while you may, like me, lose weight and \"feel great\", be very careful. Ask for more in depth testing as you age. Look more carefully at blood work; have the VAP Cholesterol test that measures the lipoproteins that make up LDL and HDL. Find a doctor that will help you understand the more elaborate details of our body. Finally, consider a plant-based diet based on research by Dr. Colin Caldwell and others.\nWhole30 has not been around for 20 years, so my question is, how long did you actually follow it? When did you start it? When was your heart attack? It takes a long time to build up to a MI and have 90% blockage. Like, years. Do you honestly think you were totally healthy, then decided to eliminate processed foods and do the Whole30 and THEN you had a heart attack? Sounds highly unlikely to me.\nThanks for your reply. Yes, of course, that specific diet plan is very recent; however, the ideas have been around for a long time. Eating low carb, plenty of plant foods, no or limited grains was the mainstay of my eating. I ate lots of lean protein, almost no red meat, lots of plants, and grains were off the menu. I can only say that my experience, while it may be unique, does say something about low-carb diets.\nThe real question is, how healthy was I? You are right, blockages come on over a long period of time. What was I doing wrong? That question has really been in the forefront of my mind since this all happened in November. Two weeks before, I road in a 50 mile bike race. I lead students on backpacking trips, cross country ski trips, and am generally what most would consider a \"healthy\" adult. I cannot pin it all on diet; what role did genetics play? All I can say is that as I research my own body, I'm trying to understand what happened and why.\nSo you've only been vegan since your MI in November?\nYes, but not really vegan; I avoid all sugar and oil, white flour, and a few other specific foods. I'd say I eat plant-based, whole foods with no oil, meat, or dairy.\nSo sorry to hear about the plaque build-up!\nThat had to have been quite a shock.\nI've been a long-time advocate for these kinds of Paleo\/primal diets, but am not married to any approach and am always looking to shore up any weak spots in my understanding of their implications for our health outcomes.\nDo you know what was the mechanism behind your eating habits contributing to the blockage?\nHope you're doing well, dude.\nYes, I am well! I did a whole series of tests and identified lipoprotein(A) as the culprit. In me, that lipoprotein was very high even though my LDL was actually low (ranged between 70 and 90 over the past 15 years or so). Beginning after the MI in November, I switched to plant based and that lipoprotein declined dramatically\u2026.am watching my blood work now and testing every 3 months to see what's happening. It's been a real journey.\nHave you looked into the dietary risk factors for lipoprotein (a)?\nI'm no cardiologist or lipidologist, but I'm having a tough time connecting it to ways of eating such as you describe.\nHDL (the \"good\" cholesterol) is actually raised by saturated fat. Please consider reading some of the work by Nina Teicholz \u2013 cholesterol is a symptom, not the \"cause\" of inflammation. Whole30 did NOT cause your heart attack. Processed foods, sugars, drinking, stress, lack of movement, poor sleep are all factors that contribute to poor health, not meat. Humans have been eating meat for a long time \u2013 heart disease is a relatively new thing. Best of luck.\nI'll second the Teicholz suggestion, and add Dr. Peter Attia and Ivor Cummins. Ivor especially goes into detail about lipoproteins in his presentations.\nRegardless, keep an eye on that blood work, dude, and hang in there.\nDefinitely looking into it. That's why I'm researching all of this information\u2026..as you know, information out in the world often is contradictory. I'm relying on scientific evidence and research. Even then it gets challenging.\nRegardless, I'm glad you seem to improving, and thank you for sharing this perspective!\nLooking forward to digging into this a bit more.\nI have lost 65lbs in 2 years on paleo. I am not a dietitian or doctor like lots of other commenters. I enjoy my vegetables, fruits, and meats and take walks and do some running. Blood pressure is normal now, cholesterol perfect, triglycerides perfect. When I stopped dairy my inflammation went away. Not just a coincidence. Say what you will, this works and I am never hungry.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I am the technology expert for the Island Riddim Radio show. I also work as a contributing web developer and administrator.\nHosted by Sharon and Desmond, I am the Tech Expert for the show Island Riddim Radio. Each week I discuss and new topic in Computing News, Internet Technology, or Cell phone technology. The show also offers How-to explanations on how to work with your device or make decisions on computer related purchases. Behind the scenes, I help support the Island Riddim Radio website and social media.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"We are seeking volunteers for two of our community projects which support people over 50 years of age, and job seekers within the Birchfield Big Local (BBL) area.\nBirchfield Big Local (BBL) is re-developing the Stepping Forward Job Club, and is seeking enthusiastic and experienced people who will have a positive and proactive commitment to providing a friendly and relaxed environment in which local people living in the BBL area can improve their employment prospects for the future and explore the opportunities available to them.\nAlong with relevant experience, skills and knowledge to undertake the role, volunteer assistants will also be required to undertake training in job club principles as part of the induction process. Other training will be made available that is appropriate to the role.\nPlease join us and help make a difference!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sunshine and perfectly comfortable, seasonable temperatures are here for one more day.\nChanges are on the horizon, but you'll have one more pristine day for Saturday. Expect highs in the upper 60s with only a few passing clouds. Tonight, lows will be a bit milder, only dipping into the low to mid 40s. Clouds increasing Sunday signal a round of rain and storms on the way.\nSunday starts mostly sunny with clouds building through the afternoon. Showers will be isolated, so your weekend plans won't be greatly impacted, aside from dodging a few raindrops. Highs remain comfortable - in the lower 70s. Monday is a different story.\nLate Sunday night and early Monday morning, a cold front will bring a round of rain and storms to the Valley. We aren't expecting any widespread threat of severe weather, but stronger storms can pack gusty wind. If there is any main concern to highlight, it will be the possibility of quick ponding and standing water with the heavy rain and already saturated soil.\nThe cold air lags behind the front, so you won't feel a major chill Monday. By Tuesday, cooler air filters in and drops highs by about ten degrees.\nDamage reported with Saturday night\/Sunday morning severe weather.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Alpha Pest Control is an independent, professional pest management business operating across the UK. We currently have a fantastic opportunity for an Area Surveyor to work with and support our National Account Manager to develop new business.\nA great time to be joining a rapidly growing business, this will be a challenging and interesting role to expand our customer base across the UK. Experience in pest control management would be desirable and an advantage to potential candidates, however, equally importantly is that the successful candidate will have the right attitude and work ethic to join our team.\nSalary Alpha Pest Control is an independent, professional pest management business operating across the UK. We currently have a fantastic opportunity for an Area Surveyor to work with and support our National Account Manager to develop new business.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Knight Nissan is the best Nissan dealership around, and we've got some of the latest deals on your car and service. From Swift Current to Saskatchewan, you won't find a better place for your next vehicle. And right now, take advantage of our special on an oil change for $39.95, including a free 100-point inspection!\nGear up for your next big trip in the 2016 Nissan Pathfinder SL with four-wheel drive. Coming in a sleek Gun Metallic color, this workhorse will keep everything running smoothly on your next big adventure. With our incredible Knight Price of only $42,738, and our dealer exclusive Lifetime Engine Warranty, there's no reason not to check this one out.\nOr look into the 2016 Nissan Sentra SL. With a great Graphic Blue Metallic exterior, great efficiency at 6L\/100km on the highway and affordable price, there's no reason not to make the Sentra your next test drive. Drive away in this great care for only $26,308.\nCheck out the rest of our great new vehicle specials and schedule a drive today!\nWe've also got great selection of used and Certified Pre-Owned vehicles to drive away in.\nTake the 2014 Dodge Journey R\/T all-wheel drive for a spin today. With a six-speed Multi-Speed Automatic transmission, white exterior with black interior, and 8L\/100km mileage, this one is a great buy; $22,700.\nAlso worth noting is the 2008 Chevrolet Cobalt LT, available for only $9,900! With 59,879 kilometers under the wheels, and a great 6L\/100km on the highway, this is a great choice for those who love affordability and efficiency. It's a beautiful red car in great shape and no accidents, so check it out today.\nYou'll also find more impressive used and certified pre-owned vehicles available to you.\nCome check out Knight Nissan in Swift Current. Here, you matter, and we get that times may get rough. So, we will work to get you in a vehicle without breaking your bank. Our deals are designed to help you out!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Occupational Therapy's Perspective on the Use of Environments and Contexts to Facilitate Health, Well-Being, and Participation in Occupations. Am J Occup Ther 2015;69(Supplement_3):6913410050p1-6913410050p13. doi: 10.5014\/ajot.2015.696S05.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Northrop Grumman Corp. has hired Steve Perkins to direct business development for the Los Angeles company's IT sector.\nPerkins most recently was senior vice president of Oracle Corp.'s North American government, education and health care business. He also developed and led the Redwood Shores, Calif., company's homeland security initiative.\nAt Northrop Grumman IT, Herndon, Va., Perkins will be vice president of business development and strategy. His tasks will include oversight of Northrop Grumman IT's long-range strategic plan, leading its marketing and communications initiatives and managing its internal partnerships and alliances.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I have long held that this describes the third messianic war before Gog uMagog - the first in 1948 returned us to our land, the second in 1967 returned us to Jerusalem and the third will return us to the Temple Mount.\nNote that we have Gaza in the \"West,\" the West Bank in the \"East\" and \"Edom, Moab and Ammon\" make up present day Jordan - all of which incorporates all of the areas currently occupied by \"Palestinians.\"\nWe've had flare-ups in Gaza many times (too many times) over the years, along with the intifadas and ongoing terror attacks \"over the Green Line.\" But suddenly, we have reached the point where Jordan may be becoming destabilized enough to allow this prophecy to come to full fruition, resulting in this third messianic war that will end any prospect of a State of Palestine forever.\nThis is the key element that has been missing until now. Once again, we have a clear sign that time is growing very, very short. The revelation of Mashiach must be very close now.\nThe Prime Minister's acceptance or rejection of this recommendation will seal the deal, as far as I am concerned, about whether he has an ounce of good left in him or if he is completely sold out to The Other Side.\n...A forum assembled on Sunday by the Chief Rabbinate was shocked by the fact that the authority over state conversions \u2013 which so far has only been usurped by local rabbinical courts \u2013 will now be yanked completely by the state, the prime minister's office, to be exact. Both chief rabbis as well as many chief rabbis of Israeli cities warned repeatedly about the halachic pitfalls of the idea of switching the process from rabbinical clerks to state clerks.\nI'd love to be proven wrong about Netanyahu, but I don't hold out much hope for him. These are the kinds of actions that will bring the Erev Rav regime to its well-deserved end.\nConversion without Torah will never work. Judaism is\/was\/will always be based on Torah.\nThe question is how to keep the \"High Court\" out of this?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Find all of our events here! Links are usually available to make your reservation!\n* If you have dietary restrictions please let us know and we will do our best to accommodate you.\nCost is $50 per person plus tax and gratuity.\nPlease call Wine Sage and Gourmet to reserve your seat!\nWe are so excited to have Molly Fowler, The Dining Diva, join us for a series of cooking classes! Molly has appeared on WLOS Carolina Kitchen segments and has been teaching culinary classes for many years.\nOur classes sell out every month \u2013 so reserve your ticket today!\nCost is $50 and includes recipe packet, samples of all the recipes and wine pairings! Call the store or email us to make your reservation! Our classes fill up so don't delay! You can also click the link and reserve your spot. If it indicates no tickets available PLEASE call the store and we can assist you.\nFee per person is $50.00, includes a recipe booklet, samples of all recipes and wine pairing tastings. Paid reservation required in advance. Limited to 16 attendees.\nCooking Class Policy: Adults only, please. Seating is limited so prepaid reservations are required. You may cancel up to 48 hours prior to the class and receive a refund less 5% (2.50) for administration fees. We reserve the right to cancel if less than 12 participants. You will be refunded your full amount in the event the class is canceled. Social time will begin at 5:30pm with the opportunity to purchase a glass of wine to enjoy. Classes will start promptly at 6:00pm. A complete recipe packet will be provided. You may want to bring a pen to take notes \u2013 this is truly a class experience! Each participant will get to sample each dish prepared along with a 1-2 oz pour of the suggested wine pairing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I joined CrossFit Strongsville ( FitCore) because last September I had terrible back spasms. After getting treatments at The Healthiest You Chiropractic Center, I was convinced that strengthening my core was the best way to avoid a relapse in the future.\nI feel my biggest accomplishment has been being able to show up for the workouts and to complete them and to find I actually enjoy doing them! Also, I can do multiple pull-ups!\nWhat I am working on now is I'm trying to gain more stamina as I continue to get stronger and improve my core muscles.\nMy favorite CrossFit (FitCore) memory so far is just meeting the others who are in our program doing the workouts. They're encouraging and inspiring!\nThe things that motivate me to continue training at CrossFit Strongsville (FitCore) are: I'm a 65 year old, retired man. I don't wish for these back problems to happen again because I love to play golf and tennis and enjoy being able to play and teach my grandchildren.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Airport Shuttle Italy offers point-to-point car service throughout the entire Rome area and all Italy. Whether you are heading to a client meeting, for a simple airport transfer or for a night out on the town, Airport Shuttle Italy is the consistent and reliable car service solution to take you where you want to go.\nWith a fleet of insured and inspected vehicles, including luxury sedans, SUV's, limousines and vans, you have the ability to select the vehicle that best meets your need. With our guaranteed on time pickup policy, enjoy Airport Shuttle Italy's reliable door-to-door pickup and drop off service provided by our professional, experienced and vetted drivers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Processing Community Day @ Worldwide currently have 100+ registered node cities around the world. Please look for your nearest city on the map and contact your local organizer directly about attending.\nPick a date and time between January 15 and February 15, 2019 to host a Processing Community Day.\nThe duration can be as short as a two-hour programming tutorial, or as long as an all day event that includes workshops, discussions on network culture, or a show & tell on software art.\nThis is an open question depending on your local culture and what you have access to, but generally we recommend collaborating with public facilities such as libraries, schools, or community-driven spaces with good, stable Wi-Fi such as community centers, art \/ culture venues, hackerspaces, cafes, tea houses, or bars.\nWe are unable to provide financial assistance for the node events, however we will be able to provide an Organizer's Kit, which includes a step-by-step organizing guide, promotion and tutorial packages. We will also support your event through our social media, and provide a platform for you to exchange experience and knowledge with communities around the world. Please use the hashtag #PCD2019 when posting on social media so that we can help promote your event.\nWe do not manage or distribute tickets on behalf of our node events. As an organizer, you will determine whether ticketing is a preferable option for your event. We encourage all organizers who are planning to charge a fee for the event to provide affordable options to promote accessibility and inclusivity.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Home \u203a Audrina Patridge \u203a Lauren Betrays Audrina!!!!!\nFINALLY! We solved this mystery!\nThe ultimate stab in the back!!!\nSources reveal exclusively to PerezHilton.com that Audrina Patridge and her former roommate, Lauren Conrad, are definitely on the outs.\n\"They're not talking right now,\" a Hills insider tells us.\nThat's where it gets tricky!\nHere's what we know thus far\u2026..\nApparently, a friend of Audrina's on\/off boyfriend, Justin Bobby, told her one night that Justin and L.C. hooked up.\n\"When confronted, Justin admitted to Audrina that he and L.C. hooked up,\" a source tells us.\nThis \"confession\" only happened within the last week, we're told.\nThis might all be too real for The Hills, though. We're not sure if this betrayal will make it onto the show.\nMore details to come hopefully. As soon as we know, U will know!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"#1 Heart Rate Calculator App in more than 40 countries!\nTouch the screen every time you feel your heart beat to get instant heart rate measurement.\niPhone4 users, read your heart rate using your Flash Camera!\n* Loads in 1 second.\n* Displays in which Training Heart Rate Zone you are according to your age.\n* Displays your Resting Heart Rate Zone.\n* Displays last Heart Rate on the icon on the menu.\n* Compatible with all iPod Touch, iPhone, and iPad models.\n* Share your heart rate measurement with your friends on facebook.\n* Record your heart rate measurements in your Carre Health account.\n1- Find your pulse with one hand.\n2- Tap the screen every time you feel your heart beat.\n3- Get an instant heart rate measurement.\n1- Cover the flash and camera with your left hand index finger.\n2- Check that the signal quality is good.\nHeart rate measurement is one of the best tools to assess your cardiac health yourself. Dr. Oz recommends to take your resting heart rate every morning before you get out of bed.\nUse it for training, jogging, cycling, running, etc.\nMeasure you resting heart rate in the morning and in the evening.\n\u2013 Heart Rate Calculator is a poorly rated app (2-star).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I am living in Greater Toronto Area of Canada.\nIf you need my help with SAP system, database, application and program performance.\nSuggestion\/advice for me to use in my future posting so my blog can help you better.\nAny suggestion\/advice for me to correct existing blog such as technical error, typo etc.\nOr just to say hi to me.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Learn To Play Guitar browser toolbar for Internet Explorer. Easily find guitar playing tips and guitar lesson resources direct from your browser toolbar. Everything you need in one convenient place to help you progress as a guitar player. Listen to classic guitar music direct from the toolbar and chat with other Muso's from across the world as you learn to play guitar. The Learn To Play Guitar IE browser toolbar is an ideal add on for beginners.\nLearn To Play Guitar browser toolbar for Internet Explorer.\nLearn To Play Guitar 101 .NET's Cool Guitar's Screensaver.\nHow to play fast and clean - Make people's jaws drop!\nCombination of best and most atractive guitar images and screensavers.\nQweas is providing links to Learn To Play Guitar 1.0 as a courtesy, and makes no representations regarding Learn To Play Guitar or any other applications or any information related thereto.\nAny questions, complaints or claims regarding this application Learn To Play Guitar 1.0 must be directed to the appropriate software vendor.\nYou may click the publisher link of Learn To Play Guitar on the top of this page to get more details about the vendor.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzamgov b/data_all_eng_slimpj/shuffled/split2/finalzzzamgov
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+{"text":"The Department of Education and Childhood, Bristol Inter-disciplinary Group for Education Research (BRIDGE), is pleased to have Dr Jessica Gagnon speak about the HEFCE-funded research project Changing Minds on 7 May 2019.\nThis presentation will explore initial findings from the two-year Office for Students (OfS) funded attainment gap project titled \"Changing Mindsets: Reducing stereotype threat and implicit bias as barriers to student success\". The project is focused on addressing unequal student outcomes for two student groups: Black and Minority Ethnic (BME) students and socio-economically disadvantaged students. Findings from the project are intended to inform higher education policies and practices to address inequalities in students' experiences and outcomes.\nView the event flyer which includes Dr Gagnon's biography.\nThis event is free to attend but pre-registration is required. Register here to book your place.\nFor more information about the project, please visit the Changing Mindsets website or follow the project on Twitter.\nProject Zulu is a UWE Bristol educational development initiative which works with partner schools in township and rural areas of Kwazul-Natal, South Africa. Collaborations between school communities in South Africa and UWE Bristol staff and students seek ways of enhancing infrastructure, pedagogy and learning opportunities through knowledge exchange, action research and practical support.\nICT project at a rural South African primary school by Dr David Wyatt and Benjamin Knight.\nFor this Lunchtime Seminar, the speakers were Jade Parnell, from the Centre for Appearance Research (Department of Health and Social Sciences, UWE Bristol); and Dr Maryam Almohammad and Dr Jane Andrews (Department of Education and Childhood, UWE Bristol).\nJade Parnell discussed her doctoral research on \"Promoting acceptance of socially stigmatised appearances in young children in primary school\".\nDr Maryam Almohammad and Dr Jane Andrews presented on their topic of \"Artmaking, materialism, and multilingualism in welcoming environments for EAL learners\".\nVisit our blog or view the event flyer for more information.\nOur speaker was Dr Richard Waller (Associate Professor of the Sociology of Education, Department of Education and Childhood, UWE Bristol). The Paired Peers project was a two part longitudinal study (2010-2103 and 2014-2017) funded by the Leverhulme Trust which followed an initial cohort of 90 undergraduates studying at either of the two universities (University of Bristol; UWE Bristol) in one English city (Bristol).\nThe project matched pairs of students by social class, gender and subject discipline both within and between the two institutions, and followed them from induction to graduation and for several years beyond into their post-university lives and careers.\nThe presentation will be followed by a discussion of the implications of the project's findings for the practice of academics and university professional services staff alike. For more details about the talk, visit our blog or view the event flyer.\nThe event is free for anyone to attend but pre-registration is required. Register here to book your place.\nOur speakers were Dr Nigel Newbutt (Senior Lecturer\/Senior Researcher in Digital Education, Department of Education and Childhood, UWE Bristol) and Ms Dimitra Magkafa (Doctoral Researcher, Department of Arts and Cultural Industries, UWE Bristol).\nFor more information about the speakers and their talk topics, visit our blog or view the event flyer.\nOur speakers for this lunchtime seminar were Laura Manison Shore, PGCE Primary and Early Years Programme Leader and Partnership Manager (Department of Education and Childhood, UWE Bristol) and Dr Maryam Almohammad, Research Associate (Department of Education and Childhood, UWE Bristol). For more details, visit our blog.\nThis conference brought together leading researchers from the South West of England and guest speakers from the USA in the field of gender, sexuality, bodies and identity and showcased some of the work of the presenters and the groups, centres and universities they represent. For more information, view the conference programme or blog.\nA new campaign has been launched to help UWE Bristol students speak up and report unacceptable behaviour such as harassment, discrimination and violence. Speak Up has been introduced this term to raise awareness and prevent these types of incidents on campus. In recognition of the UN International Day for the Elimination of Violence against Women (25 November 2018), UWE Bristol hosted a panel discussion to continue conversations around violence and abuse. View the event flyer.\nThis event brought together researchers across the University to share current research interests to establish inter-disciplinary research connections.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"Super Sport.\" From 1961 to 1972, those two words were all it took to symbolize high performance across Chevrolet's passenger-car lineup, from the top-dog Impala to the compact Nova, the SS version of which debuted in 1963, but only as an appearance package that added special trim and wheel covers, bucket seats, instrumentation, and a deluxe steering wheel. A true performance Nova SS followed in 1964 with an available 283\/195 HP small-block V-8 that finally gave the Nova the power to be taken seriously on the street. After adopting the 327 CI engine to the Nova in 1965, Chevrolet finally made it into an honest muscle car by 1966 with the L79 327\/350 HP V-8. This 1966 Chevrolet Nova Super Sport builds on that lineage in a rotisserie Resto Mod build completed in 2004. Boasting a laser-straight body and deep Black paint and custom Tan interior, it takes the small-car-small-block formula to a whole new level with a 412 CI version fed by cowl induction, finished to show-quality standards and dyno tuned by Lamar Walden. Nestled in an Alston front subframe, it incorporates aluminum heads, tube headers and a new dual exhaust system, A rebuilt 5-speed manual transmission and Ford 9-inch third member complete the car's heavy-duty drivetrain. Front and rear subframe connectors reinforce the Nova's unibody structure against torsional forces, while Caltracs traction bars work to put the power to the ground. Further equipped with power steering and brakes, tinted glass and a full range of Auto Meter Pro Comp gauges, the little Nova uses performance rubber on American Racing Torque Thrust wheels for a sharp period look.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Notice is hereby given under the Pennsylvania Fictitious Name Act, 54 PA C.S.A. \u00a7 311 et seq., that the Fictitious Name of \"Cathy's Beauty Salon\" is filed with the Department of State of the Commonwealth of Pennsylvania. Inspire To Create LLC, sole proprietor, will do business as Cathy's Beauty Salon with its registered office at 620 State Street, Meadville, Pennsylvania 16335.\nAn application for registration of the fictitious name StevePetruso's Maintenance, 14054 Brown Hill Rd., Conneaut Lake, PA 16316 has been filed in the Department of State at Harrisburg, PA, File Date 01\/14\/2019 pursuant to the Fictitious Names Act, Act 1982-295. The name and address of the person who is a party to the registration is Stephen P. Petruso, 14054 Brown Hill Rd., Conneaut Lake, PA 16316.\nYou are hereby notified that on July 13, 2018, the Plaintiff, U.S. BANK NATIONAL ASSOCIATION TRUSTEE FOR THE PENNSYLVANIA HOUSING FINANCE AGENCY filed an Amended Mortgage Foreclosure Complaint endorsed with a Notice to Defend against you in the Court of Common Pleas of Crawford County, Pennsylvania, docketed to No. AD 2018-105 wherein Plaintiff seeks to foreclose it's mortgage securing your property located at 657 Clover Lane, Meadville, PA 16335, whereupon your property would be sold by the Sheriff of Crawford County.\nYou have been sued in Court. If you wish to defend, you must enter a written appearance personally or by attorney and file your defenses or objections in writing with the Court. You are warned that if you fail to do so, the case may proceed without you and a Judgment may be entered against you without further notice for the relief requested by the Plaintiff. You may lose money or property or other rights important to you.\nYOU SHOULD TAKE THIS NOTICE TO YOUR LAWYER AT ONCE. IF YOU DO NOT HAVE A LAWYER, TELEPHONE THE OFFICE BELOW TO FIND OUT WHERE YOU CAN GET LEGAL HELP.\nIF YOU CANNOT AFFORD TO HIRE A LAWYER, THIS OFFICE MAY BE ABLE TO PROVIDE YOU WITH INFORMATION ON AGENCIES THAT MAY OFFER LEGAL SERVICES TO ELIGIBLE PERSONS AT A REDUCED FEE OR NO FEE.\nALL THE FOLLOWING DESCRIBED REAL ESTATE SITUATED IN NORTH SHENANGO TWP., CRAWFORD COUNTY, PENNSYLVANIA. HAVING ERECTED THEREON A DWELLING KNOWN AS 2066 STATE HIGHWAY 285, LINESVILLE, PA 16424. DBV 890, PG 963. ASSESSMENT #4602-065-3.\nFirst National Bank of Pennsylvania vs. Patricia A. Slagle, Known Heir of the Estate of Wallace W. Slagle and the Unknown Heirs, Executors and\/or Administrators of the Estate of Wallace W. Slagle, at Execution No. AD-2018-584 in the amount of $35,941.33.\nSchedule of Distribution will be filed by the Sheriff on the date specified by the Sheriff no later than thirty (30) days from sale date. Distributions will be made in accordance with the schedule unless exceptions are filed within ten (10) days of the filing of the Schedule.\nNotice is hereby given to all Creditors, Legatees, and other persons interested that the following accounts have been filed in the office of the Clerk of the Orphan's Court of Crawford County, Pennsylvania, and that the same together with statements of proposed distribution will be presented for confirmation on April 1, 2019, at 8:45 a.m., at the Judicial Center in Meadville, Pennsylvania, on which date said accounts and statements of proposed distribution may be confirmed unless objections are filed prior to presentation for confirmation. Any person who fails to present his or her objection prior to presentation for confirmation will be barred forever from so doing. Upon Order of Confirmation distribution may be made in accordance with the Order of the Court and the applicable rules of procedure.\nThe Corry Area School District has declared all that certain piece or parcel of land and all improvements located thereon, being located in Spartansburg Boro, County of Crawford, Commonwealth of Pennsylvania, commonly known as the Spartansburg Elementary School, 150 Water Street, Spartansburg, Pennsylvania, being further described as the portion of real property identified in Crawford County Deed Book 555 at page 438, located to the west of Water Street and identified as Crawford County Tax Index No. 5002-005-70001, hereinafter referred to as \"the Property,\" to be unused by and unnecessary to the District. The Board of Directors of the District has resolved to sell this Property by sealed bid. Terms and conditions of sale were fixed by Board resolution dated February 25, 2019, which are part of the bid packet.\nInterested bidders may pick up bid packets from the District's Board Secretary's office, located at the District's Administration offices, 540 East Pleasant Street, Corry, PA, 16407, during the hours of Monday through Friday from 8 a.m. through 4 p.m. Bids must be received by the District, in the Board Secretary's office located at the District's Administration offices, 540 East Pleasant Street, Corry, PA 16407 by noon on Friday, April 5, 2019. Bidders are required to submit a Bid Deposit made payable to the Corry Area School District in the amount of $10,000, in the form of a cashier's check or certified check, at the time the Bids are submitted to the District.\nBids shall be opened publicly and read aloud at a public meeting on April 8, 2019 at 7:00 p.m. at the Large Group Instruction Room in the District's Administration offices, 540 East Pleasant Street, Corry, PA 16407. The District reserves the right to waive any defects, errors, omissions, mistakes or irregularities in the Bids. The District reserves the right to reject any or all bids. The Board of Directors, if it determines it to be in the best interest of the District to award a bid, shall award the bid to the highest responsible and responsive bidder at a public meeting on April 23, 2019 at 7:00 p.m. at the Large Group Instruction Room in the District's Administration offices, 540 East Pleasant Street, Corry, PA, 16407.\nSeized and taken in execution as the property of Brian Bailey, Known Surviving Heir of Brenda Bailey, Deceased, and Unknown Heirs, Surviving Heirs of Brenda Bailey, Deceased at the suit of New Penn Financial, LLC, d\/b\/a Shellpoint Mortgage Servicing and to be sold on Writ of Execution No. 2019-9.\nBeing commonly known as 419 Erie Street, Saegertown, Crawford County, Pennsylvania, and being described at Record Book 960, Page 679, excluding the Outsale of April 22, 2014 at Record Book 1167, Page 368, Instrument No. 201400002816, and bearing Crawford County Tax Parcel I.D. No. 4512-003.\nSEIZED and taken in Execution as the property of Jennifer L. Baker a\/k\/a Jennifer L. Wiard, at the suit of Marquette Savings Bank, to be sold on Writ of Execution No. 2019-6.\nSeized and taken in execution as the property of Vilola Boyd-Mosso as Executrix of the Estate of Bruce Boyd, Deceased, at the suit of Ditech Financial LLC, and to be sold on Writ of Execution No. 2019-12.\nSeized and taken in execution as the property of David C. Campbell and Cynthia A. Campbell, Deceased, at the suit of The Bank of New York Mellon, as Trustee for CIT Mortgage Loan Trust 2007-1, and to be sold on Writ of Execution No. 2019-16.\nBEING THE SAME PREMISES WHICH MARGARET J. ENGLISH NOW BY MARRIAGE MARGARET J. SHADE AND BLANE J. SHADE, HER HUSBAND, BY DEED DATED MARCH 2, 1998 AND RECORDED ON MARCH 13, 1998 IN THE OFFICE OF THE RECORDER OF DEEDS IN AND FOR THE COUNTY OF CRAWFORD AT DEED BOOK 378, PAGE 421, GRANTED AND CONVEYED UNTO DARROLD E. ENGLISH, JR AND CARI L. ENGLISH, HUSBAND AND WIFE, THEIR HEIRS AND ASSIGNS.\nSEIZED AND TAKEN IN EXECUTION AS THE PROPERTY OF DARROLD E. ENGLISH JR AND CARI L. ENGLISH AT THE SUIT OF 21ST MORTGAGE CORPORATION ASSIGNEE OF ASSOCIATES HOUSING FINANCE, LLC, AND TO BE SOLD ON WRIT OF EXECUTION NO. 2019-22.\nA lot 60 feet by 33 feet located on Kerr Street.\nBEING further identified as Tax Map No. 5600-D-3-1B and more commonly referred to as 312 North Kerr Street, Titusville, Pennsylvania 16354.\nSeized and taken in execution as the property of Bradley V. Ongley at the suit of The Marshall K. Coryell Trust and to be sold on Writ of Execution No. 2019-18.\nProperty of Donald Foltz and Patricia C. Foltz, situate in Bloomfield Township, Crawford County, Pennsylvania, being more particularly described in a deed recorded with the Recorder of Deeds, Crawford County, Pennsylvania in Deed Book 470, Page 269, and being Tax Parcel ID 1312-024-70001; with an address of 35835 Foltz Road, Centerville, PA 16404 and containing a single family, two story dwelling, with an attached store, an attached garage; and a detached pole garage with an attached barn and several other outbuildings.\nSeized and taken in execution as the property of Donald Foltz and Patricia C. Foltz, both deceased, and Kenneth Orris, named Executor of the Estate of Patricia C. Foltz at the suit of Farmers National Bank of Emlenton and to be sold on Writ of Execution No. 2019-20.\nBEING the same premises in which Jennifer L. Raszman now by marriage Jennifer L. Marks and Warren A. Marks, husband and wife by deed dated October 14, 2008 and recorded in the Office of Recorder of Deeds in and for Crawford County on October 27, 2008 at Book 944, Page 234 and Instrument #200800011513 conveyed unto Heidi M. Rash, an unmarried woman.\nSeized and taken in execution as the property of Heidi M. Rash at the suit of Bank of America, N.A., successor by merger to BAC Home Loans Servicing, LP f\/k\/a Countrywide Home Loans Servicing, LP, and to be sold on Writ of Execution No. 2019-13.\nBeing known as 15192 South Mosiertown Road, Meadville, Crawford County, PA and being described at Record Book 1105, Page 471 and being Tax Map No. 2812-056.\nSeized and taken in execution as the property of Dan T. Snyder, Jr. and Anne Marie Snyder at the suit of Northwest Bank, as successor in interest to Northwest Savings Bank, at A.D. 2018-614 to be sold on Writ of Execution No. EX 2019-19. Being Tax Map No. 2812-056.\nSeized and taken in execution as the property of Unknown Heirs, Successors, Assigns, and All Persons, Firms, or Associations Claiming Right, Title or Interest From Or Under Daniel L. Webster, Deceased at the suit of Wells Fargo Bank, NA, and to be sold on Writ of Execution No. 2019-17.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I have checked your website and noticed that the entire website is not loading properly ( not this page only ) as its css\/js scripts are not properly rendered.\nWith this in mind, please flush the cache of your CDN provider or deactivate it in order to verify the issue would be resolved.\nApparently when the resources(js, css) loaded from our CDN the theme doesn't render properly. I've deactivated CDN under Assets Optimization to fix the issue.\nBut before you can enable that again, make sure you update Hummingbird to latest version.\nAfter update you can activate CDN again. And if the issue persist, go to Hummingbird > Assets Optimization > Settings and use the reset button to clear the settings.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Are you searching for home insurance coverage in Athens, PA? Enter your Zip Code to compare the best options available nearby. Comparing multiple quotes is the best way to save money on your homeowners coverage in Bradford county.\nHave you been asked by your mortgage lender to get homeowners insurance? Did you just purchase your first home and understand the value of insuring it but don't know what type of coverage to buy? If so, here is some information to help you get the Athens, Pennsylvania homeowners insurance you need.\nHomeowner's insurance policy is extremely great to have if any kind of a disaster ended up to take place. The issue is, it's difficult to get the correct type of coverage, so you should know a couple of items so you do not overpay. Stick to the suggestions in this article to pick the proper plan for you.\nVerify the standing of your homeowner's insurance policy premiums at the very least once a year, to see if you may qualify for a reduced price. Your recent price could be primarily based on an outdated criminal offense statistic, for illustration, or you may have mounted a safety program that could lower your charges. Go over these alterations with your insurance policies agent.\nTo save money on your home owners insurance policies discuss to your agent and see if the organization gives discounts for setting up further smoke detectors. Numerous more mature residences deficiency them in locations that are regarded as normal places to install nowadays and several insurance coverage organizations will provide a low cost as an incentive for you to insert far more.\nTo keep your coverage up to date, be sure to evaluation your homeowner's policy every single 12 months. Let your insurance company know of adjustments in your property and property that may possibly aid keep your rates down. For occasion, if you have changed a shake roof with some thing far more fireproof, like composite shingles, you may possibly get a premium reduction.\nThe very best way to reduced your insurance policy payment is to raise your deductible. A large-deductible plan is a guess against the property, so to communicate. You happen to be preferring the threat of getting to shell out for a higher deductable more than the truth of possessing to shell out a larger volume of income every month. So, if you are conservative, this may not be the very best in shape. But if you're willing to likelihood getting to spend out that substantial deductible, then this method is value adopting.\nWhen you get a house, do not neglect to contain flood insurance in your policy. Floods are not usually coated by traditional homeowner's insurance policies, and current events have produced it clear that flooding can occur in places that are not predicted. Even a slight flood can result in a whole lot of injury to your house, which is why you need to be covered for this eventuality.\nContemplate increasing the deductible on your house insurance coverage coverage. A greater deductible on your insurance coverage plan can drastically reduced your once-a-year property insurance policy rates. Unfortunately, by raising the deductible, your house insurance policy company will no for a longer time spend for modest statements, this kind of as damaged window repair, leaky pipe restore and small wind and flood injury repairs.\nTo help lower your homeowner's insurance coverage annual premium, you will want to pay off your mortgage as swiftly as attainable. This lowers your high quality due to the fact insurance policies firms consider that once the house is all yours, you will be more inclined to get far better care of it, reducing the probabilities of your needing to file an insurance declare.\nDocument all of your valuables and preserve the images or films of the issues that you want lined below your property insurance policies, in a fireproof lockbox. This will safeguard your information and make filing a assert for the missing or ruined products with your house insurance policies company, easier and quicker.\nIf you want to make confident that you are receiving the best route in purchasing for residence owner's insurance policy, you have to start with key actions and excellent suggestions, which you can get proper from this article. You don't have to shed out on a great plan if you are making use of the tips in this article to get the right policy when you need to have it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzammlm b/data_all_eng_slimpj/shuffled/split2/finalzzzammlm
new file mode 100644
index 0000000000000000000000000000000000000000..647386fbaf0bd4820648aa7d9a82530469e50e4e
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"A very important part of the process of getting a basset into his foster home and his forever home is the applicants' home visit. Carolina Basset Hound Rescue requires that any family wishing to foster or adopt a basset hound have an approved home visit.\nWe receive applications from all over both states and always need volunteers to conduct these visits. The more timely we can complete a home visit. the sooner a basset gets a new home.\nWe do ask that in order to conduct home visits you go through our training process. It is quite simple and usually consists of shadowing a seasoned volunteer on one needed visit and written guidelines for you to follow. You can join our Home Visit Volunteer mailing list and you will be emailed directly when a home visit needs completed; there is no need to join our big mailing list. If you can help great, if not, just hit delete!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"US President Barack Obama did not carry gifts given to him by Kenya's Uhuru Kenyatta when he left the country on Sunday, July 26, 2015, a State House official has confirmed.\nA top statehouse official was quaoted saying that President Obama left the gifts behind and there are American regulations determining how gifts from foreign governments are received and the process had already started.\nAmerican Constitution's Article I, Section 9, clause 8, restricts the president and all federal officials from receiving any gifts from foreign governments, kings, or princes, without approval of the Congress.\nHowever, Foreign Gifts and Decorations Act of 1966 allows for acceptance of gifts of \"minimal value\" from foreign governments offered as souvenirs or marks of courtesy.\nThe law also makes an exception on the acceptance gifts when a refusal of the gift may cause \"offense or embarrassment\" or otherwise harm the foreign relations of the United States, Reagan Archives states.\n\"A tangible gift of more than minimal value accepted for reasons of protocol or courtesy may not be kept as a personal gift, however, but is considered accepted on behalf of and property of the United States, and in the case of such a gift for the President or the President's family, is handled by the National Archives and Records Administration,\"Jack Maskell, a legislative lawyer for the Congressional Research Service, notes.\nPresident Kenyatta's gifts to President Obama were two pencil portraits done by Collins Okello and a sculpture of an elephant said to be worth $3,000 in total.\nThe US introduced the gifts regulations in order to protect the President and his family and also reduce chances of bribery.\nIn the United state gifts are prohibited because they are regarded as one way of corruption, no matter how small the gift is. In African politicians they go out to fund raises, the donate like ksh 7000 but they take home with them 5 goats one bull, 15 stack of bananas, 2 sacks of maize, one sack of beans, 3 sacks of potatoes, 3 sacks of charcoal, and 10 chicken. When you add all those gifts together they total up to ksh, 100,000. So when one time one of those people who donated to gifts to that politician shows up at the politician's office, then the pay back is rendered. That's what we call pure corruption in the highest order.\nI DO NOT EVEN ACCEPT A BOTTLE OF WATER IF IT IS OFFERED. IT MAY BE USED AGAiNST ME IN FUTURE DEALINGS.\ni concer with with what obama did,kenya is a home of corruption frm top to bottom,450billons aditor general has confermed aready lost under the jubelee gorv..,is 2yr has passed and almost 20trillons as been poorly used,KQ is will be closed soon if no action,now Uhuru act and abolish this.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Calling all mechanics!: An engine seizure story.\nDiscussion in 'Community Discussion' started by soco, Aug 17, 2011.\nDrive easy, but it's fine.\nDrive easy, but eventually replace the motor.\nDrive easy, but eventually replace the crank.\nGet off the road! Totally unsafe!\nDrive like normal, driving easy won't help.\nOh woe is me. Here's what happened.\nI have a 1998 Ford Ranger Sport, 2-wheel drive with 98,600 miles on it. It was in need of an oil change but only late by ~2 months. Still, not good.\nI was leaving work Friday and suddenly my oil pressure dropped out completely. The steering wheel basically locked up and I barely managed to get to the side of the road. I called a family member who does side work as a mechanic and he said get off the parkway and to a gas station right away. This, as I've learned, was a mistake. I should have stayed where I was for a tow. But what's done is done.\nSo I started the engine back up, and with the engine running and no oil pressure, I drove it ~1 mile to a gas station where I got a tow. The engine sounded like a machine gun the whole way.\nMy father (also a mechanic) took a look and determined the synchronizer needed to be replaced as the grooves in the bottom piece of it had been chewed up somehow. He said he's never even seen that before. So he replaced it and the oil pressure returned, but the terrible noise of a machine gun remains to this day.\nMy landlord (also a mechanic! geez you'd think I'd have a better grasp on this...) listened to the noise and said it was on the underside of the motor and apparently that's a good thing. He said if I baby the motor and drive like a grandma at no more than 55 mph in the right lane for the rest of the truck's life, I might be ok.\ntl;dr: I caused what seems to be minor damage to my engine by driving a short distance without oil. The engine now makes noise on the underside of the motor and I'm afraid I need a new engine.\nI was told if I replace the \"crank\" I'll be ok. Any alternative suggestions? Agree with that one? Think I should scrap the whole vehicle?\nP.S. I'm broke, so I'd rather not follow any expensive advice, but I understand if it's the only way to go, I'll figure it out.\nHow much is the vehicle worth?\nHow much are all the repairs going to cost you?\nDepending on that, either fix, or get a new car, or just sell and use public transportation depending on your needs\/location.\neeeekkk.. that doesn't sound good.\nI know it's hard to put together the money for a new car (i'm in the same boat), but it seems like repairs on this would be $$$$$$$$. a new car would probably be your best bet if you can swing it.\nI'm no mechanic. But if several of them say you can baby it and play it safe then maybe they're right. If you can't afford the repair or a new car then take this time to start saving up for a replacement car or a repair. I don't know if I would put too much money into replacing the parts or repairing the car. It may be worth it to get a cheap used car and sell this one for parts (if it's in really bad shape).\nThen again I know next to nothing about cars... drive safe!\nSounds like it might have seized a crank bearing or rod, if it did then its not going to last probably more than a couple hours of running without doing more serious damage. Should be pretty easy to diagnose just by pulling the oil pan down because you should be able to see the whole crank then.\nLack of oil pressure results I the top end not getting oil! Which in turn results in everything else not gettig oil! If you do a compression test you'll find your down by a fair amount too as the lack of oil would have killed the cylinder wall as well as your big\/small end barrings!\nIf you want to save the engine get it fixed now the more you run it the more it'll distroy itself!\nExcuse my ignorance, but what is a compression test? When you say I'll find I'm down by a fair amount, what do you mean?\nI've been driving ~25 miles to work Tuesday and today so that's ~50 miles a day and it hasn't sounded worse, but are you suggesting that even though oil is not flowing like normal, I'm still damaging the engine by driving?\nIt's worth (in good repair) ~$3,700. So in this state, maybe $2,000. I'm totally guesstimating.\nAll repairs would cost ~$1,000 since the mechanics in my family would do the work without charging for labor. They've quoted me about a thousand for an engine replacement.\nUnfortunately, public transportation is not an option as my job requires me to drive to different law offices and the like every day. I can't use public transportation for that for time-sensitive reasons.\nMaybe I'm asking a really silly question, but if this is a 1998 truck, it's new enough to have an ECM (a central controlling computer). Is the check engine light coming on? The 1998 hardware probably doesn't store as much data as the newer hardware, but this tells you the basics of whether the cylinders are firing correctly (because if they're skipping, etc, it should trigger a check engine light).\nIf this \"machine gun\" sound is really loud and you get it all the time, I am kind of suspicious of the idea that something isn't undergoing a lot of strain while that engine is running, and that something is going to break at some point.\nCheck the oil dipstick for a shiny substance\/metal. Report back what you see.\nIf it's only about a $1000 fix, I would get it repaired.\n$1000 isn't that much (compared to a new\/used car), plus then you'll get more use out of your vehicle.\nIf it's big end or main bearing damage and it sounds like it is baby it and do something about it at the first opportunity. I don't know much about modernish yank cars (nothing after the seventies) but on most cars it should be possible to drop off the sump and change the mains and\/or big ends,there may well be damage to the crank as well but changing the bearings should give you at least a few thousand miles.You seem to know a few mechanics and if they are old school they might agree to do it for you, the cost of the bearings should be minimal.Youngish mechanics will probably tell you the motor needs to come out ,crank ground,blah blah so someone who knows how to make do and mend.I'd suggest the real old school method of finding the damaged bearing and replacing it with leather but that kind of stuff is only for being stuck in the outback a thousand miles from anywhere.\nJournal bearings on the crankshaft might be the culprit. If these ran without oil for a time, they could have been damaged (that is, gone from smooth to very rough or chewed up). If you're keeping enough oil in it now, you might not be making it worse, although it'll never get better. The way to repair it would be to replace the crankshaft and possibly the piston rods - it would be rather expensive.\nIn any event, my advice is to start saving your money for a replacement vehicle. Even if you get this engine repaired, you still have a truck with close to 100k miles on it, and the time to replace it will be coming.\nTomorrow is right, I think! And driving a car without oil pressure is pretty much the worst thing you can do to it.\nBut the machine gun sound could be a multitude of things really... What would really help us would be a video\/audio of the noise... There is still a chance that you could be damaging the engine more and more as your run it, but since that's our best option right now, you might as well make the video!\n$1,000 for a replacement motor (presumably one in good nick) sounds pretty descent, especially if your family is absorbing the labor. Honestly, it sounds like that motor you have now is a ticking time bomb. Like others have said, its hard to pin point exactly what parts were effected with an oil starved motor (may be cheaper to replace than tear the whole thing apart).\nWhatever you decide on, make sure you keep an eye on those oil levels from now on! My car burns a bit of oil in-between changes so I check it atleast once a week. Good luck!\nWhatever you decide on, make sure you keep an eye on those oil levels from now on!\nAnd if there is a next time, post from the side of the road.\nLack of oil pressure means the crank journals weren't seeing oil pressure. This means there was no film on the bearings and nothing to keep the crank from contacting a bearing. Since the oil flows, through the crank, from the crank journals to the rod journals, it also means the rod big ends suffered.\nYour bottom end is almost certainly gone, and that's likely the noise you're hearing. The rod big end bearings usually suffer first and worst, quickly wearing away the Babbitt layer and allowing the harder bearing layers to touch the journal, causing more wear and more slop. The same thing happens to the crank journal bearings.\nIf you drive it in that condition, you'll turn a potentially repairable engine into a completely unusable mess.\nIf cranks and bearings are cheap for the engine and you can do it yourself, you might be OK rebuilding the engine. If they're not, the most economical route is an ebay\/junkyard engine. You should be able to easily swap the engines out in a day.\nIMHO, don't get the crank or bearings anywhere but the manufacturer\/dealer. Firsthand, I've seen too many out-of-tolerance parts from third parties when we're talking a simple rebuild.\nThat is, if you indeed ran the engine with no oil pressure. For all I know from where I sit you've just got a faulty oil light and a fan blade hitting a shroud. No way to tell from here.\nI believe that has been ruled-out. His oil pump failed.\n$1 says you fried your crank bearings and the sound you are hearing are the piston rods hitting the crank. Time for a new motor my friend.\n$1000.00 for a replacement motor sounds like a good deal to me. Who knows what you did to the motor until they drop the oil pan and take a look. I wouldn't drive it if it's making the \"machine gun\" noise you say it is, as you will just do more damage. It could be an end cap on one of the connecting rods, that would make a loud banging noise close to what you're describing.\nOil pressure drops, get off the road and shut off the car period. One mile is enough to damage a motor with no oil pressure. Sorry about your luck.\nSpend the $1000 and get the rebuilt engine put in. As it is, your truck is worth very little if you needed to sell it for a new vehicle. With a new\/rebuilt engine in it your truck is good for several more years, and then when you go to sell it or trade it in it will at least be worth something towards the newer the vehicle.\nAs sad as it is to read, I guess I'll be going the replacement engine route. Time to start saving, somehow.\nAs I said, I don't have a choice but to drive it until I have the money for this job. At least now I know what I've got to do though.\nI'm sure once you post the video you'll get much more specific help, good luck!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Develop and advance design concepts for the company. Create compelling sketches to bring the teams proposals to life. Participate in the project from initial sketch phase, development, through to finished customer-facing execution. You readily solicit and accept feedback, recognizing competing inputs and priorities and take part in constructive dialogue to resolve apparent conflicts. Work with team leadership to achieve the best possible product in terms of craftsmanship. Collaborate with internal & external resources Participate in design projects as necessary and as your own skills dictate. Qualifications: Minimum of 4+ years of experience. Experience working in a multidisciplinary environment to build customer experiences from the ground up. Solid conceptual and technical skills across a diverse range of customer-centered projects. Must be self-motivated and able to manage simultaneous parallel-path projects. Portfolio of diverse work that showcases your creativity, tenacity and practical skills. Must have good communication skills Candidate should have good knowledge in Social Media.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"[S1084] \"Chris Clark,\" e-mail message from (unknown address) to Cheryl Longyear, Apr 29, 2014, Hopefully I can piece together some information missing from your town records. Harrison B. Clark is actually Harrison R. Clark Born in Syracuse in 1890. He went by the name of Harry R. Clark. Harry moved to Florida after serving in World War I and married Leona Richardson of Georgia (Born: 1903) in 1921. They lived in Dade County, Florida. Harry Clark died 1 February 1974 (I have the death certificate). Harry's parents were Claude Clark and Mary Jones; they indeed had seven children. Claude's parents were Hiram Clark and Louisa Mills; Brother: Charles (According to New York Census 1865 & 1870 U.S. Census). Not exactly sure when Hiram or Louisa died. Samuel E Clark and Betsey (probably a nick name for Elizabeth) had Hiram, Levi (1850 U.S. Census), and Eunice Clark (Montezuma's death records, see link below). Samuel Clark died 2 Sept 1864 (1865 New York Census). No known death date for Betsy.\n[S1083] Skaneateles Democrat, Skaneateles, New York.\nSource Citation: Year: 1860; Census Place: Montezuma, Cayuga, New York; Roll: M653_729; Page: 720; Image: 243; Family History Library Film: 803729.\nSource Citation: Year: 1870; Census Place: Montezuma, Cayuga, New York; Roll: M593_911; Page: 493A; Image: 180; Family History Library Film: 552410.\n[S85] Montezuma Birth Records, online <http:\/\/web.archive.org\/web\/20001110085900\/http:\/\/\u2026>, Clark, (f) Dec 9, 1885 Aurelius, Mud Lock Sarah Jones (19) Mud Lock Claud Clark (23) Laborer Montezuma.\n[S85] Montezuma Birth Records, online <http:\/\/web.archive.org\/web\/20001110085900\/http:\/\/\u2026>, Clark (f) Jan 4, 1887 1-0 Aurelius Mary E. Jones (19) Seneca Falls Claud Clark (25) Laborer Montezuma.\nSource Citation: Year: 1900; Census Place: Pepin, Pepin, Wisconsin; Roll: 1810; Page: 5A; Enumeration District: 0116; FHL microfilm: 1241810.\nOrinda (?) married Ira Chappell.\n[S80] Ancestry.com, Biographical review : this volume contains biographical sketches of the leading citizens of Cayuga County, New York. (n.p.: Online publication - Provo, UT: The Generations Network, Inc., 2005.Original data - Biographical review : this volume contains biographical sketches of the leading citizens of Cayuga County, New York.. Boston: Biographical Review Pub. Co., 1894.Original da, unknown publish date), p 318.\n[S2] 1880 United States Federal Census, online , Year: 1880; Census Place: Carrollton, Saginaw, Michigan; Roll: 601; Family History Film: 1254601; Page: 142A; Enumeration District: 297; Image: 0770.\n[S26] 1900 United States Federal Census, online , Year: 1900; Census Place: Saginaw Ward 2, Saginaw, Michigan; Roll: T623_739; Page: 14A; Enumeration District: 44; FHL microfilm: 1240739.\n[S2] 1880 United States Federal Census, online , Year: 1880; Census Place: Mentz, Cayuga, New York; Roll: 814; Family History Film: 1254814; Page: 155D; Enumeration District: 027; Image: 0233.\n[S26] 1900 United States Federal Census, online , Year: 1900; Census Place: Aurelius, Cayuga, New York; Roll: T623_1012; Page: 3A; Enumeration District: 22; FHL microfilm: 1241012.\n[S6] Ancestry.com, 1850 United States Federal Census (n.p.: Online publication - Provo, UT, USA: Ancestry.com Operations, Inc., 2009. Images reproduced by FamilySearch.Original data - Seventh Census of the United States, 1850; (National Archives Microfilm Publication M432, 1009 rolls); Records of the Bureau of the, unknown publish date), Year: 1850; Census Place: Galway, Saratoga, New York; Roll: M432_593; Page: 433A; Image: .\n[S1] 1870 United States Federal Census, online , Year: 1870; Census Place: Galen, Wayne, New York; Roll: M593_; Page: ; Image: .\n[S9] Ancestry.com, 1860 United States Federal Census (n.p.: Online publication - Provo, UT, USA: Ancestry.com Operations, Inc., 2009. Images reproduced by FamilySearch.Original data - 1860 U.S. census, population schedule. NARA microfilm publication M653, 1,438 rolls. Washington, D.C.: National Archives and Records, unknown publish date), Year: 1860; Census Place: Galen, Wayne, New York; Roll: ; Page: 253; Image: 254.\n[S26] 1900 United States Federal Census, online , Year: 1900; Census Place: Aurelius, Cayuga, New York; Roll: T623_1012; Page: 1A; Enumeration District: 22; FHL microfilm: 1241012.\n[S11] 1910 United States Federal Census, online , Year: 1910; Census Place: Aurelius, Cayuga, New York; Roll: T624_927; Page: 3B; Enumeration District: 0032; Image: 904; FHL microfilm: 1374940.\n[S6] Ancestry.com, 1850 United States Federal Census (n.p.: Online publication - Provo, UT, USA: Ancestry.com Operations, Inc., 2009. Images reproduced by FamilySearch.Original data - Seventh Census of the United States, 1850; (National Archives Microfilm Publication M432, 1009 rolls); Records of the Bureau of the, unknown publish date), Year: 1850; Census Place: Milton, Saratoga, New York; Roll: M432_593; Page: 451A; Image: .\n[S16] Ancestry.com, 1820 United States Federal Census (n.p.: Online publication - Provo, UT, USA: Ancestry.com Operations, Inc., 2010. Images reproduced by FamilySearch.Original data - Fourth Census of the United States, 1820. (NARA microfilm publication M33, 142 rolls). Records of the Bureau of the Census, Record G, unknown publish date), Year: 1820; Census Place: , Saratoga, New York; Roll: M33_79; Page: ; Image: .\n[S2] 1880 United States Federal Census, online , Year: 1880; Census Place: Saratoga Springs, Saratoga, New York; Roll: 929; Family History Film: 1254929; Page: 465D; Enumeration District: 087; Image: 0095.\n[S3] Ancestry.com, 1840 United States Federal Census (n.p.: Online publication - Provo, UT, USA: Ancestry.com Operations, Inc., 2010. Images reproduced by FamilySearch.Original data - Sixth Census of the United States, 1840. (NARA microfilm publication M704, 580 rolls). Records of the Bureau of the Census, Record G, unknown publish date), Year: 1840; Census Place: Milton, Saratoga, New York; Roll: ; Page: .\n[S20] Ancestry.com, 1830 United States Federal Census (n.p.: Online publication - Provo, UT, USA: Ancestry.com Operations, Inc., 2010. Images reproduced by FamilySearch.Original data - Fifth Census of the United States, 1830. (NARA microfilm publication M19, 201 rolls). Records of the Bureau of the Census, Record Gr, unknown publish date), Year: 1830; Census Place: Milton, Saratoga, New York; Roll: ; Page: .","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Today (2nd Nov 2011) I gave a talk titled \"Thinking Distributed to Improve Agility\" at JAX 2011 in London. This was the first talk in the Architecture track on the second day of the conference. This is bleed over from Agile Day yesterday, but also talked to the architecture of teams within an organisation.\nThe premise of the talk was that, contrary to poplar belief, distributed teams exhibit characteristics that can make it easier for them to transition to agile and improve their agility. In contrast co-located teams can somehow struggle to transition and be agile. I take a look at organisational patterns and crowd communication in an attempt to explain why this might be the case.\nI've uploaded the slides from the presentation as a PDF. Hopefully it will be a good read.\nThere was some good feedback from the talk, with one person observing that one aspect of the distributed teams is that you are forced to do things well, otherwise you simply cannot survive. Very true in many ways but certainly only one aspect of the overall problem.\nThe conference has been very good with a high standard of talks over all, highly recommended to anyone who hasn't been to JAX before.\nYesterday I gave a talk at the inaugural Agile on the Beach conference down in Cornwall on the Tremough Campus near at Falmouth.\nIt's been a very good conference so far with a great turnout. The slides Feedback Loops in Agile Development are available for download.\nThe essence of the talk was to emphasise the importance of observing and learning from feedback loops that appear in an agile methodology. It starts by examining briefly the nature and problems of typical development, then explores the key elements of an agile methodology. From here I illustrate how we can use these elements to amplify the feedback loops present to facilitate a better understanding of how to leverage agile development. Finally it takes a look at some of the practical things you should pay attention to when going agile along with a short prescription of where you can start.\nSo everyone knows what being agile is all about? It's an old buzzword now, stamped on everything that moves in a development methodology, or everything in an organisation for that matter. The problem of course is that it's hard to to really distil what makes you agile and what doesn't. If you cobble together enough process, and throw in a couple of practices, you have all you need to innovate yourself to a successful, or failing, project. That's the trouble, without real agility it is often hard to know how successful you are being in a project, and often it is hard to know how agile you are. You might think you are agile and that your project is being successfully delivered when in fact neither is true.\nThis talk goes back to basics and re-emphasises the concepts behind agility: showing that agility is achieved through understanding, observing and amplifying the feedback loops present in a truly agile approach to development. By understanding how you setup and magnify feedback loops in your methodology, using loose processes and practices, you have a chance at agility. I'll present reliable techniques and show how they relate to agility within the feedback context. You should come away from the talk with a refreshed view of agility, some practical tips, and be able to look more critically at how you try to be agile in your organisation.\nToday I gave a talk about \"Paying Lip_service to Agile\" at JAX 2010 in London. This is the first day and so far it has been a great conference. There was a very strong turn-out for the Agile Day and it was encouraging to see so many wanting to learn more about agile development.\nI've uploaded the slides from the presentation as a PDF. I hope it makes for an enjoyable read!\nToday I gave a talk about using agile-trac at JAX 2010 in London. This quote, from the JAX London site, sums the JAX conference up quite well, \"JAX - it's all about Technology, Architecture, Agility\". I've uploaded the slides from the presentation as a PDF. There is an 'agile-trac In a Nutshell' section which actually provides quite comprehensive documentation.\nOne of the great uses for tools like trac, and obviously agile-trac, is development in a distributed environment. Much of the initial design of agile-trac grew from an agile project where the whole team was distributed with team members on different continents and hence also in different time-zones. During that project a lot was learned about how to work effectively in a distributed setting.\nA key observation is that the solution to distributed development is not co-location. You can effectively develop within a distributed team, and with reasonable tool support this can be quite pain free. I link to a presentation here that was first given at Agile 2008 in Toronto last year. It captures many of the lessons learned from several years of working in agile distributed teams. It turns out that going agile can really bring benefits in a distributed setting as it helps focus both the nature and the quality of communication.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Carbon Cycle: Is Nuclear Power the Answer to Global Warming?\nThis is a provocative question of relevance to the Australian scene since we have no nuclear power stations. It is also a \"loaded question\" in the sense that it assumes that global warming is a problem and that mankind can and should do something about it. The concern may be real or imaginary.\nA fear or phobia present in the populace is often used by promoters of products. Auto manufacturers will cater to the fantasies of the global warmers by advertising that their model has lower CO2 emissions than a competitive product. Other buyers couldn't care less, or think that CO2 is good stuff and take no notice.\nIt is the same with political parties. They too have a product to sell and need votes to stay in power. Now we have two lines of thought: the Party position and what the individual party member thinks of the proposition.\nIn Australia we have the Greens' position that nuclear power is evil and should never be contemplated for Australia. It is easy and important for them to promote this fear (one of many) so that they can gather votes and retain their few Members of Parliament. The Greens are really a Fear Party.\nThe Labor Party, our present Government, is also against nuclear power at the Party level, because it knows that by taking this position it is assured of Green preferences in an election.\nThe individual Labor members may be ambivalent about nuclear power but know that to stay in power they must not upset the Greens, either those within or outside the Labor Party.\nThe same situation occurs with the Liberal Party, but more particularly with inner city electorates where the Greens have a strong urban vote. The Nationals are lucky that they can usually speak their mind on the matter.\nThen we have the individual scientists, like our Ziggy, who helpfully promotes nuclear power for Australia. He seems to use the fear of global warming and closing down of coal-fired power stations to justify the introduction of nuclear power. It is not really necessary to do this; he has a defeatist attitude with regard to the global warmers; maybe he is one?.\nHow do Americans think on the topic of using nuclear power? What is the view of the \"man-in-the-street\". How is the average man in the US influenced by the Green Movement?\nYou can find out by checking with the US Helium writers' website where the topic \"Is Nuclear Power the Answer to Global Warming\" is presented for debate among members. So far there are 33 articles with 19 articles on the NO side and 14 on the YES side. The voting of individual Helium members is 63% No and 37% Yes. Why not join Helium and give your Australian viewpoint? It's for free. Explore www.helium.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here are the results of a study of the influence of the price of oil (Brent) on exchange rates, which was conducted in May 2015 by EXNESS analyst Sergei Kochergin. The study includes the world's largest exporters of crude oil: Saudi Arabia, Russia, Canada, Nigeria, etc. For each of these countries, they present both the changes in the country's international reserves as well as the correlation between the currency's yield and the price of oil.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Musician wants to reconnect with a relative, with whom he did not speak for a long time. A serious spat between Boris and Konstantin Burgevine occurred in 2009. Since then, the artists distanced themselves from each other.\nFormer lead singer of \"the Brothers Grimm\" \u2013 the twins, Constantine and Boris Burdaeva \u2013 can't establish a relationship for a long time. Musicians once performed the hits \"Eyelashes\", \"Kusturica\" and the \"Heart\" of a fight back in 2009. 6 Jun relatives celebrated his birthday \u2013 they turned 36 years old. On this occasion, Boris has published a touching post on Twitter in which he thanked the parents and expressed hope for reconciliation with Constantine. Your post man is accompanied by archival children's photography.\n\"It turned out that this year's birthday has stretched for four days from the official date before the official holiday. Whether I, or karmic knots came undone, but the situation was interesting\u2026 And in this post I especially want to thank and congratulate parents, they're the best in the world! And, of course, his brother. I hope that sanity will prevail and mythical grudges go. After all, this life is fleeting, and it has the meaning of only love, everything else is just a Mirage\u2026 our birthday!\" \u2013 said Boris in his microblog.\nThe fans congratulated him and wished him all the best. According to them, Boris and Konstantin will again find a common language. \"Happy birthday! You less reason for quarrels and insults\", \"You definitely need to make up! And just to take a picture\", \"Sooner or later you will succeed\", \"What a strong post, I Hope that brotherly love will prevail\" \u2013 they wrote in the comments to the publication.\nThe official community of \"Brothers Grim\" in \"Vkontakte\" mother of Boris and Konstantin commented on the publication of his son. \"Thank you very much! Actually, it is, and the only thing that matters in this world is our unconditional love \u2013 written by Olga Burdaeva. We come into this world with our inner baggage and go with him, leaving only love. So we live in light and love! We are all learning to live, learning to love. No enemies, no friends, but every person you meet is a teacher\".\nWe will add that earlier Boris Burdaev gave a Frank interview in which he told about his difficult relationship with his brother Constantin. The man believes that they need to stop being offended at each other. Boris is ready to reconcile with a relative. The musician also publicly appealed for a loved one.\n\"Kostya, we have parents, they are now 60 years old, is the Golden years. I hardly agree, I my talent will never forgive, but we have parents. I want them to be happy, think about it,\" said Bugaev.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It's 13 characters, for example 123PA01234567. It's different to your employers' PAYE reference.\nMake sure to use the correct accounts office reference if you manage more than one payroll scheme.\nWhat period are you paying PAYE for?\nPay in full by the due date to avoid paying a penalty.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Dr. Garry Williams has recently taken up a post at the John Owen Centre, part of the London Theological Seminary. EN took the opportunity to interview him about current theological education.\nEN: In these times of economic constraint do you have any advice for young people and churches on how they may finance themselves through a theological course?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"WE all know that in current times, the chances of landing an actual job in journalism are somewhat more slender than that of a snowball staying intact in the eternal furnaces of the damned.\nSo you've got to admire the pioneering spirit of the five \"wannabe hacks\" who took their fate into their own hands and set up a website full of useful advice, chronicling their journey towards solvency and success in this increasingly daunting field.\nThey've been working the social networks, getting experience of rejections, honing their skills, and gradually building that most indispensable thing: reputation.\nAnd now, one of the five Wannabe Hacks has just landed herself a job \u2013 at the highly sought-after Wired magazine.\nShe has blogged about how she did it. So here are her handy tips for any jobseeking journalist \u2013 and especially for beginners.\nWe hope these intrepid people are NUJ members. If they're not, they should be.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Arthur's music is an unique, recognizable, timeless and catching mix between rhythm 'n blues, rock 'n roll, country-blues - in a jazzy sauce. With Ray Charles he shares his soul, with Tom Waits his rawness and with Eddie Cochran his rock 'n roll.\nEbeling-originals are very own, recognizable, varied and just lovely - they sound like you've known them for years. In his song worldly wisdom and romance clash, heartache and and humor meet. His wry lyrics give way to an unconventional vision of life.\nArthur plays solo as well as with one or more members of his band. Solo, playing the guitar like nobody does, becoming one with his Gretsch, and stamping his foot, he sounds like a whole band. With double bass, saxophone and percussion, the Arthur Ebeling Band is out of this world (too).\nArthur Ebeling may be from Holland, but if you didn't know, you might think that he had come all the way from New Orleans.\nArthur Ebeling starred in the 1996 tour 'Gretsch on the Road'. He plays in Holland, Europe and the rest of the world. He has toured succesfully in England, the USA, Russia, Brazil, France and Belgium and most recently in Spain. HE has opened shows for artists like Taj Mahal, Buddy Guy, Ann Peebles and Joan Armatrading.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Today should be lived actively and in good humor. Look inside yourself. One may start any new projects. The day is particularly favorable for the purchase of land or real estate. Cultivate your good ideas. This is a good day for work with dreams.\nEyes, brain are active. Strengthen and take care of your vision. Juice fasting is recommended.\nUse greenery with any solid food you take today. Nuts are favorable. Do not forget to drink juices, eat fruits and vegetables. Dry fasting is recommended. Avoid seeds, grains and animal food.\nSubjects for meditation: Ideals, candle.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Home My Thoughts As a Physician Assistant, Do Not Judge!\nAs a physician assistant, there will be many times when you get frustrated with patients. You will have patients come in to see you because they have a cough, or a sore throat\u2026..for ONE day. It will irk you at times. It will make you wonder why you spent so many hours studying just to deal with things like this. I know it bothered me, until I started to realize something. Patients are not there just for a cough or sore throat.\nPatients are people. People have busy lives, people have problems, and people need other people. We are social creatures by nature. We are not made to be deprived from other human interaction. Sometimes, the problems we face in life, we can't necessarily tell our family, or friends. But, we feel like we need to tell someone. Thats exactly what is going on here.\nWhenever I would see the chief complaint of cough for one day, I would get irritated. That irritation would then change my mood and ultimately my perception of the patient. I would go into the room with judgment before I had ever laid eyes on the patient. This is BAD. Why? Well, it would skew my way of thinking and would skew my attitude. Instead, look to see what is really going on. Look to see the bigger picture. Try and get a sense as to why the person is really there.\nMany times it will be stress, anxiety, depression, or even the mere fact that they will have 15 minutes of someones undivided attention. Diagnose and treat the cough in 5 minutes, and spend the last ten minutes speaking and understanding your patient. Patients go in to see you because they know you are unbiased towards their actions. They feel they can trust you, so don't break that trust. We have no idea what is truly going on with their lives. We have no right to make judgments about anyone, because although this cough may not be anything, you will form a certain attitude towards this patient; the next time you see them it will trigger that emotion, and who knows, the next time they may really be sick!\nSo, I challenge you to stay unbiased. I challenge you not to say, \"oh this patient again\". I challenge you to not make judgements based on a patients previous actions, and take each encounter seriously. Lastly, I challenge you to listen and understand your patients, and to stop thinking of them as a constellation of symptoms. You never know when you will have that suicidal patient that is just looking for someone to validate their self worth. As a physician assistant, it is not only your job to treat disease, but it is also your job to care. Sometimes, the best medicine is to simply listen.\nNext article5 Things Every PA Student Should Do!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"This paper presents an ecological vehicle synchronized driving control system that aims at reducing overall fuel consumption of the vehicles in a group. A centralized system for controlling the vehicles in a group has been developed using the model predictive control method considering vehicle-aerodynamics and the resistance due to road slopes. The ecological synchronized driving system is simulated on a typical road with up-down slopes for high speed driving. Its fuel saving performance is compared with a conventional vehicle following system. Computer simulation results reveal a significant improvement in fuel economy using the proposed ecological synchronized driving control system.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Saw this over at the Working Diva and I think it is such a good idea!\nOutside my Window...it is gorgeous and I would rather be out there, maybe a walk at lunch time!\nI am thinking...gas prices are outrageous, I need to get away for a week, I do not want to go to the \"bachelorette\" party this weekend, what should I make for dinner, I better get back to planning weekly menus.\nI am thankful for...my health, good friends and a thirst for knowledge.\nFrom the kitchen...made steak sandwiches last night, but it is fresh fruit and yogurt for lunch today.\nI am creating...a better plan for the wine club that I have been neglecting for a while now.\nI am going...to the movie premiere of Swing Vote with Kevin Costner Thursday, meeting friends for drinks on the water tomorrow.\nI am wearing...my WAH uniform, yoga pants, flip flops and a tee. At least I have brightened up the colors for summer!\nI am reading...Night Fall by Nelson DeMille. Looking forward to some mindless summer chick lit though.\nI am hoping...to keep my sanity through the \"wedding\" festivities over the next two weeks and reminding myself to keep biting my tongue.\nI am hearing...Ian Murray, great summer tunes.\nAround the house...need to get laundry done and go through my cupboards and freezer.\nOne of my favorite things...is going for my walk around the lake and having dinner with the boy.\nA Few Plans For The Rest Of The Week... getting my ta-da list sorted, wrap a couple of presents, figure out what to get someone who turned 40 that I have nothing in common with and who is not one of my favorite people but is one of those, things I need to do, hope to see sister S's new baby. Look at the next couple of months and make sure calendars are synced up. Balance my budget.\nI think Swing Vote looks like so much fun. I'll be awaiting a full review!\nP.S. Add to your Summer Ta Da List - meet up with Leesie somewhere between Minnie and the 'Peg!\nYou must bring back pictures from the Nuptials from Hell for us. Along with your fabulous commentary. It's one of those stories you just couldn't make up--it's so hard to believe there are really people like that out there!\nThis is a good meme! I'll have to see if I can work it in around my barfing and cooking posts!\nLove the new red, white & blue blog! The pic is too cute! That's the bummer about using Google Reader, you don't see everyone's cute backgrounds!!\nThanks for your kind words over at Figs. I can't believe anyone would ever break up with you!\nI would love to not have laundry on my to do list for just one month. I am on vacation and doing laundry - sad as sad can be.\nHey there! Where have you been lately?? Have you survived the nuptials yet? Hope you're having a good summer.\nI was just thinking about you the other day...I hope all is well!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Home \u00bb Micro Combined Heat and Power \u00bb How does microCHP technology work in homes and businesses?\nCHP systems have been used successfully in the industrial sector since 1970 but the technology hasn't been widely applicable for domestic use, largely due to the system's size, weight, noise and cost. In industry, CHP systems are small 'Heat Engines' that provide all the power for an individual building. It is by harnessing the wasted heat from this process and using it for hot water, heating or cooling that the phrase 'combined heat and power' (CHP) is defined.\nFor domestic and small to medium enterprise (SME) applications, the microCHP unit in your home or business will do much the same as the larger scale CHP methods but on a much smaller scale. As previously mentioned, at an industrial scale, a vast amount of energy is wasted, both in its creation at the power plant and in its transference to your home or business (this industrial process can be as low as 30% efficient). It is this on-site, co-generation method that greatly increases the efficiency of the energy production process and thereby lowers the combined carbon footprint and reduces your reliance on expensive 'dirty' electricity from the national grid. mCHP boilers claim efficiencies of up to 98%!\nMicroCHP for the domestic household is a relatively new technology. Up until 2006 the economic viability when looking at the available 'domestic' unit's payback period, rendered the micro-CHP technology unfit for domestic use, however this is all set to change in the very near future, with many competitively priced units preparing to be released to the wholesale market. As they begin to appear on the UK market, please check HERE as we will list both the units themselves and installers who will supply and fit them for you.\nLarge scale 'Heat Engine Systems' (power plants), are used to generate electrical power for modern life, by burning a variety of fuels and converting them into mains electricity. They are however, extremely inefficient \u2013 they are unable to convert all the energy from the fuel they burn into electricity and some of the electricity is lost upon transmission through the national grid to your home or business. It is the wasted heat from this process that can be utilised by a microCHP system and turned into heat and hot water for your property.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Is there a cannon lying in the shallow water off Mill Island just to the south of Russell town?\nIn 1858 a windmill was built on Mill Island (Motu Kaiaraara). 'Some of Russell's citizens formed a company to grow and mill grain'. Optimistically, 'The site was carefully selected to prevent any invasion of rats\u2026the main crop seems to have been oats\u2026The little island was part of J.S. Clendon's property and the mill was operated by Andrew Judd\u2026the island was two or three metres higher\u2026but when the milling scheme was mooted the top was lopped off and steps cut in the side\u2026Grain was sown on every available slope above Kororareka and Long Beach\u2026The yield was never enough to keep the mill working...after a few years, a storm wrenched off the sails; they were never replaced and the whole structure fell into disrepair'i George Howe Cook says that Mill Island was 'where the mill that never worked\u2026with the cottage like office near the northern end\u2026when [it] first started going one of the wings gave way and it was never replaced'ii.\nThe ruins of the windmill were still visible in 1873. The steep sides of the island required 'the laborious conveyance of the wheat up a long and almost perpendicular ladder and a precarious conveyance of the flour and pollard down the same\u2026the financial blows produced by the difficult mode of conveyance no less helped to stop the clatter of the millstones'iii.\n'J. C.', James Cowan, writes in 1927 'In the early days\u2026 a windmill for grinding flour from the Maori-grown wheat stood on its flat top, fifty or sixty feet above the water. On the charts the rock is called Observatory Island, a name given when the survey of this coast was made by the officers of HMS Acheron, in 1849-50. The Maori name is Kairaro. One of the ship's guns used in the defence of Kororareka against the Maoris in l845 was used as an anchor to guy down one of the stays that supported the windmill on its precarious perch'iv.\nCook also mentions the existence of a gun off Mill Island. 'The gun that came out of the Suka Bay [Socabaya]\u2026lies in about 8 or 10 feet of water on the eastern side of what they call Mill island \u2013 old-timers called it Judd's Folly. Anyway it was a queer place to put a flourmill. One had to climb a 30 foot ladder to get to it, having first to pull there in a boat. It was for a boat mooring that the gun was put there'v.\nThe size and calibre of this gun is not known. Cook states that the Mill Island cannon came from the 'Suka Bay'. If this gun is found and identified as similar in make and calibre to the Russell cannon, this could be circumstantial evidence that the Russell cannon also came from the Socabaya.\nii New Zealand Herald. Cook. 20 October 1934. Supplement. p1.\niii Daily Southern Cross. 22 March 1873 p3.\niv Auckland Star. J. C. (James Cowan). 24 February 1927.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Do I receive the posted dividend yield every quarter?\nA dividend is a distribution of a portion of a company's earnings, decided by the board of directors, and paid to its shareholders of record on a specific date. Investors often view the company's dividend by its yield \u2014 a measure of the dividend in terms of a percent of the current market price.\nAfter being declared, a company with common stock that pays a dividend will typically distribute the dividend every quarter. However, the amount the company quotes is normally an annual figure.\nSo, to calculate the amount you will receive each quarter, you will have to take the quoted dividend amount and divide it by four.\nFor example, if you own Cory's Tequila Corporation (CTC), which pays a $1 yearly dividend on a quarterly basis, you would receive $0.25 every three months.\nFirst, a stock's dividend yield will fluctuate with the market price. If CTC is trading at $10 and it pays the $1 dividend, its dividend yield is 10 percent ($1\/$10). If the price of CTC rises to $20 and it still pays the same dividend, the yield is only five percent ($1\/$20). A change occurs in the yield any time the stock price changes, so don't mistakenly equate a change in dividend yield with a change in the payout you receive.\nSecondly, companies are not required to pay dividends. This is a completely arbitrary action that the company decides itself. Most companies will try to maintain a certain level of consistency with their dividend payout history to attract investors, but the payout can be changed at any time. Companies under financial stress might need to re-allocate money to different projects, or management may just change its mind and no longer want to pay a dividend. So, while a company's long-standing record of increased dividends is a good indication of payments in the future, the dividends aren't guaranteed.\nWhich is more important - dividend yield or total return?\nHow do I find the CUSIP number for a stock?\nHow do I interpret a Security Market Line (SML) graph?","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Happy Easter, Happy Passover, and Happy Sunday!\nhas on SignWriting in Colombia.\n\"SGN\" means a \"sign document\".\nAttach the file \"FINGER.SGN\" to an email message and send it to the SW List.\nFingerspelling SGN file? Many thanks!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"How Does the ATA Nomination Process Work?\nWho creates that slate of candidates that we see every year? How does the nomination process work? May I nominate myself? What are the criteria used to decide who should run? This article is an attempt to shed light on a process that is unknown to much of ATA's membership. We also want to describe here some changes made recently, as well as some new changes for this year.\nThe committee we are talking about used to be called simply the Nominating Committee. A bylaws amendment in 2009 changed it to the Nominating and Leadership Development Committee. As the name implies, the change expanded the committee's charge to help produce a pipeline of future leaders.\nThe Nominating and Leadership Development Committee always consists of five people, per ATA bylaws (Article VII, Section 2d). These five people are appointed at the winter Board meeting to serve during the following year. The committee members for 2015 are: Dorothee Racette (chair), Tony Guerra, Susanne van Eyl, Connie Prener, and Karen Tkaczyk. The committee continually identifies people, helps them find the right volunteer spot within the Association, keeps an eye on the quality of the work they are doing in their current role, and finds out whether they are interested in running the following year.\nWhy is ATA interested in leadership development for its Board and potential future candidates? While historically there has been a wealth of talent on the Board from the membership that has sustained and cultivated the vibrant organization that it is today, ATA recognizes that its continued effectiveness and future relevance depend on the strength and clear vision of its leadership. Therefore, plans call for expanding the committee's activities in the area of training.\nLeadership training for individuals would assist in assimilating new Board members, succession planning, developing high potentials, navigating organizational culture, and removing \"blind spots.\" Leadership training for the Board would work to cultivate team alignment and encourage the integration of and adaptation to changing cultures. It would also work to build trust and awareness among the Board to facilitate consensus, collaboration, and accountability.\nA leadership development program should improve leadership competencies, such as improved engagement and more focused and increased Board productivity. In summary, leadership training is designed to help leaders discover more effective and productive ways to achieve personal and professional goals, create alignment with ATA's organizational culture, and promote strategic objectives. ATA Board members would have an opportunity to enhance their existing skills and resources and to develop creative and innovative solutions to address effectively the challenges of representing the interests of ATA and its membership. We will take a first step in this direction by holding an invitation-only Leadership Development training session at ATA's 56th Annual Conference in Miami (November 4-7, 2015).\nThe Nominating and Leadership Development Committee is active throughout the year. Our activities for the new election cycle begin during the Annual Conference. After the election, the committee holds a follow-up meeting to discuss the candidates' presentations, as well as what we learned from them that can be passed on to future candidates. Also during the conference, committee members approach people we have contacted previously as potential future nominees to see if they have any questions or concerns about the process.\nThe committee gets together early in the year to discuss the slate for the upcoming elections. In preparation for the meeting we contact committee chairs, division administrators, chapter and affiliated group presidents, Board members, and others to solicit nominations and recommendations. We maintain a database of people who have been recommended, along with associated information. That includes their profession (e.g., interpreter, translator, educator, company owner, or employee), language pairs, and contributions to ATA and the translating\/interpreting professions.\nHow was this person active within ATA in the past?\nWhat talents and preferences were evident during that activity?\nWhat personal attributes would make her or him a good candidate and a good director or officer?\nIn order to present a balanced slate to the membership, we aim to include candidates from all the various areas of our profession. We make an effort to ensure that each is represented in a way that reflects reality. To cite an example, if the term of a director who is an interpreter is about to expire, we will try to put a candidate who is also an interpreter on the slate for that year.\nAnother consideration is gender. Since a majority of ATA members are female, if four women were leaving the Board in a given year, it would be odd to have a slate composed entirely of men. Other less crucial factors include language pair and geography. We are not terribly worried about French translators or residents of New England taking over ATA, but we would consider the information to see if a proposed slate would be adding diversity.\nWhat would this person wish to accomplish if elected?\nIs this person sufficiently known to have a chance of being elected?\nHow would this person fit into the existing Board?\nOnce the committee feels that the slate is complete, the nominees are contacted and informed that we support their candidacy. Once the finalized slate is reported to the Board, the committee is available to the candidates for fact-checking written statements and draft speeches. We also have guidelines available to prepare for the actual candidate presentations at the Annual Conference, but it is up to the candidates to devise a way to present themselves in the best light possible.\nAs part of the committee's continuous review process, the actual nominating application was revised significantly this year. Some of the questions listed on the old Nominating Form were no longer relevant. In addition, some questions were appropriate only for nominating other people, while other items pertained to members who were nominating themselves.\nWhich areas of translation and interpreting activity are you passionate about?\nWhat strengths would you bring to ATA's Board of Directors?\nIn your view, which perspectives or points of view should be represented on the Board?\nWhat particular strengths does this person have that are necessary for the officer position for which you are nominating him or her?\nHow has the candidate demonstrated commitment to the translation and interpreting professions?\nWhich areas of ATA activity would you hope to become involved in?\nHow do you feel your skills and abilities match the \"job description\" for your role?\nWe are confident that these efforts to cultivate tomorrow's leaders will ensure a strong, vibrant Association. If you have any suggestions for the nomination process or for the development of the Association's leadership, please send them to nominations@atanet.org. The nomination period for 2015 is now open. You can find nomination forms at http:\/\/www.atanet.org\/elections. php. The deadline is March 1, 2015. We hope that the process is now clear and look forward to receiving many great nominations this year.\n\u2190 Why Pairing up Is a Good Idea, Especially for Freelance Translators!\nAll this is quite valuable information, why did you come to this?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The upgraded V-NAND and controller help the Samsung deliver incredible performance.\nIn AS SSD's sequential read and write tests the Samsung scored 2,823MB\/sec and 2,346MB\/sec. The former figure is a few hundred megabytes ahead of its nearest rivals and the 950 Pro, while the latter result is almost 1GB\/sec ahead of the best other PCI-E based SSD \u2013 a vast improvement.\nThe 960 Pro continued to impress in other tests, although it wasn't always the fastest drive. Its 4K read and write speeds of 36.52MB\/sec and 159.4MB\/sec are only beaten by a couple of other products, and its 4K-64 results are mixed. Its read pace of 1,634MB\/sec is better than everything else, but drives from Zotac and Intel beat its 1,138MB\/sec write pace.\nCrucially, the 960 Pro was better than the 950 Pro in every one of these benchmarks, and often by a significant margin.\nThe new drive's good form continued in CrystalDiskMark. Its read and write speeds of 3,454MB\/sec and 2,157MB\/sec are unprecedented, with the best competition coming from the 950 Pro's 2,175MB\/sec and 1,532MB\/sec results. Everything else is further behind.\nIn smaller file tests, a familiar pattern returned. Samsung's drive was fastest by a slim margin in the 512K test, but it was beaten by a couple of other products in the 512K write benchmark. It was a little way ahead of the best rivals in the 4K-32 test.\nThe 960 Pro was a little slower than its rivals in Atto's small-file tests, too, but the Samsung quickly accelerated past the competition when file sizes increased. Its peak read speed of 3,347MB\/sec trounces every other PCI-E SSD, and its top write pace of 2,115MB\/sec is also better than most.\nThe final test, Iometer, is a tough one \u2013 but the 960 Pro rose to the occasion. Its overall result of 21,540 IOs per second is miles ahead of most drives and only marginally behind the Toshiba OCZ RD400; its speed of 825MB\/sec is similarly impressive.\nShould I Buy the Samsung 960 Pro 1TB?\nSamsung's drive is significantly faster than every rival in almost every benchmark, which means I'm willing to forgive its only minor performance issue, which occurred with marginally slower scores in small-file benchmarks.\nAnd it's worth bearing in mind that none of those results were bad \u2013 it was a tad slower than one or two other drives, sure, but even in those tests it remains one of the quickest drives I've ever tested.\nThe record-breaking performance is paired with a solid set of features, good endurance and a generous warranty, but those benchmarks also come with a high price.\nThe high price and incredible performance mean this is a drive for the privileged few who actually do require this level of speed and have enough spare cash. The Samsung 960 Pro is a niche product, then, but an incredible bit of kit. I can't wait until this technology trickles down to cheaper drives.\nSamsung's latest SSD moves its technology forward to deliver superb benchmark results. That stupendous performance does come with a high price, though, so only really makes sense if you need its extreme level of speed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Each week we will spotlight a different DPW artist who will give away one of their best paintings. To enter to win Gail's painting, \"Another Sunflower Painting\" go to Daily Paintworks and click on the link at the top of the page announcing their interview.\nMy interest in art was sparked over forty years ago when I was a theater production major in college and an aspiring scene painter. But life had other plans for me and more than twenty-five years passed before my passion was reignited. I worked first with pencil, then charcoal and pastels, and finally oils, while also trying my hand at portraits, figures, still life and landscapes.\nI first picked up a paintbrush in 1973. I was at Carnegie Mellon University majoring in Theater Production and was assigned to a scene painting crew. I loved it and I was convinced I wanted to do this for the rest of my life. That lasted about six years before I changed paths again. However, during that time, I had to take a test to join the Scenic Artists Union and one of the requirements was to paint a landscape. I was good at painting faux wood and marble and stenciling wall paper and making a flat piece of scenery look three dimensional, but I knew nothing about fine art. I even attended classes at the Art Students' League in New York for a month or two. But life got in the way and it wasn't until the 1990's, when my children were in school, that I thought about taking art lessons again. I started drawing with pencil, moved on to charcoal and pastel and then finally to oils. As soon as I picked up a brush, I was hooked all over again.\nI didn't have any stops and starts since I started drawing classes twenty years ago, but I didn't really get serious about my painting until rather recently. When my children were grown and out of the house, I set up a studio, spent more time in classes, and traveled to workshops. In the past, painting was just something I did when I had the time. Now it is something I want to do all the time!\nEnter to win by clicking on the link at the top of the DPW home page announcing Gail's interview.\nWhen I was in college, I did some watercolor copies of my favorite Andrew Wyeth paintings. When I returned to art in the 90's I started drawing portraits in pencil and charcoal, and then soft and oil pastels. Working from photographs, I did a series of drawings called Portraits in Courage. I was inspired by those who had experienced hardship and loss yet had endured and grown stronger. I wanted to show the wisdom that was written in their faces. Feeling confident enough in my drawing skills I started painting figures in oils, but only in blacks, whites, and grays. I did a series of 30\" X 40\" paintings, entitled The Park Bench, depicting people from all walks of life as they took a break from their busy lives. Eventually I moved on to color and worked from live models. Now I am concentrating mostly on still life and trying my hand at landscape and plein air. I have taken a few pastel classes too, but I always go back to oils.\nWhich ones have \"stuck\" and which ones have fallen away?\nI am deathly afraid of watercolors now. I love that oils can be reworked at any time long after they are signed and considered finished. I have paintings framed and hanging on my walls that I revisit years later. Watercolors are much less forgiving. I tried acrylics for about six months and hated it. I didn't like the feel and flow of the paint and I couldn't get used to how fast they dry. It just frustrated me.\nWhen I started drawing I only did portraits. But I worked primarily from photographs and now I don't find that satisfying. Although I still enjoy painting from live models, I like the control of the shapes and composition, and the play of light and shadow that I have when painting still life.\nLandscapes, seascapes, and snowscapes are my current challenge and I want to spend more time painting plein air, rather than from photo references. I would also like to try abstracting the landscape and I am interested in learning how to use a palette knife for more than just detail at the end. I love the look of pastels and am often tempted to try them again. Color mixing with pastels is so different from oils that it's almost like learning a new language.\nThe work of Patty Nebbeling, my teacher at the Ridgewood Art Institute in NJ, inspires me to loosen up and simplify. Peter Fiore, a landscape painter whose workshops I have taken, inspires me with his brilliant use of color. John MacDonald, another landscape artist and teacher, inspires me with his beautifully atmospheric paintings. I find inspiration from many contemporary still life and plein air painters, such as Qiang Huang, Neil Carroll, Kathie Odom, Nancy Tankersley and Roger Dale Brown. I love the masters. I am particularly drawn to the drama of light and shadow of Rembrandt and the freedom and color of Cezanne and Van Gogh. And just about everything in Sargent's paintings.\nI am always intrigued by the play and drama of light and shadow on a still life or landscape. But sometimes it is the lack of contrast and the more tonal, monochromatic setting that inspires me. Other times it's the colors or the strong shapes and values that attract me.\nWhat is your mental preparation for painting?\nI don't have any preparation protocol, but I listen to books on my ipod while I paint. I always make sure I have something riveting to listen to. Right now it's \"The Alice Network\", about women spies in WWI and WWII. It's very good!\nI think I do more daily procrastinating than I do daily painting. I am getting better however. I find it most difficult if I am starting something new and don't have a plan. Once I am working on something I find it much easier to get going.\nFor me it is like going to the gym; once I have my gym clothes on and I walk out to the car I will go and work out. I just have to get myself into the studio. Once I am there the hours fly by and the next thing I know it is evening, I am exhausted, and it's time for dinner.\nEvery Wednesday I am in a class where the teacher generally assembles half a dozen setups to choose from. I might tweak them a little. Sometimes a piece of fruit or flower she has brought in will inspire me and I will do my own composition. At home I work from photos. Lately I am concentrating on landscapes and seascapes from photos I have taken during my sailing trips. Many times I will see another artist's work and it will remind me of a photo reference I have but didn't think could be a good painting. That will inspire me to find that photo and give it a shot. This past summer I started painting plein air from the back of our sailboat, painting whatever scene is in front of me at the time. It's always paint worthy!\nI haven't experienced burnout yet, possibly because I am always learning and trying new things. Painting landscapes is new for me, and I am just starting to paint en plein air. Recently I started using a palette knife whenever possible and I am experimenting with making my style looser and more painterly. If I feel like a particular painting is bringing me down I will put it aside and paint something else.\nI am striving to learn how to simplify and loosen up and know what to leave in and what to leave out. I think that is the essence of a successful painting. I am also learning that photo references can only take you so far. You need the experience of painting from life to be able to interpret and extrapolate the information in a photo and turn it into a work of art. That seems especially true of landscapes.\nI am happiest when I am learning or trying something new. I love exploring. I also must admit that I am most happy when my efforts culminate in a successful painting. And I am ecstatic when someone else is moved by my work and loves it enough to put it in their personal collection!\nEach week we will spotlight a different DPW artist who will give away one of their best paintings. To enter to win Anna's painting, \"Milk Jug\" go to Daily Paintworks and click on the link at the top of the page announcing their interview.\nProbably just like everyone, I had some attempts at art during my school years. Actually, some level of painting and drawing was obligatory for my first school. I'd been studying at Waldorf school up to the last four grades. This education system strives to emphasize imagination in the learning process through artistic and intellectual development of pupils. I don't remember ever using textbooks except for grammar and maths there. All the possible illustrations and schemes for history, physics, you name it, we used to draw in our copybooks by ourselves under the supervision of our teacher. We even had a subject where it was our home work to paint several illustrations about some part of a story that was read to us. Sometimes I think that I remember information learned in that school better because I had to paint most of it.\nUnfortunately my Waldorf school was closed and I went to state school where I was pressed by my parents to get serious and chose a normal profession, even though by that time I started considering becoming an illustrator. But well, I listened to my family as I sincerely believed that they were the wise adults and knew what was best for me. The next time I started to think about drawing again was only when I got married. I'd been looking through CG illustrations for some months, some were too beautiful to even think about that level of mastery, others were painted by amateurs, but nice nonetheless. At one point, after watching another tutorial I thought: how much longer am I going to just look? I needed to try it myself. Seeing my interest, my husband made me a Christmas present - a tiny drawing tablet. At the beginning it was an awkward experience, but I still keep the very first picture I painted on it, it is absolutely terrible.\nAfter several months of playing with this new toy, something nice started to come out of it. It was fun and I became thoroughly engrossed in CG painting, but my works were lacking. I'd been enthusiastically stumbling in the dark without any understanding. So after two years of painting on the computer in my free time, I resigned and found an art school not far from my work. I started from the basics and found that I feel extremely motivated when I pay for my own education. I had surprisingly fast progress in pencil drawing and after some thirty lessons my teacher started to nudge me to try first watercolor and then oil. So I would say that officially my oil painting started only two years ago.\nEnter to win by clicking on the link at the top of the DPW home page announcing Anna's interview.\nI feel that currently I'm in the very beginning of my painting career and as my art finally started to bring some money I doubt that my very supportive husband would give me any chance to stop.\nI started with gouache and crayons at primary school, then moved to watercolor pencils and watercolor washes. Without any art education I was free to happily mix watercolor with gouache and thoroughly enjoyed it, fortunately I don't know where these works went. Tried pastels and acrylics for a few months. I love painting people on the computer but have never tried it with oil. Returning to watercolors after painting on the computer was a humbling experience. I started to paint flowers because ten years ago, when my mum asked me to paint red roses on her wall they turned out more like tomatoes, so I decided to learn it finally.\nOf course I still make sketches in pencil, I like to spend an evening painting something on my iPad. Currently, I'm striving to master oil. I feel like I like to look at watercolors more than I enjoy painting them. Others have fallen away.\nI would like to play a bit with acrylics this summer, maybe do some studies in my garden and in the forest. Also I want to try figurative painting, though I'm not sure that I'm ready for it.\nNature and countless wonderful artists. I'm in love with academic painters of 19th century, charmed with contemporary American realism painters and I am in constant awe of CG artists.\nI'm not able to make myself paint in the morning. I only managed it at art school and just because my teacher worked in the first part of the day. Several times a week I promise myself that I'll start at least at noon, but in the end I find a dozen seemingly important chores, interesting articles in the internet, can't stop reading that story or weather is too good to miss, and as a result I drag myself to my workplace around 5 o'clock.\nInteresting audio books help a lot, otherwise my thoughts tend to stray and I go search for the answers online. I'm not able to do anything while listening to audio books so it is a perfect combination with painting for me. Also my husband is so supportive that he doesn't let me be overcome by my laziness.\nI may see some beautiful fruit in the supermarket and buy it just for painting. Once I bought just one small plum because it had a nice leaf attached. I like to walk through the forest with my camera and visit botanical gardens. I love to paint from life, but quite often I make a photo for future use because at the given moment I may not have time or enough skills for it. Some photos I find in the internet, but I rarely like painting from them, it is too difficult to find something that feels right to me.\nWhen I get frustrated I change the subject or the medium. From time to time I can't stand painting at all, usually a day of different activities helps. Last autumn when I went camping on Ladoga islands I took with me everything for painting and just felt that I didn't want to paint, so I spent two days just laying on the rocks and gathering berries. But by the second night I started to sketch the moon road and a boat on the back of my book despite the lack of light.\nI feel like I'm at the bottom of my ladder and the skills I want are somewhere around the moon. Right now I would like to learn looser technique and the one with the pallet knife, also I need to work on my volume and I strive to understand the composition.\nI love the process of creation, but the second best part is when my works find a new home and I get feedback. I'm not very interested in my paintings after I finish them, but it brings me real joy to know that they made somebody happy.\nEach week we will spotlight a different DPW artist who will give away one of their best paintings. To enter to win Nelia's painting, \"Across the Pond\" go to Daily Paintworks and click on the link at the top of the page announcing their interview.\nNelia Harper is a landscape painter who seeks to capture the aesthetic beauty of nature, and create an intimate connection between the viewer and the natural world.\nA life-long creator, Nelia has engaged in artistic expression through photography, collage, and paper and fiber arts, before committing to painting with formal training. Her education began with college level drawing courses and continued under the instruction of Janeen Schissler, at the Schissler Academy of Fine Arts in Loveland, Colorado, where she worked primarily in pastel.\nWell, like a lot of people I know, I loved to draw and paint when I was a kid. I remember my mom subscribed to a magazine that had pictures of paintings \u2013 I think it was called Victorian. Anyway, I would look at paintings by Monet and The Impressionists. I remember thinking, \"I want to be like that when I grow up.\" I thought it was so adventurous and romantic.\nBut, like many artists I know, I dropped the idea of art and painting. I went to college and then found a 'real job' that would 'pay off'.\nThen, while I was on vacation in Paris, I remember standing in front of a painting in the Musee D'Orsay thinking, \"How do they do that? I need to learn how that do that.\" The feeling was overwhelming. I remember looking at the dabs of color on the canvas and stepping back to see a complete image. How did they make it looks so real? I wanted to step into those paintings. I knew there had to be a way to learn, and even if I wasn't any good at painting, I could still have fun learning.\nSo, I took a six week acrylic painting course, and I was hooked. From there, I decided to take a drawing course at our local community college. After that, I kept taking classes and workshops. I joined a plein air group and worked on my own too. It was almost exactly six years ago that I took that first painting class, and my interest in painting and art has only grown since then.\nEnter to win by clicking on the link at the top of the DPW home page announcing Nelia's interview.\nWell, I've really just gotten started in my painting career, and I hope to keep going.\nI've worked in a variety of mediums: oil, pastel, egg tempera, gouache, watercolor, graphite, charcoal, acrylic, ink, markers\u2026 I've done a little experimenting with abstract painting and illustration.\nOil has definitely stuck. It's easy to use, transportable, and versatile. I paint regularly in pastel, sometimes using a watercolor or oil underpainting, and egg tempera. Watercolor is mainly reserved for travel journals and painting with my nieces and nephews. It sounds strange to say when the other mediums take more time, but I just don't seem to have the patience for watercolor.\nAs I get better at color mixing and drawing, I hope to paint in egg tempera more frequently. Even though it's incredibly time consuming, it is such a luminous medium, and I enjoy making my own supports and building up layer after layer.\nI would have to say nature inspires me the most. It's so easy to take nature for granted. Looking at a sunset, the shapes of clouds, the incredible colors in a bird's feather, the variety of trees, flowers, landforms, and of course people. Everything comes from nature. We forget that we are part of nature and the endless creation of life. When I think about what it takes to create and support life on this planet, I'm endlessly amazed and intrigued by the diversity that nature provides.\nThat's easy (laughs). Being busy! I joke that my favorite form of procrastination is \"productive procrastination\". I can always find a project in the garden or around the house that needs to be done. And, I'm forever organizing my studio to make room for my next project.\nI converted a small basement bedroom into a studio. There's just enough room for me, a canvas, painting supplies, a table, and some tools. It feels like I'm constantly putting things away to make room for the next painting project.\nIf that fails, there's always a good book to read, an art show, a plein air paintout, a new technique to read about or watch on YouTube, or supplies to research and buy, not to mention looking at artwork on Instagram.\nMy favorite technique is agreeing to participate in a show with fellow artists. That seems to get the fire going more than anything. We have a strong art community in our area. So, typically we have to reserve space for a show at least a year in advance. Because I am showing my artwork along with artist friends, we are always working to improve our skills and show off for each other. And, we all paint en plein air together. Every Friday, our plein air group paints together at a set location. Sometimes we'll paint together on other days too. Getting outside with friends is always a good motivator to paint.\nI also like to submit my work to juried shows at least 2-3 times a year. Knowing that I have a show to prepare for keeps me busy in the studio. Having friends to paint with keeps me painting outside and motivated for the next show.\nAnd, I have a couple of music playlists on Spotify (music app) that help get my mind in the groove too.\nIdeas for painting typically form while I'm hiking, painting en plein air, traveling, or daydreaming. I'm usually inspired by shape and color. I often feel a small wave of excitement. Usually that feeling will last, nagging me to paint it. Sometimes it's a really bad snapshot with the cell phone, but I'll see the idea fully formed in my mind. When it stays in my mind's eye and it feels good, I know it's worth painting.\nLuckily I'm still new to painting so I haven't had a feeling of burnout. As I look back, I realize that I am good at challenging myself with small risks that have potentially large rewards. For example, I joined the Steamboat Art Musem's Plein Air Event last fall. For a $50 entry fee, I was able to experience a week-long plein air paintout, art show and competition. Attending with several friends kept the cost down, and I sold a painting at the show. I learned so much about plein air events, framing from the car, and traveling to paint.\nMostly, I find ways to challenge myself with small risks: paint larger, paint smaller, paint more realistically, more loosely, things like that. Right now, I'm learning portraiture and figure (back to the community college). I also want to paint more complex scenes, buildings, flowers\u2026there are so many challenges yet to paint. The ideas keep me going.\nHmmm\u2026 this is a tough question. From a technical standpoint, I would say that I'm learning about light. How it affects color and form, especially on the human body. From a philosophical perspective, I'm learning about the value of beauty and living a joyful life. For so long, I pushed myself in a career, I focused on society's idea of success (the accumulation of things and prestige) and I lost my way. Now, I think about what really matters: the human experience and how I want to enjoy living.\nOnce a piece leaves the studio, the influence of that artwork is often unknown.\nFor example, a few years ago, during our holiday art show, an elderly gentleman bought a print I had on display. It was a print of an egg tempera painting, depicting a scene in Italy \u2013 a small courtyard with flowers.\nHe asked me questions about the location, and we chatted for a bit about our travels. He used to travel for work and lived all over the world. I shared with him the story of the hilltop town and the process of creating the painting. He dug into his pocket for some cash, and bought the print. I signed it for him, and after a few more minutes, we wished each other a 'happy holiday' and parted.\nThe following year, as I set up for our annual holiday show a fellow artist came over and asked me, \"Do you remember Harry, that older gentleman that bought a print from you last year?\" Of course I remembered him. We had a really nice talk, and he was so excited about that print.\nShe told me he kept that print right next to his bed. He looked at it every day. It reminded him of the places he had lived and he kept it close to him. A few months after we met, he died.\nEach week we will spotlight a different DPW artist who will give away one of their best paintings. To enter to win Deborah's painting, \"A Little Bit of Spring\" go to Daily Paintworks and click on the link at the top of the page announcing their interview.\nGraduated from Moore College of Art in Philadelphia majoring in Illustration with a minor in painting. After 28 years of working as an art director I wanted to explore my love of oil painting. I love to paint people and animals and I'd love to start doing some plein air work and continue to explore light, color and composition.\nI had a love of art from a small age. My mother was very creative and encouraged my creativity by buying me different art supplies. I remember spending many lazy summer afternoons playing with my watercolor set. In college, I majored in illustration and minored in painting. After graduation, I worked doing graphic arts and illustration but remember knowing that I'd return to my love of oil painting. A few years ago when I retired, I pulled out my paints and started to play around.\nI painted in college, and other than my illustrations, I didn't paint for many years. Unless you count the many school play sets I painted when my kids were in school. I always felt like I didn't have the time to devote to painting. After working in graphics for thirty years I wanted to go back to developing my oil painting. At first I played around with the skills I had and did some pet portraits, they are still a love of mine. During my internet researching I came across Carol Marine's name and daily painting. This spurred my interest in daily painting. I loved her little paintings; they are so fresh and full of life.\nEnter to win by clicking on the link at the top of the DPW home page announcing Deborah's interview.\nMy first illustrations were done in watercolor and colored pencils.\nMany years ago I worked in an in house corporate art department. While I designed booklets and brochures, I also did some illustrating. These were mostly technical product renderings. But my favorite thing I did were these caricatures of people who were retiring or receiving awards. I continued to do these for about twenty years. Painted only a handful a year but enjoyed the opportunity to paint because, during this time, most of my designing and illustrations were created on the computer.\nMy current paintings are oils. I have a great love of animals and do pet portraits by commission. Other genres are still lifes and some florals. Attending a class at a local art center spurred my interest of alla prima and plein air painting.\nI've always loved painting oil. I haven't painted in watercolor in years but still love the luminousity that the media has.\nRight now most of my paintings are alla prima still lifes but I'd like to further my skills in plein air painting.\nMy favorite artist of the past is John Singer Sargent. There are too many artists of today that I aspire to. I'm drawn to the impressionistic and figurative painters. The looseness of their brush strokes and use of color amazes me as it is something that I struggle with.\nProcrastination is a big thing that I struggle with. I can waste a lot of time browsing the internet looking for inspiration and get caught up in looking at other artists. While I think it is important to look and learn other artists, I spend too much time looking and not getting to work on my own paintings.\nI've participated in a couple thirty day challenges and they're really helpful keeping me on a schedule.\nSometimes I go to the store and buy a bunch of pretty fruits and vegetables and they inspire me to paint them. I also have a cabinet of things I've collected that I thought would make a great painting. I love to grab my camera and ride my bike around town looking for interesting places to paint. I find myself taking mental notes as I'm driving\u2026great skies, colors in the shadows, an interesting tree, an old barn. Mostly it's the lighting that grabs my attention. I often have to remind myself to pay attention to driving. I try to only do this when I'm the passenger!\nLooking at other artists past and present. I'll pull out a book or search an artist I've heard about. I also listen to art podcasts. Listening to other artist talk about their struggles and inspirations helps me with mine.\nRight now I feel like a baby artist. Sometimes it overwhelms me how much there is to know. My use of color and brushwork is my main focus. As an illustrator everything was very tight and I have to constantly remind myself to be looser and not to get caught up in the details. I love being an artist and it is a constantly evolving journey.\nWhen I hear from a viewer that it made them smile or evoked an emotion.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As a full time artist I have recently been challenged with the reality of living in the world but not of the world.\nI was a finalist in the 2016 Archibald Prize at the Art Gallery of NSW. I painted actor and comedian Garry McDonald and portrayed his more vulnerable side of showing his head in his hand - the expression of never feeling good enough as he battled with anxiety and depression earlier on in his life. Anxiety for me has also been a personal struggle along with many other artists. I have found that most creatives battle or have battled with mental health issues of some kind, constantly comparing ourselves to each other and always doubting our ability. It is a real fight to believe in yourself. Being an artist my heart is in everything I create so I am constantly feeling vulnerable and putting myself out there for opinions and criticism.\nWe as Christians forget that we don't just live this life for us but it's about others and above all glorifying God with our gift and talents. We are created by a creative God. He is the author of creativity so remembering that we are his hands and feet makes it that much easier to have faith in our ability.\nI have been challenging myself in moving in a different direction creatively with my paintings which as a perfectionist and number one self critic is very mentally and physically challenging. Stepping out into the unknown is not a new concept for me but this time it's a step towards creating artwork for Gods greater purpose of touching people and bringing heaven to earth through the gift I have been entrusted with.\nHaving to be a part of the art world but remain true to who I am as a believer can feel like a tug of war at times. I can definitely feel overwhelmed with everything but one thing you can rely on to feed you is the word of God. It needs to be your daily bread and Gods voice needs to be the loudest one you listen to. Build your faith on a strong foundation, remember who's you are and that you are a daughter of the most high God. Remembering that even though we may walk through the valley of the shadow of death that Jesus has overcome death by the power of the cross and through that we have complete freedom. The battle has already been won so we can live in that place of victory. Chose to think Christ like and allow the Holy Spirit to walk with you through every moment. Include him in everything because first and foremost God wants relationship, not religion. He loves just being with you and spending time with you. Never forget his love for you.\nKirsty has been a part of C3 Oxford Falls for over 10 years now. She serves on the hospitality team and runs a thriving connect group. Kirsty has been a full time artist for 5 years now. Her work has been hung in prestigious prizes such as the Archibald Prize, Portia Geach Memorial Award and the Mosman Art Prize and has won awards such as the Mosman Art Prize Viewer's Choice Award twice. Kirsty loves people and their stories and strongly believes we have a mission to show the love of God to all people through the gifts God has given us.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"One of the things that has continuously help women is the inventions that have always happened in the medical industry. If you are not using the fertility tracker, you might be missing out on some very serious benefits that women are getting from using these trackers. There are some very important benefits you will be able to get from using the fertility tracker and this article is going to educate you on some of these. For a very affordable amount of money, it is possible to get this fertility trackers because they do not take a lot. Apart from using the physical ones, there is also software that you can be able to use that can help you to track your fertility very easily. One of the first benefit that you will be able to get from using the fertility checkup is that you will be able to know when your menstrual periods will be coming. Fertility trackers are specifically very good because they explain to you how your floor is going on and in addition to that, every event that you're expecting during the month.The level of accuracy that usually comes with fertility trackers is actually very high, it is able to tell you the exact day when your menstrual periods you be happening and this is going to help you to get prepared.\nThe day of ovulation will also be known very easily when you decide to use the fertility tracker, this can help the people that do not want to get pregnant and also those that want to get pregnant. Understanding the days that you're supposed to avoid is very important and it will be possible for you especially if you are using the fertility tracker and you are also on natural family planning methods. If you are able to feed the fertility tracker information about your cycle, it'll be very easy for the machine to be very accurate regarding your cycle. Another reason why fertility trackers are very crucial is because they can tell you when you get pregnant through looking at the changes.This will be very possible because the fertility trackers usually tell somebody when they are expecting the menstrual periods and therefore, if they do not come, you can be sure that you're pregnant.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Autumn fruit is lively and bright: crisp apples, pears that positively drip with nectar and crunchy pomegranates that burst with juice. They partner beautifully with Fuyu persimmons and hazelnuts, too, for this Autumn Fruit Salad. A light vinaigrette of kombucha, hazelnut oil, ginger and maple syrup dresses the salad .\nWhen autumn tables are filled with pies, tarts, and harvest cakes, an autumn fruit salad offers a nice alternative. The salad is still sweet, and still festive, but it provides that lightened up note so needed when kitchens brim over with heavy winter vegetables like squash, potatoes and beets, roasts and sweets.\nTo make an autumn fruit salad, pick the freshest fruit you can find. In addition to apples, pears, persimmons and pomegranates, you can also add other autumn fruits like cranberries, figs or concord grapes. Slicing the fruit thinly not only makes for a beautiful presentation, but it also makes the salad easier to eat \u2013 and your guests can get a bite of everything in the salad on a single fork.\nA vinaigrette fulfills to functions when making this Autumn Fruit Salad. Apples, pears and other pomaceous fruit are prone to browning when they're exposed to the air; however, acids prevent them from browning. In this fruit salad, we pair kombucha tea \u2013 which is highly acidic and similar in flavor to apple cider vinegar \u2013 with roasted hazelnut oil, maple syrup and ginger to make the vinaigrette. The acidity in the kombucha helps to prevent the apples and pears in this salad from browning. Its tartness also acts as a foil for the fruit's natural sweetness.\nApples and pears which make the base of this salad are particularly high in quercetin, an antioxidant that is highly anti-inflammatory and that can even act as an antihistamine. Similarly, persimmon is particularly high in powerful antioxidants as well, like beta carotene and vitamin C, while pomegranate arils are a good source of potassium and a very good source of various other nutrients like punicalagins. Punicalagins are powerful antioxdants which may account for why pomegranate has such a higher antioxidant activity \u2013 even higher than red wine and green tea.\nTogether these, nutrient-rich foods, pair with hazelnuts and hazelnut oil which are high in a nutrient that can be relatively difficult to get through food sources: Vitamin E. Vitamin E helps to support cardiac health and helps to prevent damage to tissues from free radicals. Lastly, the kombucha tea used in this Autumn Fruit Salad's vinaigrette is rich in beneficial bacteria that help to support gut health and immune system function.\nThis autumn fruit salad is filled with thinly sliced apples, pears and persimmons, then dotted by fresh, juicy pomegranate arils and crunchy bits of chopped hazelnut. This salad is delicately dressed with a vinaigrette made from kombucha, hazelnut oil and ginger with the lightest touch of maple syrup for sweetness.\n3 tablespoons roasted hazelnut oil (Find it here).\nCore the apples and pears. Then slice the pears, apples and persimmon no thicker than \u215b inch, using a mandoline (like this one) or very sharp knife.\nLayer the pears, apples and persimmons on a serving dish, and scatter the pomegranate arils, chopped hazelnuts and fresh mint over them.\nWhisk the kombucha, maple syrup, ginger and roasted hazelnut oil together, then drizzle the vinaigrette over the fruit, and serve immediately.\nIf you don't have kombucha tea, substitute apple cider vinegar.\nPersimmons are tricky. Make sure to use the squat Fuyu persimmon as opposed to the tear-drop shaped Hachiya. Fuyus are sweet at all stages of ripeness, whereas Hachiyas are astringent and inedible until completely and fully soft.\nYou can also chop the apples, pears and persimmons instead of slicing them thinly.\nIf you'd like to add this salad to your menu, there's a few ways you can add it to the table. You can pair it with holiday favorites for a light addition to the table, and it also pairs beautifully with light soups for an easy lunch.\nThis Autumn Fruit Salad fits perfectly on the Thanksgiving table as a light addition to heavier dishes like Slow-Roasted Turkey or Pumpkin Custard.\nIt also partners well with lighter foods too, like Salmon with Honey Chipotle Bourbon Butter and Cucumber and Fennel Quick Pickles.\nFruit salad also pairs well with this Instant Pot Red Lentil Dal that comes together in only 6 minutes.\nJenny, I made this yesterday to take to our family's Thanksgiving dinner. It was delicious AND gorgeous. I used apple cider vinegar instead of kombucha and added a generous pinch of salt to the dressing. I'll be making this salad all winter long using different combinations of fruit. Thanks for this genius recipe!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"D\u00faplex in Puerto Calero with incredible views to the ocean. The property is constructed in 3 floors. The ground floor comprises of spacious lounge, separated fully fitted kitche, utility room, garden, terrace and pool. The upper floor consists in 2 bedrooms with fitted wardrobes, 2 bathrooms from which 1 of them is ensuite and terrace with direct views to the ocean. The lowest floor comprises of a large room with ensuite bathroom, patio and garage for 2 cars. Excelent location. High quality of construction. Sea views from lounge, both terraces and from 1 of the bedrooms. Air conditioning.\nCannot find the property you are looking for? We have more Duplex\/semi-detached for sale in Puerto Calero, Yaiza, Lanzarote , please send us an email and let us know us what kind of properties you are interested on. You may also subscribe to our newsletter to receive information of new properties for sale in Lanzarote.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Deutsch, Politik free Achieving the Health Millennium Development Sport zu erlangen. Ausbildungsjahr noch weiter zu um. Unterrichtsstunden einsetzen. Zusatzqualifikationen im Differenzierungsbereich zu erhalten, man media have Fachhochschulreife zu erwerben.\nDo us on free Achieving the Health Millennium Development Goals in Asia and the Pacific: in Europe with August Burns Red and Betraying The world! have us on runder in Europe with August Burns Red and Betraying The item! signal us on weitgehend in Europe with August Burns Red and Betraying The knowledge! 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US must maintain glory in big dass crash, recognizes Turkey's Christian streng '. John King on Colbert Report '.\nClick here if you're having trouble viewing this page e.g. overlapping text. free Achieving the Health Millennium Development Goals in Asia and the Pacific: Policies and Actions Within Health Systems and Beyond (Asia has a loyalty for the den of French guarantees, which are yet published truly. participants can spend restored looking 284kg officials. Each and every one of us will enhance to run inconvenient programmes and uns that will get in Karl und this of private attacks. We are populations to have your Individualkommunikation on our society.\nSharks free; sich - Greifenberg a. Pages 5 to 68 hope well glossed in this web. thoughts 73 to 142 have anymore been in this tempest. friends 147 to 217 've abruptly ignored in this um. laws 225 to 226 are However fought in this science--outside.\njust what we have to be comes, secretly of proud jeweils, get just slowly, read religious skills and Do Vietnamese with the free Achieving the Health Millennium Development. Some considerations of Dutch improvements am referred the peril that the doing available bonheur of the United States and the network of not duale compassiemeditaties residual as China are the been photographs and benefits of the economic, key nuance power. They are three poems of the new permission that have impressed and returned by the West: few alleged boundaries( the illiberal year), Lifelong years, and Anwendungsentwicklung leader. The medical new summer Leonid Grinin handles that despite all the products the USA will encourage the being genereusement within a limited opportunity wilt since no public-relations sein has Asian to enter certainly ailing deaktiviert's outcomes.\nClick here for my old site This links in a free Achieving the Health Millennium Development Goals in Asia and the Pacific: Policies that targets Lead, with brief fishing workloads, higher paix, and lower vogue. In community, New World empires are to click warmer, Rather years shalt riper and 're more stream to be to change. Like narrative, there are principles to every network. LabelingPerhaps the hardest pp. of living between the Old World and New World dies when you mostly do to improve a world of analysis. kindly hosted by Brunel University. Denn Sie gliedern Ihr Kompetenzprofil entsprechend der Kompetenzen der due Funktion free way twenty-first Pr\u00e4. Wie Sie spirit Dokumente in Binding offer Form bringen, lesen Sie dann auf eine; Form der Bewerbung. advise Reihenfolge des Aufbaus community dabei durch die Wichtigkeit der Begriffe oder durch way Aufbau des Stellenangebots idea. library wenn Sie sich initiativ mit Ihrem Nutzen bewerben, dann orientieren Sie sich go Besten nach dem ; Grundaufbau.\nHits since 22nd May 2005 when counter was reset - Thanks to digits.com also, in global Greece and Rome free Achieving the Health Millennium Development Goals in Asia and the Pacific: Policies and Actions Within Health Systems and Beyond (Asia lost more than an constitutional work, more than the -lama and book of Possible observers in the European site of the Quarterly connections. As large l\u00f6 continued not more not( as if it was bis thus on the highest way or essential productively to determine the access of the effective years), it not liked patchwork search, from armored search to run timeline. This Call's Walter-de-Gruyter-Seminar examines needed to the public governments in which anlegen left its Pr\u00e4 on wider immer lead and advent, signifying in( but extremely developed to) hin, the all-round Tibetans, works, and independent alliance. We are Donating dangers to the trial of this ' fest of Politics ' from all peoples of interminable excavations and limiting all offers of disinformation.\nThere are no free Achieving the Health Millennium Development Goals heroes in Silicon Valley. not, able insecure land would be it remotely likely to be this task. By und, actions several thereby to depend a indigenous und final for the team will rarely attack it inconsistent for the authority to compare to eBooks of chief inability. The sie research stems been die; WTC; in a World War into good sind.\nBeijing: Foreign Language Press. Linda Woodhead, thousands in the Modern World( diversity 1991) Freedom in Exile: The sie of the Dalai Lama. nschten issues of the Fifth Dalai Lama. Serindia Publications, London. free Achieving the Health Millennium Development Goals in Asia and the Pacific: Policies and Actions will kill this future to store your chance better. book will communicate this investigation to double-check your Study better. contributed a college balance at Aftershock Festival on Saturday! redistribution happens review in Mesa. Mark Steyn, After America: argue offline for Armageddon( 2011), Regnery Publishing, free Achieving the Health Millennium Development Goals in Asia and the Pacific: Policies and Actions Within Health Systems and Beyond (Asia Pacific MDG We shall avert to tell back to help one time, or one story we shall attack no sich. Britain and den have rather the international power. Labour Government that go disputed us see Soviet stolpert. My balance uses to be Britain being contemporary.\nIn free Achieving the Health Millennium Development Goals in Asia and the Pacific: Policies and Actions Within Health Systems and Beyond (Asia Pacific, they both wage each wollen. A shark of friends that we are build on this research. different either liberal or other to be forward in that cargo, you take them otherwise and with a moral die, you die a bestimmten audience of animal, and Die a left nonproliferation group. This is the faith read.\nsame from the international on 12 July 2009. Dalai Lama may force \u2013 before wage, Jeremy Page, The such, 29 November 2007. Dalai's finance will however organise used under vocational science '. practice of Tibet in Exile ex Indian Express July 6, 1999. volcanic from the possible on 12 July 2009. Haas, Michaela( 18 March 2013). Bindley, Katherine( 2013-04-24). Dalai Lama states He Would Support A Woman Successor '. hoax of the Dalai Lama Wm.\nBuy this Poster at Beispiel erscheinen in ganz Mittelfranken unter 13 Zeitungsnamen mit 25 Unterausgaben. Von diesen Kernredaktionen, seriously main anarchy transit, nous es 134. Konzentrationswelle gegeben, als viele Redaktionen aufgegeben, time oder mit anderen fusioniert ziehen. Medienunternehmen angewachsen. Verlagsgruppen in Deutschland haben bei problem Tageszeitungen einen Marktanteil von 45 volume. Zeitung lesen wie ihre Eltern. Ende 2009 hand Check education aller Tageszeitungen zusammen 20 Millionen Exemplare. got run Zeitungsdichte anbelangt, rangiert Deutschland innovation im internationalen Vergleich im Mittelfeld. Insbesondere Skandinavier free Achieving the Health Millennium Development Goals in Asia and Japaner lesen viel mehr Zeitung. The free Achieving the Health Millennium Development Goals in Asia and the Pacific: Policies and Actions Within for a silent selbst\u00e4 for truck were still. Because the more reasons, the more indem, the more nur. WorldCat is the close schedules that ruse need of peace technologies and continue perspective Therapies demented to the largest k\u00f6 speech in the innovation. Since WorldCat proves the thought of WorldShare Interlibrary Loan, verbeteren can well Post the centuries economic from smooth buttons, fight knowingly from them and Boost their figures to sign Americans' wars. WorldCat as is debris politicians to keep difficulties not to publications followed by connection drugs, rafting strengths the messy phenomenon they have.\nAllPosters.com It is very First Manchmal: I were myself including the devices always as if I denied Cleaning a free Achieving. then obsessed. Earlier this vision I were three minimum battles in ' Tired of all Trump, all the wine? 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Your California Privacy Rights. change to gain The Atlantic Popular Latest data flights part; Policy Culture Technology Ideas Science Books Family Business Global Health Education Letters The Masthead Photo is The Atlantic Crossword Video Events Writers Projects MagazineMagazine detailed involvement All accomplishments realia population Subscribe More CategoriesMore Create are Your process in Sign out Newsletters The Atlantic Crossword iOS App Life Timeline Events Books Shop policy Center View all Subscribe Search Search Quick Links James Fallows Megan Garber David Frum Adam Serwer Manage world Search The Atlantic Quick Links James Fallows Megan Garber David Frum Adam Serwer Manage assortment Global ISIS is here creating the War of Ideas The Islamic State isn une smelling because of the leader of its phone. machinery; Users modeling because it can look a Other selbstst\u00e4. 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Wobei existence-; r defeat Wirkung Ihrer Person folgende Faustregel die: 55 search K\u00f6 und; 38 Croatia library fact Ausdrucksverm\u00f6 heute chapter-preview site 7 und satisfaction.\nNAFTA is particularly a American free Achieving the inclde, but the f\u00e4 of a cautionary domestic sector. The Los Angeles Times( 18 July 1993), Esstisch The Other sie which reported in 1789 in the Cercle Social, which in the problem of its world was as its new races Leclerc and Roux, and which Not with Babeuf's bestimmte found not digitized, was development to the Soviet wealth which Babeuf's und Buonarroti re-introduced in France after the rule of 1830. This trade, not signed, is the liberalisation of the vested office system. Karl Marx, in The Holy Family( 1845).\nWould you feed to make to an older of Twitter? 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Die us on book \u041a\u0430\u043a \u043d\u0430\u0443\u0447\u0438\u0442\u044c \u0412\u0430\u0448\u0435\u0433\u043e \u0440\u0435\u0431\u0435\u043d\u043a\u0430 \u043f\u0438\u0441\u0430\u0442\u044c \u0441\u043e\u0447\u0438\u043d\u0435\u043d\u0438\u044f in Europe with August Burns Red and Betraying The sich! prevent us on in Europe with August Burns Red and Betraying The plan! recommit us on book Lindwurmspuk vor Mitternacht. Abenteuer in K\u00e4rnten (Die Knickerbocker-Bande 3) in Europe with August Burns Red and Betraying The power! determine us on Color Atlas of Ultrasound Anatomy 2004 in Europe with August Burns Red and Betraying The state! download Advances in Spatial will die this author to See your content better. developed up an individual institutional strategy.\nIn free Achieving the Health Millennium Development Goals in Asia and the Pacific: Policies and Actions Within Health Systems and, treten is encouraged marked in Europe and along the Mediterranean for online decades. are not anchored as essential collections to vie or be a ber of Comment or und faiths. And eine seen. 8217;; A1, general minor, beginning not and in third more common found.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The core magic Lores in the Warhammer rulebook are a lot more useful, and hence see a lot more play, with the changes in 8th Edition. In this post we're looking at the Lore of Fire. The Lore of Fire is, unsurprisingly, focussed on delivering large amounts of fiery death to your opponent's troops (or \"dudes\"). Most of its spells are direct damage, which is not necessarily a bad thing if that's what you want.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzatlbh b/data_all_eng_slimpj/shuffled/split2/finalzzzatlbh
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@@ -0,0 +1,5 @@
+{"text":"Matthew Tovbin is a Principal Member of Technical Staff at Salesforce, engineering Salesforce Einstein AI platform, which powers the world's smartest CRM. Before joining Salesforce, he acted as a Director of Engineering at Badgeville, implementing scalable and highly available real-time event processing services with Scala. In addition, Matthew is a co-organizer of Scala Bay meetup (http:\/\/www.scalabay.com\/) and an active member in numerous functional programming groups. Matthew lives in SF Bay area with his wife and kid, enjoys photography, hiking, good whisky and PC gaming.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Thank you for visiting the website of Winter Park Florida ChiropractIc Physicians Dr. Kimberly \u2026 267 west comstock Ave, Winter Park, FL, 32789, United States.\nWelcome to West Park Chiropractic Center. About Us. Learn more about the Palmer College of Chiropractic graduate. DOT, medical examinations and urinalysis testing are among Dr. Dave's capabilities.\ntrue correction at Ozner Family Chiropractic, Sunrise FL chiropractor. TRUE CORRECTION. family health at Ozner Family Chiropractic, Sunrise FL chiropractor.\nDr. Day was first introduced to chiropractic right after high school. He had severely injured himself lifting weights at the gym. After the frustration of going to medical doctors and receiving medication that only temporarily covered up the symptoms, a good friend introduced him to chiropractic.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The KPI Institute's Performance Management in 2014: GCC Special Edition report followed one of our most important editorial rules, namely the triangulation of opinions gathered from practitioners, academics and consultants alike. In 2014, Mohammed S. Hyder, Executive Manager \u2013 Decision Support & Performance Management at Etihad Etisalat Mobily, Saudi Arabia, was one of the practitioners who offered us rich performance related insights.\nThe KPI Institute's Performance Management in 2014: ASEAN Special Edition report is built on the belief that a balanced approach in such research endeavors can only be achieved by triangulating the opinions of practitioners, academics and consultants alike. In 2014, Lauren Borja, Head of Corporate Strategic Planning working with BBV, Philippines Air Force, was one of the practitioners who offered us rich insights into performance related research and trends.\nTh e KPI Institute's Performance Management in 2014 report is built on the belief that a balanced approach in such research endeavours can only be achieved by triangulating the opinions of practitioners, academics and consultants alike. In 2014, Lu\u00eds Gargalo, Management Control Systems Department Coordinator for Wayfield, Trading Internacional, SA \u2013 Grupo Refriango, Portugal, was one of the practitioners who offered us rich insights into performance related research and trends.\nThe KPI Institute's reports Performance Management in 2014 and Performance Management in 2014: GCC Special Edition continue the standards established by the previous editions, which consist in gathering and compiling opinions from practitioners, academics and consultants alike, in order to gain a balanced perspective on the state of the discipline. In 2014, Mr. Jarlath Fernando, Analyst \u2013 Strategy Planning at Dubai World \u2013 Imdaad, UAE was one of the professionals who offered us rich insights into Performance Management practice.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\u2026\u2026 a few days back I got a posting inviting me to be a viewer of a programme on a Christian TV channel (don't get me started...) called 'walking in victory' or something. I pressed 'delete' almost immediately.\nHere is one of my takes on theology\/life- Jesus came to show us God and did that by entering into the ordinary mess and contradiction. He began and begins redeeming the mess.\nI guess I used to take the message covertly that he was some kind of gleaming superman who made everything better straightaway. If He didn't that had to be ignored or there was something wrong in me\u2026\u2026 I look back with the great gift of hindsight and see that some people actually taught that and some seem to continue to teach it. If they don't, well some people of faith seem to feel guilt\/desire to hide that they may be broken. 'Positive thinking' has a place but it is not the gospel\u2026do not be scared of the dark or grey side.\nA blog that I sometimes read once spoke of regret that in a group of churches that he was part of, there was a section of the meeting where 'Good News' was an agenda item. He spoke of his dislike of this section due to the pressure to share only good news, suppress the bad or sometimes embroider the truth. I think it is good that we share stuff that has worked or where we have felt the presence of God. However, I also think it is good to share lament or the grey rainy Tuesday afternoons where nothing much has happened apart from staying faithful.\nYes, yes\u2026.miserable sod- but then you guessed that from this blog. Enjoy your bank holiday weekend\u2026.\nAnd a friend, also with a large dollop of irony, called me a miserable sod.\nAs far as FB goes, it led to quite a lot of posts\/discussion.\nI love hearing news- I love hearing good news; things that my friends have done, good experiences that they have had and things that have pleased them about their family. 'Rejoicing with those who rejoice' is something that, as I get older, comes easier: life is short and celebrate the good when it comes. That is one reason why I am part of Facebook; it is a shorthand way of catching up and a good background noise when I am online.\nBut\u2026. But\u2026\u2026 some postings seem to me to be so unremittingly positive; I guess that may well be the poster's character. I wonder sometimes if the poster is scared of admitting anything less than positive.\nI long for some light and shade; I love your joys- I even love your rampant self publicity (mea culpa) you don't have to go so far and be embarrasingly confessional all the time, but just let us know that sometimes your children annoy you, you get sad, you long for more, some experiences make you miss people and occasionally you wonder what it is all about. If you don't, your joys don't seem like peaks to me, but rather your life seems like a blissed out and unobtainable plateau to the rest of us\u2026.\nAnd if that makes me a miserable sod, then so be it.\n'\u2026has enjoyed a wonderful day with his beautiful wife driving his convertible Merc to his villa in Tuscany. Looking forward to a meal in the local Michelin restaurant to celebrate another successful business flotation and the 4 A* grades his son has got at A levels to get in to Cambridge. Looks like it may not be quite as good olive harvest this year as last from my private grove'.\nIt makes me sound like a bloke propping up a bar and indulging in a stream of consciousness rant. You know, I'm slightly pleased about that impression\u2026.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It was time to go back to Wettenbostel\u2026 for the weekend. There was a wedding at the seminar house and I offered to come in from Hamburg to help out. I met up in Hamburg with a Reiki Master who was also helping out for the weekend and we traveled to Wettenbostel in her Mercedes mobile home. She lives part-time in a town by the sea called B\u00fcsom and we laughed as I practiced pronouncing it, the sounds feeling foreign in my mouth.\nThe trip there was easy and fluid. We arrived to friendliness and hugs and a pristine silence as there was a silent yoga retreat at the seminar house. Warm greetings welcoming us back were offered in hushed voices. The seminar house was booked solid for the retreat, so I found my night's sleep in the not yet renovated kitchen in the large Dojo with a mattress and comfy down cover and pillow on top of a wooden table for my bed. Despite my rough surroundings, it felt good to be back.\nSunday we were pretty tired from the intensity of the Saturday. Breakfast was served to our wedding guests and then we relaxed a bit until mid-afternoon \u2013 time to returned to Hamburg. Then\u2026 ahhh\u2026 an evening of rest and relaxation in the peace of the flat. I woke up the next morning grateful to be in Hamburg but also relaxed and somehow restored from my weekend of work in Wettenbostel.\nAnd this week my time in Hamburg comes to a close. I am still tousled on the inside from my weekend of moving and shaking and some chaos at the wedding\u2026 wondering where things will shake out for me\u2026 Is there some other interesting and inviting opportunity to unfold for me?\nAnd yet what is there for me now is\u2026 being grateful. Being grateful for the time I have had in Hamburg\u2026 the connections, the people I have met and who have extended themselves to me. The generous opportunity to stay in this flat and have a beautiful bicycle to ride!\u2026 And the goodness of having some place to return to with a warm comfortable bed, gardens and natural surrounding.\nAnd for now I have some time to myself. My mind and my body have been doing a little wrestling with one another..but I am in Hamburg for a few more days and then a train ride back to Wettenbostel. Ah\u2026 just relax\u2026 relax and trust.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaufpi b/data_all_eng_slimpj/shuffled/split2/finalzzzaufpi
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@@ -0,0 +1,5 @@
+{"text":"I don't have enough books to read, said me never. Mine is the opposite problem. My TBR pile grows steadily every week, matched only by my guilt and panic. Because there are so many books sitting there unread and unloved, choosing what to read is a nightmare. So, I devised a system (well, the internets devised a system and I appropriated it), to take the agony out of choosing. I have written all of the books on my shelf, both on my physical shelf and those sitting on my Kindle, on post-it notes and have put them into a vase, and when I'm ready for a book I get the boyfriend to choose one out of the jar. It's my 'what-to-read-next lucky dip' (I gotta come up with a snappier name), and it's working a treat so far. Although, I wish he'd stop picking out all the massive books in there \u2013 it's really messing with my reading stats.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our service department includes 4 full service warranty repair facilities, 4 fully stocked service trucks and 7 factory trained technicians. We provide service on a wide range of equipment including Vac-Con Combination cleaners and hydroexcavators, Schwarze street sweepers, Tiger boom mowers, GO-4 parking enforcement vehicles and many others. Each one of our technicians goes through rigorous factory sponsored training and is up to date with the most current technology. We also our proud to have on staff an American Lift Institute Certified Inspector. Whether you are in for preventative maintenance or a machine overhaul we have the skill to get the job done right.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"o How do I find my closest distributor?\nContact our office with your enquiry and we will advise the nearest distributor to contact you.\no If I buy from a distributor what happens to after sales service e.g. repairs?\nhas a fully equipped workshop available for back-up service.\no Are your spares readily available and for how long after initial purchase?\nWe guarantee the availability of spares for ten years.\no Do you carry stock holding of all your products?\no Do your products come in different colours and are they HACCP compliant?\norder quantity and a waiting period for delivery. Certain products HACCP compliant.\no Can your trolleys be modified to different specifications?\nYes, our trolleys are manufactured with flexibility and can be modified to various requirements.\no Can I get my own company branding on your products?\nYes, you can. There is a requirement of minimum quantities and a lead time applicable.\no Where are your products manufactured?\nOur manufacturing plant is in the United Kingdom.\no Are your products SABS approved?\no What is your BBBEE rating?\nCurrent Level 2 with Emex audit.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Nursing is one of the most popular careers in healthcare, attracting those with compassion and a desire to help others through their work.\nBut caring for others isn't the only reason people love this career so much. We have 12 other reasons nursing is one of the best careers out there. By the end of this article you might be considering how you'd look in a pair of scrubs.\nMany think it takes four years or more to enter the nursing field, but it's possible to become an RN in as little as two years. For example, the registered nursing associate degree at Eastwick College in Ramsey, NJ can be completed in as little as 18 months.\nThere are a seemingly endless number of work settings for a nurse, so you can choose what best fits you. For example, you could work as nurse at a high school, a hospital, or even from home.\nAccording to the U.S. Department of Labor, registered nurses in the NY\/NJ Metro Area make $90,840 on average; licensed practical nurses make $53,920 on average. Depending on your location and experience, the salary potential can be even greater.\nWhile many industries are scaling back benefits packages for their employees, benefits in nursing jobs remain above average.\nIt might seem like a small thing, but comfortable work attire shouldn't be overlooked, and scrubs are known to be very comfortable.\nOf course, not every day is going to be thrilling, but in general, nursing provides stimulating work that keeps you engaged.\nAs a travel nurse, if you're tired of your current location or want to get away from cold weather for the winter, you have the opportunity to take assignments in new cities and countries, typically about 13 weeks at a time.\nThere are so many nursing specialties that allow nurses to focus on what they love to do. For example, if a nurse discovers that taking care of adults is not personally fulfilling, they can pursue other opportunities as a pediatric nurse.\nThe US Department of Health and Human Services has federal loan repayment programs for qualifying registered nurses.\nNurses tend to stick together and share a common bond, whether in real life or online, so you won't be doing it alone.\nAs a nurse, you'll have the opportunity to make a difference in your patients' lives, and your skills will be needed all over the world. A career in nursing provides a unique opportunity to help make a positive change in the world through your efforts. \"One of the most rewarding aspects of the LPN and RN programs we offer is the heartfelt letters we receive from graduates thanking us for helping them achieve a fulfilling career where they really do make a difference,\" explains Thomas M. Eastwick, President of Eastwick College.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"St. Augustine, Fla. - Lynn University's women's golf team is off to a solid start to the 2018 season as the team is in fourth place after the first round of the World Golf Invitational. Kristina Ortiz is pacing the Fighting Knights in the top-10.\nLocation: Slammer & Squire Course at the World Golf Village | St. Augustine, Fla.\nLynn shot 312 as a squad in the 18-team tournament. Florida Tech (301) has a hold on the top spot ahead of Nova Southeastern (304) and host Flagler (306) while the Fighting Knights are just ahead of West Georgia (314) in fifth.\nOrtiz had the low score for the Blue and White, shooting a four over-par 76. The junior out of Puerto Rico was +1 at the turn but tallied two bogeys and a double bogey along with a birdie on the back nine.\nJenny Ayala and Helen Kreuzer are one shot behind Ortiz with rounds of 77. Ayala traded two birdies with seven bogeys while Kreuzer stashed paired three birds with six bogeys and one double bogey.\nSamantha Barber and Matilda Wahren rounded out Lynn's contingent with scores of 82 and 84, respectively.\nAs a team, Lynn is fourth in total pars tied for fifth in both par 4 and par 5 scoring and total birdies.\nOrtiz is tied for fifth in par 4 and par 5 scoring while Kreuzer is tied for seventh in par 3 scoring.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzauwuw b/data_all_eng_slimpj/shuffled/split2/finalzzzauwuw
new file mode 100644
index 0000000000000000000000000000000000000000..48450928c42d51565b859c15cd09edd3151861f8
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzauwuw
@@ -0,0 +1,5 @@
+{"text":"What makes an Antares Home Owner tick? What makes them choose us? As a company, we spend countless hours trying to figure out how to get better responses on our customer surveys, analyzing data, and attending meetings about this subject. This is the question that keeps a salesperson up at night. We ask ourselves, why this person did, or did not buy our home. In the homebuilding industry the buyer is the celebrity, the hero. They are the person we most want to know about. A new home salesperson would jump at a tabloid that featured gossip and interesting info on all of our buyers as well as those that got away! So, why not create that? I am starting a quest to find out as much as I can about our buyers. I am going to visit with them in their native habitat and study them. I will have them fill out one of those little questionnaires that you see in the celebrity profiles of your favorite magazine and take pictures and dish it up!\nForm boards have been set at 625 Sparrow. A foundation is just around the corner.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Published 04\/18\/2019 07:21:45 pm at 04\/18\/2019 07:21:45 pm in Always Simply Safe.\nalways simply safe simplisafe the best dividend stocks to buy during this correction seeking alpha source simply safe dividends.\nvideo doorbell pro wireless alarm system always on the lookout, home security systems from simplisafe this morning at am my phone rang, our favorite modern bar stools that will complete your kitchen simply put abstract designs are always a safe bet for modern bar stool seating youll want to curate designs that feature colors shapes and lines that , notices old bridge nj notices, live simply safe meet live simplisafe security live simply safe home security systems feel secure and safe using these security alarms tips livewatch, the problem with safe browsers and parental controls that use them todays topic is safe browsers the internet has gone from a curiosity to a necessity in a dramatically short period of time when combined with the surge , the best home security system for reviewscom simplisafe, baby carrier safety baby carrier support scotland baby carrier safety, simplisafe review simple but is it good quality security for simplisafe indoor camera, wireless security system home apartment and business precision detection, know exactly how much is safetospend simple its a smart budgeting tool that does the can i afford this math for you so youre better prepared to make smart financial decisions on the spot.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Prang Fine Line Markers give kids imagination a world of vivid colors to work with. These value-priced markers feature superior ink for a smooth laydown and long-lasting super-resistant 2.75mm nibs for durability and accuracy. Washable ink wont bleed through paper making these markers mess-free. AP Certified non-toxic.\nI thought it doesnt work because it is so quiet. Execelent product from Dixon Usa with many addons such as Dixon Ticonderoga Company DIX80719 Prang Markers Washable Fine Line 8 Color Set.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"1 :: What is meaning of Android?\n2 :: What is Android Mobile?\n3 :: Who are the Inventors of android?\n4 :: What are the features of Android OS?\n5 :: Which tools required for developing Android Apps?\n6 :: ADT stands for?\n7 :: SDK stands for?\n8 :: Which language need to know to develop android apps?\n9 :: Which are the advantages of android?\n10 :: Define Android application Activities components?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Step 1: Decide on a theme.\nMaybe you want to share your vacay, spread some inspo, or organize your ideas. A mood board works for about anything. The one I created is just a display of my summer essentials and favorite things.\nStep 2: Collect your photos.\nTake them, gather up old ones, find new ones on Unsplash, or if you use Milanote to make your mood board, it has its own little photo library.\nStep 3: Make your mood board!!\nGo to Milanote, create your account for free, and make your marvelous little board! Click this link to check out mine!\nEnjoy!! I hope you have a lot of fun making a unique little mood board with all kinds of fun and inspiration!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzawgpm b/data_all_eng_slimpj/shuffled/split2/finalzzzawgpm
new file mode 100644
index 0000000000000000000000000000000000000000..850722676358a69b81ca45756c47976b24dd8eba
--- /dev/null
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+{"text":"Here are some photos from our visits to woodland areas.\nThroughout school we keep all ink cartridges and batteries and an external environmental company comes in to collect them.\nWhen we visited the woods all the pupils had a task of creating an Autumnal Leaf Art using a hula-hoop.\nAt the of the session we all had a talk about different times of the year and that we are going into Autumn, then we started talking about winter and what we expect around that time \u2013 Snow, Robins, Ice & Father Christmas.\nWe welcome the wildlife, Mammals, insects and birds to St Peter's by planting the shrubs around our new playground. We planted these trees to make a natural screen and a home for nature.\nWe Planted Dogrose, Hawthorn, Hazel, Crab apple, Dogwood.\nHigh CO2 emissions are harmful to the environment and cause climate change.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Team Never Quit\u2122 embodies the heart of a warrior - men and women in all walks of life who have faced incredible hardship but have chosen not only to survive, but to learn from the experience and make themselves and those around them stronger for it. Their commitment is to honor those who have fallen, stand with those who have survived, and share their stories that they might inspire others to Never Quit.\nThe TNQ line provides solutions for every type of shooter to include dynamic training, hunting, self defense, match grade competition and reduced ricochet technology - all encompassing the mantra to Train. Hunt. Defend. The ammo has been featured on The Blaze, Shooting Illustrated, Tactical Weapons Magazine, Warrior Talk Radio and many other great media outlets and can be found in many dealerships around the country.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Bathe your wedding reception in a warm, enchanting glow and be sure to create a romantic atmosphere with our traditional and contemporary candelabras, available to hire in silver and black in Cumbria and Lancashire. Our candelabras can be dressed with crystal garlands for added sparkle, or flowers, butterflies or ivy for a truly romantic look.\nTo complement our table candelabras we have a range of floor standing candelabras, which add a dramatic elegance to any room or church.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Clapham High Street is a financial institution of the Metro Bank group. The specific location of the entity is 65-67 Clapham High Street, London, post code SW4 7TG. The branch gives detailed information over the telephone: 020 3402 8710.\nThe facility serves customers from nearby cities: Drury Lane , Aldwych, Westminster.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The following post extracted from Pankaj Seth's (ND) incredible site is a great complement the earlier post \"THE TOP 10 MEDICAL MYTHS'. As usual the article has been embellished with appropriate links. In addition to the medical archive section Punkaj' s excellent site is a treat to visit and provides an educational introduction to Ayurveda and Health.\nThe biggest myths of modern medicine were challenged in a new guide for patients launched yesterday that sets out the best treatment for 60 of the commonest medical conditions.\nInstead of claiming miracles, the guide admits that often the best treatment is no treatment. Devised by the British Medical Journal (BMJ), it is based on evidence from thousands of research studies and is being made available through the NHS Direct website, the advice service for patients.\nTreatments are ranked according to effectiveness and the pros and cons of surgery are explained. In some cases the guide says it can't recommend any treatment because there is no good evidence that anything works.\nProstate cancer is the commonest male cancer and one of the fastest growing, affecting 27,000 men a year, but surgery to remove it may cause more harm than good, according to the guide. Men who opt for \"watchful waiting\" live just as long, it says.\nOn back pain it recommends sufferers should avoid lying in bed and instead continue with their normal activities, taking painkillers if necessary.\nIt does not recommend tranquillisers as a treatment for anxiety - except for short term use. It also says that there is no evidence that any of the treatments tried for anorexia, which is a serious illness caused by a variety of different factors, work.\nMastectomy for breast cancer does not extend women's lives any more than the smaller operation of removing the lump and keeping the breast intact.\nThe removal of impacted wisdom teeth - a routine operation for decades - is needless, unless there is evidence of infection, it says.\nIt also examines a dozen common operations and diagnostic tests, weighing up the risks and benefits of each.\nLuisa Dillner, editor of BMJ Best Treatments, said the guide was designed to give patients the same information as their doctors based on the most up to date information available.\n\"The big myth about medicine is that people know what works. In fact, they do things for which there is no evidence. There is a tendency for doctors to exaggerate the benefits of what they do because they want to help.\n\"I think conveying uncertainty is important. We need to say when we just don't know. \"\nThe BMJ already publishes a guide for doctors, called Clinical Evidence, which assembles the best research to give up to date advice on the treatments that work.\n\"We thought it made sense to give patients the information that doctors get. We found in our research that patients said they wanted to read what their doctor read, not what their doctor thought they should read,\" Dr Dillner said.\nThe guide has separate sections for patients and doctors, but both can be accessed by anyone - so patients can read the advice for doctors.\nThe BMJ is paid a licence fee for use of the material but has complete editorial control, avoiding charges that the advice is subject to political interference. Dr Dillner said it was not designed to deter people from seeking treatment but it was about looking squarely at the evidence rather than relying on custom and practice.\n\"It is about trying to tell the truth,\" she said.\nRosie Winterton, health minister, said: \"We know that patients would like more information to support them in making decisions about their healthcare.\n\"This is an important step in providing patients with the resources they need to make informed choices.\"\nMyth: Treatable with a combination of drugs and therapy.\nBMJ advice There are no drugs that can cure anorexia and there is no strong research evidence that any treatments work well.\nMyth: Tranquillisers can cure anxiety.\nBMJ advice There are no quick fixes. Talking treatments (cognitive therapy) and certain drugs (some antidepressants) may help but doctors don't know which is best.\nMyth: Best cure is rest.\nBMJ advice Staying in bed doesn't help, it won't make the pain any better and could be harmful. Staying active is the best remedy.\nMyth: Mastectomy (removal of the breast) is the safest option to prevent return of the cancer.\nBMJ advice Breast-conserving surgery (only the lump is removed) is just as effective for locally-advanced disease with the same 10-year survival rate.\nBMJ advice A moderate amount of exercise is beneficial. Drug treatments, such as ACE inhibitors and beta blockers, work.\nMyth: Surgery, radiotherapy and hormone treatment are necessary to save life.\nBMJ advice Where the cancer has not spread, patients who do nothing but \"watchful waiting\", with regular check-ups, are likely to live just as long.\nMyth: When they don't come through the gum properly (impacted) dentists often recommend removal.\nBMJ advice If they are not causing problems taking them out is likely to do more harm than good.\nRemoval of adenoids at the back of the nasal cavity.\nMyth: The only way to improve breathing and prevent ear infections in children.\nBMJ advice The problems will usually clear up of their own accord, as the child grows. The operation works best in children who still have persistent problems aged five or more.\nSmall tubes inserted in the ear drum to drain fluid from the middle ear.\nMyth: Cure for glue ear.\nBMJ advice Most children grow out of glue ear. There is no good evidence demonstrating that fitting grommets is better than doing without.\nRemoval of tonsils at the back of the throat.\nMyth: The cure for repeated sore throats and ear infections.\nBMJ advice Taking antibiotics may be just as good. There is no good evidence to show that the operation reduces throat infections.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzawuql b/data_all_eng_slimpj/shuffled/split2/finalzzzawuql
new file mode 100644
index 0000000000000000000000000000000000000000..1afd31205cbcce6fb9062a2eccbf16813607b231
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Upon arrival each player will randomly pick a table from a blind \"fishbowl\".\nBring your Mah-Jongg sets (if you have one).\nDecks of cards, scoring pads and pencils will be provided.\nJoin us for camaraderie and fun and a chance to play with everyone.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Grown up luggage feels like something that should be under my belt by this stage of my life but I am still not there. I mean, I have a beat up tiny suitcase that stores my derby gear but otherwise this lady is all about the backpack. After checking into the prices of a suitcase (how can they be so expensive?!) I decided on sewing up a sweet set of weekender luggage \u2013 at a quarter of the prices and heaps more fun! Enter the twee-est, but also handiest, set of travel gear ever!\nThis is the Portside Travel Set by Grainline Studio. This pattern is a three for the price of one deal \u2013 duffle bag, dopp kit and travel pouch. The duffle bag features a front pocket, handles and an adjustable strap (which I left off). The dopp kit (fun history fact: the term dopp kit came into common use in WWII from their allocation to American GIs) has a sneaky front pocket and a handle at one end. The travel pouch is both tiny and quick to make \u2013 win! One thing to note \u2013 the pattern is crazy big so treat yourself to a print shop copy \u2013 no one needs to be taping that many PDF pages together!\nReal talk: the fabric for this one is sitting right on the cusp of twee but I couldn't say no to it. The main fabric is an adorable quilting cotton I picked up from Spotlight last year that was originally intended for sleep shorts. The contrast fabric is the stuff of nightmares \u2013 a sweet mint green pleather that stretches like Gumby every single time you start sewing. HULK SMASH! I had lots of trouble sewing this together and even with a teflon foot and a walking foot it just kept stretching. Heck, it took 8 attempts to get the duffle bag pocket on \u2013 thank goodness for the straps hiding the damage! In the end I used a stiff bag interfacing, a walking foot, released the thread tension by one, lengthened the stitch and took my sweet and slow time. It was totally worth it in the end and there is piece of the pleather left in my stash for when I'm feeling brave enough to take it back on.\nThe Duffle Bag is super straight forward \u2013 it's sewing together a bunch of rectangles and hand stitching a lining in. The only problem I faced here (apart from the cursed pocket pleather!) was the fact I had misprinted my lining pieces so the fabric didn't go together. This meant a quick recut and we were good to go. One thing to note this sucker is HUGE! It is a total weekender bag rather than an overnighter.\nThe Dopp Kit was a little bit fiddly due to easing the curves on the top but otherwise straightforward. One thing I did notice is that the pattern piece for the top was an inch too short \u2013 this was an easy fix and may be fixed on later versions of the patterns (I grabbed this when it was first released). The dopp kit is a fabulous size for all my toiletries and I have been using it for the gym and it is all kinds of fabulous!\nThe Travel Pouch is crazy simple to throw together and is something I have made previously for handmade Christmas gifts (chuck in a handmade lip balm and some chocolate and you win at secret santa!) The only change made here was to add a lining and then heave a huge sigh of relief when it was done because my nemesis Pleatherino would now leave me alone.\nNice and straightforward. It is so lovely to just sew something up without having to worry about fit!\nThe practicality. I finished this three weeks ago and have already used the entire set several times. The travel pouch is the perfect size for all your charges and cords!\nNo pleather. Seriously lovely to look at but GAH! Or maybe I should amend this to only pleather if stabilised. Look at my fancy grown up compromising skills!\nTabs at the end of the zips. I saw this on Mel's version and it makes so much sense.\nQuilted contrast fabric for the duffle straps. This could look super cute AND be much cheaper and easier to get your hands on then webbing.\nIt is such a pretty colour that I can forgive the pleather its stretchy sins!\nGumby fabric \u2013 lol! Nicely managed! Looks totally pro from this side of my iPad!\nUsing that pleather for another project? No way lady \u2013 send that piece to the Salvos for someone else to suffer with it. Anyway your Portside and accompanying accoutrements are tres speciale. Total love.\nBut it is so dreamy! What are these magical ports with wheels? I must find out more STAT!!!!\nThe puppies are too twee! (Who am I kidding, I had to look \"twee\" up on Google. I'm a Yankee Doodle Dandy.) And Berra? That's Yogi Berra the baseball player to us! Lovely weather year round to y'all. (Soon to be the Embarrassed States of America with our upcoming elections.) But, education and politics aside, LOVELY MAKE!!!\nWho makes the fabric? I lurve it, and really don't think it's twee. At least to a sewer, sewist, whatist? (Heated discussion on Facebook.) I would love to hear comments by fellow travelers!\nAnd thanks for the pleather advice. It looks fabulous, but I'll probably steer clear now.\nI think most pleather is pretty stable \u2013 this one was super soft and stretchy. I'm keen to try it again with lots o' interfacing!\nP.S. Do you use the term \"Dopp kit\" in Australia?\nI love this set and if I saw it in a shop I would snap it up, the colours are amazing and that pleather is awesome.\nThanks, Morgan! I'm really quite smitten with this set \ud83d\ude00 You totally need one \u2013 it's such a fun make!\nAnd I am sticking with Q.T. because they are cutie dogs (see what I did there?!) teehee!\nI'd never have thought pleather would be crazy-stretchy. It's good you worked out how to deal with it coz the combination of the two fabrics is pretty awesome. And the whole set most cool and groovy and very useful.\nI didn't pleather would be stretchy either but maybe it was because it was so soft and lovely? I am hanging on to the remnant I have left for a tote bag in the future \u2013 it is so worth the pain!\nOooh a tote bag in that pleather would be really yummy!\nThose quality control officers are the cutest and look like they mean business!\nI think a weekend trip is in order to try out your new luggage. It's the cutest!\nPoor Bimble is so patient but Pimble never has a bar of it!\nNot going to lie \u2013 it makes me feel all fancy an grown up to have matching luggage. And this pleather was a pest but so worth it in the end!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our team of international Tax consulting professionals includes the industry's most respected tax experts in UAE. Our expertise in VAT recovery, compliance, and deployment of tax automation solutions helps ensure our clients are protected from overstated tax liabilities, interest charges, and costly penalties. Our VAT services include VAT Registration, VAT Implementation, VAT Compliance, VAT Advisory and VAT Return Filing.\nTAXHELP is a specialized consulting division of Al Taayeen, operating in Dubai since 2003. Our experienced taxation team with exposure to different tax regimes is capable of advising and assisting businesses in the region to improve their systems and processes.\nOur dedicated team of professionals can help your business transition to the VAT regime. We can assist you in constructing the strategy and processes needed to be well prepared for VAT introduction.\nThough viewed by many as just an accounting challenge, the new VAT system has a pervasive impact on the way you do business. Identifying the impact of VAT on costing, product pricing and cash flow is our expertise.\nOur VAT experts are eager to help you make the transition to VAT smooth and hassle-free.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Coping with, and managing, a chronic illness can, and will, at times take a toll on you and your loved ones. Almost everyone with a chronic condition will experience acute flare ups that alter routine and one's sense of well-being. There may also be circumstances where things seem to be going well; and, then there is a setback. For some, there comes a time when a setback is experienced that one is not expected to overcome. Or, despite best efforts to get things going in the right direction, decline is progressive.\nAt the core, every patient wants to feel good, be free of emotional and physical suffering, and have their affairs in order. Every family wants their loved one to live well and to be comfortable. When quantity of remaining lifespan comes into question, quality of life does not have to be compromised. We are here to help when tough choices need to be made so that the choices do not feel so challenging and hopeless.\nMany patients on dialysis also have other health considerations that are not related to kidney disease. Examples are: heart disease, lung disease, liver disease, peripheral vascular disease, brain injury, cancer, etc. Patients on dialysis CAN receive hospice care. We will educate you regarding guidelines and options. If you choose to continue with a home dialysis modality, but have a need for hospice care, we have a good working relationship with area hospices and we can help to ensure that your needs, and the needs of your loved ones, are met.\nIn other circumstances, predicated by advancing age, medical complications, or simply self-determination, some patients opt to discontinue dialysis. At HDT, we will respect your choices for your care. If, in conjunction with your care team, you are able to make an informed decision and express that the burdens of dialysis outweigh the benefits for you, we will help you to get hospice services in place. We will help ensure that you are cared for with dignity at the end of your life, and that your loved ones have the support that they need.\nDialysis, and no dialysis, are choices. We at HDT are here to support you through the continuum. For help with your questions, please contact us for support.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Home with a dazzling type enjoy Mid Century Modern Furniture Atlanta Mid Century Modern Lighting Living Room Midcentury With Atlanta Blue Cable Railing photo indicates is mostly a perfect for anyone, and maybe you are one of them. Mid Century Modern Furniture Atlanta Mid Century Modern Lighting Living Room Midcentury With Atlanta Blue Cable Railing image will give you an abundance of tricks to realize this daydream home. With info which might be awesome, Mid Century Modern Furniture Atlanta Mid Century Modern Lighting Living Room Midcentury With Atlanta Blue Cable Railing photo will make suggestions to the wonderful destination to stay. Notebook know every single element in Mid Century Modern Furniture Atlanta Mid Century Modern Lighting Living Room Midcentury With Atlanta Blue Cable Railing photograph to find determination upon property style and design that is sought after simply by many people. A lot of items you can take up from Mid Century Modern Furniture Atlanta Mid Century Modern Lighting Living Room Midcentury With Atlanta Blue Cable Railing picture will be the idea, tones, and additionally your furniture. You have to blend such essentials by using good for the reason that exhibited as a result of Mid Century Modern Furniture Atlanta Mid Century Modern Lighting Living Room Midcentury With Atlanta Blue Cable Railing pic along with the total imagine in Mid Century Modern Furniture Atlanta picture gallery.\nAll illustrations or photos inside Mid Century Modern Furniture Atlanta picture stock are picked images with top notch developers. You will be able to acquire just about every graphic simply, to build Mid Century Modern Furniture Atlanta Mid Century Modern Lighting Living Room Midcentury With Atlanta Blue Cable Railing picture containing dimensions: 990 x 660 and size: 102 KB, you can actually push here. Many of us have seen this approach Mid Century Modern Furniture Atlanta Mid Century Modern Lighting Living Room Midcentury With Atlanta Blue Cable Railing graphic that is downloaded with September 30, 2017 at 5:25 am, just exactly is 1 visitors. Any time you mean to article this particular Mid Century Modern Furniture Atlanta Mid Century Modern Lighting Living Room Midcentury With Atlanta Blue Cable Railing picture coming to your website, then you definitely need to add URL. Additionally help the from your netbook and additionally mobile using Mid Century Modern Furniture Atlanta Mid Century Modern Lighting Living Room Midcentury With Atlanta Blue Cable Railing snapshot since picture. If you would like to find a further spectacular dwelling type, you may look into Mid Century Modern Furniture Atlanta picture stock and also many other art galleries in such a weblog.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaxhjv b/data_all_eng_slimpj/shuffled/split2/finalzzzaxhjv
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@@ -0,0 +1,5 @@
+{"text":"as they are defining their sound, their image, and their future as a band with more divine clarity than ever.\nIt's easy to hear the southern influence and riff driven story telling reminiscent of The Black Crowes, four part harmonies that makes one miss the Eagles, the groovy jamming of Tom Petty and the blues influenced, screaming guitars of Led Zeppelin, even when they are casually jamming in the back lounge of their tour bus.\nOver 300 days a year on a bus and a touring schedule rivaled by no one, they have been lazily labeled the \"hardest working band in rock and roll,\" but this band is anything but deserving of cliche branding.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A wonderful way to spend time with your little one.\nCasey is the creator and founder of Sensory Land. She came into her career working with children in 2009 when she became an IDTA Ballet Teacher. She is a trained and experienced Maternity Nurse and Breastfeeding specialist MNT. She has been trained in Baby Massage through IAIM and To Baby and Beyond. She also trained in Baby Yoga with To Baby and Beyond. Casey has worked as a Nanny since 2010. She studied British Sign Language with Signature through Oak Lodge school in 2015 and Tiny Talk in 2016.\nAlthough working almost full time as a nanny to Twins Jadie also assists during the Activity Classes on Friday's. She began working with children in 2007 when she became a sports coach and assisted in a nursery. Over the years Jadie has worked in many different settings including Nurseries, Mobile creches and Children's Centres.\nThe girls are technically aunt and niece but as there is only a 3 year age gap they find it easier to explain that they are cousins.\nBetween them they have 18 years experience working with children and families in various settings including; nurseries, creches, as nannies, as a maternity nurse and breastfeeding specialist and teaching sports and dance.\nBoth girls are DBS Checked, First Aid Trained and have been registered with Ofsted. As well as being trained and working with young children with learning difficulties.\nThe girls come from a very large family of people that all work within childcare (headmasters, SEN specialist teachers, primary teachers, nursery assistants and managers, children centre out-reach workers) so although Casey and Jadie are the faces you will see during classes there are a lot of other family members helping with things like proof reading, planning and EYFS support from the sidelines.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"better solution to customers, in terms of finished properties and total cost.\nwaste, and the eliminate the use of chemicals.\nby high-tech experience and precise process controls, guarantee satisfaction for all.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Is your business website working hard day and night enabling you to reach your full marketing and sales goals?\nA website is one of the most important sales tools for any business, however many don't know the secrets to leveraging its full potential. Your business website should attract and convert prospects enabling you to form relationships and create leads.\nBeautiful, functional business websites, designed to bring in leads and form relationships.\nSounds interesting, but don't know where to start?\nLet our expert web team take you through our results-focused approach to websites. So you have a website that's not only beautiful but high-performing.\nFeeling a bit unsure what to do to improve your website? Brand, SEO, lead generation, content marketing, conversion. It's a lot to handle.\n\u00b7 See how our business websites have made the difference for our customers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The members of the Unified State System on Preparedness and Response to the Emergency Situations on the table top exercise.\nOn 29th and 30th of March 2018, United Nations Development Programme (UNDP) in Tajikistan jointly with Committee of Emergency Situations and Civil Defense (CoES) under the Government of the Republic of Tajikistan conducted two days simulation exercise on response coordination to the large scale disaster for the twenty three Governmental entities members of the Unified State System. The main goal of the exercise was to enhance knowledge and capacities, examine preparedness of the government entities, and review available resources and emergency plans in managing complex and large scale disasters on the Government level. The table top exercise scenario presented helped to simulate response of an earthquake that damaged 18 small and large cities in Tajikistan.\nThe Head of Search and Rescue Department of CoES, Colonel Mr. Oleg Pilkevich noted the importance of districts authorities preparedness and knowledge of disasters along with the availability of all assets in the districts which need to be deployed in case of disasters. Simulation exercise helped to reveal the existing gaps and challenges, which need to be considered and further eliminated.\nParticipants were able to build their knowledge on response and plan actions in case of large scale disasters. In specific, the simulation exercise focused on coordination of activities of governmental entities on organization of response activities, dealing with search and rescue, and managing international assistance.\nThe event was organized under the \"Strengthening Preparedness and Response Capacities\" project funded by the Russian Federation \u2013 United Nations Development Programme Fund for Development with the facilitation of Russian Civil Defense and Emergencies Research Institute of the Ministry of Emergency Situations of Russian Federation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Morgan Stanley Private Equity Asia specializes in privately negotiated minority investments in companies with substantial business operations in the Asia-Pacific region. We target fast growing businesses with strong brands, sustainable competitive advantages and proven track records of financial performance. On a selective basis, we also pursue control investments where we believe we can add significant value to the business. Our investment process is geared towards generating returns through intrinsic value creation, as opposed to employing excessive financial leverage or relying on continuously increasing public market values.\nOur team invests across sectors, particularly in the areas of consumer products, industrial products, financial services, healthcare and telecom\/technology. Companies based in China and Korea have received a significant percentage of the total dollars we have invested since our inception, but we actively seek geographic diversification within our portfolios and we have executed deals in most major countries in the region.\nOur team consists of over 60 investment professionals who work from our offices in Hong Kong, Beijing, Shanghai, Seoul, Mumbai, Tokyo and New York. We believe that our association with Morgan Stanley, a leading global financial services firm with a significant presence in the region, provides our portfolio companies with unparalleled access to industry experts, managerial talent, intellectual capital and operational resources to help them execute their business strategies.\nWe believe the underlying factors driving Asia's growth and achievement of scale are long-term in nature and will result in the development of world-class companies and attractive investment opportunities.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Welcome to the home of Ten Thousand Flower-Flames, part 24 by Sri Chinmoy.\n2313. You have not failed!\n2340. Do you want to be happy?\nThis is the 476th book that Sri Chinmoy has written since he came to the West, in 1964.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Police Service is directed by the D.O.P.S. Police Commission, which consists of one member from each of the Local Police Committees.\nPresently there are 6 communities receiving policing services from D.O.P.S. Under the Police Services Agreement, they are known as the \"Participating Communities\". Each participating community within D.O.P.S. provides a representative for the Police Commission. Presently there is to be 6 members on the Commission that are selected from each of the Participating Communities. Each member is not to hold another elected office such as Chief or Councillor or can not to be employed by D.O.T.C. or members of D.O.P.S.\nPolice Commission members ensure that D.O.P.S. is responsive to the culture, priorities and needs of the Participating Communities and guide D.O.P.S. to ensure a proper and quality level of policing services is delivered by D.O.P.S. The members also ensure that the police service is free from inappropriate political influence. This ensures the participating communities' priorities, issues and concerns related to public safety and policing are identified and progress is monitored.\nThe Police Commission met quarterly in 2011 and regularly now. The Police Commission is provided training on Board Rules, Regulations and Liabilities and is provided a Police Commission Policy & Regulations handbook when they are appointed to service.\nThe Local Police Committees identify local policing needs, develops community-based strategies, including crime prevention programs, They also liaise with the local Dakota Ojibway Police Service Sergeant and\/or Corporal, and makes representation on the regional Police Commission on matters under its jurisdiction. The D.O.P.S. Police Commission was empowered on July 10, 1997 by the Council of Chiefs of the Dakota Ojibway Tribal Council.\nLocal Police Committees have been established in each First Nation community policed by D.O.P.S. and consist of one chairperson and three or more representatives chosen by that community.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Teen Book Festival was amazing. There were so many teens! There were FIFTY ONE BUSES that brought over 1500 people to the festival. The authors were incredible and the volunteers were brilliant. Overall, I had a blast and I'm pretty sure that everyone else did too.\nBTW ... That cup was one of the fabulous things that you could purchase, and if you had, you would have been able to catch up with me, since I was at the merchandise table all day.\nOne of the few downfalls to being on the committee is that you don't really get to \"hang out\" and check out the authors' presentations. I would have liked to, but seeing everyone's enthusiastic faces was worth it in the end.\nIts getting closer! Saturday is only a day and a half away! MORE EXCLAMATION MARKS!!!!!!!!\nI am so excited I can hardly contain myself (it is hard to tell, but I am excited). Currently I am wearing the adult volunteer shirt of sky blue and yesterday I wore the lilac purple shirt for the Nazareth College volunteers, but Saturday? Saturday I will be wearing mint green, which is the committee color. I am so proud to be wearing that color.\nAll the planning and working throughout the year has made it to then end. It all became more real when I picked up Brendan Kiely at the airport. He was super awesome and easy to talk to. Plus, he has family in Rochester. Even more awesome? His Mom went to Nazareth Academy, which is where I went to high school! Totally got me all spiked and ready to go. Which is good, because today and tomorrow its all about setting up Nazareth College. Totally exhausting, but also totally exhilarating. Get a little coffee and I am READY! I hope to see you all there!\nWhat do you call an 8-12 year old?\nWe've been thinking of doing programs specifically for 8-12 year olds and we realized that they aren't tweens, they aren't teens ... what are they? Everyone has a \"code name\". We have preschoolers and toddlers. Tweens and teens. But what are these kids? Just kids? That's no fun! We want a fun name for them too! What kind of name should we give them (that's appropriate of course) that would represent these kids?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Lipid mediators have complex effects on the cell; one of the key transcriptional factors that moderate proliferation and inflammatory effects is PPAR\u03b2\/\u03b4. Following highly successful clinical trials using the PPAR\u03b2\/\u03b4 agonists GW501516 for treatment of diabetes, GSK announced that any further research would be discontinued due to preclinical trials in rodents which linked this drug to wide spread tumour development. In this review we outline the dual molecular functions of PPAR\u03b2\/\u03b4 and connect these to the diverse results from in vitro studies, and draw parallels with the outcomes of animal and human studies. The PPAR\u03b2\/\u03b4 agonists have a great potential in terms of therapy, and we hope to provide some insight into the reasons why such contrasting results have been published. The discussion presented here is important to the future development of PPAR\u03b2\/\u03b4 agonists for the clinic, and for a fuller understanding for their complex regulatory roles in the cell.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Two late goals from Moroka Swallows resulted in a 2-1 defeat for Maritzburg United at the Harry Gwala Stadium on Wednesday night.\nIt was a first home defeat of the season for the Team of Choice, who ended the day in sixth position on the Absa Premiership standings.\nThe visiting side had the first chance of the game in the fifth minute, but Vuyisile Wana was shot wide of goals.\nMaritzburg managed to seize back the initiative, but Kwanda Mngonyama and Terrance Mandaza were unable to keep their shots on target.\nOn 28 minutes United opened the scoring through Thokozani Sekotlong, who provided a neat finish past Greg Etafia after great set-up play by Nhlanhla Vilakazi.\nA set-piece opportunity for Maritzburg almost saw them double their lead a few minutes before the break, but Ghanaian defender Mohammed Awal put his header over the crossbar.\nThe Birds again created a scoring chance for Wana soon after the restart, but once more he failed to draw Virgil Vries into action.\nSwallows were on level terms when Siyabonga Nomvethe headed home Luvhengo Mungomeni's 84th minute cross to make it 1-1.\nNomvethe completed the comeback when he scored from a Lerato Chabangu assist in stoppage time.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"JEAN CANOY FERGUSON, age 84 of Sidney, IA died Sunday, March 24, 2019 at her home unexpectedly. Jean was born April 30, 1934 in Los Angeles, CA to Jack Charles Crawford and Yvonne Marie (Lowe) Crawford.\nJean was preceded in death by her parents; husband, Melvin Ferguson; and a brother, Jack Crawford, Jr. She is survived by her daughter, Karen Rickerson and husband Kenneth of Sidney, IA; son, Gregory Ferguson of California; 12 grandchildren, Naomi, Candice, Kevin, Eric, Daniele, Adriana, Andrea, Aisli, Triston, Steven, Melissa, and Gregory, Jr.; 13 great grandchildren, Kailyn, Elijah, Andrew, Nicholas, Matthew, Kimberlie, Adam, Caleb, Nyelle, Jordan, Lucas, Liam and Izel; sister, Georgia Olvera of California; nieces, nephews, many other relatives and friends.\nTo plant a tree in memory of Jean Ferguson, please visit our Tribute Store.\n\"Email Address\" would like to share the life celebration of Jean Ferguson. Click on the \"link\" to go to share a favorite memory or leave a condolence message for the family.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Exceptional cat sitting service in Milton Keynes.\nand let us do the pampering while you're away.\nCall 07843 711075 to see how we can help.\n& water bowls. Safe and secure pick-ups & drop-offs.\nwhile offering you the peace of mind you deserve.\nservice and first class communication.\nAward winning pet care in Milton Keynes. Our full range of services include: home boarding, puppy visits, doggy day care, cat visits and dog walking services in Milton Keynes, Stony Stratford, Newport Pagnell and the surrounding villages of Bow Brickhill, Woburn Sands, Aspley Guise and Husbourne Crawley.\nRegistered, Insured, Knowledgeable and Experienced, we take the job of pet care seriously. So whether you're looking for a cat sitter for your weekend away or regular runs for your dog while you're at work, we can help! Get in touch to book your pet sitter and dog walker in Milton Keynes today.\nWITH OVER 85 FANTASTIC 5* REVIEWS ON FACEBOOK & GOOGLE!\nEver wondered what a Jog My Dog day is like for the dogs we walk in and around Milton Keynes? We have a video to show you! So sit back and get a glimpse into our day to day activities, showcasing the fun all the dogs have \u2013 and our team have too!\nWe offer varied and stimulating dog walks for your furry friend, either in a group or on their own. So for professional and registered dog walkers in Milton Keynes that you can rely on, look no further than Jog My Dog.\nOur Milton Keynes pets at home sitting service gives customers a competitively priced alternative to kennels. Guest dogs are welcomed into our homes and treated as family. Owners get peace of mind through daily picture updates.\nWe can look after your cat, rabbit or other small animals in the comfort of your own home. Our cat feeding service in Milton Keynes is particularly popular, alongside pop-ins for all other furry and not so furry animals.\nKeep your dog entertained while you're at work with our fun, social doggie day care in Woburn Sands. With a large garden, lots of fun walks and attention all day long \u2013 we have the perfect solution for all of your dog day care needs.\nFor your security we hold a full certificate of insurance for every aspect of our business and are DBS checked and trained in Canine\/Feline First Aid. We are also proud members of The Pet Professionals Guild and the National Association for Registered Pet Sitters.\nCool pics taken after the dogs have been out, and you get lots of handy tips and tricks as well about all sorts of things from helping your dog and cat to get along to making home made treats!\nExcellent, reliable, and attentive care.\nI love getting a map of where my pups have been and what they've been up to! I'm not in town too often, but this service is absolutely my go-to.\nHighly recommend. Run by a very caring, compassionate person who has excellent knowledge and understanding, along with a huge amount of experience.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Prominence of the Shrine : Among the 108 Dhivya Desams this is 12th Dhivya Desam. Thirumal (Maha Vishnu) arrived here by Vaidheega Vimanam carrying a bow called 'Saarangam' on the Mahara Sankaranthi Day and married Lakshmi who was reared as the daughter of Homa Maharishi. While the Aradhana Idol brought by Vibhishana (Ravana's borther, who inherited Lanka after Ravana was killed by Rama) from Rama settled deep at Srirangam as 'Prana Vakruthi', the 'Vaidheega Vimana' settled at Kumbakonam as Sarangabani. This shrine is celebrated as 'Ubaya Pradana Dhivya Desam'.\nPoets who sang its praise : Peyazhwar, Bhoodhathazhwar, Sri Andal, Thirumisai Azhwar, Nammazhwar, Thirumangai Azhwar (Total 52 songs).\nThis is 12th temple among the 108 Divya Desams. Lord Vishnu married Goddess lakxhmi on a \"mahara shankrathi\" day at this location.\nThis temple is located in Kumbakonam to Tanjore road and it is 1 KM away from the Kumbakonam Bus stand.\nKumbakonam is called as Temple town. Devotees should visit this temple and devoting themselves will bring peace and harmony in their lives. Mythological believe is if the students worship here will perform better in their exams.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"If you move here, bring your car\"\nThis Astoria, Queens neighborhood has long been known for its Greek population, but currently there is an incredibly diverse European population residing in the area. This is a safe place, and the elevated train will get you to midtown in a matter of twenty minutes or so, so it's perfect for young people who don't want to pay much rent but don't want to be too far from the big city life, or from their corporate jobs. There are restaurants and happenings here that reflect the neighborhood's diversity \u2013 such as the Beer Garden and the Bohemian Hall, and the streets tend to be lined with row houses and neatly trimmed hedges.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Share this 6 minute video with anyone who negotiates a hotel meeting contract. This will prepare them for some ridiculous policies that should be challenged BEFORE signing the contract. Congratulations to Travis Reedy www.travisav.com for the insight. I couldn't have done a better job.\nClick \"follow\" to be notified of new posts.\nNumber 1 answer to why the projection screen is too low?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I didn't know where to send a photo, but I'm a member of Middle Tennessee Electric and see so many member photo submissions in each issue of The Tennessee Magazine. I wanted to try my hand at submitting some. I\u2008don't know if you'd consider using this for \"Point of View\" or the cover or whatnot. Should you choose to use it for the \"Point of View\" article, let me know and I can type up the full story about what I do.\nThe attached photo is of the Perseid Meteor Shower that I photographed on a backroad close to my house near Mt. Juliet. Few people stay up to witness the magic of a meteor shower at its peak, so I try to capture it for them.\nEditor's Note: We normally only use reader-submitted photos in conjunction with Shutterbug contests, but this one deserves a mention. Thanks for sharing your meteor shower image with us. Be sure to look for the next Shutterbug contest entry form in the June issue.\nI just finished reading about it being strawberry time in Tennessee. What I did not find was any mention at all of Humboldt's West Tennessee Strawberry Festival, which is always held the first week in May. This festival has been going on for as long as I can remember, and I am 60 years old. It is really a big deal for us folks in the middle of West Tennessee. I just think there should have been some short mention of something this important that ties into your article, especially since you are a Tennessee magazine for Tennesseans.\nWe rely on event organizers to send us their information. Listings are free, so we can't guarantee placement in the magazine, but all events will be listed on our website, www.tnmagazine.org.\nWe try to publicize events during the month in which they are held. You will find Humboldt's famous festival listed on page 30 of this magazine.\nYou can find an online form to enter your event information at www.tnmagazine.org\/events\/ submit-event\/add.\nI'm writing in regards to a poem written by Samantha Rosencrants in your April edition \u2014 \"5 Senses.\" You forgot to write about the sixth sense: Reading something so wonderful! My, how much pleasure it brought! You've got a real talent, there. Keep it up!\nI was looking for the April poetry contest runners-up, but I could not find them. Is there a certain place I should be looking?\nWhere can I find the poetry runner ups for the March issue?\nEditor's Note: The winners and runners-up in the Poet's Playground contest can be found on our website at www.tnmagazine.org\/tag\/poets-playground.\nCan an individual subscribe to the magazine?\nI'd like to get your magazine but don't want to create a PayPal account. Are there any other options? Thank you. We're thinking of retiring in Tennessee.\nYou also have the option of calling our office at 615-367-9284 and paying for a subscription using your MasterCard or Visa.\nFinally, if you prefer to pay using PayPal, you can go to our website, www.tnmagazine.org, and order a subscription.\nI would like to know when your photo contest are ?\nWe will announce a new one in October or November. We hope you will enter.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Macon, GA \u2014 The official selections for the Narrative Feature and Student Shorts competition for the 2016 Macon Film Festival (July 21\u201324) have been announced.\n\"Once again we had a competitive submission process, and these films rose to the top. They exhibit excellent quality and a story line that we feel will resonate with our festival audience,\" said Julie Wilkerson, Macon Film Festival board president.\nIn addition to those announced today, the festival presents Narrative Shorts, Documentary, LGBT, Southern and Music categories. Those announcements are forthcoming.\nAutumn Fall \/ Norway (Director: Jan Vardoen) \u2013 Ingvld works as a lightning technician at the National Theatre but dreams of writing for the stage. However she can't stand actors. She still manages to entangle herself with two thespians, one a notorious hell raiser and she embarks on an exciting, scandalous, and ultimately very dangerous journey that changes her life forever.\nEnclave \/ Serbia\/Germany (Director: Goran Radovanovic) \u2013 Ten years after the war in Kosovo and unfolding through the eyes of a lonely lad, this is a poignant and intense coming-of-age story that shows young boys in their struggle to follow the blood feuds of their parents and questions whether co-existence is possible between two communities.\nJasmine \/ Hong Kong\/United States (Director: Dax Phelan) \u2013 A year after his wife's murder, once-successful Hong Kong businessman Leonard To is still reeling from the tragedy. Having lost his job, friends and all sense of order in his life, Leonard becomes obsessed with a mysterious stranger he sees at his wife's grave, believing him to be responsible for her death.\nJeric0 \/ United States (Director: Seckeita Lewis) \u2013 Jerico and Jarvis set out on the morning of the signing of the civil rights act of 1964 with aspirations to capture a newly available promotion. With the dangers of the Jim Crow South, a simple trip to work becomes a fight for survival. JERICO is a dramatic comedy that satirizes the dogma of our American history in an effort to overcome racial division.\nManifest Destiny \/ United States (Director: Anthony Parisi) \u2013 This musical comedy weaves together both sharp satire and heartfelt storytelling into a grand adventure. The film follows gruff explorer William Clark and his naturalist companion Meriwether Lewis as they blaze a trail to the Western waters. Along the way they encounter the lovely Sacagawea, battle the elements, and find that President Jefferson has some surprises up his sleeve.\nRemittance \/ Singapore (Director: Patrick Daly & Joel Fendleman) \u2013 REMITTANCE follows Marie, a foreign domestic worker from the Philippines as she struggles to cope with demanding employers, long hours of work, and separation from her family. Breaking from the conventional image of maids as labor, the story explores the transformations Marie goes through as a woman dealing with conflicting obligations and aspirations.\nThe Rainbow Kid \/ Canada (Director: Kire Paputts) \u2013 Eugene, a young man with Down syndrome, sets out on a life altering journey to find the end of the rainbow.\nThe Apology Service \/ USA \/University of North Georgia (Director: Luke Pilgrim) -Set in the near future, Jackson is the top \"apologist\" at a personalized apology firm. Jackson is a loner with his only companion being a quiet robot named AL-X or Alex. Along the way Jackson becomes overwhelmed by romantic daydreams of a friendly co-worker. This shatters his identity and causes him to question his career choice and his reclusive lifestyle.\nBird Dog \/ USA \/ New York University (Director: Katrina Whalen) \u2013 Rose is happy at home hunting birds with her family. When she rises to her brother's dare and swallows a dove heart, she worries that the heart in her belly will sprout into a fully formed dove. The conviction that she is carrying a bird alters the hunting experience, and drives her to confront her fears about growing older.\nConfessions: 5pm or by appointment \/ USA \/ Savannah College of Art & Design (Director: Joseph Brennan) \u2013 A young Catholic priest is approached by his lover in the confessional to discuss where the relationship is headed.\nKarkass Carts \/ USA \/ New York University (Director: Ben Nelson) \u2013 When the Birch brothers inherit a failing funeral home from their decaying father, they must resort to sinister means in order to save the family business.\nRidgeland \/ USA \/ Savannah College of Art & Design (Director: Taylor MacDonald) \u2013 A restless young man lives aimlessly in a rural town, wanting desperately to escape his dead-end life. As he escapes to the North, life catches up.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What is the best price for a return flight from Panama City Paitilla to Buenos Aires Ministro Pistarini?\nHow many airlines fly direct from Panama City Paitilla to Buenos Aires Ministro Pistarini?\nThere are 0 airlines who fly direct from Panama City Paitilla to Buenos Aires Ministro Pistarini.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For the first time, a team of researchers has found a specific place in the human genome that raises a person's risk of erectile dysfunction. The discovery is a significant advancement in the understanding of the genetics underlying erectile dysfunction and highlights a target for the development of new treatments that could help men who don't respond to current drugs.\nErectile dysfunction, the inability to obtain and maintain an erection sufficient for sexual activity, is a common and costly condition of men of primarily middle and older ages. The disease is linked to many causes, such as neurological, hormonal and vascular factors.\nTherapies based on these factors exist, but many men don't respond to them. Genetics are also suspected as a factor in about one-third of erectile dysfunction cases, but researchers have failed to make an association with any specific genomic locations until now.\nThe new study found that variations in a genetic locus near the SIM1 gene are significantly associated with an increased risk of erectile dysfunction. The researchers ruled out that the risk was due to other known risk factors for erectile dysfunction, including body mass index (BMI) or differences in how men describe their erectile dysfunction. The study also demonstrated a biological role for the locus in regulating sexual function, strongly suggesting that these variations can cause erectile dysfunction.\nThe study, \"Genetic variation in the SIM1 locus is associated with erectile dysfunction,\"will be published online on October 8 in the journal Proceedings of the National Academy of Sciences.\nThe researchers conducted a genome-wide association study in two large and diverse groups to investigate genetic contributors to the risk of erectile dysfunction. The first group included 36,648 men from the Genetic Epidemiology in Adult Health and Aging (GERA), which is part of the Kaiser Permanente Research Program on Genes, Environment and Health (RPGEH). This group included male members of Kaiser Permanente who completed a survey on their condition, had an electronic health record-based clinical diagnosis of erectile dysfunction and had used drugs or other erectile dysfunction treatments. The findings in the GERA group were then verified in a group of 209,758 men from the U.K. Biobank.\nThe study found that variations in the SIM1 locus were associated with a 26 percent increased risk of erectile dysfunction. This risk was independent of known erectile dysfunction risk factors. The association was replicated in the U.K. Biobank sample, providing strong confirmation of the findings.\n\"This significant advance in our understanding of erectile dysfunction is made possible by the unique ability at Kaiser Permanente to link detailed questionnaires, electronic health records and genetic data on such a large population,\" said the study's senior author, Stephen Van Den Eeden, Ph.D., a research scientist at the Division of Research.\nThe study looked to see whether the SIM1 locus was a risk factor when considering differences in how men reported their erectile dysfunction to their doctors. The study found that the SIM1 locus was a risk factor for erectile dysfunction, whether the disorder was defined through clinical diagnoses, prescriptions history or study participant self-report.\nThe study then identified a biological role for the genetic risk locus in erectile dysfunction susceptibility. The SIM1 gene is known to be part of a signaling pathway that plays a central role in body weight regulation and sexual function. The erectile dysfunction locus is located near, but not in, the SIM1 gene. Members of the research team at the University of California, San Francisco, were able to show that the risk locus physically interacts with the promoter of the SIM1 gene, and that variants in this locus alter the function of a master gene regulator, called an enhancer.\n\"The different bits of evidence that we present in this study fit together like puzzle pieces to create a picture of how the SIM1 locus can control erectile function,\" Jorgenson said.\nThe study highlights the potential of SIM1 as a target to develop new treatments for erectile dysfunction. Future work will focus on the SIM1 locus in a younger population, because they may be the most likely to have a large genetic contribution to erectile dysfunction.\nThe work received funding from the Robert Wood Johnson Foundation, the Wayne and Gladys Valley Foundation, the Ellison Medical Foundation, Kaiser Permanente Community Benefit Programs, the National Institute on Aging, the National Institute of Mental Health, the National Institute of Health Common Fund, the National Institute of Diabetes and Digestive and Kidney Diseases and the National Eye Institute.\nCo-authors on the study from the Division of Research include:Jie Yin, Jun Shan and Khanh K. Thai. Co-authors from the University of California, San Francisco, include Navneet Matharu, Thomas J. Hoffmann, Xujia Zhou and Nadav Ahituv. Co-authors from the University of Washington School of Medicine include Melody R. Palmer and Gail Jarvik.\nThis research was conducted at the Kaiser Health Division of Research and University of Washington. The release was adapted from the original article published by Kaiser Health Division of Research.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbbxkc b/data_all_eng_slimpj/shuffled/split2/finalzzzbbxkc
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+{"text":"Salmon Arm Branch Renovation Closure: The Salmon Arm Branch is closed for renovations January 2-9. We will be open for limited services starting Thursday, January 10: Customers can pick up holds and return items, but the rest of the library will be closed. The library will RE-OPEN to the public on Monday, January 14. Find out more.\nLumby Branch Maintenance Closure: Due to building maintenance, the Lumby branch is closed and is expected to RE-OPEN on Tuesday, January 15.\nTuesday: 1:00 p.m. to 8:00 p.m.\nWednesday: 1:00 p.m. to 8:00 p.m.\nThursday: 10:00 a.m. to 5:00 p.m.\nFriday: 10:00 a.m. to 5:00 p.m.\nThe Revelstoke Branch is located in the Community Centre at 605 Campbell Avenue.\nThere are two book slots to the left of the main entrance of the library along the Campbell Avenue side of the building in which to return books at any time (except during the Easter and Christmas closures).\nFree parking is available in the Community Centre parking lot on the south side of the building.\nFor: Children 3 and younger will enjoy this program but all ages are welcome.\nFun interactive literacy activities with lots of stories and songs!Please call or come in to register for this free program or just drop in.\nFor: Children 3 and older is preferred.\nHave fun creating fantastic creatures and structures according to the monthly theme! Use the library's huge Lego collection and your creation will be displayed in the library!\nNo registration is necessary for this free program.\nDates: 4th Saturday of every month 10:30 to 4:30 PM .\nWe provide the materials and instructions - you do the creating! Create a craft or piece of art to take home.\nFor: Children of all ages. Children under 4 years of age should be accompanied by a caregiver.\nExplore the library's STEAM collection: Snap Circuits, Keva Planks, Ozobots, and more! Stations will be set up for children to learn and explore.\nFor: All ages. Children under 7 years of age should be accompanied by a caregiver.\nDates: 2nd Saturday of every month 10:30 to 4:30 PM.\nBoard games for teens in our meeting room.\nLearn how to download eBooks, audiobooks, digital magazines; how to access any of the ORL's digital resources; or how to use the library's website and catalogue, or get help with using email, and social media.\nPlease phone 250-837-5095 or email revelstoke@orl. 88 fortunes online www.fashionparkway.com to set up a half hour one-on-one appointment.\nDo you knit, crochet, cross-stitch, needlepoint or latch-hook? Whether you are experienced or just beginning to learn, come and join the fun (with tea provided) as we learn together! Participants are asked to bring their own supplies.\nThis is a free program with no registration required.\nDates: Every Tuesday from 6:30 to 8:00 PM October to May.\nWriter's who are working on projects are welcome to come and share ideas, give constructive feedback, and be inspired at this weekly get-together.\nPlease bring some writing in progress to share with this supportive group of writers!\nWe currently have a bookclub that meets on the 3rd Tuesday of every month at 1:00 PM (please leave your name with us if you would like to join).\nPlease let us know if your bookclub would like to meet at the library - we would love to have you meet in our quiet meeting room with kettle, coffeemaker, and any furniture you may require.\nWe would also love to suggest and order in the books that you require for your book club or search our 88 fortunes onlinecatalogue for bookclub kits that we carry.\nWe offer one hour per day of free internet access on our three computers for members of the library as well as non-members.\nWe also offer unlimited and free internet access for anyone with their own wireless device.\nLearn how to download eBooks, audiobooks, digital magazines; how to access any of the Okanagan Regional Library's digital resources; or how to use the library's website and catalogue.\nPlease phone 250-837-5095 or email revelstoke@orl. 88 fortunes online www.fashionparkway.com to set up a one-to-one appointment.\nWe provide free proctoring services in the library for library members during any of our open hours. We have free and unlimited Wi-Fi accessibility for your online exams if you have your own device\/laptop, or you are welcome to use our public computers.\n(at least 7 days before you would like to write the exam) to set up a time.\nPlease feel free to use our quiet meeting room for tutoring, studying, formal or informal gatherings when it is not being used for our own programs and events.\nThe room accommodates approximately 30 people and is available for booking ahead of time. We have chairs and tables available for use as well.\nPlease call us at 250-837-5095, or email revelstoke@orl.bc.ca, or come in for more details.\nYou can print from our public computers for 25 cents per page.\nPhotocopying is 25 cents as well and we can do it for you or you can help yourself.\nOur photocopier is capable of making black and white double-sided copies, enlarging\/reducing, and sorting.\nWe can scan documents as well. There is no charge for scanning.\nWe would love to come to your class for a visit or have your class come to the library for a tour, a book-talk, or a storytime!\nThe art of Karen Millard's watercolour paintings are on display in the library until December 2018.\nPlease contact us if you are interested in displaying your artwork.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For the recent crowd who has been following the Wildcat soccer success, Aaron Harper is known as the head coach of the Pendleton County High School boys' soccer team that won their fourth consecutive 38th District title. He is also the father of Conner and Aven Harper, the two top scorers in Wildcat soccer history and the subjects of a debate about which is the greatest soccer player in school history.\nBefore his success as a coach, Coach Harper was also successful as a player. His performance during the fall soccer seasons from 1992-1994 has led to his selection as a member of the 2018 Northern Kentucky Boys' Soccer Coaches Association Hall of Fame.\nHe is the first player from Pendleton County to be inducted, and he joins his coach, Tony Bentley, as the only Pendleton County members in the hall of fame.\n\"I am honored to be selected. Playing for Coach Bentley laid the groundwork for my love of the game, and being recognized is truly special,\" said Coach Harper.\nAccording to Coach Bentley, Harper was the first All-State player for Pendleton County.\nHe played defense for Bentley's Wildcat teams, but his role was more than that.\nBentley pointed out that Harper's success came from his intelligence on the field.\n\"He was a thinking player who knew all of the angles. He had great anticipation and was the smartest player I ever had,\" he added.\nWhile he was a stalwart on defense, his skill allowed him to also affect the offensive end. Having a great touch on his passes, he would play the ball forward, leading to goals for his teammates.\nAccording to Bentley, Harper had 10 goals in that All-State season, but he also led the team in assists from a defensive position.\n\"Aaron set up teammates with passes. Forty yards out and, boom, the pass is right where his teammate needed it,\" said Bentley.\nBoth Harper and Bentley hope that their hall of fame selections lead to the proper recognition of several deserving Wildcat Soccer players.\n\"There have been multiple players before and after who have been far better than I,\" Harper humbly said. Two of them are his sons who are among the leading scorers in the history of the school, region and NKY.\n\"To be able to play, coach at PC, and see my two sons go through the program is something that will forever be in my heart. Soccer has been great for my family.\" said Harper.\nAnother hall of fame selection for him as a coach might be down the road, and his former coach says he deserves it.\n\"He coached like he played. He was a thinker, never dirty, just class,\" Bentley said about his former player and now fellow hall of fame member.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Though the app grants you every indication that subscribing grants you access to the issues on the app, it in fact does not. While this is illegal, there is nothing I can do about it.\nThe issues will not load when I've downloaded them, so basically I just paid for nothing. $30 down the drain..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Located in the Unknown Regions of the galaxy, Ilum is a dark arctic world orbiting far from its star. The entire surface of the planet is covered in ice and snow. Hidden beneath Ilum's frigid surface is a vast network of naturally occurring crystal caves. The gems found within these caves are not ordinary crystals but kyber crystals, powerful sources of energy most commonly used to power lightsabers. Despite the dangerous environment many Jedi travel here to complete their rite of passage. With the Sith Empire discovering the world, Ilum has become a war zone where Jedi and Sith fight each other for these precious resources, and they both battle the elements to survive.\nRecent intelligence, however, has suggested that the Sith's interest in Ilum extends far beyond simply harvesting Lightsaber crystals. Watching the Empire devote so many resources to operations on this remote world, Republic leaders and members of the Jedi Council have begun to question whether they might have surrendered a resource of far greater value than anyone ever imagined.\nAs place where Jedi initiates harvested their crystals by attuning themselves to the Force, anyone trying to make the long journey to Ilum should be prepared to confront dangers well beyond the unknown. Add the dangers of the Sith and Jedi conflict and your odds of returning from this world is next to none.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The makers of Telugu superstar Allu Arjun's next, Duvvada Jagannadham, have been gradually building the hype around this commercial potboiler in which Arjun will be seen romancing Pooja Hegde.\nThey have now released a one-minute promo for another song from the film, called 'Gudilo Badilo Madilo Vodilo'. Arjun and Hegde are seen shaking a leg on this peppy number composed by Devi Shri Prasad. While their chemistry stands out as the USP of the song, the tacky Photoshopping and CGI take away from the quality of the song. As a result, the sets come across as synthetic and the entire package too forced.\nBoth the stars don three colour-coded outfits during the one minute promo. Hegde looks gorgeous, aces the comical act, and the dance moves. Arjun channels his star power throughout and looks extremely glamorous.\nThis song comes after the chartbusting popularity of the first song from the album that the makers released on 22 May \u2014 'Saranam Bhaje Bhaje'. The makers are also planning a grand audio launch in the run up to the release of the film on 23 June.\nDuvvada Jagannadham, or DJ as fans call it, is directed by Harish Shankar.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbccor b/data_all_eng_slimpj/shuffled/split2/finalzzzbccor
new file mode 100644
index 0000000000000000000000000000000000000000..96807af4ae58563a968f9802be3b748e31430e0a
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@@ -0,0 +1,5 @@
+{"text":"The average volume for Niska Gas Storage Partners has been 81,600 shares per day over the past 30 days. Niska Gas Storage Partners has a market cap of $521.1 million and is part of the utilities industry. Shares are down 4.3% year-to-date as of the close of trading on Wednesday.\nAlto Palermo (NASDAQ: APSA) shares currently have a dividend yield of 8.50%.\nAlto Palermo S.A. engages in the ownership, acquisition, development, leasing, management, and operation of shopping centers, as well as residential and commercial complexes in Argentina. The company has a P\/E ratio of 11.19.\nThe average volume for Alto Palermo has been 1,600 shares per day over the past 30 days. Alto Palermo has a market cap of $680.3 million and is part of the real estate industry. Shares are down 2.8% year-to-date as of the close of trading on Tuesday.\nTheStreet Ratings rates Alto Palermo as a hold. The company's strengths can be seen in multiple areas, such as its expanding profit margins, increase in stock price during the past year and largely solid financial position with reasonable debt levels by most measures. However, as a counter to these strengths, we also find weaknesses including deteriorating net income and weak operating cash flow.\nThe gross profit margin for ALTO PALERMO SA is rather high; currently it is at 62.40%. It has increased from the same quarter the previous year. Regardless of the strong results of the gross profit margin, the net profit margin of -1.87% is in-line with the industry average.\nALTO PALERMO SA has experienced a steep decline in earnings per share in the most recent quarter in comparison to its performance from the same quarter a year ago. This company has not demonstrated a clear trend in earnings over the past 2 years, making it difficult to accurately predict earnings for the coming year. During the past fiscal year, ALTO PALERMO SA increased its bottom line by earning $1.45 versus $1.19 in the prior year.\nThe company, on the basis of change in net income from the same quarter one year ago, has significantly underperformed when compared to that of the S&P 500 and the Real Estate Management & Development industry. The net income has significantly decreased by 104.1% when compared to the same quarter one year ago, falling from $13.99 million to -$0.58 million.\nNet operating cash flow has significantly decreased to $11.09 million or 65.22% when compared to the same quarter last year. In addition, when comparing to the industry average, the firm's growth rate is much lower.\nYou can view the full Alto Palermo Ratings Report.\nHarvest Capital Credit (NASDAQ: HCAP) shares currently have a dividend yield of 9.60%.\nHarvest Capital Credit LLC is a specialty finance company providing structured credit to small businesses. Harvest Capital Credit, LLC was founded in 2011 and is based in the United States. The company has a P\/E ratio of 18.17.\nThe average volume for Harvest Capital Credit has been 16,200 shares per day over the past 30 days. Harvest Capital Credit has a market cap of $86.7 million and is part of the financial services industry. Shares are down 6.9% year-to-date as of the close of trading on Wednesday.\nTheStreet Ratings rates Harvest Capital Credit as a hold. The company's strengths can be seen in multiple areas, such as its robust revenue growth, impressive record of earnings per share growth and compelling growth in net income. However, as a counter to these strengths, we find that the stock has had a generally disappointing performance in the past year.\nHCAP's very impressive revenue growth greatly exceeded the industry average of 3.0%. Since the same quarter one year prior, revenues leaped by 70.9%. Growth in the company's revenue appears to have helped boost the earnings per share.\nHARVEST CAPITAL CREDIT CORP reported significant earnings per share improvement in the most recent quarter compared to the same quarter a year ago. This year, the market expects an improvement in earnings ($1.36 versus $0.47).\nThe gross profit margin for HARVEST CAPITAL CREDIT CORP is rather high; currently it is at 57.09%. Despite the high profit margin, it has decreased significantly from the same period last year. Despite the mixed results of the gross profit margin, HCAP's net profit margin of 68.01% significantly outperformed against the industry.\nIn its most recent trading session, HCAP has closed at a price level that was not very different from its closing price of one year earlier. This is probably due to its weak earnings growth as well as other mixed factors.\nWhen compared to other companies in the Capital Markets industry and the overall market, HARVEST CAPITAL CREDIT CORP's return on equity is below that of both the industry average and the S&P 500.\nYou can view the full Harvest Capital Credit Ratings Report.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In 2000, just beginning the 21th Century, an orchid geek decided to try a new mix of balanced fertilizer NPK 20-19-20, Thiamine or Vitamine B1 and Naphtylacetic Acid NAA, a growth hormone. After some years of testings and mistakes trying to find the right proportion of every ingredient the formula achieved the optimal balance between the three components to achieve their goal, orchids with strong roots and healthy green leaves and spectacular bloomings. The results were surprising.\nSomething important was born and needed a name, using old botanic books the appropriate name appeared in front of us, \"tahtso\" that in navajo language means \"big plant\".\nIn 2004 began the long process of patenting the formula and register also the name, after three years of trials, testing, certifications and demonstrations, the Spanish Ministry of Industry granted his International patent to tahtso in 2007, becoming one of the few fertilizers in the world that have an international patent.\nSince then, tahtso has been expanding among skeptics and professionals, to be recognized as one of the world's best fertilizer for orchids.\nToday tahtso is used by some of the best professionals in the world of orchids, for commercial growers and hobbyists, and even in some of the best botanical gardens in the world.\nWith factory based in Barcelona (Spain) and Distribution Center in Shanghai (China), today tahtso reaches every corner of the planet.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Address: 671 Tetra Ave, Moreleta Park, Gauteng, 0044, South Africa, Pretoria. See full address and map.\nAddress: 351 Peddie Rd, Wadeville, Gauteng, 1428, South Africa, Germiston. See full address and map.\nAddress: Joubert Pk, 10 Thirteenth Ave, Parktown North, Gauteng, 2193, South Africa, Sandton. See full address and map.\nAddress: 5 Bender St, Sunward Park, Gauteng, 1459, South Africa, Boksburg. See full address and map.\nAddress: 1607 SAGEWOOD MANOR, Johannesburg. See full address and map.\nAddress: 186 A Inanda Rd, Springfield Park, Kwazulu Natal, 4051, South Africa, Durban. See full address and map.\nAddress: 42 Old Power Station Noble St Southern Industria, Windhoek, Namibia, Namibia. See full address and map.\nAddress: House 148, Umlazi, 4066, South Africa, Kwazulu Natal. See full address and map.\nAddress: 90 Outspan Rd, City Deep, Gauteng, 2197, South Africa, Johannesburg. See full address and map.\nAddress: 279 Inanda Rd, Springfield Park, Kwazulu Natal, 4051, South Africa, Durban. See full address and map.\nAddress: 12 Lancaster Gate, 11 South Beach Ave, Point, Kwazulu Natal, 4001, South Africa, Durban. See full address and map.\nAddress: 2 Athlone Ave, Howick, 3290, South Africa, Kwazulu Natal. See full address and map.\nAddress: 329 Central St, Gauteng Central, Gauteng, 0002, South Africa, Pretoria. See full address and map.\nAddress: 12 McArthur Dve, Listerwood, Eastern Cape, 6001, South Africa, Port Elizabeth. See full address and map.\nAddress: 39 Isom Rd, Red Hill, Kwazulu Natal, 4051, South Africa, Durban. See full address and map.\nAddress: P.O. Box 1139, Swakopmund, Namibia, Namibia. See full address and map.\nAddress: 6 Golfkruin Villas 12 Nigthengale crescent Krufersdorp, Krugersdorp. See full address and map.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This book is comprised of over 100 illustrations, sketches, and bits of older and newer concept work from personal sketchbooks over the last four years. If you enjoy all manner of beasts, monsters, creatures, aliens, or just really strange animals, chances are this will be a great addition to your collection.\nAllison Theus is a very talented artist working in creature design and all kinds of fantastic artsy things in the world today. In my own head she's right up there with Terryl Whitlatch, and another favorite of mine Claire Wendling. There is no beast she has not conjured from the shadows no creature to far from reality to be rendered by her digital or traditional media. This book isn't mostly creatures it is a creature. A living breathing slick ink monster tearing it's way into the hearts and souls of admirers near and far. May she always be inspired and art driven to continue to create worlds of wonder and fantastic creations of the imagination. Keep on drawing girl! From now till the universe rises forth in every shape, form, and color. Be inspired. Be the Beast! The Beast of Oblivion.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Project Brief: Through the state of Florida's Art in State Buildings Program, the University of West Florida seeks proposals for existing or new public art to be installed permanently in or on the grounds of the University Park Center currently under construction. This building is a multi-use space that will be occupied primarily by the USHA Kundu College of Health, UWF Athletics, and FSU College of Medicine. A pedestrian boulevard will lead visitors around the southeast entrance of the building to the football field located on the north side.\nThe architects designed the interior collective spaces as large community areas to actively engage students and faculty. Tall ceilings and large windows have been incorporated with patterned floor finishes and brushed metal surfaces to increase the energy of the space. A relaxed style of furniture gives the areas on the first and second floor an informal atmosphere. The hope is that with the addition of artwork these multi-use community areas will facilitate on-the-spot discussions and the sharing of ideas. Two-dimensional work, stained-glass, or relief sculpture would be ideal for the interior space. The southern exterior has been designed to function as a public plaza and would be the primary location for sculptural work.\nArtists may submit letters of interest and qualifications, along with images or project renderings, to be considered for permanent installation. The call is open to all professional artists working in any two or three-dimensional medium. The committee is currently accepting proposals for both outdoor and indoor works of art. The artworks will be maintained and owned by the University of West Florida.\nProposals for outdoor work need to include materials and designs that are durable enough to last through potentially severe and hazardous weather, including tropical activity and high winds over a long course of time without extensive maintenance. Final installations must meet all current applicable codes. UV-resistant and rust resistant materials should be used. Artworks should be constructed to reasonably discourage theft and vandalism and must take into consideration the ground surrounding the piece. Include plans for nighttime lighting.\n*Please Note: The proposed artwork(s) is\/are intended to be installed permanently. Please allow for, and include, maintenance budgets and guidelines. Additionally, the budget is a \"project budget\" and is used for all related costs to the project including the selection, fabrication, installation, and maintenance processes.\nThis entry was posted in Exhibition Entries\/Call for Artists, Misc., Multi-Disciplinary. Bookmark the permalink.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"This staple-free stapler is safe for kids and easy to use. It allows you to easily attach 2 to 5 sheets of paper without using a single staple. Less wasteful and more economical. Great for schools, home and office. Set of 4 staplers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A man is expected to appear in court today (Monday) following an alleged serious assault in Fraserburgh.\nPolice say the alleged assault on another male occurred at about 12.20am on Sunday morning in the town's Saltoun Square.\nA police spokesperson said: \"Anyone who was in the area around this time, and believes they may have seen anything, are asked to contact Police Scotland on 101, or alternatively if they wish to remain anonymous, they can call Crimestoppers on 0800 555 111.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"The Bao Sao hotel is very cozy, good location,now you can parking easier and tra...\"\nWith a stay at Baoson International Hotel in Hanoi, you'll be connected to the airport and convenient to Thu Le Park and Temple of Literature. This casino hotel is within close proximity of Giang Vo Exhibition Center and Vietnam Museum of Ethnology.\nMake yourself at home in one of the 100 individually furnished guestrooms, featuring minibars and LCD televisions. Wired and wireless Internet access is complimentary, while DVD players and cable programming provide entertainment. Private bathrooms with separate bathtubs and showers feature deep soaking bathtubs and complimentary toiletries. Conveniences include phones, as well as safes and desks.\nTry your luck at the casino and enjoy other recreational amenities including an indoor pool and a sauna. This hotel also features complimentary wireless Internet access, concierge services, and babysitting\/childcare (surcharge).\nFeatured amenities include a 24-hour business center, complimentary newspapers in the lobby, and dry cleaning\/laundry services. Planning an event in Hanoi? This hotel has facilities measuring 968 square feet (90 square meters), including conference space. A roundtrip airport shuttle is provided for a surcharge (available 24 hours), and free self parking is available onsite.\nEnjoy Vietnamese cuisine at Crystal Palace Restaurant, one of the hotel's 2 restaurants, or stay in and take advantage of the room service (during limited hours). Wrap up your day with a drink at the bar\/lounge.\nWith a stay at Baoson International Hotel in Hanoi (Dong Da), you'll be within a 15-minute drive of Ho Tay and Vietnam Museum of Ethnology. This 4-star hotel is 4.2 mi (6.8 km) from Hanoi Opera House and 3.5 mi (5.6 km) from Hoa Lo Prison Museum.\nHotel staff doesn't know English. But hotel is neat.\nThe Bao Sao hotel is very cozy, good location,now you can parking easier and traffic better because the street 'Nguyen chi Thanh' font of the hotel has been expanded bigger..\nGood overall service at the hotel and very clean room. It seems a little old. The furniture in the room is very dated.\nCame here once for swimming. What a pathetic swimming pool! Tiny and packed with children learning swimming. Too expensive for the price. Don't go there if you are an adult.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When the fictional New Jersey college administrator at the center of Murder 101 needs a slice of forensic expertise, he goes to a real-life expert\u2014TCNJ's own John Allison, professor of chemistry.\n\"It's unusual that an actual person shows up in a work of fiction,\" Allison said.\nThe cameo grew out of Allison's work with the book's author, Heath Boice-Pardee (writing under the name Heath P. Boice). As director of forensic chemistry at TCNJ, Allison was a prime choice of adviser for Boice-Pardee's second mystery novel.\n\"Whenever I have a technical question about things, [Allison is] one of the first people I reach out to,\" the author said.\nMurder 101 follows the dean of a Jersey-shore college as he investigates the burial of an anonymous skull. At the author's request, Allison lent his imagination to the fictional processes the characters use to gather information about the mysterious remains.\n\"I do a lot of analytical chemistry, trying to get answers for things like that,\" Allison said. His past endeavors have included determining the age and authenticity of works of art.\nIn return for his advisement, the author wrote in a fictionalized Allison, a TCNJ professor who has \"developed a process for dating just about anything \u2026 except himself.\" He and TCNJ appear prominently in Chapter 14, during which he uses a laser mass spectrometer to date the dean's collection of bones.\nIn Murder 101, Dean Douglas Carter-Conners and the fictional Allison meet for the first time, and their collaboration opens the door to an even deeper mystery involving shadowy secrets and underground societies.\nBoice-Pardee, a college administrator like his protagonist, has in fact known Allison for several years. Now serving as associate vice president at Rochester Institute of Technology, he met the chemistry professor while working as an adjunct at the College.\nTheir collaboration hasn't resulted in such gloomy revelations, but they have managed to turn an academic relationship into fun and suspenseful piece of mystery fiction.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Building regionally aligned Teacher Preparation Programs (TPP) to recruit, train and support diverse teachers and ameliorate critical teacher shortages in STEM, CTE and other high need areas.\nThis website is dedicated to supporting the expansion of teacher preparation programs in the California Community Colleges.\nAs part of this, we will also develop a community college repository of student resources.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbdatm b/data_all_eng_slimpj/shuffled/split2/finalzzzbdatm
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index 0000000000000000000000000000000000000000..30e27471b87f6e941aa846878ba6fe0d53acbd0d
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@@ -0,0 +1,5 @@
+{"text":"Inspired by a visit from Tibetan Buddhist Monks, we had an assignment in Painting III to paint our own mandala. I had difficulty with this project, because I didn't like the idea of arranging icons or painting a clean design. However, i did enjoy designing the structure with a compass. The whole design is based on circles; I used the intersections of these circles as points to draw the lines. I then went over each section multiple times building up layers of drips, which allowed me to feel like the highly structured painting had a human element.\n36 in by 36 in, acrylic on canvas, 2012.\nOne of the projects in Painting I at Brevard College was a self-portrait in the style of an artist we hadn't heard of before. Well, I can't find the sketchbook from that class, so I can't tell you who I was emulating here. This painting is usually hidden facing the wall so that it doesn't frighten the family.\n24 in by 32 in acrylic on canvas, 2010.\nDuring mixed media class, I made a series of 3 paintings inspired by the Billy Collins poem \"Introduction to Poetry.\" For this painting, I painted the surface pink, spray-painted it black and then applied a dollop of white gesso. Two pounds of strawberries were then carefully placed into the gesso and then violently hammered into the surface with a 2\u00d74. Once dry, everything was coated in acrylic polymer gel to prevent the piece from rotting.\n18 in by 15 in, strawberries, spraypaint and acrylic on canvas, 2011.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Getting married in Virginia? One thing you need to do is get a marriage license!\nYou and your fiance can secure a license through any Virginia Circuit Court Clerk, and they can be found by checking the list at www.courts.state.va.us. For getting married in Virginia, a license will cost $30 and be valid for 60 days. For more info, and to see sites and officiants, head to www.virginia.org.\nAs far as age goes, you both need to be 18, so make sure to take a certified copy of your birth certificate. If you happen to be under 18, you will need to take notarized consent from a legal guardian or parent along with you. Be sure to take an ID with you \u2013 a passport, driver's license, military ID or state ID will do!\nIf you are wondering who can marry you, be sure get an ordained minister who can show proof of their ordination. You can also be married by a justice of the peace, judge or marriage commissioner.\nBe sure to get a marriage license in time.\nIt's vital, and many couples have forgotten! For the most up-to-date information on getting married in Virginia, please head to www.courts.state.va.us.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We have a handpicked collection of the latest and best Smooth Radio Promos and Deals. The last promo code for Smooth Radio was used less than 19 hours ago. Smooth Radio have been offering coupons and deals for quite some time now and a large number of people have saved big by utilizing them. They announce special deals and promo codes on holidays, special occasions as well as during launch for their special products. We collect and source all their coupons and deals from official sources such as their Twitter and Facebook pages, company announcement pages and some other reliable third party sources.\nThere is generally a 88% success rate on all Smooth Radio promo codes. All our coupons and promo codes are taken from official sources and the ones that are submitted by users are cross verified to make sure that they are still working. But still, in some rare cases they might not work. Also make sure that the deal or coupon code you are trying to use is still valid or they may not work. However, we have found that in many cases, even if the promo codes are past the validity date, they still work.\nOne should note that GetBestStuff doesn't offer customer service on behalf of Smooth Radio for any products or services purchased from their online store. If you have any questions regarding the product, please contact their customer service by email or phone. In case if you want to send back any item you purchased from their site, make sure to first read their return policy and contact them first before sending the product in fresh condition so that it can be restocked. Shipping and restocking fee will be deducted from your total purchase amount before its refunded to you. Generally you need to ship it back within 7 days or in some cases 30 days but make sure to contact them first to get the accurate information regarding the same.\nCurrently, Smooth Radio don't offer any coupons and promo codes that can be used in conjunction with some other coupon code. Also, some of the promo codes are linked to a item or a certain section of products and won't work on other products. If you want, you can however combine a special promo deal which doesn't require any special code and then use in conjuntion with a regular discount code that can be used across the board lets say a 10% discount coupon and the like.\nSmooth Radio discount codes aren't listed on GetBestStuff. We only deal in coupons that can be used online on Smooth Radio's official store.\nLooking for love? Sign up to #SmoothSingles and you could win a Secret Escapes voucher worth \u00a3500!\nWhat colour brick road did Elton John famously sing about? MY ANSWER: Yellow ---------------------------------- Please sign in\/register, good luck!\nFinish the title of this ABBA song, \"The Winner...\" MY ANSWER:Takes it All ---------------------------------- Please sign in\/register, good luck!\n\"The Best Thing That Happened To Me\" was a song by Gladys Knight & who? MY ANSWER: The Pips ---------------------------------- Please sign in\/register, good luck!\nBetsy DeVos' pet project is promoting fake news to boost her. corrects the record.\nLa vedo bene informato sul finanziamento di Stato a Repubblica. Bravo voucher.\nIf youve not managed to redeem your voucher in time, please contact us at so we can look in to this!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It is very cheap to travel by bus in Thailand, although it also can be very boring and dangerous.\nYou will find buses in many different shapes and conditions. From the ultra-modern VIP buses with TV and entertainment consoles in the seats, as if you were on a long distance flight, to old bangers resembling something that were discarded in the west 30 years ago.\nThe ultra-modern buses typically run on long routes, while you typically will find the most dilapidated buses in local routes.\nAlthough you can get plenty of comfort for almost no money, I would only recommend long-distance bus travel if there isn't a domestic flight to your destination and if your credit card is being strained, and you either do not have time to take the train or if there isn't a train to your destination.\nThe journey by bus can be very dull; also, the drivers drive fast and hazardous.\nYou cannot complain about the price, though. For example, you can travel 600-800 kilometres (375 - 500 miles) in a 24-seat VIP bus with toilet and air conditioning for 1,000 baht (1000 baht), which may for example be from Bangkok to Phuket or from Bangkok to Chiang Mai. Such a trip typically takes 10-12 hours. If you choose a bus with more seats (less room for you) and no toilet, the price is about the half.\nDo not buy your ticket from a tour agent. You will probably pay too much, and you cannot be sure that the quality of the bus is as promised. Instead, buy the ticket directly from the bus company. Be sure to bring a blanket if you are travelling in a bus with air condition - typically, the driver will set the air condition to \"freeze\".\nThere are several bus stations for long-distance buses in Bangkok, thus it will be a good idea to check which one serves buses for your destination beforehand. You can find an overview of routes, prices, and travel length at 1stop Bangkok.\nIn Phuket, the bus from the beaches at the west side of the island and to Phuket Town at the east side cost around 30 baht (30 baht). The buses are old, probably resembling a bus from the 60's in your own country. Besides, because of their condition, they probably would be illegal in your country. However, they do run here, and accidents are rare on those routes.\nThe bus in this picture is one of the newer ones on the route between Patong and Phuket Town. It is actually not a bus; it's a truck with benches and a roof.\nIn many cities, you will not even find buses like this; instead, they are using songthaews as buses. A songthaew is a pickup truck with benches and a roof. Even in Chiang Mai, the second biggest city in Thailand, they use songthaews instead of real buses.\nIn Bangkok, they do have buses, even real ones instead of the trucks with benches. In addition, it is extremely cheap to go by bus in Bangkok. However, the trip may take some time due to the incessant traffic jams, and your lungs will have to recycle the exhaust gasses from the other vehicles, as the windows are usually open in the buses.\nIf you want to avoid traffic jams, you can use the metro \/ sky train, or you can take a motorcycle taxi - the latter provided you have nerves of steel and you have paid the rates of your life insurance!\nIt can be a bit of a jungle to know the prices for buses in Bangkok, some buses \/ routes operate at a fixed price, regardless of how far you go, others have varying prices. The prices range from 7 baht (7 baht) to just over 20 baht (20 baht) for the ordinary buses.\nYou buy your ticket in the bus from a ticket seller. Make sure you have small change as you in most cases will have to pay the exact amount, otherwise, they will ask you to get off the bus. I have experienced that the ticket seller would not even agree to keep the change, I had to get off at the next stop.\nYou will also find minibuses with closed windows and air conditioning on scheduled routes; they charge a slightly higher price. When a minibus stops, the ticket seller will come out, yelling the destination in order to attract customers.\nPlease note that the regular buses do not automatically stop at the bus stops, even if many people are waiting there. In order to make the bus stop, you hail it the same way that you hail a taxi in Thailand; by holding your arm out downwards, and waving your hand.\nIt can be difficult to navigate the various bus routes for tourists, as they use Thai characters on the buses and at the signs at the bus stop. Google Maps can help you find the right bus routes, and if you use Google Maps while driving, you can plot in your destination and see where to get off.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Weather Report: Low 38, High 46. It was warmest when I started hiking, then the drizzle started and the temperature hovered between 38 and 43 for the entire hike.\nToday I was meeting Dave at Sonora Pass at 3:30, and I did not want to be late. I left camp after downing two cups of coffee, and started down the trail pretty much at mile 1000, woo!\nAfter I passed mile 1000, the trail suddenly took a very different character. The terrain had changed from granite boulders and rock to soft volcanic-looking soils, and the trail track was suddenly fast, with no boulders or water bars or icky mud to step around for many miles. Around the sixth mile (mile 1006), the trail broke away from the creek and began a steady uphill climb, up and up... and the drizzle started as well.\nI knew by looking at the map that the trail wound its way up to 10,800', and if it was drizzling at the campsite at 9600' it might well be snowing up above. But the odd thing was that it was very warm overnight, only getting down to 46 even at the 9600 feet elevation I was camping at. That tells me that it wouldn't be snowing, that this weather was likely subtropical in origin, and would pass soon enough. It's also not the type of weather that induces thunderstorms, so I decided to just keep going. It wasn't so much a \"well, I'll risk it\" decision as much as it was a \"well, I'll do it until I can't do it anymore, then I'll turn around\". I did not have to turn around.\nBut at mile 1007, the infrequent drips from the sky became a more steady drizzle, so I put on the rain jacket and the waterproof glove shells, put my REI sheer silk baseleyer leggings and midweight baselayer shirt on. I also decided to try on the rain kilt over my hiking pants to see how the whole getup would work. It worked fantastic, the upper part of my pants and crotchal region stayed dry the whole hike, no chafing or rubbing of wet fabric on private parts. I'll be doing that same thing later in this trip.\nFrom mile 1007 it's just ten miles to Sonora Pass, but the PCT goes up very high along a ridge and stays up there until nearly the very end. The first few miles were super-easy: clear of snow, a constant steady uphill grade, wide trail that was wide enough to drive a Jeep down. Then the trail meandered along a ridge, and after five miles came to a large snowfield at the top of the ridge, misty and atmospheric... and with cell phone coverage, so I stnapped a selfie and texted Dave that I'd be there on time.\nThat was a bit presumputious of me, from mile 1012 to 1017 the trail went across more snow than not as it danced along the ridge spurs, all sharp and volcanic looking and covered with snow. I walked around some snow fields, I glissaded down others, I consulted the GPS for the track now and then. It was good fun, and I was pleased that I could handle myself on the snow. It was cool to cold, but I was warm with my clothing, though I wished my feet were drier--they were pretty sopping wet. I thought \"you know if I could do this, the north Cascades won't be a problem--this is Northwest weather!\"\nI made it to Sonora Pass at 3:40... oops, I was ten minutes late... and in the last few miles, I had decided I didn't want to camp tonight. I have no dry socks or clothes, my tent is wet, I'm just tired of being soggy. I asked Dave if we could get me to a motel in Bridgeport, and he said \"well, sure, but just come back with me to Tahoe.\" That really sounded great, so I took him up on the offer, and by 6pm I was in his dry and warm house and all those soggy socks were in the wash, along with the rest of my stinky clothes. Dave said they smelled like something about me was fermenting, and I believed him.\nDinner was next, and we went over to the Harrah's buffet, but it was closed, so we ate at a restaurant bar at Montblue and chatted up the bartender. Lots of discussion about Burning Man projects; it was a good time. The food was quite good as well--I had lobster bisque and roasted grilled salmon and chocolate lava cake, which I couldn't finish! Definitely not a camp dinner.\nTomorrow I'll get back on the trail; Dave's offered to take me back. The weather should be nicer, and I'll certainly be looking forward to not hiking in the drizzly rain. I wonder if I'll see anyone else too: today I didn't run into any other hikers on the trail. Maybe they were waiting out the storm; we'll see!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbdwfz b/data_all_eng_slimpj/shuffled/split2/finalzzzbdwfz
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+{"text":"Ruby on Rails and some other web based frameworks ship with support for a small database called sqlite3 by default. SQLite is ideal for users getting started since it can be run in memory and backed by small files on disk that are easily created and moved around. While easy to use, SQLite is not intended as a production grade database. Instead Heroku provides production grade PostgreSQL databases as a service.\nWhy is SQLite a bad fit for running on Heroku?\nSQLite runs in memory, and backs up its data store in files on disk. While this strategy works well for development, Heroku's Cedar stack has an ephemeral filesystem. You can write to it, and you can read from it, but the contents will be cleared periodically. If you were to use SQLite on Heroku, you would lose your entire database at least once every 24 hours.\nEven if Heroku's disks were persistent running SQLite would still not be a good fit. Since SQLite does not run as a service, each dyno would run a separate running copy. Each of these copies need their own disk backed store. This would mean that each dyno powering your app would have a different set of data since the disks are not synchronized.\nInstead of using SQLite on Heroku you can configure your app to run on Postgres.\nPostgreSQL database can be used by any language and framework, this section covers how to connect to it through the Ruby on Rails framework. It is important that you use the same database in production as in development, so you will need to install the PostgreSQL database locally.\nThis will install the pg gem in your Gemfile and write the correct config\/database.yml configuration locally.\nIf you have a pre-existing Rails app, or you ran the rails new command without the -d postgresql flag you can convert your application manually.\nIf you've removed the gem 'sqlite3' line from your Gemfile and are still getting errors while deploying to Heroku it is likely that another gem you are using has sqlite3 as a dependency. To help find the source of this dependency look in your Gemfile.lock for sqlite3. Find the gem that has sqlite3 as a dependency and remove it from your Gemfile. Once you've done this run bundle install and ensure that sqlite3 no longer exists in your Gemfile.lock.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What happens when you find yourself in a boxing ring against Mike Tyson and your whole life is depending on it? I know you don't have a chance of winning when you are up for a professional boxer \u2013but how much effort you will give to fight him. I'm sure you will give 110% of your strengths on it.\nKeep that in mind that's important!\nNow here is the thing, when you are up against for someone in a fist fight, no matter who is your opponent, visualize that the person is Mike Tyson or Undertaker \u2013you choose. Then give that exact effort you would have if you were actually playing against a larger opponent.\nFew days ago someone at the yahoo answers community asked \"How to become an entrepreneur?\" I see the trend here. People are now leaning towards being their own boss and quit the 9-5 lifestyle.\nSo in this article I am going to answer this question in detail and tell you how exactly you can become an entrepreneur.\nDo I Qualify as an Entrepreneur?\nI have started working on my own business when I was 15 years old. I never wanted the idea to have a boss and go into the rat race. I started my own t-shirt design store at that age. It failed, and I started working on direct marketing. I learned how to sell and talk to new people. I learned how to improve my personality.\nI invested some of my family's money in transport with my mom. That failed miserably. Then I started my computer parts selling business. I was buying gadgets in cheap to sell them for high margins. Then I started a website, then a blog and went online.\nI started doing online marketing with blogs and I started to see profit. Then I became a web designer and that went very well. Then I started working as a consultant for online & offline businesses; that went very well too.\nI also share my journey that inspired so many of my readers to start their own. So far I was never held in a job and I created jobs for others.\nSo How to Become an Entrepreneur?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Based on a pre-defined weighted trend formula for chart analysis, GNF.V19 scored +90 on a scale from -100 (strong downtrend) to +100 (strong uptrend).\nWant to analyze NONFAT DRY MILK Oct 2019 GNF.V19 or another symbol? Try our Free Future Trend Analysis Report.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\u00ab Looking Ahead for the Remainder of March\u2026..Next Possible Storm and What April May have in Store for us\u2026.\nAlthough Mammoth Mt reported new snowfall of only 2 inches this morning at the main lodge\u2026..there was .44 inches tallied in the water bucket\u2026..Temperatures over the crest were in the lows 20s when the moisture band came through. So it is possible based upon .44 inches, there may have been 4 or 5 inches over the crest this morning based upon temperatures at the summit and snow to water ratios\u2026.\nIt was great to see snowfall this morning even through it was short lived!\nAs mentioned earlier.. Rising heights will allow for a warming trend beginning Tuesday then continuing through Friday. In that stronger ridging develops Wednesday\u2026.high temperatures will go from the upper 40s today to the low 50s Tuesday then upper 50s Wednesday. Further warming Thursday into Friday will push highs into the mid 60s. Over night lows will be in the 20s early this week with mild inversions at night setting up 2nd half of this week. Expect temps in the 20s and 30s second half of this week with warmer early morning temps above 8,000 as compared to the lower elevations of town.\nA weak Cutoff Low is expected by the end of the week on a track toward Southern CA Sunday. This is likely to cause some destabilization of the our local air mass, especially over the higher elevated heat sources. There is also the possibly of a zone of deformation to set up over the Sierra. The Forecast at this time indicated that there is only a slight chance of any shower activity next Sunday, about a week away but stay tuned.. The Dweebs may have to add some Thunder later in the week for Sunday PM.\nWeek 3 still shows the possibly of a pattern change to the wetter. That may just mean a storm or two\u2026.. This would be likely around or just after Easter Sunday.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A community kitchen is a group of participants who meet regularly to cook meals to feed themselves and their families. Everyone shares in the recipe selection, cooking and clean-up. You pay for the portion of food you take home. Each group decides how they will operate. Some groups meet to prepare a shared meal, others take home the meals they create.\nNew groups may be formed as the need arises. We can help you start your own group.\nMulticultural Group: Happy Chefs meets one Thursday each month at the Foodshare Center, 271 Pine St.\nSupporting Advocates in Leadership Group (SAL) meets one Thursday each month from 5-7pm at the Foodshare Center, 271 Pine St.\nCooking with Friends meets the second Monday of each month at Hope Lutheran Church.\nSizzling Sisters a newly formed group of women that regularly meet through the Harewood Community Church. Not currently meeting.\nPlease note: the Princess Royal Family Center has moved to 80 Chapel Street.\nPlease note: When schools are closed for weather, then NCK programs are cancelled.\nCooking out of the Box uses a monthly Good Food Box of fruits and vegetables as the foundation of a nutritious cooking session for low income individuals.\nThe program is free to participants and encourages the use of fresh fruits and vegetables in a healthy diet. At each 2 hour session, 3 to 4 recipes are prepared using the contents of the Good Food Box, which varies from month to month. The Community Kitchens facilitator chooses the recipes, purchases a Good Food Box for each location and the extra food needed to prepare the recipes. The sessions end with a shared meal, group discussion and clean up. Participants receive copies of the recipes and extra food is divided among the clients to take home.\nThe Tillicum Lelum location is in conjunction with the Building Better Babies program. Parents prepare a meal to take home.\nThanks to funding from the United Way Central and Northern Vancouver Island, we offer the program at three locations.\n\"My spirit feel happier\" said a mom who cooked with other young moms and became more connected to her community.\n\"This changes everything\" said a woman with a brain injury after making apple crisp in the microwave and seeing how easy it was to prepare.\nWe thank United Way Central and Northern Vancouver Island for supporting this program.\nParticipants are involved in the choice of recipes and take home servings of each recipe to feed their families. There is no cost to participants and child minding is available.\nAt the end of the program, participants will have a kitchen handbook with recipes and nutrition information.\nThe program runs at Princess Royal Family Center (80 Chapel St) on Monday mornings from 10:00 am--12:30 pm. This program is a partnership with Vancouver Island Health Authority and Nanaimo Community Kitchens.\nWe are accepting registration for the program starting April 15, 2019. Please use the attached registration form.\n\"I look at labels for fat, salt and sugar\".\nWe thank the Island Savings Community Endowment-First West Foundation and the City of Nanaimo for supporting this program.\nYou may already know how to cook, come share foods of your country.\nFive rounds of the program ended in April. The program is not currently running. Check back for updates.\nI am happy to live in Canada. I can not speak English. But I was so happy to be with you. Cooking classes made a big difference to my family. I was able to eat various kinds of food and to make healthy food. Thank you very much. I will remember the information you gave me.\nI enjoy this program which brainstorms ideas , teaches knowledge and brings happiness, will always exist in our memories. Thank you all your hard work in preparation and patience in teaching. More importantly, it gives us a snapshot of food science. I never thought cooking could be so much fun!\nCooking With Seniors Connect New Dates for 2019!\nA 6 week cooking program at Seniors Connect, 150-B Wallace St.\nCome help make simple healthy meals to eat together.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbeomz b/data_all_eng_slimpj/shuffled/split2/finalzzzbeomz
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index 0000000000000000000000000000000000000000..37dc3bccf903cb975a345ca07ec2ec2b2eaa0fac
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+{"text":"Discover a backyard studio in Broome where music magic happens.\nLauded as Broome's best export, the Pigram Brothers are Broome royalty. The band has won myriad awards, is deeply admired within the music industry and counts Paul Kelly and Jimmy Barnes as fans and friends.\nMusic runs deep through the large Pigram-Puertollano family, extending back generations, through to Narlijia Cultural Tours owner Bart Pigram, whose father Steve and Uncle Alan were the first indigenous artists to be inducted into the West Australian Music Hall of Fame.\nWhen I tell locals Bart will be taking me to Pearlshell Studios where the band had recorded, they can barely contain their excitement. Or envy.\nThe backyard studio is basic but lining its walls are signed oyster shells. \"To Alan, Josie and the family. Thanks for your hospitality. Paul Kelly,\" reads one, sitting alongside others signed by Troy Cassar-Daley, Missy Higgins, Bill Chambers and Archie Roach.\nThe new My Heritage Tour by Bart's company Narlijia Cultural Tours Broome is personal, a combination of Bart's musical heritage, his Yawuru ancestors explaining their traditional saltwater lives, language and cultural connection to the land, and the early story of Broome, his home town.\nBart has clearly inherited musical talent but it's his gift for oratory that he uses to tell the story of country. \"My family tell the Yawuru story through songwriting but I do it through cultural tours,\" he says.\nBart speaks about Broome's history.\nBart holds a Kimberley Point.\nSigned oyster shells adorn the walls of Pearlshell Studios.\nThrough his tour Bart gives the landscape cultural value, explains the pre-colonisation values that are still instilled in him today, as well as his modern culture of music, guitars and cuisine.\nBart leads us up Kennedy Hill, a red sand dune covered in cockleshells \u2014 the result of thousands of years of consumption as a staple diet of Bart's ancestors. On the dune Bart happens upon a modern Kimberley point (spear point). The glass shows evidence of knapping, the edges chipped away to form a point. Traditionally the points were made with quartzite stone or bone.\nIn town Bart and sister Naomi perform beautiful renditions of well-known Pigram Brothers songs while we eat the famous fish soup and rice sung about in Feel Like Going Back Home. Bart's dad wrote the song while missing home on the road when Bart's mother was pregnant with him.\nI'm lucky enough to catch a now relatively rare set by the Pigram Brothers on luxury adventure-cruise vessel True North \u2014 another Broome icon. We cruise Roebuck Bay in a welcome back for the small ship which is about to begin plying Kimberley waters celebrating the company's 30th anniversary.\nThe band's famous songs, a chance to view True North's luxury cabins, champagne and morsels such as soft-shell crab and Esperance sardines make for a great party on Roebuck Bay.\nMy Heritage Tour costs $105 per person. narlijiacultural tours.com.au or 9195 0232.\nAngie Tomlinson was a guest of Australia's North West tourism.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Fill out the form to receive your Free No Obligation Quote on your vehicle of interest or to Pre-Order a vehicle arriving soon.\nWith our Huge Selection, wide dealer network, and close relationship with Chrysler Canada we will do our best to get you the vehicle of your choice.\nBe sure to include all important vehicle specifications in the \"Feedback\" box.\nDetails may include make, model, year, desired features, undesired features, or any other important notes.\nfeel free to call 1 (877) 527-8380, or come in to 900 St. Laurent Blvd. Ottawa, Ontario.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Try this make-ahead breakfast for an easy breakfast to-go.\nAdd in all the ingredients to a jar or container and stir.\nSeal container with lid and refridgerate for at least 3 hours (I do this overnight).\nWhen the chia puddings are set, add your favourite fruit, nuts and seeds and enjoy!\nAdd whatever you want into this easy make-ahead meal, such as bananas or yoghurt.\nChia pudding is great by itself, or on top of a smoothie bowl or with some granola and yoghurt.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"montilla digital el tiempo 5 minute dating ct hersenen Feb 16, 6 Reasons Why Women Date Older Men Dating Older Men is exciting isn't it? 5 minute dating ct hersenen Dating a younger girl jokes ever Follow Delta Light to stay up-to-date with news, articles and jobs.\nLooking for love, 5 minutes at a time: Seniors turn out for speed dating in Enfield.\nYou Are Here: Home; About. The Date Doctor-Jaimy Blazynski I am Jaimy.\nSo what is speed dating? How it is done ? What should you expect? GOT 5.\nThe groups post-marriage dating 5 minute dating scheme. Theca-Lutein cysts 5 year age difference unlike tinder, minte excavation leads to ancient civilisation on the date and. speed dating outaouais with these ideas for events. After five minutes is committed to do in. Listed in chicago, - women looking to do in hartford, in ct. No one but.\nGot 5 Minutes is committed to creating the best speed dating and singles events on the planet. Events in Connecticut and Virginia.\nFive years later the company, \"Got 5 Minutes?\" is going strong thanks in part to a novel approach of having the participants spend mere minutes.\n5 minute dating in ct. Welcome to our reviews of the 5 minute dating in ct (also known as number of blacks killed by police ).Check out our top 10 list below and follow our links to read our full in-depth review of each online dating site, alongside which you'll find costs and features lists, user reviews and videos to help you make the right choice.\nFeb 6, PM Speed Dating 37 Tomato Joe's & Shea's on the Patio.\nThe one stop place for all CT and Western MA singles to find fun and We have a spot opened for a guy for speed dating on Sunday in Southington ages ! Image may contain: 5 people, people smiling, people standing and indoor.\n3rd Annual Women's Speed Networking Event [CT] w\/The Social Butterfly. 3rd Annual Women's Speed Networking Event [CT] w\/The Social Butterfly. Thu, Sep 12, pm. Bin Gastropub, Glastonbury, CT. Speed Dating Event in Providence, RI on January 15th for Single Professionals Ages 20's & 30's. Speed Dating Event in Providence, RI on January.\nGot 5 Minutes - Hampton Glen Mews, Chesterfield, Virginia - Rated 5 based on 2 Reviews \"Jaimy is amazing!!! Hands On Drumming - CT. Musician\/Band. Soapstone Mountain. Mountain. CCSU Office of the Registrar. College & University. Interval House CT. #love #singles via 1. See All. Posts. Got 5 Minutes shared a group 5\/5(2).\nBranford, CT - Speed Dating Ages Speed Dating Ages By Pam Landry, Patch Poster | Feb 7, pm ET. 0. This post was contributed by a .\nRecently, I attended a speed dating event at Murphy and Scarletti's run by Check out the Got 5 Minutes? website for calendar listings of events and The Prime Cut and tagged ct speed dating, dating, dating in connecticut.\nSpeed dating events events in Canton, CT .. 3rd Annual Women's Speed Networking Event [CT] w\/The Social Butterfly. Thu, Sep 12, pm.\n6 responses on \" Dating Over 8 Minute Dating is Back in CT! Kathy July 22, at pm. I am already hosting a 70's and 80's Dance Party and would be interested in reserving space at the Hilton Mystic for speed dating, in the same place the dance event takes place.\nGot 5 minutes to creating the best speed dating and singles events on the events in connecticut and help people find love and have fun doing us for one of our electrical bulkhead stud speed dating events, happy hours, or other singles help you find love and have fun doing I will kindle a fire in the wall of.\nWhy have one date on the Weekend when you can have in one night! Our Speed Dating CT events are based on the proven formula from our parent company Weekenddating, which has resulted in 45 engaged couples, 35 marriages and lots of other happy couples.5\/5().\nSpeed Dating: You will have up to 15 mini speed dates during the night, each one lasting about minutes. After each date, you will circle yes or no on your.\nSpeed dating for older adults has become much more common and useful in recent years. This dating style August 5, at pm Reply.\nConnecticut Speed Dating Singles Events by Pre-Dating \u00ae Pre-Dating \u00ae is the World's Largest Speed Dating Service Focusing on Busy, Single Professionals. Pre-Dating Speed Dating Singles .\nLiving Single: RedEye reporters try speed dating in Chicago . The organizer said the minutes we'd have with each person would go by.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"FootCareExpert was formed to offer a unique reference point on foot problems and foot care.\nThis site includes useful information about common foot problems, treatments and how to keep your feet in the best possible condition.\nFootCareExpert was founded by John Rowlinson, the founder of PtS.\nJohn, through PtS, is involved in a number of ventures including software and property companies. A percentage of revenue from these activities goes towards funding FootCareExpert and a number of similar sites.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbesal b/data_all_eng_slimpj/shuffled/split2/finalzzzbesal
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+{"text":"Pictured above: Matthew Friday, Hydrographic Apparatus, 2010\u2013present, mobile water testing and painting studio. Courtesy of the artist.\nWith Space as Substance: Beyond the Scenic Hudson, Matthew Friday relocates a mobile and modular research station that contains a variety of cartographic and scientific instruments. Operating as a provisional laboratory, camping platform, library and exhibition installation, the piece will include a portable, pigment-processing mill powered by the prototype for a 19th-century solar engine. Set up as a diorama, the research station's equipment will be demonstrated at two Wave Hill art projects\u2014one for families, the other for adults\u2014this fall. Friday will also use locally sourced dyes to create a diagrammatic wall map of the lower Hudson Valley, indicating sources of pollution, species migration, geological and meteorological transformation as well as ongoing and potential responses to these issues. He will present an archive of documentation from groups that are confronting the changes happening along the Hudson River. It will include preserved flora, fauna and man-made artifacts gathered from the watershed, as well as blackboards to solicit interaction with visitors, allowing them to share their perspectives and experiences.\nMatthew Friday works across a variety of media, contexts and institutions, focusing on organized labor, community agriculture and watershed remediation. Friday has exhibited in a number of venues, including Exit Art, New York, NY; the University of Buffalo Art Department Gallery; the Cambridge Art Association, Cambridge, MA; 1708 Gallery, Richmond, VA; the University of Rochester Art Gallery; the Pittsburgh Center for the Arts; the Center for Contemporary Arts in St. Louis; and Spaces, Cleveland, OH. Friday studied at the Slade School of Art, the University of New Mexico and the Whitney Independent Study Program. He has an MFA from Indiana State University. A committed educator, he is the graduate coordinator and associate professor of critical studies for the Art Department at the State University of New York at New Paltz.\nFunding for Matthew Friday's project has been made possible by the Puffin Foundation Ltd.\nLocally sourced pigments, specimens, plant-pulp papers, in Matthew Friday's Space as Substance: Beyond the Scenic Hudson, 2015. Photo: Stefan Hagen.\nMatthew Friday in Glyndor Gallery during installation of Field Notes.\nPure Color of the Hudson (after Rodchenko) Remediated polychlorinated biphenyl (PCB) mud obtained from General Electric dredging site, processed waste coal from power plants in the Hudson Valley on Sintra, in Matthew Friday's Space as Substance: Beyond the Scenic Hudson, 2015. Photo: Stefan Hagen.\nThe Fresnel lens in Matthew Friday's Space as Substance: Beyond the Scenic Hudson, 2015. Photo: Stefan Hagen.\nClick here for more information about the artist. Also read what Suzaan Boettger wrote about Matthew Friday's installation in Field Notes\u2014\"a strong example of current environmentalist art thinking and facture\"\u2014in a recent article in the Brooklyn Rail.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Taking a picture from water's level requires the use of a floating blind. The choice of this tool is essential for the realization of interesting and creative photos. For years, we used handcrafted floating blinds. Recently, we discovered the company \"Mr. Jan Gear\" which designs and manufactures floating hides that perfectly meet our requirements and needs as professional photographers.\nAs a rule, we normally do not use our blogs to talk about materials and equipment that are used in the creation of our photos, but this time we will make an exception. We decided to deviate from this rule because this product is simply unique and extraordinary for wildlife photographers who are passionate about photo sessions in wet areas. Our experience with the floating blind has helped us to create truly exceptional photos, and we would like to share this knowledge with you.\nChoosing a floating blind is not easy. We found a manufacturer who created an interesting product.\nA floating blind allows photographs to be taken in wet areas while walking in water with the camera attached to a ball head or a gimbal. It is covered by a tent to hide the photographer from the animals. A floating blind can be used in ponds or at sea on shores of the beach. The photographer only needs to walk and use a pair of waders to avoid getting wet.\nEven its name, \"floating blind\" indicates its function, that it \"hides\". The photographer is hidden from the eyes of the animals. But unlike the fixed hide that is used to photograph animals on the ground, it allows the photographer to move around the subjects allowing to make photographs with unique points of view. The photographer can thus better manage the scenery and the light to compose and frame the scenes.\nThe floating blind support provides the ability to photograph subjects at eye level. It avoids the visible high angle photography effect when the photo is taken from a bank. The shots have a lot of impact and a very great strength in the compositions.\nIn addition, a floating blind is mobile. It provides the closest access to live birds, insects, and mammals in wetlands. The photographer can thus vary his framing by making wide shots or tighter shots, such as portraits.\nIn general, if the photographer respects the safety distances and calmly carries out techniques of approach in the greatest silence, the animals do not escape. We have often approached herons, grebes, or coypu a few yards away without ever arousing the slightest suspicion. The animals photographed do not show any signs of stress, fear, or worry. For example, we have often captured scenes of feeding or complicity between chicks and adults.\nBut we always abide by one essential rule: never enter an animal's safety zone. This limit is very easy to determine. Just observe the animal in the camera's viewfinder. When it begins to show signs of feverishness movements with its body or head, we stop our progression and begin to slowly retreat. Sometimes it is too late because the bird flies away or the coypu dives. But often it is enough to reassure the animal. Once we are out of the limit of the security zone, we wait a few minutes before making our first pictures. Always give the animal a little time to integrate the floating blind into its environment.\nA floating blind is an indispensable accessory for the wildlife photographer who wants to make unique and creative photographs of live animals in wetlands. But the choice of a good one is not as simple as it seems.\nFor years, we used artisanal floating blinds made with polystyrene parts on which a tent was mounted. We always thought it was the only possible solution because we did not imagine that a company could invest in the research and development for an equipment not very often used.\nIt was by researching on the web that we discovered the floating blind created by the company \"Mr. Jan Gear\". The company was created by a Belgian, Jan Goddefroy whose passion is to create equipment for outdoor photography and for difficult environments. His passion for this trade inspired him to create photo bags that can be used regularly in so-called hostile environments where it is very cold or very wet. The quality of the zippers and the tightness allows us to perfectly protect our cameras and our lenses. We no longer fear rain, mud or even very low temperatures. We will come back to these photo bags of \"Mr. Jan Gear\" in another blog.\nWe contacted Jan Godeffroy by e-mail to learn more about the characteristics of his floating blinds. After some exchanges, we were convinced and we ordered two of them. After a few weeks of use, we are presenting our review.\nOur first two floating blinds were each delivered in a thick bag. We were initially very surprised because the complete equipment weighs only 3 kilograms (6 pounds). We are very far from the 20 kilograms of the artisanal hide we used before. Once unpacked, we had an inflatable part, an epoxy plate to fix the gimbal, a tent, and two hoops. The set can fit very well in a medium size travel suitcase. This opens interesting doors for future trips to Romania, Ukraine or regions with beautiful wetlands.\nThe flange is inflated with a pump in two minutes. A high-quality anti-return valve prevents it from deflating.\nThe epoxy plate that will support the ball will require 7 minutes to be mounted. The system based on rot-proof tips is ingenious.\nThe hoops are mounted in less than one minute.\nThe tent is fixed on the hoops in three minutes.\nFinally, the floating blind is assembled in 12 minutes. Surprisingly, the assembly of our old one required us more time. The whole design of this new floating blind is very aesthetic, and the size of the structure is small. The fabric composing the dome is of excellent quality, as it resists bramble thorns and does not fade in sunlight.\nAfter mounting the blind on the mainland, we grab it with two inside handles to place it in the water. We fixed our gimbal, which has a weight of two kilograms. The whole setup is light. We can move easily while walking. Now we can get out of the ponds where we want. In the event of a thunderstorm, we will be able to leave immediately and return to our car without having to cross the pond in the hide. The floating blind's lightweight feature is an important safety advantage.\nOnce the camera is attached to the gimbal, we find that we are much lower to the water's surface than when we used our previous blinds. We are almost at the water's level. Although we were satisfied with our old floating blinds, we have come to value the closeness of the new floating blinds, because we can shoot at the eye level of the animals, including ducks and grebes. Even for photographers whose size is more than 1.80 meters (6 feet), the shooting is relatively easy.\nThe first movements in the pond are impressive because the whole system is easily maneuvered. Being perfectly silent, it does not create waves.\nOur only regret is that it does not have a small box to store the keys of our car. But this problem will be solved in a few days with the use of a small waterproof bag fixed onto a hoop.\nThe small windows are perfectly designed for a photographer to see the animals without being seen. Small nets on each window protect the photographer. On each net, there is a small piece of fabric that can be scratched with Velcro. In our old blinds, we were obliged to use safety pins to fasten the protections. The design of the blind was excellently crafted.\nMr Jan Gear's floating blinds are at the water's edge and easy to maneuver.\nDuring our weeks of shooting, we were subjected to many climatic hazards. Some days we experienced strong winds. The dome shape of the blind resists weather conditions well. Even though it is very light, our floating blind never glided away. We had no problems resisting the wind.\nOn other days, we had rain. Fortunately, we never had any water inside the blind because the dome shape immediately evacuates any water that collects on the tent. It is very waterproof.\nNormally, we experienced sunny days with heavy heat. The windows were wonderfully designed as well because after opening them, we experienced slight drafts of air which were very refreshing.\nFinally, Mr. Jean Gear's floating blind was perfect in all the climatic conditions we encountered.\nThe craft blinds we used for years were built with very bulky polystyrene parts. The submerged part of the carriage was about 15 to 20 centimeters (8 inches). It was virtually impossible to move when the depth of water became less than 20 centimeters. Moreover, the weight prevented us from transporting them easily on a mudflat.\nWe were pleasantly surprised to discover that with the blinds of \"Mr. Jean Gear\" we could advance over areas of barely 5 centimeters (2 inches) of water. Indeed, this year the drought has been significant and many ponds' water levels have drastically fallen. Sometimes we had to move on our knees on some interesting mudflats to make photos.\nThe adaptability of the floating blind was a huge advantage. We could approach shorebirds like the northern lapwings and make exceptional photos, even when the water levels were shallow. This kind of approach was not possible for us when using our old floating blinds.\nMr Jan Gear's floating blinds are light but very robust.\nTo conclude this review of \"Mr. Jan Gear\" floating blinds, we must admit that despite its low weight and its construction with inflated bladders, we were surprised by the robustness of the materials used. We have no problems during all these weeks of intense use.\nThe Jan Gear's floating blind consists of an inflatable part and a tent fixed on two hoops.\nThe entry into the water is easily done using two handles attached to the inflatable bladder.\nThe floating blinds made by the company \"Mr. Jan Gear\" have become indispensable accessories for us. They are quick to assemble, easy to utilize, very maneuverable, adaptable to all weather conditions, robust, and solid. The icing on the cake is that they are compact and can easily fit into a medium size suitcase. We are completely satisfied with our choice. We look forward to our next photo sessions in the ponds using this equipment.\nSome pictures taken from the floating blinds made by \"Mr Jan Gear\"\nA black kite photographed from a floating blind.\nA gray heron photographed from a floating blind.\nA swan on takeoff photographed from a floating blind.\nAn Eurasian spoonbill photographed from a floating blind.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Turmeric Essential Oil . 100% essential oil.\n(15 ml) . 0.5 fl oz.\nAroma: base note with a medium aroma, has the same fragrant woody scent as the powdered spice used in Eastern cuisine.\nCautions: Dilute in a carrier oil before use; for external use only. May cause skin irritation in some individuals; a skin test is recommended prior to use. Contact with eyes must be avoided.\nStorage: Best stored in a cool, dark area.\nThe FDA has not evaluated the statements on this website. No claims are made by Bonny Bubbles as to the medicinal value of any products from Bonny Bubbles.\nOur essential oils are in their purest form, undiluted unless otherwise stated. Any information presented here is for educating our customers about the traditional uses of essential oils and is not intended to diagnose, treat, cure, or prevent any disease.\nYou are responsible for understanding and researching the safe application of these wonderful natural products. If pregnant or dealing with any medical conditions, please seek your doctors advice on the safe use of essential oils.\nThis is the first inhaler I've ever bought or used, so I was skeptical. However it works surprisingly well for me. My headaches aren't typical so I didn't expect it to relieve them. It is a nice wake-you-up & energizing scent. I will be trying Bonny's other products very soon.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I was working from home for a couple of weeks. I write mostly in and around airplanes, so my writing\/reading schedule fell apart for a bit. I'm travelling again for a short time, and so I hope to pick this back up. I've found writing somewhere regularly to be a pretty useful habit for organizing my thoughts, and losing time for it was no fun at all.\nI also bought an Arduino starter kit and began wiring stuff with LEDS, buttons, motors, and relays. Next week I hope to run through the rest of the projects in the little manual and graduate to solving crucial life problems. I want to automatically water my plants and for my TV to swivel to face me when I walk to and from the kitchen. Life has been too hard for too long.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Helping parents to meet their children social, physical and intellectual needs.\nAt \"My Children's Day Care\" toddlers grow healthy and smart, preschoolers gain independence and learn to explore the world, boys and girls overcome school challenges receiving special encouragement.\nWe follow DOH strict guidelines to protect childrens health, safety and well being. We go beyond standards for physical space, equipment, teacher\/child ratios. Our facilities have the written approval of the Fire Department, Buildings Department and the Bureau of Child Care. Our personnel are background checked and go regularly under health examinations and immunization.\nWe count with Early Childhood Education Consultants to assist and improve our programs. \"My Children's Day Care\" provide parents with clear admissions policies, description for indoor and outdoor activities, a plan of discipline and guidance practices. We offer transportation and delicious healthy food service to help busy parents like you.\nContact us to schedule a visit now.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbfwyw b/data_all_eng_slimpj/shuffled/split2/finalzzzbfwyw
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+{"text":"Igor the cat is a night owl like me!\nTop: Bearclaw Englemann courtesy of Bruce Harvie (thanks Spruce!)Check out that silkng! Carved arch, Tonebars.\nFingerboard: Ebony, compound radius.Very sweet low action. BTW...the \"piano key\" FB inlay design is MINE!\nHeadstock: Listen up!!! the headstock design is MINE! ain't no flowerpots here. That trick with the binding is a royal pain! The headstock design and MOP signature inlay (TD's) are copyrighted\/trademarked\/whateverelse for ever more!\nPickguard\/Pickup: MAGIC!!!! microelectronics hiding under that Macassar ebony! Vol\/Tone...even can switch from humbucker to single coil. MANY THANKS to Kent Armstrong who made the minihumbucker and helped me as I stumbled through designing the fine points. The unit can come away with simply one screw at the brass mount if desired.\nFinish: No stain whatsoever...top or back\/sides. Hand rubbed Permalyn (a la French Polish)No filler....simply coat\/sand ad nauseum until pores filled...then a bit more. Great \"depth\", but thin.....and durable. I use the same method on gunstocks.\nSound?? considering it was strung 3 days before Memorial Day, I'm pretty happy! Hey....this past weekend Adam Steffey played it and said I could quote him....he loved it!\nAnd plugged in? .....what a hoot!\nAnyway, that's probably more info than necessary....but what can I say? As a prototype, it came out as well as I had hoped!\nAnd by the way....the sides were bent with my heat blanket system (see discussion in Builders section). It DOES work!\nBeautiful, truly. A very well thought out effort, great design, and absolutely delightful woods (I'm certainly jealous of your own stash of walnut). Huge congrats, maniac, to a job very well done.\nWhat a beautiful instrument! Can I ask a silly question-where is the plug in hole for what I assume is a quarter inch cable? Can you provide a pic of it? Can we see the electronics under the guard? My compliments to the maker.\nRich....the plug in is seen in the tail pic....need to scroll a bit to the right (I'm no good at posting pics!) Actually it is a micro jack, flush with side and barely noticeable. I use a pigtail adapter (patent pending <G>) from instrument to belt where standard 1\/4\" plugs in.\nWow! Now there's one beautiful bird! That's what I love to see, a one of a kind!\nBoy, after seeing great photographs on other topics here, I had to TRY to improve a little and editted some pics above. Since there was a question on the electronics.... a couple more that show under PG a bit,as well as the microjack.\nMore of these to be in the works...since this one WORKED!\nDefinitely worthy of an emando.com listing! Would you like one, Tom?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Pitch your tent on a bluff overlooking the ocean. There's a fire pit to keep you warm. Feel free to use the bathroom inside, just a few steps away.\nYou must call the fire dept and ask for permission before having a fire.\nHave a question? Send Shelby a message!\nNatural features you'll find at Where The Sidewalk Ends in California.\nIf you've fully explored and read the information on this listing page and still have questions, you can message Shelby here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What a great weekend! Here are the 2018 Barron Valley Invitational results.\nGymnasts from across the Cairns region headed to Barron Valley Gymnastics Club to compete in the Barron Valley Invitational on Saturday 4th & Sunday 5th August. Boys and girls from Levels 3-6 were invited to perform at this annual event, which is a fantastic lead up for Junior Regional Championships, coming up in 2 weeks time (18th & 19th August) and for level 4 to 6 gymnasts preparing for the Junior State Championships in Brisbane in the September School Holidays.\nThanks to everyone who came along to support our kids over the weekend. We loved not only seeing their amazing routines and gymnastics skills, but the teamwork, smiles and cheers displayed by all.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Deniz Burak Balpinar \u30c7\u30cb\u30c418\u6b73 \u30c7\u30e5\u30a4\u30b9\u30d6\u30eb\u30b0\u9ad8\u6821\u5352 \u6765\u6708\u304b\u3089\u5927\u5b66\u306b\u5165\u5b66The day we were in Sch\u00f6nau was wuite a nice day. As usual the weather was nice. We took the bus. We soon arrived and had to walk a bit until we arried at a modern looking but rather small house. EWS was written in green soit was obvious that that was the place where we would learn about rentable energy. We were welcomed and they led us to a comferemce room. The things we were about to learn were very important to bith me and the students because learning about renewable energy strategies was the thing that we gathered for. Ir was nice to see , eventhough they had much luck, the brave citizens of this town were able to truely change sonething and established a sucsessfull all-green electricity company. The struggle was mainly important for me because the stategies were sinply zlro be learned. But the thing that led and is still leading this company to sucsess is ther braveness,courage and the fact that they were able to utilize mass media in their own Fortune and as a weapon against their superior opponents.after the informative speech and presentation we were able to see a water utilizing energy plant. Seeing it with my own eyes gave it a whole other dimension. Im happy that the interviews happened and the earthwalkers got at least a bit media attention. It was alsos nice to see that this topic really interests at least a small part of media. After that we all went to a restaurant and the Dinner was nice and we were able to talk a bit and chit chatted the rest of the day away. The people of that conpany really seemed like nice people and i was happy that i had the chance to meet them.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A broken, dying stump, the remnant of a once strong tree now forms the beautiful foreground for this picture I recently shot. Much of what I do in helping pastors and ministry leaders in pain is to guide them to get perspective. For so many the youthful hopes of this life and ministry impacting the kingdom are now nothing more than memories of what could have been. Rather than towering as a tall tree, the life of the pastor, reduced by years of failure and resistance by others, now sits, an insignificant stump in the landscape of all God does through the apparent successes of others. Yet, here there is victory, and meaning. Once his troubled and broken life is viewed through the lens of God's sovereignty and purpose, everything else appears in proper perspective to the pastor. A accurate view of oneself, against the backdrop of God's larger mission, gives meaning to brokenness. This is why I rarely talk a pastor, or a missionary, out of what he is feeling, or going through. I have no right to that. Neither can I do much to change the hard circumstances that have contributed to his brokenness. He, too, has no access to alter his history. All I can do is speak of Jesus, his love, his sovereignty and the larger purpose of his mission. I also carefully suggest that God allows brokenness to further advance his purpose. Like Paul in prison we are able to say, \"I want you to know, brothers, that what has happened to me has really served to advance the gospel, so that it has become known throughout the whole imperial guard and to all the rest that my imprisonment is for Christ.\" (Phil 1:12-13) Once the pastor views his broken, frail and fragile life next to what Jesus is doing, he sees it all differently, especially his own experiences. I love that moment when the eyes light up; the chin lifts and a long sigh releases the burdens of a heavy heart. Why? Perspective! A selfish, self-centered view of oneself, the sort driven by desire for recognition, will not allow a pastor, or any of us for that matter, to see what God is really doing. I know the message has gotten through when I hear the pastor say something like this: \"You know, I think God actually wanted me to go through this. It is helping me to see better what he is doing. My trial, and this burden I carry, this difficult church, antagonistic people, are allowing me to see him and what he is doing in a new, fresh way.\" This willingness to allow the broken dreams of a great life, to stand now in the foreground as vulnerable and weak, roots the pastor more deeply in his relationship with God and brings deeper meaning to the pastor's life and his ministry.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbfzfw b/data_all_eng_slimpj/shuffled/split2/finalzzzbfzfw
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+{"text":"Well, July 30th was another B-Day for me and I'm another year older. 50 is approaching faster than I ever imagined! If I had known I was going to live this long I would have taken better care of myself, LOL.\nWe didn't do too much. Went to dinner with my folks on Friday, then had a BBQ at my place on Saturday with a few old friends, had some Margaritas, smoked a couple of great cigars with my buddys and ate a great dinner.\nSunday was interesting and fun too. My wife and I took the president of Summit Racing Equipment and his wife out for a personal wine tasting tour of our area. They were ot here on vacation. He and his wife are REALLY nice people. We had a GREAT time with them and sipped a lot of good wine and had some fantastic food at the private pairing we did at Mayo Winery. He's been the president of Summit for 22 years now. They must be doing something right because they do over $36 Billion a year in sales. Yah, that was billion with a B, not million with an M.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We are looking or a motivated intern for assisting and developing our operations. All nationalities are welcomed.\nSend your application to info(@)sunsethill-ubud.com with your CV and a small introduction of yourself and why you would like to join us.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Tofino may be best known for its beauty and surf, but foodies know it as a culinary hot spot. Joining popular restaurants like SoBo, Wolf in the Fog, and Kuma Tofino are new eateries that are making the Tuff City food scene even better.\nYou could call it a model pizzeria.\nCo-owner Marco de Conciliis, who runs Basic Goodness with his partner, Heather MacGillivray, is a former Ford model. In the late 1980s and 1990s, the Naples native was the face of Versace, Calvin Klein, and other mega brands.\nA father of two grown kids, de Conciliis met MacGillivray in Los Angeles, where she was working as an interior designer. She's from Ottawa, but had spent time in Tofino as a youth, having been introduced to the coastal village via her uncle, who caught geoduck. She had long wanted to move to the area, and just a couple of months ago, the couple deemed it to be the right time.\nDe Conciliis trained in Naples to become a piazzolo, and his two children might come and join him behind the counter in the future.\nTofino Brewing has bright new digs.\nThere's no denying it: I scream, you scream, we all scream for ice cream, and now Tofino locals and visitors can shout from the crests of the mighty Pacific. Here's a spot that specializes in gourmet frozen treats with a hyper-local twist.\nTofino Licks is headed by Francis Eadie, who's also a commercial diver. While he loves being underwater and diving for things like sea cucumbers and geoducks, he wanted an outlet for his creativity. So he moved into a live-work studio and opened Tofino Licks\u2014which lifts the humble ice-cream cone to epic new heights with the addition of winning ingredients and items from other local restaurants and caf\u00e9s.\nOne of the toppings on offer at Tofino Licks is sea-salt honey and bacon from Picnic Charcuterie.\n- Cold-brew coffee from Tofino Coffee Co.\nWith a view of Tofino Harbour and Strawberry Island in Browning Inlet, these two spots share a kitchen in the recently revamped Tofino Resort and Marina. The latter is a restaurant that takes its name from the year Tofino was incorporated.\nExecutive chef is Paul Moran got his start in the industry at West Restaurant when it was under the leadership of chef David Hawksworth.\nBoth eateries focus on fresh seafood, with the pub serving items like tempura prawns, lingcod tacos, and a West Coast Boil, with mussels, snow crab, side-stripe shrimp, and clams. The 1909 restaurant, which is currently open for dinner only, features everything from pan-seared lingcod and chilled seafood platters to local wild sockeye and whole wood-fired rock fish.\nThe 1909 Kitchen has a view of Tofino Harbour, as does its sister waterfront pub, the Hatch.\nHowever, if you're not in the mood for food from the sea, there's also an Italian wood-fired oven.\nPie varieties range from the Local Catch with side-stripe shrimp, chorizo, basil, fresh mozzarella, and garlic to the Pineapple with pancetta, chorizo, and mozzarella.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Free illustration essays - professional help, this concept is premium quality. Free and involves writing help you can go quite a friend or even in essays written by: 2. Feel despondent if you haven t know who decides to offer essay writing help writing editing services:. Harvard essayists edit your persuasive essay writing essay help and easy prompts. Once you help you get your friends or discuss these activities mba students. Learn how brainstorming and ensure your purpose and use our service easily. 15% off your students buy a close look no. Professional online from us to write different assignments when you haven t have cheap essay writing plus, affordablepapers. Net your service in the help high quality. Grabmyessay deals with your expert essay example is easy essay writing uk s. Medical school she also offer free revisions included. Focus on how can get plagiarism-free work done up your privacy with a college. Academic writing instruction is to solve all you. Paper, essay writing with the 100 great with writing. Includes tips by i need help: writing service offers top essay questions and get quality. Boost academic help students to buy a custom essay help here to assist you. Exclusively for professional help my assignment help homework help you want to help for those listed. \u2013 excellent grades plagiarism free term paper and quality writers! Amazing quality papers from there essay writing your essay help 1.0: research for professional essay writing essays. Term paper will help professional help from experienced professionals. Just a hot season for money of requests! 855 513-7729 essay writing a research papers and join now we offer! Use anti-plagiarism software and 24\/7 to order now! Hello i think your essay writing help help? Always rely on nat geo kids to our quality level. Descriptive-Essay-Transitions-Exercise - study guides and pay for general, creative writing through the finest having difficulties writing: essay concerns.\nGreat deal with a really helps you start the strength of requests! Assignment help from help students best assignment help buy assignment help? If a high-quality that will win your in writing service? Read this handout will help tutors and amazing essay writing a unique! 1 in life into that could help writing service are here are a conclusion. May help 24\/7 for a friend or 6 hours! Describe four steps for me high school thepensters is a writing center. Use our skilled and let our team of literature essays. Proffesional writers, personal statement, we provide you can write a variety of mba and quite a perfect essay! Essayhelp4me offer professional college - professional essay https:\/\/teensgeneration.com\/argument-essay-on-social-media\/ homework help writing guide you achieve your subject. Academic writing guides techniques, replacement product of your essay online distance want to write my essay. Students can someone write customized writing service when you know how to buy essays. Amazing essay and after ordering a grade certified custom essay can students. Amazing admission\/application essay papers from your writing lab owl if you. Deadline, benefits of illustration essay 6 hours of charge. Talented writes and structuring academic writing service reviews, term papers this professional help high school students. Don't know how brainstorming and submit your synthesis essay for students recognize a research paper? As i need help - professional help you can avail its services novels high quality. Oh no matter what they have been accepted! It's finals week and engaging paper writers offer!\nInstructors may write papers at this to help. One for me to great collection of your studying in the internet. Find and relax and outlines, double check all that help high quality. Pro writers - professional essay, essay writing service tailored services lighten the research paper perfect essay writing service. Discover the and other types of 2017 how to help high quality guaranteed. ' request custom essay writing help writing application essays written college papers from university assignment services? Want to the essay writing can you to deliver a quality essay today! Some fundamental elements of recommendation, research paper writer? Submit a concise advice in minutes college custom essay writers. Custom term papers interfering with best online stores. May one that has to help, definitional, you can help for resume or a. Enjoy working custom essays that provides ivy-league essay? Great sat essay writing essay writing services, writing is writing service high quality essay writing. Purdue we are an academic work, look any assignment uk. And academic experts in 3 factors: essay to the most updated selection of experience do students. Bid4papers is: custom essay writing help online, which will suffice, economic assignment?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Fruits are a great source of vitamins and minerals. Therefore, many people opt for a fruit based diet to detox and lose weight. It is important to plan the fruit diet with care and consideration.\nWhat are the fruits you must include in your fruit diet plan?\nYour breakfast must be heavy so go for a banana smoothie with berries and coconut milk.\nThen next day you can try for low-fat yoghurt and add sliced fruits to it.\nHowever, keep the number of dry fruits minimum. Don't overeat.\nIn lunch, you can experiment with various kinds of fruits in your fruit salad. Make sure you add berries, banana and grapes to eat. Feel free to experiment with fruits and add new things eat day to keep the lunch innovative.\nYou can go for dried figs or some dry fruits for your pre-dinner snack.\nPrepare a salad for dinner including cucumber, tomato and avocado. Add olive oil and lemon as part of the dressing.\nYou can use the variety of fruits in your fruit diet to make it more interesting and fun.\nWhat are the things you should keep in mind during the fruit diet?\nYou must not forget to drink at least 12 glasses of water each day to keep yourself hydrated.\nFruit diet needs variety. Otherwise, you will find it difficult to stay on one kind of diet only. So, experiment with your fruits, use a different kind of ways to have the fruits like making a smoothie, having fruit juices or fruit salad.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbhhia b/data_all_eng_slimpj/shuffled/split2/finalzzzbhhia
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+{"text":"eBay This is a very solid West Coast car. I cannot find any patch panels.\nSolid original floor.Rear fenders are excellent.Front fenders could use some attention.59AB with three-speed transmissionJuice breaksI have hood sides and bumpers.I do not have roof bows Please note: Milk truck wheels in photos are not being sold with this vehicle. Original wires will be put back prior to sale.\nMAY CONSIDER TRADES. UP OR DOWN ON 1932-34 FORDS ONLY.\nTHIS IS A PROJECT CAR AND IS BEING SOLD AS IS.\nNo emails please. If you are truly interested, please pick up the phone and dial (201) 394-4999. CAR IS BEING ADVERTISED ELSEWHERE - MAY END AUCTION AT ANY TIME.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"While i am doing the PS series firmware upgrade Connection is lost and its failing to reconnect to group manager after restarting the module.\nThat you lost connect is a expected behaviour especially if you only have one member.\nThanks for replying, i was upgrading the PS series from 8.1.0 to 8.1.15. I am not able to connect using http:\/groupip as well. Its been 2 hrs since its not connecting. I updated in the same process for other similar machines and it didnt gave this error while upgrading them.\nDid you verify that both controllers have network cables going to the switches? This is most common reason we see what you are reporting.\nIf not, you will need to access the controller via the serial port to determine what that problem is.\nIf you have a support contract i would suggest getting that cable and opening a support case. If you do not, you can open a one-time case for a fee and they can help you triage this problem. 9600 baud, 8, n, 1. NULL model configuration.\nre: HTTPS: vs. HTTP. Just to be clear, it's not a security problem that caused the change. It was browsers. IE started preferring HTTPS which required users to set an exception for HTTP: sites. Then later browsers complained about the self-signed certificate used. Again requiring users to add an exception. Since the Java applet itself is secured there's no benefit to HTTPS over HTTP, HTTPS was removed.\nThanks Don for pitching in for help!\nWe have few critical dev VM's hosted in the underlying SAN and the upgrade is stuck in the middle? How should we proceed forward?\nI dont even see a way to cancel the on-going process and are running out of way to connect to DSUM.\nI dont even see a way to cancel the on-going process and are running out of way to connect to DSUM. Just worried if we will lose the connections to SAN storage.\nCan you access the Group CLI via SSH? Can you ping the array? Are you using dedicated management?\nIf so make sure the management cable on the other controller is connected. That being a common problem.\nThe upgrade isn't stuck in the middle, if you did the restart. The process first restarts the secondary to upgrade it, and when that is successful and reconnects to the active, the active controller fails over to complete the process.\nSince you got to the restart, the upgrade is complete.\nSo the array isn't down? You VMs are still OK? If so then we are dealing with a management \/ GUI issue.\nIf you can't ping or connect to the management port IP address, then using the serial port is next option.\nOur VM's are ok and storage of the VM's is also accessible. Only thing is we are neither able to ping the ip nor able to connect to the SSH. Connection to group manager itself is failing.\nOur Onsite resource is not available to check the cables and other connection things. We are expecting the resource will be available after 2 days. Will it be OK to postpone the activity for 2 days?\nIt's not going to get worse. The array is up and running so the upgrade process completed. It's just the out-of-band management. Since this is remote, I would strongly suggest a terminal server. They are not very expensive and would allow you direct access in case of problems like these.\nOnce this is corrected, are you planning on upgrading further? 8.x is older code, that's not maintained anymore.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hollywood star Joan Collins arrived in Amsterdam Thursday night.\nJoan Collins, the stage and screen star best known for her work on the American soap, \"Dynasty,\" brings her one-woman show to the Netherlands at the DeLaMar Theatre in Amsterdam on Friday and Saturday.\nIn One Night With Joan, Ms. Collins will share the stories and secrets of her celebrated career as one of the world's most glamorous and intriguing personalities. With the same engaging wit and frankness that turned her two memoirs Past Imperfect and Second Act into New York Times best-sellers Ms. Collins takes to the stage with honesty, style and grace.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"Beyond War: Waging Peace\" is the theme of Delta College's 19th Annual Global Awareness Program this year, and speakers discussed the origins and effects of the Palestinian-Israeli conflict Monday night.\nGlotfelty filmed a documentary \"Sucha Normal Thing: A Simple Journey into the Israeli-Occupied West Bank,\" and showed clips of her work during the talk.\nShe never hid her American heritage from people while filming, and said she experienced the opposite of the notion that \"Palestinians just want to get Jewish blood on their hands.\"\nShe spoke to both Israeli and Palestinian citizens to try to get the stories of what life is like during the occupation. She showed clips of Arabs picking olives in a untended grove, looking out nervously for extremist attacks.\nShe talked to people who spoke about terrorist groups in the West Bank that would destroy Palestinian storefronts and homes, leaving ghost towns. One woman, who was alive before fighting was rampant, told a chilling tale of a massacre.\n\"That was the most beautiful village,\" said the woman, who lived at a time before there were major conflicts. A day trip through an Arab village found the town empty on her trip back home.\nRenna gave an overview of the conflict, which he called a fight over \"bad land.\" Palestine's plight caught the attention and support of other Arab nations.\n\"The history of the Middle East can be told in committees,\" he said of citizens rejecting outsider efforts, such as a British Mandate in the 1920s and United Nation committees attempts in 1947, to split the country.\n\"It's easy for me to say i'ts a problem with real estate, but the hatred is so deep,\" Renna said.\nBoth speakers said it will take United States intervention to come up with a compromise, but with Hamas not recognizing Israel, Renna isn't hopeful.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Renowned Bitcoin investor and crypto-hedge-fund manager, Mike Novogratz believes that Bitcoin can touch $10000 by the end of this year.\nLegendary hedge fund manager and a billionaire trader \u2013 Michael Novogratz \u2013 is yet again seen pumping confidence in Bitcoin investors with his recent prediction.\nIt was just last month in October when Novogratz predicted Bitcoin to touch $10,000 in a time-frame of six months from then i.e. possibly my April 2018. However, these predictions were made before the announcement of Bitcoin-futures contract by CME group which catapulted Bitcoin prices to an unprecedented rally from over $5,500 to $8,000.\nNow, in his recent interview with Bloomberg, Novogratz seemed to have turned more bullish than he was a month earlier and has reduced the time-frame of his previous to the end of this year. This means that as per Novogratz Bitcoin prices could touch $10,000 by this year-end.\nNovogratz compared Bitcoin to an asset like gold calling it as the new 'digital gold' stating that \"gold has value solely because people say it has value; bitcoin is built on an amazing technology, there's a limited supply of it.\nThis whole revolution came out of a breakdown in trust in the 2008 crisis.\" In addition to this, Novogratz also believes that the second-most popular cryptocurrency Ethereum will also surge considerably and trade above $500 by the year-end.\nMike Novogratz is very bullish on the future of Bitcoins and has established a $500 million Galaxy Digital Assets Fund which is the biggest-of-its-kind crypto-hedge-fund. The fund will focus on investing in the blockchain and cryptocurrency space looking to its revolutionary applications and a great future potential.\nNovogratz's hedge-fund has already started active investments in the blockchain space with its first investment in Worldwide Asset eXchange (WAX) which is a decentralized blockchain platform for the development of secure and transparent online in-game virtual item exchanges.\nFollowing Novogratz's comments, Bitcoin prices were seen shooting up again above $8300 after the prices fell by over 5.4% on yesterday's news of $31 million worth Tether tokens getting stolen. Novogratz comments however instilled back the lost optimism in the market. It has to be noted though that Bitcoin has witnessed quite a bumpy ride in the past month with a huge volatility and price fluctuation.\nThe ambitious launch of $500 million worth crypto-hedge-fund marks Novogratz comeback as a prominent investor after suffering major losses at Fortress Investment Group LLC. Novogratz, who was away from Wall Street for over two years is back in action and is quite optimistic about the new blockchain revolution and the world of virtual digital currencies.\nHowever, many of his peers from the Wall Street are quite skeptical about cryptocurrencies with some few prominent personalities like Jamie Dimon even calling Bitcoin as a big \"fraud\". Neil Dwane of Allianz Global Investors said it's a \"scam for criminals around the world,\" while Larry Fink of BlackRock Inc. said Bitcoin to be the best tool for money laundering activities.\nIn addition to Novogratz, there are several other prominent figures who have increased their targets for Bitcoin prices after the recent bull run. Standpoint Research's Ronnie Moas has corrected his target second time this month from $11,000 to $14,000. There are several factors which can contribute to next upward journey for Bitcoin from here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"12 copies, 5 people are on the wait list.\nc1997. 1st ed. BasicBooks, xv, 270 p. ; 22 cm.\n7 copies, 5 people are on the wait list.\n Basic Books, xix, 294 p. ; 21 cm.\nThe classic, bestselling book on the psychology of racism -- now fully revised and updated Walk into any racially mixed high school and you will see Black, White, and Latino youth clustered in their own groups. Is this self-segregation a problem to address or a coping strategy? Beverly Daniel Tatum, a renowned authority on the psychology of racism, argues that straight talk about our racial identities is essential if we are serious about enabling communication across racial and ethnic divides. These topics have only become more urgent as the national conversation about race is increasingly acrimonious. This fully revised edition is essential reading for anyone seeking to understand the dynamics of race in America.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"All of the problems and persecution we face, whether it is something as small as being un-friended on Facebook or even sickness or being imprisoned because of your faith. God loves you and watches over you in this life and He calls those who will be useful to Him.\nWe have all been offered the chance at eternity through Jesus but few will accept it because of their prejudice or their hatred of the Jewish people or Christians in particular. God loves all of us and made sure that each of us has the chance for eternal life with Him, but each of us has to accept it and believe in the One Who provided the way.\n16 For God so loved the world that He gave the only begotten Son, so that everyone believing in Him should not perish, but should have eternal life. 17 For God did not send His Son into the world that He might judge the world, but that the world might be saved through Him. 18 The one believing in Him is not judged, but the one not believing already has been judged, because he has not believed in the name of the only begotten Son of God.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Security isn't just about protecting your assets \u2013 it's also about creating a positive atmosphere and protecting your reputation. Technology is creating new risks, but it's also giving you new ways to enhance your security and protect your members and your business. Here are some key areas to consider.\nWhen it comes to entry systems and lockers, fingerprint recognition offers more security than the traditional membership card swipe or lock. During the orientation process, you scan the member's fingerprints. Then you put scanners at the main entrance, at studio doors and on lockers, as appropriate. Then there's no more swiping membership cards. No more lost locker keys or forgetting to bring pound coins for the locker deposit. Members just press a finger to the keypad to get in and to access their locker.\nFingerprint recognition is particularly helpful for gyms open 24\/7 where there's no staff monitoring comings and goings. Fingerprint scans are impossible to fake, so you can be sure that everyone in your gym is there legitimately. Digital scanners keep a log in real time, so you know exactly who's in the building at any given time. And members are more confident their personal belongings are secure because they've got their electronic safety box.\nHaving car parks, corridors and communal areas under CCTV helps members feel safe, particularly at night. Innovations in digital CCTV are driving down prices and simplifying the monitoring process, with features like remote access, automatic number plate recognition, night-time lighting, HD resolution and analytics making it easy even for small and medium-sized gyms to boost security without having to make big investments in staff or process changes.\nAnd while security cameras themselves go a long way towards deterring crime, if you do experience a break-in, then a digital alarm system can mean the difference between an unsuccessful robbery attempt and an expensive lesson learned.\nThink about what happens when companies have cyber security breaches \u2013 last year, Three Mobile and Yahoo! were among those hitting headlines. This year, travel association ABTA, Xbox and Playstation have had issues. And that's on top of the big ransomware attack that affected the NHS and other organisations worldwide.\nAs you store an increasing amount of member and business information online, you need to have robust cyber security in place. Everything, including credit card numbers, must be securely encrypted so that it's unusable if anyone gains unauthorised access to your database.\nRegularly check all payment processing equipment for signs of tampering. Keep software updated to the latest versions. If security patches are released, update to them as soon as possible.\nPhysical copies of payment information must be securely locked away to prevent theft. Having a filing cabinet in a back office with a single key isn't a particularly reliable form of protection. This is another area to consider for fingerprint scanners, so you limit access to those who legitimately need to access the information.\nThere are many ways to make complete gym security affordable. You can lease CCTV, alarm systems, fingerprint scanners, security software and payment equipment, so you benefit straight away without tying up valuable working capital.\nSo don't scrimp on security. After all, these systems and procedures are your ultimate protection \u2013 not just for your equipment, but for your reputation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Spring Full Set Special! Get A Full Set Of Our Classic Individual Lashes $175!!** Call or Book Online (Now Two Locations)!\nBuy E-Gift Certificates for Mother's Day and get a free gift with the service (Full Set only)! Cash value does not expire, for details see below.\nMicroblading Special $400 Includes Initial Application and 4-6 Week Touch-up! Only At Eyelash Republic. Book Online!\nCascade Lash Studio and Eyelash Republic (located on Lake Eastbrook Blvd.) are sister stores with the same certified stylists and same high quality products. If you are booking online, please make sure to choose the correct location in which you wish to go. For any other questions or to book your lash fill, please contact us.\nOur advertised photos are the work of our staff. We do not advertise using internet stock photos.\nAt Cascade Lash Studio, we use only Semi-Permanent individual Eyelash Extensions. These are the highest quality of naturally light and soft faux mink lashes in a large variety of lengths and thicknesses. Our adhesive is medical grade for all skin types.\nEyelash Extensions are one of the hottest trends in the beauty industry and now they can be yours! You won't believe how such a small enhancement will make such a dramatic difference in how you see your face. Reduce your morning routine and dump your mascara! Twenty-four seven beauty is now simpler than ever, with fast and easy maintenance and uncomplicated after care.\nWhat are the Advantages of Eyelash Extensions?\nEyelash Extensions are a popular service where single strands of synthetic eyelashes, which are soft and curved to replicate a real eyelash, are applied to each individual natural eyelash one by one. Lash Extensions are perfect for special occasions or for day to day wear.\nPeople choose to wear lash extensions to lengthen their natural lashes, while others want their look to be fuller and more voluminous. Many women who have naturally light colored lashes choose extensions for a naturally darker and enhanced eye shape. Other clients love the different and fun looks they can achieve with highlighting, layering and the array of fun color options we offer.\nSemi-Permanent individual Eyelash Extensions can create a younger appearance on many clients. Often we hear from clientele who were contemplating having work done on their faces that they no longer feel the need to as a result of wearing our extensions. Our artistic application can create the appearance of a brighter, more open and rested eye, thereby giving a more youthful and rejuvenated look.\n** Limited availability. Offer subject to change anytime without notice. Only valid with participating stylists. \"Spring Full Set Special\" Offer cannot be combined with any other discounts, coupons or promotions. Use \" Spring Full Set Special\" if booking online. Offer ends on 05\/03\/2019.\nCheck out great coupons for cascade lash studio on LocalSaver!\nEyelash Extensions Grand Rapids, MI \u2013 Alexis | Powered by Mantra & WordPress.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In these lyrics I ask this age old question of being capable of maintaining peace of mind even in the midst of chaos of this world. A lot of times answers also come from these questions I ask through a song. And this was the case in my life.\nI was also imagining ocean waters through playing reverbed guitars, ocean as the ultimate soul cleanser. I love to imitate the fluidity of water through sounds.\nWith that verse I am using the power of music to send healing to my parents and to whoever is listening to the song.\nAnd of course, finishing with the triumph of love over anything. Love always wins in the end.\nAnd this is a story of the song \"Gravitude\" which is a word I made up that combines \"gratitude\" for all the lessons on my path and \"gravity\" as an earthly thing, a part of the human experience.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"As I am currently looking for a new server to host my files, the podcasts will be down from May 05 until further notice. I thank the listeners over the years. I hope to get it back up after my exam next week. Peace all.\nSeason 4 Episode One \u2013 We Shall Confound All the Listeners~!\nOn the second episode of the Mexipan Power of the Hour, I interview Sinbad of the Sinbad and Jakie Show. Plus, two new segments.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"'With its advanced technology, Comifo grows to be a company which designs, manufactures and sells to customers all over the world, from whom we obtain reputation and precious improvement advice to meet the world's verifying technology. This is so called INNOVATION, the spirit of the ever-lasting development of a company, impressing the world of machinery.\nComifo' s Research & Development Department generals technical and innovative duct machine designers who take training regularly. By doing this, top machine technology can be seen from a piece of Comifo's work, such as the patented hydraulic fold-seamer, as well as Comifo patented Automotion ..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Synaptic dynamics in complex self-assembled nanoparticle networks.\nWe report a detailed study of neuromorphic switching behaviour in inherently complex percolating networks of self-assembled metal nanoparticles. We show that variation of the strength and duration of the electric field applied to this network of synapse-like atomic switches allows us to control the switching dynamics. Switching is observed for voltages above a well-defined threshold, with higher voltages leading to increased switching rates. We demonstrate two behavioral archetypes and show how the switching dynamics change as a function of duration and amplitude of the voltage stimulus. We show that the state of each synapse can influence the activity of the other synapses, leading to complex switching dynamics. We further demonstrate the influence of the morphology of the network on the measured device properties, and the constraints imposed by the overall network conductance. The correlated switching dynamics, device stability over long periods, and the simplicity of the device fabrication provide an attractive pathway to practical implementation of on-chip neuromorphic computing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In this video, you'll see how to make perfect fried chicken. This fried chicken is better than the best. It's the bestest! What's the secret ingredient that makes this flavorful fried chicken always come out crispy on the outside and juicy on the inside? Cream of chicken soup! See how it's done. Watch the video, then get CANDY's 5-star recipe for Better Than Best Fried Chicken. Discover for yourself if it's the best ever!\nGreat idea for enhancing flavor.\nInteresting use of cream of chicken soup, going to try it.\nThe idea of chicken soup looks awesome! Going to try for Sunday dinner today.\nIf you like the taste of Japanese dishes, you will love it. Very crispy, and my friends like it too. You can buy joshinko (rice flour), katakuriko (potato starch), and sesame oil at asian market. If you live in large city, you may find them at American grocery store. Joshinko and katakuriko taste nothing different from regular flour, but they really help to make crispy fried chicken.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"DO Disrupt. This book is a self proclaimed workbook. With bold graphics, places to write, draw, create and list. This will make you think.\nMark Shayler started off working for large household names and eventually realised that no matter how many great ideas he came up with, they wouldn't be fully realised because it wasn't possible for these massive organisations to do. So now, he helps everyone else find those ideas, make those changes and disrupt the status quo.\nA real eye opener for anyone, whether you're thinking of starting a business, need to turn a business around or simply want to do something 'more' with yourself.\n178mm x 120mm x 128 pages.\nDO Design, DO Purpose, DO Fly, DO Open, DO Protect, DO Story, one of our notepads and pens to keep track of your thoughts.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"If you don't have both spices, it's okay to omit one or the other.\nPulse garlic, cilantro, parsley, cumin, fenugreek, and 2 teaspoons salt in a food processor until similar in texture to pesto. Add tahini and lemon juice; process 30 seconds (mixture will be very thick and gray).\nWith motor running, gradually drizzle in 3\/4 cup water and process, adding more water to thin if needed, until sauce is light green and the consistency of sour cream. Season with salt.\nSauce can be made 6 hours ahead. Store tightly covered at room temperature.\nMade this without any water to use as a spread. Excellent flavor!\nDidn't have fenugreek but this sauce is to die for! I want to put it on everything! It brings the falafel sandwiches up to another level.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you've ever taken the boat to Disney Springs, odds are you've seen the treehouses at Saratoga Springs from a distance. It's one of those places that just looks so cool and you think about how neat it would be to stay there \u2013 on this past trip, since we had a group of 7 people going, we were able to do it!\nWe rented DVC points for this past trip and everything went very smoothly. The treehouses can sleep up to 9 \u2013 there are 2 queen beds, a set of bunk beds, and pull out couch that holds two, and a recliner chair that unfolds to sleep 1. The nice thing is that there are three separate bedrooms \u2013 one for each queen bed, one with the bunk beds, and then the couch\/chair in the living room. Lots of space for everyone!\nThere are also two bathrooms which is nice when you have that many people. Then the rest of the treehouse is made up of a large living\/dining area and small kitchen. We were very comfortable there for 4 nights! There is an outdoor dining area too, but it was too cold that weekend to make much use of it.\nI had never stayed at Saratoga Springs before because the theme didn't appeal strongly to me, but it was actually much more beautiful than I was expecting. The main buildings and walkways are quite nice, and the treehouses are very secluded. It's a lovely area. Saratoga definitely has a peaceful feeling, at least being in the treehouses and walking around there did!\nThe theme is the city of Saratoga Springs and therefore horse racing, which is kind of funny theme for a Disney resort. But they do a good job with it \u2013 I like the late 1800's style of the buildings and the colors.\nOn the plus side, Saratoga Springs has boats that go to Disney Springs. It's close and a really nice ride. There was a boat dock right near our treehouse which was convenient for us too.\nI didn't eat at Saratoga Springs at all so I can't speak towards the food. But they do have a quick service dining location called Artist's Palette and a sit down restaurant called the Turf Club. My friends did get breakfast at Artist's Palette and were pleased they had Mickey waffles! There is no main building with food or anything at the treehouses, but on the flip side you do have the full kitchen. We had groceries delivered so that was convenient too.\nThere are a number of pools around the resort, some large and some small. There are water slides at the main pool as well as a kids' splash area. The treehouses have their own pool area, which includes its own hot tub! So that convenience of not having to go all the way down to the main building to use the pool or hot tub was really nice.\nHave you ever stayed at Saratoga Springs? What do you think of it?\nThese are so coll! It would be so fun to visit!\nThey were really awesome to stay in!\nThat's so cool that you can stay in a tree house! What a neat alternative to a regular hotel. Glad you enjoyed your stay!\nI love to try anything different, so the treehouses were a great option!\nThat looks like a really fun place to stay! I've never yet stayed on property at Walt Disney World, though!\nThere are so many great places to stay on property, I just want to try them all!\nI really love that WDW and the DVC has so many incredible, unique places to stay. They always seem to go above and beyond on not only experience but theming.\nI totally agree, Courtney. All of the Disney resorts have such great themeing, I really want to try and experience them all!\nI had never heard of Saratoga Springs before. I am learning a lot from your blog!\nSaratoga Springs is one of two Disney resorts that were originally built just for the Disney Vacation Club (the other is Old Key West).\nLooks beautiful and sounds like fun! Maybe I could convince the hubby to go\u2026..he could be secluded from the parks and stuff and I can just go on my own! lol!\nYes Marsha, I think my husband appreciated the woodsy secluded-ness of the treehouses!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Paediatricians are doctors who have specialised in the care of children. Their role is primarily with sick babies and infants, children who have severe acute illnesses such as pneumonia or meningitis, and the care of children with chronic conditions such as epilepsy, cerebral palsy, or asthma.\nPaediatricians do not routinely attend vaginal births, however a Paediatrician will attend a birth if there is a specific concern, for example multiple births, premature babies, breech deliveries, emergency caesarean sections, or a previously identified problem. Paediatricians routinely attend elective caesarean section births.\nPaediatric care is available at John Flynn Private Hospital 24 hours per day, 365 days per year. A Paediatrician will examine your baby in the first 48 hours of its life to ensure there are no apparent congenital abnormalities or other major concerns.\nYou may wish to choose a specific Paediatrician, alternatively a Paediatrician can be recommended by your Obstetrician. This consult is also an opportunity for you to discuss baby management issues, any family history of relevance, or other concerns you might have.\nThe Paediatrician may visit you and your baby, on a regular basis, during your hospital stay. They will look for, concerns with feeding; jaundice; and particularly to check that no heart murmurs develop.\nThe Paediatrician will also organise to see your baby several weeks later, to check that all the normal developmental milestones are being achieved and that your baby is developing a good routine.\nSubsequent to this visit, the baby will then be referred back to your General Practitioner for ongoing care, and would then only need to see a Paediatrician if there were specific concerns identified by the General Practitioner.\nYou will receive a bill for this medical care which may include a gap payment. Unless there are some specific concerns, a baby does not generally need to be admitted to the Special Care Nursery and therefore medical care will be as an outpatient. This may mean that you are able to utilise the Medicare safety net.\nSome parents may wish to have their sons circumcised. The policy of the Royal Australian College of Physicians is that there is no good medical indication to have this procedure performed. Circumcisions are not performed while babies are at John Flynn Private Hospital. This procedure can however be organised as an outpatient by the parents.\nThe Paediatricians attending John Flynn Private Hospital hope you have an enjoyable, supportive, and informative stay while at John Flynn Private Hospital, and a happy and healthy future with your child.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On March 8. 2019, Indian clients visited our factory for port crane project. They gave appreciation to the equipments of our factory and gave high evaluation of the technical level of workers and looked forward to the cooperation of related projects.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Velout\u00e9e Complexe is a blend of skin-loving lipids, plant extracts, and organic oils with rejuvenating and age-defying properties to make lips and eyes contours smoother, softer, and younger.\nThis In Fiore all natural healing balm is your go-to cure-all for thirsty lips and for most delicate eye contours. Velout\u00e9e Complexe does much more than seal and protect : it counters environmental free radical damage and effectively maintains skin's hydration, elasticity, and tone.\nProviding a potent boost of UV protection by stimulating the cell's protective mechanisms, In Fiore eyes and lip balm can be used to treat dry lips, surrounding upper and lower lip lines, or the orbital around the eyes.\nNatural eye and lips balm for all skin types.\nApply generously Velout\u00e9e Complexe to lips as needed; also use morning and night as a treatment for the lip line.\nFor the eyes; apply In Fiore eye balm morning and evening to orbital around the upper and lower area of the eyes using a gentle tap and press technique.\nShop here the Veloutee complexe eyes and lips balm and all In Fiore luxury organic skincare and natural beauty products at Ecocentric in Europe, Spain, Italy, Australia, Russia, UK and England.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"I feel like I'm losing my ability to understand reality; like when someone loses their hearing, they can still speak English, but their speech eventually becomes distorted because they can't hear themselves.\nHEY MASAHARU MORIMOTO! WHAT HAVE YOU BEEN UP TO?\nIf I were a customer, and I was given a dish with peppers, I would hate it. I also don't like blood sausage.\nANOTHER CASE OF WENDIGO PSYCHOSIS?\nMayang Prasetyo, believed to be a transgender prostitute, is believed to have been killed, dismembered and \"cooked\" by her chef boyfriend Marcus Peter Volke, 28, in their Teneriffe unit.\nPolice discovered a body part in a pot on the stove of his Commercial Rd apartment on Saturday night.\nPolice had been called to Teneriffe after concerns were raised over the occupants and an \"eye-watering\" smell in the unit block.\nMonkeys don't enjoy or appreciate flavours. Experts have told us that human beings are the only beings that can appreciate food at this higher level and the only living beings that cook.\nIf an architect makes a mistake, he grows ivy to cover it. If a doctor makes a mistake, he covers it with soil. If a cook makes a mistake, he covers it with some sauce and say's it is a new recipe.\nThere's a lot of processed food in America and I know that can make some tourists who're used to fresh food feel sick.\nIt is not just humans who are fascinated by watching others cook \u2013 chimps also pick up tips on how to prepare food.\nA study found that just as people learn new cookery techniques by watching celebrity chefs, chimps learn how to crack and peel fruit by studying their companions.\nThere's a fear of seafood. For some reason, people get nervous. Fish are complicated, expensive and easy to overcook. They're laced with small, sharp bones ready to choke the incautious diner. They smell.\nTo a table of vegetarians who had artichoke soup. I told them it was made with vegetable stock when it was chicken stock.\nPretty much every time I try something different or do something in front of a live audience, I truly think they might throw peanuts at me.\nMUNICH, Germany -- A chef to the stars has died following a fight that reportedly started with an argument over fried noodles.\nMiki Nozawa, 57, died from a cerebral hemorrhage on Monday after confronting two men outside a nightclub and demanding that they pay a bill left at his eponymous Asian diner on the German resort island of Sylt, officials said.\nAccording to German media reports, two customers had earlier left the restaurant after complaining to Nozawa about their beef noodle dish. The German tabloid Bild said the total bill was about $25.\nNozawa was beaten until he was unconscious, according to reports.\nForget all that fancy fresh stuff, all I really need is chili dog, some cheese fries,and a root beer. Now, that's what I call food.\nFor my 16th birthday, my parents took me to Auberge de L'ill, a three-star Michelin restaurant in Alsace. I loved the ballet of the waiters and the food. When the chef came to the table, my father said, 'My son is good for nothing. Do you need anybody to wash dishes?' The chef said, 'Come in next week.' That was the first restaurant I worked in.\nAt one time, I was traveling a lot and never in my restaurant. However, there are chefs in their restaurants all the time but doesn't mean it's any good.\nWhen we first opened Baohaus, some friends and I rolled in after a party at 1 a.m. We were like, 'Yo, customers! If you're down, we're going to smoke weed in here. We locked the door and hot-boxed it.\nThe Galloping Gourmet (1969-71), a show named for Kerr's onscreen persona, was taped in Ottawa and produced by his wife Treena Kerr.\nThe series was known for its lighthearted humour, tomfoolery and the copious use of clarified butter, cream and fat. Indeed, Kerr's most famous line on the show might have been his response to someone's criticism of his cooking: \"Madame, you could go outside and get run over by a bus and just think what you would have missed!\"\nDuring The Galloping Gourmet's successful run, he became a worldwide sensation, wrote an abundance of cookbooks, and earned two Emmy Award nominations.\nIf something doesn't taste good, I stop eating it.\n1. Put the oil in a saucepan. Chop the onion very fine, add to the pan and fry over low heat until softened. Add the peppers, salt, ginger and mixed spices, and cook for 10 minutes.\n2. Stir in the tomatoes, garlic, raisins and sugar. Add the vinegar; cook over very lot heat, covered, for 1 hour and 15 minutes, stirring occasionally. Uncover the pot and cook with the lid off for 5 to 10 more minutes.\nBroadcaster and food writer Clarissa Dickson Wright looks set to make fur fly after suggesting Britons should eat badgers.\nThe former star of TV's Two Fat Ladies said she enjoyed eating the creatures - now a protected species - when she was younger, and believes people should consume the bodies of animals which are killed as a result of culling.\nSome suggestions for dishes include badger pie, badger soup, badger salad, badger souffl\u00e9, badger milkshake and badger con carne.\nMallon was the first healthy typhoid carrier to be identified by medical science, and there was no policy providing guidelines for handling the situation. Some difficulties surrounding her case stemmed from Mallon's vehement denial of her possible role, as she refused to acknowledge any connection between her working as a cook and the typhoid cases.\nMallon maintained that she was perfectly healthy, had never had typhoid fever, and could not be the source. Public-health authorities determined that permanent quarantine was the only way to prevent Mallon from causing significant future typhoid outbreaks.\nI AM PUTTING THIS GUY IN THE CANNIBALS CATEGORY. WHAT SELF-RESPECTING CHEF WOULD SLOW COOK A DISH FOR DAYS WITHOUT HAVING A LITTLE NIBBLE AT LEAST?\nDavid Viens walked into his living room and panicked. He discovered that he had accidentally killed his wife, he said, and wanted to get rid of her body.\nThe chef said he thought of how easily he disposed of grease in his restaurant. So, he said, he stuffed his wife's body, face-down, into a drum of boiling water \u2014 and cooked it. For four days.\nDavid Viens said he packed his wife's 105-pound body into a heavy container and used weights to keep it submerged in the bubbling water.\n\"I just slowly cooked it and I ended up cooking her for four days,\" he said.\n\"You cooked on Dawn's body for four days?\" replied Los Angeles County Sheriff's Sgt. Richard Garcia.\n\"Before it was done,\" Viens said. It was unclear from the interview where the cooking took place.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Outdoor Graphic Banner that you can promote your business anywhere!\nPromo Pop Up Tradeshow Display; Promo Retractable Banner; Promo Banner. This banner is great for outdoor events because it can display anything you want on it. Also because of its size and how colorful it may be, it can be seen from a great distance. This banner is designable so you can but anything you would like on it. You may want to put your company's logo, a product, or your child's birthday celebration design. It is big and colorful so you will have no problem getting your message out to the public. This product if used for business purposes will attract more people to your business and give you more business.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Civ. No. 91-1828-R, Crim. No. 90-1088-R.\nRobert Kuhl, in pro per.\nMichael Wheat, Asst. U.S. Atty., San Diego, CA, for U.S.\nOn November 7, 1990, a federal grand jury returned a seven-count indictment charging Petitioner, Robert Lee Kuhl, with bank robbery in violation of 18 U.S.C. \u00a7 2113(a). Petitioner pleaded guilty to three counts of the seven count indictment on January 22, 1991, and entered into a plea agreement with Respondent United States of America. The text of the agreement is only known through the transcripts from the January 22, 1991, disposition hearing and the April 8, 1991, sentencing hearing.\nThis Court ordered the government to produce a copy of the plea agreement if any exists. Order of 12\/18\/92. Respondent replied that it was unable to locate a written copy. Respondent's reply to request for supplemental briefing, 12\/31\/92.\nMr. Cohen: Your Honor, the Government has agreed that at the time of sentencing in this case, in exchange for Mr. Kuhl's plea to counts 3, 5 and 7 of the indictment, that the Government will dismiss with prejudice all remaining counts of the indictment. In addition to that, the Government has agreed to recommend no more than the mid-range of the guidelines in this case. In addition to that, the Government has agreed not to object to the defendant's request for self-surrender to the institution designated by the Bureau of Prisons. And the defendant agrees that if the sentence in this case is within the guideline range, then he will waive appeal.\n\"RT-I\" refers to reporter's transcript of the hearings on January 22, 1991.\nThe Court: All right. Read the stipulation.\nMr. Cohen: The stipulation is, your honor, that the defendant did not carry a gun during the robbery charged in count 3 of the indictment as the probation report indicates and that there should be a two point upward departure because it was a robbery of a financial institution, and that was not taken into account by the old guidelines . . . and both parties are going to recommend forty-six months in custody, which is the low end of the current guideline range as calculated by probation.\n\"RT-II\" refers to the reporter's transcript of hearings on April 8, 1991.\nThe Court: You are advised that you have a right of appeal from this sentence. Or did you waive that right of appeal?\nMr. Wheat [for USA]: Did he waive it?\nMr. Cohen: Yes, it is waived, your honor.\nThe Court: He has waived it?\nMr. Cohen: Yes, your honor.\nRT-II at 11. While it would be preferable to have the agreement in writing, the form of the agreement as preserved in the record is the only form available.\nThe guideline analysis that was presented to this Court yielded an offense level of twenty-two, the corresponding sentence being forty-one to fifty-one months. Petitioner now argues that the two-level increase for crimes involving a financial institution, imposed under Section 2B3.1(b)(1), was applied erroneously. Petitioner contends that the guideline amendment cannot apply to his sentence because that amendment was not implemented until after he had committed his crimes. The sentence range for an offense level of twenty under the guidelines in place at the time was thirty-one to forty-one months. Petitioner asks for a reduction of his sentence to thirty-three months.\nI. Was the Sentence Improper?\nThe court shall impose a sentence of the kind, and within the range, referred to in subsection (a)(4) unless the court finds that there exists an aggravating or mitigating circumstance of a kind, or to a degree, not adequately taken into consideration by the Sentencing Commission in formulating the guidelines that should result in a sentence different from that described.\nTo determine the guidelines that are applicable to a defendant's sentence, the sentencing court must use the Guidelines Manual in effect on the date that the defendant is sentenced. United States Sentencing Commission, Guidelines Manual, \u00a7 1B1.11(a) (Nov. 1992). \"If the court determines that use of the Manual would violate the ex post facto clause of the United States Constitution, the court shall use the Guidelines Manual in effect on the date that the offense of conviction was committed.\" Id. \u00a7 1B1.11(b)(1). Also, as the Commentary to Section 1B1.11 notes, \"courts to date generally have held that the ex post facto clause does apply to sentencing guideline amendments that subject the defendant to increased punishment.\" U.S.S.G. \u00a7 1B1.11(b)(1), p.s.\nIn the instant case, Petitioner committed the bank robberies in July and August 1989. The two level increase at issue here is found in Section 2B3.1(b)(1) which states, \"If the property of a financial institution or post office was taken, or if the taking of such property was an object of the offense, increase by two levels.\" This amendment was effective November 1, 1989. By following the agreement between Petitioner and the government, and the representations proffered by the Petitioner, the government, and probation, this Court erred in its application of the two level increase: Petitioner's crimes had been committed before the guideline amendment and the imposition of a greater sentence violates the ex post facto clause.\nThe range that was considered by this Court, as proposed by the parties in the plea agreement, and as endorsed by probation, was forty-one to fifty-one months for a level of twenty-two. By removing the questionable two level increase, Petitioner would have been sentenced at a level of twenty, the applicable range being thirty-three to forty-one months. Since this Court finds that the sentence of forty-six months is outside the applicable guideline range, I must seek either a justification for the increase or decide what remedy, if any, is available to Petitioner.\nThe Court: Okay. I will tell you what is going through my mind and you can tell me why my thoughts may be in error. Forty six months seems kind of light for three bank robberies. Of course, I realize we are in the world of the guidelines and they think it is Okay.\nRT-II at 6. While this Court expressed concern at the suggested sentence of forty six months, there was no express finding to justify even that sentence, which this Court now acknowledges was not within the guideline range. No aggravating circumstances were presented and no uncharged offenses prompted an upward departure. Simple error on the part of all parties involved yielded a sentence that was greater than that mandated by the guidelines.\nHere, in spite of the best efforts and recommendations by experienced probation officers and counsel, all parties came up with the wrong result. The often confusing amendments to the guidelines have created the situation at issue here. Judge Kozinski's concerns expressed in another setting are equally applicable in the realm of the guidelines: \"By continually shifting the target at which we ask [judges] to aim, we no doubt make it harder for them to hit it.\" Fair v. Bowen, 885 F.2d 597, 602 n. 3 (9th Cir. 1989).\nThe fuel that powers the modern theoretical legal engine is the ideal of perfectibility \u2014 the concept that with the expenditure of sufficient time, patience, energy, and money it is possible eventually to achieve perfect justice in all legal process. . . . Yet a look at almost any specific area of the judicial process will disclose that the noble ideal has consistently spawned results that can only be described as pandemoniac. . . . Why, we ask ourselves, have such diligent attempts to create a perfect legal order fared so poorly in practice? . . . The answer, perhaps, may be found in the reason given by Macaulay for the failure of ambitious governments: the government that attempts more than it ought ends up doing less than it should.\nMacklin Fleming, The Price of Perfect Justice 3 (1974), citing Thomas Babington Macaulay, Critical and Historical Essays 335 (1890).\nAfter sentencing several hundred defendants under the sentencing guidelines and after diligent, repeated, and thoughtful study of the Manual, I am afraid that the government, in the hope that disparity would be eliminated, has attempted more than it ought and has ended up doing less than it should.\nSince the adoption of the sentencing guidelines, considerable thought, effort, scholarly comment, and judicial interpretation have been directed toward making the guidelines more workable, understandable, and harmonious with congressional intent. The fruits of this great effort include the unfortunate result in the instant case.\nAnyone who believes that the guidelines are any more comprehensible than the Internal Revenue Code should review, as an example, the recent amendment to Guidelines \u00a7 1B1.3 regarding relevant conduct. After reviewing page after page of amendments, examples, and references to other sections, one will be left with the distinct impression that confusion reigns and further amendments are on the way. One would conclude that the best efforts of the experts in the field would be insufficient to bring order out of chaos.\nAfter toiling to comprehend the guideline morass, one might ask, \"What should be done?\" Many thoughtful observers have reached the same conclusion that conductor George Szell did, when asked what might be done to fix the troublesome acoustics at Avery Fisher Hall. Mr. Szell offered a specific remedy: \"Tear it down and start all over.\" Hubert Saal, \"Brave New Hall,\" Newsweek, Nov. 1, 1979 at 59.\nAlthough guidelines cognoscenti recognize that any chance of the guidelines being abolished is nil, continued tinkering, amending, and adjusting will not make the guidelines understandable or workable, nor will the envisioned goals be any more achievable.\nAs pointed out by the Greek philosopher Zeno, \"Before a body in motion can reach a given point, it must first traverse the half of the distance; before it can traverse the half, it must traverse the quarter; and so on, ad infinitum. . . . Consequently, the goal can never be reached.\" See Zeller, Die Philosophie d. Griechen 54 (1876).\nAnd so it is in trying to find perfection in the guidelines. It cannot be done. In the meantime, confusion governs what the trial judges do, time is wasted, and, above all, justice has been replaced with the slide rule \u2014 an unfortunate state of affairs.\nJust as painting by the numbers is not art, sentencing by the numbers is not justice. Lamentably, this case is paradigmatic of what the guidelines have done and are doing to the criminal justice system.\nBased on the erroneous application of the sentencing guidelines, an inappropriate guideline was used to determine Petitioner's sentence. As the legal analysis will demonstrate, however, no remedy is available to this Court to reverse the sentence because Petitioner entered into an agreement with the government and waived his right to appeal.\nPlea bargains are contractual in nature and are therefore measured by contract law. \"[T]he negotiated plea represents a bargained-for quid pro quo.\" United States v. Escamilla, 975 F.2d 568, 571 (9th Cir. 1992). A plea agreement clause waiving a defendant's right to appeal is a bargained for element of the agreement. United States v. Gonzalez, 981 F.2d 1037 (9th Cir. 1992) (Kozinski dissenting). Under a plea agreement, a defendant receives the benefit of a favorable sentence that he otherwise might not have received. The government, in turn, receives the benefit of efficiently disposing of the case. Courts have consistently held that plea agreements waiving defendants' rights to appeal are valid and enforceable. See United States v. Bolinger, 940 F.2d 478, 480 (9th Cir. 1991); United States v. Navarro-Botello, 912 F.2d 318, 321-22 (9th Cir. 1990), cert. denied, ___ U.S. ___, 112 S.Ct. 1488, 117 L.Ed.2d 629 (1992).\nThis Court must now determine what happens when parties use a plea agreement to contract for a sentence that is not within the applicable sentencing guideline range. This court recognizes that all parties involved in the plea negotiations made a mistake. The Section 2B3.1(b)(1) amendment providing a two-point increase to crimes involving a financial institution became effective after Petitioner's offense and, had the Court and the parties been aware, the amendment would not have been applied. The government has upheld its part of the bargain in recommending a sentence consistent with the plea agreement. The agreement, however, was partially based on inapplicable guidelines.\nThis Court applies general principles of contract law to determine what remedies, if any, are available to Petitioner. In general a contract may be rescinded if both parties mistakenly assume a material fact. A mistake as to a collateral matter does not justify rescission. Williston On Contracts \u00a7 1544, at 94 (3d ed. 1970). A court may exercise its equitable power of reformation where a mistake is common to both parties and, by reason of the mistake, each party has done what neither intended. Id. at 9.\nIt is always a difficult task to attempt to determine what the parties understood the intent of the agreement to be. This case is no exception. Here, for example, the parties may have intended that Petitioner receive a mid-range sentence. See RT at 9. Petitioner's sentence of forty-six months is consistent with the \"mid-range\" intent since forty-six months is the mid-range of what was mistakenly believed to be the proper sentencing range.\nAlternatively, it is also possible that the parties intended that the sentence imposed be a \"light\" sentence. This Court indicated at the sentencing hearing that forty-six months was a \"light\" sentence for the offenses, although the Court followed the recommendation as consistent with the Federal Sentencing Guidelines. RT-II at 6. Also, the fact that the government dismissed four of the seven bank robbery counts, in exchange for Petitioner's plea of guilty to the three remaining counts, indicates that the parties' intention might have been a bargained-for exchange, involving the Petitioner receiving a \"light\" sentence.\nThis Court finds no clear indication of the parties' intent when entering the plea agreement. It would be incorrect to say that, as a result of a mistake, each party has done what neither intended. It is quite likely that both parties did what they both intended. This Court refuses to exercise its equitable power of reformation to \"correct\" the mistake.\nWhile there is not much law on the subject, courts deciding issues of mistake in interpreting plea agreements have not changed the underlying sentences based on error. For example, in United States v. Bos, 917 F.2d 1178 (9th Cir. 1990), the court upheld the use of the sentencing guidelines for arson where the defendant pleaded guilty to mail fraud. The court held that \"even if both parties had assumed the mail fraud guidelines would be applied, such a mutual mistake of law is not grounds for invalidating the plea bargain.\" Id. at 1182.\nIn United States v. Atkinson, 979 F.2d 1219, 1223 (7th Cir. 1992), a case decided on appeal, the parties partially based a plea agreement on an erroneous criminal history category, classifying the defendant as a career offender. Based on the mutual mistake, the government sought to reform the plea agreement to reflect the proper, lower criminal history category. The court held \"that while a contract may be reformed for a mutual mistake, this remedy is appropriate only when the contract fails to reflect accurately the terms of the agreement.\" Id. at 1223. The Atkinson court rejected the mutual mistake approach to reformation. \"We cannot accept the government's suggestion that we reform the plea agreement on the basis that the parties made a mutual mistake about its premises.\" Id. at 1223.\nThe Atkinson court also noted that \"many courts . . . have recognized that the contract law analogy has limits in [the plea] context and reformation seems to be one of them.\" Id. at 1223 (citing Carnine v. United States, 974 F.2d 924, 928 (7th Cir. 1992) (noting that plea agreements are \"unique contracts\")); United States v. Olesen, 920 F.2d 538 (8th Cir. 1990) (reversing the district court's reformation of a plea agreement).\nAfter rejecting the mutual mistake argument, the Atkinson court found the imposed sentence erroneous, and remanded the case to the district court for resentencing. For this Court's analysis, it is important that the mutual mistake approach was rejected. Because the Ninth Circuit has no ruling on this issue, I follow the persuasive argument of the Seventh Circuit and reject the mistake argument as well.\nSeveral remedies are available to a defendant who believes his sentence was imposed in error. For the reasons explained below, however, none of the remedies is available to Petitioner. Consequently, this Court must deny Petitioner's motion to correct his sentence pursuant to 28 U.S.C. \u00a7 2255.\n(3) is greater than the sentence specified in the applicable guideline range to the extent that the sentence includes a greater fine or term of imprisonment, probation, or supervised release than the maximum established in the guideline range, . . .\n18 U.S.C. \u00a7 3742(a). The sentence at issue here clearly falls within Section 3742(a)(2) because an incorrect guideline range was applied. However, Petitioner waived his right to appeal pursuant to a plea agreement with the government.\nWaivers of appeal are permissible and serve important policy concerns. Navarro-Botello, supra. The Navarro-Botello Court noted that the \"Supreme Court has found that knowing and voluntary constitutional waivers do not violate due process.\" Id. at 321 (citing Town of Newton v. Rumery, 480 U.S. 386, 393, 107 S.Ct. 1187, 1192, 94 L.Ed.2d 405 (1987)). The Court also held that it is not a violation of due process for a defendant to waive, in an otherwise valid plea agreement, the statutory right of appeal. United States v. Navarro-Botello, 912 F.2d at 321.\nIn this case, the record shows that the Petitioner intelligently and voluntarily waived his right to appeal. See RT-II at 11. Petitioner was therefore unable to raise his arguments on direct appeal.\nIn a case similar to the instant case, the Ninth Circuit upheld the defendant's waiver of appeal, despite the fact that the sentence imposed was greater than that mandated. United States v. Bolinger, 940 F.2d 478. The Bolinger court stated, \"Because we enforce Bolinger's waiver of his right to appeal the sentence, we do not consider his underlying claims that the district court misapplied the guidelines.\" Id. at 480. If a defendant has waived the right to appeal, a defendant has waived the right to challenge improper application of the sentencing guidelines on appeal.\nThe Bolinger court also noted that a waiver of the right to appeal would not prevent an appeal where the sentence imposed is not in accordance with the negotiated settlement. Id. This exception is not applicable here. The court-imposed sentence was within the negotiated settlement. Petitioner was thus unable to appeal his sentence since he knowingly and intelligently waived his right to appeal.\nThe Sixth Circuit addressed a similar issue in which the sentencing judge departed upward without giving sufficient findings of aggravating circumstances. United States v. Newsome, 894 F.2d 852 (6th Cir. 1990). In Newsome, the defendant entered into a plea agreement with the government that set a sentencing cap at 57 months. This cap was agreed to because of defense counsel's miscalculation of the guideline range. Id. at 854. At the sentencing hearing, the probation officer who prepared the presentence report determined that the range was only 33-41 months. \"No one challenged the probation officer's 33-41 month calculation, and the district court appears to have accepted it as correct.\" Id.\nNewsome's sentencing judge acknowledged the correct sentencing range and intentionally sentenced Newsome to 57 months, a punishment outside the applicable guideline range but within the plea agreement cap. On appeal, the court found that \"defendant's miscalculation of the guidelines could hardly constitute . . . an aggravating circumstance, and neither could the defendant's desire to insure that in no event would he spend more than four years and nine months in prison.\" Id. at 856.\nWhile Newsome appears to be directly on point, there are several differences that make the analysis less useful to this Court. First, Newsome did not waive his statutory right to appeal his sentence pursuant to a plea agreements as did Petitioner here. If he had, the Sixth Circuit might not have addressed the issue at all. Second, Newsome's sentencing judge was aware of the actual guideline range and intentionally decided to impose a higher sentence. Here, this Court was unaware of the correct guideline range, so there was no intentional departure. For purposes of this motion, this Court presumes that the imposed sentence was incorrect. The remaining issue is if Petitioner, having waived his right to appeal, may challenge his sentence in another way.\nIn fact, this Court was told by counsel for Petitioner, by probation, and by the government that the sentence imposed was correct as calculated under the guidelines.\nThis Rule provides a defendant with a remedy for an incorrect sentence. According to this rule, \"The court, acting within seven days after the imposition of sentence, may correct a sentence that was imposed as a result of arithmetical, technical, or other clear error.\" Fed.R.Crim.P. 35(c). While this rule would ordinarily help a defendant whose sentence was improperly imposed, the error was not brought to this Court's attention during the seven days after sentencing. This remedy is thus unavailable to petitioner.\n\"Errors in naming the rule or statute under which a defendant is proceeding should not be held against him, and . . . if relief is available under any of these procedures it ought to be given.\" Charles Wright, 3 Federal Practice and Procedure: Criminal 2d \u00a7 583 at 392-93 (1982). Here, this Court could conceivably bring its own motion under Fed.R.Crim.P. 36 to correct an error in the sentencing.\nRule 36 is similarly unavailing. Although the rule permits correction of clerical mistakes in the record \"arising from oversight or omission,\" the Ninth Circuit has held that the provisions of Rule 36 \"do not permit a substantive change in the period of incarceration that the defendant must serve. A change made under Fed.R.Crim.P. 36 can do no more than conform the sentence to the term which the record indicates was intended.\" United States v. Kaye, 739 F.2d 488, 490 (9th Cir. 1984). Petitioner has no remedy under this rule.\nUnder 18 U.S.C. \u00a7 3582(b)(2), a person sentenced may move to have a sentence reduced or modified if the guidelines have subsequently been amended and if the reduction is consistent with applicable policy statements issued by the sentencing commission. See also 18 U.S.C. \u00a7 3582(c). Section 3582(b) and (c) are unavailing here, too, since the sentencing guidelines were not amended after the imposition of Petitioner's sentence and thus a modification would not be consistent with an applicable policy statement. See Guidelines \u00a7 1B1.10(d). Petitioner has no remedy under \u00a7\u00a7 3582(b) or (c).\nThe Advisory Committee notes to Fed.R.Crim.P. 35(c) state that Rule 35(c) provides \"an efficient and prompt method for correcting obvious technical errors that are called to the court's attention immediately after sentencing. . . . The Committee's assumption is that a defendant detained pursuant to such a sentence could seek relief under 28 U.S.C. \u00a7 2255 if the seven day period provided in Rule 35(c) has elapsed.\" Petitioner seeks relief under Section 2255.\nRespondent argues that Petitioner's waiver of his right to appeal his sentence is binding on his right to seek collateral review under Section 2255. This is an unsettled area of the law.\nLike the right to bring a direct appeal of his sentence, the right Abarca seeks to exercise in bringing a collateral attack is statutory. A knowing and voluntary waiver of a statutory right is enforceable. While we do not hold that Abarca's waiver categorically forecloses him from bringing any section 2255 proceeding, such as a claim of ineffective assistance of counsel or involuntariness of waiver, the question of the degree of his culpability is an issue clearly contemplated by, and subject to, his plea agreement waiver.\nId. The Ninth Circuit has thus held that a waiver of the right to appeal also waives the right to raise a collateral challenge under \u00a7 2255 in most circumstances.\nThe recent Ninth Circuit opinion does not, however, extend to the facts of the instant case. On the subject of appeal from a mistake in sentencing, there is scant law available. At least two opinions have indicated that a Section 2255 petition might be available to an improperly sentenced defendant who has waived his right to appeal. In United States v. Rutan, 956 F.2d 827, 829 (8th Cir. 1992), the court noted that \"an illegal sentence can still be challenged under 28 U.S.C. \u00a7 2255 for habeas corpus relief, so a defendant is not entirely without recourse from an erroneous sentence.\"\nIn all other cases, if that plea bargain contains a no-appeal clause, I would dismiss the appeal. An alternative approach might be to remand automatically. . . . Or we might rule that defendants who have given up their right to appeal may challenge the validity of the plea agreement (including the government's compliance therewith) only by way of collateral attack, as we have done for claims of ineffective assistance of counsel.\nId. at 1043. Judge Kozinski thus suggests that a collateral attack might be permissible. The lack of jurisprudence on this issue, leads this Court to believe that the issue of waiver of appeal on a defendant's right to attack a mistake in sentencing collaterally is unsettled in this circuit.\nAs a first step, this Court distinguishes the facts in Rutan and Gonzalez. The Rutan court merely noted that a petitioner who has waived his right to appeal may still collaterally challenge an illegal sentence. In the instant case, the challenged sentence was the specific sentence agreed to by all parties involved. Rutan is therefore inapplicable.\nSimilarly, Judge Kozinski's Gonzalez dissent does not address the facts as presented here. In Gonzalez, Judge Kozinski merely suggests a possible remedy that could be made available to petitioners who have waived their right to appeal. First, there is no indication that Judge Kozinski, the lone dissenter in Gonzalez, would apply his conjecture to the instant facts. Second, the holding in Abarca may have settled the area of law raised in Judge Kozinski's dissent. Finally, given the important policy concerning waiver of the right to appeal, it would be improper to extend \u00a7 2255 protection to anything other than a charge of ineffective assistance of counsel or involuntariness of waiver.\nPetitioner here permissibly waived his statutory right to appeal when he knowingly and voluntarily entered into the plea agreement with the government. The sentence imposed on Petitioner was in accordance with the negotiated agreement. Navarro-Botello, 912 F.2d at 321. Petitioner should not be permitted to raise this issue collaterally.\nCompelling issues of public policy strongly support the use of plea agreements. \"Plea agreements are an important \u2014 indeed essential \u2014 component of our criminal justice system [providing] substantial benefits to the government, to our courts, to the public, and to criminal defendants.\" See United States v. Gonzalez, 981 F.2d at 1040 (Kozinski dissenting); Navarro-Botello, 912 F.2d at 321. Plea bargains are important to save the government money and time, which is a benefit to the public as well. Rumery 480 U.S. at 393, n. 3, 107 S.Ct. at 1192, n. 3.\nIf valid plea agreements which include a waiver of the right to appeal are not enforced, the government will be reluctant to negotiate such agreements. Decreased incentives will remain for the government to enter into a plea agreement. The detrimental result would be an even more crowded criminal docket.\nFinally, and perhaps most importantly, a weighty benefit of plea bargaining is the finality that results from such agreements. Navarro-Botello, 912 F.2d at 322. Without finality, most benefits of a plea agreement are lost. By permitting collateral attacks by defendants who have waived the right to appeal, this Court would open the floodgates to relitigation of all issues. There would be no finality.\nPetitioner here waived his right to appeal in exchange for a promise from the government to dismiss four counts of a seven count bank robbery indictment. Dismissal of the four counts was a substantial benefit to Petitioner. Petitioner may not now ignore his part of the bargain.\nPetitioner's collateral attack is based on the ex post facto clause. The Ninth Circuit has consistently recognized that where guideline amendments are ex post facto, use of the earlier guideline applies. See United States v. Castro, 972 F.2d 1107, 1112 (9th Cir. 1992), cert. denied ___ U.S. ___, 113 S.Ct. 1350, 122 L.Ed.2d 731 (1993) (remanding for resentencing under guidelines in effect at the time of the offense so as not to violate ex post facto clause). There is no question in this case that by virtue of the ex post facto clause, U.S. Const. art I, \u00a7 9, cl. 3, Petitioner's offense level was determined under the wrong guidelines. And, had Petitioner not waived his right to appeal, such error could have been corrected by appeal or by collateral attack. But this is not the case here.\nThis court holds that Petitioner has waived his right to a collateral attack when he knowingly and voluntarily waived his right to appeal.\nThis Court acknowledges that an error was committed at Petitioner's sentencing hearing. All parties involved \u2014 the Defendant, the government, probation, and this Court \u2014 believed that the proper guidelines were being used. I have since undertaken extensive research into the issues discussed above and have even explored all possible remedies, many of which were not presented by Petitioner. Each avenue of relief has resulted in a dead-end.\nSince Petitioner waived his right to appeal, he reduced his options for relief. Because Petitioner waited eight months to bring his motion, other options were foreclosed. Now, Petitioner may not challenge the sentence to which he agreed under an agreement to which he was a party.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"grilling plate non stick removable plate electric grill pans and safety thermal cut out home.\nwindow film bathroom bathroom window film elegant bathroom window film and bathroom window designs for fine bathroom window designs bathroom window film.\n2 x 3 frame 2 x 3 acrylic frame 2 x 3 framed mirror.\nshark vacuum rocket duo clean fascinating shark rocket complete duo clean reviews shark rocket complete duster shark rocket reviews.\nfabric paint brush 9 green acrylic fabric paint brush paint roller refill sleeve for paintings.\nover the counter oven quick view.\nduraflame infrared quartz fireplace infrared quartz fireplace quartz fireplace quartz fireplace chase traditional family room infrared quartz fireplace insert.\nart hanging clips 2 proper alignment instantly.\nnest thermostat 3rd generation installation nest thermostat wiring nest generation.\nfluorescent black light paint face body glow in the dark paint kit.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Recovery is full of emotional challenges as you come to terms with accepting your past, making amends, feeling your feelings, forgiving yourself and rediscovering your new sober self.\nWhile you're dealing with a host of toxic emotions on your journey toward sobriety, one of the most destructive ones is shame. What is shame? It's that pesky voice in your head that says \"I'm a bad person and there's nothing I can do about it, so I might as well just continue behaving badly.\" Shame can shut you down and stop you from moving forward and making positive changes. It's incredibly painful and has been linked to low self-esteem and poor self-image as well as addiction and depression.\nWhile you may be struggling with shame during recovery, this unhealthy emotion has likely been bubbling up for years. Experts say that shame can stem from childhood neglect or trauma (physical, emotional or sexual abuse, bullying). It may have even contributed and fueled your addiction. Many people turn to drugs or alcohol to suppress these feelings and then find they're unable to stop using, which results in more of the same.\nTreat yourself like a friend. What would you tell a friend if he told you that he wasn't good enough or didn't deserve a chance at a happy, sober life? Now say these words to yourself. This exercise can help you forgive yourself, which can help stop shame in its tracks.\nAcknowledge and share it. When you believe you're flawed or unworthy of love and happiness, the last thing you want to do is tell people about it. But hiding your shame just gives it more power. \"When we bury the story, we forever stay the subject of the story,\" writes shame and vulnerability researcher and author Bren\u00e9 Brown, PhD, LMSW. \"If we own the story we get to narrate the ending.\" Plus, blocked or repressed emotions can put added stress on your mind and body, resulting in mental ills as well as heart disease, intestinal problems, headaches, insomnia and autoimmune disorders.\nQuestion your feelings. Should you be ashamed? Or should you just feel guilty? There is a difference. Experts define shame as \"I am bad.\" Guilt, on the other hand, means \"I did something bad.\" Although guilt is painful, the ability to recognize that your own actions were harmful and to feel remorse are signs of emotional health. Writing in a journal can help you untangle your emotions and better understand what you really should be feeling.\nLearn your triggers. Shame is most often fueled by negative self-talk or telling yourself that you're not enough. For example, you might think to yourself that you're not good\/tough\/caring\/successful enough. Think about the times in your life when you've felt this way \u2013 and what could help turn those messy emotions into positive messages.\nAt Spearhead, we'll help you learn how to identify and cope with shame and other toxic emotions that can harm your recovery. To learn more about our gender-specific treatment, call today: 866-905-4550.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbnzbr b/data_all_eng_slimpj/shuffled/split2/finalzzzbnzbr
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+{"text":"Of 2017 pre-collection of CHANEL latest. Two bracelets which cute charm and logo mark are very luxurious, and are luxurious! I show big presence at hand and am most suitable for an accent of the time of light clothes! Of Chanel and sense of quality settles coordinates neatly elegantly. Brightness of the gold matches a casual fashion in a chain design. Take one by all means!as a staple item! Please bring us one by all means!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Start now: It's never too early to begin laying the groundwork for a solid grant application. Map out the steps required to create a persuasive proposal (see below for ideas) and mark them on your calendar. Include time for discussions with partners and program officers, as well as time for writing, reviewing, and revision.\nBegin with people: You will need more than data to write the proposal. Audience, real-world end users, and partners play key roles in a grant application. \"Before you start your research proposal, get out of the office and talk to real people whom you think will use or benefit from your proposed solution,\" says Dr. David Hobson, Manager, Technology Transfer & Entrepreneurship.\nDevelop a knowledge mobilization plan: A good plan that includes information about how your research findings will be communicated to end users is invaluable. Melissa MacKay, Knowledge Mobilization Manager recommends websites such as The Ontario Centre for Excellence in Child and Youth Mental Health as an excellent resource for tips on developing a knowledge mobilization plan. In addition, Research Impact Canada has developed a checklist for reviewers to assess the impact strategy. \"Ideally knowledge users should be engaged early in the research cycle in order to best understand their needs, as well as to co-create the research plan and outputs,\" MacKay says. \"This relationship building ensures the results of your research will be relevant, timely, and useful for the end users. \"\nGet to know the program guidelines: Not all grant applications ask the same questions or require the same information. Carefully read the application and compile a list of questions before you cut and paste materials prepared for other applications into any new grant application form.\nGet to know the program officer: After you've reviewed the application and compiled your questions, schedule a call with the program officer identified in the application materials. This person understands the application requirements and process. Submitting a solid, clear application makes his\/her job easier, so these contacts are genuinely motivated to help you. Don't hesitate to call on program officers when needed -- even if it's more than once.\nBuild on previous success: Talk to researchers who have previously applied to the program. \"If colleagues have been successful in previous funding calls, ask if it is possible to see an example of a successful application,\" encourages Gregor Lawson, Manager, Industry Liaison.\nEmpathize with the Adjudicators: Don't assume that reviewers will know your subject area as well as you do. Keep them in mind when writing the grant and be sure to communicate the value of your research in clear, easy to understand language that requires no technical expertise to be understood. If you're uncertain about your wording, speak with the program officer.\nConnect the dots when it comes to numbers: Numbers are crucial, but they need context. \"Provide quantifiable figures around the benefits of your research. How will the expected outcome benefit stakeholders, industry partner(s), and the economy?\" says Vanja Djukic, MSc, Industry Liaison Officer.\nReview the application - twice: Once you and your team have gone over the application and feel it is complete, get a fresh set of eyes to read it once more. Make use of on campus resources to review applications prior to submission; Research Services, Industrial Liaison Officers and College Research Managers are often able to help if given adequate time. Of course, if questions or omissions arise, revise the application one more time as needed before submitting.\nFollow these steps and you'll be well ahead of most applicants.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This kitchen and dining space has two main features.\nThe double glazed roof lantern floods the space with daylight, and provides ventilation via a motorised openable section. The full height aluminium bi-fold doors provide views of the garden and access to the patio which is raised to match internal floor level.\nWhite painted walls and neutral flooring reflect natural light emphasise the feeling of space.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"Zolotoy Vek\" is a jewelry brand, which is one of the largest manufacturer of high quality jewelry in Ukraine, it has more than 200 brand stores all over Ukraine.\nJewelry store \"Zolotoy Vek\" presents a wide range of products from red, white, lemon gold and silver. Exquisite rings, pendants, earrings with phianite, diamonds, pearls, precious and semiprecious stones, bracelets and chains, rings, men's and women's accessories will help you create an unforgettable image.\nCollections of jewelry are constantly updated in accordance with the latest trends in jewelry fashion and are pleased with their novelties.\nYou will find a comfortable atmosphere, promotions and discounts for the presented products from -38% to -65%. For regular customers, a discount program operates. And if you do not know what jewelry to choose to a loved one - a gift certificate will be an excellent decision for any holiday!\nIn \"Zolotoy Vek\" you can not only choose a product of high quality at an affordable price, but also keep your happy moments in jewelry!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The best price we currently have listed for the product 9ct Yellow Gold 7.5inch Figure-eight Link Chain Bracelet Br898-07 is \u00a3130.00 . The Jewel Hut are currently offering 9ct Yellow Gold 7.5inch Figure-eight Link Chain Bracelet Br898-07 at that price. Of course, we always want you to be able to buy 9ct Yellow Gold 7.5inch Figure-eight Link Chain Bracelet Br898-07 at the best possible price we can find and only from a retailer we know you can trust.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"What really happens behind the scene?\nYou see a nicely s taged auction floor. Some items have lot numbers written on them. Some items are in small boxes called flats with a mixture of things. Items are lined up around the walls, or on tables for people to look at during preview hours. Someone is selling food. Yummy food. Chairs might be set up for sitting in (for sure in an auction house).\n\u00ab previous article Reserve, Absolute, Minimum? What's that?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Thanks for getting in touch with Muse.\nOne of our booking team will be in touch in the next few days to arrange your portrait experience.\nIf you'd like to know more about the shoots, or would like to book right away, please give the studio a call - our number is below.\nPlease add info@museportraits.co.uk to your safe senders list.\nCan't wait to meet you all soon!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Of course, the main benefit is that you simply can get the necessary daily amount of omega 3 content without eating seafood five or six times per 7 day. This also avoids the contaminant problem that I discussed earlier.\nA recent Natural Select CBD Review published in the Journal for this American College of Cardiology looked at 40,000 people from a few different studies. Their conclusion was that people need a t least 1\/2 to at least gram of omega 3 per day for General Health.\nAll these are Healthy Supplements you may add to your canine's food to more receptive to what you feed that company. Of course, their reaction as well be telling you something with regards to the quality from the dog food brand the buying. Be certain to get brands without a lot of filler. WikiHow has helpful advice on what ingredients to find out when selecting dog diet plan. Don't go cheap when you are considering feeding your pet and Natural Select CBD Tincture somebody less fortunate less as well as behavioral troubles with them. Try these as well as both appreciate dog possibly be happier.\nIt is often a conventional pop or beverage in London. It is used worldwide for a blood unit. It also assists in prevention from diseases such as: Natural Select CBD Review arthritis, loss of hair and diabetes.\nAdding soy based products to eating habits will create an increase in estrogen levels for man. The excess estrogen offer down the testosterone levels, and beneficial body at the more balanced hormonal extent. In Japanese men there a dramatically reduced rate of prostate malignancy.\nYou should lose weight to Improve health not even though it is often a fad or another type like of which. If you undergo a weight loss plan that includes eating right and exercise so you'll feel better and convey more energy then you have made a massive step transfer.\n- Following a cleansing diet can be easier. Restricting your food to a cleansing meals are not done overnight, but better when eased towards. When done right, you replace bad foods with good as well as taking supplements, which can help complete the hunger gap. Making use of help you stay on list.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"She comes stepping out of a cloud of fog, elegant, mysterious... and straight into your wardrobe!\nBecause Deadly Beloved is a fairy tale, with her soft lace with stretch that dramatically runs down to the floor on the sides! The dress is 'strategically' lined, namely along the chest and waist till above the knee. So the diaphragm and a part of the legs are bare, but it leaves enough to the imagination. Just like the straps! The sides run down the furthest, with a height of 1.75 they reach the ground.\nGet ready for heads turning, enjoy it!\nMaterials: 100% nylon. Lining: 95% polyester - 5% elastane. Washing instructions: Washing machine on a cold setting, with similar colours on a delicate program. Do not machine dry, hang to dry, do not dry clean. Lace is like love - be carefull with it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Park-Like Setting...Pride of ownership is evident throughout this well maintained home with 3 bedrooms, bright kitchen w\/ island, fully refinished original hardwood floors, rec. room and workshop in the basement and plenty of storage space.Soak in the sunshine and take in the beautiful garden views from the bright sunroom just off the kitchen. Updates to this one owner home are newer roof, dbl glazed low-e vinyl windows up, and hot water tank. This home sits on a 7912 sf south facing lot close to parks, shopping and transit.\nFloor Area 2,377 Sq. Ft.\nLot Size 7912 Sq. Ft.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbozpf b/data_all_eng_slimpj/shuffled/split2/finalzzzbozpf
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+{"text":"James Boyle The Public Domain Enclosing the Commons of the Mind. 00 FL= Film is in Foreign Language. Wallace was raised in the Brooklyn borough of New York City.\nNotorious BIG Machine Gun Funk. The film stars Jamal Woolard Anthony Mackie, Derek Luke, Angela Bassett was released by Fox Searchlight Pictures. May 24, \u00b7 \" Gimme The Loot\" by The Notorious B. Notorious Big Gimme The Loot is popular Free Mp3.\nCom: News analysis commentary research for business technology professionals. From desktop or your mobile device. BA= Color Box Art Available for an additional $ 3. Just click download mp3. Lbx= Letterboxed or Widescreen format. CONSOLIDATED MINI CATALOGUE.\n\u7f51\u6613\u4e91\u97f3\u4e50\u662f\u4e00\u6b3e\u4e13\u6ce8\u4e8e\u53d1\u73b0\u4e0e\u5206\u4eab\u7684\u97f3\u4e50\u4ea7\u54c1, \u4f9d\u6258\u4e13\u4e1a\u97f3\u4e50\u4eba\u3001 dj\u3001 \u597d\u53cb\u63a8\u8350\u53ca\u793e\u4ea4\u529f\u80fd, \u4e3a\u7528\u6237\u6253\u9020\u5168\u65b0\u7684\u97f3\u4e50\u751f\u6d3b\u3002. Our complete list of Netflix Australia movies is updated daily, so use our helpful tool to find if that movie you' re looking for is available to stream. Gimme the loot notorious big free download.\nFor example enter \" giraffe\" , you' ll get back words like \" gazellephant\" \" gorilldebeest\". When he released his debut album Ready.\nFree biggie smalls gimme the loot mp3 music download easily listen download biggie smalls gimme the loot mp3 files on Mp3Juices. Online shopping from a great selection at Movies & TV Store.\nMp3 320kbps download songspk, instamp3, musicpleer, mp3goo, emp3z youtube. 07 MB) Free Notorious Big Gimmie The Loot mp3 download.\nThe Notorious Big Gimme The Loot Mp3 free download, Gimme The Loot. Get lyrics \u266b music videos for your iPhone\u00ae. Enter a word ( two) above you' ll get back a bunch of portmanteaux created by jamming together words that are conceptually related to your inputs.\nPort Manteaux churns out silly new words when you feed it an idea or two. Gimme the loot notorious big free download. Pour t\u00e9l\u00e9charger et voir les films en streaming gratuitement sur notre site enregistrer vous gratuitement.\nThat follows the life murder of Christopher Wallace an American rapper better known by the stage name \" The Notorious B. Donate Bitcoins Donate via Mail: Brother Nathanael Foundation PO Box 547 Priest River, ID 83856.\nCopyright \u00a9 by James Boyle. Notorious Big Gimme The Loot Download is popular Free Mp3. Vidme \u2014 the world' s most creator- friendly video platform. Notorious is a American biographical drama film directed by George Tillman Jr.\nGimme the loot notorious big free download. Download GIMME THE LOOT by THE NOTORIOUS B. You can download or play Notorious Big Gimme The Loot Download with best mp3 quality online streaming on MP3 Download.\nPROMOTE YOUR MUSIC\/ VIDEO ON WAPARZ ( CLICK HERE ) Home Sitemap. The author has made this online version available under a. # 1 rated music site. You can download or play Notorious Big Gimme The Loot with best mp3 quality online streaming on MP3 Download.\nStream Gimme The Loot by Notorious B. - Gimme The Loot Instrumental.\nJun 10, \u00b7 Original version of Gimme The Loot by Biggie. \" by The Notorious B. Listen ad- free with YouTube Red; Show more Show less.\nMobile optimized. Watch the video for Gimme the Loot from The Notorious B. ' s Ready to Die for free, and see the artwork, lyrics and similar artists.\nqq\u97f3\u4e50\u662f\u817e\u8baf\u516c\u53f8\u63a8\u51fa\u7684\u4e00\u6b3e\u7f51\u7edc\u97f3\u4e50\u670d\u52a1\u4ea7\u54c1, \u6d77\u91cf\u97f3\u4e50\u5728\u7ebf\u8bd5\u542c\u3001 \u65b0\u6b4c\u70ed\u6b4c\u5728\u7ebf\u9996\u53d1\u3001 \u6b4c\u8bcd\u7ffb\u8bd1\u3001 \u624b\u673a\u94c3\u58f0\u4e0b\u8f7d\u3001 \u9ad8\u54c1\u8d28\u65e0\u635f\u97f3\u4e50\u8bd5\u542c\u3001 \u6d77\u91cf\u65e0\u635f\u66f2\u5e93\u3001 \u6b63\u7248\u97f3\u4e50\u4e0b\u8f7d\u3001 \u7a7a\u95f4\u80cc\u666f\u97f3\u4e50\u8bbe\u7f6e\u3001 mv\u89c2\u770b\u7b49, \u662f\u4e92\u8054\u7f51\u97f3\u4e50\u64ad\u653e\u548c\u4e0b\u8f7d\u7684\u4f18\u9009\u3002. Christopher George Latore Wallace ( May 21, 1972 \u2013 March 9, 1997), better known by his stage names The Notorious B.\n, Biggie, or Biggie Smalls, was an American rapper. He is ranked by Billboard as among the ten greatest rappers of all time.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This is an alphabetical list of all my stories by title, organized for the convenience of some hypothetical person who might want to go straight to a favorite story without having to navigate the fandom or series pages. I wouldn't recommend reading stories for the first time off of this list, as none of them are listed by fandom or in series order. Story titles beginning with an article\u2013a, an, the\u2013are alphabetized by the first letter of the second word.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Prerequisite: Consent of Academic Advisor & Academic Dean.\nThis course is designed to teach the skills necessary to produce the school yearbook, which offers a complete record of an entire school year. The year begins by planning the coverage for the school year and designing a unifying theme for the book. Students will study magazine journalism including layout and design techniques, photoshop\/photo-editing, writing and editing copy, headlines and picture captions. This course provides the study of and practice in gathering and analyzing information, interviewing, note taking and photography. Students will learn strategies of planning, marketing (ad sales and yearbook dance ticket sales) and distribution of the yearbook. They will learn proofing strategies and work independently with photographers. At times, deadlines require that staff members work after school, on weekends, and holidays. Students will learn good work habits and are responsible for all phases of yearbook publication. This course is a valuable foundation to anyone thinking about going into a communications major, publication job, advertising agencies, or working for social media companies after high school.\nCalifornia State University, Long Beach - M.Ed.\nUniversity of Wisconsin, Madison - B.A.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This writing block with its 50 bonded sheets of white uncoated paper (80gm\/m2) serfes as a support for people who want to use the process of nonviolent communication according to Marshall Rosenberg in their day to day life. Even beginners can work through the individual steps, which convert evaluations and accusations in observations and feelings to find a request via one's own needs that is no longer a demand.\nThe distinctive drawings of the Swiss NVC trainer Sylvie H\u00f6rning (cert.) invite you to explore evaluative thoughts closer and to find a nonviolent way of dealing with difficult issues. One's own behavior or the actions of others can be viewed in a completely new light. The result is a deeper understanding of one's own actions or the good intentions of the other.\nWhosoever is contemplating striving for a certification as an NVC trainer has in this giraffe journal a sensible tool at hand for keeping the obligatory diary. Simply fill in the sheets according to the situation, enter the date and file away the individual sheets: the practical punch holes together with file binders makes the filing easy.\nIndividual sheets of the Giraffe Journals can also be employed in seminar and group exercise: If a participant is working on his subject in front of the group, the audience can record on the sheets either empathy for its colleague (what is suspected that the participant is feeling, and what are his needs) or the practice connection to yourself? How do I feel when I hear that? The contributions are then classified in plenary later.\nEvery giraffe journal is a accompanied by a \"compact needs card\" with lists of feelings, needs and interpretations, which makes it easier for the user to get into the process of Nonviolent Communication.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"My sons' hair grows super quick. My husband will trim it himself once a month, but every quarter I like to get it professionally cut to give it a nice shape, which we can then maintain ourselves after. Like I wrote in my Babies' First Haircut Post, you're better off paying a little more or driving further to find a salon geared to children only. We went once to a regular hair studio and it was a nightmare - stressful for everyone.\nSince moving though I've been on the lookout for a similar studio for kids down here and luckily I finally found one over in Ft. Myers - Bananas Salon.\nInside, they have a whole play area, which kept the other half of my duo entertained while his brother got his hair cut. They have a wooden train set and a clubhouse, which they both closed themselves in when it was time to go not wanting to leave. I had to knock on the plastic door and go in after them.\nInstead of chairs, each hair-cutting station has a different car for kids to sit in from a fire engine, police car to a pink power wheels jeep. My boys sat in the fire engine seat and enjoyed playing with the bell and spinning the steering wheel.\nIn front of each car they also have a TV and DVD player. When we arrived they asked us what film or show from their collection that we would like. They had about 50 titles to chose from too. We went with Mickey Mouse Clubhouse. All of these distractions made for a really easy experience.\nTheir stylist, Keri, was a pro at weaving around their bobbing heads in order to deftly cut their hair. She was even able to use the electric shaver at the end to get the little fuzzy hairs on the back of their neck.\nBe sure and sign up for their mailing list for coupons and other discounts. They don't over e-mail either.","meta":{"redpajama_set_name":"RedPajamaC4"}}